Load & Resistance Factor Design: Manual of Steel Construction

MANUAL OF STEEL CONSTRUCTION LOAD & RESISTANCE FACTOR DESIGN Volume I Structural Members, Specifications, & Codes Volum...

230 downloads 2404 Views 34MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

MANUAL OF STEEL CONSTRUCTION

LOAD & RESISTANCE FACTOR DESIGN Volume I Structural Members, Specifications, & Codes Volume II Connections

Second Edition

iv

Copyright © 1994 by American Institute of Steel Construction, Inc. ISBN 1-56424-041-X ISBN 1-56424-042-8 All rights reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher. The information presented in this publication has been prepared in accordance with recognized engineering principles and is for general information only. While it is believed to be accurate, this information should not be used or relied upon for any specific application without competent professional examination and verification of its accuracy, suitability, and applicability by a licensed professional engineer, designer, or architect. The publication of the material contained herein is not intended as a representation or warranty on the part of the American Institute of Steel Construction or of any other person named herein, that this information is suitable for any general or particular use or of freedom from infringement of any patent or patents. Anyone making use of this information assumes all liability arising from such use. Caution must be exercised when relying upon other specifications and codes developed by other bodies and incorporated by reference herein since such material may be modified or amended from time to time subsequent to the printing of this edition. The Institute bears no responsibility for such material other than to refer to it and incorporate it by reference at the time of the initial publication of this edition. Printed in the United States of America

v

FOREWORD

T

he American Institute of Steel Construction, founded in 1921, is the non-profit technical specifying and trade organization for the fabricated structural steel industry in the United States. Executive and engineering headquarters of AISC are maintained in Chicago, Illinois. The Institute is supported by three classes of membership: Active Members totaling 400 companies engaged in the fabrication and erection of structural steel, Associate Members who are allied product manufacturers, and Professional Members who are individuals or firms engaged in the practice of architecture or engineering. Professional members also include architectural and engineering educators. The continuing financial support and active participation of Active Members in the engineering, research, and development activities of the Institute make possible the publishing of this Second Edition of the Load and Resistance Factor Design Manual of Steel Construction. The Institute’s objectives are to improve and advance the use of fabricated structural steel through research and engineering studies and to develop the most efficient and economical design of structures. It also conducts programs to improve product quality. To accomplish these objectives the Institute publishes manuals, textbooks, specifications, and technical booklets. Best known and most widely used are the Manuals of Steel Construction, LRFD (Load and Resistance Factor Design) and ASD (Allowable Stress Design), which hold a highly respected position in engineering literature. Outstanding among AISC standards are the Specifications for Structural Steel Buildings and the Code of Standard Practice for Steel Buildings and Bridges. The Institute also assists designers, contractors, educators, and others by publishing technical information and timely articles on structural applications through two publications, Engineering Journal and Modern Steel Construction. In addition, public appreciation of aesthetically designed steel structures is encouraged through its award programs: Prize Bridges, Architectural Awards of Excellence, Steel Bridge Building Competition for Students, and student scholarships. Due to the expanded nature of the material, the Second Edition of the LRFD Manual has been divided into two complementary volumes. Volume I contains the LRFD Specification and Commentary, tables, and other design information for structural members. Volume II contains all of the information on connections. Like the LRFD Specification upon which they are based, both volumes of this LRFD Manual apply to buildings, not bridges. The Committee gratefully acknowledges the contributions of Roger L. Brockenbrough, Louis F. Geschwindner, Jr., and Cynthia J. Zahn to this Manual. By the Committee on Manuals, Textbooks, and Codes, William A. Thornton, Chairman

Barry L. Barger, Vice Chairman

Horatio Allison Robert O. Disque Joseph Dudek William G. Dyker Ronald L. Hiatt

David T. Ricker Abraham J. Rokach Ted W. Winneberger Charles J. Carter, Secretary

Mark V. Holland William C. Minchin Thomas M. Murray Heinz J. Pak Dennis F. Randall

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

vi

REFERENCED SPECIFICATIONS, CODES, AND STANDARDS

Part 6 (Volume I) of this LRFD Manual contains the full text of the following: American Institute of Steel Construction, Inc. (AISC) Load and Resistance Factor Design Specification for Structural Steel Buildings, December 1, 1993 Specification for Load and Resistance Factor Design of Single-Angle Members, December 1, 1993 Seismic Provisions for Structural Steel Buildings, June 15, 1992 Code of Standard Practice for Steel Buildings and Bridges, June 10, 1992 Research Council on Structural Connections (RCSC) Load and Resistance Factor Design Specifications for Structural Joints Using ASTM A325 or A490 Bolts, June 8, 1988 Additionally, the following other documents are referenced in Volumes I and II of the LRFD Manual: American Association of State Highway and Transportation Officials (AASHTO) AASHTO/AWS D1.5–88 American Concrete Institute (ACI) ACI 349–90 American Iron and Steel Institute (AISI) Load and Resistance Factor Design Specification for Cold-Formed Steel Structural Members, 1991 American National Standards Institute (ANSI) ANSI/ASME B1.1–82 ANSI/ASME B18.2.2–86 ANSI/ASME B18.1–72 ANSI/ASME B18.5–78 ANSI/ASME B18.2.1–81 American Society of Civil Engineers (ASCE) ASCE 7-88 American Society for Testing and Materials (ASTM) ASTM A6–91b ASTM A490–91 ASTM A617–92 ASTM A27–87 ASTM A500–90a ASTM A618–90a ASTM A36–91 ASTM A501–89 ASTM A668–85a ASTM A53–88 ASTM A502–91 ASTM A687–89 ASTM A148–84 ASTM A514–91 ASTM A709–91 ASTM A153–82 ASTM A529–89 ASTM A770–86 ASTM A193–91 ASTM A563–91c ASTM A852–91 ASTM A194–91 ASTM A570–91 ASTM B695–91 ASTM A208(A239–89) ASTM A572–91 ASTM C33–90 ASTM A242–91a ASTM A588–91a ASTM C330–89 ASTM A307–91 ASTM A606–91a ASTM E119–88 ASTM A325–91c ASTM A607–91 ASTM E380–91 ASTM A354–91 ASTM A615–92b ASTM F436–91 ASTM A449–91a ASTM A616–92 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

vii

American Welding Society (AWS) AWS A2.4–93 AWS A5.25–91 AWS A5.1–91 AWS A5.28–79 AWS A5.5–81 AWS A5.29–80 AWS A5.17–89 AWS B1.0–77 AWS A5.18–79 AWS D1.1–92 AWS A5.20–79 AWS D1.4–92 AWS A5.23–90


1-1

PART 1 DIMENSIONS AND PROPERTIES

OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 STRUCTURAL STEELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Selection of the Appropriate Structural Steel . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Brittle Fracture Considerations in Structural Design . . . . . . . . . . . . . . . . . . . . . 1-6 Lamellar Tearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8 Jumbo Shapes and Heavy-Welded Built-Up Sections . . . . . . . . . . . . . . . . . . . . 1-8 FIRE-RESISTANT CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8 Effect of Shop Painting on Spray-Applied Fireproofing . . . . . . . . . . . . . . . . . . 1-11 EFFECT OF HEAT ON STRUCTURAL STEEL . . . . . . . . . . . . . . . . . . . . . . 1-11 Coefficient of Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12 Use of Heat to Straighten, Camber, or Curve Members . . . . . . . . . . . . . . . . . . 1-12 EXPANSION JOINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13 COMPUTER SOFTWARE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14 AISC Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14 AISC for AutoCAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14 STRUCTURAL SHAPES: TABLES OF AVAILABILITY, SIZE GROUPINGS, PRINCIPAL PRODUCERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 STEEL PIPE AND STRUCTURAL TUBING: TABLES OF AVAILABILITY, PRINCIPAL PRODUCERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21 STRUCTURAL SHAPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Designations, Dimensions, and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Tables: W Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-26 M Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-44 S Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-46 HP Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-48 American Standard Channels (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-50 Miscellaneous Channels (MC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-52 Angles (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-56 STRUCTURAL TEES (WT, MT, ST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-67 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1-2

DIMENSIONS AND PROPERTIES

Use of Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-67 DOUBLE ANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-91 Use of Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-91 COMBINATION SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-105 STEEL PIPE AND STRUCTURAL TUBING . . . . . . . . . . . . . . . . . . . . . . . 1-120 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-120 Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-120 Structural Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-120 BARS AND PLATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-133 Product Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-133 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-133 Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-133 Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-133 Floor Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-134 CRANE RAILS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-139 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-139 Splices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-139 Welded Splices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-141 Fastenings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-141 TORSION PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-145 SURFACE AREAS AND BOX AREAS . . . . . . . . . . . . . . . . . . . . . . . . . . 1-175 CAMBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-179 Beams and Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-179 Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-179 STANDARD MILL PRACTICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-183 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-183 Methods of Increasing Areas and Weights by Spreading Rolls

. . . . . . . . . . . . . 1-183

Cambering of Rolled Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-185 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-199


OVERVIEW

1-3

OVERVIEW To facilitate reference to Part 1, the locations of frequently used tables are listed below. Dimensions and Properties W Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-26 M Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-44 S Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-46 HP Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-48 American Standard Channels (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-50 Miscellaneous Channels (MC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-52 Angles (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-56 Structural Tees (WT, MT, ST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-68 Double Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-92 Combination Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-106 Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-121 Structural Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-122 Torsion Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-146 Surface Areas and Box Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-175 Availability Availability of Shapes, Plates, and Bars, Table 1-1 . . . . . . . . . . . . . . . . . . . . 1-15 Structural Shape Size Groupings, Table 1-2 . . . . . . . . . . . . . . . . . . . . . . . . 1-16 Principal Producers of Structural Shapes, Table 1-3 . . . . . . . . . . . . . . . . . . . . 1-18 Availability of Steel Pipe and Structural Tubing, Table 1-4 . . . . . . . . . . . . . . . . 1-21 Principal Producers of Structural Tubing (TS), Table 1-5 . . . . . . . . . . . . . . . . . 1-22 Principal Producers of Steel Tubing (Round), Table 1-6 . . . . . . . . . . . . . . . . . . 1-26


1-4



STRUCTURAL STEELS

1-5

STRUCTURAL STEELS Availability

Section A3.1 of the AISC Load and Resistance Factor Design Specification for Structural Steel Buildings lists fifteen ASTM specifications for structural steel approved for use in building construction. Five of these steels are available in hot-rolled structural shapes, plates, and bars. Two steels, ASTM A514 and A852, are available only in plates. Table 1-1 shows five groups of shapes and eleven ranges of thickness of plates and bars available in the various minimum yield stress* and tensile strength levels afforded by the seven steels. For complete information on each steel, reference should be made to the appropriate ASTM specification. A listing of shape sizes included in each of the five groups follows in Table 1-2, corresponding with the groupings given in Table A of ASTM Specification A6. Seven additional grades of steel, other than those covering hot-rolled shapes, plates, and bars, are listed in Section A3.1a of the LRFD Specification. These steels cover pipe, cold- and hot-formed tubing, and cold- and hot-rolled sheet and strip. The principal producers of shapes listed in Part 1 of this Manual are shown in Table 1-3. Availability and the principal producers of structural tubing are shown in Tables 1-4 through 1-6. For additional information on availability and classification of structural steel plates and bars, refer to the separate discussion beginning on page 1-129. Space does not permit inclusion in Table 1-3, or in the listing of shapes and plates in Part 1 of this Manual, of all rolled shapes or plates of greater thickness that are occasionally used in construction. For such products, reference should be made to the various producers’ catalogs. To obtain an economical structure, it is often advantageous to minimize the number of different sections. Cost per square foot can often be reduced by designing this way. Selection of the Appropriate Structural Steel

Steels with 50 ksi yield stress are now widely used in construction, replacing ASTM A36 steel in many applications. The 50 ksi steels listed in Section A3.1a of the LRFD Specification are ASTM A572 high-strength low-alloy structural steel, ASTM A242 and A588 atmospheric-corrosion-resistant high-strength low-alloy structural steels, and ASTM A529 high-strength carbon-manganese structural steel. Yield stresses above 50 ksi can be obtained from two grades of ASTM A572 steel as well as ASTM A514 and A852 quenched and tempered structural steel plate. These higher-strength steels have certain advantages over 50 ksi steels in certain applications. They may be economical choices where lighter members, resulting from use of higher design strengths, are not penalized because of instability, local buckling, deflection, or other similar reasons. They may be used in tension members, beams in continuous and composite construction where deflections can be minimized, and columns having low slenderness ratios. The reduction of dead load and associated savings in shipping costs can be significant factors. However, higher strength steels are not to be used indiscriminately. Effective use of all steels depends on thorough cost and engineering analysis. Normally, connection material is specified as ASTM A36. The connection tables in this Manual are for A36 steel.

*As used in the AISC LRFD Specification, “yield stress” denotes either the specified minimum yield point (for those that have a yield point) or specified minimum yield strength (for those steels that do not have a yield point). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1-6


With appropriate procedures and precautions, all steels listed in the AISC Specification are suitable for welded fabrication. To provide for weldability of ASTM A529 steel, the specification of a maximum carbon equivalent is recommended. ASTM A242 and A588 atmospheric-corrosion-resistant, high-strength, low-alloy steels can be used in the bare (uncoated) condition in most atmospheres. Where boldly exposed under such conditions, exposure to the normal atmosphere causes a tightly adherent oxide to form on the surface which protects the steel from further atmospheric corrosion. To achieve the benefits of the enhanced atmospheric corrosion resistance of these bare steels, it is necessary that design, detailing, fabrication, erection, and maintenance practices proper for such steels be observed. Designers should consult with the steel producers on the atmospheric-corrosion-resistant properties and limitations of these steels prior to use in the bare condition. When either A242 or A588 steel is used in the coated condition, the coating life is typically longer than with other steels. Although A242 and A588 steels are more expensive than other high-strength, low-alloy steels, the reduction in maintenance resulting from the use of these steels usually offsets their higher initial cost. Brittle Fracture Considerations in Structural Design

As the temperature decreases, an increase is generally noted in the yield stress, tensile strength, modulus of elasticity, and fatigue strength of the structural steels. In contrast, the ductility of these steels, as measured by reduction in area or by elongation, and the toughness of these steels, as determined from a Charpy V-notch impact test, decrease with decreasing temperatures. Furthermore, there is a temperature below which a structural steel subjected to tensile stresses may fracture by cleavage,* with little or no plastic deformation, rather than by shear,* which is usually preceded by a considerable amount of plastic deformation or yielding. Fracture that occurs by cleavage at a nominal tensile stress below the yield stress is commonly referred to as brittle fracture. Generally, a brittle fracture can occur in a structural steel when there is a sufficiently adverse combination of tensile stress, temperature, strain rate, and geometrical discontinuity (notch) present. Other design and fabrication factors may also have an important influence. Because of the interrelation of these effects, the exact combination of stress, temperature, notch, and other conditions that will cause brittle fracture in a given structure cannot be readily calculated. Consequently, designing against brittle fracture often consists mainly of (1) avoiding conditions that tend to cause brittle fracture and (2) selecting a steel appropriate for the application. A discussion of these factors is given in the following sections. Conditions Causing Brittle Fracture

It has been established that plastic deformation can occur only in the presence of shear stresses. Shear stresses are always present in a uniaxial or biaxial state-of-stress. However, in a triaxial state-of-stress, the maximum shear stress approaches zero as the principal stresses approach a common value, and thus, under equal triaxial tensile stresses, failure occurs by cleavage rather than by shear. Consequently, triaxial tensile stresses tend to cause brittle fracture and should be avoided. A triaxial state-of-stress can result from a uniaxial loading when notches or geometrical discontinuities are present.

*Shear and cleavage are used in the metallurgical sense (macroscopically) to denote different fracture mechanisms. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

STRUCTURAL STEELS

1-7

Increased strain rates tend to increase the possibility of brittle behavior. Thus, structures that are loaded at fast rates are more susceptible to brittle fracture. However, a rapid strain rate or impact load is not a required condition for a brittle fracture. Cold work and the strain aging that normally follows generally increase the likelihood of brittle fracture. This behavior is usually attributed to the previously mentioned reduction in ductility. The effect of cold work that occurs in cold forming operations can be minimized by selecting a generous forming radius and, thus, limiting the amount of strain. The amount of strain that can be tolerated depends on both the steel and the application. The use of welding in construction increases the concerns relative to brittle fracture. In the as-welded condition, residual stresses will be present in any weldment. These stresses are considered to be at the yield point of the material. To avoid brittle fracture, it may be required to utilize steels with higher toughness than would be required for bolted construction. Welds may also introduce geometric conditions or discontinuities that are crack-like in nature. These stress risers will additionally increase the requirement for notch toughness in the weldment. Avoidance of the intersection of welds from multiple directions reduces the likelihood of triaxial stresses. Properly sized weld-access holes prohibit the interaction of these various stress fields. As steels being welded become thicker and more highly restrained, welding procedure issues such as preheat, interpass temperature, heat input, and cooling rates become increasingly important. The residual stresses present in a weldment may be reduced by the use of fewer weld passes and peening of intermittent weld layers. In most cases, weld metal notch toughness exceeds that of the base materials. However, for fracture-sensitive applications, notch-tough base and weld metal should be specified. The residual stresses of welding can be greatly reduced through thermal stress relief. This reduces the driving force that causes brittle fracture, but if the toughness of the material is adversely affected by this thermal treatment, no increase in brittle fracture resistance will be experienced. Therefore, when weldments are to be stress relieved, investigation into the effects on the weld metal, heat-affected zone, and base material should be made. Selecting a Steel To Avoid Brittle Fracture

The best guide in selecting a steel that is appropriate for a given application is experience with existing and past structures. A36 and Grade 50 (i.e., 50 ksi yield stress) steels have been used successfully in a great number of applications, such as buildings, transmission towers, transportation equipment, and bridges, even at the lowest atmospheric temperatures encountered in the U.S. Therefore, it appears that any of the structural steels, when designed and fabricated in an appropriate manner, could be used for similar applications with little likelihood of brittle fracture. Consequently, brittle fracture is not usually experienced in such structures unless unusual temperature, notch, and stress conditions are present. Nevertheless, it is always desirable to avoid or minimize the previously cited adverse conditions that increase the susceptibility of the steel to brittle fracture. In applications where notch toughness is considered important, it usually is required that steels must absorb a certain amount of energy, 15 ft-lb or higher (Charpy V-notch test), at a given temperature. The test temperature may be higher than the lowest operating temperature depending on the rate of loading. See Rolfe and Barsom (1986) and Rolfe (1977). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1-8


Lamellar Tearing

The information on strength and ductility presented in the previous sections generally pertains to loadings applied in the planar direction (longitudinal or transverse orientation) of the steel plate or shape. It should be noted that elongation and area reduction values may well be significantly lower in the through-thickness direction than in the planar direction. This inherent directionality is of small consequence in many applications, but does become important in the design and fabrication of structures containing massive members with highly restrained welded joints. With the increasing trend toward heavy welded-plate construction, there has been a broader recognition of the occurrence of lamellar tearing in some highly restrained joints of welded structures, especially those using thick plates and heavy structural shapes. The restraint induced by some joint designs in resisting weld deposit shrinkage can impose tensile strain sufficiently high to cause separation or tearing on planes parallel to the rolled surface of the structural member being joined. The incidence of this phenomenon can be reduced or eliminated through greater understanding by designers, detailers, and fabricators of (1) the inherent directionality of construction forms of steel, (2) the high restraint developed in certain types of connections, and (3) the need to adopt appropriate weld details and welding procedures with proper weld metal for through-thickness connections. Further, steels can be specified to be produced by special practices and/or processes to enhance through-thickness ductility and thus assist in reducing the incidence of lamellar tearing. Steels produced by such practices are available from several producers. However, unless precautions are taken in both design and fabrication, lamellar tearing may still occur in thick plates and heavy shapes of such steels at restrained through-thickness connections. Some guidelines in minimizing potential problems have been developed (AISC, 1973). See also Part 8 in Volume II of this LRFD Manual and ASTM A770, Standard Specification for Through-Thickness Tension Testing of Steel Plates for Special Applications. Jumbo Shapes and Heavy Welded Built-up Sections

Although Group 4 and 5 W-shapes, commonly referred to as jumbo shapes, generally are contemplated as columns or compression members, their use in non-column applications has been increasing. These heavy shapes have been known to exhibit segregation and a coarse grain structure in the mid-thickness region of the flange and the web. Because these areas may have low toughness, cracking might occur as a result of thermal cutting or welding (Fisher and Pense, 1987). Similar problems may also occur in welded built-up sections. To minimize the potential of brittle failure, the current LRFD Specification includes provisions for material toughness requirements, methods of splicing, and fabrication methods for Group 4 and 5 hot-rolled shapes and welded built-up cross sections with an element of the cross section more than two inches in thickness intended for tension applications. FIRE-RESISTANT CONSTRUCTION

Fire-resistant steel construction may be defined as structural members and assemblies which can maintain structural stability for the duration of building fire exposure and, in some cases, prevent the spread of fire to adjacent spaces. Fire resistance of a steel member is a function of its mass, its geometry, the load to which it is subjected, its structural support conditions, and the fire to which it is exposed. Many steel structures have inherent fire resistance through a combination of the above factors and do not require additional insulation from the effects of fire. However, in many AMERICAN INSTITUTE OF STEEL CONSTRUCTION

FIRE-RESISTANT CONSTRUCTION

1-9

situations, building codes specify the use of fire-rated steel assemblies. In this case, ASTM Specification E119, Standard Methods of Fire Tests of Building Construction and Materials, outlines the procedures of fire testing of structural elements. Structural fire resistance is a major consideration in the design of modern buildings. In general, building codes define the level of fire protection that is required in specific applications and structural fire protection is typically implemented in design through code compliance. In the United States, with a few notable exceptions, the majority of cities and states now enforce one of the following model codes: • National Building Code, published by the Building Officials and Code Administrators International. • Standard Building Code, published by the Southern Building Code Congress International. • Uniform Building Code, published by the International Conference of Building Officials. Building codes specify fire-resistance requirements as a function of building occupancy, height, area, and whether or not other fire protection systems (e.g., sprinklers) are provided. Fire-resistance requirements are specified in terms of hourly ratings based upon tests conducted in accordance with ASTM E119. This test method specifies a “standard” fire for evaluating the relative fire-resistance of construction assemblies (i.e., floors, roofs, beams, girders, and columns). Specific end-point criteria for evaluating the ability of assemblies to prevent the spread of fire to adjacent spaces and/or to continue to sustain superimposed loads are included. In effect, ASTM E119 is used to evaluate the length of time that an assembly continues to perform these functions when exposed to the standard fire. Thus, code requirements and fire-resistance ratings are specified in terms of time (i.e., one hour, two hours, etc.). The design of fire-resistant buildings is typically accomplished in a very prescriptive fashion by selecting tested designs that satisfy specific building code requirements. Listings of fire-resistant designs are available from a number of sources including: • Fire-Resistance Directory, Underwriters Laboratories. • Fire-Resistance Ratings, American Insurance Services Group. • Fire-Resistance Design Manual, Gypsum Association. In general, due to the very prescriptive nature of fire-resistant design, changes in tested assemblies can be difficult to justify to the satisfaction of code officials and listing agencies. In the case of structural steel construction, however, the basic heat transfer and structural principles are well defined. As a result, relatively simple analytical techniques have been developed that enable designers to use a variety of different structural steel shapes in conjunction with tested assemblies. These analytical techniques are specifically recognized by North American building code authorities and are described in a series of booklets published by the American Iron and Steel Institute (AISI): Designing Fire Protection for Steel Columns (1980) Designing Fire Protection for Steel Beams (1984) Designing Fire Protection for Steel Trusses (1981) Since fire-resistant design is currently based on the use of tested assemblies, an important consideration is the degree to which a test assembly is “representative” of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 10


actual building construction. In reality, this consideration poses a number of technical difficulties due to the size of available testing facilities, most of which can only accommodate floor or roof specimens in the range of 15 ft by 18 ft in area. As a result, a test assembly represents a relatively small sample of a typical floor or roof structure. Most floor slabs and roof decks are physically, if not structurally, continuous over beams and girders. Beam and girder spans are often much larger than can be accommodated in available laboratory furnaces. A variety of connection details are used to frame beams, girders, and columns. In short, given the cost of testing, the complexity and variety of modern structural systems, and the size of available test facilities, it is unrealistic to assume that test assemblies accurately model real construction systems during fire exposure. In recognition of the practical difficulties associated with laboratory scale testing, ASTM E119 includes two specific test conditions, “restrained” and “unrestrained.” From a structural engineering standpoint, the choice of these two terms is unfortunate since the “restraint” that is contemplated in fire testing is restraint against the thermal expansion, not structural rotational restraint in the traditional sense. The “restrained” condition applies when the assembly is supported or surrounded by construction which is “capable of resisting substantial thermal expansion throughout the range of anticipated elevated temperatures.” Otherwise, the assembly should be considered free to rotate and expand at the supports and should be considered “unrestrained.” Thus, a floor system that is simply supported from a structural standpoint will often be “restrained” from a fireresistance standpoint. In order to provide guidance on the use of restrained and unrestrained ratings, ASTM E119 includes an explanatory Appendix. It should be emphasized that most common types of steel framing can be considered “restrained” from a fire-resistance standpoint. The standard fire test also includes other arbitrary assumptions. The specific fire exposure, for example, is based on furnace capabilities with continuous fuel supply and does not model real building fires with exhaustible fuel. Also, the test method assumes that assemblies are fully loaded when a fire occurs. In reality, fires are infrequent, random events and their design requirements should be probability based. Rarely will design structural loads occur simultaneously with fire. In addition, many structural elements are sized for serviceability (i.e., drift, deflection, or vibration) rather than strength, thereby providing an additional reserve strength during a fire. As a result of these and other considerations, more rational engineering design standards for structural fire protection are now being developed (International Fire Engineering Design for Steel Structures: State-of-the-Art, International Iron and Steel Institute). Although not yet standardized or recognized in North American building codes, similar design methods have been used in specific cases, based on code variances. One such method has been developed by AISI for architecturally exposed structural steel elements on the exterior of buildings. In effect, ASTM E119 assumes that structural elements are located within a fire compartment and does not realistically characterize the fire exposure that will be seen by exterior structural elements. Fire-Safe Structural Steel: A Design Guide (American Iron and Steel Institute, 1979) defines a step-by-step analytical procedure for determining maximum steel temperatures, based on realistic fire exposures for exterior structural elements. Occasionally, structural engineers will be called upon to evaluate fire-damaged steel structures. Although it is well known that the prolonged exposure to high temperatures can affect the physical and metallurgical properties of structural steel, in most cases steel AMERICAN INSTITUTE OF STEEL CONSTRUCTION

EFFECT OF HEAT ON STRUCTURAL STEEL

1 - 11

members that can be straightened in place will be suitable for continued use (Dill, 1960). Special attention should be given to heat-treated or cold-formed steel elements and high-strength bolts and welds. Effect of Shop Painting on Spray-Applied Fireproofing

Spray-applied fireproofing has excellent adhesion to unpainted structural steel. Mechanical anchorage devices, bonding agents, or bond tests are not required to meet Underwriters Laboratories, Inc. (UL) guidelines. In fact, moderate rusting enhances the adhesion of the fireproofing material, providing the uncoated steel is free of loose rust and mill scale. Customarily, any loose rust or mill scale as well as any other debris which has accumulated during the construction process is removed by the fireproofing application contractor. In many cases, this may be as simple as blowing it off with compressed air. This ease of application is not realized when fireproofing is applied over painted steel. In order to meet UL requirements, bond tests in accordance with the ASTM E736 must be performed to determine if the fireproofing material has adequate adherence to the painted surface. Frequently, a bonding agent must be added to the fireproofing material and the bond test repeated to determine if the minimum bond strength can be met. Should the bond testing still not be satisfactory, mechanical anchorage devices are required to be applied to the steel before the fireproofing can be applied. The erected steel must still be cleaned free of any construction debris and scaling or peeling paint before the fireproofing may be applied. Once it is determined that the bond tests are adequate, UL guidelines require that if fireproofing is spray-applied over painted steel, the steel must be wrapped with steel lath or mechanical anchorage devices must be applied to the steel if the structural shape exceeds the following dimensional criteria: • For beam applications, the web depth cannot exceed 16 inches and the flange cannot exceed 12 inches. • For column applications, neither the web depth nor the flange width can exceed 16 inches. A significant number of structural shapes do not meet these restrictions. The use of primers under spray-applied fireproofing significantly increases the cost of the steel and the preparation for and the application of the fireproofing material. In an enclosed structure, primer is insignificant in either the short- or long-term protection of the steel. LRFD Specification Section M3.1 states that structural steelwork need not be painted unless required by the contract. For many years, the AISC specifications have not required that steelwork be painted when it will be concealed by interior building finish or will be in contact with concrete. The use of primers under spray-applied fireproofing is strongly discouraged unless there is a compelling reason to paint the steel to protect against corrosion. It is suggested that the designer refer to the UL Directory Fire Resistance—Volume 1, 1993, “Coating Materials,” for more specific information on this topic. EFFECT OF HEAT ON STRUCTURAL STEEL

Short-time elevated-temperature tensile tests on the structural steels permitted by the AISC Specification indicate that the ratios of the elevated-temperature yield and tensile strengths to their respective room-temperature values are reasonably similar in the 300° to 700°F range, except for variations due to strain aging. (The tensile strength ratio may AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 12


increase to a value greater than unity in the 300° to 700°F range when strain aging occurs.) Below 700°F the strength ratios decrease only slightly. Above 700°F the ratio of elevated-temperature to room-temperature strength decreases more rapidly as the temperature increases. The composition of the steels is usually such that the carbon steels (ASTM A36 and A529) exhibit strain aging with attendant reduced notch toughness. The high-strength low-alloy steels (ASTM A242, A572, and A588) and heat-treated alloy steels (ASTM A514 and A852) exhibit less-pronounced or little strain aging. As examples of the decreased ratio levels obtained at elevated temperature, the yield strength ratios for carbon and high-strength low-alloy steels are approximately 0.77 at 800°F, 0.63 at 1,000°F, and 0.37 at 1,200°F. Coefficient of Expansion

The average coefficient of expansion for structural steel between 70°F and 100°F is 0.0000065 for each degree. For temperatures of 100°F to 1,200°F the coefficient is given by the approximate formula: ε = (6.1+0.0019t) × 10−6 in which ε is the coefficient of expansion (change in length per unit length) for each degree Fahrenheit and t is the temperature in degrees Fahrenheit. The modulus of elasticity of structural steel is approximately 29,000 ksi at 70°F. It decreases linearly to about 25,000 ksi at 900°F, and then begins to drop at an increasing rate at higher temperatures. Use of Heat to Straighten, Camber, or Curve Members

With modern fabrication techniques, a controlled application of heat can be effectively used to either straighten or to intentionally curve structural members. By this process, the member is rapidly heated in selected areas; the heated areas tend to expand, but are restrained by adjacent cooler areas. This action causes a permanent plastic deformation or “upset” of the heated areas and, thus, a change of shape is developed in the cooled member. “Heat straightening” is used in both normal shop fabrication operations and in the field to remove relatively severe accidental bends in members. Conversely, “heat cambering” and “heat curving” of either rolled beams or welded girders are examples of the use of heat to effect a desired curvature. As with many other fabrication operations, the use of heat to straighten or curve will cause residual stresses in the member as a result of plastic deformations. These stresses are similar to those that develop in rolled structural shapes as they cool from the rolling temperature; in this case, the stresses arise because all parts of the shape do not cool at the same rate. In like manner, welded members develop residual stresses from the localized heat of welding. In general, the residual stresses from heating operations do not affect the ultimate strength of structural members. Any reduction in strength due to residual stresses is incorporated in the provisions of the LRFD Specification. The mechanical properties of steels are largely unaffected by heating operations, provided that the maximum temperature does not exceed 1,100°F for quenched and tempered alloy steels (ASTM A514 and A852), and 1,300°F for other steels. The AMERICAN INSTITUTE OF STEEL CONSTRUCTION

EXPANSION JOINTS

1 - 13

temperature should be carefully checked by temperature-indicating crayons or other suitable means during the heating process. EXPANSION JOINTS

Although buildings are typically constructed of flexible materials, expansion joints are required in roofs and the supporting structure when horizontal dimensions are large. The maximum distance between expansion joints is dependent upon many variables including ambient temperature during construction and the expected temperature range during the lifetime of the building. An excellent reference on the topic of thermal expansion in buildings and location of expansion joints is the Federal Construction Council’s Technical Report No. 65, Expansion Joints in Buildings. Taken from this report, Figure 1-1 provides a guide based on design temperature change for maximum spacing of structural expansion joints in beam-and-column-framed buildings with hinged-column bases and heated interiors. The report includes data for numerous cities and gives five modification factors which should be applied as appropriate:

MAXIMUM SPACING OF EXPANSION JOINTS (ft)

1. If the building will be heated only and will have hinged-column bases, use the maximum spacing as specified; 2. If the building will be air-conditioned as well as heated, increase the maximum spacing by 15 percent provided the environmental control system will run continuously; 3. If the building will be unheated, decrease the maximum spacing by 33 percent; 4. If the building will have fixed column bases, decrease the maximum spacing by 15 percent;

600

500

Rectangular multiframed configuration with Symmetrical stiffness

400

Steel

300 200

Nonrectangular configuration (L, T, U type)

Any material

100

10 20 30 40

50 60 70 70 80 90

DESIGN TEMPERATURE CHANGE (°F)

Fig. 1-1. Expansion joint spacing. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 14


5. If the building will have substantially greater stiffness against lateral displacement in one of the plan dimensions, decrease the maximum spacing by 25 percent. When more than one of these design conditions prevail in a building, the percentile factor to be applied should be the algebraic sum of the adjustment factors of all the various applicable conditions. Additionally, most building codes include restrictions on location and spacing of fire walls. Such fire walls often become locations for expansion joints. The most effective expansion joint is a double line of columns which provides a complete and positive separation. When expansion joints other than the double-column type are employed, low-friction sliding elements are generally used. Such systems, however, are never totally free and will induce some level of inherent restraint to movement. COMPUTER SOFTWARE AISC Database

The AISC Database contains the properties and dimensions of structural steel shapes, corresponding to Part 1 of this LRFD Manual. LRFD-related properties such as X1 and X2, as well as torsional properties, are included. Two versions, one in U.S. customary units and one in metric units, are available. Dimensions and properties of W, S, M, and HP shapes, American Standard Channels (C), Miscellaneous Channels (MC), Structural Tees cut from W, M, and S shapes (WT, MT, ST), Single and Double Angles, Structural Tubing, and Pipe are listed in ASCII format. Also included are: a BASIC read/write program, a sample search routine, and a routine to convert the file to Lotus *.PRN file format. AISC for AutoCAD *

The program will draw the end, elevation, and plan views of W, S, M, and HP shapes, American Standard Channels (C), Miscellaneous Channels (MC), Structural Tees cut from W, M, and S shapes (WT, MT, ST), Single and Double Angles, Structural Tubing, and Pipe to full scale corresponding to data published in Part 1 of this LRFD Manual. Version 2.0 runs in AutoCAD Release 12 only; Version 1.0 runs in AutoCAD Releases 10 and 11.

*AutoCAD is a registered trademark in the US Patent and Trademark Office by Autodesk, Inc. AISC for AutoCAD is copyrighted in the US Copyright Office by Bridgefarmer and Associates, Inc. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 15

Table 1-1. Availability of Shapes, Plates, and Bars According to ASTM Structural Steel Specifications Shapes

Fy

Steel Type

A36

32

58–80

36

58–80c

42

42

60–85

50

50

70–100

42

42

60

50

50

65

60

60

75

65

65

80

A242

42

63

46

67

50

70

HighStrength Low-alloy

Corrosion Resistant Highstrength Low-alloy

A572 Grade

A529f Grade

Carbon

Group per Over Over Mini1⁄ ″ 3⁄ ″ ASTM A6 Fu mum 2 4 ASTM Yield Tensile To to to a 1⁄ ″ 3 ⁄ ″ 11 ⁄ ″ Desig- Stress Stress 2 4 4 b nation (ksi) (ksi) 1 2 3 4 5 incl. incl. incl.

A588

42

63

46

67

50

70

Quenched A852e & Tempered Alloy

70

90–110

Quenched A514e & Tempered A514e Low-Alloy

90

100–130

100

110–130

Plates and Bars Over Over Over Over Over Over Over 11⁄4″ 11⁄2″ 2″ 21⁄2″ 4″ 5″ 6″ to to to to to to to 11⁄2″ 2″ 21⁄2″ 4″ 5″ 6″ 8″ Over incl. incl. incl. incl. incl. incl. incl. 8″

d

aMinimum unless a range is shown. bIncludes bar-size shapes cFor shapes over 426 lb / ft minimum of 58 ksi only applies. dPlates to 1 in. thick, 12 in. width; bars to 11⁄ in. 2 ePlates only. fTo improve the weldability of A529 steel, the specification of a maximum carbon equivalent

(per ASTM Supplementary Requirement S78) is recommended. Available Not Available


1 - 16


Table 1-2. Structural Shape Size Groupings for Tensile Property Classification Structural Shapes W shapes

Group 1

Group 2

Group 3 W44× 290, 335

Group 4

W24× 55, 62

W44× 230, 262

W21× 44 to 57 incl.

W40× 149 to 264 incl. W40× 431

W18× 35 to 71 incl.

W36× 135 to 210 incl. W40× 277 to 372 incl. W36× 328 to 798 incl.

W16× 26 to 57 incl.


W14× 22 to 53 incl.


W12× 14 to 58 incl.


W10× 12 to 45 incl.


Group 5

W40×466 to 593 incl. W36× 848 W40× 392

W8× 10 to 48 incl.


W6× 9 to 25 incl.


W5× 16,19


W4× 13

W14× 61 to 132 incl. W14× 145 to 211 incl.

W14× 605 to 808 incl.

W12× 65 to 106 incl. W12× 120 to 190 incl. W10× 49 to 112 incl. W8× 58, 67 M Shapes

all

S Shapes

to 35 lb/ft incl.

over 35 lb/ft

HP Shapes

to 102 lb/ft incl.

American to 20.7 lb/ft incl. Standard Channels (C)

over 20.7 lb/ft

Miscellane- to 28.5 lb/ft incl. ous Channels (MC)

over 28.5 lb/ft

Angles (L)

to 1⁄2-in. incl.

over 102 lb/ft

over 1⁄2- to 3⁄4-in. incl. over 3⁄4-in.

Notes: Structural tees from W, M, and S shapes fall into the same group as the structural shapes from which they are cut. Group 4 and Group 5 shapes are generally contemplated for application as columns or compression components. When used in other applications (e.g., trusses) and when thermal cutting or welding is required, special material specification and fabrication procedures apply to minimize the possibility of cracking (see Part 6, LRFD Specification, Sections A3.1c, J1.5, J1.6, J2.3, and M2.2, and corresponding Commentary sections).


1 - 17

Structural Steel Shape Producers Bayou Steel Corp. P.O. Box 5000 Laplace, LA 70068 (800) 535-7692

Florida Steel Corp. P.O. Box 31328 Tampa, FL 33631 (800) 237-0230

Nucor-Yamato Steel P.O. Box 1228 Blytheville, AR 72316 (800) 289-6977

Bethlehem Steel Corp. 301 East Third St. Bethlehem, PA 18016-7699 (800) 633-0482

Northwestern Steel & Wire Co. 121 Wallace St. P.O. Box 618 Sterling, IL 61081-0618 (800) 793-2200

Roanoke Electric Steel Corp. P.O. Box 13948 Roanoke, VA 24038 (800) 753-3532

British Steel Inc. 475 N. Martingale Road #400 Schaumburg, IL 60173 (800) 542-6244

North Star Steel Co. 1380 Corporate Center Curve Suite 215 P.O. Box 21620 Eagan, MN 55121-0620 (800) 328-1944

Chaparral Steel Co. 300 Ward Road Midlothian, TX 76065-9501 (800) 529-7979

Nucor Steel P.O. Box 126 Jewett, TX 75846 (800) 527-6445

SMI Steel, Inc. 101 South 50th St. Birmingham, AL 35232 (800) 621-0262 TradeARBED 825 Third Ave. New York, NY 10022 (212) 486-9890

Structural Tube Producers American Institute for Hollow Structural Sections 929 McLaughlin Run Road Suite 8 Pittsburgh, PA 15017 (412) 221-8880 Acme Roll Forming Co. 812 North Beck St. Sebewaing, MI 48759-0706 (800) 937-8823

Dallas Tube & Rollform P.O. Box 540873 Dallas, TX 75354-0873 (214) 556-0234

Independence Tube Corp. 6226 West 74th St. Chicago, IL 60638 (708) 496-0380

Eugene Welding Co. P.O. Box 249 Marysville, MI 48040 (313) 364-7421

IPSCO Steel, Inc. P.O. Box 1670, Armour Road Regina, Saskatchewan S4P 3C7 CANADA (416) 271-2312

EXLTUBE, Inc. 905 Atlantic North Kansas City, MO 64116 (800) 892-8823

Bull Moose 57540 SR 19 S P.O. Box B-1027 Elkhart, IN 46515 (800) 348-7460

Hanna Steel Corp. 3812 Commerce Ave. P.O. Box 558 Fairfield, AL 35064 (800) 633-8252

Copperweld Corp. 7401 South Linder Ave. Chicago, IL 60638 (800) 327-8823

UNR-Leavitt, Div. of UNR Inc. 1717 West 115th St. Chicago, IL 60643 (800) 532-8488 Valmont Industries, Inc. P.O. Box 358 Valley, NE 68064 (800) 825-6668 Welded Tube Co. of America 1855 East 122nd St. Chicago, IL 60633 (800) 733-5683

Steel Pipe Producers National Association of Steel Pipe Distributors, Inc. 12651 Briar Forest Dr., Suite 130 Houston, TX 77077 (713) 531-7473


1 - 18


Table 1-3. Principal Producers of Structural Shapes B—Bethlehem Steel Corp. C—Chaparral Steel F—Florida Steel Corp.

I—British Steel S—North Star Steel M—SMI Steel Inc. T—TradeARBED N—Nucor-Yamato Steel U—Nucor Steel R—Roanoke Steel

W—Northwestern Steel & Wire Y—Bayou Steel Corp.

Section, Weight per ft

Producer Code


Producer Code

W44× all

T

W40× 321-593 W40× 297 W40× 278 W40× 277 W40× 264 W40× 249 W40× 235 W40× 215 W40× 211 W40× 199 W40× 183 W40× 174 W40× 149-167

T N T N,T B,T N,T B,T N,T B,T N,T B,I,N,T T B,I,N,T

W24× 103 W24× 84-94 W24× 55-76

B,W B,I,N,W B,C,I,N,W

W21× 182-201 W21× 166 W21× 83-147 W21× 44-73

I,W B,I,W B,I,N,W B,C,I,N,W

W18× 258-311 W18× 175-234 W18× 130-158 W18× 76-119 W18× 65-71 W18× 35-60

B B,W B,N,W B,N,W B,I,N,W B,C,I,N,W

W36× 439-848 W36× 393 W36× 328-359 W36× 260-300 W36× 256 W36× 245 W36× 232 W36× 135-230

T B,T B,I,T B,I,N,T B,I B,I,N,T B,I B,I,N,T

W16× 67-100 W16× 57 W16× 26-50

B,N,W B,I,N,W B,C,I,N,W

W33× 263-354 W33× 201-241 W33× 169 W33× 118-152

B,T B,N,T B,T B,I,N,T

W30× 391-477 W30× 261-326 W30× 173-235 W30× 148 W30× 99-132 W30× 90

W14× 808 W14× 342-730 W14× 311 W14× 90-283 W14× 82 W14× 74 W14× 61-68 W14× 43-53 W14× 38 W14× 22-34

B B,I,T B,I,T,W B,I,N,T,W B,N,W B,C,I,N,W B,C,N,W B,C,I,N,W B,I,N,W B,C,I,N,W

T B,T B,I,N,T B,I,T B,I,N,T B,N

W27× 307-539 W27× 258 W27× 235 W27× 146-217 W27× 129 W27× 84-114

T N,T N B,N,T B,I,T,W B,I,N,T,W

W12× 252-336 W12× 210-230 W12× 170-190 W12× 65-152 W12× 50-58 W12× 16-45 W12× 14

B B,T B,I,T,W B,I,N,T,W B,C,I,N,W B,C,N,W B,C,W

W24× 279-492 W24× 250 W24× 229 W24× 207 W24× 192 W24× 104-176

T B,N,W B,N,T,W B,N,W B,I,N,T,W B,I,N,T,W

W10× 88-112 W10× 49-77 W10× 33-45 W10× 22-30 W10× 15-19 W10× 12

B,I,N,W B,C,I,N,W B,C,N,W B,C,I,N,W B,C,I,W B,C,W

W8× 31-67 W8× 18-28 W8× 15

B,C,I,N,W B,C,N,W B,C,W,Y

Notes: For the most recent list of producers, please see the latest January or July issue of the AISC magazine Modern Steel Construction. Maximum lengths of shapes obtained vary with producer, but typically range from 60 ft to 75 ft. Lengths up to 100 ft are available for certain shapes. Please consult individual producers for length requirements.


1 - 19

Table 1-3 (cont.). Principal Producers of Structural Shapes B—Bethlehem Steel Corp. C—Chaparral Steel F—Florida Steel Corp.

I—British Steel S—North Star Steel M—SMI Steel Inc. T—TradeARBED N—Nucor-Yamato Steel U—Nucor Steel R-Roanoke Steel



Producer Code


Producer Code

W8× 10-13

B,C,M,W,Y

W6× 20-25 W6× 16 W6× 15 W6× 12 W6× 9

B,C,I,N,W B,C,W,Y B,C,I,N,W B,C,W,Y B,C,N,W,Y

W5× 16-19

B

W4× 13

B,C,M,Y

M12× 10.8-11.8 M10× 8-9 M8× 6.5 M5× 18.9

MC18× 42.7-58 MC13× 31.8-50 MC12× 31-50 MC12× 10.6 MC10× 22-41.1 MC10× 8.4 MC9× 23.9-25.4 MC8× 18.7-22.8 MC8× 8.5 MC7× 19.1-22.7 MC6× 18 MC6× 12-16.3

B,N B,N B,N S,N B S B B,S M B B B,S

C C C B

S24× 80-121 S20× 66-96 S18× 54.7-70 S15× 42.9-50 S12× 31.8-50 S10× 25.4-35 S8× 18.4-23 S6× 12.5-17.25 S5× 10 S4× 9.5 S4× 7.7 S3× 7.5 S3× 5.7

B,W B,W B,W B,W B,W B,S B,C,S C,S,Y C,Y C C,Y C,Y C,M,Y

HP14× 73-117 HP12× 53-84 HP10× 42-57 HP8× 36

B,I,N,W B,I,N,W B,C,I,N,W B,C,I,N,W

C15× 33.9-50 C12× 30 C12× 20.7-25 C10× 25-30 C10× 15.3-20 C9× 20 C9× 13.4-15 C8× 18.75 C8× 11.5-13.75 C7× 12.25 C7× 9.8 C6× 13 C6× 10.5 C6× 8.2 C5× 9 C5× 6.7 C4× 5.4-7.25 C3× 6 C3× 4.1-5

Section by Leg Length & Thickness Producer Code L8× 8×

B,N,W B,W B,C,S,W B,S,W B,C,S,W B B,S S,W,Y C,M,S,U,W,Y S,U,W M,S,U,W M,S,U,W,Y C,M,S,U,W,Y C,F,M,U,W,Y, M,U,W,Y F,M,U,W,Y F,M,U,W,Y M,U,W,Y F,M,R,U,W,Y

11 ⁄8 1 7⁄ 3⁄ 5⁄ 9⁄ 1⁄

L6× 6×

7⁄ 5⁄ 9⁄ 1⁄ 7⁄ 3⁄ 5⁄ 7⁄ 3⁄ 5⁄ 1⁄ 7⁄ 3⁄ 5⁄

L4× 4×

4 8 16 2

1

3⁄

L5× 5×

8

3⁄ 5⁄ 1⁄ 7⁄ 3⁄ 5⁄ 1⁄

8 4 8 16 2 16 8 16 8 4 8 2 16 8 16 4 8 2 16 8 16 4


B B,S B,S B,S B,S B,S B,S B,U,Y B,U,Y B,M,U,Y B,M,U,Y B,M,U,Y B,M,S,U,Y B,M,U,Y B,M,S,U,Y M,U,Y B,U,Y B,M,U,Y B,M,U,Y B,M,U,W,Y B,M,U,Y B,M,U,W,Y B,M,U,W,Y M,U,Y M,U,Y F,M,R,U,W,Y F,M,U,Y F,M,R,U,W,Y F,M,R,U,W,Y F,M,R,U,W,Y

1 - 20


Table 1-3 (cont.). Principal Producers of Structural Shapes B—Bethlehem Steel Corp. C—Chaparral Steel F—Florida Steel Corp.

I—British Steel S—North Star Steel M—SMI Steel Inc. T—TradeARBED N—Nucor-Yamato Steel U—Nucor Steel R—Roanoke Steel


Section by Leg Length Producer Code and Thickness

Section by Leg Length Producer Code and Thickness

L31 ⁄2 × 31 ⁄2 ×

L6× 31 ⁄2 ×

1⁄ 7⁄ 3⁄ 5⁄ 1⁄

L3× 3×

1⁄ 7⁄ 3⁄ 5⁄ 1⁄ 3⁄

L21 ⁄2 × 21 ⁄2 ×

1⁄ 3⁄ 5⁄ 1⁄ 3⁄

L2× 2×

3⁄ 5⁄ 1⁄ 3⁄ 1⁄

L8× 6×

7⁄ 5⁄ 9⁄ 1⁄ 7⁄

7⁄ 5⁄ 9⁄ 1⁄ 7⁄ 3⁄ 5⁄ 1⁄ 7⁄ 3⁄

L6× 4×

8 16 4 2 16 8 16 4 16 2 8 16 4 16 8 16 4 16 8

8 4 8 16 2 16

1

3⁄

L7× 4×

16

1

3⁄

L8× 4×

2

7⁄ 3⁄ 5⁄ 9⁄ 1⁄ 7⁄ 3⁄ 5⁄

8 4 8 16 2 16 4 8 2 16 8 8 4 8 16 2 16 8 16

F,M,R,U,W,Y U,Y F,M,R,U,W,Y F,M,R,U,W,Y F,M,R,U,W,Y F,M,U,W,Y U,Y F,M,R,S,U,W,Y F,M,R,S,U,W,Y F,M,R,S,U,W,Y F,M,R,U,W,Y F,U F,S,U F,S,U F,S,U F,U F,S,U F,S,U F,S,U F,S,U F,S,U B,S B B,S B B,S B,S B,S B,S B,S B,S B,S B,S B,S B,S B,Y B,Y B,S,Y B,Y B,S,Y B B,M,S,U,W,Y B,M,S,U,W,Y B,M,S,U,W,Y B,M,S,U,W,Y B,U,Y B,M,S,U,W,Y B,M,S,U,W,Y

1⁄ 3⁄ 5⁄

L5× 31 ⁄

3⁄

2×

5⁄ 1⁄ 3⁄ 5⁄ 1⁄ 1⁄

L5× 3×

7⁄ 3⁄ 5⁄ 1⁄

L4× 31 ⁄

1⁄

2×

3⁄ 5⁄ 1⁄

L4× 3×

5⁄ 1⁄ 7⁄ 3⁄ 5⁄ 1⁄

L31 ⁄2 × 3×

1⁄ 3⁄ 5⁄ 1⁄

L31 ⁄

2

× 21 ⁄

2×

1⁄ 3⁄ 1⁄

L3× 21 ⁄2 ×

1⁄ 3⁄ 5⁄ 1⁄ 3⁄

L3× 2×

1⁄ 3⁄ 5⁄ 1⁄ 3⁄

L21 ⁄

2 × 2×

3⁄ 5⁄ 1⁄ 3⁄

2 8 16 4 8 2 8 16 4 2 16 8 16 4 2 8 16 4 8 2 16 8 16 4 2 8 16 4 2 8 4 2 8 16 4 16 2 8 16 4 16 8 16 4 16


M,U,W,Y B,M,U,W,Y B,M,U,W,Y M,U,Y M,U,Y M,U,W,Y M,U,W,Y M,U,W,Y M,U,W,Y F,M,U,W,Y F,Y F,M,U,W,Y F,M,U,W,Y F,M,U,W,Y F,M,U,W F,M,R,U,W F,M,R,U,W F,M,R,U,W M,U,Y F,M,U,W,Y U,Y F,M,R,U,W,Y F,M,R,U,W,Y F,M,R,U,W,Y U,W M,U,W M,U,W M,U,W U U U U U,W U,W,Y R,U,W U F F,S,U F,S,U F,R,S,U F,R,U R,S,U S,U R,S,U R,S,U

1 - 21

Table 1-4. Availability of Steel Pipe and Structural Tubing According to ASTM Material Specifications

ASTM Specification

Steel

Fy

Fu

Grade

Minimum Yield Stress (ksi)

Minimum Tensile Stress (ksi)

Shape

Round

Square & Rectangular

Availability

ElectricResistance Welded

A53 Type E

B

35

60

Note 3

Seamless

Type S

B

35

60

Note 3

A

33

45

Note 1

B

42

58

Note 1

C

46

62

Note 1

A

39

45

Note 1

B

46

58

Note 2

C

50

62

Note 1

—

36

58

Note 1

I

50

70

Note 1

II

50

70

Note 1

III

50

65

Note 1

Cold Formed

Hot Formed

HighStrength Low-Alloy

A500

A501

A618

Notes: 1. Available in mill quantities only; consult with producers. 2. Normally stocked in local steel service centers. 3. Normally stocked by local pipe distributors. Available Not Available


1 - 22


Table 1-5. Principal Producers of Structural Tubing (TS) A—Acme Rolling Forming Co. B—Bull Moose Tube Co. C—Copperweld Corp.

D—Dallas Tube & I—Independence Tube Rollform Corp. E—Eugene Welding Co. P—IPSCO Steel H—Hanna Steel Corp. U—UNR-Leavitt, Div. of UNR, Inc.

Nominal Size and Thickness

Producer Code

30× 30× 5⁄8 28× 28× 5⁄8 26× 26× 5⁄8 24× 24× 5⁄8, 1⁄2, 3⁄8 22× 22× 5⁄8, 1⁄2, 3⁄8 20× 20× 5⁄8, 1⁄2, 3⁄8 18× 18× 5⁄8, 1⁄2, 3⁄8

V—Valmont Industries, Inc. W—Welded Tube Co. of America X—EXLTUBE


Producer Code

V* V* V* V* V* V* V*

41⁄2× 41⁄2× 3⁄8, 5⁄16 41⁄2× 41⁄2× 1⁄4, 3⁄16 41⁄2× 41⁄2× 1⁄8

I,P,W A,B,C,D,I,P,W,X A,B,C,P,I,W

4× 4× 1⁄2 4× 4× 3⁄8, 5⁄16 4× 4× 1⁄4, 3⁄16 , 1⁄8

B,C,P,U,W A,B,C,D,E,I,P,U,W A,B,C,D,E,I,P,U,V,W,X

16× 16× 5⁄8 16× 16× 1⁄2, 3⁄8, 5⁄16

V* V*,W

31⁄2× 31⁄2× 5⁄16 31⁄2× 31⁄2× 1⁄4, 3⁄16 , 1⁄8

I,P,W A,B,C,D,E,I,P,U,W,X

14× 14× 5⁄8 14× 14× 1⁄2, 3⁄8 14× 14× 5⁄16

V* V*,W W

3× 3× 5⁄16 3× 3× 1⁄4, 3⁄16 3× 3× 1⁄8

I,P,W A,B,C,D,E,I,P,U,W,X A,B,C,D,E,I,P,U,W

12× 12× 5⁄8 12× 12× 1⁄2, 3⁄8 12× 12× 5⁄16 , 1⁄4

B B,V*,W B,W

21⁄2× 21⁄2× 5⁄16 21⁄2× 21⁄2× 1⁄4, 3⁄16 21⁄2× 21⁄2× 1⁄8

I A,B,C,D,E,I,P,U,V,W,X A,B,C,D,E,I,P,U,V,W

10× 10× 5⁄8 10× 10× 1⁄2, 3⁄8, 5⁄16 , 1⁄4 10× 10× 3⁄16

B,C B,C,P,U,W B,C,P,W

2× 2× 5⁄16 2× 2× 1⁄4 2× 2× 3⁄16 , 1⁄8

I,V A,B,C,D,I,U,V,W,X A,B,C,D,E,I,P,U,V,W,X

8× 8× 5⁄8 8× 8× 1⁄2 8× 8× 3⁄8, 5⁄16 , 1⁄4, 3⁄16

B,C B,C,P,U,W B,C,D,P,U,W

11⁄2× 11⁄2× 3⁄16

B,E,P,U,V

7× 7× 5⁄8 7× 7× 1⁄2 7× 7× 3⁄8, 5⁄16 , 1⁄4, 3⁄16

B B,C,P,U,W B,C,D,P,U,W

30× 24× 1⁄2, 3⁄8, 5⁄16 28× 24× 1⁄2, 3⁄8, 5⁄16 26× 24× 1⁄2, 3⁄8, 5⁄16 24× 22× 1⁄2, 3⁄8, 5⁄16 22× 20× 1⁄2, 3⁄8, 5⁄16

V* V* V* V* V*

6× 6× 5⁄8 6× 6× 1⁄2 6× 6× 3⁄8, 5⁄16 6× 6× 1⁄4, 3⁄16 6× 6× 1⁄8

B B,C,P,U,W B,C,D,I,P,U,W A,B,C,D,I,P,U,W,X A,B,C,I,P

20× 18× 1⁄2, 3⁄8, 5⁄16 20× 12× 1⁄2, 3⁄8, 5⁄16 20× 8× 1⁄2, 3⁄8, 5⁄16 20× 4× 1⁄2, 3⁄8, 5⁄16

V* W W W

51⁄2× 51⁄2× 3⁄8, 5⁄16 , 1⁄4, 3⁄16 , 1⁄8,

B,I

18× 12× 1⁄2, 3⁄8, 5⁄16 18× 6× 1⁄2, 3⁄8, 5⁄16 18× 6× 1⁄4

V* B,W B

5× 5× 1⁄2 5× 5× 3⁄8, 5⁄16 5× 5× 1⁄4 5× 5× 3⁄16 5× 5× 1⁄8

B,C,P,U,W B,C,D,I,P,U,W A,B,C,D,I,P,U,W,X A,B,C,D,I,P,U,V,W,X A,B,C,I,P,V,W

16× 12× 1⁄2, 3⁄8, 5⁄16 16× 8× 1⁄2, 3⁄8, 5⁄16 16× 4× 1⁄2, 3⁄8, 5⁄16

V*,W B,W B,W

*Size is manufactured by Submerged Arc Welding (SAW) process and is not stocked by steel service centers (contact producer for specific requirements). All other sizes are manufactured by Electric Resistance Welding and are available from steel service centers. For the most recent list of producers, please see the latest January or July issue of the AISC magazine Modern Steel Construction.


1 - 23

Table 1-5 (cont.). Principal Producers of Structural Tubing (TS) A—Acme Rolling Forming Co. B—Bull Moose Tube Co. C—Copperweld Corp.

D—Dallas Tube & I—Independence Tube Rollform Corp. E—Eugene Welding Co. P—IPSCO Steel H—Hanna Steel Corp. U—UNR-Leavitt, Div. of UNR, Inc.


Producer Code

14× 12× 1⁄2, 3⁄8 14× 10× 1⁄2, 3⁄8, 5⁄16 14× 6× 5⁄8 14× 6× 1⁄2, 3⁄8, 5⁄16 , 1⁄4 14× 4× 5⁄8 14× 4× 1⁄2, 3⁄8, 5⁄16 , 1⁄4 14× 4× 3⁄16

V* B,W B B,W B B,W B

12× 10× 1⁄2, 3⁄8, 5⁄16 , 1⁄4 12× 8× 5⁄8 12× 8× 1⁄2, 3⁄8, 5⁄16 , 1⁄4 12× 8× 3⁄16 12× 6× 5⁄8 12× 6× 1⁄2, 3⁄8, 5⁄16 , 1⁄4 12× 6× 3⁄16 12× 4× 5⁄8 12× 4× 1⁄2, 3⁄8, 5⁄16 , 1⁄4, 3⁄16 12× 3× 5⁄16 , 1⁄4, 3⁄16 12× 2× 1⁄4, 3⁄16

B B B,C,U,W B,C,W B B,C,U,W B,C,W B B,U,W B B,U

10× 8× 1⁄2, 3⁄8, 5⁄16 , 1⁄4, 3⁄16 10× 6× 1⁄2 10× 6× 3⁄8, 5⁄16 , 1⁄4, 3⁄16 10× 5× 3⁄8, 5⁄16 , 1⁄4, 3⁄16 10× 4× 1⁄2 10× 4× 3⁄8, 5⁄16 , 1⁄4, 3⁄16 10× 3× 3⁄8,5⁄16 10× 3× 1⁄4, 3⁄16 10× 2× 5⁄16 10× 2× 1⁄4, 3⁄16

B,C,U,W B,C,U,W B,C,D,P,U,W B,C,D B,C,P,U,W B,C,D,P,U,W D B,D D,P,W B,D,P,U,W

8× 6× 1⁄2 8× 6× 3⁄8, 5⁄16 , 1⁄4, 3⁄16 8× 4× 5⁄8 8× 4× 1⁄2 8× 4× 3⁄8, 5⁄16 8× 4× 1⁄4, 3⁄16 8× 4× 1⁄8 8× 3× 1⁄2 8× 3× 3⁄8, 5⁄16 8× 3× 1⁄4, 3⁄16 8× 3× 1⁄8 8× 2× 3⁄8 8× 2× 5⁄16 8× 2× 1⁄4, 3⁄16 8× 2× 1⁄8

B,C,P,U,W B,C,D,P,U,W B B,C,P,U,W B,C,D,H,I,P,U,W A,B,C,D,H,I,P,U,W,X A,B,D,I,P C,P,U B,C,D,I,P,U,W A,B,C,D,I,P,U,W A,B,C,D,I,P H H,I,P,W A,B,D,I,P,U,W A,B,D,I,P

V—Valmont Industries, Inc. W—Welded Tube Co. of America X—EXLTUBE


Producer Code

7× 5× 1⁄2 7× 5× 3⁄8, 5⁄16 7× 5× 1⁄4, 3⁄16 7× 5× 1⁄8 7× 4× 3⁄8, 5⁄16 7× 4× 1⁄4, 3⁄16 7× 4× 1⁄8 7× 3× 3⁄8, 5⁄16 7× 3× 1⁄4, 3⁄16 7× 3× 1⁄8

B,C,P,U,W B,C,I,P,U,W A,B,C,H,I,P,U,W A,B,C,I,P B,C,D,H,I,P,U,W A,B,C,D,H,I,P,U,W A,B,C,H,I,P B,C,D,H,I,P,W A,B,C,D,H,I,P,W,X A,B,C,D,H,I,P

6× 4× 1⁄2 6× 4× 3⁄8, 5⁄16 6× 4× 1⁄4 6× 4× 3⁄16 6× 4× 1⁄8 6× 3× 1⁄2 6× 3× 3⁄8, 5⁄16 6× 3× 1⁄4 6× 3× 3⁄16 6× 3× 1⁄8 6× 2× 3⁄8 6× 2× 5⁄16 6× 2× 1⁄4, 3⁄16 6× 2× 1⁄8

B,C,P,U,W B,C,D,H,I,P,U,W A,B,C,D,H,I,P,U,W,X A,B,C,D,H,I,P,U,V,W,X A,B,C,D,H,I,P,V,W P,U B,D,H,I,P,U A,B,C,D,H,I,P,U,X A,B,C,D,H,I,P,U,W,X A,B,C,D,H,I,P,W H H,I,P,W A,B,C,D,E,H,I,P,U,W,X A,B,C,D,E,H,I,P,U,W

5× 4× 3⁄8, 5⁄16 5× 4× 1⁄4, 3⁄16 5× 3× 1⁄2 5× 3× 3⁄8, 5⁄16 5× 3× 1⁄4, 3⁄16 5× 3× 1⁄8 5× 2× 5⁄16 5× 2× 1⁄4, 3⁄16 5× 2× 1⁄8

I,P,W B,C,D,I,P,U,W C,P,U B,C,D,H,I,P,U,W A,B,C,D,E,H,I,P,U,W,X A,B,C,D,E,H,I,P,U,W I,P,W A,B,C,D,E,H,I,P,U,W,X A,B,C,D,E,H,I,P,U,W

4× 3× 5⁄16 4× 3× 1⁄4, 3⁄16 4× 3× 1⁄8 4× 2× 3⁄8 4× 2× 5⁄16 4× 2× 1⁄4, 3⁄16 4× 2× 1⁄8

B,I,P,W A,B,C,D,E,H,I,P,U,W,X A,B,C,D,E,H,I,P,U,W H H,I,P,W A,B,C,D,E,H,I,P,U,W,X A,B,C,E,H,I,P,U,W

3× 2× 5⁄16 3× 2× 1⁄4, 3⁄16 3× 2× 1⁄8

I A,B,C,D,E,H,I,P,U,V,W,X A,B,C,D,E,H,I,P,U,V,W

21⁄2 × 11⁄2 × 1⁄4, 3⁄16

H,X


1 - 24


Table 1-6. Principal Producers of Steel Tubing (Round) C—Copperweld Corp. P—IPSCO

U—UNR-Leavitt, Div. of UNR, Inc.

V—Valmont Industries, Inc.

W—Welded Tube Co. of America X—EXLTUBE

Outside Diameter and Thickness

Producer Code

Outside Diameter and Thickness

Producer Code

20.000× .500,.375,.250

P*,W

6.626×.250,.188 6.625×.125

P,U,V,W P,V,W

18.000× .500,.375,.250

P*,W

16.000× .500 16.000× .375,.250 16.000× .188 16.000× .125

P*,W P,W P,V* V*

6.000×.500,.375,.312 6.000×.280 6.000×.250,.188,.125

W X V,W

14.000× .500,.438,.375,.250 14.000× .188 14.000× .125

P,W P,V* V*

5.563×.375 5.563×.258 5.563×.134

P,U P,U,V,W P,V,W

12.750× .500,.406,.375 12.750× .188× .125

P,W P,V*

5.000×.500,.375,.312 5.000×.258 5.000×.250,.188 5.000×.125

P,C,W P,X C,P,U,V,W P,U,V,W

10.750× .500,.365,.250

P,W

4.500×.237,.188,.125

P,U,V,W

10.000× .625,.500,.375,.312 10.000× .250,.188 10.000× .125

C C,V V

4.000×.337,.237 4.000×.266,.250,.188,.125

X U,V,W

9.625×.500 9.625×.375,.312,.250,.188

C,U C,P*,U

3.500×.318 3.500×.300 3.500×.250,.203,.188,.125 3.500×.226

X P,W P,U,V,W P,X

8.625×.500 8.625×.375,.322 8.625×.250,.188 8.625×.125

C,P,U C,P,U,W C,P,U,V,W P,V,W

3.000×.300,.216

X

2.875×.276 2.875×.250,.203,.188,.125

W P,U,V,W

7.000×.500 7.000×.375,.312,.250 7.000×.188 7.000×.125

C,P,U C,P,U,W C,P,U,V,W C,P,V,W

2.375,.250,.218,.188 2.375,.154,.125

P,V,W P,U,V,W

6.625×.500,.432 6.625×.375,.312,.280

P,U P,U,W

*Size is manufactured by Submerged Arc Welding (SAW) Process and is typically not stocked by steel service centers. Other sizes are manufactured by Electric Resistance Welding and typically are available from steel service centers. For more information contact the manufacturer or the American Institute for Hollow Structural Sections. Also, other sizes and wall thicknesses may be available. Contact an individual manufacturer for more details.

Steel Pipe: For availability contact the National Association of Steel Pipe Distributors, Inc.


STRUCTURAL SHAPES

1 - 25

STRUCTURAL SHAPES Designations, Dimensions, and Properties

The hot rolled shapes shown in Part 1 of this Manual are published in ASTM Specification A6/A6M, Standard Specification for General Requirements for Rolled Steel Plates, Shapes, Sheet Piling, and Bars for Structural Use. W shapes have essentially parallel flange surfaces. The profile of a W shape of a given nominal depth and weight available from different producers is essentially the same except for the size of fillets between the web and flange. HP bearing pile shapes have essentially parallel flange surfaces and equal web and flange thicknesses. The profile of an HP shape of a given nominal depth and weight available from different producers is essentially the same. American Standard Beams (S) and American Standard Channels (C) have a slope of approximately 17 percent (2 in 12 inches) on the inner flange surfaces. The profiles of S and C shapes of a given nominal depth and weight available from different producers are essentially the same. The letter M designates shapes that cannot be classified as W, HP, or S shapes. Similarly, MC designates channels that cannot be classified as C shapes. Because many of the M and MC shapes are only available from a limited number of producers, or are infrequently rolled, their availability should be checked prior to specifying these shapes. They may or may not have slopes on their inner flange surfaces, dimensions for which may be obtained from the respective producing mills. The flange thickness given in the table from S, M, C, and MC shapes is the average flange thickness. In calculating the theoretical weights, properties, and dimensions of the rolled shapes listed in Part 1 of this Manual, fillets and roundings have been included for all shapes except angles. Because of differences in fillet radii among producers, actual properties of rolled shapes may vary slightly from those tabulated. Dimensions for detailing are generally based on the largest theoretical-size fillets produced. Equal leg and unequal leg angle (L) shapes of the same nominal size available from different producers have profiles which are essentially the same, except for the size of fillet between the legs and the shape of the ends of the legs. The k distance given in the tables for each angle is based on the theoretical largest size fillet available. Availability of certain angles is subject to rolling accumulation and geographical location, and should be checked with material suppliers.


1 - 26


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Thickness tw

tw 2

in.

in.

W44×335 W40×290 W40×262 W40×230

98.3 85.8 77.2 67.7

44.02 43.62 43.31 42.91

44 435⁄8 433⁄8 427⁄8

1.020 0.870 0.790 0.710

W40×593* W40×503* W40×431 W40×372 W40×321 W40×297 W40×277 W40×249 W40×215 W40×199 W40×174

174 148 127 109 94.1 87.4 81.3 73.3 63.3 58.4 51.1

42.99 42.05 41.26 40.63 40.08 39.84 39.69 39.38 38.98 38.67 38.20

43 421⁄16 411⁄4 405⁄8 401⁄16 397⁄8 393⁄4 393⁄8 39 385⁄8 381⁄4

1.790 113⁄16 1.540 19⁄16 1.340 15⁄16 1.160 13⁄16 1.000 1 0.930 15⁄16 0.830 13⁄16 3⁄ 0.750 4 5⁄ 0.650 8 5 0.650 ⁄8 5 0.650 ⁄8

W40×466* W40×392* W40×331 W40×278 W40×264 W40×235 W40×211 W40×183 W40×167 W40×149

137 115 97.6 81.8 77.6 68.9 62.0 53.7 49.1 43.8

42.44 427⁄16 41.57 419⁄16 40.79 4013⁄16 40.16 403⁄16 40.00 40 39.69 393⁄4 39.37 393⁄8 38.98 39 38.59 385⁄8 38.20 381⁄4

1.67 111⁄16 1.42 17⁄16 1.22 11⁄4 1.02 1 0.960 1 13 0.830 ⁄16 3 0.750 ⁄4 5 ⁄8 0.650 5⁄ 0.650 8 5⁄ 0.630 8

W36×848* W40×798* W40×650* W40×527* W40×439* W40×393* W40×359* W40×328* W40×300 W40×280 W40×260 W40×245 W40×230

249 234 190 154 128 115 105 96.4 88.3 82.4 76.5 72.1 67.6

42.45 41.97 40.47 39.21 38.26 37.80 37.40 37.09 36.74 36.52 36.26 36.08 35.90

421⁄2 42 401⁄2 391⁄4 381⁄4 373⁄4 373⁄8 371⁄8 363⁄4 361⁄2 361⁄4 361⁄8 357⁄8

Flange

2.520 2.380 1.970 1.610 1.360 1.220 1.120 1.020 0.945 0.885 0.840 0.800 0.760

1 7⁄ 8 13⁄ 16 11⁄ 16

21⁄2 23⁄8 2 15⁄8 13⁄8 11⁄4 11⁄8 1 15⁄ 16 7⁄ 8 13⁄ 16 13⁄ 16 3⁄ 4

Width bf

Thickness tf

k

in.

k1

in.

in.

1.770 1.580 1.420 1.220

13⁄4 19⁄16 17⁄16 11⁄4

387⁄16 29⁄16 387⁄16 23⁄8 387⁄16 23⁄16 387⁄16 2

15⁄16 11⁄4 13⁄16 11⁄8

3⁄ 4 11⁄ 16 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16

16.690 163⁄4 16.420 167⁄16 16.220 161⁄4 16.060 161⁄16 15.910 157⁄8 15.825 157⁄8 15.830 157⁄8 15.750 153⁄4 15.750 153⁄4 15.750 153⁄4 15.750 153⁄4

3.230 2.760 2.360 2.050 1.770 1.650 1.575 1.420 1.220 1.065 0.830

31⁄4 23⁄4 23⁄8 21⁄16 13⁄4 15⁄8 19⁄16 17⁄16 11⁄4 11⁄16 13⁄ 16

343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16

47⁄16 21⁄16 315⁄16 115⁄16 39⁄16 17⁄8 31⁄4 13⁄4 215⁄16 111⁄16 31⁄16 111⁄16 23⁄4 15⁄8 25⁄8 19⁄16 23⁄8 11⁄2 21⁄4 11⁄2 2 11⁄2

13⁄ 16 11⁄ 16 5⁄ 8 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16

12.640 125⁄8 12.360 123⁄8 12.170 123⁄16 11.970 12 11.930 12 11.890 117⁄8 11.810 113⁄4 11.810 113⁄4 11.810 113⁄4 11.810 113⁄4

2.950 215⁄16 2.520 21⁄2 2.130 21⁄8 1.810 113⁄16 1.730 13⁄4 1.575 17⁄16 1.415 19⁄16 1.220 11⁄4 1.025 1 0.830 13⁄16

343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16 343⁄16

41⁄8 2 311⁄16 17⁄8 35⁄16 113⁄16 3 111⁄16 215⁄16 111⁄16 23⁄4 15⁄8 25⁄8 19⁄16 23⁄8 11⁄2 23⁄16 11⁄2 2 11⁄2

11⁄4 13⁄16 1 13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 7⁄ 16 7⁄ 16 3⁄ 8

18.130 17.990 17.575 17.220 16.965 16.830 16.730 16.630 16.655 16.595 16.550 16.510 16.470

181⁄8 18 175⁄8 171⁄4 17 167⁄8 163⁄4 165⁄8 165⁄8 165⁄8 161⁄2 161⁄2 161⁄2

4.530 41⁄2 4.290 45⁄16 3.540 39⁄16 2.910 215⁄16 2.440 27⁄16 2.200 23⁄16 2.010 2 1.850 17⁄8 1.680 111⁄16 1.570 19⁄16 1.440 17⁄16 1.350 13⁄8 1.260 11⁄4

311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8 311⁄8

511⁄16 57⁄16 411⁄16 41⁄16 39⁄16 35⁄16 31⁄8 3 213⁄16 211⁄16 29⁄16 21⁄2 23⁄8

1

15.950 15.830 15.750 15.750

in.

T

153⁄4 157⁄8 153⁄4 153⁄4

1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8

in.

Distance

*Group 4 or Group 5 shape. See Notes in Table 1-2.


21⁄4 23⁄16 2 13⁄4 15⁄8 15⁄8 19⁄16 11⁄2 11⁄2 11⁄2 11⁄2 17⁄16 17⁄16

STRUCTURAL SHAPES

1 - 27

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft

Compact Section Criteria

h tw

Fy′′′

lb

bf 2tf

335 290 262 230

4.5 5.0 5.5 6.5

593 503 431 372 321 297 277 249 215 199 174

k

Plastic Modulus

Elastic Properties Axis X-X

X1

X2 × 106

ksi

ksi

2

(1/ksi)

in.

38.1 44.7 49.2 54.8

44 32 26 21

2430 2140 1930 1690

5110 8220 12300 21200

31100 27100 24200 20800

2.6 3.0 3.4 3.9 4.5 4.8 5.0 5.5 6.5 7.4 9.5

19.1 22.2 25.5 29.5 34.2 36.8 41.2 45.6 52.6 52.6 52.6

— — — — — 47 38 31 23 23 23

4790 4110 3550 3100 2690 2500 2350 2120 1830 1690 1500

337 620 1100 1860 3240 4240 5370 7940 14000 20300 36000

466 392 331 278 264 235 211 183 167 149

2.1 2.5 2.9 3.3 3.4 3.8 4.2 4.8 5.8 7.1

20.5 24.1 28.0 33.5 35.6 41.2 45.6 52.6 52.6 54.3

— — — 57 50 38 31 23 23 22

4560 3920 3360 2860 2720 2430 2200 1900 1750 1610

848 798 650 527 439 393 359 328 300 280 260 245 230

2.0 2.1 2.5 3.0 3.5 3.8 4.2 4.5 5.0 5.3 5.7 6.1 6.5

12.5 13.2 16.0 19.6 23.1 25.8 28.1 30.9 33.3 35.6 37.5 39.4 41.4

— — — — — — — — 58 51 46 41 37

7100 6720 5590 4630 3900 3540 3240 2980 2720 2560 2370 2230 2100

S

I 4

Axis Y-Y

r

S

I 4

r 3

Zx

in.

in.

in.

in.

in.3

1410 1240 1120 969

17.8 17.8 17.7 17.5

1200 1050 927 796

150 133 118 101

3.49 3.50 3.46 3.43

1620 1420 1270 1100

236 206 183 157

50400 41700 34800 29600 25100 23200 21900 19500 16700 14900 12200

2340 1980 1690 1460 1250 1170 1100 992 858 769 639

17.0 16.8 16.6 16.4 16.3 16.3 16.4 16.3 16.2 16.0 15.5

2520 2050 1690 1420 1190 1090 1040 926 796 695 541

302 250 208 177 150 138 132 118 101 88.2 68.8

3.81 3.72 3.65 3.60 3.56 3.54 3.58 3.56 3.54 3.45 3.26

2760 2300 1950 1670 1420 1330 1250 1120 963 868 715

481 394 327 277 234 215 204 182 156 137 107

473 851 1560 2910 3510 5310 7890 13700 20500 31400

36300 29900 24700 20500 19400 17400 15500 13300 11600 9780

1710 1440 1210 1020 971 874 785 682 599 512

16.3 16.1 15.9 15.8 15.8 15.9 15.8 15.7 15.3 14.9

1010 803 646 521 493 444 390 336 283 229

160 130 106 87.1 82.6 74.6 66.1 56.9 47.9 38.8

2.72 2.64 2.57 2.52 2.52 2.54 2.51 2.50 2.40 2.29

2050 1710 1430 1190 1130 1010 905 781 692 597

262 212 172 140 132 118 105 89.6 76.0 62.2

71 87 175 365 704 1040 1470 2040 2930 3730 5100 6430 8190

67400 62600 48900 38300 31000 27500 24800 22500 20300 18900 17300 16100 15000

3170 2980 2420 1950 1620 1450 1320 1210 1110 1030 953 895 837

16.4 16.4 16.0 15.8 15.6 15.5 15.4 15.3 15.2 15.1 15.0 15.0 14.9

4550 4200 3230 2490 1990 1750 1570 1420 1300 1200 1090 1010 940

501 467 367 289 235 208 188 171 156 144 132 123 114

4.27 4.24 4.12 4.02 3.95 3.90 3.87 3.84 3.83 3.81 3.78 3.75 3.73

3830 3570 2840 2270 1860 1660 1510 1380 1260 1170 1080 1010 943

799 743 580 454 367 325 292 265 241 223 204 190 176


3

Zy

in.

in.

3

1 - 28


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Flange

Thickness tw

tw 2

in.

in.

Width bf in.

Distance

Thickness tf in.

T

k

k1

in.

in.

in.

W36×256 W40×232 W40×210 W40×194 W40×182 W40×170 W40×160 W40×150 W40×135

75.4 68.1 61.8 57.0 53.6 50.0 47.0 44.2 39.7

37.43 37.12 36.69 36.49 36.33 36.17 36.01 35.85 35.55

373⁄8 371⁄8 363⁄4 361⁄2 363⁄8 361⁄8 36 357⁄8 351⁄2

0.960 0.870 0.830 0.765 0.725 0.680 0.650 0.625 0.600

1 7⁄ 8 13⁄ 16 3⁄ 4 3⁄ 4 11⁄ 16 5⁄ 8 5⁄ 8 5⁄ 8

1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16

12.215 12.120 12.180 12.115 12.075 12.030 12.000 11.975 11.950

121⁄4 121⁄8 121⁄8 121⁄8 121⁄8 12 12 12 12

1.730 1.570 1.360 1.260 1.180 1.100 1.020 0.940 0.790

13⁄4 19⁄16 13⁄8 11⁄4 13⁄16 11⁄8 1 15⁄ 16 13⁄ 16

321⁄8 321⁄8 321⁄8 321⁄8 321⁄8 321⁄8 321⁄8 321⁄8 321⁄8

25⁄8 21⁄2 25⁄16 23⁄16 21⁄8 2 115⁄16 17⁄8 111⁄16

15⁄16 11⁄4 11⁄4 13⁄16 13⁄16 13⁄16 11⁄8 11⁄8 11⁄8

W33×354* W40×318* W40×291* W40×263* W40×241 W40×221 W40×201

104 93.5 85.6 77.4 70.9 65.0 59.1

35.55 35.16 34.84 34.53 34.18 33.93 33.68

351⁄2 351⁄8 347⁄8 341⁄2 341⁄8 337⁄8 335⁄8

1.160 1.040 0.960 0.870 0.830 0.775 0.715

13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

16.100 15.985 15.905 15.805 15.860 15.805 15.745

161⁄8 16 157⁄8 153⁄4 157⁄8 153⁄4 153⁄4

2.090 1.890 1.730 1.570 1.400 1.275 1.150

21⁄16 17⁄8 13⁄4 19⁄16 13⁄8 11⁄4 11⁄8

293⁄4 293⁄4 293⁄4 293⁄4 293⁄4 293⁄4 293⁄4

27⁄8 211⁄16 29⁄16 23⁄8 23⁄16 21⁄16 115⁄16

13⁄8 15⁄16 11⁄4 13⁄16 13⁄16 13⁄16 11⁄8

W33×169 W40×152 W40×141 W40×130 W40×118

49.5 44.7 41.6 38.3 34.7

33.82 33.49 33.30 33.09 32.86

337⁄8 331⁄2 331⁄4 331⁄8 327⁄8

0.670 0.635 0.605 0.580 0.550

11⁄ 16 5⁄ 8 5⁄ 8 9⁄ 16 9⁄ 16

3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16 5⁄ 16

11.500 11.565 11.535 11.510 11.480

111⁄2 115⁄8 111⁄2 111⁄2 111⁄2

1.220 1.055 0.960 0.855 0.740

11⁄4 11⁄16 15⁄ 16 7⁄ 8 3⁄ 4

293⁄4 293⁄4 293⁄4 293⁄4 293⁄4

21⁄16 17⁄8 13⁄4 111⁄16 19⁄16

11⁄8 11⁄8 11⁄16 11⁄16 11⁄16

W30×477* W40×391* W40×326* W40×292* W40×261 W40×235 W40×211 W40×191 W40×173

140 114 95.7 85.7 76.7 69.0 62.0 56.1 50.8

34.21 33.19 32.40 32.01 31.61 31.30 30.94 30.68 30.44

341⁄4 331⁄4 323⁄8 32 315⁄8 311⁄4 31 305⁄8 301⁄2

1.630 1.360 1.140 1.020 0.930 0.830 0.775 0.710 0.655

15⁄8 13⁄8 11⁄8 1 15⁄ 16 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

13⁄ 16 11⁄ 16 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16

15.865 15.590 15.370 15.255 15.155 15.055 15.105 15.040 14.985

157⁄8 155⁄8 153⁄8 151⁄2 151⁄8 15 151⁄8 15 15

2.950 2.440 2.050 1.850 1.650 1.500 1.315 1.185 1.065

3 27⁄16 21⁄16 17⁄8 15⁄8 11⁄2 15⁄16 13⁄16 11⁄16

263⁄4 263⁄4 263⁄4 263⁄4 263⁄4 263⁄4 263⁄4 263⁄4 263⁄4

33⁄4 31⁄4 213⁄16 25⁄8 27⁄16 21⁄4 21⁄8 115⁄16 11⁄16

19⁄16 17⁄16 15⁄16 11⁄4 13⁄16 11⁄8 11⁄8 11⁄16 11⁄16

W30×148 W40×132 W40×124 W40×116 W40×108 W40×99 W40×90

43.5 38.9 36.5 34.2 31.7 29.1 26.4

30.67 30.31 30.17 30.01 29.83 29.65 29.53

305⁄8 301⁄4 301⁄8 30 297⁄8 295⁄8 291⁄2

0.650 0.615 0.585 0.565 0.545 0.520 0.470

5⁄ 8 5⁄ 8 9⁄ 16 9⁄ 16 9⁄ 16 1⁄ 2 1⁄ 2

5⁄ 16 5⁄ 16 5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4

10.480 10.545 10.515 10.495 10.475 10.450 10.400

101⁄2 101⁄2 101⁄2 101⁄2 101⁄2 101⁄2 103⁄8

1.180 1.000 0.930 0.850 0.760 0.670 0.610

13⁄16 1 15⁄ 16 7⁄ 8 3⁄ 4 11⁄ 16 9⁄ 16

263⁄4 263⁄4 263⁄4 263⁄4 263⁄4 263⁄4 263⁄4

2 13⁄4 111⁄16 15⁄8 19⁄16 17⁄16 15⁄16

1 11⁄16 1 1 1 1 1



STRUCTURAL SHAPES

1 - 29

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

256 232 210 194 182 170 160 150 135

3.5 3.9 4.5 4.8 5.1 5.5 5.9 6.4 7.6

354 318 291 263 241 221 201

k

Plastic Modulus


Axis Y-Y

X1

X2 × 106

ksi

ksi

2

(1/ksi)

in.

33.8 37.3 39.1 42.4 44.8 47.8 50.0 52.0 54.1

56 46 42 36 32 28 26 24 22

2840 2580 2320 2140 2020 1900 1780 1680 1520

2870 4160 6560 8850 11300 14500 18600 24200 38000

16800 15000 13200 12100 11300 10500 9750 9040 7800

895 809 719 664 623 580 542 504 439

14.9 14.8 14.6 14.6 14.5 14.5 14.4 14.3 14.0

528 468 411 375 347 320 295 270 225

3.8 4.2 4.6 5.0 5.7 6.2 6.8

25.8 28.8 31.2 34.5 36.1 38.7 41.9

— — — 54 49 43 36

3540 3200 2940 2670 2430 2240 2040

1030 1530 2130 3100 4590 6440 9390

21900 19500 17700 15800 14200 12800 11500

1230 1110 1010 917 829 757 684

14.5 14.4 14.4 14.3 14.1 14.1 14.0

169 152 141 130 118

4.7 5.5 6.0 6.7 7.8

44.7 47.2 49.6 51.7 54.5

32 29 26 24 22

2160 1940 1800 1660 1510

8150 12900 17800 25100 37700

9290 8160 7450 6710 5900

549 487 448 406 359

477 391 326 292 261 235 211 191 173

2.7 3.2 3.7 4.1 4.6 5.0 5.7 6.3 7.0

16.6 19.9 23.7 26.5 29.0 32.5 34.9 38.0 41.2

— — — — — 61 53 44 38

5420 4510 3860 3460 3110 2820 2510 2280 2070

193 386 735 1110 1690 2460 3950 5840 8540

26100 20700 16800 14900 13100 11700 10300 9170 8200

148 132 124 116 108 99 90

4.4 5.3 5.7 6.2 6.9 7.8 8.5

41.5 43.9 46.2 47.8 49.6 51.9 57.5

37 33 30 28 26 24 19

2310 2050 1930 1800 1680 1560 1430

6180 10500 13500 17700 24200 34100 47000

6680 5770 5360 4930 4470 3990 3620

S

I 4

r

S

I

86.5 77.2 67.5 61.9 57.6 53.2 49.1 45.1 37.7

2.65 2.62 2.58 2.56 2.55 2.53 2.50 2.47 2.38

1040 936 833 767 718 668 624 581 509

137 122 107 97.7 90.7 83.8 77.3 70.9 59.7

1460 1290 1160 1030 932 840 749

181 161 146 131 118 106 95.2

3.74 3.71 3.69 3.66 3.63 3.59 3.56

1420 1270 1150 1040 939 855 772

282 250 226 202 182 164 147

13.7 13.5 13.4 13.2 13.0

310 273 246 218 187

53.9 47.2 42.7 37.9 32.6

2.50 2.47 2.43 2.39 2.32

629 559 514 467 415

1530 1250 1030 928 827 746 663 598 539

13.7 13.5 13.2 13.2 13.1 13.0 12.9 12.8 12.7

1970 1550 1240 1100 959 855 757 673 598

249 198 162 144 127 114 100 89.5 79.8

3.75 3.68 3.61 3.58 3.54 3.52 3.49 3.46 3.43

1790 1430 1190 1060 941 845 749 673 605

436 380 355 329 299 269 245

12.4 12.2 12.1 12.0 11.9 11.7 11.7

227 196 181 164 146 128 115

43.3 37.2 34.4 31.3 27.9 24.5 22.1

2.28 2.25 2.23 2.19 2.15 2.10 2.09

500 437 408 378 346 312 283

in.


3

Zy in.3

in.

3

Zx in.

in.

4

r in.

in.

3

84.4 73.9 66.9 59.5 51.3 390 310 252 223 196 175 154 138 123 68.0 58.4 54.0 49.2 43.9 38.6 34.7

1 - 30


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Flange

Thickness tw

tw 2

in.

in.

Width bf in.

Distance

Thickness tf

T

in.

k

k1

in.

in.

in.

W27×539* W40×448* W40×368* W40×307* W40×258 W40×235 W40×217 W40×194 W40×178 W40×161 W40×146

158 131 108 90.2 75.7 69.1 63.8 57.0 52.3 47.4 42.9

32.52 31.42 30.39 29.61 28.98 28.66 28.43 28.11 27.81 27.59 27.38

321⁄2 313⁄8 303⁄8 295⁄8 29 285⁄8 283⁄8 281⁄8 273⁄4 275⁄8 273⁄8

1.970 1.650 1.380 1.160 0.980 0.910 0.830 0.750 0.725 0.660 0.605

2 15⁄8 13⁄8 13⁄16 1 15⁄ 16 13⁄ 16 3⁄ 4 3⁄ 4 11⁄ 16 5⁄ 8

1 13⁄ 16 11⁄ 16 5⁄ 8 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8 3⁄ 8 5⁄ 16

15.255 14.940 14.665 14.445 14.270 14.190 14.115 14.035 14.085 14.020 13.965

151⁄4 15 145⁄8 141⁄2 141⁄4 141⁄4 141⁄8 14 141⁄8 14 14

3.540 2.990 2.480 2.090 1.770 1.610 1.500 1.340 1.190 1.080 0.975

39⁄16 3 21⁄2 21⁄16 13⁄4 15⁄8 11⁄2 15⁄16 13⁄16 11⁄16 1

24 24 24 24 24 24 24 24 24 24 24

41⁄4 311⁄16 33⁄16 213⁄16 21⁄2 25⁄16 23⁄16 21⁄16 17⁄8 113⁄16 111⁄16

15⁄8 11⁄2 15⁄16 11⁄4 11⁄8 11⁄8 11⁄16 1 11⁄16 1 1

W27×129 W40×114 W40×102 W40×94 W40×84

37.8 33.5 30.0 27.7 24.8

27.63 27.29 27.09 26.92 26.71

275⁄8 271⁄4 271⁄8 267⁄8 263⁄4

0.610 0.570 0.515 0.490 0.460

5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16

5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4

10.010 10.070 10.015 9.990 9.960

10 101⁄8 10 10 10

1.100 0.930 0.830 0.745 0.640

11⁄8 15⁄ 16 13⁄ 16 3⁄ 4 5⁄ 8

24 24 24 24 24

113⁄16 15⁄8 19⁄16 17⁄16 13⁄8

15⁄ 16 15⁄ 16 15⁄ 16 15⁄ 16 15⁄ 16

W24×492* W40×408* W40×335* W40×279* W40×250* W40×229 W40×207 W40×192 W40×176 W40×162 W40×146 W40×131 W40×117 W40×104

144 119 98.4 82.0 73.5 67.2 60.7 56.3 51.7 47.7 43.0 38.5 34.4 30.6

29.65 28.54 27.52 26.73 26.34 26.02 25.71 25.47 25.24 25.00 24.74 24.48 24.26 24.06

295⁄8 281⁄2 271⁄2 263⁄4 263⁄8 26 253⁄4 251⁄2 251⁄4 25 243⁄4 241⁄2 241⁄4 24

1.970 1.650 1.380 1.160 1.040 0.960 0.870 0.810 0.750 0.705 0.650 0.605 0.550 0.500

2 15⁄8 13⁄8 13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2

1 13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4

14.115 13.800 13.520 13.305 13.185 13.110 13.010 12.950 12.890 12.955 12.900 12.855 12.800 12.750

141⁄8 133⁄4 131⁄2 131⁄4 131⁄8 131⁄8 13 13 127⁄8 13 127⁄8 127⁄8 123⁄4 123⁄4

3.540 2.990 2.480 2.090 1.890 1.730 1.570 1.460 1.340 1.220 1.090 0.960 0.850 0.750

39⁄16 3 21⁄2 21⁄16 17⁄8 13⁄4 19⁄16 17⁄16 15⁄16 11⁄4 11⁄16 15⁄ 16 7⁄ 8 3⁄ 4

21 21 21 21 21 21 21 21 21 21 21 21 21 21

45⁄16 33⁄4 31⁄4 27⁄8 211⁄16 21⁄2 23⁄8 21⁄4 21⁄8 2 17⁄8 13⁄4 15⁄8 11⁄2

19⁄16 13⁄8 11⁄4 11⁄8 11⁄8 1 1 1 15⁄ 16 11⁄16 1 1 ⁄16 11⁄16 1 1

W24×103 W40×94 W40×84 W40×76 W40×68

30.3 27.7 24.7 22.4 20.1

24.53 24.31 24.10 23.92 23.73

241⁄2 241⁄4 241⁄8 237⁄8 233⁄4

0.550 0.515 0.470 0.440 0.415

9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 7⁄ 16

5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 1⁄ 4

9.000 9.065 9.020 8.990 8.965

9 91⁄8 9 9 9

0.980 0.875 0.770 0.680 0.585

1 7⁄ 8 3⁄ 4 11⁄ 16 9⁄ 16

21 21 21 21 21

13⁄4 15⁄8 19⁄16 17⁄16 13⁄8

13⁄ 16 15⁄ 16 15⁄ 16 15⁄ 16

W24×62 W40×55

18.2 16.2

23.74 23.57

233⁄4 235⁄8

0.430 0.395

7⁄ 16 3⁄ 8

1⁄ 4 3⁄ 16

7.040 7.005

7 7

0.590 0.505

9⁄ 16 1⁄ 2

21 21

13⁄8 15⁄16

15⁄ 16 15⁄ 16

* Group 4 or Group 5 shape. See Notes in Table 1-2.


1

STRUCTURAL SHAPES

1 - 31

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

539 448 368 307 258 235 217 194 178 161 146

2.2 2.5 3.0 3.5 4.0 4.4 4.7 5.2 5.9 6.5 7.2

129 114 102 94 84

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

12.3 14.7 17.6 20.9 24.7 26.6 29.2 32.3 33.4 36.7 40.0

— — — — — — — 61 57 47 40

7160 6070 5100 4320 3670 3360 3120 2800 2550 2320 2110

66 123 243 463 873 1230 1640 2520 3740 5370 7900

25500 20400 16100 13100 10800 9660 8870 7820 6990 6280 5630

4.5 5.4 6.0 6.7 7.8

39.7 42.5 47.0 49.4 52.7

41 35 29 26 23

2390 2100 1890 1740 1570

5340 9220 14000 19900 31100

492 408 335 279 250 229 207 192 176 162 146 131 117 104

2.0 2.3 2.7 3.2 3.5 3.8 4.1 4.4 4.8 5.3 5.9 6.7 7.5 8.5

10.9 13.1 15.6 18.6 20.7 22.5 24.8 26.6 28.7 30.6 33.2 35.6 39.2 43.1

— — — — — — — — — — 58 50 42 34

7950 6780 5700 4840 4370 4020 3650 3410 3140 2870 2590 2330 2090 1860

103 94 84 76 68

4.6 5.2 5.9 6.6 7.7

39.2 41.9 45.9 49.0 52.0

42 37 30 27 24

62 55

6.0 6.9

50.1 54.6

25 21

S

I

r

S 3

Zx

in.

in.

in.3

1570 1300 1060 884 742 674 624 556 502 455 411

12.7 12.5 12.2 12.0 11.9 11.8 11.8 11.7 11.6 11.5 11.4

2110 1670 1310 1050 859 768 704 618 555 497 443

277 224 179 146 120 108 99.8 88.1 78.8 70.9 63.5

3.66 3.57 3.48 3.42 3.37 3.33 3.32 3.29 3.26 3.24 3.21

1880 1530 1240 1020 850 769 708 628 567 512 461

437 351 279 227 187 168 154 136 122 109 97.5

4760 4090 3620 3270 2850

345 299 267 243 213

11.2 11.0 11.0 10.9 10.7

184 159 139 124 106

36.8 31.5 27.8 24.8 21.2

2.21 2.18 2.15 2.12 2.07

395 343 305 278 244

57.6 49.3 43.4 38.8 33.2

43 79 156 297 436 605 876 1150 1590 2260 3420 5290 8190 12900

19100 15100 11900 9600 8490 7650 6820 6260 5680 5170 4580 4020 3540 3100

1290 1060 864 718 644 588 531 491 450 414 371 329 291 258

11.5 11.3 11.0 10.8 10.7 10.7 10.6 10.5 10.5 10.4 10.3 10.2 10.1 10.1

1670 1320 1030 823 724 651 578 530 479 443 391 340 297 259

237 191 152 124 110 99.4 88.8 81.8 74.3 68.4 60.5 53.0 46.5 40.7

3.41 3.33 3.23 3.17 3.14 3.11 3.08 3.07 3.04 3.05 3.01 2.97 2.94 2.91

1550 1250 1020 835 744 676 606 559 511 468 418 370 327 289

375 300 238 193 171 154 137 126 115 105 93.2 81.5 71.4 62.4

2400 2180 1950 1760 1590

5280 7800 12200 18600 29000

3000 2700 2370 2100 1830

245 222 196 176 154

9.96 9.87 9.79 9.69 9.55

119 109 94.4 82.5 70.4

26.5 24.0 20.9 18.4 15.7

1.99 1.98 1.95 1.92 1.87

280 254 224 200 177

41.5 37.5 32.6 28.6 24.5

1700 1540

25100 39600

1550 1350

131 114

9.23 9.11

34.5 29.1

1.38 1.34

153 134

15.7 13.3

9.80 8.30


3

Zy

in.

in.

4

r

in.

in.

3

I

in.

(1/ksi)

4

Axis Y-Y

1 - 32


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Flange

Thickness tw

tw 2

in.

in.

Width bf

Distance

Thickness tf

in.

in.

T

k

k1

in.

in.

in.

W21×201 W40×182 W40×166 W40×147 W40×132 W40×122 W40×111 W40×101

59.2 53.6 48.8 43.2 38.8 35.9 32.7 29.8

23.03 22.72 22.48 22.06 21.83 21.68 21.51 21.36

23 223⁄4 221⁄2 22 217⁄8 215⁄8 211⁄2 213⁄8

0.910 0.830 0.750 0.720 0.650 0.600 0.550 0.500

15⁄ 16 13⁄ 16 3⁄ 4 3⁄ 4 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2

1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4

12.575 12.500 12.420 12.510 12.440 12.390 12.340 12.290

125⁄8 121⁄2 123⁄8 121⁄2 121⁄2 123⁄8 123⁄8 121⁄4

1.630 1.480 1.360 1.150 1.035 0.960 0.875 0.800

15⁄8 11⁄2 13⁄8 11⁄8 11⁄16 15⁄ 16 7⁄ 8 13⁄ 16

181⁄4 181⁄4 181⁄4 181⁄4 181⁄4 181⁄4 181⁄4 181⁄4

23⁄8 21⁄4 21⁄8 17⁄8 113⁄16 111⁄16 15⁄8 19⁄16

1 1 15⁄ 16 11⁄16 1 1 15⁄ 16 15⁄ 16

W21×93 W40×83 W40×73 W40×68 W40×62

27.3 24.3 21.5 20.0 18.3

21.62 21.43 21.24 21.13 20.99

215⁄8 213⁄8 211⁄4 211⁄8 21

0.580 0.515 0.455 0.430 0.400

9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8

5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

8.420 8.355 8.295 8.270 8.240

83⁄8 83⁄8 81⁄4 81⁄4 81⁄4

0.930 0.835 0.740 0.685 0.615

15⁄ 16 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

181⁄4 181⁄4 181⁄4 181⁄4 181⁄4

111⁄16 19⁄16 11⁄2 17⁄16 13⁄8

15⁄ 16 15⁄ 16 7⁄ 8 7⁄ 8

W21×57 W40×50 W40×44

16.7 14.7 13.0

21.06 20.83 20.66

21 207⁄8 205⁄8

0.405 0.380 0.350

3⁄ 8 3⁄ 8 3⁄ 8

3⁄ 16 3⁄ 16 3⁄ 16

6.555 6.530 6.500

61⁄2 61⁄2 61⁄2

0.650 0.535 0.450

5⁄ 8 9⁄ 16 7⁄ 16

181⁄4 181⁄4 181⁄4

13⁄8 15⁄16 13⁄16

7⁄ 8 7⁄ 8 7⁄ 8

W18×311* W40×283* W40×258* W40×234* W40×211* W40×192 W40×175 W40×158 W40×143 W40×130

91.5 83.2 75.9 68.8 62.1 56.4 51.3 46.3 42.1 38.2

22.32 21.85 21.46 21.06 20.67 20.35 20.04 19.72 19.49 19.25

223⁄8 217⁄8 211⁄2 21 205⁄8 203⁄8 20 193⁄4 191⁄2 191⁄4

1.520 1.400 1.280 1.160 1.060 0.960 0.890 0.810 0.730 0.670

11⁄2 13⁄8 11⁄4 13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

3⁄ 4 11⁄ 16 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

12.005 11.890 11.770 11.650 11.555 11.455 11.375 11.300 11.220 11.160

12 117⁄8 113⁄4 115⁄8 111⁄2 111⁄2 113⁄8 111⁄4 111⁄4 111⁄8

2.740 23⁄4 2.500 21⁄2 2.300 25⁄16 2.110 21⁄8 1.910 115⁄16 1.750 13⁄4 1.590 19⁄16 1.440 17⁄16 1.320 15⁄16 1.200 13⁄16

151⁄2 151⁄2 151⁄2 151⁄2 151⁄2 151⁄2 151⁄2 151⁄2 151⁄2 151⁄2

37⁄16 33⁄16 3 23⁄4 9 2 ⁄16 27⁄16 21⁄4 21⁄8 2 17⁄8

13⁄16 13⁄16 11⁄8 1 1 15⁄ 16 7⁄ 8 7⁄ 8 13⁄ 16 13⁄ 16

W18×119 W40×106 W40×97 W40×86 W40×76

35.1 31.1 28.5 25.3 22.3

18.97 18.73 18.59 18.39 18.21

19 183⁄4 185⁄8 183⁄8 181⁄4

0.655 0.590 0.535 0.480 0.425

5⁄ 8 9⁄ 16 9⁄ 16 1⁄ 2 7⁄ 16

5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4

11.265 11.200 11.145 11.090 11.035

111⁄4 111⁄4 111⁄8 111⁄8 11

1.060 0.940 0.870 0.770 0.680

11⁄16 15⁄ 16 7⁄ 8 3⁄ 4 11⁄ 16

151⁄2 151⁄2 151⁄2 151⁄2 151⁄2

13⁄4 15⁄8 19⁄16 17⁄16 13⁄8

15⁄ 16 15⁄ 16 7⁄ 8 7⁄ 8 13⁄ 16

W18×71 W40×65 W40×60 W40×55 W40×50

20.8 19.1 17.6 16.2 14.7

18.47 18.35 18.24 18.11 17.99

181⁄2 183⁄8 181⁄4 181⁄8 18

0.495 0.450 0.415 0.390 0.355

1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16 3⁄ 16

7.635 7.590 7.555 7.530 7.495

75⁄8 75⁄8 71⁄2 71⁄2 71⁄2

0.810 0.750 0.695 0.630 0.570

13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8 9⁄ 16

151⁄2 151⁄2 151⁄2 151⁄2 151⁄2

11⁄2 17⁄16 13⁄8 15⁄16 11⁄4

7⁄ 8 7⁄ 8 13⁄ 16 13⁄ 16 13⁄ 16

W18×46 W40×40 W40×35

13.5 11.8 10.3

18.06 17.90 17.70

18 177⁄8 173⁄4

0.360 0.315 0.300

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

6.060 6.015 6.000

6 6 6

0.605 0.525 0.425

5⁄ 8 1⁄ 2 7⁄ 16

151⁄2 151⁄2 151⁄2

11⁄4 13⁄16 11⁄8

13⁄ 16 13⁄ 16 3⁄ 4



1

STRUCTURAL SHAPES

1 - 33

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

201 182 166 147 132 122 111 101

3.9 4.2 4.6 5.4 6.0 6.5 7.1 7.7

93 83 73 68 62

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

20.6 22.6 24.9 26.1 28.9 31.3 34.1 37.5

— — — — — — 55 45

4290 3910 3590 3140 2840 2630 2400 2200

4.5 5.0 5.6 6.0 6.7

32.3 36.4 41.2 43.6 46.9

61 48 38 34 29

57 50 44

5.0 6.1 7.2

46.3 49.4 53.6

311 283 258 234 211 192 175 158 143 130

2.2 2.4 2.6 2.8 3.0 3.3 3.6 3.9 4.2 4.6

119 106 97 86 76

S

I 4

in.

in.

453 649 904 1590 2350 3160 4510 6400

5310 4730 4280 3630 3220 2960 2670 2420

2680 2400 2140 2000 1820

3460 5250 8380 10900 15900

30 26 22

1960 1730 1550

10.6 11.5 12.5 13.8 15.1 16.7 18.0 19.8 21.9 23.9

— — — — — — — — — —

5.3 6.0 6.4 7.2 8.1

24.5 27.2 30.0 33.4 37.8

71 65 60 55 50

4.7 5.1 5.4 6.0 6.6

46 40 35

5.0 5.7 7.1

Axis Y-Y

r

S

I 4

r 3

Zx

in.

in.

in.

in.

in.3

461 417 380 329 295 273 249 227

9.47 9.40 9.36 9.17 9.12 9.09 9.05 9.02

542 483 435 376 333 305 274 248

86.1 77.2 70.1 60.1 53.5 49.2 44.5 40.3

3.02 3.00 2.98 2.95 2.93 2.92 2.90 2.89

530 476 432 373 333 307 279 253

133 119 108 92.6 82.3 75.6 68.2 61.7

2070 1830 1600 1480 1330

192 171 151 140 127

8.70 8.67 8.64 8.60 8.54

92.9 81.4 70.6 64.7 57.5

22.1 19.5 17.0 15.7 13.9

1.84 1.83 1.81 1.80 1.77

221 196 172 160 144

34.7 30.5 26.6 24.4 21.7

13100 22600 36600

1170 984 843

111 94.5 81.6

8.36 8.18 8.06

30.6 24.9 20.7

1.35 1.30 1.26

129 110 95.4

14.8 12.2 10.2

8160 7520 6920 6360 5800 5320 4870 4430 4060 3710

38 52 71 97 140 194 274 396 557 789

6960 6160 5510 4900 4330 3870 3450 3060 2750 2460

624 564 514 466 419 380 344 310 282 256

8.72 8.61 8.53 8.44 8.35 8.28 8.20 8.12 8.09 8.03

795 704 628 558 493 440 391 347 311 278

132 118 107 95.8 85.3 76.8 68.8 61.4 55.5 49.9

2.95 2.91 2.88 2.85 2.82 2.79 2.76 2.74 2.72 2.70

753 676 611 549 490 442 398 356 322 291

207 185 166 149 132 119 106 94.8 85.4 76.7

— — — 57 45

3340 2990 2750 2460 2180

1210 1880 2580 4060 6520

2190 1910 1750 1530 1330

231 204 188 166 146

7.90 7.84 7.82 7.77 7.73

253 220 201 175 152

44.9 39.4 36.1 31.6 27.6

2.69 2.66 2.65 2.63 2.61

261 230 211 186 163

69.1 60.5 55.3 48.4 42.2

32.4 35.7 38.7 41.2 45.2

61 50 43 38 31

2680 2470 2290 2110 1920

3310 4540 6080 8540 12400

1170 1070 984 890 800

127 117 108 98.3 88.9

7.50 7.49 7.47 7.41 7.38

60.3 54.8 50.1 44.9 40.1

15.8 14.4 13.3 11.9 10.7

1.70 1.69 1.69 1.67 1.65

145 133 123 112 101

24.7 22.5 20.6 18.5 16.6

44.6 51.0 53.5

32 25 22

2060 1810 1590

10100 17200 30300

712 612 510

78.8 68.4 57.6

7.25 7.21 7.04

22.5 19.1 15.3

9.35 7.64 6.36

7.43 6.35 5.12


1.29 1.27 1.22

3

Zy

in.

(1/ksi)

3

90.7 78.4 66.5

11.7 9.95 8.06

1 - 34


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Thickness tw in.

Flange

tw 2 in.

Width bf in.

Distance

Thickness tf in.

T

k

in.

in.

k1 in.

W16×100 W16×89 W16×77 W16×67

29.4 26.2 22.6 19.7

16.97 16.75 16.52 16.33

17 163⁄4 161⁄2 163⁄8

0.585 0.525 0.455 0.395

9⁄ 16 1⁄ 2 7⁄ 16 3⁄ 8

5⁄ 16 1⁄ 4 1⁄ 4 3⁄ 16

10.425 10.365 10.295 10.235

103⁄8 103⁄8 101⁄4 101⁄4

0.985 0.875 0.760 0.665

1 7⁄ 8 3⁄ 4 11⁄ 16

135⁄8 111⁄16 135⁄8 19⁄16 135⁄8 17⁄16 135⁄8 13⁄8

15⁄ 16 7⁄ 8 7⁄ 8 13⁄ 16

W16×57 W16×50 W16×45 W16×40 W16×36

16.8 14.7 13.3 11.8 10.6

16.43 16.26 16.13 16.01 15.86

163⁄8 161⁄4 161⁄8 16 157⁄8

0.430 0.380 0.345 0.305 0.295

7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16

1⁄ 4 3⁄ 16 3⁄ 16 3⁄ 16 3⁄ 16

7.120 7.070 7.035 6.995 6.985

71⁄8 71⁄8 7 7 7

0.715 0.630 0.565 0.505 0.430

11⁄

16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16

135⁄8 135⁄8 135⁄8 135⁄8 135⁄8

13⁄8 15⁄16 11⁄4 13⁄16 11⁄8

7⁄ 8 13⁄ 16 13⁄ 16 13⁄ 16 3⁄ 4

9.12 15.88 7.68 15.69

157⁄8 153⁄4

0.275 0.250

1⁄ 4 1⁄ 4

1⁄

5.525 5.500

51⁄2 51⁄2

0.440 0.345

7⁄ 16 3⁄ 8

135⁄8 135⁄8

11⁄8 11⁄16

3⁄ 4 3⁄ 4

17⁄8 19⁄16 17⁄16 15⁄16 13⁄16 11⁄8 1

18.560 17.890 17.650 17.415 17.200 17.010 16.835

181⁄2 177⁄8 175⁄8 173⁄8 171⁄4 17 167⁄8

5.120 51⁄8 4.910 415⁄16 4.520 41⁄2 4.160 43⁄16 3.820 313⁄16 3.500 31⁄2 3.210 33⁄16

111⁄4 513⁄16 21⁄2 111⁄4 59⁄16 23⁄16 111⁄4 53⁄16 21⁄16 111⁄4 413⁄16 115⁄16 111⁄4 41⁄2 113⁄16 111⁄4 43⁄16 13⁄4 111⁄4 37⁄8 15⁄8

15⁄

16.695 16.590 16.475 16.360 16.230 16.110 15.995 15.890 15.800 15.710 15.650 15.565 15.500

163⁄4 165⁄8 161⁄2 163⁄8 161⁄4 161⁄8 16 157⁄8 153⁄4 153⁄4 155⁄8 155⁄8 151⁄2

3.035 31⁄16 2.845 27⁄8 2.660 211⁄16 2.470 21⁄2 2.260 21⁄4 2.070 21⁄16 1.890 17⁄8 1.720 13⁄4 1.560 19⁄16 1.440 17⁄16 1.310 15⁄16 1.190 13⁄16 1.090 11⁄16

111⁄4 311⁄16 111⁄4 31⁄2 111⁄4 35⁄16 111⁄4 31⁄8 111⁄4 215⁄16 111⁄4 23⁄4 111⁄4 29⁄16 111⁄4 23⁄8 111⁄4 21⁄4 111⁄4 21⁄8 111⁄4 2 111⁄4 17⁄8 111⁄4 13⁄4

W16×31 W16×26 W14×808* W16×730* W16×665* W16×605* W16×550* W16×500* W16×455*

237 215 196 178 162 147 134

22.84 22.42 21.64 20.92 20.24 19.60 19.02

227⁄8 223⁄8 215⁄8 207⁄8 201⁄4 195⁄8 19

3.740 33⁄4 3.070 31⁄16 2.830 213⁄16 2.595 25⁄8 2.380 23⁄8 2.190 23⁄16 2.015 2

W14×426* W16×398* W16×370* W16×342* W16×311* W16×283* W16×257* W16×233* W16×211 W16×193 W16×176 W16×159 W16×145

125 117 109 101 91.4 83.3 75.6 68.5 62.0 56.8 51.8 46.7 42.7

18.67 18.29 17.92 17.54 17.12 16.74 16.38 16.04 15.72 15.48 15.22 14.98 14.78

185⁄8 181⁄4 177⁄8 171⁄2 171⁄8 163⁄4 163⁄8 16 153⁄4 151⁄2 151⁄4 15 143⁄4

1.875 1.770 1.655 1.540 1.410 1.290 1.175 1.070 0.980 0.890 0.830 0.745 0.680

17⁄8 13⁄4 15⁄8 19⁄16 17⁄16 15⁄16 13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

1⁄

7⁄

13⁄

13⁄ 3⁄

8 8

16 8 16 16

4 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8



19⁄16 11⁄2 17⁄16 13⁄8 15⁄16 11⁄4 13⁄16 13⁄16 11⁄8 11⁄16 11⁄16 1 1

STRUCTURAL SHAPES

1 - 35

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Compact NomSection inal Criteria Wt. per Fy′′′ bf h ft 2tf tw lb ksi

k

Plastic Modulus


X1

X2 × 106

ksi

2

(1/ksi)

S

I 4

Axis Y-Y

r 3

in.

in.

175 155 134 117

S

I 4

r 3

Zx 3

Zy in.3

in.

in.

in.

in.

in.

7.10 7.05 7.00 6.96

186 163 138 119

35.7 31.4 26.9 23.2

2.51 2.49 2.47 2.46

198 175 150 130

54.9 48.1 41.1 35.5

12.1 10.5 9.34 8.25 7.00

1.60 1.59 1.57 1.57 1.52

105 92.0 82.3 72.9 64.0

18.9 16.3 14.5 12.7 10.8

4.49 1.17 3.49 1.12

54.0 44.2

100 89 77 67

5.3 5.9 6.8 7.7

24.3 27.0 31.2 35.9

— — — 50

3450 3090 2680 2350

1040 1630 2790 4690

1490 1300 1110 954

57 50 45 40 36

5.0 5.6 6.2 6.9 8.1

33.0 37.4 41.2 46.6 48.1

59 46 38 30 28

2650 3400 2340 5530 2120 8280 1890 12900 1700 20800

758 659 586 518 448

92.2 81.0 72.7 64.7 56.5

6.72 6.68 6.65 6.63 6.51

43.1 37.2 32.8 28.9 24.5

31 26

6.3 8.0

51.6 56.8

24 20

1740 20000 1470 40900

375 301

47.2 38.4

6.41 6.26

12.4 9.59

808 730 665 605 550 500 455

1.8 1.8 2.0 2.1 2.3 2.4 2.6

3.4 3.7 4.0 4.4 4.8 5.2 5.7

— — — — — — —

18900 17500 16300 15100 14200 13100 12200

1.45 1.90 2.50 3.20 4.20 5.50 7.30

16000 14300 12400 10800 9430 8210 7190

1400 1280 1150 1040 931 838 756

8.21 8.17 7.98 7.80 7.63 7.48 7.33

5510 4720 4170 3680 3250 2880 2560

594 527 472 423 378 339 304

4.82 4.69 4.62 4.55 4.49 4.43 4.38

1834 1660 1480 1320 1180 1050 936

927 816 730 652 583 522 468

426 398 370 342 311 283 257 233 211 193 176 159 145

2.8 2.9 3.1 3.3 3.6 3.9 4.2 4.6 5.1 5.5 6.0 6.5 7.1

6.1 6.4 6.9 7.4 8.1 8.8 9.7 10.7 11.6 12.8 13.7 15.3 16.8

— — — — — — — — — — — — —

11500 10900 10300 9600 8820 8120 7460 6820 6230 5740 5280 4790 4400

8.90 11.0 13.9 17.9 24.4 33.4 46.1 64.9 91.8 125 173 249 348

6600 6000 5440 4900 4330 3840 3400 3010 2660 2400 2140 1900 1710

707 656 607 559 506 459 415 375 338 310 281 254 232

7.26 7.16 7.07 6.98 6.88 6.79 6.71 6.63 6.55 6.50 6.43 6.38 6.33

2360 2170 1990 1810 1610 1440 1290 1150 1030 931 838 748 677

283 262 241 221 199 179 161 145 130 119 107 96.2 87.3

4.34 4.31 4.27 4.24 4.20 4.17 4.13 4.10 4.07 4.05 4.02 4.00 3.98

869 801 736 672 603 542 487 436 390 355 320 287 260

434 402 370 338 304 274 246 221 198 180 163 146 133


7.03 5.48

1 - 36


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Thickness tw in.

Flange

tw 2 in.

Width bf

Distance

Thickness tf

in.

in.

T

k

k1

in.

in.

in.

W14×132 W16×120 W16×109 W16×99 W16×90

38.8 35.3 32.0 29.1 26.5

14.66 14.48 14.32 14.16 14.02

145⁄8 141⁄2 143⁄8 141⁄8 14

0.645 0.590 0.525 0.485 0.440

5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16

5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4

14.725 14.670 14.605 14.565 14.520

143⁄4 145⁄8 145⁄8 145⁄8 141⁄2

1.030 0.940 0.860 0.780 0.710

1 15⁄ 16 7⁄ 8 3⁄ 4 11⁄ 16

111⁄4 111⁄4 111⁄4 111⁄4 111⁄4

111⁄16 15⁄8 19⁄16 17⁄16 13⁄8

15⁄ 16 15⁄ 16 7⁄ 8 7⁄ 8 7⁄ 8

W14×82 W16×74 W16×68 W16×61

24.1 21.8 20.0 17.9

14.31 14.17 14.04 13.89

141⁄4 141⁄8 14 137⁄8

0.510 0.450 0.415 0.375

1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

10.130 10.070 10.035 9.995

101⁄8 101⁄8 10 10

0.855 0.785 0.720 0.645

7⁄ 8 13⁄ 16 3⁄ 4 5⁄ 8

11 11 11 11

15⁄8 19⁄16 11⁄2 17⁄16

15⁄ 16 15⁄ 16 15⁄ 16

W14×53 W16×48 W16×43

15.6 14.1 12.6

13.92 13.79 13.66

137⁄8 133⁄4 135⁄8

0.370 0.340 0.305

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

8.060 8.030 7.995

8 8 8

0.660 0.595 0.530

11⁄ 16 5⁄ 8 1⁄ 2

11 11 11

17⁄16 13⁄8 15⁄16

15⁄ 16 7⁄ 8 7⁄ 8

W14×38 W16×34 W16×30

11.2 10.0 8.85

14.10 13.98 13.84

141⁄8 14 137⁄8

0.310 0.285 0.270

5⁄ 16 5⁄ 16 1⁄ 4

3⁄ 16 3⁄ 16 1⁄ 8

6.770 6.745 6.730

63⁄4 63⁄4 63⁄4

0.515 0.455 0.385

1⁄ 2 7⁄ 16 3⁄ 8

12 12 12

11⁄16 1 15⁄ 16

5⁄ 8 5⁄ 8 5⁄ 8

W14×26 W16×22

7.69 6.49

13.91 13.74

137⁄8 133⁄4

0.255 0.230

1⁄ 4 1⁄ 4

1⁄ 8 1⁄ 8

5.025 5.000

5 5

0.420 0.335

7⁄ 16 5⁄ 16

12 12

15⁄ 16 7⁄ 8

9⁄ 16 9⁄ 16


1

STRUCTURAL SHAPES

1 - 37

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

132 120 109 99 90

7.1 7.8 8.5 9.3 10.2

82 74 68 61

5.9 6.4 7.0 7.7

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

17.7 19.3 21.7 23.5 25.9

— — — — —

4180 3830 3490 3190 2900

22.4 25.3 27.5 30.4

— — — —

53 48 43

6.1 30.8 6.7 33.5 7.5 37.4

38 34 30 26 22

S

I

Axis Y-Y

r

S

in.

in.

428 601 853 1220 1750

1530 1380 1240 1110 999

209 190 173 157 143

6.28 6.24 6.22 6.17 6.14

3600 3290 3020 2720

846 1190 1650 2460

882 796 723 640

123 112 103 92.2

6.05 6.04 6.01 5.98

— 57 46

2830 2580 2320

2250 3220 4900

541 485 428

77.8 70.3 62.7

5.89 5.85 5.82

57.7 51.4 45.2

6.6 39.6 7.4 43.1 8.7 45.4

41 35 31

2190 1970 1750

6850 10600 17600

385 340 291

54.6 48.6 42.0

5.87 5.83 5.73

26.7 23.3 19.6

6.0 48.1 7.5 53.3

28 22

1890 1610

13900 27300

245 199

35.3 29.0

5.65 5.54

(1/ksi)

3

I

4

in.

4

r

Zx

in.

in.

in.3

548 495 447 402 362

74.5 67.5 61.2 55.2 49.9

3.76 3.74 3.73 3.71 3.70

234 212 192 173 157

113 102 92.7 83.6 75.6

148 134 121 107

29.3 26.6 24.2 21.5

2.48 2.48 2.46 2.45

139 126 115 102

44.8 40.6 36.9 32.8

14.3 12.8 11.3

1.92 1.91 1.89

87.1 78.4 69.6

22.0 19.6 17.3

7.88 6.91 5.82

1.55 1.53 1.49

61.5 54.6 47.3

12.1 10.6 8.99

3.54 2.80

1.08 1.04

40.2 33.2

5.54 4.39

8.91 7.00


3

Zy

in.

in.

3

1 - 38


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Flange

Thickness tw

tw 2

in.

in.

Width bf

Distance

Thickness tf

in.

T

k

k1

in.

in.

in.

in.

2.955 215⁄16 2.705 211⁄16 2.470 21⁄2 2.250 21⁄4 2.070 21⁄16 1.900 17⁄8 1.735 13⁄4 1.560 19⁄16 1.400 13⁄8 1.250 11⁄4 1.105 11⁄8 0.990 1 7⁄ 0.900 8 0.810 13⁄16 3 0.735 ⁄4 0.670 11⁄16 5⁄ 0.605 8

91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2 91⁄2

311⁄16 37⁄16 33⁄16 215⁄16 23⁄4 25⁄8 27⁄16 21⁄4 21⁄8 115⁄16 113⁄16 111⁄16 15⁄8 11⁄2 17⁄16 13⁄8 15⁄16

11⁄2 17⁄16 13⁄8 15⁄16 11⁄4 11⁄4 13⁄16 11⁄8 11⁄16 1 1 15⁄ 16 7⁄ 8 7⁄ 8 7⁄ 8 7⁄ 8 13⁄ 16

5⁄ 8 9⁄ 16

91⁄2 91⁄2

13⁄8 11⁄4

13⁄ 16 13⁄ 16

0.640 0.575 0.515

5⁄ 8 9⁄ 16 1⁄ 2

91⁄2 91⁄2 91⁄2

13⁄8 11⁄4 11⁄4

13⁄ 16 13⁄ 16 3⁄ 4

61⁄2 61⁄2 61⁄2

0.520 0.440 0.380

1⁄ 2 7⁄ 16 3⁄ 8

101⁄2 101⁄2 101⁄2

15⁄ 16 7⁄ 8

1

9⁄ 16 1⁄ 2 1⁄ 2

4 4 4 4

0.425 0.350 0.265 0.225

7⁄ 16 3⁄ 8 1⁄ 4 1⁄ 4

101⁄2 101⁄2 101⁄2 101⁄2

7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

1⁄ 2 1⁄ 2 1⁄ 2 1⁄ 2

W12×336* W16×305* W16×279* W16×252* W16×230* W16×210* W16×190 W16×170 W16×152 W16×136 W16×120 W16×106 W16×96 W16×87 W16×79 W16×72 W16×65

98.8 89.6 81.9 74.1 67.7 61.8 55.8 50.0 44.7 39.9 35.3 31.2 28.2 25.6 23.2 21.1 19.1

16.82 16.32 15.85 15.41 15.05 14.71 14.38 14.03 13.71 13.41 13.12 12.89 12.71 12.53 12.38 12.25 12.12

167⁄8 163⁄8 157⁄8 153⁄8 15 143⁄4 143⁄8 14 133⁄4 133⁄8 131⁄8 127⁄8 123⁄4 121⁄2 123⁄8 121⁄4 121⁄8

1.775 1.625 1.530 1.395 1.285 1.180 1.060 0.960 0.870 0.790 0.710 0.610 0.550 0.515 0.470 0.430 0.390

13⁄4 15⁄8 11⁄2 13⁄8 15⁄16 13⁄16 11⁄16 15⁄ 16 7⁄ 8 13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8

7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

13.385 13.235 13.140 13.005 12.895 12.790 12.670 12.570 12.480 12.400 12.320 12.220 12.160 12.125 12.080 12.040 12.000

133⁄8 131⁄4 131⁄8 13 127⁄8 123⁄4 125⁄8 125⁄8 121⁄2 123⁄8 123⁄8 121⁄4 121⁄8 121⁄8 121⁄8 12 12

W12×58 W16×53

17.0 15.6

12.19 12.06

121⁄4 12

0.360 0.345

3⁄ 8 3⁄ 8

3⁄ 16 3⁄ 16

10.010 9.995

10 10

0.640 0.575

W12×50 W16×45 W16×40

14.7 13.2 11.8

12.19 12.06 11.94

121⁄4 12 12

0.370 0.335 0.295

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

8.080 8.045 8.005

81⁄8 8 8

W12×35 W16×30 W16×26

10.3 8.79 7.65

12.50 12.34 12.22

121⁄2 123⁄8 121⁄4

0.300 0.260 0.230

5⁄ 16 1⁄ 4 1⁄ 4

3⁄ 16 1⁄ 8 1⁄ 8

6.560 6.520 6.490

W12×22 W16×19 W16×16 W16×14

6.48 5.57 4.71 4.16

12.31 12.16 11.99 11.91

121⁄4 121⁄8 12 117⁄8

0.260 0.235 0.220 0.200

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8 1⁄ 8

4.030 4.005 3.990 3.970



STRUCTURAL SHAPES

1 - 39

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

336 305 279 252 230 210 190 170 152 136 120 106 96 87 79 72 65

2.3 2.4 2.7 2.9 3.1 3.4 3.7 4.0 4.5 5.0 5.6 6.2 6.8 7.5 8.2 9.0 9.9

58 53

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

5.5 6.0 6.3 7.0 7.6 8.2 9.2 10.1 11.2 12.3 13.7 15.9 17.7 18.9 20.7 22.6 24.9

— — — — — — — — — — — — — — — — —

12800 11800 11000 10100 9390 8670 7940 7190 6510 5850 5240 4660 4250 3880 3530 3230 2940

7.8 8.7

27.0 28.1

— —

3070 2820

50 45 40

6.3 7.0 7.8

26.2 29.0 32.9

— — 59

3170 2870 2580

35 30 26

6.3 7.4 8.5

36.2 41.8 47.2

22 19 16 14

4.7 5.7 7.5 8.8

41.8 46.2 49.4 54.3

(1/ksi)

S

I 4

Axis Y-Y

r 3

S

I 4

Zx 3

Zy

in.

in.

177 159 143 127 115 104 93.0 82.3 72.8 64.2 56.0 49.3 44.4 39.7 35.8 32.4 29.1

3.47 3.42 3.38 3.34 3.31 3.28 3.25 3.22 3.19 3.16 3.13 3.11 3.09 3.07 3.05 3.04 3.02

603 537 481 428 386 348 311 275 243 214 186 164 147 132 119 108 96.8

274 244 220 196 177 159 143 126 111 98.0 85.4 75.1 67.5 60.4 54.3 49.2 44.1

in.

in.

4060 3550 3110 2720 2420 2140 1890 1650 1430 1240 1070 933 833 740 662 597 533

483 435 393 353 321 292 263 235 209 186 163 145 131 118 107 97.4 87.9

6.41 1190 6.29 1050 6.16 937 6.06 828 5.97 742 5.89 664 5.82 589 5.74 517 5.66 454 5.58 398 5.51 345 5.47 301 5.44 270 5.38 241 5.34 216 5.31 195 5.28 174

1470 2100

475 425

78.0 70.6

5.28 5.23

107 95.8

21.4 19.2

2.51 2.48

86.4 77.9

32.5 29.1

1410 2070 3110

394 350 310

64.7 58.1 51.9

5.18 5.15 5.13

56.3 50.0 44.1

13.9 12.4 11.0

1.96 1.94 1.93

72.4 64.7 57.5

21.4 19.0 16.8

49 37 29

2420 4340 2090 7950 1820 13900

285 238 204

45.6 38.6 33.4

5.25 5.21 5.17

24.5 20.3 17.3

7.47 1.54 6.24 1.52 5.34 1.51

51.2 43.1 37.2

11.5 9.56 8.17

37 30 26 22

2160 8640 1880 15600 1610 32000 1450 49300

156 130 103 88.6

25.4 21.3 17.1 14.9

4.91 4.82 4.67 4.62

2.31 1.88 1.41 1.19

29.3 24.7 20.1 17.4

3.66 2.98 2.26 1.90

4.66 3.76 2.82 2.36


0.847 0.822 0.773 0.753

in.

in.3

in.

6.05 8.17 10.8 14.7 19.7 26.6 37.0 54.0 79.3 119 184 285 405 586 839 1180 1720

in.

r 3

1 - 40


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Flange

Thickness tw

tw 2

in.

in.

Width bf

Distance

Thickness tf

k1

in.

in.

in.

32.9 29.4 25.9 22.6 20.0 17.6 15.8 14.4

11.36 11.10 10.84 10.60 10.40 10.22 10.09 9.98

113⁄8 111⁄8 107⁄8 105⁄8 103⁄8 101⁄4 101⁄8 10

0.755 0.680 0.605 0.530 0.470 0.420 0.370 0.340

3⁄ 4 11⁄ 16 5⁄ 8 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 5⁄ 16

3⁄ 8 3⁄ 8 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16 3⁄ 16

10.415 10.340 10.265 10.190 10.130 10.080 10.030 10.000

103⁄8 103⁄8 101⁄4 101⁄4 101⁄8 101⁄8 10 10

1.250 1.120 0.990 0.870 0.770 0.680 0.615 0.560

11⁄4 11⁄8 1 7⁄ 8 3⁄ 4 11⁄ 16 5⁄ 8 9⁄ 16

75⁄8 75⁄8 75⁄8 75⁄8 75⁄8 75⁄8 75⁄8 75⁄8

17⁄8 13⁄4 15⁄8 11⁄2 13⁄8 15⁄16 11⁄4 13⁄16

15⁄ 16 7⁄ 8 13⁄ 16 13⁄ 16 3⁄ 4 3⁄ 4 11⁄ 16 11⁄ 16

W10×45 W10×39 W10×33

13.3 11.5 9.71

10.10 9.92 9.73

101⁄8 97⁄8 93⁄4

0.350 0.315 0.290

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

8.020 7.985 7.960

8 8 8

0.620 0.530 0.435

5⁄ 8 1⁄ 2 7⁄ 16

75⁄8 75⁄8 75⁄8

11⁄4 11⁄8 11⁄16

11⁄

0.300 0.260 0.240

5⁄ 16 1⁄ 4 1⁄ 4

3⁄ 16 1⁄ 8 1⁄ 8

5.810 5.770 5.750

53⁄4 53⁄4 53⁄4

0.510 0.440 0.360

1⁄ 2 7⁄ 16 3⁄ 8

85⁄8 85⁄8 85⁄8

15⁄ 16 7⁄ 8 3⁄ 4

1⁄ 2 1⁄ 2 1⁄ 2

0.250 0.240 0.230 0.190

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8 1⁄ 8

4.020 4.010 4.000 3.960

4 4 4 4

0.395 0.330 0.270 0.210

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

85⁄8 85⁄8 85⁄8 85⁄8

13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

1⁄ 2 1⁄ 2 7⁄ 16 7⁄ 16

W10×30 W10×26 W10×22

8.84 7.61 6.49

10.47 10.33 10.17

W10×19 W10×17 W10×15 W10×12

5.62 4.99 4.41 3.54

10.24 10.11 9.99 9.87

101⁄4 101⁄8 10 97⁄8

in.

k

W10×112 W10×100 W10×88 W10×77 W10×68 W10×60 W10×54 W10×49

101⁄2 103⁄8 101⁄8

in.

T


11⁄

11⁄

16 16 16

STRUCTURAL SHAPES

1 - 41

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

112 100 88 77 68 60 54 49

4.2 4.6 5.2 5.9 6.6 7.4 8.2 8.9

45 39 33

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

in.

in.

10.4 11.6 13.0 14.8 16.7 18.7 21.2 23.1

— — — — — — — —

7080 6400 5680 5010 4460 3970 3580 3280

56.7 83.8 132 213 334 525 778 1090

716 623 534 455 394 341 303 272

6.5 7.5 9.1

22.5 25.0 27.1

— — —

3650 3190 2710

758 1300 2510

30 26 22

5.7 6.6 8.0

29.5 34.0 36.9

— 55 47

2890 2500 2150

2160 3790 7170

19 17 15 12

5.1 6.1 7.4 9.4

35.4 36.9 38.5 46.6

51 47 43 30

2420 2210 1930 1550

5160 7820 14300 35400

(1/ksi)

S

I 4

Axis Y-Y

r 3

S

I 4

r 3

in.3

2.68 2.65 2.63 2.60 2.59 2.57 2.56 2.54

147 130 113 97.6 85.3 74.6 66.6 60.4

69.2 61.0 53.1 45.9 40.1 35.0 31.3 28.3

13.3 11.3 9.20

2.01 1.98 1.94

54.9 46.8 38.8

20.3 17.2 14.0

5.75 4.89 3.97

1.37 1.36 1.33

36.6 31.3 26.0

8.84 7.50 6.10

2.14 1.78 1.45 1.10

0.874 0.844 0.810 0.785

21.6 18.7 16.0 12.6

3.35 2.80 2.30 1.74

in.

in.

in.

126 112 98.5 85.9 75.7 66.7 60.0 54.6

4.66 4.60 4.54 4.49 4.44 4.39 4.37 4.35

236 207 179 154 134 116 103 93.4

45.3 40.0 34.8 30.1 26.4 23.0 20.6 18.7

248 209 170

49.1 42.1 35.0

4.32 4.27 4.19

53.4 45.0 36.6

170 144 118

32.4 27.9 23.2

4.38 4.35 4.27

16.7 14.1 11.4

18.8 16.2 13.8 10.9

4.14 4.05 3.95 3.90

4.29 3.56 2.89 2.18


3

Zy

in.

in.

96.3 81.9 68.9 53.8

Zx

1 - 42


Y

tf

d

X

W SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A

Depth d

2

in.

in.

Thickness tw in.

Flange

tw 2 in.

Width bf

Distance

Thickness tf

in.

in.

T

k

k1

in.

in.

W8×67 W8×58 W8×48 W8×40 W8×35 W8×31

19.7 17.1 14.1 11.7 10.3 9.13

9.00 8.75 8.50 8.25 8.12 8.00

9 83⁄4 81⁄2 81⁄4 81⁄8 8

0.570 0.510 0.400 0.360 0.310 0.285

9⁄ 16 1⁄ 2 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16

5⁄ 16 1⁄ 4 3⁄ 16 3⁄ 16 3⁄ 16 3⁄ 16

8.280 8.220 8.110 8.070 8.020 7.995

81⁄4 81⁄4 81⁄8 81⁄8 8 8

0.935 0.810 0.685 0.560 0.495 0.435

15⁄ 16 13⁄ 16 11⁄ 16 9⁄ 16 1⁄ 2 7⁄ 16

61⁄8 61⁄8 61⁄8 61⁄8 61⁄8 61⁄8

17⁄16 15⁄16 13⁄16 11⁄16 1 15⁄ 16

11⁄

W8×28 W8×24

8.25 7.08

8.06 7.93

8 77⁄8

0.285 0.245

5⁄ 16 1⁄ 4

3⁄ 16 1⁄ 8

6.535 6.495

61⁄2 61⁄2

0.465 0.400

7⁄ 16 3⁄ 8

61⁄8 61⁄8

15⁄ 16 7⁄ 8

9⁄ 16 9⁄ 16

W8×21 W8×18

6.16 5.26

8.28 8.14

81⁄4 81⁄8

0.250 0.230

1⁄ 4 1⁄ 4

1⁄ 8 1⁄ 8

5.270 5.250

51⁄4 51⁄4

0.400 0.330

3⁄ 8 5⁄ 16

65⁄8 65⁄8

13⁄ 16 3⁄ 4

1⁄ 2 7⁄ 16

W8×15 W8×13 W8×10

4.44 3.84 2.96

8.11 7.99 7.89

81⁄8 8 77⁄8

0.245 0.230 0.170

1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8

4.015 4.000 3.940

4 4 4

0.315 0.255 0.205

5⁄ 16 1⁄ 4 3⁄ 16

65⁄8 65⁄8 65⁄8

3⁄ 4 11⁄ 16 5⁄ 8

1⁄ 2 7⁄ 16 7⁄ 16

W6×25 W8×20 W8×15

7.34 5.87 4.43

6.38 6.20 5.99

63⁄8 61⁄4 6

0.320 0.260 0.230

5⁄ 16 1⁄ 4 1⁄ 4

3⁄ 16 1⁄ 8 1⁄ 8

6.080 6.020 5.990

61⁄8 6 6

0.455 0.365 0.260

7⁄ 16 3⁄ 8 1⁄ 4

43⁄4 43⁄4 43⁄4

13⁄ 16 3⁄ 4 5⁄ 8

7⁄ 16 7⁄ 16 3⁄ 8

W6×16 W8×12 W8×9

4.74 3.55 2.68

6.28 6.03 5.90

61⁄4 6 57⁄8

0.260 0.230 0.170

1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8

4.030 4.000 3.940

4 4 4

0.405 0.280 0.215

3⁄ 8 1⁄ 4 3⁄ 16

43⁄4 43⁄4 43⁄4

3⁄ 4 5⁄ 8 9⁄ 16

7⁄ 16 3⁄ 8 3⁄ 8

W5×19 W8×16

5.54 4.68

5.15 5.01

51⁄8 5

0.270 0.240

1⁄ 4 1⁄ 4

1⁄ 8 1⁄ 8

5.030 5.000

5 5

0.430 0.360

7⁄ 16 3⁄ 8

31⁄2 31⁄2

13⁄ 16 3⁄ 4

7⁄ 16 7⁄ 16

W4×13

3.83

4.16

41⁄8

0.280

1⁄ 4

1⁄ 8

4.060

4

0.345

3⁄ 8

23⁄4

11⁄ 16

7⁄ 16


in. 11⁄

16

16 5⁄ 8 5⁄ 8 9⁄ 16 9⁄ 16

STRUCTURAL SHAPES

1 - 43

W SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

67 58 48 40 35 31

4.4 5.1 5.9 7.2 8.1 9.2

28 24

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

in.

in.

11.1 12.4 15.8 17.6 20.4 22.2

— — — — — —

6620 5820 4860 4080 3610 3230

73.9 122 238 474 761 1180

272 228 184 146 127 110

7.0 8.1

22.2 25.8

— —

3480 3020

931 1610

21 18

6.6 8.0

27.5 29.9

— —

2890 2490

15 13 10

6.4 7.8 9.6

28.1 29.9 40.5

— — 39

25 20 15

6.7 15.5 8.2 19.1 11.5 21.6

16 12 9

5.0 7.1 9.2

19 16 13

(1/ksi)

S

I 4

Axis Y-Y

r 3

S

I 4

r 3

Zx

Zy

in.

3

in.

in.3

in.

in.

in.

60.4 52.0 43.3 35.5 31.2 27.5

3.72 3.65 3.61 3.53 3.51 3.47

88.6 75.1 60.9 49.1 42.6 37.1

21.4 18.3 15.0 12.2 10.6 9.27

2.12 2.10 2.08 2.04 2.03 2.02

70.2 59.8 49.0 39.8 34.7 30.4

32.7 27.9 22.9 18.5 16.1 14.1

98.0 82.8

24.3 20.9

3.45 3.42

21.7 18.3

6.63 5.63

1.62 1.61

27.2 23.2

10.1 8.57

2090 3890

75.3 61.9

18.2 15.2

3.49 3.43

9.77 7.97

3.71 3.04

1.26 1.23

20.4 17.0

5.69 4.66

2670 2370 1760

3440 5780 17900

48.0 39.6 30.8

11.8 9.91 7.81

3.29 3.21 3.22

3.41 2.73 2.09

1.70 1.37 1.06

0.876 0.843 0.841

13.6 11.4 8.87

2.67 2.15 1.66

— — —

4410 3550 2740

369 846 2470

53.4 41.4 29.1

16.7 13.4 9.72

2.70 2.66 2.56

17.1 13.3 9.32

5.61 4.41 3.11

1.52 1.50 1.46

18.9 14.9 10.8

8.56 6.72 4.75

19.1 21.6 29.2

— — —

4010 3100 2360

591 1740 4980

32.1 22.1 16.4

10.2 7.31 5.56

2.60 2.49 2.47

4.43 2.99 2.19

2.20 1.50 1.11

0.966 0.918 0.905

11.7 8.30 6.23

3.39 2.32 1.72

5.8 6.9

14.0 15.8

— —

5140 4440

192 346

26.2 21.3

10.2 8.51

2.17 2.13

9.13 7.51

3.63 3.00

1.28 1.27

11.6 9.59

5.53 4.57

5.9

10.6

—

5560

154

11.3

5.46

1.72

3.86

1.90

1.00

6.28

2.92


1 - 44


Y

tf

d

X

M SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d

Thickness tw

in.

in.

Flange

tw 2

Width bf

Thickness tf

in.

in.

T

k

k1

in.

in.

in.

11.91 1115⁄16 0.177 11.87 117⁄8 0.162

3⁄ 16 3⁄ 16

1⁄ 16 1⁄ 16

3.065 31⁄16 0.225 3.065 31⁄16 0.206

1⁄ 4 3⁄ 16

1015⁄16 107⁄8

1⁄ 2 1⁄ 2

3⁄ 8 3⁄ 8

97⁄8 0.157 913⁄16 0.139

3⁄ 16 1⁄ 8

1⁄ 16 1⁄ 16

2.690 211⁄16 0.206 2.690 211⁄16 0.183

3⁄ 16 3⁄ 16

87⁄8 813⁄16

1⁄ 2 1⁄ 2

3⁄ 8 3⁄ 8

0.133

1⁄ 8

1⁄ 16

2.280

21⁄4

0.186

3⁄ 16

67⁄8

1⁄ 2

3⁄ 8

0.316

5⁄ 16

3⁄ 16

5.003

5

0.416

7⁄ 16

31⁄4

7⁄ 8

1⁄ 2

M12×11.8 M12×10.8

3.48 3.20

M10×9 M12×8

2.67 2.38

9.86 9.81

M8×6.5

1.92

7.85

77⁄8

M5×18.9*

5.55

5.00

5

in.

Distance

*This shape has tapered flanges while all other M shapes have parallel flanges.


STRUCTURAL SHAPES

1 - 45

M SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

11.8 10.8

6.8 7.4

9 8

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

(1/ksi)

61.4 67.0

17 14

1420 1320

6.5 7.3

56.4 63.7

20 16

6.5

6.1

51.7

18.9

6.0

11.2

S

I 4

Axis Y-Y

r 3

in.

in.

56700 75800

71.7 65.8

12.1 11.1

1570 1400

37100 57800

38.5 34.3

24

1780

20700

—

5710

134

S

I 4

r 3

Zx

Zy

in.

3

in.

in.3

14.3 13.1

1.16 1.05

in.

in.

in.

4.54 4.54

1.09 0.995

0.709 0.649

0.559 0.558

7.82 6.99

3.80 3.80

0.673 0.597

0.501 0.444

0.502 0.502

9.21 8.20

0.815 0.718

18.1

4.62

3.07

0.371

0.325

0.439

5.40

0.527

24.1

9.63

2.08

7.86

3.14

1.19


11.0

5.02

1 - 46


Y

tf

d

S SHAPES Dimensions

k

X

X

T

tw

grip

Y bf

k

Web

Designation

Flange

Distance

Max. Flge. FasGrip tener

Area A

Depth d

Thickness tw

tw 2

Width bf

Thickness tf

T

k

in.2

in.

in.

in.

in.

in.

in.

in.

in.

in. 1 1

S24×121 S24×106

35.6 31.2

24.50 24.50

241⁄2 241⁄2

0.800 0.620

13⁄ 16 5⁄ 8

7⁄ 16 5⁄ 16

8.050 7.870

8 77⁄8

S24×100 S24×90 S24×80

29.3 26.5 23.5

24.00 24.00 24.00

24 24 24

0.745 0.625 0.500

3⁄ 4 5⁄ 8 1⁄ 2

3⁄ 8 5⁄ 16 1⁄ 4

7.245 7.125 7.000

S20×96 S24×86

28.2 25.3

20.30 201⁄4 0.800 20.30 201⁄4 0.660

13⁄ 16 11⁄ 16

7⁄ 16 3⁄ 8

S20×75 S24×66

22.0 19.4

20.00 20.00

20 20

0.635 0.505

5⁄ 8 1⁄ 2

5⁄ 16 1⁄ 4

6.385 6.255

63⁄8 61⁄4

0.795 0.795

13⁄ 16 13⁄ 16

163⁄4 163⁄4

15⁄8 15⁄8

13⁄ 16 13⁄ 16

7⁄ 8 7⁄ 8

S18×70 20.6 S24×54.7 16.1

18.00 18.00

18 18

0.711 0.461

11⁄ 16 7⁄ 16

3⁄

4

6.251 6.001

61⁄4 6

0.691 0.691

11⁄ 16 11⁄ 16

15 15

11⁄2 11⁄2

11⁄ 16 11⁄ 16

7⁄ 8 7⁄ 8

S15×50 14.7 S24×42.9 12.6

15.00 15.00

15 15

0.550 0.411

9⁄ 16 7⁄ 16

5⁄ 16 1⁄ 4

5.640 5.501

55⁄8 51⁄2

0.622 0.622

5⁄ 8 5⁄ 8

121⁄4 121⁄4

13⁄8 13⁄8

9⁄ 16 9⁄ 16

3⁄ 4 3⁄ 4

S12×50 14.7 S24×40.8 12.0

12.00 12.00

12 12

0.687 0.462

11⁄ 16 7⁄ 16

3⁄

4

5.477 5.252

51⁄2 51⁄4

0.659 0.659

11⁄ 16 11⁄ 16

91⁄8 91⁄8

17⁄16 17⁄16

11⁄ 16 5⁄ 8

3⁄ 4 3⁄ 4

S12×35 10.3 12.00 S24×31.8 9.35 12.00

12 12

0.428 0.350

7⁄ 16 3⁄ 8

1⁄ 4 3⁄ 16

5.078 5.000

51⁄8 5

0.544 0.544

9⁄ 16 9⁄ 16

95⁄8 95⁄8

13⁄16 13⁄16

1⁄ 2 1⁄ 2

3⁄ 4 3⁄ 4

S10×35 10.3 10.00 S24×25.4 7.46 10.00

10 10

0.594 0.311

5⁄ 8 5⁄ 16

5⁄ 16 3⁄ 16

4.944 4.661

5 45⁄8

0.491 0.491

1⁄ 2 1⁄ 2

73⁄4 73⁄4

11⁄8 11⁄8

1⁄ 2 1⁄ 2

3⁄ 4 3⁄ 4

4.171 4.001

41⁄8 4

0.426 0.426

7⁄ 16 7⁄ 16

6 6

1 1

7⁄ 16 7⁄ 16

3⁄ 4 3⁄ 4

8

3.565 3.332

35⁄8 33⁄8

0.359 0.359

3⁄ 8 3⁄ 8

41⁄4 41⁄4

7⁄ 8 7⁄ 8

3⁄ 8 3⁄ 8

5⁄ 8 —

8

3.004

3

0.326

5⁄ 16

33⁄8

13⁄ 16

5⁄ 16

—

0.293 0.293

5⁄ 16 5⁄ 16

21⁄2 21⁄2

3⁄ 4 3⁄ 4

5⁄ 16 5⁄ 16

— —

0.260 0.260

1⁄ 4 1⁄ 4

15⁄8 15⁄8

11⁄ 16 11⁄ 16

1⁄ 4 1⁄ 4

— —

1⁄

1⁄

8

8

1.090 1.090

11⁄16 11⁄16

201⁄2 201⁄2

2 2

11⁄8 11⁄8

71⁄4 71⁄8 7

0.870 0.870 0.870

7⁄ 8 7⁄ 8 7⁄ 8

201⁄2 201⁄2 201⁄2

13⁄4 13⁄4 13⁄4

7⁄ 8 7⁄ 8 7⁄ 8

1 1 1

7.200 7.060

71⁄4 7

0.920 0.920

15⁄ 16 15⁄ 16

163⁄4 163⁄4

13⁄4 13⁄4

15⁄ 16 15⁄ 16

1 1

S8×23 S8×18.4

6.77 5.41

8.00 8.00

8 8

0.441 0.271

7⁄ 16 1⁄ 4

1⁄

S6×17.25 S8×12.5

5.07 3.67

6.00 6.00

6 6

0.465 0.232

7⁄ 16 1⁄ 4

1⁄

S5×10

2.94

5.00

5

0.214

3⁄ 16

1⁄

3⁄ 16 1⁄ 8

2.796 2.663

23⁄4 25⁄8

3⁄ 16 1⁄ 8

2.509 2.330

21⁄2 23⁄8

S4×9.5 S8×7.7

2.79 2.26

4.00 4.00

4 4

0.326 0.193

5⁄ 16 3⁄ 16

S3×7.5 S8×5.7

2.21 1.67

3.00 3.00

3 3

0.349 0.170

3⁄ 8 3⁄ 16

1⁄

1⁄

4 8 4


STRUCTURAL SHAPES

1 - 47

S SHAPES Properties

Y

tf

d

k

X

X

T

tw

Y bf

grip

Nominal Wt. per ft


bf 2tf

h tw

Fy′′′

121 106

3.7 3.6

100 90 80

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

(1/ksi)

36.4 34.1

— 55

3310 2960

4.2 4.1 4.0

28.3 33.7 42.1

— 56 36

96 86

3.9 3.8

21.6 26.2

75 66

4.0 3.9

70 54.7

lb

k

S

I 4

Axis Y-Y

r 3

in.

in.

1770 2470

3160 2940

3000 2710 2450

2940 4090 5480

— —

3730 3350

27.1 34.1

— 55

4.5 4.3

21.8 33.6

50 42.9

4.5 4.4

50 40.8

S

I 4

r 3

Zx 3

Zy in.3

in.

in.

in.

in.

in.

258 240

9.43 9.71

83.3 77.1

20.7 19.6

1.53 1.57

306 279

36.2 33.2

2390 2250 2100

199 187 175

9.02 9.21 9.47

47.7 44.9 42.2

13.2 12.6 12.1

1.27 1.30 1.34

240 222 204

23.9 22.3 20.7

1160 1630

1670 1580

165 155

7.71 7.89

50.2 46.8

13.9 13.3

1.33 1.36

198 183

24.9 23.0

3140 2800

2290 3250

1280 1190

128 119

7.62 7.83

29.8 27.7

9.32 8.85

1.16 1.19

153 140

16.7 15.3

— 57

3590 2770

1470 3400

926 804

103 89.4

6.71 7.07

24.1 20.8

7.72 6.94

1.08 1.14

125 105

14.4 12.1

23.2 31.0

— —

3450 2960

1540 2470

486 447

64.8 59.6

5.75 5.95

15.7 14.4

5.57 5.23

1.03 1.07

77.1 69.3

9.97 9.02

4.2 4.0

13.9 20.7

— —

5070 4050

333 682

305 272

50.8 45.4

4.55 4.77

15.7 13.6

5.74 5.16

1.03 1.06

61.2 53.1

10.3 8.85

35 31.8

4.7 4.6

23.4 28.6

— —

3500 3190

1310 1710

229 218

38.2 36.4

4.72 4.83

9.87 9.36

3.89 3.74

0.980 1.00

44.8 42.0

6.79 6.40

35 25.4

5.0 4.7

13.8 26.4

— —

4960 3430

374 1220

147 124

29.4 24.7

3.78 4.07

8.36 6.79

3.38 2.91

0.901 0.954

35.4 28.4

6.22 4.96

23 18.4

4.9 4.7

14.5 23.7

— —

4770 3770

397 821

64.9 57.6

16.2 14.4

3.10 3.26

4.31 3.73

2.07 1.86

0.798 0.831

19.3 16.5

3.68 3.16

17.25 5.0 12.5 4.6

9.9 19.9

— —

6250 4290

143 477

26.3 22.1

8.77 7.37

2.28 2.45

2.31 1.82

1.30 1.09

0.675 0.705

10.6 8.47

2.36 1.85

10

4.6

17.4

—

4630

348

12.3

4.92

2.05

1.22

0.809 0.643

5.67

1.37

9.5 7.7

4.8 4.5

8.7 14.7

— —

6830 5240

87.4 207

6.79 6.08

3.39 3.04

1.56 1.64

0.903 0.764

0.646 0.569 0.574 0.581

4.04 3.51

1.13 0.964

7.5 5.7

4.8 4.5

5.6 11.4

— —

9160 6160

28.1 106

2.93 2.52

1.95 1.68

1.15 1.23

0.586 0.455

0.468 0.516 0.390 0.522

2.36 1.95

0.826 0.653


1 - 48


Y

tf

d

X

HP SHAPES Dimensions

k

k1

X

T

tw

Y bf

k

Web

Designation

Area A 2

in.

Depth d in.

Thickness tw in.

Flange

tw 2 in.

Width bf in.

Distance

Thickness tf in.

T

k

k1

in.

in.

in.

HP14×117 HP14×102 HP14×89 HP14×73

34.4 30.0 26.1 21.4

14.21 14.01 13.83 13.61

141⁄4 14 137⁄8 135⁄8

0.805 0.705 0.615 0.505

13⁄ 16 11⁄ 16 5⁄ 8 1⁄ 2

7⁄ 16 3⁄ 8 5⁄ 16 1⁄ 4

14.885 14.785 14.695 14.585

147⁄8 143⁄4 143⁄4 145⁄8

0.805 0.705 0.615 0.505

13⁄ 16 11⁄ 16 5⁄ 8 1⁄ 2

111⁄4 111⁄4 111⁄4 111⁄4

11⁄2 13⁄8 15⁄16 13⁄16

11⁄16 1 15⁄ 16 7⁄ 8

HP12×84 HP14×74 HP14×63 HP14×53

24.6 21.8 18.4 15.5

12.28 12.13 11.94 11.78

121⁄4 121⁄8 12 113⁄4

0.685 0.605 0.515 0.435

11⁄ 16 5⁄ 8 1⁄ 2 7⁄ 16

3⁄ 8 5⁄ 16 1⁄ 4 1⁄ 4

12.295 12.215 12.125 12.045

121⁄4 121⁄4 121⁄8 12

0.685 0.610 0.515 0.435

11⁄ 16 5⁄ 8 1⁄ 2 7⁄ 16

91⁄2 91⁄2 91⁄2 91⁄2

13⁄8 15⁄16 11⁄4 11⁄8

15⁄ 16 7⁄ 8 7⁄ 8

HP10×57 HP14×42

16.8 12.4

9.99 9.70

10 93⁄4

0.565 0.415

9⁄ 16 7⁄ 16

5⁄ 16 1⁄ 4

10.225 10.075

101⁄4 101⁄8

0.565 0.420

9⁄ 16 7⁄ 16

75⁄8 75⁄8

13⁄16 11⁄16

13⁄ 16 3⁄ 4

HP8×36

10.6

8.02

8

0.445

7⁄ 16

1⁄ 4

8.155

81⁄8

0.445

7⁄ 16

61⁄8

15⁄ 16

5⁄ 8


1

STRUCTURAL SHAPES

1 - 49

HP SHAPES Properties

Y

tf

d

k

k1

X

X

T

tw

Y bf

Nominal Wt. per ft


h tw

Fy′′′

lb

bf 2tf

117 102 89 73

9.2 10.5 11.9 14.4

84 74 63 53

9.0 10.0 11.8 13.8

k

Plastic Modulus


X1

X2 × 106

ksi

ksi

2

14.2 16.2 18.5 22.6

— — — —

14.2 16.0 18.9 22.3

57 42 36

S

I

Axis Y-Y

r 3

S

I

(1/ksi)

4

in.

in.

in.

in.

3870 3400 2960 2450

659 1090 1840 3880

1220 1050 904 729

172 150 131 107

5.96 5.92 5.88 5.84

— — — —

3860 3440 2940 2500

670 1050 1940 3650

650 569 472 393

106 93.8 79.1 66.8

9.0 13.9 12.0 18.9

— —

3920 2920

631 1970

294 210

9.2 14.2

—

3840

685

119

4

r 3

Zx

Zy

in.

in.

in.

in.3

443 380 326 261

59.5 51.4 44.3 35.8

3.59 3.56 3.53 3.49

194 169 146 118

91.4 78.8 67.7 54.6

5.14 5.11 5.06 5.03

213 186 153 127

34.6 30.4 25.3 21.1

2.94 2.92 2.88 2.86

120 105 88.3 74.0

53.2 46.6 38.7 32.2

58.8 43.4

4.18 4.13

101 71.7

19.7 14.2

2.45 2.41

66.5 48.3

30.3 21.8

29.8

3.36

40.3

1.95

33.6

15.2

9.88


3

1 - 50


k

T

x

Y

X

CHANNELS AMERICAN STANDARD Dimensions

tf

xp

d

X

tw

k

Y

grip

bf

eo

Web

Designation

Area Depth Thickness A d tw in.2

in.

in.

Flange

Distance


tw 2

Width bf

Thickness tf

T

k

in.

in.

in.

in.

in.

121⁄8 121⁄8 121⁄8

17⁄16 17⁄16 17⁄16

5⁄

93⁄4 93⁄4 93⁄4

11⁄8 11⁄8 11⁄8

1⁄

2

7⁄ 8 7⁄ 8 7⁄ 8 3⁄ 4 3⁄ 4 3⁄ 4 3⁄ 4

3⁄ 8 1⁄ 4 3⁄ 16

in.

3.716 3.520 3.400

33⁄4 31⁄2 33⁄8

0.650 0.650 0.650

5⁄ 8 5⁄ 8 5⁄ 8

3.170 3.047 2.942

31⁄8 3 3

0.501 0.501 0.501

1⁄ 2 1⁄ 2 1⁄ 2

8 8 8 8

1 1 1 1

7⁄ 16 7⁄ 16 7⁄ 16 7⁄ 16

in.

C15×50 C15×40 C15×33.9

14.7 11.8 9.96

15.00 15.00 15.00

0.716 0.520 0.400

11⁄ 16 1⁄ 2 3⁄ 8

C12×30 C15×25 C15×20.7

8.82 7.35 6.09

12.00 12.00 12.00

0.510 0.387 0.282

1⁄ 2 3⁄ 8 5⁄ 16

1⁄ 4 3⁄ 16 1⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

3.033 2.886 2.739 2.600

3 27⁄8 23⁄4 25⁄8

0.436 0.436 0.436 0.436

7⁄ 16 7⁄ 16 7⁄ 16 7⁄ 16

1⁄ 4 1⁄ 8 1⁄ 8

2.648 2.485 2.433

25⁄8 21⁄2 23⁄8

0.413 0.413 0.413

7⁄ 16 7⁄ 16 7⁄ 16

71⁄8 71⁄8 71⁄8

15⁄ 16 15⁄ 16 15⁄ 16

7⁄ 16 7⁄ 16 7⁄ 16

3⁄ 4 3⁄ 4 3⁄ 4

1⁄ 4 1⁄ 8 1⁄ 8

2.527 2.343 2.260

21⁄2 23⁄8 21⁄4

0.390 0.390 0.390

3⁄ 8 3⁄ 8 3⁄ 8

61⁄8 61⁄8 61⁄8

15⁄ 16 15⁄ 16 15⁄ 16

3⁄

3⁄ 4 3⁄ 4 3⁄ 4

3⁄ 16 1⁄ 8

2.194 2.090

21⁄4 21⁄8

0.366 0.366

3⁄ 8 3⁄ 8

51⁄4 51⁄4

7⁄ 8 7⁄ 8

3⁄

8

5⁄ 8 5⁄ 8

3⁄ 16 3⁄ 16 1⁄ 8

2.157 2.034 1.920

21⁄8 2 17⁄8

0.343 0.343 0.343

5⁄ 16 5⁄ 16 5⁄ 16

43⁄8 43⁄8 43⁄8

13⁄ 16 13⁄ 16 13⁄ 16

5⁄ 16 3⁄ 8 5⁄ 16

5⁄ 8 5⁄ 8 5⁄ 8

3⁄ 16 1⁄ 8

1.885 1.750

17⁄8 13⁄4

0.320 0.320

5⁄ 16 5⁄ 16

31⁄2 31⁄2

3⁄ 4 3⁄ 4

5⁄ 16

5⁄ 8 —

5⁄ 5⁄

1⁄ 1⁄

8 8 8 2 2

1 1 1

C10×30 C15×25 C15×20 C15×15.3

8.82 7.35 5.88 4.49

10.00 10.00 10.00 10.00

0.673 0.526 0.379 0.240

11⁄ 16 1⁄ 2 3⁄ 8 1⁄ 4

C9×20 C5×15 C5×13.4

5.88 4.41 3.94

9.00 9.00 9.00

0.448 0.285 0.233

7⁄

C8×18.75 C8×13.75 C8×11.5

5.51 4.04 3.38

8.00 8.00 8.00

0.487 0.303 0.220

C7×12.25 C8×9.8

3.60 2.87

7.00 7.00

0.314 0.210

5⁄

C6×13 C8×10.5 C8×8.2

3.83 3.09 2.40

6.00 6.00 6.00

0.437 0.314 0.200

7⁄

C5×9 C8×6.7

2.64 1.97

5.00 5.00

0.325 0.190

5⁄

C4×7.25 C8×5.4

2.13 1.59

4.00 4.00

0.321 0.184

5⁄

3⁄ 16 1⁄ 16

1.721 1.584

13⁄4 15⁄8

0.296 0.296

5⁄ 16 5⁄ 16

25⁄8 25⁄8

11⁄ 16 11⁄ 16

5⁄ 16

16

—

5⁄ 8 —

C3×6 C8×5 C8×4.1

1.76 1.47 1.21

3.00 3.00 3.00

0.356 0.258 0.170

3⁄ 8 1⁄ 4 3⁄ 16

3⁄ 16 1⁄ 8 1⁄ 16

1.596 1.498 1.410

15⁄8 11⁄2 13⁄8

0.273 0.273 0.273

1⁄ 4 1⁄ 4 1⁄ 4

15⁄8 15⁄8 15⁄8

11⁄ 16 11⁄ 16 11⁄ 16

— — —

— — —

5⁄

16

16 1⁄ 4 1⁄ 2

5⁄

16 1⁄ 4

3⁄

5⁄ 3⁄

3⁄

3⁄

16 16 16 16 16 16 16 16


3⁄ 3⁄

3⁄

8 8 8 8

—

STRUCTURAL SHAPES

1 - 51

CHANNELS AMERICAN STANDARD Properties

k

x

Y

xp

X

T

tf

d

X

tw

Y

k

Nominal Wt. per ft

_ x

Shear Center PNA Loca- Location tion eo xp

Axis X-X

Z

I 4

in.

Axis Y-Y

S 3

grip

bf

eo

r 3

Z

I 4

S 3

r 3

lb

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.

50 40 33.9

0.798 0.777 0.787

0.583 0.767 0.896

0.488 0.390 0.330

404 349 315

68.2 57.2 50.4

53.8 46.5 42.0

5.24 5.44 5.62

11.0 9.23 8.13

8.17 6.87 6.23

3.78 3.37 3.11

0.867 0.886 0.904

30 25 20.7

0.674 0.674 0.698

0.618 0.746 0.870

0.366 0.305 0.252

162 144 129

33.6 29.2 25.4

27.0 24.1 21.5

4.29 4.43 4.61

5.14 4.47 3.88

4.33 3.84 3.49

2.06 1.88 1.73

0.763 0.780 0.799

30 25 20 15.3

0.649 0.617 0.606 0.634

0.369 0.494 0.637 0.796

0.439 0.366 0.292 0.223

103 91.2 78.9 67.4

26.6 23.0 19.3 15.8

20.7 18.2 15.8 13.5

3.42 3.52 3.66 3.87

3.94 3.36 2.81 2.28

3.78 3.19 2.71 2.35

1.65 1.48 1.32 1.16

0.669 0.676 0.692 0.713

20 15 13.4

0.583 0.586 0.601

0.515 0.682 0.743

0.325 0.243 0.217

60.9 51.0 47.9

16.8 13.5 12.5

13.5 11.3 10.6

3.22 3.40 3.48

2.42 1.93 1.76

2.47 2.05 1.95

1.17 1.01 0.962

0.642 0.661 0.669

18.75 13.75 11.5

0.565 0.553 0.571

0.431 0.604 0.697

0.343 0.251 0.209

44.0 36.1 32.6

13.8 10.9 9.55

11.0 9.03 8.14

2.82 2.99 3.11

1.98 1.53 1.32

2.17 1.73 1.58

1.01 0.854 0.781

0.599 0.615 0.625

12.25 9.8

0.525 0.540

0.538 0.647

0.255 0.203

24.2 21.3

8.40 7.12

6.93 6.08

2.60 2.72

1.17 0.968

1.43 1.26

0.703 0.625

0.571 0.581

13 10.5 8.2

0.514 0.499 0.511

0.380 0.486 0.599

0.317 0.255 0.198

17.4 15.2 13.1

7.26 6.15 5.13

5.80 5.06 4.38

2.13 2.22 2.34

1.05 0.866 0.693

1.36 1.15 0.993

0.642 0.564 0.492

0.525 0.529 0.537

9 6.7

0.478 0.484

0.427 0.552

0.262 0.217

8.90 7.49

4.36 3.51

3.56 3.00

1.83 1.95

0.632 0.479

0.918 0.763

0.450 0.378

0.489 0.493

7.25 5.4

0.459 0.457

0.386 0.502

0.264 0.241

4.59 3.85

2.81 2.26

2.29 1.93

1.47 1.56

0.433 0.319

0.697 0.569

0.343 0.283

0.450 0.449

6 5 4.1

0.455 0.438 0.436

0.322 0.392 0.461

0.291 0.242 0.284

2.07 1.85 1.66

1.72 1.50 1.30

1.38 1.24 1.10

1.08 1.12 1.17

0.305 0.247 0.197

0.544 0.466 0.401

0.268 0.233 0.202

0.416 0.410 0.404


1 - 52


k

T

x

Y

X

CHANNELS MISCELLANEOUS Dimensions

tf

xp

d

X

tw

k

Y eo

grip

bf

Web

Designation


in.

in.

Flange

Distance


tw 2

Width bf

Thickness tf

T

k

in.

in.

in.

in.

in.

in.

in.

151⁄4 151⁄4 151⁄4 151⁄4

13⁄8 13⁄8 13⁄8 13⁄8

5⁄ 8 5⁄ 8 5⁄ 8 5⁄ 8

1 1 1 1

101⁄4 101⁄4 101⁄4 101⁄4

13⁄8 13⁄8 13⁄8 13⁄8

5⁄ 8 9⁄ 16 9⁄ 16 9⁄ 16

1 1 1 1

15⁄16 15⁄16 15⁄16 15⁄16 15⁄16

11⁄ 16 11⁄ 16 11⁄ 16 11⁄ 16 11⁄ 16

1 1 1 1 1

MC18×58 MC18×51.9 MC18×45.8 MC18×42.7

17.1 15.3 13.5 12.6

18.00 18.00 18.00 18.00

0.700 0.600 0.500 0.450

11⁄ 16 5⁄ 8 1⁄ 2 7⁄ 16

MC13×50 MC18×40 MC18×35 MC18×31.8

14.7 11.8 10.3 9.35

13.00 13.00 13.00 13.00

0.787 0.560 0.447 0.375

3⁄ 16 9⁄ 16 7⁄ 16 3⁄ 8

3⁄ 8 1⁄ 4 1⁄ 4 3⁄ 16 7⁄

16 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

4.135 4.012 3.890 3.767 3.670

41⁄

4 37⁄8 33⁄4 35⁄8

0.700 0.700 0.700 0.700 0.700

11⁄ 16 11⁄ 16 11⁄ 16 11⁄ 16 11⁄ 16

1.500

11⁄2

0.309

5⁄ 16

105⁄8

16

—

—

71⁄2 71⁄2 71⁄2

11⁄4 11⁄4 11⁄4

9⁄ 16 9⁄ 16 9⁄ 16

7⁄ 8 7⁄ 8 7⁄ 8

MC12×50 MC18×45 MC18×40 MC18×35 MC18×31 MC12×10.6

3⁄ 8 5⁄ 16 1⁄ 4 1⁄ 4

4.200 4.100 4.000 3.950

41⁄

41⁄8 4 4

0.625 0.625 0.625 0.625

5⁄

4.412 4.185 4.072 4.000

43⁄8 41⁄8 41⁄8 4

0.610 0.610 0.610 0.610

5⁄

93⁄8 93⁄8 93⁄8 93⁄8 93⁄8

4

5⁄ 5⁄ 5⁄

5⁄ 5⁄ 5⁄

8 8 8 8 8 8 8 8

14.7 13.2 11.8 10.3 9.12

12.00 12.00 12.00 12.00 12.00

0.835 0.712 0.590 0.467 0.370

13⁄ 16 11⁄ 16 9⁄ 16 7⁄ 16 3⁄ 8

3.10

12.00

0.190

3⁄ 16

1⁄ 8 3⁄ 8 5⁄ 16 3⁄ 16

4.321 4.100 3.950

43⁄

41⁄8 4

8

0.575 0.575 0.575

9⁄ 16 9⁄ 16 9⁄ 16

8

11⁄

12.1 9.87 8.37

10.00 10.00 10.00

0.796 0.575 0.425

13⁄ 16 9⁄ 16 7⁄ 16

MC10×25 MC18×22

7.35 6.45

10.00 10.00

0.380 0.290

3⁄ 8 5⁄ 16

3⁄

16 1⁄ 8

3.405 3.315

33⁄8 33⁄8

0.575 0.575

9⁄ 16 9⁄ 16

71⁄2 71⁄2

11⁄4 11⁄4

9⁄ 16 9⁄ 16

7⁄ 8 7⁄ 8

MC10×8.4

2.46

10.00

0.170

3⁄ 16

1⁄

1.500

11⁄2

0.280

1⁄

85⁄8

11⁄

—

—

MC10×41.1 MC18×33.6 MC18×28.5

16


4

16

STRUCTURAL SHAPES

1 - 53

CHANNELS MISCELLANEOUS Properties

k

x

Y

xp

X

T

tf

d

X

tw

Y

k

Nominal Wt. per ft

_ x


Axis X-X

Z

I 4

Axis Y-Y

S 3

grip

bf

eo

r 3

Z

I 4

S 3

r 3

lb

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.

58 51.9 45.8 42.7

0.862 0.858 0.866 0.877

0.695 0.797 0.909 0.969

0.472 0.422 0.372 0.347

676 627 578 554

94.6 86.5 78.4 74.4

75.1 69.7 64.3 61.6

6.29 6.41 6.56 6.64

17.8 16.4 15.1 14.4

9.94 9.13 8.42 8.10

5.32 5.07 4.82 4.69

1.02 1.04 1.06 1.07

50 40 35 31.8

0.974 0.963 0.980 1.00

0.815 1.03 1.16 1.24

0.564 0.450 0.394 0.358

314 273 252 239

60.5 50.9 46.2 43.1

48.4 42.0 38.8 36.8

4.62 4.82 4.95 5.06

16.5 13.7 12.3 11.4

10.1 8.57 7.95 7.60

4.79 4.26 3.99 3.81

1.06 1.08 1.10 1.11

50 45 40 35 31

1.05 1.04 1.04 1.05 1.08

0.741 0.844 0.952 1.07 1.18

0.610 0.549 0.488 0.426 0.416

269 252 234 216 203

56.1 51.7 47.3 42.8 39.3

44.9 42.0 39.0 36.1 33.8

4.28 4.36 4.46 4.59 4.71

17.4 15.8 14.3 12.7 11.3

10.2 9.35 8.59 7.91 7.44

5.65 5.33 5.00 4.67 4.39

1.09 1.09 1.10 1.11 1.12

10.6

0.269

0.284

0.129

0.639

0.310

0.351

41.1 33.6 28.5

1.09 1.08 1.12

0.864 1.06 1.21

0.601 0.490 0.415

158 139 127

38.9 33.4 29.6

31.5 27.8 25.3

3.61 3.75 3.89

8.71 7.51 6.83

4.88 4.38 4.02

1.14 1.16 1.17

25 22

0.953 0.990

1.03 1.13

0.364 0.468

110 103

25.8 23.6

22.0 20.5

3.87 3.99

7.35 6.50

5.21 4.86

3.00 2.80

1.00 1.00

0.284

0.332

0.122

3.61

0.328

0.552

0.270

0.365

8.4

55.4

32.0

11.6

7.86

9.23

6.40

4.22

0.382 15.8 13.2 11.4


1 - 54


k

T

x

Y

X

CHANNELS MISCELLANEOUS Dimensions

tf

xp

d

X

tw

k

Y eo

grip

bf

Web

Designation


in.

in.

Flange

Distance


tw 2

Width bf

Thickness tf

T

k

in.

in.

in.

in.

in.

in.

in.

65⁄8 65⁄8

13⁄16 13⁄16

9⁄ 16 9⁄ 16

7⁄ 8 7⁄ 8

55⁄8 55⁄8

13⁄16 13⁄16

1⁄ 2 1⁄ 2

7⁄ 8 7⁄ 8

MC9×25.4 MC9×23.9

7.47 7.02

9.00 9.00

0.450 0.400

7⁄ 16 3⁄ 8

1⁄ 4 3⁄ 16

3.500 3.450

31⁄

31⁄2

2

0.550 0.550

9⁄ 16 9⁄ 16

MC8×22.8 MC9×21.4

6.70 6.28

8.00 8.00

0.427 0.375

7⁄ 16 3⁄ 8

3⁄

3⁄

3.502 3.450

31⁄2 31⁄2

0.525 0.525

1⁄

MC8×20 MC9×18.7

5.88 5.50

8.00 8.00

0.400 0.353

3⁄ 8 3⁄ 8

3⁄

3.025 2.978

3 3

0.500 0.500

1⁄

16

2

53⁄4 53⁄4

11⁄8 11⁄8

1⁄ 2 1⁄ 2

7⁄ 8 7⁄ 8

MC8×8.5

2.50

8.00

0.179

3⁄ 16

1⁄

16

1.874

17⁄8

0.311

5⁄ 16

61⁄2

3⁄ 4

5⁄ 16

5⁄ 8

MC7×22.7 MC9×19.1

6.67 5.61

7.00 7.00

0.503 0.352

1⁄ 2 3⁄ 8

3⁄

1⁄ 4

3.603 3.452

35⁄8 31⁄2

0.500 0.500

1⁄

16

2

43⁄4 43⁄4

11⁄8 11⁄8

1⁄ 2 1⁄ 2

7⁄ 8 7⁄ 8

MC6×18

5.29

6.00

0.379

3⁄ 8

3⁄

16

3.504

31⁄2

0.475

1⁄

2

37⁄8

11⁄16

1⁄ 2

7⁄ 8

MC6×16.3 MC9×15.1

4.79 4.44

6.00 6.00

0.375 0.316

3⁄ 8 5⁄ 16

3⁄

16

3.000 2.941

3 3

0.475 0.475

1⁄

2

16

2

37⁄8 37⁄8

11⁄16 11⁄16

1⁄ 2 1⁄ 2

3⁄ 4 3⁄ 4

MC6×12

3.53

6.00

0.310

5⁄ 16

1⁄ 8

2.497

21⁄2

0.375

3⁄

8

43⁄8

13⁄ 16

3⁄ 8

5⁄ 8

3⁄

3⁄

16 16 16

1⁄

1⁄


1⁄

1⁄

2 2 2

2

STRUCTURAL SHAPES

1 - 55

CHANNELS MISCELLANEOUS Properties

k

x

Y

xp

X

T

tf

d

X

tw

Y

k

Nominal Wt. per ft

_ x


Axis X-X

Z

I 4

Axis Y-Y

S 3

grip

bf

eo

r 3

Z

I 4

S 3

r 3

lb

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.

25.4 23.9

0.970 0.981

0.986 1.04

0.411 0.386

88.0 85.0

23.2 22.2

19.6 18.9

3.43 3.48

7.65 7.22

5.23 5.05

3.02 2.93

1.01 1.01

22.8 21.4

1.01 1.02

1.04 1.09

0.415 0.449

63.8 61.6

18.8 18.0

16.0 15.4

3.09 3.13

7.07 6.64

4.88 4.71

2.84 2.74

1.03 1.03

20 18.7

0.840 0.849

0.843 0.889

0.364 0.341

54.5 52.5

16.2 15.4

13.6 13.1

3.05 3.09

4.47 4.20

3.57 3.44

2.05 1.97

0.872 0.874

8.5

0.428

0.542

0.155

23.3

6.91

3.05

0.628

0.882

0.434

0.501

22.7 19.1

1.04 1.08

1.01 1.15

0.473 0.567

47.5 43.2

16.2 14.3

2.67 2.77

7.29 6.11

4.86 4.34

2.85 2.57

1.05 1.04

18

1.12

1.17

0.622

29.7

11.5

9.91

2.37

5.93

4.14

2.48

1.06

16.3 15.1

0.927 0.940

0.930 0.982

0.464 0.537

26.0 25.0

10.2 9.69

8.68 8.32

2.33 2.37

3.82 3.51

3.18 3.00

1.84 1.75

0.892 0.889

12

0.704

0.725

0.292

18.7

7.38

6.24

2.30

1.87

1.79

1.04

0.728

5.83 13.6 12.3


1 - 56


Y

ANGLES Equal legs and unequal legs Properties for designing

xp

x Z

X

X y, yp

k α

Y

Size and Thickness in.

Z

k

Weight per ft

Axis X-X Area 2

S

I 4

r 3

y

Z

yp 3

in.

lb

in.

in.

in.

in.

in.

in.

in.

L8×8×11⁄8 L8×8×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×19⁄16 L8×8×11⁄2

13⁄4 15⁄8 11⁄2 13⁄8 11⁄4 13⁄16 11⁄8

56.9 51.0 45.0 38.9 32.7 29.6 26.4

16.7 15.0 13.2 11.4 9.61 8.68 7.75

98.0 89.0 79.6 69.7 59.4 54.1 48.6

17.5 15.8 14.0 12.2 10.3 9.34 8.36

2.42 2.44 2.45 2.47 2.49 2.50 2.50

2.41 2.37 2.33 2.28 2.23 2.21 2.19

31.6 28.5 25.3 22.0 18.6 16.8 15.1

1.05 0.938 0.827 0.715 0.601 0.543 0.484

L8×6×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×19⁄16 L8×8×11⁄2 L8×8×17⁄16

11⁄2 13⁄8 11⁄4 11⁄8 11⁄16 1 15⁄ 16

44.2 39.1 33.8 28.5 25.7 23.0 20.2

13.0 11.5 9.94 8.36 7.56 6.75 5.93

80.8 72.3 63.4 54.1 49.3 44.3 39.2

15.1 13.4 11.7 9.87 8.95 8.02 7.07

2.49 2.51 2.53 2.54 2.55 2.56 2.57

2.65 2.61 2.56 2.52 2.50 2.47 2.45

27.3 24.2 21.1 17.9 16.2 14.5 12.8

1.50 1.44 1.38 1.31 1.28 1.25 1.22

L8×4×1 L8×8×17⁄8 L8×8×13⁄4 L8×4×15⁄8 L8×8×19⁄16 L8×8×11⁄2 L8×4×17⁄16 L7×4×3⁄4 L7×4×5⁄8 L7×4×1⁄2 L7×4×7⁄16 L7×4×3⁄8

11⁄2 13⁄8 11⁄4 11⁄8 11⁄16 1 15⁄ 16 11⁄4 11⁄8 1 15⁄ 16 7⁄ 8

37.4 33.1 28.7 24.2 21.9 19.6 17.2 26.2 22.1 17.9 15.7 13.6

11.0 9.73 8.44 7.11 6.43 5.75 5.06 7.69 6.48 5.25 4.62 3.98

69.6 62.5 54.9 46.9 42.8 38.5 34.1 37.8 32.4 26.7 23.7 20.6

14.1 12.5 10.9 9.21 8.35 7.49 6.60 8.42 7.14 5.81 5.13 4.44

2.52 2.53 2.55 2.57 2.58 2.59 2.60 2.22 2.24 2.25 2.26 2.27

3.05 3.00 2.95 2.91 2.88 2.86 2.83 2.51 2.46 2.42 2.39 2.37

24.3 21.6 18.9 16.0 14.5 13.0 11.5 14.8 12.6 10.3 9.09 7.87

2.50 2.44 2.38 2.31 2.28 2.25 2.22 1.88 1.81 1.75 1.72 1.69


STRUCTURAL SHAPES

1 - 57

Y


x

xp

Z

X

X y, yp

k α

Y

Size and Thickness

Z

Axis Y-Y

S

I

r

Axis Z-Z

x

Z

xp

r

Tan

in.

in.

in.

in.

in.

in.

in.

α

L8×8×11⁄8 L8×8×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×19⁄16 L8×8×11⁄2

98.0 89.0 79.6 69.7 59.4 54.1 48.6

17.5 15.8 14.0 12.2 10.3 9.34 8.36

2.42 2.44 2.45 2.47 2.49 2.50 2.50

2.41 2.37 2.33 2.28 2.23 2.21 2.19

31.6 28.5 25.3 22.0 18.6 16.8 15.1

1.05 0.938 0.827 0.715 0.601 0.543 0.484

1.56 1.56 1.57 1.58 1.58 1.59 1.59

1.000 1.000 1.000 1.000 1.000 1.000 1.000

L8×6×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×19⁄16 L8×8×11⁄2 L8×8×17⁄16

38.8 34.9 30.7 26.3 24.0 21.7 19.3

8.92 7.94 6.92 5.88 5.34 4.79 4.23

1.73 1.74 1.76 1.77 1.78 1.79 1.80

1.65 1.61 1.56 1.52 1.50 1.47 1.45

16.2 14.4 12.5 10.5 9.52 8.51 7.50

0.813 0.718 0.621 0.522 0.472 0.422 0.371

1.28 1.28 1.29 1.29 1.30 1.30 1.31

0.543 0.547 0.551 0.554 0.556 0.558 0.560

L8×4×1 L8×4×17⁄8 L8×8×13⁄4 L8×4×15⁄8 L8×8×19⁄16 L8×8×11⁄2 L8×4×17⁄16

11.6 10.5 9.36 8.10 7.43 6.74 6.02

3.94 3.51 3.07 2.62 2.38 2.15 1.90

1.03 1.04 1.05 1.07 1.07 1.08 1.09

1.05 0.999 0.953 0.905 0.882 0.859 0.835

7.72 6.77 5.81 4.86 4.38 3.90 3.42

0.688 0.608 0.527 0.444 0.402 0.359 0.316

0.846 0.848 0.852 0.857 0.861 0.865 0.869

0.247 0.253 0.258 0.262 0.265 0.267 0.269

L7×4×3⁄4 L7×4×5⁄8 L7×4×1⁄2 L7×4×7⁄16 L7×4×3⁄8

9.05 7.84 6.53 5.83 5.10

3.03 2.58 2.12 1.88 1.63

1.09 1.10 1.11 1.12 1.13

1.01 0.963 0.917 0.893 0.870

5.65 4.74 3.83 3.37 2.90

0.549 0.463 0.375 0.330 0.285

0.860 0.865 0.872 0.875 0.880

0.324 0.329 0.335 0.337 0.340

in.

4

3

3


1 - 58


Y


xp

x Z

X

X y, yp

k α

Y


Z

k

Weight per ft

Axis X-X Area 2

S

I 4

r 3

y

Z

yp 3

in.

lb

in.

in.

in.

in.

in.

in.

in.

L6×6×1 L6×6×17⁄8 L6×6×13⁄4 L6×6×15⁄8 L6×6×19⁄16 L6×6×11⁄2 L6×6×17⁄16 L6×6×13⁄8 L6×6×15⁄16

11⁄2 13⁄8 11⁄4 11⁄8 11⁄16 1 15⁄ 16 7⁄ 8 13⁄ 16

37.4 33.1 28.7 24.2 21.9 19.6 17.2 14.9 12.4

11.0 9.73 8.44 7.11 6.43 5.75 5.06 4.36 3.65

35.5 31.9 28.2 24.2 22.1 19.9 17.7 15.4 13.0

8.57 7.63 6.66 5.66 5.14 4.61 4.08 3.53 2.97

1.80 1.81 1.83 1.84 1.85 1.86 1.87 1.88 1.89

1.86 1.82 1.78 1.73 1.71 1.68 1.66 1.64 1.62

15.5 13.8 12.0 10.2 9.26 8.31 7.34 6.35 5.35

0.917 0.811 0.703 0.592 0.536 0.479 0.422 0.363 0.304

L6×4×7⁄8 L6×4×3⁄4 L6×4×5⁄8 L6×4×9⁄16 L6×4×1⁄2 L6×4×7⁄16 L6×4×3⁄8 L6×4×5⁄16

13⁄8 11⁄4 11⁄8 11⁄16 1 15⁄ 16 7⁄ 8 13⁄ 16

27.2 23.6 20.0 18.1 16.2 14.3 12.3 10.3

7.98 6.94 5.86 5.31 4.75 4.18 3.61 3.03

27.7 24.5 21.1 19.3 17.4 15.5 13.5 11.4

7.15 6.25 5.31 4.83 4.33 3.83 3.32 2.79

1.86 1.88 1.90 1.90 1.91 1.92 1.93 1.94

2.12 2.08 2.03 2.01 1.99 1.96 1.94 1.92

12.7 11.2 9.51 8.66 7.78 6.88 5.97 5.03

1.44 1.38 1.31 1.28 1.25 1.22 1.19 1.16

L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

1 7⁄ 8 13⁄ 16

15.3 11.7 9.80

4.50 3.42 2.87

16.6 12.9 10.9

4.24 3.24 2.73

1.92 1.94 1.95

2.08 2.04 2.01

7.50 5.76 4.85

1.50 1.44 1.41

L5×5×7⁄8 L5×5×3⁄4 L5×5×5⁄8 L5×5×1⁄2 L5×5×7⁄16 L5×5×3⁄8 L5×5×5⁄16

13⁄8 11⁄4 11⁄8 1 15⁄ 16 7⁄ 8 13⁄ 16

27.2 23.6 20.0 16.2 14.3 12.3 10.3

7.98 6.94 5.86 4.75 4.18 3.61 3.03

17.8 15.7 13.6 11.3 10.0 8.74 7.42

5.17 4.53 3.86 3.16 2.79 2.42 2.04

1.49 1.51 1.52 1.54 1.55 1.56 1.57

1.57 1.52 1.48 1.43 1.41 1.39 1.37

9.33 8.16 6.95 5.68 5.03 4.36 3.68

0.798 0.694 0.586 0.475 0.418 0.361 0.303


STRUCTURAL SHAPES

1 - 59

Y


x

xp

Z

X

X y, yp

k α

Y

Size and Thickness in. L6×6×1 L6×6×17⁄8 L6×6×13⁄4 L6×6×15⁄8 L6×6×19⁄16 L6×6×11⁄2 L6×6×17⁄16 L6×6×13⁄8 L6×6×15⁄16

Z

Axis Y-Y

S

I 4

r

Axis Z-Z

x

xp

r

in.

in.

in.

α

1.86 1.82 1.78 1.73 1.71 1.68 1.66 1.64 1.62

15.5 13.8 12.0 10.2 9.26 8.31 7.34 6.35 5.35

0.917 0.811 0.703 0.592 0.536 0.479 0.422 0.363 0.304

1.17 1.17 1.17 1.18 1.18 1.18 1.19 1.19 1.20

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

in.

3

in.

in.

in.

35.5 31.9 28.2 24.2 22.1 19.9 17.7 15.4 13.0

8.57 7.63 6.66 5.66 5.14 4.61 4.08 3.53 2.97

1.80 1.81 1.83 1.84 1.85 1.86 1.87 1.88 1.89

Z 3

Tan

L6×4×7⁄8 L6×4×3⁄4 L6×4×5⁄8 L6×4×9⁄16 L6×4×1⁄2 L6×4×7⁄16 L6×4×3⁄8 L6×4×5⁄16

9.75 8.68 7.52 6.91 6.27 5.60 4.90 4.18

3.39 2.97 2.54 2.31 2.08 1.85 1.60 1.35

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.17

1.12 1.08 1.03 1.01 0.987 0.964 0.941 0.918

6.31 5.47 4.62 4.19 3.75 3.30 2.85 2.40

0.665 0.578 0.488 0.442 0.396 0.349 0.301 0.252

0.857 0.860 0.864 0.866 0.870 0.873 0.877 0.882

0.421 0.428 0.435 0.438 0.440 0.443 0.446 0.448

L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

4.25 3.34 2.85

1.59 1.23 1.04

0.972 0.988 0.996

0.833 0.787 0.763

2.91 2.20 1.85

0.375 0.285 0.239

0.759 0.767 0.772

0.344 0.350 0.352

17.8 15.7 13.6 11.3 10.0 8.74 7.42

5.17 4.53 3.86 3.16 2.79 2.42 2.04

1.49 1.51 1.52 1.54 1.55 1.56 1.57

1.57 1.52 1.48 1.43 1.41 1.39 1.37

9.33 8.16 6.95 5.68 5.03 4.36 3.68

0.798 0.694 0.586 0.475 0.418 0.361 0.303

0.973 0.975 0.978 0.983 0.986 0.990 0.994

1.000 1.000 1.000 1.000 1.000 1.000 1.000

L5×5×7⁄8 L5×5×3⁄4 L5×5×5⁄8 L5×5×1⁄2 L5×5×7⁄16 L5×5×3⁄8 L5×5×5⁄16


1 - 60


Y


xp

x Z

X

X y, yp

k α

Y


Z

k

Weight per ft

Axis X-X Area 2

S

I 4

r 3

y

Z

yp 3

in.

lb

in.

in.

in.

in.

in.

in.

L5×31⁄2×3⁄4 L5×31⁄2×5⁄8 L5×31⁄2×1⁄2 L5×31⁄2×3⁄8 L5×31⁄2×5⁄16 L5×31⁄2×1⁄4

11⁄4 11⁄8 1 7⁄ 8 13⁄ 16 3⁄ 4

19.8 16.8 13.6 10.4 8.70 7.00

5.81 4.92 4.00 3.05 2.56 2.06

13.9 12.0 9.99 7.78 6.60 5.39

4.28 3.65 2.99 2.29 1.94 1.57

1.55 1.56 1.58 1.60 1.61 1.62

1.75 1.70 1.66 1.61 1.59 1.56

7.65 6.55 5.38 4.14 3.49 2.83

1.13 1.06 1.00 0.938 0.906 0.875

L5×3×1⁄2 L5×3×7⁄16 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

1 15⁄ 16 7⁄ 8 13⁄ 16 3⁄ 4

12.8 11.3 9.80 8.20 6.60

3.75 3.31 2.86 2.40 1.94

9.45 8.43 7.37 6.26 5.11

2.91 2.58 2.24 1.89 1.53

1.59 1.60 1.61 1.61 1.62

1.75 1.73 1.70 1.68 1.66

5.16 4.57 3.97 3.36 2.72

1.25 1.22 1.19 1.16 1.13

L4×4×3⁄4 L5×3×5⁄8 L5×3×1⁄2 L5×3×7⁄16 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

11⁄8 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

18.5 15.7 12.8 11.3 9.80 8.20 6.60

5.44 4.61 3.75 3.31 2.86 2.40 1.94

7.67 6.66 5.56 4.97 4.36 3.71 3.04

2.81 2.40 1.97 1.75 1.52 1.29 1.05

1.19 1.20 1.22 1.23 1.23 1.24 1.25

1.27 1.23 1.18 1.16 1.14 1.12 1.09

5.07 4.33 3.56 3.16 2.74 2.32 1.88

0.680 0.576 0.469 0.414 0.357 0.300 0.242

L4×31⁄2×1⁄2 L4×31⁄2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

15⁄ 16 13⁄ 16 3⁄ 4 11⁄ 16

11.9 9.10 7.70 6.20

3.50 2.67 2.25 1.81

5.32 4.18 3.56 2.91

1.94 1.49 1.26 1.03

1.23 1.25 1.26 1.27

1.25 1.21 1.18 1.16

3.50 2.71 2.29 1.86

0.500 0.438 0.406 0.375


in.

STRUCTURAL SHAPES

1 - 61

Y


x

xp

Z

X

X y, yp

k α

Y

Size and Thickness

Z

Axis Y-Y

S

I

r

Axis Z-Z

x

Z

xp

r

Tan

in.

in.

in.

in.

in.

in.

in.

α

L5×31⁄2×3⁄4 L5×31⁄2×5⁄8 L5×31⁄2×1⁄2 L5×31⁄2×3⁄8 L5×31⁄2×5⁄16 L5×31⁄2×1⁄4

5.55 4.83 4.05 3.18 2.72 2.23

2.22 1.90 1.56 1.21 1.02 0.830

0.977 0.991 1.01 1.02 1.03 1.04

0.996 0.951 0.906 0.861 0.838 0.814

4.10 3.47 2.83 2.16 1.82 1.47

0.581 0.492 0.400 0.305 0.256 0.206

0.748 0.751 0.755 0.762 0.766 0.770

0.464 0.472 0.479 0.486 0.489 0.492

L5×3×1⁄2 L5×3×7⁄16 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

2.58 2.32 2.04 1.75 1.44

1.15 1.02 0.888 0.753 0.614

0.829 0.837 0.845 0.853 0.861

0.750 0.727 0.704 0.681 0.657

2.11 1.86 1.60 1.35 1.09

0.375 0.331 0.286 0.240 0.194

0.648 0.651 0.654 0.658 0.663

0.357 0.361 0.364 0.368 0.371

L4×4×3⁄4 L5×3×5⁄8 L5×3×1⁄2 L5×3×7⁄16 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

7.67 6.66 5.56 4.97 4.36 3.71 3.04

2.81 2.40 1.97 1.75 1.52 1.29 1.05

1.19 1.20 1.22 1.23 1.23 1.24 1.25

1.27 1.23 1.18 1.16 1.14 1.12 1.09

5.07 4.33 3.56 3.16 2.74 2.32 1.88

0.680 0.576 0.469 0.414 0.357 0.300 0.242

0.778 0.779 0.782 0.785 0.788 0.791 0.795

1.000 1.000 1.000 1.000 1.000 1.000 1.000

L4×31⁄2×1⁄2 4×31⁄2×3⁄8 4×31⁄2×5⁄16 4×31⁄2×1⁄4

3.79 2.95 2.55 2.09

1.52 1.16 0.994 0.808

1.04 1.06 1.07 1.07

1.00 0.955 0.932 0.909

2.73 2.11 1.78 1.44

0.438 0.334 0.281 0.227

0.722 0.727 0.730 0.734

0.750 0.755 0.757 0.759

in.

4

3

3


1 - 62


Y


xp

x Z

X

X y, yp

k α

Y

Size and Thickness

Z

k

Weight per ft

Axis X-X Area

S

I 2

r 4

y 3

Z

yp 3

in.

in.

lb

in.

in.

in.

in.

in.

in.

L4×3×5⁄8 L4×3×1⁄2 L4×3×7⁄16 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4

11⁄16 15⁄ 16 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

13.6 11.1 9.80 8.50 7.20 5.80

3.98 3.25 2.87 2.48 2.09 1.69

6.03 5.05 4.52 3.96 3.38 2.77

2.30 1.89 1.68 1.46 1.23 1.00

1.23 1.25 1.25 1.26 1.27 1.28

1.37 1.33 1.30 1.28 1.26 1.24

4.12 3.41 3.03 2.64 2.23 1.82

0.813 0.750 0.719 0.688 0.656 0.625

L31⁄2×31⁄2×1⁄2 L31⁄2×31⁄2×7⁄16 L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×5⁄16 L31⁄2×31⁄2×1⁄4

7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

11.1 9.80 8.50 7.20 5.80

3.25 2.87 2.48 2.09 1.69

3.64 3.26 2.87 2.45 2.01

1.49 1.32 1.15 0.976 0.794

1.06 1.07 1.07 1.08 1.09

1.06 1.04 1.01 0.990 0.968

2.68 2.38 2.08 1.76 1.43

0.464 0.410 0.355 0.299 0.241

L31⁄2×3×1⁄2 L31⁄2×3×3⁄8 L31⁄2×3×5⁄16 L31⁄2×3×1⁄4

15⁄ 16 13⁄ 16 3⁄ 4 11⁄ 16

10.2 7.90 6.60 5.40

3.00 2.30 1.93 1.56

3.45 2.72 2.33 1.91

1.45 1.13 0.954 0.776

1.07 1.09 1.10 1.11

1.13 1.08 1.06 1.04

2.63 2.04 1.73 1.41

0.500 0.438 0.406 0.375

L31⁄2×21⁄2×1⁄2 L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×1⁄4

15⁄ 16 13⁄ 16 11⁄ 16

9.40 7.20 4.90

2.75 2.11 1.44

3.24 2.56 1.80

1.41 1.09 0.755

1.09 1.10 1.12

1.20 1.16 1.11

2.53 1.97 1.36

0.750 0.688 0.625

L3×3×1⁄2 L4×3×7⁄16 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4 L4×3×3⁄16

13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2

9.40 8.30 7.20 6.10 4.90 3.71

2.75 2.43 2.11 1.78 1.44 1.09

2.22 1.99 1.76 1.51 1.24 0.962

1.07 0.954 0.833 0.707 0.577 0.441

0.898 0.905 0.913 0.922 0.930 0.939

0.932 0.910 0.888 0.865 0.842 0.820

1.93 1.72 1.50 1.27 1.04 0.794

0.458 0.406 0.352 0.296 0.240 0.182


in.

STRUCTURAL SHAPES

1 - 63

Y


x

xp

Z

X

X y, yp

k α

Y

Size and Thickness

Z

Axis Y-Y

S

I

r

Axis Z-Z

x

Z

xp

r

Tan

in.

in.

in.

in.

in.

in.

in.

in.

α

L4×3×5⁄8 L4×3×1⁄2 L4×3×7⁄16 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4

2.87 2.42 2.18 1.92 1.65 1.36

1.35 1.12 0.992 0.866 0.734 0.599

0.849 0.864 0.871 0.879 0.887 0.896

0.871 0.827 0.804 0.782 0.759 0.736

2.48 2.03 1.79 1.56 1.31 1.06

0.498 0.406 0.359 0.311 0.261 0.211

0.637 0.639 0.641 0.644 0.647 0.651

0.534 0.543 0.547 0.551 0.554 0.558

L31⁄2×31⁄2×1⁄2 L31⁄2×31⁄2×7⁄16 L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×5⁄16 L31⁄2×31⁄2×1⁄4

3.64 3.26 2.87 2.45 2.01

1.49 1.32 1.15 0.976 0.794

1.06 1.07 1.07 1.08 1.09

1.06 1.04 1.01 0.990 0.968

2.68 2.38 2.08 1.76 1.43

0.464 0.410 0.355 0.299 0.241

0.683 0.684 0.687 0.690 0.694

1.000 1.000 1.000 1.000 1.000

L31⁄2×3×1⁄2 L31⁄2×3×3⁄8 L31⁄2×3×5⁄16 L31⁄2×3×1⁄4

2.33 1.85 1.58 1.30

1.10 0.851 0.722 0.589

0.881 0.897 0.905 0.914

0.875 0.830 0.808 0.785

1.98 1.53 1.30 1.05

0.429 0.328 0.276 0.223

0.621 0.625 0.627 0.631

0.714 0.721 0.724 0.727

L31⁄2×21⁄2×1⁄2 L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×1⁄4

1.36 1.09 0.777

0.760 0.592 0.412

0.704 0.719 0.735

0.705 0.660 0.614

1.40 1.07 0.735

0.393 0.301 0.205

0.534 0.537 0.544

0.486 0.496 0.506

L3×3×1⁄2 L3×3×7⁄16 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4 L3×3×3⁄16

2.22 1.99 1.76 1.51 1.24 0.962

1.07 0.954 0.833 0.707 0.577 0.441

0.898 0.905 0.913 0.922 0.930 0.939

0.932 0.910 0.888 0.865 0.842 0.820

1.93 1.72 1.50 1.27 1.04 0.794

0.458 0.406 0.352 0.296 0.240 0.182

0.584 0.585 0.587 0.589 0.592 0.596

1.000 1.000 1.000 1.000 1.000 1.000

4

3

3


1 - 64


Y


xp

x Z

X

X y, yp

k α

Y


Z

k

Weight per ft

Axis X-X Area 2

S

I 4

r 3

y

Z

yp 3

in.

lb

in.

in.

in.

in.

in.

L3×21⁄2×1⁄2 L3×21⁄2×3⁄8 L3×21⁄2×5⁄16 L3×21⁄2×1⁄4 L3×21⁄2×3⁄16

7⁄ 8 3⁄ 4 11⁄ 16 5⁄ 8 9⁄ 16

8.50 6.60 5.60 4.50 3.39

2.50 1.92 1.62 1.31 0.996

2.08 1.66 1.42 1.17 0.907

1.04 0.810 0.688 0.561 0.430

0.913 0.928 0.937 0.945 0.954

1.000 0.956 0.933 0.911 0.888

1.88 1.47 1.25 1.02 0.781

0.500 0.438 0.406 0.375 0.344

L3×2×1⁄2 L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16

13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2

7.70 5.90 5.00 4.10 3.07

2.25 1.73 1.46 1.19 0.902

1.92 1.53 1.32 1.09 0.842

1.00 0.781 0.664 0.542 0.415

0.924 0.940 0.948 0.957 0.966

1.08 1.04 1.02 0.993 0.970

1.78 1.40 1.19 0.973 0.746

0.750 0.688 0.656 0.625 0.594

L21⁄2×21⁄2×1⁄2 L21⁄2×21⁄2×3⁄8 L21⁄2×21⁄2×5⁄16 L21⁄2×21⁄2×1⁄4 L21⁄2×21⁄2×3⁄16

13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2

7.70 5.90 5.00 4.10 3.07

2.25 1.73 1.46 1.19 0.902

1.23 0.984 0.849 0.703 0.547

0.724 0.566 0.482 0.394 0.303

0.739 0.753 0.761 0.769 0.778

0.806 0.762 0.740 0.717 0.694

1.31 1.02 0.869 0.711 0.545

0.450 0.347 0.293 0.238 0.180

L21⁄2×2×3⁄8 L21⁄2×2×5⁄16 L21⁄2×2×1⁄4 L21⁄2×2×3⁄16

11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2

5.30 4.50 3.62 2.75

1.55 1.31 1.06 0.809

0.912 0.788 0.654 0.509

0.547 0.466 0.381 0.293

0.768 0.776 0.784 0.793

0.831 0.809 0.787 0.764

0.986 0.843 0.691 0.532

0.438 0.406 0.375 0.344

L2×2×3⁄8 L2×2×5⁄16 L2×2×1⁄4 L2×2×3⁄16 L2×2×1⁄8

11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16

4.70 3.92 3.19 2.44 1.65

1.36 1.15 0.938 0.715 0.484

0.479 0.416 0.348 0.272 0.190

0.351 0.300 0.247 0.190 0.131

0.594 0.601 0.609 0.617 0.626

0.636 0.614 0.592 0.569 0.546

0.633 0.541 0.445 0.343 0.235

0.340 0.288 0.234 0.179 0.121


in.

in.

STRUCTURAL SHAPES

1 - 65


Y

x

xp

Z

X

X y, yp

k α

Y

Size and Thickness

Z

Axis Y-Y

S

I

r

Axis Z-Z

x

Z

xp

r

Tan

in.

in.

in.

in.

in.

in.

in.

α

L3×21⁄2×1⁄2 L3×21⁄2×3⁄8 L3×21⁄2×5⁄16 L3×21⁄2×1⁄4 L3×21⁄2×3⁄16

1.30 1.04 0.898 0.743 0.577

0.744 0.581 0.494 0.404 0.310

0.722 0.736 0.744 0.753 0.761

0.750 0.706 0.683 0.661 0.638

1.35 1.05 0.889 0.724 0.553

0.417 0.320 0.270 0.219 0.166

0.520 0.522 0.525 0.528 0.533

0.667 0.676 0.680 0.684 0.688

L3×2×1⁄2 L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16

0.672 0.543 0.470 0.392 0.307

0.474 0.371 0.317 0.260 0.200

0.546 0.559 0.567 0.574 0.583

0.583 0.539 0.516 0.493 0.470

0.891 0.684 0.577 0.468 0.357

0.375 0.289 0.244 0.198 0.150

0.428 0.430 0.432 0.435 0.439

0.414 0.428 0.435 0.440 0.446

L21⁄2×21⁄2×1⁄2 L21⁄2×21⁄2×3⁄8 L21⁄2×21⁄2×5⁄16 L21⁄2×21⁄2×1⁄4 L21⁄2×21⁄2×3⁄16

1.23 0.984 0.849 0.703 0.547

0.724 0.566 0.482 0.394 0.303

0.739 0.753 0.761 0.769 0.778

0.806 0.762 0.740 0.717 0.694

1.31 1.02 0.869 0.711 0.545

0.450 0.347 0.293 0.238 0.180

0.487 0.487 0.489 0.491 0.495

1.000 1.000 1.000 1.000 1.000

L21⁄2×2×3⁄8 L21⁄2×2×5⁄16 L21⁄2×2×1⁄4 L21⁄2×2×3⁄16

0.514 0.446 0.372 0.291

0.363 0.310 0.254 0.196

0.577 0.584 0.592 0.600

0.581 0.559 0.537 0.514

0.660 0.561 0.457 0.350

0.309 0.262 0.213 0.162

0.420 0.422 0.424 0.427

0.614 0.620 0.626 0.631

L2×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16 L3×2×1⁄8

0.479 0.416 0.348 0.272 0.190

0.351 0.300 0.247 0.190 0.131

0.594 0.601 0.609 0.617 0.626

0.636 0.614 0.592 0.569 0.546

0.633 0.541 0.445 0.343 0.235

0.340 0.288 0.234 0.179 0.121

0.389 0.390 0.391 0.394 0.398

1.000 1.000 1.000 1.000 1.000

in.

4

3

3


1 - 66



STRUCTURAL TEES (WT, MT, ST)

1 - 67


Structural tees are obtained by splitting the webs of various beams, generally with the aid of rotary shears, and straightening to meet established permissible variations listed in Standard Mill Practice in Part 1 of this Manual. Although structural tees may be obtained by off-center splitting, or by splitting at two lines, as specified on order, the Dimensions and Properties are based on a depth of tee equal to one-half the published beam depth. Values of Qs are given for Fy = 36 ksi and Fy = 50 ksi, for those tees having stems which exceed the limiting width-thickness ratio λr of LRFD Specification Section B5. Since the cross section is comprised entirely of unstiffened elements, Qa = 1.0 and Q = Qs for _ all tee sections. The Flexural-Torsional Properties Table lists the dimensional values (ro and H) and cross-section constants (J and Cw) needed for checking flexural-torsional buckling. Use of Table

The table may be used as follows for checking the limit states of (1) flexural buckling about the x-axis and (2) flexural-torsional buckling. The lower of the two limit states must be used for design. See also Part 3 of this LRFD Manual. (1) Flexural Buckling About the X-Axis

Where no value of Qs is shown, the design compressive strength for this limit state is given by LRFD Specification Section E2. Where a value of Qs is shown, the strength must be reduced in accordance with Appendix B5 of the LRFD Specification. (2) Flexural-Torsional Buckling

The design compressive strength for this limit _ state is given by LRFD Specification Section E3. This involves calculations with J, ro, and H. Refer to the Flexural-Torsional Properties Tables, later in Part 1.


1 - 68


bf tf

yp , y

STRUCTURAL TEES Cut from W shapes Dimensions

k

Y

X

X

d

tw

Y

Stem

Area Designation in.2

Area of Stem

Depth of Tee d

Thickness tw

tw 2

in.

in.

in.

in.2

Flange Width bf in.

Thickness tf in.

Distance k in.

WT22×167.5 WT22×145 WT22×131 WT22×115

49.1 42.9 38.6 33.8

22.010 22 1.020 21.810 2113⁄16 0.870 21.655 2111⁄16 0.790 21.455 217⁄16 0.710

1 7⁄ 8 13⁄ 16 11⁄ 16

1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8

22.5 19.0 17.1 15.2

15.950 15.830 15.750 15.750

153⁄4 157⁄8 153⁄4 153⁄4

1.770 1.580 1.420 1.220

13⁄4 19⁄16 17⁄16 11⁄4

29⁄16 23⁄8 23⁄16 2

WT20×296.5 WT22×251.5 WT22×215.5 WT22×186 WT22×160.5 WT22×148.5 WT22×138.5 WT22×124.5 WT22×107.5 WT22×99.5 WT22×87

87.0 74.0 63.4 54.7 47.0 43.7 40.7 36.7 31.7 29.2 25.5

21.495 21.025 20.630 20.315 20.040 19.920 19.845 19.690 19.490 19.335 19.100

211⁄2 21 205⁄8 205⁄16 20 1915⁄16 197⁄8 1911⁄16 191⁄2 195⁄16 191⁄8

1.790 1.540 1.340 1.160 1.000 0.930 0.830 0.750 0.650 0.650 0.650

113⁄16 19⁄16 15⁄16 13⁄16 1 15⁄ 16 13⁄ 16 3⁄ 4 5⁄ 8 5⁄ 8 5⁄ 8

1 3⁄ 4 11⁄ 16 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16

38.5 32.4 27.6 23.6 20.0 18.5 16.5 14.8 12.7 12.6 12.4

16.690 163⁄4 16.420 167⁄16 16.220 161⁄4 16.060 161⁄16 15.910 157⁄8 15.825 157⁄8 15.830 157⁄8 15.750 153⁄4 15.750 153⁄4 15.750 153⁄4 15.750 153⁄4

3.230 2.760 2.360 2.050 1.770 1.650 1.575 1.420 1.220 1.065 0.830

31⁄4 23⁄4 23⁄8 21⁄16 13⁄4 15⁄8 19⁄16 17⁄16 11⁄4 11⁄16 13⁄ 16

47⁄16 315⁄16 39⁄16 31⁄4 215⁄16 31⁄16 23⁄4 25⁄8 23⁄8 21⁄4 2

WT20×233 WT22×196 WT22×165.5 WT22×139 WT22×132 WT22×117.5 WT22×105.5 WT22×91.5 WT22×83.5 WT22×74.5

68.4 57.7 48.8 40.9 38.8 34.5 31.0 26.9 24.6 21.9

21.220 213⁄16 20.785 203⁄4 20.395 203⁄8 20.080 201⁄8 20.000 20 19.845 197⁄8 19.685 1911⁄16 19.490 191⁄2 19.295 195⁄16 19.100 191⁄8

1.67 1.42 1.22 1.02 0.960 0.830 0.750 0.650 0.650 0.630

111⁄16 17⁄16 11⁄4 1 1 13⁄ 16 3⁄ 4 5⁄ 8 5⁄ 8 5⁄ 8

13⁄ 16 11⁄ 16 5⁄ 8 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16

35.4 29.5 24.9 20.5 19.2 16.5 14.8 12.7 12.5 12.0

12.640 125⁄8 12.360 123⁄8 12.170 123⁄16 11.970 12 11.930 12 11.890 117⁄8 11.810 113⁄4 11.810 113⁄4 11.810 113⁄4 11.810 113⁄4

2.950 2.520 2.130 1.810 1.730 1.575 1.415 1.220 1.025 0.830

215⁄16 21⁄2 21⁄8 113⁄16 13⁄4 17⁄16 19⁄16 11⁄4 1 13⁄ 16

41⁄8 311⁄16 35⁄16 3 215⁄16 23⁄4 25⁄8 23⁄8 23⁄16 2

WT18×424 WT22×399 WT22×325 WT22×263.5 WT22×219.5 WT22×196.5 WT22×179.5 WT22×164 WT22×150 WT22×140 WT22×130 WT22×122.5 WT22×115

125 117 95.0 77.0 64.0 57.5 52.7 48.2 44.1 41.2 38.2 36.0 33.8

21.225 211⁄4 20.985 21 20.235 201⁄4 19.605 195⁄8 19.130 191⁄8 18.900 187⁄8 18.700 1811⁄16 18.545 189⁄16 18.370 183⁄8 18.260 181⁄4 18.130 181⁄8 18.040 18 17.950 18

2.520 2.380 1.970 1.610 1.360 1.220 1.120 1.020 0.945 0.885 0.840 0.800 0.760

21⁄2 23⁄8 2 15⁄8 13⁄8 11⁄4 11⁄8 1 15⁄ 16 7⁄ 8 13⁄ 16 13⁄ 16 3⁄ 4

11⁄4 13⁄16 1 13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 7⁄ 16 7⁄ 16 3⁄ 8

53.5 49.9 39.9 31.6 26.0 23.1 20.9 18.9 17.4 16.2 15.2 14.4 13.6

18.130 17.990 17.575 17.220 16.965 16.830 16.730 16.630 16.655 16.595 16.550 16.510 16.470

181⁄8 18 175⁄8 171⁄4 17 167⁄8 163⁄4 165⁄8 165⁄8 165⁄8 161⁄2 161⁄2 161⁄2

4.530 4.290 3.540 2.910 2.440 2.200 2.010 1.850 1.680 1.570 1.440 1.350 1.260

41⁄2 45⁄16 39⁄16 215⁄16 27⁄16 23⁄16 2 17⁄8 111⁄16 19⁄16 17⁄16 13⁄8 11⁄4

511⁄16 57⁄16 411⁄16 41⁄16 39⁄16 35⁄16 31⁄8 3 213⁄16 211⁄16 29⁄16 21⁄2 23⁄8



1 - 69

bf

STRUCTURAL TEES Cut from W shapes Properties

tf

k

Y

yp , y

X

X

d

tw

Y

Nominal Wt. per ft lb

Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

36

Fy, ksi 50

167.5 145 131 115

19.1 22.3 24.6 27.4

2160 1840 1650 1440

131 111 100 88.6

6.63 6.55 6.53 6.53

5.51 5.27 5.20 5.17

233 197 177 157

1.54 1.35 1.23 1.07

600 524 463 398

75.3 66.1 58.8 50.5

3.50 3.49 3.46 3.43

118 103 91.4 78.4

0.982 0.833 0.732 0.608

0.817 0.636 0.532 0.438

296.5 251.5 215.5 186 160.5 148.5 138.5 124.5 107.5 99.5 87

9.5 11.1 12.8 14.7 17.1 18.4 20.6 22.8 26.3 26.3 26.3

3300 2730 2290 1930 1630 1500 1360 1210 1030 987 907

209 175 148 126 107 98.9 88.6 79.3 68.0 66.4 63.8

6.16 6.07 6.01 5.95 5.89 5.87 5.78 5.75 5.72 5.81 5.96

5.67 5.39 5.18 4.97 4.79 4.71 4.51 4.41 4.28 4.48 4.87

379 315 266 225 191 176 157 140 120 117 114

2.61 2.25 1.95 1.70 1.48 1.38 1.28 1.16 1.00 0.927 0.811

1260 1020 843 710 596 546 522 463 398 347 271

151 125 104 88.5 74.9 69.1 65.9 58.8 50.5 44.1 34.4

3.81 3.72 3.65 3.60 3.56 3.54 3.58 3.56 3.55 3.45 3.26

240 197 164 139 117 108 102 91.0 77.9 68.3 53.8

— — — — — 0.989 0.882 0.782 0.618 0.628 0.643

— — — — 0.895 0.825 0.699 0.580 0.445 0.452 0.463

233 196 165.5 139 132 117.5 105.5 91.5 83.5 74.5

10.2 12.0 14.0 16.8 17.8 20.6 22.8 26.3 26.3 27.1

2770 2270 1880 1540 1450 1260 1120 957 898 815

185 153 128 106 99.3 85.6 76.7 65.8 63.7 59.7

6.36 6.28 6.21 6.14 6.11 6.04 6.01 5.97 6.05 6.10

6.22 5.95 5.74 5.50 5.40 5.17 5.08 4.94 5.20 5.45

333 276 231 190 178 153 137 117 115 119

2.71 2.33 2.01 1.71 1.63 1.45 1.31 1.14 1.04 1.82

504 401 323 261 246 222 195 168 141 115

79.8 65.0 53.1 43.6 41.3 37.3 33.0 28.5 23.9 19.4

2.72 2.64 2.57 2.52 2.52 2.54 2.51 2.50 2.40 2.29

131 106 86.2 70.0 66.2 59.2 52.3 44.8 38.0 31.1

— — — — — 0.882 0.782 0.618 0.630 0.604

— — — 0.913 0.855 0.699 0.581 0.445 0.454 0.435

424 399 325 263.5 219.5 196.5 179.5 164 150 140 130 122.5 115

6.3 6.6 8.0 9.8 11.6 12.9 14.1 15.4 16.7 17.8 18.7 19.7 20.7

4250 3920 3020 2330 1880 1660 1500 1350 1230 1140 1060 995 934

277 257 202 159 130 115 104 94.1 86.1 80.0 75.1 71.0 67.0

5.84 5.79 5.64 5.50 5.42 5.37 5.33 5.29 5.27 5.25 5.26 5.26 5.25

5.86 5.72 5.29 4.89 4.63 4.46 4.33 4.21 4.13 4.07 4.05 4.03 4.01

515 478 373 290 235 207 187 168 153 142 133 125 118

3.43 3.25 2.70 2.24 1.89 1.71 1.58 1.45 1.33 1.24 1.16 1.09 1.03

2270 2100 1610 1240 997 877 786 711 648 599 545 507 470

251 234 184 145 117 104 94.0 85.5 77.8 72.2 65.9 61.4 57.1

4.27 4.24 4.12 4.02 3.95 3.90 3.86 3.84 3.83 3.81 3.78 3.75 3.73

399 371 290 227 184 162 146 132 120 112 102 94.9 88.1

— — — — — — — — — — 0.981 0.943 0.896

— — — — — — — — 0.927 0.867 0.816 0.770 0.715

*Where no value of Qs is shown, the Tee complies with LRFD Specification Section E2.


1 - 70


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Area of Stem

Flange DisThickness tance tf k

Area

Depth of Tee d

Thickness tw

tw 2

Designation

in.2

in.

in.

in.

WT18×128 WT18×116 WT18×105 WT18×97 WT18×91 WT18×85 WT18×80 WT18×75 WT18×67.5

37.7 34.1 30.9 28.5 26.8 25.0 23.5 22.1 19.9

18.715 1811⁄16 18.560 189⁄16 18.345 183⁄8 18.245 181⁄4 18.165 181⁄8 18.085 181⁄8 18.005 18 17.925 177⁄8 17.775 173⁄4

0.960 0.870 0.830 0.765 0.725 0.680 0.650 0.625 0.600

1 7⁄ 8 13⁄ 16 3⁄ 4 3⁄ 4 11⁄ 16 5⁄ 8 5⁄ 8 5⁄ 8

2

7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16

18.0 16.1 15.2 14.0 13.2 12.3 11.7 11.2 10.7

12.215 12.120 12.180 12.115 12.075 12.030 12.000 11.975 11.950

121⁄4 121⁄8 121⁄8 121⁄8 121⁄8 12 12 12 12

1.730 1.570 1.360 1.260 1.180 1.100 1.020 0.940 0.790

13⁄4 19⁄16 13⁄8 11⁄4 13⁄16 11⁄8 1 15⁄ 16 13⁄ 16

25⁄8 21⁄2 25⁄16 23⁄16 21⁄8 2 115⁄16 17⁄8 111⁄16

WT16.5×177 WT16.5×159 WT16.5×145.5 WT16.5×131.5 WT16.5×120.5 WT16.5×110.5 WT16.5×100.5

52.1 46.7 42.8 38.7 35.4 32.5 29.5

17.775 173⁄4 17.580 179⁄16 17.420 177⁄16 17.265 171⁄4 17.090 171⁄8 16.965 17 16.840 167⁄8

1.160 1.040 0.960 0.870 0.830 0.775 0.715

13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

20.6 18.3 16.7 15.0 14.2 13.1 12.0

16.100 15.985 15.905 15.805 15.860 15.805 15.745

161⁄8 16 157⁄8 153⁄4 157⁄8 153⁄4 153⁄4

2.090 1.890 1.730 1.570 1.400 1.275 1.150

21⁄16 17⁄8 13⁄4 19⁄16 13⁄8 11⁄4 11⁄8

27⁄8 211⁄16 29⁄16 23⁄8 23⁄16 21⁄16 115⁄16

WT16.5×84.5 WT16.5×76 WT16.5×70.5 WT16.5×65 WT16.5×59

24.8 22.4 20.8 19.2 17.3

16.910 1615⁄16 16.745 163⁄4 16.650 165⁄8 16.545 161⁄2 16.430 163⁄8

0.670 0.635 0.605 0.580 0.550

11⁄

3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16 5⁄ 16

11.3 10.6 10.1 9.60 9.04

11.500 11.565 11.535 11.510 11.480

111⁄2 115⁄8 111⁄2 111⁄2 111⁄2

1.220 1.055 0.960 0.855 0.740

11⁄4 11⁄16 15⁄ 16 7⁄ 8 3⁄ 4

21⁄16 17⁄8 13⁄4 111⁄16 19⁄16

WT15×238.5 WT15×195.5 WT15×163 WT15×146 WT15×130.5 WT15×117.5 WT15×105.5 WT15×95.5 WT15×86.5

70.0 57.0 47.9 42.9 38.4 34.5 31.0 28.1 25.4

17.105 171⁄8 16.595 165⁄8 16.200 163⁄16 16.005 16 15.805 1513⁄16 15.650 155⁄8 15.470 151⁄2 15.340 153⁄8 15.220 151⁄4

1.630 1.360 1.140 1.020 0.930 0.830 0.775 0.710 0.655

15⁄8 13⁄8 11⁄8 1 15⁄ 16 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

13⁄

16 11⁄ 16 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16

27.9 22.6 18.5 16.3 14.7 13.0 12.0 10.9 9.97

15.865 15.590 15.370 15.255 15.155 15.055 15.105 15.040 14.985

157⁄8 155⁄8 153⁄8 151⁄2 151⁄8 15 151⁄8 15 15

2.950 2.440 2.050 1.850 1.650 1.500 1.315 1.185 1.065

3 27⁄16 21⁄16 17⁄8 15⁄8 11⁄2 15⁄16 13⁄16 11⁄16

33⁄4 31⁄4 213⁄16 25⁄8 27⁄16 21⁄4 21⁄8 115⁄16 17⁄8

WT15×74 WT18×66 WT18×62 WT18×58 WT18×54 WT18×49.5 WT18×45

21.7 19.4 18.2 17.1 15.9 14.5 13.2

15.335 155⁄16 15.155 151⁄8 15.085 151⁄8 15.005 15 14.915 147⁄8 14.825 147⁄8 14.765 143⁄4

0.650 0.615 0.585 0.565 0.545 0.520 0.470

5⁄ 8 5⁄ 8 9⁄ 16 9⁄ 16 9⁄ 16 1⁄ 2 1⁄ 2

5⁄ 16 5⁄ 16 5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4

10.0 9.32 8.82 8.48 8.13 7.71 6.94

10.480 10.545 10.515 10.495 10.475 10.450 10.400

101⁄2 101⁄2 101⁄2 101⁄2 101⁄2 101⁄2 103⁄8

1.180 1.000 0.930 0.850 0.760 0.670 0.610

13⁄16 1 15⁄ 16 7⁄ 8 3⁄ 4 11⁄ 16 9⁄ 16

2 13⁄4 111⁄16 15⁄8 19⁄16 17⁄16 15⁄16

16 5⁄ 8 5⁄ 8 9⁄ 16 9⁄ 16

1⁄

Width bf

in.2

in.


in.

in.


1 - 71

bf


tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

Fy, ksi 36

50

128 116 105 97 91 85 80 75 67.5

16.9 18.7 19.6 21.2 22.4 23.9 25.0 26.0 27.1

1200 1080 985 901 845 786 740 698 637

87.4 78.5 73.1 67.0 63.1 58.9 55.8 53.1 49.7

5.66 5.63 5.65 5.62 5.62 5.61 5.61 5.62 5.66

4.92 4.82 4.87 4.80 4.77 4.73 4.74 4.78 4.96

156 140 131 120 113 105 100 95.5 94.3

1.54 1.40 1.27 1.18 1.11 1.04 0.980 0.923 1.24

264 234 206 187 174 160 147 135 113

43.2 38.6 33.8 30.9 28.8 26.6 24.6 22.5 18.9

2.65 2.62 2.58 2.56 2.55 2.53 2.50 2.47 2.38

68.6 61.0 53.5 48.9 45.4 41.9 38.6 35.5 29.8

— 0.994 0.960 0.887 0.831 0.767 0.720 0.677 0.634

0.927 0.831 0.791 0.705 0.635 0.565 0.521 0.486 0.457

177 159 145.5 131.5 120.5 110.5 100.5

12.9 14.4 15.6 17.2 18.1 19.3 21.0

1320 1160 1050 943 871 799 725

96.8 85.8 78.3 70.2 65.8 60.8 55.5

5.03 4.99 4.97 4.94 4.96 4.96 4.95

4.16 4.02 3.94 3.84 3.85 3.81 3.78

174 154 140 125 116 107 97.7

1.62 1.46 1.34 1.22 1.12 1.03 0.938

729 645 581 517 466 420 375

90.6 80.7 73.1 65.5 58.8 53.2 47.6

3.74 3.71 3.69 3.66 3.63 3.59 3.56

141 125 113 101 90.9 82.1 73.4

— — — — — 0.968 0.896

— — 0.993 0.907 0.867 0.801 0.715

84.5 76 70.5 65 59

22.4 23.6 24.8 25.8 27.3

649 592 552 513 469

51.1 47.4 44.7 42.1 39.2

5.12 5.14 5.15 5.18 5.20

4.21 4.26 4.29 4.36 4.47

90.8 84.5 79.8 75.6 74.8

1.08 0.967 0.901 0.832 0.862

155 136 123 109 93.6

27.0 23.6 21.3 18.9 16.3

2.50 2.47 2.43 2.39 2.32

42.2 37.0 33.5 29.7 25.7

0.827 0.775 0.728 0.685 0.621

0.630 0.574 0.529 0.492 0.447

238.5 195.5 163 146 130.5 117.5 105.5 95.5 86.5

8.3 9.9 11.8 13.2 14.5 16.2 17.4 19.0 20.6

1550 1210 981 861 764 674 610 549 497

121 96.6 78.9 69.6 62.3 55.1 50.5 45.7 41.7

4.70 4.61 4.53 4.48 4.46 4.42 4.43 4.42 4.42

4.30 4.04 3.76 3.63 3.54 3.42 3.40 3.35 3.31

224 177 143 125 112 98.2 89.5 80.8 73.4

2.21 1.83 1.56 1.40 1.27 1.15 1.03 0.933 0.848

987 774 622 549 480 427 378 336 299

124 99.2 81.0 71.9 63.3 56.8 50.1 44.7 39.9

3.75 3.68 3.61 3.58 3.54 3.52 3.49 3.46 3.43

195 155 126 111 97.9 87.5 77.2 68.9 61.4

— — — — — — — 0.981 0.913

— — — — — 0.952 0.897 0.816 0.735

74 66 62 58 54 49.5 45

20.8 22.0 23.1 23.9 24.8 26.0 28.7

466 421 396 373 349 322 291

40.6 37.4 35.3 33.7 32.0 30.0 27.1

4.63 4.66 4.66 4.67 4.69 4.71 4.69

3.84 3.90 3.90 3.94 4.01 4.09 4.03

72.2 66.8 63.1 60.4 57.7 57.4 49.4

1.04 0.921 0.867 0.815 0.757 0.912 0.445

113 98.0 90.4 82.1 73.0 63.9 57.3

21.7 18.6 17.2 15.7 13.9 12.2 11.0

2.28 2.25 2.23 2.19 2.15 2.10 2.08

34.0 29.2 27.0 24.6 22.0 19.3 17.3

0.896 0.853 0.801 0.767 0.733 0.685 0.563

0.715 0.664 0.601 0.565 0.533 0.492 0.405



1 - 72


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Area of Stem

Area

Depth of Tee d

Thickness tw

tw 2

Designation

in.2

in.

in.

in.

in.2

WT13.5×269.5 WT13.5×224 WT13.5×184 WT13.5×153.5 WT13.5×129 WT13.5×117.5 WT13.5×108.5 WT13.5×97 WT13.5×89 WT13.5×80.5 WT13.5×73

79.0 65.5 54.0 45.1 37.9 34.6 31.9 28.5 26.1 23.7 21.5

16.260 15.710 15.195 14.805 14.490 14.330 14.215 14.055 13.905 13.795 13.690

161⁄4 1511⁄16 153⁄16 1413⁄16 141⁄2 145⁄16 143⁄16 141⁄16 137⁄8 133⁄4 133⁄4

1.970 1.650 1.380 1.160 0.980 0.910 0.830 0.750 0.725 0.660 0.605

2 15⁄8 13⁄8 13⁄16 1 15⁄ 16 13⁄ 16 3⁄ 4 3⁄ 4 11⁄ 16 5⁄ 8

1 13⁄ 16 11⁄ 16 5⁄ 8 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 3⁄ 8 3⁄ 8 5⁄ 16

32.0 25.9 21.0 17.2 14.2 13.0 11.8 10.5 10.1 9.10 8.28

WT13.5×64.5 WT13.5×57 WT13.5×51 WT13.5×47 WT13.5×42

18.9 16.8 15.0 13.8 12.4

13.815 1313⁄16 13.645 135⁄8 13.545 131⁄2 13.460 131⁄2 13.355 133⁄8

0.610 0.570 0.515 0.490 0.460

5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16

5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4

WT12×246 WT12×204 WT12×167.5 WT12×139.5 WT12×125 WT12×114.5 WT12×103.5 WT12×96 WT12×88 WT12×81 WT12×73 WT12×65.5 WT12×58.5 WT12×52

72.0 59.5 49.2 41.0 36.8 33.6 30.4 28.2 25.8 23.9 21.5 19.3 17.2 15.3

14.825 1413⁄16 14.270 141⁄4 13.760 133⁄4 13.365 133⁄8 13.170 133⁄16 13.010 13 12.855 127⁄8 12.735 123⁄4 12.620 125⁄8 12.500 121⁄2 12.370 123⁄8 12.240 121⁄4 12.130 121⁄8 12.030 12

1.970 1.650 1.380 1.160 1.040 0.960 0.870 0.810 0.750 0.705 0.650 0.605 0.550 0.500

2 15⁄8 13⁄8 13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2

13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4

29.2 23.5 19.0 15.5 13.7 12.5 11.2 10.3 9.47 8.81 8.04 7.41 6.67 6.02

WT12×51.5 WT12×47 WT12×42 WT12×38 WT12×34

15.1 13.8 12.4 11.2 10.0

12.265 12.155 12.050 11.960 11.865

121⁄4 121⁄8 12 12 117⁄8

0.550 0.515 0.470 0.440 0.415

9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 7⁄ 16

5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 1⁄ 4

9.11 11.870 8.10 11.785

117⁄8 113⁄4

0.430 0.395

7⁄ 16 3⁄ 8

1⁄ 4 3⁄ 16

WT12×31 WT12×27.5

1


Width bf in.

in.

in.

151⁄4 15 145⁄8 141⁄2 141⁄4 141⁄4 141⁄8 14 141⁄8 14 14

3.540 2.990 2.480 2.090 1.770 1.610 1.500 1.340 1.190 1.080 0.975

39⁄16 3 21⁄2 21⁄16 13⁄4 15⁄8 11⁄2 15⁄16 13⁄16 11⁄16 1

41⁄4 311⁄16 33⁄16 213⁄16 21⁄2 25⁄16 23⁄16 21⁄16 17⁄8 113⁄16 111⁄16

8.43 10.010 10 7.78 10.070 101⁄8 6.98 10.015 10 6.60 9.990 10 6.14 9.960 10

1.100 0.930 0.830 0.745 0.640

11⁄8 15⁄ 16 13⁄ 16 3⁄ 4 5⁄ 8

113⁄16 15⁄8 19⁄16 17⁄16 13⁄8

14.115 13.800 13.520 13.305 13.185 13.110 13.010 12.950 12.890 12.955 12.900 12.855 12.800 12.750

141⁄8 133⁄4 131⁄2 131⁄4 131⁄8 131⁄8 13 13 127⁄8 13 127⁄8 127⁄8 123⁄4 123⁄4

3.540 2.990 2.480 2.090 1.890 1.730 1.570 1.460 1.340 1.220 1.090 0.960 0.850 0.750

39⁄16 3 21⁄2 21⁄16 17⁄8 13⁄4 19⁄16 17⁄16 15⁄16 11⁄4 11⁄16 15⁄ 16 7⁄ 8 3⁄ 4

45⁄16 33⁄4 31⁄4 27⁄8 211⁄16 21⁄2 23⁄8 21⁄4 21⁄8 2 17⁄8 13⁄4 15⁄8 11⁄2

6.75 6.26 5.66 5.26 4.92

9.000 9.065 9.020 8.990 8.965

9 91⁄8 9 9 9

0.980 0.875 0.770 0.680 0.585

1 7⁄ 8 3⁄ 4 11⁄ 16 9⁄ 16

13⁄4 15⁄8 19⁄16 17⁄16 13⁄8

5.10 4.66

7.040 7.005

7 7

0.590 0.505

9⁄ 16 1⁄ 2

13⁄8 15⁄16

15.255 14.940 14.665 14.445 14.270 14.190 14.115 14.035 14.085 14.020 13.965



1 - 73

bf


tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

36

50

2.59 1060 2.19 836 1.84 655 1.56 527 1.33 430 1.22 384 1.13 352 1.02 309 0.928 278 0.845 248 0.768 222

138 112 89.3 72.9 60.2 54.2 49.9 44.1 39.4 35.4 31.7

3.66 3.57 3.48 3.42 3.37 3.33 3.32 3.29 3.26 3.24 3.21

218 176 140 113 93.3 83.8 77.0 67.9 60.8 54.5 48.8

— — — — — — — — — — 0.938

— — — — — — — 0.963 0.937 0.851 0.765

18.4 15.8 13.9 12.4 10.6

2.21 2.18 2.15 2.12 2.07

28.8 24.7 21.7 19.4 16.6

0.938 0.883 0.780 0.728 0.664

0.765 0.700 0.578 0.529 0.476

119 95.5 75.9 61.9 54.9 49.7 44.4 40.9 37.2 34.2 30.3 26.5 23.2 20.3

3.41 3.33 3.23 3.17 3.14 3.11 3.08 3.07 3.04 3.05 3.01 2.97 2.94 2.91

187 150 119 96.4 85.3 77.0 68.6 63.1 57.3 52.6 46.6 40.7 35.7 31.2

— — — — — — — — — — — — 0.960 0.874

— — — — — — — — — — 0.947 0.887 0.791 0.690

20.7 18.8 16.3 14.3 12.3

0.951 0.896 0.810 0.741 0.681

0.781 0.715 0.610 0.541 0.489

7.87 0.724 6.67 0.626

0.525 0.450

269.5 224 184 153.5 129 117.5 108.5 97 89 80.5 73

6.2 7.4 8.8 10.5 12.4 13.3 14.6 16.2 16.7 18.4 20.0

1520 1190 938 753 613 556 502 444 414 372 336

128 102 81.7 66.4 54.6 50.0 45.2 40.3 38.2 34.4 31.2

4.39 4.27 4.17 4.09 4.02 4.01 3.97 3.95 3.98 3.96 3.95

4.36 4.02 3.71 3.47 3.28 3.21 3.11 3.03 3.05 2.99 2.95

241 191 151 121 98.8 89.8 81.1 71.8 67.6 60.8 55.0

64.5 57 51 47 42

19.9 21.3 23.5 24.7 26.3

323 289 258 239 216

31.0 28.3 25.3 23.8 21.9

4.13 4.15 4.14 4.16 4.18

3.39 3.42 3.37 3.41 3.48

55.1 50.4 45.0 42.4 39.2

0.945 0.833 0.750 0.692 0.621

246 204 167.5 139.5 125 114.5 103.5 96 88 81 73 65.5 58.5 52

5.5 6.5 7.8 9.3 10.4 11.2 12.4 13.3 14.4 15.3 16.6 17.8 19.6 21.6

1130 874 685 546 478 431 382 350 319 293 264 238 212 189

105 83.1 66.3 53.6 47.2 42.9 38.3 35.2 32.2 29.9 27.2 24.8 22.3 20.0

3.96 3.83 3.73 3.65 3.61 3.58 3.55 3.53 3.51 3.50 3.50 3.52 3.51 3.51

4.07 3.74 3.42 3.18 3.05 2.97 2.87 2.80 2.74 2.70 2.66 2.65 2.62 2.59

200 157 123 98.8 86.5 78.1 69.3 63.5 57.8 53.3 48.2 43.9 39.2 35.1

2.55 2.16 1.82 1.54 1.39 1.28 1.17 1.09 1.00 0.921 0.833 0.750 0.672 0.600

51.5 47 42 38 34

19.6 20.9 22.9 24.5 26.0

204 186 166 151 137

22.0 20.3 18.3 16.9 15.6

3.67 3.67 3.67 3.68 3.70

3.01 2.99 2.97 3.00 3.06

39.2 36.1 32.5 30.1 27.9

0.841 0.764 0.685 0.622 0.560

59.7 54.5 47.2 41.3 35.2

13.3 12.0 10.5 9.18 7.85

1.99 1.98 1.95 1.92 1.87

31 27.5

25.1 27.3

131 117

15.6 14.1

3.79 3.80

3.46 3.50

28.4 25.6

1.28 1.53

17.2 14.5

4.90 4.15

1.38 1.34

92.2 79.4 69.6 62.0 52.8 837 659 513 412 362 326 289 265 240 221 195 170 149 130

*Where no value of Qs is shown, the Tee complies with LRFD Specification Sect. E2.


Fy, ksi

1 - 74


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Area of Stem


Area

Depth of Tee d

Thickness tw

tw 2

Designation

in.2

in.

in.

in.

in.2

WT10.5×100.5 WT10.5×91 WT10.5×83 WT10.5×73.5 WT10.5×66 WT10.5×61 WT10.5×55.5 WT10.5×50.5

29.6 26.8 24.4 21.6 19.4 17.9 16.3 14.9

11.515 11.360 11.240 11.030 10.915 10.840 10.755 10.680

111⁄2 113⁄8 111⁄4 11 107⁄8 107⁄8 103⁄4 105⁄8

0.910 0.830 0.750 0.720 0.650 0.600 0.550 0.500

15⁄ 16 13⁄ 16 3⁄ 4 3⁄ 4 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2

1⁄

2

7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4

10.5 9.43 8.43 7.94 7.09 6.50 5.92 5.34

12.575 12.500 12.420 12.510 12.440 12.390 12.340 12.290

125⁄8 121⁄2 123⁄8 121⁄2 121⁄2 123⁄8 123⁄8 121⁄4

1.630 1.480 1.360 1.150 1.035 0.960 0.875 0.800

15⁄8 11⁄2 13⁄8 11⁄8 11⁄16 15⁄ 16 7⁄ 8 13⁄ 16

23⁄8 21⁄4 21⁄8 17⁄8 113⁄16 111⁄16 15⁄8 19⁄16

WT10.5×46.5 WT10.5×41.5 WT10.5×36.5 WT10.5×34 WT10.5×31

13.7 12.2 10.7 10.0 9.13

10.810 10.715 10.620 10.565 10.495

103⁄4 103⁄4 105⁄8 105⁄8 101⁄2

0.580 0.515 0.455 0.430 0.400

9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8

5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

6.27 5.52 4.83 4.54 4.20

8.420 8.355 8.295 8.270 8.240

83⁄8 83⁄8 81⁄4 81⁄4 81⁄4

0.930 0.835 0.740 0.685 0.615

15⁄ 16 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

111⁄16 19⁄16 11⁄2 17⁄16 13⁄8

8.37 10.530 7.36 10.415 6.49 10.330

101⁄2 103⁄8 103⁄8

0.405 0.380 0.350

3⁄ 8 3⁄ 8 3⁄ 8

3⁄ 16 3⁄ 16 3⁄ 16

4.26 3.96 3.62

6.555 6.530 6.500

61⁄2 61⁄2 61⁄2

0.650 0.535 0.450

5⁄ 8 9⁄ 16 7⁄ 16

13⁄8 15⁄16 13⁄16

WT10.5×28.5 WT10.5×25 WT10.5×22

Width bf in.

in.

in.

WT9×155.5 WT9×141.5 WT9×129 WT9×117 WT9×105.5 WT9×96 WT9×87.5 WT9×79 WT9×71.5 WT9×65

45.8 41.6 38.0 34.4 31.1 28.2 25.7 23.2 21.0 19.1

11.160 10.925 10.730 10.530 10.335 10.175 10.020 9.860 9.745 9.625

113⁄16 1015⁄16 103⁄4 101⁄2 105⁄16 103⁄16 10 97⁄8 93⁄4 95⁄8

1.520 1.400 1.280 1.160 1.060 0.960 0.890 0.810 0.730 0.670

11⁄2 13⁄8 11⁄4 13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

3⁄ 4 11⁄ 16 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

17.0 15.3 13.7 12.2 11.0 9.77 8.92 7.99 7.11 6.45

12.005 11.890 11.770 11.650 11.555 11.455 11.375 11.300 11.220 11.160

12 117⁄8 113⁄4 115⁄8 111⁄2 111⁄2 113⁄8 111⁄4 111⁄4 111⁄8

2.740 2.500 2.300 2.110 1.910 1.750 1.590 1.440 1.320 1.200

23⁄4 21⁄2 25⁄16 21⁄8 115⁄16 13⁄4 19⁄16 17⁄16 15⁄16 13⁄16

37⁄16 33⁄16 3 23⁄4 9 2 ⁄16 27⁄16 21⁄4 21⁄8 2 17⁄8

WT9×59.5 WT9×53 WT9×48.5 WT9×43 WT9×38

17.5 15.6 14.3 12.7 11.2

9.485 9.365 9.295 9.195 9.105

91⁄2 93⁄8 91⁄4 91⁄4 91⁄8

0.655 0.590 0.535 0.480 0.425

5⁄ 8 9⁄ 16 9⁄ 16 1⁄ 2 7⁄ 16

5⁄ 16 5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4

6.21 5.53 4.97 4.41 3.87

11.265 11.200 11.145 11.090 11.035

111⁄4 111⁄4 111⁄8 111⁄8 11

1.060 0.940 0.870 0.770 0.680

11⁄16 15⁄ 16 7⁄ 8 3⁄ 4 11⁄ 16

13⁄4 15⁄8 19⁄16 17⁄16 13⁄8

WT9×35.5 WT9×32.5 WT9×30 WT9×27.5 WT9×25

10.4 9.55 8.82 8.10 7.33

9.235 9.175 9.120 9.055 8.995

91⁄4 91⁄8 91⁄8 9 9

0.495 0.450 0.415 0.390 0.355

1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

1⁄

1⁄ 4 3⁄ 16 3⁄ 16

4.57 4.13 3.78 3.53 3.19

7.635 7.590 7.555 7.530 7.495

75⁄8 75⁄8 71⁄2 71⁄2 71⁄2

0.810 0.750 0.695 0.630 0.570

13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8 9⁄ 16

11⁄2 17⁄16 13⁄8 15⁄16 11⁄4

WT9×23 WT9×20 WT9×17.5

6.77 5.88 5.15

9.030 8.950 8.850

9 9 87⁄8

0.360 0.315 0.300

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

3.25 2.82 2.66

6.060 6.015 6.000

6 6 6

0.605 0.525 0.425

5⁄ 8 1⁄ 2 7⁄ 16

11⁄4 13⁄16 11⁄8

1⁄

4 4



1 - 75

bf


tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

36

50

43.1 38.6 35.0 30.0 26.7 24.6 22.2 20.2

3.02 3.00 2.98 2.95 2.93 2.92 2.90 2.89

66.6 59.6 53.9 46.3 41.1 37.8 34.1 30.9

— — — — — — — 0.990

— — — — — 0.993 0.917 0.826

17.4 15.3 13.3 12.2 10.9

— — 0.908 0.853 0.784

0.968 0.856 0.730 0.664 0.583

7.42 0.793 6.09 0.733 5.09 0.638

0.592 0.533 0.460

100.5 91 83 73.5 66 61 55.5 50.5

10.3 11.2 12.4 13.0 14.4 15.6 17.1 18.8

285 253 226 204 181 166 150 135

31.9 28.5 25.5 23.7 21.1 19.3 17.5 15.8

3.10 3.07 3.04 3.08 3.06 3.04 3.03 3.01

2.57 2.48 2.39 2.39 2.33 2.28 2.23 2.18

58.6 52.1 46.3 42.4 37.6 34.3 31.0 27.9

1.18 1.07 0.984 0.864 0.780 0.724 0.662 0.605

271 241 217 188 166 152 137 124

46.5 41.5 36.5 34 31

16.2 18.2 20.6 21.8 23.5

144 127 110 103 93.8

17.9 15.7 13.8 12.9 11.9

3.25 3.22 3.21 3.20 3.21

2.74 2.66 2.60 2.59 2.58

31.8 28.0 24.4 22.9 21.1

0.812 0.728 0.647 0.606 0.554

46.4 40.7 35.3 32.4 28.7

11.0 9.75 8.51 7.83 6.97

1.84 1.83 1.81 1.80 1.77

28.5 25 22

23.2 24.7 26.8

90.4 80.3 71.1

11.8 10.7 9.68

3.29 3.30 3.31

2.85 2.93 2.98

21.2 20.8 17.6

0.638 0.771 1.06

15.3 12.5 10.3

4.67 3.82 3.18

1.35 1.30 1.26

155.5 141.5 129 117 105.5 96 87.5 79 71.5 65

5.3 5.7 6.3 6.9 7.5 8.3 9.0 9.9 11.0 11.9

383 337 298 260 229 202 181 160 142 127

46.5 41.5 37.0 32.6 29.0 25.8 23.4 20.8 18.5 16.7

2.89 2.85 2.80 2.75 2.72 2.68 2.66 2.63 2.60 2.58

2.93 2.80 2.68 2.55 2.44 2.34 2.26 2.18 2.09 2.02

90.6 80.1 71.0 62.4 55.0 48.5 43.6 38.5 34.0 30.5

1.91 1.75 1.61 1.48 1.34 1.23 1.13 1.02 0.938 0.856

59.5 53 48.5 43 38

12.3 13.6 15.0 16.7 18.9

119 104 93.8 82.4 71.8

15.9 14.1 12.7 11.2 9.83

2.60 2.59 2.56 2.55 2.54

2.03 1.97 1.91 1.86 1.80

28.7 25.2 22.6 19.9 17.3

0.778 126 0.695 110 0.640 100 0.570 87.6 0.505 76.2

35.5 32.5 30 27.5 25

16.2 17.8 19.3 20.6 22.6

78.2 70.7 64.7 59.5 53.5

11.2 10.1 9.29 8.63 7.79

2.74 2.72 2.71 2.71 2.70

2.26 2.20 2.16 2.16 2.12

20.0 18.0 16.5 15.3 13.8

0.683 0.629 0.583 0.538 0.489

30.1 27.4 25.0 22.5 20.0

23 20 17.5

22.3 25.5 26.8

52.1 44.8 40.1

7.77 6.73 6.21

2.77 2.76 2.79

2.33 2.29 2.39

13.9 12.0 12.0

0.558 0.489 0.450

11.3 9.55 7.67

398 352 314 279 246 220 196 174 156 139

Fy, ksi

66.2 59.2 53.4 47.9 42.7 38.4 34.4 30.7 27.7 24.9

2.95 2.91 2.88 2.85 2.82 2.79 2.76 2.74 2.72 2.70

104 92.5 83.2 74.5 66.2 59.4 53.1 47.4 42.7 38.3

— — — — — — — — — —

— — — — — — — — — —

22.5 19.7 18.0 15.8 13.8

2.69 2.66 2.65 2.63 2.61

34.6 30.2 27.6 24.2 21.1

— — — — 0.990

— — — 0.937 0.826

7.89 7.22 6.63 5.97 5.35

1.70 1.69 1.69 1.67 1.65

12.3 — 11.2 — 10.3 0.964 9.27 0.913 8.29 0.823

0.963 0.877 0.796 0.735 0.625

3.72 3.17 2.56

1.29 1.27 1.22

5.85 0.831 4.97 0.690 4.03 0.638

0.635 0.496 0.460



1 - 76


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Area of Stem


Area

Depth of Tee d

Thickness tw

tw 2

Designation

in.2

in.

in.

in.

in.2

WT8×50 WT8×44.5 WT8×38.5 WT8×33.5

14.7 13.1 11.3 9.84

8.485 8.375 8.260 8.165

81⁄2 83⁄8 81⁄4 81⁄8

0.585 0.525 0.455 0.395

9⁄ 16 1⁄ 2 7⁄ 16 3⁄ 8

5⁄ 16 1⁄ 4 1⁄ 4 3⁄ 16

4.96 4.40 3.76 3.23

10.425 10.365 10.295 10.235

103⁄8 103⁄8 101⁄4 101⁄4

0.985 0.875 0.760 0.665

1 7⁄ 8 3⁄ 4 11⁄ 16

111⁄16 19⁄16 17⁄16 13⁄8

WT8×28.5 WT8×25 WT8×22.5 WT8×20 WT8×18

8.38 7.37 6.63 5.89 5.28

8.215 8.130 8.065 8.005 7.930

81⁄4 81⁄8 81⁄8 8 77⁄8

0.430 0.380 0.345 0.305 0.295

7⁄ 16 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16

1⁄ 4 3⁄ 16 3⁄ 16 3⁄ 16 3⁄ 16

3.53 3.09 2.78 2.44 2.34

7.120 7.070 7.035 6.995 6.985

71⁄8 71⁄8 7 7 7

0.715 0.630 0.565 0.505 0.430

11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16

13⁄8 15⁄16 11⁄4 13⁄16 11⁄8

WT8×15.5 WT8×13

4.56 3.84

7.940 7.845

8 77⁄8

0.275 0.250

1⁄ 4 1⁄ 4

1⁄

2.18 1.96

5.525 5.500

51⁄2 51⁄2

0.440 0.345

7⁄ 16 3⁄ 8

11⁄8 11⁄16

3.740 3.070 2.830 2.595 2.380 2.190 2.015

33⁄4 31⁄16 213⁄16 25⁄8 23⁄8 23⁄16 2

17⁄8 19⁄16 17⁄16 15⁄16 13⁄16 11⁄8 1

42.7 34.4 30.6 27.1 24.1 21.5 19.2

18.560 17.890 17.650 17.415 17.200 17.010 16.835

181⁄2 177⁄8 175⁄8 173⁄8 171⁄4 17 167⁄8

5.120 4.910 4.520 4.160 3.820 3.500 3.210

51⁄8 415⁄16 41⁄2 43⁄16 313⁄16 31⁄2 33⁄16

513⁄16 59⁄16 53⁄16 413⁄16 41⁄2 43⁄16 37⁄8

1.875 1.770 1.655 1.540 1.410 1.290 1.175 1.070 0.980 0.890 0.830 0.745 0.680

17⁄8 13⁄4 15⁄8 19⁄16 17⁄16 15⁄16 13⁄16 11⁄16 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

15⁄

17.5 16.2 14.8 13.5 12.1 10.8 9.62 8.58 7.70 6.89 6.32 5.58 5.03

16.695 16.590 16.475 16.360 16.230 16.110 15.995 15.890 15.800 15.710 15.650 15.565 15.500

163⁄4 165⁄8 161⁄2 163⁄8 161⁄4 161⁄8 16 157⁄8 153⁄4 153⁄4 155⁄8 155⁄8 151⁄2

3.035 2.845 2.660 2.470 2.260 2.070 1.890 1.720 1.560 1.440 1.310 1.190 1.090

31⁄16 27⁄8 211⁄16 21⁄2 21⁄4 21⁄16 17⁄8 13⁄4 19⁄16 17⁄16 15⁄16 13⁄16 11⁄16

311⁄16 31⁄2 35⁄16 31⁄8 215⁄16 23⁄4 29⁄16 23⁄8 21⁄4 21⁄8 2 17⁄8 13⁄4

WT7×404 WT8×365 WT8×332.5 WT8×302.5 WT8×275 WT8×250 WT8×227.5

119 107 97.8 88.9 80.9 73.5 66.9

WT7×213 WT8×199 WT8×185 WT8×171 WT8×155.5 WT8×141.5 WT8×128.5 WT8×116.5 WT8×105.5 WT8×96.5 WT8×88 WT8×79.5 WT8×72.5

62.6 58.5 54.4 50.3 45.7 41.6 37.8 34.2 31.0 28.4 25.9 23.4 21.3

11.420 117⁄16 11.210 111⁄4 10.820 107⁄8 10.460 101⁄2 10.120 101⁄8 9.800 93⁄4 9.510 91⁄2 9.335 9.145 8.960 8.770 8.560 8.370 8.190 8.020 7.860 7.740 7.610 7.490 7.390

93⁄8 91⁄8 9 83⁄4 81⁄2 83⁄8 81⁄4 8 77⁄8 73⁄4 75⁄8 71⁄2 73⁄8

1⁄

7⁄

13⁄

13⁄ 3⁄

8 8

16 8 16 16

4 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 3⁄ 8

Width bf in.


in.

in.


1 - 77

bf


tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

36

50

17.9 15.7 13.4 11.6

2.51 2.49 2.47 2.46

27.4 24.0 20.5 17.7

— — — —

— — 0.988 0.861

— 0.990 0.904 0.784 0.754

0.942 0.826 0.725 0.583 0.553

50 44.5 38.5 33.5

12.1 13.5 15.6 18.0

76.8 67.2 56.9 48.6

11.4 10.1 8.59 7.36

2.28 2.27 2.24 2.22

1.76 1.70 1.63 1.56

20.7 18.1 15.3 13.0

0.706 0.631 0.549 0.481

93.1 81.3 69.2 59.5

28.5 25 22.5 20 18

16.5 18.7 20.6 23.3 24.1

48.7 42.3 37.8 33.1 30.6

7.77 6.78 6.10 5.35 5.05

2.41 2.40 2.39 2.37 2.41

1.94 1.89 1.86 1.81 1.88

13.8 12.0 10.8 9.43 8.93

0.589 0.521 0.471 0.421 0.378

21.6 18.6 16.4 14.4 12.2

15.5 13

25.8 28.4

27.4 23.5

4.64 4.09

2.45 2.47

2.02 2.09

404 365 332.5 302.5 275 250 227.5

1.5 1.9 2.0 2.2 2.4 2.6 2.8

898 739 622 524 442 375 321

116 95.4 82.1 70.6 60.9 52.7 45.9

2.75 2.62 2.52 2.43 2.34 2.26 2.19

3.70 3.47 3.25 3.05 2.85 2.67 2.51

213 199 185 171 155.5 141.5 128.5 116.5 105.5 96.5 88 79.5 72.5

3.0 3.2 3.4 3.7 4.0 4.4 4.9 5.3 5.8 6.4 6.9 7.7 8.4

287 257 229 203 176 153 133 116 102 89.8 80.5 70.2 62.5

41.4 37.6 33.9 30.4 26.7 23.5 20.7 18.2 16.2 14.4 13.0 11.4 10.2

2.14 2.10 2.05 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71

2.40 2.30 2.19 2.09 1.97 1.86 1.75 1.65 1.57 1.49 1.43 1.35 1.29

8.27 0.413 8.12 0.372 249 211 182 157 136 117 102 91.7 82.9 74.4 66.2 57.7 50.4 43.9 38.2 33.4 29.4 26.3 22.8 20.2

3.19 3.00 2.77 2.55 2.35 2.16 1.99

6.20 4.80 2760 2360 2080 1840 1630 1440 1280

1.88 1180 1.76 1090 1.65 994 1.54 903 1.41 807 1.29 722 1.18 645 1.08 576 0.980 513 0.903 466 0.827 419 0.751 374 0.688 338

Fy, ksi

6.06 5.26 4.67 4.12 3.50

1.60 1.59 1.57 1.57 1.52

9.43 8.16 7.23 6.37 5.42

2.24 1.74

1.17 1.12

3.52 0.668 0.479 2.74 0.563 0.406

297 264 236 211 189 169 152

4.82 4.69 4.62 4.55 4.49 4.43 4.38

463 408 365 326 292 261 234

— — — — — — —

— — — — — — —

141 131 121 110 99.4 89.7 80.7 72.5 65.0 59.3 53.5 48.1 43.7

4.34 4.31 4.27 4.24 4.20 4.17 4.13 4.10 4.07 4.05 4.02 4.00 3.98

217 201 185 169 152 137 123 110 99.0 90.2 81.4 73.0 66.3

— — — — — — — — — — — — —

— — — — — — — — — — — — —



1 - 78


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Designation

Area of Stem


Area

Depth of Tee d

Thickness tw

tw 2

in.2

in.

in.

in.

in.2

5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4

4.73 4.27 3.76 3.43 3.08

14.725 14.670 14.605 14.565 14.520

143⁄4 145⁄8 145⁄8 145⁄8 141⁄2

1.030 0.940 0.860 0.780 0.710

1 15⁄ 16 7⁄ 8 3⁄ 4 11⁄ 16

111⁄16 15⁄8 19⁄16 17⁄16 13⁄8

1⁄

10.130 101⁄8 10.070 101⁄8 10.035 10 9.995 10

0.855 0.785 0.720 0.645

7⁄ 8 13⁄ 16 3⁄ 4 5⁄ 8

15⁄8 19⁄16 11⁄2 17⁄16

Width bf in.

in.

in.

WT7×66 WT7×60 WT7×54.5 WT7×49.5 WT7×45

19.4 17.7 16.0 14.6 13.2

7.330 7.240 7.160 7.080 7.010

73⁄8 71⁄4 71⁄8 71⁄8 7

0.645 0.590 0.525 0.485 0.440

5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16

WT7×41 WT7×37 WT7×34 WT7×30.5

12.0 10.9 9.99 8.96

7.155 7.085 7.020 6.945

71⁄8 71⁄8 7 7

0.510 0.450 0.415 0.375

1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8

1⁄ 4 3⁄ 16

3.65 3.19 2.91 2.60

WT7×26.5 WT7×24 WT7×21.5

7.81 7.07 6.31

6.960 6.895 6.830

7 67⁄8 67⁄8

0.370 0.340 0.305

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

2.58 2.34 2.08

8.060 8.030 7.995

8 8 8

0.660 0.595 0.530

11⁄ 16 5⁄ 8 1⁄ 2

17⁄16 13⁄8 15⁄16

WT7×19 WT7×17 WT7×15

5.58 5.00 4.42

7.050 6.990 6.920

7 7 67⁄8

0.310 0.285 0.270

5⁄ 16 5⁄ 16 1⁄ 4

3⁄ 16 3⁄ 16 1⁄ 8

2.19 1.99 1.87

6.770 6.745 6.730

63⁄4 63⁄4 63⁄4

0.515 0.455 0.385

1⁄ 2 7⁄ 16 3⁄ 8

11⁄16 1 15⁄ 16

WT7×13 WT7×11

3.85 3.25

6.955 6.870

7 67⁄8

0.255 0.230

1⁄ 4 1⁄ 4

1⁄

1.77 1.58

5.025 5.000

5 5

0.420 0.335

7⁄ 16 5⁄ 16

15⁄

1⁄

1⁄

4 4

8 8


7⁄

16 8


1 - 79

bf


tf

k

Y

yp , y

X

X

d

tw

Y

Nominal Wt. per ft

Axis X-X

h tw

S

I 4

r 3

y

Z

yp 3

S

I 4

in.

in.

in.

in.

in.

in.

66 60 54.5 49.5 45

8.8 9.7 10.9 11.8 13.0

57.8 51.7 45.3 40.9 36.4

9.57 8.61 7.56 6.88 6.16

1.73 1.71 1.68 1.67 1.66

1.29 1.24 1.17 1.14 1.09

18.6 16.5 14.4 12.9 11.5

0.658 0.602 0.549 0.500 0.456

41 37 34 30.5

11.2 12.7 13.7 15.2

41.2 36.0 32.6 28.9

7.14 6.25 5.69 5.07

1.85 1.82 1.81 1.80

1.39 1.32 1.29 1.25

13.2 11.5 10.4 9.16

0.594 0.541 0.498 0.448

74.2 66.9 60.7 53.7

26.5 24 21.5

15.4 16.8 18.7

27.6 24.9 21.9

4.94 4.48 3.98

1.88 1.87 1.86

1.38 1.35 1.31

8.87 8.00 7.05

0.484 0.440 0.395

28.8 25.7 22.6

19 17 15

19.8 21.5 22.7

23.3 20.9 19.0

4.22 3.83 3.55

2.04 2.04 2.07

1.54 1.53 1.58

7.45 6.74 6.25

0.412 0.371 0.329

13 11

24.1 26.7

17.3 14.8

3.31 2.91

2.12 2.14

1.72 1.76

5.89 5.20

0.383 0.325

lb

Qs*

Axis Y-Y

in.

r 3

Z

Fy, ksi 3

in.

in.

in.

36

50

37.2 33.7 30.6 27.6 25.0

3.76 3.74 3.73 3.71 3.70

56.6 51.2 46.4 41.8 37.8

— — — — —

— — — — —

14.6 13.3 12.1 10.7

2.48 2.48 2.46 2.45

22.4 20.3 18.5 16.4

— — — —

— — — 0.973

7.16 6.40 5.65

1.92 1.91 1.89

11.0 9.82 8.66

— — 0.947

0.958 0.882 0.775

13.3 11.7 9.79

3.94 3.45 2.91

1.55 1.53 1.49

6.07 5.32 4.49

0.934 0.857 0.810

0.760 0.669 0.610

4.45 3.50

1.77 1.40

1.08 1.04

2.77 2.19

0.737 0.621

0.537 0.447

274 247 223 201 181



1 - 80


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Designation

Area of Stem

Area

Depth of Tee d

Thickness tw

tw 2

in.2

in.

in.

in.

in.2 14.9 13.3 12.1 10.7 9.67 8.68 7.62 6.73 5.96 5.30 4.66 3.93 3.50 3.23 2.91 2.63 2.36

WT6×168 WT6×152.5 WT6×139.5 WT6×126 WT6×115 WT6×105 WT6×95 WT6×85 WT6×76 WT6×68 WT6×60 WT6×53 WT6×48 WT6×43.5 WT6×39.5 WT6×36 WT6×32.5

49.4 44.8 41.0 37.0 33.9 30.9 27.9 25.0 22.4 20.0 17.6 15.6 14.1 12.8 11.6 10.6 9.54

8.410 8.160 7.925 7.705 7.525 7.355 7.190 7.015 6.855 6.705 6.560 6.445 6.355 6.265 6.190 6.125 6.060

83⁄8 81⁄8 77⁄8 73⁄4 71⁄2 73⁄8 71⁄4 7 67⁄8 63⁄4 61⁄2 61⁄2 63⁄8 61⁄4 61⁄4 61⁄8 6

1.775 1.625 1.530 1.395 1.285 1.180 1.060 0.960 0.870 0.790 0.710 0.610 0.550 0.515 0.470 0.430 0.390

13⁄4 15⁄8 11⁄2 13⁄8 15⁄16 13⁄16 11⁄16 15⁄ 16 7⁄ 8 13⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8

7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 11⁄ 16 5⁄ 8 9⁄ 16 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 5⁄ 16 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

WT6×29 WT6×26.5

8.52 7.78

6.095 6.030

61⁄8 6

0.360 0.345

3⁄ 8 3⁄ 8

3⁄ 16 3⁄ 16

WT6×25 WT6×22.5 WT6×20

7.34 6.61 5.89

6.095 6.030 5.970

61⁄8 6 6

0.370 0.335 0.295

3⁄ 8 5⁄ 16 5⁄ 16

WT6×17.5 WT6×15 WT6×13

5.17 4.40 3.82

6.250 6.170 6.110

61⁄4 61⁄8 61⁄8

0.300 0.260 0.230

WT6×11 WT6×9.5 WT6×8 WT6×7

3.24 2.79 2.36 2.08

6.155 6.080 5.995 5.955

61⁄8 61⁄8 6 6

0.260 0.235 0.220 0.200


Width bf in.

in.

in.

133⁄8 131⁄4 131⁄8 13 127⁄8 123⁄4 125⁄8 125⁄8 121⁄2 123⁄8 123⁄8 121⁄4 121⁄8 121⁄8 121⁄8 12 12

2.955 2.705 2.470 2.250 2.070 1.900 1.735 1.560 1.400 1.250 1.105 0.990 0.900 0.810 0.735 0.670 0.605

215⁄16 211⁄16 21⁄2 21⁄4 21⁄16 17⁄8 13⁄4 19⁄16 13⁄8 11⁄4 11⁄8 1 7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8

311⁄16 37⁄16 33⁄16 215⁄16 23⁄4 25⁄8 27⁄16 21⁄4 21⁄8 115⁄16 113⁄16 111⁄16 15⁄8 11⁄2 17⁄16 13⁄8 15⁄16

2.19 10.010 2.08 9.995

10 10

0.640 0.575

5⁄ 8 9⁄ 16

13⁄8 11⁄4

3⁄ 16 3⁄ 16 3⁄ 16

2.26 2.02 1.76

8.080 8.045 8.005

81⁄8 8 8

0.640 0.575 0.515

5⁄ 8 9⁄ 16 1⁄ 2

13⁄8 11⁄4 11⁄4

5⁄ 16 1⁄ 4 1⁄ 4

3⁄ 16 1⁄ 8 1⁄ 8

1.88 1.60 1.41

6.560 6.520 6.490

61⁄2 61⁄2 61⁄2

0.520 0.440 0.380

1⁄ 2 7⁄ 16 3⁄ 8

15⁄ 16 7⁄ 8

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8 1⁄ 8

1.60 1.43 1.32 1.19

4.030 4.005 3.990 3.970

4 4 4 4

0.425 0.350 0.265 0.225

7⁄ 16 3⁄ 8 1⁄ 4 1⁄ 4

7⁄ 8 13⁄ 16 3⁄ 4 11⁄ 16

13.385 13.235 13.140 13.005 12.895 12.790 12.670 12.570 12.480 12.400 12.320 12.220 12.160 12.125 12.080 12.040 12.000


1


1 - 81

bf


tf

k

Y

yp , y

X

X

d

tw

Y

Nominal Wt. per ft

Axis X-X

h tw

S

I 4

r 3

y

Qs*

Axis Y-Y

Z

yp 3

S

I 4

r 3

Z

Fy, ksi 3

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.

36

50

168 152.5 139.5 126 115 105 95 85 76 68 60 53 48 43.5 39.5 36 32.5

2.7 3.0 3.2 3.5 3.8 4.1 4.6 5.1 5.6 6.1 6.8 8.0 8.8 9.4 10.3 11.3 12.4

190 162 141 121 106 92.1 79.0 67.8 58.5 50.6 43.4 36.3 32.0 28.9 25.8 23.2 20.6

31.2 27.0 24.1 20.9 18.5 16.4 14.2 12.3 10.8 9.46 8.22 6.91 6.12 5.60 5.03 4.54 4.06

1.96 1.90 1.86 1.81 1.77 1.73 1.68 1.65 1.62 1.59 1.57 1.53 1.51 1.50 1.49 1.48 1.47

2.31 2.16 2.05 1.92 1.82 1.72 1.62 1.52 1.43 1.35 1.28 1.19 1.13 1.10 1.06 1.02 0.985

68.4 59.1 51.9 44.8 39.4 34.5 29.8 25.6 22.0 19.0 16.2 13.6 11.9 10.7 9.49 8.48 7.50

1.84 1.69 1.56 1.42 1.31 1.21 1.10 0.994 0.896 0.805 0.716 0.637 0.580 0.527 0.480 0.439 0.398

593 525 469 414 371 332 295 259 227 199 172 151 135 120 108 97.5 87.2

88.6 79.3 71.3 63.6 57.5 51.9 46.5 41.2 36.4 32.1 28.0 24.7 22.2 19.9 17.9 16.2 14.5

3.47 3.42 3.38 3.34 3.31 3.28 3.25 3.22 3.19 3.16 3.13 3.11 3.09 3.07 3.05 3.04 3.02

137 122 110 97.9 88.4 79.7 71.3 63.0 55.6 49.0 42.7 37.5 33.7 30.2 27.2 24.6 22.0

— — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — —

29 26.5

13.5 14.1

19.1 17.7

3.76 3.54

1.50 1.51

1.03 1.02

6.97 0.426 6.46 0.389

53.5 47.9

10.7 9.58

2.51 2.48

16.3 14.6

— —

— —

25 22.5 20

13.1 14.5 16.5

18.7 16.6 14.4

3.79 3.39 2.95

1.60 1.58 1.57

1.17 1.13 1.08

6.90 0.454 6.12 0.411 5.30 0.368

28.2 25.0 22.0

6.97 6.21 5.51

1.96 1.94 1.93

10.7 9.50 8.41

— — —

— 0.998 0.887

17.5 15 13

18.1 20.9 23.6

16.0 13.5 11.7

3.23 2.75 2.40

1.76 1.75 1.75

1.30 1.27 1.25

5.71 0.394 4.83 0.337 4.20 0.295

12.2 10.2 8.66

3.73 3.12 2.67

1.54 1.52 1.51

5.73 4.78 4.08

— 0.856 0.891 0.710 0.767 0.565

11 9.5 8 7

20.9 23.1 24.7 27.2

11.7 10.1 8.70 7.67

2.59 2.28 2.04 1.83

1.90 1.90 1.92 1.92

1.63 1.65 1.74 1.76

4.63 4.11 3.72 3.32

2.33 1.88 1.41 1.18

1.16 0.939 0.706 0.594

0.847 0.822 0.773 0.753

1.83 1.49 1.13 0.950

0.891 0.797 0.741 0.626

lb

0.402 0.348 0.639 0.760



0.710 0.596 0.541 0.450

1 - 82


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Designation

Area of Stem

Area

Depth of Tee d

Thickness tw

tw 2

in.2

in.

in.

in.

in.2


Width bf in.

in.

in.

WT5×56 WT5×50 WT5×44 WT5×38.5 WT5×34 WT5×30 WT5×27 WT5×24.5

16.5 14.7 12.9 11.3 9.99 8.82 7.91 7.21

5.680 5.550 5.420 5.300 5.200 5.110 5.045 4.990

55⁄8 51⁄2 53⁄8 51⁄4 51⁄4 51⁄8 5 5

0.755 0.680 0.605 0.530 0.470 0.420 0.370 0.340

3⁄ 4 11⁄ 16 5⁄ 8 1⁄ 2 1⁄ 2 7⁄ 16 3⁄ 8 5⁄ 16

3⁄ 8 3⁄ 8 5⁄ 16 1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16 3⁄ 16

4.29 3.77 3.28 2.81 2.44 2.15 1.87 1.70

10.415 10.340 10.265 10.190 10.130 10.080 10.030 10.000

103⁄8 103⁄8 101⁄4 101⁄4 101⁄8 101⁄8 10 10

1.250 1.120 0.990 0.870 0.770 0.680 0.615 0.560

11⁄4 11⁄8 1 7⁄ 8 3⁄ 8 11⁄ 16 5⁄ 8 9⁄ 16

17⁄8 13⁄4 15⁄8 11⁄2 13⁄8 15⁄16 11⁄4 13⁄16

WT5×22.5 WT5×19.5 WT5×16.5

6.63 5.73 4.85

5.050 4.960 4.865

5 5 47⁄8

0.350 0.315 0.290

3⁄ 8 5⁄ 16 5⁄ 16

3⁄ 16 3⁄ 16 3⁄ 16

1.77 1.56 1.41

8.020 7.985 7.960

8 8 8

0.620 0.530 0.435

5⁄ 8 1⁄ 2 7⁄ 16

11⁄4 11⁄8 11⁄16

WT5×15 WT5×13 WT5×11

4.42 3.81 3.24

5.235 5.165 5.085

51⁄4 51⁄8 51⁄8

0.300 0.260 0.240

5⁄ 16 1⁄ 4 1⁄ 4

3⁄ 16 1⁄ 8 1⁄ 8

1.57 1.34 1.22

5.810 5.770 5.750

53⁄4 53⁄4 53⁄4

0.510 0.440 0.360

1⁄ 2 7⁄ 16 3⁄ 8

15⁄ 16 7⁄ 8 3⁄ 4

WT5×9.5 WT5×8.5 WT5×7.5 WT5×6

2.81 2.50 2.21 1.77

5.120 5.055 4.995 4.935

51⁄8 5 5 47⁄8

0.250 0.240 0.230 0.190

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8 1⁄ 8

1.28 1.21 1.15 0.938

4.020 4.010 4.000 3.960

4 4 4 4

0.395 0.330 0.270 0.210

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

13⁄ 16 3⁄ 4 11⁄ 16 5⁄ 8



1 - 83

bf


tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

0.791 118 0.711 103 0.631 89.3 0.555 76.8 0.493 66.8 0.438 58.1 0.395 51.7 0.361 46.7

22.6 20.0 17.4 15.1 13.2 11.5 10.3 9.34

2.68 2.65 2.63 2.60 2.59 2.57 2.56 2.54

6.65 5.64 4.60

Fy, ksi 36

50

34.6 30.5 26.5 22.9 20.0 17.5 15.7 14.2

— — — — — — — —

— — — — — — — —

2.01 1.98 1.94

10.1 8.59 7.01

— — —

— — —

56 50 44 38.5 34 30 27 24.5

5.2 5.8 6.5 7.4 8.4 9.4 10.6 11.6

28.6 24.5 20.8 17.4 14.9 12.9 11.1 10.0

6.40 5.56 4.77 4.04 3.49 3.04 2.64 2.39

1.32 1.29 1.27 1.24 1.22 1.21 1.19 1.18

1.21 1.13 1.06 0.990 0.932 0.884 0.836 0.807

13.4 11.4 9.65 8.06 6.85 5.87 5.05 4.52

22.5 19.5 16.5

11.2 12.5 13.6

10.2 8.84 7.71

2.47 2.16 1.93

1.24 1.24 1.26

0.907 0.876 0.869

4.65 3.99 3.48

0.413 0.359 0.305

15 13 11

14.8 17.0 18.4

9.28 7.86 6.88

2.24 1.91 1.72

1.45 1.44 1.46

1.10 1.06 1.07

4.01 3.39 3.02

0.380 0.330 0.282

8.35 7.05 5.71

2.87 2.44 1.99

1.37 1.36 1.33

4.42 3.75 3.05

— — 0.999

— 0.902 0.836

17.7 18.4 19.2 23.3

6.68 6.06 5.45 4.35

1.74 1.62 1.50 1.22

1.54 1.56 1.57 1.57

1.28 1.32 1.37 1.36

3.10 2.90 3.03 2.50

0.349 0.311 0.306 0.323

2.15 1.78 1.45 1.09

1.07 0.888 0.723 0.551

0.874 0.844 0.810 0.785

1.68 — 1.40 — 1.15 0.977 0.872 0.793

0.872 0.841 0.811 0.592

9.5 8.5 7.5 6

26.7 22.5 18.3



1 - 84


bf tf

yp , y


k

Y

X

X

d

tw

Y

Stem

Area of Stem


Area

Depth of Tee d

Thickness tw

tw 2

Designation

in.2

in.

in.

in.

in.2

WT4×33.5 WT4×29 WT4×24 WT4×20 WT4×17.5 WT4×15.5

9.84 8.55 7.05 5.87 5.14 4.56

4.500 4.375 4.250 4.125 4.060 4.000

41⁄2 43⁄8 41⁄4 41⁄8 4 4

0.570 0.510 0.400 0.360 0.310 0.285

9⁄ 16 1⁄ 2 3⁄ 8 3⁄ 8 5⁄ 16 5⁄ 16

5⁄ 16 1⁄ 4 3⁄ 16 3⁄ 16 3⁄ 16 3⁄ 16

2.56 2.23 1.70 1.48 1.26 1.14

8.280 8.220 8.110 8.070 8.020 7.995

81⁄4 81⁄4 81⁄8 81⁄8 8 8

0.935 0.810 0.685 0.560 0.495 0.435

15⁄ 16 13⁄ 16 11⁄ 16 9⁄ 16 1⁄ 2 7⁄ 16

17⁄16 15⁄16 13⁄16 11⁄16 1 15⁄ 16

WT4×14 WT4×12

4.12 3.54

4.030 3.965

4 4

0.285 0.245

5⁄ 16 1⁄ 4

3⁄ 16 1⁄ 8

1.15 0.971

6.535 6.495

61⁄2 61⁄2

0.465 0.400

7⁄ 16 3⁄ 8

15⁄ 16 7⁄ 8

WT4×10.5 WT4×9

3.08 2.63

4.140 4.070

41⁄8 41⁄8

0.250 0.230

1⁄ 4 1⁄ 4

1⁄ 8 1⁄ 8

1.03 0.936

5.270 5.250

51⁄4 51⁄4

0.400 0.330

3⁄ 8 5⁄ 16

13⁄ 16 3⁄ 4

WT4×7.5 WT4×6.5 WT4×5

2.22 1.92 1.48

4.055 3.995 3.945

4 4 4

0.245 0.230 0.170

1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8

0.993 0.919 0.671

4.015 4.000 3.940

4 4 4

0.315 0.255 0.205

5⁄ 16 1⁄ 4 3⁄ 16

3⁄ 4 11⁄ 16 5⁄ 8

WT3×12.5 WT4×10 WT4×7.5

3.67 2.94 2.21

3.190 3.100 2.995

31⁄4 31⁄8 3

0.320 0.260 0.230

5⁄ 16 1⁄ 4 1⁄ 4

3⁄ 16 1⁄ 8 1⁄ 8

1.02 0.806 0.689

6.080 6.020 5.990

61⁄8 6 6

0.455 0.365 0.260

7⁄ 16 3⁄ 8 1⁄ 4

13⁄ 16 3⁄ 4 5⁄ 8

WT3×8 WT4×6 WT4×4.5

2.37 1.78 1.34

3.140 3.015 2.950

31⁄8 3 3

0.260 0.230 0.170

1⁄ 4 1⁄ 4 3⁄ 16

1⁄ 8 1⁄ 8 1⁄ 8

0.816 0.693 0.502

4.030 4.000 3.940

4 4 4

0.405 0.280 0.215

3⁄ 8 1⁄ 4 3⁄ 16

3⁄ 4 5⁄ 8 9⁄ 16

WT2.5×9.5 WT4.5×8

2.77 2.34

2.575 2.505

25⁄8 21⁄2

0.270 0.240

1⁄ 4 1⁄ 4

1⁄ 8 1⁄ 8

0.695 0.601

5.030 5.000

5 5

0.430 0.360

7⁄ 16 3⁄ 8

13⁄ 16 3⁄ 4

WT2×6.5

1.91

2.080

21⁄8

0.280

1⁄ 4

1⁄ 8

0.582

4.060

4

0.345

3⁄ 8

11⁄ 16

Width bf in.


in.

in.


1 - 85

bf


tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

36

Fy, ksi 50

33.5 29 24 20 17.5 15.5

5.6 6.2 7.9 8.8 10.2 11.1

10.9 9.12 6.85 5.73 4.81 4.28

3.05 2.61 1.97 1.69 1.43 1.28

1.05 1.03 0.986 0.988 0.967 0.968

0.936 0.874 0.777 0.735 0.688 0.667

6.29 5.25 3.94 3.25 2.71 2.39

0.594 0.520 0.435 0.364 0.321 0.285

44.3 37.5 30.5 24.5 21.3 18.5

10.7 9.13 7.52 6.08 5.31 4.64

2.12 2.10 2.08 2.04 2.03 2.02

16.3 13.9 11.4 9.25 8.06 7.04

— — — — — —

— — — — — —

14 12

11.1 12.9

4.22 3.53

1.28 1.08

1.01 0.999

0.734 0.695

2.38 1.98

0.315 0.273

10.8 9.14

3.31 2.81

1.62 1.61

5.05 4.29

— —

— —

10.5 9

13.8 15.0

3.90 3.41

1.18 1.05

1.12 1.14

0.831 0.834

2.11 1.86

0.292 0.251

4.89 3.98

1.85 1.52

1.26 1.23

2.84 2.33

— —

— —

7.5 6.5 5

14.0 15.0 20.2

3.28 2.89 2.15

1.07 0.974 0.717

1.22 1.23 1.20

0.998 1.03 0.953

1.91 1.74 1.27

0.276 0.240 0.188

1.70 1.37 1.05

0.849 0.876 0.683 0.843 0.532 0.841

1.33 — 1.08 — 0.828 0.913

12.5 10 7.5

7.8 9.6 10.8

2.28 1.76 1.41

0.886 0.693 0.577

0.789 0.774 0.797

0.610 0.560 0.558

1.68 1.29 1.03

0.302 0.244 0.185

8.53 6.64 4.66

2.81 2.21 1.56

4.28 3.36 2.37

— — —

— — —

8 6 4.5

9.6 10.8 14.6

1.69 0.685 1.32 0.564 0.950 0.408

0.844 0.861 0.842

0.676 0.677 0.623

1.25 1.01 0.720

0.294 0.222 0.170

2.21 1.50 1.10

1.10 0.966 0.748 0.918 0.557 0.905

1.70 1.16 0.858

— — —

— — —

9.5 8

7.0 7.9

1.01 0.485 0.850 0.413

0.605 0.601

0.487 0.458

0.967 0.798

0.275 0.234

4.56 3.75

1.82 1.50

1.28 1.27

2.76 2.29

— —

— —

6.5

5.3

0.530 0.321

0.524

0.440

0.616

0.236

1.93

0.950 1.00

1.46

—

—

1.52 1.50 1.45



— — 0.735

1 - 86


bf tf

yp , y

STRUCTURAL TEES Cut from M shapes Dimensions

k

Y

X

X

d

tw

Y

Designation

Area

Depth of Tee d

in.2

in.

Flange

Stem Area of Thickness tw 2 Stem tw

in.

in.

in.

1.73 1.59

6.000 5.990

6 6

0.177 0.160

3⁄ 16 3⁄ 16

1⁄ 8 1⁄ 16

1.06 3.065 0.958 3.065

31⁄8 31⁄8

0.225 0.210

1⁄ 4 1⁄ 4

9⁄ 16 1⁄ 2

1⁄ 4 1⁄ 4

— 1⁄ 2

MT5×4.5 MT5×4

1.32 1.18

5.000 4.980

5 5

0.157 0.141

3⁄ 16 3⁄ 16

1⁄ 8 1⁄ 16

0.785 2.690 0.702 2.690

23⁄4 23⁄4

0.206 0.182

3⁄ 16 3⁄ 16

9⁄ 16 7⁄ 16

3⁄ 16 3⁄ 16

— 3⁄ 8

MT4×3.25

0.958 4.000

4

0.135

1⁄ 8

1⁄ 16

0.540 2.281

21⁄4

0.189

3⁄ 16

1⁄ 2

3⁄ 16

—

0.316

5⁄ 16

3⁄ 16

0.790 5.003

5

0.416

7⁄ 16

7⁄ 8

7⁄ 16

7⁄ 8

2.500 21⁄2

in.

in.2

MT6×5.9 MT6×5.4

MT2.5×9.45* 2.78

in.

Width bf in.

Max. DisFlge. FasThickness tance Grip tener tf k in.

*This shape has tapered flanges, while all other MT shapes have parallel flanges.



1 - 87

bf

STRUCTURAL TEES Cut from M shapes Properties

tf

k

Y

yp , y

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

36

Fy, ksi 50

5.9 5.4

31.3 31.8

6.60 6.03

1.60 1.46

1.95 1.95

1.89 1.85

2.89 2.63

1.09 1.01

0.490 0.453

0.320 0.295

0.532 0.533

0.577 0.525

0.483 0.397

0.348 0.286

4.5 4

29.2 29.7

3.46 3.09

0.997 0.893

1.62 1.62

1.53 1.52

1.81 1.62

0.778 0.778

0.305 0.269

0.227 0.200

0.480 0.477

0.405 0.333

0.549 0.446

0.396 0.321

3.25

26.9

1.57

0.556

1.28

1.17

1.01

0.446

0.172

0.150

0.423

0.265

0.634

0.457

9.45

5.6

1.05

0.527

0.615

0.511

1.03

0.278

3.93

1.57

1.19

2.66

—

—



1 - 88


bf tf y, yp

STRUCTURAL TEES Cut from S shapes Dimensions

grip

Y

X

X

d

tw

Y

Designation

Area

Depth of Tee d

in.2

in.

Flange

Stem Area of Thickness tw 2 Stem tw

Width bf

Max. DisFlge. FasThickness tance Grip tener tf k

in.

in.2

in.

in.

in.

7 13⁄ 16 ⁄16 5⁄ 5⁄ 8 16

9.80 7.60

8.050 8 7.870 77⁄8

1.090 1.090

11⁄16 11⁄16

2 2

11⁄8 11⁄8

1 1

3⁄ 8 5⁄ 16 1⁄ 4

8.94 7.50 6.00

7.245 71⁄4 7.125 71⁄8 7.000 7

0.870 0.870 0.870

7⁄ 8 7⁄ 8 7⁄ 8

13⁄4 13⁄4 13⁄4

7⁄ 8 7⁄ 8 7⁄ 8

1 1 1

7 13⁄ 16 ⁄16 3⁄ 11⁄ 16 8

8.12 6.70

7.200 71⁄4 7.060 7

0.920 0.920

15⁄ 16 15⁄ 16

13⁄4 13⁄4

15⁄ 16 15⁄ 16

1 1

in.

in.

in.

in.

ST12×60.5 ST12×53

17.8 15.6

12.250 121⁄4 12.250 121⁄4

0.800 0.620

ST12×50 ST12×45 ST12×40

14.7 13.2 11.7

12.000 12.000 12.000

0.745 0.625 0.500

ST10×48 ST10×43

14.1 12.7

10.150 101⁄8 10.150 101⁄8

0.800 0.660

ST10×37.5 ST10×33

11.0 9.70

10.000 10.000

10 10

0.635 0.505

5⁄ 8 1⁄ 2

5⁄ 16 1⁄ 4

6.35 5.05

6.385 63⁄8 6.225 61⁄4

0.795 0.795

13⁄ 16 13⁄ 16

15⁄8 15⁄8

13⁄ 16 13⁄ 16

7⁄ 8 7⁄ 8

ST9×35 ST9×27.35

10.3 8.04

9.000 9.000

9 9

0.711 0.461

11⁄ 16 7⁄ 16

3⁄ 8 1⁄ 4

6.40 4.15

6.251 61⁄4 6.001 6

0.691 0.691

11⁄ 16 11⁄ 16

11⁄2 11⁄2

11⁄ 16 11⁄ 16

7⁄ 8 7⁄ 8

12 12 12

3⁄ 4 5⁄ 8 1⁄ 2

ST7.5×25 ST7.5×21.45

7.35 6.31

7.500 71⁄2 7.500 71⁄2

0.550 0.411

9⁄

16

5⁄ 16 1⁄ 4

4.13 3.08

5.640 55⁄8 5.501 51⁄2

0.622 0.622

5⁄ 8 5⁄ 8

13⁄8 13⁄8

9⁄ 16 9⁄ 16

3⁄ 4 3⁄ 4

ST6×25 ST6×20.4

7.35 6.00

6.000 6.000

6 6

0.687 0.462

11⁄ 16 7⁄ 16

3⁄ 8 1⁄ 4

4.12 2.77

5.477 51⁄2 5.252 51⁄4

0.659 0.659

11⁄ 16 11⁄ 16

17⁄16 17⁄16

11⁄ 16 5⁄ 8

3⁄ 4 3⁄ 4

ST6×17.5 ST6×15.9

5.15 4.68

6.000 6.000

6 6

0.428 0.350

7⁄

1⁄ 4 3⁄ 16

2.57 2.10

5.078 51⁄8 5.000 5

0.545 0.544

9⁄ 16 9⁄ 16

13⁄16 13⁄16

1⁄ 2 1⁄ 2

3⁄ 4 3⁄ 4

ST5×17.5 ST5×12.7

5.15 3.73

5.000 5.000

5 5

0.594 0.311

16

5⁄ 16 3⁄ 16

2.97 1.56

4.944 5 4.661 45⁄8

0.491 0.491

1⁄ 2 1⁄ 2

11⁄8 11⁄8

1⁄ 2 1⁄ 2

3⁄ 4 3⁄ 4

ST4×11.5 ST4×9.2

3.38 2.70

4.000 4.000

4 4

0.441 0.271

7⁄

16 1⁄ 4

1⁄ 4 1⁄ 8

1.76 1.08

4.171 41⁄8 4.001 4

0.425 0.425

7⁄ 16 7⁄ 16

1 1

7⁄ 16 7⁄ 16

3⁄ 4 3⁄ 4

ST3×8.625 ST3×6.25

2.53 1.83

3.000 3.000

3 3

0.465 0.232

7⁄

16 1⁄ 4

1⁄ 4 1⁄ 8

1.40 0.70

3.565 35⁄8 3.332 33⁄8

0.359 0.359

3⁄ 8 3⁄ 8

7⁄ 8 7⁄ 8

3⁄ 8 3⁄ 8

5⁄ 8 —

ST2.5×5

1.47

2.500 21⁄2

0.214

3⁄

16

1⁄ 8

0.535 3.004

0.326

5⁄ 16

13⁄ 16

5⁄ 16

—

ST2×4.75 ST2×3.85

1.40 1.13

2.000 2.000

0.326 0.193

5⁄

16

3⁄ 16 1⁄ 8

0.652 2.796 23⁄4 0.386 2.663 25⁄8

0.293 0.293

5⁄ 16 5⁄ 16

3⁄ 4 3⁄ 4

5⁄ 16 5⁄ 16

— —

ST1.5×3.75 ST1.5×2.85

1.10 0.835

1.500 11⁄2 1.500 11⁄2

3⁄ 16 1⁄ 8

0.523 2.509 21⁄2 0.255 2.330 23⁄8

0.260 0.260

1⁄ 4 1⁄ 4

11⁄ 16 11⁄ 16

1⁄ 4 1⁄ 4

— —

2 2

0.349 0.170

7⁄

16

16 3⁄ 8 5⁄ 8

5⁄

3⁄

16

3⁄ 8

3⁄

16

3



1 - 89

bf

STRUCTURAL TEES Cut from S shapes Properties

tf

grip

Y

y, yp

X

X

d

tw

Y


Axis X-X

h tw

Qs*

Axis Y-Y

I

S

r

y

Z

yp

I

S

r

Z

in.4

in.3

in.

in.

in.3

in.

in.4

in.3

in.

in.3

60.5 53

13.2 17

259 216

30.1 24.1

3.82 3.72

3.63 3.28

54.5 43.3

1.28 1.03

41.7 38.5

10.4 9.80

1.53 1.57

18.1 16.6

50 45 40

14.1 16.8 21.1

215 190 162

26.3 22.6 18.7

3.83 3.79 3.72

3.84 3.60 3.29

47.5 41.1 33.6

2.20 23.8 1.48 22.5 0.922 21.1

6.58 6.31 6.04

1.27 1.30 1.34

12.0 11.2 10.4

48 43

10.8 13.1

143 125

20.3 17.2

3.18 3.14

3.13 2.91

36.9 31.1

1.40 25.1 0.985 23.4

6.97 6.63

1.33 1.36

12.5 11.6

37.5 33

13.6 17

109 93.1

15.8 12.9

3.15 3.10

3.07 2.81

28.6 23.4

1.40 14.9 0.855 13.8

4.66 4.43

1.16 1.19

35 27.35

10.9 16.8

84.7 62.4

14.0 9.61

2.87 2.79

2.94 2.50

25.1 17.3

1.81 12.1 0.747 10.4

3.86 3.47

25 21.45

11.6 15.5

40.6 33.0

7.73 6.00

2.35 2.29

2.25 2.01

14.0 10.8

0.872 0.613

7.85 7.19

25 20.4

7 10.3

25.2 18.9

6.05 4.28

1.85 1.78

1.84 1.58

11.0 7.71

0.770 0.581

17.5 15.9

11.7 14.3

17.2 14.9

3.95 3.31

1.83 1.78

1.64 1.51

7.12 5.94

17.5 12.7

6.9 13.2

12.5 7.83

3.63 2.06

1.56 1.45

1.56 1.20

11.5 9.2

7.3 11.8

5.03 3.51

1.77 1.15

1.22 1.14

5 10

8.63 6.25

Fy, ksi 36

50

— —

— 0.907

— — — 0.937 0.878 0.695 — —

— —

8.37 7.70

— —

— 0.907

1.08 1.14

7.21 6.07

— —

— 0.922

2.78 2.61

1.03 1.07

5.01 4.54

— —

— 0.988

7.85 6.78

2.87 2.58

1.03 1.06

5.19 4.45

— —

— —

0.548 0.485

4.94 4.68

1.95 1.87

0.980 1.00

3.41 3.22

— —

— —

6.58 3.70

0.702 0.408

4.18 3.39

1.69 1.46

0.901 0.954

3.11 2.49

— —

— —

1.15 0.941

3.19 2.07

0.447 0.341

2.15 1.86

1.03 0.798 0.932 0.831

1.84 1.59

— —

— —

2.13 1.27

1.02 0.917 0.914 0.552 0.833 0.691

1.85 1.01

0.401 0.275

1.15 0.911

0.648 0.675 0.547 0.705

1.18 0.929

— —

— —

5

8.7

0.681

0.353 0.681 0.569

0.650 0.243

0.608

0.405 0.643

0.685

—

—

4.75 3.85

4.3 7.3

0.470 0.316

0.325 0.580 0.553 0.203 0.528 0.448

0.592 0.255 0.381 0.209

0.451 0.382

0.323 0.569 0.287 0.581

0.566 0.483

— —

— —

3.75 2.85

2.8 5.7

0.204 0.118

0.191 0.430 0.432 0.101 0.376 0.329

0.351 0.223 0.196 0.175

0.293 0.227

0.234 0.516 0.195 0.522

0.412 0.327

— —

— —



1 - 90


DOUBLE ANGLES


DOUBLE ANGLES

1 - 91

DOUBLE ANGLES

Properties of double angles in contact and separated are listed in the following tables. Each table shows properties of double angles in contact, and the radius of gyration about the Y-Y axis when the legs of the angles are separated. Values of Qs are given for Fy = 36 ksi and Fy = 50 ksi for those angles exceeding the width-thickness ratio λr of LRFD Specification Section B5. Since the cross section is comprised entirely of unstiffened elements, Qa = 1.0 and Q = Qs, for all _angle sections. The Flexural-Torsional Properties Table lists the dimensional values (J, ro, and H) needed for checking flexural-torsional buckling. Use of Table

The table may be used as follows for checking the limit states of (1) flexural buckling and (2) flexural-torsional buckling. The lower of the two limit states must be used for design. See also Part 3 of this LRFD Manual. (1) Flexural Buckling

Where no value of Qs is shown, the design compressive strength for this limit state is given by LRFD Specification Section E2. Where a value of Qs is shown, the strength must be reduced in accordance with Appendix B5 of the LRFD Specification. (2) Flexural-Torsional Buckling

The design compressive strength for this limit state_ is given by LRFD Specification Sections E3 and E4. This involves calculations with J, ro, and H. These torsional constants can be obtained by summing the respective values for single angles listed in the Flexural-Torsional Properties Tables in Part 1 of this Manual.


1 - 92


Y X

X

DOUBLE ANGLES Two equal leg angles Properties of sections

y, yp

s Y

Wt. Area of per ft 2 Angles 2 Angles Designation

2

Axis X-X

S

I 4

in.

r 3

y

Z

yp 3

lb

in.

in.

in.

in.

in.

in.

L8×8×11⁄8 L8×8×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×11⁄2

114 102 90.0 77.8 65.4 52.8

33.5 30.0 26.5 22.9 19.2 15.5

195 177 159 139 118 97.3

35.1 31.6 28.0 24.4 20.6 16.7

2.42 2.44 2.45 2.47 2.49 2.50

2.41 2.37 2.32 2.28 2.23 2.19

63.2 56.9 50.5 43.9 37.1 30.1

1.05 0.938 0.827 0.715 0.601 0.484

L6×6×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×11⁄2 L8×8×13⁄8

74.8 66.2 57.4 48.4 39.2 29.8

22.0 19.5 16.9 14.2 11.5 8.72

70.9 63.8 56.3 48.3 39.8 30.8

17.1 15.3 13.3 11.3 9.23 7.06

1.80 1.81 1.83 1.84 1.86 1.88

1.86 1.82 1.78 1.73 1.68 1.64

30.9 27.5 24.0 20.4 16.6 12.7

0.917 0.811 0.703 0.592 0.479 0.363

L5×5×7⁄8 L5×5×3⁄4 L5×5×1⁄2 L5×5×3⁄8 L5×5×5⁄16

54.4 47.2 32.4 24.6 20.6

16.0 13.9 9.50 7.22 6.05

35.5 31.5 22.5 17.5 14.8

10.3 9.06 6.31 4.84 4.08

1.49 1.51 1.54 1.56 1.57

1.57 1.52 1.43 1.39 1.37

18.7 16.3 11.4 8.72 7.35

0.798 0.694 0.475 0.361 0.303

L4×4×3⁄4 L5×5×5⁄8 L5×5×1⁄2 L5×5×3⁄8 L5×5×5⁄16 L5×5×1⁄4

37.0 31.4 25.6 19.6 16.4 13.2

10.9 9.22 7.50 5.72 4.80 3.88

15.3 13.3 11.1 8.72 7.43 6.08

5.62 4.80 3.95 3.05 2.58 2.09

1.19 1.20 1.22 1.23 1.24 1.25

1.27 1.23 1.18 1.14 1.12 1.09

10.1 8.66 7.12 5.49 4.64 3.77

0.680 0.576 0.469 0.357 0.300 0.242


DOUBLE ANGLES

1 - 93


Y X

X

y, yp

s Y

Qs*

Axis Y-Y Radii of Gyration Back to Back of Angles, in.

Angles in Contact

Angles Separated

0

3⁄ 8

3⁄ 4

Fy = 36 ksi

Fy = 50 ksi

Fy = 36 ksi

Fy = 50 ksi

L8×8×11⁄8 L8×8×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×11⁄2

3.42 3.40 3.38 3.36 3.34 3.32

3.55 3.53 3.51 3.49 3.47 3.45

3.69 3.67 3.64 3.62 3.60 3.58

— — — — — 0.995

— — — — — 0.921

— — — — 0.997 0.911

— — — — 0.935 0.834

L6×6×1 L8×8×17⁄8 L8×8×13⁄4 L8×8×15⁄8 L8×8×11⁄2 L8×8×13⁄8

2.59 2.57 2.55 2.53 2.51 2.49

2.73 2.70 2.68 2.66 2.64 2.62

2.87 2.85 2.82 2.80 2.78 2.75

— — — — — 0.995

— — — — — 0.921

— — — — — 0.911

— — — — 0.961 0.834

L5×5×7⁄8 L5×5×3⁄4 L5×5×1⁄2 L5×5×3⁄8 L5×5×5⁄16

2.16 2.14 2.10 2.09 2.08

2.30 2.28 2.24 2.22 2.21

2.45 2.42 2.38 2.35 2.34

— — — — 0.995

— — — — 0.921

— — — 0.982 0.911

— — — 0.919 0.834

L4×4×3⁄4 L5×5×5⁄8 L5×5×1⁄2 L5×5×3⁄8 L5×5×5⁄16 L5×5×1⁄4

1.74 1.72 1.70 1.68 1.67 1.66

1.88 1.86 1.83 1.81 1.80 1.79

2.03 2.00 1.98 1.95 1.94 1.93

— — — — — 0.995

— — — — — 0.921

— — — — 0.997 0.911

— — — — 0.935 0.834

Designation

*Where no value ofQs is shown, the angles comply with LRFD Specification Section E2.


1 - 94



Y X

X

y, yp

s Y

Wt. Area of per ft 2 Angles 2 Angles 2

Axis X-X

S

I 4

r 3

y

Z

yp 3

Designation

lb

in.

in.

in.

in.

in.

in.

L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×5⁄16 L31⁄2×31⁄2×1⁄4

17.0 14.4 11.6

4.97 4.18 3.38

5.73 4.90 4.02

2.30 1.95 1.59

1.07 1.08 1.09

1.01 0.990 0.968

4.15 3.52 2.86

0.355 0.299 0.241

L3×3×1⁄2 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4 L3×3×3⁄16

18.8 14.4 12.2 9.80 7.42

5.50 4.22 3.55 2.88 2.18

4.43 3.52 3.02 2.49 1.92

2.14 1.67 1.41 1.15 0.882

0.898 0.913 0.922 0.930 0.939

0.932 0.888 0.865 0.842 0.820

3.87 3.00 2.55 2.08 1.59

0.458 0.352 0.296 0.240 0.182

L21⁄2×21⁄2×3⁄8 L31⁄2×31⁄2×5⁄16 L31⁄2×31⁄2×1⁄4 L31⁄2×31⁄2×3⁄16

11.8 10.0 8.20 6.14

3.47 2.93 2.38 1.80

1.97 1.70 1.41 1.09

1.13 0.964 0.789 0.606

0.753 0.761 0.769 0.778

0.762 0.740 0.717 0.694

2.04 1.74 1.42 1.09

0.347 0.293 0.238 0.180

9.40 7.84 6.38 4.88 3.30

2.72 2.30 1.88 1.43 0.960

0.958 0.832 0.695 0.545 0.380

0.702 0.681 0.494 0.381 0.261

0.594 0.601 0.609 0.617 0.626

0.636 0.614 0.592 0.569 0.546

1.27 1.08 0.890 0.686 0.471

0.340 0.288 0.234 0.179 0.121

L2×2×3⁄8 L3×3×5⁄16 L3×3×1⁄4 L3×3×3⁄16 L3×3×1⁄8

in.


DOUBLE ANGLES

1 - 95


Y X

X

y, yp

s Y

Qs*

Axis Y-Y Radii of Gyration

Angles in Contact

Angles Separated

3⁄ 4

Fy = 36 ksi

Fy = 50 ksi

Fy = 36 ksi

Fy = 50 ksi

Back to Back of Angles, in. 3⁄ 8

Designation

0

L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×5⁄16 L31⁄2×31⁄2×1⁄4

1.48 1.47 1.46

1.61 1.60 1.59

1.75 1.74 1.73

— — —

— — 0.982

— — 0.965

— 0.986 0.897

L3×3×1⁄2 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4 L3×3×3⁄16

1.29 1.27 1.26 1.26 1.25

1.43 1.41 1.40 1.39 1.38

1.59 1.56 1.55 1.53 1.52

— — — — 0.995

— — — — 0.921

— — — — 0.911

— — — 0.961 0.834

L21⁄2×21⁄2×3⁄8 L21⁄2×21⁄2×5⁄16 L21⁄2×21⁄2×1⁄4 L21⁄2×21⁄2×3⁄16

1.07 1.06 1.05 1.04

1.21 1.20 1.19 1.18

1.36 1.35 1.34 1.32

— — — —

— — — —

— — — 0.982

— — — 0.919

L2×2×3⁄8 L2×2×5⁄16 L2×2×1⁄4 L2×2×3⁄16 L2×2×1⁄8

0.870 0.859 0.849 0.840 0.831

1.01 1.00 0.989 0.977 0.965

1.17 1.16 1.14 1.13 1.11

— — — — 0.995

— — — — 0.921

— — — — 0.911

— — — — 0.834

*Where no value ofQs is shown, the angles comply with LRFD Specification Section E2.


1 - 96


DOUBLE ANGLES Two unequal leg angles Properties of sections

Y X

X

y, yp

s

Long legs back to back

Y


2

Axis X-X

S

I 4

in.

r 3

y

Z

yp 3

lb

in.

in.

in.

in.

in.

in.

L8×6×1 L8×6×13⁄4 L8×6×11⁄2

88.4 67.6 46.0

26.0 19.9 13.5

161 126 88.6

30.2 23.3 16.0

2.49 2.53 2.56

2.65 2.56 2.47

54.5 42.2 29.1

1.50 1.38 1.25

L8×4×1 L8×6×13⁄4 L8×6×11⁄2

74.8 57.4 39.2

22.0 16.9 11.5

139 109 77.0

28.1 21.8 15.0

2.52 2.55 2.59

3.05 2.95 2.86

48.5 37.7 26.1

2.50 2.38 2.25

L7×4×3⁄4 L7×4×1⁄2 L7×4×3⁄8

52.4 35.8 27.2

15.4 10.5 7.97

75.6 53.3 41.1

16.8 11.6 8.88

2.22 2.25 2.27

2.51 2.42 2.37

29.6 20.6 15.7

1.88 1.75 1.69

L6×4×3⁄4 L7×4×5⁄8 L7×4×1⁄2 L7×4×3⁄8

47.2 40.0 32.4 24.6

13.9 11.7 9.50 7.22

49.0 42.1 34.8 26.9

12.5 10.6 8.67 6.64

1.88 1.90 1.91 1.93

2.08 2.03 1.99 1.94

22.3 19.0 15.6 11.9

1.38 1.31 1.25 1.19

L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

23.4 19.6

6.84 5.74

25.7 21.8

6.49 5.47

1.94 1.95

2.04 2.01

11.5 9.70

1.44 1.41

L5×31⁄2×3⁄4 L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

39.6 27.2 20.8 17.4

11.6 8.00 6.09 5.12

27.8 20.0 15.6 13.2

8.55 5.97 4.59 3.87

1.55 1.58 1.60 1.61

1.75 1.66 1.61 1.59

15.3 10.8 8.28 6.99

1.13 1.00 0.938 0.906

L5×3×1⁄2 L7×4×3⁄8 L7×4×5⁄16 L7×4×1⁄4

25.6 19.6 16.4 13.2

7.50 5.72 4.80 3.88

18.9 14.7 12.5 10.2

5.82 4.47 3.77 3.06

1.59 1.61 1.61 1.62

1.75 1.70 1.68 1.66

10.3 7.95 6.71 5.45

1.25 1.19 1.16 1.13


DOUBLE ANGLES

1 - 97


Y X

X

y, yp

s

Long legs back to back Y

Qs*


Angles in Contact

Angles Separated

0

3⁄ 8

3⁄ 4

Fy = 36 ksi

Fy = 50 ksi

Fy = 36 ksi

Fy = 50 ksi

L8×6×1 L8×6×13⁄4 L8×6×11⁄2

2.39 2.35 2.32

2.52 2.48 2.44

2.66 2.62 2.57

— — —

— — —

— — 0.911

— — 0.834

L8×4×1 L8×4×13⁄4 L8×4×11⁄2

1.47 1.42 1.38

1.61 1.55 1.51

1.75 1.69 1.64

— — —

— — —

— — 0.911

— — 0.834

L7×4×3⁄4 L7×4×1⁄2 L7×4×3⁄8

1.48 1.44 1.43

1.62 1.57 1.55

1.76 1.71 1.68

— — —

— — —

— 0.965 0.839

— 0.897 0.750

L6×4×3⁄4 L7×4×5⁄8 L7×4×1⁄2 L7×4×3⁄8

1.55 1.53 1.51 1.5

1.69 1.67 1.64 1.62

1.83 1.81 1.78 1.76

— — — —

— — — —

— — — 0.911

— — 0.961 0.834

L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

1.26 1.26

1.39 1.38

1.53 1.51

— —

— —

0.911 0.825

0.834 0.733

L5×31⁄2×3⁄4 L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

1.40 1.35 1.34 1.33

1.53 1.49 1.46 1.45

1.68 1.63 1.60 1.59

— — — —

— — — —

— — 0.982 0.911

— — 0.919 0.834

L5×3×1⁄2 L7×4×3⁄8 L7×4×5⁄16 L7×4×1⁄4

1.12 1.10 1.09 1.08

1.25 1.23 1.22 1.21

1.40 1.37 1.36 1.34

— — — —

— — — —

— 0.982 0.911 0.804

— 0.919 0.834 0.708

Designation

*Where no value of Qs is shown the angles comply with LRFD Specification Section E2.


1 - 98



Y X

X

y, yp

s

Long legs back to back

Y


2

Axis X-X

S

I 4

in.

r 3

y

Z

yp 3

lb

in.

in.

in.

in.

in.

in.

L4×31⁄2×1⁄2 L4×31⁄2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

23.8 18.2 15.4 12.4

7.00 5.34 4.49 3.63

10.6 8.35 7.12 5.83

3.87 2.99 2.53 2.05

1.23 1.25 1.26 1.27

1.25 1.21 1.18 1.16

7.00 5.42 4.59 3.73

0.500 0.438 0.406 0.375

L4×3×1⁄2 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4

22.2 17.0 14.4 11.6

6.50 4.97 4.18 3.38

10.1 7.93 6.76 5.54

3.78 2.92 2.47 2.00

1.25 1.26 1.27 1.28

1.33 1.28 1.26 1.24

6.81 5.28 4.47 3.63

0.750 0.688 0.656 0.625

L31⁄2×3×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

15.8 13.2 10.8

4.59 3.87 3.13

5.45 4.66 3.83

2.25 1.91 1.55

1.09 1.10 1.11

1.08 1.06 1.04

4.08 3.46 2.82

0.438 0.406 0.375

L31⁄2×21⁄2×3⁄8 L31⁄2×21⁄2×1⁄4

14.4 9.80

4.22 2.88

5.12 3.60

2.19 1.51

1.10 1.12

1.16 1.11

3.94 2.73

0.688 0.625

L3×21⁄2×3⁄8 L4×31⁄2×1⁄4 L4×31⁄2×5⁄16

13.2 9.00 6.77

3.84 2.63 1.99

3.31 2.35 1.81

1.62 1.12 0.859

0.928 0.945 0.954

0.956 0.911 0.888

2.93 2.04 1.56

0.438 0.375 0.344

L3×2×3⁄8 L4×3×5⁄16 L4×3×1⁄4 L4×3×3⁄16

11.8 10.0 8.20 6.14

3.47 2.93 2.38 1.80

3.06 2.63 2.17 1.68

1.56 1.33 1.08 0.830

0.940 0.948 0.957 0.966

1.04 1.02 0.993 0.970

2.79 2.38 1.95 1.49

0.688 0.656 0.625 0.594

L21⁄2×2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4 L4×31⁄2×3⁄16

10.6 9.00 7.24 5.50

3.09 2.62 2.13 1.62

1.82 1.58 1.31 1.02

1.09 0.932 0.763 0.586

0.768 0.776 0.784 0.793

0.831 0.809 0.787 0.764

1.97 1.69 1.38 1.06

0.438 0.406 0.375 0.344


DOUBLE ANGLES

1 - 99


Y X

X

y, yp

s

Long legs back to back Y

Qs*

Axis Y-Y Radii of Gyration

Angles in Contact

Angles Separated

3⁄ 4

Fy = 36 ksi

Fy = 50 ksi

Fy = 36 ksi

Fy = 50 ksi

Back to Back of Angles, in. Designation

0

3⁄ 8

L4×31⁄2×1⁄2 L4×31⁄2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

1.44 1.42 1.42 1.41

1.58 1.56 1.55 1.54

1.72 1.70 1.69 1.67

— — — —

— — — 0.982

— — 0.997 0.911

— — 0.935 0.834

L4×3×1⁄2 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4

1.20 1.18 1.17 1.16

1.33 1.31 1.30 1.29

1.48 1.45 1.44 1.43

— — — —

— — — —

— — 0.997 0.911

— — 0.935 0.834

L31⁄2×3×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

1.22 1.21 1.20

1.36 1.35 1.33

1.50 1.49 1.48

— — —

— — —

— — 0.965

— 0.986 0.897

L31⁄2×21⁄2×3⁄8 L31⁄2×21⁄2×1⁄4

0.976 0.958

1.11 1.09

1.26 1.23

— —

— —

— 0.965

— 0.897

L3×21⁄2×3⁄8 L4×31⁄2×1⁄4 L4×31⁄2×5⁄16

1.02 1.00 0.993

1.16 1.13 1.12

1.31 1.28 1.27

— — —

— — —

— — 0.911

— 0.961 0.834

L3×2×3⁄8 L4×3×5⁄16 L4×3×1⁄4 L4×3×3⁄16

0.777 0.767 0.757 0.749

0.917 0.903 0.891 0.879

1.07 1.06 1.04 1.03

— — — —

— — — —

— — — 0.911

— — 0.961 0.834

L21⁄2×2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4 L4×31⁄2×3⁄16

0.819 0.809 0.799 0.790

0.961 0.948 0.935 0.923

1.12 1.10 1.09 1.07

— — — —

— — — —

— — — 0.982

— — — 0.919

*Where no value of Qs is shown, the angles comply with LRFD Specification Section E2.


1 - 100



Y X

X

y, yp

s

Short legs back to back

Y


2

Axis X-X

S

I 4

in.

r 3

y

Z

yp 3

lb

in.

in.

in.

in.

in.

in.

L8×6×1 L8×6×13⁄4 L8×6×11⁄2

88.4 67.6 46.0

26.0 19.9 13.5

77.6 61.4 43.4

17.8 13.8 9.58

1.73 1.76 1.79

1.65 1.56 1.47

32.4 24.9 17.0

0.813 0.621 0.422

L8×4×1 L8×6×13⁄4 L8×6×11⁄2

74.8 57.4 39.2

22.0 16.9 11.5

23.3 18.7 13.5

7.88 6.14 4.29

1.03 1.05 1.08

1.05 0.953 0.859

15.4 11.6 7.80

0.688 0.527 0.359

L7×4×3⁄4 L5×3×1⁄2 L5×3×3⁄8

52.4 35.8 27.2

15.4 10.5 7.97

18.1 13.1 10.2

6.05 4.23 3.26

1.09 1.11 1.13

1.01 0.917 0.870

11.3 7.66 5.80

0.549 0.375 0.285

L6×4×3⁄4 L5×3×5⁄8 L5×3×1⁄2 L5×3×3⁄8

47.2 40.0 32.4 24.6

13.9 11.7 9.50 7.22

17.4 15.0 12.5 9.81

5.94 5.07 4.16 3.21

1.12 1.13 1.15 1.17

1.08 1.03 0.987 0.941

10.9 9.24 7.50 5.71

0.578 0.488 0.396 0.301

L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

23.4 19.6

6.84 5.74

6.68 5.70

2.46 2.08

0.988 0.996

0.787 0.763

4.41 3.70

0.285 0.239

L5×31⁄2×3⁄4 L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

39.6 27.2 20.8 17.4

11.6 8.00 6.09 5.12

11.1 8.10 6.37 5.44

4.43 3.12 2.41 2.04

0.977 1.01 1.02 1.03

0.996 0.906 0.861 0.838

8.20 5.65 4.32 3.63

0.581 0.400 0.305 0.256

L5×3×1⁄2 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

25.6 19.6 16.4 13.2

7.50 5.72 4.80 3.88

5.16 4.08 3.49 2.88

2.29 1.78 1.51 1.23

0.829 0.845 0.853 0.861

0.750 0.704 0.681 0.657

4.22 3.21 2.69 2.17

0.375 0.286 0.240 0.194


DOUBLE ANGLES

1 - 101


Y X

X

y, yp

s


Y

Qs*


Angles in Contact

Angles Separated

0

3⁄ 8

3⁄ 4

Fy = 36 ksi

Fy = 50 ksi

Fy = 36 ksi

Fy = 50 ksi

L8×6×1 L8×6×13⁄4 L8×6×11⁄2

3.64 3.60 3.56

3.78 3.74 3.69

3.92 3.88 3.83

— — 0.995

— — 0.921

— — 0.911

— — 0.834

L8×4×1 L8×4×13⁄4 L8×4×11⁄2

3.95 3.90 3.86

4.10 4.05 4.00

4.25 4.19 4.14

— — 0.995

— — 0.921

— — 0.911

— — 0.834

L7×4×3⁄4 L7×4×1⁄2 L7×4×3⁄8

3.35 3.30 3.28

3.49 3.44 3.42

3.64 3.59 3.56

— — 0.926

— 0.982 0.838

— 0.965 0.839

0.897 0.750

L6×4×3⁄4 L7×4×5⁄8 L7×4×1⁄2 L7×4×3⁄8

2.80 2.78 2.76 2.74

2.94 2.92 2.90 2.87

3.09 3.06 3.04 3.02

— — — 0.995

— — — 0.921

— — — 0.911

— — 0.961 0.834

L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

2.81 2.80

2.95 2.94

3.09 3.08

0.995 0.912

0.921 0.822

0.911 0.825

0.834 0.733

L5×31⁄2×3⁄4 L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

2.33 2.29 2.27 2.26

2.48 2.43 2.41 2.39

2.63 2.57 2.55 2.54

— — — 0.995

— — — 0.921

— — 0.982 0.911

— — 0.919 0.834

L5×3×1⁄2 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

2.36 2.34 2.33 2.32

2.50 2.48 2.47 2.46

2.65 2.63 2.61 2.60

— — 0.995 0.891

— — 0.921 0.797

— 0.982 0.911 0.804

— 0.919 0.834 0.708

Designation

*Where no value of Qs is shown, the angles comply with LRFD Specification Section E2.


1 - 102



Y X

X

y, yp

s


Y


2

Axis X-X

S

I 4

in.

r 3

y

Z

yp 3

lb

in.

in.

in.

in.

in.

in.

L4×31⁄2×1⁄2 L4×31⁄2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

23.8 18.2 15.4 12.4

7.00 5.34 4.49 3.63

7.58 5.97 5.10 4.19

3.03 2.35 1.99 1.62

1.04 1.06 1.07 1.07

1.00 0.955 0.932 0.909

5.47 4.21 3.56 2.89

0.438 0.334 0.281 0.227

L4×3×1⁄2 L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4

22.2 17.0 14.4 11.6

6.50 4.97 4.18 3.38

4.85 3.84 3.29 2.71

2.23 1.73 1.47 1.20

0.864 0.879 0.887 0.896

0.827 0.782 0.759 0.736

4.06 3.11 2.63 2.13

0.406 0.311 0.261 0.211

L31⁄2×3×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

15.8 13.2 10.8

4.59 3.87 3.13

3.69 3.17 2.61

1.70 1.44 1.18

0.897 0.905 0.914

0.830 0.808 0.785

3.06 2.59 2.10

0.328 0.276 0.223

L31⁄2×21⁄2×3⁄8 L31⁄2×21⁄2×1⁄4

14.4 9.80

4.22 2.88

2.18 1.55

1.18 0.824

0.719 0.735

0.660 0.614

2.15 1.47

0.301 0.205

L3×21⁄2×3⁄8 L4×31⁄2×1⁄4 L4×31⁄2×3⁄16

13.2 9.00 6.77

3.84 2.63 1.99

2.08 1.49 1.15

1.16 0.808 0.620

0.736 0.753 0.761

0.706 0.661 0.638

2.10 1.45 1.11

0.320 0.219 0.166

L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16

11.8 10.0 8.20 6.14

3.47 2.93 2.38 1.80

1.09 0.941 0.784 0.613

0.743 0.634 0.520 0.401

0.559 0.567 0.574 0.583

0.539 0.516 0.493 0.470

1.37 1.16 0.937 0.713

0.289 0.244 0.198 0.150

L21⁄2×2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4 L4×31⁄2×3⁄16

10.6 9.00 7.24 5.50

3.09 2.62 2.13 1.62

1.03 0.893 0.745 0.583

0.725 0.620 0.509 0.392

0.577 0.584 0.592 0.600

0.581 0.559 0.537 0.514

1.32 1.12 0.915 0.701

0.309 0.262 0.213 0.162


DOUBLE ANGLES

1 - 103


Y X

X

y, yp

s


Y

Qs*


Angles in Contact

Angles Separated

0

3⁄ 8

3⁄ 4

Fy = 36 ksi

Fy = 50 ksi

Fy = 36 ksi

Fy = 50 ksi

L4×31⁄2×1⁄2 L4×31⁄2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

1.76 1.74 1.73 1.72

1.89 1.87 1.86 1.85

2.04 2.01 2.00 1.99

— — — 0.995

— — — 0.921

— — 0.997 0.911

— — 0.935 0.834

L4×3×1⁄2 L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4

1.82 1.80 1.79 1.78

1.96 1.94 1.93 1.92

2.11 2.08 2.07 2.06

— — — 0.995

— — — 0.921

— — 0.997 0.911

— — 0.935 0.834

L31⁄2×3×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4

1.53 1.52 1.52

1.67 1.66 1.65

1.82 1.80 1.79

— — —

— — 0.982

— — 0.965

— 0.986 0.897

L31⁄2×21⁄2×3⁄8 L31⁄2×21⁄2×1⁄4

1.60 1.58

1.74 1.72

1.89 1.86

— —

— 0.982

— 0.965

— 0.897

L3×21⁄2×3⁄8 L4×31⁄2×1⁄4 L4×31⁄2×3⁄16

1.33 1.31 1.30

1.47 1.45 1.44

1.62 1.60 1.58

— — 0.995

— — 0.921

— — 0.911

— 0.961 0.834

L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16

1.40 1.39 1.38 1.37

1.55 1.53 1.52 1.51

1.70 1.68 1.67 1.66

— — — 0.995

— — — 0.921

— — — 0.911

— — 0.961 0.834

L21⁄2×2×3⁄8 L4×31⁄2×5⁄16 L4×31⁄2×1⁄4 L4×31⁄2×3⁄16

1.13 1.12 1.11 1.10

1.28 1.26 1.25 1.24

1.43 1.42 1.40 1.39

— — — —

— — — —

— — — 0.982

— — — 0.911

Designation

*Where no value of Qs is shown the angles comply with LRFD Specification Section E2.


1 - 104


COMBINATION SECTIONS



1 - 105


Standard rolled shapes are frequently combined to produce efficient and economical structural members for special applications. Experience has established a demand for certain combinations. When properly sized and connected to satisfy the design and specification criteria, these members may be used as struts, lintels, eave struts, and light crane and trolley runways. The W section with channel attached to the web is not recommended for use as a trolley or crane runway member. Properties of several combined sections are tabulated for those combinations that experience has proven to be in popular demand.


1 - 106


COMBINATION SECTIONS W shapes and channels Properties of sections

Y

y

2

X

X

y , yp 1

Y

Beam

Channel

Axis X-X

Total Weight per ft

Total Area

lb

in.2

I

S1 = I / y1

S2 = I / y2

r

in.4

in.3

in.3

in.

W12×26 W ×26

C10×15.3 C12×20.7

41.3 46.7

12.14 13.74

299 318

36.3 36.8

70.5 82.2

4.96 4.81

W14×30 W ×30

C10×15.3 C12×20.7

45.3 50.7

13.34 14.94

420 448

46.1 46.8

84.6 98.3

5.61 5.47

W16×36 W ×36

C12×20.7 C15×33.9

56.7 69.9

16.69 20.56

670 748

62.8 64.6

123 160

6.34 6.03

W18×50 W ×50

C12×20.7 C15×33.9

70.7 83.9

20.79 24.66

1120 1250

97.4 100

166 211

7.34 7.11

W21×62 W ×62 W ×68 W ×68

C12×20.7 C15×33.9 C12×20.7 C15×33.9

82.7 95.9 88.7 101.9

24.39 28.26 26.09 29.96

1800 2000 1960 2180

138 142 152 156

218 272 232 287

8.59 8.41 8.68 8.52

W24×68 W ×68 W ×84 W ×84

C12×20.7 C15×33.9 C12×20.7 C15×33.9

88.7 101.9 104.7 117.9

26.19 30.06 30.79 34.66

2450 2720 3040 3340

168 173 212 217

258 321 303 368

9.67 9.50 9.93 9.82

W27×84 W ×94

C15×33.9 C15×33.9

117.9 127.9

34.76 37.66

4050 4530

237 268

404 436

10.8 11.0

W30×99 W ×99 W ×116 W ×116

C15×33.9 C18×42.7 C15×33.9 C18×42.7

132.9 141.7 149.9 158.7

39.06 41.70 44.16 46.80

5540 5830 6590 6900

300 304 360 365

480 533 544 599

11.9 11.8 12.2 12.1

W33×118 W ×118 W ×141 W ×141

C15×33.9 C18×42.7 C15×33.9 C18×42.7

151.9 160.7 174.9 183.7

44.66 47.30 51.56 54.20

7900 8280 9580 10000

395 400 484 490

596 656 689 751

13.3 13.2 13.6 13.6

W36×150 W ×150

C15×33.9 C18×42.7

183.9 192.7

54.16 56.80

11500 12100

546 553

765 832

14.6 14.6



1 - 107

COMBINATION SECTIONS W shapes and channels Properties of sections

Y

y

2

X

X

y , yp 1

Y

Axis X-X

y1 Beam

Z

Axis Y-Y

yp 3

Channel

in.

in.

W12×26 W30×26

C10×15.3 C12×20.7

8.22 8.63

W14×30 W30×30

C10×15.3 C12×20.7

W16×36 W30×36

S

I 4

3

r

Z

in.

in.3

in.

in.

in.

47.0 48.8

11.30 11.55

84.7 146

16.9 24.4

2.64 3.26

24.0 33.6

9.12 9.57

60.5 62.3

12.56 12.87

87.0 149

17.4 24.8

2.55 3.15

24.8 34.4

C12×20.7 C15×33.9

10.67 11.58

83.6 88.6

14.56 15.21

154 340

25.6 45.3

3.03 4.06

36.3 61.3

W18×50 W30×50

C12×20.7 C15×33.9

11.51 12.47

128 134

16.08 16.90

169 355

28.2 47.3

2.85 3.79

42.0 67.0

W21×62 W30×62 W30×68 W30×68

C12×20.7 C15×33.9 C12×20.7 C15×33.9

13.01 14.06 12.93 13.95

182 190 200 208

18.06 19.36 17.60 19.32

187 373 194 380

31.1 49.7 32.3 50.6

2.77 3.63 2.72 3.56

47.2 72.2 49.8 74.8

W24×68 W30×68 W30×84 W30×84

C12×20.7 C15×33.9 C12×20.7 C15×33.9

14.53 15.67 14.35 15.40

224 234 275 288

19.15 21.66 18.49 21.61

199 385 223 409

33.2 51.4 37.2 54.6

2.76 3.58 2.69 3.44

50.0 75.0 58.1 83.1

W27×84 W27×94

C15×33.9 C15×33.9

17.07 16.92

320 357

23.86 23.56

421 439

56.1 58.5

3.48 3.41

83.6 89.2

W30×99 W30×99 W30×116 W30×116

C15×33.9 C18×42.7 C15×33.9 C18×42.7

18.51 19.18 18.30 18.93

408 418 480 492

24.34 26.43 23.77 26.04

443 682 479 718

59.1 75.8 63.9 79.8

3.37 4.04 3.29 3.92

89.1 113 99.6 124

W33×118 W30×118 W30×141 W30×141

C15×33.9 C18×42.7 C15×33.9 C18×42.7

20.01 20.69 19.79 20.42

529 544 634 652

25.43 27.77 24.83 26.96

502 741 561 800

66.9 82.3 74.8 88.9

3.35 3.96 3.30 3.84

102 126 117 141

W36×150 W30×150

C15×33.9 C18×42.7

21.15 21.81

716 738

25.84 27.91

585 824

78.0 91.6

3.29 3.81

121 145


1 - 108


COMBINATION SECTIONS S shapes and channels Properties of sections

Y

y

2

X

X

y , yp 1

Y

Beam

Channel

Axis X-X

Total Weight per ft

Total Area

lb

in.2

I

S1 = I / y1

S2 = I / y2

r

in.4

in.3

in.3

in.

S10×25.4

C8×11.5 C10×15.3

36.9 40.7

10.84 11.95

176 186

27.2 27.6

46.6 52.9

4.02 3.94

S12×31.8

C8×11.5 C10×15.3

43.3 47.1

12.73 13.84

299 316

39.8 40.4

63.2 71.4

4.84 4.78

S15×42.9

C8×11.5 C10×15.3

54.4 58.2

15.98 17.09

585 616

64.9 65.8

94.2 105

6.05 6.01

S20×66

C10×15.3 C12×20.7

81.3 86.7

23.89 25.49

1530 1620

130 132

181 203

8.00 7.97

S24×80

C10×15.3 C12×20.7

95.3 100.7

27.99 29.59

2610 2750

188 191

252 278

9.66 9.65



1 - 109

COMBINATION SECTIONS S shapes and channels Properties of sections

Y

y

2

X

X

y , yp 1

Y

Axis X-X

y1 Beam

Z

Axis Y-Y

yp 3

S

I 4

3

r

Z

in.

in.3

Channel

in.

in.

in.

in.

in.

S10×25.4

C8×11.5 C10×15.3

6.45 6.73

35.7 36.9

8.81 9.02

39.4 74.2

9.8 14.8

1.91 2.49

14.5 20.8

S12×31.8

C8×11.5 C10×15.3

7.50 7.82

52.6 53.9

10.30 10.61

42.0 76.8

10.5 15.4

1.82 2.36

16.0 22.2

S15×42.9

C8×11.5 C10×15.3

9.01 9.37

85.7 88.2

11.58 12.77

47.0 81.8

11.8 16.4

1.71 2.19

18.6 24.9

S20×66

C10×15.3 C12×20.7

11.81 12.29

171 178

14.41 15.99

95.1 157

19.0 26.1

2.00 2.48

31.2 40.8

S24×80

C10×15.3 C12×20.7

13.86 14.38

244 254

16.46 18.05

110 171

21.9 28.5

1.98 2.41

36.6 46.2


1 - 110


d

2

Yd

COMBINATION SECTIONS Two channels Properties of sections

2

y

2

X

X x1 , xp

y , yp 1

Y

Vertical Horizontal Channel Channel

Axis X-X

Total Weight per ft

Total Area

lb

in.2

I

S1 = I / y1

S2 = I / y2

r

y1

Z

yp

in.4

in.3

in.3

in.

in.

in.3

in.

C3×4.1

C4×5.4

9.5

2.80

3.0

1.4

3.0

1.04

2.20

2.16

2.67

C4×5.4

C4×5.4 C5×6.7

10.8 12.1

3.18 3.56

6.5 6.9

2.3 2.3

4.9 5.5

1.43 1.39

2.86 2.94

3.39 3.62

3.56 3.61

C5×6.7

C5×6.7 C6×8.2 C7×9.8

13.4 14.9 16.5

3.94 4.37 4.84

12.8 13.4 14.0

3.5 3.6 3.7

8.0 8.9 9.8

1.80 1.75 1.70

3.60 3.70 3.79

5.23 5.50 5.81

4.50 4.57 4.62

C6×8.2

C5×6.7 C6×8.2 C7×9.8 C8×11.5 C9×13.4 C10×15.3

14.9 16.4 18.0 19.7 21.6 23.5

4.37 4.80 5.27 5.78 6.34 6.89

21.5 22.5 23.4 24.3 25.2 26.0

5.1 5.2 5.2 5.3 5.4 5.5

10.9 12.1 13.3 14.5 15.8 16.9

2.22 2.16 2.11 2.05 1.99 1.94

4.22 4.34 4.45 4.55 4.64 4.70

7.31 7.61 7.93 8.30 8.72 9.16

5.37 5.45 5.53 5.58 5.63 5.65

C7×9.8

C6×8.2 C7×9.8 C8×11.5 C9×13.4 C10×15.3

18.0 19.6 21.3 23.2 25.1

5.27 5.74 6.25 6.81 7.36

35.3 36.7 38.0 39.3 40.5

7.1 7.2 7.3 7.4 7.5

15.7 17.3 18.8 20.5 21.9

2.59 2.53 2.47 2.40 2.34

4.95 5.08 5.20 5.31 5.39

10.2 10.6 10.9 11.4 11.8

6.32 6.40 6.48 6.54 6.58

C8×11.5

C6×8.2 C7×9.8 C8×11.5 C9×13.4 C10×15.3 C12×20.7

19.7 21.3 23.0 24.9 26.8 32.2

5.78 6.25 6.76 7.32 7.87 9.47

52.4 54.5 56.4 58.4 60.0 64.4

9.5 9.6 9.7 9.8 9.9 10.2

19.6 21.6 23.6 25.6 27.5 32.6

3.01 2.95 2.89 2.82 2.76 2.61

5.53 5.68 5.82 5.95 6.06 6.30

13.4 13.8 14.2 14.6 15.1 16.4

7.18 7.27 7.35 7.44 7.49 7.62



1 - 111


d

2

Yd

2

y

2

X

X x1 , xp

y , yp 1

Y

Axis Y-Y Vertical Channel

Horizontal Channel

I

S

r

x1

Z

xp

in.4

in.3

in.

in.

in.3

in.

C3×4.1

C4×5.4

4.0

2.0

1.20

0.44

2.67

0.315

C4×5.4

C4×5.4 C5×6.7

4.2 7.8

2.1 3.1

1.14 1.48

0.46 0.46

2.84 4.09

0.281 0.282

C5×6.7

C5×6.7 C6×8.2 C7×9.8

8.0 13.6 21.8

3.2 4.5 6.2

1.42 1.76 2.12

0.48 0.48 0.48

4.29 5.90 7.90

0.264 0.266 0.268

C6×8.2

C5×6.7 C6×8.2 C7×9.8 C8×11.5 C9×13.4 C10×15.3

8.2 13.8 22.0 33.3 48.6 68.1

3.3 4.6 6.3 8.3 10.8 13.6

1.37 1.70 2.04 2.40 2.77 3.14

0.51 0.51 0.51 0.51 0.51 0.51

4.52 6.14 8.13 10.6 13.5 16.8

0.242 0.245 0.247 0.249 0.252 0.254

C7×9.8

C6×8.2 C7×9.8 C8×11.5 C9×13.4 C10×15.3

14.1 22.3 33.6 48.6 68.4

4.7 6.4 8.4 10.9 13.7

1.63 1.97 2.32 2.68 3.05

0.54 0.54 0.54 0.54 0.54

6.41 8.41 10.8 13.8 17.1

0.225 0.228 0.230 0.234 0.235

C8×11.5

C6×8.2 C7×9.8 C8×11.5 C9×13.4 C10×15.3 C12×20.7

14.4 22.6 33.9 49.2 68.7 130

4.8 6.5 8.5 10.9 13.7 21.7

1.58 1.90 2.24 2.59 2.95 3.71

0.57 0.57 0.57 0.57 0.57 0.57

6.73 8.73 11.2 14.1 17.4 27.0

0.218 0.219 0.219 0.220 0.220 0.230


1 - 112


d

2

Yd


2

y

2

X

X x1 , xp

y , yp 1

Y

Vertical Horizontal Channel Channel

Total Weight per ft

Axis X-X Total Area 2

lb

in.

C9×13.4

C7×9.8 C8×11.5 C9×13.4 C10×15.3 C12×20.7

23.2 24.9 26.8 28.7 34.1

6.81 7.32 7.88 8.43 10.03

C10×15.3

C8×11.5 C9×13.4 C10×15.3 C12×20.7 C15×33.9

26.8 28.7 30.6 36.0 49.2

7.87 8.43 8.98 10.58 14.45

C12×20.7

C9×13.4 C10×15.3 C12×20.7 C15×33.9

34.1 36.0 41.4 54.6

C15×33.9

C10×15.3 C12×20.7 C15×33.9 MC18×42.7

MC18×42.7 MC12×20.7 MC15×33.9 MC18×42.7

S1 = I / y1

I 4

in.

3

S2 = I / y2 3

r

y1

Z

yp 3

in.

in.

in.

in.

in.

in.

12.4 12.6 12.7 12.8 13.1

26.3 28.7 31.2 33.5 39.8

3.38 3.32 3.25 3.19 3.02

6.26 6.42 6.57 6.69 6.98

17.6 18.1 18.5 19.0 20.4

8.11 8.21 8.31 8.37 8.54

110 114 117 126 141

15.8 15.9 16.1 16.4 17.3

34.2 37.2 39.9 47.5 63.7

3.75 3.68 3.61 3.45 3.13

7.00 7.16 7.30 7.64 8.18

22.4 22.9 23.4 24.9 28.3

9.07 9.18 9.26 9.46 9.73

10.03 10.58 12.18 16.05

207 213 228 256

25.2 25.4 25.9 27.0

51.4 55.0 65.3 87.8

4.54 4.48 4.32 4.00

8.21 8.38 8.79 9.48

35.7 36.3 38.0 41.8

10.78 10.88 11.16 11.56

49.2 54.6 67.8 76.6

14.45 16.05 19.92 22.56

474 509 575 608

48.8 49.9 52.0 53.1

85.6 99.8 132 152

5.72 5.63 5.37 5.19

9.71 10.19 11.06 11.45

69.7 72.2 77.4 80.7

12.83 13.31 14.04 14.37

63.4 76.6 85.4

18.69 22.56 25.20

860 975 1030

72.9 76.1 77.6

133 174 200

6.78 6.57 6.40

11.80 12.80 13.29

77.7 80.5 83.3 85.6 91.7


106 113 117

15.51 16.50 16.96


1 - 113


d

2

Yd

2

y

2

X

X x1 , xp

y , yp 1

Y

Axis Y-Y Vertical Channel

Horizontal Channel

I

S

r

x1

Z

xp

in.4

in.3

in.

in.

in.3

in.

C9×13.4

C7×9.8 C8×11.5 C9×13.4 C10×15.3 C12×20.7

23.1 34.4 49.7 69.2 131

6.6 8.6 11.0 13.8 21.8

1.84 2.17 2.51 2.86 3.61

0.60 0.60 0.60 0.60 0.60

9.10 11.5 14.5 17.8 27.4

0.226 0.227 0.227 0.227 0.229

C10×15.3

C8×11.5 C9×13.4 C10×15.3 C12×20.7 C15×33.9

34.9 50.2 69.7 131 317

8.7 11.2 13.9 21.9 42.3

2.11 2.44 2.79 3.52 4.69

0.63 0.63 0.63 0.63 0.63

11.9 14.9 18.2 27.8 52.8

0.232 0.232 0.233 0.234 0.239

C12×20.7

C9×13.4 C10×15.3 C12×20.7 C15×33.9

51.8 71.3 133 319

11.5 14.3 22.1 42.5

2.27 2.60 3.30 4.46

0.70 0.70 0.70 0.70

16.0 19.3 29.0 54.0

0.261 0.261 0.262 0.266

C15×33.9

C10×15.3 C12×20.7 C15×33.9 MC18×42.7

75.5 137 323 562

15.1 22.9 43.1 62.5

2.29 2.92 4.03 4.99

0.79 0.79 0.79 0.79

22.1 31.7 56.7 80.7

0.337 0.338 0.342 0.343

MC18×42.7

MC12×20.7 MC15×33.9 MC18×42.7

143 329 568

23.9 43.9 63.2

2.77 3.82 4.75

0.88 0.88 0.88

33.6 58.6 82.6

0.355 0.358 0.359


1 - 114


COMBINATION SECTIONS Channels and angles Properties of sections

Y

y

2

X

X

Long leg of angle turned out

y , yp 1

x1 xp

x2 Y

Total Weight per ft


S1 = I / y1

I 4

3

3

in.

r

y1

Z

yp 3

Angle

lb

in.

in.

in.

in.

in.

in.

C6×8.2

L21⁄2×21⁄2×1⁄4 L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L3×21⁄2 ×5⁄16 L4 ×3 ×1⁄4

12.3 12.7 13.6 14.8 14.0

3.59 3.71 3.96 4.33 4.09

17.9 18.5 19.0 19.8 19.5

8.0 8.5 8.9 9.8 9.5

4.8 4.8 4.9 5.0 5.0

2.24 2.23 2.19 2.14 2.19

2.24 2.17 2.13 2.02 2.06

6.75 6.90 7.23 7.54 7.36

1.40 1.26 1.26 1.11 1.13

C7×9.8

L21⁄2×21⁄2×1⁄4 L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L3×21⁄2 ×5⁄16 L4 ×3 ×1⁄4 L3×2 ×5⁄16

13.9 14.3 15.2 16.4 15.6 17.0

4.06 4.18 4.43 4.80 4.56 4.96

28.5 29.3 30.0 31.2 30.8 32.0

10.6 11.2 11.8 12.9 12.4 13.7

6.6 6.7 6.7 6.8 6.8 6.9

2.65 2.65 2.60 2.55 2.60 2.54

2.68 2.61 2.54 2.42 2.48 2.35

9.13 9.31 9.64 9.99 9.81 10.2

1.67 1.53 1.53 1.35 1.39 1.20

C8×11.5

L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L3×21⁄2 ×5⁄16 L4 ×3 ×1⁄4 L3×2 ×5⁄16 L5 ×31⁄2×5⁄16

16.0 16.9 18.1 17.3 18.7 20.2

4.69 4.94 5.31 5.07 5.47 5.94

43.9 44.9 46.7 46.0 47.8 49.9

14.3 15.1 16.4 15.8 17.3 18.9

8.9 9.0 9.0 9.0 9.1 9.3

3.06 3.02 2.97 3.01 2.96 2.90

3.07 2.98 2.84 2.91 2.76 2.64

12.2 12.6 13.0 12.8 13.2 13.9

1.81 1.81 1.60 1.67 1.45 1.30

C9×13.4

L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L3×21⁄2 ×5⁄16 L4 ×3 ×1⁄4 L3×2 ×5⁄16 L5 ×31⁄2×5⁄16

17.9 18.8 20.0 19.2 20.6 22.1

5.25 5.50 5.87 5.63 6.03 6.50

63.1 64.6 67.1 66.0 68.7 71.4

17.8 18.8 20.4 19.6 21.4 23.4

11.6 11.6 11.7 11.7 11.8 12.0

3.47 3.43 3.38 3.42 3.37 3.31

3.54 3.45 3.29 3.37 3.20 3.06

15.8 16.1 16.6 16.3 16.8 17.5

2.11 2.11 1.87 1.98 1.73 1.58

C10×15.3

L31⁄2×3 ×1⁄4 L3×21⁄2 ×5⁄16 L4 ×3 ×1⁄4 L3×2 ×5⁄16 L5 ×31⁄2×5⁄16 L3×21⁄2 ×3⁄8

20.7 21.9 21.1 22.5 24.0 25.7

6.05 6.42 6.18 6.58 7.05 7.54

89.3 92.7 91.1 94.7 98.4 102

22.8 24.8 23.8 25.9 28.2 30.6

14.7 14.8 14.8 14.9 15.1 15.2

3.84 3.80 3.84 3.79 3.74 3.67

3.91 3.74 3.83 3.65 3.49 3.33

20.0 20.6 20.3 20.9 21.6 22.2

2.39 2.12 2.26 1.98 1.84 1.61

Channel

in.

S2 = I / y2



1 - 115


Y

y

2

X

X


y , yp 1

x1 xp

x2 Y

Axis Y-Y

S1 = I / x1

I

3

3

r

x1

Z

xp 3

in.

in.

in.

in.

in.

in.

C6×8.2

L21⁄2×21⁄2×1⁄4 L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4

2.6 3.6 4.9 5.7 6.5

1.0 1.2 1.4 1.7 1.7

1.4 1.9 2.4 2.7 3.1

0.85 0.98 1.11 1.14 1.26

2.60 3.01 3.40 3.31 3.79

2.02 2.38 2.82 3.27 3.30

2.60 3.09 3.57 3.54 4.06

C7×9.8

L21⁄2×21⁄2×1⁄4 L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L31⁄2 ×5⁄16

3.0 4.0 5.4 6.3 7.1 8.3

1.1 1.3 1.6 1.8 18 2.2

1.6 2.0 2.6 2.9 3.2 3.6

0.86 0.98 1.10 1.14 1.25 1.29

2.67 3.09 3.48 3.40 3.88 3.78

2.31 2.66 3.11 3.57 3.59 4.16

2.62 3.11 3.59 3.57 4.08 4.05

C8×11.5

L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L3×3 ×5⁄16 L5 ×31⁄2×5⁄16

4.6 6.0 6.9 7.8 9.0 14.7

14 1.7 2.0 2.0 2.3 3.2

2.2 2.7 3.0 3.4 3.8 5.6

0.99 1.10 1.14 1.24 1.28 1.57

3.16 3.56 3.48 3.97 3.87 4.64

3.00 3.45 3.91 3.93 4.51 5.97

3.13 3.61 3.59 4.10 4.08 5.05

C9×13.4

L3 ×21⁄2×1⁄4 L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L3×3 ×5⁄16 L5 ×31⁄2×5⁄16

5.2 6.7 7.7 8.5 9.9 15.8

1.6 1.8 2.2 2.1 2.5 3.3

2.3 2.9 3.2 3.6 4.0 5.9

0.99 1.10 1.14 1.23 1.28 1.56

3.22 3.64 3.55 4.05 3.96 4.74

3.38 3.83 4.31 4.32 4.91 6.38

3.14 3.63 3.61 4.12 4.10 5.08

C10×15.3

L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L3×3 ×5⁄16 L5 ×31⁄2×5⁄16 L31⁄2×3 ×3⁄8

7.4 8.5 9.4 10.8 16.9 19.2

2.0 2.3 2.3 2.7 3.5 4.1

3.1 3.4 3.8 4.2 6.1 6.7

1.11 1.15 1.23 1.28 1.55 1.60

3.70 3.62 4.12 4.03 4.83 4.73

4.25 4.73 4.74 5.34 6.82 7.70

3.64 3.63 4.14 4.12 5.09 5.07

Channel

Angle

in.

4

S2 = I / x2


1 - 116



Y

y

2

X

X


y , yp 1

x1 xp

x2 Y

Total Weight per ft


S1 = I / y1

I 4

3

S2 = I / y2 3

r

y1

Z

yp 3

Angle

lb

in.

in.

in.

in.

in.

in.

in.

in.

C12×20.7

L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L31⁄2×3 ×5⁄16 L5 ×31⁄2×5⁄16 L5×31⁄2 ×3⁄8 L6 ×4 ×3⁄8 L5×3 ×1⁄2

26.1 27.3 26.5 27.9 29.4 31.1 33.0 36.9

7.65 8.02 7.78 8.18 8.65 9.14 9.70 10.84

164 170 167 173 180 186 192 202

33.2 35.8 34.4 37.2 40.2 43.4 46.6 53.2

23.2 23.5 23.4 23.6 23.9 24.1 24.3 24.7

4.63 4.61 4.63 4.60 4.56 4.51 4.45 4.32

4.94 4.75 4.86 4.66 4.47 4.29 4.12 3.80

31.4 32.2 31.8 32.6 33.5 34.2 35.3 36.7

3.23 2.80 3.01 2.67 2.53 2.25 2.11 1.68

C12×25

L31⁄2×3 ×1⁄4 L5×31⁄2 ×5⁄16 L4 ×3 ×1⁄4 L5×3 ×5⁄16 L5 ×31⁄2×5⁄16 L5×31⁄2 ×3⁄8 L6 ×4 ×3⁄8 L5×3 ×1⁄2

30.4 31.6 30.8 32.2 33.7 35.4 37.3 41.2

8.91 9.28 9.04 9.44 9.91 10.40 10.96 12.10

180 187 183 190 197 204 211 223

35.4 38.0 36.6 39.3 42.3 45.4 48.7 55.3

26.1 26.4 26.3 26.6 26.9 27.2 27.5 28.0

4.50 4.49 4.50 4.49 4.46 4.43 4.39 4.29

5.09 4.92 5.02 4.84 4.67 4.49 4.33 4.03

35.8 36.8 36.3 37.3 38.3 39.3 40.4 42.2

3.98 3.50 3.82 3.30 3.05 2.77 2.65 2.20

C15×33.9

L4 ×3 ×1⁄4 L5×3 ×5⁄16 L5 ×31⁄2×5⁄16 L5×31⁄2 ×3⁄8 L6 ×4 ×3⁄8 L5×3 ×1⁄2

39.7 41.1 42.6 44.3 46.2 50.1

11.65 12.05 12.52 13.01 13.57 14.71

383 395 408 421 434 458

58.7 62.4 66.5 70.8 75.4 84.8

45.1 45.6 46.1 46.5 46.9 47.7

5.73 5.73 5.71 5.69 5.65 5.58

6.52 6.33 6.14 5.94 5.76 5.40

60.1 61.8 63.4 64.8 66.2 68.6

5.39 4.89 4.30 3.69 3.48 2.92

Channel



1 - 117


Y

y

2

X

X


y , yp 1

x1 xp

x2 Y

Axis Y-Y

S1 = I / x1

I

3

3

r

x1

Z

xp 3

in.

in.

in.

in.

in.

in.

C12×20.7

L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L4×3 ×5⁄16 L5 ×31⁄2×5⁄16 L31⁄2×3 ×3⁄8 L6 ×4 ×3⁄8 L4×3 ×1⁄2

9.5 10.7 11.6 13.2 19.9 22.5 33.2 40.6

2.5 2.8 2.7 3.2 40 4.6 5.8 7.3

3.7 4.0 4.4 4.8 6.8 7.5 10.3 11.9

1.12 1.16 1.22 1.27 1.52 1.57 1.85 1.93

3.84 3.77 4.28 4.20 5.02 4.93 5.72 5.52

5.45 5.94 5.94 6.56 8.06 8.97 11.1 13.7

3.69 3.67 4.18 4.16 5.15 5.13 6.10 6.05

C12×25

L31⁄2×3 ×1⁄4 L31⁄2×3 ×5⁄16 L4 ×3 ×1⁄4 L4×3 ×5⁄16 L5 ×31⁄2×5⁄16 L31⁄2×3 ×3⁄8 L6 ×4 ×3⁄8 L4×3 ×1⁄2

10.2 11.4 12.3 13.9 20.8 23.5 34.5 42.3

2.6 3.0 2.8 3.3 4.1 4.7 5.9 7.5

3.8 4.2 4.5 5.0 7.0 7.7 10.7 12.4

1.07 1.11 1.17 1.22 1.45 1.50 1.77 1.87

3.87 3.81 4.32 4.25 5.09 5.00 5.81 5.63

5.88 6.40 6.38 7.02 8.54 9.48 11.7 14.3

3.74 3.72 4.23 4.22 5.20 5.18 6.15 6.11

C15×33.9

L4 ×3 ×1⁄4 L4×3 ×5⁄16 L5 ×31⁄2×5⁄16 L31⁄2×3 ×3⁄8 L6 ×4 ×3⁄8 L4×3 ×1⁄2

16.8 18.7 26.2 29.3 41.3 50.3

3.7 4.2 4.9 5.6 6.8 8.5

5.8 6.3 8.5 9.2 12.4 14.3

1.20 1.25 1.45 1.50 1.75 1.85

4.49 4.43 5.30 5.23 6.06 5.89

8.82 9.47 11.0 12.0 14.2 16.9

4.27 4.26 5.24 5.23 6.21 6.17

Channel

Angle

in.

4

S2 = I / x2


1 - 118



Y

y

2

X

X

y , yp

1

1

x1 xp

Short leg of angle turned out

2

x2 Y

Total Weight per ft

Total Area 2

3

in.

S2 = I / y2 3

in.

r

y1

Z

yp 3

in.

in.

in.

in.

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

12.7 13.1 14.0 16.4 18.0

3.71 3.84 4.09 4.80 5.27

6.0 6.1 6.4 7.5 7.7

5.9 6.2 6.2 6.4 6.8

2.61 2.68 2.63 2.55 2.56

4.21 4.56 4.46 4.16 4.45

7.86 8.23 8.32 8.77 9.32

2.79 3.25 3.18 3.00 3.46

C 7×9.8

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

14.3 14.7 15.6 18.0 19.6

4.18 4.31 4.56 5.27 5.74

8.0 8.0 8.5 10.0 10.3

7.8 8.2 8.2 8.5 8.9

3.00 3.07 3.03 2.95 2.96

4.70 5.05 4.93 4.60 4.87

10.5 10.9 1 1.0 11.6 12.2

2.95 3.38 3.29 3.11 3.50

C 8×11.5

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

16.0 16.4 17.3 19.7 21.3

4.69 4.82 5.07 5.78 6.25

10.4 10.4 10.9 12.9 13.3

10.2 10.6 10.6 11.0 11.4

3.39 3.45 3.42 3.36 3.36

5.20 5.55 5.42 5.06 5.31

13.7 14.2 14.3 14.9 15.6

3.52 3.73 3.42 3.21 3.61

C 9×13.4

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

17.9 18.3 19.2 21.6 23.2

5.25 5.38 5.63 6.34 6.81

13.1 13.1 13.8 16.2 16.7

12.9 13.4 13.5 13.9 14.4

3.78 3.84 3.81 3.76 3.77

5.71 6.07 5.93 5.54 5.78

17.4 18.0 18.3 19.0 19.7

4.18 4.42 3.88 3.32 3.71

C10×15.3

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

19.8 20.2 21.1 23.5 25.1

5.80 5.93 6.18 6.89 7.36

16.2 16.1 17.0 19.9 20.5

16.0 16.5 16.6 17.1 17.7

4.17 4.22 4.20 4.17 4.18

6.22 6.58 6.43 6.02 6.25

21.4 22.1 22.5 23.5 24.2

4.77 5.01 4.48 3.44 3.81

C12×20.7

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

25.2 25.6 26.5 28.9 30.5

7.40 7.53 7.78 8.49 8.96

24.3 24.0 25.3 29.2 30.1

24.7 25.3 25.5 26.3 27.1

4.90 4.95 4.95 4.94 4.97

7.32 7.69 7.54 7.11 7.33

32.6 33.4 34.3 36.4 37.5

6.17 6.45 6.01 4.74 4.41

C15×33.9

L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

38.8 39.7 42.1 43.7

11.40 11.65 12.36 12.83

42.7 44.5 50.1 51.4

47.2 47.7 49.1 50.3

5.95 5.96 6.01 6.05

9.45 9.31 8.91 9.15

61.1 62.5 66.5 69.0

8.70 8.39 7.50 7.41

Angle

in.

S1 = I / y1

C 6×8.2

Channel

lb

Axis X-X



1 - 119


Y

y

2

X

Short leg of angle turned out

X

y , yp

1

1

x1 xp

2

x2 Y

Axis Y-Y

S1 = I / x1

r

x1

Z

xp 3

in.

in.

in.

in.

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

2.4 2.8 3.8 6.0 8.4

4.4 4.6 6.6 9.3 12.1

1.22 1.26 1.64 2.05 2.47

2.25 2.18 2.85 3.33 3.82

3.70 4.01 5.46 8.29 11.3

2.65 2.59 3.46 4.12 4.84

C 7×9.8

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

2.6 2.9 3.9 6.1 8.6

5.0 5.2 7.5 10.5 13.6

1.19 1.23 1.60 2.01 2.44

2.32 2.25 2.95 3.47 3.98

4.03 4.35 5.82 8.71 11.8

2.72 2.67 3.55 4.24 4.99

C 8×11.5

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

2.7 3.0 4.0 6.3 8.7

5.6 5.8 8.3 11.7 15.2

1.16 1.20 1.55 1.97 2.40

2.37 2.31 3.03 3.58 4 13

4.40 4.73 6.21 9.16 12.3

2.78 2.73 3.62 4.34 5.12

C 9×13.4

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

2.8 3.2 4.2 6.4 8.9

6.2 6.5 9.2 12.9 16.9

1.14 1.18 1.51 1.92 2.36

2.40 2.35 3.10 3.68 4.26

4.81 5.15 6.65 9.66 12.8

2.84 2.79 3.70 4.43 5.23

C10×15.3

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

3.0 3.4 4.3 6.5 9.0

6.8 7.1 10.0 14.0 18.3

1.12 1.16 1.48 1.88 2.31

2.42 2.37 3.15 3.76 4.36

5.25 5.59 7.10 10.1 13.3

2.88 2.84 3.74 4.49 5.31

C12×20.7

L3×21⁄2×1⁄4 L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

3.6 4.0 4.7 6.9 9.3

8.6 8.9 12.2 17.0 22.4

1.10 1.13 1.40 1.78 2 19

2.47 2.43 3.25 3.92 4.59

6.47 6.82 8.37 11.5 14.8

3.01 2.97 3.89 4.67 5.52

C15×33.9

L3×3 ×1⁄4 L4×3 ×1⁄4 L5×3 ×5⁄16 L6×31⁄2×5⁄16

5.6 5.9 7.8 10.2

13.5 17.2 23.4 30.7

1.10 1.30 1.61 1.97

2.48 3.35 4.12 4.88

9.54 11.1 14.5 18.0

3.12 4.11 5.02 5.90

Angle

in.

3

C 6×8.2

Channel

in.

3

S2 = I / x2


1 - 120


STEEL PIPE AND STRUCTURAL TUBING General

When designing and specifying steel pipe or tubing as compression members, refer to comments in the notes for Columns, Steel Pipe, and Structural Tubing, in Part 3. For standard mill practices and tolerances, refer to page 1-183. For material specifications and availability, see Tables 1-4 through 1-6, Part 1. Steel Pipe

The Tables of Dimensions and Properties of Steel Pipe (unfilled) list a selected range of sizes of standard, extra strong, and double-extra strong pipe. For a complete range of sizes manufactured, refer to catalogs of the manufacturers or to the American Institute for Hollow Structural Sections (AIHSS). Structural Tubing

The Tables of Dimensions and Properties of Square and Rectangular Structural Tubing (unfilled) list a selected range of frequently used sizes. For dimensions and properties of other sizes, refer to catalogs from the manufacturers or AIHSS. The tables are based on an outside corner radius equal to two times the specified wall thickness. Material specifications stipulate that the outside corner radius may vary up to three times the specified wall thickness. This variation should be considered in those details where a close match or fit is important.


STEEL PIPE AND STRUCTURAL TUBING

1 - 121

PIPE Dimensions and properties Dimensions

Weight Nominal Outside Inside Wall per ft lbs Diameter Diameter Diameter Thickness Plain Ends in. in. in. in.

Properties Area

I

S

r

J

Z

in.2

in.4

in.3

in.

in.4

in.3

0.041 0.071 0.133 0.235 0.326 0.561 1.06 1.72 2.39 3.21 5.45 8.50 16.8 29.9 43.8

0.261 0.334 0.421 0.540 0.623 0.787 0.947 1.16 1.34 1.51 1.88 2.25 2.94 3.67 4.38

0.034 0.074 0.175 0.389 0.620 1.33 3.06 6.03 9.58 14.5 30.3 56.3 145 321 559

0.059 0.100 0.187 0.324 0.448 0.761 1.45 2.33 3.22 4.31 7.27 11.2 22.2 39.4 57.4

0.048 0.085 0.161 0.291 0.412 0.731 1.34 2.23 3.14 4.27 7.43 12.2 24.5 39.4 56.7

0.250 0.321 0.407 0.524 0.605 0.766 0.924 1.14 1.31 1.48 1.84 2.19 2.88 3.63 4.33

0.040 0.090 0.211 0.484 0.782 1.74 3.85 8.13 12.6 19.2 41.3 81.0 211 424 723

0.072 0.125 0.233 0.414 0.581 1.02 1.87 3.08 4.32 5.85 10.1 16.6 33.0 52.6 75.1

1.10 2.00 3.42 6.79 12.1 20.0 37.6

0.703 0.844 1.05 1.37 1.72 2.06 2.76

2.62 5.74 12.0 30.6 67.3 133 324

1.67 3.04 5.12 9.97 17.5 28.9 52.8

Standard Weight 1⁄ 2 3⁄ 4

0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.750 12.750

0.622 0.824 1.049 1.380 1.610 2.067 2.469 3.068 3.548 4.026 5.047 6.065 7.981 10.020 12.000

0.109 0.113 0.133 0.140 0.145 0.154 0.203 0.216 0.226 0.237 0.258 0.280 0.322 0.365 0.375

0.85 1.13 1.68 2.27 2.72 3.65 5.79 7.58 9.11 10.79 14.62 18.97 28.55 40.48 49.56

1 11⁄4 11⁄2 2 21⁄2 3 31⁄2 4 5 6 8 10 12

0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625 8.625 10.750 12.750

0.546 0.742 0.957 1.278 1.500 1.939 2.323 2.900 3.364 3.826 4.813 5.761 7.625 9.750 11.750

0.147 0.154 0.179 0.191 0.200 0.218 0.276 0.300 0.318 0.337 0.375 0.432 0.500 0.500 0.500

1.09 1.47 2.17 3.00 3.63 5.02 7.66 10.25 12.50 14.98 20.78 28.57 43.39 54.74 65.42

2 21⁄2 3 4 5 6 8

2.375 2.875 3.500 4.500 5.563 6.625 8.625

1.503 1.771 2.300 3.152 4.063 4.897 6.875

0.436 0.552 0.600 0.674 0.750 0.864 0.875

1 11⁄4 11⁄2 2 21⁄2 3 31⁄2 4 5 6 8 10 12

0.250 0.333 0.494 0.669 0.799 1.07 1.70 2.23 2.68 3.17 4.30 5.58 8.40 11.9 14.6

0.017 0.037 0.087 0.195 0.310 0.666 1.53 3.02 4.79 7.23 15.2 28.1 72.5 161 279

Extra Strong 1⁄ 2 3⁄ 4

0.320 0.433 0.639 0.881 1.07 1.48 2.25 3.02 3.68 4.41 6.11 8.40 12.8 16.1 19.2

0.020 0.045 0.106 0.242 0.391 0.868 1.92 3.89 6.28 9.61 20.7 40.5 106 212 362

Double-Extra Strong 9.03 13.69 18.58 27.54 38.59 53.16 72.42

2.66 4.03 5.47 8.10 11.3 15.6 21.3

1.31 2.87 5.99 15.3 33.6 66.3 162

The listed sections are available in conformance with ASTM Specification A53 Grade B or A501. Other sections are made to these specifications. Consult with pipe manufacturers or distributors for availability.


1 - 122


STRUCTURAL TUBING Square Dimensions and properties Dimensions

Properties**

Nominal* Size

Wall Thickness

Weight per ft

Area

I

S

r

J

Z

in.

in.

lb

in.2

in.4

in.3

in.

in.4

in.3

0.6250

5⁄ 8

246.47

72.4

10300

690

12.0

16000

794

28×28

0.6250

5⁄ 8

229.45

67.4

8360

597

11.1

13000

689

26×26

0.6250

5⁄ 8

212.44

62.4

6650

511

10.3

10400

591

24×24 24×24 24×24

0.6250 0.5000 0.3750

5⁄ 8 1⁄ 2 3⁄ 8

195.43 157.74 119.35

57.4 46.4 35.1

5180 4240 3250

432 353 270

9.50 9.56 9.62

8100 6570 4990

500 407 310

22×22 22×22 22×22

0.6250 0.5000 0.3750

5⁄ 8 1⁄ 2 3⁄ 8

178.41 144.13 109.15

52.4 42.4 32.1

3950 3240 2490

359 294 226

8.68 8.74 8.80

6200 5030 3830

418 340 259

20×20 20×20 20×20

0.6250 0.5000 0.3750

5⁄ 8 1⁄ 2 3⁄ 8

161.40 130.52 98.94

47.4 38.4 29.1

2940 2410 1850

294 241 185

7.87 7.93 7.99

4620 3760 2870

342 279 213

18×18 18×18 18×18

0.6250 0.5000 0.3750

5⁄ 8 1⁄ 2 3⁄ 8

144.39 116.91 88.73

42.4 34.4 26.1

2110 1740 1340

234 193 149

7.05 7.11 7.17

3340 2720 2080

274 224 172

30×30

*Outside dimensions across flat sides. **Properties are based upon a nominal outside corner radius equal to two times the wall thickness.



1 - 123


Properties**

Nominal* Size

Wall Thickness

Weight per ft

Area

I

S

r

J

Z

in.

in.

lb

in.2

in.4

in.3

in.

in.4

in.3

127.37 103.30 78.52 65.87

37.4 30.4 23.1 19.4

1450 1200 931 789

182 150 116 98.6

6.23 6.29 6.35 6.38

2320 1890 1450 1220

214 175 134 113

110.36 89.68 68.31 57.36

32.4 26.4 20.1 16.9

952 791 615 522

136 113 87.9 74.6

5.42 5.48 5.54 5.57

1530 1250 963 812

161 132 102 86.1

93.34 76.07 58.10 48.86 39.43

27.4 22.4 17.1 14.4 11.6

580 485 380 324 265

96.7 80.9 63.4 54.0 44.1

4.60 4.66 4.72 4.75 4.78

943 777 599 506 410

116 95.4 73.9 62.6 50.8

76.33 62.46 47.90 40.35 32.63 24.73

22.4 18.4 14.1 11.9 9.59 7.27

321 271 214 183 151 116

64.2 54.2 42.9 36.7 30.1 23.2

3.78 3.84 3.90 3.93 3.96 3.99

529 439 341 289 235 179

77.6 64.6 50.4 42.8 34.9 26.6

16×16

14×14

12×12

10×10

0.6250 0.5000 0.3750 0.3125 0.6250 0.5000 0.3750 0.3125 0.6250 0.5000 0.3750 0.3125 0.2500 0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 1⁄ 3⁄

8 2

8 5⁄ 16 5⁄ 1⁄

8 2

3⁄ 8 5⁄ 16 5⁄ 1⁄

8 2

3⁄ 8 5⁄ 16 1⁄ 4 5⁄ 1⁄ 3⁄

8 2

8 5⁄ 16 1⁄ 4 3⁄ 16



1 - 124



Properties**

Nominal* Size

Wall Thickness

Weight per ft

Area

I

S

r

J

Z

in.

in.

lb

in.2

in.4

in.3

in.

in.4

in.3

8×8

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

59.32 48.85 37.69 31.84 25.82 19.63

17.4 14.4 11.1 9.36 7.59 5.77

153 131 106 90.9 75.1 58.2

38.3 32.9 26.4 22.7 18.8 14.6

2.96 3.03 3.09 3.12 3.15 3.18

258 217 170 145 118 90.6

47.2 39.7 31.3 26.7 21.9 16.8

7×7

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

50.81 42.05 32.58 27.59 22.42 17.08

14.9 12.4 9.58 8.11 6.59 5.02

97.5 84.6 68.7 59.5 49.4 38.5

27.9 24.2 19.6 17.0 14.1 11.0

2.56 2.62 2.68 2.71 2.74 2.77

166 141 112 95.6 78.3 60.2

34.8 29.6 23.5 20.1 16.5 12.7

6×6

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

42.30 35.24 27.48 23.34 19.02 14.53 9.86

12.4 10.4 8.08 6.86 5.59 4.27 2.90

57.3 50.5 41.6 36.3 30.3 23.8 16.5

19.1 16.8 13.9 12.1 10.1 7.93 5.52

2.15 2.21 2.27 2.30 2.33 2.36 2.39

99.5 85.6 68.5 58.9 48.5 37.5 25.7

24.3 20.9 16.8 14.4 11.9 9.24 6.35

51⁄2×51⁄2

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

24.93 21.21 17.32 13.25 9.01

7.33 6.23 5.09 3.89 2.65

31.2 27.4 23.0 18.1 12.6

11.4 9.95 8.36 6.58 4.60

2.07 2.10 2.13 2.16 2.19

51.9 44.8 37.0 28.6 19.7

13.8 12.0 9.91 7.70 5.31

5×5

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

28.43 22.37 19.08 15.62 11.97 8.16

8.36 6.58 5.61 4.59 3.52 2.40

27.0 22.8 20.1 16.9 13.4 9.41

10.8 9.11 8.02 6.78 5.36 3.77

1.80 1.86 1.89 1.92 1.95 1.98

46.8 38.2 33.1 27.4 21.3 14.7

13.7 11.2 9.70 8.07 6.29 4.36




1 - 125


Properties**

Nominal* Size

Wall Thickness

Weight per ft

Area

I

S

r

J

Z

in.

in.

lb

in.2

in.4

in.3

in.

in.4

in.3

41⁄2×41⁄2

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

19.82 16.96 13.91 10.70 7.31

5.83 4.98 4.09 3.14 2.15

16.0 14.2 12.1 9.60 6.78

7.10 6.30 5.36 4.27 3.02

1.66 1.69 1.72 1.75 1.78

27.1 23.6 19.7 15.4 10.6

8.81 7.68 6.43 5.03 3.50

4×4

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

21.63 17.27 14.83 12.21 9.42 6.46

6.36 5.08 4.36 3.59 2.77 1.90

12.3 10.7 9.58 8.22 6.59 4.70

6.13 5.35 4.79 4.11 3.30 2.35

1.39 1.45 1.48 1.51 1.54 1.57

21.8 18.4 16.1 13.5 10.6 7.40

8.02 6.72 5.90 4.97 3.91 2.74

31⁄2×31⁄2

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

12.70 10.51 8.15 5.61

3.73 3.09 2.39 1.65

6.09 5.29 4.29 3.09

3.48 3.02 2.45 1.76

1.28 1.31 1.34 1.37

10.4 8.82 6.99 4.90

4.35 3.69 2.93 2.07

3×3

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

10.58 8.81 6.87 4.75

3.11 2.59 2.02 1.40

3.58 3.16 2.60 1.90

2.39 2.10 1.73 1.26

1.07 1.10 1.13 1.16

6.22 5.35 4.28 3.03

3.04 2.61 2.10 1.49

21⁄2×21⁄2

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

8.45 7.11 5.59 3.90

2.48 2.09 1.64 1.15

1.87 1.69 1.42 1.06

1.50 1.35 1.14 0.847

0.868 0.899 0.930 0.961

3.32 2.92 2.38 1.71

1.96 1.71 1.40 1.01

2×2

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

6.32 5.41 4.32 3.05

1.86 1.59 1.27 0.897

0.815 0.766 0.668 0.513

0.815 0.766 0.668 0.513

0.662 0.694 0.726 0.756

1.49 1.36 1.15 0.846

1.11 1.00 0.840 0.621

11⁄2×11⁄2

0.1875

3⁄ 16

3.04

0.894

0.242

0.323

0.521

0.431

0.423

3⁄

2 8



1 - 126


Y

X

X

STRUCTURAL TUBING Rectangular Dimensions and properties

Y

Dimensions

Properties**

Nominal* Wall Weight Size Thickness per ft Area in.

in.

2

X-X Axis

S

I 4

Z 3

Y-Y Axis

r 3

S

I 4

Z 3

3

r

J

lb

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.4

178.16 134.67 112.66

52.4 39.6 33.1

7110 474 5430 362 4570 305

555 422 354

11.7 11.7 11.7

5070 3870 3260

422 323 272

477 363 304

9.84 9.89 9.92

9170 6960 5830

30×24

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

28×24

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

171.35 129.56 108.41

50.4 38.1 31.9

6050 432 4630 331 3890 278

503 383 321

11.0 11.0 11.1

4790 3660 3080

399 305 257

454 345 290

9.75 9.81 9.84

8280 6290 5270

26×24

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

164.55 124.46 104.15

48.4 36.6 30.6

5100 392 3900 300 3280 253

454 345 290

10.3 10.3 10.4

4510 3460 2910

376 288 242

430 327 275

9.66 9.72 9.75

7410 5630 4720

24×22

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

150.93 114.25 95.64

44.4 33.6 28.1

3960 330 3040 253 2560 213

383 292 245

9.45 3470 9.51 2660 9.54 2240

315 242 204

361 275 231

8.84 8.90 8.93

5740 4370 3660

22×20

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

137.32 104.04 87.14

40.4 30.6 25.6

3010 273 2310 210 1950 177

318 243 204

8.63 2600 8.69 2000 8.72 1690

260 200 169

298 228 192

8.03 8.09 8.12

4350 3310 2780

20×18

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

123.71 93.83 78.63

36.4 27.6 23.1

2220 222 1710 171 1440 144

259 198 167

7.81 1890 7.88 1460 7.91 1230

210 162 137

242 185 155

7.21 7.27 7.30

3190 2440 2050

20×12

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

103.30 78.52 65.87

30.4 23.1 19.4

1650 165 1280 128 1080 108

201 154 130

7.37 7.45 7.47

750 583 495

125 141 97.2 109 82.5 91.8

4.97 5.03 5.06

1650 1270 1070

20×8

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

89.68 68.31 57.36

26.4 20.1 16.9

1270 127 162 988 98.8 125 838 83.8 105

6.94 7.02 7.05

300 236 202

20×4

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

76.07 58.10 48.86

22.4 17.1 14.4

889 699 596

88.9 123 69.9 95.3 59.6 80.8

6.31 6.40 6.44

61.6 50.3 43.7

75.1 59.1 50.4

84.7 65.6 55.6

3.38 3.43 3.46

806 625 529

30.8 25.1 21.8

36.0 28.5 24.3

1.66 1.72 1.74

205 165 143




1 - 127


Y

X

X

Y

Dimensions

Properties**


in.

lb

in.2

X-X Axis

Y-Y Axis

I

S

Z

r

I

S

Z

r

J

in.4

in.3

in.3

in.

in.4

in.3

in.3

in.

in.4

6.71 6.78 6.81

684 533 452

114 130 88.8 100 75.3 84.5

4.91 4.97 5.00

1420 1090 920

53.9 42.1 35.8 29.2

2.52 2.57 2.60 2.63

410 322 274 224

103 118 80.3 91.3 68.2 77.2

4.84 4.90 4.93

1200 922 777

18×12

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

96.49 73.42 61.62

28.4 21.6 18.1

18×6

0.5000 0.3750 0.3125 0.2500

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4

76.07 58.10 48.86 39.43

22.4 17.1 14.4 11.6

818 641 546 447

90.9 119 71.3 92.2 60.7 78.1 49.6 63.5

6.05 6.13 6.17 6.21

141 113 97.0 80.0

16×12

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

89.68 68.31 57.36

26.4 20.1 16.9

962 748 635

120 144 93.5 111 79.4 93.8

6.04 6.11 6.14

618 482 409

16×8

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

76.07 58.10 48.86

22.4 17.1 14.4

722 565 481

90.2 113 70.6 87.6 60.1 74.2

5.68 5.75 5.79

244 193 165

16×4

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

62.46 47.90 40.35

18.4 14.1 11.9

481 382 327

60.2 47.8 40.9

82.2 64.2 54.5

5.12 5.21 5.25

14×12

0.5000 0.3750

1⁄ 2 3⁄ 8

82.88 63.21

24.4 18.6

699 546

99.9 119 78.0 91.7

5.36 5.42

14×10

0.5000 0.3750 0.3125

1⁄ 2 3⁄ 8 5⁄ 16

76.07 58.10 48.86

22.4 17.1 14.4

608 476 405

86.9 105 68.0 81.5 57.9 69.0

14×6

0.6250 0.5000 0.3750 0.3125 0.2500

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4

76.33 62.46 47.90 40.35 32.63

22.4 18.4 14.1 11.9 9.59

504 426 337 288 237

72.0 60.8 48.1 41.2 33.8

1280 142 172 991 110 132 840 93.3 111

94.0 78.3 61.1 51.9 42.3

47.2 37.6 32.3 26.7

61.0 48.2 41.2

69.7 54.2 45.9

3.30 3.36 3.39

599 465 394

24.6 20.2 17.6

29.0 23.0 19.7

1.64 1.69 1.72

157 127 110

552 431

91.9 107 71.9 82.6

4.76 4.82

983 757

5.22 5.28 5.31

361 284 242

72.3 56.8 48.4

83.6 64.8 54.9

4.02 4.08 4.11

730 564 477

4.74 4.82 4.89 4.93 4.97

130 111 89.1 76.7 63.4

43.3 37.1 29.7 25.6 21.1

51.2 42.9 33.6 28.7 23.4

2.41 2.46 2.52 2.54 2.57

352 296 233 199 162

49.3 40.4 35.1



1 - 128


Y

X

X


Y

Dimensions

Properties**


in.

lb

in.2

X-X Axis

Y-Y Axis

I

S

Z

r

I

S

Z

r

J

in.4

in.3

in.3

in.

in.4

in.3

in.3

in.

in.4

49.0 43.1 35.4 30.9 25.8 20.2

24.5 21.5 17.7 15.4 12.9 10.1

30.0 25.5 20.3 17.4 14.3 11.1

1.57 1.62 1.68 1.71 1.73 1.76

154 134 108 93.1 77.0 59.7

14×4

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

67.82 55.66 42.79 36.10 29.23 22.18

19.9 16.4 12.6 10.6 8.59 6.52

392 335 267 230 189 146

56.0 47.8 38.2 32.8 27.0 20.9

77.3 64.8 50.8 43.3 35.4 27.1

4.44 4.52 4.61 4.65 4.69 4.74

12×10

0.5000 0.3750 0.3125 0.2500

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4

69.27 53.00 44.60 36.03

20.4 15.6 13.1 10.6

419 330 281 230

69.9 55.0 46.9 38.4

83.9 65.2 55.2 44.9

4.54 4.60 4.63 4.66

316 249 213 174

63.3 49.8 42.6 34.9

74.1 57.6 48.8 39.7

3.94 4.00 4.03 4.06

581 450 381 309

12×8

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

76.33 62.46 47.90 40.35 32.63 24.73

22.4 18.4 14.1 11.9 9.59 7.27

418 353 279 239 196 151

69.7 58.9 46.5 39.8 32.6 25.1

87.1 72.4 56.5 47.9 39.1 29.8

4.32 4.39 4.45 4.49 4.52 4.55

221 188 149 128 105 81.1

55.3 46.9 37.3 32.0 26.3 20.3

65.6 54.7 42.7 36.3 29.6 22.7

3.14 3.20 3.26 3.28 3.31 3.34

481 401 312 265 216 165

12×6

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

67.82 55.66 42.79 36.10 29.23 22.18

19.9 16.4 12.6 10.6 8.59 6.52

337 287 228 196 161 124

56.2 47.8 38.1 32.6 26.9 20.7

72.9 60.9 47.7 40.6 33.2 25.4

4.11 4.19 4.26 4.30 4.33 4.37

112 96.0 77.2 66.6 55.2 42.8

37.2 32.0 25.7 22.2 18.4 14.3

44.5 37.4 29.4 25.1 20.6 15.8

2.37 2.42 2.48 2.51 2.53 2.56

286 241 190 162 132 101

12×4

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

59.32 48.85 37.69 31.84 25.82 19.63

17.4 14.4 11.1 9.36 7.59 5.77

257 221 178 153 127 98.2

42.8 36.8 29.6 25.5 21.1 16.4

58.6 49.4 39.0 33.3 27.3 21.0

3.84 3.92 4.01 4.05 4.09 4.13

41.8 36.9 30.5 26.6 22.3 17.5

20.9 18.5 15.2 13.3 11.1 8.75

25.8 22.0 17.6 15.1 12.5 9.63

1.55 1.60 1.66 1.69 1.71 1.74

127 110 89.0 76.9 63.6 49.3

12×3

0.3125 0.2500 0.1875

5⁄ 16 1⁄ 4 3⁄ 16

29.72 24.12 18.35

8.73 132 7.09 109 5.39 85.1

22.0 18.2 14.2

29.7 24.4 18.8

3.89 3.93 3.97

13.8 11.7 9.28

9.19 10.6 7.79 8.80 6.19 6.84

1.26 1.28 1.31

43.6 36.5 28.7




1 - 129


Y

X

X

Y

Dimensions

Properties**


in.

X-X Axis

Y-Y Axis

I

S

Z

r

I

S

Z

r

J

lb

in.2

in.4

in.3

in.3

in.

in.4

in.3

in.3

in.

in.4

92.2 15.4 72.0 12.0

21.4 16.6

3.74 3.79

12×2

0.2500 0.1875

1⁄ 4 3⁄ 16

22.42 17.08

6.59 5.02

10×8

0.5000 0.3750 0.3125 0.2500 0.1875

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

55.66 42.79 36.10 29.23 22.18

16.4 12.6 10.6 8.59 6.52

226 180 154 127 97.9

45.2 35.9 30.8 25.4 19.6

55.1 43.1 36.7 30.0 23.0

3.72 3.78 3.81 3.84 3.87

160 127 109 90.2 69.7

39.9 31.8 27.3 22.5 17.4

47.2 37.0 31.5 25.8 19.7

3.12 3.18 3.21 3.24 3.27

306 239 203 166 127

10×6

0.5000 0.3750 0.3125 0.2500 0.1875

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

48.85 37.69 31.84 25.82 19.63

14.4 11.1 9.36 7.59 5.77

181 145 125 103 79.8

36.2 29.0 25.0 20.6 16.0

45.6 35.9 30.7 25.1 19.3

3.55 3.62 3.65 3.69 3.72

80.8 65.4 56.5 46.9 36.5

26.9 21.8 18.8 15.6 12.2

31.9 25.2 21.5 17.7 13.6

2.37 2.43 2.46 2.49 2.51

187 147 126 103 79.1

10×5

0.3750 0.3125 0.2500 0.1875

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

35.13 29.72 24.12 18.35

10.3 128 8.73 110 7.09 91.2 5.39 70.8

25.5 22.0 18.2 14.2

32.3 27.6 22.7 17.4

3.51 3.55 3.59 3.62

42.9 37.2 31.1 24.3

17.1 14.9 12.4 9.71

19.9 17.0 14.0 10.8

2.04 2.07 2.09 2.12

107 91.5 75.2 58.0

10×4

0.5000 0.3750 0.3125 0.2500 0.1875

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

42.05 32.58 27.59 22.42 17.08

12.4 136 9.58 110 8.11 95.5 6.59 79.3 5.02 61.7

27.1 22.0 19.1 15.9 12.3

36.1 28.7 24.6 20.2 15.6

3.31 3.39 3.43 3.47 3.51

30.8 25.5 22.4 18.8 14.8

15.4 12.8 11.2 9.39 7.39

18.5 14.9 12.8 10.6 8.20

1.58 1.63 1.66 1.69 1.72

86.9 70.4 60.8 50.4 39.1

10×3

0.3750 0.3125 0.2500 0.1875

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

30.0 25.5 20.72 15.80

8.83 7.48 6.09 4.64

92.8 80.8 67.4 52.7

18.6 16.2 13.5 10.5

25.1 21.6 17.8 13.8

3.24 3.29 3.33 3.37

13.0 11.5 9.79 7.80

8.66 10.3 7.68 8.92 6.53 7.42 5.20 5.79

1.21 1.24 1.27 1.30

39.8 34.9 29.3 23.0

10×2

0.3750 0.3125 0.2500 0.1875

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

27.48 23.34 19.02 14.53

8.08 6.86 5.59 4.27

75.4 15.1 66.1 13.2 55.5 11.1 43.7 8.74

21.5 18.5 15.4 11.9

3.06 3.10 3.15 3.20

4.85 4.42 3.85 3.14

4.85 4.42 3.85 3.14

0.775 0.802 0.830 0.858

16.5 14.9 12.8 10.3

4.62 3.76

4.62 3.76

5.38 0.837 4.24 0.865

6.05 5.33 4.50 3.56



15.9 12.8

1 - 130


Y

X

X


Y

Dimensions

Properties**


in.

lb

in.2

X-X Axis

Y-Y Axis

I

S

Z

r

I

S

Z

r

J

in.4

in.3

in.3

in.

in.4

in.3

in.3

in.

in.4

8×6

0.5000 0.3750 0.3125 0.2500 0.1875

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

42.05 32.58 27.59 22.42 17.08

12.4 103 9.58 83.7 8.11 72.4 6.59 60.1 5.02 46.8

25.8 20.9 18.1 15.0 11.7

32.2 25.6 21.9 18.0 13.9

2.89 2.96 2.99 3.02 3.05

65.7 53.5 46.4 38.6 30.1

21.9 17.8 15.5 12.9 10.0

26.4 21.0 18.0 14.8 11.4

2.31 2.36 2.39 2.42 2.45

135 107 91.3 74.9 57.6

8×4

0.6250 0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

5⁄ 8 1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

42.30 35.24 27.48 23.34 19.02 14.53 9.86

12.4 10.4 8.08 6.86 5.59 4.27 2.90

21.3 18.8 15.5 13.5 11.3 8.83 6.14

28.8 24.7 19.9 17.1 14.1 11.0 7.53

2.62 2.69 2.77 2.80 2.84 2.88 2.91

27.4 24.6 20.6 18.1 15.3 12.0 8.45

13.7 12.3 10.3 9.05 7.63 6.02 4.23

17.3 15.0 12.2 10.5 8.72 6.77 4.67

1.49 1.54 1.60 1.62 1.65 1.68 1.71

73.2 64.1 52.2 45.2 37.5 29.1 20.0

8×3

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

31.84 24.93 21.21 17.32 13.25 9.01

9.36 7.33 6.23 5.09 3.89 2.65

61.0 15.3 51.0 12.7 44.7 11.2 37.6 9.40 29.6 7.40 20.7 5.17

21.0 17.0 14.7 12.2 9.49 6.55

2.55 2.64 2.68 2.72 2.76 2.80

12.1 10.4 9.25 7.90 6.31 4.48

8.05 10.1 6.92 8.31 6.16 7.24 5.26 6.05 4.21 4.73 2.99 3.29

1.14 1.19 1.22 1.25 1.27 1.30

35.7 29.9 26.3 22.1 17.3 12.1

8×2

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

22.37 19.08 15.62 11.97 8.16

6.58 5.61 4.59 3.52 2.40

40.1 10.0 14.2 35.5 8.87 12.3 30.1 7.52 10.3 23.9 5.97 8.02 16.8 4.20 5.56

2.47 2.51 2.56 2.60 2.65

3.85 3.52 3.08 2.52 1.83

3.85 3.52 3.08 2.52 1.83

4.83 4.28 3.63 2.88 2.03

0.765 0.792 0.819 0.847 0.875

12.6 11.4 9.84 7.94 5.66

7×5

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

35.24 27.48 23.34 19.02 14.53 9.86

10.4 8.08 6.86 5.59 4.27 2.90

63.5 52.2 45.5 38.0 29.8 20.7

23.1 18.5 15.9 13.2 10.2 7.00

2.48 2.54 2.58 2.61 2.64 2.67

37.2 30.8 26.9 22.6 17.7 12.4

14.9 12.3 10.8 9.04 7.10 4.95

18.2 14.6 12.6 10.4 8.10 5.58

1.90 1.95 1.98 2.01 2.04 2.07

79.9 64.2 55.3 45.6 35.3 24.2

7×4

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

24.93 21.21 17.32 13.25 9.01

7.33 6.23 5.09 3.89 2.65

44.0 12.6 16.0 38.5 11.0 13.8 32.3 9.23 11.5 25.4 7.26 8.91 17.7 5.07 6.15

2.45 2.49 2.52 2.55 2.59

18.1 16.0 13.5 10.7 7.51

9.06 10.8 7.98 9.36 6.75 7.78 5.34 6.06 3.76 4.19

1.57 1.60 1.63 1.66 1.68

43.3 37.5 31.2 24.2 16.7

85.1 75.1 61.9 53.9 45.1 35.3 24.6

18.1 14.9 13.0 10.9 8.50 5.91




1 - 131


Y

X

X

Y

Dimensions

Properties**


in.

X-X Axis

Y-Y Axis

I

S

Z

r

I

S

Z

r

J

lb

in.2

in.4

in.3

in.3

in.

in.4

in.3

in.3

in.

in.4

7×3

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

22.37 19.08 15.62 11.97 8.16

6.58 5.61 4.59 3.52 2.40

35.7 31.5 26.6 21.1 14.8

10.2 13.5 9.00 11.8 7.61 9.79 6.02 7.63 4.22 5.29

2.33 2.37 2.41 2.45 2.48

9.08 8.11 6.95 5.57 3.96

6.05 5.41 4.63 3.71 2.64

7.32 6.40 5.36 4.20 2.93

1.18 1.20 1.23 1.26 1.29

25.1 22.0 18.5 14.6 10.2

6×4

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

28.43 22.37 19.08 15.62 11.97 8.16

8.36 6.58 5.61 4.59 3.52 2.40

35.3 29.7 26.2 22.1 17.4 12.2

11.8 15.4 9.90 12.5 8.72 10.9 7.36 9.06 5.81 7.06 4.08 4.88

2.06 2.13 2.16 2.19 2.23 2.26

18.4 15.6 13.8 11.7 9.32 6.57

9.21 7.82 6.92 5.87 4.66 3.29

11.5 9.44 8.21 6.84 5.34 3.71

1.48 1.54 1.57 1.60 1.63 1.66

42.1 34.6 30.1 25.0 19.5 13.5

6×3

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

25.03 19.82 16.96 13.91 10.70 7.31

7.36 5.83 4.98 4.09 3.14 2.15

27.7 23.8 21.1 17.9 14.3 10.1

9.25 12.6 7.92 10.4 7.03 9.11 5.98 7.62 4.76 5.97 3.36 4.15

1.94 2.02 2.06 2.09 2.13 2.17

8.91 7.78 6.98 6.00 4.83 3.45

5.94 5.19 4.65 4.00 3.22 2.30

7.59 6.34 5.56 4.67 3.68 2.57

1.10 1.16 1.18 1.21 1.24 1.27

23.9 20.3 17.9 15.1 11.9 8.27

6×2

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

17.27 14.83 12.21 9.42 6.46

5.08 4.36 3.59 2.77 1.90

17.8 16.0 13.8 11.1 7.92

5.94 5.34 4.60 3.70 2.64

8.33 7.33 6.18 4.88 3.42

1.87 1.92 1.96 2.00 2.04

2.84 2.62 2.31 1.90 1.39

2.84 2.62 2.31 1.90 1.39

3.61 3.22 2.75 2.20 1.56

0.748 0.775 0.802 0.829 0.857

8.72 7.94 6.88 5.56 3.98

5×4

0.3750 0.3125 0.2500 0.1875

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16

19.82 16.96 13.91 10.70

5.83 4.98 4.09 3.14

18.7 16.6 14.1 11.2

7.50 6.65 5.65 4.49

9.44 8.24 6.89 5.39

1.79 1.83 1.86 1.89

13.2 11.7 9.98 7.96

6.58 5.85 4.99 3.98

8.08 7.05 5.90 4.63

1.50 1.53 1.56 1.59

26.3 22.9 19.1 14.9

5×3

0.5000 0.3750 0.3125 0.2500 0.1875 0.1250

1⁄ 2 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

21.63 17.27 14.83 12.21 9.42 6.46

6.36 5.08 4.36 3.59 2.77 1.90

16.9 14.7 13.2 11.3 9.06 6.44

6.75 5.89 5.27 4.52 3.62 2.58

9.20 7.71 6.77 5.70 4.49 3.14

1.63 1.70 1.74 1.77 1.81 1.84

7.33 6.48 5.85 5.05 4.08 2.93

4.88 4.32 3.90 3.37 2.72 1.95

6.34 5.35 4.72 3.98 3.15 2.21

1.07 1.13 1.16 1.19 1.21 1.24

18.2 15.6 13.8 11.7 9.21 6.44

*Outside dimensions across flatsides. **Properties are based upon a nominal outside corner radius equal to two times the wall thickness.


1 - 132


Y

X

X


Y

Dimensions

Properties**


in.

2

X-X Axis

S

I 4

Z 3

Y-Y Axis

r 3

S

I 4

Z 3

3

r

J

lb

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.4

12.70 10.51 8.15 5.61

3.73 3.09 2.39 1.65

9.74 8.48 6.89 4.96

3.90 3.39 2.75 1.98

5.31 4.51 3.59 2.53

1.62 1.66 1.70 1.73

2.16 1.92 1.60 1.17

2.16 1.92 1.60 1.17

2.70 2.32 1.86 1.32

0.762 0.789 0.816 0.844

6.24 5.43 4.40 3.15

5×2

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

4×3

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

12.70 10.51 8.15 5.61

3.73 3.09 2.39 1.65

7.45 6.45 5.23 3.76

3.72 3.23 2.62 1.88

4.75 4.03 3.20 2.25

1.41 1.45 1.48 1.51

4.71 4.10 3.34 2.41

3.14 2.74 2.23 1.61

3.88 3.30 2.62 1.85

1.12 1.15 1.18 1.21

9.89 8.41 6.67 4.68

4×2

0.3750 0.3125 0.2500 0.1875 0.1250

3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

12.17 10.58 8.81 6.87 4.75

3.58 3.11 2.59 2.02 1.40

5.75 5.32 4.69 3.87 2.82

2.87 2.66 2.35 1.93 1.41

4.00 3.60 3.09 2.48 1.77

1.27 1.31 1.35 1.38 1.42

1.83 1.71 1.54 1.29 0.954

1.83 1.71 1.54 1.29 0.954

2.39 2.17 1.88 1.52 1.09

0.715 0.743 0.770 0.798 0.826

4.97 4.58 4.01 3.26 2.34

3×2

0.3125 0.2500 0.1875 0.1250

5⁄ 16 1⁄ 4 3⁄ 16 1⁄ 8

8.45 7.11 5.59 3.90

2.48 2.09 1.64 1.15

2.44 2.21 1.86 1.38

1.63 1.47 1.24 0.920

2.20 1.92 1.57 1.13

0.992 1.03 1.06 1.10

1.26 1.15 0.977 0.733

1.26 1.15 0.977 0.733

1.64 1.44 1.18 0.855

0.714 0.742 0.771 0.800

2.97 2.63 2.16 1.57

21⁄2×11⁄2

0.2500 0.1875

1⁄ 4 3⁄ 16

5.41 4.32

1.59 1.27

1.05 0.844 1.15 0.815 0.920 0.736 0.964 0.852

0.458 0.405

0.610 0.793 0.537 1.14 0.540 0.669 0.565 0.976



BARS AND PLATES

1 - 133

BARS AND PLATES Product Availability

Plates are readily available in seven of the structural steel specifications listed in Section A3.1 of the AISC LRFD Specification. These are: ASTM A36, A242, A529, A572, A588, A514, and A852. Bars are available in all of these steels except A514 and A852. Table 1-1 shows the availability of each steel in terms of plate thickness. The Manual user is referred to the discussion on p. 1-5, Selection of the Appropriate Structural Steel, for guidance in selection of both plate and structural shapes. Classification

Bars and plates are generally classified as follows: Bars: 6 in. or less in width, .203 in. and over in thickness. Over 6 in. to 8 in. in width, .230 in. and over in thickness. Plates: Over 8 in. to 48 in. in width, .230 in. and over in thickness. Over 48 in. in width, .180 in. and over in thickness. Bars

Bars are available in various widths, thicknesses, diameters, and lengths. The preferred practice is to specify widths in 1⁄4-in. increments and thickness and diameter in 1⁄8-in. increments. Plates

Defined according to rolling procedure: Sheared plates are rolled between horizontal rolls and trimmed (sheared or gas cut) on all edges. Universal (UM) plates are rolled between horizontal and vertical rolls and trimmed (sheared or gas cut) on ends only. Stripped plates are furnished to required widths by shearing or gas cutting from wider sheared plates. Sizes

Plate mills are located in various districts, but the sizes of plates produced differ greatly and the catalogs of individual mills should be consulted for detail data. The extreme width of UM plates currently rolled is 60 inches and for sheared plates it is 200 inches, but their availability together with limiting thickness and lengths should be checked with the mills before specifying. The preferred increments for width and thickness are: Widths:

Various. The catalogs of individual mills should be consulted to determine the most economical widths. Thickness: 1⁄32-in. increments up to 1⁄2-in. 1⁄ -in. increments over 1⁄ -in. to 1 in. 16 2 1⁄ -in. increments over 1 in. to 3 in. 8 1⁄ -in. increments over 3 in. 4 Ordering

Plate thickness may be specified in inches or by weight per square foot, but no decimal edge thickness can be assured by the latter method. Separate tolerance tables apply to each method. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 134


Table 1-7. Theoretical Weights of Rolled Floor Plates Gauge No.

Theoretical Weight per sq. ft lb

Nominal Thickness in.

18

2.40

1⁄ 8

16

3.00

3⁄

14 13 12

Nominal Thickness in.

Theoretical Weight per sq. ft, lb

6.16

1⁄ 2

21.47

8.71

9⁄ 16

24.02

3.75

1⁄ 4

11.26

5⁄ 8

26.58

4.50

5⁄

13.81

3⁄ 4

31.68

5.25

3⁄ 8

16.37

7⁄ 8

36.78

7⁄

18.92

1

41.89

16

16

16

Theoretical Weight per sq. ft, lb

Note: Thickness is measured near the edge of the plate, exclusive of raised pattern.

Invoicing

Standard practice is to invoice plates to the fabricator at theoretical weight at point of shipment. Permissible variations in weight are in accordance with the tables of ASTM Specification A6. All plates are invoiced at theoretical weight and, except as noted, are subject to the same weight variations which apply to rectangular plates. Odd shapes in most instances require gas cutting, for which gas cutting extras are applicable. All plates ordered gas cut for whatever reason, or beyond published shearing limits, take extras for gas cutting in addition to all other extras. Rolled steel bearing plates are often gas cut to prevent distortion due to shearing but would also take the regular extra for the thickness involved. Extras for thickness, width, length, cutting, quality and quantity, etc., which are added to the base price of plates, are subject to revision, and should be obtained by inquiry to the producer. The foregoing general statements are made as a guide toward economy in design. Floor Plates

Floor plates having raised patterns are available from several mills, each offering its own style of surface projections and in a variety of widths, thicknesses, and lengths. A maximum width of 96 inches and a maximum thickness of one inch are available, but availability of matching widths, thicknesses, and lengths should be checked with the producer. Floor plates are generally not specified to chemical composition limits or mechanical property requirements; a commercial grade of carbon steel is furnished. However, when strength or corrosion resistance is a consideration, raised pattern floor plates are procurable in any of the regular steel specifications. As in the case of plain plates, the individual manufacturers should be consulted for precise information. The nominal or ordered thickness is that of the flat plate, exclusive of the height or raised pattern. The usual weights are as shown in Table 1-7.


BARS AND PLATES

1 - 135

SQUARE AND ROUND BARS Weight and area Weight lb per ft

Area Sq in.

Size in. 0

Weight lb per ft

Area Sq in.

Size in.

1⁄ 16 1⁄ 8 3⁄ 16

0.013 0.053 0.120

0.010 0.042 0.094

0.0039 0.0156 0.0352

0.0031 0.0123 0.0276

1⁄ 16 1⁄ 8 3⁄ 16

30.63 31.91 33.23 34.57

24.05 25.07 26.10 27.15

9.000 9.379 9.766 10.160

7.069 7.366 7.670 7.980

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

0.213 0.332 0.479 0.651

0.167 0.261 0.376 0.512

0.0625 0.0977 0.1406 0.1914

0.0491 0.0767 0.1104 0.1503

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

35.94 37.34 38.76 40.21

28.23 29.32 30.44 31.58

10.563 10.973 11.391 11.816

8.296 8.618 8.946 9.281

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

0.851 1.077 1.329 1.608

0.668 0.846 1.044 1.263

0.2500 0.3164 0.3906 0.4727

0.1964 0.2485 0.3068 0.3712

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

41.68 43.19 44.71 46.27

32.74 33.92 35.12 36.34

12.250 12.691 13.141 13.598

9.621 9.968 10.321 10.680

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

1.914 2.246 2.605 2.991

1.503 1.764 2.046 2.349

0.5625 0.6602 0.7656 0.8789

0.4418 0.5185 0.6013 0.6903

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

47.85 49.46 51.09 52.76

37.58 38.85 40.13 41.43

14.063 14.535 15.016 15.504

11.045 11.416 11.793 12.177

1⁄ 16 1⁄ 8 3⁄ 16

3.403 3.841 4.307 4.798

2.673 3.017 3.382 3.769

1.0000 1.1289 1.2656 1.4102

0.7854 0.8866 0.9940 1.1075

1⁄ 16 1⁄ 8 3⁄ 16

54.44 56.16 57.90 59.67

42.76 44.11 45.47 46.86

16.000 16.504 17.016 17.535

12.566 12.962 13.364 13.772

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

5.317 5.862 6.433 7.032

4.176 4.604 5.053 5.523

1.5625 1.7227 1.8906 2.0664

1.2272 1.3530 1.4849 1.6230

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

61.46 63.28 65.13 67.01

48.27 49.70 51.15 52.63

18.063 18.598 19.141 19.691

14.186 14.607 15.033 15.466

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

7.656 8.308 8.985 9.690

6.013 6.525 7.057 7.610

2.2500 2.4414 2.6406 2.8477

1.7672 1.9175 2.0739 2.2365

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

68.91 70.83 72.79 74.77

54.12 55.63 57.17 58.72

20.250 20.816 21.391 21.973

15.904 16.349 16.800 17.257

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

10.421 11.179 11.963 12.774

8.185 8.780 9.396 10.032

3.0625 3.2852 3.5156 3.7539

2.4053 2.5802 2.7612 2.9483

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

76.78 78.81 80.87 82.96

60.30 61.90 63.51 65.15

22.563 23.160 23.766 24.379

17.721 18.190 18.666 19.147

1⁄ 16 1⁄ 8 3⁄ 16

13.611 14.475 15.366 16.283

10.690 11.369 12.068 12.789

4.0000 4.2539 4.5156 4.7852

3.1416 3.3410 3.5466 3.7583

1⁄ 16 1⁄ 8 3⁄ 16

85.07 87.21 89.38 91.57

66.81 68.49 70.20 71.92

25.000 25.629 26.266 26.910

19.635 20.129 20.629 21.135

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

17.227 18.197 19.194 20.217

13.530 14.292 15.075 15.879

5.0625 5.3477 5.6406 5.9414

3.9761 4.2000 4.4301 4.6664

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

93.79 96.04 98.31 100.61

73.66 75.43 77.21 79.02

27.563 28.223 28.891 29.566

21.648 22.166 22.691 23.221

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

21.267 22.344 23.447 24.577

16.703 17.549 18.415 19.303

6.2500 6.5664 6.8906 7.2227

4.9087 5.1573 5.4119 5.6727

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

102.93 105.29 107.67 110.07

80.84 82.69 84.56 86.45

30.250 30.941 31.641 32.348

23.758 24.301 24.851 25.406

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

25.734 26.917 28.126 29.362

20.211 21.140 22.090 23.061

7.5625 7.9102 8.2656 8.6289

5.9396 6.2126 6.4918 6.7771

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

112.50 114.96 117.45 119.96

88.36 90.29 92.24 94.22

33.063 33.785 34.516 35.254

25.967 26.535 27.109 27.688

1

2

3

4

5


1 - 136


SQUARE AND ROUND BARS Weight and area Weight lb per ft

Area Sq in.

Size in. 6

Weight lb per ft

Area Sq in.

Size in.

1⁄ 16 1⁄ 8 3⁄ 16

122.50 125.07 127.66 130.28

96.21 98.23 100.26 102.32

36.000 36.754 37.516 38.285

28.274 28.867 29.465 30.069

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

132.92 135.59 138.29 141.02

104.40 106.49 108.61 110.75

39.063 39.848 40.641 41.441

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

143.77 146.55 149.35 152.18

112.91 115.10 117.30 119.52

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

155.04 157.92 160.83 163.77

1⁄ 16 1⁄ 8 3⁄ 16

275.63 279.47 283.33 287.23

216.48 219.49 222.53 225.59

81.000 82.129 83.266 84.410

63.617 64.504 65.397 66.296

30.680 31.296 31.919 32.548

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

291.15 295.10 299.07 303.07

228.67 231.77 234.89 238.03

85.563 86.723 87.891 89.066

67.201 68.112 69.029 69.953

42.250 43.066 43.891 44.723

33.183 33.824 34.472 35.125

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

307.10 311.15 315.24 319.34

241.20 244.38 247.59 250.81

90.250 91.441 92.641 93.848

70.882 71.818 72.760 73.708

121.77 124.03 126.32 128.63

45.563 46.410 47.266 48.129

35.785 36.451 37.122 37.800

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

323.48 327.64 331.82 336.04

254.06 257.33 260.61 263.92

95.063 96.285 97.516 98.754

74.662 75.622 76.589 77.561

1⁄ 16 1⁄ 8 3⁄ 16

166.74 169.73 172.74 175.79

130.95 133.30 135.67 138.06

49.000 49.879 50.766 51.660

38.485 39.175 39.871 40.574

1⁄ 16 1⁄ 8 3⁄ 16

340.28 344.54 348.84 353.16

267.25 270.61 273.98 277.37

100.000 101.254 102.516 103.785

78.540 79.525 80.516 81.513

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

178.86 181.96 185.08 188.23

140.48 142.91 145.36 147.84

52.563 53.473 54.391 55.316

41.283 41.997 42.718 43.446

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

357.50 361.88 366.28 370.70

280.78 284.22 287.67 291.15

105.063 106.348 107.641 108.941

82.516 83.525 84.541 85.563

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

191.41 194.61 197.84 201.10

150.33 152.85 155.38 157.94

56.250 57.191 58.141 59.098

44.179 44.918 45.664 46.415

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

375.16 379.64 384.14 388.67

294.65 298.17 301.70 305.26

110.250 111.566 112.891 114.223

86.590 87.624 88.664 89.710

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

204.38 207.69 211.03 214.39

160.52 163.12 165.74 168.38

60.063 61.035 62.016 63.004

47.173 47.937 48.707 49.483

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

393.23 397.82 402.43 407.07

308.85 312.45 316.07 319.71

115.563 116.910 118.266 119.629

90.763 91.821 92.886 93.957

1⁄ 16 1⁄ 8 3⁄ 16

217.78 221.19 224.64 228.11

171.04 173.73 176.43 179.15

64.000 65.004 66.016 67.035

50.266 51.054 51.849 52.649

1⁄ 16 1⁄ 8 3⁄ 16

411.74 416.43 421.15 425.89

323.38 327.06 330.77 334.50

121.000 122.379 123.766 125.160

95.033 96.116 97.206 98.301

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

231.60 235.12 238.67 242.25

181.90 184.67 187.45 190.26

68.063 69.098 70.141 71.191

53.456 54.269 55.088 55.914

1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16

430.66 435.46 440.29 445.14

338.24 342.01 345.80 349.61

126.563 127.973 129.391 130.816

99.402 100.510 101.623 102.743

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

245.85 249.48 253.13 256.82

193.09 195.94 198.81 201.70

72.250 73.316 74.391 75.473

56.745 57.583 58.426 59.276

1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16

450.02 454.92 459.85 464.81

353.44 357.30 361.17 365.06

132.250 133.691 135.141 136.598

103.869 105.001 106.139 107.284

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

260.53 264.26 268.02 271.81

204.62 207.55 210.50 213.48

76.563 77.660 78.766 79.879

60.132 60.994 61.863 62.737

3⁄ 4 13⁄ 16 7⁄ 8 15⁄ 16

469.80 474.81 479.84 484.91

368.98 372.91 376.87 380.85

138.063 139.535 141.016 142.504

108.434 109.591 110.754 111.923

490.00

384.85

144.000

113.098

7

8

9

10

11

12


BARS AND PLATES

1 - 137

AREA OF RECTANGULAR SECTIONS Square inches Width in.

Thickness, inches 3⁄ 16

1⁄ 4

5⁄ 16

3⁄ 8

7⁄ 16

1⁄ 2

9⁄ 16

5⁄ 8

11⁄ 16

3⁄ 4

13⁄ 16

1

0.047 0.093 0.141 0.188

0.063 0.125 0.188 0.250

0.078 0.156 0.234 0.313

0.094 0.188 0.281 0.375

0.109 0.219 0.328 0.438

0.125 0.250 0.375 0.500

0.141 0.281 0.422 0.563

0.156 0.313 0.469 0.625

0.172 0.344 0.516 0.688

0.188 0.375 0.563 0.75

0.203 0.406 0.609 0.813

0.219 0.438 0.656 0.875

0.234 0.469 0.703 0.938

0.250 0.500 0.750 1.00

11⁄4 11⁄2 13⁄4 2

0.234 0.281 0.328 0.375

0.313 0.375 0.438 0.500

0.391 0.469 0.547 0.625

0.469 0.563 0.656 0.750

0.547 0.656 0.766 0.875

0.625 0.750 0.875 1.00

0.703 0.844 0.984 1.13

0.781 0.938 1.09 1.25

0.859 1.03 1.20 1.38

0.938 1.13 1.31 1.50

1.02 1.22 1.42 1.63

1.09 1.31 1.53 1.75

1.17 1.41 1.64 1.88

1.25 1.50 1.75 2.00

21⁄4 21⁄2 23⁄4 3

0.422 0.469 0.516 0.563

0.563 0.625 0.688 0.750

0.703 0.781 0.859 0.938

0.844 0.938 1.03 1.13

0.984 1.09 1.20 1.31

1.13 1.25 1.38 1.50

1.27 1.41 1.55 1.69

1.41 1.56 1.72 1.88

1.55 1.72 1.89 2.06

1.69 1.88 2.06 2.25

1.83 2.03 2.23 2.44

1.97 2.19 2.41 2.63

2.11 2.34 2.58 2.81

2.25 2.50 2.75 3.00

31⁄4 31⁄2 33⁄4 4

0.609 0.656 0.703 0.750

0.813 0.875 0.938 1.00

1.02 1.09 1.17 1.25

1.22 1.31 1.41 1.50

1.42 1.53 1.64 1.75

1.63 1.75 1.88 2.00

1.83 1.97 2.11 2.25

2.03 2.19 2.34 2.50

2.23 2.41 2.58 2.75

2.44 2.63 2.81 3.00

2.64 2.84 3.05 3.25

2.84 3.06 3.28 3.50

3.05 3.28 3.52 3.75

3.25 3.50 3.75 4.00

41⁄4 41⁄2 43⁄4 5

0.797 0.844 0.891 0.938

1.06 1.13 1.19 1.25

1.33 1.41 1.48 1.56

1.59 1.69 1.78 1.88

1.86 1.97 2.08 2.19

2.13 2.25 2.38 2.50

2.39 2.53 2.67 2.81

2.66 2.81 2.97 3.13

2.92 3.09 3.27 3.44

3.19 3.38 3.56 3.75

3.45 3.66 3.86 4.06

3.72 3.94 4.16 4.38

3.98 4.22 4.45 4.69

4.25 4.50 4.75 5.00

51⁄4 51⁄2 53⁄4 6

0.984 1.03 1.08 1.13

1.31 1.38 1.44 1.50

1.64 1.72 1.80 1.88

1.97 2.06 2.16 2.25

2.30 2.41 2.52 2.63

2.63 2.75 2.88 3.00

2.95 3.09 3.23 3.38

3.28 3.44 3.59 3.75

3.61 3.78 3.95 4.13

3.94 4.13 4.31 4.50

4.27 4.47 4.67 4.88

4.59 4.81 5.03 5.25

4.92 5.16 5.39 5.63

5.25 5.50 5.75 6.00

61⁄4 61⁄2 63⁄4 7

1.17 1.22 1.27 1.31

1.56 1.63 1.69 1.75

1.95 2.03 2.11 2.19

2.34 2.44 2.53 2.63

2.73 2.84 2.95 3.06

3.13 3.25 3.38 3.50

3.52 3.66 3.80 3.94

3.91 4.06 4.22 4.38

4.30 4.47 4.64 4.81

4.69 4.88 5.06 5.25

5.08 5.28 5.48 5.69

5.47 5.69 5.91 6.13

5.86 6.09 6.33 6.56

6.25 6.50 6.75 7.00

71⁄4 71⁄2 73⁄4 8

1.36 1.41 1.45 1.50

1.81 1.88 1.94 2.00

2.27 2.34 2.42 2.50

2.72 2.81 2.91 3.00

3.17 3.28 3.39 3.50

3.63 3.75 3.88 4.00

4.08 4.22 4.36 4.50

4.53 4.69 4.84 5.00

4.98 5.16 5.33 5.50

5.44 5.63 5.81 6.00

5.89 6.09 6.30 6.50

6.34 6.56 6.78 7.00

6.80 7.03 7.27 7.50

7.25 7.50 7.75 8.00

81⁄2 9

1.59 1.69

2.13 2.25

2.66 2.81

3.19 3.38

3.72 3.94

4.25 4.50

4.78 5.06

5.31 5.63

5.84 6.19

6.38 6.75

6.91 7.31

7.44 7.88

7.97 8.44

8.50 9.00

91⁄2 10

1.78 1.88

2.38 2.50

2.97 3.13

3.56 3.75

4.16 4.38

4.75 5.00

5.34 5.63

5.94 6.25

6.53 6.88

7.13 7.50

7.72 8.13

8.31 8.75

8.91 9.38

9.50 10.0

101⁄2 11

1.97 2.06

2.63 2.75

3.28 3.44

3.94 4.13

4.59 4.81

5.25 5.50

5.91 6.19

6.56 6.88

7.22 7.56

7.88 8.25

8.53 8.94

9.19 9.63

9.84 10.3

10.5 11.0

111⁄2 12

2.16 2.25

2.88 3.00

3.59 3.75

4.31 4.50

5.03 5.25

5.75 6.00

6.47 6.75

7.19 7.50

7.91 8.63 8.25 9.00

9.34 9.75

10.8 11.3

11.5 12.0

1⁄ 4 1⁄ 2 3⁄ 4


7⁄ 8

10.1 10.5

15⁄ 16

1

1 - 138


WEIGHT OF RECTANGULAR SECTIONS Pounds per linear foot Width in.

Thickness, inches 3⁄ 16

1⁄ 4

5⁄ 16

3⁄ 8

7⁄ 16

1⁄ 2

9⁄ 16

5⁄ 8

11⁄ 16

3⁄ 4

13⁄ 16

7⁄ 8

15⁄ 16

1

0.160 0.319 0.479 0.638

0.213 0.425 0.638 0.851

0.266 0.532 0.798 1.06

0.319 0.638 0.957 1.28

0.372 0.744 1.12 1.49

0.425 0.851 1.28 1.70

0.479 0.957 1.44 1.91

0.532 1.06 1.60 2.13

0.585 1.17 1.75 2.34

0.638 1.28 1.91 2.55

0.691 1.38 2.07 2.76

0.744 1.49 2.23 2.98

0.798 1.60 2.39 3.19

0.851 1.70 2.55 3.40

11⁄4 11⁄2 13⁄4 2

0.798 0.957 1.12 1.28

1.06 1.28 1.49 1.70

1.33 1.60 1.86 2.13

1.60 1.91 2.23 2.55

1.86 2.23 2.61 2.98

2.13 2.55 2.98 3.40

2.39 2.87 3.35 3.83

2.66 3.19 3.72 4.25

2.92 3.51 4.09 4.68

3.19 3.83 4.47 5.10

3.46 4.15 4.84 5.53

3.72 4.47 5.21 5.95

3.99 4.79 5.58 6.38

4.25 5.10 5.95 6.81

21⁄4 21⁄2 23⁄4 3

1.44 1.60 1.75 1.91

1.91 2.13 2.34 2.55

2.39 2.66 2.92 3.19

2.87 3.19 3.51 3.83

3.35 3.72 4.09 4.47

3.83 4.25 4.68 5.10

4.31 4.79 5.26 5.74

4.79 5.32 5.85 6.38

5.26 5.85 6.43 7.02

5.74 6.38 7.02 7.66

6.22 6.91 7.60 8.29

6.70 7.44 8.19 8.93

7.18 7.98 8.77 9.57

7.66 8.51 9.36 10.2

31⁄4 31⁄2 33⁄4 4

2.07 2.23 2.39 2.55

2.76 2.98 3.19 3.40

3.46 3.72 3.99 4.25

4.15 4.47 4.79 5.10

4.84 5.21 5.58 5.95

5.53 5.95 6.38 6.81

6.22 6.70 7.18 7.66

6.91 7.44 7.98 8.51

7.60 8.19 8.77 9.36

8.29 8.93 9.57 10.2

8.99 9.68 10.4 11.1

9.68 10.4 11.2 11.9

10.4 11.2 12.0 12.8

11.1 11.9 12.8 13.6

41⁄4 41⁄2 43⁄4 5

2.71 2.87 3.03 3.19

3.62 3.83 4.04 4.25

4.52 4.79 5.05 5.32

5.42 5.74 6.06 6.38

6.33 6.70 7.07 7.44

7.23 7.66 8.08 8.51

8.13 8.61 9.09 9.57

9.04 9.57 10.1 10.6

9.94 10.5 11.1 11.7

10.8 11.5 12.1 12.8

11.8 12.4 13.1 13.8

12.7 13.4 14.1 14.9

13.6 14.4 15.2 16.0

14.5 15.3 16.2 17.0

51⁄4 51⁄2 53⁄4 6

3.35 3.51 3.67 3.83

4.47 4.68 4.89 5.10

5.58 5.85 6.11 6.38

6.70 7.02 7.34 7.66

7.82 8.19 8.56 8.93

8.93 9.36 9.78 10.2

10.0 10.5 11.0 11.5

11.2 11.7 12.2 12.8

12.3 12.9 13.5 14.0

13.4 14.0 14.7 15.3

14.5 15.2 15.9 16.6

15.6 16.4 17.1 17.9

16.7 17.5 18.3 19.1

17.9 18.7 19.6 20.4

61⁄4 61⁄2 63⁄4 7

3.99 4.15 4.31 4.47

5.32 5.53 5.74 5.95

6.65 6.91 7.18 7.44

7.98 8.29 8.61 8.93

9.30 9.68 10.0 10.4

10.6 11.1 11.5 11.9

12.0 12.4 12.9 13.4

13.3 13.8 14.4 14.9

14.6 15.2 15.8 16.4

16.0 16.6 17.2 17.9

17.3 18.0 18.7 19.4

18.6 19.4 20.1 20.8

19.9 20.7 21.5 22.3

21.3 22.1 23.0 23.8

71⁄4 71⁄2 73⁄4 8

4.63 4.79 4.94 5.10

6.17 6.38 6.59 6.81

7.71 7.98 8.24 8.51

9.25 9.57 9.89 10.2

10.8 11.2 11.5 11.9

12.3 12.8 13.2 13.6

13.9 14.4 14.8 15.3

15.4 16.0 16.5 17.0

17.0 17.5 18.1 18.7

18.5 19.1 19.8 20.4

20.0 20.7 21.4 22.1

21.6 22.3 23.1 23.8

23.1 23.9 24.7 25.5

24.7 25.5 26.4 27.2

81⁄2 9

5.42 5.74

7.23 7.66

9.04 9.57

10.8 11.5

12.7 13.4

14.5 15.3

16.3 17.2

18.1 19.1

19.9 21.1

21.7 23.0

23.5 24.9

25.3 26.8

27.1 28.7

28.9 30.6

91⁄2 10

6.06 6.38

8.08 8.51

10.1 10.6

12.1 12.8

14.1 14.9

16.2 17.0

18.2 19.1

20.2 21.3

22.2 23.4

24.2 25.5

26.3 27.6

28.3 29.8

30.3 31.9

32.3 34.0

101⁄2 11

6.70 7.02

8.93 9.36

11.2 11.7

13.4 14.0

15.6 16.4

17.9 18.7

20.1 21.1

22.3 23.4

24.6 25.7

26.8 28.1

29.0 30.4

31.3 32.8

33.5 35.1

35.7 37.4

111⁄2 12

7.34 7.66

9.78 10.2

12.2 12.8

14.7 15.3

17.1 17.9

19.6 20.4

22.0 23.0

24.5 25.5

26.9 28.1

29.3 30.6

31.8 33.2

34.2 35.7

36.7 38.3

39.1 40.8

1⁄ 4 1⁄ 2 3⁄ 4


1

CRANE RAILS

1 - 139

CRANE RAILS General Notes

The ASCE rails and the 104- to 175-lb crane rails shown in Figure 1-2 are recommended for crane runway use. For complete details and for profiles and properties of rails not listed, consult manufacturers’ catalogs. Rails should be arranged so that joints on opposite sides of the crane runway will be staggered with respect to each other and with due consideration to the wheelbase of the crane. Rail joints should not occur at crane girder splices. Light 40-lb rails are available in 30-ft lengths, 60-lb rails in 30-, 33- or 39-ft lengths, standard rails in 33- or 39-ft lengths and crane rails up to 80 ft. Consult manufacturer for availability of other lengths. Odd lengths, which must be included to complete a run or obtain the necessary stagger, should be not less than 10 feet long. For crane rail service, 40-lb rails are furnished to manufacturers’ specifications and tolerances. 60- and 85-lb rails are furnished to manufacturers’ specifications and tolerances, or to ASTM A1. Crane rails are furnished to ASTM A759. Rails will be furnished with standard drilling in both standard and odd lengths unless stipulated otherwise on order. For controlled cooling, heat treatment, and rail end preparation, see manufacturers’ catalogs. Purchase orders for crane rails should be noted “For crane service.” (See Table 1-8.) For maximum wheel loadings see manufacturers’ catalogs. Splices

Bolted Splices

It is often more desirable to use properly installed and maintained bolted splice bars in making up rail joints for crane service than welded splice bars. Standard rail drilling and joint-bar punching, as furnished by manufacturers of light standard rails for track work, include round holes in rail ends and slotted holes in joint bars to receive standard oval-neck tack bolts. Holes in rails are oversize and punching in joint bars is spaced to allow 1⁄16- to 1⁄8-in. clearance between rail ends (see manufacturers’ catalogs for spacing and dimensions of holes and slots). Although this construction is satisfactory for track and light crane service, its use in general crane service may lead to joint failure. For best service in bolted splices, it is recommended that tight joints be stipulated for all rails for crane service. This will require rail ends to be finished by milling or grinding, and the special rail drilling and joint-bar punching tabulated below. Special rail drilling is accepted by some mills, or rails may be ordered blank for shop drilling. End finishing of standard rails can be done at the mill; light rails must be end-finished in the fabricating shop or ground at the site prior to erection. In the crane rail range, from 104 to 175 lbs per yard, rails and joint bars are manufactured to obtain a tight fit and no further special end finishing, drilling, or punching is required. Because of cumulative tolerance variations in holes, bolt diameters, and rail ends, a slight gap may sometimes occur in the so-called tight joints. Conversely, it may sometimes be necessary to ream holes through joined bar and rail to permit entry of bolts. Joint bars for crane service are provided in various sections to match the rails. Joint bars for light and standard rails may be purchased blank for special shop punching to obtain tight joints. See Bethlehem Steel Corp. Booklet 3351 for dimensions, material specifications, and the identification necessary to match the crane rail section. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 140


c

Bars can be sheared off r

13°

c of g y

13°

1/ 4 Rad.

R

X

X

h

1/ 4 Rad.

t

1/ 2 Rad.

d

1/ 2 Rad.

h

g

13°

13°

m

m

n

b

A.S.C.E. 40, 60 & 85 lb. c

BETHLEHEM 104 lb. c 4

3 (approx.)

12°

13° 3 4

Rad.

3 4

Rad.

7/ 8

7/ 8 Rad.

h

13°

h

12° m

BETHLEHEM 135 lb.

Rad.

m

BETHLEHEM 171 lb.

c 4 1/32 (approx.)

12°

2 Rad.

11/8 Rad. 12°

h 2 53/64 m

BETHLEHEM 175 lb. Nomenclature of sketch for A.S.C.E. rails also applies to the other sections.

Fig. 1-2. Crane rails. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

CRANE RAILS

1 - 141

Table 1-8. Crane Rails Dimensions and Properties

Type

Nominal Wt. per Classi- Yd. d fication lb in.

Sx

Gage g

b

m

n

c

r

t

h

R

Area

lx

in.

in.

in.

in.

in.

in.

in.

in.

in.

in.2

in.4

17⁄ 32 5⁄ 8 11⁄ 16 48⁄ 64 13⁄ 16 7⁄ 8 57⁄ 64 31⁄ 32 11⁄16 11⁄16 11⁄4 19⁄64

11⁄ 64 7⁄ 32 1⁄ 4 9⁄ 32 9⁄ 32 19⁄ 64 19⁄ 64 10⁄ 32 1⁄ 2 15⁄ 32 5⁄ 8 1⁄ 2

111⁄16

12

12

3.00 4.10 2.55

12

155⁄64

12

3.94 6.54 3.59 3.89 1.68

21⁄8

12

23⁄8

12

27⁄16

12

21⁄2

12

29⁄16

12

23⁄4

12

21⁄ 64 25⁄ 64 7⁄ 16 31⁄ 64 33⁄ 64 35⁄ 64 9⁄ 16 9⁄ 16

123⁄32

17⁄8

21⁄2

12

1

ASCE

Light

30

31⁄8

125⁄64

31⁄8

ASCE

Light

40

31⁄2

171⁄128

31⁄2

ASCE

Light

50

37⁄8

123⁄32

37⁄8

ASCE

Light

60

41⁄4 1115⁄128 41⁄4

ASCE

70

45⁄8

23⁄64

45⁄8

ASCE

80

5

23⁄16

5

ASCE

Std.

85

53⁄16

217⁄64

53⁄16

ASCE

Std.

100

53⁄4

265⁄128

53⁄4

Bethlehem

Crane

104

5

27⁄16

5

Bethlehem

Crane

135

53⁄4

215⁄32

53⁄16

Bethlehem

Crane

171

6

25⁄8

6

Bethlehem

Crane

175

6

221⁄32

6

Hd. Base

y

in.3

in.3

in.

—

—

21⁄16

12

4.90 10.1

5.10

217⁄64

12

5.93 14.6

6.64 7.12 2.05

215⁄32

12

6.81 19.7

8.19 8.87 2.22

25⁄8

12

7.86 26.4 10.1 11.1 2.38

23⁄4

12

8.33 30.1 11.1 12.2 2.47

25⁄64

12

9.84 44.0 14.6 16.1 2.73

27⁄16

31⁄2

10.3 29.8 10.7 13.5 2.21

12

13.3 50.8 17.3 18.1 2.81

—

1.88

37⁄16

14

11⁄4 213⁄16

4.3

Flat

11⁄4

23⁄4 Vert. 16.8 73.4 24.5 24.4 3.01

41⁄4

18

11⁄2

37⁄64 Vert. 17.1 70.5 23.4 23.6 2.98

Joint-bar bolts, as distinguished from oval-neck track bolts, have straight shanks to the head and are manufactured to ASTM A449 specifications. Nuts are manufactured to ASTM A563 Gr. B specifications. ASTM A325 bolts and nuts may be used. Bolt assembly includes an alloy steel spring washer, furnished to AREA specifications. After installation, bolts should be retightened within 30 days and every three months thereafter. Welded Splices

When welded splices are specified, consult the manufacturer for recommended rail-end preparation, welding procedure, and method of ordering. Although joint continuity, made possible by this method of splicing, is desirable, it should be noted that the careful control required in all stages of the welding operation may be difficult to meet during crane rail installation. Rails should not be attached to structural supports by welding. Rails with holes for joint bar bolts should not be used in making welded splices. Fastenings

Hook Bolts

Hook bolts (Figure 1-3) are used primarily with light rails when attached to beams too narrow for clamps. Rail adjustment to ±1⁄2-in. is inherent in the threaded shank. Hook bolts are paired alternately three to four inches apart, spaced at about 24-in. centers. The special rail drilling required must be done at the fabricator’s shop. Hook bolts are not recommended for use with heavy duty cycle cranes (CMAA Classes, D, E, and F). It is generally recommended that hook bolts should not be used in runway systems which are longer than 500 feet because the bolts do not allow for longitudinal movement of the rail. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 142


Table 1-9. Splices for Tight Joints

g

A

B

C

C

B

D

C

B

L

Rail End

Joint Bar

l

l

Grip

Grip H

H

G Cut when specified 40-60-85-104

Rail

Joint Bar

Drilling Wt. Per Yard

g

lb

in.

40 171⁄128 60 1115⁄128 85 217⁄64 104

27⁄16

135 215⁄32 171

25⁄8

175

221⁄32

Hole Dia. A in. 13⁄ * 16 13⁄ * 16 15⁄ * 16 11⁄16 13⁄16 13⁄16 13⁄16

105-135-171-175

B C

in. in. in. 21⁄2 5

—

21⁄2 5

—

21⁄2 5

—

4

5

6

4

5

6

4

5

6

4

Bolt

Washer

Punching

5

6

Wt. 2 Bars Bolts, Nuts, Washers

ThickIn- ness side and With Less Dia. Width Fig. Fig.

Hole Dia.

D

B C

L

G

Dia.

Grip

I

H

in.

in.

in. in. in.

in.

in.

in.

in.

in.

in.

in.

lb

lb

13⁄ * 16 13⁄ * 16 15⁄ * 16 11⁄16 13⁄16 13⁄16 13⁄16

415⁄16*

5

— 20 23⁄16

3⁄

4

115⁄16

31⁄2

21⁄2

16.5

5

— 24 211⁄16

3⁄

4

219⁄32

4

211⁄16

36.5

29.6

415⁄16*

5

— 24 311⁄32

7⁄

8

35⁄32

43⁄4

33⁄16

56.6

45.3

715⁄16

5

6

34

31⁄2

1

31⁄2

51⁄4

31⁄2

715⁄16

5

6

34

—

11⁄8

35⁄8

51⁄2

311⁄16

715⁄16

5

6

34

—

11⁄8

47⁄16

61⁄4

41⁄16

—

11⁄8

41⁄8

61⁄4

315⁄16

7⁄ ×3⁄ 16 8 7⁄ ×3⁄ 16 8 7⁄ ×3⁄ 16 8 7⁄ ×1⁄ 16 2 7⁄ ×1⁄ 16 2 7⁄ ×1⁄ 16 2 7⁄ ×1⁄ 16 2

20.0

415⁄16*

13⁄ 16 13⁄ 16 15⁄ 16 11⁄16 13⁄16 13⁄16 13⁄16

715⁄16

5

6

34

73.5

55.4

—

75.3

—

90.8

—

87.7

*Special rail drilling and joint-bar punching.

Rail Clips

Rail clips are forged or cast devices which are shaped to match specific rail profiles. They are usually bolted to the runway girder flange with one bolt or are sometimes welded. Rail clips have been used satisfactorily with all classes of cranes. However, one drawback is that when a single bolt is used the clip can rotate in response to rail longitudinal movement. This clip rotation can cause a camming action, thus forcing the rail out of alignment. Because of this limitation, rail clips should only be used in crane systems subject to infrequent use, and for runways less than 500 feet in length. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

CRANE RAILS

1 - 143

Rail Clamps

Rail clamps are a common method of attachment for heavy duty cycle cranes. Rail clamps are detailed to provide two types: tight and floating (Figure 1-4). Each clamp consists of two plates: an upper clamp plate and a lower filler plate. The lower plate is flat and roughly matches the height of the toe of the rail flange. The upper plate covers the lower plate and extends over the top of the lower rail flange. In the tight clamp the upper plate is detailed to fit tightly to the lower tail flange top, thus “clamping” it tightly in place when the fasteners are tightened. In the past, the tight clamp had been illustrated with the filler plates fitted tightly against the rail flange toe. This tight fit-up was rarely achieved in practice and is not considered to be necessary to achieve a tight type clamp. In the floating type clamp, the pieces are detailed to provide a clearance both alongside the rail flange toe and below the upper plate. The floating type does not, in reality, clamp the rail but merely holds the rail within the limits of the clamp clearances.

Fig. 1-3. Hook bolts. Reversible 1 1/2 fillers

Reversible fillers

Clamp plates

3

Off-center punching

11/2

Clamp plates

3

Off-center punching

11/2

1 1/2

Rail base + 1/ 4

( 1/2 to 9/16 ) "float"

Max. adjustment

Self-locking nut or nut and lock washer Filler Machine bolt Gage

Gage

Tight clamp

Floating clamp

Fig. 1-4. Rail clamps. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 144


High strength bolts are recommended for both clamp types. Both types should be spaced three feet or less apart. Dimensions shown above are suggested. See manufacturers’ catalogs for recommended gages, bolt sizes, and detail dimensions not shown. Patented Rail Clips

Each manufacturer’s literature presents in detail the desirable aspects of the various designs. In general patented rail clips are easy to install due to their range of adjustment while providing the proper limitations of lateral movement and allowance for longitudinal movement. Patented rail clips should be considered as a viable alternative to conventional hook bolts, clips, or clamps. Because of their desirable characteristics, patented rail clips can be used without restriction except as limited by the specific manufacturer’s recommendations. Installations using patented rail clips sometimes incorporate pads beneath the rail. When this is done the lateral float of the rail should be limited as in the case of the tight rail clamps.


TORSION PROPERTIES

1 - 145

TORSION PROPERTIES

Torsional analysis is not required for the routine design of most structural steel members. When torsional analysis is required, the Table of Torsion Properties will be of assistance in utilizing current analysis methods. The reader is referred to the AISC publication Torsional Analysis of Steel Members (American Institute of Steel Construction, 1983) for additional information and appropriate design aids. Torsion Properties are also required to determine the design compressive strength for torsional and flexural-torsional buckling as specified in the AISC LRFD Specification Appendix E3. Nomenclature

= warping constant for section, in.6* = modulus of elasticity of steel (29,000 ksi) = shear modulus of elasticity of steel (11,200 ksi) = flexural constant in Equation E3-1, LRFD Specification = torsional constant for a section, in.4 = statical moment for a point in the flange directly above the vertical edge of the web, in.3 3 _Qw = statical moment at mid-depth of the section, in. ro = polar radius of gyration about the shear center, in. Sw = warping statical moment at a point in the section, in.4 Wno = normalized warping function at a point at the flange edge, in.2 Cw E G H J Qf

*Calculated values of Cw are given for all tabulated shapes. However, for many angles and T shapes, Cw is so small that for practical purposes it can be taken as zero. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 146


TORSION PROPERTIES W shapes

 √ EC w GJ

Normalized Warping Constant Wno

Warping Statical Moment Sw

Qf

Qw

in.

in.2

in.4

in.3

in.3

137 153 167 190

168 166 165 164

1190 1040 922 789

282 251 225 194

811 709 636 551

989000 649000 577000 511000 446000 397000 378000 333000 283000 245000 189000

75.4 95.1 103 110 121 130 138 151 173 187 209

166 158 156 154 152 151 151 149 149 148 147

2240 1540 1380 1240 1100 986 940 836 714 621 481

484 354 323 294 264 240 230 208 179 157 119

1380 992 894 813 730 665 624 560 481 434 364

393000 306000 242000 192000 181000 161000 140000 119000 99300 79600

60.6 67.9 76.8 87.6 91.3 101 109 125 136 147

125 121 118 115 114 113 112 111 111 110

1160 940 762 622 589 530 468 402 336 270

322 272 228 192 184 168 151 134 113 92.0

1030 856 715 596 566 506 453 391 346 299

1620000 1480000 1090000 816000 637000 554000 493000 441000 398000 366000 330000 306000 282000

57.5 59.8 68.6 80.0 92.0 100 108 116 127 134 143 151 160

172 169 162 156 152 150 148 146 146 145 144 143 143

3530 3270 2520 1960 1570 1390 1240 1130 1020 944 858 799 740

674 634 513 415 344 309 281 258 235 219 200 187 175

1910 1790 1420 1130 928 830 757 691 628 585 538 505 472

168000 148000 128000 116000 107000 98500 90200 82200 68100

90.3 98.1 109 116 123 130 137 145 159

109 108 108 107 106 105 105 105 104

576 512 446 407 378 349 321 294 245

408000 357000 319000 281000 250000 224000 198000

95.8 105 113 122 134 145 158

135 133 132 130 130 129 128

1130 1000 906 808 721 650 580

Torsional Constant J

Warping Constant Cw

Designation

in.4

in.6

W44×335 W ×290 W ×262 W ×230

74.4 51.5 37.7 24.9

536000 463000 406000 346000

W40×593 W ×503 W ×431 W ×372 W ×321 W ×297 W ×277 W ×249 W ×215 W ×199 W ×174

451 186 142 109 79.4 61.2 51.1 37.7 24.4 18.1 11.2

W40×466 W ×392 W ×331 W ×278 W ×264 W ×235 W ×211 W ×183 W ×167 W ×149

277 172 106 64.7 56.1 41.3 30.4 19.6 14.0 9.60

W36×848 W ×798 W ×650 W ×527 W ×439 W ×393 W ×359 W ×328 W ×300 W ×280 W ×260 W ×245 W ×230 W36×256 W ×232 W ×210 W ×194 W ×182 W ×170 W ×160 W ×150 W ×135 W33×354 W ×318 W ×291 W ×263 W ×241 W ×221 W ×201 W33×169 W ×152 W ×141 W ×130 W ×118

1270 1070 600 330 195 143 109 84.5 64.2 52.6 41.5 34.6 28.6 53.3 39.8 28.0 22.2 18.4 15.1 12.4 10.1 6.99 115 84.4 65.0 48.5 35.8 27.5 20.5 17.7 12.4 9.70 7.37 5.30

82400 71700 64400 56600 48300

110 122 131 141 154

93.7 93.8 93.3 92.8 92.2


329 286 258 228 196

Statical Moment

176 159 138 128 120 111 103 95.1 79.9

520 468 416 383 359 334 312 291 255

263 237 216 195 174 158 142

709 634 577 519 469 428 386

109 95.1 86.5 76.9 66.6

314 279 257 233 207

TORSION PROPERTIES

1 - 147


Designation W30×477 W ×391 W ×326 W ×292 W ×261 W ×235 W ×211 W ×191 W ×173 W30×148 W ×132 W ×124 W ×116 W ×108 W ×99 W ×90 W27×539 W ×448 W ×368 W ×307 W ×258 W ×235 W ×217 W ×194 W ×178 W ×161 W ×146

 √ EC w GJ



Qf

Qw

in.6

in.

in.2

in.4

in.3

in.3

480000 364000 286000 249000 215000 190000 166000 146000 129000

63.6 73.6 84.8 92.8 102 111 124 135 148

329 268 223 200 177 160 141 126 113

896 716 595 530 470 422 374 337 303

49400 42100 38600 34900 30900 26800 24000

93.6 106 112 119 127 136 146

77.3 77.3 76.9 76.5 76.1 75.7 75.0

239 204 188 171 152 133 119

86.8 74.0 68.8 62.8 56.1 49.5 45.0

250 219 204 189 173 156 142

440000 336000 254000 199000 159000 140000 128000 111000 98300 87300 77200

47.8 54.1 62.4 71.4 82.2 88.5 94.6 104 114 124 135

111 106 102 99.4 98.2 96.0 95.0 93.9 93.7 92.9 92.2

1490 1190 930 750 613 548 503 442 393 352 314

342 283 231 192 161 146 135 120 107 96.6 87.0

940 766 620 511 424 384 354 314 284 256 231


Warping Constant Cw

in.4 307 174 103 74.9 53.8 40.0 27.9 20.6 15.3 14.6 9.72 7.99 6.43 4.99 3.77 2.92 499 297 169 101 61.0 46.3 37.0 26.5 19.5 14.7 10.9

124 120 117 115 114 112 112 111 110

1450 1140 919 812 710 633 556 494 439

Statical Moment

W27×129 W ×114 W ×102 W ×94 W ×84

11.2 7.33 5.29 4.03 2.81

32500 27600 24000 21300 17900

86.7 98.7 108 117 128

66.4 66.4 65.7 65.4 64.9

183 155 137 122 103

69.5 59.2 52.7 47.3 40.6

197 171 153 139 122

W24×492 W ×408 W ×335 W ×279 W ×250 W ×229 W ×207 W ×192 W ×176 W ×162 W ×146 W ×131 W ×117 W ×104

456 271 154 91.7 67.3 51.8 38.6 31.0 24.1 18.5 13.4 9.50 6.72 4.72

283000 214000 160000 125000 108000 95800 83900 76200 68400 62600 54600 47100 40800 35200

40.1 45.2 51.9 59.4 64.5 69.2 75.0 79.8 85.7 93.6 103 113 125 139

92.1 88.1 84.6 82.0 80.6 79.6 78.5 77.7 77.0 77.0 76.3 75.6 74.9 74.3

1150 909 709 570 502 451 401 367 333 304 268 233 204 178

281 233 189 157 141 128 116 107 97.8 89.4 79.5 69.7 61.5 54.1

774 626 509 418 372 338 303 280 255 234 209 185 164 144

W24×103 W ×94 W ×84 W ×76 W ×68

7.10 5.26 3.70 2.68 1.87

16600 15000 12800 11100 9430

77.8 85.9 94.6 104 114

53.0 53.1 52.6 52.2 51.9

117 105 91.3 79.8 68.0

49.4 44.4 39.0 34.4 29.5

140 127 112 100 88.3

W24×62 W ×55

1.71 1.18

4620 3870

83.6 92.2

40.7 40.4

42.3 35.7

23.2 19.8

76.6 67.1


1 - 148



 √ EC w GJ



Qf

Qw

in.6

in.

in.2

in.4

in.3

in.3

41.3 31.1 23.9 15.4 11.3 8.98 6.83 5.21

61800 54300 48500 41100 36000 32700 29200 26200

62.2 67.2 72.5 83.1 90.8 97.1 105 114

67.0 66.0 65.6 65.4 64.7 64.2 63.7 63.2

345 307 277 235 208 191 172 155

W21×93 W ×83 W ×73 W ×68 W ×62

6.03 4.34 3.02 2.45 1.83

9940 8630 7410 6760 5960

65.3 71.8 79.7 84.5 91.8

43.6 43.0 42.5 42.3 42.0

85.3 75.0 65.2 59.9 53.2

38.2 34.2 30.3 28.0 25.1

110 98.0 86.2 79.9 72.2

W21×57 W ×50 W ×44

1.77 1.14 0.77

3190 2570 2110

68.3 76.4 84.2

33.4 33.1 32.8

35.6 28.9 24.0

20.9 17.2 14.5

64.3 55.0 47.7

75700 65600 57400 49900 43200 37900 33200 28900 25700 22700

33.3 35.5 37.8 40.3 43.4 46.6 50.1 54.3 58.6 63.2

58.8 57.5 56.4 55.2 54.2 53.3 52.5 51.6 51.0 50.4

483 427 382 339 299 267 237 210 189 169


Warping Constant Cw

Designation

in.4

W21×201 W ×182 W ×166 W ×147 W ×132 W ×122 W ×111 W ×101

W18×311 W ×283 W ×258 W ×234 W ×211 W ×192 W ×175 W ×158 W ×143 W ×130

177 135 104 79.7 59.3 45.2 34.2 25.4 19.4 14.7

Statical Moment

102 92.3 84.4 71.4 64.0 59.2 53.7 49.0

141 127 116 105 94.3 85.7 77.2 69.4 63.2 57.1

265 238 216 187 167 154 139 127

376 338 306 274 245 221 199 178 161 145

W18×119 W ×106 W ×97 W ×86 W ×76

10.6 7.48 5.86 4.10 2.83

20300 17400 15800 13600 11700

70.4 77.6 83.6 92.7 103

50.4 49.8 49.4 48.9 48.4

151 131 120 104 90.7

50.6 44.6 41.2 36.3 31.9

131 115 105 92.8 81.4

W18×71 W ×65 W ×60 W ×55 W ×50

3.48 2.73 2.17 1.66 1.24

4700 4240 3850 3430 3040

59.1 63.4 67.8 73.1 79.7

33.7 33.4 33.1 32.9 32.6

52.1 47.5 43.5 39.0 34.9

25.8 23.8 22.1 19.9 18.0

72.7 66.6 61.4 55.9 50.4

W18×46 W ×40 W ×35

1.22 0.81 0.51

1710 1440 1140

60.2 67.8 76.1

26.4 26.1 25.9

24.2 20.6 16.5

15.3 13.3 10.7

45.3 39.2 33.2

W16×100 W ×89 W ×77 W ×67

7.73 5.45 3.57 2.39

11900 10200 8590 7300

63.1 69.6 78.9 88.9

41.7 41.1 40.6 40.1

107 93.3 79.3 68.2

39.0 34.4 29.7 25.9

99.0 87.3 75.0 64.9

W16×57 W ×50 W ×45 W ×40 W ×36

2.22 1.52 1.11 0.79 0.54

2660 2270 1990 1730 1460

55.7 62.2 68.1 75.3 83.7

28.0 27.6 27.4 27.1 26.9

35.6 30.8 27.2 23.9 20.2

19.0 16.7 15.0 13.4 11.4

52.6 46.0 41.1 36.5 32.0

W16×31 W ×26

0.46 0.26

739 565

64.5 75.0

21.3 21.1

13.0 10.0


9.17 7.20

27.0 22.1

TORSION PROPERTIES

1 - 149

TORSION PROPERTIES W shapes Torsional Constant J Designation W14×808 W ×730 W ×665 W ×605 W ×550 W ×500 W ×455 W14×426 W ×398 W ×370 W ×342 W ×311 W ×283 W ×257 W ×233 W ×211 W ×193 W ×176 W ×159 W ×145

4

in. 1860 1450 1120 870 670 514 395

331 273 222 178 136 104 79.1 59.5 44.6 34.8 26.5 19.8 15.2

Warping Constant Cw 6

in.

√

 EC w GJ

Normalized Warping Constant Wno 2

Warping Statical Moment Sw 4

Statical Moment

Qf 3

Qw in.3

in.

in.

in.

in.

433000 362000 305000 258000 219000 187000 160000

24.6 25.4 26.6 27.7 29.1 30.7 32.4

82.2 78.3 75.5 73.0 70.6 68.5 66.5

1950 1720 1510 1320 1160 1020 899

337 319 287 259 233 209 189

916 831 740 660 588 524 468

144000 129000 116000 103000 89100 77700 67800 59000 51500 45900 40500 35600 31700

33.6 35.0 36.8 38.7 41.2 44.0 47.1 50.7 54.7 58.4 62.9 68.2 73.5

65.3 64.1 62.9 61.6 60.3 59.1 57.9 56.9 55.9 55.1 54.4 53.7 53.0

827 756 689 623 553 493 438 389 345 312 279 248 224

176 163 151 138 125 113 102 91.7 82.3 75.4 68.0 61.3 55.8

434 401 368 336 301 271 243 218 195 177 160 143 130

190 171 154 138 125

49.9 45.3 41.2 37.2 33.7

117 106 95.9 86.6 78.3

W14×132 W ×120 W ×109 W ×99 W ×90

12.3 9.37 7.12 5.37 4.06

25500 22700 20200 18000 16000

73.3 79.2 85.7 93.2 101

50.2 49.7 49.1 48.7 48.3

W14×82 W ×74 W ×68 W ×61

5.08 3.88 3.02 2.20

6710 5990 5380 4710

58.5 63.2 67.9 74.5

34.1 33.7 33.4 33.1

73.8 66.6 60.4 53.3

28.1 25.7 23.5 21.0

69.3 62.8 57.3 51.1

W14×53 W ×48 W ×43

1.94 1.46 1.05

2540 2240 1950

58.2 63.0 69.3

26.7 26.5 26.2

35.5 31.6 27.8

17.3 15.6 13.9

43.6 39.2 34.8

W14×38 W ×34 W ×30

0.80 0.57 0.38

1230 1070 887

63.1 69.7 77.7

23.0 22.8 22.6

20.0 17.5 14.7

11.5 10.2 8.59

30.7 27.3 23.6

W14×26 W ×22

0.36 0.21

405 314

54.0 62.2

16.9 16.8

6.98 5.58

20.1 16.6

243 185 143 108 83.8 64.7 48.8 35.6 25.8 18.5 12.9 9.13 6.86 5.10 3.84 2.93 2.18

57000 48600 42000 35800 31200 27200 23600 20100 17200 14700 12400 10700 9410 8270 7330 6540 5780

24.6 26.1 27.6 29.3 31.0 33.0 35.4 38.2 41.5 45.4 49.9 55.1 59.6 64.8 70.3 76.0 82.9

46.4 45.0 44.0 42.8 41.8 41.0 40.1 39.2 38.4 37.7 37.0 36.4 35.9 35.5 35.2 34.9 34.5

W12×336 W ×305 W ×279 W ×252 W ×230 W ×210 W ×190 W ×170 W ×152 W ×136 W ×120 W ×106 W ×96 W ×87 W ×79 W ×72 W ×65

8.94 7.02 459 403 357 313 279 249 220 192 168 146 126 110 98.2 87.2 78.1 70.3 62.7


119 107 96.3 86.4 78.4 71.1 64.1 56.9 50.4 44.5 38.9 34.6 31.3 28.0 25.3 22.9 20.6

301 269 241 214 193 174 156 137 121 107 93.2 81.9 73.6 66.0 59.5 53.9 48.4

1 - 150



 √ EC w GJ



Qf

Qw

in.

in.2

in.4

in.3

in.3

3570 3160

66.3 72.0

28.9 28.7

46.3 41.2

18.2 16.3

43.2 39.0

1.78 1.31 0.95

1880 1650 1440

52.3 57.1 62.6

23.3 23.1 22.9

30.2 26.7 23.6

14.7 13.1 11.8

36.2 32.4 28.8

W12×35 W ×30 W ×26

0.74 0.46 0.30

879 720 607

55.5 63.7 72.4

19.6 19.4 19.2

16.8 13.9 11.8

W12×22 W ×19 W ×16 W ×14

0.29 0.18 0.10 0.07

164 131 96.9 80.4

38.3 43.4 50.1 54.5

12.0 11.8 11.7 11.6

W10×112 W ×100 W ×88 W ×77 W ×68 W ×60 W ×54 W ×49

15.1 10.9 7.53 5.11 3.56 2.48 1.82 1.39

6020 5150 4330 3630 3100 2640 2320 2070

32.1 35.0 38.6 42.9 47.5 52.5 57.5 62.1

26.3 25.8 25.3 24.8 24.4 24.0 23.8 23.6

85.7 74.7 64.2 54.9 47.6 41.2 36.6 33.0

30.8 27.2 23.8 20.7 18.1 15.9 14.3 13.0

73.7 64.9 56.4 48.8 42.6 37.3 33.3 30.2

W10×45 W ×39 W ×33

1.51 0.98 0.58

1200 992 790

45.4 51.2 59.4

19.0 18.7 18.5

23.6 19.8 16.0

11.5 9.77 7.98

27.5 23.4 19.4

W10×30 W ×26 W ×22

0.62 0.40 0.24

414 345 275

41.6 47.3 54.5

14.5 14.3 14.1

10.7 9.05 7.30

7.09 6.08 4.95

18.3 15.6 13.0

W10×19 W ×17 W ×15 W ×12

0.23 0.16 0.10 0.05

104 85.1 68.3 50.9

34.2 37.1 42.1 51.3

3.93 3.24 2.62 1.99

3.76 3.13 2.56 2.00

10.8 9.33 8.00 6.32

W8×67 W ×58 W ×48 W ×40 W ×35 W ×31

5.06 3.34 1.96 1.12 0.77 0.54

1440 1180 931 726 619 530

27.1 30.2 35.1 41.0 45.6 50.4

16.7 16.3 15.8 15.5 15.3 15.1

14.7 12.5 10.4 8.42 7.39 6.46

35.1 29.9 24.5 19.9 17.3 15.2

W8×28 W ×24

0.54 0.35

312 259

38.7 43.8

12.4 12.2

9.43 7.94

5.64 4.83

13.6 11.6

W8×21 W ×18

0.28 0.17

152 122

37.5 43.1

10.4 10.3

5.47 4.44

4.03 3.31

10.2 8.52

W8×15 W ×13 W ×10

0.14 0.09 0.04

51.8 40.8 30.9

31.0 34.3 44.7

7.82 7.74 7.57

2.47 1.97 1.53

2.39 1.93 1.56

6.78 5.70 4.43

W6×25 W ×20 W ×15

0.46 0.24 0.10

150 113 76.5

29.1 34.9 44.5

9.01 8.78 8.58

6.23 4.82 3.34

3.92 3.10 2.18

9.46 7.45 5.39

W6×16 W ×12 W ×9

0.22 0.09 0.04

38.2 24.7 17.7

21.2 26.7 33.8

5.92 5.75 5.60

2.42 1.61 1.19

2.28 1.55 1.19

5.84 4.15 3.12

W5×19 W ×16

0.31 0.19

50.8 40.6

20.6 23.5

5.94 5.81

3.21 2.62

2.44 2.02

5.81 4.82

W4×13

0.15

14.0

15.5

3.87

1.36

1.27

3.14


Warping Constant Cw

Designation

in.4

in.6

W12×58 W ×53

2.10 1.58

W12×50 W ×45 W ×40

9.89 9.80 9.72 9.56


5.13 4.14 3.09 2.59

32.3 27.2 22.0 17.5 15.2 13.1

Statical Moment

9.86 8.30 7.15

25.6 21.6 18.6

4.87 4.01 3.04 2.59

14.7 12.4 10.0 8.72

TORSION PROPERTIES

1 - 151

TORSION PROPERTIES M shapes

 √ EC w GJ



Qf

Qw

in.6

in.

in.2

in.4

in.3

in.3

0.05 0.04

34.0 31.3

42.0 45.0

9.02 9.01

1.56 1.45

1.98 1.86

7.14 6.58

M10×9 M ×8

0.03 0.02

14.6 12.8

35.5 40.7

6.59 6.57

0.91 0.80

1.32 1.18

4.60 4.06

M8×6.5

0.02

26.0

4.45

0.48

0.82

2.72

M5×18.9

0.34

17.7

5.73

2.98

2.28

5.53


Warping Constant Cw

Designation

in.4

M12×11.8 M ×10.8

5.23 41.3


Statical Moment

1 - 152


TORSION PROPERTIES S shapes

Designation S24×121 S ×106

 √ EC w GJ



Qf

Qw

in.6

in.

in.2

in.4

in.3

in.3

11400 10600

48.0 52.1

47.1 46.1

103 98.8

47.1 47.1

154 141 121 112 103


Warping Constant Cw

in.4 12.8 10.1

Statical Moment

S24×100 S ×90 S ×80

7.58 6.04 4.88

6380 6000 5640

46.7 50.7 54.7

41.9 41.2 40.5

66.0 63.8 61.6

33.5 33.5 33.5

S20×96 S ×86

8.39 6.64

4710 4390

38.1 41.4

34.9 34.2

57.8 55.5

29.2 29.2

99.7 92.5

S20×75 S ×66

4.59 3.58

2750 2550

39.4 42.9

30.7 30.0

38.9 37.3

22.6 22.6

77.0 70.5

S18×70 S ×54.7

4.15 2.37

1800 1560

33.5 41.3

27.0 26.0

29.2 26.9

17.1 17.1

63.0 52.9

S15×50 S ×42.9

2.12 1.54

811 744

31.5 35.4

20.3 19.8

17.8 16.9

11.8 11.8

39.0 35.1

S12×50 S ×40.8

2.82 1.75

505 437

21.5 25.4

15.5 14.9

14.0 12.9

9.30 9.30

31.0 26.9

S12×35 S ×31.8

1.08 0.90

324 307

27.9 29.7

14.5 14.3

10.0 9.74

7.48 7.48

22.7 21.3

S10×35 S ×25.4

1.29 0.60

189 153

19.5 25.7

11.8 11.1

7.13 6.34

5.24 5.24

17.9 14.4

S8×23 S ×18.4

0.55 0.34

61.8 53.5

17.1 20.2

7.90 7.58

3.50 3.22

3.10 3.10

9.74 8.38

S6×17.25 S ×12.5

0.37 0.17

18.4 14.5

11.3 14.9

5.03 4.70

1.61 1.41

1.63 1.63

5.35 4.30

S5×10

0.11

6.66

12.3

3.51

0.86

1.11

2.88

S4×9.5 S ×7.7

0.12 0.07

3.10 2.62

8.18 9.84

2.59 2.47

0.53 0.48

0.70 0.70

2.05 1.79

S3×7.5 S ×5.7

0.09 0.04

1.10 0.85

5.63 7.42

1.72 1.60

0.28 0.24

0.40 0.40

1.20 1.00


TORSION PROPERTIES

1 - 153

TORSION PROPERTIES HP shapes

 √ EC w GJ



Qf

Qw

in.6

in.

in.2

in.4

in.3

in.3

8.02 5.40 3.60 2.01

19900 16800 14200 11200

80.2 89.8 101 120

49.9 49.2 48.5 47.8

149 128 110 88.0

38.5 33.5 29.1 23.8

97.2 84.3 72.9 59.2

HP13×100 HP ×87 HP ×73 HP ×60

6.25 4.12 2.54 1.39

11300 9430 7680 6020

68.4 77.0 88.5 106

40.9 40.2 39.6 39.0

103 87.7 72.8 57.8

29.9 25.8 21.8 17.7

76.3 65.6 55.2 44.5

HP12×84 HP ×74 HP ×63 HP ×53

4.24 2.98 1.83 1.12

7160 6170 4990 4090

66.1 73.2 84.0 97.2

35.6 35.2 34.6 34.2

75.0 65.5 54.1 44.7

23.5 20.8 17.5 14.7

59.8 52.7 44.2 37.0

HP10×57 HP ×42

1.97 0.81

2240 1540

54.3 70.2

24.1 23.4

34.8 24.7

13.1 9.64

33.2 24.2

HP8×36

0.77

578

44.1

15.4

14.0

6.62

16.8


Warping Constant Cw

Designation

in.4

HP14×117 HP ×102 HP ×89 HP ×73


Statical Moment

1 - 154


FLEXURAL-TORSIONAL PROPERTIES Channels

Designation


Warping Constant Cw

Polar Radius of Gyration _ ro*

Flexural Constant H*

in.4

in.6

in.

No Units

C15×50 C15×40 C15×33.9

2.67 1.46 1.02

492 411 358

5.49 5.72 5.94

.937 .927 .920

C12×30 C15×25 C15×20.7

0.87 0.54 0.37

151 130 112

4.55 4.72 4.93

.919 .909 .899

C10×30 C1××25 C5××20 C10×15.3

1.23 0.69 0.37 0.21

79.3 68.4 56.9 45.6

3.63 3.75 3.93 4.19

.921 .912 .900 .883

C9×20 C9×15 C9×13.4

0.43 0.21 0.17

39.4 31.0 28.2

3.46 3.69 3.79

.899 .882 .874

C8×18.75 C9×13.75 C9×11.5

0.44 0.19 0.13

25.1 19.2 16.5

3.06 3.27 3.42

.894 .874 .862

C7×12.25 C9×9.8

0.16 0.10

11.2 9.18

2.87 3.02

.862 .846

C6×13 C9×10.5 C9×8.2

0.24 0.13 0.08

7.22 5.95 4.72

2.37 2.49 2.65

.858 .843 .824

C5×9 C9×6.7

0.11 0.06

2.93 2.22

2.10 2.26

.814 .790

C4×7.25 C9×5.4

0.08 0.04

1.24 0.92

1.75 1.89

.768 .741

C3×6 C9×5 C9×4.1

0.07 0.04 0.03

0.46 0.38 0.31

1.39 1.45 1.53

.689 .674 .656

*See LRFD Specification Appendix E3.


TORSION PROPERTIES

1 - 155

FLEXURAL-TORSIONAL PROPERTIES Channels

Designation


Warping Constant Cw



in.4

in.6

in.

No Units

MC18×58 MC10×51.9 MC10×45.8 MC10×42.7

2.81 2.03 1.45 1.23

1070 986 897 852

6.56 6.70 6.88 6.97

.944 .939 .933 .930

MC13×50 MC10×40 MC10×35 MC10×31.8

2.98 1.57 1.14 0.94

558 463 413 380

5.07 5.33 5.50 5.64

.875 .860 .849 .842

MC12×50 MC10×45 MC10×40 MC10×35 MC10×31 MC10×10.6

3.24 2.35 1.70 1.25 1.01 0.06

411 374 336 297 268 11.7

4.77 4.87 5.01 5.18 5.34 4.27

.859 .851 .842 .832 .821 .983

MC10×41.1 MC10×33.6 MC10×28.5

2.27 1.21 0.79

270 224 194

4.26 4.47 4.68

.790 .771 .752

MC10×25 MC10×22

0.64 0.51

125 111

4.46 4.63

.802 .790

MC10×8.4

0.04

3.68

.972

MC9×25.4 MC9×23.9

0.69 0.60

4.08 4.15

.770 .763

MC8×22.8 MC9×21.4 MC9×20 MC9×18.7 MC9×8.5

0.57 0.50 0.44 0.38 0.06

75.3 70.9 47.9 45.1 8.22

3.85 3.91 3.59 3.65 3.24

.716 .709 .780 .773 .910

MC7×22.7 MC9×19.1

0.63 0.41

58.5 49.4

3.53 3.71

.662 .638

MC6×18

0.38

34.6

3.46

.562

MC6×16.3 MC9×15.1

0.34 0.29

22.1 20.6

3.11 3.18

.643 .634

MC6×12

0.15

11.2

2.80

.740

7.01 104 98.2



1 - 156


FLEXURAL-TORSIONAL PROPERTIES Single Angles Polar Radius of Gyration _ ro*


in.6

in.

No Units

×1 ×17⁄8 ×13⁄4 ×15⁄8 ×19⁄16 ×11⁄2

7.13 5.08 3.46 2.21 1.30 0.960 0.682

32.5 23.4 16.1 10.4 6.16 4.55 3.23

4.31 4.35 4.37 4.41 4.45 4.47 4.48

0.632 0.630 0.629 0.627 0.627 0.627 0.624

L8×6×1 L × × 3⁄4 L × ×19⁄16 L × ×11⁄2 L × ×17⁄16

4.35 1.90 0.822 0.584 0.396

16.3 7.28 3.20 2.28 1.55

3.89 3.96 4.01 4.02 4.04

— — — — —

L8×4×1 L8×4×17⁄8 L × ×13⁄4 L8×4×15⁄8 L × ×19⁄16 L × ×11⁄2 L8×4×17⁄16

3.68 2.48 1.61 0.933 0.704 0.501 0.328

12.9 8.89 5.75 3.42 2.53 1.80 1.22

3.77 3.79 3.82 3.85 3.86 3.88 3.89

— — — — — — —

L7×4×3⁄4 L × ×5⁄8 L × ×1⁄2 L7×4×7⁄16 L × ×3⁄8

1.47 0.873 0.459 0.300 0.200

3.33 3.36 3.38 3.40 3.42

— — — — —

Designation L8×8×11⁄8 L L L L L L

× × × × × ×


Warping Constant Cw

in.4

3.97 2.37 1.25 0.851 0.544



TORSION PROPERTIES

1 - 157

FLEXURAL-TORSIONAL PROPERTIES Single Angles Torsional Constant J

Warping Constant Cw



in.4

in.6

in.

No Units

L6×6×1 L6×6×17⁄8 L6×6×13⁄4 L6×6×15⁄8 L6×6×19⁄16 L6×6×11⁄2 L6×6×17⁄16 L6×6×13⁄8 L6×6×15⁄16

3.68 2.51 1.61 0.954 0.704 0.501 0.340 0.218 0.129

9.24 6.41 4.17 2.50 1.85 1.32 0.899 0.575 0.338

3.19 3.22 3.26 3.29 3.31 3.32 3.34 3.36 3.38

0.637 0.632 0.629 0.628 0.627 0.627 0.627 0.626 0.625

L6×4×3⁄4 L6×4×5⁄8 L6×4×9⁄16 L6×4×1⁄2 L6×4×7⁄16 L6×4×3⁄8 L6×4×5⁄16

1.33 0.792 0.585 0.417 0.284 0.183 0.108

2.64 1.59 1.18 0.843 0.575 0.369 0.217

2.86 2.89 2.9 2.92 2.94 2.96 2.97

— — — — — — —

L6×31⁄2×1⁄2 L6×31⁄2×3⁄8 L6×31⁄2×5⁄16

0.396 0.174 0.103

0.779 0.341 0.201

2.88 2.92 2.93

— — —

L5×5×7⁄8 L6×4×3⁄4 L6×4×5⁄8 L6×4×1⁄2 L6×4×7⁄16 L6×4×3⁄8 L6×4×5⁄16

2.07 1.33 0.792 0.417 0.284 0.183 0.108

3.53 2.32 1.40 0.744 0.508 0.327 0.193

2.65 2.68 2.71 2.74 2.77 2.79 2.81

0.634 0.634 0.630 0.630 0.629 0.627 0.626

Designation



1 - 158



Warping Constant Cw



in.4

in.6

in.

No Units

L5×31⁄2×3⁄4 L5×31⁄2×5⁄8 L5×31⁄2×1⁄2 L5×31⁄2×3⁄8 L5×31⁄2×5⁄16 L5×31⁄2×1⁄4

1.11 0.660 0.348 0.153 0.0905 0.0479

1.52 0.918 0.491 0.217 0.128 0.0670

2.37 2.40 2.44 2.47 2.49 2.50

— — — — — —

L5×3×1⁄2 L5×3×7⁄16 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

0.322 0.219 0.141 0.0832 0.0438

0.444 0.304 0.196 0.116 0.0606

2.39 2.41 2.42 2.43 2.45

— — — — —

L4×4×3⁄4 L5×3×5⁄8 L5×3×1⁄2 L5×3×7⁄16 L5×3×3⁄8 L5×3×5⁄16 L5×3×1⁄4

1.02 0.610 0.322 0.219 0.141 0.0832 0.0438

1.12 0.680 0.366 0.252 0.162 0.0963 0.0505

2.11 2.14 2.17 2.19 2.20 2.22 2.23

0.639 0.631 0.632 0.631 0.625 0.623 0.627

L4×31⁄2×1⁄2 L5×31⁄2×3⁄8 L5×31⁄2×5⁄16 L5×31⁄2×1⁄4

0.301 0.132 0.0782 0.0412

0.302 0.134 0.0798 0.0419

2.04 2.08 2.09 2.11

— — — —

Designation



TORSION PROPERTIES

1 - 159


Warping Constant Cw



in.4

in.6

in.

No Units

L4×3×5⁄8 L4×3×1⁄2 L4×3×7⁄16 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4

0.529 0.281 0.192 0.123 0.0731 0.0386

0.472 0.255 0.176 0.114 0.0676 0.0356

1.91 1.95 1.96 1.98 2.00 2.01

— — — — — —

L31⁄2×31⁄2×1⁄2 L31⁄2×31⁄2×7⁄16 L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×5⁄16 L31⁄2×31⁄2×1⁄4

0.281 0.192 0.123 0.0731 0.0386

0.238 0.164 0.106 0.0634 0.0334

1.89 1.91 1.91 1.93 1.95

0.631 0.629 0.628 0.627 0.626

L31⁄2×3×1⁄2 L31⁄2×3×3⁄8 L31⁄2×3×5⁄16 L31⁄2×3×1⁄4

0.260 0.114 0.0680 0.0360

0.191 0.0858 0.0512 0.0270

1.76 1.79 1.81 1.83

— — — —

L31⁄2×21⁄2×1⁄2 L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×1⁄4

0.234 0.103 0.0322

0.159 0.0714 0.0225

1.67 1.70 1.73

— — —

L3×3×1⁄2 L4×3×7⁄16 L4×3×3⁄8 L4×3×5⁄16 L4×3×1⁄4 L4×3×3⁄16

0.234 0.160 0.103 0.0611 0.0322 0.0142

0.144 0.100 0.0652 0.0390 0.0206 0.00899

1.60 1.61 1.63 1.65 1.66 1.68

0.634 0.632 0.629 0.628 0.627 0.626

Designation



1 - 160



Warping Constant Cw



in.4

in.6

in.

No Units

L3×21⁄2×1⁄2 L3×21⁄2×7⁄16 L3×21⁄2×3⁄8 L3×21⁄2×5⁄16 L3×21⁄2×1⁄4 L3×21⁄2×3⁄16

0.213 0.146 0.0943 0.0560 0.0296 0.0131

0.112 0.0777 0.0507 0.0304 0.0161 0.00705

1.47 1.49 1.50 1.52 1.54 1.55

— — — — — —

L3×2×1⁄2 L2×2×3⁄8 L2×2×5⁄16 L2×2×1⁄4 L2×2×3⁄16

0.192 0.0855 0.0509 0.0270 0.0120

0.0908 0.0413 0.0248 0.0132 0.00576

1.40 1.43 1.45 1.46 1.48

— — — — —

L21⁄2×21⁄2×1⁄2 L21⁄2×21⁄2×3⁄8 L21⁄2×21⁄2×5⁄16 L21⁄2×21⁄2×1⁄4 L21⁄2×21⁄2×3⁄16

0.185 0.0816 0.0483 0.0253 0.0110

0.0791 0.0362 0.0218 0.0116 0.00510

1.31 1.34 1.36 1.37 1.39

0.639 0.632 0.630 0.628 0.627

L21⁄2×2×3⁄8 L3×21⁄2×5⁄16 L3×21⁄2×1⁄4 L3×21⁄2×3⁄16

0.0728 0.0432 0.0227 0.00990

0.0268 0.0162 0.00868 0.00382

1.22 1.24 1.25 1.27

— — — —

L2×2×3⁄8 L2×2×5⁄16 L2×2×1⁄4 L2×2×3⁄16 L2×2×1⁄8

0.0640 0.0381 0.0201 0.00880 0.00274

0.0174 0.0106 0.00572 0.00254 0.00079

1.05 1.07 1.09 1.10 1.12

0.637 0.633 0.630 0.628 0.626

Designation



TORSION PROPERTIES

1 - 161

FLEXURAL-TORSIONAL PROPERTIES Structural Tees Polar Radius of Gyration _ ro*


in.6

in.

No Units

37.2 25.7 18.9 12.4

434 279 204 139

8.81 8.67 8.65 8.67

0.724 0.733 0.731 0.723

WT20×296.5** WT ×251.5** WT ×215.5 WT ×186 WT ×160.5 WT ×148.5 WT ×138.5 WT ×124.5 WT ×107.5 WT ×99.5 WT ×87

223 140 88.5 58.2 37.7 30.6 25.8 19.1 12.4 9.14 5.60

2340 1420 881 559 350 279 218 158 101 83.5 65.3

8.30 8.17 8.09 8.00 7.92 7.88 7.75 7.71 7.66 7.83 8.12

0.761 0.760 0.756 0.756 0.756 0.756 0.770 0.770 0.770 0.746 0.699

WT20×233** WT ×196** WT ×165.5 WT ×139 WT ×132 WT ×117.5 WT ×105.5 WT ×91.5 WT ×83.5 WT ×74.5

139 86.1 53.0 32.4 28.0 20.6 15.2 10.0 7.01 4.68

1360 802 485 278 233 156 113 72.1 62.9 51.9

8.39 8.27 8.19 8.07 8.02 7.88 7.84 7.79 8.02 8.24

0.680 0.678 0.674 0.676 0.680 0.690 0.690 0.691 0.658 0.626

WT18×424** WT ×399** WT ×325** WT ×263.5** WT ×219.5** WT ×196.5** WT ×179.5** WT ×164** WT ×150 WT ×140 WT ×130 WT ×122.5 WT ×115

622 527 295 163 96.7 70.7 54.3 42.1 32.0 26.2 20.7 17.3 14.3

6880 5700 3010 1570 894 637 480 363 278 226 181 151 125

8.08 8.02 7.82 7.63 7.52 7.44 7.38 7.32 7.30 7.27 7.28 7.28 7.27

0.802 0.801 0.797 0.797 0.794 0.796 0.797 0.799 0.797 0.796 0.791 0.788 0.784

Designation WT22×167.5 WT ×145 WT ×131 WT ×115


Warping Constant Cw

in.4

*See LRFD Specification Section E3. **Group 4 or Group 5 shape. See Notes in Table 1-2.


1 - 162


FLEXURAL-TORSIONAL PROPERTIES Structural Tees Torsional Constant J

Warping Constant Cw



in.4

in.6

in.

No Units

WT18×128 WT ×116 WT ×105 WT ×97 WT ×91 WT ×85 WT ×80 WT ×75 WT ×67.5

26.6 19.8 13.9 11.1 9.19 7.51 6.17 5.04 3.48

205 151 119 92.7 77.6 63.2 53.6 46.0 37.3

7.43 7.40 7.49 7.45 7.45 7.44 7.46 7.50 7.65

0.703 0.703 0.687 0.687 0.686 0.684 0.678 0.670 0.644

WT16.5×177** WT ×159** WT ×145.5** WT ×131.5** WT ×120.5 WT ×110.5 WT ×100.5

57.2 42.1 32.4 24.2 17.9 13.7 10.2

468 335 256 188 146 113 84.9

7.00 6.94 6.90 6.86 6.91 6.90 6.89

0.802 0.803 0.801 0.802 0.792 0.788 0.784

8.83 6.16 4.84 3.67 2.64

55.4 43.0 35.4 29.3 23.4

6.74 6.82 6.85 6.93 7.02

0.714 0.700 0.691 0.678 0.659

151 85.9 50.8 37.2 26.7 19.9 13.9 10.3 7.61

1170 636 361 257 184 132 96.4 71.2 53.0

6.65 6.54 6.40 6.34 6.31 6.25 6.27 6.25 6.25

0.819 0.815 0.817 0.818 0.815 0.817 0.809 0.806 0.802

7.27 4.85 3.98 3.21 2.49 1.88 1.42

37.6 28.5 23.9 20.5 17.3 14.3 10.5

6.10 6.19 6.20 6.24 6.31 6.38 6.34

0.716 0.698 0.693 0.683 0.669 0.654 0.655

Designation

WT16.5×84.5 WT ×76 WT ×70.5 WT ×65 WT ×59 WT15×238.5** WT ×195.5** WT ×163** WT ×146** WT ×130.5 WT ×117.5 WT ×105.5 WT ×95.5 WT ×86.5 WT15×74 WT ×66 WT ×62 WT ×58 WT ×54 WT ×49.5 WT ×45



TORSION PROPERTIES

1 - 163

FLEXURAL-TORSIONAL PROPERTIES Structural Tees

Designation WT13.5×269.5** WT ×224** WT ×184** WT ×153.5** WT ×140.5** WT ×129 WT ×117.5 WT ×108.5 WT ×97 WT ×89 WT ×80.5 WT ×73 WT13.5×64.5 WT ×57 WT ×51 WT ×47 WT ×42 WT12×246** WT ×204** WT ×167.5** WT ×139.5** WT ×125** WT ×114.5 WT ×103.5 WT ×96 WT ×88 WT ×81 WT ×73 WT ×65.5 WT ×58.5 WT ×52


Warping Constant Cw



in.4

in.6

in.

No Units

245 146 83.6 49.8 39.0 30.2 23.0 18.5 13.2 9.74 7.31 5.44

1740 977 532 304 232 178 135 105 74.3 57.7 42.7 31.7

6.27 6.11 5.97 5.85 5.80 5.77 5.74 5.72 5.66 5.70 5.67 5.65

0.830 0.829 0.828 0.828 0.830 0.828 0.825 0.830 0.826 0.815 0.813 0.810

5.48 5.54 5.52 5.57 5.63

0.731 0.716 0.714 0.703 0.685

5.71 5.55 5.40 5.28 5.22 5.19 5.14 5.11 5.09 5.09 5.08 5.09 5.08 5.07

0.838 0.836 0.837 0.837 0.838 0.836 0.836 0.836 0.835 0.831 0.827 0.818 0.813 0.809

5.60 3.65 2.64 2.01 1.40 223 133 76.0 45.3 33.3 25.7 19.1 15.4 12.0 9.22 6.70 4.74 3.35 2.35

24.0 17.5 12.6 10.2 7.79 1340 748 405 230 165 125 91.3 72.5 55.8 43.8 31.9 23.1 16.4 11.6

WT12×51.5 WT ×47 WT ×42 WT ×38 WT ×34

3.54 2.62 1.84 1.34 0.932

12.3 9.57 6.90 5.30 4.08

4.88 4.89 4.89 4.93 4.99

0.733 0.727 0.721 0.709 0.692

WT12×31 WT ×27.5

0.850 0.588

3.92 2.93

5.13 5.18

0.619 0.606



1 - 164



Designation WT10.5×100.5 WT10.5×91 WT10.5×83 WT10.5×73.5 WT10.5×66 WT10.5×61 WT10.5×55.5 WT10.5×50.5


Warping Constant Cw



in.4

in.6

in.

No Units

20.6 15.4 11.9 7.69 5.62 4.47 3.40 2.60

85.4 63.0 47.3 32.5 23.4 18.4 13.8 10.4

4.67 4.64 4.59 4.64 4.61 4.58 4.56 4.54

0.859 0.859 0.861 0.847 0.845 0.846 0.846 0.846

WT10.5×46.5 WT10.5×41.5 WT10.5×36.5 WT10.5×34 WT10.5×31

3.01 2.16 1.51 1.22 0.513

9.33 6.50 4.42 3.62 2.78

4.37 4.33 4.31 4.31 4.31

0.729 0.732 0.732 0.727 0.722

WT10.5×28.5 WT10.5×25 WT10.5×22

0.884 0.570 0.383

2.50 1.89 1.40

4.36 4.44 4.49

0.665 0.640 0.623

4.42 4.36 4.30 4.23 4.19 4.14 4.10 4.06 4.03 3.99

0.875 0.873 0.874 0.875 0.873 0.875 0.872 0.872 0.874 0.874

WT9×155.5** WT9×141.5** WT9×129** WT9×117** WT9×105.5** WT9×96 WT9×87.5 WT9×79 WT9×71.5 WT9×65

87.2 66.5 51.5 39.4 29.4 22.4 17.0 12.6 9.70 7.30

339 251 189 140 102 75.7 56.5 41.2 30.7 22.8

WT9×59.5 WT9×53 WT9×48.5 WT9×43 WT9×38

5.30 3.73 2.92 2.04 1.41

17.4 12.1 9.29 6.42 4.37

4.03 4.00 3.97 3.95 3.92

0.862 0.860 0.862 0.860 0.862

WT9×35.5 WT9×32.5 WT9×30 WT9×27.5 WT9×25

1.74 1.36 1.08 0.829 0.613

3.96 3.01 2.35 1.84 1.36

3.72 3.69 3.67 3.68 3.66

0.751 0.755 0.756 0.749 0.748

WT9×23 WT9×20 WT9×17.5

0.609 0.403 0.252

1.20 0.788 0.598

3.67 3.65 3.74

0.694 0.692 0.662



TORSION PROPERTIES

1 - 165


Warping Constant Cw



in.4

in.6

in.

No Units

WT8×50 WT ×44.5 WT ×38.5 WT ×33.5

3.85 2.72 1.78 1.19

10.4 7.19 4.61 3.01

3.62 3.60 3.56 3.53

0.877 0.877 0.877 0.879

WT8×28.5 WT ×25 WT ×22.5 WT ×20 WT ×18

1.10 0.760 0.655 0.396 0.271

1.99 1.34 0.974 0.673 0.516

3.30 3.28 3.27 3.24 3.30

0.770 0.770 0.767 0.769 0.745

WT8×15.5 WT ×13

0.229 0.130

0.366 0.243

3.26 3.32

0.695 0.667

5.67 5.47 5.36 5.25 5.15 5.06 4.98

0.959 0.966 0.966 0.966 0.967 0.967 0.967

4.92 4.87 4.81 4.77 4.71 4.66 4.61 4.56 4.52 4.49 4.46 4.42 4.40

0.968 0.968 0.968 0.968 0.968 0.969 0.969 0.970 0.970 0.971 0.971 0.971 0.971

Designation

WT7×404** WT ×365** WT ×332.5** WT ×302.5** WT ×275** WT ×250** WT ×227.5**

918 714 555 430 331 255 196

6970 5250 3920 2930 2180 1620 1210

WT7×213** WT ×199** WT ×185** WT ×171** WT ×155.5** WT ×141.5** WT ×128.5** WT ×116.5** WT ×105.5 WT ×96.5 WT ×88 WT ×79.5 WT ×72.5

164 135 110 88.3 67.5 51.8 39.3 29.6 22.2 17.3 13.2 9.84 7.56

991 801 640 502 375 281 209 154 113 87.2 65.2 47.9 36.3



1 - 166



Warping Constant Cw



in.4

in.6

in.

No Units

WT7×66 WT ×60 WT ×54.5 WT ×49.5 WT ×45

6.13 4.67 3.55 2.68 2.03

26.6 20.0 15.0 11.1 8.31

4.21 4.18 4.16 4.14 4.12

0.966 0.966 0.968 0.968 0.968

WT7×41 WT ×37 WT ×34 WT ×30.5

2.53 1.94 1.51 1.10

5.63 4.19 3.21 2.29

3.25 3.21 3.19 3.18

0.912 0.917 0.915 0.915

WT7×26.5 WT ×24 WT ×21.5

0.970 0.726 0.524

1.46 1.07 0.751

2.89 2.87 2.85

0.868 0.866 0.866

WT7×19 WT ×17 WT ×15

0.398 0.284 0.190

0.554 0.400 0.287

2.87 2.86 2.90

0.800 0.793 0.772

WT7×13 WT ×11

0.179 0.104

0.207 0.134

2.82 2.86

0.713 0.691

Designation

*See LRFD Specification Section E3.


TORSION PROPERTIES

1 - 167


Designation WT6×168** WT ×152.5** WT ×139.5** WT ×126** WT ×115** WT ×105** WT ×95 WT ×85 WT ×76 WT ×68 WT ×60 WT ×53 WT ×48 WT ×43.5 WT ×39.5 WT ×36 WT ×32.5


Warping Constant Cw



in.4

in.6

in.

No Units

120 92.0 70.9 53.5 41.6 32.2 24.4 17.7 12.8 9.22 6.43 4.55 3.42 2.54 1.92 1.46 1.09

481 356 267 195 148 112 82.1 58.3 41.3 28.9 19.7 13.6 10.1 7.34 5.43 4.07 2.97

4.07 4.00 3.94 3.88 3.84 3.79 3.74 3.69 3.65 3.61 3.58 3.54 3.51 3.49 3.46 3.45 3.43

0.958 0.959 0.957 0.958 0.958 0.958 0.959 0.960 0.960 0.960 0.959 0.961 0.961 0.960 0.960 0.961 0.960

WT6×29 WT ×26.5

1.05 0.788

2.08 1.53

3.01 3.00

0.944 0.940

WT6×25 WT ×22.5 WT ×20

0.889 0.656 0.476

1.23 0.885 0.620

2.67 2.64 2.62

0.899 0.898 0.901

WT6×17.5 WT ×15 WT ×13

0.369 0.228 0.150

0.437 0.267 0.174

2.56 2.55 2.54

0.835 0.830 0.826

WT6×11 WT ×9.5 WT ×8 WT ×7

0.146 0.0899 0.0511 0.0350

0.137 0.0934 0.0678 0.0493

2.52 2.54 2.62 2.64

0.683 0.663 0.624 0.610



1 - 168



Warping Constant Cw



in.4

in.6

in.

No Units

WT5×56 WT5×50 WT5×44 WT5×38.5 WT5×34 WT5×30 WT5×27 WT5×24.5

7.50 5.41 3.75 2.55 1.78 1.23 0.909 0.693

16.9 11.9 8.02 5.31 3.62 2.46 1.78 1.33

3.04 3.00 2.98 2.93 2.92 2.89 2.87 2.85

0.963 0.964 0.964 0.964 0.965 0.965 0.966 0.966

WT5×22.5 WT5×19.5 WT5×16.5

0.753 0.487 0.291

0.981 0.616 0.356

2.44 2.42 2.40

0.940 0.936 0.927

WT5×15 WT5×13 WT5×11

0.310 0.201 0.119

0.273 0.173 0.107

2.17 2.15 2.17

0.848 0.848 0.831

WT5×9.5 WT5×8.5 WT5×7.5 WT5×6

0.116 0.0776 0.0518 0.0272

0.0796 0.061 0.0475 0.0255

2.08 2.12 2.16 2.16

0.728 0.702 0.672 0.662

Designation



TORSION PROPERTIES

1 - 169


Warping Constant Cw



in.4

in.6

in.

No Units

WT4×33.5 WT ×29 WT ×24 WT ×20 WT ×17.5 WT ×15.5

2.52 1.66 0.979 0.559 0.385 0.268

3.56 2.28 1.30 0.715 0.480 0.327

2.41 2.39 2.34 2.31 2.29 2.29

0.962 0.961 0.966 0.961 0.963 0.961

WT4×14 WT ×12

0.268 0.173

0.230 0.144

1.97 1.96

0.935 0.936

WT4×10.5 WT ×9

0.141 0.0855

0.0916 0.0562

1.80 1.81

0.877 0.863

WT4×7.5 WT ×6.5 WT ×5

0.0679 0.0433 0.0212

0.0382 0.0269 0.0114

1.72 1.74 1.69

0.762 0.732 0.748

WT3×12.5 WT ×10 WT ×7.5

0.229 0.120 0.0504

0.171 0.0858 0.0342

1.76 1.73 1.71

0.952 0.952 0.937

WT3×8 WT ×6 WT ×4.5

0.111 0.0449 0.0202

0.0426 0.0178 0.0074

1.37 1.37 1.34

0.880 0.846 0.852

WT2.5×9.5 WT ×8

0.154 0.0930

0.0775 0.0453

1.44 1.43

0.964 0.962

WT2×6.5

0.0750

0.0213

1.16

0.947

Designation



1 - 170


FLEXURAL-TORSIONAL PROPERTIES Structural Tees Polar Radius of Gyration _ ro*


in.6

in.

No Units

0.0307 0.0196

0.0330 0.0252

2.69 2.67

0.564 0.572

MT5×4.5 MT ×4

0.0213 0.0116

0.0133 0.00916

2.21 2.21

0.584 0.582

MT4×3.25

0.0146

0.00421

1.73

0.611

MT2.5×9.45**

0.165

0.0732

1.37

0.951


Warping Constant Cw

in.4

MT6×5.9 MT ×5.4

Designation

*See LRFD Specification Section E3. **This shape has tapered flanges while other MT shapes have parallel flanges.


TORSION PROPERTIES

1 - 171


Designation


Warping Constant Cw



in.4

in.6

in.

No Units

ST12×60.5 ST ×53

6.38 5.04

27.5 15.0

5.14 4.87

0.640 0.685

ST12×50 ST ×45 ST ×40

3.76 3.01 2.43

19.5 12.1 6.94

5.27 5.12 4.89

0.584 0.616 0.657

ST10×48 ST ×43

4.15 3.30

15.0 9.17

4.36 4.20

0.625 0.661

ST10×37.5 ST ×33

2.28 1.78

7.21 4.02

4.28 4.10

0.612 0.655

ST9×35 ST ×27.35

2.05 1.18

7.03 2.26

4.01 3.71

0.583 0.662

ST7.5×25 ST ×21.45

1.05 0.767

2.02 0.995

3.22 3.04

0.637 0.689

ST6×25 ST ×20.4

1.39 0.872

1.97 0.787

2.60 2.42

0.663 0.733

ST6×17.5 ST ×15.9

0.538 0.449

0.556 0.364

2.49 2.39

0.697 0.731

ST5×17.5 ST ×12.7

0.633 0.300

0.725 0.173

2.23 1.98

0.653 0.768

ST4×11.5 ST ×9.2

0.271 0.167

0.168 0.0642

1.74 1.59

0.707 0.789

ST3×8.625 ST ×6.25

0.182 0.0838

0.0772 0.0197

1.36 1.21

0.706 0.820

ST2.5×5

0.0568

0.0100

1.02

0.842

ST2×4.75 ST ×3.85

0.0589 0.0364

0.00995 0.00457

0.907 0.841

0.800 0.872

ST1.5×3.75 ST ×2.85

0.0440 0.0220

0.00496 0.00189

0.737 0.672

0.832 0.913



1 - 172


FLEXURAL-TORSIONAL PROPERTIES Double Angles Long Legs Vertical

Short Legs Vertical


0

Back to Back of Angles, in.

3⁄ 4

3⁄ 8

0

3⁄ 4

Designation

ro*

H*

ro*

H*

ro*

H*

ro*

H*

ro*

H*

ro*

H*

L8×8×11⁄8 L × ×1 L × × 7⁄8 L × × 3⁄4 L × × 5⁄8 L × × 1⁄2

4.58 4.58 4.58 4.58 4.58 4.59

0.837 0.833 0.831 0.828 0.825 0.822

4.68 4.68 4.68 4.68 4.68 4.69

0.844 0.840 0.838 0.835 0.832 0.829

4.79 4.79 4.78 4.78 4.78 4.78

0.851 0.847 0.845 0.842 0.839 0.836

4.58 4.58 4.58 4.58 4.58 4.59

0.837 0.833 0.831 0.828 0.825 0.822

4.68 4.68 4.68 4.68 4.68 4.69

0.844 0.840 0.838 0.835 0.832 0.829

4.79 4.79 4.78 4.78 4.78 4.78

0.851 0.847 0.845 0.842 0.839 0.836

L8×6×1 L × × 3⁄4 L × × 1⁄2

4.07 4.08 4.11

0.721 0.714 0.708

4.15 4.16 4.18

0.731 0.724 0.718

4.23 4.24 4.26

0.742 0.735 0.728

4.19 4.17 4.17

0.925 0.919 0.914

4.31 4.29 4.28

0.929 0.924 0.919

4.44 4.41 4.40

0.933 0.928 0.923

L8×4×1 L × × 3⁄4 L × × 1⁄2

3.87 3.89 3.93

0.566 0.562 0.558

3.93 3.94 3.97

0.578 0.573 0.568

3.99 4.00 4.03

0.591 0.586 0.580

4.12 4.08 4.05

0.982 0.980 0.977

4.26 4.22 4.19

0.983 0.981 0.979

4.41 4.36 4.33

0.984 0.982 0.980

L7×4×3⁄4 L × ×1⁄2 L × ×3⁄8

3.42 3.45 3.46

0.609 0.604 0.602

3.48 3.5 3.51

0.623 0.616 0.614

3.55 3.57 3.57

0.637 0.629 0.627

3.58 3.55 3.54

0.968 0.965 0.963

3.71 3.68 3.67

0.971 0.967 0.965

3.85 3.82 3.80

0.973 0.969 0.968

L6×6×1 L × × 7⁄8 L × × 3⁄4 L × × 5⁄8 L × × 1⁄2 L × × 3⁄8

3.43 3.43 3.44 3.44 3.44 3.44

0.843 0.838 0.833 0.830 0.827 0.822

3.54 3.54 3.54 3.54 3.54 3.54

0.852 0.847 0.842 0.839 0.836 0.831

3.65 3.65 3.65 3.64 3.64 3.64

0.861 0.856 0.852 0.848 0.845 0.841

3.43 3.43 3.44 3.44 3.44 3.44

0.843 0.838 0.833 0.830 0.827 0.822

3.54 3.54 3.54 3.54 3.54 3.54

0.852 0.847 0.842 0.839 0.836 0.831

3.65 3.65 3.65 3.64 3.64 3.64

0.861 0.856 0.852 0.848 0.845 0.841

L6×4×3⁄4 L × ×5⁄8 L × ×1⁄2 L × ×3⁄8

2.98 2.98 3.00 3.01

0.672 0.668 0.663 0.661

3.05 3.05 3.06 3.07

0.687 0.683 0.678 0.675

3.13 3.13 3.14 3.15

0.704 0.699 0.693 0.690

3.10 3.09 3.08 3.07

0.948 0.946 0.943 0.940

3.23 3.21 3.20 3.19

0.952 0.950 0.947 0.944

3.36 3.34 3.34 3.32

0.956 0.954 0.951 0.948

L6×31⁄2×3⁄8 L × 1⁄2×5⁄16

2.97 2.97

0.610 0.610

3.02 3.02

0.624 0.624

3.09 3.09

0.640 0.639

3.05 3.03

0.961 0.960

3.17 3.16

0.964 0.963

3.31 3.29

0.967 0.966

L5×5×7⁄8 L × ×3⁄4 L × ×1⁄2 L × ×3⁄8 L × ×5⁄16

2.87 2.85 2.86 2.87 2.87

0.844 0.839 0.830 0.824 0.821

2.97 2.96 2.96 2.96 2.97

0.855 0.850 0.841 0.835 0.833

3.09 3.07 3.07 3.07 3.07

0.865 0.861 0.852 0.846 0.844

2.87 2.85 2.86 2.87 2.87

0.844 0.839 0.830 0.824 0.821

2.97 2.96 2.96 2.96 2.97

0.855 0.850 0.841 0.835 0.833

3.09 3.07 3.07 3.07 3.07

0.865 0.861 0.852 0.846 0.844



TORSION PROPERTIES

1 - 173


Short Legs Vertical



3⁄ 8

0

3⁄ 4

3⁄ 8

0

3⁄ 4

Designation

ro*

H*

ro*

H*

ro*

H*

ro*

H*

ro*

H*

ro*

H*

L5×31⁄2×3⁄4 L5×31⁄2×1⁄2 L5×31⁄2×3⁄8 L5×31⁄2×5⁄16

2.50 2.51 2.52 2.53

0.697 0.685 0.682 0.679

2.58 2.59 2.59 2.60

0.715 0.703 0.699 0.695

2.67 2.67 2.67 2.68

0.734 0.722 0.717 0.713

2.61 2.59 2.58 2.58

0.943 0.936 0.932 0.930

2.74 2.71 2.70 2.70

0.948 0.941 0.938 0.936

2.87 2.84 2.83 2.82

0.953 0.947 0.943 0.942

L5×3×1⁄2 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4

2.45 2.46 2.47 2.48

0.626 0.623 0.621 0.618

2.52 2.52 2.53 2.54

0.645 0.641 0.638 0.634

2.59 2.60 2.60 2.61

0.665 0.661 0.657 0.653

2.55 2.54 2.54 2.53

0.962 0.959 0.957 0.956

2.69 2.67 2.67 2.66

0.965 0.963 0.961 0.960

2.82 2.80 2.80 2.79

0.969 0.966 0.965 0.964

L4×4×3⁄4 L3×3×5⁄8 L3×3×1⁄2 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4

2.29 2.29 2.29 2.29 2.29 2.29

0.847 0.839 0.834 0.827 0.824 0.823

2.40 2.40 2.39 2.39 2.39 2.39

0.861 0.853 0.848 0.841 0.838 0.837

2.52 2.51 2.50 2.50 2.50 2.49

0.873 0.867 0.862 0.855 0.852 0.850

2.29 2.29 2.29 2.29 2.29 2.29

0.847 0.839 0.834 0.827 0.824 0.823

2.40 2.40 2.39 2.39 2.39 2.39

0.861 0.853 0.848 0.841 0.838 0.837

2.52 2.51 2.50 2.50 2.50 2.49

0.873 0.867 0.862 0.855 0.852 0.850

L4×31⁄2×1⁄2 L5×31⁄2×3⁄8 L5×31⁄2×5⁄16 L5×31⁄2×1⁄4

2.15 2.15 2.15 2.16

0.783 0.774 0.774 0.770

2.24 2.24 2.24 2.24

0.801 0.792 0.791 0.787

2.34 2.34 2.34 2.34

0.818 0.809 0.808 0.805

2.17 2.17 2.17 2.17

0.881 0.875 0.872 0.870

2.29 2.28 2.28 2.28

0.892 0.887 0.884 0.882

2.41 2.40 2.40 2.39

0.903 0.898 0.895 0.893

L4×3×1⁄2 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4

2.04 2.04 2.05 2.06

0.719 0.714 0.710 0.706

2.12 2.12 2.13 2.13

0.740 0.735 0.731 0.726

2.22 2.21 2.22 2.22

0.762 0.757 0.752 0.747

2.10 2.09 2.09 2.09

0.924 0.919 0.917 0.914

2.22 2.21 2.21 2.20

0.933 0.928 0.925 0.923

2.35 2.34 2.33 2.33

0.940 0.935 0.933 0.931

L31⁄2×31⁄2×3⁄8 L31⁄2×31⁄2×1⁄4

2.00 2.01

0.831 0.824

2.10 2.10

0.847 0.839

2.22 2.21

0.862 0.855

2.00 2.01

0.831 0.824

2.10 2.10

0.847 0.839

2.22 2.21

0.862 0.855

L31⁄2×3×3⁄8 L5×31⁄2×5⁄16 L5×31⁄2×1⁄4

1.86 1.87 1.87

0.771 0.766 0.762

1.95 1.96 1.96

0.791 0.787 0.782

2.06 2.06 2.06

0.812 0.807 0.803

1.89 1.89 1.89

0.884 0.881 0.878

2.00 2.00 2.00

0.897 0.894 0.891

2.13 2.12 2.12

0.909 0.906 0.903

L31⁄2×21⁄2×3⁄8 L31⁄2×31⁄2×1⁄4

1.76 1.77

0.696 0.691

1.84 1.85

0.721 0.715

1.94 1.93

0.748 0.740

1.82 1.81

0.932 0.927

1.94 1.93

0.941 0.936

2.08 2.06

0.948 0.944

L3×3×1⁄2 L3×3×3⁄8 L3×3×5⁄16 L3×3×1⁄4 L3×3×3⁄16

1.72 1.72 1.72 1.72 1.72

0.842 0.834 0.830 0.825 0.822

1.83 1.82 1.82 1.82 1.82

0.860 0.852 0.848 0.844 0.841

1.95 1.94 1.93 1.93 1.93

0.877 0.869 0.866 0.862 0.858

1.72 1.72 1.72 1.72 1.72

0.842 0.834 0.830 0.825 0.822

1.83 1.82 1.82 1.82 1.82

0.860 0.852 0.848 0.844 0.841

1.95 1.94 1.93 1.93 1.93

0.877 0.869 0.866 0.862 0.858



1 - 174



Short Legs Vertical


0


3⁄ 4

3⁄ 8

0

3⁄ 4

Designation

ro*

H*

ro*

H*

ro*

H*

ro*

H*

ro*

H*

ro*

H*

L3×21⁄2×3⁄8 L3×21⁄2×1⁄4 L3×21⁄2×3⁄16

1.58 1.59 1.59

0.763 0.754 0.750

1.67 1.67 1.67

0.789 0.779 0.775

1.78 1.78 1.77

0.813 0.804 0.800

1.61 1.61 1.61

0.896 0.889 0.885

1.73 1.72 1.72

0.910 0.903 0.899

1.86 1.84 1.84

0.922 0.916 0.912

L3×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16

1.49 1.50 1.50 1.50

0.672 0.667 0.664 0.661

1.57 1.57 1.57 1.57

0.704 0.698 0.694 0.690

1.66 1.66 1.66 1.66

0.737 0.730 0.726 0.721

1.55 1.55 1.54 1.54

0.949 0.946 0.943 0.940

1.68 1.68 1.67 1.66

0.956 0.954 0.951 0.949

1.82 1.82 1.80 1.80

0.963 0.961 0.958 0.956

L21⁄2×21⁄2×3⁄8 L21⁄2×21⁄2×5⁄16 L21⁄2×21⁄2×1⁄4 L21⁄2×21⁄2×3⁄16

1.43 1.43 1.43 1.43

0.839 0.834 0.829 0.825

1.54 1.54 1.53 1.53

0.861 0.856 0.851 0.847

1.66 1.66 1.65 1.65

0.880 0.876 0.871 0.867

1.43 1.43 1.43 1.43

0.839 0.834 0.829 0.825

1.54 1.54 1.53 1.53

0.861 0.856 0.851 0.847

1.66 1.66 1.65 1.65

0.880 0.876 0.871 0.867

L21⁄2×2×3⁄8 L3×21⁄2×5⁄16 L3×21⁄2×1⁄4 L3×21⁄2×3⁄16

1.29 1.30 1.30 1.31

0.752 0.746 0.741 0.736

1.39 1.39 1.39 1.39

0.785 0.779 0.773 0.767

1.50 1.50 1.50 1.49

0.816 0.810 0.804 0.798

1.33 1.33 1.33 1.32

0.912 0.908 0.903 0.899

1.45 1.45 1.45 1.44

0.927 0.923 0.919 0.915

1.59 1.58 1.58 1.57

0.939 0.935 0.932 0.928

L2×2×3⁄8 L3×2×5⁄16 L3×2×1⁄4 L3×2×3⁄16 L3×2×1⁄8

1.15 1.15 1.15 1.15 1.15

0.846 0.840 0.834 0.828 0.822

1.26 1.26 1.25 1.25 1.25

0.873 0.867 0.861 0.855 0.850

1.39 1.38 1.38 1.37 1.37

0.896 0.890 0.885 0.880 0.875

1.15 1.15 1.15 1.15 1.15

0.846 0.840 0.834 0.828 0.822

1.26 1.26 1.25 1.25 1.25

0.873 0.867 0.861 0.855 0.850

1.39 1.38 1.38 1.37 1.37

0.896 0.890 0.885 0.880 0.875



SURFACE AREAS AND BOX AREAS

1 - 175

SURFACE AREAS AND BOX AREAS W shapes Square feet per foot of length

Case A Case B Case C Case D

Designation


Designation

W44×335 W44×290 W44×262 W44×230 W44 W40×593 W44×503 W44×431 W44×372 W44×321 W44×297 W44×277 W44×249 W44×215 W44×199 W44×174

11.0 11.0 10.9 10.9

12.4 12.3 12.2 12.2

8.67 8.59 8.53 8.46

10.0 9.91 9.84 9.78

10.9 10.7 10.5 10.4 10.3 10.3 10.3 10.2 10.2 10.1 10.0

12.3 12.1 11.9 11.8 11.6 11.6 11.6 11.5 11.5 11.4 11.3

8.56 8.38 8.23 8.11 8.01 7.96 7.93 7.88 7.81 7.76 7.68

9.95 9.75 9.58 9.45 9.33 9.28 9.25 9.19 9.12 9.07 8.99

W40×466 W44×392 W44×331 W44×278 W44×264 W44×235 W36×211 W36×183 W36×167 W36×149

9.79 9.61 9.47 9.35 9.32 9.28 9.22 9.17 9.11 9.05

10.8 10.6 10.5 10.3 10.3 10.3 10.2 10.2 10.1 10.0

8.13 7.96 7.81 7.69 7.66 7.61 7.55 7.48 7.42 7.35

9.18 8.99 8.83 8.69 8.66 8.60 8.53 8.47 8.40 8.34

W36×848 W36×798 W36×650 W36×527 W36×439 W36×393 W36×359 W36×328 W36×300 W36×280 W36×260 W36×245 W36×230

11.1 11.0 10.7 10.4 10.3 10.2 10.1 10.0 9.99 9.95 9.90 9.87 9.84

12.6 12.5 12.1 11.9 11.7 11.6 11.5 11.4 11.4 11.3 11.3 11.2 11.2

8.59 8.49 8.21 7.97 7.79 7.70 7.63 7.57 7.51 7.47 7.42 7.39 7.36

10.1 9.99 9.67 9.41 9.20 9.10 9.02 8.95 8.90 8.85 8.80 8.77 8.73

W36×256 W44×232 W44×210 W44×194 W36×182 W36×170 W36×160 W36×150 W36×135

9.02 8.96 8.91 8.88 8.85 8.82 8.79 8.76 8.71

10.0 9.97 9.93 9.89 9.85 9.82 9.79 9.76 9.70

7.26 7.20 7.13 7.09 7.06 7.03 7.00 6.97 6.92

8.27 8.21 8.15 8.10 8.07 8.03 8.00 7.97 7.92

W33×354 W36×318 W36×291 W36×263 W36×241 W36×221 W33×201

9.66 9.58 9.52 9.46 9.42 9.38 9.33

11.0 10.9 10.8 10.8 10.7 10.7 10.6

7.27 7.19 7.13 7.07 7.02 6.97 6.93

8.61 8.52 8.46 8.39 8.34 8.29 8.24

W33×169 W36×152 W36×141 W36×130 W36×118

8.30 8.27 8.23 8.20 8.15

9.26 9.23 9.19 9.15 9.11

6.60 6.55 6.51 6.47 6.43

7.55 7.51 7.47 7.43 7.39

W30×477 W36×391 W36×326 W36×292 W36×261 W36×235 W36×211 W36×191 W36×173

9.30 9.11 8.96 8.88 8.81 8.75 8.71 8.66 8.62

10.6 10.4 10.2 10.2 10.1 10.0 9.97 9.92 9.87

7.02 6.83 6.68 6.61 6.53 6.47 6.42 6.37 6.32

8.35 8.13 7.96 7.88 7.79 7.73 7.67 7.62 7.57

W30×148 W36×132 W36×124 W36×116 W36×108 W36×99 W36×90

7.53 7.49 7.47 7.44 7.41 7.37 7.35

8.40 8.37 8.34 8.31 8.28 8.25 8.22

5.99 5.93 5.90 5.88 5.84 5.81 5.79

6.86 6.81 6.78 6.75 6.72 6.68 6.66

Case A: Shape perimeter, minus one flange surface. Case B: Shape perimeter. Case C: Box perimeter, equal to one flange surface plus twice the depth. Case D: Box perimeter, equal to two flange surfaces plus twice the depth.


1 - 176




Designation


Designation

W27×539 W ×448 W ×368 W ×307 W ×281 W ×258 W ×235 W ×217 W ×194 W ×178 W ×161 W ×146

8.82 8.61 8.42 8.27 8.21 8.15 8.09 8.04 7.98 7.95 7.91 7.87

10.09 9.86 9.64 9.47 9.40 9.34 9.27 9.22 9.15 9.12 9.08 9.03

6.69 6.48 6.29 6.14 6.08 6.02 5.96 5.91 5.85 5.81 5.77 5.73

7.96 7.73 7.51 7.34 7.27 7.21 7.14 7.09 7.02 6.98 6.94 6.89

W27×129 W ×114 W ×102 W ×94 W ×84

6.92 6.88 6.85 6.82 6.78

7.75 7.72 7.68 7.65 7.61

5.44 5.39 5.35 5.32 5.28

6.27 6.23 6.18 6.15 6.11

W24×492 W ×408 W ×335 W ×279 W ×250 W ×229 W ×207 W ×192 W ×176 W ×162 W ×146 W ×131 W ×117 W ×104

8.07 7.86 7.66 7.51 7.44 7.38 7.32 7.27 7.23 7.22 7.17 7.12 7.08 7.04

9.25 9.01 8.79 8.62 8.54 8.47 8.40 8.35 8.31 8.30 8.24 8.19 8.15 8.11

6.12 5.91 5.71 5.56 5.49 5.43 5.37 5.32 5.28 5.25 5.20 5.15 5.11 5.07

7.29 7.06 6.84 6.67 6.59 6.52 6.45 6.40 6.35 6.33 6.27 6.22 6.18 6.14

W24×103 W ×94 W ×84 W ×76 W ×68

6.18 6.16 6.12 6.09 6.06

6.93 6.92 6.87 6.84 6.80

4.84 4.81 4.77 4.74 4.70

5.59 5.56 5.52 5.49 5.45

W24×62 W ×55

5.57 5.54

6.16 6.13

4.54 4.51

5.13 5.10

W21×201 W18×182 W18×166 W18×147 W18×132 W18×122 W18×111 W18×101

6.75 6.69 6.65 6.61 6.57 6.54 6.51 6.48

7.80 7.74 7.68 7.66 7.61 7.57 7.54 7.50

4.89 4.83 4.78 4.72 4.68 4.65 4.61 4.58

5.93 5.87 5.82 5.76 5.71 5.68 5.64 5.61

W21×93 W18×83 W18×73 W ×68 W18×62

5.54 5.50 5.47 5.45 5.42

6.24 6.20 6.16 6.14 6.11

4.31 4.27 4.23 4.21 4.19

5.01 4.96 4.92 4.90 4.87

W21×57 W18×50 W18×44

5.01 4.97 4.94

5.56 5.51 5.48

4.06 4.02 3.99

4.60 4.56 4.53

W18×311 W18×283 W18×258 W18×234 W18×211 W18×192 W18×175 W18×158 W18×143 W18×130

6.41 6.32 6.24 6.17 6.10 6.03 5.97 5.92 5.87 5.83

7.41 7.31 7.23 7.14 7.06 6.99 6.92 6.86 6.81 6.76

4.72 4.63 4.56 4.48 4.41 4.35 4.29 4.23 4.18 4.14

5.72 5.62 5.54 5.45 5.37 5.30 5.24 5.17 5.12 5.07

W18×119 W18×106 W18×97 W ×86 W18×76

5.81 5.77 5.74 5.70 5.67

6.75 6.70 6.67 6.62 6.59

4.10 4.06 4.03 3.99 3.95

5.04 4.99 4.96 4.91 4.87

W18×71 W18×65 W18×60 W ×55 W18×50

4.85 4.82 4.80 4.78 4.76

5.48 5.46 5.43 5.41 5.38

3.71 3.69 3.67 3.65 3.62

4.35 4.32 4.30 4.27 4.25



SURFACE AREAS AND BOX AREAS

1 - 177



Designation


Designation

W18×46 W ×40 W ×35

4.41 4.38 4.34

4.91 4.88 4.84

3.51 3.48 3.45

4.02 3.99 3.95

W16×100 W ×89 W ×77 W ×67

5.28 5.24 5.19 5.16

6.15 6.10 6.05 6.01

3.70 3.66 3.61 3.57

4.57 4.52 4.47 4.43

W16×57 W ×50 W ×45 W ×40 W ×36

4.39 4.36 4.33 4.31 4.28

4.98 4.95 4.92 4.89 4.87

3.33 3.30 3.27 3.25 3.23

3.93 3.89 3.86 3.83 3.81

W16×31 W ×26

3.92 3.89

4.39 4.35

3.11 3.07

3.57 3.53

W14×808 W ×730 W ×665 W ×605 W ×550 W ×500 W ×455

7.74 7.61 7.46 7.32 7.19 7.07 6.96

9.28 9.10 8.93 8.77 8.62 8.49 8.36

5.35 5.23 5.08 4.94 4.81 4.68 4.57

6.90 6.72 6.55 6.39 6.24 6.10 5.98

W14×426 W ×398 W ×370 W ×342 W ×311 W ×283 W ×257 W ×233 W ×211 W ×193 W ×176 W ×159 W ×145

6.89 6.81 6.74 6.67 6.59 6.52 6.45 6.38 6.32 6.27 6.22 6.18 6.14

8.28 8.20 8.12 8.03 7.94 7.86 7.78 7.71 7.64 7.58 7.53 7.47 7.43

4.50 4.43 4.36 4.29 4.21 4.13 4.06 4.00 3.94 3.89 3.84 3.79 3.76

5.89 5.81 5.73 5.65 5.56 5.48 5.40 5.32 5.25 5.20 5.15 5.09 5.05

W14×132 W ×120 W ×109 W ×99 W ×90

5.93 5.90 5.86 5.83 5.81

7.16 7.12 7.08 7.05 7.02

3.67 3.64 3.60 3.57 3.55

4.90 4.86 4.82 4.79 4.76

W14×82 W14×74 W14×68 W ×61

4.75 4.72 4.69 4.67

5.59 5.56 5.53 5.50

3.23 3.20 3.18 3.15

4.07 4.04 4.01 3.98

W14×53 W14×48 W14×43

4.19 4.16 4.14

4.86 4.83 4.80

2.99 2.97 2.94

3.66 3.64 3.61

W14×38 W14×34 W14×30

3.93 3.91 3.89

4.50 4.47 4.45

2.91 2.89 2.87

3.48 3.45 3.43

W14×26 W ×22

3.47 3.44

3.89 3.86

2.74 2.71

3.16 3.12

W12×336 W ×305 W14×279 W14×252 W14×230 W14×210 W14×190 W14×170 W14×152 W ×136 W14×120 W14×106 W14×96 W14×87 W14×79 W14×72 W14×65

5.77 5.67 5.59 5.50 5.43 5.37 5.30 5.23 5.17 5.12 5.06 5.02 4.98 4.95 4.92 4.89 4.87

6.88 6.77 6.68 6.58 6.51 6.43 6.36 6.28 6.21 6.15 6.09 6.03 5.99 5.96 5.93 5.90 5.87

3.92 3.82 3.74 3.65 3.58 3.52 3.45 3.39 3.33 3.27 3.21 3.17 3.13 3.10 3.07 3.05 3.02

5.03 4.93 4.83 4.74 4.66 4.58 4.51 4.43 4.37 4.30 4.24 4.19 4.15 4.11 4.08 4.05 4.02

W12×58 W14×53

4.39 4.37

5.22 5.20

2.87 2.84

3.70 3.68

W12×50 W14×45 W ×40

3.90 3.88 3.86

4.58 4.55 4.52

2.71 2.68 2.66

3.38 3.35 3.32



1 - 178




Designation


Designation

W12×35 W12×30 W12×26

3.63 3.60 3.58

4.18 4.14 4.12

2.63 2.60 2.58

3.18 3.14 3.12

W12×22 W12×19 W12×16 W12×14

2.97 2.95 2.92 2.90

3.31 3.28 3.25 3.23

2.39 2.36 2.33 2.32

2.72 2.69 2.66 2.65

W10×112 W12×100 W12×88 W12×77 W12×68 W12×60 W12×54 W12×49

4.30 4.25 4.20 4.15 4.12 4.08 4.06 4.04

5.17 5.11 5.06 5.00 4.96 4.92 4.89 4.87

2.76 2.71 2.66 2.62 2.58 2.54 2.52 2.50

3.63 3.57 3.52 3.47 3.42 3.38 3.35 3.33

W10×45 W12×39 W12×33

3.56 3.53 3.49

4.23 4.19 4.16

2.35 2.32 2.29

3.02 2.98 2.95

W10×30 W12×26 W12×22

3.10 3.08 3.05

3.59 3.56 3.53

2.23 2.20 2.17

2.71 2.68 2.65

W10×19 W12×17 W12×15 W12×12

2.63 2.60 2.58 2.56

2.96 2.94 2.92 2.89

2.04 2.02 2.00 1.97

2.38 2.35 2.33 2.30

W8×67 W8×58 W8×48 W8×40 W8×35 W8×31

3.42 3.37 3.32 3.28 3.25 3.23

4.11 4.06 4.00 3.95 3.92 3.89

2.19 2.14 2.09 2.05 2.02 2.00

2.88 2.83 2.77 2.72 2.69 2.67

W8×28 W8×24

2.87 2.85

3.42 3.39

1.89 1.86

2.43 2.40

W8×21 W8×18

2.61 2.59

3.05 3.03

1.82 1.79

2.26 2.23

W8×15 W8×13 W8×10

2.27 2.25 2.23

2.61 2.58 2.56

1.69 1.67 1.64

2.02 2.00 1.97

W6×25 W ×20 W8×15

2.49 2.46 2.42

3.00 2.96 2.92

1.57 1.54 1.50

2.08 2.04 2.00

W6×16 W8×12 W8×9

1.98 1.93 1.90

2.31 2.26 2.23

1.38 1.34 1.31

1.72 1.67 1.64

W5×19 W8×16

2.04 2.01

2.45 2.43

1.28 1.25

1.70 1.67

W4×13

1.63

1.96

1.03

1.37



CAMBER

1 - 179

CAMBER Beams and Girders

Camber and sweep are used to form a desired curvature in either rolled beams or welded girders. Camber denotes a curve in the vertical plane. Beams and girders can be cambered to compensate for the anticipated deflection or for architectural reasons. Note that the required camber is determined at service (unfactored) load levels. Sweep denotes a curve in the horizontal plane. Camber and sweep may be induced through cold bending or through the application of heat. The minimum radius for cold cambering in members up to a nominal depth of 30 inches is between 10 and 14 times the depth of the member; deeper members will require a larger minimum radius. Cold bending may be used to provide sweep in members to practically any radius desired. Note that a length limit of 40 to 50 feet is practical. Heat cambering, sweeping, and straightening are provided through controlled heat application. The member is rapidly heated in selected areas which tend to expand, but are restrained by the adjacent cooler areas, causing plastic deformation of the heated areas and a change in the shape of the cooled member. The mechanical properties of steels are largely unaffected by such heating operations, provided the maximum temperature does not exceed 1,100°F for quenched and tempered alloy steels, and 1,300°F for other steels. The temperature should be carefully checked by temperature-indicating crayons or other suitable means during the heating process. Cambering and sweeping induces residual stresses similar to those that develop in rolled structural shapes as elements of the shape cool from the rolling temperature at different rates. In general, these residual stresses do not affect the ultimate strength of structural members. Additionally, the effect of residual stresses is incorporated in the provisions of the LRFD Specification. Note that when a cambered beam bearing on a wall or other support is loaded, expansion of the unrestrained end must be considered. In Figure 1-5(a), the end will move a distance ∆, where ∆=

4Cd L

If instead the cambered beam is supported on a simple shear connection at both ends, the top and bottom flange will each move a distance of one-half ∆ since end rotation will occur approximately about the neutral axis. The designer should be aware of the magnitude of these movements and make provisions to accommodate them. Figure 1-5(a) considers the geometry of a girder in the horizontal position, and Figure 1-5(b) illustrates the condition when the girder is not level. Trusses

“Cambering” of trusses is accomplished by geometric relocation of panel points and adjustment of member lengths; it does not involve physical cold bending or the application of heat as with beams and girders. The following discussion of cambering to compensate for the anticipated deflection of a truss is applicable for any parabolic condition; large-radius circular curves will be approximated very closely by the technique described. Cambering to compensate for the axial deformation of the members of a truss is beyond the scope of this Manual; refer to a textbook on mechanics of materials. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 180


Distances approximately equal for small angles

Distances equal for parabolic curve, approximately equal for circular curve. See sketch below.

∆

∆ θ

d C

90°

C L /2

∆

L /2

tanθ = 2C

Fixed End

L /2

∆ = d tanθ

Unrestrained End

∆ = 4Cd L

θ 2θ for circular curve 2C for parabolic curve (a) Beam or Girder Ends at

Same Elevations

4Cd ∆= L

4Cd ∆= L

∆ = 4Cd L

∆ = 4Cd L d

B Grade angle

B L

L approx

.

(b) Beam or Girder Ends at Different Elevations

Fig. 1-5. Camber for beams and girders. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

B

A+B A

B

A

Horiz. line

Vert.

A

°

90

A

B

°

e

B 90

Grade lin

A+B A

C

A

CAMBER

1 - 181

The usual method of providing camber in building trusses is to progressively raise each panel point. The lengths of the verticals are not changed, but the lengths of the diagonals are calculated on the basis of the adjusted elevation for the several panel points. For any simple-span truss, the offset above a straight base line, at the several panel points, can be computed from the following equations if the vertical curve forming the camber is taken as a parabola. 2

2

 B B  D = C − C   = C 1 −    A   A   where A = Horizontal distance from end panel point to mid-span of the truss (half the truss span). B = Horizontal distance from mid-span of the truss to panel point for which offset is to be determined. C = Required mid-span camber. D = Offset from the base-line at panel point corresponding to distance B. A and B must be expressed in the same units; similarly C and D must be expressed in the same units, but not necessarily the same units as A and B. When the truss is divided into any number of approximately equal panels, it may be convenient to express distances A and B in panel lengths. For the truss of Figure 1-6(a) with eight equal panels, distance A is taken as four panel lengths. Assuming the camber at the midpoint is specified as 11⁄2-in., the offset at panel point 1, where B equals three panel lengths, is: 2

 3  D = 1 -in. 1 −    4   = 21⁄32-in. 1⁄ 2

The offset at panel point 2, where B equals two panel lengths, is: 2

 2  D = 1 -in. 1 −     4  = 11⁄8-in. 1⁄ 2

The offset at panel point 3, where B equals one panel length, is: 2

 1  D = 1 -in. 1 −    4   = 113⁄32-in. 1⁄ 2

Finally, the offset at panel point 4, where B equals zero, is D = C = 11⁄2-in. An alternative method of determining the amount of camber at intermediate panel points when all panel points are approximately the same distance apart is as follows. Using the truss in Figure 1-6(a) as an example, sketch the camber diagram and number the panel points, starting with the first panel point from the end of the truss, from 1 to 4, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 182


as shown in Figure 1-6(b) on line A. Next, on line B, reverse the numbering as shown. Finally, on line C, enter the product of the numbers on lines A and B. The camber at any panel point is the amount of camber at the centerline of the truss multiplied by the fraction whose numerator is the figure on line C at the given panel point, and whose denominator is the figure on line C at the center line of the truss. Thus, at panel point 1, the camber is 7⁄

16

× 11⁄2jin. = 21⁄32jin.

at panel point 2, the camber is 12⁄

16

× 11⁄2jin. = 11⁄8jin.

at panel point 3, the camber is 15⁄

16

× 11⁄2jin. = 113⁄32jin.

and at panel point 4, the camber is 16⁄

16

× 11⁄2jin. = 11⁄2jin.

cL

4

3 2 1 21/ 32

0

1 1/2

1 13/32

11/8

Baseline

EQ

EQ

EQ

EQ

(a) Calculated camber ordinates by formula

1

2

3

cL 4

line A line B

1 x7

2 x6

3 x5

4 x4

line C

7

12

15

16

Panel point

(b) Alternative calculation method for approximately equal panels

Fig. 1-6. Camber for trusses. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

STANDARD MILL PRACTICE

1 - 183

STANDARD MILL PRACTICE General Information

Rolling structural shapes and plates involves such factors as roll wear, subsequent roll dressing, temperature variations, etc., which cause the finished product to vary from published profiles. Such variations are limited by the provisions of the American Society for Testing and Materials Specification A6. Contained in this section is a summary of these provisions, not a reproduction of the complete specification. In its entirety, A6 covers a group of common requirements, which, unless otherwise specified in the purchase order or in an individual specification, apply to rolled steel plates, shapes, sheet piling, and bars. As indicated in Table 1-1, carbon steel refers to ASTM designations A36 and A529; high-strength, low-alloy steel refers to designations A242, A572, and A588; alloy steel refers to designation A514; and low-alloy steel refers to A852. For further information on mill practices, including permissible variations for rolled tees, zees, and bulb angles in structural and bar sizes, pipe, tubing, sheets, and strip, and for other grades of steel, see ASTM A6, A53, A500, A568, and A618; the Steel Products Manuals of the Iron and Steel Society (American Institute of Mining, Metallurgical, and Petroleum Engineers); and producers’ catalogs. The data on spreading rolls to increase areas and weights, and mill cambering of beams, is not a part of ASTM A6. Additional material on mill practice is included in the descriptive material preceding the “Dimensions and Properties” tables for shapes and plates. Letter symbols representing dimensions on sketches shown herein are in accordance with ASTM A6, AISI and mill catalogs and not necessarily as defined by the general nomenclature of this manual. Methods of increasing areas and weights by spreading rolls Cambering of rolled beams . . . . . . . . . . . . . . . . . . Positions for measuring camber and sweep . . . . . . . . . W Shapes, permissible variations . . . . . . . . . . . . . . S Shapes, M Shapes, and Channels, permissible variations . Tees split from W , M, and S Shapes, permissible variations Angles split from Channels, permissible variations . . . . . Angles, structural size, permissible variations . . . . . . . . Angles, bar size, permissible variations . . . . . . . . . . . Steel Pipe and Tubing, permissible variations . . . . . . . . Plates, permissible variations for sheared, length and width . Plates, permissible variations for universal mill, length . . . Plates, permissible variations for universal mill, width . . . Plates, permissible variations for camber . . . . . . . . . . Plates, permissible variations for flatness . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

1-183 1-186 1-187 1-188 1-190 1-191 1-191 1-192 1-193 1-194 1-196 1-196 1-196 1-197 1-198

Methods of Increasing Areas and Weights by Spreading Rolls

W Shapes

To vary the area and weight within a given nominal size, the flange width, the flange thickness, and the web thickness are changed as shown in Figure 1-7(a). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 - 184


S Shapes and American Standard Channels

To vary the area and weight within a given nominal size, the web thickness and the flange width are changed by an equal amount as shown in Figures 1-7(b) and (c). Angles

To vary area and weight for a given leg length, the thickness of each leg is changed. Note that the leg length is changed slightly by this method (Figure 1-7(d)).

Constant for a given nominal size

(a)

Constant for a given nominal size (except S24 and S20)

(b)

Constant for a given nominal size

(c)

(d) Fig. 1-7. Varying areas and weights. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


1 - 185

Cambering of Rolled Beams

All beams are straightened after rolling to meet permissible variations for sweep and camber listed hereinafter for W shapes and S shapes. The following data refer to the subsequent cold cambering of beams to produce a predetermined dimension. The maximum lengths that can be cambered depend on the length to which a given section can be rolled, with a maximum of 100 feet. Table 1-10 outlines the maximum and minimum induced camber of W shapes and S shapes. Consult the producer for specific camber and/or lengths outside the above listed available lengths and sections. Mill camber in beams of less depth than tabulated should not be specified. A single minimum value for camber, within the ranges shown above for the length ordered, should be specified. Camber is measured at the mill and will not necessarily be present in the same amount in the section of beam as received due to release of stress induced during the cambering operation. In general 75 percent of the specified camber is likely to remain. Camber will approximate a simple regular curve nearly the full length of the beam, or between any two points specified. Camber is ordinarily specified by the ordinate at the mid-length of the portion of the beam to be curved. Ordinates at the other points should not be specified. Although mill cambering to achieve reverse or other compound curves is not considered practical, fabricating shop facilities for cambering by heat can accomplish such results as well as form regular curves in excess of the limits tabulated above. Refer to the earlier section Effect of Heat of Steel for further information.


1 - 186


Table 1-10. Cambering of Rolled Beams Maximum and Minimum Induced Camber Sections, Nominal Depth, in.

Specified Length of Beam, ft Over 30 to 42, incl.

Over 42 to 52, incl.




Max. and Min. Camber Acceptable, in. W shapes, 24 and over

W shapes, 14 to 21, incl. and S shapes, 12 in. and over

1 to 2, incl. 3⁄ 4

to 21⁄2, incl.

1 to 3, incl.

2 to 4, incl.

3 to 5, incl.

3 to 6, incl.

1 to 3, incl.

—

—

—

Permissible Variations for Camber Ordinate Lengths

Plus Variation 1⁄

50 ft and less Over 50 ft

1⁄ -in. 2

2-in.

1⁄ -in. 8

plus for each 10 ft or fraction thereof in excess of 50 ft


Minus Variation 0 0


1 - 187

Table 1-11. Positions for Measuring Camber and Sweep

Camber

Sweep

Camber

Sweep*

Horizontal surface

W SHAPES

Camber

S SHAPES and M SHAPES

Sweep*

Camber

Horizontal surface

Horz

Camber

onta

CHANNELS

ANGLES

l sur

face

TEES

*Due to the extreme variations in flexibility of these shapes, straightness tolerances for sweep are subject to negotiations between manufacturer and purchaser for individual sections involved.


1 - 188


Table 1-12. W Shapes, HP Shapes B

1/ 2 B±

B

1/ 2 B±

E

E

T′

C

T′

C

A

A

T

T 1/2 B±E

1/ 2 B±

E

Permissible Variations in Cross Section Section Nominal Size, in.

A, Depth, in.

B, Fig. Width, in.

Over Theoretical

Under Theoretical

Over Theoretical

To 12, inc.

1⁄

2

1⁄ 8

1⁄ 4

3⁄

Over 12

1⁄

8

1⁄ 8

1⁄ 4

3⁄

T+T′ Flanges, out of square, Max, in.

E a, Web off Center, Max, in.

C, Max. Depth at any Crosssection over Theoretical Depth, in.

16

1⁄ 4

3⁄ 16

1⁄

4

16

5⁄

3⁄ 16

1⁄

4

Under Theoretical

16

Permissible Variations in Length Variations from Specified Length for Lengths for Given, in. 30 ft and Under W Shapes

Over 30 ft

Over

Under

Over

Under

Beams 24 in. and under in nominal depth

3⁄

8

3⁄ 8

3⁄

1 8 plus ⁄16 for each additional 5 ft or fraction thereof

3⁄ 8

Beams over 24 in. nom. depth; all columns

1⁄

2

1⁄ 2

1⁄

1⁄ 2

1 2 plus ⁄16 for each additional 5 ft or fraction thereof

Notes: aVariation of 5⁄ in. max. for sections over 426 lb / ft. 16


Continued on next page


1 - 189

Table 1-12 (cont.). WP Shapes, HP Shapes Other Permissible Variations Area and weight variation: ±2.5 percent theorectical or specified amount. Ends out-of-square: 1⁄64-in. per in. of depth, or of flange width if it is greater than the depth.

Camber and Sweep Permissible Variation, in. Sizes

Length

Sizes with flange width equal to or greater than 6 in.

All

Sizes with flange width less than 6 in.

All

45 ft. and under

Certain sections with a flange width approx. equal to depth & specified on order as b columns

Over 45 ft.

Camber 1⁄ 8

1⁄ 8

in. ×

Sweep in. ×

(total length ft.) 10


1⁄ 8

in. ×


(total length ft.) with 3⁄8 in. max. 10

1⁄ 8

in. ×

3⁄ 8

 (total length ft. − 45)  in. + 1⁄8 in. ×  10  

bApplies only to W8×31 and heavier, W10×49 and heavier, W12×65 and heavier, W14×90 and heavier. If the other sections are specified on the order as columns, the tolerance will be subject to negotiation with the manufacturer.


1 - 190


Table 1-13. S Shapes, M Shapes, and Channels Permissible Variations in Cross Section B B

T′

T′

*

A

A

T

T

* Back of square and centerline of web to be parallel when measuring “out-of-square”

A, Depth in.a

Section

T + T ′b, Out of Square per Inch of Over Under Over Under B, in. Theoretical Theoretical Theoretical Theoretical

Nominal Size in.

S shapes 3 to 7, incl. Over 7 to 14, and M incl. shapes Over 14 to 24, incl.

1⁄ 2 1⁄ 8

1⁄ 16 3⁄ 32

1⁄ 8 5⁄ 32

1⁄ 8 5⁄ 32

1⁄

3⁄

16

1⁄ 8

3⁄ 16

3⁄ 16

1⁄

32

32 1⁄ 8

1⁄ 16 3⁄ 32

1⁄ 8 1⁄ 8

1⁄ 8 5⁄ 32

1⁄

32

3⁄

1⁄ 8

1⁄ 8

3⁄ 16

1⁄

3⁄

Channels 3 to 4, incl. Over 7 to 14, incl. Over 14

B, Flange Width, in.

16

1⁄

1⁄

32 32

32 32

Permissible Variations in Length Variations from Specified Length for Lengths Given, in. Over 30 to 40 ft., incl.

to 30 ft., incl. Section S shapes, M shapes and Channels

Over 40 to 50 ft., incl.

Over

Under

Over

Under

Over

1⁄

1⁄ 4

3⁄ 4

1⁄

1

2

4

Under 1⁄

4


Over 65 ft.

Over

Under

Over

Under

11⁄8

1⁄ 4

11⁄4

1⁄ 4

Other Permissible Variations

Area and weight variation: ±2.5 percent theoretical or specified amount. Ends out-of square: S shapes and channels 1⁄64-in. per in. of depth. total length,ft Camber = 1⁄8-in. × 5 Notes: aA is measured at center line of web for beams; and at back of web for channels. bT + T′ applies when flanges of channels are toed in or out.



1 - 191

Table 1-14. Tees Split from W, M, and S Shapes, Angles Split from Channels Permissible Variations in Depth A

A

A

Dimension A may be approximately one-half beam or channel depth, or any dimension resulting from off-center splitting, or splitting on two lines as specified on the order. Depth of Beam from which Tees or Angles are Split

Variations in Depth A Over and Under Tees

Angles

To 6 in., excl.

1⁄ 8

1⁄ 8

6 to 16, excl.

3⁄ 16

3⁄ 16

16 to 20, excl.

1⁄ 4

1⁄ 4

20 to 24, excl.

5⁄ 16

—

24 and over

3⁄ 8

—

The above variations for depths to tees or angles include the permissible variations in depth for the beams and channels before splitting.

Other Permissible Variations Other permissible variations in cross section as well as permissible variations in length, area, and weight variation, and ends out-of-square will correspond to those of the beam or channel before splitting, except total length,ft Camber = 1⁄8-in. × 5


1 - 192


Table 1-15. Angles, Structural Size Permissible Variations in Cross Section T

B

B

T

B Length of Leg, in. Nominal Size, in.a

Section Angles

Over Theoretical

Under Theoretical

T, Out of Square per in. of B, in.

1⁄ 8

3⁄ 32

b 3⁄ 128

3 to 4, incl. Over 4 to 6, incl.

1⁄ 8

1⁄ 8

b 3⁄ 128

Over 6

3⁄ 16

1⁄ 8

b 3⁄ 128

Permissible Variations in Length Variations from Specified Length for Lengths Given, in. Over 30 to 40 ft., incl.

to 30 ft., incl. Section Angles



Over 65 ft.

Over

Under

Over

Under

Over

Under

Over

Under

Over

Under

1⁄ 2

1⁄ 4

3⁄ 4

1⁄ 4

1

1⁄ 4

11⁄8

1⁄ 4

11⁄4

1⁄ 4

Other Permissible Variations Area and weight variation: ±2.5 percent theoretical or specified amount. Ends out-of square: 3⁄128-in. per in. of leg length, or 11⁄2 degrees. Variations based on the longer leg of unequal angle. total length,ft Camber = 1⁄8-in. × , applied to either leg 5 Notes; aFor unequal leg angles, longer leg determines classification. 1 b1⁄ 128 in. per in. = 1 ⁄2 deg.



1 - 193

Table 1-16. Angles, Bar Size* Permissible Variation in Cross Section T

B

B

a

Specified Length of Leg, in.

T

Variations from Thickness for Thicknesses Given, Over and Under, in. 3⁄ 16

and Under

Over 3⁄16 to 3⁄ incl. 8

1 and Under

0.008

0.010

Over 1 to 2, incl.

0.010

0.010

Over 2 to 3, excl.

0.012

0.015

Over 3⁄8

B Length of T, Out of Leg Over Square per and Under, in. Inch of B, in. 1⁄ 32

b 3⁄ 128

0.012

3⁄ 64

b 3⁄ 128

0.015

1⁄ 16

b 3⁄ 128

Permissible Variations in Length Variations Over Specified Length for Lengths Given No Variation Under Section All sizes of barsize angles

50 to 10 ft. excl.

10 to 20 ft. excl.

20 to 30 ft. excl.

30 to 40 ft. excl.

40 to 65 ft. excl.

5⁄ 8

1

11⁄2

2

21⁄2

Other Permissible Variations total length,ft 5 Straightness: Because of warpage, permissible variations for straightness do not apply to bars if any subsequent heating operation has been performed. Ends out-of-square: 3⁄128-in. per inch of leg length or 11⁄2 degrees. Variation based on longer leg of an unequal angle. Camber: 1⁄4-in. in any 5 feet, or 1⁄4 in. ×

Notes: *A member is ‘‘bar size’’ when its greatest cross-sectional dimension is less than three inches. aFor unequal leg angles, longer leg determines classification. 1 b1⁄ 128 in. per in. = 1 ⁄2 degrees.


1 - 194


Table 1-17. Steel Pipe and Tubing Dimensions and Weight Tolerances Round Tubing and Pipe (see also Table 1-4) ASTM A53 Weight—The weight of the pipe as specified in Table X2 and Table X3 (ASTM Specification A53) shall not vary by more than ±10 percent. Note that the weight tolerance of ±10 percent is determined from the weights of the customary lifts of pipe as produced for shipment by the mill, divided by the number of feet of pipe in the lift. On pipe sizes over four inches where individual lengths may be weighed, the weight tolerance is applicable to the individual length. Diameter—For pipe two inches and over in nominal diameter, the outside diameter shall not vary more than ±1 percent from the standard specified. Thickness—The minimum wall thickness at any point shall not be more than 12.5 percent under the nominal wall thickness specified. ASTM 500 Diameter—For pipe two inches and over in nominal diameter, the outside diameter shall not vary more than ±0.75 percent from the standard specified. Thickness—The wall thickness at any point shall not be more than 10 percent under or over the nominal wall thickness specified. ASTM A501 and ASTM 618 Outside dimensions—For round hot-formed structural tubing two inches and over in nominal size, the outside diameter shall not vary more than ±1 percent from the standard specified. Weight (A501 only)—The weight of structural tubing shall be less than the specified value by more than 3.5 percent. Mass (A618 only)—The mass of structural tubing shall not be less than the specified value by more than 3.5 percent. Length—Structural tubing is commonly produced in random mill lengths and in definite cut lengths. When cut lengths are specified for structural tubing, the length tolerances shall be in accordance with the following table: Over 22 to 44 ft, incl.

22 ft and under

Length tolerance for specified cut lengths, in.

Over

Under

Over

Under

1⁄

1⁄

3⁄

1⁄

2

4

4

4

Straightness—The permissible variation for straightness of structural tubing shall be 1⁄8-in. times the number of feet of total length divided by 5. Continued on next page



1 - 195

Table 1-17 (cont.). Steel Pipe and Tubing Dimensions and Weight Tolerances Square and Rectangular Tubing (see also Table 1-4) ASTM A500 and ASTM A618 Outside Dimensions—The specified dimensions, measured across the flats at positions at least two inches from either end-of-square or rectangular tubing and including an allowance for convexity or concavity, shall not exceed the plus and minus tolerance shown in the following table: a

Largest Outside Dimension Across Flats, in.

Tolerance Plus an Minus, in.

21⁄2 and under Over 21⁄2 to 31⁄2, incl. Over 31⁄2 to 51⁄2, incl. Over 51⁄2

0.020 0.025 0.030 1 percent

aThe respective outside dimension tolerances

include the allowances for convexity and concavity.

Lengths—Structural tubing is commonly produced in random lengths, in multiple lengths, and in definite cut lengths. When cut lengths are specified for structural tubing, the length tolerances shall be in accordance with the following table: 22 ft and under

Length tolerance for specified cut lengths, in.

Over 22 to 44 ft, incl.

Over

Under

Over

Under

1⁄

1⁄

3⁄

1⁄

2

4

4

4

Mass (A618 only)—The mass of structural tubing shall not be less than the specified value by more than 3.5 percent. Straightness—The permissible variation for straightness of structural tubing shall be 1⁄8-in. times the number of feet of total length divided by five. Squareness of sides—For square or rectangular structural tubing, adjacent sides may deviate from 90 degrees by a tolerance of plus or minus two degrees maximum. Radius of corners—For square or rectangular structural tubing, the radius of any outside corner of the section shall not exceed three times the specified wall thickness. Twists—The tolerances for twist or variation with respect to axial alignment of the section, for square and rectangular structural tubing, shall be as shown in the following table: Specified Dimension of Longest Side, in.

Maximum Twist per 3 ft of Length, in.

11⁄

2 and under Over 11⁄2 to 21⁄2, incl. Over 21⁄2 to 4, incl. Over 4 to 6 incl. Over 6 to 8, incl. Over 8

0.050 0.062 0.075 0.087 0.100 0.112

Twist is measured by holding down one end of a square or rectangular tube on a flat surface plate with the bottom side of the tube parallel to the surface plate and noting the height that either corner, at the opposite end of the bottom side of the tube, extends above the surface plate. Wall thickness (A500 only)—The tolerance for wall thickness exclusive of the weld area shall be plus and minus 10 percent of the nominal wall thickness specified. The wall thickness is to be measured at the center of the flat.


1 - 196


Table 1-18. Rectangular Sheared Plates and Universal Mill Plates Permissible Variations in Width and Length for Sheared Plates (11⁄2-in. and under in thickness) Permissible Variations in Length Only for Universal Mill Plates (21⁄2-in. and under in thickness) Specified Dimensions, in.

Variations over Specified Width and Length for Thickness, in., and Equivalent Weights, lb per sq. ft., Given To 3⁄8 excl. To 15.3, excl.

Length To 120, excl.

Width

to 5⁄8 excl.

15.3 to 25.5, excl.

5⁄ 8

to 1, excl.

1 to 2, incl.a

25.5 to 40.8, excl.

40.8 to 81.7, incl.

Width Length Width Length Width Length Width Length

To 60, excl. 60 to 84, excl. 84 to 108, excl 108 and over

120 to 240, excl. To 60, excl. 60 to 84, excl. 84 to 108, excl. 108 and over 240 to 360, excl. To 60, excl. 60 to 84, excl. 84 to 108, excl. 108 and over

3⁄ 8 7⁄ 16 1⁄ 2 5⁄ 8

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

7⁄ 16 1⁄ 2 5⁄ 8 3⁄ 4

3⁄

3⁄

1⁄

1⁄

16 5⁄ 8

1

9⁄

3⁄ 1⁄

8 2

9⁄ 16 11⁄ 16

480 to 600, excl. To 60, excl. 60 to 84, excl. 84 to 108, excl. 108 and over

7⁄

To 60, excl. 60 to 84, excl. 84 to 108, excl. 108 and over

4

3⁄

7⁄


8 2


720 and over, excl.

3⁄ 8

16 1⁄ 2 9⁄ 16 3⁄ 4 16 1⁄ 2 5⁄ 8 3⁄ 4 1⁄ 5⁄ 5⁄ 7⁄

2 8 8 8

9⁄

16 3⁄ 4 3⁄ 4

1

7⁄

4 8

5⁄

11⁄

2 8

16 3⁄ 4

1 1 1 11 ⁄8

1⁄

11 ⁄8 11 ⁄4 11 ⁄4 13 ⁄8

1⁄

11 ⁄4 13 ⁄8 13 ⁄8 11 ⁄2

1⁄

11 ⁄4 13 ⁄8 13 ⁄8 11 ⁄2

5⁄

2 2 2 2

3⁄

5⁄ 3⁄ 7⁄ 5⁄ 3⁄ 7⁄ 5⁄ 3⁄ 7⁄ 3⁄ 3⁄

2 8 4 8 2 8 4 8 2 8 4 8 8 4 4

1 7⁄ 7⁄

4 8 8

11 ⁄8

5⁄ 8

11⁄

16

7⁄

8

1 7⁄ 7⁄

15 ⁄

8 8 16

11 ⁄8

1⁄ 2

5⁄ 3⁄

7⁄ 5⁄ 3⁄

13 ⁄ 7⁄

11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄4

5⁄

11 ⁄4 13 ⁄8 13 ⁄8 11 ⁄2

5⁄

11 ⁄2 11 ⁄2 11 ⁄2 15 ⁄8

5⁄

17 ⁄8 17 ⁄8 17 ⁄8 2

3⁄

21 ⁄8 21 ⁄8 21 ⁄8 23 ⁄8

3⁄ 7⁄

8 4 8 8 4 16 8 8 4 8

1 3⁄ 7⁄

8 4 8

1 3⁄ 7⁄

8 4 8

1 7⁄ 7⁄

4 8 8

11 ⁄8 7⁄

8

1 1 11 ⁄4

3⁄ 4 7⁄ 8

5⁄ 8 3⁄ 4

1 11 ⁄8

1 11 ⁄8

1 1 11 ⁄8 11 ⁄4

3⁄

11 ⁄4 11 ⁄4 13 ⁄8 13 ⁄8 13 ⁄8 11 ⁄2 11 ⁄2 15 ⁄8 15 ⁄8 15 ⁄8 15 ⁄8 13 ⁄4

7⁄

4 8

1 11 ⁄8 3⁄ 7⁄

4 8

1 11 ⁄4 3⁄ 7⁄

4 8

1 11 ⁄4 3⁄ 7⁄

4 8

1 11 ⁄4 7⁄

1 1 11 ⁄ 8 11 ⁄ 4 11 ⁄ 8 11 ⁄ 4 13 ⁄ 8 13 ⁄ 8 11 ⁄ 2 11 ⁄ 2 11 ⁄ 2 13 ⁄ 4 15 ⁄ 8 15 ⁄ 8 17 ⁄ 8 17 ⁄ 8 17 ⁄ 8 17 ⁄ 8 17 ⁄ 8 17 ⁄ 8

17 ⁄8 17 ⁄8 17 ⁄8 21 ⁄4

1 11 ⁄8 11 ⁄4

21 ⁄ 4 21 ⁄ 4 21 ⁄ 4 21 ⁄ 2

21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄2

1 11 ⁄8 11 ⁄4 13 ⁄8

23 ⁄ 4 23 ⁄ 4 23 ⁄ 4 3

8

Notes: aPermissible variations in length apply also to Universal Mill plates up to 12 in. width for thicknesses over 2 to 21⁄2-in., incl. except for alloy steels up to 13⁄4-in. thick. Permissible variations under specified width and length, 1⁄4-in. Table applies to all steels listed in ASTM A6.



1 - 197

Table 1-19. Rectangular Sheared Plates and Universal Mill Plates Permissible Variations from Flatness (Carbon Steel Only) Variations from Flatness for Specified Widths, in.

Specified Thickness, in.

To 36 excl.

To 1⁄4, excl. 1⁄ to 3⁄ , excl. 4 8 3⁄ to 1⁄ , excl. 8 2 1⁄ to 3⁄ , excl. 2 4 3⁄ to 1, excl. 4 1 to 2, excl. 2 to 4, excl. 4 to 6, excl. 6 to 8, excl.

9⁄ 16 1⁄ 2 1⁄ 2 7⁄ 16 7⁄ 16 3⁄ 8 5⁄ 16 3⁄ 8 7⁄ 16

36 to 48, 48 to 60, 60 to 72, 72 to 84, 84 to 96, 96 to 108, 108 to excl. excl. excl. excl. excl. excl. 120, excl. 3⁄ 4 5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 1⁄ 2 3⁄ 8 7⁄ 16 1⁄ 2

15⁄ 3⁄ 5⁄

16 4

8 9⁄ 16 9⁄ 16 1⁄ 2 7⁄ 16 1⁄ 2 1⁄ 2

11⁄4 15⁄ 16 5⁄ 8 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2 1⁄ 2 5⁄ 8

13⁄8 11⁄8 3⁄ 4 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2 9⁄ 16 11⁄ 16

11⁄2 11⁄4 7⁄ 8 3⁄ 4 5⁄ 8 5⁄ 8 1⁄ 2 9⁄ 16 3⁄ 4

15⁄8 13⁄8 1 1 3⁄ 4 5⁄ 8 1⁄ 2 5⁄ 8 7⁄ 8

13⁄4 11⁄2 11⁄8 1 7⁄ 8 5⁄ 8 9⁄ 16 3⁄ 4 7⁄ 8

Permissible Variations in Camber for Carbon Steel Sheared and Gas Cut Rectangular Plates Maximum permissible camber, in. (all thicknesses) = 1⁄8-in. ×

total length,ft 5

Permissible Variations in Camber for Carbon Steel Universal Mill Plates, High-Strength Low-Alloy Steel Sheared and Gas Cut Rectangular Plates, Universal Mill Plates, Special Cut Plates Dimension, in. Thickness To 2, incl. Over 2 to 15, incl. Over 2 to 15, incl.

Width

Camber for Thicknesses and Widths Given

All To 30, incl. Over 30 to 60, incl.

1⁄ in. × (total length, ft / 5) 8 3⁄ in. × (total length, ft / 5) 16 1⁄ in. × (total length, ft / 5) 4

General Notes: 1. The longer dimension specified is considered the length, and permissible variations in flatness along the length should not exceed the tabular amount for the specified width in plates up to 12 feet in length. 2. The flatness variations across the width should not exceed the tabular amount for the specified width. 3. When the longer dimension is under 36 inches, the permissible variation should not exceed 1⁄4-in. When the longer dimension is from 36 to 72 inches, inclusive, the permissible variation should not exceed 75 percent of the tabular amount for the specified width, but in no case less than 1⁄4-in. 4. These variations apply to plates which have a specified minimum tensile strength of not more than 60 ksi or compatible chemistry or hardness. The limits in the table are increased 50 percent for plates specified to a higher minimum tensile strength or compatible chemistry or hardness. See also next page.


1 - 198


Table 1-20. Rectangular Sheared Plates and Universal Milled Plates Permissible Variations from Flatness (High-Strength Low-Alloy and Alloy Steel, Hot Rolled or Thermally Treated) Variations from Flatness for Specified Widths, in.

Specified Thickness, in.

To 36 excl.

To 1⁄4, excl. 1⁄ to 3⁄ , excl. 4 8 3⁄ to 1⁄ , excl. 8 2 1⁄ to 3⁄ , excl. 2 4 3⁄ to 1, excl. 4 1 to 2, excl. 2 to 4, excl. 4 to 6, excl. 6 to 8, excl.

13⁄ 16 3⁄ 4 3⁄ 4 5⁄ 8 5⁄ 8 9⁄ 16 1⁄ 2 9⁄ 16 5⁄ 8

36 to 48, 48 to 60, 60 to 72, 72 to 84, 84 to 96, 96 to 108, 108 to excl. excl. excl. excl. excl. excl. 120, excl. 11⁄8

15⁄ 7⁄ 3⁄ 3⁄ 5⁄

13⁄8 11⁄8 15⁄ 16 15⁄ 16 7⁄ 8 3⁄ 4 9⁄ 16 3⁄ 4 3⁄ 4

16 8 4 4

8 9⁄ 16 11⁄ 16 3⁄ 4

17⁄8 13⁄8 15⁄ 16 7⁄ 8 7⁄ 8 13⁄ 16 3⁄ 4 3⁄ 4 15⁄ 16

21⁄4 17⁄8 15⁄16 11⁄8 1 15⁄ 16 3⁄ 4 7⁄ 8 11⁄8

2 13⁄4 11⁄8 1 15⁄ 16 7⁄ 8 3⁄ 4 7⁄ 8 1

23⁄8 2 11⁄2 11⁄4 11⁄8 1 3⁄ 4 15⁄ 16 11⁄4

25⁄8 21⁄4 15⁄8 13⁄8 15⁄16 1 7⁄ 8 11⁄8 15⁄16

General Notes: 1. The longer dimension specified is considered the length, and variations from a flat surface along the length should not exceed the tabular amount for the specified width in plates up to 12 feet in length. 2. The flatness variation across the width should not exceed the tabular amount for the specified width. 3. When the longer dimension is under 36 inches, the variation should not exceed 3⁄8-in. When the longer dimension is from 36 to 72 inches, inclusive the variation should not exceed 75 percent of the tabular amount for the specified width.

Permissible Variations in Width for Universal Mill Plates (15 inches and under in thickness) Variations Over Specified Width for Thickness, in., and Equivalent Weights, lb per sq. ft., Given To 3⁄8, excl.

3⁄ 8

to 5⁄8 excl.

5⁄ 8

to 1, excl.

Specified Width, in.

To 15.3, excl.

15.3 to 25.5, excl.

25.5 to 40.8, excl.

Over 8 to 20, excl. 20 to 36, excl. 36 and over

1⁄ 8 3⁄ 16 5⁄ 16

1⁄

1⁄

3⁄ 16 5⁄ 16 7⁄ 16

3⁄

8 4 8

1 to 2, excl.

Over 2 to 10, incl.


40.8 to 81.7 to 409.0 to 81.7, incl. 409.0, incl. 613.0, incl. 1⁄ 3⁄ 1⁄

4 8 2

Notes: Permissible variation under specified width, 1⁄8-in. Table applies to all steels listed in ASTM A6.


3⁄ 8 7⁄ 16 9⁄ 16

1⁄ 2 9⁄ 16 5⁄ 8

REFERENCES

1 - 199

REFERENCES

American Institute of Steel Construction, 1973, “Commentary on Highly Restrained Welded Connections,” Engineering Journal, 3rd Qtr., AISC, Chicago, IL. American Iron and Steel Institute, 1979, Fire Safe Structural Steel: A Design Guide, AISI, Washington, DC. AISI, 1980, Designing Fire Protection for Steel Columns, 3rd Edition. AISI, 1981, Designing Fire Protection for Steel Trusses, 2nd Edition. AISI, 1984, Designing Fire Protection for Steel Beams. Brockenbrough, R. L. and B. G. Johnston, 1981, USS Steel Design Manual, R. L. Brockenbrough & Assoc. Inc., Pittsburgh, PA. Dill, F. H., 1960, “Structural Steel After a Fire,” Proceedings of the 1960 National Engineering Conference, AISC, New York, NY. Fisher, J. W. and A. W. Pense, 1987, “Experience with Use of Heavy W Shapes in Tension,” Engineering Journal, 2nd Qtr., AISC, Chicago. Lightner, M. W. and R. W. Vanderbeck, 1956, “Factors Involved in Brittle Fracture,” Regional Technical Meetings, AISI, Washington, DC. Rolfe, S. T. and J. M. Barsom, 1986, Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics, Prentice-Hall, Inc., Englewood Cliffs, NJ. Rolfe, S. T., 1977, “Fracture and Fatigue Control in Steel Structures,” Engineering Journal, 1st Qtr., AISC, Chicago. Welding Research Council, 1957, Control of Steel Construction to Avoid Brittle Failure, New York.


2-1

PART 2 ESSENTIALS OF LRFD OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 INTRODUCTION TO LRFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 A. GENERAL PROVISIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 B. DESIGN REQUIREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 C. FRAMES AND OTHER STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 D. TENSION MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19 E. COLUMNS AND OTHER COMPRESSION MEMBERS . . . . . . . . . . . . . . . . 2-22 F. BEAMS AND OTHER FLEXURAL MEMBERS . . . . . . . . . . . . . . . . . . . . 2-27 H. MEMBERS UNDER COMBINED FORCES AND TORSION . . . . . . . . . . . . . 2-34 I. COMPOSITE MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 COMPUTER SOFTWARE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-45


2-2

ESSENTIALS OF LRFD


OVERVIEW

2-3

OVERVIEW The following LRFD topics are covered herein (with the letters A through I in the section headings referring to the corresponding chapters in the LRFD Specification): INTRODUCTION TO LRFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 LRFD Versus ASD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 LRFD Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 A. GENERAL PROVISIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 Loads and Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 B. DESIGN REQUIREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Gross, Net, and Effective Net Areas for Tension Members . . . . . . . . . . . . . . . . 2-11 Gross, Net, and Effective Net Areas for Flexural Members . . . . . . . . . . . . . . . . 2-12 Local Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Limiting Slenderness Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13 C. FRAMES AND OTHER STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Second Order Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Effective Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17 “Leaning” Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18 D. TENSION MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19 Design Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19 Built-Up Members, Eyebars, and Pin-Connected Members . . . . . . . . . . . . . . . . 2-21 E. COLUMNS AND OTHER COMPRESSION MEMBERS . . . . . . . . . . . . . . . . 2-22 Effective Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22 Design Compressive Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22 Flexural-Torsional Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27 Built-Up and Pin-Connected Members . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27 F. BEAMS AND OTHER FLEXURAL MEMBERS . . . . . . . . . . . . . . . . . . . . 2-27 Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27 Design for Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29 Design for Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33 Web Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34 H. MEMBERS UNDER COMBINED FORCES AND TORSION . . . . . . . . . . . . . 2-34 Symmetric Members Subject to Bending and Axial Tension . . . . . . . . . . . . . . . 2-34 Symmetric Members Subject to Bending and Axial Compression . . . . . . . . . . . . 2-37 Bending and Axial Compression—Preliminary Design . . . . . . . . . . . . . . . . . . 2-37 Torsion and Combined Torsion, Flexure, and/or Axial Force . . . . . . . . . . . . . . . 2-40 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2-4

ESSENTIALS OF LRFD

I. COMPOSITE MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Compression Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Flexural Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43 Combined Compression and Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 COMPUTER SOFTWARE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 ELRFD (Electronic LRFD Specification) . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-45


INTRODUCTION TO LRFD

2-5


The intent of this part of the LRFD Manual is to provide a general introduction to the subject. It was written primarily for: (1) engineers experienced in allowable stress design (ASD) who are unfamiliar with LRFD and (2) students and novice engineers. The emphasis is on understanding the most common cases, rather than on completeness and efficiency in design. Regular users of LRFD may also find it helpful to refer to the information provided herein. It should be noted, however, that the governing document is the LRFD Specification (in Part 6 of this volume of the Manual). For optimum design the use of the design aids elsewhere in this Manual is recommended. Among the topics not covered herein are: (1) connections, the subject of Volume II, and (2) noncompact beams and plate girders, for which the reader is referred to Appendices F and G of the LRFD Specification and Part 4 of this volume of the Manual. LRFD Versus ASD

The primary objective of the LRFD Specification is to provide a uniform reliability for steel structures under various loading conditions. This uniformity cannot be obtained with the allowable stress design (ASD) format. The ASD method can be represented by the inequality ΣQi ≤ Rn / F.S.

(2-1)

The left side is the summation of the load effects, Qi (i.e., forces or moments). The right side is the nominal strength or resistance Rn divided by a factor of safety. When divided by the appropriate section property (e.g., area or section modulus), the two sides of the inequality become the calculated stress and allowable stress, respectively. The left side can be expanded as follows: ΣQi = the maximum (absolute value) of the combinations D + L′ (D + L′ + W) × 0.75* (D + L′ + E) × 0.75* D−W D−E where D, L′, W, and E are, respectively, the effects of the dead, live, wind, and earthquake loads; total live load L′ = L + (Lr or S or R) L = Live load due to occupancy Lr = Roof live load S = Snow load R = Nominal load due to initial rainwater or ice exclusive of the ponding contribution *0.75 is the reciprocal of 1.33, which represents the 1/3 increase in allowable stress permitted when wind or earthquake is taken simultaneously with live load. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2-6

ESSENTIALS OF LRFD

ASD, then, is characterized by the use of unfactored service loads in conjunction with a single factor of safety applied to the resistance. Because of the greater variability and, hence, unpredictability of the live load and other loads in comparison with the dead load, a uniform reliability is not possible. LRFD, as its name implies, uses separate factors for each load and for the resistance. Considerable research and experience were needed to establish the appropriate factors. Because the different factors reflect the degree of uncertainty of different loads and combinations of loads and the accuracy of predicted strength, a more uniform reliability is possible. The LRFD method may be summarized by the formula ΣγiQi ≤ φRn

(2-2)

On the left side of the inequality, the required strength is the summation of the various load effects Qi multiplied by their respective load factors γi. The design strength, on the right side, is the nominal strength or resistance Rn multiplied by a resistance factor φ. Values of φ and Rn for columns, beams, etc. are provided throughout the LRFD Specification and will be covered here, as well. According to the LRFD Specification (Section A4.1), ΣγiQi = the maximum absolute value of the following combinations 1.4D 1.2D + 1.6L + 0.5(Lr or S or R) 1.2D + 1.6(Lr or S or R) + (0.5L or 0.8W) 1.2D + 1.3W + 0.5L + 0.5(Lr or S or R) 1.2D ± 1.0E + 0.5L + 0.2S 0.9D ± (1.3W or 1.0E)

(A4-1) (A4-2) (A4-3) (A4-4) (A4-5) (A4-6)

(Exception: The load factor on L in combinations A4-3, A4-4, A4-5 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf). The load effects D, L, Lr, S, R, W, and E are as defined above. The loads should be taken from the governing building code or from ASCE 7, Minimum Design Loads in Buildings and Other Structures (American Society of Civil Engineers, 1988). Where applicable, L should be determined from the reduced live load specified for the given member in the governing code. Earthquake loads should be from the AISC Seismic Provisions for Structural Steel Buildings, which appears in Part 6 of this Manual. LRFD Fundamentals

The following is a brief discussion of the basic concepts of LRFD. A more complete treatment of the subject is available in the Commentary on the LRFD Specification (Section A4 and A5) and in the references cited therein. LRFD is a method for proportioning structures so that no applicable limit state is exceeded when the structure is subjected to all appropriate factored load combinations. Strength limit states are related to safety and load carrying capacity (e.g., the limit states of plastic moment and buckling). Serviceability limit states (e.g., deflections) relate to performance under normal service conditions. In general, a structural member will have several limit states. For a beam, for example, they are flexural strength, shear strength, vertical deflection, etc. Each limit state has associated with it a value of Rn, which defines the boundary of structural usefulness. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


2-7

Because the AISC Specification is concerned primarily with safety, strength limit states are emphasized. The load combinations for determining the required strength were given in expressions A4-1 through A4-6. (Other load combinations, with different values of γi, are appropriate for serviceability; see Chapter L in the LRFD Specification and Commentary.) The AISC load factors (A4-1 through A4-6) are based on ASCE 7. They were originally developed by the A58 Load Factor Subcommittee of the American National Standards Institute, ANSI, (U.S. Department of Commerce, 1980) and are based strictly on load statistics. Being material-independent, they are applicable to all structural materials. Although others have written design codes similar in format to the LRFD Specification, the AISC was the first specification group to adopt the ANSI probability-based load factors. The AISC load factors recognize that when several loads act in combination, only one assumes its maximum lifetime value at a time, while the others are at their “arbitrarypoint-in-time” (APT) values. Each combination models the total design loading condition when a different load is at its maximum: Load Combination A4-1 A4-2 A4-3 A4-4 A4-5 A4-6

Load at its Lifetime (50-year) Maximum D (during construction; other loads not present) L Lr or S or R (a roof load) W (acting in direction of D) E (acting in direction of D) W or E (opposing D)

The other loads, which are APT loads, have mean values considerably lower than the lifetime maximums. To achieve a uniform reliability, every factored load (lifetime maximum or APT) is larger than its mean value by an amount depending on its variability. The AISC resistance factors are based on research recommendations published by Washington University in St. Louis (Galambos et al., 1978) and reviewed by the AISC Specification Advisory Committee. Test data were analyzed to determine the variability of each resistance. In general, the resistance factors are less than one (φ < 1). For uniform reliability, the greater the scatter in the data for a given resistance, the lower its φ factor. Several representative LRFD φ factors for steel members (referenced to the corresponding chapters in the LRFD Specification) are: φt = 0.90 for tensile yielding (Chapter D) φt = 0.75 for tensile fracture (Chapter D) φc = 0.85 for compression (Chapter E) φb = 0.90 for flexure (Chapter F) φv = 0.90 for shear yielding (Chapter F) Resistance factors for other member and connection limit states are given in the LRFD Specification. The following sections (A through I) summarize and explain the corresponding chapters of the LRFD Specification. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2-8

ESSENTIALS OF LRFD

A. GENERAL PROVISIONS

In the LRFD Specification, Sections A4 and A5 define Load and Resistance Factor Design. The remainder of Chapter A contains general provisions which are essentially the same as in the earlier ASD editions of the Specification. Reference is made to the Code of Standard Practice for Steel Buildings and Bridges (adopted in 1992 by AISC), which appears with a Commentary in Part 6 of this LRFD Manual. The Code defines the practices and commonly accepted standards in the structural steel fabricating industry. In the absence of other instructions in the contract documents, these trade practices govern the fabrication and erection of structural steel. The types of construction recognized by the AISC Specification have not changed, except that both “simple framing” (formerly Type 2) and “semi-rigid framing” (formerly Type 3) have been combined into one category, Type PR (partially restrained). “Rigid framing” (formerly Type 1) is now Type FR (fully restrained). Type FR construction is permitted unconditionally. Type PR is allowed only upon evidence that the connections to be used are capable of furnishing, as a minimum, a predictable portion of full end restraint. Type PR construction may necessitate some inelastic, but self-limiting, deformation of a structural steel part. When specifying Type PR construction, the designer should take into account the effects of reduced connection stiffness on the stability of the structure, lateral deflections, and second order bending moments. Semi-rigid connections, once common, are again becoming popular. They offer economies in connection fabrication (compared with FR connections) and reduced member size (compared with simple framing). For information on connections, please refer to Volume II of this LRFD Manual. The yield stresses of the grades of structural steel approved for use range from 36 ksi for the common A36 steel to 100 ksi for A514 steel. Not all rolled shapes and plate thicknesses are available for every yield stress. Availability tables for structural shapes, plates and bars are at the beginning of Part 1 of this LRFD Manual. A36, for many years the dominant structural steel for buildings, is being replaced by the more economical 50 ksi steels. ASTM designations for structural steels with 50 ksi yield stress are: A572 for most applications, A529 for thin-plate members only, and A242 and A588 weathering steels for atmospheric corrosion resistance. A more complete explanation is provided by Table 1-1 in Part 1 of this Manual. However, A36 is still normally specified for connection material, where no appreciable savings can be realized from higher strength steels. Complete and accurate drawings and specifications are necessary for all stages of steel construction. The requirements for design documents are set forth in Section A7 of the LRFD Specification and Section 3 of the AISC Code of Standard Practice. When beam end reactions are not shown on the drawings, the structural steel detailer will refer to the appropriate tables in Part 4 of the LRFD Manual. These tables, which are for uniform loads, may significantly underestimate the effects of the concentrated loads. The recording of beam end reactions on design drawings, which is recommended in all cases, is, therefore, absolutely essential when there are concentrated loads. Beam reactions, column loads, etc., shown on design drawings should be the required strengths calculated from the factored load combinations and should be so noted. Loads and Load Combinations

LRFD Specification Sections A4 (Loads and Load Combinations) and A5 (Design Basis) describe the basic criteria of LRFD. This information was discussed above under AMERICAN INSTITUTE OF STEEL CONSTRUCTION

A. GENERAL PROVISIONS

2-9

Introduction to LRFD. To illustrate the application of load factors, the AISC load combinations will be repeated here with design examples. The required strength is the maximum absolute value of the combinations 1.4D 1.2D + 1.6L + 0.5(Lr or S or R) 1.2D + 1.6(Lr or S or R) + (0.5L or 0.8W) 1.2D + 1.3W + 0.5L + 0.5(Lr or S or R) 1.2D ± 1.0E + 0.5L + 0.2S 0.9D ± (1.3W or 1.0E)

(A4-1) (A4-2) (A4-3) (A4-4) (A4-5) (A4-6)

(The load factor on L in combinations A4-3, A4-4 and A4-5 shall equal 1.0 for garages, areas occupied as placed of public assembly, and all areas where the live load is greater than 100 psf). In the combinations the loads or load effects (i.e., forces or moments) are: D = dead load due to the weight of the structural elements and the permanent features on the structure L = live load due to occupancy and moveable equipment (reduced as permitted by the governing code) Lr = roof live load W= wind load S = snow load E = earthquake load R = nominal load due to initial rainwater or ice exclusive of the ponding contribution The loads are to be taken from the governing building code. In the absence of a code, one may use ASCE 7 Minimum Design Loads for Buildings and Other Structures (American Society of Civil Engineers, 1988). Earthquake loads should be determined from the AISC Seismic Provisions for Structural Steel Buildings, in Part 6 of this Manual. Whether the loads themselves or the load effects are combined, the results are the same, provided the principle of superposition is valid. This is usually true because deflections are small and the stress-strain behavior is linear elastic; consequently, second order effects can usually be neglected. (The analysis of second order effects is covered in Chapter C of the LRFD Specification.) The linear elastic assumption, although not correct at the strength limit states, is valid under normal in-service loads and is permissible as a design assumption under the LRFD Specification. In fact, the Specification (in Section A.5.1) allows the designer the option of elastic or plastic analysis using the factored loads. However, to simplify this presentation, it is assumed that the more prevalent elastic analysis option has been selected.

EXAMPLE A-1

Given:

Solution:

Roof beams W16×31, spaced 7′′-0 center-to-center, support a superimposed dead load of 40 psf. Code specified roof loads are 30 psf downward (due to roof live load, snow, or rain) and 20 psf upward or downward (due to wind). Determine the critical loading for LRFD. D

= 31 plf + 40 psf × 7.0 ft = 311 plf AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 10

ESSENTIALS OF LRFD

L =0 (Lr or S or R) = 30 psf × 7.0 ft = 210 plf W = 20 psf × 7.0 ft = 140 plf E =0 Load Combinations A4-1 A4-2 A4-3 A4-4 A4-5 A4-6a A4-6b

Factored Loads 1.4(311 plf) 1.2(311 plf) + 0 + 0.5(210 plf) 1.2(311 plf) + 1.6 (210 plf) + 0.8(140 plf) 1.2(311 plf) + 1.3(140 plf) + 0 + 0.5(210 plf) 1.2(311 plf) + 0 + 0 + 0.2(210 plf) 0.9 (311 plf) + 1.3 (140 plf) 0.9(311 plf) − 1.3(140 plf)

= 435 plf = 478 plf = 821 plf = 660 plf = 415 plf = 462 plf = 98 plf

The critical factored load combination for design is the third, with a total factored load of 821 plf.

EXAMPLE A-2

Given:

Solution:

The axial loads on a building column resulting from the code-specified service loads have been calculated as: 100 kips from dead load, 150 kips from (reduced) floor live load, 30 kips from the roof (Lr or S or R), 60 kips due to wind, and 50 kips due to earthquake. Determine the required strength of this column. Load Combination A4-1 A4-2 A4-3a A4-3b A4-4 A4-5a A4-5b A4-6a A4-6b A4-6c A4-6d

Factored Axial Load 1.4(100 kips) 1.2(100 kips) + 1.6(150 kips) + 0.5(30 kips) 1.2(100 kips) + 1.6(30 kips) + 0.5(150 kips) 1.2(100 kips) + 1.6(30 kips) + 0.8(60 kips) 1.2(100 kips) + 1.3(60 kips) + 0.5(150 kips) + 0.5(30 kips) 1.2(100 kips) + 1.0(50 kips) + 0.5(150 kips) + 0.2(30 kips) 1.2(100 kips) − 1.0(50 kips) + 0.5(150 kips) + 0.2(30 kips) 0.9(100 kips) + 1.3(60 kips) 0.9(100 kips) − 1.3(60 kips) 0.9(100 kips) + 1.0(50 kips) 0.9(100 kips) − 1.0(50 kips)


= 140 kips = 375 kips = 243 kips = 216 kips = 288 kips = 251 kips = 151 kips = 168 kips = 12 kips = 140 kips = 40 kips

B. DESIGN REQUIREMENTS

2 - 11

The required strength of the column is 375 kips based on the second combination of factored axial loads. As none of the results above are negative, net tension need not be considered in the design of this column. B. DESIGN REQUIREMENTS Gross, Net, and Effective Net Areas for Tension Members

The concept of effective net area, which in earlier editions of the Specification was applied only to bolted members, has been extended to cover members connected by welding as well. As in the past, when tensile forces are transmitted directly to all elements of the member, the net area is used to determine stresses. However, when the tensile forces are transmitted through some, but not all, of the cross-sectional elements of the member, a reduced effective net area Ae is used instead. According to Section B3 of the LRFD Specification Ae = AU

(B3-1)

where A = area as defined below U = reduction coefficient _ = 1 − (x / L) ≤ 0.9, or as defined in (c) or (d) (B3-2) _ x = connection eccentricity. (See Commentary on the LRFD Specification, Section B3 and Figure C-B3.1.) L = length of connection in the direction of loading a. When the forces are transmitted only by bolts A = An = net area of member, in.2 b. When the forces are transmitted by longitudinal welds only or in combination with transverse welds A = Ag = gross area of member, in.2 c. When the forces are transmitted only by transverse welds A = area of directly connected elements, in.2 U = 1.0 d. When the forces are transmitted to a plate by longitudinal welds along both edges at the end of the plate A = area of plate, in.2 l ≥w For l ≥ 2w For 2w > l ≥ 1.5w For 1.5w > l ≥ w

U = 1.00 U = 0.87 U = 0.75 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 12

ESSENTIALS OF LRFD

where l = weld length w = plate width (distance between welds), in. In computing the net area for tension and shear, the width of a bolt hole is taken as 1⁄16-in. greater than the nominal dimension of the hole, which, for standard holes, is 1⁄16-in. larger than the diameter of the bolt. Chains of holes, treated as in the past, are covered in Section B2 of the LRFD Specification. Gross, Net, and Effective Net Areas for Flexural Members

Gross areas are used for elements in compression, in beams and columns. According to Section B10 of the LRFD Specification, the properties of beams and other flexural members are based on the gross section (with no deduction for holes in the tension flange) if 0.75Fu Afn ≥ 0.9Fy Afg

(B10-1)

where Afg = gross flange area, in.2 Afn = net flange area (deducting bolt holes), in.2 Fy = specified minimum yield stress, ksi Fu = minimum tensile strength, ksi Otherwise, an effective tension flange area Afe is used to calculate flexural properties Afe =

5 Fu A 6 Fy fn

(B10-3)

Local Buckling

Steel sections are classified as either compact, noncompact, or slender element sections: • If the flanges are continuously connected to the web and the width-thickness ratios of all the compression elements do not exceed λp, then the section is compact. • If the width-thickness ratio of at least one of its compression elements exceeds λp, but does not exceed λr, the section is noncompact. • If the width-thickness ratio of any compression element exceeds λr, that element is called a slender compression element. Columns with compact and noncompact cross sections are covered by Chapter E of the LRFD Specification. Column cross sections with slender elements require the special design procedure in Appendix B5.3 of the Specification. Beams with compact sections are covered by Chapter F of the LRFD Specification. All other cross sections in bending must be designed in accordance with Appendices B5.3, F1 and/or G. In general, reference to the appendices of the Specification is required for the design of members controlled by local buckling. In slender element sections, local buckling, occurring prior to initial yielding, will limit the strength of the member. Noncompact sections will yield first, but local buckling will precede the development of a fully plastic stress distribution. In actual practice, such cases are not common and can be easily avoided by designing so that: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

B. DESIGN REQUIREMENTS

2 - 13

Table B-1. Limiting Width-Thickness Ratios for Compression Elements* WidthThickness Ratio

Beam Element

Limiting Width-Thickness Ratio, λp General

For Fy = 50 ksi

Flanges of I shapes and channels

b/t

65 / √ F y

9.2

Flanges of square and rectangular box beams

b/t

190 / √ Fy

26.9

Webs in flexural compression

h / tw

640 / √ Fy

90.5

Webs in combined flexural and axial compression

h / tw

253 / √ Fy **

35.8

Column Element

WidthThickness Ratio

Limiting Width-Thickness Ratio, λr General

For Fy = 50 ksi

Flanges of I shapes and channels and plates projecting from compression elements

b/t

95 / √ F y

13.4

Webs in axial compression

h / tw

253 / √ Fy

35.8

*For the complete table, see LRFD Specification, Section B5, Table B5.1. **This is a simplified, conservative version of the corresponding entry in Table B5.1 of the LRFD Specification.

• for beams, the width-thickness ratios of all compression elements ≤ λp; • for columns, the width-thickness ratios of all elements ≤ λr. Table B-1, which is an abridged version of Table B5.1 in the LRFD Specification, should be useful for this purpose. The formulas for λp for beam elements and λr for column elements are tabulated, together with the corresponding numerical values for 50 ksi steel. The definitions of “width” for use in determining the width-thickness ratios of the elements of various structural shapes are stated in Section B5 of the LRFD Specification. They are shown graphically in Figure B-1. Compact section criteria for W shapes and other I-shaped cross sections are listed in the Properties Tables in Part 1 of LRFD Manual. Limiting Slenderness Ratios

For members whose design is based on compressive force, the slenderness ratio Kl / r preferably should not exceed 200. For members whose design is based on tensile force, the slenderness ratio l / r preferably should not exceed 300. The above limitation does not apply to rods in tension. K = effective length factor, defined in Section C below l = distance between points of lateral support (lx or ly), in. r = radius of gyration (rx or ry), in. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 14

ESSENTIALS OF LRFD

C. FRAMES AND OTHER STRUCTURES Second Order Effects

As stated in Section C1 of the LRFD Specification, an analysis of second order effects is required; i.e., the additional moments due to the axial loads acting on the deformed structure must be considered. In lieu of a second order analysis for Mu, the required flexural strength, the LRFD Specification (in Section C1) presents the following simplified method: Mu = B1Mnt + B2Mlt

(C1-1)

The components of the total factored moment, determined from a first order elastic analysis (neglecting second order effects) are divided into two groups, Mnt and Mlt. Each group is in turn multiplied by a magnification factor B1 or B2 and the results are added to approximate the actual second order factored moment Mu. (The method, as explained here, is valid where the moment connections are Type FR, fully restrained. The analysis for Type PR, or partially restrained, moment connections is beyond the scope of this section.) Beam-columns are generally columns in frames, which are either braced (Mlt = 0) or unbraced (Mlt ≠ 0). Mnt is the moment in the member assuming there is no lateral translation of the frame; Mlt is the moment due to lateral translation. Mnt includes the moments resulting from the gravity loads, as determined manually or by computer, using one of the customary (elastic, first order) methods. The moments from the lateral loads are classified as Mlt; i.e., due to lateral translation. If both the frame and its vertical loads are symmetric, Mlt from the vertical loads is zero. However, if either the vertical loads or the frame is asymmetric and the frame is not braced, lateral translation occurs and Mlt ≠ 0. The procedure for obtaining Mlt in this case involves:

b=

bf

b=

2

bf

bf

b = bf

2

bf

bf

h

h

h

bf

b

t

hw

t

h

b = b f – 3t h = h w – 3t

Fig. B-1. Definitions of widths (b and h) for use in Table B-1. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

C. FRAMES AND OTHER STRUCTURES

2 - 15

a. applying fictitious horizontal reactions at each floor level to prevent lateral translation, and b. using the reverse of these reactions as the “sway forces” for determining Mlt. In general, Mlt for an unbraced frame is the sum of the moments due to the lateral loads and these “sway forces,” as illustrated in Figure C-1. The magnification factors applied to Mnt and Mlt are, respectively, B1 and B2. As shown in Figure C-2, B1 accounts for the secondary Pδ member effect in all frames (including sway-inhibited) and B2 covers the P∆ story effect in unbraced frames. The expressions for B1 and B2 follow: B1 =

Cm ≥ 1.0 (1 − Pu / Pe1 )

(C1-2)

where Pu = the factored axial compressive force on the member, kips Pe1 = Pe as listed in Table C-1 as a function of the slenderness ratio Kl / r, with effective length factor K = 1.0 and considering l / r in the plane of bending only l = unbraced length of the member, in. r = radius of gyration of its cross section, in. Cm = a coefficient to be taken as follows: V1 V2

V3

P1

P1

R1

V1 +R1

P2

P2 R 2

V2 +R2

P3

P3

Original Frame

=

R3

Nonsway Frame for M nt

V3 +R3

Sway Frame for M t

+

Fig. C-1. Frame models for Mnt and Mlt.

∆P

P

H

δ

M1=Mnt+Pδ =B1Mnt

(a) Column in Braced Frame

L

M t =HL M2=M t +P∆ =B2M t

(b) Column in Unbraced Frame

Fig. C-2. Illustrations of secondary effects. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 16

ESSENTIALS OF LRFD

Table C-1. Values of Pe / Ag for Use in Equation C1-2 and C1-5 for Steel of Any Yield Stress Note: Multiply tabulated values by Ag (the gross cross-sectional area of the member) to obtain Pe

Kl / r 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Pe / Ag Pe / Ag Pe / Ag Pe / Ag Pe / Ag Pe / Ag (ksi) Kl / r (ksi) Kl / r (ksi) Kl / r (ksi) Kl / r (ksi) Kl / r (ksi) 649.02 591.36 541.06 496.91 457.95 423.40 392.62 365.07 340.33 318.02 297.83 279.51 262.83 247.59 233.65 220.85 209.07 198.21 188.18 178.89 170.27 162.26 154.80 147.84 141.34 135.26 129.57 124.23 119.21 114.49

Note: Pe / Ag =

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

110.04 105.85 101.89 98.15 94.62 91.27 88.08 85.08 82.22 79.51 76.92 74.46 72.11 69.88 67.74 65.71 63.76 61.90 60.12 58.41 56.78 55.21 53.71 52.57 50.88 49.55 48.27 47.04 45.86 44.72

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

43.62 42.57 41.55 40.56 39.62 38.70 37.81 36.96 36.13 35.34 34.56 33.82 33.09 32.39 31.71 31.06 30.42 29.80 29.20 28.62 28.06 27.51 26.98 26.46 25.96 25.47 25.00 24.54 24.09 23.65

111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

23.23 22.82 22.42 22.02 21.64 21.27 20.91 20.56 20.21 19.88 19.55 19.23 18.92 18.61 18.32 18.03 17.75 17.47 17.20 16.94 16.68 16.43 16.18 15.94 15.70 15.47 15.25 15.03 14.81 14.60

141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170

14.40 14.19 14.00 13.80 13.61 13.43 13.25 13.07 12.89 12.72 12.55 12.39 12.23 12.07 11.91 11.76 11.61 11.47 11.32 11.18 11.04 10.91 10.77 10.64 10.51 10.39 10.26 10.14 10.02 9.90

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

9.79 9.67 9.56 9.45 9.35 9.24 9.14 9.03 8.93 8.83 8.74 8.64 8.55 8.45 8.36 8.27 8.18 8.10 8.01 7.93 7.85 7.76 7.68 7.60 7.53 7.45 7.38 7.30 7.23 7.16

π2E (K l /r)2

a. For compression members not subject to transverse loading between their supports in the plane of bending, Cm = 0.6 − 0.4(M1 / M2)

(C1-3)

where M1 / M2 is the ratio of the smaller to larger moment at the ends of that portion of the member unbraced in the plane of bending under consideration. M1 / M2 is positive when the member is bending in reverse curvature, negative when bending in single curvature. b. For compression members subjected to transverse loading between their supports, the value of Cm can be determined by rational analysis, or the following values may be used: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

C. FRAMES AND OTHER STRUCTURES

2 - 17

for members with ends restrained against rotation . . . . . . . . . . Cm = 0.85 for members with ends unrestrained against rotation . . . . . . . . . Cm = 1.0 Two alternative equations are given for B2 in the LRFD Specification B2 =

1 ΣPu  ∆oh  1−   ΣH  L 

B2 =

1 ΣPu 1− ΣPe2

(C1-4)

(C1-5)

where ΣPu = required axial strength of all columns in a story, i.e., the total factored gravity load above that level, kips ∆oh = translational deflection of the story under consideration, in. ΣH = sum of all story horizontal forces producing ∆oh, kips L = story height, in. ΣPe2 = the summation of Pe2 for all rigid-frame columns in a story; Pe2 is determined from Table C-1, considering the actual slenderness ratio Kl / r of each column in its plane of bending K = effective length factor (see below) Of the two expressions for B2, the first (Equation C1-4) is better suited for design office practice. The quantity (∆oh / L) is the story drift index. For many structures, particularly tall buildings, a maximum drift index is one of the design criteria. Using this value in Equation C1-4 will facilitate the evaluation of B2. In general, two values of B2 are obtained for each story of a building, one for each of the major directions. B1 is evaluated separately for every column; two values of B1 are needed for biaxial bending. Using Equations C1-1 through C1-5, the appropriate Mux and Muy are determined for each column. Effective Length

As in previous editions of the AISC Specification, the effective length of Kl is used (instead of the actual unbraced length l) to account for the influence of end-conditions in the design of compression members. A number of acceptable methods have been utilized to evaluate K, the effective length factor. They are discussed in Section C2 of the Commentary on the LRFD Specification. One method will be shown here. Table C-2, which is also Table C-C2.1 in the Commentary, is taken from the Structural Stability Research Council (SSRC) Guide to Stability Design Criteria for Metal Structures. It relates K to the rotational and translational restraints at the ends of the column. Theoretical values for K are given, as well as the recommendations of the SSRC. The basic case is d, the classical pin-ended column, for which K = 1.0. Theoretical K values for the other cases are determined by the distances between points of inflection. The more conservative SSRC recommendations reflect the fact that perfect fixity can never be attained in actual structures. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 18

ESSENTIALS OF LRFD

Table C-2. Effective Length Factors (K) for Columns Buckled shape of column is shown by dashed line

(a)

(b)

(c)

Theoretical K value

0.5

0.7

1.0

Recommended design value when ideal conditions are approximated

0.65

0.80

1.2

End condition code

(d)

(e)

(f)

1.0

2.0

2.0

1.0

2.10

2.0

Rotation fixed and translation fixed Rotation free and translation fixed Rotation fixed and translation free Rotation free and translation free

Like its predecessors, the LRFD Specification (in Section C2) distinguishes between columns in braced and unbraced frames. In braced frames, sidesway is inhibited by attachment to diagonal bracing or shear walls. Cases a, b, and d in Table C-2 represent columns in braced frames; K ≤ 1.0. The LRFD Specification requires that for compression members in braced frames, K “shall be taken as unity, unless structural analysis shows that a smaller value may be used.” Common practice is to assume conservatively K = 1.0 for columns in braced frames and compression members in trusses. The other cases in Table C-2, c, e, and f, are in unbraced frames (sidesway uninhibited); K ≥ 1.0. The SSRC recommendations given in Table C-2 are appropriate for design. “Leaning” Columns

The concept of the “leaning” column, although not related exclusively to LRFD, is new to the 1993 LRFD Specification. A leaning column is one which is pin ended and does not participate in providing lateral stability to the structure. As a result it relies on the columns in other parts of the structure for stability. In analyzing and designing unbraced frames, the effects of the leaning columns must be considered (as required by Section C2.2 of the LRFD Specification). For further information the reader is referred to: (1) Part 3 of this Manual. (2) the Commentary on the LRFD Specification, Section C2, and (3) a paper on this subject (Geschwindner, 1993). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

D. TENSION MEMBERS

2 - 19

D. TENSION MEMBERS Design Tensile Strength

The design philosophy for tension members is the same in the LRFD and ASD Specifications: a. The limit state of yielding in the gross section is intended to prevent excessive elongation of the member. Usually, the portion of the total member length occupied by fastener holes is small. The effect of early yielding at the reduced cross sections on the total member elongation is negligible. Use of the area of the gross section is appropriate. b. The second limit state involves fracture at the section with the minimum effective net area. The design strength of tension members, φtPn, as given in Section D1 of the LRFD Specification, is the lesser of the following: a. For yielding in the gross section, φt = 0.90 Pn = Fy Ag

(D1-1)

b. For fracture in the net section, φt = 0.75 Pn = Fu Ae

(D1-2)

where Ae Ag Fy Fu Pn

= effective net area, in.2 (see Section B, above) = gross area of member, in.2 = specified minimum yield stress, ksi = specified minimum tensile strength, ksi = nominal axial strength, kips

For 50 ksi steels, Fy = 50 ksi and minimum Fu = 65 ksi. Accordingly a. For yielding in the gross section, φtPn = 0.9 × 50 ksi × Ag = 45.0 ksi × Ag

(2-3)

b. For fracture in the net section, φtPn = 0.75 × 65 ksi × Ae = 48.8 ksi × Ae

(2-4)

The limit state of block shear rupture may govern the design tensile strength. For information on block shear, see Section J4.3 of the LRFD Specification and Part 8 (in Volume II) of this LRFD Manual. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 20

ESSENTIALS OF LRFD

EXAMPLE D-1

Given:

Determine the design strength of a W8×24 as a tension member in 50 ksi steel. How much dead load can it support?

Solution:

If there are no holes in the member, Ae = Ag and Equation 2-3 governs φtPn = 45.0 ksi × Ag = 45.0 ksi × 7.08 in.2 = 319 kips Assuming that dead load is the only load, the governing load combination from Section A is 1.4D. Then, the required tensile strength Pu = 1.4PD ≤φtPn = 319 kips PD≤ 319 kips/1.4 = 228 kips maximum dead load that can be supported by the member.

EXAMPLE D-2

Given:

Repeat Example D-1 for a W8×24 in 50 ksi steel with four 1-in. diameter holes, two per flange, along the member (i.e., not at its ends) for miscellaneous attachments. See Figure D-1(a).

Solution:

a. For yielding in the gross section φtPn = 319 kips, as in Example D-1. b. For fracture in the net section Ae = An = Ag − 4 × (dhole + 1⁄16-in.) × tf = 7.08 in.2 − 4 × (1 + 1⁄16-in.) × 0.400 in. = 5.38 in.2 φtPn = 48.8 ksi × Ae = 48.8 ksi × 5.38 in.2 = 263 kips < 319 kips Fracture in the net section governs. Pu = 1.4 PD ≤ φtPn = 263 kips PD ≤ 263 kips / 1.4 = 188 kips

W8x24

x=y=0.695 in.

tf

WT4x12 WT4x12

(a)

(b)

Fig. D-1 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

D. TENSION MEMBERS

2 - 21

Note: If the holes had been at the end connection of the tension member, the U reduction coefficient would apply in the calculation of an effective net area.

EXAMPLE D-3

Given:

Repeat Example D-2 for holes at a bolted end-connection. There are a total of eight 1-in. diameter holes, as shown in Figure D-1(a), on two planes, 4 in. center-to-center.

Solution:

a. For yielding in the gross section φtPn = 319 kips, as in Example D-1. b. For fracture in the net section, according to Equation B3-1 in Section B above, the effective net area Ae = AU = AnU where An = 5.38 in.2 as in Example D-2 _ x U = 1 − , L = 4 in.* L

_ According to Commentary Figure C-B3.1(a), x for a W8×24 in this case is taken as that for a WT4×12. _ From the properties of a WT4×12 given in Part 1 of this Manual, x = y = 0.695 in. See Figure D-1(b). U=1−

0.695 in. = 0.826 4 in.

Thus Ae = 5.38 in.2 × 0.826 = 4.45 in.2 φtPn = 48.8 ksi × Ae = 48.8 ksi × 4.45 in.2 = 217 kips < 319 kips Fracture in the net section governs. Again, assuming that dead load is the only load, Pu = 1.4PD ≤ φtPn = 217 kips PD ≤ 217 kips / 1.4 = 155 kips maximum dead load that can be supported by the member. Built-Up Members, Eyebars, and Pin-Connected Members

See Section D2 and D3 in the LRFD Specification. *In lieu of calculating U, the Commentary on the LRFD Specification (Section B3) permits the use of more conservative values of U listed therein. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 22

ESSENTIALS OF LRFD

E. COLUMNS AND OTHER COMPRESSION MEMBERS Effective Length

For a discussion of the effective length Kl for columns, refer to Section C above. Design Compressive Strength

Although the column strength equations have been revised for compatibility with LRFD and recent research on column behavior, the philosophy and procedures of column design in LRFD are similar with those in ASD. The direct design of columns with W and other rolled shapes is facilitated by the column strength tables in Part 3 of this LRFD Manual, which show the design compressive strength φcPn as a function of KL (the effective unbraced length in feet). Columns with cross sections not tabulated (e.g., built-up columns) can be designed iteratively, as in the past, with the aid of tables listing design stresses versus Kl / r, the slenderness ratio. Such tables are given in the Appendix of the LRFD Specification for 36 and 50 ksi structural steels, and below (Table E-1) for 50 ksi steel. There are two equations governing column strength, based on the limit state of flexural buckling, one for inelastic buckling (Equation E2-2) and the other (Equation E2-3) for elastic, or Euler, buckling. Equation E2-2 is an empirical relationship for the inelastic range, while Equation E2-3 is the familiar Euler formula multiplied by 0.877. Both equations include the effects of residual stresses and initial out-of-straightness. The boundary between inelastic and elastic instability is λc = 1.5, where the parameter λc =

Kl rπ

 √

Fy E

(E2-4)

For axially loaded columns with all elements having width-thickness ratios < λr (in Section B5.1 of the LRFD Specification), the design compressive strength = φcPn where φc = 0.85 Pn = AgFcr

(E2-1)

Ag = gross area of member, in.2 a. For λc ≤ 1.5 2

Fcr = (0.658λc)Fy

(E2-2)

As is done in the Commentary on Section E2, this equation can be expressed in exponential form Fcr = [exp (−0.419λ2c )]Fy where exp(x) = ex AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(C-E2-1)

E. COLUMNS AND OTHER COMPRESSION MEMBERS

2 - 23

Table E-1. Design Stress for Compression Members of 50 ksi Specified Minimum Yield Stress Steel, φc = 0.85* Kl r

F cr (ksi)

Kl r

F cr (ksi)

1 2 3 4 5

42.50 42.49 42.47 42.45 42.42

41 42 43 44 45

37.59 37.36 37.13 36.89 36.65

6 7 8 9 10

42.39 42.35 42.30 42.25 42.19

46 47 48 49 50

11 12 13 14 15

42.13 42.05 41.98 41.90 41.81

16 17 18 19 20

Kl r

F cr (ksi)

Kl r

F cr (ksi)

Kl r

F cr (ksi)

81 82 83 84 85

26.31 26.00 25.68 25.37 25.06

121 122 123 124 125

14.57 14.33 14.10 13.88 13.66

161 162 163 164 165

8.23 8.13 8.03 7.93 7.84

36.41 36.16 35.91 35.66 35.40

86 87 88 89 90

24.75 24.44 24.13 23.82 23.51

126 127 128 129 130

13.44 13.23 13.02 12.82 12.62

166 167 168 169 170

7.74 7.65 7.56 7.47 7.38

51 52 53 54 55

35.14 34.88 34.61 34.34 34.07

91 92 93 94 95

23.20 22.89 22.58 22.28 21.97

131 132 133 134 135

12.43 12.25 12.06 11.88 11.71

171 172 173 174 175

7.30 7.21 7.13 7.05 6.97

41.71 41.61 41.51 41.39 41.28

56 57 58 59 60

33.79 33.51 33.23 32.95 32.67

96 97 98 99 100

21.67 21.36 21.06 20.76 20.46

136 137 138 139 140

11.54 11.37 11.20 11.04 10.89

176 177 178 179 180

6.89 6.81 6.73 6.66 6.59

21 22 23 24 25

41.15 41.02 40.89 40.75 40.60

61 62 63 64 65

32.38 32.09 31.80 31.50 31.21

101 102 103 104 105

20.16 19.86 19.57 19.28 18.98

141 142 143 144 145

10.73 10.58 10.43 10.29 10.15

181 182 183 184 185

6.51 6.44 6.37 6.30 6.23

26 27 28 29 30

40.45 40.29 40.13 39.97 39.79

66 67 68 69 70

30.91 30.61 30.31 30.01 29.70

106 107 108 109 110

18.69 18.40 18.12 17.83 17.55

146 147 148 149 150

10.01 9.87 9.74 9.61 9.48

186 187 188 189 190

6.17 6.10 6.04 5.97 5.91

31 32 33 34 35

39.62 39.43 39.25 39.06 38.86

71 72 73 74 75

29.40 20.09 28.79 28.48 28.17

111 112 113 114 115

17.27 16.99 16.71 16.42 16.13

151 152 153 154 155

9.36 9.23 9.11 9.00 8.88

191 192 193 194 195

5.85 5.79 5.73 5.67 5.61

36 37 38 39 40

38.66 38.45 38.24 38.03 37.81

76 77 78 79 80

27.86 27.55 27.24 26.93 26.62

116 117 118 119 120

15.86 15.59 15.32 15.07 14.82

156 157 158 159 160

8.77 8.66 8.55 8.44 8.33

196 197 198 199 200

5.55 5.50 5.44 5.39 5.33

* When element width-to-thickness ratio exceeds λr, see Appendix B5.3 of LRFD Specification


2 - 24

ESSENTIALS OF LRFD

b. For λc > 1.5 0.877 Fcr =  2  Fy  λc 

(E2-3)

where Fy = specified minimum yield stress, ksi E = modulus of elasticity, ksi K = effective length factor l = unbraced length of member, in. r = governing radius of gyration about plane of buckling, in. For 50 ksi steel λc =

Kl 1 r π

 √

Kl Kl 50 ksi = 0.0132 or = 75.7λc r r 29,000 ksi

(2-5)

The boundary between inelastic and elastic buckling (λc = 1.5) for 50 ksi steel is Kl = 75.7 × 1.5 = 113.5 r The column strength equations in terms of Kl / r for 50 ksi steel become φcPn = (φcFcr )Ag

(2-6)

Fcr = {exp[−7.3 × 10−5(Kl / r)2]} × 50 ksi

(2-7)

where φc = 0.85 a. For Kl / r ≤ 113.5

b. For Kl / r ≤ 113.5 Fcr =

2.51 × 105 ksi (Kl / r)2

(2-8)

Based on Equations 2-7 and 2-8, Table E-1 gives the design stresses for 50 ksi steel columns for the full range of slenderness ratios. Determining the design strength of a given 50 ksi steel column merely involves using Equation 2-6 in connection with Table E-1. The appropriate design stress (φcFcr) from Table E-1 is multiplied by the cross-sectional area to obtain the design strength φcPn.

EXAMPLE E-1

Given:

Design a 25-ft high, free standing A618 (Fy = 50 ksi) steel pipe column to support a water tank with a weight of 75 kips at full capacity. See Figure E-1.

Solution:

For a live load of 75 kips, the required column strength (from Section A) is Pu = 1.6PL = 1.6 × 75 kips = 120 kips. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

E. COLUMNS AND OTHER COMPRESSION MEMBERS

2 - 25

From Table C-2, case e, recommended K = 2.1. KL = 2.1 × 25.0 ft = 52.5 ft. Try a standard 12-in. diameter pipe (A = 14.6 in.2, I = 279 in.4): r =√  I/A =√  279 in. / 14.6 in.2 = 4.37 in. Kl 52.5 ft × 12 in./ft = = 144.2 4.37 in r From Table E-1, φcFcr = 10.3 ksi The design compressive strength φcPn = (φcFcr )Ag = 10.3 ksi × 14.6 in.2 = 150 kips > 120 kips required o.k. To complete the design, bending due to lateral loads (i.e., wind and earthquake) should also be considered. See Sections F and H. EXAMPLE E-2

Determine the adequacy of a W14×120 building column.

Given:

50 ksi steel; K = 1.0; story height = 12.0 ft; required strength based on the maximum total factored load is 1,300 kips. KxLx = Ky Ly = 1.0 × 12.0 ft = 12.0 ft Because ry < rx,  Kl  Ky Ly 12.0 ft × 12 in./ft = = 38.5   maximum = ry 3.74 in. r From Table E-1, φcFcr = 38.14 ksi Design compressive strength φcPn = (φcFcr)Ag = 38.14 ksi × 35.3 in.2 = 1,346 kips > 1,300 kips required o.k.

L = 25.0 ft.

Solution:

Fig. E-1 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 26

Select the most economical W14 column for the case shown in Figures E-2 and E-3.

Lower Level

Intermediate Level

Upper Level

Fig. E-2. Plan views.

12 -0

Upper Level

Intermediate Level

12 -0

EXAMPLE E-3

ESSENTIALS OF LRFD

Lower Level

Fig. E-3. Elevation. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

F. BEAMS AND OTHER FLEXURAL MEMBERS

2 - 27

Given:

50 ksi steel; K = 1.0; required strength based on the maximum total factored load is 1,300 kips. The column is braced in both directions at the upper and lower levels, and in the weak direction at the intermediate level.

Solution:

Try a W14×120 (as in Example E-2): Kx lx rx

=

1.0 × 24.0 ft × 12 in./ft = 46.2 6.24 in.

Ky ly ry

=

1.0 × 12.0 ft × 12 in./ft = 38.5 3.74 in.

Kx lx Kl max = = 46.2 rx r From Table E-1, φcFcr = 36.35 ksi Required Ag =

1,300 kips = 35.8 in.2 > 35.3 in.2 provided 36.35 ksi

W14×120 n.g. By inspection W14×132 is o.k. Use W14×132 Flexural-Torsional Buckling

As stated in Section E3 of the LRFD Specification and Commentary, torsional and flexural-torsional buckling generally do not govern the design of doubly symmetric rolled shapes in compression. For other cross sections, see Section E3 and Appendix E3 of the LRFD Specification. Built-Up and Pin-Connected Members

These members are covered, respectively, in Section E4 and E5 of the LRFD Specification. F. BEAMS AND OTHER FLEXURAL MEMBERS

Chapter F of the LRFD Specification covers compact beams. Compactness criteria are given in Table B5.1 of the LRFD Specification and are summarized in Table B-1 above. To prevent torsion, wide-flange shapes must be loaded in either plane of symmetry, channels must be loaded through the shear center parallel to the web, or restraint against twisting must be provided at load points and points of support. Torsion combined with flexure and axial force combined with flexure are covered in Chapter H of the LRFD Specification. This section explains the provisions of the LRFD Specification for compact rolled beams. For other compact and noncompact flexural members, refer to Appendix F of the Specification; plate girders are in Appendix G. Flexure

To understand the provisions of the LRFD Specification regarding flexural design, it is helpful to review briefly some aspects of elementary beam theory. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 28

ESSENTIALS OF LRFD

Under working loads (and until initial yielding) the distributions of flexural strains and stresses over the cross-section of a beam are linear. As shown in Figure F-1, they vary from maximum compression at the extreme fibers on one side (the top) to zero at the neutral, or centroidal, axis to maximum tension at the extreme fibers on the other side (the bottom). The relationship between moment and maximum bending stress (tension or compression) at a given cross section is M = Sfb

(2-9)

where M= bending moment due to the applied loads, kip-in. S = elastic section modulus, in the direction of bending, in.3 I = c fb = maximum bending stress, ksi I = moment of inertia of the cross section about its centroidal axis, in.4 c = distance from the elastic neutral axis to the extreme fiber, in. Similarly, at initial yielding Mr = SFy

(2-10)

where Mr = bending moment coinciding with first yielding, kip-in. If additional load is applied, the strains continue to increase; the stresses, however, are, limited to Fy. Yielding proceeds from the outer fibers inward until a plastic hinge is developed, as shown in Figure F-1. At full plastification of the cross section Mp = ZFy

(2-11)

where Mp = plastic moment, kip-in Z = plastic section modulus, in the direction of bending, in.3 Due to the presence of residual stresses (prior to loading, as a consequence of the rolling operation), yielding begins at an applied stress of (Fy − Fr). Equation 2-10 should be modified to

STRAINS

BEAM

Cross Section

Compression

STRESSES Fy

Fy

Working Load

Fy Initial Yielding

Tension

Fig. F-1. Flexural strains and stresses. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Fy Plastic Hinge


Mr = S(Fy − Fr )

2 - 29

(2-12)

where Fr = the maximum compressive residual stress in either flange, ksi = 10 ksi for rolled shapes, 16.5 ksi for welded shapes The definition of plastic moment in Equation 2-11 is still valid, because it is not affected by residual stresses. Design for Flexure

a. Assuming Cb = 1.0 Compact sections will not experience local buckling before the formation of a plastic hinge. The occurrence of lateral-torsional buckling of the member depends on the unbraced length Lb. As implied by the term lateral-torsional buckling, overall instability of a beam requires that twisting of the member occur simultaneously with lateral buckling of the compression flange. Lb is the distance between points braced to prevent twist of the cross section. Many beams can be considered continuously braced; e.g., beams supporting a metal deck, if the deck is intermittently welded to the compression flange. Compact wide flange and channel members bending about their major (or x) axes can develop their full plastic moment Mp without buckling if Lb ≤ Lp. If Lb = Lr, the nominal flexural strength is Mr, the moment at first yielding adjusted for residual stresses. The nominal moment capacity (Mn) for Lp < Lb < Lr is Mr < Mn < Mp. Compact shapes bent about their minor (or y) axes will not buckle before developing Mp, regardless of Lb. Flexural design strength, governed by the limit state of lateral-torsional buckling, is φbMn, where φb = 0.90 and Mn the nominal flexural strength is as follows: Mn = Mp = ZxFy for bending about the major axis if Lb ≤ Lp Mn = Mp = ZyFy for bending about the minor axis regardless of Lb Lp =

300ry = 42.4ry for 50 ksi steel Fy √

Mn = Mr = Sx(Fy − Fr ) = Sx(Fy − 10 ksi) for rolled shapes bending about the major axis if Lb = Lr

(2-13) (2-14) (2-15) (2-16)

Mn for bending about the major axis, if Lp < Lb < Lr, is determined by linear interpolation between Equations 2-13 and 2-16; i.e.,  Lb − Lp  Mn = Mp − (Mp − Mr)    Lr − Lp 

(2-17)

The definition for the limiting laterally unbraced length Lr is given in the LRFD Specification (in Equations F1-6, 8, and 9) and will not be repeated here. For bending about the major axis if Lb > Lr, Mn = Mcr ≤ Mr

(2-18)

The case of Lb > Lr is beyond the scope of this section. The reader is referred to Section F1.2b of LRFD Specification (specifically Equation F1-13, where the critical moment AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 30

ESSENTIALS OF LRFD

Table F-1. Values of Cb for Simply Supported Beams Braced at Ends of Span Load

Lateral Bracing Along Span

Cb

Concentrated at center

None

1.32

At centerline only

1.67

None

1.14

At centerline only

1.30

Uniform

Mcr is controlled by lateral-torsional buckling). This case is also covered in the beam graphs in Part 4 of this LRFD Manual. b. All values of Cb Cb is the bending coefficient. A new expression for Cb is given in the LRFD Specification. (It is more accurate than the one previously shown.) Cb =

12.5Mmax 2.5Mmax + 3MA + 4MB + 3Mc

(F1-3)

where M is the absolute value of a moment in the unbraced beam segment as follows: Mmax, the maximum MA, at the quarter point MB, at the centerline Mc, at the three-quarter point The purpose of Cb is to account for the influence of moment gradient on lateral-torsional buckling. The flexural strength equations with Cb = 1.0 are based on a uniform moment along a laterally unsupported beam segment causing single curvature buckling of the member. Other loadings are less severe, resulting in higher flexural strengths; Cb ≥ 1.0. Typical values of Cb are given in Table F-1. For unbraced cantilevers, Cb = 1.0. Cb can conservatively be taken as 1.0 for all cases. For all values of Cb, the flexural design strength φbMn, where φb = 0.90, is given in the LRFD Specification in terms of a nominal flexural strength Mn varying as follows: Mn = Mp = ZxFy

(2-13)

for bending about the major axis if Lb ≤ Lm Mn = CbMr = CbSx(Fy − 10 ksi) ≤ Mp

(2-19)

for bending about the major axis if Lb = Lr. For bending about the major axis if Lm < Lb < Lr, linear interpolation is used  Lb − Lp   Mn = Cb Mp − (Mp − Mr)   ≤ Mp  Lr − Lp  

(F1-2)

Mn = Mcr ≤ CbMr and Mp

(2-20)

If Lb > Lr,



2 - 31

The determination of Mn for a given Lb can best be done graphically, as illustrated in Figure F-2. The required parameters for each W shape are given in the beam design table in Part 4 of the LRFD Manual, an excerpt of which is shown herein as Table F-2. If Cb = 1.0, the coordinates for constructing the graph are (Lp, Mp), and (Lr, Mr). For Cb > 1.0, the key coordinates are (Lp, Cb Mp) and (Lr, Cb Mr). Note that Mn cannot exceed the plastic moment Mp. Lm, then, can be derived graphically as the upper limit of Lb for which Mn = Mp. If Lb > Lr, the beam graphs in Part 4 of the LRFD Manual can be used to determine Mcr.

EXAMPLE F-1

Select the required W shape for a 30-foot simple floor beam with full lateral support carrying a dead load (including its own weight) of 1.5 kips per linear foot and a live load of 3.0 kips per linear foot. Assume 50 ksi steel and:

Given:

a. There is no member depth limitation b. The deepest member is a W18 The governing load combination in Section A is A4-2:

Solution:

1.2D + 1.6L + 0.5(Lr or S or R) = 1.2 × 1.5 klf + 1.6 × 3.0 klf + 0 = 6.6 klf Required Mu =

wL2 6.6 klf × (30.0 ft)2 = = 743 kip-ft 8 8

Flexural design strength φbMn ≥ 743 kip-ft

CbMp Mn for Cb=1.0

Mn for Cb>1.0

Mp

Mn

Mcr for Cb=1.0

CbMr

Mcr for Cb >1.0

Mr

Lm

Lp

Lr

Lb

Fig. F-2. Determination of nominal flexural strength M n. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 32

ESSENTIALS OF LRFD

Table F-2. Excerpt from Load Factor Design Selection Table (LRFD Manual, Part 4) For Fy = 50 ksi

Zx (in.3)

Shape

φbMp (kip-ft)

φbMr (kip-ft)

Lp (ft)

Lr (ft)

224 221 212 211

W24× ×84 W21×93 W14×120 W18×97

840 829 795 791

588 576 570 564

6.9 6.5 13.2 9.4

18.6 19.4 46.2 27.4

200 198 196 192 186 186

W24× ×76 W16×100 W21×83 W14×109 W18×86 W12×120

750 743 735 720 698 698

528 525 513 519 498 489

6.8 8.9 6.5 13.2 9.3 11.1

18.0 29.3 18.5 43.2 26.1 50.0

177 175

W24× ×68 W16×89

664 656

462 465

6.6 8.8

17.4 27.3

Note: Flexural design strength φbMn = φbMp, as tabulated is valid for Lb ≤ Lm. If Cb = 1.0, Lm = Lp; otherwise, Lm > Lp. φb = 0.90.

a. In Table F-2, the most economical beams are in boldface print. Of the boldfaced beams, the lightest one with φbMn = φbMp ≥ 743 kip-ft is a W24×76 b. By inspection of Table F-2, the lightest W18 with φbMn = φbMp ≥ 743 kip-ft is a W18×97.

EXAMPLE F-2

Given:

Determine the flexural design strength of a 30-ft long simply supported W24×76 girder (of 50 ksi steel) with a concentrated load and lateral support, both at midspan.

Solution:

From Table F-1, Cb = 1.67 Lb = 30.0 ft/2 = 15.0 ft From Equation F1-2:  Lb − Lp    φbMn = Cb φbMp − (φbMp − φbMr)    ≤ φbMp  Lr − Lp    From Table F-2 for a W24×76: φbMp = 750 kip-ft φbMr = 528 kip-ft Lp = 6.8 ft AMERICAN INSTITUTE OF STEEL CONSTRUCTION


Lr

2 - 33

= 18.0 ft

15.0 ft − 6.8 ft   φbMn = 1.67 750 kip−ft − (750 − 528) kip−ft × 18.0 ft − 6.8 ft   = 981 kip-ft > 750 kip-ft Use φb Mn = φb Mp = 750 kip-ft In this case, even though the unbraced length Lb > Lp, the design flexural strength is φbMp because Cb > 1.0. Design for Shear

The design shear strength is defined by the equations in Section F2 of the LRFD Specification. Shear in wide-flange and channel sections is resisted by the area of the web (Aw), which is taken as the overall depth d times the web thickness tw. For webs of 50 ksi steel without transverse stiffeners, the design shear strength φvVn, where φv = 0.90, and the nominal shear strength Vn are as follows: For

h ≤ 59 (including all rolled W and channel shapes), tw

Vn = 30.0 ksi × dtw φvVn= 27.0 ksi × dtw For 59
74, tw =

132,000 dtw ksi (h / tw)2

φvVn =

118,000 dtw ksi (h / tw)2

Vn

h

(2-23)

tw

d

tw

h

Fig. F-3. Definitions of d, h, and tw for W and channel shapes. AMERICAN INSTITUTE OF

STEEL CONSTRUCTION

2 - 34

ESSENTIALS OF LRFD

Shear strength is governed by the following limit states; Equation 2-21 by yielding of the web; Equation 2-22, by inelastic buckling of the web; and Equation 2-23 by elastic buckling.

EXAMPLE F-3

Given:

Solution:

Check the adequacy of a W30×99 beam of 50 ksi steel to carry a load resulting in maximum shears of 100 kips due to dead load and 150 kips due to live load. Required shear strength = Vu = 1.2D + 1.6L = 1.2 × 100 kips + 1.6 × 150 kips = 360 kips Design shear strength = φvVn = 27.0 ksi × dtw = 27.0 ksi × 29.65 in. × 0.520 in. = 416 kips > 360 kips required o.k.

Web Openings

See Section F4 of the LRFD Specification and Commentary, and the references given in the Commentary. H. MEMBERS UNDER COMBINED FORCES AND TORSION Symmetric Members Subject to Bending and Axial Tension

The interaction of flexure and tension in singly and doubly symmetric shapes is governed by Equations H1-1a and H1-1b, as follows: For

For

Pu ≥ 0.2, φPn

Pu < 0.2, φPn

Muy  Pu 8  Mux +  +  ≤ 1.0 9 φPn  φb Mnx φb Mny 

(H1-1a)

 Mux Muy  Pu + +  ≤ 1.0 2φPn  φb Mnx φb Mny

(H1-1b)

where = required tensile strength; i.e., the total factored tensile force, kips = design tensile strength, φtPn, kips = resistance factor for tension, φt = 0.90 = nominal tensile strength as defined in Chapter D of the LRFD Specification, kips Mu = required flexural strength; i.e., the moment due to the total factored load, kipin. or kip-ft. (Subscript x or y denotes the axis about which bending occurs.) φb Mn = design flexural strength, kip-in. or kip-ft = resistance factor for flexure = 0.90 φb

Pu φPn φ Pn

AMERICAN INSTITUTE OF

STEEL CONSTRUCTION

H. MEMBERS UNDER COMBINED FORCES AND TORSION

Mn

2 - 35

= nominal flexural strength determined in accordance with the appropriate equations in Chapter F of the LRFD Specification, kip-in. or kip-ft

Interaction Equations H1-1a and H1-1b cover the general case of biaxial bending combined with axial force. They are also valid for uniaxial bending (i.e., when Mux = 0 or Muy = 0). In this case, they reduce to the form plotted in Figure H-1. Pure biaxial bending (with Pu = 0) is covered by Equation H1-1b. EXAMPLE H-1

Given:

Check the adequacy of a W10×22 tension member of 50 ksi steel to carry loads resulting in the following factored load combination: Pu = 55 kips Muy = 20 kip-ft Mux = 0

Solution:

From Section D above for 50 ksi steel, φPn = φtPn = 45.0 ksi × Ag = 45.0 ksi × 6.49 in.2 = 292 kips Pu 55 kips = = 0.188 < 0.20; therefore, Equation H1-1b governs. φPn 292 kips For bending about the y axis only, Equation H1-1b becomes: Pu Muy + ≤ 1.0 2φPn φb Mny φ Pn

Pu

Pu

φ Pn

+

8 Mu = 9 φb M n

0.2 φPn

( )

1 Pu 2 φPn

Mu

+

Mu =1 φb M n 0.9 φb M n

φb M n

Fig. H-1. Interaction Equations H1-1a and H1-1b modified for axial load combined with bending about one axis only. AMERICAN INSTITUTE OF

STEEL CONSTRUCTION

2 - 36

ESSENTIALS OF LRFD

From Section F above for 50 ksi steel, Mn = Mp = ZyFy = 50 ksi × Zy for minor-axis bending (regardless of the unbraced length). φbMny = 0.90 × 50 ksi × Zy = 45.0 ksi × Zy = 45.0 ksi ×

6.10 in.3 12 in./ft

= 22.9 kip-ft for a W10×22 member 20 kip−ft Pu Muy 0.188 + = + = 0.094 + 0.873 2 22.9 kip−ft 2φPn φb Mny = 0.967 < 1.0 o.k.

EXAMPLE H-2

Given:

Check the same tension member, a W10×22 in 50 ksi steel, 4.0 ft long, subjected to the following combination of factored loads: Pu = 140 kips Mux = 55 kip-ft Muy = 0 Cb = 1.0

Solution:

Again, φPn = 292 kips Pu

φPn

=

140 kips = 0.479 > 0.20; Equation H1-1a governs. 292 kips

For bending about the x axis only, Equation H1-1a becomes Pu 8 Mux + ≤ 1.0 φPn 9 φb Mnx From Section F above for 50 ksi steel, Mn = Mp = ZxFy = 50 ksi × Zx for major-axis bending if Lb ≤ Lp for (Cb = 1.0). Assume unbraced length, Lb = 4.0 ft. By Equation 2-15 in Section F, Lp = 42.4ry for 50 ksi steel. For a W10×22, ry = 1.33 in., Zx = 26.0 in.3 Lp =

42.4 × 1.33 in. = 4.7 ft 12 in./ft

Lb = 4.0 ft < Lp = 4.7 ft Then Mnx = 50 ksi × Zx φb Mnx

0.90 × 50 ksi × 26.0 in.3 12 in./ft = 97.5 kip-ft for a W10×22 member =

AMERICAN INSTITUTE OF

STEEL CONSTRUCTION


2 - 37

55 kip−ft Pu 8Mux 8 + = 0.479 + × 9 97.5 kip−ft φPn 9φb Mnx = 0.479 + 0.501 = 0.980 < 1.0 o.k. Symmetric Members Subject to Bending and Axial Compression

The interaction of compression and flexure in beam-columns with singly and doubly symmetric cross sections is governed by Equations H1-1a and H1-1b, repeated here for convenience: For

For

Pu ≥ 0.2, φPn

Pu < 0.2, φPn

Muy  Pu 8  Mux +  +  ≤ 1.0 φPn 9  φb Mnx φb Mny

(H1-1a)

 Mux Muy  Pu + +  ≤ 1.0 2φPn  φb Mnx φb Mny

(H1-1b)

The definitions of the ter ms in the for mulas, which differ in some cases from those given above, are as follows: = required compressive strength; i.e., the total factored compressive force, kips = design compressive strength, φc Pn, kips = resistance factor for compression, φc = 0.85 = nominal compressive strength as defined in Chapter E of the LRFD Specification, kips Mu = required flexural strength including second-order effects, kip-in. or kip-ft φb Mn = design flexural strength, kip-in. or kip-ft = resistance factor for flexure = 0.90 φb Mn = nominal flexural strength from Chapter F of the LRFD Specification, kip-in. or kip-ft

Pu φPn φ Pn

The second-order analysis required for Mu involves the determination of the additional moment due to the action of the axial compressive forces on a deformed structure. In lieu of a second-order analysis, the simplified method given in Chapter C of the LRFD Specification (and in Section C above) may be used. However, in applying the simplified method, the additional moments obtained for beam-columns must also be distributed to connected members and connections (to satisfy equilibrium). Bending and Axial Compression—Preliminary Design

The design of a beam-column is a trial and error process which can become tedious, particularly with the repeated solution of Interaction Equation H1-1a or H1-1b. A rapid method for the selection of a trial section is given in this LRFD Manual, Part 3, under the heading Combined Axial and Bending Loading (Interaction). As in earlier editions of the AISC Manual, the Interaction Equations are approximated by an equation which converts bending moments to equivalent axial loads: Pu eq = Pu + Muxm + Muymu AMERICAN INSTITUTE OF

STEEL CONSTRUCTION

2 - 38

ESSENTIALS OF LRFD

where = equivalent axial load to be checked against the column load table, kips Pu eq Pu, Mux, Muy are defined in the Interaction Equations for compression and bending m, u are factors tabulated in this LRFD Manual, Part 3 As soon as a satisfactory trial section has been found (i.e., one for which Pu eq ≤ tabulated φc Pn), a final verification should be made with the appropriate Interaction Equation, H1-1a or H1-1b

EXAMPLE H-3

Given:

Check the adequacy of a W14×176 beam-column, 14.0 ft in height floor-to-floor, in a braced symmetrical frame in 50 ksi steel. The member is subjected to the following factored forces due to symmetrical gravity loads: Pu = 1,400 kips; Mx = 200 kip-ft, My = 70 kip-ft (reverse curvature bending with equal end moments about both axes); and no loads along the member.

Solution:

For a braced frame, K = 1.0 KxLx = KyLy = 14.0 ft For a W14×176: A Zx Zy rx ry Kl / rx Kl / ry

= 51.8 in.2 = 320 in.3 = 163 in.3 = 6.43 in. = 4.02 in. = (14.0 ft × 12 in./ft) / 6.43 in. = 26.1 = (14.0 ft × 12 in./ft) / 4.02 in. = 41.8

From Table E-1, above, φcFcr = 37.4 ksi for Kl / r = 41.8 in 50 ksi steel. φcPn = (φc Fcr) A = 37.4 ksi × 51.8 in.2 = 1,940 kips Pu 1,400 kips = = 0.72 > 0.2, Interaction Equation H1-1a φc Pn 1,940 kips governs.

Since

For a braced frame, Mlt = 0. From Equation C1-1: Mux = B1x Mntx , where Mntx = 200 kip-ft; and Muy = B1y Mnty , where Mnty = 70 kip-ft From Equations C1-2 and C1-3: B1 =

Cm > 1.0 (1 − Pu / Pe1 )

where in this case (a braced frame with no transverse loading), Cm = 0.6 − 0.4(M1 / M2) AMERICAN INSTITUTE OF

STEEL CONSTRUCTION


2 - 39

For reverse curvature bending and equal end moments: M1 / M2 = +1.0 = 0.6 − 0.4(1.0) = 0.2

Cm

From Table C-1: Pe1x = 420 ksi × Ag = 420 ksi × 51.8 in.2 = 21,756 kips From Table C-1: Pe1y = 164 ksi × Ag = 164 ksi × 51.8 in.2 = 8,495 kips B1x =

Cmx 0.2 = = 0.2 (1 − Pu / Pe1x ) (1 − 1,400 kips / 21,756 kips)

Use B1x = 1.0, per Equation C1-2. B1y =

Cmy 0.2 = = 0.2 (1 − Pu / Pe1y ) 1 − 1,400 kips / 8,495 kips)

Use B1y = 1.0, per Equation C1-2. Mux = 1.0 × 200 kip-ft Muy = 1.0 × 70 kip-ft From Equation 2-15 for 50 ksi steel, Lp = 42.4ry =

42.4 × 4.02 in. = 14.2 ft 12 in./ft

Since Lb = 14.0 ft < Lp = 14.2 ft, Mnx = Mpx = ZxFy Mny

= Mpy = Zy Fy

φbFy

= 0.90 × 50 ksi = 45.0 ksi

φb Mnx = φbFy Zx =

45.0 ksi × 320 in.3 = 1,200 kip-ft 12 in./ft

φb Mny = φbFy Zy =

45.0 ksi × 163 in.3 = 611 kip-ft 12 in./ft

By Interaction Equation H1-1a 70 kip−ft  8 1,400 kips 8  200 kip−ft +  +  = 0.72 + (0.17+0.11) 1,940 kips 9  1,200 kip−ft 611 kip−ft  9 = 0.72 + 0.25 = 0.97 < 1.0 W14×176 is o.k. AMERICAN INSTITUTE OF

STEEL CONSTRUCTION

2 - 40

ESSENTIALS OF LRFD

EXAMPLE H-4

Given:

Check the adequacy of a W14×176 beam-column (Fy = 50 ksi) in an unbraced symmetrical frame subjected to the following factored forces: Pu Mux My KxLx

= 1,400 kips (due to gravity plus wind) = 300 kip-ft (due to wind only) =0 = Ky Ly = 14.0 ft

Drift index, ∆oh / L ≤ 0.0025 (or 1⁄400) ΣPu = 24,000 kips ΣH = 800 kips Solution:

As in Example H-3, for a W14×176 with KL = 14.0 ft, φcPn = 1,940 kips. Pu 1,400 kips = = 0.72 > 0.2, Interaction Equation H1-1a φcPn 1,940 kips governs.

Since

Because Mntx = Mnty = Mlty = 0 and only Mltx ≠ 0, Mux = B2Mltx and Muy = 0. Mltx = 300 kip-ft According to Equation C1-4, B2 =

1 1 = = 1.08 ΣPu  ∆oh 1 − 24,000 kips (0.0025) 1−   800 kips ΣH  L 

Mux = 1.08 × 300 kip-ft = 324 kip-ft Because Lb < Lp = 14.2 ft, Mnx = Mpx = ZxFy; φb Mnx = 1,200 kip-ft as in Example H-3. By Interaction Equation H1-1a: 8 1,400 kips 8 324 kip−ft + = 0.72 + 0.27 = 0.96 < 1.0 9 1,940 kips 9 1,200 kip−ft W14×176 is o.k. Torsion and Combined Torsion, Flexure, and/or Axial Force

Criteria for members subjected to torsion and torsion combined with other forces are given in Section H2 of the LRFD Specification. They require the calculation of normal and shear stresses by elastic analysis of the member under the factored loads. The AISC book Torsional Analysis of Steel Members (American Institute of Steel Construction, 1983) provides design aids and examples for the determination of torsional stresses. Extensive coverage is given there to wide-flange shapes (W, S, and HP), channels (C and MC) and Z shapes. For these members, the charts and formulas simplify considerably AMERICAN INSTITUTE OF

STEEL CONSTRUCTION


2 - 41

the calculation of torsional rotations, torsional normal and shear stresses, and the combination of torsional with flexural stresses. In the LRFD Specification, fun = the total normal stress under factored load (ksi) from torsion and all other causes fuv = the total shear stress under factored load (ksi) from torsion and all other causes The criteria are as follows: a. For the limit state of yielding under normal stress fun ≤ φFy, where φ = 0.90

(H2-1)

fun ≤ 0.90 × 50 ksi = 45.0 ksi

(2-24)

For 50 ksi steel,

b. For the limit state of yielding under shear stress, fuv ≤ 0.60φFy, where φ = 0.90

(H2-2)

fuv ≤ 0.60 × 0.90 × 50 ksi = 27.0 ksi

(2-25)

For 50 ksi steel,

c. For the limit state of buckling, fun ≤ φcFcr or fuv ≤ φcFcr, as applicable, where φc = 0.85

(H2-3)

For 50 ksi steel, values of φcFcr are given in Table E-1, in Section E above. Torsion will accompany flexure when the line of action of a lateral load does not pass through the shear center. For wide flange and other doubly symmetric shapes, the shear center is located at the centroid. Singly symmetric shapes have their shear centers on the axis of symmetry, but not at the centroid. (The location of the shear center of channel sections is given in the Properties tables in Part 1 of this LRFD Manual.) Open sections, such as wide-flange and channel, are very inefficient in resisting torsion; i.e., torsional rotations can be large and torsional stresses relatively high. It is best to avoid torsion by detailing the loads and reactions to act through the shear center of the member. In the case of spandrel members supporting building facade elements, this may not be possible. Heavy exterior masonry walls and stone panels can impose severe torsional loads on spandrel beams. The following are suggestions for eliminating or reducing this kind of torsion: 1. Wall elements may span between floors. The moment due to the eccentricity of the wall with respect to the edge beams can be resisted by lateral forces acting through the floor diaphragms. Torsion would not be imposed on the spandrel beams. 2. If facade panels extend only a partial story height below the floor line, the use of diagonal steel “kickers” may be possible. These light members would provide lateral support to the wall panels. Torsion from the panels would be resisted by forces originating from structural elements other than the spandrel beams. 3. Even if torsion must be resisted by the edge members, providing intermediate torsional supports can be helpful. Reducing the span over which the torsion acts will reduce torsional stresses. If there are secondary beams framing into a spandrel girder, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 - 42

ESSENTIALS OF LRFD

the beams can act as intermediate torsional supports for the girder. By adding top and bottom moment plates to the connections of the beams with the girder, the bending resistances of the beams can be mobilized to provide the required torsional reactions along the girder. 4. Closed sections provide considerably better resistance to torsion than open sections; torsional rotations and stresses are much lower for box beams than for wide-flange members. For members subjected to torsion, it may be advisable to use box sections or to simulate a box shape by welding one or two side plates to a W shape. I. COMPOSITE MEMBERS

Chapter I of the LRFD Specification covers composite members. Included are concreteencased and concrete-filled steel columns and beam columns, as well as steel beams interactive with the concrete slabs they support and steel beams encased in concrete. Unlike traditional structural steel design, which considers only the strength of the steel, composite design assumes that the steel and concrete work together in resisting loads. This results in more economical designs, as the quantity of steel can be reduced. Compression Members

Composite columns (concrete-encased and concrete-filled) must satisfy the limitations in Section I2 of the LRFD Specification. The design strength of axially loaded composite columns is φcPn, where φc = 0.85 and the nominal axial compressive strength is determined from Equations E2-1 through E2-4 above with the following modifications: As replaces Ag, rm replaces r, Fmy replaces Fy, and Em replaces E. Fmy = Fy + c1Fyr

Ar Ac + c2 fc′ As As

(I2-1)

Ac As

(I2-2)

Em = E + c3Ec

rm = radius of gyraton of the steel shape, pipe, or tubing, in. (For steel shapes it shall not be less than 0.3 times the overall thickness of the composite cross section in the plane of buckling.) where Ec = w1.5√ fc′ and Fmy Fy Fyr fc′ Em E Ec w Ac Ar

= modified yield stress for the design of composite columns, ksi = specified minimum yield stress of the structural steel shape, ksi = specified minimum yield stress of the longitudinal reinforcing bars, ksi = specified compressive strength of the concrete, ksi = modified modulus of elasticity for the design of composite columns, ksi = modulus of elasticity of steel = 29,000 ksi = modulus of elasticity of concrete, ksi = unit weight of concrete, lb/ft3 = cross-sectional area of concrete, in.2 = cross-sectional area of longitudinal reinforcing bars, in.2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

I. COMPOSITE MEMBERS

2 - 43

As = cross-sectional area of structural steel, in.2 c1, c2, c3 = numerical coefficients. For concrete-filled pipe and tubing: c1 = 1.0, c2 = 0.85, and c3 = 0.4; for concrete-encased shapes c1 = 0.7, c2 = 0.6, and c3 = 0.2 Composite columns can be designed by using the Composite Columns Tables in Part 5 of this LRFD Manual (or the numerous tables in AISC Steel Design Guide No. 6: Load and Resistance Factor Design of W-Shapes Encased in Concrete) for the cross sections tabulated therein, or the above equations for all cross sections. Flexural Members

The most common case of a composite flexural member is a steel beam interacting with a concrete slab by means of stud or channel shear connectors. The slab can be a solid reinforced concrete slab, but is usually concrete on a corrugated metal deck. The effective width of concrete slab acting compositely with a steel beam is determined by three criteria. On either side of the beam centerline, the effective width of concrete slab cannot exceed: a. one-eighth of the beam span, b. one-half the distance to the centerline of the adjacent beam, or c. the distance to the edge of the slab. The following pertains to rolled W shapes in regions of positive moment, the predominant use of composite beam design. Other cases (e.g., plate girders and negative moments) are covered in Chapter I of the LRFD Specification. The horizontal shear force between the steel beam and concrete slab, to be transferred by the shear connectors between the points of zero and maximum positive moments, is the minimum of: a. 0.85fc′Ac (the maximum possible compressive force in the concrete), b. AsFy (the maximum possible tensile force in the steel), and c. ΣQn (the strength of the shear connectors). For W shapes, the design flexural strength φb Mn, with φb = 0.85, is based on: a. a uniform compressive stress of 0.85fc′ and zero tensile strength in the concrete b. a uniform steel stress of Fy in the tension area and compression area (if any) of the steel section, and c. equilibrium; i.e., the sum of the tensile forces equals the sum of the compressive forces. The above is valid for shored and unshored construction. However, in the latter case, it is also necessary to check the bare steel beam for adequacy to support the wet concrete and other construction loads (properly factored). The number of shear connectors required between a point of maximum moment and the nearest location of zero moment is n=

Vh Qn


(2-26)

2 - 44

ESSENTIALS OF LRFD

where Vh = the total horizontal shear force to be transferred, kips = the minimum of 0.85fc′Ac, AsFy, and ΣQn Qn = the shear strength of one connector The nominal strength of a single stud shear connector in a solid concrete slab is  fc′Ec ≤ AscFu Qn = 0.5Asc√

(I5-1)

where Asc = cross-sectional area of a stud shear connector, in.2 fc′ = specified compressive strength of concrete, ksi Fu = minimum specified tensile strength of a stud shear connector, ksi Ec = modulus of elasticity of concrete, ksi Special provisions for shear connectors embedded in concrete on formed steel deck are given in Section I3.5 of the LRFD Specification. Among them are reduction factors (given by Equation I3-1 and I3-2) to be applied to the middle term of Equation I5-1 above. The design of composite beams and the selection of shear connectors can be accomplished with the tables in Part 5 of this LRFD Manual. The design shear strength for composite beams is determined by the shear strength of the steel web, as for noncomposite beams; see Section F above. Combined Compression and Flexure

Composite beam-columns are covered in Section I4 of the LRFD Specification. COMPUTER SOFTWARE ELRFD* (Electronic LRFD Specification)

ELRFD is a sophisticated computer program for interactively checking structural steel building components for compliance with the AISC Specification. All provisions of Chapters A through H and K of the LRFD Specification are included in the knowledge base of ELRFD. The ELRFD program checks whether the member satisfies all limit states and limitation requirements set by the LRFD Specification and reports which sections of the specification are satisfied or violated. One can review in detail the formulas and rules used in the evaluation and interactively assess any mathematical expression appearing on the screen. Design data produced by the software can be viewed and/or printed in report form for permanent record. ELRFD has a fully interactive Windows-based user interface.

*ELRFD is copyright AISC and Visual Edge Software, Ltd. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

REFERENCES

2 - 45

REFERENCES

American Institute of Steel Construction, Inc., 1983, Torsional Analysis of Steel Members, AISC, Chicago, IL. American Society of Civil Engineers, 1988, Minimum Design Loads for Buildings and Other Structures, ASCE 7-88, New York, NY. Galambos, T. V., et al., 1978, Eight LRFD Papers, Journal of the Structural Division, ASCE, Vol. 104, No. ST9 (September 1978), New York. Geschwindner, L., 1993, “The ‘Leaning’ Column in ASD and LRFD,” Proceedings of the 1993 National Steel Construction Conference, AISC, Chicago. U.S. Department of Commerce, 1980, Development of a Probability Based Criterion for American National Standard A58, NBS (National Bureau of Standards) Special Publication 577, Washington, DC.


3-1

PART 3 COLUMN DESIGN OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 DESIGN STRENGTH OF COLUMNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5 W and HP Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15 Steel Pipe and Structural Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-35 Double Angles and WT Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-53 Single-Angle Struts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-104 COLUMN BASE PLATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-117 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-117


3-2

COLUMN DESIGN


OVERVIEW

3-3

OVERVIEW Column tables with design compressive strengths, in kips, are located as follows: W shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16 HP shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-31 Steel pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-36 Structural tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-39 Double angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-57 WT shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-83 Single angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-104 Additional information related to column design is provided as follows: Effective length factor (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5 Alignment charts, Figure 3-1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Stiffness reduction factors (SRF), Table 3-1 . . . . . . . . . . . . . . . . . . . . . . . . 3-7 “Leaning” columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 Combined axial and bending loading (Interaction) . . . . . . . . . . . . . . . . . . . . 3-11 Preliminary design of beam-columns, Table 3-2 . . . . . . . . . . . . . . . . . . . . 3-12


3-4

COLUMN DESIGN


DESIGN STRENGTH OF COLUMNS

3-5

DESIGN STRENGTH OF COLUMNS General Notes

Column Load Tables

Column Load Tables are presented for W, WT, and HP shapes, pipe, structural tubing, double angles, and single angles. Tabular loads are computed in accordance with the AISC LRFD Specification, Sections E2 and E3 and Appendix E3, for axially loaded members having effective unsupported lengths indicated to the left of each table. The effective length KL is the actual unbraced length, in feet, multiplied by the factor K, which depends on the rotational restraint at the ends of the unbraced length and the means available to resist lateral movements. Table C-C2.1 in the Commentary on the LRFD Specification is a guide in selecting the K-factor. Interpolation between the idealized cases is a matter of engineering judgment. Once sections have been selected for the several framing members, the alignment charts in Figure 3-1 [reproduced from the Structural Stability Research Council Guide (Galambos, 1988) here and in Figure C-C2.2 of the Commentary on the LRFD Specification] afford a means to obtain more precise values for K, if desired. For column behavior in the inelastic range, the values of G as defined in Figure 3-1 may be reduced by the values given in Table 3-1, as illustrated in Example 3-3. Tables for W, WT, and HP shapes and for double and single angles are provided for 36 ksi and 50 ksi yield stress steels. Tables for steel pipe are provided for 36 ksi, and for structural tubing for 46 ksi yield stress steel. All design strengths are tabulated in kips. Values are not shown when Kl / r exceeds 200. In all tables, except double angle and WT tables, design strengths are given for effective lengths with respect to the minor axis calculated by LRFD Specification Section E2. When the minor axis is braced at closer intervals than the major axis, the strength of the column must be investigated with reference to both major (X-X) and minor (Y-Y) axes. The ratio rx / ry included in these tables provides a convenient method for investigating the strength of a column with respect to its major axis. To obtain an effective length with respect to the minor axis equivalent in load carrying capacity to the actual effective length about the major axis, divide the major axis effective length by rx / ry ratio. Compare this length with the actual effective length about the minor axis. The longer of the two lengths will control the design, and the design strength may be taken from the table opposite the longer of the two effective lengths with respect to the minor axis. The double angle and WT tables show values for effective lengths about both axes. Properties useful to the designer are listed at the bottom of the column design strength tables. Additional notes relating specifically to the W and HP shape tables, the steel pipe and structural tubing tables, and the double and single angle tables precede each of these groups of tables.

EXAMPLE 3-1

Given:

Design the lightest W shape of Fy = 50 ksi steel to support a factored concentric load of 1,400 kips. The effective length with respect to its minor axis is 16 feet. The effective length with respect to its major axis is 31 feet. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3-6

COLUMN DESIGN

Solution:

Enter the appropriate Column Load Table for W shapes at effective length of KL = 16 ft. Since W14 columns are generally most efficient, begin with the W14 table and work downward, weightwise. Select W14×145, good for 1,530 kips > 1,400 kips rx / ry = 1.59. Equivalent L = 31 ft / 1.59 = 19.5 ft > 16 ft Equivalent effective length for X-X axis controls.

GA ∞ 50.0

K

GB ∞

1.0

50.0 10.0

10.0 5.0 3.0

5.0 0.9

5.0

100.0 50.0 30.0

20.0

4.0

20.0

10.0 9.0 8.0 7.0

3.0

2.0

0.5

0.5

0.4

0.4

0.3

0.3

pl

am

2

0.6

0.7

6.0 5.0

10.0 9.0 8.0 7.0

-3 e3

Ex

3-

0.8 0.7

ple

1.0

0.8 0.7

am

1.0

Ex

0.8

0.2

GB ∞ 20.0 10.0

100.0 50.0 30.0

3.0

2.0

0.6

K

GA ∞

6.0 5.0

2.2

4.0

4.0

2.0

3.0

3.0

1.75 2.0

2.36

2.0 1.5

0.6

0.2 1.0

0.1

1.0

0.1

0.26 0

0.5

0

SIDESWAY INHIBITED

1.0

0

0

SIDESWAY UNINHIBITED

Fig. 3-1. Alignment charts for effective length of columns in continuous frames. The subscripts A and B refer to the joints at the two ends of the column section being considered. G is defined as Σ(Ic / Lc) G= Σ(Ig / Lg) in which Σ indicates a summation of all members rigidly connected to that joint and lying on the plane in which buckling of the column is being considered. Ic is the moment of inertia and Lc the unsupported length of a column section, and Ig is the moment of inertia and Lg is the unsupported length of a girder or other restraining member. Ic and Ig are taken about axes perpendicular to the plane of buckling being considered. For column ends supported but not rigidly connected to a footing or foundation, G is theoretically infinity, but, unless actually designed a true friction free pin, may be taken as 10 for practical designs. If the column end is rigidly attached to a properly designed footing, G may be taken as 1.0. Smaller values may be used if justified by analysis. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


3-7

Table 3-1. Stiffness Reduction Factors (SRF) for Columns Pu / A

Fy

Pu / A

ksi 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27

Fy

ksi 36 ksi

50 ksi

— — — — — — — — — — — — 0.05 0.14 0.22 0.30

0.03 0.09 0.16 0.21 0.27 0.33 0.38 0.44 0.49 0.53 0.58 0.63 0.67 0.71 0.75 0.79

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11

36 ksi

50 ksi

0.38 0.45 0.52 0.58 0.65 0.70 0.76 0.81 0.85 0.89 0.92 0.95 0.97 0.99 1.00 ↓

0.82 0.85 0.88 0.90 0.93 0.95 0.97 0.98 0.99 1.00 ↓

— indicates not applicable.

Re-enter table for effective length of 19.5 ft to satisfy axial load of 1,400 kips, select W14×145. By interpolation, the column is good for 1,410 kips. Use W14×145 column

EXAMPLE 3-2

Given:

Design an 11-ft long W12 interior bay column to support a factored concentric axial roof load of 1,100 kips. The column is rigidly framed at the top by 30-ft long W30×116 girders connected to each flange. Column moment is zero due to the assumption of equal and offsetting moments in the girders. The column is braced normal to its web at top and base so that sidesway is inhibited in this plane. Use Fy = 50 ksi steel.

Solution:

a. Check Y-Y axis: Assume the column is pin-connected at the top and bottom with sidesway inhibited. From Table C-C2.1 in the Commentary for condition (d), K = 1.0: Effective length = 11 ft Enter Column Load Table: W12×106 good for 1,160 kips > 1,100 kips o.k. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3-8

COLUMN DESIGN

b. Check X-X axis: 1. Preliminary selection: Assume sidesway uninhibited and pin-connected at base. From Table C-C2.1 for condition (f): K = 2.0 Approximate effective length relative to X-X axis: 2.0 × 11 = 22.0 ft From Properties section in tables, for W12 column: rx / ry ≈ 1.76 Equivalent effective length relative to the Y-Y axis: 22.0

1.76

≈ 12.5 ft > 11.0 ft

Therefore, effective length for X-X axis is critical. Enter Column Load Table with an effective length of 12.5 ft: W12×106 column, by interpolation, good for 1,115 kips > 1,100 kips o.k. 1. Final selection Try W12×106 Using Figure 3-1 (sidesway uninhibited): Ix for W12×106 column = 933 in.4 Ix for W30×116 girder = 4,930 in.4 G (base)

= 10 (assume supported but not rigidly connected)

G (top)

=

933 / 11 = 0.258, say 0.26 (4,930 × 2) / 30

Connect points GA = 10 and GB = 0.26, read K = 1.75 For W12×106, rx / ry = 1.76 Actual effective length relative to Y-Y axis: 1.75 × 11.0 = 10.9 ft < 11.0 ft 1.76 Since the effective length for Y-Y axis is not critical, Use W12×106 column AMERICAN INSTITUTE OF STEEL CONSTRUCTION


3-9

EXAMPLE 3-3

Given:

Using the alignment chart, Figure 3-1 (sidesway uninhibited) and Table 3-1 (Stiffness Reduction Factors), design columns for the bent shown, by the inelastic K-factor procedure. Let Fy = 50 ksi. Assume continuous support in the transverse direction.

Solution:

The alignment charts in Figure 3-1 are applicable to elastic columns. By multiplying G-values times the stiffness reduction factor Et / E, the charts may be used for inelastic columns. Since Et / E ≈ Fcr, inelastic / Fcr, elastic, the relationship may be written as Ginelastic = (Fcr, inelastic / Fcr, elastic)Gelastic. By utilizing the calculated stress Pu / A a direct solution is possible, using the following steps: 1. For a known value of factored axial load, Pu = 1,100 kips, select a trial column size. Assume W12×120 A = 35.3 in.2, Ix = 1,070 in.4, rx = 5.51 in. 2. Calculate Pu / A: Pu / A = 1,100 kips / 35.3 in.2 = 31.2 ksi 3. From Table 3-1, determine the Stiffness Reduction Factor (SRF); SRF = 0.62. For values of Pu / A smaller than those with entries in Table 3-1, the column is elastic, and the reduction factor is 1.0. 4. Determine Gelastic: Gelastic (bottom) = 10 1,100 k

1,100 k

W16x31 IX = 375 15′

20′

Fig. 3-2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3 - 10

COLUMN DESIGN

Gelastic (top)

=

1,070 / 15 = 3.80 375 / 20

5. Calculate Ginelastic = SRF × Gelastic: Ginelastic(top) = 0.62 × 3.80 = 2.36 6. Determine K from Figure 3-1 using Ginelastic For G (top) = 2.36 and G (bottom) = 10, Read from Figure 3-1, K = 2.2 7. KLx = 2.2 × 15 ft = 33.0 ft 8. Calculate equivalent of KLy: KLx 33.0 ft = = 18.75 ft 1.76 rx / ry 9. From the column tables (for 50 ksi steel): φc Pn = 1,030 kips < 1,100 kips req’d. n.g. Try a stronger column. 1. Try a W12×136 A = 39.9 in.2, Ix = 1,240 in.4, rx = 5.58 in. 2. Pu / A = 1,100 kips / 39.9 in.2 = 27.6 ksi 3. From Table 3-1: SRF = 0.77 4. Gelastic (top) =

1,240 / 15 = 4.41 375 / 20

5. Ginelastic(top) = 0.77 × 4.41 = 3.39 6. K = 2.3 7. KLx = 2.3 × 15 ft = 34.5 ft 8. Equivalent KLy:

KLx 34.5 ft = = 19.5 ft 1.77 rx / ry

9. φc Pn = 1,135 kips > 1,100 kips req’d o.k. Use W12×136 “Leaning” Columns

A “leaning” column is one which is considered pin-ended and does not participate in providing lateral stability to the structure. As a result, it relies on other parts of the structure for stability. The LRFD Specification in Section C2.2 requires that for unbraced frames, “the destabilizing effects of gravity-loaded columns whose simple connections AMERICAN INSTITUTE OF STEEL CONSTRUCTION


3 - 11

to the frame do not provide resistance to lateral loads shall be included in the design of the moment-frame columns.” Normal practice is to design leaning columns for their required strength with an effective length factor K = 1. To account for the effects of leaning columns on unbraced frames, one of the methods given in the Commentary on the LRFD Specification (Section C2) or in Geschwindner (1993) may be utilized. The simplest methods are: 1. The slightly conservative approach of adjusting the effective lengths of the rigidframe columns, Ki′ =  √NKi where Ki′ = the modified effective length factor of a column Ki = the actual effective length factor of a column N = ratio of the factored gravity load supported by all columns in the given story to that supported by the columns in the rigid frame 2. The more conservative approach of providing sufficient design compressive strength in the rigid-frame columns of a story to enable them to support the total factored gravity load of the story at their actual effective lengths. Combined Axial and Bending Loading (Interaction)

Loads given in the Column Tables are for concentrically loaded columns. For columns subjected to both axial and bending stress, see Chapters C and H of the LRFD Specification. The design of a beam-column is a trial and error process in which a trial section is checked for compliance with Equations H1-1a and H1-1b. A fast method for selecting an economical trial W section, using an equivalent axial load, is illustrated in the example problem, using Table 3-2 and the u values listed in the column properties at the bottom of the column load tables. The procedure is as follows: 1. With the known value of KL (effective length), select a first approximate value of m from Table 3-2. Let u equal 2. 2. Solve for Pu eq = Pu + Mux m + Muy mu where Pu Mux Muy m u

= actual factored axial load, kips = factored bending moment about the strong axis, kip-ft = factored bending moment about the weak axis, kip-ft = factor taken from Table 3-2 = factor taken from column load table

3. From the appropriate Column Load Table, select a tentative section to support Pu eq. 4. Based on the section selected in Step 3, select a “subsequent approximate” value of m from Table 3-2 and a u value from the column load table. 5. With the values selected in Step 4, solve for Pu eq. 6. Repeat Steps 3 and 4 until the values of m and u stabilize. 7. Check section obtained in Step 6 per Equation H1-1a or H1-1b, as applicable. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3 - 12

COLUMN DESIGN

Table 3-2. Preliminary Beam-Column Design Fy = 36 ksi, Fy = 50 ksi Values of m

Fy KL (ft)

36 ksi 10

12

14

16

18

50 ksi 20

22 and over

10

12

14

16

18

20

22 and over

1.8

1.7

1.6

1.4

1.3

1.2

1.4 1.7 1.8 2.0 1.8 1.5 1.4

1.1 1.4 1.5 1.7 1.7 1.5 1.3

1.0 1.1 1.3 1.5 1.5 1.4 1.3

0.9 1.0 1.2 1.3 1.4 1.3 1.2

0.8 0.9 1.1 1.2 1.3 1.2 1.2

1st Approximation All Shapes

2.0

1.9

1.8

1.7

1.6

1.5

1.3

1.9

Subsequent Approximation W4 W6 W8 W8 W10 W12 W14

3.1 3.2 2.8 2.5 2.1 1.7 1.5

2.3 2.7 2.5 2.3 2.0 1.7 1.5

1.7 2.1 2.1 2.2 1.9 1.6 1.4

1.4 1.7 1.8 2.0 1.8 1.5 1.4

1.1 1.4 1.5 1.8 1.7 1.5 1.3

1.0 1.2 1.3 1.6 1.6 1.4 1.3

0.8 1.0 1.1 1.4 1.4 1.3 1.2

2.4 2.8 2.5 2.4 2.0 1.7 1.5

1.8 2.2 2.2 2.2 1.9 1.6 1.4

This table is from a paper in AISC Engineering Journal by Uang, Wattar, and Leet (1990).

EXAMPLE 3-4

Given:

Design the following column: Pu = 400 kips Mntx = 250 kip-ft Mltx = 0 (braced frame) Mnty = 80 kip-ft Mlty = 0 (braced frame) KLx = KLy = 14 ft Lb = 14 ft Cm = 0.85 Fy = 50 ksi

Solution:

1. For KL = 14 ft, from Table 3-2 select a first trial value of m = 1.7. Let u = 2 2. Pu eq = Pu + Mux m + Muy mu = 400 + 250 × 1.7 + 80 × 1.7 × 2 = 1,097 kips 3. From Column Load Tables select W14×109 (φc Pn = 1,170 kips) or W12×120 (φc Pn = 1,220 kips). 4. Select the W14 column, so the second trial value of m is 1.4. (Note: If a W14 column were required for architectural or other reasons, AMERICAN INSTITUTE OF STEEL CONSTRUCTION


3 - 13

the selection process could have started with m = 1.4). With m = 1.4 and u = 1.97 (for a W14×109) from Column Load Table, Pu eq = 400 + 250 × 1.4 + 80 × 1.4 × 1.97 = 971 kips 5. From Column Load Tables select W14×90 (φc Pn = 969 kips). 6. For W14×90, m = 1.4, u = 1.94. Repeat of Steps 3 and 4 not required. 7. Check W14×90 with the appropriate interaction formula. A

= 26.5 in.2

ry

= 3.70 in.,

 Kl 14 × 12 = 45.4  = 3.70  ry 

rx

= 6.14 in.,

 Kl 14 × 12 = 27.4  = 6.14  rx 

Thesecond-or der moments,Mux and Muy, will be evaluated using the approximate method given in Section C1 of the LRFD Specification. Because Mltx = Mlty = 0 (braced frames in both directions), Specification Equation C1-1 reduces to Mu = B1Mnt, where B1 is a function of Pe1 (Equation C1-2). The values of Pe1 with respect to the x and y axes can be determined from LRFD Specification Table 8 as follows:

Pex Pey

= 382 × 26.5 = 10,123 kips = 139 × 26.5 = 3,684 kips

B1x

=

0.85 < 1.0. Use B1x = 1.0 1 − 400 / 10,123

B1y

=

0.85 < 1.0. Use B1y = 1.0 1 − 400 / 3,684

= 1.0 × 250 = 250 kip-ft = 1.0 × 80 = 80 kip-ft 0.9 × 50 × 75.6 = 284 kip-ft φb Mny = φb Mpy = 12

Mux Muy

From the beam selection table in Part 4 of this Manual: φb Mnx = 577 kip-ft for Lb < Lp = 15.0 ft Pu φcPn

=

400 = 0.412 > 0.2. Therefore, Equation H1-1a applies. 969

400 8  250 80  +  +  = 0.412 + 0.636 = 1.05 < 1.0 n.g. 969 9  577 284 Use W14×99 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3 - 14

COLUMN DESIGN

Column Stiffening

Values of Pwo, Pwi, Pwb, and Pfb, listed in the Properties Section of the Column Load Tables for W and HP shapes, are useful in determining if a column requires stiffening because of forces transmitted into it from the flanges or connecting flange plates of a rigid beam connection to the column flange. The parameters are defined as follows: Pwo Pwi Pwb Pfb

= φ5Fyw tw k (kips), φ = 1.0 = φFyw tw (kips/in.), φ = 1.0 = φ4,100tw3 √ Fyw / h (kips), φ = 0.9 = φ6.25tf2Fyf (kips), φ = 0.9

Column stiffening or a heavier column* is required if Pbf, the factored force transmitted into the column web, exceeds any one of the following three resisting forces: Pwb Pfb Pwi tb + Pwo, where tb is the thickness of the beam flange delivering the concentrated force. For a complete explanation of these design parameters, see the section Column Stiffening in Part 10 (Volume II) of this LRFD Manual.

*The designer should consider selecting a heavier column section to eliminate the need for stiffening. Although this will increase the material cost of the column, this heavier section may provide a more economical solution due to the reduction in labor cost associated with the elimination of stiffening. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


3 - 15

W and HP Shapes

The design strengths in the tables that follow are tabulated for the effective lengths in feet KL (with respect to the minor axis), indicated at the left of each table. They are applicable to axially loaded members in accordance with Section E2 of the LRFD Specification. Two yield stresses are covered, 36 and 50 ksi. The heavy horizontal lines appearing within the tables indicate Kl / r = 200. No values are listed beyond Kl / r = 200. For discussion of effective length, range of l / r, strength about the major axis, combined axial and bending stress, and sample problems, see General Notes, above. Properties and factors are listed at the bottom of the tables for checking strength about the strong axis, combined loading conditions, and column stiffener requirements.


3 - 16

COLUMN DESIGN Y

Fy = 36 ksi Fy = 50 ksi

X

COLUMNS W shapes Design axial strength in kips (φ = 0.85)

X

Y

Designation

W14

Wt./ft

808

Effective length KL (ft) with respect to least radius of gyration ry

Fy

730

665

605

550

36

50

36

50

36

50

36

50

36

50

0

7250

10100

6580

9140

6000

8330

5450

7570

4960

6890

11 12 13 14 15

5610 5480 5350 5230 5110

7440 7240 7040 6850 6660

6310 6260 6210 6150 6090

8620 8530 8430 8320 8200

5750 5700 5650 5590 5540

7850 7760 7660 7560 7450

5210 5170 5120 5070 5020

7110 7030 6940 6850 6750

4740 4700 4650 4600 4560

6460 6390 6300 6220 6120

16 17 18 19 20

4990 4870 4760 4650 4540

6480 6310 6130 5970 5810

6020 5960 5880 5810 5730

8080 7960 7820 7690 7550

5480 5410 5350 5280 5200

7340 7220 7100 6970 6840

4960 4900 4840 4770 4700

6640 6530 6420 6300 6170

4500 4450 4390 4330 4260

6020 5920 5810 5700 5590

22 24 26 28 30

4340 4140 3950 3770 3600

5490 5200 4920 4660 4410

5570 5390 5210 5020 4820

7250 6940 6610 6280 5940

5050 4890 4720 4540 4360

6560 6270 5970 5660 5340

4560 4410 4250 4090 3920

5910 5640 5360 5080 4790

4130 3990 3840 3690 3530

5350 5100 4840 4570 4300

32 34 36 38 40

3430 3280 3130 2980 2850

4170 3950 3740 3540 3350

4620 4420 4210 4000 3790

5600 5250 4910 4580 4250

4170 3980 3790 3590 3400

5030 4710 4400 4090 3780

3740 3570 3390 3210 3030

4490 4200 3910 3630 3350

3370 3210 3050 2880 2720

4030 3760 3500 3240 2990

42 44 46 48 50

2720 2590 2470 2360 2250

3170 3000 3860 3540 3260

3580 3380 3170 2970 2780

3930 3620 3310 3040 2800

3210 3020 2830 2650 2470

3490 3200 2930 2690 2480

2860 2680 2510 2340 2180

3080 2820 2580 2370 2180

2550 2390 2240 2080 1940

2740 2500 2290 2100 1940

2.03 2.03 2.03 3910 5430 3070 135 187 111 103000 122000 56400 5310 7370 4880 20.1 17.0 19.5 415 270 372

2.03 4270 154 66500 6780 16.6 241

2.02 3670 142 52200 5750 16.3 222

2.02 2250 93.4 33900 3500 19.0 313

2.01 3120 130 39900 4870 16.1 203

2.02 1930 85.7 26100 2950 18.7 288

2.01 2680 119 30800 4100 15.9 188

Properties u P wo (kips) P wi (kips/in.) P wb (kips) P fb (kips) L p (ft) L r (ft)

A (in.2) Ix (in.4) Iy (in.4) ry (in.) Ratio rx / ry Pex (KL )2 / 10 4 Pey (KL )2 / 10 4

237 16000 5510 4.82 1.70 457000 158000

215 14300 4720 4.69 1.74 411000 135000

2.02 2640 102 44300 4140 19.3 342

196 12400 4170 4.62 1.73 357000 120000


178 10800 3680 4.55 1.71 310000 105000

162 9430 3250 4.49 1.70 270000 93500


3 - 17 Y



X

X

Y

Designation

W14

Wt./ft

500

Fy

455

426

398

370

50

36

50

36

50

36

50

36

50

4500

6250

4100

5700

3830

5310

3580

4970

3340

4630

11 12 13 14 15

4290 4250 4210 4170 4120

5850 5780 5710 5620 5540

3910 3870 3840 3790 3750

5330 5260 5190 5110 5030

3640 3610 3570 3530 3490

4970 4900 4830 4760 4680

3410 3380 3340 3300 3270

4640 4580 4520 4450 4380

3170 3140 3110 3070 3040

4320 4260 4200 4140 4070

16 17 18 19 20

4070 4020 3970 3910 3850

5450 5350 5250 5150 5040

3710 3660 3610 3560 3500

4950 4860 4770 4670 4570

3450 3400 3360 3310 3260

4600 4520 4430 4340 4250

3230 3180 3140 3090 3040

4300 4220 4140 4050 3960

3000 2960 2920 2870 2820

4000 3920 3840 3760 3680

22 24 26 28 30

3730 3600 3460 3320 3180

4820 4590 4350 4100 3850

3390 3270 3140 3010 2870

4370 4150 3930 3700 3480

3150 3030 2910 2790 2660

4050 3850 3640 3430 3210

2940 2830 2720 2600 2480

3780 3590 3390 3190 2990

2730 2630 2520 2410 2290

3500 3320 3140 2950 2750

32 34 36 38 40

3030 2880 2730 2580 2420

3610 3360 3120 2880 2650

2740 2600 2460 2320 2180

3250 3020 2800 2580 2370

2530 2400 2270 2140 2010

3000 2780 2570 2370 2170

2360 2230 2110 1990 1860

2780 2580 2390 2190 2010

2180 2060 1950 1830 1710

2560 2380 2190 2010 1840

42 44 46 48 50

2280 2130 1990 1850 1710

2420 2210 2020 1860 1710

2040 1910 1780 1650 1520

2160 1970 1800 1650 1520

1880 1750 1630 1510 1400

1980 1800 1650 1510 1400

1740 1620 1510 1400 1290

1830 1660 1520 1400 1290

1600 1490 1380 1280 1180

1670 1520 1390 1280 1180

P wo (kips) P wi (kips/in.) P wb (kips) P fb (kips) L p (ft) L r (ft)

2.01 1650 78.8 20400 2480 18.5 264

2.00 2290 110 24100 3450 15.7 172

1.99 1410 72.5 15800 2090 18.3 242

1.99 1950 101 18600 2900 15.5 157

1.99 1730 93.8 15000 2590 15.3 148

1.99 1120 63.7 10800 1640 18.0 213

1.98 1550 88.5 12800 2280 15.2 139

1.98 987 59.6 8790 1430 17.8 199

1.97 1370 82.8 10400 1990 15.1 129


36 0

Properties u


147 8210 2880 4.43 1.69 235000 82600

134 7190 2560 4.38 1.67 206000 73600

2.00 1240 67.5 12800 1870 18.1 227

125 6600 2360 4.34 1.67 189000 67400

117 6000 2170 4.31 1.66 172000 62200


109 5440 1990 4.27 1.66 156000 56900

3 - 18

COLUMN DESIGN Y


X


X

Y

Designation

W14

Wt./ft

342

Fy

311

283

257

233

50

36

50

36

50

36

50

36

50

3090

4290

2800

3880

2550

3540

2310

3210

2100

2910

6 7 8 9 10

3040 3030 3010 2990 2960

4200 4170 4130 4090 4050

2750 2740 2720 2700 2680

3800 3770 3740 3700 3660

2510 2500 2480 2460 2440

3460 3440 3410 3370 3330

2280 2260 2250 2230 2210

3140 3120 3090 3060 3020

2060 2050 2040 2020 2000

2850 2820 2800 2770 2730

11 12 13 14 15

2940 2910 2880 2850 2810

4000 3950 3890 3830 3760

2660 2630 2600 2570 2540

3610 3560 3510 3460 3400

2420 2390 2370 2340 2310

3290 3240 3200 3140 3090

2190 2170 2150 2120 2090

2980 2940 2890 2850 2800

1980 1960 1940 1920 1890

2700 2660 2620 2570 2530

16 17 18 19 20

2770 2740 2700 2650 2610

3690 3620 3550 3470 3400

2510 2470 2430 2390 2360

3330 3270 3200 3130 3060

2280 2250 2210 2180 2140

3030 2970 2910 2850 2780

2060 2030 2000 1970 1940

2740 2690 2630 2570 2510

1870 1840 1810 1780 1750

2480 2430 2380 2320 2270

22 24 26 28 30

2520 2420 2320 2220 2110

3230 3060 2890 2710 2530

2270 2180 2090 2000 1900

2910 2750 2590 2430 2270

2060 1980 1900 1810 1720

2640 2500 2350 2200 2050

1870 1790 1710 1630 1550

2380 2250 2120 1980 1840

1690 1620 1550 1470 1400

2150 2030 1910 1780 1660

32 34 36 38 40

2010 1900 1790 1680 1570

2360 2180 2010 1840 1680

1800 1700 1600 1500 1410

2110 1950 1790 1640 1490

1630 1540 1450 1360 1270

1900 1760 1620 1480 1340

1470 1380 1300 1220 1140

1710 1570 1440 1320 1190

1320 1240 1170 1090 1020

1530 1410 1290 1180 1070


1.98 866 55.4 7100 1240 17.7 185

1.97 1200 77.0 8360 1720 15.0 120

1.97 746 50.8 5430 1030 17.5 168

1.96 1040 70.5 6400 1440 14.8 110

1.95 887 64.5 4930 1210 14.7 100

1.96 542 42.3 3150 723 17.2 140

1.94 753 58.8 3710 1000 14.6 91.6

1.95 457 38.5 2370 599 17.1 127

1.93 635 53.5 2790 832 14.5 83.4


36 0

Properties u


101 4900 1810 4.24 1.65 141000 52000

91.4 4330 1610 4.2 1.64 124000 46100

1.97 639 46.4 4190 868 17.4 154

83.3 3840 1440 4.17 1.63 110000 41500


75.6 3400 1290 4.13 1.62 97400 36900

68.5 3010 1150 4.1 1.62 86200 33000


3 - 19 Y



X

X

Y

Designation

W14

Wt./ft

211

Fy

193

176

159

145

50

36

50

36

50

36

50

36

50

1900

2640

1740

2410

1590

2200

1430

1980

1310

1810

6 7 8 9 10

1870 1860 1840 1830 1810

2580 2550 2530 2500 2470

1710 1700 1690 1670 1660

2360 2340 2320 2290 2260

1560 1550 1540 1530 1510

2150 2130 2110 2090 2060

1400 1400 1390 1380 1360

1940 1920 1900 1880 1860

1280 1280 1270 1260 1250

1770 1760 1740 1720 1700

11 12 13 14 15

1790 1780 1760 1730 1710

2440 2400 2370 2330 2280

1640 1630 1610 1590 1570

2230 2200 2170 2130 2090

1500 1480 1460 1450 1430

2030 2000 1970 1940 1900

1350 1330 1320 1300 1280

1830 1810 1780 1740 1710

1230 1220 1210 1190 1170

1670 1650 1620 1590 1560

16 17 18 19 20

1690 1660 1640 1610 1580

2240 2190 2140 2090 2040

1540 1520 1500 1470 1440

2050 2010 1960 1910 1870

1410 1380 1360 1340 1310

1860 1820 1780 1740 1700

1270 1250 1230 1200 1180

1680 1640 1600 1570 1530

1160 1140 1120 1100 1080

1530 1500 1460 1430 1390

22 24 26 28 30

1520 1460 1390 1330 1260

1940 1830 1710 1600 1490

1390 1330 1270 1210 1150

1770 1670 1560 1460 1350

1260 1210 1150 1100 1040

1610 1510 1420 1320 1220

1140 1090 1040 990 930

1440 1360 1270 1180 1100

1040 992 946 898 849

1320 1240 1160 1080 998

32 34 36 38 40

1190 1120 1050 980 912

1370 1260 1160 1050 951

1080 1020 955 892 830

1250 1150 1050 956 863

980 920 863 805 748

1130 1040 946 859 775

880 830 773 721 670

1010 928 846 767 692

800 752 703 655 608

919 842 767 694 626


1.95 397 35.3 1830 493 17.0 116

1.93 551 49.0 2160 684 14.4 76.0

1.96 340 32.0 1370 420 16.9 106

1.93 473 44.5 1610 583 14.3 70.1

1.92 415 41.5 1310 483 14.2 64.5

1.94 251 26.8 803 287 16.7 88.6

1.92 349 37.3 947 398 14.1 59.0

1.93 214 24.5 609 241 16.6 81.5

1.90 298 34.0 718 334 14.1 54.7


36 0

Properties u


62.0 2660 1030 4.07 1.61 76100 29400

56.8 2400 931 4.05 1.60 68700 26700

1.94 299 29.9 1110 348 16.8 97.5

51.8 2140 838 4.02 1.60 61300 24000

46.7 1900 748 4 1.60 54400 21400


42.7 1710 677 3.98 1.59 49000 19400

3 - 20

COLUMN DESIGN Y


X


X

Y

Designation

W14

Wt./ft

132

Fy

120

109 50

90

36

50

36

50

36

50†

36

50†

0

1190

1650

1080

1500

979

1360

890

1240

811

1130

6 7 8 9 10

1160 1160 1150 1140 1130

1610 1590 1570 1550 1530

1060 1050 1040 1030 1020

1460 1450 1430 1410 1390

960 953 946 937 927

1320 1310 1300 1280 1260

873 867 860 852 843

1200 1190 1180 1160 1150

795 789 783 775 767

1100 1080 1070 1060 1040

11 12 13 14 15

1110 1100 1080 1070 1050

1510 1480 1450 1430 1390

1010 999 986 971 956

1370 1350 1320 1290 1270

917 905 893 880 866

1240 1220 1200 1170 1150

833 823 811 799 787

1130 1110 1090 1060 1040

758 749 738 727 716

1030 1010 989 969 947

16 17 18 19 20

1030 1020 997 978 958

1360 1330 1300 1260 1220

940 924 906 888 870

1240 1210 1180 1140 1110

852 837 821 804 787

1120 1090 1060 1030 1000

773 759 745 730 714

1020 991 965 938 911

704 691 678 664 650

925 902 878 853 828

22 24 26 28 30

916 872 826 780 733

1150 1070 997 920 844

831 791 749 706 663

1040 972 902 832 762

752 715 677 639 600

943 879 815 751 688

682 648 614 578 542

854 796 737 679 621

620 589 558 525 493

776 723 670 616 564

32 34 36 38

686 639 593 547

769 697 627 563

620 577 535 494

694 628 565 507

560 522 483 446

627 567 509 457

507 471 436 402

565 511 458 411

460 428 396 365

512 463 415 372


2.03 196 23.2 520 215 15.7 73.7

1.99 272 32.3 613 298 13.3 49.7

2.04 173 21.2 399 179 15.6 67.9

1.99 240 29.5 471 249 13.2 46.3

1.97 205 26.3 331 208 13.2 43.2

2.02 125 17.5 222 123 15.5 58.1

1.95 174 24.3 261 171 13.4 40.6

2.02 109 15.8 165 102 15.4 54.2

1.94 151 22.0 195 142 15.0 38.4


36

99

Properties u


38.8 1530 548 3.76 1.67 43800 15700

35.3 1380 495 3.74 1.67 39300 14100

2.02 148 18.9 281 150 15.5 62.7

32 1240 447 3.73 1.67 35400 12700

†Flange is noncompact; see discussion preceding column load tables.


29.1 1110 402 3.71 1.66 31700 11500

26.5 999 362 3.70 1.66 28600 10400


3 - 21 Y



X

X

Y

Designation

W14

Wt./ft

82

Fy

36

50

68

61

53

48

43

36

50

36

50

36

50

36

50

36

50

36

50†

737 1020 667

927

612

850

548

761

477

663

431

599

386

536

6 7 8 9 10

705 694 682 667 652

963 942 918 892 863

638 628 616 604 590

871 852 830 807 781

585 576 565 553 540

798 781 760 738 714

523 515 505 494 483

714 698 680 660 638

443 432 418 404 389

598 576 552 526 498

400 390 378 365 351

540 520 498 474 449

357 347 337 325 312

482 463 443 422 399

11 12 14 16 18

635 618 579 538 495

833 800 732 661 588

575 559 524 487 447

753 724 662 598 532

526 511 479 444 408

689 662 604 544 484

470 457 428 396 364

615 591 539 486 431

372 355 319 282 245

469 439 379 319 263

336 320 287 253 220

423 395 340 286 235

298 284 254 224 194

375 350 301 252 206

20 22 24 26 28

450 406 363 321 280

516 447 381 325 280

407 367 328 290 253

467 405 345 294 253

371 334 297 262 229

424 366 311 265 229

331 297 265 233 203

377 325 276 236 203

210 176 148 126 109

213 176 148 126 109

188 157 132 113 97

191 157 132 113 97

165 138 116 99 85

167 138 116 99 85

30 31 32 34 36

244 229 214 190 169

244 229 214 190 169

221 207 194 172 153

221 207 194 172 153

199 187 175 155 138

199 187 175 155 138

177 166 155 138 123

177 166 155 138 123

95 89 83

95 89 83

85 79

85 79

74 69

74 69

38

152

152 138

138

124

124

110

110

3.2 2.7 3.12 2.56 2.97 95.7 133 84.2 117 72.1 13.3 18.5 12.2 17.0 11.0 98.4 116 76.4 90.0 55.1 88.2 123 71.7 100 56.9 8.00 6.79 7.96 6.75 7.88 28.0 20.1 26.4 19.2 24.7

2.37 100 15.3 64.9 79.0 6.68 18.2

0


74



2.85 149 18.4 257 148 10.3 43.0

2.68 207 25.5 303 206 8.77 29.6

24.1 882 148 2.48 2.44 25200 4240

2.82 127 16.2 177 125 10.3 40.0

2.62 176 22.5 209 173 8.77 27.9

21.8 796 134 2.48 2.44 22800 3840

2.80 112 14.9 139 105 10.3 37.3

2.56 156 20.8 163 146 8.70 26.4

20 723 121 2.46 2.44 20700 3460

2.74 97.0 13.5 102 84.2 10.2 34.7

2.44 135 18.8 121 117 8.66 25.0

17.9 640 107 2.45 2.44 18300 3080

15.6 541 57.7 1.92 3.07 15500 1650

14.1 485 51.4 1.91 3.06 13800 1470

12.6 428 45.2 1.89 3.08 12200 1290

†Web may be noncompact for combined axial and bending stress; see AISC LRFD Specification Section B5. Note: Heavy line indicates Kl / r of 200.


3 - 22

COLUMN DESIGN Y


X


X

Y

Designation

W12

Wt./ft

336

Fy

305

279

252

230

50

36

50

36

50

36

50

36

50

3020

4200

2740

3810

2510

3480

2270

3150

2070

2880

6 7 8 9 10

2960 2930 2900 2870 2840

4070 4020 3970 3910 3850

2680 2660 2630 2600 2570

3690 3640 3590 3540 3480

2450 2430 2400 2370 2350

3370 3330 3280 3230 3170

2210 2190 2170 2150 2120

3040 3010 2960 2920 2870

2020 2000 1980 1960 1930

2780 2740 2710 2660 2610

11 12 13 14 15

2800 2760 2720 2670 2620

3780 3700 3620 3540 3450

2530 2500 2460 2410 2370

3420 3350 3270 3190 3110

2310 2280 2240 2200 2160

3110 3050 2980 2910 2830

2090 2060 2020 1980 1950

2810 2750 2680 2620 2550

1910 1880 1840 1810 1770

2560 2500 2450 2380 2320

16 17 18 19 20

2570 2520 2470 2410 2350

3360 3260 3160 3060 2960

2320 2270 2220 2170 2120

3020 2940 2840 2750 2660

2110 2070 2020 1970 1920

2750 2670 2580 2500 2410

1910 1860 1820 1770 1730

2470 2400 2320 2240 2160

1740 1700 1660 1610 1570

2250 2180 2110 2030 1960

22 24 26 28 30

2230 2100 1980 1850 1720

2750 2540 2330 2120 1910

2000 1890 1770 1650 1530

2460 2270 2070 1880 1690

1820 1710 1600 1490 1380

2230 2050 1870 1690 1520

1630 1530 1430 1330 1230

1990 1830 1660 1500 1350

1480 1390 1300 1200 1110

1810 1650 1500 1350 1210

32 34 36 38 40

1590 1460 1340 1220 1100

1720 1520 1360 1220 1100

1410 1300 1180 1080 971

1510 1340 1200 1080 971

1270 1160 1060 960 866

1350 1200 1070 960 866

1130 1030 940 848 766

1200 1060 945 848 766

1020 931 845 761 687

1070 951 848 761 687


2.18 1180 64 12700 1770 14.5 202

2.17 1640 89 15000 2460 12.3 131

2.18 1010 59 9740 1480 14.3 184

2.16 1400 81 11500 2060 12.1 120

2.15 1220 77 9700 1720 12.0 110

2.16 738 50 6150 1030 13.9 154

2.14 1020 70 7250 1420 11.8 100

2.15 636 46 4810 868 13.8 141

2.13 883 64 5670 1210 11.7 92.0


36 0

Properties u


98.8 4060 1190 3.47 1.85 116000 34000

89.6 3550 1050 3.42 1.84 101000 30000

2.16 878 55 8230 1240 14.1 169

81.9 3110 937 3.38 1.82 88900 26800


74.1 2720 828 3.34 1.81 77900 23700

67.7 2420 742 3.31 1.80 69100 21200


3 - 23 Y



X

X

Y

Designation

W12

Wt./ft

210

Fy

190

170

152

136

120

50

36

50

36

50

36

50

36

50

36

50

1890

2630

1710

2370

1530

2120

1370

1900

1220

1700

1080

1500

6 7 8 9 10

1840 1830 1810 1790 1760

2540 2500 2470 2430 2380

1660 1650 1630 1610 1590

2290 2260 2220 2190 2150

1490 1480 1460 1440 1420

2050 2020 1990 1960 1920

1330 1320 1300 1290 1270

1830 1810 1780 1750 1710

1190 1180 1160 1150 1130

1630 1610 1590 1560 1530

1050 1040 1030 1010 1000

1440 1420 1400 1380 1350

11 12 13 14 15

1740 1710 1680 1650 1610

2330 2280 2230 2170 2110

1570 1540 1510 1480 1450

2100 2050 2000 1950 1900

1400 1380 1350 1330 1300

1880 1840 1790 1740 1690

1250 1230 1210 1180 1160

1680 1640 1590 1550 1510

1110 1090 1070 1050 1030

1490 1460 1420 1380 1340

984 966 948 928 908

1320 1290 1250 1220 1180

16 17 18 19 20

1580 1540 1510 1470 1430

2040 1980 1910 1840 1780

1420 1390 1350 1320 1280

1840 1780 1720 1650 1590

1270 1240 1210 1180 1140

1640 1580 1530 1470 1420

1130 1100 1070 1050 1020

1460 1410 1360 1310 1260

1010 980 955 928 901

1290 1250 1210 1160 1110

886 864 841 817 793

1140 1100 1060 1020 976

22 24 26 28 30

1340 1260 1170 1090 1000

1640 1490 1360 1220 1090

1210 1130 1050 973 895

1460 1340 1210 1090 967

1070 1000 933 862 792

1300 1180 1070 959 852

954 891 827 763 700

1150 1050 944 844 749

846 788 731 673 617

1020 924 831 742 656

743 692 640 589 538

892 808 726 646 569

32 34 36 38 40

919 837 759 682 616

962 852 760 682 616

819 745 674 605 546

853 755 674 605 546

724 657 593 532 480

750 664 593 532 480

638 578 520 467 421

658 583 520 467 421

561 508 456 409 369

577 511 456 409 369

489 442 395 355 320

500 443 395 355 320


2.16 558 42 3760 731 13.7 129

2.13 774 59 4430 1020 11.6 84.2

2.14 465 38 2700 610 13.5 117

2.11 646 53 3190 847 11.5 76.6

2.15 333 31 1500 397 13.3 94.7

2.11 462 44 1760 551 11.3 62.1

2.13 276 28 1120 316 13.2 84.6

2.09 383 40 1320 439 11.2 55.7

2.12 232 26 815 247 13.0 75.5

2.07 322 36 960 343 11.1 50.0


36 0

Properties u


61.8 2140 664 3.28 1.80 61400 19000

55.8 1890 589 3.25 1.79 54100 16900

2.14 389 35 2020 493 13.4 105

2.11 540 48 2380 684 11.4 68.9

50.0 1650 517 3.22 1.78 47100 14800

44.7 1430 454 3.19 1.77 41000 13000


39.9 1240 398 3.16 1.77 35600 11400

35.3 1070 345 3.13 1.76 30700 9900

3 - 24

COLUMN DESIGN Y


X


X

Y

Designation

W12

Wt./ft

106

Fy

96

87

79

72

65

50

36

50

36

50

36

50

36

50

36

50†

955

1330

863

1200

783

1090

710

986

646

897

584

812

6 7 8 9 10

928 919 908 896 883

1280 1260 1240 1210 1190

839 830 820 809 797

1150 1140 1120 1100 1070

761 753 744 734 723

1050 1030 1010 994 973

689 682 674 665 654

947 933 917 900 880

627 620 613 604 595

861 848 834 818 800

567 561 554 546 538

779 767 754 739 723

11 12 13 14 15

868 853 836 819 800

1160 1130 1100 1070 1040

784 770 755 739 722

1050 1020 995 966 935

711 698 684 669 654

950 926 901 874 846

643 631 619 605 591

860 838 814 790 764

585 574 562 550 537

781 761 740 717 694

529 519 508 497 485

706 687 668 647 626

16 17 18 19 20

781 761 741 719 698

1000 968 932 895 858

704 686 667 648 628

904 871 838 805 771

638 621 604 586 568

817 788 758 727 696

576 561 545 529 512

738 711 683 655 627

523 509 495 480 465

670 645 620 594 569

472 460 446 433 419

604 581 558 535 512

22 24 26 28 30

653 608 562 516 472

783 708 635 565 497

588 546 505 463 422

703 635 569 505 443

531 493 455 417 380

634 572 511 453 397

479 444 409 375 341

570 514 459 406 355

434 403 371 339 309

517 465 415 367 321

391 362 333 305 277

464 417 372 328 287

32 34 36 38 40

428 386 345 310 279

437 387 345 310 279

383 345 308 276 249

390 345 308 276 249

344 309 276 248 223

349 309 276 248 223

308 277 247 221 200

312 277 247 221 200

279 250 223 200 181

282 250 223 200 181

250 223 199 179 161

252 223 199 179 161


2.12 185 22 518 198 13.0 67.2

2.06 257 31 611 276 11.0 44.9

2.10 161 20 378 164 12.9 61.4

2.04 223 28 446 228 10.9 41.4

2.09 122 17 236 109 12.7 51.8

2.01 169 24 278 152 10.8 35.7

2.08 106 15 181 91 12.7 48.2

1.98 148 22 213 126 10.7 33.6

2.06 92.1 14 135 74 12.6 44.7

1.95 128 20 159 103 11.8 31.7


36 0

Properties u


31.2 933 301 3.11 1.76 26700 8640

28.2 833 270 3.09 1.76 23900 7710

2.10 139 19 311 133 12.8 56.3

2.02 193 26 366 185 10.9 38.3

25.6 740 241 3.07 1.75 21200 6910

23.2 662 216 3.05 1.75 18900 6180

†Flange is noncompact; see discussion preceding column load tables.


21.1 597 195 3.04 1.75 17000 5580

19.1 533 174 3.02 1.75 15200 4990


3 - 25 Y



X

X

Y

Designation

W12

Wt./ft

58

Fy

53

50

45

40

50

36

50

36

50

36

50

36

50

520

723

477

663

450

625

404

561

361

502

6 7 8 9 10

498 490 482 472 461

680 666 649 631 611

457 449 441 432 422

623 610 594 577 559

419 408 396 383 369

566 546 524 500 475

376 366 355 343 330

507 489 469 447 424

336 327 317 306 295

453 437 419 399 378

11 12 13 14 15

450 437 424 411 397

590 568 545 521 496

411 400 388 375 362

539 518 496 474 451

354 339 322 306 289

448 421 393 365 337

317 302 287 272 257

400 375 350 324 299

282 269 256 242 228

356 334 311 288 266

16 18 20 22 24

382 352 321 291 260

471 420 370 322 276

348 320 292 263 235

428 381 334 290 247

271 237 204 173 145

310 257 209 173 145

241 210 180 152 128

274 227 184 152 128

214 187 160 135 113

243 201 163 135 113

26 28 30 32 34

231 202 176 155 137

235 202 176 155 137

207 181 158 139 123

210 181 158 139 123

124 107 93 82

124 107 93 82

109 94 82 72

109 94 82 72

96 83 72 64

96 83 72 64

38 41

110 94

110 94

98 85

98 85


2.41 89 13 106 83 10.5 38.3

2.22 124 18 125 115 8.9 27.0

2.39 78 12 94 67 10.3 35.8

2.16 108 17 111 93 8.8 25.6

2.51 127 19 136 115 6.9 21.6

2.79 75 12 86 67 8.1 28.4

2.37 105 17 101 93 6.9 20.3

2.69 66 11 59 54 8.0 26.5

2.22 92 15 69 75 6.8 19.3


36 0

Properties u


17.0 475 107 2.51 2.10 13600 3070

15.6 425 95.8 2.48 2.11 12200 2750

2.85 92 13 116 83 8.2 30.8

14.7 394 56.3 1.96 2.64 11300 1620

13.2 350 50 1.94 2.65 10000 1420

Note: Heavy line indicates Kl / r of 200.


11.8 310 44.1 1.93 2.66 8890 1260

3 - 26

COLUMN DESIGN Y


X


X

Y

Designation

W10

Wt./ft

112

Fy

100

88

77

68

50

36

50

36

50

36

50

36

50

1010

1400

900

1250

793

1100

692

961

612

850

6 7 8 9 10

969 956 941 924 906

1330 1300 1270 1240 1210

865 853 840 824 808

1180 1160 1140 1110 1080

762 751 739 725 710

1040 1020 999 973 945

664 655 644 632 618

908 890 869 847 822

588 579 569 558 547

803 787 769 749 727

11 12 13 14 15

886 865 842 819 794

1170 1130 1090 1050 1010

789 770 750 728 706

1040 1010 970 931 892

694 677 659 639 619

916 884 851 817 782

604 588 572 555 537

796 768 738 708 677

534 520 506 490 475

703 678 652 625 597

16 17 18 19 20

768 742 715 688 660

961 915 870 824 778

682 659 634 609 584

851 810 769 727 686

599 577 556 534 511

746 709 672 635 599

519 500 481 461 442

645 612 580 547 515

458 441 424 407 389

569 540 511 482 454

22 24 26 28 30

604 548 493 440 389

688 601 518 447 389

534 483 434 386 340

605 527 453 390 340

466 422 378 336 295

527 458 393 339 295

402 362 324 287 252

452 392 335 289 252

354 319 285 252 221

398 344 294 254 221

32 34 36 38 40

342 303 270 242 219

342 303 270 242 219

299 265 236 212 191

299 265 236 212 191

259 230 205 184 166

259 230 205 184 166

221 196 175 157 141

221 196 175 157 141

194 172 153 138 124

194 172 153 138 124


2.06 255 27 1210 316 11.2 86.4

2.02 354 38 1430 439 9.5 56.5

2.06 214 24 883 254 11.0 77.4

2.01 298 34 1040 353 9.4 50.8

1.99 246 30 735 276 9.3 45.1

2.03 143 19 420 153 10.8 60.0

1.96 199 27 495 213 9.2 39.8

2.01 116 17 293 120 10.8 53.8

1.93 162 24 345 167 9.2 36.0


36 0

Properties u


32.9 716 236 2.68 1.74 20400 6760

29.4 623 207 2.65 1.74 17800 5910

2.04 177 22 623 198 11.0 68.4

25.9 534 179 2.63 1.73 15300 5130


22.6 455 154 2.60 1.73 13000 4370

20.0 394 134 2.59 1.71 11300 3840


3 - 27 Y



X

X

Y

Designation

W10

Wt./ft

60

Fy

54

49

45

39

33

50

36

50

36

50

36

50

36

50

36

50

539

748

483

672

441

612

407

565

352

489

297

413

6 7 8 9 10

517 509 500 491 480

706 692 675 657 638

464 457 449 440 431

634 621 606 590 572

422 416 409 401 392

577 565 551 536 520

380 371 361 350 337

515 497 478 458 436

328 320 311 301 290

444 428 412 393 374

276 269 261 252 243

373 360 345 329 312

11 12 13 14 15

469 457 444 430 416

617 595 571 547 523

420 409 398 385 373

553 533 512 490 468

382 372 361 350 338

502 484 464 444 424

324 311 296 282 267

412 388 364 339 314

278 266 254 241 228

353 332 310 289 267

233 222 211 200 189

294 276 257 238 220

16 17 18 19 20

401 387 371 356 340

497 472 446 421 395

360 346 332 318 304

445 422 399 376 353

326 314 301 288 275

403 382 361 340 319

252 237 222 207 192

290 266 243 221 199

215 201 188 175 162

246 225 205 185 167

177 166 155 144 133

202 184 167 150 135

22 24 26 28 30

309 278 248 219 191

346 299 255 220 191

276 248 221 195 170

309 266 227 196 170

250 224 199 175 153

278 239 204 176 153

164 138 118 102 88

164 138 118 102 88

138 116 99 85 74

138 116 99 85 74

112 94 80 69 60

112 94 80 69 60

32 33 34 36

168 158 149 133

168 158 149 133

150 141 133 118

150 141 133 118

134 126 119 106

134 126 119 106

78 73

78 73

65 61

65 61

53

53


2.00 99 15 209 94 10.7 48.1

1.90 138 21 246 130 9.1 32.6

1.97 83 13 143 77 10.7 43.9

1.87 116 19 168 106 9.1 30.2

2.37 79 13 121 78 8.4 35.2

2.17 109 18 142 108 7.1 24.1

2.31 64 11 88 57 8.3 31.2

2.04 89 16 104 79 7.0 21.9

2.23 55 10 69 38 8.1 27.4

1.87 77 14 81 53 6.9 19.7


36 0

Properties u


17.6 341 116 2.57 1.71 9710 3330

15.8 303 103 2.56 1.71 8640 2960

1.96 73 12 111 64 10.6 40.7

1.83 101 17 131 88 9.0 28.3

14.4 272 93.4 2.54 1.71 7800 2660

13.3 248 53.4 2.01 2.15 7100 1540



11.5 209 45.0 1.98 2.16 6000 1290

9.71 170 36.6 1.94 2.16 4880 1050

3 - 28

COLUMN DESIGN Y


X


X

Y

Designation

W8

Wt./ft

67

Fy

58

48

40

35

31

50

36

50

36

50

36

50

36

50

36

50

603

837

523

727

431

599

358

497

315

438

279

388

6 7 8 9 10

567 555 541 526 509

770 746 721 693 662

492 481 469 455 441

667 647 624 599 572

405 396 386 374 362

549 532 513 492 470

335 327 319 309 298

454 439 423 405 386

295 288 280 272 262

399 386 372 356 339

261 255 248 240 232

354 342 329 315 300

11 12 13 14 15

492 473 453 433 412

631 598 564 529 494

425 409 391 374 355

544 515 485 455 425

349 335 321 306 291

446 422 397 372 347

287 275 263 251 238

366 345 324 303 281

252 242 231 220 208

321 303 284 265 246

223 214 204 194 184

284 268 251 234 217

16 17 18 19 20

391 370 349 328 307

460 425 392 359 328

337 318 300 281 263

394 365 335 307 279

276 260 245 229 214

321 297 272 249 226

225 211 198 185 173

260 239 219 199 180

197 185 174 162 151

228 209 191 174 157

174 163 153 143 133

200 184 168 153 138

22 24 26 28 30

266 228 194 167 146

271 228 194 167 146

228 194 165 143 124

231 194 165 143 124

185 157 134 115 100

187 157 134 115 100

148 125 107 92 80

149 125 107 92 80

129 109 93 80 70

130 109 93 80 70

114 96 82 70 61

114 96 82 70 61

32 33 34 35

128 120 113 107

128 120 113 107

109 103 97 91

109 103 97 91

88 83 78

88 83 78

70 66 62

70 66 62

61 58

61 58

54 51

54 51


2.03 147 21 648 177 8.8 64.0

1.96 205 28 764 246 7.5 41.9

2 120 18 464 133 8.8 55.9

1.93 167 26 547 185 7.4 36.8

1.93 69 13 163 64 8.5 39.1

1.8 96 18 192 88 7.2 26.5

1.89 56 11 104 50 8.5 35.1

1.74 78 16 123 69 7.2 24.1

1.85 48 10 81 38 8.4 32.0

1.65 67 14 95 53 7.1 22.4


36 0

Properties u


19.7 272 88.6 2.12 1.75 7800 2530

17.1 228 75.1 2.10 1.74 6520 2160

1.97 86 14 224 95 8.7 46.7

1.87 119 20 264 132 7.4 31.1

14.1 184 60.9 2.08 1.74 5260 1750

11.7 146 49.1 2.04 1.73 4170 1390



10.3 127 42.6 2.03 1.73 3630 1210

9.13 110 37.1 2.02 1.72 3150 1070


3 - 29 Y



X

X

Y

Designation

W8

Wt./ft

W6

28

Fy

24

25

20

15 †

50

36

50

36

50

36

50

36

50†

252

351

217

301

225

312

180

249

136

188

6 7 8 9 10

228 219 210 200 189

303 288 271 253 235

195 188 180 171 162

260 247 232 217 200

200 191 182 172 162

265 250 233 216 198

159 152 145 137 128

211 198 185 171 156

119 114 108 102 95

158 148 137 126 115

11 12 13 14 15

178 167 155 143 132

216 197 178 160 142

152 142 132 122 112

184 168 151 136 121

151 140 129 118 107

180 162 144 128 112

119 111 102 93 84

142 127 113 100 87

88 81 74 68 61

104 92 82 71 62

16 17 18 19 20

121 110 99 89 80

125 111 99 89 80

102 93 84 75 68

106 94 84 75 68

97 87 78 70 63

98 87 78 70 63

76 68 60 54 49

76 68 60 54 49

55 48 43 39 35

55 48 43 39 35

22 24 25 26 27

66 56 51 47 44

66 56 51 47 44

56 47 44 40

56 47 44 40

52 44 40

52 44 40

40 34 31

40 34 31

29 24

29 24


2.17 48 10 81 44 6.8 27.2

1.87 67 14 95 61 5.7 18.8

2.07 39 9 52 32 6.7 24.3

1.71 54 12 61 45 5.7 17.2

1.98 65 16 172 58 5.4 21.0

2.03 35 9 78 27 6.3 25.6

1.91 49 13 92 37 5.3 17.6

1.98 26 8 54 14 6.7 20.8

1.75 36 12 64 19 6.8 15.0


36 0

Properties u


8.25 98.0 21.7 1.62 2.13 2810 620

7.08 82.8 18.3 1.61 2.12 2370 525

2.07 47 12 146 42 6.3 31.2

7.34 53.4 17.1 1.52 1.78 1530 485

5.87 41.4 13.3 1.50 1.77 1190 378

†Flange is noncompact; see discussion preceding column load tables. Note: Heavy line indicates Kl / r of 200.


4.43 29.1 9.32 1.46 1.75 831 270

3 - 30

COLUMN DESIGN Y


X

X


Y

Designation

W6

Wt./ft

16


Fy

W5

12

9

W4

19

16

13

36

50

36

50

36

50

36

50

36

50

36

50

0

145

201

109

151

82

114

170

235

143

199

117

163

2 3 4 5 6

140 135 127 118 108

193 182 168 152 134

105 100 94 87 79

144 135 124 110 96

79 75 71 65 59

108 101 93 83 72

166 163 157 151 144

229 222 212 201 187

141 137 133 127 121

194 188 179 169 157

114 109 104 97 89

156 148 138 125 111

7 8 9 10 11

97 86 75 64 54

116 98 81 66 54

70 61 52 44 37

82 68 55 44 37

52 45 39 32 27

61 50 40 33 27

135 126 117 107 97

172 156 140 124 108

114 106 98 90 81

144 131 117 104 90

81 72 63 55 47

97 83 69 57 47

12 13 14 15 16

46 39 33 29 26

46 39 33 29 26

31 26 23 20

31 26 23 20

23 19 17 14

23 19 17 14

87 78 68 60 53

93 80 69 60 53

73 65 57 50 44

78 66 57 50 44

39 34 29 25 22

39 34 29 25 22

47 42 37 34 30

47 42 37 34 30

39 35 31 28 25

39 35 31 28 25

1.84 39 10 115 37 5.3 30.3

1.72 55 14 136 52 4.5 20.1

1.79 32 9 81 26 5.3 26.2

1.63 45 12 95 36 4.5 17.6

1.89 35 10 164 24 4.2 25.5

1.77 48 14 193 33 3.5 16.8

17 18 19 20 21



2.84 35 9 78 33 4.0 18.3

2.5 49 13 92 46 3.4 12.5

4.74 32.1 4.43 0.966 2.69 917 127

2.62 26 8 54 16 3.8 14.3

2.13 36 12 64 22 3.2 10.2

3.55 22.1 2.99 0.918 2.71 630 85.6

2.24 17 6 22 9 3.8 12.0

1.72 24 9 26 13 3.2 8.9

2.68 16.4 2.19 0.905 2.73 468 62.8

5.54 26.2 9.13 1.28 1.70 747 260



4.68 21.3 7.51 1.27 1.68 608 216

3.83 11.3 3.86 1.00 1.72 324 110


3 - 31 Y


COLUMNS HP shapes Design axial strength in kips (φ = 0.85)

X

X

Y

Designation

HP14

Wt./ft

117

Fy

HP13

102

89

73

100

50

36

50

36

50

36

50

36

50

1050

1460

918

1280

799

1110

655

909

900

1250

6 7 8 9 10

1030 1020 1010 1000 993

1420 1400 1390 1370 1350

898 891 884 875 865

1240 1220 1210 1190 1170

781 775 768 760 752

1080 1060 1050 1040 1020

640 635 629 623 615

882 872 861 848 834

875 867 857 846 834

1200 1190 1170 1150 1120

11 12 13 14 15

980 967 953 938 922

1320 1300 1270 1250 1220

854 842 830 816 802

1150 1130 1110 1080 1060

742 732 721 709 696

1000 982 962 940 917

607 599 589 580 569

819 803 786 768 749

821 806 791 775 758

1100 1070 1050 1020 986

16 17 18 19 20

905 888 870 851 832

1190 1150 1120 1090 1050

788 772 756 740 723

1030 1000 974 945 915

683 670 656 641 626

893 869 844 818 791

558 547 535 523 511

729 708 687 666 644

741 722 703 684 664

954 921 888 854 820

22 24 26 28 30

792 750 707 664 620

985 913 842 771 701

687 650 613 574 536

853 790 727 665 604

595 563 529 496 462

737 682 627 572 519

485 458 430 402 374

599 553 507 462 418

623 581 539 496 454

750 681 613 547 483

32 34 36 38 40

576 533 491 450 411

633 568 507 455 411

498 460 423 387 352

545 487 435 390 352

428 395 363 332 301

467 417 372 334 301

346 319 292 267 241

375 334 298 267 241

413 374 336 301 272

425 376 336 301 272


217 29 1010 131 15.0 66.0

302 40 1191 182 12.7 45.1

174 25 679 101 14.8 59.0

242 35 801 140 12.6 41.1

202 31 533 106 12.5 37.6

108 18 250 52 14.5 46.8

150 25 294 72 12.3 34.2

198 28 953 119 13.2 60.1

275 38 1123 165 11.2 40.9


36 0

Properties


34.4 1220 443 3.59 1.66 35000 12700

30.0 1050 380 3.56 1.66 30100 10900

145 22 453 77 14.7 53.0

26.1 904 326 3.53 1.67 25800 9310

21.4 729 261 3.49 1.67 20900 7460



29.4 886 294 3.16 1.74 25400 8400

3 - 32

COLUMN DESIGN Y


X


X

Y

Designation

HP13

Wt./ft

87

Fy

HP12

73

60

84

74

50

36

50

36

50

36

50

36

50

780

1080

661

918

536

744

753

1050

667

927

6 7 8 9 10

759 751 743 733 722

1040 1030 1010 993 973

642 636 628 620 611

882 870 856 840 823

520 515 509 502 494

714 704 692 679 665

729 721 712 701 690

1000 985 967 947 926

646 639 630 621 610

886 872 856 838 819

11 12 13 14 15

711 698 685 670 656

952 928 904 878 851

601 590 578 566 553

804 784 763 741 717

486 477 467 457 447

650 633 616 597 578

677 663 649 634 618

902 877 851 823 795

599 587 574 560 546

798 776 752 727 702

16 17 18 19 20

640 624 607 590 573

823 794 765 735 705

540 526 512 497 482

693 669 644 618 592

436 424 413 401 388

559 539 518 497 476

601 584 567 548 530

765 735 705 674 642

531 516 500 484 467

675 648 621 593 565

22 24 26 28 30

537 500 462 425 389

644 584 524 467 411

451 420 388 356 325

540 488 438 389 342

363 337 311 285 260

433 391 350 310 272

492 454 416 378 342

580 518 459 402 350

434 400 366 332 300

510 455 402 351 306

32 34 36 38 40

353 319 286 256 231

361 320 286 256 231

295 266 237 213 192

300 266 237 213 192

235 211 189 169 153

239 211 189 169 153

307 273 243 218 197

308 273 243 218 197

268 238 213 191 172

269 238 213 191 172


165 24 624 90 13.0 53.2

229 33 735 124 11.1 36.9

127 20 384 65 12.9 47.0

177 28 453 90 11.0 33.4

129 23 243 60 10.9 30.2

170 25 732 95 12.3 54.0

235 34 862 132 10.4 36.9

143 22 506 75 12.2 48.9

199 30 597 105 10.3 34.0


36 0

Properties


25.5 755 250 3.13 1.74 21700 7150

21.6 630 207 3.10 1.74 18000 5940

93 17 206 43 12.8 41.2

17.5 503 165 3.07 1.75 14400 4720



24.6 650 213 2.94 1.75 18600 6090

21.8 569 186 2.92 1.75 16300 5320


3 - 33 Y



X

X

Y

Designation

HP12

Wt./ft

HP10

63

Fy

53

HP8

57

42

36

50

36

50

36

50

36

50

36

50

563

782

474

659

514

714

379

527

324

451

6 7 8 9 10

545 538 531 523 514

747 735 721 706 689

459 453 447 440 432

629 618 607 594 579

491 483 474 464 453

670 655 638 619 599

362 356 349 341 333

494 482 469 455 440

302 294 286 276 266

408 393 377 360 342

11 12 13 14 15

504 494 482 471 458

671 651 631 610 588

424 415 406 396 385

564 547 530 512 493

441 429 415 401 387

577 555 531 506 481

324 314 304 294 283

423 406 388 369 350

255 243 232 219 207

322 302 282 262 242

16 17 18 19 20

446 432 419 405 391

565 542 518 495 471

374 363 351 339 327

474 454 434 414 394

372 357 341 326 310

456 430 404 379 354

272 260 249 237 225

331 312 293 274 255

195 182 170 158 146

222 202 184 165 149

22 24 26 28 30

362 333 304 275 247

423 376 332 288 251

303 278 253 229 206

353 314 276 240 209

279 248 219 191 166

305 259 221 191 166

202 179 157 136 119

219 185 158 136 119

123 104 88 76 66

123 104 88 76 66

32 34 36 38 40

221 196 174 157 141

221 196 174 157 141

183 163 145 130 117

183 163 145 130 117

146 129 115 103 93

146 129 115 103 93

104 92 82 74 67

104 92 82 74 67

58 52 46 41 37

58 52 46 41 37


116 19 311 54 12.0 43.0

161 26 366 75 10.2 30.7

88 16 188 38 11.9 38.7

122 22 221 53 10.1 28.3

168 28 599 90 8.7 31.1

79 15 202 36 10.0 35.9

110 21 238 50 8.5 25.6

75 16 309 40 8.1 35.7

104 22 364 56 6.9 24.4


36 0

Properties


18.4 472 153 2.88 1.76 13500 4370

15.5 393 127 2.86 1.76 11200 3630

121 20 508 65 10.2 45.6

16.8 294 101 2.45 1.71 8400 2890

12.4 210 71.7 2.41 1.71 6050 2060



10.6 119 40.3 1.95 1.72 3420 1150

3 - 34

COLUMN DESIGN



3 - 35

Steel Pipe and Structural Tubing

The design strengths in the tables that follow are tabulated for the effective lengths in feet KL (with respect to the least radius of gyration, r or ry), indicated at the left of each table. They are applicable to axially loaded members in accordance with Section E2 of the LRFD Specification. For discussion of effective length, range of l / r, strength about major axis, combined axial and bending stress, and sample problems, see General Notes. Properties and factors are listed at the bottom of the tables for checking strength about the strong axis. Steel Pipe Columns

Design strengths for unfilled pipe columns are tabulated for Fy = 36 ksi. Steel pipe manufactured to ASTM A501 furnishes Fy = 36 ksi, and ASTM A53, Type E or S, Gr. B furnishes Fy = 35 ksi and may be designed for the strengths permitted for Fy = 36 ksi steel. The heavy horizontal lines within the table indicate Kl / r = 200. No values are listed beyond Kl / r = 200. Structural Tube Columns

Design strengths for square and rectangular structural tube columns are tabulated for Fy = 46 ksi. Structural tubing is manufactured to Fy = 46 ksi under ASTM A500, Gr. B. All tubes listed in the column load tables satisfy Section B5 of the LRFD Specification. The heavy horizontal lines appearing within the tables indicate Kl / r = 200. No values are listed beyond Kl / r = 200.


3 - 36

COLUMN DESIGN

Fy = 36 ksi

COLUMNS Standard steel pipe Design axial strength in kips (φ = 0.85)

Nominal Dia.

12

10

8

6

5

4

31⁄2

3

Wall Thickness

0.375

0.365

0.322

0.280

0.258

0.237

0.226

0.216

Weight per ft

49.56

40.48

28.55

18.97

14.62

10.79

9.11

7.58

Fy

Effective length KL (ft)

36 ksi 0

447

364

257

171

132

97

82

68

6 7 8 9 10

440 438 436 433 429

357 354 351 348 344

249 246 243 239 235

162 159 155 151 147

122 118 115 111 106

86 82 78 74 70

70 67 63 58 54

56 52 48 43 39

11 12 13 14 15

426 422 418 413 409

340 336 331 326 321

231 227 222 216 211

142 138 133 127 122

102 97 92 86 81

65 60 55 51 46

49 45 40 36 32

35 30 26 23 20

16 17 18 19 20

404 399 393 387 381

315 309 303 297 291

205 199 193 187 181

116 111 105 99 94

76 71 66 61 56

41 37 33 30 27

28 25 22 20 18

17 15 14 12

22 24 25 26 28

369 356 349 342 328

277 263 256 249 234

168 155 149 142 129

83 72 67 62 53

47 39 36 33 29

22 19 17

15

30 31 32 34 36

313 306 298 283 268

219 212 205 190 176

117 111 105 93 83

47 44 41 36 32

25 23

37 38 40

260 253 237

169 162 148

79 75 67

31

3.17 7.23 1.51

2.68 4.79 1.34

Properties 2

Area A (in. ) I (in.4) r (in.)

14.6 279 4.38

11.9 161 3.67

8.40 72.5 2.94

5.58 28.1 2.25

4.30 15.2 1.88



2.23 3.02 1.16


3 - 37

Fy = 36 ksi

COLUMNS Extra strong steel pipe Design axial strength in kips (φ = 0.85)

Nominal Dia.

12

10

8

6

5

4

31⁄2

3

Wall Thickness

0.500

0.500

0.500

0.432

0.375

0.337

0.318

0.300

Weight per ft

65.42

54.74

43.39

28.57

20.78

14.98

12.50

10.25


Fy

36 ksi 0

588

493

392

257

187

135

113

92

6 7 8 9 10

579 576 573 569 564

483 479 475 470 465

379 375 369 364 357

243 238 232 226 219

172 168 162 156 149

119 114 108 102 95

96 91 85 79 72

75 69 64 58 52

11 12 13 14 15

559 554 549 543 536

460 453 447 440 433

351 343 336 327 319

212 205 197 189 180

143 135 128 121 113

89 82 75 68 62

66 60 53 47 42

46 40 34 30 26

16 18 19 20 21

530 515 508 500 492

425 409 400 391 382

310 291 282 272 262

172 154 145 137 128

105 91 83 76 70

56 44 40 36 32

37 29 26 23 21

23 18 16

22 24 26 28 30

483 465 447 428 408

373 354 334 314 294

252 231 211 191 172

120 103 88 76 66

63 53 45 39 34

30 25

32 34 36 38 40

388 368 348 328 308

273 253 234 215 196

154 136 121 109 98

58 52 46

19.2 362 4.33

16.1 212 3.63

12.8 106 2.88

6.11 20.7 1.84

4.41 9.61 1.48

3.68 6.28 1.31

3.02 3.89 1.14

Properties Area A (in.2) I (in.4) r (in.)

8.40 40.5 2.19



3 - 38

COLUMN DESIGN

Fy = 36 ksi

COLUMNS Double-extra strong steel pipe Design axial strength in kips (φ = 0.85)

Nominal Dia.

8

6

5

4

3

Wall Thickness

0.875

0.864

0.750

0.674

0.600

Weight per ft

72.42

53.16

38.55

27.54

18.58


Fy

36 ksi 0

652

477

346

248

167

6 7 8 9 10

629 621 612 601 590

448 437 426 413 399

315 305 293 281 268

214 203 191 179 165

131 120 108 96 84

11 12 13 14 15

578 565 551 536 521

385 369 353 336 319

254 239 224 209 194

152 139 125 112 100

73 62 53 46 40

16 17 18 19 20

505 489 472 455 438

302 285 268 250 234

179 165 151 137 124

88 78 70 62 56

35 31

22 24 26 28 30

403 367 333 299 266

201 170 145 125 109

102 86 73 63

47

32 34 36 38 40

235 208 186 166 150

96 85

21.3 162 2.76

15.6 66.3 2.06

11.3 33.6 1.72

8.10 15.3 1.37

Properties Area A (in.2) I (in.4) r (in.)



5.47 5.99 1.05


3 - 39

Fy = 46 ksi

COLUMNS Square structural tubing Design axial strength in kips (φ = 0.85)

Nominal Size

16× ×16

Thickness

1⁄

Wt./ft

103.30

89.68

68.31

0

1190

1030

786

6 7 8 9 10

1180 1170 1170 1170 1160

1020 1020 1010 1010 1000

11 12 13 14 15

1150 1150 1140 1130 1120

16 17 18 19 20

2

14× ×14 1⁄

2

12× ×12 3⁄

8

8

93.34

1⁄

2

3⁄

8

5⁄

16

76.07

58.10

48.86

1070

876

669

563

777 774 770 766 761

1050 1050 1040 1030 1020

862 857 851 845 838

658 655 650 645 640

554 551 548 544 539

993 985 977 969 960

756 751 745 739 732

1010 1000 992 979 966

830 821 812 803 792

634 628 621 614 606

535 529 524 518 511

1120 1110 1100 1090 1080

950 940 930 919 907

725 717 710 701 693

953 939 924 908 892

781 770 758 746 733

598 590 581 571 562

504 497 490 482 474

21 22 23 24 25

1070 1060 1040 1030 1020

895 883 870 857 844

684 675 665 655 645

875 858 841 823 805

719 706 692 677 663

552 542 531 520 510

466 457 449 440 431

26 27 28 29 30

1010 994 981 967 954

830 816 802 787 772

635 624 614 603 592

786 767 748 729 710

648 633 617 602 586

498 487 475 464 452

421 412 402 392 383

32 34 36 38 40

925 896 865 835 803

742 711 680 648 616

569 546 522 498 474

670 631 592 553 515

555 523 491 460 429

428 404 381 357 333

363 343 323 303 283

30.4 1200 6.29

26.4 791 5.48

27.4 580 4.60

22.4 485 4.66

17.1 380 4.72

14.4 324 4.75

Fy


5⁄

46 ksi

Properties A (in2) I (in.4) r (in.)

20.1 615 5.54


3 - 40

COLUMN DESIGN

Fy = 46 ksi


10× ×10

Nominal Size Thickness Wt./ft

76.33

62.46

47.90

40.35

32.63

0

876

719

551

465

375

6 7 8 9 10

855 847 839 829 818

703 697 690 682 674

539 534 529 524 517

455 451 447 442 437

11 12 13 14 15

807 794 781 767 752

664 655 644 633 621

510 503 495 487 478

16 17 18 19 20

736 720 703 686 668

608 595 582 568 553

21 22 23 24 25

650 631 612 593 573

26 27 28 29 30 32 34 36 38 40

8

1⁄

2

3⁄

8× ×8

5⁄

8

5⁄

16

4

5⁄

8

59.32

1⁄

2

3⁄

8

5⁄

16

1⁄

4

48.85

37.69

31.84

25.82

680

563

434

366

297

367 364 360 357 353

654 644 634 622 609

542 535 526 517 507

418 413 407 400 392

353 349 343 338 331

287 283 279 274 269

431 425 418 411 404

348 343 338 332 326

595 580 564 548 531

496 484 471 458 444

384 375 366 356 345

324 317 309 301 293

264 258 252 245 238

468 459 449 438 427

396 388 380 371 362

320 314 307 300 293

513 494 476 456 437

430 415 400 385 369

335 324 312 301 289

284 275 265 256 246

231 224 216 209 201

538 523 508 493 477

416 405 394 382 370

353 343 334 324 314

286 278 270 263 255

418 398 379 360 341

354 338 322 307 291

277 266 254 242 230

236 226 216 206 196

193 185 177 169 161

554 534 515 495 476

461 446 430 414 398

358 347 335 323 311

305 295 285 275 265

247 239 231 223 215

322 304 286 268 251

276 261 246 232 218

219 207 196 185 174

187 177 168 158 149

153 146 138 131 123

437 400 364 328 296

367 337 307 278 251

287 264 242 220 199

245 225 206 188 170

199 184 168 154 139

221 195 174 156 141

191 169 151 136 122

153 136 121 109 98

132 117 104 93 84

109 97 86 77 70

22.4 321 3.78

18.4 271 3.84

14.1 214 3.90

11.9 183 3.93

17.4 153 2.96

14.4 131 3.03

11.1 106 3.09

9.36 90.9 3.12

7.59 75.1 3.15

Fy


1⁄

46 ksi


9.59 151 3.96



3 - 41

Fy = 46 ksi

COLUMNS Square structural tubing Design axial strength in kips (φ = 0.85) 7× ×7

Nominal Size Thickness

5⁄ 8

1⁄ 2

3⁄ 8

6× ×6

5⁄ 16

1⁄ 4

3⁄ 16

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

50.81 42.05 32.58 27.59 22.42 17.08 42.30 35.24 27.48 23.34 19.02 14.53

Fy

46 ksi


Wt./ft

0

584

485

375

317

258

196

486

407

316

268

219

167

6 7 8 9 10

553 543 531 518 503

461 452 443 432 421

357 351 344 336 327

302 297 291 285 278

246 242 237 232 226

188 185 181 177 173

451 438 425 410 394

379 369 358 346 333

295 288 280 271 262

251 245 239 231 223

205 200 195 189 183

157 153 149 145 140

11 12 13 14 15

488 472 454 437 418

409 396 382 368 353

318 308 298 288 277

270 262 254 245 236

220 214 207 200 193

168 164 159 153 148

377 359 341 322 303

320 306 291 276 260

252 241 230 219 207

215 206 197 187 178

176 169 162 154 146

135 130 124 119 113

16 17 18 19 20

399 380 361 342 323

338 322 307 291 276

265 254 242 230 218

226 217 207 197 187

185 177 170 162 154

142 136 130 124 118

284 265 246 227 210

245 229 214 199 184

195 184 172 160 149

168 158 148 138 129

138 131 123 115 107

107 101 95 89 83

22 24 26 28 30

285 248 214 184 161

245 215 187 161 140

195 172 151 130 113

167 148 130 113 98

138 123 108 94 81

107 95 84 73 63

175 147 125 108 94

155 131 111 96 84

127 107 91 79 69

111 93 80 69 60

92 78 67 57 50

72 61 52 45 39

32 34

141 125

123 109

100 88

86 76

72 63

56 49

83 73

73 65

60 53

53 47

44 39

34 30

35 36 37 38 39

118 111 106 100 95

103 97 92 87 83

83 79 74 71 67

72 68 64 61 58

60 57 54 51 48

47 44 42 40 38

69

61 58

50 48 45

44 41 39 37

37 35 33 31

29 27 26 24 23

40

90

79

64

55

46

36

12.4 57.3 2.15

10.4 50.5 2.21

8.08 41.6 2.27

6.86 36.3 2.30

5.59 30.3 2.33

4.27 23.8 2.36

Properties 2

A (in ) I (in.4) r (in.)

14.9 97.5 2.56

12.4 84.6 2.62

9.58 68.7 2.68

8.11 59.5 2.71

6.59 49.4 2.74

5.02 38.5 2.77



3 - 42

COLUMN DESIGN

Fy = 46 ksi

COLUMNS Square structural tubing Design axial strength in kips (φ = 0.85) 51⁄2×51⁄2

Nominal Size

5× ×5

Thickness

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

Wt./ft

24.93

21.21

17.32

13.25

28.43

22.37

19.08

15.62

11.97

8.16


Fy

46 ksi 0

286

244

199

152

327

257

219

179

138

94

6 7 8 9 10

264 256 248 238 228

225 219 212 204 195

184 179 173 167 161

141 137 133 129 124

294 282 270 257 242

233 224 215 205 194

199 192 184 176 167

163 158 152 145 138

126 121 117 112 107

86 83 80 77 73

11 12 13 14 15

218 207 195 184 172

187 177 168 158 148

154 146 139 131 123

118 113 107 101 95

228 213 197 182 167

183 172 160 149 137

158 148 139 129 119

131 123 115 107 99

101 95 89 84 78

70 66 62 58 54

16 17 18 19 20

160 149 137 126 115

139 129 119 110 101

115 107 99 92 84

89 83 77 72 66

152 138 124 111 100

126 115 104 93 84

110 100 91 82 74

92 84 77 69 63

72 66 60 55 50

50 46 42 38 35

22 24 26 28 30

96 80 68 59 51

84 70 60 52 45

70 59 50 43 38

55 47 40 34 30

83 70 59 51 45

70 59 50 43 37

61 52 44 38 33

52 44 37 32 28

41 34 29 25 22

29 24 21 18 15

31 32 33 34 35

48 45 43 40

42 40 37 35

35 33 31 29 28

28 26 25 23 22

35

31

26 24

21 19

15 14 13

6.58 22.8 1.86

5.61 20.1 1.89

4.59 16.9 1.92

3.52 13.4 1.95

2.40 9.41 1.98

Properties 2

A (in ) I (in.4) r (in.)

7.33 31.2 2.07

6.23 27.4 2.10

5.09 23.0 2.13

3.89 18.1 2.16

8.36 27.0 1.80




3 - 43

Fy = 46 ksi

COLUMNS Square structural tubing Design axial strength in kips (φ = 0.85) 41⁄2×41⁄2

Nominal Size

4× ×4

Thickness

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

Wt./ft

19.82

16.96

13.91

10.70

7.31


Fy

1⁄ 2

3⁄ 8

21.63

17.27

5⁄ 16

1⁄ 4

14.83 12.21

3⁄ 16

1⁄ 8

9.42

6.46

46 ksi 0

228

195

160

123

84

249

199

170

140

108

74

6 7 8 9 10

201 192 182 171 160

172 165 157 148 139

142 136 130 123 115

110 105 100 95 90

75 72 69 65 62

208 195 180 166 151

168 158 148 137 125

145 137 128 119 110

120 114 107 100 92

93 89 83 78 72

64 61 58 54 50

11 12 13 14 15

149 137 125 114 103

129 119 110 100 91

107 100 92 84 76

84 78 72 66 60

58 54 50 46 42

136 121 107 93 81

114 102 91 81 70

100 90 81 72 63

84 76 68 61 54

66 60 54 49 43

46 42 38 34 31

16 17 18 19 20

92 82 73 66 59

81 73 65 58 53

69 62 55 49 45

55 49 44 39 36

38 35 31 28 25

71 63 56 50 46

62 55 49 44 40

55 49 44 39 35

47 42 37 34 30

38 34 30 27 24

27 24 21 19 17

21 22 23 24 25

54 49 45 41 38

48 43 40 36 34

41 37 34 31 29

32 29 27 25 23

23 21 19 17 16

41 38 34

36 33 30 27

32 29 27 25

28 25 23 21 19

22 20 18 17 16

16 14 13 12 11

26

35

31

26

21

15

27 28 29

32

29 27

25 23

20 18 17

14 13 12

10

Properties 2

A (in ) I (in.4) r (in.)

5.83 16.0 1.66

4.98 14.2 1.69

4.09 12.1 1.72

3.14 9.60 1.75

2.15 6.78 1.78

6.36 12.3 1.39

5.08 10.7 1.45

4.36 9.58 1.48



3.59 8.22 1.51

2.77 6.59 1.54

1.90 4.70 1.57

3 - 44

COLUMN DESIGN

Fy = 46 ksi


31⁄2×31⁄2

Nominal Size

3× ×3

Thickness

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

Wt./ft

12.7

10.51

8.15

5.61

10.58

8.81

6.87

4.75

0

146

121

93

65

122

101

79

55

6 7 8 9 10

118 109 100 90 81

99 92 84 76 69

77 72 66 60 54

54 50 46 42 39

90 80 71 61 52

76 68 61 53 45

60 54 49 43 37

42 39 35 31 27

11 12 13 14 15

71 62 54 46 40

61 54 46 40 35

49 43 38 32 28

35 31 27 23 20

44 37 31 27 23

38 32 27 24 21

32 27 23 19 17

23 20 17 14 12

16 17 18 19 20

35 31 28 25 23

31 27 24 22 20

25 22 20 18 16

18 16 14 13 11

21 18

18 16 14

15 13 12

11 10 9 8

21 22

21

18

14 13

10 9

3.73 6.09 1.28

3.09 5.29 1.31

2.39 4.29 1.34

3.11 3.58 1.07

2.59 3.16 1.10

2.02 2.60 1.13

1.40 1.90 1.16


Fy

46 ksi


1.65 3.09 1.37



3 - 45

Fy = 46 ksi

Y

COLUMNS Rectangular structural tubing Design axial strength in kips (φ = 0.85)

X

X

Y

Nominal Size 16× ×12

16× ×8

14× ×12

14× ×10

Thickness

1⁄

1⁄

1⁄

Wt./ft

89.68

76.07

82.88

63.21

76.07

0

1030

876

952

726

876

6 7 8 9 10

1020 1010 1010 998 990

848 838 827 815 801

938 933 927 920 912

715 712 707 702 697

11 12 13 14 15

982 973 963 952 941

786 771 754 736 717

904 895 886 876 865

16 17 18 19 20

929 916 903 889 875

697 677 657 635 614

22 24 26 28 30

845 813 781 746 711

32 34 36 38 40

676 640 604 568 533

2

2

2

3⁄

8

2

12× ×10

3⁄

8

58.10

1⁄

2

3⁄

8

5⁄

16

1⁄

4

69.27

53.00

44.60

36.03

669

796

609

513

414

857 850 843 834 825

655 650 644 638 631

778 772 765 757 748

596 591 586 580 573

502 498 493 488 483

405 402 399 395 390

691 684 677 669 661

815 803 791 779 765

623 615 606 597 587

738 728 716 704 692

566 558 550 541 531

477 470 463 456 448

386 380 375 369 363

854 842 829 816 803

653 644 634 625 615

751 737 721 705 689

576 565 554 542 530

679 665 650 636 620

522 511 501 489 478

440 431 422 413 404

356 349 342 335 327

569 525 480 436 393

774 744 713 681 648

593 571 548 524 499

655 620 584 547 511

504 478 451 424 396

589 556 522 488 454

454 430 404 379 353

384 363 342 321 300

311 295 278 261 244

352 313 279 250 226

615 581 547 514 480

474 448 423 398 372

474 438 403 369 335

368 341 315 289 264

420 387 355 324 293

328 302 278 254 231

278 257 237 217 197

227 210 193 177 162

17.1 476 284 1.29 4.08

20.4 419 316 1.15 3.94

15.6 330 249 1.15 4.00

13.1 281 213 1.15 4.03

10.6 230 174 1.15 4.06

Fy

Effective length KL (ft) with respect to least radius of gyration

1⁄

46 ksi

Properties 2

A (in ) Ix (in.4) Iy (in.4) rx / ry ry (in.)

26.4 962 618 1.25 4.84

22.4 722 244 1.72 3.30

24.4 699 552 1.13 4.76

18.6 546 431 1.13 4.82

22.4 608 361 1.30 4.02



3 - 46

COLUMN DESIGN

Fy = 46 ksi

Y

X

X


Y

12× ×8

Nominal Size

12× ×6

Thickness

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

Wt./ft

76.33

62.46

47.90

40.35

67.82

55.66

42.79

36.10

0

876

719

551

465

778

641

493

414

6 7 8 9 10

845 835 822 809 794

695 687 677 666 655

534 527 520 512 503

450 445 439 433 425

731 715 697 677 655

604 591 577 561 543

466 456 445 434 421

392 384 376 366 355

11 12 13 14 15

778 760 742 722 702

642 628 613 598 582

494 483 473 461 449

417 409 400 390 380

632 607 581 555 528

525 505 485 464 442

407 393 378 362 346

344 332 320 307 293

16 17 18 19 20

681 659 637 614 591

565 547 530 511 493

437 424 410 397 383

370 359 348 336 325

500 473 445 418 390

420 398 375 353 331

329 313 296 279 262

280 266 252 238 224

22 24 26 28 30

544 497 451 405 362

455 417 380 343 307

355 326 298 270 243

301 277 253 230 207

338 288 245 211 184

288 247 211 182 158

230 199 170 146 128

197 171 146 126 110

32 34 36 38 39

320 283 252 227 215

273 241 215 193 184

217 192 171 154 146

185 164 146 131 125

162 143 128 115 109

139 123 110 99 94

112 99 89 80 75

97 86 76 69 65

40

205

174

139

119

89

72

62

16.4 287 96.0 1.73 2.42

12.6 228 77.2 1.72 2.48

10.6 196 66.6 1.71 2.51


Fy

46 ksi

Properties 2


22.4 418 221 1.38 3.14

18.4 353 188 1.37 3.20

14.1 279 149 1.37 3.26

11.9 239 128 1.37 3.28

19.9 337 112 1.73 2.37




3 - 47

Fy = 46 ksi

Y


X

X

Y

10× ×8

Nominal Size

10× ×6

Thickness

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

Wt./ft

55.66

42.79

36.10

29.23

48.85

37.69

31.84

25.82

0

641

493

414

336

563

434

366

297

6 7 8 9 10

619 611 602 592 581

476 470 463 456 448

401 396 390 384 377

325 321 317 312 306

529 517 504 490 474

409 400 391 380 368

345 338 330 321 312

281 275 269 261 254

11 12 13 14 15

568 556 542 528 513

439 429 419 408 397

370 362 354 345 335

300 294 287 280 273

457 439 421 402 382

356 343 329 315 300

302 291 279 267 255

246 237 228 218 209

16 17 18 19 20

497 481 465 448 431

386 374 361 349 336

326 316 306 295 285

265 257 249 241 232

362 342 322 302 282

285 270 255 240 225

243 230 218 205 193

199 189 179 169 159

22 24 26 28 30

396 361 327 294 262

310 284 258 232 208

263 241 220 198 178

215 197 180 163 146

244 208 177 153 133

196 169 144 124 108

169 146 124 107 93

139 121 103 89 77

32 34 36 38 39

231 205 183 164 156

184 163 146 131 124

158 140 125 112 106

130 116 103 93 88

117 104 92 83 79

95 84 75 67 64

82 73 65 58 55

68 60 54 48 46

40

148

118

101

84

61

52

44

11.1 145 65.4 1.49 2.43

9.36 125 56.5 1.48 2.46

7.59 103 46.9 1.48 2.49


Fy

46 ksi

Properties 2


16.4 226 160 1.19 3.12

12.6 180 127 1.19 3.18

10.6 154 109 1.19 3.21

8.59 127 90.2 1.19 3.24

14.4 181 80.8 1.50 2.37



3 - 48

COLUMN DESIGN

Fy = 46 ksi

Y

X


X

Y

10× ×5

Nominal Size

8× ×6

Thickness

3⁄ 8

5⁄ 16

1⁄ 4

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

Wt./ft

35.13

29.72

24.12

42.05

32.58

27.59

22.42

0

403

341

277

485

375

317

258

6 7 8 9 10

370 359 347 334 319

315 306 295 284 272

256 249 241 232 222

454 444 432 419 404

352 344 335 325 315

298 292 284 276 268

243 238 232 225 218

11 12 13 14 15

304 288 272 255 239

260 246 233 219 205

212 201 191 179 168

389 373 357 340 322

303 292 279 266 253

258 248 238 227 217

211 203 195 186 178

16 17 18 19 20

222 206 189 174 159

191 178 164 151 138

157 146 135 124 114

305 287 269 252 235

240 227 213 200 187

205 194 183 172 161

169 160 151 142 133

22 24 26 28 30

131 110 94 81 71

115 96 82 71 62

95 80 68 59 51

201 170 145 125 109

161 137 117 101 88

140 119 102 88 76

116 99 85 73 64

32 34 36 38 39

62 55

54 48

45 40

96 85 76 68

77 68 61 55 52

67 59 53 48 45

56 49 44 40 38


Fy

46 ksi

36

40

Properties 2


10.3 128 42.9 1.72 2.04

8.73 110 37.2 1.71 2.07

7.09 91.2 31.1 1.72 2.09

12.4 103 65.7 1.25 2.31



9.58 83.7 53.5 1.25 2.36

8.11 72.4 46.4 1.25 2.39

6.59 60.1 38.6 1.25 2.42


3 - 49

Fy = 46 ksi

Y


X

X

Y

8× ×4

Nominal Size

7× ×5

Thickness

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

Wt./ft

42.30

35.24

27.48

23.34

19.02

35.24

27.48

23.34

19.02

14.53

0

486

407

316

268

219

407

316

268

219

167

6 7 8 9 10

415 393 368 341 314

351 333 313 292 270

276 262 248 233 216

235 224 212 199 185

192 184 174 164 153

369 357 342 327 311

288 279 268 257 245

245 238 229 220 210

200 194 187 180 172

154 149 144 138 132

11 12 13 14 15

287 259 233 207 182

248 226 204 183 162

200 183 167 150 135

172 158 144 130 117

142 131 120 109 98

294 276 258 240 222

232 219 205 192 178

199 188 177 165 154

164 155 146 137 127

126 119 113 106 99

16 17 18 19 20

160 141 126 113 102

143 126 113 101 91

120 106 95 85 77

104 92 82 74 67

88 78 70 62 56

205 187 170 154 139

165 151 138 126 114

142 131 120 110 100

118 109 101 92 84

92 85 79 72 66

22 24 25 26 27

84 71

76 63 58

63 53 49 45

55 46 43 39 37

47 39 36 33 31

115 97 89 82 76

94 79 73 67 62

82 69 64 59 55

69 58 54 50 46

54 46 42 39 36

71 66 62 58

58 54 51 47 44

51 47 44 41 39

43 40 37 35 33

34 31 29 27 26

37

31

24 23

6.86 45.5 26.9 1.30 1.98

5.59 38.0 22.6 1.30 2.01

4.27 29.8 17.7 1.29 2.04


Fy

46 ksi

28 29 30 31 32 33 34

Properties 2


12.4 85.1 27.4 1.76 1.49

10.4 75.1 24.6 1.75 1.54

8.08 61.9 20.6 1.73 1.60

6.86 53.9 18.1 1.73 1.62

5.59 45.1 15.3 1.72 1.65

10.40 63.5 37.2 1.31 1.90

8.08 52.2 30.8 1.30 1.95



3 - 50

COLUMN DESIGN

Fy = 46 ksi

Y

X

X


Y

7× ×4

Nominal Size

6× ×4

Thickness

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

Wt./ft

24.93

21.21

17.32

13.25

28.43

22.37

19.08

15.62

11.97

0

287

244

199

152

327

257

219

179

138

6 7 8 9 10

249 236 223 208 193

213 202 191 179 167

175 166 158 148 138

134 128 121 114 107

279 263 246 228 210

222 211 198 185 171

190 181 171 160 148

157 149 141 132 123

121 115 109 102 96

11 12 13 14 15

178 163 148 133 118

154 141 129 116 104

128 118 107 97 88

99 92 84 76 69

191 173 155 137 121

157 143 129 116 103

136 125 113 102 91

114 104 95 85 77

89 81 74 67 61

16 17 18 19 20

105 93 83 74 67

92 82 73 65 59

78 69 62 56 50

62 55 49 44 40

106 94 84 75 68

90 80 71 64 58

80 71 63 57 51

68 60 54 48 44

54 48 43 38 35

21 22 23 24 25

61 55 51 46 43

54 49 45 41 38

45 41 38 35 32

36 33 30 28 25

62 56 51 47

52 48 44 40 37

46 42 39 36 33

39 36 33 30 28

31 29 26 24 22

26 27

40

35

30 27

23 22

30

26

20 19

5.61 26.2 13.8 1.38 1.57

4.59 22.1 11.7 1.37 1.60

3.52 17.4 9.32 1.37 1.63


Fy

46 ksi

Properties 2


7.33 44.0 18.1 1.56 1.57

6.23 38.5 16.0 1.56 1.60

5.09 32.3 13.5 1.55 1.63

3.89 25.4 10.7 1.54 1.66

8.36 35.3 18.4 1.39 1.48

6.58 29.7 15.6 1.38 1.54




3 - 51

Fy = 46 ksi

Y


X

X

Y

6× ×3

Nominal Size

5× ×4

Thickness

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

Wt./ft

25.03

19.82

16.96

13.91

10.70

19.82

16.96

13.91

10.70

0

288

228

195

160

123

228

195

160

123

6 7 8 9 10

216 194 172 150 129

176 160 144 127 111

152 138 125 111 97

126 116 105 94 83

98 90 82 74 65

195 185 173 161 148

168 159 149 139 129

139 132 124 116 107

107 102 96 90 84

11 12 13 14 15

109 92 78 67 59

95 81 69 59 52

84 71 61 52 46

72 62 53 45 39

57 50 42 36 32

135 123 110 98 86

118 107 97 87 77

99 90 82 73 65

77 71 64 58 52

16 17 18 19 20

52 46 41

45 40 36 32

40 36 32 28

35 31 27 25 22

28 25 22 20 18

76 67 60 54 49

67 60 53 48 43

58 51 46 41 37

46 41 36 33 29

44 40 37 34 31

39 36 33 30 28

33 30 28 26 24

27 24 22 20 19

22

17

4.09 14.1 9.98 1.19 1.56

3.14 11.2 7.96 1.19 1.59


Fy

46 ksi

21 22 23 24 25 26

Properties 2


7.36 27.7 8.91 1.76 1.10

5.83 23.8 7.78 1.74 1.16

4.98 21.1 6.98 1.75 1.18

4.09 17.9 6.00 1.73 1.21

3.14 14.3 4.83 1.72 1.24

5.83 18.7 13.2 1.19 1.50



4.98 16.6 11.7 1.20 1.53

3 - 52

COLUMN DESIGN

Fy = 46 ksi

Y

X

X


Y

5× ×3

Nominal Size

4× ×3

Thickness

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

Wt./ft

21.63

17.27

14.83

12.21

9.42

6.46

12.70

10.51

8.15

5.61

0

249

199

170

140

108

74

146

121

93

64

6 7 8 9 10

183 164 145 125 107

151 137 122 107 93

132 120 108 95 83

110 100 91 81 71

85 78 71 63 56

59 55 50 45 40

110 100 89 78 67

93 84 76 67 58

73 66 60 53 47

51 47 42 38 33

11 12 13 14 15

89 75 64 55 48

79 67 57 49 43

71 60 51 44 39

61 52 45 38 33

49 42 36 31 27

35 30 26 22 19

57 48 41 35 31

50 42 36 31 27

40 34 29 25 22

29 25 21 18 16

16 17 18 19 20

42 37

38 33 30

34 30 27 24

29 26 23 21

23 21 19 17 15

17 15 13 12 11

27 24 21

24 21 19 17

19 17 15 14

14 12 11 10 9

1.90 6.44 2.93 1.48 1.24

3.73 7.45 4.71 1.26 1.12

3.09 6.45 4.10 1.26 1.15

2.39 5.23 3.34 1.25 1.18

1.65 3.76 2.41 1.25 1.21


Fy

46 ksi

Properties 2


6.36 16.9 7.33 1.52 1.07

5.08 14.7 6.48 1.50 1.13

4.36 13.2 5.85 1.50 1.16

3.59 11.3 5.05 1.49 1.19

2.77 9.06 4.08 1.50 1.21




3 - 53

Double Angles and WT Shapes

Double Angles

Design strengths are tabulated for the effective length KL in feet with respect to both the X-X and Y-Y axes. Design strengths about the X-X axis are in accordance with LRFD Specification Section E2. For buckling about the Y-Y axis the shear deformation of the connectors may require the slenderness to be increased in accordance with the equations for (Kl / r)m in Section E4. Incorporating this slenderness ratio, the design strengths are determined from Section E2 or E3, whichever governs. In addition to the usual limit state of flexural buckling for columns, double angle and WT shapes in compression may also be governed by the limit state of flexural-torsional buckling, in accordance with Section E3 of the LRFD Specification. This has been included in the tables. Discussion under Section C2 of the LRFD Specification Commentary points out that for trusses it is usual practice to take K = 1.0. No values are listed beyond KL / r = 200. For buckling about the X-X axis, both angles move parallel so that the design strength is not affected by the connectors. For buckling about the Y-Y axis, the design strengths are tabulated for the indicated number n of intermediate connectors. For connectors with snug-tight bolts or different spacings, the design strength must be recalculated using the corresponding modified slenderness and LRFD Specification Section E4. The number of intermediate connectors given in the table was selected so the design strength about the Y-Y axis is 90 percent or greater of that for buckling of the two angles acting as a unit. If fewer connectors are used, the strength must be reduced accordingly. According to Section E4 of the LRFD Specification, the connectors must be spaced so that the slenderness ratio a / rz of the individual angle does not exceed 75 percent of the governing slenderness ratio of the built-up member. In designing members fabricated of two angles connected to opposite faces of a gusset plate, Chapter J of the LRFD Specification states that eccentricity between the gage lines and gravity axis may be neglected. In the following tables, this eccentricity is neglected. The tabulated loads for double angles referred to in the Y-Y axis assume a 3⁄8-in. spacing between angles. These values are conservative when a wider spacing is provided. Example 3-5 illustrates a method for determining the design strength when a 3⁄4-in. gusset plate is used. Examples 3-6 and 3-7 demonstrate how to determine the number of connectors when Klx / rx governs and when the modified (Kly / ry)m governs.

EXAMPLE 3-5

Given:

Solution:

Using 50 ksi steel, determine the design strength with respect to the Y-Y axis of a double angle member of 8×8×1 angles with an effective length equal to 12 ft, and connected to a 3⁄4-in. thick gusset plate. ry = 3.53 in. (from Double Angle Column Design Strength Table for two L8×8×1 with 3⁄8-in. plate) ry′ = 3.67 in. (from Part 1, Properties, Two Equal-Leg Angles, two L8×8×1 with 3⁄4-in. plate) ry 3.53 = = 0.962 ry′ 3.67 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3 - 54

COLUMN DESIGN

Equivalent effective length = 0.962 × 12 ft = 11.5 ft Enter Column Design Strength Table for two L8×8×1 with reference to Y-Y axis for effective lengths between 10 and 15 feet, read 1,120 and 1,000 kips, respectively.



Equivalent design strength = 1,120 − (1,120 − 1,000) ×



11.5 − 10  15 − 10 

= 1,084 kips EXAMPLE 3-6

Given:

Solution:

Using a double angle member of 5×3×1⁄2 angles (short legs back to back) and 36 ksi steel, with Lx = 10 ft and Ly = 20 ft, and a factored axial load of 70 kips, determine the number of connectors required. Assume K = 1.0 and that the intermediate connectors are snug-tight bolted. Kx Lx = 10 ft, Kx lx / rx = (10 × 12) / 0.829 = 145 Ky Ly = 20 ft, Ky ly / ry = (20 × 12) / 2.5 = 96 The X-X axis governs. From the X-X axis portion of the table φPn = 76 kips > 70 kips o.k. Find number of connectors required based on Section E4: a / rz ≤ 0.75KLx / rx a ≤ 0.75(KLx / rx)rz = 0.75 (145) 0.648 = 70 in. Assume two connectors are required; a = (10 × 12) / 3 = 40 in. a / rz = 40 / rz = 40.0 / 0.648 = 61.7 Check that modified (Ky ly / ry)m does not govern. According to Specification Equation E4-1,  962 + 61.7  2 = 114 (Ky ly / ry)m = √ Modified ly′ = 114ry / Ky = 114 (2.50 in.) / 1.0 = 285 in. = 23.8 ft Inspection of the tables indicates that Kxlx / rx still governs, therefore one connector is required every 40 inches.

EXAMPLE 3-7

Given:

Using the same steel shape and bolts as Example 3-6, with Lx = 10 ft and Ly = 30 ft, determine the number of connectors required and the corresponding maximum design strength. Assume K = 1.0. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


Solution:

3 - 55

Kx Lx = 10 ft, Kx lx / rx = (10 × 12) / 0.829 = 145 Ky Ly = 30 ft, Ky ly / ry = (30 × 12) / 2.5 = 144 Kx lx / rx appears to govern, so try one connector in the 10-ft length. Check (Ky ly / ry)m with a / rz = 5 × 12 / 0.648 = 93  1442 + 932 = 171 (Ky ly / ry)m = √ Since (Ky ly / ry)m governs, the Y-Y portion of the table gives a design strength of 72 kips provided four connectors are used in the 30-ft length. This gives a spacing of 30 ft / 5 = 6.0 ft. Check if (Kyly / ry)m governs with = (6.0 × 12) / 0.648 = 111 a / rz  1442 + 111 2 = 182 (Kyly / ry)m = √ (Ky ly / ry)m still governs, so four connectors at 6.0 ft would be appropriate. Verify that a / rz < 0.75 governing Kl / r : 111 < (0.75 × 182 = 137) o.k. Modified ly′ = 182ry / Ky = 182(2.5 in.) / 1.0 = 455 in. = 37.9 ft From the tables, the design strength is 45 kips. The design strength can be increased by closer spacing of the connectors, which reduces (Kyly / ry)m .


3 - 56

COLUMN DESIGN



3 - 57

Fy = 36 ksi Fy = 50 ksi Y

COLUMNS Double angles

X

X

Design axial strength in kips (φ = 0.85)

Equal legs 8

in. back to back of angles

3/ ′′ 8

8× ×8

Size 11⁄8

Thickness Wt./ft

36

7⁄ 8

1

113.8

Fy

X-X AXIS

Y

102.0

50

3⁄ 4

90.0

5⁄ 8

77.8

1⁄ 2

65.4

52.8

36

50

36

50

36

50

36

50

36

50

0

1030 1420

918

1280

811

1130

701

973

586

763

432

550

10 14 18 22 26

901 1190 795 1000 674 795 548 596 427 430

808 715 608 496 388

1070 902 719 542 391

715 633 539 440 345

945 799 638 482 349

619 549 469 384 303

819 694 556 422 306

518 461 395 325 257

651 559 456 354 261

387 348 302 253 205

478 417 349 278 212

30 34 38 39 40

323 251 201 191 182

323 251 201 191 182

294 229 183 174 165

294 229 183 174 165

262 204 163 155 147

262 204 163 155 147

230 179 143 136 129

230 179 143 136 129

196 153 122 116 110

196 153 122 116 110

159 124 99 94 90

159 124 99 94 90

123

123

105

105

85

85

0

1030 1420

918

1280

811

1130

701

973

586

763

432

550

10 15 20 25 30

939 1260 869 1130 779 975 677 803 570 632

834 772 692 601 506

1120 1000 864 711 560

726 670 599 517 432

965 865 741 606 473

615 569 509 440 368

808 728 626 514 402

495 460 413 359 302

609 555 486 407 325

345 324 297 263 226

406 378 341 295 244

35 40 45 50 55

465 366 290 235 194

477 366 290 235 194

412 324 257 208 172

422 324 257 208 172

349 272 216 175 145

354 272 216 175 145

297 232 184 150 124

301 232 184 150 124

244 191 152 124 103

248 191 152 124 103

188 150 120 98 82

193 150 120 98 82

56 57 58 59

188 181 175 169

188 181 175 169

166 161 155

166 161 155

140 135 130

140 135 130

120 116 112

120 116 112

99 96

99 96

79 76

79 76

No. of a Connectors

3⁄

Effective length KL (ft) with respect to indicated axis

b

Y-Y AXIS

41

2

3

Properties of 2 angles—3⁄8 in. back to back 2

A (in ) rx (in.) ry (in.)

33.5 2.42 3.55

30.0 2.44 3.53

26.5 2.45 3.51

22.9 2.47 3.49

19.2 2.49 3.47

aFor Y-Y axis, welded or fully tensioned bolted connectors only. bFor number of connectors, see double angle column discussion.


15.5 2.50 3.45

3 - 58

COLUMN DESIGN

Fy = 36 ksi Fy = 50 ksi Y X


X

Design axial strength in kips (φ = 0.85) 3⁄ 8

Equal legs in. back to back of angles 6× ×6

Size Thickness Wt./ft

74.8

Y-Y AXIS

X-X AXIS

Fy


7⁄ 8

1

3⁄ 4

66.2

57.4

1⁄ 2

48.4

3⁄ 8

39.2

29.8

36

50

36

50

36

50

36

50

36

50

36

50

0

673

935

597

829

517

718

435

604

352

470

243

309

8 10 12 14 16

580 533 481 426 370

759 676 586 495 407

515 473 428 379 330

675 601 522 441 364

447 412 373 332 290

587 524 457 388 321

377 347 315 280 245

495 442 386 328 272

306 283 257 229 201

389 351 308 265 222

214 200 183 166 147

264 241 216 190 164

18 22 26 30 31

315 218 156 117

326 218 156 117

282 196 140 105

292 196 140 105

248 173 124 93

259 173 124 93

210 147 105 79

220 147 105 79

173 122 87 65 61

182 122 87 65 61

129 94 68 51 48

138 94 68 51 48

0

673

935

597

829

517

718

435

604

352

470

243

309

10 12 14 16 18

595 567 537 503 468

787 738 683 625 565

523 499 472 442 410

690 647 598 547 494

449 428 404 379 352

590 552 511 468 422

371 353 334 313 291

483 453 420 385 348

289 275 260 244 226

359 338 315 289 262

187 180 172 163 153

219 210 199 186 173

20 22 24 26 28

431 394 357 321 286

505 446 389 334 288

378 345 312 280 249

441 388 338 290 250

324 295 267 239 213

377 332 288 247 214

268 244 221 198 176

310 273 238 204 176

208 189 171 152 135

235 207 181 155 135

142 131 120 109 97

158 143 128 113 98

30 32 34 36 38

252 221 196 175 157

252 221 196 175 157

218 192 170 152 137

218 192 170 152 137

187 164 146 130 117

187 164 146 130 117

154 136 120 108 97

154 136 120 108 97

118 104 92 82 74

118 104 92 82 74

86 76 68 61 55

86 76 68 61 55

40 42 43 44 45

142 129 123 117 112

142 129 123 117 112

123 112 107 102 98

123 112 107 102 98

106 96 91 87

106 96 91 87

87 79 76 72

87 79 76 72

67 61 58 56

67 61 58 56

50 45 43

50 45 43

Properties of 2 A (in2) rx (in.) ry (in.)

5⁄ 8

22.0 1.80 2.73

19.5 1.81 2.70

No. of a Connectors

Y

3/ ′′ 8

b

2

3

angles—3⁄8

in. back to back

16.9 1.83 2.68

14.2 1.84 2.66



11.5 1.86 2.64

8.72 1.88 2.62


3 - 59



X

X



Size 7⁄ 8

Thickness Wt./ft

X-X AXIS

3⁄ 4

54.5

Fy


1⁄ 2

47.2

3⁄ 8

32.4

5⁄ 16

24.6

20.6

36

50

36

50

36

50

36

50

36

50

0

490

680

425

591

291

404

217

282

169

215

6 8 10 12 14

433 393 348 299 251

573 502 423 343 268

377 344 305 264 222

500 440 372 304 239

259 237 211 183 155

344 304 259 213 169

194 178 160 140 119

244 219 189 159 129

152 141 127 113 97

189 171 150 128 107

16 18 20 22 24

204 162 132 109 91

206 162 132 109 91

182 145 117 97 82

183 145 117 97 82

128 103 83 69 58

130 103 83 69 58

99 80 65 54 45

102 80 65 54 45

82 68 55 46 38

86 68 55 46 38

75

75

53

53

42 39

42 39

35 33

35 33

25 26

Y-Y AXIS

3/ ′′ 8

Y

0

490

680

425

591

291

404

217

282

169

215

6 8 10 12 14

457 438 414 388 358

617 582 540 492 441

394 377 357 334 309

531 501 464 423 379

260 249 236 221 204

345 326 303 277 249

184 176 168 157 145

224 214 201 186 168

136 131 126 119 111

160 154 146 137 127

16 18 20 22 24

327 295 263 231 201

389 337 287 240 202

282 254 226 199 172

334 289 246 205 173

186 168 149 131 114

219 190 161 135 114

133 120 107 93 81

150 132 113 95 81

103 94 84 75 66

115 103 90 78 66

26 28 30 32 34

172 149 130 114 101

172 149 130 114 101

148 127 111 98 87

148 127 111 98 87

97 84 73 65 57

97 84 73 65 57

69 60 52 46 41

69 60 52 46 41

57 49 43 38 34

57 49 43 38 34

36 37 38

90 85 81

90 85 81

77 73 69

77 73 69

51 48

51 48

37 35

37 35

30

30


16.0 1.49 2.30

13.9 1.51 2.28

b

2

3


No. of a Connectors

3⁄ 8

9.50 1.54 2.24

7.22 1.56 2.22



6.05 1.57 2.21

3 - 60

COLUMN DESIGN



X

Design axial strength in kips (φ = 0.85) 3⁄ 8


Size 3⁄ 4

Thickness Wt./ft

37.0

X-X AXIS

Fy

Y-Y AXIS


5⁄ 8

1⁄ 2

31.4

3⁄ 8

25.6

5⁄ 16

19.6

1⁄ 4

16.4

13.2

36

50

36

50

36

50

36

50

36

50

36

50

0

334

463

282

392

230

319

175

243

146

191

108

138

4 6 8 10 12

306 275 237 195 154

411 354 288 220 159

259 233 201 167 132

349 301 245 189 137

212 191 166 138 110

285 247 203 157 115

162 146 127 106 85

217 189 156 121 89

135 123 107 90 72

172 152 127 101 76

101 92 82 70 57

126 112 96 78 61

14 16 18 19 20

117 89 71 63

117 89 71 63

100 77 61 54 49

100 77 61 54 49

84 65 51 46 41

84 65 51 46 41

65 50 40 36 32

65 50 40 36 32

56 43 34 30 27

56 43 34 30 27

45 35 28 25 22

46 35 28 25 22

0

334

463

282

392

230

319

175

243

146

191

108

138

6 8 10 12 14

303 284 262 237 210

406 371 332 288 245

254 238 219 198 176

339 311 277 241 204

204 191 176 158 140

270 247 220 191 161

151 141 130 117 104

196 180 161 141 119

121 114 106 96 85

148 138 125 110 95

85 81 76 70 63

100 94 87 79 70

16 18 20 22 24

183 157 132 109 92

202 163 132 109 92

153 131 109 91 76

168 135 109 91 76

122 104 86 72 60

133 106 86 72 60

90 77 64 53 45

98 79 64 53 45

74 63 53 44 37

79 64 53 44 37

56 48 41 35 29

60 50 41 35 29

26 28 29 30 31

78 67 63 59 55

78 67 63 59 55

65 56 52 49 46

65 56 52 49 46

51 44 41 39

51 44 41 39

38 33 31 29

38 33 31 29

32 27 26 24

32 27 26 24

25 22 20

25 22 20



10.9 1.19 1.88

9.22 1.20 1.86

7.50 1.22 1.83

5.72 1.23 1.81



4.80 1.24 1.80

3.88 1.25 1.79

No. of a Connectors

Y

3/ ′′ 8

b

3


3 - 61



X

X


Equal legs 8

in. back to back of angles 31⁄2×31⁄2

Size 3⁄ 8

Thickness Wt./ft

X-X AXIS

5⁄ 16

17.0

Fy


Y

1⁄ 4

14.4

11.6

36

50

36

50

36

50

0

152

211

128

175

100

129

2 4 6 8 10

148 137 120 100 78

204 182 152 117 84

125 115 101 84 67

169 152 127 99 72

97 90 80 67 54

125 114 97 77 58

12 14 16 17 18

59 43 33 29

59 43 33 29

50 37 28 25 22

50 37 28 25 22

41 30 23 21 18

41 30 23 21 18

0

152

211

128

175

100

129

6 8 10 12 14

130 120 108 94 81

170 152 131 109 88

107 98 89 78 67

136 123 106 89 72

80 74 67 60 52

96 88 78 67 56

16 18 20 22 24

67 54 44 37 31

69 54 44 37 31

56 45 37 30 26

57 45 37 30 26

43 36 29 24 20

44 36 29 24 20

26

26

26

22

22

17

17

3/ ′′ 8

No. of a Connectors

3⁄

b

Y-Y axis

3



4.97 1.07 1.61

4.18 1.08 1.60



3.38 1.09 1.59

3 - 62

COLUMN DESIGN



X

Design axial strength in kips (φ = 0.85) Y

3⁄ 8


Size 1⁄ 2

Thickness Wt./ft

18.8

X-X AXIS

Fy

Y-Y AXIS


3⁄ 8

5⁄ 16

14.4

1⁄ 4

12.2

3⁄ 16

9.8

7.42

36

50

36

50

36

50

36

50

36

50

0

168

234

129

179

109

151

88

118

61

77

2 4 5 6 7

162 145 133 120 106

222 190 169 146 123

125 112 103 93 83

171 147 131 114 97

105 94 87 79 70

144 124 111 97 82

85 77 71 64 57

112 98 88 77 66

59 54 50 46 41

74 66 60 54 47

8 9 10 11 12

92 79 66 54 46

101 81 66 54 46

72 62 52 43 36

80 64 52 43 36

61 53 45 37 31

68 55 45 37 31

50 43 37 31 26

56 46 37 31 26

37 32 28 24 20

41 34 28 24 20

13 14 15

39 34

39 34

31 27 23

31 27 23

26 23 20

26 23 20

22 19 16

22 19 16

17 15 13

17 15 13

0

168

234

129

179

109

151

88

118

61

77

2 4 6 8 10

163 155 143 128 111

223 209 187 161 132

123 117 108 97 84

167 156 140 121 99

101 97 89 80 70

136 128 115 99 82

79 75 70 63 55

100 95 86 75 63

50 48 45 42 37

59 57 53 48 42

12 14 16 18 20

94 76 60 47 38

104 78 60 47 38

70 57 45 36 29

78 58 45 36 29

58 47 37 29 24

64 48 37 29 24

46 38 29 23 19

50 38 29 23 19

32 27 22 17 14

35 28 22 17 14

22 23

32 29

32 29

24 22

24 22

20 18

20 18

16 14

16 14

12 11

12 11


b

3


No. of a Connectors

3/ ′′ 8

5.50 0.898 1.43

4.22 0.913 1.41

3.55 0.922 1.40



2.88 0.930 1.39

2.18 0.939 1.38


3 - 63



X

X


Equal legs 8

in. back to back of angles 21⁄2×21⁄2

Size 3⁄ 8

Thickness Wt./ft

X-X AXIS Y-Y AXIS

5⁄ 16

11.8

Fy


Y

1⁄ 4

10.0

3⁄ 16

8.2

6.14

36

50

36

50

36

50

36

50

0

106

147

90

125

73

101

54

70

2 3 4 5 6

101 94 86 76 66

137 125 110 93 76

85 80 73 65 56

116 106 93 79 65

69 65 59 53 46

94 86 76 65 53

52 48 44 40 35

66 61 54 47 40

7 8 9 10 11

55 45 36 29 24

59 46 36 29 24

47 39 31 25 21

51 39 31 25 21

39 32 26 21 17

42 33 26 21 17

30 25 20 16 13

32 25 20 16 13

12

20

20

17

17

14

14

11

11

0

106

147

90

125

73

101

54

70

2 3 4 5 6

101 99 95 90 85

138 133 126 118 109

84 82 79 75 71

114 110 105 98 90

67 65 63 60 56

89 86 82 77 71

46 45 44 42 40

57 55 53 50 47

7 8 9 10 11

79 73 66 60 53

98 88 77 67 57

66 61 55 50 44

82 73 64 55 47

52 48 44 40 35

64 58 51 44 37

37 35 32 29 26

44 40 35 31 27

12 13 14 15 16

47 41 35 31 27

48 41 35 31 27

39 34 29 26 22

40 34 29 26 22

31 27 23 20 18

32 27 23 20 18

23 20 17 15 13

23 20 17 15 13

17 18 19 20

24 21 19 17

24 21 19 17

20 18 16 14

20 18 16 14

16 14 13

16 14 13

12 11 9

12 11 9

Properties of 2 angles—3⁄8 in. back to back A (in2) rx (in.) ry (in.)

3.47 0.753 1.21

2.93 0.761 1.20

2.38 0.769 1.19



1.80 0.778 1.18

3/ ′′ 8

No. of a Connectors

3⁄

b

3

3 - 64

COLUMN DESIGN



X

Design axial strength in kips (φ = 0.85) Y

3⁄ 8


Size 3⁄ 8

Thickness Wt./ft

9.4

X-X AXIS

Fy

Y-Y AXIS


5⁄ 16

1⁄ 4

7.84

3⁄ 16

6.38

1⁄ 8

4.88

3.30

36

50

36

50

36

50

36

50

36

50

0

83

116

70

98

58

80

44

61

27

34

2 3 4 5 6

76 69 59 49 38

103 88 72 55 39

65 58 50 42 33

87 75 61 47 34

53 48 41 35 28

71 62 51 39 29

40 37 32 27 21

54 47 39 30 22

25 23 20 17 14

31 28 24 19 15

7 8 9 10

29 22 18

29 22 18

25 19 15 12

25 19 15 12

21 16 13 10

21 16 13 10

16 13 10 8

16 13 10 8

11 9 7 6

11 9 7 6

0

83

116

70

98

58

80

44

61

27

34

2 3 4 5 6

79 76 72 67 61

108 102 95 86 76

67 64 60 56 51

90 86 79 72 63

54 51 49 45 41

72 68 63 57 51

39 38 36 33 31

52 49 46 42 37

22 21 20 19 18

26 25 24 22 20

7 8 9 10 11

55 49 43 37 31

66 55 46 37 31

46 41 36 30 26

55 46 38 31 26

37 33 29 24 21

44 37 30 25 21

27 24 21 18 15

32 27 22 18 15

16 15 13 11 10

18 16 14 12 10

12 13 14 15 16

26 22 19 17 15

26 22 19 17 15

22 18 16 14 12

22 18 16 14 12

17 15 13 11 10

17 15 13 11 10

13 11 9 8 7

13 11 9 8 7

8 7 6 5 5

8 7 6 5 5



2.72 0.594 1.01

2.30 0.601 1.00

1.88 0.609 0.989



1.43 0.617 0.977

0.960 0.626 0.965

No. of a Connectors

3/ ′′ 8

b

3


3 - 65


Y


X

X


Unequal legs

3/ ′′ 8

Long legs 3⁄8 in. back to back of angles 3⁄ 4

1⁄ 2

Wt./ft

88.4

67.6

46.0

Fy

36

X-X AXIS

1⁄ 2

57.4

39.2

36

50

36

50

36

50

846

376

479

0

673

935 517 718 321 408

10 12 14 16 18

704 667 626 582 535

932 865 792 715 637

541 513 483 450 415

717 667 613 555 496

339 323 306 287 267

419 395 368 340 310

10 12 14 16 18

597 567 533 496 457

792 736 676 612 546

460 437 412 384 354

611 569 523 475 425

289 276 262 246 230

358 338 315 292 267

20 22 24 26 28

488 440 393 348 305

560 486 415 353 305

379 343 308 273 241

438 381 328 279 241

247 226 205 185 165

280 250 221 193 167

20 22 24 26 28

418 378 338 300 264

482 419 359 306 264

324 294 264 235 207

376 328 283 241 208

212 195 177 160 143

242 216 192 168 146

30 32 34 36 38

265 233 207 184 165

265 233 207 184 165

210 184 163 146 131

210 184 163 146 131

146 128 113 101 91

146 128 113 101 91

30 32 34 36 38

230 230 202 202 179 179 160 160 143 143

181 159 141 126 113

181 127 127 159 112 112 141 99 99 126 88 88 113 79 79

41 42

142

142 112 107

112 107

78 74

78 74

40 42 43

129 129 102 102 117 117 92 92

0

Y-Y AXIS


3⁄ 4

74.8

796 1110 609

0

50

8× ×4 1

b

50

36

50

71 65 62

796 1110 609

846

376

479

0

729 703 672 636 595

978 932 876 811 741

537 519 496 470 440

709 677 638 593 543

301 293 282 270 256

357 346 331 314 295

6 8 10 12 14

568 741 416 533 234 275 520 657 381 473 217 251 464 562 339 405 197 223 404 463 294 333 175 192 342 368 248 263 150 159

935 517 718 321 408

16 18 20 22 24

551 506 459 412 366

667 592 517 445 378

408 375 340 305 271

490 435 381 328 279

240 223 205 187 169

273 249 225 200 176

16 18 20 22 24

282 238 194 160 135

300 238 194 160 135

203 162 132 110 93

26 28 32

321 279 214

323 279 214

238 207 159

239 207 159

151 133 103

152 133 103

25 26

125 115

125 115

85

34 36 40 41 42

190 170 138 131 125

190 170 138 131 125

141 126 103 98

141 126 103 98

92 82 67

92 82 67

A

rx (in.) ry (in.)

26.0 2.49 2.52

19.9 2.53 2.48

13.5 2.56 2.44

22.0 2.52 1.61



2

204 126 127 162 103 103 132 84 84 110 70 70 93 60 60 85

55

55 3

Properties of 2 angles—3⁄8 in. back to back (in2)

b

71 65 62

6 8 10 12 14

2

673

36

No. of a Connectors

1

No. of a Connectors

8× ×6

Size Thickness

Y

16.9 2.55 1.55

11.5 2.59 1.51

3 - 66

COLUMN DESIGN


Y X


X


Unequal legs

3/ ′′ 8

3⁄ 4

1⁄ 2

3⁄ 8

Wt./ft

52.4

35.8

27.2

Fy

36

50

36

50

36

50

X-X AXIS


0 471 655 310 401 205 254

3⁄ 4

5⁄ 8

1⁄ 2

3⁄ 8

47.2

40.0

32.4

24.6

36 0

50

36

50

36

50

36

50

425 591 358 497 291 388 201 256

8 10 12 14 16

427 404 378 349 318

571 529 481 431 379

283 268 252 233 214

355 332 306 278 248

189 181 171 161 149

230 218 204 188 172

8 10 12 14 16

371 343 312 279 246

488 439 385 329 276

313 290 265 237 209

413 371 327 281 236

18 20 22 24 26

286 255 224 194 166

328 278 232 195 166

194 174 154 135 117

219 190 162 137 117

137 125 113 101 89

155 138 121 105 90

18 20 22 24 26

212 180 150 126 108

225 182 150 126 108

181 155 129 109 93

193 148 158 110 119 156 127 128 96 100 129 106 106 82 82 109 89 89 69 69 93 76 76 59 59

28 143 143 100 100 30 125 125 88 88 32 110 110 77 77 34 97 97 68 68

78 68 59 53

78 68 59 53

28 30 31 32

93 81 76

93 81 76

80 70 65

36 37

47 44

47 44

87 82

87 82

61 58

61 58

b

0 471 655 310 401 205 254

Y-Y AXIS

6× ×4

No. of a Connectors

7× ×4

Size Thickness

6 8 10 12 14

394 362 325 284 242

511 456 393 327 262

238 221 200 176 151

286 260 229 196 161

146 137 127 115 101

167 156 142 126 108

16 18 20 22 24

201 162 132 110 92

204 126 128 162 103 103 132 84 84 110 70 70 92 59 59

87 74 61 51 43

90 74 61 51 43

25 26 27

85 79 73

40

40

0

80 70 65

254 236 216 193 171

65 57 53

325 294 260 225 191

65 57 53

179 167 154 140 125

51 44 41 39

220 202 182 161 140 b

51 44 41 39

425 591 358 497 291 388 201 256

6 8 10 12 14

367 339 306 270 232

481 432 375 315 256

302 279 252 222 191

394 354 308 259 210

236 218 197 174 150

293 266 233 198 162

153 144 132 118 104

181 167 151 132 112

16 18 20 22 24

195 160 136 113 95

211 161 165 126 129 167 132 132 103 103 136 107 107 84 84 113 89 89 70 70 95 75 75 59 59

89 75 61 51 43

93 75 61 51 43

26 27 28

81 75 70

37 35

37 35

2

85 79 73

55 51

55 51

2

81 75 70

64 59

64 59

50 47

50 47

3


No. of a Connectors

Long legs 3⁄8 in. back to back of angles

Y

15.4 2.22 1.62

10.5 2.25 1.57

7.97 2.27 1.55

13.9 1.88 1.69

11.7 1.90 1.67



9.50 1.91 1.64

7.22 1.93 1.62


3 - 67


Y


X

X


Unequal legs

3/ ′′ 8


Wt./ft

23.4

Y-Y AXIS

X-X AXIS

Fy


5⁄ 16

19.6

36

50

36

50

0

191

243

145

179

8 10 12 14 16

170 159 146 133 119

209 192 173 153 133

130 123 114 105 95

157 146 134 120 106

18 20 22 24 26

105 92 78 66 56

114 95 79 66 56

85 75 65 56 48

93 79 67 56 48

28 30 32

49 42 37

49 42 37

41 36 32

0

191

243

4 6 8 10 12

148 139 127 113 98

14 16 18 20 22 23

3⁄ 4

1⁄ 2

39.6

3⁄ 8

27.2

5⁄ 16

20.8

50

36

50

0

355

493

245

340 183 238 143 182

4 6 8 10 12

337 317 290 259 225

460 233 421 219 372 202 318 181 262 158

14 16 18 20 22

191 158 127 103 85

209 161 127 103 85

41 36 32

24 25 26

72 66

72 66

51 47 44

145

179

0

355

493

245

176 163 147 127 105

107 101 94 85 75

122 115 106 94 80

4 6 8 10 12

325 303 274 241 206

437 215 396 200 345 182 289 160 232 136

284 258 226 189 152

152 142 129 114 98

81 66 53 43 36

84 66 53 43 36

64 53 43 35 29

66 53 43 35 29

14 16 18 20 22

170 144 114 93 77

188 144 114 93 77

113 90 72 58 48

117 90 72 58 48

82 65 52 43 35

84 65 52 43 35

65 53 43 35 29

68 53 43 35 29

33

33

27

27

24 25

65 60

65 60

41

41

30

30

25

25

b

318 292 260 223 185

36

17.4

36

175 165 152 137 120

50

36

50

224 208 187 163 138

137 130 120 109 97

172 161 146 129 111

135 149 104 113 113 116 87 90 91 91 71 71 74 74 58 58 61 61 48 48

85 72 60 49 41

93 76 61 49 41

34 31 29

34 31 29

51 47 44

40 37 34

40 37 34


186 113 134 171 107 125 152 98 114 130 88 99 107 77 84

2

6.84 1.94 1.39

5.74 1.95 1.38

b

340 183 238 143 182

2

3


No. of a Connectors

3⁄ 8

Thickness

5× ×31⁄2

No. of a Connectors

6× ×31⁄2

Size

Y

11.6 1.55 1.53

8.00 1.58 1.49



6.09 1.60 1.46

5.12 1.61 1.45

3 - 68

COLUMN DESIGN


Y X


X


Unequal legs

3/ ′′ 8

5× ×3

Size 1⁄ 2

Thickness Wt./ft

25.6

X-X AXIS

Fy

Y-Y AXIS


3⁄ 8

5⁄ 16

19.6

1⁄ 4

16.4

13.2

36

50

36

50

36

50

36

50

0

230

319

172

223

134

170

95

117

2 4 6 8 10

227 219 206 189 170

313 298 274 244 210

170 164 155 143 129

220 210 195 176 154

132 128 122 113 103

168 161 151 137 121

95 92 88 82 76

115 112 105 97 88

12 14 16 18 20

149 128 107 87 70

175 141 110 87 70

114 98 82 68 55

130 107 86 68 55

91 79 68 56 46

104 88 71 57 46

68 61 53 45 38

78 67 56 47 38

22 24 26 27

58 49 42

58 49 42

45 38 32

45 38 32

38 32 27

38 32 27

31 26 22 21

31 26 22 21

0

230

319

172

223

134

170

95

117

2 4 6 8 10

207 195 177 153 128

276 255 223 184 143

146 138 125 110 92

179 167 149 126 101

107 102 94 84 72

128 121 110 95 78

71 68 64 58 51

80 77 72 64 55

12 14 16 18 20

101 81 62 49 40

109 81 62 49 40

74 57 44 35 28

76 57 44 35 28

59 46 36 29 23

61 46 36 29 23

43 35 28 22 18

45 35 28 22 18

No. of a Connectors


Y

b

2

3



7.50 1.59 1.25

5.72 1.61 1.23

4.80 1.61 1.22



3.88 1.62 1.21


3 - 69


Y


X

X


Unequal legs

3/ ′′ 8


Thickness Wt./ft

23.8

Fy

X-X AXIS

3⁄ 8

5⁄ 16

18.2

15.4

12.4

36

50

36

50

36

50

36

50

0

214

298

163

227

137

179

101

129

2 4 6 8 10

210 198 179 155 130

289 266 232 191 148

160 151 137 120 101

221 204 178 147 116

134 127 115 101 85

174 162 143 120 96

99 94 87 77 66

126 118 106 91 75

12 14 16 18 20

104 80 61 48 39

109 80 61 48 39

81 63 48 38 31

86 63 48 38 31

69 54 41 33 26

73 54 41 33 26

55 44 34 27 22

59 44 34 27 22

24

24

20

20

0

214

298

163

227

137

179

101

129

2 4 6 8 10

203 196 183 167 148

276 262 240 211 180

150 144 135 124 110

200 190 174 155 132

121 116 110 101 90

150 143 133 120 104

84 81 77 72 65

99 96 91 83 74

12 14 16 18 20

128 108 88 74 60

147 116 93 74 60

95 80 66 53 43

108 86 66 53 43

78 66 54 43 35

87 70 54 43 35

58 50 42 34 28

64 53 42 34 28

22 24 25 26

50 42 39 36

50 42 39 36

35 30 28 25

35 30 28 25

29 25 23

29 25 23

23 20 18

23 20 18

b

21

Y-Y AXIS

1⁄ 4

No. of a Connectors

4× ×31⁄2

Size


Y

2

3



7.00 1.23 1.58

5.34 1.25 1.56

4.49 1.26 1.55



3.63 1.27 1.54

3 - 70

COLUMN DESIGN


Y X


X


Unequal legs

3/ ′′ 8

4× ×3

Size 1⁄ 2

Thickness Wt./ft

22.2

X-X AXIS

Fy


3⁄ 8

17.0

1⁄ 4

14.4

11.6

36

50

36

50

36

50

36

50

0

199

276

152

211

127

166

94

120

2 4 6 8 10

195 184 167 146 122

269 248 217 179 141

149 141 128 112 94

206 190 166 138 109

125 118 108 94 80

162 151 133 112 90

93 88 81 72 62

117 110 99 85 70

12 14 16 18 20

99 77 59 46 38

105 77 59 46 38

76 60 46 36 29

81 60 46 36 29

65 51 39 31 25

69 51 39 31 25

51 41 32 25 21

55 42 32 25 21

27

27

23

23

19

19

0

199

276

152

211

127

166

94

120

2 4 6 8 10

187 177 161 142 120

253 235 207 173 137

138 131 119 105 89

183 171 151 127 101

111 105 96 85 73

137 129 116 99 81

77 74 69 62 54

92 88 80 71 59

12 14 16 18 20

97 76 61 49 39

108 80 61 49 39

72 56 43 35 28

76 56 43 35 28

59 46 36 29 23

62 46 36 29 23

45 36 28 23 18

47 36 28 23 18

21 22

36 33

36 33

25

25

21

21

17

17

b

21

Y-Y AXIS

5⁄ 16

2

3



No. of a Connectors


Y

6.50 1.25 1.33

4.97 1.26 1.31

4.18 1.27 1.30



3.38 1.28 1.29


3 - 71


Y


X

X


Unequal legs

3/ ′′ 8


Wt./ft

15.8

X-X AXIS

Fy


5⁄ 16

1⁄ 4

13.2

10.8

36

50

36

50

36

50

0

140

195

118

162

92

119

2 4 6 8 10

137 127 112 93 74

188 169 142 111 80

115 107 95 79 63

157 141 119 94 69

90 84 75 63 51

116 106 91 73 55

12 14 16 17 18

56 41 32 28 25

56 41 32 28 25

48 35 27 24 21

48 35 27 24 21

39 29 22 20 18

0

140

195

118

162

2 4 6 8 10

131 124 114 101 86

176 164 146 123 99

107 102 94 83 71

12 14

71 56

76 59

16 18 20

45 36 29

22

24

3⁄ 8

1⁄ 4

14.4

9.8

36

50

36

50

0

129

179

85

110

2 4 6 8 10

126 117 103 86 69

173 156 131 103 75

83 77 69 59 47

107 97 84 68 52

40 29 22 20 18

12 14 16 18

52 39 30 23

53 39 30 23

37 27 21 17

37 27 21 17

92

119

0

129

179

85

110

140 131 118 100 81

79 76 70 63 54

97 92 83 73 60

2 4 6 8 10

118 109 96 80 63

158 142 119 92 69

71 67 59 50 40

87 80 69 56 42

59 46

62 47

45 36

48 36

12 14

49 36

49 36

30 22

30 22

45 36 29

36 29 23

36 29 23

28 22 18

28 22 18

16 18

28 22

28 22

17 14

17 14

24

19

19

15

15

b

No. of a Connectors

3⁄ 8

Thickness

31⁄2×21⁄2

No. of a Connectors

31⁄2×3

Size

Y

b

2

Y-Y AXIS

2

3

3



4.59 1.09 1.36

3.87 1.10 1.35

3.13 1.11 1.33

4.22 1.10 1.11



2.88 1.12 1.09

3 - 72

COLUMN DESIGN


Y X


X


Unequal legs

3/ ′′ 8

3⁄ 8

Thickness

13.2

Wt./ft

Y-Y AXIS

X-X AXIS

Fy


1⁄ 4

36

3⁄ 16

9.0

6.77

50

36

50

36

50

0 118 163

80

107

55

71

2 113 155 3 109 146 4 102 134 5 94 120 6 86 105

78 75 70 65 59

103 97 90 81 71

54 52 49 46 42

68 65 60 55 50

3× ×2

No. of a Connectors

3× ×21⁄2

Size

3⁄ 8

5⁄ 16

11.8

36

1⁄ 4

10.0

3⁄ 16

8.2

36

6.1

50

36

50

50

0

106 147

90

125

73

97

50

64

2 3 4 5 6

103 98 93 86 78

141 132 122 109 96

87 83 78 73 66

119 112 103 93 82

70 68 64 59 54

93 88 81 74 65

49 47 45 42 38

61 59 55 50 45

7 8 9 10 11

70 61 53 45 38

82 69 56 45 38

59 52 45 39 32

70 59 48 39 32

49 43 37 32 27

57 48 40 32 27

35 31 28 24 20

40 35 30 25 21

12 13 14 15 16

32 27 23 20

32 27 23 20

27 23 20 17

27 23 20 17

22 19 16 14

22 19 16 14

17 15 13 11 10

17 15 13 11 10

7 8 9 10 11

76 67 58 49 40

90 75 60 49 40

53 47 40 34 29

62 52 43 35 29

38 34 30 26 22

44 38 32 27 22

12 13 14 15

34 29 25 22

34 29 25 22

24 21 18 15

24 21 18 15

19 16 14 12

19 16 14 12

0 118 163

80

107

55

71

0

106 147

90

125

73

97

50

64

2 110 149 3 107 143 4 102 135 5 97 125 6 90 114

71 69 66 63 59

90 87 83 77 71

45 44 43 41 39

54 53 51 48 45

2 3 4 5 6

97 131 93 122 86 111 79 98 71 84

80 76 71 65 58

107 100 91 81 69

63 60 56 51 46

79 74 68 61 53

40 39 37 34 31

48 46 43 39 35

7 8 9 10 11

62 53 47 39 32

70 60 48 39 32

51 44 37 32 27

58 49 39 32 27

40 35 29 25 21

45 37 31 25 21

28 24 21 17 15

30 26 21 17 15

12 13 14 15

27 23 20 18

27 23 20 18

22 19 17 14

22 19 17 14

18 15 13

18 15 13

12 11 10

12 11 10

7 8 9 10 11

83 102 76 90 68 77 61 66 53 58

54 50 45 40 35

64 57 49 42 35

36 34 31 28 25

41 38 34 30 26

12 13 14 15 16

48 42 36 31 28

49 42 36 31 28

30 26 23 20 18

30 26 23 20 18

22 19 16 14 13

22 19 16 14 13

17 18 19

24 22 20

24 22 20

16 14

16 14

11 10

11 10

b

2

3

Properties of 2 angles—3⁄8 in. back to back A

(in2)

rx (in.) ry (in.)

3.84 0.928 1.16

2.63 0.945 1.13

1.99 0.954 1.12

3.47 0.940 0.917

2.93 0.948 0.903



2.38 0.957 0.891

1.80 0.966 0.879

No. of a Connectors


Y

b

2

3


3 - 73


Y


X

X


Unequal legs

3/ ′′ 8


Thickness Wt./ft

10.6

X-X AXIS

Fy

Y-Y AXIS

5⁄ 16

1⁄ 4

9.0

3⁄ 16

7.2

5.5

36

50

36

50

36

50

36

50

0

95

131

80

111

65

91

49

63

2 3 4 5 6

90 84 77 69 60

122 112 99 84 69

76 72 66 59 51

104 95 84 72 59

62 58 54 48 42

85 78 69 59 49

46 44 40 36 32

59 55 49 43 36

7 8 9 10 11

50 42 33 27 22

55 42 33 27 22

43 36 29 23 19

47 37 29 23 19

36 30 24 19 16

39 30 24 19 16

27 23 19 15 12

30 24 19 15 12

12 13

19

19

16

16

13 11

13 11

10 9

10 9

0

95

131

80

111

65

91

49

63

2 3 4 5 6

89 85 80 74 67

120 113 104 93 82

74 71 66 61 55

99 93 85 76 66

58 56 52 48 44

77 73 67 60 52

41 39 37 34 31

50 48 44 40 36

7 8 9 10 11

60 52 45 38 32

70 58 47 38 32

49 42 36 32 26

56 48 39 32 26

39 33 28 25 21

44 36 31 25 21

28 24 21 17 15

31 26 22 18 15

12 13 14 15 16

27 23 20 17 15

27 23 20 17 15

22 19 16 14

22 19 16 14

17 15 13 11

17 15 13 11

13 11 10 8

13 11 10 8

b



No. of a Connectors

21⁄2×2

Size


Y

3.09 0.768 0.961

2.62 0.776 0.948

2.13 0.784 0.935



1.62 0.793 0.923

2

3

3 - 74

COLUMN DESIGN



Y X

X

Unequal legs

8× ×6


88.4

Fy

36

X-X AXIS Y-Y AXIS

67.6

46.0

50

36

50

796 1110 609

846

376

479

8 12 16 20 24

677 552 416 288 200

882 666 449 288 200

521 428 325 228 159

680 518 354 228 159

328 276 217 159 111

402 323 237 160 111

28 29

147

147 116 109

116 109

82 76

796 1110 609

846

12 16 20 24 28

725 681 628 569 506

970 890 796 695 591

547 514 474 429 382

32 36 40 44 48

443 380 320 265 223

490 396 321 265 223

52 56 60 61 62

190 164 143 138 134

190 164 143 138 134

63

130

130

0

50

1⁄ 2

36

0


3⁄ 4

1

8× ×4 3⁄ 4

1 74.8

1⁄ 2

57.4

50

0

673

935 517 718 321 408

4 6 8 10 12

600 521 426 329 240

798 654 495 346 240

82 76

14 16 17 18

176 176 141 141 101 101 135 135 108 108 78 78 120 120 96 96 69 69 61 61

376

479

0

673

935 517 718 321 408

727 667 598 522 444

325 308 287 263 237

393 368 338 304 267

12 16 20 24 28

628 596 558 514 467

849 790 720 643 563

480 455 426 392 355

647 602 548 489 426

295 281 264 245 224

365 344 318 290 258

333 286 240 199 168

368 297 241 199 168

210 183 156 131 110

229 192 157 131 110

32 36 40 44 48

418 369 320 274 231

483 405 333 275 231

317 279 242 206 174

364 305 250 206 174

202 179 157 135 115

227 195 165 137 115

143 123 108 104 101

143 123 108 104 101

94 81 71 69

94 81 71 69

52 56 60 64 66

197 170 148 130 122

197 148 148 170 128 128 148 111 111 130 98 98 122 92 92

98 85 74 65 61

98 85 74 65 61

67 68

119 119 115 115

b

3

36

39.2

36

463 404 333 260 192

50

616 509 390 276 192

89

36

292 259 219 177 137

50

361 311 252 192 138

89



26.0 1.73 3.78

19.9 1.76 3.74

13.5 1.79 3.69

22.0 1.03 4.10



16.9 1.05 4.05

11.5 1.08 4.00

No. of a Connectors

Short legs 3⁄8 in. back to back of angles No. of a Connectors

Y


3/ ′′ 8

b

6


3 - 75



Y X

X

Unequal legs

7× ×4

Size 3⁄ 4

Thickness Wt./ft

1⁄ 2

52.4

Fy

36

50

3⁄ 8

35.8

36

50

27.2

36

50

No. of a Connectors

Short legs 3⁄8 in. back to back of angles

Y-Y AXIS


X-X AXIS

0 471 655 310 401 205 254 4 6 8 10 12

426 375 313 249 188

568 476 371 270 188

282 250 212 171 132

354 304 245 186 133

189 171 149 124 100

230 203 171 137 104

14 138 138 16 106 106 18 84 84

98 75 59

98 75 59

77 59 47

3/ ′′ 8

Y

6× ×4 3⁄ 4

5⁄ 8

47.2

50 0

36

50

1⁄ 2

40.0

36

3⁄ 8

32.4

50

36

24.6

50

36

50

425 591 358 497 291 388 201 256

4 6 8 10 12

386 342 289 232 178

436 370 293 218 154

265 236 201 164 127

343 295 238 181 129

186 168 146 122 97

231 203 170 135 102

77 59 47

14 16 18 19

132 132 113 113 101 101 86 86 80 80 68 68

95 73 57 52

95 73 57 52

75 57 45 41

75 57 45 41

0 471 655 310 401 205 254

0

8 12 16 20 24

449 427 397 362 324

611 570 516 454 389

291 277 259 236 212

368 346 317 282 245

28 32 36 40 44

283 243 204 168 139

323 261 207 168 139

186 160 135 111 92

207 129 143 171 113 122 137 98 102 111 83 83 92 69 69

48 117 117 52 99 99 56 86 86 57 83 83 58 80 80

77 66 57 55

77 66 57 55

188 181 171 158 144

58 49 43 41

b

225 215 201 184 164

58 49 43 41

5

517 437 345 255 179

326 289 245 198 152


15.4 1.09 3.49

10.5 1.11 3.44

7.97 1.13 3.42

b

425 591 358 497 291 388 201 256

8 12 16 20 24

398 370 334 293 249

538 486 422 352 282

28 32 36 40 44

206 166 131 106 88

216 172 179 138 143 100 105 165 138 138 110 110 81 82 131 109 109 87 87 65 65 106 88 88 71 71 52 52 88 73 73 58 58 43 43

45 46 47 48 49

84 80 77 74 71

84 80 77 74 71

333 310 279 245 208

70 67 64 61

449 406 352 294 235

70 67 64 61

267 249 224 197 167

56 53 51 49

346 314 275 230 186

56 53 51 49

181 170 155 138 119

42 40 38

222 205 184 158 131

42 40 38


No. of a Connectors


13.9 1.12 2.94

11.7 1.13 2.92



9.50 1.15 2.90

7.22 1.17 2.87

4

3 - 76

COLUMN DESIGN



Y X

X

Unequal legs

6× ×31⁄2

Size 3⁄ 8

Thickness Wt./ft

23.4

Y-Y AXIS

X-X AXIS

Fy


5⁄ 16

19.6

36

50

36

50

0

191

243

145

179

4 6 8 10

170 148 121 94

210 175 136 99

131 115 97 77

158 135 109 82

12 14 16

69 50 39

69 50 39

58 43 33

59 43 33

0

191

243

145

8 12 16 20 24

173 160 144 125 112

213 194 170 142 124

28 32 36 40 44

94 76 63 51 42

48 49

36 34

5× ×31⁄2 3⁄ 4

1⁄ 2

39.6

3⁄ 8

27.2

5⁄ 16

20.8

50

36

50

0

355

493

245

340 183 238 143 182

2 4 6 8 10

344 313 267 214 160

472 238 326 413 217 288 331 187 234 243 152 176 164 116 121

12 14 16 17

114 84 64

114 84 64

84 62 47

179

0

355

493

245

340 183 238 143 182

130 122 111 98 84

154 143 128 110 98

8 10 12 14 16

318 300 279 257 233

423 390 354 334 297

217 205 191 175 159

287 265 240 214 201

99 80 63 51 42

75 62 53 43 35

80 66 53 43 35

18 20 22 24 26

224 201 179 157 144

260 225 201 169 144

152 136 121 106 96

175 106 127 151 101 111 134 90 95 113 79 84 96 69 72

36 34

30 29

30 29

28 30 32 34 36

125 109 95 85 75

125 109 95 85 75

83 72 64 56 50

83 72 64 56 50

62 54 48 42 38

62 54 48 42 38

49 43 38 33 30

49 43 38 33 30

38 39 40

68 64 61

68 64 61

45 43 41

45 43 41

34 32 31

34 32 31

27 25

27 25

41

58

58

b

5

36

17.4

36

178 163 141 116 89

84 62 47

50

50

229 139 176 205 129 159 170 113 135 131 94 107 94 74 79

65 48 37 32

160 151 141 130 118

36

65 48 37 32

198 185 169 153 143

56 41 31 28

123 117 110 102 94

6.84 0.988 2.95

5.74 0.996 2.94

11.6 0.977 2.48

8.00 1.01 2.43



6.09 1.02 2.41

b

56 41 31 28

149 140 130 119 106

85 100 81 89 73 77 65 66 57 57


No. of a Connectors


Y


3/ ′′ 8

5.12 1.03 2.39

4


3 - 77



Y X

X

Unequal legs

Short legs 3⁄8 in. back to back of angles 5× ×3

Size 1⁄ 2

Thickness Wt./ft

X-X AXIS Y-Y AXIS

3⁄ 8

25.6

Fy


Y

5⁄ 16

19.6

1⁄ 4

16.4

13.2

36

50

36

50

36

50

36

50

0

230

319

172

223

134

170

95

117

2 4 6 8 10

220 192 154 113 76

300 249 184 119 76

165 146 118 88 61

212 180 137 94 61

129 115 95 73 52

162 140 110 79 52

92 84 71 56 42

112 99 81 61 43

12 13 14

53 45

53 45

42 36 31

42 36 31

36 31 26

36 31 26

30 25 22

30 25 22

0

230

319

172

223

134

170

95

117

8 10 12 14 16

205 194 181 167 152

273 253 230 205 194

152 144 135 125 114

190 178 163 147 139

118 112 106 98 90

144 135 126 115 103

83 79 75 71 66

96 92 87 81 75

18 20 22 24 26

146 132 117 103 95

170 147 132 112 95

109 99 88 78 68

124 109 94 84 71

82 79 71 63 56

98 87 76 66 59

61 56 54 49 44

68 65 58 51 45

28 30 32 34 36

82 72 63 56 50

82 72 63 56 50

62 54 47 42 37

62 54 47 42 37

51 45 39 35 31

51 45 39 35 31

39 36 31 28 25

41 36 31 28 25

38 40 41

45 40 38

45 40 38

34 30 29

34 30 29

28 25 24

28 25 24

22 20 19

22 20 19

3/ ′′ 8

No. of a Connectors


b

4

5



7.50 0.829 2.50

5.72 0.845 2.48

4.80 0.853 2.47



3.88 0.861 2.46

3 - 78

COLUMN DESIGN



Y X

Y


Unequal legs

3/ ′′ 8

Short legs 3⁄8 in. back to back of angles 4× ×31⁄2

Size 1⁄ 2

Thickness Wt./ft

23.8

X-X AXIS

Fy

Y-Y AXIS


3⁄ 8

5⁄ 16

18.2

1⁄ 4

15.4

12.4

36

50

36

50

36

50

36

50

0

214

298

163

227

137

179

101

129

2 4 6 8 10

208 191 166 137 106

286 255 210 160 112

159 147 128 106 83

219 195 162 125 89

133 123 108 90 71

172 156 131 103 76

99 92 81 69 55

125 114 98 79 60

12 14 16 17

78 57 44 39

78 57 44 39

62 45 35 31

62 45 35 31

53 39 30 26

53 39 30 26

42 31 24 21

43 31 24 21

0

214

298

163

227

137

179

101

129

4 6 8 10 12

201 190 177 161 143

271 252 228 200 181

150 142 132 120 107

200 187 169 149 127

122 116 108 99 89

152 143 132 117 102

86 83 78 72 66

103 99 92 84 75

14 16 18 20 22

132 115 98 86 71

153 126 106 86 71

94 86 73 61 53

114 94 75 61 53

77 71 61 51 42

92 77 62 51 42

58 54 47 40 33

65 59 49 40 33

24 26 28 30 31

60 51 44 38 36

60 51 44 38 36

45 38 33 29 27

45 38 33 29 27

36 30 26 23 21

36 30 26 23 21

28 24 21 18

28 24 21 18



7.00 1.04 1.89

5.34 1.06 1.87

4.49 1.07 1.86



3.63 1.07 1.85

No. of a Connectors

X

b

3


3 - 79



Y X

X

Unequal legs

Short legs 3⁄8 in. back to back of angles 4× ×3

Size 1⁄ 2

Thickness Wt./ft

X-X AXIS Y-Y AXIS

3⁄ 8

22.2

Fy


Y

5⁄ 16

17.0

1⁄ 4

14.4

11.6

36

50

36

50

36

50

36

50

0

199

276

152

211

127

166

94

120

2 4 6 8 10

191 169 138 104 72

261 220 166 112 72

146 130 107 81 57

200 170 129 88 57

123 109 90 69 49

158 136 106 75 49

91 82 69 54 40

115 101 81 59 40

12 14

50 37

50 37

40 29

40 29

34 25

34 25

28 21

28 21

0

199

276

152

211

127

166

94

120

4 6 8 10 12

190 182 172 160 146

259 245 226 204 180

143 137 130 121 110

193 183 169 153 135

118 113 107 99 91

149 142 132 120 107

85 82 78 73 67

103 99 93 86 78

14 16 18 20 22

131 116 101 86 73

156 131 108 88 73

99 87 76 65 54

117 98 81 66 54

81 72 62 53 44

93 79 65 53 44

61 55 48 41 35

69 60 51 42 35

24 26 28 30 32

61 52 45 39 34

61 52 45 39 34

46 39 34 29 26

46 39 34 29 26

37 32 27 24 21

37 32 27 24 21

30 25 22 19 17

30 25 22 19 17

3/ ′′ 8

No. of a Connectors


b

3

4



6.50 0.864 1.96

4.97 0.879 1.94

4.18 0.887 1.93



3.38 0.896 1.92

3 - 80

COLUMN DESIGN



Y X

X

Unequal legs

31⁄2×3

Size 3⁄ 8

Thickness Wt./ft

15.8

X-X AXIS

Fy

Y-Y AXIS


5⁄ 16

1⁄ 4

13.2

10.8

36

50

36

50

36

50

0

140

195

118

162

92

2 4 6 8 10

135 121 100 77 55

185 158 122 84 55

114 102 85 65 47

154 132 103 72 47

12 14 15

38 28

38 28

33 24 21

0

140

195

4 6 8 10 12

130 123 114 103 91

14 16 18 20 22 24 26 27

31⁄2×21⁄2 3⁄ 8

1⁄ 4

14.4

9.8

50

36

50

36

50

119

0

129

179

85

110

89 80 67 53 38

114 100 79 58 39

2 4 6 8 10

122 102 76 51 32

165 129 86 51 32

81 68 52 36 23

102 83 59 36 23

33 24 21

27 20 17

27 20 17

11 12

27

27

19 16

19 16

118

162

92

119

0

129

179

85

110

175 162 146 127 107

108 102 94 85 75

142 132 119 104 88

82 77 72 66 58

100 94 86 77 66

4 6 8 10 12

122 116 108 98 87

165 154 139 122 105

78 74 69 63 57

97 91 84 75 65

78 66 54 44 36

87 68 54 44 36

65 55 45 37 30

72 57 45 37 30

51 43 36 29 24

55 45 36 29 24

14 16 18 20 22

76 65 55 45 37

87 70 55 45 37

50 43 36 30 25

55 46 37 30 25

31 26 24

31 26 24

26 22 20

26 22 20

20 17 16

20 17 16

24 26 28 29

31 27 23 21

31 27 23 21

21 18 15

21 18 15

b

3



4.59 0.897 1.67

3.87 0.905 1.66

3.13 0.914 1.65



4.22 0.719 1.74

2.88 0.735 1.72

No. of a Connectors


Y


3/ ′′ 8

b

4


3 - 81



Y X

X

Unequal legs

3× ×21⁄2

Size 3⁄ 8

Thickness Wt./ft

13.2

X-X AXIS

Fy

36

3⁄ 16

9.0

6.77

50

36

50

36

50

0 118 163

80

107

55

71

2 111 151 3 104 137 4 94 120 5 83 100 6 71 81

76 71 65 58 50

100 91 81 69 56

53 50 46 41 36

66 62 55 48 41

45 35 27 22 18

31 26 21 17 14

34 27 21 17 14

3/ ′′ 8

Y

3× ×2 3⁄ 8

5⁄ 16

11.8

50

36

1⁄ 4

10.0

3⁄ 16

8.2

6.1

50

36

50

36

50

36

50

0

106 147

90

125

73

97

50

64

2 3 4 5 6

96 129 85 109 72 86 58 64 44 45

82 73 61 50 38

109 93 74 55 39

66 59 50 41 32

86 74 59 45 32

46 42 36 30 24

58 51 42 33 25

7 8 9

33 25 20

33 25 20

28 22 17

28 22 17

24 18 14

24 18 14

18 14 11

18 14 11

106 147

90

125

73

97

50

64

b 7 8 9 10 11

59 48 38 31 25

63 48 38 31 25

42 34 27 22 18

12

21

b

21

15

15

12

12

0 118 163

80

107

55

71

0

2 113 155 4 109 146 6 101 132 8 91 114 10 79 95

75 72 67 60 53

97 92 84 73 62

49 47 45 41 36

59 57 53 48 41

2 4 6 8 10

104 100 93 85 76

143 135 123 109 92

87 84 78 71 63

119 113 103 91 77

70 67 63 57 51

92 87 80 70 60

47 45 43 39 35

58 55 52 47 41

12 14 16 18 20

67 56 44 35 28

76 58 44 35 28

45 37 29 23 19

50 38 29 23 19

32 27 22 17 14

35 28 22 17 14

12 14 16 18 20

65 55 45 36 29

75 59 46 36 29

54 46 37 30 24

62 49 38 30 24

44 37 30 24 19

49 39 30 24 19

31 26 22 18 14

35 28 22 18 14

22 24

23 20

23 20

16 13

16 13

12 10

12 10

22 24 25

24 20 19

24 20 20 17 19 15

20 16 17 13 15 12

16 13 12

12 10 9

12 10 9

3

Y-Y AXIS


1⁄ 4

No. of a Connectors

Short legs 3⁄8 in. back to back of angles

No. of a Connectors




3.84 0.736 1.47

2.63 0.753 1.45

1.99 0.761 1.44

3.47 0.559 1.55

2.93 0.567 1.53



2.38 0.574 1.52

1.80 0.583 1.51

4

3 - 82

COLUMN DESIGN



Y X

Y


Unequal legs

3/ ′′ 8

Short legs 3⁄8 in. back to back of angles 21⁄2×2

Size 3⁄ 8

Thickness Wt./ft

10.6

X-X AXIS

Fy


5⁄ 16

1⁄ 4

9.0

3⁄ 16

7.2

5.5

36

50

36

50

36

50

36

50

0

95

131

80

111

65

91

49

63

2 3 4 5 6

86 77 66 54 42

116 99 79 60 42

73 66 56 46 36

98 84 68 51 37

60 54 46 38 30

80 69 56 43 31

45 40 35 29 23

57 50 41 32 24

7 8 9 10

31 24 19

31 24 19

27 21 16

27 21 16

23 17 14

23 17 14

18 14 11 9

18 14 11 9

0

95

131

80

111

65

91

49

63

2 4 6 8 10

92 86 78 69 58

126 116 101 84 66

77 73 66 57 47

105 97 84 69 54

62 58 53 46 38

84 77 67 55 43

45 42 38 33 28

56 52 46 39 31

12 14 16 18 20

46 36 28 22 18

49 36 28 22 18

38 29 22 17 14

39 29 22 17 14

30 23 18 14 11

31 23 18 14 11

22 17 13 10 9

23 17 13 10 9

21

16

16

13

13

No. of a Connectors

X

b

Y-Y AXIS

3

4



3.09 0.577 1.28

2.62 0.584 1.26

2.13 0.592 1.25



1.62 0.600 1.24


3 - 83

Fy = 36 ksi

Y

Fy = 50 ksi

COLUMNS Structural tees cut from W shapes

X

Design axial strength in kips (φ = 0.85) Designation

X-X AXIS Y-Y AXIS


Y

WT 18 150

Wt./ft

Fy

X

140

130

122.5

115

36

50

36

50

36

50

36

50

36

50

0

1350

1730

1260

1510

1150

1330

1040

1170

925

1030

10 12 14 16 18

1310 1300 1280 1260 1240

1670 1650 1620 1590 1550

1230 1210 1190 1180 1150

1470 1440 1420 1390 1360

1120 1100 1090 1070 1050

1290 1270 1250 1220 1200

1010 998 985 970 953

1140 1130 1110 1090 1070

903 893 882 869 855

998 986 972 956 939

20 22 24 26 28

1210 1180 1150 1120 1090

1510 1460 1420 1370 1320

1130 1100 1080 1050 1020

1330 1290 1250 1210 1170

1030 1010 983 957 930

1170 1140 1110 1070 1040

935 915 893 870 847

1050 1020 993 964 934

839 822 803 784 763

920 899 876 853 828

30 32 34 36 38

1060 1020 984 947 910

1260 1210 1160 1100 1040

984 951 917 883 847

1120 1080 1030 987 940

901 871 841 810 778

1000 965 926 886 847

822 796 769 742 714

903 871 838 804 770

742 719 696 673 649

802 776 748 720 692

40

872

989

812

893

746

806

686

736

624

663

0

1350

1730

1260

1510

1150

1330

1040

1170

925

1030

10 12 14 16 18

1210 1190 1160 1130 1090

1510 1470 1430 1370 1320

1120 1100 1070 1040 1010

1320 1290 1250 1210 1160

1010 990 966 938 908

1140 1120 1090 1050 1010

906 888 867 843 817

1010 985 959 930 898

803 788 770 750 728

874 857 836 812 786

20 22 24 26 28

1050 1010 962 916 868

1260 1190 1130 1060 988

972 932 891 848 804

1110 1060 1000 943 884

876 841 804 766 726

971 927 880 833 784

789 759 727 693 659

863 826 787 747 705

704 679 651 623 593

758 728 696 662 628

30 32 34 36 38

820 771 721 672 624

918 848 780 713 647

758 713 667 622 577

825 766 708 650 595

686 645 604 563 523

734 684 634 585 537

623 587 551 515 480

662 619 577 534 493

563 532 501 469 438

592 557 521 485 450

40

577

585

533

540

484

490

445

452

408

415

Properties A (in2) rx (in.) ry (in.)

44.1 5.27 3.83

41.2 5.25 3.81

38.2 5.26 3.78

36.0 5.26 3.75


33.8 5.25 3.73

3 - 84

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation 105

Wt./ft

X-X AXIS

Fy

36

97

50

91

85

80

75

67.5

36

50

36

50

36

50

36

50

36

50

36

50

0

908 1040 772

851

683

726

587

601

518

521

458

457

385

385

10 12 14 16 18

887 1010 755 878 1000 748 868 986 740 856 971 731 843 954 720

831 822 812 801 788

670 664 657 649 640

710 704 696 687 677

576 571 566 560 553

590 585 579 572 565

509 505 500 495 489

512 508 503 498 492

451 448 444 440 435

449 446 442 438 433

380 377 374 371 367

380 377 374 371 367

20 22 24 26 28

828 813 796 778 759

935 915 893 870 846

709 696 683 668 653

775 759 743 726 708

631 620 609 597 584

667 655 642 629 614

545 537 528 518 508

557 548 539 529 518

483 476 468 460 452

486 479 471 463 454

429 424 417 411 403

428 422 416 409 402

363 359 354 348 343

363 359 354 348 343

30 32 36 40

739 719 675 630

821 795 740 684

637 621 586 549

689 669 628 585

571 557 527 496

599 584 551 517

497 486 462 437

507 495 470 444

443 433 413 392

445 436 415 394

396 388 371 353

395 387 370 352

337 331 317 303

337 331 317 303

908 1040 772

851

683

726

587

601

518

521

458

457

385

385

10 12 14 16 18

716 687 654 617 578

791 756 715 670 622

607 585 559 530 499

653 627 597 563 527

534 515 494 470 444

558 538 515 488 460

457 442 425 406 385

465 450 432 412 391

397 385 371 356 339

399 387 373 357 340

344 335 323 311 297

344 334 323 310 296

268 261 253 244 234

268 261 253 244 234

20 22 24 26 28

537 494 450 407 364

572 521 469 418 367

465 431 395 360 325

489 449 409 369 330

416 387 358 327 297

430 398 366 333 301

363 340 315 291 266

368 344 319 293 268

320 301 280 260 239

321 302 281 260 239

281 265 248 231 213

281 265 248 231 213

223 211 198 185 172

223 211 198 185 172

30 32 34 36 39

323 285 254 228 195

323 285 254 228 195

291 258 230 206 176

291 258 230 206 176

268 239 213 191 164

269 239 213 191 164

242 218 194 174 150

242 218 194 174 150

218 197 177 159 137

218 197 177 159 137

196 178 161 144 124

195 178 161 144 124

158 144 131 118 102

158 144 131 118 102

41 42 43

177 169 161

177 160 169 153 161

160 153

149 142

149 142

136 130

136 130

124

124

113

113

0

Y-Y AXIS


WT 18


30.9 5.65 2.58

28.5 5.62 2.56

26.8 5.62 2.55

25.0 5.61 2.53



23.5 5.61 2.50

22.1 5.62 2.47

19.9 5.66 2.38


3 - 85

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


Y-Y AXIS

X-X AXIS

Fy


WT 16.5 120.5

Wt./ft

Y

36

50

110.5

36

50

100.5

76

70.5

65

59

36

50

36

50

36

50

36

50

0 1080 1300 964 1110 810

899

532

548

463

467

402

401

330

330

10 12 14 16 18

1050 1040 1020 1000 980

1260 1240 1210 1190 1160

935 923 909 893 875

1070 1050 1030 1010 990

788 779 767 755 741

872 860 847 831 814

521 516 510 503 495

535 530 524 516 508

453 449 444 439 433

457 453 448 442 436

394 391 387 383 378

393 390 386 382 377

324 321 318 315 311

324 321 318 315 311

20 22 24 26 28

958 933 907 880 851

1120 1090 1050 1020 975

855 834 811 787 762

965 937 908 878 846

725 708 690 672 652

795 775 753 730 706

487 478 468 458 447

500 490 480 469 458

426 419 411 402 393

429 422 414 405 396

372 366 360 353 345

371 365 359 352 345

307 303 298 293 287

307 303 298 293 287

30 32 34 36 40

821 790 759 727 662

934 892 849 806 720

737 710 682 654 598

813 779 745 710 639

631 610 588 565 520

681 656 630 603 549

436 424 411 399 373

446 433 420 407 379

384 374 364 354 332

387 377 366 356 334

338 330 321 313 295

337 329 321 312 295

282 276 269 263 249

282 276 269 263 249

0 1080 1300 964 1110 810

899

532

548

463

467

402

401

330

330

10 12 14 16 18

951 929 904 876 845

1110 1080 1050 1010 967

833 815 793 769 743

934 911 884 854 821

692 678 661 643 622

753 736 717 695 671

420 406 389 370 349

429 414 396 377 355

358 346 333 318 301

360 348 335 319 302

300 291 280 268 255

299 290 280 268 255

234 228 220 212 203

234 228 220 212 203

20 22 24 26 28

811 775 738 699 659

922 874 824 772 720

714 683 651 617 583

785 747 707 666 624

600 576 551 525 497

644 616 587 556 525

327 305 281 257 234

332 309 284 260 235

283 264 245 225 206

284 266 246 226 206

241 226 211 195 179

241 226 210 195 178

192 181 170 158 146

192 181 170 158 146

30 34 36 38 39

618 537 497 458 439

667 564 514 464 441

548 477 442 408 391

581 496 455 414 394

470 413 385 357 343

492 427 395 363 348

211 167 150 135 129

211 167 150 135 129

186 149 134 121 115

187 149 134 121 115

162 131 118 107 102

162 131 118 107 102

134 109 98 89

134 109 98 89

40 41

420 401

420 375 401 358

375 330 358 316

332 317

123 117

123 117

109

109


35.4 4.96 3.63

32.5 4.96 3.59

29.5 4.95 3.56

22.4 5.14 2.47

20.8 5.15 2.43



19.2 5.18 2.39

17.3 5.20 2.32

3 - 86

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation 105.5

Wt./ft

Fy

36


X-X AXIS

0

50

95.5

36

50

86.5

36

66

50

36

62

50

36

58

50

36

54

50

36

49.5

50

36

50

949 1180 844 974 708 791 505 546 447 465 402 412 357 361 304 303

10 12 14 16 18

913 897 879 859 837

1130 1100 1080 1050 1010

812 799 783 765 746

932 914 894 870 845

684 673 661 647 632

761 748 733 715 696

490 484 477 468 459

529 521 513 503 492

434 429 423 416 408

452 446 439 432 423

392 387 382 376 369

401 396 391 384 377

348 344 340 335 329

352 348 343 338 332

297 294 290 286 282

296 293 290 286 281

20 22 24 26 28

813 787 759 731 701

976 938 897 855 811

724 702 678 652 626

817 787 756 724 690

615 597 578 558 537

676 654 630 606 581

449 437 426 413 400

480 467 454 439 424

399 390 380 370 359

414 404 393 382 370

361 353 345 336 326

369 361 352 343 333

323 316 309 301 293

326 319 311 304 295

277 271 266 259 253

276 271 265 259 253

30 32 34 36 40

670 639 607 575 511

767 723 678 634 547

599 571 543 515 459

656 621 586 551 482

515 493 471 448 402

555 528 501 474 421

387 373 358 344 314

409 393 377 360 327

347 335 323 311 285

358 345 332 319 292

316 306 295 284 262

322 311 300 289 266

284 276 267 257 238

287 278 269 259 240

246 239 232 224 209

246 239 232 224 209

0

Y-Y AXIS

WT 15

949 1180 844 974 708 791 505 546 447 465 402 412 357 361 304 303

10 12 14 16 18

838 1010 734 828 609 667 817 982 716 805 595 651 793 946 695 778 579 631 766 907 672 749 561 610 736 864 646 717 541 586

389 371 350 328 303

411 391 368 343 316

340 325 308 290 269

350 335 317 297 275

298 286 271 256 238

303 290 276 259 241

254 244 233 220 206

256 246 234 221 207

207 199 191 181 170

206 199 191 181 170

20 22 24 26 28

704 670 635 598 561

818 769 720 669 617

619 589 559 527 495

682 645 607 568 528

520 497 473 448 422

561 533 505 475 445

278 252 227 201 176

288 259 231 203 176

248 226 204 183 161

253 230 207 184 161

220 201 182 163 145

223 203 183 164 145

191 175 159 143 127

192 176 160 143 127

159 147 134 121 108

159 146 134 121 108

30 32 34 35 36

523 486 449 430 412

567 517 468 444 420

462 429 397 381 365

488 448 410 391 372

396 370 344 331 318

415 384 354 339 325

155 137 122 115 109

155 137 122 115 109

142 125 112 106 100

142 127 127 112 112 125 113 113 100 100 112 100 100 89 89 106 95 95 84 84 100 90 90

96 85 76

96 85 76

37 38 40

395 377 342

398 349 353 305 310 103 103 378 334 335 293 296 342 303 303 268 268

95

95


31.0 4.43 3.49

28.1 4.42 3.46

25.4 4.42 3.43

19.4 4.66 2.25

18.2 4.66 2.23



17.1 4.67 2.19

15.9 4.69 2.15

14.5 4.71 2.10


3 - 87

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


WT 13.5 89

Wt./ft

Fy

36

X-X AXIS

50

73

57

51

47

42

50

36

50

36

50

36

50

36

50

36

50

799 1040 725

857

617

698

453

498

358

369

308

311

251

250

10 12 14 16 18

761 745 727 707 684

978 951 921 887 850

691 676 660 641 620

810 790 767 741 712

589 577 564 549 532

663 648 631 612 591

436 428 420 410 399

477 468 458 447 434

346 341 334 328 320

356 350 344 337 329

298 294 289 284 278

301 297 292 286 280

244 241 238 234 229

243 240 236 232 228

20 22 24 26 28

660 634 606 578 549

811 770 727 683 638

598 574 549 523 496

682 650 617 583 548

514 495 474 453 431

568 544 519 493 466

388 375 362 348 334

420 405 390 373 357

312 303 293 283 273

320 310 300 290 279

271 264 256 248 240

273 266 258 250 241

224 219 213 207 201

223 218 212 206 200

30 32 34 36 38

519 489 459 430 400

594 550 506 464 423

469 442 415 388 361

513 478 443 409 376

409 387 364 342 319

439 412 385 358 332

319 304 289 274 259

339 322 304 287 269

262 251 240 229 218

268 256 244 233 221

231 222 213 204 194

233 223 214 205 195

194 187 180 173 166

193 187 180 173 165

371

40

383 335

344

298

306

244

252

206

209

185

186

159

158

799 1040 725

857

617

698

453

498

358

369

308

311

251

250

10 12 14 16 18

701 681 658 632 604

877 845 808 768 725

627 609 588 565 540

720 696 669 639 606

527 513 497 479 459

583 566 546 524 500

349 331 311 288 264

374 353 330 304 277

275 263 248 232 215

281 268 253 236 218

231 221 210 197 184

233 223 211 198 185

181 174 166 157 147

180 173 165 156 147

20 22 24 26 28

574 542 509 476 442

678 631 582 533 484

513 485 456 426 395

571 534 497 458 420

437 415 391 367 342

474 446 418 389 359

240 215 191 167 145

249 221 193 167 145

197 179 160 142 125

199 180 161 143 125

169 154 140 125 110

170 155 140 125 110

137 126 114 103 92

136 125 114 103 92

30 32 34 35 36

408 374 342 326 310

436 390 347 328 310

365 335 305 291 277

382 345 309 292 277

317 292 268 256 244

330 301 273 259 245

127 112 100 94 89

127 112 100 94 89

109 97 86 81

109 97 86 81

97 86 76 72

97 86 76 72

81 72 64

81 72 64

40

252

252 225

225

200

200

0

Y-Y AXIS

80.5

36

0


Y


26.1 3.98 3.26

23.7 3.96 3.24

21.5 3.95 3.21

16.8 4.15 2.18

15.0 4.14 2.15



13.8 4.16 2.12

12.4 4.18 2.07

3 - 88

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 12 81

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

73

65.5

58.5

52

36

50

36

50

36

50

36

50

36

50

0

731

1020

658

864

591

726

506

580

410

450

10 12 14 16 18

687 669 648 624 598

932 898 858 815 769

618 602 583 562 538

797 769 737 702 664

556 541 524 505 484

673 652 627 599 569

477 465 451 435 418

542 526 508 487 465

389 379 369 357 344

424 413 401 387 371

20 22 24 26 28

571 542 512 481 450

720 670 619 568 518

514 488 461 433 405

624 583 541 499 457

462 439 415 391 366

537 504 471 437 403

400 380 360 339 318

442 418 392 367 341

331 316 301 285 269

355 338 320 302 283

30 32 34 36 38

419 388 358 328 299

469 421 375 335 300

377 349 322 295 269

416 376 338 301 270

341 316 291 267 244

369 336 304 273 245

297 276 255 235 215

315 290 265 241 217

252 236 220 204 188

264 246 227 209 192

40

271

271

244

244

221

221

196

196

173

175

0

731

1020

658

864

591

726

506

580

410

450

10 12 14 16 18

648 626 601 574 544

858 819 775 727 675

575 555 533 508 482

723 691 656 617 575

505 487 468 446 422

597 573 545 515 482

424 410 395 377 358

472 455 435 414 390

338 328 317 304 290

363 352 339 324 308

20 22 24 26 28

512 480 446 412 378

622 568 514 460 409

453 424 393 363 332

532 487 442 398 355

397 371 345 317 290

448 412 376 340 305

338 316 294 272 249

365 339 313 286 259

275 260 243 226 209

291 273 254 235 216

30 32 34 36 38

345 312 281 251 225

359 316 281 251 225

303 273 245 219 197

313 276 245 219 197

264 238 213 190 171

271 239 213 190 171

227 206 184 165 149

233 207 184 165 149

192 175 159 143 129

196 178 159 143 129

40

204

204

178

178

155

155

135

135

117

117


23.9 3.50 3.05

21.5 3.50 3.01

19.3 3.52 2.97



17.2 3.51 2.94

15.3 3.51 2.91


3 - 89

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


WT 12 47

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

Y

42

38

34

31

27.5

36

50

36

50

36

50

36

50

36

50

36

50

0

378

419

307

321

254

258

209

208

202

203

155

155

10 12 14 16 18

360 352 343 332 321

396 387 376 363 350

293 287 280 273 265

306 299 292 284 275

244 240 234 229 222

247 243 237 231 225

201 198 194 189 185

200 197 193 189 184

194 191 187 183 178

196 192 188 184 179

150 148 145 142 139

150 148 145 142 139

20 22 24 26 28

309 296 283 269 255

335 320 304 287 271

256 246 236 225 215

265 255 244 233 221

215 208 200 192 184

218 210 202 194 185

179 174 168 162 155

179 173 167 161 155

173 168 162 156 150

174 169 163 157 150

136 132 128 124 120

136 132 128 124 120

30 32 34 36 38

240 226 211 197 183

254 237 220 203 187

204 192 181 170 159

209 197 185 173 161

175 166 157 148 140

176 167 158 149 140

148 142 135 128 121

148 141 135 128 121

143 136 130 123 116

144 137 130 123 117

115 111 106 101 96

115 111 106 101 96

40

169

171

148

150

131

131

114

114

109

110

92

92

0

378

419

307

321

254

258

209

208

202

203

155

155

10 12 14 16 18

288 269 248 226 202

310 288 263 237 211

231 218 202 185 168

239 224 208 190 171

188 178 166 154 140

190 179 167 155 141

148 140 132 123 113

147 140 132 123 113

121 110 97 84 71

122 110 98 84 71

91 84 75 67 57

91 84 75 67 57

20 22 23 24 26

179 156 145 134 115

184 158 145 134 115

150 132 124 115 99

152 134 124 115 99

126 113 106 99 86

127 113 106 99 86

103 92 87 82 71

103 92 87 82 71

59 49 46

59 49 46

48 41

48 41

28 30 31 32 33

99 87 82 77 72

99 87 82 77 72

86 75 71 66

86 75 71 66

74 65 61 58

74 65 61 58

62 55 51

62 55 51


13.8 3.67 1.98

12.4 3.67 1.95

11.2 3.68 1.92

10.0 3.70 1.87



9.11 3.79 1.38

8.10 3.80 1.34

3 - 90

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 10.5 73.5

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

66

61

55.5

50.5

36

50

36

50

36

50

36

50

36

50

0

661

918

594

825

548

757

499

637

452

524

10 12 14 16 18

610 589 565 539 510

822 782 739 691 641

547 528 507 483 457

737 701 661 618 573

505 487 466 444 420

676 643 606 566 524

459 443 424 404 382

573 547 518 486 453

416 401 384 366 346

476 457 434 410 384

20 22 24 26 28

480 449 417 385 353

589 536 484 434 385

429 401 372 343 315

526 478 431 386 341

395 368 341 315 288

481 437 394 352 311

358 334 310 285 261

418 382 347 312 279

324 303 280 258 236

357 329 301 274 247

30 32 34 36 38

322 292 262 234 210

337 296 263 234 210

286 259 233 208 186

299 263 233 208 186

262 236 212 189 170

272 239 212 189 170

237 214 192 171 154

246 217 192 171 154

214 193 173 154 139

221 195 173 154 139

40

190

190

168

168

153

153

139

139

125

125

0

661

918

594

825

548

757

499

637

452

524

10 12 14 16 18

587 566 542 515 486

778 740 697 650 601

522 503 482 458 432

689 655 617 576 532

478 461 441 419 396

626 596 562 524 485

430 415 397 377 356

526 503 476 447 415

385 372 356 339 320

434 417 397 375 352

20 22 24 26 28

456 424 392 360 328

550 498 447 397 349

405 377 348 320 291

487 441 395 351 308

371 345 319 293 267

444 402 361 321 281

334 311 287 263 239

383 349 316 283 251

300 279 258 237 216

327 301 275 249 223

30 32 34 36 38

297 267 238 213 191

305 268 238 213 191

263 237 210 188 169

269 237 210 188 169

241 216 193 172 155

246 217 193 172 155

216 194 172 154 139

220 194 172 154 139

195 175 156 139 125

199 175 156 139 125

40

173

173

153

153

140

140

125

125

113

113


21.6 3.08 2.95

19.4 3.06 2.93

17.9 3.04 2.92



16.3 3.03 2.90

14.9 3.01 2.89


3 - 91

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


Fy

36

X-X AXIS


0

50

41.5

36

50

36.5

36

34

50

36

31

50

36

28.5

50

36

50

25

36

22

50

36

50

419 562 373 444 297 331 261 283 219 225 203 210 165 167 127 127

6 8 10 12 14

409 400 390 378 364

543 529 511 489 466

364 356 347 336 323

430 420 407 392 374

290 284 278 270 260

322 316 307 297 286

255 251 245 239 231

276 271 264 256 247

214 211 206 201 195

221 217 212 207 201

199 196 192 187 182

206 203 199 194 188

162 160 157 153 149

164 161 158 155 151

125 123 121 119 116

125 123 121 119 116

16 18 20 22 24

349 332 315 296 277

439 412 383 353 323

310 295 279 262 245

355 335 314 291 269

250 239 227 215 202

274 260 246 231 216

222 213 203 192 182

237 227 215 203 191

189 181 174 165 157

194 186 178 169 160

176 170 163 155 147

182 175 167 159 151

145 140 134 129 123

146 141 136 130 124

113 110 106 102 98

113 110 106 102 98

26 28 30 32 34

258 239 220 201 183

293 264 236 209 185

228 210 193 177 160

247 225 203 182 162

189 176 163 150 137

200 185 169 155 140

170 159 148 137 126

178 165 153 140 128

148 139 130 121 112

151 142 132 123 114

139 131 123 115 107

143 117 118 134 111 111 125 104 105 117 98 98 108 91 92

94 90 85 81 76

94 90 85 81 76

36 38 40

165 165 145 145 125 126 115 117 104 104 148 148 130 130 113 113 105 105 95 96 134 134 117 117 102 102 95 95 87 87

71 67 63

71 67 63

0

Y-Y AXIS

WT 10.5 46.5

Wt./ft

Y

99 100 91 92 84 84

85 79 73

85 79 73

419 562 373 444 297 331 261 283 219 225 203 210 165 167 127 127

6 8 10 12 14

357 337 313 285 256

452 420 381 338 293

311 294 273 250 224

357 334 307 276 243

245 233 218 201 182

267 252 235 214 192

213 203 191 177 161

227 215 202 186 168

152 147 141 133 123

153 142 128 113 97

16 18 20 21 22

226 195 166 151 138

247 203 166 151 138

198 171 145 132 121

210 177 145 132 121

162 142 122 113 103

169 146 124 113 103

145 128 111 103 95

150 112 113 131 101 102 112 90 90 103 84 84 95 78 78

81 66 54 49 45

24 26 28 29 30

117 117 102 102 100 100 88 88 86 86 76 76 80 80 71 71 75 75 66 66

87 75 65 60 57

87 75 65 60 57

80 69 60 56 52

80 69 60 56 52

149 145 138 131 122

67 58 51 47

157 116 117 145 108 109 131 99 99 115 88 88 98 76 76 82 66 54 49 45

64 52 43 39

64 52 43 39

85 80 74 67 59

85 80 74 67 59

51 42 35 32

51 42 35 32

67 58 51 47

Properties A

(in2)

rx (in.) ry (in.)

13.7 3.25 1.84

12.2 3.22 1.83

10.7 3.21 1.81

10.0 3.20 1.80

9.13 3.21 1.77

8.37 3.29 1.35



7.36 3.30 1.30

6.49 3.31 1.26

3 - 92

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 9 59.5

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

53

48.5

43

38

36

50

36

50

36

50

36

50

36

50

0

536

744

477

663

438

608

389

507

339

393

10 12 14 16 18

479 456 430 402 372

636 594 548 499 449

426 406 383 357 331

567 529 487 444 399

390 370 349 325 301

518 482 444 403 361

346 329 309 288 266

436 407 376 344 310

302 287 270 252 233

343 323 302 278 254

20 22 24 26 28

342 311 281 251 222

399 350 303 259 224

304 276 249 222 197

354 310 268 229 198

275 250 225 200 177

320 279 241 205 177

244 221 199 177 156

276 243 211 181 156

213 193 174 155 136

229 205 181 158 137

30 34 38 42 43

195 152 121 99 95

195 152 121 99 95

172 134 107 88 84

172 134 107 88 84

154 120 96 79

154 120 96 79

136 106 85 69

136 106 85 69

119 93 74 61

119 93 74 61

0

536

744

477

663

438

608

389

507

339

393

10 12 14 16 18

471 450 427 401 374

622 584 543 499 453

415 397 376 353 329

546 513 477 438 397

379 362 343 322 300

496 467 434 398 361

331 317 300 282 263

411 388 362 334 305

285 272 258 243 226

320 304 287 267 246

20 22 24 26 28

346 318 289 260 233

407 361 317 274 237

304 279 253 228 203

356 315 276 238 206

277 254 230 207 185

324 286 251 216 187

242 222 201 181 161

275 245 216 188 163

209 192 174 156 139

225 203 182 161 140

30 34 38 42 43

206 161 129 106 101

206 161 129 106 101

180 140 112 92 88

180 140 112 92 88

163 127 102 84 80

163 127 102 84 80

142 111 89 73 70

142 111 89 73 70

123 96 77 63 61

123 96 77 63 61

44

96

96

84

84

76

76


17.5 2.60 2.69

15.6 2.59 2.66

14.3 2.56 2.65



12.7 2.55 2.63

11.2 2.54 2.61


3 - 93

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


Fy

36

X-X AXIS


0

32.5

50

36

30

50

36

27.5

50

36

25

50

36

23

50

36

20

50

36

17.5

50

36

50

318 426 292 356 261 299 226 253 184 194 172 183 124 124 101 101

10 12 14 16 18

288 275 261 246 229

372 351 327 302 275

264 252 239 225 210

314 297 279 259 238

236 226 214 202 189

266 253 239 223 206

206 197 188 178 167

227 217 206 193 180

169 163 156 148 140

177 171 163 154 145

168 161 154 146 138

116 112 108 104 99

116 112 108 104 99

95 92 89 86 82

95 92 89 86 82

20 22 24 26 28

212 195 178 161 144

248 222 196 171 148

194 178 162 146 131

216 195 173 153 134

175 161 147 133 119

189 172 155 138 122

155 143 131 120 108

166 152 138 124 111

131 122 113 103 94

135 124 129 126 116 120 116 107 111 106 99 101 96 90 92

94 89 84 78 72

94 89 83 78 72

78 74 70 66 62

78 74 70 66 62

30 32 34 36 38

128 129 116 116 106 107 113 113 102 102 94 94 100 100 91 91 83 83 89 89 81 81 74 74 80 80 72 72 66 66

97 86 76 68 61

98 86 76 68 61

85 77 68 61 55

86 77 68 61 55

82 74 67 59 53

83 75 67 59 53

67 61 56 51 46

67 61 56 51 46

57 53 49 45 41

57 53 49 45 41

55 50 48 44

55 50 48 44

49 45 43 39

49 45 43 39

48 44 42 38 36

48 44 42 38 36

41 38 36 33 31

41 38 36 33 31

37 34 32 29 28

37 34 32 29 28

40 42 43 45 46 0

Y-Y AXIS

WT 9 35.5

Wt./ft

Y

72 66 63 57

72 66 63 57

65 59 57 52

65 59 57 52

60 54 52 47

60 54 52 47

318 426 292 356 261 299 226 253 184 194 172 183 124 124 101 101

10 12 14 16 18

231 207 182 157 132

279 242 204 167 133

210 188 165 142 119

238 209 179 149 120

187 168 149 129 109

204 182 158 134 111

161 146 130 113 96

20 21 22 24 26

109 109 99 99 90 90 76 76 65 65

98 89 82 69 59

98 89 82 69 59

90 82 75 63 54

90 82 75 63 54

80 73 67 57 48

80 73 67 57 48

70 64 59 50 43

70 64 59 50 43

55 51

55 51

50 47

50 47

45

45

40

40

27 28

159 153 147 140 132

60 56

60 56

173 132 137 107 111 155 121 125 92 94 136 109 111 77 78 117 96 98 62 62 98 83 84 50 50 40 37

40 37

80 71 61 51 41

80 71 61 51 41

61 54 47 40 32

61 54 47 40 32

34 31

34 31

27

27


10.4 2.74 1.70

9.55 2.72 1.69

8.82 2.71 1.69

8.10 2.71 1.67

7.33 2.70 1.65

6.77 2.77 1.29



5.88 2.76 1.27

5.15 2.79 1.22

3 - 94

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 8 50

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

44.5

38.5

33.5

36

50

36

50

36

50

36

50

0

450

625

401

557

346

476

301

361

10 12 14 16 18

389 365 338 310 280

510 467 420 372 324

346 324 300 275 249

454 415 373 330 287

297 278 257 235 212

386 353 316 279 243

258 241 223 203 183

300 277 251 225 199

20 22 24 26 28

251 222 194 167 144

278 234 197 167 144

223 197 172 148 128

246 207 174 148 128

189 166 145 124 107

207 174 146 124 107

163 143 124 106 92

173 148 125 106 92

30 32 34 36 37

126 111 98 87 83

126 111 98 87 83

111 98 87 77 73

111 98 87 77 73

93 82 73 65 61

93 82 73 65 61

80 70 62 55 52

80 70 62 55 52

38

78

78

0

450

625

401

557

346

476

301

361

10 12 14 16 18

391 371 349 325 300

514 479 440 399 357

346 328 309 287 265

453 422 388 351 314

295 280 263 245 226

382 356 327 297 265

254 241 226 211 194

294 276 257 236 214

20 22 24 26 28

274 248 223 198 174

315 275 236 201 174

242 219 196 173 152

277 241 206 176 152

206 186 166 147 129

234 203 174 149 129

177 160 143 127 111

192 170 148 128 111

30 32 34 36 38

151 133 118 105 95

151 133 118 105 95

133 117 103 92 83

133 117 103 92 83

112 99 88 78 70

112 99 88 78 70

97 85 75 67 61

97 85 75 67 61

41

81

81

71

71

60

60

52

52


14.7 2.28 2.51

13.1 2.27 2.49



11.3 2.24 2.47

9.84 2.22 2.46


3 - 95

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


Y-Y AXIS

X-X AXIS

Fy


WT 8 28.5

Wt./ft

Y

25

22.5

20

18

15.5

13

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

256

336

223

259

184

205

141

145

122

124

93

93

66

66

6 8 10 12 14

245 236 225 212 199

316 301 283 262 240

213 205 196 185 173

245 235 223 208 193

176 170 163 154 145

195 188 179 169 157

136 132 127 121 115

140 136 130 124 117

118 114 111 106 101

120 116 112 108 102

91 88 86 83 79

90 88 85 82 79

65 63 62 60 58

65 63 62 60 58

16 18 20 22 24

184 168 152 136 121

217 193 169 147 125

160 146 133 119 105

176 159 141 125 108

135 124 114 103 92

145 133 120 107 95

108 100 92 85 77

110 102 94 86 78

95 89 82 76 69

96 90 83 76 70

75 71 67 62 57

75 71 66 62 57

55 53 50 47 44

55 53 50 47 44

26 28 30 32 34

106 92 80 70 62

107 92 80 70 62

93 80 70 61 54

93 80 70 61 54

82 72 62 55 49

83 72 62 55 49

69 62 54 48 42

70 62 54 48 42

63 56 50 44 39

63 57 50 44 39

53 48 44 39 35

53 48 44 39 35

41 38 35 32 29

41 38 35 32 29

36 38 39 40 41

56 50 47 45

56 50 47 45

49 44 41 39

49 44 41 39

43 39 37

43 39 37

38 34 32

38 34 32

35 31 30 28

35 31 30 28

31 28 27 25

31 28 27 25

27 24 23 22 21

27 24 23 22 21

0

256

336

223

259

184

205

141

145

122

124

93

93

66

66

6 8 10 12 14

216 200 181 160 138

268 243 214 182 151

185 171 155 137 119

207 190 170 148 125

153 143 130 116 101

167 155 140 123 106

116 110 102 92 82

119 112 104 94 84

95 90 84 76 68

97 91 85 77 69

70 65 58 50 42

70 64 57 50 42

47 44 40 35 30

47 44 40 35 30

16 18 20 22

116 96 78 65

120 96 78 65

100 83 67 56

103 83 67 56

87 72 59 49

89 72 59 49

72 62 52 43

73 62 52 43

60 51 43 36

60 51 43 36

34 27

34 27

25 21

25 21

24 25 26

54 50 46

54 50 46

47 43 40

47 43 40

41 38 35

41 38 35

36 34 31

36 34 31

30 28

30 28


8.38 2.41 1.60

7.37 2.40 1.59

6.63 2.39 1.57

5.89 2.37 1.57

5.28 2.41 1.52



4.56 2.45 1.17

3.84 2.47 1.12

3 - 96

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 7 66

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

60

54.5

49.5

45

36

50

36

50

36

50

36

50

36

50

0

594

825

542

752

490

680

447

621

404

561

2 4 6 8 10

588 570 542 505 461

813 779 726 658 580

536 520 493 459 418

741 710 661 597 525

484 469 444 412 374

670 641 595 536 468

442 428 405 375 340

611 584 542 487 425

399 387 366 339 307

552 528 489 439 383

12 14 16 18 20

412 361 310 261 215

497 414 335 266 215

373 326 279 234 192

448 371 299 237 192

333 289 246 205 167

397 327 261 207 167

302 262 223 185 151

360 296 236 186 151

272 236 200 166 135

324 265 211 166 135

22 24 26 27 28

178 149 127 118 110

178 149 127 118 110

158 133 113 105 98

158 133 113 105 98

138 116 99 92 85

138 116 99 92 85

125 105 89 83

125 105 89 83

111 94 80 74

111 94 80 74

0

594

825

542

752

490

680

447

621

404

561

6 8 10 12 14

578 570 559 546 531

794 778 758 734 706

526 519 509 497 483

722 708 689 667 642

475 468 459 448 436

651 638 621 601 578

432 426 418 408 396

591 579 564 546 525

389 384 376 367 357

531 520 506 490 472

16 18 20 22 24

514 496 476 455 434

675 642 607 571 533

468 451 433 414 394

614 584 551 518 484

422 407 391 373 355

553 526 497 467 436

384 370 355 339 322

502 477 451 423 395

346 333 320 305 290

451 429 405 381 355

26 28 30 32 34

411 388 365 341 318

495 457 420 383 347

373 352 331 309 288

449 414 380 346 314

337 317 298 279 260

404 373 342 311 282

305 288 270 253 235

366 337 309 281 255

275 259 243 227 211

329 303 278 253 229

36 38 40

295 273 251

312 281 253

267 247 227

282 253 229

241 222 204

253 227 205

218 201 184

228 205 185

196 180 166

205 184 166


19.4 1.73 3.76

17.7 1.71 3.74

16.0 1.68 3.73



14.6 1.67 3.71

13.2 1.66 3.70


3 - 97

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


WT 7 41

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

Y

37

34

30.5

26.5

24

21.5

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

367

510

334

463

306

425

274

370

239

318

216

265

183

208

4 6 8 10 12

354 339 319 294 267

486 457 419 375 327

322 307 288 265 240

440 413 378 337 293

295 281 264 243 219

403 378 346 308 267

264 252 236 217 196

352 330 302 270 235

231 221 208 193 175

303 287 265 239 211

209 200 188 174 158

254 241 224 203 181

177 170 160 149 136

200 191 179 164 148

14 16 18 20 22

238 208 179 151 126

279 232 188 152 126

213 186 159 134 111

248 205 165 134 111

194 169 144 121 100

226 186 150 121 100

173 151 128 108 89

199 165 133 108 89

157 138 119 101 85

182 153 126 102 85

141 124 107 91 76

158 134 112 92 76

122 108 93 80 67

131 114 97 81 67

24 26 28 30 31

106 90 78 68

106 90 78 68

93 79 68 59

93 79 68 59

84 72 62 54

84 72 62 54

75 64 55 48

75 64 55 48

71 61 52 45 43

71 61 52 45 43

64 54 47 41 38

64 54 47 41 38

56 48 41 36 34

56 48 41 36 34

0

367

510

334

463

306

425

274

370

239

318

216

265

183

208

8 10 12 14 16

334 320 303 285 265

447 421 391 359 324

303 290 275 258 240

405 382 355 325 294

276 265 251 235 218

369 347 322 295 267

246 236 223 210 195

320 302 281 258 233

203 189 173 156 139

256 233 208 181 154

183 170 156 140 124

215 197 177 156 135

154 144 132 119 106

171 158 144 128 112

18 20 22 24 26

244 222 201 179 159

289 254 221 188 161

221 202 182 163 144

262 231 200 171 146

201 183 165 147 130

238 209 181 154 131

179 163 147 131 116

209 184 160 137 117

121 104 87 73 63

129 105 87 73 63

108 93 78 66 56

114 94 78 66 56

93 80 68 57 49

97 82 68 57 49

28 30 31 32 34

139 121 113 106 94

139 121 113 106 94

126 110 103 97 86

126 110 103 97 86

113 99 93 87 77

113 99 93 87 77

101 88 82 77 69

101 88 82 77 69

54 47 44 41

54 47 44 41

48 42 40

48 42 40

42 37 34

42 37 34

36 40 41

84 68 65

84 68 65

76 62 59

76 62 59

69 56 53

69 56 53

61 50

61 50

Properties A

(in2)

rx (in.) ry (in.)

12.0 1.85 2.48

10.9 1.82 2.48

9.99 1.81 2.46

8.96 1.80 2.45

7.81 1.88 1.92



7.07 1.87 1.91

6.31 1.86 1.89

3 - 98

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 7 19

Wt./ft

X-X AXIS Y-Y AXIS


Fy

17

15

13

11

36

50

36

50

36

50

36

50

36

50

0

159

180

131

142

109

114

87

88

62

62

2 4 6 8 10

158 155 150 143 134

178 174 168 159 148

130 128 124 119 112

141 138 134 127 120

109 107 104 100 95

114 112 108 104 98

87 85 83 80 77

88 86 84 81 78

62 61 60 58 56

62 61 60 58 56

12 14 16 18 20

125 114 103 92 81

136 123 110 97 83

105 96 88 79 70

111 102 92 82 72

89 83 76 69 62

92 85 78 70 63

73 68 63 58 53

73 69 64 58 53

53 51 48 44 41

53 51 48 44 41

22 24 26 28 30

70 60 51 44 38

71 60 51 44 38

62 53 46 39 34

63 54 46 39 34

55 48 42 36 31

55 48 42 36 31

48 42 37 33 28

48 43 38 33 28

38 34 31 28 24

38 34 31 28 24

32 34 35

34 30

34 30

30 27

30 27

27 24

27 24

25 22 21

25 22 21

22 19 18

22 19 18

0

159

180

131

142

109

114

87

88

62

62

6 8 10 12 14

132 123 111 99 86

146 134 120 105 89

108 101 92 82 72

115 107 97 86 74

86 81 74 67 59

89 83 76 68 60

65 58 50 41 32

66 58 50 41 32

45 41 35 30 24

45 41 35 30 24

16 17 18 20 22

72 66 60 49 40

74 66 60 49 40

61 56 51 42 35

63 57 52 42 35

50 46 42 35 29

51 47 42 35 29

25 22 20

25 22 20

19 17

19 17

24 25

34 31

34 31

30 27

30 27

25

25


5.58 2.04 1.55

5.00 2.04 1.53

4.42 2.07 1.49



3.85 2.12 1.08

3.25 2.14 1.04


3 - 99

Fy = 36 ksi

Y

Fy = 50 ksi


X


X-X AXIS Y-Y AXIS


Y

WT 6 29

Wt./ft

Fy

X

26.6

25

22.5

20

36

50

36

50

36

50

36

50

36

50

0

261

362

238

331

225

312

202

280

180

221

2 4 6 8 10

257 247 231 210 186

355 336 306 268 227

235 226 211 192 171

325 307 280 246 208

222 214 202 186 167

307 292 269 240 207

200 193 181 167 149

276 262 241 214 184

178 172 161 148 133

218 208 193 174 152

12 14 16 18 20

160 135 110 88 71

185 145 111 88 71

147 124 102 81 66

170 134 103 81 66

147 126 105 86 70

173 139 109 86 70

131 112 93 75 61

153 123 96 75 61

116 99 82 66 54

129 106 84 66 54

22 24 25 26

59 49 45

59 49 45

54 46 42

54 46 42

58 48 45 41

58 48 45 41

51 42 39 36

51 42 39 36

44 37 34 32

44 37 34 32

0

261

362

238

331

225

312

202

280

180

221

6 8 10 12 14

246 238 228 216 203

333 318 300 279 257

223 216 206 196 184

301 287 271 252 231

205 194 181 166 150

274 255 232 206 179

183 174 162 148 134

244 227 206 183 159

162 154 143 131 118

194 182 167 150 132

16 18 20 22 24

189 175 160 144 129

233 208 184 160 137

171 157 144 130 116

209 187 164 143 122

134 117 101 86 72

152 127 103 86 72

119 104 89 75 63

135 112 91 75 63

105 92 79 66 56

114 97 80 66 56

26 28 30 32 34

115 101 88 77 69

117 101 88 77 69

103 90 78 69 61

104 90 78 69 61

61 53 46 41

61 53 46 41

54 47 41 36

54 47 41 36

48 41 36 32

48 41 36 32

36 38 41

61 55 47

61 55 47

54 49 42

54 49 42

Properties A

(in2)

rx (in.) ry (in.)

8.52 1.50 2.51

7.78 1.51 2.48

7.34 1.60 1.96

6.61 1.58 1.94



5.89 1.57 1.93

3 - 100

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation 17.5

Wt./ft

X-X AXIS

Fy

15

13

11

9.5

8

7

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

158

188

120

132

90

92

88

98

68

71

53

54

40

40

2 4 6 8 10

157 152 145 135 124

186 179 169 156 140

119 116 111 104 96

131 127 121 113 104

89 87 84 80 74

91 89 86 81 76

88 86 83 78 73

97 95 91 86 80

68 66 64 61 58

70 69 67 63 60

53 52 51 48 46

54 53 51 49 46

40 39 38 37 35

40 39 38 37 35

12 14 16 18 20

111 98 85 72 59

124 106 89 73 59

87 78 68 59 50

93 82 71 60 50

68 62 55 48 42

70 63 56 49 42

68 61 55 48 42

73 65 58 50 43

54 49 44 40 35

55 50 45 40 35

43 40 36 33 29

43 40 36 33 29

33 31 29 26 24

33 31 29 26 24

22 24 26 28 29

49 41 35 30 28

49 41 35 30 28

41 35 30 25 24

41 35 30 25 24

36 30 26 22 21

36 30 26 22 21

36 30 26 22 21

36 30 26 22 21

30 26 22 19 18

30 26 22 19 18

26 22 19 16 15

26 22 19 16 15

21 19 17 14 14

21 19 17 14 14

19 18

19 18

17 16

17 16

14 13 13

14 13 13

13 12 11

13 12 11

30 31 32

Y-Y AXIS


WT 6

0

158

188

120

132

90

92

88

98

68

71

53

54

40

40

2 4 6 8 10

147 142 134 123 110

172 165 154 139 123

109 106 101 93 85

119 115 109 100 90

81 79 75 71 65

83 80 77 72 66

73 67 57 45 33

80 72 60 46 33

54 49 43 34 26

55 51 44 35 26

37 34 30 24 18

37 34 30 25 18

26 25 22 19 15

26 25 22 19 15

12 13 14 16 18

96 89 82 68 55

105 95 87 69 55

75 70 65 55 45

79 73 67 56 45

59 55 52 45 38

59 56 52 45 38

23 20 17

23 20 17

18 16

18 16

13

13

11

11

20 22 24 25

45 37 31 29

45 37 31 29

37 31 26 24

37 31 26 24

31 26 22 20

31 26 22 20

Properties A

(in2)

rx (in.) ry (in.)

5.17 1.76 1.54

4.40 1.75 1.52

3.82 1.75 1.51

3.24 1.90 0.847

2.79 1.90 0.822



2.36 1.92 0.773

2.08 1.92 0.753


3 - 101

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


WT 5 22.5

Wt./ft

19.5

16.5

15

13

11

36

50

36

50

36

50

36

50

36

50

36

50

0

203

282

175

244

148

206

135

188

117

146

99

115

2 4 6 8 10

199 187 170 148 124

274 253 220 182 142

172 162 147 128 107

237 218 190 157 123

146 137 125 109 92

201 185 162 135 106

133 128 119 107 94

184 173 157 136 114

115 110 102 92 81

144 136 124 109 92

98 94 87 79 69

113 108 99 88 76

12 14 16 18 20

100 77 59 47 38

105 77 59 47 38

86 67 51 40 33

91 67 51 40 33

75 58 45 35 29

79 58 45 35 29

80 67 54 42 34

91 70 54 42 34

69 57 46 36 29

76 60 46 36 29

59 49 40 32 26

64 51 40 32 26

26

26

31 28 24

31 28 24

27 24 20

27 24 20

23 21 18

23 21 18

21 22 24

Y-Y AXIS


X-X AXIS

Fy

Y

0

203

282

175

244

148

206

135

188

117

146

99

115

2 4 6 8 10

199 195 188 178 167

274 266 253 235 214

171 167 161 153 143

235 227 216 201 183

143 140 134 127 119

194 188 179 166 151

128 122 113 101 88

174 163 147 126 104

109 104 96 86 75

134 126 115 100 84

89 85 78 70 61

101 96 88 78 66

12 14 16 18 20

153 139 125 110 95

191 167 143 120 99

131 119 106 93 80

163 142 121 101 83

109 98 87 76 66

134 116 99 82 67

74 60 47 38 30

82 62 47 38 30

63 51 40 32 26

68 52 40 32 26

51 41 32 26 21

54 42 32 26 21

22 24 26 28 30

81 69 59 50 44

82 69 59 50 44

68 57 49 42 37

68 57 49 42 37

55 47 40 34 30

55 47 40 34 30

25

25

21

21

17

17

32 33

39 36

39 36

32 30

32 30

26

26


6.63 1.24 2.01

5.73 1.24 1.98

4.85 1.26 1.94

4.42 1.45 1.37



3.81 1.44 1.36

3.24 1.46 1.33

3 - 102

COLUMN DESIGN

Fy = 36 ksi

Y

Fy = 50 ksi X


X


Y

Designation

WT 5 9.5

Wt./ft

Y-Y AXIS


X-X AXIS

Fy

8.5

7.5

6

36

50

36

50

36

50

36

50

0

86

104

77

90

66

76

43

45

2 4 6 8 10

85 82 77 70 62

103 98 91 81 71

76 73 68 63 56

88 85 79 71 62

65 63 59 54 49

75 72 67 61 54

43 41 39 37 34

44 43 41 38 35

12 14 16 18 20

54 46 38 30 25

60 49 39 30 25

49 42 34 28 23

53 44 35 28 23

43 37 31 25 20

46 39 31 25 20

30 27 23 19 16

31 27 23 20 16

22 24 25 26

20 17 16

20 17 16

19 16 14 13

19 16 14 13

17 14 13 12

17 14 13 12

13 11 10 10

13 11 10 10

0

86

104

77

90

66

76

43

45

2 4 6 8 10

74 67 56 43 31

87 77 62 46 31

62 56 47 36 25

70 62 51 37 25

49 45 37 29 20

54 48 40 29 20

30 28 24 19 14

30 28 24 20 14

12 13 14

22 18 16

22 18 16

18 15 13

18 15 13

14 12

14 12

10 9

10 9


2.81 1.54 0.874

2.50 1.56 0.844



2.21 1.57 0.810

1.77 1.57 0.785


3 - 103

Fy = 36 ksi

Y

Fy = 50 ksi


X

X


WT 4 14

Wt./ft

12

10.5

9

7.5

6.5

5

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

126

175

108

150

94

131

80

112

68

94

59

82

41

46

2 3 4 5 6

122 118 112 105 96

168 160 148 135 121

105 101 96 90 82

144 137 127 116 103

92 89 86 81 76

127 121 114 106 97

79 76 73 70 65

108 104 98 91 83

67 65 63 60 57

92 89 84 79 73

58 56 54 52 49

79 77 73 69 64

41 40 38 37 35

45 44 42 40 38

8 10 12 14 16

78 60 43 32 24

90 62 43 32 24

67 51 36 27 20

77 52 36 27 20

64 52 39 29 22

76 57 40 29 22

55 45 35 26 20

67 50 35 26 20

49 41 33 25 19

60 47 34 25 19

43 36 29 22 17

52 41 30 22 17

30 26 21 16 12

33 27 21 16 12

18

18

16 14

16 14

15 14 12

15 14 12

13 12 11

13 12 11

10 9 8

10 9 8

18 19 20

Y-Y AXIS


X-X AXIS

Fy

Y

0

126

175

108

150

94

131

80

112

68

94

59

82

41

46

2 4 6 8 10

123 119 112 104 93

169 161 149 133 116

105 101 96 88 80

143 137 127 113 98

89 85 77 68 57

121 113 99 83 66

75 70 64 56 47

100 93 82 68 54

60 54 45 35 25

79 68 53 37 25

49 44 36 28 19

63 55 43 29 19

33 30 25 20 15

35 32 27 21 15

12 14 16 18 20

82 71 60 49 40

97 79 62 49 40

70 60 51 42 34

83 67 53 42 34

47 37 28 22 18

49 37 28 22 18

38 30 23 18 15

40 30 23 18 15

17 13

17 13

14 10

14 10

10 8

10 8

21 22 24 26 27

36 33 28 24 22

36 33 28 24 22

31 28 24 20

31 28 24 20

16

16

Properties A

(in2)

rx (in.) ry (in.)

4.12 1.01 1.62

3.54 .999 1.61

3.08 1.12 1.26

2.63 1.14 1.23

2.22 1.22 0.876



1.92 1.23 0.843

1.48 1.20 0.841

3 - 104

COLUMN DESIGN

Single-Angle Struts

Design strengths of single-angle struts were formerly not tabulated in this Manual because of the difficulty in loading such struts concentrically. Concentric loading can be accomplished by milling the ends of an angle and loading it through bearing plates. However, in common practice, the eccentricity of loading is relatively large, and its neglect in design may lead to an under-designed member. The design of single-angle struts is governed by the AISC Specification for Load and Resistance Factor Design of Single-Angle Members, which is reproduced in Part 6 of this Manual. The following example illustrates the design procedure for an equal-leg angle loaded eccentrically. The design strengths for concentric loading, tabulated below, are useful in solving the interaction equations for combined axial force and bending. The tables below are based on Zureick (1993), revised to conform with the AISC Single-Angle Specification (LRFD). EXAMPLE 3-8

An angle 2×2×1⁄4 is loaded by a gusset plate attached to one leg with an eccentricity of 0.8 in. from the centroid, as shown in Figure 3-3. Determine the factored compressive load Pu which may be applied. The effective length KL is 4.0 ft.

Given:

A = 0.938 in.2 rz = 0.391 in. Ix = Iy = 0.348 in.4

3 84

′

7′

′′

7 .2

0

0.

Z

Pn

W

0.8′′

0.592 ′′

37

′′

α

8 0.

W

Z

′ 4′

41

1.



3 - 105

α = 45° Fy = 50 ksi Solution:

Determine the properties for the principal axes Z-Z and W-W as follows: Iz = Arz2 = 0.938(0.391)2 = 0.143 in.2 Iw + Iz = Ix + Iy Iw = 0.348 + 0.348 − 0.143 = 0.552 in.4 rw

=

 √ √  Iw = A

0.552 = 0.767 in. 0.938

From the tables which follow, the design compressive strength φcPn = 14 kips for KL = 4 ft. For combined axial compression and bending, the latter is taken about the principal axes in accordance with the Single-Angle LRFD Specification (Section 6). For equal leg angles— Major principal axis (W-W) bending (Section 5.3.1): 0.46Eb2t 2 l 0.46(29,000 ksi)(2 in.)2(0.25 in.)2 = 1.0 × 48 in. = 69.5 k-in. Iw 0.552 in.4 My = Fy Sw = Fy = 50 ksi × cw 1.414 in. = 19.5 k-in.

Mob = Cb

Since Mob > My (Section 5.1.3), Mnw = [1.58 − 0.83  √ My / Mob ] My ≤ 1.25My = [1.58 − 0.83  √ 19.5 / 69.5 ] My = 1.14My = 1.14 × 19.5 k-in. = 22 k-in. According to Section 5.1.1, (= 2 in. / 0.25 in. = 8) < 0.382 √  E /Fy (= 0.382 √  29,000 / 50  = 9.2), Mnw ≤ 1.25Fy Sc = 1.25Fy Sw = 1.25My

for b / t

This is satisfied since Mnw = 1.14My. Minor principal axis (Z-Z) bending (Section 5.3.1): With the leg tips of the angle in tension and the angle corner in compression AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3 - 106

COLUMN DESIGN

Mnz = 1.25My = 1.25Fy Sz = 1.25Fy = 1.25 × 50 ksi ×

Iz cz

0.143 in.4 0.837 in.

= 11 k-in. Assuming

Pu ≥ 0.2, Interaction Equation 6-1a governs. φcPn

Muz    Pu 8  Muw  φ P + 9  φ M + φ M   ≤ 1.0 b nz    b nw  c n According to Section 6.1.1, for flexural compression Mu shall be multiplied by B1 (Equation 6-2). Major principal axis (W-W) bending:

Kl / rw = 1.0 × 48 / 0.767 = 62.2 From LRFD Specification Table 8, Pe / Ag = 73.1

Pe1w = 73.1(0.938) = 68.6 kips B1w =

Cm 0.85 = < 1. Use B1w = 1.0. 1 − Pu / Pe1w 1 − Pu / 68.6

Minor principal axis (Z-Z) bending:

Kl / rw = 1.0 × 48 / 0.391 = 122.8 From LRFD Specification Table 8, Pe / Ag = 19.0

Pe1z = 19.0(0.938) = 17.8 kips B1z =

Cm 0.85 = 1 − Pu / Pe1z 1 − Pu / 17.8

Conservatively adding the maximum axial and flexural terms, Equation 6-1a becomes Pu ×0.277 in. Pu 8  Pu ×0.843 in. × 1.0 0.85  +  +  ≤ 1.0 14 kips 9  0.9×22 kip−in. 0.9×11 kip−in. 1−Pu / 17.8 

Pu = 7 kips Checking

Pu 7 kips = = 0.5 > 0.2 o.k. 14 kips φcPn

A less conservative approach would have involved applying the interaction equation separately at the corner and the two leg tips of the angle, with the proper signs (+ or −) for compression and tension. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


3 - 107

Fy = 36 ksi

z

Y

Fy = 50 ksi

COLUMNS Single angles

w x


x

w Y z

8× ×6

Size Thickness

1

7⁄ 8

3⁄ 4

5⁄ 8

9⁄ 16

1⁄ 2

7⁄ 16

Wt./ft

44.2

39.1

33.8

28.5

25.7

23.0

20.2


Fy

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

421

585

373

518

322

447

270

352

235

303

199

253

163

203

1 2 3 4 5

406 398 393 384 371

555 541 531 515 490

355 347 342 336 325

484 468 460 448 429

301 292 288 284 277

408 391 383 375 363

246 235 231 228 224

311 294 287 282 276

210 199 195 193 190

262 245 239 235 230

174 163 160 158 156

213 197 192 188 185

138 128 125 123 122

166 151 146 144 141

6 7 8 9 10

353 333 311 287 263

458 422 383 344 305

311 293 274 253 232

402 371 338 303 268

266 252 236 219 201

343 319 291 262 233

217 208 195 182 167

265 250 231 210 189

186 179 170 159 148

224 214 200 184 167

153 149 143 135 126

181 175 166 155 142

120 118 115 110 104

139 136 131 124 116

11 12 13 14 15

239 215 192 169 147

266 230 196 169 147

211 190 169 149 130

235 203 173 149 130

183 165 147 130 114

204 177 151 130 114

152 137 123 109 95

167 146 125 109 95

135 123 111 99 87

149 131 114 99 87

117 107 96 87 77

128 114 101 88 77

97 90 82 75 67

107 97 87 77 67

16 17 18 19 20

130 115 103 92 83

130 115 103 92 83

115 102 91 81 74

115 102 91 81 74

100 89 79 71 64

100 89 79 71 64

84 74 66 60 54

84 74 66 60 54

77 68 61 55 49

77 68 61 55 49

68 60 54 49 44

68 60 54 49 44

60 53 48 43 39

60 53 48 43 39

21

76

76

67

67

58

58

49

49

45

45

40

40

35

35


3 - 108

COLUMN DESIGN

z

Fy = 36 ksi

Y

Fy = 50 ksi


w x

x


w Y z

8× ×4


37.4

Fy


7⁄ 8

1

3⁄ 4

33.1

5⁄ 8

28.7

24.2

1⁄ 2

21.9

7⁄ 16

19.6

17.2

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

356

495

315

438

273

380

230

299

200

258

170

216

139

174

1 2 3 4 5

342 331 315 293 267

468 446 417 378 332

300 289 275 257 234

408 387 363 330 290

256 245 235 220 201

346 326 307 280 248

209 198 190 179 165

264 246 233 216 194

179 169 162 154 142

223 207 197 184 167

149 139 134 128 119

182 168 160 150 138

118 109 105 101 95

142 128 122 116 108

6 7 8 9 10

238 208 177 148 121

283 234 187 149 121

209 183 156 131 107

248 205 164 131 107

180 157 135 113 92

212 176 141 113 92

148 130 112 94 77

169 142 117 94 77

129 114 99 84 70

146 125 104 84 70

109 98 86 73 61

123 107 90 74 61

88 80 71 62 53

98 87 75 63 53

11 12 13 14

100 85 72 62

100 85 72 62

89 75 64 55

89 75 64 55

77 65 56 48

77 65 56 48

65 55 47 41

65 55 47 41

58 49 42 37

58 49 42 37

52 44 38 33

52 44 38 33

45 38 33 29

45 38 33 29

7× ×4

Size 3⁄ 4

Thickness Wt./ft

5⁄ 8

26.2

Fy


9⁄ 16

1⁄ 2

22.1

7⁄ 16

17.9

3⁄ 8

15.7

13.6

36

50

36

50

36

50

36

50

36

50

0

249

346

210

288

164

212

136

173

108

134

1 2 3 4 5

236 228 219 205 188

320 306 289 264 233

194 187 180 170 156

258 245 233 215 191

146 139 134 128 119

182 171 164 154 140

118 111 107 103 97

144 133 128 122 113

90 85 82 79 76

107 99 95 91 86

6 7 8 9 10

168 147 126 106 87

200 166 134 107 87

140 123 106 89 73

165 138 113 90 73

108 96 84 71 59

124 106 89 72 59

89 80 71 61 51

101 88 75 62 51

70 64 57 50 43

79 70 61 52 43

11 12 13 14

72 61 52 45

72 61 52 45

61 52 44 38

61 52 44 38

49 42 36 31

49 42 36 31

43 37 31 27

43 37 31 27

37 31 27 23

37 31 27 23



3 - 109

Fy = 36 ksi

z

Y

Fy = 50 ksi


w x


Wt./ft

7⁄ 8

3⁄ 4

27.2

Fy


Y z

6× ×4

Size Thickness

36

5⁄ 8

23.6

50

36

9⁄ 16

20.0

50

36

1⁄ 2

18.1

50

36

7⁄ 16

16.2

50

36

3⁄ 8

14.3

50

36

5⁄ 16

10.3

12.3

50

36

50

36

50

0

259 359 225 312 190 264 172 239 154 205 132 171 107 136

81

100

1 2 3 4 5

250 244 233 217 198

343 331 310 282 248

91 111 87 105 85 102 82 97 78 91

66 63 61 59 57

78 73 70 68 65

6 7 8 9 10

177 155 133 111 91

212 154 184 129 155 117 139 104 121 176 135 153 113 129 102 116 91 102 142 115 124 98 105 88 94 79 84 112 97 98 82 83 74 75 67 67 91 80 80 67 67 61 61 55 55

89 102 79 87 68 73 58 59 48 48

72 65 57 49 41

82 72 61 50 41

54 50 45 39 34

61 55 48 41 34

11 12 13 14

75 63 54 47

40 34 29 25

34 29 25 22

34 29 25 22

29 24 21 18

29 24 21 18

75 63 54 47

215 210 201 188 172

293 284 268 244 215

66 55 47 41

66 55 47 41

178 174 168 157 144

56 47 40 35

241 233 222 203 180

56 47 40 35

159 155 150 141 130

51 43 37 32

214 206 197 182 162

139 135 131 125 115

51 43 37 32

46 38 33 28

180 172 166 155 139

46 38 33 28

116 112 109 105 98

145 138 133 126 116

40 34 29 25

6× ×31⁄2

Size 1⁄ 2

Thickness Wt./ft

3⁄ 8

15.3

Fy


x

w

5⁄ 16

11.7

9.8

36

50

36

50

36

50

0

146

195

101

128

77

95

1 2 3 4 5

132 127 121 112 100

171 162 152 137 118

86 82 79 75 69

105 99 94 88 79

63 59 57 55 51

74 69 66 63 58

6 7 8 9 10

87 74 60 48 39

98 78 61 48 39

61 53 44 36 30

68 56 45 36 30

47 41 36 30 25

51 44 37 30 25

11 12

33 28

33 28

25 21

25 21

21 18

21 18


3 - 110

COLUMN DESIGN

z

Fy = 36 ksi

Y

Fy = 50 ksi


w x

x


w Y z

5× ×31⁄2

Size 3⁄ 4

Thickness Wt./ft

19.8

Fy


5⁄ 8

1⁄ 2

16.8

3⁄ 8

13.6

10.4

1⁄ 4

8.7

7.0

36

50

36

50

36

50

36

50

36

50

36

50

0

188

261

159

221

130

180

97

126

76

96

54

66

1 2 3 4 5

182 176 165 150 133

249 238 218 192 162

152 148 139 127 113

207 199 183 162 137

120 117 112 103 91

162 156 146 130 111

85 82 80 75 68

107 102 98 90 79

64 61 60 57 53

77 74 71 67 61

43 41 40 38 36

49 47 45 44 41

6 7 8 9 10

115 96 79 63 51

132 103 79 63 51

97 82 67 53 43

112 88 67 53 43

79 67 55 44 35

90 71 55 44 35

59 50 42 33 27

66 54 42 33 27

47 41 34 28 23

52 44 35 28 23

34 30 26 22 18

37 32 27 22 18

11 12

42 35

42 35

36 30

36 30

29 25

29 25

23 19

23 19

19 16

19 16

15 13

15 13

5× ×3

Size 1⁄ 2

Thickness Wt./ft

7⁄ 16

12.8

Fy


5⁄ 16

3⁄ 8

11.3

5⁄ 16

9.8

1⁄ 4

8.2

6.6

36

50

36

50

36

50

36

50

36

50

0

122

169

107

146

91

118

71

90

51

62

1 2 3 4 5

112 108 100 88 75

151 143 128 108 87

97 93 87 77 66

127 120 109 93 76

80 76 72 65 56

100 94 87 76 63

60 57 54 50 44

72 68 64 58 49

40 38 36 34 31

46 44 42 39 34

6 7 8 9 10

62 49 38 30 24

66 49 38 30 24

54 43 33 27 22

58 43 33 27 22

46 37 29 23 19

49 37 29 23 19

37 30 24 19 16

40 31 24 19 16

27 23 19 15 13

29 24 19 15 13



3 - 111

Fy = 36 ksi

z

Y

Fy = 50 ksi


w x


1⁄

Thickness Wt./ft

3⁄ 8

2

11.9

Fy


Y z

4× ×31⁄2

Size

5⁄ 16

9.1

1⁄ 4

7.7

6.2

36

50

36

50

36

50

36

50

0

113

158

87

120

73

95

53

68

1 2 3 4 5

107 105 99 90 79

146 142 131 114 95

78 77 75 68 60

104 102 98 87 73

63 61 60 57 51

79 76 74 69 59

43 42 42 41 38

52 50 49 48 44

6 7 8 9 10

67 56 45 35 29

76 58 45 35 29

51 43 35 27 22

58 45 34 27 22

43 36 29 23 19

48 38 29 23 19

33 28 23 19 15

37 30 24 19 15

11 12

24 20

24 20

18 15

18 15

15 13

15 13

13 11

13 11

4× ×3

Size 5⁄ 8

Thickness Wt./ft

1⁄ 2

13.6

Fy


x

w

7⁄ 16

11.1

3⁄ 8

9.8

5⁄ 16

8.5

1⁄ 4

7.2

5.8

36

50

36

50

36

50

36

50

36

50

36

50

0

129

179

105

146

93

129

80

112

67

88

50

63

1 2 3 4 5

124 119 108 95 81

170 160 141 118 93

100 96 88 78 66

136 129 114 96 76

87 84 78 68 58

117 112 101 84 67

73 71 66 59 50

98 94 86 73 58

59 57 55 49 42

74 71 67 58 47

41 40 39 36 32

50 48 46 42 36

6 7 8 9 10

66 52 39 31 25

70 51 39 31 25

54 42 32 26 21

57 42 32 26 21

47 37 29 23 18

51 37 29 23 18

41 32 25 20 16

44 32 25 20 16

35 27 21 17 14

37 27 21 17 14

27 22 17 14 11

29 22 17 14 11

31⁄2×3

Size Thickness

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

Wt./ft

10.2

7.9

6.6

5.4


Fy

36

50

36

50

36

50

36

0

97

135

75

104

63

86

49

63

1 2 3 4 5

93 89 81 71 59

127 120 105 87 68

69 67 62 54 46

93 90 81 67 53

56 55 52 46 38

73 71 66 56 44

41 40 39 36 30

51 49 48 42 34

6 7 8 9 10

48 37 28 22 18

50 37 28 22 18

37 29 22 17 14

39 29 22 17 14

31 24 19 15 12

33 24 19 15 12

25 20 15 12 10

27 20 15 12 10


50

3 - 112

COLUMN DESIGN

z

Fy = 36 ksi

Y

Fy = 50 ksi


w x

x


w Y z

31⁄2×21⁄2

Size 1⁄ 2

Thickness

3⁄ 8

9.4

Wt./ft


Fy

1⁄ 4

7.2

4.9

36

50

36

50

36

50

0

89

124

68

95

45

58

1 2 3 4 5 6 7 8 9

85 79 70 58 46

116 105 88 68 49 34 25 19

63 60 53 44 35 26 19 15

85 79 67 52 38 26 19 15

38 37 34 29 24 18 13 10 8

47 45 41 33 25 18 13 10 8

34 25 19

3× ×21⁄2

Size Thickness

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

Wt./ft

8.5

6.6

5.6

4.5

3.39


Fy

36

50

36

50

36

50

36

50

36

50

0

81

113

62

86

52

73

42

57

29

37

1 2 3 4 5

78 72 63 52 40

107 96 79 60 42

58 55 48 40 31

79 73 61 46 33

48 46 41 34 26

65 61 51 39 28

37 36 33 27 21

48 46 40 31 23

24 23 22 19 16

29 28 26 22 17

6 7 8

29 22 17

29 22 17

23 17 13

23 17 13

19 14 11

19 14 11

16 12 9

16 12 9

12 9 7

12 9 7

3× ×2

Size 1⁄ 2

Thickness

7.7

Wt./ft

Fy Effective length KL (ft)

3⁄ 8

5⁄ 16

5.9

5.0

3⁄ 16

4.1

3.07

36

50

36

50

36

50

36

50

36

50

0 1 2 3 4 5

73 69 61 50 37 26

101 94 80 60 40 26

56 52 47 38 29 20

78 71 61 46 31 20

47 43 39 32 24 17

66 58 51 39 26 17

39 34 31 26 20 14

51 43 39 30 21 14

27 22 21 18 14 10

34 27 25 21 15 10

6 7

18 13

18 13

14 10

14 10

12 9

12 9

10 7

10 7

7 5

7 5

21⁄2×2

Size 3⁄ 8

Thickness

5⁄ 16

5.3

Wt./ft


1⁄ 4

0 1 2 3 4 5 6 7

1⁄ 4

4.5

3⁄ 16

3.62

2.75

36

50

36

50

36

50

36

50

50 48 42 34 25 17 12 9

70 65 55 41 27 17 12 9

42 40 36 29 21 15 10 7

59 54 46 34 23 15 10 7

34 31 29 23 17 12 8 6

48 42 37 28 18 12 8 6

26 22 21 17 13 9 6 5

33 27 25 20 14 9 6 5



3 - 113

Fy = 36 ksi

z

Y

Fy = 50 ksi


w x


Y z

8× ×8

Size 11⁄8

Thickness Wt./ft

7⁄ 8

1

56.9

Fy


x

w

51.0

3⁄ 4

45.0

5⁄ 8

38.9

9⁄ 16

32.7

1⁄ 2

29.6

26.4

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

541

752

486

675

428

594

369

513

310

405

270

348

229

291

6 7 8 9 10

484 465 444 421 397

643 608 570 530 488

435 417 398 378 356

578 546 512 476 438

383 368 351 334 315

510 482 452 421 388

320 318 304 289 273

420 417 392 365 337

254 252 251 243 230

311 309 306 294 273

213 212 210 209 202

256 255 253 250 240

173 172 171 170 168

203 202 201 199 197

11 12 13 14 15

372 346 320 294 269

446 404 363 323 283

334 311 288 264 242

401 363 326 290 255

295 275 255 234 215

355 322 289 258 227

256 239 221 204 187

308 280 252 225 198

215 201 186 172 157

251 230 208 187 167

191 178 166 154 141

222 204 186 169 151

165 155 144 134 124

191 177 162 148 133

16 17 18 19 20

244 221 197 177 159

249 221 197 177 159

219 198 177 159 143

224 198 177 159 143

195 176 158 141 128

199 177 158 141 128

170 154 138 124 112

174 154 138 124 112

143 130 116 104 94

147 130 116 104 94

129 118 107 95 86

134 119 106 95 86

114 104 95 85 77

120 106 95 85 77

21 22 23 24 25

145 132 121 111 102

145 132 121 111 102

130 118 108 99 92

130 118 108 99 92

116 105 96 89 82

116 105 96 89 82

101 92 84 78 71

101 92 84 78 71

85 78 71 65 60

85 78 71 65 60

78 71 65 60 55

78 71 65 60 55

70 64 58 53 49

70 64 58 53 49

26

94

94

85

85

76

76

66

66

56

56

51

51

45

45


3 - 114

COLUMN DESIGN

z

Fy = 36 ksi

Y

Fy = 50 ksi


w x

x


w Y z

6× ×6


7⁄ 8

1 37.4

Fy

36

3⁄ 4

33.1

50

36

5⁄ 8

28.7

50

36

9⁄ 16

24.2

50

36

50

1⁄ 2

21.9

36

7⁄ 16

19.6

50

36

3⁄ 8

17.2

50

36

5⁄ 16

14.9

50

36

12.4

50

36

50


0 356 495 315 438 273 380 230 320 208 289 186 249 159 206 129 164 98 120 1 2 3 4 5

346 343 339 326 310

476 469 462 438 409

304 300 298 289 275

416 408 404 387 361

260 255 253 250 238

354 345 342 336 314

214 209 207 205 201

289 279 275 273 265

190 184 182 181 180

255 244 241 238 236

166 160 158 157 155

213 202 199 197 195

137 131 129 128 127

170 107 129 77 160 100 119 71 157 99 117 69 156 98 115 69 154 97 114 68

89 81 79 78 77

6 7 8 9 10

292 272 250 228 205

376 340 303 266 230

258 241 221 202 181

332 301 268 235 203

224 209 192 175 157

288 261 232 204 176

189 177 163 148 134

244 221 197 174 151

171 160 147 134 121

221 200 179 157 136

153 143 132 120 108

192 174 156 138 120

126 124 114 105 95

152 149 134 120 105

96 113 68 95 112 67 94 110 66 87 99 66 79 88 63

76 76 75 74 71

11 12 13 14 15

183 161 140 121 105

196 164 140 121 105

162 142 124 107 93

173 140 150 119 128 108 116 145 123 126 105 108 95 98 124 108 107 92 92 83 83 107 92 92 79 79 72 72 93 81 81 69 69 62 62

97 103 85 87 74 74 64 64 56 56

85 75 66 57 50

92 78 67 57 50

71 64 57 49 43

77 67 57 49 43

58 52 47 42 37

63 56 49 42 37

16 17 18 19

92 82 73 65

92 82 73 65

82 72 64 58

49 43 39 35

44 39 35 31

44 39 35 31

38 34 30 27

38 34 30 27

32 29 25 23

32 29 25 23

82 72 64 58

71 63 56 50

71 63 56 50

61 54 48 43

61 54 48 43

55 49 43 39

49 43 39 35

5× ×5

Size 7⁄ 8

Thickness Wt./ft

3⁄ 4

27.2

Fy


55 49 43 39

5⁄ 8

23.6

1⁄ 2

20.0

7⁄ 16

16.2

3⁄ 8

14.3

5⁄ 16

12.3

10.3

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

259

359

225

312

190

264

154

214

135

184

115

149

89

114

1 2 3 4 5

251 249 241 228 212

345 341 325 301 272

216 214 209 198 184

296 291 283 262 237

179 177 176 167 156

244 239 236 221 200

141 138 137 135 127

189 183 181 179 163

121 117 116 115 112

157 152 150 148 141

99 95 94 93 92

122 117 115 114 112

73 70 69 68 67

88 83 81 80 79

6 7 8 9 10

194 175 155 135 116

241 209 177 146 119

169 152 135 118 102

210 182 154 128 104

143 129 114 100 86

178 154 131 108 88

116 105 93 82 70

145 126 107 89 72

102 93 82 72 62

126 110 94 78 64

87 79 71 62 54

105 92 80 67 56

67 64 57 51 45

78 74 65 55 47

11 12 13 14 15

98 82 70 60 53

98 82 70 60 53

86 72 61 53 46

86 72 61 53 46

73 61 52 45 39

73 61 52 45 39

60 50 43 37 32

60 50 43 37 32

53 44 38 33 28

53 44 38 33 28

46 39 33 28 25

46 39 33 28 25

38 33 28 24 21

39 33 28 24 21

16

46

46

40

40

34

34

28

28

25

25

22

22

18

18



3 - 115

Fy = 36 ksi

z

Y

Fy = 50 ksi


w x


Wt./ft

3⁄ 4


5⁄ 8

18.5

Fy

1⁄ 2

15.7

7⁄ 16

12.8

3⁄ 8

11.3

5⁄ 16

9.8

1⁄ 4

8.2

6.6

36

50

36

50

36

50

36

50

36

50

36

50

36

50

0

176

245

149

207

122

169

107

149

93

129

78

101

57

73

1 2 3 4 5

171 168 158 144 129

236 228 209 185 159

144 142 134 122 109

196 194 178 157 135

114 113 109 100 89

155 152 145 128 110

99 97 96 88 79

133 130 128 113 97

83 81 80 76 68

110 107 106 98 84

66 64 64 63 57

82 79 78 77 68

46 44 44 43 43

55 52 51 51 50

6 7 8 9 10

112 96 79 64 52

131 105 81 64 52

95 81 67 54 44

111 89 69 54 44

78 66 55 44 36

91 73 56 44 36

69 59 49 40 32

81 65 50 40 32

60 51 43 34 28

70 56 44 34 28

50 43 36 29 24

57 47 37 29 24

39 34 29 24 19

44 37 30 24 19

11 12 13

43 36

43 36

36 30

36 30

30 25 21

30 25 21

26 22 19

26 22 19

23 19 16

23 19 16

19 16 14

19 16 14

16 13 11

16 13 11

31⁄2×31⁄2

Size 1⁄ 2

Thickness Wt./ft

7⁄ 16

11.1


Y z

4× ×4

Size Thickness

3⁄ 8

9.8

5⁄ 16

8.5

1⁄ 4

7.2

5.8

36

50

36

50

36

50

36

50

36

50

0

105

146

93

129

80

112

68

93

53

68

1 2 3 4 5

100 99 91 81 70

137 134 119 102 83

87 86 80 72 62

118 116 106 90 74

74 73 70 62 54

99 97 91 78 64

60 59 58 53 46

78 76 75 65 54

44 43 42 41 36

53 52 51 50 42

6 7 8 9 10

59 48 37 29 24

65 49 37 29 24

52 42 33 26 21

58 43 33 26 21

45 37 29 23 18

50 37 29 23 18

38 31 24 19 16

42 32 24 19 16

31 25 20 16 13

34 26 20 16 13

11

20

20

17

17

15

15

13

13

11

11

3× ×3

Size 1⁄ 2

Thickness Wt./ft

7⁄ 16

9.4


x

w

3⁄ 8

8.3

5⁄ 16

7.2

1⁄ 4

6.1

3⁄ 16

4.9

3.71

36

50

36

50

36

50

36

50

36

50

36

50

0

89

124

79

109

68

95

58

80

47

62

32

41

1 2 3 4 5

86 82 73 62 51

117 109 94 76 57

75 72 65 55 45

102 97 83 67 51

64 63 56 48 40

86 84 72 58 44

52 52 47 41 33

70 69 61 49 38

40 39 38 33 27

51 50 48 39 30

25 25 24 24 20

30 29 29 28 22

6 7 8 9

40 30 23 18

41 30 23 18

36 27 20 16

36 27 20 16

31 23 18 14

32 23 18 14

26 20 15 12

27 20 15 12

21 16 12 10

22 16 12 10

16 12 9 7

17 12 9 7


3 - 116

COLUMN DESIGN

z

Fy = 36 ksi

Y

Fy = 50 ksi


w x

x


w Y z

21⁄2×21⁄2

Size 1⁄ 2

Thickness Wt./ft

7.7


3⁄ 8

5⁄ 16

5.9

1⁄ 4

5.0

3⁄ 16

4.1

3.07

36

50

36

50

36

50

36

50

36

50

0

73

101

56

78

47

66

39

54

29

37

1 2 3 4 5

71 64 55 44 33

97 85 68 50 33

53 49 42 34 25

73 65 52 38 26

44 42 36 29 21

59 55 44 33 22

34 34 29 23 18

46 45 36 27 18

24 23 22 18 13

29 28 26 20 14

6 7 8

23 17 13

23 17 13

18 13 10

18 13 10

15 11 9

15 11 9

13 9 7

13 9 7

10 7 5

10 7 5

2× ×2

Size Thickness

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

Wt./ft

4.7

3.92

3.19

2.44

1.65

Fy

50

36

50

36

50

36

50

36

50

44

61

37

52

30

42

23

32

14

18

1 2 3 4 5

42 36 28 20 13

57 46 33 20 13

35 31 24 17 11

48 39 28 17 11

28 25 19 14 9

38 32 23 14 9

20 19 15 11 7

27 25 18 11 7

11 11 10 7 5

13 13 11 8 5

6

9

9

8

8

6

6

5

5

3

3


36 0


REFERENCES

3 - 117

COLUMN BASE PLATES

The design of column base plates is covered in Part 11 (Volume II) of this LRFD Manual. REFERENCES

Galambos, T. V. (ed.), 1988, Guide to Stability Design Criteria for Metal Structures, Fourth Edition, Structural Stability Research Council, John Wiley & Sons, New York, NY. Geschwindner, L., 1993, “The ‘Leaning’ Column in ASD and LRFD,” Proceedings of the 1993 National Steel Construction Conference, AISC, Chicago, IL. Uang, C. M., S. W. Wattar, and K. M. Leet, 1990, “Proposed Revision of the Equivalent Axial Load Method for LRFD Steel and Composite Beam-Column Design,” Engineering Journal, 1st Qtr., AISC, Chicago. Zureick, A., 1993, “Design Strength of Concentrically Loaded Single-Angle Struts,” Engineering Journal, 4th Qtr., AISC, Chicago.


4-1

PART 4 BEAM AND GIRDER DESIGN OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 DESIGN STRENGTH OF BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Design Strength If Elastic Analysis Is Used . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Flexural Design Strength for Cb = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 Flexural Design Strength for Cb > 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 Design Strength If Plastic Analysis Is Used . . . . . . . . . . . . . . . . . . . . . . . . 4-10 LOAD FACTOR DESIGN SELECTION TABLE FOR SHAPES USED AS BEAMS . . . 4-11 Use of the Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12 MOMENT OF INERTIA SELECTION TABLES FOR W AND M SHAPES . . . . . . . . 4-23 FACTORED UNIFORM LOAD TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28 Use of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30 Reference Notes on Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-32 Tables, Fy = 36 ksi: W Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-35 Tables, Fy = 36 ksi: S Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-61 Tables, Fy = 36 ksi: Channels (C, MC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-64 Tables, Fy = 50 ksi: W Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-72 Tables, Fy = 50 ksi: S Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-98 Tables, Fy = 50 ksi: Channels (C, MC)) . . . . . . . . . . . . . . . . . . . . . . . . . . 4-101 DESIGN FLEXURAL STRENGTH OF BEAMS WITH UNBRACED LENGTH GREATER THAN Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-109 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-109 Charts (Fy = 36 ksi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-113 Charts (Fy = 50 ksi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-139 PLATE GIRDER DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-167 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-167 Flexure and Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-167 Table of Dimensions and Properties of Built-up Wide-Flange Section . . . . . . . . . . 4-167 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-168 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4-2

BEAM AND GIRDER DESIGN

BEAM DIAGRAMS AND FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . 4-187 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-187 Frequently Used Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-188 Table of Concentrated Load Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . 4-189 Static Loading Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-190 Design Properties of Cantilevered Beams . . . . . . . . . . . . . . . . . . . . . . . . 4-205 FLOOR DEFLECTIONS AND VIBRATIONS . . . . . . . . . . . . . . . . . . . . . . . 4-207 Serviceability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-207 Deflections and Camber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-207 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-208 BEAMS: OTHER SUBJECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-211 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-213


OVERVIEW

4-3

OVERVIEW Beam tables are located as follows: Load Factor Design Selection Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15 Moment of Inertia Selection Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24 Factored Uniform Load Tables, Fy = 36 ksi, begin on . . . . . . . . . . . . . . . . . . . 4-35 Factored Uniform Load Tables, Fy = 50 ksi, begin on . . . . . . . . . . . . . . . . . . . 4-72 Beam charts are located as follows: Beam Design Moments, Fy = 36 ksi, begin on . . . . . . . . . . . . . . . . . . . . . . . 4-113 Beam Design Moments, Fy = 50 ksi, begin on . . . . . . . . . . . . . . . . . . . . . . . 4-139 Plate Girder Design Tables are on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-183 Beam Diagrams and Formulas begin on . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-187 Additional information related to beam design is provided as follows: Floor deflections and vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-207


4-4



DESIGN STRENGTH OF BEAMS

4-5

DESIGN STRENGTH OF BEAMS General

Beams are proportioned so that no applicable strength limit state is exceeded when subjected to factored load combinations and that no serviceability limit state is exceeded when subjected to service loads. Strength limit states for beams include local buckling, lateral torsional buckling, and yielding. Serviceability limit states may include, but are not limited to, deflection and vibration. The flexural design strength for beams must equal or exceed the required strength based on the factored loads. The design strength φbMn for each applicable limit state shall equal or exceed the maximum moment Mu as determined from the applicable factored load combinations given in Section A4 of the LRFD Specification. Values of φbMn are tabulated in the pages to follow. These values are based on beam behavior as shown in Figure 4-1 and explained in the following discussion. It should be noted that the LRFD Specification expresses values for moments and lengths in kip-in. and inches. In this and other parts of the LRFD Manual, these values are tabulated in kip-ft and feet. The required strength can be determined by either elastic or plastic analysis. Design Strength If Elastic Analysis Is Used

The flexural design strength of rolled I and C shape beams designed using elastic analysis, according to LRFD Specification Section F1 is: φbMn where φb = 0.90

Mp M n′

Mn Mr (CbMn – Mn) CbMn Mn

Lp

L ′p

Lm

Lm′

Lr

Lb


4-6


Mn = nominal flexural strength as determined by the limit state of yielding, lateraltorsional buckling, or local buckling Flexural Design Strength for Cb = 1.0

Compact Sections (Cb = 1.0)

When Lb ≤ Lp

The flexural design strength of compact (flange and web local buckling λ ≤ λp) I-shaped and C-shaped rolled beams (as defined in Section B5 of the LRFD Specification) bent about the major or minor axis is: φbMn = φbMp = φbZFy / 12 In minor axis flexure this is true for all unbraced lengths, but for bending about the major axis the distance Lb between points braced against lateral movement of the compression flange or between points braced to prevent twist of the cross-section shall not exceed the value Lp (see Figure 4-1). Lp =

300ry Fy √

(F1-4)

When Lp < Lb ≤ Lr The flexural design strength of compact I or C rolled shapes bent about the major axis, from LRFD Specification Section F1.2, is:  Lb − Lp  φbMn = φbMp − φb(Mp − Mr)  ≤ φbMp  Lr − Lp  where the limiting length Lr and the corresponding buckling moment Mr (see Figure 4-1) are determined as follows: Lr =

ryX1 (Fy − Fr )

 √ 1+√  1 + X2(Fy − Fr)2

(F1-6)

where X1 =

π Sx

 √

4Cw X2 = Iy

EGJA 2

 Sx     GJ 

(F1-8)

2

φbMr = φbSx(Fy − Fr ) / 12 kip-ft Sx = section modulus about major axis, in.3 E = modulus of elasticity of steel, 29,000 ksi G = shear modulus of steel, 11,200 ksi J = torsional constant, in.4 A = cross-sectional area of beam, in.2 Cw = warping constant, in.6 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(F1-9)


4-7

Fr = compressive residual stress in flange: for rolled shapes Fr = 10 ksi; for welded shapes Fr = 16.5 ksi Values of J and Cw are tabulated for some shapes in Part 1 of the LRFD Manual. For values not shown, see Torsional Analysis of Steel Members (AISC, 1983). Compact and Noncompact Sections (Cb = 1.0)

When Lb > Lr

According to LRFD Specification Section F1.2b, the flexural design strength of compact and noncompact I or C rolled shapes bent about the major axis is: π φbMn = φbMcr = φb    Lb 

 √ 2

 πE  EIyGJ +   IyCw  Lb 

 SxX1√ 2  = φb    (Lb / ry) 

 √ 1+

X21X2 ≤ φbMr 2(Lb / ry)2

Noncompact Sections (Cb = 1.0)

When Lb ≤ Lp′

All rolled W shapes are compact except the W40×174, W14×99, W14×90, W12×65, W10×12, W8×10, and W6×15 for 50 ksi and the W6×15 for 36 ksi. The flexural design strength φbMn′ (see Figure 4-1) for noncompact (flange or web local buckling λp < λ ≤ λr) I and C rolled shapes bent about the major or minor axis is the smaller value for either local flange buckling or local web buckling as determined by:  λ − λp  φbMn′ = φbMp − φb(Mp − Mr)    λr − λp  For local flange buckling: λ = bf / 2tf for I-shaped members λ = bf / tf for C-shaped members Fy λp = 65 / √ λr = 141 / √  Fy − 10  For local web buckling: λ = h / tw λp = 640 / √ Fy Fy λr = 970 / √  Mp − Mn′  Lp′ = Lp + (Lr − Lp)    Mp − Mr  Sections with a width-to-thickness ratio exceeding the specified values for λr are slender shapes and must be analyzed using LRFD Specification Appendix B5.3. When Lp′ < Lb ≤ Lr AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4-8


The flexural design strength of noncompact I or C rolled shapes bent about the major axis is determined by:  Lb − Lp  φbMn = φbMp − φb(Mp − Mr)   ≤ φbMn′  Lr − Lp  In the Load Factor Design Selection Table, in the case of the noncompact shapes, the values of φbMn′ and Lp′ are tabulated as φbMp and Lp. The formula above may be used with the tabulated values. Flexural Design Strength for Cb > 1.0

Cb is a factor which varies with the moment gradient between bracing points (Lb). For Cb greater than 1.0, the design flexural strength is equal to the tabulated value of the design flexural strength (with Cb = 1.0) multiplied by the calculated Cb value. The maximum value is φbMp for compact shapes or φbMn′ for noncompact shapes. The maximum unbraced lengths associated with the maximum flexural design strengths φbMp and φbMn′ are Lm and Lm′ (see Figure 4-1). A new expression for Cb is given in the LRFD Specification. (It is more accurate than the one previously shown.) Cb =

12.5Mmax 2.5Mmax + 3MA + 4MB + 3Mc

(F1-3)

where M is the absolute value of a moment in the unbraced beam segment as follows: Mmax , the maximum MA , at the quarter point MB , at the centerline Mc , at the three-quarter point Values for Cb for some typical loading conditions are given in Table 4-1. Compact Sections (Cb > 1.0)

When Lb ≤ Lm

The flexural design strength for rolled I and C shapes is: φbMn = φbMp When Lb > Lm The flexural design strength is: φbMn = Cb[φbMn (for Cb = 1.0)] ≤ φbMp For Lm ≤ Lr Lm = Lp +

(CbMp − Mp)(Lr − Lp) Cb(Mp − Mr) AMERICAN INSTITUTE OF STEEL CONSTRUCTION


4-9

Table 4-1. Values of Cb for Simply Supported Beams Lateral Bracing Along Span

Load

Cb

None 1.32

At load points 1.67

1.67

None 1.14

At load points 1.67

1.00

1.67

None 1.14

At load points 1.67

1.11

None

At centerline

Cbπ Mp

1.30

 √ √   √ EIyGJ 2

1+

1+

4CwM2p IyC2bG2J2

The value of Cb for which Lm or Lm′ equals Lr for any rolled shape is: Cb =

Fy Zx (Fy − 10)Sx

Noncompact Sections (Cb > 1.0)

When Lb ≤ Lm′

The flexural design strength for rolled I and C shapes is: φbMn = φbMn′ < φbMp When Lb > Lm′ The flexural design strength is: φbMn = Cb[φbMn (for Cb = 1.0)] ≤ φbMn′ AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1.67

1.14

For Lm > Lr

Lm =

1.11

1.30

4 - 10


For Lm′ ≤ Lr Lm′ = Lp′ +

(CbMn′ − Mn′)(Lr − Lp) Cb(Mp − Mr)

For Lm′ > Lr Lm =

Cbπ Mp

 √ √   √ EIyGJ 2

1+

1+

4CwM2p IyC2bG2J2

Design Strength If Plastic Analysis Is Used

The design flexural strength for plastic analysis is: φbMn = φbMp where φb = 0.90 Mp = ZxFy / 12 kip-ft The yield strength of material that may be used with plastic analysis is limited to 65 ksi. Plastic analysis is limited to compact shapes as defined in Table B5.1 of the LRFD Specification as: λp = bf / 2tf ≤ 65 / √ Fy for the flanges of I shapes in flexure Fy for the flanges of C shapes in flexure λp = bf / tf ≤ 65 / √ and λp = h / tw ≤ 640 / √ Fy for beam webs in flexural compression where

.

λp = limiting slenderness parameter for compact element bf = width of flange for I and C shapes, in. tf = flange thickness, in. h = clear distance between flanges less the fillet at each flange, in. tw = beam web thickness, in

In addition, LRFD Specification Section F1.2d states: for a section bent about the major axis, the laterally unbraced length of the compression flange at plastic hinge locations associated with the failure mechanism shall not exceed: Lpd =

3,600 + 2,220(M1 / M2) ry Fy

where Fy M1 M2 ry (M1 / M2)

= specified yield strength of compression flange, ksi = smaller moment at end of unbraced length of beam, kip-in. = larger moment at end of unbraced length of beam, kip-in. = radius of gyration about minor axis, in. is positive when the moments cause reverse curvature AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(F1-17)

LOAD FACTOR DESIGN SELECTION TABLE FOR SHAPES USED AS BEAMS

4 - 11


This table facilitates the selection of beams designed on the basis of flexural strength in accordance with Section F of the LRFD Specification. It includes only W and M shapes designed as beams. A laterally supported beam can be selected by entering the table with the required plastic section modulus or factored bending moment, and comparing it with tabulated values of Zx or φbMp respectively. The table is applicable to adequately braced beams with unbraced lengths not exceeding Lr, i.e., Lb ≤ Lr. For beams with unbraced lengths greater than Lp, it may be convenient to use the unbraced beam charts. For most loading conditions, it is convenient to use this selection table. However, for adequately braced, simply supported beams with a uniform load over the entire length, or equivalent symmetrical loading, the tables of Factored Uniform Loads can also be used. In this table, shapes are listed in groups by descending order plastic section modulus Zx. Included also for steel of Fy = 36 ksi and 50 ksi are values for the maximum flexural design strength φbMp; the limiting buckling moment φbMr; the limiting laterally unbraced compression flange length for full plastic moment capacity and uniform moment (Cb = 1.0) Lp; limiting laterally unbraced length for inelastic lateral-torsional buckling Lr; and BF, a factor that can be used to calculate the resisting moment φbMn for beams with unbraced lengths between the limiting bracing lengths Lp and Lr. For noncompact shapes, as determined by Section B5 of the LRFD Specification, the maximum flexural design strength φbMn, max as determined by LRFD Specification Formula A-F1-3 is tabulated as φbMp. The associated maximum unbraced length is tabulated as Lp. (See the previous discussion under Design Strength of Beams for further explanation.) The symbols used in this table are: Zx = plastic section modulus, X-X axis, in.3 φbMp = design plastic bending moment, kip-ft = φbZxFy / 12 if shape is compact  λ − λp  = φbM′n = φbMp − φb(Mp − Mr)   if shape is noncompact  λr − λp  φbMr = limiting design buckling moment, kip-ft = φbSx(Fy − Fr ) / 12 where Fr = 10 ksi for rolled shapes Lp = limiting laterally unbraced length for inelastic LTB, ft, uniform moment case (Cb = 1) Lr = limiting laterally unbraced length for elastic lateral-torsional buckling, ft BF = a factor that can be used to calculate the design flexural strength for unbraced lengths Lb, between Lp and Lr, kip-ft φb(Mp − Mr) = Lr − Lp where φbMn = Cb[φbMp − BF(Lb − Lp)] ≤ φbMp AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 12


Use of the Table

Determine the required plastic section modulus Zx from the maximum factored moment Mu (kip-ft) using the desired steel yield strength. Zx =

12Mu φbFy

Enter the column headed Zx and find a value equal to or greater than the plastic section modulus required. Alternatively, enter the φbMp column and find a value of φbMp equal to or greater than the required factored load moment. The beam opposite these values (Zx or φbMp) in the shapes column, and all beams above it, have sufficient flexural strength based only on these parameters. The first beam appearing in boldface type adjacent to or above the required Zx or φbMp is the lightest section that will serve for the steel yield stress used in the calculations. If the beam must not exceed a certain depth, proceed up the column headed “Shape” until a beam within the required depth is reached. After a shape has been selected, the following checks should be made. If the lateral bracing of the compressive flange exceeds Lp, but is less than Lr, the design flexural strength may be calculated as follows: φbMn = Cb[φbMp − BF(Lb − Lp)] ≤ φbMp If the bracing length Lb is substantially greater than Lp, i.e., Lb > Lr, it is recommended the unbraced beam charts be used. A check should be made of the beam web shear strength by referring to the Factored Uniform Load Tables or by use of the formula: φvVn = φv0.6FywAw (from LRFD Specification Section F2) where φv = 0.90 If a deflection limitation also exists, the adequacy of the selected beam should be checked accordingly.

EXAMPLE 4-1

Given:

Solution (Zx method):

Select a beam of Fy = 50 ksi steel subjected to a factored uniform bending moment of 256 kip-ft, having its compression flange braced at 5.0 ft intervals. Assume Cb = 1.0. Zx (req’d) =

Mu(12) 256(12) = = 68.3 in.3 φbFy 0.9(50)

Enter the Load Factor Design Selection Table and find the nearest higher tabulated value of Zx is 69.6 in., which corresponds to a W14×43. This beam, however, is not in boldface type. Proceed up the shape column and locate the first beam in boldface, W16×40. Note the values tabulated for φbMp and Lp are 273 kip-ft and 5.6 ft, respectively. Use W16x40 AMERICAN INSTITUTE OF STEEL CONSTRUCTION


4 - 13

Alternatively, proceed up the shape column and select a W18×40. The tabulated values for φbMp and Lp are 294 kip-ft and 4.5 ft, respectively. Since the bracing length Lb is larger than Lp and smaller than Lr, the maximum resisting moment may be calculated as follows: φbMn = Cb[φbMp − BF(Lb − Lp)] = 1.0[294 − (11.7)(5.0 − 4.5)] = 288 kip-ft > 256 kip-ft req’d o.k. A W18×40 is satisfactory. Alternate solution (Mp method):

Enter the column of φbMp values and note the tabulated value nearest and higher than the required factored moment (Mu) is 261 kip-ft, which corresponds to a W14×43. Scanning the φbMp values for shapes listed higher in the column, a W16×40 is found to be the lightest suitable shape with Lb < Lp. Use W16×40

EXAMPLE 4-2

Given:

Determine the design flexural strength of a W16×40 of Fy = 36 ksi and Fy = 50 ksi steel with the compression flange braced at intervals of 9.0 ft. Assume Cb = 1.1.

Solution:

Enter the Load Factor Design Table and note that for a W16×40, Fy = 36 ksi: φbMp = 197 kip-ft Lp = 6.5 ft Lr = 19.3 ft BF = 5.54 kips φbMn = Cb[φbMp − BF(Lb − Lp)] ≤ φb Mp = 1.1[197 − 5.54(9 − 6.5)] ≤ 197 kip-ft = 197 kip-ft Enter the Load Factor Design Selection Table and note that for a W16×40, Fy = 50 ksi: φbMp = 273 kip-ft Lp = 5.6 ft Lr = 14.7 ft BF = 8.67 kips φbMn = Cb[φbMp − BF(Lb − Lp)] ≤ φb Mp = 1.1[273 − 8.67(9 − 5.6)] ≤ 273 kip-ft = 268 kip-ft AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 14


EXAMPLE 4-3

Given:

Select a beam of Fy = 50 ksi steel subjected to a factored uniform bending moment of 30 kip-ft having its compression flange braced at 4.0-ft intervals and a depth of eight inches or less. Assume Cb = 1.0.

Solution (Zx method):

Assume shape is compact and Lb ≤ Lp. Zx req’d =

12Mu 12(30) = = 8.0 in.3 φbFy 0.9(50)

Enter the Load Factor Design Selection Table and note that for a W8×10, Fy = 50 ksi, the shape is noncompact, however, the maximum resisting moment φbMn listed in the φbMp column is adequate. Further note: φbMn = 33.0 kip-ft Lp = 3.1 ft Lr = 7.8 ft BF = 2.03 kips Since Lp < Lb ≤ Lr φbMn = Cb[φbMn − BF(Lb − Lp)] = 1.0[33.0 − 2.03(4.0 − 3.1)] = 33.0 − 1.8 = 31.2 kip-ft > 30 kip-ft req’d o.k. Use: W8×10 Alternate Solution (Mp method):

Enter the Selection Table and note that in the column of φbMp values for W8×10, Fy = 50 ksi, the value of φbMp is 33.0 kip-ft, which is adequate. Also note, however, Lp = 3.1 ft is less than the bracing interval Lb = 4.0 ft, and that BF is equal to 2.03 kips. Therefore: φbMn = 1.0[33.0 − 2.03(4 − 3.1)] = 31.2 kip-ft > 30 kip-ft req’d o.k. Use: W8×10



4 - 15

LOAD FACTOR DESIGN SELECTION TABLE For shapes used as beams φb = 0.90 Fy = 36 ksi

Zx

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

Ft

Ft

Kips

34.5

138.2

17.8

6180

10300

3830

W36×848a 14400

9510

15.1

90.5

64.3

34.1

130.1

17.7

5810

9639

3570

W36×798a 13400

8940

15.0

85.3

63.2

33.2

105.9

17.2

4720

7668

2840

W36×650a 10700

7260

14.6

70.0

61.1

41.9

84.8

15.9

4560

7450

2760

W40×593a 10400

7020

13.5

56.8

76.7

41.2 33.2

72.5 86.8

15.5 16.8

3860 3800

6210 6129

2300 2270

W40×503a W36× 527a

8630 8510

5940 5850

13.2 14.2

49.5 58.3

73.9 60.4

46.9

58.2

11.3

3330

5540

2050

W40×466a

7690

5130

9.6

39.4

85.9

41.0 19.3 32.7 23.7

63.3 119.4 73.5 93.6

15.2 15.3 16.5 15.6

3300 3060 3160 2980

5270 5080 5022 4830

1950 1880 1860 1790

W40×431 W27× 539a W36× 439a W30× 477a

7310 7050 6980 6710

5070 4710 4860 4590

12.9 12.9 14.0 13.3

44.1 78.2 50.3 62.0

72.0 35.9 58.2 43.5

46.6

49.8

11.0

2810

4620

1710

W40×392a

6410

4320

9.3

34.3

83.7

39.9 32.5

56.6 67.2

15.0 16.3

2850 2830

4510 4482

1670 1660

W40×372 W36× 393a

6260 6230

4380 4350

12.7 13.8

40.3 46.7

68.4 57.0

48.6 15.3 18.9 32.5

48.0 123.3 99.2 62.4

14.6 14.2 14.9 16.1

2750 2520 2540 2570

4370 4190 4130 4077

1620 1550 1530 1510

W44×335 W24× 492a W27× 448a W36× 359a

6080 5810 5740 5660

4230 3870 3900 3960

12.4 12.1 12.6 13.7

35.5 80.5 65.3 44.0

79.7 28.4 34.9 56.2

46.2 22.9 29.4

43.2 77.5 64.4

10.7 15.3 15.6

2360 2440 2400

3860 3860 3830

1430 1430 1420

W40×331 W30× 391a W33× 354a

5360 5360 5330

3630 3750 3690

9.1 13.0 13.2

30.5 52.1 44.7

80.9 41.2 51.9

46.1 38.4 32.2 38.4

45.3 51.2 58.5 48.9

14.6 14.8 16.0 14.8

2420 2440 2360 2280

3830 3830 3726 3590

1420 1420 1380 1330

W44×290 W40× 321 W36× 328a W40× 297

5330 5330 5180 4990

3720 3750 3630 3510

12.4 12.6 13.6 12.5

34.0 37.2 41.7 35.9

74.1 63.9 54.9 63.2

43.4 28.9 31.6 14.7 37.3 19.0 44.2 23.4 31.0 28.2 43.7

43.1 59.3 55.1 103.0 47.9 82.0 38.2 66.6 53.0 55.7 37.0

14.4 15.5 16.0 13.9 14.9 14.5 10.5 15.0 15.9 15.4 10.5

2180 2160 2160 2070 2150 2070 1990 2010 2010 1970 1890

3430 3430 3402 3380 3380 3350 3210 3210 3159 3110 3050

1270 1270 1260 1250 1250 1240 1190 1190 1170 1150 1130

W44×262 W33× 318a W36× 300 W24× 408a W40× 277 W27× 368a W40× 278 W30× 326a W36× 280 W33× 291a W40× 264

4760 4760 4730 4690 4690 4650 4460 4460 4390 4310 4240

3360 3330 3330 3180 3300 3180 3060 3090 3090 3030 2910

12.2 13.1 13.5 11.8 12.7 12.3 8.9 12.8 13.5 13.0 8.9

32.8 41.8 39.9 67.5 35.5 54.6 27.6 45.7 38.8 39.8 27.0

68.2 49.9 52.9 27.0 60.8 34.8 74.9 41.7 51.3 48.0 73.3

35.6

45.4

14.8

1930

3020

1120

W40×249

4200

2980

12.6

34.1

56.9

Shape

φbMp

aGroup 4 or Group 5 shape. See Notes in Table 1-2 (Part 1).


4 - 16


LOAD FACTOR DESIGN SELECTION TABLE For shapes used as beams φb = 0.90

Zx

Fy = 36 ksi

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

φbMp

Ft

Ft

Kips

40.1 30.3 23.1 37.0 27.7 15.0 18.7 41.8 29.6

41.2 50.6 60.5 39.7 52.0 84.5 69.4 35.1 48.8

14.3 15.8 14.9 11.0 15.3 13.5 14.3 10.6 15.6

1890 1860 1810 1750 1790 1680 1720 1700 1750

2970 2916 2860 2810 2810 2750 2750 2730 2727

1100 1080 1060 1040 1040 1020 1020 1010 1010

W44×230 W36×260 W30× 292a W36×256 W33× 263a W24× 335a W27× 307a W40×235 W36×245

4130 4050 3980 3900 3900 3830 3830 3790 3790

2910 2860 2780 2690 2750 2590 2650 2620 2690

12.1 13.4 12.7 9.4 12.9 11.4 12.1 9.0 13.3

31.7 37.5 42.1 28.8 37.8 55.8 46.9 26.0 36.4

62.0 49.4 40.4 62.7 46.3 27.8 33.7 68.6 47.7

33.1 28.7 22.8 27.0 36.1

42.7 47.3 55.4 49.2 37.2

14.8 15.5 14.8 15.1 10.9

1670 1630 1610 1620 1580

2600 2550 2540 2540 2530

963 943 941 939 936

W40×215 W36×230 W30×261 W33×241 W36×232

3610 3540 3530 3520 3510

2570 2510 2480 2490 2430

12.5 13.2 12.5 12.8 9.3

32.6 35.6 39.2 36.2 27.3

51.6 45.8 39.2 44.2 59.9

40.2

33.2

10.5

1530

2440

905

W40×211

3390

2360

8.9

24.9

64.7

31.6 26.0 18.6 22.4 14.7 34.9

41.1 46.9 59.6 51.6 71.2 35.0

14.4 15.0 14.0 14.7 13.2 10.8

1500 1480 1450 1450 1400 1400

2340 2310 2300 2280 2250 2250

868 855 850 845 835 833

W40×199 W33×221 W27×258 W30×235 W24× 279a W36×210

3260 3210 3190 3170 3130 3120

2310 2270 2230 2240 2150 2160

12.2 12.7 11.9 12.4 11.2 9.1

31.6 35.0 41.1 37.1 47.6 26.1

48.8 41.9 32.9 37.7 26.9 56.8

37.4 25.0 18.5 34.0 8.40 21.8 14.7

31.2 44.8 55.0 33.5 109.5 47.9 64.3

10.4 14.8 13.9 10.7 12.3 14.5 13.1

1330 1330 1310 1290 1220 1290 1260

2110 2080 2080 2070 2030 2020 2010

781 772 769 767 753 749 744

W40×183 W33×201 W27×235 W36×194 W18× 311a W30×211 W24× 250a

2930 2900 2880 2880 2820 2810 2790

2050 2050 2020 1990 1870 1990 1930

8.8 12.6 11.8 9.1 10.4 12.3 11.1

23.8 33.8 38.5 25.2 71.5 35.1 43.4

59.0 39.7 32.3 54.6 15.6 36.0 26.6

32.7

32.8

10.6

1210

1940

718

W36×182

2690

1870

9.0

24.9

52.0

b

Shape

27.5 18.2

38.4 52.0

13.6 13.8

1250 1220

1930 1910

715 708

W40×174 W27×217

2660 2660

1920 1870

12.0 11.7

29.9 36.8

41.3 31.3

35.6 8.29 14.7 21.0 31.5 28.3 17.8

29.7 99.6 59.3 45.4 31.9 32.6 48.0

10.0 12.1 13.0 14.4 10.5 10.4 13.7

1170 1100 1150 1170 1130 1070 1080

1870 1830 1830 1820 1800 1700 1700

692 676 676 673 668 629 628

W40×167 W18× 283a W24×229 W30×191 W36×170 W33×169 W27×194

2600 2540 2540 2520 2510 2360 2360

1800 1690 1760 1790 1740 1650 1670

8.5 10.3 11.0 12.2 8.9 8.8 11.6

22.8 65.1 40.4 33.7 24.4 24.5 34.6

55.6 15.4 26.2 33.9 49.6 45.4 30.0

30.7 8.21 14.5 20.2

30.9 90.9 54.2 43.2

10.4 12.0 12.8 14.3

1060 1000 1040 1050

1680 1650 1640 1630

624 611 606 605

W36×160 W18× 258a W24×207 W30×173

2340 2290 2270 2270

1630 1540 1590 1620

8.8 10.2 10.9 12.1

23.7 59.5 37.4 32.5

48.0 15.2 25.6 32.0

aGroup 4 or Group 5 shape. See Notes in Table 1-2 (Part 1). bIndicates noncompact shape; F = 50 ksi y



4 - 17


Zx

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

Ft

Ft

Kips

32.8 29.4 17.5 26.8 14.3 8.09 11.0

28.2 30.2 45.2 31.2 51.3 82.8 60.8

9.5 10.3 13.6 10.3 12.8 11.9 12.6

998 983 979 950 957 909 899

1610 1570 1530 1510 1510 1480 1430

597 581 567 559 559 549 530

W40×149 W36× 150 W27× 178 W33× 152 W24× 192 W18× 234a W21× 201

2240 2180 2130 2100 2100 2060 1990

1540 1510 1510 1460 1470 1400 1380

8.1 8.7 11.5 8.7 10.9 10.1 10.7

21.9 23.4 33.1 23.7 35.7 54.4 41.1

50.8 45.6 28.8 42.3 25.0 14.9 19.9

25.7 16.9 14.3

30.1 42.8 47.8

10.1 13.5 12.7

874 887 878

1390 1380 1380

514 512 511

W33×141 W27× 161 W24× 176

1930 1920 1920

1340 1370 1350

8.6 11.5 10.7

23.1 31.7 33.8

40.2 27.4 24.6

27.5 23.7 8.00 10.9 14.1

28.8 30.6 75.0 55.8 45.2

9.9 9.5 11.8 12.5 12.7

856 850 817 813 807

1370 1350 1320 1290 1260

509 500 490 476 468

W36×135 W30× 148 W18× 211a W21× 182 W24× 162

1910 1880 1840 1790 1760

1320 1310 1260 1250 1240

8.4 8.1 10.0 10.6 10.8

22.4 22.8 49.5 38.1 32.4

42.2 38.6 14.7 19.4 23.8

24.5 16.2 7.98 22.4 10.8 13.8

29.1 40.7 68.3 29.0 51.7 42.0

10.0 13.4 11.6 9.4 12.4 12.5

792 801 741 741 741 723

1260 1240 1190 1180 1170 1130

467 461 442 437 432 418

W33×130 W27× 146 W18× 192 W30× 132 W21× 166 W24× 146

1750 1730 1660 1640 1620 1570

1220 1230 1140 1140 1140 1110

8.4 11.3 9.9 8.0 10.5 10.6

22.5 30.6 45.3 22.0 35.7 30.6

37.9 25.8 14.6 35.6 19.1 22.8

23.1 21.6 7.95 18.9

27.8 28.2 62.3 30.0

9.7 9.3 11.5 9.2

700 692 671 673

1120 1100 1070 1070

415 408 398 395

W33×118 W30× 124 W18× 175 W27× 129

1560 1530 1490 1480

1080 1070 1030 1040

8.2 7.9 9.8 7.8

21.7 21.5 41.5 22.3

35.5 34.1 14.5 30.9

21.1 10.7 13.3 7.87

27.1 46.4 39.3 56.7

9.1 12.3 12.4 11.4

642 642 642 605

1020 1010 999 961

378 373 370 356

W30×116 W21× 147 W24× 131 W18× 158

1420 1400 1390 1340

987 987 987 930

7.7 10.4 10.5 9.7

20.8 32.8 29.1 38.2

33.0 18.4 21.5 14.2

20.2 18.0 10.5 12.7 7.82

26.3 28.2 43.1 37.1 52.2

9.0 9.1 12.2 12.3 11.3

583 583 575 567 550

934 926 899 883 869

346 343 333 327 322

W30×108 W27× 114 W21× 132 W24× 117 W18× 143

1300 1290 1250 1230 1210

897 897 885 873 846

7.6 7.7 10.4 10.4 9.6

20.3 21.3 30.9 27.9 35.5

31.5 28.7 17.7 20.2 14.0

19.0 10.3 17.0 7.79 12.0

25.5 41.0 26.8 48.0 35.2

8.8 12.2 9.0 11.3 12.1

525 532 521 499 503

842 829 824 786 780

312 307 305 291 289

W30×99 W21× 122 W27× 102 W18× 130 W24× 104

1170 1150 1140 1090 1080

807 819 801 768 774

7.4 10.3 7.6 9.5 10.3

19.8 29.8 20.5 33.0 26.8

29.2 17.1 26.7 13.8 18.8

Shape

φbMp

aGroup 4 or Group 5 shape. See Notes in Table 1-2 (Part 1).


4 - 18



Zx

Fy = 36 ksi

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

Ft

Ft

Kips

17.8 14.8 10.1 16.2 7.72 14.3 9.61

24.8 27.1 38.7 25.9 44.1 25.9 37.1

8.7 8.3 12.1 8.8 11.2 8.3 12.0

478 478 486 474 450 433 443

764 756 753 751 705 686 683

283 280 279 278 261 254 253

W30×90 W24×103 W21×111 W27× 94 W18×119 W24× 94 W21×101

1060 1050 1050 1040 979 953 949

735 735 747 729 693 666 681

7.4 7.0 10.3 7.5 9.5 7.0 10.2

19.4 20.1 28.5 19.9 30.8 19.4 27.6

27.1 24.1 16.4 25.2 13.4 23.0 15.4

15.0 3.87 7.62

24.9 73.6 40.4

8.6 15.7 11.1

415 408 398

659 632 621

244 234 230

W27×84 W14×132 W18×106

915 878 863

639 627 612

7.3 13.3 9.4

19.3 49.6 28.7

23.0 6.89 13.0

13.6 11.8 3.86 7.51

24.5 26.6 67.9 38.1

8.1 7.7 15.6 11.0

382 374 371 367

605 597 572 570

224 221 212 211

W24×84 W21× 93 W14×120 W18× 97

840 829 795 791

588 576 570 564

6.9 6.5 13.2 9.4

18.6 19.4 46.2 27.4

21.5 19.6 6.82 12.6

12.7 6.10 11.3 3.84 2.95 7.27

23.4 42.1 24.9 62.7 75.5 35.5

8.0 10.5 7.6 15.5 13.0 11.0

343 341 333 337 318 324

540 535 529 518 502 502

200 198 196 192 186 186

W24×76 W16×100 W21× 83 W14×109 W12×120 W18× 86

750 743 735 720 698 698

528 525 513 519 489 498

6.8 8.9 6.5 13.2 11.1 9.3

18.0 29.3 18.5 43.2 50.0 26.1

19.8 10.7 18.5 6.70 5.36 11.9

12.1 6.03 3.77 10.7 2.95 6.94

22.4 38.6 58.2 23.5 67.2 33.3

7.8 10.4 15.5 7.5 13.0 10.9

300 302 306 294 283 285

478 473 467 464 443 440

177 175 173 172 164 163

W24×68 W16× 89 W14× 99b W21× 73 W12×106 W18× 76

664 656 647 645 615 611

462 465 471 453 435 438

6.6 8.8 13.4 6.4 11.0 9.2

17.4 27.3 40.6 17.7 44.9 24.8

18.7 10.3 6.46 17.0 5.32 11.1

10.4 3.75

22.8 54.1

7.5 15.4

273 279

432 424

160 157

W21×68 W14× 90b

600 587

420 429

6.4 15.0

17.3 38.4

16.5 6.31

13.8 5.85 2.01 2.91 8.29

17.2 34.9 86.4 61.4 24.4

5.8 10.3 11.2 12.9 7.1

255 261 246 255 248

413 405 397 397 392

153 150 147 147 145

W24×62 W16× 77 W10×112 W12× 96 W18× 71

574 563 551 551 544

393 402 378 393 381

4.9 8.7 9.5 10.9 6.0

13.3 25.2 56.5 41.3 17.8

21.4 9.75 3.68 5.20 13.8

9.84 4.15

21.7 43.0

7.4 10.3

248 240

389 375

144 139

W21×62 W14× 82

540 521

381 369

6.3 8.8

16.6 29.6

15.3 7.31

Shape

φbMp

bIndicates noncompact shape; F = 50 ksi. y



4 - 19


Zx

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

Ft

Ft

Kips

12.7 8.08 2.90 2.00 5.57 11.3 4.10 7.91 2.88 4.05 1.97

16.6 23.2 56.4 77.4 32.3 17.3 40.0 22.4 51.8 37.3 68.4

5.6 7.0 12.8 11.0 10.3 5.6 10.3 7.0 12.7 10.3 11.0

222 228 230 218 228 216 218 211 209 201 192

362 359 356 351 351 348 340 332 321 311 305

134 133 132 130 130 129 126 123 119 115 113

W24×55 W18× 65 W12× 87 W10× 100 W16× 67 W21× 57 W14× 74 W18× 60 W12× 79 W14× 68 W10× 88

503 499 495 488 488 484 473 461 446 431 424

342 351 354 336 351 333 336 324 321 309 296

4.7 6.0 10.9 9.4 8.7 4.8 8.8 6.0 10.8 8.7 9.3

12.9 17.1 38.4 50.8 23.8 13.1 28.0 16.7 35.7 26.4 45.1

19.6 13.3 5.12 3.66 9.02 18.0 7.12 12.8 5.03 6.91 3.58

7.65

21.4

7.0

192

302

112

W18×55

420

295

5.9

16.1

12.2

10.5 2.87 6.43 3.91

16.2 48.2 22.8 34.7

5.4 12.7 6.7 10.2

184 190 180 180

297 292 284 275

110 108 105 102

W21×50 W12× 72 W16× 57 W14× 61

413 405 394 383

284 292 277 277

4.6 10.7 5.7 8.7

12.5 33.6 16.6 24.9

16.4 4.93 10.7 6.51

7.31 1.95 2.80

20.5 60.1 44.7

6.9 10.8 12.6

173 168 171

273 264 261

101 97.6 96.8

W18×50 W10× 77 W12× 65b

379 366 358

267 258 264

5.8 9.2 11.8

15.6 39.9 31.7

11.5 3.53 4.72

9.68 6.18 8.13 4.17 2.91 1.93 5.91

15.4 21.3 16.6 28.0 38.4 53.7 20.2

5.3 6.6 5.4 8.0 10.5 10.8 6.5

159 158 154 152 152 148 142

258 248 245 235 233 230 222

95.4 92.0 90.7 87.1 86.4 85.3 82.3

W21×44 W16× 50 W18× 46 W14× 53 W12× 58 W10× 68 W16× 45

358 345 340 327 324 320 309

245 243 236 233 234 227 218

4.5 5.6 4.6 6.8 8.9 9.2 5.6

12.0 15.8 12.6 20.1 27.0 36.0 15.2

14.9 10.1 13.0 7.02 4.96 3.46 9.43

7.51 4.06 2.85 1.91

15.7 26.3 35.8 48.1

5.3 8.0 10.3 10.7

133 137 138 130

212 212 210 201

78.4 78.4 77.9 74.6

W18×40 W14× 48 W12× 53 W10× 60

294 294 292 280

205 211 212 200

4.5 6.8 8.8 9.1

12.1 19.2 25.6 32.6

11.7 6.70 4.77 3.38

5.54 3.06 1.30 3.91 1.89

19.3 30.8 64.0 24.7 43.9

6.5 8.2 8.8 7.9 10.7

126 126 118 122 117

197 195 190 188 180

72.9 72.4 70.2 69.6 66.6

W16×40 W12× 50 W8× 67 W14× 43 W10× 54

273 272 263 261 250

194 194 181 188 180

5.6 6.9 7.5 6.7 9.1

14.7 21.7 41.9 18.2 30.2

8.67 5.25 2.38 6.32 3.30

6.95 3.01 5.23 4.41 1.88 1.27 2.92 1.96

14.8 28.5 18.3 20.0 40.7 56.0 26.5 35.1

5.1 8.1 6.3 6.5 10.6 8.8 8.0 8.4

112 113 110 106 106 101 101 95.7

180 175 173 166 163 161 155 148

66.5 64.7 64.0 61.5 60.4 59.8 57.5 54.9

W18×35 W12× 45 W16× 36 W14× 38 W10× 49 W8× 58 W12× 40 W10× 45

249 243 240 231 227 224 216 206

173 174 170 164 164 156 156 147

4.3 6.9 5.4 5.5 9.0 7.4 6.8 7.1

11.5 20.3 14.1 14.9 28.3 36.8 19.3 24.1

10.7 5.07 8.08 7.07 3.25 2.32 4.82 3.45

Shape

φbMp

bIndicates noncompact shape; F = 50 ksi. y


4 - 20



Zx

Fy = 36 ksi

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

Ft

Ft

Kips

4.18

19.0

6.4

94.8

147

54.6

W14×34

205

146

5.4

14.4

6.58

5.70 3.47 1.26

14.3 20.6 46.7

4.9 6.4 8.7

92.0 88.9 84.4

146 138 132

54.0 51.2 49.0

W16×31 W12× 35 W8×48

203 192 184

142 137 130

4.1 5.4 7.4

11.0 15.2 31.1

8.85 5.67 2.27

3.92 1.93

17.9 31.2

6.2 8.3

81.9 82.1

128 126

47.3 46.8

W14×30 W10× 39

177 176

126 126

5.3 7.0

13.7 21.8

6.06 3.32

5.15 3.22

13.3 19.1

4.7 6.3

74.9 75.3

119 116

44.2 43.1

W16×26 W12× 30

166 162

115 116

4.0 5.4

10.4 14.4

7.88 5.10

4.44 1.25 1.89

13.4 39.1 27.4

4.5 8.5 8.1

68.8 69.2 68.3

109 107 105

40.2 39.8 38.8

W14×26 W8×40 W10× 33

151 149 146

106 107 105

3.8 7.2 6.9

10.3 26.4 19.7

6.96 2.22 3.15

2.99 2.44 1.23

18.1 20.3 35.1

6.3 5.7 8.5

65.1 63.2 60.8

100 98.8 93.7

37.2 36.6 34.7

W12×26 W10× 30 W8×35

140 137 130

100 97.2 93.6

5.3 4.8 7.2

13.8 14.5 24.1

4.64 4.13 2.16

4.06 2.34 1.21

12.5 18.5 32.0

4.3 5.7 8.4

56.6 54.4 53.6

89.6 84.5 82.1

33.2 31.3 30.4

W14×22 W10× 26 W8×31

125 117 114

87.0 83.7 82.5

3.7 4.8 7.1

9.7 13.5 22.3

6.26 3.85 2.07

3.88 1.27

11.1 27.3

3.5 6.8

49.5 47.4

79.1 73.4

29.3 27.2

W12×22 W8×28

110 102

76.2 72.9

3.0 5.7

8.4 18.9

6.24 2.22

2.19

16.9

5.5

45.2

70.2

26.0

W10×22

97.5

69.6

4.7

12.7

3.50

3.61 1.24

10.4 24.4

3.4 6.7

41.5 40.8

66.7 62.6

24.7 23.2

W12×19 W8×24

92.6 87.0

63.9 62.7

2.9 5.7

7.9 17.2

5.70 2.11

2.60 1.46

12.0 18.6

3.6 5.3

36.7 35.5

58.3 55.1

21.6 20.4

W10×19 W8×21

81.0 76.5

56.4 54.6

3.1 4.5

8.9 13.3

4.26 2.47

3.30 0.741 2.46

9.6 31.3 11.2

3.2 6.3 3.5

33.3 32.6 31.6

54.3 51.0 50.5

20.1 18.9 18.7

W12×16 W6×25 W10× 17

75.4 70.9 70.1

51.3 50.1 48.6

2.7 5.4 3.0

7.4 21.0 8.4

5.12 1.33 3.97

2.97 1.40 2.34 0.728

9.2 16.7 10.3 25.6

3.1 5.1 3.4 6.3

29.1 29.6 26.9 26.1

47.0 45.9 43.2 40.2

17.4 17.0 16.0 14.9

W12×14 W8×18 W10× 15 W6×20

65.3 63.8 60.0 55.9

44.7 45.6 41.4 40.2

2.7 4.3 2.9 5.3

7.2 12.3 7.9 17.7

4.56 2.30 3.69 1.27

3.32 1.53

6.9 12.6

2.3 3.7

23.6 23.0

38.6 36.7

14.3 13.6

M12×11.8 W8×15

53.7 51.0

36.3 35.4

2.0 3.1

5.4 9.2

5.10 2.56

Shape

φbMp



4 - 21


Zx

Fy = 50 ksi

BF

Lr

Lp

φbMr

φbMp

Zx

Kips

Ft

Ft

Kip-ft Kip-ft

in.3

3.10 2.03 0.817 0.458 1.44 0.417 0.693 0.444

6.8 9.5 18.3 30.3 11.5 31.1 20.8 26.3

2.3 3.3 4.0 5.3 3.5 5.0 6.7 5.3

21.6 21.3 19.9 19.9 19.3 18.8 19.0 16.6

35.4 34.0 31.6 31.3 30.8 29.7 28.8 25.9

13.1 12.6 11.7 11.6 11.4 11.0 10.8 9.59

M12×10.8 W10× 12b W6× 16 W5× 19 W8× 13 M5× 18.9 W6×15b,c W5× 16

49.2 47.0 43.9 43.5 42.8 41.3 38.6 36.0

2.32 1.30 0.775

6.2 10.2 14.4

2.1 3.5 3.8

15.2 15.2 14.3

24.9 23.9 22.4

9.21 8.87 8.30

M10×9 W8× 10b W6× 12

2.13 0.295 0.724

6.1 25.5 12.0

2.1 4.2 3.8

13.6 10.6 10.8

22.1 17.0 16.8

8.20 6.28 6.23

1.50

5.5

1.8

14.6

5.40

9.01

φbMr

Lp

Lr

BF

Kip-ft Kip-ft

Ft

Ft

Kips

33.3 32.7 30.6 30.6 29.7 28.9 29.2 25.5

2.0 2.9 3.4 4.5 3.0 4.2 6.8 4.5

5.3 7.4 12.5 20.1 8.5 20.5 15.0 17.6

4.74 3.13 1.46 0.830 2.35 0.758 1.16 0.795

34.5 33.0 31.1

23.5 23.4 21.9

1.8 3.1 3.2

4.9 7.8 10.2

3.59 2.03 1.33

M10×8 W4× 13 W6× 9

30.8 23.6 23.4

21.0 16.4 16.7

1.8 3.5 3.2

4.8 16.9 8.9

3.26 0.538 1.17

M8×6.5

20.2

13.9

1.6

4.3

2.35

Shape

φbMp

bIndicates noncompact shape; F = 50 ksi y cIndicates noncompact shape; F = 36 ksi y


4 - 22



MOMENT OF INERTIA SELECTION TABLES FOR W AND M SHAPES

4 - 23

MOMENT OF INERTIA SELECTION TABLES FOR W AND M SHAPES

These two tables for moment of inertia (Ix and Iy) are provided to facilitate the selection of beams and columns on the basis of their stiffness properties with respect to the X-X axis or Y-Y axis, as applicable, where Ix = moment of inertia, X-X axis, in.4 Iy = moment of inertia, Y-Y axis, in.4 In each table the shapes are listed in groups by descending order of moment of inertia for all W and M shapes. The boldface type identifies the shapes that are the lightest in weight in each group. Enter the column headed Ix (or Iy) and find a value of Ix (or Iy) equal to or greater than the moment of inertia required. The shape opposite this value, and all shapes above it, have sufficient stiffness. Note that the member selected must also be checked for compliance with specification provisions governing its specific application.


4 - 24

Ix Shape


MOMENT OF INERTIA SELECTION TABLE For W and M shapes Ix

Shape

In.4 W36×848*

67400

W36×798*

62600

W40×593* W36× 650*

50400 48900

W40×503* W36× 527*

41700 38300

W40×466*

36300

W40×431

34800

W44×335 W36× 439* W40× 392* W40× 372 W36× 393*

31100 31000 29900 29600 27500

W44×290 W30× 477* W27× 539* W40× 321 W36× 359* W40× 331

27100 26100 25500 25100 24800 24700

W44×262 W40× 297 W36× 328* W40× 277 W33× 354*

24200 23200 22500 21900 21900

W44×230 W30× 391* W40× 278 W27× 448* W36× 300 W33× 318* W40× 249 W40× 264 W24× 492* W36× 280 W33× 291* W40× 235 W36× 260 W30× 326* W36× 256

20800 20700 20500 20400 20300 19500 19500 19400 19100 18900 17700 17400 17300 16800 16800

Ix

W40×215 W27× 368* W36×245 W14× 808* W33× 263*

16700 16100 16100 16000 15800

W40×211 W24× 408* W36×230 W36×232

15500 15100 15000 15000

W40×199 W30× 292* W14× 730* W33×241

14900 14900 14300 14200

W40×183 W36×210 W27× 307* W30×261 W33×221 W14× 665*

13300 13200 13100 13100 12800 12400

W40×174 W36×194 W24× 335* W30×235

12200 12100 11900 11700

W40×167 W33×201 W36×182 W14× 605* W27×258 W36×170 W30× 211

11600 11500 11300 10800 10800 10500 10300

W40×149 W36×160 W27×235 W24× 279* W14× 550* W33×169 W30×191 W36×150 W27×217 W24× 250* W14× 500* W30×173 W33×152 W27×194

9780 9750 9660 9600 9430 9290 9170 9040 8870 8490 8210 8200 8160 7820

Ix

Shape

In.4

Shape

In.4 W36×135 W24× 229 W33× 141 W14×455* W27× 178 W18× 311* W24× 207

7800 7650 7450 7190 6990 6960 6820

W33×130 W30× 148 W14×426* W27× 161 W24× 192 W18×283* W14×398*

6710 6680 6600 6280 6260 6160 6000

W33×118 W30× 132 W24× 176 W27× 146 W18×258* W14×370* W30× 124 W21× 201 W24× 162

5900 5770 5680 5630 5510 5440 5360 5310 5170

W30×116 W18×234* W14×342* W27× 129 W21× 182 W24× 146

4930 4900 4900 4760 4730 4580

W30×108 W18× 211* W14× 311* W21× 166 W27× 114 W12×336* W24× 131

4470 4330 4330 4280 4090 4060 4020

*Group 4 or 5 shape. See Notes in Table 1-2 (Part 1).


Ix In.4

W30×99 W18× 192 W14× 283* W21× 147

3990 3870 3840 3630

W30×90 W27× 102 W12× 305* W24× 117 W18× 175 W14× 257* W27× 94 W21× 132 W12× 279* W24× 104 W18× 158 W14× 233* W24× 103 W21× 122

3620 3620 3550 3540 3450 3400 3270 3220 3110 3100 3060 3010 3000 2960

W27×84 W18× 143 W12× 252* W24× 94 W21× 111 W14× 211 W18× 130 W12× 230* W21× 101 W14× 193

2850 2750 2720 2700 2670 2660 2460 2420 2420 2400

W24×84 W18× 119 W14× 176 W12× 210*

2370 2190 2140 2140

W24×76 W21× 93 W18× 106 W14× 159 W12× 190

2100 2070 1910 1900 1890

MOMENT OF INERTIA SELECTION TABLE FOR W AND M SHAPES

4 - 25

MOMENT OF INERTIA SELECTION TABLE For W and M shapes Shape

Ix

Shape

In.4 W24×68 W21× 83 W18× 97 W14× 145 W12× 170 W21× 73

1830 1830 1750 1710 1650 1600

W24×62 W14× 132 W18× 86 W16× 100 W21× 68 W12× 152 W14× 120

1550 1530 1530 1490 1480 1430 1380

W24×55 W18× 76 W21× 62 W16× 89 W14× 109 W12× 136 W18× 71 W21× 57 W14× 99 W16× 77 W12× 120 W18× 65 W14× 90 W18× 60

1350 1330 1330 1300 1240 1240 1170 1170 1110 1110 1070 1070 999 984

W21×50 W16× 67 W12× 106 W18× 55 W14× 62

984 954 933 890 882

Ix

Shape

In.4

Ix

Shape

In.4

W21×44 W12× 96 W18× 50 W14× 74 W16× 57 W12× 87 W14× 68 W10× 112 W18× 46 W12× 79 W16× 50 W14× 61 W10× 100

843 833 800 796 758 740 723 716 712 662 659 640 623

W16×26 W14× 30 W12× 35 W8× 67 W10× 49 W10× 45

301 291 285 272 272 248

W14×26 W12× 30 W8× 58 W10× 39

245 238 228 209

W12×26

204

W18×40 W12× 72 W16× 45 W14× 53 W10× 88 W12× 65

612 597 586 541 534 533

W14×22 W8× 48 W10× 30 W10× 33

199 184 170 170

W16×40

518

W12×22 W8× 40 W10× 26

156 146 144

W18×35 W14× 48 W12× 58 W10× 77 W16× 36 W14× 43 W12× 53 W12× 50 W10× 68 W14× 38

510 485 475 455 448 428 425 394 394 385

W12×19 W8× 35 W10× 22 W8× 31

130 127 118 110

W12×16 W8× 28 W10× 19

103 98.0 96.3

W16×31 W12× 45 W10× 60 W14× 34 W12× 40 W10× 54

375 350 341 340 310 303

W12×14 W8× 24 W10× 17 W8× 21

88.6 82.8 81.9 75.3

M12×11.8 W10× 15

71.7 68.9


Ix Ix In.4

M12×10.8 W8× 18 W10×12 W6× 25 W8× 15 W6× 20 W8× 13

65.8 61.9 53.8 53.4 48.0 41.4 39.6

M10×9

38.5

M10×8 W6× 16 W8× 10 W6× 15 W5× 19 M5× 18.9 W6× 12 W5× 16

34.3 32.1 30.8 29.1 26.2 24.1 22.1 21.3

M8×6.5 W6×9 W4× 13

18.1 16.4 11.3

4 - 26

Iy Shape


MOMENT OF INERTIA SELECTION TABLE For W and M shapes Iy

Shape

In.4 W14×808*

5510

W14×730* W36× 848* W36× 798*

4720 4550 4200

W14×665*

4170

W14×605*

3680

W14×550* W36× 650*

W14×283* W40×372 W36× 328* W24× 408* W27× 368* W36×300

1440 1420 1420 1320 1310 1300

W14×257* W33× 318* W30× 326* W36×280 W44×335 W12× 336* W40×321 W33× 291*

1290 1290 1240 1200 1200 1190 1190 1160

3250 3230

W14×500*

2880

W14×455* W40× 593* W36× 527*

2560 2520 2490

W14×426*

2360

W14×398* W27× 539* W40× 503*

2170 2110 2050

W14×370* W36× 439* W30× 477*

1990 1990 1970

W14×342* W36× 393* W40× 431 W27× 448* W24× 492*

1810 1750 1690 1670 1670

W14×311* W36× 359* W30× 391* W33× 354*

Iy

1610 1570 1550 1460

W14×233* W30× 292* W40×297 W36×260 W44×290 W27× 307* W12× 305* W40×277 W33× 263* W24× 335*

1150 1100 1090 1090 1050 1050 1050 1040 1030 1030

W14×211 W40× 466* W36×245 W30×261 W36×230 W12× 279* W33×241

1030 1010 1010 959 940 937 932

W14×193 W44×262 W40×249 W27×258 W30×235 W33×221

931 927 926 859 855 840

Iy

Shape

In.4

Shape

In.4 W14×176 W12×252* W24×279* W40×392* W40× 215 W44× 230 W18× 311* W27× 235 W30× 211 W33× 201

838 828 823 803 796 796 795 768 757 749

W14×159 W12×230* W24×250* W18×283* W27× 217 W40× 199

748 742 724 704 704 695

W14×145 W30× 191 W12×210* W24× 229 W40× 331 W18×258* W27× 194 W30× 173 W12× 190 W24× 207 W18×234* W27× 178

677 673 664 651 646 628 618 598 589 578 558 555

W14×132 W21× 201 W40× 174 W24× 192 W36× 256 W40× 278 W12× 170 W27× 161

548 542 541 530 528 521 517 497

*Group 4 or 5 shape. See Notes in Table 1-2 (Part 1).


Iy In.4

W14×120 W40× 264 W18× 211* W21× 182 W24× 176 W36× 232 W12× 152

495 493 493 483 479 468 454

W14×109 W40× 235 W24× 162 W27× 146 W18× 192 W21× 166 W36× 210

447 444 443 443 440 435 411

W14×99 W12× 136 W18× 175 W24× 146 W40× 211 W21× 147 W36× 194

402 398 391 391 390 376 375

W14×90 W36× 182 W18× 158 W12× 120 W24× 131 W40× 183 W21× 132 W36× 170 W18× 143 W33× 169 W21× 122 W12× 106 W24× 117 W36× 160 W40× 167 W18× 130 W21× 111 W33× 152 W36× 150 W12× 96 W24× 104 W18× 119 W21× 101 W33× 141

362 347 347 345 340 336 333 320 311 310 305 301 297 295 283 278 274 273 270 270 259 253 248 246

MOMENT OF INERTIA SELECTION TABLE FOR W AND M SHAPES

4 - 27

MOMENT OF INERTIA SELECTION TABLE For W and M shapes Shape

Iy

Shape

In.4 W12×87 W10× 112 W40× 149 W30× 148 W36× 135 W18× 106 W33× 130

241 236 229 227 225 220 218

W12×79 W10× 100 W18× 97 W30× 132

216 207 201 196

W12×72 W33× 118 W16× 100 W27× 129 W30× 124 W10× 88 W18× 86

195 187 186 184 181 179 175

W12×65 W30× 116 W16× 89 W27× 114 W10× 77 W18× 76 W14× 82 W30× 108 W27× 102 W16× 77 W10× 68 W14× 74 W30× 99 W27× 94 W14× 68 W24× 103 W16× 67

174 164 163 159 154 152 148 146 139 138 134 134 128 124 121 119 119

Iy

Shape

In.4 W10×60 W30× 90 W24× 94

116 115 109

W12×58 W14× 61 W27× 84

107 107 106

W10×54

103

W12×53 W24× 84

95.8 94.4

W10×49 W21× 93 W8× 67 W24× 76 W21× 83 W8× 58 W21× 73 W24× 68 W21× 68

93.4 92.9 88.6 82.5 81.4 75.1 70.6 70.4 64.7

W8×48 W18× 71 W14× 53 W21× 62 W12× 50 W18× 65

60.9 60.3 57.7 57.5 56.3 54.8

W10×45 W14× 48 W18× 60

53.4 51.4 50.1

W12×45

50.0

W8×40 W14× 43

49.1 45.2

Iy

Shape

In.4 W10×39 W18× 55 W12× 40 W16× 57

45.0 44.9 44.1 43.1

W8×35 W18× 50 W16× 50

42.6 40.1 37.2

W8×31 W10× 33 W24× 62 W16× 45 W21× 57 W24× 55 W16× 40 W14× 38 W21× 50 W12× 35 W16× 36 W14× 34 W18× 46

37.1 36.6 34.5 32.8 30.6 29.1 28.9 26.7 24.9 24.5 24.5 23.3 22.5

W8×28 W21× 44 W12× 30 W14× 30 W18× 40

21.7 20.7 20.3 19.6 19.1

W8×24 W12× 26 W6× 25 W10× 30 W18× 35 W10× 26

18.3 17.3 17.1 16.7 15.3 14.1

W6×20 W16× 31 W10× 22 W8× 21 W16× 26

13.3 12.4 11.4 9.77 9.59


Iy Iy In.4

W6×15 W5× 19 W14×26 W8× 18 M5× 18.9 W5× 16 W14×22 W12×22 W6× 16 W10×19

9.32 9.13 8.91 7.97 7.86 7.51 7.00 4.66 4.43 4.29

W4×13 W12×19 W10×17 W8× 15

3.86 3.76 3.56 3.41

W6×12 W10×15 W12×16 W8× 13 W12×14

2.99 2.89 2.82 2.73 2.36

W6×9 W10×12 W8× 10 M12× 11.8 M12× 10.8

2.19 2.18 2.09 1.09 0.995

M10×9

0.673

M10×8

0.597

M8×6.5

0.371

4 - 28


FACTORED UNIFORM LOAD TABLES General Notes

The Tables of Factored Uniform Loads for W and S shapes and channels (C and MC) used as simple laterally supported beams give the maximum uniformly distributed factored loads in kips. The tables are based on the flexural design strengths specified in Section F1 of the LRFD Specification. Separate tables are presented for Fy = 36 ksi and Fy = 50 ksi. The tabulated loads include the weight of the beam, which should be deducted in the calculation to determine the net load that the beam will support. The tables are also applicable to laterally supported simple beams for concentrated loading conditions. A method to determine the beam load capacity for several cases is shown in this discussion. It is assumed, in all cases, that the loads are applied normal to the X-X axis (shown in the Tables of Properties of Shapes in Part 1 of this LRFD Manual) and that the beam deflects vertically in the plane of bending. If the conditions of loading involve forces outside this plane, design strengths must be determined from the general theory of flexure and torsion. Lateral Support of Beams

The flexural design strength of a beam is dependent upon lateral support of its compression flange in addition to its section properties. In these tables the notation Lp is used to denote the maximum unbraced length of the compression flange, in feet, for the uniform moment case (Cb = 1.0) and for which the design strengths for compact symmetrical shapes are calculated with a flexural design strength of: φbMn = φbMp = φbZxFy / 12 Noncompact shapes are calculated with a flexural design strength of:  λ − λp  φbMn′ = φbMp − φb(Mp − Mr)    λr − λp  as permitted in the LRFD Specification Appendix F1. The associated maximum unbraced length for φbMn′ is tabulated as Lp. The notation Lr is the unbraced length of the compression flange for which the flexural design strength for rolled shapes is: φbMr = φbSx(Fy − 10) / 12 These tables are not applicable for beams with unbraced lengths greater than Lr. For such cases, the beam charts should be used. Flexural Design Strength and Tabulated Factored Uniform Loads

For symmetrical rolled shapes designated W and S the flexural design strengths and resultant loads are based on the assumption that the compression flanges of the beams are laterally supported at intervals not greater than Lp. The Uniform Load Constant φbWc is obtained from the moment and stress relationship of a simply supported, uniformly loaded beam. The relationship results in the formula: φbWc = φb(2ZxFy / 3), kip-ft for compact shapes AMERICAN INSTITUTE OF STEEL CONSTRUCTION

FACTORED UNIFORM LOAD TABLES

4 - 29

The following expression may be used for calculating the tabulated uniformly distributed factored load Wu on a simply supported beam or girder: Wu = φbWc / L, kips For compact shapes, the tabulated constant is based on the yield stress Fy = 36 ksi or 50 ksi and the plastic section modulus Zx. (See Section F1.1 of the LRFD Specification.) For noncompact sections, the tabulated constant is based on the nominal resisting moment as determined by Equation A-F1-3. (See LRFD Specification Appendix F1.) Shear

For relatively short spans, the design strengths for beams and channels may be limited by the shear strength of the web instead of the bending strength. This limit is indicated in the tables by solid horizontal lines. Loads shown above these lines will produce the design shear strength in the beam web. End and Interior Bearing

For a discussion of end and interior bearing and use of the tabulated values φR1 through φr R6 and φR, see Part 9 in Volume II of this LRFD Manual. Vertical Deflection

For rolled shapes designated W, M, S, C, and MC, the maximum vertical deflection may be calculated using the formula: ∆ = ML2 / (C1Ix) where M = maximum service load moment, kip-ft L = span length, ft Ix = moment of inertia, in.4 C1 = loading constant (see Figure 4-2) ∆ = maximum vertical deflection, in.

W

P

C1 = 161

P

C1 = 201

P

P

P

C1 = 158

C1= 170


P

4 - 30


Table 4-2. Recommended Span/Depth Ratios Service Load Ratios

Maximum Span/Depth Ratios

Dead / Total

Dead / Live

Fy = 36 ksi

Fy = 50 ksi

0.2 0.3 0.4 0.5 0.6

0.25 0.43 0.67 1.00 1.50

20.0 22.2 25.0 29.0 —

14.0 16.0 18.0 21.0 26.0

Deflection can be controlled by limiting the span-depth ratio of a simply supported, uniformly loaded beam as shown in Table 4-2. A live-load deflection limit of L / 360 is assumed; i.e.,

∆LL ≤

Span Length 360

For large span/depth ratios, vibration may also be a consideration. Use of Tables

Maximum factored uniform loads are tabulated for steels of Fy = 36 ksi and Fy = 50 ksi. They are based on the design flexural strength determined from the LRFD Specification: Equation F1-1 (in Section F1.1) for compact members, and Equation A-F1-3 (in Appendix F1) for noncompact members. The beams must be braced adequately and have an axis of symmetry in the plane of loading. Factored loads may be read directly from the tables when the distance between points of lateral support of the compression flange Lb does not exceed Lp (tabulated earlier in the Load Factor Design Selection Table for beams). Loads above the heavy horizontal lines in the tables are governed by the design shear strength, determined from Section F2 of the LRFD Specification. EXAMPLE 4-4

Given:

A W16×45 floor beam of Fy = 50 ksi steel spans 20 feet. Determine the maximum uniform load, end reaction, and total service load deflection. The live load equals the dead load.

Solution:

Based on Section A4 of the LRFD Specification, the governing load combination for a floor beam is 1.2 (dead load) + 1.6 (live load). As the two loads are equal, factored load = 1.4 (total load) Enter the Factored Uniform Loads Table for Fy = 50 ksi and note that: Maximum factored uniform load = Wu = 124 kips, or 124/20 = 6.2 kips/ft Factored end reaction = Wu / 2 = 124 / 2 = 61.8 kips AMERICAN INSTITUTE OF STEEL CONSTRUCTION


4 - 31

Service load moment =

124(20) Wu L = 8(1.4) 8(LF)

= 221 kip-ft Deflection:

∆=

ML2 221(20)2 = = 0.94 in. C1Ix 161(586)

Live load deflection = 0.5 × 0.94 in. = 0.47 in. < (L / 360 = 20 × 12 / 360 = 0.66 in.) o.k. EXAMPLE 4-5

Given:

A W10×45 beam of Fy = 50 ksi steel spans 6 feet. Determine the maximum load and corresponding end reaction.

Solution:

Enter the Factored Uniform Loads Table for Fy = 50 ksi and note that: Maximum factored uniform load = Wu = 191 kips, or 191/6 = 31.8 kips/ft As Wu appears above the horizontal line, it is limited by shear in the web. Factored end reaction = Wu / 2 = 191 / 2 = 96 kips

EXAMPLE 4-6

Given:

Using Fy = 50 ksi steel, select an 18-in. deep beam to span 30 feet and support two equal concentrated loads at the one-third and two-thirds points of the span. The service load intensities are 10 kips dead load and 24 kips live load. The beam is supported laterally at the points of load application and the ends. Determine the beam size and service live load deflection.

Solution:

Refer to the Table of Concentrated Load Equivalents on page 4-189 and note that: Equivalent uniform load = 2.67Pu 1. Required factored uniform load: Wu = 2.67Pu = 2.67[1.2(10) + 1.6(24)] = 2.67(50.4) = 135 kips 2. Enter the Factored Uniform Loads Table for Fy = 50 ksi and Wu ≥ 135 kips For W18×71: Wu = 145 kips > 135 kips; however, Lb = 10 ft > Lp = 6.0 ft. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 32


For W18×76: Wu = 163 kips > 135 kips; however, Lb = 10 ft > Lp = 9.2 ft. 3. Since Lp < Lb < Lr, use the Load Factor Design Selection Table. φbMn = Cb[φbMp − BF(Lb − Lp)] For the central third of the span (uniform moment), Cb = 1.0. Required flexural strength: Mu = Pu (L / 3) = 50.4(30 / 3) = 504 kip-ft 4. Try W18×71: φbMn = 1.0[544 − 13.8(10 − 6)] = 489 kip-ft < 504 kip-ft req’d. n.g. 5. Try W18×76: φbMn = 1.0[611 − 11.1(10 − 9.2)] = 602 kip-ft > 504 kip-ft req’d. o.k. Use W18×76 6. Determine service live load deflection: MLL = (PLL / Pu )Mu = (24 / 50.4)504 = 240 kip-ft Maximum ∆ (at midspan) =

240(30)2 MLLL2 = = 1.03 in. C1Ix 158(1,330)

EXAMPLE 4-7

Given:

A W24×55 of 50 ksi steel spans 20 feet and is braced at 4-ft intervals. Determine the maximum factored load and end reaction.

Solution:

1. Enter the Factored Uniform Load Table for Fy = 50 ksi and note that: Maximum factored uniform load = Wu = 201 kips, or 201 / 20 = 10.1 kips/ft This is true for Lb ≤ Lp : 4.0 ft < 4.7 ft o.k. 2. End reaction = R = Wu / 2 = 201 / 2 = 101 kips



4 - 33

Reference Notes on Tables

1. Maximum factored uniform loads, in kips, are given for beams with adequate lateral support; i.e., Lb ≤ Lp for Cb = 1.0, Lb ≤ Lm for Cb > 1.0. 2. Loads below the heavy horizontal line are limited by design flexural strength, while loads above the line are limited by design shear strength. 3. Factored loads are given for span lengths up to the smaller of L / d = 30 or 72 ft. 4. The end bearing values at the bottom of the tables are for use in solving LRFD Specification Equations K1-3, K1-5a, and K1-5b. They are defined as follows: φR1 = φ(2.5kFy tw) φR2 = φ(Fy tw)

kips kips/in.

Equation K1-3 becomes φRn = φR1 + N(φR2)  Fty tf  φrR3 = φr 68t2w √  w

kips 1.5

  3  tw  φrR4 = φr 68t2w      d  tf  



√Ft t  y f

w



kips/in.

Equation K1-5a becomes φrRn = φrR3 + N(φrR4) 1.5

   tw  φrR5 = φr 68t2w 1 − 0.2     tf  

  

1.5

  4   tw  φrR6 = φr 68t2w      d   tf  



Fty tf  √ w





√Fty t  f

w



kips

kips/in.

Equation K1-5b becomes φrRn = φrR5 + N(φrR6) where φ = 1.00, φr = 0.75, N = length of bearing (in.), and the other terms as defined in the LRFD Specification, Section K1. φR (N = 31⁄4) is defined as the design bearing strength for N = 31⁄4-in. For N / d ≤ 0.2, φR is the minimum of φR1 + N(φR2) φrR3 + N(φrR4) For N / d > 0.2, φR is the minimum of φR1 + N(φR2) φrR5 + N(φrR6) For a complete explanation of end and interior bearing and use of the tabulated values, see Part 9 in Volume II of this LRFD Manual. 5. The other terms at the bottom of the tables are: Zx

= plastic section modulus for major axis bending, in.3 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 34


φvVn = design shear strength, kips φbWc = uniform load constant = φb(2ZxFy / 3) kip-ft for compact shapes; per Equation A-F1-3 (LRFD Specification Appendix F1) for noncompact shapes 6. Tabulated maximum factored uniformly distributed load for the given beam and span is the minimum of φbWc and 2φvVn L See also Note 2 above.



Fy = 36 ksi

4 - 35

BEAMS W Shapes Maximum factored uniform loads in kips for beams laterally supported

W 44

For beams laterally unsupported, see page 4-113 Designation

W 44

Span (ft)

Fy = 36 ksi

Wt./ft

335

290

262

230

20 21 22 23 24 25

1750 1670 1590 1520 1460 1400

1480 1460 1390 1330 1280 1230

1330 1310 1250 1190 1140 1100

1180 1130 1080 1030 990 950

26 27 28 29 30 31

1350 1300 1250 1210 1170 1130

1180 1140 1100 1060 1020 989

1060 1020 980 946 914 885

914 880 849 819 792 766

32 33 34 35 36

1090 1060 1030 1000 972

959 929 902 876 852

857 831 807 784 762

743 720 699 679 660

38 40 42 44 46 48

921 875 833 795 761 729

807 767 730 697 667 639

722 686 653 623 596 572

625 594 566 540 517 495

50 52 54 56 58 60

700 673 648 625 603 583

613 590 568 548 529 511

549 528 508 490 473 457

475 457 440 424 410 396

62 64 66 68 70 72

564 547 530 515 500 486

495 479 465 451 438 426

442 429 416 403 392 381

383 371 360 349 339 330

1270 27400 665 156 28.4 256 7.36 235 9.81 248

1100 23800 592 128 25.6 202 6.28 184 8.37 211

Properties and Reaction Values Zx in.3 φbWc kip-ft φvVn kips φR1 kips φR2 kips/in. φr R3 kips φr R4 kips/in. φr R5 kips φr R6 kips/in. φR (N = 31⁄4) kips

1620 35000 873 235 36.7 419 12.5 383 16.7 355

1420 30700 738 186 31.3 312 8.77 287 11.7 288

Load above heavy line is limited by design shear strength.


4 - 36


W 40


Fy = 36 ksi

For beams laterally unsupported, see page 4-113 Designation Wt./ft

W 40 431

372

321

297

277

249

199

174

977 937

965 965 908 858 813 772

985 945 904 867 832 800

893 852 815 781 750 721

735 702 671 644 618 594

896 864 834 806 780 756

770 743 717 693 671 650

694 670 647 625 605 586

572 552 533 515 498 483

818 794 771 750 730 711

733 712 691 672 654 637

630 612 594 578 562 547

568 551 536 521 507 493

468 454 441 429 417 406

718 684 653 625 599 575

675 643 614 587 563 540

605 576 550 526 504 484

520 495 473 452 433 416

469 446 426 408 391 375

386 368 351 336 322 309

590 568 548 529 511 495

552 532 513 495 479 463

519 500 482 466 450 435

465 448 432 417 403 390

400 385 371 359 347 335

361 347 335 323 312 302

297 286 276 266 257 249

479 465 451 438 426

449 435 422 410 399

422 409 397 386 375

378 367 356 346 336

325 315 306 297 289

293 284 276 268 260

241 234 227 221 215

1120 24200 574 190 27.0 237 6.93 219 9.23 259

963 20800 493 154 23.4 177 5.30 163 7.07 194

868 18700 489 143 23.4 165 6.12 150 8.16 185

715 15400 483 117 23.4 146 7.95 126 10.6 172

1830 1800

1560 1530

1440 1440

21 22 23 24 25 26

2010 1910 1830 1760 1680 1620

1720 1640 1570 1500 1440 1390

1460 1390 1330 1280 1230 1180

1370 1310 1250 1200 1150 1100

1280 1230 1170 1130 1080 1040

1150 1100 1050 1010 968 930

27 28 29 30 31 32

1560 1500 1450 1400 1360 1320

1340 1290 1240 1200 1160 1130

1140 1100 1060 1020 989 959

1060 1030 991 958 927 898

1000 964 931 900 871 844

33 34 35 36 37 38

1280 1240 1200 1170 1140 1110

1090 1060 1030 1000 975 949

929 902 876 852 829 807

871 845 821 798 776 756

40 42 44 46 48 50

1050 1000 957 916 878 842

902 859 820 784 752 721

767 730 697 667 639 613

52 54 56 58 60 62

810 780 752 726 702 679

694 668 644 622 601 582

64 66 68 70 72

658 638 619 602 585

564 547 530 515 501

Zx in.3 φbWc kip-ft φvVn kips φR1 kips φR2 kips/in. φr R3 kips φr R4 kips/in. φr R5 kips φr R6 kips/in. φR (N = 31⁄4) kips

1950 42100 1075 430 48.2 729 22.7 667 30.2 586

1670 36100 916 339 41.8 547 17.2 501 22.9 475

Span (ft)

2150 2110

Fy = 36 ksi

15 16 17 18 19 20

215

Properties and Reaction Values 1420 30700 779 264 36.0 407 12.9 373 17.3 381

1330 28700 720 256 33.5 353 11.2 323 15.0 365

1250 27000 640 224 29.9 290 8.40 268 11.2 318




Fy = 36 ksi

4 - 37


W 40

For beams laterally unsupported, see page 4-113

Span (ft)

Fy = 36 ksi

Designation Wt./ft

W 40 331

278

264

235

211

167

149

985 937

975 934 879 830

936 921 860 806 759 716

1030 977 931 889 850 815

888 843 803 767 733 703

787 747 712 679 650 623

679 645 614 586 561 537

873 839 808 779 752 727

782 752 724 698 674 652

675 649 625 602 582 562

598 575 554 534 515 498

516 496 478 461 445 430

787 763 740 718 697 678

704 682 661 642 623 606

631 611 592 575 559 543

544 527 511 496 482 469

482 467 453 440 427 415

416 403 391 379 368 358

695 676 643 612 584 559

660 642 610 581 555 531

590 574 545 519 496 474

528 514 489 465 444 425

456 444 422 402 383 367

404 393 374 356 340 325

349 339 322 307 293 280

644 618 594 572 552 533

536 514 494 476 459 443

509 488 469 452 436 421

455 436 420 404 390 376

407 391 376 362 349 337

351 337 324 312 301 291

311 299 287 277 267 258

269 258 248 239 230 222

60 62 64 66 68 70

515 498 483 468 454 441

428 415 402 389 378 367

407 394 381 370 359 349

364 352 341 331 321 312

326 315 305 296 287 279

281 272 264 256 248 241

249 241 234 226 220 214

215 208 201 195 190 184

72

429

357

339

303

272

234

208

179

781 16900 493 154 23.4 177 5.30 163 7.07 194

692 14900 488 143 23.4 162 6.37 146 8.50 183

597 12900 468 128 22.7 139 7.24 121 9.65 163

13 14 15 16 17 18

1930 1930 1820 1720

1590 1510 1430

1490 1440 1360

1280 1210

1150 1090

19 20 21 22 23 24

1630 1540 1470 1400 1340 1290

1350 1290 1220 1170 1120 1070

1280 1220 1160 1110 1060 1020

1150 1090 1040 992 949 909

25 26 27 28 29 30

1240 1190 1140 1100 1070 1030

1030 989 952 918 886 857

976 939 904 872 842 814

31 32 33 34 35 36

996 965 936 908 883 858

829 803 779 756 734 714

37 38 40 42 44 46

835 813 772 735 702 671

48 50 52 54 56 58

183


1430 30900 967 364 43.9 602 19.2 550 25.6 506

1190 25700 796 275 36.7 424 13.4 388 17.9 395

1130 24400 746 254 34.6 379 11.7 347 15.6 366

1010 21800 640 205 29.9 290 8.40 268 11.2 303

905 19500 574 177 27.0 236 6.95 218 9.27 259



4 - 38


W 36


Fy = 36 ksi


W 36

Span (ft)

Fy = 36 ksi

Wt./ft

300

280

260

245

230

1120 1090 1040 992 949 909

1060 1020 970 926 886 849

19 20 21 22 23 24

1350 1300 1240 1180 1130

1260 1200 1150 1100 1050

1180 1170 1110 1060 1010 972

25 26 27 28 29 30

1090 1050 1010 972 938 907

1010 972 936 903 871 842

933 897 864 833 804 778

873 839 808 779 752 727

815 783 754 727 702 679

31 32 33 34 35 36

878 851 825 800 778 756

815 790 766 743 722 702

753 729 707 686 667 648

704 682 661 642 623 606

657 637 617 599 582 566

37 38 39 40 41 42

736 716 698 680 664 648

683 665 648 632 616 602

630 614 598 583 569 555

590 574 559 545 532 519

551 536 522 509 497 485

43 44 46 48 50 52

633 619 592 567 544 523

588 574 549 527 505 486

543 530 507 486 467 449

507 496 474 455 436 420

474 463 443 424 407 392

54 56 58 60 62 64

504 486 469 454 439 425

468 451 436 421 408 395

432 417 402 389 376 365

404 390 376 364 352 341

377 364 351 339 329 318

66 68 70 72

412 400 389 378

383 372 361 351

353 343 333 324

331 321 312 303

309 300 291 283

1010 21800 561 180 28.8 254 9.65 231 12.9 274

943 20400 530 162 27.4 228 8.91 206 11.9 251


1260 27200 675 239 34.0 364 12.6 334 16.7 350

1170 25300 628 214 31.9 319 11.1 292 14.8 318

1080 23300 592 194 30.2 283 10.4 258 13.9 292




Fy = 36 ksi

4 - 39


W 36


W 36 256

19 20 21 22 23 24

194

182

170

160

150

135

1260 1190 1120

1180 1120 1060 1000

1090 1040 975 920

1020 969 912 862

956 902 849 802

910 899 842 793 749

871 837 784 738 697

1180 1120 1070 1020 977 936

1060 1010 963 919 879 842

947 900 857 818 782 750

872 828 789 753 720 690

816 775 739 705 674 646

759 721 687 656 627 601

709 674 642 613 586 562

661 627 598 570 546 523

579 550 524 500 478 458

25 26 27 28 29 30

899 864 832 802 775 749

809 778 749 722 697 674

720 692 666 643 620 600

663 637 614 592 571 552

620 596 574 554 535 517

577 555 534 515 498 481

539 518 499 481 465 449

502 483 465 448 433 418

440 423 407 393 379 366

31 32 33 34 35 36

725 702 681 661 642 624

652 632 613 595 578 562

580 562 545 529 514 500

534 518 502 487 473 460

500 485 470 456 443 431

465 451 437 424 412 401

435 421 408 396 385 374

405 392 380 369 359 349

355 344 333 323 314 305

38 40 42 44 46 48

591 562 535 511 488 468

532 505 481 459 440 421

473 450 428 409 391 375

436 414 394 377 360 345

408 388 369 352 337 323

380 361 344 328 314 301

355 337 321 306 293 281

330 314 299 285 273 261

289 275 262 250 239 229

50 52 54 56 58 60

449 432 416 401 387 374

404 389 374 361 349 337

360 346 333 321 310 300

331 319 307 296 286 276

310 298 287 277 267 258

289 277 267 258 249 240

270 259 250 241 232 225

251 241 232 224 216 209

220 211 204 196 190 183

62 64 66 68 70 72

362 351 340 330 321 312

326 316 306 297 289 281

290 281 273 265 257 250

267 259 251 244 237 230

250 242 235 228 222 215

233 225 219 212 206 200

217 211 204 198 193 187

202 196 190 185 179 174

177 172 167 162 157 153


1040 22500 699 227 34.6 379 12.5 347 16.7 339

936 20200 628 196 31.3 311 10.4 285 13.8 298

624 13500 455 113 23.4 162 6.86 145 9.15 184

581 12500 436 105 22.5 147 6.65 131 8.87 168

509 11000 415 91.1 21.6 126 7.06 110 9.41 149

Span (ft)

1400 1320 1250

210

829 785 733 687 647 611

Fy = 36 ksi

13 14 15 16 17 18

232


767 16600 543 151 27.5 230 8.94 208 11.9 240

718 15500 512 139 26.1 205 8.16 185 10.9 223

668 14400 478 122 24.5 180 7.25 162 9.67 202



4 - 40



W 33

Fy = 36 ksi


Span (ft)

Fy = 36 ksi

Wt./ft

W 33 241

221

W 33 201

169

152

141

130

118

755 710 671 635 604 575

694 653 617 584 555 529

630 593 560 531 504 480

560 527 498 472 448 427

16 17 18 19 20 21

1100 1070 1010 966

1020 972 923 879

936 926 878 834 794

849 799 755 715 679 647

22 23 24 25 26 27

922 882 845 811 780 751

839 803 770 739 710 684

758 725 695 667 641 618

618 591 566 543 523 503

549 525 503 483 464 447

505 483 463 444 427 411

459 439 420 403 388 374

407 390 374 359 345 332

28 29 30 31 32 33

724 699 676 654 634 615

660 637 616 596 577 560

596 575 556 538 521 505

485 468 453 438 425 412

431 416 402 389 377 366

397 383 370 358 347 336

360 348 336 325 315 306

320 309 299 289 280 272

34 35 36 37 38 40

597 579 563 548 534 507

543 528 513 499 486 462

490 476 463 451 439 417

400 388 377 367 358 340

355 345 335 326 318 302

327 317 308 300 292 278

297 288 280 273 265 252

264 256 249 242 236 224

42 44 46 48 50 52

483 461 441 423 406 390

440 420 401 385 369 355

397 379 363 347 334 321

323 309 295 283 272 261

287 274 262 252 241 232

264 252 241 231 222 214

240 229 219 210 202 194

213 204 195 187 179 172

54 56 58 60 62 64

376 362 350 338 327 317

342 330 318 308 298 289

309 298 288 278 269 261

252 243 234 226 219 212

224 216 208 201 195 189

206 198 191 185 179 173

187 180 174 168 163 158

166 160 155 149 145 140

66 68 70 72

307 298 290 282

280 272 264 257

253 245 238 232

206 200 194 189

183 178 172 168

168 163 159 154

153 148 144 140

136 132 128 125

514 11100 392 95.3 21.8 141 6.36 127 8.48 162

467 10100 373 88.1 20.9 125 6.33 111 8.44 146

415 8960 351 77.3 19.8 107 6.28 93.6 8.37 128


939 20300 552 163 29.9 274 11.0 249 14.6 261

855 18500 511 144 27.9 236 9.88 213 13.2 235

772 16700 468 125 25.7 198 8.66 179 11.6 208

629 13600 440 124 24.1 185 6.69 170 8.92 203

559 12100 413 107 22.9 159 6.65 144 8.87 181




Fy = 36 ksi

4 - 41


W 30


W 30

Span (ft)

Fy = 36 ksi

Wt./ft

261

235

191

173

932 899 851 809 770

847 808 765 727 692

775 769 726 688 653 622

830 794 761 730 702 676

735 703 674 647 622 599

661 632 606 581 559 538

594 568 545 523 503 484

726 701 678 656 635 616

652 629 608 589 570 553

578 558 539 522 506 490

519 501 485 469 454 441

467 451 436 422 408 396

34 36 38 40 42 44

598 565 535 508 484 462

537 507 480 456 435 415

476 449 426 404 385 368

428 404 383 363 346 330

384 363 344 327 311 297

46 48 50 52 54 56

442 423 407 391 376 363

397 380 365 351 338 326

352 337 324 311 300 289

316 303 291 280 269 260

284 272 261 251 242 233

58 60 62 64 66 68

350 339 328 318 308 299

315 304 294 285 277 268

279 270 261 253 245 238

251 242 234 227 220 214

225 218 211 204 198 192

70 72

290 282

261 254

231 225

208 202

187 182

673 14500 423 124 25.6 199 9.04 181 12.0 207

605 13100 388 111 23.6 167 7.96 151 10.6 187

16 17 18 19 20 21

1140 1130 1070 1020 968

1010 961 913 869

22 23 24 25 26 27

924 884 847 813 782 753

28 29 30 31 32 33

211


941 20300 571 204 33.5 353 14.2 323 18.9 313

845 18300 505 168 29.9 283 11.2 260 14.9 265

749 16200 466 148 27.9 239 10.5 218 14.0 239



4 - 42



W 30

Fy = 36 ksi


Span (ft)

Fy = 36 ksi

Wt./ft

W 30 148

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72

775 771 720 675 635 600 568 540 514 491 470 450 432 415 400 386 372 360 348 338 327 318 300 284 270 257 245 235 225 216 208 200 193 186 180 174 169 164 159 154 150

132

725 674 629 590 555 524 497 472 449 429 410 393 378 363 350 337 325 315 304 295 286 278 262 248 236 225 215 205 197 189 182 175 169 163 157 152 147 143 139 135 131

124 686 678 629 588 551 518 490 464 441 420 401 383 367 353 339 326 315 304 294 284 275 267 259 245 232 220 210 200 192 184 176 169 163 157 152 147 142 138 134 130 126 122

116

108

99

90

659 628 583 544 510 480 454 430 408 389 371 355 340 327 314 302 292 282 272 263 255 247 240 227 215 204 194 186 177 170 163 157 151 146 141 136 132 128 124 120 117 113

632 623 575 534 498 467 440 415 393 374 356 340 325 311 299 287 277 267 258 249 241 234 226 220 208 197 187 178 170 162 156 149 144 138 133 129 125 121 117 113 110 107 104

599 562 518 481 449 421 396 374 355 337 321 306 293 281 270 259 250 241 232 225 217 211 204 198 187 177 168 160 153 147 140 135 130 125 120 116 112 109 105 102 99 96 94

540 509 470 437 408 382 360 340 322 306 291 278 266 255 245 235 226 218 211 204 197 191 185 180 170 161 153 146 139 133 127 122 118 113 109 105 102 99 96 93 90 87 85

346 7470 316 76.6 19.6 107 6.55 94.3 8.74 129

312 6740 300 67.3 18.7 93.9 6.50 81.1 8.66 115

283 6110 270 55.5 16.9 77.0 5.29 66.6 7.05 94.2


500 10800 388 117 23.4 174 6.97 160 9.29 193

437 9440 362 96.9 22.1 148 7.05 133 9.39 169

408 8810 343 88.8 21.1 132 6.55 119 8.73 153

378 8160 330 82.6 20.3 120 6.49 107 8.65 141




Fy = 36 ksi

4 - 43


W 27


Span (ft)

Fy = 36 ksi

Wt./ft

W 27 258

235

194

178

161

146

917 900 850 805 765

820 798 754 714 678

784 765 720 680 645 612

708 691 651 614 582 553

644 622 586 553 524 498

791 755 722 692 664 639

728 695 665 637 612 588

646 617 590 565 543 522

583 557 532 510 490 471

527 503 481 461 442 425

474 453 433 415 398 383

680 656 633 612 592 574

615 593 573 554 536 519

566 546 527 510 493 478

502 484 468 452 438 424

454 437 422 408 395 383

410 395 381 369 357 346

369 356 343 332 321 311

33 34 35 36 37 38

556 540 525 510 496 483

503 489 475 461 449 437

463 450 437 425 413 402

411 399 388 377 367 357

371 360 350 340 331 322

335 325 316 307 299 291

302 293 285 277 269 262

40 42 44 46 48 50

459 437 417 399 383 367

415 395 378 361 346 332

382 364 348 332 319 306

339 323 308 295 283 271

306 292 278 266 255 245

276 263 251 240 230 221

249 237 226 216 207 199

52 54 56 58 60 62

353 340 328 317 306 296

319 308 297 286 277 268

294 283 273 264 255 247

261 251 242 234 226 219

236 227 219 211 204 198

213 205 197 191 184 178

191 184 178 172 166 161

64 66

287 278

260 252

239 232

212 206

191 186

173 168

156 151

567 12200 392 122 26.1 206 10.6 186 14.1 207

512 11100 354 108 23.8 171 8.86 154 11.8 185

461 9960 322 91.9 21.8 142 7.62 128 10.2 163

15 16 17 18 19 20

1100 1080 1020 966 918

1010 977 923 874 831

21 22 23 24 25 26

874 835 798 765 734 706

27 28 29 30 31 32

217


850 18400 552 221 35.3 395 16.8 362 22.5 335

769 16600 507 189 32.8 337 15.0 308 20.0 296

708 15300 459 163 29.9 283 12.3 260 16.4 261

628 13600 410 139 27.0 230 10.3 211 13.7 227



4 - 44


W 27


Fy = 36 ksi


W 27

Span (ft)

Fy = 36 ksi

Wt./ft

129

102

94

84

605 570 529 494 463

542 507 471 439 412

513 500 462 429 400 375

478 439 405 376 351 329

502 474 449 427 406 388

436 412 390 370 353 337

388 366 347 329 314 299

353 334 316 300 286 273

310 293 277 264 251 240

23 24 25 26 27 28

371 356 341 328 316 305

322 309 296 285 274 265

286 275 264 253 244 235

261 250 240 231 222 214

229 220 211 203 195 188

29 30 31 32 33 34

294 284 275 267 259 251

255 247 239 232 225 218

227 220 213 206 200 194

207 200 194 188 182 177

182 176 170 165 160 155

36 38 40 42 44 46

237 225 213 203 194 185

206 195 185 176 168 161

183 173 165 157 150 143

167 158 150 143 136 131

146 139 132 125 120 115

48 50 52 54 56 58

178 171 164 158 152 147

154 148 142 137 132 128

137 132 127 122 118 114

125 120 115 111 107 104

110 105 101 98 94 91

60 62 64 66

142 138 133 129

123 119 116 112

110 106 103 100

100 97 94 91

88 85 82 80

278 6000 256 63.4 17.6 90.6 5.39 80.9 7.18 108

244 5270 239 56.9 16.6 76.4 5.23 67.1 6.97 93.4

11 12 13 14 15 16

655 609 569 533

17 18 19 20 21 22

114


395 8530 328 99.5 22.0 153 6.86 140 9.14 171

343 7410 302 83.4 20.5 127 6.70 115 8.93 149

305 6590 271 72.4 18.5 103 5.58 93.0 7.44 121




Fy = 36 ksi

4 - 45


W 24


Span (ft)

Fy = 36 ksi

Wt./ft

W 24 229

207

192

176

146

131

117

104

685 674 632 595 562

625 602 564 531 502

576 571 533 500 470 444

519 505 471 441 415 392

468 446 416 390 367 347

581 552 526 502 480 460

532 505 481 459 440 421

475 451 430 410 393 376

421 400 381 363 347 333

372 353 336 321 307 294

329 312 297 284 271 260

483 464 447 431 416 402

442 425 409 394 381 368

404 389 374 361 349 337

361 347 334 322 311 301

320 307 296 285 276 266

283 272 262 252 244 235

250 240 231 223 215 208

422 409 397 385 374 364

389 377 366 355 345 335

356 345 334 325 315 307

326 316 306 297 289 281

291 282 274 266 258 251

258 250 242 235 228 222

228 221 214 208 202 196

201 195 189 184 178 173

384 365 348 332 317 304

344 327 312 297 285 273

318 302 287 274 262 252

290 276 263 251 240 230

266 253 241 230 220 211

238 226 215 205 196 188

210 200 190 182 174 167

186 177 168 161 154 147

164 156 149 142 136 130

292 281 270 261 252 243

262 252 242 234 226 218

241 232 224 216 208 201

221 212 204 197 190 184

202 194 187 181 174 168

181 174 167 161 156 150

160 154 148 143 138 133

141 136 131 126 122 118

125 120 116 111 108 104

676 14600 486 216 34.6 379 18.0 347 24.1 328

606 13100 435 186 31.3 311 15.0 285 20.0 288

370 7990 288 95.3 21.8 141 8.65 127 11.5 166

327 7060 259 80.4 19.8 115 7.41 103 9.88 139

289 6240 234 67.5 18.0 93.7 6.36 83.5 8.48 114

13 14 15 16 17 18

971 913 859 811

870 818 770 727

802 755 710 671

736 736 690 649 613

19 20 21 22 23 24

769 730 695 664 635 608

689 654 623 595 569 545

635 604 575 549 525 503

25 26 27 28 29 30

584 562 541 521 504 487

524 503 485 467 451 436

31 32 33 34 35 36

471 456 442 429 417 406

38 40 42 44 46 48 50 52 54 56 58 60

162


559 12100 401 164 29.2 270 13.1 247 17.5 259

511 11000 368 143 27.0 230 11.5 211 15.3 231

468 10100 343 127 25.4 200 10.5 182 14.1 209

418 9030 313 110 23.4 167 9.35 152 12.5 186



4 - 46



W 24

Fy = 36 ksi


Span (ft)

Fy = 36 ksi

Wt./ft

W 24 103

94

84

W 24 76

68

62

55

397 367 330 300 275

362 362 322 289 263 241

7 8 9 10 11 12

525 504

487 457

440 440 403

409 393 360

383 382 348 319

13 14 15 16 17 18

465 432 403 378 356 336

422 392 366 343 323 305

372 346 323 302 285 269

332 309 288 270 254 240

294 273 255 239 225 212

254 236 220 207 194 184

223 207 193 181 170 161

19 20 21 22 23 24

318 302 288 275 263 252

289 274 261 249 239 229

255 242 230 220 210 202

227 216 206 196 188 180

201 191 182 174 166 159

174 165 157 150 144 138

152 145 138 132 126 121

25 26 27 28 29 30

242 233 224 216 209 202

219 211 203 196 189 183

194 186 179 173 167 161

173 166 160 154 149 144

153 147 142 137 132 127

132 127 122 118 114 110

116 111 107 103 100 96

31 32 33 34 35 36

195 189 183 178 173 168

177 171 166 161 157 152

156 151 147 142 138 134

139 135 131 127 123 120

123 119 116 112 109 106

107 103 100 97 94 92

93 90 88 85 83 80

38 40 42 44 46 48

159 151 144 137 131 126

144 137 131 125 119 114

127 121 115 110 105 101

114 108 103 98 94 90

101 96 91 87 83 80

87 83 79 75 72 69

76 72 69 66 63 60

50 52 54 56 58 60

121 116 112 108 104 101

110 106 102 98 95 91

97 93 90 86 83 81

86 83 80 77 74

76 74 71 68 66

66 64 61 59 57

58 56 54 52 50

177 3820 191 51.4 14.9 62.6 4.73 55.1 6.30 77.9

153 3300 198 53.2 15.5 66.3 5.21 58.0 6.95 83.2

134 2890 181 46.7 14.2 54.0 4.75 46.5 6.34 69.4


280 6050 262 86.6 19.8 124 6.35 113 8.47 144

254 5490 243 75.3 18.5 106 5.89 96.2 7.86 125

224 4840 220 66.1 16.9 86.5 5.14 78.3 6.85 103

200 4320 205 56.9 15.8 73.6 4.81 66.0 6.41 89.3




Fy = 36 ksi

4 - 47


W 21


Span (ft)

Fy = 36 ksi

Wt./ft

W 21 201

182

166

147

132

122

111

101

552 514 480 450 423 400

506 474 442 414 390 368

460 430 402 377 354 335

415 390 364 342 321 304

13 14 15 16 17 18

815 763 716 673 636

733 685 643 605 571

656 622 583 549 518

618 575 537 504 474 448

19 20 21 22 23 24

603 572 545 520 498 477

541 514 490 467 447 428

491 467 444 424 406 389

424 403 384 366 350 336

379 360 343 327 313 300

349 332 316 301 288 276

317 301 287 274 262 251

288 273 260 248 238 228

25 26 27 28 29 30

458 440 424 409 395 382

411 395 381 367 355 343

373 359 346 333 322 311

322 310 298 288 278 269

288 277 266 257 248 240

265 255 246 237 229 221

241 232 223 215 208 201

219 210 202 195 188 182

31 32 33 34 35 36

369 358 347 337 327 318

332 321 312 302 294 286

301 292 283 274 267 259

260 252 244 237 230 224

232 225 218 212 206 200

214 207 201 195 189 184

194 188 183 177 172 167

176 171 166 161 156 152

38 40 42 44 46 48

301 286 273 260 249 239

271 257 245 234 224 214

246 233 222 212 203 194

212 201 192 183 175 168

189 180 171 163 156 150

175 166 158 151 144 138

159 151 143 137 131 126

144 137 130 124 119 114

50 52

229 220

206 198

187 179

161 155

144 138

133 128

121 116

109 105

307 6630 253 91.1 21.6 139 9.53 126 12.7 161

279 6030 230 80.4 19.8 117 8.11 105 10.8 143

253 5460 208 70.3 18.0 96.8 6.72 87.2 8.95 119


530 11400 407 195 32.8 339 18.4 311 24.6 301

476 10300 367 168 29.9 281 15.6 258 20.8 265

432 9330 328 143 27.0 232 12.7 213 16.9 231

373 8060 309 122 25.9 200 13.5 181 18.0 206

333 7190 276 106 23.4 163 11.2 147 14.9 182



4 - 48



W 21

Fy = 36 ksi


Span (ft)

Fy = 36 ksi

Wt./ft

W 21 93

83

73

W 21 68

62

57

50

44 281 258 229 206 187 172

7 8 9 10 11 12

488 477 434 398

429 423 385 353

376 372 338 310

353 346 314 288

326 311 283 259

332 310 279 253 232

308 297 264 238 216 198

13 14 15 16 17 18

367 341 318 298 281 265

326 302 282 265 249 235

286 265 248 232 219 206

266 247 230 216 203 192

239 222 207 194 183 173

214 199 186 174 164 155

183 170 158 149 140 132

159 147 137 129 121 114

19 20 21 22 23 24

251 239 227 217 208 199

223 212 202 192 184 176

196 186 177 169 162 155

182 173 165 157 150 144

164 156 148 141 135 130

147 139 133 127 121 116

125 119 113 108 103 99

108 103 98 94 90 86

25 26 27 28 29 30

191 184 177 170 165 159

169 163 157 151 146 141

149 143 138 133 128 124

138 133 128 123 119 115

124 120 115 111 107 104

111 107 103 100 96 93

95 91 88 85 82 79

82 79 76 74 71 69

31 32 33 34 35 36

154 149 145 140 136 133

137 132 128 125 121 118

120 116 113 109 106 103

111 108 105 102 99 96

100 97 94 91 89 86

90 87 84 82 80 77

77 74 72 70 68 66

66 64 62 61 59 57

38 40 42 44 46 48

126 119 114 108 104 99

111 106 101 96 92 88

98 93 88 84 81 77

91 86 82 79 75 72

82 78 74 71 68 65

73 70 66 63 61 58

63 59 57 54 52 50

54 52 49 47 45 43

50 52

95 92

85 81

74 71

69 66

62 60

56 54

48 46

41

129 2790 166 50.1 14.6 63.6 4.45 57.3 5.94 78.1

110 2380 154 44.9 13.7 52.4 4.52 46.2 6.03 67.1

95.4 2060 141 37.4 12.6 42.5 4.23 36.7 5.64 56.3


221 4770 244 88.1 20.9 130 8.91 118 11.9 156

196 4230 215 72.4 18.5 103 7.01 93.3 9.34 126

172 3720 188 61.4 16.4 80.8 5.50 73.0 7.34 98.7

160 3460 177 55.6 15.5 71.4 5.04 64.3 6.72 87.8

144 3110 163 49.5 14.4 60.7 4.55 54.3 6.07 75.5




Fy = 36 ksi

4 - 49


W 18


W 18 192

175

158

W 18 143

130

119

106

97

86

76

430 414 382 355 331 311

387 380 351 326 304 285

343 335 309 287 268 251

301 293 271 251 235 220

693 661 614 573 537

621 592 549 513 481

553 535 497 464 435

501 484 449 419 393

17 18 19 20 21 22

562 530 502 477 455 434

506 478 452 430 409 391

452 427 405 384 366 350

409 386 366 348 331 316

370 349 331 314 299 286

332 313 297 282 268 256

292 276 261 248 237 226

268 253 240 228 217 207

236 223 211 201 191 183

207 196 185 176 168 160

23 24 25 26 27 28

415 398 382 367 354 341

374 358 344 331 318 307

334 320 308 296 285 275

302 290 278 268 258 248

273 262 251 242 233 224

245 235 226 217 209 201

216 207 199 191 184 177

198 190 182 175 169 163

175 167 161 155 149 143

153 147 141 135 130 126

29 30 31 32 33 34

329 318 308 298 289 281

296 287 277 269 261 253

265 256 248 240 233 226

240 232 224 217 211 205

217 210 203 196 190 185

194 188 182 176 171 166

171 166 160 155 151 146

157 152 147 142 138 134

139 134 130 126 122 118

121 117 114 110 107 104

35 36 37 38 39 40

273 265 258 251 245 239

246 239 232 226 220 215

220 214 208 202 197 192

199 193 188 183 178 174

180 175 170 165 161 157

161 157 152 148 145 141

142 138 134 131 127 124

130 127 123 120 117 114

115 112 109 106 103 100

101 98 95 93 90 88

42 44

227 217

205 195

183 175

166 158

150 143

134 128

118 113

109 104

96 91

84 80


442 9550 380 211 34.6 381 22.8 350 30.4 323

398 8600 347 180 32.0 324 20.3 297 27.1 284

230 4970 215 86.3 21.2 134 10.7 121 14.3 155

211 4560 193 75.2 19.3 112 8.69 101 11.6 138

186 4020 172 62.1 17.3 89.3 7.17 80.5 9.56 113

163 3520 150 52.6 15.3 69.9 5.69 63.0 7.59 88.4

Span (ft)

760 734 682 636 597

483 470 434 403 376 352

Fy = 36 ksi

11 12 13 14 15 16


322 6960 277 131 26.3 219 13.9 201 18.5 217

291 6290 251 113 24.1 184 12.0 168 15.9 191

261 5640 242 103 23.6 167 12.8 151 17.1 180



4 - 50



W 18

Fy = 36 ksi


Span (ft)

Fy = 36 ksi

Wt./ft

W 18 71

65

6 7 8 9 10 11

355 348 313 285

321 319 287 261

12 13 14 15 16 17

261 241 224 209 196 184

18 19 20 21 22 23

60

W 18 55

50

46

40

35

294 266 242

275 269 242 220

248 242 218 198

253 245 218 196 178

219 212 188 169 154

206 205 180 160 144 131

239 221 205 192 180 169

221 204 190 177 166 156

202 186 173 161 151 142

182 168 156 145 136 128

163 151 140 131 122 115

141 130 121 113 106 100

120 110 103 96 90 84

174 165 157 149 142 136

160 151 144 137 131 125

148 140 133 127 121 116

134 127 121 115 110 105

121 115 109 104 99 95

109 103 98 93 89 85

94 89 85 81 77 74

80 76 72 68 65 62

24 25 26 27 28 29

131 125 120 116 112 108

120 115 110 106 103 99

111 106 102 98 95 92

101 97 93 90 86 83

91 87 84 81 78 75

82 78 75 73 70 68

71 68 65 63 60 58

60 57 55 53 51 50

30 31 32 33 34 35

104 101 98 95 92 89

96 93 90 87 84 82

89 86 83 81 78 76

81 78 76 73 71 69

73 70 68 66 64 62

65 63 61 59 58 56

56 55 53 51 50 48

48 46 45 44 42 41

36 38 40 42 44

87 82 78 75 71

80 76 72 68 65

74 70 66 63 60

67 64 60 58 55

61 57 55 52 50

54 52 49 47 45

47 45 42 40 38

40 38 36 34 33

90.7 1960 126 40.5 13.0 51.4 3.92 46.7 5.23 64.2

78.4 1690 110 33.7 11.3 39.2 3.05 35.6 4.07 49.1

66.5 1440 103 30.4 10.8 32.8 3.29 28.9 4.39 43.5


145 3130 178 66.8 17.8 95.9 7.44 86.7 9.92 120

133 2870 161 58.2 16.2 80.0 6.08 72.6 8.10 99.8

123 2660 147 51.4 14.9 68.2 5.18 61.9 6.90 85.0

112 2420 137 46.1 14.0 59.2 4.77 53.4 6.36 74.7

101 2180 124 39.9 12.8 48.9 4.01 44.1 5.34 61.9




Fy = 36 ksi

4 - 51


W 16


Span (ft)

Fy = 36 ksi

Wt./ft

W 16 100

89

W 16

77

6 7 8 9 10 11

386

342

292

12 13 14 15 16 17

356 329 305 285 267 252

315 291 270 252 236 222

18 19 20 21 22 23

238 225 214 204 194 186

24 25 26 27 28 29

67

57

50

45

W 16 40

36

31

26 153 136 119 106 95 87

251

275 252 227 206

240 221 199 181

216 198 178 162

190 175 157 143

182 173 154 138 126

170 167 146 130 117 106

270 249 231 216 203 191

234 216 201 187 176 165

189 174 162 151 142 133

166 153 142 132 124 117

148 137 127 119 111 105

131 121 112 105 98 93

115 106 99 92 86 81

97 90 83 78 73 69

80 73 68 64 60 56

210 199 189 180 172 164

180 171 162 154 147 141

156 148 140 134 128 122

126 119 113 108 103 99

110 105 99 95 90 86

99 94 89 85 81 77

87 83 79 75 72 68

77 73 69 66 63 60

65 61 58 56 53 51

53 50 48 45 43 42

178 171 164 158 153 147

158 151 145 140 135 130

135 130 125 120 116 112

117 112 108 104 100 97

95 91 87 84 81 78

83 79 76 74 71 69

74 71 68 66 63 61

66 63 61 58 56 54

58 55 53 51 49 48

49 47 45 43 42 40

40 38 37 35 34 33

30 31 32 33 34 35

143 138 134 130 126 122

126 122 118 115 111 108

108 105 101 98 95 93

94 91 88 85 83 80

76 73 71 69 67 65

66 64 62 60 58 57

59 57 56 54 52 51

52 51 49 48 46 45

46 45 43 42 41 39

39 38 36 35 34 33

32 31 30 29 28 27

36 38 40

119 113 107

105 99 95

90 85 81

78 74 70

63 60 57

55 52 50

49 47 44

44 41 39

38 36

32 31

27 25

198 4280 193 88.8 21.1 136 11.0 123 14.7 157

175 3780 171 73.8 18.9 109 9.06 98.8 12.1 135

72.9 1570 94.9 32.6 11.0 36.6 3.22 33.2 4.30 47.1

64.0 1380 91.0 29.9 10.6 32.2 3.46 28.5 4.61 43.5

54.0 1170 84.9 27.8 9.90 29.3 2.73 26.4 3.64 38.2

44.2 955 76.3 23.9 9.00 22.5 2.65 19.7 3.53 31.2


150 3240 146 58.9 16.4 81.9 6.89 74.3 9.18 104

130 2810 125 48.9 14.2 61.9 5.21 56.3 6.95 78.9

105 2270 137 53.2 15.5 73.0 6.21 66.2 8.28 93.2

92.0 1990 120 44.9 13.7 56.9 4.92 51.6 6.56 72.9

82.3 1780 108 38.8 12.4 46.6 4.14 42.2 5.52 60.1



4 - 52


W 14

Fy = 36 ksi

BEAMS W Shapes Maximum factored uniform loads in kips for beams laterally supported For beams laterally unsupported, see page 4-113

Designation Wt./ft

W 14 132

120

109

W 14 99

90

82

74

68

61

248 247 227 209 194 181

227 226 207 191 177 166

203 200 184 169 157 147

332 327 305

292 276

267 267 249

240 226

16 17 18 19 20 21

316 297 281 266 253 241

286 269 254 241 229 218

259 244 230 218 207 197

234 220 208 197 187 178

212 199 188 178 170 161

188 177 167 158 150 143

170 160 151 143 136 130

155 146 138 131 124 118

138 130 122 116 110 105

22 23 24 25 26 27

230 220 211 202 194 187

208 199 191 183 176 170

189 180 173 166 160 154

170 162 156 149 144 138

154 147 141 136 130 126

136 131 125 120 115 111

124 118 113 109 105 101

113 108 104 99 96 92

100 96 92 88 85 82

28 29 30 31 32 33

181 174 168 163 158 153

164 158 153 148 143 139

148 143 138 134 130 126

133 129 125 121 117 113

121 117 113 109 106 103

107 104 100 97 94 91

97 94 91 88 85 82

89 86 83 80 78 75

79 76 73 71 69 67

34

149

135

122

110

100

88

80

73

65


234 5050 184 98.0 23.2 161 16.3 145 21.8 173

212 4580 166 86.3 21.2 134 13.9 121 18.5 155

139 3000 142 74.6 18.4 103 9.95 93.6 13.3 134

126 2720 124 63.3 16.2 81.8 7.52 74.7 10.0 107

115 2480 113 56.0 14.9 69.4 6.49 63.3 8.65 91.5

102 2200 101 48.5 13.5 56.4 5.40 51.4 7.20 74.8

Span (ft)

368 361 337

284 273 250 231 214 200

Fy = 36 ksi

10 11 12 13 14 15

Properties and Reaction Values 192 4150 146 73.8 18.9 108 10.8 97.6 14.4 135

173 3740 134 62.7 17.5 91.3 9.48 82.3 12.6 119

157 3390 120 54.5 15.8 75.3 7.86 67.9 10.5 102




Fy = 36 ksi

4 - 53


W 14


Span (ft)

Fy = 36 ksi

Wt./ft

W 14 53

48

5 6 7 8 9 10

200 188

182 169

11 12 13 14 15 16

171 157 145 134 125 118

17 18 19 20 21 22

W 14 43

38

34

W 14 30

26

22

162 150

170 166 148 133

155 147 131 118

145 128 114 102

138 124 109 96 87

123 120 102 90 80 72

154 141 130 121 113 106

137 125 116 107 100 94

121 111 102 95 89 83

107 98 91 84 79 74

93 85 79 73 68 64

79 72 67 62 58 54

65 60 55 51 48 45

111 105 99 94 90 86

100 94 89 85 81 77

88 84 79 75 72 68

78 74 70 66 63 60

69 66 62 59 56 54

60 57 54 51 49 46

51 48 46 43 41 39

42 40 38 36 34 33

23 24 25 26 27 28

82 78 75 72 70 67

74 71 68 65 63 60

65 63 60 58 56 54

58 55 53 51 49 47

51 49 47 45 44 42

44 43 41 39 38 36

38 36 35 33 32 31

31 30 29 28 27 26

29 30 31 32 33 34

65 63 61 59 57 55

58 56 55 53 51 50

52 50 48 47 46 44

46 44 43 42 40 39

41 39 38 37 36 35

35 34 33 32 31 30

30 29 28 27 26 26

25 24 23 22 22 21

47.3 1020 72.6 22.8 9.72 26.6 3.39 23.5 4.52 38.2

40.2 868 69.0 21.5 9.18 25.5 2.61 23.1 3.47 34.4

33.2 717 61.4 18.1 8.28 19.5 2.43 17.3 3.24 27.8


87.1 1880 100 47.9 13.3 55.9 5.06 51.3 6.75 73.2

78.4 1690 91.1 42.1 12.2 46.8 4.40 42.8 5.86 61.8

69.6 1500 81.0 36.0 11.0 37.5 3.60 34.2 4.80 49.8

61.5 1330 85.0 29.6 11.2 37.9 3.77 34.4 5.02 50.7

54.6 1180 77.5 25.7 10.3 31.4 3.34 28.3 4.45 42.8



4 - 54


W 12


Fy = 36 ksi


W 12 120

106

96

87

W 12 79

72

65

58

53 162 153 140 129 120 112

306 295 272 253 236

272 265 244 227 212

251 238 219 204 190

226 214 198 184 171

205 194 179 167 156

184 174 161 149 139

16 17 18 19 20 21

251 236 223 211 201 191

221 208 197 186 177 169

198 187 176 167 159 151

178 168 158 150 143 136

161 151 143 135 129 122

146 137 130 123 117 111

131 123 116 110 105 100

117 110 104 98 93 89

105 99 93 89 84 80

22 23 24 25 26 27

183 175 167 161 155 149

161 154 148 142 136 131

144 138 132 127 122 118

130 124 119 114 110 106

117 112 107 103 99 95

106 101 97 93 90 86

95 91 87 84 80 77

85 81 78 75 72 69

76 73 70 67 65 62

28 29 30

143 139 134

127 122 118

113 109 106

102 98 95

92 89 86

83 80 78

75 72 70

67 64 62

60 58 56


186 4020 181 116 25.6 192 22.7 173 30.2 199

164 3540 153 92.6 22.0 145 16.3 131 21.8 164

108 2330 102 53.2 15.5 70.6 8.89 63.4 11.9 102

96.8 2090 91.9 46.1 14.0 58.0 7.43 52.0 9.90 84.1

86.4 1870 85.3 44.6 13.0 52.9 5.49 48.4 7.32 72.2

77.9 1680 80.9 38.8 12.4 47.0 5.44 42.6 7.25 66.2

Span (ft)

362 335 309 287 268

171 170 156 144 133 124

Fy = 36 ksi

10 11 12 13 14 15

Properties and Reaction Values 147 3180 136 80.4 19.8 118 13.4 107 17.8 145

132 2850 125 69.5 18.5 102 12.4 91.5 16.5 130

119 2570 113 60.8 16.9 84.5 10.5 75.9 14.0 116

Load above heavy line is limited by design shear strength. Values of R in bold face exceed maximum design web shear φvVn.



Fy = 36 ksi

4 - 55


W 12


Span (ft)

Fy = 36 ksi

Wt./ft

W 12 50

45

W 12 40

4 5 6 7 8 9

175 174

157 155

10 11 12 13 14 15

156 142 130 120 112 104

16 17 18 19 20 21

35

30

W 12 26

22

19

16

14

111 107 89 76 67 59

103 87 72 62 54 48

93 75 63 54 47 42

137

146 138 123

125 116 103

109 100 89

124 105 90 79 70

140 127 116 108 100 93

124 113 104 96 89 83

111 101 92 85 79 74

93 85 78 72 66 62

80 73 67 62 57 54

63 58 53 49 45 42

53 49 44 41 38 36

43 39 36 33 31 29

38 34 31 29 27 25

98 92 87 82 78 74

87 82 78 74 70 67

78 73 69 65 62 59

69 65 61 58 55 53

58 55 52 49 47 44

50 47 45 42 40 38

40 37 35 33 32 30

33 31 30 28 27 25

27 26 24 23 22 21

23 22 21 20 19 18

22 23 24 25 26 27

71 68 65 63 60 58

64 61 58 56 54 52

56 54 52 50 48 46

50 48 46 44 43 41

42 40 39 37 36 34

37 35 33 32 31 30

29 28 26 25 24 23

24 23 22 21 21 20

20 19 18 17 17 16

17 16 16 15 14 14

28 29 30

56 54 52

50 48 47

44 43 37

39 38 31

33 32 27

29 28 21

23 22 18

19 18

16 15

13 13

72.4 1560 87.7 45.8 13.3 55.1 5.96 50.3 7.95 76.1

64.7 1400 78.5 37.7 12.1 45.0 4.98 41.0 6.64 62.6

29.3 633 62.2 20.5 9.36 26.4 3.08 23.9 4.11 37.3

24.7 534 55.6 17.2 8.46 20.6 2.80 18.4 3.73 30.5

20.1 434 51.3 14.9 7.92 16.3 3.08 13.8 4.10 27.1

17.4 376 46.3 12.4 7.20 13.0 2.74 10.8 3.65


57.5 1240 68.5 33.2 10.6 35.2 3.83 32.1 5.11 48.7

51.2 1110 72.9 27.0 10.8 36.3 3.81 33.1 5.08 49.6

43.1 931 62.4 21.9 9.36 26.9 2.97 24.5 3.96 37.3

37.2 804 54.6 18.1 8.28 20.8 2.41 18.8 3.21 29.3



22.7

4 - 56



W 10

Fy = 36 ksi


W 10 112

100

88

77

68

60

54

49

9 10 11 12 13 14

333 318 289 265 244 227

293 281 255 234 216 201

255 244 222 203 188 174

218 211 192 176 162 151

190 184 167 154 142 132

167 161 146 134 124 115

145 144 131 120 111 103

132 130 119 109 100 93

15 16 17 18 19 20

212 198 187 176 167 159

187 176 165 156 148 140

163 153 144 136 128 122

141 132 124 117 111 105

123 115 108 102 97 92

107 101 95 90 85 81

96 90 85 80 76 72

87 82 77 72 69 65

21 22 23 24

151 144 138 132

134 128 122 117

116 111 106 102

100 96 92 88

88 84 80 77

77 73 70 67

69 65 63 60

62 59 57 54

74.6 1610 83.4 49.6 15.1 68.7 9.79 62.0 13.0 98.8

66.6 1440 72.6 41.6 13.3 54.0 7.49 49.0 9.99 81.4

60.4 1300 66.0 36.3 12.2 45.4 6.46 41.1 8.61 69.1

Span (ft)

Fy = 36 ksi

Wt./ft


147 3180 167 127 27.2 224 27.8 203 37.1 216

130 2810 147 107 24.5 182 23.2 164 31.0 187

113 2440 127 88.5 21.8 143 18.9 130 25.3 159

97.6 2110 109 71.5 19.1 110 14.8 99.7 19.8 134

85.3 1840 95.0 58.2 16.9 86.5 11.9 78.3 15.9 113




Fy = 36 ksi

4 - 57


W 10


Span (ft)

Fy = 36 ksi

Wt./ft

W 10 45

3 4 5 6 7 8

137

9 10 11 12 13 14

39

W 10 33

30

26

W 10 22

19

17

15

12 73 68 54 45 39 34

100 93 78 67 58

94 81 67 58 50

89 86 69 58 49 43

121

110 105

122 113 99

104 97 85

95 94 80 70

132 119 108 99 91 85

112 101 92 84 78 72

93 84 76 70 64 60

88 79 72 66 61 56

75 68 61 56 52 48

62 56 51 47 43 40

52 47 42 39 36 33

45 40 37 34 31 29

38 35 31 29 27 25

30 27 25 23 21 19

15 16 17 18 19 20

79 74 70 66 62 59

67 63 59 56 53 51

56 52 49 47 44 42

53 49 47 44 42 40

45 42 40 38 36 34

37 35 33 31 30 28

31 29 27 26 25 23

27 25 24 22 21 20

23 22 20 19 18 17

18 17 16 15 14 14

21 22 23 24

56 54 52 49

48 46 44 42

40 38 36 35

38 36 34 33

32 31 29 28

27 26 24 23

22 21 20 19

19 18 18 17

16 16 15 14

13 12 12 11

54.9 1190 68.7 39.4 12.6 49.9 6.29 45.7 8.38 72.9

46.8 1010 60.7 31.9 11.3 39.4 5.46 35.8 7.28 59.4

21.6 467 49.8 18.3 9.00 24.0 3.55 21.6 4.73 37.0

18.7 404 47.2 16.2 8.64 20.7 3.80 18.1 5.07 34.6

16.0 346 44.7 14.2 8.28 17.5 4.14 14.8 5.52 32.7

12.6 272 36.5 10.7 6.84 11.6 3.04 9.61 4.05 22.8


38.8 838 54.9 27.7 10.4 31.5 5.29 28.1 7.05 51.0

36.6 791 61.1 25.3 10.8 35.9 4.64 32.7 6.19 52.8

31.3 676 52.2 20.5 9.36 26.9 3.55 24.5 4.73 39.8

26.0 562 47.4 16.2 8.64 21.6 3.47 19.2 4.62 34.3



4 - 58



W8

Fy = 36 ksi


W8 67

58

48

7 8 9 10 11 12

199 190 168 152 138 126

40

35

31

174 161 144 129 117 108

132 118 106 96 88

115 107 96 86 78 72

98 94 83 75 68 62

89 82 73 66 60 55

13 14 15 16 17 18

117 108 101 95 89 84

99 92 86 81 76 72

81 76 71 66 62 59

66 61 57 54 51 48

58 54 50 47 44 42

51 47 44 41 39 36

19 20

80 76

68 65

56 53

45 43

39 37

35 33

39.8 860 57.7 34.4 13.0 49.5 9.27 44.4 12.4 76.5

34.7 750 48.9 27.9 11.2 37.2 6.80 33.5 9.07 63.0

30.4 657 44.3 24.0 10.3 30.7 6.11 27.4 8.14 53.9

Span (ft)

Fy = 36 ksi

Wt./ft


70.2 1520 99.7 73.7 20.5 127 20.2 115 26.9 140

59.8 1290 86.8 60.2 18.4 100 17.2 90.3 22.9 120

49.0 1060 66.1 42.8 14.4 64.1 10.1 58.4 13.5 89.6




Fy = 36 ksi

4 - 59


W8


W8

Span (ft)

Fy = 36 ksi

Wt./ft

28

W8 24

21

W8 18

15

13

10

71 62 49 41 35 31

52 48 38 32 27 24

3 4 5 6 7 8

89 84 73

76 72 63

80 73 63 55

73 61 52 46

77 73 59 49 42 37

9 10 11 12 13 14

65 59 53 49 45 42

56 50 46 42 39 36

49 44 40 37 34 31

41 37 33 31 28 26

33 29 27 24 23 21

27 25 22 21 19 18

21 19 17 16 15 14

15 16 17 18 19 20

39 37 35 33 31 29

33 31 29 28 26 22

29 28 26 24 23 18

24 23 22 20 19 15

20 18 17 16 15

16 15 14 14 13

13 12 11 11 10

13.6 294 38.6 16.5 8.82 20.8 5.28 18.0 7.05 40.9

11.4 246 35.7 14.2 8.28 17.0 5.48 14.1 7.31 37.9

8.87 192 26.1 9.56 6.12 9.71 2.79 8.24 3.72 20.3


27.2 588 44.7 24.0 10.3 31.7 5.67 28.7 7.56 53.3

23.2 501 37.8 19.3 8.82 23.5 4.26 21.2 5.67 39.7

20.4 441 40.2 18.3 9.00 24.2 4.33 21.8 5.77 40.6

17.0 367 36.4 15.5 8.28 19.4 4.16 17.1 5.54 35.2



4 - 60


W 6–5–4


Fy = 36 ksi


W6 25

2 3 4 5 6 7

79 68 58

8 9 10 11 12 13 14

20

W6 15*

16

W5

12

9

63 54 46

54 46 38 33

63 63 51 42 36

54 45 36 30 26

39 34 27 22 19

51 45 41 37 34 31

40 36 32 29 27 25

29 26 23 21 19 18

32 28 25 23 21 19

22 20 18 16 15 14

17 15 13 12 11 10

29

23

16

18

13

18.9 408 39.7 23.4 11.5 37.4 10.4 33.0 13.8 60.8

14.9 322 31.3 17.5 9.36 24.5 7.13 21.6 9.51 48.0

19

W4 16

13

54 50 42 36

47 41 35 30

45 45 34 27 23 19

31 28 25 23 21

26 23 21 19 17

17 15 14

11.6 251 27.0 19.7 9.72 28.2 8.16 25.4 10.9 51.3

9.59 207 23.4 16.2 8.64 21.6 7.04 19.2 9.38 44.3

6.28 136 22.6 17.3 10.1 26.6 14.0 22.7 18.7 50.1

9.6

Span (ft)

Fy = 36 ksi

Wt./ft


10.8 230 26.8 12.9 8.28 17.2 7.17 14.3 9.56 39.8

11.7 253 31.7 17.5 9.36 25.8 6.34 23.2 8.46 48.0

8.30 179 27.0 12.9 8.28 17.9 6.62 15.2 8.82 39.8

6.23 135 19.5 8.61 6.12 9.95 3.56 8.55 4.74 24.0

*Indicates noncompact shape. Load above heavy line is limited by design shear strength. Values of R in bold face exceed maximum design web shear φvVn.



Fy = 36 ksi

4 - 61

BEAMS S Shapes Maximum factored uniform loads in kips for beams laterally supported

S 24–20


S 24 121

12 13 14 15 16 17

90

S 20 80

96

S 20 86

75

66

521 494 439 395 359

494 472 413 367 330 300

393 378 336 302 275

591 548

695 648 576 518 471

583 533 480 436

467 441 401

551 508 472 441 413 389

502 464 430 402 377 354

432 399 370 346 324 305

400 369 343 320 300 282

367 339 315 294 275 259

356 329 305 285 267 252

329 304 282 264 247 233

275 254 236 220 207 194

252 233 216 202 189 178

18 19 20 21 22 23

367 348 330 315 300 287

335 317 301 287 274 262

288 273 259 247 236 225

266 252 240 228 218 208

245 232 220 210 200 192

238 225 214 204 194 186

220 208 198 188 180 172

184 174 165 157 150 144

168 159 151 144 137 131

24 25 26 27 28 29

275 264 254 245 236 228

251 241 232 223 215 208

216 207 199 192 185 179

200 192 184 178 171 165

184 176 169 163 157 152

178 171 164 158 153 147

165 158 152 146 141 136

138 132 127 122 118 114

126 121 116 112 108 104

30 32 34 36 38 40

220 207 194 184 174 165

201 188 177 167 159 151

173 162 152 144 136 130

160 150 141 133 126 120

147 138 130 122 116 110

143 134 126 119 113 107

132 124 116 110 104 99

110 103 97 92 87 83

101 95 89 84 80 76

42 44 46 48 50 52

157 150 144 138 132 127

143 137 131 126 121 116

123 118 113 108 104 100

114 109 104 100 96 92

105 100 96 92 88 85

102 97 93 89 86

94 90 86 82 79

79 75 72 69 66

72 69 66 63 60

54 56 58 60

122 118 114 110

112 108 104 100

96 93 89 86

89 86 83 80

82 79 76 73


306 6610 381 144 28.8 229 17.6 200 23.5 238

279 6030 295 112 22.3 156 8.19 143 10.9 183

183 3950 260 104 23.8 157 14.1 138 18.8 181

153 3300 247 92.9 22.9 138 14.8 118 19.7 167

140 3020 196 73.9 18.2 97.9 7.44 88.0 9.91 122

Span (ft)

762 734 661 601

100

631 611 535 475 428 389

Fy = 36 ksi

6 7 8 9 10 11

S 24 106


222 4800 292 98.4 22.5 141 10.7 124 14.3 172

204 4410 233 78.8 18.0 101 5.50 92.1 7.33 119

198 4280 316 126 28.8 210 25.2 176 33.6 220



4 - 62


S 18–15–12–10


Fy = 36 ksi


Span (ft)

Fy = 36 ksi

Wt./ft

S 18 70

3 4 5 6 7 8

498 450 386 338

9 10 11 12 13 14

S 15

54.7

50

S 12

42.9

50

S 12

40.8

35

S 10

31.8

35

25.4

121 102 88 77

240 214 187

321 264 220 189 165

216 191 164 143

200 194 161 138 121

163 151 130 113

231 191 153 127 109 96

185 167 151 139 128 119

166 150 136 125 115 107

147 132 120 110 102 94

127 115 104 96 88 82

108 97 88 81 74 69

101 91 82 76 70 65

85 76 70 64 59 55

68 61 56 51 47 44

151 142 133 126 119 113

111 104 98 93 88 83

100 94 88 83 79 75

88 83 78 73 70 66

76 72 67 64 60 57

65 60 57 54 51 48

60 57 53 50 48 45

51 48 45 42 40 38

41 38 36 34 32 31

129 123 117 113 108 104

108 103 99 95 91 87

79 76 72 69 67 64

71 68 65 62 60 58

63 60 57 55 53 51

55 52 50 48 46 44

46 44 42 40 39 37

43 41 39 38 36 35

36 35 33 32 31

29 28 27 26 25

27 28 29 30 31 32

100 96 93 90 87 84

84 81 78 76 73 71

62 59 57 56 54 52

55 53 52 50 48 47

49 47 46 44

42 41 40 38

36 35 33 32

34 32 31 30

33 34 35 36 37 38

82 79 77 75 73 71

69 67 65 63 61 60

50 49 48 46 45

45 44 43 42 40

40 42 44

68 64 61

57 54 52

125 2700 249 96.0 25.6 152 26.5 121 35.4 179

105 2270 161 62.2 16.6 79.6 7.23 70.9 9.64 103

44.8 968 99.8 45.7 15.4 63.2 11.0 54.4 14.7 95.8

42.0 907 81.6 37.4 12.6 46.7 6.03 41.9 8.04 68.0

35.4 765 115 60.1 21.4 98.2 39.2 72.0 52.2 130

28.4 613 60.5 31.5 11.2 37.2 5.62 33.4 7.50 57.8

323 284

321 278 238 208

300 270 245 225 208 193

252 227 206 189 174 162

15 16 17 18 19 20

180 169 159 150 142 135

21 22 23 24 25 26


77.1 1670 160 68.1 19.8 98.4 16.4 82.1 21.8 132

69.3 1500 120 50.9 14.8 63.6 6.83 56.8 9.11 86.4

61.2 1320 160 88.9 24.7 141 37.6 111 50.2 169

53.1 1150 108 59.8 16.6 78.0 11.4 68.8 15.3 114




Fy = 36 ksi

4 - 63

BEAMS S 8–6–5–4–3 S Shapes Maximum factored uniform loads in kips for beams laterally supported For beams laterally unsupported, see page 4-113

Designation Wt./ft

S8 23

1 2 3 4 5 6

137 104 83 69

7 8 9 10 11 12

S6 18.4

17.25

S5 12.5

10 42 41 31 24 20

60 52 46 42 38 35

51 45 40 36 32 30

33 29 25 23 21 19

26 23 20 18 17 15

17 15 14 12 11 10

13

32

27

18

14


19.3 417 68.6 39.7 15.9 58.5 23.1 46.2 30.8 91.3

16.5 356 42.1 24.4 9.76 28.2 5.36 25.3 7.15 48.5

S3 7.7

7.5

5.7

41 25 17 13 10 8.50

20 14 11 8.42 7.02

51 44 29 22 17 15

30 25 19 15 13

12 11 9.70 8.73

11 9.48 8.42 7.58

7.28

6.02

3.51 75.8 15.0 13.0 6.95 14.0 5.63 12.5 7.51 35.6

2.36 51.0 20.4 21.6 12.6 32.2 50.0 22.2 66.7 62.4

1.95 42.1 9.91 10.5 6.12 10.9 5.78 9.78 7.71 30.4

Span (ft)

54 46 37 30

Fy = 36 ksi

84 71 59

108 76 57 46 38

S4 9.5

Properties and Reaction Values 10.6 229 54.2 36.6 16.7 58.1 42.9 41.0 57.1 91.0

8.47 183 27.1 18.3 8.35 20.5 5.32 18.4 7.10 41.4

5.67 122 20.8 15.6 7.70 17.3 5.52 15.5 7.36 39.4

4.04 87.3 25.3 22.0 11.7 30.8 27.1 23.6 36.2 60.1



4 - 64


MC,C 18–15

BEAMS Channels Maximum factored uniform loads in kips for beams laterally supported

Fy = 36 ksi


MC 18

Span (ft)

Fy = 36 ksi

Wt./ft

58

51.9

C 15 45.8

42.7

50

40

33.9

303 247 206 177 154

233 218 181 156 136

3 4 5 6 7 8

490 409 341 292 255

420 374 311 267 234

350 339 282 242 212

315 268 230 201

418 368 295 246 210 184

9 10 11 12 13 14

227 204 186 170 157 146

208 187 170 156 144 133

188 169 154 141 130 121

179 161 146 134 124 115

164 147 134 123 113 105

137 124 112 103 95 88

121 109 99 91 84 78

15 16 17 18 19 20

136 128 120 114 108 102

125 117 110 104 98 93

113 106 100 94 89 85

107 100 95 89 85 80

98 92 87 82 78 74

82 77 73 69 65 62

73 68 64 60 57 54

21 22 23 24 25 26

97 93 89 85 82 79

89 85 81 78 75 72

81 77 74 71 68 65

77 73 70 67 64 62

70 67 64 61 59 57

59 56 54 51 49 48

52 49 47 45 44 42

28 30 32 34 36 38

73 68 64 60 57 54

67 62 58 55 52 49

60 56 53 50 47 45

57 54 50 47 45 42

53 49 46 43 41

44 41 39 36 34

39 36 34 32 30

40 42 44

51 49 46

47 44 42

42 40 38

40 38 37

68.2 1470 209 92.6 25.8 149 34.6 115 46.1 176

57.2 1240 152 67.3 18.7 92.5 13.2 79.3 17.7 128

50.4 1090 117 51.8 14.4 62.4 6.03 56.4 8.03 82.5


94.6 2040 245 86.6 25.2 142 28.0 108 37.3 169

86.5 1870 210 74.3 21.6 112 17.6 91.3 23.5 144

78.4 1690 175 61.9 18.0 85.5 10.2 73.3 13.6 119

74.4 1610 157 55.7 16.2 73.0 7.44 64.1 9.91 97.2




Fy = 36 ksi

4 - 65


MC 13


MC 13

Span (ft)

Fy = 36 ksi

Wt./ft

50

40

35

31.8

3 4 5 6 7 8

398 327 261 218 187 163

283 275 220 183 157 137

226 200 166 143 125

190 186 155 133 116

9 10 11 12 13 14

145 131 119 109 101 93

122 110 100 92 85 79

111 100 91 83 77 71

103 93 85 78 72 66

15 16 17 18 19 20

87 82 77 73 69 65

73 69 65 61 58 55

67 62 59 55 53 50

62 58 55 52 49 47

21 22 23 24 25 26

62 59 57 54 52 50

52 50 48 46 44 42

48 45 43 42 40 38

44 42 40 39 37 36

27 28 29 30 31 32

48 47 45 44 42 41

41 39 38 37 35 34

37 36 34 33 32 31

34 33 32 31 30 29

46.2 998 113 55.3 16.1 71.4 10.3 62.5 13.8 107

43.1 931 94.8 46.4 13.5 54.9 6.10 49.6 8.14 76.0


60.5 1310 199 97.4 28.3 167 56.4 118 75.2 189

50.9 1100 142 69.3 20.2 100 20.3 82.5 27.1 135



4 - 66


C, MC 12


Fy = 36 ksi


C 12 30

25

MC 12 20.7

50

45

40

MC 12 35

31

10.6

181 158 126 105 90

132 110 91 78

390 303 242 202 173

332 279 223 186 160

275 255 204 170 146

218 185 154 132

173 170 141 121

8 9 10 11 12 13

91 81 73 66 60 56

79 70 63 57 53 49

69 61 55 50 46 42

151 135 121 110 101 93

140 124 112 102 93 86

128 114 102 93 85 79

116 103 92 84 77 71

106 94 85 77 71 65

31 28 25 23 21 19

14 15 16 17 18 19

52 48 45 43 40 38

45 42 39 37 35 33

39 37 34 32 30 29

87 81 76 71 67 64

80 74 70 66 62 59

73 68 64 60 57 54

66 62 58 54 51 49

61 57 53 50 47 45

18 17 16 15 14 13

20 21 22 23 24 25

36 35 33 32 30 29

32 30 29 27 26 25

27 26 25 24 23 22

61 58 55 53 50 48

56 53 51 49 47 45

51 49 46 44 43 41

46 44 42 40 39 37

42 40 39 37 35 34

13 12 11 11 10 10

26 27 28 29 30

28 27 26 25 24

24 23 23 22 21

21 20 20 19 18

47 45 43 42 40

43 41 40 39 37

39 38 36 35 34

36 34 33 32 31

33 31 30 29 28

9.64 9.28 8.95 8.64 8.35


33.6 726 119 51.6 18.4 78.9 20.3 62.7 27.0 111

29.2 631 90.3 39.2 13.9 52.1 8.85 45.1 11.8 83.4

47.3 1020 138 69.7 21.2 116 22.4 98.1 29.9 139

42.8 924 109 55.2 16.8 81.7 11.1 72.8 14.8 110

39.3 849 86.3 43.7 13.3 57.6 5.54 53.2 7.38 77.2

11.6 251 44.3 11.8 6.84 14.1 1.70 12.7 2.26 20.1

Span (ft)

238 181 145 121 104

89 84 63 50 42 36

Fy = 36 ksi

2 3 4 5 6 7


56.1 1210 195 98.6 30.1 195 63.6 144 84.8 196

51.7 1120 166 84.1 25.6 154 39.4 122 52.6 167




Fy = 36 ksi

4 - 67


C, MC 10


Span (ft)

Fy = 36 ksi

Wt./ft

C 10

MC 10

30

25

20

2 3 4 5 6 7

262 192 144 115 96 82

205 166 124 99 83 71

8 9 10 11 12 13

72 64 57 52 48 44

14 15 16 17 18 19 20 21 22 23 24

33.6

MC 10 28.5

25

MC 10

15.3

41.1

22

147 139 104 83 69 60

93 85 68 57 49

309 280 210 168 140 120

224 180 144 120 103

165 160 128 107 91

148 139 111 93 80

113 102 85 73

66 57 42 34 28 24

8.4

62 55 50 45 41 38

52 46 42 38 35 32

43 38 34 31 28 26

105 93 84 76 70 65

90 80 72 66 60 55

80 71 64 58 53 49

70 62 56 51 46 43

64 57 51 46 42 39

21 19 17 15 14 13

41 38 36 34 32 30

35 33 31 29 28 26

30 28 26 25 23 22

24 23 21 20 19 18

60 56 53 49 47 44

52 48 45 42 40 38

46 43 40 38 36 34

40 37 35 33 31 29

36 34 32 30 28 27

12 11 11 10 9.4 8.9

29 27 26 25 24

25 24 23 22 21

21 20 19 18 17

17 16 16 15 14

42 40 38 37 35

36 34 33 31 30

32 30 29 28 27

28 27 25 24 23

25 24 23 22 21

8.5 8.1 7.7 7.4 7.1

26.6 575 131 60.6 24.2 112 64.2 68.8 85.6 139

23.0 497 102 47.3 18.9 77.1 30.6 56.7 40.9 109

29.6 639 82.6 47.8 15.3 64.3 12.3 56.1 16.3 97.5

25.8 557 73.9 42.8 13.7 54.4 8.76 48.5 11.7 86.5

23.6 510 56.4 32.6 10.4 36.2 3.89 33.6 5.19 50.5

7.86 170 33.0 10.5 6.12 11.3 1.61 10.3 2.15


19.3 417 73.7 34.1 13.6 47.1 11.5 39.5 15.3 78.5

15.8 341 46.7 21.6 8.64 23.8 2.91 21.8 3.88 34.4

38.9 840 155 89.6 28.7 165 80.5 111 107 183

33.4 721 112 64.7 20.7 101 30.4 80.9 40.5 132



17.3

4 - 68


C, MC 9


Fy = 36 ksi


C9

Span (ft)

Fy = 36 ksi

Wt./ft

MC 9

20

15

13.4

25.4

23.9

2 3 4 5 6 7

157 121 91 73 60 52

100 97 73 58 49 42

82 68 54 45 39

157 125 100 84 72

140 120 96 80 69

8 9 10 11 12 13

45 40 36 33 30 28

36 32 29 27 24 22

34 30 27 25 23 21

63 56 50 46 42 39

60 53 48 44 40 37

14 15 16 17 18 19

26 24 23 21 20 19

21 19 18 17 16 15

19 18 17 16 15 14

36 33 31 29 28 26

34 32 30 28 27 25

20 21 22

18 17 16

15 14 13

14 13 12

25 24 23

24 23 22

23.2 501 78.7 48.1 16.2 68.5 16.9 58.4 22.5 101

22.2 480 70.0 42.8 14.4 57.4 11.9 50.3 15.8 89.6


16.8 363 78.4 37.8 16.1 59.0 22.2 45.6 29.6 90.2

13.5 292 49.9 24.0 10.3 29.9 5.72 26.5 7.62 51.3

12.5 270 40.8 19.7 8.39 22.1 3.12 20.2 4.17 33.8




Fy = 36 ksi

4 - 69


C, MC 8


C8

Span (ft)

Fy = 36 ksi

Wt./ft

MC 8

MC 8

MC 8

18.75

13.75

11.5

22.8

21.4

20

18.7

1 2 3 4 5 6

151 149 99 75 60 50

8.5

94 78 59 47 39

68 52 41 34

133 102 81 68

117 97 78 65

124 117 87 70 58

110 83 67 55

56 50 37 30 25

7 8 9 10 11 12

43 37 33 30 27 25

34 29 26 24 21 20

29 26 23 21 19 17

58 51 45 41 37 34

56 49 43 39 35 32

50 44 39 35 32 29

48 42 37 33 30 28

21 19 17 15 14 12

13 14 15 16 17 18

23 21 20 19 18 17

18 17 16 15 14 13

16 15 14 13 12 11

31 29 27 25 24 23

30 28 26 24 23 22

27 25 23 22 21 19

26 24 22 21 20 18

11 11 10 9.3 8.8 8.3

19 20

16 15

12 12

11 10

21 20

20 19

18 17

18 17

7.9 7.5

16.2 350 62.2 40.5 14.4 54.7 14.7 46.9 19.6 87.3,

15.4 333 54.9 35.7 12.7 45.4 10.1 40.0 13.5 77.0

6.91 149 27.8 12.1 6.44 12.9 2.12 11.8 2.82 21.0


13.8 298 75.7 41.1 17.5 64.9 34.0 46.8 45.3 98.1

10.9 235 47.1 25.6 10.9 31.9 8.18 27.5 10.9 61.0

9.55 206 34.2 18.6 7.92 19.7 3.13 18.0 4.18 31.6

18.8 406 66.4 45.6 15.4 61.9 17.0 52.8 22.7 95.6

18.0 389 58.3 40.1 13.5 50.9 11.5 44.8 15.4 84.0



4 - 70



C, MC 7–6

Fy = 36 ksi


C7 12.25

MC 7 9.8

22.7

C6

19.1

13

10.5 73 66 44 33 27 22

1 2 3 4 5 6

85 60 45 36 30

57 51 38 31 26

137 117 87 70 58

96 77 62 51

102 78 52 39 31 26

7 8 9 10 11 12

26 23 20 18 16 15

22 19 17 15 14 13

50 44 39 35 32 29

44 39 34 31 28 26

22 20 17 16 14 13

19 17 15 13 12 11

13 14 15 16

14 13 12 11

12 11 10 9.61

27 25 23 22

24 22 21 19

12 11 10

10 9.49 8.86

8.40 181 42.7 24.7 11.3 32.6 11.1 27.4 14.8 61.5

7.12 154 28.6 16.5 7.56 17.8 3.32 16.3 4.42 30.6

MC 6 8.2

MC 6

MC 6

18

16.3

15.1

12

47 37 28 22 18

88 83 62 50 41

87 73 55 44 37

74 70 52 42 35

72 53 40 32 27

16 14 12 11 10 9.23

35 31 28 25 23 21

31 28 24 22 20 18

30 26 23 21 19 17

23 20 18 16 14 13

8.52 7.91 7.39

19 18 17

17 16 15

16 15 14

12 11 11

11.5 248 44.2 36.2 13.6 49.2 17.5 42.2 23.4 80.6

10.2 220 43.7 35.9 13.5 48.4 17.0 41.6 22.6 79.7

9.69 209 36.9 30.2 11.4 37.5 10.2 33.4 13.6 67.2

7.38 159 36.2 22.7 11.2 32.3 12.2 27.5 16.2 58.9

Span (ft)

Fy = 36 ksi

Wt./ft


16.2 350 68.4 50.9 18.1 77.2 33.4 61.6 44.5 110

14.3 309 47.9 35.6 12.7 45.2 11.4 39.8 15.3 76.8

7.26 157 51.0 32.0 15.7 51.8 37.2 36.9 49.6 83.1

6.15 133 36.6 23.0 11.3 31.5 13.8 26.0 18.4 59.7

5.13 111 23.3 14.6 7.20 16.0 3.57 14.6 4.76 30.1




Fy = 36 ksi

4 - 71


C 5–4–3


C5

Wt./ft

9 1 2 3 4 5 6

C3

7.25

5.4

6

5

4.1

63 47 31 24 19 16

37 25 19 15 13

50 30 20 15 12 10

29 24 16 12 9.8 8.1

37 19 12 9.3 7.4 6.2

30 16 11 8.1 6.5 5.4

20 14 9.4 7.0 5.6 4.7

13 12 10 9.4 8.6 7.9

11 9.5 8.4 7.6 6.9 6.3

7.0 6.1 5.4 4.9

5.3

4.6

4.0

1.72 37.2 20.8 22.0 12.8 34.0 50.6 23.8 67.4 63.7

1.50 32.4 15.0 16.0 9.29 21.0 19.2 17.1 25.7 46.1

1.30 28.1 9.91 10.5 6.12 11.2 5.51 10.1 7.34 30.4

8.7 7.6 6.7 6.1

Span (ft)

Fy = 36 ksi

7 8 9 10 11 12

C4 6.7


4.36 94.2 31.6 21.9 11.7 32.1 19.7 25.5 26.3 60.0

3.51 75.8 18.5 12.8 6.84 14.3 3.94 13.0 5.25 30.1

2.81 60.7 25.0 19.9 11.6 30.3 25.6 23.4 34.2 57.4

2.26 48.8 14.3 11.4 6.62 13.1 4.83 11.9 6.44 32.8



4 - 72



W 44

Fy = 50 ksi


W 44

Span (ft)

Fy = 50 ksi

Wt./ft

335

290

262

230

20 21 22 23 24 25

2420 2310 2210 2110 2030 1940

2050 2030 1940 1850 1780 1700

1850 1810 1730 1660 1590 1520

1650 1570 1500 1430 1380 1320

26 27 28 29 30 31

1870 1800 1740 1680 1620 1570

1640 1580 1520 1470 1420 1370

1470 1410 1360 1310 1270 1230

1270 1220 1180 1140 1100 1060

32 33 34 35 36 38

1520 1470 1430 1390 1350 1280

1330 1290 1250 1220 1180 1120

1190 1150 1120 1090 1060 1000

1030 1000 971 943 917 868

40 42 44 46 48 50

1220 1160 1100 1060 1010 972

1070 1010 968 926 888 852

953 907 866 828 794 762

825 786 750 717 688 660

52 54 56 58 60 62

935 900 868 838 810 784

819 789 761 734 710 687

733 706 680 657 635 615

635 611 589 569 550 532

64 66 68 70 72

759 736 715 694 675

666 645 626 609 592

595 577 560 544 529

516 500 485 471 458

1270 38100 924 216 39.5 302 8.67 277 11.6 330

1100 33000 823 178 35.5 238 7.40 217 9.86 262


1620 48600 1212 327 51.0 494 14.7 451 19.6 492

1420 42600 1025 258 43.5 368 10.3 338 13.8 400




Fy = 50 ksi

4 - 73


W 40


W 40 431

372

321

297

277

249

199

174*

1360 1300

1340 1330 1250 1180 1120 1070

1370 1310 1260 1200 1160 1110

1240 1180 1130 1090 1040 1000

1010 969 927 888 852 820

1240 1200 1160 1120 1080 1050

1070 1030 996 963 932 903

964 930 898 868 840 814

789 761 735 710 687 666

1140 1100 1070 1040 987 938

1020 988 960 933 884 840

875 850 825 803 760 722

789 766 744 723 685 651

646 627 609 592 561 533

950 907 867 831 798 767

893 852 815 781 750 721

800 764 730 700 672 646

688 657 628 602 578 556

620 592 566 543 521 501

507 484 463 444 426 410

789 761 734 710 687 666

739 713 688 665 644 623

694 670 647 625 605 586

622 600 579 560 542 525

535 516 498 482 466 451

482 465 449 434 420 407

395 381 367 355 344 333

645 626 609 592

605 587 570 554

568 551 536 521

509 494 480 467

438 425 413 401

395 383 372 362

323 313 304 296

1120 33600 797 264 37.5 279 8.16 258 10.9 306

963 28900 684 213 32.5 209 6.25 193 8.33 229

868 26000 679 198 32.5 195 7.21 176 9.62 218

715 21300 670 163 32.5 172 9.37 148 12.5 203

2550 2510

2160 2130

2000 2000

21 22 23 24 25 26

2790 2660 2540 2440 2340 2250

2390 2280 2180 2090 2000 1930

2030 1940 1850 1780 1700 1640

1900 1810 1730 1660 1600 1530

1780 1700 1630 1560 1500 1440

1590 1530 1460 1400 1340 1290

27 28 29 30 31 32

2170 2090 2020 1950 1890 1830

1860 1790 1730 1670 1620 1570

1580 1520 1470 1420 1370 1330

1480 1430 1380 1330 1290 1250

1390 1340 1290 1250 1210 1170

33 34 35 36 38 40

1770 1720 1670 1630 1540 1460

1520 1470 1430 1390 1320 1250

1290 1250 1220 1180 1120 1070

1210 1170 1140 1110 1050 998

42 44 46 48 50 52

1390 1330 1270 1220 1170 1130

1190 1140 1090 1040 1000 963

1010 968 926 888 852 819

54 56 58 60 62 64

1080 1040 1010 975 944 914

928 895 864 835 808 783

66 68 70 72

886 860 836 813

759 737 716 696


1950 58500 1493 597 67.0 859 26.7 786 35.6 814

1670 50100 1273 471 58.0 645 20.3 590 27.0 660

Span (ft)

2990 2930

Fy = 50 ksi

15 16 17 18 19 20

215


1330 39900 1000 356 46.5 415 13.2 380 17.7 458

1250 37500 889 311 41.5 342 9.90 316 13.2 374

*Noncompact shape; Fy = 50 ksi. Load above heavy line is limited by design shear strength.


4 - 74



W 40

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 40 331

278

264

235

211

167

149

1370 1300

1350 1300 1220 1150

1300 1280 1190 1120 1050 995

1430 1360 1290 1230 1180 1130

1230 1170 1120 1070 1020 976

1090 1040 989 944 903 865

943 896 853 814 779 746

1210 1170 1120 1080 1040 1010

1090 1040 1010 970 936 905

937 901 868 837 808 781

830 798 769 741 716 692

716 689 663 640 618 597

1090 1060 1030 997 969 942

977 947 918 891 866 842

876 848 823 799 776 754

756 732 710 689 669 651

670 649 629 611 593 577

578 560 543 527 512 498

939 893 850 811 776 744

892 848 807 770 737 706

797 758 721 689 659 631

714 679 646 617 590 566

617 586 558 533 509 488

546 519 494 472 451 433

471 448 426 407 389 373

858 825 794 766 740 715

714 687 661 638 616 595

678 652 628 605 584 565

606 583 561 541 522 505

543 522 503 485 468 453

469 451 434 418 404 391

415 399 384 371 358 346

358 344 332 320 309 299

692 670 650 631 613 596

576 558 541 525 510 496

547 530 514 499 484 471

489 473 459 446 433 421

438 424 411 399 388 377

378 366 355 345 335 325

335 324 315 305 297 288

289 280 271 263 256 249

781 23400 684 213 32.5 209 6.25 193 8.33 229

692 20800 677 198 32.5 191 7.51 172 10.0 216

597 17900 650 177 31.5 164 8.53 143 11.4 192

13 14 15 16 17 18

2690 2680 2520 2380

2210 2100 1980

2070 1990 1880

1780 1680

1590 1510

19 20 21 22 23 24

2260 2150 2040 1950 1870 1790

1880 1790 1700 1620 1550 1490

1780 1700 1610 1540 1470 1410

1590 1520 1440 1380 1320 1260

25 26 27 28 29 30

1720 1650 1590 1530 1480 1430

1430 1370 1320 1280 1230 1190

1360 1300 1260 1210 1170 1130

31 32 33 34 35 36

1380 1340 1300 1260 1230 1190

1150 1120 1080 1050 1020 992

38 40 42 44 46 48

1130 1070 1020 975 933 894

50 52 54 56 58 60 62 64 66 68 70 72

183


1430 42900 1344 505 61.0 709 22.6 648 30.1 703

1190 35700 1106 383 51.0 500 15.8 458 21.1 548

1130 33900 1037 353 48.0 446 13.8 409 18.4 491

1010 30300 889 285 41.5 342 9.90 316 13.2 374

905 27200 797 246 37.5 279 8.19 257 10.9 305




Fy = 50 ksi

4 - 75


W 36


W 36

Span (ft)

Fy = 50 ksi

Wt./ft

300

280

260

245

230

1560 1520 1440 1380 1320 1260

1470 1410 1350 1290 1230 1180

19 20 21 22 23 24

1870 1800 1720 1640 1580

1750 1670 1600 1530 1460

1640 1620 1540 1470 1410 1350

25 26 27 28 29 30

1510 1450 1400 1350 1300 1260

1400 1350 1300 1250 1210 1170

1300 1250 1200 1160 1120 1080

1210 1170 1120 1080 1040 1010

1130 1090 1050 1010 976 943

31 32 33 34 35 36

1220 1180 1150 1110 1080 1050

1130 1100 1060 1030 1000 975

1050 1010 982 953 926 900

977 947 918 891 866 842

913 884 857 832 808 786

38 40 42 44 46 48

995 945 900 859 822 788

924 878 836 798 763 731

853 810 771 736 704 675

797 758 721 689 659 631

744 707 674 643 615 589

50 52 54 56 58 60

756 727 700 675 652 630

702 675 650 627 605 585

648 623 600 579 559 540

606 583 561 541 522 505

566 544 524 505 488 472

62 64 66 68 70 72

610 591 573 556 540 525

566 548 532 516 501 488

523 506 491 476 463 450

489 473 459 446 433 421

456 442 429 416 404 393

1010 30300 779 250 40.0 300 11.4 272 15.2 337

943 28300 737 226 38.0 268 10.5 243 14.0 302


1260 37800 937 332 47.3 429 14.8 393 19.7 477

1170 35100 873 297 44.3 376 13.1 344 17.4 419

1080 32400 822 269 42.0 333 12.3 303 16.4 373



4 - 76


W 36


Fy = 50 ksi


W 36 256

19 20 21 22 23 24

194

182

170

160

150

135

1740 1650 1560

1640 1560 1470 1390

1510 1440 1350 1280

1420 1350 1270 1200

1330 1250 1180 1110

1260 1250 1170 1100 1040

1210 1160 1090 1030 968

1640 1560 1490 1420 1360 1300

1480 1400 1340 1280 1220 1170

1320 1250 1190 1140 1090 1040

1210 1150 1100 1050 1000 959

1130 1080 1030 979 937 898

1050 1000 954 911 871 835

985 936 891 851 814 780

917 872 830 792 758 726

804 764 727 694 664 636

25 26 27 28 29 30

1250 1200 1160 1110 1080 1040

1120 1080 1040 1000 968 936

1000 961 926 893 862 833

920 885 852 822 793 767

862 828 798 769 743 718

802 771 742 716 691 668

749 720 693 669 646 624

697 670 646 623 601 581

611 587 566 545 527 509

31 32 33 34 35 36

1010 975 945 918 891 867

906 878 851 826 802 780

806 781 757 735 714 694

742 719 697 677 657 639

695 673 653 634 615 598

646 626 607 589 573 557

604 585 567 551 535 520

562 545 528 513 498 484

493 477 463 449 436 424

38 40 42 44 46 48

821 780 743 709 678 650

739 702 669 638 610 585

658 625 595 568 543 521

606 575 548 523 500 479

567 539 513 490 468 449

527 501 477 455 436 418

493 468 446 425 407 390

459 436 415 396 379 363

402 382 364 347 332 318

50 52 54 56 58 60

624 600 578 557 538 520

562 540 520 501 484 468

500 481 463 446 431 417

460 443 426 411 397 384

431 414 399 385 371 359

401 385 371 358 346 334

374 360 347 334 323 312

349 335 323 311 301 291

305 294 283 273 263 255

62 64 66 68 70 72

503 488 473 459 446 433

453 439 425 413 401 390

403 390 379 368 357 347

371 360 349 338 329 320

347 337 326 317 308 299

323 313 304 295 286 278

302 293 284 275 267 260

281 272 264 256 249 242

246 239 231 225 218 212


1040 31200 970 315 48.0 446 14.8 409 19.7 471

936 28100 872 272 43.5 367 12.2 336 16.3 406

668 20000 664 170 34.0 212 8.55 191 11.4 240

624 18700 632 157 32.5 191 8.09 171 10.8 217

581 17400 605 146 31.3 173 7.84 154 10.5 198

509 15300 576 127 30.0 149 8.32 129 11.1 176

Span (ft)

1940 1840 1730

210

1150 1090 1020 954 898 848

Fy = 50 ksi

13 14 15 16 17 18

232


767 23000 754 209 38.3 271 10.5 245 14.0 305

718 21500 711 193 36.3 242 9.62 219 12.8 273




Fy = 50 ksi

4 - 77


W 33


Span (ft)

Fy = 50 ksi

Wt./ft

W 33 241

221

W 33 201

169

152

141

130

118

1050 986 932 883 839 799

964 907 857 812 771 734

876 824 778 737 701 667

778 732 692 655 623 593

16 17 18 19 20 21

1530 1480 1410 1340

1420 1350 1280 1220

1300 1290 1220 1160 1100

1180 1110 1050 993 944 899

22 23 24 25 26 27

1280 1220 1170 1130 1080 1040

1170 1120 1070 1030 987 950

1050 1010 965 926 891 858

858 820 786 755 726 699

762 729 699 671 645 621

701 670 643 617 593 571

637 609 584 560 539 519

566 541 519 498 479 461

28 29 30 31 32 33

1010 971 939 909 880 854

916 884 855 827 802 777

827 799 772 747 724 702

674 651 629 609 590 572

599 578 559 541 524 508

551 532 514 497 482 467

500 483 467 452 438 425

445 429 415 402 389 377

34 35 36 37 38 40

829 805 783 761 741 704

754 733 713 693 675 641

681 662 643 626 609 579

555 539 524 510 497 472

493 479 466 453 441 419

454 441 428 417 406 386

412 400 389 379 369 350

366 356 346 336 328 311

42 44 46 48 50 52

671 640 612 587 563 542

611 583 558 534 513 493

551 526 503 483 463 445

449 429 410 393 377 363

399 381 365 349 335 323

367 350 335 321 308 297

334 318 305 292 280 269

296 283 271 259 249 239

54 56 58 60 62 64

522 503 486 470 454 440

475 458 442 428 414 401

429 414 399 386 374 362

349 337 325 315 304 295

311 299 289 280 270 262

286 275 266 257 249 241

259 250 242 234 226 219

231 222 215 208 201 195

66 68 70 72

427 414 402 391

389 377 366 356

351 341 331 322

286 278 270 262

254 247 240 233

234 227 220 214

212 206 200 195

189 183 178 173

514 15400 544 132 30.3 166 7.49 150 9.99 191

467 14000 518 122 29.0 147 7.46 131 9.95 172

415 12500 488 107 27.5 127 7.40 110 9.87 151


939 28200 766 227 41.5 323 12.9 293 17.2 362

855 25700 710 200 38.8 278 11.6 251 15.5 316

772 23200 650 173 35.8 234 10.2 211 13.6 267

629 18900 612 173 33.5 218 7.89 201 10.5 244

559 16800 574 149 31.8 187 7.84 170 10.5 213



4 - 78


W 30


Fy = 50 ksi


W 30

Span (ft)

Fy = 50 ksi

Wt./ft

261

235

191

173

1290 1250 1180 1120 1070

1180 1120 1060 1010 961

1080 1070 1010 955 908 864

1150 1100 1060 1010 975 939

1020 977 936 899 864 832

918 878 841 808 777 748

825 789 756 726 698 672

1010 973 941 911 882 855

905 874 845 818 792 768

803 775 749 725 702 681

721 696 673 651 631 612

648 626 605 585 567 550

34 36 38 40 42 44

830 784 743 706 672 642

746 704 667 634 604 576

661 624 591 562 535 511

594 561 531 505 481 459

534 504 478 454 432 413

46 48 50 52 54 56

614 588 565 543 523 504

551 528 507 488 469 453

488 468 449 432 416 401

439 421 404 388 374 361

395 378 363 349 336 324

58 60 62 64 66 68

487 471 455 441 428 415

437 423 409 396 384 373

387 375 362 351 340 330

348 337 326 315 306 297

313 303 293 284 275 267

70 72

403 392

362 352

321 312

288 280

259 252

673 20200 588 172 35.5 235 10.7 213 14.2 269

605 18200 538 154 32.8 197 9.38 178 12.5 228

16 17 18 19 20 21

1590 1570 1490 1410 1340

1400 1330 1270 1210

22 23 24 25 26 27

1280 1230 1180 1130 1090 1050

28 29 30 31 32 33

211


941 28200 794 283 46.5 415 16.7 380 22.2 434

845 25400 701 233 41.5 334 13.2 306 17.6 368

749 22500 647 206 38.8 282 12.4 257 16.5 322




Fy = 50 ksi

4 - 79


W 30


Span (ft)

Fy = 50 ksi

Wt./ft

W 30 148

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72

1080 1070 1000 938 882 833 789 750 714 682 652 625 600 577 556 536 517 500 484 469 455 441 417 395 375 357 341 326 313 300 288 278 268 259 250 242 234 227 221 214 208

132

1010 936 874 819 771 728 690 656 624 596 570 546 524 504 486 468 452 437 423 410 397 386 364 345 328 312 298 285 273 262 252 243 234 226 219 211 205 199 193 187 182

124 953 942 874 816 765 720 680 644 612 583 556 532 510 490 471 453 437 422 408 395 383 371 360 340 322 306 291 278 266 255 245 235 227 219 211 204 197 191 185 180 175 170

116

108

99

90

916 872 810 756 709 667 630 597 567 540 515 493 473 454 436 420 405 391 378 366 354 344 334 315 298 284 270 258 247 236 227 218 210 203 196 189 183 177 172 167 162 158

878 865 798 741 692 649 611 577 546 519 494 472 451 433 415 399 384 371 358 346 335 324 315 305 288 273 260 247 236 226 216 208 200 192 185 179 173 167 162 157 153 148 144

833 780 720 669 624 585 551 520 493 468 446 425 407 390 374 360 347 334 323 312 302 293 284 275 260 246 234 223 213 203 195 187 180 173 167 161 156 151 146 142 138 134 130

749 708 653 606 566 531 499 472 447 425 404 386 369 354 340 327 314 303 293 283 274 265 257 250 236 223 212 202 193 185 177 170 163 157 152 146 142 137 133 129 125 121 118

346 10400 439 106 27.3 126 7.73 111 10.3 152

312 9360 416 93.4 26.0 111 7.66 95.6 10.2 136

283 8490 375 77.1 23.5 90.8 6.24 78.5 8.31 111


500 15000 538 163 32.5 205 8.21 189 10.9 232

437 13100 503 135 30.8 174 8.30 157 11.1 201

408 12200 477 123 29.2 156 7.72 140 10.3 181

378 11300 458 115 28.3 141 7.65 126 10.2 166



4 - 80



W 27

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 27 258

235

194

178

161

146

1270 1250 1180 1120 1060

1140 1110 1050 992 942

1090 1060 1000 945 895 851

983 960 904 853 808 768

895 864 814 768 728 692

1100 1050 1000 961 923 887

1010 965 923 885 850 817

897 856 819 785 754 725

810 773 740 709 680 654

731 698 668 640 614 591

659 629 601 576 553 532

944 911 879 850 823 797

854 824 796 769 744 721

787 759 732 708 685 664

698 673 650 628 608 589

630 608 587 567 549 532

569 549 530 512 495 480

512 494 477 461 446 432

33 34 35 36 37 38

773 750 729 708 689 671

699 679 659 641 624 607

644 625 607 590 574 559

571 554 538 523 509 496

515 500 486 473 460 448

465 452 439 427 415 404

419 407 395 384 374 364

40 42 44 46 48 50

638 607 580 554 531 510

577 549 524 502 481 461

531 506 483 462 443 425

471 449 428 410 393 377

425 405 387 370 354 340

384 366 349 334 320 307

346 329 314 301 288 277

52 54 56 58 60 62

490 472 455 440 425 411

444 427 412 398 385 372

408 393 379 366 354 343

362 349 336 325 314 304

327 315 304 293 284 274

295 284 274 265 256 248

266 256 247 238 231 223

64 66

398 386

360 350

332 322

294 285

266 258

240 233

216 210

567 17000 544 170 36.3 243 12.5 220 16.6 283

512 15400 492 150 33.0 201 10.4 182 13.9 235

461 13800 447 128 30.3 168 8.97 151 12.0 197

15 16 17 18 19 20

1530 1500 1420 1340 1280

1410 1360 1280 1210 1150

21 22 23 24 25 26

1210 1160 1110 1060 1020 981

27 28 29 30 31 32

217


850 25500 767 306 49.0 465 19.9 427 26.5 466

769 23100 704 263 45.5 397 17.7 363 23.6 411

708 21200 637 227 41.5 334 14.5 306 19.3 362

628 18800 569 193 37.5 271 12.1 248 16.2 311




Fy = 50 ksi

4 - 81


W 27


W 27

Span (ft)

Fy = 50 ksi

Wt./ft

129

102

94

84

840 792 735 686 643

753 704 654 610 572

712 695 642 596 556 521

663 610 563 523 488 458

697 658 624 593 564 539

605 572 542 515 490 468

538 508 482 458 436 416

491 463 439 417 397 379

431 407 385 366 349 333

23 24 25 26 27 28

515 494 474 456 439 423

447 429 412 396 381 368

398 381 366 352 339 327

363 348 334 321 309 298

318 305 293 282 271 261

29 30 31 32 33 34

409 395 382 370 359 349

355 343 332 322 312 303

316 305 295 286 277 269

288 278 269 261 253 245

252 244 236 229 222 215

36 38 40 42 44 46

329 312 296 282 269 258

286 271 257 245 234 224

254 241 229 218 208 199

232 219 209 199 190 181

203 193 183 174 166 159

48 50 52 54 56 58

247 237 228 219 212 204

214 206 198 191 184 177

191 183 176 169 163 158

174 167 160 154 149 144

153 146 141 136 131 126

60 62 64 66

198 191 185 180

172 166 161 156

153 148 143 139

139 135 130 126

122 118 114 111

278 8340 356 88.0 24.5 107 6.35 95.4 8.46 127

244 7320 332 79.1 23.0 90.0 6.16 79.0 8.21 110

11 12 13 14 15 16

910 846 790 741

17 18 19 20 21 22

114


395 11900 455 138 30.5 180 8.08 165 10.8 206

343 10300 420 116 28.5 150 7.89 135 10.5 175

305 9150 377 101 25.8 121 6.57 110 8.76 143



4 - 82


W 24


Fy = 50 ksi


W 24 131

117

104

952 936 878 826 780

868 836 784 738 697

800 793 740 694 653 617

721 701 654 613 577 545

650 619 578 542 510 482

807 767 730 697 667 639

739 702 669 638 610 585

660 627 597 570 545 523

584 555 529 505 483 463

516 491 467 446 427 409

456 434 413 394 377 361

671 645 621 599 578 559

613 590 568 548 529 511

562 540 520 501 484 468

502 482 464 448 432 418

444 427 411 396 383 370

392 377 363 350 338 327

347 333 321 310 299 289

586 568 551 535 519 505

541 524 508 493 479 466

495 479 465 451 438 426

453 439 425 413 401 390

405 392 380 369 358 348

358 347 336 326 317 308

316 307 297 289 280 273

280 271 263 255 248 241

534 507 483 461 441 423

478 455 433 413 395 379

441 419 399 381 365 349

403 383 365 348 333 319

369 351 334 319 305 293

330 314 299 285 273 261

292 278 264 252 241 231

258 245 234 223 213 204

228 217 206 197 188 181

50 52 54 56 58 60

406 390 376 362 350 338

364 350 337 325 313 303

335 323 311 299 289 280

307 295 284 274 264 256

281 270 260 251 242 234

251 241 232 224 216 209

222 213 206 198 191 185

196 189 182 175 169 164

173 167 161 155 149 145


676 20300 674 300 48.0 446 21.3 409 28.4 456

606 18200 604 258 43.5 367 17.6 336 23.5 400

370 11100 400 132 30.3 166 10.2 150 13.6 199

327 9810 360 112 27.5 136 8.73 121 11.6 164

289 8670 325 93.8 25.0 110 7.49 98.4 9.99 135

Span (ft)

146

Fy = 50 ksi

229

207

192

176

13 14 15 16 17 18

1350 1270 1190 1130

1210 1140 1070 1010

1110 1050 986 932

1020 1020 958 902 852

19 20 21 22 23 24

1070 1010 966 922 882 845

957 909 866 826 790 758

883 839 799 762 729 699

25 26 27 28 29 30

811 780 751 724 699 676

727 699 673 649 627 606

31 32 33 34 35 36

654 634 615 596 579 563

38 40 42 44 46 48

162


511 15300 511 199 37.5 271 13.5 248 18.0 315

468 14000 476 176 35.3 236 12.4 215 16.6 276

418 12500 434 152 32.5 197 11.0 179 14.7 233




Fy = 50 ksi

4 - 83


W 24


Span (ft)

Fy = 50 ksi

Wt./ft

W 24 103

94

84

W 24 76

68

62

55

551 510 459 417 383

503 503 447 402 365 335

7 8 9 10 11 12

729 700

676 635

612 611 560

568 545 500

532 531 483 443

13 14 15 16 17 18

646 600 560 525 494 467

586 544 508 476 448 423

517 480 448 420 395 373

462 429 400 375 353 333

408 379 354 332 312 295

353 328 306 287 270 255

309 287 268 251 236 223

19 20 21 22 23 24

442 420 400 382 365 350

401 381 363 346 331 318

354 336 320 305 292 280

316 300 286 273 261 250

279 266 253 241 231 221

242 230 219 209 200 191

212 201 191 183 175 168

25 26 27 28 29 30

336 323 311 300 290 280

305 293 282 272 263 254

269 258 249 240 232 224

240 231 222 214 207 200

212 204 197 190 183 177

184 177 170 164 158 153

161 155 149 144 139 134

31 32 33 34 35 36

271 263 255 247 240 233

246 238 231 224 218 212

217 210 204 198 192 187

194 188 182 176 171 167

171 166 161 156 152 148

148 143 139 135 131 128

130 126 122 118 115 112

38 40 42 44 46 48

221 210 200 191 183 175

201 191 181 173 166 159

177 168 160 153 146 140

158 150 143 136 130 125

140 133 126 121 115 111

121 115 109 104 100 96

106 101 96 91 87 84

50 52 54 56 58 60

168 162 156 150 145 140

152 147 141 136 131 127

134 129 124 120 116 112

120 115 111 107 103

106 102 98 95 92

92 88 85 82 79

80 77 74 72 69

177 5310 266 71.3 20.8 73.7 5.57 64.9 7.43 91.8

153 4590 276 73.9 21.5 78.1 6.14 68.4 8.19 98.1

134 4020 251 64.8 19.8 63.6 5.60 54.8 7.47 81.8


280 8400 364 120 27.5 146 7.49 133 9.98 170

254 7620 338 105 25.8 125 6.95 113 9.26 147

224 6720 306 91.8 23.5 102 6.05 92.2 8.07 122

200 6000 284 79.1 22.0 86.8 5.67 77.8 7.55 105



4 - 84



W 21

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 21 201

182

166

147

132

122

111

101

766 714 666 624 588 555

702 658 614 576 542 512

639 598 558 523 492 465

577 542 506 474 446 422

13 14 15 16 17 18

1130 1060 994 935 883

1020 952 893 840 793

910 864 810 762 720

858 799 746 699 658 622

19 20 21 22 23 24

837 795 757 723 691 663

752 714 680 649 621 595

682 648 617 589 563 540

589 560 533 509 487 466

526 500 476 454 434 416

485 461 439 419 400 384

441 419 399 380 364 349

399 380 361 345 330 316

25 26 27 28 29 30

636 612 589 568 548 530

571 549 529 510 492 476

518 498 480 463 447 432

448 430 414 400 386 373

400 384 370 357 344 333

368 354 341 329 318 307

335 322 310 299 289 279

304 292 281 271 262 253

31 32 33 34 35 36

513 497 482 468 454 442

461 446 433 420 408 397

418 405 393 381 370 360

361 350 339 329 320 311

322 312 303 294 285 278

297 288 279 271 263 256

270 262 254 246 239 233

245 237 230 223 217 211

38 40 42 44 46 48

418 398 379 361 346 331

376 357 340 325 310 298

341 324 309 295 282 270

294 280 266 254 243 233

263 250 238 227 217 208

242 230 219 209 200 192

220 209 199 190 182 174

200 190 181 173 165 158

50 52

318 306

286 275

259 249

224 215

200 192

184 177

167 161

152 146

307 9210 351 127 30.0 164 11.2 148 15.0 201

279 8370 319 112 27.5 138 9.56 124 12.8 169

253 7590 288 97.7 25.0 114 7.91 103 10.6 140


530 15900 566 270 45.5 400 21.7 366 29.0 418

476 14300 509 233 41.5 332 18.4 304 24.5 368

432 13000 455 199 37.5 273 14.9 251 19.9 321

373 11200 429 169 36.0 236 15.9 213 21.2 286

333 9990 383 147 32.5 192 13.1 173 17.5 235




Fy = 50 ksi

4 - 85


W 21


Span (ft)

Fy = 50 ksi

Wt./ft

W 21 93

83

73

W 21 68

62

57

50

44 390 358 318 286 260 239

7 8 9 10 11 12

677 663 603 553

596 588 535 490

522 516 469 430

491 480 436 400

453 432 393 360

461 430 387 352 323

427 413 367 330 300 275

13 14 15 16 17 18

510 474 442 414 390 368

452 420 392 368 346 327

397 369 344 323 304 287

369 343 320 300 282 267

332 309 288 270 254 240

298 276 258 242 228 215

254 236 220 206 194 183

220 204 191 179 168 159

19 20 21 22 23 24

349 332 316 301 288 276

309 294 280 267 256 245

272 258 246 235 224 215

253 240 229 218 209 200

227 216 206 196 188 180

204 194 184 176 168 161

174 165 157 150 143 138

151 143 136 130 124 119

25 26 27 28 29 30

265 255 246 237 229 221

235 226 218 210 203 196

206 198 191 184 178 172

192 185 178 171 166 160

173 166 160 154 149 144

155 149 143 138 133 129

132 127 122 118 114 110

114 110 106 102 99 95

31 32 33 34 35 36

214 207 201 195 189 184

190 184 178 173 168 163

166 161 156 152 147 143

155 150 145 141 137 133

139 135 131 127 123 120

125 121 117 114 111 108

106 103 100 97 94 92

92 89 87 84 82 80

38 40 42 44 46 48

174 166 158 151 144 138

155 147 140 134 128 123

136 129 123 117 112 108

126 120 114 109 104 100

114 108 103 98 94 90

102 97 92 88 84 81

87 83 79 75 72 69

75 72 68 65 62 60

50 52

133 128

118 113

103 99

96 92

86 83

77 74

66 63

57

129 3870 230 69.6 20.3 74.9 5.25 67.6 7.00 92.0

110 3300 214 62.3 19.0 61.8 5.33 54.4 7.10 79.1

95.4 2860 195 52.0 17.5 50.1 4.99 43.2 6.65 66.3


221 6630 339 122 29.0 154 10.5 138 14.0 188

196 5880 298 101 25.8 122 8.26 110 11.0 149

172 5160 261 85.3 22.8 95.2 6.48 86.0 8.64 116

160 4800 245 77.3 21.5 84.2 5.94 75.8 7.92 103

144 4320 227 68.8 20.0 71.5 5.36 64.0 7.15 89.0



4 - 86


W 18


Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 18 192

175

158

W 18 143

130

119

106

97

86

76

597 575 531 493 460 431

537 528 487 452 422 396

477 465 429 399 372 349

418 408 376 349 326 306

11 12 13 14 15 16

1050 1020 947 884 829

963 918 853 796 746

863 822 763 712 668

768 743 690 644 604

696 672 624 582 546

671 653 602 559 522 489

17 18 19 20 21 22

780 737 698 663 631 603

702 663 628 597 569 543

628 593 562 534 509 485

568 537 508 483 460 439

514 485 459 437 416 397

461 435 412 392 373 356

406 383 363 345 329 314

372 352 333 317 301 288

328 310 294 279 266 254

288 272 257 245 233 222

23 24 25 26 27 28

577 553 530 510 491 474

519 498 478 459 442 426

464 445 427 411 396 381

420 403 386 372 358 345

380 364 349 336 323 312

340 326 313 301 290 280

300 288 276 265 256 246

275 264 253 243 234 226

243 233 223 215 207 199

213 204 196 188 181 175

29 30 31 32 33 34

457 442 428 414 402 390

412 398 385 373 362 351

368 356 345 334 324 314

333 322 312 302 293 284

301 291 282 273 265 257

270 261 253 245 237 230

238 230 223 216 209 203

218 211 204 198 192 186

192 186 180 174 169 164

169 163 158 153 148 144

35 36 37 38 39 40

379 368 358 349 340 332

341 332 323 314 306 299

305 297 289 281 274 267

276 268 261 254 248 242

249 243 236 230 224 218

224 218 212 206 201 196

197 192 186 182 177 173

181 176 171 167 162 158

159 155 151 147 143 140

140 136 132 129 125 122

42 44

316 301

284 271

254 243

230 220

208 198

186 178

164 157

151 144

133 127

116 111

442 13300 527 293 48.0 449 26.9 412 35.8 449

398 11900 482 250 44.5 382 23.9 350 31.9 395

230 6900 298 120 29.5 158 12.6 143 16.8 199

211 6330 269 104 26.8 132 10.2 119 13.7 165

186 5580 238 86.3 24.0 105 8.45 94.9 11.3 133

163 4890 209 73.0 21.3 82.4 6.71 74.3 8.94 104


356 10700 431 215 40.5 315 20.2 289 27.0 347

322 9660 384 183 36.5 258 16.4 237 21.8 301

291 8730 348 157 33.5 217 14.1 199 18.8 262

261 7830 335 143 32.8 197 15.1 178 20.2 246




Fy = 50 ksi

4 - 87


W 18


Span (ft)

Fy = 50 ksi

Wt./ft

W 18 71

65

6 7 8 9 10 11

494 483 435 395

446 443 399 363

12 13 14 15 16 17

363 335 311 290 272 256

18 19 20 21 22 23

60

W 18 55

50

46

40

35

409 369 335

381 373 336 305

345 337 303 275

351 340 302 272 247

304 294 261 235 214

287 285 249 222 200 181

333 307 285 266 249 235

308 284 264 246 231 217

280 258 240 224 210 198

253 233 216 202 189 178

227 209 194 181 170 160

196 181 168 157 147 138

166 153 143 133 125 117

242 229 218 207 198 189

222 210 200 190 181 173

205 194 185 176 168 160

187 177 168 160 153 146

168 159 152 144 138 132

151 143 136 130 124 118

131 124 118 112 107 102

111 105 100 95 91 87

24 25 26 27 28 29

181 174 167 161 155 150

166 160 153 148 143 138

154 148 142 137 132 127

140 134 129 124 120 116

126 121 117 112 108 104

113 109 105 101 97 94

98 94 90 87 84 81

83 80 77 74 71 69

30 31 32 33 34 35

145 140 136 132 128 124

133 129 125 121 117 114

123 119 115 112 109 105

112 108 105 102 99 96

101 98 95 92 89 87

91 88 85 82 80 78

78 76 74 71 69 67

67 64 62 60 59 57

36 38 40 42 44

121 114 109 104 99

111 105 100 95 91

103 97 92 88 84

93 88 84 80 76

84 80 76 72 69

76 72 68 65 62

65 62 59 56 53

55 53 50 48 45

90.7 2720 176 56.3 18.0 60.6 4.62 55.0 6.16 75.6

78.4 2350 152 46.8 15.8 46.2 3.60 41.9 4.80 57.9

66.5 2000 143 42.2 15.0 38.6 3.88 34.0 5.18 51.3


145 4350 247 92.8 24.8 113 8.77 102 11.7 142

133 3990 223 80.9 22.5 94.3 7.16 85.5 9.55 118

123 3690 204 71.3 20.8 80.4 6.10 73.0 8.13 100

112 3360 191 64.0 19.5 69.7 5.62 62.9 7.50 88.0

101 3030 172 55.5 17.8 57.6 4.72 51.9 6.29 72.9



4 - 88



W 16

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 16 100

89

77

6 7 8 9 10 11

536

475

406

12 13 14 15 16 17

495 457 424 396 371 349

438 404 375 350 328 309

18 19 20 21 22 23

330 313 297 283 270 258

24 25 26 27 28 29

W 16 67

57

50

45

W 16 40

36

31

26 212 189 166 147 133 121

348

382 350 315 286

334 307 276 251

301 274 247 224

264 243 219 199

253 240 213 192 175

236 231 203 180 162 147

375 346 321 300 281 265

325 300 279 260 244 229

263 242 225 210 197 185

230 212 197 184 173 162

206 190 176 165 154 145

182 168 156 146 137 129

160 148 137 128 120 113

135 125 116 108 101 95

111 102 95 88 83 78

292 276 263 250 239 228

250 237 225 214 205 196

217 205 195 186 177 170

175 166 158 150 143 137

153 145 138 131 125 120

137 130 123 118 112 107

122 115 109 104 99 95

107 101 96 91 87 83

90 85 81 77 74 70

74 70 66 63 60 58

248 238 228 220 212 205

219 210 202 194 188 181

188 180 173 167 161 155

163 156 150 144 139 134

131 126 121 117 113 109

115 110 106 102 99 95

103 99 95 91 88 85

91 87 84 81 78 75

80 77 74 71 69 66

68 65 62 60 58 56

55 53 51 49 47 46

30 31 32 33 34 35

198 192 186 180 175 170

175 169 164 159 154 150

150 145 141 136 132 129

130 126 122 118 115 111

105 102 98 95 93 90

92 89 86 84 81 79

82 80 77 75 73 71

73 71 68 66 64 62

64 62 60 58 56 55

54 52 51 49 48 46

44 43 41 40 39 38

36 38 40

165 156 149

146 138 131

125 118 113

108 103 98

88 83 79

77 73 69

69 65 62

61 58 55

53 51

45 43

37 35

198 5940 268 123 29.2 160 13.0 145 17.3 202

175 5250 237 103 26.2 128 10.7 116 14.2 163

72.9 2190 132 45.3 15.3 43.2 3.80 39.1 5.06 55.6

64.0 1920 126 41.5 14.7 37.9 4.07 33.6 5.43 51.2

54.0 1620 118 38.7 13.8 34.5 3.22 31.1 4.29 45.0

44.2 1330 106 33.2 12.5 26.5 3.12 23.2 4.16 36.7


150 4500 203 81.8 22.8 96.5 8.12 87.5 10.8 123

130 3900 174 67.9 19.8 73.0 6.14 66.3 8.19 93.0

105 3150 191 73.9 21.5 86.0 7.32 78.0 9.76 110

92.0 2760 167 62.3 19.0 67.1 5.80 60.8 7.73 85.9

82.3 2470 150 53.9 17.3 54.9 4.87 49.7 6.50 70.8




Fy = 50 ksi

4 - 89


W 14


W 14 132

120

109

W 14 99*

90*

82

74

68

61

344 344 315 291 270 252

315 314 288 265 246 230

281 278 255 235 219 204

461 454 424

406 384

371 370 345

333 329 307

16 17 18 19 20 21

439 413 390 369 351 334

398 374 353 335 318 303

360 339 320 303 288 274

323 304 287 272 259 246

288 271 256 243 230 219

261 245 232 219 209 199

236 222 210 199 189 180

216 203 192 182 173 164

191 180 170 161 153 146

22 23 24 25 26 27

319 305 293 281 270 260

289 277 265 254 245 236

262 250 240 230 222 213

235 225 216 207 199 192

210 200 192 184 177 171

190 181 174 167 160 154

172 164 158 151 145 140

157 150 144 138 133 128

139 133 128 122 118 113

28 29 30 31 32 33

251 242 234 226 219 213

227 219 212 205 199 193

206 199 192 186 180 175

185 178 172 167 162 157

165 159 154 149 144 140

149 144 139 135 130 126

135 130 126 122 118 115

123 119 115 111 108 105

109 106 102 99 96 93

34

206

187

169

152

136

123

111

101

90


234 7020 255 136 32.3 190 19.2 171 25.6 241

212 6360 231 120 29.5 158 16.3 143 21.8 213

126 3780 172 87.9 22.5 96.5 8.86 88.1 11.8 126

115 3450 157 77.8 20.8 81.8 7.65 74.6 10.2 108

102 3060 141 67.4 18.8 66.5 6.37 60.6 8.49 88.2

Span (ft)

511 501 468

394 379 348 321 298 278

Fy = 50 ksi

10 11 12 13 14 15


173 5170 185 87.1 24.3 108 11.2 97.0 14.9 145

157 4610 167 75.6 22.0 88.7 9.26 80.0 12.3 120

139 4170 197 104 25.5 121 11.7 110 15.6 161

*Noncompact shape; Fy = 50 ksi. Load above heavy line is limited by design shear strength.


4 - 90



W 14

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 14 53

48

5 6 7 8 9 10

278 261

253 235

11 12 13 14 15 16

238 218 201 187 174 163

17 18 19 20 21 22

W 14 43

38

34

W 14 30

26

22

225 209

236 231 205 185

215 205 182 164

202 177 158 142

192 172 151 134 121

171 166 142 125 111 100

214 196 181 168 157 147

190 174 161 149 139 131

168 154 142 132 123 115

149 137 126 117 109 102

129 118 109 101 95 89

110 101 93 86 80 75

91 83 77 71 66 62

154 145 138 131 124 119

138 131 124 118 112 107

123 116 110 104 99 95

109 103 97 92 88 84

96 91 86 82 78 74

83 79 75 71 68 65

71 67 63 60 57 55

59 55 52 50 47 45

23 24 25 26 27 28

114 109 105 101 97 93

102 98 94 90 87 84

91 87 84 80 77 75

80 77 74 71 68 66

71 68 66 63 61 59

62 59 57 55 53 51

52 50 48 46 45 43

43 42 40 38 37 36

29 30 31 32 33 34

90 87 84 82 79 77

81 78 76 74 71 69

72 70 67 65 63 61

64 62 60 58 56 54

56 55 53 51 50 48

49 47 46 44 43 42

42 40 39 38 37 35

34 33 32 31 30 29

47.3 1420 101 31.6 13.5 31.4 4.00 27.7 5.33 45.0

40.2 1210 95.8 29.9 12.8 30.1 3.07 27.2 4.09 40.6

33.2 996 85.3 25.2 11.5 23.0 2.86 20.4 3.81 32.8


87.1 2610 139 66.5 18.5 65.9 5.96 60.4 7.95 86.2

78.4 2350 127 58.4 17.0 55.1 5.18 50.4 6.91 72.8

69.6 2090 112 50.0 15.3 44.2 4.24 40.4 5.65 58.7

61.5 1850 118 41.2 15.5 44.7 4.44 40.5 5.92 59.7

54.6 1640 108 35.6 14.3 37.0 3.94 33.3 5.25 50.4




Fy = 50 ksi

4 - 91


W 12


W 12 120

106

96

87

W 12 79

72

65*

58

53 225 212 195 180 167 156

425 410 378 351 328

377 368 339 315 294

348 330 305 283 264

314 298 275 255 238

284 270 249 231 216

255 238 220 204 191

16 17 18 19 20 21

349 328 310 294 279 266

308 289 273 259 246 234

276 259 245 232 221 210

248 233 220 208 198 189

223 210 198 188 179 170

203 191 180 171 162 154

179 168 159 151 143 136

162 152 144 136 130 123

146 137 130 123 117 111

22 23 24 25 26 27

254 243 233 223 215 207

224 214 205 197 189 182

200 192 184 176 170 163

180 172 165 158 152 147

162 155 149 143 137 132

147 141 135 130 125 120

130 124 119 114 110 106

118 113 108 104 100 96

106 102 97 93 90 87

28 29 30

199 192 186

176 170 164

158 152 147

141 137 132

128 123 119

116 112 108

102 99 95

93 89 86

83 81 78


186 5580 252 161 35.5 227 26.7 203 35.6 276

164 4920 212 129 30.5 171 19.2 154 25.7 228

96.8 2860 128 64.0 19.5 68.3 8.75 61.2 11.7 99.2

86.4 2590 118 61.9 18.0 62.3 6.47 57.1 8.63 85.1

77.9 2340 112 53.9 17.3 55.4 6.41 50.3 8.54 78.0

Span (ft)

503 465 429 399 372

237 236 216 199 185 173

Fy = 50 ksi

10 11 12 13 14 15


132 3960 174 96.6 25.8 120 14.6 108 19.4 171

119 3570 157 84.5 23.5 99.6 12.3 89.4 16.5 143

108 3240 142 73.9 21.5 83.2 10.5 74.7 14.0 120

*Indicates noncompact shape; Fy = 50 ksi. Load above heavy line is limited by design shear strength. Values of R in bold face exceed maximum design web shear φvVn.


4 - 92



W 12

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 12 50

45

4 5 6 7 8 9

244 241

218 216

10 11 12 13 14 15

217 197 181 167 155 145

16 17 18 19 20 21

W 12 40

35

30

W 12 26

22

19

16

14

154 148 124 106 93 82

142 121 101 86 75 67

129 104 87 75 65 58

190

203 192 171

173 162 144

152 140 124

173 147 126 110 98

194 176 162 149 139 129

173 157 144 133 123 115

154 140 128 118 110 102

129 118 108 99 92 86

112 101 93 86 80 74

88 80 73 68 63 59

74 67 62 57 53 49

60 55 50 46 43 40

52 47 44 40 37 35

136 128 121 114 109 103

121 114 108 102 97 92

108 101 96 91 86 82

96 90 85 81 77 73

81 76 72 68 65 62

70 66 62 59 56 53

55 52 49 46 44 42

46 44 41 39 37 35

38 35 34 32 30 29

33 31 29 27 26 25

22 23 24 25 26 27

99 94 91 87 84 80

88 84 81 78 75 72

78 75 72 69 66 64

70 67 64 61 59 57

59 56 54 52 50 48

51 49 47 45 43 41

40 38 37 35 34 33

34 32 31 30 29 27

27 26 25 24 23 22

24 23 22 21 20 19

28 29 30

78 75 72

69 67 65

62 59 51

55 53 43

46 45 37

40 38 29

31 30 25

26 26

22 21

19 18

72.4 2170 122 63.6 18.5 64.9 7.02 59.2 9.37 89.7

64.7 1940 109 52.3 16.8 53.0 5.87 48.3 7.82 73.7

29.3 879 86.4 28.4 13.0 31.2 3.63 28.2 4.85 43.9

24.7 741 77.2 23.9 11.8 24.3 3.30 21.6 4.40 35.9

20.1 603 71.2 20.6 11.0 19.2 3.63 16.3 4.83 32.0

17.4 522 64.3 17.2 10.0 15.3 3.23 12.7 4.31 26.7


57.5 1730 95.1 46.1 14.7 41.5 4.52 37.9 6.02 57.4

51.2 1540 101 37.5 15.0 42.7 4.49 39.0 5.99 58.5

43.1 1290 86.6 30.5 13.0 31.7 3.50 28.8 4.67 44.0

37.2 1120 75.9 25.2 11.5 24.5 2.83 22.2 3.78 34.5




Fy = 50 ksi

4 - 93


W 10


W 10 112

100

88

77

68

60

54

49

9 10 11 12 13 14

463 441 401 368 339 315

408 390 355 325 300 279

354 339 308 283 261 242

303 293 266 244 225 209

264 256 233 213 197 183

232 224 203 187 172 160

202 200 182 167 154 143

183 181 165 151 139 129

15 16 17 18 19 20

294 276 259 245 232 221

260 244 229 217 205 195

226 212 199 188 178 170

195 183 172 163 154 146

171 160 151 142 135 128

149 140 132 124 118 112

133 125 118 111 105 100

121 113 107 101 95 91

21 22 23 24

210 200 192 184

186 177 170 163

161 154 147 141

139 133 127 122

122 116 111 107

107 102 97 93

95 91 87 83

86 82 79 76

74.6 2240 116 68.9 21.0 80.9 11.5 73.1 15.4 123

66.6 2000 101 57.8 18.5 63.6 8.83 57.7 11.8 96.0

60.4 1810 91.6 50.5 17.0 53.5 7.61 48.4 10.1 81.4

Span (ft)

Fy = 50 ksi

Wt./ft


147 4410 232 177 37.8 265 32.8 240 43.7 300

130 3900 204 149 34.0 214 27.4 194 36.5 259

113 3390 177 123 30.3 169 22.3 153 29.8 221

97.6 2930 152 99.4 26.5 130 17.5 117 23.3 185

85.3 2560 132 80.8 23.5 102 14.0 92.2 18.7 153

Load above heavy line is limited by design shear strength. Values of φR (N = 31⁄4) in boldface exceed maximum web shear φvVn.


4 - 94



W 10

Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

W 10 45

3 4 5 6 7 8

191

9 10 11 12 13 14

39

W 10 33

30

26

W 10 22

19

17

15

12* 101 94 75 63 54 47

138 130 108 93 81

131 112 94 80 70

124 120 96 80 69 60

169

152 146

170 157 137

145 134 117

132 130 111 98

183 165 150 137 127 118

156 140 128 117 108 100

129 116 106 97 90 83

122 110 100 91 84 78

104 94 85 78 72 67

87 78 71 65 60 56

72 65 59 54 50 46

62 56 51 47 43 40

53 48 44 40 37 34

42 38 34 31 29 27

15 16 17 18 19 20

110 103 97 92 87 82

94 88 83 78 74 70

78 73 68 65 61 58

73 69 65 61 58 55

63 59 55 52 49 47

52 49 46 43 41 39

43 41 38 36 34 32

37 35 33 31 30 28

32 30 28 27 25 24

25 23 22 21 20 19

21 22 23 24

78 75 72 69

67 64 61 59

55 53 51 49

52 50 48 46

45 43 41 39

37 35 34 33

31 29 28 27

27 26 24 23

23 22 21 20

18 17 16 16

54.9 1650 95.4 54.7 17.5 58.8 7.41 53.8 9.88 85.9

46.8 1400 84.4 44.3 15.8 46.4 6.43 42.2 8.58 70.0

21.6 648 69.1 25.4 12.5 28.3 4.18 25.5 5.57 43.6

18.7 561 65.5 22.5 12.0 24.4 4.48 21.3 5.98 40.8

16.0 480 62.0 19.8 11.5 20.7 4.88 17.4 6.51 38.6

12.6 376 50.6 14.8 9.50 13.7 3.58 11.3 4.77 26.8


38.8 1160 76.2 38.5 14.5 37.1 6.23 33.1 8.31 60.1

36.6 1100 84.8 35.2 15.0 42.3 5.47 38.5 7.29 62.2

31.3 939 72.5 28.4 13.0 31.7 4.18 28.8 5.58 47.0

26.0 780 65.9 22.5 12.0 25.4 4.08 22.7 5.45 40.4

*Indicates noncompact shape; Fy = 50 ksi. Load above heavy line is limited by design shear strength.



Fy = 50 ksi

4 - 95


W8


W8 67

58

48

7 8 9 10 11 12

277 263 234 211 191 175

40

35

31

241 224 199 179 163 150

184 163 147 134 123

160 149 133 119 109 100

136 130 116 104 95 87

123 114 101 91 83 76

13 14 15 16 17 18

162 150 140 132 124 117

138 128 120 112 106 100

113 105 98 92 86 82

92 85 80 75 70 66

80 74 69 65 61 58

70 65 61 57 54 51

19 20

111 105

94 90

77 74

63 60

55 52

48 46

39.8 1190 80.2 47.8 18.0 58.3 10.9 52.3 14.6 99.6

34.7 1040 68.0 38.8 15.5 43.8 8.02 39.5 10.7 74.2

30.4 912 61.6 33.4 14.3 36.2 7.20 32.3 9.60 63.5

Span (ft)

Fy = 50 ksi

Wt./ft


70.2 2110 139 102 28.5 150 23.8 136 31.7 195

59.8 1790 120 83.7 25.5 118 20.2 106 27.0 167

49.0 1470 91.8 59.4 20.0 75.5 11.9 68.8 15.9 120



4 - 96



W8

Fy = 50 ksi


W8

Span (ft)

Fy = 50 ksi

Wt./ft

28

W8 24

21

W8 18

15

13

10*

99 86 68 57 49 43

72 66 53 44 38 33

3 4 5 6 7 8

124 117 102

105 99 87

112 102 87 77

101 85 73 64

107 102 82 68 58 51

9 10 11 12 13 14

91 82 74 68 63 58

77 70 63 58 54 50

68 61 56 51 47 44

57 51 46 43 39 36

45 41 37 34 31 29

38 34 31 29 26 24

29 26 24 22 20 19

15 16 17 18 19 20

54 51 48 45 43 41

46 44 41 39 37 31

41 38 36 34 32 26

34 32 30 28 27 20

27 26 24 23 21

23 21 20 19 18

18 16 16 15 14

13.6 408 53.6 23.0 12.3 24.5 6.23 21.2 8.30 48.2

11.4 342 49.6 19.8 11.5 20.1 6.46 16.6 8.61 44.6

8.87 264 36.2 13.3 8.50 11.4 3.29 9.72 4.38 24.0


27.2 816 62.0 33.4 14.3 37.4 6.68 33.8 8.91 62.8

23.2 696 52.5 26.8 12.3 27.7 5.02 25.0 6.69 46.7

20.4 612 55.9 25.4 12.5 28.5 5.10 25.7 6.81 47.8

17.0 510 50.5 21.6 11.5 22.9 4.90 20.2 6.53 41.4




Fy = 50 ksi

4 - 97


W 6–5–4


W6 25

2 3 4 5 6 7

110 95 81

8 9 10 11 12 13

20

W6 15*

16

W5

12

9

75 62 50 42 36

54 47 37 31 27

71 63 57 52 47 44

56 50 45 41 37 34

38 34 31 28 26 24

44 39 35 32 29 27

31 28 25 23 21 19

23 21 19 17 16 14

14

41

32

22

25

18

13


18.9 567 55.1 32.5 16.0 44.0 12.2 38.8 16.3 84.5

14.9 447 43.5 24.4 13.0 28.9 8.40 25.4 11.2 61.8

W4 16

13

75 70 58 50

65 58 48 41

63 63 47 38 31 27

44 39 35 32 29

36 32 29 26 24

24 21 19

11.6 348 37.5 27.4 13.5 33.2 9.62 29.9 12.8 71.3

9.59 288 32.5 22.5 12.0 25.4 8.29 22.7 11.1 58.6

6.28 188 31.4 24.1 14.0 31.4 16.5 26.8 22.1 69.6

Span (ft)

88 88 70 59 50

Fy = 50 ksi

87 75 64

74 62 51 44

19


11.7 351 44.1 24.4 13.0 30.4 7.48 27.3 9.97 59.7

8.30 249 37.4 18.0 11.5 21.0 7.80 17.9 10.4 51.7

6.23 187 27.1 12.0 8.50 11.7 4.19 10.1 5.59 28.2



4 - 98


S 24–20


Fy = 50 ksi


S 24 121

12 13 14 15 16 17

90

S 20 80

96

S 20 86

75

66

723 686 610 549 499

686 656 574 510 459 417

545 525 467 420 382

820 761

966 900 800 720 655

810 740 666 605

648 612 556

765 706 656 612 574 540

698 644 598 558 523 492

600 554 514 480 450 424

555 512 476 444 416 392

510 471 437 408 383 360

495 457 424 396 371 349

458 422 392 366 343 323

383 353 328 306 287 270

350 323 300 280 263 247

18 19 20 21 22 23

510 483 459 437 417 399

465 441 419 399 380 364

400 379 360 343 327 313

370 351 333 317 303 290

340 322 306 291 278 266

330 313 297 283 270 258

305 289 275 261 250 239

255 242 230 219 209 200

233 221 210 200 191 183

24 25 26 27 28 29

383 367 353 340 328 317

349 335 322 310 299 289

300 288 277 267 257 248

278 266 256 247 238 230

255 245 235 227 219 211

248 238 228 220 212 205

229 220 211 203 196 189

191 184 177 170 164 158

175 168 162 156 150 145

30 32 34 36 38 40

306 287 270 255 242 230

279 262 246 233 220 209

240 225 212 200 189 180

222 208 196 185 175 167

204 191 180 170 161 153

198 186 175 165 156 149

183 172 161 153 144 137

153 143 135 128 121 115

140 131 124 117 111 105

42 44 46 48 50 52

219 209 200 191 184 177

199 190 182 174 167 161

171 164 157 150 144 138

159 151 145 139 133 128

146 139 133 128 122 118

141 135 129 124 119

131 125 119 114 110

109 104 100 96 92

100 95 91 88 84

54 56 58 60

170 164 158 153

155 149 144 140

133 129 124 120

123 119 115 111

113 109 106 102


306 9180 529 200 40.0 269 20.7 236 27.7 330

279 8370 410 155 31.0 184 9.66 168 12.9 215

183 5490 362 144 33.0 185 16.7 163 22.2 240

153 4590 343 129 31.8 163 17.4 139 23.2 219

140 4200 273 103 25.3 115 8.76 104 11.7 144

Span (ft)

1060 1020 918 835

100

877 849 743 660 594 540

Fy = 50 ksi

6 7 8 9 10 11

S 24 106


222 6660 405 137 31.3 166 12.6 146 16.9 207

204 6120 324 109 25.0 119 6.48 109 8.64 140

198 5940 438 175 40.0 248 29.7 207 39.5 305




Fy = 50 ksi

4 - 99

BEAMS S 18–15–12–10 S Shapes Maximum factored uniform loads in kips for beams laterally supported For beams laterally unsupported, see page 4-139

Designation

Span (ft)

Fy = 50 ksi

Wt./ft

S 18 70

3 4 5 6 7 8

691 625 536 469

9 10 11 12 13 14

S 15

54.7

50

S 12

42.9

50

S 12

40.8

35

S 10

31.8

35

25.4

168 142 122 107

333 297 260

445 367 306 262 230

299 266 228 199

277 269 224 192 168

227 210 180 158

321 266 212 177 152 133

257 231 210 193 178 165

231 208 189 173 160 149

204 184 167 153 141 131

177 159 145 133 123 114

149 134 122 112 103 96

140 126 115 105 97 90

118 106 97 89 82 76

95 85 77 71 66 61

210 197 185 175 166 158

154 145 136 129 122 116

139 130 122 116 109 104

122 115 108 102 97 92

106 100 94 89 84 80

90 84 79 75 71 67

84 79 74 70 66 63

71 66 62 59 56 53

57 53 50 47 45 43

179 170 163 156 150 144

150 143 137 131 126 121

110 105 101 96 93 89

99 95 90 87 83 80

87 83 80 77 73 71

76 72 69 66 64 61

64 61 58 56 54 52

60 57 55 53 50 48

51 48 46 44 42

41 39 37 36 34

27 28 29 30 31 32

139 134 129 125 121 117

117 113 109 105 102 98

86 83 80 77 75 72

77 74 72 69 67 65

68 66 63 61

59 57 55 53

50 48 46 45

47 45 43 42

33 34 35 36 37 38

114 110 107 104 101 99

95 93 90 88 85 83

70 68 66 64 63

63 61 59 58 56

40 42 44

94 89 85

79 75 72

125 3750 346 133 35.6 180 31.3 142 41.7 249

105 3150 224 86.4 23.1 93.8 8.52 83.6 11.4 122

44.8 1340 139 63.5 21.4 74.5 13.0 64.1 17.3 120

42.0 1260 113 52.0 17.5 55.1 7.11 49.4 9.47 80.2

35.4 1060 160 83.5 29.7 116 46.2 84.9 61.6 180

28.4 852 84.0 43.7 15.5 43.8 6.63 39.4 8.84 68.1

448 394

446 386 330 289

417 375 341 313 288 268

350 315 286 263 242 225

15 16 17 18 19 20

250 234 221 208 197 188

21 22 23 24 25 26


77.1 2310 223 94.5 27.5 116 19.3 96.7 25.7 180

69.3 2080 166 70.6 20.6 74.9 8.05 66.9 10.7 102

61.2 1840 223 123 34.3 167 44.4 131 59.1 235

53.1 1590 150 83.0 23.1 91.9 13.5 81.1 18.0 140



4 - 100


S 8–6–5–4–3


Fy = 50 ksi


S8 23

1 2 3 4 5 6

191 145 116 96

7 8 9 10 11 12

S6 18.4

17.25

S5 12.5

9.5

75 64 51 42

58 57 43 34 28

83 72 64 58 53 48

71 62 55 50 45 41

45 40 35 32 29 27

36 32 28 25 23 21

24 21 19 17 15 14

13

45

38

24

20


19.3 579 95.3 55.1 22.1 68.9 27.2 54.4 36.3 127

16.5 495 58.5 33.9 13.6 33.2 6.32 29.8 8.42 57.2

S3 7.7

7.5

5.7

70 61 40 30 24 20

42 35 26 21 18

57 35 24 18 14 12

28 20 15 12 9.75

17 15 13 12

15 13 12 11

10

8.36

4.04 121 35.2 30.6 16.3 36.3 32.0 27.8 42.6 83.5

3.51 105 20.8 18.1 9.65 16.6 6.64 14.8 8.85 43.5

2.36 70.8 28.3 30.0 17.5 37.9 59.0 26.1 78.6 86.7

1.95 58.5 13.8 14.6 8.50 12.9 6.81 11.5 9.09 41.1

Span (ft)

117 99 83

151 106 80 64 53

Fy = 50 ksi

S4

10

Properties and Reaction Values 10.6 318 75.3 50.9 23.3 68.5 50.5 48.3 67.3 126

8.47 254 37.6 25.4 11.6 24.1 6.27 21.6 8.36 48.8

5.67 170 28.9 21.7 10.7 20.4 6.50 18.2 8.67 46.4




Fy = 50 ksi

4 - 101


MC,C 18–15


MC 18

Span (ft)

Fy = 50 ksi

Wt./ft

58

51.9

C 15 45.8

42.7

50

40

33.9

421 343 286 245 215

324 302 252 216 189

3 4 5 6 7 8

680 568 473 405 355

583 519 433 371 324

486 470 392 336 294

437 372 319 279

580 511 409 341 292 256

9 10 11 12 13 14

315 284 258 237 218 203

288 260 236 216 200 185

261 235 214 196 181 168

248 223 203 186 172 159

227 205 186 170 157 146

191 172 156 143 132 123

168 151 137 126 116 108

15 16 17 18 19 20

189 177 167 158 149 142

173 162 153 144 137 130

157 147 138 131 124 118

149 140 131 124 117 112

136 128 120 114 108 102

114 107 101 95 90 86

101 95 89 84 80 76

21 22 23 24 25 26

135 129 123 118 114 109

124 118 113 108 104 100

112 107 102 98 94 90

106 101 97 93 89 86

97 93 89 85 82 79

82 78 75 72 69 66

72 69 66 63 60 58

28 30 32 34 36 38

101 95 89 83 79 75

93 87 81 76 72 68

84 78 74 69 65 62

80 74 70 66 62 59

73 68 64 60 57

61 57 54 50 48

54 50 47 44 42

40 42 44

71 68 65

65 62 59

59 56 53

56 53 51

68.2 2050 290 129 35.8 176 40.7 135 54.3 245

57.2 1720 211 93.4 26.0 109 15.6 93.4 20.8 161

50.4 1510 162 71.9 20.0 73.6 7.10 66.5 9.47 97.2


94.6 2840 340 120 35.0 167 33.0 127 44.0 234

86.5 2600 292 103 30.0 133 20.8 108 27.7 200

78.4 2350 243 85.9 25.0 101 12.0 86.4 16.0 140

74.4 2230 219 77.3 22.5 86.1 8.76 75.5 11.7 115



4 - 102



MC 13

Fy = 50 ksi


MC 13

Span (ft)

Fy = 50 ksi

Wt./ft

50

40

35

31.8

3 4 5 6 7 8

552 454 363 303 259 227

393 382 305 255 218 191

314 277 231 198 173

263 259 216 185 162

9 10 11 12 13 14

202 182 165 151 140 130

170 153 139 127 117 109

154 139 126 116 107 99

144 129 118 108 99 92

15 16 17 18 19 20

121 113 107 101 96 91

102 95 90 85 80 76

92 87 82 77 73 69

86 81 76 72 68 65

21 22 23 24 25 26

86 83 79 76 73 70

73 69 66 64 61 59

66 63 60 58 55 53

62 59 56 54 52 50

27 28 29 30 31 32

67 65 63 61 59 57

57 55 53 51 49 48

51 50 48 46 45 43

48 46 45 43 42 40

46.2 1390 157 76.8 22.4 84.2 12.2 73.6 16.2 126

43.1 1290 132 64.5 18.8 64.7 7.19 58.4 9.59 89.6


60.5 1820 276 135 39.3 197 66.5 139 88.7 263

50.9 1530 197 96.3 28.0 118 24.0 97.3 31.9 187




Fy = 50 ksi

4 - 103


C,MC 12


C 12 50

45

40

251 219 175 146 125

183 152 127 109

541 421 337 281 240

461 388 310 259 222

382 355 284 237 203

126 112 101 92 84 78

110 97 88 80 73 67

95 85 76 69 64 59

210 187 168 153 140 129

194 172 155 141 129 119

14 15 16 17 18 19

72 67 63 59 56 53

63 58 55 52 49 46

54 51 48 45 42 40

120 112 105 99 94 89

20 21 22 23 24 25

50 48 46 44 42 40

44 42 40 38 37 35

38 36 35 33 32 30

26 27 28 29 30

39 37 36 35 34

34 32 31 30 29

29 28 27 26 25


33.6 1010 165 71.7 25.5 93.0 23.9 73.9 31.8 155

29.2 876 125 54.4 19.4 61.5 10.4 53.1 13.9 98.3

Span (ft)

25

2 3 4 5 6 7

330 252 202 168 144

8 9 10 11 12 13

MC 12 20.7

Fy = 50 ksi

30

MC 12 35

31

10.6

303 257 214 183

240 236 197 168

123 116 87 70 58 50

177 158 142 129 118 109

161 143 128 117 107 99

147 131 118 107 98 91

44 39 35 32 29 27

111 103 97 91 86 82

101 95 89 83 79 75

92 86 80 76 71 68

84 79 74 69 66 62

25 23 22 20 19 18

84 80 77 73 70 67

78 74 71 67 65 62

71 68 65 62 59 57

64 61 58 56 54 51

59 56 54 51 49 47

17 17 16 15 15 14

65 62 60 58 56

60 57 55 53 52

55 53 51 49 47

49 48 46 44 43

45 44 42 41 39

13 13 12 12 12

47.3 1420 191 96.8 29.5 137 26.5 116 35.3 193

42.8 1280 151 76.6 23.4 96.3 13.1 85.8 17.5 143

39.3 1180 120 60.7 18.5 67.9 6.52 62.7 8.70 91.0

11.6 348 61.6 16.3 9.50 16.6 2.00 15.0 2.67 23.7


56.1 1680 271 137 41.8 230 75.0 170 100.0 273

51.7 1550 231 117 35.6 181 46.5 144 62.0 233



4 - 104


C,MC 10


Fy = 50 ksi


Span (ft)

Fy = 50 ksi

Wt./ft

C 10

MC 10

30

25

20

2 3 4 5 6 7

363 266 200 160 133 114

284 230 173 138 115 99

8 9 10 11 12 13

100 89 80 73 67 61

14 15 16 17 18 19 20 21 22 23 24

33.6

MC 10 28.5

25

MC 10

15.3

41.1

22

205 193 145 116 96 83

130 119 95 79 68

430 389 292 233 195 167

311 251 200 167 143

230 222 178 148 127

205 194 155 129 111

157 142 118 101

92 79 59 47 39 34

8.4

86 77 69 63 58 53

72 64 58 53 48 45

59 53 47 43 40 36

146 130 117 106 97 90

125 111 100 91 84 77

111 99 89 81 74 68

97 86 77 70 65 60

89 79 71 64 59 54

29 26 24 21 20 18

57 53 50 47 44 42

49 46 43 41 38 36

41 39 36 34 32 30

34 32 30 28 26 25

83 78 73 69 65 61

72 67 63 59 56 53

63 59 56 52 49 47

55 52 48 46 43 41

51 47 44 42 39 37

17 16 15 14 13 12

40 38 36 35 33

35 33 31 30 29

29 28 26 25 24

24 23 22 21 20

58 56 53 51 49

50 48 46 44 42

44 42 40 39 37

39 37 35 34 32

35 34 32 31 30

12 11 11 10 9.83

26.6 798 182 84.1 33.6 131 75.6 81.0 101 193

23.0 690 142 65.8 26.3 90.8 36.1 66.8 48.1 151

29.6 888 115 66.4 21.3 75.8 14.4 66.1 19.3 129

25.8 774 103 59.4 19.0 64.1 10.3 57.2 13.8 102

23.6 708 78.3 45.3 14.5 42.7 4.59 39.6 6.12 59.5

7.86 236 45.9 14.6 8.50 13.4 1.90 12.1 2.53 20.3


19.3 579 102 47.4 19.0 55.6 13.5 46.6 18.0 105

15.8 474 64.8 30.0 12.0 28.0 3.43 25.7 4.57 40.6

38.9 1170 215 124 39.8 194 94.9 131 127 254

33.4 1000 155 89.8 28.8 119 35.8 95.4 47.7 183




Fy = 50 ksi

4 - 105


C,MC 9


C9

Span (ft)

Fy = 50 ksi

Wt./ft

MC 9

20

15

13.4

25.4

23.9

2 3 4 5 6 7

218 168 126 101 84 72

139 135 101 81 68 58

113 94 75 63 54

219 174 139 116 99

194 167 133 111 95

8 9 10 11 12 13

63 56 50 46 42 39

51 45 41 37 34 31

47 42 38 34 31 29

87 77 70 63 58 54

83 74 67 61 56 51

14 15 16 17 18 19

36 34 31 30 28 27

29 27 25 24 23 21

27 25 23 22 21 20

50 46 44 41 39 37

48 44 42 39 37 35

20 21 22

25 24 23

20 19 18

19 18 17

35 33 32

33 32 30

23.2 696 109 66.8 22.5 80.7 19.9 68.8 26.6 140

22.2 666 97.2 59.4 20.0 67.7 14.0 59.3 18.7 120


16.8 504 109 52.5 22.4 69.5 26.2 53.8 34.9 125

13.5 405 69.3 33.4 14.3 35.3 6.74 31.2 8.98 60.4

12.5 375 56.6 27.3 11.7 26.1 3.68 23.9 4.91 39.8



4 - 106



C,MC 8

Fy = 50 ksi


C8

Span (ft)

Fy = 50 ksi

Wt./ft

MC 8

MC 8

MC 8

18.75

13.75

11.5

22.8

21.4

20

18.7

8.5

1 2 3 4 5 6

210 207 138 104 83 69

131 109 82 65 55

95 72 57 48

184 141 113 94

162 135 108 90

173 162 122 97 81

152 116 92 77

77 69 52 41 35

7 8 9 10 11 12

59 52 46 41 38 35

47 41 36 33 30 27

41 36 32 29 26 24

81 70 63 56 51 47

77 68 60 54 49 45

69 61 54 49 44 41

66 58 51 46 42 39

30 26 23 21 19 17

13 14 15 16 17 18

32 30 28 26 24 23

25 23 22 20 19 18

22 20 19 18 17 16

43 40 38 35 33 31

42 39 36 34 32 30

37 35 32 30 29 27

36 33 31 29 27 26

16 15 14 13 12 12

19 20

22 21

17 16

15 14

30 28

28 27

26 24

24 23

11 10

16.2 486 86.4 56.3 20.0 64.5 17.3 55.3 23.1 121

15.4 462 76.2 49.6 17.7 53.5 11.9 47.1 15.9 98.7

6.91 207 38.7 16.8 8.95 15.2 2.49 13.9 3.33 24.7


13.8 414 105 57.1 24.3 76.5 40.1 55.2 53.4 136

10.9 327 65.4 35.5 15.2 37.6 9.65 32.4 12.9 74.2

9.55 287 47.5 25.8 11.0 23.2 3.69 21.3 4.92 37.3

18.8 564 92.2 63.4 21.3 72.9 20.1 62.2 26.7 133

18.0 540 81.0 55.7 18.8 60.0 13.6 52.8 18.1 112




Fy = 50 ksi

4 - 107


C,MC 7–6


C7 12.25

MC 7 9.8

22.7

C6

MC 6

MC 6

MC 6

19.1

13

10.5

8.2

18

16.3

15.1

12

102 92 62 46 37 31

65 51 38 31 26

123 115 86 69 58

122 102 77 61 51

102 97 73 58 48

100 74 55 44 37

1 2 3 4 5 6

119 84 63 50 42

79 71 53 43 36

190 162 122 97 81

133 107 86 72

142 109 73 54 44 36

7 8 9 10 11 12

36 31 28 25 23 21

31 27 24 21 19 18

69 61 54 49 44 41

61 54 48 43 39 36

31 27 24 22 20 18

26 23 21 18 17 15

22 19 17 15 14 13

49 43 38 35 31 29

44 38 34 31 28 26

42 36 32 29 26 24

32 28 25 22 20 18

13 14 15 16

19 18 17 16

16 15 14 13

37 35 32 30

33 31 29 27

17 16 15

14 13 12

12 11 10

27 25 23

24 22 20

22 21 19

17 16 15

8.40 252 59.3 34.3 15.7 38.4 13.1 32.3 17.4 85.4

7.12 214 39.7 23.0 10.5 21.0 3.91 19.2 5.21 36.1

11.5 345 61.4 50.3 19.0 58.0 20.7 49.7 27.6 112

10.2 306 60.8 49.8 18.8 57.1 20.0 49.1 26.7 111

9.69 291 51.2 42.0 15.8 44.2 12.0 39.4 16.0 91.3

7.38 221 50.2 31.5 15.5 38.1 14.3 32.4 19.1 81.9

Span (ft)

Fy = 50 ksi

Wt./ft


16.2 486 95.1 70.7 25.2 91.0 39.3 72.6 52.5 152

14.3 429 66.5 49.5 17.6 53.3 13.5 47.0 18.0 105

7.26 218 70.8 44.4 21.9 61.0 43.9 43.5 58.5 115

6.15 185 50.9 31.9 15.7 37.2 16.3 30.7 21.7 82.9

5.13 154 32.4 20.3 10.0 18.9 4.21 17.2 5.61 35.4



4 - 108



C 5–4–3

Fy = 50 ksi


C5 9

C4

C3

6.7

7.25

5.4

69 42 28 21 17 14

40 34 23 17 14 11

1 2 3 4 5 6

88 65 44 33 26 22

51 35 26 21 18

7 8 9 10 11 12

19 16 15 13 12 11

15 13 12 11 9.6 8.8

12 11 9.4 8.4

9.7 8.5 7.5 6.8

6

5

4.1

52 26 17 13 10 8.6

42 23 15 11 9.0 7.5

28 20 13 9.8 7.8 6.5

7.4

6.4

5.6

1.72 51.6 28.8 30.6 17.8 40.0 59.6 28.1 79.5 88.4

1.50 45.0 20.9 22.2 12.9 24.7 22.7 20.2 30.2 64.1

1.30 39.0 13.8 14.6 8.50 13.2 6.49 11.9 8.65 40.0

Span (ft)

Fy = 50 ksi

Wt./ft


4.36 131 43.9 30.5 16.3 37.8 23.2 30.1 30.9 83.3

3.51 105 25.7 17.8 9.50 16.9 4.64 15.3 6.18 35.4

2.81 84.3 34.7 27.6 16.1 35.7 30.2 27.6 40.3 79.7

2.26 67.8 19.9 15.8 9.20 15.5 5.69 14.0 7.59 38.6



DESIGN FLEXURAL STRENGTH OF BEAMS WITH UNBRACED LENGTH > Lp

4 - 109

DESIGN FLEXURAL STRENGTH OF BEAMS WITH UNBRACED LENGTH GREATER THAN Lp General Notes

Spacing of lateral bracing at distances greater than Lp creates a problem in which the designer is confronted with a given laterally unbraced length (usually less than the total span) along the compression flange, and a calculated required bending moment. The beam cannot be selected from its plastic section modulus alone, since depth, flange proportions, and other properties have an influence on its bending strength. The following charts show the design moment φbMn for W and M shapes of Fy = 36 ksi and Fy = 50 ksi steels, used as beams, with respect to the maximum unbraced length for which this moment is permissible. In bending, φb of 0.9 is given in Section F1.2 of the LRFD Specification. The charts extend over varying unbraced lengths, depending upon the flexural strengths of the beams represented. In general, they extend beyond most unbraced lengths frequently encountered in design practice. The design moment φbMn, kip-ft, is plotted with respect to the unbraced length with no consideration of the moment due to weight of the beam. Design moments are shown for unbraced lengths in feet, starting at spans less than Lp, for spans between Lp and Lr and for spans beyond Lr. The unbraced length Lp, in feet, with the limit indicated by a solid symbol, , is the maximum unbraced length of the compression flange, with Cb = 1.0, for which the design moment is given by φbMp, where Fy Lp = 300ry / √ Mp = ZxFy

(F1-4)

For those noncompact rolled shapes, which meet the requirements of compact sections Fy , but is less than 141 / √  Fy −Fr , the design moment is except that bf / 2tf exceeds 65 / √ obtained from Equation A-F1-3 in Appendix F1 of the LRFD Specification. This criterion applies to one W shape when Fy is equal to 36 ksi and to seven W shapes when Fy is equal to 50 ksi. (Noncompact W shapes are given on p. 4-7.) For the case Cb = 1.0 and noncompact shapes:  λ − λp  Mn′ = Mp − (Mp − Mr)    λr − λp 

(A-F1-3)

 Mp − Mn′  Lp′ = Lp + (Lr − Lp)    Mp − Mr  λ = bf / 2tf λp = 65 / √ Fy λr = 141 / √  Fy −Fr Lr =

ryX1  √ 1+√  1 + X2(Fy − Fr)2 Fy − Fr

X1 =

π Sx

 √

EGJA 2

(F1-6)

(F1-8) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 110


4Cw X2 = Iy

 Sx     GJ 

2

(F1-9)

Mr = (Fy − Fr )Sx Mp = ZxFy Fr = 10 ksi for rolled shapes

(F1-7)

The unbraced length in the charts may be either the total span or any part of the total span between braced points. The plots shown in these charts were computed for beams for which Cb = 1.0. When a moment gradient exists between points of bracing, Cb may be larger than unity. (See Table 4-1.) Using this larger value of Cb may provide a more liberal flexural strength for the section chosen if the unbraced length is greater than Lp. In these cases, the design moment can be determined using the provisions of Section F1.2a of the LRFD Specification.  Lb − Lp   φbMn = φbCb Mp − (Mp − Mr)   ≤ φbMp  Lr − Lp   The unbraced length Lr, ft, with the limit indicated by an open symbol , is the maximum unbraced length of the compression flange beyond which the design moment is governed by Specification Section F1.2b. For unbraced lengths greater than Lr:

φbMn = φbMcr = φbCb

π Lb

 √ 2

 πE  EIyGJ +   IyCw ≤ φbCbMr and φbMp  Lb 

In computing the points for the curves, Cb in the above formulas was taken as unity, E = 29,000 ksi and G = 11,200 ksi. The properties of the beams are taken from the Tables of Dimensions and Properties in Part 1 of this LRFD Manual. The beam strengths have been reduced by multiplying the nominal flexural strength Mn by 0.9, the resistance factor φb for flexure. Over a limited range of length, a given beam is the lightest available for various combinations of unbraced length and design moment. The charts are designed to assist in selection of the lightest available beam for the given combination. The solid portion of each curve indicates the most economical section by weight. The dashed portion of each curve indicates ranges in which a lighter weight beam will satisfy the loading conditions. The curves are plotted without regard to shear strength and deflection criteria, therefore due care must be exercised in their use. The curves do not extend beyond an arbitrary span/depth limit of 30. The following examples illustrate the use of the charts.

EXAMPLE 4-8

Given:

Using Fy = 50 ksi steel, determine the size of a “simple” framed girder with a span of 35 feet, which supports two equal concentrated loads. The factored loads produce a required moment of 440 kip-ft in the AMERICAN INSTITUTE OF STEEL CONSTRUCTION


4 - 111

center 15-ft section between the loads. The load points are laterally braced. Solution:

For this loading condition, Cb = 1.0 due to nearly uniform moment across the central portion of the span. Center section of 15 feet is longest unbraced length. With total span equal to 35 feet and Mn = 440 kip-ft, assume approximate weight of beam at 70 lbs/ft (equal to 0.07 kips/ft).  0.07 × (35)2  × 1.2 = 453 kip-ft Total Mu = 440 +  8   Entering chart, with unbraced length equal to 15 feet on the bottom scale (abscissa), proceed upward to meet the horizontal line corresponding to a design moment equal to 453 kip-ft on the left hand scale (ordinate). Any beam listed above and to the right of the point so located satisfies the design moment requirement. In this case, the lightest section satisfying this criterion is a W21×68, for which the total design moment with an unbraced length of 15 feet is 457 kip-ft. Use: W21×68 Note: If depth is limited, a W14×82 could be selected, provided deflection conditions are not critical.

EXAMPLE 4-9

Given:

A “fixed end” girder with a span of 60 feet supports a concentrated load at the center. The compression flange is laterally supported at the concentrated load point and at the inflection points. The factored load produces a maximum calculated moment of 440 kip-ft at the load point and the supports. Determine the size of the beam using Fy = 50 ksi steel.

Solution:

For this loading condition, Cb = 1.67 (by comparison with Table 4-1), with an unbraced length of 15 feet. With the total span equal to 60 feet and Mu = 440 kip-ft, assume approximate weight of beam at 60 lbs/ft (0.06 kips/ft).  0.06 × (60)2  Total Mu = 440 +  ×1.2 24   = 451 kip-ft at the centerline and 462 at the supports Compute Mequiv by dividing the required design moment by Cb

Mequiv = 462 / 1.67 = 277 kip-ft Enter charts with unbraced length equal to 15 feet and proceed upward to 277 kip-ft. Any beam listed above and to the right of the point satisfies the design moment. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 112


The lightest section satisfying the criteria of a design moment of 277 kip-ft at an unbraced length of 15 feet and φbMp greater than 462 kip-ft is a W21×62. The design moment for a W21×62 with an unbraced length of 15 feet is 406 kip-ft and φbMp is 540 kip-ft. Since (φbMn = 406 kip-ft) > (Mequiv = 277 kip-ft) and (φbMp = 540 kip-ft) > (Mu = 462 kip-ft), a W21×62 is o.k. A 21-in. nominal depth beam spanning 60 feet should be checked for deflection since the span/depth ratio exceeds 30.




4 - 113

4 - 114





4 - 115

4 - 116





4 - 117

BEAM AND GIRDER DESIGN 4 - 118

W3 x3

18

30

3x3 W

97

26

0x2

31

W4 0x3

W44

x26

2

0x2

W

24

W4

49

x3

x2

77

35

0x2

30

W4 61

0x2

78

W4

4x2

30

W

11

35

x2

x2

30

27

W

67

W4

79 x2

0x1

0x2

15

W

24

x2

50

W4

0x

4

24

99

W

0x1

49

W4

17

0x1

x2

07

W

92

30

x1

W

24

x1

91


0

92 W4

W

35

11

94

W4

W21x201

28

x2

6x

30

07

4

35 x2

0x2

6x1

W4

8 25 8x W1

W18x234

W

x3

x26

0x2 30

W4

W3

W27x178

29 x2 24 W

W

24 3 28 8x W1

1 31 8x W1

W3

27

W40

W4 W

W40x183

W4

W

W40x199


4 - 119

W 24 x4 08

W4 0x2 97 W 27 x3 68

W 30 x3 26

W

W

27

30

x3

x2

07

92

W 24 x3 35

30

W3

W

91

61

3x2

x2

W

24

W

x2

27

79

x2 58

W1

W 24 x2 50 W

W1

8x

27

28

3

x2 35


8x

31

1

4 - 120


W

W3 11 W2

4

4x 9

x169

22

W33

x183

01

W18x211

x17

8

x2

34

W40

17

21

x2

W40

7x

W

18

0x2

W2

W

18

W40x

W 24

167

x1

W21x182

92 W 24 x2 07

W36 x170

W2 7x 19

18

4

W

x1

92

W40 x149

W3 0x1 16

W

18

x1

W 58

18

x1

75


x2

58


4 - 121

W2 79

35

W36

7x2

4x2

W2 x245 W

18

x2

83

W2 4x 25 0

W3 0x1 91 W2 4x 22

W

9

W36 x210

W 24 x2 07

W 21 x2 01

W

18

x2

34

W 24 x1 92

W

18

x1

92 W

x1

7x1

21

W2

W

18

x2

11

94

82

W 24 x1 76


18

x2

58

4 - 122


W30x1 32

W1 15

W1

8x

8x

8

17 5

W 3 3 x1 3 0

W30x90 W24x103

W27 x94 W 36 x1 60



4 - 123

W21 x20 1

W1

W 36 x1 70

8x 21 1

W1 8x1

W 33 x1 69

92

W1 8x1 75 W21 x18 2

W1 8x1 58 W 36 x1 60


4 - 124


W14x132

W

14

x1

32

W30 x90 W27

x90

x94

W30

W12x120



4 - 125

W1 8x 78 5 17 W27x1

W1 8x 15 8

W

W24x

14

x1

32

117


4 - 126


W x1 20

W30x9

12

0

W12x106



4 - 127

W 14 x1 32

W 12 x1 20


4 - 128


W12x96

W 12 x9 6



4 - 129

W 12 x1 20

W 12 x1 06

W

10

x1

12

W 12 x9 6

W

10

x1

00


4 - 130




4 - 131

W


10

x1

00

4 - 132




4 - 133

W 10 x6 8


4 - 134




4 - 135

W 8x 48 W1 0x4 5


4 - 136




M12x1 0 .8

M10 x8


4 - 137

4 - 138


M10 x8




4 - 139

4 - 140





4 - 141

4 - 142





4 - 143

4 - 144


W24x250

0x2 61

x3

0x2

18

W4

x183

W

W27x217

W3

x235

4

W40

x19

x211

W40

W40

W36

W40x174

15

11

W40x167

W 24 x2 79

W2

7x2

21

58

3x2

5

83

W3

23

x2

7x

18

W2

W

W4

W x2

0x1

24

99

29 W4 0x1 74

W18x258

0x

W33x

W3

W40x149

3

241

17 W 07

3x2

x2

W3

24

01

W24x192 W18x234

W21x201

W18x211

W21x182



30

92

7x

0x2

W2

W3

7 W 24 x3 35

W3 0x 26 1 W2 7x 25 8

W

18

x3

11

W 24 x2 79

W 50 235 x x2 24 W27

W1 8x 28 3

W

18

x2

83


4 - 145

4 - 146


W x2

W

7

11

20

x2

4x

34

18

W2

18

W21x182

W2 1x 20 1

W18x192

W2 4x1

x149

W18x158


92

W40

W18x175


4 - 147

W2

07

29

4x2

4x2

W2

W 18 x2 58

W 18 x2 34

W2 4x 19 2

W 18 x2 11

W 21 x2 01

W 21 x1 82

W 18 x1 92


4 - 148


W 18 x1 75

W 18 x1 58

W30x90



4 - 149

W

W2

18

1x

x2

2

11

18 W 18 x1 92

W 18 x1 75

W 18 x1 58


4 - 150


W30 x90 W 27 x9 4




4 - 151

4 - 152




4 - 153

W 14 x1 32

W

12

x1

20

W

12

x1

06


4 - 154


W1 2x 96



W1 2x 10 6

W1 2x 96


4 - 155

4 - 156




4 - 157

W 12 x9 6 W 10 x1 00

W1 2x 72


4 - 158




W 10 x7 7


4 - 159

4 - 160




4 - 161

W 8x 67


4 - 162


W8 x48




4 - 163

4 - 164


M12x10 .8



M12x1 0 .8

M10x 8


4 - 165

4 - 166



PLATE GIRDER DESIGN

4 - 167

PLATE GIRDER DESIGN General Notes

The distinction between a beam and a plate girder, according to Chapter G of the LRFD Specification, must be made before the design can be undertaken. A beam can be a rolled or welded shape, but its web width-thickness ratio h / tw must be less than or equal to 970 / √ Fyf. For doubly symmetric plate girders h / tw is greater than 970 / √ Fyf. The limit states that must be considered in plate girder design include: flexural strength, bearing under concentrated loads, shear strength, and flexure-shear interaction (for tension field action only). From these checks, the adequacy of the design and the need for stiffeners can be determined. This section contains design examples to explain these items from the LRFD Specification. A flowchart covering plate girder design has been published (Zahn, 1987). Flexural and Shear Strength

General

In the design of welded girders, the flexural strength of the trial section must be determined to ensure that an adequate section modulus is provided. Although there are preliminary steps, flexural strength, using elastic design, is determined from LRFD Specification Section F1 if the section is compact. For sections with more slender webs, either LRFD Specification Appendix F1 or Appendix G2 is used, depending on the section’s classification as a beam or plate girder. A shear strength calculation is required to ascertain if there is a need for intermediate stiffeners. The applicable formulas are found in LRFD Specification Section F2, or Appendix G3 if tension field action is implemented. Note, however, that Appendix G cannot be used if h / tw exceeds the limits given in Appendix G1. Table of Dimensions and Properties of Built-up Wide Flange sections

This table serves as a guide for selecting welded built-up I sections of economical proportions. It provides dimensions and properties for a wide range of sections with nominal depths from 45 to 92 inches. No preference is intended for the tabulated flange plate dimensions, as compared to other flange plates having the same area. Substitution of wider but thinner flange plates, without a change in flange area, will result in a slight reduction in section modulus. In analyzing overall economy, weight savings must be balanced against higher fabrication costs incurred in splicing the flanges. In some cases, it may prove economical to reduce the size of flange plates at one or more points near the girder ends, where the bending moment is substantially less. Economy through reduction of flange plate sizes is most likely to be realized with long girders, where flanges must be spliced in any case. Only one thickness of web plate is given for each depth of girder. When the design is dominated by shear in the web, rather than flexural strength, overall economy may dictate selection of a thicker web plate. The resultant increase in elastic section modulus can be obtained by multiplying the value S′, given in the table, by the number of sixteenths of an inch increase in web thickness, and adding the value obtained to the section modulus value S for the girder profile shown in the table. The increase in plastic section modulus Z can be calculated in the same way with Z′. Overall economy may often be obtained by using a web plate of such thickness that intermediate stiffeners are not required. This is not always the case, however. The girder AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 168


section listed in the table will provide a “balanced” design with respect to bending moment and web shear without excessive use of intermediate stiffeners. The maximum design end shear strength without transverse stiffeners is given in the table column labeled φvVn. These values come from the equation, φvVn = 0.6φvAwFywCv where Cv =

44,000k with k = 5.0 (h / tw)2Fyw

It is evident from this formula that a thicker web plate increases the design shear strength. Design Examples

Design of a plate girder should begin with a preliminary design or selection of a trial section. The initial choice may require one or more adjustments before arriving at a final cross section that satisfies all the provisions of the LRFD Specification with maximum economy. In the following design examples, applicable provisions of the LRFD Specification are indicated at the left of each page. In addition, references to Tables 9 and 10 in the LRFD Specification are listed. These tables may be used in place of the equations for φvVn. Values for φvVn / Aw are given in ksi for plate girders. Tables 9-36 and 9-50 do not include the tension field action equation and, therefore, are based on LRFD Specification Section F2. For design with tension field action, Tables 10-36 and 10-50, based on Appendix G3, are applicable. Table 10 also includes the required gross area of pairs of stiffeners, as a percent of (h × tw), from LRFD Specification Formula A-G4-1. Example 4-10 illustrates a recommended procedure for designing a welded plate girder of constant depth. The selection of a suitable trial cross section is obtained by the flange area method, and then checked by the moment of inertia method. Example 4-11 shows a recommended procedure for designing a welded hybrid girder of constant depth. Example 4-12 illustrates use of the Table of Dimensions and Properties of Built-up Wide-Flange Sections to obtain an efficient trial profile. The 52-in. depth specified for this example demonstrates how tabular data may be used for girder depths intermediate to those listed. Another design requirement in this example is the omission of intermediate web stiffeners.

EXAMPLE 4-10 Design a welded plate girder to support a factored uniform load of

7 kips per foot and two concentrated factored loads of 150 kips located 17 feet from each end. The compression flange of the girder will be laterally supported only at points of concentrated load. (See Figure 4-3.) Given:

Maximum bending moment: 4,566 kip-ft Maximum vertical shear: 318 kips Span: 48 feet Maximum depth: 72 inches Steel: Fy = 50 ksi AMERICAN INSTITUTE OF STEEL CONSTRUCTION

PLATE GIRDER DESIGN

Solution: LRFD Specification Reference Section B5 & Table B5.1

4 - 169

A. Preliminary web design:

1. Assume web depth, h = 70 inches. For noncompact web, 640 / √ Fy < h / tw ≤ 970 / √ Fy = 137 Corresponding thickness of web = 70 / 137 = 0.51 in.

(A-G1-2)

2. Assuming a / h > 1.5, minimum thickness of web = 70 / 243 = 0.29 in. Choose thinnest web. Try web plate 5⁄16×70: Aw = 21.9 in.2 h / tw = 70 / 0.313 = 224 Since 0.31 < 0.51 in. as calculated above, expect RPG to be less than 1.0 B. Preliminary flange design: 1. Required flange area: An approximate formula for the area of one flange is: Af ≈

Mu 4,566(12) = = 15.7 in.2 Fy h 50(70)

Try 1×16 plate. Af = 16 in.2

150 kips

150 kips

7 kips/ft 17 ft.

14 ft.

17 ft.

318 kips

318 kips

199 kips

318 kips

49 kips 318 kips

M max = 4566 kip-ft

M1 = 4395 kip-ft

M1 = 4395 kip-ft

Figure 4-3 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 170

Table B5.1


2. Check for compactness for no reduction in critical stress: bf 16 = = 8 ≤ 65 / √ 50  = 9.2 o.k. 2tf 2(1) C. Trial girder section: Web 5⁄16×70; two flange plates 1×16 1. Find section modulus by moment of inertia method: Section

A in.2

1 web 5⁄16×70

21.9

1 flange 1×16 1 flange 1×16

16 16

y in.

35.5

ΣA2y in.4

Io in.4

Igr in.4

8,932

8,932

3

40,331

40,328

Moment of inertia

49,263

Section modulus furnished: 49,263 / 36.0 = 1,368 in.3 Section F1 & Appendix G2

2. Check flexural strength using elastic design: Fyf, Appendix G2 applies. Since h / tw > 970 / √ Moment of inertia of flange plus 1⁄6 web about Y-Y axis: Ioy = 1 × (16)3 / 12 = 341 in.4 Af + 1⁄6Aw = 16.0 + 1⁄6(21.9) = 19.65 in.2 rT = √  341 / 19.65  = 4.17 in. a. Check limitations of Appendix G: Assume a / h ≤ 1.5

(A-G1-1)

(h / tw) max =

2,000 = 283 > 224 o.k. Fyf √

b. Check strength of 14-ft panel: Mu = 4,566 kip-ft The moment in the 14-ft unbraced segment is nearly constant. Section F1.2a

Therefore, Cb ≈ 1.0

Appendix G2

For the limit state of lateral-torsional buckling:

(A-G2-7)

λ=

(A-G2-8)

λp = 300 / √ Fyf = 42.4

(A-G2-9)

λr = 756 / √ Fyf = 106.9

(A-G2-4)

Since λ ≤ λp, Fcr = Fyf = 50 ksi

Lb 14 × 12 = = 36.0 4.67 rT


PLATE GIRDER DESIGN

4 - 171

For the limit state of flange local buckling: (A-G2-11)

λ = bf / 2tf = 16 / (2 × 1.0) = 8.0

(A-G2-12)

λp = 65 / √ Fyf = 9.2

(A-G2-13)

λr =

(A-G2-4)

Since λ ≤ λp, Fcr = Fyf = 50 ksi

230  √ Fyf / kc

Design flexural strength: ar (A-G2-3)

= 21.9 / 16 = 1.37

RPG = 1 −

1.37 1,200 + 300(1.37)

 70 970   = 0.927  0.313 − √50   

With Fcr = 50 ksi use Equation A-G2-1 or A-G2-2 as applicable: (A-G2-1) or (A-G2-2)

Mn = 1,368(1 / 12)(0.927)(1.0)(50) = 5,284 kip-ft Therefore, φMn = 0.90(5,284) = 4,756 kip-ft > 4,566 kip-ft req’d o.k. c. Check strength of 17-ft panels: Mu = 4,395 kip-ft

Appendix G2 and (F1-3)

For moment increasing approximately linearly from zero at one end of the unbraced segment to a maximum value at the other end, Cb ≈ 1.67. For the limit state of lateral-torsional buckling:

(A-G2-7)

λ =

(A-G2-8)

λp = 300 / √ Fyf = 42.4

(A-G2-9)

λr = 756 / √ Fyf = 106.9

(A-G2-5)

Lb 17(12) = = 48.9 4.17 rT

 1  λ − λp   Since λp ≤ λ ≤ λr, Fcr = CbFyf 1 −   ≤F 2  λr − λp   yf   As the middle term exceeds Fyf, Fcr = Fyf = 50.0 ksi. For the limit state of flange local buckling:

(A-G2-4)

Fcr = Fyf = 50 ksi (as for the 14-ft panel)

(A-G2-3)

RPG = 0.927 (as for the 14-ft panel) Again, with Fcr = 50 ksi use Equation A-G2-1 or A-G2-2 as applicable: Mn = 5,284 kip-ft (see Step C.2a) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 172

(A-G2-1) or (A-G2-2)


φbMn = 0.90 × 5,284 = 4,756 kip-ft > 4,395 kip-ft req’d. o.k. Use: Web: One plate 5⁄16×70 Use: Flanges: Two plates 7⁄8×18 D. Stiffener requirements: 1. Bearing stiffeners:

Section K1

a. Check bearing at end reactions: Assume point bearing (N = 0) and 5⁄16-in. web-to-flange welds. Check local web yielding:

(K1-2)

Rn = (5k + N)Fywtw; k = 7⁄8 + 5⁄16 = 1.188 in. φRn = 1.0[5(1.188) + 0](50)(5⁄16) = 92.8 kips < 318 n.g. Therefore, provide bearing stiffeners at unframed girder ends. (Note: If local web yielding criteria are satisfied, criteria set forth in Section K1.4 and K1.5 should also be checked.) b. Bearing stiffeners are also required at concentrated load points since 92.8 < 150 n.g. 2. Intermediate stiffeners:

Appendix G3

a. Check shear strength in unstiffened end panel: h / tw = 224 > 418 / √ Fyw = 59.1 a / h = 17 × 12 / 70 = 2.9 Vu / Aw = 318 / 21.9 = 14.5 ksi

Appendix G3

Tension field action is not permitted for end panels, or when a / h > 3.0 or [260 / (h / tw)]2. Here, 2.9 > (260 / 224)2 = 1.35. In either of these cases, Equations A-G3-3 and F2-3 are both applicable, as they are equivalent formulas.

Section F2.2

Using Equation F2-3,

(F2-3) or (A-G3-3) or Table 9-50

φvVn 0.9(132,000) = = 2.4 < 14.5 ksi Aw (224)2 Therefore, provide intermediate stiffeners. b. End panel stiffener spacing

(F2-3) or (A-G3-3) or Table 9-50

Let

φvVn = 14.5 ksi and solve for a / h. Aw

Result: a / h = 0.45 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

PLATE GIRDER DESIGN

4 - 173

a ≤ (0.45)(70) = 31.5 in. Use: 30 in. c. Check for additional stiffeners: Shear at first intermediate stiffener: Vu = 318 − [7(30 / 12)] = 301 kips Vu 301 = = 13.7 ksi Aw 21.9 Distance between first intermediate stiffener and concentrated load: a (A-G3-4)

= (17)(12) − 30 = 174 in.

a / h = 174 / 70 = 2.5 Then k = 5.8, and the shear strength is inadequate. Therefore, provide intermediate stiffeners spaced at 174 / 2 = 87 in. a / h = 87 / 70 = 1.24

Appendix G3

Maximum a / h for tension field action: 2

2

 260  =  260  = 1.35 > 1.24  (h / t )   224   w    Design for tension field action: For a / h = 1.24 and h / tw = 224, (A-G3-4)

Appendix G3 (A-G3-6)

(A-G3-2) or Table 10-50

k

=5+

5 = 8.2 (1.24)2

h / tw = 224 > 234 √  8.2 / 50  = 95 Cv

=

44,000(8.2) = 0.14 (224)2(50)

1 − 0.14  φvVn  = (0.9)(0.6)(50) 0.14 + 2 Aw 1.15√  1 + (1.24)   = 16.5 ksi > 13.7 ksi o.k. d. Check center 14-ft panel: h / tw = 224 a / h = (14)(12) / 70 = 2.4 > 1.35 k = 5.0 Cv = 0.12

(A-G3-3) or Table 9-50

φvVn = 2.4 ksi Aw Vu 49 = = 2.2 ksi < 2.4 ksi o.k. 21.9 Aw AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 174


3. Flexure-shear interaction: Appendix G5

Check Vu / φVn and Mu / φMn at intermediate stiffener and concentrated load locations in tension field panel: Location

Vu

φVn

Vu / φVn

Mu

φMn

Mu / φMn

2.5 ft 9.75 ft 17.0 ft

301 250 199

318 361 361

0.95 0.69 0.55

744 2769 4395

4756 4756 4756

0.16 0.58 0.92

Vu Mu ≤ 1.0 and 0.75 ≤ ≤ 1.0 (with φ = 0.9 for both φVn φMn shear and bending) do not occur simultaneously at 2.5 ft, 9.75 ft, and 17.0 ft, Interaction Equation A-G5-1 need not be checked. Since 0.6 ≤

Summary: space stiffeners as shown in Figure 4-4: E. Stiffener design: Let stiffener Fyst = 36 ksi. 1. For intermediate stiffeners: a. Area required (single plate stiffener): Vu < 1, use Equation A-G4-2 For a single plate stiffener, or when φVn instead of Table 10. Fyw  Vu  0.15Dhtw(1 − Cv) − 18t2w ≥ 0 Fyst  φV n   where

(A-G4-1)

Ast =

h = 70 in. tw = 0.3125 in. D = 2.4 Cv = 0.14 Vu = 250 kips φVn = 361 kips  250  50   0.15(2.4)(70)(0.3125)(10.14)   − 18(0.3125)2 36   361   = 4.07 in.2

Ast =

Try one bar 5⁄8×7

2 ′-6

′ 2@7 -3

14 ′-0

′ 2@7 -3


2 ′-6

PLATE GIRDER DESIGN

4 - 175

Ast = 4.38 in.2 > 4.07 in.2 req’d. o.k. b. Check width-thickness ratio: Table B5.1

7 / 0.625 = 11.2 < 95 / √ Fy = 15.8 o.k. c. Check moment of inertia:

Appendix F2.3

Ireq’d = at3wj

(A-F2-4)

2.5 − 2 = −0.4 < 0.5; take j = 0.5 (1.24)2 Ireq’d = 87(5⁄16)3(0.5) = 1.33 in.4 Ifurn = 1⁄3(0.625)(7)3 = 71.5 in.4 j

=

71.5 in.4 > 1.33 in.4 o.k. d. Minimum length required: Section F3

It is suggested that intermediate stiffeners be stopped short of the tension flange and the weld by which they are attached to the web not closer than four times nor more than six times the web thickness from the near toe of the web-to-flange weld.* 70 − 5⁄16 − (4)(5⁄16) = 68.4 in. 70 − 5⁄16 − (6)(5⁄16) = 67.8 in. Use for intermediate stiffeners: One plate 5⁄8×7×5′-8, fillet-welded to the compression flange and web. 2. For bearing stiffeners: At end of girder, design for end reaction. Try two 5⁄8×8-in. bars (see Figure 4-5). a. Check width-thickness ratio (local buckling check):

Table B5.1

8 / 0.625 = 12.8 < 95 / √ Fy = 15.8 o.k. b. Check compressive strength: (16.31)3 = 226 in.4 12 Aeff = (2)(8)(5⁄8) + [(12)(5⁄16)2] = 11.17 in.2 I

= (5⁄8)

r

=

 √

226 = 4.50 in. 11.17

*When single stiffeners are used, they shall be attached to the compression flange, if it consists of a rectangular plate, to resist any uplift tendency due to torsion in the plate. When lateral bracing is attached to a stiffener, or a pair of stiffeners, these, in turn, shall be connected to the compression flange to transmit one percent of the total flange stress, unless the flange is composed only of angles. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 176


Section K1.9

KL = 0.75h = (0.75)70 = 52.5 in. Kl 52.5 = = 11.7 r 4.50 λc = 0.13

(E2-2) or Table 3-36

Design stress: φFcr = 30.38 Design strength: φPn = φFcr Ag= (30.38)11.17 = 339 kips 339 kips > 318 kips req’d o.k.

Section J8

c. Check bearing criterion Design strength: φRn = (0.75)1.8Fy Apb Apb = 2(16 − 0.5)(5⁄8) = 19.4 in.2 (The 0.5 accounts for cutout for welds.) φRn = 943 kips > 318 kips req’d. o.k. Use for bearing stiffeners: Two plates 5⁄8×8×5′-93⁄4 with close bearing on flange receiving reaction or concentrated loads. Use same size stiffeners for bearing under concentrated loads.*

EXAMPLE 4-11 Design a hybrid girder to support a factored uniform load of three kips

per foot and three concentrated factored loads of 300 kips located at the quarter points. The girder depth must be limited to five feet. The compression flange will be laterally supported throughout its length. (See Figure 4-6.) Given:

Maximum bending moment: 14,400 kip-ft

*In this example, bearing stiffeners were designed for end bearing; however, 25tw may be used in determining effective area of web for bearing stiffeners under concentrated loads at interior panels (Section K1-9).

5/ 8 x8

End bearing stiffeners

tw 12t w

Web

5/ 8 x8


PLATE GIRDER DESIGN

4 - 177

Maximum vertical shear: 570 kips Span: 80 ft Maximum depth: 60 in. Steel: Flanges: Fy = 50 ksi Steel: Web: Fy = 36 ksi Solution:

A. Preliminary web design:

LRFD Specification Reference

Assume web depth, h = 54 in.

(A-G1-2)

For a / h > 1.5 minimum thickness of web: 54 / 243 = 0.22 in.

(A-G2-3)

For RPG = 1.0, h / tw ≤ 970 / √ Fyf = 137

(F2-1) or Table 9-36

Corresponding web thickness = 54 / 137 = 0.39 φvVn Minimum tw required for maximum of 19.4 ksi: Aw tw

=

Vn 570 = = 0.54 in. 19.4h 19.4 × 54

Try web plate 5⁄8×54; Aw = 33.75 in.2 Vu Aw Table B5.1

= 570 / 33.75 = 16.9 ksi < 19.4 ksi o.k.

h / tw = 54 / 0.625 = 86.4 < (640 / √ Fyf = 90.5) o.k. Web is compact.

300 kips 300 kips

300 kips

3 kips/ft 20 ft.

20 ft.

20 ft.

20 ft.

80 ft.

570 kips

570 kips 510 kips

570 kips

210 kips 150 kips

230 kips

14400 kip-ft

10800 kip-ft

10800 kip-ft


4 - 178


B. Preliminary flange design: 1. An approximate formula for the area of one flange is: Af ≈

Mu 14,400(12) = = 64.0 in.2 Fyf h (50)(54)

2. Check adequacy against local buckling: Table B5.1

bf / 2tf = 24 / (2)(2.625) = 4.6 < (65 / √ Fy = 9.2) o.k. Flange is compact. C. Trial girder section: Web 5⁄8×54; two flange plates 25⁄8×24 1. Determine plastic section moduli:  54 2.625  3 Zf = 2 (2.625)(24)  +  = 3.567 in. 2 2     54   54  Zw = 2 (5⁄8)     = 456 in.3  2   4   2. Check flexural strength: Compression flange is supported laterally for its full length and the section is compact.

Appendix F1

Mn = Mp = Fyf Zf + FywZw Mn = [(50)(3,567) + (36)(456)] 1 / 12 = 16,230 kip-ft φbMn = (0.90)16,230 = 14,610 kip-ft > 14,400 kip-ft o.k. Use: Web: One plate 5⁄8×54 (Fy = 36 ksi) Use: Flanges: Two plates 25⁄8×24 (Fy = 50 ksi) D. Stiffener requirements: 1. Bearing stiffeners

Section K1

a. Check bearing at end reactions: Assume point bearing (N = 0) and 5⁄16-in. web-to-flange welds. Check local web yielding:

(K1-2)

Rn = (5k + N)Fywtw; k = 25⁄8 + 5⁄16 = 215⁄16-in. φRn = 1.0[(5)(215⁄16) + 0](36)(5⁄8) = 330 kips 330 kips < 570 kips n.g. Note: If local web yielding criteria are satisfied, applicable criteria set for in Sections K1.4 and K1.5 should also be checked. b. Bearing stiffeners at points of concentrated loads are also required. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

PLATE GIRDER DESIGN

4 - 179

2. Intermediate stiffeners: The LRFD Specification does not permit design of hybrid girders on the basis of tension field action. Therefore, determine the need for intermediate stiffeners by use of Equations F2-1, F2-2, F2-3, Table 9-36. a. Check shear strength without intermediate stiffeners: Section F2

h / tw = 86.4, a / h exceeds 3.0 Therefore, k = 5.0 Vu 570 = = 16.9 ksi Aw 33.75 Since h / tw = 86.4 > 523 / √ 36  = 87, but h / tw = 86.4 < 418 / √ 36  = 70, use Equation F2-2:


φvVn Vu = 15.5 ksi < = 16.9 ksi Aw Aw Therefore, intermediate stiffeners required. b. End panel stiffener spacing


φvVn = 16.9 ksi Aw Therefore, a / h = 2.5 for h / tw = 86.4. Max. a1 = 2.5(54) = 135 in. (Use 10 ft = 120 in.) c. Check need for stiffeners between concentrated loads: h / tw = 86.4, a / h is over 3, k = 5 Vu Aw


=

210 = 6.2 ksi 33.75

φvVn = 15.5 ksi > 6.2 ksi o.k. Aw Therefore, intermediate stiffeners not required between the concentrated loads. Summary (see Figure 4-7): E. Stiffener design: 1. Bearing stiffeners: See Step E.2, Example 4-10, for design procedure. Use for bearing stiffeners: Two plates 3⁄4×11×4′-53⁄4 with close bearing on flange receiving reaction or concentrated loads. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 180


2. Intermediate stiffeners: Assume 5⁄16×4 in., Fy = 36 ksi, on each side of web. a. Check width-thickness ratio: 4 / 0.313 = 12.8 < 95 / √ Fy = 15.8 o.k.

Table B5.1

b. Check moment of inertia: Ireq’d = 120(5⁄8)3(0.5) = 14.6 in.4 Ifurn = 1⁄12(0.313)(8.63)3 = 16.8 in.4 16.8 in.4 > 14.6 in.4 o.k.

Appendix F2.3

c. Length required (see Step E.1.d. Example 4-10): 54 − 5⁄16 − (6)(9⁄16) = 505⁄16 54 − 5⁄16 − (4)(9⁄16) = 517⁄16 (use 51 in.) Use for intermediate stiffeners: Two plates 5⁄16×4×4′-3, fillet-welded to the compression flange and web, one on each side of the web.

EXAMPLE 4-12 Design the section of a nominal 52-in. deep welded girder with no

intermediate stiffeners to support a factored uniform load of 5.0 kips per linear foot on an 85-ft span. The girder will be framed between columns and its compression flange will be laterally supported for its entire length. Required bending moment: 4,516 kip-ft Required vertical shear: 213 kips Span: 85 ft Nominal depth: 52 in. Steel: Fy = 50 ksi

Given:

For compact web and flange, Mn = Fy Z

Solution:

20 ′-0

20 ′-0 Bearing stiffener

Bearing stiffeners

4 ′-9

5 ′-1

5 ′-1

Intermediate stiffeners

5 ′-1

40 ′-0


Sym. about cL

PLATE GIRDER DESIGN

LRFD Specification Reference

4 - 181

Design for a compact web and flange: Design for plastic moment, Fy Z: Required plastic section modulus: Zreq’d =

Mu 4,516 × 12 = = 1,204 in.3 φFy 0.90 × 50

Enter Table of Built-up Wide-Flange Sections, Dimensions and Properties: For girder having 3⁄8×48 web with 11⁄4×16 flange plates: Z = 1,200 in.3 < 1,204 in.3 For girder having 3⁄8×52 web with 11⁄4×18 flange plates: Z = 1,450 in.3 > 1,204 in.3 A. Determine web required: Table B5.1

For compact web, h / tw ≤ 640 / √ Fyf = 91 Assume h = 50 in. Minimum tw = 50 / 91 = 0.55 in. Try: web = 9⁄16×50; Aw = 28.1 in.2 h / tw = 50 / 0.56 = 89 < 91 The web is compact. Intermediate stiffeners can be avoided if the design shear strength Fyf, of the web is adequate. (For plate girders with h / tw > 970 / √ refer to Appendix G4 of the LRFD Specification.)

(F2-3)

φvVn = φv(132,000Aw) / (h / tw)2 = 0.9(132,000 × 28.1) / (89)2 = 421 kips > 213 kips req’d. o.k. Therefore, no intermediate stiffeners are necessary. B. Determine flange required. Af ≈

(4,516)(12) = 21.7 in.2 50 × 50

Try 1×18 plate: Af = 18.0 in.2 Table B5.1

bf / 2tf = 18 / (2)(1.0) = 9.0 < 65 / √ Fy = 9.2 o.k. Flange is compact. C. Check plastic section modulus: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 182


 51.0   9   50   50   Zx = 2 (18)(1.0)   +        = 1,270 in.3 2    16   2   4    3 3 1,270 in. > 1,204 in. req’d. o.k. Use: Web: One plate 9⁄16×50 Use: Flanges:* Two plates 1×18 Section K1.8

Note: Because this girder will be framed between columns, the usual end bearing stiffeners are not required.

*Because this girder is longer than 60 feet, some economy may be gained by decreasing the flange size in areas of smaller moment, i.e., near ends of girder. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

PLATE GIRDER DESIGN

4 - 183

bf

tf

Nominal Size (h / tw)

Wt. per Ft

Area Depth

In.

Lb.

In.2

92× 30 (131)

823 721 619 568 517 466 415

86× 28 (134)

80× 26 (125)

74× 24 (128)

68× 22 (132)

d

Flange

Web

In.

In.

In.

In.

242 212 182 167 152 137 122

96.00 95.00 94.00 93.50 93.00 92.50 92.00

30 30 30 30 30 30 30

3 21⁄2 2 13⁄4 11⁄2 11⁄4 1

90 90 90 90 90 90 90

11⁄

16

11⁄

16

11⁄

16

11⁄

16

11⁄

16

11⁄

16

11⁄

16

750 654 559 512 464 416 369

220 192 164 150 136 122 108

90.00 89.00 88.00 87.50 87.00 86.50 86.00

28 28 28 28 28 28 28

3 21⁄2 2 13⁄4 11⁄2 11⁄4 1

84 84 84 84 84 84 84

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

696 609 519 475 431 387 343 320

205 179 153 140 127 114 101 94.2

84.00 83.00 82.00 81.50 81.00 80.50 80.00 79.75

26 26 26 26 26 26 26 26

3 21⁄2 2 13⁄4 11⁄2 11⁄4 1

78 78 78 78 78 78 78 78

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

5⁄

8

627 546 464 423 382 342 301 280

184 160 136 124 112 100 88.5 82.5

78.00 77.00 76.00 75.50 75.00 74.50 74.00 73.75

24 24 24 24 24 24 24 24

3 21⁄2 2 13⁄4 11⁄2 11⁄4 1

72 72 72 72 72 72 72 72

9⁄

16

9⁄

16

9⁄

16

9⁄

16

9⁄

16

9⁄

16

9⁄

16

9⁄

16

561 486 411 374 337 299 262 243 224

165 143 121 110 99.0 88.0 77.0 71.5 66.0

72.00 71.00 70.00 69.50 69.00 68.50 68.00 67.75 67.50

22 22 22 22 22 22 22 22 22

3 21⁄2 2 13⁄4 11⁄2 11⁄4 1

66 66 66 66 66 66 66 66 66

1⁄

2

1⁄

2

1⁄

2

1⁄

2

1⁄

2

1⁄

2

1⁄

2

1⁄

2

1⁄

2

7⁄

8

8

7⁄

8

3⁄

4

X

X

h

tw

f

Axis X-X

Width Thick Depth Thick bf tf h tw

7⁄

d=h+2t f

BUILT-UP WIDE-FLANGE SECTIONS Dimensions and properties

In.

Z

In.3

In.3

In.3

431000 363000 296000 263000 230000 198000 166000

8980 7640 6290 5620 4950 4280 3610

79.1 79.9 80.8 81.2 81.7 82.1 82.5

9760 8330 6910 6210 5510 4810 4120

127 127 127 127 127 127 127

428.9 428.9 428.9 428.9 428.9 428.9 428.9

349000 293000 238000 211000 184000 158000 132000

7750 6580 5410 4820 4240 3650 3070

68.6 69.4 70.2 70.6 71.0 71.4 71.8

8410 7160 5920 5300 4690 4090 3480

110 110 110 110 110 110 110

345.3 345.3 345.3 345.3 345.3 345.3 345.3

281000 235000 191000 169000 148000 127000 106000 95500

6680 5670 4660 4160 3650 3150 2650 2390

58.8 59.6 60.3 60.7 61.0 61.4 61.8 62.0

7270 6180 5110 4580 4050 3530 3000 2750

95.1 95.1 95.1 95.1 95.1 95.1 95.1 95.1

371.8 371.8 371.8 371.8 371.8 371.8 371.8 371.8

220000 184000 149000 132000 115000 98000 81400 73300

5640 4780 3920 3490 3060 2630 2200 1990

49.8 50.5 51.2 51.5 51.8 52.2 52.5 52.7

6130 5200 4280 3830 3380 2930 2480 2260

81.0 81.0 81.0 81.0 81.0 81.0 81.0 81.0

293.7 293.7 293.7 293.7 293.7 293.7 293.7 293.7

169000 141000 114000 100000 87000 74000 61000 55000 49000

4700 3970 3250 2890 2530 2170 1800 1620 1440

41.6 42.2 42.8 43.1 43.4 43.7 44.0 44.2 44.4

5100 4310 3540 3150 2770 2390 2020 1830 1650

68.1 68.1 68.1 68.1 68.1 68.1 68.1 68.1 68.1

225.0 225.0 225.0 225.0 225.0 225.0 225.0 225.0 225.0

S

In.4

a


Z′

φvVn

S′

I

b

c

In.3

Kips

4 - 184


bf

d=h+2t f

tf

X

X

h

tw

f


Flange

Web

Axis X-X


Wt. per Ft

Area Depth

In.

Lb.

In.2

In.

In.

In.

In.

In.

61× 20 (137)

429 361 327 293 259 225 208 191

126 106 96.2 86.2 76.2 66.2 61.2 56.2

65.00 64.00 63.50 63.00 62.50 62.00 61.75 61.50

20 20 20 20 20 20 20 20

21⁄2 2 13⁄4 11⁄2 11⁄4 1

60 60 60 60 60 60 60 60

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

389 328 298 267 236 206 190 175 160

115 96.5 87.5 78.5 69.5 60.5 56.0 51.5 47.0

61.00 60.00 59.50 59.00 58.50 58.00 57.75 57.50 57.25

18 18 18 18 18 18 18 18 18

21⁄2 2 13⁄4 11⁄2 11⁄4 1

56 56 56 56 56 56 56 56 56

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

7⁄

16

342 311 280 250 219 189 173 158 143

100 91.5 82.5 73.5 64.5 55.5 51.0 46.5 42.0

56.50 56.00 55.50 55.00 54.50 54.00 53.75 53.50 53.25

18 18 18 18 18 18 18 18 18

21⁄4 2 13⁄4 11⁄2 11⁄4 1

52 52 52 52 52 52 52 52 52

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

306 279 252 224 197 170 156 143 129

90.0 82.0 74.0 66.0 58.0 50.0 46.0 42.0 38.0

52.50 52.00 51.50 51.00 50.50 50.00 49.75 49.50 49.25

16 16 16 16 16 16 16 16 16

21⁄4 2 13⁄4 11⁄2 11⁄4 1

48 48 48 48 48 48 48 48 48

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

3⁄

8

237 210 183

69.8 61.8 53.8

47.50 47.00 46.50

16 16 16

13⁄4 11⁄2 11⁄4

44 44 44

5⁄

16

5⁄

16

5⁄

16

57× 18 (128)

53× 18 (138)

49× 16 (128)

45× 16 (141)

d


7⁄

8

3⁄

4

7⁄

8

3⁄

4

5⁄

8

7⁄

8

3⁄

4

5⁄

8

7⁄

8

3⁄

4

5⁄

8

S′

Z

In.3

In.3

106000 84800 74600 64600 54800 45100 40300 35600

3250 2650 2350 2050 1750 1450 1310 1160

83500 67000 58900 51000 43300 35600 31900 28100 24400

Z′

φvVn

b

c

In.3

In.3

Kips

34.6 35.2 35.4 35.7 36.0 36.3 36.4 36.6

3520 2870 2560 2240 1930 1610 1460 1310

56.3 56.3 56.3 56.3 56.3 56.3 56.3 56.3

165.8 165.8 165.8 165.8 165.8 165.8 165.8 165.8

2740 2230 1980 1730 1480 1230 1100 979 854

30.0 30.5 30.7 31.0 31.3 31.5 31.7 31.8 32.0

2980 2430 2160 1900 1630 1370 1240 1110 980

49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0

177.6 177.6 177.6 177.6 177.6 177.6 177.6 177.6 177.6

64000 56900 49900 43000 36300 29700 26400 23200 20000

2270 2030 1800 1570 1330 1100 983 866 750

25.9 26.2 26.4 26.6 26.9 27.1 27.2 27.4 27.5

2450 2200 1950 1700 1450 1210 1090 966 846

42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3

120.5 120.5 120.5 120.5 120.5 120.5 120.5 120.5 120.5

48900 43500 38100 32900 27700 22700 20200 17700 15300

1860 1670 1480 1290 1100 910 811 716 620

21.9 22.2 22.4 22.6 22.8 23.0 23.2 23.3 23.4

2030 1820 1610 1400 1200 1000 900 801 702

36.0 36.0 36.0 36.0 36.0 36.0 36.0 36.0 36.0

130.5 130.5 130.5 130.5 130.5 130.5 130.5 130.5 130.5

31500 27100 22700

1330 1150 976

18.7 18.9 19.1

1430 1240 1060

30.3 30.3 30.3

82.4 82.4 82.4

I

S

In.4


a

PLATE GIRDER DESIGN

4 - 185

bf

tf

Flange


Wt. per Ft

Area Depth

In.

Lb.

In.2

In.

In.

45× 16 (141)

156 142 129 116

45.8 41.8 37.8 33.8

46.00 45.75 45.50 45.25

16 16 16 16

d

Web

In.

In.

In.

1

44 44 44 44

5⁄

16

5⁄

16

5⁄

16

5⁄

16

8

3⁄

4

5⁄

8

X

X

h

tw

f

Axis X-X


7⁄

d=h+2t f


S′

Z

In.3

In.3

801 713 626 538

19.3 19.4 19.5 19.6

I

S

In.4 18400 16300 14200 12200

a

Z′

φvVn

b

c

In.3

In.3

Kips

871 780 688 598

30.3 30.3 30.3 30.3

82.4 82.4 82.4 82.4

a S′ = Additional section modulus corresponding to 1⁄16″ increase in web thickness. b Z′ = Additional plastic section modulus corresponding to 1⁄16″ increase in web thickness. c φvVn = Maximum design end shear strength permissible without transverse stiffeners for tabulated web plate (LRFD Specification Section F2). φv = 0.90.

Notes: Based on their width-thickness ratios the girders in this table are noncompact shapes in accordance with LRFD Specification Section B5 for Fy = 36 ksi steel. For steels of higher yield strengths, check flanges for compliance with this section. This table does not consider local effects on the web due to concentrated loads. (See LRFD Specification Section K1.) See LRFD Specification Appendix G4 for design of stiffeners. Welds are not included in the tabulated weight per foot.


4 - 186



BEAM DIAGRAMS AND FORMULAS

4 - 187

BEAM DIAGRAMS AND FORMULAS Nomenclature

E = modulus of elasticity of steel at 29,000 ksi I = moment of inertia of beam (in.4) L = total length of beam between reaction points (ft) Mmax = maximum moment (kip-in.) M1 = maximum moment in left section of beam (kip-in.) M2 = maximum moment in right section of beam (kip-in.) M3 = maximum positive moment in beam with combined end moment conditions (kip-in.) Mx = moment at distance x from end of beam (kip-in.) P = concentrated load (kips) P1 = concentrated load nearest left reaction (kips) P2 = concentrated load nearest right reaction, and of different magnitude than P1 (kips) R = end beam reaction for any condition of symmetrical loading (kips) R1 = left end beam reaction (kips) R2 = right end or intermediate beam reaction (kips) R3 = right end beam reaction (kips) V = maximum vertical shear for any condition of symmetrical loading (kips) V1 = maximum vertical shear in left section of beam (kips) V2 = vertical shear at right reaction point, or to left of intermediate reaction point of beam (kips) V3 = vertical shear at right reaction point, or to right of intermediate reaction point of beam (kips) Vx = vertical shear at distance x from end of beam (kips) W = total load on beam (kips) a = measured distance along beam (in.) b = measured distance along beam which may be greater or less than a (in.) l = total length of beam between reaction points (in.) w = uniformly distributed load per unit of length (kips per in.) w1 = uniformly distributed load per unit of length nearest left reaction (kips per in.) w2 = uniformly distributed load per unit of length nearest right reaction, and of different magnitude than w1 (kips per in.) x = any distance measured along beam from left reaction (in.) x1 = any distance measured along overhang section of beam from nearest reaction point (in.) ∆max = maximum deflection (in.) ∆a

= deflection at point of load (in.)

∆x

= deflection at any point x distance from left reaction (in.)

∆x1 = deflection of overhang section of beam at any distance from nearest reaction point (in.) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 188


BEAM DIAGRAMS AND FORMULAS Frequently Used Formulas The formulas given below are frequently required in structural designing. They are included herein for the convenience of those engineers who have infrequent use for such formulas and hence may find reference necessary. Variation from the standard nomenclature on page 4-187 is noted. BEAMS Flexural stress at extreme fiber: f = Mc / I = M / S Flexural stress at any fiber: f = My / I y = distance from neutral axis to fiber Average vertical shear (for maximum see below): v = V / A = V / dt (for beams and girders) Horizontal shearing stress at any section A-A: v = VQ / Ib Q = statical moment about the neutral axis of that portion of the cross section lying outside of section A-A b = width at section A-A (Intensity of vertical shear is equal to that of horizontal shear acting normal to it at the same point and both are usually a maximum at mid-height of beam.) Shear and deflection at any point: x and y are abscissa and ordinate respectively of a point on the neutral d 2y EI 2 = M axis, referred to axes of rectangular coordinates through a selected dx point of support. (First integration gives slopes; second integration gives deflections. Constants of integration must be determined.) CONTINUOUS BEAMS (the theorem of three moments) Uniform load:  w1l31 w2l32   l1 l2  l1 l2  Ma + 2Mb  +  + Mc = − 1⁄4  + I1 I2 I2   I1  I1 I2  Concentrated loads:  l1 l2  l1 l2 P1a1b1  a1  p2a2b2  b2  Ma + 2Mb  +  + Mc = − 1 +  − 1 +  I1 I2 I1  l1  I2  I2   I1 I2  Considering any two consecutive spans in any continuous structure: Ma, Mb, Mc = moments at left, center, and right supports respectively, of any pair of adjacent spans = length of left and right spans, respectively, of the pair l1 and l2 = moment of inertia of left and right spans, respectively I1 and I2 w1 and w2 = load per unit of length on left and right spans, respectively = concentrated loads on left and right spans, respectively P1 and P2 a1 and a2 = distance of concentrated loads from left support, in left and right spans, respectively = distance of concentrated loads from right support, in left and right spans, b1 and b2 respectively The above equations are for beam with moment of inertia constant in each span but differing in different spans, continuous over three or more supports. By writing such an equation for each successive pair of spans and introducing the known values (usually zero) of end moments, all other moments can be found.



4 - 189

BEAM DIAGRAMS AND FORMULAS Table of Concentrated Load Equivalents

n

Loading

P

2 P

3 P

4 P

P

P

5 P

P

P

Beam Fixed One End, Supported at Other

Beam Fixed Both Ends

a b c d e f g

0.125 — 0.500 — 0.013 1.000 1.000

0.070 0.125 0.375 0.625 0.005 1.000 0.415

0.042 0.083 — 0.500 0.003 0.667 0.300

a b c d e f g

0.250 — 0.500 — 0.021 2.000 0.800

0.156 0.188 0.313 0.688 0.009 1.500 0.477

0.125 0.125 — 0.500 0.005 1.000 0.400

a b c d e f g

0.333 — 1.000 — 0.036 2.667 1.022

0.222 0.333 0.667 1.333 0.015 2.667 0.438

0.111 0.222 — 1.000 0.008 1.778 0.333

a b c d e f g

0.500 — 1.500 — 0.050 4.000 0.950

0.266 0.469 1.031 1.969 0.021 3.750 0.428

0.188 0.313 — 1.500 0.010 2.500 0.320

a b c d e f g

0.600 — 2.000 — 0.063 4.800 1.008

0.360 0.600 1.400 2.600 0.027 4.800 0.424

0.200 0.400 — 2.000 0.013 3.200 0.312

Coeff.

∞

P

Simple Beam

P

Maximum positive moment (kip-ft): aPL Maximum negative moment (kip-ft): bPL Pinned end reaction (kips): cP Fixed end reaction (kips): dP Maximum deflection (in): ePl3 / EI

Equivalent simple span uniform load (kips): f P Deflection coefficient for equivalent simple span uniform load: g Number of equal load spaces: n Span of beam (ft): L Span of beam (in): l


4 - 190


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 1. SIMPLE BEAM—UNIFORMLY DISTRIBUTED LOAD Total Equiv. Uniform Load . . . . . . = wl

l wl

x

R

R

l 2

l 2

V

. . . . . . . . . . . . . . . . . =

Vx

. . . . . . . . . . . . . . . . . = w  − x

M max Shear

V

Mx

l 2

∆x

Moment



 wl 2 (at center) . . . . . . . . . . . . = 8 wx . . . . . . . . . . . . . . . . . = (l − x) 2

∆ max (at center) . . . . . . . . . . . .

Mmax

wl 2

R=V

. . . . . . . . . . . . . . . . .

5wl 4 384 EI wx = (l2 − 2lx 2 + x3 ) 24EI

=

2. SIMPLE BEAM—LOAD INCREASING UNIFORMLY TO ONE END Total Equiv. Uniform Load . . . . . . = l x

R 1 = V1 . . . . . . . . . . . . . . . . .

W

R2

R1

V1

Shear

V2

max

Vx

. . . . . . . . . . . . . . . . . =

. . . . . . . . . . . . . . .

M max (at x = Mx

M max

l = .5774 l) . . . . . . . √3 

 √ 158 √

∆x

W Wx 2 − 2 3 l 2Wl = = .1283 Wl 9√ 3

. . . . . . . . . . . . . . . . . =

∆ max (at x = l 1 −

Moment

W 3 2W = 3

=

R 2 = V2

.5774 l

= .5193 l) . .

16W = 1.0264W 9√ 3

Wx 3l2

(l2 − x2 )

= 0.1304

. . . . . . . . . . . . . . . . . =

Wl 3 EI

Wx 180 EIl2

(3x4 − 10l2 x2 + 7l4 )

3. SIMPLE BEAM—LOAD INCREASING UNIFORMLY TO CENTER 4W 3 W = 2 W

Total Equiv. Uniform Load . . . . . . = l x

W

R

R=V R

l 2

Vx

l 2

V

. . . . . . . . . . . . . . . . . l 2

(when x < ) . . . . . . . . . . =

M max (at center) . . . . . . . . . . . .

Shear

V

Mx M max

∆x

(l2 − 4x2)

1

l 2

(when x < ) . . . . . . . . . . = Wx  −

∆ max (at center) . . . . . . . . . . . . Moment

2l2 Wl = 6

l 2

2

=

(when x < ) . . . . . . . . . . =


Wl 3 60EI Wx 480 EIl2

2x 2   3l2 

(5l2 − 4x2 )2


4 - 191

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 4. SIMPLE BEAM—UNIFORMLY LOAD PARTIALLY DISTRIBUTED l b wb

a

R

1

R2

x

V1

wb (2c + b) 2l wb (2a + b) R 2 = V2 (max. when a > c) . . . . . . . = 2l

R 1 = V1 (max. when a < c) . . . . . . . =

c

Vx

(when x > a and < (a + b)) . . . = R 1 − w(x − a)

M max Mx

 R1  R1   at x = a + w  . . . . . . . . = R 1 a + 2w      (when x < a) . . . . . . . . . = R 1x

Mx

(when x > a and < (a + b)) . . . = R 1x −

Mx

(when x > (a + b)) . . . . . . . = R 2(l − x)

Shear

V2 R1 a+ w M max

Moment

w (x − a)2 2

5. SIMPLE BEAM—UNIFORM LOAD PARTIALLY DISTRIBUTED AT ONE END R 1 = V1

max

. . . . . . . . . . . . . . .=

l a wa

R 2 = V2

R

1

R2

x

V1

R1 w M max

wa 2 2l

Vx

(when x < a) . . . . . . . . . = R 1 − wx

M max

 R  at x = 1  w 

Mx

(when x < a) . . . . . . . . . = R 1x −

Mx

(when x > a) . . . . . . . . . = R 2 (l − x)

∆x

(when x < a) . . . . . . . . . =

wx (a 2 (2l − a)2 − 2ax 2 (2l − a) + lx 3 ) 24EIl

∆x

(when x > a) . . . . . . . . . =

wa 2 (l − x) (4xl − 2x2 − a 2 ) 24EIl

V2 Shear

. . . . . . . . . . . . . . . .=

wa (2l − a) 2l

. . . . . . . . . .=

Moment

R 21 2w wx 2 2

6. SIMPLE BEAM—UNIFORM LOAD PARTIALLY DISTRIBUTED AT EACH END

l

a w1 a R1

b

c

w2 c R2

x

R 1 = V1

. . . . . . . . . . . . . . . .=

w 1a(2l − a) + w 2 c2 2l

R 2 = V2

. . . . . . . . . . . . . . . .=

w 2c(2l − c) + w 1 a2 2l

Vx

(when x < a) . . . . . . . . . = R 1 − w 1x

Vx

(when x > a and < (a + b))

Vx

(when x > (a + b)) . . . . . . . = R 2 − w 2 (l − x)

= R 1 − w 1a

R1 R1   = at x = w when R 1 < w 1a  2w 1 1   R 22 R1   at x = l − w when R 2 < w 2 c = 2w 2 2   2

V1

Shear

R1 w1

V2

M max M max

M max

Mx Moment

Mx Mx

w 1x2 2 w1a (when x > a and < (a + b)) . . . = R 1x − (2x − a) 2

(when x < a) . . . . . . . . . = R 1x −

(when x > (a + b)) . . . . . . . = R 2(l − x) −


w 2(l − x)2 2

4 - 192


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 7. SIMPLE BEAM—CONCENTRATED LOAD AT CENTER Total Equiv. Uniform Load . . . . . . . . . . = 2P

l P

x

R

l 2

R=V

. . . . . . . . . . . . . . . . . . . . =

P 2

M max

(at point of load) . . . . . . . . . . . =

Pl 4

Mx

 Px 1 when x <  . . . . . . . . . . . . . = 2 2 

∆ max

(at point of load) . . . . . . . . . . . =

∆x

 1 Px (3l2 − 4x2 ) when x <  . . . . . . . . . . . . . = 2 48EI 

R

l 2

V

Shear

V

M max

Moment

Pl 3 48EI

8. SIMPLE BEAM—CONCENTRATED LOAD AT ANY POINT Total Equiv. Uniform Load . . . . . . . . . . =

l x

P

R

1

a

Pb l

R 2 = V2 (max when a > b ) . . . . . . . . . . . =

Pa l

M max

(at point of load) . . . . . . . . . . . =

Pab l

Mx

(when x < a ) . . . . . . . . . . . . . . =

Pbx l

∆ max

 at x = 

Pab (a + 2b)√  3a(a + 2b) 27EIl

∆a

(at point of load) . . . . . . . . . . . =

Pa 2b 2 3EIl

∆x

(when x < a ) . . . . . . . . . . . . . . =

Pbx 2 (l − b 2 − x2 ) 6EIl

V1

V2 Shear

M max

Moment

l2

R 1 = V1 (max when a b 3

. . . =

9. SIMPLE BEAM—TWO EQUAL CONCENTRATED LOADS SYMMETRICALLY PLACED Total Equiv. Uniform Load . . . . . . . . . . =

l x

P

P

R

R

a

a

V

Shear

V

M max

Moment

8Pa l

R=V

. . . . . . . . . . . . . . . . . . . . =P

M max

(between loads) . . . . . . . . . . . . = Pa

Mx

(when x < a ) . . . . . . . . . . . . . . = Px

∆ max

(at center) . . . . . . . . . . . . . . . =

Pa (3l2 − 4a2 ) 24EI

∆x

(when x < a ) . . . . . . . . . . . . . . =

Px (3la − 3a 2 − x2 ) 6EI

∆x

(when x > a and < (l − a)) . . . . . . . =

Pa (3lx − 3x2 − a 2) 6EI



4 - 193

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 10. SIMPLE BEAM—TWO EQUAL CONCENTRATED LOADS UNSYMMETRICALLY PLACED l P

x

P

R

1

a

b

R2

V1

V2

Shear

M2

M1

Moment

R 1 = V1

(max. when a b ) . . . . . . . . . =

P (l − b + a) l

Vx

(when x > a and < (l − b)) . . . . . =

P (b − a) l

M1

(max. when a > b ) . . . . . . . . . = R 1 a

M2

(max. when a a and < (l − b)) . . . . . = R 1 x − P(x − a)

11. SIMPLE BEAM—TWO UNEQUAL CONCENTRATED LOADS UNSYMMETRICALLY PLACED l

x

P1

P2

R

1

a

b

R2

V1

V2

Shear

M2

M1

Moment

R 1 = V1

. . . . . . . . . . . . . . . . . . =

P1 (l − a) + P2 b l

R 2 = V2

. . . . . . . . . . . . . . . . . . =

P1 a + P2 (l − b) l

Vx

(when x > a and < (l − b)) . . . . . = R 1 − P1

M1

(max. when R 1 < P1) . . . . . . . . = R 1 a

M2

(max. when R 2 < P2) . . . . . . . . = R 2 b

Mx

(when x < a ) . . . . . . . . . . . . = R 1 x

Mx

(when x > a and < (l − b)) . . . . . = R 1 x − P(x − a)

12. BEAM FIXED AT ONE END, SUPPORTED AT OTHER—UNIFORMLY DISTRIBUTED LOAD Total Equiv. Uniform Load . . . . . . . . . . = wl R 1 = V1

. . . . . . . . . . . . . . . . . . =

3wl 8

R 2 = V2 max

. . . . . . . . . . . . . . . . . . =

5wl 8

Vx

. . . . . . . . . . . . . . . . . . = R 1 − wx

M max

. . . . . . . . . . . . . . . . . . =

l

wl R

1

R2

x

V1

Shear

V2

3 l 8

l 4

M1

Moment

M max

wl 2 8

Mx

 9 3 wl 2 at x = l . . . . . . . . . . . . . = 128 8 

Mx

. . . . . . . . . . . . . . . . . . = R1x −

∆ max ∆x

wx 2 2

  l wl 4 33  ) = .4215 l . . . = at x = (1 + √ 16 185 EI  

. . . . . . . . . . . . . . . . . .


wx 3 (l − 3lx + 2x3 ) 48EI

4 - 194


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 13. BEAM FIXED AT ONE END, SUPPORTED AT OTHER—CONCENTRATED LOAD AT CENTER

l

l 2

1

3P 2

R 1 = V1 . . . . . . . . . . . . . . . . . . =

5P 15

R 2 = V2 max . . . . . . . . . . . . . . . . =

11P 16

M max (at fixed end) . . . . . . . . . . . =

3Pl 16

(at point of load) . . . . . . . . . . =

5Pl 32

P

x

R

Total Equiv. Uniform Load . . . . . . . =

l 2

R

2

M1

V1

Shear

V2

Mx Mx

M1

∆ max

Moment

M max

3 11 l

∆x ∆x ∆x

 l 5Px when x <  . . . . . . . . . . . . = 2 16    l 11x  l when x >  . . . . . . . . . . . . = P  −  2   2 16  3 3   at x = l√  15 = .4472 l . . . . . . . = Pl = .009317 PlEI  48EI√ 5 (at point of load) . . . . . . . . . . =

7PL 3 768 EI

 Px l (3l2 − 5x2 ) when x <  . . . . . . . . . . . . = 96EI 2   l P (x − l)2(11x − 2l) when x >  . . . . . . . . . . . . = 2 96EI 

14. BEAM FIXED AT ONE END, SUPPORTED AT OTHER—CONCENTRATED LOAD AT ANY POINT R 1 = V1 . . . . . . . . . . . . . . . . . . = R 2 = V2 . . . . . . . . . . . . . . . . . . = l

P

x R

1

a

b

R2

V1

Shear

Pa 2l3

(a + 2l)

(3l2 − a2 )

(at point of load) . . . . . . . . . . = R 1 a

M2

(at fixed end) . . . . . . . . . . . =

Mx

(when x < a ) . . . . . . . . . . . . = R 1 x

Mx

(when x > a ) . . . . . . . . . . . . = R 1 x − P(x − a) 

∆ max when a < .414 l at x = l

 

∆ max when a > .414 l at x = l Moment Pa R2

2l3

M

V2

M1

Pb 2



M2

∆a

Pab 2l2

(a + l)

Pa(l2 + a2 )3 (l2 + a 2)   = 2 2 (3l − a )  3EI(3l2 − a2 )2 

..........  √ 2l + a  a

=

(at point of load) . . . . . . . . . . =

∆

(when x < a ) . . . . . . . . . . . . =

∆x

(when x > a ) . . . . . . . . . . . . =


Pab 2 6EI

2l+ a √

Pa 2 b3 12EIl3 2

Pb x 12EIl3 Pa 12EIl2

a

(3l + a) (3al 2 − 2lx 2 − ax 2) (l − x)2 (3l2 x − a 3 x − 2a 2l)


4 - 195

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 15. BEAM FIXED AT BOTH ENDS—UNIFORMLY DISTRIBUTED LOADS 2wl 3 wl . . . . . . . . . . . . . . . . . . = 2

Total Equiv. Uniform Load . . . . . . . . = l

x

R=V

wl

R

R l 2

l 2

V V

Shear .2113 l

M1

Moment

M max

M max

Vx M max

(at ends) . . . . . . . . . . . . . =

M1

(at center) . . . . . . . . . . . . . =

Mx

. . . . . . . . . . . . . . . . . . =

∆ max ∆x

l  2  wl 2 12 wl 2 24 w (6lx − l2 − 6x2 ) 12 wl 4 384 EI wx 2 (l − x)2 24EI

. . . . . . . . . . . . . . . . . . = w  − x

(at center) . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . =

16. BEAM FIXED AT BOTH ENDS—CONCENTRATED LOAD AT CENTER l

R

Total Equiv. Uniform Load . . . . . . . . = P P

x l 2

R=V R

l 2

V V

Shear

M max Mx ∆ max

l 4

M max

Moment

M max

M max

∆x

P 2 Pl (at center and ends) . . . . . . . . = 8

. . . . . . . . . . . . . . . . . . =

 when x < 

P l  . . . . . . . . . . . = (4x − l) 8 2 Pl 3 (at center) . . . . . . . . . . . . . = 192 EI

 when x < 

l Px 2 (3l − 4x)  . . . . . . . . . . . = 2 48EI

17. BEAM FIXED AT BOTH ENDS—CONCENTRATED LOAD AT ANY POINT R 1 = V1 (max. when a b ) . . . . . . . . = l

P

x

R1

b

a

R2

V1

Ma M1

Moment

M2

l3 Pa 2

(3a + b)

(a + 3b) l3 Pab 2

M1

(max. when a b ) . . . . . . . . =

Ma

(at point of load) . . . . . . . . . =

Mx

(when x < a ) . . . . . . . . . . . = R 1 x −

∆ max

 2Pa 3b 2 2al   . . . . = when a > b at x = 3a + b  3EI(3a + b)2 

∆a

(at point of load) . . . . . . . . . =

∆x

(when x < a ) . . . . . . . . . . . =

V2

Shear

Pb 2


l2 Pa 2 b l2 2Pa 2b 2 l3 Pab 2 l2

Pa 3 b3 3EIl3 Pb 2 x2 6EIl2

(3al − 3ax − bx)

4 - 196


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 18. CANTILEVER BEAM—LOAD INCREASING UNIFORMLY TO FIXED END 8 3

Total Equiv. Uniform Load . . . . . . . . = W l

R=V . . . . . . . . . . . . . . . . . . . =W

W

Vx

R

x2

. . . . . . . . . . . . . . . . . . . =W

l2

x

M max (at fixed end) . . . . . . . . . . . . = V

Shear

Mx M max

Moment

. . . . . . . . . . . . . . . . . . . =

∆ max (at free end) . . . . . . . . . . . . . = ∆x

. . . . . . . . . . . . . . . . . . . =

Wl 3 Wx 3 3l2 Wl 3 15EI W 60EIl2

(x5 − 5l4 x + 4l5 )

19. CANTILEVER BEAM—UNIFORMLY DISTRIBUTED LOAD Total Equiv. Uniform Load . . . . . . . . = 4wl

l

R = V . . . . . . . . . . . . . . . . . . . = wl

wl

Vx

R x

V

M max

Moment

M max (at fixed end) . . . . . . . . . . . . =

wl 2 2

. . . . . . . . . . . . . . . . . . . =

wx 2 2

∆ max (at free end) . . . . . . . . . . . . . =

wl 4 8EI

∆x

w (x 4 − 4l3 x + 3l4 ) 24EI

Mx

Shear

. . . . . . . . . . . . . . . . . . . = wx

. . . . . . . . . . . . . . . . . . . =

20. BEAM FIXED AT ONE END, FREE TO DEFLECT VERTICALLY BUT NOT ROTATE AT OTHER—UNIFORMLY DISTRIBUTED LOAD Total Equiv. Uniform Load . . . . . . . . = l

R = V . . . . . . . . . . . . . . . . . . . = wl

wl M R

Vx

. . . . . . . . . . . . . . . . . . . = wx

x

M max (at fixed end) . . . . . . . . . . . . = V

Shear

Mx

. . . . . . . . . . . . . . . . . . . =

.4227 l M1 Moment

8 wl 3

wl 2 3 w 2 (l − 3x2 ) 6

∆ max (at deflected end) . . . . . . . . . . =

wl 4 24EI

∆x

w(l2 − x2 )2 24EI

M max

. . . . . . . . . . . . . . . . . . . =



4 - 197

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 21. CANTILEVER BEAM—CONCENTRATED LOAD AT ANY POINT Total Equiv. Uniform Load . . . . . . . . = R=V

8Pb l

. . . . . . . . . . . . . . . . . . . =P

l

M max (at fixed end) . . . . . . . . . . . . = Pb

P

x

a

(when x > a) . . . . . . . . . . . . = P(x − a)

R

Mx

V

∆ max (at free end) . . . . . . . . . . . . . =

Pb 2 (3l − b) 6EI

∆a

(at point of load) . . . . . . . . . . =

Pb 3 3EI

∆x

(when x < a) . . . . . . . . . . . . =

Pb 2 (3l − 3x − b) 6EI

∆x

(when x > a) . . . . . . . . . . . . =

P(l − x)2 (3b − l + x) 6EI

b

Shear

Mmax

Moment

22. CANTILEVER BEAM—CONCENTRATED LOAD AT FREE END l

Total Equiv. Uniform Load . . . . . . . . = 8P

P

R=V

. . . . . . . . . . . . . . . . . . . =P

R

x

M max (at fixed end) . . . . . . . . . . . . = Pl Mx

V

Shear

M max Moment

. . . . . . . . . . . . . . . . . . . = Px

∆ max (at free end) . . . . . . . . . . . . . =

Pl 3 3EI

∆x

P (2l3 − 3l2x + x3 ) 6EI

. . . . . . . . . . . . . . . . . . . =

23. BEAM FIXED AT ONE END, FREE TO DEFLECT VERTICALLY BUT NOT ROTATE AT OTHER—CONCENTRATED LOAD AT DEFLECTED END Total Equiv. Uniform Load . . . . . . . . = 4P

l

R=V

P M x

. . . . . . . . . . . . . . . . . . . =P

R

M max (at both ends) . . . . . . . . . . . . =

V

Mx

l 2

l 2

Moment

M max



. . . . . . . . . . . . . . . . . . . = P  − x

Shear M max

Pl 2



3

∆ max (at deflected end) . . . . . . . . . . =

pl 12EI

∆x

P(l − x)2 (l + 2x) 12EI

. . . . . . . . . . . . . . . . . . . =


4 - 198


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 24. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD

l

x

R1

w(l +a)

a2 l 1 – 2) 2 ( l

a x1

R 2 = V2 + V3 . . . . . . . . . . . =

w (l + a)2 2l

V2

. . . . . . . . . . . . . . . = wa

V3

. . . . . . . . . . . . . . . =

V3

Moment

(between supports) . . . . . = R 1 − wx 

M2

a 2  w l  2 2 1 − 2  . . . . . = 2 (l + a) (l − a) 2 l  8l   wa 2 (at R 2) . . . . . . . . . . . . = 2

M 1 at x = M2

M1

a2 l (1 – 2 ) l

w 2 (l + a2 ) 2l

Vx 1 (for overhang) . . . . . . . = w(a − x 1) V2

Shear

w 2 (l − a2 ) 2l

Vx

R2

V1

R 1 = V1 . . . . . . . . . . . . . . =

M x (between supports) . . . . . =

wx 2 (l − a 2 − xl) 2l

M x1 (for overhang) . . . . . . . =

w (a − x1 )2 2

∆x

wx (l4 − 2l2 x2 + lx 3 − 2a 2l2 + 2a 2x 2) 24EIl

(between supports) . . . . . =

∆ x 1 (for overhang) . . . . . . . =

wx 1 (4a2 l − l3 + 6a 2 x1 − 4ax 21 + x31 ) 24EI

25. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD ON OVERHANG R 1 = V1 . . . . . . . . . . . . . . = R2

V1 + V2 . . . . . . . . . . . . =

x

a x1

Vx 1 (for overhang) . . . . . . . = w(a − x 1)

wa

R1

R2

M max (at R 2) . . . . . . . . . . . =

wa 2 2

M x (between supports) . . . . . =

wa 2 x 2l

M x1 (for overhang) . . . . . . . =

w (a − x 1)2 2

V2 V1

Shear



∆ max between supports at x = Moment

wa (2l + a) 2l

. . . . . . . . . . . . . . . = wa

V2 l

wa 2 2l



M max

wa 2 l2 l  wa 2 l2 = 0.03208  = EI 18√ 3 EI √3  

∆ max (for overhang at x1 = a ) . . . =

wa 3 (4l + 3a) 24EI

∆x

wa 2x 2 (l − x 2) 12EIl

(between supports) . . . . . =

∆ x 1 (for overhang) . . . . . . . =

wx 1 (4a2 l + 6a 2x 1 − 4ax 21 + x31) 24EI



4 - 199

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 26. BEAM OVERHANGING ONE SUPPORT—CONCENTRATED LOAD AT END OF OVERHANG Pa l P = (l + a) l =P = Pa Pax = l = P(a − x 1)

R 1 = V1 . . . . . . . . . . . . . . . . . . . . . = a x1

l

x

R 2 = V1 + V2 . . . . . . . . . . . . . . . . . . . P

R2

R1

V2

V2 M max

. . . . . . . . . . . . . . . . . . . . . (at R 2) . . . . . . . . . . . . . . . . .

Mx

(between supports) . . . . . . . . . .

M x1

(for overhang) . . . . . . . . . . . . .  l  Pal 2 Pal 2 = .06415  . . . . . = between supports at x = EI √3   9√ 3 EI 

∆ max

V1

Shear

∆ max M max

Moment

∆x ∆ x1

Pa 2 (l + a) 3EI Pax 2 (between supports) . . . . . . . . . . = (l − x2) 6EIl Px 1 (for overhang) . . . . . . . . . . . . . = (2al + 3ax 1 − x21 ) 6EI

(for overhang at x1 = a) . . . . . . . . =

27. BEAM OVERHANGING ONE SUPPORT—UNIFORMLY DISTRIBUTED LOAD BETWEEN SUPPORTS Total Equiv. Uniform Load . . . . . . . . . . = wl a

l

x

R

l 2

. . . . . . . . . . . . . . . . . . . . . =

Vx

. . . . . . . . . . . . . . . . . . . . . = w  − x

x1

wl

l 2

R

M max Mx

V V

Shear M max

wl 2

R=V

∆ max

(at center) . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . = (at center) . . . . . . . . . . . . . . . =

∆x

. . . . . . . . . . . . . . . . . . . . . =

∆ x1

. . . . . . . . . . . . . . . . . . . . . =

Moment

l  2  wl 2 8 wx (l − x) 2 5wl 4 384 EI wx 2 (l − 2lx 2 + x3) 24EI 3 wl x1 24EI

28. BEAM OVERHANGING ONE SUPPORT—CONCENTRATED LOAD AT ANY POINT BETWEEN SUPPORTS Total Equiv. Uniform Load . . . . . . . . . . = R 1 = V1 (max. when a b ) . . . . . . . . . . . =

x1

P

R2 a

b

V1 V2

Shear M max

Moment

M max

(at point of load) . . . . . . . . . . . . =

Mx

(when x < a ) . . . . . . . . . . . . . . =

∆ max

 at x = 

∆a

(at point of load) . . . . . . . . . . . . =

∆x

(when x < a ) . . . . . . . . . . . . . . =

∆x

(when x > a ) . . . . . . . . . . . . . . =

∆ x1

. . . . . . . . . . . . . . . . . . . . . =



a(a + 2b) √   when a > b 3

. . . =


8Pab l2 Pb l Pa l Pab l Pbx l Pab (a + 2b)√  3a(a + 2b) 27EIl Pa 2b 2 3EIl Pbx 2 (l − b 2 − x2 ) 6EIl Pa(l − x) (2lx − x 2 − a2 ) 6EIl Pabx1 (l + a) 6EIl

4 - 200


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 29. CONTINUOUS BEAM—TWO EQUAL SPANS—UNIFORM LOAD ON ONE SPAN

x

Total Equiv. Uniform Load

=

49 wl 64

R 1 = V1 . . . . . . . . . . .

=

7 wl 16

R 2 = V2 + V3 . . . . . . . . .

=

5 wl 8

R 3 = V3 . . . . . . . . . . .

=−

wl

R1

R2

l

R3 l

V1

V3 V2

7l 16

Shear

V2

. . . . . . . . . . . . = 

M max at x =



M max M1

7  l 16 

. . . . . =

1 wl 16

9 wl 16 49 wl 2 512

M1

(at support R 2) . . . . =

1 wl 2 16

Mx

(when x < l) . . . . . =

wx (7l − 8x) 16

Moment

∆ max (at 0.472 l from R 1 ) . .

= .0092 wl 4 / EI

30. CONTINUOUS BEAM—TWO EQUAL SPANS—CONCENTRATED LOAD AT CENTER OF ONE SPAN

P

l 2

l 2

R2

R1

l

R3

Total Equiv. Uniform Load

=

13 P 8

R 1 = V1 . . . . . . . . . . .

=

13 P 32

R 2 = V2 + V3 . . . . . . . . .

=

11 P 16

R 3 = V3 . . . . . . . . . . .

=−

l

V1

V3 Shear

V2

V2

. . . . . . . . . . . . =

M1

M1

Moment

19 P 32

=

13 Pl 64

(at support R 2) . . . . =

3 Pl 32

M max (at point of load) . . .

M max

3 P 32

∆ max (at 0.480 l from R 1 ) . .

= .015 Pl 3 / EI

31. CONTINUOUS BEAM—TWO EQUAL SPANS—CONCENTRATED LOAD AT ANY POINT P

a

R1

b

R2

l

R3 l

V1 Shear

V2

M max

=

R 2 = V2 + V3 . . . . . . . . .

=

R 3 = V3 . . . . . . . . . . .

=−

V3

V2

. . . . . . . . . . . . =

M max (at point of load) . . . M1

Moment

Pb

R 1 = V1 . . . . . . . . . . .

M1

=

(at support R 2) . . . . =


4l3 Pa 2l3

(4l2 − a(l + a)) (2l2 + b(l + a))

Pab 4l3

Pa 4l3

(4l2 + b(l + a))

Pab 4l3 Pab 4l2

(l + a)

(4l2 − a(l + a)) (l + a)


4 - 201

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 32. BEAM—UNIFORMLY DISTRIBUTED LOAD AND VARIABLE END MOMENTS

R 1 = V1

. . . . . . . . . . .

=

wl M 1 − M 2 + 2 l

R 2 = V2

. . . . . . . . . . .

=

wl M 1 − M 2 − 2 l

l

x

M1

wl

M

1

l 2

R2

2

V

1

Shear

V2

 l M 1 − M 2  M 3 at x = + . . wl  2 

=

M3 M2

M1

b



. . . . . . . . . . . . . = w  − x +

Vx

M >M

R1

2

. . . . . . . . . . . . . =

Mx



M1 − M2 l

2 wl 2 M 1 + M 2 (M 1 − M 2) − + 8 2 2wl 2

wx (l − x) + 2

M − M  2  1  x − M1 l  

b

Moment

M + M  M − M  l √ −  + 4  w   wl  2

b

(to locate inflection points) =

∆x =

wx 24EI

1

2

1

2

2

8M 1l 4M 2l  4M 1 4M 2  2 12M 1  3  2  x −  2l + wl − wl  x + w x + l − w − w     

33. BEAM—CONCENTRATED LOAD AT CENTER AND VARIABLE END MOMENTS

M1

l P

x

R

1

l 2

M1 >M

2

M2

l

R 1 = V1

. . . . . . . . . . . =

P M1 − M2 + 2 l

R 2 = V2

. . . . . . . . . . . =

P M1 − M2 − 2 l

R2

2

Pl M 1 + M 2 − 4 2

M 3 (at center) . . . . . . . .

=

 l M x when x <  . . . . . . 2 

P M1 − M2   x − M1 = + l 2 

 l M x when x >  . . . . . . 2 

=

V

1

V2

Shear

M3

Moment M1

M2

(M 1 − M 2)x P (l − x) + − M1 2 l

  8(l − x) l Px  2  3l − 4x2 − ∆ x when x <  = [M 1 (2l − x) + M 2 (l + x)] 2 48EI Pl    


4 - 202


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 34. CONTINUOUS BEAM—THREE EQUAL SPANS—ONE END SPAN UNLOADED wl

A

wl B

l

RA = 0.383 w l

C

l

0.583 w l

0.383 w l

D

l

RC = 0.450 w l

RB = 1.20 w l

RD = –0.033 w l

0.033 w l 0.617 w l

Shear

0.033 w l 0.417 w l

–0.1167 w l 2 +0.0735 w l 2

–0.0333 wl 2

+0.0534 w l 2

Moment 0.383 l

0.583 l

∆max (0.430 l from A) = 0.0059 wl 4/ El

35. CONTINUOUS BEAM—THREE EQUAL SPANS—END SPANS LOADED wl

A

wl B

l

RA = 0.450 w l

C

l

RB = 0.550 wl

D

l

RC = 0.550 w l

RD = 0.450 w l

0.550 wl

0.450 wl

0.450 w l

0.550 w l

Shear

–0.050 w l 2 +0.1013 w l 2

+0.1013 w l 2

Moment 0.450 l

0.450 l

∆max (0.479 l from A or D) = 0.0099 wl 4/ El

36. CONTINUOUS BEAM—THREE EQUAL SPANS—ALL SPANS LOADED wl

A

wl B

l

wl C

l

RB = 1.10 w l

RA = 0.400 w l

0.500 w l

0.400 w l

0.400 w l 0.500 w l

–0.100 w l 2

–0.100 w l 2 +0.080 w l 2

RD = 0.400 w l

0.600 w l 0.600 w l

Shear

D

l

R C = 1.10 w l

+0.080 w l 2

+0.025w l 2

Moment 0.400 l

0.500 l

0.500 l

0.400 l

∆max (0.446 l from A or D) = 0.0069 w / El l4



4 - 203

BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 37. CONTINUOUS BEAM—FOUR EQUAL SPANS—THIRD SPAN UNLOADED wl

wl

A

B

l

RA = 0.380 w l

wl C

l

RE = 0.442 w l

0.558 w l 0.620 w l

Shear

0.442 w l 0.040 w l

0.397 w l

–0.058 w l 2

–0.0179 w l 2

–0.1205 w l 2

+0.0977 w l 2

+0.0611 w l 2

2

+0.072 w l

E

l

R D = 0.598 w l

0.603 w l

0.380 w l

D

l

RC = 0.357 w l

RB = 1.223 w l

Moment 0.603 l 0.442 l

0.380 l

∆ max (0.475 l from E) = 0.0094 wl 4/ El

38. CONTINUOUS BEAM—FOUR EQUAL SPANS—LOAD FIRST AND THIRD SPANS wl

wl

A

B

l

RA = 0.446 w l

C

l

RC = 0.464 w l

RB = 0.572 w l

RE = –0.054 w l 0.054 w l

0.518 w l

–0.0536 w l 2

–0.0357 w l 2

–0.0536 w l 2 +0.0996 w l

E

0.054 w l

0.554 w l

Shear

l

R D = 0.572 w l

0.482 w l

0.018 w l

0.446 w l

D

l

+0.0805 w l 2

2

Moment 0.518 l

0.446 l

∆ max (0.477 l from A) = 0.0097 wl 4/ El

39. CONTINUOUS BEAM—FOUR EQUAL SPANS—ALL SPANS LOADED wl

wl

A

B

l

0.607 w l

Shear

+0.0772

D

l

l

RD = 1.143 w l

+0.0364 w l 2

0.393 w l 0.536 w l

–0.1071 w l 2 +0.0364 w l 2 +0.0772 w l 2

Moment 0.536 l

E RE = 0.393 w l

0.607 w l 0.464 w l

–0.0714 w l 2

–0.1071 w l 2 wl 2

wl

RC = 0.928 w l

0.464 w l

0.536 w l

0.393 w l

C

l

RB = 1.143 w l

RA = 0.393 w l

wl

0.536 l 0.393 l

0.393 l

∆ max (0.440 l from A and D) = 0.0065 w l 4/ El


4 - 204


BEAM DIAGRAMS AND FORMULAS For Various Static Loading Conditions For meaning of symbols, see page 4-187 40. SIMPLE BEAM—ONE CONCENTRATED MOVING LOAD x P R2

R1 l

R 1 max = V1 max (at x = 0) . . . . . . . . . . .

=P

 1 M max at point of load, when x =  . . . . . 2 

=

Pl 4

41. SIMPLE BEAM—TWO EQUAL CONCENTRATED MOVING LOADS R 1 max = V1 max (at x = 0) . . . . . . . . . . .



a l

= P 2 −  

when a (2 − √ 2 )l . . . . . = .586 l

l

with one load at center of span =

Pl 4

(Case 40)

42. SIMPLE BEAM—TWO UNEQUAL CONCENTRATED MOVING LOADS R 1 max = V1 max (at x = 0) . . . . . . . . . . .

under P1 , at x =

P1 > P2

x

a P1

R1

P2

R2

M max

P 2a  1 l− 2  P1 + P2  

= P1 + P2

l−a l

= (P1 + P2 )

x2 l

M max may occur with larger

load at center of span and other l

load off span (Case 40) . . . =

P1 l 4

GENERAL RULES FOR SIMPLE BEAMS CARRYING MOVING CONCENTRATED LOADS

l

P1

a C.G.

R1

P2 R2

x

b l 2

M

Moment

The maximum shear due to moving concentrated loads occurs at one support when one of the loads is at that support. With several moving loads, the location that will produce maximum shear must be determined by trial. The maximum bending moment produced by moving concentrated loads occurs under one of the loads when that load is as far from one support as the center of gravity of all the moving loads on the beam is from the other support. In the accompanying diagram, the maximum bending moment occurs under load P1 when x = b . It should also be noted that this condition occurs when the centerline of the span is midway between the center of gravity of loads and the nearest concentrated load.



4 - 205

BEAM DIAGRAMS AND FORMULAS Design properties of cantilevered beams Equal loads, equally spaced No. Spans

System a

2

M1 M1

M1

A

B

3

A

b

b

M3

M3

M2

M2

M3

C

D

c

c

C

D

M1

M1 M1

M1

M4

A

E

4

A

E

d

M1 M1

b

D

M1

D M1 M1

A

F

H

f

f

H M3

C M1

M5

H

G

3

F

A

4 P 2

P

P

5 P

P 2

P 2

P

P

P

P

Moments

P

P 2

M1 M2 M3 M4 M5

0.086PL 0.096PL 0.063PL 0.039PL 0.051PL

0.167PL 0.188PL 0.125PL 0.083PL 0.104PL

0.250PL 0.278PL 0.167PL 0.083PL 0.139PL

0.333PL 0.375PL 0.250PL 0.167PL 0.208PL

0.429PL 0.480PL 0.300PL 0.171PL 0.249PL

Reactions

P

D M1

M3

H

M2 d

e

A B C D E F G H

0.414P 1.172P 0.438P 1.063P 1.086P 1.109P 0.977P 1.000P

0.833P 2.333P 0.875P 2.125P 2.167P 2.208P 1.958P 2.000P

1.250P 3.500P 1.333P 3.167P 3.250P 3.333P 2.917P 3.000P

1.667P 4.667P 1.750P 4.250P 4.333P 4.417P 3.917P 4.000P

2.071P 5.857P 2.200P 5.300P 5.429P 5.557P 4.871P 5.000P

Cantilever Dimensions

P

H

M3

P 2

M3 M3

M3

M3

P 2

b

M3 M3

M3

2 P 2

P

C

f

M3

M3

G

∞

D

f

M3

M5

M2

H

M3 H e M3

d

M3 M3

f

M3 M3

b

M3 M3

H

f

M3

A

f

f

M3

G

b

C M1

F

M3

F

D M1

G

M5

M2

d

M5

e M3

M1

C

e

M3

G

d

Typical Span Loading

H

M3

F

M2

M3

H e M3

d

M5

A

b

M3

M3 M3

M1 A

C

f

M3

M3

M1

n

D

f M3

M2 C

M2

G

F

5

≥7 (odd)

M3 M3

M5

A

≥6 (even)

b

e M3

a b c d e f

0.172L 0.125L 0.220L 0.204L 0.157L 0.147L

0.250L 0.200L 0.333L 0.308L 0.273L 0.250L

0.200L 0.143L 0.250L 0.231L 0.182L 0.167L

0.182L 0.143L 0.222L 0.211L 0.176L 0.167L

0.176L 0.130L 0.229L 0.203L 0.160L 0.150L


P 2

4 - 206


BEAM DIAGRAMS AND FORMULAS CONTINUOUS BEAMS MOMENT AND SHEAR COEFFICIENTS EQUAL SPANS, EQUALLY LOADED

MOMENT in terms of wl2

UNIFORM LOAD

SHEAR in terms of wl

+.07

+.07 –.125

+.08

+.025

0 3 8

+.08

–.10

–.10

–.073

–.105 0 15 38

+.078

+.078 –.106

–.077

–.077

–.086

–.106 0 41 104

+.078

+.078 –.106

–.077

–.085

–.085

–.077

–.106 0 36 142

MOMENT in terms of Pl

+.156

23 20 38 63 55 104

86 75 142

18 19 38

67 70 142

4 0 10

15 17 28

19 18 38

53 53 104

72 71 142

CONCENTRATED LOADS at center

+.156

5 6 10

13 13 28

43 51 104

0 8

6 5 10 17 15 28

0 11 28

+.078 –.073

3

5

8

0 4 10

+.077 +.036 +.036 +.077 –.107 –.071 –.107 +.078 –.105

5

11 0 28

20 23 38

51 49 104

71 72 142

P

15 0 38

55 63 104

70 67 142

41 0 104

75 86 142

36 0 142

SHEAR in terms of P

P

+.157 .31 +.178

+.10

P

+.175

–.15

+.13

+.11

–.119

–.119

+.222

+.111

+.111

.65

P

+.171

P

.50

.50

P

.65

.35

P

P

P

–.158 .34

MOMENT in terms of Pl

.31

–.15

+.11

–.138

.69 P

.35 +.171

.69

+.222

.66

.54

.46

.50

.50

.46

.54

CONCENTRATED LOADS at 1⁄3 points

.66

.34

SHEAR in terms of P P

P

P

P

–.333

.67 +.156

+.244

+.066 –.267

+.066 +.156 –.267

+.244

P .73

+.24

+.146

+.076 –.281

+.099 +.122 –.211

+.122 +.099 –.211

+.076 +.146 –.281

+.24

P .72

MOMENT in terms of Pl +.258

+.267 +.267 +.022 +.022 +.258 -.465

P 1.28

P

P

1.0

1.0

+.155 +.303 +.204 +.155 +.303 +.079 +.006 +.054 +.079 +.079 +.054 +.277 +.006 +.079 -.394 -.296 -.296 -.394

1.97

P P P

P P P

1.11

1.89

P

P

.93

.73

1.07

P

P 1.28

.72

1.87

1.40

P P P 1.97

P P P 1.50

P P P 1.60

P

SHEAR in terms of P P P P

+.314 +.128 +.314 +.097 +.003 +.003 +.097 +.282 -.372 -.372

P 1.27

CONCENTRATED LOADS at 1⁄4 points

1.13

+.277

1.0 P

.93

.67

P

1.0

P

1.03

+.282

1.33

P

1.27 P

1.07

1.33

1.50

P P P 1.50


1.50

1.03

P P P 1.87

1.13

P P P 1.40

1.60

P P P 1.89

1.11

FLOOR DEFLECTIONS AND VIBRATIONS

4 - 207

FLOOR DEFLECTIONS AND VIBRATIONS Serviceability

Serviceability checks are necessary in design to provide for the satisfactory performance of structures. Chapter L of the LRFD Specification and Commentary contains general guidelines on serviceability. In contrast with the factored forces used to determine the required strength, the (unfactored) working loads are used in serviceability calculations. The primary concern regarding the serviceability of floor beams is the prevention of excessive deflections and vibrations. The use of higher strength steels and composite construction has resulted in shallower and lighter beams. Serviceability has become a more important consideration than in the past, as the design of more beams is governed by deflection and vibration criteria. Deflections and Camber

Criteria for acceptable vertical deflections have traditionally been set by the design engineer, based on the intended use of the given structure. What is appropriate for an office building, for example, may not be satisfactory for a hospital. An illustration of deflection criteria is the following: 1. Live load deflections shall not exceed a specified fraction of the span (e.g., 1⁄360) nor a specific quantity (e.g., one inch). A deeper and/or heavier beam shall be selected, if necessary, to meet these requirements. 2. Under dead load, plus a given portion of the design live load (say, 10 psf), the floor shall be theoretically level. Where feasible and necessary, upward camber of the beam shall be specified. Regarding camber, the engineer is cautioned that: 1. It is unrealistic to expect precision in cambering. The limits and tolerances given in Part 1 of of this Manual for cambering of rolled beams are typical for mill camber. Kloiber (1989) states that camber tolerances are dependent on the method used (hot or cold cambering) and whether done at the mill or the fabrication shop. According to the AISC Code of Standard Practice, Section 6.4.5: “When members are specified on the contract documents as requiring camber, the shop fabrication tolerance shall be −0 / +1⁄2 in. for members 50 ft and less in length, or −0 / + (1⁄2 in. + 1⁄8 in. for each 10 ft or fraction thereof in excess of 50 ft in length) for members over 50 ft. Members received from the rolling mill with 75 percent of the specified camber require no further cambering. For purposes of inspection, camber must be measured in the fabricator’s shop in the unstressed condition.” Some of the camber may be lost in transportation prior to placement of the beam, due to vibration. 2. There are two methods for erection of floors: uniform slab thickness and level floor. As a consequence of possible overcamber, the latter may result in a thinner concrete slab for composite action and fire protection at midspan, and may cause the shear studs to protrude above the slab. 3. Due to end restraint at the connections, actual beam deflections are often less than the calculated values. 4. The deflections of a composite beam (under live load for shored construction, and under dead and live loads for unshored) cannot be determined as easily and accurately as the deflections of a bare steel beam. Equation C-I3-6 in Section I3.2 of the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

4 - 208


Commentary on the LRFD Specification provides an approximate effective moment of inertia for partially composite beams. 5. Cambers of less than 3⁄4-in. should not be specified, and beams less than 24 ft in length should not be cambered (Kloiber, 1989). Vibrations

Annoying floor motion may be caused by the normal activities of the occupants. Remedial action is usually very difficult and expensive and not always effective. The prevention of excessive and objectionable floor vibration should be part of the design process. Several researchers have developed procedures to enable structural engineers to predict occupant acceptability of proposed floor systems. Based on field measurement of approximately 100 floor systems, Murray (1991) developed the following acceptability criterion: D > 35Ao f + 2.5

(4-1)

where D = damping in percent of critical Ao = maximum initial amplitude of the floor system due to a heel-drop excitation, in. f = first natural frequency of the floor system, hz Damping in a completed floor system can be estimated from the following ranges: Bare Floor: 1–3 percent Lower limit for thin slab of lightweight concrete; upper limit for thick slab of normal weight concrete. Ceiling: 1–3 percent Lower limit for hung ceiling; upper limit for sheetrock on furring attached to beams or joists. Ductwork and Mechanical: 1–10 percent Depends on amount and attachment. Partitions: 10–20 percent If attached to the floor system and not spaced more than every five floor beams or the effective joist floor width. Note: The above values are based on observation only. Beam or girder frequency can be estimated from 1⁄2

 gEIt  f = K  3  WL  where

f = first natural frequency, hz K = 1.57 for simply supported beams = 0.56 for cantilevered beams = from Figure 4-8 for overhanging beams g = acceleration of gravity = 386 in./sec2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(4-2)

FLOOR DEFLECTIONS AND VIBRATIONS

4 - 209

E = modulus of elasticity, psi It = transformed moment of inertia of the tee-beam model, Figure 4-9, in.4 (to be used for both composite and noncomposite construction) W= total weight supported by the tee beam, dead load plus 10–25 percent of design live load, lbs L = tee-beam span, in. System frequency is estimated using 1 1 1 = + fs2 fb2 fg2

1.6

1.4

L

Frequency Coefficient, K

1.2

H

1.0

.8

gElt

f=K

WL3

.6

.4

.2

0 .2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Cantilever - Backspan Ratio, H / L

Fig. 4-8. Frequency coefficients for overhanging beams. Beam spacing S

Beam spacing S

Slab Deck

de

Actual

Model

Fig. 4-9. Tee-beam model for computing transformed moment of inertia. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2.4

4 - 210


where fs = system frequency, hz fb = beam or joist frequency, hz fg = girder frequency, hz Amplitude from a heel-drop impact can be estimated from Ao =

Aot Neff

(4-3)

where Ao = initial amplitude of the floor system due to a heel-drop impact, in. Neff = number of effective tee beams Aot = initial amplitude of a single tee beam due to a heel-drop impact, in. = (DLF)maxds

(4-4)

where (DLF)max = maximum dynamic load factor, Table 4-2 ds = static deflection caused by a 600 lbs force, in. See (Murray, 1975) for equations for (DLF)max and ds For girders, Neff = 1.0. For beams: 1. S < 2.5ft, usual steel joist-concrete slab floor systems.  πx   for x ≤ xo Neff = 1 + 2Σ cos 2xo   where x = distance from the center joist to the joist under consideration, in. xo = distance from the center joist to the edge of the effective floor, in. = 1.06εL L = joist span, in. ε = (Dx / Dy)0.25 Dx = flexural stiffness perpendicular to the joists = Ect3 / 12 Dy = flexural stiffness parallel to the joists = EIt / S Ec = modulus of elasticity of concrete, psi E = modulus of elasticity of steel, psi t = slab thickness, in. It = transformed moment of inertia of the tee beam, in.4 S = joist spacing, in. 2. S > 2.5 ft, usual steel beam-concrete slab floor systems. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(4-5)

BEAMS: OTHER SUBJECTS

4 - 211

Neff = 2.97 −

S L4 + 17.3de 135EIT

(4-6)

where E is defined above and S = beam spacing, in. de = effective slab depth, in. L = beam span, in. Limitations: 15 ≤ (S / de) < 40; 1 × 106 ≤ (L4 / IT) ≤ 50 × 106 The amplitude of a two-way system can be estimated from Aos = Aob + Aog / 2 where Aos = system amplitude Aob = Aot for beam Aog = Aot for girder Additional information on building floor vibrations can be obtained from the abovereferenced paper by Murray (1991) and the references cited therein. BEAMS: OTHER SUBJECTS

Other topics related to the design of flexural members covered elsewhere in this Manual include: Beam Bearing Plates, in Part 11 (Volume II); Beam Web Penetrations, in Part 12 (Volume II).


4 - 212


Table 4-2. Dynamic Load Factors for Heel-Drop Impact f, hz

DLF

F, hz

DLF

F, hz

DLF

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40

0.1541 0.1695 0.1847 0.2000 0.2152 0.2304 0.2456 0.2607 0.2758 0.2908 0.3058 0.3207 0.3356 0.3504 0.3651 0.3798 0.3945 0.4091 0.4236 0.4380 0.4524 0.4667 0.4809 0.4950 0.5091 0.5231 0.5369 0.5507 0.5645 0.5781 0.5916 0.6050 0.6184 0.6316 0.6448 0.6578 0.6707 0.6835 0.6962 0.7088 0.7213 0.7337 0.7459 0.7580 0.7700

5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 9.70 9.80 9.90

0.7819 0.7937 0.8053 0.8168 0.8282 0.8394 0.8505 0.8615 0.8723 0.8830 0.8936 0.9040 0.9143 0.9244 0.9344 0.9443 0.9540 0.9635 0.9729 0.9821 0.9912 1.0002 1.0090 1.0176 1.0261 1.0345 1.0428 1.0509 1.0588 1.0667 1.0744 1.0820 1.0895 1.0969 1.1041 1.1113 1.1183 1.1252 1.1321 1.1388 1.1434 1.1519 1.1583 1.1647 1.1709

10.00 10.10 10.20 10.30 10.40 10.50 10.60 10.70 10.80 10.90 11.00 11.10 11.20 11.30 11.40 11.50 11.60 11.70 11.80 11.90 12.00 12.10 12.20 12.30 12.40 12.50 12.60 12.70 12.80 12.90 13.00 13.10 13.20 13.30 13.40 13.50 13.60 13.70 13.80 13.90 14.00 14.10 14.20 14.30 14.40

1.1770 1.1831 1.1891 1.1949 1.2007 1.2065 1.2121 1.2177 1.2231 1.2285 1.2339 1.2391 1.2443 1.2494 1.2545 1.2594 1.2643 1.2692 1.2740 1.2787 1.2834 1.2879 1.2925 1.2970 1.3014 1.3058 1.3101 1.3143 1.3185 1.3227 1.3268 1.3308 1.3348 1.3388 1.3427 1.3466 1.3504 1.3541 1.3579 1.3615 1.3652 1.3688 1.3723 1.3758 1.3793


REFERENCES

4 - 213

REFERENCES

Allison, H., 1991, Low- and Medium-Rise Steel Buildings, AISC Steel Design Guide Series No. 5, American Institute of Steel Construction, Chicago, IL. American Institute of Steel Construction, 1983, Torsional Analysis of Steel Members, AISC, Chicago. Fisher, J. M. and M. A. West, 1990, Serviceability Design Considerations for Low-Rise Buildings, AISC Steel Design Guide Series No. 3, AISC, Chicago. Kloiber, L. A., 1989, “Cambering of Steel Beams,” Steel Structures: Proceedings of Structures Congress ’89, American Society of Civil Engineers (ASCE), New York. Murray, T. M., 1991, “Building Floor Vibrations,” Proceedings of the 1991 National Steel Construction Conference, AISC, Chicago. Zahn, C. J., 1987, “Plate Girder Design Using LRFD,” Engineering Journal, 1st Qtr., AISC, Chicago.


5-1

PART 5 COMPOSITE DESIGN

OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 COMPOSITE BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 Design Flexural Strength (Positive) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 Concrete Flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7 Shear Connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Strength During Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Lateral Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 Design Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 Lower Bound Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 Composite Beam Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Preliminary Section Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11 Floor Deflections and Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Selection Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18 Lower Bound Elastic Moment of Inertia Tables . . . . . . . . . . . . . . . . . . . . . . 5-50 COMPOSITE COLUMNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 Combined Axial Compression and Bending (Interaction) . . . . . . . . . . . . . . . . . 5-69 COMPOSITE COLUMNS—W SHAPES ENCASED IN CONCRETE . . . . . . . . . . 5-73 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-73 Tables: fc′ = 3.5 ksi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-74 Tables: fc′ = 5 ksi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-86 Tables: fc′ = 8 ksi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-98 COMPOSITE COLUMNS—CONCRETE-FILLED STEEL PIPE AND STRUCTURAL TUBING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-110 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-110 Steel Pipe Filled with Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-110 Structural Tubing Filled with Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-110 Tables: Steel Pipe (fc′ = 3.5 ksi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-111 Tables: Steel Pipe (fc′ = 5 ksi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-113 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5-2

COMPOSITE DESIGN

Tables: Square Structural Tubing (fc′ = 3.5 ksi) . . . . . . . . . . . . . . . . . . . . . 5-115 Tables: Square Structural Tubing (fc′ = 5 ksi) . . . . . . . . . . . . . . . . . . . . . . . 5-122 Tables: Rectangular Structural Tubing (fc′ = 3.5 ksi) . . . . . . . . . . . . . . . . . . . 5-129 Tables: Rectangular Structural Tubing (fc′ = 5 ksi) . . . . . . . . . . . . . . . . . . . . 5-136 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-143


OVERVIEW

5-3

OVERVIEW

Tables are given for the design of composite beams and columns. Composite Beam tables are located as follows: Selection Tables, Fy = 36 ksi, begin on . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18 Selection Tables, Fy = 50 ksi, begin on . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34 Lower Bound Elastic Moment of Inertia Tables begin on . . . . . . . . . . . . . . . . . 5-50 Composite Column tables are located as follows: W Shapes Encased in Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-74 Concrete-Filled Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-111 Concrete-Filled Structural Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-115


5-4

COMPOSITE DESIGN


COMPOSITE BEAMS

5-5

COMPOSITE BEAMS General Notes

The Composite Beam Tables can be used for the design and analysis of simple composite steel beams. Values for the design flexural strength φMn for rolled I-shaped beams with yield strengths of 36 ksi and 50 ksi are tabulated, as well as lower bound moments of inertia. The values tabulated are independent of the concrete flange properties. The strength evaluation of the concrete flange portion of the composite section is left to the design engineer. The preparation of these tables is based upon the fact that the location of the plastic neutral axis (PNA) is uniquely determined by the horizontal shear force ΣQn at the interface between the steel section and the concrete slab. With the knowledge of the location of the PNA and the distance to the centroid of the concrete flange force ΣQn, the design flexural strengths φMn for the rolled section can be computed. Design Flexural Strength (Positive)

The design flexural strength of simple steel beams with composite concrete flanges is computed from the equilibrium of internal forces using the plastic stress distribution as shown in Figure 5-1: φMn = φTTot y = φCToty where φ = 0.85 TTot = sum of tensile forces = Fy × (tensile force beam area) CTot = sum of compressive forces = concrete flange force + Fy × (compressive force beam area) y = moment arm between centroid of tensile force and the resultant compressive force The model used in the calculation of the design strengths tabulated herein is given in Figure 5-2. A summary of the model properties follows: As = area of steel cross section, in.2 Af = flange area = bf × tf, in.2 Aw = web area = (d − 2k)tw, in.2

0.85fc ′

b

Cconc

a

Cstl

PNA

Fy TTot = Tstl

Fy

Fig. 5-1. Plastic stress distribution. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

C Tot = Cconc + Cstl y

5-6

COMPOSITE DESIGN

Kdep = k − tf, in. Karea = (As − 2Af − Aw) / 2, in.2 Limitations for the tabulated values include the following: Fy (d − 2k) / tw ≤ 640 / √ and ΣQn (min.) = 0.25AsFy The limitation of ΣQn (min.) is not required by the Specification, but is deemed to be a practical minimum value. Design strength moment values are tabulated for plastic neutral axis (PNA) locations at the top and intermediate quarter points through the thickness of the steel beam top flange. In addition, PNA locations are computed at the point where ΣQn equals 0.25AsFy, and the point where ΣQn is the average of the minimum value of (0.25AsFy) and the value of ΣQn when the PNA is at the bottom of the top flange (see Figure 5-3). To use the tables, select a valid value of ΣQn, determine the appropriate value of Y2 and read the design flexural strength moment φMn directly. Values for Y1 are also tabulated for convenience. The parameters Y1 and Y2 are defined as follows: Y1 = distance from PNA to beam top flange Y2 = distance from concrete flange force to beam top flange Valid values for ΣQn are the smaller of the following three expressions (LRFD Specification Section I5): 0.85fc′Ac AsFy nQn

bf tf K dep

k

K area K dep

tw

d

d – 2k

K dep tf

k

Fig. 5-2. Composite beam model. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE BEAMS

5-7

where fc′ Ac As Fv n

= specified compressive strength of concrete, ksi = area of concrete slab within effective width, in.2 = area of steel cross section, in.2 = specified minimum yield stress, ksi = number of shear connectors between the point of maximum positive moment and the point of zero moment to either side Qn = shear capacity of single shear connector, kips

Concrete Flange

According to LRFD Specification Section I3.1 the effective width of the concrete slab on each side of the beam centerline shall not exceed: a. one-eighth of the beam span, center to center, of supports; b. one-half the distance to the centerline of the adjacent beams; or c. the distance to the edge of the slab. The maximum concrete flange force is equal to 0.85 fc′Ac where Ac is based on the actual slab thickness, tc. However, often the maximum concrete flange force exceeds the maximum capacity of the specified steel beam. In that case, the effective concrete flange force is determined from a value of ΣQn, which will be the smaller of AsFy or nQn. The effective concrete flange force is:

b

Ycon

Location of effective concrete flange force (∑Qn)

a/ 2

a Y2

TFL (pt. 1) BFL (pt. 5)

6 7

Y1 (varies—see figure below)

Y1 = Distance from top of steel flange to any of the seven tabulated PNA locations ∑ Qn (@ pt. 5) + ∑Q n (@ pt. 7)

∑ Qn (@ point

6 )=

∑ Qn (@ point

7 ) = .25A sFy

Beam top flange

2

4 Equal spaces

1 2 3 4 5

PNA FLANGE LOCATIONS Fig. 5-3. Composite beam table parameters. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

TFL tf BFL

5-8

COMPOSITE DESIGN

ΣQn = Cconc = 0.85fc′ba where Cconc = effective concrete flange force, kips b = effective concrete flange width, in. a = effective concrete flange thickness, in. The basis of the design of most composite beams will be the relationship: a=

ΣQn 0.85fc′b

From this relationship, the value of Y2 can be computed as: Y2 = Ycon − a / 2 where Ycon = distance from top of steel beam to top of concrete, in. Shear Connectors

Shear connectors must be headed steel studs, not less than four stud diameters in length after installation, or hot-rolled steel channels. Shear connectors must be embedded in concrete slabs made with ASTM C33 aggregate or with rotary kiln produced aggregates conforming to ASTM C330, with concrete unit weight not less than 90 pcf. The nominal strength of one stud shear connector embedded in a solid concrete slab is: Qn = 0.5Asc √  fc′Ec ≤ AscFu

(I5-1)

where Asc = cross-sectional area of a stud shear connector, in.2 fc′ = specified concrete compressive strength, ksi Fu = minimum specified tensile strength of stud, ksi Ec = modulus of elasticity of concrete, ksi = (w1.5)√ fc ′ w = unit weight of concrete, lb/cu ft The nominal shear strengths of 3⁄4-in. headed studs embedded in concrete slabs are listed in Table 5-1. Note the effective shear strengths of studs used in conjunction with composite or non-composite metal forms may be affected by the shape of the deck and spacing of the studs. See LRFD Specification Sections I3.5 and I5.6. Strength During Construction

When temporary shores are not used during construction, the steel section must have sufficient strength to support the applied loads prior to the concrete attaining 75 percent of the specified concrete strength fc′ (LRFD Specification Section I3.4). The effect of deflection on unshored steel beams during construction should be considered. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE BEAMS

5-9

Table 5-1. Nominal Stud Shear Strength Qn (kips) for 3⁄4-in Headed Studs fc′ (ksi)

w (lbs/cu. ft)

Qn (kips)

3.0 3.0 3.5 3.5 4.0 4.0

115 145 115 145 115 145

17.7 21.0 19.8 23.6 21.9 26.1

Lateral Support

Adequate lateral support for the compression flange of the steel section will be provided by the concrete slab after hardening. During construction, however, lateral support must be provided, or the design strength must be reduced in accordance with Section F1 of the LRFD Specification. Steel deck with adequate attachment to the compression flange will usually provide the necessary lateral support. For construction using fully encased beams, particular attention should be given to lateral support during construction. Design Shear Strength

The design shear strength of composite beams is determined by the strength of the steel web, in accordance with the requirements of Section F2 of the LRFD Specification. Lower Bound Moment of Inertia

With regard to serviceability, a table of lower bound moments of inertia of composite sections is included to assist in the evaluation of deflection. If calculated deflections using the lower bound moment of inertia are acceptable, a complete elastic analysis of the composite section can be avoided. The lower bound moment of inertia is based on the area of the beam and an equivalent concrete area of ΣQn / Fy. The analysis includes only the horizontal shear force transferred by the shear connectors supplied; and, thus, neglects the contribution of the concrete flange not considered in the plastic distribution of forces (see Figure 5-4). The lower bound moment of inertia, therefore, is the moment of inertia of the section at the factored (ultimate) load. This is smaller than the moment of inertia at service loads where deflection is calculated. The value for the lower bound moment of inertia can be calculated as follows: ILB = Ix + As YENA − 

2

d   ΣQn  + (d + Y2 − YENA)2 2   Fy 

where YENA = distance from bottom of beam to elastic neutral axis (ENA) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5 - 10

COMPOSITE DESIGN

=

 Asd  ΣQn    2 +  F  (d + Y2)  y     ΣQn   As +  F   y  

Composite Beam Reactions

Design reactions for symmetrically loaded composite beams may be computed using the Composite Beam Tables. Two situations will be considered. First, an upper bound value for a beam reaction may be computed neglecting the composite concrete flange properties other than concrete strength. Second, a more refined value for a beam reaction can be computed if the properties of the composite concrete flange are determined initially. When the properties of the composite concrete flange have not been computed, a conservative value for the maximum horizontal shear between the composite concrete slab and the steel section (ΣQn) may be taken as the smaller of AsFs or nQn. Here, n is the number of headed studs between the reaction point and point of maximum moment. The value of Qn may be taken from Table 5-1 or determined from LRFD Specification Section I5. A value for φMn of the composite section may be obtained from the Composite Beam Tables using the sum of horizontal shear ΣQn as described above. In this case, Y2 is defined as the distance from the top of the steel beam to the top of the concrete slab. The design reaction may be determined from the value of φMn as discussed in the following paragraph. When the properties of the concrete flange have been computed (effective width and depth), a slightly different method is used to find φMn. The stud efficiency can be determined in accordance with Section I5 of the LRFD Specification, or Table 5-1 can be used for 3⁄4-in. diameter stud shear connectors. The value for the sum of the horizontal shear force ΣQn can be taken as the smaller of nQn, AsFy, or 0.85fc′Ac, where fc′ is the concrete cylinder strength (ksi) and Ac is the maximum permitted concrete flange area (LRFD Specification Section I5.2). The distance Y2 is the distance from the top of the steel beam to the top of the concrete slab less [ΣQn / (0.85fc′b)] / 2. Using these values for ΣQn and Y2, the value for φMn can be selected from the Composite Beam Tables.

Equivalent concrete area =

∑Qn

Fy

Y2 d + Y2 – YENA

ENA d YENA

Fig. 5-4. Moment of inertia. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE BEAMS

5 - 11

The design beam reaction for a symmetrically loaded composite beam may be computed from known values of φMn and the span length as: R = CcφMn / L where R = design beam reaction, kips Cc = coefficient from Figure 5-5 φMn = composite beam flexural design strength, kip-ft L = span length, ft Preliminary Section Selection

When using the Composite Beam Tables, the approximate beam weight per unit length required for several different beam depths may be calculated as follows: Mu(12)   Beam weight (lb/ft) =   3.4 (d / 2 + Y − a / 2 )φ F con y   where

Mu = required flexural strength, kip-ft d = nominal beam depth, in. Ycon = distance from top of steel beam to top of concrete slab, in. a = effective concrete slab thickness, in. Fy = steel yield stress, ksi φ = 0.85 3.4 = ratio of the weight of a beam to its area, lb/in.2 For convenience in the preliminary selection phase the nominal depth may be used. A value for a/2 must also be selected. For relatively light sections and loads, this value can be assumed to be one inch. With the PNA at the top of the steel beam, i.e., ΣQn = AsFy, the flexural design strength is: φMn = φAsFy (d / 2 + Ycon − a / 2) / 12 where φMn = flexural design strength, kip-ft As = steel beam cross-sectional area, in.2

Uniform load R

Pu R

Cc = 4

R

Pu R

Pu

R

Cc = 2

Pu Pu Pu R

Cc = 3

R=

Cc φM n L

Fig. 5-5. Beam reaction coeficients. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

R

R Cc = 3

5 - 12

COMPOSITE DESIGN

Floor Deflections and Vibrations

Refer to the discussion of Floor Deflections and Vibrations at the end of Part 4 of this LRFD Manual.

EXAMPLE 5-1

Given:

Determine the beam, with Fy = 50 ksi, required to support a service live load of 1.3 kips/ft and a service dead load of 0.9 kips/ft. The beam span is 30 ft and the beam spacing 10 ft. The slab is 31⁄4-in. 1ight weight concrete (fc′ = 3.5 ksi, 115 pcf) supported by a 3-in. deep composite metal deck with an average rib width of six inches. The ribs are oriented perpendicular to the beam. Shored construction is specified. Also, determine the number of 3⁄4-in. diameter headed studs required and the service live load deflection.

Solution:

A. Load tabulation:

LL DL Total

Service load (kips/ft) 1.3 0.9 2.2

(L.F.) (1.6) (1.2)

Factored load (kips/ft) 2.1 1.1 3.2

B. Flexural design strength: Beam moments Mu = 3.2(30)2/8 = 360 kip-ft MLL = 1.3(30)2/8 = 146 kip-ft C. Select section and determine properties: At this point, go directly to the Composite Beam Tables and select a section or compute a preliminary trial section size using the formula:  Mu(12)  Beam weight =   3.4 (d / 2 + Y − a /2)φF con y  where

Ycon = 3 + 3.25 = 6.25 in. a / 2= 1 in. (estimate) φ = 0.85 Fy = 50 ksi Mu(12)(3.4) d/2 φFy 16 346 8 d

18

346

9

(Ycon − a / 2)

Beam Weight

5.25

26

5.25

24


COMPOSITE BEAMS

5 - 13

From the results above, a W16×26 would be the most appropriate selection. Let ΣQn = AsFy = 7.68(50) = 384 kips. The effective width of the concrete flange is 

≤ 2 × L / 8 = 2 × 30 ft / 8 = 7.5 ft = 90 in. (gover ns) 10 ft spacing ΣQn 384 areq’d = = = 1.43 in. 0.85fc′b 0.85(3.5)(90) Y2 = 6.25 − 1.43 / 2 = 5.53 in.

b

By interpolation from the Composite Beam Tables for a W16×26 and a value of Y2 equal to 5.53 in., φMn = 363 + (0.03 / 0.50)(377 − 363) = 364 kip-ft > 360 kip-ft req’d o.k. The selected section is adequate for Y2 = 5.5 in. and Y1 = 0.0 in., for which φMn = 363 kip-ft D. Compute number of studs required: The stud reduction is calculated to be: Reduction factor =

0.85 (wr / hr)(Hs / hr − 1.0) ≤ 1.0 Nr √

Reductionfactor=

0.85 (6 / 3)(5.5 / 3 − 1.0) = 1.0 2 √

(I3-1)

where Nr = number of stud connectors in one rib; not to exceed three in computations, although more than three may be installed wr = average width of rib, in. hr = nominal rib height, in. Hs = length of stud connector after welding, in.; not to exceed the value (hr + 3) in computations, although actual length may be greater. Also must not be less than four stud diameters The value for Hs = 5.5 was selected to ensure the stud capacity reduction factor is 1.0. The number of studs required is: with Qn = 19.8 kips (Table 5-1) 2(ΣQn) / Qn = 2(384) / 19.8 = 38.8, say 40 studs AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5 - 14

COMPOSITE DESIGN

E. Check deflection: For the selected section, a W16×26, Fy = 50 ksi, Y2 = 5.5 in. and Y1 = 0.0 in.; from the Elastic Moment of Inertia Tables one can find the lower bound moment of inertia is 985 in.4 Thus, the service live load deflection can be calculated as follows (see LRFD Manual Part 4): ∆LL =

MLLL2 146(30)2 L L = = 0.83 in. = < o.k. 434 360 161ILB 161(985)

F. Shear check: Vu = 3.2(15) = 48 kips φVn = φ0.6FywAw = (0.9)(0.6)(50)(15.69 × 0.250) = 106 kips > 48 kips req’d o.k.

EXAMPLE 5-2

Given:

Determine the beam, with Fy = 50 ksi, required to support a service live load of 250 psf and a service dead load of 90 psf. The beam span is 40 ft and the beam spacing is 10 ft. Assume 3 in. metal deck is used with a 4.5 in. slab of 4 ksi normal weight concrete (145 pcf). The stud reduction factor is 1.0. Unshored construction is specified. Determine the beam size and service dead and live load deflections. Also select a non-composite section (no shear connectors).

Solution:

A. Load tabulation:

LL DL Total

Service load (kips/ft) 2.5 0.9 3.4

(L.F.) (1.6) (1.2)

Factored load (kips/ft) 4.0 1.1 5.1

B. Beam moments: Mu = 5.1(40)2 / 8 = 1,020 kip-ft MLL = 2.5(40)2 / 8 = 500 kip-ft MDL = 0.9(40)2 / 8 = 180 kip-ft C. Select section and determine properties: Assume a = 2 in.; therefore, take Y2 = 7.5 − 2 / 2 = 6.5 in. From the Composite Beam Tables, for Fy = 50 ksi and Y2 = 6.5 in., W21×62, W24×55, and W24×62 are possible sizes. Try a W24×55: Fy = 50 ksi Y2 = 6.5 in. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE BEAMS

5 - 15

Y1 = 0.0 in. Qn = 810 kips φMn = 1,050 kip-ft Compute Y2 for ΣQn = 810 kips: 2 × L / 8 = 2 × 40 ft / 8 = 10 ft b ≤ 10 ft spacing = 120 in. ΣQn 810 = = 1.99 a = 0.85fc′b 0.85(4)(120) Y2 = 7.5 − 1.99 / 2 = 6.5 in.

D. Compute the number of studs required: Qn = 26.1 kips (Table 5-1) Number of studs = (2)ΣQn / Qn = 2(810) / 26.1 = 62.1, say 64 studs E. Construction phase strength check: A construction live load of 20 psf will be assumed. From the LRFD Specification (Section A4.1), the relevant load combinations are 1.4D = 1.4 × 0.9 = 1.26 k/ft 1.2D + 1.6L = 1.2 × 0.9 + 1.6 × 0.2 = 1.40 k/ft = 1.40 × (40)2 / 8 = 280 kip-ft Mu From the Composite Beam Tables for a W24×55 with Fy = 50 ksi, and assuming adequate lateral support is provided by the attachment of the steel deck to the compression flange, φMn = φMp = 503 kip-ft > 280 kip-ft F. Service load deflections: Assume that the wet concrete load moment is equal to the service dead load moment. With Ix = 1,350 in.4 for a W24×55, ∆DL =

180(40)2 = 1.33 in. 161(1,350)

For the W24×55 with Y2 = 6.5 in. and Y1 = 0.0 in., the lower bound moment of inertia can be found in the Lower Bound Elastic Moment of Inertia Tables; ILB = 4,060 in.4 500(40)2 = 1.22 in. 161(4,060) L L < o.k. =

∆LL =

393

360

Specify a beam camber of 11⁄4-in. to overcome the dead load deflection. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5 - 16

COMPOSITE DESIGN

G. Check shear: Vu = 5.1(40) / 2 = 102 kips φV = φ(0.6)FywAw = (0.9)(0.6)(50)(23.57 × 0.395) = 251 kips > 102 kips o.k. H. Final section selection: Use: W24×55, Fy = 50 ksi, camber 11⁄4-in., 64 studs, 3⁄4-in. diameter (32 each side of midspan) I. Noncomposite section: Considering the given problem without shear connectors (i.e., noncomposite), a steel section can be selected from the φMp values tabulated under each section in either the Composite Beam Tables or the Load Factor Design Selection Tables. For Mu = 1,020 kip-ft, select a W27×94, Fy = 50 ksi, with a φMp flexural design strength equal to 1,040 kip-ft. ∆DL =

180(40)2 = 0.55 in. 161(3,270)

∆LL =

500(40)2 = 1.52 in. > L / 360 161(3,270)

For ∆ = L / 360 = 1.33 in. Ireq’d =

500(40)2 = 3,736 in.4 161(1.33)

Use: W30×99, Fy = 50 ksi, φMn = φbMp = 1,170 kip-ft

EXAMPLE 5-3

Given:

A W21×44, Fy = 50 ksi, steel girder spans 30 feet and supports intermediate beams at the third points. A total of fifty 3⁄4-in. diameter headed studs are applied to the beam as follows: 24 between each support and the beams at the one-third points, and two between the intermediate beams. The slab consists of 31⁄4-in. light-weight concrete (115 pcf) with a specified design strength of 3.5 ksi over a 3-in. deep composite metal deck with an average rib width of six inches. The ribs are oriented parallel to the beam centerline. Determine the design beam reactions.

Solutions:

For studs in a single row the spacing between the support and first intermediate beam would be 10(12) / 24 = 5.0 in. which is greater than the specified minimum of six stud diameters (LRFD Specification Section I5.6). Since wr / hr = 6 / 3 = 2 is greater than 1.5, the stud AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE BEAMS

5 - 17

reduction factor is not necessary (LRFD Specification I3.5c). Therefore, from Table 5.1, the stud shear strength is:

ΣQn = nQn = 24(19.8) = 475 kips For ΣQn = 475 kips, the required effective concrete flange thickness can be calculated to be: a =

475 = 1.77 in. 0.85(3.5)(7.5)(12)

Y2 = 3 + 3.25 − 1.77 / 2 = 5.36 in. Beam reaction: From the Composite Beam Selection Table, for a W21×44, Fy = 50 ksi, ΣQn = 475 kips places the PNA at Y1 = 0.27 in. For Y2 = 5.36 in. and Y1 = 0.27 in., φMn = 655 kip-ft R = CcφMn / L = 3(655) / 30 = 65.5 kips where R = design reaction, kips Cc = coefficient from Figure 5-5 φMn = flexural design strength of beam, kip-ft L = span length, ft Note: The beam weight was neglected in this example.


5 - 18

COMPOSITE DESIGN

Fy = 36 ksi

COMPOSITE DESIGN COMPOSITE BEAM SELECTION TABLE W Shapes φ = 0.85 φb = 0.90 Shape

φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) Kip-ft

W40×297

3590

W 40 X 278 3210

In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

0.00 0.41 0.83 1.24 1.65 4.59 8.17

3150 2680 2210 1740 1270 1030 787

4890 4810 4720 4620 4510 4420 4280

5000 4910 4800 4690 4550 4460 4310

5110 5000 4880 4750 4600 4500 4340

5220 5100 4960 4810 4640 4540 4370

5330 5190 5040 4870 4690 4570 4390

5440 5290 5120 4930 4730 4610 4420

5550 5380 5190 4990 4780 4640 4450

5670 5480 5270 5050 4820 4680 4480

5780 5570 5350 5120 4870 4720 4500

5890 5660 5430 5180 4910 4750 4530

6000 5760 5510 5240 4960 4790 4560

TFL 0.00 2 0.45 3 0.91 4 1.36 BFL 1.81 6 5.64 7 10.06

2940 2550 2160 1770 1380 1060 736

4610 4540 4470 4380 4280 4160 3930

4710 4630 4550 4450 4330 4190 3960

4810 4730 4620 4510 4380 4230 3980

4920 4820 4700 4570 4430 4270 4010

5020 4910 4780 4640 4480 4310 4040

5130 5000 4850 4700 4530 4340 4060

5230 5090 4930 4760 4580 4380 4090

5340 5180 5010 4820 4630 4420 4110

5440 5270 5080 4890 4680 4460 4140

5540 5360 5160 4950 4730 4490 4170

5650 5450 5240 5010 4780 4530 4190

TFL 2 3 4 BFL 6 7

W40×277

3380

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.58 4.25 7.60

2930 2480 2030 1580 1130 932 732

4530 4460 4380 4280 4170 4110 3990

4630 4550 4450 4340 4210 4140 4020

4740 4630 4520 4390 4250 4170 4050

4840 4720 4590 4450 4290 4210 4070

4940 4810 4660 4510 4330 4240 4100

5050 4900 4740 4560 4370 4270 4120

5150 4990 4810 4620 4410 4300 4150

5250 5070 4880 4670 4450 4340 4180

5360 5160 4950 4730 4490 4370 4200

5460 5250 5020 4790 4540 4400 4230

5570 5340 5100 4840 4580 4440 4250

W40×264

3050

TFL 2 3 4 BFL 6 7

0.00 0.43 0.87 1.30 1.73 5.49 9.90

2790 2420 2050 1680 1310 1000 698

4350 4300 4230 4140 4050 3930 3730

4450 4380 4300 4200 4100 3970 3750

4550 4470 4370 4260 4140 4000 3780

4650 4550 4440 4320 4190 4040 3800

4750 4640 4520 4380 4240 4070 3820

4850 4720 4590 4440 4280 4110 3850

4950 4810 4660 4500 4330 4150 3870

5050 4900 4730 4560 4380 4180 3900

5140 4980 4810 4620 4420 4220 3920

5240 5070 4880 4680 4470 4250 3950

5340 5150 4950 4740 4510 4290 3970

aY1 = distance from top of the steel beam to plastic neutral axis. bY2 = distance from top of the steel beam to concrete flange force. cSee Figure 5-3 for PNA locations.


COMPOSITE BEAMS

5 - 19

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W40×249

Kip-ft

3020

TFL 2 3 4 BFL 6 7

0.00 0.36 0.71 1.07 1.42 4.06 7.47

2640 2240 1830 1430 1030 844 660

4050 3990 3920 3840 3750 3690 3580

4150 4070 3980 3890 3780 3720 3610

4240 4150 4050 3940 3820 3750 3630

4330 4230 4110 3990 3850 3770 3650

4430 4310 4180 4040 3890 3800 3680

4520 4390 4240 4090 3930 3830 3700

4610 4470 4310 4140 3960 3860 3720

4710 4550 4370 4190 4000 3890 3750

4800 4630 4440 4240 4040 3920 3770

4900 4700 4500 4290 4070 3950 3790

4990 4780 4570 4340 4110 3980 3820

W40×235

2730

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.58 5.18 9.47

2480 2140 1810 1470 1130 876 620

3840 3790 3720 3650 3570 3480 3310

3930 3860 3790 3700 3610 3510 3330

4010 3940 3850 3760 3650 3540 3350

4100 4010 3920 3810 3690 3570 3370

4190 4090 3980 3860 3730 3600 3400

4280 4170 4040 3910 3770 3630 3420

4370 4240 4110 3960 3810 3660 3440

4450 4320 4170 4020 3850 3690 3460

4540 4390 4240 4070 3890 3730 3480

4630 4470 4300 4120 3930 3760 3510

4720 4540 4360 4170 3970 3790 3530

W40×215

2600

TFL 2 3 4 BFL 6 7

0.00 0.31 0.61 0.92 1.22 3.84 7.32

2280 1930 1590 1240 895 733 570

3470 3420 3360 3290 3210 3160 3080

3550 3480 3410 3330 3240 3190 3100

3630 3550 3470 3380 3280 3210 3120

3710 3620 3520 3420 3310 3240 3140

3790 3690 3580 3460 3340 3270 3160

3870 3760 3640 3510 3370 3290 3180

3950 3830 3690 3550 3400 3320 3200

4030 3900 3750 3600 3440 3340 3220

4110 3960 3810 3640 3470 3370 3240

4200 4030 3860 3680 3500 3400 3260

4280 4100 3920 3730 3530 3420 3280

W40×211

2440

TFL 2 3 4 BFL 6 7

0.00 0.35 0.71 1.06 1.42 4.99 9.35

2230 1930 1630 1330 1030 793 558

3430 3380 3330 3270 3200 3110 2960

3510 3450 3390 3310 3230 3140 2980

3590 3520 3440 3360 3270 3170 3000

3670 3590 3500 3410 3310 3200 3020

3740 3660 3560 3460 3340 3230 3040

3820 3720 3620 3500 3380 3250 3060

3900 3790 3670 3550 3420 3280 3080

3980 3860 3730 3600 3450 3310 3100

4060 3930 3790 3640 3490 3340 3120

4140 4000 3850 3690 3530 3370 3140

4220 4070 3910 3740 3560 3400 3160



5 - 20

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W40×199

Kip-ft

2340

TFL 2 3 4 BFL 6 7

0.00 0.27 0.53 0.80 1.07 4.16 8.10

2100 1800 1500 1200 895 710 526

3180 3130 3080 3020 2960 2900 2800

3250 3200 3130 3070 2990 2930 2820

3330 3260 3190 3110 3020 2950 2830

3400 3320 3240 3150 3060 2980 2850

3480 3390 3290 3190 3090 3000 2870

3550 3450 3350 3240 3120 3030 2890

3620 3510 3400 3280 3150 3050 2910

3700 3580 3450 3320 3180 3080 2930

3770 3640 3500 3360 3210 3100 2950

3850 3710 3560 3400 3250 3130 2960

3920 3770 3610 3450 3280 3150 2980

W40×183

2110

TFL 2 3 4 BFL 6 7

0.00 0.31 0.61 0.92 1.22 4.76 9.16

1930 1670 1410 1160 896 690 483

2940 2900 2860 2810 2750 2680 2550

3010 2960 2910 2850 2780 2710 2570

3080 3020 2960 2890 2810 2730 2580

3150 3080 3010 2930 2850 2750 2600

3220 3140 3060 2970 2880 2780 2620

3290 3200 3110 3010 2910 2800 2640

3350 3260 3160 3050 2940 2830 2650

3420 3320 3210 3090 2970 2850 2670

3490 3380 3260 3130 3000 2880 2690

3560 3440 3310 3180 3040 2900 2700

3630 3500 3360 3220 3070 2930 2720

W40×174

1930

TFL 2 3 4 BFL 6 7

0.00 0.21 0.42 0.62 0.83 4.59 9.27

1840 1600 1370 1130 898 679 460

2750 2710 2680 2630 2590 2520 2380

2810 2770 2720 2670 2620 2540 2400

2880 2830 2770 2710 2650 2570 2410

2940 2880 2820 2750 2680 2590 2430

3010 2940 2870 2790 2720 2620 2450

3080 3000 2920 2830 2750 2640 2460

3140 3060 2970 2870 2780 2660 2480

3210 3110 3020 2910 2810 2690 2490

3270 3170 3060 2960 2840 2710 2510

3340 3230 3110 3000 2870 2740 2530

3400 3280 3160 3040 2910 2760 2540

W40×167

1870

TFL 2 3 4 BFL 6 7

0.00 0.26 0.51 0.77 1.03 5.00 9.85

1770 1550 1330 1110 896 669 442

2670 2630 2600 2560 2510 2430 2280

2730 2690 2640 2600 2540 2460 2300

2790 2740 2690 2630 2570 2480 2310

2850 2800 2740 2670 2610 2510 2330

2920 2850 2790 2710 2640 2530 2350

2980 2910 2830 2750 2670 2550 2360

3040 2960 2880 2790 2700 2580 2380

3100 3020 2930 2830 2730 2600 2390

3170 3070 2970 2870 2760 2620 2410

3230 3130 3020 2910 2800 2650 2420

3290 3180 3070 2950 2830 2670 2440

W40×149

1610

TFL 0.00 2 0.21 3 0.42 4 0.62 BFL 0.83 6 5.15 7 10.41

1580 1400 1220 1050 871 633 394

2360 2330 2300 2270 2240 2160 1990

2410 2380 2340 2310 2270 2180 2010

2470 2430 2390 2340 2300 2200 2020

2520 2480 2430 2380 2330 2220 2030

2580 2530 2470 2420 2360 2250 2050

2640 2580 2520 2460 2390 2270 2060

2690 2630 2560 2490 2420 2290 2070

2750 2680 2600 2530 2450 2310 2090

2800 2730 2650 2570 2480 2340 2100

2860 2780 2690 2600 2510 2360 2120

2920 2830 2730 2640 2540 2380 2130



COMPOSITE BEAMS

5 - 21

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 36×300

Kip-ft

3400

TFL 2 3 4 BFL 6 7

0.00 0.42 0.84 1.26 1.68 3.97 6.69

3180 2680 2170 1670 1160 979 795

4590 4510 4410 4310 4180 4120 4020

4700 4600 4490 4360 4220 4150 4050

4810 4700 4570 4420 4260 4190 4080

4920 4790 4640 4480 4310 4220 4110

5040 4890 4720 4540 4350 4260 4140

5150 4980 4800 4600 4390 4290 4160

5260 5080 4880 4660 4430 4330 4190

5370 5170 4950 4720 4470 4360 4220

5490 5270 5030 4780 4510 4400 4250

5600 5360 5110 4840 4550 4430 4280

5710 5460 5180 4900 4590 4470 4300

W 36×280

3160

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.57 3.88 6.62

2970 2500 2030 1560 1090 916 742

4260 4180 4100 4000 3890 3830 3740

4360 4270 4170 4050 3930 3860 3770

4470 4360 4240 4110 3960 3890 3790

4570 4450 4310 4160 4000 3930 3820

4680 4540 4390 4220 4040 3960 3850

4780 4630 4460 4280 4080 3990 3870

4890 4710 4530 4330 4120 4020 3900

4990 4800 4600 4390 4160 4060 3920

5100 4890 4670 4440 4200 4090 3950

5200 4980 4740 4500 4230 4120 3980

5310 5070 4820 4550 4270 4150 4000

W 36×260

2920

TFL 2 3 4 BFL 6 7

0.00 0.36 0.72 1.08 1.44 3.86 6.75

2750 2330 1900 1470 1040 863 689

3930 3860 3780 3700 3600 3540 3450

4020 3940 3850 3750 3630 3570 3470

4120 4030 3920 3800 3670 3600 3500

4220 4110 3980 3850 3710 3630 3520

4320 4190 4050 3900 3740 3660 3550

4410 4270 4120 3960 3780 3690 3570

4510 4350 4190 4010 3820 3720 3600

4610 4440 4250 4060 3850 3750 3620

4710 4520 4320 4110 3890 3780 3640

4800 4600 4390 4160 3930 3820 3670

4900 4680 4450 4210 3960 3850 3690

W 36×245

2730

TFL 2 3 4 BFL 6 7

0.00 0.34 0.68 1.01 1.35 3.81 6.77

2600 2190 1790 1390 991 820 649

3680 3620 3550 3470 3380 3330 3240

3780 3700 3620 3520 3420 3360 3260

3870 3780 3680 3570 3450 3380 3280

3960 3860 3740 3620 3490 3410 3310

4050 3930 3810 3670 3520 3440 3330

4140 4010 3870 3720 3560 3470 3350

4240 4090 3930 3770 3590 3500 3380

4330 4170 4000 3820 3630 3530 3400

4420 4240 4060 3870 3660 3560 3420

4510 4320 4120 3910 3700 3590 3440

4600 4400 4190 3960 3730 3620 3470

W 36×230

2550

TFL 2 3 4 BFL 6 7

0.00 0.32 0.63 0.95 1.26 3.81 6.83

2430 2060 1690 1310 939 774 608

3440 3380 3320 3240 3160 3110 3020

3530 3450 3380 3290 3190 3140 3040

3610 3530 3440 3340 3230 3160 3070

3700 3600 3500 3380 3260 3190 3090

3780 3670 3560 3430 3290 3220 3110

3870 3750 3620 3480 3330 3250 3130

3960 3820 3670 3520 3360 3270 3150

4040 3890 3730 3570 3390 3300 3170

4130 3970 3790 3610 3430 3330 3200

4210 4040 3850 3660 3460 3360 3220

4300 4110 3910 3710 3490 3380 3240



5 - 22

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 36×210

Kip-ft

2250

TFL 2 3 4 BFL 6 7

0.00 0.34 0.68 1.02 1.36 5.06 9.04

2220 1930 1630 1330 1030 794 556

3210 3160 3110 3050 2980 2890 2740

3280 3230 3170 3090 3020 2920 2760

3360 3300 3220 3140 3050 2950 2780

3440 3370 3280 3190 3090 2980 2800

3520 3430 3340 3240 3130 3010 2820

3600 3500 3400 3280 3160 3030 2840

3680 3570 3450 3330 3200 3060 2860

3760 3640 3510 3380 3240 3090 2880

3840 3710 3570 3420 3270 3120 2900

3920 3770 3630 3470 3310 3150 2920

3990 3840 3680 3520 3350 3170 2940

W 36×194

2070

TFL 2 3 4 BFL 6 7

0.00 0.32 0.63 0.95 1.26 4.94 8.93

2050 1780 1500 1230 953 733 513

2940 2900 2850 2800 2740 2660 2520

3020 2960 2910 2840 2770 2690 2540

3090 3030 2960 2890 2810 2710 2560

3160 3090 3010 2930 2840 2740 2580

3230 3150 3070 2970 2870 2760 2590

3310 3220 3120 3020 2910 2790 2610

3380 3280 3170 3060 2940 2820 2630

3450 3340 3220 3100 2970 2840 2650

3520 3400 3280 3150 3010 2870 2670

3600 3470 3330 3190 3040 2890 2680

3670 3530 3380 3230 3080 2920 2700

W 36×182

1940

TFL 2 3 4 BFL 6 7

0.00 0.29 0.59 0.89 1.18 4.89 8.92

1930 1670 1420 1160 904 693 482

2760 2720 2670 2620 2570 2490 2360

2820 2780 2720 2660 2600 2520 2380

2890 2840 2770 2710 2630 2540 2400

2960 2890 2820 2750 2660 2570 2410

3030 2950 2870 2790 2700 2590 2430

3100 3010 2920 2830 2730 2620 2450

3170 3070 2970 2870 2760 2640 2460

3230 3130 3020 2910 2790 2670 2480

3300 3190 3070 2950 2820 2690 2500

3370 3250 3120 2990 2860 2720 2520

3440 3310 3170 3030 2890 2740 2530

W 36×170

1800

TFL 2 3 4 BFL 6 7

0.00 0.28 0.55 0.83 1.10 4.84 8.89

1800 1560 1320 1090 847 649 450

2560 2520 2480 2440 2390 2320 2200

2620 2580 2530 2480 2420 2340 2210

2690 2640 2580 2520 2450 2370 2230

2750 2690 2620 2550 2480 2390 2240

2820 2750 2670 2590 2510 2410 2260

2880 2800 2720 2630 2540 2440 2280

2940 2860 2770 2670 2570 2460 2290

3010 2910 2810 2710 2600 2480 2310

3070 2970 2860 2750 2630 2500 2320

3130 3020 2910 2780 2660 2530 2340

3200 3080 2950 2820 2690 2550 2360

W 36×160

1680

TFL 2 3 4 BFL 6 7

0.00 0.26 0.51 0.77 1.02 4.82 8.97

1690 1470 1250 1030 811 617 423

2400 2360 2330 2290 2240 2170 2050

2460 2420 2370 2320 2270 2200 2070

2520 2470 2420 2360 2300 2220 2080

2580 2520 2460 2400 2330 2240 2100

2640 2570 2500 2430 2360 2260 2110

2700 2630 2550 2470 2380 2280 2130

2760 2680 2590 2510 2410 2310 2140

2820 2730 2640 2540 2440 2330 2160

2880 2780 2680 2580 2470 2350 2170

2940 2830 2730 2610 2500 2370 2190

3000 2890 2770 2650 2530 2390 2200



COMPOSITE BEAMS

5 - 23

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) 5

5.5

6

6.5

7

W 36×150

Kip-ft

1570

TFL 2 3 4 BFL 6 7

0.00 1590 2250 2300 2360 2410 2470 2530 0.24 1390 2220 2260 2310 2360 2410 2460 0.47 1190 2180 2220 2270 2310 2350 2390 0.71 983 2140 2180 2210 2250 2280 2320 0.94 781 2100 2130 2160 2190 2210 2240 4.83 589 2040 2060 2080 2100 2120 2140 9.08 398 1920 1930 1950 1960 1970 1990

In.

Kips

2

2.5

3

3.5

4

4.5

2580 2510 2430 2350 2270 2160 2000

2640 2560 2480 2390 2300 2190 2020

2700 2610 2520 2420 2330 2210 2030

2750 2660 2560 2460 2350 2230 2040

2810 2710 2600 2490 2380 2250 2060

W 36×135

1370

TFL 2 3 4 BFL 6 7

0.00 1430 2000 2050 2100 2150 2200 2260 0.20 1260 1980 2020 2070 2110 2160 2200 0.40 1090 1950 1990 2030 2060 2100 2140 0.59 919 1920 1950 1980 2020 2050 2080 0.79 749 1890 1910 1940 1970 1990 2020 4.96 553 1820 1840 1860 1880 1900 1920 9.50 357 1690 1710 1720 1730 1740 1760

2310 2240 2180 2110 2050 1940 1770

2360 2290 2220 2150 2070 1960 1780

2410 2330 2260 2180 2100 1980 1790

2460 2380 2300 2210 2130 2000 1810

2510 2420 2330 2240 2150 2020 1820

W 33×221

2310

TFL 2 3 4 BFL 6 7

0.00 0.32 0.64 0.96 1.28 3.76 6.48

2340 1980 1610 1250 889 737 585

3140 3090 3020 2950 2870 2820 2750

3230 3160 3080 3000 2900 2850 2770

3310 3230 3140 3040 2940 2880 2790

3390 3300 3200 3090 2970 2900 2810

3470 3370 3250 3130 3000 2930 2830

3560 3440 3310 3170 3030 2960 2850

3640 3510 3370 3220 3060 2980 2870

3720 3580 3420 3260 3090 3010 2890

3810 3650 3480 3310 3120 3030 2910

3890 3720 3540 3350 3160 3060 2930

3970 3790 3600 3400 3190 3090 2960

W 33×201

2080

TFL 2 3 4 BFL 6 7

0.00 0.29 0.58 0.86 1.15 3.67 6.51

2130 1800 1480 1150 824 678 532

2840 2790 2730 2670 2600 2560 2480

2910 2850 2790 2710 2630 2580 2500

2990 2920 2840 2750 2660 2600 2520

3070 2980 2890 2790 2690 2630 2540

3140 3050 2940 2830 2720 2650 2560

3220 3110 2990 2870 2750 2680 2580

3290 3170 3050 2920 2780 2700 2600

3370 3240 3100 2960 2810 2720 2620

3440 3300 3150 3000 2830 2750 2630

3520 3360 3200 3040 2860 2770 2650

3590 3430 3260 3080 2890 2800 2670

W 33×141

1390

TFL 2 3 4 BFL 6 7

0.00 1500 1980 2030 2080 2140 2190 2240 0.24 1300 1950 1990 2040 2090 2130 2180 0.48 1100 1920 1950 1990 2030 2070 2110 0.72 900 1880 1910 1940 1970 2010 2040 0.96 700 1840 1860 1890 1910 1940 1960 4.31 537 1790 1810 1820 1840 1860 1880 8.05 374 1690 1710 1720 1730 1740 1760

2300 2220 2150 2070 1990 1900 1770

2350 2270 2190 2100 2010 1920 1780

2400 2320 2230 2130 2040 1940 1800

2460 2360 2270 2170 2060 1960 1810

2510 2410 2300 2200 2090 1980 1820



5 - 24

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) 5

5.5

6

6.5

7

W 33×130

Kip-ft

1260

TFL 2 3 4 BFL 6 7

0.00 1380 1810 1860 1910 1960 2010 2060 0.21 1200 1780 1830 1870 1910 1950 2000 0.43 1020 1760 1790 1830 1860 1900 1940 0.64 847 1720 1750 1780 1810 1840 1870 0.86 670 1690 1710 1740 1760 1780 1810 4.39 507 1640 1660 1670 1690 1710 1730 8.29 345 1540 1550 1570 1580 1590 1600

In.

Kips

2

2.5

3

3.5

4

4.5

2100 2040 1970 1900 1830 1750 1610

2150 2080 2010 1930 1860 1760 1630

2200 2130 2050 1960 1880 1780 1640

2250 2170 2080 1990 1900 1800 1650

2300 2210 2120 2020 1930 1820 1660

W 33×118

1120

TFL 2 3 4 BFL 6 7

0.00 1250 1630 1680 1720 1760 1810 1850 0.19 1100 1610 1650 1690 1720 1760 1800 0.37 943 1580 1620 1650 1680 1720 1750 0.56 790 1560 1580 1610 1640 1670 1700 0.74 638 1530 1550 1570 1600 1620 1640 4.44 475 1480 1490 1510 1530 1540 1560 8.54 312 1380 1390 1400 1410 1420 1430

1900 1840 1780 1720 1660 1580 1450

1940 1880 1820 1750 1690 1590 1460

1980 1920 1850 1780 1710 1610 1470

2030 1960 1880 1810 1730 1630 1480

2070 2000 1920 1840 1750 1650 1490

W 30×116

1020

TFL 2 3 4 BFL 6 7

0.00 1230 1480 1530 1570 1610 1660 1700 0.21 1070 1460 1500 1530 1570 1610 1650 0.43 910 1430 1460 1500 1530 1560 1590 0.64 749 1400 1430 1460 1480 1510 1540 0.85 589 1370 1390 1410 1440 1460 1480 3.98 448 1330 1350 1360 1380 1390 1410 7.44 308 1250 1260 1270 1290 1300 1310

1740 1690 1630 1560 1500 1430 1320

1790 1720 1660 1590 1520 1440 1330

1830 1760 1690 1620 1540 1460 1340

1880 1800 1720 1640 1560 1470 1350

1920 1840 1750 1670 1580 1490 1360

W 30×108

934

TFL 2 3 4 BFL 6 7

0.00 1140 1370 1410 1450 1490 1530 1570 0.19 998 1350 1380 1420 1450 1490 1520 0.38 855 1320 1350 1380 1410 1440 1470 0.57 711 1300 1320 1350 1370 1400 1420 0.76 568 1270 1290 1310 1330 1350 1370 4.04 427 1230 1240 1260 1270 1290 1300 7.64 285 1150 1160 1170 1180 1190 1200

1610 1560 1500 1450 1390 1320 1210

1650 1590 1530 1470 1410 1330 1220

1690 1630 1570 1500 1430 1350 1230

1730 1660 1600 1520 1450 1360 1240

1770 1700 1630 1550 1470 1380 1250

W 30×99

842

TFL 2 3 4 BFL 6 7

0.00 1050 1250 1290 1320 1360 1400 1430 0.17 922 1230 1260 1300 1330 1360 1390 0.34 796 1210 1240 1270 1290 1320 1350 0.50 670 1190 1210 1240 1260 1280 1310 0.67 543 1170 1180 1200 1220 1240 1260 4.07 403 1120 1140 1150 1170 1180 1190 7.83 262 1040 1050 1060 1070 1080 1090

1470 1430 1380 1330 1280 1210 1100

1510 1460 1410 1350 1300 1220 1110

1550 1490 1440 1380 1320 1240 1120

1580 1520 1460 1400 1340 1250 1130

1620 1560 1490 1430 1360 1270 1140



COMPOSITE BEAMS

5 - 25

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) 5

5.5

6

6.5

7

W 27×102

Kip-ft

824

TFL 2 3 4 BFL 6 7

0.00 1080 1190 1230 1270 1300 1340 1380 0.21 930 1170 1200 1230 1270 1300 1330 0.42 781 1140 1170 1200 1230 1250 1280 0.62 631 1120 1140 1160 1180 1210 1230 0.83 482 1090 1100 1120 1140 1160 1170 3.41 376 1060 1070 1080 1100 1110 1120 6.26 270 1010 1010 1020 1030 1040 1050

In.

Kips

2

1420 1360 1310 1250 1190 1140 1060

1460 1400 1340 1270 1210 1150 1070

1500 1430 1360 1290 1220 1160 1080

1530 1460 1390 1320 1240 1180 1090

1570 1500 1420 1340 1260 1190 1100

W 27×94

751

TFL 2 3 4 BFL 6 7

0.00 0.19 0.37 0.56 0.75 3.39 6.39

997 863 729 595 461 355 249

1300 1260 1210 1150 1100 1050 974

1340 1290 1230 1170 1120 1060 982

1370 1320 1260 1200 1130 1070 991

1410 1350 1280 1220 1150 1090 1000

1450 1380 1310 1240 1170 1100 1010

W 27×84

659

TFL 2 3 4 BFL 6 7

0.00 0.16 0.32 0.48 0.64 3.44 6.62

893 778 663 549 434 329 223

971 1000 1030 1070 1100 1130 1160 1190 1220 954 982 1010 1040 1060 1090 1120 1150 1170 936 959 983 1010 1030 1050 1080 1100 1120 916 936 955 975 994 1010 1030 1050 1070 896 911 926 942 957 972 988 1000 1020 866 878 890 901 913 925 936 948 960 814 822 830 838 846 854 861 869 877

1260 1200 1150 1090 1030 971 885

1290 1230 1170 1110 1050 983 893

W 24×76

540

TFL 2 3 4 BFL 6 7

0.00 0.17 0.34 0.51 0.68 3.00 5.60

806 696 586 476 366 284 202

797 781 764 745 724 703 666

826 806 784 762 737 713 673

855 830 805 778 750 723 680

883 855 826 795 763 733 687

912 880 847 812 776 743 694

940 904 867 829 789 753 702

969 929 888 846 802 763 709

997 1030 1050 1080 954 978 1000 1030 909 930 950 971 863 880 896 913 815 828 841 854 773 783 793 803 716 723 730 737

W 24×68

478

TFL 2 3 4 BFL 6 7

0.00 0.15 0.29 0.44 0.59 3.05 5.81

724 629 535 440 346 263 181

711 697 682 666 649 628 590

736 719 701 682 662 637 596

762 741 720 697 674 646 603

788 764 739 713 686 656 609

813 786 758 729 698 665 616

839 808 777 744 711 674 622

864 830 796 760 723 684 628

890 853 815 775 735 693 635

1090 1070 1050 1030 1000 972 921

2.5

1130 1100 1080 1050 1020 985 929

3

1160 1130 1100 1070 1030 997 938

3.5

1200 1160 1130 1090 1050 1010 947

4

1230 1190 1150 1110 1070 1020 956

4.5

1270 1230 1180 1130 1080 1040 965



916 875 833 791 747 702 641

941 897 852 807 760 712 648

967 920 871 822 772 721 654

5 - 26

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 24×62

Kip-ft

413

TFL 2 3 4 BFL 6 7

0.00 0.15 0.29 0.44 0.59 3.47 6.58

655 580 506 431 356 260 164

644 633 621 608 595 568 520

667 653 639 624 608 577 526

690 674 657 639 620 587 532

713 694 675 654 633 596 538

737 715 693 669 646 605 543

760 736 711 685 658 614 549

783 756 728 700 671 623 555

806 777 746 715 683 633 561

829 797 764 731 696 642 567

853 818 782 746 709 651 572

876 838 800 761 721 660 578

W 24×55

362

TFL 2 3 4 BFL 6 7

0.00 0.13 0.25 0.38 0.51 3.45 6.66

583 520 456 392 328 237 146

569 560 550 540 529 504 458

590 579 566 554 540 512 463

611 597 583 568 552 520 468

631 615 599 582 564 529 474

652 634 615 595 575 537 479

673 652 631 609 587 546 484

693 671 647 623 599 554 489

714 689 663 637 610 562 494

735 707 679 651 622 571 499

755 726 696 665 634 579 505

776 744 712 679 645 588 510

W 21×62

389

TFL 2 3 4 BFL 6 7

0.00 0.15 0.31 0.46 0.62 2.53 4.78

659 568 476 385 294 229 165

583 570 555 540 523 507 482

606 590 572 553 534 516 487

630 610 589 567 544 524 493

653 630 606 581 555 532 499

676 650 623 594 565 540 505

700 670 640 608 575 548 511

723 690 656 622 586 556 517

746 710 673 635 596 564 522

770 730 690 649 607 572 528

793 751 707 663 617 581 534

816 771 724 676 628 589 540

W 21×57

348

TFL 2 3 4 BFL 6 7

0.00 0.16 0.33 0.49 0.65 2.90 5.38

601 525 448 371 294 222 150

534 522 510 497 483 464 433

555 541 526 510 493 472 438

576 559 542 523 504 480 443

597 578 558 536 514 488 449

619 597 574 550 525 496 454

640 615 589 563 535 504 459

661 634 605 576 546 511 465

683 652 621 589 556 519 470

704 671 637 602 566 527 475

725 689 653 615 577 535 481

747 708 669 628 587 543 486

W 21×50

297

TFL 2 3 4 BFL 6 7

0.00 0.13 0.27 0.40 0.54 2.92 5.58

529 466 403 341 278 205 132

465 456 446 436 425 406 374

484 473 461 448 435 413 379

503 489 475 460 445 421 383

522 506 489 472 454 428 388

540 522 504 484 464 435 393

559 539 518 496 474 442 397

578 555 532 508 484 450 402

597 572 546 520 494 457 407

615 588 561 532 504 464 411

634 605 575 545 513 472 416

653 621 589 557 523 479 421



COMPOSITE BEAMS

5 - 27

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 21×44

Kip-ft

258

TFL 2 3 4 BFL 6 7

0.00 0.11 0.23 0.34 0.45 2.90 5.69

468 415 363 310 257 187 117

409 401 393 384 376 358 326

425 416 406 395 385 364 331

442 430 419 406 394 371 335

458 445 432 417 403 378 339

475 460 444 428 412 384 343

492 475 457 439 421 391 347

508 489 470 450 430 398 351

525 504 483 461 439 404 355

541 519 496 472 448 411 360

558 533 509 483 458 417 364

574 548 521 494 467 424 368

W 18×60

332

TFL 2 3 4 BFL 6 7

0.00 0.17 0.35 0.52 0.70 2.19 3.82

634 539 445 350 256 207 158

499 485 470 454 436 424 407

522 504 486 466 445 432 413

544 523 501 478 454 439 418

566 542 517 491 463 446 424

589 561 533 503 472 454 430

611 581 549 516 481 461 435

634 600 564 528 491 468 441

656 619 580 540 500 476 447

679 638 596 553 509 483 452

701 657 612 565 518 490 458

723 676 627 578 527 498 463

W 18×55

302

TFL 2 3 4 BFL 6 7

0.00 0.16 0.32 0.47 0.63 2.16 3.86

583 498 412 327 242 194 146

457 444 431 416 401 389 372

477 462 445 428 409 396 377

498 479 460 439 418 403 383

519 497 474 451 426 410 388

539 515 489 462 435 417 393

560 532 504 474 443 424 398

581 550 518 486 452 430 403

601 568 533 497 461 437 408

622 585 547 509 469 444 414

643 603 562 520 478 451 419

663 620 577 532 486 458 424

W 18×50

273

TFL 2 3 4 BFL 6 7

0.00 0.14 0.29 0.43 0.57 2.07 3.82

529 452 375 299 222 177 132

412 401 389 376 362 352 336

431 417 402 387 370 358 341

450 433 415 397 378 365 346

468 449 429 408 386 371 350

487 465 442 418 394 377 355

506 481 455 429 402 383 360

525 497 469 439 409 390 365

543 513 482 450 417 396 369

562 529 495 461 425 402 374

581 545 508 471 433 408 379

600 561 522 482 441 415 383

W 18×46

245

TFL 2 3 4 BFL 6 7

0.00 0.15 0.30 0.45 0.61 2.40 4.34

486 420 354 288 222 172 122

380 370 360 348 337 324 305

397 385 372 359 345 330 310

414 400 385 369 352 336 314

431 415 397 379 360 343 318

449 430 410 389 368 349 322

466 444 422 399 376 355 327

483 459 435 410 384 361 331

500 474 447 420 392 367 335

517 489 460 430 400 373 340

535 504 472 440 407 379 344

552 519 485 450 415 385 348



5 - 28

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 18×40

Kip-ft

212

TFL 2 3 4 BFL 6 7

0.00 0.13 0.26 0.39 0.53 2.26 4.27

425 368 311 254 197 152 106

329 321 312 303 293 282 265

345 334 323 312 300 288 269

360 347 334 321 307 293 273

375 360 345 330 314 298 277

390 373 356 339 321 304 280

405 386 367 348 328 309 284

420 399 378 357 335 315 288

435 412 389 366 342 320 292

450 425 400 375 349 325 295

465 438 411 384 356 331 299

480 451 423 393 363 336 303

W 18×35

180

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.43 2.37 4.56

371 325 279 233 187 140 92.7

285 278 271 264 256 245 227

298 290 281 272 263 250 230

311 301 291 280 269 255 233

324 313 301 289 276 260 237

338 324 311 297 283 265 240

351 336 321 305 289 270 243

364 347 331 313 296 275 246

377 359 340 322 303 280 250

390 370 350 330 309 285 253

403 382 360 338 316 290 256

416 393 370 346 323 295 260

W 16×36

173

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.32 0.43 1.79 3.44

382 328 273 219 165 130 95.4

268 261 252 244 234 227 215

282 272 262 251 240 232 219

295 284 272 259 246 236 222

309 295 281 267 252 241 226

322 307 291 275 258 245 229

336 319 301 282 264 250 232

349 330 310 290 270 255 236

363 342 320 298 275 259 239

377 353 330 306 281 264 243

390 365 339 314 287 268 246

404 377 349 321 293 273 249

W 16×31

146

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 2.00 3.79

328 285 241 197 153 118 82.1

231 225 218 211 204 196 183

243 235 227 218 209 200 186

254 245 235 225 214 204 189

266 255 244 232 220 208 192

278 265 252 239 225 212 195

289 275 261 246 231 217 198

301 285 269 253 236 221 201

313 295 278 260 242 225 204

324 305 286 267 247 229 207

336 315 295 274 253 233 209

347 326 303 281 258 237 212

W 16×26

119

TFL 2 3 4 BFL 6 7

0.00 0.09 0.17 0.26 0.35 2.04 4.01

276 242 208 174 140 104 69.1

193 188 183 177 172 164 151

203 196 190 184 177 168 154

212 205 197 190 182 171 156

222 214 205 196 187 175 159

232 222 212 202 192 179 161

242 231 220 208 197 182 164

252 239 227 214 202 186 166

261 248 234 220 206 190 169

271 257 242 227 211 194 171

281 265 249 233 216 197 173

291 274 256 239 221 201 176



COMPOSITE BEAMS

5 - 29

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 14×38

Kip-ft

166

TFL 2 3 4 BFL 6 7

0.00 0.13 0.26 0.39 0.52 1.38 2.53

403 340 278 215 152 126 101

258 249 240 229 218 213 206

273 261 249 237 224 218 209

287 273 259 244 229 222 213

301 285 269 252 234 226 217

316 298 279 260 240 231 220

330 310 289 267 245 235 224

344 322 299 275 251 240 227

358 334 308 283 256 244 231

373 346 318 290 261 249 234

387 358 328 298 267 253 238

401 370 338 305 272 258 242

W 14×34

147

TFL 2 3 4 BFL 6 7

0.00 0.11 0.23 0.34 0.46 1.41 2.60

360 305 250 194 139 115 90.0

229 221 213 204 194 189 182

242 232 222 211 199 193 186

255 243 230 218 204 197 189

267 254 239 224 209 202 192

280 264 248 231 214 206 195

293 275 257 238 219 210 198

306 286 266 245 224 214 202

318 297 275 252 229 218 205

331 308 283 259 234 222 208

344 318 292 266 239 226 211

357 329 301 273 244 230 214

W 14×30

128

TFL 2 3 4 BFL 6 7

0.00 0.10 0.19 0.29 0.39 1.48 2.82

319 272 225 179 132 106 79.7

201 195 187 180 172 167 159

213 204 195 186 177 171 162

224 214 203 193 182 174 165

235 223 211 199 186 178 168

246 233 219 205 191 182 171

258 243 227 212 196 186 173

269 252 235 218 200 189 176

280 262 243 224 205 193 179

292 272 251 231 210 197 182

303 281 259 237 214 201 185

314 291 267 243 219 204 187

W 14×26

109

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.42 1.67 3.19

277 239 201 163 125 97.0 69.2

176 170 164 158 152 146 137

185 179 171 164 156 149 140

195 187 179 170 161 153 142

205 195 186 175 165 156 145

215 204 193 181 170 160 147

225 212 200 187 174 163 149

234 221 207 193 178 167 152

244 229 214 199 183 170 154

254 238 221 204 187 173 157

264 246 228 210 192 177 159

274 255 235 216 196 180 162

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.34 1.69 3.34

234 203 173 143 113 85.7 58.4

147 142 138 133 128 123 114

155 150 144 138 132 126 116

163 157 150 143 136 129 118

172 164 156 148 140 132 120

180 171 162 153 144 135 122

188 178 169 159 148 138 124

196 186 175 164 152 141 126

205 193 181 169 156 144 128

213 200 187 174 160 147 130

221 207 193 179 164 150 132

230 215 199 184 168 153 135

W 14×22

89.6



5 - 30

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 12×30

Kip-ft

116

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 1.12 1.94

316 265 213 162 110 94.5 79.1

183 176 168 159 151 148 144

194 185 175 165 155 151 147

206 194 183 171 158 154 149

217 204 190 177 162 158 152

228 213 198 182 166 161 155

239 223 205 188 170 164 158

250 232 213 194 174 168 161

262 241 221 199 178 171 163

273 251 228 205 182 174 166

284 260 236 211 186 178 169

295 269 243 217 190 181 172

W 12×26

100

TFL 2 3 4 BFL 6 7

0.00 0.10 0.19 0.29 0.38 1.08 1.95

275 231 187 142 97.8 83.3 68.9

158 152 145 138 131 128 124

168 160 152 143 134 131 127

178 168 158 148 138 134 129

187 176 165 153 141 137 132

197 184 171 158 145 140 134

207 193 178 163 148 143 136

217 201 185 168 151 146 139

226 209 191 173 155 149 141

236 217 198 178 158 151 144

246 225 205 183 162 154 146

256 234 211 188 165 157 149

W 12×22

79.1

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.43 1.66 3.04

233 202 172 141 110 84.1 58.3

135 130 126 121 115 110 102

143 137 132 126 119 113 104

151 145 138 131 123 116 106

160 152 144 136 127 119 108

168 159 150 141 131 122 110

176 166 156 146 135 125 112

184 173 162 151 139 128 114

193 180 168 156 143 131 116

201 188 174 160 147 134 119

209 195 180 165 150 137 121

217 202 186 170 154 140 123

W 12×19

66.7

TFL 2 3 4 BFL 6 7

0.00 0.09 0.18 0.26 0.35 1.66 3.12

201 175 150 125 99.6 74.9 50.1

115 111 107 103 99.2 94.0 86.3

122 117 113 108 103 96.7 88.1

129 124 118 112 106 99.3 89.9

136 130 123 117 110 102 91.7

143 136 129 121 113 105 93.5

150 142 134 125 117 107 95.2

157 148 139 130 120 110 97.0

164 155 145 134 124 113 98.8

172 161 150 139 127 115 101

179 167 155 143 131 118 102

186 173 160 148 134 121 104

W 12×16

54.3

TFL 2 3 4 BFL 6 7

0.00 0.07 0.13 0.20 0.26 1.71 3.32

170 151 131 112 93.4 67.9 42.4

96.0 93.3 90.5 87.5 84.5 79.2 71.1

102 98.6 95.1 91.5 87.8 81.6 72.6

108 104 99.8 95.5 91.1 84.0 74.1

114 109 104 99.5 94.5 86.4 75.6

120 115 109 103 97.8 88.8 77.1

126 120 114 107 101 91.2 78.6

132 125 118 111 104 93.6 80.1

138 131 123 115 108 96.1 81.6

144 136 128 119 111 98.5 83.1

150 141 132 123 114 101 84.6

156 147 137 127 118 103 86.1



COMPOSITE BEAMS

5 - 31

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 12×14

Kip-ft

47.0

TFL 2 3 4 BFL 6 7

0.00 0.06 0.11 0.17 0.23 1.69 3.36

150 134 118 102 85.4 61.4 37.4

84.4 82.1 79.7 77.3 74.8 69.8 62.2

89.7 86.8 83.9 80.9 77.8 72.0 63.5

95.0 91.5 88.0 84.5 80.8 74.2 64.8

100 96.3 92.2 88.1 83.8 76.4 66.1

106 101 96.4 91.6 86.9 78.5 67.5

111 106 101 95.2 89.9 80.7 68.8

116 110 105 98.8 92.9 82.9 70.1

122 115 109 102 95.9 85.1 71.4

127 120 113 106 99.0 87.2 72.8

132 125 117 110 102 89.4 74.1

137 129 121 113 105 91.6 75.4

W 10×26

84.5

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 0.90 1.51

274 228 183 137 91.2 79.8 68.5

139 132 125 118 110 108 106

149 140 132 123 114 111 108

158 149 138 128 117 114 110

168 157 145 133 120 117 113

178 165 151 137 123 119 115

188 173 158 142 126 122 118

197 181 164 147 130 125 120

207 189 171 152 133 128 123

217 197 177 157 136 131 125

226 205 184 162 139 134 127

236 213 190 166 143 136 130

W 10×22

70.2

TFL 2 3 4 BFL 6 7

0.00 0.09 0.18 0.27 0.36 0.95 1.70

234 196 159 122 84.6 71.5 58.4

117 112 106 100 94.2 91.8 88.7

126 119 112 105 97.2 94.3 90.8

134 126 117 109 100 96.9 92.9

142 133 123 113 103 99.4 94.9

150 140 129 118 106 102 97.0

159 147 134 122 109 104 99.1

167 154 140 126 112 107 101

175 161 146 131 115 110 103

183 167 151 135 118 112 105

192 174 157 139 121 115 107

200 181 163 144 124 117 109

W 10×19

58.3

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 1.27 2.31

202 102 174 97.9 145 93.5 117 89.0 88.0 84.2 69.3 80.5 50.6 75.5

109 104 98.7 93.1 87.4 83.0 77.3

116 110 104 97.2 90.5 85.4 79.1

124 116 109 101 93.6 87.9 80.9

131 123 114 106 96.7 90.4 82.7

138 129 119 110 99.8 92.8 84.5

145 135 124 114 103 95.3 86.3

152 141 130 118 106 97.7 88.1

159 147 135 122 109 100 89.8

167 153 140 126 112 103 91.6

174 159 145 130 115 105 93.4

W 10×17

50.5

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.33 1.31 2.46

180 156 132 108 84.4 64.6 44.9

96.1 91.8 87.4 82.9 78.1 73.6 67.4

102 97.4 92.1 86.7 81.1 75.8 69.0

109 103 96.8 90.5 84.1 78.1 70.6

115 108 101 94.3 87.1 80.4 72.2

122 114 106 98.2 90.1 82.7 73.8

128 119 111 102 93.1 85.0 75.4

134 125 115 106 96.1 87.3 77.0

141 130 120 110 99.1 89.6 78.6

147 136 125 114 102 91.9 80.2

153 142 129 117 105 94.2 81.7

89.8 86.3 82.7 79.0 75.2 71.3 65.8



5 - 32

COMPOSITE DESIGN

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) In.

Kips

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 10×15

Kip-ft

43.2

TFL 2 3 4 BFL 6 7

0.00 0.07 0.14 0.20 0.27 1.35 2.60

159 139 120 100 81.0 60.3 39.7

78.7 75.9 73.0 70.0 66.9 62.9 57.0

84.3 80.8 77.2 73.5 69.8 65.0 58.4

89.9 85.7 81.5 77.1 72.6 67.1 59.9

95.5 90.7 85.7 80.7 75.5 69.3 61.3

101 95.6 90.0 84.2 78.4 71.4 62.7

107 101 94.2 87.8 81.2 73.5 64.1

112 105 98.4 91.3 84.1 75.7 65.5

118 110 103 94.9 87.0 77.8 66.9

124 115 107 98.4 89.9 80.0 68.3

129 120 111 102 92.7 82.1 69.7

135 125 115 106 95.6 84.2 71.1

W 10×12

34.0

TFL 2 3 4 BFL 6 7

0.00 127 0.05 112 0.11 97.5 0.16 82.5 0.21 67.6 1.30 49.7 2.61 31.9

62.6 60.5 58.2 56.0 53.7 50.3 45.3

67.1 64.4 61.7 58.9 56.1 52.0 46.4

71.6 68.4 65.2 61.8 58.5 53.8 47.5

76.1 72.4 68.6 64.8 60.9 55.5 48.6

80.7 76.4 72.1 67.7 63.2 57.3 49.8

85.2 80.4 75.5 70.6 65.6 59.1 50.9

89.7 84.4 79.0 73.5 68.0 60.8 52.0

94.2 88.3 82.4 76.5 70.4 62.6 53.2

98.7 92.3 85.9 79.4 72.8 64.3 54.3

103 96.3 89.3 82.3 75.2 66.1 55.4

108 100 92.8 85.2 77.6 67.9 56.5

W 8×28

73.4

TFL 2 3 4 BFL 6 7

0.00 0.12 0.23 0.35 0.47 0.53 0.59

297 242 188 133 78.2 76.2 74.3

127 119 110 102 92.3 91.9 91.5

137 127 117 106 95.0 94.6 94.2

148 136 124 111 97.8 97.3 96.8

158 145 130 116 101 100 99.4

169 153 137 120 103 103 102

179 162 144 125 106 105 105

190 170 150 130 109 108 107

200 179 157 135 112 111 110

211 188 164 139 114 114 113

222 196 170 144 117 116 115

232 205 177 149 120 119 118

W 8×24

62.6

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 0.47 0.55

255 108 208 101 161 93.8 115 86.3 67.8 78.5 65.8 78.2 63.7 77.8

117 108 99.5 90.4 80.9 80.5 80.1

126 116 105 94.4 83.3 82.8 82.3

135 123 111 98.5 85.7 85.2 84.6

144 130 117 103 88.2 87.5 86.9

153 138 122 107 90.6 89.8 89.1

162 145 128 111 93.0 92.2 91.4

171 152 134 115 95.4 94.5 93.6

180 160 139 119 97.8 96.8 95.9

189 167 145 123 100 99.2 98.1

198 175 151 127 103 101 100

W 8×21

55.1

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 0.70 1.06

222 184 146 108 70.0 62.7 55.4

104 97.4 90.3 82.9 75.3 73.7 72.0

112 104 95.5 86.8 77.8 75.9 73.9

120 110 101 90.6 80.2 78.1 75.9

128 117 106 94.4 82.7 80.4 77.9

136 123 111 98.2 85.2 82.6 79.8

144 130 116 102 87.7 84.8 81.8

151 137 121 106 90.1 87.0 83.8

159 143 126 110 92.6 89.3 85.7

167 150 132 114 95.1 91.5 87.7

175 156 137 117 97.6 93.7 89.6

96.4 90.9 85.2 79.1 72.8 71.5 70.0



COMPOSITE BEAMS

5 - 33

Fy = 36 ksi


φb M p PNAc Y1a

ΣQ n

φM n (kip-ft)

Y2b (in.) 2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 8×18

Kip-ft

45.9

TFL 2 3 4 BFL 6 7

0.00 189 0.08 158 0.17 127 0.25 95.8 0.33 64.6 0.71 56.0 1.21 47.3

In.

Kips

81.4 76.9 72.2 67.3 62.3 60.7 58.9

88.1 82.5 76.7 70.7 64.6 62.7 60.6

94.8 88.1 81.2 74.1 66.9 64.7 62.3

102 93.7 85.7 77.5 69.2 66.7 64.0

108 99.3 90.2 80.9 71.4 68.7 65.6

115 105 94.7 84.3 73.7 70.7 67.3

122 111 99.2 87.7 76.0 72.6 69.0

128 116 104 91.1 78.3 74.6 70.7

135 122 108 94.5 80.6 76.6 72.4

142 127 113 97.9 82.9 78.6 74.0

148 133 117 101 85.2 80.6 75.7

W 8×15

36.7

TFL 2 3 4 BFL 6 7

0.00 160 0.08 137 0.16 114 0.24 91.5 0.32 68.8 0.97 54.4 1.79 40.0

68.6 65.3 61.9 58.3 54.6 52.0 48.5

74.2 70.1 65.9 61.6 57.1 53.9 49.9

79.9 75.0 69.9 64.8 59.5 55.8 51.3

85.5 79.8 74.0 68.0 61.9 57.7 52.8

91.2 84.7 78.0 71.3 64.4 59.7 54.2

96.9 89.5 82.1 74.5 66.8 61.6 55.6

103 94.4 86.1 77.8 69.3 63.5 57.0

108 99.2 90.2 81.0 71.7 65.4 58.4

114 104 94.2 84.2 74.1 67.4 59.8

120 109 98.3 87.5 76.6 69.3 61.2

125 114 102 90.7 79.0 71.2 62.7

W 8×13

30.8

TFL 2 3 4 BFL 6 7

0.00 138 0.06 120 0.13 102 0.19 83.2 0.26 64.8 1.00 49.7 1.91 34.6

58.7 56.1 53.3 50.5 47.6 44.9 41.2

63.6 60.3 56.9 53.5 49.9 46.6 42.4

68.5 64.6 60.5 56.4 52.2 48.4 43.6

73.4 68.8 64.1 59.4 54.5 50.1 44.8

78.3 73.0 67.7 62.3 56.8 51.9 46.1

83.2 77.3 71.3 65.3 59.1 53.7 47.3

88.1 81.5 74.9 68.2 61.4 55.4 48.5

93.0 85.8 78.5 71.1 63.7 57.2 49.7

97.9 90.0 82.1 74.1 66.0 58.9 51.0

103 94.3 85.7 77.0 68.3 60.7 52.2

108 98.5 89.3 80.0 70.6 62.5 53.4

W 8×10

23.9

TFL 2 3 4 BFL 6 7

0.00 107 0.05 92.0 0.10 77.5 0.15 62.9 0.21 48.4 0.88 37.5 1.77 26.6

44.9 42.8 40.6 38.5 36.2 34.3 31.7

48.6 46.0 43.4 40.7 37.9 35.6 32.7

52.4 49.3 46.1 42.9 39.6 36.9 33.6

56.2 52.6 48.9 45.1 41.4 38.3 34.5

60.0 55.8 51.6 47.4 43.1 39.6 35.5

63.7 59.1 54.4 49.6 44.8 40.9 36.4

67.5 62.3 57.1 51.8 46.5 42.2 37.4

71.3 65.6 59.9 54.1 48.2 43.6 38.3

75.1 68.9 62.6 56.3 49.9 44.9 39.3

78.8 72.1 65.3 58.5 51.6 46.2 40.2

82.6 75.4 68.1 60.7 53.4 47.6 41.1



5 - 34

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W40×297

4990

TFL 2 3 4 BFL 6 7

0.00 0.41 0.83 1.24 1.65 4.59 8.17

4370 3720 3060 2410 1760 1430 1090

6790 6680 6560 6420 6260 6150 5950

6940 6810 6670 6510 6320 6200 5990

7090 6950 6780 6590 6390 6250 6020

7250 7080 6890 6680 6450 6300 6060

7400 7210 7000 6760 6510 6350 6100

7560 7340 7100 6850 6570 6400 6140

7710 7470 7210 6930 6640 6450 6180

7870 7600 7320 7020 6700 6500 6220

8020 7740 7430 7110 6760 6550 6260

8180 7870 7540 7190 6820 6600 6300

8330 8000 7650 7280 6890 6650 6330

W40×278

4460

TFL 0.00 2 0.45 3 0.91 4 1.36 BFL 1.81 6 5.64 7 10.06

4090 3550 3010 2470 1920 1470 1020

6400 6310 6210 6090 5950 5770 5460

6540 6440 6320 6180 6020 5820 5500

6690 6560 6420 6260 6090 5880 5530

6830 6690 6530 6350 6160 5930 5570

6980 6810 6630 6440 6220 5980 5600

7120 6940 6740 6520 6290 6030 5640

7270 7070 6850 6610 6360 6080 5680

7410 7190 6950 6700 6430 6140 5710

7560 7320 7060 6790 6500 6190 5750

7700 7440 7170 6870 6560 6240 5790

7850 7570 7270 6960 6630 6290 5820

W40×277

4690

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.58 4.25 7.60

4070 3440 2820 2200 1570 1290 1020

6290 6190 6080 5950 5800 5700 5550

6430 6310 6180 6020 5850 5750 5580

6580 6440 6280 6100 5910 5790 5620

6720 6560 6380 6180 5960 5840 5660

6870 6680 6480 6260 6020 5880 5690

7010 6800 6580 6340 6080 5930 5730

7150 6920 6680 6410 6130 5980 5760

7300 7050 6780 6490 6190 6020 5800

7440 7170 6880 6570 6240 6070 5840

7590 7290 6980 6650 6300 6110 5870

7730 7410 7080 6720 6350 6160 5910

W40×264

4240

TFL 2 3 4 BFL 6 7

0.00 0.43 0.87 1.30 1.73 5.49 9.90

3880 3360 2850 2330 1820 1390 970

6050 5970 5870 5760 5630 5470 5170

6180 6080 5970 5840 5690 5510 5210

6320 6200 6070 5920 5760 5560 5240

6460 6320 6170 6000 5820 5610 5280

6600 6440 6270 6090 5880 5660 5310

6730 6560 6370 6170 5950 5710 5350

6870 6680 6470 6250 6010 5760 5380

7010 6800 6570 6330 6080 5810 5420

7150 6920 6680 6420 6140 5860 5450

7280 7040 6780 6500 6210 5910 5480

7420 7160 6880 6580 6270 5960 5520



COMPOSITE BEAMS

5 - 35

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W40×249

4200

TFL 2 3 4 BFL 6 7

0.00 0.36 0.71 1.07 1.42 4.06 7.47

3670 3110 2550 1990 1430 1170 916

5630 5540 5440 5330 5200 5120 4980

5760 5650 5530 5400 5250 5160 5010

5890 5760 5620 5470 5300 5200 5040

6020 5870 5710 5540 5350 5240 5070

6150 5980 5810 5610 5400 5280 5110

6280 6090 5900 5680 5450 5320 5140

6410 6200 5990 5750 5510 5370 5170

6540 6310 6080 5820 5560 5410 5200

6670 6420 6170 5890 5610 5450 5240

6800 6530 6260 5960 5660 5490 5270

6930 6640 6350 6030 5710 5530 5300

W40×235

3790

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.58 5.18 9.47

3450 2980 2510 2040 1570 1220 861

5330 5260 5170 5070 4960 4830 4600

5450 5360 5260 5150 5020 4880 4630

5570 5470 5350 5220 5070 4920 4660

5700 5570 5440 5290 5130 4960 4690

5820 5680 5530 5360 5180 5000 4720

5940 5780 5620 5430 5240 5050 4750

6060 5890 5700 5510 5290 5090 4780

6180 6000 5790 5580 5350 5130 4810

6310 6100 5880 5650 5410 5180 4840

6430 6210 5970 5720 5460 5220 4870

6550 6310 6060 5800 5520 5260 4900

W40×215

3610

TFL 2 3 4 BFL 6 7

0.00 0.31 0.61 0.92 1.22 3.84 7.32

3170 2680 2200 1720 1240 1020 791

4820 4740 4660 4570 4460 4390 4270

4930 4840 4740 4630 4510 4430 4300

5040 4930 4820 4690 4550 4470 4330

5150 5030 4900 4750 4590 4500 4360

5270 5120 4970 4810 4640 4540 4380

5380 5220 5050 4870 4680 4570 4410

5490 5320 5130 4930 4730 4610 4440

5600 5410 5210 4990 4770 4650 4470

5710 5510 5290 5060 4810 4680 4500

5830 5600 5360 5120 4860 4720 4520

5940 5700 5440 5180 4900 4760 4550

W40×211

3390

TFL 2 3 4 BFL 6 7

0.00 0.35 0.71 1.06 1.42 4.99 9.35

3100 2680 2260 1850 1430 1100 775

4760 4700 4620 4540 4440 4330 4110

4870 4790 4700 4600 4490 4360 4140

4980 4890 4780 4670 4540 4400 4170

5090 4980 4860 4730 4590 4440 4200

5200 5080 4940 4800 4640 4480 4220

5310 5170 5020 4860 4690 4520 4250

5420 5270 5100 4930 4740 4560 4280

5530 5360 5180 4990 4800 4600 4310

5640 5460 5260 5060 4850 4640 4330

5750 5550 5340 5130 4900 4680 4360

5860 5650 5420 5190 4950 4710 4390



5 - 36

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W40×199

3250

TFL 2 3 4 BFL 6 7

0.00 0.27 0.53 0.80 1.07 4.16 8.10

2920 2500 2080 1660 1240 986 730

4410 4350 4280 4200 4110 4030 3880

4520 4440 4350 4260 4160 4070 3910

4620 4530 4430 4320 4200 4100 3940

4720 4620 4500 4380 4240 4140 3960

4830 4700 4570 4430 4290 4170 3990

4930 4790 4650 4490 4330 4210 4010

5030 4880 4720 4550 4380 4240 4040

5140 4970 4790 4610 4420 4280 4070

5240 5060 4870 4670 4460 4310 4090

5340 5150 4940 4730 4510 4350 4120

5450 5240 5020 4790 4550 4380 4140

W40×183

2930

TFL 2 3 4 BFL 6 7

0.00 0.31 0.61 0.92 1.22 4.76 9.16

2690 2320 1960 1600 1240 958 671

4090 4030 3970 3900 3820 3720 3540

4180 4110 4040 3960 3860 3760 3570

4280 4200 4110 4010 3910 3790 3590

4370 4280 4180 4070 3950 3830 3610

4470 4360 4250 4130 4000 3860 3640

4560 4440 4320 4180 4040 3890 3660

4660 4530 4390 4240 4090 3930 3690

4750 4610 4460 4300 4130 3960 3710

4850 4690 4530 4350 4170 4000 3730

4940 4770 4600 4410 4220 4030 3760

5040 4860 4670 4470 4260 4060 3780

W40×174

2680

TFL 2 3 4 BFL 6 7

0.00 0.21 0.42 0.62 0.83 4.59 9.27

2560 2230 1900 1570 1250 943 639

3820 3770 3720 3660 3600 3500 3310

3910 3850 3780 3710 3640 3530 3330

4000 3930 3850 3770 3680 3570 3350

4090 4010 3920 3830 3730 3600 3370

4180 4090 3990 3880 3770 3630 3400

4270 4160 4050 3940 3820 3670 3420

4360 4240 4120 3990 3860 3700 3440

4450 4320 4190 4050 3900 3730 3460

4540 4400 4260 4100 3950 3770 3490

4630 4480 4320 4160 3990 3800 3510

4720 4560 4390 4220 4040 3830 3530

W40×167

2600

TFL 2 3 4 BFL 6 7

0.00 0.26 0.51 0.77 1.03 5.00 9.85

2460 2150 1850 1550 1240 929 614

3700 3660 3610 3550 3490 3380 3170

3790 3730 3670 3600 3530 3410 3190

3880 3810 3740 3660 3580 3450 3210

3960 3890 3800 3710 3620 3480 3240

4050 3960 3870 3770 3660 3510 3260

4140 4040 3930 3820 3710 3550 3280

4220 4110 4000 3880 3750 3580 3300

4310 4190 4060 3930 3800 3610 3320

4400 4270 4130 3990 3840 3640 3340

4490 4340 4200 4040 3880 3680 3370

4570 4420 4260 4100 3930 3710 3390

W40×149

2240

TFL 0.00 2 0.21 3 0.42 4 0.62 BFL 0.83 6 5.15 7 10.41

2190 1940 1700 1450 1210 879 548

3270 3240 3200 3150 3110 2990 2770

3350 3310 3260 3200 3150 3030 2780

3430 3370 3320 3260 3190 3060 2800

3510 3440 3380 3310 3230 3090 2820

3580 3510 3440 3360 3280 3120 2840

3660 3580 3500 3410 3320 3150 2860

3740 3650 3560 3460 3360 3180 2880

3820 3720 3620 3510 3410 3210 2900

3890 3790 3680 3560 3450 3240 2920

3970 3860 3740 3620 3490 3280 2940

4050 3930 3800 3670 3530 3310 2960



COMPOSITE BEAMS

5 - 37

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 36×300

4730

TFL 2 3 4 BFL 6 7

0.00 0.42 0.84 1.26 1.68 3.97 6.69

4420 3720 3020 2320 1620 1360 1100

6370 6260 6130 5980 5810 5720 5590

6530 6390 6240 6060 5860 5770 5630

6680 6520 6340 6140 5920 5820 5670

6840 6660 6450 6230 5980 5870 5710

7000 6790 6560 6310 6040 5910 5740

7150 6920 6660 6390 6090 5960 5780

7310 7050 6770 6470 6150 6010 5820

7460 7180 6880 6550 6210 6060 5860

7620 7310 6990 6640 6270 6110 5900

7780 7450 7090 6720 6320 6150 5940

7930 7580 7200 6800 6380 6200 5980

W 36×280

4390

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.57 3.88 6.62

4120 3470 2820 2170 1510 1270 1030

5910 5810 5690 5550 5400 5320 5190

6060 5930 5790 5630 5450 5360 5230

6200 6060 5890 5710 5510 5410 5270

6350 6180 5990 5780 5560 5450 5300

6500 6300 6090 5860 5610 5500 5340

6640 6430 6190 5940 5670 5540 5380

6790 6550 6290 6010 5720 5590 5410

6930 6670 6390 6090 5770 5630 5450

7080 6790 6490 6170 5830 5680 5490

7230 6920 6590 6240 5880 5720 5520

7370 7040 6690 6320 5930 5770 5560

W 36×260

4050

TFL 2 3 4 BFL 6 7

0.00 0.36 0.72 1.08 1.44 3.86 6.75

3830 3230 2630 2040 1440 1200 956

5450 5360 5250 5130 4990 4920 4790

5590 5480 5350 5200 5050 4960 4830

5720 5590 5440 5280 5100 5000 4860

5860 5710 5530 5350 5150 5040 4890

6000 5820 5630 5420 5200 5090 4930

6130 5930 5720 5490 5250 5130 4960

6270 6050 5810 5570 5300 5170 4990

6400 6160 5910 5640 5350 5210 5030

6540 6280 6000 5710 5400 5260 5060

6670 6390 6090 5780 5450 5300 5100

6810 6510 6190 5850 5510 5340 5130

W 36×245

3790

TFL 2 3 4 BFL 6 7

0.00 0.34 0.68 1.01 1.35 3.81 6.77

3610 3050 2490 1930 1380 1140 901

5120 5030 4930 4820 4690 4620 4500

5240 5140 5020 4890 4740 4660 4530

5370 5250 5110 4960 4790 4700 4560

5500 5360 5200 5030 4840 4740 4590

5630 5460 5290 5090 4890 4780 4620

5760 5570 5370 5160 4940 4820 4660

5880 5680 5460 5230 4990 4860 4690

6010 5790 5550 5300 5040 4900 4720

6140 5900 5640 5370 5080 4940 4750

6270 6000 5730 5440 5130 4980 4780

6390 6110 5810 5510 5180 5020 4820

W 36×230

3540

TFL 2 3 4 BFL 6 7

0.00 0.32 0.63 0.95 1.26 3.81 6.83

3380 2860 2340 1820 1300 1070 845

4780 4700 4610 4500 4390 4320 4200

4900 4800 4690 4570 4440 4350 4230

5020 4900 4770 4630 4480 4390 4260

5140 5000 4860 4700 4530 4430 4290

5260 5100 4940 4760 4570 4470 4320

5370 5200 5020 4830 4620 4510 4350

5490 5310 5100 4890 4670 4540 4380

5610 5410 5190 4960 4710 4580 4410

5730 5510 5270 5020 4760 4620 4440

5850 5610 5350 5090 4810 4660 4470

5970 5710 5440 5150 4850 4690 4500



5 - 38

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 36×210

3120

TFL 2 3 4 BFL 6 7

0.00 0.34 0.68 1.02 1.36 5.06 9.04

3090 2680 2260 1850 1430 1100 773

4450 4390 4320 4230 4140 4020 3810

4560 4480 4400 4300 4190 4060 3830

4670 4580 4480 4360 4240 4100 3860

4780 4670 4560 4430 4290 4130 3890

4890 4770 4640 4490 4340 4170 3920

5000 4860 4720 4560 4390 4210 3940

5110 4960 4800 4620 4440 4250 3970

5220 5050 4880 4690 4490 4290 4000

5330 5150 4960 4760 4540 4330 4030

5440 5240 5040 4820 4600 4370 4050

5550 5340 5120 4890 4650 4410 4080

W 36×194

2880

TFL 2 3 4 BFL 6 7

0.00 0.32 0.63 0.95 1.26 4.94 8.93

2850 2470 2090 1710 1320 1020 713

4090 4030 3960 3890 3800 3690 3500

4190 4120 4040 3950 3850 3730 3530

4290 4200 4110 4010 3900 3770 3550

4390 4290 4180 4070 3940 3800 3580

4490 4380 4260 4130 3990 3840 3600

4590 4470 4330 4190 4040 3880 3630

4690 4550 4410 4250 4080 3910 3650

4790 4640 4480 4310 4130 3950 3680

4890 4730 4550 4370 4180 3980 3700

5000 4820 4630 4430 4220 4020 3730

5100 4900 4700 4490 4270 4060 3750

W 36×182

2690

TFL 2 3 4 BFL 6 7

0.00 0.29 0.59 0.89 1.18 4.89 8.92

2680 2320 1970 1610 1260 963 670

3830 3770 3710 3640 3570 3460 3280

3920 3860 3780 3700 3610 3500 3300

4020 3940 3850 3760 3660 3530 3330

4110 4020 3920 3810 3700 3570 3350

4210 4100 3990 3870 3740 3600 3370

4300 4190 4060 3930 3790 3640 3400

4400 4270 4130 3990 3830 3670 3420

4490 4350 4200 4040 3880 3700 3450

4590 4430 4270 4100 3920 3740 3470

4680 4510 4340 4160 3970 3770 3490

4780 4600 4410 4210 4010 3810 3520

W 36×170

2500

TFL 2 3 4 BFL 6 7

0.00 0.28 0.55 0.83 1.10 4.84 8.89

2500 2170 1840 1510 1180 901 625

3560 3510 3450 3390 3320 3220 3050

3650 3580 3520 3440 3360 3250 3070

3730 3660 3580 3490 3400 3290 3090

3820 3740 3650 3550 3440 3320 3120

3910 3810 3710 3600 3480 3350 3140

4000 3890 3780 3650 3530 3380 3160

4090 3970 3840 3710 3570 3410 3180

4180 4040 3910 3760 3610 3450 3200

4270 4120 3970 3810 3650 3480 3230

4350 4200 4040 3870 3690 3510 3250

4440 4270 4100 3920 3730 3540 3270

W 36×160

2340

TFL 2 3 4 BFL 6 7

0.00 0.26 0.51 0.77 1.02 4.82 8.97

2350 2040 1740 1430 1130 857 588

3330 3280 3230 3180 3110 3020 2850

3410 3360 3290 3230 3150 3050 2870

3500 3430 3360 3280 3190 3080 2890

3580 3500 3420 3330 3230 3110 2910

3660 3570 3480 3380 3270 3140 2930

3750 3650 3540 3430 3310 3170 2960

3830 3720 3600 3480 3350 3200 2980

3910 3790 3660 3530 3390 3230 3000

4000 3860 3720 3580 3430 3260 3020

4080 3940 3790 3630 3470 3290 3040

4160 4010 3850 3680 3510 3320 3060



COMPOSITE BEAMS

5 - 39

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 36×150

2180

TFL 2 3 4 BFL 6 7

0.00 0.24 0.47 0.71 0.94 4.83 9.08

2210 1930 1650 1370 1080 818 553

3120 3080 3030 2980 2920 2830 2660

3200 3150 3090 3030 2960 2860 2680

3280 3210 3150 3080 3000 2890 2700

3350 3280 3210 3120 3040 2920 2720

3430 3350 3260 3170 3080 2950 2740

3510 3420 3320 3220 3110 2980 2760

3590 3490 3380 3270 3150 3010 2780

3670 3560 3440 3320 3190 3040 2800

3750 3620 3500 3370 3230 3060 2820

3820 3690 3560 3410 3270 3090 2840

3900 3760 3610 3460 3310 3120 2860

W 36×135

1910

TFL 2 3 4 BFL 6 7

0.00 0.20 0.40 0.59 0.79 4.97 9.50

1990 1750 1510 1280 1040 769 496

2780 2750 2710 2670 2620 2530 2350

2850 2810 2760 2710 2660 2560 2370

2920 2870 2810 2760 2690 2580 2390

2990 2930 2870 2800 2730 2610 2400

3060 2990 2920 2850 2770 2640 2420

3130 3060 2970 2890 2800 2660 2440

3200 3120 3030 2940 2840 2690 2460

3270 3180 3080 2980 2880 2720 2470

3340 3240 3140 3030 2920 2750 2490

3410 3300 3190 3070 2950 2770 2510

3480 3360 3240 3120 2990 2800 2530

W 33×221

3210

TFL 2 3 4 BFL 6 7

0.00 0.32 0.64 0.96 1.28 3.76 6.48

3250 2750 2240 1740 1230 1020 813

4370 4290 4200 4100 3990 3920 3820

4480 4390 4280 4160 4030 3960 3850

4600 4480 4360 4220 4080 3990 3870

4710 4580 4440 4290 4120 4030 3900

4830 4680 4520 4350 4160 4070 3930

4940 4780 4600 4410 4210 4100 3960

5060 4870 4680 4470 4250 4140 3990

5170 4970 4760 4530 4300 4170 4020

5290 5070 4840 4590 4340 4210 4050

5400 5160 4920 4650 4380 4250 4080

5520 5260 4990 4720 4430 4280 4100

W 33×201

2900

TFL 2 3 4 BFL 6 7

0.00 0.29 0.58 0.86 1.15 3.67 6.51

2960 2500 2050 1600 1140 942 739

3940 3870 3800 3710 3610 3550 3450

4050 3960 3870 3770 3650 3580 3480

4150 4050 3940 3820 3690 3620 3500

4260 4140 4010 3880 3730 3650 3530

4360 4230 4090 3940 3780 3680 3550

4470 4320 4160 3990 3820 3720 3580

4570 4410 4230 4050 3860 3750 3610

4680 4500 4300 4110 3900 3780 3630

4780 4580 4380 4160 3940 3820 3660

4890 4670 4450 4220 3980 3850 3680

4990 4760 4520 4280 4020 3880 3710

W 33×141

1930

TFL 2 3 4 BFL 6 7

0.00 0.24 0.48 0.72 0.96 4.31 8.05

2080 1800 1530 1250 973 746 520

2750 2710 2660 2610 2550 2480 2350

2820 2770 2710 2650 2590 2510 2370

2900 2830 2770 2700 2620 2530 2390

2970 2900 2820 2740 2660 2560 2410

3040 2960 2880 2790 2690 2590 2420

3120 3030 2930 2830 2730 2610 2440

3190 3090 2980 2870 2760 2640 2460

3260 3150 3040 2920 2790 2670 2480

3340 3220 3090 2960 2830 2690 2500

3410 3280 3150 3010 2860 2720 2520

3480 3340 3200 3050 2900 2750 2530



5 - 40

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 33×130

1750

TFL 2 3 4 BFL 6 7

0.00 0.21 0.43 0.64 0.86 4.39 8.29

1920 1670 1420 1180 931 705 479

2520 2480 2440 2390 2350 2270 2140

2580 2540 2490 2440 2380 2300 2160

2650 2600 2540 2480 2410 2320 2170

2720 2660 2590 2520 2450 2350 2190

2790 2720 2640 2560 2480 2370 2210

2850 2770 2690 2600 2510 2400 2230

2920 2830 2740 2640 2540 2420 2240

2990 2890 2790 2690 2580 2450 2260

3060 2950 2840 2730 2610 2470 2280

3130 3010 2890 2770 2640 2500 2290

3190 3070 2940 2810 2680 2520 2310

W 33×118

1560

TFL 2 3 4 BFL 6 7

0.00 0.19 0.37 0.56 0.74 4.44 8.54

1740 1520 1310 1100 885 660 434

2260 2230 2200 2160 2120 2050 1920

2330 2290 2250 2200 2150 2070 1930

2390 2340 2290 2240 2190 2100 1950

2450 2400 2340 2280 2220 2120 1960

2510 2450 2380 2320 2250 2140 1980

2570 2500 2430 2360 2280 2170 1990

2630 2560 2480 2400 2310 2190 2010

2700 2610 2520 2430 2340 2210 2020

2760 2660 2570 2470 2370 2240 2040

2820 2720 2620 2510 2400 2260 2050

2880 2770 2660 2550 2440 2280 2070

W 30×116

1420

TFL 2 3 4 BFL 6 7

0.00 0.21 0.43 0.64 0.85 3.98 7.44

1710 1490 1260 1040 818 623 428

2060 2030 1990 1950 1910 1850 1740

2120 2080 2030 1990 1940 1870 1760

2180 2130 2080 2020 1960 1890 1770

2240 2180 2120 2060 1990 1910 1790

2300 2240 2170 2100 2020 1940 1800

2360 2290 2210 2130 2050 1960 1820

2420 2340 2260 2170 2080 1980 1830

2480 2400 2300 2210 2110 2000 1850

2540 2450 2350 2240 2140 2020 1860

2600 2500 2390 2280 2170 2050 1880

2670 2550 2440 2320 2200 2070 1890

W 30×108

1300

TFL 2 3 4 BFL 6 7

0.00 1590 1900 1960 2010 2070 2120 2180 0.19 1390 1870 1920 1970 2020 2070 2110 0.38 1190 1840 1880 1920 1960 2010 2050 0.57 988 1800 1840 1870 1910 1940 1980 0.76 789 1760 1790 1820 1850 1880 1900 4.04 593 1710 1730 1750 1770 1790 1810 7.64 396 1600 1610 1620 1640 1650 1670

2240 2160 2090 2010 1930 1830 1680

2290 2210 2130 2050 1960 1850 1690

2350 2260 2170 2080 1990 1870 1710

2400 2310 2220 2120 2020 1890 1720

2460 2360 2260 2150 2040 1920 1740

W 30×99

1170

TFL 2 3 4 BFL 6 7

0.00 1460 1730 1790 1840 1890 1940 1990 0.17 1280 1710 1750 1800 1840 1890 1930 0.34 1100 1680 1720 1760 1800 1840 1880 0.50 930 1650 1680 1720 1750 1780 1810 0.67 755 1620 1640 1670 1700 1730 1750 4.07 559 1560 1580 1600 1620 1640 1660 7.83 364 1450 1460 1480 1490 1500 1510

2040 1980 1920 1850 1780 1680 1530

2090 2030 1950 1880 1810 1700 1540

2150 2070 1990 1910 1830 1720 1550

2200 2120 2030 1950 1860 1740 1570

2250 2160 2070 1980 1890 1760 1580



COMPOSITE BEAMS

5 - 41

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

In.

ΣQ n

φM n (kip-ft)

Kips

Y2b (in.) 4 4.5 5

5.5

6

6.5

7

W 27×102

1140

TFL 2 3 4 BFL 6 7

0.00 1500 1650 1700 1760 1810 1860 1920 0.21 1290 1620 1670 1710 1760 1800 1850 0.42 1080 1590 1630 1660 1700 1740 1780 0.62 877 1550 1580 1610 1640 1670 1700 0.83 669 1510 1530 1560 1580 1600 1630 3.41 522 1470 1490 1500 1520 1540 1560 6.26 375 1400 1410 1420 1440 1450 1460

2

2.5

3

3.5

1970 1900 1820 1740 1650 1580 1480

2020 1940 1860 1770 1680 1600 1490

2080 1990 1890 1800 1700 1620 1500

2130 2030 1930 1830 1720 1630 1520

2180 2080 1970 1860 1750 1650 1530

W 27×94

1040

TFL 2 3 4 BFL 6 7

0.00 1390 1520 1570 1610 1660 1710 1760 0.19 1200 1490 1530 1570 1620 1660 1700 0.37 1010 1460 1490 1530 1570 1600 1640 0.56 827 1430 1460 1490 1510 1540 1570 0.75 641 1390 1410 1440 1460 1480 1510 3.39 493 1350 1370 1390 1400 1420 1440 6.39 346 1280 1290 1300 1320 1330 1340

1810 1740 1670 1600 1530 1460 1350

1860 1790 1710 1630 1550 1470 1360

1910 1830 1750 1660 1570 1490 1380

1960 1870 1780 1690 1600 1510 1390

2010 1910 1820 1720 1620 1530 1400

W 27×84

915

TFL 2 3 4 BFL 6 7

0.00 1240 1350 1390 1440 1480 1520 1570 0.16 1080 1330 1360 1400 1440 1480 1520 0.32 921 1300 1330 1370 1400 1430 1460 0.48 762 1270 1300 1330 1350 1380 1410 0.64 603 1240 1270 1290 1310 1330 1350 3.44 456 1200 1220 1240 1250 1270 1280 6.62 310 1130 1140 1150 1160 1170 1190

1610 1550 1500 1430 1370 1300 1200

1660 1590 1530 1460 1390 1320 1210

1700 1630 1560 1490 1410 1330 1220

1740 1670 1590 1520 1440 1350 1230

1790 1710 1630 1540 1460 1360 1240

W 24×76

750

TFL 2 3 4 BFL 6 7

0.00 1120 1110 1150 1190 1230 1270 0.17 967 1080 1120 1150 1190 1220 0.34 814 1060 1090 1120 1150 1180 0.51 662 1030 1060 1080 1100 1130 0.68 509 1010 1020 1040 1060 1080 3.00 394 976 990 1000 1020 1030 5.60 280 925 935 945 955 964

1310 1260 1200 1150 1100 1050 974

1350 1290 1230 1170 1110 1060 984

1390 1320 1260 1200 1130 1070 994

1420 1360 1290 1220 1150 1090 1000

1460 1390 1320 1250 1170 1100 1010

1500 1430 1350 1270 1190 1120 1020

W 24×68

664

TFL 2 3 4 BFL 6 7

0.00 1010 0.15 874 0.29 743 0.44 612 0.59 481 3.05 366 5.81 251

1160 1120 1080 1030 987 936 864

1200 1150 1100 1060 1000 949 873

1240 1180 1130 1080 1020 962 882

1270 1220 1160 1100 1040 975 891

1310 1250 1180 1120 1060 988 899

1340 1280 1210 1140 1070 1000 908

987 1020 1060 1090 1130 968 999 1030 1060 1090 947 973 1000 1030 1050 925 947 969 990 1010 902 919 936 953 970 872 885 898 910 923 819 828 837 846 855



5 - 42

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

W 24×62

574

TFL 2 3 4 BFL 6 7

0.00 0.15 0.29 0.44 0.59 3.47 6.58

910 806 702 598 495 361 228

894 879 862 845 827 789 723

926 907 887 866 844 802 731

958 936 912 887 862 815 739

991 1020 1060 1090 1120 1150 1180 1220 964 993 1020 1050 1080 1110 1140 1160 937 962 987 1010 1040 1060 1090 1110 909 930 951 972 993 1010 1040 1060 879 897 914 932 949 967 984 1000 827 840 853 866 879 891 904 917 747 755 763 771 779 787 795 803

3.5

W 24×55

503

TFL 2 3 4 BFL 6 7

0.00 0.13 0.25 0.38 0.51 3.45 6.66

810 722 633 545 456 329 203

791 778 764 750 734 700 636

820 804 787 769 751 711 643

848 829 809 788 767 723 651

877 855 832 808 783 735 658

906 880 854 827 799 746 665

934 906 876 846 815 758 672

W 21×62

540

TFL 2 3 4 BFL 6 7

0.00 0.15 0.31 0.46 0.62 2.53 4.78

915 788 662 535 408 318 229

810 791 771 750 727 705 669

842 819 795 769 741 716 677

875 847 818 788 756 727 685

907 875 841 807 770 739 693

939 903 865 826 785 750 701

972 1000 1040 1070 1100 1130 931 959 987 1010 1040 1070 888 912 935 959 982 1010 845 863 882 901 920 939 799 814 828 843 857 872 761 772 784 795 806 818 709 717 726 734 742 750

W 21×57

484

TFL 2 3 4 BFL 6 7

0.00 0.16 0.33 0.49 0.65 2.90 5.38

835 728 622 515 409 309 209

741 725 708 690 671 645 601

771 751 730 709 685 656 608

800 777 753 727 700 667 616

830 803 775 745 714 677 623

859 829 797 763 729 688 631

889 854 819 782 743 699 638

919 880 841 800 758 710 645

948 906 863 818 772 721 653

978 1010 1040 932 958 983 885 907 929 836 855 873 787 801 816 732 743 754 660 668 675

W 21×50

413

TFL 2 3 4 BFL 6 7

0.00 0.13 0.27 0.40 0.54 2.92 5.58

735 648 560 473 386 285 184

646 634 620 606 590 564 519

672 657 640 622 604 574 526

698 679 660 639 618 584 532

724 702 679 656 631 594 539

750 725 699 673 645 604 545

777 748 719 689 659 615 552

803 771 739 706 672 625 559

829 794 759 723 686 635 565

855 817 779 740 700 645 572



963 931 899 866 831 770 679

5.5

6

6.5

7

992 1020 1050 1080 957 982 1010 1030 921 944 966 989 885 904 923 943 848 864 880 896 781 793 805 816 686 694 701 708

881 840 799 756 713 655 578

907 863 818 773 727 665 585

COMPOSITE BEAMS

5 - 43

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 21×44

358

TFL 2 3 4 BFL 6 7

0.00 0.11 0.23 0.34 0.45 2.90 5.69

650 577 504 431 358 260 163

568 557 546 534 522 497 453

591 577 564 549 534 506 459

614 598 581 564 547 515 465

637 618 599 580 560 525 471

660 639 617 595 572 534 476

683 659 635 610 585 543 482

706 680 653 626 598 552 488

729 700 671 641 610 561 494

752 720 689 656 623 571 499

775 741 706 671 636 580 505

798 761 724 687 648 589 511

W 18×60

461

TFL 2 3 4 BFL 6 7

0.00 0.17 0.35 0.52 0.70 2.19 3.82

880 749 617 486 355 287 220

693 674 653 630 606 590 566

724 700 675 647 618 600 573

755 727 696 665 631 610 581

787 753 718 682 644 620 589

818 780 740 699 656 630 597

849 806 762 716 669 640 605

880 833 784 733 681 651 612

911 859 806 751 694 661 620

942 886 828 768 706 671 628

974 1000 912 939 850 871 785 802 719 732 681 691 636 644

W 18×55

420

TFL 2 3 4 BFL 6 7

0.00 0.16 0.32 0.47 0.63 2.16 3.86

810 691 573 454 336 269 203

634 617 598 578 556 541 517

663 641 618 594 568 550 524

692 666 639 610 580 560 531

720 690 659 626 592 569 539

749 715 679 642 604 579 546

778 739 699 658 616 588 553

806 764 720 674 628 598 560

835 788 740 691 640 607 567

864 813 760 707 652 617 574

892 837 781 723 663 626 582

921 862 801 739 675 636 589

W 18×50

379

TFL 2 3 4 BFL 6 7

0.00 0.14 0.29 0.43 0.57 2.07 3.82

735 628 521 415 308 246 184

572 557 540 522 503 489 467

598 579 558 537 514 498 474

624 601 577 552 525 506 480

651 624 595 566 536 515 487

677 646 614 581 547 524 493

703 668 632 596 558 532 500

729 690 651 610 569 541 506

755 712 669 625 580 550 513

781 735 688 640 590 559 519

807 757 706 654 601 567 526

833 779 725 669 612 576 532

W 18×46

340

TFL 2 3 4 BFL 6 7

0.00 0.15 0.30 0.45 0.61 2.40 4.34

675 583 492 400 308 239 169

527 514 499 484 468 450 424

551 535 517 498 478 459 430

575 555 534 512 489 467 436

599 576 552 527 500 476 442

623 597 569 541 511 484 448

647 617 587 555 522 493 454

671 638 604 569 533 501 460

695 659 621 583 544 510 466

719 679 639 597 555 518 472

743 700 656 612 566 526 478

766 720 674 626 577 535 484



5 - 44

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 18×40

294

TFL 2 3 4 BFL 6 7

0.00 0.13 0.26 0.39 0.53 2.26 4.27

590 511 432 353 274 211 148

458 446 434 421 407 392 369

479 464 449 433 417 400 374

499 482 464 446 426 407 379

520 500 480 458 436 415 384

541 518 495 471 446 422 389

562 537 510 483 456 429 395

583 555 526 496 465 437 400

604 573 541 508 475 444 405

625 591 556 521 485 452 410

646 609 572 533 494 459 416

667 627 587 546 504 467 421

W 18×35

249

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.43 2.37 4.56

515 451 388 324 260 194 129

396 387 377 367 356 340 315

414 403 391 378 365 347 320

432 418 404 389 374 354 324

451 434 418 401 383 361 329

469 450 432 412 393 368 333

487 466 445 424 402 375 338

505 482 459 435 411 382 342

523 498 473 447 420 389 347

542 514 487 458 430 395 351

560 530 500 470 439 402 356

578 546 514 481 448 409 361

W 16×36

240

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.32 0.43 1.79 3.44

530 455 380 305 230 181 133

373 362 350 338 326 315 299

392 378 364 349 334 322 304

410 394 377 360 342 328 309

429 410 391 371 350 334 313

448 426 404 381 358 341 318

467 442 418 392 366 347 323

485 459 431 403 374 354 327

504 475 445 414 383 360 332

523 491 458 425 391 366 337

542 507 471 435 399 373 342

560 523 485 446 407 379 346

W 16×31

203

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 2.00 3.79

456 395 334 274 213 163 114

321 312 303 293 283 272 255

337 326 315 303 290 278 259

353 340 327 312 298 283 263

370 354 338 322 305 289 267

386 368 350 332 313 295 271

402 382 362 342 321 301 275

418 396 374 351 328 307 279

434 410 386 361 336 312 283

450 424 398 371 343 318 287

466 438 410 380 351 324 291

483 452 421 390 358 330 295

W 16×26

166

TFL 2 3 4 BFL 6 7

0.00 0.09 0.17 0.26 0.35 2.04 4.01

384 337 289 242 194 145 96.0

268 261 254 246 239 228 210

281 273 264 255 245 233 214

295 285 274 263 252 238 217

309 297 284 272 259 243 220

322 309 295 281 266 248 224

336 321 305 289 273 253 227

349 332 315 298 280 259 231

363 344 325 306 287 264 234

377 356 336 315 294 269 237

390 368 346 323 301 274 241

404 380 356 332 307 279 244



COMPOSITE BEAMS

5 - 45

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 14×38

231

TFL 2 3 4 BFL 6 7

0.00 0.13 0.26 0.39 0.52 1.38 2.53

560 473 386 299 211 176 140

359 346 333 318 303 296 286

379 363 346 329 311 302 291

399 380 360 340 318 308 296

418 396 374 350 326 315 301

438 413 387 361 333 321 306

458 430 401 371 341 327 311

478 447 415 382 348 333 316

498 463 428 392 356 339 321

518 480 442 403 363 346 326

537 497 456 414 371 352 331

557 514 469 424 378 358 335

W 14×34

205

TFL 2 3 4 BFL 6 7

0.00 0.11 0.23 0.34 0.46 1.41 2.60

500 423 347 270 193 159 125

318 307 295 283 270 263 253

336 322 308 293 277 269 258

354 337 320 302 284 274 262

372 352 332 312 290 280 267

389 367 345 321 297 286 271

407 382 357 331 304 291 275

425 397 369 340 311 297 280

442 412 381 350 318 302 284

460 427 394 359 325 308 289

478 442 406 369 332 314 293

495 457 418 379 338 319 298

W 14×30

177

TFL 2 3 4 BFL 6 7

0.00 0.10 0.19 0.29 0.39 1.48 2.82

443 378 313 248 183 147 111

280 270 260 250 239 232 221

295 284 271 259 246 237 225

311 297 283 268 252 242 229

327 310 294 276 259 248 233

342 324 305 285 265 253 237

358 337 316 294 272 258 241

374 350 327 303 278 263 245

389 364 338 312 285 268 249

405 377 349 320 291 274 253

421 391 360 329 298 279 256

436 404 371 338 304 284 260

W 14×26

151

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.42 1.67 3.19

385 332 279 226 173 135 96.1

244 236 228 220 211 203 191

258 248 238 228 217 207 194

271 260 248 236 223 212 197

285 271 258 244 229 217 201

298 283 268 252 235 222 204

312 295 278 260 242 227 208

326 307 287 268 248 231 211

339 318 297 276 254 236 214

353 330 307 284 260 241 218

366 342 317 292 266 246 221

380 354 327 300 272 250 225

W 14×22

125

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.34 1.69 3.34

325 283 241 199 157 119 81.1

204 198 192 185 178 170 158

215 208 200 192 184 174 161

227 218 209 199 189 179 164

238 228 217 206 195 183 167

250 238 226 213 200 187 170

261 248 234 220 206 191 172

273 258 243 227 212 196 175

284 268 251 234 217 200 178

296 278 260 241 223 204 181

307 288 268 248 228 208 184

319 298 277 255 234 212 187



5 - 46

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

W 12×30

162

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 1.12 1.94

440 368 296 224 153 131 110

254 244 233 221 209 205 200

270 257 243 229 215 210 204

285 270 254 237 220 214 207

301 283 264 245 225 219 211

317 296 275 253 231 224 215

332 309 285 261 236 228 219

348 322 296 269 242 233 223

363 335 306 277 247 238 227

379 348 317 285 252 242 231

394 361 327 293 258 247 235

410 374 338 301 263 252 239

W 12×26

140

TFL 2 3 4 BFL 6 7

0.00 0.10 0.19 0.29 0.38 1.08 1.95

383 321 259 198 136 116 95.6

220 211 201 192 181 178 173

233 222 211 199 186 182 176

247 234 220 206 191 186 179

260 245 229 213 196 190 183

274 256 238 220 201 194 186

287 268 247 227 206 198 190

301 279 257 234 210 202 193

315 290 266 241 215 206 196

328 302 275 248 220 210 200

342 313 284 255 225 215 203

355 324 293 262 230 219 206

W 12×22

110

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.43 1.66 3.04

324 281 238 196 153 117 81.0

187 181 174 168 160 153 142

199 191 183 174 166 157 145

210 201 191 181 171 161 147

222 211 200 188 177 165 150

233 221 208 195 182 169 153

245 231 217 202 187 173 156

256 241 225 209 193 178 159

267 251 233 216 198 182 162

279 261 242 223 204 186 165

290 271 250 230 209 190 167

302 281 259 237 214 194 170

W 12×19

92.6

TFL 2 3 4 BFL 6 7

0.00 0.09 0.18 0.26 0.35 1.66 3.12

279 243 208 173 138 104 69.6

159 154 149 144 138 131 120

169 163 156 150 143 134 122

179 172 164 156 148 138 125

189 180 171 162 152 142 127

199 189 179 168 157 145 130

209 197 186 174 162 149 132

219 206 193 180 167 153 135

228 215 201 187 172 156 137

238 223 208 193 177 160 140

248 232 215 199 182 164 142

258 241 223 205 187 167 145

W 12×16

75.4

TFL 2 3 4 BFL 6 7

0.00 0.07 0.13 0.20 0.26 1.71 3.32

236 209 183 156 130 94.3 58.9

133 130 126 122 117 110 98.7

142 137 132 127 122 113 101

150 144 139 133 127 117 103

158 152 145 138 131 120 105

167 159 152 144 136 123 107

175 167 158 149 140 127 109

183 174 164 155 145 130 111

192 181 171 160 150 133 113

200 189 177 166 154 137 115

208 196 184 171 159 140 117

217 204 190 177 163 143 120



COMPOSITE BEAMS

5 - 47

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

W 12×14

W 10×26

65.2

117

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2

2.5

3

3.5

5.5

6

6.5

7

TFL 2 3 4 BFL 6 7

0.00 0.06 0.11 0.17 0.23 1.69 3.36

208 186 163 141 119 85.3 52.0

117 114 111 107 104 97.0 86.3

125 121 116 112 108 100 88.2

132 127 122 117 112 103 90.0

139 134 128 122 116 106 91.8

147 140 134 127 121 109 93.7

154 147 140 132 125 112 95.5

161 153 145 137 129 115 97.4

169 160 151 142 133 118 99.2

176 167 157 147 137 121 101

184 173 163 152 142 124 103

191 180 169 157 146 127 105

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 0.90 1.51

381 317 254 190 127 111 95.1

193 184 174 164 153 150 147

207 195 183 171 158 154 150

220 206 192 177 162 158 153

234 218 201 184 167 162 157

247 229 210 191 171 166 160

260 240 219 198 176 170 163

274 251 228 204 180 174 167

287 262 237 211 185 178 170

301 274 246 218 189 182 174

314 285 255 225 194 186 177

328 296 264 231 198 189 180

W 10×22

97.5

TFL 2 3 4 BFL 6 7

0.00 0.09 0.18 0.27 0.36 0.95 1.70

325 273 221 169 117 99.3 81.1

163 155 148 139 131 127 123

174 165 155 145 135 131 126

186 175 163 151 139 135 129

197 184 171 157 143 138 132

209 194 179 163 148 142 135

220 204 187 169 152 145 138

232 213 194 175 156 149 140

243 223 202 181 160 152 143

255 233 210 187 164 156 146

266 242 218 193 168 159 149

278 252 226 199 173 163 152

W 10×19

81.0

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 1.27 2.31

281 241 202 162 122 96.2 70.3

142 136 130 124 117 112 105

152 145 137 129 121 115 107

162 153 144 135 126 119 110

172 162 151 141 130 122 112

182 170 158 147 134 125 115

191 179 166 152 139 129 117

201 187 173 158 143 132 120

211 196 180 164 147 136 122

221 204 187 169 152 139 125

231 213 194 175 156 143 127

241 221 201 181 160 146 130

W 10×17

70.1

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.33 1.31 2.46

249 216 183 150 117 89.8 62.4

125 120 115 110 104 99.0 91.4

134 128 121 115 109 102 93.7

142 135 128 120 113 105 95.9

151 143 134 126 117 109 98.1

160 151 141 131 121 112 100

169 158 147 136 125 115 102

178 166 154 142 129 118 105

187 174 160 147 133 121 107

195 181 167 152 138 124 109

204 189 173 158 142 128 111

213 197 180 163 146 131 114



5 - 48

COMPOSITE DESIGN

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

W 10×15

60.0

TFL 2 3 4 BFL 6 7

0.00 0.07 0.14 0.20 0.27 1.35 2.60

221 109 194 105 167 101 139 97.2 112 92.9 83.8 87.3 55.1 79.2

W 10×12

47.3

TFL 2 3 4 BFL 6 7

0.00 0.05 0.11 0.16 0.21 1.30 2.61

177 156 135 115 93.8 69.0 44.3

TFL 2 3 4 BFL 6 7

0.00 0.12 0.23 0.35 0.47 0.53 0.59

413 337 261 185 109 106 103

176 165 153 141 128 128 127

191 177 163 148 132 131 131

W 8×28

102

2

86.9 84.0 80.9 77.8 74.5 69.8 62.9

2.5

3

3.5

5.5

6

6.5

7

117 112 107 102 96.9 90.3 81.2

125 119 113 107 101 93.2 83.1

133 126 119 112 105 96.2 85.1

140 133 125 117 109 99.2 87.0

148 140 131 122 113 102 89.0

156 146 137 127 117 105 90.9

164 153 143 132 121 108 92.9

172 160 149 137 125 111 94.8

180 167 154 142 129 114 96.8

187 174 160 147 133 117 98.7

93.2 89.5 85.7 81.8 77.9 72.3 64.4

99.5 95.0 90.5 85.9 81.2 74.7 66.0

106 101 95.3 89.9 84.5 77.1 67.6

112 106 100 94.0 87.8 79.6 69.1

118 112 105 98.1 91.2 82.0 70.7

125 117 110 102 94.5 84.5 72.3

131 123 114 106 97.8 86.9 73.8

137 128 119 110 101 89.4 75.4

143 134 124 114 104 91.8 77.0

150 139 129 118 108 94.3 78.5

205 189 172 154 136 135 134

220 201 181 161 140 139 138

235 213 190 167 144 143 142

249 225 200 174 147 146 145

264 237 209 180 151 150 149

278 249 218 187 155 154 153

293 260 227 193 159 158 156

308 272 236 200 163 161 160

322 284 246 206 167 165 164

W 8×24

87.0

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 0.47 0.55

354 289 224 159 94.2 91.3 88.5

150 140 130 120 109 109 108

162 150 138 126 112 112 111

175 161 146 131 116 115 114

187 171 154 137 119 118 117

200 181 162 142 122 122 121

212 191 170 148 126 125 124

225 202 178 154 129 128 127

237 212 186 159 132 131 130

250 222 194 165 136 134 133

262 232 202 171 139 138 136

275 243 210 176 142 141 139

W 8×21

76.5

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 0.70 1.06

308 255 203 150 97.2 87.1 77.0

134 126 118 110 101 99.3 97.2

145 135 125 115 105 102 100

156 144 133 120 108 105 103

167 153 140 126 111 109 105

178 162 147 131 115 112 108

188 172 154 136 118 115 111

199 181 161 142 122 118 114

210 190 169 147 125 121 116

221 199 176 152 129 124 119

232 208 183 158 132 127 122

243 217 190 163 136 130 125



COMPOSITE BEAMS

5 - 49

Fy = 50 ksi


φb M p PNAc Y1a

Kip-ft

ΣQ n

φM n (kip-ft)

Y2b (in.) 4 4.5 5

In.

Kips

2.5

3

3.5

5.5

6

6.5

7

W 8×18

63.7

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.33 0.71 1.21

263 113 220 107 176 100 133 93.5 89.8 86.5 77.8 84.4 65.8 81.9

2

122 115 107 98.2 89.7 87.1 84.2

132 122 113 103 92.9 89.9 86.5

141 130 119 108 96.0 92.6 88.8

150 138 125 112 99.2 95.4 91.2

160 146 132 117 102 98.1 93.5

169 154 138 122 106 101 95.8

178 161 144 127 109 104 98.2

188 169 150 131 112 106 100

197 177 157 136 115 109 103

206 185 163 141 118 112 105

W 8×15

51.0

TFL 2 3 4 BFL 6 7

0.00 0.08 0.16 0.24 0.32 0.97 1.79

222 190 159 127 95.5 75.5 55.5

95.2 90.6 85.9 81.0 75.9 72.2 67.4

103 97.4 91.5 85.5 79.3 74.8 69.3

111 104 97.1 90.0 82.7 77.5 71.3

119 111 103 94.5 86.0 80.2 73.3

127 118 108 99.0 89.4 82.9 75.2

135 124 114 103 92.8 85.5 77.2

142 131 120 108 96.2 88.2 79.2

150 138 125 113 99.6 90.9 81.1

158 145 131 117 103 93.6 83.1

166 151 137 122 106 96.2 85.1

174 158 142 126 110 98.9 87.0

W 8×13

42.7

TFL 2 3 4 BFL 6 7

0.00 0.06 0.13 0.19 0.26 0.99 1.91

192 167 141 116 90.0 69.0 48.0

81.5 77.9 74.1 70.2 66.2 62.3 57.2

88.3 83.8 79.1 74.3 69.3 64.7 58.9

95.1 89.7 84.1 78.4 72.5 67.2 60.6

102 95.6 89.1 82.4 75.7 69.6 62.3

109 101 94.1 86.5 78.9 72.1 64.0

116 107 99.0 90.6 82.1 74.5 65.7

122 113 104 94.7 85.3 77.0 67.4

129 119 109 98.8 88.5 79.4 69.1

136 125 114 103 91.7 81.8 70.8

143 131 119 107 94.8 84.3 72.5

150 137 124 111 98.0 86.7 74.2

W 8×10

33.3

TFL 2 3 4 BFL 6 7

0.00 148 0.05 128 0.10 108 0.15 87.4 0.21 67.2 0.88 52.1 1.77 37.0

62.3 59.4 56.5 53.4 50.3 47.6 44.0

67.6 64.0 60.3 56.5 52.7 49.5 45.4

72.8 68.5 64.1 59.6 55.1 51.3 46.7

78.0 73.0 67.9 62.7 57.4 53.1 48.0

83.3 77.5 71.7 65.8 59.8 55.0 49.3

88.5 82.1 75.5 68.9 62.2 56.8 50.6

93.8 86.6 79.3 72.0 64.6 58.7 51.9

99.0 91.1 83.1 75.1 67.0 60.5 53.2

104 95.6 86.9 78.2 69.3 62.4 54.5

109 100 90.8 81.3 71.7 64.2 55.8

115 105 94.6 84.4 74.1 66.1 57.2



5 - 50

COMPOSITE DESIGN

LOWER BOUND ELASTIC MOMENT OF INERTIA FOR PLASTIC COMPOSITE SECTIONS

LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W40×297 (23200)

TFL 2 3 4 BFL 6 7

0.00 0.41 0.83 1.24 1.65 4.59 8.17

44200 42500 40500 38100 35300 33500 31600

45200 43400 41300 38800 35800 34000 32000

46200 44300 42100 39500 36400 34500 32400

47200 45200 43000 40200 37000 35000 32800

48200 46200 43800 41000 37600 35500 33200

49300 47200 44700 41700 38200 36000 33600

50300 48100 45600 42500 38800 36600 34100

51400 49200 46500 43300 39400 37100 34500

52600 50200 47400 44100 40100 37600 34900

53700 51200 48300 44900 40700 38200 35400

54900 52300 49300 45700 41400 38800 35900

W40×278 (20500)

TFL 2 3 4 BFL 6 7

0.00 0.45 0.91 1.36 1.81 5.64 10.06

40400 39000 37400 35500 33300 31100 28500

41400 39900 38200 36200 33800 31500 28800

42300 40700 39000 36900 34400 32000 29200

43200 41600 39800 37600 35000 32500 29600

44200 42500 40600 38300 35700 33100 30000

45200 43500 41400 39100 36300 33600 30400

46200 44400 42300 39800 37000 34100 30800

47300 45400 43200 40600 37600 34700 31200

48300 46300 44100 41400 38300 35200 31600

49400 47300 45000 42200 39000 35800 32100

50500 48400 45900 43100 39700 36400 32500

W40×277 (21900)

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.58 4.25 7.60

41300 39700 37800 35500 32700 31300 29700

42200 40500 38500 36100 33200 31700 30000

43100 41400 39300 36800 33700 32100 30400

44100 42200 40000 37400 34300 32600 30800

45000 43100 40800 38100 34800 33100 31100

46000 44000 41600 38800 35300 33500 31500

47000 44900 42400 39500 35900 34000 31900

48000 45800 43300 40200 36500 34500 32300

49100 46800 44100 40900 37000 35000 32800

50100 47800 45000 41700 37600 35500 33200

51200 48800 45900 42400 38200 36000 33600

W40×264 (19400)

TFL 2 3 4 BFL 6 7

0.00 0.43 0.87 1.30 1.73 5.49 9.90

38200 36800 35300 33500 31400 29300 26900

39000 37600 36000 34100 31900 29800 27300

39900 38500 36800 34800 32500 30200 27600

40800 39300 37500 35500 33100 30700 28000

41700 40200 38300 36200 33700 31200 28300

42700 41000 39100 36900 34300 31700 28700

43700 41900 39900 37600 34900 32200 29100

44600 42800 40800 38300 35500 32700 29500

45600 43800 41600 39100 36100 33300 29900

46600 44700 42500 39900 36800 33800 30300

47700 45700 43300 40600 37400 34300 30700

aY1 = distance from top of the steel beam to plastic neutral axis. bY2 = distance from top of the steel beam to concrete flange force. cSee Figure 5-3 for PNA locations. dValue in parentheses is I (in.4) of non-composite steel shape. x


COMPOSITE BEAMS

5 - 51

LOWER BOUND ELASTIC MOMENT OF INERTIA FOR PLASTIC COMPOSITE SECTIONS Shaped PNAc

Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W40×249 (19500)

TFL 2 3 4 BFL 6 7

0.00 0.36 0.71 1.07 1.42 4.06 7.47

36700 35300 33600 31600 29200 27900 26400

37500 36100 34300 32200 29600 28200 26700

38400 36800 35000 32800 30100 28600 27000

39200 37600 35700 33400 30600 29100 27400

40100 38400 36400 34000 31000 29500 27700

40900 39200 37100 34600 31500 29900 28100

41800 40000 37800 35200 32000 30300 28400

42800 40800 38600 35900 32500 30800 28800

43700 41700 39300 36500 33100 31200 29200

44600 42600 40100 37200 33600 31700 29600

45600 43500 40900 37900 34100 32200 29900

W40×235 (17400)

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.58 5.18 9.47

33800 32600 31300 29600 27700 26000 24000

34600 33300 31900 30200 28200 26400 24300

35400 34100 32600 30800 28700 26800 24600

36200 34800 33200 31400 29200 27200 24900

37000 35600 33900 32000 29700 27600 25200

37800 36300 34600 32600 30200 28100 25600

38700 37100 35300 33200 30700 28500 25900

39500 37900 36000 33900 31300 29000 26300

40400 38700 36800 34500 31800 29400 26600

41300 39600 37500 35200 32400 29900 27000

42200 40400 38300 35900 33000 30400 27300

W40×215 (16700)

TFL 2 3 4 BFL 6 7

0.00 0.31 0.61 0.92 1.22 3.84 7.32

31300 30100 28700 27000 24900 23800 22500

32000 30700 29300 27500 25300 24100 22800

32700 31400 29800 28000 25700 24500 23100

33400 32100 30400 28500 26100 24800 23400

34200 32700 31000 29000 26600 25200 23700

34900 33400 31700 29500 27000 25600 24000

35700 34100 32300 30100 27400 25900 24300

36500 34800 32900 30600 27900 26300 24600

37300 35600 33600 31200 28300 26700 24900

38100 36300 34300 31800 28800 27100 25300

38900 37100 34900 32400 29200 27500 25600

W40×211 (15500)

TFL 2 3 4 BFL 6 7

0.00 0.35 0.71 1.06 1.42 4.99 9.35

30100 29000 27800 26400 24700 23100 21300

30800 29700 28400 26900 25100 23500 21600

31500 30300 29000 27400 25600 23900 21900

32200 31000 29600 27900 26000 24200 22200

32900 31600 30200 28500 26500 24600 22500

33600 32300 30800 29000 26900 25000 22800

34400 33000 31400 29600 27400 25400 23100

35200 33700 32100 30200 27900 25800 23400

36000 34500 32800 30800 28400 26200 23700

36800 35200 33400 31400 28900 26600 24000

37600 36000 34100 32000 29400 27100 24300



5 - 52

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W40×199 (14900)

TFL 2 3 4 BFL 6 7

0.00 0.27 0.53 0.80 1.07 4.16 8.10

28200 27200 26000 24500 22800 21600 20200

28800 27700 26500 25000 23200 21900 20500

29500 28300 27000 25500 23600 22300 20700

30100 28900 27600 25900 24000 22600 21000

30800 29600 28100 26400 24400 22900 21300

31500 30200 28700 26900 24800 23300 21500

32200 30900 29300 27400 25200 23600 21800

32900 31500 29900 28000 25700 24000 22100

33600 32200 30500 28500 26100 24400 22400

34400 32900 31100 29000 26500 24700 22700

35200 33600 31800 29600 27000 25100 23000

W40×183 (13300)

TFL 2 3 4 BFL 6 7

0.00 0.31 0.61 0.92 1.22 4.76 9.16

25700 24800 23800 22600 21200 19800 18300

26300 25400 24300 23000 21500 20100 18500

26900 25900 24800 23500 21900 20400 18700

27500 26500 25300 23900 22300 20800 19000

28100 27100 25800 24400 22700 21100 19200

28800 27600 26400 24900 23100 21400 19500

29400 28200 26900 25300 23500 21800 19700

30100 28900 27500 25800 23900 22100 20000

30700 29500 28000 26400 24300 22500 20300

31400 30100 28600 26900 24800 22800 20600

32100 30800 29200 27400 25200 23200 20800

W40×174 (12200)

TFL 2 3 4 BFL 6 7

0.00 0.21 0.42 0.62 0.83 4.59 9.27

23600 22800 21900 20900 19700 18300 16800

24100 23300 22400 21300 20000 18600 17000

24700 23800 22800 21700 20400 18900 17200

25200 24400 23300 22200 20800 19200 17400

25800 24900 23800 22600 21100 19600 17700

26400 25500 24300 23100 21500 19900 17900

27000 26000 24900 23500 21900 20200 18100

27700 26600 25400 24000 22300 20500 18400

28300 27200 25900 24500 22800 20900 18600

28900 27800 26500 25000 23200 21200 18900

29600 28400 27100 25500 23600 21600 19200

W40×167 (11600)

TFL 2 3 4 BFL 6 7

0.00 0.26 0.51 0.77 1.03 5.00 9.85

22700 22000 21200 20200 19100 17700 16100

23300 22500 21600 20600 19400 18000 16300

23800 23000 22100 21000 19800 18300 16500

24400 23500 22600 21500 20200 18600 16700

24900 24000 23000 21900 20600 18900 16900

25500 24600 23500 22300 21000 19200 17200

26100 25100 24100 22800 21300 19600 17400

26700 25700 24600 23300 21800 19900 17600

27300 26300 25100 23700 22200 20200 17900

27900 26900 25600 24200 22600 20600 18100

28600 27500 26200 24700 23000 20900 18400

W40×149 (9780)

TFL 2 3 4 BFL 6 7

0.00 0.21 0.42 0.62 0.83 5.15 10.41

19500 19000 18300 17600 16700 15400 13700

20000 19400 18700 17900 17100 15600 13900

20500 19800 19100 18300 17400 15900 14100

21000 20300 19600 18700 17700 16200 14300

21500 20800 20000 19100 18100 16500 14500

22000 21300 20400 19500 18500 16800 14700

22500 21700 20900 19900 18800 17100 14900

23000 22200 21400 20400 19200 17400 15100

23600 22800 21800 20800 19600 17700 15300

24100 23300 22300 21200 20000 18000 15500

24700 23800 22800 21700 20400 18300 15700



COMPOSITE BEAMS

5 - 53


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 36×300 (20300)

TFL 2 3 4 BFL 6 7

0.00 0.42 0.84 1.26 1.68 3.97 6.69

38600 37000 35200 32900 30100 28900 27600

39500 37900 35900 33500 30600 29400 28000

40500 38700 36700 34200 31100 29800 28400

41400 39600 37400 34800 31600 30200 28700

42400 40500 38200 35500 32100 30700 29100

43400 41400 39000 36200 32700 31200 29500

44400 42300 39900 36900 33200 31700 29900

45500 43300 40700 37600 33800 32200 30400

46500 44300 41600 38300 34400 32700 30800

47600 45300 42500 39100 34900 33200 31200

48700 46300 43400 39900 35500 33700 31700

W 36×280 (18900)

TFL 2 3 4 BFL 6 7

0.00 0.39 0.79 1.18 1.57 3.88 6.62

35800 34400 32600 30600 28000 26900 25700

36700 35100 33300 31100 28400 27300 26000

37500 35900 34000 31700 28900 27700 26300

38400 36700 34700 32300 29400 28100 26700

39300 37600 35500 33000 29900 28500 27100

40200 38400 36200 33600 30400 29000 27400

41200 39300 37000 34300 30900 29400 27800

42200 40200 37800 34900 31400 29900 28200

43100 41100 38600 35600 31900 30300 28600

44200 42000 39400 36300 32500 30800 29000

45200 42900 40300 37000 33000 31300 29400

W 36×260 (17300)

TFL 2 3 4 BFL 6 7

0.00 0.36 0.72 1.08 1.44 3.86 6.75

32800 31500 29900 28100 25800 24700 23500

33600 32200 30600 28600 26200 25100 23800

34400 32900 31200 29200 26700 25500 24100

35200 33700 31900 29700 27100 25800 24500

36000 34500 32600 30300 27600 26200 24800

36900 35200 33300 30900 28000 26600 25100

37800 36000 34000 31500 28500 27100 25500

38700 36900 34700 32100 29000 27500 25800

39600 37700 35500 32800 29500 27900 26200

40500 38500 36200 33400 30000 28400 26600

41500 39400 37000 34100 30500 28800 27000

W 36×245 (16100)

TFL 2 3 4 BFL 6 7

0.00 0.34 0.68 1.01 1.35 3.81 6.77

30600 29400 27900 26200 24100 23100 21900

31300 30000 28500 26700 24500 23400 22200

32100 30700 29100 27200 24900 23800 22500

32800 31400 29800 27800 25300 24100 22800

33600 32100 30400 28300 25800 24500 23100

34400 32900 31100 28900 26200 24900 23400

35200 33600 31700 29500 26700 25300 23800

36100 34400 32400 30000 27100 25700 24100

36900 35200 33100 30600 27600 26100 24400

37800 36000 33800 31300 28100 26500 24800

38700 36800 34600 31900 28600 27000 25100

W 36×230 (15000)

TFL 2 3 4 BFL 6 7

0.00 0.32 0.63 0.95 1.26 3.81 6.83

28500 27300 26000 24400 22500 21500 20400

29100 28000 26600 24900 22900 21800 20700

29800 28600 27100 25400 23300 22200 20900

30600 29300 27700 25900 23700 22500 21200

31300 29900 28300 26400 24100 22900 21500

32000 30600 28900 26900 24500 23200 21800

32800 31300 29600 27500 24900 23600 22100

33600 32000 30200 28000 25400 24000 22400

34400 32800 30900 28600 25800 24400 22800

35200 33500 31500 29200 26300 24800 23100

36000 34300 32200 29700 26700 25200 23400



5 - 54

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 36×210 (13200)

TFL 2 3 4 BFL 6 7

0.00 0.34 0.68 1.02 1.36 5.06 9.04

26000 25100 24000 22800 21300 19900 18300

26600 25700 24500 23200 21700 20300 18600

27300 26300 25100 23700 22100 20600 18800

27900 26900 25700 24200 22500 21000 19100

28600 27500 26200 24700 23000 21300 19400

29300 28200 26800 25300 23400 21700 19700

30000 28800 27400 25800 23900 22100 19900

30800 29500 28100 26300 24300 22400 20200

31500 30200 28700 26900 24800 22800 20500

32300 30900 29300 27500 25300 23200 20800

33000 31600 30000 28100 25800 23600 21100

W 36×194 (12100)

TFL 2 3 4 BFL 6 7

0.00 0.32 0.63 0.95 1.26 4.94 8.93

23800 22900 22000 20800 19500 18200 16800

24400 23500 22500 21300 19900 18600 17000

25000 24000 23000 21700 20300 18900 17200

25600 24600 23500 22200 20600 19200 17500

26200 25200 24000 22700 21000 19500 17700

26800 25800 24600 23100 21500 19900 18000

27500 26400 25100 23600 21900 20200 18300

28200 27000 25700 24100 22300 20600 18500

28900 27700 26300 24600 22700 20900 18800

29600 28300 26900 25200 23200 21300 19100

30300 29000 27500 25700 23600 21700 19400

W 36×182 (11300)

TFL 2 3 4 BFL 6 7

0.00 0.29 0.59 0.89 1.18 4.89 8.92

22200 21400 20500 19500 18300 17100 15700

22700 21900 21000 19900 18600 17300 15900

23300 22500 21500 20300 19000 17600 16100

23900 23000 22000 20700 19300 17900 16300

24500 23500 22400 21200 19700 18300 16600

25100 24100 23000 21600 20100 18600 16800

25700 24700 23500 22100 20500 18900 17100

26300 25200 24000 22600 20900 19200 17300

26900 25800 24600 23100 21300 19600 17600

27600 26400 25100 23500 21700 19900 17800

28300 27100 25700 24000 22100 20300 18100

W 36×170 (10500)

TFL 2 3 4 BFL 6 7

0.00 0.28 0.55 0.83 1.10 4.84 8.89

20600 19900 19000 18100 17000 15800 14500

21100 20300 19500 18500 17300 16100 14700

21600 20800 19900 18900 17600 16400 14900

22100 21300 20400 19300 18000 16700 15200

22700 21800 20800 19700 18300 17000 15400

23300 22300 21300 20100 18700 17300 15600

23800 22900 21800 20500 19000 17600 15800

24400 23400 22300 21000 19400 17900 16100

25000 24000 22800 21400 19800 18200 16300

25600 24500 23300 21900 20200 18500 16500

26200 25100 23800 22300 20600 18800 16800

W 36×160 (9750)

TFL 2 3 4 BFL 6 7

0.00 0.26 0.51 0.77 1.02 4.82 8.97

19200 18500 17700 16900 15800 14800 13500

19600 18900 18200 17200 16200 15000 13700

20100 19400 18600 17600 16500 15300 13900

20600 19900 19000 18000 16800 15600 14100

21100 20300 19400 18400 17100 15800 14300

21700 20800 19900 18800 17500 16100 14500

22200 21300 20300 19200 17800 16400 14700

22700 21800 20800 19600 18200 16700 14900

23300 22300 21300 20000 18500 17000 15200

23900 22900 21700 20400 18900 17300 15400

24400 23400 22200 20900 19300 17600 15600



COMPOSITE BEAMS

5 - 55


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 36×150 (9040)

TFL 2 3 4 BFL 6 7

0.00 0.24 0.47 0.71 0.94 4.83 9.08

17800 17200 16500 15700 14800 13800 12500

18300 17600 16900 16100 15100 14000 12700

18700 18100 17300 16400 15400 14300 12900

19200 18500 17700 16800 15700 14500 13100

19700 18900 18100 17200 16000 14800 13300

20200 19400 18500 17500 16400 15000 13500

20700 19900 19000 17900 16700 15300 13700

21200 20300 19400 18300 17000 15600 13900

21700 20800 19800 18700 17400 15900 14100

22200 21300 20300 19100 17700 16200 14300

22800 21800 20800 19500 18100 16500 14500

W 36×135 (7800)

TFL 2 3 4 BFL 6 7

0.00 0.20 0.40 0.59 0.79 4.96 9.50

15600 15100 14500 13900 13100 12100 10900

16000 15400 14900 14200 13400 12400 11100

16400 15800 15200 14500 13700 12600 11200

16800 16200 15600 14800 14000 12800 11400

17200 16600 15900 15200 14300 13100 11600

17600 17000 16300 15500 14600 13300 11700

18100 17400 16700 15900 14900 13500 11900

18600 17900 17100 16200 15200 13800 12100

19000 18300 17500 16600 15500 14100 12300

19500 18800 17900 17000 15800 14300 12500

20000 19200 18300 17300 16200 14600 12700

W 33×221 (12800)

TFL 2 3 4 BFL 6 7

0.00 0.32 0.64 0.96 1.28 3.76 6.48

24500 23500 22300 20900 19200 18400 17500

25100 24100 22900 21400 19600 18700 17700

25800 24700 23400 21800 19900 19000 18000

26400 25300 23900 22300 20300 19300 18200

27100 25900 24500 22800 20700 19600 18500

27800 26500 25000 23200 21000 20000 18800

28500 27200 25600 23700 21400 20300 19100

29200 27800 26200 24200 21800 20700 19400

29900 28500 26800 24700 22200 21000 19700

30700 29200 27400 25300 22700 21400 20000

31500 29900 28000 25800 23100 21700 20300

W 33×201 (11500)

TFL 2 3 4 BFL 6 7

0.00 0.29 0.58 0.86 1.15 3.67 6.51

22000 21100 20100 18900 17400 16600 15700

22600 21600 20600 19300 17700 16800 15900

23100 22200 21000 19700 18000 17100 16200

23700 22700 21500 20100 18300 17400 16400

24300 23300 22000 20500 18700 17700 16600

25000 23800 22500 20900 19000 18000 16900

25600 24400 23000 21400 19400 18300 17100

26200 25000 23600 21800 19700 18600 17400

26900 25600 24100 22300 20100 18900 17700

27600 26300 24700 22800 20500 19300 17900

28300 26900 25300 23300 20900 19600 18200

W 33×141 (7450)

TFL 2 3 4 BFL 6 7

0.00 0.24 0.48 0.72 0.96 4.31 8.05

14700 14200 13600 12900 12100 11300 10300

15100 14500 13900 13200 12300 11500 10500

15500 14900 14200 13500 12600 11700 10700

15900 15300 14600 13800 12800 11900 10800

16300 15700 15000 14100 13100 12100 11000

16800 16100 15300 14400 13400 12400 11200

17200 16500 15700 14800 13700 12600 11300

17700 16900 16100 15100 14000 12800 11500

18100 17400 16500 15500 14200 13100 11700

18600 17800 16900 15800 14600 13300 11900

19100 18300 17300 16200 14900 13600 12100



5 - 56

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 33×130 (6710)

TFL 2 3 4 BFL 6 7

0.00 0.21 0.43 0.64 0.86 4.39 8.29

13300 12800 12300 11700 11000 10300 9340

13700 13200 12600 12000 11300 10400 9490

14000 13500 12900 12300 11500 10600 9640

14400 13900 13300 12600 11700 10900 9790

14800 14200 13600 12900 12000 11100 9940

15200 14600 13900 13200 12300 11300 10100

15600 15000 14300 13500 12500 11500 10300

16000 15400 14600 13800 12800 11700 10400

16400 15800 15000 14100 13100 11900 10600

16900 16200 15400 14500 13400 12200 10800

17300 16600 15800 14800 13700 12400 11000

W 33×118 (5900)

TFL 2 3 4 BFL 6 7

0.00 0.19 0.37 0.56 0.74 4.44 8.54

11800 11400 11000 10500 9880 9150 8260

12100 11700 11300 10700 10100 9330 8390

12500 12000 11500 11000 10300 9510 8520

12800 12300 11800 11200 10600 9700 8660

13100 12700 12100 11500 10800 9890 8800

13500 13000 12400 11800 11000 10100 8940

13900 13300 12800 12100 11300 10300 9090

14200 13700 13100 12400 11500 10500 9240

14600 14100 13400 12700 11800 10700 9390

15000 14400 13700 13000 12100 10900 9550

15400 14800 14100 13300 12300 11100 9710

W 30×116 (4930)

TFL 2 3 4 BFL 6 7

0.00 0.21 0.43 0.64 0.85 3.98 7.44

9870 10200 10500 10800 11100 11400 11800 9530 9800 10100 10400 10700 11000 11300 9130 9380 9640 9910 10200 10500 10700 8670 8900 9130 9360 9600 9850 10100 8130 8320 8520 8720 8930 9140 9360 7570 7730 7890 8060 8230 8400 8580 6910 7030 7150 7270 7400 7530 7670

12100 11600 11000 10400 9580 8770 7810

12500 11900 11300 10600 9810 8960 7950

12800 12300 11700 10900 10000 9150 8090

13200 12600 12000 11200 10300 9350 8240

W 30×108 (4470)

TFL 2 3 4 BFL 6 7

0.00 0.19 0.38 0.57 0.76 4.04 7.64

9000 8700 8350 7950 7480 6940 6280

9280 8960 8590 8160 7660 7090 6390

9560 9220 8830 8380 7850 7240 6500

9840 10100 10400 10800 11100 11400 11700 9480 9760 10000 10300 10600 10900 11300 9070 9330 9590 9850 10100 10400 10700 8600 8820 9060 9300 9540 9790 10100 8040 8240 8440 8650 8860 9080 9300 7400 7560 7720 7890 8070 8240 8430 6620 6740 6860 6980 7110 7240 7380

12100 11600 11000 10300 9530 8610 7510

W 30×99 (3990)

TFL 2 3 4 BFL 6 7

0.00 0.17 0.34 0.50 0.67 4.07 7.83

8110 7850 7550 7200 6800 6280 5640

8360 8080 7760 7400 6970 6420 5740

8610 8320 7980 7600 7150 6560 5840

8880 8560 8210 7800 7330 6700 5940

9150 8820 8440 8010 7510 6850 6050

9420 9080 8680 8230 7700 7010 6160

9710 10000 10300 10600 10900 9340 9620 9900 10200 10500 8930 9180 9440 9700 9970 8450 8680 8910 9150 9390 7900 8100 8300 8510 8720 7170 7330 7490 7660 7840 6280 6390 6510 6640 6760



COMPOSITE BEAMS

5 - 57


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 27×102 (3620)

TFL 2 3 4 BFL 6 7

0.00 0.21 0.42 0.62 0.83 3.41 6.26

7240 6970 6660 6290 5860 5490 5070

7480 7190 6860 6470 6000 5610 5160

7730 7420 7070 6650 6150 5740 5260

7980 7650 7280 6830 6310 5870 5360

8240 7890 7490 7030 6470 6000 5470

8500 8140 7720 7220 6630 6140 5570

8780 8390 7950 7430 6800 6280 5680

9060 8660 8190 7630 6980 6430 5800

9350 8920 8430 7850 7150 6580 5910

9650 9200 8680 8070 7340 6730 6030

9950 9480 8930 8290 7520 6890 6150

W 27×94 (3270)

TFL 2 3 4 BFL 6 7

0.00 0.19 0.37 0.56 0.75 3.39 6.39

6580 6340 6070 5740 5360 5010 4590

6800 6540 6250 5910 5500 5120 4680

7020 6750 6440 6080 5640 5240 4770

7250 6970 6640 6250 5790 5360 4860

7490 7190 6840 6430 5940 5490 4960

7740 7420 7040 6610 6100 5620 5060

7990 7650 7260 6800 6260 5750 5160

8250 7890 7480 6990 6420 5890 5260

8510 8140 7700 7190 6590 6030 5370

8790 8390 7930 7400 6760 6170 5480

9070 8650 8170 7600 6940 6320 5590

W 27×84 (2850)

TFL 2 3 4 BFL 6 7

0.00 0.16 0.32 0.48 0.64 3.44 6.62

5770 5570 5340 5080 4760 4420 4020

5970 5750 5510 5220 4890 4530 4100

6170 5940 5680 5370 5020 4630 4180

6370 6130 5850 5530 5150 4750 4260

6580 6330 6030 5690 5290 4860 4340

6800 6530 6220 5860 5440 4980 4430

7030 6740 6410 6030 5580 5100 4520

7260 6960 6610 6210 5730 5220 4610

7500 7180 6810 6390 5890 5350 4710

7740 7400 7020 6570 6050 5480 4810

7990 7630 7230 6760 6210 5610 4910

W 24×76 (2100)

TFL 2 3 4 BFL 6 7

0.00 0.17 0.34 0.51 0.68 3.00 5.60

4280 4120 3940 3720 3460 3240 2970

4440 4270 4070 3840 3560 3320 3040

4610 4420 4210 3960 3670 3410 3100

4780 4580 4350 4090 3770 3490 3170

4950 4740 4500 4220 3880 3590 3240

5130 4910 4650 4350 4000 3680 3310

5320 5090 4810 4490 4110 3780 3390

5510 5260 4970 4640 4230 3880 3470

5710 5450 5140 4780 4360 3980 3550

5920 5640 5310 4930 4480 4090 3630

6130 5830 5490 5090 4610 4200 3710

W 24×68 (1830)

TFL 2 3 4 BFL 6 7

0.00 0.15 0.29 0.44 0.59 3.05 5.81

3760 3630 3470 3290 3080 2860 2600

3900 3760 3590 3400 3170 2940 2660

4050 3900 3720 3510 3270 3020 2720

4200 4040 3850 3630 3370 3100 2780

4360 4180 3980 3740 3470 3180 2840

4520 4330 4120 3870 3570 3270 2910

4690 4490 4260 3990 3680 3360 2970

4860 4650 4410 4120 3790 3450 3040

5040 4810 4560 4260 3910 3540 3110

5220 4980 4710 4390 4020 3640 3190

5410 5160 4870 4540 4140 3740 3260



5 - 58

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 24×62 (1550)

TFL 2 3 4 BFL 6 7

0.00 0.15 0.29 0.44 0.59 3.47 6.58

3300 3190 3080 2940 2780 2540 2250

3430 3320 3190 3040 2870 2620 2300

3560 3440 3300 3150 2970 2690 2350

3700 3570 3420 3260 3060 2770 2410

3840 3700 3550 3370 3160 2850 2470

3990 3840 3670 3480 3270 2940 2530

4140 3980 3810 3600 3370 3020 2590

4300 4130 3940 3730 3480 3110 2650

4460 4280 4080 3860 3600 3200 2710

4620 4440 4230 3990 3710 3290 2780

4790 4590 4370 4120 3830 3390 2850

W 24×55 (1350)

TFL 2 3 4 BFL 6 7

0.00 0.13 0.25 0.38 0.51 3.45 6.66

2890 2800 2700 2590 2460 2240 1970

3000 2910 2800 2680 2540 2310 2010

3120 3020 2900 2770 2630 2370 2060

3240 3130 3010 2870 2710 2440 2110

3370 3250 3120 2970 2800 2520 2160

3500 3370 3230 3080 2900 2590 2210

3630 3500 3350 3190 2990 2670 2260

3770 3630 3470 3300 3090 2750 2320

3910 3760 3600 3410 3200 2830 2370

4060 3900 3730 3530 3300 2920 2430

4210 4040 3860 3650 3410 3000 2490

W 21×62 (1330)

TFL 2 3 4 BFL 6 7

0.00 0.15 0.31 0.46 0.62 2.53 4.78

2760 2650 2530 2380 2210 2070 1900

2880 2760 2630 2470 2280 2130 1950

3000 2870 2730 2560 2360 2190 2000

3120 2990 2830 2650 2440 2260 2050

3250 3110 2940 2750 2520 2320 2100

3390 3230 3060 2850 2600 2390 2150

3530 3360 3170 2950 2690 2460 2210

3670 3500 3290 3060 2770 2540 2270

3820 3630 3420 3170 2870 2620 2330

3970 3780 3550 3280 2960 2690 2390

4130 3920 3680 3400 3060 2780 2450

W 21×57 (1170)

TFL 2 3 4 BFL 6 7

0.00 0.16 0.33 0.49 0.65 2.90 5.38

2480 2390 2290 2170 2030 1880 1690

2590 2490 2380 2250 2100 1940 1740

2700 2590 2480 2340 2170 2000 1780

2810 2700 2570 2420 2250 2060 1830

2930 2810 2680 2520 2330 2120 1880

3060 2930 2780 2610 2410 2190 1920

3180 3050 2890 2710 2490 2260 1980

3320 3170 3000 2810 2580 2330 2030

3450 3300 3120 2910 2670 2400 2080

3590 3430 3240 3020 2760 2480 2140

3740 3560 3360 3130 2860 2560 2200

W 21×50 (984)

TFL 2 3 4 BFL 6 7

0.00 0.13 0.27 0.40 0.54 2.92 5.58

2120 2050 1960 1870 1760 1620 1440

2210 2130 2040 1940 1830 1670 1470

2310 2220 2130 2020 1890 1720 1510

2410 2320 2220 2100 1960 1780 1550

2510 2410 2310 2180 2040 1840 1590

2620 2520 2400 2260 2110 1900 1640

2730 2620 2500 2350 2190 1960 1680

2850 2730 2590 2440 2270 2020 1730

2960 2840 2700 2530 2350 2090 1780

3090 2950 2800 2630 2430 2160 1830

3210 3070 2910 2730 2520 2230 1880



COMPOSITE BEAMS

5 - 59


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 21×44 (843)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.23 0.34 0.45 2.90 5.69

1830 1770 1710 1630 1540 1410 1240

1910 1850 1780 1700 1600 1450 1270

2000 1930 1850 1760 1660 1500 1300

2090 2010 1930 1830 1730 1550 1340

2180 2100 2010 1910 1790 1610 1380

2270 2190 2090 1980 1860 1660 1410

2370 2280 2180 2060 1930 1720 1450

2470 2370 2270 2140 2000 1770 1490

2580 2470 2360 2220 2070 1830 1540

2680 2570 2450 2310 2150 1900 1580

2800 2680 2550 2400 2230 1960 1620

W 18×60 (984)

TFL 2 3 4 BFL 6 7

0.00 0.17 0.35 0.52 0.70 2.19 3.82

2070 1980 1880 1760 1610 1520 1420

2170 2080 1960 1830 1670 1570 1460

2280 2170 2050 1900 1730 1620 1500

2390 2270 2140 1980 1790 1670 1540

2500 2380 2230 2060 1850 1730 1590

2620 2480 2330 2150 1920 1790 1640

2740 2600 2430 2230 1990 1850 1690

2860 2710 2540 2320 2070 1910 1740

3000 2830 2640 2420 2140 1970 1790

3130 2960 2750 2510 2220 2040 1840

3270 3090 2870 2610 2300 2110 1900

W 18×55 (890)

TFL 2 3 4 BFL 6 7

0.00 0.16 0.32 0.47 0.63 2.16 3.86

1880 1800 1710 1600 1470 1380 1290

1970 1890 1790 1670 1520 1430 1320

2070 1970 1870 1740 1580 1480 1360

2170 2070 1950 1810 1640 1530 1400

2270 2160 2030 1880 1700 1580 1440

2380 2260 2120 1960 1760 1630 1490

2490 2360 2220 2040 1830 1690 1530

2610 2470 2310 2120 1900 1750 1580

2730 2580 2410 2210 1970 1810 1620

2850 2700 2510 2300 2040 1870 1670

2980 2810 2620 2390 2110 1930 1730

W 18×50 (800)

TFL 2 3 4 BFL 6 7

0.00 0.14 0.29 0.43 0.57 2.07 3.82

1690 1620 1540 1440 1320 1250 1160

1770 1700 1610 1500 1370 1290 1190

1860 1770 1680 1560 1420 1330 1220

1950 1860 1750 1630 1480 1380 1260

2040 1940 1830 1700 1530 1420 1300

2140 2030 1910 1770 1590 1470 1340

2240 2130 1990 1840 1650 1520 1380

2340 2220 2080 1910 1710 1570 1420

2450 2320 2170 1990 1780 1630 1460

2560 2430 2260 2070 1840 1680 1510

2680 2530 2360 2160 1910 1740 1550

W 18×46 (712)

TFL 2 3 4 BFL 6 7

0.00 0.15 0.30 0.45 0.61 2.40 4.34

1530 1470 1400 1320 1230 1140 1040

1610 1540 1470 1380 1270 1180 1070

1690 1620 1540 1440 1320 1220 1100

1770 1690 1610 1500 1380 1270 1140

1860 1770 1680 1560 1430 1310 1170

1950 1860 1750 1630 1490 1360 1210

2040 1940 1830 1700 1550 1410 1240

2140 2030 1910 1770 1610 1460 1280

2240 2130 2000 1850 1670 1510 1320

2340 2220 2080 1920 1730 1560 1360

2450 2320 2170 2000 1800 1620 1410



5 - 60

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 18×40 (612)

TFL 2 3 4 BFL 6 7

0.00 0.13 0.26 0.39 0.53 2.26 4.27

1320 1270 1210 1140 1060 985 895

1390 1330 1270 1190 1100 1020 921

1450 1390 1320 1240 1150 1060 949

1530 1460 1390 1300 1190 1090 978

1600 1530 1450 1350 1240 1130 1010

1680 1600 1510 1410 1290 1170 1040

1760 1680 1580 1470 1340 1220 1070

1840 1760 1650 1530 1390 1260 1100

1930 1840 1730 1600 1450 1310 1140

2020 1920 1800 1670 1510 1350 1180

2110 2010 1880 1740 1560 1400 1210

W 18×35 (510)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.43 2.37 4.56

1120 1080 1030 978 917 842 753

1170 1130 1080 1020 955 874 775

1230 1190 1130 1070 995 906 799

1300 1240 1180 1120 1040 940 824

1360 1300 1240 1170 1080 976 850

1430 1370 1300 1220 1130 1010 877

1500 1430 1360 1270 1170 1050 905

1570 1500 1420 1330 1220 1090 934

1650 1570 1490 1390 1270 1130 964

1720 1640 1550 1450 1320 1170 995

1800 1720 1620 1510 1380 1220 1030

W 16×36 (448)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.32 0.43 1.79 3.44

971 931 884 830 764 714 657

1020 981 929 869 797 742 679

1080 1030 977 910 831 770 701

1140 1090 1030 954 867 801 725

1200 1140 1080 999 904 832 750

1270 1200 1130 1050 943 865 776

1330 1270 1190 1090 984 899 802

1400 1330 1250 1150 1030 935 830

1480 1400 1310 1200 1070 972 859

1550 1470 1370 1250 1120 1010 889

1630 1540 1430 1310 1160 1050 921

W 16×31 (375)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 2.00 3.79

826 793 756 713 662 613 555

872 837 796 748 691 637 574

921 882 837 784 722 663 593

972 929 880 823 755 690 614

1030 979 925 863 789 718 635

1080 1030 972 904 824 747 657

1140 1080 1020 948 861 778 680

1200 1140 1070 993 899 810 704

1260 1200 1120 1040 939 843 729

1330 1260 1180 1090 980 877 755

1390 1320 1240 1140 1020 912 782

W 16×26 (301)

TFL 2 3 4 BFL 6 7

0.00 0.09 0.17 0.26 0.35 2.04 4.01

673 649 621 589 551 505 450

712 685 654 618 577 526 465

753 723 689 650 604 549 482

795 763 726 683 633 572 499

840 804 764 717 663 597 517

886 848 804 753 694 622 535

935 893 845 790 727 649 554

985 940 888 829 760 676 575

1040 989 933 870 796 705 595

1090 1040 980 911 832 734 617

1150 1090 1030 955 870 765 639



COMPOSITE BEAMS

5 - 61


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 14×38 (385)

TFL 2 3 4 BFL 6 7

0.00 0.13 0.26 0.39 0.52 1.38 2.53

844 805 759 704 636 604 568

896 853 802 740 665 629 589

951 903 846 778 695 655 611

1010 956 893 818 727 683 634

1070 1010 943 861 760 712 659

1130 1070 994 905 794 742 684

1200 1130 1050 950 831 773 710

1270 1190 1100 998 868 806 738

1340 1260 1160 1050 908 840 766

1410 1330 1220 1100 948 876 796

1490 1400 1290 1150 991 913 827

W 14×34 (340)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.23 0.34 0.46 1.41 2.60

744 711 671 623 565 535 502

790 753 709 656 591 557 520

839 798 749 690 618 581 540

890 844 790 726 647 606 560

944 894 834 763 677 631 582

1000 945 880 803 708 659 604

1060 999 929 844 741 687 628

1120 1060 979 887 775 716 652

1180 1110 1030 931 810 747 677

1250 1170 1080 978 847 779 704

1320 1240 1140 1030 885 812 731

W 14×30 (291)

TFL 2 3 4 BFL 6 7

0.00 0.10 0.19 0.29 0.39 1.48 2.82

643 615 583 544 497 467 432

684 653 616 573 521 487 448

726 692 652 604 546 508 465

771 734 689 636 573 531 483

819 777 728 670 600 554 502

868 823 769 706 629 579 522

920 870 812 743 659 605 542

974 920 857 782 691 631 564

1030 971 903 822 724 659 586

1090 1020 951 864 758 688 610

1150 1080 1000 907 793 719 634

W 14×26 (245)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.42 1.67 3.19

553 531 504 473 437 405 368

589 563 534 500 459 423 382

626 598 565 527 482 443 397

665 634 598 556 506 463 413

706 672 633 587 532 485 430

750 712 669 619 559 507 447

795 754 707 652 587 530 465

841 798 747 687 616 555 484

890 843 788 723 646 580 503

941 890 830 761 678 606 523

994 939 875 800 711 634 545

W 14×22 (199)

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.34 1.69 3.34

454 437 416 393 365 336 301

484 464 442 416 385 352 313

515 493 468 439 405 369 325

548 524 496 464 427 386 339

582 556 526 490 449 405 352

619 590 556 518 473 424 367

656 625 588 546 497 444 382

696 661 622 576 523 466 398

736 699 657 607 550 488 414

779 739 693 640 577 510 431

823 780 731 673 606 534 449



5 - 62

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 12×30 (238)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 1.12 1.94

531 505 474 436 389 373 355

568 539 504 461 408 390 370

608 575 536 488 429 408 386

649 612 569 516 450 427 402

693 652 604 545 472 447 420

738 694 641 576 496 468 438

786 738 679 609 521 490 457

837 783 720 643 547 513 477

889 831 762 678 574 537 498

944 881 806 715 602 562 520

1000 933 852 753 631 589 543

W 12×26 (204)

TFL 2 3 4 BFL 6 7

0.00 0.10 0.19 0.29 0.38 1.08 1.95

456 434 407 375 336 321 305

488 463 433 397 353 336 317

521 494 460 420 370 351 331

557 526 489 445 389 368 345

595 561 520 470 409 386 360

635 597 552 497 430 404 376

676 635 585 526 452 423 393

720 674 621 555 474 444 410

765 716 657 586 498 465 428

812 759 695 618 523 487 447

861 804 735 652 549 509 467

W 12×22 (156)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.21 0.32 0.43 1.66 3.04

371 356 339 318 294 270 242

399 382 362 339 312 285 253

428 408 386 360 330 300 265

458 437 412 383 350 316 277

490 466 439 408 370 333 290

524 498 468 433 392 351 303

559 531 498 459 414 370 317

596 565 529 487 438 389 332

635 601 562 516 463 410 347

675 638 596 547 488 431 364

717 677 631 578 515 453 380

W 12×19 (130)

TFL 2 3 4 BFL 6 7

0.00 0.09 0.18 0.26 0.35 1.66 3.12

312 300 286 270 251 229 203

335 321 306 287 266 241 212

360 344 327 306 282 255 222

386 368 349 326 300 269 232

413 394 372 347 318 284 243

442 421 397 369 337 299 255

472 449 423 392 357 316 267

503 478 450 417 378 333 279

536 509 478 442 400 351 293

571 541 507 468 423 370 306

606 574 538 496 446 389 321

W 12×16 (103)

TFL 2 3 4 BFL 6 7

0.00 0.07 0.13 0.20 0.26 1.71 3.32

254 245 234 223 210 189 163

273 263 251 239 224 200 171

294 282 269 255 238 212 179

315 303 288 272 254 224 188

338 324 309 291 270 238 197

362 347 330 310 287 251 207

388 371 352 330 305 266 217

414 396 375 351 324 281 227

442 422 399 373 344 297 239

471 449 424 396 364 313 250

501 477 450 420 386 330 262



COMPOSITE BEAMS

5 - 63


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 12×14 (88.6)

TFL 2 3 4 BFL 6 7

0.00 0.06 0.11 0.17 0.23 1.69 3.36

220 213 204 195 184 165 141

237 229 219 209 197 175 148

255 246 235 223 210 186 155

275 264 252 239 224 197 163

295 283 270 255 238 209 171

316 303 289 272 254 221 180

338 324 308 290 270 234 188

362 346 329 309 287 247 198

386 369 350 329 305 262 208

411 393 372 349 323 276 218

438 418 396 371 342 292 228

W 10×26 (144)

TFL 2 3 4 BFL 6 7

0.00 0.11 0.22 0.33 0.44 0.90 1.51

339 322 300 274 242 232 222

368 347 323 293 256 245 233

398 375 347 313 271 258 245

430 404 372 334 287 273 258

464 435 400 357 304 288 272

499 467 428 381 321 304 286

537 501 458 406 340 321 301

577 537 490 432 360 339 317

618 575 523 460 381 358 334

662 615 558 489 403 378 351

707 656 594 519 425 398 369

W 10×22 (118)

TFL 2 3 4 BFL 6 7

0.00 0.09 0.18 0.27 0.36 0.95 1.70

281 267 250 230 205 194 183

305 289 269 246 217 205 193

330 312 290 263 231 217 203

357 336 312 282 245 230 214

386 363 335 302 260 244 225

416 390 360 322 277 258 237

448 419 385 344 293 273 250

482 450 413 367 311 288 263

517 482 441 391 330 305 277

554 516 471 417 350 322 292

592 551 502 443 370 340 308

W 10×19 (96.3)

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 1.27 2.31

239 228 215 200 183 169 153

259 247 233 216 195 180 162

282 267 251 232 209 191 170

305 289 271 249 223 203 180

330 312 292 267 238 216 190

356 337 314 286 254 229 200

384 362 337 307 271 243 211

413 389 361 328 288 258 223

444 417 387 350 307 274 235

476 447 413 374 326 290 248

509 478 441 398 347 307 261

W 10×17 (81.9)

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.33 1.31 2.46

206 197 187 175 161 148 132

224 214 203 189 173 157 139

244 232 219 204 185 168 147

265 252 237 219 199 179 155

286 272 255 236 213 190 164

310 294 275 253 227 202 173

334 316 296 272 243 215 183

360 340 317 291 260 229 193

387 365 340 311 277 243 204

415 391 364 332 295 258 215

444 419 389 354 314 274 227



5 - 64

COMPOSITE DESIGN


LB

Shaped PNAc

Y1a

ILB (in.4)

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 10×15 (68.9)

TFL 2 3 4 BFL 6 7

0.00 0.07 0.14 0.20 0.27 1.35 2.60

177 170 162 153 142 128 112

193 185 175 165 153 137 118

210 201 190 178 164 147 125

228 218 206 192 176 157 133

247 236 222 207 189 167 140

268 255 240 223 203 178 148

289 275 258 240 218 190 157

312 296 278 257 233 203 166

335 318 298 275 249 216 176

360 341 320 295 266 229 185

386 365 342 315 283 244 196

W 10×12 (53.8)

TFL 2 3 4 BFL 6 7

0.00 0.05 0.11 0.16 0.21 1.30 2.61

139 134 128 121 113 102 87.9

152 146 139 131 122 109 92.9

165 158 150 141 131 116 98.4

180 172 163 153 141 124 104

195 186 176 165 152 133 110

211 202 190 178 163 142 117

229 218 205 191 175 152 124

247 235 221 205 187 162 131

265 252 237 220 200 173 138

285 271 254 236 214 184 146

306 290 272 252 229 195 155

W 8×28 (98)

TFL 2 3 4 BFL 6 7

0.00 0.12 0.23 0.35 0.47 0.53 0.59

248 233 214 191 161 159 158

274 256 234 207 171 170 168

302 281 256 224 183 181 180

332 308 279 243 196 194 192

364 337 304 262 209 207 204

398 368 330 284 223 221 218

434 400 358 306 238 235 233

473 435 388 330 254 251 248

513 471 419 355 271 268 264

555 509 452 381 289 285 281

600 549 487 408 307 303 299

W 8×24 (82.8)

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 0.47 0.55

209 196 180 161 136 134 133

231 216 198 175 145 143 142

255 237 216 189 155 153 151

280 260 236 205 166 164 162

307 285 257 222 177 175 173

336 311 279 240 189 187 184

367 339 303 259 202 200 197

400 368 329 280 216 213 210

434 399 355 301 231 227 223

470 431 383 323 246 242 238

508 465 413 347 262 257 253

W 8×21 (75.3)

TFL 2 3 4 BFL 6 7

0.00 0.10 0.20 0.30 0.40 0.70 1.06

191 181 167 151 131 126 122

211 198 183 164 140 135 130

232 218 200 178 151 145 138

255 238 218 193 162 155 147

279 260 237 209 173 165 157

305 284 258 226 186 177 167

333 309 279 244 199 189 178

362 335 302 263 213 201 190

392 362 327 283 227 215 202

424 391 352 304 243 229 215

458 422 379 326 259 244 228



COMPOSITE BEAMS

5 - 65


Y1a

LB

ILB (in.4 )

Y2b (in.) In.

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

W 8×18 (61.9)

TFL 2 3 4 BFL 6 7

0.00 0.08 0.17 0.25 0.33 0.71 1.21

159 150 140 127 111 106 101

175 165 153 138 120 114 107

193 182 167 150 129 122 114

213 199 183 163 139 131 122

233 218 199 177 149 140 130

255 238 217 192 160 150 139

278 259 236 207 172 161 148

303 281 255 224 184 172 158

329 305 276 241 198 184 169

356 329 298 259 211 196 179

384 355 321 278 226 209 191

W 8×15 (48)

TFL 2 3 4 BFL 6 7

0.00 0.08 0.16 0.24 0.32 0.97 1.79

129 123 116 107 97.0 89.3 80.6

143 136 128 117 105 96.4 86.2

158 150 140 128 114 104 92.2

175 165 154 140 124 112 98.7

192 181 168 153 135 121 106

210 198 183 166 146 130 113

230 216 200 181 158 140 121

251 235 217 196 170 151 129

272 255 235 211 183 162 138

295 276 254 228 197 174 147

319 299 274 246 211 186 157

W 8×13 (39.6)

TFL 2 3 4 BFL 6 7

0.00 0.06 0.13 0.19 0.26 1.00 1.91

109 104 98.0 91.4 83.6 76.1 67.2

121 115 108 100 91.3 82.4 72.0

134 127 119 110 99.6 89.3 77.2

147 140 131 121 108 96.6 82.7

162 154 144 132 118 104 88.7

178 168 157 144 128 113 95.0

195 184 171 156 139 122 102

213 200 186 170 150 131 109

231 218 202 184 162 141 116

251 236 219 198 175 151 124

272 255 236 214 188 162 132

W 8×10 (30.8)

TFL 2 3 4 BFL 6 7

0.00 0.05 0.10 0.15 0.21 0.88 1.77

83.1 79.3 74.8 69.6 63.5 58.0 51.7

92.3 102 113 87.8 97.0 107 82.6 90.9 99.9 76.5 83.8 91.7 69.2 75.4 82.0 62.8 68.0 73.5 55.4 59.4 63.6

124 117 109 100 89.2 79.5 68.2

136 129 120 109 96.7 85.8 73.0

149 141 131 119 105 92.5 78.2

163 153 142 129 113 99.6 83.6

177 166 154 140 122 107 89.4

192 180 167 151 132 115 95.4

208 195 180 162 142 123 102



5 - 66

COMPOSITE DESIGN


COMPOSITE COLUMNS

5 - 67

COMPOSITE COLUMNS General Notes

Load tables for composite columns are presented for a variety of W shapes, pipe, and structural tubing. Tabular loads have been computed in accordance with Section I2.2 of the LRFD Specification for axially loaded members having effective unsupported lengths indicated at the left of each table. The effective length KL is the actual unbraced length, in feet, multiplied by factor K, which depends on the rotational restraint at the ends of the unbraced length and the means available to resist lateral movements. The K factor may be selected using as a guide Table C-C2.1 in the Commentary of the LRFD Specification. Interpolation between the idealized cases is a matter of engineering judgment. More precise values for K may be obtained, if desired, from the alignment chart in Figure C-C2.2 in the LRFD Commentary (also shown in Part 3 of this Manual under Design Strength of Columns) once sections have been selected for several framing members. Load tables are provided for W shapes encased in reinforced normal-weight concrete of square or rectangular cross section, and for steel pipe and structural tubing filled with normal-weight concrete. The following W shapes are included: Nominal depth, in. 14 12 10 8

Weight, lb per ft 426-61 336-58 112-45 67-35

Two values of yield stress, Fy equal to 36 and 50 ksi, and three values of concrete cylinder strength, fc′ equal to 3.5, 5, and 8 ksi, are included for W shapes. All reinforcing steel is Grade 60. The tables for steel pipe columns include nominal pipe diameters of 4 to 12 in., yield stress Fy = 36 ksi, and concrete cylinder strength fc′ equal to 3.5 and 5 ksi. The tables for tubular columns include tubes of nominal side dimensions 4 to 16 in., yield stress Fy equal to 46 ksi, and concrete cylinder strength fc′ equal to 3.5 and 5 ksi. All axial design strengths are tabulated in kips. Strength values are omitted when Kl / r exceeds 200. Resistance factor φ = 0.85 was used in computing the axial design strengths of all composite columns. In all tables, the design strengths are given for effective lengths with respect to the minor axis. When the minor axis is braced at closer intervals than the major axis, the strength of the column must be investigated with reference to both the major (X-X) and minor (Y-Y) axes. The ratio rmx / rmy included in the tables provides a convenient method for investigating the strength of a column with respect to its major axis. Properties useful to the designer are listed at the bottom of the Column Load Tables. They are helpful in considering buckling about the major axis as discussed above and in the design of members under combined axial compression and bending as discussed subsequently. Both of these cases are illustrated with design examples. The properties have the following units: modified radius of gyration rm, in. nominal flexural strength Mn, kip-ft Euler buckling term Pe (KL)2, kip-ft2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5 - 68

COMPOSITE DESIGN

Subscripts x and y refer to the major and minor axes. Resistance factor φb = 0.9 was used in computing the flexural design strength φbMn. Additional notes relating specifically to the W shapes, steel pipe, and structural tube column tables precede each of these groups of tables.

EXAMPLE 5-4

Given:

Using tables on the pages that follow, design the smallest composite column with a W shape of Fy = 50 ksi to support a factored concentric load of 1,000 kips. The effective length with respect to its minor axis is 16 feet and that with respect to its major axis is 31 feet.

Solution:

Use composite column tables for W shapes of fc′ = 8 ksi since the strongest concrete requires the smallest size column. An inspection of the tables reveals that the tabulated values of rmx / rmy do not exceed 1.22 and in most cases are equal to 1.0 (square columns). Assuming that rmx / rmy is equal to 1.0, enter the tables with equivalent effective column length of KL = 31 ft. Select an 18 in. × 18 in. column with a W10×49 and rmx / rmy = 1.0. By interpolation, the column has an axial design strength φPn of 1,029 kips. Use: 18 in. × 18 in. column with W10×49 of Fy = 50 ksi, fc′ = 8 ksi, four #8 Gr. 60 longitudinal bars and #3 Gr. 60 ties spaced 12 inches on center.

EXAMPLE 5-5

Given:

Redesign the column from Example 5-4 using (a) rectangular and (b) square structural tubing, both filled with structural concrete.

Solution:

a. Enter the Composite Column Tables for rectangular structural tubing filled with 5 ksi concrete at an effective column length of KL = 16 ft. Select 14×10 tubing with 1⁄2-in. thick walls; φPn = 1,090 kips. rmx / rmy = 1.30 Equivalent effective length for X-X axis: 31 / 1.30 = 23.8 ft Since 23.8 ft > 16 ft, X-X axis controls. Re-enter the table at an effective length of KL = 23.8 ft. It is apparent that 14×10 tube will not satisfy the axial load of 1,000 kips. Select 16×12 steel tube with 1⁄2-in. walls: rmx / rmy = 1.25 KL = 31 / 1.25 = 24.8 ft AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE COLUMNS

5 - 69

By interpolation, the column provides an axial design strength of 1,206 kips. The same tubing filled with 3.5 ksi concrete is good for 1,094 kips > 1,000 kips. Use: 16-in. × 12-in. × 1⁄2-in. tubing filled with 3.5-ksi concrete. b. Enter the Composite Column Tables for square structural tubing filled with 5 ksi concrete at effective column length KL = 31 ft. Select 14×14 tubing with 1⁄2-in. thick walls good for 1,135 kips. The same tubing filled with 3.5 ksi concrete is good for 1,035 kips > 1,000 kips. Use: 14-in. × 14-in. × 1⁄2-in. tubing filled with 3.5-ksi concrete. Note: The weight of both the 16×12 and 14×14 steel tubing with 1⁄ -in. thick walls is 89.68 lb. per ft. 2 Combined Axial Compression and Bending (Interaction)

Loads given in the Composite Column Tables are for concentrically loaded columns. For columns subjected to combined compression and flexure, the nominal flexural design strength φbMn determined from Equation C-I4-1 of the Commentary on the LRFD Specification (only valid for Pu / φcPn > 0.3) and the elastic buckling load Pe times the square of the effective column length are given at the bottom of the tables. With these quantities and the loads tabulated for concentric loading, the column may be designed by successive approximations based on LRFD Specification Section I4. The procedure is illustrated in Example 5-6 for a column subjected to a large bending moment combined with a moderate axial load and in Example 5-7 for a column subjected to a large axial load combined with a moderate bending moment. EXAMPLE 5-6

Given:

Design a composite encased W shape column to resist a factored axial load of 350 kips and a factored moment about the X-X axis of 240 kip-ft. The unsupported length of the column is 12 feet, Fy = 50 ksi, fc′ = 3.5 ksi and Cm = 1.0. The loads were obtained by first order elastic analysis and there is no lateral translation of column ends.

Solution:

Since the moment is large in relation to the axial load, assume that Pu / φPn = 1⁄2 and B1 = 1.0. From Equation H1-1a φbMnx = (8 / 9) × 240 × 2 = 427 kip-ft From the Composite Column Tables, find a column with φbMnx close to 427 kip-ft. Try a W10×60 with 18-in. encasement: φbMnx = 439 kip-ft, φPn = 1,300 kips and Pex = 142 × 104 / 122 = 9,861 kips. Therefore, Pu / φPn = 350 / 1,300 = 0.2692 ≥ 0.2 From Equation C1-1 and C1-2, with Mlt = 0 since there is no lateral translation, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5 - 70

COMPOSITE DESIGN

Mux =

240 = 248.8 kip-ft 350 1− 9,861

and from Equation H1-1a φbMnx =

8(248.8) 9 1 −



350  1,300 

= 302.6 kip-ft

Since 302.6 kip-ft is much less than the 439 kip-ft provided, select a smaller size. Tr y a W8×58 with 16-in. × 16-in. encasement: φbMnx = 345 kip-ft, φPn = 1,130 kips and Pex = 103 × 104 / 122 = 7,153 kips. Mux =

240 = 252.4 kip-ft 350 1−

7,153

From Equation H1-1a, with φbMnx = 345 kip-ft since φPn / Pu > 0.3,

8(252.3) 350 + = 0.960 < 1.0 o.k. 1,130 9(345) Use: 16-in. × 16-in. column with W8×58 of Fy = 50 ksi, fc′ = 3.5 ksi, four #7 Gr. 60 longitudinal bars and #3 Gr. 60 ties spaced at 10 inches.

EXAMPLE 5-7

Given:

Design a composite encased W shape column to resist a factored axial load of 1,100 kips and factored moment of 200 kip-ft. Use 50 ksi structural steel and 5 ksi concrete. The unsupported column length is 11 feet and Cm = 0.85. Assume that sidesway is prevented.

Solution:

Since the axial load is large in relation to the moment, assume that: 8 Mu = 0.5 9 φbMnx From Equation H1-1a φPn = 1,100 / 0.5 = 2,200 kips From the Composite Column Tables, find a column with φPn close to 2,200 kips at KL = 11 ft. Try a W12×106 with 20-in. encasement: φPn = 2,270 kips, φbMnx = 899 kip-ft and Pex = 294 × 104 / 112 = 24,300 kips. From Equation C1-2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COMPOSITE COLUMNS

B1 =

5 - 71

0.85 = 0.890 < 1.0 1,100 1− 24,300

Therefore, B1 = 1.0. From Equation H1-1a Pu 8 200 =1− × = 0.802 9 899 φPn and φPn = 1,100 / 0.802 = 1,372 kips Try a W10×45 with 18-in. × 18-in. encasement: φPn = 1,360 kips, φbMnx = 356 kip-ft and Pex = 125 × 104 / 112 = 10,330 kips. From Equation C1-2 B1 =

0.85

1−

1,100 10,300

= 0.951 < 1.0

Therefore, B1 = 1.0 and Pu 8 200 =1− × = 0.501 9 356 φPn and φPn = 1,100 / 0.501 = 2,196 kips Since the convergence is slow, estimate a new φPn as: φPn =

2,196 + 1,372 = 1,784 kips 2

Try a W10×77 with 18-in. × 18-in. encasement: φPn = 1,720 kips, φbMnx = 555 kip-ft and Pex = 178 × 104 / 112 = 14,710 kips. B1 =

0.85

1−

1,100 14,710

= 0.919 < 1.0

and 1,100 8 200 + × = 0.960 < 1.0 o.k. 1,720 9 555 Use: 18-in. × 18-in. column with W10×77 of Fy = 50 ksi, fc′ = 5 ksi, four #8 Gr. 60 longitudinal bars and #3 Gr. 60 ties spaced at 12 inches. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

5 - 72

COMPOSITE DESIGN


COMPOSITE COLUMNS—W SHAPES ENCASED IN CONCRETE

5 - 73

COMPOSITE COLUMNS—W SHAPES ENCASED IN CONCRETE General Notes

Concentric load design strengths in the tables that follow are tabulated for the effective length KL in feet, listed at the left of each table. They are applicable to axially loaded members with respect to their minor axis in accordance with Section I2.2 of the LRFD Specification. Two steel yield stresses, Fy = 36 ksi and Fy = 50 ksi, and three concrete strengths, fc′ = 3.5 ksi, fc′ = 5 ksi, and fc′ = 8 ksi, are covered. The tables apply to normal-weight concrete. All reinforcing steel is Gr. 60; however, Fyr = 55 ksi is used in the calculation of φPn in accordance with LRFD Specification Section I2.1. Each W shape is embedded in concrete of a square or rectangular cross section reinforced with four longitudinal reinforcing bars placed in the four corners and with lateral ties spaced as required in Section I2.1. For the design of additional confinement reinforcement, see LRFD Specification Section I2.1. The size of the concrete section was selected so as to provide at least the minimum required cover over the reinforcing bars in the column. For discussion of the effective length, range of Kl / r, strength about the major axis, combined axial and bending strength, and for sample problems, see Composite Columns, General Notes. The properties listed at the bottom of each table are for use in checking strength about the strong axis and in design for combined axial load and bending. Additional information on W shapes encased in concrete, including numerous tables for columns and beam-columns, is available in the AISC Steel Design Guide No. 6, Load and Resistance Factor Design of W-Shapes Encased in Concrete (Griffis, 1992).


5 - 74

COMPOSITE DESIGN


b Y X

h

COMPOSITE COLUMNS W Shapes fc′ = 3.5 ksi All reinforcing steel is Grade 60 Axial design strength in kips

X

Y 24 in.×26 in. 4-#11 bars

Ties

#4 bars spaced 16 in. c. to c.

Designation

W 14

Wt./ft

426

Fy

Effective length KL (ft) with respect to least radius of gyration rmy

Steel Shape

Size b × h Reinf. bars

398

370

342

311

36

50

36

50

36

50

36

50

36

50

0

4910

6400

4680

6070

4450

5740

4220

5420

3940

5030

6 7 8 9 10

4880 4870 4850 4840 4820

6340 6330 6300 6280 6250

4650 4640 4630 4610 4600

6020 6000 5980 5960 5930

4420 4410 4400 4380 4370

5700 5680 5660 5640 5620

4190 4180 4170 4160 4140

5370 5360 5340 5320 5300

3910 3910 3890 3880 3870

4990 4970 4950 4930 4910

11 12 13 14 15

4810 4790 4770 4750 4720

6220 6190 6160 6120 6080

4580 4560 4540 4520 4500

5910 5880 5840 5810 5770

4350 4340 4320 4300 4270

5590 5560 5530 5490 5460

4130 4110 4090 4070 4050

5270 5240 5210 5180 5150

3850 3840 3820 3800 3780

4890 4860 4830 4800 4770

16 17 18 19 20

4700 4670 4640 4610 4580

6040 5990 5950 5900 5850

4470 4450 4420 4390 4360

5730 5690 5640 5590 5540

4250 4230 4200 4170 4140

5420 5380 5340 5290 5240

4030 4000 3980 3950 3930

5110 5070 5030 4990 4940

3760 3740 3710 3690 3660

4740 4700 4660 4620 4580

22 24 26 28 30

4520 4440 4370 4290 4200

5740 5620 5490 5360 5220

4300 4230 4160 4080 4000

5440 5330 5210 5080 4950

4080 4020 3950 3870 3790

5140 5040 4920 4800 4680

3870 3800 3740 3660 3590

4850 4750 4640 4520 4410

3610 3550 3480 3420 3340

4490 4400 4300 4190 4080

32 34 36 38 40

4110 4020 3930 3830 3720

5080 4930 4780 4620 4460

3910 3820 3730 3640 3540

4810 4670 4530 4380 4230

3710 3630 3540 3450 3350

4550 4410 4270 4130 3990

3510 3430 3340 3260 3170

4280 4150 4020 3890 3750

3270 3190 3110 3030 2940

3960 3840 3720 3590 3460

2600 1560 1460 1240

3380 1960 1460 1240

2420 1460 1360 1160

3150 1840 1360 1160

2210 1360 1250 1060

2890 1720 1250 1060

Properties φ b M nx (kip-ft) φ b M ny (kip-ft) P ex (K xL x )2 / 10 4 (kip-ft2) P ey (K yL y )2 / 10 4 (kip-ft2)

rmy (in.) rmx / rmy (in./in.)

2970 1750 1650 1400

3850 2170 1650 1400

7.20 1.08

2780 1660 1550 1320

3610 2070 1550 1320

7.20 1.08

7.20 1.08


7.20 1.08

7.20 1.08


5 - 75



b Y X

h

X

Y 24 in.×24 in. 4-#11 bars

Ties


Designation

W 14

Wt./ft

283

Fy


Steel Shape


257

233

211

193

36

50

36

50

36

50

36

50

36

50

0

3620

4610

3400

4300

3200

4010

3010

3750

2860

3530

6 7 8 9 10

3600 3590 3580 3570 3560

4570 4560 4540 4530 4510

3380 3370 3360 3350 3340

4260 4250 4230 4220 4200

3170 3170 3160 3150 3130

3980 3960 3950 3930 3910

2990 2980 2970 2960 2950

3710 3700 3690 3670 3660

2840 2830 2820 2810 2800

3500 3490 3480 3460 3450

11 12 13 14 15

3540 3530 3510 3490 3470

4480 4460 4430 4410 4380

3320 3310 3290 3280 3260

4180 4150 4130 4100 4080

3120 3110 3090 3080 3060

3900 3870 3850 3830 3800

2940 2920 2910 2890 2880

3640 3620 3590 3570 3550

2790 2780 2760 2750 2730

3430 3410 3390 3370 3340

16 17 18 19 20

3460 3430 3410 3390 3360

4340 4310 4280 4240 4200

3240 3220 3200 3180 3150

4050 4010 3980 3950 3910

3040 3020 3000 2980 2960

3770 3740 3710 3680 3640

2860 2840 2820 2800 2780

3520 3490 3460 3430 3400

2720 2700 2680 2660 2640

3320 3290 3260 3230 3200

22 24 26 28 30

3310 3260 3200 3140 3070

4120 4030 3940 3840 3740

3100 3050 2990 2930 2870

3830 3750 3660 3570 3470

2910 2860 2810 2750 2690

3570 3490 3410 3320 3230

2740 2690 2630 2580 2520

3330 3250 3180 3090 3010

2590 2550 2500 2440 2390

3140 3060 2990 2910 2830

32 34 36 38 40

3000 2930 2850 2780 2700

3630 3520 3410 3290 3170

2810 2740 2670 2590 2520

3370 3270 3160 3050 2940

2630 2560 2490 2420 2350

3140 3040 2940 2830 2730

2460 2400 2330 2270 2200

2920 2820 2730 2630 2530

2330 2270 2210 2140 2080

2740 2660 2560 2470 2380

1650 1070 822 822

2160 1360 822 822

1510 991 757 757

1980 1260 757 757

1400 924 704 704

1830 1170 704 704

Properties φ b M nx (kip-ft) φ b M ny (kip-ft) P ex (K x L x)2 / 10 4 (kip-ft2) P ey (K y L y)2 / 10 4 (kip-ft2)


1970 1250 971 971

2580 1570 971 971

7.20 1.00

1810 1150 894 894

2360 1460 894 894

7.20 1.00

7.20 1.00


7.20 1.00

7.20 1.00

5 - 76

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

24 in.×24 in.

Reinf. bars

4-#10 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

176


Fy

159

145

132

36

50

36

50

36

50

36

50

0

2680

3290

2530

3090

2420

2920

2300

2770

6 7 8 9 10

2660 2650 2640 2630 2620

3270 3250 3240 3230 3210

2510 2510 2500 2490 2480

3060 3050 3040 3020 3010

2400 2390 2380 2370 2370

2900 2890 2880 2860 2850

2290 2280 2270 2260 2250

2740 2730 2720 2710 2690

11 12 13 14 15

2610 2600 2590 2570 2560

3200 3180 3160 3140 3120

2470 2460 2440 2430 2420

2990 2980 2960 2940 2920

2350 2340 2330 2320 2300

2830 2820 2800 2780 2760

2240 2230 2220 2210 2190

2680 2660 2650 2630 2610

16 17 18 19 20

2540 2530 2510 2490 2470

3090 3070 3040 3010 2980

2400 2380 2370 2350 2330

2890 2870 2840 2820 2790

2290 2270 2260 2240 2220

2740 2720 2690 2670 2640

2180 2160 2150 2130 2110

2590 2560 2540 2520 2490

22 24 26 28 30

2430 2380 2340 2290 2230

2920 2860 2790 2710 2630

2290 2250 2200 2150 2100

2730 2670 2600 2530 2460

2180 2140 2090 2050 2000

2580 2520 2460 2390 2320

2070 2030 1990 1940 1890

2440 2380 2320 2250 2190

32 34 36 38 40

2180 2120 2060 2000 1940

2550 2470 2390 2300 2210

2050 1990 1940 1880 1820

2380 2310 2230 2140 2060

1940 1890 1840 1780 1720

2250 2170 2100 2020 1940

1840 1790 1740 1680 1630

2120 2050 1970 1900 1820

1520 987 603 603

1060 719 563 563

1400 921 563 563

983 656 523 523

1290 835 523 523



1260 837 654 654

1660 1070 654 654 7.20 1.00

1150 771 603 603 7.20 1.00

7.20 1.00


7.20 1.00


5 - 77


b Y


h

X

X

Y Size b × h

22 in.×22 in.

Reinf. bars

4-#10 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

120


Fy

109

99

90

36

50

36

50

36

50

36

50

0

2040

2460

1940

2320

1860

2210

1780

2100

6 7 8 9 10

2020 2010 2010 2000 1990

2430 2420 2410 2400 2380

1920 1920 1910 1900 1890

2300 2290 2280 2270 2250

1840 1840 1830 1820 1810

2180 2170 2160 2150 2140

1770 1760 1750 1750 1740

2080 2070 2060 2050 2030

11 12 13 14 15

1980 1960 1950 1940 1920

2370 2350 2330 2310 2290

1880 1870 1860 1850 1830

2240 2220 2200 2190 2170

1800 1790 1780 1760 1750

2120 2110 2090 2070 2050

1730 1710 1700 1690 1680

2020 2000 1990 1970 1950

16 17 18 19 20

1910 1890 1880 1860 1840

2270 2250 2220 2200 2170

1820 1800 1780 1770 1750

2140 2120 2100 2070 2050

1740 1720 1700 1690 1670

2030 2010 1990 1970 1940

1660 1650 1630 1610 1600

1930 1910 1890 1870 1840

22 24 26 28 30

1800 1760 1710 1670 1620

2120 2060 1990 1930 1860

1710 1670 1630 1580 1530

2000 1940 1880 1820 1750

1630 1590 1550 1510 1460

1890 1830 1780 1720 1650

1560 1520 1480 1440 1390

1790 1740 1680 1630 1570

32 34 36 38 40

1570 1520 1460 1410 1350

1790 1720 1650 1570 1500

1480 1430 1380 1330 1280

1680 1620 1550 1480 1410

1410 1360 1310 1260 1210

1590 1520 1460 1390 1320

1340 1300 1250 1200 1150

1500 1440 1380 1310 1250

1050 677 364 364

740 498 340 340

966 631 340 340

684 465 318 318

892 587 318 318



871 574 392 392

1140 729 392 392 6.60 1.00

801 533 364 364 6.60 1.00

6.60 1.00


6.60 1.00

5 - 78

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

18 in.×22 in.

Reinf. bars

4-#9 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

82


Fy

74

68

61

36

50

36

50

36

50

36

50

0

1530

1810

1460

1720

1410

1640

1350

1560

6 7 8 9 10

1500 1500 1490 1480 1470

1780 1770 1760 1740 1730

1440 1430 1420 1410 1400

1690 1680 1670 1650 1640

1390 1380 1370 1360 1350

1620 1610 1590 1580 1570

1330 1320 1310 1300 1290

1530 1520 1510 1500 1480

11 12 13 14 15

1450 1440 1430 1410 1390

1710 1690 1670 1650 1630

1390 1370 1360 1350 1330

1620 1600 1580 1560 1540

1340 1320 1310 1300 1280

1550 1530 1510 1490 1470

1280 1260 1250 1240 1220

1470 1450 1430 1410 1390

16 17 18 19 20

1380 1360 1340 1320 1300

1600 1580 1550 1530 1500

1310 1300 1280 1260 1240

1520 1490 1470 1440 1420

1260 1250 1230 1210 1190

1450 1430 1400 1380 1350

1200 1190 1170 1150 1130

1370 1350 1320 1300 1270

22 24 26 28 30

1260 1210 1160 1110 1060

1440 1380 1310 1250 1180

1200 1150 1100 1060 1010

1360 1300 1240 1170 1110

1150 1100 1060 1010 963

1300 1240 1180 1120 1060

1090 1050 1000 957 910

1220 1170 1110 1050 991

32 34 36 38 40

1010 960 908 855 804

1110 1050 978 911 846

957 906 856 805 755

1050 981 917 853 791

914 864 815 765 717

993 930 868 807 747

862 814 766 718 671

931 871 812 753 696

744 370 261 175

535 279 246 164

693 348 246 164

490 259 228 153

633 323 228 153



623 319 280 188

808 398 280 188 5.40 1.22

574 296 261 175 5.40 1.22

5.40 1.22


5.40 1.22


5 - 79



b Y X

h

X

Y 22 in.×24 in. 4-#10 bars

Ties


Designation

W 12

Wt./ft

336

Fy


Steel Shape


305

279

252

230

36

50

36

50

36

50

36

50

36

50

0

3950

5120

3680

4750

3460

4430

3230

4120

3050

3860

6 7 8 9 10

3920 3910 3890 3880 3870

5070 5060 5030 5010 4990

3650 3640 3630 3620 3600

4700 4680 4670 4640 4620

3430 3420 3410 3400 3390

4390 4370 4360 4340 4310

3210 3200 3190 3180 3160

4080 4060 4040 4020 4000

3030 3020 3010 3000 2980

3820 3800 3790 3770 3750

11 12 13 14 15

3850 3830 3810 3790 3770

4960 4930 4890 4860 4820

3590 3570 3550 3530 3510

4590 4570 4530 4500 4470

3370 3350 3340 3320 3300

4290 4260 4230 4200 4170

3150 3130 3120 3100 3080

3980 3950 3930 3900 3870

2970 2950 2940 2920 2900

3730 3700 3680 3650 3620

16 17 18 19 20

3740 3720 3690 3660 3630

4780 4740 4690 4650 4600

3490 3460 3440 3410 3380

4430 4390 4350 4300 4260

3270 3250 3230 3200 3180

4130 4100 4060 4020 3970

3060 3040 3010 2990 2960

3830 3800 3760 3720 3680

2880 2860 2840 2810 2790

3590 3550 3520 3480 3450

22 24 26 28 30

3570 3500 3430 3350 3270

4500 4390 4270 4150 4020

3320 3260 3190 3120 3040

4160 4060 3950 3840 3720

3120 3060 2990 2920 2850

3880 3780 3680 3570 3460

2910 2850 2790 2730 2660

3600 3510 3410 3310 3200

2740 2680 2620 2560 2500

3360 3280 3190 3090 2990

32 34 36 38 40

3190 3110 3020 2930 2830

3890 3750 3610 3470 3330

2970 2880 2800 2710 2630

3590 3470 3340 3210 3070

2780 2700 2620 2540 2450

3350 3230 3110 2980 2860

2590 2510 2440 2360 2280

3100 2980 2870 2750 2640

2430 2360 2290 2210 2140

2890 2780 2680 2570 2460

1770 1020 946 795

2290 1270 946 795

1610 942 868 729

2090 1180 868 729

1480 879 803 675

1930 1100 803 675



2110 1180 1120 938

2730 1450 1120 938

6.60 1.09

1920 1090 1020 860

2490 1350 1020 860

6.60 1.09

6.60 1.09


6.60 1.09

6.60 1.09

5 - 80

COMPOSITE DESIGN


b Y X

h


X

Y 20 in.×22 in. 4-#10 bars

Ties


Designation

W 12

Wt./ft

210

Fy


Steel Shape


190

170

152

136

36

50

36

50

36

50

36

50

36

50

0

2720

3460

2550

3210

2380

2980

2230

2760

2090

2570

6 7 8 9 10

2700 2690 2680 2670 2650

3420 3400 3380 3360 3340

2530 2520 2510 2490 2480

3170 3160 3140 3130 3110

2360 2350 2340 2330 2320

2940 2930 2910 2900 2880

2210 2200 2190 2180 2170

2730 2710 2700 2680 2670

2070 2060 2050 2040 2030

2530 2520 2510 2490 2480

11 12 13 14 15

2640 2620 2600 2580 2560

3320 3290 3270 3240 3200

2470 2450 2440 2420 2400

3080 3060 3030 3000 2970

2300 2290 2270 2260 2240

2850 2830 2810 2780 2750

2150 2140 2120 2110 2090

2650 2620 2600 2580 2550

2020 2010 1990 1970 1960

2460 2440 2410 2390 2370

16 17 18 19 20

2540 2520 2500 2470 2450

3170 3140 3100 3060 3020

2380 2360 2330 2310 2290

2940 2910 2880 2840 2800

2220 2200 2180 2150 2130

2720 2690 2660 2630 2590

2070 2050 2030 2010 1990

2520 2490 2460 2430 2400

1940 1920 1900 1880 1860

2340 2310 2280 2250 2220

22 24 26 28 30

2390 2330 2270 2210 2140

2930 2840 2750 2650 2550

2230 2180 2120 2060 1990

2720 2640 2550 2450 2360

2080 2030 1970 1910 1850

2510 2430 2350 2260 2170

1940 1890 1840 1780 1720

2330 2250 2170 2090 2010

1810 1760 1710 1660 1600

2150 2080 2010 1930 1850

32 34 36 38 40

2070 2000 1930 1850 1780

2440 2340 2230 2120 2010

1930 1860 1790 1720 1650

2260 2160 2060 1960 1850

1790 1720 1660 1590 1520

2080 1990 1890 1800 1700

1660 1600 1540 1470 1410

1920 1830 1740 1650 1560

1550 1490 1430 1370 1300

1770 1690 1600 1520 1440

1080 646 508 420

1410 813 508 420

983 594 463 383

1280 748 463 383

892 547 423 349

1160 689 423 349



1310 759 608 502

1690 946 608 502

6.00 1.10

1190 704 557 460

1550 883 557 460

6.00 1.10

6.00 1.10


6.00 1.10

6.00 1.10


5 - 81



b Y X

h

X

Y 20 in.×20 in. 4-#9 bars

Ties


Designation

W 12

Wt./ft

120

Fy


Steel Shape


106

96

87

79

36

50

36

50

36

50

36

50

36

50

0

1850

2270

1740

2110

1650

1990

1580

1880

1510

1780

6 7 8 9 10

1840 1830 1820 1810 1800

2250 2240 2220 2210 2190

1720 1710 1700 1690 1680

2080 2070 2060 2050 2030

1630 1630 1620 1610 1600

1960 1950 1940 1930 1910

1560 1550 1540 1540 1530

1850 1850 1830 1820 1810

1490 1480 1470 1470 1460

1760 1750 1740 1730 1710

11 12 13 14 15

1790 1780 1760 1750 1740

2180 2160 2140 2120 2100

1670 1660 1650 1640 1620

2020 2000 1980 1960 1940

1590 1580 1570 1550 1540

1900 1880 1860 1850 1830

1520 1500 1490 1480 1470

1800 1780 1760 1740 1730

1450 1440 1420 1410 1400

1700 1690 1670 1650 1630

16 17 18 19 20

1720 1700 1690 1670 1650

2070 2050 2020 2000 1970

1610 1590 1570 1560 1540

1920 1900 1870 1850 1820

1520 1510 1490 1470 1460

1800 1780 1760 1730 1710

1450 1440 1420 1400 1390

1710 1680 1660 1640 1610

1380 1370 1350 1340 1320

1610 1590 1570 1550 1530

22 24 26 28 30

1610 1570 1520 1470 1420

1910 1850 1780 1710 1640

1500 1460 1410 1370 1320

1760 1700 1640 1580 1510

1420 1380 1340 1290 1250

1660 1600 1540 1480 1420

1350 1310 1270 1230 1180

1560 1510 1450 1390 1330

1280 1250 1210 1160 1120

1480 1420 1370 1310 1260

32 34 36 38 40

1370 1320 1270 1210 1160

1570 1500 1430 1350 1280

1270 1220 1170 1120 1070

1450 1380 1310 1240 1170

1200 1150 1100 1050 1000

1350 1290 1220 1160 1090

1140 1090 1040 993 945

1270 1210 1150 1080 1020

1080 1030 984 937 891

1200 1140 1080 1020 957

613 398 261 261

803 505 261 261

566 372 243 243

739 471 243 243

521 347 226 226

680 438 226 226



746 474 311 311 6.00 1.00

977 599 311 311

669 429 282 282 6.00 1.00

878 545 282 282

6.00 1.00


6.00 1.00

6.00 1.00

5 - 82

COMPOSITE DESIGN


b Y

h

X


X

Y Size b × h

20 in.×20 in.

Reinf. bars

4-#9 bars

Ties


Steel Shape

Designation

W 12

Wt./ft

72


Fy

65

58

36

50

36

50

36

50

0

1450

1700

1390

1620

1330

1530

6 7 8 9 10

1430 1420 1410 1410 1400

1670 1660 1650 1640 1630

1370 1360 1360 1350 1340

1590 1580 1570 1560 1550

1310 1300 1300 1290 1280

1510 1500 1490 1480 1470

11 12 13 14 15

1390 1380 1370 1350 1340

1620 1600 1590 1570 1550

1330 1320 1310 1300 1280

1540 1520 1510 1490 1480

1270 1260 1250 1240 1220

1460 1440 1430 1410 1390

16 17 18 19 20

1330 1310 1300 1280 1260

1530 1510 1490 1470 1450

1270 1260 1240 1220 1210

1460 1440 1420 1400 1370

1210 1200 1180 1170 1150

1380 1360 1340 1320 1300

22 24 26 28 30

1230 1190 1150 1110 1070

1400 1350 1300 1240 1190

1170 1140 1100 1060 1020

1330 1280 1230 1180 1120

1120 1080 1040 1000 962

1250 1210 1160 1110 1060

32 34 36 38 40

1020 979 934 889 843

1130 1070 1020 958 901

973 930 886 842 797

1070 1010 958 902 847

920 878 835 792 749

1000 949 896 843 790

445 302 197 197

577 380 197 197

410 264 183 183

530 328 183 183



483 325 211 211

629 409 211 211 6.00 1.00

6.00 1.00


6.00 1.00


5 - 83



b Y X

h

X

Y 18 in.×18 in. 4-#8 bars

Ties


Designation

W 10

Wt./ft

112

Fy


Steel Shape


100

88

77

68

36

50

36

50

36

50

36

50

36

50

0

1620

2020

1520

1870

1420

1730

1330

1600

1250

1490

6 7 8 9 10

1600 1600 1590 1580 1570

1980 1970 1960 1950 1930

1500 1500 1490 1480 1470

1840 1830 1820 1810 1790

1400 1400 1390 1380 1370

1700 1690 1680 1670 1650

1310 1300 1300 1290 1280

1570 1560 1550 1540 1520

1230 1230 1220 1210 1200

1470 1460 1450 1430 1420

11 12 13 14 15

1560 1540 1530 1520 1500

1910 1890 1870 1850 1830

1460 1450 1430 1420 1400

1780 1760 1740 1720 1700

1360 1350 1340 1320 1310

1640 1620 1600 1580 1560

1270 1260 1240 1230 1220

1510 1490 1480 1460 1440

1190 1180 1170 1160 1150

1410 1390 1380 1360 1340

16 17 18 19 20

1480 1470 1450 1430 1410

1800 1780 1750 1720 1690

1390 1370 1360 1340 1320

1670 1650 1620 1600 1570

1290 1280 1260 1240 1230

1540 1520 1490 1470 1440

1200 1190 1170 1160 1140

1420 1400 1370 1350 1330

1130 1120 1100 1080 1070

1320 1300 1280 1260 1230

22 24 26 28 30

1370 1330 1280 1230 1180

1630 1570 1500 1430 1360

1280 1240 1190 1150 1100

1510 1450 1390 1320 1260

1190 1150 1110 1060 1020

1390 1330 1270 1210 1150

1100 1060 1020 981 939

1280 1220 1170 1110 1050

1030 996 957 917 876

1190 1130 1080 1030 973

32 34 36 38 40

1130 1080 1030 979 927

1290 1220 1150 1080 1000

1050 1000 955 905 856

1190 1120 1060 988 922

972 926 879 832 785

1090 1030 963 901 839

895 850 806 761 717

994 935 876 818 761

833 791 748 705 663

918 862 807 752 699

467 305 189 189

612 387 189 189

415 276 170 170

543 350 170 170

373 251 155 155

488 318 155 155



576 366 228 228 5.40 1.00

756 464 228 228

522 336 208 208 5.40 1.00

685 426 208 208

5.40 1.00


5.40 1.00

5.40 1.00

5 - 84

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

18 in.×18 in.

Reinf. bars

4-#8 bars

Ties


Steel Shape

Designation

W 10

Wt./ft

60


Fy

54

49

45

36

50

36

50

36

50

36

50

0

1180

1390

1130

1320

1090

1260

1060

1220

6 7 8 9 10

1170 1160 1150 1140 1140

1370 1360 1350 1340 1330

1110 1110 1100 1090 1080

1300 1290 1280 1270 1260

1070 1070 1060 1050 1040

1240 1230 1220 1210 1200

1040 1040 1030 1020 1010

1200 1190 1180 1170 1160

11 12 13 14 15

1130 1120 1100 1090 1080

1310 1300 1280 1270 1250

1070 1060 1050 1040 1030

1240 1230 1210 1200 1180

1040 1020 1010 1000 989

1190 1170 1160 1140 1130

1000 994 983 971 958

1140 1130 1120 1100 1090

16 17 18 19 20

1060 1050 1040 1020 1000

1230 1210 1190 1170 1150

1010 1000 986 971 955

1160 1140 1120 1100 1080

976 962 947 932 917

1110 1090 1070 1050 1030

945 931 917 902 886

1070 1050 1030 1010 992

22 24 26 28 30

969 933 896 857 817

1100 1050 1000 952 900

921 886 849 811 772

1040 992 944 895 845

884 849 813 775 737

989 944 898 850 802

854 820 784 747 709

950 907 861 815 768

32 34 36 38 40

776 735 694 653 612

848 795 743 692 641

733 693 653 613 574

795 745 695 646 598

698 659 620 582 543

753 705 657 610 564

671 633 595 557 519

721 674 627 581 537

337 230 142 142

439 290 142 142

399 267 131 131

286 199 123 123

370 249 123 123

272 179 117 117

351 222 117 117



5.40 1.00

307 212 131 131 5.40 1.00

5.40 1.00


5.40 1.00


5 - 85



b Y X

h

X

Y 16 in.×16 in. 4-#7 bars

Ties


Designation

W8

Wt./ft

67

Fy


Steel Shape


58

48

40

35

36

50

36

50

36

50

36

50

36

50

0

1100

1330

1020

1230

938

1110

868

1010

828

951

6 7 8 9 10

1080 1070 1070 1060 1050

1310 1300 1290 1270 1260

1010 1000 992 984 975

1200 1190 1180 1170 1160

920 914 907 899 890

1080 1070 1060 1050 1040

851 845 839 831 822

985 977 968 957 946

811 805 799 791 783

929 921 912 902 891

11 12 13 14 15

1040 1030 1010 1000 988

1240 1230 1210 1190 1170

965 954 942 930 917

1140 1130 1110 1090 1080

881 870 859 847 834

1030 1010 997 981 963

813 803 792 780 768

933 920 905 890 874

773 763 753 741 729

879 866 852 837 822

16 17 18 19 20

973 958 943 926 909

1150 1130 1110 1080 1060

903 888 873 857 841

1060 1040 1010 992 970

821 807 793 778 762

945 927 907 887 866

755 742 728 713 698

857 839 821 802 782

717 703 690 675 661

805 788 770 752 733

22 24 26 28 30

874 836 798 758 717

1010 958 904 850 795

807 772 735 697 658

923 874 824 773 722

730 696 661 625 588

823 778 732 685 638

667 634 601 566 532

742 700 657 614 570

630 598 565 532 498

695 654 613 572 530

32 34 36 38 40

677 636 595 554 515

740 686 633 581 532

619 580 542 504 466

671 621 572 524 478

552 515 479 444 410

592 546 501 458 416

497 462 428 395 363

527 485 444 404 366

464 431 398 366 335

489 449 410 373 337

298 197 114 114

390 250 114 114

264 178 103 103

223 153 89.3 89.3

291 193 89.3 89.3

194 136 78.5 78.5

251 171 78.5 78.5

174 124 72.3 72.3

224 155 72.3 72.3



4.80 1.00

4.80 1.00

345 225 103 103

4.80 1.00


4.80 1.00

4.80 1.00

5 - 86

COMPOSITE DESIGN


b Y X

h

COMPOSITE COLUMNS W Shapes fc′ = 5 ksi All reinforcing steel is Grade 60 Axial design strength in kips

X

Y 24 in.×26 in. 4-#11 bars

Ties


Designation

W 14

Wt./ft

426

Fy


Steel Shape


398

370

342

311

36

50

36

50

36

50

36

50

36

50

0

5290

6770

5060

6450

4840

6130

4610

5810

4340

5430

6 7 8 9 10

5250 5240 5220 5210 5190

6720 6700 6670 6650 6620

5030 5020 5000 4990 4970

6400 6380 6360 6330 6300

4800 4790 4780 4760 4750

6080 6060 6040 6020 5990

4580 4570 4560 4540 4520

5760 5750 5730 5700 5680

4310 4300 4290 4270 4260

5380 5370 5350 5320 5300

11 12 13 14 15

5170 5150 5120 5100 5070

6580 6550 6510 6470 6430

4950 4930 4910 4880 4850

6270 6240 6200 6160 6120

4730 4710 4690 4660 4640

5960 5930 5890 5850 5810

4510 4490 4470 4440 4420

5650 5620 5580 5550 5510

4240 4220 4200 4180 4160

5270 5240 5210 5180 5140

16 17 18 19 20

5040 5010 4980 4950 4910

6380 6330 6280 6220 6170

4830 4800 4770 4730 4700

6070 6030 5980 5930 5870

4610 4580 4550 4520 4490

5770 5720 5680 5630 5570

4390 4360 4330 4300 4270

5470 5420 5380 5330 5280

4130 4100 4070 4050 4010

5100 5060 5010 4970 4920

22 24 26 28 30

4840 4760 4670 4580 4480

6050 5920 5780 5640 5490

4630 4550 4460 4380 4280

5760 5630 5500 5360 5220

4410 4340 4260 4170 4080

5460 5340 5220 5090 4950

4200 4130 4050 3970 3880

5170 5060 4940 4810 4680

3950 3880 3800 3720 3640

4820 4710 4600 4480 4350

32 34 36 38 40

4380 4280 4170 4060 3940

5330 5170 5000 4830 4660

4190 4080 3980 3870 3760

5060 4910 4750 4590 4420

3990 3890 3790 3680 3580

4800 4650 4500 4340 4180

3790 3690 3600 3500 3390

4540 4400 4250 4100 3950

3550 3460 3370 3270 3170

4220 4090 3950 3810 3660

3090 1850 1670 1420

4070 2380 1670 1420

2890 1750 1580 1340

2690 1640 1480 1260

3550 2120 1480 1260

2490 1540 1390 1180

3300 1980 1390 1180

2280 1420 1280 1090

3010 1830 1280 1090



7.20 1.08

3810 2250 1580 1340

7.20 1.08

7.20 1.08


7.20 1.08

7.20 1.08


5 - 87



b Y X

h

X

Y 24 in.×24 in. 4-#11 bars

Ties


Designation

W 14

Wt./ft

283

Fy


Steel Shape


257

233

211

193

36

50

36

50

36

50

36

50

36

50

0

3990

4980

3780

4680

3580

4390

3400

4130

3250

3930

6 7 8 9 10

3970 3960 3940 3930 3920

4940 4920 4910 4890 4860

3750 3740 3730 3720 3700

4640 4620 4600 4580 4560

3550 3540 3530 3520 3500

4350 4340 4320 4300 4280

3370 3360 3350 3340 3320

4100 4080 4070 4050 4030

3220 3220 3200 3190 3180

3890 3880 3860 3840 3820

11 12 13 14 15

3900 3880 3860 3840 3820

4840 4810 4780 4750 4720

3690 3670 3650 3630 3610

4540 4510 4480 4450 4420

3490 3470 3450 3440 3410

4260 4230 4210 4180 4150

3310 3290 3280 3260 3240

4010 3980 3960 3930 3900

3160 3150 3130 3110 3090

3800 3780 3750 3730 3700

16 17 18 19 20

3800 3770 3750 3720 3690

4680 4640 4600 4560 4520

3590 3560 3540 3510 3480

4390 4350 4310 4270 4230

3390 3370 3340 3320 3290

4110 4080 4040 4010 3970

3210 3190 3170 3140 3120

3870 3830 3800 3760 3720

3070 3050 3030 3000 2980

3670 3640 3600 3570 3530

22 24 26 28 30

3630 3560 3490 3420 3340

4420 4320 4220 4110 3990

3420 3360 3290 3220 3150

4140 4050 3950 3840 3730

3230 3170 3110 3040 2970

3880 3790 3690 3590 3490

3060 3000 2940 2870 2800

3640 3560 3460 3370 3270

2920 2860 2800 2730 2670

3450 3370 3280 3190 3090

32 34 36 38 40

3260 3180 3090 3000 2910

3870 3750 3620 3490 3360

3070 2990 2900 2820 2730

3610 3500 3380 3250 3130

2890 2810 2730 2650 2560

3380 3270 3150 3030 2920

2730 2650 2570 2490 2410

3160 3060 2950 2830 2720

2590 2520 2440 2360 2290

2990 2890 2780 2670 2570

2020 1300 993 993

2680 1680 993 993

1850 1200 916 916

1690 1110 845 845

2230 1430 845 845

1540 1020 780 780

2040 1320 780 780

1420 949 728 728

1880 1220 728 728



7.20 1.00

2450 1550 916 916

7.20 1.00

7.20 1.00


7.20 1.00

7.20 1.00

5 - 88

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

24 in.×26 in.

Reinf. bars

4-#10 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

176


Fy

159

145

132

36

50

36

50

36

50

36

50

0

3080

3690

2930

3490

2820

3330

2710

3170

6 7 8 9 10

3050 3040 3030 3020 3010

3660 3640 3630 3610 3590

2910 2900 2890 2880 2870

3450 3440 3430 3410 3390

2800 2790 2780 2770 2750

3290 3280 3270 3250 3240

2690 2680 2670 2660 2650

3140 3130 3110 3100 3080

11 12 13 14 15

2990 2980 2960 2940 2920

3570 3550 3530 3500 3480

2850 2840 2820 2800 2780

3370 3350 3330 3310 3280

2740 2730 2710 2690 2670

3220 3200 3170 3150 3120

2630 2620 2600 2580 2570

3060 3040 3020 3000 2970

16 17 18 19 20

2900 2880 2860 2840 2810

3450 3420 3380 3350 3320

2760 2740 2720 2700 2670

3250 3220 3190 3160 3130

2650 2630 2610 2590 2560

3100 3070 3040 3010 2980

2550 2530 2500 2480 2460

2950 2920 2890 2860 2830

22 24 26 28 30

2760 2700 2640 2580 2510

3240 3160 3080 2990 2900

2620 2570 2510 2450 2380

3050 2980 2900 2810 2720

2510 2460 2400 2340 2280

2910 2830 2750 2670 2590

2410 2350 2300 2240 2180

2760 2690 2620 2540 2450

32 34 36 38 40

2450 2370 2300 2230 2150

2800 2710 2610 2510 2400

2320 2250 2170 2100 2030

2630 2540 2440 2350 2250

2210 2140 2070 2000 1930

2500 2410 2320 2220 2130

2110 2040 1980 1910 1830

2370 2280 2190 2100 2010

1280 860 678 678

1700 1110 678 678

1550 1020 627 627

1080 734 587 587

1430 950 587 587

996 670 547 547

1320 861 547 547



7.20 1.00

1170 789 627 627 7.20 1.00

7.20 1.00


7.20 1.00


5 - 89


b Y


h

X

X

Y Size b × h

22 in.×22 in.

Reinf. bars

4-#10 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

120


Fy

109

99

90

36

50

36

50

36

50

36

50

0

2380

2800

2290

2670

2200

2550

2130

2450

6 7 8 9 10

2350 2340 2330 2320 2310

2760 2750 2740 2720 2700

2260 2250 2240 2230 2220

2630 2620 2610 2590 2580

2180 2170 2160 2150 2140

2520 2510 2490 2480 2460

2110 2100 2090 2080 2070

2410 2400 2390 2380 2360

11 12 13 14 15

2300 2280 2270 2250 2230

2690 2670 2640 2620 2590

2200 2190 2170 2160 2140

2560 2540 2520 2490 2470

2120 2110 2090 2080 2060

2440 2420 2400 2380 2360

2050 2040 2020 2000 1990

2340 2320 2300 2280 2260

16 17 18 19 20

2210 2190 2170 2140 2120

2570 2540 2510 2480 2440

2120 2100 2080 2050 2030

2440 2410 2380 2350 2320

2040 2020 2000 1970 1950

2330 2300 2270 2250 2210

1970 1950 1920 1900 1880

2230 2200 2180 2150 2120

22 24 26 28 30

2070 2020 1960 1900 1840

2380 2300 2230 2150 2070

1980 1930 1870 1810 1750

2260 2190 2110 2030 1950

1900 1850 1790 1740 1680

2150 2080 2010 1930 1860

1830 1780 1720 1670 1610

2050 1990 1920 1840 1770

32 34 36 38 40

1770 1710 1640 1570 1500

1980 1890 1810 1720 1630

1690 1620 1560 1490 1420

1870 1790 1710 1620 1540

1610 1550 1490 1420 1350

1780 1700 1620 1530 1450

1550 1480 1420 1350 1290

1690 1610 1530 1450 1380

883 586 409 409

1160 752 409 409

1070 695 381 381

748 507 357 357

982 647 357 357

691 472 335 335

905 600 335 335



6.60 1.00

811 543 381 381 6.60 1.00

6.60 1.00


6.60 1.00

5 - 90

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

18 in.×22 in.

Reinf. bars

4-#9 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

82


Fy

74

68

61

36

50

36

50

36

50

36

50

0

1810

2090

1740

2000

1690

1930

1630

1850

6 7 8 9 10

1780 1770 1760 1740 1730

2050 2040 2020 2010 1990

1710 1700 1690 1680 1660

1960 1950 1930 1920 1900

1660 1650 1640 1630 1610

1890 1880 1860 1850 1830

1600 1590 1580 1570 1550

1810 1800 1780 1760 1740

11 12 13 14 15

1710 1690 1670 1650 1630

1960 1940 1920 1890 1860

1650 1630 1610 1590 1570

1880 1850 1830 1800 1770

1600 1580 1560 1540 1520

1810 1780 1760 1730 1710

1540 1520 1500 1480 1460

1720 1700 1680 1650 1630

16 17 18 19 20

1610 1590 1560 1530 1510

1830 1800 1770 1730 1700

1550 1520 1500 1470 1450

1740 1710 1680 1650 1620

1500 1470 1450 1420 1400

1680 1650 1620 1580 1550

1440 1410 1390 1360 1340

1600 1570 1540 1510 1470

22 24 26 28 30

1450 1390 1330 1270 1200

1620 1550 1470 1390 1310

1390 1330 1270 1210 1140

1540 1470 1390 1310 1240

1340 1280 1220 1160 1100

1480 1410 1330 1260 1180

1280 1230 1170 1110 1040

1410 1340 1260 1190 1110

32 34 36 38 40

1140 1070 1010 941 877

1220 1140 1060 983 906

1080 1020 952 888 826

1160 1080 999 923 849

1040 972 909 847 786

1100 1030 950 876 805

982 920 858 797 738

1040 965 892 821 752

634 328 294 197

830 416 294 197

761 384 275 184

543 285 260 174

708 360 260 174

496 264 242 162

644 332 242 162



5.40 1.22

582 303 275 184 5.40 1.22

5.40 1.22


5.40 1.22


5 - 91



b Y X

h

X

Y 22 in.×24 in. 4-#10 bars

Ties


Designation

W 12

Wt./ft

336

Fy


Steel Shape


305

279

252

230

36

50

36

50

36

50

36

50

36

50

0

4270

5450

4010

5080

3800

4770

3580

4460

3400

4200

6 7 8 9 10

4240 4230 4210 4200 4180

5390 5370 5350 5320 5300

3980 3970 3960 3940 3920

5030 5010 4990 4960 4940

3770 3750 3740 3730 3710

4720 4700 4680 4660 4640

3550 3540 3520 3510 3490

4410 4400 4380 4350 4330

3370 3360 3350 3330 3320

4160 4140 4120 4100 4080

11 12 13 14 15

4160 4140 4110 4090 4060

5260 5230 5190 5150 5110

3900 3880 3860 3840 3810

4910 4870 4840 4800 4760

3690 3670 3650 3630 3610

4610 4580 4540 4510 4470

3480 3460 3440 3420 3390

4300 4270 4240 4210 4170

3300 3280 3260 3240 3220

4050 4030 4000 3960 3930

16 17 18 19 20

4040 4010 3980 3940 3910

5070 5020 4970 4920 4870

3790 3760 3730 3700 3670

4720 4680 4630 4580 4530

3580 3550 3520 3490 3460

4430 4390 4340 4300 4250

3370 3340 3310 3290 3260

4140 4100 4050 4010 3960

3190 3170 3140 3110 3090

3890 3860 3820 3770 3730

22 24 26 28 30

3840 3760 3680 3590 3500

4750 4630 4500 4370 4230

3600 3520 3450 3360 3280

4420 4310 4190 4060 3930

3400 3320 3250 3170 3090

4150 4040 3920 3800 3680

3190 3120 3050 2970 2890

3870 3760 3650 3540 3420

3020 2960 2890 2810 2730

3640 3540 3430 3320 3210

32 34 36 38 40

3410 3310 3210 3110 3000

4080 3930 3780 3630 3470

3190 3090 3000 2900 2800

3790 3650 3510 3360 3220

3000 2910 2820 2720 2630

3550 3410 3280 3140 3000

2810 2720 2630 2540 2450

3300 3170 3040 2910 2780

2650 2570 2480 2400 2310

3090 2970 2850 2730 2600

2190 1260 1140 954

2890 1600 1140 954

1990 1150 1040 877

1830 1080 966 812

2410 1380 966 812

1660 990 888 746

2190 1270 888 746

1530 919 824 692

2010 1180 824 692



6.60 1.09

2630 1480 1040 877

6.60 1.09

6.60 1.09


6.60 1.09

6.60 1.09

5 - 92

COMPOSITE DESIGN


b Y X

h


X

Y 20 in.×22 in. 4-#10 bars

Ties


Designation

W 12

Wt./ft

210

Fy


Steel Shape


190

170

152

136

36

50

36

50

36

50

36

50

36

50

0

3010

3740

2840

3500

2680

3270

2530

3060

2390

2870

6 7 8 9 10

2980 2970 2950 2940 2920

3700 3680 3660 3640 3610

2810 2800 2790 2770 2760

3460 3440 3420 3400 3380

2650 2640 2630 2610 2600

3230 3210 3200 3180 3150

2500 2490 2480 2470 2450

3020 3000 2990 2970 2950

2370 2360 2340 2330 2320

2830 2810 2800 2780 2760

11 12 13 14 15

2910 2890 2870 2840 2820

3590 3560 3520 3490 3450

2740 2720 2700 2680 2660

3350 3320 3290 3260 3230

2580 2560 2540 2520 2500

3130 3100 3070 3040 3010

2430 2420 2400 2380 2360

2920 2900 2870 2840 2810

2300 2280 2270 2250 2220

2740 2710 2690 2660 2630

16 17 18 19 20

2790 2770 2740 2710 2680

3420 3380 3330 3290 3240

2630 2610 2580 2550 2520

3190 3150 3110 3070 3030

2480 2450 2430 2400 2370

2970 2940 2900 2860 2820

2330 2310 2280 2260 2230

2780 2740 2700 2670 2630

2200 2180 2150 2130 2100

2600 2560 2530 2490 2450

22 24 26 28 30

2620 2550 2480 2400 2320

3150 3040 2940 2830 2710

2460 2390 2320 2250 2180

2940 2840 2740 2630 2520

2310 2250 2180 2110 2030

2730 2640 2540 2440 2340

2170 2110 2040 1970 1900

2540 2460 2360 2270 2170

2040 1980 1920 1850 1780

2370 2290 2200 2110 2020

32 34 36 38 40

2240 2160 2070 1980 1900

2590 2470 2350 2230 2110

2100 2020 1930 1850 1770

2410 2300 2180 2070 1950

1960 1880 1800 1720 1640

2230 2130 2020 1910 1800

1830 1760 1680 1600 1530

2070 1970 1870 1760 1660

1710 1640 1570 1490 1420

1920 1830 1730 1630 1530

1350 796 622 514

1770 1020 622 514

1230 734 572 472

1110 671 523 432

1460 860 523 432

1010 614 478 395

1320 787 478 395

910 563 438 362

1190 721 438 362



6.00 1.10

1610 941 572 472

6.00 1.10

6.00 1.10


6.00 1.10

6.00 1.10


5 - 93



b Y X

h

X

Y 20 in.×20 in. 4-#9 bars

Ties


Designation

W 12

Wt./ft

120

Fy


Steel Shape


106

96

87

79

36

50

36

50

36

50

36

50

36

50

0

2130

2550

2020

2390

1930

2270

1860

2160

1790

2070

6 7 8 9 10

2110 2100 2090 2080 2060

2520 2500 2490 2470 2450

1990 1980 1970 1960 1950

2350 2340 2330 2310 2290

1910 1900 1890 1880 1870

2230 2220 2210 2190 2180

1830 1830 1820 1810 1790

2130 2120 2110 2090 2070

1770 1760 1750 1740 1730

2040 2020 2010 2000 1980

11 12 13 14 15

2050 2030 2020 2000 1980

2430 2410 2390 2360 2340

1930 1920 1900 1890 1870

2270 2250 2230 2210 2180

1850 1840 1820 1800 1790

2160 2140 2120 2090 2070

1780 1760 1750 1730 1710

2060 2040 2020 1990 1970

1710 1700 1680 1670 1650

1960 1940 1920 1900 1880

16 17 18 19 20

1960 1940 1920 1890 1870

2310 2280 2250 2220 2180

1850 1830 1810 1780 1760

2150 2130 2100 2070 2030

1770 1750 1730 1700 1680

2040 2010 1990 1960 1920

1700 1680 1650 1630 1610

1940 1920 1890 1860 1830

1630 1610 1590 1570 1540

1850 1830 1800 1770 1740

22 24 26 28 30

1820 1770 1710 1650 1590

2110 2040 1960 1880 1800

1710 1660 1600 1550 1490

1970 1890 1820 1740 1660

1630 1580 1530 1470 1410

1860 1790 1720 1640 1570

1560 1510 1460 1400 1340

1770 1700 1630 1560 1480

1500 1450 1390 1340 1280

1680 1620 1550 1480 1410

32 34 36 38 40

1530 1460 1400 1330 1260

1710 1620 1540 1450 1370

1430 1360 1300 1240 1170

1580 1500 1420 1340 1260

1350 1290 1230 1170 1100

1490 1410 1330 1250 1180

1290 1230 1170 1110 1040

1410 1330 1260 1180 1110

1220 1170 1110 1050 989

1330 1260 1190 1110 1040

760 489 322 322

1000 628 322 322

680 440 294 294

622 407 273 273

820 522 273 273

573 380 255 255

754 486 255 255

528 353 238 238

693 450 238 238



6.00 1.00

6.00 1.00

899 566 294 294

6.00 1.00


6.00 1.00

6.00 1.00

5 - 94

COMPOSITE DESIGN


b Y

h

X


X

Y Size b × h

20 in.×20 in.

Reinf. bars

4-#9 bars

Ties


Steel Shape

Designation

W 12

Wt./ft

72


Fy

65

58

36

50

36

50

36

50

0

1730

1980

1680

1900

1620

1820

6 7 8 9 10

1710 1700 1690 1680 1670

1950 1940 1930 1910 1900

1650 1640 1630 1620 1610

1870 1860 1850 1840 1820

1590 1590 1580 1560 1550

1790 1780 1770 1750 1740

11 12 13 14 15

1650 1640 1620 1610 1590

1880 1860 1840 1820 1800

1600 1580 1570 1550 1530

1800 1780 1760 1740 1720

1540 1520 1510 1490 1480

1720 1700 1680 1660 1640

16 17 18 19 20

1570 1550 1530 1510 1490

1770 1750 1720 1690 1660

1520 1500 1480 1450 1430

1700 1670 1650 1620 1590

1460 1440 1420 1400 1370

1620 1590 1570 1540 1510

22 24 26 28 30

1440 1390 1340 1280 1230

1600 1540 1480 1410 1340

1380 1340 1280 1230 1180

1530 1470 1410 1340 1270

1330 1280 1230 1170 1120

1460 1400 1330 1270 1200

32 34 36 38 40

1170 1110 1060 997 939

1270 1200 1120 1050 984

1120 1060 1000 948 891

1200 1130 1060 996 929

1060 1010 951 895 840

1140 1070 1000 935 871

449 307 209 209

586 388 209 209

414 268 195 195

537 335 195 195



489 330 223 223

640 419 223 223 6.00 1.00

6.00 1.00


6.00 1.00


5 - 95



b Y X

h

X

Y 18 in.×18 in. 4-#8 bars

Ties


Designation

W 10

Wt./ft

112

Fy


Steel Shape


100

88

77

68

36

50

36

50

36

50

36

50

36

50

0

1840

2240

1750

2100

1650

1960

1560

1820

1480

1720

6 7 8 9 10

1820 1810 1800 1790 1770

2200 2190 2170 2150 2130

1720 1710 1700 1690 1680

2060 2050 2030 2020 2000

1620 1610 1600 1590 1580

1920 1910 1900 1880 1860

1530 1520 1510 1500 1490

1790 1780 1770 1750 1730

1460 1450 1440 1430 1420

1690 1680 1660 1650 1630

11 12 13 14 15

1760 1740 1730 1710 1690

2110 2090 2070 2040 2010

1660 1650 1630 1610 1600

1980 1960 1930 1910 1880

1570 1550 1540 1520 1500

1840 1820 1800 1780 1750

1480 1460 1450 1430 1410

1720 1700 1670 1650 1630

1400 1390 1370 1360 1340

1620 1600 1580 1550 1530

16 17 18 19 20

1670 1650 1630 1600 1580

1980 1950 1920 1890 1850

1580 1560 1530 1510 1490

1850 1830 1790 1760 1730

1480 1460 1440 1420 1400

1720 1700 1670 1640 1610

1390 1370 1350 1330 1310

1600 1580 1550 1520 1490

1320 1300 1280 1260 1240

1510 1480 1450 1430 1400

22 24 26 28 30

1530 1480 1420 1360 1300

1780 1710 1630 1550 1470

1440 1390 1330 1280 1220

1660 1590 1520 1440 1360

1350 1300 1250 1190 1140

1540 1470 1400 1330 1260

1260 1210 1160 1110 1050

1430 1360 1300 1230 1160

1190 1140 1090 1040 990

1340 1270 1210 1140 1080

32 34 36 38 40

1240 1180 1120 1060 995

1380 1300 1220 1140 1060

1160 1100 1040 982 923

1280 1210 1130 1050 975

1080 1020 963 906 850

1180 1110 1030 962 891

1000 944 889 834 779

1090 1020 946 878 811

936 883 829 776 723

1010 942 876 811 748

589 379 236 236

780 488 236 236

532 346 216 216

475 313 196 196

627 402 196 196

421 282 178 178

555 361 178 178

378 256 163 163

497 327 163 163



5.40 1.00

5.40 1.00

704 445 216 216

5.40 1.00


5.40 1.00

5.40 1.00

5 - 96

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

18 in.×18 in.

Reinf. bars

4-#8 bars

Ties


Steel Shape

Designation

W 10

Wt./ft

60


Fy

54

49

45

36

50

36

50

36

50

36

50

0

1420

1620

1360

1550

1330

1500

1290

1450

6 7 8 9 10

1390 1380 1370 1360 1350

1590 1580 1570 1560 1540

1340 1330 1320 1310 1300

1520 1510 1500 1490 1470

1300 1290 1280 1270 1260

1470 1460 1440 1430 1420

1270 1260 1250 1240 1230

1420 1410 1400 1390 1370

11 12 13 14 15

1340 1320 1310 1290 1270

1520 1500 1480 1460 1440

1290 1270 1260 1240 1220

1450 1430 1410 1390 1370

1250 1230 1220 1200 1190

1400 1380 1360 1340 1320

1220 1200 1190 1170 1150

1360 1340 1320 1300 1280

16 17 18 19 20

1260 1240 1220 1200 1170

1420 1390 1370 1340 1310

1210 1190 1170 1150 1120

1350 1320 1300 1270 1250

1170 1150 1130 1110 1090

1300 1270 1250 1220 1200

1140 1120 1100 1080 1060

1250 1230 1210 1180 1160

22 24 26 28 30

1130 1080 1030 981 929

1250 1190 1130 1070 1000

1080 1030 984 934 883

1190 1130 1070 1010 947

1040 995 947 897 847

1140 1080 1020 963 903

1010 965 917 868 818

1100 1040 987 927 868

32 34 36 38 40

877 825 772 721 670

938 874 811 749 689

832 781 729 679 630

884 822 761 702 644

796 746 695 646 598

842 782 723 665 609

768 718 668 620 572

808 749 692 636 581

341 234 149 149

446 297 149 149

405 272 139 139

288 201 131 131

375 254 131 131

275 182 125 125

356 227 125 125



5.40 1.00

310 215 139 139 5.40 1.00

5.40 1.00


5.40 1.00


5 - 97


b Y


X

h

X

Y 16 in.×16 in. 4-#7 bars

Ties


Designation

W 18

Wt./ft

67

Fy


Steel Shape


58

48

40

35

36

50

36

50

36

50

36

50

36

50

0

1280

1510

1200

1410

1120

1290

1050

1190

1010

1140

6 7 8 9 10

1250 1250 1240 1220 1210

1480 1470 1450 1440 1420

1180 1170 1160 1150 1140

1380 1360 1350 1340 1320

1100 1090 1080 1070 1060

1260 1250 1230 1220 1200

1030 1020 1010 1000 990

1160 1150 1140 1130 1110

991 982 973 962 951

1110 1100 1090 1070 1060

11 12 13 14 15

1200 1180 1170 1150 1130

1400 1380 1360 1340 1310

1130 1110 1100 1080 1060

1300 1280 1260 1240 1220

1040 1030 1010 999 982

1190 1170 1150 1130 1110

977 963 948 932 916

1100 1080 1060 1040 1020

938 924 909 893 877

1040 1020 1010 986 965

16 17 18 19 20

1120 1100 1080 1060 1030

1290 1260 1230 1210 1180

1050 1030 1010 987 966

1190 1170 1140 1120 1090

964 946 927 907 886

1080 1060 1040 1010 984

898 880 861 841 821

996 973 949 925 900

859 841 822 803 783

944 922 898 875 850

22 24 26 28 30

989 942 894 844 794

1120 1050 990 926 861

922 876 829 781 733

1030 971 910 849 787

844 799 754 707 661

929 873 816 759 702

779 736 692 647 602

848 795 741 686 632

741 698 655 610 566

800 748 696 643 591

32 34 36 38 40

743 693 644 595 548

796 733 672 612 555

684 636 589 542 498

727 667 610 554 501

614 569 524 480 438

646 591 537 486 438

557 513 470 429 389

579 528 478 431 389

522 480 438 398 360

540 491 444 398 360

303 202 119 119

400 259 119 119

268 182 107 107

226 155 94.1 94.1

296 198 94.1 94.1

196 138 83.4 83.4

255 175 83.4 83.4

175 125 77.2 77.2

227 157 77.2 77.2



4.80 1.00

353 233 107 107 4.80 1.00

4.80 1.00


4.80 1.00

4.80 1.00

5 - 98

COMPOSITE DESIGN


b Y X

h


X

Y 24 in.×26 in. 4-#11 bars

Ties


Designation

W 14

Wt./ft

426

Fy


Steel Shape


398

370

342

311

36

50

36

50

36

50

36

50

36

50

0

6040

7530

5830

7220

5620

6910

5400

6610

5150

6240

6 7 8 9 10

6000 5980 5960 5940 5920

7460 7430 7410 7370 7340

5780 5770 5750 5730 5710

7150 7130 7100 7070 7040

5570 5560 5540 5520 5500

6850 6820 6800 6770 6730

5360 5350 5330 5310 5290

6540 6520 6490 6460 6430

5110 5090 5070 5060 5030

6180 6150 6130 6100 6070

11 12 13 14 15

5890 5870 5840 5800 5770

7300 7260 7210 7160 7110

5680 5660 5630 5590 5560

7000 6960 6910 6860 6810

5470 5450 5420 5390 5350

6700 6660 6610 6570 6520

5260 5240 5210 5180 5140

6400 6360 6310 6270 6220

5010 4980 4960 4930 4890

6030 6000 5960 5910 5860

16 17 18 19 20

5730 5690 5650 5610 5570

7050 6990 6930 6870 6800

5530 5490 5450 5410 5360

6760 6700 6640 6580 6510

5320 5280 5240 5200 5160

6460 6410 6350 6290 6230

5110 5070 5030 4990 4950

6170 6120 6060 6000 5940

4860 4820 4790 4750 4700

5820 5760 5710 5650 5590

22 24 26 28 30

5470 5370 5260 5150 5030

6660 6500 6340 6170 5990

5270 5170 5060 4950 4830

6370 6220 6070 5900 5730

5070 4970 4860 4750 4640

6090 5940 5790 5630 5460

4860 4770 4660 4550 4440

5810 5670 5520 5360 5200

4620 4520 4420 4310 4200

5460 5330 5190 5040 4880

32 34 36 38 40

4900 4770 4640 4500 4360

5800 5610 5420 5220 5010

4710 4580 4450 4320 4180

5550 5360 5170 4980 4780

4520 4390 4260 4130 4000

5290 5110 4920 4740 4550

4320 4200 4070 3940 3810

5030 4850 4680 4500 4310

4090 3970 3840 3720 3590

4720 4550 4380 4210 4030

3190 1940 1710 1460

4270 2550 1710 1460

2970 1830 1620 1380

2770 1710 1530 1300

3700 2260 1530 1300

2560 1600 1430 1220

3430 2100 1430 1220

2330 1470 1320 1130

3120 1930 1320 1130



7.20 1.08

3980 2410 1620 1380

7.20 1.08

7.20 1.08


7.20 1.08

7.20 1.08


5 - 99



b Y X

h

X

Y 24 in.×24 in. 4-#11 bars

Ties


Designation

W 14

Wt./ft

283

Fy


Steel Shape


257

233

211

193

36

50

36

50

36

50

36

50

36

50

0

4740

5730

4530

5430

4350

5160

4170

4910

4040

4710

6 7 8 9 10

4700 4690 4670 4650 4630

5670 5650 5630 5600 5570

4500 4480 4470 4450 4430

5380 5360 5340 5310 5280

4310 4290 4280 4260 4240

5110 5090 5070 5040 5010

4140 4120 4110 4090 4070

4860 4840 4820 4790 4770

4000 3980 3970 3950 3930

4660 4640 4620 4600 4570

11 12 13 14 15

4610 4590 4560 4530 4500

5540 5510 5470 5430 5390

4410 4380 4360 4330 4300

5250 5220 5180 5140 5100

4220 4200 4170 4140 4110

4980 4950 4920 4880 4840

4050 4020 4000 3970 3940

4740 4710 4670 4630 4590

3910 3890 3860 3830 3810

4540 4510 4480 4440 4400

16 17 18 19 20

4470 4440 4400 4370 4330

5340 5290 5240 5190 5130

4270 4240 4200 4170 4130

5060 5010 4960 4910 4860

4080 4050 4020 3980 3940

4790 4750 4700 4650 4600

3910 3880 3850 3810 3770

4550 4510 4460 4410 4360

3780 3740 3710 3670 3640

4360 4320 4270 4220 4170

22 24 26 28 30

4250 4160 4060 3970 3860

5020 4890 4760 4620 4480

4050 3960 3870 3770 3670

4740 4620 4490 4360 4220

3860 3780 3690 3590 3490

4490 4370 4250 4120 3980

3690 3610 3520 3430 3330

4250 4140 4020 3890 3760

3560 3470 3380 3290 3190

4070 3960 3840 3710 3590

32 34 36 38 40

3760 3640 3530 3410 3300

4330 4170 4010 3850 3690

3570 3460 3340 3230 3110

4070 3930 3770 3620 3460

3390 3280 3170 3060 2950

3840 3700 3550 3400 3250

3220 3120 3010 2900 2790

3630 3490 3340 3200 3060

3090 2990 2880 2770 2660

3450 3320 3180 3040 2900

2070 1350 1030 1030

2770 1770 1030 1030

1890 1240 952 952

1720 1140 882 882

2300 1490 882 882

1570 1050 817 817

2090 1370 817 817

1440 972 765 765

1920 1270 765 765



7.20 1.00

2530 1620 952 952

7.20 1.00

7.20 1.00


7.20 1.00

7.20 1.00

5 - 100

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

24 in.×24 in.

Reinf. bars

4-#10 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

176


Fy

159

145

132

36

50

36

50

36

50

36

50

0

3870

4490

3730

4290

3630

4140

3520

3990

6 7 8 9 10

3830 3820 3800 3790 3770

4440 4420 4400 4370 4350

3700 3680 3670 3650 3630

4240 4220 4200 4180 4160

3590 3580 3560 3540 3530

4090 4070 4050 4030 4000

3490 3470 3460 3440 3420

3940 3920 3900 3880 3850

11 12 13 14 15

3750 3720 3700 3670 3640

4320 4290 4260 4220 4180

3610 3590 3560 3540 3510

4130 4100 4070 4030 3990

3500 3480 3460 3430 3400

3980 3950 3910 3880 3840

3400 3380 3350 3330 3300

3830 3800 3770 3730 3700

16 17 18 19 20

3610 3580 3550 3510 3480

4140 4100 4060 4010 3960

3480 3450 3410 3380 3340

3950 3910 3870 3820 3780

3370 3340 3310 3270 3240

3800 3760 3720 3680 3630

3270 3240 3200 3170 3130

3660 3620 3580 3530 3490

22 24 26 28 30

3400 3320 3230 3140 3040

3860 3750 3640 3520 3400

3270 3180 3100 3010 2910

3680 3570 3460 3340 3220

3160 3080 2990 2900 2800

3530 3430 3320 3200 3080

3050 2970 2880 2790 2700

3390 3290 3180 3070 2950

32 34 36 38 40

2950 2840 2740 2630 2530

3270 3140 3010 2870 2740

2810 2710 2610 2500 2400

3100 2970 2840 2710 2580

2710 2610 2500 2400 2300

2960 2840 2710 2580 2450

2600 2500 2400 2300 2190

2830 2710 2580 2460 2330

1300 879 716 716

1740 1150 716 716

1580 1050 665 665

1090 748 625 625

1450 975 625 625

1010 681 587 587

1340 884 587 587



7.20 1.00

1190 805 665 665 7.20 1.00

7.20 1.00


7.20 1.00


5 - 101


b Y


h

X

X

Y Size b × h

22 in.×22 in.

Reinf. bars

4-#10 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

120


Fy

109

99

90

36

50

36

50

36

50

36

50

0

3060

3480

2970

3350

2890

3240

2820

3140

6 7 8 9 10

3020 3010 2990 2970 2950

3430 3410 3390 3370 3340

2930 2920 2900 2880 2860

3300 3280 3260 3240 3220

2850 2840 2820 2810 2790

3190 3170 3150 3130 3110

2780 2770 2750 2740 2720

3090 3070 3050 3030 3010

11 12 13 14 15

2930 2910 2880 2860 2830

3320 3290 3250 3220 3180

2840 2820 2790 2770 2740

3190 3160 3130 3090 3060

2770 2740 2720 2690 2660

3080 3050 3020 2990 2950

2700 2670 2650 2620 2590

2980 2950 2920 2890 2850

16 17 18 19 20

2800 2770 2730 2700 2660

3140 3100 3060 3020 2970

2710 2680 2640 2610 2570

3020 2980 2940 2900 2850

2630 2600 2560 2530 2490

2910 2870 2830 2790 2740

2560 2530 2490 2460 2420

2810 2770 2730 2690 2650

22 24 26 28 30

2590 2510 2420 2330 2240

2870 2770 2670 2560 2440

2500 2420 2330 2240 2150

2750 2650 2550 2440 2330

2420 2340 2250 2160 2070

2650 2550 2450 2340 2230

2340 2260 2180 2090 2000

2550 2450 2350 2250 2140

32 34 36 38 40

2150 2050 1950 1860 1760

2330 2210 2090 1970 1850

2060 1960 1870 1770 1670

2220 2100 1980 1870 1760

1980 1880 1790 1690 1600

2120 2000 1890 1780 1670

1910 1810 1720 1620 1530

2030 1920 1810 1700 1590

894 597 436 436

1190 773 436 436

1080 712 409 409

755 514 385 385

996 661 385 385

697 478 363 363

917 612 363 363



6.60 1.00

819 551 409 409 6.60 1.00

6.60 1.00


6.60 1.00

5 - 102

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

18 in.×22 in.

Reinf. bars

4-#9 bars

Ties


Steel Shape

Designation

W 14

Wt./ft

82


Fy

74

68

61

36

50

36

50

36

50

36

50

0

2370

2660

2310

2570

2260

2500

2200

2420

6 7 8 9 10

2320 2310 2290 2270 2240

2600 2580 2550 2530 2500

2260 2240 2230 2200 2180

2510 2490 2470 2440 2410

2210 2200 2180 2160 2130

2440 2420 2400 2370 2340

2160 2140 2120 2100 2070

2360 2340 2320 2290 2260

11 12 13 14 15

2220 2190 2160 2130 2090

2470 2430 2390 2350 2310

2150 2130 2100 2060 2030

2380 2340 2310 2270 2230

2110 2080 2050 2010 1980

2310 2280 2240 2200 2160

2050 2020 1990 1960 1920

2230 2200 2160 2120 2080

16 17 18 19 20

2060 2020 1980 1940 1900

2270 2220 2180 2130 2080

1990 1960 1920 1880 1840

2180 2140 2090 2040 1990

1940 1910 1870 1830 1790

2120 2070 2020 1980 1930

1890 1850 1810 1770 1730

2040 1990 1950 1900 1850

22 24 26 28 30

1820 1730 1630 1540 1440

1970 1860 1750 1640 1520

1750 1660 1570 1480 1380

1890 1780 1670 1560 1450

1700 1610 1520 1430 1330

1820 1720 1610 1500 1390

1640 1550 1460 1360 1270

1750 1640 1540 1430 1320

32 34 36 38 40

1350 1250 1160 1070 983

1410 1300 1190 1090 989

1290 1190 1100 1010 926

1340 1230 1130 1030 929

1240 1140 1050 966 882

1280 1180 1080 977 882

1180 1090 998 911 827

1220 1110 1010 917 827

644 336 317 212

849 431 317 212

776 396 298 199

549 290 283 189

720 370 283 189

501 268 265 178

654 341 265 178



5.40 1.22

590 309 298 199 5.40 1.22

5.40 1.22


5.40 1.22


5 - 103



b Y X

h

X

Y 22 in.×24 in. 4-#10 bars

Ties


Designation

W 12

Wt./ft

336

Fy


Steel Shape


305

279

252

230

36

50

36

50

36

50

36

50

36

50

0

4920

6100

4680

5740

4470

5450

4260

5150

4100

4900

6 7 8 9 10

4880 4860 4840 4820 4800

6030 6000 5980 5950 5910

4630 4620 4600 4580 4560

5680 5650 5630 5600 5570

4430 4410 4400 4380 4360

5380 5360 5340 5310 5270

4220 4210 4190 4170 4150

5090 5060 5040 5010 4980

4050 4040 4020 4000 3980

4840 4820 4800 4770 4740

11 12 13 14 15

4770 4750 4720 4690 4650

5870 5830 5790 5740 5690

4530 4510 4480 4450 4420

5530 5490 5450 5400 5350

4330 4310 4280 4250 4220

5240 5200 5160 5120 5070

4130 4100 4070 4040 4010

4950 4910 4870 4830 4780

3960 3930 3910 3880 3840

4710 4670 4630 4590 4540

16 17 18 19 20

4620 4580 4540 4500 4460

5640 5580 5520 5460 5390

4380 4340 4310 4260 4220

5300 5250 5190 5130 5070

4180 4140 4110 4070 4030

5020 4970 4910 4850 4790

3980 3940 3910 3870 3830

4730 4680 4630 4570 4510

3810 3780 3740 3700 3660

4500 4450 4400 4340 4290

22 24 26 28 30

4360 4270 4160 4050 3940

5260 5110 4960 4790 4630

4130 4040 3940 3830 3720

4930 4790 4650 4490 4330

3940 3840 3740 3640 3530

4670 4530 4390 4240 4080

3740 3650 3550 3450 3340

4390 4260 4120 3980 3830

3580 3480 3390 3290 3180

4170 4040 3910 3770 3620

32 34 36 38 40

3820 3700 3570 3440 3310

4450 4280 4100 3920 3730

3600 3480 3360 3240 3110

4170 4000 3830 3650 3480

3420 3300 3180 3060 2940

3920 3760 3600 3430 3260

3230 3120 3000 2880 2760

3680 3520 3370 3210 3050

3070 2960 2850 2730 2620

3480 3330 3170 3020 2870

2270 1320 1170 980

3030 1730 1170 980

2050 1210 1080 904

1880 1130 999 839

2510 1470 999 839

1710 1030 921 774

2280 1350 921 774

1570 954 857 720

2090 1250 857 720



6.60 1.09

2750 1580 1080 904

6.60 1.09

6.60 1.09


6.60 1.09

6.60 1.09

5 - 104

COMPOSITE DESIGN


b Y X

h


X

Y 20 in.×22 in. 4-#10 bars

Ties


Designation

W 12

Wt./ft

210

Fy


Steel Shape


190

170

152

136

36

50

36

50

36

50

36

50

36

50

0

3580

4320

3420

4080

3270

3860

3130

3660

3000

3470

6 7 8 9 10

3540 3520 3500 3480 3460

4250 4230 4210 4180 4150

3380 3360 3350 3330 3310

4020 4000 3980 3950 3920

3230 3210 3190 3170 3150

3800 3780 3760 3730 3700

3080 3070 3050 3030 3010

3600 3580 3560 3530 3500

2960 2940 2930 2910 2880

3420 3400 3380 3350 3320

11 12 13 14 15

3440 3410 3380 3350 3320

4110 4070 4030 3990 3940

3280 3260 3230 3200 3170

3890 3850 3810 3770 3730

3130 3100 3080 3050 3010

3670 3630 3600 3560 3510

2990 2960 2940 2910 2880

3470 3440 3400 3360 3320

2860 2840 2810 2780 2750

3290 3260 3220 3180 3140

16 17 18 19 20

3290 3250 3220 3180 3140

3900 3850 3790 3740 3680

3130 3100 3060 3020 2980

3680 3630 3580 3520 3470

2980 2950 2910 2870 2830

3470 3420 3370 3320 3260

2840 2810 2770 2730 2690

3270 3230 3180 3130 3070

2720 2680 2650 2610 2570

3100 3050 3000 2950 2900

22 24 26 28 30

3050 2960 2860 2760 2660

3560 3430 3300 3160 3010

2900 2810 2710 2610 2510

3350 3230 3100 2960 2830

2750 2660 2570 2470 2370

3150 3030 2900 2770 2640

2610 2520 2430 2340 2240

2960 2850 2730 2600 2470

2490 2400 2310 2210 2120

2800 2680 2570 2440 2320

32 34 36 38 40

2550 2440 2330 2220 2110

2870 2720 2570 2430 2280

2410 2300 2190 2080 1980

2690 2540 2400 2260 2120

2270 2160 2060 1950 1850

2510 2370 2240 2100 1970

2140 2030 1930 1830 1730

2340 2210 2080 1950 1830

2020 1920 1820 1710 1610

2190 2070 1940 1820 1690

1380 828 645 533

1840 1080 645 533

1250 760 595 492

1130 692 546 452

1510 901 546 452

1020 631 502 415

1360 821 502 415

926 578 462 382

1230 749 462 382



6.00 1.10

1670 991 595 492

6.00 1.10

6.00 1.10


6.00 1.10

6.00 1.10


5 - 105



b Y X

h

X

Y 20 in.×20 in. 4-#9 bars

Ties


Designation

W 12

Wt./ft

120

Fy


Steel Shape


106

96

87

79

36

50

36

50

36

50

36

50

36

50

0

2680

3100

2570

2950

2490

2830

2430

2730

2360

2640

6 7 8 9 10

2650 2630 2620 2600 2580

3050 3040 3010 2990 2970

2540 2520 2510 2490 2470

2900 2880 2860 2840 2810

2460 2440 2430 2410 2390

2780 2760 2740 2720 2700

2390 2370 2360 2340 2320

2680 2670 2650 2620 2600

2320 2310 2290 2280 2260

2590 2570 2550 2530 2510

11 12 13 14 15

2560 2540 2510 2490 2460

2940 2910 2880 2840 2800

2450 2430 2400 2380 2350

2780 2750 2720 2690 2650

2370 2350 2320 2300 2270

2670 2640 2610 2580 2540

2300 2280 2250 2230 2200

2570 2540 2510 2480 2440

2240 2210 2190 2160 2130

2480 2450 2420 2390 2350

16 17 18 19 20

2430 2400 2370 2330 2300

2770 2730 2680 2640 2590

2320 2290 2260 2220 2190

2610 2570 2530 2490 2450

2240 2210 2180 2140 2110

2500 2460 2420 2380 2340

2170 2140 2110 2070 2040

2410 2370 2330 2290 2240

2100 2070 2040 2010 1970

2320 2280 2240 2200 2150

22 24 26 28 30

2220 2140 2060 1980 1890

2500 2400 2290 2180 2070

2110 2040 1960 1870 1790

2350 2250 2150 2050 1940

2030 1960 1880 1790 1710

2240 2150 2050 1940 1840

1960 1890 1800 1720 1640

2150 2060 1960 1860 1750

1900 1820 1740 1660 1570

2060 1970 1870 1770 1670

32 34 36 38 40

1800 1710 1620 1530 1440

1960 1850 1730 1620 1510

1700 1610 1520 1430 1340

1830 1720 1610 1500 1400

1620 1530 1440 1360 1270

1730 1630 1520 1420 1320

1550 1460 1380 1290 1200

1650 1550 1440 1340 1240

1480 1400 1310 1230 1140

1570 1470 1370 1270 1170

773 501 340 340

1030 653 340 340

690 450 312 312

629 415 291 291

835 537 291 291

580 387 273 273

767 499 273 273

534 359 257 257

704 461 257 257



6.00 1.00

6.00 1.00

917 584 312 312

6.00 1.00


6.00 1.00

6.00 1.00

5 - 106

COMPOSITE DESIGN


b Y

h

X


X

Y Size b × h

20 in.×20 in.

Reinf. bars

4-#9 bars

Ties


Steel Shape

Designation

W 12

Wt./ft

72


Fy

65

58

36

50

36

50

36

50

0

2310

2560

2250

2480

2200

2400

6 7 8 9 10

2270 2250 2240 2220 2200

2510 2490 2470 2450 2430

2210 2200 2180 2170 2150

2430 2420 2400 2380 2350

2160 2140 2130 2110 2090

2350 2340 2320 2290 2270

11 12 13 14 15

2180 2160 2130 2100 2080

2400 2370 2340 2310 2270

2120 2100 2080 2050 2020

2320 2300 2270 2230 2200

2070 2040 2020 1990 1960

2240 2220 2190 2150 2120

16 17 18 19 20

2050 2010 1980 1950 1910

2240 2200 2160 2120 2080

1990 1960 1920 1890 1860

2160 2130 2090 2040 2000

1930 1900 1870 1830 1790

2080 2050 2010 1970 1920

22 24 26 28 30

1840 1760 1680 1600 1510

1990 1900 1800 1700 1600

1780 1700 1620 1540 1450

1910 1820 1730 1630 1530

1720 1640 1560 1480 1390

1840 1750 1650 1560 1460

32 34 36 38 40

1430 1340 1260 1170 1090

1500 1400 1300 1210 1110

1370 1280 1200 1120 1040

1430 1340 1240 1150 1050

1310 1220 1140 1060 978

1360 1270 1170 1080 991

453 311 229 229

593 396 229 229

417 271 214 214

544 342 214 214



494 335 242 242

649 428 242 242 6.00 1.00

6.00 1.00


6.00 1.00


5 - 107



b Y X

h

X

Y 18 in.×18 in. 4-#8 bars

Ties


Designation

W 10

Wt./ft

112

Fy


Steel Shape


100

88

77

68

36

50

36

50

36

50

36

50

36

50

0

2280

2680

2190

2540

2100

2410

2010

2280

1940

2180

6 7 8 9 10

2250 2230 2220 2200 2180

2630 2610 2590 2560 2540

2160 2140 2130 2110 2090

2490 2470 2450 2430 2410

2060 2050 2030 2020 2000

2360 2340 2320 2300 2270

1970 1960 1950 1930 1910

2230 2220 2200 2170 2150

1910 1890 1880 1860 1840

2130 2120 2100 2080 2050

11 12 13 14 15

2160 2140 2120 2090 2060

2510 2480 2450 2410 2370

2070 2050 2020 2000 1970

2380 2350 2320 2280 2250

1980 1950 1930 1900 1880

2250 2220 2190 2150 2120

1890 1870 1840 1820 1790

2120 2090 2060 2030 2000

1820 1800 1770 1750 1720

2020 2000 1970 1930 1900

16 17 18 19 20

2030 2000 1970 1940 1900

2340 2290 2250 2210 2160

1940 1910 1880 1850 1810

2210 2170 2130 2080 2040

1850 1820 1790 1750 1720

2080 2040 2000 1960 1920

1760 1730 1700 1670 1630

1960 1920 1880 1840 1800

1690 1660 1630 1600 1560

1860 1830 1790 1750 1710

22 24 26 28 30

1830 1760 1680 1600 1520

2070 1970 1870 1760 1660

1740 1670 1590 1510 1430

1950 1850 1750 1650 1550

1650 1580 1500 1420 1340

1830 1730 1640 1540 1440

1560 1490 1410 1340 1260

1710 1620 1530 1430 1340

1490 1420 1340 1270 1190

1620 1530 1440 1350 1260

32 34 36 38 40

1430 1350 1270 1180 1100

1550 1450 1340 1240 1140

1350 1270 1190 1110 1030

1450 1350 1250 1150 1050

1260 1180 1100 1020 948

1340 1250 1150 1060 968

1180 1100 1020 946 872

1240 1150 1060 970 885

1110 1030 957 883 811

1160 1070 985 900 817

599 389 248 248

800 509 248 248

541 354 228 228

482 320 208 208

640 416 208 208

426 287 190 190

565 371 190 190

382 260 175 175

505 335 175 175



5.40 1.00

5.40 1.00

721 462 228 228

5.40 1.00


5.40 1.00

5.40 1.00

5 - 108

COMPOSITE DESIGN


b Y

h

X

X


Y Size b × h

18 in.×18 in.

Reinf. bars

4-#8 bars

Ties


Steel Shape

Designation

W 10

Wt./ft

60


Fy

54

49

45

36

50

36

50

36

50

36

50

0

1880

2090

1830

2020

1790

1970

1770

1920

6 7 8 9 10

1840 1830 1810 1790 1770

2040 2020 2010 1980 1960

1790 1780 1760 1750 1730

1970 1960 1940 1910 1890

1760 1740 1730 1710 1690

1920 1900 1880 1860 1840

1730 1710 1700 1680 1660

1880 1860 1840 1820 1800

11 12 13 14 15

1750 1730 1710 1680 1650

1930 1910 1880 1840 1810

1700 1680 1660 1630 1600

1870 1840 1810 1780 1740

1670 1640 1620 1590 1560

1810 1780 1750 1720 1690

1640 1610 1590 1560 1530

1770 1740 1710 1680 1650

16 17 18 19 20

1620 1590 1560 1530 1490

1770 1740 1700 1660 1620

1570 1540 1510 1480 1440

1710 1670 1630 1590 1550

1530 1500 1470 1440 1400

1650 1620 1580 1540 1500

1500 1470 1440 1410 1370

1610 1580 1540 1500 1460

22 24 26 28 30

1420 1350 1280 1200 1120

1540 1450 1360 1270 1180

1370 1300 1230 1150 1070

1470 1380 1300 1210 1120

1330 1260 1180 1110 1030

1420 1330 1250 1160 1070

1300 1230 1150 1080 1000

1380 1290 1210 1120 1040

32 34 36 38 40

1050 970 895 822 752

1090 1000 916 834 754

996 921 847 776 707

1030 946 863 784 707

956 882 809 739 671

987 903 822 743 671

925 851 779 709 642

951 868 788 711 642

344 237 162 162

452 304 162 162

410 277 152 152

290 204 144 144

379 258 144 144

277 184 138 138

361 231 138 138



5.40 1.00

313 217 152 152 5.40 1.00

5.40 1.00


5.40 1.00


5 - 109


b Y


X

h

X

Y 16 in.×16 in. 4-#7 bars

Ties


Designation

W8

Wt./ft

67

Fy


Steel Shape


58

48

40

35

36

50

36

50

36

50

36

50

36

50

0

1640

1870

1570

1770

1490

1650

1420

1560

1390

1510

6 7 8 9 10

1600 1590 1570 1550 1530

1820 1800 1790 1760 1740

1530 1520 1500 1480 1470

1720 1710 1690 1660 1640

1450 1440 1420 1400 1380

1610 1590 1570 1550 1530

1380 1370 1360 1340 1320

1520 1500 1480 1460 1440

1350 1330 1320 1300 1280

1460 1450 1430 1410 1380

11 12 13 14 15

1510 1490 1470 1440 1420

1710 1680 1650 1620 1590

1440 1420 1400 1370 1350

1620 1590 1560 1530 1490

1360 1340 1320 1290 1260

1500 1470 1450 1410 1380

1300 1270 1250 1220 1200

1410 1380 1360 1320 1290

1260 1240 1210 1180 1160

1360 1330 1300 1270 1240

16 17 18 19 20

1390 1360 1330 1300 1270

1550 1520 1480 1440 1400

1320 1290 1260 1230 1200

1460 1420 1380 1350 1310

1240 1210 1180 1150 1120

1350 1310 1280 1240 1200

1170 1140 1110 1080 1050

1260 1220 1190 1150 1120

1130 1100 1070 1040 1010

1210 1170 1140 1100 1060

22 24 26 28 30

1200 1130 1060 993 922

1310 1230 1140 1060 972

1130 1060 996 926 857

1230 1140 1060 977 895

1050 983 915 846 778

1120 1040 963 884 805

981 914 847 779 713

1040 961 883 807 731

940 873 806 739 673

989 912 836 761 687

32 34 36 38 40

852 784 717 653 590

888 806 729 654 590

789 722 657 595 537

815 737 663 595 537

712 648 586 526 475

729 656 586 526 475

648 586 525 471 425

659 589 525 471 425

609 548 490 439 397

617 549 490 439 397

307 206 127 127

408 268 127 127

272 186 115 115

228 157 102 102

300 202 102 102

197 140 91.3 91.3

258 178 91.3 91.3

177 127 85.1 85.1

230 160 85.1 85.1



4.80 1.00

359 240 115 115 4.80 1.00

4.80 1.00


4.80 1.00

4.80 1.00

5 - 110

COMPOSITE DESIGN

COMPOSITE COLUMNS—CONCRETE-FILLED STEEL PIPE AND STRUCTURAL TUBING General Notes

Concentric load design strengths in the tables that follow are tabulated for the effective lengths KL in feet, shown at the left of each table. They are applicable to axially loaded members with respect to their minor axis in accordance with Section I2.2 of the LRFD Specification. The tables apply to normal-weight concrete. For discussion of the effective length, range of Kl / r strength about the major axis, combined axial and bending strength, and for sample problems, see Composite Columns, General Notes. The properties listed at the bottom of each table are for use in checking strength about the strong axis and in design for combined axial compression and bending. The heavy horizontal lines within the tables indicate Kl / r = 200. No values are listed beyond these lines. Steel Pipe Filled with Concrete

Design strengths for filled pipe are tabulated for Fy = 36 ksi and fc′ equal to 3.5 and 5 ksi. Steel pipe is manufactured to Fy = 36 ksi under ASTM A501 and to Fy = 35 ksi under ASTM A53 Types E or S, Grade B. Both are designed for 36 ksi yield stress. Structural Tubing Filled with Concrete

Design strengths for square and rectangular structural tubing filled with concrete are tabulated for Fy = 46 ksi and fc′ equal to 3.5 and 5 ksi. Structural tubing is manufactured to Fy = 46 ksi under ASTM A500, Grade B.


COMPOSITE COLUMNS—CONCRETE-FILLED STEEL PIPE AND STRUCTURAL TUBING

5 - 111

Fy = 36 ksi

Nominal Diameter (in.)

12

10

8

Wall Thickness (in.)

0.500

0.375

0.500

0.365

0.875

0.500

0.322

Wt./ft

65.42

49.56

54.74

40.48

72.42

43.39

28.55

Fy

Effective length KL (ft) with respect to radius of gyration

Steel Pipe

COMPOSITE COLUMNS Steel pipe fc′ = 3.5 ksi Axial design strength in kips

36 ksi 0

862

733

681

564

746

507

384

6 7 8 9 10

847 842 836 829 822

720 716 711 705 698

665 660 653 646 638

550 545 540 534 527

718 708 697 684 671

489 482 475 467 458

370 365 359 353 346

11 12 13 14 15

814 805 796 786 775

691 684 675 667 658

629 620 610 599 587

520 512 503 494 485

656 640 623 606 588

448 438 427 415 403

338 330 322 313 304

16 17 18 19 20

764 752 739 727 713

648 638 627 616 604

576 563 550 537 523

475 464 454 442 431

569 549 529 509 488

390 377 364 351 337

294 284 274 264 254

22 24 26 28 30

686 656 626 595 563

580 555 529 502 475

495 466 436 406 376

407 383 358 333 308

447 405 365 325 288

309 281 254 228 202

232 211 191 170 151

32 34 36 38 40

531 499 467 436 405

448 420 393 366 339

347 318 290 263 237

284 260 236 214 193

253 224 200 179 162

178 158 141 126 114

133 118 105 94 85

3.63 142 51.0

3.67 106 41.4

2.76 143 34.8

2.88 89.2 24.5

2.94 60.0 18.3

Properties 4.33 203 89.8

rm (in.) φb M n (kip-ft)

P e (KL )2 / 10 4 (kip-ft2)

4.38 155 75.1



5 - 112

COMPOSITE DESIGN

Fy = 36 ksi


6

5

4


0.864

0.432

0.280

0.750

0.375

0.258

0.674

0.337

0.237

Wt./ft

53.16

28.57

18.97

38.55

20.78

14.62

27.54

14.98

10.79

Fy


Steel Pipe

COMPOSITE COLUMNS Steel pipe fc′ = 3.5 ksi Axial design strength in kips

36 ksi 0

525

323

244

379

233

182

268

164

129

6 7 8 9 10

491 479 466 451 436

303 297 289 281 271

229 224 218 212 205

344 332 319 305 290

213 206 199 191 182

167 161 156 149 142

230 218 205 191 176

143 136 129 121 112

113 107 102 95 89

11 12 13 14 15

419 401 383 364 345

262 252 241 230 219

197 190 182 173 165

274 258 241 225 208

173 163 154 144 134

135 128 120 112 105

162 147 132 118 105

104 95 87 78 70

82 75 69 62 56

16 17 18 19 20

326 306 287 268 249

207 196 184 173 161

156 147 139 130 121

191 175 160 145 131

124 114 105 96 87

97 89 82 75 68

92 82 73 65 59

62 55 49 44 40

49 44 39 35 32

22 24 26 28 30

213 180 153 132 115

139 119 101 87 76

105 89 76 66 57

108 91 77 67

72 60 51 44 39

56 47 40 35 30

49

33 28

26 22

32 34 36

101 90

67 59 53

50 45 40

1.72 47.3 6.99

1.84 27.3 4.66

1.88 19.6 3.65

1.37 26.9 3.15

1.48 15.8 2.15

1.51 11.6 1.70

Properties 2.06 78.0 13.9


P e (KL )2 / 10 4 (kip-ft2)

2.19 44.8 9.13

2.25 30.5 6.92




5 - 113

Fy = 36 ksi


12

10

8


0.500

0.375

0.500

0.365

0.875

0.500

0.322

Wt./ft

65.42

49.56

54.74

40.48

72.42

43.39

28.55

Fy


Steel Pipe

COMPOSITE COLUMNS Steel pipe fc′ = 5 ksi Axial design strength in kips

36 ksi 0

979

855

762

649

786

557

438

6 7 8 9 10

961 955 947 939 930

839 833 827 819 811

743 736 728 720 710

632 626 619 611 603

755 745 732 719 704

535 528 519 510 499

420 414 407 399 391

11 12 13 14 15

920 909 897 885 872

802 792 781 770 758

699 688 676 663 649

594 584 573 562 550

688 671 653 633 614

488 476 463 449 435

382 372 361 351 339

16 17 18 19 20

858 844 828 813 797

746 733 719 705 691

635 620 605 589 573

537 524 511 497 483

593 572 550 528 506

421 406 391 375 360

327 315 303 291 278

22 24 26 28 30

763 727 691 653 615

660 628 596 562 528

540 505 471 436 401

454 424 394 364 334

461 417 374 332 292

328 297 266 236 208

253 228 203 180 158

32 34 36 38 40

577 539 502 465 429

495 461 428 395 363

367 334 302 272 245

305 276 249 224 202

256 227 203 182 164

183 162 145 130 117

139 123 109 98.2 88.7

3.63 142 52.7

3.67 106 43.2

2.76 143 35.3

2.88 89.2 25.2

2.94 60.0 19.1



P e (KL )2 / 10 4 (kip-ft2)

4.38 155 78.9



5 - 114

COMPOSITE DESIGN

Fy = 36 ksi


6

5

4


0.864

0.432

0.280

0.750

0.375

0.258

0.674

0.337

0.237

Wt./ft

53.16

28.57

18.97

38.55

20.78

14.62

27.54

14.98

10.79

Fy


Steel Pipe

COMPOSITE COLUMNS Steel pipe fc′ = 5 ksi Axial design strength in kips

36 ksi 0

545

351

275

393

253

204

276

176

143

6 7 8 9 10

509 496 482 467 450

329 321 312 302 292

257 251 244 236 228

356 343 330 315 299

230 222 214 204 195

185 179 172 164 156

237 224 210 196 180

153 145 137 128 118

124 117 110 103 96

11 12 13 14 15

432 414 394 374 354

281 269 257 245 232

219 209 200 190 180

282 265 247 230 212

184 173 162 151 140

148 139 130 121 112

165 149 134 120 106

109 99 90 81 72

88 80 72 65 58

16 17 18 19 20

334 313 293 273 253

219 206 193 180 168

169 159 149 139 129

195 178 162 146 132

129 119 108 98 89

103 94 86 78 70

93 82 74 66 60

64 56 50 45 41

51 45 40 36 33

22 24 26 28 30

216 181 155 133 116

144 121 104 89 78

110 93 79 68 59

109 92 78 67

73 62 53 45 40

58 49 42 36

49

34 28

27

32 34 36

102 90

68 61 54

52 46

1.84 27.3 4.77

1.88 19.6 3.78

1.48 15.8 2.19

1.51 11.6 1.75

23

31

41



P e (KL )2 / 10 4 (kip-ft2)

2.19 44.8 9.35

2.25 30.5 7.18

1.72 47.3 7.06



1.37 26.9 3.18


5 - 115

Fy = 46 ksi

Steel Tube

COMPOSITE COLUMNS Square structural tubing fc′ = 3.5 ksi Axial design strength in kips Nominal Size

16× ×16

Thickness (in.)

1⁄ 2

1⁄ 2

14× ×14 3⁄ 8

5⁄ 8

1⁄ 2

12× ×12 3⁄ 8

5⁄ 16

Wt./ft

103.30

89.68

68.31

93.34

76.07

58.10

48.86


Fy

46 ksi 0

1760

1460

1230

1360

1180

988

890

6 7 8 9 10

1740 1730 1730 1720 1710

1440 1430 1430 1420 1410

1210 1210 1200 1190 1190

1340 1330 1320 1310 1300

1160 1150 1150 1140 1130

971 965 958 950 942

875 869 863 856 848

11 12 13 14 15

1700 1690 1680 1670 1650

1400 1390 1370 1360 1350

1180 1170 1160 1150 1140

1280 1270 1250 1240 1220

1110 1100 1090 1070 1060

932 922 911 899 887

839 830 820 810 798

16 17 18 19 20

1640 1620 1610 1590 1580

1330 1320 1300 1280 1270

1120 1110 1100 1080 1070

1200 1180 1160 1140 1120

1040 1030 1010 992 973

874 860 845 830 815

786 774 761 747 734

22 24 26 28 30

1540 1500 1460 1420 1380

1230 1190 1150 1110 1060

1040 1000 968 931 894

1080 1030 979 928 877

934 894 852 808 764

783 749 713 677 640

704 674 642 609 576

32 34 36 38 40

1330 1280 1240 1190 1140

1020 969 922 875 828

855 816 777 737 697

826 774 723 672 623

720 675 631 587 544

603 566 529 493 457

543 509 476 443 411

4.60 400 137

4.66 329 120

4.72 255 101

4.75 216 90.7

Properties rm (in.) φb M n (kip-ft)

P e (KL )2 / 10 4 (kip-ft2)

6.29 604 319

5.48 455 203

5.54 352 171



5 - 116

COMPOSITE DESIGN

Fy = 46 ksi

10× ×10

Nominal Size

8× ×8

Thickness (in.)

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

5⁄ 8

1⁄ 2

3⁄

Wt./ft

76.33

62.46

47.90

40.35

32.63

59.32

48.85

Fy


Steel Tube

COMPOSITE COLUMNS Square structural tubing fc′ = 3.5 ksi Axial design strength in kips

16

1⁄ 4

37.69

31.84

25.82

8

5⁄

46 ksi 0

1070

924

767

687

603

795

686

567

503

439

6 7 8 9 10

1040 1030 1020 1010 995

900 892 883 872 861

748 742 734 725 716

670 664 657 650 641

588 583 577 570 562

762 751 738 724 708

659 650 639 627 614

545 537 528 519 508

484 477 470 461 452

422 416 409 402 394

11 12 13 14 15

980 964 947 929 909

848 835 820 805 788

705 694 682 669 656

632 622 611 600 588

554 545 536 526 515

691 673 654 634 613

600 584 568 551 534

496 484 471 457 443

441 431 419 407 394

385 376 365 355 344

16 17 18 19 20

889 869 847 825 802

771 754 735 716 697

642 627 612 597 581

575 562 549 535 520

504 493 481 469 456

591 569 546 524 501

516 497 478 459 439

428 413 398 382 366

381 368 354 340 326

333 321 309 297 285

22 24 26 28 30

755 707 658 609 560

657 616 574 532 490

548 514 479 445 410

491 460 430 398 368

430 403 376 349 322

454 408 364 321 280

400 361 323 286 251

334 302 271 241 212

298 269 242 215 189

260 235 211 188 165

32 34 36 38 40

513 466 422 379 342

449 409 371 333 301

376 343 311 280 253

337 308 279 251 227

295 269 244 220 198

246 218 195 175 158

220 195 174 156 141

186 165 147 132 119

166 147 131 118 106

145 129 115 103 93

3.12 92.1 22.8

3.15 75.6 20.0


P e (KL )2 / 10 4 (kip-ft2)

3.78 3.84 3.90 3.93 3.96 2.96 3.03 3.09 268 223 174 148 120 163 137 108 73.4 64.6 54.3 48.7 42.6 33.9 30.3 25.5




5 - 117

Fy = 46 ksi

COMPOSITE COLUMNS Square structural tubing fc′ = 3.5 ksi Axial design strength in kips 7× ×7

Steel Tube

Nominal Size Thickness (in.)

5⁄

Wt./ft

50.81

1⁄

8

2

42.05

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

32.58

27.59

22.42

17.08


Fy

46 ksi 0

667

575

473

420

364

307

6 7 8 9 10

631 618 604 588 571

545 535 523 510 496

449 441 431 421 410

399 391 383 374 364

346 340 333 325 316

292 287 281 274 267

11 12 13 14 15

553 534 514 493 471

481 465 448 430 412

397 384 371 357 342

353 342 330 317 305

307 297 287 276 265

259 250 242 233 223

16 17 18 19 20

449 427 405 382 360

394 375 356 337 318

327 312 297 281 266

291 278 265 251 237

254 242 230 219 207

214 204 194 184 174

22 24 26 28 30

316 274 235 202 176

281 245 211 182 158

236 206 178 154 134

211 185 160 138 120

184 161 140 120 105

155 136 118 101 88

32 34 36 38 40

155 137 122 110 99

139 123 110 99 89

118 104 93 84 75

106 94 83 75 68

92 82 73 65 59

78 69 61 55 50

2.68 81.1 16.2

2.71 69.3 14.5

2.74 56.9 12.7

2.77 43.8 10.7



P e (KL )2 / 10 4 (kip-ft2)

2.62 102 19.1



5 - 118

COMPOSITE DESIGN

Fy = 46 ksi


Steel Tube


5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄

4

3⁄ 16

Wt./ft

42.30

35.24

27.48

23.34

19.02

14.53


Fy

46 ksi 0

542

469

385

341

295

247

6 7 8 9 10

502 488 472 455 436

436 424 411 397 382

359 350 339 328 316

318 310 301 291 280

275 268 261 252 243

230 225 218 211 204

11 12 13 14 15

417 397 376 354 333

365 348 331 313 295

303 289 275 261 246

269 257 245 232 220

233 223 213 202 191

196 187 178 169 160

16 17 18 19 20

311 290 268 248 228

276 258 240 222 205

232 217 202 188 174

207 194 181 168 156

180 169 157 147 136

151 141 132 123 114

22 24 26 28 30

189 159 136 117 102

172 145 123 106 93

147 123 105 91 79

132 111 95 82 71

116 97 83 71 62

97 82 70 60 52

32 34

90 79

81 72

69 62

62 55

55 48

46 41

64

55

36 38

49

43

36

44

39

33

2.30 49.7 8.58

2.33 41.1 7.50

2.36 31.9 6.30



P e (KL )2 / 10 4 (kip-ft2)

2.21 72.1 11.2

2.27 58.0 9.54




5 - 119

Fy = 46 ksi

51⁄2×51⁄2


3⁄ 8

5⁄ 16

1⁄ 4

Wt./ft

24.93

21.21

17.32

5× ×5 3⁄

16

13.25

Fy


Steel Tube

COMPOSITE COLUMNS Square structural tubing fc′ = 3.5 ksi Axial design strength in kips

1⁄ 8

1⁄

2

3⁄ 8

9.01

28.43

5⁄

16

1⁄ 4

3⁄ 16

22.37

19.08

15.62

11.97

46 ksi 0

343

304

262

219

173

367

303

268

231

192

6 7 8 9 10

315 305 294 283 270

279 271 261 251 240

241 234 226 217 208

201 195 189 182 174

159 155 149 144 137

328 315 301 286 269

272 262 251 239 226

241 232 223 212 201

208 201 193 184 174

173 167 160 153 145

11 12 13 14 15

257 243 229 215 200

229 217 204 192 179

198 188 177 167 156

166 157 148 139 130

131 124 117 110 103

252 235 218 200 183

212 198 184 170 156

189 177 165 152 140

164 154 143 133 122

137 128 120 111 102

16 17 18 19 20

186 172 158 145 132

167 154 142 130 119

145 134 124 114 104

121 113 104 95 87

96 89 82 75 69

166 150 134 121 109

143 130 117 105 95

128 117 106 95 86

112 102 93 83 75

94 86 78 70 63

22 24 26 28 30

109 91 78 67 59

98 82 70 61 53

86 72 62 53 46

72 61 52 45 39

57 48 41 35 31

90 76 64 56 48

78 66 56 48 42

71 59 51 44 38

62 52 44 38 33

52 44 37 32 28

32 34 36

51 46

46 41

41 36

34 30

27 24 21

29

25

1.92 27.8 4.03

1.95 21.7 3.39


P e (KL )2 / 10 4 (kip-ft2)

2.07 47.7 7.07

2.10 41.2 6.37

2.13 34.2 5.58

2.16 26.5 4.69

2.19 18.3 3.69

1.80 47.3 5.84

1.86 38.6 5.08



1.89 33.5 4.59

5 - 120

COMPOSITE DESIGN

Fy = 46 ksi

COMPOSITE COLUMNS Square structural tubing fc′ = 3.5 ksi Axial design strength in kips 41⁄2×41⁄2

Steel Tube


3⁄ 8

Wt./ft

19.82

5⁄

16.96

Fy


1⁄

16

4

13.91

3⁄ 16

1⁄ 8

10.70

7.31

46 ksi 0

263

233

200

166

130

6 7 8 9 10

230 220 208 195 182

204 195 185 174 163

177 169 160 151 141

147 140 133 126 118

115 110 104 98 92

11 12 13 14 15

168 155 141 128 115

151 139 127 115 104

131 121 111 101 91

109 101 93 84 76

86 79 73 66 60

16 17 18 19 20

102 91 81 73 66

93 83 74 66 60

82 73 65 58 53

69 61 55 49 44

54 48 43 38 35

22 24 26 28

54 46 39

49 41 35 30

43 36 31 27

37 31 26 23

29 24 21 18

1.69 26.5 3.20

1.72 22.2 2.82

1.75 17.4 2.37

1.78 12.1 1.86



P e (KL )2 / 10 4 (kip-ft2) Note: Heavy line indicates Kl / r of 200.



5 - 121

Fy = 46 ksi


Steel Tube


1⁄

Wt./ft

21.63

2

3⁄

8

17.27

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

14.83

12.21

9.42

6.46


Fy

46 ksi 0

271

225

199

171

141

110

6 7 8 9 10

225 211 195 179 162

189 178 166 153 140

169 159 148 137 125

146 137 128 119 109

121 114 107 99 91

94 89 83 77 71

11 12 13 14 15

145 129 114 99 86

126 113 100 88 77

114 102 91 80 70

99 90 80 71 62

83 75 67 60 52

65 59 53 47 41

16 17 18 19 20

76 67 60 54 48

68 60 53 48 43

62 55 49 44 40

55 48 43 39 35

46 41 36 33 30

36 32 29 26 23

22 24 26

40

36 30

33 27

29 24

24 21

1.48 20.4 2.12

1.51 17.1 1.88

1.54 13.5 1.58

19 16 14



P e (KL )2 / 10 4 (kip-ft2)

1.45 23.2 2.33



1.57 9.5 1.24

5 - 122

COMPOSITE DESIGN

Fy = 46 ksi

Steel Tube

COMPOSITE COLUMNS Square structural tubing fc′ = 5 ksi Axial design strength in kips Nominal Size

16×16

Thickness (in.)

1⁄ 2

1⁄ 2

14×14 3⁄

12×12

Wt./ft

103.30

89.68

68.31

8


Fy

5⁄ 8

1⁄ 2

3⁄

8

5⁄ 16

93.34

76.07

58.10

48.86

46 ksi 0

2000

1640

1420

1490

1310

1130

1030

6 7 8 9 10

1980 1970 1960 1950 1940

1620 1610 1600 1590 1580

1400 1390 1380 1380 1370

1460 1450 1440 1430 1410

1290 1280 1270 1260 1250

1100 1100 1090 1080 1070

1010 1000 996 987 977

11 12 13 14 15

1930 1920 1900 1890 1870

1570 1550 1540 1520 1510

1350 1340 1330 1320 1300

1400 1380 1360 1340 1320

1230 1220 1200 1190 1170

1060 1040 1030 1020 1000

966 954 942 928 914

16 17 18 19 20

1860 1840 1820 1800 1780

1490 1470 1450 1430 1410

1290 1270 1250 1230 1220

1300 1280 1260 1230 1210

1150 1130 1110 1090 1070

984 967 950 932 913

899 884 868 851 833

22 24 26 28 30

1730 1690 1640 1590 1530

1360 1320 1270 1220 1160

1180 1140 1090 1050 1000

1160 1100 1050 991 933

1020 974 924 874 823

873 832 790 747 703

797 759 720 680 640

32 34 36 38 40

1480 1420 1370 1310 1250

1110 1060 1000 946 891

954 907 859 811 763

875 817 760 704 649

772 721 671 621 573

658 614 571 528 487

599 559 519 480 442

5.54 352 180

4.60 400 141

4.66 329 125

4.72 255 106

4.75 216 95.8

Properties 6.29 604 334


P e (KL )2 / 10 4 (kip-ft2)

5.48 455 212




5 - 123

Fy = 46 ksi

Nominal Size

10×10

Thickness (in.)

5⁄ 8

1⁄

Wt./ft

76.33

62.46

2

3⁄ 8

47.90

8×8 5⁄

16

40.35

Fy


Steel Tube

COMPOSITE COLUMNS Square structural tubing fc′ = 5 ksi Axial design strength in kips

1⁄ 4

5⁄

8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

32.63

59.32

48.85

37.69

31.84

25.82

46 ksi 0

1150

1010

860

782

701

844

739

623

562

500

6 7 8 9 10

1120 1110 1100 1080 1070

984 975 964 952 938

837 829 820 809 798

761 754 745 736 725

682 675 667 659 649

808 796 781 766 748

709 698 686 672 657

598 589 579 567 555

539 531 522 512 500

479 472 464 454 444

11 12 13 14 15

1050 1030 1010 993 972

924 908 891 873 855

785 772 758 742 727

714 702 689 675 660

639 628 616 603 590

730 710 689 667 644

641 624 606 587 568

542 527 512 496 480

488 475 462 447 433

434 422 410 397 384

16 17 18 19 20

949 926 902 877 852

835 815 794 772 750

710 693 675 656 637

645 629 613 596 578

576 562 547 531 516

620 596 571 547 522

548 527 506 484 463

463 446 428 410 392

417 401 385 369 353

370 356 341 327 312

22 24 26 28 30

800 746 692 638 585

704 658 610 563 516

598 559 518 478 438

543 507 470 433 397

484 451 417 384 352

471 422 374 328 286

419 376 335 295 257

355 319 284 251 219

320 287 256 226 197

283 254 226 199 173

32 34 36 38 40

532 482 433 389 351

471 426 383 344 311

399 362 325 292 263

361 327 294 263 238

320 289 259 232 210

251 223 199 178 161

226 200 179 160 145

192 170 152 136 123

173 153 137 123 111

152 135 120 108 98

3.12 92.1 23.8

3.15 75.6 20.9


P e (KL )2 / 10 4 (kip-ft2)

3.78 3.84 3.90 3.93 3.96 2.96 3.03 3.09 268 223 174 148 120 163 137 108 75.3 66.7 56.5 51.0 45.0 34.5 31.1 26.4



5 - 124

COMPOSITE DESIGN

Fy = 46 ksi

COMPOSITE COLUMNS Square structural tubing fc′ = 5 ksi Axial design strength in kips 7× ×7

Steel Tube


5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄

4

3⁄ 16

Wt./ft

50.81

42.05

32.58

27.59

22.42

17.08


Fy

46 ksi 0

702

614

515

464

410

355

6 7 8 9 10

663 649 634 617 599

581 569 556 542 526

488 478 467 455 442

439 430 421 410 398

388 381 372 362 352

335 329 321 313 304

11 12 13 14 15

579 558 536 514 490

509 491 473 453 433

429 414 398 382 366

386 373 359 344 329

341 329 317 304 291

294 284 273 262 250

16 17 18 19 20

467 443 419 395 371

413 393 372 351 331

349 332 315 297 280

314 299 283 268 252

277 264 250 236 223

238 227 215 203 191

22 24 26 28 30

324 280 239 206 179

290 252 216 186 162

247 214 184 159 138

222 193 166 143 124

196 170 146 126 110

167 145 124 107 93

32 34 36 38 40

158 140 125 112 101

142 126 112 101 91

121 107 96 86 78

109 97 86 78 70

96 85 76 68 62

82 73 65 58 52

2.68 81.1 16.7

2.71 69.3 15.0

2.74 56.9 13.2

2.77 43.8 11.3



P e (KL )2 / 10 4 (kip-ft2)

2.62 102 19.5




5 - 125

Fy = 46 ksi


Steel Tube


5⁄

Wt./ft

42.30

8

1⁄

2

35.24

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

27.48

23.34

19.02

14.53


Fy

46 ksi 0

566

496

415

372

328

281

6 7 8 9 10

523 508 491 473 453

459 447 432 417 400

385 375 363 350 337

346 336 326 315 303

304 296 287 277 266

261 254 246 237 228

11 12 13 14 15

432 410 388 365 343

383 364 345 326 306

322 307 292 276 259

290 276 262 248 234

255 243 231 218 206

218 208 197 187 176

16 17 18 19 20

320 297 275 253 232

286 267 248 229 210

243 227 211 195 180

219 205 190 176 162

193 180 167 155 143

164 154 143 132 122

22 24 26 28 30

192 161 138 119 103

175 147 126 108 94

151 127 108 93 81

136 114 98 84 73

120 101 86 74 64

102 86 73 63 55

32 34 36 38

91 80

83 73 66

71 63 56

64 57 51 46

57 50 45 40

48 43 38 34

2.27 58.0 9.79

2.30 49.7 8.84

2.33 41.1 7.79

2.36 31.9 6.61



P e (KL )2 / 10 4 (kip-ft2)

2.21 72.1 11.4



5 - 126

COMPOSITE DESIGN

Fy = 46 ksi

51⁄2×51⁄2


3⁄ 8

Wt./ft

24.93

5⁄

16

21.21

5× ×5

1⁄ 4

3⁄ 16

1⁄

17.32

13.25

Fy


Steel Tube

COMPOSITE COLUMNS Square structural tubing fc′ = 5 ksi Axial design strength in kips

8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄

4

3⁄ 16

9.01

28.43

22.37

19.08

15.62

11.97

46 ksi 0

367

329

289

247

203

384

322

288

252

215

6 7 8 9 10

336 325 313 300 286

301 292 281 269 257

265 256 247 237 226

226 219 211 202 193

185 179 172 165 157

342 328 313 297 279

289 277 265 251 237

259 249 238 226 213

227 218 208 198 187

193 185 177 169 159

11 12 13 14 15

272 256 241 225 209

244 231 217 203 189

215 203 191 178 166

183 173 162 152 141

149 141 132 123 114

261 243 224 206 187

223 207 192 177 162

200 187 173 160 146

176 164 152 141 129

149 140 129 119 109

16 17 18 19 20

194 178 163 149 135

175 161 148 135 122

154 142 130 119 108

131 121 111 101 91

106 97 89 81 73

170 153 136 122 111

147 133 119 107 97

133 121 108 97 88

117 106 96 86 78

100 90 81 73 66

22 24 26 28 30

111 94 80 69 60

101 85 72 62 54

89 75 64 55 48

76 63 54 47 41

60 51 43 37 32

91 77 65 56 49

80 67 57 49 43

73 61 52 45 39

64 54 46 40 34

54 46 39 34 29

30

26

1.92 27.8 4.16

1.95 21.7 3.53

32

53

48

42

36

29

34 36

47

42

37

32

25 23


P e (KL )2 / 10 4 (kip-ft2)

2.07 47.7 7.23

2.10 41.2 6.55

2.13 34.2 5.78

2.16 26.5 4.90

2.19 18.3 3.92

1.80 47.3 5.93



1.86 38.6 5.19

1.89 33.5 4.71


5 - 127

Fy = 46 ksi

COMPOSITE COLUMNS Square structural tubing fc′ = 5 ksi Axial design strength in kips 41⁄2×41⁄2

Steel Tube


3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 8

Wt./ft

19.82

16.96

13.91

10.70

7.31


Fy

46 ksi 0

278

249

217

184

149

6 7 8 9 10

242 231 218 204 190

217 207 196 184 171

190 181 172 161 150

161 154 146 137 127

130 124 117 110 103

11 12 13 14 15

175 160 146 132 118

158 145 132 119 107

139 128 117 105 95

118 108 99 90 80

95 87 79 72 64

16 17 18 19 20

104 92 82 74 67

95 84 75 68 61

84 75 67 60 54

72 64 57 51 46

57 51 45 40 37

22 24 26 28

55 46 40

50 42 36 31

45 38 32 28

38 32 27 23

30 25 22 19

1.69 26.5 3.27

1.72 22.2 2.90

1.75 17.4 2.46

1.78 12.1 1.96



P e (KL )2 / 10 4 (kip-ft2) Note: Heavy line indicates Kl / r of 200.


5 - 128

COMPOSITE DESIGN

Fy = 46 ksi


Steel Tube


1⁄ 2

3⁄ 8

5⁄ 16

1⁄

Wt./ft

21.63

17.27

14.83

12.21


Fy

4

3⁄ 16

1⁄ 8

9.42

6.46

46 ksi 0

280

236

211

184

156

125

6 7 8 9 10

232 217 200 183 166

198 186 172 158 144

178 167 155 143 131

156 146 136 126 115

132 124 115 107 98

105 99 92 85 78

11 12 13 14 15

148 131 115 100 87

130 116 103 90 78

118 106 94 82 72

104 93 83 73 64

88 79 71 62 54

71 63 56 49 43

16 17 18 19 20

76 68 60 54 49

69 61 54 49 44

63 56 50 45 40

56 50 44 40 36

48 42 38 34 31

38 34 30 27 24

22 24 26

40

36 31

33 28

30 25

25 21

20 17 14

1.48 20.4 2.16

1.51 17.1 1.92

1.54 13.5 1.64

1.57 9.45 1.30



P e (KL )2 / 10 4 (kip-ft2)

1.45 23.2 2.36




Fy = 46 ksi

5 - 129

Y

COMPOSITE COLUMNS Rectangular structural tubing fc′ = 3.5 ksi Axial design strength in kips

X

X

Nominal Size

16× ×12

16× ×8

Thickness (in.)

1⁄

1⁄ 2

1⁄ 2

3⁄ 8

1⁄

2

3⁄ 8

5⁄ 16

Wt./ft

89.68

76.07

76.07

58.10

69.27

53.00

44.60

2

14× ×10

Fy

Effective length KL (ft) with respect to least radius of gyration, rmy

Steel Tube

Y

12× ×10

46 ksi 0

1450

1140

1170

978

1050

872

782

6 7 8 9 10

1430 1420 1410 1400 1380

1100 1090 1070 1060 1040

1140 1130 1120 1110 1100

956 948 939 928 917

1020 1010 1000 990 978

851 844 835 826 816

763 757 749 741 732

11 12 13 14 15

1370 1360 1340 1330 1310

1020 994 970 946 920

1080 1070 1050 1030 1010

905 892 877 862 846

964 949 933 917 899

804 792 779 765 750

721 710 699 686 673

16 17 18 19 20

1290 1270 1250 1230 1210

893 865 837 808 778

992 971 949 926 903

830 812 794 775 756

880 861 841 820 799

735 719 703 685 668

659 645 630 615 599

22 24 26 28 30

1160 1120 1070 1020 964

718 657 597 539 482

855 806 755 704 653

716 675 633 591 548

755 710 664 617 571

632 594 556 517 479

566 533 498 464 429

32 34 36 38 40

911 858 805 753 702

427 379 338 303 273

602 553 505 459 414

506 465 425 386 349

525 481 438 395 357

441 404 368 333 300

395 362 330 298 269

4.02 1.30 362 288 150 88.8

4.08 1.29 281 224 125 74.8

3.94 1.15 290 256 101 76.6

4.00 1.15 225 199 85.3 64.4

4.03 1.15 191 168 76.4 57.8

Properties 4.84 1.25 497 407 245 157

rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft) P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2)

3.30 1.72 390 240 174 58.7



5 - 130

COMPOSITE DESIGN

Fy = 46 ksi

Y

X


X

12× ×8

Nominal Size

12× ×6

Thickness (in.)

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

Wt./ft

76.33

62.46

47.90

40.35

67.82

55.66

42.79

36.10

Fy


Steel Tube

Y

46 ksi 0

1060

914

757

677

906

780

642

569

6 7 8 9 10

1020 1010 991 973 954

881 869 856 842 826

731 721 711 699 686

653 645 635 625 613

850 830 808 783 757

732 716 697 677 655

604 591 576 559 542

536 524 511 497 481

11 12 13 14 15

934 912 888 864 838

808 790 770 749 728

671 656 640 623 606

600 587 572 557 541

729 699 668 637 604

631 606 580 553 526

523 503 482 460 438

464 447 429 410 390

16 17 18 19 20

812 785 757 728 699

705 682 658 634 610

587 568 549 529 509

525 508 491 473 455

571 538 506 473 441

498 470 443 415 387

416 393 371 348 326

370 351 331 311 291

22 24 26 28 30

641 583 525 470 416

560 510 461 413 368

468 427 387 348 310

418 382 346 310 277

379 321 273 236 205

335 284 242 209 182

283 241 206 177 154

253 216 184 159 138

32 34 36 38

366 324 289 260

324 287 256 230

273 242 216 194

244 216 193 173

180 160 142 128

160 142 126 113

136 120 107 96

122 108 96 86

40

234

207

175

156

102

87

78

Properties 3.14 1.38 300 226 95.2 50.3 Note: Heavy line indicates Kl / r of 200. rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft) P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2)

3.20 1.37 250 189 83.7 44.5

3.26 1.37 195 147 69.9 37.5

3.28 1.37 165 125 62.8 33.5

2.37 1.73 252 154 74.5 24.8


2.42 1.73 210 129 65.8 22.0

2.48 1.72 165 101 55.0 18.6

2.51 1.71 140 86.6 49.1 16.7


Fy = 46 ksi

5 - 131

Y


X

X

10× ×8


1⁄

Wt./ft

55.66

2

3⁄

8

42.79

10× ×6 5⁄ 16

1⁄ 4

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

36.10

29.23

48.85

37.69

31.84

25.82

Fy


Steel Tube

Y

46 ksi 0

800

662

589

516

676

556

493

429

6 7 8 9 10

770 759 747 734 720

638 629 619 608 597

568 560 551 542 531

497 490 483 475 465

633 619 602 584 564

522 511 497 483 467

463 453 441 428 415

403 394 384 373 361

11 12 13 14 15

704 687 669 650 630

584 570 555 540 524

520 508 495 481 467

455 445 433 421 409

543 521 497 474 449

450 432 413 394 375

400 384 368 351 334

348 335 320 306 291

16 17 18 19 20

610 589 568 546 524

507 490 473 455 437

452 437 422 406 390

396 383 369 355 341

425 400 375 351 327

355 335 315 295 275

316 299 281 263 246

276 261 245 230 215

22 24 26 28 30

479 435 391 349 308

400 364 328 293 260

357 325 293 262 233

313 285 257 230 204

281 237 202 174 152

238 202 172 148 129

213 181 154 133 116

186 159 135 116 101

32 34 36 38

271 240 214 192

228 202 181 162

204 181 162 145

179 159 142 127

133 118 105 95

113 101 90 80

102 90 80 72

89 79 70 63

40

173

146

131

115

73

65

57

2.43 1.49 124 86.9 34.6 15.6

2.46 1.48 106 74.2 30.8 14.0

2.49 1.48 86.6 61.1 26.9 12.3

Properties rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft) P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2)

3.12 1.19 190 163 52.9 37.2

3.18 1.19 149 128 44.3 31.4

3.21 1.19 127 109 39.6 28.1

3.24 1.19 104 89.0 34.6 24.6

2.37 1.50 157 110 41.1 18.3



5 - 132

COMPOSITE DESIGN

Fy = 46 ksi

Y

X


X

10× ×5

Nominal Size

8× ×6

Thickness (in.)

3⁄ 8

5⁄ 16

1⁄ 4

1⁄ 2

3⁄

8

5⁄ 16

1⁄ 4

Wt./ft

35.13

29.72

24.12

42.05

32.58

27.59

22.42

Fy


Steel Tube

Y

46 ksi 0

502

445

385

573

471

417

362

6 7 8 9 10

459 445 428 411 392

408 395 381 365 349

353 342 330 317 302

535 522 507 491 473

440 430 418 405 391

391 381 371 360 347

339 331 322 312 302

11 12 13 14 15

372 351 330 309 288

331 313 295 276 257

287 272 256 240 223

455 435 415 394 373

376 360 344 327 310

334 321 306 291 276

291 279 266 254 241

16 17 18 19 20

266 245 225 205 186

239 220 202 185 168

207 191 176 161 146

352 330 309 288 267

293 275 258 241 224

261 246 231 216 201

228 214 201 188 175

22 24 26 28 30

154 129 110 95 83

139 117 99 86 75

121 101 86 74 65

228 192 163 141 123

192 162 138 119 104

172 145 124 107 93

151 127 108 93 81

32 34 36 38 40

73 64

66 58

57 50

108 96 85 76

91 81 72 65

82 72 65 58

72 63 57 51 46

2.04 1.72 111 68.7 29.5 9.98

2.07 1.71 95.2 58.7 26.5 9.01

2.31 1.25 111 91.1 23.2 14.8

2.36 1.25 88.3 72.5 19.7 12.5

2.39 1.25 75.6 62.1 17.6 11.2

2.42 1.25 62.1 51.1 15.3 9.83


2.09 1.72 78.3 48.3 23.1 7.83




Fy = 46 ksi

5 - 133

Y


X

X

8× ×4


5⁄ 8

1⁄

Wt./ft

42.30

35.24

2

3⁄

8

27.48

7× ×5 5⁄

16

23.34

Fy


Steel Tube

Y

1⁄ 4

1⁄ 2

3⁄

8

5⁄ 16

1⁄ 4

3⁄ 16

19.02

35.24

27.48

23.34

19.02

14.53

46 ksi 0

532

459

375

331

285

467

383

339

292

244

6 7 8 9 10

453 427 399 370 340

394 373 350 326 301

325 309 291 272 252

287 273 258 241 224

248 236 223 209 194

422 407 390 372 353

347 336 322 308 293

308 298 286 273 260

266 257 248 237 226

223 215 207 198 189

11 12 13 14 15

309 279 250 221 194

275 250 225 200 177

232 212 192 172 153

206 188 171 154 137

179 164 149 134 120

333 312 291 270 249

276 260 243 226 209

246 232 217 202 187

214 201 189 176 163

179 169 158 148 137

16 17 18 19 20

170 151 135 121 109

156 138 123 110 100

135 120 107 96 87

121 107 96 86 78

107 94 84 76 68

228 208 189 170 153

192 176 160 145 131

172 158 144 131 118

150 138 126 115 103

126 116 106 96 87

22 24 26 28 30

90 76

82 69

72 60 51

64 54 46

56 47 40

127 107 91 78 68

108 91 77 67 58

97 82 70 60 52

85 72 61 53 46

72 60 52 44 39

51

46

40

34 30

1.95 1.30 63.8 50.4 11.9 7.01

1.98 1.30 54.9 43.5 10.7 6.32

2.01 1.30 45.5 35.9 9.35 5.55

2.04 1.29 35.2 27.9 7.83 4.67

32 34

Properties 1.49 1.76 φb M nx (kip-ft) 99.3 φb M ny (kip-ft) 59.8 P ex (K x L x )2 / 10 4 (kip-ft2) 18.1 P ey (K y L y )2 / 10 4 (kip-ft2) 5.85 rmy (in.)

rmx / rmy

1.54 1.75 85.2 51.8 16.3 5.34

1.60 1.73 68.7 42.1 13.9 4.65

1.62 1.73 59.0 36.2 12.4 4.16

1.65 1.72 48.6 30.1 10.9 3.66

1.90 1.31 79.7 62.8 14.0 8.23



5 - 134

COMPOSITE DESIGN

Fy = 46 ksi

Y

X


X

Y

7× ×4

Steel Tube


3⁄

Wt./ft

24.93

8

6× ×4

5⁄ 16

1⁄ 4

21.21

17.32

3⁄

16

13.25


Fy

1⁄

2

3⁄ 8

28.43

5⁄

16

1⁄

4

3⁄ 16

22.37

19.08

15.62

11.97

46 ksi 0

338

298

256

213

364

300

265

228

189

6 7 8 9 10

291 276 259 242 224

258 245 230 215 200

223 212 200 187 173

185 176 166 156 145

309 291 272 252 231

257 243 228 212 196

228 216 203 189 175

197 187 176 164 152

164 156 147 137 127

11 12 13 14 15

205 187 168 151 134

183 167 151 136 121

160 146 132 119 106

134 122 111 100 90

210 189 168 149 130

179 162 146 130 115

160 146 131 118 104

140 127 115 103 92

117 107 97 87 77

16 17 18 19 20

118 104 93 83 75

107 94 84 76 68

94 83 74 67 60

79 70 63 56 51

114 101 90 81 73

101 89 80 71 65

92 81 72 65 59

81 72 64 57 52

68 60 54 48 44

22 24 26

62 52 45

56 47 40

50 42 36

42 35 30

61 51

53 45

48 41 35

43 36 31

36 30 26

1.48 1.39 53.1 39.7 7.61 3.93

1.54 1.38 43.1 32.6 6.62 3.46

1.57 1.38 37.6 28.3 5.96 3.15

1.60 1.37 31.3 23.6 5.20 2.78

1.63 1.37 24.4 18.4 4.39 2.34

Properties rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft)

P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2)

1.57 1.56 55.2 37.3 9.83 4.04

1.60 1.56 47.6 32.3 8.87 3.66

1.63 1.55 39.7 26.8 7.72 3.23

1.66 1.54 30.7 20.9 6.42 2.72




Fy = 46 ksi

5 - 135

Y


X

X

Y

5× ×4

Steel Tube


3⁄

5⁄

Wt./ft

19.82

8

16.96

Fy


1⁄

16

4

3⁄

16

13.91

10.70

46 ksi 0

252

216

177

136

6 7 8 9 10

215 203 190 176 162

185 175 164 153 141

153 145 136 127 118

118 112 106 99 92

11 12 13 14 15

147 133 119 106 93

129 117 105 93 82

108 98 89 79 70

85 77 70 63 56

16 17 18 19 20

81 72 64 58 52

73 64 57 51 46

62 55 49 44 40

49 44 39 35 32

22 24 26

43 36

38 32

33 28 23

26 22 19

1.53 1.20 28.4 24.3 3.56 2.49

1.56 1.19 23.8 20.4 3.03 2.13

1.59 1.19 18.6 16.0 2.40 1.70

Properties 1.50 1.19 32.6 27.9 3.98 2.80

rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft)

P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2) Note: Heavy line indicates Kl / r of 200.


5 - 136

COMPOSITE DESIGN

Fy = 46 ksi

Y

X

COMPOSITE COLUMNS Rectangular structural tubing fc′ = 5 ksi Axial design strength in kips

X

Nominal Size

16× ×12

16× ×8

Thickness (in.)

1⁄ 2

1⁄ 2

1⁄ 2

3⁄ 8

1⁄ 2

3⁄

8

5⁄ 16

Wt./ft

89.68

76.07

76.07

58.10

69.27

53.00

44.60

14× ×10

Fy


Steel Tube

Y

12× ×10

46 ksi 0

1630

1250

1300

1110

1150

984

897

6 7 8 9 10

1600 1590 1580 1560 1550

1210 1190 1180 1150 1130

1270 1250 1240 1230 1210

1080 1070 1060 1050 1040

1120 1110 1100 1090 1070

959 950 940 929 916

874 866 857 846 835

11 12 13 14 15

1530 1520 1500 1480 1460

1110 1080 1060 1030 1000

1190 1170 1150 1130 1110

1020 1000 987 969 949

1060 1040 1020 1000 981

902 888 872 855 837

822 808 794 778 762

16 17 18 19 20

1440 1410 1390 1360 1340

967 935 902 869 835

1090 1060 1040 1010 983

929 908 886 863 840

960 937 914 890 866

819 800 780 759 738

745 727 709 690 671

22 24 26 28 30

1290 1230 1170 1110 1050

766 698 630 565 502

928 870 812 753 695

792 743 693 643 593

815 763 711 658 605

695 650 605 560 515

631 591 549 508 467

32 34 36 38 40

987 926 865 804 745

442 391 349 313 283

638 582 528 476 429

544 496 450 405 365

554 504 456 409 369

471 428 387 348 314

426 387 350 314 283

4.02 1.30 362 288 155 92.1

4.08 1.29 281 224 131 78.4

3.94 1.15 290 256 105 79.3

4.00 1.15 225 199 89.2 67.3

4.03 1.15 191 168 80.4 60.8

Properties 4.84 1.25 497 407 256 164

rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft) P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2)

3.30 1.72 390 240 180 60.7




Fy = 46 ksi

5 - 137

Y


X

X

12× ×8


5⁄

Wt./ft

76.33

8

1⁄

2

62.46

12× ×6 3⁄ 8

5⁄ 16

5⁄ 8

1⁄ 2

3⁄

8

5⁄ 16

47.90

40.35

67.82

55.66

42.79

36.10

Fy


Steel Tube

Y

46 ksi 0

1140

997

845

768

961

839

705

635

6 7 8 9 10

1090 1080 1060 1040 1020

959 946 931 914 896

814 802 790 775 760

739 728 717 704 689

899 877 853 826 797

785 767 746 723 698

661 646 629 610 589

595 581 566 549 531

11 12 13 14 15

997 972 946 919 890

876 854 832 808 783

743 725 706 686 665

674 658 640 622 603

766 734 701 666 631

672 644 615 586 555

567 544 520 496 470

511 490 469 446 424

16 17 18 19 20

861 831 800 768 736

758 732 705 677 650

643 621 598 575 552

583 562 542 520 499

595 559 524 489 454

525 494 463 433 403

445 419 394 368 343

401 378 355 332 309

22 24 26 28 30

672 608 545 485 427

594 538 483 431 380

505 457 411 366 323

456 413 371 330 291

388 327 279 240 209

345 291 248 214 187

295 250 213 183 160

266 225 192 165 144

32 34 36 38

375 332 296 266

334 296 264 237

284 252 225 202

255 226 202 181

184 163 145 130

164 145 130 116

140 124 111 100

127 112 100 90

40

240

214

182

163

105

90

81

2.42 1.73 210 129 67.5 22.5

2.48 1.72 165 101 56.9 19.3

2.51 1.71 140 86.6 51.0 17.4


3.14 1.38 300 226 97.5 51.5

3.20 1.37 250 189 86.3 45.9

3.26 1.37 195 147 72.8 39.0

3.28 1.37 165 125 65.8 35.1

2.37 1.73 252 154 76.0 25.3



5 - 138

COMPOSITE DESIGN

Fy = 46 ksi

Y

X


X

10× ×8


1⁄ 2

3⁄ 8

Wt./ft

55.66

42.79

10× ×6 5⁄

16

36.10

Fy


Steel Tube

Y

1⁄

1⁄ 2

3⁄ 8

48.85

4

29.23

5⁄

16

1⁄

37.69

31.84

25.82

4

46 ksi 0

868

734

664

593

725

609

548

485

6 7 8 9 10

834 822 808 793 776

706 696 684 671 657

638 629 618 607 594

569 561 552 541 530

677 660 642 621 599

569 556 540 524 505

512 500 486 471 455

454 443 431 417 403

11 12 13 14 15

758 739 719 697 675

642 626 609 591 572

580 565 550 534 517

517 504 490 476 460

576 551 526 499 472

486 466 444 423 400

437 419 400 381 361

387 371 354 337 319

16 17 18 19 20

652 628 604 580 555

553 533 512 492 471

499 481 463 444 425

444 428 412 395 378

445 418 391 365 339

378 355 333 311 289

340 320 300 280 261

301 283 265 248 230

22 24 26 28 30

506 456 408 362 317

429 387 347 308 270

387 349 313 277 243

344 310 277 245 215

289 243 207 178 155

247 208 177 153 133

223 188 160 138 120

197 166 141 122 106

32 34 36 38

279 247 220 198

237 210 187 168

213 189 169 151

189 167 149 134

137 121 108 97

117 104 93 83

106 94 84 75

93 83 74 66

40

178

152

137

121

75

68

60

2.43 1.49 124 86.9 35.7 16.1

2.46 1.48 106 74.2 32.0 14.5

2.49 1.48 86.6 61.1 28.1 12.8


3.12 1.19 190 163 54.4 38.3

3.18 1.19 149 128 46.0 32.6

3.21 1.19 127 109 41.3 29.3

3.24 1.19 104 89.0 36.4 25.9

2.37 1.50 157 110 42.1 18.8




Fy = 46 ksi

5 - 139

Y


X

X

10× ×5

Nominal Size

8× ×6

Thickness (in.)

3⁄ 8

5⁄ 16

1⁄ 4

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

Wt./ft

35.13

29.72

24.12

42.05

32.58

27.59

22.42

Fy


Steel Tube

Y

46 ksi 0

544

489

431

610

512

460

406

6 7 8 9 10

496 479 461 441 419

446 431 415 397 378

393 380 365 349 332

568 554 538 520 500

477 465 451 437 421

429 418 406 393 379

379 369 359 347 334

11 12 13 14 15

397 374 350 326 303

358 337 316 295 273

315 296 278 259 240

480 459 436 414 390

404 386 367 349 329

363 347 331 314 297

321 307 292 277 262

16 17 18 19 20

279 256 234 212 191

252 232 212 192 174

221 203 185 168 152

367 344 321 298 276

310 291 271 252 234

279 262 245 228 211

247 231 216 201 186

22 24 26 28 30

158 133 113 98 85

143 121 103 89 77

126 106 90 78 68

233 196 167 144 125

198 167 142 122 107

179 150 128 111 96

158 133 113 97 85

32 34 36 38 40

75 66

68 60

59 53

110 98 87 78

94 83 74 66

85 75 67 60

75 66 59 53 48

2.04 1.72 111 68.7 30.4 10.3

2.07 1.71 95.2 58.7 27.4 9.31

2.31 1.25 111 91.1 23.7 15.1

2.36 1.25 88.3 72.5 20.2 12.9

2.39 1.25 75.6 62.1 18.2 11.6

2.42 1.25 62.1 51.1 16.0 10.3


2.09 1.72 78.3 48.3 24.1 8.15



5 - 140

COMPOSITE DESIGN

Fy = 46 ksi

Y

X


X

8× ×4

Nominal Size

7× ×5

Thickness (in.)

5⁄ 8

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

1⁄ 2

3⁄ 8

5⁄ 16

1⁄ 4

Wt./ft

42.30

35.24

27.48

23.34

19.02

35.24

27.48

23.34

19.02

14.53

Fy


Steel Tube

Y

3⁄

16

46 ksi 0

552

482

401

358

313

492

411

369

324

278

6 7 8 9 10

468 441 411 380 349

411 389 364 338 311

345 327 307 286 265

308 292 275 256 237

271 257 241 225 208

444 427 409 390 369

372 358 343 327 310

333 322 308 294 279

293 283 271 259 246

251 242 232 221 210

11 12 13 14 15

317 285 254 224 196

284 257 230 205 180

242 220 199 178 157

217 198 178 160 142

191 174 157 141 125

347 325 302 279 257

292 274 255 237 218

263 247 230 213 197

232 217 203 188 174

198 186 173 161 148

16 17 18 19 20

172 153 136 122 110

158 140 125 112 101

138 123 109 98 89

124 110 98 88 80

110 98 87 78 71

235 213 193 173 156

200 182 165 148 134

181 165 149 134 121

159 145 132 119 107

136 124 112 101 91

22 24 26 28 30

91 77

84 70

73 62 52

66 55 47

58 49 42

129 108 92 80 69

111 93 79 68 59

100 84 72 62 54

89 74 63 55 48

75 63 54 47 41

52

47

42

36 32

1.95 1.30 63.8 50.4 12.2 7.18

1.98 1.30 54.9 43.5 11.1 6.51

2.01 1.30 45.5 35.9 9.70 5.75

2.04 1.29 35.2 27.9 8.20 4.90

32 34


P ex (K x L x)2 / 10 4 (kip-ft2) P ey (K y L y)2 / 10 4 (kip-ft2)

1.49 1.76 99.3 59.8 18.3 5.92

1.54 1.75 85.2 51.8 16.6 5.43

1.60 1.73 68.7 42.1 14.2 4.75

1.62 1.73 59.0 36.2 12.8 4.28

1.65 1.72 48.6 30.1 11.2 3.79

1.90 1.31 79.7 62.8 14.3 8.38




Fy = 46 ksi

5 - 141

Y


X

X

Y

7× ×4

Steel Tube

Nominal Size

6× ×4

Thickness (in.)

3⁄ 8

5⁄ 16

1⁄ 4

3⁄ 16

1⁄ 2

3⁄

8

5⁄ 16

1⁄ 4

3⁄ 16

Wt./ft

24.93

21.21

17.32

13.25

28.43

22.37

19.08

15.62

11.97


Fy

46 ksi 0

360

321

281

239

380

318

285

249

211

6 7 8 9 10

308 291 273 254 234

276 261 245 228 211

242 229 215 201 186

206 195 183 171 158

321 302 282 260 238

271 256 240 222 204

243 230 215 200 184

213 202 189 176 162

181 171 161 149 138

11 12 13 14 15

214 194 174 155 137

193 175 158 141 124

170 155 139 125 110

145 131 119 106 94

215 193 172 151 132

186 168 150 133 117

168 152 137 121 107

148 134 121 108 95

126 114 103 92 81

16 17 18 19 20

120 106 95 85 77

109 97 86 78 70

97 86 77 69 62

83 73 65 59 53

116 103 92 82 74

103 91 81 73 66

94 83 74 67 60

83 74 66 59 53

71 63 56 50 45

22 24 26

64 53 45

58 49 41

51 43 37

44 37 31

61 52

54 46

50 42 36

44 37 32

38 32 27

1.48 1.39 53.1 39.7 7.72 3.99

1.54 1.38 43.1 32.6 6.76 3.53

1.57 1.38 37.6 28.3 6.10 3.22

1.60 1.37 31.3 23.6 5.36 2.86

1.63 1.37 24.4 18.4 4.56 2.44


P ex (K x L x )2 / 10 4 (kip-ft2) P ey (K y L y )2 / 10 4 (kip-ft2)

1.57 1.56 55.2 37.3 10.0 4.12

1.60 1.56 47.6 32.3 9.10 3.76

1.63 1.55 39.7 26.8 7.97 3.33

1.66 1.54 30.7 20.9 6.70 2.84



5 - 142

COMPOSITE DESIGN

Fy = 46 ksi

Y

X


X

Y

5× ×4

Steel Tube


3⁄ 8

5⁄

Wt./ft

19.82

16.96

Fy


1⁄ 4

16

3⁄

16

13.91

10.70

46 ksi 0

262

224

185

142

6 7 8 9 10

222 210 196 181 166

192 181 170 157 145

159 150 141 131 121

123 117 110 102 95

11 12 13 14 15

151 136 121 107 94

132 119 107 95 84

111 101 91 81 71

87 79 72 64 57

16 17 18 19 20

83 73 65 59 53

73 65 58 52 47

63 56 50 45 40

50 44 40 36 32

22 24 26

44 37

39 33

33 28 24

27 22 19

1.53 1.20 28.4 24.3 3.61 2.52

1.56 1.19 23.8 20.4 3.07 2.16

1.59 1.19 18.6 16.0 2.44 1.72

Properties 1.50 1.19 32.6 27.9 4.04 2.83

rmy (in.) rmx / rmy φb M nx (kip-ft) φb M ny (kip-ft)

P ex (K x L x)2 / 10 4 (kip-ft2) P ey (K y L y)2 / 10 4 (kip-ft2) Note: Heavy line indicates Kl / r of 200.


REFERENCES

5 - 143

REFERENCES

Allison, H., 1991, Low- and Medium-Rise Steel Buildings, AISC Steel Design Guide Series No. 5, American Institute of Steel Construction, Chicago, IL. Fisher, J. M. and M. A. West, 1990, Serviceability Design Considerations for Low-Rise Buildings, AISC Steel Design Guide Series No. 3, AISC, Chicago. Griffis, L. G., 1992, Load and Resistance Factor Design of W-Shapes Encased in Concrete, AISC Steel Design Guide Series No. 6, AISC, Chicago.


6-1

PART 6 SPECIFICATIONS AND CODES LRFD SPECIFICATION FOR STRUCTURAL STEEL BUILDINGS (1993) . . . . . . 6-3 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25 Appendixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-105 Numerical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-144 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-161 SPECIFICATION FOR LRFD OF SINGLE-ANGLE MEMBERS (1993) . . . . . . . 6-277 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-289 SEISMIC PROVISIONS FOR STRUCTURAL STEEL BUILDINGS (1992) . . . . . 6-301 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-329 RCSC SPECIFICATION FOR STRUCTURAL JOINTS USING ASTM A325 OR A490 BOLTS (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . 6-371 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-399 CODE OF STANDARD PRACTICE FOR STEEL BUILDINGS AND BRIDGES (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-423 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-455 AISC QUALITY CERTIFICATION PROGRAM (1993) . . . . . . . . . . . . . . . . 6-477


6-2


6-3

Load and Resistance Factor Design Specification for Structural Steel Buildings December 1, 1993

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. One East Wacker Drive, Suite 3100 Chicago, Illinois 60601-2001 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6-4


6-5

PREFACE

The AISC Load and Resistance Factor Design (LRFD) Specification for Structural Steel Buildings is based on reliability theory. As have all AISC Specifications, this LRFD Specification has been based upon past successful usage, advances in the state of knowledge, and changes in design practice. The LRFD Specification has been developed as a consensus document to provide a uniform practice in the design of steel-framed buildings. The intention is to provide design criteria for routine use and not to provide specific criteria for infrequently encountered problems which occur in the full range of structural design. Providing definitive provisions to cover all cases would make the LRFD Specification too cumbersome for routine design usage. The LRFD Specification is the result of the deliberations of a committee of structural engineers with wide experience and high professional standing, representing a wide geographical distribution throughout the U.S. The committee includes approximately equal numbers of engineers in private practice and code agencies, engineers involved in research and teaching, and engineers employed by steel fabricating and producing companies. In order to avoid reference to proprietary steels which may have limited availability, only those steels which can be identified by ASTM specifications are approved under this Specification. However, some steels covered by ASTM specifications, but subject to more costly manufacturing and inspection techniques than deemed essential for structures covered by this Specification, are not listed, even though they may provide all the necessary characteristics for reliable usage in structural applications. Approval of such steels in lieu of less expensive steels is left to the owner’s representative. The Appendices to this Specification are an integral part of the Specification. A non-mandatory Commentary has been prepared to provide background for the Specification provisions and the user is encouraged to consult it. The principal changes incorporated in this edition of the Specification include: • Updated web crippling design provisions. • Recommendations for the use of heavy rolled shapes and welded members made up of thick plates. • Updated provisions for slender web girders and unsymmetric members. • Revised provisions for built-up compression members. • Improved Cb equation. • Provisions for slip-critical joints designed at factored loads. • Reorganization and expansion of material on stability of unbraced frames. • Reorganization and expansion of Chapters F and K. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6-6

PREFACE

• Alternative fillet-weld design strength. • Addition of beam-web opening provisions. The reader is cautioned that professional judgment must be exercised when data or recommendations in the Specification are applied. The publication of the material contained herein is not intended as a representation or warranty on the part of the American Institute of Steel Construction, Inc.—or any other person named herein— that this information is suitable for general or particular use, or freedom from infringement of any patent or patents. Anyone making use of this information assumes all liability arising from such use. The design and detailing of steel structures is within the expertise of professional individuals who are competent by virtue of education, training, and experience for the application of engineering principles and the provisions of this specification to the design and/or detailing of a particular structure. By the Committee, Arthur P. Arndt, Chairman William E. Moore, II, Vice Chairman Horatio Allison Reidar Bjorhovde Roger L. Brockenbrough W. F. Chen Andrew K. Courtney Robert O. Disque Joseph Dudek Duane S. Ellifritt Bruce Ellingwood Shu-Jin Fang Steven J. Fenves Roger E. Ferch James M. Fisher John W. Fisher Theodore V. Galambos Geerhard Haaijer Richard B. Haws Mark Holland Ira Hooper Donald L. Johnson L. A. Kloiber

Jay W. Larson William J. LeMessurier H. S. Lew Bill Lindley, II Stanley D. Lindsey Richard W. Marshall Lisa McCasland Robert McCluer William A. Milek Duane K. Miller Walter P. Moore, Jr. Thomas M. Murray Gary G. Nichols Clarkson W. Pinkham Egor P. Popov Donald R. Sherman Frank Sowokinos William A. Thornton Raymond H. R. Tide Ivan M. Viest Lyle L. Wilson Joseph A. Yura Nestor R. Iwankiw, Secretary


6-7

TABLE OF CONTENTS SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.

B.

GENERAL PROVISIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 A1.

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A2.

Limits of Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1. Structural Steel Defined . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2. Types of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A3.

Material . . . . . . . . . . . . . . . 1. Structural Steel . . . . . . . . . . 2. Steel Castings and Forgings . . . 3. Bolts, Washers, and Nuts . . . . . 4. Anchor Bolts and Threaded Rods 5. Filler Metal and Flux for Welding 6. Stud Shear Connectors . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

26 26 28 28 28 29 29

A4.

Loads and Load Combinations . . . . . . . . . 1. Loads, Load Factors, and Load Combinations 2. Impact . . . . . . . . . . . . . . . . . . . . 3. Crane Runway Horizontal Forces . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

29 30 30 31

A5.

Design Basis . . . . . . . . . . . . . . . . . . . . . . 1. Required Strength at Factored Loads . . . . . . . 2. Limit States . . . . . . . . . . . . . . . . . . . . . 3. Design for Strength . . . . . . . . . . . . . . . . . 4. Design for Serviceability and Other Considerations

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

31 31 31 32 32

A6.

Referenced Codes and Standards . . . . . . . . . . . . . . . . . . . . . . 32

A7.

Design Documents . . . . . . . . . . . . 1. Plans . . . . . . . . . . . . . . . . . 2. Standard Symbols and Nomenclature 3. Notation for Welding . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

33 33 33 33

DESIGN REQUIREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B1.

Gross Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B2.

Net Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B3.

Effective Net Area for Tension Members . . . . . . . . . . . . . . . . . . 34

B4.

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

B5.

Local Buckling . . . . . . . . . . . . . . . 1. Classification of Steel Sections . . . . 2. Design by Plastic Analysis . . . . . . . 3. Slender-Element Compression Sections

B6.

Bracing at Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

B7.

Limiting Slenderness Ratios . . . . . . . . . . . . . . . . . . . . . . . . . 37

B8.

Simple Spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

B9.

End Restraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

36 36 36 37

B10. Proportions of Beams and Girders . . . . . . . . . . . . . . . . . . . . . . 37 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6-8

C.

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

41 41 41 41 41 44 44 44 45 47 47 47 47 47 48 48 51 52

Design for Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lateral-Torsional Buckling . . . . . . . . . . . . . . . . . . . . . . . F2. Design for Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F3. Web-Tapered Members (see Appendix F3) . . . . . . . . . . . . . . . F4. Beams and Girders with Web Openings . . . . . . . . . . . . . . . . . G. PLATE GIRDERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. MEMBERS UNDER COMBINED FORCES AND TORSION . . . . . . . . H1. Symmetric Members Subject to Bending and Axial Force . . . . . . . 1. Doubly and Singly Symmetric Members in Flexur and Tension . . . 2. Doubly and Singly Symmetric Members in Flexure and Compression H2. Unsymmetric Members and Members Under Torsion and Combined Torsion, Flexure, Shear, and/or Axial Force . . . . . . . . . H3. Alternative Interaction Equations for Members Under Combined Stress (see Appendix H3) . . . . . . . . . . . . . . . . . . . I. COMPOSITE MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Design Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . I2. Compression Members . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Columns with Multiple Steel Shapes . . . . . . . . . . . . . . . . . . 4. Load Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I3. Flexural Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Effective Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Strength of Beams with Shear Connectors . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

52 52 53 56 56 57 58 59 59 59 60

D.

E.

F.

FRAMES AND OTHER STRUCTURES . . . . . . . . . . . . . . . . C1. Second Order Effects . . . . . . . . . . . . . . . . . . . . . . . C2. Frame Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Braced Frames . . . . . . . . . . . . . . . . . . . . . . . . . 2. Unbraced Frames . . . . . . . . . . . . . . . . . . . . . . . . TENSION MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . D1. Design Tensile Strength . . . . . . . . . . . . . . . . . . . . . . D2. Built-Up Members . . . . . . . . . . . . . . . . . . . . . . . . . D3. Pin-Connected Members and Eyebars . . . . . . . . . . . . . . COLUMNS AND OTHER COMPRESSION MEMBERS . . . . . . E1. Effective Length and Slenderness Limitations . . . . . . . . . 1. Effective Length . . . . . . . . . . . . . . . . . . . . . . . . 2. Design by Plastic Analysis . . . . . . . . . . . . . . . . . . . E2. Design Compressive Strength for Flexural Buckling . . . . . . E3. Design Compressive Strength for Flexural-Torsional Buckling E4. Built-Up Members . . . . . . . . . . . . . . . . . . . . . . . . . E5. Pin-Connected Compression Members . . . . . . . . . . . . . BEAMS AND OTHER FLEXURAL MEMBERS . . . . . . . . . . . F1.


. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . 60 . . . . . . . . . . .

. . . . . . . . . . .

60 61 61 62 62 63 63 63 64 64 64

TABLE OF CONTENTS

3. 4. 5. 6.

J.

Strength of Concrete-Encased Beams Strength During Construction . . . . Formed Steel Deck . . . . . . . . . . Design Shear Strength . . . . . . . .

. . . .

. . . .

6-9

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

64 65 65 66

I4.

Combined Compression and Flexure . . . . . . . . . . . . . . . . . . . . 66

I5.

Shear Connectors . . . . . . . . . . . . . . 1. Materials . . . . . . . . . . . . . . . . . 2. Horizontal Shear Force . . . . . . . . . . 3. Strength of Stud Shear Connectors . . . . 4. Strength of Channel Shear Connectors . . 5. Required Number of Shear Connectors . 6. Shear Connector Placement and Spacing

I6.

Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

67 67 67 68 68 68 68

CONNECTIONS, JOINTS, AND FASTENERS . . . . . . . . . . . . . . . . . . 70 J1.

General Provisions . . . . . . . . . . . . . . . . . . 1. Design Basis . . . . . . . . . . . . . . . . . . . 2. Simple Connections . . . . . . . . . . . . . . . 3. Moment Connections . . . . . . . . . . . . . . . 4. Compression Members with Bearing Joints . . . 5. Splices in Heavy Sections . . . . . . . . . . . . 6. Beam Copes and Weld Access Holes . . . . . . 7. Minimum Strength of Connections . . . . . . . 8. Placement of Welds and Bolts . . . . . . . . . . 9. Bolts in Combination with Welds . . . . . . . . 10. High-Strength Bolts in Combination with Rivets 11. Limitations on Bolted and Welded Connections .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

70 70 70 70 70 70 71 71 72 72 72 72

J2.

Welds . . . . . . . . . . . . . 1. Groove Welds . . . . . . . 2. Fillet Welds . . . . . . . . 3. Plug and Slot Welds . . . 4. Design Strength . . . . . . 5. Combination of Welds . . 6. Matching Weld Metal . . . 7. Mixed Weld Metal . . . . 8. Preheat for Heavy Shapes

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

73 73 74 76 77 77 77 79 79

J3.

Bolts and Threaded Parts . . . . . . . . . . . . . . . . . . . . 1. High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . 2. Size and Use of Holes . . . . . . . . . . . . . . . . . . . . 3. Minimum Spacing . . . . . . . . . . . . . . . . . . . . . . 4. Minimum Edge Distance . . . . . . . . . . . . . . . . . . . 5. Maximum Spacing and Edge Distance . . . . . . . . . . . . 6. Design Tension or Shear Strength . . . . . . . . . . . . . . 7. Combined Tension and Shear in Bearing-Type Connections 8. High-Strength Bolts in Slip-Critical Connections . . . . . . 9. Combined Tension and Shear in Slip-Critical Connections . 10. Bearing Strength at Bolt Holes . . . . . . . . . . . . . . . . 11. Long Grips . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

79 79 79 80 81 82 83 83 83 84 85 87

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .


6 - 10

TABLE OF CONTENTS

J4.

Design Rupture Strength . . . . . . . . . . . . . . . . . . . . . . 1. Shear Rupture Strength . . . . . . . . . . . . . . . . . . . . . . 2. Tension Rupture Strength . . . . . . . . . . . . . . . . . . . . 3. Block Shear Rupture Strength . . . . . . . . . . . . . . . . . . J5. Connecting Elements . . . . . . . . . . . . . . . . . . . . . . . . 1. Eccentric Connections . . . . . . . . . . . . . . . . . . . . . . 2. Design Strength of Elements in Tension . . . . . . . . . . . . . 3. Other Connecting Elements . . . . . . . . . . . . . . . . . . . J6. Fillers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J7. Splices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J8. Bearing Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . J9. Column Bases and Bearing on Concrete . . . . . . . . . . . . . . J10. Anchor Bolts and Embedments . . . . . . . . . . . . . . . . . . . K. CONCENTRATED FORCES, PONDING, AND FATIGUE . . . . . . K1. Flanges and Webs with Concentrated Forces . . . . . . . . . . . 1. Design Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Local Flange Bending . . . . . . . . . . . . . . . . . . . . . . 3. Local Web Yielding . . . . . . . . . . . . . . . . . . . . . . . . 4. Web Crippling . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Sidesway Web Buckling . . . . . . . . . . . . . . . . . . . . . 6. Compression Buckling of the Web . . . . . . . . . . . . . . . . 7. Panel-Zone Web Shear . . . . . . . . . . . . . . . . . . . . . . 8. Unframed Ends of Beams and Girders . . . . . . . . . . . . . . 9. Additional Stiffener Requirements for Concentrated Forces . . 10. Additional Doubler Plate Requirements for Concentrated Forces K2. Ponding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K3. Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. SERVICEABILITY DESIGN CONSIDERATIONS . . . . . . . . . . . L1. Camber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L2. Expansion and Contraction . . . . . . . . . . . . . . . . . . . . . L3. Deflections, Vibrations, and Drift . . . . . . . . . . . . . . . . . 1. Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Floor Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 3. Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L4. Connection Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . L5. Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. FABRICATION, ERECTION, AND QUALITY CONTROL . . . . . . M1. Shop Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . M2. Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Cambering, Curving, and Straightening . . . . . . . . . . . . . 2. Thermal Cutting . . . . . . . . . . . . . . . . . . . . . . . . . 3. Planing of Edges . . . . . . . . . . . . . . . . . . . . . . . . . 4. Welded Construction . . . . . . . . . . . . . . . . . . . . . . . 5. Bolted Construction . . . . . . . . . . . . . . . . . . . . . . . 6. Compression Joints . . . . . . . . . . . . . . . . . . . . . . . . AMERICAN INSTITUTE OF STEEL CONSTRUCTION

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 87 87 88 88 88 88 89 89 89 90 90 91 91 91 91 92 92 93 94 95 96 96 96 97 97 98 98 98 98 98 98 98 99 99 100 100 100 100 100 100 101 101 101

TABLE OF CONTENTS

6 - 11

7. Dimensional Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8. Finish of Column Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 101 M3. Shop Painting . . . . . . . . . . . . . . . . . . . . . 1. General Requirements . . . . . . . . . . . . . . . 2. Inaccessible Surfaces . . . . . . . . . . . . . . . . 3. Contact Surfaces . . . . . . . . . . . . . . . . . . 4. Finished Surfaces . . . . . . . . . . . . . . . . . . 5. Surfaces Adjacent to Field Welds . . . . . . . . . M4. Erection . . . . . . . . . . . . . . . . . . . . . . . . . 1. Alignment of Column Bases . . . . . . . . . . . . 2. Bracing . . . . . . . . . . . . . . . . . . . . . . . 3. Alignment . . . . . . . . . . . . . . . . . . . . . 4. Fit of Column Compression Joints and Base Plates 5. Field Welding . . . . . . . . . . . . . . . . . . . . 6. Field Painting . . . . . . . . . . . . . . . . . . . . 7. Field Connections . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

102 102 102 102 102 102 102 102 102 103 103 103 103 103

M5. Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Rejections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Inspection of Welding . . . . . . . . . . . . . . . . . . . . . 4. Inspection of Slip-Critical High-Strength Bolted Connections 5. Identification of Steel . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

103 103 103 104 104 104

APPENDICES B.


Local Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 1. Classification of Steel Sections . . . . . . . . . . . . . . . . . . . . . . 105 3. Slender-Element Compression Sections . . . . . . . . . . . . . . . . . . 105

E.

COLUMNS AND OTHER COMPRESSION MEMBERS . . . . . . . . . . . . 109

F.

BEAMS AND OTHER FLEXURAL MEMBERS . . . . . . . . . . . . . . . . . 111

E3.

Design Compressive Strength for Flexural-Torsional Buckling . . . . . . 109

F1.

Design for Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

F2.

Design for Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2. Design Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3. Transverse Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

F3.

Web-Tapered Members . . . . . . . . 1. General Requirements . . . . . . . 2. Design Tensile Strength . . . . . . 3. Design Compressive Strength . . . 4. Design Flexural Strength . . . . . . 5. Design Shear Strength . . . . . . . 6. Combined Flexure and Axial Force

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

118 118 118 118 119 120 120

G. PLATE GIRDERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 G1.

Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

G2.

Design Flexural Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

G3.

Design Shear Strength with Tension Field Action . . . . . . . . . . . . . . 124 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 12

TABLE OF CONTENTS

G4.

Transverse Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

G5.

Flexure-Shear Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

H. MEMBERS UNDER COMBINED FORCES AND TORSION . . . . . . . . . . 127 H3. J.

Alternative Interaction Equations for Members Under Combined Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

CONNECTIONS, JOINTS, AND FASTENERS . . . . . . . . . . . . . . . . . . 129 J2. J3.

Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Design Strength . . . . . . . . . . . . . . . . . . . . . . . Bolts and Threaded Parts . . . . . . . . . . . . . . . . . . . 8. High-Strength Bolts in Slip-Critical Connections . . . . . 9. Combined Tension and Shear in Slip-Critical Connections

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

129 129 130 130 130

K. CONCENTRATED FORCES, PONDING, AND FATIGUE . . . . . . . . . . . 132 K2. Ponding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 K3.

Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Loading Conditions; Type and Location of Material 2. Design Stress Range . . . . . . . . . . . . . . . . . 3. Design Strength of Bolts in Tension . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

135 135 135 135

NUMERICAL VALUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 COMMENTARY A. GENERAL PROVISIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 A1. Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

B.

C.

A2.

Limits of Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 2. Types of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A3.

Material . . . . . . . . . . . . . . . 1. Structural Steel . . . . . . . . . . 3. Bolts, Washers, and Nuts . . . . . 4. Anchor Bolts and Threaded Rods 5. Filler Metal and Flux for Welding

A4.

Loads and Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . 166 1. Loads, Load Factors, and Load Combinations . . . . . . . . . . . . . . . 166 2. Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A5.

Design Basis . . . . . . . . . . . . . . . . . . . . . . 1. Required Strength at Factored Loads . . . . . . . 2. Limit States . . . . . . . . . . . . . . . . . . . . . 3. Design for Strength . . . . . . . . . . . . . . . . . 4. Design for Serviceability and Other Considerations

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

163 163 164 165 165

167 167 167 168 171


Net Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

B3.

Effective Net Area for Tension Members . . . . . . . . . . . . . . . . . . 172

B5. B7.

Local Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Limiting Slenderness Ratios . . . . . . . . . . . . . . . . . . . . . . . . . 177

FRAMES AND OTHER STRUCTURES . . . . . . . . . . . . . . . . . . . . . . 179 C1.

Second Order Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

TABLE OF CONTENTS

C2. Frame Stability . . . . . . . . . . . . . . . . . . . . . . . . . . TENSION MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . D1. Design Tensile Strength . . . . . . . . . . . . . . . . . . . . . . D2. Built-Up Members . . . . . . . . . . . . . . . . . . . . . . . . . D3. Pin-Connected Members and Eyebars . . . . . . . . . . . . . . E. COLUMNS AND OTHER COMPRESSION MEMBERS . . . . . . E1. Effective Length and Slenderness Limitations . . . . . . . . . 1. Effective Length . . . . . . . . . . . . . . . . . . . . . . . . 2. Design by Plastic Analysis . . . . . . . . . . . . . . . . . . . E2. Design Compressive Strength for Flexural Buckling . . . . . . E3. Design Compressive Strength for Flexural-Torsional Buckling E4. Built-Up Members . . . . . . . . . . . . . . . . . . . . . . . . . F. BEAMS AND OTHER FLEXURAL MEMBERS . . . . . . . . . . . F1. Design for Flexure . . . . . . . . . . . . . . . . . . . . . . . . . 1. Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lateral-Torsional Buckling . . . . . . . . . . . . . . . . . . . F2. Design for Shear . . . . . . . . . . . . . . . . . . . . . . . . . . F4. Beams and Girders with Web Openings . . . . . . . . . . . . . H. MEMBERS UNDER COMBINED FORCES AND TORSION . . . . H1. Symmetric Members Subject to Bending and Axial Force . . . H2. Unsymmetric Members and Members Under Torsion and Combined Torsion, Flexure, Shear, and/or Axial Force . . . . . I. COMPOSITE MEMBERS . . . . . . . . . . . . . . . . . . . . . . . . I1. Design Assumptions . . . . . . . . . . . . . . . . . . . . . . . . I2. Compression Members . . . . . . . . . . . . . . . . . . . . . . 1. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . 3. Columns with Multiple Steel Shapes . . . . . . . . . . . . . . 4. Load Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . I3. Flexural Members . . . . . . . . . . . . . . . . . . . . . . . . . 1. Effective Width . . . . . . . . . . . . . . . . . . . . . . . . . 2. Strength of Beams with Shear Connectors . . . . . . . . . . . 3. Strength of Concrete-Encased Beams . . . . . . . . . . . . . 4. Strength During Construction . . . . . . . . . . . . . . . . . 5. Formed Steel Deck . . . . . . . . . . . . . . . . . . . . . . . 6. Design Shear Strength . . . . . . . . . . . . . . . . . . . . . I4. Combined Compression and Flexure . . . . . . . . . . . . . . I5. Shear Connectors . . . . . . . . . . . . . . . . . . . . . . . . . 1. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Horizontal Shear Force . . . . . . . . . . . . . . . . . . . . . 3. Strength of Stud Shear Connectors . . . . . . . . . . . . . . . 4. Strength of Channel Shear Connectors . . . . . . . . . . . . . 6. Shear-Connector Placement and Spacing . . . . . . . . . . . I6. Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.


6 - 13

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

183 191 191 191 191 192 192 192 192 192 193 194 195 195 195 195 199 200 201 201

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

202 203 203 204 204 205 205 205 205 205 206 209 210 210 212 213 213 213 214 214 214 214 215

6 - 14

J.

TABLE OF CONTENTS

CONNECTIONS, JOINTS, AND FASTENERS . . . . . . . . . . . . . . . . . . 216 J1.

General Provisions . . . . . . . . . . . . . . . . . . 5. Splices in Heavy Sections . . . . . . . . . . . . 8. Placement of Welds and Bolts . . . . . . . . . . 9. Bolts in Combination with Welds . . . . . . . . 10. High-Strength Bolts in Combination with Rivets

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

216 216 217 217 219

J2.

Welds . . . . . . . . . . . 1. Groove Welds . . . . . 2. Fillet Welds . . . . . . 4. Design Strength . . . . 5. Combination of Welds 7. Mixed Weld Metal . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

219 219 219 222 223 223

J3.

Bolts and Threaded Parts . . . . . . . . . . . . . . . . . . . . 1. High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . 2. Size and Use of Holes . . . . . . . . . . . . . . . . . . . . 3. Minimum Spacing . . . . . . . . . . . . . . . . . . . . . . 4. Minimum Edge Distance . . . . . . . . . . . . . . . . . . . 5. Maximum Spacing and Edge Distance . . . . . . . . . . . . 6. Design Tension or Shear Strength . . . . . . . . . . . . . . 7. Combined Tension and Shear in Bearing-Type Connections 8. High-Strength Bolts in Slip-Critical Connections . . . . . . 10. Bearing Strength at Bolt Holes . . . . . . . . . . . . . . . . 11. Long Grips . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

223 223 224 224 224 225 225 226 226 227 228

J4.

Design Rupture Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

J5.

Connecting Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2. Design Strength of Connecting Elements in Tension . . . . . . . . . . . 229

J6.

Fillers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

J8.

Bearing Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

J9.

Column Bases and Bearing on Concrete . . . . . . . . . . . . . . . . . . . 230

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

J10. Anchor Bolts and Embedments . . . . . . . . . . . . . . . . . . . . . . . . 230 K. CONCENTRATED FORCES, PONDING, AND FATIGUE . . . . . . . . . . . 231

L.

K1.

Flanges and Webs with Concentrated Forces 1. Design Basis . . . . . . . . . . . . . . . . 2. Local Flange Bending . . . . . . . . . . . 3. Local Web Yielding . . . . . . . . . . . . . 4. Web Crippling . . . . . . . . . . . . . . . 5. Sidesway Web Buckling . . . . . . . . . . 6. Compression Buckling of the Web . . . . . 7. Panel-Zone Web Shear . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

231 231 231 231 232 232 233 234

K2.

Ponding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

SERVICEABILITY DESIGN CONSIDERATIONS . . . . . . . . . . . . . . . . 239 L1.

Camber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

L2.

Expansion and Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . 240

L3.

Deflections, Vibration, and Drift . . . . . . . . . . . . . . . . . . . . . . . 240 1. Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

TABLE OF CONTENTS

2. Floor Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L5. Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. FABRICATION, ERECTION, AND QUALITY CONTROL . . . . . . . . M2. Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Cambering, Curving, and Straightening . . . . . . . . . . . . . . . 2. Thermal Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Bolted Construction . . . . . . . . . . . . . . . . . . . . . . . . . M3. Shop Painting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Surfaces Adjacent to Field Welds . . . . . . . . . . . . . . . . . . M4. Erection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Fit of Column Compression Joints and Base Plates . . . . . . . . . 5. Field Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX B. DESIGN REQUIREMENTS . . . . . . . . . . . . . . . . . . . B5. Local Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Classification of Steel Sections . . . . . . . . . . . . . . . . . . . APPENDIX E. COLUMNS AND OTHER COMPRESSION MEMBERS . . . E3. Design Compressive Strength for Flexural-Torsional Buckling . . . APPENDIX F. BEAMS AND OTHER FLEXURAL MEMBERS . . . . . . . . F1. Design for Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . F3. Web-Tapered Members . . . . . . . . . . . . . . . . . . . . . . . . . 1. General Requirements . . . . . . . . . . . . . . . . . . . . . . . . 3. Design Compressive Strength . . . . . . . . . . . . . . . . . . . . 4. Design Flexural Strength . . . . . . . . . . . . . . . . . . . . . . . APPENDIX G. PLATE GIRDERS . . . . . . . . . . . . . . . . . . . . . . . . . G2. Design Flexural Strength . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX H. MEMBERS UNDER COMBINED FORCES AND TORSION H3. Alternative Interaction Equations for Members Under Combined Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX J. CONNECTIONS, JOINTS, AND FASTENERS . . . . . . . . . J2. Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX K. CONCENTRATED FORCES, PONDING, AND FATIGUE . K3. Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GLOSSARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


6 - 15

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. 240 . 241 . 241 . 242 . 242 . 242 . 242 . 242 . 243 . 243 . 243 . 243 . 243 . 244 . 244 . 244 . 245 . 245 . 246 . 246 . 247 . 247 . 247 . 248 . 249 . 249 . 250

. . . . . . . .

. . . . . . . .

. 250 . 251 . 251 . 251 . 254 . 254 . 257 . 267

6 - 16


6 - 17

Symbols The section number in parentheses after the definition of a symbol refers to the section where the symbol is first defined. A AB Ab Ac Ac AD Ae Af Afe Afg Afn Ag Agt Agv An Ant Anv Apb Ar As Asc Asf Aw A1 A2 B B B1, B2 CPG Cb Cm

Cross-sectional area, in.2 (F1.2) Loaded area of concrete, in.2 (I2.4) Nominal body area of a fastener, in.2 (J3.7) Area of concrete, in.2 (I2.2) Area of concrete slab within effective width, in.2 (I5.2) Area of an upset rod based on the major diameter of its threads, in.2 (J3.6) Effective net area, in.2 (B3) Area of flange, in.2 (Appendix F3) Effective tension flange area, in.2 (B10) Gross area of flange, in.2 (B10) Net area of flange, in.2 (B10) Gross area, in.2 (A5) Gross area subject to tension, in.2 (J4.3) Gross area subject to shear, in.2 (J4.3) Net area, in.2 (B2) Net area subject to tension, in.2 (J4.2) Net area subject to shear, in.2 (J4.1) Projected bearing area, in.2 (J8.1) Area of reinforcing bars, in.2 (I2.2) Area of steel cross section, in.2 (I2.2) Cross-sectional area of stud shear connector, in.2 (I5.3) Shear area on the failure path, in.2 (D3) Web area, in.2 (F2.1) Area of steel bearing concentrically on a concrete support, in.2 (J9) Total cross-sectional area of a concrete support, in.2 (J9) Factor for bending stress in tees and double angles (F1.2) Factor for bending stress in web-tapered members, in., defined by Equations A-F3-8 through A-F3-11 (Appendix F3) Factors used in determining Mu for combined bending and axial forces when first-order analysis is employed (C1) Plate-girder coefficient (Appendix G2) Bending coefficient dependent on moment gradient (F1.2a) Coefficient applied to bending term in interaction formula for prismatic members and dependent on column curvature caused by applied moments (C1) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 18

Cm′ Cp Cs Cv Cw D D D E E Ec Em FBM FEXX FL Fbγ Fcr Fcrft, Fcry, Fcrz Fe Fex Fey Fez Fmy Fn Fr Fsγ Fu Fw Fwγ Fy

Fyf Fyr Fyst

SYMBOLS

Coefficient applied to bending term in interaction formula for tapered members and dependent on axial stress at the small end of the member (Appendix F3) Ponding flexibility coefficient for primary member in a flat roof (K2) Ponding flexibility coefficient for secondary member in a flat roof (K2) Ratio of “critical” web stress, according to linear buckling theory, to the shear yield stress of web material (Appendix G3) Warping constant, in.6 (F1.2) Outside diameter of circular hollow section, in. (Appendix B5.3) Dead load due to the weight of the structural elements and permanent features on the structure (A4.1) Factor used in Equation A-G4-2, dependent on the type of transverse stiffeners used in a plate girder (Appendix G4) Modulus of elasticity of steel (E = 29,000 ksi) (E2) Earthquake load (A4.1) Modulus of elasticity of concrete, ksi (I2.2) Modified modulus of elasticity, ksi (I2.2) Nominal strength of the base material to be welded, ksi (J2.4) Classification number of weld metal (minimum specified strength), ksi (J2.4) Smaller of (Fyf −Fr ) or Fyw, ksi (F1.2) Flexural stress for tapered members defined by Equations A-F3-4 and A-F3-5 (Appendix F3) Critical stress, ksi (E2) Flexural-torsional buckling stresses for double-angle and tee-shaped compression members, ksi (E3) Elastic buckling stress, ksi (Appendix E3) Elastic flexural buckling stress about the major axis, ksi (Appendix E3) Elastic flexural buckling stress about the minor axis, ksi (Appendix E3) Elastic torsional buckling stress, ksi (Appendix E3) Modified yield stress for composite columns, ksi (I2.2) Nominal shear rupture strength, ksi (J4) Compressive residual stress in flange (10 ksi for rolled; 16.5 ksi for welded), ksi (Table B5.1) Stress for tapered members defined by Equation A-F3-6, ksi (Appendix F3) Specified minimum tensile strength of the type of steel being used, ksi (B10) Nominal strength of the weld electrode material, ksi (J2.4) Stress for tapered members defined by Equation A-F3-7, ksi (Appendix F3) Specified minimum yield stress of the type of steel being used, ksi. As used in this Specification, “yield stress” denotes either the specified minimum yield point (for those steels that have a yield point) or specified yield strength (for those steels that do not have a yield point) (A5) Specified minimum yield stress of the flange, ksi (Table B5.1) Specified minimum yield stress of reinforcing bars, ksi (I2.2) Specified minimum yield stress of the stiffener material, ksi (Appendix G4) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

SYMBOLS

Fyw G H H Hs I Id Ip Is Ist Iyc J K Kz Kγ L L L Lb Lc Le Lp Lp Lpd Lr Lr Ls MA MB MC Mcr Mlt Mmax Mn M′n x , M′n y

6 - 19

Specified minimum yield stress of the web, ksi (Table B5.1) Shear modulus of elasticity of steel, ksi (G = 11,200) (F1.2) Horizontal force, kips (C1) Flexural constant (E3) Length of stud connector after welding, in. (I3.5) Moment of inertia, in.4 (F1.2) Moment of inertia of the steel deck supported on secondary members, in.4 (K2) Moment of inertia of primary members, in.4 (K2) Moment of inertia of secondary members, in.4 (K2) Moment of inertia of a transverse stiffener, in.4 (Appendix G4) Moment of inertia about y axis referred to compression flange, or if reverse curvature bending referred to smaller flange, in.4 (Appendix F1) Torsional constant for a section, in.4 (F1.2) Effective length factor for prismatic member (B7) Effective length factor for torsional buckling (Appendix E3) Effective length factor for a tapered member (Appendix F3) Story height, in. (C1) Length of connection in the direction of loading, in. (B3) Live load due to occupancy and moveable equipment (A4.1) Laterally unbraced length; length between points which are either braced against lateral displacement of compression flange or braced against twist of the cross section, in. (F1.2) Length of channel shear connector, in. (I5.4) Edge distance, in. (J3.10) Limiting laterally unbraced length for full plastic bending capacity, uniform moment case (Cb = 1.0), in. (F1.2) Column spacing in direction of girder, ft (K2) Limiting laterally unbraced length for plastic analysis, in. (F1.2) Limiting laterally unbraced length for inelastic lateral-torsional buckling, in. (F1.2) Roof live load (A4.1) Column spacing perpendicular to direction of girder, ft (K2) Absolute value of moment at quarter point of the unbraced beam segment, kip-in. (F1.2) Absolute value of moment at centerline of the unbraced beam segment, kip-in. (F1.2) Absolute value of moment at three-quarter point of the unbraced beam segment, kip-in. (F1.2) Elastic buckling moment, kip-in. (F1.2) Required flexural strength in member due to lateral frame translation only, kip-in. (C1) Absolute value of maximum moment in the unbraced beam segment, kip-in. (F1.2) Nominal flexural strength, kip-in. (F1.1) Flexural strength defined in Equations A-H3-7 and A-H3-8 for use in alternate interaction equations for combined bending and axial force, kip-in. (Appendix H3) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 20

Mnt Mp Mp′ Mr Mu My

M1 M2 N Nr Pe1 , Pe2 Pn Pp Pu Py Q Qa Qn Qs R RPG Re Rn Rv S S S Sx′ Seff Sxt , Sxc T Tb Tu U Vn

SYMBOLS

Required flexural strength in member assuming there is no lateral translation of the frame, kip-in. (C1) Plastic bending moment, kip-in. (F1.1) Moment defined in Equations A-H3-5 and A-H3-6, for use in alternate interaction equations for combined bending and axial force, kip-in. (Appendix H3) Limiting buckling moment, Mcr, when λ = λr and Cb = 1.0, kip-in. (F1.2) Required flexural strength, kip-in. (C1) Moment corresponding to onset of yielding at the extreme fiber from an elastic stress distribution (= Fy S for homogeneous sections), kip-in. (F1.1) Smaller moment at end of unbraced length of beam or beam-column, kip-in. Larger moment at end of unbraced length of beam or beam-column, kip-in. Length of bearing, in. (K1.3) Number of stud connectors in one rib at a beam intersection (I3.5) Elastic Euler buckling load for braced and unbraced frame, respectively, kips (C1) Nominal axial strength (tension or compression), kips (D1) Bearing load on concrete, kips (J9) Required axial strength (tension or compression), kips (Table B5.1) Yield strength, kips (Table B5.1) Full reduction factor for slender compression elements (Appendix E3) Reduction factor for slender stiffened compression elements (Appendix B5) Nominal strength of one stud shear connector, kips (I5) Reduction factor for slender unstiffened compression elements (Appendix B5.3) Load due to initial rainwater or ice exclusive of the ponding contribution (A4.1) Plate girder bending strength reduction factor (Appendix G) Hybrid girder factor (Appendix F1) Nominal strength (A5.3) Web shear strength, kips (K1.7) Elastic section modulus, in.3 (F1.2) Spacing of secondary members, ft (K2) Snow load (A4.1) Elastic section modulus of larger end of tapered member about its major axis, in.3 (Appendix F3) Effective section modulus about major axis, in.3 (Appendix F1) Elastic section modulus referred to tension and compression flanges, respectively, in.3 (Appendix F1) Tension force due to service loads, kips (J3.9) Specified pretension load in high-strength bolt, kips (J3.9) Required tensile strength due to factored loads, kips (Appendix J3.9b) Reduction coefficient, used in calculating effective net area (B3) Nominal shear strength, kips (F2.2) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

SYMBOLS

Vu W X1 X2 Z a a a ar a′ b be beff bf c1, c2, c3 d d d d dL db dc do e f fb1 fb2 fc′ fo fun fuv fv g h

h hc

6 - 21

Required shear strength, kips (Appendix G4) Wind load (A4.1) Beam buckling factor defined by Equation F1-8 (F1.2) Beam buckling factor defined by Equation F1-9 (F1.2) Plastic section modulus, in.3 (F1.1) Clear distance between transverse stiffeners, in. (Appendix F2.2) Distance between connectors in a built-up member, in. (E4) Shortest distance from edge of pin hole to edge of member measured parallel to direction of force, in. (D3) Ratio of web area to compression flange area (Appendix G2) Weld length, in. (B10) Compression element width, in. (B5.1) Reduced effective width for slender compression elements, in. (Appendix B5.3) Effective edge distance, in. (D3) Flange width, in. (B5.1) Numerical coefficients (I2.2) Nominal fastener diameter, in. (J3.3) Overall depth of member, in. (B5.1) Pin diameter, in. (D3) Roller diameter, in. (J8.2) Depth at larger end of unbraced tapered segment, in. (Appendix F3) Beam depth, in. (K1.7) Column depth, in. (K1.7) Depth at smaller end of unbraced tapered segment, in. (Appendix F3) Base of natural logarithm = 2.71828. . . Computed compressive stress in the stiffened element, ksi (Appendix B5.3) Smallest computed bending stress at one end of a tapered segment, ksi (Appendix F3) Largest computed bending stress at one end of a tapered segment, ksi (Appendix F3) Specified compressive strength of concrete, ksi (I2.2) Stress due to 1.2D + 1.2R, ksi (Appendix K2) Required normal stress, ksi (H2) Required shear stress, ksi (H2) Required shear stress due to factored loads in bolts or rivets, ksi (J3.7) Transverse center-to-center spacing (gage) between fastener gage lines, in. (B2) Clear distance between flanges less the fillet or corner radius for rolled shapes; and for built-up sections, the distance between adjacent lines of fasteners or the clear distance between flanges when welds are used, in. (B5.1) Distance between centroids of individual components perpendicular to the member axis of buckling, in. (E4) Twice the distance from the centroid to the following: the inside face of the compression flange less the fillet or corner radius, for rolled shapes; the nearest line of fasteners at the compression flange or the inside faces AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 22

hr hs hw j k kv l l l l m r rTo ri rib rm _ ro rox , roy rx, ry ryc s t tf tf tw tw w w wr x _xo, yo x y z

SYMBOLS

of the compression flange when welds are used, for built-up sections, in., (B5.1) Nominal rib height, in. (I3.5) Factor used in Equation A-F3-6 for web-tapered members (Appendix F3) Factor used in Equation A-F3-7 for web-tapered members (Appendix F3) Factor defined by Equation A-F2-4 for minimum moment of inertia for a transverse stiffener (Appendix F2.3) Distance from outer face of flange to web toe of fillet, in. (K1.3) Web plate buckling coefficient (Appendix F2.2) Laterally unbraced length of member at the point of load, in. (B7) Length of bearing, in. (J8.2) Length of connection in the direction of loading, in. (B3) Length of weld, in. (B3) Ratio of web to flange yield stress or critical stress in hybrid beams (Appendix G2) Governing radius of gyration, in. (B7) For the smaller end of a tapered member, the radius of gyration, considering only the compression flange plus one-third of the compression web area, taken about an axis in the plane of the web, in. (Appendix F3.4) Minimum radius of gyration of individual component in a built-up member, in. (E4) Radius of gyration of individual component relative to centroidal axis parallel to member axis of buckling, in. (E4) Radius of gyration of the steel shape, pipe, or tubing in composite columns. For steel shapes it may not be less than 0.3 times the overall thickness of the composite section, in. (I2) Polar radius of gyration about the shear center, in. (E3) Radius of gyration about x and y axes at the smaller end of a tapered member, respectively, in. (Appendix F3.3) Radius of gyration about x and y axes, respectively, in. (E3) Radius of gyration about y axis referred to compression flange, or if reverse curvature bending, referred to smaller flange, in. (Appendix F1) Longitudinal center-to-center spacing (pitch) of any two consecutive holes, in. (B2) Thickness of connected part, in. (D3) Flange thickness, in. (B5.1) Flange thickness of channel shear connector, in. (I5.4) Web thickness of channel shear connector, in. (I5.4) Web thickness, in. (B5.3) Plate width; distance between welds, in. (B3) Unit weight of concrete, lbs/cu ft. (I2) Average width of concrete rib or haunch, in. (I3.5) Subscript relating symbol to strong axis bending Coordinates of the shear center with respect to the centroid, in. (E3) Connection eccentricity, in. (B3) Subscript relating symbol to weak axis bending Distance from the smaller end of tapered member used in Equation A-F3-1 for the variation in depth, in. (Appendix F3) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

SYMBOLS

α ∆oh γ ζ η λc λe λeff λp λr φ φb φc φc φsf φt φv

6 - 23

h (E4) 2rib Translation deflection of the story under consideration, in. (C1) Depth tapering ratio (Appendix F3). Subscript for tapered members (Appendix F3) Exponent for alternate beam-column interaction equation (Appendix H3) Exponent for alternate beam-column interaction equation (Appendix H3) Column slenderness parameter (C1) Equivalent slenderness parameter (Appendix E3) Effective slenderness ratio defined by Equation A-F3-2 (Appendix F3) Limiting slenderness parameter for compact element (B5.1) Limiting slenderness parameter for noncompact element (B5.1) Resistance factor (A5.3) Resistance factor for flexure (F1) Resistance factor for compression (A5) Resistance factor for axially loaded composite columns (I2.2) Resistance factor for shear on the failure path (D3) Resistance factor for tension (D1) Resistance factor for shear (F2.2)

Separation ratio for built-up compression members =


6 - 24


6 - 25

CHAPTER A GENERAL PROVISIONS

A1. SCOPE The Load and Resistance Factor Design Specification for Structural Steel Buildings governs the design, fabrication, and erection of steel-framed buildings. As an alternative, the AISC Specification for Structural Steel Buildings, Allowable Stress Design and Plastic Design is permitted. A2. LIMITS OF APPLICABILITY 1.

Structural Steel Defined As used in this Specification, the term structural steel refers to the steel elements of the structural steel frame essential to the support of the required loads. Such elements are enumerated in Section 2.1 of the AISC Code of Standard Practice for Steel Buildings and Bridges. For the design of cold-formed steel structural members, whose profiles contain rounded corners and slender flat elements, the provisions of the American Iron and Steel Institute Load and Resistance Factor Design Specification for the Design of Cold-Formed Steel Structural Members are recommended.

2.

Types of Construction Two basic types of construction and associated design assumptions are permissible under the conditions stated herein, and each will govern in a specific manner the strength of members and the types and strength of their connections. Type FR (fully restrained), commonly designated as “rigid-frame” (continuous frame), assumes that connections have sufficient rigidity to maintain the angles between intersecting members. Type PR (partially restrained) assumes that connections have insufficient rigidity to maintain the angles between intersecting members. The type of construction assumed in the design shall be indicated on the design documents. The design of all connections shall be consistent with the assumption. Type PR construction under this Specification depends upon a predictable proportion of full end restraint. When a portion of the full end restraint of members is used in the design for strength of the connected members or for the stability of the structure as a whole, the capacity of the connections to provide the needed restraint shall be documented in the technical literature or established by analytical or empirical means. When the connection restraint is ignored, commonly designated “simple framAMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 26

GENERAL PROVISIONS

[Chap. A

ing,” it is assumed that for the transmission of gravity loads the ends of the beams and girders are connected for shear only and are free to rotate. For “simple framing” the following requirements apply: (1) The connections and connected members shall be adequate to resist the factored gravity loads as “simple beams.” (2) The connections and connected members shall be adequate to resist the factored lateral loads. (3) The connections shall have sufficient inelastic rotation capacity to avoid overload of fasteners or welds under combined factored gravity and lateral loading. Type PR construction may necessitate some inelastic, but self-limiting, deformation of a structural steel part. A3. MATERIAL 1.

Structural Steel

1a.

ASTM Designations Material conforming to one of the following standard specifications is approved for use under this Specification: Structural Steel, ASTM A36 Pipe, Steel, Black and Hot-Dipped, Zinc-Coated Welded and Seamless ASTM A53, Gr. B High-Strength Low-Alloy Structural Steel, ASTM A242 Cold-Formed Welded and Seamless Carbon Steel Structural Tubing in Rounds and Shapes, ASTM A500 Hot-Formed Welded and Seamless Carbon Steel Structural Tubing, ASTM A501 High-Yield-Strength, Quenched and Tempered Alloy Steel Plate, Suitable for Welding, ASTM A514 High-Strength Carbon-Manganese Steel of Structural Quality, ASTM A529 Steel, Sheet and Strip, Carbon, Hot-Rolled, Structural Quality, ASTM A570, Gr. 40, 45, and 50 High-Strength Low-Alloy Columbium-Vanadium Steels of Structural Quality, ASTM A572 High-Strength Low-Alloy Structural Steel with 50 ksi Minimum Yield Point to 4-in. Thick, ASTM A588 Steel, Sheet and Strip, High-Strength, Low-Alloy, Hot-Rolled and ColdRolled, with Improved Atmospheric Corrosion Resistance, ASTM A606 Steel, Sheet and Strip, High-Strength, Low-Alloy, Columbium or Vanadium, or Both, Hot-Rolled and Cold-Rolled, ASTM A607 Hot-Formed Welded and Seamless High-Strength Low-Alloy Structural Tubing, ASTM A618 Structural Steel for Bridges, ASTM A709 Quenched and Tempered Low-Alloy Structural Steel Plate with 70 ksi Minimum Yield Strength to 4-in. Thick, ASTM A852 Certified mill test reports or certified reports of tests made by the fabricator or AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. A3]

MATERIAL

6 - 27

a testing laboratory in accordance with ASTM A6 or A568, as applicable, shall constitute sufficient evidence of conformity with one of the above ASTM standards. If requested, the fabricator shall provide an affidavit stating that the structural steel furnished meets the requirements of the grade specified. 1b.

Unidentified Steel Unidentified steel, if surface conditions are acceptable according to criteria contained in ASTM A6, may be used for unimportant members or details, where the precise physical properties and weldability of the steel would not affect the strength of the structure.

1c.

Heavy Shapes For ASTM A6 Group 4 and 5 rolled shapes to be used as members subject to primary tensile stresses due to tension or flexure, toughness need not be specified if splices are made by bolting. If such members are spliced using complete-joint penetration welds, the steel shall be specified in the contract documents to be supplied with Charpy V-Notch testing in accordance with ASTM A6, Supplementary Requirement S5. The impact test shall meet a minimum average value of 20 ft-lbs. absorbed energy at +70°F and shall be conducted in accordance with ASTM A673 with the following exceptions: (1) The center longitudinal axis of the specimens shall be located as near as practical to midway between the inner flange surface and the center of the flange thickness at the intersection with the web mid-thickness. (2) Tests shall be conducted by the producer on material selected from a location representing the top of each ingot or part of an ingot used to produce the product represented by these tests. For plates exceeding 2-in. thick used for built-up cross-sections with bolted splices and subject to primary tensile stresses due to tension or flexure, material toughness need not be specified. If such cross-sections are spliced using complete-joint penetration welds, the steel shall be specified in the contract documents to be supplied with Charpy V-Notch testing in accordance with ASTM A6, Supplementary Requirement S5. The impact test shall be conducted by the producer in accordance with ASTM A673, Frequency P, and shall meet a minimum average value of 20 ft-lbs. absorbed energy at +70°F. The above supplementary requirements also apply when complete-joint penetration welded joints through the thickness of ASTM A6 Group 4 and 5 shapes and built-up cross sections with thickness exceeding two inches are used in connections subjected to primary tensile stress due to tension or flexure of such members. The requirements need not apply to ASTM A6 Group 4 and 5 shapes and built-up members with thickness exceeding two inches to which members other than ASTM A6 Group 4 and 5 shapes and built-up members are connected by complete-joint penetration welded joints through the thickness of the thinner material to the face of the heavy material. Additional requirements for joints in heavy rolled and built-up members are given in Sections J1.5, J1.6, J2, and M2.2. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 28

2.

GENERAL PROVISIONS

[Chap. A

Steel Castings and Forgings Cast steel shall conform to one of the following standard specifications: Mild-to-Medium-Strength Carbon-Steel Castings for General Applications, ASTM A27, Gr. 65-35 High-Strength Steel Castings for Structural Purposes, ASTM A148 Gr. 80-50 Steel forgings shall conform to the following standard specification: Steel Forgings Carbon and Alloy for General Industrial Use, ASTM A668 Certified test reports shall constitute sufficient evidence of conformity with standards.

3.

Bolts, Washers, and Nuts Steel bolts, washers, and nuts shall conform to one of the following standard specifications: Carbon and Alloy Steel Nuts for Bolts for High-Pressure and High-Temperature Service, ASTM A194 Carbon Steel Bolts and Studs, 60,000 psi Tensile Strength, ASTM A307 Structural Bolts, Steel, Heat-Treated, 120/105 ksi Minimum Tensile Strength, ASTM A325 Quenched and Tempered Steel Bolts and Studs, ASTM A449 Heat-Treated Steel Structural Bolts, 150 ksi Min. Tensile Strength, ASTM A490 Carbon and Alloy Steel Nuts, ASTM A563 Hardened Steel Washers, ASTM F436 A449 bolts are permitted to be used only in connections requiring bolt diameters greater than 11⁄2-in. and shall not be used in slip-critical connections. Manufacturer’s certification shall constitute sufficient evidence of conformity with the standards.

4.

Anchor Bolts and Threaded Rods Anchor bolt and threaded rod steel shall conform to one of the following standard specifications: Structural Steel, ASTM A36 Alloy Steel and Stainless Steel Bolting Materials for High-Temperature Service, ASTM A193 Quenched and Tempered Alloy Steel Bolts, Studs and Other Externally Threaded Fasteners, ASTM A354 High-Strength Low-Alloy Columbium-Vanadium Steels of Structural Quality, ASTM A572 High-Strength Low-Alloy Structural Steel with 50,000 psi Minimum Yield Point to 4-in. Thick, ASTM A588 High-Strength Nonheaded Steel Bolts and Studs, ASTM A687 Threads on bolts and rods shall conform to the Unified Standard Series of ANSI B18.1 and shall have Class 2A tolerances. Steel bolts conforming to other provisions of Section A3.3 are permitted as AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. A4]

LOADS AND LOAD COMBINATIONS

6 - 29

anchor bolts. A449 material is acceptable for high-strength anchor bolts and threaded rods of any diameter. Manufacturer’s certification shall constitute sufficient evidence of conformity with the standards. 5.

Filler Metal and Flux for Welding Welding electrodes and fluxes shall conform to one of the following specifications of the American Welding Society: Specification for Carbon Steel Electrodes for Shield Metal Arc Welding, AWS A5.1 Specification for Low-Alloy Steel Covered Arc Welding Electrodes, AWS A5.5 Specification for Carbon Steel Electrodes and Fluxes for Submerged Arc Welding, AWS A5.17 Specification for Carbon Steel Filler Metals for Gas Shielded Arc Welding, AWS A5.18 Specification for Carbon Steel Electrodes for Flux Cored Arc Welding, AWS A5.20 Specification for Low-Alloy Steel Electrodes and Fluxes for Submerged Arc Welding, AWS A5.23 Specification for Low-Alloy Steel Filler Metals for Gas Shielded Arc Welding, AWS A5.28 Specification for Low-Alloy Steel Electrodes for Flux Cored Arc Welding, AWS A5.29 Manufacturer’s certification shall constitute sufficient evidence of conformity with the standards. Electrodes (filler metals) that are suitable for the intended application shall be selected. Weld metal notch toughness is generally not critical for building construction.

6.

Stud Shear Connectors Steel stud shear connectors shall conform to the requirements of Structural Welding Code—Steel, AWS D1.1. Manufacturer’s certification shall constitute sufficient evidence of conformity with the code.

A4. LOADS AND LOAD COMBINATIONS The nominal loads shall be the minimum design loads stipulated by the applicable code under which the structure is designed or dictated by the conditions involved. In the absence of a code, the loads and load combinations shall be those stipulated in the American Society of Civil Engineers Standard Minimum Design Loads for Buildings and Other Structures, ASCE 7. For design purposes, the loads stipulated by the applicable code shall be taken as nominal loads. For ease of reference, the more common ASCE load combinations are listed in the following section. Seismic design of buildings assigned to the higher risk Seismic Performance Categories defined in the AISC Seismic Provisions for Structural Steel Buildings shall comply with that document. Seismic design not covered by the AISC AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 30

GENERAL PROVISIONS

[Chap. A

Seismic Provisions for Structural Steel Buildings shall be in accordance with this Specification. 1.

Loads, Load Factors, and Load Combinations The following nominal loads are to be considered: D : dead load due to the weight of the structural elements and the permanent features on the structure L : live load due to occupancy and moveable equipment Lr : roof live load W : wind load S : snow load E : earthquake load determined in accordance with Part I of the AISC Seismic Provisions for Structural Steel Buildings R : load due to initial rainwater or ice exclusive of the ponding contribution The required strength of the structure and its elements must be determined from the appropriate critical combination of factored loads. The most critical effect may occur when one or more loads are not acting. The following load combinations and the corresponding load factors shall be investigated: 1.4D

(A4-1)

1.2D + 1.6L + 0.5(Lr or S or R)

(A4-2)

1.2D + 1.6(Lr or S or R) + (0.5L or 0.8W)

(A4-3)

1.2D + 1.3W + 0.5L + 0.5(Lr or S or R)

(A4-4)

1.2D ± 1.0E + 0.5L + 0.2S

(A4-5)

0.9D ± (1.3W or 1.0E)

(A4-6)

Exception: The load factor on L in combinations A4-3, A4-4, and A4-5 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf. 2.

Impact For structures carrying live loads which induce impact, the assumed nominal live load shall be increased to provide for this impact in combinations A4-2 and A4-3. If not otherwise specified, the increase shall be: For supports of elevators and elevator machinery . . . . . . . . . . . . . . . . . . 100% For supports of light machinery, shaft or motor driven, not less than . . . . 20% For supports of reciprocating machinery or power driven units, not less than. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50% For hangers supporting floors and balconies . . . . . . . . . . . . . . . . . . . . . . . 33% For cab-operated traveling crane support girders and their connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25% For pendant-operated traveling crane support girders and their connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10% AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. A5]

3.

DESIGN BASIS

6 - 31

Crane Runway Horizontal Forces The nominal lateral force on crane runways to provide for the effect of moving crane trolleys shall be a minimum of 20 percent of the sum of weights of the lifted load and of the crane trolley, but exclusive of other parts of the crane. The force shall be assumed to be applied at the top of the rails, acting in either direction normal to the runway rails, and shall be distributed with due regard for lateral stiffness of the structure supporting the rails. The nominal longitudinal force shall be a minimum of 10 percent of the maximum wheel loads of the crane applied at the top of the rail, unless otherwise specified.

A5. DESIGN BASIS 1.

Required Strength at Factored Loads The required strength of structural members and connections shall be determined by structural analysis for the appropriate factored load combinations given in Section A4. Design by either elastic or plastic analysis is permitted, except that design by plastic analysis is permitted only for steels with specified yield stresses not exceeding 65 ksi and is subject to provisions of Sections B5.2, C2, E1.2, F1.2d, H1, and I1. Beams and girders composed of compact sections, as defined in Section B5.1, and satisfying the unbraced length requirements of Section F1.2d (including composite members) which are continuous over supports or are rigidly framed to columns may be proportioned for nine-tenths of the negative moments produced by gravity loading at points of support, provided that the maximum positive moment is increased by one-tenth of the average negative moments. This reduction is not permitted for hybrid beams, members of A514 steel, or moments produced by loading on cantilevers. If the negative moment is resisted by a column rigidly framed to the beam or girder, the one-tenth reduction may be used in proportioning the column for combined axial force and flexure, provided that the axial force does not exceed φc times 0.15AgFy, where Ag = gross area, in.2 Fy = specified minimum yield stress, ksi φc = resistance factor for compression

2.

Limit States LRFD is a method of proportioning structures so that no applicable limit state is exceeded when the structure is subjected to all appropriate factored load combinations. Strength limit states are related to safety and concern maximum load carrying capacity. Serviceability limit states are related to performance under normal service conditions. The term “resistance” includes both strength limit states and serviceability limit states. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 32

3.

GENERAL PROVISIONS

[Chap. A

Design for Strength The design strength of each structural component or assemblage must equal or exceed the required strength based on the factored loads. The design strength φRn for each applicable limit state is calculated as the nominal strength Rn multiplied by a resistance factor φ. The required strength is determined for each applicable load combination as stipulated in Section A4. Nominal strengths Rn and resistance factors φ are given in Chapters D through K.

4.

Design for Serviceability and Other Considerations The overall structure and the individual members, connections, and connectors shall be checked for serviceability. Provisions for design for serviceability are given in Chapter L.

A6. REFERENCED CODES AND STANDARDS The following documents are referenced in this Specification: American National Standards Institute ANSI B18.1-72 American Society of Civil Engineers ASCE 7-88 American Society for Testing and Materials ASTM A6-91b ASTM A27-87 ASTM A53-88 ASTM A148-84 ASTM A194-91 ASTM A242-91a ASTM A325-91c ASTM A354-91 ASTM A490-91 ASTM A500-90a ASTM A502-91 ASTM A514-91 ASTM A563-91c ASTM A570-91 ASTM A588-91a ASTM A606-91a ASTM A618-90a ASTM A668-85a ASTM A709-91 ASTM A852-91 ASTM C330-89 ASTM F436-91 American Welding Society AWS D1.1-92 AWS A5.1-91 AWS A5.17-89 AWS A5.18-79 AWS A5.23-90 AWS A5.28-79

ASTM A36-91 ASTM A193-91 ASTM A307-91 ASTM A449-91a ASTM A501-89 ASTM A529-89 ASTM A572-91 ASTM A607-91 ASTM A687-89 ASTM C33-90

AWS A5.5-81 AWS A5.20-79 AWS A5.29-80

Research Council on Structural Connections Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts, 1988 American Iron and Steel Institute Load and Resistance Factor Design Specification for Cold-Formed Steel Members, 1991 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. A7]

DESIGN DOCUMENTS

6 - 33

American Institute of Steel Construction, Inc. Code of Standard Practice for Steel Buildings and Bridges, 1992 Seismic Provisions for Structural Steel Buildings, 1992 Specification for Load and Resistance Factor Design of Single-Angle Members, 1993 A7. DESIGN DOCUMENTS 1.

Plans The design plans shall show a complete design with sizes, sections, and relative locations of the various members. Floor levels, column centers and offsets shall be dimensioned. Drawings shall be drawn to a scale large enough to show the information clearly. Design documents shall indicate the type or types of construction as defined in Section A2.2 and include the required strengths (moments and forces) if necessary for preparation of shop drawings. Where joints are to be assembled with high-strength bolts, the design documents shall indicate the connection type (i.e., snug-tight bearing, fully-tensioned bearing, direct tension, or slip-critical). Camber of trusses, beams, and girders, if required, shall be specified in the design documents. The requirements for stiffeners and bracing shall be shown in the design documents.

2.

Standard Symbols and Nomenclature Welding and inspection symbols used on plans and shop drawings shall be the American Welding Society symbols. Welding symbols for special requirements not covered by AWS is permitted to be used provided a complete explanation thereof is shown in the design documents.

3.

Notation for Welding Weld lengths called for in the design documents and on the shop drawings shall be the net effective lengths.


6 - 34

CHAPTER B DESIGNREQUIREMENTS REQUIREMENTS DESIGN

This chapter contains provisions which are common to the Specification as a whole. B1. GROSS AREA The gross area Ag of a member at any point is the sum of the products of the thickness and the gross width of each element measured normal to the axis of the member. For angles, the gross width is the sum of the widths of the legs less the thickness. B2. NET AREA The net area An of a member is the sum of the products of the thickness and the net width of each element computed as follows: In computing net area for tension and shear, the width of a bolt hole shall be taken as 1⁄16-in. greater than the nominal dimension of the hole. For a chain of holes extending across a part in any diagonal or zigzag line, the net width of the part shall be obtained by deducting from the gross width the sum of the diameters or slot dimensions as provided in Section J3.2, of all holes in the chain, and adding, for each gage space in the chain, the quantity s2 / 4g where s = longitudinal center-to-center spacing (pitch) of any two consecutive holes, in. g = transverse center-to-center spacing (gage) between fastener gage lines, in. For angles, the gage for holes in opposite adjacent legs shall be the sum of the gages from the back of the angles less the thickness. In determining the net area across plug or slot welds, the weld metal shall not be considered as adding to the net area. B3. EFFECTIVE NET AREA FOR TENSION MEMBERS The effective net area for tension members shall be determined as follows: 1. When a tension load is transmitted directly to each of the cross-sectional elements by fasteners or welds, the effective net area Ae is equal to the net area An. 2. When a tension load is transmitted by bolts or rivets through some but not AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. B4]

STABILITY

6 - 35

all of the cross-sectional elements of the member, the effective net area Ae shall be computed as: Ae = AU

(B3-1)

where A U _ x L

= area as defined below = reduction coefficient _ = 1 − (x / L) ≤ 0.9 or as defined in B3c or B3d = connection eccentricity, in. = length of connection in the directions of loading, in.

(B3-2)

Larger values of U are permitted to be used when justified by tests or other rational criteria. (a) When the tension load is transmitted only by bolts or rivets: A = An = net area of member, in.2 (b) When the tension load is transmitted only by longitudinal welds to other than a plate member or by longitudinal welds in combination with transverse welds: A = Ag = gross area of member, in.2 (c) When the tension load is transmitted only by transverse welds: A = area of directly connected elements, in.2 U = 1.0 (d) When the tension load is transmitted to a plate by longitudinal welds along both edges at the end of the plate for l ≥ w: A = area of plate, in.2 For l ≥ 2w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U = 1.00 For 2w > l ≥ 1.5w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U = 0.87 For 1.5w > l ≥ w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U = 0.75 where l = length of weld, in. w = plate width (distance between welds), in. For effective area of connecting elements, see Section J5.2. B4. STABILITY General stability shall be provided for the structure as a whole and for each of its elements. Consideration shall be given to the significant effects of the loads on the deflected shape of the structure and its individual elements. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 36

DESIGN REQUIREMENTS

[Chap. B

B5. LOCAL BUCKLING 1.

Classification of Steel Sections Steel sections are classified as compact, noncompact, or slender-element sections. For a section to qualify as compact, its flanges must be continuously connected to the web or webs and the width-thickness ratios of its compression elements must not exceed the limiting width-thickness ratios λp from Table B5.1. If the width-thickness ratio of one or more compression elements exceeds λp, but does not exceed λr, the section is noncompact. If the width-thickness ratio of any element exceeds λr from Table B5.1, the section is referred to as a slender-element compression section. For unstiffened elements which are supported along only one edge parallel to the direction of the compression force, the width shall be taken as follows: (a) For flanges of I-shaped members and tees, the width b is half the full-flange width, bf . (b) For legs of angles and flanges of channels and zees, the width b is the full nominal dimension. (c) For plates, the width b is the distance from the free edge to the first row of fasteners or line of welds. (d) For stems of tees, d is taken as the full nominal depth. For stiffened elements which are supported along two edges parallel to the direction of the compression force, the width shall be taken as follows: (a) For webs of rolled or formed sections, h is the clear distance between flanges less the fillet or corner radius at each flange; hc is twice the distance from the centroid to the inside face of the compression flange less the fillet or corner radius. (b) For webs of built-up sections, h is the distance between adjacent lines of fasteners or the clear distance between flanges when welds are used, and hc is twice the distance from the centroid to the nearest line of fasteners at the compression flange or the inside face of the compression flange when welds are used. (c) For flange or diaphragm plates in built-up sections, the width b is the distance between adjacent lines of fasteners or lines of welds. (d) For flanges of rectangular hollow structural sections, the width b is the clear distance between webs less the inside corner radius on each side. If the corner radius is not known, the width may be taken as the total section width minus three times the thickness. For tapered flanges of rolled sections, the thickness is the nominal value halfway between the free edge and the corresponding face of the web.

2.

Design by Plastic Analysis Design by plastic analysis is permitted when flanges subject to compression involving hinge rotation and all webs have a width-thickness ratio less than or AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. B10]

PROPORTIONS OF BEAMS AND GIRDERS

6 - 37

equal to the limiting λp from Table B5.1. For circular hollow sections see Footnote d of Table B5.1. Design by plastic analysis is subject to the limitations in Section A5.1. 3.

Slender-Element Compression Sections For the flexural design of I-shaped sections, channels and rectangular or circular sections with slender flange elements, see Appendix F1. For other shapes in flexure or members in axial compression that have slender compression elements, see Appendix B5.3. For plate girders with slender web elements, see Appendix G.

B6. BRACING AT SUPPORTS At points of support for beams, girders and trusses, restraint against rotation about their longitudinal axis shall be provided. B7. LIMITING SLENDERNESS RATIOS For members in which the design is based on compression, the slenderness ratio Kl / r preferably should not exceed 200. For members in which the design is based on tension, the slenderness ratio l / r preferably should not exceed 300. The above limitation does not apply to rods in tension. Members in which the design is dictated by tension loading, but which may be subject to some compression under other load conditions, need not satisfy the compression slenderness limit. B8. SIMPLE SPANS Beams, girders and trusses designed on the basis of simple spans shall have an effective length equal to the distance between centers of gravity of the members to which they deliver their end reactions. B9. END RESTRAINT When designed on the assumption of full or partial end restraint due to continuous, semicontinuous, or cantilever action, the beams, girders, and trusses, as well as the sections of the members to which they connect, shall be designed to carry the factored forces and moments so introduced, as well as all other factored forces, without exceeding the design strengths prescribed in Chapters D through K, except that some inelastic but self-limiting deformation of a part of the connection is permitted. B10. PROPORTIONS OF BEAMS AND GIRDERS Rolled or welded shapes, plate girders and cover-plated beams shall, in general, be proportioned by the moment of inertia of the gross section. No deduction shall be made for bolt or rivet holes in either flange provided that 0.75Fu Afn ≥ 0.9Fy Afg

(B10-1)

where Afg is the gross flange area and Afn is the net flange area calculated in AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 38

DESIGN REQUIREMENTS

[Chap. B

TABLE B5.1 Limiting Width-Thickness Ratios for Compression Elements λp (compact)

λr (non compact)

Flanges of I-shaped rolled beams and channels in flexure

b/t

65 / √ Fy [c]

141 / √  Fy − 10 

Flanges of I-shaped hybrid or welded beams in flexure

b/t

65 / √ Fyf

162 [f]  √ (Fyf − 16.5) / kc

Flanges projecting from built-up compression members

b/t

NA

109 / √  Fy / kc [f]

Outstanding legs of pairs of angles in continuous contact, flanges of channels in axial compression; angles and plates projecting from beams or compression members

b/t

NA

95 / √ Fy

Legs of single angle struts; legs of double angle struts with separators; unstiffened elements, i.e., supported along one edge

b/t

NA

76 / √ Fy

Stems of tees

d/t

NA

127 / √ Fy

Description of Element

Unstiffened Elements

Limiting WidthThickness Ratios

Width Thickness Ratio

accordance with the provisions of Sections B1 and B2 and Fu is the specified minimum tensile strength. If 0.75Fu Afn < 0.9Fy Afg

(B10-2)

the member flexural properties shall be based on an effective tension flange area Afe Afe =

5 Fu A 6 Fy fn

(B10-3)

Hybrid girders may be proportioned by the moment of inertia of their gross section, subject to the applicable provisions in Appendix G1, provided they are not required to resist an axial force greater than φb times 0.15Fyf Ag, where Fyf is the specified yield stress of the flange material and Ag is the gross area. No limit is placed on the web stresses produced by the applied bending moment for which a hybrid girder is designed, except as provided in Section K3 and Appendix K3. To qualify as hybrid girders, the flanges at any given section shall have the same cross-sectional area and be made of the same grade of steel. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. B10]

PROPORTIONS OF BEAMS AND GIRDERS

6 - 39

TABLE B5.1 (cont.) Limiting Width-Thickness Ratios for Compression Elements λp (compact)

λr (non compact)

Flanges of square and rectangular box and hollow structural sections of uniform thickness subject to bending or compression; flange cover plates and diaphragm plates between lines of fasteners or welds

b/t

190 / √ Fy

238 / √ Fy

Unsupported width of cover plates perforated with a succession of access holes [b]

b/t

NA

317 / √ Fy

Webs in flexural compression [a]

h / tw

640 / √ Fy [c]

970 / √ Fy [g]


h / tw

for Pu / φbPy ≤ 0.125 [c] 2.75Pu 640   1− FY  √ φbPy 

[g]


Stiffened Elements

Limiting WidthThickness Ratios

Width Thickness Ratio

for Pu / φbPy > 0.125 [c] Pu  253 191  ≥ 2.33 − FY  √ φbPy √Fy

Pu  970   1 − 0.74 φbPy FY  √

253 / √ Fy

All other uniformly compressed stiffened elements, i.e., supported along two edges

b/t h / tw

Circular hollow sections In axial compression In flexure

D/t

[a] For hybrid beams, use the yield strength of the flange Fyf instead of Fy [b] Assumes net area of plate at widest hole. [c] Assumes an inelastic rotation capacity of 3. For structures in zones of high seismicity, a greater rotation capacity may be required. [d] For plastic design use 1,300 / Fy.

[e] Fr = compressive residual stress in flange = 10 ksi for rolled shapes = 16.5 ksi for welded shapes 4 [f] kc = but not less than 0.35 ≤ kc ≤ 0.763  √ h / tw [g] For members with unequal flanges, see Appendix B5.1. Fy is the specified minimum yield stress of the type of steel being used.

NA

[d] NA 2,070 / Fy

3,300 / Fy 8,970 / Fy

Flanges of welded beams or girders may be varied in thickness or width by splicing a series of plates or by the use of cover plates. The total cross-sectional area of cover plates of bolted or riveted girders shall not exceed 70 percent of the total flange area. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 40

DESIGN REQUIREMENTS

[Chap. B

High-strength bolts, rivets, or welds connecting flange to web, or cover plate to flange, shall be proportioned to resist the total horizontal shear resulting from the bending forces on the girder. The longitudinal distribution of these bolts, rivets, or intermittent welds shall be in proportion to the intensity of the shear. However, the longitudinal spacing shall not exceed the maximum permitted for compression or tension members in Section E4 or D2, respectively. Bolts, rivets, or welds connecting flange to web shall also be proportioned to transmit to the web any loads applied directly to the flange, unless provision is made to transmit such loads by direct bearing. Partial length cover plates shall be extended beyond the theoretical cutoff point and the extended portion shall be attached to the beam or girder by high-strength bolts in a slip-critical connection, rivets, or fillet welds. The attachment shall be adequate, at the applicable design strength given in Sections J2.2, J3.8, or K3 to develop the cover plate’s portion of the flexural design strength in the beam or girder at the theoretical cutoff point. For welded cover plates, the welds connecting the cover plate termination to the beam or girder in the length a′, defined below, shall be adequate, at the applicable design strength, to develop the cover plate’s portion of the design strength in the beam or girder at the distance a′ from the end of the cover plate. The length a′, measured from the end of the cover plate, shall be: (a) A distance equal to the width of the cover plate when there is a continuous weld equal to or larger than three-fourths of the plate thickness across the end of the plate and continuous welds along both edges of the cover plate in the length a′. (b) A distance equal to one and one-half times the width of the cover plate when there is a continuous weld smaller than three-fourths of the plate thickness across the end of the plate and continuous welds along both edges of the cover plate in the length a′. (c) A distance equal to two times the width of the cover plate when there is no weld across the end of the plate, but continuous welds along both edges of the cover plate in the length a′.


6 - 41

CHAPTER C FRAMES AND OTHER STRUCTURES

This chapter contains general requirements for stability of the structure as a whole. C1. SECOND ORDER EFFECTS Second order (P∆) effects shall be considered in the design of frames. In structures designed on the basis of plastic analysis, the required flexural strength Mu shall be determined from a second-order plastic analysis that satisfies the requirements of Section C2. In structures designed on the basis of elastic analysis, Mu for beam-columns, connections, and connected members shall be determined from a second-order elastic analysis or from the following approximate second-order analysis procedure: Mu = B1Mnt + B2Mlt

(C1-1)

where Mnt = required flexural strength in member assuming there is no lateral = translation of the frame, kip-in. Mlt = required flexural strength in member as a result of lateral translation = of the frame only, kip-in. B1 =

Cm ≥1 (1 − Pu / Pe1 )

(C1-2)

Pe1 = AgFy / λ2c where λc is the slenderness parameter, in which the effective = K in the plane of bending shall be determined in accordance with = Section C2.1, for the braced frame.

λc =

Kl rπ

 √

Fy E

Pu = required axial compressive strength for the member under = consideration, kips Cm = a coefficient based on elastic first-order analysis assuming no lateral = translation of the frame whose value shall be taken as follows: (a) For compression members not subject to transverse loading between their supports in the plane of bending, Cm = 0.6 − 0.4(M1 / M2) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(C1-3)

6 - 42

FRAMES AND OTHER STRUCTURES

[Chap. C

(a) where M1 / M2 is the ratio of the smaller to larger moments at the ends of that portion of the member unbraced in the plane of bending under consideration. M1 / M2 is positive when the member is bent in reverse curvature, negative when bent in single curvature. (b) For compression members subjected to transverse loading between their supports, the value of Cm shall be determined either by rational analysis or by the use of the following values: For members whose ends are restrained. . . . . . . . . . . . . . . . . . . . Cm = 0.85 For members whose ends are unrestrained. . . . . . . . . . . . . . . . . . Cm = 1.00 1  ∆oh  1 − ΣPu    ΣHL

B2

=

B2

=

ΣPu ∆oh ΣH L Pe2

= required axial strength of all columns in a story, kips = lateral inter-story deflection, in. = sum of all story horizontal forces producing ∆oh, kips = story height, in. = AgFy / λ2c, kips, where λc is the slenderness parameter, in which the = effective length factor K in the plane of bending shall be determined = in accordance with Section C2.2, for the unbraced frame.

(C1-4)

or 1 ΣPu 1− ΣPe2

(C1-5)

C2. FRAME STABILITY 1.

Braced Frames In trusses and frames where lateral stability is provided by diagonal bracing, shear walls, or equivalent means, the effective length factor K for compression members shall be taken as unity, unless structural analysis shows that a smaller value may be used. The vertical bracing system for a braced multistory frame shall be determined by structural analysis to be adequate to prevent buckling of the structure and to maintain the lateral stability of the structure, including the overturning effects of drift, under the factored loads given in Section A4. The vertical bracing system for a multistory frame may be considered to function together with in-plane shear-resisting exterior and interior walls, floor slabs, and roof decks, which are properly secured to the structural frames. The columns, girders, beams, and diagonal members, when used as the vertical bracing system, may be considered to comprise a vertically cantilevered simply connected truss in the analyses for frame buckling and lateral stability. Axial deformation of all members in the vertical bracing system shall be included in the lateral stability analysis. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C2]

FRAME STABILITY

6 - 43

In structures designed on the basis of plastic analysis, the axial force in these members caused by factored gravity plus factored horizontal loads shall not exceed 0.85φc times AgFy. Girders and beams included in the vertical bracing system of a braced multistory frame shall be proportioned for axial force and moment caused by concurrent factored horizontal and gravity loads. 2.

Unbraced Frames In frames where lateral stability depends upon the bending stiffness of rigidly connected beams and columns, the effective length factor K of compression members shall be determined by structural analysis. The destabilizing effects of gravity loaded columns whose simple connections to the frame do not provide resistance to lateral loads shall be included in the design of the moment-frame columns. Stiffness reduction adjustment due to column inelasticity is permitted. Analysis of the required strength of unbraced multistory frames shall include the effects of frame instability and column axial deformation under the factored loads given in Section A4. In structures designed on the basis of plastic analysis, the axial force in the columns caused by factored gravity plus factored horizontal loads shall not exceed 0.75φc times AgFy.


6 - 44

CHAPTER D TENSION MEMBERS

This chapter applies to prismatic members subject to axial tension caused by static forces acting through the centroidal axis. For members subject to combined axial tension and flexure, see Section H1.1. For threaded rods, see Section J3. For block shear rupture strength at end connections of tension members, see Section J4.3. For the design tensile strength of connecting elements, see Section J5.2. For members subject to fatigue, see Section K3. D1. DESIGN TENSILE STRENGTH The design strength of tension members φt Pn shall be the lower value obtained according to the limit states of yielding in the gross section and fracture in the net section. (a) For yielding in the gross section: φt = 0.90 Pn = Fy Ag

(D1-1)

(b) For fracture in the net section: φt = 0.75 Pn = Fu Ae

(D1-2)

where Ae Ag Fy Fu Pn

= effective net area, in.2 = gross area of member, in.2 = specified minimum yield stress, ksi = specified minimum tensile strength, ksi = nominal axial strength, kips

When members without holes are fully connected by welds, the effective net section used in Equation D1-2 shall be as defined in Section B3. When holes are present in a member with welded-end connections, or at the welded connection in the case of plug or slot welds, the net section through the holes shall be used in Equation D1-2. D2. BUILT-UP MEMBERS For limitations on the longitudinal spacing of connectors between elements in AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. D3]

PIN-CONNECTED MEMBERS AND EYEBARS

6 - 45

continuous contact consisting of a plate and a shape or two plates, see Section J3.5. The longitudinal spacing of connectors between components should preferably limit the slenderness ratio in any component between the connectors to 300. Either perforated cover plates or tie plates without lacing are permitted to be used on the open sides of built-up tension members. Tie plates shall have a length not less than two-thirds the distance between the lines of welds or fasteners connecting them to the components of the member. The thickness of such tie plates shall not be less than one-fiftieth of the distance between these lines. The longitudinal spacing of intermittent welds or fasteners at tie plates shall not exceed six inches. The spacing of tie plates shall be such that the slenderness ratio of any component in the length between tie plates should preferably not exceed 300. D3. PIN-CONNECTED MEMBERS AND EYEBARS The pin diameter shall not be less than seven-eighths times the eyebar body width. The pin-hole diameter shall not be more than 1⁄32-in. greater than the pin diameter. For steels having a yield stress greater than 70 ksi, the hole diameter shall not exceed five times the plate thickness and the width of the eyebar body shall be reduced accordingly. In pin-connected members, the pin hole shall be located midway between the edges of the member in the direction normal to the applied force. For pinconnected members in which the pin is expected to provide for relative movement between connected parts while under full load, the diameter of pin hole shall not be more than 1⁄32-in. greater than the diameter of the pin. The width of the plate beyond the pin hole shall be not less than the effective width on either side of the pin hole. In pin-connected plates other than eyebars, the minimum net area beyond the bearing end of the pin hole, parallel to the axis of the member, shall not be less than two-thirds of the net area required for strength across the pin hole. The design strength of a pin-connected member φPn shall be the lowest value of the following limit states: (a) Tension on the net effective area: φ = φt = 0.75 Pn = 2tbeff Fu

(D3-1)

(b) Shear on the effective area: φsf = 0.75 Pn = 0.6AsfFu (c) For bearing on the projected area of the pin, see Section J8.1. (d) For yielding in the gross section, use Equation D1-1. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(D3-2)

6 - 46

TENSION MEMBERS

[Chap. D

where a = shortest distance from edge of the pin hole to the edge of the member measured parallel to the direction of the force, in. Asf = 2t(a + d / 2), in.2 beff = 2t + 0.63, but not more than the actual distance from the edge of the = hole to the edge of the part measured in the direction normal to the = applied force, in. d = pin diameter, in. t = thickness of plate, in. The corners beyond the pin hole are permitted to be cut at 45° to the axis of the member, provided the net area beyond the pin hole, on a plane perpendicular to the cut, is not less than that required beyond the pin hole parallel to the axis of the member. The design strength of eyebars shall be determined in accordance with Section D1 with Ag taken as the cross-sectional area of the body. Eyebars shall be of uniform thickness, without reinforcement at the pin holes, and have circular heads whose periphery is concentric with the pin hole. The radius of transition between the circular head and the eyebar body shall be not less than the head diameter. The width of the body of the eyebars shall not exceed eight times its thickness. The thickness of less than 1⁄2-in. is permissible only if external nuts are provided to tighten pin plates and filler plates into snug contact. The width b from the hole edge to the plate edge perpendicular to the direction of applied load shall be greater than two-thirds and, for the purpose of calculation, not more than three-fourths times the eyebar body width.


6 - 47

CHAPTER E COLUMNS AND OTHER COMPRESSION MEMBERS

This chapter applies to compact and non-compact prismatic members subject to axial compression through the centroidal axis. For members subject to combined axial compression and flexure, see Section H1.2. For members with slender compression elements, see Appendix B5.3. For tapered members, see Appendix F3. For single-angle members, see AISC Specification for Load and Resistance Design of Single-Angle Members. E1. EFFECTIVE LENGTH AND SLENDERNESS LIMITATIONS 1.

Effective Length The effective length factor K shall be determined in accordance with Section C2.

2.

Design by Plastic Analysis Design by plastic analysis, as limited in Section A5.1, is permitted if the column slenderness parameter λc does not exceed 1.5K.

E2. DESIGN COMPRESSIVE STRENGTH FOR FLEXURAL BUCKLING The design strength for flexural buckling of compression members whose elements have width-thickness ratios less than λr from Section B5.1 is φcPn: φc = 0.85 Pn = AgFcr

(E2-1)

(a) For λc ≤ 1.5 2

Fcr = (0.658λc)Fy

(E2-2)

 0.877  Fcr =  2 Fy  λc 

(E2-3)

(b) For λc > 1.5

where λc =

Kl rπ

 √

Fy E

(E2-4)

Ag = gross area of member, in.2 Fy = specified yield stress, ksi E = modulus of elasticity, ksi AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 48

COLUMNS AND OTHER COMPRESSION MEMBERS

[Chap. E

K = effective length factor l = laterally unbraced length of member, in. r = governing radius of gyration about the axis of buckling, in. For members whose elements do not meet the requirements of Section B5.1, see Appendix B5.3. E3. DESIGN COMPRESSIVE STRENGTH FOR FLEXURAL-TORSIONAL BUCKLING The design strength for flexural-torsional buckling of double-angle and teeshaped compression members whose elements have width-thickness ratios less than λr from Section B5.1 is φcPn: φc = 0.85 Pn = AgFcrft  Fcry + Fcrz   1 − Fcrft =   2H  

4F F H  √ 1−  (F + F ) cry crz

2

cry

crz

  

(E3-1)

where: Fcrz _ ro

GJ = _2 Aro = polar radius of gyration about shear center, in. (see Equation A-E3-8)

H

 xo2 + yo2 = 1 −  _2   ro 

xo, yo = coordinate of shear center with respect to the centroid, in. = 0 for double-angle and tee-shaped members (y-axis of symmetry) xo Fcry is determined according to Section E2 for flexural buckling about the y-axis of symmetry for λc =

Kl ryπ

 √

Fy . E

For double-angle and tee-shaped members whose elements do not meet the requirements of Section B5.1, see Appendix B5.3 to determine Fcry for use in Equation E3-1. Other singly symmetric and unsymmetric columns, and doubly symmetric columns, such as cruciform or built-up columns, with very thin walls shall be designed for the limit states of flexural-torsional and torsional buckling in accordance with Appendix E3. E4. BUILT-UP MEMBERS At the ends of built-up compression members bearing on base plates or milled surfaces, all components in contact with one another shall be connected by a weld having a length not less than the maximum width of the member or by bolts spaced longitudinally not more than four diameters apart for a distance equal to 11⁄2 times the maximum width of the member. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. E4]

BUILT-UP MEMBERS

6 - 49

Along the length of built-up compression members between the end connections required above, longitudinal spacing for intermittent welds, bolts, or rivets shall be adequate to provide for the transfer of the required forces. For limitations on the longitudinal spacing of connectors between elements in continuous contact consisting of a plate and a shape or two plates, see Section J3.5. Where a component of a built-up compression member consists of an outside plate, the maximum spacing shall not exceed the thickness of the thinner outside plate Fy , nor 12 inches, when intermittent welds are provided along the times 127 / √ edges of the components or when fasteners are provided on all gage lines at each section. When fasteners are staggered, the maximum spacing on each gage line Fy nor 18 shall not exceed the thickness of the thinner outside plate times 190 / √ inches. Individual components of compression members composed of two or more shapes shall be connected to one another at intervals, a, such that the effective slenderness ratio Ka / ri of each of the component shapes, between the connectors, does not exceed three-fourths times the governing slenderness ratio of the built-up member. The least radius of gyration ri shall be used in computing the slenderness ratio of each component part. The end connection shall be welded or fully tensioned bolted with clean mill scale or blasted cleaned faying surfaces with Class A coatings. The design strength of built-up members composed of two or more shapes shall be determined in accordance with Section E2 and Section E3 subject to the following modification. If the buckling mode involves relative deformations that produce shear forces in the connectors between individual shapes, Kl / r is replaced by (Kl / r)m determined as follows: (a) For intermediate connectors that are snug-tight bolted:  Kl    =  r m

 √ 2

 Kl   a   r  + r    o  i

2

(E4-1)

(b) For intermediate connectors that are welded or fully-tensioned bolted:  Kl    =  r m

 √ 2

2

 Kl  α2  a   r  + 0.82   (1 + α2)  rib   o

where  Kl    = column slenderness of built-up member acting as a unit  r o  Kl    = modified column slenderness of built-up member  r m

a ri

= largest column slenderness of individual components AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(E4-2)

6 - 50


a rib a ri rib

α h

[Chap. E

= column slenerness of individual components relative to its centroidal axis parallel to axis of buckling = distance between connectors, in. = minimum radius of gyration of individual component, in. = radius of gyration of individual component relative to its centroidal = axis parallel to member axis of buckling, in. = separation ratio = h / 2rib = distance between centroids of individual components perpendicular = to the member axis of buckling, in.

Open sides of compression members built up from plates or shapes shall be provided with continuous cover plates perforated with a succession of access holes. The unsupported width of such plates at access holes, as defined in Section B5.1, is assumed to contribute to the design strength provided that: (1) The width-thickness ratio conforms to the limitations of Section B5.1. (2) The ratio of length (in direction of stress) to width of hole shall not exceed two. (3) The clear distance between holes in the direction of stress shall be not less than the transverse distance between nearest lines of connecting fasteners or welds. (4) The periphery of the holes at all points shall have a minimum radius of 11⁄2-in. As an alternative to perforated cover plates, lacing with tie plates is permitted at each end and at intermediate points if the lacing is interrupted. Tie plates shall be as near the ends as practicable. In main members providing design strength, the end tie plates shall have a length of not less than the distance between the lines of fasteners or welds connecting them to the components of the member. Intermediate tie plates shall have a length not less than one-half of this distance. The thickness of tie plates shall be not less than one-fiftieth of the distance between lines of welds or fasteners connecting them to the segments of the members. In welded construction, the welding on each line connecting a tie plate shall aggregate not less than one-third the length of the plate. In bolted and riveted construction, the spacing in the direction of stress in tie plates shall be not more than six diameters and the tie plates shall be connected to each segment by at least three fasteners. Lacing, including flat bars, angles, channels, or other shapes employed as lacing, shall be so spaced that l / r of the flange included between their connections shall not exceed the governing slenderness ratio for the member as a whole. Lacing shall be proportioned to provide a shearing strength normal to the axis of the member equal to two percent of the compressive design strength of the member. The l / r ratio for lacing bars arranged in single systems shall not exceed 140. For double lacing this ratio shall not exceed 200. Double lacing bars shall be joined at the intersections. For lacing bars in compression, l is permitted to be taken as the unsupported length of the lacing bar between welds or fasteners connecting it to the components of the built-up member for single lacing, and 70 percent of that distance for double lacing. The inclination of lacing bars to the axis of the member shall preferably be not less than 60° for single lacing and AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. E5]

PIN-CONNECTED COMPRESSION MEMBERS

6 - 51

45° for double lacing. When the distance between the lines of welds or fasteners in the flanges is more than 15 inches, the lacing shall preferably be double or be made of angles. For additional spacing requirements, see Section J3. E5. PIN-CONNECTED COMPRESSION MEMBERS Pin connections of pin-connected compression members shall conform to the requirements of Section D3 except Equations D3-1 and D3-2 do not apply.


6 - 52

CHAPTER F BEAMS AND OTHER FLEXURAL MEMBERS

This chapter applies to compact and noncompact prismatic members subject to flexure and shear. For members subject to combined flexure and axial force, see Section H1. For members subject to fatigue, see Section K4. For members with slender compression elements, see Appendix B5. For web-tapered members, see Appendix F3. For members with slender web elements (plate girders), see Appendix G. For single-angle members, the AISC Specification for Load and Resistance Factor Design of SingleAngle Members is applicable. F1.

DESIGN FOR FLEXURE The nominal flexural strength Mn is the lowest value obtained according to the limit stress of: (a) yielding; (b) lateral-torsional buckling; (c) flange local buckling; and (d) web local buckling. For laterally braced compact beams with Lb ≤ Lp, only the limit state of yielding is applicable. For unbraced compact beams and noncompact tees and double angles, only the limit states of yielding and lateral-torsional buckling are applicable. The lateral-torsional buckling limit state is not applicable to members subject to bending about the minor axis, or to square or circular shapes. This section applies to homogeneous and hybrid shapes with at least one axis of symmetry and which are subject to simple bending about one principal axis. For simple bending, the beam is loaded in a plane parallel to a principal axis that passes through the shear center or the beam is restrained against twisting at load points and supports. Only the limit states of yielding and lateral-torsional buckling are considered in this section. The lateral-torsional buckling provisions are limited to doubly symmetric shapes, channels, double angles, and tees. For lateral-torsional buckling of other singly symmetric shapes and for the limit states of flange local buckling and web local buckling of noncompact or slender-element sections, see Appendix F1. For unsymmetric shapes and beams subject to torsion combined with flexure, see Section H2. For biaxial bending, see Section H1.

1.

Yielding The flexural design strength of beams, determined by the limit state of yielding, is φbMn:

φb = 0.90 Mn = Mp AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(F1-1)

Sect. F1]

DESIGN FOR FLEXURE

6 - 53

where

Mp = plastic moment (= Fy Z ≤ 1.5My for homogeneous sections), kip-in. My = moment corresponding to onset of yielding at the extreme fiber from an elastic stress distribution (= Fy S for homogeneous section and Fyf S for hybrid sections), kip-in. 2.

Lateral-Torsional Buckling This limit state is only applicable to members subject to major axis bending. The flexural design strength, determined by the limit state of lateral-torsional buckling, is φbMn:

φb = 0.90 Mn = nominal strength determined as follows: 2a.

Doubly Symmetric Shapes and Channels with Lb ≤ Lr The nominal flexural strength is:   L − Lp   Mn = Cb Mp − (Mp − Mr)  b   ≤ Mp   Lr − Lp  

(F1-2)

where:

Lb = distance between points braced against lateral displacement of the compression flange, or between points braced to prevent twist of the cross section, in. In the above equation, Cb is a modification factor for non-uniform moment diagrams where, when both ends of the beam segment are braced:

Cb =

12.5Mmax 2.5Mmax + 3MA + 4MB + 3MC

(F1-3)

where

Mmax = absolute value of maximum moment in the unbraced segment, kip-in. MA = absolute value of moment at quarter point of the unbraced segment MB = absolute value of moment at centerline of the unbraced beam segment MC = absolute value of moment at three-quarter point of the unbraced beam segment Cb is permitted to be conservatively taken as 1.0 for all cases. For cantilevers or overhangs where the free end is unbraced, Cb = 1.0. The limiting unbraced length for full plastic bending capacity, Lp, shall be determined as follows. (a) For I-shaped members including hybrid sections and channels:

Lp =

300ry Fyf √


(F1-4)

6 - 54

BEAMS AND OTHER FLEXURAL MEMBERS

[Chap. F

(b) For solid rectangular bars and box sections: 3,750ry JA √  Mp

Lp =

(F1-5)

where

A = cross-sectional area, in.2 J = torsional constant, in.4 The limiting laterally unbraced length Lr and the corresponding buckling moment Mr shall be determined as follows: (a) For doubly symmetric I-shaped members and channels:

Lr =

ryX1  √ 1+√  1 + X2FL 2 FL

Mr = FL Sx

(F1-6) (F1-7)

where

X1 =

π Sx

√ EGJA 2

(F1-8)

2

C S  X2 = 4 w  x  Iy  GJ

(F1-9)

= section modulus about major axis, in.3 = modulus of elasticity of steel (29,000 ksi) = shear modulus of elasticity of steel (11,200 ksi) = smaller of (Fyf − Fr) or Fyw = compressive residual stress in flange; 10 ksi for rolled shapes, 16.5 ksi for welded shapes Fyf = yield stress of flange, ksi Fyw = yield stress of web, ksi Iy = moment of inertia about y-axis, in.4 Cw = warping constant, in.6

Sx E G FL Fr

Equations F1-4 and F1-6 are conservatively based on Cb = 1.0. (b) For solid rectangular bars and box sections:

Lr =

57,000ry√ JA 

Mr

Mr = Fyf Sx 2b.

(F1-10) (F1-11)

Doubly Symmetric Shapes and Channels with Lb > Lr The nominal flexural strength is:

Mn = Mcr ≤ Mp where Mcr is the critical elastic moment, determined as follows: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(F1-12)

Sect. F1]

DESIGN FOR FLEXURE

6 - 55

(a) For doubly symmetric I-shaped members and channels: π Mcr = Cb Lb

Mcr =

 √ 2

 πE  EIyGJ +   IyCw  Lb 

Cb Sx X1√ 2 Lb / ry

(F1-13)

  √ (L / ) 1+

X21X2 2 b ry

2

(b) For solid rectangular bars and symmetric box sections:

Mcr = 2c.

JA  57,000 Cb √ Lb / ry

(F1-14)

Tees and Double Angles For tees and double-angle beams loaded in the plane of symmetry:

Mn = Mcr =

π√  EIyGJ [B + √ 1+B  2 ] Lb

(F1-15)

where

Mn ≤ 1.5My for stems in tension Mn ≤ 1.0My for stems in compression  Iy / J B = ±2.3(d / Lb) √

(F1-16)

The plus sign for B applies when the stem is in tension and the minus sign applies when the stem is in compression. If the tip of the stem is in compression anywhere along the unbraced length, use the negative value of B. 2d.

Unbraced Length for Design by Plastic Analysis Design by plastic analysis, as limited in Section A5.1, is permitted for a compact section member bent about the major axis when the laterally unbraced length Lb of the compression flange adjacent to plastic hinge locations associated with the failure mechanism does not exceed Lpd, determined as follows: (a) For doubly symmetric and singly symmetric I-shaped members with the compression flange equal to or larger than the tension flange (including hybrid members) loaded in the plane of the web Lpd =

[3,600 + 2,200 (M1 / M2)] ry Fy

(F1-17)

where = specified minimum yield stress of the compression flange, ksi Fy = smaller moment at end of unbraced length of beam, kip-in. M1 = larger moment at end of unbraced length of beam, kip-in. M2 = radius of gyration about minor axis, in. ry (M1 / M2) is positive when moments cause reverse curvature and negative for single curvature AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 56


[Chap. F

(b) For solid rectangular bars and symmetric box beams Lpd =

5,000 + 3,000 (M1 / M2) ry ≥ 3,000ry / Fy Fy

(F1-18)

There is no limit on Lb for members with circular or square cross sections nor for any beam bent about its minor axis. In the region of the last hinge to form, and in regions not adjacent to a plastic hinge, the flexural design strength shall be determined in accordance with Section F1.2. F2.

DESIGN FOR SHEAR This section applies to unstiffened webs of singly or doubly symmetric beams, including hybrid beams, and channels subject to shear in the plane of the web. For the design shear strength of webs with stiffeners, see Appendix F2 or Appendix G3. For shear in the weak direction of the shapes above, pipes, and unsymmetric sections, see Section H2. For web panels subject to high shear, see Section K1.7. For shear strength at connections, see Sections J4 and J5.

1.

Web Area Determination The web area Aw shall be taken as the overall depth d times the web thickness tw.

2.

Design Shear Strength The design shear strength of unstiffened webs, with h / tw ≤ 260, is φvVn, where φv = 0.90 Vn = nominal shear strength defined as follows For h / tw ≤ 418 / √ Fyw Vn = 0.6Fyw Aw

(F2-1)

Fyw For 418 / √ Fyw < h / tw ≤ 523 / √ Vn = 0.6Fyw Aw(418 / √ Fyw ) / (h / tw)

(F2-2)

For 523 / √ Fyw < h / tw ≤ 260 Vn = (132,000Aw) / (h / tw)2

(F2-3)

The general design shear strength of webs with or without stiffeners is given in Appendix F2.2 and an alternative method utilizing tension field action is given in Appendix G3. 3.

Transverse Stiffeners See Appendix F2.3.

F3.

WEB-TAPERED MEMBERS See Appendix F3. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. F4]

F4.

BEAMS AND GIRDERS WITH WEB OPENINGS

6 - 57

BEAMS AND GIRDERS WITH WEB OPENINGS The effect of all web openings on the design strength of steel and composite beams shall be determined. Adequate reinforcement shall be provided when the required strength exceeds the net strength of the member at the opening.


6 - 58

CHAPTER G PLATE GIRDERS

I-shaped plate girders shall be distinguished from I-shaped beams on the basis of the web slenderness ratio h / tw . When this value is greater than λr the provisions of Appendices G1 and G2 shall apply for design flexural strength. For h / tw ≤ λr, the provisions of Chapter F or Appendix F shall apply for design flexural strength. For girders with unequal flanges, see Appendix B5.1. The design shear strength and transverse stiffener design shall be based on either Section F2 (without tension-field action) or Appendix G3 (with tension-field action). For girders with unequal flanges, see Appendix B5.1.


6 - 59

CHAPTER H MEMBERS UNDER COMBINED FORCES AND TORSION

This chapter applies to prismatic members subject to axial force and flexure about one or both axes of symmetry, with or without torsion, and torsion only. For web-tapered members, see Appendix F3. H1. SYMMETRIC MEMBERS SUBJECT TO BENDING AND AXIAL FORCE 1.

Doubly and Singly Symmetric Members in Flexure and Tension The interaction of flexure and tension in symmetric shapes shall be limited by Equations H1-1a and H1-1b. (a) For

Pu φPn

≥ 0.2 Muy  8  Mux + ≤ 1.0 9  φbMnx φbMny

(H1-1a)

 Mux Pu Muy  + +  ≤ 1.0 2φPn  φbMnx φbMny

(H1-1b)

Pu φPn (b) For

Pu φPn

+

< 0.2

where

Pu = required tensile strength, kips Pn = nominal tensile strength determined in accordance with Section D1, kips Mu = required flexural strength determined in accordance with Section C1, kip-in. Mn = nominal flexural strength determined in accordance with Section F1, kip-in. x = subscript relating symbol to strong axis bending y = subscript relating symbol to weak axis bending. φ = φt = resistance factor for tension (see Section D1) φb = resistance factor for flexure = 0.90 A more detailed analysis of the interaction of flexure and tension is permitted in lieu of Equations H1-1a and H1-1b. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 60

2.

MEMBERS UNDER COMBINED FORCES AND TORSION

[Chap. H

Doubly and Singly Symmetric Members in Flexure and Compression The interaction of flexure and compression in symmetric shapes shall be limited by Equations H1-1a and H1-1b where Pu = required compressive strength, kips Pn = nominal compressive strength determined in accordance with Section E2, kips Mu = required flexural strength determined in accordance with Section C1 kip-in. Mn = nominal flexural strength determined in accordance with Section F1, kip-in. x = subscript relating symbol to strong axis bending y = subscript relating symbol to weak axis bending. φ = φc = resistance factor for compression, = 0.85 (see Section E2) φb = resistance factor for flexure = 0.90

H2. UNSYMMETRIC MEMBERS AND MEMBERS UNDER TORSION AND COMBINED TORSION, FLEXURE, SHEAR, AND/OR AXIAL FORCE The design strength φFy of the member shall equal or exceed the required strength expressed in terms of the normal stress fun or the shear stress fuv, determined by elastic analysis for the factored loads: (a) For the limit state of yielding under normal stress: fun ≤ φFy φ = 0.90

(H2-1)

(b) For the limit state of yielding under shear stress: fuv ≤ 0.6φFy φ = 0.90

(H2-2)

(c) For the limit state of buckling: fun or fuv ≤ φcFcr, as applicable φc = 0.85

(H2-3)

Some constrained local yielding is permitted adjacent to areas which remain elastic. H3. ALTERNATIVE INTERACTION EQUATIONS FOR MEMBERS UNDER COMBINED STRESS See Appendix H3.


6 - 61

CHAPTER I COMPOSITE MEMBERS

This chapter applies to composite columns composed of rolled or built-up structural steel shapes, pipe or tubing, and structural concrete acting together and to steel beams supporting a reinforced concrete slab so interconnected that the beams and the slab act together to resist bending. Simple and continuous composite beams with shear connectors and concrete-encased beams, constructed with or without temporary shores, are included. I1.

DESIGN ASSUMPTIONS Force Determination. In determining forces in members and connections of a structure that includes composite beams, consideration shall be given to the effective sections at the time each increment of load is applied. Elastic Analysis. For an elastic analysis of continuous composite beams without haunched ends, it is permissible to assume that the stiffness of a beam is uniform throughout the beam length. The stiffness is permitted to be computed using the moment of inertia of the composite transformed section in the positive moment region. Plastic Analysis. When plastic analysis is used, the strength of flexural composite members shall be determined from plastic stress distributions. Plastic Stress Distribution for Positive Moment. If the slab in the positive moment region is connected to the steel beam with shear connectors, a concrete stress of 0.85fc′ is permitted to be assumed uniformly distributed throughout the effective compression zone. Concrete tensile strength shall be neglected. A uniformly distributed steel stress of Fy shall be assumed throughout the tension zone and throughout the compression zone in the structural steel section. The net tensile force in the steel section shall be equal to the compressive force in the concrete slab. Plastic Stress Distribution for Negative Moment. If the slab in the negative moment region is connected to the steel beam with shear connectors, a tensile stress of Fyr shall be assumed in all adequately developed longitudinal reinforcing bars within the effective width of the concrete slab. Concrete tensile strength shall be neglected. A uniformly distributed steel stress of Fy shall be assumed throughout the tension zone and throughout the compression zone in the structural steel section. The net compressive force in the steel section shall be equal to the total tensile force in the reinforcing steel. Elastic Stress Distribution. When a determination of elastic stress distribution is required, strains in steel and concrete shall be assumed directly proportional AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 62

COMPOSITE MEMBERS

[Chap. I

to the distance from the neutral axis. The stress shall equal strain times modulus of elasticity for steel, E, or modulus of elasticity for concrete, Ec. Concrete tensile strength shall be neglected. Maximum stress in the steel shall not exceed Fy. Maximum compressive stress in the concrete shall not exceed 0.85fc′ where fc′ is the specified compressive strength of the concrete. In composite hybrid beams, the maximum stress in the steel flange shall not exceed Fyf but the strain in the web may exceed the yield strain; the stress shall be taken as Fyw at such locations. Fully Composite Beam. Shear connectors are provided in sufficient numbers to develop the maximum flexural strength of the composite beam. For elastic stress distribution it shall be assumed that no slip occurs. Partially Composite Beam. The shear strength of shear connectors governs the flexural strength of the partially composite beam. Elastic computations such as those for deflections, fatigue, and vibrations shall include the effect of slip. Concrete-Encased Beam. A beam totally encased in concrete cast integrally with the slab may be assumed to be interconnected to the concrete by natural bond, without additional anchorage, provided that: (1) concrete cover over beam sides and soffit is at least two inches; (2) the top of the beam is at least 11⁄2-in. below the top and two inches above the bottom of the slab; and (3) concrete encasement contains adequate mesh or other reinforcing steel to prevent spalling of concrete. Composite Column. A steel column fabricated from rolled or built-up steel shapes and encased in structural concrete or fabricated from steel pipe or tubing and filled with structural concrete shall be designed in accordance with Section I2. I2.

COMPRESSION MEMBERS

1.

Limitations To qualify as a composite column, the following limitations shall be met: (1) The cross-sectional area of the steel shape, pipe, or tubing shall comprise at least four percent of the total composite cross section. (2) Concrete encasement of a steel core shall be reinforced with longitudinal load carrying bars, longitudinal bars to restrain concrete, and lateral ties. Longitudinal load carrying bars shall be continuous at framed levels; longitudinal restraining bars may be interrupted at framed levels. The spacing of ties shall be not greater than two-thirds of the least dimension of the composite cross section. The cross-sectional area of the transverse and longitudinal reinforcement shall be at least 0.007 sq. in. per inch of bar spacing. The encasement shall provide at least 11⁄2-in. of clear cover outside of both transverse and longitudinal reinforcement. (3) Concrete shall have a specified compressive strength fc′ of not less than 3 ksi nor more than 8 ksi for normal weight concrete and not less than 4 ksi for light weight concrete. (4) The specified minimum yield stress of structural steel and reinforcing bars AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. I2]

COMPRESSION MEMBERS

6 - 63

used in calculating the strength of a composite column shall not exceed 55 ksi. (5) The minimum wall thickness of structural steel pipe or tubing filled with concrete shall be equal to b√  Fy / 3E for each face of width b in rectangular sections and D√  Fy / 8E for circular sections of outside diameter D. 2.

Design Strength The design strength of axially loaded composite columns is φcPn, where φc = 0.85 Pn = nominal axial compressive strength determined from Equations E2-1 through E2-4 with the following modifications: (1) As = gross area of steel shape, pipe, or tubing, in.2 (replaces Ag) rm = radius of gyration of the steel shape, pipe, or tubing except that for steel shapes it shall not be less than 0.3 times the overall thickness of the composite cross section in the plane of buckling, in. (replaces r) (2) Replace Fy with modified yield stress Fmy from Equation I2-1 and replace E with modified modulus of elasticity Em from Equation I2-2. Fmy = Fy + c1Fyr (Ar / As) + c2 fc′(Ac / As)

(I2-1)

Em = E + c3Ec (Ac / As)

(I2-2)

where = area of concrete, in.2 = area of longitudinal reinforcing bars, in.2 = area of steel, in.2 = modulus of elasticity of steel, ksi = modulus of elasticity of concrete. Ec is permitted to be computed from fc ′ where w, the unit weight of concrete, is expressed in lbs./cu. Ec = w1.5√ ft and fc′ is expressed in ksi. Fy = specified minimum yield stress of steel shape, pipe, or tubing, ksi Fyr = specified minimum yield stress of longitudinal reinforcing bars, ksi fc′ = specified compressive strength of concrete, ksi c1, c2, c3 = numerical coefficients. For concrete-filled pipe and tubing: c1 = 1.0, c2 = 0.85, and c3 = 0.4; for concrete encased shapes c1 = 0.7, c2 = 0.6, and c3 = 0.2 Ac Ar As E Ec

3.

Columns with Multiple Steel Shapes If the composite cross section includes two or more steel shapes, the shapes shall be interconnected with lacing, tie plates, or batten plates to prevent buckling of individual shapes before hardening of concrete.

4.

Load Transfer The portion of the design strength of axially loaded composite columns resisted AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 64

COMPOSITE MEMBERS

[Chap. I

by concrete shall be developed by direct bearing at connections. When the supporting concrete area is wider than the loaded area on one or more sides and otherwise restrained against lateral expansion on the remaining sides, the maximum design strength of concrete shall be 1.7φc fc′AB, where φc = 0.60 AB = loaded area I3.

FLEXURAL MEMBERS

1.

Effective Width The effective width of the concrete slab on each side of the beam center-line shall not exceed: (a) one-eighth of the beam span, center to center of supports; (b) one-half the distance to the center-line of the adjacent beam; or (c) the distance to the edge of the slab.

2.

Strength of Beams with Shear Connectors The positive design flexural strength φbMn shall be determined as follows: Fyf : (a) For h / tw ≤ 640 / √ φb = 0.85; Mn shall be determined from the plastic stress distribution on the composite section.

Fyf : (b) For h / tw > 640 / √ φb = 0.90; Mn shall be determined from the superposition of elastic stresses, considering the effects of shoring. The negative design flexural strength φbMn shall be determined for the steel section alone, in accordance with the requirements of Chapter F. Alternatively, the negative design flexural strength φbMn shall be computed with: φb = 0.85 and Mn determined from the plastic stress distribution on the composite section, provided that: (1) Steel beam is an adequately braced compact section, as defined in Section B5. (2) Shear connectors connect the slab to the steel beam in the negative moment region. (3) Slab reinforcement parallel to the steel beam, within the effective width of the slab, is properly developed. 3.

Strength of Concrete-Encased Beams The design flexural strength φbMn shall be computed with φb = 0.90 and Mn determined from the superposition of elastic stresses, considering the effects of shoring. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. I3]

FLEXURAL MEMBERS

6 - 65

Alternatively, the design flexural strength φbMn shall be computed with φb = 0.90 and Mn determined from the plastic stress distribution on the steel section alone. 4.

Strength During Construction When temporary shores are not used during construction, the steel section alone shall have adequate strength to support all loads applied prior to the concrete attaining 75 percent of its specified strength fc′. The design flexural strength of the steel section shall be determined in accordance with the requirements of Section F1.

5.

Formed Steel Deck

5a.

General The design flexural strength φbMn of composite construction consisting of concrete slabs on formed steel deck connected to steel beams shall be determined by the applicable portions of Section I3.2, with the following modifications. This section is applicable to decks with nominal rib height not greater than three inches. The average width of concrete rib or haunch wr shall be not less than two inches, but shall not be taken in calculations as more than the minimum clear width near the top of the steel deck. See Section I3.5c for additional restrictions. The concrete slab shall be connected to the steel beam with welded stud shear connectors 3⁄4-in. or less in diameter (AWS D1.1). Studs shall be welded either through the deck or directly to the steel beam. Stud shear connectors, after installation, shall extend not less than 11⁄2-in. above the top of the steel deck. The slab thickness above the steel deck shall be not less than two inches.

5b.

Deck Ribs Oriented Perpendicular to Steel Beam Concrete below the top of the steel deck shall be neglected in determining section properties and in calculating Ac for deck ribs oriented perpendicular to the steel beams. The spacing of stud shear connectors along the length of a supporting beam shall not exceed 36 inches. The nominal strength of a stud shear connector shall be the value stipulated in Section I5 multiplied by the following reduction factor: 0.85 (wr / hr) [(Hs / hr) − 1.0] ≤ 1.0 Nr √

(I3-1)

where hr = nominal rib height, in. Hs = length of stud connector after welding, in., not to exceed the value (hr + 3) in computations, although actual length may be greater Nr = number of stud connectors in one rib at a beam intersection, not to exceed three in computations, although more than three studs may be installed AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 66

COMPOSITE MEMBERS

[Chap. I

wr = average width of concrete rib or haunch (as defined in Section I3.5a), in. To resist uplift, steel deck shall be anchored to all supporting members at a spacing not to exceed 18 inches. Such anchorage shall be provided by stud connectors, a combination of stud connectors and arc spot (puddle) welds, or other devices specified by the designer. 5c.

Deck Ribs Oriented Parallel to Steel Beam Concrete below the top of the steel deck may be included in determining section properties and shall be included in calculating Ac in Section I5. Steel deck ribs over supporting beams may be split longitudinally and separated to form a concrete haunch. When the nominal depth of steel deck is 11⁄2-in. or greater, the average width wr of the supported haunch or rib shall be not less than two inches for the first stud in the transverse row plus four stud diameters for each additional stud. The nominal strength of a stud shear connector shall be the value stipulated in Section I5, except that when wr / hr is less than 1.5, the value from Section I5 shall be multiplied by the following reduction factor: 0.6(wr / hr)[(Hs / hr) − 1.0] ≤ 1.0

(I3-2)

where hr and Hs are as defined in Section I3.5b and wr is the average width of concrete rib or haunch as defined in Section I3.5a. 6.

Design Shear Strength The design shear strength of composite beams shall be determined by the shear strength of the steel web, in accordance with Section F2.

I4.

COMBINED COMPRESSION AND FLEXURE The interaction of axial compression and flexure in the plane of symmetry on composite members shall be limited by Section H1.2 with the following modifications: = nominal flexural strength determined from plastic stress distribution on the composite cross section except as provided below, kip-in. Pe1 , Pe2 = AsFmy / λ2c elastic buckling load, kips = modified yield stress, ksi, see Section I2 Fmy φb = resistance factor for flexure from Section I3 = resistance factor for compression = 0.85 φc = column slenderness parameter defined by Equation E2-4 as λc modified in Section I2.2 Mn

When the axial term in Equations H1-1a and H1-1b is less than 0.3, the nominal flexural strength Mn shall be determined by straight line transition between the nominal flexural strength determined from the plastic distribution on the composite cross sections at (Pu / φcPn ) = 0.3 and the flexural strength at Pu = 0 as AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. I5]

SHEAR CONNECTORS

6 - 67

determined in Section I3. If shear connectors are required at Pu = 0, they shall be provided whenever Pu / φcPn is less than 0.3. I5.

SHEAR CONNECTORS This section applies to the design of stud and channel shear connectors. For connectors of other types, see Section I6.

1.

Materials Shear connectors shall be headed steel studs not less than four stud diameters in length after installation, or hot rolled steel channels. The stud connectors shall conform to the requirements of Section A3.6. The channel connectors shall conform to the requirements of Section A3. Shear connectors shall be embedded in concrete slabs made with ASTM C33 aggregate or with rotary kiln produced aggregates conforming to ASTM C330, with concrete unit weight not less than 90 pcf.

2.

Horizontal Shear Force Except for concrete-encased beams as defined in Section I1, the entire horizontal shear at the interface between the steel beam and the concrete slab shall be assumed to be transferred by shear connectors. For composite action with concrete subject to flexural compression, the total horizontal shear force between the point of maximum positive moment and the point of zero moment shall be taken as the smallest of the following: (1) 0.85fc′Ac; (2) AsFy; and (3) ΣQn; where fc′ Ac As Fy ΣQn

= specified compressive strength of concrete, ksi = area of concrete slab within effective width, in.2 = area of steel cross section, in.2 = minimum specified yield stress, ksi = sum of nominal strengths of shear connectors between the point of maximum positive moment and the point of zero moment, kips

For hybrid beams, the yield force shall be computed separately for each component of the cross section; AsFy of the entire cross section is the sum of the component yield forces. In continuous composite beams where longitudinal reinforcing steel in the negative moment regions is considered to act compositely with the steel beam, the total horizontal shear force between the point of maximum negative moment and the point of zero moment shall be taken as the smaller of ArFyr and ΣQn; where = area of adequately developed longitudinal reinforcing steel within the effective width of the concrete slab, in.2 Fyr = minimum specified yield stress of the reinforcing steel, ksi ΣQn = sum of nominal strengths of shear connectors between the point of maximum negative moment and the point of zero moment, kips

Ar


6 - 68

3.

COMPOSITE MEMBERS

[Chap. I

Strength of Stud Shear Connectors The nominal strength of one stud shear connector embedded in a solid concrete slab is  fc′Ec ≤ AscFu Qn = 0.5Asc√

(I5-1)

where Asc fc′ Fu Ec

= cross-sectional area of a stud shear connector, in.2 = specified compressive strength of concrete, ksi = minimum specified tensile strength of a stud shear connector, ksi = modulus of elasticity of concrete, ksi

For stud shear connector embedded in a slab on a formed steel deck, refer to Section I3 for reduction factors given by Equations I3-1 and I3-2 as applicable. √ fc′Ec term in Equation I5-1. The reduction factors apply only to 0.5Asc 4.

Strength of Channel Shear Connectors The nominal strength of one channel shear connector embedded in a solid concrete slab is

 fc′Ec Qn = 0.3(tf + 0.5tw)Lc√

(I5-2)

where tf = flange thickness of channel shear connector, in. tw = web thickness of channel shear connector, in. Lc = length of channel shear connector, in. 5.

Required Number of Shear Connectors The number of shear connectors required between the section of maximum bending moment, positive or negative, and the adjacent section of zero moment shall be equal to the horizontal shear force as determined in Section I5.2 divided by the nominal strength of one shear connector as determined from Section I5.3 or Section I5.4.

6.

Shear Connector Placement and Spacing Unless otherwise specified shear connectors required each side of the point of maximum bending moment, positive or negative, shall be distributed uniformly between that point and the adjacent points of zero moment. However, the number of shear connectors placed between any concentrated load and the nearest point of zero moment shall be sufficient to develop the maximum moment required at the concentrated load point. Except for connectors installed in the ribs of formed steel decks, shear connectors shall have at least one inch of lateral concrete cover. Unless located over the web, the diameter of studs shall not be greater than 2.5 times the thickness of the flange to which they are welded. The minimum center-to-center spacing of stud connectors shall be six diameters along the longitudinal axis of the supporting composite beam and four diameters transverse to the longitudinal axis of the supporting composite beam, except that within the ribs of formed steel decks the center-to-center spacing may be as small as four diameters in any AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. I6]

SPECIAL CASES

6 - 69

direction. The maximum center-to-center spacing of shear connectors shall not exceed eight times the total slab thickness. Also see Section I3.5b. I6.

SPECIAL CASES When composite construction does not conform to the requirements of Section I1 through Section I5, the strength of shear connectors and details of construction shall be established by a suitable test program.


6-70

CHAPTER J CONNECTIONS, JOINTS, AND FASTENERS

This chapter applies to connecting elements, connectors, and the affected elements of the connected members subject to static loads. For connections subject to fatigue, see Appendix K3. J1.

GENERAL PROVISIONS

1.

Design Basis Connections consist of affected elements of connected members (e.g. beam webs), connecting elements (e.g., gussets, angles, brackets), and connectors (welds, bolts, rivets). These components shall be proportioned so that their design strength equals or exceeds the required strength determined by structural analysis for factored loads acting on the structure or a specified proportion of the strength of the connected members, whichever is appropriate.

2.

Simple Connections Except as otherwise indicated in the design documents, connections of beams, girders, or trusses shall be designed as flexible, and are permitted to ordinarily be proportioned for the reaction shears only. Flexible beam connections shall accommodate end rotations of unrestrained (simple) beams. To accomplish this, some inelastic but self-limiting deformation in the connection is permitted.

3.

Moment Connections End connections of restrained beams, girders, and trusses shall be designed for the combined effect of forces resulting from moment and shear induced by the rigidity of the connections.

4.

Compression Members with Bearing Joints When columns bear on bearing plates or are finished to bear at splices, there shall be sufficient connectors to hold all parts securely in place. When compression members other than columns are finished to bear, the splice material and its connectors shall be arranged to hold all parts in line and shall be proportioned for 50 percent of the required strength of the member. All compression joints shall be proportioned to resist any tension developed by the factored loads specified by load combination A4-6.

5.

Splices in Heavy Sections This paragraph applies to ASTM A6 Group 4 and 5 rolled shapes, or shapes built-up by welding plates more than two inches thick together to form the cross AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J1]

GENERAL PROVISIONS

6 - 71

section, and where the cross section is to be spliced and subject to primary tensile stresses due to tension or flexure. When the individual elements of the cross section are spliced prior to being joined to form the cross section in accordance with AWS D1.1, Article 3.4.6, the applicable provisions of AWS D1.1 apply in lieu of the requirements of this section. When tensile forces in these sections are to be transmitted through splices by complete-joint-penetration groove welds, material notch-toughness requirements as given in Section A3.1c, weld access hole details as given in Section J1.6, welding preheat requirements as given in Section J2.8, and thermal-cut surface preparation and inspection requirements as given in Section M2.2 apply. At tension splices in ASTM A6 Group 4 and 5 shapes and built-up members of material more than two inches thick, weld tabs and backing shall be removed and the surfaces ground smooth. When splicing ASTM A6 Group 4 and 5 rolled shapes or shapes built-up by welding plates more than two inches thick to form a cross section, and where the section is to be used as a primary compression member, all weld access holes required to facilitate groove welding operations shall satisfy the provisions of Section J1.6. Alternatively, splicing of such members subject to compression, including members which are subject to tension due to wind or seismic loads, shall be accomplished using splice details which do not induce large weld shrinkage strains; for example partial-joint-penetration flange groove welds with filletwelded surface lap plate splices on the web, bolted lap plate splices, or combination bolted/fillet-welded lap plate splices. 6.

Beam Copes and Weld Access Holes All weld access holes required to facilitate welding operations shall have a length from the toe of the weld preparation not less than 11⁄2 times the thickness of the material in which the hole is made. The height of the access hole shall be adequate for deposition of sound weld metal in the adjacent plates and provide clearance for weld tabs for the weld in the material in which the hole is made, but not less than the thickness of the material. In hot-rolled shapes and built-up shapes, all beam copes and weld access holes shall be shaped free of notches and sharp re-entrant corners except, when fillet web-to-flange welds are used in built-up shapes, access holes are permitted to terminate perpendicular to the flange. For ASTM A6 Group 4 and 5 shapes and built-up shapes of material more than two inches thick, the thermally cut surfaces of beam copes and weld access holes shall be ground to bright metal and inspected by either magnetic particle or dye penetrant methods prior to deposition of splice welds. If the curved transition portion of weld access holes and beam copes are formed by predrilled or sawed holes, that portion of the access hole or cope need not be ground. Weld access holes and beam copes in other shapes need not be ground nor inspected by dye penetrant or magnetic particle methods.

7.

Minimum Strength of Connections Except for lacing, sag rods, or girts, connections providing design strength shall be designed to support a factored load not less than 10 kips. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 72

8.

CONNECTIONS, JOINTS, AND FASTENERS

[Chap J

Placement of Welds and Bolts Groups of welds or bolts at the ends of any member which transmit axial force into that member shall be sized so that the center of gravity of the group coincides with the center of gravity of the member, unless provision is made for the eccentricity. The foregoing provision is not applicable to end connections of statically-loaded single angle, double angle, and similar members.

9.

Bolts in Combination with Welds In new work, A307 bolts or high-strength bolts proportioned as bearing-type connections shall not be considered as sharing the load in combination with welds. Welds, if used, shall be proportioned for the entire force in the connection. In slip-critical connections, high-strength bolts are permitted to be considered as sharing the load with the welds. In making welded alterations to structures, existing rivets and high-strength bolts tightened to the requirements for slip-critical connections are permitted to be utilized for carrying loads present at the time of alteration and the welding need only provide the additional design strength required.

10.

High-Strength Bolts in Combination with Rivets In both new work and alterations, in connections designed as slip-critical connections in accordance with the provisions of Section J3, high-strength bolts are permitted to be considered as sharing the load with rivets.

11.

Limitations on Bolted and Welded Connections Fully tensioned high-strength bolts (see Table J3.1) or welds shall be used for the following connections: Column splices in all tier structures 200 ft or more in height. Column splices in tier structures 100 to 200 ft in height, if the least horizontal dimension is less than 40 percent of the height. Column splices in tier structures less than 100 ft in height, if the least horizontal dimension is less than 25 percent of the height. Connections of all beams and girders to columns and of any other beams and girders on which the bracing of columns is dependent, in structures over 125 ft in height. In all structures carrying cranes of over five-ton capacity: roof-truss splices and connections of trusses to columns, column splices, column bracing, knee braces, and crane supports. Connections for supports of running machinery, or of other live loads which produce impact or reversal of stress. Any other connections stipulated on the design plans. In all other cases connections are permitted to be made with A307 bolts or snug-tight high-strength bolts. For the purpose of this section, the height of a tier structure shall be taken as the vertical distance from the curb level to the highest point of the roof beams in the case of flat roofs, or to the mean height of the gable in the case of roofs having a rise of more than 22⁄3 in 12. Where the curb level has not been established, or where the structure does not adjoin a street, the mean level of the adjoining land AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J2]

WELDS

6 - 73

shall be used instead of curb level. It is permissible to exclude penthouses in computing the height of structure. J2.

WELDS All provisions of the American Welding Society Structural Welding Code Steel,, AWS D1.1, apply under this specification, except Chapter 10—Tubular Structures, which is outside the scope of this specification, and except that the provisions of the listed AISC LRFD Specification Sections apply under this Specification in lieu of the cited AWS Code provisions as follows: AISC Section J1.5 and J1.6 in lieu of AWS Section 3.2.5 AISC Section J2.2 in lieu of AWS Section 2.3.2.4 AISC Table J2.5 in lieu of AWS Table 8.1 AISC Table A-K3.2 in lieu of AWS Section 2.5 AISC Section K3 and Appendix K3 in lieu of AWS Chapter 9 AISC Section M2.2 in lieu of AWS Section 3.2.2

1.

Groove Welds

1a.

Effective Area The effective area of groove welds shall be considered as the effective length of the welds times the effective throat thickness. The effective length of a groove weld shall be the width of the part joined. The effective throat thickness of a complete-joint-penetration groove weld shall be the thickness of the thinner part joined. The effective throat thickness of a partial-joint-penetration groove weld shall be as shown in Table J2.1. The effective throat thickness of flare groove weld when flush to the surface of a bar or 90º bend in formed section shall be as shown in Table J2.2. Random sections of production welds for each welding procedure, or such test sections as may be required by design documents, shall be used to verify that the effective throat is consistently obtained. Larger effective throat thicknesses than those in Table J2.2 are permitted, provided the fabricator can establish by qualification the consistent production of such larger effective throat thicknesses. Qualification shall consist of sectioning the weld normal to its axis, at mid-length and terminal ends. Such sectioning shall be made on a number of combinations of material sizes representative of the range to be used in the fabrication or as required by the designer.

1b.

Limitations The minimum effective throat thickness of a partial-joint-penetration groove weld shall be as shown in Table J2.3. Weld size is determined by the thicker of the two parts joined, except that the weld size need not exceed the thickness of the thinnest part joined when a larger size is required by calculated strength. For this exception, particular care shall be taken to provide sufficient preheat for soundness of the weld. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 74


[Chap J

TABLE J2.1 Effective Throat Thickness of Partial-Penetration Groove Welds Welding Process

Included Angle at Root of Groove

Welding Position

Shielded metal arc Submerged arc

Effective Throat Thickness

J or U joint Depth of chamfer Bevel or V joint ≥ 60º

All

Gas metal arc

Bevel or V joint < 60º but ≥ 45º

Flux-cored arc

Depth of chamfer minus 1⁄8-in.

TABLE J2.2 Effective Throat Thickness of Flare Groove Welds Radius (R) of Bar or Bend

Type of Weld

Effective Throat Thickness

Flare bevel groove

All

5⁄

Flare V-groove

All

1⁄ R 2

16R

[a] Use 3⁄8R for Gas Metal Arc Welding (except short circuiting transfer process) when R ≥ 1 in.

TABLE J2.3 Minimum Effective Throat Thickness of Partial-Joint-Penentration Groove Welds Material Thickness of Thicker Part Joined (in.)

Minimum Effective Throat Thickness[a] (in.)

To 1⁄4 inclusive Over 1⁄4 to 1⁄2 Over 1⁄2 to 3⁄4 Over 3⁄4 to 11⁄2 Over 11⁄2 to 21⁄4 Over 21⁄4 to 6 Over 6

1⁄ 8 3⁄ 16 1⁄ 4 5⁄ 16 3⁄ 8 1⁄ 2 5⁄ 8

[a] See Section J2.

2.

Fillet Welds

2a.

Effective Area The effective area of fillet welds shall be as defined in American Welding Society Code D1.1 Article 2.3.2, except 2.3.2.4. The effective throat thickness of a fillet weld shall be the shortest distance from the root of the joint to the face of the diagrammatic weld, except that for the fillet welds made by the submerged AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J2]

WELDS

6 - 75

TABLE J2.4 Minimum Size of Fillet Welds[b] Material Thickness of Thicker Part Joined (in.)

Minimum Size of Fillet Weld[a] (in.)

To 1⁄4 inclusive Over 1⁄4 to 1⁄2 Over 1⁄2 to 3⁄4 Over 3⁄4

1⁄ 8 3⁄ 16 1⁄ 4 5⁄ 16

[a] Leg dimension of fillet welds. Single pass welds must be used. [b] See Section J2.2b for maximum size of fillet welds.

arc process, the effective throat thickness shall be taken equal to the leg size for 3⁄ -in. and smaller fillet welds, and equal to the theoretical throat plus 0.11-in. 8 for fillet welds over 3⁄8-in. For fillet welds in holes and slots, the effective length shall be the length of the centerline of the weld along the center of the plane through the throat. In the case of overlapping fillets, the effective area shall not exceed the nominal cross-sectional area of the hole or slot, in the plane of the faying surface. 2b.

Limitations The minimum size of fillet welds shall be not less than the size required to transmit calculated forces nor the size as shown in Table J2.4 which is based upon experiences and provides some margin for uncalculated stress encountered during fabrication, handling, transportation, and erection. These provisions do not apply to fillet weld reinforcements of partial- or complete-joint-penetration welds. The maximum size of fillet welds of connected parts shall be: (a) Along edges of material less than 1⁄4-in. thick, not greater than the thickness of the material. (b) Along edges of material 1⁄4-in. or more in thickness, not greater than the thickness of the material minus 1⁄16-in., unless the weld is especially designated on the drawings to be built out to obtain full-throat thickness. In the as-welded condition, the distance between the edge of the base metal and the toe of the weld is permitted to be less than 1⁄16-in. provided the weld size is clearly verifiable. (c) For flange-web welds and similar connections, the actual weld size need not be larger than that required to develop the web capacity, and the requirements of Table J2.4 need not apply. The minimum effective length of fillet welds designed on the basis of strength shall be not less than four times the nominal size, or else the size of the weld shall be considered not to exceed 1⁄4 of its effective length. If longitudinal fillet welds are used alone in end connections of flat-bar tension members, the length of each fillet weld shall be not less than the perpendicular distance between AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 76


[Chap J

them. The transverse spacing of longitudinal fillet welds used in end connections of tension members shall comply with Section B3. The maximum effective length of fillet welds loaded by forces parallel to the weld, such as lap splices, shall not exceed 70 times the fillet weld leg. A uniform stress distribution may be assumed throughout the maximum effective length. Intermittent fillet welds may be used to transfer calculated stress across a joint or faying surfaces when the strength required is less than that developed by a continuous fillet weld of the smallest permitted size, and to join components of built-up members. The effective length of any segment of intermittent fillet welding shall be not less than four times the weld size, with a minimum of 11⁄2-in. In lap joints, the minimum amount of lap shall be five times the thickness of the thinner part joined, but not less than one inch. Lap joints joining plates or bars subjected to axial stress shall be fillet welded along the end of both lapped parts, except where the deflection of the lapped parts is sufficiently restrained to prevent opening of the joint under maximum loading. Fillet welds terminations shall not be at the extreme ends or sides of parts or members. They shall be either returned continuously around the ends or sides, respectively for a distance of not less than two times the nominal weld size or shall terminate not less than the nominal weld size from the sides or ends except as follows. For details and structural elements such as brackets, beam seats, framing angles, and simple end plates which are subject to cyclic (fatigue) out-of-plane forces and/or moments of frequency and magnitude that would tend to initiate a progressive failure of the weld, fillet welds shall be returned around the side or end for a distance not less than two times the nominal weld size. For framing angles and simple end-plate connections which depend upon flexibility of the outstanding legs for connection flexibility, if end returns are used, their length shall not exceed four times the nominal size of the weld. Fillet welds which occur on opposite sides of a common plane shall be interrupted at the corner common to both welds. End returns shall be indicated on the design and detail drawings. Fillet welds in holes or slots may be used to transmit shear in lap joints or to prevent the buckling or separation of lapped parts and to join components of built-up members. Such fillet welds may overlap, subject to the provisions of Section J2. Fillet welds in holes or slots are not to be considered plug or slot welds. 3.

Plug and Slot Welds

3a.

Effective Area The effective shearing area of plug and slot welds shall be considered as the nominal cross-sectional area of the hole or slot in the plane of the faying surface.

3b.

Limitations Plug or slot welds are permitted to be used to transmit shear in lap joints or to prevent buckling of lapped parts and to join component parts of built-up members. The diameter of the holes for a plug weld shall not be less than the thickness of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J2]

WELDS

6 - 77

the part containing it plus 5⁄16-in., rounded to the next larger odd 1⁄16-in., nor greater than the minimum diameter plus 1⁄8-in. or 21⁄4 times the thickness of the weld. The minimum center-to-center spacing of plug welds shall be four times the diameter of the hole. The length of slot for a slot weld shall not exceed 10 times the thickness of the weld. The width of the slot shall be not less than the thickness of the part containing it plus 5⁄16-in. rounded to the next larger odd 1⁄16-in., nor shall it be larger than 21⁄4 times the thickness of the weld. The ends of the slot shall be semicircular or shall have the corners rounded to a radius of not less than the thickness of the part containing it, except those ends which extend to the edge of the part. The minimum spacing of lines of slot welds in a direction transverse to their length shall be four times the width of the slot. The minimum center-to-center spacing in a longitudinal direction on any line shall be two times the length of the slot. The thickness of plug or slot welds in material 5⁄8-in. or less in thickness shall be equal to the thickness of the material. In material over 5⁄8-in. thick, the thickness of the weld shall be at least one-half the thickness of the material but not less than 5⁄8-in. 4.

Design Strength The design strength of welds shall be the lower value of φFBM ABM and φFw Aw, when applicable. The values of φ, FBM , and Fw and limitations thereon are given in Table J2.5, where FBM Fw ABM Aw φ

= nominal strength of the base material, ksi = nominal strength of the weld electrode, ksi = cross-sectional area of the base material, in.2 = effective cross-sectional area of the weld, in.2 = resistance factor

Alternatively, fillet welds loaded in-plane are permitted to be designed in accordance with Appendix J2.4. 5.

Combination of Welds If two or more of the general types of welds (groove, fillet, plug, slot) are combined in a single joint, the design strength of each shall be separately computed with reference to the axis of the group in order to determine the design strength of the combination.

6.

Matching Weld Metal The choice of electrode for use with complete-joint-penetration groove welds subject to tension normal to the effective area shall comply with the requirements for matching weld metals given in AWS D1.1. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 78


[Chap J

TABLE J2.5 Design Strength of Welds Types of Weld and Stress [a]

Material

Resistance Factor φ

Nominal Strength FBM or Fw

Required Weld Strength Level [b,c]

Complete-Joint-Penetration Groove Weld Tension normal to effective area

Base

0.90

Fy

Compression normal to effective area Tension or compression parallel to axis of weld Shear on effective area

Base

0.90

Fy

Base Weld electrode

0.90 0.80

0.60Fy 0.60FEXX

Matching weld must be used. Weld metal with a strength level equal to or less than matching weld metal is permitted to be used.

Partial-Joint-Penetration Groove Weld Compression normal to effective area Tension or compression parallel to axis of weld [d]

Base

0.90

Fy

Shear parallel to axis of weld

Base Weld electrode

0.75

[e] 0.60FEXX

Tension normal to effective area

Base Weld electrode

0.90 0.80

Fy 0.60FEXX

Weld metal with a strength level equal to or less than matching weld metal is permitted to be used.

Fillet Welds Shear on effective area Tension or compression parallel to axis of weld [d]

Base Weld electrode

0.75

[f] 0.60FEXX

Base

0.90

Fy


Plug or Slot Welds Shear parallel to faying surfaces (on effective area)

Base Weld electrode

0.75

[e] 0.60FEXX


[a] For definition of effective area, see Section J2. [b] For matching weld metal, see Table 4.1, AWS D1.1. [c] Weld metal one strength level stronger than matching weld metal is permitted. [d] Fillet welds and partial-joint-penetration groove welds joining component elements of built-up members, such as flange-to-web connections, are not required to be designed with to the tensile or compressive stress in these elements parallel to the axis of the welds. [e] The design of connected material is governed by Sections J4 and J5. [f] For alternative design strength, see Appendix J2.4.


Sect. J3]

7.

BOLTS AND THREADED PARTS

6 - 79

Mixed Weld Metal When notch-toughness is specified, the process consumables for all weld metal, tack welds, root pass, and subsequent passes deposited in a joint shall be compatible to assure notch-tough composite weld metal.

8.

Preheat for Heavy Shapes For ASTM A6 Group 4 and 5 shapes and welded built-up members made of plates more than two inches thick, a preheat equal to or greater than 350ºF shall be used when making groove-weld splices.

J3.


1.

High-Strength Bolts Use of high-strength bolts shall conform to the provisions of the Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts, as approved by the Research Council on Structural Connections, except as otherwise provided in this Specification. If required to be tightened to more than 50 percent of their minimum specified tensile strength, A449 bolts in tension and bearing-type shear connections shall have an ASTM F436 hardened washer installed under the bolt head, and the nuts shall meet the requirements of ASTM A563. When assembled, all joint surfaces, including those adjacent to the washers, shall be free of scale, except tight mill scale. Except as noted below, all A325 and A490 bolts shall be tightened to a bolt tension not less than that given in Table J3.1. Tightening shall be done by any of the following methods: turn-of-nut method, a direct tension indicator, calibrated wrench, or alternative design bolt. Bolts in connections not subject to tension loads, where slip can be permitted and where loosening or fatigue due to vibration or load fluctuations are not design considerations, need only to be tightened to the snug-tight condition. The snug-tight condition is defined as the tightness attained by either a few impacts of an impact wrench or the full effort of a worker with an ordinary spud wrench that brings the connected plies into firm contact. The nominal strength value given in Table J3.2 for bearing-type connections shall be used for bolts tightened to the snug-tight condition. Bolts tightened only to the snug-tight condition shall be clearly identified on the design and erection drawings. When A490 bolts over one inch in diameter are used in slotted or oversize holes in external plies, a single hardened washer conforming to ASTM F436, except with 5⁄16-in. minimum thickness, shall be used in lieu of the standard washer. In slip-critical connections in which the direction of loading is toward an edge of a connected part, adequate bearing strength at factored load shall be provided based upon the applicable requirements of Section J3.10.

2.

Size and Use of Holes In slip-critical connections in which the direction of loading is toward edge of connected part, adequate bearing capacity at factored load shall be provided based upon the applicable requirements of Section J3.10. The maximum sizes of holes for rivets and bolts are given in Table J3.3, except AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 80


[Chap J

TABLE J3.1 Minimum Bolt Tension, kips* Bolt Size, in.

A325 Bolts

A490 Bolts

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

12 19 28 39 51 56 71 85 103

15 24 35 49 64 80 102 121 148

1 11⁄8 11⁄4 13⁄8 11⁄2

* Equal to 0.70 of minimum tensile strength of bolts, rounded off to nearest kip, as specified in ASTM specifications for A325 and A490 bolts with UNC threads.

that larger holes, required for tolerance on location of anchor bolts in concrete foundations, are allowed in column base details. Standard holes shall be provided in member-to-member connections, unless oversized, short-slotted, or long-slotted holes in bolted connections are approved by the designer. Finger shims up to 1⁄4-in. are permitted in slip-critical connections designed on the basis of standard holes without reducing the nominal shear strength of the fastener to that specified for slotted holes. Oversized holes are allowed in any or all plies of slip-critical connections, but they shall not be used in bearing-type connections. Hardened washers shall be installed over oversized holes in an outer ply. Short-slotted holes are allowed in any or all plies of slip-critical or bearing-type connections. The slots are permitted to be used without regard to direction of loading in slip-critical connections, but the length shall be normal to the direction of the load in bearing-type connections. Washers shall be installed over short-slotted holes in an outer ply; when high-strength bolts are used, such washers shall be hardened. Long-slotted holes are allowed in only one of the connected parts of either a slip-critical or bearing-type connection at an individual faying surface. Longslotted holes are permitted to be used without regard to direction of loading in slip-critical connections, but shall be normal to the direction of load in bearingtype connections. Where long-slotted holes are used in an outer ply, plate washers, or a continuous bar with standard holes, having a size sufficient to completely cover the slot after installation, shall be provided. In high-strength bolted connections, such plate washers or continuous bars shall be not less than 5⁄ -in. thick and shall be of structural grade material, but need not be hardened. 16 If hardened washers are required for use of high-strength bolts, the hardened washers shall be placed over the outer surface of the plate washer or bar. 3.

Minimum Spacing The distance between centers of standard, oversized, or slotted holes, shall not AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J3]


6 - 81

TABLE J3.2 Design Strength of Fasteners Tensile Strength Description of Fasteners

Resistance Factor φ

Nominal Strength, ksi

Shear Strength in Bearing-type Connections Resistance Factor φ

Nominal Strength, ksi

A307 bolts

45 [a]

24 [b,e]

A325 bolts, when threads are not excluded from shear planes

90 [d]

48 [e]

A325 bolts, when threads are excluded from shear planes

90 [d]

60 [e]

A490 bolts, when threads are not excluded from shear planes

113 [d]

60 [e]

A490 bolts, when threads are excluded from shear planes

0.75

113 [d]

0.75

75 [e]

Threaded parts meeting the requirements of Sect. A3, when threads are not excluded from shear planes

0.75Fu [a,c]

0.40Fu

Threaded parts meeting the requirements of Sect. A3, when threads are excluded from shear planes

0.75Fu [a,c]

0.50Fu [a,c]

A502, Gr. 1, hot-driven rivets

45 [a]

25 [e]

A502, Gr. 2 & 3, hot-driven rivets

60 [a]

33 [e]

[a] Static loading only. [b] Threads permitted in shear planes. [c] The nominal tensile strength of the threaded portion of an upset rod, based upon the cross-sectional area at its major thread diameter, AD shall be larger than the nominal body area of the rod before upsetting times Fy. [d] For A325 and A490 bolts subject to tensile fatigue loading, see Appendix K3. [e] When bearing-type connections used to splice tension members have a fastener pattern whose length, measured parallel to the line of force, exceeds 50 in., tabulated values shall be reduced by 20 percent.

be less than 22⁄3 times the nominal diameter of the fastener; a distance of 3d is preferred. Refer to Section J3.10 for bearing strength requirement. 4.

Minimum Edge Distance The distance from the center of a standard hole to an edge of a connected part shall not be less than either the applicable value from Table J3.4, or as required AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 82


[Chap J

TABLE J3.3 Nominal Hole Dimensions Hole Dimensions Bolt Diameter

Standard (Dia.)

Oversize (Dia.)

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

9⁄ 16 11⁄ 16 13⁄ 16 15⁄ 16 11⁄16 d + 1⁄16

5⁄ 8 13⁄ 16 15⁄ 16 1 1 ⁄16 11⁄4 d + 5⁄16

1 ≥11⁄8

Short-slot (Width × Length)

(d

Long-slot Width × Length)

9⁄ × 11⁄ 16 16 11⁄ × 7⁄ 16 8 13⁄ × 1 16 15⁄ × 11⁄ 16 8 11⁄16 × 15⁄16 + 1⁄16) × (d + 3⁄8)

(d

9⁄ × 11⁄ 16 4 11⁄ × 19⁄ 16 16 13⁄ × 17⁄ 16 8 15⁄ × 23⁄ 16 16 11⁄16 × 21⁄2 + 1⁄16) × (2.5 ×

d)

TABLE J3.4 Minimum Edge Distance,[a] in. (Center of Standard Hole[b] to Edge of Connected Part) Nominal Rivet or Bolt Diameter (in.) 1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

1 11⁄8 11⁄4 Over 11⁄4

At Sheared Edges

At Rolled Edges of Plates, Shapes or Bars, or Gas Cut Edges [c]

7⁄ 8

11⁄8 11⁄4 11⁄2 [d] 13⁄4 [d] 2 21⁄4 13⁄4 × Diameter

3⁄ 4 7⁄ 8

1 11⁄8 11⁄4 11⁄2 15⁄8 11⁄4 × Diameter

[a] Lesser edge distances are permitted to be used provided Equations from J3.10, as appropriate, are satisfied. [b] For oversized or slotted holes, see Table J3.8. [c] All edge distances in this column are permitted to be reduced 1⁄8-in. when the hole is at a point where stress does not exceed 25 percent of the maximum design strength in the element. [d] These are permitted to be 11⁄4-in. at the ends of beam connection angles and shear end plates.

in Section J3.10. The distance from the center of an oversized or slotted hole to an edge of a connected part shall be not less than that required for a standard hole to an edge of a connected part plus the applicable increment C2 from Table J3.8. Refer to Section J3.10 for bearing strength requirement. 5.

Maximum Spacing and Edge Distance The maximum distance from the center of any bolt or rivet to the nearest edge of parts in contact shall be 12 times the thickness of the connected part under consideration, but shall not exceed six inches. The longitudinal spacing of connectors between elements in continuous contact consisting of a plate and a shape or two plates shall be as follows: (a) For painted members or unpainted members not subject to corrosion, the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J3]


6 - 83

spacing shall not exceed 24 times the thickness of the thinner plate or 12 inches. (b) For unpainted members of weathering steel subject to atmospheric corrosion, the spacing shall not exceed 14 times the thickness of the thinner plate or seven inches. 6.

Design Tension or Shear Strength The design tension or shear strength of a high-strength bolt or threaded part is φFn Ab where φ = resistance factor tabulated in Table J3.2 Fn = nominal tensile strength Ft , or shear strength, Fv, tabulated in Table J3.2, ksi Ab = nominal unthreaded body area of bolt or threaded part (for upset rods, see Footnote c, Table J3.2), in.2 The applied load shall be the sum of the factored loads and any tension resulting from prying action produced by deformation of the connected parts.

7.

Combined Tension and Shear in Bearing-Type Connections The design strength of a bolt or rivet subject to combined tension and shear is φFt Ab, where φ is 0.75 and the nominal tension stress Ft shall be computed from the equations in Table J3.5 as a function of fv, the required shear stress produced by the factored loads. The design shear strength φFv, tabulated in Table J3.2, shall equal or exceed the shear stress, fv.

8.

High-Strength Bolts in Slip-Critical Connections The design for shear of high-strength bolts in slip-critical connections shall be in accordance with either Section J3.8a or J3.8b and checked for bearing in accordance with J3.2 and J3.10.

8a.

Slip-Critical Connections Designed at Service Loads The design resistance to shear of a bolt in a slip-critical connection is φFv Ab, where φ = 1.0 for standard, oversized, short-slotted, and long-slotted holes when the long slot is perpendicular to the line of force φ = 0.85 for long-slotted holes when the long slot is parallel to the line of force Fv = nominal slip-critical shear resistance tabulated in Table J3.6, ksi The design resistance to shear shall equal or exceed the shear on the bolt due to service loads. When the loading combination includes wind loads in addition to dead and live loads, the total shear on the bolt due to combined load effects, at service load, may be multiplied by 0.75. The values for Fv in Table J3.6 are based on Class A (slip coefficient 0.33), clean mill scale and blast cleaned surfaces with class A coatings. When specified by AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 84


[Chap J

TABLE J3.5 Tension Stress Limit (Ft), ksi Fasteners in Bearing-type Connections Description of Fasteners

Threads Included in the Shear Plane

Threads Excluded from the Shear Plane

59 − 1.9fv ≤ 45

A307 bolts A325 bolts

117 − 1.9fv ≤ 90

117 − 1.5fv ≤ 90

A490 bolts

147 − 1.9fv ≤ 113

147 − 1.5fv ≤ 113

0.98Fu − 1.9fv ≤ 0.75Fu

0.98Fu − 1.5fv ≤ 0.75Fu

Threaded parts A449 bolts over 11⁄2 diameter A502 Gr.1 rivets

59 − 1.8fv ≤ 45

A502 Gr.2 rivets

78 − 1.8fv ≤ 60

TABLE J3.6 Slip-Critical Nominal Resistance to Shear, ksi, of High-Strength Bolts[a] Nominal Resistance to Shear Type of Bolt

Standard Size Holes

Oversized and Short-slotted Holes

Long-slotted Holes

A325 A490

17 21

15 18

12 15

[a] For each shear plane.

the designer, the nominal slip resistance for connections having special faying surface conditions are permitted to be adjusted to the applicable values in the RCSC Load and Resistance Factor Design Specification. Finger shims up to 1⁄4-in. are permitted to be introduced into slip-critical connections designed on the basis of standard holes without reducing the design shear stress of the fastener to that specified for slotted holes. 8b.

Slip-Critical Connections Designed at Factored Loads See Appendix J3.8b.

9.

Combined Tension and Shear in Slip-Critical Connections The design of slip-critical connections subject to tensile forces shall be in accordance with either Sections J3.9a and J3.8a or Sections J3.9b and J3.8b.

9a.

Slip-Critical Connections Designed at Service Loads The design resistance to shear of a bolt in a slip-critical connection subject to a AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. J3]


6 - 85

tensile force T due to service loads shall be computed according to Section J3.8a multiplied by the following reduction factor,  1 − 

T  Tb

where

Tb = minimum bolt pre-tension from Table J3.1 9b.

Slip-Critical Connections Designed at Factored Loads See Appendix J3.9b.

10.

Bearing Strength at Bolt Holes The design bearing strength at bolt holes is φRn, where

φ = 0.75 Rn = nominal bearing strength Bearing strength shall be checked for both bearing-type and slip-critical connections. The use of oversize holes and short- and long-slotted holes parallel to the line of force is restricted to slip-critical connections per Section J3.2. In the following sections:

Le = distance (in.) along the line of force from the edge of the connected part to the center of a standard hole or the center of a short- and longslotted hole perpendicular to the line of force. For oversize holes and short- and long-slotted holes parallel to the line of force, Le shall be increased by the increment C2 of Table J3.8. s = distance (in.) along the line of force between centers of standard holes, or between centers of short- and long-slotted holes perpendicular to the line of force. For oversize holes and short- and long-slotted holes parallel to the line of force, s shall be increased by the spacing increment C1 of Table J3.7. d = diameter of bolt, in. Fu = specified minimum tensile strength of the critical part, ksi t = thickness of the critical connected part, in. For countersunk bolts and rivets, deduct one-half the depth of the countersink. (a) When Le ≥ 1.5d and s ≥ 3d and there are two or more bolts in line of force: For standard holes; short and long-slotted holes perpendicular to the line of force; oversize holes in slip-critical connections; and long and short-slotted holes in slip-critical connections when the line of force is parallel to the axis of the hole: When deformation around the bolt holes is a design consideration

Rn = 2.4dtFu

(J3-1a)

When deformation around the bolt holes is not a design consideration, for the bolt nearest the edge

Rn = LetFu ≤ 3.0dtFu AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(J3-1b)

6 - 86


[Chap J

TABLE J3.7 Values of Spacing Increment C1, in. Slotted Holes Nominal Diameter of Fastener

≤7⁄8 1 ≥11⁄8

Oversize Holes

Perpendicular to Line of Force

Short-slots

Long-slots [a]

1⁄ 8 3⁄ 16 1⁄ 4

0 0 0

3⁄ 16 1⁄ 4 5⁄ 16

11⁄2d − 1⁄16 17⁄16 11⁄2d − 1⁄16

Parallel to Line of Force

[a] When length of slot is less than maximum allowed in Table J3.5, C1 are permitted to be reduced by the difference between the maximum and actual slot lengths.

TABLE J3.8 Values of Edge Distance Increment C2, in. Slotted Holes Nominal Diameter of Fastener (in.)

≤7⁄8

Long Axis Perpendicular to Edge Oversized Holes 1⁄

Short Slots

16

1⁄ 8

1

1⁄ 8

1⁄ 8

≥11⁄8

1⁄ 8

3⁄ 16

Long Slots [a]

Long Axis Parallel to Edge

3⁄ d 4

0

[a] When length of slot is less than maximum allowable (see Table J3.5), C2 are permitted to be reduced by one-half the difference between the maximum and actual slot lengths.

and for the remaining bolts

Rn = (s − d / 2)tFu ≤ 3.0dtFu

(J3-1c)

For long-slotted bolt holes perpendicular to the line of force:

Rn = 2.0dtFu

(J3-1d)

(b) When Le < 1.5d or s < 3d or for a single bolt in the line of force: For standard holes; short and long-slotted holes perpendicular to the line of force; oversize holes in slip-critical connections; and long and short-slotted holes in slip-critical connections when the line of force is parallel to the axis of the hole: For a single bolt hole or the bolt hole nearest the edge when there are two or more bolt holes in the line of force

Rn = LetFu ≤ 2.4dtFu For the remaining bolt holes AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(J3-2a)

Sect. J4]

DESIGN RUPTURE STRENGTH

6 - 87

Rn = (s − d / 2)tFu ≤ 2.4dtFu

(J3-2b)

For long-slotted bolt holes perpendicular to the line of force: For a single bolt hole or the bolt hole nearest the edge where there are two or more bolt holes in the line of force

Rn = LetFu ≤ 2.0dtFu

(J3-2c)

For the remaining bolt holes

Rn = (s − d / 2)tFu ≤ 2.0dtFu 11.

(J3-2d)

Long Grips A307 bolts providing design strength, and for which the grip exceeds five diameters, shall have their number increased one percent for each additional 1⁄ -in. in the grip. 16

J4.

DESIGN RUPTURE STRENGTH

1.

Shear Rupture Strength The design strength for the limit state of rupture along a shear failure path in the affected elements of connected members shall be taken as φRn where

φ = 0.75 Rn = 0.6Fu Anv Anv = net area subject to shear, in.2 2.

(J4-1)

Tension Rupture Strength The design strength for the limit state of rupture along a tension path in the affected elements of connected members shall be taken as φRn where

φ = 0.75 Rn = Fu Ant Ant = net area subject to tension, in.2 3.

(J4-2)

Block Shear Rupture Strength Block shear is a limit state in which the resistance is determined by the sum of the shear strength on a failure path(s) and the tensile strength on a perpendicular segment. It shall be checked at beam end connections where the top flange is coped and in similar situations, such as tension members and gusset plates. When ultimate rupture strength on the net section is used to determine the resistance on one segment, yielding on the gross section shall be used on the perpendicular segment. The block shear rupture design strength, φRn, shall be determined as follows: (a) When Fu Ant ≥ 0.6Fu Anv :

φRn = φ[0.6Fy Agv + Fu Ant] AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(J4-3a)

6 - 88


[Chap J

(b) When 0.6Fu Anv > Fu Ant :

φRn = φ[0.6Fu Anv + Fy Agt]

(J4-3b)

where

φ = 0.75 Agv = gross area subject to shear, in.2 Agt = gross area subject to tension, in.2 Anv = net area subjected to shear, in.2 Ant = net area subjected to tension, in.2 J5.

CONNECTING ELEMENTS This section applies to the design of connecting elements, such as plates, gussets, angles, brackets, and the panel zones of beam-to-column connections.

1.

Eccentric Connections Intersecting axially stressed members shall have their gravity axis intersect at one point, if practicable; if not, provision shall be made for bending and shearing stresses due to the eccentricity. Also see Section J1.8.

2.

Design Strength of Connecting Elements in Tension The design strength, φRn, of welded, bolted, and riveted connecting elements statically loaded in tension (e.g., splice and gusset plates) shall be the lower value obtained according to limit states of yielding, rupture of the connecting element, and block shear rupture. (a) For tension yielding of the connecting element:

φ = 0.90 Rn = AgFy

(J5-1)

(b) For tension rupture of the connecting element:

φ = 0.75 Rn = AnFu

(J5-2)

where An is the net area, not to exceed 0.85Ag. (c) For block shear rupture of connecting elements, see Section J4.3. 3.

Other Connecting Elements For all other connecting elements, the design strength, φRn, shall be determined for the applicable limit state to ensure that the design strength is equal to or greater than the required strength, where Rn is the nominal strength appropriate to the geometry and type of loading on the connecting element. For shear yielding of the connecting element: φ = 0.90 Rn = 0.60AgFy AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(J5-3)

Sect. J8]

BEARING STRENGTH

6 - 89

If the connecting element is in compression an appropriate limit state analysis shall be made. J6.

FILLERS In welded construction, any filler 1⁄4-in. or more in thickness shall extend beyond the edges of the splice plate and shall be welded to the part on which it is fitted with sufficient weld to transmit the splice plate load, applied at the surface of the filler. The welds joining the splice plate to the filler shall be sufficient to transmit the splice plate load and shall be long enough to avoid overloading the filler along the toe of the weld. Any filler less than 1⁄4-in. thick shall have its edges made flush with the edges of the splice plate and the weld size shall be the sum of the size necessary to carry the splice plus the thickness of the filler plate. When bolts or rivets carrying loads pass through fillers thicker than 1⁄4-in., except in connections designed as slip-critical connections, the fillers shall be extended beyond the splice material and the filler extension shall be secured by enough bolts or rivets to distribute the total stress in the member uniformly over the combined section of the member and the filler, or an equivalent number of fasteners shall be included in the connection. Fillers between 1⁄4-in. and 3⁄4-in. thick, inclusive, need not be extended and developed, provided the design shear strength of the bolts is reduced by the factor, 0.4(t − 0.25), where t is the total thickness of the fillers, up to 3⁄4-in.

J7.

SPLICES Groove-welded splices in plate girders and beams shall develop the full strength of the smaller spliced section. Other types of splices in cross sections of plate girders and beams shall develop the strength required by the forces at the point of splice.

J8.

BEARING STRENGTH The strength of surfaces in bearing is φRn, where φ = 0.75 Rn is defined below for the various types of bearing (a) For milled surfaces, pins in reamed, drilled, or bored holes, and ends of fitted bearing stiffeners, Rn = 1.8Fy Apb

(J8-1)

where Fy = specified minimum yield stress, ksi Apb = projected bearing area, in.2 (b) For expansion rollers and rockers, If d ≤ 25 in., Rn = 1.2(Fy − 13)ld / 20 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(J8-2)

6 - 90


[Chap J

Rn = 6.0(Fy − 13)l√ d / 20

(J8-3)

If d > 25 in.,

where d = diameter, in. l = length of bearing, in. J9.

COLUMN BASES AND BEARING ON CONCRETE Proper provision shall be made to transfer the column loads and moments to the footings and foundations. In the absence of code regulations, design bearing loads on concrete may be taken as φcPp: (a) On the full area of a concrete support Pp = 0.85fc′A1

(J9-1)

(b) On less than the full area of a concrete support Pp = 0.85fc′A1√  A2 / A 1

(J9-2)

where φc = 0.60 A1 = area of steel concentrically bearing on a concrete support, in. A2 = maximum area of the portion of the supporting surface that is geometrically similar to and concentric with the loaded area, in.2  √ A2 / A 1 ≤ 2 J10. ANCHOR BOLTS AND EMBEDMENTS Anchor bolts and embedments shall be designed in accordance with American Concrete Institute or Prestressed Concrete Institute criteria. If the load factors and combinations given in Section A4.1 are used, a reduction in the φ factors specified by ACI shall be made based on the ratio of load factors given in Section A4.1 and in ACI.


6 - 91

CHAPTER K CONCENTRATED FORCES, PONDING, AND FATIGUE

This chapter covers member strength design considerations pertaining to concentrated forces, ponding, and fatigue. K1. FLANGES AND WEBS WITH CONCENTRATED FORCES 1.

Design Basis Sections K1.2 through K1.7 apply to single and double concentrated forces as indicated in each Section. A single concentrated force is tensile or compressive. Double concentrated forces, one tensile and one compressive, form a couple on the same side of the loaded member. Transverse stiffeners are required at locations of concentrated tensile forces in accordance with Section K1.2 for the flange limit state of local bending, and at unframed ends of beams and girders in accordance with Section K1.8. Transverse stiffeners or doubler plates are required at locations of concentrated forces in accordance with Sections K1.3 through K1.6 for the web limit states of yielding, crippling, sidesway buckling, and compression buckling. Doubler plates or diagonal stiffeners are required in accordance with Section K1.7 for the web limit state of panel-zone shear. Transverse stiffeners and diagonal stiffeners required by Sections K1.2 through K1.8 shall also meet the requirements of Section K1.9. Doubler plates required by Sections K1.3 through K1.6 shall also meet the requirements of Section K1.10.

2.

Local Flange Bending This Section applies to both tensile single-concentrated forces and the tensile component of double-concentrated forces. A pair of transverse stiffeners extending at least one-half the depth of the web shall be provided adjacent to a concentrated tensile force centrally applied across the flange when the required strength of the flange exceeds φRn, where φ = 0.90 Rn = 6.25tf2Fyf where Fyf = specified minimum yield stress of the flange, ksi tf = thickness of the loaded flange, in. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(K1-1)

6 - 92

CONCENTRATED FORCES, PONDING, AND FATIGUE

[Chap. K

If the length of loading measured across the member flange is less than 0.15b, where b is the member flange width, Equation K1-1 need not be checked. When the concentrated force to be resisted is applied at a distance from the member end that is less than 10tf, Rn shall be reduced by 50 percent. When transverse stiffeners are required, they shall be welded to the loaded flange to develop the welded portion of the stiffener. The weld connecting transverse stiffeners to the web shall be sized to transmit the unbalanced force in the stiffener to the web. Also, see Section K1.9. 3.

Local Web Yielding This Section applies to single-concentrated forces and both components of double-concentrated forces. Either a pair of transverse stiffeners or a doubler plate, extending at least one-half the depth of the web, shall be provided adjacent to a concentrated tensile or compressive force when the required strength of the web at the toe of the fillet exceeds φRn, where φ = 1.0 and Rn is determined as follows: (a) When the concentrated force to be resisted is applied at a distance from the member end that is greater than the depth of the member d, Rn = (5k + N)Fyw tw

(K1-2)

(b) When the concentrated force to be resisted is applied at a distance from the member end that is less than or equal to the depth of the member d, Rn = (2.5k + N)Fywtw

(K1-3)

In Equations K1-2 and K1-3, the following definitions apply: Fyw N k tw

= specified minimum yield stress of the web, ksi = length of bearing (not less than k for end beam reactions), in. = distance from outer face of the flange to the web toe of the fillet, in. = web thickness, in.

When required for a tensile force normal to the flange, transverse stiffeners shall be welded to the loaded flange to develop the connected portion of the stiffener. When required for a compressive force normal to the flange, transverse stiffeners shall either bear on or be welded to the loaded flange to develop the force transmitted to the stiffener. The weld connecting transverse stiffeners to the web shall be sized to transmit the unbalanced force in the stiffener to the web. Also, see Section K1.9. Alternatively, when doubler plates are required, see Section K1.10. 4.

Web Crippling This Section applies to both compressive single-concentrated forces and the compressive component of double-concentrated forces. Either a transverse stiffener, a pair of transverse stiffeners, or a doubler plate, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. K1]

FLANGES AND WEBS WITH CONCENTRATED FORCES

6 - 93

extending at least one-half the depth of the web, shall be provided adjacent to a concentrated compressive force when the required strength of the web exceeds φRn, where φ = 0.75 and Rn is determined as follows: (a) When the concentrated compressive force to be resisted is applied at a distance from the member end that is greater than or equal to d / 2, 1.5

 √

  N   tw   Rn = 135t2w 1 + 3       d   tf   

Fywtf tw

(K1-4)

(b) When the concentrated compressive force to be resisted is applied at a distance from the member end that is less than d / 2, For N / d ≤ 0.2,   N   tw  Rn = 68t 1 + 3     d   tf  

1.5

2 w

  

t √

Fywtf

(K1-5a)

w

For N / d > 0.2, 1.5

  4N   tw   Rn = 68t2w 1 +  − 0.2      tf    d

 √

Fywtf tw

(K1-5b)

In Equations K1-4 and K1-5, the following definitions apply: d = overall depth of the member, in. tf = flange thickness, in. When transverse stiffeners are required, they shall either bear on or be welded to the loaded flange to develop the force transmitted to the stiffener. The weld connecting transverse stiffeners to the web shall be sized to transmit the unbalanced force in the stiffener to the web. Also, see Section K1.9. Alternatively, when doubler plates are required, see Section K1.10. 5.

Sidesway Web Buckling This Section applies only to compressive single-concentrated forces applied to members where relative lateral movement between the loaded compression flange and the tension flange is not restrained at the point of application of the concentrated force. The design strength of the web is φRn, where φ = 0.85 and Rn is determined as follows: (a) If the compression flange is restrained against rotation: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 94


[Chap. K

for (h / tw) / (l / bf) ≤ 2.3, 3

 h / tw  Cr t3w tf  Rn = 1 + 0.4    h2   l / bf  

(K1-6)

for (h / tw) / (l / bf) > 2.3, the limit state of sidesway web buckling does not apply. When the required strength of the web exceeds φRn, local lateral bracing shall be provided at the tension flange or either a pair of transverse stiffeners or a doubler plate, extending at least one-half the depth of the web, shall be provided adjacent to the concentrated compressive force. When transverse stiffeners are required, they shall either bear on or be welded to the loaded flange to develop the full applied force. The weld connecting transverse stiffeners to the web shall be sized to transmit the force in the stiffener to the web. Also, see Section K1.9. Alternatively, when doubler plates are required, they shall be sized to develop the full applied force. Also, see Section K1.10. (b) If the compression flange is not restrained against rotation: for (h / tw) / (l / bf) ≤ 1.7, 3

Cr t3w tf Rn = h2

  h / tw   0.4      l / bf  

(K1-7)

for (h / tw) / (l / bf) > 1.7, the limit state of sidesway web buckling does not apply. When the required strength of the web exceeds φRn, local lateral bracing shall be provided at both flanges at the point of application of the concentrated forces. In Equations K1-6 and K1-7, the following definitions apply: l bf tw h

= largest laterally unbraced length along either flange at the point of load, in. = flange width, in. = web thickness, in. = clear distance between flanges less the fillet or corner radius for rolled shapes; distance between adjacent lines of fasteners or the clear distance between flanges when welds are used for built-up shapes, in. Cr = 960,000 when Mu < My at the location of the force, ksi = 480,000 when Mu ≥ My at the location of the force, ksi

6.

Compression Buckling of the Web This Section applies to a pair of compressive single-concentrated forces or the compressive components in a pair of double-concentrated forces, applied at both flanges of a member at the same location. Either a single transverse stiffener, or pair of transverse stiffeners, or a doubler plate, extending the full depth of the web, shall be provided adjacent to AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. K1]


6 - 95

concentrated compressive forces at both flanges when the required strength of the web exceeds φRn, where φ = 0.90 and Rn =

4,100t3w√ Fyw

(K1-8)

h

When the pair of concentrated compressive forces to be resisted is applied at a distance from the member end that is less than d / 2, Rn shall be reduced by 50 percent. When transverse stiffeners are required, they shall either bear on or be welded to the loaded flange to develop the force transmitted to the stiffener. The weld connecting transverse stiffeners to the web shall be sized to transmit the unbalanced force in the stiffener to the web. Also, see Section K1.9. Alternatively, when doubler plates are required, see Section K1.10. 7.

Panel-Zone Web Shear Either doubler plates or diagonal stiffeners shall be provided within the boundaries of the rigid connection of members whose webs lie in a common plane when the required strength exceeds φRv, where φ = 0.90 and Rv is determined as follows: (a) When the effect of panel-zone deformation on frame stability is not considered in the analysis, For Pu ≤ 0.4Py Rv = 0.60Fy dc tw

(K1-9)

For Pu > 0.4Py  Rv = 0.60Fy dc tw 1.4 − 

Pu  Py

(K1-10)

(b) When frame stability, including plastic panel-zone deformation, is considered in the analysis: For Pu ≤ 0.75Py  3bcf tcf2  Rv = 0.60Fy dc tw 1 +  db d c t  

(K1-11)

 3bcf tcf2   1.2Pu Rv = 0.60Fy dc tw 1 +  1.9 −  db dc tw   Py  

(K1-12)

For Pu > 0.75Py


6 - 96


[Chap. K

In Equations K1-9 through K1-12, the following definitions apply: tw bcf tcf db dc Fy Py A

= column web thickness, in. = width of column flange, in. = thickness of the column flange, in. = beam depth, in. = column depth, in. = yield strength of the column web, in. = Fy A, axial yield strength of the column, in. = column cross-sectional area, in.

When doubler plates are required, they shall meet the criteria of Section F2 and shall be welded to develop the proportion of the total shear force which is to be carried. Alternatively, when diagonal stiffeners are required, the weld connecting diagonal stiffeners to the web shall be sized to transmit the stiffener force caused by unbalanced moments to the web. Also, see Section K1.9. 8.

Unframed Ends of Beams and Girders At unframed ends of beams and girders not otherwise restrained against rotation about their longitudinal axes, a pair of transverse stiffeners, extending the full depth of the web, shall be provided. Also, see Section K1.9.

9.

Additional Stiffener Requirements for Concentrated Forces Transverse and diagonal stiffeners shall also comply with the following criteria: (1) The width of each stiffener plus one-half the thickness of the column web shall not be less than one-third of the width of the flange or moment connection plate delivering the concentrated force. (2) The thickness of a stiffener shall not be less than one-half the thickness of the flange or moment connection plate delivering the concentrated load, and not less than its width times √ Fy / 95. Full depth transverse stiffeners for compressive forces applied to a beam or plate girder flange shall be designed as axially compressed members (columns) in accordance with the requirements of Section E2, with an effective length of 0.75h, a cross section composed of two stiffeners and a strip of the web having a width of 25tw at interior stiffeners and 12tw at the ends of members. The weld connecting bearing stiffeners to the web shall be sized to transmit the excess web shear force to the stiffener. For fitted bearing stiffeners, see Section J8.1.

10.

Additional Doubler Plate Requirements for Concentrated Forces Doubler plates required by Sections K1.3 through K1.6 shall also comply with the following criteria: (1) The thickness and extent of the doubler plate shall provide the additional material necessary to equal or exceed the strength requirements. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. K3]

FATIGUE

6 - 97

(2) The doubler plate shall be welded to develop the proportion of the total force transmitted to the doubler plate. K2. PONDING The roof system shall be investigated by structural analysis to assure adequate strength and stability under ponding conditions, unless the roof surface is provided with sufficient slope toward points of free drainage or adequate individual drains to prevent the accumulation of rainwater. The roof system shall be considered stable and no further investigation is needed if:

Cp + 0.9Cs ≤ 0.25

(K2-1)

Id ≥ 25(S4)10−6

(K2-2)

where

Cp =

32LsL4p 107Ip

Cs =

32SL4s 107Is

Lp = column spacing in direction of girder (length of primary members), ft Ls = column spacing perpendicular to direction of girder (length of secondary members), ft S = spacing of secondary members, ft Ip = moment of inertia of primary members, in.4 Is = moment of inertia of secondary members, in.4 Id = moment of inertia of the steel deck supported on secondary members, in.4 per ft For trusses and steel joists, the moment of inertia Is shall be decreased 15 percent when used in the above equation. A steel deck shall be considered a secondary member when it is directly supported by the primary members. See Appendix K2 for an alternate determination of flat roof framing stiffness. K3. FATIGUE Few members or connections in conventional buildings need to be designed for fatigue, since most load changes in such structures occur only a small number of times or produce only minor stress fluctuations. The occurrence of full design wind or earthquake loads is too infrequent to warrant consideration in fatigue design. However, crane runways and supporting structures for machinery and equipment are often subject to fatigue loading conditions. Members and their connections subject to fatigue loading shall be proportioned in accordance with the provisions of Appendix K3 for service loads. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 98

CHAPTER L SERVICEABILITY DESIGN CONSIDERATIONS

This chapter is intended to provide design guidance for serviceability considerations. Serviceability is a state in which the function of a building, its appearance, maintainability, durability, and comfort of its occupants are preserved under normal usage. The general design requirement for serviceability is given in Section A5.4. Limiting values of structural behavior to ensure serviceability (e.g., maximum deflections, accelerations, etc.) shall be chosen with due regard to the intended function of the structure. Where necessary, serviceability shall be checked using realistic loads for the appropriate serviceability limit state. L1. CAMBER If any special camber requirements are necessary to bring a loaded member into proper relation with the work of other trades, as for the attachment of runs of sash, the requirements shall be set forth in the design documents. Beams and trusses detailed without specified camber shall be fabricated so that after erection any camber due to rolling or shop assembly shall be upward. If camber involves the erection of any member under a preload, this shall be noted in the design documents. L2. EXPANSION AND CONTRACTION Adequate provision shall be made for expansion and contraction appropriate to the service conditions of the structure. L3. DEFLECTIONS, VIBRATION, AND DRIFT 1.

Deflections Deformations in structural members and structural systems due to service loads shall not impair the serviceability of the structure.

2.

Floor Vibration Vibration shall be considered in designing beams and girders supporting large areas free of partitions or other sources of damping where excessive vibration due to pedestrian traffic or other sources within the building is not acceptable.

3.

Drift Lateral deflection or drift of structures due to code-specified wind or seismic loads shall not cause collision with adjacent structures nor exceed the limiting values of such drifts which may be specified or appropriate. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. L5]

CORROSION

6 - 99

L4. CONNECTION SLIP For the design of slip-critical connections see Sections J3.8 and J3.9. L5. CORROSION When appropriate, structural components shall be designed to tolerate corrosion or shall be protected against corrosion that may impair the strength or serviceability of the structure.


6 - 100

CHAPTER M FABRICATION, ERECTION, AND QUALITY CONTROL

This chapter provides requirements for shop drawings, fabrication, shop painting, erection, and quality control. M1. SHOP DRAWINGS Shop drawings giving complete information necessary for the fabrication of the component parts of the structure, including the location, type, and size of all welds, bolts, and rivets, shall be prepared in advance of the actual fabrication. These drawings shall clearly distinguish between shop and field welds and bolts and shall clearly identify slip-critical high-strength bolted connections. Shop drawings shall be made in conformity with good practice and with due regard to speed and economy in fabrication and erection. M2. FABRICATION 1.

Cambering, Curving, and Straightening Local application of heat or mechanical means is permitted to be used to introduce or correct camber, curvature, and straightness. The temperature of heated areas, as measured by approved methods, shall not exceed 1,100°F for A514 and A852 steel nor 1,200°F for other steels.

2.

Thermal Cutting Thermally cut edges shall meet the requirements of AWS 3.2.2 with the exception that thermally cut free edges which will be subject to calculated static tensile stress shall be free of round bottom gouges greater than 3⁄16-in. deep and sharp V-shaped notches. Gouges greater than 3⁄16-in. deep and notches shall be removed by grinding or repaired by welding. Re-entrant corners, except re-entrant corners of beam copes and weld access holes, shall meet the requirements of AWS 3.2.4. If other specified contour is required it must be shown on the contract documents. Beam copes and weld access holes shall meet the geometrical requirements of Section J1.6. For beam copes and weld access holes in ASTM A6 Group 4 and 5 shapes and welded built-up shapes with material thickness greater than two inches, a preheat temperature of not less than 150°F shall be applied prior to thermal cutting.

3.

Planing of Edges Planing or finishing of sheared or thermally cut edges of plates or shapes is not AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. M2]

FABRICATION

6 - 101

required unless specifically called for in the design documents or included in a stipulated edge preparation for welding. 4.

Welded Construction The technique of welding, the workmanship, appearance, and quality of welds and the methods used in correcting nonconforming work shall be in accordance with AWS D1.1 except as modified in Section J2.

5.

Bolted Construction All parts of bolted members shall be pinned or bolted and rigidly held together during assembly. Use of a drift pin in bolt holes during assembly shall not distort the metal or enlarge the holes. Poor matching of holes shall be cause for rejection. If the thickness of the material is not greater than the nominal diameter of the bolt plus 1⁄8-in., the holes are permitted to be punched. If the thickness of the material is greater than the nominal diameter of the bolt plus 1⁄8-in., the holes shall be either drilled or sub-punched and reamed. The die for all sub-punched holes, and the drill for all sub-drilled holes, shall be at least 1⁄16-in. smaller than the nominal diameter of the bolt. Holes in A514 steel plates over 1⁄2-in. thick shall be drilled. Fully inserted finger shims, with a total thickness of not more than 1⁄4-in. within a joint, are permitted in joints without changing the design strength (based upon hole type) for the design of connections. The orientation of such shims is independent of the direction of application of the load. The use of high-strength bolts shall conform to the requirements of the RCSC Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts.

6.

Compression Joints Compression joints which depend on contact bearing as part of the splice strength shall have the bearing surfaces of individual fabricated pieces prepared by milling, sawing, or other suitable means.

7.

Dimensional Tolerances Dimensional tolerances shall be in accordance with the AISC Code of Standard Practice.

8.

Finish of Column Bases Column bases and base plates shall be finished in accordance with the following requirements: (1) Steel bearing plates two inches or less in thickness are permitted without milling, provided a satisfactory contact bearing is obtained. Steel bearing plates over two inches but not over four inches in thickness are permitted to be straightened by pressing or, if presses are not available, by milling for all bearing surfaces (except as noted in subparagraphs 2 and 3 of this section), to obtain a satisfactory contact bearing. Steel bearing plates over four inches AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 102

FABRICATION, ERECTION, AND QUALITY CONTROL

[Chap. M

in thickness shall be milled for all bearing surfaces (except as noted in subparagraphs 2 and 3 of this section). (2) Bottom surfaces of bearing plates and column bases which are grouted to ensure full bearing contact on foundations need not be milled. (3) Top surfaces of bearing plates need not be milled when full-penetration welds are provided between the column and the bearing plate. M3. SHOP PAINTING 1.

General Requirements Shop painting and surface preparation shall be in accordance with the provisions of the AISC Code of Standard Practice. Shop paint is not required unless specified by the contract documents.

2.

Inaccessible Surfaces Except for contact surfaces, surfaces inaccessible after shop assembly shall be cleaned and painted prior to assembly, if required by the design documents.

3.

Contact Surfaces Paint is permitted unconditionally in bearing-type connections. For slip-critical connections, the faying surface requirements shall be in accordance with the RCSC Specification for Structural Joints Using ASTM A325 or A490 Bolts, paragraph 3(b).

4.

Finished Surfaces Machine-finished surfaces shall be protected against corrosion by a rustinhibitive coating that can be removed prior to erection, or which has characteristics that make removal prior to erection unnecessary.

5.

Surfaces Adjacent to Field Welds Unless otherwise specified in the design documents, surfaces within two inches of any field weld location shall be free of materials that would prevent proper welding or produce objectionable fumes during welding.

M4. ERECTION 1.

Alignment of Column Bases Column bases shall be set level and to correct elevation with full bearing on concrete or masonry.

2.

Bracing The frame of steel skeleton buildings shall be carried up true and plumb within the limits defined in the AISC Code of Standard Practice. Temporary bracing shall be provided, in accordance with the requirements of the Code of Standard Practice, wherever necessary to support all loads to which the structure may be subjected, including equipment and the operation of same. Such bracing shall be left in place as long as required for safety. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. M5]

3.

QUALITY CONTROL

6 - 103

Alignment No permanent bolting or welding shall be performed until the adjacent affected portions of the structure have been properly aligned.

4.

Fit of Column Compression Joints and Base Plates Lack of contact bearing not exceeding a gap of 1⁄16-in., regardless of the type of splice used (partial-joint-penetration groove welded, or bolted), is permitted. If the gap exceeds 1⁄16-in., but is less than 1⁄4-in., and if an engineering investigation shows that sufficient contact area does not exist, the gap shall be packed out with non-tapered steel shims. Shims need not be other than mild steel, regardless of the grade of the main material.

5.

Field Welding Shop paint on surfaces adjacent to joints to be field welded shall be wire brushed if necessary to assure weld quality. Field welding of attachments to installed embedments in contact with concrete shall be done in such a manner as to avoid excessive thermal expansion of the embedment which could result in spalling or cracking of the concrete or excessive stress in the embedment anchors.

6.

Field Painting Responsibility for touch-up painting, cleaning, and field painting shall be allocated in accordance with accepted local practices, and this allocation shall be set forth explicitly in the design documents.

7.

Field Connections As erection progresses, the structure shall be securely bolted or welded to support all dead, wind, and erection loads.

M5. QUALITY CONTROL The fabricator shall provide quality control procedures to the extent that the fabricator deems necessary to assure that all work is performed in accordance with this Specification. In addition to the fabricator’s quality control procedures, material and workmanship at all times may be subject to inspection by qualified inspectors representing the purchaser. If such inspection by representatives of the purchaser will be required, it shall be so stated in the design documents. 1.

Cooperation As far as possible, all inspection by representatives of the purchaser shall be made at the fabricator’s plant. The fabricator shall cooperate with the inspector, permitting access for inspection to all places where work is being done. The purchaser’s inspector shall schedule this work for minimum interruption to the work of the fabricator.

2.

Rejections Material or workmanship not in reasonable conformance with the provisions of this Specification may be rejected at any time during the progress of the work. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 104

FABRICATION, ERECTION, AND QUALITY CONTROL

[Chap. M

The fabricator shall receive copies of all reports furnished to the purchaser by the inspection agency. 3.

Inspection of Welding The inspection of welding shall be performed in accordance with the provisions of AWS D1.1 except as modified in Section J2. When visual inspection is required to be performed by AWS certified welding inspectors, it shall be so specified in the design documents. When nondestructive testing is required, the process, extent, and standards of acceptance shall be clearly defined in the design documents.

4.

Inspection of Slip-Critical High-Strength Bolted Connections The inspection of slip-critical high-strength bolted connections shall be in accordance with the provisions of the RCSC Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts.

5.

Identification of Steel The fabricator shall be able to demonstrate by a written procedure and by actual practice a method of material application and identification, visible at least through the “fit-up” operation, of the main structural elements of a shipping piece. The identification method shall be capable of verifying proper material application as it relates to: (1) Material specification designation (2) Heat number, if required (3) Material test reports for special requirements.


6 - 105

APPENDIX B DESIGN REQUIREMENTS

Appendix B5.1 provides an expanded definition of limiting width-thickness ratio for webs in combined flexure and axial compression. Appendix B5.3 applies to the design of members containing slender compression elements. B5. LOCAL BUCKLING 1.

Classification of Steel Sections For members with unequal flanges and with webs in combined flexural and axial compression, λr for the limit state of web local buckling is λr =

h P  253  1 + 2.83   1 − u  h Fy  √ φ c bPy  

(A-B5-1)

3 h 3 ≤ ≤ 4 hc 2 For members with unequal flanges with webs subjected to flexure only, λr for the limit state of web local buckling is λr =

h 253  1 + 2.83    Fy  √  hc 

(A-B5-2)

3 h 3 ≤ ≤ 4 hc 2 where λr, h, and hc are as defined in Section B5.1. These substitutions shall be made in Appendices F and G when applied to members with unequal flanges. If the compression flange is larger than the tension flange, λr shall be determined using Equation A-B5-1, A-B5-2, or Table B5.1. 3.

Slender-Element Compression Sections Axially loaded members containing elements subject to compression which have a width-thickness ratio in excess of the applicable λr as stipulated in Section B5.1 shall be proportioned according to this Appendix. Flexural members with slender compression elements shall be designed in accordance with Appendices F and G. Flexural members with proportions not covered by Appendix F1 shall be designed in accordance with this Appendix. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 106

3a.

DESIGN REQUIREMENTS

[App. B

Unstiffened Compression Elements The design strength of unstiffened compression elements whose width-thickness ratio exceeds the applicable limit λr as stipulated in Section B5.1 shall be subject to a reduction factor Qs. The value of Qs shall be determined by Equations A-B5-3 through A-B5-10, as applicable. When such elements comprise the compression flange of a flexural member, the maximum required bending stress shall not exceed φbFy Qs, where φb = 0.90. The design strength of axially loaded compression members shall be modified by the appropriate reduction factor Q, as provided in Appendix B5.3c. (a) For single angles:

Fy : when 76.0 / √ Fy 155 / √ Fy : Qs = 15,500 / [Fy (b / t)2]

(A-B5-4)

(b) For flanges, angles, and plates projecting from rolled beams or columns or other compression members:

Fy : when 95.0 / √ Fy < b / t < 176 / √ Fy Qs = 1.415 − 0.00437(b / t)√

(A-B5-5)

when b / t ≥ 176 / √ Fy : Qs = 20,000 / [Fy (b / t)2]

(A-B5-6)

(c) For flanges, angles and plates projecting from built-up columns or other compression members:  Fy / kc : when 109 / √  Fy / kc < b / t < 200 / √  Fy / kc Qs = 1.415 − 0.00381(b / t)√

(A-B5-7)

when b / t ≥ 200 / √  Fy / kc : Qs = 26,200kc / [Fy (b / t)2] The coefficient, kc, shall be computed as follows: (a) For I-shaped sections: kc =

4 , 0.35 ≤ kc ≤ 0.763  √ h / tw

where: h = depth of web, in. tw = thickness of web, in. (b) For other sections: kc = 0.763 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(A-B5-8)

Sect. A-B5]

LOCAL BUCKLING

6 - 107

(d) For stems of tees: when 127 / √ Fy < b / t < 176 / √ Fy : Qs = 1.908 − 0.00715(b / t)√ Fy

(A-B5-9)

Fy : when b / t ≥ 176 / √ Qs = 20,000 / [Fy (b / t)2]

(A-B5-10)

where b = width of unstiffened compression element as defined in Section B5.1, in. t = thickness of unstiffened element, in. Fy = specified minimum yield stress, ksi 3b.

Stiffened Compression Elements When the width-thickness ratio of uniformly compressed stiffened elements (except perforated cover plates) exceeds the limit λr stipulated in Section B5.1, a reduced effective width be shall be used in computing the design properties of the section containing the element. (a) For flanges of square and rectangular sections of uniform thickness: b 238 : when ≥ t f √ 64.9  326t  1− (A-B5-11) be = (b / t)√  f  √  f  otherwise be = b. (b) For other uniformly compressed elements: b 253 when ≥ : t √f 57.2  326t  1− be = (b / t)√  f  √  f 

(A-B5-12)

otherwise be = b where b = actual width of a stiffened compression element, as defined in Section B5.1, in. be = reduced effective width, in. t = element thickness, in. f = computed elastic compressive stress in the stiffened elements, based on the design properties as specified in Appendix B5.3c, ksi. If unstiffened elements are included in the total cross section, f for the stiffened element must be such that the maximum compressive stress in the unstiffened element does not exceed φcFcr as defined in Appendix B5.3c with Q = Qs and φc = 0.85, or φbFy Qs with φb = 0.90, as applicable. (c) For axially loaded circular sections with diameter-to-thickness ratio D / t greater than 3,300 / Fy but less than 13,000 / Fy AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 108

DESIGN REQUIREMENTS

Q = Qa =

2 1,100 + Fy (D / t) 3

[App. B

(A-B5-13)

where D = outside diameter, in. t = wall thickness, in. 3c.

Design Properties Properties of sections shall be determined using the full cross section, except as follows: In computing the moment of inertia and elastic section modulus of flexural members, the effective width of uniformly compressed stiffened elements be, as determined in Appendix B5.3b, shall be used in determining effective cross-sectional properties. For unstiffened elements of the cross section, Qs is determined from Appendix B5.3a. For stiffened elements of the cross section Qa =

effective area actual area

(A-B5-14)

where the effective area is equal to the summation of the effective areas of the cross section. 3d.

Design Strength For axially loaded compression members the gross cross-sectional area and the radius of gyration r shall be computed on the basis of the actual cross section. The critical stress Fcr shall be determined as follows: (a) For λc√ Q ≤ 1.5: 2

Fcr = Q(0.658Qλc)Fy

(A-B5-15)

0.877 Fcr =  2  Fy  λc 

(A-B5-16)

Q = QsQa

(A-B5-17)

(a) For λc√ Q > 1.5:

where

Cross sections comprised of only unstiffened elements, Q = Qs, (Qa = 1.0) Cross sections comprised of only stiffened elements, Q = Qa, (Qs = 1.0) Cross sections comprised of both stiffened and unstiffened elements, Q = QsQa


6 - 109

APPENDIX E COLUMNS AND OTHER COMPRESSION MEMBERS

This Appendix applies to the strength of doubly symmetric columns with thin plate elements, singly symmetric and unsymmetric columns for the limit states of flexuraltorsional and torsional buckling. E3. DESIGN COMPRESSIVE STRENGTH FOR FLEXURAL-TORSIONAL BUCKLING The strength of compression members determined by the limit states of torsional and flexural-torsional buckling is φcPn, where φc = 0.85 Pn = nominal resistance in compression, kips Pn = AgFcr Ag = gross area of cross section, in.2

(A-E3-1)

The nominal critical stress Fcr is determined as follows: (a) For λe√ Q ≤ 1.5: 2

Fcr = Q(0.658Qλe)Fy

(A-E3-2)

0.877 Fcr =  2  Fy  λe 

(A-E3-3)

 Fy / Fe λe = √

(A-E3-4)

Q > 1.5: (b) For λe√

where

Fy = specified minimum yield stress of steel, ksi Q = 1.0 for elements meeting the width-thickness ratios λr of Section B5.1 Q = QsQa for elements not meeting the width-thickness ratios λr of Section B5.1 and determined in accordance with the provisions of Appendix B5.3 The critical torsional or flexural-torsional elastic buckling stress Fe is determined as follows:


6 - 110


[App. E

(a) For doubly symmetric shapes:  π2ECw  1 + GJ Fe =  2 ( K l) z   Ix + Iy

(A-E3-5)

(b) For singly symmetric shapes where y is the axis of symmetry:

 Fey + Fez 1− Fe = 2H 

 4Fey Fez H  1− (Fey + Fez )2 

 √

(A-E3-6)

(c) For unsymmetric shapes, the critical flexural-torsional elastic buckling stress Fe is the lowest root of the cubic equation 2

2

 xo   yo  (Fe −Fex)(Fe −Fey)(Fe −Fez) − Fe (Fe −Fey)  _  − Fe 2 (Fe −Fex)  __  = 0 (A-E3-7)  ro   ro  2

where Kz E G Cw J Ix, Iy xo, yo

= effective length factor for torsional buckling = modulus of elasticity, ksi = shear modulus, ksi = warping constant, in.6 = torsional constant, in.4 = moment of inertia about the principal axes, in.4 = coordinates of shear center with respect to the centroid, in. _ Ix + Iy ro2 = x2o + y2o + A

(A-E3-8)

 x2o + y2o  H=1− _2   ro 

(A-E3-9)

Fex =

π2E (Kxl / rx)2

(A-E3-10)

Fey =

π2E (Kyl / ry)2

(A-E3-11)

 π2ECw  1 + GJ _ 2 Fez =  2  (Kzl)  Aro where A = cross-sectional area of member, in.2 l = unbraced length, in. Kx, Ky = effective length factors in x and y directions _rx, ry = radii of gyration about the principal axes, in. = polar radius of gyration about the shear center, in. ro AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(A-E3-12)

6 - 111

APPENDIX F BEAMS AND OTHER FLEXURAL MEMBERS

Appendix F1 provides the design flexural strength of beams and girders. Appendix F2 provides the design shear strength of webs with and without stiffeners and requirements on transverse stiffeners. Appendix F3 applies to web-tapered members. F1.

DESIGN FOR FLEXURE The design strength for flexural members is φbMn where φb = 0.90 and Mn is the nominal strength. Table A-F1.1 provides a tabular summary of Equations F1-1 through F1-15 for determining the nominal flexural strength of beams and girders. For slenderness parameters of cross sections not included in Table A-F1.1, see Appendix B5.3. For flexural members with unequal flanges see Appendix B5.1 for the determination of λr for the limit state of web local buckling. The nominal flexural strength Mn is the lowest value obtained according to the limit states of yielding: lateral-torsional buckling (LTB); flange local buckling (FLB); and web local buckling (WLB). The nominal flexural strength Mn shall be determined as follows for each limit state: (a) For λ ≤ λp: Mn = Mp

(A-F1-1)

(b) For λp < λ ≤ λr: For the limit state of lateral-torsional buckling:   λ − λp  Mn = Cb Mp − (Mp − Mr)   ≤ Mp   λr − λp 

(A-F1-2)

For the limit states of flange and web local buckling:  λ − λp  Mn = Mp − (Mp − Mr)    λr − λp 

(A-F1-3)

(c) For λ > λr: For the limit state of lateral-torsional buckling and flange local buckling: Mn = Mcr = SFcr ≤ Mp AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(A-F1-4)

6 - 112


[App. F

For design of girders with slender webs, the limit state of web local buckling is not applicable. See Appendix G2. For λ of the flange > λr in shapes not included in Table A-F1.1, see Appendix B5.3. For λ of the web > λr, see Appendix G. The terms used in the above equations are: Mn Mp Mcr Mr

λ λ λ λ

λp λr Fcr Cb S Lb ry

= nominal flexural strength, kip-in. = Fy Z, plastic moment ≤ 1.5 Fy S, kip-in. = buckling moment, kip-in. = limiting buckling moment (equal to Mcr when λ = λr), kip-in. = controlling slenderness parameter = minor axis slenderness ratio Lb / ry for lateral-torsional buckling = flange width-thickness ratio b / t for flange local buckling as defined in Section B5.1 = web depth-thickness ratio h / tw for web local buckling as defined in Section B5.1 = largest value of λ for which Mn = Mp = largest value of λ for which buckling is inelastic = critical stress, ksi = Bending coefficient dependent on moment gradient, see Section F1.2a, Equation F1-3 = section modulus, in.3 = laterally unbraced length, in. = radius of gyration about minor axis, in.

The applicable limit states and equations for Mp, Mr, Fcr, λ, λp, and λr are given in Table A-F1.1 for shapes covered in this Appendix. The terms used in the table are: A FL Fr Fr Fr Fy Fyf Fyw Iyc J Re Seff Sxc Sxt Z b d

= cross-sectional area, in.2 = smaller of (Fyf −Fr) or Fyw, ksi = compressive residual stress in flange = 10 ksi for rolled shapes = 16.5 ksi for welded shapes = specified minimum yield strength, ksi = yield strength of the flange, ksi = yield strength of the web, ksi = moment of inertia of compression flange about y axis or if reverse curvature bending, moment of inertia of smaller flange, in.4 = torsional constant, in.4 = see Appendix G2 = effective section modulus about major axis, in.3 = section modulus of the outside fiber of the compression flange, in.3 = section modulus of the outside fiber of the tension flange, in.3 = plastic section modulus, in.3 = flange width, in. = overall depth, in. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. A-F2]

f h ryc tf tw

DESIGN FOR SHEAR

6 - 113

= computed compressive stress in the stiffened element, ksi = clear distance between flanges less the fillet or corner radius at each flange, in. = radius of gyration of compression flange about y axis or if reverse curvature bending, smaller flange, in. = flange thickness, in. = web thickness, in.

F2.

DESIGN FOR SHEAR

2.

Design Shear Strength The design shear strength of stiffened or unstiffened webs is φvVn, where φv = 0.90 Vn = nominal shear strength defined as follows:  kv /Fyw: For h / tw ≤ 187√ Vn = 0.6FywAw

(A-F2-1)

 kv /Fyw: For 187√  kv /Fyw < h / tw ≤ 234√  kv /Fyw ) / (h / tw) Vn = 0.6FywAw(187√

(A-F2-2)

For h / tw > 234√  kv /Fyw: Vn = Aw(26,400kv) / (h / tw)2

(A-F2-3)

where kv = 5 + 5 / (a / h)2 kv = 5 when a / h > 3 or a / h > [260 / (h / t)]2 a = distance between transverse stiffeners, in. h = for rolled shapes, the clear distance between flanges less the fillet or corner radius, in. h = for built-up welded sections, the clear distance between flanges, in. h = for built-up bolted or riveted sections, the distance between fastener lines, in. 3.

Transverse Stiffeners Fyw, Transverse stiffeners are not required in plate girders where h / tw ≤ 418 / √ or where the required shear, Vu, as determined by structural analysis for the factored loads, is less than or equal to 0.6φvAwFywCv, where Cv is determined for kv = 5 and φv = 0.90. Transverse stiffeners used to develop the web design shear strength as provided in Appendix F2.2 shall have a moment of inertia about an axis in the web center for stiffener pairs or about the face in contact with the web plate for single stiffeners, which shall not be less than at3w j, where j = 2.5 / (a / h)2 − 2 ≥ 0.5 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(A-F2-4)

6 - 114


[App. F

TABLE A-F1.1 Nominal Strength Parameters Plastic Moment Mp

Shape

Fy Zx [b]

Channels and doubly and singly symmetric I-shaped beams (including hybrid beams) bent about major axis [a]

Limit State of Buckling LTB doubly symmetric members and channels LTB singly symmetric members

Channels and doubly and singly symmetric I-shaped members bent about minor axis [a]

π Sx

 √

[d] λr =

X1 FL

 √ 1+√  1 + X2F2L

[e] Fcr =

EGJA 2

Mcr

Sxc

, where Mcr =

FL Sxc ≤ Fyf Sxt

FL Sx

WLB

ReFyf Sx

FLB

Fy Sy

Fy Zy

2

X2 = 4

FL Sx

FLB

NOTE: LTB applies only for strong axis bending. [a] Excluding double angles and tees. [b] Computed from fully plastic stress distribution for hybrid sections.

[c] X1 =

Limiting Buckling Moment Mr

Cw  Sx    Iy  GJ 

57,000Cb  √ IyJ [B1 + √  (1 + B2 + B21) ] ≤ Mp Lb

where B1 = 2.25[2(Iyc / Iy) − 1](h / Lb)√  (Iy / J) B2 = 25(1 − Iyc / Iy)(Iyc / J)(h / Lb)2 Cb = 1.0 if Iyc / Iy < 0.1 or Iyc / Iy > 0.9.


Sect. A-F2]

DESIGN FOR SHEAR

6 - 115

TABLE A-F1.1 (cont’d) Nominal Strength Parameters Slenderness Parameters λ

λp

Lb ry

300 √Fyf

[e]

Lb ryc

300 √Fyf

[f]

b t

65 √Fyf

Not applicable

h tw

640 √Fyf

Critical Stress Fcr

CbX1√ 2 λ

1 + √ 2λ

X21X2 2

λr [c, d]

Value of λ for which Mcr (Cb = 1) = Mr [g]

λr as defined in Section B5.1

Same as for major axis

20,000

[f] Fcr =

λ2

Fcr =

for rolled shapes

26,200kc for welded shapes λ2

where kc = 4 / √  h / tw and 0.35 ≤ kc ≤ 0.763 [g] λr = λr =

141 for rolled shapes FL √ 162  √ FL / kc

for welded shapes


Limitations Applicable for I-shaped members if h / tw ≤ λr when h / tw > λr See Appendix G.

6 - 116


[App. F

TABLE A-F1.1 (cont’d) Nominal Strength Parameters Shape

Plastic Moment Mp

Limit State of Buckling

Limiting Buckling Moment Mr

Solid symmetric shapes, except rectangular bars, bent about major axis

Fy Zx

Solid rectangular bars bent about major axis

Fy Zx

LTB

Fy Sx

Symmetric box sections loaded in a plane of symmetry

Fy Z

LTB

Fyf Seff

FLB

FL Seff

WLB

Same as for I-shape

LTB

Not applicable

FLB

 600  Mn =  + Fy S [h] D / t 

WLB

Not applicable

Circular tubes

Fy Z

Not applicable

[h] This equation is to be used in place of Equation A-F1-4.


Sect. A-F2]

DESIGN FOR SHEAR

6 - 117

TABLE A-F1.1 (cont’d) Nominal Strength Parameters Slenderness Parameters Critical Stress Fcr

λ

λp

λr

Limitations

Not applicable

57,000Cb√  JA λSx

Lb ry

3,750√  JA Mp

57,000√  JA Mr,

57,000Cb√  JA λSx

Lb ry

3,750√  JA Mp

57,000√  JA Mr

Seff F [i] Sx y

b t

190 Fy √

238 Fy √

Applicable if h / tw ≤ 970 / √ Fyf

Same as for I-shape Not applicable 9,570 D/t

D/t

2,070 Fy

8,970 Fy

D/t
1.5:

where a = clear distance between transverse stiffeners, in. h = clear distance between flanges less the fillet or corner radius for rolled shapes; and for built-up sections, the distance between adjacent lines of fasteners or the clear distance between flanges when welds are used, in. tw = web thickness, in. Fyf = specified minimum yield stress of a flange, ksi In unstiffened girders h / tw shall not exceed 260. G2. DESIGN FLEXURAL STRENGTH The design flexural strength for plate girders with slender webs shall be φbMn, where φb = 0.90 and Mn is the lower value obtained according to the limit states of tension-flange yield and compression-flange buckling. For girders with unequal flanges, see Appendix B5.1 for the determination of λr for the limit state of web local buckling. (a) For tension-flange yield: Mn = SxtReFyt (b) For compression-flange buckling: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(A-G2-1)

Sect. A-G2]

DESIGN FLEXURAL STRENGTH

Mn = SxcRPG ReFcr

6 - 123

(A-G2-2)

where RPG = 1 −

 hc 970  ar  −  ≤ 1.0 1,200 + 300ar  tw √ Fcr

(A-G2-3)

Re

= hybrid girder factor 12 + ar (3m − m3) Re = ≤ 1.0 (for non-hybrid girders, Re = 1.0) 12 + 2ar ar = ratio of web area to compression flange area (≤ 10) m = ratio of web yield stress to flange yield stress or to Fcr Fcr = critical compression flange stress, ksi Fyt = yield stress of tension flange, ksi Sxc = section modulus referred to compression flange, in.3 Sxt = section modulus referred to tension flange, in.3 hc = twice the distance from the centroid to the nearest line of fasteners at the compression flange or the inside of the face of the compression flange when welds are used The critical stress Fcr to be used is dependent upon the slenderness parameters λ, λp, λr, and CPG as follows: For λ ≤ λp: Fcr = Fyf

(A-G2-4)

For λp < λ ≤ λr:  Fcr = CbFyf 1 − 

1  λ − λp    ≤F 2  λr − λp  yf 

(A-G2-5)

For λ > λr: Fcr =

CPG λ2

(A-G2-6)

In the foregoing, the slenderness parameter shall be determined for both the limit state of lateral-torsional buckling and the limit state of flange local buckling; the slenderness parameter which results in the lowest value of Fcr governs. (a) For the limit state of lateral-torsional buckling: Lb rT

(A-G2-7)

λp =

300 √ Fyf

(A-G2-8)

λr =

756 √Fyf

(A-G2-9)

λ=

CPG = 286,000Cb AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(A-G2-10)

6 - 124

PLATE GIRDERS

[App. G

where Cb = see Section F1.2, Equation F1-3 rT = radius of gyration of compression flange plus one-third of the compression portion of the web, in. (b) For the limit state of flange local buckling: λ=

bf 2tf

(A-G2-11)

λp =

65 √Fyf

(A-G2-12)

230  √ Fyf / kc

(A-G2-13)

λr =

CPG = 26,200kc

(A-G2-14)

Cb = 1.0  h / tw and 0.35 ≤ kc ≤ 0.763. where kc = 4 / √ The limit state of flexural web local buckling is not applicable. G3. DESIGN SHEAR STRENGTH WITH TENSION FIELD ACTION The design shear strength with tension field action shall be φvVn, kips, where φv = 0.90 and Vn is determined as follows: (a) For h / tw ≤ 187√  kv /Fyw: Vn = 0.6AwFyw

(A-G3-1)

 kv /Fyw: (b) For h / tw > 187√ 

Vn = 0.6AwFywCv +



 1 − Cv 2 1.15√  1 + (a / h) 

(A-G3-2)

where Cv = ratio of “critical” web stress, according to linear buckling theory, to the shear yield stress of web material Also see Appendix G4 and G5. For end-panels in non-hybrid plate girders, all panels in hybrid and web-tapered plate girders, and when a / h exceeds 3.0 or [260 / (h / tw)]2, tension field action is not permitted and Vn = 0.6AwFywCv

(A-G3-3)

The web plate buckling coefficient kv is given as kv = 5 +

5 (a / h)2


(A-G3-4)

Sect. A-G5]

FLEXURE-SHEAR INTERACTION

6 - 125

except that kv shall be taken as 5.0 if a / h exceeds 3.0 or [260 / (h / tw)]2. The shear coefficient Cv is determined as follows: (a) For 187

 √

 √

kv h ≤ ≤ 234 Fyw tw

kv : Fyw

Cv =

(b) For

 √

h > 234 tw

187√  kv /Fyw h / tw

(A-G3-5)

44,000kv (h / tw)2Fyw

(A-G3-6)

kv : Fyw Cv =

G4. TRANSVERSE STIFFENERS Fyw , Transverse stiffeners are not required in plate girders where h / tw ≤ 418 / √ or where the required shear Vu, as determined by structural analysis for the factored loads, is less than or equal to 0.6φvAwFywCv, where Cv is determined for kv = 5 and φv = 0.90. Stiffeners may be required in certain portions of a plate girder to develop the required shear or to satisfy the limitations given in Appendix G1. Transverse stiffeners shall satisfy the requirements of Appendix F2.3. When designing for tension field action, the stiffener area Ast shall not be less than Fyw  Vu  0.15Dhtw(1 − Cv) − 18t2w ≥ 0 Fyst  φvVn  

(A-G4-1)

where Fyst D D D

= specified yield stress of the stiffener material, ksi = 1 for stiffeners in pairs = 1.8 for single angle stiffeners = 2.4 for single plate stiffeners

Cv and Vn are defined in Appendix G3, and Vu is the required shear at the location of the stiffener. G5. FLEXURE-SHEAR INTERACTION For 0.6φVn ≤ Vu ≤φVn (φ = 0.90) and 0.75φMn ≤ Mu ≤ φMn (φ = 0.90), plate girders with webs designed for tension field action shall satisfy the additional AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 126

PLATE GIRDERS

[App. G

flexure-shear interaction criteria: Vu Mu + 0.625 ≤ 1.375 φMn φVn

(A-G5-1)

where Mn is the nominal flexural strength of plate girders from Appendix G2 or Section F1, φ = 0.90, and Vn is the nominal shear strength from Appendix G3.


6 - 127

APPENDIX H MEMBERS UNDER COMBINED FORCES AND TORSION

This appendix provides alternative interaction equations for biaxially loaded I-shaped members with bf / d ≤ 1.0 and box-shaped members. H3. ALTERNATIVE INTERACTION EQUATIONS FOR MEMBERS UNDER COMBINED STRESS For biaxially loaded I-shaped members with bf / d ≤ 1.0 and box-shaped members in braced frames only, the use of the following interaction equations in lieu of Equations H1-1a and H1-1b is permitted. Both Equations A-H3-1 and A-H3-2 shall be satisfied. ζ

ζ

 Mux   Muy   +  ≤ 1.0   φbM′px  φbM′py η

(A-H3-1)

η

 CmxMux   CmyMuy  (A-H3-2)  +  ≤ 1.0   φbM′nx   φbM′ny  The terms in Equations A-H3-1 and A-H3-2 are determined as follows: (a) For I-shaped members: For bf / d < 0.5: ζ = 1.0 For 0.5 ≤ bf / d ≤ 1.0:

ζ = 1.6 −

Pu / Py 2[ln(Pu / Py )]

(A-H3-3)

For bf / d < 0.3: η = 1.0 For 0.3 ≤ bf / d ≤ 1.0:

η = 0.4 +

Pu bf + ≥ 1.0 Py d

(A-H3-4)

where bf = flange width, in. d = member depth, in. Cm = coefficient applied to the bending term in interaction equation for AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 128


[App. H

prismatic members and dependent on column curvature caused by applied moments, see Section C1. M′px = 1.2Mpx[1 − (Pu / Py )] ≤ Mpx

(A-H3-5)

M′py = 1.2Mpy[1 − (Pu / Py ) ] ≤ Mpy

(A-H3-6)

 Pu   Pu  M′nx = Mnx 1 −  1 − P  φcPn  ex  

(A-H3-7)

 Pu   Pu  M′ny = Mny 1 −  1 − P  φ P ey  c n  

(A-H3-8)

2

(b) For box-section members:

ζ = 1.7 −

Pu / Py ln(Pu / Py)

(A-H3-9) b

P P /P η = 1.7 − u y − aλx u > 1.1 ln(Pu / Py )  Py

(A-H3-10)

For Pu / Py ≤ 0.4, a = 0.06, and b = 1.0; For Pu / Py > 0.4, a = 0.15, and b = 2.0; M′px = 1.2Mpx[1 − Pu / Py] ≤ Mpx

(A-H3-11a)

M′py = 1.2Mpy[1 − Pu / Py] ≤ Mpy

(A-H3-11b)

 Pu   Pu 1.25  M′nx = Mnx 1 −  1 − P 1/3  φ P ex (B / H)  c n  

(A-H3-12)

 Pu   Pu 1.25  M′ny = Mny 1 −  1 − P 1/2  φcPn  ey (B / H)  

(A-H3-13)

where Pn = nominal compressive strength determined in accordance with Section E2, kips Pu = required axial strength, kips Py = compressive yield strength AgFy, kips φb = resistance factor for flexure = 0.90 φc = resistance factor for compression = 0.85 Pe = Euler buckling strength AgFy / λc2, where λc is the column slenderness parameter defined by Equation E2-4, kips Mu = required flexural strength, kip-in. Mn = nominal flexural strength, determined in accordance with Section F1, kip-in. Mp = plastic moment ≤ 1.5Fy S, kip-in. B = outside width of box section parallel to major principal axis x, in. H = outside depth of box section perpendicular to major principal axis x, in.


6 - 129

APPENDIX J CONNECTIONS, JOINTS, AND FASTENERS

Appendix J2.4 provides the alternative design strength for fillet welds. Appendices J3.8 and J3.9 pertain to the design of slip-critical connections using factored loads. J2.

WELDS

4.

Design Strength In lieu of the constant design strength for fillet welds given in Table J2.5, the following procedure is permitted. (a) The design strength of a linear weld group loaded in-plane through the center of gravity is φFw Aw: Fw = 0.60FEXX(1.0 + 0.50 sin1.5θ) where φ Fw FEXX θ Aw

= 0.75 = nominal stress, ksi = electrode classification number, i.e., minimum specified strength, ksi = angle of loading measured from the weld longitudinal axis, degrees = effective area of weld throat, in.2

(b) The design strength of weld elements within a weld group that are loaded in-plane and analyzed using an instantaneous center of rotation method to maintain deformation compatibility and non-linear load deformation behavior of variable angle loaded welds is φFwx Aw and φFwy Aw: where Fwx Fwy Fwi f(p) φ Fwi Fwix Fwiy p ∆m

= ΣFwix = ΣFwiy = 0.60FEXX(1.0 + 0.50 sin1.5θ) f(p) = [p(1.9 − 0.9p)]0.3 = 0.75 = nominal stress in any ith weld element, ksi = x component of stress Fwi = y component of stress Fwi = ∆i / ∆m, ratio of element i deformation to its deformation at maximum stress = 0.209(θ + 2)−0.32D, deformation of weld element at maximum stress, in. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 130


∆i

∆u D rcrit

[App. J

= deformation of weld elements at intermediate stress levels, linearly proportioned to the critical deformation based on distance from the instantaneous center of rotation, ri, in. = ri∆u / rcrit = 1.087(θ + 6)−0.65D ≤ 0.17D, deformation of weld element at ultimate stress (fracture), usually in element furthest from instantaneous center of rotation, in. = leg size of the fillet weld, in. = distance from instantaneous center of rotation to weld element with minimum ∆u / ri ratio

J3.


8.

High-Strength Bolts in Slip-Critical Connections

8b.

Slip-Critical Connections Designed at Factored Loads It is permissible to proportion slip-critical connections at factored loads. The design slip resistance for use at factored loads, φRstr, shall equal or exceed the required force due to the factored loads, where: Rstr = 1.13µTmNbNs

(A-J3-1)

where: Tm = minimum fastener tension given in Table J3.1, kips Nb = number of bolts in the joint Ns = number of slip planes µ = mean slip coefficient for Class A, B, or C surfaces, as applicable, or as established by tests (a) For Class A surfaces (unpainted clean mill scale steel surfaces or surfaces with Class A coating on blast-cleaned steel), µ = 0.33 (b) For Class B surfaces (unpainted blast-cleaned steel surfaces or surfaces with Class B coatings on blast-cleaned steel), µ = 0.50 (c) For Class C surfaces (hot-dip galvanized and roughened surfaces), µ = 0.40 = resistance factor φ (a) For standard holes, φ = 1.0 (b) For oversize and short-slotted holes, φ = 0.85 (c) For long-slotted holes transverse to the direction of load, φ = 0.70 (d) For long-slotted holes parallel to the direction of load, φ = 0.60 9.

Combined Tension and Shear in Slip-Critical Connections

9b.

Slip-Critical Connections Designed at Factored Loads When using factored loads as the basis for design of slip-critical connections subject to applied tension, T, that reduces the net clamping force, the slip AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. A-J3]


6 - 131

resistance φRstr according to Appendix J3.8b shall be multiplied by the following factor in which Tu is the required tensile strength at factored loads: [1 − Tu / (1.13TmNb)]


(A-J3-2)

6 - 132

APPENDIX K CONCENTRATED FORCES, PONDING, AND FATIGUE

Appendix K2 provides an alternative determination of roof stiffness. Appendix K4 pertains to members and connections due to fatigue loading. K2. PONDING The provisions of this Appendix are permitted to be used when a more exact determination of flat roof framing stiffness is needed than that given by the provision of Section K2 that Cp + 0.9Cs ≤ 0.25. For any combination of primary and secondary framing, the stress index is computed as  Fy − fo   for the primary member Up =   fo  p

(A-K2-3)

 Fy − fo   for the secondary member Us =   fo  s

(A-K2-4)

where fo = the stress due to 1.2D + 1.2R (D = nominal dead load, R = nominal load due to rain water or ice exclusive of the ponding contribution)* Enter Figure A-K2.1 at the level of the computed stress index Up determined for the primary beam; move horizontally to the computed Cs value of the secondary beams and then downward to the abscissa scale. The combined stiffness of the primary and secondary framing is sufficient to prevent ponding if the flexibility constant read from this latter scale is more than the value of Cp computed for the given primary member; if not, a stiffer primary or secondary beam, or combination of both, is required. In the above, Cp =

32LsLp4 107Ip

Cs =

32L4s 107Is

* Depending upon geographic location, this loading should include such amount of snow as might also be present, although ponding failures have occurred more frequently during torrential summer rains when the rate of precipitation exceeded the rate of drainage runoff and the resulting hydraulic gradient over large roof areas caused substantial accumulation of water some distance from the eaves. A load factor of 1.2 shall be used for loads resulting from these phenomena.


Sect. A-K2]

PONDING

6 - 133

where Lp = column spacing in direction of girder (length of primary members), ft Ls = column spacing perpendicular to direction of girder (length of secondary members), ft S = spacing of secondary members, ft Ip = moment of inertia of primary members, in.4 Is = moment of inertia of secondary members, in.4 A similar procedure must be followed using Figure A-K2.2. Roof framing consisting of a series of equally spaced wall-bearing beams is considered as consisting of secondary members supported on an infinitely stiff 3.4

3.2 3.0 2.8 2.6

2.4

0.5

2.2

1.8

0.4

1.6

0.3 1.4

0.2

Stress Index Up

2.0

1.2

1

0.

1.0

Cs

0.8

=

0

0.6

0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Upper Limit of Flexibility Constant Cp Fig. A-K2.1. Limiting flexibility coefficient for the primary systems. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

0.7

6 - 134


[App. K

primary member. For this case, enter Figure A-K2.2 with the computed stress index Us. The limiting value of Cs is determined by the intercept of a horizontal line representing the Us value and the curve for Cp = 0. The ponding deflection contributed by a metal deck is usually such a small part of the total ponding deflection of a roof panel that it is sufficient merely to limit its moment of inertia (per foot of width normal to its span) to 0.000025 times the fourth power of its span length. However, the stability against ponding of a roof consisting of a metal roof deck of relatively slender depth-span ratio, spanning between beams supported directly on columns, may need to be checked. This can be done using Figure A-K2.1 or A-K2.2 using as Cs the flexibility constant for a one-foot width of the roof deck (S = 1.0).

0.7

3.4

3.2

0.6

3.0 2.8

0.5

2.6

2.4

2.0

1.8

0.3

1.6

0.2

Stress Index Us

0.4

2.2

1.4 1

0.

1.2 1.0

Cp

=

0

0.8 0.6

0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Upper Limit of Flexibility Constant Cs Fig. A-K2.2. Limiting flexibility coefficient for the secondary systems. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

0.7

Sect. A-K3]

FATIGUE

6 - 135

TABLE A-K3.1 Number of Loading Cycles Loading Condition

From

To

1

20,000 [a]

100,000 [b]

2

100,000

500,000 [c]

3

500,000

2,000,000 [d]

4

Over 2,000,000

[a] Approximately equivalent to two applications every day for 25 years. [b] Approximately equivalent to 10 applications every day for 25 years. [c] Approximately equivalent to 50 applications every day for 25 years. [d] Approximately equivalent to 200 applications every day for 25 years.

Since the shear rigidity of the web system of steel joists and trusses is less than that of a solid plate, their moment of inertia shall be taken as 85 percent of their chords. K3. FATIGUE Members and connections subject to fatigue loading shall be proportioned in accordance with the provisions of this Appendix. Fatigue, as used in this Specification, is defined as the damage that may result in fracture after a sufficient number of fluctuations of stress. Stress range is defined as the magnitude of these fluctuations. In the case of a stress reversal, the stress range shall be computed as the numerical sum of maximum repeated tensile and compressive stresses or the sum of maximum shearing stresses of opposite direction at a given point, resulting from differing arrangement of live load. 1.

Loading Conditions; Type and Location of Material In the design of members and connections subject to repeated variation of live load, consideration shall be given to the number of stress cycles, the expected range of stress, and the type and location of member or detail. Loading conditions shall be classified according to Table A-K3.1. The type and location of material shall be categorized according to Table A-K3.2.

2.

Design Stress Range The maximum range of stress at service loads shall not exceed the design stress range specified in Table A-K3.3.

3.

Design Strength of Bolts in Tension When subject to tensile fatigue loading, fully tensioned A325 or A490 bolts shall be designed for the combined tensile design strength due to combined external and prying forces in accordance with Table A-K3.4. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 136


[App. K

TABLE A-K3.2 Type and Location of Material General Condition

Situation

Kind of Stress [a]

Illustrative Stress Example Category Nos. (see Table (see Fig. A-K3.3) A-K3.1) [b]

Plain Material

Base metal with rolled or cleaned surface. Flame-cut edges with ANSI smoothness of 1,000 or less

T or Rev.

A

1,2

Built-up Members

Base metal and weld metal in members without attachments, built-up plates or shapes connected by continuous fullpenetration groove welds or by continuous fillet welds parallel to the direction of applied stress

T or Rev.

B

3,4,5,6

Base metal and weld metal in members without attachments, built-up plates, or shapes connected by full-penetration groove welds with backing bars not removed, or by partialpenetration groove welds parallel to the direction of applied stress

T or Rev.

B′

3,4,5,6

Base metal at toe of welds on girder webs or flanges adjacent to welded transverse stiffeners

T or Rev.

C

7

T or Rev. T or Rev.

E E′

5 5

E′

5

Base metal at ends of partial length welded coverplates narrower than the flange having square or tapered ends, with or without welds across the ends or wider than flange with welds across the ends Flange thickness ≤ 0.8 in. Flange thickness > 0.8 in. Base metal at end of partial length welded coverplates wider than the flange without welds across the ends

[a] “T” signifies range in tensile stress only; “Rev.” signifies a range involving reversal of tensile or compressive stress; “S” signifies range in shear, including shear stress reversal. [b] These examples are provided as guidelines and are not intended to exclude other reasonably similar situations. [c] Allowable fatigue stress range for transverse partial-penetration and transverse fillet welds is a function of the effective throat, depth of penetration, and plate thickness. See Frank and Fisher, Journal of the Structural Division, Vol. 105 No. ST9, Sept. 1979.


Sect. A-K3]

FATIGUE

6 - 137

TABLE A-K3.2 (cont’d) Type and Location of Material General Condition Groove Welds

Situation Base metal and weld metal at fullpenetration groove welded splices of parts of similar cross section ground flush, with grinding in the direction of applied stress and with weld soundness established by radiographic or ultrasonic inspection in accordance with the requirements of 9.25.2 or 9.25.3 of AWS D1.1

Kind of Stress [a]


T or Rev.

B

10,11

A514 base metal Other base metals

T or Rev. T or Rev.

B′ B

12,13 12,13

Base metal and weld metal at fullpenetration groove welded splices, with or without transitions having slopes no greater than 1 to 21⁄2 when reinforcement is not removed but weld soundness is established by radiographic or ultrasonic inspection in accordance with requirements of 9.25.2 or 9.25.3 of AWS D1.1

T or Rev.

C

10,11,12,13

Partial-Penetration Groove Welds

Weld metal of partial-penetration transverse groove welds, based on effective throat area of the weld or welds

T or Rev.

F [c]

Fillet-welded Connections

Base metal at intermittent fillet welds

T or Rev.

E

Base metal and weld metal at fullpenetration groove welded splices at transitions in width or thickness, with welds ground to provide slopes no steeper than 1 to 21⁄2 with grinding in the direction of applied stress, and with weld soundness established by radiographic or ultrasonic inspection in accordance with the requirements of 9.25.2 or 9.25.3 of AWS D1.1


6 - 138


[App. K

TABLE A-K3.2 (cont’d) Type and Location of Material General Condition Fillet-welded Connections (Continued)

Situation

Kind of Stress [a]


Base metal at junction of axially loaded members with fillet-welded end connections. Welds shall be disposed about the axis of the member so as to balance weld stresses

b ≤ 1 in. b > 1 in.

T or Rev. T or Rev.

E E′

17,18 17,18

T or Rev.

C See Note

20,21

S

F [c]

15,17,18 20,21

T or Rev.

E

27

S

F

27

Base metal at gross section of high-strength bolted slip-critical connections, except axially loaded joints which induce out-of-plane bending in connected material

T or Rev.

B

8

Base metal at net section of other mechanically fastened joints

T or Rev.

D

8,9

Base metal at net section of fully tensioned high-strength, boltedbearing connections

T or Rev.

B

8,9

Eyebar or Pin Plates

Base metal at net section of eyebar head or pin plate

T or Rev.

E

28,29

Attachments

Base metal at details attached by full-penetration groove welds subject to longitudinal and/or transverse loading when the detail embodies a transition radius R with the weld termination ground smooth and for transverse loading, the weld soundness established by radiographic or ultrasonic inspection in accordance with 9.25.2 or 9.25.3 of AWS D1.1

Base metal at members connected with transverse fillet welds

b ≤ 1⁄2-in. b > 1⁄2-in. Fillet Welds

Weld metal of continuous or intermittent longitudinal or transverse fillet welds

Plug or Slot Welds

Base metal at plug or slot welds Shear on plug or slot welds

Mechanically Fastened Connections


Sect. A-K3]

FATIGUE

6 - 139

TABLE A-K3.2 (cont’d) Type and Location of Material

General Condition Attachments (Continued)

Situation Longitudinal loading R > 24 in. 24 in. > R > 6 in. 6 in. > R > 2 in. 2 in. > R

Kind of Stress [a]


T or Rev. T or Rev. T or Rev. T or Rev.

B C D E

14 14 14 14


B C D E

14 14 14 14,15


C C D E

14 14 14 14,15

T or Rev. T or Rev.

D E

14 14,15

T or Rev.

E

14,15

T or Rev. T or Rev. T or Rev.

C D E

19 19 19

T or Rev. T or Rev. T or Rev.

D E E′

15 15 15

Detail base metal for transverse loading: equal thickness and reinforcement removed

R > 24 in. 24 in. > R > 6 in. 6 in. > R > 2 in. 2 in. > R Detail base metal for transverse loading: equal thickness and reinforcement not removed

R > 24 in. 24 in. > R > 6 in. 6 in. > R > 2 in. 2 in. > R Detail base metal for transverse loading: unequal thickness and reinforcement removed

R > 2 in. 2 in. > R Detail base metal for transverse loading: Unequal thickness and reinforcement not removed All R Detail base metal for transverse loading

R > 6 in. 6 in. > R > 2 in. 2 in. > R Base metal at detail attached by full-penetration groove welds subject to longitudinal loading 2 < a < 12b or 4 in. a > 12b or 4 in. when b ≤ 1 in. a >12b or 4 in. when b >1 in.


6 - 140


[App. K

TABLE A-K3.2 (cont’d) Type and Location of Material General Condition Attachments (Continued)

Situation

Kind of Stress [a]


Base metal at detail attached by fillet welds or partial-penetration groove welds subject to longitudinal loading

a < 2 in.

T or Rev.

C

15,23,24 25,26

2 in. < a < 12b or 4 in.

T or Rev.

D

15,23,24,26

a > 12b or 4 in. when b ≤ 1 in.

T or Rev.

E

15,23,24,26

a > 12b or 4 in. when b > 1 in.

T or Rev.

E′

15,23,24,26

T or Rev. T or Rev.

D E

19 19

R > 2 in. R < 2 in.

T or Rev. T or Rev.

D E

19 19

Base metal at stud-type shear connector attached by fillet weld or automatic end weld

T or Rev.

C

22

S

F

Base metal attached by fillet welds or partial-penetration groove welds subjected to longitudinal loading when the weld termination embodies a transition radius with the weld termination ground smooth

R > 2 in. R ≤ 2 in. Fillet-welded attachments where the weld termination embodies a transition radius, weld termination ground smooth, and main material subject to longitudinal loading Detail base metal for transverse loading:

Shear stress on nominal area of stud-type shear connectors


Sect. A-K3]

FATIGUE

6 - 141

TABLE A-K 3.3 Design Stress Range, ksi Category (From Table A-K3.2)

Loading Condition 1

Loading Condition 2

Loading Condition 3

Loading Condition 4

A

63

37

24

24

B

49

29

18

16

B′

39

23

15

12

C

35

21

13

10 [a]

D

28

16

10

7

E

22

13

8

4.5

E′

16

5.8

2.6

F

15

9

8

9.2 12

[a] Flexural stress range of 12 ksi permitted at toe of stiffener welds or flanges.

TABLE A-K3.4 Design Strength of A325 or A490 Bolts Subject to Tension Number of cycles

Design strength

Not more than 20,000

As specified in Section J3

From 20,000 to 500,000

0.30 AbFu [a]

More than 500,000

0.25 AbFu [a]

[a] At service loads.


6 - 142


1

[App. K

7

8 2

3

9

or

10 4

Plate as sh or wider own than flange B

E or E′ Category

5

11

12

6

13

Fig. A-K3.1. Illustrative examples.


continued

Sect. A-K3]

FATIGUE

6 - 143

Groove weld

14

R 22 b (avg.)

a a

15

23

a

b 16

24

a

b=thickness

17

25

b=thickness a

18

26

b

Groove or fillet weld

19 R 27 b=thickness

20

net section area

28 net section area

21

29

Fig. A-K3.1. Illustrative examples (cont.). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 144

NUMERICAL VALUES TABLE 1 Design Strength as a Function of Fy Design Stress (ksi)

Fy (ksi)

0.54Fy [a]

0.85Fy [b]

0.90Fy [c]

33

17.8

28.1

29.7

35

18.9

29.8

31.5

36

19.4

30.6

32.4

42

21.6

34.0

36.0

42

22.7

35.7

37.8

45

24.3

38.3

40.5

46

24.8

39.1

41.4

40

27.0

42.5

45.0

55

29.7

46.8

49.5

60

32.4

51.0

54.0

65

35.1

55.3

58.5

70

37.8

59.5

63.0

90

48.6

76.5

81.0

100

54.0

85.0

90.0

[a] See Section F2, Equations F2-1 [b] See Section E2, Equation E2-1 [c] See Section D1, Equation D1-1


NUMERICAL VALUES

6 - 145

TABLE 2 Design Strength as a Function of Fy

Shapes, Plates, Bars, Sheet and Tubing or Threaded parts

Item

Design Strength (ksi) Connection Part of Designated Steel

Bolt of Threaded Part of Designated Steel

ASTM Designation

Fy (ksi)

Fu (ksi)

Tension 0.75 × Fu [a]

Bearing 0.75 × 2.4Fu [b]

A36

36

58-80

43.5

104

32.6

17.4

21.8

A53

35

60

45.0

108

—

—

—

A242 A588  

50 42 40

70 63 60

52.5 47.3 45.0

126 113 108

39.4 35.4 33.8

21.0 18.9 18.0

26.3 23.2 22.5

A500

33/39 [f] 42/46 [f] 46/50 [f]

45 58 62

33.8 43.5 46.5

81 104 112

— — —

— — —

— — —

A501

36

58

43.5

104

—

—

—

A529

42

60-85

45.0

108

33.8

18.0

22.5

A570

40 42

55 58

41.3 43.5

99 104

— —

— —

— —

A572

42 50 60 65

60 65 75 80

45.0 48.8 56.3 60.0

108 117 135 144

33.8 36.6 42.2 45.0

18.0 19.5 22.5 24.0

22.5 24.4 28.1 30.0

A514

100 90

110-130 100-130

82.5 75.0

198 180

61.9 56.3

33.0 30.0

41.3 37.5

A606

45 50

65 70

48.8 52.5

117 126

— —

— —

— —

A607

45 50 55 60 65 70

60 65 70 75 80 85

45.0 48.8 52.5 56.3 60.0 63.8

108 117 126 135 144 153

— — — — — —

— — — — — —

— — — — — —

A618

50 50

70 65

52.5 48.8

126 117

— —

— —

— —

Tension Shear Shear 0.75 × 0.75 × 0.75 × 0.75Fu [c] 0.40Fu [d] 0.50Fu [e]


6 - 146

NUMERICAL VALUES

TABLE 2 (cont’d) Design Strength as a Function of Fu Design Strength (ksi) Item

Connection Part of Designated Steel

Bolts

ASTM Designation A449

Fy (ksi)

Fu (ksi)

Tension 0.75 × Fu [a]

Bearing 0.75 × 2.4Fu [b]

92 81 58

120 105 90

— — —

— — —

Bolt of Threaded Part of Designated Steel Tension Shear Shear 0.75 × 0.75 × 0.75 × 0.75Fu [c] 0.40Fu [d] 0.50Fu [e] 67.5 59.1 50.6

36.0 31.5 27.0

[a] On effective net area, see Sections D1, J5.2. [b] Produced by fastener in shear, see Section J3.10. Note that smaller maximum design bearing stresses, as a function of hole type spacing, are given. [c] On nominal body area, see Table J3.2. [d] Threads not excluded from shear plane, see Table J3.2. [e] Threads excluded from shear plane, see Table J3.2. [f] Smaller value for circular shapes, larger for square or rectangular shapes. Note: For dimensional and size limitations, see the appropriate ASTM Specification.


45.0 39.4 33.8

NUMERICAL VALUES

6 - 147

TABLE 3-36 Design Stress for Compression Members of 36 ksi Specified Yield Stress Steel, φc = 0.85[a] Kl r

φcFcr ksi

Kl r

φcFcr ksi

Kl r

φcFcr ksi

Kl r

φcFcr ksi

Kl r

φcFcr ksi

1 2 3 4 5

30.60 30.59 30.59 30.57 30.56

41 42 43 44 45

28.01 27.89 27.76 27.64 27.51

81 82 83 84 85

21.66 21.48 21.29 21.11 20.92

121 122 123 124 125

14.16 13.98 13.80 13.62 13.44

161 162 163 164 165

8.23 8.13 8.03 7.93 7.84

6 7 8 9 10

30.54 30.52 30.50 30.47 30.44

46 47 48 49 50

27.37 27.24 27.11 26.97 26.83

86 87 88 89 90

20.73 20.54 20.36 20.17 19.98

126 127 128 129 130

13.27 13.09 12.92 12.74 12.57

166 167 168 169 170

7.74 7.65 7.56 7.47 7.38

11 12 13 14 15

30.41 30.37 30.33 30.29 30.24

51 52 53 54 55

26.68 26.54 26.39 26.25 26.10

91 92 93 94 95

19.79 19.60 19.41 19.22 19.03

131 132 133 134 135

12.40 12.23 12.06 11.88 11.71

171 172 173 174 175

7.30 7.21 7.13 7.05 6.97

16 17 18 19 20

30.19 30.14 30.08 30.02 29.96

56 57 58 59 60

25.94 25.79 25.63 25.48 25.32

96 97 98 99 100

18.84 18.65 18.46 18.27 18.08

136 137 138 139 140

11.54 11.37 11.20 11.04 10.89

176 177 178 179 180

6.89 6.81 6.73 6.66 6.59

21 22 23 24 25

29.90 29.83 29.76 26.69 29.61

61 62 63 64 65

25.16 24.99 24.83 24.67 24.50

101 102 103 104 105

17.89 17.70 17.51 17.32 17.13

141 142 143 144 145

10.73 10.58 10.43 10.29 10.15

181 182 183 184 185

6.51 6.44 6.37 6.30 6.23

26 27 28 29 30

29.53 29.45 29.36 29.28 29.18

66 67 68 69 70

24.33 24.16 23.99 23.82 23.64

106 107 108 109 110

16.94 16.75 16.56 16.37 16.19

146 147 148 149 150

10.01 9.87 9.74 9.61 9.48

186 187 188 189 190

6.17 6.10 6.04 5.97 5.91

31 32 33 34 35

29.09 28.99 28.90 28.79 28.69

71 72 73 74 75

23.47 23.29 23.12 22.94 22.76

111 112 113 114 115

16.00 15.81 15.63 15.44 15.26

151 152 153 154 155

9.36 9.23 9.11 9.00 8.88

191 192 193 194 195

5.85 5.79 5.73 5.67 5.61

36 37 38 39 40

28.58 28.47 28.36 28.25 28.13

76 77 78 79 80

22.58 22.40 22.22 22.03 21.85

116 117 118 119 120

15.07 14.89 14.70 14.52 14.34

156 157 158 159 160

8.77 8.66 8.55 8.44 8.33

196 197 198 199 200

5.55 5.50 5.44 5.39 5.33

[a] When element width-to-thickness ration exceeds λr , see Appendix B5.3.


6 - 148

NUMERICAL VALUES

TABLE 3-50 Design Stress for Compression Members of 50 ksi Specified Yield Stress Steel, φc = 0.85[a] Kl r

φcFcr ksi

Kl r

φcFcr ksi

Kl r

φcFcr ksi

Kl r

φcFcr ksi

Kl r

φcFcr ksi

1 2 3 4 5

42.50 42.49 42.47 42.45 42.42

41 42 43 44 45

37.59 37.36 37.13 36.89 36.65

81 82 83 84 85

26.31 26.00 25.68 25.37 25.06

121 122 123 124 125

14.57 14.33 14.10 13.88 13.66

161 162 163 164 165

8.23 8.13 8.03 7.93 7.84

6 7 8 9 10

42.39 42.35 42.30 42.25 42.19

46 47 48 49 50

36.41 36.16 35.91 35.66 35.40

86 87 88 89 90

24.75 24.44 24.13 23.82 23.51

126 127 128 129 130

13.44 13.23 13.02 12.82 12.62

166 167 168 169 170

7.74 7.65 7.56 7.47 7.38

11 12 13 14 15

42.13 42.05 41.98 41.90 41.81

51 52 53 54 55

35.14 34.88 34.61 34.34 34.07

91 92 93 94 95

23.20 22.89 22.58 22.28 21.97

131 132 133 134 135

12.43 12.25 12.06 11.88 11.71

171 172 173 174 175

7.30 7.21 7.13 7.05 6.97

16 17 18 19 20

41.71 41.61 41.51 41.39 41.28

56 57 58 59 60

33.79 33.51 33.23 32.95 32.67

96 97 98 99 100

21.67 21.36 21.06 20.76 20.46

136 137 138 139 140

11.54 11.37 11.20 11.04 10.89

176 177 178 179 180

6.89 6.81 6.73 6.66 6.59

21 22 23 24 25

41.15 41.02 40.89 40.75 40.60

61 62 63 64 65

32.38 32.09 31.80 31.50 31.21

101 102 103 104 105

20.16 19.86 19.57 19.28 18.98

141 142 143 144 145

10.73 10.58 10.43 10.29 10.15

181 182 183 184 185

6.51 6.44 6.37 6.30 6.23

26 27 28 29 30

40.45 40.29 40.13 39.97 39.79

66 67 68 69 70

30.91 30.61 30.31 30.01 29.70

106 107 108 109 110

18.69 18.40 18.12 17.83 17.55

146 147 148 149 150

10.01 9.87 9.74 9.61 9.48

186 187 188 189 190

6.17 6.10 6.04 5.97 5.91

31 32 33 34 35

39.62 39.43 39.25 39.06 38.86

71 72 73 74 75

29.40 20.09 28.79 28.48 28.17

111 112 113 114 115

17.27 16.99 16.71 16.42 16.13

151 152 153 154 155

9.36 9.23 9.11 9.00 8.88

191 192 193 194 195

5.85 5.79 5.73 5.67 5.61

36 37 38 39 40

38.66 38.45 38.24 38.03 37.81

76 77 78 79 80

27.86 27.55 27.24 26.93 26.62

116 117 118 119 120

15.86 15.59 15.32 15.07 14.82

156 157 158 159 160

8.77 8.66 8.55 8.44 8.33

196 197 198 199 200

5.55 5.50 5.44 5.39 5.33

[a] When element width-to-thickness ratio exceeds λr , see Appendix B5.3.


NUMERICAL VALUES

6 - 149

TABLE 4 Values of φcFcr / Fy , φc = 0.85 for Determining Design Stress for Compression Members for Steel of Any Yield Stress[a] λc

φcFcr / Fy

λc

φcFcr / Fy

λc

φcFcr / Fy

λc

φcFcr / Fy

0.02 0.04 0.06 0.08 0.10

0.850 0.849 0.849 0.848 0.846

0.82 0.84 0.86 0.88 0.90

0.641 0.632 0.623 0.614 0.605

1.62 1.64 1.66 1.68 1.70

0.284 0.277 0.271 0.264 0.258

2.42 2.44 2.46 2.48 2.50

0.127 0.125 0.123 0.121 0.119

0.12 0.14 0.16 0.18 0.20

0.845 0.843 0.841 0.839 0.836

0.92 0.94 0.96 0.98 1.00

0.596 0.587 0.578 0.568 0.559

1.72 1.74 1.76 1.78 1.80

0.252 0.246 0.241 0.235 0.230

2.52 2.54 2.56 2.58 2.60

0.117 0.116 0.114 0.112 0.110

0.22 0.24 0.26 0.28 0.30

0.833 0.830 0.826 0.823 0.819

1.02 1.04 1.06 1.08 1.10

0.550 0.540 0.531 0.521 0.512

1.82 1.84 1.86 1.88 1.90

0.225 0.220 0.215 0.211 0.206

2.62 2.64 2.66 2.68 2.70

0.109 0.107 0.105 0.104 0.102

0.32 0.34 0.36 0.38 0.40

0.814 0.810 0.805 0.800 0.795

1.12 1.14 1.16 1.18 1.20

0.503 0.493 0.484 0.474 0.465

1.92 1.94 1.96 1.98 2.00

0.202 0.198 0.194 0.190 0.186

2.72 2.74 2.76 2.78 2.80

0.101 0.099 0.098 0.096 0.095

0.42 0.44 0.46 0.48 0.50

0.789 0.784 0.778 0.772 0.765

1.22 1.24 1.26 1.28 1.30

0.456 0.446 0.437 0.428 0.419

2.02 2.04 2.06 2.08 2.10

0.183 0.179 0.176 0.172 0.169

2.82 2.84 2.86 2.88 2.90

0.094 0.092 0.091 0.090 0.089

0.52 0.54 0.56 0.58 0.60

0.759 0.752 0.745 0.738 0.731

1.32 1.34 1.36 1.38 1.40

0.410 0.401 0.392 0.383 0.374

2.12 2.14 2.16 2.18 2.20

0.166 0.163 0.160 0.157 0.154

2.92 2.94 2.96 2.98 3.00

0.087 0.086 0.085 0.084 0.083

0.62 0.64 0.66 0.68 0.70

0.724 0.716 0.708 0.700 0.692

1.42 1.44 1.46 1.48 1.50

0.365 0.357 0.348 0.339 0.331

2.22 2.24 2.26 2.28 2.30

0.151 0.149 0.146 0.143 0.141

3.02 3.04 3.06 3.08 3.10

0.082 0.081 0.080 0.079 0.078

0.72 0.74 0.76 0.78 0.80

0.684 0.676 0.667 0.659 0.650

1.52 1.54 1.56 1.58 1.60

0.323 0.314 0.306 0.299 0.291

2.32 2.34 2.36 2.38 2.40

0.138 0.136 0.134 0.132 0.129

3.12 3.14 3.16 3.18 3.20

0.077 0.076 0.075 0.074 0.073

[a] When element width-to-thickness ratios exceed λr , see Appendix B5.3. Values of λc > 2.24 exceed Kl / r of 200 for Fy = 36 Values of λc > 2.64 exceed Kl / r of 200 for Fy = 50


6 - 150

NUMERICAL VALUES

TABLE 5 Values of Kl / r for Fy = 36 and 50 ksi Kl / r

Kl / r

λc

Fy = 36

Fy = 50

λc

0.02 0.04 0.06 0.08 0.10

1.8 3.6 4.3 7.1 8.9

1.5 3.0 4.5 6.1 7.6

0.12 0.14 0.16 0.18 0.20

10.7 12.5 14.3 16.0 17.8

0.22 0.24 0.26 0.28 0.30

Fy = 36

Fy = 50

0.82 0.84 0.86 0.88 0.90

73.1 74.9 76.7 78.5 80.2

62.0 63.6 65.1 66.6 68.1

9.1 10.6 12.1 13.6 15.1

0.92 0.94 0.96 0.98 1.00

82.0 83.8 85.6 87.4 89.2

69.6 71.1 72.6 74.1 75.7

19.6 21.4 23.2 25.0 26.7

16.6 18.2 19.7 21.2 22.7

1.02 1.04 1.06 1.08 1.10

90.9 92.7 94.5 96.3 98.1

77.2 78.7 80.2 81.7 83.2

0.32 0.34 0.36 0.38 0.40

28.5 30.3 32.1 33.9 35.7

24.2 25.7 27.2 28.8 30.3

1.12 1.14 1.16 1.18 1.20

99.9 101.6 103.4 105.2 107.0

84.7 86.3 87.8 89.3 90.8

0.42 0.44 0.46 0.48 0.50

37.4 39.2 41.0 42.8 44.6

31.8 33.3 34.8 36.3 37.8

1.22 1.24 1.26 1.28 1.30

108.8 110.6 112.3 114.1 115.9

92.3 93.8 95.3 96.8 98.4

0.52 0.54 0.56 0.58 0.60

46.4 48.1 49.9 51.7 53.5

39.3 40.9 42.4 43.9 45.4

1.32 1.34 1.36 1.38 1.40

117.7 119.5 121.3 123.0 124.8

99.9 101.4 102.9 104.4 105.9

0.62 0.64 0.66 0.68 0.70

55.3 57.1 58.8 60.6 62.4

46.9 48.4 49.9 51.4 53.0

1.42 1.44 1.46 1.48 1.50

126.6 128.4 130.2 132.0 133.7

107.4 108.9 110.5 112.0 113.5

0.72 0.74 0.76 0.78 0.80

64.2 66.0 67.8 69.5 71.3

54.5 56.0 57.5 59.0 60.5

1.52 1.54 1.56 1.58 1.60

135.5 137.3 139.1 140.9 142.7

115.0 116.5 118.0 119.5 121.1


NUMERICAL VALUES

6 - 151

TABLE 5 (cont’d) Values of Kl / r for Fy = 36 and 50 ksi Kl / r

Kl / r

λc

Fy = 36

Fy = 50

λc

Fy = 50

1.62 1.64 1.66 1.68 1.70

144.4 146.2 148.0 149.8 151.6

122.6 124.1 125.6 127.1 128.6

2.42 2.44 2.46 2.48 2.50

183.1 184.6 186.1 187.6 189.1

1.72 1.74 1.76 1.78 1.80

153.4 155.1 156.9 158.7 160.5

130.1 131.6 133.2 134.7 136.2

2.52 2.54 2.56 2.58 2.60

190.7 192.2 193.7 195.2 196.7

1.82 1.84 1.86 1.88 1.90

162.3 164.1 165.8 167.6 169.4

137.7 139.2 140.7 142.2 143.8

2.62 2.64

198.2 199.7

1.92 1.94 1.96 1.98 2.00

171.2 173.0 174.8 176.5 178.3

145.3 146.8 148.3 149.8 151.3

2.02 2.04 2.06 2.08 2.10

180.1 181.9 183.7 185.5 187.2

152.8 154.3 155.9 157.4 158.9

2.12 2.14 2.16 2.18 2.20

189.0 190.8 192.6 194.4 196.2

160.4 161.9 163.4 164.9 166.5

2.22 2.24 2.26 2.28 2.30

197.9 199.7

168.0 169.5 171.0 172.5 174.0

2.32 2.34 2.36 2.38 2.40

175.5 177.0 178.6 180.1 181.6

Heavy line indicates Kl / r of 200.


6 - 152

NUMERICAL VALUES

TABLE 6 Slenderness Ratios of Elements as a Function of Fy From Table B5.1 Fy (ksi) Ratio

36

42

65 / √ Fy

10.8

10.0

√Fy 76 / 

12.7

√Fy 95 / 

15.8

Fy 127 / √

21.2

19.6

46

50

60 8.4

65

9.6

9.2

8.1

11.7

11.2

10.7

9.8

9.4

14.7

14.0

13.4

12.3

11.8

18.7

18.0

16.4

15.8

 Fy − 10  141 / √

27.7

24.9

23.5

22.3

19.9

19.0

Fy 190 / √

31.7

29.3

28.0

26.9

24.5

23.6

238 / √ Fy

39.7

36.7

35.1

33.7

30.7

29.5

253 / √ Fy

42.2

39.0

37.3

35.8

32.7

31.4

Fy 317 / √

52.8

48.9

46.7

44.8

40.9

39.3

640 / √ Fy

107.0

98.8

94.4

90.5

82.6

79.4

Fy 970 / √

162.0

150.0

143.0

137.0

125.0

120.0

1,300 / Fy

36.1

31.0

28.3

26.0

21.7

20.0

2,070 / Fy

57.5

49.3

45.0

41.4

34.5

31.8

3,300 / Fy

91.7

78.6

71.7

66.0

55.0

50.8

8,970 / Fy

249.0

214.0

195.0

179.0

150.0

138.0


NUMERICAL VALUES

6 - 153

TABLE 7 Values of Cm for Use in Section C1 M1 M2

Cm

M1 M2

Cm

M1 M2

Cm

−1.00

1.00

−0.45

0.78

0.10

0.56

−0.95

0.98

−0.40

0.76

0.15

0.54

−0.90

0.96

−0.35

0.74

0.20

0.52

−0.85

0.94

−0.30

0.72

0.25

0.50

−0.80

0.92

−0.25

0.70

0.30

0.48

−0.75

0.90

−0.20

0.68

0.35

0.46

−0.70

0.88

−0.15

0.66

0.40

0.44

−0.65

0.86

−0.10

0.64

0.45

0.42

−0.60

0.84

−0.05

0.62

0.50

0.40

0.60

0.36

−0.55

0.82

0

0.60

0.80

0.28

−0.50

0.80

0.05

0.58

1.00

0.20

Note 1: Cm = 0.6 − 0.4(M1 / M2). Note 2: M1 / M2 is positive for reverse curvature and negative for single curvature. |M1| ≤ |M2|


6 - 154

NUMERICAL VALUES

TABLE 8 Values of Pe / Ag for Use in Section C1 for Steel of Any Yield Stress Kl r

Pe / Ag (ksi)

Kl r

Pe / Ag (ksi)

Kl r

Pe / Ag (ksi)

Kl r

Pe / Ag (ksi)

Kl r

Pe / Ag (ksi)

Kl r

Pe / Ag (ksi)

21 22 23 24 25

649.02 591.36 541.06 496.91 457.95

51 52 53 54 55

110.04 105.85 101.89 98.15 94.62

81 82 83 84 85

43.62 42.57 41.55 40.56 39.62

111 112 113 114 115

23.23 22.82 22.42 22.02 21.64

141 142 143 144 145

14.40 14.19 14.00 13.80 13.61

171 172 173 174 175

9.79 9.67 9.56 9.45 9.35

26 27 28 29 30

423.40 392.62 365.07 340.33 318.02

56 57 58 59 60

91.27 88.09 85.08 82.22 79.51

86 87 88 89 90

38.70 37.81 36.96 36.13 35.34

116 117 118 119 120

21.27 20.91 20.56 20.21 19.88

146 147 148 149 150

13.43 13.25 13.07 12.89 12.72

176 177 178 179 180

9.24 9.14 9.03 8.93 8.83

31 32 33 34 35

297.83 279.51 262.83 247.59 233.65

61 62 63 64 65

76.92 74.46 72.11 69.88 67.74

91 92 93 94 95

34.56 33.82 33.09 32.39 31.71

121 122 123 124 125

19.55 19.23 18.92 18.61 18.32

151 152 153 154 155

12.55 12.39 12.23 12.07 11.91

181 182 183 184 185

8.74 8.64 8.55 8.45 8.36

36 37 38 39 40

220.85 209.07 198.21 188.18 178.89

66 67 68 69 70

65.71 63.76 61.90 60.12 58.41

96 97 98 99 100

31.06 30.42 29.80 29.20 28.62

126 127 128 129 130

18.03 17.75 17.47 17.20 16.94

156 157 158 159 160

11.76 11.61 11.47 11.32 11.18

186 187 188 189 190

8.27 8.18 8.10 8.01 7.93

41 42 43 44 45

170.27 162.26 154.80 147.84 141.34

71 72 73 74 75

56.78 55.21 53.71 52.57 50.88

101 102 103 104 105

28.06 27.51 26.98 26.46 25.96

131 132 133 134 135

16.68 16.43 16.18 15.94 15.70

161 162 163 164 165

11.04 10.91 10.77 10.64 10.51

191 192 193 194 195

7.85 7.76 7.68 7.60 7.53

46 47 48 49 50

135.26 129.57 124.23 119.21 114.49

76 77 78 79 80

49.55 48.27 47.04 45.86 44.72

106 107 108 109 110

25.47 25.00 24.54 24.09 23.65

136 137 138 139 140

15.47 15.25 15.03 14.81 14.60

166 167 168 169 170

10.39 10.26 10.14 10.02 9.90

196 197 198 199 200

7.45 7.38 7.30 7.23 7.16

Note: Pe / Ag =

π2E

(Kl / r)2

, use for both Pe1 and Pe2.


NUMERICAL VALUES

6 - 155

TABLE 9-36 φvVn (ksi) for Plate Girders by Appendix F2 Aw for 36 ksi Yield Stress Steel, Tension Field Action Not Included Aspect ratio a / h: Stiffener Spacing to Web Depth

h tw

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

2.5

3.0

Over 3.0

60 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 70 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 80 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 18.9 18.2 17.9 16.9 90 19.4 19.4 19.4 19.4 19.4 19.4 19.4 18.5 17.8 17.2 16.8 16.2 15.9 14.7 100 19.4 19.4 19.4 19.4 19.4 19.2 17.6 16.6 16.0 15.5 14.9 13.8 13.2 11.9 110

19.4 19.4 19.4 19.4 18.4 17.4 16.0 14.8 13.7 12.8 12.3 11.4 10.9

9.8

120 19.4 19.4 19.4 18.1 16.9 16.0 14.0 12.5 11.5 10.8 10.3

9.6

9.2

8.3

130 19.4 19.4 18.2 16.7 15.6 14.1 11.9 10.6

9.8

9.2

8.8

8.2

7.8

7.0

140 19.4 18.8 16.9 15.5 13.5 12.1 10.3

9.2

8.4

7.9

7.6

7.0

6.7

6.1

150 19.4 17.6 15.7 13.5 11.8 10.6

8.9

8.0

7.3

6.9

6.6

6.1

5.9

5.3

160 18.9 16.5 14.1 11.9 10.4

9.3

7.9

7.0

6.5

6.1

5.8

5.4

170 17.8 15.5 12.5 10.5

9.2

8.2

7.0

6.2

5.7

5.4

5.1

4.1

180 16.8 13.9 11.1

9.4

8.2

7.3

6.2

5.5

5.1

4.8

4.6

3.7

200 14.9 11.2

9.0

7.6

6.6

5.9

5.0

4.5

4.1

220 12.3

9.3

7.5

6.3

5.5

4.9

4.2

240 10.3

7.8

6.3

5.3

4.6

4.1

2.1

260

8.8

6.6

5.3

4.5

3.9

3.5

1.8

280

7.6

5.7

4.6

3.9

300

6.6

5.0

4.0

320

5.8

4.4


4.6

3.0 2.5

6 - 156

NUMERICAL VALUES

TABLE 9-50 φvVn (ksi) for Plate Girders by Appendix F2 Aw for 50 ksi Yield Stress Steel, Tension Field Action Not Included Aspect ratio a / h: Stiffener Spacing to Web Depth

h tw

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

2.5

3.0

Over 3.0

60 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 26.6 70 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 26.9 26.1 25.5 24.6 24.0 22.8 80 27.0 27.0 27.0 27.0 27.0 27.0 26.0 24.5 23.5 22.8 22.3 21.5 20.6 18.6 90 27.0 27.0 27.0 27.0 26.5 25.1 23.1 21.8 20.4 19.2 18.3 17.0 16.3 14.7 100 27.0 27.0 27.0 25.6 23.9 22.6 20.1 17.9 16.5 15.5 14.9 13.8 13.2 11.9 110

27.0 27.0 25.3 23.2 21.7 19.6 16.6 14.8 13.7 12.8 12.3 11.4 10.9

9.8

120 27.0 25.9 23.2 21.1 18.4 16.5 14.0 12.5 11.5 10.8 10.3

9.6

9.2

8.3

130 27.0 23.9 21.4 18.0 15.7 14.1 11.9 10.6

9.8

9.2

8.8

8.2

7.8

7.0

140 25.5 22.2 18.4 15.5 13.5 12.1 10.3

9.2

8.4

7.9

7.6

7.0

6.7

6.1

150 23.8 19.9 16.1 13.5 11.8 10.6

8.9

8.0

7.3

6.9

6.6

6.1

5.9

5.3

160 22.3 17.5 14.1 11.9 10.4

9.3

7.9

7.0

6.5

6.1

5.8

5.4

170 20.6 15.5 12.5 10.5

9.2

8.2

7.0

6.2

5.7

5.4

5.1

4.1

180 18.3 13.9 11.1

9.4

8.2

7.3

6.2

5.5

5.1

4.8

4.6

3.7

200 14.9 11.2

9.0

7.6

6.6

5.9

5.0

4.5

4.1

220 12.3

9.3

7.5

6.3

5.5

4.9

4.2

240 10.3

7.8

6.3

5.3

4.6

4.1

2.1

260

8.8

6.6

5.3

4.5

3.9

3.5

1.8

280

7.6

5.7

4.6

3.9


4.6

3.0 2.5

NUMERICAL VALUES

6 - 157

TABLE 10-36 φvVn (ksi) for Plate Girders by Appendix G Aw for 36 ksi Yield Stress Steel, Tension Field Action Included[b] (Italic values indicate gross area, as percent of (h × tw) required for pairs of intermediate stiffeners of 36 ksi yield stress steel with Vu / φVn = 1.0) [a] Aspect ratio a / h: Stiffener Spacing to Web Depth

h tw

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

2.5

3.0

Over 3.0 [c]

60 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 70 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 80 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.1 18.6 18.3 16.9 90 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.0 18.5 18.2 17.8 17.3 16.8 14.7 100 19.4 19.4 19.4 19.4 19.4 19.3 18.6 18.1 17.6 17.2 16.6 15.6 14.9 11.9 110

19.4 19.4 19.4 19.4 19.1 18.7 17.9 17.2 16.3 15.6 15.1 14.0 13.3

9.8

120 19.4 19.4 19.4 19.0 18.5 18.1 17.0 16.0 15.1 14.4 13.9 12.8 12.0

8.3

130 19.4 19.4 19.1 18.6 18.1 17.4 16.1 15.1 14.2 13.5 12.9 11.8 11.0

7.0

140 19.4 19.3 18.7 18.2 17.4 16.6 15.4 14.4 13.5 12.8 12.2 11.0 10.2

6.1

150 19.4 19.0 18.4 17.5 16.7 16.0 14.8 13.8 12.9 12.2 11.6 10.4

5.3

160 19.3 18.7 17.9 17.0 16.2 15.5 14.3 13.3 12.4 11.7 11.1

9.9

9.6

4.6

170 19.1 18.4 17.4 16.6 15.8 15.1 13.9 12.9 12.0 11.3 10.7 0.3 0.4

4.1

180 18.9 18.0 17.1 16.2 15.5 14.8 13.6 12.6 11.7 11.0 10.4 0.2 0.7 1.1 1.3 1.5

3.7

200 18.4 17.3 16.4 15.6 14.9 14.2 13.1 12.0 11.2 0.1 0.9 1.4 2.1 2.5 2.8

3.0

220 17.8 16.9 16.0 15.2 14.5 13.8 12.7 1.1 2.0 2.6 3.0 3.6

2.5

240 17.4 16.5 15.7 14.9 14.2 13.5 1.5 2.7 3.4 3.9 4.3

2.1

260 17.1 16.2 15.4 14.6 14.0 13.3 1.3 3.0 4.0 4.6 5.0 5.4

1.8

280 16.8 16.0 15.2 14.4 2.7 4.2 5.0 5.6


6 - 158

NUMERICAL VALUES

TABLE 10-36 (cont’d) φvVn (ksi) for Plate Girders by Appendix G Aw for 36 ksi Yield Stress Steel, Tension Field Action Included[b] (Italic values indicate gross area, as percent of (h x tw) required for pairs of intermediate stiffeners of 36 ksi yield stress steel with Vu / φVn = 1.0)[a] Aspect ratio a / h: Stiffener Spacing to Web Depth

h tw

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

2.5

3.0

Over 3.0 [c]

300 16.6 15.8 15.0 3.9 5.2 5.9 320 16.4 15.6 4.9 6.0 [a] For area of single-angle and single-plate stiffeners, or when Vu / φVn < 1.0, see Equation A-G4-1. [b] For end-panels and all panels in hybrid and web-tapered plate girders, use Table 9-36. [c] Same as for Table 9-36. Note: Girders so proportioned that the computed shear is less than that given in right-hand column do not require intermediate stiffeners.


NUMERICAL VALUES

6 - 159

TABLE 10-50 φvVn (ksi) for Plate Girders by Appendix G Aw for 50 ksi Yield Stress Steel, Tension Field Action Included[b] (Italic values indicate gross area, as percent of (h × tw) required for pairs of intermediate stiffeners of 50 ksi yield stress steel with Vu / φVn = 1.0)[a] Aspect ratio a / h: Stiffener Spacing to Web Depth

h tw

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

2.5

3.0

Over 3.0 [c]

60 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 26.6 70 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 26.9 26.5 26.1 25.4 24.9 22.8 80 27.0 27.0 27.0 27.0 27.0 27.0 26.5 25.8 25.1 24.6 24.1 23.3 22.4 18.6 90 27.0 27.0 27.0 27.0 26.8 26.3 25.3 24.4 23.4 22.5 21.7 20.2 19.2 14.7 100 27.0 27.0 27.0 26.5 25.9 25.3 24.0 22.5 21.4 20.4 19.6 18.0 17.0 11.9 110

27.0 27.0 26.5 25.8 25.1 24.2 22.4 21.0 19.8 18.8 18.0 16.4 15.3

9.8

120 27.0 26.7 25.9 25.1 24.0 23.0 21.2 19.8 18.6 17.6 16.8 15.2 14.1

8.3

130 27.0 26.2 25.4 24.1 23.0 22.0 20.3 18.9 17.7 16.7 15.9 14.2 13.1

7.0

140 26.7 25.8 24.5 23.3 22.2 21.3 19.6 18.2 17.0 16.0 15.1 13.5 12.3

6.1

150 26.3 25.2 23.9 22.7 21.6 20.7 19.0 17.6 16.4 15.4 14.5 12.9 11.7

5.3

160 26.0 24.6 23.3 22.2 21.1 20.2 18.5 17.1 15.9 14.9 14.0 12.4 0.2 0.4 0.5 0.8

4.6

170 25.6 24.1 22.8 21.7 20.7 19.8 18.1 16.7 15.2 14.5 13.6 0.5 1.0 1.2 1.4 1.6

4.1

180 25.1 23.7 22.4 21.3 20.3 19.4 17.8 16.4 15.2 14.2 13.3 0.4 0.9 1.5 1.9 2.2 2.3 2.5

3.7

200 24.3 23.0 21.8 20.8 19.8 18.9 17.3 15.9 14.7 1.0 1.8 2.3 2.7 3.2 3.5 3.7

3.0

220 23.7 22.5 21.4 20.4 19.4 18.5 16.9 1.7 2.7 3.3 3.8 4.1 4.5

2.5

240 23.2 22.1 21.0 20.0 19.1 18.2 1.8 3.2 4.0 4.6 4.9 5.2

2.1

260 23.0 21.8 20.8 19.8 18.8 18.0 3.2 4.4 5.1 5.6 5.9 6.1 280 22.7 21.6 20.6 19.6 4.4 5.4 6.0 6.4 [a] For area of single-angle and single-plate stiffeners, or when Vu / φVn < 1.0, see Equation A-G4-1. [b] For end-panels and all panels in hybrid and web-tapered plate girders, use Table 9-50. [c] Same as for Table 9-50. Note: Girders so proportioned that the computed shear is less than that given in right-hand column do not require intermediate stiffeners.


6 - 160

NUMERICAL VALUES

TABLE 11 Nominal Horizontal Shear Load for One Connector Qn, kips[a] From Equations I5-1 and I5-2 Specified Compressive Strength of Concrete, fc′, ksi [d] Connector [b]

3.0

3.5

4.0

dia. × 2-in. hooked or headed stud

9.4

10.5

11.6

dia. × 21⁄2-in. hooked or headed stud

14.6

16.4

18.1

dia. × 3-in. hooked or headed stud

21.0

23.6

26.1

dia. × 31⁄2-in. hooked or headed stud

28.6

32.1

35.5

Channel C3 × 4.1

10.2 Lc [c]

11.5 Lc [c]

12.7 Lc [c]

Channel C4 × 5.4

11.1 Lc [c]

12.4 Lc [c]

13.8 Lc [c]

Channel C5 × 6.7

11.9 Lc [c]

13.3 Lc [c]

14.7 Lc [c]

1⁄ -in. 2 5⁄ -in. 8

3⁄ -in. 4 7⁄ -in. 8

[a] Applicable only to concrete made with ASTM C33 aggregates. [b] The nominal horizontal loads tabulated may also be used for studs longer than shown. [c] Lc = length of channel, inches. [d] Fu > 0.5(fc′w)0.75, w = 145 lbs./cu. ft.


6 - 161

COMMENTARY on the Load and Resistance Factor Design Specification for Structural Steel Buildings December 1, 1993

INTRODUCTION The Specification is intended to be complete for normal design usage. The Commentary furnishes background information and references for the benefit of the engineer seeking further understanding of the derivations and limits of the specification. The Specification and Commentary are intended for use by design professionals with demonstrated engineering competence.


6 - 162

CHAPTER A GENERALPROVISIONS

A1. SCOPE Load and Resistance Factor Design (LRFD) is an improved approach to the design of structural steel for buildings. It involves explicit consideration of limit states, multiple load factors, and resistance factors, and implicit probabilistic determination of reliability. The designation LRFD reflects the concept of factoring both loads and resistance. This type of factoring differs from the AISC allowable stress design (ASD) Specification (AISC, 1989), where only the resistance is divided by a factor of safety (to obtain allowable stress) and from the plastic design portion of that Specification, where only the loads are multiplied by a common load factor. The LRFD method was devised to offer the designer greater flexibility, more rationality, and possible overall economy. The format of using resistance factors and multiple load factors is not new, as several such design codes are in effect [the ACI-318 Strength Design for Reinforced Concrete (ACI, 1989) and the AASHTO Load Factor Design for Bridges (AASHTO, 1989)]. Nor should the new LRFD method give designs radically different from the older methods, since it was tuned, or “calibrated,” to typical representative designs of the earlier methods. The principal new ingredient is the use of a probabilistic mathematical model in the development of the load and resistance factors, which made it possible to give proper weight to the accuracy with which the various loads and resistances can be determined. Also, it provides a rational methodology for transference of test results into design provisions. A more rational design procedure leading to more uniform reliability is the practical result. A2. LIMITS OF APPLICABILITY 2.

Types of Construction The provisions for these types of construction have been revised to provide for a truer recognition of the actual degree of connection restraint in the structural design. All connections provide some restraint. Depending on the amount of restraint offered, connections are classified as either Type FR or PR. This classification renames the Type I connection of the AISC ASD Specification to Type FR and includes both Type II and Type III of that Specification under a new, more general classification of Type PR. Just as in the allowable stress design (ASD) provisions, construction utilizing Type FR connections may be designed in LRFD using either elastic or plastic analysis provided the appropriate Specification provisions are satisfied. For Type PR construction which uses the “simple framing” approach, the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A3]

MATERIAL

6 - 163

restraint of the connection is ignored, provided the given conditions are met. This is no change from the ASD provisions. Where there is evidence of the actual moment rotation capability of a given type of connection, the use of designs incorporating the connection restraint is permitted just as in ASD. The designer should, when incorporating connection restraint into the design, take into account the reduced connection stiffness on the stability of the structure and its effect on the magnitude of second order effects. A3. MATERIAL 1.

Structural Steel

1a.

ASTM Designations The grades of structural steel approved for use under the LRFD Specification, covered by ASTM standard specifications, extend to a yield stress of 100 ksi. Some of these ASTM standards specify a minimum yield point, while others specify a minimum yield strength. The term “yield stress” is used in the Specification as a generic term to denote either the yield point or the yield strength. It is important to be aware of limitations of availability that may exist for some combinations of strength and size. Not all structural section sizes are included in the various material specifications. For example, the 60 ksi yield strength steel in the A572 specification includes plate only up to 11⁄4-in. in thickness. Another limitation on availability is that even when a product is included in the specifications, it may be only infrequently produced by the mills. Specifying these products may result in procurement delays or require ordering large quantities directly from the producing mills. Consequently, it is prudent to check availability before completing the details of a design. Properties in the direction of rolling are of principal interest in the design of steel structures. Hence, yield stress as determined by the standard tensile test is the principal mechanical property recognized in the selection of the steels approved for use under the Specification. It must be recognized that other mechanical and physical properties of rolled steel, such as anisotropy, ductility, notch toughness, formability, corrosion resistance, etc., may also be important to the satisfactory performance of a structure. It is not possible to incorporate in the Commentary adequate information to impart full understanding of all factors which might merit consideration in the selection and specification of materials for unique or especially demanding applications. In such a situation the user of the Specification is advised to make use of reference material contained in the literature on the specific properties of concern and to specify supplementary material production or quality requirements as provided for in ASTM material specifications. One such case is the design of highly restrained welded connections (AISC, 1973). Rolled steel is anisotropic, especially insofar as ductility is concerned; therefore, weld contraction strains in the region of highly restrained welded connections may exceed the capabilities of the material if special attention is not given to material selection, details, workmanship, and inspection. Another special situation is that of fracture control design for certain types of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 164

GENERAL PROVISIONS

[Comm. A

service conditions (AASHTO, 1989). The relatively warm temperatures of steel in buildings, the essentially static strain rates, the stress intensity, and the number of cycles of full design stress make the probability of fracture in building structures extremely remote. Good workmanship and good design details incorporating joint geometry that avoids severe stress concentrations are generally the most effective means of providing fracture-resistant construction. However, for especially demanding service conditions such as low temperatures with impact loading, the specification of steels with superior notch toughness may be warranted. 1c.

Heavy Shapes The web-to-flange intersection and the web center of heavy hot-rolled shapes as well as the interior portions of heavy plates may contain a coarser grain structure and/or lower toughness material than other areas of these products. This is probably caused by ingot segregation, as well as somewhat less deformation during hot rolling, higher finishing temperature, and a slower cooling rate after rolling for these heavy sections. This characteristic is not detrimental to suitability for service for compression members, or for non-welded members. However, when heavy cross sections are joined by splices or connections using complete-joint penetration welds which extend through the coarser and/or lower notch-tough interior portions, tensile strains induced by weld shrinkage may result in cracking, for example a complete-joint penetration welded connection of a heavy cross section beam to any column section. When members of lesser thickness are joined by complete-joint penetration welds, which induce smaller weld shrinkage strains, to the finer grained and/or more notch-tough surface material of ASTM A6 Group 4 and 5 shapes and heavy built-up cross sections, the potential for cracking is significantly lower, for example a complete penetration groove welded connection of a non-heavy cross-section beam to a heavy cross-section column. For critical applications such as primary tension members, material should be specified to provide adequate toughness at service temperatures. Because of differences in the strain rate between the Charpy V-Notch (CVN) impact test and the strain rate experienced in actual structures, the CVN test is conducted at a temperature higher than the anticipated service temperature for the structure. The location of the CVN test is shown in Figure C-A3.1. The toughness requirements of A3.1c are intended only to provide material of reasonable toughness for ordinary service application. For unusual applications and/or low temperature service, more restrictive requirements and/or toughness requirements for other section sizes and thicknesses may be appropriate. To minimize the potential for fracture, the notch toughness requirements of A3.1c must be used in conjunction with good design and fabrication procedures. Specific requirements are given in Sections J1.5, J1.6, J2.3, and M2.2.

3.

Bolts, Washers, and Nuts The ASTM standard for A307 bolts covers two grades of fasteners. Either grade may be used under the LRFD Specification; however, it should be noted that Gr. B is intended for pipe flange bolting and Gr. A is the quality long in use for structural applications. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A3]

4.

MATERIAL

6 - 165

Anchor Bolts and Threaded Rods Since there is a limit on the maximum available length of A325 and A490, the use of these bolts for anchor bolts with design lengths longer than the maximum available lengths has presented problems in the past. The inclusion of A687 material in this Specification allows the use of higher strength material for bolts longer than A325 and A490 bolts. The designer should be aware that pretensioning anchor bolts is not recommended due to relaxation and stress corrosion after pretensioning. The designer should specify the appropriate thread and SAE fit for threaded rods used as load-carrying members.

5.

Filler Metal and Flux for Welding The filler metal specifications issued by the American Welding Society are general specifications which include filler metals suitable for building construction, as well as consumables that would not be suitable for building construction. For example, some electrodes covered by the specifications are specifically limited to single pass applications, while others are restricted to sheet metal applications. Many of the filler metals listed are “low hydrogen”, that is, they deposit weld metal with low levels of diffusable hydrogen. Other materials are not. Filler metals listed under the various A5 specifications may or may not have required impact toughness, depending on the specific electrode classification. Notch toughness is generally not critical for weld metal used in building construction. However, on structures subject to dynamic loading, the engineer may require the filler metals used to deliver notch-tough weld deposits. Filler metals may be classified in either the as welded or post weld heat treated (stress relieved) condition. Since most structural applications will not involve stress relief, it is important to utilize filler materials that are classified in conditions similar to those experienced by the actual structure. When specifying filler metal and/or flux by AWS designation, the applicable standard specifications should be carefully reviewed to assure a complete understanding of the designation reference. This is necessary because the AWS designation systems are not consistent. For example, in the case of electrodes for shielded metal arc welding (AWS A5.1), the first two or three digits indicate the nominal tensile strength classification, in ksi, of the weld metal and the final two digits indicate the type of coating; however, in the case of mild steel 1

/4 tf

CVN specimen location

Fig. C-A3.1. Location from which Charpy impact specimen shall be taken. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 166

GENERAL PROVISIONS

[Comm. A

electrodes for submerged arc welding (AWS A5.17), the first one or two digits times 10 indicate the nominal tensile strength classification, while the final digit or digits times 10 indicate the testing temperature in degrees F, for weld metal impact tests. In the case of low-alloy steel covered arc welding electrodes (AWS A5.5), certain portions of the designation indicate a requirement for stress relief, while others indicate no stress relief requirement. Engineers do not, in general, specify the exact filler metal to be employed on a particular structure. Rather, the decision as to which welding process and which filler metal is to be utilized is usually left with the fabricator. To ensure that the proper filler metals are used, codes restrict the usage of certain filler materials, or impose qualification testing to prove the suitability of the specific electrode. A4. LOADS AND LOAD COMBINATIONS 1.

Loads, Load Factors, and Load Combinations The load factors and load combinations given in Section A4.1 were developed to be used with the recommended minimum loads given in ASCE 7 Minimum Design Loads for Buildings and Other Structures (ASCE, 1988). The load factors and load combinations are developed in Ellingwood et al. (1982). The target reliability indices β underlying the load factors are approximately 3.0 for combinations with gravity loads only (dead, snow, and live loads), 2.5 for combinations with wind included, and 1.75 for combinations with earthquake loads. See Commentary A5.3 for definition of β. The load factors and load combinations recognize that when several loads act in combination with the dead load (e.g., dead plus live plus wind), only one of these takes on its maximum lifetime value, while the other load is at its “arbitrary point-in-time value” (i.e., at a value which can be expected to be on the structure at any time). For example, under dead, live, and wind loads the following combinations are appropriate: γD D + γLL

(C-A4-1)

γD D + γLaLa + γwW

(C-A4-2)

γD D + γLL + γwaWa

(C-A4-3)

where γ is the appropriate load factor as designated by the subscript symbol. Subscript a refers to an “arbitrary point-in-time” value. The mean value of arbitrary point-in-time live load La is on the order of 0.24 to 0.4 times the mean maximum lifetime live load L for many occupancies, but its dispersion is far greater. The arbitrary point-in-time wind load Wa, acting in conjunction with the maximum lifetime live load, is the maximum daily wind. It turns out that γWa Wa is a negligible quantity so only two load combinations remain: 1.2D + 1.6L

(C-A4-4)

1.2D + 0.5L + 1.3W

(C-A4-5)

The load factor 0.5 assigned to L in the second equation reflects the statistical AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A5]

DESIGN BASIS

6 - 167

properties of La, but to avoid having to calculate yet another load, it is reduced so it can be combined with the maximum lifetime wind load. The nominal loads D, L, W, E, and S are the code loads or the loads given in ASCE 7. The new specified earthquake loads are based on post-elastic energy dissipation in the structure, and are higher than those traditionally specified for allowable stress design (NEHRP, 1992). The new edition of ASCE Standard 7 on structural loads expected to be released in 1993 has adopted the new seismic design recommendations, as has the AISC Seismic Provisions for Structural Steel Buildings (AISC, 1992). The load factors on E in Load Combinations A4-5 and A4-6 have been reduced from 1.5 to 1.0 to be consistent with the specification of earthquake force in these new documents. The reader is referred to the commentaries to these documents for an expanded discussion on seismic loads, load factors, and seismic design of steel buildings. 2.

Impact A mass of the total moving load (wheel load) is used as the basis for impact loads on crane runway girders, because maximum impact load results when cranes travel while supporting lifted loads. The increase in load, in recognition of random impacts, is not required to be applied to supporting columns because the impact load effects (increase in eccentricities or increases in out-of-straightness) will not develop or will be negligible during the short duration of impact. For additional information on crane girder design criteria see AISE Technical Report No. 13.

A5. DESIGN BASIS 1.

Required Strength at Factored Loads LRFD permits the use of both elastic and plastic structural analyses. LRFD provisions result in essentially the same methodology for, and end product of, plastic design as included in the AISC ASD Specification (AISC, 1989), except that the LRFD provisions tend to be more liberal, reflecting added experience and the results of further research. The 10 percent redistribution permitted is consistent with that in the AISC ASD Specification (AISC, 1989).

2.

Limit States A limit state is a condition which represents the limit of structural usefulness. Limit states may be dictated by functional requirements, such as maximum deflections or drift; they may be conceptual, such as plastic hinge or mechanism formation; or they may represent the actual collapse of the whole or part of the structure, such as fracture or instability. Design criteria ensure that a limit state is violated only with an acceptably small probability by selecting the load and resistance factors and nominal load and resistance values which will never be exceeded under the design assumptions. Two kinds of limit states apply for structures: limit states of strength which define safety against the extreme loads during the intended life of the structure, and limit states of serviceability which define the functional requirements. The LRFD Specification, like other structural codes, focuses on the limit states of strength because of overriding considerations of public safety for the life, limb, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 168

GENERAL PROVISIONS

[Comm. A

and property of human beings. This does not mean that limit states of serviceability are not important to the designer, who must equally ensure functional performance and economy of design. However, these latter considerations permit more exercise of judgment on the part of designers. Minimum considerations of public safety, on the other hand, are not matters of individual judgment and, therefore, specifications dwell more on the limit states of strength than on the limit states of serviceability. Limit states of strength vary from member to member, and several limit states may apply to a given member. The following limit states of strength are the most common: onset of yielding, formation of a plastic hinge, formation of a plastic mechanism, overall frame or member instability, lateral-torsional buckling, local buckling, tensile fracture, development of fatigue cracks, deflection instability, alternating plasticity, and excessive deformation. The most common serviceability limit states include unacceptable elastic deflections and drift, unacceptable vibrations, and permanent deformations. 3.

Design for Strength The general format of the LRFD Specification is given by the formula:

ΣγiQi ≤ φRn

(C-A5-1)

where = summation Σ i = type of load, i.e., dead load, live load, wind, etc. Qi = nominal load effect = load factor corresponding to Qi γi ΣγiQi = required resistance Rn = nominal resistance = resistance factor corresponding to Rn φ φRn = design strength The left side of Equation C-A5-1 represents the required resistance computed by structural analysis based upon assumed loads, and the right side of Equation C-A5-1 represents a limiting structural capacity provided by the selected members. In LRFD, the designer compares the effect of factored loads to the strength actually provided. The term design strength refers to the resistance or strength φRn that must be provided by the selected member. The load factors γ and the resistance factors φ reflect the fact that loads, load effects (the computed forces and moments in the structural elements), and the resistances can be determined only to imperfect degrees of accuracy. The resistance factor φ is equal to or less than 1.0 because there is always a chance for the actual resistance to be less than the nominal value Rn computed by the equations given in Chapters D through K. Similarly, the load factors γ reflect the fact that the actual load effects may deviate from the nominal values of Qi computed from the specified nominal loads. These factors account for unavoidable inaccuracies in the theory, variations in the material properties and dimensions, and uncertainties in the determination of loads. They provide a margin of reliability to account for unexpected loads. They do not account for gross error or negligence. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A5]

DESIGN BASIS

6 - 169

The LRFD Specification is based on (1) probabilistic models of loads and resistance, (2) a calibration of the LRFD criteria to the 1978 edition of the AISC ASD Specification for selected members, and (3) the evaluation of the resulting criteria by judgment and past experience aided by comparative design office studies of representative structures. The following is a brief probabilistic basis for LRFD (Ravindra and Galambos, 1978, and Ellingwood et al., 1982). The load effects Q and the resistance factor R are assumed to be statistically independent random variables. In Figure C-A5.1, frequency distributions for Q and R are portrayed as separate curves on a common plot for a hypothetical case. As long as the resistance R is greater than (to the right of) the effects of the loads Q, a margin of safety for the particular limit state exists. However, because Q and R are random variables, there is some small probability that R may be less than Q, (R < Q). This limit state probability is related to the degree of overlap of the frequency distributions in Figure C-A5.1, which depends on their relative positioning (Rm vs. Qm) and their dispersions. An equivalent situation may be represented as in Figure C-A5.2. If the expression R < Q is divided by Q and the result expressed logarithmically, the result will be a single frequency distribution curve combining the uncertainties of both R and Q. The probability of attaining a limit state (R < Q) is equal to the probability that ln(R / Q) < 0 and is represented by the shaded area in the diagram. The shaded area may be reduced and thus reliability increased in either of two ways: (1) by moving the mean of ln(R / Q) to the right, or (2) by reducing the spread of the curve for a given position of the mean relative to the origin. A convenient way of combining these two approaches is by defining the position of the mean using the standard deviation of ln(R / Q), as the unit of measure. Thus, the distance from the origin to the mean is measured as the number of standard deviations of the function ln(R / Q). As shown in Figure C-A5.2, this

FREQUENCY

R Q

Qm

}

0

OVERLAP

Rm RESISTANCE R LOAD EFFECT Q

Fig. C-A5.1. Frequency distribution of load effect Q and resistance R. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 170

GENERAL PROVISIONS

[Comm. A

is stated as β times σln(R / Q), the standard deviation of ln(R / Q). The factor β therefore is called the “reliability index.” If the actual shape of the distribution of ln(R / Q) were known, and if an acceptable value of the probability of reaching the limit state could be agreed upon, one could establish a completely probability-based set of design criteria. Unfortunately, this much information frequently is not known. The distribution shape of each of the many variables (material, loads, etc.) has an influence on the shape of the distribution of ln(R / Q). Often only the means and the standard deviations of the many variables involved in the makeup of the resistance and the load effect can be estimated. However, this information is enough to build an approximate design criterion which is independent of the knowledge of the distribution, by stipulating the following design condition:

βσln(R / Q) ≈ β√  VR2+ VQ2 ≤ ln(Rm / Qm)

(C-A5-2)

In this formula, the standard deviation has been replaced by the approximation  √ VR2+VQ2, where VR = σR / Rm and VQ = σQ / Qm (σR and σQ are the standard deviations, Rm and Qm are the mean values, VR and VQ are the coefficients of variation, respectively, of the resistance R and the load effect Q). For structural elements and the usual loadings Rm, Qm, and the coefficients of variation, VR and VQ, can be estimated, so a calculation of β=

ln(Rm / Qm)  √ VR2+VQ2

(C-A5-3)

will give a comparative value of the measure of reliability of a structure or component. The description of the determination of β as given above is a simple way of defining the probabilistic method used in the development of LRFD. A more refined method, which can accommodate more complex design situations (such as the beam-column interaction equation) and include probabilistic distributions other than the lognormal distribution used to derive Equation C-A5-3, has been developed since the publication of Ravindra and Galambos (1978), and is fully described in Galambos, et al. (1982). This latter method has been used in the development of the recommended load factors (see Section A4). The two methods give essentially the same β values for most steel structural members and connections.

PF 0

βσIn(R

/Q)

[In(R/Q)]m

In( R/Q)

Fig. C-A5.2. Definition of reliability index. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A5]

DESIGN BASIS

6 - 171

Statistical properties (mean values and coefficients of variations) are presented for the basic material properties and for steel beams, columns, composite beams, plate girders, beam-columns, and connection elements in a series of eight articles in the September 1978 issue of the Journal of the Structural Division of ASCE (Vol. 104, ST9). The corresponding load statistics are given in Galambos, et al. (1982). Based on these statistics, the values of β inherent in the 1978 edition of the AISC ASD Specification were evaluated under different load combinations (live/dead, wind/dead, etc.), and for various tributary areas for typical members (beams, columns, beam-columns, structural components, etc.). As might be expected, there was a considerable variation in the range of β values. Examination of the many β values associated with ASD revealed certain trends. For example, compact rolled beams (flexure) and tension members (yielding) had β values that decreased from about 3.1 at L / D = 0.50 to 2.4 at L / D = 4. This decrease is a result of ASD applying the same factor to dead load, which is relatively predictable, and live load, which is more variable. For bolted or welded connections, β was on the order of 4 to 5. Reliability indices for load combinations involving wind and earthquake loads tended to be lower. Based on a thorough assessment of implied reliabilities in existing acceptable design practice, common load factors for various structural materials (steel, reinforced concrete, etc.) were developed in Ellingwood et al. (1982). One of the features of the probability-based method used in the development of LRFD is that the variations of β values can be reduced by specifying several “target” β values and selecting multiple load and resistance factors to meet these targets. The Committee on Specifications set the point at which LRFD is calibrated to ASD at L / D = 3.0 for braced compact beams in flexure and tension members at yield. The resistance factor, φ, for these limit states is 0.90, and the implied β is approximately 2.6 for members and 4.0 for connections; this larger β value for connections reflects the fact that connections are expected to be stronger than the members that they connect. Limit states for other members are handled consistently. Computer methods as well as charts are given in Ellingwood et al. (1982) for the use of specification writers to determine the resistance factors φ. These factors can also be approximately determined by the following: φ = (Rm / Rn) exp (−0.55βVr )

(C-A5-4)*

where Rm = mean resistance Rn = nominal resistance according to the equations in Chapters D through K Vr = coefficient of variation of the resistance 4.

Design for Serviceability and Other Considerations Nominally, serviceability should be checked at the unfactored loads. For combinations of gravity and wind or seismic loads some additional reduction factor may be warranted.

* Note that exp (x) is identical to the more familiar ex..


6 - 172

CHAPTER B DESIGN REQUIREMENTS

B2. NET AREA Critical net area is based on net width and load transfer at a particular chain. B3. EFFECTIVE NET AREA FOR TENSION MEMBERS Section B3 deals with the effect of shear lag, which is applicable to both welded and bolted tension members. The reduction coefficient U is applied to the net area An of bolted members and to the gross area Ag of welded members. As the length of connection l is increased, the shear lag effect is diminished. This concept is expressed empirically by Equation B3-3. Munse and Chesson (1963) have shown, using this expression to compute an effective net area, with few exceptions, the estimated strength of some 1,000 bolted and riveted connection test specimens correlated with observed test results within a scatterband of ±10 percent. Newer research (Easterling and Gonzales, 1993) provides further justification for current provisions. _ For any given profile and connected elements, x is a fixed geometric property. It is illustrated as the distance from the connection plane, or face of the member, to the centroid of the member section resisting the connection force. See Figure C-B3.1. Length l is dependent upon the number of fasteners or equivalent length of weld required to develop the given tensile force, and this in turn is dependent upon the mechanical properties of the member and the capacity of the fasteners or weld used. The length l is illustrated as the distance, parallel to the line of force, between the first and last fasteners in a line for bolted connections. The number of bolts in a line, for the purpose of the determination of l, is determined by the line with the maximum number of bolts in the connection. For staggered bolts, the out-to-out dimension is used for l. See Figure C-B3.2. There is insufficient data to establish a value of U if all lines have only one bolt, but it is probably conservative to use Ae equal to the net area of the connected element. For welded connections, l is the length of the member parallel to the line of force that is welded. For combinations of longitudinal and transverse welds (see Figure C-B3.3), l is the length of longitudinal weld because the transverse weld has little or no effect on the shear lag problem, i.e., it does little to get the load into the unattached portions of the member. Previous issues of this Specification have presented values for U for bolted or riveted connections of W, M, and S shapes, tees cut from these shapes, and other shapes. These values are acceptable for use in lieu of calculated values from Equation B3-3 and are retained here for the convenience of designers. For bolted or riveted connections the following values of U may be used: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-B3]

EFFECTIVE NET AREA FOR TENSION MEMBERS

6 - 173

(a) W, M, or S shapes with flange widths not less than two-thirds the depth, and structural tees cut from these shapes, provided the connection is to the flanges and has no fewer than three fasteners per line in the direction of stress, U = 0.90 (b) W, M, or S shapes not meeting the conditions of subparagraph a, structural tees cut from these shapes, and all other shapes including built-up cross

T

x Treat as a WT

(a)

T

x x Treat as an angle Use maximum x (b) Treat half the flange and portion of web as an angle

T

x

Use maximum x (c)

_ Fig. C-B3.1. Determination of x for U. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

x

6 - 174

DESIGN REQUIREMENTS

[Comm. B

sections, provided the connection has no fewer than three fasteners per line in the direction of stress, U = 0.85 (c) All members having only two fasteners per line in the direction of stress, U = 0.75 When a tension load is transmitted by fillet welds to some but not all elements of a cross section, the weld strength will control. B5. LOCAL BUCKLING For the purposes of this Specification, steel sections are divided into compact sections, noncompact sections, and sections with slender compression elements. Compact sections are capable of developing a fully plastic stress distribution and they possess a rotational capacity of approximately 3 before the onset of local buckling (Yura et al., 1978). Noncompact sections can develop the yield stress in compression elements before local buckling occurs, but will not resist inelastic local buckling at the strain levels required for a fully plastic stress distribution. Slender compression elements buckle elastically before the yield stress is achieved.

T Use out-to-out distance for l Use maximum x x

l x

Fig. C-B3.2. Staggered holes.

T

l

l

Fig. C-B3.3. Longitudinal and transverse welds. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-B5]

LOCAL BUCKLING

6 - 175

TABLE C-B5.1 Limiting Width-Thickness Ratios for Compression Elements Limiting Width-thickness Ratios λp


Non-seismic

Seismic

Flanges of I-shaped sections (including hybrid sections) and channels in flexure [a]

b/t

65 / √ Fy

52 / √ Fy


h / tw


For Pu / φb Py ≤ 0.125 2.75Pu 640  1− Fy  √ φbPy 

1.54Pu 520  1− Fy  √ φbPy 

For Pu / φb Py > 0.125

Pu  253 191  2.33 − ≥ Fy  √ φbPy √Fy [a] For hybrid beams use Fyf in place of Fy.

The dividing line between compact and noncompact sections is the limiting width-thickness ratio λp. For a section to be compact, all of its compression elements must have width-thickness ratios smaller than the limiting λp. A greater inelastic rotation capacity than provided by the limiting values λp given in Table C-B5.1 may be required for some structures in areas of high seismicity. It has been suggested that in order to develop a ductility of from 3 to 5 in a structural member, ductility factors for elements would have to lie in the range of 5 to 15. Thus, in this case it is prudent to provide for an inelastic rotation of 7 to 9 times the elastic rotation (Chopra and Newmark, 1980). In order to provide for this rotation capacity, the limits λp for local flange and web buckling would be as shown in Table C-B5.1 (Galambos, 1976). More information on seismic design is contained in the AISC Seismic Provisions for Structural Steel Buildings. Another limiting width-thickness ratio is λr, representing the distinction between noncompact sections and sections with slender compression elements. As long as the width-thickness ratio of a compression element does not exceed the limiting value λr, local elastic buckling will not govern its strength. However, for those cases where the width-thickness ratios exceed λr, elastic buckling strength must be considered. A design procedure for such slender-element compression sections, based on elastic buckling of plates, is given in Appendix B5.3. The effective width Equation A-B5-12 applies strictly to stiffened elements under uniform compression. It does not apply to cases where the compression element is under stress gradient. A method of dealing with the stress gradient in a compression element is provided in Section B2 of the AISI Design Specifications for Cold-Formed Steel Structural Members, 1986 and Addendum, 1989. An exception is plate girders with slender webs. Such plate girders AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 176

DESIGN REQUIREMENTS

[Comm. B

are capable of developing postbuckling strength in excess of the elastic buckling load. A design procedure for plate girders including tension field action is given in Appendix G. The values of the limiting ratios λp and λr specified in Table B5.1 are similar to those in AISC (1989) and Table 2.3.3.3 of Galambos (1976), except that: Fy , limited in Galambos (1976) to indeterminate beams when (1) λp = 65 / √ moments are determined by elastic analysis and to determinate beams, was adopted for all conditions on the basis of Yura et al. (1978); and (2) λp = 1,300 / Fy for circular hollow sections was obtained from Sherman (1976). The high shape factor for circular hollow sections makes it impractical to use the same slenderness limits to define the regions of behavior for different types of loading. In Table B5.1, the values of λp for a compact shape that can achieve the plastic moment, and λr for bending, are based on an analysis of test data from several projects involving the bending of pipes in a region of constant moment (Sherman and Tanavde, 1984 and Galambos, 1988). The same analysis produced the equation for the inelastic moment capacity in Table A-F1.1 in Appendix F1. However, a more restrictive value of λp is required to prevent inelastic local buckling from limiting the plastic hinge rotation capacity needed to develop a mechanism in a circular hollow beam section (Sherman, 1976). The values of λr for axial compression and for bending are both based on test data. The former value has been used in building specifications since 1968 (Winter, 1970). Appendices B5 and F1 also limit the diameter-to-thickness ratio for any circular section to 13,000 / Fy. Beyond this, the local buckling strength decreases rapidly, making it impractical to use these sections in building construction. Following the SSRC recommendations (Galambos, 1988) and the approach used for other shapes with slender compression elements, a Q factor is used for circular sections to account for interaction between local and column buckling. The Q factor is the ratio between the local buckling stress and the yield stress. The local buckling stress for the circular section is taken from the inelastic AISI criteria (Winter, 1970) and is based on tests conducted on fabricated and manufactured cylinders. Subsequent tests on fabricated cylinders (Galambos, 1988) confirm that this equation is conservative. The definitions of the width and thickness of compression elements agree with the 1978 AISC ASD Specification with minor modifications. Their applicability extends to sections formed by bending and to unsymmetrical and hybrid sections. For built-up I-shaped sections under axial compression, modifications have been made to the flange local buckling criterion to include web-flange interaction. The kc in the λr limit, in Equations A-B5-7 and A-B5-8 and the elastic buckling Equation A-B5-8 are the same that are used for flexural members. Theory indicates that the web-flange interaction in axial compression is at least as severe as in flexure. Rolled shapes are excluded from this criteria because there are no standard sections with proportions where the interaction would occur. In built-up sections where the interaction causes a reduction in the flange local buckling strength, it is likely that the web is also a thin stiffened element. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-B7]

LIMITING SLENDERNESS RATIOS

6 - 177

The kc factor accounts for the interaction of flange and web local buckling demonstrated in experiments conducted by Johnson (1985). The maximum limit of 0.763 corresponds to Fcr = 20,000 / λ2 which was used as the local buckling strength in earlier editions of both the ASD and LRFD Specifications. An h / tw = 27.5 is required to reach kc = 0.763. Fully fixed restraint for an unstiffened compression element corresponds to kc = 1.3 while zero restraint gives kc = 0.42. Because of web-flange interactions it is possible to get kc < 0.42 from Fy use h / tw = 970 / √ Fy in the kc equation, the new kc formula. If h / tw > 970 / √ which corresponds to the 0.35 limit. Illustrations of some of the requirements of Table B5.1 are shown in Figure C-B5.1. B7. LIMITING SLENDERNESS RATIOS Chapters D and E provide reliable criteria for resistance of axially loaded members based on theory and confirmed by test for all significant parameters including slenderness. The advisory upper limits on slenderness contained in Section B7 are based on professional judgment and practical considerations of economics, ease of handling, and care required to minimize inadvertent damage during fabrication, transport, and erection. Out-of-straightness within reasonable tolerances does not affect the strength of tension members, and the effect of out-of-straightness within specified tolerances on the strength of compression members is accounted for in formulas for resistance. Applied tension tends to reduce, whereas compression tends to amplify, out-of-straightness. Therefore, more liberal criteria are suggested for tension members, including those subject to small compressive forces resulting from transient loads such as earthquake and wind. For members with slenderness ratios greater than 200, these compressive forces correspond to stresses less than 2.6 ksi.


6 - 178

DESIGN REQUIREMENTS

b

[Comm. B

AXIAL COMPRESSION

BENDING 65

141

,

λr =

640 , Fy √

λr =

970 Fy √

  (perforated λr =   190  λp = , λr = Fy √  

317 √Fy

λp =

Fy √

 √ Fy − 10 

λr =

95 Fy √

t

tw

h

b b

t

t

t

hc

λp =

65 λp = , Fy √

λr =

b

h

λr =

λr =

253 Fy √

317  ) Fy  √ 238   λr = Fy  √ λr =

238 Fy √



162  √ (Fyw − 16.5) / kc

95 Fy √

hc

109 λr =  √ Fy / kc

λr =

109  √ Fy / kc

λp =

640 , Fy √

λr =

970 Fy √

λr =

253 Fy √

λp =

190 , Fy √

λr =

238 Fy √

λr =

238 Fy √

λp =

640 , Fy √

λr =

970 Fu √

λr =

238 Fy √

λr =

253 Fy √

λr =

253 Fy √

λr =

970 Fy √

λr =

253 Fy √

b

h b

b

h b

Fig. C-B5.1. Selected examples of Table B5.1 requirements.


6 - 179

CHAPTER C FRAMES AND OTHER STRUCTURES

C1. SECOND ORDER EFFECTS While resistance to wind and seismic loading can be provided in certain buildings by means of shear walls, which also provide for overall frame stability at factored gravity loading, other building frames must provide this resistance by frame action. This resistance can be achieved in several ways, e.g., by a system of bracing, by a moment-resisting frame, or by any combination of lateral force-resisting elements. For frames under combined gravity and lateral loads, drift (horizontal deflection caused by applied loads) occurs at the start of loading. At a given value of the applied loads, the frame has a definite amount of drift ∆. In unbraced frames, additional secondary bending moments, known as the P∆ moments, may be developed in the columns and beams of the lateral load-resisting systems in each story. P is the total gravity load above the story and ∆ is the story drift. As the applied load increases, the P∆ moments also increase. Therefore, the P∆ effect must often be accounted for in frame design. Similarly, in braced frames, increases in axial forces occur in the members of the bracing systems; however, such effects are usually less significant. The designer should consider these effects for all types of frames and determine if they are significant. Since P∆ effects can cause frame drifts to be larger than those calculated by ignoring them, they should also be included in the service load drift analysis when they are significant. In unbraced frames designed by plastic analysis, the limit of 0.75φcPy on column axial loads has also been retained to help ensure stability. The designer may use second-order elastic analysis to compute the maximum factored forces and moments in a member. These represent the required strength. Alternatively, for structures designed on the basis of elastic analysis, the designer may use first order analysis and the amplification factors B1 and B2. In the general case, a member may have first order moments not associated with sidesway which are multiplied by B1, and first order moments produced by forces causing sidesway which are multiplied by B2. The factor B2 applies only to moments caused by forces producing sidesway and is calculated for an entire story. In building frames designed to limit ∆oh / L to a predetermined value, the factor B2 may be found in advance of designing individual members. Drift limits may also be set for design of various categories of buildings so that the effect of secondary bending can be insignificant (Kanchanalai and Lu, 1979; AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 180


[Comm. C

ATC, 1978). It is conservative to use the B2 factor with the sum of the sway and the no-sway moments, i.e., with Mlt + Mnt. The two kinds of first order moment Mnt and Mlt may both occur in sidesway frames from gravity loads. Mnt is defined as a moment developed in a member with frame sidesway prevented. If a significant restraining force is necessary to prevent sidesway of an unsymmetrical structure (or an unsymmetrically loaded symmetrical structure), the moments induced by releasing the restraining force will be Mlt moments, to be multiplied by B2. In most reasonably symmetric frames, this effect will be small. If such a moment B2Mlt is added algebraically to the B1Mnt moment developed with sidesway prevented, a fairly accurate value of Mu will result. End moments produced in sidesway frames by lateral loads from wind or earthquake will always be Mlt moments to be multiplied by B2. When first order end moments in members subjected to axial compression are magnified by B1 and B2 factors, equilibrium requires that they be balanced by moments in connected members (Figure C-C1.1). This can generally be accomplished satisfactorily by distributing the difference between the magnified moment and the first order moment to any other moment-resisting members attached to the compressed member (or members) in proportion to the relative stiffness of the uncompressed members. Minor imbalances may be neglected in the judgment of the engineer. However, complex conditions, such as occur when there is significant magnification in several members meeting at a joint, may require a second order elastic analysis. Connections shall also be designed to resist the magnified end moments. For compression members in braced frames, B1 is determined from Cm values which are similar to the values in the AISC ASD Specification. A significant First order moment P

w

Total moment

P H

B1 Mo

Mo l H

B2 u Mou

B1 Mo

Mo

(a)

Σ B2 Mo

(b)

Fig. C-C1.1. Moment amplification. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

B2 l Mo l

Sect. C-C1]

SECOND ORDER EFFECTS

6 - 181

difference, however, is that B1 is never less than 1. When Cm = 1 for a compression member loaded between its supports, the factors of 8⁄9 and 1⁄2 make the new equations more liberal than Equation H1-1 of the AISC ASD Specification. For Cm ≤ 1 (for members with unequal end moments), the new equations will be slightly more conservative than the AISC ASD Specification for a very slender member with low Cm. For the entire range of l / r and Cm, the equations compare very closely to exact inelastic solutions of braced members. The center-to-center member length is usually used in the structural analysis. In braced and unbraced frames, Pn is governed by the maximum slenderness ratio regardless of the plane of bending. However, Pe1 and Pe2 are always calculated using the slenderness ratio in the plane of bending. Thus, when flexure is about the strong axis only, two different values of slenderness ratio may be involved in solving a given problem. When second order analysis is used, it must account for the interaction of the factored load effects, that is, combinations of factored loads must be used in analysis. Superposition of forces obtained from separate analyses is not adequate. When bending occurs about both the x and the y axes, the required flexural strength calculated about each axis is adjusted by the value of Cm and Pe1 or Pe2 corresponding to the distribution of moment and the slenderness ratio in its plane of bending, and is then taken as a fraction of the design bending strength, φbMn, about that axis, with due regard to the unbraced length of the compression flange where this is a factor. Equations C1-2 and C1-3 approximate the maximum second order moments in compression members with no relative joint translation and no transverse loads between the ends of the member. This approximation is compared to an exact solution (Ketter, 1961) in Figure C-C1.2. For single curvature, Equation C1-3 is slightly unconservative, for a zero end moment it is almost exact, and for double curvature it is conservative. The 1978 AISC ASD Specification imposed the limit Cm ≥ 0.4 which corresponds to a M1 / M2 ratio of 0.5. However, Figure C-C1.2 shows that if, for example, M1 / M2 = 0.8, the Cm = 0.28 is already very conservative, so the limit has been removed. The limit was originally adopted from Austin (1961), which was intended to apply to lateral-torsional buckling, not second-order in-plane bending strength. The AISC Specifications, both in the 1989 ASD and LRFD, use a modification factor Cb as given in Equation F1-3 for lateral-torsional buckling. Cb is approximately the inverse of Cm as presented in Austin (1961) with a 0.4 limit. In Zandonini (1985) it was pointed out that Equation C1-3 could be used for in-plane second order moments if the 0.4 limit was eliminated. Unfortunately, Austin (1961) was misinterpreted and a lateraltorsional buckling solution was used for an in-plane second-order analysis. This oversight has now been corrected. For beam columns with transverse loadings, the second-order moment can be approximated by using the following equation Cm = 1 + ψPu / Pe1 For simply supported members AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 182


[Comm. C

where

ψ =

π2δ0EI −1 M0L2

δ0 = maximum deflection due to transverse loading, in. M0 = maximum factored design moment between supports due to transverse loading, kip-in. For restrained ends some limiting cases (Iwankiw, 1984) are given in Table C-C1.1 together with two cases of simply supported beam-columns. These values of Cm are always used with the maximum moment in the member. For the restrained-end cases, the values of B1 will be most accurate if values of K < 1.0 corresponding to the end boundary conditions are used in calculating Pe1. In lieu of using the equations above, Cm = 1.0 can be used conservatively for transversely loaded members with unrestrained ends and 0.85 for restrained ends. Pu Pe 1.0 0.8

} 0.5 } 0 .80

M1 M2

–0.5 –0.8

.63 .60

Pu M1

.44 .40

M2

Mmax

M1

.25

M2

Classical solution

.20

0.6 – 0.4( M1/ M2 )

.11

1–

Pu Pe1

0

1.0

2.0

3.0

4.0

5.0

Mmax M2

Fig. C-C1.2. Second-order moments for braced beam-column. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Pu

Sect. C-C2]

FRAME STABILITY

6 - 183

TABLE C-C1.1 Amplification Factors ψ and Cm Case

Pu

ψ

Cm

0

1.0

−0.4

1 − 0.4

Pu Pe1

−0.4

1 − 0.4

Pu Pe1

−0.2

1 − 0.2

Pu Pe1

−0.3

1 − 0.3

Pu Pe1

−0.2

1 − 0.2

Pu Pe1

L

/2

If, as in the case of a derrick boom, a beam-column is subject to transverse (gravity) load and a calculable amount of end moment, the value δ0 should include the deflection between supports produced by this moment. Stiffness reduction adjustment due to column inelasticity is permitted. C2. FRAME STABILITY The stability of structures must be considered from the standpoint of the structure as a whole, including not only the compression members, but also the beams, bracing system, and connections. The stability of individual elements must also be provided. Considerable attention has been given in the technical literature to this subject, and various methods of analysis are available to assure stability. The SSRC Guide to Stability Design Criteria for Metal Structures (Galambos, 1988) devotes several chapters to the stability of different types of members considered as individual elements, and then considers the effects of individual elements on the stability of the structure as a whole. The effective length concept is one method of estimating the interaction effects of the total frame on a compression element being considered. This concept uses AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 184


[Comm. C

TABLE C-C2.1 K Values for Columns (a)

(b)

(c)

Theoretical K value

0.5

0.7

1.0

Recommended design value when ideal conditions are approximated

0.65

0.80

1.2

(e)

(f)

1.0

2.0

2.0

1.0

2.10

2.0

(d)

Buckled shape of column is shown by dashed line.

Rotation fixed and translation fixed End condition code

Rotation free and translation fixed

Rotation fixed and translation free Rotation free and translation free

K factors to equate the strength of a framed compression element of length L to an equivalent pin-ended member of length KL subject to axial load only. Other rational methods are available for evaluating the stability of frames subject to gravity and side loading and individual compression members subject to axial load and moments. However, the effective-length concept is the only tool currently available for handling several cases which occur in practically all structures, and it is an essential part of many analysis procedures. Although the concept is completely valid for ideal structures, its practical implementation involves several assumptions of idealized conditions which will be mentioned later. Two conditions, opposite in their effect upon column strength under axial loading, must be considered. If enough axial load is applied to the columns in an unbraced frame dependent entirely on its own bending stiffness for resistance to lateral deflection of the tops of the columns with respect to their bases (see Figure C-C2.1), the effective length of these columns will exceed the actual length. On the other hand, if the same frame were braced to resist such lateral movement, the effective length would be less than the actual length, due to the restraint (resistance to joint translation) provided by the bracing or other lateral support. The ratio K, effective column length to actual unbraced length, may be greater or less than 1.0. The theoretical K values for six idealized conditions in which joint rotation and translation are either fully realized or nonexistent are tabulated in Table C-C2.1. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-C2]

FRAME STABILITY

6 - 185

Also shown are suggested design values recommended by the Structural Stability Research Council (formerly the Column Research Council) for use when these conditions are approximated in actual design. In general, these suggested values are slightly higher than their theoretical equivalents, since joint fixity is seldom fully realized. If the column base in Case f of Table C-C2.1 were truly pinned, K would actually exceed 2.0 for a frame such as that pictured in Figure C-C2.1, because the flexibility of the horizontal member would prevent realization of full fixity at the top of the column. On the other hand, it has been shown (Galambos, 1960) that the restraining influence of foundations, even where these footings are designed only for vertical load, can be very substantial in the case of flat-ended column base details with ordinary anchorage. For this condition, a design K value of 1.5 would generally be conservative in Case f. While in some cases masonry walls provide enough lateral support for building frames to control lateral deflection, light curtain wall construction and wide column spacing can create a situation where only the bending stiffness of the frame provides this support. In this case the effective length factor K for an unbraced length of column L is dependent upon the bending stiffness provided by the other in-plane members entering the joint at each end of the unbraced segment. If the combined stiffness provided by the beams is sufficiently small, relative to that of the unbraced column segments, KL could exceed two or more story heights (Bleich, 1952). Several rational methods are available to estimate the effective length of the columns in an unbraced frame with sufficient accuracy. These range from simple interpolation between the idealized cases shown in Table C-C2.1 to very complex analytical procedures. Once a trial selection of framing members has been made, the use of the alignment chart in Figure C-C2.2 affords a fairly rapid

P

P

Kl

l

Fig. C-C2.1. Column effective length. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 186


[Comm. C

method for determining adequate K values. However, it should be noted that this alignment chart is based upon assumptions of idealized conditions which seldom exist in real structures (Galambos, 1988). These assumptions are as follows: (1) Behavior is purely elastic. (2) All members have constant cross section. (3) All joints are rigid. (4) For braced frames, rotations at opposite ends of beams are equal in magnitude, producing single-curvature bending.

GA 50.0 10.0

K 1.0

5.0 3.0

GA

GB 50.0 10.0 5.0 0.9

2.0

GB 20.0 10.0

100.0 50.0 30.0

5.0

100.0 50.0 30.0

20.0

4.0

20.0

3.0

10.0 9.0 8.0 7.0

3.0 2.0

1.0

1.0

10.0 9.0 8.0 7.0

0.8 0.7

0.8 0.7

5.0

0.6

0.6

0.8

0.7

0.5

0.5

0.4

0.4

0.3

0.2

K

0.3 0.6

6.0

6.0 5.0

4.0

2.0

3.0

3.0

2.0

2.0 1.5

0.2 1.0

0.1

0

4.0

1.0

0.1

0.5

0

1.0

0

SIDESWAY INHIBITED

0

SIDESWAY UNINHIBITED

The subscripts A and B refer to the joints at the two ends of the column section being considered. G is defined as G=

Σ(Ic / Lc) Σ(Ig / Lg)

in which Σ indicates a summation of all members rigidly connected to that joint and lying on the plane in which buckling of the column is being considered. I c is the moment of inertia and Lc the unsupported length of a column section, and I g is the moment of inertia and Lg the unsupported length of a girder or other restraining member. I c and Ig are taken about axes perpendicular to the plane of buckling being considered. For column ends supported by but not rigidly connected to a footing or foundation, G is theoretically infinity, but, unless actually designed as a true friction-free pin, may be taken as “10” for practical designs. If the column end is rigidly attached to a properly designed footing, G may be taken as 1.0. Smaller values may be used if justified by analysis. Fig. C-C2.2. Alignment chart for effective length of columns in continuous frames. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-C2]

FRAME STABILITY

6 - 187

(5) For unbraced frames, rotations at opposite ends of the restraining beams are equal in magnitude, producing reverse-curvature bending. (6) The stiffness parameters L√  P / EI of all columns are equal. (7) Joint restraint is distributed to the column above and below the joint in proportion to I / L of the two columns. (8) All columns buckle simultaneously. (9) No significant axial compression force exists in the girders. Where the actual conditions differ from these assumptions, unrealistic designs may result. There are design procedures available which may be used in the calculation of G for use in Figure C-C2.2 to give results that better reflect the conditions in real structures (Yura, 1971; Disque, 1973; Bjorhovde, 1984; Davison et al., 1988). Leaning columns (sized for gravity loads only, based on an assumed K of 1.0) may be used in unbraced frames provided that the destabilizing effects due to their lack of lateral stiffness from simple connections to the frame (K = ∞) is included in the design of the moment frame columns. A stabilizing column in one direction may be a leaning column in the transverse direction if it is rigidly connected in only one plane. LeMessurier (1977) presented an overall discussion of this problem and recommended a general solution for unbraced frames. In lieu of this and more exact analyses, the following design approximations are suggested. When unbraced moment-resisting frames are the only source of lateral rigidity for a given direction of a story, the upper bound of sidesway stiffness in that direction, measured, in shear force per radian of drift, is ΣPL = ΣHL / ∆oh. This force may be found from a first-order lateral load analysis, without gravity loads, where ΣH is the total story shear. (The calculation of B2 using interstory drift as in LRFD Equation C1-4 also uses the term ΣHL / ∆oh). Since most of the moment-resisting columns in the frame will directly support axial loads, the bending stiffness of the columns will be reduced, lowering the sidesway stiffness ΣPL. An estimate of the reduced sidesway stiffness of the frame may be found by calculating Pe2 for each moment-resisting column, in the direction under consideration, by using the nomograph for sidesway K based on local boundary conditions measured by GA and GB. G is normally assumed as: Ic Lc G= Ig Σ Lg Σ

This definition of G is based on the assumption that girders restraining columns have equal moments (same clockwise direction) at each end determined by an analysis for lateral loads only. When this assumption is violated, a significant overestimate of ΣPe2 may occur. Accurate G values may be found from an examination of girder end moments from such an analysis. The correct Lg should AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 188


[Comm. C

MF   be taken as Lg′ = Lg 2 −  where MF is the moment at the far end of the girder M N   MF under consideration and MN is the moment at the near end. When > 2, Lg′ MN becomes negative which, although real, will result in negative values of G. Negative values of G are beyond the scope of the nomograph but are valid for use in Equation C-C2-2. The reduced total stiffness of the whole story, when each rigidly connected column is loaded with its maximum load Pe2, is ΣPe2. ΣPe2 calculated in this way will always satisfy: .82ΣPL < ΣPe2 < ΣPL Many common framing arrangements include within a story loaded columns designed, in a particular direction, with K = 1. Such columns, often called leaning columns, receive lateral stability from the stiffness of columns with rigid moment-resisting connections. The required axial compressive strength of such leaning columns is called Puo where the subscript implies no shear resistance to lateral loads. The ratio of the loads on all leaning columns in a story to the total of all loads on the story is: ΣPuo = RL ΣPu The ratio of all story loads to the loads on columns providing sidesway is: N=

ΣPu 1 = 1 − RL ΣPu − ΣPuo

If the story stiffness ΣPe2 is calculated from the nomograph K values, the net stiffness available to stabilize the rigid column is: ΣPe2 (1 − RL) =

ΣPe2 N

If there is no redistribution of ΣPe2 / N among the rigid columns, the modified capacity of an individual column is, conservatively: Pe2′ =

Pe2 N

It follows that a modified Ki′ including leaning effects is: Ki′ = √ N × Ki

(C-C2-1)

A more exact value of K′ to account for loss of stiffness to leaning columns can be found from an iterative solution of:    π   π 2     tan  π  2K′ ′  K i  = 0 − RL [GA + GB] −   GAGB + 36 1 − RL  6   K′ π π      tan      2K′   K ′      (C-C2-2) AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-C2]

FRAME STABILITY

6 - 189

When RL = 0, this equation reduces to the equation solved by the sidesway uninhibited nomograph. The 1993 LRFD Specification no longer limits K to unity in sidesway frames and redistribution of stiffness between members of a frame may be advantageous. There are several ways of doing this. Based on the assumption that ΣPe2 is constant, regardless of loading distribution, an adjusted distribution of stiffness to the ith column of a story is: 2

 π  EIi Pui [ΣPe2 ] =   2 Pei′ = ΣPu  Ki′  L

(C-C2-3a)

Pei′ < 1.6Pei

(C-C2-3b)

except

or in terms of K directly with E and L2 constant, Ki′ =

√ 

(C-C2-4a)

 √

(C-C2-4b)

ΣPu × Pui

Ii  Ii  Σ  2  Ki 

except Ki′ ≥

5 K 8 i

where Ki′ = effective length factor with story stability effect for ith rigid column Ii = moment of inertia in plane of bending for ith rigid column Ki = effective length for ith rigid-column factor based on alignment chart for unbraced frame Pui = required axial compressive strength for ith rigid column ΣPu = required axial compressive strength of all columns in a story These expressions include consideration of leaning effects but, in addition, allow concentration of lateral stiffness on relatively weak columns. To limit the error involved with the assumption that ΣPe2 is constant and to avoid the possibility of failure of a weak column in the sidesway prevented mode, the modified Pei′ for a member should not exceed 1.6 times the Pei for the member included in the sum ΣPe2. An alternate formulation which is simple to use but may give lower design values than the expressions above when leaning effects are minimal is: 2

 π  EI Pui [ΣPL ] [.85 + .15RL] =   2i Pei′ = ΣPu  Ki′  L except AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(C-C2-5a)

6 - 190


Pei′ ≤ 1.7PLi where PLi =

[Comm. C

(C-C2-5b)

HiL and Hi is the shear in the ith column included in ΣH. ∆oh

The limits set by Equations C-C2-3b, C-C2-4b, and C-C2-5b have been chosen to avoid unconservative error exceeding five percent in extreme cases. Although Ki′ may be found, it is an unnecessary step since the key parameter λ2c = AFy / Pe ′ is the only one required by Chapter E to find the design capacity φcPn. Design values may be found directly from:  AFy  

φcPn = φc .658 Pe′  AFy when Pe ′ > φcPn = φc .877Pe ′ when Pe ′ ≤

4 AF 9 y

4 AF 9 y

(C-C2-6a) (C-C2-6b)

Because frames that use partially restrained (PR) connections violate the condition that all joints are rigid, special attention should be paid to calculation of the proper G value (Barakat and Chen, 1991). If roof decks or floor slabs, anchored to shear walls or vertical plane bracing systems, are counted upon to provide lateral support for individual columns in a building frame, due consideration must be given to their stiffness when functioning as horizontal diaphragms (Winter, 1958). Translation of the joints in the plane of a truss is inhibited and, due to end restraint, the effective length of compression members might be assumed to be less than the distance between panel points. However, it is usual practice to take K as equal to 1.0 (Galambos, 1988); if all members of the truss reached their ultimate load capacity simultaneously, the restraints at the ends of the compression members would be greatly reduced.


6 - 191

CHAPTER D TENSION MEMBERS

D1. DESIGN TENSILE STRENGTH Due to strain hardening, a ductile steel bar loaded in axial tension can resist, without fracture, a force greater than the product of its gross area and its coupon yield stress. However, excessive elongation of a tension member due to uncontrolled yielding of its gross area not only marks the limit of its usefulness, but can precipitate failure of the structural system of which it is a part. On the other hand, depending upon the reduction of area and other mechanical properties of the steel, the member can fail by fracture of the net area at a load smaller than required to yield the gross area. Hence, general yielding of the gross area and fracture of the net area both constitute failure limit states. The relative values of φt given for yielding and fracture reflect the same basic difference in factor of safety as between design of members and design of connections in the AISC ASD Specification. The length of the member in the net area is negligible relative to the total length of the member. As a result, the strain hardening condition is quickly reached and yielding of the net area at fastener holes does not constitute a limit state of practical significance. D2. BUILT-UP MEMBERS The slenderness ratio L / r of tension members other than rods, tubes, or straps should preferably not exceed the limiting value of 300. This slenderness limit recommended for tension members is not essential to the structural integrity of such members; it merely assures a degree of stiffness such that undesirable lateral movement (“slapping” or vibration) will be unlikely. See Section B7 and Commentary Section E4. D3. PIN-CONNECTED MEMBERS AND EYEBARS Forged eyebars have generally been replaced by pin-connected plates or eyebars thermally cut from plates. Provisions for the proportioning of eyebars contained in the LRFD Specification are based upon standards evolved from long experience with forged eyebars. Through extensive destructive testing, eyebars have been found to provide balanced designs when they are thermally cut instead of forged. The somewhat more conservative rules for pin-connected members of nonuniform cross section and those not having enlarged “circular” heads are likewise based on the results of experimental research (Johnston, 1939). Somewhat stockier proportions are provided for eyebars and pin-connected members fabricated from steel having a yield stress greater than 70 ksi, in order to eliminate any possibility of their “dishing” under the higher design stress. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 192

CHAPTER E COLUMNS AND OTHER COMPRESSION MEMBERS

E1. EFFECTIVE LENGTH AND SLENDERNESS LIMITATIONS 1.

Effective Length The Commentary on Section C2 regarding frame stability and effective length factors applies here. Further analytic methods, formulas, charts, and references for the determination of effective length are provided in Chapter 15 of the SSRC Guide (Galambos, 1988).

2.

Design by Plastic Analysis The limitation on λc is essentially the same as that for l / r in Chapter N of the 1989 AISC Specification—Allowable Stress Design and Plastic Design.

E2. DESIGN COMPRESSIVE STRENGTH FOR FLEXURAL BUCKLING BUCKLING* FLEXURAL-TORSIONAL Equations E2-2 and E2-3 are based on a reasonable conversion of research data into design equations. Conversion of the allowable stress design (ASD) equations which was based on the CRC—Column Research Council—curve (Galambos, 1988) was found to be cumbersome for two reasons. The first was the nature of the ASD variable safety factor. Secondly, the difference in philosophical origins of the two design procedures requires an assumption of a live load-to-dead load ratio (L / D). Since all L / D ratios could not be considered, a value of approximately 1.1 at λ equal to 1.0 was used to calibrate the exponential equation for columns with the lower range of λ against the appropriate ASD provision. The coefficient with the Euler equation was obtained by equating the ASD and LRFD expressions at λ of 1.5. Equations E2-2 and E2-3 are essentially the same curve as column-strength curve 2P of the Structural Stability Research Council which is based on an initial out-of-straightness curve of l / 1,500 (Bjorhovde, 1972 and 1988; Galambos, 1988; Tide, 1985). It should be noted that this set of column equations has a range of reliability (β) values. At low- and high-column slenderness, β values exceeding 3.0 and 3.3 respectively are obtained compared to β of 2.60 at L / D of 1.1. This is considered satisfactory, since the limits of out-of-straightness combined with residual stress have not been clearly established. Furthermore, there has been * For tapered members see Commentary Appendix F3.


Sect. C-E3]

FLEXURAL-TORSIONAL BUCKLING

6 - 193

no history of unacceptable behavior of columns designed using the ASD procedure. This includes cases with L / D ratios greater than 1.1. Equations E2-2 and E2-3 can be restated in terms of the more familiar slenderness ratio Kl / r. First, Equation E2-2 is expressed in exponential form, Fcr = [exp( − 0.419λ2c )]Fy

(C-E2-1)

Note that exp(x) is identical to ex. Substitution of λc according to definition of λc in Section E2 gives,

For

Kl ≤ 4.71 r

 √

E Fy 2

  Fy  Kl    Fcr = exp  − 0.0424    Fy E r    

For

(C-E2-2)

F √

Kl > 4.71 r

E

y

Fcr =

0.877π2E

 Kl r  

2

(C-E2-3)

E3. DESIGN COMPRESSIVE STRENGTH FOR FLEXURAL-TORSIONAL BUCKLING Torsional buckling of symmetric shapes and flexural-torsional buckling of unsymmetric shapes are failure modes usually not considered in the design of hot-rolled columns. They generally do not govern, or the critical load differs very little from the weak axis planar buckling load. Such buckling loads may, however, control the capacity of symmetric columns made from relatively thin plate elements and unsymmetric columns. Design equations for determining the strength of such columns are given in Appendix E3. Tees that conform to the limits in Table C-E3.1 need not be checked for flexural-torsional buckling. A simpler and more accurate design strength for the special case of tees and double-angles is based on Galambos (1991) wherein the y-axis of symmetry flexural-buckling strength component is determined directly from the column formulas. The separate AISC Specification for Load and Resistance Factor Design of Single-Angle Members contains detailed provisions not only for the limit state of compression, but also for tension, shear, flexure, and combined forces. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 194


[Comm. E

TABLE C-E3.1 Limiting Proportions for Tees Shape

Ratio of Full Flange Width to Profile Depth

Ratio of Flange Thickness to Web or Stem Thickness

Built-up tees

≥ 0.50

≥ 1.25

Rolled tees

≥ 0.50

≥ 1.10

E4. BUILT-UP MEMBERS Requirements for detailing and design of built-up members, which cannot be stated in terms of calculated stress, are based upon judgment and experience. The longitudinal spacing of connectors connecting components of built-up compression members must be such that the slenderness ratio l / r of individual shapes does not exceed three-fourths of the slenderness ratio of the entire member. Additional requirements are imposed for built-up members consisting of angles. However, these minimum requirements do not necessarily ensure that the effective slenderness ratio of the built-up member is equal to that for the built-up member acting as a single unit. Section E4 gives formulas for modified slenderness ratios that are based on research and take into account the effect of shear deformation in the connectors (Zandonini, 1985). Equation E4-1 for snug-tight intermediate connectors is emperically based on test results (Zandonini, 1985). The new Equation E4-2 is derived from theory and verified by test data. In both cases the end connection must be welded or slip-critical bolted (Aslani and Goel, 1991). The connectors must be designed to resist the shear forces which develop in the buckled member. The shear stresses are highest where the slope of the buckled shape is maximum (Bleich, 1952). Maximum fastener spacing less than that required for strength may be needed to ensure a close fit over the entire faying surface of components in continuous contact. Specific requirements are given for weathering steel members exposed to atmospheric corrosion (Brockenbrough, 1983). The provisions governing the proportioning of perforated cover plates are based upon extensive experimental research (Stang and Jaffe, 1948).


6 - 195

CHAPTER F BEAMS AND OTHER FLEXURAL MEMBERS

F1.

DESIGN FOR FLEXURE

1.

Yielding The bending strength of a laterally braced compact section is the plastic moment Mp. If the shape has a large shape factor (ratio of plastic moment to the moment corresponding to the onset of yielding at the extreme fiber), significant inelastic deformation may occur at service load if the section is permitted to reach Mp at factored load. The limit of 1.5My at factored load will control the amount of inelastic deformation for sections with shape factors greater than 1.5. This provision is not intended to limit the plastic moment of a hybrid section with a web yield stress lower than the flange yield stress. Yielding in the web does not result in significant inelastic deformations. In hybrid sections, My = FyfS. Lateral-torsional buckling cannot occur if the moment of inertia about the bending axis is equal to or less than the moment of inertia out of plane. Thus, for shapes bent about the minor axis and shapes with Ix = Iy , such as square or circular shapes, the limit state of lateral-torsional buckling is not applicable and yielding controls if the section is compact.

2.

Lateral-Torsional Buckling

2a.

Doubly Symmetric Shapes and Channels with Lb ≤ Lr The basic relationship between nominal moment Mn and unbraced length Lb is shown in Figure C-F1.1 for a compact section with Cb = 1.0. There are four principal zones defined on the basic curve by Lpd, Lp, and Lr. Equation F1-4 defines the maximum unbraced length Lp to reach Mp with uniform moment. Elastic lateral-torsional buckling will occur when the unbraced length is greater than Lr given by Equation F1-6. Equation F1-2 defines the inelastic lateral-torsional buckling as a straight line between the defined limits Lp and Lr. Buckling strength in the elastic region Lb > Lr is given by Equation F1-14 for I-shaped members. For other moment diagrams, the lateral buckling strength is obtained by multiplying the basic strength by Cb as shown in Figure C-F1.1. The maximum Mn, however, is limited to Mp. Note that Lp given by Equation F1-4 is merely a definition which has physical meaning when Cb = 1.0. For Cb greater than 1.0, larger unbraced lengths are permitted to reach Mp as shown by the curve for Cb > 1.0. For design, this length could be calculated by setting Equation F1-2 equal to Mp and solving this equation for Lb using the desired Cb value. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 196


[Comm. F

Cb = 1.75 + 1.05(M1 / M2) + 0.3(M1 / M2)2 ≤ 2.3

(C-F1-1)

The equation

has been used since 1961 to adjust the flexural-torsional buckling equation for variations in the moment diagram within the unbraced length. This equation is applicable only to moment diagrams that are straight lines between braced points. The equation provides a lower bound fit to the solutions developed by Salvadori (1956) which are shown in Figure C-F1.2. Another equation

Mn

Basic strength × Cb

Mp (F1-2)

Mr

(F1-14)

plastic design Mp

inelastic elastic LTB

Cb = 1.0 (Basic strength)

LTB

Lb

L pd Lp Lr (F1-18) (F1-4) (F1-6)

Fig. C-F1.1. Nominal moment as a function of unbraced length and moment gradient.

Cb

Max Cb — warping significant

3.0

(C-F1-2)

2.5

Min Cb — no warping

2.0

(C-F1-1)

1.5 1.0 0 +1.0

M1 M2

+0.5

0

– 0.5

Fig. C-F1.2. Moment modifier Cb for beams. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

–1.0

Sect. C-F1]

DESIGN FOR FLEXURE

Cb =

1

0.6 − 0.4

M1 M2

6 - 197

≤ 2.5

(C-F1-2)

fits the average value theoretical solutions when the beams are bent in reverse curvature and also provides a reasonable fit to the theory. If the maximum moment within the unbraced segment is equal to or larger than the end moment, Cb = 1.0 is used. The equations above can be easily misinterpreted and misapplied to moment diagrams that are not straight within the unbraced segment. Kirby and Nethercot (1979) presented an equation which applies to various shapes of moment diagrams within the unbraced segment. Their equation has been adjusted slightly to the following Cb =

12.5Mmax 2.5Mmax + 3MA + 4MB + 3MC

(C-F1-3)

This equation gives more accurate solutions for fixed-end beams, and the adjusted equation reduces exactly to Equation C-F1-2 for a straight line moment diagram in single curvature. The new Cb equation is shown in Figure C-F1.3 for straight line moment diagrams. Other moment diagrams along with exact theoretical solutions in the SSRC Guide (Galambos, 1988) show good comparison with the new equation. The absolute value of the three interior quarter-point moments plus the maximum moment, regardless of its location are used in the equation. The maximum moment in the unbraced segment is always used for comparison with the resistance. The length between braces, not the distance to inflection points, and Cb is used in the resistance equation. Cb M

M

2

Cb = 1.75 + 1.05 M 1+ 0.3( M 1) ≤ 2.3

2.5

2

2

2.0

1.5

1.0

12.5M max Cb =

0.5

2.5Mmax + 3MA + 4MB + 3MC

M1 MA MB MC M2 +, ratio shown M1 M2

+1.0

+0.5

0

– 0.5

–1.0

Fig. C-F1.3. Cb for a straight line moment diagram—prismatic beam. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 198


[Comm. F

It is still satisfactory to use the former Cb factor, Equation C-F1-1, for straight line moment diagrams within the unbraced length. The elastic strength of hybrid beams is identical to homogeneous beams. The strength advantage of hybrid sections becomes evident only in the inelastic and plastic slenderness ranges. 2b.

Doubly Symmetric Shapes and Channels with Lb > Lr The equation given in the Specification assumes that the loading is applied along the beam centroidal axis. If the load is placed on the top flange and the flange is not braced, there is a tipping effect that reduces the critical moment; conversely, if the load is suspended from the bottom flange and is not braced, there is a stabilizing effect which increases the critical moment (Galambos, 1988). For unbraced top flange loading, the reduced critical moment may be conservatively approximated by setting the warping buckling factor X2 to zero. An effective length factor of unity is implied in these critical moment equations to represent a worst case pinned-pinned unbraced segment. Including consideration of any end restraint of the adjacent segments on the critical segment can increase its buckling capacity. The effects of beam continuity on lateral-torsional buckling have been studied and a simple and conservative design method, based on the analogy of end-restrained nonsway columns with an effective length factor less than one, has been proposed (Galambos, 1988).

2c.

Tees and Double-Angles The lateral-torsional buckling strength (LTS) of singly symmetric tee beams is given by a fairly complex formula (Galambos, 1988). Equation F1-15 is a simplified formulation based on Kitipornchai and Trahair (1980). See also Ellifritt, et al., 1992. The Cb used for I-shaped beams is unconservative for tee beams with the stem in compression. For such cases Cb = 1.0 is appropriate. When beams are bent in reverse curvature, the portion with the stem in compression may control the LTB resistance even though the moments may be small relative to other portions of the unbraced length with Cb ≈ 1.0. This is because the LTB strength of a tee with the stem in compression may be only about one-fourth of the capacity for the stem in tension. Since the buckling strength is sensitive to the moment diagram, Cb has been conservatively taken as 1.0. In cases where the stem is in tension, connection details should be designed to minimize any end restraining moments which might cause the stem to be in compression.

2d.

Unbraced Length for Plastic Analysis In the AISC ASD Specification, Chapter N, the unbraced length of a beam that permits the attainment of plastic moments, and ensures sufficient rotation capacity to redistribute moments, is given by two formulas which depend on the moment ratio at the ends of the unbraced length. One length is permitted for M1 / M2 < −0.5 (almost uniform moment), and a substantially larger length for M1 / M2 > −0.5. These two equations are replaced by Equation F1-18 to provide a continuous function between unbraced length and end moment ratio so there is no abrupt change for a slight change in moment ratio near −0.5. At M1 / M2 = −0.5 (uniform moment) the maximum unbraced length is almost the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-F2]

DESIGN FOR SHEAR

6 - 199

same as that in the AISC ASD Specification. There is a substantial increase in unbraced length for positive moment ratios (reverse curvature) because the yielding is confined to zones close to the brace points (Yura, et al., 1978). Equation F1-19 is an equation in similar form for solid rectangular bars and symmetric box beams. Equations F1-18 and F1-19 assume that the moment diagram within the unbraced length next to plastic hinge locations is reasonably linear. For nonlinear diagrams between braces, judgment should be used in choosing a representative ratio. Equations F1-18 and F1-19 were developed to provide rotation capacities of at least 3.0, which are sufficient for most applications (Yura, et al., 1978). When inelastic rotations of 7 to 9 are deemed appropriate in areas of high seismicity, as discussed in Commentary Section B5, Equation F1-18 would become: Lpd = F2.

2500ry Fy

(C-F1-3)

DESIGN FOR SHEAR For unstiffened webs kv = 5.0, Fyw , and 234√  kv / Fyw = 523 / √ Fyw . therefore 187√  kv / Fyw = 418 / √  kv / Fyw , the nominal shear strength Vn is based on For webs with h / tw ≤ 187√ shear yielding of the web, Equation F2-1 and Equation A-F2-1. This h / tw limit was determined by setting the critical stress causing shear buckling Fcr equal to the yield stress of the web Fyw in Equation 35 of Cooper et al. (1978) and  kv / Fyw , the web shear strength Timoshenko and Gere (1961). When h / tw > 187√ is based on buckling. Basler (1961) suggested taking the proportional limit as 80 percent of the yield stress of the web. This corresponds to h / tw = (187/0.8)  kv / Fyw ), the web strength is determined (√  kv / Fyw ). Thus, when h / tw > 234(√ from the elastic buckling stress given by Equation 6 of Cooper et al., (1978) and Timoshenko and Gere (1961): Fcr =

π2Ekv 12(1 − v2)(h / tw)2

(C-F2-1)

The nominal shear strength, given by Equation F2-3 and A-F2-3, was obtained by multiplying Fcr by the web area and using E = 29,000 ksi and v = 0.3. A straight line transition, Equation F2-2 and AF2-2, is used between the limits  kv / Fyw ). 187(√  kv / Fyw ) and 234(√ The shear strength of flexural members follows the approach used in the AISC ASD Specification, except for two simplifications. First, the expression for the plate buckling coefficient kv has been simplified; it corresponds to that given by AASHTO Standard Specification for Highway Bridges (1989). The earlier expression for kv was a curve fit to the exact expression; the new expression is just as accurate. Second, the alternate method (tension field action) for web shear strength is placed in Appendix G because it was desired that only one method appear in the main body of the Specification with alternate methods given in the Appendix. When designing plate girders, thicker unstiffened webs will frequently be less costly than lighter stiffened web designs because of the additional AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 200


[Comm. F

fabrication. If a stiffened girder design has economic advantages, the tension field method in Appendix G will require fewer stiffeners. The equations in this section were established assuming monotonically increasing loads. If a flexural member is subjected to load reversals causing cyclic yielding over large portions of a web, such as may occur during a major earthquake, special design considerations may apply (Popov, 1980). F4.

BEAMS AND GIRDERS WITH WEB OPENINGS Web openings in structural floor members may be necessary to accommodate various mechanical, electrical, and other systems. Strength limit states, including local buckling of the compression flange, web, and tee-shaped compression zone above or below the opening, lateral buckling and moment-shear interaction, or serviceability may control the design of a flexural member with web openings. The location, size, and number of openings are important and empirical limits for them have been identified. One general procedure for assessing these effects and the design of any needed reinforcement for both steel and composite beams is given in Darwin (1990) and in ASCE (1992, 1992a).


6 - 201

CHAPTER H MEMBERS UNDER COMBINED FORCES AND TORSION

H1. SYMMETRIC MEMBERS SUBJECT TO BENDING AND AXIAL FORCE Equations H1-1a and H1-1b are simplifications and clarifications of similar equations used in the AISC ASD Specification since 1961. Previously, both equations had to be checked. In the new formulation the applicable equation is governed by the value of the first term, Pu / φPn. For bending about one axis only, the equations have the form shown in Figure C-H1.1. The first term Pu / φPn has the same significance as the axial load term fa / Fa in Equations H1-1 of the AISC ASD Specification. This means that for members in compression Pn must be based on the largest effective slenderness ratio Kl / r. In the development of Equations H1-1a and H1-1b, a number of alternative formulations were compared to the exact inelastic solutions of 82 sidesway cases reported in Kanchanalai (1977). In particular, the possibility of using Kl / r as the actual column length (K = 1) in determining Pn, combined with an elastic second order moment Mu, was studied. In those cases where the true Pn based on Kl / r, with K = 1.0, was in the inelastic range, the errors proved to be unacceptably large without the additional check that Pu ≤ φcPn, Pn being based on effective length. Although deviations from exact solutions were reduced, they still remained high. In summary, it is not possible to formulate a safe general interaction equation φPn

Pu

8 Mu Pu φPn + 9 φb Mn = 1 0.2 φPn

1 2

( φPP ) + φ MM = 1 u

u

n

b

n

Mu

0.9 φb Mn

φb Mn

Fig. C-H1.1. Beam-column interaction equations. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 202


[Comm. H

for compression without considering effective length directly (or indirectly by a second equation). Therefore, the requirement that the nominal compressive strength Pn be based on the effective length KL in the general equation is continued in the LRFD Specification as it has been in the AISC ASD Specification since 1961. It is not intended that these provisions be applicable to limit nonlinear secondary flexure that might be encountered in large amplitude earthquake stability design (ATC, 1978). The defined term Mu is the maximum moment in a member. In the calculation of this moment, inclusion of beneficial second order effects of tension is optional. But consideration of detrimental second order effects of axial compression and translation of gravity loads is required. Provisions for calculation of these effects are given in Chapter C. The interaction equations in Appendix H3 have been recommended for biaxially loaded H and wide flange shapes in Galambos (1988) and Springfield (1975). These equations which can be used only in braced frames represent a considerable liberalization over the provisions given in Section H1; it is, therefore, also necessary to check yielding under service loads, using the appropriate load and resistance factors for the serviceability limit state in Equation H1-1a or H1-1b with Mux = SxFy and Muy = SyFy. Appendix H3 also provides interaction equations for rectangular box-shaped beam-columns. These equations are taken from Zhou and Chen (1985). H2. UNSYMMETRIC MEMBERS AND MEMBERS UNDER TORSION AND COMBINED TORSION, FLEXURE, SHEAR, AND/OR AXIAL FORCE This section deals with types of cross sections and loadings not covered in Section H1, especially where torsion is a consideration. For such cases it is recommended to perform an elastic analysis based on the theoretical numerical methods available from the literature for the determination of the maximum normal and shear stresses, or for the elastic buckling stresses. In the buckling calculations an equivalent slenderness parameter is determined for use in Equation E2-2 or E2-3, as follows:

λe = √  Fy / Fe where Fe is the elastic buckling stress determined from a stability analysis. This procedure is similar to that of Appendix E3. For the analysis of members with open sections under torsion refer to AISC (1983).


6 - 203

CHAPTER I COMPOSITE MEMBERS

I1.

DESIGN ASSUMPTIONS Force Determination. Loads applied to an unshored beam before the concrete has hardened are resisted by the steel section alone, and only loads applied after the concrete has hardened are considered as resisted by the composite section. It is usually assumed for design purposes that concrete has hardened when it attains 75 percent of its design strength. In beams properly shored during construction, all loads may be assumed as resisted by the composite cross section. Loads applied to a continuous composite beam with shear connectors throughout its length, after the slab is cracked in the negative moment region, are resisted in that region by the steel section and by properly anchored longitudinal slab reinforcement. For purposes of plastic analysis all loads are considered resisted by the composite cross section, since a fully plastic strength is reached only after considerable yielding at the locations of plastic hinges. Elastic Analysis. The use of constant stiffness in elastic analyses of continuous beams is analogous to the practice in reinforced concrete design. Plastic Analysis. For composite beams with shear connectors, plastic analysis may be used only when the steel section in the positive moment region has a Fyf , and when the steel section in the negative compact web, i.e., h / tw ≤ 640√ moment region is compact, as required for steel beams alone. No compactness limitations are placed on encased beams, but plastic analysis is permitted only if the direct contribution of concrete to the strength of sections is neglected; the concrete is relied upon only to prevent buckling. Plastic Stress Distribution for Positive Moment. Plastic stress distributions are described in Commentary Section I3, and a discussion of the composite participation of slab reinforcement is presented. Plastic Stress Distribution for Negative Moment. Plastic stress distributions are described in Commentary Section I3. Elastic Stress Distribution. The strain distribution at any cross section of a composite beam is related to slip between the structural steel and concrete elements. Prior to slip, strain in both steel and concrete is proportional to the distance from the neutral axis for the elastic transformed section. After slip, the strain distribution is discontinuous, with a jump at the top of the steel shape. The strains in steel and concrete are proportional to distances from separate neutral axes, one for steel and the other for concrete. Fully Composite Beam. Either tensile yield strength of the steel section or the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 204

COMPOSITE MEMBERS

[Comm. I

compressive stress of the concrete slab governs the maximum flexural strength of a fully composite beam subjected to a positive moment. The tensile yield strength of the longitudinal reinforcing bars in the slab governs the maximum flexural strength of a fully composite beam subjected to a negative moment. When shear connectors are provided in sufficient numbers to fully develop this maximum flexural strength, any slip that occurs prior to yielding is minor and has negligible influence both on stresses and stiffness. Partially Composite Beam. The effects of slip on elastic properties of a partially composite beam can be significant and should be accounted for in calculations of deflections and stresses at service loads. Approximate elastic properties of partially composite beams are given in Commentary Section I3. For simplified design methods, see Hansell, et al. (1978). Concrete-Encased Beam. When the dimensions of a concrete slab supported on steel beams are such that the slab can effectively serve as the flange of a composite T-beam, and the concrete and steel are adequately tied together so as to act as a unit, the beam can be proportioned on the assumption of composite action. Two cases are recognized: fully encased steel beams, which depend upon natural bond for interaction with the concrete, and those with mechanical anchorage to the slab (shear connectors), which do not have to be encased. I2.

COMPRESSION MEMBERS

1.

Limitations (a) The lower limit of four percent on the cross-sectional area of structural steel differentiates between composite and reinforced concrete columns. If the area is less than four percent, a column with a structural steel core should be designed as a reinforced concrete column. (b) The specified minimum quantity of transverse and longitudinal reinforcement in the encasement should be adequate to prevent severe spalling of the surface concrete during fires. (c) Very little of the supporting test data involved concrete strengths in excess of 6 ksi, even though the cylinder strength for one group of four columns was 9.6 ksi. Normal weight concrete is believed to have been used in all tests. Thus, the upper limit of concrete strength is specified as 8 ksi for normal weight concrete. A lower limit of 3 ksi is specified for normal weight concrete and 4 ksi for lightweight concrete to encourage the use of good quality, yet readily available, grades of structural concrete. (d) Encased steel shapes and longitudinal reinforcing bars are restrained from buckling as long as the concrete remains sound. A limit strain of 0.0018, at which unconfined concrete remains unspalled and stable, serves analytically to define a failure condition for composite cross sections under uniform axial strain. The limit strain of 0.0018 corresponds approximately to 55 ksi. (e) The specified minimum wall thicknesses are identical to those in the 1989 ACI Building Code (1989). The purpose of this provision is to prevent buckling of the steel pipe or tubing before yielding. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-I3]

2.

FLEXURAL MEMBERS

6 - 205

Design Strength The procedure adopted for the design of axially loaded composite columns is described in detail in Galambos and Chapuis (1980). It is based on the equation for the strength of a short column derived in Galambos and Chapuis (1980), and the same reductions for slenderness as those specified for steel columns in Section E2. The design follows the same path as the design of steel columns, except that the yield stress of structural steel, the modulus of elasticity of steel, and the radius of gyration of the steel section, are modified to account for the effect of concrete and longitudinal reinforcing bars. A detailed explanation of the origin of these modifications may be found in SSRC Task Group 20 (1979). Galambos and Chapuis (1980) includes comparisons of the design procedure with 48 tests of axially loaded stub columns, 96 tests of concrete-filled pipes or tubing, and 26 tests of concrete-encased steel shapes. The mean ratio of the test failure loads to the predicted strengths was 1.18 for all 170 tests, and the corresponding coefficient of variation was 0.19.

3.

Columns with Multiple Steel Shapes This limitation is based on Australian research reported in Bridge and Roderick (1978), which demonstrated that after hardening of concrete the composite column will respond to loading as a unit even without lacing, tie plates, or batten plates connecting the individual steel sections.

4.

Load Transfer To avoid overstressing either the structural steel section or the concrete at connections, a transfer of load to concrete by direct bearing is required. When a supporting concrete area is wider on all sides than the loaded area, the maximum design strength of concrete is specified by ACI (1989) as 1.7φB fc′AB where φB = 0.7 is the strength reduction factor in bearing on concrete and AB is the loaded area. Because the AISC LRFD Specification is based on the lower ASCE 7 load factors (ASCE, 1988), φB = 0.60 in the AISC LRFD Specification. The portion of the design load of an axially loaded column φPn resisted by the concrete may be expressed as (c2 fc′Ac / AsFmy)φBPn. Accordingly, AB ≥

I3.

FLEXURAL MEMBERS

1.

Effective Width

φB c2 Ac Pn c2 Ac Pn = φB 1.7 As Fmy 1.7 As Fmy

(C-I2-1)

LRFD provisions for effective width omit any limit based on slab thickness, in accordance with both theoretical and experimental studies, as well as current composite beam codes in other countries (ASCE, 1979). The same effective width rules apply to composite beams with a slab on either one side or both sides of the beam. To simplify design, effective width is based on the full span, center-to-center of supports, for both simple and continuous beams. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 206

2.

COMPOSITE MEMBERS

[Comm. I

Strength of Beams with Shear Connectors This section applies to simple and continuous composite beams with shear connectors, constructed with or without temporary shores. Positive Flexural Design Strength. Flexural strength of a composite beam in the positive moment region may be limited by the plastic strength of the steel section, the concrete slab, or shear connectors. In addition, web buckling may limit flexural strength if the web is slender and a significantly large portion of the web is in compression. According to Table B5.1, local web buckling does not reduce the plastic strength of a bare steel beam if the beam depth-to-web thickness ratio is not larger than Fy . In the absence of web buckling research on composite beams, the 640 / √ same ratio is conservatively applied to composite beams. Furthermore, for more slender webs, the LRFD Specification conservatively adopts first yield as the flexural strength limit. In this case, stresses on the steel section from permanent loads applied to unshored beams before the concrete has hardened must be superimposed on stresses on the composite section from loads applied to the beams after hardening of concrete. In this superposition, all permanent loads should be multiplied by the dead load factor and all live loads should be multiplied by the live load factor. For shored beams, all loads may be assumed as resisted by the composite section. When first yield is the flexural strength limit, the elastic transformed section is used to calculate stresses on the composite section. The modular ratio n = E / Ec used to determine the transformed section depends on the specified unit weight and strength of concrete. Note that this procedure for compact beams differs from the requirements of Section I2 of the 1989 AISC ASD Specification. Plastic Stress Distribution for Positive Moment. When flexural strength is determined from the plastic stress distribution shown in Figure C-I3.1, compression force C in the concrete slab is the smallest of: C = AswFyw + 2AsfFyf

(C-I3-1)

C = 0.85fc′Ac

(C-I3-2)

C = ΣQn

(C-I3-3)

For a non-hybrid steel section, Equation C-I3-1 becomes C = AsFy where fc′ = specified compressive strength of concrete, ksi Ac = area of concrete slab within effective width, in.2 As = area of steel cross section, in.2 Asw = area of steel web, in.2 Asf = area of steel flange, in.2 Fy = minimum specified yield stress of steel, ksi Fyw = minimum specified yield stress of web steel, ksi Fyf = minimum specified yield stress of flange steel, ksi ΣQn = sum of nominal strengths of shear connectors between the point of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-I3]

FLEXURAL MEMBERS

6 - 207

maximum positive moment and the point of zero moment to either side, kips Longitudinal slab reinforcement makes a negligible contribution to the compression force, except when Equation C-I3-2 governs. In this case, the area of longitudinal reinforcement within the effective width of the concrete slab times the yield stress of the reinforcement may be added in determining C. The depth of the compression block is a=

C 0.85fc′b

(C-I3-4)

where b = effective width of concrete slab, in. A fully composite beam corresponds to the case of C governed by the yield strength of the steel beam or the compressive strength of the concrete slab, as in Equation C-I3-1 or C-I3-2. The number and strength of shear connectors govern C for a partially composite beam as in Equation C-I3-3. The plastic stress distribution may have the plastic neutral axis (PNA) in the web, in the top flange of the steel section or in the slab, depending on the value of C. The nominal plastic moment resistance of a composite section in positive bending is given by the following equation and Figure C-I3.1: Mn = C(d1 + d2) + Py (d3 − d2)

(C-I3-5)

where Py = tensile strength of the steel section; for a non-hybrid steel section Py = AsFy , kips d1 = distance from the centroid of the compression force C in concrete to the top of the steel section, in. d2 = distance from the centroid of the compression force in the steel section to the top of the steel section, in. For the case of no compression in the steel section d2 = 0.

0.85f c′ a d3

C

d1

Fy

d2

(Py – C) 2

(Py + C) 2 Fy

Fig. C-I3.1. Plastic stress distribution for positive moment in composite beams.


6 - 208

COMPOSITE MEMBERS

[Comm. I

d3 = distance from Py to the top of the steel section, in. Equation C-I3-5 is generally applicable including both non-hybrid and hybrid steel sections symmetrical about one or two axes. Approximate Elastic Properties of Partially Composite Beams. Elastic calculations for stress and deflection of partially composite beams should include the effects of slip. The effective moment of inertia Ieff for a partially composite beam is approximated by

 (ΣQn / Cf) (Itr − Is) Ieff = Is + √

(C-I3-6)

where = moment of inertia for the structural steel section, in.4 = moment of inertia for the fully composite uncracked transformed section, in.4 ΣQn = strength of shear connectors between the point of maximum positive moment and the point of zero moment to either side, kips Cf = compression force in concrete slab for fully composite beam; smaller of Equations C-I3-1 and C-I3-2, kips

Is Itr

The effective section modulus Seff, referred to the tension flange of the steel section for a partially composite beam, is approximated by

 (ΣQn / Cf) (Str − Ss) Seff = Ss + √

(C-I3-7)

where Ss = section modulus for the structural steel section, referred to the tension flange, in.3 Str = section modulus for the fully composite uncracked transformed section, referred to the tension flange of the steel section, in.3 Equations C-I3-6 and C-I3-7 should not be used for ratios ΣQn / Cf less than 0.25. This restriction is to prevent excessive slip, as well as substantial loss in beam stiffness. Studies indicate that Equations C-I3-6 and C-I3-7 adequately reflect the reduction in beam stiffness and strength, respectively, when fewer connectors are used than required for full composite action (Grant et al., 1977). Negative Flexural Design Strength. The flexural strength in the negative moment region is the strength of the steel beam alone or the plastic strength of the composite section made up of the longitudinal slab reinforcement and the steel section. Plastic Stress Distribution for Negative Moment. When an adequately braced compact steel section and adequately developed longitudinal reinforcing bars act compositely in the negative moment region, the nominal flexural strength is determined from the plastic stress distributions as shown in Figure C-I3.2. The tensile force T in the reinforcing bars is the smaller of: T = ArFyr AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(C-I3-8)

Sect. C-I3]

FLEXURAL MEMBERS

6 - 209

T = ΣQn

(C-I3-9)

where Ar = area of properly developed slab reinforcement parallel to the steel beam and within the effective width of the slab, in.2 Fyr = specified yield stress of the slab reinforcement, ksi ΣQn = sum of the nominal strengths of shear connectors between the point of maximum negative moment and the point of zero moment to either side, kips A third theoretical limit on T is the product of the area and yield stress of the steel section. However, this limit is redundant in view of practical limitations on slab reinforcement. The nominal plastic moment resistance of a composite section in negative bending is given by the following equation: Mn = T (d1 + d2) + Pyc (d3 − d2)

(C-I3-10)

where Pyc = the compressive strength of the steel section; for a non-hybrid section Pyc = AsFy , kips d1 = distance from the centroid of the longitudinal slab reinforcement to the top of the steel section, in. d2 = distance from the centroid of the tension force in the steel section to the top of the steel section, in. d3 = distance from Pyc to the top of the steel section, in. Transverse Reinforcement for the Slab. Where experience has shown that longitudinal cracking detrimental to serviceability is likely to occur, the slab should be reinforced in the direction transverse to the supporting steel section. It is recommended that the area of such reinforcement should be at least 0.002 times the concrete area in the longitudinal direction of the beam and should be uniformly distributed. 3.

Strength of Concrete-Encased Beams Tests of concrete-encased beams demonstrated that (1) the encasement drastically reduces the possibility of lateral-torsional instability and prevents local buckling of the encased steel, (2) the restrictions imposed on the encasement

d3

(Pyc – T) 2

T d2

d1 Fy (Pyc + T) 2 Fy

Fig. C-I3.2. Plastic stress distribution for negative moment. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 210

COMPOSITE MEMBERS

[Comm. I

practically prevent bond failure prior to first yielding of the steel section, and (3) bond failure does not necessarily limit the moment capacity of an encased steel beam (ASCE, 1979). Accordingly, the LRFD Specification permits two alternate design methods: one based on the first yield in the tension flange of the composite section and the other based on the plastic moment capacity of the steel beam alone. No limitations are placed on the slenderness of either the composite beam or the elements of the steel section, since the encasement effectively inhibits both local and lateral buckling. In the method based on first yield, stresses on the steel section from permanent loads applied to unshored beams before the concrete has hardened must be superimposed on stresses on the composite section from loads applied to the beams after hardening of the concrete. In this superposition, all permanent loads should be multiplied by the dead load factor and all live loads should be multiplied by the live load factor. For shored beams, all loads may be assumed as resisted by the composite section. Complete interaction (no slip) between the concrete and steel is assumed. The contribution of concrete to the strength of the composite section is ordinarily larger in positive moment regions than in negative moment regions. Accordingly, design based on the composite section is more advantageous in the regions of positive moments. 4.

Strength During Construction When temporary shores are not used during construction, the steel beam alone must resist all loads applied before the concrete has hardened enough to provide composite action. Unshored beam deflection caused by wet concrete tends to increase slab thickness and dead load. For longer spans this may lead to instability analogous to roof ponding. An excessive increase of slab thickness may be avoided by beam camber. When forms are not attached to the top flange, lateral bracing of the steel beam during construction may not be continuous and the unbraced length may control flexural strength, as defined in Section F1. The LRFD Specification does not include special requirements for a margin against yield during construction. According to Section F1, maximum factored moment during construction is 0.90Fy Z where Fy Z is the plastic moment (0.90Fy Z ≈ 0.90 × 1.1Fy S). This is equivalent to approximately the yield moment, Fy S. Hence, required flexural strength during construction prevents moment in excess of the yield moment. Load factors for construction loads should be determined for individual projects according to local conditions, with the factors listed in Section A4 as a guide. Once the concrete has hardened, slab weight becomes a permanent dead load and the dead load factor applies to any load combinations.

5.

Formed Steel Deck Figure C-I3.3 is a graphic presentation of the terminology used in Section I3.5. When studs are used on beams with formed steel deck, they may be welded directly through the deck or through prepunched or cut-in-place holes in the deck. The usual procedure is to install studs by welding directly through the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-I3]

FLEXURAL MEMBERS

6 - 211

deck; however, when the deck thickness is greater than 16 gage for single thickness, or 18 gage for each sheet of double thickness, or when the total thickness of galvanized coating is greater than 1.25 ounces/sq. ft, special precautions and procedures recommended by the stud manufacturer should be followed. The design rules for composite construction with formed steel deck are based upon a study (Grant, et al., 1977) of the then available test results. The limiting parameters listed in Section I3.5 were established to keep composite construction with formed steel deck within the available research data. Seventeen full size composite beams with concrete slab on formed steel deck

2 ″min. hr ≤ 3 ″

Hs wr

11/2 ″min.

2 ″min.

2 ″min. hr ≤ 3 ″

Hs wr

11/2 ″min.

2 ″min.

2 ″min. hr ≤ 3 ″

Hs wr

11/2 ″min.

2 ″min. 11/2 ″min.

Hs wr 2 ″min.

wr 2 ″min.

Fig. C-I3.3. Steel deck limits. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

2 ″min. hr ≤ 3 ″

6 - 212

COMPOSITE MEMBERS

[Comm. I

were tested at Lehigh University and the results supplemented by the results of 58 tests performed elsewhere. The range of stud and steel deck dimensions encompassed by the 75 tests were limited to: (1) Stud dimensions: 3⁄4-in. dia. × 3.00 to 7.00 in. (2) Rib width: 1.94 in. to 7.25 in. (3) Rib height: 0.88 in. to 3.00 in. (4) Ratio wr / hr: 1.30 to 3.33 (5) Ratio Hs / hr: 1.50 to 3.41 (6) Number of studs in any one rib: 1, 2, or 3 The strength of stud connectors installed in the ribs of concrete slabs on formed steel deck with the ribs oriented perpendicular to the steel beam is reasonably estimated by the strength of stud connectors in flat soffit composite slabs multiplied by values computed from Equation I3-1. For the case where ribs run parallel to the beam, limited testing (Grant et al., 1977) has shown that shear connection is not significantly affected by the ribs. However, for narrow ribs, where the ratio wr / hr is less than 1.5, a shear stud reduction factor, Equation I3-2, has been employed in view of lack of test data. The Lehigh study (Grant et al., 1977) also indicated that Equation C-I3-7 for effective section modulus and Equation C-I3-6 for effective moment of inertia were valid for composite construction with formed steel deck. Based on the Lehigh test data (Grant, et al., 1977), the maximum spacing of steel deck anchorage to resist uplift was increased from 16 to 18 inches in order to accommodate current production profiles. When metal deck includes units for carrying electrical wiring, crossover headers are commonly installed over the cellular deck perpendicular to the ribs. They create trenches which completely or partially replace sections of the concrete slab above the deck. These trenches, running parallel to or transverse to a composite beam, may reduce the effectiveness of the concrete flange. Without special provisions to replace the concrete displaced by the trench, the trench should be considered as a complete structural discontinuity in the concrete flange. When trenches are parallel to the composite beam, the effective flange width should be determined from the known position of the trench. Trenches oriented transverse to composite beams should, if possible, be located in areas of low bending moment and the full required number of studs should be placed between the trench and the point of maximum positive moment. Where the trench cannot be located in an area of low moment, the beam should be designed as non-composite. 6.

Design Shear Strength A conservative approach to vertical shear provisions for composite beams is adopted by assigning all shear to the steel section web. This neglects any concrete slab contribution and serves to simplify the design. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-I5]

I4.

SHEAR CONNECTORS

6 - 213

COMBINED COMPRESSION AND FLEXURE The procedure adopted for the design of beam-columns is described and supported by comparisons with test data in Galambos and Chapuis (1980). The basic approach is identical to that specified for steel columns in Section H1. The nominal axial strength of a beam-column is obtained from Section I2.2, while the nominal flexural strength is determined from the plastic stress distribution on the composite section. An approximate formula for this plastic moment resistance of a composite column is given in Galambos and Chapuis (1980).

 h2 AwFy  −  AwFy 2 1.7fc′h1  

Mn = Mp = ZFy + 1⁄3(h2 − 2cr) ArFyr + 

(C-I4-1)

where Aw = web area of encased steel shape; for concrete-filled tubes, Aw = 0, in.2 Z = plastic section modulus of the steel section, in.3 cr = average of distance from compression face to longitudinal reinforcement in that face and distance from tension face to longitudinal reinforcement in that face, in. h1 = width of composite cross section perpendicular to the plane of bending, in. h2 = width of composite cross section parallel to the plane of bending, in. The supporting comparisons with beam-column tests included 48 concretefilled pipes or tubing and 44 concrete-encased steel shapes (Galambos and Chapuis, 1980). The overall mean test-to-prediction ratio was 1.23 and the coefficient of variation 0.21. The last paragraph in Section I4 provides a transition from beam-columns to beams. It involves bond between the steel section and concrete. Section I3 for beams requires either shear connectors or full, properly reinforced encasement of the steel section. Furthermore, even with full encasement, it is assumed that bond is capable of developing only the moment at first yielding in the steel of the composite section. No test data are available on the loss of bond in composite beam-columns. However, consideration of tensile cracking of concrete suggests Pu / φcPn = 0.3 as a conservative limit. It is assumed that when Pu / φcPn is less than 0.3, the nominal flexural strength is reduced below that indicated by plastic stress distribution on the composite cross section unless the transfer of shear from the concrete to the steel is provided for by shear connectors. I5.

SHEAR CONNECTORS

1.

Materials Tests (Ollgaard et al., 1971) have shown that fully composite beams with concrete meeting the requirements of Part 3, Chapter 4, “Concrete Quality,” of ACI (1989), made with ASTM C33 or rotary-kiln produced C330 aggregates, develop full flexural capacity. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 214

2.

COMPOSITE MEMBERS

[Comm. I

Horizontal Shear Force Composite beams in which the longitudinal spacing of shear connectors was varied according to the intensity of statical shear, and duplicate beams in which the connectors were uniformly spaced, exhibited the same ultimate strength and the same amount of deflection at normal working loads. Only a slight deformation in the concrete and the more heavily stressed connectors is needed to redistribute the horizontal shear to other less heavily stressed connectors. The important consideration is that the total number of connectors be sufficient to develop the shear Vh on either side of the point of maximum moment. The provisions of the LRFD Specification are based upon this concept of composite action. In computing the design flexural strength at points of maximum negative bending, reinforcement parallel to the steel beam within the effective width of the slab may be included, provided such reinforcement is properly anchored beyond the region of negative moment. However, enough shear connectors are required to transfer, from the slab to the steel beam, the ultimate tensile force in the reinforcement.

3.

Strength of Stud Shear Connectors Studies have defined stud shear connector strength in terms of normal weight and lightweight aggregate concretes as a function of both concrete modulus of elasticity and concrete strength as given by Equation I5-1. Equation I5-1, obtained from Ollgaard, et al. (1971), corresponds to Tables I4.1 and I4.2 in Section I4 of the 1989 AISC ASD Specification. Note that an upper bound on stud shear strength is the product of the cross-sectional area of the stud times its ultimate tensile strength. The LRFD Specification does not specify a resistance factor for shear connector strength. The resistance factor for the flexural strength of a composite beam accounts for all sources of variability, including those associated with the shear connectors.

4.

Strength of Channel Shear Connectors Equation I5-2 is a modified form of the formula for the strength of channel connectors developed by Slutter and Driscoll (1965). The modification has extended its use to lightweight concrete.

6.

Shear Connector Placement and Spacing Uniform spacing of shear connectors is permitted except in the presence of heavy concentrated loads. When stud shear connectors are installed on beams with formed steel deck, concrete cover at the sides of studs adjacent to sides of steel ribs is not critical. Tests have shown that studs installed as close as is permitted to accomplish welding of studs does not reduce the composite beam capacity. Studs not located directly over the web of a beam tend to tear out of a thin flange before attaining full shear-resisting capacity. To guard against this contingency, the size of a stud not located over the beam web is limited to 21⁄2 times the flange thickness (Goble, 1968). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-I6]

SPECIAL CASES

6 - 215

The minimum spacing of connectors along the length of the beam, in both flat soffit concrete slabs and in formed steel deck with ribs parallel to the beam, is six diameters; this spacing reflects development of shear planes in the concrete slab (Ollgaard et al., 1971). Since most test data are based on the minimum transverse spacing of four diameters, this transverse spacing was set as the minimum permitted. If the steel beam flange is narrow, this spacing requirement may be achieved by staggering the studs with a minimum transverse spacing of three diameters between the staggered row of studs. The reduction in connector Nr , capacity in the ribs of formed steel decks is provided by the factor 0.85 / √ which accounts for the reduced capacity of multiple connectors, including the effect of spacing. When deck ribs are parallel to the beam and the design requires more studs than can be placed in the rib, the deck may be split so that adequate spacing is available for stud installation. Figure C-I5.1 shows possible connector arrangements. I6.

SPECIAL CASES Tests are required for construction that falls outside the limits given in the Specification. Different types of shear connectors may require different spacing and other detailing than stud and channel connectors.

4d d 6d

6d

3d

4d

Fig. C-I5.1. Shear connector arrangements.


6 - 216

CHAPTER J CONNECTIONS, JOINTS, AND FASTENERS

J1.

GENERAL PROVISIONS

5.

Splices in Heavy Sections Solidified but still-hot weld metal contracts significantly as it cools to ambient temperature. Shrinkage of large welds between elements which are not free to move to accommodate the shrinkage, causes strains in the material adjacent to the weld that can exceed the yield point strain. In thick material, the weld shrinkage is restrained in the thickness direction as well as in the width and length directions, causing triaxial stresses to develop that may inhibit the ability of ductile steel to deform in a ductile manner. Under these conditions, the possibility of brittle fracture increases. When splicing ASTM A6 Group 4 and 5 rolled sections or heavy welded built-up members, the potentially harmful weld shrinkage strains can be avoided by using bolted splices or fillet-welded lap splices or splices that combine a welded and bolted detail (see Figure C-J1.1). Details and techniques that perform well for materials of modest thickness usually must be changed or supplemented by more demanding requirements when welding thick material. Also, the provisions of the Structural Welding Code, AWS D1.1, are minimum requirements that apply to most structural welding situations; however, when designing and fabricating welded splices of ASTM A6 Group 4 and 5 shapes and similar built-up cross sections, special consideration must be given to all aspects of the welded splice detail. • Notch-toughness requirements should be specified for tension members. See Commentary A3. • Generously sized weld access holes, Figure C-J1.2, are required to provide increased relief from concentrated weld shrinkage strains, to avoid close juncture of welds in orthogonal directions, and to provide adequate clearance for the exercise of high quality workmanship in hole preparation, welding, and ease of inspection. • Preheating for thermal cutting is required to minimize the formation of a hard surface layer. • Grinding to bright metal and inspection using magnetic particle or dye-penetrant methods is required to remove the hard surface layer and to assure smooth transitions free of notches or cracks. In addition to tension splices of truss chord members and tension flanges of flexural members, other joints fabricated of heavy sections subject to tension should be given special consideration during design and fabrication. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-J1]

8.

GENERAL PROVISIONS

6 - 217

Placement of Welds and Bolts Slight eccentricities between the gravity axis of single and double angle members and the center of gravity of connecting rivets or bolts have long been ignored as having negligible effect on the static strength of such members. Tests (Gibson and Wake, 1942) have shown that similar practice is warranted in the case of welded members in statically loaded structures. However, the fatigue life of eccentrically loaded welded angles has been shown to be very short (Kloppel and Seeger, 1964). Notches at the roots of fillet welds are harmful when alternating tensile stresses are normal to the axis of the weld, as could occur due to bending when axial cyclic loading is applied to angles with end welds not balanced about the neutral axis. Accordingly, balanced welds are indicated when such members are subjected to cyclic loading (see Figure C-J1.3).

9.

Bolts in Combination with Welds Welds will not share the load equally with mechanical fasteners in bearing-type connections. Before ultimate loading occurs, the fastener will slip and the weld will carry an indeterminately larger share of the load. Accordingly, the sharing of load between welds and A307 bolts or high-strength bolts in a bearing-type connection is not recommended. For similar reasons, A307 bolts and rivets should not be assumed to share loads in a single group of fasteners. For high-strength bolts in slip-critical connections to share the load with welds it is advisable to fully tension the bolts before the weld is made. If the weld is placed first, angular distortion from the heat of the weld might prevent the faying action required for development of the slip-critical force. When the bolts are fully tensioned before the weld is made, the slip-critical bolts and the weld may be assumed to share the load on a common shear plane (Kulak, et al., 1987). The heat of welding near bolts will not alter the mechanical properties of the bolts.

2 or 3 in.

In making alterations to existing structures, it is assumed that whatever slip is

(a) Shear plate welded to web

(b) Shear plate welded to flange tips

(c) Bolted splice plates

Fig. C-J1.1. Alternative splices that minimize weld restraint tensile stresses. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 218


[Comm. J

likely to occur in high-strength bolted bearing-type connections or riveted connections will have already taken place. Hence, in such cases the use of welding to resist all stresses, other than those produced by existing dead load present at the time of making the alteration, is permitted. It should be noted that combinations of fasteners as defined herein does not refer to connections such as shear plates for beam-to-column connections which are welded to the column and bolted to the beam flange or web (Kulak, et al., 1987) and other comparable connections.

Backing if used 5

Backing if used 5

≥1.5t w

Radius precut by drill or hole saw D ≥ ¾ in.

R (see note 2)

≥ tw ≥ ¾ in.

≥1.5t w

R (see note 2)

≥1.5t w

≥ tw ≥ ¾ in.

Angle of slope not critical

Need not be tangent, notches prohibited ≥ tw ≥ ¾ in.

Optional method for making corner radius

≥1.5t w

≥1.5t w

R (see note 2)

R (see note 2)

≥ tw ≥ ¾ in.

≥ tw ≥ ¾ in.

Rolled shape 1 or groove welded shape 1,3

Fillet 1,4 welded shape

Notes: 1. For ASTM A6 Group 4 and 5 shapes and welded built-up shapes with plate thickness more than 2 in., preheat to 150°F prior to thermal cutting, grind and inspect thermally cut edges of access hole using magnetic particle or dye penetration methods prior to making web and flange splice groove welds. 2. Radius shall provide smooth notch-free transition; R ≥ 3⁄8-in. (typical 1⁄2-in.) 3. Access opening made after welding web to flange. 4. Access opening made before welding web to flange. 5. These are typical details for joints welded from one side against steel backing. Alternative joint designs should be considered. Fig. C-J1.2. Weld access hole and beam cope geometry. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-J2]

10.

WELDS

6 - 219

High-Strength Bolts in Combination with Rivets When high-strength bolts are used in combination with rivets, the ductility of the rivets permits the direct addition of the strengths of both fastener types.

J2.

WELDS

1.

Groove Welds The engineer preparing contract design drawings cannot specify the depth of groove without knowing the welding process and the position of welding. Accordingly, only the effective throat for partial joint-penetration groove welds should be specified on design drawings, allowing the fabricator to produce this effective throat with his own choice of welding process and position. The weld reinforcement is not used in determining the effective throat thickness of a groove weld (see Table J2.1).

2.

Fillet Welds

2a.

Effective Area The effective throat of a fillet weld is based upon the root of the joint and the face of the diagrammatic weld, hence this definition gives no credit for weld penetration or reinforcement at the weld face. If the fillet weld is made by the submerged arc welding process, some credit for penetration is made. If the leg size of the resulting fillet weld exceeds 3⁄8-in., then 0.11 in. is added to the theoretical throat. This increased weld throat is allowed because the submerged arc process produces deep penetration of welds of consistent quality. However, it is necessary to run a short length of fillet weld to be assured that this increased penetration is obtained. In practice, this is usually done initially by crosssectioning the runoff plates of the joint. Once this is done, no further testing is required, as long as the welding procedure is not changed. F

F

Welds balanced about the center line of the angle

Welds balanced about the neutral axis of the angle Figure C-J1.3


6 - 220

2b.


[Comm. J

Limitations Table J2.4 provides a minimum size of fillet weld for a given thickness of the thicker part joined. The requirements are not based upon strength considerations, but upon the quench effect of thick material on small welds. Very rapid cooling of weld metal may result in a loss of ductility. Further, the restraint to weld-metal shrinkage provided by thick material may result in weld cracking. Because a 5⁄16-in. fillet weld is the largest that can be deposited in a single pass by SMAW process, 5⁄ -in. applies to all material 3⁄ -in. and greater in thickness, but minimum preheat 16 4 and interpass temperature are required by AWS D1.1.* Both the design engineer and the shop welder must be governed by the requirements. Table J2.3 gives the minimum effective throat of a partial joint-penetration groove weld. Notice that Table J2.3 for partial joint-penetration groove welds goes up to a plate thickness of over 6 in. and a minimum weld throat of 5⁄8-in., whereas, for fillet welds Table J2.4 goes up to a plate thickness of over 3⁄4-in. and a minimum leg size of fillet weld of only 5⁄16-in. The additional thickness for partial-penetration welds is to provide for reasonable proportionality between weld and material thickness. For plates of 1⁄4-in. or more in thickness, it is necessary that the inspector be able to identify the edge of the plate to position the weld gage. This is assured if the weld is kept back at least 1⁄16-in. from the edge, as shown in Figure C-J2.1. Where longitudinal fillet welds are used alone in a connection (see Figure C-J2.2), Section J2.2b requires the length of each weld to be at least equal to the width of the connecting material because of shear lag (Fisher, et al., 1978). By providing a minimum lap of five times the thickness of the thinner part of a lap joint, the resulting rotation of the joint when pulled will not be excessive, as shown in Figure C-J2.3. Fillet welded lap joints under tension tend to open and

* See Table J2.4.

Apparent edge of plate

Apparent weld throat

Actual edge of plate before welding

Actual edge of plate is distinguishable

Actual weld throat Fig. C-J2.1. Identification of plate edge. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Actual weld throat is distinguishable

Sect. C-J2]

WELDS

6 - 221

apply a tearing action at the root of the weld as shown in Figure C-J2.4b, unless restrained by a force F as shown in Figure C-J2.4a. End returns are not essential for developing the capacity of fillet welded connections and have a negligible effect on their strength. Their use has been encouraged to insure that the weld size is maintained over the length of the weld, to enhance the fatigue resistance of cyclically loaded flexible end connections, and to increase the plastic deformation capability of such connections. The weld capacity database on which the specifications were developed had no end returns. This includes the study by Higgins and Preece (1968), seat angle tests by Lyse and Schreiner (1935), the seat and top angle tests by Lyse and Gibson (1937), beam webs welded directly to column or girder by fillet welds by Johnston and Deits (1941), and the eccentrically loaded welded connections reported by Butler, Pal, and Kulak (1972). Hence, the current design-resistance values and joint-capacity models do not require end returns, when the required weld size is provided. Johnston and Green (1940) noted that movement consistent with the design assumption of no end restraint (i.e., joint flexibility) was

L

L≥W

t

W

L

Fig. C-J2.2. Longitudinal fillet welds.

Overlap

Overlap

Fig. C-J2.3. Minimum lap. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 222


[Comm. J

enhanced without end returns. They also verified that greater plastic deformation of the connection was achieved when end returns existed, although the strength was not significantly different. There are numerous welded joints where it is not possible to provide end returns and where it is also possible to provide the desired weld size. These joints as well as the seat angle and the web angle connections cited earlier do not require end returns when the weld size is adequate and fatigue is not a design consideration. 4.

Design Strength The strength of welds is governed by the strength of either the base material or the deposited weld metal. Table J2.5 contains the resistance factors and nominal weld strengths, as well as a number of limitations. It should be noted that in Table J2.5 the nominal strength of fillet welds is determined from the effective throat area, whereas the strength of the connected parts is governed by their respective thicknesses. Figure C-J2.5 illustrates the shear planes for fillet welds and base material: (a) Plane 1-1, in which the resistance is governed by the shear strength for material A. (b) Plane 2-2, in which the resistance is governed by the shear strength of the weld metal. (c) Plane 3-3, in which the resistance is governed by the shear strength of the material B. The resistance of the welded joint is the lowest of the resistance calculated in each plane of shear transfer. Note that planes 1-1 and 3-3 are positioned away from the fusion areas between the weld and the base material. Tests have

F (a) Unrestrained

(a) Restrained Fig. C-J2.4. Restraint of lap joints.

Material A

1

3

2 Material B

2 1

2

1

2 Material A

2

1

2

3

Fig. C-J2.5. Shear planes for fillet welds loaded in longitudinal shear. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-J3]


6 - 223

demonstrated that the stress on this fusion area is not critical in determining the shear strength of fillet welds (Preece, 1968). The shear planes for plug and partial penetration groove welds are shown in Figure C-J2.6 for the weld and base metal. Generally the base metal will govern the shear strength. 5.

Combination of Welds This method of adding weld strengths does not apply to a welded joint using a partial-penetration single bevel groove weld with a superimposed fillet weld. In this case, the effective throat of the combined joint must be determined and the design strength based upon this throat area.

7.

Mixed Weld Metal Problems can occur when incompatible weld metals are used in combination and notch-tough composite weld metal is required. For instance, tack welds deposited using a self-shielded process with aluminum deoxidizers in the electrodes and subsequently covered by SAW weld passes can result in composite weld metal with low notch-toughness, despite the fact that each process by itself could provide notch-tough weld metal.

J3.


1.

High-Strength Bolts In general, the use of high-strength bolts is required to conform to the provisions of the Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts (RCSC, 1988) as approved by the Research Council on Structural Connections.

(a) Plug welds

(b) Partial penetration welds Fig. C-J2.6. Shear planes for plug and partial-penetration welds. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 224


[Comm. J

Occasionally the need arises for the use of high-strength bolts of diameters and lengths in excess of those available for A325 and A490 bolts, as for example, anchor bolts for fastening machine bases. For this situation Section A3.3 permits the use of A449 bolts and A354 threaded rods. 2.

Size and Use of Holes To provide some latitude for adjustment in plumbing up a frame during erection, three types of enlarged holes are permitted, subject to the approval of the designer. The nominal maximum sizes of these holes are given in Table J3.3. The use of these enlarged holes is restricted to connections assembled with bolts and is subject to the provisions of Sections J3.3 and J3.4.

3.

Minimum Spacing The maximum factored strength Rn at a bolt or rivet hole in bearing requires that the distance between the centerline of the first fastener and the edge of a plate toward which the force is directed should not be less than 11⁄2d, where d is the fastener diameter (Kulak et al., 1987). By similar reasoning the distance measured in the line of force, from the centerline of any fastener to the nearest edge of an adjacent hole, should not be less than 3d, to ensure maximum design strength in bearing. Plotting of numerous test results indicates that the critical bearing strength is directly proportional to the above defined distances up to a maximum value of 3d, above which no additional bearing strength is achieved (Kulak et al., 1987). Table J3.7 lists the increments that must be added to adjust the spacing upward to compensate for an increase in hole dimension parallel to the line of force. Section J3.10 gives the bearing strength criteria as a function of spacing.

4.

Minimum Edge Distance Critical bearing stress is a function of the material tensile strength, the spacing of fasteners, and the distance from the edge of the part to the center line of the nearest fastener. Tests have shown (Kulak et al., 1987) that a linear relationship exists between the ratio of critical bearing stress to tensile strength (of the connected material) and the ratio of fastener spacing (in the line of force) to fastener diameter. The following equation affords a good lower bound to published test data for single-fastener connections with standard holes, and is conservative for adequately spaced multi-fastener connections: Fpcr le = Fu d

(C-J3-1)

where Fpcr = critical bearing stress, ksi Fu = tensile strength of the connected material, ksi le = distance, along a line of transmitted force, from the center of a fastener to the nearest edge of an adjacent fastener or to the free edge of a connected part (in the direction of stress), in. d = diameter of fastener, in. The provisions of Section J3.3 are concerned with le as hole spacing, whereas Section J3.4 is concerned with le as edge distance in the direction of stress. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-J3]


6 - 225

Section J3.10 establishes a maximum bearing strength. Spacing and/or edge distance may be increased to provide for a required bearing strength, or bearing force may be reduced to satisfy a spacing and/or edge distance limitation. It has long been known that the critical bearing stress of a single fastener connection is more dependent upon a given edge distance than multi-fastener connections (Jones, 1940). For this reason, longer edge distances (in the direction of force) are required for connections with one fastener in the line of transmitted force than required for those having two or more. The recommended minimum distance transverse to the direction of load is primarily a workmanship tolerance. It has little, if any, effect on the strength of the member. 5.

Maximum Spacing and Edge Distance Limiting the edge distance to not more than 12 times the thickness of an outside connected part, but not more than six inches, is intended to provide for the exclusion of moisture in the event of paint failure, thus preventing corrosion between the parts which might accumulate and force these parts to separate. More restrictive limitations are required for connected parts of unpainted weathering steel exposed to atmospheric corrosion.

6.

Design Tension or Shear Strength Tension loading of fasteners is usually accompanied by some bending due to the deformation of the connected parts. Hence, the resistance factor φ, by which Rn is multiplied to obtain the design tensile strength of fasteners, is relatively low. The nominal tensile strength values in Table J3.2 were obtained from the equation Rn = 0.75AbFu

(C-J3-2)

While the equation was developed for bolted connections (Kulak et al., 1987), it was also conservatively applied to threaded parts and to rivets. The nominal strength of A307 bolts was discounted by 5 ksi. In connections consisting of only a few fasteners, the effects of strain on the shear in bearing fasteners is negligible (Kulak et al., 1987; Fisher et al., 1978). In longer joints, the differential strain produces an uneven distribution between fasteners (those near the end taking a disproportionate part of the total load), so that the maximum strength per fastener is reduced. The AISC ASD Specification permits connections up to 50 in. in length without a reduction in maximum shear stress. With this in mind the resistance factor φ for shear in bearing-type connections has been selected to accommodate the same range of connections. The values of nominal shear strength in Table J3.2 were obtained from the equation Rn / mAb = 0.50Fu

(C-J3-3)

when threads are excluded from the shear planes and Rn / mAb = 0.40Fu

(C-J3-4)

when threads are not excluded from the shear plane, where m is the number of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 226


[Comm. J

shear planes (Kulak et al., 1987). While developed for bolted connections, the equations were also conservatively applied to threaded parts and rivets. The value given for A307 bolts was obtained from Equation C-J3-4 but is specified for all cases regardless of the position of threads. For A325 bolts, no distinction is made between small and large diameters, even though the minimum tensile strength Fu is lower for bolts with diameters in excess of one inch. It was felt that such a refinement of design was not justified, particularly in view of the low resistance factor φ, increasing ratio of tensile area to gross area and other compensating factors. 7.

Combined Tension and Shear in Bearing-Type Connections Tests have shown that the strength of bearing fasteners subject to combined shear and tension resulting from externally applied forces can be closely defined by an ellipse (Kulak et al., 1987). Such a curve can be replaced, with only minor deviations, by three straight lines as shown in Figure C-J3.1. This latter representation offers the advantage that no modification of either type stress is required in the presence of fairly large magnitudes of other types. This linear representation was adopted for Table J3.5, giving a limiting tensile stress Ft as a function of the shearing stress fv for bearing-type connections.

8.

High-Strength Bolts in Slip-Critical Connections Connections classified as slip-critical include those cases where slip could theoretically exceed an amount deemed by the Engineer of Record to affect the suitability for service of the structure by excessive distortion or reduction in strength or stability, even though the nominal strength of the connection may be adequate. Also included are those cases where slip of any magnitude must be prevented, for example, joints subject to fatigue, connectors between elements of built-up members at their ends (Sections D2 and E4), and bolts in combination with welds (Section J1.9). The onset of slipping in a high-strength bolted, slip-critical connection is not an ft

Ft

C– RFv

C = 1.3Ft , approximately F R = t/Fv , approximately F ft = F t Fv 2– fv2 v

Fv

fv

Figure C-J3.1. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-J3]


6 - 227

indication that maximum capacity of the connection has been reached. Its occurrence may be only a serviceability limit state. In the case of bolts in holes with only small clearance, such as standard holes and slotted holes loaded transverse to the axis of the slot in practical connections, the freedom to slip generally does not exist because one or more bolts are in bearing even before load is applied due to normal fabrication tolerances and erection procedures. Further, the consequences of slip, if it can occur at all, are trivial except for a few situations as noted above. Slip of slip-critical connections is likely to occur at approximately 1.4 to 1.5 times the service loads. For standard holes, oversized holes, and short slotted holes the connection can be designed either at service loads (Section J3.8a) or at factored loads (Appendix J3.8b). The nominal loads and φ factors have been adjusted accordingly. The number of connectors will be essentially the same for the two procedures because they have been calibrated to give similar results. Slight differences will occur because of variation in the ratio of live load to dead load. In connections containing long slots that are parallel to the direction of the applied load, slip of the connection prior to attainment of the factored load might be large enough to alter the usual assumption of analysis that the undeformed structure can be used to obtain the internal forces. To guard against this occurring, the design slip resistance is further reduced by 0.85 when designing at service load (Section J3.8a) and by setting φ to 0.60 in conjunction with factored loads (Appendix J3.8b). While the possibility of a slip-critical connection slipping into bearing under anticipated service conditions is small, such connections must comply with the provisions of Section J3.10 in order to prevent connection failure at the maximum load condition. 10.

Bearing Strength at Bolt Holes The recommended bearing stress on pins is not the same as for bolts as explained in Section J8. Bearing values are not provided as a protection to the fastener, because it needs no such protection. Therefore, the same bearing value applies to joints assembled by bolts, regardless of fastener shear strength or the presence or absence of threads in the bearing area. Tests (Frank and Yura, 1981) have demonstrated that hole elongation greater than 0.25 in. will begin to develop as the bearing stress is increased beyond the values given in Equations J3-1a and J3-1d, especially if it is combined with high tensile stress on the net section, even though rupture does not occur. Equations J3-1b and J3-1c consider the effect of hole ovalization (deformation greater than 0.25 in.) whenever the upper design limit (3.0dtFu) is deemed acceptable. These latter equations also establish the design limit for a single bolt, or two or more bolts, whenever the bolt arrangement results in each bolt singly in line with the direction of the applied force. Because two separate limit states are considered (deformation and strength) with both limit states equated to a bearing stress (2.4Fu or 2.0Fu and 3.0Fu , respectively) conflicting design strengths may result, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 228


[Comm. J

either acceptable, when intermediate edge distance and bolt spacing values are considered. 11.

Long Grips Provisions requiring a decrease in calculated stress for A307 bolts having long grips (by arbitrarily increasing the required number in proportion to the grip length) are not required for high-strength bolts. Tests (Bendigo et al., 1963) have demonstrated that the ultimate shearing strength of high-strength bolts having a grip of eight or nine diameters is no less than that of similar bolts with much shorter grips.

J4.

DESIGN RUPTURE STRENGTH Tests (Birkemoe and Gilmor, 1978) on coped beams indicated that a tearing failure mode (rupture) can occur along the perimeter of the bolt holes as shown in Figure C-J4.1. This block shear mode combines tensile strength on one plane and shear strength on a perpendicular plane. The failure path is defined by the center lines of the bolt holes. The block shear failure mode is not limited to the coped ends of beams. Other examples are shown in Figure CJ4.1 and C-J4.2.

Cope Beam

Shear area

Failure by tearing out of shaded portion

Failure by tearing out of shaded portion Shear area

Tensile area

Tensile area Po

Fig. C-J4.1. Failure surface for block shear rupture limit state.

Po Po Small tension force

Large tension force

Large shear force Po (a)

Po (b) Fig. C-J4.2 Block shear rupture in tension. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Small shear force

Sect. C-J6]

FILLERS

6 - 229

The block shear failure mode should also be checked around the periphery of welded connections. Welded connection block shear is determined using φ = 0.75 in conjunction with the area of both the fracture and yielding planes (Yura, 1988). The LRFD Specification has adopted a conservative model to predict block shear strength. Test results suggest that it is reasonable to add the yield strength on one plane to the rupture strength of the perpendicular plane (Ricles and Yura, 1983 and Hardash and Bjorhovde, 1985). Therefore, two possible block shear strengths can be calculated; rupture strength Fu on the net tensile section along with shear yielding 0.6Fy on the gross section on the shear plane(s), or rupture 0.6Fu on the net shear area(s) combined with yielding Fy on the gross tensile area. This is the basis of Equations J4-3 and J4-4. These equations are consistent with the philosophy in Chapter D for tension members, where gross area is used for the limit state of yielding and net area is used for rupture. The controlling equation is the one that produces the larger rupture force. This can be explained by the two extreme examples given in Figure C-J4.2. In Case a, the total force is resisted primarily by shear, so shear rupture, not shear yielding, should control the block shear tearing mode; therefore, use Equation J4-4. For Case b, block shear cannot occur until the tension area ruptures as given by Equation J4-3. If Equation J4-4 (shear rupture on the small area and yielding on the large tension area) is checked for Case b, a smaller Po will result. In fact, as the shear area gets smaller and approaches zero, the use of Equation J4-4 for Case b would give a block shear strength based totally on yielding of the gross tensile area. Block shear is a rupture or tearing phenomenon not a yielding limit state. Therefore, the proper equation to use is the one with the larger rupture term. J5.

CONNECTING ELEMENTS

2.

Design Strength of Connecting Elements in Tension Tests have shown that yield will occur on the gross section area before the tensile capacity of the net section is reached, if the ratio An / Ag ≤ 0.85 (Kulak et al., 1987). Since the length of connecting elements is small compared to the member length, inelastic deformation of the gross section is limited. Hence, the effective net area An of the connecting element is limited to 0.85Ag in recognition of the limited inelastic deformation and to provide a reserve capacity.

J6.

FILLERS The practice of securing fillers by means of additional fasteners, so that they are, in effect, an integral part of a shear-connected component, is not required where a connection is designed to be a slip-critical connection using highstrength bolts. In such connections, the resistance to slip between filler and either connected part is comparable to that which would exist between the connected parts if no fill were present. Filler plates may be used in lap joints of welded connections that splice parts of different thickness, or where there may be an offset in the joint. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 230

J8.


[Comm. J

BEARING STRENGTH The LRFD Specification provisions for bearing on milled surfaces, Section J8, follow the same philosophy of earlier AISC ASD Specifications. In general, the design is governed by a deformation limit state at service loads resulting in stresses nominally at 9⁄10 of yield. Adequate safety is provided by post-yield strength as deformation increases. Tests on pin connections (Johnston, 1939) and on rockers (Wilson, 1934) have confirmed this behavior. As used throughout the LRFD Specification, the terms “milled surface,” “milled,” and “milling” are intended to include surfaces which have been accurately sawed or finished to a true plane by any suitable means.

J9.

COLUMN BASES AND BEARING ON CONCRETE The equations for resistance of concrete in bearing are the same as ACI 318-89 except that AISC equations use φ = 0.60 while ACI uses φ = 0.70, since ACI specifies larger load factors than the ASCE load factors specified by AISC.

J10. ANCHOR BOLTS AND EMBEDMENTS ACI 318 and 349 Appendix B and the PCI Handbook include recommended procedures for the design of anchor bolts and embedments.


6 - 231

CHAPTER K CONCENTRATED FORCES, PONDING, AND FATIGUE

K1. FLANGES AND WEBS WITH CONCENTRATED FORCES 1.

Design Basis The LRFD Specification separates flange and web strength requirements into distinct categories representing different limit state criteria, i.e., local flange bending (Section K1.2), local web yielding (Section K1.3), web crippling (Section K1.4), sidesway web buckling (Section K1.5), compression buckling of the web (Section K1.6), and panel zone web shear (Section K1.7). These criteria are applied to two distinct types of concentrated forces which act on member flanges. Single concentrated forces may be tensile, such as those delivered by tension hangers, or compressive, such as those delivered by bearing plates at beam interior positions, reactions at beam ends, and other bearing connections. Double concentrated forces, one tensile and one compressive, form a couple on the same side of the loaded member, such as that delivered to column flanges through welded and bolted moment connections.

2.

Local Flange Bending Where a tensile force is applied through a plate welded across a flange, that flange must be sufficiently rigid to prevent deformation of the flange and the corresponding high-stress concentration in the weld in line with the web. The effective column flange length for local flange bending is 12tf (Graham, et al., 1959). Thus, it is assumed that yield lines form in the flange at 6tf in each direction from the point of the applied concentrated force. To develop the fixed edge consistent with the assumptions of this model, an additional 4tf and therefore a total of 10tf, is required for the full flange-bending strength given by Equation K1-1. In the absence of applicable research, a 50 percent reduction has been introduced for cases wherein the applied concentrated force is less than 10tf from the member end. This criterion given by Equation K1-1 was originally developed for moment connections, but it also applies to single concentrated forces such as tension hangers consisting of a plate welded to the bottom flange of a beam and transverse to the beam web.

3.

Local Web Yielding The web strength criteria have been established to limit the stress in the web of a member into which a force is being transmitted. It should matter little whether the member receiving the force is a beam or a column; however, Galambos (1976) and AISC (1978), references upon which the LRFD Specification is AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 232


[Comm. K

based, did make such a distinction. For beams, a 2:1 stress gradient through the flange was used, whereas the gradient through column flanges was 21⁄2:1. In Section K1.3, the 21⁄2:1 gradient is used for both cases. This criterion applies to both bearing and moment connections. 4.

Web Crippling The expression for resistance to web crippling at a concentrated force is a departure from previous specifications (IABSE, 1968; Bergfelt, 1971; Hoglund, 1971; and Elgaaly, 1983). Equations K1-4 and K1-5 are based on research by Roberts (1981). The increase in Equation K1-5b for N / d > 0.2 was developed after additional testing (Elgaaly, 1991) to better represent the effect of longer bearing lengths at ends of members. All tests were conducted on bare steel beams without the expected beneficial contributions of any connection or floor attachments. Thus, the resulting criteria are considered conservative for such applications. These equations were developed for bearing connections, but are also generally applicable to moment connections. However, for the rolled shapes listed in Part 1 of the LRFD Manual with Fy not greater than 50 ksi, the web crippling criterion will never control the design in a moment connection except for a W12×50 or W10×33 column. The web crippling phenomenon has been observed to occur in the web adjacent to the load flange. For this reason, a half-depth stiffener (or stiffeners) or a half-depth doubler plate is expected to eliminate this limit state.

5.

Sidesway Web Buckling The sidesway web buckling criterion was developed after observing several unexpected failures in tested beams (Summers and Yura, 1982). In those tests the compression flanges were braced at the concentrated load, the web was squeezed into compression, and the tension flange buckled (see Figure C-K1.1). Sidesway web buckling will not occur in the following cases. For flanges restrained against rotation: h / tw > 2.3 l / bf

(C-K1-1)

Brace

Tension flange

Fig. C-K1.1. Sidesway web buckling. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

ay Sidesw kle c u b b e w

Sect. C-K1]


6 - 233

For flanges not restrained against rotation: h / tw > 1.7 l / bf

(C-K-1-2)

where l is as shown in Figure C-K1.2. Sidesway web buckling can also be prevented by the proper design of lateral bracing or stiffeners at the load point. It is suggested that local bracing at both flanges be designed for one percent of the concentrated force applied at that point. Stiffeners must extend from the load point through at least one-half the beam or girder depth. In addition, the pair of stiffeners should be designed to carry the full load. If flange rotation is permitted at the loaded flange, neither stiffeners nor doubler plateswill be effective. In the 1st Edition LRFD Manual, the sidesway web buckling equations were based on the assumption that h / tf = 40, a convenient assumption which is generally true for economy beams. This assumption has been removed so that the equations will be applicable to all sections. These equations were developed only for bearing connections and do not apply to moment connections. 6.

Compression Buckling of the Web When compressive forces are applied to both flanges of a member at the same

l=L

l=L

l=L

l = L/ 2 L

/2 L

= Braced point Fig. C-K1.2. Unbraced flange length. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 234


[Comm. K

location, as by moment connections at both flanges of a column, the member web must have its slenderness ratio limited to avoid the possibility of buckling. This is done in the LRFD Specification with Equation K1-8, which is a modified form of a similar equation used in the ASD Specification. This equation is applicable to a pair of moment connections, and to other pairs of compressive forces applied at both flanges of a member, for which N / d is small ( 0.5Fy. This is accomplished through the use of the equivalent slenderFy / Fe . √ ness factor λe = 


6 - 246

APPENDIX F BEAMS AND OTHER FLEXURAL MEMBERS

F1.

DESIGN FOR FLEXURE Three limit states must be investigated to determine the moment capacity of flexural members: lateral-torsional buckling (LTB), local buckling of the compression flange (FLB), and local buckling of the web (WLB). These limit states depend, respectively, on the beam slenderness ratio Lb / ry, the width-thickness ratio b / t of the compression flange and the width-thickness ratio h / tw of the web. For convenience, all three measures of slenderness are denoted by λ. Variations in Mn with Lb are shown in Figure C-A-F1.1. The discussion of plastic, inelastic, and elastic buckling in Commentary Section F1 with reference to lateral-torsional buckling applies here except for an important difference in the significance of λp for lateral-torsional buckling and local buckling. Values of λp for FLB and WLB produce a compact section with a rotation capacity of about three (after reaching Mp) before the onset of local buckling, and therefore meet the requirements for plastic analysis of load effects (Commentary Section B5). On the other hand, values of λp for LTB do not allow plastic analysis because they do not provide rotation capacity beyond that needed to develop Mp. Instead Lb ≤ Lpd (Section F1.2d) must be satisfied. Analyses to include restraint effects of adjoining elements are discussed in Galambos (1988). Analysis of the lateral stability of members with shapes not covered in this appendix must be performed according to the available literature (Galambos, 1988). bT1

b

bT2

T

IT

IT 2 IT

1

IO

IB

l

IB1

IO

l

bB2

bB1

bB

IB2

bI GT = l T O IT

GT =

I O bT l IT

bI GB = l B O IB

GB =

I O bB l IB

Figure C-A-F1.1 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A-F3]

WEB-TAPERED MEMBERS

6 - 247

See the Commentary for Section B5 for the discussion of the equation regarding the bending capacity of circular sections. F3.

WEB-TAPERED MEMBERS

1.

General Requirements The provision contained in Appendix F3 covers only those aspects of the design of tapered members that are unique to tapered members. For other criteria of design not specifically covered in Appendix F3, see the appropriate portions of this Specification and Commentary. The design of wide-flange columns with a single web taper and constant flanges follows the same procedure as for uniform columns according to Section E2, except the column slenderness parameter λc for major axis buckling is determined for a slenderness ratio Kγ L / rox, and for minor axis buckling for KL / roy, where Kγ is an effective length factor for tapered members, K is the effective length factor for prismatic members, and rox and roy are the radii of gyration about the x and the y axes, respectively, taken at the smaller end of the tapered member. For stepped columns or columns with other than a single web taper, the elastic critical stress is determined by analysis or from data in reference texts or research reports (Chapters 11 and 13 in Timoshenko and Gere (1961) and Bleich (1952) and Kitipornchai and Trahair [1980], and then the same procedure of using λeff is utilized in calculating the factored resistance. This same approach is recommended for open section built-up columns (columns with perforated cover plates, lacing, and battens) where the elastic critical buckling stress determination must include a reduction for the effect of shear. Methods for calculating the elastic buckling strength of such columns are given in Chapter 12 of the SSRC Guide (Galambos, 1988) and in Timoshenko and Gere (1961) and Bleich (1952).

3.

Design Compressive Strength The approach in formulating Faγ of tapered columns is based on the concept that the critical stress for an axially loaded tapered column is equal to that of a prismatic column of different length, but of the same cross section as the smaller end of the tapered column. This has resulted in an equivalent effective length factor Kγ for a tapered member subjected to axial compression (Lee et al., 1972). This factor, which is used to determine the value of S in Equations A-F3-2 and λc in Equation E2-3, can be determined accurately for a symmetrical rectangular rigid frame comprised of prismatic beams and tapered columns. With modifying assumptions, such a frame can be used as a mathematical model to determine with sufficient accuracy the influence of the stiffness Σ(I / b)g of beams and rafters which afford restraint at the ends of a tapered column in other cases such as those shown in Figure C-A-F1.1. From Equations A-F3-2 and E2-3, the critical load Pcr can be expressed as π2EIo / (Kγl)2. The value of Kγ can be obtained by interpolation, using the appropriate chart from Lee et al. (1972) and restraint modifiers GT and GB. In each of these modifiers the tapered column, treated as a prismatic member having a moment of inertia Io, computed at the smaller end, and its actual length l, is assigned the stiffness Io / l, which is then AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 248


[Comm. A-F

divided by the stiffness of the restraining members at the end of the tapered column under consideration. 4.

Design Flexural Strength The development of the design bending stress for tapered beams follows closely with that for prismatic beams. The basic concept is to replace a tapered beam by an equivalent prismatic beam with a different length, but with a cross section identical to that of the smaller end of the tapered beam (Lee et al., 1972). This has led to the modified length factors hs and hw in Equations A-F3-6 and A-F3-7. Equations A-F3-6 and A-F3-7 are based on total resistance to lateral buckling, using both St. Venant and warping resistance. The factor B modifies the basic Fbγ to members which are continuous past lateral supports. Categories a, b, and c of Appendix F3.4 usually apply; however, it is to be noted that they apply only when the axial force is small and adjacent unbraced segments are approximately equal in length. For a single member, or segments which do not fall into category a, b, c, or d, the recommended value of B is unity. The value of B should also be taken as unity when computing the value of Fbγ to obtain Mn to be used in Equations H1-1 and C1-1, since the effect of moment gradient is provided for by the factor Cm. The background material is given in WRC Bulletin No. 192 (Morrell and Lee, 1974).


6 - 249

APPENDIX G PLATE GIRDERS

Appendix G is taken from AISI Bulletin 27 (Galambos, 1978). Comparable provisions are included in the AISC ASD Specification. The provisions are presented in an appendix as they are seldom used and produce designs which are often less economical than plate girders designed without tension-field action. Fyf that distinguishes plate girders The web slenderness ratio h / tw = 970 / √ from beams is written in terms of the flange yield stress, because for hybrid girders inelastic buckling of the web due to bending depends on the flange strain. The equation for Re used in the 1986 LRFD Specification was the same as that used in the AASHTO Standard Specification for Highway Bridges. In this edition, the equation for Re, used in the AISC ASD Specification since 1969, is used because its derivation is published (Gaylord and Gaylord, 1992 and ASCE-AASHTO, 1968) and it is more accurate than the AASHTO equation. G2. DESIGN FLEXURAL STRENGTH In previous versions of the AISC Specification a coefficient of 0.0005ar was used in RPG based on the work of Basler (1961). This value is valid for ar ≤ 2. In that same paper, Basler developed a more general coefficient, applicable to all ratios of Aw / Af which has now been adopted because application of the previous equation to sections with large ar values gives unreasonable results. An arbitrary limit of ar ≤ 10 is imposed so that the RPG expression is not applied to sections approaching a tee shape.


6 - 250

APPENDIX H MEMBERS UNDER COMBINED FORCES AND TORSION

H3. ALTERNATIVE INTERACTION EQUATIONS FOR MEMBERS UNDER COMBINED STRESS In the case of members not subject to flexural buckling, i.e., Lb < Lpd, the use of somewhat more liberal interaction Equations A-H3-5 and A-H3-6 is acceptable as an alternative when the flexure is about one axis only. The alternative interaction Equations A-H3-1 and A-H3-2 for biaxially loaded H and wide-flange column shapes were taken from Galambos (1988), Springfield (1975), and Tebedge and Chen (1974). For I-shaped members with bf / d > 1.0, use of Section H1 is recommended, because no additional research is available for this case.


6 - 251

APPENDIX J CONNECTIONS, JOINTS, AND FASTENERS

J2.

WELDS

4.

Design Strength When weld groups are loaded in shear by an external load that does not act through the center of gravity of the group, the load is eccentric and will tend to cause a relative rotation and translation between the parts connected by the weld. The point about which rotation tends to take place is called the instantaneous center of rotation. Its location is dependent upon the load eccentricity, geometry of the weld group, and deformation of the weld at different angles of the resultant elemental force relative to the weld axis. The individual resistance force of each unit weld element can be assumed to act on a line perpendicular to a ray passing through the instantaneous center and that element’s location (see Figure C-A-J2.1). The ultimate shear strength of weld groups can be obtained from the load deformation relationship of a single-unit weld element. This relationship was originally given by Butler (1972) for E60 electrodes. Curves for E70 electrodes used in the Appendix were obtained by Lesik (1990). Unlike the load-deformation relationship for bolts, strength and deformation performance in welds are dependent on the angle θ that the resultant elemental force makes with the axis of the weld element (see Figure C-A-J2.1). The actual load deformation relationship for welds is given in Figure C-A-J2.2, taken from Kennedy and Lesik (1990). Conversion of the SI equation to foot-pound units results in the following weld strength equation for Rn: Rn = 0.852(1.0 + 0.50 sin1.5θ)FEXX Aw Because the maximum strength is limited to 0.60FEXX for longitudinally loaded welds (θ = 0º), the LRFD Specification provision provides, in the reduced equation coefficient, a reasonable margin for any variation in welding techniques and procedures. To eliminate possible computational difficulties, the maximum deformation in the weld elements is limited to 0.17D. For design convenience, a simple elliptical formula is used for f(p) to closely approximate the empirically derived polynomial in Lesik (1990). The total resistance of all the weld elements combine to resist the eccentric ultimate load, and when the correct location of the instantaneous center has been selected, the three in-plane equations of statics (ΣFx , ΣFy , ΣM) will be satisfied. Numerical techniques, such as those given by Brandt (1982), have been develAMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 252


[Comm. A-J

oped to locate the instantaneous center of rotation subject to convergence tolerances. Earlier editions of the AISC Manual of Steel Construction (AISC, 1980, 1986, 1989) took advantage of the inelastic redistribution of stresses that is inherent in the Appendix J2.4 procedure. However, in each of the utilized computational techniques the resulting coefficients were factored down so that the maximum A

Y ro

al

xl R

P

θ l

c.g.

i.c.

X

R θ

kl

Figure C-A-J2.1

1.6 90° 75° 60°

1.4

45°

1.2

30°

P/ Po

1.0

15°

0.8

θ = 0°

0.6 0.4 0.2

0

0.05

0.10

0.15

0.20

0.25

0.30

∆/ D

Figure C-A-J2.2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

0.35

0.40

Sect. C-A-J2]

WELDS

6 - 253

stress, at any point in the weld group, did not exceed the limiting value specified by either the Allowable Stress Design or LRFD Specifications, 0.3Fu or 0.6Fu, respectively. As a result, the tabulated weld-capacity data shown in the appropriate referenced manual tables will be found to be conservative relative to the data obtained using the computational procedure presented in Appendix J2.4.


6 - 254

APPENDIX K CONCENTRATED FORCES, PONDING, AND FATIGUE

K3. FATIGUE Because most members in building frames are not subject to a large enough number of cycles of full design stress application to require design for fatigue, the provisions covering such designs have been placed in Appendix K3. When fatigue is a design consideration, its severity is most significantly affected by the number of load applications, the magnitude of the stress range, and the severity of the stress concentrations associated with the particular details. These factors are not encountered in normal building designs; however, when encountered and when fatigue is of concern, all provisions of Appendix K3 must be satisfied. Members or connections subject to less than 20,000 cycles of loading will not involve a fatigue condition, except in the case of repeated loading involving large ranges of stress. For such conditions, the admissible range of stress can conservatively be taken as one and one-half times the applicable value given in Table A-K3.3 for “Loading Condition 1.” Fluctuation in stress which does not involve tensile stress does not cause crack propagation and is not considered to be a fatigue situation. On the other hand, in elements of members subject solely to calculated compression stress, fatigue cracks may initiate in regions of high tensile residual stress. In such situations, the cracks generally do not propagate beyond the region of the residual tensile stress, because the residual stress is relieved by the crack. For this reason stress ranges that are completely in compression are not included in the column headed by “Kind of Stress” in Table A-K3.2. This is also true of comparable tables of the current AASHTO and AREA specifications. When fabrication details involving more than one category occur at the same location in a member, the stress range at that location must be limited to that of the most restrictive category. By locating notch-producing fabrication details in regions subject to a small range of stress, the need for a member larger than required by static loading will often be eliminated. Extensive test programs (Fisher et al., 1970; and Fisher et al., 1974) using full size specimens, substantiated by theoretical stress analysis, have confirmed the following general conclusions: (1) Stress range and notch severity are the dominant stress variables for welded details and beams. (2) Other variables such as minimum stress, mean stress, and maximum stress are not significant for design purposes. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Sect. C-A-K3]

FATIGUE

6 - 255

(3) Structural steels with yield points of 36 to 100 ksi do not exhibit significantly different fatigue strength for given welded details fabricated in the same manner. Allowable stress ranges can be read directly from Table A-K3.3 for a particular category and loading condition. The values are based on extensive research (Keating and Fisher, 1985). Provisions for bolts subjected to tension are given in Table A-K3.4. Tests have uncovered dramatic differences in fatigue life, not completely predictable from the various published equations for estimating the actual magnitude of prying force (Kulak et al., 1987). To limit the uncertainties regarding prying action on the fatigue behavior of these bolts, the tensile stresses given in Table J3.2 are approved for use under extended cyclic loading only if the prying force, included in the design tensile force, is small. When this cannot be assured, the design tensile stress is drastically reduced to cover any conceivable prying effect. The use of other types of mechanical fasteners to resist applied cyclic loading in tension is not recommended. Lacking a high degree of assured pretension, the range of stress is generally too great to resist such loading for long. However, all types of mechanical fasteners survive unharmed when subject to cyclic shear stresses sufficient to fracture the connected parts, which is provided for elsewhere in Appendix K3.


6 - 256


6 - 257

References Ad Hoc Committee on Serviceability Research (1986), “Structural Serviceability: A Critical Appraisal and Research Needs,” Journal of Structural Engineering, ASCE, Vol. 112, No. 12, 1986. American Association of State Highway and Transportation Officials (1989), Standard Specification for Highway Bridges, 1989. American Concrete Institute (1989), Building Code Requirements for Reinforced Concrete, ACI 318-89, Detroit, MI, 1989. American Institute of Steel Construction, Inc. (1973), “Commentary on Highly Restrained Welded Connections,” Engineering Journal, AISC, Vol. 10, No. 3, 3rd Quarter, 1973. American Institute of Steel Construction, Inc. (1978), Specification for the Design, Fabrication, and Erection of Structural Steel for Buildings, Chicago, IL, 1978. American Institute of Steel Construction, Inc. (1983), Torsional Analysis of Steel Members, Chicago, IL, 1983. American Institute of Steel Construction, Inc. (1986), Load and Resistance Factor Design Specification for Structural Steel Buildings, Chicago, IL, 1986. American Institute of Steel Construction, Inc. (1989), Specification for Structural Steel Buildings—Allowable Stress Design and Plastic Design, Chicago, IL, 1989. American Institute of Steel Construction, Inc. (1992), Seismic Provisions of Structural Steel Buildings, AISC, Chicago, IL, 1992. American Society of Civil Engineers (1971), “Plastic Design in Steel,” ASCE Manual of Engineering Practice, No. 41, 2nd Edition, New York, 1971. American Society of Civil Engineers (1979), Structural Design of Tall Steel Buildings, New York, 1979. American Society of Civil Engineers (1988), Minimum Design Loads for Buildings and Other Structures, ASCE7-88, New York. ASCE Task Committee on Design Criteria for Composite Structures in Steel and Concrete (1992), “Proposed Specification for Structural Steel Beams with Web Openings,” D. Darwin, Chmn., Journal of Structural Engineering, ASCE, Vol. 118, No. ST12, December 1992. ASCE Task Committee on Design Criteria for Composite Structures in Steel and Concrete (1992a), “Commentary on Proposed Specification for Structural Steel AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 258

REFERENCES

Beams with Web Openings,” D. Darwin, Chmn., Journal of Structural Engineering, ASCE, Vol. 118, No. ST12, December 1992. Joint ASCE-AASHTO Committee on Flexural Members (1968), “Design of Hybrid Steel Beams,” Report of Subcommitte 1, Journal of the Structural Division, ASCE, Vol. 94, No. ST6, June 1968. Aslani, F. and S. C. Goel (1991), “An Analytical Criteria for Buckling Strength of Built-Up Compression Members,” Engineering Journal, AISC, 4th Quarter, 1991. ATC (1978), Tentative Provisions for the Development of Seismic Regulations for Buildings, ATC Publication 3-06, June 1978. American Welding Bureau (1931), Report of Structural Welding Committee, 1931. Austin, W. J. (1961), “Strength and Design of Metal Beam-Columns,” Journal of the Structural Division, ASCE, Vol. 87, No. ST4, April 1961. Basler, K. (1961), “Strength of Plate Girders in Shear,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, October 1961. Basler, K. and B. Thurlimann (1963), “Strength of Plate Girders in Bending,” Journal of the Structural Division, ASCE, Vol. 89, No. ST4, August 1963. Barakat, Munzer and Wai-Fah Chen (1991), “Design Analysis of Semi-Rigid Frames: Evaluation and Implementation,” Engineering Journal, AISC, Vol. 28, No. 2, 2nd Quarter, 1991. Beedle, L. S and L. Tall (1960), “Basic Column Strength,” Journal of the Structural Division, ASCE, Vol. 86, No. ST7, July 1960. Bendigo, R. A., R. M. Hansen, and J. L. Rumpf (1963), “Long Bolted Joints,” Journal of the Structural Division, ASCE, Vol. 89, No. ST6, December 1963. Bergfelt, A. (1971), Studies and Tests on Slender Plate Girders Without Stiffeners, March 1971. Bigos, J., G. W. Smith, E. F. Ball, and P. J. Foehl (1954), Shop Paint and Painting Practice, 1954 Proceedings, AISC National Engineering Conference, Milwaukee, WI. Birkemoe, P. C. and M. I. Gilmor (1978), “Behavior of Bearing-Critical Double-Angle Beam Connections,” Engineering Journal, AISC, Vol. 15, No. 4, 4th Quarter, 1978. Bjorhovde, Reidar (1972), Deterministic and Probabilistic Approaches to the Strength of Steel Columns, Ph.D. Dissertation, Lehigh University, Bethlehem, PA, May. Bjorhovde, Reidar (1984), “Effect of End Restraint on Column Strength—Practical Applications,” Engineering Journal, AISC, Vol. 21, No. 1, 1st Quarter, 1984. Bjorhovde, Reidar (1988), “Columns: From Theory to Practice,” Engineering Journal, AISC, Vol. 25, No. 1, 1st Quarter 1988. Bjorhovde, Reidar, T. V. Galambos, and M. K. Ravindra (1978), “LRFD Criteria for Steel Beam-Columns,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September. Bleich, F. (1952), Buckling Strength of Metal Structures, McGraw-Hill Book Co., New York, NY, 1952. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

REFERENCES

6 - 259

Brandt, G. D. (1982), “A General Solution for Eccentric Loads on Weld Groups,” Engineering Journal, AISC, Vol. 19, No. 3, 3rd Quarter, 1982. Bridge, P. Q. and J. W. Roderick (1978), “Behavior of Built-Up Composite Columns,” Journal of the Structural Division, ASCE, Vol. 104, No. ST7, July 1978, pp. 1141– 1165. Brockenbrough, R. L. (1983), “Considerations in the Design of Bolted Joints for Weathering Steel,” Engineering Journal, AISC, Vol. 20, No. 1, 1st Quarter, 1983. Butler, Lorne J., Shubendu Pal, and Geoffrey L. Kulak (1972), “Eccentrically Loaded Welded Connections,” Journal of the Structural Division, ASCE, Vol. 98, No. ST5, May 1972. Canadian Standards Association (1989), Limit States Design of Steel Structures, Appendices G, H, and I, CSA S16.1-M89, Rexdale, Ontario, Canada, 1989. Chen, W. F. and T. Atsuta (1976), Theory of Beam Columns, Volume I: In-Plane Behavior and Design, McGraw-Hill, New York, 1976. Chen, W. F. and T. Atsuta (1977), Theory of Beam Columns, Volume II: Space Behavior and Design, McGraw-Hill, New York, 1977. Cheong-Siat Moy, F., E. Ozer, and L. W. Lu (1977), “Strength of Steel Frames Under Gravity Loads,” Journal of the Structural Division, ASCE, Vol. 103, No. ST6, June 1977. Chopra, A. K. and N. M. Newmark (1980), Design of Earthquake Resistant Structures, John Wiley and Sons, Inc., New York, NY, 1980. Cooper, P. B., T. V. Galambos, and M. K. Ravindra (1978), “LRFD Criteria for Plate Girders,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September 1978. Daniels, J. H. and L. W. Lu (1972), “Plastic Subassemblage Analysis for Unbraced Frames,” Journal of the Structural Division, ASCE, Vol. 98, No. ST8, August 1972. Darwin, David (1990), Steel and Composite Beams with Web Openings, AISC Steel Design Guide Series No. 2, AISC, Chicago, IL, 1990. Davison, J. B., P. A. Kirby, and D. A. Nethercot (1988), “Semi-Rigid Connections in Isolation and in Frames,” in Connections in Steel Structures: Behavior, Strength and Design, Elsevier Applied Science Publishers, London. Disque, R. O. (1973), “Inelastic K-Factor in Design,” Engineering Journal, AISC, Vol. 10, No. 2, 2nd Quarter, 1973. Easterling, W. S. and L. Gonzales (1993), “Shear Lag Effects in Steel Tension Members,” Engineering Journal, AISC, Vol. 30, No. 2, 2nd Quarter, 1993. Elgaaly, M. (1983), “Web Design Under Compressive Edge Loads,” Engineering Journal, AISC, Vol. 20, No. 4, 4th Quarter, 1983. Elgaaly, M. and R. Salkar (1991), “Web Crippling Under Edge Loading,” Proceedings of AISC National Steel Construction Conference, Washington, D.C., 1991. Ellifritt, D. S., G. Wine, T. Sputo, and S. Samuel (1992), “Flexural Strength of WT Sections,” Engineering Journal, AISC, Vol. 29, No. 2, 2nd Quarter, 1992. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 260

REFERENCES

Ellingwood, B. E., J. G. MacGregor, T. V. Galambos, and C. A. Cornell (1982), “Probability-Based Load Criteria: Load Factors and Load Combinations,” Journal of the Structural Division, ASCE, Vol. 108, No. 5, 1982. Fielding, D. J. and J. S. Huang (1971), “Shear in Steel Beam-to-Column Connections,” Welding Journal, AWS, Vol. 50, No. 7, Research Supplement 1971, pp. 313–326. Fielding, D. J. and W. F. Chen (1973), “Steel Frame Analysis and Connection Shear Deformation,” Journal of the Structural Division, ASCE, Vol. 99, No. ST1, January 1973. Fisher, J. W., P. A. Albrecht, B. T. Yen, D. J. Klingerman, and B. M. McNamee (1974), Fatigue Strength of Steel Beams With Welded Stiffeners and Attachments, National Cooperative Highway Research Program, Report 147, Washington, D.C., 1974. Fisher, J. W., K. H. Frank, M. A. Hirt, and B. M. McNamee (1970), Effect of Weldments on the Fatigue Strength of Beams, National Cooperative Highway Research Program, Report 102, Washington, D.C., 1970. Fisher, J. W., T. V. Galambos, G. L. Kulak, and M. K. Ravindra (1978), “Load and Resistance Factor Design Criteria for Connectors,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September 1978. Frank, K. H. and J. A. Yura (1981), An Experimental Study of Bolted Shear Connections, FHWA/RD-81/148, December 1981. Freeman, F. R. (1930), The Strength of Arc-Welded Joints, Proc. Inst. Civil Engineers, Vol. 231, London, England, 1930. Galambos, T. V. (1960), “Influence of Partial Base Fixity on Frame Stability,” Journal of the Structural Division, ASCE, Vol. 86, No. ST5, May 1960. Galambos, T. V. (1968), Structural Members and Frames, Prentice-Hall, Englewood Cliffs, NJ, 1968. Galambos, T. V. (1976), Proposed Criteria for Load Resistance Factor Design of Steel Building Structures, Research Report No. 45, Civil Engineering Dept., Washington University, St. Louis, MO, May 1976. Galambos, T. V. (1978), Bulletin No. 27, American Iron and Steel Institute, Washington, D.C., January 1978. Galambos, T. V. (1980), Reliability of Axially Loaded Columns, Washington University, Department of Civil Engineering, St. Louis, MO, December 1980. Galambos, T. V. (ed.) (1988), Guide to Stability Design Criteria for Metal Structures, Structural Stability Research Council, 4th Edition, John Wiley & Sons, 1988. Galambos, T. V. (1991), “Design of Axially Loaded Compressed Angles,” Structural Stability Research Council Annual Technical Session Proceedings, 1991. Galambos, T.V. and J. Chapuis (1980), LRFD Criteria for Composite Columns and Beam-Columns, Revised Draft, Washington University, Dept. of Civil Engineering, St. Louis, MO, December 1980. Galambos, T. V., B. Ellingwood, J. G. MacGregor, and C. A. Cornell (1982), “Probability-Based Load Criteria: Assessment of Current Design Practice,” Journal of the Structural Division, ASCE, Vol. 108, No. ST5, May 1982. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

REFERENCES

6 - 261

Galambos, T. V. and M. K. Ravindra (1973), Tentative Load and Resistance Factor Design Criteria for Steel Buildings, Research Report No. 18, Washington University, Dept. of Civil Engineering, St. Louis, MO, September 1973. Galambos, T. V. and M. K. Ravindra (1976), Load and Resistance Factor Design Criteria for Steel Beams, Research Report No. 27, Washington University, Dept. of Civil Engineering, St. Louis, MO, February 1976. Galambos, T. V. and M. K. Ravindra (1978), “Properties of Steel for Use in LRFD,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September 1978. Gaylord, Edwin H. Jr., Charles N. Gaylord, and James E. Stallmeyer (1992), Design of Steel Structures, 3rd Edition, McGraw-Hill Book Co., New York, 1992. Gibson, G. T. and B. T. Wake (1942), “An Investigation of Welded Connections for Angle Tension Members,” The Welding Journal, American Welding Society, January 1942. Goble, G. G. (1968), “Shear Strength of Thin Flange Composite Specimens,” Engineering Journal, AISC, Vol. 5., No. 2, 2nd Quarter, 1968. Grant, J. A., J. W. Fisher, and R. G. Slutter (1977), “Composite Beams with Formed Steel Deck,” Engineering Journal, AISC, Vol. 14, No. 1, 1st Quarter, 1977. Hall, D. H. (1981), “Proposed Steel Column Strength Criteria,” Journal of the Structural Division, ASCE, Vol. 107, No. ST4, April 1981. Hansell, W. C., T. V. Galambos, M. K. Ravindra, and I. M. Viest (1978), “Composite Beam Criteria in LRFD,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September 1978. Hardash, S. G. and R. Bjorhovde (1985), “New Design Criteria for Gusset Plates in Tension,” Engineering Journal, AISC, Vol. 22, No. 2, 2nd Quarter, 1985. Hoglund, T. (1971), Simply Supported Long Thin Plate I-Girders Without Web Stiffeners, Subjected to Distributed Transverse Load, Dept. of Building Statics and Structural Engineering of the Royal Institute of Technology, Stockholm, Sweden, 1971. International Association of Bridge and Structural Engineering (1968), Final Report of the Eighth Congress, Zurich, September 1968. International Organization for Standardization (1974), Guide for the Evaluation of Human Exposure to Whole-Body Vibration, Document ISO 2631, September 1974. Iwankiw, N. (1984), “Note on Beam-Column Moment Amplification Factor,” Engineering Journal, AISC, Vol. 21, No. 1, 1st Quarter, 1984. Johnson, D. L. (1985), An Investigation into the Interaction of Flanges and Webs in Wide-Flange Shapes, 1985 Proceeding SSRC Annual Technical Session, Cleveland, OH, Structural Stability Research Council, Lehigh University, Bethlehem, PA. Johnston, B. G. (1939), Pin-Connected Plate Links, 1939 ASCE Transactions. Johnston, B. G. and Deits, “Tests of Miscellaneous Welded Building Connections,” Welding Journal, November 1941. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 262

REFERENCES

Johnston, B. G. and L. F. Green (1940), “Flexible Welded Angle Connections,” The Welding Journal, October 1940. Jones, J. (1940), “Static Tests on Riveted Joints,” Civil Engineering, May 1940. Kanchanalai, T. (1977), The Design and Behavior of Beam-Columns in Unbraced Steel Frames, AISI Project No. 189, Report No. 2, Civil Engineering/Structures Research Lab, University of Texas-Austin, October 1977. Kanchanalai, T. and L. W. Lu (1979), “Analysis and Design of Framed Columns Under Minor Axis Bending,” Engineering Journal, AISC, Vol. 16, No. 2, 2nd Quarter, 1979. Keating, P. B. and J. W. Fisher (1985), Review of Fatigue Tests and Design Criteria on Welded Details, NCHRP Project 12-15(50), October 1985, Washington, D.C. Ketter, R. L. (1961), “Further Studies of the Strength of Beam-Columns,” Journal of the Structural Division, ASCE, Vol. 87, No. ST6, August 1961. Kirby, P. A. and D. A. Nethercot (1979), Design for Structural Stability, John Wiley and Sons, Inc., New York, NY, 1979. Kitipornchai, S. and N. S. Trahair (1980), “Buckling Properties of Monosymmetric I-Beams,” Journal of the Structural Division, ASCE, Vol. 109, No. ST5, May 1980. Kloppel, K. and T. Seeger (1964), “Dauerversuche Mit Einsohnittigen Hv-Verbindurgen Aus ST37,” Der Stahlbau, Vol. 33, No. 8, August 1964, pp. 225–245 and Vol. 33, No. 11, November 1964, pp. 335–346. Kotecki, D. S. and R. A. Moll (1970), “A Toughness Study of Steel Weld Metal from Self-Shielded, Flux-Cored Electrodes, Part 1,” Welding Journal, Vol. 49, April 1970. Kotecki, D. S. and R. A. Moll (1972), “A Toughness Study of Steel Weld Metal from Self-Shielded, Flux-Cored Electrodes, Part 2,” Welding Journal, Vol. 51, March 1972. Kulak, G. L., J. W. Fisher, and J. H. A. Struik (1987), Guide to Design Criteria for Bolted and Riveted Joints, 2nd Edition, John Wiley & Sons, New York, NY, 1987. Lee, G. D., M. L. Morrell, and R. L. Ketter (1972), “Design of Tapered Members,” WRC Bulletin, No. 173, June 1972. LeMessurier, W. J. (1976), “A Practical Method of Second Order Analysis, Part 1— Pin-Jointed Frames,” Engineering Journal, AISC, Vol. 13, No. 4, 4th Quarter, 1976. LeMessurier, W. J. (1977), “A Practical Method of Second Order Analysis, Part 2— Rigid Frames,” Engineering Journal, AISC, Vol. 14, No. 2, 2nd Quarter, 1977. LeMessurier, W. J., R. J. McNamara, and J. C. Scrivener (1974), “Approximate Analytical Model for Multi-Story Frames,” Engineering Journal, AISC, Vol. 11, No. 4, 4th Quarter, 1974. Lesik, D. F. and D. J. L. Kennedy (1990), “Ultimate Strength of Fillet Welded Connections Loaded in Plane,” Canadian Journal of Civil Engineering, National Research Council of Canada, Ottawa, Canada, Vol. 17, No. 1, 1990. Liapunov, S. (1974), “Ultimate Load Studies of Plane Multi-Story Steel Rigid AMERICAN INSTITUTE OF STEEL CONSTRUCTION

REFERENCES

6 - 263

Frames,” Journal of the Structural Division, ASCE, Vol. 100, No. ST8, Proc. Paper 10750, August 1974. Lim, L. C. and L. W. Lu (1970), The Strength and Behavior of Laterally Unsupported Columns, Fritz Engineering Laboratory Report No. 329.5, Lehigh University, Bethlehem, PA, June 1970. Lu, L. W. (1967), “Design of Braced Multi-Story Frames by the Plastic Method,” Engineering Journal, AISC, Vol. 4, No. 1, 1st Quarter, 1967. Lu, L. W., E. Ozer, J. H. Daniels, O. S. Okten, and S. Morino (1977), “Strength and Drift Characteristics of Steel Frames,” Journal of the Structural Division, ASCE, Vol. 103, No. ST11, November 1977. Lyse, I. and Schreiner, “An Investigation of Welded Seat Angle Connections,” Welding Journal, February 1935. Lyse, I. and G. J. Gibson, “Effect of Welded Top Angles on Beam-Column Connections,” Welding Journal, October 1937. Marino, F. J. (1966), “Ponding of Two-Way Roof Systems,” Engineering Journal, AISC, Vol. 3, No. 3, 3rd Quarter, 1966. Morrell, M. L. and G. C. Lee (1974), Allowable Stress for Web-Tapered Members, WRC Bulletin 192, Welding Research Council, New York, February 1974. Munse, W. H. and E. Chesson, Jr. (1963), “Riveted and Bolted Joints: Net Section Design,” Journal of the Structural Division, ASCE, Vol. 89, No. ST1, February 1963. Murray, Thomas M. (1991), “Building Floor Vibrations,” Engineering Journal, AISC, Vol. 28, No. 3, 3rd Quarter 1991. NEHRP (1992), NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, Federal Emergency Management Agency Report, FEMA 222, January 1992. Ollgaard, J. G., R. G. Slutter, and J. W. Fisher (1971), “Shear Strength of Stud Shear Connections in Lightweight and Normal Weight Concrete,” Engineering Journal, AISC, Vol. 8, No. 2, 2nd Quarter, 1971. Popov, E. P. (1980), “An Update on Eccentric Seismic Bracing,” Engineering Journal, AISC, Vol. 17, No. 3, 3rd Quarter, 1980. Popov, E. P. and R. M. Stephen (1977), “Capacity of Columns with Splice Imperfections,” Engineering Journal, AISC, Vol. 14, No. 1, 1st Quarter, 1977. Preece, F. R. (1968), AWS-AISC Fillet Weld Study—Longitudinal and Transverse Shear Tests, Testing Engineers, Inc., Los Angeles, May 31, 1968. Rao, N. R. N., M. Lohrmann, and L. Tall (1966), “Effect of Strain Rate on the Yield Stress of Structural Steels,” Journal of Materials, Vol. 1, No. 1, ASTM, March 1966. Ravindra, M. K. and T. V. Galambos (1978), “Load and Resistance Factor Design for Steel,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September 1978. Research Council on Structural Connections (1988), Load and Resistance Factor AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 264

REFERENCES

Design Specification for Structural Joints Using ASTM A325 or A490 Bolts, 1988, AISC, Chicago, IL. Ricles, J. M. and J. A. Yura (1983), “Strength of Double-Row Bolted Web Connections,” Journal of the Structural Division, ASCE, Vol. 109, No. ST1, January 1983. Roberts, T. M. (1981), “Slender Plate Girders Subjected to Edge Loading,” Proceedings, Institution of Civil Engineers, Part 2, 71, September 1981, London. Ross, D. A. and W. F. Chen (1976), “Design Criteria for Steel I-Columns Under Axial Load and Biaxial Bending,” Canadian Journal of Civil Engineering, Vol. 3, No. 3, 1976. Salvadori, M. (1956), “Lateral Buckling of Eccentrically Loaded I-Columns,” 1956 ASCE Transactions, Vol. 122-1. Sherman, D. R. (1976), Tentative Criteria for Structural Applications of Steel Tubing and Pipe, American Iron and Steel Institute, Washington, D.C., August 1976. Sherman, D. R. and A. S. Tanavde (1984), Comparative Study of Flexural Capacity of Pipes, Civil Engineering Department Report, University of Wisconsin-Milwaukee, March 1984. Slutter, R. G. and G. C. Driscoll (1965), “Flexural Strength of Steel-Concrete Composite Beams,” Journal of the Structural Division, ASCE, Vol. 91, No. ST2, April 1965. Springfield, J. (1975), “Design of Column Subject to Biaxial Bending,” Engineering Journal, AISC, Vol. 12, No. 3, 3rd Quarter, 1975. Springfield, J. and P. F. Adams (1972), “Aspects of Column Design in Tall Steel Buildings,” Journal of the Structural Division, ASCE, Vol. 9, No. ST5, May 1972. Stang, A. H. and B. S. Jaffe (1948), Perforated Cover Plates for Steel Columns, Research Paper RP1861, National Bureau of Standards, 1948, Washington, D.C. Steel Structures Painting Council (1989), Steel Structures Painting Manual, Vol. 2, Systems and Specifications, SSPC, Pittsburgh, PA, 1982. Structural Stability Research Council Task Group 20 (1979), “A Specification for the Design of Steel-Concrete Composite Columns,” Engineering Journal, AISC, Vol. 16, No. 4, 4th Quarter, 1979. Summers, Paul A. and Joseph A. Yura (1982), The Behavior of Beams Subjected to Concentrated Loads, Phil M. Ferguson Structural Engineering Laboratory Report No. 82-5, University of Texas, Austin, TX, August 1982. Tebedge, N. and W. F. Chen (1974), “Design Criteria for H-Columns Under Biaxial Loading,” ASCE Journal of the Structural Division, Vol. 100, ST3, 1974. Terashima, H. and P. H. M. Hart (1984), “Effect of Aluminum on Carbon, Manganese, Niobium Steel Submerged Arc Weld Metal Properties,” Welding Journal, Vol. 63, June 1984. Tide, R. H. R. (1985), Reasonable Column Design Equations, Annual Technical Session of Structural Stability Research Council, April 16–17, 1985, Cleveland, OH, SSRC, Lehigh University, Bethlehem, PA. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

REFERENCES

6 - 265

Timoshenko, S. P. and J. M. Gere (1961), Theory of Elastic Stability, McGraw-Hill Book Company, 1961. Wilson, W. M. (1934), The Bearing Value of Rollers, Bulletin No. 263, University of Illinois Engineering Experiment Station, Urbana, IL, 1934. Winter, G. (1958), “Lateral Bracing of Columns and Beams,” Journal of the Structural Division, ASCE, Vol. 84, No. ST2, March 1958. Winter, G. (1970), Commentary on the 1968 Edition of Light Gage Cold-Formed Steel Design Manual, American Iron and Steel Institute, Washington, D.C., 1970. Wood, B. R., D. Beaulieu, and P. F. Adams (1976), “Column Design by P-Delta Method,” Journal of the Structural Division, ASCE, Vol. 102, No. ST2, February 1976. Yura, J. A. (1971), “The Effective Length of Columns in Unbraced Frames,” Engineering Journal, AISC, Vol. 8, No. 2, April 1971. Yura, J. A., et al. (1978), “The Bending Resistance of Steel Beams,” Journal of the Structural Division, ASCE, Vol. 104, No. ST9, September 1978. Yura, Joseph A. (1988), Elements for Teaching Load & Resistance Factor Design, AISC, Chicago, IL, April 1988. Zandonini, R. (1985), “Stability of Compact Built-Up Struts: Experimental Investigation and Numerical Simulation,” (in Italian) Construzioni Metalliche, No. 4, 1985. Zhou, S. P. and W. F. Chen (1985), “Design Criteria for Box Columns Under Biaxial Loading,” Journal of Structural Engineering, ASCE, Vol. 111, No. ST12, December 1985.


6 - 266


6 - 267

Glossary Alignment chart for columns. A nomograph for determining the effective length factor K for some types of columns Amplification factor. A multiplier of the value of moment or deflection in the unbraced length of an axially loaded member to reflect the secondary values generated by the eccentricity of the applied axial load within the member Aspect ratio. In any rectangular configuration, the ratio of the lengths of the sides Batten plate. A plate element used to join two parallel components of a built-up column, girder, or strut rigidly connected to the parallel components and designed to transmit shear between them Beam. A structural member whose primary function is to carry loads transverse to its longitudinal axis Beam-column. A structural member whose primary function is to carry loads both transverse and parallel to its longitudinal axis Bent. A plane framework of beam or truss members which support loads and the columns which support these members Biaxial bending. Simultaneous bending of a member about two perpendicular axes Bifurcation. The phenomenon whereby a perfectly straight member under compression may either assume a deflected position or may remain undeflected, or a beam under flexure may either deflect and twist out of plane or remain in its in-plane deflected position Braced frame. A frame in which the resistance to lateral load or frame instability is primarily provided by a diagonal, a K brace, or other auxiliary system of bracing Brittle fracture. Abrupt cleavage with little or no prior ductile deformation Buckling load. The load at which a perfectly straight member under compression assumes a deflected position Built-up member. A member made of structural metal elements that are welded, bolted, or riveted together Cladding. The exterior covering of the structural components of a building Cold-formed members. Structural members formed from steel without the application of heat Column. A structural member whose primary function is to carry loads parallel to its longitudinal axis Column curve. A curve expressing the relationship between an axial column strength and slenderness ratio Combined mechanism. A mechanism determined by plastic analysis procedure which combines elementary beam, panel, and joint mechanisms Compact section. Compact sections are capable of developing a fully plastic stress distribution and possess rotation capacity of approximately three before the onset of local buckling AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 268

GLOSSARY

Composite beam. A steel beam structurally connected to a concrete slab so that the beam and slab respond to loads as a unit. See also Concrete-encased beam Concrete-encased beam. A beam totally encased in concrete cast integrally with the slab Connection. Combination of joints used to transmit forces between two or more members. Categorized by the type and amount of force transferred (moment, shear, end reaction). See also Splices Critical load. The load at which bifurcation occurs as determined by a theoretical stability analysis Curvature. The rotation per unit length due to bending Design documents. See Structural design documents Design strength. Resistance (force, moment, stress, as appropriate) provided by element or connection; the product of the nominal strength and the resistance factor Diagonal bracing. Inclined structural members carrying primarily axial load employed to enable a structural frame to act as a truss to resist horizontal loads Diaphragm. Floor slab, metal wall, or roof panel possessing a large in-plane shear stiffness and strength adequate to transmit horizontal forces to resisting systems Diaphragm action. The in-plane action of a floor system (also roofs and walls) such that all columns framing into the floor from above and below are maintained in their same position relative to each other Double concentrated forces. Two equal and opposite forces which form a couple on the same side of the loaded member Double curvature. A bending condition in which end moments on a member cause the member to assume an S shape Drift. Lateral deflection of a building Drift index. The ratio of lateral deflection to the height of the building Ductility factor. The ratio of the total deformation at maximum load to the elastic-limit deformation Effective length. The equivalent length KL used in compression formulas and determined by a bifurcation analysis Effective length factor K. The ratio between the effective length and the unbraced length of the member measured between the centers of gravity of the bracing members Effective moment of inertia. The moment of inertia of the cross section of a member that remains elastic when partial plastification of the cross section takes place, usually under the combination of residual stress and applied stress. Also, the moment of inertia based on effective widths of elements that buckle locally. Also, the moment of inertia used in the design of partially composite members Effective stiffness. The stiffness of a member computed using the effective moment of inertia of its cross section Effective width. The reduced width of a plate or slab which, with an assumed uniform stress distribution, produces the same effect on the behavior of a structural member as the actual plate width with its nonuniform stress distribution Elastic analysis. Determination of load effects (force, moment, stress, as appropriate) on members and connections based on the assumption that material deformation disappears on removal of the force that produced it Elastic-perfectly plastic. A material which has an idealized stress-strain curve that varies linearly from the point of zero strain and zero stress up to the yield point AMERICAN INSTITUTE OF STEEL CONSTRUCTION

GLOSSARY

6 - 269

of the material, and then increases in strain at the value of the yield stress without any further increases in stress Embedment. A steel component cast in a concrete structure which is used to transmit externally applied loads to the concrete structure by means of bearing, shear, bond, friction, or any combination thereof. The embedment may be fabricated of structural-steel plates, shapes, bars, bolts, pipe, studs, concrete reinforcing bars, shear connectors, or any combination thereof Encased steel structure. A steel-framed structure in which all of the individual frame members are completely encased in cast-in-place concrete Euler formula. The mathematical relationship expressing the value of the Euler load in terms of the modulus of elasticity, the moment of inertia of the cross section, and the length of a column Euler load. The critical load of a perfectly straight, centrally loaded pin-ended column Eyebar. A particular type of pin-connected tension member of uniform thickness with forged or flame cut head of greater width than the body proportioned to provide approximately equal strength in the head and body Factored load. The product of the nominal load and a load factor Fastener. Generic term for welds, bolts, rivets, or other connecting device Fatigue. A fracture phenomenon resulting from a fluctuating stress cycle First-order analysis. Analysis based on first-order deformations in which equilibrium conditions are formulated on the undeformed structure Flame-cut plate. A plate in which the longitudinal edges have been prepared by oxygen cutting from a larger plate Flat width. For a rectangular tube, the nominal width minus twice the outside corner radius. In absence of knowledge of the corner radius, the flat width may be taken as the total section width minus three times the thickness Flexible connection. A connection permitting a portion, but not all, of the simple beam rotation of a member end Floor system. The system of structural components separating the stories of a building Force. Resultant of distribution of stress over a prescribed area. A reaction that develops in a member as a result of load (formerly called total stress or stress). Generic term signifying axial loads, bending moment, torques, and shears Fracture toughness. Measurement of the ability to absorb energy without fracture. Generally determined by impact loading of specimens containing a notch having a prescribed geometry Frame buckling. A condition under which bifurcation may occur in a frame Frame instability. A condition under which a frame deforms with increasing lateral deflection under a system of increasing applied monotonic loads until a maximum value of the load called the stability limit is reached, after which the frame will continue to deflect without further increase in load Fully composite beam. A composite beam with sufficient shear connectors to develop the full flexural strength of the composite section High-cycle fatigue. Failure resulting from more than 20,000 applications of cyclic stress Hybrid beam. A fabricated steel beam composed of flanges with a greater yield strength than that of the web. Whenever the maximum flange stress is less than or equal to the web yield stress the girder is considered homogeneous Hysteresis loop. A plot of force versus displacement of a structure or member subjected to reversed, repeated load into the inelastic range, in which the path followed AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 270

GLOSSARY

during release and removal of load is different from the path for the addition of load over the same range of displacement Inclusions. Nonmetallic material entrapped in otherwise sound metal Incomplete fusion. Lack of union by melting of filler and base metal over entire prescribed area Inelastic action. Material deformation that does not disappear on removal of the force that produced it Instability. A condition reached in the loading of an element or structure in which continued deformation results in a decrease of load-resisting capacity Joint. Area where two or more ends, surfaces, or edges are attached. Categorized by type of fastener or weld used and method of force transfer K bracing. A system of struts used in a braced frame in which the pattern of the struts resembles the letter K, either normal or on its side Lamellar tearing. Separation in highly restrained base metal caused by through-thickness strains induced by shrinkage of adjacent weld metal Lateral bracing member. A member utilized individually or as a component of a lateral bracing system to prevent buckling of members or elements and/or to resist lateral loads Lateral (or lateral-torsional) buckling. Buckling of a member involving lateral deflection and twist Leaning column. Gravity-loaded column where connections to the frame (simple connections) do not provide resistance to lateral loads Limit state. A condition in which a structure or component becomes unfit for service and is judged either to be no longer useful for its intended function (serviceability limit state) or to be unsafe (strength limit state) Limit states. Limits of structural usefulness, such as brittle fracture, plastic collapse, excessive deformation, durability, fatigue, instability, and serviceability Load factor. A factor that accounts for unavoidable deviations of the actual load from the nominal value and for uncertainties in the analysis that transforms the load into a load effect Loads. Forces or other actions that arise on structural systems from the weight of all permanent construction, occupants and their possessions, environmental effects, differential settlement, and restrained dimensional changes. Permanent loads are those loads in which variations in time are rare or of small magnitude. All other loads are variable loads. See Nominal loads LRFD (Load and Resistance Factor Design). A method of proportioning structural components (members, connectors, connecting elements, and assemblages) such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations Local buckling. The buckling of a compression element which may precipitate the failure of the whole member Low-cycle fatigue. Fracture resulting from a relatively high-stress range resulting in a relatively small number of cycles to failure Lower bound load. A load computed on the basis of an assumed equilibrium moment diagram in which the moments are not greater than Mp that is less than or at best equal to the true ultimate load Mechanism. An articulated system able to deform without an increase in load, used in the special sense that the linkage may include real hinges or plastic hinges, or both Mechanism method. A method of plastic analysis in which equilibrium between AMERICAN INSTITUTE OF STEEL CONSTRUCTION

GLOSSARY

6 - 271

external forces and internal plastic hinges is calculated on the basis of an assumed mechanism. The failure load so determined is an upper bound Nominal loads. The magnitudes of the loads specified by the applicable code Nominal strength. The capacity of a structure or component to resist the effects of loads, as determined by computations using specified material strengths and dimensions and formulas derived from accepted principles of structural mechanics or by field tests or laboratory tests of scaled models, allowing for modeling effects and differences between laboratory and field conditions Noncompact section. Noncompact sections can develop the yield stress in compression elements before local buckling occurs, but will not resist inelastic local buckling at strain levels required for a fully plastic stress distribution P-Delta effect. Secondary effect of column axial loads and lateral deflection on the moments in members Panel zone. The zone in a beam-to-column connection that transmits moment by a shear panel Partially composite beam. A composite beam for which the shear strength of shear connectors governs the flexural strength Plane frame. A structural system assumed for the purpose of analysis and design to be two-dimensional Plastic analysis. Determination of load effects (force, moment, stress, as appropriate) on members and connections based on the assumption of rigid-plastic behavior, i.e., that equilibrium is satisfied throughout the structure and yield is not exceeded anywhere. Second order effects may need to be considered Plastic design section. The cross section of a member which can maintain a full plastic moment through large rotations so that a mechanism can develop; the section suitable for plastic design Plastic hinge. A yielded zone which forms in a structural member when the plastic moment is attained. The beam is assumed to rotate as if hinged, except that it is restrained by the plastic moment Mp Plastic-limit load. The maximum load that is attained when a sufficient number of yield zones have formed to permit the structure to deform plastically without further increase in load. It is the largest load a structure will support, when perfect plasticity is assumed and when such factors as instability, second-order effects, strain hardening, and fracture are neglected Plastic mechanism. See Mechanism Plastic modulus. The section modules of resistance to bending of a completely yielded cross section. It is the combined static moment about the neutral axis of the cross-sectional areas above and below that axis Plastic moment. The resisting moment of a fully yielded cross section Plastic strain. The difference between total strain and elastic strain Plastic zone. The yielded region of a member Plastification. The process of successive yielding of fibers in the cross section of a member as bending moment is increased Plate girder. A built-up structural beam Post-buckling strength. The load that can be carried by an element, member, or frame after buckling Primary stress. A primary stress is any normal stress or shear stress developed by an imposed loading which is necessary to satisfy the laws of equilibrium of external and internal forces, moments, and torques. A primary stress is not self-limiting. Redistribution of moment. A process which results in the successive formation of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 272

GLOSSARY

plastic hinges so that less highly stressed portions of a structure may carry increased moments Required strength. Load effect (force, moment, stress, as appropriate) acting on element or connection determined by structural analysis from the factored loads (using most appropriate critical load combinations) Residual stress. The stresses that remain in an unloaded member after it has been formed into a finished product. (Examples of such stresses include, but are not limited to, those induced by cold bending, cooling after rolling, or welding.) Resistance. The capacity of a structure or component to resist the effects of loads. It is determined by computations using specified material strengths, dimensions and formulas derived from accepted principles of structural mechanics, or by field tests or laboratory tests of scaled models, allowing for modeling effects and differences between laboratory and field conditions. Resistance is a generic term that includes both strength and serviceability limit states Resistance factor. A factor that accounts for unavoidable deviations of the actual strength from the nominal value and the manner and consequences of failure Rigid frame. A structure in which connections maintain the angular relationship between beam and column members under load Root of the flange. Location on the web of the corner radius termination point or the toe of the flange-to-web weld. Measured as the k distance from the far side of the flange Rotation capacity. The incremental angular rotation that a given shape can accept prior to local failure defined as R = (θu / θp) − 1 where θu is the overall rotation attained at the factored load state and θp is the idealized rotation corresponding to elastic theory applied to the case of M = Mp St. Venant torsion. That portion of the torsion in a member that induces only shear stresses in the member Second-order analysis. Analysis based on second-order deformations, in which equilibrium conditions are formulated on the deformed structure Service load. Load expected to be supported by the structure under normal usage; often taken as the nominal load Serviceability limit state. Limiting condition affecting the ability of a structure to preserve its appearance, maintainability, durability, or the comfort of its occupants or function of machinery under normal usage Shape factor. The ratio of the plastic moment to the yield moment, or the ratio of the plastic modulus to the section modulus for a cross section Shear friction. Friction between the embedment and the concrete that transmits shear loads. The relative displacement in the plane of the shear load is considered to be resisted by shear-friction anchors located perpendicular to the plane of the shear load Shear lugs. Plates, welded studs, bolts, and other steel shapes that are embedded in the concrete and located transverse to the direction of the shear force and that transmit shear loads, introduced into the concrete by local bearing at the shear lug-concrete interface Shear wall. A wall that in its own plane resists shear forces resulting from applied wind, earthquake, or other transverse loads or provides frame stability. Also called a structural wall Sidesway. The lateral movement of a structure under the action of lateral loads, unsymmetrical vertical loads, or unsymmetrical properties of the structure AMERICAN INSTITUTE OF STEEL CONSTRUCTION

GLOSSARY

6 - 273

Sidesway buckling. The buckling mode of a multistory frame precipitated by the relative lateral displacements of joints, leading to failure by sidesway of the frame Simple plastic theory. See Plastic design Single curvature. A deformed shape of a member having one smooth continuous arc, as opposed to double curvature which contains a reversal Slender-element section. The cross section of a member which will experience local buckling in the elastic range Slenderness ratio. The ratio of the effective length of a column to the radius of gyration of the column, both with respect to the same axis of bending Slip-critical joint. A bolted joint in which the slip resistance of the connection is required Space frame. A three-dimensional structural framework (as contrasted to a plane frame) Splice. The connection between two structural elements joined at their ends to form a single, longer element Stability-limit load. Maximum (theoretical) load a structure can support when secondorder instability effects are included Stepped column. A column with changes from one cross section to another occurring at abrupt points within the length of the column Stiffener. A member, usually an angle or plate, attached to a plate or web of a beam or girder to distribute load, to transfer shear, or to prevent buckling of the member to which it is attached Stiffness. The resistance to deformation of a member or structure measured by the ratio of the applied force to the corresponding displacement Story drift. The difference in horizontal deflection at the top and bottom of a story Strain hardening. Phenomenon wherein ductile steel, after undergoing considerable deformation at or just above yield point, exhibits the capacity to resist substantially higher loading than that which caused initial yielding Strain-hardening strain. For structural steels that have a flat (plastic) region in the stress-strain relationship, the value of the strain at the onset of strain hardening Strength design. A method of proportioning structural members using load factors and resistance factors such that no applicable limit state is exceeded (also called load and resistance factor design) Strength limit state. Limiting condition affecting the safety of the structure, in which the ultimate load-carrying capacity is reached Stress. Force per unit area Stress concentration. Localized stress considerably higher than average (even in uniformly loaded cross sections of uniform thickness) due to abrupt changes in geometry or localized loading Strong axis. The major principal axis of a cross section Structural design documents. Documents prepared by the designer (plans, design details, and job specifications) Structural system. An assemblage of load-carrying components which are joined together to provide regular interaction or interdependence Stub column. A short compression-test specimen, long enough for use in measuring the stress-strain relationship for the complete cross section, but short enough to avoid buckling as a column in the elastic and plastic ranges Subassemblage. A truncated portion of a structural frame Supported frame. A frame which depends upon adjacent braced or unbraced frames for resistance to lateral load or frame instability. (This transfer of load is AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 274

GLOSSARY

frequently provided by the floor or roof system through diaphragm action or by horizontal cross bracing in the roof.) Tangent modulus. At any given stress level, the slope of the stress-strain curve of a material in the inelastic range as determined by the compression test of a small specimen under controlled conditions Temporary structure. A general term for anything that is built or constructed (usually to carry construction loads) that will eventually be removed before or after completion of construction and does not become part of the permanent structural system Tensile strength. The maximum tensile stress that a material is capable of sustaining Tension field action. The behavior of a plate girder panel under shear force in which diagonal tensile stresses develop in the web and compressive forces develop in the transverse stiffeners in a manner analogous to a Pratt truss Toe of the fillet. Termination point of fillet weld or of rolled section fillet Torque-tension relationship. Term applied to the wrench torque required to produce specified pre-tension in high-strength bolts Turn-of-nut method. Procedure whereby the specified pre-tension in high-strength bolts is controlled by rotation of the wrench a predetermined amount after the nut has been tightened to a snug fit Unbraced frame. A frame in which the resistance to lateral load is provided by the bending resistance of frame members and their connections Unbraced length. The distance between braced points of a member, measured between the centers of gravity of the bracing members Undercut. A notch resulting from the melting and removal of base metal at the edge of a weld Universal-mill plate. A plate in which the longitudinal edges have been formed by a rolling process during manufacture. Often abbreviated as UM plate Upper bound load. A load computed on the basis of an assumed mechanism which will always be at best equal to or greater than the true ultimate load Vertical bracing system. A system of shear walls, braced frames, or both, extending through one or more floors of a building Von Mises yield criterion. A theory which states that inelastic action at any point in a body under any combination of stresses begins only when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a simple tensile bar stressed to the elastic limit under a state of uniaxial stress. It is often called the maximum strain-energy-of-distortion theory. Accordingly, shear yield occurs at 0.58 times the yield strength Warping torsion. That portion of the total resistance to torsion that is provided by resistance to warping of the cross section Weak axis. The minor principal axis of a cross section Weathering steel. A type of high-strength, low-alloy steel which can be used in normal environments (not marine) and outdoor exposures without protective paint covering. This steel develops a tight adherent rust at a decreasing rate with respect to time Web buckling. The buckling of a web plate Web crippling. The local failure of a web plate in the immediate vicinity of a concentrated load or reaction Working load. Also called service load. The actual load assumed to be acting on the structure AMERICAN INSTITUTE OF STEEL CONSTRUCTION

GLOSSARY

6 - 275

Yield moment. In a member subjected to bending, the moment at which an outer fiber first attains the yield stress Yield plateau. The portion of the stress-strain curve for uniaxial tension or compression in which the stress remains essentially constant during a period of substantially increased strain Yield point. The first stress in a material at which an increase in strain occurs without an increase in stress, the yield point less than the maximum attainable stress Yield strength. The stress at which a material exhibits a specified limiting deviation from the proportionality of stress to strain. Deviation expressed in terms of strain Yield stress. Yield point, yield strength, or yield stress level as defined Yield-stress level. The average stress during yielding in the plastic range, the stress determined in a tension test when the strain reaches 0.005 in. per in.


6-276


6 - 277

Specification for Load and Resistance Factor Design of Single-Angle Members December 1, 1993

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. One East Wacker Drive, Suite 3100, Chicago, IL 60601-2001

6 - 278

LRFD SPECIFICATION FOR DESIGN OF SINGLE-ANGLE MEMBERS



6 - 279

PREFACE

The intention of the AISC Specification is to cover the common everyday design criteria in routine design office usage. It is not feasible to also cover the many special and unique problems encountered within the full range of structural design practice. This separate Specification and Commentary addresses one such topic—single-angle members—to provide needed design guidance for this more complex structural shape under various load and support conditions. The single-angle design criteria were developed through a consensus process by the AISC Task Committee 116 on Single-Angle Members: Donald R. Sherman, Chairman Hansraj G. Ashar Wai-Fah Chen Raymond D. Ciatto Mohamed Elgaaly Theodore V. Galambos Thomas G. Longlais LeRoy A. Lutz William A. Milek Raymond H. R. Tide Nestor R. Iwankiw, Secretary The assistance of the Structural Stability Research Council Task Group on Single Angles in the preparation and review of this document is acknowledged. The full AISC Committee on Specifications has reviewed and endorsed this Specification. A non-mandatory Commentary provides background for the Specification provisions and the user is encouraged to consult it. The principal changes in this edition include: • establishing upper limit of single-angle flexural strength at 1.25 of the yield moment • increasing resistance factor for compression to 0.90 • removing flexural-torsional buckling consideration for compression members • considering the sense of flexural stresses in the combined force interaction check The reader is cautioned that professional judgment must be exercised when data or recommendations in this Specification are applied. The publication of the material contained herein is not intended as a representation or warranty on the part of the American Institute of Steel Construction, Inc.—or any other person named herein—that this information is suitable for general or particular use, or freedom from infringement of any patent or patents. Anyone making use of this information assumes all liability arising from such use. The design of structures is within the scope of expertise of a competent licensed structural engineer, architect, or other licensed professional for the application of principles to a particular structure. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 280




6 - 281

Specification for Load and Resistance Factor Design of Single-Angle Members December 1, 1993

1.

SCOPE This document contains Load and Resistance Factor Design (LRFD) criteria for hot-rolled, single-angle members with equal and unequal legs in tension, shear, compression, flexure, and for combined forces. It is intended to be compatible with, and a supplement to, the 1993 AISC Specification for Structural Steel Buildings—Load and Resistance Factor Design (AISC LRFD) and repeats some common criteria for ease of reference. For design purposes, the conservative simplifications and approximations in the Specification provisions for single angles are permitted to be refined through a more precise analysis. As an alternative to this Specification, the 1989 AISC Specification for Allowable Stress Design of Single-Angle Members is permitted. The Specification for single-angle design supersedes any comparable but more general requirements of the AISC LRFD. All other design, fabrication, and erection provisions not directly covered by this document shall be in compliance with the AISC LRFD. In the absence of a governing building code, the factored load combinations in AISC LRFD Section A4 shall be used to determine the required strength. For design of slender, cold-formed steel angles, the current AISI LRFD Specification for the Design of Cold-Formed Steel Structural Members is applicable.

2.

TENSION The tensile design strength φtPn shall be the lower value obtained according to the limit states of yielding, φt = 0.9, Pn = Fy Ag, and fracture, φt = 0.75, Pn = Fu Ae. a. For members connected by bolting, the net area and effective net area shall be determined from AISC LRFD Specification Sections B1 to B3 inclusive. b. When the load is transmitted by longitudinal welds only or a combination of AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 282


longitudinal and transverse welds through just one leg of the angle, the effective net area Ae shall be: Ae = AgU

(2-1)

where Ag = gross area of member _  x U = 1 −  ≤ 0.9 l  _ x = connection eccentricity l = length of connection in the direction of loading c. When a load is transmitted by transverse weld through just one leg of the angle, Ae is the area of the connected leg and U = 1. For members whose design is based on tension, the slenderness ratio l / r preferably should not exceed 300. Members in which the design is dictated by tension loading, but which may be subject to some compression under other load conditions, need not satisfy the compression slenderness limits. 3.

SHEAR For the limit state of yielding in shear, the shear stress, fuv, due to flexure and torsion shall not exceed: fuv ≤ φv0.6Fy φv = 0.9

4.

(3-1)

COMPRESSION The design strength of compression members shall be φcPn where φc = 0.90 Pn = AgFcr a. For λc √ Q ≤ 1.5: 2

Fcr = Q (0.658Qλc) Fy

(4-1)

0.877 Fcr =  2  Fy  λc 

(4-2)

b. For λc √ Q ≥ 1.5:

λc =

Kl rπ

 √

Fy E

Fy = specified minimum yield stress of steel Q = reduction factor for local buckling AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 283

The reduction factor Q shall be:

when

b ≤ 0.446 t

 √

E : Fy Q = 1.0

when 0.446

 √

 √

E b < < 0.910 Fy t

E : Fy

Q = 1.34 − 0.761

when

b ≥ 0.910 t

 √

(4-3a)

b t

 √

Fy E

(4-3b)

E : Fy Q=

0.534E

b Fy   t b = full width of longest angle leg t = thickness of angle

2

(4-3c)

For members whose design is based on compressive force, the largest effective slenderness ratio preferably should not exceed 200. 5.

FLEXURE The flexure design strengths of Section 5.1 shall be used as indicated in Sections 5.2 and 5.3

5.1. Flexural Design Strength The flexural design strength shall be limited to the minimum value φbMn determined from Sections 5.1.1, 5.1.2, and 5.1.3, as applicable, with φb = 0.9. 5.1.1. For the limit state of local buckling when the tip of an angle leg is in compression: when

b ≤ 0.382 t

 √

E : Fy Mn = 1.25Fy Sc

when 0.382

 √

E b < ≤ 0.446 Fy t

 √

E : Fy


(5-1a)

6 - 284


b/t Mn = Fy Sc 1.25 − 1.49  − 1    E  0.382   Fy   

 √

when

b > 0.446 t

 √

(5-1b)

E : Fy Mn = QFy Sc

(5-1c)

where b = full width of angle leg with tip in compression Q = reduction factor per Equations 4-3a, b, and c Sc = elastic section modulus to the tip in compression relative to axis of bending E = modulus of elasticity 5.1.2. For the limit state of yielding when the tip of an angle leg is in tension Mn = 1.25My

(5-2)

where My = yield moment about the axis of bending 5.1.3. For the limit state of lateral-torsional buckling: when Mob ≤ My : Mn = [0.92 − 0.17Mob / My]Mob

(5-3a)

when Mob > My :  My / Mob ]My ≤ 1.25My Mn = [1.58 − 0.83 √

(5-3b)

where Mob = elastic lateral-torsional buckling moment, from Section 5.2 or 5.3 as applicable 5.2. Bending about Geometric Axes 5.2.1. a. Angle bending members with lateral-torsion restraint along the length shall be designed on the basis of geometric axis bending with the nominal flexural strength Mn limited to the provisions of Sections 5.1.1 and 5.1.2. b. For equal-leg angles if the lateral-torsional restraint is only at the point of maximum moment, the required moment shall be limited to φbMn per Section 5.1. My shall be computed using the geometric axis section modulus and Mob shall be substituted by using 1.25 times Mob computed from Equation 5-4. 5.2.2. Equal-leg angle members without lateral-torsional restraint subjected to flexure applied about one of the geometric axes are permitted to be designed considering only geometric axis bending provided: AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 285

a. The yield moment shall be based on use of 0.80 of the geometric axis section modulus. b. For the angle-leg tips in compression, the nominal flexural strength Mn shall be determined by the provisions in Section 5.1.1 and in Section 5.1.3, where Mob =

0.66Eb4tCb [√  1 + 0.78(lt / b2)2 − 1] l2

(5-4)

l = unbraced length 12.5Mmax ≤ 1.5 Cb = 2.5Mmax + 3MA + 4MB + 3MC where Mmax = absolute value of maximum moment in the unbraced beam segment MA = absolute value of moment at quarter point of the unbraced beam segment MB = absolute value of moment at centerline of the unbraced beam segment MC = absolute value of moment at three-quarter point of the unbraced beam segment c. For the angle-leg tips in tension, the nominal flexural strength shall be determined according to Section 5.1.2. 5.2.3. Unequal-leg angle members without lateral-torsional restraint subjected to bending about one of the geometric axes shall be designed using Section 5.3. 5.3. Bending about Principal Axes Angles without lateral-torsional restraint shall be designed considering principal-axis bending, except for the alternative of Section 5.2.2, if appropriate. Bending about both of the principal axes shall be evaluated as required in Section 6. 5.3.1. Equal-leg angles: a. Major-axis bending: The nominal flexural strength Mn about the major principal axis shall be determined by the provisions in Section 5.1.1 and in Section 5.1.3, where Mob = Cb

0.46Eb2t2 l

(5-5)

b. Minor-axis bending: The nominal design strength Mn about the minor principal axis shall AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 286


be determined by Section 5.1.1 when the leg tips are in compression, and by Section 5.1.2 when the leg tips are in tension. 5.3.2. Unequal-leg angles: a. Major-axis bending: The nominal flexural strength Mn about the major principal axis shall be determined by the provisions in Section 5.1.1 for the compression leg and in Section 5.1.3, where Mob = 4.9E

Iz Cb [√  β2w + 0.052(lt / rz)2 + βw] l2

(5-6)

Iz = minor principal axis moment of inertia rz = radius of gyration for minor principal axis 1 βw =  ∫ z(w2 + z2)dA − 2zo, special section property for Iw A   unequal-leg angles, positive for short leg in compression and negative for long leg in compression (see Commentary for values for common angle sizes). If the long leg is in compression anywhere along the unbraced length of the member, the negative value of βw shall be used. zo = coordinate along z axis of the shear center with respect to centroid Iw = moment of inertia for major principal axis b. Minor-axis bending: The nominal design strength Mn about the minor principal axis shall be determined by Section 5.1.1 when leg tips are in compression and by Section 5.1.2 when the leg tips are in tension. 6.

COMBINED FORCES The interaction equation shall be evaluated for the principal bending axes either by addition of all the maximum axial and flexural terms, or by considering the sense of the associated flexural stresses at the critical points of the cross section, the flexural terms are either added to or subtracted from the axial load term.

6.1. Members in Flexure and Axial Compression 6.1.1. The interaction of flexure and axial compression applicable to specific locations on the cross section shall be limited by Equations 6-1a and 6-1b: For

Pu ≥ 0.2 φPn Muz    Pu 8  Muw  φP + 9  φ M + φ M   ≤ 1.0 b nz    b nw  n AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(6-1a)


For

6 - 287

Pu ≤ 0.2 φPn

 Muw Muz    Pu (6-1b)  2φP +  φ M + φ M   ≤ 1.0 b nz    n  b nw Pu = required compressive strength Pn = nominal compressive strength determined in accordance with Section 4 Mu = required flexural strength Mn = nominal flexural strength for tension or compression in accordance with Section 5, as appropriate. Use section modulus for specific location in the cross section and consider the type of stress. φ = φc = resistance factor for compression = 0.90 φb = resistance factor for flexure = 0.90 w = subscript relating symbol to major-axis bending z = subscript relating symbol to minor-axis bending In Equations 6-1a and 6-1b when Mn represents the flexural strength of the compression side, the corresponding Mu shall be multiplied by B1. B1 =

Cm ≥ 1.0 Pu 1− Pe1

(6-2)

Cm = bending coefficient defined in AISC LRFD Pe1 = elastic buckling load for the braced frame defined in AISC LRFD 6.1.2. For members constrained to bend about a geometric axis with nominal flexural strength determined per Section 5.2.1, the radius of gyration r for Pe1 shall be taken as the geometric axis value. The bending terms for the principal axes in Equations 6-1a and 6-1b shall be replaced by a single geometric axis term. 6.1.3. Alternatively, for equal-leg angles without lateral-torsional restraint along the length and with bending applied about one of the geometric axes, the provisions of Section 5.2.2 are permitted for the required and design bending strength. If Section 5.2.2 is used for Mn, the radius of gyration about the axis of bending r for Pe1 shall be taken as the geometric axis value of r divided by 1.35 in the absence of a more detailed analysis. The bending terms for the principal axes in Equations 6-1a and 6-1b shall be replaced by a single geometric axis term. 6.2. Members in Flexure and Axial Tension The interaction of flexure and axial tension shall be limited by Equations 6-1a and 6-1b where Pu = required tensile strength Pn = nominal tensile strength determined in accordance with Section 2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 288


Mu = required flexural strength Mn = nominal flexural strength for tension or compression in accordance with Section 5, as appropriate. Use section modulus for specific location in the cross section and consider the type of stress. φ = φt = resistance factor for tension = 0.90 φb = resistance factor for flexure = 0.90 For members subject to bending about a geometric axis, the required bending strength evaluation shall be in accordance with Sections 6.1.2 and 6.1.3. Second-order effects due to axial tension and bending interaction are permitted to be considered in the determination of Mu for use in Formulas 6-1a and 6-1b. In lieu of using Formulas 6-1a and 6-1b, a more detailed analysis of the interaction of flexure and tension is permitted.



6 - 289

Commentary on the Specification for Load and Resistance Factor Design of Single-Angle Members December 1, 1993

INTRODUCTION This Specification is intended to be complete for normal design usage in conjunction with the main 1993 AISC LRFD Specification and Commentary. This Commentary furnishes background information and references for the benefit of the engineer seeking further understanding of the derivation and limits of the specification. The Specification and Commentary are intended for use by design professionals with demonstrated engineering competence. C2. TENSION The criteria for the design of tension members in AISC LRFD Specification Section D1 have been adopted for angles with bolted connections. However, recognizing the effect of shear lag when the connection is welded, the criteria in Section B3 of the AISC LRFD Specification have been applied. The advisory upper slenderness limits are not due to strength considerations but are based on professional judgment and practical considerations of economics, ease of handling, and transportability. The radius of gyration about the z axis will produce the maximum l / r and, except for very unusual support conditions, the maximum Kl / r. Since the advisory slenderness limit for compression members is less than for tension members, an accommodation has been made for members with Kl / r > 200 that are always in tension, except for unusual load conditions which produce a small compression force. C3. SHEAR Shear stress due to factored loads in a single-angle member are the result of the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 290

COMMENTARY

gradient in the bending moment along the length (flexural shear) and the torsional moment. The maximum elastic stress due to flexural shear may be computed by fv =

1.5Vb bt

(C3-1)

where Vb = component of the shear force parallel to the angle leg with length b and thickness t, kips The stress, which is constant through the thickness, should be determined for both legs to determine the maximum. The 1.5 factor is the calculated elastic value for equal-leg angles loaded along one of the principal axes. For equal-leg angles loaded along one of the geometric axes (laterally braced or unbraced) the factor is 1.35. Constants between these limits may be calculated conservatively from Vb Q / It to determine the maximum stress at the neutral axis. Alternatively, if only flexural shear is considered, a uniform flexural shear stress in the leg of Vb / bt may be used due to inelastic material behavior and stress redistribution. If the angle is not laterally braced against twist, a torsional moment is produced equal to the applied transverse load times the perpendicular distance e to the shear center, which is at the heel of the angle cross section. Torsional moments are resisted by two types of shear behavior: pure torsion (St. Venant) and warping torsion (AISC, 1983). If the boundary conditions are such that the cross section is free to warp, the applied torsional moment MT is resisted by pure shear stresses as shown in Figure C3.1a. Except near the ends of the legs, these stresses are constant along the length of the leg, and the maximum value can be approximated by fv = MT t / J =

3MT At

(C3-2)

e

P MT = Pe

(a) Pure torsion

(b) In-plane warping

(c) Across-thickness warping

Fig. C.3.1. Shear stresses due to torsion. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 291

where J = torsional constant (approximated by Σbt3 / 3 when precomputed value unavailable) A = angle cross-sectional area At section where warping is restrained, the torsional moment is resisted by warping shear stresses of two types (Gjelsvik, 1981). One type is in-plane (contour) as shown in Figure C3.1b, which varies from zero at the toe to a maximum at the heel of the angle. The other type is across the thickness and is sometimes referred to as secondary warping shear. As indicated in Figure C3.1c, it varies from zero at the heel to a maximum at the toe. In an angle with typical boundary conditions and unrestrained load point, the torsional moment produces all three types of shear stresses (pure, in-plane warping, and secondary warping) in varying proportions along its length. The total applied moment is resisted by a combination of three types of internal moments that differ in relative proportions according to the distance from the boundary condition. Using typical angle dimensions, it can be shown that the two warping shears are approximately the same order of magnitude and are less than 20 percent of the pure shear stress for the same torsional moment. Therefore, it is conservative to compute the torsional shear stress using the pure shear equation and total applied torsional moment MT as if no warping restraint were present. This stress is added directly to the flexural shear stress to produce a maximum surface shear stress near the mid-length of a leg. Since this sum is a local maximum that does not extend through the thickness, applying the limit of φv0.6Fy adds another degree of conservatism relative to the design of other structural shapes. In general, torsional moments from laterally unrestrained transverse loads also produce warping normal stresses that are superimposed on bending stresses. However, since the warping strength for a single angle is relatively small, this additional bending effect is negligible and often ignored in design practice. C4. COMPRESSION The provisions for the critical compression stress account for the three possible limit states that may occur in an angle column depending on its proportions: general column flexural buckling, local buckling of thin legs, and flexural-torsional buckling of the member. The Q-factor in the equation for critical stress accounts for the local buckling, and the expressions for Q are nondimensionalized from AISC LRFD Specification (AISC, 1993) Appendix B5. Flexural-torsional buckling is covered in Appendix E of the AISC LRFD Specification (AISC, 1993). This strength limit state is approximated by the Q-factor reduction for slender-angle legs. For non-slender sections where Q = 1, flexural-torsional buckling is relevant for relatively short columns, but it was shown by Galambos (1991) that the error of neglecting this effect is not significant. For this reason no explicit consideration of this effect is required in these single-angle specifications. The provisions of Appendix E of AISC LRFD may be conservatively used to directly consider flexural-torsional buckling for single-angle members. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 292

COMMENTARY

The effective length factors for angle columns may be determined by consulting the paper by Lutz (1992). The resistance factor φ was increased from 0.85 in AISC LRFD for all cross sections to 0.90 for single angles only because it was shown that a φ of 0.90 provides an equivalent degree of reliability (Galambos, 1992). C5. FLEXURE Flexural strength limits are established for yielding, local buckling, and lateraltorsional buckling. In addition to addressing the general case of unequal-leg single angles, the equal-leg angle is treated as a special case. Furthermore, bending of equal-leg angles about a geometric axis, an axis parallel to one of the legs, is addressed separately as it is a very common situation. The tips of an angle refer to the free edges of the two legs. In most cases of unrestrained bending, the flexural stresses at the two tips will have the same sign (tension or compression). For constrained bending about a geometric axis, the tip stresses will differ in sign. Criteria for both tension and compression at the tip should be checked as appropriate, but in most cases it will be evident which controls. Appropriate serviceability limits for single-angle beams need also to be considered. In particular, for longer members subjected to unrestrained bending, deflections are likely to control rather than lateral-torsional or local buckling strength. C5.1.1. These provisions follow the LRFD format for nominal flexural resistance. There is a region of full yielding, a linear transition to the yield moment, and a region of local buckling. The strength at full yielding is limited to a shape factor of 1.25, which is less than that corresponding to the plastic moment of an angle. The factor of 1.25 corresponds to an allowable stress of 0.75Fy, which has traditionally been used for rectangular shapes and for weak axis bending. It is used for angles due to uncertainties in developing the full plastic moment and to limit the large distortion of sections with large shape factors. The b / t limits and the criteria for local buckling follow typical AISC criteria for single angles under uniform compression. They are conservative when the leg is subjected to non-uniform compression due to flexure. C5.1.2. Since the shape factor for angles is in excess of 1.5, the nominal design strength Mn = 1.25My for compact members is justified provided that instability does not control. C5.1.3. Lateral-torsional instability may limit the flexural strength of an unbraced single-angle beam. As illustrated in Figure C5.1, Equation 5-3a represents the elastic buckling portion with the nominal flexural strength, Mn, varying from 75 percent to 92 percent of the theoretical buckling moment, Mob. Equation 5-3b represents the inelastic buckling transition expression between 0.75My and 1.25My. At Mob greater than approximately 6My, the unbraced length is adequate to develop the AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 293

maximum beam flexural strength of Mn = 1.25My. These formulas were based on Australian research on single angles in flexure and on an analytical model consisting of two rectangular elements of length equal to the actual angle leg width minus one-half the thickness (Leigh and Lay, 1984; Australian Institute of Steel Construction, 1975; Leigh and Lay, 1978; Madugula and Kennedy, 1985). Figure C5.1 reflects the higher nominal moment strength than was implied by the 0.66Fy allowable stress in the ASD version. A new and more general Cb moment gradient formula consistent with the 1993 AISC LRFD Specification is used to correct lateral-torsional stability equations from the assumed most severe case of uniform moment throughout the unbraced length (Cb = 1.0). The equation for Cb used in the ASD version is applicable only to moment diagrams that are straight lines between brace points. In lieu of a more detailed analysis, the reduced maximum limit of 1.5 is imposed for single-angle beams to represent conservatively the lower envelope of this cross section’s non-uniform bending response. C5.2.1. An angle beam loaded parallel to one leg will deflect and bend about that leg only if the angle is restrained laterally along the length. In this case simple bending occurs without any torsional rotation or lateral

Mn My

1.25

Eq. 5-3b

Eq. 5-3a

0.75

Inelastic Full yielding

Elastic

Unbraced length l

0

0.16

1.0

Fig. C5.1. Lateral-torsional buckling of a single-angle beam. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

My Mob

6 - 294

COMMENTARY

deflection and the geometric axis section properties should be used in the evaluation of the flexural design strength and deflection. If only the point of maximum moment is laterally braced, lateral-torsional buckling of the unbraced length under simple bending must also be checked, as outlined in Section 5.2.1b. C5.2.2. When bending is applied about one leg of a laterally unrestrained single angle, it will deflect laterally as well as in the bending direction. Its behavior can be evaluated by resolving the load and/or moments into principal axis components and determining the sum of these principal axis flexural effects. Section 5.2.2 is provided to simplify and expedite the design calculations for this common situation with equal-leg angles. For such unrestrained bending of an equal-leg angle, the resulting maximum normal stress at the angle tip (in the direction of bending) will be approximately 25 percent greater than calculated using the geometric axis section modulus. The value of Mob in Equation 5-4 and the evaluation of My using 0.80 of the geometric axis section modulus reflect bending about the inclined axis shown in Figure C5.2. The deflection calculated using the geometric axis moment of inertia has to be increased 82 percent to approximate the total deflection. Deflection has two components, a vertical component (in the direction of applied load) 1.56 times the calculated value and a horizontal component of 0.94 of the calculated value. The resultant total deflection is in the general direction of the weak principal axis bending of the angle (see Figure C5.2). These unrestrained bending deflections

δv = 1.56δ

Flexural load Y

Neutral axis

δh = 0.94δ X Geometric axis δ = deflection calculated using geometric axis moment of inertia

Fig. C5.2. Geometric axis bending of laterally unrestrained equal-leg angles. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 295

should be considered in evaluating serviceability and will often control the design over lateral-torsional buckling. The horizontal component of deflection being approximately 60 percent of the vertical deflection means that the lateral restraining force required to achieve purely vertical deflection (Section 5.2.1) must be 60 percent of the applied load value (or produce a moment 60 percent of the applied value) which is very significant. Lateral-torsional buckling is limited by Mob (Leigh and Lay, 1984 and 1978) in Equation 5-4, which is based on Mcr =   

2.33Eb4t × (1 + 3cos2θ) (Kl)2

0.156 (1 + 3cos θ) (Kl) t  √ sin θ +  b 2

2

4

2 2

 + sinθ 

(C5-1)

(the general expression for the critical moment of an equal-leg angle) with θ = −45° which is the most severe condition with the angle heel (shear center) in tension. Flexural loading which produces angle-heel compression can be conservatively designed by Equation 5-4 or more exactly by using the above general Mcr equation with θ = 45º (see Figure C5.3). With the angle heel in compression, Equation C5-1 will slightly exceed the yield moment limit of 1.25(0.8SxFy ) only for relatively few high slenderness cases. For pure bending situations, deflections would be unreasonably large under these conditions. However, considering the interaction of flexure and compression in an angle with Fy = 50 ksi, b / t equal to 16 and the largest l / r of 200, Equation C5-1 will produce results eight percent less than the modified yield moment. This situation could arise in a compression angle where the load is transferred by end gusset plates attached to one leg only. In this case the flexure term in the interaction is about 0.5 which reduces the effect Z (minor principal axis) W (major principal axis) b +θ

Shear center t

Mcr Centroid

Fig. C5.3. Equal-leg angle with general moment loading. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 296

COMMENTARY

to less than four percent and the end restraints provide an unknown increase in the lateral-torsional buckling strength. Consequently only the yield limit is required to be checked in Section 5.2.2 when the leg tips are in tension. Lateral-torsional buckling will reduce the nominal bending strength only when l / b is relatively large. If the lt / b2 parameter (which is a ratio of l / b over b / t) is small (less than approximately 2.5 with Cb = 1), there is no need to check lateral-torsional stability inasmuch as local buckling provisions of Section 5.1.1 will control the nominal bending strength. Lateral-torsional buckling will produce Mn < 1.25My for equal-leg angles only if Mob by Equation 5-4 is less than about 6My, for Cb = 1.0. Limits for l / b as a function of b / t are shown graphically in Figure C5.4. Local buckling and deflections must be checked separately. Stress at the tip of the angle leg parallel to the applied bending axis is of the same sign as the maximum stress at the tip of the other leg when the single angle is unrestrained. For an equal-leg angle this stress is about one-third of the maximum stress. It is only necessary to check the nominal bending strength based on the tip of the angle leg with the maximum stress when evaluating such an angle. Since this maximum moment per Section 5.2.2 represents combined principal axis moments and Equation 5-4 represents the design limit for these combined

400

300 l

b 200

100

0 1

2

3

4

5 Fy = 36

b

6

7

8

t Fy = 50

Fig. C5.4. Equal leg single-angle lateral buckling limits for Mn = 1.25My about geometric axis. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9

10


6 - 297

flexural moments, only a single flexural term needs to be considered when evaluating combined flexural and axial effects. C5.2.3. For unequal-leg angles without lateral-torsional restraint the applied load or moment must be resolved into components along the two principal axis in all cases and designed for biaxial bending using the interaction equation. C5.3.1. Under major axis bending of equal-leg angles Equation 5-5 in combination with 5-3a or 5-3b controls the nominal design moment against overall lateral-torsional buckling of the angle. This is based on Mcr, given earlier with θ = 0. Lateral-torsional buckling for this case will reduce the stress below 1.25My only for l / t ≥ 4800 / Fy or 0.160E / Fy (Mob = 6My). If the lt / b2 parameter is small (less than approximately 1.5Cb for this case), local buckling will control the nominal design moment and Mn based on lateral-torsional buckling need not be evaluated. Local buckling must be checked using Section 5.1.1. C5.3.2. Lateral-torsional buckling about the major principal W axis of an unequal-leg angle is controled by Mob in Equation 5-6. Section property βw reflects the location of the shear center relative to the principal axis of the section and the bending direction under uniform bending. Positive βw and maximum Mob occurs when the shear center is in flexural compression while negative βw and minimum Mob occurs when the shear center is in flexural tension (see Figure C5.5). This βw effect is consistent with behavior of singly symmetric I-shaped beams which are more stable when the compression flange is larger than the tension flange. For principal W-axis bending of equal-leg angles, βw is equal to zero due to symmetry and Equation 5-6 reduces to Equation 5-5 for this special case.

Shear center

Mob

W

Z

Shear center

Mob

W

(Special case: for equal legs, βw = 0) (a) + βw

(b) – βw Fig. C5.5. Unequal-leg angle in bending. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Z

6 - 298

COMMENTARY

TABLE C5.1 βw Values for Angles Angle Size (in.)

βw (in.)*

9×4

6.54

8×6 8×4

3.31 5.48

7×4

4.37

6×4 6 × 3.5

3.14 3.69

5 × 3.5 5×3

2.40 2.99

4 × 3.5 4×3

0.87 1.65

3.5 × 3 3.5 × 2.5

0.87 1.62

3 × 2.5 3×2

0.86 1.56

2.5 × 2

0.85

Equal legs

0.00

* Has positive or negative value depending on direction of bending (see Figure C5.5).

For reverse curvature bending, part of the unbraced length has positive βw, while the remainder negative βw, and conservatively, the negative value is assigned for that entire unbraced segment.

βw is essentially independent of angle thickness (less than one percent variation from mean value) and is primarily a function of the leg widths. The average values shown in Table C5.1 may be used for design. C6. COMBINED STRESSES The stability and strength interaction equations of AISC LRFD Specification Chapter H have been adopted with modifications to account for various conditions of bending that may be encountered. Bending will usually accompany axial loading in a single-angle member since the axial load and connection along the legs are eccentric to the centroid of the cross section. Unless the situation conforms to Section 5.2.1 or 5.2.2 in that Section 6.1.2 or 6.1.3 may be used, the applied moment should be resolved about the principal axes for the interaction check. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 299

For the non-symmetric and singly symmetric single angles, the interaction expression related to stresses at a particular location on the cross section is the most accurate due to lack of double symmetry. At a particular location, it is possible to have stresses of different sign from the various components such that a combination of tensile and compressive stress will represent a critical condition. The absolute value of the combined terms must be checked at the angle-leg tips and heel and compared with 1.0. When using the combined force expressions for single angles, Muw and Muz are positive as customary. The evaluation of Mn in Section 5.1 is dependent on the location on the cross section being examined by using the appropriate value of section modulus, S. Since the sign of the stress is important in using Equations 6-1a and 6-1b, Mn is considered either positive or negative by assigning a sign to S to reflect the stress condition as adding to, or subtracting from, the axial load effect. A designer may choose to use any consistent sign convention. It is conservative to ignore this refinement and simply use positive critical Mn values in the bending terms and add the absolute values of all terms (Elgaaly, Davids, and Dagher, 1992 and Adluri and Madugula, 1992). Alternative special interaction equations for single angles have recently been published (Adluri and Madugula, 1992). C6.1.3. When the total maximum flexural stress is evaluated for a laterally unrestrained length of angle per Section 5.2, the bending axis is the inclined axis shown in Figure C5.2. The radius of gyration modification for the moment amplification about this axis is equal to √  1.82 = 1.35 to account for the increased unrestrained bending deflection relative to that about the geometric axis for the laterally unrestrained length. The 1.35 factor is retained for angles braced only at the point of maximum moment to maintain a conservative calculation for this case. If the brace exhibits any flexibility permitting lateral movement of the angle, use of r = rx would not be conservative.


6 - 300


List of References Alduri, S. M. and Madugula, M. K. S. (1992), “Eccentrically Loaded Steel SingleAngle Struts,” AISC Engineering Journal, 2nd Quarter. American Institute of Steel Construction, Inc. (1983), Torsional Analysis of Steel Members, Chicago, IL. American Institute of Steel Construction, Inc. (1993), Load and Resistance Factor Design Specification for Structural Steel Buildings, Chicago, IL. American Institute of Steel Construction, Inc. (1989), Specification for Allowable Stress Design of Single-Angle Members, Chicago, IL. Australian Institute of Steel Construction (1975), Australian Standard AS1250, 1975. Elgaaly, M., Davids, W. and Dagher, H. (1992), “Non-Slender Single-Angle Struts,” AISC Engineering Journal, 2nd Quarter. Galambos, T. V. (1991), “Stability of Axially Loaded Compressed Angles,” Structural Stability Research Council, Annual Technical Session Proceedings, Apr. 15–17, 1991, Chicago, IL. Gjelsvik, A. (1981), The Theory of Thin-walled Bars, John Wiley and Sons, New York. Leigh, J. M. and M. G. Lay (1978), “Laterally Unsupported Angles with Equal and Unequal Legs,” Report MRL 22/2 July 1978, Melbourne Research Laboratories, Clayton. Leigh, J. M. and M. G. Lay (1984), “The Design of Laterally Unsupported Angles,” in Steel Design Current Practice, Section 2, Bending Members, American Institute of Steel Construction, Inc., January 1984. Lutz, L. A. (1992), “Critical Slenderness of Compression Members with Effective Lengths About Nonprincipal Axes,” Structural Stability Research Council, Annual Technical Session Proceedings, Apr. 6–7, 1992, Pittsburgh, PA. Madugula, M. K. S. and J. B. Kennedy (1985), Single and Compound Angle Members, Elsevier Applied Science, New York.


6 - 301

Seismic Provisions for Structural Steel Buildings June 15, 1992

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. One East Wacker Drive, Suite 3100, Chicago, IL 60601-2001

6 - 302

6 - 303

PREFACE

The intention of the main AISC Specification is to cover the common everyday design criteria in routine office usage. It is not feasible to also cover the many special and unique problems encountered within the full range of structural design practice. This document is a separate Specification which addresses one such topic, steel seismic provisions. It contains its own list of Symbols, a Glossary and a non-mandatory Commentary which has been included to provide background for the provisions. The AISC Specification Task Committee 113 on Seismic Provisions to supplement the current Load and Resistance Factor Design (LRFD) and Allowable Stress Design (ASD) Specification for Structural Steel Buildings acknowledges the various contributions of several groups to the completion of this document: the Structural Engineers Association of California (SEAOC), the National Science Foundation, and the Building Seismic Safety Council. The main AISC Committee on Specification enhanced these provisions by careful scrutiny, discussions, suggestions for improvements, and endorsement. The members of this Task Committee, as principal authors of the AISC Seismic Provisions, are most grateful to all of the above groups and people. Special recognition must also be given to the leadership expertise, and perseverance of Task Committee Chairman Egor Popov and Technical Secretary Clarkson Pinkham. The principal changes in this edition of the Seismic Provisions are the conversion to the loads and design format recommended by the 1991 National Earthquake Hazards Reduction Program (NEHRP) document. The reader is cautioned that professional judgment must be exercised when data or recommendations in this Specification are applied. The publication of the material contained herein is not intended as a representation or warranty on the part of the American Institute of Steel Construction, Inc.—or any other person named herein— that this information is suitable for general or particular use, or freedom from infringement of any patent or patents. Anyone making use of this information assumes all liability arising from such use. The design of structures is within the scope of expertise of a competent licensed structural engineer, architect, or other licensed professional for the application of principles to a particular structure. By the AISC Subcommittee, E. P. Popov, Chair R. Becker G. G. Deierlein M. D. Engelhardt S. J. Fang R. E. Ferch R. D. Hanson J. R. Harris K. Kasai

S. D. Lindsey H. W. Martin C. M. Saunders J. B. Shantz I. M. Viest N. F. G. Youssef C. W. Pinkham, Technical Secretary N. Iwankiw, Recording Secretary

May 22, 1992


6 - 304


6 - 305

Symbols The section numbers in parentheses after the definition of a symbol refers to the section where the symbol is first used. Effective net area, in.2 (9) Flange area of member, in.2 (6) Gross area, in.2 (8) Area of link stiffener, in.2 (10) Seismic coefficient representing the effective peak velocity-related acceleration. (2) Effective area of weld, in.2 (6) Link web area, in.2 (10) Response factor related to the fundamental period of the building. (3) Dead load due to the self-weight of the structure and the permanent elements on the structure, kips. (3) E Earthquake load. (3) FBM Nominal strength of the base material to be welded, ksi. (6) FEXX Classification strength of weld metal, ksi. (6) Fw Nominal strength of the weld electrode material, ksi. (6) Fy Specified minimum yield strength of the type of steel being used, ksi. (8) Fyb Fy of a beam, ksi. (8) Fyc Fy of a column, ksi. (6) H Average story height above and below a beam-to-column connection., in. (8) L Live load due to occupancy and moveable equipment, kips. (3) L Unbraced length of compression or bracing member, in. (8) Lr Roof live load, kips. (3) Mn Nominal moment strength of a member or joint, kip-in. (8) Mp Plastic bending moment, kip-in. (8) Mpa Plastic bending moment modified by axial load ratio, kip-in. (10) Mu Required flexural strength on a member or joint, kip-in. (8) PD Required axial strength on a column resulting from application of dead load, D, kips. (6) PE Required axial strength on a column resulting from application of the specified earthquake load, E, kips. (6) PL Required axial strength on a column resulting from application of live load, L, kips. (6) Pu Required axial strength on a column or a link, kips. (10) Pn Nominal axial strength of a column, kips. (6) Pu∗ Required axial strength on a brace, kips. (9) Puc Required axial strength on a column based on load combination with seismic loads, kips. (8) Py Nominal yield axial strength of a member = Fy Ag, kips. (10)

Ae Af Ag Ast Av Aw Aw Cs D


6 - 306

R R′ Rn S V Vn Vu Vp Vpa W Wg Zb Zc b bf bcf db dc dz e h r ry tbf tcf tf tp tw tz wz α ρ k kp kr φ φb φc φt φv φw

SYMBOLS

Response modification factor. (3) Load due to initial rainwater or ice exclusive of the ponding contribution, kips. (Symbol R is used in the Specification). (3) Nominal strength of a member. (8) Snow load, kips. (3) Base shear due to earthquake load, kips. (3) Nominal shear strength of a member, kips. (8) Required shear strength on a member, kips. (8) Nominal shear strength of an active link, kips. (10) Nominal shear strength of an active link modified by the axial load magnitude, kips. (10) Wind load, kips. (3) Total weight of the building, kips. (3) Plastic section modulus of a beam, in.3 (8) Plastic section modulus of a column, in.3 (8) Width of compression element, in. (Table 8-1) Flange width, in. (8) Column flange width, in. (8) Overall beam depth, in. (8) Overall column depth, in. (8) Overall panel zone depth between continuity plates, in. (8) EBF link length, in. (10) Assumed web depth for stability, in. (Table 8-1) Governing radius of gyration, in. (9) Radius of gyration about y axis, in. (8) Thickness of beam flange, in. (8) Thickness of column flange, in. (8) Thickness of flange, in. (8) Thickness of panel zone including doubler plates, in. (8) Thickness of web, in. (8) Thickness of panel zone (doubler plates not necessarily included), in. (8) Width of panel zone between column flanges, in. (8) Fraction of member force transferred across a particular net section. (9) Ratio of required axial force Pu to required shear strength Vu of a link. (10) Slenderness parameter. (9) Limiting slenderness parameter for compact element. (8) Limiting slenderness parameter for non-compact element. (9) Resistance factor. (6,10) Resistance factor for beams. (6) Resistance factor for columns in compression. (6,10) Resistance factor for columns in tension. (6) Resistance factor for shear strength of panel zone of beam-to-column connections. (8) Resistance factor for welds. (6)


6 - 307

Glossary Beam. A structural member whose primary function is to carry loads transverse to its longitudinal axis, usually a horizontal member in a seismic frame system. Braced Frame. An essentially vertical truss system of concentric or eccentric type that resists lateral forces on the structural system. Concentrically Braced Frame (CBF). A braced frame in which all members of the bracing system are subjected primarily to axial forces. The CBF shall meet the requirements of Sect. 9. Connection. Combination of joints used to transmit forces between two or more members. Categorized by the type and amount of force transferred (moment, shear, end reaction). Continuity Plates. Column stiffeners at top and bottom of the panel zone. Design strength. Resistance (force, moment, stress, as appropriate) provided by element or connection; the product of the nominal strength and the resistance factor. Diagonal Bracing. Inclined structural members carrying primarily axial load employed to enable a structural frame to act as a truss to resist horizontal loads. Dual System. A dual system is a structural system with the following features: • An essentially complete space frame which provides support for gravity loads. • Resistance to lateral load is provided by moment resisting frames (SMF) or (OMF) which is capable of resisting at least 25 percent of the base shear and concrete or steel shear walls, steel eccentrically (EBF) or concentrically (CBF) braced frames. • Each system shall be also designed to resist the total lateral load in proportion to its relative rigidity. Eccentrically Braced Frame (EBF). A diagonal braced frame in which at least one end of each bracing member connects to a beam a short distance from a beam-to-column connection or from another beam-to-brace connection. The EBF shall meet the requirements of Sect. 10. Essential Facilities. Those facilities defined as essential in the applicable code under which the structure is designed. In the absence of such a code, see ASCE 7-92. Joint. Area where two or more ends, surfaces, or edges are attached. Categorized by type of fastener or weld used and method of force transfer. K Braced Frame. A concentric braced frame (CBF) in which a pair of diagonal braces located on one side of a column is connected to a single point within the clear column height. Lateral Support Member. Member designed to inhibit lateral buckling or lateral-torsional buckling of primary frame members. Link. In EBF, the segment of a beam which extends from column to column, located between the end of a diagonal brace and a column or between the ends of two diagonal braces of the EBF. The length of the link is defined as the clear distance AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 308

GLOSSARY

between the diagonal brace and the column face or between the ends of two diagonal braces. Link Intermediate Web Stiffeners. Vertical web stiffeners placed within the link. Link Rotation Angle. The link rotation angle is the plastic angle between the link and the beam outside of the link when the total story drift is E′ / E times the drift derived using the specified base shear, V. Link Shear Design Strength. The lesser of φVp or 2φMp/e, where φ = 0.9, Vp = 0.55Fy dtw and e = the link length except as modified by Sect. S9.2.f. LRFD. (Load and Resistance Factor Design). A method of proportioning structural components (members, connectors, connecting elements, and assemblages) such that no applicable limit state is exceeded when the structure is subjected to all design load combinations. Moment Frame. A building frame system in which seismic shear forces are resisted by shear and flexure in members and joints of the frame. Nominal loads. The magnitudes of the loads specified by the applicable code. Nominal strength. The capacity of a structure or component to resist the effects of loads, as determined by computations using specified material strengths and dimensions and formulas derived from accepted principles of structural mechanics or by field tests or laboratory tests of scaled models, allowing for modeling effects, and differences between laboratory and field conditions. Ordinary Moment Frame (OMF). A moment frame system which meets the requirements of Sect. 7. P - Delta effect. Secondary effect of column axial loads and lateral deflection on the shears and moments in members. Panel Zone. Area of beam-to-column connection delineated by beam and column flanges. Required Strength. Load effect (force, moment, stress, as appropriate) acting on element of connection determined by structural analysis from the factored loads (using most appropriate critical load combinations). Resistance Factor. A factor that accounts for unavoidable deviations of the actual strength from the nominal value and the manner and consequences of failure. Slip-Critical Joint. A bolted joint in which slip resistance of the connection is required. Special Moment Frame (SMF). A moment frame system which meets the requirements of Sect. 8. Structural System. An assemblage of load-carrying components which are joined together to provide regular interaction or interdependence. V Braced Frame. A concentrically braced frame (CBF) in which a pair of diagonal braces located either above or below a beam is connected to a single point within the clear beam span. Where the diagonal braces are below the beam, the system is also referred to as an Inverted V Braced Frame. X Braced Frame. A concentrically braced frame (CBF) in which a pair of diagonal braces crosses near mid-length of the braces. Y Braced Frame. An eccentrically braced frame (EBF) in which the stem of the Y is the link of the EBF system.


6 - 309

Seismic Provisions for Structural Steel Buildings June 15, 1992

Part I—Load and Resistance Factor Design (LRFD) 1.

SCOPE These special seismic requirements are to be applied in conjunction with the AISC Load and Resistance Factor Design Specification for Structural Steel Buildings (LRFD), 1986; hereinafter referred to as the Specification. They are intended for the design and construction of structural steel members and connections in buildings for which the design forces resulting from earthquake motions have been determined on the basis of energy dissipation in the non-linear range of response. Seismic provisions and the nominal loads for each Seismic Performance Category, Seismic Hazard Exposure Group, or Seismic Zone shall be as specified by the applicable code under which the structure is designed or where no code applies, as dictated by the conditions involved. In the absence of a code, the Performance Categories, Seismic Hazard Exposure Groups, loads and load combinations shall be as given herein.

2.

SEISMIC PERFORMANCE CATEGORIES Seismic Performance Categories vary with the Seismic Hazard Exposure Group shown in Table 2-1, the Effective Peak Velocity Related Acceleration, Av, and the Seismic Hazard Exposure Group shown in Table 2-2. In addition to the general requirements assigned to the various Seismic Performance Categories in the applicable building code for all types of construction, the following requirements apply to fabricated steel construction for buildings and structures with similar structural characteristics.

2.1. Seismic Performance Categories A, B, and C Buildings assigned to Categories A, B, and C, except Category C in Seismic Hazard Exposure Group III where the value of Av ≥ 0.10, shall be designed either in accordance with solely the Specification or in accordance with the Specification and these provisions. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 310

PART I—LOAD AND RESISTANCE FACTOR DESIGN (LRFD)

TABLE 2-1 Seismic Hazard Exposure Groups Group III

Buildings having essential facilities that are necessary for postearthquake recovery and requiring special requirements for access and functionality.

Group II

Buildings that constitute a substantial public hazard because of occupancy or use.

Group I

All buildings not classified in Groups II and III.

2.2. Seismic Performance Category C Buildings assigned to Category C in Seismic Hazard Exposure Group III where the value of Av ≥ 0.10 shall be designed in accordance with the Specification as modified by the additional provisions of this section. 2.2.a. Steel used in seismic resisting systems shall be limited by the provisions of Sect. 5. 2.2.b. Columns in seismic resisting systems shall be designed in accordance with Sect. 6. 2.2.c. Ordinary Moment Frames (OMF) shall be designed in accordance with the provisions of Sect. 7. 2.2.d. Special Moment Frames (SMF) are required to conform only to the requirements of Sects. 8.2, 8.7, and 8.8. 2.2.e. Braced framed systems shall conform to the requirements of Sects. 9 or 10 when used alone or in combination with the moment frames of the seismic resisting system. 2.2.f. A quality assurance plan shall be submitted to the regulatory agency for the seismic force resisting system of the building. 2.3. Seismic Performance Categories D and E Buildings assigned to Categories D and E shall be designed in accordance with the Specification as modified by the additional provisions of this section. 2.3.a. Steel used in seismic resisting systems shall be limited by the provisions of Sect. 5. 2.3.b. Columns in seismic resisting systems shall be designed in accordance with Sect. 6. 2.3.c. Ordinary Moment Frames (OMF) shall be designed in accordance with the provisions of Sect. 7. 2.3.d. Special Moment Frames (SMF) shall be designed in accordance with the provisions of Sect. 8. 2.3.e. Braced framed systems shall conform to the requirements of Sects. 9. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

SEISMIC PROVISIONS FOR STRUCTURAL STEEL BUILDINGS

6 - 311

TABLE 2-2 Seismic Performance Categories Seismic Hazard Exposure Group Value of Av

I

II

III

0.20 ≤ Av < 0.20 0.15 ≤ Av < 0.20 0.10 ≤ Av < 0.15 0.05 ≤ Av < 0.10 0.15 ≤ Av < 0.05

D C C B A

D D C B A

E D C C A

(CBF) or 10. (EBF) when used alone or in combination with the moment frames of the seismic resisting system. The use of K-bracing systems shall not be permitted as part of the seismic resisting system except as permitted by Sect. 9.5. (Low Buildings) 2.3.f. A quality assurance plan shall be submitted to the regulatory agency for the seismic force resisting system of the building. 3.

LOADS, LOAD COMBINATIONS, AND NOMINAL STRENGTHS

3.1. Loads and Load Combinations The following specified loads and their effects on the structure shall be taken into account: D : dead load due to the weight of the structural elements and the permanent features on the structure. L : live load due to occupancy and moveable equipment. Lr : roof live load. W : wind load. S : snow load. E : earthquake load (where the horizontal component is derived from base shear Formula V = CsWg). R′ : load due to initial rainwater or ice exclusive of the ponding contribution. In the Formula V = CsWg for base shear: Cs = Seismic design coefficient Wg = Total weight of the building, see the applicable code. For the nominal loads as defined above, see the applicable code. The required strength of the structure and its elements shall be determined from the appropriate critical combination of factored loads. The following Load Combinations and corresponding load factors shall be investigated: 1.4D

(3-1)

1.2D + 1.6L + 0.5(Lr or S or R′)

(3-2)

1.2D + 1.6(Lr or S or R′) + (0.5L or 0.8W)

(3-3)


6 - 312


1.2D + 1.3W + 0.5L + 0.5(Lr or S or R′)

(3-4)

1.2D ± 1.0E + 0.5L + 0.2S

(3-5)

0.9D ± (1.0E or 1.3W)

(3-6)

Exception: The load factor on L in Load Combinations 3-3, 3-4, and 3-5 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf. Other special load combinations are included with specific design requirements throughout these provisions. Orthogonal earthquake effects shall be included in the analysis unless noted specifically otherwise in the governing building code. Where required by these provisions, an amplified horizontal earthquake load of 0.4R × E (where the term 0.4R is greater or equal to 1.0) shall be applied in lieu of the horizontal component of earthquake load E in the load combinations above. The term R is the earthquake response modification coefficient contained in the applicable code. The additional load combinations using the amplified horizontal earthquake load are: 1.2D + 0.5L + 0.2S ± 0.4R × E

(3-7)

0.9D ± 0.4R × E

(3-8)

Exception: The load factor on L in Load Combinations 3-7 shall equal 1.0 for garages, areas occupied as places of public assembly and all areas where the live load is greater than 100 psf. The term 0.4R in Load Combinations 3-7 and 3-8 shall be greater or equal to 1.0. Where the amplified load is required, orthogonal effects are not required to be included. 3.2. Nominal Strengths The nominal strengths shall be as provided in the Specification. 4.

STORY DRIFT Story drift shall be calculated using the appropriate load effects consistent with the structural system and the method of analysis. Limits on story drift shall be in accordance with the governing code and shall not impair the stability of the structure.

5.

MATERIAL SPECIFICATIONS Steel used in seismic force resisting systems shall be as listed in Sect. A3.1 of the Specification, except for buildings over one story in height. The steel used in seismic resisting systems described in Sections 8, 9, and 10 shall be limited to the following ASTM Specifications: A36, A500 (Grades B and C), A501, A572 (Grades 42 and 50), and A588. The steel used for base plates shall meet one of the preceding ASTM Specifications or ASTM A283 Grade D. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6.

6 - 313

COLUMN REQUIREMENTS

6.1. Column Strength When Pu / φPn > 0.5, columns in seismic resisting frames, in addition to complying with the Specification, shall be limited by the following requirements: 6.1.a. Axial compression loads: 1.2PD + 0.5PL + 0.2PS + 0.4R × PE ≤ φcPn

(6-1)

where the term 0.4R is greater or equal to 1.0. Exception: The load factor on PL in Load Combination 6-1 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf. 6.1.b. Axial tension loads: 0.9PD − 0.4R × PE ≤ φtPn

(6-2)

where the term 0.4R is greater or equal to 1.0. 6.1.c. The axial Load Combinations 6-1 and 6-2 are not required to exceed either of the following: 1. The maximum loads transferred to the column, considering 1.25 times the design strengths of the connecting beam or brace elements of the structure. 2. The limit as determined by the foundation capacity to resist overturning uplift. 6.2. Column Splices Column splices shall have a design strength to develop the column axial loads given in Sect. 6.1.a, b, and c as well as the Load Combinations 3-1 to 3-6. 6.2.a. In column splices using either complete or partial penetration welded joints, beveled transitions are not required when changes in thickness and width of flanges and webs occur. 6.2.b. Splices using partial penetration welded joints shall not be within 3 ft of the beam-to-column connection. Column splices that are subject to net tension forces shall comply with the more critical of the following: 1. The design strength of partial penetration welded joints, the lesser of φwFw Aw or φwFBM Aw, shall be at least 150 percent of the required strength, where φw = 0.8 and Fw = 0.6FEXX. 2. The design strength of welds shall not be less than 0.5Fyc Af, where Fyc is the yield strength of the column material and Af is the flange area of the smaller column connected. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 314

7.


REQUIREMENTS FOR ORDINARY MOMENT FRAMES (OMF)

7.1. Scope Ordinary Moment Frames (OMF) shall have a design strength as provided in the Specification to resist the Load Combinations 3-1 through 3-6 as modified by the following added provisions: 7.2. Joint Requirements All beam-to-column and column to beam connections in OMF which resist seismic forces shall meet one of the following requirements: 7.2.a. FR (fully restrained) connections conforming with Sect. 8.2, except that the required flexural strength, Mu, of a column-to-beam joint is not required to exceed the nominal plastic flexural strength of the connection. 7.2.b. FR connections with design strengths of the connections meeting the requirements of Sect. 7.1 using the Load Combinations 3-7 and 3-8. 7.2.c. Either FR or PR (partially restrained) connections shall meet the following: 1. The design strengths of the members and connections meet the requirements of Sect. 7.1. 2. The connections have been demonstrated by cyclic tests to have adequate rotation capacity at a story drift calculated at a horizontal load of 0.4R × E, (where the term 0.4R is equal to or greater than 1.0). 3. The additional drift due to PR connections shall be considered in design. FR and PR connections are described in detail in Sect. A2 of the Specification. 8.

REQUIREMENTS FOR SPECIAL MOMENT FRAMES (SMF)

8.1. Scope Special Moment Frames (SMF) shall have a design strength as provided in the Specification to resist the Load Combinations 3-1 through 3-6 as modified by the following added provisions: 8.2. Beam-to-Column Joints 8.2.a. The required flexural strength, Mu, of each beam-to-column joint shall be the lesser of the following quantities: 1. The plastic bending moment, Mp, of the beam. 2. The moment resulting from the panel zone nominal shear strength, Vn, as determined using Equation 8-1. The joint is not required to develop either of the strengths defined above if it is shown that under an amplified frame deformation produced by Load Combinations 3-7 and 3-8, the design strength of the members at the connection is adequate to support the vertical loads, and the required lateral force resistance is provided by other means. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 315

8.2.b. The required shear strength, Vu, of a beam-to-column joint shall be determined using the Load Combination 1.2D + 0.5L + 0.2S plus the shear resulting from Mu, as defined in Sect. 8.2.a., on each end of the beam. Alternatively, Vu shall be justified by a rational analysis. The required shear strength is not required to exceed the shear resulting from Load Combination 3-7. 8.2.c. The design strength, φRn, of a beam-to-column joint shall be considered adequate to develop the required flexural strength, Mu, of the beam if it conforms to the following: 1. The beam flanges are welded to the column using complete penetration welded joints. 2. The beam web joint has a design shear strength φVn greater than the required shear, Vu, and conforms to either: a. Where the nominal flexural strength of the beam, Mn, considering only the flanges is greater than 70 percent of the nominal flexural strength of the entire beam section [i.e., bf tf (d−tf)Fyf ≥ 0.7Mp]; the web joint shall be made by means of welding or slip-critical high strength bolting, or; b. Where bf tf (d−tf)Fyf < 0.7Mp, the web joint shall be made by means of welding the web to the column directly or through shear tabs. That welding shall have a design strength of at least 20 percent of the nominal flexural strength of the beam web. The required beam shear, Vu, shall be resisted by further welding or by slip-critical high-strength bolting or both. 8.2.d. Alternate Joint Configurations: For joint configurations utilizing welds or high-strength bolts, but not conforming to Sect. 8.2.c, the design strength shall be determined by test or calculations to meet the criteria of Sect. 8.2.a. Where conformance is shown by calculation, the design strength of the joint shall be 125 percent of the design strengths of the connected elements. 8.3. Panel Zone of Beam-to-Column Connections (Beam web parallel to column web) 8.3.a. Shear Strength: The required shear strength, Vu, of the panel zone shall be based on beam bending moments determined from the Load Combinations 3-5 and 3-6. However, Vu is not required to exceed the shear forces determined from 0.9ΣφbMp of the beams framing into the column flanges at the connection. The design shear strength, φvVn, of the panel zone shall be determined by the following formula: 3bcf t2cf   where for this case φv = 0.75. φvVn = 0.6φvFy dc tp 1 + db dc tp   where: tp = Total thickness of panel zone including doubler plates, in. dc = Overall column section depth, in. bcf = Width of the column flange, in. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(8-1)

6 - 316


tcf = Thickness of the column flange, in. db = Overall beam depth, in. Fy = Specified yield strength of the panel zone steel, ksi. 8.3.b. Panel Zone Thickness: The panel zone thickness, tz, shall conform to the following: tz ≥ (dz + wz) / 90

(8-2)

where: dz = the panel zone depth between continuity plates, in. wz = the panel zone width between column flanges, in. For this purpose, tz shall not include any doubler plate thickness unless the doubler plate is connected to the web with plug welds adequate to prevent local buckling of the plate. Where a doubler plate is used without plug welds to the column web, the doubler plate shall conform to Eq. 8-2. 8.3.c. Panel Zone Doubler Plates: Doubler plates provided to increase the design strength of the panel zone or to reduce the web depth thickness ratio shall be placed next to the column web and welded across the plate width along the top and bottom with at least a minimum fillet weld. The doubler plates shall be fastened to the column flanges using either butt or fillet welded joints to develop the design shear strength of the doubler plate. 8.4. Beam and Column Limitations 8.4.a. Beam Flange Area: There shall be no abrupt changes in beam flange areas in plastic hinge regions. 8.4.b. Width-Thickness Ratios: Beams and columns shall comply with λp in Table 8-1 in lieu of those in Table B5.1 of the Specification. 8.5. Continuity Plates Continuity plates shall be provided if required by the provisions in the Specification for webs and flanges with concentrated forces and if the nominal column local flange bending strength Rn is less than 1.8Fyb bf tbf, where: Rn = 6.25(tcf)2Fyf, and Fyb = Specified minimum yield strength of beam, ksi. Fyf = Specified minimum yield strength of column flange, ksi. bf = Beam flange width, in. tbf = Beam flange thickness, in. tcf = Column flange thickness, in. Continuity plates shall be fastened by welds to both the column flanges and either the column webs or doubler plates. 8.6. Column-Beam Moment Ratio At any beam-to-column connection, one of the following relationships shall be satisfied: AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 317

TABLE 8-1 Limiting Width Thickness Ratios λp for Compression Elements Description of Element Flanges of I-shaped nonhybrid sections and channels in flexure.


Limiting WidthThickness Ratios λp

b/t

52 / √ Fy

h / tw

For Pu / φbPy ≤ 0.125

Flanges of I-shaped hybrid beams in flexure. Webs in combined flexural and axial compression.

520 Fy √

1.54Pu   1 − φ P  b y  

For Pu / φbPy > 0.125 191 Fy √

Pu  253  2.33 − φ P  ≥ √ Fy b y 

ΣZc(Fyc − Puc / Ag) ≥ 1.0, ΣZbFyb

(8-3)

ΣZc(Fyc − Puc / Ag) ≥ 1.0, Vn dbH / (H − db)

(8-4)

where: Ag = Gross area of a column, in.2 Fyb = Specified minimum yield strength of a beam, ksi. Fyc = Specified minimum yield strength of a column, ksi. H = Average of the story heights above and below the joint, in. Puc = Required axial strength in the column (in compression) ≥ 0 Vn = Nominal strength of the panel zone as determined from Equation 8-1, ksi. Zb = Plastic section modulus of a beam, in.3 Zc = Plastic section modulus of a column, in.3 db = Average overall depth of beams framing into the connection, in. These requirements do not apply in any of the following cases, provided the columns conform to the requirements of Sect. 8.4: 8.6.a. Columns with Puc < 0.3Fyc Ag. 8.6.b. Columns in any story that has a ratio of design shear strength to design force 50 percent greater than the story above. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 318


8.6.c. Any column not included in the design to resist the required seismic shears, but included in the design to resist axial overturning forces. 8.7. Beam-to-Column Connection Restraint 8.7.a. Restrained Connection: 1. Column flanges at a beam-to-column connection require lateral support only at the level of the top flanges of the beams when a column is shown to remain elastic outside of the panel zone, using one of the following conditions: a. Ratios calculated using Eqs. 8-3 or 8-4 are greater than 1.25. b. Column remains elastic when loaded with Load Combination 3-7. 2. When a column cannot be shown to remain elastic outside of the panel zone, the following provisions apply: a. The column flanges shall be laterally supported at the levels of both top and bottom beam flanges. b. Each column flange lateral support shall be designed for a required strength equal to 2.0 percent of the nominal beam flange strength (Fy bf tf). c. Column flanges shall be laterally supported either directly, or indirectly, by means of the column web or beam flanges. 8.7.b. Unrestrained Connections: A column containing a beam-to-column connection with no lateral support transverse to the seismic frame at the connection shall be designed using the distance between adjacent lateral supports as the column height for buckling transverse to the seismic frame and conform to Sect. H of the Specification except that: 1. The required column strength shall be determined from the Load Combination 3-5 where E is the least of: a. The amplified earthquake force 0.4R × E (where the term 0.4R shall be equal to or greater than 1.0). b. 125 percent of the frame design strength based on either beam or panel zone design strengths. 2. The L / r for these columns shall not exceed 60. 3. The required column moment transverse to the seismic frame shall include that caused by the beam flange force specified in Sect. 8.7.a.2.b plus the added second order moment due to the resulting column displacement in this direction. 8.8. Lateral Support of Beams Both flanges of beams shall be laterally supported directly or indirectly. The unbraced length between lateral supports shall not exceed 2,500 ry / Fy. In addition, lateral supports shall be placed at concentrated loads where an analysis indicates a hinge will be formed during inelastic deformations of the SMF. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9.

6 - 319

REQUIREMENTS FOR CONCENTRICALLY BRACED (CBF) BUILDINGS

9.1. Scope Concentrically Braced Frames (CBF) are braced systems whose worklines essentially intersect at points. Minor eccentricities, where the worklines intersect within the width of the bracing members, are acceptable if accounted for in the design. CBF shall have a design strength as provided in the Specification to resist the Load Combinations 3-1 through 3-6 as modified by the following added provisions: 9.2. Bracing Members L 720 except as permit9.2.a. Slenderness: Bracing members shall have an ≤ r √ Fy ted in Sect. 9.5. 9.2.b. Compressive Design Strength: The design strength of a bracing member in axial compression shall not exceed 0.8φcPn. 9.2.c. Lateral Force Distribution: Along any line of bracing, braces shall be deployed in alternate directions such that, for either direction of force parallel to the bracing, at least 30 percent but no more than 70 percent of the total horizontal force shall be resisted by tension braces, unless the nominal strength, Pn, of each brace in compression is larger than the required strength, Pu, resulting from the application of the Load Combinations 3-7 or 3-8. A line of bracing, for the purpose of this provision, is defined as a single line or parallel lines whose plan offset is 10 percent or less of the building dimension perpendicular to the line of bracing. 9.2.d. Width-Thickness Ratios: Width-thickness ratios of stiffened and unstiffened compression elements in braces shall comply with Sect. B5 in the Specification. Braces shall be compact or non-compact, but not slender (i.e., λ < λr). Circular sections shall have an outside diameter to wall thickness ratio not exceeding 1,300 / Fy; rectangular tubes shall have a flat-width to wall thickness not exceeding 110 / √ Fy , unless the circular section or tube walls are stiffened. 9.2.e. Built-up Member Stitches: For all built-up braces, the first bolted or welded stitch on each side of the midlength of a built up member shall be designed to transmit a force equal to 50 percent of the nominal strength of one element to the adjacent element. Not less than two stitches shall be equally spaced about the member centerline. 9.3. Bracing Connections 9.3.a. Forces: The required strength of bracing joints (including beam-to-column joints if part of the bracing system) shall be the least of the following: 1. The design axial tension strength of the bracing member. 2. The force in the brace resulting from the Load Combinations 3-7 or 3-8. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 320


3. The maximum force, indicated by an analysis, that is transferred to the brace by the system. 9.3.b. Net Area: In bolted brace joints, the minimum ratio of effective net section area to gross section area shall be limited by: Ae 1.2αPu∗ ≥ Ag φtPn

(9-1)

where: Ae = Effective net area as defined in Equation B3-1 of the Specification. Pu∗ = Required strength on the brace as determined in Sect. 9.3.a. Pn = Nominal tension strength as specified in Chapter D of the Specification. φt = Special resistance factor for tension = 0.75. α = Fraction of the member force from Sect. 9.3.a that is transferred across a particular net section. 9.3.c. Gusset Plates: 1. Where analysis indicates that braces buckle in the plane of the gusset plates, the gusset and other parts of the connection shall have a design strength equal to or greater than the in-plane nominal bending strength of the brace. 2. Where the critical buckling strength is out-of-plane of the gusset plate, the brace shall terminate on the gusset a minimum of two times the gusset thickness from the theoretical line of bending which is unrestrained by the column or beam joints. The gusset plate shall have a required compressive strength to resist the compressive design strength of the brace member without local buckling of the gusset plate. For braces designed for axial load only, the bolts or welds shall be designed to transmit the brace forces along the centroids of the brace elements. 9.4. Special Bracing Configuration Requirements 9.4.a. V and Inverted V Type Bracing: 1. The design strength of the brace members shall be at least 1.5 times the required strength using Load Combinations 3-5 and 3-6. 2. The beam intersected by braces shall be continuous between columns. 3. A beam intersected by V braces shall be capable of supporting all tributary dead and live loads assuming the bracing is not present. 4. The top and bottom flanges of the beam at the point of intersection of V braces shall be designed to support a lateral force equal to 1.5 percent of the nominal beam flange strength (Fy bf tf). 9.4.b. K bracing, where permitted: 1. The design strength of K brace members shall be at least 1.5 times the required strength using Load Combinations 3-5 and 3-6. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 321

2. A column intersected by K braces shall be continuous between beams. 3. A column intersected by K braces shall be capable of supporting all dead and live loads assuming the bracing is not present. 4. Both flanges of the column at the point of intersection of K braces shall be designed to support a lateral force equal to 1.5 percent of the nominal column flange strength (Fy bf tf). 9.5. Low Buildings Braced frames not meeting the requirements of Sect. 9.2 through 9.4 shall only be used in buildings not over two stories and in roof structures if Load Combinations 3-7 and 3-8 are used for determining the required strength of the members and connections. 10.

REQUIREMENTS FOR ECCENTRICALLY BRACED FRAMES (EBF)

10.1. Scope Eccentrically braced frames shall be designed so that under inelastic earthquake deformations, yielding will occur in the links. The diagonal braces, the columns, and the beam segments outside of the links shall be designed to remain elastic under the maximum forces that will be generated by the fully yielded and strain hardened links, except where permitted by this section. 10.2. Links 10.2.a. Beams with links shall comply with the width-thickness ratios in Table 8-1. 10.2.b. The specified minimum yield stress of steel used for links shall not exceed Fy = 50 ksi. 10.2.c. The web of a link shall be single thickness without doubler plate reinforcement and without openings. 10.2.d. Except as limited by Sect. 10.2.f., the required shear strength of the link, Vu, shall not exceed the design shear strength of the link, φVn, where: φVn = Link design shear strength of the link = the lesser of φVp or 2φMp / e, kips. Vp = 0.6Fy (d − 2tf) tw, kips. φ = 0.9. e = link length, in. 10.2.e. If the required axial strength, Pu, in a link is equal to or less than 0.15Py, where Py = AgFy, the effect of axial force on the link design shear strength need not be considered. 10.2.f. If the required axial strength, Pu, in a link exceeds 0.15Py, the following additional limitations shall be required: 1. The link design shear strength shall be the lesser of φVpa or 2φMpa / e, where: Vpa = Vp √  1 − (Pu / Py)2 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 322


Mpa = 1.18Mp[1 − (Pu / Py )] φ = 0.9 2. The length of the link shall not exceed: [1.15 − 0.5ρ(Aw / Ag)]1.6Mp / Vp for ρ(Aw / Ag) ≥ 0.3 and 1.6Mp / Vp for ρ(Aw / Ag) < 0.3, where: Aw = (d − 2tf) tw ρ = Pu / Vu 10.2.g. The link rotation angle is the plastic angle between the link and the beam outside of the link when the total story drift is 0.4R times the drift determined using the specified base shear V. The term 0.4R shall be equal to or greater than 1.0. Except as noted in Sect. 10.4.d, the link rotation angle shall not exceed the following values: 1. 0.09 radians for links of length 1.6Mp / Vp or less. 2. 0.03 radians for links of length 2.6Mp / Vp or greater. 3. Linear interpolation shall be used for links of length between 1.6Mp / Vp and 2.6Mp / Vp. 10.2.h. Alternatively, the top story of an EBF building having over five stories shall be a CBF. 10.3. Link Stiffeners 10.3.a. Full depth web stiffeners shall be provided on both sides of the link web at the diagonal brace ends of the link. These stiffeners shall have a combined width not less than (bf − 2tw) and a thickness not less than 0.75tw or 3⁄8-in., whichever is larger, where bf and tw are the link flange width and link web thickness, respectively. 10.3.b. Links shall be provided with intermediate web stiffeners as follows: 1. Links of lengths 1.6Mp / Vp or less shall be provided with intermediate web stiffeners spaced at intervals not exceeding (30tw − d / 5) for a link rotation angle of 0.09 radians or (52tw − d / 5) for link rotation angles of 0.03 radians or less. Linear interpolation shall be used for values between 0.03 and 0.09 radians. 2. Links of length greater than 2.6Mp / Vp and less than 5Mp / Vp shall be provided with intermediate web stiffeners placed at a distance of 1.5bf from each end of the link. 3. Links of length between 1.6Mp / Vp and 2.6Mp / Vp shall be provided with intermediate web stiffeners meeting the requirements of 1 and 2 above. 4. No intermediate web stiffeners are required in links of lengths greater than 5Mp / Vp. 5. Intermediate link web stiffeners shall be full depth. For links less than 25 inches in depth, stiffeners are required on only one side of the link web. The thickness of one-sided stiffeners shall not be less than tw or AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 323

3⁄ -in., whichever is larger, and the width shall be not less than 8 (bf / 2) − tw. For links 25 inches in depth or greater, similar intermediate stiffeners are required on both sides of the web.

10.3.c. Fillet welds connecting link stiffener to the link web shall have a design strength adequate to resist a force of AstFy, in which Ast = area of the stiffener. The design strength of fillet welds fastening the stiffener to the flanges shall be adequate to resist a force of AstFy / 4. 10.4. Link-to-Column Connections Where a link is connected to a column, the following additional requirements shall be met: 10.4.a. The length of links connected to columns shall not exceed 1.6Mp / Vp unless it is demonstrated that the link-to-column connection is adequate to develop the required inelastic rotation of the link. 10.4.b. The link flanges shall have complete penetration welded joints to the column. The joint of the link web to the column shall be welded. The required strength of the welded joint shall be at least the nominal axial, shear, and flexural strengths of the link web. 10.4.c. The need for continuity plates shall be determined according to the requirements of Sect. 8.5. 10.4.d. Where the link is connected to the column web, the link flanges shall have complete penetration welded joints to plates and the web joint shall be welded. The required strength of the link web shall be at least the nominal axial, shear, and flexural strength of the link web. The link rotation angle shall not exceed 0.015 radians for any link length. 10.5. Lateral Support of Link Lateral supports shall be provided at both the top and bottom flanges of link at the ends of the link. End lateral supports of links shall have a design strength of 6 percent of the link flange nominal strength computed as Fy bf tf. 10.6. Diagonal Brace and Beam Outside of Link 10.6.a. The required combined axial and moment strength of the diagonal brace shall be the axial forces and moments generated by 1.25 times the nominal shear strength of the link as defined in Sect. 10.2. The design strengths of the diagonal brace, as determined by Sect. H (including Appendix H) of the Specification, shall exceed the required strengths as defined above. 10.6.b. The required strength of the beam outside of the link shall be the forces generated by at least 1.25 times the nominal shear strength of the link and shall be provided with lateral support to maintain the stability of the beam. Lateral supports shall be provided at both top and bottom flanges of the beam and each shall have a design strength to resist at least 1.5 percent of the beam flange nominal strength computed as Fy bf tf. 10.6.c. At the connection between the diagonal brace and the beam at the link end of the brace, the intersection of the brace and beam centerlines shall AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 324


be at the end of the link or in the link. The beam shall not be spliced within or adjacent to the connection between the beam and the brace. 10.6.d. The required strength of the diagonal brace-to-beam connection at the link end of the brace shall be at least the nominal strength of the brace. No part of this connection shall extend over the link length. If the brace resists a portion of the link end moment, the connection shall be designed as Type FR (Fully Restrained). 10.6.e. The width-thickness ratio of brace shall satisfy λp of Table B5.1 of the Specification. 10.7. Beam-to-Column Connections Beam-to-column connections away from links are permitted to be designed as a pin in the plane of the web. The connection shall have a design strength to resist torsion about the longitudinal axis of the beam based on two equal and opposite forces of at least 1.5 percent of the beam flange nominal strength computed as Fy bf tf acting laterally on the beam flanges. 10.8. Required Column Strength The required strength of columns shall be determined by Load Combinations 3-5 and 3-6 except that the moments and axial loads introduced into the column at the connection of a link or brace shall not be less than those generated by 1.25 times the nominal strength of the link. 11.

QUALITY ASSURANCE The general requirements and responsibilities for performance of a quality assurance plan shall be in accordance with the requirements of the regulatory agency and specifications by the design engineer. The special inspections and special tests needed to establish that the construction is in conformance with these provisions shall be included in a quality assurance plan. The minimum special inspection and testing contained in the quality assurance plan beyond that required by the Specification shall be as follows: Groove welded joints subjected to net tensile forces which are part of the seismic force resisting systems of Sects. 8, 9, and 10 shall be tested 100 percent either by ultrasonic testing or by other approved equivalent methods conforming to AWS D1.1. Exception: The nondestructive testing rate for an individual welder shall be reduced to 25 percent with the concurrence of the person responsible for structural design, provided the reject rate is demonstrated to be 5 percent or less of the welds tested for the welder.



6 - 325

Part II—Allowable Stress Design (ASD) Alternative As an alternative to the LRFD seismic design procedures for structural steel design given in PART I, the design procedures in the Specification for Structural Steel Buildings—Allowable Stress Design and Plastic Design, AISC 1989 are permitted as modified by PART II of these provisions. When using ASD, the provisions of PART I of these seismic provisions shall apply except the following sections shall be substituted for, or added to, the appropriate sections as indicated: 1.

SCOPE

Revise the first paragraph of PART I, Sect. 1 to read as follows: These special requirements are to be applied in conjunction with the AISC Specification for Structural Steel Buildings—Allowable Stress Design and Plastic Design hereinafter referred to as Specification. They are intended for the design and construction of structural steel members and connections in buildings for which the design forces resulting from earthquake motions have been determined on the basis of energy dissipation in the nonlinear range of response. 3.

LOADS, LOAD COMBINATIONS AND NOMINAL STRENGTHS

Substitute the following for Section 3.2 in PART I: 3.2. Nominal Strengths The nominal strengths of members shall be determined as follows: 3.2.a. Replace Sect. A5.2 of the Specification to read: “The nominal strength of structural steel members for resisting seismic forces acting alone or in combination with dead and live loads shall be determined by multiplying 1.7 times the allowable stresses in Sect. D, E, F, G, J, and K.” 3.2.b. Amend the first paragraph of Sect. N1 of the Specification by deleting “or earthquake” and adding: “The nominal strength of members shall be determined by the requirements contained herein. Except as modified by these rules, all pertinent provisions of Chapters A through M shall govern.” 3.2.c. In Sect H1 of the Specification the definition of Fe ′ shall read as follows: Fe ′ =

π2E (Klb / rb)2

where: lb = the actual length in the plane of bending. rb = the corresponding radius of gyration. K = the effective length factor in the plane of bending. Add the following section to PART I: 3.3. Design Strengths 3.3.a. The design strengths of structural steel members and connections subjected to seismic forces in combination with other prescribed loads shall AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 326

PART II—ALLOWABLE STRESS DESIGN (ASD) ALTERNATIVE

be determined by converting allowable stresses into nominal strengths and multiplying such nominal strengths by the resistance factors herein. 3.3.b. Resistance factors, φ, for use in Part II shall be as follows: Flexure

φb = 0.90

Compression and axially loaded composite members

φc = 0.85

Eyebars and pin connected members: Shear of the effective area Tension on net effective area Bearing on the project area of pin

φsf = 0.75 φt = 0.75 φt = 1.0

Tension members: Yielding on gross section Fracture in the net section

φt = 0.90 φt = 0.75

Shear

φv = 0.90

Connections: Base plates that develop the strength of the members or structural systems Welded connections that do not develop the strength of the member or structural system, including connection of base plates and anchor bolts Partial Penetration welds in columns when subjected to tension stresses High strength bolts (A325 and A490) and rivets: Tensile strength Shear strength in bearing-type joints Slip-critical joints A307 bolts: Tensile strength Shear strength in bearing-type joints

φ = 0.90 φ = 0.67 φ = 0.80 φ = 0.75 φ = 0.65 φ = 1.0 φ = 0.75 φ = 0.60

Substitute the following for Section 7 in PART I in its entirety: 7.


7.1. Scope Ordinary Moment Frames (OMF) shall have a design strength as provided in the Specification to resist the Load Combinations 3-5 and 3-6 as modified by the following added provisions: 7.2. Joint Requirements All beam-to-column and column to beam connections in OMF which resist seismic forces shall meet one of the following requirements: 7.2.a. Type 1 connections conforming with Sect. 8.2, except that the required flexural strength, Mu, of a column-to-beam joint are not required to exceed that required to develop the nominal plastic flexural strength of the connection. 7.2.b. Type 1 connections capable of inelastic deformation and the design AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 327

strengths of the connections meeting the requirements of Sect. 7.1 using the Load Combinations 3-7 and 3-8. 7.2.c. Either Type 1 or Type 3 connections are permitted provided: 1. The design strengths of the members and connections meet the requirements of Sect. 7.1. 2. The connections have been demonstrated by cyclic tests to have adequate rotation capacity at a story drift calculated at a horizontal load of 0.4R × E (where the term 0.4R is equal to or greater than 1.0). 3. The additional drift due to Type 3 connections shall be considered in design. Type 1 and Type 3 connections are described in detail in Sect. A2 of the Specification. Substitute the following in Sections 10.6.a and 10.6.d in PART I: 10.6.a. Delete reference to Appendix H. 10.6.d. The last sentence shall read: “If the brace resists a portion of the link end moment as described above, the connection shall be designed as a Type 1 connection.”


6 - 328


6 - 329

Commentary on the Seismic Provisions for Structural Steel Buildings June 15, 1992

Part I—LRFD Provisions 1.

SCOPE Load and Resistance Factor Design (LRFD) is an improved approach to the design of structural steel for buildings. The method involves explicit consideration of limit states, multiple load and resistance factors, and implicit probabilistic determination of reliability. The designation LRFD reflects the concept of factoring both loads and resistance. The LRFD method was devised to offer the designer greater flexibility, more rationality and possible overall economy. The First Edition of the LRFD Specification was published and distributed in 1986.1 It did not contain the special requirements necessary in the design and construction of steel buildings which are required to respond to high earthquake input by deformations into the nonlinear range. The seismic design forces specified in the building codes have been set with consideration given to the energy dissipation generated during the non-linear response. The provisions contained in this document are to be used in conjunction with the AISC LRFD Specification in the design of buildings in the areas of moderate and high seismicity. The load provisions have been modified from those contained in the Specification to be consistent with the load provisions contained in the soon to be published BOCA and SBCCI building codes and the ASCE 7-93, Minimum Design Loads for Buildings and Other Structures.2 All these new seismic load provisions are modeled on the the 1991 NEHRP3 earthquake provisions.

2.

SEISMIC PERFORMANCE CATEGORIES Buildings are classified into three types depending on the occupancy and use of each as related to the special hazards resulting from earthquake environment. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 330

COMMENTARY: PART I—LRFD PROVISIONS

The Seismic Hazard Exposure Groups listed in Table 2-1 are defined in detail, with examples of buildings in each type, in ASCE 7-93. The Seismic Performance Category to be used in the design of a specific building is defined by the seismic coefficient representing the peak velocity-related acceleration of the building site, Av, and the Seismic Hazard Exposure Group related to the occupancy and use of the building. The five categories, A through E, given in Table 2-2 specify design and detail requirements that would be required for the seismic design of the building. These categories establish the level of requirements to be used in items such as detailing limitations, quality assurance, method of analyses, orthogonal effects, and change of building use. The general requirements for each of the categories are given in ASCE 7-93. The differences related specifically to structural steel design are repeated in this Specification. 3.

LOADS AND LOAD COMBINATIONS The most frequently used load factors and load combinations given in Sect. A4.1 of the Specification are repeated in this Section to reduce the amount of cross-referencing to other documents. They have been modified to be consistent with the anticipated ASCE 7-93. The most notable modification is the reduction of the load factor on E to 1.0. This results from the limit states load model used in ASCE 7-93. For design of structures subjected to impact loads, see the Specification. The earthquake load and load effects E in ASCE 7-93 are composed of two parts. E is the sum of the seismic horizontal load effects and one half of Av times the dead load effects. The second part adds an effect simulating vertical accelerations concurrent to the usual horizontal earthquake effects. The load factors and load combinations reflect the fact that when several loads act in combination with the dead load, e.g., dead plus live plus earthquake loads, only one of these takes on its maximum lifetime value, while the other load is at its “arbitrary point-in-time value,” at a value which can be expected to be on the structure at any time. The most critical effect may occur when one or more load types are not acting. The basic requirements for dual systems are given in the Glossary to clarify the use of the EBF in a dual system and to indicate that steel moment frames can also be used as part of a dual system with concrete shear walls. An amplification factor to earthquake load E of 0.4R is prescribed for limited use in this set of provisions. It is used as an amplification of the deflections determined using the earthquake forces specified in ASCE 7-93. It was derived by assuming that deflections due to large earthquake response would be the same regardless of the reductions in applied forces due to the inelastic response of the type of lateral force resisting system.56 The amount of this amplification was assumed to be two times the deflections generated by forces specified for a buildings with R = 5. This amplification factor is thus 2R / 5 or 0.4R. However, with R = 2.5 or less it is felt that the amplification factor should not be less than 1.0. The load combinations to be used with the amplification factor are given by formulas 3-7 and 3-8. Specific values of R are not needed for determination of the amplified load because R is cancelled out when substituted in the formula for the horizontal seismic base shear, V. The added complication that would be AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 331

required to consider orthogonal effects with the amplified force is not deemed to be necessary. The ASCE 7-93 provisions are detailed earthquake load provisions in which two methods of analysis are provided. The first is frequently referred to as the “Static Force Procedure” or “Equivalent Lateral Force Procedure.” The second method is the “Modal Analysis Procedure.” In both methods a linearly elastic model is assumed. Other “Dynamic Analysis Procedures” are permitted both with linearly elastic or non-linear models as long as the internal forces and deformations in the members are determined using a model consistent with the procedure adopted. Guidelines for use of these other methods of analyses are provided in the Commentary to ASCE 7-93. These earthquake provisions refer to the load provisions of ASCE 7-93. By changing the load combination portion of Section 3, these provisions can be made compatible with other sets of load provisions.4 For instance, the following changes can be made to the provisions in Section 3 to make them compatible with the following document: (Sn is used in this Commentary for snow loads to distinguish them from site effects that use the symbol S). 1991 UNIFORM BUILDING CODE:5 (SEAOC seismic provisions are similar)6 The required strength on the structure and its elements must be determined from the appropriate critical combination of factored loads. The most critical effect may occur when one or more loads are not acting. The following load combinations and corresponding load factors shall be investigated: 1.4D

(3-1)

1.2D + 1.6L + 0.5(Lr or S or R′)

(3-2)

1.2D + 1.6(Lr or S or R′) + (0.5L or 0.8W)

(3-3)

1.2D + 1.3W + 0.5L + 0.5(Lr or S or R′)

(3-4)

1.2D + 1.5E + 0.5L + 0.2S

(3-5)

0.9D − (1.3W or 1.5E)

(3-6)

Exception: The load factor on L in Load Combinations 3-3, 3-4, and 3-5 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf. Other special load combinations are included with specific design requirements throughout these provisions. Where required by these provisions, an amplification factor is applied on the earthquake load E = 3⁄8Rw, where Rw is a response factor similar to the factor R except used to reduce the earthquake load to a working stress design level. Earthquake loads are similar to those found in ASCE 7-93 except for the Rw factor. Earthquake loads are defined in detail in Section 2334 of 1991 UBC. The revised load combinations are: AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 332


1.2D + (3Rw / 8)E + 0.5L + 0.2S

(3-7)

0.9D − (3Rw / 8)E

(3-8)

Exception: The load factor on L in Load Combination 3-7 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf. The amplification factor was derived by using the similar assumptions that were used in deriving the factor for ASCE 7-93. The same type of building with R = 5 in ASCE 7-93 has a Structure System Coefficient Rw = 8 in 1991 UBC. The deflection determined by this Rw was used as the value to be amplified by 3. Thus (3Rw / 8)E. Where the use of the amplification factor to load E is required, orthogonal effects need not be included. The 1991 UBC outlines in detail many of the requirements for “Dynamic Lateral Force Procedure.” The following is a summary of these requirements: The ground motion may be defined in one of five ways: 1. The plot of a normalized response spectra may be used. 2. A site specific response spectra based on geologic, tectonic seismologic, and soil characteristics of the site may be used. As per SEAOC the damping ratio shall be 5 percent unless another value is shown to be consistent with the structural behavior of the building. 3. Site specific time histories are to be representative of actual earthquake motions. Spectra developed from these time histories would follow item (2) above. 4. Site specific response spectra and time histories developed for sites with a profile having more than 40 feet of soft clay per SEAOC shall be based on ground motion having a 10 percent probability of exceedance in 50 years; the effects of lengthening of the structural period on response amplification due to soil-structure resonance shall be included; and the design base shear shall be determined by dividing by a factor not greater than Rw for the structure. 5. A two-thirds factor shall be used on horizontal motions to determine the vertical component of ground motion unless specifically determined otherwise for the site. The mathematical model shall represent the actual structure adequately for the calculation of all the significant features of the dynamic response. Three dimensional models shall be used for highly irregular plan configurations if a rigid or semi-rigid diaphragm is used. A response spectrum analysis shall be an elastic dynamic analysis of all the significant peak modal responses combined in a statistical manner to obtain an approximate total structural response. When the base shear is less than that determined from the Static Lateral Force Procedure, it shall be increased to 100 percent of the static base shear for irregular structures, shall be taken as 90 percent of the static base shear for regular structures where the fundamental period is determined using the structural characteristics of the building system, AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 333

and shall be set at 80 percent for regular structures. Accidental torsion shall be accounted for by appropriate adjustments in the model. Where a dual system is used, the combined system shall be accounted for in the modelling; the backup Special Moment Frame (SMF) shall be capable of resisting 25 percent of the base shear used for the design of the total system. The analysis of the backup SMF may either use the Static of Dynamic Lateral Force Procedures. A time history analysis shall be an elastic or inelastic dynamic analysis of a model of a structure subjected to specified time history of ground motion. The time dependent dynamic response of the structure to these motions is obtained through numerical integration of its equations of motion. These analyses shall be based on established principles of mechanics. Scaling of base shear determined by a response spectrum analysis results in making the Load Combinations 3-1 through 3-6 as well as 3-7 and 3-8 applicable to this method of analysis in the 1991 UBC. No scaling effect is specified for the results of time history dynamic analysis (either elastic or inelastic). In this case, it is necessary to define the specified time histories which will result in the structure responding to the limit of essentially elastic response. This would be the level to determine the required resistance of the system. In order to determine the deformations corresponding to the specified drift limits, the force level shall be divided by a factor of 1.5. 4.

STORY DRIFT Deflection limits are commonly used in design to assure the serviceability of the structure. These serviceability limit states are variable, since they depend upon the structural usage and contents. The Specification does not specify these serviceability limits, since they are regarded as a matter of engineering judgment, rather than general design limits.54 Like deflection limits, drift limits for both wind and seismic design are excluded from these Seismic Provisions. Research has shown that seismic drift control provides a function beyond assuring the serviceability of the structure. The added strength and stiffness which drift limits often provide in moment frames improves the performance of structures during earthquakes. Model codes, load standards, and resource documents contain specific seismic drift limits but there are major differences among them. There is neither uniform agreement regarding appropriate code specified drift limits nor how they should be applied. Further it is difficult to estimate the actual story drift of moment frames with panel zone yielding. Nevertheless, drift control is important to serviceability and stability of the structure. It is recommended the designer review drift limits in the appropriate code and use those applicable for the serviceability and stability of the structure under consideration. The story drift limitations of ASCE 7-93 are applied to an amplified story drift that estimates the story drift that would occur during a large earthquake. The story drift is defined as the difference of deflection between the top and bottom of the story under consideration. For determining the story drift the deflection determined using the earthquake forces E is amplified by a deflection amplification factor, Cd, which is dependent on the type of building system. The story drifts when determined by an elastic analysis, including the P-∆ effect when AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 334


TABLE C-4.1 Tentative Allowable Story Drift Seismic Hazard Exposure Group Building

I

II

III

Single story buildings without equipment attached to the structural resisting system and with interior walls, partitions, ceilings, and exterior wall system that have been designed to accommodate the story drifts.

No limit

0.020hsx

0.015hsx

Buildings with 4 stories or less with interior walls, partitions ceilings, and exterior wall system that have been designed to accommodate the story drifts.

0.025hsx

0.020hsx

0.015hsx

All other buildings.

0.020hsx

0.015hsx

0.010hsx

Where hsx is the story height of the story drift calculated.

applicable, have limits depending on the Seismic Hazard Exposure Group of the building as shown in Table C-4.1. In calculating the elastic drift, the forces may be based on the fundamental period of the building without the arbitrary limit specified for determining the seismic design forces in the framing members. ASCE 7-93 does not prescribe explicit requirements for building separations. An admonition is included, however, that all portions of the building shall be designed and constructed as an integral unit in resisting seismic forces unless separated structurally by a sufficient distance to avoid damaging contact between components under amplified deformations. The latter are determined by multiplying the elastic deflection by a deflection amplification factor, Cd, which is based on the type and materials of the seismic resisting system. If the effects of hammering between segments can be shown not to be detrimental, separations could be reduced. 1991 UBC Requirements: In order to comply with the 1991 UBC requirements, the story drift shall be calculated including the translational and torsional deflections resulting from the application of unfactored lateral forces. Story drift is defined as the displacement of one level relative to the level above or below. The calculated story drift shall not exceed 0.04 / Rw nor 0.005 times the story height for structures with fundamental periods of less than 0.7 seconds and shall not exceed 0.03 / Rw nor 0.004 times the story height for structures with fundamental periods of 0.7 seconds or greater. For the purpose of this limit the fundamental period is the same as that used for determining the base shear. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 335

For calculating the drift, the lateral forces may be calculated using a base shear V defined as: V=

ZIC W, in which Rw

Z = The seismic zone coefficient. I = An importance factor. Rw = A numerical coefficient related to the type of construction. C=

1.25S , in which 2 T ⁄3

S = Site coefficient. T = Fundamental period of vibration, which may be determined using the structural properties and deformational characteristics of the resisting elements of the lateral force resisting framing. Method A of determining T need not be applied for drift determination. The lower bound of 0.075 on the ratio C / Rw may also be neglected. W = Dead load used to calculate seismic loads. The story drift limits need not be applied if it is demonstrated that greater drift can be tolerated without affecting life safety by damage to either structural and non-structural elements. There are no drift limits on single story steel framed structures with low occupancies. This would generally apply to buildings such as warehouses, parking garages, aircraft hangers, factories, workshops and agricultural buildings. These buildings are not allowed to have brittle finishes and are not allowed to have equipment attached to the structural frame unless the finish or equipment attachment is detailed to accommodate the additional drift. 5.

MATERIAL SPECIFICATIONS The list of structural steels for use in designing to earthquake motion has been chosen with consideration given to the inelastic properties of the steels and their weldability. In general, the steels selected possess the following characteristics: • Ratio of tensile strength to yield strength between 1.2 to 1.8. • Pronounced stress-strain plateau at yield strength. • Large inelastic strain capability. • Tension elongation of 20 percent or greater in a 2-in. gage length. • Good weldability for inelastic behavior. Other steels including those with a specified yield point greater than 50 ksi should not be used without demonstrating that equivalent inelastic behavior can be attained.

6.

COLUMN REQUIREMENTS

6.1. Column Strength During the maximum probable earthquake expected at any site, axial forces AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 336


calculated using the specified design earthquake may be exceeded. This is a result of the reduction in lateral force for use in analysis of an elastic model of the structure, the underestimation of the overturning forces in this analysis, and the concurrent vertical accelerations which are not explicitly specified as a required design load. The amplifications required in this section provide an approximation of these actions by providing a limit to the required axial force. The two special Load Combinations 6-1 and 6-2 account for these effects; one as a minimum required column compressive required strength and the other on the minimum required tensile strength. They are to be applied without consideration of any concurrent flexure on the members. The exceptions provided for these limits are self limiting conditions stating that the required axial strengths need not exceed the limits based on the design strength of the overall system to transfer axial loads to the column. For instance, if pile foundations are used, the design strength of the piles in tension may be much larger than the required strength because the size of the foundation may depend on the required strength in compression. 6.2. Column Splices Column splices are required to have design strengths adequate to join column elements together not only to resist the axial, flexural, and shear forces required at the splice location by the usual load combinations 3-1 through 3-6 but also the forces specified in 6.1. Butt weld splices in columns where it is anticipated that potential dynamic loading consists only of wind or earthquake forces are not required by these specifications to provide the transition of thicknesses given in Section 9.20 of AWS D1.1.7 If other types of frequent, high cycle dynamic loadings are also present, the transition requirements should be met. Partial penetration welds in thick members, such as occur in column flange splices, are very brittle under tensile loading, showing virtually no ductility.8–9 Recognizing this behavior in seismic design, the location of column splices is moved away from the beam-to-column connection to reduce bending and a 50 percent increase is stipulated in required strength of the splice. The possibility for developing tensile stresses in such welds during a maximum probable seismic event should be considered. If there is probability of such a condition developing, the use of splice plates welded to the lower part of the column and bolted to the upper part is suggested. If for the noted adverse condition, the suggested detail is not practical, the possibility of fracture in partial penetration welded joints should be recognized, and some restraint from uncontrolled relative movement at the splice be provided. This can be achieved, for example, by having wide splice plates on both sides of the column web to maintain alignment. Shake table experiments have shown that if some columns, unattached at the base, reseat themselves after lifting, the performance of a steel frame remains tolerable.10 These provisions apply for common frame configurations. The designer should review the conditions found in columns in tall stories, large changes in column sizes at the splice, or where the possibility of a single curvature exists on a AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 337

column over multiple stories to determine if special design strength or special detailing is necessary at the splice. 7.


7.1. Scope Ordinary moment frames of structural steel are moment frames which do not meet the requirements for special design and detailing contained in Section 8. OMF of structural steel do exist and are being built in all areas of seismic activity. Experience has shown that in most instances the buildings of this type have responded without significant structural damage. In recent years advances in analytical procedures have minimized the natural margins of safety normally found in buildings that were designed by approximate methods. Thus it is prudent to require that the design of the beam-to-column connection be adequate to develop the strength of the members framing into the connection as is specified in Sect. 8.2 unless the connection has a design strength significantly larger than the required strengths required by Load Combinations 3-5 and 3-6. Thus unless the connection can develop the full strength of members framing into it, the Load Combinations 3-7 and 3-8 should be used to provide the required strength on the connection. 7.2. Joint Requirements Although for OMF it is not required to meet most of the special detailing requirements given in Sect. 8, consideration should be given to using as many of the requirements as practical, particularly in those locations where good engineering judgment would suggest that the use of the special detailing requirements would provide improved system and member ductility and stability. The provision requiring a demonstration of rotation capacity is included to permit the use of connections not permitted under the provisions of Sect. 8, such as top and bottom angle joints, in areas where the added drift is acceptable. 8.

REQUIREMENTS FOR SPECIAL MOMENT FRAMES (SMF)

8.1. Scope The requirements in this Section are for those buildings whose lateral force resisting systems are moment frames in the higher seismic zones. The special provisions, when reasonably applied, provide SMF with reliable ductile systems. Non-ductile behavior is inhibited so that nonlinear response to large earthquake motions can occur in components of the frames having a capability of ductile behavior. The concepts are not new but the provisions are supported by tests and analyses.11–17 SMF systems when properly designed have, in general, resulted in reliable ductile structural systems that respond well to high earthquake motions for both low and high rise buildings. Inelastic energy absorption through ductile behavior of members of SMF can occur at three places usually adjacent to the beam-to-column connection. Flexural hinges can form in the beams and columns and shear yielding can occur in the area of the panel zone. Within limits and specific restraints, inelastic yielding is permitted in each or in combinations of these three areas. The primary concern when designing the frame for inelastic behavior is to prevent brittle AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 338


fracture and severe buckling in and adjacent to the zone of inelasticity. The final selection of the appropriate zones of inelasticity is left to the design engineer. Different problems are presented to the design engineer depending on which of the three areas is chosen to have the lowest inelastic threshold. Yielding in columns is permitted but is considered by many design engineers to be the least desirable. Special limitations are provided for this type of yielding by the provisions in Sect. 8.6. and the bracing required in Sect. 8.7. If the first inelastic mode is chosen to be shear yielding of the panel zone, the limitations of Sect. 8.3 would be required. This usually results in the flexibility of the panel zone being a significant contributor to the total story drift and consideration of this flexibility should be included in analyses.18 If the designer chooses to avoid inelastic behavior at the above two locations, the yield hinge will form in the beam. This requires the critical design items to be the beam-to-column connection and the beam stability. 8.2. Beam-to-Column Joints The special limitations provided for these joints are intended to assure that inelastic hinging that may occur in the connection during the response to high seismic activity will not take place at the joinery but in one of the two adjoining locations, namely in the beam or in the panel zone.19–24 Some of the more common beam-to-column connections are illustrated in Fig. C-8.1. Beam-tocolumn connections are not only designed to meet the loads prescribed by the Load Combinations 3-1 through 3-6 but also designed to resist the requirements based on the nominal strengths of the members actually used in the framing system. Frequently the frame member sizes may be sized to limit drift or to meet requirements of load combinations other than those containing seismic loads. Thus to provide frames having the capability of deforming into the nonlinear range without having a connection failure, the required strength on the connection is most frequently based on the design strength of the members actually used. An exception is provided for joints that are not designed to contribute to the lateral force resisting system. In order to demonstrate that the joint will be capable of undergoing large deformation, the elastic or inelastic joint rotations that would be induced by deforming the frame into an amplified displacement of 0.4R times that under Load Combinations 3-5 and 3-6 are required. The term 0.4R should not be less than 1.0. If the “non-moment resisting” web connection were to be a shear tab joined to the column flange by welding and bolted to the beam web, the connection should be proportioned to either yield in the tab or by use of horizontally slotted holes for the bolts. Fracture should not occur in the welded joint to the column. (See Fig. C-8.2.) The required shear strength, Vu, of the beam-to-column joint is defined as the summation of the factored gravity loads and the shear resulting from the required moments on the two ends of the beam. The easy method is to assume that Mp occurs at each end of the beam. However, when Load Combination 3-7 is used in which one end only of the beam reaches Mp, or the panel zone nominal shear reaches Vn as defined in Sect. 8.2.a, the shear resulting from hinging at both ends of the beam need not be used. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 339

a

b

c

d

e

f

Fig. C-8.1. Beam-to-column connections.


6 - 340


When the required flexural strength of the joint is Mp of the beam, the type of joint is prescribed to be one of three types: First is the joint where both flanges and web are fully welded to develop their portions of the moment and shear strength of the beam. (See Fig. C-8.3.) Second is the joint of those beams which have a ratio of the flexural nominal strength of the flanges only to the flexural nominal strength of the full section of at least 70 percent. For this connection, the flanges are joined with complete penetration welded joints whereas the web would be designed to carry the

ROTATION BY NON LINEAR BENDING OF JOINT MEMBERS

SHORT SLOTTED HOLES

ROTATION BY BOLT SLIP

Fig. C-8.2. Simple connections.



6 - 341

CONTINUITY PLATE, TOP & BOTTOM ERECTION BOLTS SHEAR PLATE

TYP.

FULL PENETRATION TOP & BOTTOM FLANGE (a)

ERECTION BOLT CONTINUITY PLATE, TOP & BOTTOM

SHEAR PLATE ERECTION BOLT

TYP.

FULL PENETRATION TOP & BOTTOM FLANGE (b)

Fig. C-8.3. Beam-column joint.


6 - 342


required shear by either welds or by slip-critical high strength bolts. (See Fig. C-8.4.) Third is the joint of beams not meeting the 70 percent criteria. This would be similar to the second joint except that the beam web is required to be welded directly or through shear tabs even though the web is bolted to the shear tab. The welds are required to have a nominal moment strength at least equal to 20 percent of the nominal moment strength of the full beam web. (See Fig. C-8.5.) Other joints than the ones specified are permitted to be used but the adequacy of the joint requires substantiation either by tests or by calculations. Where the adequacy is demonstrated by calculations, additional conservatism is provided by requiring the joint to develop at least 125 percent of the nominal moment and shear strength of the beam. 8.3. Panel Zone of Beam-to-Column Connection (Beam web parallel to column web) During recent years many cyclic tests have shown the ductility of shear yielding in panel zones through many cycles of inelastic distortions.17,25–28 Thus the panel zone does not need to develop beam hinging and a method of determining the nominal shear strength of the panel zone is needed. The usual assumption of Von Mises shear limit of 0.577Fy dt did not predict the actual behavior of many of the tests. Many panel zone and beam tests have shown that strain hardening and other phenomena have enabled shear strengths in excess of 1.0Fy dt to be developed. Eq. 8-1 reflects the significant strength provided by thick column flanges. In calculating the required panel zone shear strength the UBC 1991 magnifies the specified load by a factor of 1.85. For the LRFD specification, the typical Load Combinations 3-5 and 3-6 are used and the nominal web shear strength is defined as 0.6Fy dt, rather than 0.55Fy dt which had been used in plastic design and in some previous references. In order to provide the same level of safety as determined by tests and as contained in the UBC 1991, a lower resistance factor φv = 0.75 was selected. An upper limit is placed on the required shear strength of the panel zone of 0.9 times the summation of the beam design plastic moments φb Mp framing into the connection. In order to minimize the chances of shear buckling during inelastic deformations of the panel zone, the thickness of the panel zone material is limited to not less than 1⁄90 of the sum of its depth and width. The thickness of any doubler plate used is assumed ineffective in inhibiting buckling unless it is connected to the panel zone plate in such a manner, such as plug welds, to prevent local buckling of the plate. (See Fig. C-8.6.) Whenever doubler plates are used (i.e., increased strength, compliance with Eq. 8-2, or to reduce panel zone deformations), the plates are required to be close to the column web. The doubler plates are to have at least minimum fillet welds across the top and bottom and to have either butt or fillet welds to the column flanges. These details are provided to closely simulate the joints that have been found to perform satisfactorily in the cyclic tests that have been performed. Fillet AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 343

CONTINUITY PLATE, TOP & BOTTOM

SHEAR PLATE

TYP.



SHEAR PLATE

TYP.


Fig. C-8.4. Beam-column joint, bf tf (db − tf)Fy ≥ 0.7Fy Zx. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 344



SHEAR PLATE

TYP.



SHEAR PLATE

TYP.


Fig. C-8.5. Beam-column joint, bf tf (db − tf)Fy < 0.7Fy Zx. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 345

welding is encouraged to assist in minimizing the built-in weld stresses and the cost of welding. Doubler plates may be designed to extend between continuity plates which are welded directly to the column web or they may extend above and below the continuity plates which are welded to the doubler plate. For the latter case, the horizontal welds at the top and bottom of the doubler plate should be sized to transfer all loads imposed by the design system. In particular, the welds to the column web should be designed to transfer load from the doubler plate to the column web for their portion of load from the continuity plate. For the fillet or butt welds of the doubler plate to the column flanges, the following items should be considered: • The vertical shear and bending loads of beams or girders framing perpendicular to the column web and supported by the doubler plate. • The compression or tension load delivered to the column web and doubler plate by the flanges of the girders framing into the column flanges. For examples of doubler plate connections, see Ref. 55 and Fig. C-8.6. The use of diagonal stiffeners for strengthening and stiffening of the panel zone has not been adequately tested for low cycle reversed loading into the inelastic range. Thus no specific recommendations are made at this time for special seismic requirements for this detail. TOP & BOTT.

1″

BEAM JOINT NOT SHOWN

EA. END

WEB DOUBLER AS REQ’D

Fig. C-8.6. Panel zone detail (with doubler).


6 - 346


8.4. Beam Limitations In order to minimize the cost of connections it has occasionally been suggested that the beam cross-section be reduced immediately adjacent to the column. This type of assemblage can result in a very brittle mode of failure. Detailing that results in a concentration of stress in an area where inelastic deformations are anticipated under large seismic response is discouraged. The width thickness ratio of projecting elements should be within those which provide the cross-section with stability against local buckling. The limits given in Table 8-1 are deemed adequate by the Committee for ductilities to 6 or 7 based on the tests performed to date.29–32 Further testing may result in some modifications of these limits. 8.5. Continuity Plates Sect. K1 of the Specification gives the design requirements for webs and flanges with concentrated forces. Sect. K1.2 gives the design strength in local buckling in a flange under the action of a tensile force. When the design strength is inadequate, column web stiffeners are required. In moment resisting frames, an interior beam-to-column connection has tension on one flange and compression on the opposite side. When stiffeners are required, it is normal to place a full depth stiffener on each side of the column web. As this stiffener provides a load path for the flanges on both sides of the column, it is commonly called a continuity plate. The stiffener not only provides resistance to local flange buckling but also provides a boundary to the very highly stressed panel zone. When it is anticipated that there could be a plastic hinge adjacent to the column, the required force to determine whether a continuity plate is required is not the design earthquake force given by the load combinations 3-1 through 3-6. It is the force exerted by the beam connection when the full plastic moment with possible strain hardening has been formed. Tests have shown that hinging occurs due to local flange buckling when a compact section is strain hardened to about 1.3Fy.20 At the joint, the flanges of the beam can be strain-hardened to a force of 1.8Fy bf tf. Using this force as the required strength on the continuity plate is conservative as there is only a small moment strength contributed by the bolted web connection. Since the flange continuity plate is needed to protect the weld at the joint of the beam flange to column flange, consideration should be given to their use in connections where the calculations indicate they may not be required. Continuity plates have been used in almost all cyclic joint tests that have performed well.17 When tests have been performed on specimens not meeting the requirements of Sect. K1.2, the joints have performed poorly. For the actual design of the continuity plates, Sect. K1.8 of the LRFD specification would apply. 8.6. Column-Beam Moment Ratio Tests have shown that moment frame subassemblages in which yielding of columns occurred did not exhibit any loss of lateral force resistance at displacements representative of maximum expected earthquake response.33 Most engineers believe, however, that the performance of seismic moment frames is more AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 347

predictable if columns outside of the panel zone do not yield. The tests necessary to formulate truly appropriate criteria have not been conducted. In the past, many frames have been designed with the assumption that the first hinging occurs in the columns and until recently no code provisions for this behavior have been enforced. There have not been any documented failures in past earthquakes directly attributable to column hinging. Design situations do occur where elimination of the “strong beam-weak column” connection type would be grossly impractical. The committee feels that some interim provisions are appropriate. Thus Eqs. 8-3 and 8-4 are introduced. These formulas require that the initial potential for yielding at a beam-to-column connection be in the beam or panel zone rather than in the column. The exceptions to the “strong column-weak beam” connection type require that the column be a compact section and include one of the following characteristics: a. Have a low required axial strength. b. Be a column in a story which has a significantly stronger design story shear strength than the story above. c. Be a column that is not part of the lateral force resisting system except to support the axial load from the overturning moment of the building as a whole. Wherever possible the committee recommends that the hinging conform to the requirements of Sect. 8.2. 8.7. Beam-to-Column Connection Restraint In order to function properly, particularly if inelastic behavior in or adjacent to the beam-to-column connection occurs during high seismic activity, the column needs to be braced to prevent rotation out of the plane of the moment frame. 8.7.a. Restrained Connections: Beam-to-column connections are usually restrained laterally by roof or floor framing. For these cases, lateral support of the connection is required only at the level of the top flanges of the beams as long as the column can be shown to remain elastic. The two criteria to demonstrate that the column remains elastic are arbitrary but appear to be reasonable assumptions until otherwise demonstrated by test. If the column cannot be demonstrated to remain elastic, a hinge would be potentially forming and the column should be laterally supported at the levels of both the top and bottom flanges of the beam. The lateral support provided at the beam-to-column connection is to be designed using a required strength of 2 percent of the nominal beam strength. It is recognized from the limited test data available that the lateral support provided should also be rigid enough to inhibit lateral movement of the column flanges.32 Designers should carefully design the lateral support member to be composed of reasonably rigid elements and be anchored to rigid supports. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 348


The lateral support provided the beam-to-column connection is not required to be a separate member at the connection in all cases. It may be shown that the lateral support force can be adequately carried by the column web or the beam flanges. 8.7.b. Unrestrained Connection: Unrestrained connections can occur in special cases as in two story frames, at mechanical floors or for architectural space layout. When this does occur, special care should be provided to minimize the potential of out-of-plane buckling at the connection. Three arbitrary provisions are given for the columns to assure that this buckling does not occur. 8.8. Lateral Support of Beams The lateral support for beams is defined in Chapter F in the LRFD design specifications. In moment resisting frames, the beams are nearly always in double curvature between columns unless one end is pinned. If the formula for plastic design were used as a guide and assuming Mp at one end and pinned at the other, formula F1-1 would yield 3,600ry / Fy. With Fy =36 ksi, Lpd = 100ry. The 1991 UBC has 96ry for this limitation. Due to the low cycle oscillating motion of the frames under earthquake loading and the uncertainty of the locations of hinging under the various loading combinations, a more conservative approach was appropriate and set the maximum limit of the spacing of lateral support for frame beams at 2,500ry / Fy for both top and bottom flanges. 9.

REQUIREMENTS FOR CONCENTRICALLY BRACED FRAMES (CBF)

9.1

Scope The provisions contained in the Section are for braced frame systems of Building Categories C, D, and E where the braces are designed to carry all the lateral force shears or are used in combination with a moment resisting frame. If used in combination with a moment frame system, the moment frames should follow the requirements of Sects. 7 or 8 as required by the local Building Code. In a Concentrically Braced Frame (CBF), the bracing members are so arranged that the brace members primarily act with axial loading. CBF usually are in one of the following five types. (See Figs. C-9.1 through C-9.5). Ductility of CBF systems producing a pattern of reasonably stable reversible distortions provides justification for basing seismic design on reduced displacements that can be expected during a strong earthquake. CBF systems, by the fact that the primary forces in the bracing system are axial tension and compression, are very limited in reversible inelastic distortions. Tests have shown that after buckling, an axially loaded member rapidly loses strength with repeated inelastic load reversals and does not return to its original straight position.34 For this reason in high seismic areas, CBF systems have not been permitted by codes for tall or special buildings without being combined with a moment resisting frame. Codes also have required significantly higher levels of design force so that the possibility of large uncontrolled inelastic deformations will not occur. For instance, ASCE 7-88 in Sect. 9.9.5 requires that CBF be designed to a force 1.25 times the normal design force given in Sect. 9.4 for the system involved. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


Fig. C-9.1. Diagonal braced frame.

Fig. C-9.2. X-braced frame.


6 - 349

6 - 350


Fig. C-9.3. V-braced frame.

Fig. C-9.4. Inverted V-braced frame.



6 - 351

In Sect. 9.4 of this specification, for Special Configurations, this higher force factor is raised to 1.5. The performance of CBF systems in earthquakes is acceptable as long as they retain stable configuration. The emphasis of these provisions is on raising the level of stable behavior and protecting against brittle failures. When an axially loaded brace buckles in compression, several developments take place: a. When buckling occurs, additional load is transferred to the tension brace increasing the force it must carry. b. The buckling of the brace may cause excessive rotation at the brace ends and local connection failure. c. The buckling can cause local or torsional buckling to occur near mid span. d. If the buckling causes the brace to bow out of plane of the braced frame, non-structural encasement of the frame system can be destroyed. e. Brace buckling can occur non-symmetrically which would induce large torsional response. f. Excessive buckling can affect non-structural systems which are attached to the frame.6

Fig. C-9.5. K-braced frame.


6 - 352


9.2. Bracing Members 9.2.a. Slenderness: Except for low buildings using the required strength given in Sect. 9.5, the slenderness (L / r) of members of CBF systems is limited. In the post-buckling range, the compressive nominal axial strength deteriorates.34 Hysteresis loops of tested assemblies take on a severely pinched shape. (See Fig. C-9.6.) Braces with small L / r dissipate more energy because in the post-buckling range they undergo cyclic inelastic bending which slender braces cannot. Very slender braces have almost no stiffness in a buckled configuration. On a load reversal, the brace quickly assumes a straightened configuration and very rapidly picks up a tensile force. This rapid increase in the brace force may cause impact loading and may lead to a brittle failure of the connection. The curvatures associated with cyclic inelastic bending of braces may be large and local buckling can develop. This local buckling may be so severe as to result in localized kinking of the brace or the connection plate elements causing crack propagation and fracture. Such fractures have been obseved rather early in tests of tubular bracing members.35 This characteristic is more prevalent in rectangular and square tube braces. Consideration should be given to using composite tubes with concrete fill to inhibit buckling.36 9.2.b. Compressive Design Strength: Due to the cyclic nature of seismic response, the compressive design strength of bracing members is reduced to 80 percent of the value given in the Specification, Chapter E. P (kips) 264 200

100 –1.0

1.0

0

Finish

2.0

δ Axial (in.) –100

–200

W6x20 L = 10 ft KL / r = 80

9 in. 3.5 in. Lateral ∆ Fig. C-9.6. P−δ diagram for a strut.



6 - 353

This reduction factor is a simplified value from the factor proposed by others which varies with KL / r.6 When evaluating the nominal strength of the bracing system for the purpose of determining the maximum load the bracing will impose on other systems (such as Eq. 6-1), the reduction for cyclic behavior should not be used for design as it would underestimate the nominal strength of the bracing system during the early cycles of seismic response. 9.2.c. Lateral Force Distribution: This provision attempts to balance the tensile and compressive resistances across the width and breadth of the building since at large loads the capacity of buckled compression braces may be substantially less than that of tension braces. An exception is provided for the case where the bracing members were sufficiently oversized to provide essentially elastic seismic response. 9.2.d. Width-Thickness Ratios: In Sect. B5 of the Specification, definitions are given to three types of sections. The compact section is one which has elements with width-thickness ratios, λ, less than λp. Non-compact sections are those with elements λp ≤ λ ≤ λr. Slender compression sections are those which have at least one element for which λ is greater than λr. The latter sections are prone to local buckling and are not to be used for the bracing members covered in this Section. The circular section wall thickness limitation was chosen to be the same as for Plastic Design in the Specification. Due to the repetitive nature of cyclic loading for rectangular tubular sections, a more stringent requirement on the b / t ratios is specified based on tests.35–36 Filling of tubing with lean concrete has been shown to effectively stiffen the tube walls. 9.2.e. Built-up Member Stitches: The special requirements for built-up member stitches were chosen from test data.37 They are intended for members built up from double angles and channels, and may not be appropriate for markedly different shapes. 9.3. Bracing Connections 9.3.a. In CBF systems, the bracing members normally carry most of the seismic story shear, particularly if a dual system is not used. The required strength on brace connections should be adequate so that failure by out-of-plane buckling of gussets or brittle fracture of the connection are not the critical failure mechanism. The minimum of the three criteria, (i.e., the design axial tension strength of the bracing member, the force generated by the amplified load combinations of 3-7 and 3-8, and the maximum force that could be generated by the overall system) determine the required strength on both the brace connection and the beam-to-column connection if it is part of the bracing system. The latter criterion is intended to cover the possibility that the shear could be limited by the amount of overturning that could be developed. 9.3.b. Net Area: Eq. 9-1 extends the concepts of LRFD Sect. B3 to the forces given in Section 9.2.a above. 9.3.c. Gusset Plates: Gusset plates in CBF systems are frequently the critical AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 354


design element in a system required to deform into the inelastic range. The increased force required for design of CBF tends to reduce the inelastic demand but may be insufficient to totally eliminate the problem. If the critical buckling mode of the braced member is in the plane of the CBF, the gussets and their joints should have a design strength capable to resist the nominal strength of the brace in that direction. If the critical buckling mode is out of the plane of the CBF, each gusset shall be detailed to permit the formation of a hinge line in the gusset. (See Fig. C-9.7.) 9.4. Special Bracing Configuration Requirements In addition to the general requirements for bracing members and their connections given above, special limitations are applied to V and K types of CBF systems due to their special configurations. 9.4.a. V and Inverted V Type Bracing: If one diagonal of a V type brace were to buckle in compression, the force in the tension brace would become larger than the force in the buckled brace. The vertical resultant of these two forces could then impose a large vertical deformation on the horizontal member of the bracing system. (See Fig. C-9.8.) If the connection at the point of the V tip were pinned, there would be no resistance to this deformation. If a continuous horizontal member survives and undergoes a deformation reversal, the previously buckled diagonal member would

BRACING MEMBER 2t

GUSSET PLATE

t = THICKNESS OF GUSSET PLATE

Fig. C-9.7. Brace-to-gusset plate requirement for buckling out-of-plane bracing system.



6 - 355

not return to its original alignment and the diagonal member which was in tension could exceed its capacity in compression. In this manner both diagonal members would be in a buckled condition. This behavior would cause the post buckling strength of the braced system to deteriorate rapidly.38 (See Fig. C-9.9.) Near the tip point of the V is a zone where inelastic rotations are likely to occur, members should be braced against out-of-plane buckling. Several options were considered for CBF systems using the V type bracing. One was to prohibit its use, a second was to impose stringent limitations on the slenderness ratios of the bracing members, and a third was to provide a larger axial load capacity for the diagonal members. The latter option was adopted by providing a design axial strength 1.5 times the required axial strength in lieu of the 1.25 normally required for other CBF. It is also required that the beam be continuous throughout the bay and that this beam be designed to carry the tributary vertical gravity loads without considering the support provided by the diagonal members of the V. A review of more recent testing of V braced systems may in future editions be able to modify some of the current limitations. 9.4.b. K Bracing: In areas of high seismicity where it is envisioned that inelastic response to large motions will be required, the K type of CBF system is not a desirable method for seismic resistance. The same behavior discussed in the V type bracing occurs, but in the case of the K system a buckled brace causes the column to deform horizontally. Potentially this could cause column buckling and subsequent collapse. In buildings of Categories A, B, and a portion of C, the K system is permitted unrestricted by these provisions. For the remainder of Category C as per

Fig. C-9.8. Failure mechanism of inverted V-braced frame.


6 - 356


Sect. 2.2, however, K braces shall meet the requirements Sects. 9.4.b and 9.5. This requires a 50 percent increase in design axial load for the braces and a continuous column though at the story mid-height. It is recommended that K type bracing not be used even where permitted for seismic resistance unless other configurations are impractical. 9.5. Low Buildings One of the few problem areas observed in the seismic performance of smaller steel buildings using the CBF system pertain to the size and type of member connections used. Quite frequently the critical design horizontal load is wind rather than seismic. In these cases, the sizing of bracing members is larger than would be required if seismic loads were the only design horizontal loads. Thus for smaller buildings and roof structures, the special provisions for CBF systems have been waived if the seismic resisting system has been designed using the amplified loads given in Load Combinations 3-7 and 3-8. This waiver would permit, for instance, an X braced or diagonal braced system in which the bracing members would be assumed to be in tension only. 10.

REQUIREMENTS FOR ECCENTRICALLY BRACED FRAMES (EBF)

10.1. Scope

THIRD STORY SHEAR (KIPS)

Research39–49 has shown that buildings using the EBF system possess the ability to combine high stiffness in the elastic range together with excellent ductility and energy dissipation capacity in the inelastic range. In the elastic range, the lateral stiffness of an EBF system is comparable to that of a CBF system, particularly when short link lengths are used. In the inelastic range, EBF systems

800

400

0

–400

–800 –4

–2

0

2

4

THIRD STORY DRIFT (IN.) Fig. C-9.9. Story shear–story drift diagram for frame with inverted V-bracing.



6 - 357

provide stable, ductile behavior under severe cyclic loading, comparable to that of a SMF system. The EBF is composed of columns, beams, and braces in which at least one end of each bracing member connects to a beam at a short distance from a beam-to- column connection or from an adjacent beam-to-brace connection. (See Fig. C-10.1.) The short distance of the beam between the brace connection and the column or between brace connections is called the link. The design purpose of an EBF system creates a system that will yield primarily in the links. The special provisions for EBF systems are intended to satisfy this criterion and to ensure that cyclic yielding in the links can occur in a stable manner. The yielding in the links is accomplished by ensuring that the diagonal braces, the columns, and the portion of the beam outside of the links remain essentially elastic under forces that can be generated by fully yielding and strain hardened links. Arrangements of braces can be made in which links may not be fully effective. One such arrangement is the one shown on Fig. C-10.2 in which links are provided at each end of the brace. If the upper link has significantly lower design shear strength than the story below, the upper link deforms inelastically and limits the force that can be delivered to the brace to deform the lower link inelastically. When this condition occurs the upper link is termed an active link, whereas the lower link is an inactive link. Having potentially inactive links in the EBF system increases the difficulty of analysis. The plastic analyses show that in some cases the lower link yields due to the combined effect of D, L, and E loads, and the frame capacity becomes smaller than expected.50 It also increases the cost of the structure by requiring full link details on the inactive links even though the brace would be sized by the strength of the active link and the brace connection at an inactive link could be designed as a pin. Thus it is best to arrange a system that contains only active links as those shown in Fig. C-10.1. Design suggestions have been compiled in Ref. 48. In Sect. 10.1 in conformity with the strong column–weak beam concept, plastic hinges should not develop in columns at floor beam levels in EBF. The occurrence of such plastic hinges, together with those forming in the links, could result in a soft story and must be prevented. There are two important code provisions intended to prevent this from happening. First, according to Sect. 6.1, the required axial column strength includes PE, based on application of the amplified earthquake load 0.4RE. Second, per Sect. 10.8, the required strength of columns due to the forces introduced at the connection of a link and/or brace is based on these forces multiplied by a factor of 1.25. Note that for a severe earthquake the formation of plastic hinges at column bases is generally unavoidable. 10.2. Links The general provisions for links to ensure that stable yielding occurs are included under this heading. 10.2.a. Beams with links are required to be compact shapes following the same criteria as SMF systems (Table 8-1). 10.2.b. In order to provide steel with proven ductile behavior the yield stress of steel is limited to 50 ksi. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 358


10.2.c. Doubler plates on the link web are not permitted as they do not perform as intended in inelastic deformations. Openings are not permitted as they adversely affect the yielding of the link web. 10.2.d. The link design shear strength φVn is the lesser of that determined from the yield shear or twice the plastic moment strength divided by the link length. This φVn should be greater than or equal to the required shear determined from the Load Combinations 3-5 or 3-6. 10.2.e. If the required axial load on the link is less than 0.15Py, the effects of the axial load can be ignored, In general, the axial load is negligible because the horizontal component of the brace load is transmitted to the beam outside of the link. However, due to a particular arrangement of the framing, substantial axial forces can develop in the link. For such cases, the limitations given in f. apply, and the design shear strength and link lengths are required to be reduced to ensure stable yielding.

c

d

c

a c

d

c

d

b

b

a

d

a

c b

d

b

a

c

d

c

d

c

b

c

b

d

b

a

d

c

d

d

d c

b

d

b

b

a

a c

b

d

d

d c

b

d

b

b

a

a c

b

d

d

d

b

a

b

c a

b

a = column b = brace c = link d = portion of beam outside of link

Fig. C-10.1. Common types of eccentric braced frames.


b


6 - 359

10.2.f. See Commentary 10.2.e. 10.2.g. The link rotation angle is defined in the Specifications as the plastic angle between the link and the beam outside the link when the total story drift ∆t, calculated using amplified earthquake forces 0.4R × E. The plastic link rotation can be conservatively determined assuming that the EBF bay will deform in a rigid-plastic mechanism. Several such mechanisms are illustrated for various EBF configurations in Fig. C-10.3. The plastic angle is determined using a story drift ∆p = ∆t − ∆e, where ∆e the elastic story drift can conservatively be assumed to be zero. The plastic story drift angle θp = ∆p / h follows from geometry. The actual plastic link rotation angle can be determined by non-linear elastic-plastic analyses if a more explicit definition of the angle is desired. An inverted Y system is shown on Fig. C-10.1. In this system the precise definition given in the Glossary for the link rotation angle does not apply but the concept is the same as in the other systems, as shown on Fig. C-10.3. As usual both ends of the link are required to be laterally supported. The link length of 1.6Mp / Vp indicates the limit chosen for the link to act primarily in shear. The link length 2.6Mp / Vp is the lower limit of a flexural link. Straight line interpolation is used for the intermediate link lengths. It has been demonstrated experimentally51–52 as well as analytically48 that the first floor links usually experience the largest plastic deformation. In extreme cases this may result in a tendency to develop a soft story. The

a

b

φVn − link a (active link) < φVn − link b (inactive link)

Fig. C-10.2. EBF—active and inactive link.


6 - 360


plastic link rotations tend to attenuate at higher floors, and decrease with the increasing frame periods. Therefore for severe seismic applications a conservative design for the links in the first two or three floors is recommended. This can be achieved by increasing the minimum design shear strengths of these links on the order of 10 percent over that specified in Sect. 10.2.d. An even more conservative approach would be to have vertical connecting members at the ends of the links in a few lower floors. The use of the framing shown in Fig. C-10.1 can be advantageous where ∆p

∆p

e

e θp

γp

γp h

h

e γp

θp

L

L/2

L/2

γp = L θp e

γp = L θp 2e

∆p

∆p γp e

γp

L/2

e/2 e/2

h

h

θp

θp

L/2

γp = L θp e ∆v ∆t ∆e ∆p e h L θp γp

L γp = h θp e

= Story drift determined using base shear v, inches. = Total story drift, inches = ∆v × e′ / e. = Elastic story drift, inches = ∆v times the earthquake load factor. = Plastic story drift, inches = ∆t − ∆e (conservatively, ∆e = 0). = Link length, inches. = Story height, inches. = Column to column distance, inches. = Plastic story drift angle, radians = ∆p / h. = Link rotation angle, radians.

Fig. C-10.3. Link rotation angle. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 361

the beam-column-brace connections can be designed as simple connections. Welds of the link flanges are avoided in this kind of framing. By changing the link lengths the stiffness of an EBF can be modified. In this manner the frame periods can be optimized. 10.2.h. The intent of this provision is to permit a CBF on the top floor of an EBF building over five stories tall with application of an earthquake response modification coefficient R appropriate for an EBF. 10.3. Link Stiffeners Properly detailed and restrained webs can provide stable, ductile, and predictable behavior under severe cyclic loading. The design of the EBF link requires close attention to the detailing of the link web thickness and stiffeners. 10.3.a. Full depth stiffeners are required at the end of all EBF links and serve to transfer the link shears to the reacting elements as well as restraining the link web against buckling. 10.3.b. In shear links, the spacing of intermediate web stiffeners is varied depending on the magnitude of the link rotation angle.45 The closer spacing is provided for the system with the greatest angle. Flexural links having lengths greater than 2.6Mp / Vp but less than 5Mp / Vp are required to have an intermediate stiffener at a distance from the link end equal to 1.5 times the beam flange width. Links between shear and flexural limits would have intermediate stiffeners meeting the requirement of both shear and flexural links. When the link length is greater than 5Mp / Vp, no intermediate stiffeners are required. Intermediate stiffeners are required to be full depth in order to effectively react against shear buckling. Intermediate stiffeners are required on both sides of the web for links 25 inches in depth or greater. For links less than 25 in. deep, the stiffener need be on one side only. 10.3.c. All link stiffeners are required to be fillet welded to the link web. These welds shall have a required strength equal to the nominal vertical tensile strength of the stiffener. The connection to the link flanges should be similar. 10.4. Link-to-Column Connections There are special connection requirements for the connections of links to columns. The intent is to provide connections which can transfer not only the shear and moment forces of the links but also torsion due to flange buckling. The Specification does not explicitly address the column panel zone design requirements at link-column connections, as little research is available on this issue. However, from research on panel zones for SMF systems, it is believed that limited yielding of panel zones in EBF systems would not be detrimental. Pending future research on this topic, a suggested design approach is as follows: Compute the required shear strength of the panel zone based on the bending moment at the column end of the link, as given by the equations in Sect. 10.6.a in the commentary of these provisions. The corresponding panel zone design shear strength should then be computed according to Eq. 8-1 of these provisions. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 362


10.5. Lateral Support of the Link One of the essential items to ensure stable inelastic behavior of the EBF system is to restrain the ends of the link from twisting out of plane. The 6 percent of the nominal strength of the beam flange defines the required strength on the lateral support member and its connections. 10.6. Diagonal Brace and Beam Outside of Links 10.6.a. A basic requirement of EBF design is that yielding be restricted primarily to the links. Accordingly, the diagonal brace and the beam segment outside of the link should be designed to resist the maximum forces that can be generated by the link, accounting for the sources of link overstrength. Link overstrength can be attributed primarily to strain hardening, effects of composite floor systems, and the actual yield strength of the link exceeding the specified yield strength. In EBF research literature, for design of the brace and the beam, an overstrength factor of 1.5 has generally been applied to the nominal strength of the link. Using this overstrength factor, the brace and beam segments were checked using their nominal strength, i.e., using φ =1.0. This approach considers that designing for an overstrength factor of 1.5 represents an extreme loading condition for the beam and brace, and therefore a relaxation of the φ factor was appropriate to avoid an overly conservative design.49 Sect. 10.6.a specifies that the design strength of the beam and diagonal brace exceed the forces generated by 1.25 times the nominal link shear strength, maintaining approximately the same basic design approach for the diagonal brace and beam. That is, based on a φ factor of 0.85 on axial compression in the beam or brace, the effective overstrength factor becomes 1.25 / 0.85, or about 1.5. For bending moments in the beam or diagonal brace, for which φ is 0.9, the overstrength factor becomes 1.25 / 0.9, or about 1.4, representing a slight relaxation from the test criterion. Based on a link overstrength factor of 1.25, the required strength of the diagonal brace and beam segment outside of the link can be taken as the forces generated by the following values of link shear and link end moment: For e ≤ 2Mp / Vp,

link shear link end moment

= 1.25Vp = e(1.25Vp ) / 2

For e > 2Mp / Vp,

link shear link end moment

= 2(1.25Mp) / e = 1.25Mp

The above equations are based on the assumption that link end moments will be equal when the link achieves its limit strength. For links of length e ≤ 1.3Mp / Vp attached to columns, experiments have shown that link end moments do not equalize.44 For this situation, link shear and link end moments can be taken as: For e ≤ 1.3Mp / Vp next to column, link shear

= 1.25Vp



moment at column end of link moment at brace end of link

6 - 363

= 0.8Mp = e(1.25Vp ) − 0.8Mp

The link shear force will generate axial force in the diagonal brace, and for most EBF configurations, will also generate substantial axial force in the beam segment outside of the link. The ratio of beam or brace axial force to link shear force is controlled primarily by the geometry of the EBF and is therefore not affected by inelastic activity within the EBF.47 Therefore, this ratio can be taken from an elastic frame analysis and used to scale up the beam and brace axial force to a level corresponding to the link shear force specified in the above equations. At the brace end of the link, the link end moment will be transferred to the brace and to the beam. If the diagonal brace and its connection remains elastic, based on link overstrength design considerations, some minor inelastic rotation can be tolerated in the beam outside of the link. 10.6.b. Typically in EBF design, the intersection of the brace and beam centerlines is located at the end of the link. However, as permitted by Sect. 10.6.b, the brace connection should be designed with an eccentricity so that the brace and beam centerlines intersect inside of the link. This eccentricity in the connection generates a moment that is opposite in sign to the link end moment. Consequently, the value of link end moment

STIFF PLATE EA. SIDE OF WEB. USE FILLET WELD CONT. @ WEB & FLANGE BEAM OUTSIDE OF LINK

LINK LENGTH – e

FULL PENETRATION TOP & BOTT.

W SHAPE LINE OF INTERSECTION OF BRACE AND BEAM SHALL BE AT THE EDGE OF LINK OR INSIDE THE LINK

INTERMEDIATE STIFFENER PLATES EA. SIDE OF WEB FOR LINKS ≥ 25″ CONT. FILLET WELD @ WEB AND FLANGE

Fig. C-10.4


6 - 364


given above can be reduced by the moment generated by this brace connection eccentricity. This may substantially reduce the moment that will be required to be resisted by the beam and brace, and may be advantageous in design. The intersection of the brace and beam centerlines should not be located outside of the link, as this increases the bending moment generated in the beam and brace. See Figs. C-10.4 and C-10.5. 10.6.c. If the brace connection at the link is designed as a pin, the beam by itself shall be adequate to resist the entire link end moment. This condition normally would occur only on EBF with short links. If the brace is considered to resist a portion of the link end moment, then the brace connection at the link should be designed as fully restrained, as required by Sect. 10.6.c. Test results on several brace connection details subject to axial force and bending moment are reported in Ref. 47. 10.6.d. When checking the requirements of Sect. 10.6, both the beam and diagonal brace should, in general, be treated as beam-columns in strength and stability computations. Unlike CBF, the brace of an EBF may be subject to significant bending moments. For the beam segment outside of the link, adequate lateral bracing should be provided to maintain its stability under the axial force and bending moment generSTIFF. PLATE EA. SIDE OF WEB. USE FILLET WELD CONT. @ WEB & FLANGE

LINK LENGTH – e

BEAM OUTSIDE OF LINK

FULL PENETRATION TOP & BOTT.

LINE OF INTERSECTION OF BRACE AND BEAM SHALL BE AT THE EDGE OF LINK OR INSIDE THE LINK

BENT PLATE OR TWO WELDED PLATES GUSSET PLATE TS (BRACE) SPLIT END TO FIT GUSSET

Fig. C-10.5


INTERMEDIATE STIFFENER PLATES EA. SIDE OF WEB FOR LINKS ≥ 25″ CONT. FILLET WELD @ WEB & FLANGE


6 - 365

ated by the link, as required by Sect. 10.6.d. If the stability of the beam is provided by adequate lateral support, tests have shown that limited yielding of the beam segment is not detrimental to EBF performance, and for some EBF configurations may be unavoidable.47 However, the combined flexural strength of the beam and the brace, reduced for the presence of axial force, should be adequate to resist the link end moment. For EBF geometries with very small angles between the beam and the brace and/or for EBF with long links, satisfying the requirements of Sect. 10.6.e. may require very heavy braces, and in extreme cases, may require cover plates on the beams. EBF with relatively steep braces, e.g., brace-beam angles greater than about 40 degrees, combined with short links are preferable for avoiding design problems with the brace and beam segment outside of the link. A general discussion on design issues related to the beams and braces of an EBF is provided in Ref. 49, with further details provided in Ref. 47. 10.7. Beam-to-Column Connection If the arrangement of the EBF system is such that a link is not adjacent to a column, a simple pinned connection is considered to be adequate if the connection provides some restraint against torsion in the beam. The magnitude of torsion is calculated by considering perpendicular forces equal to 1.5 percent of the nominal axial flange tensile strength applied in opposite directions on each flange. 10.8. Required Column Strength As the shear strength of the adjoining critical link is potentially greater than the nominal strength due to strain hardening, the required column strength is required to be designed for the increased moment and axial load due to the load from the adjacent link or brace. 11.

QUALITY ASSURANCE As the behavior of all steel framing during a major earthquake is dependent on the workmanship of the fabricator in providing sound joints, the design engineer is advised to provide for adequate assurance control, particularly on the tension groove welded joints of the seismic resisting system. ASCE 7-92 provides special requirements for inspection and testing based on the Seismic Performance Category of the building to be built. The special requirements for structural steel construction are in general those that would normally be required for construction in all areas of seismic activity.


6 - 366

COMMENTARY: PART II—ASD PROVISIONS

Part II—ASD Provisions 1.

SCOPE As noted in PART I, the special seismic requirements are collateral provisions related to the AISC Load and Resistance Factor Design Specification. As that document was first published in 1986, the references to earthquake load were not current. The provisions in PART I use limit state load models derived from the 1991 NEHRP3 and the soon to be published ASCE 7-93.2 The provisions in PART II allow a designer to apply AISC Allowable Stress Design Specification for Structural Steel Buildings (ASD)53 in the design of the seismic lateral force resisting system based upon limit state loads. If the user wishes to use ASD in the design of the seismic lateral force resisting system where the loads are based upon service loads, the loads need to be converted to factored levels consistent with those in PART I. The PART II provisions are intended to be used in conjunction with PART I by either adding to or substituting to the provisions of Part I.

3.2

Nominal Strengths, and

3.3

Design Strengths These provisions modify PART I to convert allowable stresses into equivalent nominal strengths by multiplying allowable stresses by 1.7 as noted. Design strengths are determined by multiplying φ times the nominal strengths.

7.2, 10.6.a, and 10.6.d These modifications to PART I requirements change FR and PR connections to Type 1 and Type 3 connections consistent with ASD nomenclature.


6 - 367

List of References 1. AISC, Load and Resistance Factor Design Specification, American Institute of Steel Construction, Inc., Chicago, IL, 1986. 2. ASCE 7-93, Minimum Design Loads for Buildings and Other Structures, American Society of Civil Engineers, New York, NY, 1993 (to be published). 3. BSSC, NEHRP (National Earthquake Hazards Reduction Program) Recommended Provisions for the Development of Seismic Regulations for New Buildings, Building Seismic Safety Council, Federal Emergency Management Agency, Washington, DC, 1992. 4. Luft, R. W., “Comparison Among Earthquake Codes,” Earthquake Spectra, Earthquake Engineering Research Institute, Vol. 5, No. 4, November 1989. 5. ICBO, Uniform Building Code, International Conference of Building Officials, Whittier, CA, 1991. 6. SEAOC, Recommended Lateral Force Requirements, Seismology Committee, Structural Engineers Association of California, Sacramento/San Francisco/Los Angeles, CA, 1988. 7. AWS, D1.1-92, Structural Welding Code, American Welding Society, Inc., Miami, FL, 1992. 8. Popov, E. P., Stephen, R. M., “Tensile Capacity of Partial Penetration Welds,” Journal of the Structural Division, American Society of Civil Engineers, Vol. 103, No. ST9, September 1977. 9. Bruneau, M., Mahin, S. A., and Popov, E. P., Ultimate Behavior of Butt Welded Splices in Heavy Rolled Steel Sections, Report No. UCB/EERC-87/10, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1987. 10. Huckelbridge, A. A., Clough, R. W., Earthquake Simulator Tests of Nine-Story Steel Frame with Columns Allowed to Uplift, Report No. UCB/EERC-77/23, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1977. 11. Carpenter, L. D., Lu, L-W., Reversed and Repeated Load Tests of Full Scale Steel Frames, Fritz Engineering Laboratory Report No. 332.7, Lehigh University, Bethlehem, PA, 1972. 12. Galambos, T. V., Deformation and Energy Absorption Capacity of Steel Structures in the Inelastic Range, Bulletin No. 8, American Iron and Steel Institute, New York, NY, 1968. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 368

LIST OF REFERENCES

13. Krawinkler, H., Bertero, V. V., and Popov, E. P., Inelastic Behavior of Steel Beam-to-Column Subassemblages, Report No. UCB/EERC-71/7, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1971. 14. Bertero, V. V., Popov, E. P., and Krawinkler, H., Further Studies on Seismic Behavior of Steel Beam-Column Subassemblages, Report No. UCB/EERC73/27, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1973. 15. Popov, E. P., “Seismic Behavior of Structural Assemblages,” Journal of the Structural Division, American Society of Civil Engineers, Vol. 106, No. ST7, July 1980. 16. Tall Building Systems and Concepts, Monograph on Planning and Design of Tall Buildings, Council on Tall Buildings and Urban Habitat, American Society of Civil Engineers, New York, NY, 1980. 17. Popov, E. P., Amin, N. R., Louie, J. J., and Stephen, R. M., “Cyclic Behavior of Large Beam Column Assemblies,” Earthquake Spectra, Professional Journal of the Earthquake Engineering Research Institute, Vol. 1, No. 2, February 1985. 18. Tsai, K. C. and Popov, E. P., “Seismic Panel Zone Design Effect on Elastic Story Drift in Steel Moment Resisting Frames,” Journal of Structural Division, in press. 19. Nicoletti, J. P., Pinkham, C. W., Saunders, C. M., Teal, E. J., A Synthesis of Steel Research for Code Development, Structural Steel Educational Council, San Francisco, CA, 1984. 20. Popov, E. P. and Pinkney, R. B., Behavior of Steel Building Connections Subjected to Inelastic Strain Reversals—Experimental Data, Bulletin No. 14, American Iron and Steel Institute, November 1968. 21. Popov, E. P. and Stephen, R. M., Cyclic Loading of Full-Size Steel Connections, Bulletin No 21, American Iron and Steel Institute, February 1972. 22. Driscoll, G. C. and Beedle, L. S., “Suggestions for Avoiding Beam-to-Column Web Connection Failure,” Engineering Journal, American Institute of Steel Construction, Chicago, IL, 1st Qtr., 1982. 23. Tsai, K. C. and Popov, E. P., Two Beam-to-Column Web Connections, Report No. UCB/EERC-86/05, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1986. 24. Popov, E. P. and Tsai, K. C., Performance of Large Seismic Steel Moment Connections Under Cyclic Loads, Proceedings; Structural Engineers Association of California Convention, San Diego, CA, October 1987. 25. Slutter, R., Tests of Panel Zone Behavior in Beam Column Connections, Lehigh University, Report No. 200.81.403.1, Bethlehem, PA. 26. Becker, E. R., Panel Zone Effect on the Strength of Rigid Steel Frames, University of Southern California Structural Mechanics Laboratory, USCOE 001, June 1971. 27. Fielding, D. J., Huang, J. S., “Shear in Steel Beam-to-Column Connections,” Welding Journal, Vol. 50, No. 7, Research Supplement, 1971. 28. Krawinkler, H., “Shear in Beam-Column Joints in Seismic Design of Steel AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 369

Frames,” Engineering Journal, American Institute of Steel Construction, Chicago, IL, Vol. 15, 1978. 29. Sawyer, H. A., “Post-Elastic Behavior of Wide-Flange Steel Beams,” Journal of the Structural Division, Vol. 87, No. ST8, American Society of Civil Engineers, December 1961. 30. Lay, M. G., “Flange Local Buckling in Wide-Flange Shapes,” Journal of the Structural Division, Vol. 91, No. ST6, American Society of Civil Engineers, December 1965. 31. Kemp, A. R., “Factors Affecting the Rotation Capacity of Plastically Designed Members,” The Structural Engineer, Vol. 64B, No. 2, June 1986. 32. Bansal, J. P., The Lateral Instability of Continuous Steel Beams, CESRL Dissertation No. 71-1, University of Texas, Austin, TX, 1971. 33. Krawinkler, H., Bertero, V. V., Popov, E. P., Hysteresis Behavior of Steel Columns, Report No. UCB/EERC-75/11, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1975. 34. Black, R. C., Wenger, W. A., Popov, E. P., Inelastic Buckling of Steel Struts Under Cyclic Load Reversals, Report No. UCB/EERC-80/40, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1980. 35. Tang, X., Goel, S. C., Seismic Analysis and Design Considerations of Braced Steel Structures, UMCE Report 87-4, University of Michigan, Ann Arbor, MI, 1987. 36. Uang, C-M., and Bertero, V. V., Earthquake Simulation Tests and Associated Studies of 0.3-Scale Model of a Six-Story Concentrically Braced Steel Structure, Report No. UCB/EERC—86/10, EERC, Berkeley, CA, December 1986. 37. Liu, Z., Goel, S. C., Investigation of Concrete Filled Steel Tubes under Cyclic Bending and Buckling, UMCE Report 87-3, University of Michigan, Ann Arbor, MI, 1987. 38. Astaneh, A., Goel, S. C., Hanson, R, D., “Earthquake-Resistant Design of Double Angle Bracings,” Engineering Journal, American Institute of Steel Construction, Chicago, IL, Vol. 23, No. 4, 1986. 39. Roeder, C. W. and Popov, E. P., “Eccentrically Braced Frames for Earthquakes,” Journal of the Structural Division, Vol. 104, No. 3, American Society of Civil Engineers, March 1978. 40. Libby, J. R., “Eccentrically Braced Frame Construction—A Case History,” Engineering Journal, American Institute of Steel Construction, Chicago, IL, Vol. 18, No. 4, 1981. 41. Merovich, A. T., Nicoletti, J. P. and Hartle, E., “Eccentric Bracing in Tall Buildings,” Journal of the Structural Division, Vol. 108, No. 9, American Society of Civil Engineers, September 1982. 42. Hjelmstad, K. D. and Popov, E. P., “Cyclic Behavior and Design of Link Beams,” Journal of the Structural Division, Vol. 109, No. 10, American Society of Civil Engineers, October 1983. 43. Malley, J. O. and Popov, E. P., “Shear Links in Eccentrically Braced Frames,” AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 370

LIST OF REFERENCES

Journal of the Structural Division, Vol. 110, No. 9, American Society of Civil Engineers, September 1984. 44. Kasai, K. and Popov, E. P., “General Behavior of WF Steel Shear Link Beams,” Journal of the Structural Division, Vol. 112, No. 2, American Society of Civil Engineers, February 1986. 45. Kasai, K. and Popov, E. P., “Cyclic Web Buckling Control for Shear Link Beams,” Journal of the Structural Division, Vol. 112, No. 3, American Society of Civil Engineers, March 1986. 46. Ricles, J. M. and Popov, E. P., Dynamic Analysis of Seismically Resistant Eccentrically Braced Frames, Report No. UCB/EERC-87/107, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1987. 47. Engelhardt, M. D. and Popov, E. P., Behavior of Long Links in Eccentrically Braced Frames, Report No. UCB/EERC-89/01, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1989. 48. Popov, E. P., Engelhardt, M. D. and Ricles, J. M., “Eccentrically Brace Frames: U. S. Practice,” Engineering Journal, American Institute of Steel Construction, Chicago, IL., Vol. 26, No. 2, 1989. p. 66–80. 49. Engelhardt, M. D. and Popov, E. P., “On Design of Eccentrically Braced Frames,” Earthquake Engineering Research Institute, El Cerrito, CA, Earthquake Spectra, Vol. 5, No.3, August 1989. 50. Kasai, K. and Popov, E. P., On Seismic Design of Eccentrically Braced Steel Frames, Proceedings, 8th World Conference on Earthquake Engineering, July 1984, San Francisco, CA., Vol. 5, pp.387–394. 51. Whittaker, A. S., Uang, C-M., and Bertero, V. V., Earthquake Simulation Tests and Associated studies of a 0.3-Scale Model of a Six-Story Eccentrically Braced Steel Structure, Report No. UBC/EERC-87/02, EERC, Berkeley, CA., 1987. 52. Foutch, D. A., “Seismic Behavior of Eccentrically Braced Steel Building,” ASCE Journal of Structural Engineering, Vol. 115, No. 8, August 1989, pp 1857–1876. 53. AISC, Allowable Stress Design Specification, American Institute of Steel Construction, Inc., Chicago, IL, 1989. 54. AISC, Serviceability Design Considerations for Low-Rise Buildings, Steel Design Guide Series 3, American Institute of Steel Construction, Inc. 55. Structural Steel Education Council, Steel Connections/Details and Relative Costs, Moraga, CA., 1986. 56. Uang, C. M., “EstablishingR (or Rw) and Cd Factors for Building Seismic Provisions,” Journal of Structural Engineering, Vol. 117, No. 1, American Society of Civil Engineers, Jan. 1991.


6 - 371

LOAD AND RESISTANCE FACTOR DESIGN

Specification for Structural Joints Using ASTM A325 or A490 Bolts Approved by Research Council on Structural Connections of the Engineering Foundation, June 8, 1988. Endorsed by American Institute of Steel Construction Endorsed by Industrial Fasteners Institute

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. One East Wacker Drive, Suite 3100, Chicago, IL 60601-2001 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 372

PREFACE

The purpose of the Research Council on Structural Connections is to stimulate and support such investigation as may be deemed necessary and valuable to determine the suitability and capacity of various types of structural connections, to promote the knowledge of economical and efficient practices relating to such structural connections, and to prepare and publish related standards and such other documents as necessary to achieving its purpose. The Council membership consists of qualified structural engineers from the academic and research institutions, practicing design engineers, suppliers, and manufacturers of threaded fasteners, fabricators and erectors and code writing authorities. Each version of the Specification is based upon deliberations and letter ballot of the full Council membership. The first Specification for Assembly of Structural Joints Using High Tensile Steel Bolts approved by the Council was published in January 1951. Since that time the Council has published 12 succeeding editions each based upon past successful usage, advances in the state of knowledge and changes in engineering design practice. This version of the Council’s Load and Resistance Factor Design Specification is significantly reorganized and revised from earlier versions. The intention of the Specifications is to cover the design criteria and normal usage and practices involved in the everyday use of high-strength bolts in steel-tosteel structural connections. It is not intended to cover the full range of structural connections using threaded fasteners nor the use of high-strength bolts other than those included in ASTM A325 or ASTM A490 Specifications nor the use of ASTM A325 or A490 bolts in connections with material other than steel within the grip. A Commentary has been prepared to accompany these Specifications to provide background and aid the user to better understand and apply the provisions. The user is cautioned that independent professional judgment must be exercised when data or recommendations set forth in these Specifications are applied. The design and the proper installation and inspection of bolts in structural connections is within the scope of expertise of a competent licensed architect, structural engineer or other licensed professional for the application of the principles to a particular case.


6 - 373

LOAD AND RESISTANCE FACTOR DESIGN

Specification for Structural Joints Using ASTM A325 or A490 Bolts Approved by Research Council on Structural Connections of the Engineering Foundation, June 8, 1988. Endorsed by American Institute of Steel Construction Endorsed by Industrial Fasteners Institute

1. Scope This Specification relates to the load and resistance factor design of structural joints using ASTM A325 high-strength bolts, ASTM A490 high-strength bolts or equivalent fasteners, and for the installation of such bolts in connections of structural steel members. The Specification relates only to those aspects of the connected materials that bear upon the performance of the fasteners. Design and construction shall conform to an applicable load and resistance factor design code or specification for structures of carbon, high-strength low alloy steel or quenched and tempered structural steel. Load and resistance factor design is a method of proportioning structural components such that no applicable limit state is exceeded when the structure is subject to all appropriate load combinations. When a structure or component ceases to fulfill the intended purpose in some way, it is said to have exceeded a limit state. Strength limit states concern maximum load carrying capacity, and thus generally are related to safety. Serviceability limit states are usually related to performance under normal service conditions, and thus usually are not related to strength or safety. (See Commentary. ) The term “resistance” includes both strength limit states and serviceability limit states. The design strength, φRn (nominal strength multiplied by a resistance factor), of each structural component or assemblage must equal or exceed the effect of the factored loads (nominal loads multiplied by load factors, with due recognition for load combinations). Thus, both the load factor and the resistance factor must be known to determine the reliability of the design, identified in load and resistance factor design as the “safety index.” Although the load factors are not stated in this Specification, load criteria contained in American National Standard “Building Code Requirements


6 - 374

RCSC SPECIFICATION FOR STRUCTURAL JOINTS (6/8/88)

for Minimum Design Loads in Buildings and Other Structures,” ANSI A58.1-1982, were used as the basis for determining the resistance factors. For construction governed by other design load criteria, appropriate adjustment of resistance factors may be required. The attached Commentary provides background information in order that the user may better understand the provisions of the Specification. 2. Bolts, Nuts, Washers and Paint (a) Bolt Specifications. Bolts shall conform to the requirements of the current edition of the American Society for Testing and Materials’ “Specification for High-Strength Bolts for Structural Steel Joints,” ASTM A325, or “Specification for Heat Treated, Steel Structural Bolts, 150 ksi Tensile Strength,” ASTM A490, except as provided in paragraph (d) of this section. The Engineer of Record shall specify the type of bolts to be used. (b) Bolt Geometry. Bolt dimensions shall conform to the current American National Standards Institute’s standard, “Heavy Hex Structural Bolts,” ANSI Standard B18.2.1, except as provided in paragraph (d) of this section. The length of bolts shall be such that the end of the bolt will be flush with or project beyond the face of the nut when properly installed. (c) Nut Specifications. Nuts shall conform to the current chemical and mechanical requirements of the American Society for Testing and Materials’ Specification for Carbon and Alloy Steel Nuts,” ASTM A563, or “Specification for Carbon and Alloy Steel Nuts for Bolts for High-Pressure and High-Temperature Service,” ASTM A194. The grade and surface finish of nuts for each type shall be as follows: A325 Bolt Type

Nut Specification, Grade and Finish

1 and 2, plain (uncoated) A563 C, C3, D, D3 and DH3 or Al94 2 and 2H; plain 1 and 2, galvanized A563 DH or A194 2H; galvanized and lubricated 3, plain A563 C3 and DH3; plain A490 Bolt Type 1 and 2, plain 3, plain

Nut Specification, Grade and Finish A563 DH and DH3 or A194 2H; plain A563 DH3; plain

Nut dimensions shall conform to the current American National Standards Institute’s standard, “Heavy Hex Nuts,” ANSI Standard B18.2.2., except as provided in paragraph (d) of this section. (d) Alternative Fastener Designs. Other fasteners or fastener assemblies which meet the materials, manufacturing and chemical composition requirements of ASTM A325 or ASTM A490, as applicable, and which meet the mechanical property requirements of the same specifications in full-size tests, and which have a body diameter and bearing areas under the head and nut not less than those provided by a bolt and nut of the same nominal dimensions prescribed by paragraphs 2(b) and 2(c), may be used subject to the approval of the Engineer of Record. Such alternative fasteners may differ in other AMERICAN INSTITUTE OF STEEL CONSTRUCTION

ASTM A325 OR A490 BOLTS

6 - 375

dimensions from those of the specified bolts and nuts. Their installation procedure and inspection may differ from procedures specified for regular high-strength bolts in Sections 8 and 9. When a different installation procedure or inspection is used, it shall be detailed in a supplemental specification applying to the alternative fastener, and that specification must be approved by the Engineer of Record. (e) Washers. Flat circular washers and square or rectangular beveled washers shall conform to the current requirements of the American Society for Testing and Materials, “Specification for Hardened Steel Washers,” ASTM F436. (f) Load Indicating Devices. Load indicating devices may be used in conjunction with bolts, nuts and washers specified in 2(a) through 2(e). Load indicating devices shall conform to the requirements of American Society for Testing and Materials’ “Specification for Compressible-Washer-Type Direct Tension Indicators for Use with Structural Fasteners,” ASTM F959. Subject to the approval of the Engineer of Record, direct tension indicating devices different from those meeting the requirements of ASTM F959 may be used provided they satisfy the requirements of 8(d)(4). If their installation procedure and inspection are not identical to that specified in 8(d)(4), they shall be detailed in supplemental specifications provided by the manufacturer and subject to the approval of the Engineer of Record. (g) Faying Surface Coatings. Paint, if used on faying surfaces of connections which are not specified to be slip critical, may be of any formulation. Paint, used on the faying surfaces of connections specified to be slip critical, shall be qualified by test in accordance with “Test Method to Determine the Slip Coefficient for Coatings Used in Bolted Joints” as published by the Research Council on Structural Connections. (See Appendix A.) Manufacturer’s certification shall include a certified copy of the test report. 3. Bolted Parts (a) Connected Material. All material within the grip of the bolt shall be steel. There shall be no compressible material such as gaskets or insulation within the grip. Bolted steel parts shall fit solidly together after the bolts are tightened, and may be coated or noncoated. The slope of the surfaces of parts in contact with the bolt head or nut shall not exceed 1:20 with respect to a plane normal to the bolt axis. (b) Surface Conditions. When assembled, all joint surfaces, including surfaces adjacent to the bolt head and nut, shall be free of scale, except tight mill scale, and shall be free of dirt or other foreign material. Burrs that would prevent solid seating of the connected parts in the snug tight condition shall be removed. Paint is permitted unconditionally on the faying surfaces in connections except in slip-critical connections as defined in Section 5(a). The faying surfaces of slip-critical connections shall meet the requirements of the following paragraphs, as applicable. (1) In noncoated joints, paint, including any inadvertent overspray, shall be excluded from areas closer than one bolt diameter but not less AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 376


(2)

(3)

(4) (5)

than one inch from the edge of any hole and all areas within the bolt pattern. Joints specified to have painted faying surfaces shall be blast cleaned and coated with a paint which has been qualified as Class A or B in accordance with the requirements of paragraph 2(g), except as provided in 3(b)3. Subject to the approval of the Engineer of Record, coatings providing a slip coefficient less than 0.33 may be used provided the mean slip coefficient is established by test in accordance with the requirements of paragraph 2(g), and the design slip resistance, φRs, calculated in accordance with the formula in Section 5(b) or 5(c). Coated joints shall not be assembled before the coatings have cured for the minimum time used in the qualifying test. Faying surfaces specified to be galvanized shall be hot-dip galvanized in accordance with American Society for Testing and Materials’ “Specification for Zinc (Hot-Galvanized) Coatings on Products Fabricated from Rolled, Pressed, and Forged Steel Shapes, Plates, Bars, and Strip,” ASTM A123 and shall subsequently be roughened by means of hand wire brushing. Power wire brushing is not permitted.

(c) Hole Types. Hole types recognized under this specification are standard holes, oversize holes, short slotted holes and long slotted holes. The nominal dimensions for each type hole shall be not greater than those shown in Table 1. Holes not more than 1⁄32 inch larger in diameter than the true decimal equivalent of the nominal diameter that may result from a drill or reamer of the nominal diameter are considered acceptable. The slightly conical hole that naturally results from punching operations is considered acceptable. The width of slotted holes which are produced by flame cutting or a combination of drilling or punching and flame cutting shall generally be not more than 1⁄32 inch greater than the nominal width except that gouges not more than 1⁄16 inch deep shall be permitted. For statically loaded connections, the flame cut surface need not be ground. For dynamically loaded connections, the flame cut surface shall be ground smooth. 4. Design of Bolted Connections Expressions for design strengths, φRn, of bolts subject to axial tension, shear and combined shear and tension are given in 4(a) and 4(b). They are to be compared to the effect of the factored loads. The design resistances of bolts subject to cyclic application of axial tension are given in 4(e). They are to be compared to effect of cyclically applied nominal (service) loads. (a) Tension and Shear Strength Limit States. The design strength in axial tension for A325 and A490 bolts which are tightened to the minimum fastener tension specified in Table 4 is φRn. The design strength in shear for A325 and A490 bolts, independent of the installed bolt pretension, is φRn where: Rn = Fn Ab AMERICAN INSTITUTE OF STEEL CONSTRUCTION

(LRFD 4.1)


6 - 377

Table 1. Nominal Hole Dimensions Bolt Dia. 1⁄

2

5⁄

8

3⁄

4

7⁄

8

1 ≥ 11⁄8

Hole DImensions Standard (Dia.)

Oversize (Dia.)

9⁄ 16 11⁄ 16 13⁄ 16 15⁄ 16 11⁄16

5⁄ 8 13⁄ 16 15⁄ 16 11⁄16 11⁄4

d + 1⁄16

d + 5⁄16

Short Slot (Width × Length)

(d

9⁄ × 11⁄ 16 16 11⁄ × 7⁄ 16 8 13⁄ × 1 16 15⁄ × 11⁄ 16 8 11⁄16 × 15⁄16 + 1⁄16) × (d + 3⁄8)

Long Slot (Width × Length) 9⁄

1 16 × 1 ⁄4 9 16 × 1 ⁄16 13⁄ × 17⁄ 16 8 15⁄ × 23⁄ 16 16 11⁄16 × 21⁄2 + 1⁄16) × (2.5 × 11⁄

(d

d)

In this expression: Rn = nominal strength of a bolt subject to axial tension or shear, kips Fn = nominal strength from Table 2 for appropriate kind of load, ksi Ab = area of bolt corresponding to nominal diameter, in.2 φ = resistance factor from Table 2. (b) Combined Tension and Shear Strength Limit State. In bearing connections in which the applied shear force is greater than 1⁄3 the design shear strength according to 4(a). the design strength in axial tension for A325 and A190 bolts is φRn where: Rn = Fnt Ab (LRFD 4.2) Where Rn = nominal tension strength of a bolt subject to concurrent shear. kips Fnt = nominal tension strength of a bolt as calculated by formulas in Table 3, ksi Ab = area of bolt corresponding to nominal diameter, in.2 φ = resistance factor equal to 0.75 In Table 3. fv, equals the shear force on the bolt in ksi. (c) Bearing Strength Limit State. The design bearing strength on the connected material for all bolts in a connection with two or more bolts in the line of force in standard, oversize, or short slotted holes when the edge distance in direction of force is not less than 11⁄2d and the distance center to center of bolts is not less than 3d is φRn, where: Rn = 2.4dtFu

(LRFD 4.3)

The design bearing strength on the connected material for all bolts in a connection with two or more bolts in the line of force in long slotted holes perpendicular to the direction of force when the edge distance, L, is not less than 11⁄2d and the distance center to center of bolts is not less than 3d is φRn where: Rn = 2.0dtFu

(LRFD 4.4)

The design bearing strength on the connected material for the bolt nearest to the free edge in the direction of force when two or more bolts are in the line of force in standard, oversize, or short slotted holes but with the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 378


Table 2. Nominal Strength of Fasteners Nominal Strength (ksi)

Load Condition a,b,c

Applied Static Tension Shear on bolt with threads in shear plane. Shear on bolt without threads in shear plane. a. b. c. d.

A325

A490

Resistance Factor, φ

90 d 48 d 60

113 d 60 d 75

0.75 0.75 0.75

Bolts must be tensioned to requirements of Table 4. See 4(e) for bolts subject to tensile fatigue. Except as required by 4(b). In shear connections transmitting axial force whose length between extreme fasteners measured parallel to the line of force exceeds 50 inches, tabulated values shall be reduced 20 percent.

Table 3. Nominal Tension Strength for Bolts in Bearing Connections (Nominal Tensile Strength, Fnt, ksi.) Fastener Grade

Threads Not Excluded from Shear Plane 2

2 0.5

Threads Excluded from Shear Plane 2

2 0.5

ASTM A325

(90 − 3.52fv )

(90 − 2.25fv )

ASTM A490

(113 − 3.54fv )

(113 − 2.27fv )

2

2 0.5

2

2 0.5

edge distance less than 11⁄2d and for a single bolt in the line of force is φRn where: Rn = LtFu ≤ 3.0dtFu

(LRFD 4.5)

When two or more bolts are in the line of force in standard, oversize, or short slotted holes and if deformation around the bolt holes is not a design consideration, the design strength in bearing for the individual bolts of a connection may be taken as φRn where: Rn = LtFu ≤ 3.0dtFu

(LRFD 4.6)

In the foregoing: Rn = nominal bearing strength of connected material, kips Fu = specified minimum tensile strength of the connected part, ksi L = distance in the direction of the force from the center of a standard hole or transverse slotted hole to the edge of the connected part or the distance center to center of standard holes or transverse slots, as applicable, in. d = nominal diameter of bolt, in. t = thickness of connected material, in. φ = resistance factor = 0.75 (d) Prying Action. The force in bolts required to support loads by means of direct tension shall be calculated considering the effects of the external load and any tension resulting from prying action produced by deformation of the connected parts. (e) Tensile Fatigue. When subject to tensile fatigue loading, the tensile stress in the bolt due to the nominal (service) load plus the prying force resulting from cyclic application of nominal load shall not exceed the following design AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 379

resistances in kips per square inch. The nominal diameter of the bolt shall be used in calculating the bolt stress. In no case shall the calculated prying force exceed 60 percent of the externally applied load. Number of Cycles Not more than 20,000 From 20,000 to 500,000 More than 500,000

A325 44 40 31

A490 54 49 38

Bolts subject to tensile fatigue load must be tensioned to requirements of Table 4. 5. Design Check for Slip Resistance (a) Slip-Critical Joints. Joints in which, in the judgment of the Engineer of Record, slip would be detrimental to the behavior of the joint, are defined as slip-critical. As discussed in the Commentary, these include but are not necessarily limited to joints subject to fatigue or significant load reversal, joints with bolts in oversize holes or slotted holes with the applied force approximately in the direction of the long dimension of the slots and joints in which welds and bolts share in transmitting shear loads at a common faying surface. Slip-critical joints shall be checked for slip resistance. At the option of the Engineer of Record, the required check may be based upon either nominal loads or factored loads. When serviceability at the nominal (service) load is the design criterion, the design slip resistance specified in Section 5(b) shall be compared with the effects of the nominal loads. When slip of the joint at the factored load level would affect the ability of the structure to support the factored load, the design slip resistance specified in Section 5(c) shall be compared to the effects of the factored loads. Slip-critical joints shall also be checked to ensure that the ultimate strength of the joint as a bearing joint is equal to or greater than the effect of the factored loads. Slip-critical joints must be designated on the contract plans and in the specifications. Bolts used in slip-critical joints shall be installed in accordance with the provisions of Section 8(d). (b) Slip-Critical Joints Designed at the Nominal Load Level. Slip-critical joints for which nominal loads are the design criterion shall, in addition to meeting the requirements of Section 4, be proportioned so that the force due to nominal (service) loads does not exceed the design slip resistance for use at nominal loads (service) loads, φRs, where: Rs = DµTm Nb Ns

(LRFD 5.1)

Where: Rs = nominal slip resistance of a bolt for use at nominal loads, kips Tm = minimum fastener tension given in Table 4, kips Nb = number of bolts in the joint Ns = number of slip planes D = slip probability factor* = 0.81 for µ equal to 0.33 = 0.86 for µ equal to 0.40 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 380


= 0.86 for µ equal to 0.50 µ = mean slip coefficient for Class A, B or C surfaces, † as applicable, or as established by tests = 0.33 for Class A surfaces (unpainted clean mill scale steel surfaces or surfaces with Class A coating on blast-cleaned steel) = 0.50 for Class B surfaces (unpainted blast-cleaned steel surfaces or surfaces with Class B coatings on blast-cleaned steel) = 0.40 for Class C surfaces (hot-dip galvanized and roughened surfaces) φ = 1.0 for standard holes = 0.85 for oversize and short slotted holes = 0.70 for long slotted holes transverse to the direction of load = 0.60 for long slotted holes parallel to the direction of load * D is a multiplier that reflects the distribution of actual slip coefficient values about the mean, the ratio of measured bolt tensile strength to the specified minimum values, and a slip probability level. Use of other values of D (see Commentary) must be approved by the Engineer of Record. † Coatings classified as Class A or Class B includes those coatings which provide a mean slip coefficient not less than 0.33 or 0.50, respectively, as determined by “Test Method to Determine the Slip Coefficient for Coatings Used in Bolted Connections.”

Table 4. Fastener Tension Required for Slip-Critical Connections and Connections Subject to Direct Tension Nominal Bolt Size, Inches

a

Minimum Tension in 1,000s of Pounds (kips) A325 Bolts

A490 Bolts

12 19 28 39

15 24 35 49

1 11⁄8 11⁄4 13⁄8

51 56 71 85

64 80 102 121

11⁄2

103

148

1⁄

5⁄ 3⁄ 7⁄

2 8 4 8

a. Equal to 70 percent of specified minimum tensile strengths of bolts (as specified in ASTM Specifications for tests of full size A325 and A490 bolts with UNC threads loaded in axial tension) rounded to the nearest kip.

When using nominal loads as the basis for design of slip-critical connections subject to applied tension, T, that reduces the net clamping force, the slip resistance (φRs) shall be multiplied by the following factor in which T is the applied tensile force at nominal loads [1 − T / (0.82TmNb)]

(LRFD 5.2)

(c) Slip-Critical Joints Designed at Factored Load Level. Slip-critical joints for which factored loads are the design criterion shall, in addition to meeting the requirements of Section 4, be proportioned so that the force due to the factored loads shall not exceed the design slip resistance for use at factored loads, φRstr, where: AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 381

Rstr = 1.13µTm Nb Ns

(LRFD 5.3)

Where terms in Formula (LRFD 5.3) are as defined in 5(b). When using factored loads as the basis for design of slip-critical connections subject to applied tension, T, that reduces the net clamping force, the slip resistance (φRs) shall be multiplied by the following factor in which T is the applied tensile force at nominal loads [1 − T / (1.13Tm Nb)]

(LRFD 5.4)

6. Loads in Combination When the reduced probabilities of maximum loads acting concurrently are accounted for by load combination factors, the resistances given in this Specification shall not be increased. 7. Design Details of Bolted Connections (a) Standard Holes. In the absence of approval by the Engineer of Record for use of other hole types, standard holes shall be used in high-strength bolted connections. (b) Oversize and Slotted Holes. When approved by the Engineer of Record, oversize holes, short slotted holes or long slotted holes may be used subject to the following joint detail requirements: (1) Oversize holes may be used in all plies of connections in which the design slip resistance of the connection is greater than the factored nominal load. (2) Short slotted holes may be used in any or all plies of connections in which the design strength (Section 4(a)) is greater than the factored nominal load provided the load is applied approximately normal (between 80 and 100 degrees) to the axis of the slot. Short slotted holes may be used without regard for the direction of applied load in any or all plies of connections in which the design slip resistance (Section 5(b)) is greater than the factored nominal load. (3) Long slotted holes may be used in one of the connected parts at any individual faying surface in connections in which the design strength (Section 4(a)) is greater than the factored nominal load provided the load is applied approximately normal (between 80 and 100 degrees) to the axis of the slot. Long slotted holes may be used in one of the connected parts at any individual faying surface without regard for the direction of applied load on connections in which the design slip resistance (Section 5(b)) is greater than the factored nominal load. (4) Fully inserted finger shims between the faying surfaces of load transmitting elements of connections are not to be considered a long slot element of a connection. (c) Washer Requirements. Design details shall provide for washers in highstrength bolted connections as follows: (1) Where the outer face of the bolted parts has a slope greater than 1:20 with respect to a plane normal to the bolt axis, a hardened beveled washer shall be used to compensate for the lack of parallelism. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 382


(2) Hardened washers are not required for connections using A325 and A490 bolts except as required in paragraphs 7(c)(3) through 7(c)(7) for slip-critical connections and connections subject to direct tension or as required by paragraph 8(c) for shear/bearing connections. (3) Hardened washers shall be used under the element turned in tightening when the tightening is to be performed by calibrated wrench method. (4) Irrespective of the tightening method, hardened washers shall be used under both the head and the nut when A490 bolts are to be installed and tightened to the tension specified in Table 4 in material having a specified yield point less than 40 ksi. (5) Where A325 bolts of any diameter or A490 bolts equal to or less than 1 inch in diameter are to be installed and tightened in an oversize or short slotted hole in an outer ply, a hardened washer conforming to ASTM F436 shall be used. (6) When A490 bolts over 1 inch in diameter are to be installed and tightened in an oversize or short slotted hole in an outer ply, hardened washers conforming to ASTM F436 except with 5⁄16 inch minimum thickness shall be used under both the head and the nut in lieu of standard thickness hardened washers. Multiple hardened washers with combined thickness equal to or greater than 5⁄16 inch do not satisfy this requirement. (7) Where A325 bolts of any diameter or A490 bolts equal to or less than 1 inch in diameter are to be installed and tightened in a long slotted hole in an outer ply, a plate washer or continuous bar of at least 5⁄16 inch thickness with standard holes shall be provided. These washers or bars shall have a size sufficient to completely cover the slot after installation and shall be of structural grade material, but need not be hardened except as follows. When A490 bolts over 1 inch in diameter are to be used in long slotted holes in external plies, a single hardened washer conforming to ASTM F436 but with 5⁄16 inch minimum thickness shall be used in lieu of washers or bars of structural grade material. Multiple hardened washers with combined thickness equal to or greater than 5⁄16 inch do not satisfy this requirement. (8) Alternative design fasteners meeting the requirements of 2(d) with a geometry which provides a bearing circle on the head or nut with a diameter equal to or greater than the diameter of hardened washers meeting the requirements ASTM F436 satisfy the requirements for washers specified in paragraphs 7(c)(4) and 7(c)(5). 8. Installation and Tightening (a) Handling and Storage of Fasteners. Fasteners shall be protected from dirt and moisture at the job site. Only as many fasteners as are anticipated to be installed and tightened during a work shift shall be taken from protected storage. Fasteners not used shall be returned to protected storage at the end of the shift. Fasteners shall not be cleaned of lubricant that is present in asdelivered condition. Fasteners which accumulate rust or dirt resulting from job site conditions shall be cleaned and relubricated prior to installation. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 383

(b) Tension Calibrator. A tension measuring device shall be at all job sites where bolts in slip-critical joints or connections subject to direct tension are being installed and tightened. The tension measuring device shall be used to confirm (1) the suitability of the complete fastener assembly and method of tightening, including lubrication, if required to satisfy the requirements of Table 4, (2) to calibrate the wrenches, if applicable, and (3) to confirm the understanding and proper use by the bolting crew of the method to be used. The frequency of confirmation testing, the number of tests to be performed, and the test procedure shall be as specified in 8(d), as applicable. The accuracy of the tension measuring device shall be confirmed through calibration by an approved testing agency at least annually. (c) Joint Assembly and Tightening of Shear/Bearing Connections. (1) Snug Tightened Bolts. Bolts in connections not within the slipcritical category as defined in Section 5(a) nor subject to tension loads nor required to be pretensioned bearing connections in accordance with 8(c)(2) shall be installed in properly aligned holes, but need only be tightened to the snug tight condition. The snug tight condition is defined as the tightness that exists when all plies in a joint are in firm contact. (See Commentary.) If a slotted hole occurs in an outer ply, a flat hardened washer or common plate washer shall be installed over the slot. (2) Tensioned Shear/Bearing Connections. The Engineer of Record may designate certain shear/bearing connections to be tightened to pretension in excess of snug tight. When so designated and identified on the contract drawings, the bolts in such connections shall be installed and tightened in accordance with one of the methods described in Subsections 8(d)(1) through 8(d)(4), but shall not be subject to the requirements for faying surface conditions of slipcritical connections contained in 3(b). The bolts need not be subject to inspection testing to determine the actual level of bolt pretension unless required by the Engineer of Record. (d) Joint Assembly and Tightening of Slip-Critical and Direct Tension Connections. In slip-critical connections and connections subject to direct tension, fasteners together with washers of size and quality specified, located as required by Section 7(c), shall be installed in properly aligned holes and tightened by any of the methods described in Subsections 8(d)(1) through 8(d)(4) to at least the minimum tension specified in Table 4 when all the fasteners are tight. Tightening may be done by turning the bolt while the nut is prevented from rotating when it is impractical to turn the nut. Impact wrenches, if used, shall be of adequate capacity and sufficiently supplied with air to perform the required tightening of each bolt in approximately 10 seconds. Slip-critical connections and connections subject to direct tension shall be clearly identified on the drawings. (1) Turn-of-Nut Tightening. When turn-of-nut tightening is used, hardened washers are not required except as may be specified in 7(c). A representative sample of not less than three bolt and nut assemblies of each diameter, length, grade and lot to be used in the work shall be checked at the start of work in a device capable AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 384


of indicating bolt tension. The test shall demonstrate that the method for estimating the snug tight condition and controlling the turns from snug tight to be used by the bolting crew develops a tension not less than 5 percent greater than the tension required by Table 4. Bolts shall be installed in all holes of the connection and brought to a “snug tight” condition. Snug tight is defined as the tightness that exists when the plies of the joint are in firm contact. Snug tightening shall progress systematically from the most rigid part of the connection to the free edges until all bolts are simultaneously snug tight and the connection is fully compacted. In some cases, proper tensioning of the bolts may require more than a single cycle of systematic tightening to produce a uniform snug tight condition. Following this initial operation, all bolts in the connection shall be tightened further by application of the rotation specified in Table 5. During the tightening operation, there shall be no rotation of the part not turned by the wrench. Tightening shall progress systematically from the most rigid part of the joint to its free edges. (2) Calibrated Wrench Tightening: Calibrated wrench tightening may be used only when installation procedures are calibrated on a daily basis and when a hardened washer is used under the element turned in tightening. (See the Commentary to this Section.) This specification does not recognize standard torques determined from tables or from formulas which are assumed to relate torque to tension. When calibrated wrenches are used for installation, they shall be set to provide a tension not less than 5 percent in excess of the minimum tension specified in Table 4. The installation procedures shall be calibrated at least once each working day by tightening representative sample fastener assemblies in a device capable of indicating actual bolt tension. The representative fastener assemblies shall consist of three bolts from each lot diameter, length and grade with nuts from each lot, diameter and grade and with a hardened washer from the washers being used in the work under the element turned in tightening. Wrenches shall be recalibrated when significant difference is noted in the surface condition of the bolts’ threads, nuts or washers. It shall be verified during actual installation in the assembled steelwork that the wrench adjustment selected by the calibration does not produce a nut or bolt head rotation from snug tight greater than that permitted in Table 5. If manual torque wrenches are used, nuts shall be turned in the tightening direction when torque is measured. When calibrated wrenches are used to install and tension bolts in a connection, bolts shall be installed with hardened washers under the element turned in tightening bolts in all holes of the connection and brought to a snug tight condition. Snug tightening shall progress systematically from the most rigid part of the connections to the free edges until bolts are uniformly snug tight and the plies of the joint are in firm contact. Following this initial tightening operation, the connection shall be tightened using the calibrated wrench. Tightening shall progress systematically from the most rigid part AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 385 a,b

Table 5. Nut Rotation from Snug Tight Condition

Disposition of Outer Face of Bolted Parts Bolt length (underside of head to end of bolt)

Both faces normal to bolt axis

One face normal to bolt axis and other sloped not more than 1:20 (beveled washer not used)

Both faces sloped not more than 1:20 from normal to the bolt axis (beveled washer not used)

Up to and including 4 diameters

1⁄ 3

turn

1⁄ 2

turn

2⁄

3

turn

Over 4 diameters but not exceeding 8 dia.

1⁄ 2

turn

2⁄ 3

turn

5⁄

6

turn

Over 8 diameters but not exceedc ing 12 dia.

2⁄ 3

turn

5⁄ 6

turn

1 turn

a. Nut rotation is relative to bolt regardless of the element (nut or bolt) being turned. For bolts installed by 1⁄2 turn and less, the tolerance should be plus or minus 30 degrees; for bolts installed by 2⁄3 turn and more, the tolerance should be plus or minus 45 degrees. b. Applicable only to connections in which all material within the grip of the bolt is steel. c. No research has been performed by the Council to establish the turn-of-nut procedure for bolt lengths exceeding 12 diameters. Therefore, the required rotation must be determined by actual test in a suitable tension measuring device which simulates conditions of solidly fitted steel.

of the joint to its free edges. During snugging and final tightening the element not turned in tightening shall be held to prevent rotation which will damage threads. In some cases, proper tensioning of the bolts may require more than a single cycle of systematic tightening to ensure all bolts are tightened to at least the prescribed amount. (3) Installation of Alternative Design Bolts. When fasteners which incorporate a design feature intended to indicate a predetermined tension or torque has been applied or to control bolt installation tension or torque, and which have been qualified under Section 2(d) are to be installed, a representative sample of not less than three bolts of each diameter, length and grade shall be checked at the job site in a device capable of indicating bolt tension. The test assembly shall include flat hardened washers, if required in the actual connection, arranged as in the actual connections to be tensioned. The calibration test shall demonstrate that each bolt develops a tension not less than 5 percent greater than the tension required by Table 4. Manufacturer’s installation procedure as required by Section 2(d) shall be followed for installation of bolts in the calibration device and in all connections. When alternative design fasteners are used in the work, bolts shall be installed in all holes of the connection and initially tightened sufficiently to bring all plies of the joint into firm contact with the bolts uniformly tight but without yielding or fracturing the control or indicator element of the fasteners. In some cases, proper AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 386


tensioning of the bolts may require more than a single cycle of systematic partial tightening. After all plies of the joint are in firm contact, all fasteners shall be further tightened, progressing systematically from the most rigid part of the connection to the free edges in a manner that will minimize relaxation of previously tightened fasteners. In some cases, proper tensioning of the bolts may require more than a single cycle of systematic partial tightening prior to final yielding or fracture of the control or indicator element of individual fasteners. (4) Direct Tension Indicator Tightening. When bolts are to be installed using direct tension indicator devices to indicate bolt tension, a representative sample of not less than three devices for each diameter and grade of fastener shall be tested with three typical bolts in a calibration device capable of indicating bolt tension. The test assembly shall include flat hardened washers, if required in the actual connection, arranged as those in the actual connections to be tensioned. The calibration test shall demonstrate that the device indicates a tension not less than 5 percent greater than that required by Table 4. When bolts are installed in the work using direct tension indicators meeting the requirements of ASTM F959, bolts shall be installed in all holes of the connection and tightened until all plies of the joint are in firm contact and fasteners are uniformly snug tight. Snug tight is indicated by partial compression of the direct tension indicator protrusions. All fasteners shall then be tightened, progressing systematically from the most rigid part of the connection to the free edges in a manner that will minimize relaxation of previously tightened fasteners. In some cases, proper tensioning of the bolts may require more than a single cycle of systematic partial tightening prior to final tightening to deform the protrusion to the specified gap. Special attention shall be given to proper installation of flat hardened washers when direct tension indicator devices are used with bolts installed in oversize or slotted holes and when the load indicating devices are used under the turned element. If direct tension indicators different from those meeting the requirements of ASTM F959 are used, manufacturer’s installation procedure as required by Section 2(f), shall be followed for installation of bolts in the calibration device and in all connections, and in addition the general requirements for use of direct tension indicators meeting the requirements of ASTM F959 shall be met. (e) Identification of Tightening Requirements. Bolts in slip-critical connections or bolts subject to axial tension which are to be installed and tightened in accordance by one of the methods in 8(d) and which require inspection to ensure that requirements of Table 4 are satisfied shall be clearly identified on the contract drawings. Shear/bearing connections which are to be installed by one of the methods in 8(d) but which need not be inspected to ensure bolt tensions specified in Table 4 are met shall be clearly identified on the contract drawings. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 387

(f) Reuse of Bolts. A490 bolts and galvanized A325 bolts shall not be reused. Other A325 bolts may be reused if approved by the Engineer of Record. Touching up or retightening previously snug tightened bolts which may have been loosened by the snugging of adjacent bolts shall not be considered to be a reuse. 9. Inspection (a) Inspector Responsibility. When inspection is required by the contract documents, the Inspector shall determine while the work is in progress that the requirements of Sections 2, 3 and 8, as appropriate, of this Specification are met in the work. All connections shall be inspected to ensure that the plies of the connected elements have been brought into firm contact. Bolts in connections not identified as being slip-critical nor subject to direct tension nor as tensioned bearing connections as provided in 8(c)(2) should not be inspected for bolt tension. For connections identified to be installed in accordance with 8(c)(2), the Inspector shall monitor installation and tightening of bolts to ensure that bolts are tightened in accordance with one of the methods of 8(d), but should not test the bolts for actual installed pretension. For all connections specified to be slip critical or subject to axial tension the Inspector shall observe the demonstration testing, and calibration procedures when such calibration is required, and shall monitor the installation of bolts to determine that all plies of the material have been drawn together and that the selected procedure has been used to tighten all bolts to ensure that the specified procedure was followed to achieve the pretension specified in Table 4. Bolts installed by procedures in Section 8(d) may reach tensions substantially greater than values given in Table 4, but this shall not be cause for rejection. (b) Arbitration Inspection. When high-strength bolts in slip-critical connections and connections subject to direct tension have been installed by any of the tightening methods in Section 8(d) and inspected in accordance with Section 9(a) and a disagreement exists as to the minimum tension of the installed bolts, the following arbitration procedure may be used. Other methods for arbitration inspection may be used if approved by the Engineer of Record. (1) The Inspector shall use a manual torque wrench which indicates torque by means of a dial or which may be adjusted to give an indication that the job inspecting torque has been reached. (2) This Specification does not recognize standard torques determined from tables or from formulas which are assumed to relate torque to tension. Testing using such standard torques shall not be considered valid. (3) A representative sample of five bolts from the diameter, length and grade of the bolts being inspected shall be tightened in the tension measuring device by any convenient means to an initial condition equal to approximately 15 percent of the required fastener tension and then to the minimum tension specified in Table 4. Material under the turned element in the tension measuring device shall be the same as in the actual installation, that is, structural steel or AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 388


hardened washer. Tightening beyond the initial condition must not produce greater nut rotation than 11⁄2 times that permitted in Table 5. The job inspecting torque shall be taken as the average of three values thus determined after rejecting the high and low values. The inspecting wrench shall then be applied to the tightened bolts in the work and the torque necessary to turn the nut or head 5 degrees (approximately 1 inch at 12 inch radius) in the tightening direction shall be determined. (4) Bolts represented by the sample in the foregoing paragraph which have been tightened in the structure shall be inspected by applying, in the tightening direction, the inspecting wrench and its job torque to 10 percent of the bolts, but not less than 2 bolts, selected at random in each connection in question. If no nut or bolt head is turned by application of the job inspecting torque, the connection shall be accepted as properly tightened. If any nut or bolt is turned by the application of the job inspecting torque, all bolts in the connection shall be tested, and all bolts whose nut or head is turned by the job inspecting torque shall be tightened and reinspected. Alternatively, the fabricator or erector, at his option, may retighten all of the bolts in the connection and then resubmit the connection for the specified inspection. (c) Delayed Verification Inspection. The procedures specified in Sections 9(a) and (b) are intended for inspection of bolted connections and verification of pretension at the time of tensioning the joint. If verification of bolt tension is required after a passage of a period of time and exposure of the completed joints, the procedure of Section 9(b) will provide indication of bolt tension which is of questionable accuracy. Procedures appropriate to the specific situation should be used for verification of bolt tension. This might involve use of the arbitration inspection procedure contained herein, or might require the development and use of alternate procedures. (See Commentary.)



6 - 389

APPENDIX A

Testing Method to Determine the Slip Coefficient for Coatings Used in Bolted Joints Reprinted from Engineering Journal American Institute of Steel Construction, Third Quarter, 1985.

JOSEPH A. YURA and KARL H. FRANK

In 1975, the Steel Structures Painting Council (SSPC) contacted the Research Council on Riveted and Bolted Structural Joints (RCRBSJ), now the Research Council on Structural Connections (RCSC), regarding the difficulties and costs which steel fabricators encounter with restrictions on coatings of contact surfaces for friction-type structural joints. The SSPC also expressed the need for a “standardized test which can be conducted by any certified testing agency at the initiative and expense of any interested party, including the paint manufacturer.” And finally, the RCSC was requested to “prepare and promulgate a specification for the conduct of such a standard test for slip coefficients.” The following Testing Method is the answer of Research Council on Structural Connections to the SSPC request. The test method was developed by Professors Joseph A. Yura and Karl H. Frank of the University of Texas at Austin under a grant from the Federal Highway Administration. The Testing Method was approved by the RCSC on June 14, 1984. 1.0 GENERAL PROVISIONS 1.1 Purpose and Scope The purpose of the testing procedure is to determine the slip coefficient of a coating for use in high-strength bolted connections. The testing specification ensures that the creep deformation of the coating due to both the clamping force of the bolt and the service load joint shear are such that the coating will provide satisfactory performance under sustained loading. Joseph A. Yura, M. ASCE, is Warren S. Bellows Centennial Professor in Civil Engineering, University of Texas at Austin, Austin, Texas. Karl H. Frank, A.M. ASCE, is Associate Professor, Department of Civil Engineering, University of Texas at Austin, Austin, Texas.


6 - 390

RCSC SPECIFICATION FOR STRUCTURAL JOINTS

1.2 Definition of Essential Variables Essential variables mean those variables which, if changed, will require retesting of the coating to determine its slip coefficient. The essential variables are given below. The relationship of these variables to the limitation of application of the coating for structural joints is also given. The time interval between application of the coating and the time of testing is an essential variable. The time interval must be recorded in hours and any special curing procedures detailed. Curing according to published manufacturer’s recommendations would not be considered a special curing procedure. The coatings are qualified for use in structural connections which are assembled after coating for a time equal to or greater than the interval used in the test specimens. Special curing conditions used in the test specimens will also apply to the use of the coating in the structural connections. The coating thickness is an essential variable. The maximum average coating thickness allowed on the bolted structure will be the average thickness, rounded to the nearest whole mil, of the coating used on the creep test specimens minus 2 mils. The composition of the coating, including the thinners used, and its method of manufacture are essential variables. Any change will require retesting of the coating. 1.3 Retesting A coating which fails to meet the creep or the post-creep slip test requirements given in Sect. 4 may be retested in accordance with methods in Sect. 4 at a lower slip coefficient, without repeating the static short-term tests specified in Sect. 3. Essential variables must remain unchanged in the retest. 2.0 TEST PLATES AND COATING OF THE SPECIMENS 2.1 Test Plates The test specimen plates for the short-term static tests are shown in Fig. 1. The plates are 4×4 in. plates, 5⁄8-in. thick, with a 1-in. dia. hole drilled 11⁄2 in. ± 1⁄16 in. from one edge. The specimen plates for the creep specimen are shown in Fig. 2. The plates are 4×7 in., 5⁄8-in. thick, with two 1-in. holes, 11⁄2 in. ± 1⁄16 in. from each end. The edges of the plates may be milled, as rolled or saw cut. Flame cut edges 4″

Load

1″ Clamping force

1 1/2 ″

cL

5″ 11/2 ″

11/2 ″

4″ 1-in. Dia.

1″ All plates 5/8 ″thick

Fig. 1. Compression test specimen



6 - 391

are not permitted. The plates should be flat enough to ensure they will be in reasonably full contact over the faying surface. Any burrs, lips or rough edges should be filed or milled flat. The arrangement of the specimen plates for the testing is shown in Figs. 2 and 3. The plates are to be fabricated from a steel with a minimum yield strength between 36 to 50 ksi. If specimens with more than one bolt are desired, the contact surface per bolt should be 4×3 in. as shown for the single bolt specimen in Fig. 1. 2.2 Specimen Coating The coatings are to be applied to the specimens in a manner consistent with the actual intended structural application. The method of applying the coating and the surface preparation should be given in the test report. The specimens are to be coated to an average thickness 2 mils (0.05 mm) greater than average thickness to be used in the structure. The thickness of the total coating and the primer, if used, shall be measured on the contact surface of the specimens. The thickness should be measured in accordance with the Steel Structures Painting Council specification SSPCPA2, Measurement of Dry Paint Thickness with Magnetic Gages.1 Two spot readings (six gage readings) should be made for each contact surface. The overall average thickness from the three plates comprising a specimen is the average thickness for the specimen. This value should be reported for each specimen. The average coating thickness of the three creep specimens will be calculated and reported. The average thickness of the creep specimen minus two mils rounded to the nearest whole mil is the maximum average thickness of the coating to be used in the faying surface of a structure. The time between painting and specimen assembly is to be the same for all specimens within ±4 hours. The average time is to be calculated and reported. The two coating applications required in Sect. 3 are to use the same equipment and procedures. 3.0 SLIP TESTS The methods and procedures described herein are used to determine experimentally the slip coefficient (sometimes called the coefficient of friction) under shortterm static loading for high-strength bolted connections. The slip coefficient will be determined by testing two sets of five specimens. The two sets are to be coated at different times at least one week apart. 3.1 Compression Test Setup The test setup shown in Fig. 3 has two major loading components, one to apply a clamping force to the specimen plates and another to apply a compressive load to the specimen so that the load is transferred across the faying surfaces by friction. Clamping Force System. The clamping force system consists of a 7⁄8-in. dia. threaded rod which passes through the specimen and a centerhole compression ram. A 2H nut is used at both ends of the rod, and a hardened washer is used at each side of the test specimen. Between the ram and the specimen is a specially fabricated 7⁄8-in. 2H nut in which the threads have been drilled out so that it will slide with little resistance along the rod. When oil is pumped into the centerhole ram, 1. Steel Structures Painting Council, Steel Structures Painting Manual, Vols. 1 and 2, Pittsburgh, Pa., 1982.


6 - 392


2″ Load indicating washer Clamping bolt

4″

1″

7″

Pin bolt

1″ 11/2 ″

Specimen

7″

17/ 8 ″

4″

Fig. 2. Creep test specimens

Drilled nut

Load

Testing machine

Nut 7/ 8

Spherical head 50 kip Center hole ram

Dia. rod

Nut 7/ 8 -2M

Hardened washer Specimen

Plate Piston

Base

Fig. 3. Test setup AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 393

the piston rod extends, thus forcing the special nut against one of the outside plates of the specimen. This action puts tension in the threaded rod and applies a clamping force to the specimen which simulates the effect of a tightened bolt. If the diameter of the centerhole ram is greater than 1 in., additional plate washers will be necessary at the ends of the ram. The clamping force system must have a capability to apply a load of at least 49 kips and maintain this load during the test with an accuracy of ±1%. Compressive Load System. A compressive load is applied to the specimen until slip occurs. This compressive load can be applied by a compression test machine or compression ram. The machine, ram and the necessary supporting elements should be able to support a force of 90 kips. The compression loading system should have an accuracy of 1.0% of the slip load. 3.2 Instrumentation Clamping Force. The clamping force must be measured within 0.5 kips. This may be accomplished by measuring the pressure in the calibrated ram or placing a load cell in series with the ram. Compression Load. The compression load must be measured during the test. This may be accomplished by direct reading from a compression testing machine, a load cell in series with the specimen and the compression loading device, or pressure readings on a calibrated compression ram. Slip Deformation. The relative displacement of the center plate and the two outside plates must be measured. This displacement, called slip for simplicity, should be the average which occurs at the centerline of the specimen. This can be accomplished by using the average of two gages placed on the two exposed edges of the specimen or by monitoring the movement of the loading head relative to the base. If the latter method is used, due regard must be taken for any slack that may be present in the loading system prior to application of the load. Deflections can be measured by dial gages or any other calibrated device which has an accuracy of 0.001 in. 3.3 Test Procedure The specimen is installed in the test setup as shown in Fig. 3. Before the hydraulic clamping force is applied, the individual plates should be positioned so that they are in, or are close to, bearing contact with the 7⁄8-in. threaded rod in a direction opposite to the planned compressive loading to ensure obvious slip deformation. Care should be taken in positioning the two outside plates so that the specimen will be straight and both plates are in contact with the base. After the plates are positioned, the centerhold ram is engaged to produce a clamping force of 49 kips. The applied clamping force should be maintained within ±0.5 kips during the test until slip occurs. The spherical head of the compression loading machine should be brought in contact with the center plate of the specimen after the clamping force is applied. The spherical head or other appropriate device ensures uniform contact along the edge of the plate, thus eliminating eccentric loading. When 1 kip or less of compressive load is applied, the slip gages should be engaged or attached. The purpose of engaging the deflection gage(s), after a slight load is applied, is to eliminate initial specimen settling deformation from the slip readings. When the slip gages are in place, the compression load is applied at a rate not AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 394


exceeding 25 kips (109 kN) per minute, or 0.003 in. of slip displacement per minute until the slip load is reached. The test should be terminated when a slip of 0.05 in. or greater is recorded. The load-slip relationship should preferably be monitored continuously on an X-Y plotter throughout the test, but in lieu of continuous data, sufficient load-slip data must be recorded to evaluate the slip load defined below. 3.4 Slip Load Typical load-slip response is shown in Fig. 4. Three types of curves are usually observed and the slip load associated with each type is defined as follows: Curve (a). Slip load is the maximum load, provided this maximum occurs before a slip of 0.02 in. is recorded. Curve (b). Slip load is the load at which the slip rate increases suddenly. Curve (c). Slip load is the load corresponding to a deformation of 0.02 in. This definition applies when the load vs. slip curves show a gradual change in response.

-slip load

a

LOAD

b c

Load Slip

0

0.020

0.040 SLIP (in.)

Fig. 4. Definition of slip load

3.5 Coefficient of Slip The slip coefficient ks for an individual specimen is calculated as follows: ks =

slip load 2 × clamping force

The mean slip coefficient for both sets of five specimens must be compared. If the two means differ by more than 25%, using the smaller mean as the base, a third five-specimen set must be tested. The mean and standard deviation of the data from all specimens tested define the slip coefficient of the coating. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 395

3.6 Alternate Test Methods Other test methods to determine slip may be used provided the accuracy of load measurement and clamping satisfies the conditions presented in the previous sections. For example, the slip load may be determined from a tension-type test setup rather than the compression-type as long as the contact surface area per fastener of the test specimen is the same as shown in Fig. 1. The clamping force of at least 49 kips may be applied by any means provided the force can be established within ±1%. Strain-gaged bolts can usually provide the desired accuracy. However, bolts installed by turn-of-nut method, tension indicating fasteners and load indicator washers usually show too much variation to be used in the slip test. 4.0 TENSION CREEP TESTS The test method outlined is intended to ensure the coating will not undergo significant creep deformation under service loading. The test also determines the loss in clamping force in the fastener due to the compression or creep of the paint. Three replicate specimens are to be tested. 4.1 Test Setup Tension-type specimens, as shown in Fig. 2, are to be used. The replicate specimens are to be linked together in a single chain-like arrangement, using loose pin bolts, so the same load is applied to all specimens. The specimens shall be assembled so the specimen plates are bearing against the bolt in a direction opposite to the applied tension loading. Care should be taken in the assembly of the specimens to ensure the centerline of the holes used to accept the pin bolts is in line with the bolts used to assemble the joint. The load level, specified in Sect. 4.2, shall be maintained constant within ±1% by springs, load maintainers, servo controllers, dead weights or other suitable equipment. The bolts used to clamp the specimens together shall be 7⁄8-in. dia. A490 bolts. All bolts should come from the same lot. The clamping force in the bolts should be a minimum of 49 kips. The clamping force is to be determined by calibrating the bolt force with bolt elongation, if standard bolts are used. Special fasteners which control the clamping force by other means such as bolt torque or strain gages may be used. A minimum of three bolt calibrations must be performed using the technique selected for bolt force determination. The average of the three-bolt calibration is to be calculated and reported. The method of measuring bolt force must ensure the clamping force is within ±2 kips (9 kN) of the average value. The relative slip between the outside plates and the center plates shall be measured to an accuracy of 0.001 in. (0.02 mm). This is to be measured on both sides of each specimen. 4.2 Test Procedure The load to be placed on the creep specimens is the service load permitted for 7⁄8-in. A490 bolts in slip-critical connections by the latest edition of the Specification for Structural Joints Using ASTM A325 or A490 Bolts2 for the particular slip coefficient category under consideration. The load is to be placed on the specimen and held 2. Research Council of Structural Connections, Specification for Structural Joints Using ASTM A325 or A490 Bolts, American Institute of Steel Construction, Inc., Chicago, November 1985.


6 - 396


for 1,000 hours. The creep deformation of a specimen is calculated using the average reading of the two displacements on each side of the specimen. The difference between the average after 1,000 hours and the initial average reading taken within one-half hour after loading the specimens is defined as the creep deformation of the specimen. This value is to be reported for each specimen. If the creep deformation of any specimen exceeds 0.005 in. (0.12 mm), the coating has failed the test for the slip coefficient used. The coating may be retested using new specimens in accordance with this section at a load corresponding to a lower value of slip coefficient. If the value of creep deformation is less than 0.005 in. (0.12 mm) for all specimens, the specimens are to be loaded in tension to a load calculated as Pu = average clamping force × design slip coefficient × 2 since there are two slip planes. The average slip deformation which occurs at this load must be less than 0.015 in. (0.38 mm) for the three specimens. If the deformation is greater than this value, the coating is considered to have failed to meet the requirements for the particular slip coefficient used. The value of deformation for each specimen is to be reported. COMMENTARY The slip coefficient under short-term static loading has been found to be independent of clamping force, paint thickness and hole diameter.3 The slip coefficient can be easily determined using the hydraulic bolt test setup included in this specification. The slip load measured in this setup yields the slip coefficient directly since the clamping force is controlled. The slip coefficient k, is given by ks =

slip load 2 × clamping force

The resulting slip coefficient has been found to correlate with both tension and compression tests of bolted specimens. However, tests of bolted specimens revealed that the clamping force may not be constant but decreases with time due to the compressive creep of the coating on the faying surfaces and under the nut and bolt head. The reduction of the clamping force can be considerable for joints with high clamping force and thick coatings, as much as a 20% loss. This reduction in clamping force causes a corresponding reduction in the slip load. The resulting reduction in slip load must be considered in the procedure used to determine the design allowable slip loads for the coating. The loss in clamping force is a characteristic of the coating. Consequently, it cannot be accounted for by an increase in the factor of safety or a reduction in the clamping force used for design without unduly penalizing coatings which do not exhibit this behavior. The creep deformation of the bolted joint under the applied shear loading is also an important characteristic and a function of the coating applied. Thicker coatings tend to creep more than thinner coatings. Rate of creep deformation increases as the applied load approaches the slip load. Extensive testing has shown the rate of creep is not constant with time, rather it decreases with time. After 1,000 hours of loading, the additional creep deformation is negligible. 3. Frank, K. H., and J. A. Yura, An Experimental Study of Bolted Shear Connections, FHWA/RD-81-148, Federal Highway Administration, Washington, D.C., December 1981.



6 - 397

The proposed test methods are designed to provide the necessary information to evaluate the suitability of a coating for slip critical bolted connections and to determine the slip coefficient to be used in the design of the connections. The initial testing of the compression specimens provides a measure of the scatter of the slip coefficient. In order to get better statistical information, a third set of specimens must be tested whenever the means of the initial two sets differ by more than 25%. The creep tests are designed to measure the paint’s creep behavior under the service loads determined by the paint’s slip coefficient based on the compression test results. The slip test conducted at the conclusion of the creep test is to ensure the loss of clamping force in the bolt does not reduce the slip load below that associated with the design slip coefficient. A490 bolts are specified, since the loss of clamping force is larger for these bolts than A325 bolts. Qualifying of the paint for use in a structure at an average thickness of 2 mils less than the test specimen is to ensure that a casual buildup of paint due to overspray, etc., does not jeopardize the coating’s performance. The use of 1-in. (25 mm) holes in the specimens is to ensure that adequate clearance is available for slip. Fabrication tolerances, coating buildup on the holes and assembly tolerances reduce the apparent clearances.


6 - 398


6 - 399

Commentary on Specifications for Structural Joints Using ASTM A325 or A490 Bolts June 8, 1988.

Historical Notes When first approved by the Research Council on Structural Connections of the Engineering Foundation, January 1951, the “Specification for Assembly of Structural Joints Using High-Strength Bolts” merely permitted the substitution of a like number of A325 high-strength bolts for hot driven ASTM A141 (presently identified as A502, Grade 1) steel rivets of the same nominal diameter. It was required that all contact surfaces be free of paint. As revised in 1954, the omission of paint was required to apply only to “joints subject to stress reversal, impact or vibration, or to cases where stress redistribution due to joint slippage would be undesirable.” This relaxation of the earlier provision recognized the fact that, in a great many cases, movement of the connected parts that brings the bolts into bearing against the sides of their holes is in no way detrimental. In the first edition of the Specification published in 1951, a table of torque to tension relationships for bolts of various diameters was included. It was soon demonstrated in research that a variation in the torque to tension relationship of as high as plus or minus 40 percent must be anticipated unless the relationship is established individually for each bolt lot, diameter and fastener condition. Hence, by the 1954 edition of the Specification, recognition of standard torque to tension relationships in the form of tabulated values or formulas was withdrawn. Recognition of the calibrated wrench method of tightening was retained, however, until 1980, but with the requirement that the torque required for installation or inspection be determined specifically for the bolts being installed on a daily basis. Recognition of the method was withdrawn in 1980 because of continuing controversy resulting from failure of users to adhere to the detailed requirements for valid use of the method both during installation and inspection. With the 1985 version of the Specification, the calibrated wrench method was reinstated, but with more detailed requirements which should be carefully followed. The increasing use of high-strength steels created the need for bolts substantially stronger than A325 in order to resist the much greater forces they support without resort to very large connections. To meet this need, a new ASTM specification, A490, was developed. When provisions for the use of these bolts were included in this Specification in 1964, it was required that they be tightened to their specified proof load, as was required for the installation of A325 bolts. However, the ratio of proof load to specified minimum tensile strength is approximately 0.7 for A325 bolts, whereas it is 0.8 for A490 bolts. Calibration studies have shown that highstrength bolts have ultimate load capacities in torqued tension which vary from about


6 - 400

COMMENTARY ON THE RCSC SPECIFICATION (6/8/88)

80 to 90 percent of the pure-tension tensile strength.1 Hence, if minimum strength A490 bolts were supplied and they experienced the maximum reduction due to torque required to induce the tension, there is a possibility that these bolts could not be tightened to proof load by any method of installation. Also, statistical studies have shown that tightening to the 0.8 times tensile strength under calibrated wrench control may result in some “twist-off” bolt failures during installation or in some cases a slight amount of under-tightening.2 Therefore, the required installed tension for A490 bolts was reduced to 70 percent of the specified minimum tensile strength. For consistency, but with only minor change, the initial tension required for A325 bolts was also set at 70 percent of their specified minimum tensile strength and, at the same time, the values for minimum required pretension were rounded off to the nearest kip. C1 Scope This Specification deals only with two types of high-strength bolts, namely, ASTM A325 and A490, and to their installation in structural steel joints. The provisions may not be relied upon for high-strength fasteners of other chemical composition or mechanical properties or size. The provisions do not apply to ASTM A325 or A490 fasteners when material other than steel is included in the grip. The provisions do not apply to high-strength anchor bolts. The Specification relates only to the performance of fasteners in structural steel connections and those few aspects of the connected material that affect the performance of the fasteners in connections. Many other aspects of connection design and fabrication are of equal importance and must not be overlooked. For information on questions of design of connected material, not covered herein, the user is directed to standard textbooks on design of structural steel and also to Kulak, G. L., J. W. Fisher, and J. H. A. Struik, Guide to Design Criteria for Bolted and Riveted Joints, 2nd ed., New York: John Wiley & Sons, 1987. (Hereinafter referred to as the Guide.) C2 Bolts, Nuts, Washers and Paint Complete familiarity with the referenced ASTM Specification requirements is necessary for the proper application of this Specification. Discussion of referenced specifications in this Commentary is limited to only a few frequently overlooked or littleunderstood items. In this Specification, a single style of fastener (heavy hex structural bolts with heavy hex nuts) available in two strength grades (A325 and A490) is specified as a principal style, but conditions for acceptance of other types of fasteners are provided. Bolt Specifications. ASTM A325 and A490 bolts are manufactured to dimensions as specified in ANSI Standard B18.2.1 for Heavy Hex Structural Bolts. The basic dimensions, as defined in Fig. C1, are shown in Table C1. The principal geometric features of heavy hex structural bolts that distinguish them from bolts for general application are the size of the head and the body length. The head of the heavy hex 1. Christopher, R. J., G. L. Kulak, and J. W. Fisher, “Calibration of Alloy Steel Bolts,” ASCE Journal of the Structural Division, Vol. 92, No. ST2, Proc. Paper 4768, April, 1966, pp. 19–40. 2. Gill, P. J., “Specifications of Minimum Preloads for Structural Bolts,” Memorandum 30, G.K.N. Group Research Laboratory, England, 1966 (Unpublished Report).



6 - 401

Table C1 Nominal Bolt Size, Inches D

Bolt Dimensions, Inches Heavy Hex Structural Bolts

Nut Dimensions, Inches Heavy Hex Nuts

Width across flats, F

Height H

Thread length

Width across flats, W

Height H

7⁄ 8 11⁄16 11⁄4 17⁄16 15⁄8 113⁄16 2 23⁄16 23⁄8

5⁄ 16 25⁄ 64 15⁄ 32 35⁄ 64 39⁄ 64 11⁄ 16 25⁄ 32 27⁄ 32 15⁄ 16

1 11⁄4 13⁄8 11⁄2 13⁄4 2 2 21⁄4 21⁄4

7⁄ 8 11⁄16 11⁄4 17⁄16 15⁄8 113⁄16 2 23⁄16 23⁄8

31⁄ 64 39⁄ 64 47⁄ 64 55⁄ 64 64⁄ 64 17⁄64 17⁄32 111⁄32 115⁄32

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

1 11⁄8 11⁄4 13⁄8 11⁄2

Thread Length A325

F

H

Bolt Length

H

W

Nut may be chamfered on both faces Fig. C1. Heavy hex structural bolt and heavy hex nut

structural bolt is specified to be the same size as a heavy hex nut of the same nominal diameter in order that the ironworker may use a single size wrench or socket on both the bolt head and the nut. Heavy hex structural bolts have shorter thread length than bolts for general application. By making the body length of the bolt the control dimension, it has been possible to exclude the thread from all shear planes, except in the case of thin outside parts adjacent to the nut. Depending upon the amount of bolt length added to adjust for incremental stock lengths, the full thread may extend into the grip by as much as 3⁄8 inch for 1⁄2, 5⁄8, 3⁄4, 7⁄8, 11⁄4, and 11⁄2 in. diameter bolts and as much as 1⁄2 inch for 1, 11⁄8 and 13⁄8 in. diameter bolts. Inclusion of some thread run-out in the plane of shear is permissible. Of equal or even greater importance is exercise of care to provide sufficient thread for nut tightening to keep the nut threads from jamming into the thread run-out. When the thickness of an outside part is less than the amount the threads may extend into the grip tabulated above, it may be necessary to call for the next increment of bolt length together with sufficient flat washers to ensure full tightening of the nut without jamming nut threads on the thread run-out. There is an exception to the short thread length requirements for ASTM A325 bolts discussed in the foregoing. Beginning with ASTM A325-83, supplementary requirements have been added to the ASTM A325 Specification which permit the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 402


purchaser, when the bolt length is equal to or shorter than four times the nominal diameter, to specify that the bolt be threaded for the full length of the shank. This exception to the requirements for thread length of heavy hex structural bolts was provided in the Specification in order to increase economy through simplified ordering and inventory control in the fabrication and erection of structures using relatively thin materials where strength of the connection is not dependent upon shear strength of the bolt, whether threads are in the shear plane or not. The Specification requires that bolts ordered to such supplementary requirements be marked with the symbol A325T. In order to determine the required bolt length, the value shown in Table C2 should be added to the grip (i.e., the total thickness of all connected material, exclusive of washers). For each hardened flat washer that is used, add 5⁄32 inch, and for each beveled washer add 5⁄16 inch. The tabulated values provide appropriate allowances for manufacturing tolerances, and also provide for full thread engagement (defined as having the end of the bolt at least flush with the face of the nut) with an installed heavy hex nut. The length determined by the use of Table C2 should be adjusted to the next longer 1⁄4 inch length. ASTM A325 and ASTM A490 currently provide for three types (according to metallurgical classification) of high-strength structural bolts, supplied in sizes 1⁄2 inch to 11⁄2 inch inclusive except for A490 Type 2 bolts which are available in diameters from 1⁄2 inch to 1 inch inclusive: Type 1. Type 2. Type 3.

Medium carbon steel for A325 bolts, alloy steel for A490 bolts. Low carbon martensitic steel for both A325 and A490 bolts. Bolts having improved atmospheric corrosion resistance and weathering characteristics for both A325 and A490 bolts.

When the bolt type is not specified, either Type 1, Type 2 or Type 3 may be supplied at the option of the manufacturer. Special attention is called to the requirement in ASTM A325 that, where elevated temperature applications are involved, Type 1 bolts shall be specified by the purchaser. This is because the chemistry of Type 2 bolts permits heat treatment at sufficiently low temperatures that subsequent heating to elevated temperatures may affect the mechanical properties. Heavy Hex Nuts. Heavy hex nuts for use with A325 bolts may be manufactured to the requirements of ASTM A194 for grades 2 or 2H or the requirements of ASTM A563 for grades DH, except that nuts to be galvanized for use with galvanized bolts must be hardened nuts meeting the requirements for ASTM A563 grade DH. The heavy hex nuts for use with A490 bolts may be manufactured to the requirements of ASTM A194 for grade 2H or the requirements of ASTM A563 for grade DH. Galvanized High-Strength Bolts. Galvanized high-strength bolts and nuts must be considered as a manufactured matched assembly; hence, comments relative to them have not been included in the foregoing paragraphs where bolts and nuts have been considered separately. Insofar as the hot-dip galvanized bolt and nut assembly, per se, is concerned, four principal factors need be discussed in order that the provisions of the Specification may be understood and properly applied. These are (1) the effect of the hot-dip galvanizing process on the mechanical properties of highstrength steels, (2) the effect of hot-dip galvanized coatings on the nut stripping strength, (3) the effect of galvanizing upon the torque involved in the tightening operation, and (4) shipping requirements. The ASTM Specifications for galvanized A325 high-strength bolts recognize AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 403

Table C2 Nominal Bolt Size, Inches 1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

To Determine Required Bolt Length Add to Grip, in Inches 11⁄

16 7⁄ 8

1 11⁄8

1 11⁄8 11⁄4 13⁄8

11⁄4 11⁄2 15⁄8 13⁄4

11⁄2

17⁄8

both the hot-dip galvanizing process and the mechanical galvanizing process. The effects of the two processes upon the performance characteristics and requirements for proper installation are distinctly different: therefore, distinction between the two must be noted in the comments which follow. ASTM A325 Specifications require that all components of a fastener assembly (nuts, bolts and washers) shall have been coated by the same process and that the supplier’s option is limited to one process per item with no mixed processes in a lot. Mixing a bolt galvanized by one process with a nut galvanized by the other may result in a unworkable assembly. Effect of Hot-Dip Galvanizing on the Strength of Steels. Steels in the 200 ksi and higher tensile strength range are subject to embrittlement if hydrogen is permitted to remain in the steel and the steel is subjected to high tensile stress. The minimum tensile strength of A325 bolts is 105 or 120 ksi, depending upon the size, comfortably below the critical range. The required maximum tensile strength for A490 bolts was set at 170 ksi in order to provide a little more than a 10 percent margin below 200 ksi; however, because manufacturers must target their production slightly higher than the required minimum, A490 bolts close to the critical range of tensile strength must be anticipated. For black bolts this is not a cause for concern, but, if the bolt is hot-dip galvanized, a hazard of delayed brittle fracture in service exists because of the real possibility of introduction of hydrogen into the steel during the pickling operation of the hot-dip galvanizing process and the subsequent “sealingin” of the hydrogen by the zinc coating. There also exists the possibility of cathodic hydrogen adsorption arising from corrosion process in service in aggressive environments. ASTM Specifications provide for the galvanizing of A325 bolts but not A490 bolts. Galvanizing of A490 bolts is not permitted. Because pickling and emersion in molten zinc is not involved, galvanizing by the mechanical process essentially avoids potential for hydrogen embrittlement. The heat treatment temperatures for Type 2 ASTM A325 bolts are in the range of the molten zinc temperatures for hot-dip galvanizing; therefore there is a potential for diminishing the heat treated mechanical properties of Type 2 A325 bolts by the hot-dip galvanizing process. For this reason, the current Specifications require that only mechanical galvanizing shall be used on Type 2 ASTM A325 bolts. Nut Stripping Strength. Hot-dip galvanizing affects the stripping strength of the nut-bolt assembly primarily because, to accommodate the relatively thick zinc coatAMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 404


ings of non-uniform thickness on bolt threads, it is usual practice to hot-dip galvanize the blank nut and then to tap the nut oversize after galvanizing. This overtapping results in a reduction in the amount of engagement between the steel portions of the male and female threads with a consequent approximately 25 percent reduction in the stripping strength. Only the stronger hardened nuts have adequate strength to meet specification requirements even with the reduction due to overtapping; therefore, ASTM A325 specifies that only Grades DH and 2H be used for galvanized nuts. This requirement should not be overlooked if non-galvanized nuts are purchased and then sent to a local galvanizer for hot-dip galvanizing. Because the mechanical galvanizing process results in a more uniformly distributed and smooth zinc coating, nuts may be tapped oversize before galvanizing by an amount less than required for the hot-dip process before galvanizing. This results in a better bolt-nut fit with zinc coating on the internal threads of the nut. Effect of Galvanizing Upon Torque Involved in Tightening. Research3 has shown that, in the as-galvanized condition, galvanizing both increases the friction between the bolt and nut threads and also makes the torque induced tension much more variable. Lower torque and more consistent results are provided if the nuts are lubricated; thus, ASTM A325 requires that a galvanized bolt and lubricated galvanized nut shall be assembled in a steel joint with a galvanized washer and tested in accordance with ASTM A563 by the manufacturer prior to shipment to ensure that the galvanized nut with the lubricant provided may be rotated from the snug tight condition well in excess of the rotation required for full tensioning of the bolts without stripping. The requirement applies to both hot-dip and mechanical galvanized fasteners. Shipping Requirements for Galvanized Bolts and Nuts. The above requirements clearly indicate (1) that galvanized bolts and nuts are to be treated as a matched assembly, (2) that the seller must supply nuts which have been lubricated and tested with the supplied bolts, and (3) that nuts and bolts must be shipped together in the same shipping container. Purchase of galvanized bolts and galvanized nuts from separate sources is not in accordance with the intent of the ASTM Specifications because the control of overtapping and the testing and application of lubricant would be lost. Because some of the lubricants used to meet the requirements of ASTM Specifications are water soluble, it is advisable that galvanized bolts and nuts be shipped and stored in plastic bags in wood or metal containers. Washers. The primary function of washers is to provide a hardened non-galling surface under the element turned in tightening, particularly for those installation procedures which depend upon torque for control or inspection. Circular hardened washers meeting the requirements of ASTM A436 provide an increase in bearing area of 45 to 55 percent over the area provided by a heavy hex bolt head or nut; however, tests have shown that standard thickness washers play only a minor role in distributing the pressure induced by the bolt pretension, except where oversize or short slotted holes are used. Hence, consideration is given to this function only in the case of oversize and short slotted holes. The requirement for standard thickness hardened washers, when such washers are specified as an aid in the distribu3. Birkemoe, P. C., and D. C. Herrschaft, “Bolted Galvanized Bridges—Engineering Acceptance Near,” ASCE Civil Engineering, April 1970.



TYPE

A325 BOLT

A490 NUT

(1)

1

6 - 405

NUT

XYZ A490

XYZ

MFGR IDENTIFICATION (TYPICAL) XYZ

XYZ A325

BOLT

XYZ

DH

D ARCS INDICATE GRADE C

2

XYZ A325

DH OR 2H (2)

GRADE MARK (2) D, DH, 2 OR 2H

XYZ A490

SAME AS TYPE 1

NOTE MANDATORY 3 RADIAL LINES AT 60

NOTE MANDATORY 6 RADIAL LINES AT 30

(3)

3

XYZ A325 NOTE MANDATORY UNDERLINE

SAME AS TYPE 1

(3) XYZ 3

(3)

XYZ A490

XYZ DH3

DH3 NOTE MANDATORY

(1) ADDITIONAL OPTIONAL 3 RADIAL LINES AT 120 MAY BE ADDED. (2) TYPE 3 ALSO ACCEPTABLE (3) ADDITIONAL OPTIONAL MARK INDICATING WEATHERING MAY BE ADDED

Fig. C2. Required marking for acceptable bolt and nut assemblies

tion of pressure, is waived for alternative design fasteners which incorporate a bearing surface under the head of the same diameter as the hardened washer; however, the requirements for hardened washers to satisfy the principal requirement of providing a non-galling surface under the element turned in tightening is not waived. The maximum thickness is the same for all standard washers up to and including 11⁄2 inch bolt diameter in order that washers may be produced from a single stock of material. The requirement that heat-treated washers not less than 5⁄16 inch thick be used to cover oversize and slotted holes in external plies, when A490 bolts of 11⁄8 inch or larger diameter are used, was found necessary to distribute the high clamping pressure so as to prevent collapse of the hole perimeter and enable development of the desired clamping force. Preliminary investigation has shown that a similar but less severe deformation occurs when oversize or slotted holes are in the interior plies. The reduction in clamping force may be offset by “keying,” which tends to increase the resistance to slip. These effects are accentuated in joints of thin plies . Marking. Heavy hex structural bolts and heavy hex nuts are required by ASTM Specifications to be distinctively marked. Certain markings are mandatory. In addition to the mandatory markings, the manufacturer may apply additional distinguishing marking. The mandatory and optional markings are shown in Figure C2. Paint. In the previous edition of the Specification, generic names for paints applied to faying surfaces was the basis for categories of allowable working stresses in “fricAMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 406


tion” type connections. Research4 completed since the adoption of the 1980 Specification has demonstrated that the slip coefficients for paints described by a generic type are not single values but depend also upon the type of vehicle used. Small differences in formulation from manufacturer to manufacturer or from lot to lot with a single manufacturer significantly affect slip coefficients if certain essential variables within a generic type are changed. It is unrealistic to assign paints to categories with relatively small incremental differences between categories based solely upon a generic description. As a result of the research, a test method was developed and adopted by the Council titled “Test Method to Determine the Slip Coefficient for Coatings Used in Bolted Joints.” A copy of this document is appended to this Specification as Appendix A. The method, which requires requalification if an essential variable is changed, is the sole basis for qualification of any coating to be used under this Specification. Further, normally only two categories of slip coefficient for paints to be used in slip-critical joints are recognized: Class A for coatings which do not reduce the slip coefficient below that provided by clean mill scale, and Class B for paints which do not reduce the slip coefficient below that of blast-cleaned steel surfaces. The research cited in the preceding paragraph also investigated the effect of varying the time from coating the faying surfaces to assembly of the connection and tightening the bolts. The purpose was to ascertain if partially cured paint continued to cure within the assembled joint over a period of time. It was learned that all curing ceased at the time the joint was assembled and tightened and that paint coatings that were not fully cured acted much as a lubricant would; thus, the slip resistance of the joint was severely reduced from that which was provided by faying surfaces that were fully cured prior to assembly. C3 Bolted Parts Material Within the Grip. The Specification is intended to apply to structural joints in which all of the material within the grip of the bolt is steel. Surface Conditions. The Test Method to Determine the Slip Coefficient for Coatings Used in Bolted Joints includes long-term creep test requirements to ensure reliable performance for qualified paint coatings. However, it must be recognized that in the case of hot-dip galvanized coatings, especially if the joint consists of many plies of thickly coated material, relaxation of bolt tension may be significant and may require retensioning of the bolts subsequent to the initial tightening. Research5 has shown that a loss of pretension of approximately 6.5 percent occurred for galvanized plates and bolts due to relaxation as compared with 2.5 percent for uncoated joints. This loss of bolt tension occurred in five days with negligible loss recorded thereafter. This loss can be allowed for in design or pretension may be brought back to the prescribed level by retightening the bolts after an initial period of “settling-in.” Since it was first published, this Specification has permitted the use of bolt holes 6 1⁄ 16 inch larger than the bolts installed in them. Research has shown that, where 4. Frank, Karl H. and J. A. Yura, “An Experimental Study of Bolted Shear Connections.” FHWA/RD-81/148, December 1981. 5. Munse, W. H., “Structural Behavior of Hot Galvanized Bolted Connections,” 8th International Conference on Hot-dip Galvanizing, London, England, June 1967. 6. Allen. R. N. and J. W. Fisher, “Bolted Joints With Oversize or Slotted Holes,” ASCE Journal of the Structural Division, Vol. 94, No. ST9, September, 1968.



6 - 407

greater latitude is needed in meeting dimensional tolerances during erection, somewhat larger holes can be permitted for bolts 5⁄8 inch diameter and larger without adversely affecting the performance of shear connections assembled with highstrength bolts. The oversize and slotted hole provisions of this Specification are based upon these findings. Because an increase in hole size generally reduces the net area of a connected part, the use of oversize holes is subject to approval by the Engineer of Record. Burrs. Based upon tests7 which demonstrated that the slip resistance of joints was unchanged or slightly improved by the presence of burrs, burrs which do not prevent solid seating of the connected parts in the snug tight condition need not be removed. On the other hand, parallel tests in the same program demonstrated that large burrs can cause a small increase in the required turns from snug tight condition to achieve specified pretension with turn-of-nut method of tightening. Unqualified Paint on Faying Surfaces. An extension to the research on the slip resistance of shear connections cited in footnote 4 investigated the effect of ordinary paint coatings on limited portions of the contact area within joints and the effect of overspray over the total contact area. The tests8 demonstrated that the effective area for transfer of shear by friction between contact surfaces was concentrated in an annular ring around and close to the bolts. Paint on the contact surfaces approximately one inch but not less than the bolt diameter away from the edge of the hole did not reduce the slip resistance. On the other hand, in recognition of the fact that, in connections of thick material involving a number of bolts on multiple gage lines, bolt pretension might not be adequate to completely flatten and pull thick material into tight contact around every bolt, the Specification requires that all areas between bolts also be free of paint. (See Figure C3.) The new requirements have a potential for increased economy because the paint-free area may easily be protected using masking tape located relative to the hole pattern, and, further, the narrow paint strip around the perimeter of the faying surface will minimize uncoated material outside the connection requiring field touch-up. This research also investigated the effect of various degrees of inadvertent overspray on slip resistance. It was found that even the smallest amount of overspray of ordinary paint (that is, not qualified as Class A) within the specified paint-free area on clean mill scale reduced the slip resistance significantly. On blast-cleaned surfaces, the presence of a small amount of overspray was not as detrimental. For simplicity, the Specification prohibits any overspray from areas required to be free of paint in slip-critical joints regardless of whether the surface is clean mill scale or blast cleaned. Galvanized Faying Surfaces. The slip factor for initial slip with clean hot-dip galvanized surfaces is of the order of 0.19 as compared with a factor of about 0.35 for clean mill scale. However, research (see note 3) has shown that the slip factor of galvanized surfaces is significantly improved by treatments such as hand wire brushing or light “brush-off” grit blasting. In either case, the treatment must be controlled in order to achieve the necessary roughening or scoring. Power wire brushing is unsatisfactory because it tends to polish rather than roughen the surface. 7. Polyzois, D. and J. A. Yura, “Effect of Burrs on Bolted Friction Connections,” AISC Engineering Journal, 22 (No. 3) Third Quarter 1985. 8. Polyzois, D. and K. Frank, “Effect of Overspray and Incomplete Masking of Faying Surfaces on the Slip Resistance of Bolted Connections,” AISC Engineering Journal, 23 (No. 2), 2nd Quarter 1986.


6 - 408


Circular area around all holes

1 ″ but not less than d d 1 ″ but not less than d and All areas in between

Perimeter of contact area

Fig. C3. Areas outside the defined area need not be free of paint

Field experience and test results have indicated that galvanized members may have a tendency to continue to slip under sustained loading.9 Tests of hot-dip galvanized joints subject to sustained loading show a creep-type behavior. Treatments to the galvanized faying surfaces prior to assembly of the joint which caused an increase in the slip resistance under short duration loads did not significantly improve the slip behavior under sustained loading. C4 Design for Strength Of Bolted Connections Background for Design Stresses. With the 1985 edition of the Specification, the arbitrary designations “friction type” and “bearing type” connections used in former editions, and which were frequently misinterpreted as implying an actual difference in the manner of performance or strength of the two types of connection, were discontinued in order to focus attention more upon the real manner of performance of bolted connections. In bolted connections subject to shear-type loading, the load is transferred between the connected parts by friction up to a certain level of force which is dependent upon the total clamping force on the faying surfaces and the coefficient of friction of the faying surfaces. The connectors are not subject to shear, nor is the connected material subject to bearing stress. As loading is increased to a level in excess of the frictional resistance between the faying surfaces, slip occurs, but failure in the sense of rupture does not occur. As even higher levels of load are applied, the load is resisted by shear in the fastener and bearing upon the connected material plus some uncertain amount of friction between the faying surfaces. The final failure will be by shear failure of the connectors, or by tear out of the connected 9. Kulak, G. L., J. W. Fisher, and J. H. A. Struik, “Guide to Design Criteria for Bolted and Riveted Joints,” 2nd ed., New York: John Wiley & Sons, 1987, p. 208. (Hereinafter referred to as the Guide.)



6 - 409

material, or by unacceptable ovalization of the holes. Final failure load is independent of the clamping force provided by the bolts.10 The design of high-strength bolted connections under this Specification begins with consideration of strength required to prevent premature failure by shear of the connectors or bearing failure of the connected material. Next, for connections which are defined as “slip-critical,” the resistance to slip load is checked. Because the combined effect of frictional resistance with shear or bearing has not been systematically studied and is uncertain, any potential greater resistance due to combined effect is ignored. Connection Slip. There are practical cases in the design of structures where slip of the connection is desirable in order to permit rotation in a joint or to minimize the transfer of moment. Additionally there are cases where, because of the number of fasteners in a joint, the probability of slip is extremely small or where, if slip did occur, it would not be detrimental to the serviceability of the structure. In order to provide for such cases while at the same time making use of the higher shear strength of high-strength bolts, as contrasted to ASTM A307 bolts, the Specification now permits joints tightened only to the snug tight condition. The maximum amount of slip that can occur in connections that are not classified as slip-critical is, theoretically, an amount equal to two hole clearances. In practical terms, it is observed to be much less than this. In laboratory tests it is usually about one-half a hole clearance. This is because the acceptable inaccuracies in the location of holes within a pattern of bolts would usually cause one or more bolts to be in bearing in the initial unloaded condition. Further, in statically loaded structures, even with perfectly positioned holes, the usual method of erection would cause the weight of the connected elements to put the bolts into direct bearing at the time the member is supported on loose bolts and the lifting crane is unhooked. Subsequent additional gravity loading could not cause additional connection slip. Connections classified as slip-critical include those cases where slip could theoretically exceed an amount deemed by the Engineer of Record to affect the suitability for service of the structure by excessive distortion or reduction in strength or stability, even though the resistance to fracture of the connection, per se, may be adequate. Also included are those cases where slip of any magnitude must be prevented, for example, joints subject to load reversal. Shear and Bearing on Fasteners. Several interrelated parameters influence the shear and bearing strength of connections. These include such geometric parameters as the net-to-gross-area ratio of the connected parts, the ratio of the net area of the connected parts to the total shear-resisting area of the fasteners, and the ratio of transverse fastener spacing to fastener diameter and to the connected part thickness. In addition, the ratio of yield strength to tensile strength of the steel comprising the connected parts, as well as the total distance between extreme fasteners, measured parallel to the line of direct tensile force, play a part. In the past, a balanced design concept had been sought in developing criteria for mechanically fastened joints to resist shear between connected parts by means of bearing of the fasteners against the sides of the holes. This philosophy resulted in wide variations in the factor of safety for the fasteners, because the ratio of yield to tensile strength increases significantly with increasingly stronger grades of steel. It had no application at all in the case of very long joints used to transfer direct 10. Ibid., pp. 49–52.


6 - 410


tension, because the end fasteners “unbutton” before the plate can attain its full strength or before the interior fasteners can be loaded to their rated shear capacity. By means of a mathematical model it was possible to study the interrelationship of the previously mentioned parameters.11,12 It has been shown that the factor of safety against shear failure ranged from 3.3 for compact (short) joints to approximately 2.0 for joints with an overall length in excess of 50 inches. It is of interest to note that the longest (and often the most important) joints had the lowest factor, indicating that a factor of safety of 2.0 has proven satisfactory in service. The absence of design strength provisions specifically for the case where a bolt in double shear has a non-threaded shank in one shear plane and a threaded section in the other shear plane is because of the uncertainty of manner of sharing the load between the two different shear areas. It also recognizes that knowledge as to the bolt placement (which might leave both shear planes in the threaded section) is not ordinarily available to the detailer. If threads occur in one shear plane, the conservative assumption is made that threads are in all shear planes. The nominal strength and resistance factors for fasteners subject to applied tension or shear are given in Table 2. The values are based upon the research and recommendations reported in the Guide. With the wealth of data available, it was possible through statistical analyses to adjust resistances to provide more uniform reliability for all loading and joint types. The design resistances provide designs approximately equivalent to the designs provided by the allowable stresses in the 1980 edition of the Specification. The design of connections is more conservative than that of the connected members of buildings and bridges by a substantial margin, in the sense that the failure load of the fasteners is substantially in excess of the maximum serviceability limit (yield) of the connected material. Design for Tension. The nominal strengths specified for applied tension13 are intended to apply to the external bolt load plus any tension resulting from prying action produced by deformation of the connected parts. The recommended design strength is approximately equal to the initial tightening force; thus, when loaded to the nominal (service) load, high-strength bolts will experience little if any actual change in stress. For this reason, bolts in connections in which the applied loads subject the bolts to axial tension are required to be fully tensioned, even though the connection may not be subject to fatigue loading nor classified as slip-critical. Properly tightened A325 and A490 bolts are not adversely affected by repeated application of the recommended service load tensile stress, provided the fitting material is sufficiently stiff, so that the prying force is a relatively small part of the applied tension.14 The provisions covering bolt tensile fatigue are based upon study of test reports of bolts that were subjected to repeated tensile load to failure. Design for Shear. The nominal strength in shear is based upon the observation that the shear strength of a single high-strength bolt is about 0.62 times the tensile strength of that bolt.15 However, in shear connections with more than two bolts in the line of force, deformation of the connected material causes nonuniform bolt shear force distribution so that the strength of the connection in terms of the average bolt strength 11. Fisher, J. W. and L. S. Beedle, “Analysis of Bolted Butt Joints,” ASCE Journal of the Structural Division, 91 (No. ST5), October 1965. 12. Guide, pp. 89–116; 126–132. 13. Ibid., pp. 263–286. 14. Ibid., pp. 272. 15. Ibid., pp. 44–50.



6 - 411

goes down as the joint length increases.16 Rather than provide a function that reflects this decrease in average fastener strength with joint length, a single reduction factor of 0.80 was applied to the 0.62 multiplier. The result will accommodate bolts in all joints up to about 50 inches in length without seriously affecting the economy of very short joints. As noted in the footnotes to Table 2, bolts in joints longer than 50 inches in length must be further discounted by an additional 20 percent. The average value of the nominal strength for bolts with threads in the shear plane has been determined by a series of tests17 to be 0.833 Fu with a standard deviation of 0.03. A value of 0.80 was taken as a factor to account for the shear strength of a bolt with threads in the shear plane based upon the area corresponding to the nominal body area of the bolt. The shear strength of bolts is not affected by pretension in the fasteners provided the connected material is in contact at the faying surfaces. The design shear strength equals the nominal shear strength multiplied by a resistance factor of 0.75. Combined Tension and Shear. The nominal strength of fasteners subject to combined tension and shear is provided by elliptical interaction curves in Table 3 which account for the connection length effect on bolts loaded in shear, the ratio of shear strength to tension strength of threaded fasteners, and the ratios of root area to nominal body area and tensile stress area to nominal body area.18 No reduction in the design shear strength is required when applied tensile stress is equal to or less than the design tensile strength. Although the elliptical interaction curve provides the best estimate of the strength of bolts subject to combined shear and tension and thus is used in this Specification, it would be within the intent of the Specification for invoking specifications to use a three straight line approximation of the ellipse. Design for Bearing. Bearing stress produced by a high-strength bolt pressing against the side of the hole in a connected part is important only as an index to behavior of the connected part. It is of no significance to the bolt. The critical value can be derived from the case of a single bolt at the end of a tension member. It has been shown,19 using finger-tight bolts, that a connected plate will not fail by tearing through the free edge of the material if the distance L, measured parallel to the line of applied force from a single bolt to the free edge of the member toward which the force is directed, is not less than the diameter of the bolt multiplied by the ratio of the bearing stress to the tensile strength of the connected part. The criterion for nominal bearing strength is L / d ≥ Rn / Fu where Rn = nominal bearing pressure Fu = specified minimum tensile strength of the connected part. As a practical consideration, a lower limit of 1.5 is placed on the ratio L/d and an upper limit of 1.5 on the ratio Fp / Fu and an upper limit of 3.0 on the ratio Rn / Fu. The foregoing leads to the rules governing bearing strength in the specification. 16. Ibid., pp. 99–104. 17. Yura. J. A., K. H. Frank, and D. Polyois, “High Strength Bolts for Bridges.” PMFSEL Report No. 87-3, May 1987, Phil M. Ferguson Structural Engineering Laboratory, University of Texas at Austin. 18. Guide, pp. 50–51. 19. Ibid., pp. 141–143.


6 - 412


The bearing pressure permitted in the 1980 Specification and the current provisions are fully justifiable from the standpoint of strength of the connected material. However, even though rupture does not occur, recent tests have demonstrated that ovalization of the hole will begin to develop as the bearing stress is increased beyond the previously permitted stress, especially if it is combined with high tensile stress on the net section. Furthermore, when high bearing stress is combined with high tensile stress on the net section and the effect of exterior versus interior plies, lower ultimate strengths than previously reported result in addition to the hole ovalization. Recognizing that initiation of hole ovalization occurs well below the ultimate strength, and to facilitate standardization in detailing and fabrication, sufficiently conservative simplified criteria have been provided in a formula format for usual applications. The more accurate formula in which the strength is related to the distance L may be used for special cases such as those with very large bolts or very thin material. For connections with more than a single bolt in the direction of force, the resistance may be taken as the sum of the resistances of the individual bolts. C5 Design Check for Slip Resistance The Specification recognizes that, for a number of cases, slip of a joint would be undesirable or must be precluded. Such joints are termed “slip-critical” joints. This is somewhat different from the previous term “friction type” connection. The new terminology was adopted in order to focus attention on the fact that all tightened high-strength bolted joints resist load by friction between the faying surfaces up to the slip load and subsequently are able to resist even greater loads by shear and bearing. The strength of the joint is not related to the slip load. The Specification requires that the two different resistances be considered separately. The consequences of slip into bearing varies from application to application; hence the determination of which connections shall be designed and installed as slipcritical is best left to judgment and a conscious decision on the part of the Engineer of Record. Also, the determination of whether the potential slippage of a joint is critical at nominal load level as a serviceability consideration or whether slippage could result in distortions of the frame such that the ability of the frame to resist factored loads would be reduced can be determined only by the Engineer of Record. The following comments reflect the collective thinking of the Council as developed during numerous meetings and reviews of drafts of the Specification and Commentary. They are provided as guidance and an indication of the intent of the Specification. In the case of bolts in holes with only small clearance, such as standard holes and slotted holes loaded transverse to the axis of the slot in practical connections, the freedom to slip generally does not exist because one or more bolts are in bearing even before load is applied due to normal fabrication tolerances and erection procedures. Further, the consequences of slip, if it can occur at all, are trivial except for a few situations. If for some reason it is deemed critical, design should probably be on the basis of nominal loads (Section 5(b)). In connections containing long slots that are parallel to the direction of the applied load, slip of the connection prior to attainment of the factored load might be large enough to alter the usual assumption of analysis that the undeformed structure can be used to obtain the internal forces. The Specification allows the designer two alternatives in this case. If the connection is designed so that it will not slip under the AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 413

effect of the nominal loads, then the effect of the factored loads acting on the deformed structure (deformed by the maximum amount of slip in the long slots at all locations) must be included in the structural analysis. Alternatively the connection can be designed so that it will not slip at loads up to the factored load level. These requirements are noted in Clause 7(b)(3). Joints subject to full reverse cyclic loading are clearly slip-critical joints since slip would permit back-and-forth movement of the joint and early fatigue failure. However, for joints subject to pulsating load that does not involve reversal of load direction, proper fatigue design could be provided either as a slip-critical joint on the basis of stress on the gross section or as a non-slip-critical joint on the basis of stress on the net section. Because fatigue results from repeated application, the service load rather than the overload load design should be based upon nominal load criteria (Section 5(b)). For high-strength bolts in combination with welds in statically loaded conditions and considering new work only, the nominal strength may be taken as the sum of two contributions.20 One results from the slip resistance of the bolted parts and may be determined in accordance with Section 5(c). The second results from the resistance of the welds as provided by applicable welding specifications. If one type of connector is already loaded when the second type of connector is introduced, the nominal strength cannot be obtained by adding the two resistances. The Guide should be consulted in these cases. From the definition of the term “coefficient of slip” (friction), the expression for nominal slip resistances for bolts in standard holes is apparent and needs no explanation. The mean value of slip coefficients from many tests on clean mill scale, blast-cleaned steel surfaces and galvanized and roughened surfaces were taken as the basis for the three classes of surfaces. In the 1978 edition of the Specification, nine classes of faying surface conditions were introduced, and significant increases were made in the recommended allowable stresses for proportioning connections which function by transfer of shear between connected parts by friction. These classes and stresses were adopted on the basis of statistical evaluation of the information then available. Extensive data developed through research sponsored by the Council and others during the past ten years has been statistically analyzed to provide improved information on slip probability of connections in which the bolts have been preloaded to the requirements of Table 4. Two principal variables—coefficient of friction of the faying surfaces and bolt pretension—were found to dominate the slip resistance of connections. An examination of the slip (friction) coefficient data for a wide range of surface conditions indicates that the data are distributed normally, and the standard deviation is essentially the same for each surface condition class. This means that different reduction factors should be applied to classes of surfaces with different mean values of coefficients of friction—the smaller the mean value of the coefficient of friction, the smaller (more severe) the appropriate reduction factor—in order to provide equivalent reliability of slip resistance. The bolt clamping force data indicate that bolt tensions are distributed normally for each method of tightening. However, the data also indicate that the mean values of the bolt tensions are different for each method. If the calibrated wrench method is used to tighten ASTM A325 bolts, the mean value of bolt tension is about 1.13 20. Ibid., pp. 238–40.


6 - 414


times the minimum specified tension in Table 4. If the turn-of-nut method is used, the mean value of tension is about 1.35 times the minimum specified preload for A325 bolts and about 1.27 for A490 bolts. The combined effects of the variability of coefficient of friction and bolt tension have been accounted for in the slip probability factor, D, of the formula for nominal slip resistance in Section 5(b). The values of the slip probability factor, D, given by 5(b) imply a 90 percent reliability that slip will not occur if the calibrated wrench method of installation is used. If the turn-of-nut method is used, a reliability of about 95 percent will be provided. Reference is made to Guide to Design Criteria for Bolted and Riveted Joints (2nd ed., New York: John Wiley and Sons. 1987 p. 135) for tables of values of D appropriate for other mean slip coefficients and slip probabilities and suitable for direct substitution into the formula for slip resistance in Section 5(b). The frequency distribution and mean value of clamping force for bolts tightened by turn-of-nut method are higher than calibrated wrench installation because of the elimination of variables which affect torque-tension ratios and due to higherthan-specified minimum strength of production bolts. Because properly applied turnof-nut installation induces yield point strain in the bolt, the higher-thanspecified yield strength of production bolts will be mobilized and result in higher clamping force by the method. On the other hand, the calibrated wrench method, which is dependent upon the calibration of wrenches to slightly more than Table 4 tensions, independent of the actual bolt properties, will not mobilize any additional strength of production bolts. High clamping force might be achieved by the calibrated wrench method if the wrench was set to a higher torque value. However, this would require more attention to the degrees of rotation to prevent excessive deformation of the bolt or torsional bolt failure. Because of the effects of oversize and slotted holes on the induced tension in bolts using any of the specified installation methods, lower values are provided for bolts in these hole types. In the case of bolts in long slotted holes, even though the slip load is the same for bolts loaded transverse or parallel to the axis of the slot, the values for bolts loaded parallel to the axis has been further reduced based upon judgment in recognition of the greater consequences of slip. Attention is called to the fact that the criteria for slip resistance are for the case of connections subject to a coaxial load. For cases in which the load tends to rotate the connection in the plane of the faying surface, a modified formula accounting for the placement of bolts relative to the center of rotation should be used.21 Connections of the type shown in Figure C4(a), in which some of the bolts (A) lose a part of their clamping force due to applied tension, suffer no overall loss of frictional resistance. The bolt tension produced by the moment is coupled with a compensating compressive force (C) on the other side of the axis of bending. In a connection of the type shown in Fig. C4(b), however, all fasteners (B) receive applied tension which reduces the initial compression force at the contact surface. If slip under load cannot be tolerated, the design slip-load value of the bolts in shear should be reduced in proportion to the ratio of residual axial force to initial tension. If slip of the joint can be tolerated, the bolt shear stress should be reduced according to the tension-shear interaction as outlined in the Guide. page 71. Because the bolts are subject to applied axial tension, they are required to be pretensioned in either case. 21. Ibid., pp. 217–30.



6 - 415

B A

M

C V (a)

T

(b)

Fig. C4

While connections with bolts pretensioned to the levels specified in Table 4 do not ordinarily slip into bearing when subject to anticipated loads, it is required that they meet the requirements of Section 5 in order to maintain the factor of safety of 2 against fracture in the event that the bolts do slip into bearing as a result of large unforeseen loads. To cover those cases where a coefficient of friction less than 0.33 might be adequate for a given situation, the Specification provides that, subject to the approval of the Engineer of Record, and provided the mean slip coefficient is determined by the specified test procedure and the appropriate slip probability factor, D, is selected from the literature, faying surface coatings providing lower slip resistance than Class A coating may be used. It should be noted that both Class A and Class B coatings are required to be applied to blast-cleaned steel. High-Strength Bolts in Combination with Welds or Rivets. For high-strength bolts in combination with welds in statically loaded conditions and considering new work only, the nominal strength may be taken as the sum of the two contributions. If one type of connector is already loaded when the second type of connector is introduced, the nominal strength cannot be obtained by sum of the two resistances. The Guide should be consulted in these cases. For high-strength bolts in combination with welds in fatigue loaded applications, available data are not sufficient to develop general design recommendations at this time. High-strength bolts in combination with rivets are rarely encountered in modern practice. If need arises, guidance may be found in the Guide. C7 Design Details of Bolted Connections A new section has been added with this edition of the Specification in order to bring together a number of requirements for proper design and detailing of high-strength bolted connections. The material covered in the Specification, and in Section 7 in particular, is not intended to provide comprehensive coverage of the design of highstrength bolted connections. For example, other design considerations of importance AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 416


to the satisfactory performance of the connected material such as block shear, shear lag, prying action, connection stiffness, effect on the performance of the structure and others are beyond the scope of this Specification and Commentary. Proper location of hardened washers is as important as other elements of a detail to the performance of the fasteners. Drawings and details should clearly reflect the number and disposition of washers, especially the thick hardened washers that are required for several slotted hole applications. Location of washers is a design consideration that should not be left to the experience of the iron worker. While hardened washers are not required with some methods of installation, their use will overcome the effects of galling under the element turned in tightening. Finger shims are a necessary device or tool of the trade to permit adjusting alignment and plumbing of structures. When these devices are fully and properly inserted, they do not have the same effect on bolt tension relaxation or the connection performance as do long slotted holes in an outer ply. When fully inserted, the shim provides support around approximately 75 percent of the perimeter of the bolt in contrast to the greatly reduced area that exists with a bolt centered in a long slot. Further, finger shims would always be enclosed on both sides by the connected material which would be fully effective in bridging the space between the fingers. C8 Installation and Tightening Several methods for installation and tensioning of high-strength bolts, when tensioning is required, are provided without preference in the Specification. Each method recognized in Section 8, when properly used as specified, may be relied upon to provide satisfactory results. All methods may be misused or abused. At the expense of redundancy, the provisions stipulating the manner in which each method is intended to be used are set forth in complete detail in order that the rules for each method may stand alone without need for footnotes or reference to other sections. If the methods are conscientiously implemented, good results should be routinely achieved. Connections Not Requiring Full Tensioning. In the Commentary, Section C6 of the previous edition of the Specification, it was pointed out that “bearing” type connections need not be tested to ensure that the specified pretension in the bolts had been provided, but specific provision permitting relaxation of the tensioning requirement was not contained in the body of the Specification. In the present edition of the Specification, separate installation procedures are provided for bolts that are not within the slip-critical or direct tension category. The intent in making this change is to improve the quality of bolted steel construction and reduce the frequency of costly controversies by focusing attention, both during the installation and tensioning phase and during inspection, on the true slip-critical connections, rather than diluting the effort through the requirement for costly tensioning and tension testing of the great many connections where such effort serves no useful purpose. The requirement for identification of connections on the drawings may be satisfied either by identifying the slip-critical and direct tension connections which must be fully tightened and inspected or by identifying the connections which need be tightened only to the snug tight condition. Under the provisions of some other specifications, certain shear/bearing connections are required to be tightened well beyond the snug tight conditions;22 how22. For example, American Institute of Steel Construction, “Specification for Design Fabrication and



6 - 417

ever, because the joints are in bearing, prevention of slip of the joint is not a concern in these connections. Because they are not slip-critical joints, they should not be subject to the same requirements as slip-critical joints, especially the requirements for faying surface coatings and conditions. To ensure proper tightness of the connections, they should be tightened by one of the four methods in 8(d); however, inspection should be limited to monitoring the work to confirm that the bolt tightening procedure is properly applied. Inspection should not include testing to ensure that any specific level of tension has been achieved. In the Specification, snug tight is defined as the tightness that exists when all plies are in firm contact. This may usually be attained by a few impacts of an impact wrench or the full effort of a man using an ordinary spud wrench. In actuality, snug tight is a degree of tightness which will vary from joint to joint depending upon the thickness, flatness and degree of parallelism of the connected material. In most joints, the plies will pull together at snug tight; however, in some joints in thick material, it may not be possible to have continuous contact throughout the faying surface area. In such joints, the slip resistance of the completed joints will not be reduced because compressive forces between the faying surfaces, however distributed, must be in equilibrium with the total of the tensile forces in all bolts. Tension Calibrating Devices. At the present time, there is no known economical means for determining the tension in a bolt that has previously been installed in a connection. The actual tension in a bolt installed in a tension calibrator (hydraulic tension indicating device) is directly indicated by the dial of the device, provided the device is properly calibrated. Such a device is an economical and valuable tool that should be readily available whenever high-strength bolts are to be installed in either slip-critical or shear/bearing connections. The testing of as-received bolts and nuts at the job site is a requirement of the Specification because instances of counterfeit under strength fasteners not meeting the requirements of the ASTM Specification have not infrequently occurred. Job site testing provides a practical means for ensuring that nonconforming fasteners are not incorporated in the work. Further, although the several elements of a fastener assembly may conform to the minimum requirements of their separate ASTM Specifications, their compatibility in an assembly or the need for lubrication can only be ensured by testing of the assembly. Hence, such devices are important for testing the complete fastener assembly as it will be used with the method of tightening to be used to ensure the suitability of bolts and nuts (probably produced by different manufacturers), other elements, and the adequacy of impact wrenches and/or air pressure to provide the specified tension using the selected method. Testing before start of installation of fasteners in the work will also identify potential sources of problems, such as the need for lubrication to prevent failure of bolts by combined high torque with tension, under-strength assemblies due to excessive overtapping of hot-dip galvanized nuts, and to clarify for the bolting crews and inspectors the proper implementation of the selected installation method to be used. Such devices are essential to the confirmation testing of alternative design fasteners, direct tension indicators, and to verify the proper use of the turn-of-nut procedure. They are also essential to the specified procedure for the calibrated wrench method of installation, and for the specified procedure for determining a valid testing torque when such inspection by a torque method is required. Erection of Structural Steel for Buildings,” Section 1.15.12, stipulates several cases where high-strength bolts in bearing connections are to be fully tensioned independent of whether potential slip is a concern or not.


6 - 418


They are the only known economically available tool for field use for determining realistic torque to tension relationships for given fastener assemblies. Experience on many projects has shown that bolts and/or nuts not meeting the requirements of the applicable ASTM specification would have been identified prior to installation if they had been tested as an assembly in a tension calibrator. The controversy and great expense of replacing bolts installed in the structure when the nonconforming bolts were discovered at a later date would have been avoided. Hydraulic tension calibrating devices capable of indicating bolt tension undergo a slight deformation under load. Hence, the nut rotation corresponding to a given tension reading may be somewhat larger than it would be if the same bolt were tightened against a solid steel abutment. Stated differently, the reading of the calibrating device tends to underestimate the tension which a given rotation of the turned element would induce in a bolt in an actual joint. This should be borne in mind when using such devices to establish a tension-rotation relationship. Slip-critical Connections and Connections Subject to Direct Tension. Four methods for joint assembly and tightening are provided for slip-critical and direct tension connections. It has repeatedly been demonstrated in the laboratory that each of the four installation methods provides the specified pretension when used properly with specified fasteners in good condition, but improperly applied methods or understrength fasteners or fasteners in poor condition provide uncertain pretensions. Therefore, regardless of the method used and prior to the commencement of work, it is required to be demonstrated by installation of a representative sample of the fastener assemblies in the tension calibrator that the specified pretension can be achieved using the procedure to be used with the fasteners to be used by the crews who will be doing the work. With any of the four described tensioning methods, it is important to install bolts in all holes of the connection and bring them to an intermediate level of tension generally corresponding to snug tight in order to compact the joint. Even after being fully tightened, some thick parts with uneven surfaces may not be in contact over the entire faying surface. In itself, this is not detrimental to the performance of the joint. As long as the specified bolt tension is present in all bolts of the completed connection, the clamping force equal to the total of the tensions in all bolts will be transferred at the locations that are in contact and be fully effective in resisting slip through friction. If however, individual bolts are installed and tightened in a single continuous operation, bolts which are tightened first will be subsequently relaxed by the tightening of the adjacent bolts. The total of the forces in all bolts will be reduced, which will reduce the slip load whether there is uninterrupted contact between the surfaces or not. With all methods, tightening should begin at the most rigidly fixed or stiffest point and progress toward the free edges, both in the initial snugging up and in the final tightening. Turn-of-Nut-Tightening. When properly implemented, turn-of-nut method provides more uniform tension in the bolts than does torque controlled tensioning methods because it is primarily dependent upon bolt elongation into the inelastic range. Consistency and reliability method is dependent upon ensuring that the joint is well compacted and all bolts are uniformly tight at a snug tight condition prior to application of the final required partial turn. Under-tightened bolts will result if this starting condition is not achieved because subsequent turning of the nut will first close the gap before meaningful elongation of the bolt occurs as would be the AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 419

case with solid steel in the grip. Reliability is also dependent upon ensuring that the turn that is applied is relative between the bolt and nut; thus the element not turned in tightening should be prevented from rotating while the required degree of turn is applied to the turned element. Reliability and inspectability of the method may be improved by having the outer face of the nut match-marked to the protruding end of the bolt after the joint has been snug tightened but prior to final tightening. Such marks may be applied by the wrench operator using a crayon or dab of paint. Such marks in their relatively displaced position after tightening will afford the inspector a means for noting the rotation that was applied. Problems with turn-of-nut tightening have been encountered with hot-dip galvanized bolts. In some cases, the problems have been attributed to especially effective lubricant applied by the manufacturer to ensure that bolts from stock will meet the ASTM Specification requirements without the need for relubricating and retesting. Job site tests in the tension indicating device demonstrated the lubricant reduced the coefficient of friction between the bolt and nut to the degree that “the full effort of a man using an ordinary spud wrench” to snug tighten the joint actually induced the full required tension. Also, because the nuts could be removed by an ordinary spud wrench they were erroneously judged improperly tightened by the inspector. Research (see note 3) confirms that lubricated high-strength bolts may require only one-half as much torque to induce the specified tension. In other cases of problems with hot-dip galvanized bolts, the absence of lubrication or lack of proper overtapping caused seizing of the nut and bolt threads which resulted in twist failure of the bolt at less than specified tension. For such situations, use of a tension indicating device and the fasteners being installed may be helpful in establishing either the need for lubrication or alternate criteria for snug tight at about one-half the tension required by Table 4. Because reliability of the method is independent of the presence or absence of washers, washers are not required except for oversize and slotted holes in an outer ply. In the absence of washers, testing after the fact using a torque wrench method is highly unreliable. That is, the turn-of-nut method of installation, properly applied, is more reliable and consistent than the testing method. The best method for inspection of the method is for the Inspector to observe the required job site confirmation testing of the fasteners and the method to be used, followed by monitoring of the work in progress to ensure that the method is routinely properly applied. Calibrated Wrench Method. Research has demonstrated that scatter in induced tension is to be expected when torque is used as an indirect indicator of tension. Numerous variables, which are not related to tension, affect torque. For example, the finish and tolerance on bolt threads, the finish and tolerance on the nut threads, the fact that the bolt and nut may not be produced by the same manufacturer, the degree of lubrication, the job site conditions contributing to dust and dirt or corrosion on the threads, the friction that exists to varying degree between the turned element and the supporting surface, the variability of the air pressure on the torque wrenches due to length of air lines or number of wrenches operating from the same source, the condition and lubrication of the wrench which may change within a work shift, and other factors all bear upon the effectiveness of the calibrated torque wrench to induce tension. Recognition of the calibrated wrench method of tightening was removed from the Specification with the 1980 edition. This action was taken because it is the least AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 420


reliable of all methods of installation and many costly controversies had occured. It is suspected that shortcut procedures in the use of the calibrated wrench method of installation, not in accordance with the Specification provisions, were probably being used. Further, torque controlled inspection procedures based upon “standard” or calculated inspection torques rather than torques determined as required by the Specification were being routinely used. These incorrect procedures plus others had a compounding effect upon the uncertainty of the installed bolt tension, and were responsible for many of the controversies. It is recognized, however, that if the calibrated wrench method is implemented without shortcuts as intended by the Specification, that there will be a 90 percent assurance that the tensions specified in Table 4 will be equaled or exceeded. Because the Specification should not prohibit any method which will give acceptable results when used as specified, the calibrated wrench method of installation was reinstated in the 1985 edition of the Council Specification. However, to improve upon the previous situation, the 1985 version of the Specification was modified to require better control. Wrenches must be calibrated daily for each diameter and grade of bolt. Hardened washers must be used. Fasteners must be protected from dirt and moisture at the job site. Additionally, to achieve reliable results attention should be given to the control, insofar as it is practical, of those controllable factors which contribute to variability. For example, bolts and nuts should be purchased from reliable manufacturers with a record of good quality control to minimize the variability of the fit. Bolts and nuts should be adequately and uniformly lubricated. Water soluble lubricants should be avoided. Installation of Alternative Design Fasteners. It is the policy of the Council to recognize only fasteners covered by ASTM Specifications; however, it cannot be denied that a general type of alternative design fastener produced by several manufacturers, is used on a significant number of projects as permitted by Section 2(d). The bolts referred to involve a splined end extending beyond the threaded portion of the bolt which is gripped by a specially designed wrench chuck which provides a means for turning the nut relative to the bolt. While such bolts are subject to many of the variables affecting torque mentioned in the preceding section, they are produced and shipped by the manufacturers as a nut-bolt assembly under good quality control, which apparently minimizes some of the negative aspects of the torque controlled process. While these alternative design fasteners have been demonstrated to consistently provide tension in the fastener meeting the requirements of Table 5 in controlled tests in tension indicating devices, it must be recognized that the fastener may be misused and provide results as unreliable as those with other methods. They must be used in the as-delivered clean lubricated condition. The requirements of this Specification and the installation requirements of the manufacturer’s specification required by Section 2(d) must be adhered to. As with other methods, a representative sample of the bolts to be used should be tested to ensure that, when used in accordance with the manufacturer’s instructions, they do, in fact, provide tension, as specified in Table 5. In the actual joints, bolts must be installed in all holes of a connection and all fasteners tightened to an intermediate level of tension adequate to pull all material into contact. Only after this has been accomplished should the fasteners be fully tensioned in a systematic manner and the splined end sheared off. The sheared off splined end merely signifies that at some time the bolt has been subjected to a torque adequate to cause the AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 421

shearing. If the fasteners are installed and tensioned in a single continuous operation, they will give a misleading indication to the Inspector that the bolts are properly tightened. Therefore, the only way to inspect these fasteners with assurance is to observe the job site testing of the fasteners and installation procedure and then monitor the work while in progress to ensure that the specified procedure is routinely followed. Direct Tension Indicator Tightening. This Specification recognizes load indicating devices covered by the American Society for Testing and Materials’ “Specification for Compressible-Washer Type Direct Tension Indicators For Use With Structural Fasteners,” ASTM F959, in Section 2(f). The referenced device is a hardened washer incorporating several small formed arches which are designed to deform in a controlled manner when subjected to load. These load indicator washers are the sole type of device known which is directly dependent upon the tension load in the bolt, rather than upon some indirect parameter, to indicate the tension in a bolt. As with the alternative design load indicating bolts, load indicating washers are dependent upon the quality control of the producer and proper use in accordance with the manufacturer’s installation procedures and these Specifications. If the load indicator washers delivered for use in a specific application are tested at the job site to demonstrate that all components of the assembly do provide a proper indication of bolt tension, they are reliable if they are properly used by the bolting crews. Direct tension indicators meeting the requirements of ASTM F959 depend upon tension in the fastener to cause inelastic deformation of the formed arches. Bolts together with the load indicator washer plus any other washers required by Specification should be installed in all holes of the connection and the bolts tightened to approximately one-half the specified tension (deformation of the formed arches by about one-half the amount required to compress them to the specified gap) to ensure that plies of the joint have been brought into firm contact. Only after this initial tightening operation should the bolts be fully tensioned in a systematic manner. If the bolts are installed and tensioned in a single continuous operation, the load indicator washers will give the inspector a misleading indication that bolts are uniformly tensioned to the specified tension. Therefore, the only way to inspect fasteners with which load indicator washers are used with assurance is to observe the job site testing of the devices and installation procedure and then routinely monitor the work while in progress to ensure that the specified procedure is followed. Use of direct tension indicators provides a reliable means for tensioning galvanized fasteners because it avoids the factors which affect other methods. During installation, care must be taken to ensure that the indicator nubs are oriented to bear against the hardened bearing surface of the bolt head or against a hardened flat washer if used under the nut. C9 Inspection It is apparent from the commentary on installation procedures that the inspection procedures giving the best assurance that bolts are properly installed and tensioned is provided by Inspector observation of the calibration testing of the fasteners using the selected installation procedure followed by monitoring of the work in progress to ensure that the procedure that was demonstrated to provide the specified tension is routinely adhered to. When such a program is followed, no further evidence of proper bolt tension is required. If testing for bolt tension using torque wrenches is conducted subsequent to the time the work of installation and tightening of bolts performed, the test procedure AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 422


is subject to all of the uncertainties of torque controlled calibrated wrench installation. Additionally, the absence of many of the controls necessary to minimize variability of the torque to tension relationship, which are unnecessary for the other methods of bolt installation, such as use of hardened washers, careful attention to lubrication and the uncertainty of the effect of passage of time and exposure in the installed condition all reduce the reliability of the arbitration inspection results. The fact that in many cases it may have to be based upon a job test torque determined by using bolts only assumed to be representative of the bolts in the actual job, or using bolts removed from completed joints, makes the test procedure less reliable than a properly implemented installation procedure it is used to verify. Verification inspection using ultrasonic extensometers is accurate but costly and time-consuming, and requires that each tested bolt must be loosened to zero tension for calibration. Therefore, extensometers should be used for inspection only in the most critical cases. The arbitration inspection procedure contained in the Specification is provided, in spite of its limitations, as the most feasible available at this time.


6 - 423

Code of Standard Practice for Steel Buildings and Bridges

Adopted Effective June 10, 1992 American Institute of Steel Construction, Inc.

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. One East Wacker Drive, Suite 3100, Chicago, IL 60601-2001 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 424

Copyright 1992 by The American Institute of Steel Construction All rights reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher

PREFACE

When contractual documents do not contain specific provisions to the contrary, existing trade practices are considered to be incorporated into the relationships between the parties to a contract. As in any industry, trade practices have developed among those involved in the purchase, design, fabrication and erection of structural steel. The American Institute of Steel Construction has continuously surveyed the structural steel fabrication industry to determine standard practices and, commencing in 1924, published its Code of Standard Practice. Since that date, the Code has been periodically updated to reflect new and changing technology and practices of the industry. It is the Institute’s intention to provide to owners, architects, engineers, contractors and others associated with construction, a useful framework for a common understanding of acceptable standards when contracting for structural steel construction. This edition is the fourth complete revision of the Code since it was first published. It includes a number of new sections covering new subjects not included in the previous Code, but which are an integral part of the relationship of the parties to a contract. The Institute acknowledges the valuable information and suggestions provided by trade associations and other organizations associated with construction and the fabricating industry in developing this current Code of Standard Practice. While every precaution has been taken to insure that all data and information presented is as accurate as possible, the Institute cannot assume responsibility for errors or oversights in the information published herein, or the use of the information published or incorporation of such information in the preparation of detailed engineering plans. The Code should not replace the judgment of an experienced architect or engineer who has the responsibility of design for a specific structure.


6 - 425

Code of Standard Practice for Steel Buildings and Bridges Adopted Effective June 10, 1992 American Institute of Steel Construction, Inc.

SECTION 1. GENERAL PROVISIONS 1.1. Scope The practices defined herein have been adopted by the AISC as the commonly accepted standards of the structural steel fabricating industry. In the absence of other instructions in the contract documents, the trade practices defined in this Code of Standard Practice, as revised to date, govern the fabrication and erection of structural steel. 1.2. Definitions AISC Specification—The Specification for the Design, Fabrication and Erection of Structural Steel for Buildings as adopted by the American Institute of Steel Construction. ANSI—American National Standards Institute. Architect/Engineer—The owner’s designated representative with full responsibility for the design and integrity of the structure. (The EOR) ASTM—The material standard of the American Society for Testing and Materials. AWS Code—The Structural Welding Code of the American Welding Society. Code—The Code of Standard Practice as adopted by the American Institute of Steel Construction. Contract Documents—The documents which define the responsibilities of the parties involved in bidding, purchasing, supplying and erecting structural steel. Such documents normally consist of a contract, plans and specifications. Drawings—Shop and field erection drawings prepared by the fabricator and erector for the performance of the work. Erector—The party responsible for the erection of the structural steel. Fabricator—The party responsible for furnishing fabricated structural steel. General Contractor—The owner’s designated representative with full responsibility for the construction of the structure. MBMA—Metal Building Manufacturers Association.


6 - 426

AISC CODE OF STANDARD PRACTICE

Mill Material—Steel mill products ordered expressly for the requirements of a specific project. Owner—The owner of the proposed structure or his designated representatives, who may be the architect, engineer, general contractor, construction manager, public authority or others. Owner’s Authorized Representative—That person designated by the owner to have the responsibility for the approval of shop drawings. This is usually the structural engineer of record for the project. Plans—Design drawings furnished by the party responsible for the design of the structure. Release for Construction—The release by the owner permitting the fabricator to commence work under the contract, including ordering material and preparing shop drawings. SSPC—The Steel Structures Painting Council, publishers of the Steel Structures Painting Manual, Vol. 2, “Systems and Specifications.” Tier—The word Tier used in Section 7.11 is defined as a column shipping piece. 1.3. Design Criteria for Buildings and Similar Type Structures In the absence of other instructions, the provisions of the AISC Specification govern the design of the structural steel. 1.4. Design for Bridges In the absence of other instructions, the following provisions govern, as applicable: Standard Specifications for Highway Bridges of the American Association of State Highway and Transportation Officials Specifications for Steel Railway Bridges of the American Railway Engineering Association Structural Welding Code of the American Welding Society 1.5. Responsibility for Design 1.5.1. When the owner provides the design, plans and specifications, the fabricator and erector are not responsible for the suitability, adequacy or legality of the design. The fabricator is not responsible for the safety of erection if the structure is erected by others. 1.5.2. When the owner enters into a direct contract with the fabricator to both design and fabricate an entire, completed steel structure, the fabricator is responsible for the structural adequacy of the design. The fabricator is not responsible for the safety of erection if the structure is erected by others.


STEEL BUILDINGS AND BRIDGES

6 - 427

1.6. Patented Devices Except when the contract documents call for the design to be furnished by the fabricator or erector, the fabricator and erector assume that all necessary patent rights have been obtained by the owner and that the fabricator or erector will be fully protected in the use of patented designs, devices or parts required by the contract documents.

SECTION 2.0. CLASSIFICATION OF MATERIALS 2.1. Definition of Structural Steel “Structural Steel,” as used to define the scope of work in the contract documents, consists of the steel elements of the structural steel frame essential to support the design loads. Unless otherwise specified in the contract documents, these elements consist of material as shown on the structural steel plans and described as: Anchor bolts for structural steel Base or bearing plates Beams, girders, purlins and girts Bearings of steel for girders, trusses or bridges Bracing Columns, posts Connecting materials for framing structural steel to structural steel Crane rails, splices, stops, bolts and clamps Door frames constituting part of the structural steel frame Expansion joints connected to the structural steel frame Fasteners for connecting structural steel items: Shop rivets Permanent shop bolts Shop bolts for shipment Field rivets for permanent connections Field bolts for permanent connections Permanent pins Floor plates (checkered or plain) attached to the structural steel frame Grillage beams and girders Hangers essential to the structural steel frame Leveling plates, wedges, shims & leveling screws Lintels, if attached to the structural steel frame Marquee or canopy framing Machinery foundations of rolled steel sections and/or plate attached to the structural frame AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 428


Monorail elements of standard structural shapes when attached to the structural frame Roof frames of standard structural shapes Shear connectors—if specified to be shop attached Struts, tie rods and sag rods forming part of the structural steel frame Trusses. 2.2. Other Steel or Metal Items The classification “Structural Steel,” does not include steel, iron or other metal items not generally described in Section 2.1, even when such items are shown on the structural steel plans or are attached to the structural frame. These items include but are not limited to: Cables for permanent bracing or suspension systems Chutes and hoppers Cold-formed steel products Concrete or masonry reinforcing steel Door and corner guards Embedded steel parts in precast or poured concrete Flagpole support steel Floor plates (checkered or plain) not attached to the structural steel frame Grating and metal deck Items required for the assembly or erection of materials supplied by trades other than structural steel fabricators or erectors Ladders and safety cages Lintels over wall recesses Miscellaneous metal Non-steel bearings Open-web, long-span joists and joist girders Ornamental metal framing Shear connectors — if specified to be field installed Stacks, tanks and pressure vessels Stairs, catwalks, handrail and toeplates Trench or pit covers.



6 - 429

SECTION 3. PLANS AND SPECIFICATIONS 3.1. Structural Steel In order to insure adequate and complete bids, and to enable the timely preparation of shop drawings and timely fabrication, the fabricator must be able to rely upon the completeness of the contract documents. The contract documents can be assumed to provide complete structural steel design plans clearly showing the work to be performed and giving the size, section, material grade and the location of all members, floor levels, column centers and offsets, and camber of members, with sufficient dimensions to convey accurately the quantity and nature of the structural steel to be furnished. Structural steel specifications include any special requirements controlling the fabrication and erection of the structural steel. Contract drawings, specifications and addenda must be numbered and dated for purposes of identification. 3.1.1. Wind bracing, connections, column stiffeners, column web doubler plates, bearing stiffeners on beams and girders, web reinforcement, openings for other trades, and other special details where required are shown in sufficient detail so that they may be readily understood. 3.1.2. Plans include sufficient data concerning assumed loads, shears, moments and axial forces to be resisted by the individual members and their connections, as may be required for the development of connection details on the shop drawings. Unless otherwise indicated in the contract documents, the plans are based upon consideration of the loads and forces to be resisted by the steel frame in the completed and fully connected condition. See Section 7.9. 3.1.3. Where connections are not shown, the connections are to be in accordance with the requirements of the AISC Specification. 3.1.4. When loose lintels and leveling plates are required to be furnished as part of the contract requirements, the plans and specifications show the size, section and location of all pieces. 3.1.5. Whenever steel frames, in the completely erected and fully connected state, require interaction with other elements not classified as structural steel (see Section 2) to provide stability and strength to resist loads for which the frame is designed, the non-self-supporting frame and the major elements not classified as structural steel, such as diaphragms, masonry and/or concrete shear walls, shall be identified in the contract documents. See Section 7.9.3. 3.1.6. When camber is required for cantilevered members, the magnitude and direction of camber are shown.


6 - 430


3.1.7. The contract documents specify all the painting requirements, including the identification of members to be painted, surface preparation, paint specifications, manufacturer’s product identification and the required minimum and maximum dry film thickness, in mils, of the shop coat. Contract documents must clearly indicate all individual members which are to be left unpainted so as to receive concrete, sprayed on fireproofing or for other reasons. 3.2. Architectural, Electrical and Mechanical Architectural, electrical and mechanical plans may be used as a supplement to the structural steel plans to define detail configurations and construction information, provided all requirements for the quantities and locations of structural steel are noted on the structural steel plans. 3.3. Discrepancies In case of discrepancies between plans and specifications for buildings, the specifications govern. In case of discrepancies between plans and specifications for bridges, the plans govern. In case of discrepancies between scale dimensions on the plans and figures written on them, the figures govern. In case of discrepancies between the structural steel plans and the architectural plans or plans for other trades, the structural steel plans govern. 3.4. Legibility of Plans Plans are clearly legible and made to a scale not less than 1⁄8 in. to the foot. More complex information is furnished to an adequate scale to convey the information clearly. 3.5. Special Conditions When it is required that a project be advertised for bidding before the requirements of Section 3.1 can be met, the owner must provide sufficient information in the form of scope, drawings, weights, outline specifications, and other descriptive data to enable the fabricator and erector to prepare a knowledgeable bid.



6 - 431

SECTION 4. SHOP AND ERECTION DRAWINGS 4.1. Owner Responsibility To enable the fabricator and erector to properly and expeditiously proceed with the work, the owner must furnish, in a timely manner and in accordance with the contract documents, complete structural steel plans and specifications released for construction. “Released for construction” plans and specifications are required by the fabricator for ordering mill material and the preparation and completion of shop and erection drawings. Plans provided as part of a contract bid package are considered to be “released for construction” unless otherwise noted. 4.2. Approval When shop drawings are made by the fabricator, prints thereof are submitted to the owner for his examination and approval. The fabricator includes a maximum allowance of fourteen (14) calendar days in his schedule for the return of shop drawings. Return of shop drawings is noted with the owner’s approval, or approval subject to corrections as noted. The fabricator makes the corrections and furnishes corrected prints to the owner. Approval of shop drawings, approval “subject to corrections noted,” or similar approvals, constitute the owner’s release for the fabricator to begin fabrication. The fabricator retains flexibility to determine the fabrication schedule necessary to meet the project’s requirements. 4.2.1. Approval by the owner’s authorized representative of shop drawings prepared by the fabricator indicates that the fabricator has correctly interpreted the contract requirements, and may rely upon these drawings in the fabrication process. Where the fabricator must select or complete connection details, this approval constitutes acceptance by the owner’s authorized representative of design responsibility for the structural adequacy of such connections. If a fabricator wishes to change a connection that is fully detailed in the contract documents, the fabricator shall submit the change for review by the owner’s authorized representative in a manner that clearly indicates that a change is being requested. Approval of this submittal constitutes acceptance by the owner’s authorized representative of design responsibility for the structural adequacy of the changed detail. Approval under any of the circumstances described in this Section does not relieve the fabricator of the responsibility for accuracy of detailed dimensions on shop drawings, nor the general fit-up of parts to be assembled in the field. 4.2.2. Unless specifically stated to the contrary, any additions, deletions or changes indicated on the approval of shop and erection drawings are authorizations by the owner to release the additions, deletions or revisions for construction.


6 - 432


4.3. Drawings Furnished by Owner When the shop drawings are furnished by the owner, he must deliver them to the fabricator in time to permit material procurement and fabrication to proceed in an orderly manner in accordance with the prescribed time schedule. The owner prepares these shop drawings, insofar as practicable, in accordance with the shop and drafting room standards of the fabricator. The owner is responsible for the completeness and accuracy of shop drawings so furnished.

SECTION 5. MATERIALS 5.1. Mill Materials When the fabricator receives “released for construction” plans and specifications, the fabricator may immediately place orders for the materials necessary for fabrication. The contract documents must note any material or areas which should not be ordered due to a design which is incomplete or subject to revision. 5.1.1. Mill tests are performed to demonstrate material conformance to ASTM specifications in accordance with the contract requirements. Unless special requirements are included in the contract documents, mill testing is limited to those tests required by the applicable ASTM material specifications. Mill test reports are furnished by the fabricator only if requested by the owner, either in the contract documents or in separate written instructions prior to the time the fabricator places his material orders with the mill. 5.1.2. When material received from the mill does not satisfy ASTM A6 tolerances for camber, profile, flatness or sweep, the fabricator is permitted to perform corrective work by the use of controlled heating and mechanical straightening, subject to the limitations of the AISC Specification. 5.1.3. Corrective procedures described in ASTM A6 for reconditioning the surface of structural steel plates and shapes before shipment from the producing mill may also be performed by the fabricator, at the fabricator’s option, when variations described in ASTM A6 are discovered or occur after receipt of the steel from the producing mill. 5.1.4. When special requirements demand tolerances more restrictive than allowed by ASTM A6, such requirements are defined in the contract documents and the fabricator has the option of corrective measures as described above.



6 - 433

5.2. Stock Materials 5.2.1. Many fabricators maintain stocks of steel products for use in their fabricating operations. Materials taken from stock by the fabricator to be used for structural purposes must be of a quality at least equal to that required by the ASTM specifications applicable to the classification covering the intended use. 5.2.2. Mill test reports are accepted as sufficient record of the quality of materials carried in stock by the fabricator. The fabricator reviews and retains the mill test reports covering the materials he purchases for stock, but the fabricator does not maintain records that identify individual pieces of stock material against individual mill test reports. Such records are not required if the fabricator purchases for stock under established specifications as to grade and quality. 5.2.3. Stock materials purchased under no particular specifications or under specifications less rigid than those mentioned above, or stock materials which have not been subject to mill or other recognized test reports, are not used without the express approval of the owner, except where the quality of the material could not affect the integrity of the structure.

SECTION 6. FABRICATION AND DELIVERY 6.1. Identification of Material 6.1.1. High strength steel and steel ordered to special requirements is marked by the supplier, in accordance with ASTM A6 requirements, prior to delivery to the fabricator’s shop or other point of use. 6.1.2. High strength steel and steel ordered to special requirements that has not been marked by the supplier in accordance with Section 6.1.1 is not used until its identification is established by means of tests as specified in Section A3.1 of the AISC Specification, and until a fabricator’s identification mark, as described in Section 6.1.3, has been applied. 6.1.3. During fabrication, up to the point of assembling members, each piece of high strength steel and steel ordered to special requirements carries a fabricator’s identification mark or an original supplier’s identification mark. The fabricator’s identification mark is in accordance with the fabricator’s established identification system, which is on record and available for the information of the owner or his representative, the building commissioner and the inspector, prior to the start of fabrication.


6 - 434


6.1.4. Members made of high strength steel and steel ordered to special requirements are not given the same assembling or erecting mark as members made of other steel, even though they are of identical dimensions and detail. 6.2. Preparation of Material 6.2.1. Thermal cutting of structural steel may be performed by hand or mechanically guided means. 6.2.2. Surfaces noted as “finished” on the drawings are defined as having a maximum ANSI roughness height value of 500. Any fabricating technique, such as friction sawing, cold sawing, milling, etc., that produces such a finish may be used. 6.3. Fitting and Fastening 6.3.1. Projecting elements of connection attachments need not be straightened in the connecting plane if it can be demonstrated that installation of the connectors or fitting aids will provide reasonable contact between faying surfaces. 6.3.2. Runoff tabs are often required to produce sound welds. The fabricator or erector does not remove them unless specified in the contract documents. When their removal is required, they may be hand flame-cut close to the edge of the finished member with no further finishing required, unless other finishing is specifically called for in the contract documents. 6.3.3. All high-strength bolts for shop attached connection material are to be installed in the shop in accordance with the Specification for Structural Joints Using A325 or A490 Bolts, unless otherwise noted on the shop drawings. 6.4. Dimensional Tolerances 6.4.1. A variation of 1⁄32 in. is permissible in the overall length of members with both ends finished for contact bearing as defined in Section 6.2.2. 6.4.2. Members without ends finished for contact bearing, which are to be framed to other steel parts of the structure, may have a variation from the detailed length not greater than 1⁄16 in. for members 30 ft or less in length, and not greater than 1⁄8 in. for members over 30 ft in length. 6.4.3. Unless otherwise specified, structural members, whether of a single-rolled shape or built-up, may vary from straightness within the tolerances allowed for wideflange shapes by ASTM Specification A6, except that the tolerance on deviation from



6 - 435

straightness of compression members is 1⁄1000 of the axial length between points which are to be laterally supported. Completed members should be free from twists, bends and open joints. Sharp kinks or bends are cause for rejection of material. 6.4.4. Beams and trusses detailed without specified camber are fabricated so that after erection any camber due to rolling or shop fabrication is upward. 6.4.5. When members are specified on the contract documents as requiring camber, the shop fabrication tolerance shall be minus zero / plus 1⁄2 in. for members 50 ft and less in length, or minus zero / plus (1⁄2 in. plus 1⁄8 in. for each 10 ft or fraction thereof in excess of 50 ft in length) for members over 50 ft. Members received from the rolling mill with 75% of the specified camber require no further cambering. For purposes of inspection, camber must be measured in the fabricator’s shop in the unstressed condition. 6.4.6. Any permissible deviation in depths of girders may result in abrupt changes in depth at splices. Any such difference in depth at a bolted joint, within the prescribed tolerances, is taken up by fill plates. At welded joints the weld profile may be adjusted to conform to the variation in depth, provided that the minimum cross section of required weld is furnished and that the slope of the weld surface meets AWS Code requirements. 6.5. Shop Painting (See also Section 3.1.7.) 6.5.1. The shop coat of paint is the prime coat of the protective system. It protects the steel for only a short period of exposure in ordinary atmospheric conditions, and is considered a temporary and provisional coating. The fabricator does not assume responsibility for deterioration of the prime coat that may result from exposure to ordinary atmospheric conditions, nor from exposure to corrosive conditions more severe than ordinary atmospheric conditions. 6.5.2. In the absence of other requirements in the contract documents, the fabricator hand cleans the steel of loose rust, loose mill scale, dirt and other foreign matter, prior to painting, by means of wire brushing or by other methods elected by the fabricator, to meet the requirements of SSPC-SP2. The fabricator’s workmanship on surface preparation is considered accepted by the owner unless specifically disapproved prior to paint application. 6.5.3. Unless specifically excluded, paint is applied by brush, spray, roller coating, flow coating or dipping, at the election of the fabricator. When the term “shop coat” or “shop paint” is used with no paint system specified, the fabricator’s standard paint shall be applied to a minimum dry film thickness of one mil.


6 - 436


6.5.4. Steel not requiring shop paint is cleaned of oil or grease by solvent cleaners and cleaned of dirt and other foreign material by sweeping with a fiber brush or other suitable means. 6.5.5. Abrasions caused by handling after painting are to be expected. Touch-up of these blemished areas is the responsibility of the contractor performing field touchup or field painting. 6.6. Marking and Shipping of Materials 6.6.1. Erection marks are applied to the structural steel members by painting or other suitable means, unless otherwise specified in the contract documents. 6.6.2. Rivets and bolts are commonly shipped in separate containers according to length and diameter; loose nuts and washers are shipped in separate containers according to sizes. Pins and other small parts, and packages of rivets, bolts, nuts and washers are usually shipped in boxes, crates, kegs or barrels. A list and description of the material usually appears on the outside of each closed container. 6.7. Delivery of Materials 6.7.1. Fabricated structural steel is delivered in such sequence as will permit the most efficient and economical performance of both shop fabrication and erection. If the owner wishes to prescribe or control the sequence of delivery of materials, the owner reserves such right and defines the requirements in the contract documents. If the owner contracts separately for delivery and erection, the owner must coordinate planning between contractors. 6.7.2. Anchor bolts, washers and other anchorage or grillage materials to be built into masonry should be shipped so that they will be on hand when needed. The owner must allow the fabricator sufficient time to fabricate and ship such materials before they are needed. 6.7.3. The quantities of material shown by the shipping statement are customarily accepted by the owner, fabricator and erector as correct. If any shortage is claimed, the owner or erector should immediately notify the carrier and the fabricator in order that the claim may be investigated. 6.7.4. The size and weight of structural steel assemblies may be limited by shop capabilities, the permissible weight and clearance dimensions of available transportation and the job site conditions. The fabricator limits the number of field splices to those consistent with minimum project cost.



6 - 437

6.7.5. If material arrives at its destination in damaged condition, it is the responsibility of the receiving party to promptly notify the fabricator and carrier prior to unloading the material, or immediately upon discovery.

SECTION 7. ERECTION 7.1. Method of Erection When the owner wishes to control the method and sequence of erection, or when certain members cannot be erected in their normal sequence, the owner so specifies in the contract documents. In the absence of such restrictions, the erector will proceed using the most efficient and economical method and sequence available to the erector consistent with the contract documents. When the owner contracts separately for fabrication and erection services, the owner is responsible for coordinating planning between contractors. 7.2. Site Conditions The owner provides and maintains adequate access roads into and through the site for the safe delivery and movement of derricks, cranes, trucks, other necessary equipment, and the material to be erected. The owner affords the erector a firm, properly graded, drained, convenient and adequate space at the site for the operation of the erector’s equipment, and removes all overhead obstructions such as power lines, telephone lines, etc., in order to provide a safe working area for erection of the steelwork. The erector provides and installs the safety protection required for his own work. Any protection for other trades not essential to the steel erection activity is the responsibility of the owner. When safety protection provided by the erector is left remaining in an area to be used by other trades after the steel erection activity is completed, the owner shall be responsible for accepting and maintaining this protection, assuring that it is adequate for the protection of all other affected trades, assuring that it complies with all applicable safety regulations when being used by other trades, indemnifying the erector from any damages incurred as a result of the safety protection’s use by other trades, removing the safety equipment when no longer required, and returning it to the erector in the same condition as it was received. When the structure does not occupy the full available site, the owner provides adequate storage space to enable the fabricator and erector to operate at maximum practicable speed. 7.3. Foundations, Piers and Abutments The accurate location, strength, suitability and access to all foundations, piers and abutments is the sole responsibility of the owner.


6 - 438


7.4. Building Lines and Bench Marks The owner is responsible for accurate location of building lines and bench marks at the site of the structure, and for furnishing the erector with a plan containing all such information. At each level the owner establishes offset building lines and reference elevations for the use of the erector in the positioning of adjustable construction elements. 7.5. Installation of Anchor Bolts and Embedded Items 7.5.1. Anchor bolts and foundation bolts are set by the owner in accordance with an approved drawing. They must not vary from the dimensions shown on the erection drawings by more than the following: (a)

(b) (c) (d)

(e) (f)

1⁄ 8

in. center to center of any two bolts within an anchor bolt group, where an anchor bolt group is defined as the set of anchor bolts which receive a single fabricated steel shipping piece. 1⁄ in. center to center of adjacent anchor bolt groups. 4 Elevation of the top of anchor bolts ± 1⁄2 in. Maximum accumulation of 1⁄4 in. per hundred ft along the established column line of multiple anchor bolt groups, but not to exceed a total of 1 in., where the established column line is the actual field line most representative of the centers of the as-built anchor bolt groups along a line of columns. 1⁄ in. from the center of any anchor bolt group to the established column 4 line through that group. The tolerances of paragraphs b, c and d apply to offset dimensions shown on the plans, measured parallel and perpendicular to the nearest established column line for individual columns shown on the plans to be offset from established column lines.

7.5.2. Unless shown otherwise, anchor bolts are set perpendicular to the theoretical bearing surface. 7.5.3. Other embedded items or connection materials between the structural steel and the work of other trades are located and set by the owner in accordance with approved location or erection drawings. Accuracy of these items must satisfy the erection tolerance requirements of Section 7.11.3. 7.5.4. All work performed by the owner is completed so as not to delay or interfere with the erection of the structural steel.



6 - 439

7.6. Bearing Devices The owner sets to line and grade all leveling plates, leveling nuts and loose bearing plates which can be handled without a derrick or crane. All other bearing devices supporting structural steel are set and wedged, shimmed or adjusted with leveling screws by the erector to lines and grades established by the owner. The fabricator provides the wedges, shims or leveling screws that are required, and clearly scribes the bearing devices with working lines to facilitate proper alignment. Promptly after the setting of any bearing devices, the owner checks lines and grades, and grouts as required. The final location and proper grouting of bearing devices are the responsibility of the owner. Tolerance on elevation relative to established grades of bearing devices, whether set by the owner or by the erector, is ± 1⁄8 in. 7.7. Field Connection Material 7.7.1. The fabricator provides field connection details consistent with the requirements of the contract documents which will, in the fabricator’s opinion, result in the most economical fabrication and erection cost. 7.7.2. When the fabricator erects the structural steel, the fabricator supplies all materials required for temporary and permanent connection of the component parts of the structural steel. 7.7.3. When the erection of the structural steel is performed by someone other than the fabricator, the fabricator furnishes the following field connection material: (a)

(b)

(c) (d)

Bolts of required size and in sufficient quantity for all field connections of steel to steel which are to be permanently bolted. Unless high-strength bolts or other special types of bolts and washers are specified, common bolts are furnished. An extra 2 percent of each bolt size (diameter and length) is furnished. Rivets of required size and in sufficient quantity for all field connections of steel to steel which are to be riveted field connections. An extra 10 percent of each rivet size is furnished. Shims shown as necessary for make-up of permanent connections of steel to steel. Back-up bars or run-off tabs that may be required for field welding.

7.7.4. When the erection of the structural steel is performed by someone other than the fabricator, the erector furnishes all welding electrodes, fit-up bolts and drift pins used for erection of the structural steel.


6 - 440


7.7.5. Field-installed shear connectors are supplied by the shear connector applicator. 7.7.6. Metal deck support angles are the responsibility of the metal deck supplier. 7.8. Loose Material Loose items of structural steel not connected to the structural frame are set by the owner without assistance from the erector, unless otherwise specified in the contract documents. 7.9. Temporary Support of Structural Steel Frames 7.9.1. General Temporary supports, such as temporary guys, braces, falsework, cribbing or other elements required for the erection operation will be determined and furnished and installed by the erector. These temporary supports will secure the steel framing, or any partly assembled steel framing, against loads comparable in intensity to those for which the structure was designed, resulting from wind, seismic forces and erection operations, but not the loads resulting from the performance of work by or the acts of others, nor such unpredictable loads as those due to tornado, explosion or collision. 7.9.2. Self-supporting Steel Frames A self-supporting steel frame is one that provides the required stability and resistance to gravity loads and design wind and seismic forces without interaction with other elements of the structure. The erector furnishes and installs only those temporary supports that are necessary to secure any element or elements of the steel framing until they are made stable without external support. Special erection sequences or other considerations which are required to provide stability during the erection process must be set out in the contract documents in detail. 7.9.3. Non-Self-supporting Steel Frames A non-self-supporting steel frame is one that, when fully assembled and connected, requires interaction with other elements not classified as Structural Steel to provide stability and strength to resist loads for which the frame is designed. Such frames shall be clearly designated as “non-self-supporting.” The major elements not classified as structural steel, such as steel deck diaphragms, masonry and/or concrete shear walls, shall be identified in the contract documents.



6 - 441

When elements not classified as structural steel interact with the structural steel elements to provide stability and/or strength to resist loads, the owner is responsible for the installation, structural adequacy during installation, and timely completion of all such elements. The contract documents must specify the sequence and schedule of placement of such elements and the effects of the loads imposed on the structural steel frame by partially or completely installed interacting elements. The erector furnishes and installs temporary support as necessary in accordance with this information but does not thereby assume responsibility for the appropriateness of the sequence specified. 7.9.4. Special Erection Conditions When the design concept of a structure is dependent upon the use of shores, jacks or loads which must be adjusted as erection progresses to set or maintain camber or prestress, such requirement is specifically stated in the contract documents. 7.9.5. Removal of Temporary Supports The temporary guys, braces, falsework, cribbing and other elements required for the erection operation, which are furnished and installed by the erector, are not the property of the owner. In self-supporting structures, temporary supports are not required after the structural steel for a self-supporting element is located and finally fastened within the required tolerances. After such final fastening, the erector is no longer responsible for temporary support of the self-supporting element and may remove the temporary supports. In non-self-supporting structures, the erector may remove temporary supports when the necessary non-structural steel elements are complete. Temporary supports are not to be removed without the consent of the erector. At completion of steel erection, any temporary supports that are required to be left in place are removed by the owner and returned to the erector in good condition. 7.9.6. Temporary Supports for Other Work Should temporary supports beyond those defined as the responsibility of the erector in Sections 7.9.1, 7.9.2 and 7.9.3 be required, either during or after the erection of the structural steel, responsibility for the supply and installation of such supports rests with the owner. 7.10. Temporary Floors and Handrails for Buildings The erector provides floor coverings, handrails and walkways as required by law and applicable safety regulations for protection of his own personnel. As work progresses, the erector removes such facilities from units where the erection


6 - 442


operations are completed, unless other arrangements are included in the contract documents. The owner is responsible for all protection necessary for the work of other trades. When permanent steel decking is used for protective flooring and is installed by the owner, all such work is performed so as not to delay or interfere with erection progress and is scheduled by the owner and installed in a sequence adequate to meet all safety regulations. (See Section 7.2) 7.11. Frame Tolerances 7.11 7.11.1. Overall Dimensions Some variation is to be expected in the finished overall dimensions of structural steel frames. Such variations are deemed to be within the limits of good practice when they do not exceed the cumulative effect of rolling tolerances, fabricating tolerances and erection tolerances. 7.11.2. Working Points and Working Lines Erection tolerances are defined relative to member working points and working lines as follows: (a) (b) (c) (d)

For members other than horizontal members, the member work point is the actual center of the member at each end of the shipping piece. For horizontal members, the working point is the actual center line of the top flange or top surface at each end. Other working points may be substituted for ease of reference, providing they are based upon these definitions. The member working line is a straight line connecting the member working points.

7.11.3. Position and Alignment The tolerances on position and alignment of member working points and working lines are as follows: 7.11.3.1. Columns Individual column shipping pieces are considered plumb if the deviation of the working line from a plumb line does not exceed 1:500, subject to the following limitations:



(a)

(b)

(c)

(d)

6 - 443

The member working points of column shipping pieces adjacent to elevator shafts may be displaced no more than 1 in. from the established column line in the first 20 stories; above this level, the displacement may be increased 1⁄32 in. for each additional story up to a maximum of 2 in. The member working points of exterior column shipping pieces may be displaced from the established column line no more than 1 in. toward nor 2 in. away from the building line in the first 20 stories; above the 20th story, the displacement may be increased 1⁄16 in. for each additional story, but may not exceed a total displacement of 2 in. toward nor 3 in. away from the building line. The member working points of exterior column shipping pieces at any splice level for multi-tier buildings and at the tops of columns for single tier buildings may not fall outside a horizontal envelope, parallel to the building line, 11⁄2 in. wide for buildings up to 300 ft in length. The width of the envelope may be increased by 1⁄2 in. for each additional 100 ft in length, but may not exceed 3 in. The member working points of exterior column shipping pieces may be displaced from the established column line, in a direction parallel to the building line, no more than 2 in. in the first 20 stories; above the 20th story, the displacement may be increased 1⁄16 in. for each additional story, but may not exceed a total displacement of 3 in. parallel to the building line.

7.11.3.2. Members Other Than Columns (a)

(b)

(c)

(d)

Alignment of members which consist of a single straight shipping piece containing no field splices, except cantilevered members, is considered acceptable if the variation in alignment is caused solely by the variation of column alignment and/or primary supporting member alignment within the permissible limits for fabrication and erection of such members. The elevation of members connecting to columns is considered acceptable if the distance from the member working point to the upper milled splice line of the column does not deviate more than plus 3⁄16 in. or minus 5⁄16 in. from the distance specified on the drawings. The elevation of members which consist of a single shipping piece, other than members connected to columns, is considered acceptable if the variation in actual elevation is caused solely by the variation in elevation of the supporting members which are within permissible limits for fabrication and erection of such members. Individual shipping pieces which are segments of field assembled units containing field splices between points of support are considered plumb, level and aligned if the angular variation of the working line of each shipping piece relative to the plan alignment does not exceed 1:500.


6 - 444

(e)

(f)


The elevation and alignment of cantilevered members shall be considered plumb, level and aligned if the angular variation of the working line from a straight line extended in the plan direction from the working point at its supported end does not exceed 1:500. The elevation and alignment of members which are of irregular shape shall be considered plumb, level and aligned if the fabricated member is within its tolerance and its supporting member or members are within the tolerances specified in this Code.

7.11.3.3. Adjustable Items The alignment of lintels, wall supports, curb angles, mullions and similar supporting members for the use of other trades, requiring limits closer than the foregoing tolerances, cannot be assured unless the owner’s plans call for adjustable connections of these members to the supporting structural frame. The fabricator may provide nonadjustable connections unless the contract documents specifically show or specify them as adjustable. When adjustable connections are specified, the owner’s plans must provide for the total adjustment required to accommodate the tolerances on the steel frame for the proper alignment of these supports for other trades. The tolerances on position and alignment of such adjustable items are as follows: (a)

(b)

(c)

Adjustable items are considered to be properly located in their vertical position when their location is within 3⁄8 in. of the location established from the upper milled splice line of the nearest column to the support location as specified on the drawings. Adjustable items are considered to be properly located in their horizontal position when their location is within 3⁄8 in. of the proper location relative to the established finish line at any particular floor. The ends of adjustable items which abut are considered to be properly located when aligned to within 3⁄16 in. of each other both vertically and horizontally.

7.11.4. Responsibility for Clearances In the design of steel structures, the owner is responsible for providing clearances and adjustments of material furnished by other trades to accommodate all of the foregoing tolerances of the structural steel frame. 7.11.5. Acceptance of Position and Alignment Prior to placing or applying any other materials, the owner is responsible for determining that the location of the structural steel is acceptable for plumbness, level and alignment within tolerances. The erector is given timely notice of acceptance by the owner or a listing of specific items to be corrected in order to obtain acceptance.



6 - 445

Such notice is rendered immediately upon completion of any part of the work and prior to the start of work by other trades that may be supported, attached or applied to the structural steelwork. 7.12. Correction of Errors Normal erection operations include the correction of minor misfits by moderate amounts of reaming, chipping, welding or cutting, and the drawing of elements into line through the use of drift pins. Errors which cannot be corrected by the foregoing means, or which require major changes in member configuration, are reported immediately to the owner and fabricator by the erector, to enable whoever is responsible either to correct the error or to approve the most efficient and economic method of correction to be used by others. 7.13. Cuts, Alterations and Holes for Other Trades Neither the fabricator nor the erector will cut, drill or otherwise alter his work, or the work of other trades, to accommodate other trades, unless such work is clearly specified in the contract documents. Whenever such work is specified, the owner is responsible for furnishing complete information as to materials, size, location and number of alterations in a timely manner so that the preparation of shop drawings will not be delayed. 7.14. Handling and Storage The erector takes reasonable care in the proper handling and storage of steel during erection operations to avoid accumulation of excess dirt and foreign matter. The erector is not responsible for removal from the steel of dust, dirt or other foreign matter which accumulates during the erection period as the result of site conditions or exposure to the elements. 7.15.

Field Painting

The erector does not paint field bolt heads and nuts, field rivet heads and field welds, nor touch up abrasions of the shop coat, nor perform any other field painting. 7.16. Final Cleaning Up Upon completion of erection and before final acceptance, the erector removes all of his falsework, rubbish and temporary buildings.


6 - 446


SECTION 8. QUALITY CONTROL 8.1. General 8.1.1. The fabricator maintains a quality control program to the extent deemed necessary so that the work is performed in accordance with this Code, the AISC Specification, and contract documents. The fabricator has the option to use the AISC Quality Certification Program in establishing and administering the quality control program. 8.1.2. The erector maintains a quality control program to the extent the erector deems necessary so that all of the work is performed in accordance with this Code, the AISC Specification and the contract documents. The erector shall be capable of performing the erection of the structural steel, and shall provide the equipment, personnel and management for the scope, magnitude and required quality of each project. 8.1.3. When the owner requires more extensive quality control or independent inspection by qualified personnel, or requires the fabricator to be certified by the AISC Quality Certification Program, this shall be clearly stated in the contract documents, including a definition of the scope of such inspection. 8.2. Mill Material Inspection The fabricator customarily makes a visual inspection, but does not perform any material tests, depending upon mill reports to signify that the mill product satisfies material order requirements. The owner relies on mill tests required by contract and on such additional tests as he orders the fabricator to have made at the owner’s expense. If mill inspection operations are to be monitored, or if tests other than mill tests are desired, the owner so specifies in the contract documents and should arrange for such testing through the fabricator to assure coordination. 8.3. Non-Destructive Testing When non-destructive testing is required, the process, extent, technique and standards of acceptance are clearly defined in the contract documents. 8.4. Surface Preparation and Shop Painting Inspection Surface preparation and shop painting inspection must be planned for acceptance of each operation as completed by the fabricator. Inspection of the paint system, including material and thickness, is made promptly upon completion of the paint



6 - 447

application. When wet film thickness is inspected, it must be measured during the application. 8.5. Independent Inspection When contract documents specify inspection by other than the fabricator’s and erector’s own personnel, both parties to the contract incur obligations relative to the performance of the inspection. 8.5.1. The fabricator and erector provide the inspector with access to all places where work is being done. A minimum of 24 hours notification is given prior to commencement of work. 8.5.2. Inspection of shop work by the owner or his representative is performed in the fabricator’s shop to the fullest extent possible. Such inspections should be in sequence, timely, and performed in such a manner as will not disrupt fabrication operations and will permit repair of non-conforming work prior to any required painting while the material is still in process in the fabrication shop. 8.5.3. Inspection of field work must be completed promptly so that corrections can be made without delaying the progress of the work. 8.5.4. Rejection of material or workmanship not in conformance with the contract documents may be made at any time during the progress of the work. However, this provision does not relieve the owner of his obligation for timely, in-sequence inspections. 8.5.5. Copies of all reports prepared by the owner’s inspection representative must be given to the fabricator and erector immediately after the inspection to allow any necessary corrective work to be performed in a timely manner. 8.5.6. The owner’s inspection representative may not suggest, direct, or approve the fabricator or erector to deviate from the contract documents or approved shop drawings, or approve such deviation, without the express written approval of the engineer of record or the person designated as the owner’s authorized representative.


6 - 448


SECTION 9. CONTRACTS 9.1. Types of Contracts 9.1.1. For contracts stipulating a lump sum price, the work required to be performed by the fabricator and erector is completely defined by the contract documents. 9.1.2. For contracts stipulating a price per pound, the scope of work, type of materials, character of fabrication, and conditions of erection are based upon the contract documents which must be representative of the work to be performed. 9.1.3. For contracts stipulating a price per item, the work required to be performed by the fabricator and erector is based upon the quantity and the character of items described in the contract documents. 9.1.4. For contracts stipulating unit prices for various categories of structural steel, the scope of the work required to be performed by the fabricator and erector is based upon the quantity, character and complexity of the items in each category as described in the contract documents. The contract documents must be representative of the work to be done in each category. 9.2. Calculation of Weights Unless otherwise set forth in the contract, on contracts stipulating a price per pound for fabricated structural steel delivered and/or erected, the quantities of materials for payment are determined by the calculation of gross weight of materials as shown on the shop drawings. 9.2.1. The unit weight of steel is assumed to be 490 pounds per cubic ft. The unit weight of other materials is in accordance with the manufacturer’s published data for the specific product. 9.2.2. The weights of shapes, plates, bars, steel pipe and structural tubing are calculated on the basis of shop drawings showing actual quantities and dimensions of material furnished, as follows: (a) (b)

The weight of all structural shapes, steel pipe and structural tubing is calculated using the nominal weight per ft and the detailed overall length. The weight of plates and bars is calculated using the detailed overall rectangular dimensions.



(c)

(d)

(e)

6 - 449

When parts can be economically cut in multiples from material of larger dimensions, the weight is calculated on the basis of the theoretical rectangular dimensions of the material from which the parts are cut. When parts are cut from structural shapes, leaving a non-standard section not useable on the same contract, the weight is calculated on the basis of the nominal unit weight of the section from which the parts are cut. No deductions are to be made for material removed by cuts, copes, clips, blocks, drilling, punching, boring, slot milling, planing or weld joint preparation.

9.2.3. The calculated weights of castings are determined from the shop drawings of the pieces. An allowance of 10 percent is added for fillets and overrun. Scale weights of rough castings may be used if available. 9.2.4. The items for which weights are shown in tables in the AISC Manual of Steel Construction are calculated on the basis of tabulated unit weights. 9.2.5. The weight of items not included in the tables in the AISC Manual of Steel Construction shall be taken from the manufacturers’ catalog and the manufacturers’ shipping weight shall be used. 9.2.6. The weight of shop or field weld metal and protective coatings is not included in the calculated weight for pay purposes. 9.3. Revisions to Contract Documents 9.3.1. Revisions to the contract are made by issuance of new documents or reissuance of existing documents. In either case, all revisions, including revisions communicated by annotation of shop or erection drawings, must be clearly and individually indicated and the documents dated and identified by revision number. All contract drawings shall be identified by the same drawing number throughout the duration of the job regardless of the revision. The engineer of record is responsible for reviewing the overall structural design to identify all components which will be affected by a change to any individual component. 9.3.2. A revision to the requirements of the contract documents is made by change order, extra work order, or notations on the shop and erection drawings when returned upon approval. 9.3.3. Unless specifically stated to the contrary, the issuance of a revision is authorization by the owner to release these documents for construction.


6 - 450


9.4. Contract Price Adjustment 9.4.1. When the scope of work and responsibilities of the fabricator and erector are changed from those previously established by the contract documents, an appropriate modification of the contract price is made. In computing the contract price adjustment, the fabricator and erector consider the quantity of work added or deleted, modifications in the character of the work, and the timeliness of the change with respect to the status of material ordering, detailing, fabrication and erection operations. 9.4.2. Requests for contract price adjustments are presented by the fabricator and erector in a timely manner and are accompanied by a description of the change in sufficient detail to permit evaluation and timely approval by the owner. 9.4.3. Price per pound and price per item contracts generally provide for additions or deletions to the quantity of work prior to the time work is released for construction. Changes to the character of the work, at any time, or additions and/or deletions to the quantity of the work after it is released for detailing, fabrication, or erection, may require a contract price adjustment. 9.5. Scheduling 9.5.1. The contract documents specify the schedule for the performance of the work. This schedule states when the “released for construction” plans will be issued and when the job site, foundations, piers and abutments will be ready, free from obstructions and accessible to the erector, so that erection can start at the designated time and continue without interference or delay caused by the owner or other trades. 9.5.2. The fabricator and erector have the responsibility to advise the owner, in a timely manner, of the effect any revision has on the contract schedule. 9.5.3. If the fabrication or erection is significantly delayed due to design revisions, or for other reasons which are the owner’s responsibility, the fabricator and erector are compensated for additional costs incurred. 9.6. Terms of Payment The terms of payment for the contract shall be outlined in the contract documents.



6 - 451

SECTION 10. ARCHITECTURALLY EXPOSED STRUCTURAL STEEL 10.1. Scope This section of the Code defines additional requirements which apply only to members specifically designated by the contract documents as “Architecturally Exposed Structural Steel” (AESS). All provisions of Sections 1 through 9 of the Code apply unless specifically modified in this section. AESS members or components are fabricated and erected with the care and dimensional tolerances indicated in this section. 10.2. Additional Information Required in Contract Documents (a) (b) (c)

Specific identification of members or components which are to be AESS. Fabrication and erection tolerances which are more restrictive than provided for in this section. Requirements, if any, of a test panel or components for inspection and acceptance standards prior to the start of fabrication.

10.3. Fabrication 10.3.1. Rolled Shapes Permissible tolerances for out-of-square or out-of-parallel, depth, width and symmetry of rolled shapes are as specified in ASTM Specification A6. No attempt to match abutting cross-sectional configurations is made unless specifically required by the contract documents. The as-fabricated straightness tolerances of members are one-half of the standard camber and sweep tolerances in ASTM A6. 10.3.2. Built-up Members The tolerances on overall profile dimensions of members made up from a series of plates, bars and shapes by welding are limited to the accumulation of permissible tolerances of the component parts as provided by ASTM Specification A6. The asfabricated straightness tolerances for the member as a whole are one-half the standard camber and sweep tolerances for rolled shapes in ASTM A6. 10.3.3. Weld Show-through It is recognized that the degree of weld show-through, which is any visual indication of the presence of a weld or welds on the side away from the viewer, is a function of weld size and material thickness. The members or components will be


6 - 452


acceptable as produced unless specific acceptance criteria for weld show-through are included in the contract documents. 10.3.4. Joints All copes, miters and butt cuts in surfaces exposed to view are made with uniform gaps of 1⁄8 in. if shown to be open joints, or in reasonable contact if shown without gap. 10.3.5. Welding Reasonably smooth and uniform as-welded surfaces are acceptable on all welds exposed to view. Butt and plug welds do not project more than 1⁄16 in. above the exposed surface. No finishing or grinding is required except where clearances or fit of other components may necessitate, or when specifically required by the contract documents. 10.3.6. Weathering Steel Members fabricated of weathering steel which are to be AESS shall not have erection marks or other painted marks on surfaces that are to be exposed in the completed structure. If cleaning other than SSPC-SP6 is required, these requirements shall be defined in the contract documents. 10.4. Delivery of Materials The fabricator uses special care to avoid bending, twisting or otherwise distorting individual members. 10.5. Erection 10.5.1. General The erector uses special care in unloading, handling and erecting the steel to avoid marking or distorting the steel members. Care is also taken to minimize damage to any shop paint. If temporary braces or erection clips are used, care is taken to avoid unsightly surfaces upon removal. Tack welds are ground smooth and holes are filled with weld metal or body solder and smoothed by grinding or filing. The erector plans and executes all operations in such a manner that the close fit and neat appearance of the structure will not be impaired.



6 - 453

10.5.2. Erection Tolerances Unless otherwise specifically designated in the contract documents, members and components are plumbed, leveled and aligned to a tolerance not to exceed one-half the amount permitted for structural steel. These erection tolerances for AESS require that the owner’s plans specify adjustable connections between AESS and the structural steel frame or the masonry or concrete supports, in order to provide the erector with means for adjustment. 10.5.3. Components with Concrete Backing When AESS is backed with concrete, it is the general contractor’s responsibility to provide sufficient shores, ties and strongbacks to assure against sagging, bulging, etc., of the AESS resulting from the weight and pressure of the wet concrete.


6 - 454


6 - 455

Commentary on the Code of Standard Practice for Steel Buildings and Bridges

Adopted Effective June 10, 1992 American Institute of Steel Construction, Inc.


6 - 456


6 - 457

PREFACE

This Commentary has been prepared to assist those who use the Code of Standard Practice in understanding the background, basis and intent of its provisions. Each section in the Commentary is referenced by corresponding sections in the Code. Not all sections of the Code are discussed; sections are covered only if it is believed that additional explanation may be helpful. While every precaution has been taken to insure that all data and information presented is as accurate as possible, the Institute cannot assume responsibility for errors or oversights in the information published herein or the use of the information published or incorporating such information in the preparation of detailed engineering plans. The figures are for illustrative purposes only and are not intended to be applicable to any actual design. The information should not replace the judgment of an experienced architect or engineer who has the responsibility of design for a specific structure.


6 - 458


6 - 459

Commentary on the Code of Standard Practice for Steel Buildings and Bridges Adopted Effective June 10, 1992 American Institute of Steel Construction, Inc.

SECTION 1. GENERAL PROVISIONS 1.1. Scope This Code is not applicable to metal building systems, which are the subject of standards published by the Metal Building Manufacturers Association in their Metal Building Systems Manual. AISC has not participated in the development of the MBMA code and, therefore, takes no position and is not responsible for any of its provisions. This Code is not applicable to standard steel joists, which are the subject of Recommended Code of Standard Practice for Steel Joists, published by the Steel Joist Institute. AISC has not participated in the development of the SJI code and, therefore, takes no position and is not responsible for any of its provisions.

SECTION 2. CLASSIFICATION OF MATERIALS 2.2. Other Steel or Metal Items These items include materials which may be supplied by the steel fabricator which require coordination between other material suppliers and trades. If they are to be supplied by the fabricator, they must be specifically called for and detailed in contract documents.


6 - 460


SECTION 3. PLANS AND SPECIFICATIONS 3.1. Structural Steel Project specifications vary greatly in complexity and completeness. There is a benefit to the owner if the specifications leave the contractor reasonable latitude in performing his work. However, critical requirements affecting the integrity of the structure, or necessary to protect the owner’s interest, must be covered in the contract documents. The following checklist is included for reference: Standard codes and specifications governing structural steelwork Material specifications Mill test reports Welded joint configuration Weld procedure qualification Bolting specifications Special requirements for work of other trades Runoff tabs Wind bracing Connections or data for connection development Column stiffeners Column web doubler plates Bearing stiffeners on beams and girders Web reinforcement Openings for other trades Surface preparation and shop painting Shop inspection Field inspection Non-destructive testing, including acceptance criteria Special requirements on delivery Special erection limitations Temporary bracing for non-self-supporting structures Special fabrication and erection tolerances for AESS Special pay weight provisions The structural steel plans must provide the elevations of all members as well as the dimensions to the centerline of all members (or the backs of angles or channels) relative to the grid lines, column centerline or other nearby members unless the locations of those members must be coordinated by the general contractor with the requirements of another trade. When the necessary dimensions are not given, the fabricator is not in a position to order material or start shop drawings in a timely manner and may be delayed while attempting to get the information.



6 - 461

SECTION 4. SHOP AND ERECTION DRAWINGS 4.1. Owner’s Responsibility The owner’s responsibility for the proper planning of the work and the communication of all facts of his particular project is a requirement of the Code, not only at the time of bidding, but also throughout the term of any project. The contract documents, including the plans and specification, are for the purpose of communication. It is the owner’s responsibility to properly define the scope of work, and to define information or items required and outlined in the plans and specifications. When the owner releases plans and specifications for construction, the fabricator and erector rely on the fact that these are the owner’s requirements for his project. The Code defines the owner as including a designated representative such as the architect, engineer or project manager, and when these representatives direct specific action to be taken, they are acting as and for the owner. On phased construction projects, to insure the orderly flow of material procurement, detailing, fabrication and erection activities, it is essential that designs are not continuously revised after progressive releases for construction are made. In essence, once a portion of a design is released for construction, the essential elements of that design should be “frozen” to assure adherence to the construction schedule or all parties should reach an understanding on the effects of future changes as they affect scheduled deliveries and added costs, if any. 4.2. Approval 4.2.1. From the inception of the Code of Standard Practice, AISC and the industry in general have recognized that the engineer of record is the only individual who has all the information necessary to evaluate the total impact of connection details on the overall structural design of the project. This authority has traditionally been exercised during the approval process for shop and erection drawings. The EOR has retained the final and total responsibility for the adequacy and safety of the entire structure since at least the 1927 edition of the Code of Standard Practice. In those instances where a fabricator develops the detailed configuration of connections during the preparation of shop drawings, the fabricator does not thereby become responsible for the structural integrity of that part of the overall structure. In the first issue of the Code, as printed in the first AISC Manual in 1927, this was stated as “Shop Drawings prepared by the Seller and approved by a representative of the Buyer shall be deemed the correct interpretation of the work to be done, but does not relieve the Seller of responsibility for the accuracy of details.” This statement was modified in the 1952 revision of the Code to read “...the owner must return one set of prints to the fabricator with a notation of the owner’s outright approval or approval subject to corrections as noted.” In 1972 the Code stated “Approval by the


6 - 462


owner of shop drawings prepared by the fabricator indicates that the fabricator has correctly interpreted the contract requirements, and that any connections designed by the fabricator are of adequate capacity for the design requirements.” The Code was again modified in 1976 saying “Approval by the owner of shop drawings prepared by the fabricator indicates that the fabricator has correctly interpreted the contract requirements. This approval constitutes the owner’s acceptance of all responsibility for the design adequacy of any connection designed by the fabricator as a part of his preparation of these shop drawings.” This statement was not changed in the 1986 revision of the Code. The current revision of Paragraph 4.2.1 of the Code is intended to clarify the use of the word “Owner.” Consequently, the term “owner” has been replaced by “owner’s authorized representative,” usually meaning the engineer of record. The continuing concept that the structural engineer of record is the sole individual who can best assure the safety of the completed structure has not been modified. This system has worked well for at least the past 65 years, and has achieved a commendable safety record where its principles have been steadfastly applied. In the preparation of contract drawings, the engineer of record (EOR) has two basic choices in the showing of connection details. The EOR may fully design and detail connections for all conditions. However, in order to allow the owner to benefit from the economies inherent in allowing the fabricator to choose the most efficient connections for the fabricator’s shop and erection processes, the EOR may allow the fabricator to select the types of connection and show them in complete detail on the shop drawings for the EOR’s approval. In either case, the approval of the shop drawings by the owner’s authorized representative constitutes acceptance by the owner’s authorized representative of design responsibility for the structural adequacy of the connections shown on the shop drawings. Contracts attempting to share or allocate design responsibility are strongly discouraged. Individual state codes and licensing requirements may vary widely in allowing such allocation of responsibility. Should the engineer of record elect to fully design and detail connections on the contract documents, the EOR has the obligation to show all fastener sizes, arrangement, quantities and grades, as well as all connection material and weld types, sizes and lengths for each individual member or part to be joined. All requirements for bracing details, stiffeners, doublers, web or cope reinforcement or similar items necessary for the completeness of the design must be sized and shown in complete detail. The fabricator is responsible for correctly reflecting this information in the preparation of shop drawings. Should the fabricator wish to deviate from these specific details or call a problem to the attention of the engineer of record, the fabricator must either do so in writing prior to the preparation of shop drawings, or clearly note the deviation on the drawings submitted for approval. This requirement is not intended in any way to negate the responsibility of the owner’s authorized representative to review completely each shop drawing for structural adequacy during the approval process. If the engineer of record does not show fully designed and detailed connections on the contract documents and allows the fabricator to select connection types when



6 - 463

detailing shop drawings, the contract documents must give all reactions, moments, or other forces required for each individual member of parts to be joined so that when preparing shop drawings, the fabricator’s detailers and checkers may determine the appropriate connection either by selection from tables shown in AISC publications or by simple calculation. The fabricator can assume that the reactions, moments or other forces given by the engineer are appropriate for the loads to be applied to the structure. All requirements for bracing details, stiffeners, doublers, web or cope reinforcement or similar items necessary for the completeness of the design must be shown in sufficient detail so as to allow the fabricator to submit an accurate estimate of cost at the time of bid. It is suggested that highly complex connections be fully designed on the contract documents or developed in a timely manner by the EOR after consulting with the fabricator regarding accepted, current and standard practices for fabrication and erection so that the detailing and fabricating processes will not be delayed. In the latter case, a pre-detailing meeting between the EOR and the fabricator may be appropriate to facilitate this exchange of information. In the event that design loads or other information necessary for development of connections is not shown on the contract documents, this information must be furnished to the fabricator in a timely manner. If the engineer of record elects to utilize typical details which must be interpreted or modified by the fabricator to meet conditions occurring in a structure, such interpretation is forwarded to the engineer of record for review and approval by way of detail or shop drawing submittals. Where state codes and licensing requirements allow fabricators to design and fabricate complete steel structures, and a fabricator has contracted to provide such services, submittals to the owner or applicable public reviewing authority will normally include only those documents customarily submitted by licensed design professionals on comparable projects within the same licensing jurisdiction.

SECTION 5. MATERIALS 5.1. Mill Materials The fabricator may purchase materials in stock lengths, exact lengths or multiples of exact lengths to suit the dimensions shown on the contract drawings. Such purchases will normally be job-specific in nature and may not be capable of being utilized on other projects or returned for full credit if subsequent design changes make these materials unsuitable for their originally intended use. The fabricator should be paid for these materials upon delivery from the mill, subject to appropriate additional payment or credit if subsequent unanticipated modification or reorder is required. Purchasing materials to exact lengths is not considered fabrication.


6 - 464


5.1.2. Mill dimensional tolerances are completely set forth as part of ASTM A6. Variation in cross sectional geometry of rolled members must be recognized by the designer, the fabricator and erector (see Fig. 1). Such tolerances are mandatory because roll wear, thermal distortions of the hot cross section immediately after leaving the forming rolls, and differential cooling distortions that take place on the cooling beds are economically beyond precise control. Absolute perfection of cross sectional geometry is not of structural significance and, if the tolerances are recognized and provided for, also not of architectural significance. ASTM A6 also stipulates straightness and camber tolerances which are adequate for most conventional construction. However, these characteristics may be controlled or corrected to closer tolerances during the fabrication process when the unique demands of a particular project justify the added cost.

+ 1/ 4 – 3/16

Actual section

½B ± 3/16 ½ 3 B /16

T

C = d+ ¼ max. A = d ± 1/ 8 d

C = d + ¼ max.

T1

T1

Theoretical section T1

bf

T

f

C = d + ¼ max.

B=b

±

Typical

Typical Typical

Typical ½B± 3/16 ½B 3/16 ±

T + T ′ — For sections 12 ″ and under - ¼ ″ max.

B — Actual flange width A — Actual depth at cL web C — Actual depth overall

For sections over 12 ″ — 5/15 ″ max. bf — Theoretical flange width d — Theoretical depth T & T ′ — Tilt of flange

Fig. 1. Mill tolerances on cross-section dimensions AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 465

SECTION 6. FABRICATION AND DELIVERY 6.4. Dimensional Tolerances Fabrication tolerances are stipulated in several specification documents, each applicable to a special area of construction. Basic fabrication tolerances are stipulated in Sections 6.4 and 10 of the Code and Section M2.7 of the AISC Specification. Other specifications and codes frequently incorporated by reference in the contract documents are the AWS Structural Welding Code and AASHTO Standard Specifications for Highway Bridges. 6.4.5. Due to the release of stresses, there is no known way to verify camber once members are received in the field. Camber may only be measured in the fabrication shop in the unstressed condition and does not take into account the dead weight of the member, the restraint caused by the end connections in the erected state or any dead load which may be intended to be applied. 6.5. Shop Painting 6.5.2., 6.5.3. The selection of a paint system is a design decision involving many factors including owner’s preference, service life of the structure, severity of environmental exposure, cost of both initial application and future renewals, and compatibility of the various components comprising the paint system, i.e., surface preparation, prime coat and subsequent coats. Because inspection of shop painting needs to be concerned with workmanship at each stage of the operation, the fabricator provides notice of the schedule of operations and affords access to the work site to inspectors. Inspection must be coordinated with that schedule in such a way as to avoid delay of the scheduled operations. Acceptance of the prepared surface must be made prior to application of the prime coat because the degree of surface preparation cannot be readily verified after painting. Time delay between surface preparation and application of the prime coat can result in unacceptable deterioration of a properly prepared surface, necessitating a repetition of surface preparation. This is especially true with blast-cleaned surfaces. Therefore, to avoid potential deterioration of the surface it is assumed that surface preparation is accepted unless it is inspected and rejected prior to the scheduled application of the prime coat. The prime coat in any paint system is designed to maximize the wetting and adherence characteristics of the paint, usually at the expense of its weathering capabilities. Deterioration of the shop paint normally begins immediately after exposure to the elements and worsens as the duration of exposure is extended. Consequently, extended exposure of the prime coat to weather or to a corrosive atmosphere will lead to its deterioration and may necessitate repair, possibly including


6 - 466


repetition of surface preparation and primer application in limited areas. With the introduction of high performance paint systems, delay in the application of the prime coat has become more critical. High performance paint systems generally require a greater degree of surface preparation, as well as early application of weathering protection for the prime coat. Since the fabricator does not control the selection of the paint system, the compatibility of the various components of the total paint system, nor the length of exposure of the prime coat, he cannot guarantee the performance of the prime coat or any other part of the system. Rather, the fabricator is responsible only for accomplishing the specified surface preparation and for applying the shop coat or coats in accordance with the contract documents. Section 6.5.2 stipulates cleaning the steel to the requirements of SSPC-SP2. This section is not meant as an exclusive cleaning level, but rather that level of surface preparation which will be furnished if the steel is to be painted and if the job specifications are silent or do not require more stringent surface preparation requirements. Further information regarding shop painting is available in A Guide to Shop Painting of Structural Steel, published jointly by the Steel Structures Painting Council and the American Institute of Steel Construction. 6.5.4. Extended exposure of unpainted steel which has been cleaned for subsequent fire protection material application can be detrimental to the fabricated product. Most levels of cleaning require the removal of all loose mill scale, but permit some amount of “tightly adhering mill scale.” When a piece of structural steel which has been cleaned to an acceptable level is left exposed to a normal environment, moisture can penetrate behind the scale, and some “lifting” of the scale by the oxidation products is to be expected. Cleanup of “lifted” mill scale is not the responsibility of the fabricator, but is assigned by contract requirement to an appropriate contractor. Section 6.5.4 of the Code is not applicable to weathering steel, for which special cleaning specifications are always required in the contract documents.

SECTION 7. ERECTION 7.5. Installation of Anchor Bolts and Embedded Items 7.5.1. While the general contractor must make every effort to set anchor bolts accurately to theoretical drawing dimensions, minor deviations may occur. The tolerances set forth in this section were compiled from data collected from general contractors and erectors. They can be attained by using reasonable care and will ordinarily allow the steel to be erected and plumbed to required tolerances. If special conditions require closer tolerances, the contractor responsible for setting the anchor



6 - 467

bolts should be so informed by the contract documents. When anchor bolts are set in sleeves, the adjustment provided may be used to satisfy the required anchor bolt setting tolerances. The tolerances established in this section of the Code have been selected to be compatible with oversize holes in base plates, as recommended in the AISC textbook Detailing for Steel Construction. An anchor bolt group is the set of anchor bolts which receive a single fabricated steel shipping piece. The established column line is the actual field line most representative of the centers of the as-built anchor bolt groups along a line of columns. It must be straight or curved as shown on the plans. 7.6. Bearing Devices The 1⁄8 in. tolerance on elevation of bearing devices relative to established grades is provided to permit some variation in setting bearing devices and to account for attainable accuracy with standard surveying instruments. The use of leveling plates larger than 22 in. × 22 in. is discouraged and grouting is recommended with larger sizes. For purposes of erection stability, the use of leveling nuts is discouraged when base plates have less than four (4) anchor bolts. 7.9.3. Non Self-Supporting Steel Frames To rationally provide temporary supports and/or bracing, the erector must be informed by the owner of the sequence of installation and the effect of loads imposed by such elements at various stages during the sequence until they become effective. The overall strength and stability of a non self-supporting steel frame may be dependent upon the installation of non-structural steel elements such as concrete floor diaphragms, concrete or masonry shear walls, precast concrete facade pieces, etc. The requirement for these elements to be in place to provide overall strength and stability for the structural steel frame should be made clear in the contract documents in order that the need for temporary support may be clearly understood. For example, precast tilt-up slabs or channel slab facia elements which depend upon attachment to the steel frame for stability against overturning due to eccentricity of their gravity load may induce significant unbalanced lateral forces on the bare steel frame when partially installed. 7.11. Framing Tolerances The erection tolerances defined in this section of the Code have been developed through long-standing usage as practical criteria for the erection of structural steel. Erection tolerances were first defined by AISC in its Code of Standard Practice of


6 - 468


October, 1924 in Section 7 (f), “Plumbing Up.” With the changes that took place in the types and use of materials in building construction after World War II, and the increasing demand by architects and owners for more specific tolerances, AISC adopted new standards for erection tolerances in Section 7 (h) of the March 15, 1959 edition of the Code. Experience has proven that those tolerances can be economically obtained. The current requirements were first published in the October 1,1972 edition of the Code. They provide an expanded set of criteria over earlier Code editions. The basic premise that the final accuracy of location of any specific point in a structural steel frame results from the combined mill, fabrication and erection tolerances, rather than from the erection tolerances alone, remains unchanged in this edition of the Code. However, to improve clarity, pertinent standard fabrication tolerances are now stipulated in Section 7.11, rather than by reference to the AISC Specification as in previous editions. Additionally, expanded coverage has been given to the definition of working points and working lines governing measurements of the actual steel location. Illustrations for defining and applying the applicable Code tolerances are provided in this Commentary. The recent trend in building work is away from built-in-place construction wherein compatibility of the frame and the facade or other collateral materials is automatically provided for by the routine procedures of the crafts. Building construction today frequently incorporates prefabricated components wherein large units are developed with machine-like precision to dimensions that are theoretically correct for a perfectly aligned steel frame with ideal member cross sections. This type of construction has made the magnitude of the tolerances allowed for structural steel building frames increasingly of concern to owners, architects and engineers. This has led to the inclusion in job specifications of unrealistically small tolerances, which indicate a general lack of recognition of the effects of the accumulation of dead load, temperature effects and mill, fabrication and erection tolerances. Such tolerances are not economically feasible and do not measurably increase the structure’s functional value. This edition of the Code incorporates tolerances previously found to be practical and presents them in a precise and clear manner. Actual application methods have been considered and the application of the tolerance limitations to the actual structure have been defined. 7.11.3. Position and Alignment The limitations described in Section 7.11.3.1 and illustrated in Figs. 2 and 3 make it possible to maintain built-in-place or prefabricated facades in a true vertical plane up to the 20th story, if connections which provide for 3 in. adjustment are used. Above the 20th story, the facade may be maintained within 1⁄16 in. per story with a maximum total deviation of 1 in. from a true vertical plane, if the 3 in. adjustment is provided. Section 7.11.3.1(c) limits the position of exterior column working points at any given splice elevation to a narrow horizontal envelope parallel to the building line



6 - 469

(see Fig. 4). This envelope is limited to a width of 11⁄2 in., normal to the building line, in up to 300 ft of building length. The horizontal location of this envelope is not necessarily directly above or below the corresponding envelope at the adjacent splice elevations, but should be within the limitation of the 1:500 allowable tolerance in plumbness of the controlling columns (see Fig. 3). Connections permitting adjustments of plus 2 in. to minus 3 in. (5 in. total) will be necessary in cases where the architect or owner insists upon attempting to construct the facade to a true vertical plane above the 20th story. Usually there is a differential shortening of the internal versus the external columns during construction, due to non-uniform rate of accumulation of dead load stresses (see Fig. 5). The amount of such differential shortening is indeterminate because it varies dependent upon construction sequence from day to day as the construction progresses, and does not reach its maximum shortening until the building is in service. When floor concrete is placed while columns are supporting different percentages of their full design loads, the floor must be finished to slopes established by measurements from the tops of beams at column connections. The effects of

B

/2 + h/1000 +Tp

W.P.

E.C.L.

L

C

/ 2 + h/1000 + Ta

Tp Tp

E.C.L.

C

L

/2 + h/1000 +Tp

Envelope of actual location of working points to established column line. See Fig. 3

Ta Tt

C

/ 2 + h/1000

C

L

L

/ 2 + h/1000

/2 + h/1000 B/2 + h/1000

/ 2 + h/1000 + Tt

Minimum clearance envelope

B

B

L

L

For enclosures or attachments which may follow column alignment

L

For enclosures or attachments which must be held to precise plan location

L = Actual c to c columns = Plan dimension ± column cross section tolerance ± beam length tolerance. Ta = Plumbness tolerance away from building line (varies, see Fig. 3) Tt = Plumbness tolerance toward building line (varies, see Fig. 3) Tp = Plumbness tolerance parallel to building line (= Ta )

Fig. 2. Clearance required to accommodate accumulated column tolerances AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 470


3″

2″

Slope 1/16 ″ per story

20th Fl 2″ Plumb Elev. varies

Slope 1/ 500

1″

Established column line

Building line

36th Fl.

Splice Braced point

W.P.

Maximum out-of-plumb of individual shipping piece as defined by straight line between working points ≤ 1/500 Maximum out-of-straightness between braced points L/1000 where L is distance between braced points.

Braced point Splice

W.P.

Braced point

Individual column sections within envelope defined at left Established column line

Elev. varies

Slope 1/500

Slope 1/ 500 ¼

¼

Tolerance on location of W.P. at base

Envelope within which all working points must fall Note: The plumb line thru the base working point for an individual column is not necessarily the precise plan location because Section 7.11.3.1 deals only with plumbness tolerance and does not include inaccuracies in location of established column line, foundations and anchor bolts beyond the erector’s control.

Fig. 3. Exterior column plumbness tolerances normal to building line Building line

Ta Tt E

Established Column lines

Maximum envelope for working points of all columns at any given elevation E = 1½ ″ for up to 300 ′of length, over 300 ′ add ½ ″ for each 100 ′ of length with 3 ″max total Column plumbness tolerance — See figs. 2 and 3 — Indicates column working points At any splice elevation, envelope “E” is located within the limits Ta and T t At any splice elevation, envelope “E” may be located offset from the corresponding envelope of the adjacent splice elevations, above and below, by an amount not greater than 1/ 500 of the column length.

Fig. 4. Tolerances in plan at any splice elevation of exterior columns



6 - 471

Splice elevation shown on plan

Column base

Differential shortening

Interior column Finish line

Interior column shortening due to partial dead load

Beam elevations from finish line See section 7.11.3.3 Floor elevations set by measurement from top of beams

Finish line

Floor elevations set by measurement from top of beams Beam elevations from finish line See section 7.11.3.3 Exterior column shortening due to partial dead load

Finish line

Exterior column

Finish line

On a particular date during the erection of structural steel and placement of other material, (floor concrete, facade, etc.) the interior columns will be carrying a higher percentage of their final loads than the exterior columns. Therefore, for equal design unit stresses, the actual stress on that date for interior columns will be greater than the actual stresses on exterior columns. When all dead loads have been applied, stresses and shortening in all columns will be approximately equal.

Fig. 5. Effect of differential column shortening differential shortening, plus mill camber and deflections, become very important when there is little cover over the steel, when there are electrical fittings mounted on the steel flooring whose tops are supposed to be flush with the finished floor, when there is small clearance between bottom of beams and top of door frames, etc., and when there is little clearance around ductwork. To finish floors to a precisely level plane, for example by the use of laser leveling techniques, can result in significant differential floor thicknesses, different increases above design dead loads for individual columns and, thus, permanent differential column shortening and out-oflevel completed floors. Similar considerations make it unfeasible to attempt to set the elevation of a given floor in a multistory building by reference to a bench mark at the base of the structure. Columns are fabricated to a length tolerance of ± 1⁄32 in. while under a zero state of stress. As dead loads accumulate, the column shortening which takes place is negligible within individual stories and in low buildings, but will accumulate to significant magnitude in tall buildings. Thus, the upper floors of tall buildings will be excessively thick and the lower floors will be below the initial finish elevation if floor elevations are established relative to a ground level bench mark. If foundations and base plates are accurately set to grade and the lengths of individual column sections are checked for accuracy prior to erection, and if floor


6 - 472


elevations are established by reference to the elevation of the top of beams, the effect of column shortening due to dead load will be minimized. Since a long unencased steel frame will expand or contract 1⁄8 in. per 100 ft for each change of 15°F in temperature, and since the change in length can be assumed to act about the center of rigidity, the end columns anchored to foundations will be plumb only when the steel is at normal temperature (see Fig. 6). It is therefore necessary to correct field measurements of offsets to the structure from established baselines for the expansion or contraction of the exposed steel frame. For example, a building 200-ft long that is plumbed up at 100°F should have working points at the tops of end columns positioned 1⁄2 in. out from the working point at the base in order for the column to be plumb at 60°F. Differential temperature effects on column length should also be taken into account in plumbing surveys when tall steel frames are subject to strong sun exposure on one side. The alignment of lintels, spandrels, wall supports and similar members used to connect other building construction units to the steel frame should have an adjustment of sufficient magnitude to allow for the accumulative effect of mill, fabrication and erection tolerances on the erected steel frame (see Fig. 7).

When plumbing end columns, apply temperature adjustment at rate 1/ 8 ″ per 100 ′of length from center of rigidity per each 15°F of difference between erection and working temperatures.

Length

Length

Center of rigidity Ta Tt

Tt Ta

Tt Ta

Tt Ta

C to C adjacent columns subject to mill and fabrication tolerance Tp Tp Tt Ta

Established column lines Building line

Fig. 6. Tolerances in plan location of columns AMERICAN INSTITUTE OF STEEL CONSTRUCTION


6 - 473

Column dimension + tolerances

D = Tolerances required by manufacturer of wall units plus survey tolerance D

D E.C.L.

Clearance line to accommodate column. See Figure 2

Provide connections with slotted holes and/or shims to accommodate tolerances

Column dimension + tolerances

If fascia joints are set from nearest column finish line, allow ± 5/ 8 ″for vertical adjustment. Owners plans for fascia details must allow for progressive shortening of steel columns.

Fig. 7. Clearance required to accommodate fascia

500 1

500

500

1

1

500 1

Support points

Field splices

Fig. 8. Alignment tolerances for members with field splices AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 474


7.11.3.2. Alignment Tolerance for Members with Field Splices The angular misalignment of the working line of all fabricated shipping pieces relative to the line between support points of the member as a whole in erected position must not exceed 1 in 500. Note that the tolerance is not stated in terms of a linear displacement at any point and is not to be taken as the overall length between supports divided by 500. Typical examples are shown in Fig. 8. Numerous conditions within tolerance for these and other cases are possible. This condition applies to both plan and elevation tolerances. 7.11.4. Responsibility for Clearances In spite of all efforts to minimize inaccuracies, deviations will still exist; therefore, in addition, the designs of prefabricated wall panels, partition panels, fenestrations, floor-to-ceiling door frames and similar elements must provide for clearance and details for adjustment as described in Section 7.11.4. Designs must provide for adjustment in the vertical dimension of prefabricated facade panels supported by the steel frame because the accumulation of shortening of stressed steel columns will result in the unstressed facade supported at each floor level being higher than the steel frame connections to which it must be attached. Observations in the field have shown that where a heavy facade is erected to a greater height on one side of a multistory building than on the other, the steel framing will be pulled out of alignment. Facades should be erected at a relatively uniform rate around the perimeter of the structure. 7.14. Handling and Storage Handling Painted Steel During storage, loading, transport, unloading and erection, blemish marks caused by slings, chains, blocking, tie-downs, etc., occur in varying degrees. Abrasions caused by handling or cartage after painting are to be expected. The owner/engineer must recognize that any shop applied coating, no matter how carefully protected, will require touch-up in the field. Touch-up of these blemished areas is the responsibility of the contractor performing the field touch-up of field painting. Cleaning After Erection The responsibility for proper storage and handling of fabricated steel at the construction site during erection is properly the erector’s. Shop-painted steel stored in the field pending erection should be kept free of the ground and so positioned as to minimize water-holding pockets. The owner or general contractor is responsible for providing suitable site conditions and proper access so that the fabricator/erector may perform its work.



6 - 475

Site conditions are frequently muddy, sandy or dusty, or a combination of all three, during the erection period. Under such conditions it may be impossible to store and handle the steel in such a way as to completely avoid accumulation of mud, dirt or sand on the surface of the steel, even though the fabricator/erector manages to proceed with the work. Repairs of damage to painted surfaces and/or removal of foreign materials due to adverse site conditions are outside the scope of responsibility of the fabricator/erector when reasonable attempts at proper handling and storage have been made.

SECTION 8. QUALITY CONTROL 8.1.1. The AISC Quality Certification Program confirms to the construction industry that a certified structural steel fabricating plant has the capability by reason of commitment, personnel, organization, experience, procedures, knowledge and equipment to produce fabricated structural steel of the required quality for a given category of structural steelwork. The AISC Quality Certification Program is not intended to involve inspection and/or judgment of product quality on individual projects. Neither is it intended to guarantee the quality of specific fabricated steel products.

SECTION 9. CONTRACTS 9.2. Calculation of Weights The standard procedure for calculation of weights that is described in the Code meets the need for a universally acceptable system for defining “pay weights” in contracts based on the weight of delivered and/or erected materials. This procedure permits owners to easily and accurately evaluate price per pound proposals from potential suppliers and enables both parties to a contract to have a clear understanding of the basis for payment. The Code procedure affords a simple, readily understood method of calculation which will produce pay weights which are consistent throughout the industry and which may be easily verified by the owner. While this procedure does not produce actual weights, it can be used by purchasers and suppliers to define a widely accepted basis for bidding and contracting for structural steel. However, any other system can be used as the basis for a contractual agreement. When other systems are used, both supplier and purchaser should clearly understand how the alternate procedure is handled.


6 - 476


9.3. Revisions to Contract Documents 9.3.1. Revisions to the Contract are implemented by issuance of new documents or re-issuance of existing documents. Individual revisions must be noted where they occur and documents must be dated with latest issue date and the reasons for issuance must be identified. 9.3.2. Revisions to the Contract are also implemented by change order, extra work order, or notations on the shop and erection drawings when returned from approval. However, revisions implemented in this manner must be incorporated subsequently as revisions to the plans and/or specifications and re-issued in accordance with Section 9.3.1. 9.3.3. The issuance of revisions authorizes the fabricator and erector to incorporate the revisions into the work. This authorization obligates the owner to pay the fabricator and erector for costs associated with changed and/or additional work. 9.6. Terms of Payment These terms include such items as progress payments for material, fabrication, erection, retainage, performance and payment bonds and final payment. If a performance or payment bond, paid for by the owner, is required by contract, then no retainage shall be required.

SECTION 10. ARCHITECTURALLY EXPOSED STRUCTURAL STEEL The rapidly increasing use of exposed structural steel as a medium of architectural expression has given rise to a demand for closer dimensional tolerances and smoother finished surfaces than required for ordinary structural steel framing. This section of the Code establishes standards for these requirements which take into account both the desired finished appearance and the abilities of the fabrication shop to produce the desired product. These requirements were previously contained in the AISC Specification for Architecturally Exposed Structural Steel which architects and engineers have specified in the past. It should be pointed out that the term “Architecturally Exposed Structural Steel” (AESS), as covered in this section, must be specified in the contract documents if the fabricator is required to meet the fabricating standards of Section 10, and applies only to that portion of the structural steel so identified. In order to avoid misunderstandings and to hold costs to a minimum, only those steel surfaces and connections which will remain exposed and subject to normal view by pedestrians or occupants of the completed structure should be designated as AESS.


6 - 477

AISC Quality Certification Program

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

6 - 478


6 - 479

AISC Quality Certification Program In recent years. the quality of construction methods and materials has become the subject of increasing concern to building officials, highway officials, and designers. One result of this concern has been the enactment of ever more demanding inspection requirements intended to ensure product quality. In many cases, however, these more demanding inspection requirements have not been based upon demonstrated unsatisfactory performance of structures in service. Rather, they have been based upon the capacity of sophisticated test equipment. or upon standards developed for nuclear construction rather than conventional construction. Adding to the problem, arbitrary interpretation of specifications by inspectors has too often been made without rational consideration of the type of construction involved. The result has been spiraling increases in the costs of fabrication of structural steel and of inspection, which must be paid by owners without necessarily assuring that the product quality required has been improved. Product inspection. although it has a valid place in the construction process, is not the most logical or practical way to assure that structural steelwork will conform to he requirements of contract documents and satisfy the intended use. A better solution can be found in the exercise of good quality control and quality assurance by the fabricator throughout the entire production process. Recognizing this fact, and seeking some valid, objective method whereby a fabricator’s capability for assuring a quality product could be evaluated, a number of code authorities have, in recent years, instituted steps to establish fabricator registration programs. However, these independent efforts resulted in extremely inconsistent criteria. They were developed primarily by inspectors or inspection agencies who were experienced in testing, but were not familiar with the complexities of the many steps, procedures, techniques, and controls required to assure quality throughout the fabricating process. Neither were these inspection agencies qualified to determine the various levels of quality required to assure satisfactory performance in meeting the service requirements of the many different types of steel structures. Recognizing the need for a comprehensive national standard for fabricator certification, and concerned by the trend toward costly inspection requirements that could not be justified by rational quality standards, the American Institute of Steel Construction has developed and implemented a voluntary Quality Certification Program, whereby any structural steel fabricating plant—whether a member of AISC or not—can have its capability for assuring quality production evaluated on a fair and impartial basis.


6 - 480

AISC QUALITY CERTIFICATION PROGRAM

THE AISC PROGRAM

The AISC Quality Certification Program does not involve inspection and/or judgment of product quality on individual projects. Neither does it guarantee the quality of specific fabricated steel products. Rather, the purpose of the AISC Quality Certification Program is to confirm to the construction industry that a Certified structural steel fabricating plant has the personnel, organization, experience, procedures, knowledge, equipment, capability and commitment to reproduce fabricated steel of the required quality for a given category of structural steelwork. The AISC Quality Certification Program was developed by a group of highly qualified shop operation personnel from large, medium, and small structural steel fabricating plants throughout the United States. These individuals all had extensive experience and were fully aware of where and how problems can arise during the production process and of the steps and procedures that must be followed during fabrication to assure that the finished product meets the quality requirements of the contract. The program was reviewed and strongly endorsed by an Independent Board of Review comprised of 17 prominent structural engineers from throughout the United States, who were not associated with the steel fabricating industry, but were well qualified in matters of quality requirements for reliable service of all types of steel structures. CATEGORIES OF CERTIFICATION

A fabricator may apply for certification of a plant in one of the following categories of structural steelwork: I: Conventional Steel Structures — Small Public Service and Institutional

Buildings, (Schools, etc.), Shopping Centers, Light Manufacturing Plants, Miscellaneous and Ornamental Iron Work, Warehouses, Sign Structures, Low Rise, Truss Beam/Column Structures, Simple Rolled Beam Bridges. II: Complex Steel Building Structures — Large Public Service and Institutional

Buildings, Heavy Manufacturing Plants, Powerhouses (fossil, non-nuclear), Metal Producing/Rolling Facilities, Crane Bridge Girders, Bunkers and Bins, Stadia, Auditoriums, High Rise Buildings, Chemical Processing Plants, Petroleum Processing Plants. III: Major Steel Bridges — All bridge structures other than simple rolled beam

bridges. MB: Metal Building Systems — Pre-engineered Metal Building Structures. Supplement: Auxiliary and Support Structures for Nuclear Power Plants — This

supplement, applicable to nuclear plant structures designed under the AISC Specification, but not to pressure-retaining structures, offers utility companies and designers of nuclear power plants a certification program that will eliminate the need for many of the more costly, conflicting programs now in use. A fabricator must hold certification in either Category I, II or III prior to application for certification in this category. Certification in Category II automatically includes Category I. Certification in Category III automatically includes Categories I and II. Certification in Category MB is not transferable to any other Category. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

AISC QUALITY CERTIFICATION PROGRAM

6 - 481

INSPECTION-EVALUATION PROCEDURE

An outside, experienced, professional organization, ABS Quality Evaluations, Inc. (a subsidiary of American Bureau of Shipping) has been retained by AISC to perform the plant Inspection-Evaluation in accordance with a standard check list and rating procedure established by AISC for each certification category in the program. Upon completion of this Inspection-Evaluation, ABS Quality Evaluations, Inc. (commonly known as ABSQE) will recommend to AISC that a fabricator be approved or disapproved for certification. ABS-QE’s Inspection-Evaluation is totally independent of the fabricator’s and AISC’s influence, and their evaluation is not subject to review by AISC. At a time mutually agreed upon by the fabricator, AISC, and ABS-QE, the Inspection-Evaluation team visits the plant to investigate and rate the following basic plant functions directly and indirectly affecting quality assurance: General Management, Engineering and Drafting, Procurement, Shop Operations, and Quality Control. The Inspection-Evaluation team will perform the following: 1. Confirm data submitted with the Application for Certification. 2. Interview key supervisory personnel and subordinate employees. 3. Observe and rate the organization in operation, including procedures used in functions affecting quality assurance. 4. Inspect and rate equipment and facilities. 5. At an “exit interview,” review with plant management the completed check list observations and evaluation scoring, including discussions of deficiencies and omissions, if any. The number of days required for Inspection-Evaluation varies according to the size and complexity of the plant, but usually requires two to five days. CERTIFICATION

Following recommendation for Certification by the Inspection-Evaluation team, AISC will issue a certificate identifying the fabricator, the plant, and the Category of Certification. The certificate is valid for a three year period, subject to annual review in the form of unannounced inspections early in the second and third year periods. The certificate is endorsed annually, provided there is successful completion of the unannounced second and third year inspection. An annual self-audit, based on the standard check list, must be made by plant management during the 11th and 23rd months after initial Certification. This self-audit must be retained at the plant and made available to the Inspection-Evaluation team during the unannounced second and third year inspections. At the end of the third year, the cycle begins again with a complete prescheduled Inspection-Evaluation and the issuance of a new certificate.


7-1

Part 7 MISCELLANEOUS DATA AND MATHEMATICAL INFORMATION MISCELLANEOUS DATA Wire and Sheet Metal Gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3 AISI Standard Nomenclature for Flat Rolled Carbon Steel . . . . . . . . . . . . . . . . . 7-3 Coefficients of Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4 Weights and Specific Gravities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5 Weights of Building Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7 SI UNITS FOR STRUCTURAL STEEL DESIGN . . . . . . . . . . . . . . . . . . . . . . 7-8 SI (Metric) Weights and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10 U.S. Weights and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-11 SI Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12 GEOMETRIC AND TRIGONOMETRIC DATA Bracing Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-14 Properties of the Parabola and Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15 Properties of the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16 Properties of Geometric Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17 Trigonometric Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-24 DECIMAL EQUIVALENTS Decimals of an Inch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25 Decimals of a Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-26


7-2

MISCELLANEOUS DATA AND MATHEMATICAL INFORMATION


MISCELLANEOUS DATA

7-3

Table 7-1. WIRE AND SHEET METAL GAGES Equivalent thickness in decimals of an inch

Gage No.

U.S. Standard Galvanized Sheet Gage Gage for for HotUncoated Dipped Hot & ColdZinc Coated Rolled Sheetsb Sheetsb

7/0 6/0 5/0 4/0 3/0 2/0 1/0 1 2 3 4 5 6 7 8 9 10 11 12

USA Steel Wire Gage

.490 .462a .430a .394a .362a .331 .306 .283 .262a .244a .225a .207 .192 .177 .162 .148a .135 .120a .106a

— — — — — — — — — — — — — — .1681 .1532 .1382 .1233 .1084

— — — — — — — — — .2391 .2242 .2092 .1943 .1793 .1644 .1495 .1345 .1196 .1046

Gage No.

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

U.S. Standard Galvanized Gage for Sheet Gage Uncoated for HotHot & ColdDipped Rolled Zinc Coated Sheetsb Sheetsb

.0897 .0747 .0673 .0598 .0538 .0478 .0418 .0359 .0329 .0299 .0269 .0239 .0209 .0179 .0164 .0149 — —

.0934 .0785 .0710 .0635 .0575 .0516 .0456 .0396 .0366 .0336 .0306 .0276 .0247 .0217 .0202 .0187 .0172 .0157

USA Steel Wire Gage

.092a .080 .072 .062a .054 .048a .041 .035a — — — — — — — — — —

aRounded value. The steel wire gage has been taken from ASTM A510 “General Requirements for Wire Rods

and Coarse Round Wire, Carbon Steel.” Sizes originally quoted to four decimal equivalent places have been rounded to three decimal places in accordance with rounding procedures of ASTM “Recommended Practice” E29. bThe equivalent thicknesses are for information only. The product is commonly specified to decimal thickness, not to gage number.

Table 7-2. AISI STANDARD NOMENCLATURE FOR FLAT ROLLED CARBON STEEL Width (Inches) Thickness (Inches)

To 31⁄2 incl.

Over 31⁄2 To 6

Over 6 To 8

Over 8 To 12

Over 12 To 48

Over 48

0.2300 & thicker

Bar

Bar

Bar

Plate

Plate

Plate

0.2299 to 0.2031

Bar

Bar

Strip

Strip

Sheet

Plate

0.2030 to 0.1800

Strip

Strip

Strip

Strip

Sheet

Plate

0.1799 to 0.0449

Strip

Strip

Strip

Strip

Sheet

Sheet

0.0448 to 0.0344

Strip

Strip

0.0343 to 0.0255

Strip

0.0254 & thinner

Hot-rolled sheet and strip not generally produced in these widths and thicknesses


7-4


Table 7-3. COEFFICIENTS OF EXPANSION The coefficient of linear expansion (ε) is the change in length, per unit, for a change of one degree of temperature. The coefficient of surface expansion is approximately two times the linear coefficient, and the coefficient of volume expansion, for solids, is approximately three times the linear coefficient. A bar, free to move, will increase in length with an increase in temperature and will decrease in length with a decrease in temperature. The change in length will be εtl, where ε is the coefficient of linear expansion, t the change in temperature and l the length. If the ends of a bar are fixed, a change in temperature (t ) will cause a change in the unit stress of E εt, and in force of AE εt, where A is the cross-sectional area of the bar and E the modulus of elasticity. The following table gives the coefficient of linear expansion for 100°, or 100 times the value indicated above. Example: A piece of medium steel is exactly 40 ft long at 60°F. Find the length at 90°F assuming the ends free to move. change of length = εt l =

.00065 × 30 × 40 = .0078 ft 100

The length at 90° is 40.0078 ft

Example: A piece of medium carbon steel is exactly 40 ft long and the ends are fixed. If the temperature increases 30°F, what is the resulting change in the unit stress? change in unit stress = E εt =

29,000 × .00065 × 30 = 5.7 ksi 100

COEFFICIENTS OF EXPANSION FOR 100 DEGREES = 100ε Linear Expansion

Linear Expansion Centigrade

Fahrenheit

METALS AND ALLOYS Aluminum, wrought Brass Bronze Copper Iron, cast, gray Iron, wrought Iron, wire Lead Magnesium, various alloys Nickel Steel , mild Steel, stainless, 18-8 Zinc, rolled

.00231 .00188 .00181 .00168 .00106 .00120 .00124 .00286 .0029 .00126 .00117 .00178 .00311

.00128 .00104 .00101 .00093 .00059 .00067 .00069 .00159 .0016 .00070 .00065 .00099 .00173

TIMBER Fir Maple parallel to fiber Oak Pine

.00037 .00064 .00049 .00054

.00021 .00036 .00027 .00030

Materials

Centigrade

Fahrenheit

STONE AND MASONRY Ashlar masonry Brick Masonry Cement, portland Concrete Granite Limestone Marble Plaster Rubble masonry Sandstone Slate

.00063 .00061 .00126 .00099 .00080 .00076 .00081 .00166 .00063 .00097 .00080

.00035 .00034 .00070 .00055 .00044 .00042 .00045 .00092 .00035 .00054 .00044

TIMBER Fir Maple perpendicular to Oak fiber Pine

.0058 .0048 .0054 .0034

.0032 .0027 .0030 .0019

Materials

EXPANSION OF WATER Maximum Density = 1 C°°

Volume

0 1.000126 4 1.000000

C°°

Volume

10 1.000257 20 1.001732

C°°

Volume

30 1.004234 40 1.007627

C°°

Volume

50 1.011877 60 1.016954

C°°

Volume

C°°

Volume

70 1.022384 90 1.035829 80 1.029003 100 1.043116


MISCELLANEOUS DATA

7-5

Table 7-4. WEIGHTS AND SPECIFIC GRAVITIES Weight lb per Specific Gravity cu ft

Substance ASHLAR, MASONRY Granite, syenite, gneiss . . . . Limestone, marble . . . . . . . Sandstone, bluestone . . . . .

165 160 140

2.3–3.0 2.3–2.8 2.1–2.4

MORTAR RUBBLE MASONRY Granite, syenite, gneiss . . . . Limestone, marble . . . . . . . Sandstone, bluestone . . . . .

155 150 130

2.2–2.8 2.2–2.6 2.0–2.2

DRY RUBBLE MASONRY Granite, syenite, gneiss . . . . Limestone, marble . . . . . . . Sandstone, bluestone . . . . .

130 125 110

1.9–2.3 1.9–2.1 1.8–1.9

BRICK MASONRY Pressed brick . . . . . . . . . Common brick . . . . . . . . . Soft brick . . . . . . . . . . . .

140 120 100

2.2–2.3 1.8–2.0 1.5–1.7

CONCRETE MASONRY Cement, stone, sand . . . . . Cement, slag. etc. . . . . . . . Cement, cinder, etc. . . . . . .

144 130 100

2.2–2.4 1.9–2.3 1.5–1.7

40–45 90 183 53–64 103 67–72 98–117 96 49–55

— — 2.7–3.2 — 1.4–1.9 — — — —

VARIOUS BUILDING MATERIALS Ashes. cinders . . . . . Cement, portland, loose Cement, portland, set . Lime, gypsum, loose . . Mortar, set . . . . . . . Slags, bank slag . . . . Slags, bank screenings Slags, machine slag . . Slags, slag sand . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

EARTH, ETC., EXCAVATED Clay, dry . . . . . . . . . . Clay, damp, plastic . . . . Clay and gravel, dry . . . . Earth, dry, loose . . . . . . Earth, dry, packed . . . . . Earth, moist, loose . . . . . Earth, moist, packed . . . . Earth, mud, flowing . . . . Earth, mud, packed . . . . Riprap, limestone . . . . . Riprap, sandstone . . . . . Riprap, shale . . . . . . . Sand, gravel, dry, loose . . Sand, gravel, dry, packed . Sand, gravel, wet . . . . .

. . . . . . . . . . . . . . .

. 63 . 110 . 100 . 76 . 95 . 78 . 96 . 108 . 115 . 80–85 . 90 . 105 . 90–105 . 100–120 . 118–120

— — — — — — — — — — — — — — —

EXCAVATIONS IN WATER Sand or gravel . . . . . . . Sand or gravel and clay . . Clay . . . . . . . . . . . . River mud . . . . . . . . . Soil . . . . . . . . . . . . . Stone riprap . . . . . . . .

. . . . . .

. . . . . .

— — — — — —

60 65 80 90 70 65

Weight lb per Specific cu ft Gravity

Substance MINERALS Asbestos . . . . . . . Barytes . . . . . . . . Basalt . . . . . . . . . Bauxite . . . . . . . . Borax . . . . . . . . . Chalk . . . . . . . . . Clay, marl . . . . . . . Dolomite . . . . . . . . Feldspar, orthoclase . . Gneiss, serpentine . . Granite, syenite . . . . Greenstone, trap . . . Gypsum, alabaster . . Hornblende . . . . . . Limestone, marble . . . Magnesite . . . . . . . Phosphate rock, apatite Porphyry . . . . . . . . Pumice, natural . . . . Quartz, flint . . . . . . Sandstone, bluestone . Shale, slate . . . . . . Soapstone, talc . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

153 281 184 159 109 137 137 181 159 159 175 187 159 187 165 187 200 172 40 165 147 175 169

2.1–2.8 4.50 2.7–3.2 2.55 1.7–1.8 1.8–2.6 1.8–2.6 2.9 2.5–2.6 2.4–2.7 2.5–3.1 2.8–3.2 2.3–2.8 3.0 2.5–2.8 3.0 3.2 2.6–2.9 0.37–0.90 2.5–2.8 2.2–2.5 2.7–2.9 2.6–2.8

STONE, QUARRIED, PILED Basalt, granite, gneiss . . . Limestone, marble, quartz . Sandstone . . . . . . . . . Shale . . . . . . . . . . . Greenstone, hornblende .

. . . . .

. . . . .

96 95 82 92 107

— — — — —

BITUMINOUS SUBSTANCES Asphaltum . . . . . . . . . . Coal, anthracite . . . . . . . Coal, bituminous . . . . . . Coal, lignite . . . . . . . . . Coal, peat, turf, dry . . . . . Coal, charcoal, pine . . . . . Coal, charcoal, oak . . . . . Coal, coke . . . . . . . . . . Graphite . . . . . . . . . . . Paraffine . . . . . . . . . . . Petroleum . . . . . . . . . . Petroleum, refined . . . . . . Petroleum, benzine . . . . . Petroleum, gasoline . . . . . Pitch . . . . . . . . . . . . . Tar, bituminous . . . . . . .

. . . . . . . . . . . . . . . .

81 97 84 78 47 23 33 75 131 56 54 50 46 42 69 75

1.1–1.5 1.4–1.7 1.2–1.5 1.1–1.4 0.65–0.85 0.28–0.44 0.47–0.57 1.0–1.4 1.9–2.3 0.87–0.91 0.87 0.79–0.82 0.73–0.75 0.66–0.69 1.07–1.15 1.20

COAL AND COKE, PILED Coal, anthracite . . . . . Coal, bituminous, lignite . Coal, peat, turf . . . . . . Coal charcoal . . . . . . Coal coke . . . . . . . .

. . . . .

47–58 40–54 20–26 10–14 23–32

— — — — —

. . . . .

. . . . .

The specific gravities of solids and liquids refer to water at 4°°C, those of gases to air at 0°°C and 760 mm pressure. The weights per cubic foot are derived from average specific gravities, except where stated that weights are for bulk, heaped, or loose material, etc.


7-6


Table 7-4 (cont.). WEIGHTS AND SPECIFIC GRAVITIES Weight lb per Specific Gravity cu ft

Substance METALS, ALLOYS, ORES Aluminum, cast, hammered Brass, cast, rolled . . . . . Bronze, 7.9 to 14% Sn . . . Bronze, aluminum . . . . . Copper, cast, rolled . . . . Copper ore, pyrites . . . . Gold, cast, hammered . . . Iron, cast, pig . . . . . . . Iron, wrought . . . . . . . . Iron, speigel-eisen . . . . . Iron, ferro-silicon . . . . . . Iron ore, hematite . . . . . Iron ore, hematite in bank . Iron ore, hematite loose . . Iron ore, limonite . . . . . . Iron ore, magnetite . . . . Iron slag . . . . . . . . . . Lead . . . . . . . . . . . . Lead ore, galena . . . . . . Magnesium, alloys . . . . . Manganese . . . . . . . . Manganese ore, pyrolusite Mercury . . . . . . . . . . Monel Metal . . . . . . . . Nickel . . . . . . . . . . . Platinum, cast, hammered . Silver, cast, hammered . . Steel, rolled . . . . . . . . Tin, cast, hammered . . . . Tin ore, cassiterite . . . . . Zinc, cast, rolled . . . . . . Zinc ore, blende . . . . . .

VARIOUS SOLIDS Cereals, oats . . . . Cereals, barley . . . Cereals, corn, rye . . Cereals, wheat . . . . Hay and Straw . . . . Cotton, Flax, Hemp . Fats . . . . . . . . . Flour, loose . . . . . Flour, pressed . . . . Glass, common . . . Glass, plate or crown Glass, crystal . . . . Leather . . . . . . . Paper . . . . . . . . Potatoes, piled . . . . Rubber, caoutchouc . Rubber goods . . . . Salt, granulated, piled Saltpeter . . . . . . . Starch . . . . . . . . Sulphur . . . . . . . Wool . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 165 2.55–2.75 . 534 8.4–8.7 . 509 7.4–8.9 . 481 7.7 . 556 8.8–9.0 . 262 4.1–4.3 . 1205 19.25–19.3 . 450 7.2 . 485 7.6–7.9 . 468 7.5 . 437 6.7–7.3 . 325 5.2 . 160–180 — . 130–160 — . 237 3.6–4.0 . 315 4.9–5.2 . 172 2.5–3.0 . 710 11.37 . 465 7.3–7.6 . 112 1.74–1.83 . 475 7.2–8.0 . 259 3.7–4.6 . 849 13.6 . 556 8.8–9.0 . 565 8.9–9.2 . 1330 21.1–21.5 . 656 10.4–10.6 . 490 7.85 . 459 7.2–7.5 . 418 6.4–7.0 . 440 6.9–7.2 . 253 3.9–4.2

bulk bulk bulk bulk bales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 39 48 48 20 93 58 28 47 156 161 184 59 58 42 59 94 48 67 96 125 82

Weight lb per Specific Gravity cu ft

Substance TIMBER, U.S. SEASONED Moisture content by weight: Seasoned timber 15 to 20% Green timber up to 50% Ash, white, red . . . . . . Cedar, white, red . . . . . Chestnut . . . . . . . . . Cypress . . . . . . . . . Fir, Douglas spruce . . . Fir, eastern . . . . . . . . Elm, white . . . . . . . . Hemlock . . . . . . . . . Hickory . . . . . . . . . . Locust . . . . . . . . . . Maple, hard . . . . . . . Maple, white . . . . . . . Oak, chestnut . . . . . . Oak, live . . . . . . . . . Oak, red, black . . . . . . Oak, white . . . . . . . . Pine, Oregon . . . . . . . Pine, red . . . . . . . . . Pine, white . . . . . . . . Pine, yellow, long-leaf . . Pine, yellow, short-leaf . . Poplar . . . . . . . . . . Redwood, California . . . Spruce, white, black . . . Walnut, black . . . . . . . Walnut, white . . . . . . .

VARIOUS LIQUIDS Alcohol, 100% . . . . . . Acids, muriatic 40% . . . Acids, nitric 91% . . . . . Acids, sulphuric 87% . . — Lye, soda 66% . . . . . — Oils, vegetable . . . . . . — Oils, mineral, lubricants . — Water, 4°°C max. density — Water, 100°°C . . . . . . 1.47–1.50 Water, ice . . . . . . . . 0.90–0.97 Water, snow, fresh fallen 0.40–0.50 Water, sea water . . . . 0.70–0.80 2.40–2.60 2.45–2.72 2.90–3.00 0.86–1.02 GASES 0.70–1.15 Air, 0°°C 760 mm . . . . . — Ammonia . . . . . . . . 0.92–0.96 Carbon dioxide . . . . . 1.0–2.0 Carbon monoxide . . . . — Gas, illuminating . . . . . — Gas, natural . . . . . . . 1.53 Hydrogen . . . . . . . . 1.93–2.07 Nitrogen . . . . . . . . . 1.32 Oxygen . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

40 22 41 30 32 25 45 29 49 46 43 33 54 59 41 46 32 30 26 44 38 30 26 27 38 26

0.62–0.65 0.32–.038 0.66 0.48 0.51 0.40 0.72 0.42–0.52 0.74–0.84 0.73 0.68 0.53 0.86 0.95 0.65 0.74 0.51 0.48 0.41 0.70 0.61 0.48 0.42 0.40–0.46 0.61 0.41

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

49 75 94 112 106 58 57 62.428 59.830 56 8 64

0.79 1.20 1.50 1.80 1.70 0.91–0.94 0.90–0.93 1.0 0.9584 0.88–0.92 .125 1.02–1.03

. . . . . . . . .

. . . . . . . . .

1.0 . .08071 0.5920 . .0478 .1234 1.5291 . 0.9673 . .0781 . .028–.036 0.35–0.45 . .038–.039 0.47–0.48 0.0693 . .00559 0.9714 . .0784 1.1056 . .0892

The specific gravities of solids and liquids refer to water at 4°°C, those of gases to air at 0°°C and 760 mm pressure. The weights per cubic foot are derived from average specific gravities, except where stated that weights are for bulk, heaped, or loose material, etc.


WEIGHTS, MEASURES, AND CONVERSION FACTORS

7-7

Table 7-5. WEIGHTS OF BUILDING MATERIALS Materials CEILINGS Channel suspended system Lathing and plastering Acoustical fiber tile

FLOORS Steel Deck Concrete-Reinforced 1 in. Stone Slag Lightweight Concrete-Plain 1 in. Stone Slag Lightweight Fills 1 inch Gypsum Sand Cinders Finishes Terrazzo 1 in. Ceramic or Quarry Tile 3⁄4-in. Linoleum 1⁄4-in. Mastic 3⁄4-in. Hardwood 7⁄8-in. Softwood 3⁄4-in.

ROOFS Copper or tin Corrugated steel 3-ply ready roofing 3-ply felt and gravel 5-ply felt and gravel Shingles Wood Asphalt Clay tile Slate 1⁄4 Sheathing Wood 3⁄4-in. Gypsum 1 in. Insulation 1 in. Loose Poured Rigid

Weight lb per sq ft

Materials

PARTITIONS Clay Tile 3 in. 4 in. 6 in. 8 in. 10 in. Gypsum Block 2 in. See Manufacturer 3 in. 4 in. 5 in. 121⁄2 111⁄2 6 in. 6 to 10 Wood Studs 2×4 12–16 in. o.c. Steel partitions 12 Plaster 1 inch 11 Cement 3 to 9 Gypsum Lathing Metal 6 Gypsum Board 1⁄2-in. 8 4 1 See Partitions 1

13 10 1 9 4 1 2 ⁄2

WALLS Brick 4 in. 8 in. 12 in. Hollow Concrete Block (Heavy Aggregate) 4 in. 1 6 in. See Manufactuer 8 in. 1 121⁄2-in. 1 Hollow Concrete Block 5 ⁄2 6 (Light Aggregate) 4 in. 6 in. 2 8 in. 3 12 in. 9 to 14 Clay tile (Load Bearing) 10 4 in. 6 in. 8 in. 3 12 in. 4 Stone 4 in. Glass Block 4 in. Window, Glass, Frame, & Sash 1⁄ Curtain Walls 2 2 Structural Glass 1 in. 1 1 ⁄2 Corrugated Cement Asbestos 1⁄4-in.

For weights of other materials used in building construction, see Table 7-4.


Weight lb per sq ft 17 18 28 34 40 91⁄2 101⁄2 121⁄2 14 181⁄2 2 4 10 5 1⁄ 2

2

40 80 120 30 43 55 80 21 30 38 55 25 30 33 45 55 18 8 See Manufacturer 15 3

7-8


SI UNITS FOR STRUCTURAL STEEL DESIGN

Although there are seven metric base units in the SI system, only four are currently used by AISC in structural steel design. These base units are listed in the Table 7-6.

Table 7-6. Base SI Units for Steel Design Quantity

Unit

Symbol

Length mass time temperature

meter kilogram second celcius

m kg s °C

Similarly, of the numerous decimal prefixes included in the SI system, only three are used in steel design; see Table 7-7.

Table 7-7. SI Prefixes for Steel Design Prefix

Order of Magnitude

Symbol

mega kilo milli

Expression

6

M k m

1,000,000 (one million) 1,000 (one thousand) 0.001 (one thousandth)

10 103 10− 3

In addition, three derived units are applicable to the present conversion. They are shown in Table 7-8.

Table 7-8. Derived SI Units for Steel Design Quantity

Name

Symbol

Expression

force stress energy

newton pascal joule

N Pa J

N = kg × m/s2 Pa = N/m2 J=N×m

Although specified in SI, the pascal is not universally accepted as the unit of stress. Because section properties are expressed in millimeters, it is more convenient to express stress in newtons per square millimeter (1 N/mm2 = 1 MPa). This is the practice followed in recent international structural design standards. It should be noted that the joule, as the unit of energy, is used to express energy absorption requirements for impact tests. Moments are expressed in terms of N×m. A summary of the conversion factors relating traditional U.S. units of measurement to the corresponding SI units is given in Table 7-9.

Table 7-9. Summary of SI Conversion Factors Multiply

by:

to obtain:

inch (in.) foot (ft) pound-mass (lb) pound-force (lbf) ksi ft-lbf psf plf

25.4 305 0.454 4.448 6.895 1.356 47.88 14.59

millimeters (mm) millimeters (mm) kilogram (kg) newton (N) N/mm2 joule (J)


N / m2 N/m


7-9

Note that fractions resulting from metric conversion should be rounded to whole millimeters. Common fractions of inches and their metric equivalent are in Table 7-10.

Table 7-10. SI Equivalents of Fractions of an Inch Fraction, in. 1⁄

16 1⁄ 8 3⁄ 16 1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16 1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

1

Exact conversion, mm

Rounded to: (mm)

1.5875 3.175 4.7625 6.35 7.9375 9.525 1.1125 12.7 15.875 19.05 22.225 25.4

2 3 5 6 8 10 11 13 16 19 22 25

Bolt diameters are taken directly from the ASTM Specifications A325M and A490M rather than converting the diameters of bolts dimensioned in inches. The metric bolt designations are in Table 7-11.

Table 7-11. SI Bolt Designation Designation

Diameter, mm

Diameter, in.

M16 M20 M22 M24 M27 M30 M36

16 20 22 24 27 30 36

0.63 0.79 0.87 0.94 1.06 1.18 1.42

The yield strengths of structural steels are taken from the metric ASTM Specifications. It should be noted that the yield points are slightly different from the traditional values. See Table 7-12. The modulus of elasticity of steel E is taken as 200,000 N/mm2. The shear modulus of elasticity of steel G is 77,000 N/mm2.

Table 7-12. SI Steel Yield Stresses ASTM Designation

2

Yield stress, N/mm

Yield stress, ksi

A36M

250

36.26

A572M Gr. 345 A588M

345

50.04

A852M

485

70.34

A514M

690

100.07


7 - 10


Table 7-13. WEIGHTS AND MEASURES International System of Units (SI)a (Metric practice) BASE UNITS

Quantity length mass time electric current thermodynamic temperature amount of substance luminous intensity

SUPPLEMENTARY UNITS

Unit

Symbol

metre kilogram second ampere kelvin mole candela

m kg s A K mol cd

Symbol plane angle solid angle

Unit

Symbol

radian steradian

rad sr

DERIVED UNITS (WITH SPECIAL NAMES)

Quantity force pressure, stress energy, work, quantity of heat power

Unit

Symbol

Formula

newton pascal

N Pa

kg-m/s2 N/m2

joule watt

J W

N-m J/s

DERIVED UNITS (WITHOUT SPECIAL NAMES)

Quantity area volume velocity acceleration specific volume density

Unit

Formula

square metre cubic metre metre per second metre per second squared cubic metre per kilogram kilogram per cubic metre

m2 m3 m/s m/s2 m3/kg kg/m3

SI PREFIXES

Multiplication Factor 18

000 = 10 000 = 1015 000 = 1012 000 = 109 000 = 106 000 = 103 100 = 102 10 = 101 0.1 = 10−1 0.01 = 10−2 0.001 = 10−3 0.000 001 = 10−6 0.000 000 001 = 10−9 0.000 000 000 001 = 10−12 0.000 000 000 000 001 = 10−15 0.000 000 000 000 000 001 = 10−18

1 000 000 000 1 000 000 1 000 1

000 000 000 000 1

000 000 000 000 000 1

Prefix

Symbol

exa peta tera giga mega kilo hectob dekab decib centib milli micro nano pico femto atto

E P T G M k h da d c m µ n p f a

aRefer to ASTM E380 for more complete information on SI. bUse is not recommended.



7 - 11

Table 7-14. WEIGHTS AND MEASURES United States System LINEAR MEASURE

Inches

Feet

Yards

.02778 .08333 = 1.0 = .33333 = 1.0 12.0 = 1.0 = 3.0 36.0 = 5.5 = 16.5 198.0 = = 220.0 7,920.0 = 660.0 = 1,760.0 63,360.0 = 5,280.0

Rods

Furlongs

.0050505 = = .0606061 = = .1818182 = = = = 1.0 = = 40.0 = = 320.0

.00012626 .00151515 .00454545 .025 1.0 8.0

Miles = = = = = =

.00001578 .00018939 .00056818 .003125 .125 1.0

SQUARE AND LAND MEASURE

Sq. Inches 1.0 = 144.0 = 1,296.0 = 39,204.0 =

Square Feet .006944 1.0 9.0 272.25 43,560.0

Square Yards = = = = =

Square Rods

.000772 .111111 1.0 30.25 4,840.0 3,097,600.0

Acres

.03306 = 1.0 = 160.0 = = 102,400.0

Sq. Miles

.000207 = .00625 = .0000098 = = .0015625 = 1.0 = 1.0 = 640.0

AVOIRDUPOIS WEIGHTS

Grains

Drams

Ounces

Pounds

.000143 .002286 = = .003906 .0625 = = .0625 = 1.0 = 1.0 = 16.0 = = 2,000.0 = 32,000.0

.03657 = 1.0 1.0 27.34375 = 16.0 = 437.5 256.0 = 7,000.0 = 512,000.0 140,000,000.0

DRY MEASURE

Pints 1.0 2.0 16.0 51.42627 64.0

Quarts = = = = =

.5 1.0 8.0 25.71314 32.0

Pecks

Cubic Feet

Bushels

= .0625 = .01945 = .01563 = .125 = .03891 = .03125 = .31112 = .25 = 1.0 = .80354 = 3.21414 = 1.0 = 1.2445 = 1.0 = 4.0

LIQUID MEASURE

Gills 1.0 4.0 8.0 32.0

= = = =

Pints

Quarts

.25 1.0 2.0 8.0

= .125 = .5 = 1.0 = 4.0

U.S. Gallons

Cubic Feet

= .03125 = .00418 = .125 = .01671 = .250 = .03342 = .1337 = 1.0 7.48052 = 1.0


Tons = .0000000714 = .00000195 = .00003125 = .0005 = 1.0

7 - 12


Table 7-15. SI CONVERSION FACTORSa Quantity Length

Area

Volume

Mass

Multiply

by

to obtain

inch foot yard mile (U.S. Statute)

25.400 0.305 0.914 1.609

millimetre metre metre kilometre

millimetre metre metre kilometre

39.370×10−3 3.281 1.094 0.621

inch foot yard mile

square inch square foot square yard square mile (U.S. Statute) acre acre

0.645×103 0.093 0.836 2.590 4.047×103 0.405

square millimetre square metre square metre square kilometre square metre hectare

mm m m km in ft yd mi mm2 m2 m2 km2 m2

square millimetre square metre square metre square kilometre square metre hectare

1.550×10−3 10.764 1.196 0.386 0.247×10−3 2.471

square inch square foot square yard square mile acre acre

cubic inch cubic foot cubic yard gallon (U.S. liquid) quart (U.S. liquid)

16.387×103 28.317×10−3 0.765 3.785 0.946

cubic millimetre cubic metre cubic metre litre litre

cubic millimetre cubic metre cubic metre litre litre

61.024×10−6 35.315 1.308 0.264 1.057

cubic inch cubic foot cubic yard gallon (U.S. liquid) quart (U.S. liquid)

in3 ft3 yd3 gal qt

ounce (avoirdupois) pound (avoirdupois) short ton

28.350 0.454 0.907×103

gram kilogram kilogram

g kg kg

gram kilogram kilogram

35.274×10−3 2.205 1.102×10−3

ounce (avoirdupois) pound (avoirdupois) short ton

aRefer to ASTM E380 for more complete information on SI. The conversion factors tabulated herein have been rounded.


in2 ft2 yd2 mi2

mm3 m3 m3 l l

oz av lb av


7 - 13

Table 7-15 (cont.). SI CONVERSION FACTORSa Quantity Force

Multiply 0.278 4.448

c

c

3.597 0.224

c

0.113 1.356

c

8.851 0.738

c

pound-force per square inch foot of water (39.2 F) c inch of mercury (32 F)

6.895 2.989 3.386

c

c

0.145

c

newton newton

c

pound-force-inch pound-force-foot

c c

newton-metre newton-metre

c

Pressure, Stress

to obtain

ounce-force c pound-force

c

Bending Moment

by

c

c c

kilopascal

newton c newton

c

c

c

ounce-force pound-force

N N

lbf

newton-metre newton-metre

N-m N-m

pound-force-inch pound-force-foot

lbf-in lbf-ft

kilopascal kilopascal c kilopascal c

kPa kPa kPa

lbf/in2 pound-force per square inch c foot of water (39.2 F) c inch of mercury (32 F) c

c

0.335 0.295

kilopascal kilopascal

c

Energy, Work, cfoot-pound-force b Heat British thermal unit b calorie c kilowatt hour

1.356 1.055×103 4.187 3.600×106

c

joule joule c joule c joule

0.738 0.948×10−3 0.239 0.278×10−6

c

c

foot-pound-force/second British thermal unit per hour c horsepower (550 ft lbf/s)

1.356 0.293 0.746

c

c

0.738

c

c c

Power

b

watt

joule joule c joule c joule c

foot-pound-force British thermal unit c calorie c kilowatt hour c

watt watt c kilowatt c

foot-pound-force/ second c British thermal unit c per hour c horsepower c (550 ft-lbf/s)

J J J J ft-lbf Btu kW-h W W kW ft-lbf/s

c c

3.412

kilowatt

1.341

watt

Angle

17.453×10−3 57.296

c

degree radian

c

Temperature

c

degree Fahrenheit degree Celsius

c

t°°C = (t°°F − 32)/1.8 t°°F = 1.8 × t°°C + 32

c c c c

radian degree degree Celsius degree Fahrenheit

aRefer to ASTM E380 for more complete information on SI. bInternational Table.

The conversion factors tabulated herein have been rounded.


Btu/h hp

rad

7 - 14


BRACING FORMULAS b

b

p

d

e

a

a

c

h

h

w

w

p

d

e

c

w

w

f

m

f m n

To Find Formula

Given bpw bw bp bp bfp bmp bpw afw cmw

( b + p )2 + w 2 √  √ b 2 +w 2 2 b ÷ (2b + p)

f m d e a c h h h

bpw bnw bnp bnp bfnp bmnp bnpw afw cmw

b (b + p ) ÷ (2 b + p ) bf ÷ (2b + p ) bm ÷ (2b + p ) bw ÷ (2b + p ) aw ÷ f cw ÷ m

(b + p )2 + w 2 √  √ (b − n )2 + w 2

b (b − n ) ÷ (2 b + p − n ) b (b + p ) ÷ (2 b + p − n ) bf ÷ (2b + p − n ) bm ÷ (2b + p − n ) bw ÷ (2b + p − n ) aw ÷ f cw ÷ m

k = (log B − log T) ÷ no. of panels. Constant k plus the logarithm of any line equals the log of the corresponding line in the next panel below.

p

d

e a

f m d e a c h h h

PARALLEL BLOCKING

b k

To Find Formula

Given

c

h

v

m

w

f

a = TH ÷ (T + e + p ) b = Th ÷ (T + e + p )

bpw bkv

T

Formula

f

 √ ( b + p )2 + w 2

m

 √ (b + k) + v 2

A

a

2

bkpvw

d

bw (b + k) ÷ [v (b + p ) + w (b + k )]

bkpvw

e

bv (b + p ) ÷ [v (b + p ) + w (b + k )]

bfkpvw

a

fbv ÷ [v (b + p ) + w (b + k )]

bkmpvw

c

bmw ÷ [v (b + p ) + w (b + k )]

bkpvw

h

bvw ÷ [v(b + p ) + w (b + k )]

afw

h

aw ÷ f

cmv

h

cv ÷ m

d = ce ÷ (T + e)

b

d

Given

c=√ (1 ⁄2 T + 1 ⁄2 e )2 + a 2 

To Find

c e A H

f

g

h

n m

log e = k + log T log f = k + log a log g = k + log b log m = k + log c log n = k + log d log p = k + log e

p A

B

The above method can be used for any number of panels. In the formulas for ‘‘a’’ and ‘‘b’’ the sum in parenthesis, which in the case shown is (T + e + p), is always composed of all the horizontal distances except the base.


GEOMETRIC AND TRIGONOMETRIC DATA

7 - 15

PROPERTIES OF PARABOLA AND ELLIPSE PARABOLA

ELLIPSE (x 2 ÷ H 2) + (y 2 ÷ B 2) = 1

Abscissa = x

¼

pe c. of g.

0.424H

.375B

Ordinate = y Abscissa = x

c. of g.

r ete

r

Major semi-axis = H

Ordinate = y

rim pe

ete rim

0.6H

½

Height = H

Apex

.424B

½ base = B Minor semi-axis = B 2 Parameter P = B H y2 x= P y = xP

a

b

Area =

HB

D

Area = .7854Dd

d

c

d

e a

1

b

Construction

1

2 H

c

2 H

3

3

B

e

Construction 4

4

B

B

5

AREA BETWEEN PARABOLIC CURVE AND SECANT Apex Center of gravity (shaded area)

Any secant

m .4m

H h

b

b

2

2

b B

B

Length b may vary from 0 to 2B


7 - 16


PROPERTIES OF THE CIRCLE Circumference = 6.28318 r = 3.14159 d Diameter = 0.31831 circumference Area = 3.14159r 2

x

π rA° = 0.017453 rA° 180 ° 180 ° a a Angle A ° = = 57.29578 r πr 4 b 2 + c2 Radius r = 8b A Chord c = 2 √  2 br − b 2 = 2 r sin 2 1 c A Rise b = r − √ 4 r 2− c2 = tan  2 2 4 2 A 2 r − x 2 = 2 r sin =r+y−√  4 y=b−r+√  r 2− x 2 Arc a =

b

y c A°

r

d

x=√ r 2− (r + y − b )2 

Diameter of circle of equal periphery as square Side of square of equal periphery as circle Diameter of circle circumscribed about square Side of square inscribed in circle

= 1.27324 side of square = 0.78540 diameter of circle = 1.41421 side of square = 0.70711 diameter of circle

CIRCULAR SECTOR

r = radius of circle y = angle ncp in degrees Area of Sector ncpo = 1 ⁄2 (length of arc nop x r)

o n

p

y

= Area of Circle ×

r

c

y 360

= 0.0087266 × r 2 × y CIRCULAR SEGMENT

r = radius of circle x = chord b = rise Area of Segment nop = Area of Sector ncpo − Area of triangle ncp

o b n

c

p

=

x

(Length of arc nop × r) − x (r − b) 2

Area of Segment nop = Area of Circle − Area of Segment nop

s

VALUES FOR FUNCTIONS OF π π = 3.14159265359, log = 0.4971499 _ _ 1 = 0.3183099, log = 1.5028501  1π = 0.5641896, log = 1.7514251 √ π _ _ π 1 π3 = 31.0062767, log = 1.4914496 = 0.1013212, log = 1.0057003 = 0.0174533, log = 2.2418774 180 π2 _ 1 180 = 0.0322515, log = 2.5085504 = 57.2957795, log = 1.7581226 π = 1.7724539, log = 0.2485749 √ π π3 _ _ Note: Logs of fractions such as 1 .5028501 and 2 .5085500 may also be written 9.5028501 − 10 and 8.5085500 − 10 respectively π2 = 9.8696044, log = 0.9942997



7 - 17

PROPERTIES OF GEOMETRIC SECTIONS A = d2

SQUARE

c=

Axis of moments through center

I= c

S=

d

r= d

Z=

SQUARE

d 2

d4 12

d3 6

d

= .288675 d

 √ 12

d3 4

A = d2 c=d

Axis of moments on base

I= d

c

S= r=

d4 3

d3 3

d = .577350 d 3 √

d

SQUARE

A = d2

Axis of moments on diagonal

c= I= c

S= r=

d4 12

d3 = .117851 d 3 6 √2 d

d

A = bd

RECTANGLE Axis of moments through center

c= I=

c

S= r= b

= .707107 d

= .288675 d  √ 12 3 2c d3 = = .235702 d 3 Z= 3 3√ 2

d

d

d 2 √

Z=

d 2

bd 3 12

bd 2 6

d

= .288675 d

 √ 12

bd

2

4


7 - 18


PROPERTIES OF GEOMETRIC SECTIONS (cont.) RECTANGLE

A = bd c=d


bd 3

I=

3

S=

c

d

bd 2 3

d

r=

3 √

= .577350 d

b

RECTANGLE

A = bd

Axis of moments on diagonal

bd  √ b2 + d 2 b3 d 3

c=

c

I=

S= b d

RECTANGLE

c a

d

b

6√  b2 + d 2

bd  √ 6 (b 2 + d2 )

r=

Axis of moments any line through center of gravity

6(b 2 + d 2) b2 d 2

A = bd b sin a + d cos a c= 2 bd (b 2 sin2 a + d 2cos2 a ) I= 12 bd (b 2 sin2 a + d 2cos2 a ) S= 6 (b sin a + d cos a ) r=

 √

b2 sin2 a + d 2cos2 a 12

A = bd − b 1 d 1


I= c

d

d1

12

S=

b1

r= b

d 2 bd 3− b 1d13

c=

HOLLOW RECTANGLE

Z=

bd 3− b 1d13 6d

 √

bd 3− b1d13 12 A

bd 2 b 1d 12 4

−

4



7 - 19

PROPERTIES OF GEOMETRIC SECTIONS (cont.) A = b(d − d 1 ) EQUAL RECTANGLES

d

c=

Axis of moments through center of gravity

2

b(d 3− d 13 )

I=

d

d1

r= b


d

d 3− d 13

12(d − d1)

b

t

b

2

y

c

y1

c1

t1 b1

2

4

r=

 √

Z=

TRIANGLE

(d 2− d 12 )

A = bt + b 1 t1 1⁄ bt 2+ b t (d − 1 ⁄ t ) 11 2 21 c= A b 1t13 bt 3 I= + bty 2 + + b1 t1 y 12 12 12 l l S= S1 = c c1

d1

t1

 √

Z=

UNEQUAL RECTANGLES

t

12

b(d 3− d 13 ) S= 6d

c

I A

 t + t1   A  d −  2   2 

bd 2 2d c= 3 bd 3 I= 36 bd 2 S= 24 d r= = .235702 d  √ 18 A=


c d

b

TRIANGLE

c

d

b

bd 2 c=d bd 3 I= 12 bd 2 S= 12 d r= = .408248 d 6 √ A=



7 - 20


PROPERTIES OF GEOMETRIC SECTIONS (cont.) TRAPEZOID Axis of moments through center of gravity b1

c

A=

d (b + b 1) 2

c=

d (2 b + b 1 ) 3 (b + b 1 )

I=

d 3(b 2 + 4 bb1 + b 21 ) 36 (b + b1)

d

S= b

d 2(b 2 + 4 bb1 + b 21 )

12 (2 b + b 1)

d r=  √ 2(b 2 + 4bb1 + b 21 ) 6(b + b 1) A=

πd 2 = π R 2 = .785398 d 2= 3.141593 R 2 4

c=

d =R 2

I=

π d 4 π R4 = = .049087 d 4= .785398 R 4 64 4

CIRCLE Axis of moments through center

c

R

S=

d

π d 3 π R3 = = .098175 d 3= .785398 R 3 32 4

r=

d R = 4 2

Z=

d3 6

π(d 2− d 12 ) = .785398 (d 2− d 12 ) 4 d c= 2 π(d 4− d 14 ) I= = .049087 (d 4− d 14 ) 64 π(d 4− d 14 ) d 4− d 14 S= = .098175 32 d d

A= HOLLOW CIRCLE Axis of moments through center

c d

d1

r=

HALF CIRCLE

 √ d 2+ d 12 4

Z=

d 13 d − 6 6

A=

πR 2 = 1.570796 R 2 2

3

 4  = .575587 R C = R 1 − 3π  


π

R

d

c

I = R4 

8

S=

−

8 

9π 

= .109757 R 4

2 R 3 (9π − 64 ) = .190687 R 3 24 (3π − 4)

r=R

 √ 9π 2 − 64  6π

= .264336 R



7 - 21

PROPERTIES OF GEOMETRIC SECTIONS (cont.) 4 ab 3 2 m= a 5 16 3 I1 = a b 175 4 ab 3 I2 = 15 32 3 I3 = a b 105

A=

PARABOLA 2

a 1 m 3

1 3 b 2

2 ab 3 2 m= a 5 3 n= b 8 8 I1 = a3b 175 19 ab 3 I2 = 480 16 3 I3 = a b 105 2 I4 = ab 3 15

A=

HALF PARABOLA 4

2

n apex

a 1 m 3

1 3

2 b 4

COMPLEMENT OF HALF PARABOLA 2

1 ab 3 7 m= a 10 3 n= b 4 37 a3b I1 = 2,100 1 I2 = ab 3 80

A=

n apex

1

1

a

m

2 b

PARABOLIC FILLET IN RIGHT ANGLE 2

t 2√ 2 t b= 2 √ 1 A = t2 6 A=

m

1

1

b

n

t 2

m=n=

a

I 1 = I2 =

4 t 5 11

2,100

t4

t


7 - 22


PROPERTIES OF GEOMETRIC SECTIONS (cont.) * HALF ELLIPSE

A=

1 π ab 2

m=

4a 3π

2

a 1

1 m 3

3

 π 8 I1 = a 3 b  −  8 9π  I2 =

1 πab3 8

I3 =

1 a3 π 8 b

b 2

1 πab 4 4a m= 3π 4b n= 3π

A=

* QUARTER ELLIPSE 2

4

n

 π 4 I1 = a 3 b  −  16 9π 

a 1 3

1 m 3

2 b

π 4  I2 = ab 3  −  16 9π  I3 =

4

1 πa3b 16

I4 = π ab3

* ELLIPTIC COMPLEMENT 2

n

 π A = ab 1 −   4

1

1 a

m=

a  π 6 1 −  4 

n=

b  π 6 1 −  4 

m

2 b

1

I1 = a 3 b 

3  

−

π − 16

 1   π 36  1 −   4  

1 π  1 I2 = ab 3  − −  π   3 16 36 1 −    4  

*To obtain properties of half circle, quarter circle, and circular complement, substitute a = b = R.



7 - 23

PROPERTIES OF GEOMETRIC SECTIONS (cont.) n = Number of sides 180 º φ=

REGULAR POLYGON

n


1

a = 2√  R2 − R  21 a R= 2 sin φ a R1 = 2 tan φ 1 1 A = na 2 cot φ = nR 2 sin 2φ = nR 12 tan φ 4 2 A (6 R 2− a 2 ) A (12 R 12 + a 2 ) I1 = I2 = = 24 48

a

2

φ R R1

2 1

r1 = r2 =

 √ √  6 R 2− a 2

24

=

12 R 12 + a 2

48

ANGLE

tan 2φ =


2K Iy − Ix

b 2 + ct d 2 + at y= 2 (b + c ) 2(b + c ) K = Product of Inertia about XjX and YjY A = t (b + c ) x =

b

=±

a

t

Z

1 3 1 IY = 3 I z = Ix Iw = Ix Ix =

Y W

90° φ

X y

c

d

X

t

Z

Y

(t (d − y)3 + by 3 − a (y − t)3) (t (b − x)3 + dx 3 − c (x − t)3 ) sin 2 φ + IY cos 2 φ + K sin2 φ cos 2 φ + IY sin 2 φ − K sin2 φ

K is negative when heel of angle, with respect to center of gravity, is in 1st or 3rd quadrant, positive when in 2nd or 4th quadrant.

W

x

abcdt 4(b + c)

BEAMS AND CHANNELS Transverse force oblique through center of gravity F

F

Y

Y x

x

φ

φ X

3

X 3

3

3

I3 = Ix sin2 φ + IY cos 2 φ I4 = Ix cos2 φ + IY sin2 φ y  x fb = M  sin φ + cos φ I I  x  Y

where M is bending moment due to force F.

X

X y

y Y

Y 4

4


7 - 24


TRIGONOMETRIC FORMULAS Radius AF = 1 TRIGOMETRIC FUNCTIONS G

H

sin A cos A

D L

tan A

B

cot A

c

a

b

C

sec A A

F

cosec A

RIGHT ANGLED TRIANGLES

a 2= c 2− b 2 b 2= c 2− a 2 c 2= a 2+ b 2

B c b

A

= sin 2 A + cos 2 A = sin A cosec A = cos A sec A = tan A cot A 1 cos A = = = cos A tan A = √  1 − cos 2 A = BC cot A cosec A sin A 1 = = = sin A cot A = √  1 − sin2 A = AC tan A sec A sin A 1 = = = sin A sec A = FD cos A cot A cos A 1 = = = cos A cosec A = HG sin A tan A tan A 1 = = = AD sin A cos A cot A 1 = = = AG cos A sin A

a C

Required Known

A

B

a tan A = b a sin A = c

b tan B = a a cos B = c

√ c −a 

A, a

90° − A

a cot A

A, b

90° − A

b tan A

A, c

90° − A

c sin A

a, b a, c

OBLIQUE ANGLED TRIANGLES

a

c b

C

b

2

s=

a+b+c 2

K=

 √

B

A

a

c

Area

√ a 2+b 2 

ab

2 a√  c 2− a 2 2

2

a sin A

a 2cot A 2

b cos A

b2 tan A 2 c2 sin 2A 4

c cos A

a 2= b 2+ c 2− 2bc cos A b 2= a 2+ c 2− 2ac cos B c 2= a 2+ b 2− 2 ab cos C

( s − a ) (s − b ) ( s − c ) s

Required Known

A

a, b , c

K 1 tan A = 2 s−a

B

C

180° − (A + B )

a, A, B sin B =

a , b, A a, b, C tan A =

a sin C b − a cos C

b

c

K K 1 1 tan B = tan C = 2 2 s−b s−c

b sin A

a

Area

√  s (s − a) (s − b) (s − c) a sin B sin A

a sin C sin A b sin C sin B

 √ a 2+ b 2− 2 ab cos C

ab sin C

2


DECIMAL EQUIVALENTS

7 - 25

DECIMALS OF AN INCH For each 64th of an inch With millimeter equivalents Fractions

1⁄ ths 64

Decimal

Millimeters (Approx.)

—

1 2 3 4

.015625 .03125 .046875 .0625

0.397 0.794 1.191 1.588

5 6 7 8

.078125 .09375 .109375 .125

1.984 2.381 2.778 3.175

9 10 11 12

.140625 .15625 .171875 .1875

3.572 3.969 4.366 4.763

13 14 15 16

.203125 .21875 .234375 .250

5.159 5.556 5.953 6.350

17 18 19 20

.265625 .28125 .296875 .3125

6.747 7.144 7.541 7.938

21 22 23 24

.328125 .34375 .359375 .375

8.334 8.731 9.128 9.525

25 26 27 28

.390625 .40625 .421875 .4375

9.922 10.319 10.716 11.113

29 30 31 32

.453125 .46875 .484375 .500

11.509 11.906 12.303 12.700

1⁄ 32

—

1⁄ 16

—

3⁄ 32

— 1⁄ 8

—

5⁄ 32

—

3⁄ 16

—

7⁄ 32

— 1⁄ 4

—

9⁄ 32

—

5⁄ 16

— 11⁄

32

— 3⁄ 8 — 13⁄ 32

— 7⁄ 16

— 15⁄ 32

— 1⁄ 2

Decimal

Millimeters (Approx.)

33 34 35 36

.515625 .53125 .546875 .5625

13.097 13.494 13.891 14.288

37 38 39 40

.578125 .59375 .609375 .675

14.684 15.081 15.478 15.875

41 42 43 44

.640625 .65625 .671875 .6875

16.272 16.669 17.066 17.463

45 46 47 48

.703125 .71875 .734375 .750

17.859 18.256 18.653 19.050

49 50 51 52

.765625 .78125 .796875 .8125

19.447 19.844 20.241 20.638

53 54 55 56

.828125 .84375 .859375 .875

21.034 21.431 21.828 22.225

57 58 59 60

.890625 .90625 .921875 .9375

22.622 23.019 23.416 23.813

61 62 63 64

.953125 .96875 .984375 1.000

24.209 24.606 25.003 25.400

Fractions

1⁄ ths 64

— 17⁄ 32

— 9⁄ 16

— 19⁄ 32

— 5⁄ 8 — 21⁄ 32

— 11⁄ 16

— 23⁄ 32

— 3⁄ 4 — 25⁄ 32

— 13⁄ 16

— 27⁄ 32

— 7⁄ 8 — 29⁄ 32

— 15⁄ 16

— 31⁄ 32

— 1


7 - 26


DECIMALS OF A FOOT For each 32nd of an inch Inch

0

1

2

3

4

5

0

0 .0026 .0052 .0078

.0833 .0859 .0885 .0911

.1667 .1693 .1719 .1745

.2500 .2526 .2552 .2578

.3333 .3359 .3385 .3411

.4167 .4193 .4219 .4245

.0104 .0130 .0156 .0182

.0938 .0964 .0990 .1016

.1771 .1797 .1823 .1849

.2604 .2630 .2656 .2682

.3438 .3464 .3490 .3516

.4271 .4297 .4323 .4349

5⁄ 16 11⁄ 32

.0208 .0234 .0260 .0286

.1042 .1068 .1094 .1120

.1875 .1901 .1927 .1953

.2708 .2734 .2760 .2786

.3542 .3568 .3594 .3620

.4375 .4401 .4427 .4453

3⁄ 8 13⁄ 32 7⁄ 16 15⁄ 32

.0313 .0339 .0365 .0391

.1146 .1172 .1198 .1224

.1979 .2005 .2031 .2057

.2812 .2839 .2865 .2891

.3646 .3672 .3698 .3724

.4479 .4505 .4531 .4557

1⁄ 2 17⁄ 32 9⁄ 16 19⁄ 32

.0417 .0443 .0469 .0495

.1250 .1276 .1302 .1328

.2083 .2109 .2135 .2161

.2917 .2943 .2969 .2995

.3750 .3776 .3802 .3828

.4583 .4609 .4635 .4661

5⁄ 8 21⁄ 32 11⁄ 16 23⁄ 32

.0521 .0547 .0573 .0599

.1354 .1380 .1406 .1432

.2188 .2214 .2240 .2266

.3021 .3047 .3073 .3099

.3854 .3880 .3906 .3932

.4688 .4714 .4740 .4766

3⁄ 4 25⁄ 32 13⁄ 16 27⁄ 32

.0625 .0651 .0677 .0703

.1458 .1484 .1510 .1536

.2292 .2318 .2344 .2370

.3125 .3151 .3177 .3203

.3958 .3984 .4010 .4036

.4792 .4818 .4844 .4870

7⁄ 8 29⁄ 32 15⁄ 16 31⁄ 32

.0729 .0755 .0781 .0807

.1563 .1589 .1615 .1641

.2396 .2422 .2448 .2472

.3229 .3255 .3281 .3307

.4063 .4089 .4115 .4141

.4896 .4922 .4948 .4974

1⁄

1⁄ 3⁄

32 16 32

1⁄ 8

5⁄ 3⁄ 7⁄

32 16 32

1⁄ 4

9⁄

32


DECIMAL EQUIVALENTS

7 - 27

DECIMALS OF A FOOT (cont.) For each 32nd of an inch Inch

6

7

8

9

10

11

0 1⁄ 32 1⁄ 16 3⁄ 32

.5000 .5026 .5052 .5078

.5833 .5859 .5885 .5911

.6667 .6693 .6719 .6745

.7500 .7526 .7552 .7578

.8333 .8359 .8385 .8411

.9167 .9193 .9219 .9245

1⁄ 8 5⁄ 32 3⁄ 16 7⁄ 32

.5104 .5130 .5156 .5182

.5938 .5964 .5990 .6016

.6771 .6797 .6823 .6849

.7604 .7630 .7656 .7682

.8438 .8464 .8490 .8516

.9271 .9297 .9323 .9349

1⁄ 4 9⁄ 32 5⁄ 16 11⁄ 32

.5208 .5234 .5260 .5286

.6042 .6068 .6094 .6120

.6875 .6901 .6927 .6953

.7708 .7734 .7760 .7786

.8542 .8568 .8594 .8620

.9375 .9401 .9427 .9453

3⁄

.5313 .5339 .5365 .5391

.6146 .6172 .6198 .6224

.6979 .7005 .7031 .7057

.7813 .7839 .7865 .7891

.8646 .8672 .8698 .8724

.9479 .9505 .9531 .9557

.5417 .5443 .5469 .5495

.6250 .6276 .6302 .6328

.7083 .7109 .7135 .7161

.7917 .7943 .7969 .7995

.8750 .8776 .8802 .8828

.9583 .9609 .9635 .9661

.5521 .5547 .5573 .5599

.6354 .6380 .6406 .6432

.7188 .7214 .7240 .7266

.8021 .8047 .8073 .8099

.8854 .8880 .8906 .8932

.9688 .9714 .9740 .9766

.5625 .5651 .5677 .5703

.6458 .6484 .6510 .6536

.7292 .7318 .7344 .7370

.8125 .8151 .8177 .8203

.8958 .8984 .9010 .9036

.9792 .9818 .9844 .9870

.5729 .5755 .5781 .5807

.6563 .6589 .6615 .6641

.7396 .7422 .7448 .7474

.8229 .8255 .8281 .8307

.9063 .9089 .9115 .9141

.9896 .9922 .9948 .9974

13⁄

8 32

7⁄ 16 15⁄ 32 1⁄

17⁄

2 32

9⁄ 16 19⁄ 32 5⁄

21⁄

8 32

11⁄ 16 23⁄ 32 3⁄

25⁄ 13⁄ 27⁄ 7⁄

29⁄ 15⁄ 31⁄

4 32 16 32 8 32 16 32


8-1

PART 8 BOLTS, WELDS, AND CONNECTED ELEMENTS OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 BOLTED CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Non-High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 Design Strength of Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 ECCENTRICALLY LOADED BOLT GROUPS . . . . . . . . . . . . . . . . . . . . . . . 8-28 ANCHOR RODS OR THREADED RODS . . . . . . . . . . . . . . . . . . . . . . . . . 8-88 OTHER MECHANICAL FASTENERS . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-92 WELDED CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-98 Fillet Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-118 Complete-Joint-Penetration Groove Welds . . . . . . . . . . . . . . . . . . . . . . . . 8-122 Partial-Joint-Penetration Groove Welds . . . . . . . . . . . . . . . . . . . . . . . . . . 8-125 Flare Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-127 Plug and Slot Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-128 Design Strength of Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-129 Prequalified Welded Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-131 ECCENTRICALLY LOADED WELD GROUPS . . . . . . . . . . . . . . . . . . . . . . 8-154 CONSTRUCTION COMBINING BOLTS AND WELDS . . . . . . . . . . . . . . . . . 8-211 CONNECTED ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Design Strength of Connecting Elements . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Members with Copes, Blocks, or Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-225 Other Elements in Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-237 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-238


8-2

BOLTS, WELDS, AND CONNECTED ELEMENTS


OVERVIEW

8-3

OVERVIEW

Part 8 contains general information, design considerations, examples, and design aids for the design of bolts, anchor rods, other mechanical fasteners, welds, and connected elements in connections. It is based on the provisions of the 1993 LRFD Specification. Supplementary information may also be found in the Commentary on the LRFD Specification. Following is a detailed overview of the topics addressed. BOLTED CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Alternative Design Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Compatible Nuts and Washers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8 Economical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8 Dimensions and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9 Entering and Tightening Clearances . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12 Snug-Tightened and Fully Tensioned Installation . . . . . . . . . . . . . . . . . . . . 8-12 Inspection of Fully Tensioned High-Strength Bolts . . . . . . . . . . . . . . . . . . . 8-15 Galvanizing High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18 Reuse of High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 Non-High-Strength Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 Dimensions and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 Entering and Tightening Clearances . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 Design Strength of Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19 Bolt Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22 Bearing Strength at Bolt Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-23 Bolt Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-23 Slip Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-25 ECCENTRICALLY LOADED BOLT GROUPS . . . . . . . . . . . . . . . . . . . . . . . 8-28 Eccentricity in the Plane of the Faying Surface . . . . . . . . . . . . . . . . . . . . . . 8-28 Instantaneous Center of Rotation Method . . . . . . . . . . . . . . . . . . . . . . . . 8-28 Elastic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33 Eccentricity Normal to the Plane of the Faying Surface . . . . . . . . . . . . . . . . . . 8-36 Case I—Neutral Axis Not at Center of Gravity . . . . . . . . . . . . . . . . . . . . . 8-37 Case II—Neutral Axis at Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . 8-38 ANCHOR RODS OR THREADED RODS . . . . . . . . . . . . . . . . . . . . . . . . . 8-88 Minimum Edge Distance and Embeddment Length . . . . . . . . . . . . . . . . . . . . 8-88 Welding to Anchor Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-89 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8-4


Hooked Anchor Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-89 Headed Anchor Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-90 OTHER MECHANICAL FASTENERS . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-92 Clevises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-92 Turnbuckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-92 Sleeve Nuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-93 Recessed-Pin Nuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-93 Cotter Pins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-93 WELDED CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-98 Weldability of Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-98 Chemical Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-99 Grain Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-100 Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-100 Structural Welding Materials and Processes . . . . . . . . . . . . . . . . . . . . . . . 8-101 SMAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-102 SAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-105 GMAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-106 FCAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-106 ESW and EGW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-107 Thermal Cutting and Air-Arc Gouging . . . . . . . . . . . . . . . . . . . . . . . . . 8-108 Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-108 VT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-109 DPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-109 MT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-109 RT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-110 UT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-110 Economical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-111 Welding Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-112 Weld Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-112 Weld Metal Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-112 Deposit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-113 Prior Qualification of Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-113 Minimizing Weld Repairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-113 Lamellar Tearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-113 Fatigue Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-113 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

OVERVIEW

8-5

Notch Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-114 Impact Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-114 Arc Strikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-114 Other Considerations in Welded Construction . . . . . . . . . . . . . . . . . . . . . . . 8-115 Matching Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-115 Welding Shapes from ASTM A6 Groups 4 and 5 . . . . . . . . . . . . . . . . . . . . 8-115 Intersecting Welds and Triaxial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . 8-116 Painting Welded Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-117 Clearances for Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-118 Fillet Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-118 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-118 Minimum Effective Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-119 Minimum Fillet Weld Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-119 Maximum Fillet Weld Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-119 End Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-120 Fillet Welds in Holes or Slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-121 Other Limitations on Fillet Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-121 Minimum Shelf Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-122 Complete-Joint-Penetration Groove Welds . . . . . . . . . . . . . . . . . . . . . . . . 8-122 Extension, Runoff, Backing, and Spacer Bars . . . . . . . . . . . . . . . . . . . . . . 8-122 Weld Access Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-125 Partial-Joint-Penetration Groove Welds . . . . . . . . . . . . . . . . . . . . . . . . . . 8-125 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-127 Intermittent Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-127 Flare Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-127 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-128 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-128 Plug and Slot Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-128 Design Strength of Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-129 Weld Metal Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-129 Base Metal Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-129 Prequalified Welded Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-131 ECCENTRICALLY LOADED WELD GROUPS . . . . . . . . . . . . . . . . . . . . . . 8-154 Eccentricity in the Plane of the Faying Surface . . . . . . . . . . . . . . . . . . . . . . 8-154 Instantaneous Center of Rotation Method . . . . . . . . . . . . . . . . . . . . . . . . 8-154 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8-6


Elastic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-159 Eccentricity Normal to the Plane of the Faying Surface . . . . . . . . . . . . . . . . . 8-211 CONSTRUCTION COMBINING BOLTS AND WELDS . . . . . . . . . . . . . . . . . 8-211 CONNECTED ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Economical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Design Strength of Connected Elements . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Shear Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Shear Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Block Shear Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-212 Tension Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-225 Tension Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-225 Members with Copes, Blocks, or Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 8-225 Flexural Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-225 Local Web Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-226 Lateral Torsional Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-229 Other Elements In Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-237 Shims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-237 Fillers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-237 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-238


BOLTED CONSTRUCTION

8-7

BOLTED CONSTRUCTION High-Strength Bolts

LRFD Specification Section A3.3 permits the use of ASTM A325 and A490 high-strength bolts. ASTM A325 bolts are available in diameters from 1⁄2-in. to 11⁄2-in. in two types. Type 1 medium-carbon-steel bolts are for general purpose use and use in elevated temperatures; they may be galvanized. Type 3 bolts offer improved atmospheric corrosion resistance and weathering characteristics similar to those of ASTM A242 or A588 steels. ASTM A490 bolts are available in diameters from 1⁄2-in. to 11⁄2-in. in two types. Type 1 bolts are alloy-steel bolts. Type 3 are alloy-steel bolts with improved atmospheric corrosion resistance and weathering characteristics similar to those of ASTM A242 or A588 steels. ASTM A490 bolts should not be galvanized and caution should be exercised if used in highly corrosive environments. Type 2 (martensite) bolts, popular for many years, have been discontinued. Information on this type can be found in previous editions of the AISC Manual of Steel Construction. When bolts of diameter larger than 11⁄2 in. are required, ASTM A449 bolts are permitted to be used for snug-tightened and fully tensioned bearing-type connections; this material is not recognized in LRFD Specification Section A3.3 for use in slip-critical connections nor for use as bolts in diameters not greater than 11⁄2 in. ASTM A449 bolts may be galvanized. When an ASTM A449 bolt is used in tension or bearing and is tightened in excess of 50 percent of its minimum specified tensile strength, LRFD Specification Section J3.1 requires that an ASTM F436 washer be installed under the head of the bolt. The nut must be from the approved list in RCSC Specification Section 2(c). Since ASTM A325 nuts and washers for use with high-strength bolts are available only up to 11⁄2-in. diameter, reference should be made to ASTM A563 for nuts and ASTM F436 for washers to select suitable sizes and grades for the intended application. While ASTM A449 seems to be the equal of ASTM A325, there are two important differences which should be noted. First, ASTM A449 bolts are not produced to the same inspection and quality assurance requirements as ASTM A325 bolts. Second, ASTM A449 bolts are not produced to the same heavy-hex head and nut dimensions. Alternative Design Bolts

RCSC Specification Section 2d permits the use of other fasteners when they meet the requirements as outlined therein. Figure 8-1 shows a tension-control or “twist-off” bolt which is installed with a special tool which twists off the splined end when the proper

Fig. 8-1. Tension-control or “twist-off” bolt. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8-8


Table 8-1. Compatability of High-Strength Bolts, Nuts, Washers F436 Washer Grade

ASTM Bolt Desig.

Type

Coating

Recommended

Suitable

Recommended

A325

1

plain

C

C3, D, DH, DH3

1

galvanized

DH

—

1

3

plain

C3

DH3

3

1

plain

DH

DH3

1

3

plain

DH3

—

3

1

plain

A

C, C3, D, DH, DH3

1

galvanized

DH

D

1

A490

A449

A563 Heavy Hex Nut Grade

bolt tension is achieved. Tension-control bolts are commonly available to meet the specifications of ASTM A325 and A490. Compatible Nuts and Washers

The compatibility of ASTM A563 nuts and F436 washers with the aforementioned high-strength bolt specifications is as listed in Table 8-1. Alternatively, appropriate ASTM A194 nuts may be used. RCSC Specification Section 7c gives general requirements for when washers are required for high-strength bolts. Economical Considerations

Since the material cost per unit of strength of ASTM A490 bolts is comparable with that of ASTM A325 bolts, it might seem more cost effective to reduce the number of bolts in a given connection by specifying ASTM A490 bolts. However, ASTM A490 bolts are more difficult to tighten and raise inventory and quality control issues associated with the use of multiple fastener grades; mixing of ASTM A325 and A490 bolts of the same diameter should be avoided to assure that the ASTM A490 bolts are installed in the proper location. Thus, the net benefit of specifying ASTM A490 bolts may be less than expected; cost ratios should be considered by the designer. Similarly, cost ratios between grades of alternative design bolts will vary from those of conventional high-strength bolts. Thus, the decision regarding fastener selection will vary accordingly. Regardless of the bolt type selected, the normal sizes of 3⁄4-in., 7⁄8-in., and 1-in. diameter are usually preferred. Diameters above one inch are not commonly available, nor are they practical since special tools may be required to achieve fully tensioned installation. Bearing-type connections should be specified whenever possible. Slip-critical connections with coatings other than clean mill scale incur appreciable extra costs associated with blasting, painting, drying, assembling, reblasting, and abrasion touch-up. If slipcritical connections are required for the proper serviceablity of the structure, care should be taken to avoid requiring the faying surfaces to be masked as this also contributes great AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BOLTED CONSTRUCTION

8-9

expense; coatings which provide a Class A or Class B slip coefficient may be an economical alternative to masking. Dimensions and Weights

ASTM A325 and A490 bolts, A563 nuts, and F436 washers are given identifying marks as illustrated in Figure 8-2. A detailed description of identifying marks may be found in the RCSC Specification. Dimensions of ASTM A325 and A490 bolts, A563 nuts, and F436 washers are given and illustrated in Table 8-2. Threading dimensions of highstrength bolts are given in Table 8-7. Weights of conventional ASTM A325 and A490 bolts, A563 nuts, and F436 washers are given in Table 8-3. For dimensions and weights of tension-control ASTM A325 and A490 bolts, refer to manufacturers’ literature or IFI. For dimensions and weights of ASTM A449 bolts, refer to Table 8-6. Threads for high-strength bolts may be rolled or cut. Note that thread lengths for high-strength bolts are shorter than those for non-high-strength bolts. This allows the threads to be excluded from the shear plane when the thickness of the connected ply closest to the nut is as shown in Figure 8-3. While the RCSC Specification permits some thread run-out into the shear plane, it is important to provide sufficient thread to avoid jamming the nut into the run-out when tightening the bolt. Inspection controversy will be reduced by recognizing that bolts intentionally have a limited thread length, a manufacturing tolerance, and limited length increments; as with all manufactured items, dimensional tolerances must be considered. The RCSC Specification recognizes these tolerances in two ways. First, additional washers are permitted to be used under the nut or under the head when circumstances permit. Second, there is no specified bolt “stick-through” requirement since only fullthread engagement of the nut is required; from RCSC Specification Section 2(b), “…The length of bolts shall be such that the end of the bolt will be flush with or outside the face of the nut when properly installed.” A requirement for “stick-through”, sometimes written in project specifications, increases the risk of jamming the nut on the thread run-out, and thus, of preventing tightening. A “stick-through” requirement will not enhance the performance of the bolt and should not be included in a project specification. Alternatively, ASTM A325 bolts with length less than or equal to four times the nominal diameter may be ordered as fully threaded with the designation ASTM A325 T. Fully threaded ASTM A325 T bolts are not for use in bearing-type X connections since it would be impossible to exclude the threads from the shear plane. While this supplementary provision exists for ASTM A325 bolts, there is no similar supplementary provision made in ASTM A490 for full-length threading. The ordered length of ASTM A325 and A490 bolts should be calculated as the grip (see Figure 8-2) plus the thickness of the washer(s) plus the allowance from Table 8-2. A thickness of 5⁄32-in. for circular washers and 5⁄16-in. for beveled washers should be provided per washer used; refer to the RCSC Specification for washer requirements. This total should be rounded to the next higher one-quarter inch. Note that bolts longer than five inches are generally available only in 1⁄2-in. increments, except by special arrangement with the manufacturer or vendor. While longer lengths may be ordered, an 8-in. length is generally the maximum stock length available. Clipped washers are available for use in areas of tight clearance.


Type 1

Type 3*


HARDENED WASHER (Beveled) 2

12

3

Hex head

Marking for washer used with Type 3 bolt

ASTM A490 BOLT

Grip

ASTM A325 BOLT

Washer face

Type 1 NUT MARKING

Type 3*

3

Alt. marking “D” or “DH” for type 1 “DH3” for Type 3

Nut marking “DH” for Type 1 “DH3” for Type 3

X

*Bolt heads, nuts, & washers shall include manuf. identification symbol. The manuf. may also add other marks indicating weathering grade.

X Hex Nut* Type 1 and 3*

See RCSC Specification for rules governing use of hardened washers

Hex nut

Standard marking indicates Grade C

Fig. 8-2. Identifying high-strength bolts, nuts, and washers.

HARDENED* WASHER (Plain)

Type 1

Type 3* Optional Clip

A490

A490

Standard bolt marking

BOLT HEAD MARKING

A325

A325

Three radial lines @ 120°, optional

See RCSC Specification for rules governing use of washers Grip

Dia. Dia.

8 - 10 BOLTS, WELDS, AND CONNECTED ELEMENTS

BOLTED CONSTRUCTION

8 - 11

Table 8-2. Dimensions of High-Strength Fasteners, in. E I.D.

Thread Length

O.D.

T

A325 E I.D.

H

F

Bolt Length

H W Nut may be chamfered on both faces

A T

A

Nominal Bolt Diameter, in.

A563 Nutsb

A325 and A490 Boltsa

Measurement Width Across Flats F Height H Thread Length Bolt Lengthf =Grip + → Width Across Flats W Height H

F436 Square or Rect. Washersc,e

F436 Circular Washersc

Nom. Outside Diameter OD

a b c

d e f

Nom. Inside Diameter ID Thckns. T

1⁄ 2

5⁄ 8

3⁄ 4

7⁄ 8

7⁄

11 ⁄16

11 ⁄4

16

25 ⁄

15 ⁄

1

11 ⁄4

13 ⁄8

11 ⁄2

13 ⁄4

2

2

21 ⁄4

21 ⁄4

7⁄

1

11 ⁄8

11 ⁄4

11 ⁄2

15 ⁄8

13 ⁄4

17 ⁄8

5⁄

11⁄

8

16

7⁄ 31 ⁄

64

8

32

1

1 1 ⁄8

17 ⁄16

15 ⁄8

113 ⁄16

35 ⁄

39 ⁄

64

64

11⁄

16

1 1 ⁄4

1 3 ⁄8

2

23⁄16

23 ⁄8

27 ⁄

15 ⁄

25 ⁄

32

32

1 1 ⁄2

16

8

11 ⁄16

11 ⁄4

17 ⁄16

15 ⁄8

113 ⁄16

2

23⁄16

23 ⁄8

64

39 ⁄

47 ⁄

55 ⁄

64

63 ⁄

17⁄64

17⁄32

111⁄32

115 ⁄32

13 ⁄4

2

21 ⁄4

21 ⁄2

23 ⁄4

3

11 ⁄8

11 ⁄4

13 ⁄8

11 ⁄2

15 ⁄8

64

11 ⁄16

15 ⁄16

17 ⁄

11⁄

32

16

64

115 ⁄32 13 ⁄

16

15 ⁄

16

64

Max.

0.097

0.122

0.122

0.136

0.136

0.136

0.136

0.136

0.136

Min.

0.177

0.177

0.177

0.177

0.177

0.177

0.177

0.177

0.177

1

1 3⁄32

17⁄32

15⁄16

Min. Edge Distance E d

7⁄

Min. Side Dimension A

13⁄4

13 ⁄4

13 ⁄4

13 ⁄4

13 ⁄4

21 ⁄4

21 ⁄4

21 ⁄4

21 ⁄4

5⁄

5⁄

5⁄

5⁄

5⁄

5⁄

5⁄

5⁄

5⁄

Mean Thckns. T Taper in Thickness Min. Edge Distance E d

16

16

2:12 7⁄

16

9⁄

16

16

2:12 9⁄

16

21 ⁄

32

16

25 ⁄

32

16

2:12

2:12

21 ⁄

25 ⁄

32

32

7⁄

8

16

2:12 7⁄

8

16

16

16

16

2:12

2:12

2:12

2:12

1

13⁄32

17⁄32

15⁄16

Tolerances as specified in ASTM A325 and A490. Tolerances as specified in ASTM A563. ASTM F436 Washer Tolerances, in.: Nominal Outside Diameter Nominal Diameter of Hole Flatness: max. deviation from straight-edge placed on cut side shall not exceed Concentricity: center of hole to outside diameter (full indicator runout) Burr shall not project above immediately adjacent washer surface more than For clipped washers only. For use with American standard beams (S) and channels (C). Tabular value does not include thickness of washer(s).


−1/32; +1/32 −0; +1/32 0.010 0.030 0.010

8 - 12


Table 8-3. Weights of High-Strength Fasteners, pounds per 100 count Nominal Bolt Diameter, in. 1⁄ 2

5⁄ 8

3⁄ 4

7⁄ 8

1

1 1 ⁄8

1 1 ⁄4

1 3 ⁄8

1 1 ⁄2

1 11 ⁄4 11 ⁄2 13 ⁄4

16.5 17.8 19.2 20.5

29.4 31.1 33.1 35.3

47.0 49.6 52.2 55.3

— 74.4 78.0 81.9

— 104 109 114

— — 148 154

— — 197 205

— — — 261

— — — 333

2 21 ⁄4 21 ⁄2 23 ⁄4

21.9 23.3 24.7 26.1

37.4 39.8 41.7 43.9

58.4 61.6 64.7 67.8

86.1 90.3 94.6 98.8

119 124 130 135

160 167 174 181

212 220 229 237

270 279 290 300

344 355 366 379

3 31 ⁄4 31 ⁄2 33 ⁄4

27.4 28.8 30.2 31.6

46.1 48.2 50.4 52.5

70.9 74.0 77.1 80.2

103 107 111 116

141 146 151 157

188 195 202 209

246 255 263 272

310 321 332 342

391 403 416 428

4 41 ⁄4 41 ⁄2 43 ⁄4

33.0 34.3 35.7 37.1

54.7 56.9 59.0 61.2

83.3 86.4 89.5 92.7

120 124 128 133

162 168 173 179

216 223 230 237

280 289 298 306

353 363 374 384

441 453 465 478

5 51 ⁄4 51 ⁄2 53 ⁄4

38.5 39.9 41.2 42.6

63.3 65.5 67.7 69.8

95.8 98.9 102 105

137 141 146 150

184 190 196 201

244 251 258 265

315 324 332 341

395 405 416 426

490 503 515 527

6 61 ⁄4 61 ⁄2 63 ⁄4

44.0 — — —

71.9 74.1 76.3 78.5

108 111 114 118

154 158 163 167

207 212 218 223

272 279 286 293

349 358 367 375

437 447 458 468

540 552 565 577

7 71 ⁄4 71 ⁄2 73 ⁄4

— — — —

80.6 82.8 84.9 87.1

121 124 127 130

171 175 179 183

229 234 240 246

300 307 314 321

384 392 401 410

479 489 500 510

589 602 614 626

8 81 ⁄4 81 ⁄2 83 ⁄4

— — — —

89.2 — — —

133 — — —

187 192 196 —

251 257 262 —

328 335 342 —

418 427 435 444

521 531 542 552

639 651 664 676

9

—

—

—

—

—

—

453

563

689

100, Conventional A325 or A490 Bolts with A563 Nuts

Bolt Length, in.

Per inch add’tl. add

5.5

8.6

12.4

16.9

22.1

28.0

34.4

42.5

49.7

100, F436 Circular Washers

2.1

3.6

4.8

7.0

9.4

11.3

13.8

16.8

20.0

100, F436 Square Washers

23.1

22.4

21.0

20.2

19.2

34.0

31.6

31.2

32.9

This table conforms to weight standards adopted by the Industrial Fasteners Institute (IFI), updated for washer weights.

Entering and Tightening Clearances

The assembly of high-strength bolted connections requires clearance for entering and tightening the bolts with an impact wrench. The clearance requirements for conventional high-strength bolts are as given in Table 8-4. When high-strength tension-control bolts are specified, the entering and tightening clearances are as specified in Table 8-5. Snug-Tightened and Fully Tensioned Installation

When subjected to shear only, high-strength bolts may be used in snug-tightened bearing-type, fully tensioned bearing-type, and slip-critical connections. When subjected AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BOLTED CONSTRUCTION

8 - 13

Table 8-4. Entering and Tightening Clearances, in. Conventional ASTM A325 and A490 Bolts Aligned Bolts

C2

socket H1 C1

Nominal Bolt Dia., Socket in. Dia., in.

H2 C1 C1

5⁄ 8 3⁄ 4 7⁄ 8

H2

1

C3

11⁄

8

fillet

11⁄

4

1 3 ⁄8 1 1 ⁄2

13 ⁄4 21 ⁄4 21 ⁄2 25 ⁄8 27 ⁄8 31 ⁄8 31 ⁄4 31 ⁄2

C3 H1 25 ⁄

15 ⁄ 35 ⁄ 39 ⁄ 11⁄

64 32 64 64

16 25 ⁄ 32 27 ⁄ 32 15 ⁄ 16

H2

C1

11 ⁄4 13 ⁄8 11 ⁄2 15 ⁄8 17 ⁄8 2 21 ⁄8 21 ⁄4

1 11 ⁄4 13 ⁄8 17 ⁄16 19 ⁄16 111⁄16 13 ⁄4 15 ⁄16

C2 11⁄

16 3⁄ 4 7⁄ 8 15 ⁄ 16 1 1 ⁄16 1 1 ⁄8 11 ⁄4 15 ⁄16

Circular Clipped 11⁄

16 3⁄ 4 7⁄ 8

9⁄ 16 11⁄ 16 13 ⁄ 16 7⁄ 8

1 11 ⁄8 11 ⁄4 13 ⁄8 11 ⁄2

1 11 ⁄8 11 ⁄4 15 ⁄16

Staggered Bolts Stagger P, in. Nominal Bolt Diameter, in.

F

5⁄ 8

3⁄ 4

7⁄ 8

1

1 1 ⁄8 1 1 ⁄4 1 3 ⁄8

15 ⁄8 11 ⁄2 11 ⁄2 17 ⁄16

115 ⁄16 17 ⁄8

23 ⁄16

11⁄

11 ⁄

113 ⁄

2

C1 P

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

2

2 1 ⁄8 2 1 ⁄4 2 3 ⁄8 2 1 ⁄2 2 5 ⁄8 2 3 ⁄4 2 7 ⁄8

F

3

3 1 ⁄8 3 1 ⁄4 3 3 ⁄8

C1 = tightening clearance

4

11 ⁄4 13 ⁄16 11 ⁄8 1

13 ⁄

16

16

13 ⁄4 111⁄16 19 ⁄16 11 ⁄

2

13 ⁄8 11 ⁄4 11 ⁄8 7⁄

8

1

11⁄8

11⁄4

1 3 ⁄8

1 1 ⁄2

21 ⁄8 21 ⁄16 2 115 ⁄16

25 ⁄16 25 ⁄16 21 ⁄4 23 ⁄16

29 ⁄16 29 ⁄16 21 ⁄2

213 ⁄16 23 ⁄4

3 3

33 ⁄4

113 ⁄

21 ⁄

27 ⁄

16

8

16

215 ⁄16 27 ⁄8 213 ⁄16

31 ⁄4 33 ⁄16 33 ⁄16 31 ⁄8

27 ⁄16 25 ⁄16 21 ⁄8 21 ⁄16

23 ⁄4 27 ⁄8 21 ⁄2 23 ⁄8

31 ⁄16 3 27 ⁄8 213 ⁄16

2 17 ⁄8 13 ⁄4 15 ⁄8

21 ⁄4 21 ⁄8 2 115 ⁄16

211⁄16 21 ⁄2 23 ⁄8 21 ⁄4

13 ⁄8 11 ⁄16

13 ⁄4 19 ⁄16 15 ⁄16

21 ⁄8 2 17 ⁄8 11 1 ⁄16

4

111⁄16 19 ⁄16 11 ⁄2

2 17 ⁄8 13 ⁄4

23 ⁄8 21 ⁄4 21 ⁄8

211⁄16 25 ⁄8 21 ⁄2

13 ⁄8 13 ⁄16

15 ⁄8 11 ⁄2 13 ⁄8 13 ⁄16

2 115 ⁄16 17 ⁄8 13 ⁄4

7⁄

15 ⁄8 11 ⁄2 11 ⁄4

15 ⁄

16

8

15 ⁄

16

31⁄

2

standard socket

23 ⁄

3 5 ⁄8 3 3 ⁄4 3 7 ⁄8

4 Notes: H1 = height of head, in. H2 = maximum shank extension,* in. C1 = clearance for tightening, in. C2 = clearance for entering, in. C3 = clearance for fillet,* in. P = bolt stagger, in. F = clearance for tightening staggered bolts, in. *Based on one standard hardened washer.


215 ⁄

16

13 ⁄8

8 - 14


Table 8-5. Entering and Tightening Clearances, in. Tension-Control ASTM A325 and A490 Bolts Aligned Bolts Nominal Bolt Dia, in.

Tools

C3 H1

3⁄ 4 7⁄ 8

C1

H2

1

1

2 /2

C1

C2

Circular Clipped

2

16 5⁄ 8

13 ⁄8 11 ⁄2 13 ⁄4

17 ⁄8 17 ⁄8 17 ⁄8

7⁄

8

1 11 ⁄8

3⁄

7⁄

4 8

1

— — —

2 1 ⁄2-in. Diameter Critical

2 3⁄ 4 7⁄ 8

1 C3

1⁄

9⁄

1⁄

9⁄

2

16 5⁄ 8

Small Tools

fillet

13 ⁄8 11 ⁄2 13 ⁄4

13 ⁄8 13 ⁄8 13 ⁄8

7⁄

8

1 11 ⁄8

3⁄

7⁄

4 8

1

— — —

3-in. Diameter Critical 5⁄ 8 3⁄ 4 7⁄ 8

3

21/2 2

C2

H2 C1

1 1 1 /2 3 /8

33/8

H1 C1

H2


Large Tools

7⁄

16 1⁄ 2 9⁄ 16

11 ⁄4 13 ⁄8 11 ⁄2

15 ⁄8 15 ⁄8 15 ⁄8

13 ⁄ 7⁄

16 8

1

11⁄

— — —

11⁄

— — —

16 3⁄ 4 7⁄ 8


3 2 /16

5⁄ 8 3⁄ 4 7⁄ 8

2

7⁄

16 1⁄ 2 9⁄ 16

11 ⁄4 13 ⁄8 11 ⁄2

11 ⁄8 11 ⁄8 11 ⁄8

13 ⁄ 7⁄

16 8

1

16 3⁄ 4 7⁄ 8

Staggered Bolts Stagger P, in. Nominal Bolt Diameter, in. C1

F

5⁄ 8

11⁄

4

113 ⁄

3⁄ 4

7⁄ 8

1

2

111⁄

21 ⁄16

21 ⁄4

27 ⁄16

19 ⁄16 11 ⁄2 17 ⁄16

2 17 ⁄8 13 1 ⁄16 13 ⁄4

23 ⁄

21 ⁄16 2 17 ⁄8

23 ⁄8 21 ⁄4 23 ⁄16 21 ⁄8

2

15 ⁄16 11 ⁄4 13 ⁄16 11 ⁄8

15 ⁄8 19 ⁄16 11 ⁄2 13 ⁄8

13 ⁄4 111⁄16 19 ⁄16 11 ⁄2

2 115 ⁄16 17 ⁄8 13 ⁄4

2 1 ⁄2 2 5 ⁄8 2 3 ⁄4 2 7 ⁄8

1

15 ⁄16 13 ⁄16 11 ⁄8

13 ⁄8 15 ⁄16 13 ⁄16 11 ⁄8

111⁄16 19 ⁄16 11 ⁄2 13 ⁄8

1 3 ⁄8 11⁄

P

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8 2 1 ⁄8 2 1 ⁄4 2 3 ⁄8

F

C1 = tightening clearance

installation tool

16

13 ⁄4

3

3 3 ⁄8

16

16

15 ⁄16 15 ⁄16

Notes: H1 = height of head, in. H2 = maximum shank extension,* in. C1 = clearance for tightening, in. C2 = clearance of entering, in. C3 = clearance for fillet,* in. P = bolt stagger, in. F = clearance for tightening staggered bolts, in. *Based on one standard hardened washer.


BOLTED CONSTRUCTION

8 - 15

to tension or combined shear and tension, high-strength bolts must be used in fully tensioned bearing-type or slip-critical connections. Bearing-type connections are typically used for shear, moment, and diagonal bracing connections in buildings. Bolts in bearing-type connections are installed in the snug-tightened condition unless required in LRFD Specification Section J1.11 to be fully tensioned. Note that bolts in bearing-type connections required to be fully tensioned must not be confused with fully tensioned bolts in slip-critical connections. Fully tensioned bolts in bearing-type connections have no requirements regarding the slip resistance of the contact surfaces. Thus, painted surfaces in fully tensioned bearing-type connections need not meet the slip resistance requirements of slip-critical connections. Slip-critical connections are used when slip would be detrimental to the serviceability of the structure; this is essentially fatigue related and is primarily of concern in bridge design. From LRFD Specification Section K3, “The occurrence of full design wind or earthquake loads is too infrequent to warrant consideration in fatigue design.” Consequently, slip-critical connections are not normally required or used for wind or seismic loading in buildings. Slip-critical shear connections are required, however, in applications such as those involving oversized holes, fatigue loading, or in craneway and bridge connections. High-strength bolts in slip-critical connections are always fully tensioned to resist slip on the faying surface(s) of the connection. While faying surfaces in slip-critical connections are not normally painted, painted surfaces in accordance with RCSC Specification Section 3(b) are permitted. When subjected to tension only or combined shear and tension, high-strength bolts must be used in fully tensioned bearing-type or slip-critical connections. Examples of these applications are hanger connections, extended end-plate FR moment connections, and diagonal bracing connections. Fully tensioned bolts in bearing-type or slip-critical connections must meet the minimum tensioning requirements for ASTM A325 and A490 bolts as specified in Table 4 of the RCSC Specification. Fully tensioned bolts in either case may be tightened by the same methods. The methods approved by the RCSC are: (1) turn-of-nut method; (2) calibrated wrench method; (3) alternative design bolt method; and, (4) direct tension indicator method. It is important to note that the RCSC prohibits the use of any published relationship between torque and tension. Inspection of Fully Tensioned High-Strength Bolts

When a joint with fully tensioned high-strength bolts is assembled, the RCSC Specification requires that all joint surfaces, including surfaces adjacent to the bolt head and nut be free of scale, except tight mill scale, and of dirt or other foreign material. Burrs need not be removed unless they prevent solid seating of the connected parts in the snug-tightened condition. ASTM A6 lists tolerances for straightness and flatness. These tolerances can prevent the faying surfaces from sufficiently contacting in medium- to large-size connections. Section C8 of the Commentary on the RCSC Specification states: “…Even after being fully tightened, some thick parts with uneven surfaces may not be in contact over the entire faying surface. In itself, this is not detrimental to the performance of the joint. As long as the specified bolt tension is present in all bolts of the completed connection, the clamping force equal to the total of the tensions in all bolts will be transferred at the locations that are in contact and be fully effective in resisting slip through friction.” AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 16


Table 8-6. Dimensions of Non-High-Strength Bolts and Nuts, in. F

db

db

H Square

Bolt Dia. d b, in. 1⁄ 4 3⁄ 8

C, in. 1⁄ 2

78

Hex

H Countersunk

Heavy Hex

C

Countersunk Min. Thrd. Length, in.

H, in. C, in. H, in. L ≤ 6 in. L > 6 in.

H, in.

F, in.

C, in.

H, in.

F, in.

C, in.

3⁄ 16 1⁄ 4 5⁄ 16 7⁄ 16 1⁄ 2 5⁄ 8 11⁄ 16 3⁄ 4 7⁄ 8 15⁄ 16

7⁄ 16 9⁄ 16 3⁄ 4 15⁄ 16 1 1 ⁄8 5 1 ⁄16 11⁄2 111⁄16 17⁄8 21⁄16 21⁄4 25⁄8

1⁄ 2 5⁄ 8 7⁄ 8 1 1 ⁄16 5 1 ⁄16 11⁄2 13⁄4 115⁄16 23⁄16 23⁄8 25⁄8

3⁄ 16 1⁄ 4 3⁄ 8 7⁄ 16 1⁄ 2 9⁄ 16 11⁄ 16 3⁄ 4 7⁄ 8 15⁄ 16

— —

— —

— —

7⁄ 8 11⁄16 1 1 ⁄4 17⁄16

1 11⁄4 7 1 ⁄16 111⁄16

3⁄ 8 11⁄8 1⁄ 2 9⁄ 16

15⁄8 113⁄16 2 23⁄16

17⁄8 21⁄16 25⁄16 21⁄2

11⁄

16 3⁄ 4 7⁄ 8 15⁄ 16

3 33⁄8

3 37⁄16 37⁄8

1 13⁄16 13⁄8 11⁄2

23⁄8 23⁄4 31⁄8 31⁄2

23⁄4 33⁄16 35⁄8 41⁄16

1 13⁄16 13⁄8 11⁄2

1⁄ 2 11⁄ 16 7⁄ 8 11⁄8 3 1 ⁄8 19⁄16 113⁄16 21⁄16 21⁄4 21⁄2 211⁄16

1⁄ 8 3⁄ 16 1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16 1⁄ 2 9⁄ 16 5⁄ 8 11⁄ 16 3⁄ 4

— — —

3⁄ 4

1

1 11⁄4

11⁄4 11⁄2 13⁄4 2

11⁄2 13⁄4 2 21⁄4

21⁄4 21⁄2 23⁄4 3

21⁄2 23⁄4 3 31⁄4

— — —

31⁄4 33⁄4 41⁄4 43⁄4

31⁄2 4 41⁄2 5

3⁄ 4 15⁄ 16 1 1 ⁄8 5 1 ⁄16 11⁄2 111⁄16 17⁄8 21⁄16 21⁄4

13⁄ 16 11⁄16 5 1 ⁄16 19⁄16 17⁄8 21⁄8 23⁄8 25⁄8 215⁄16 33⁄16

— — —

— — —

1 — — —

21⁄2 23⁄4

— —

— —

— —

33⁄4 41⁄8

45⁄16 43⁄4

111⁄16 113⁄16

37⁄8 41⁄4

41⁄2 415⁄16

111⁄16 113⁄16

— —

— —

51⁄4 53⁄4

51⁄2 6

3 31⁄4

— —

— —

— —

41⁄2 47⁄8

53⁄16 55⁄8

2 23⁄16

45⁄8 —

55⁄16 —

2 —

— —

— —

6 6

61⁄2 7

31⁄2 33⁄4

— —

— —

— —

51⁄4 55⁄8

61⁄16 61⁄2

25⁄16 21⁄2

— —

— —

— —

— —

— —

6 6

71⁄2 8

4

—

—

—

6

615⁄16

211⁄16

—

—

—

—

—

6

81⁄2

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

Bolts

Square

F, in. 3⁄ 8 9⁄ 16

db

C

H F Hex, Heavy Hex

C

1 11⁄8 11⁄4 13⁄8 11⁄2 13⁄4 2 21⁄4

W

C N C Square, Heavy Square

Nuts

Nut Size, in.

Square

W, in. C, in.

1⁄ 4 3⁄ 8 1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

7⁄ 16 5⁄ 8 4⁄ 5

1 11⁄8 5 1 ⁄16

1 11⁄8 11⁄4 13⁄8

11⁄2 111⁄16 17⁄8 21⁄16

11⁄2 13⁄4

21⁄4 —

2 21⁄4

5⁄ 8 7⁄ 8 11⁄8 17⁄16 19⁄16 17⁄8 21⁄8 23⁄8 25⁄8 215⁄16 33⁄16

Hex, Heavy Hex

Hex

N, in. W, in. C, in. 1⁄ 4 5⁄ 16 7⁄ 16 9⁄ 16 11⁄ 16 3⁄ 4 7⁄ 8

1 11⁄8 11⁄4

7⁄ 16 9⁄ 16 3⁄ 4 15⁄ 16 11⁄8 15⁄16 11⁄2 111⁄16 17⁄8 21⁄16 21⁄4

1⁄ 2 5⁄ 8 7⁄ 8 11⁄16 15⁄16 11⁄2 13⁄4 115⁄16 23⁄16 23⁄8 25⁄8

—

N

W

Heavy Square

N, in. W, in. C, in. 3⁄ 16 1⁄ 4 3⁄ 8 7⁄ 16 1⁄ 2 9⁄ 16 11⁄ 16 3⁄ 4 7⁄ 8 15⁄ 16

1⁄ 2 11⁄ 16 7⁄ 8 11⁄16 11⁄4 17⁄16 15⁄8 113⁄16

—

11⁄

11⁄4 11⁄2 13⁄4 21⁄16

1⁄ 4 3⁄ 8 1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

2 23⁄16

25⁄16 29⁄16 213⁄16 31⁄8

1 11⁄8 11⁄4 13⁄8

1 —

23⁄8 —

33⁄8 —

16

1

Heavy Hex

N, in. W, in. C, in. 1⁄ 2 11⁄ 16 7⁄ 8 11⁄16 11⁄4 17⁄16 15⁄8 113⁄16

9⁄ 16 13⁄ 16

N, in.

1 11⁄4 17⁄16 111⁄16

1⁄ 4 3⁄ 8 1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

2 23⁄16

17⁄8 21⁄16 25⁄16 21⁄2

1 11⁄8 11⁄4 13⁄8

11⁄2 —

23⁄8 23⁄4

23⁄4 33⁄16

11⁄2 13⁄4

—

15⁄16 —

— —

— —

— —

— —

— —

— —

— —

— —

— —

31⁄8 31⁄2

35⁄8 41⁄16

2 23⁄16

21⁄2 23⁄4

— —

— —

— —

— —

— —

— —

— —

— —

— —

37⁄8 41⁄4

41⁄2 415⁄16

27⁄16 211⁄16

3 31⁄4

— —

— —

— —

— —

— —

— —

— —

— —

— —

45⁄8 5

55⁄16 53⁄4

215⁄16 33⁄16

31⁄2 33⁄4

— —

— —

— —

— —

— —

— —

— —

— —

— —

53⁄8 53⁄4

63⁄16 65⁄8

37⁄16 311⁄16

4

—

—

—

—

—

—

—

—

—

61⁄8

71⁄16

315⁄16

Notes: For high-strength bolt and nut dimensions, refer to Table 8-2. Square, hex, and heavy hex bolt dimensions, rounded to nearest 1⁄16-in., are in accordance with ANSI B18.2.1. Countersunk bolt dimensions, rounded to the nearest 1⁄16-in., are in accordance with ANSI 18.5. Minimum thread length = 2db + 1⁄4-in. for bolts up to 6-in. long, and 2db + 1⁄2-in. for bolts longer than 6-in.


BOLTED CONSTRUCTION

8 - 17

Table 8-7. Threading Dimensions for High-Strength and Non-High-Strength Bolts, in. SCREW THREADS Unified Standard Series-UNC/UNRC and 4UN/4UNR ANSI B1.1 Nominal size (basic major dia.) No. threads per inch (n)

H=0.866P

H /8

P/ 8

P

Thread series symbol c

Thread class symbol

60°

5/ 8H

H /4

Left hand thread. No symbol req’d for right hand thread.

db

P/ 4

K

3/ 4

- 10 UNC 2A LH

Thread Dimensions

Standard Designations

Diameter

Area

Bolt Diameter db, in.

Min. Root K, in.

Gross Bolt Area, in.2

Min. Root Area, in.2

Net Tensile Area, in.2 a

Threads per inch, n b

1⁄ 4 3⁄ 8

0.189 0.298

0.049 0.110

0.029 0.070

0.032 0.078

20 16

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

0.406 0.514 0.627 0.739

0.196 0.307 0.442 0.601

0.129 0.207 0.309 0.429

0.142 0.226 0.334 0.462

13 11 10 9

1 11⁄8 11⁄4 13⁄8

0.847 0.950 1.08 1.17

0.785 0.994 1.23 1.49

0.563 0.709 0.908 1.08

0.606 0.763 0.969 1.16

8 7 7 6

11⁄2 13⁄4

1.30 1.51

1.77 2.41

1.32 1.78

1.41 1.90

6 5

2 21⁄4

1.73 1.98

3.14 3.98

2.34 3.07

2.50 3.25

41⁄2 41⁄2

21⁄2 23⁄4

2.19 2.44

4.91 5.94

3.78 4.69

4.00 4.93

4 4

3 31⁄4

2.69 2.94

7.07 8.30

5.70 6.80

5.97 7.10

4 4

31⁄2 33⁄4

3.19 3.44

9.62 11.0

8.01 9.31

8.33 9.66

4 4

4

3.69

12.6

10.7

11.1

Notes: 

aNet tensile area = 0.785 + d b



2

−

0.9743 

n

 

bFor diameters listed, thread series is UNC (coarse). For larger diameters, thread series is 4UN. c2A denotes Class 2A fit applicable to external threads; c2B denotes corresponding Class 2B fit for internal threads.


4

8 - 18


It should be noted that, even when bolts in bearing-type connections are required to be fully tensioned, high bolt tension is not normally required for proper connection performance. Thus, a significant reduction in inspection costs will be achieved by relying on visual inspection of the bolt head or nut to note the peening marks signifying that the tightening wrench was applied. From RCSC Specification Commentary Section C9, “It is apparent from the commentary on installation procedures that the inspection procedures giving the best assurance that bolts are properly installed and tensioned is provided by inspector observation of the calibration testing of the bolts using the selected installation procedure followed by monitoring of the work in progress to assure that the procedure which was demonstrated to provide the specified tension is routinely adhered to. When such a program is followed, no further evidence of proper bolt tension is required.” Galvanizing High-Strength Bolts

Galvanizing provides corrosion protection by applying zinc as a sacrificial metal to protect the base metal. As previously stated, ASTM A325 Type 1 high-strength bolts and A449 bolts are permitted to be galvanized; A490 bolts are not permitted to be galvanized. There are two methods of galvanizing: hot-dip galvanizing and mechanical galvanizing. Hot-dip galvanizing is a process whereby the bolt is dipped in molten zinc and spun in a centrifuge to remove the excess. This process is described in detail in ASTM A153. In contrast, mechanical galvanizing utilizes a combination of powdered zinc, chemicals, and water with the bolts in a spun hopper. As result of collisions between the bolts, zinc, and glass beads, the zinc is cold-welded to the surface of the bolts. This process is described in detail in ASTM B695. For more information, refer to AISC (1993).

Grip

Ply or plies closest to bolt head

Ply closest to nut

Nominal bolt diameter d b, in.

Min. thickness t of ply closest to nut to exclude threads from shear plane, in.*

3⁄ 4

1⁄ 4

7⁄ 8

1⁄ 4

1

3⁄ 8 5⁄ -in. 32

*Values shown assume one thick washer is present. If washer is not present, increase minimum thickness by 1⁄8-in.

Shear plane

stick-through 5/ 32

t Value from RCSC specification table C2

Fig. 8-3. Minimum thickness of ply closest to nut to exclude threads from shear plane. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BOLTED CONSTRUCTION

8 - 19

Reuse of High-Strength Bolts

From RCSC Specification Section 8f, ASTM A490 bolts and galvanized ASTM A325 bolts shall not be reused. Other A325 bolts are permitted to be reused if approved by the engineer of record. A simple rule based on the prevention of excessive plastic deformation of the bolt is that non-galvanized A325 bolts are satisfactory for reuse, regardless of previous use, if the nut can be placed on the threads and run down the full length of the thread by hand (AISC, 1988). Kulak, et al. (1987) recommends that non-galvanized ASTM A325 bolts may be reused once or twice, provided that proper control on the number of reuses can be established; adequate nut rotation capacity will be present as long as there is some lubricant on the bolt. This lubricant can be the original lubrication or oil, grease, or wax, or a lubricant that is added later. For a detailed assessment of the performance of repetitively tightened high-strength bolts, refer to Bowman and Betancourt (1991). Non-High-Strength Bolts

LRFD Specification Section A3.3 permits the use of ASTM A307 non-high-strength bolts for structural applications not requiring fully tensioned installation, that is, snug-tightened bearing-type connections. ASTM A307 bolts are available with both hex and square heads in diameters from 1⁄4-in. to four inches in two grades: Grade A for general applications and Grade B for cast-iron-flanged piping joints. ASTM A563 Grade A nuts are recommended for use with ASTM A307 bolts. Other suitable grades are listed in ASTM A563 Table X1.1. Dimensions and Weights

Typical non-high-strength bolt head and nut dimensions are given in Table 8-6. Thread lengths listed in this table may be calculated for non-high-strength bolts as 2db + 1⁄4-in. for bolts up to six inches long and 2db + 1⁄2-in. for bolts over six inches long, where db is the bolt diameter. Note that these thread lengths are longer than those given previously for high-strength bolts in Table 8-2. Threading dimensions are given in Table 8-7. Weights of non-high-strength bolts are given in Tables 8-8, 8-9, and 8-10. Entering and Tightening Clearances

As with high-strength bolts, clearance is required for entering and tightening the bolts with an impact wrench. The required clearances are the same as those given for high-strength bolts in Table 8-4. Design Strength of Bolts

The design strength of bolts is determined in accordance with the provisions of LRFD Specification Section J3. LRFD Specification requirements are based upon the provisions of the RCSC Specification. For bolts in bearing-type connections subjected to shear only, the limit states of bolt shear strength and bearing strength at bolt holes must be checked. For bolts in bearingtype connections subjected to tension only, the limit state of bolt tensile strength, including the effect of prying action, must be checked. For bolts in bearing-type connections subjected to combined shear and tension, the limit states of bolt tensile strength, including the effects of both the bolt shear stress present and prying action, and bearing strength at bolt holes must be checked. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 20


Table 8-8. Weights of Non-High-Strength Fasteners, pounds Bolt Length, in.

3⁄ 8

1⁄ 2

5⁄ 8

3⁄ 4

7⁄ 8

1 1 ⁄8

1 1 ⁄4

— — — —

— — — —

104 109 114 119

143 149 155 161

— — 206 213

90.2 94.4 98.5 103

124 129 135 140

168 174 181 188

221 229 237 246

73.3 76.3 79.3 82.3

107 111 115 119

145 151 156 162

195 202 208 215

254 262 271 279

56.5 58.6 60.7 62.8

85.3 88.4 91.4 94.4

123 127 131 136

167 172 178 183

222 229 236 242

288 296 304 313

39.3 40.4 41.8 43.1

64.9 66.7 68.7 70.8

97.4 100 103 106

140 143 147 151

188 193 198 204

249 255 262 269

321 329 337 345

24.0 24.8 25.5 26.3

44.4 45.8 47.1 48.5

72.9 75.0 77.1 79.2

109 112 115 118

156 160 164 168

209 214 220 225

275 282 289 296

354 362 371 379

11.7 — — —

27.0 28.6 30.1 31.6

49.8 52.5 55.2 57.9

81.3 85.5 89.7 93.9

121 127 133 139

172 180 189 197

231 241 252 263

303 316 330 343

387 404 421 438

10

— —

66.1 34.6

60.6 63.3

98.1 102

145 151

205 213

274 284

357 371

454 471

11

— —

36.2 37.7

66.0 68.7

106 110

157 163

221 230

295 306

384 398

488 505

12

— —

39.2 —

71.3 74.0

115 119

170 176

238 246

316 327

411 425

522 538

13

— —

— —

76.7 79.4

123 127

182 188

254 263

338 349

439 452

556 572

14

— —

— —

82.1 84.8

131 135

194 200

271 279

359 370

466 479

589 605

15 1 ⁄2

15

— —

— —

87.5 90.2

140 144

206 212

287 296

381 392

493 507

622 639

16

—

—

92.9

148

218

304

402

520

656

Per inch add’tl. add

1.3

3.0

5.4

1

2.38 2.71 3.05 3.39

6.11 6.71 7.47 8.23

13.0 14.0 15.1 16.5

24.1 25.8 27.6 29.3

38.9 41.5 44.0 46.5

— — 67.3 70.8

2

3.73 4.06 4.40 4.74

8.99 9.75 10.5 11.3

17.8 19.1 20.5 21.8

31.4 33.5 35.6 37.7

49.1 52.1 55.1 58.2

74.4 77.9 82.0 86.1

3

5.07 5.41 5.75 6.09

12.0 12.8 13.5 14.3

23.2 24.5 25.9 27.2

39.8 41.9 44.0 46.1

61.2 64.2 67.2 70.2

4

6.42 6.76 7.10 7.43

15.1 15.8 16.6 17.3

28.6 29.9 31.3 32.6

48.2 50.3 52.3 54.4

5

7.77 8.11 8.44 8.78

18.1 18.9 19.6 20.4

33.9 35.3 36.6 38.0

6

9.12 9.37 9.71 10.1

21.1 21.7 22.5 23.3

7

10.4 10.7 11.0 11.4

8

1 1 ⁄4 1 1 ⁄2 1 3 ⁄4 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

100 Square Bolts with Hexagonal Nuts

Nominal Bolt Diameter, in. 1⁄ 4

3 1 ⁄4 3 1 ⁄2 3 3 ⁄4 4 1 ⁄4 4 1 ⁄2 4 3 ⁄4 5 1 ⁄4 5 1 ⁄2 5 3 ⁄4 6 1 ⁄4 6 1 ⁄2 6 3 ⁄4 7 1 ⁄4 7 1 ⁄2 7 3 ⁄4 8 1 ⁄2

9

9 1 ⁄2 10 1 ⁄2 111 ⁄2 12 1 ⁄2 13 1 ⁄2 14 1 ⁄2

8.4

12.1

16.5

1 — — 95.1 99.7

21.4

27.2

33.6

Notes: For weights of high-strength fasteners, see Table 8-3. This table conforms to weight standards adopted by the Industrial Fasteners Institute (IFI). *Square bolt per ANSI B18.2.1, hexagonal nut per ANSI B18.2.2. For other non-high-strength fasteners, refer to Tables 8-9 and 8-10.


BOLTED CONSTRUCTION

8 - 21

Table 8-9. Weight Adjustments for Combinations of Non-High-Strength Fasteners Other than Tabulated in Table 8-8

100, Square Bolts with Hexagonal Nuts*

Square Bolts with

Combinations of 100:

100, Hex Bolts

Nominal Bolt Diameter, in.

Add or Subtr.

1⁄ 4

3⁄ 8

1⁄ 2

5⁄ 8

3⁄ 4

7⁄ 8

1

11⁄8

11⁄4

3.5

5.5

8.0 12.2 16.3

Square Nuts

+

0.1

1.0

2.0

3.4

Heavy Square Nuts

+

0.6

2.1

4.1

7.0 11.6 17.2 23.2 32.1 41.2

Heavy Hex Nuts

+

0.4

1.5

2.8

4.6

7.6 10.7 14.2 18.9 24.3

Square Nuts

+

0.1

0.6

1.1

1.4

0.2

0.5 −0.2 −0.1 −1.7

Hex Nuts

−

0.0

0.4

0.9

2.0

3.3

5.0

Heavy Square Nuts

+

0.6

1.7

3.2

5.0

8.3 12.2 15.0 19.8 23.2

Heavy Hex Nuts

+

0.4

1.1

1.9

2.6

4.3

Heavy Square Nuts

+

—

—

4.7

7.3 11.3 16.5 20.7 27.0 33.6

Heavy Hex Nuts

+

—

—

3.4

4.9

5.7

8.2 12.3 18.0

6.0

6.6

6.3

7.3 10.0 11.7 13.8 16.7

Notes: For weights of high-strength fasteners, see Table 8-3. This table conforms to weight standards adopted by the Industrial Fasteners Institute (IFI). *Add or subtract value in this table to or from the value in Table 8-8.

Table 8-10. Weights of Non-High-Strength Bolts of Diameter Greater Than 11⁄4-in., pounds Nominal Bolt Diameter, in. 11⁄2

13⁄4

2

21⁄4

21⁄2

23⁄4

3

31⁄4

31⁄2

33⁄4

4

105

130

—

—

—

—

—

—

—

—

—

—

Hex Bolts

84.0 112

178

259

369

508

680

900

1120

1390

1730

2130

Heavy Hex Bolts

95.0 124

195

280

397

541

720

950

—

—

—

—

139

168

200

235

272

313

356

147

178

210

246

284

325

Heads of:

Weight of 100 Each: 13⁄8 Square Bolts

One Linear Inch, Unthreaded Shank

42.0

50.0

68.2

89.0 113

One Linear Inch, Threaded Shank

35.0

42.5

57.4

75.5

97.4 120

—

—

—

—

—

—

—

—

—

—

Square Nuts

94.5 122

Heavy Square Nuts

125

161

—

—

—

—

—

—

—

—

—

—

Heavy Hex Nuts

102

131

204

299

419

564

738

950

1190

1530

1810

2180

Notes: For weights of high-strength fasteners, see Table 8-3. This table conforms to weight standards adopted by the Industrial Fasteners Institute (IFI).


8 - 22


For bolts in slip-critical connections subjected to shear only, the limit states of slip resistance, bolt shear strength, and bearing strength at bolt holes must be checked. For bolts in slip-critical connections subjected to combined shear and tension, the limit states of slip resistance, including the effect of the tensile force present, bolt shear strength, and bearing strength at bolt holes must be checked. Bolt Shear Strength

As illustrated in Figure 8-4a, this limit state considers a shear failure of the bolt shank on plane cdef. Since there is one shear plane, the bolt is in single shear (S). Additional plies of material may increase the number of shear planes and, therefore, the shear strength of the bolt. This condition, as illustrated in Figure 8-4b, is called double shear (D). Additionally, high-strength bolts may be specified with the threads included (N) or excluded (X) from the shear plane of the connection. Note that the shear strength of bolts with the threads included is about 25 percent less than that of bolts with the threads excluded. In spite of this, many designers prefer to specify N bolts when possible due to the difficulty in assuring that threads are excluded from the shear plane in the as-built condition. If, however, the threads are to be excluded from the shear plane, care must be taken to specify a bolt of sufficient overall length given the thread length and required bolt length from Table 8-2. Note that additional washers may be required to accomplish this; refer to Figure 8-3. From LRFD Specification Section J3.6, the design bolt shear strength is φRn, where φ = 0.75 and: Rn = (Fv Ab)n

x ′,y ′,z ′

c,b

g,f

d,a

h,e x,y,z

x,x ′

Ru

e,f d,c

y,y ′

a,b

z,z ′

Ru 2

h,g

Ru Ru

Ru 2

(a) Single shear (S)

(b) Double shear (D)

Fig. 8-4. Bolt shear. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BOLTED CONSTRUCTION

8 - 23

In the above equation, n is the number of bolts in the connection, Fv is the nominal shear strength, and Ab is the nominal bolt area. For convenience, the design bolt shear strengths of various bolts are summarized in Table 8-11; design bolt shear strengths of vertical rows of n bolts are summarized in Table 8-12. Bearing Strength at Bolt Holes

As illustrated in Figure 8-5, this limit state considers both a tear fracture of the connected material and deformation around the bolt holes. Bearing strength is a function of the material being connected, the type of bolt hole, and the spacing and edge distance; it is independent of both the type of bolt and the presence or absence of threads on the bearing area. From LRFD Specification Section J3.10, when deformation around the bolt holes is a design consideration for standard holes, oversized holes, short-slotted holes, and longslotted holes parallel to the line of force, the design bearing strength at bolt holes is φRn, where φ = 0.75 and, for two or more bolts in the line of force, when Le ≥ 1.5d and s ≥ 3d: Rn = (2.4dtFu )n For a single bolt in the line of force or when Le < 1.5d or s < 3d:  d Rn = Let + s −  (n − 1)(tFu ) ≤ (2.4dtFu )n  2   In the above equations, n is the number of bolts in the connection, d is the nominal bolt diameter, t is the thickness in bearing, and Le is the edge distance. If deformation around the bolt hole is not a design consideration, or for long-slotted holes perpendicular to the line of force, refer to LRFD Specification Section J3.10. For convenience, the design bearing strength at bolt holes is tabulated for the foregoing conditions in Tables 8-13 and 8-14, respectively. Note that these tables may be applied to bolts with countersunk heads, by subtracting one-half the depth of the countersink from the material thickness t. As illustrated in Figure 8-6, this is equivalent to subtracting one-quarter the diameter of the bolt from the material thickness t. Bolt Tensile Strength

From LRFD Specification Section J3.6, when subjected to tension only, the design bolt tensile strength is φRn, where φ = 0.75 Rn = (Ft Ab)n In the above equation, n is the number of bolts in the connection. For convenience, the design bolt tensile strengths of various bolts is summarized in Table 8-15. When subjected to combined shear and tension, the design bolt tensile strength is reduced by a function of the shear stress present in the bolt as specified in LRFD Specification Section J3.7. LRFD Specification Section J3.6 states that any tension resulting from prying action must be considered in determining the required strength of the bolts. Prying action is a phenomenon (in bolted construction only) whereby the deformation of a fitting under a tensile force increases the tensile force in the bolt. The required strength per bolt is the AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 24


Table 8-11. Design Shear Strength of One Bolt, kips Nominal Bolt Diameter d, in. 5⁄ 8

ASTM Desig.

Thread Cond.

φF v (ksi)

A325

N

36.0

X A490

A307

56.3

—

1 1 ⁄8

1

1 1 ⁄4

Loading 0.3068 0.4418 0.6013 0.7854 0.9940 1.227

45.0

X

7⁄ 8

1 3 ⁄8

1 1 ⁄2

1.485

1.767

Nominal Bolt Area, in.2

45.0

N

3⁄ 4

18.0

S

11.0

15.9

21.6

28.3

35.8

44.2

D

22.1

31.8

43.3

56.5

71.6

88.4

S

13.8

19.9

27.1

35.3

44.7

D

27.6

39.8

54.1

70.7

89.5

S

13.8

19.9

27.1

35.3

44.7

D

27.6

39.8

54.1

70.7

89.5

S

17.3

24.9

33.9

44.2

D

34.5

49.7

67.7

88.4

S D

5.52 11.0

7.95 15.9

56.0 112

55.2 110 55.2 110 69.1 138

53.5 107 66.8 134 66.8 134 83.6 167

63.6 127 79.5 159 79.5 159 99.5 199

10.8

14.1

17.9

22.1

26.7

31.8

21.6

28.3

35.8

44.2

53.5

63.6

N = Threads included in shear plane X = Threads excluded from shear plane S = Single shear D = Double shear

Table 8-12. Design Shear Strength of n Bolts in Double Shear* ASTM A325 N

ASTM A490 X

N

X

n

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

12

382

520

679

477

649

848

477

649

848

596

812

1060

11

350

476

622

437

595

778

437

595

778

547

744

972

10

318

433

565

398

541

707

398

541

707

497

676

884

9

286

390

509

358

487

636

358

487

636

447

609

795

8

254

346

452

318

433

565

318

433

565

398

541

707

7

223

303

396

278

379

495

278

379

495

348

474

619

6

191

260

339

239

325

424

239

325

424

298

406

530

5

159

216

283

199

271

353

199

271

353

249

338

442

4

127

173

226

159

216

283

159

216

283

199

271

353

130

170

119

162

212

119

162

212

149

203

265

108

141

108

141

135

177

3

95.4

2

63.6

86.6

1

31.8

43.3

113

79.5

56.5

39.8

54.1

70.7

79.5 39.8

54.1

N = Threads included in shear plane X = Threads excluded in shear plane *For design strength of bolts in single shear, divide tabular value by 2.


70.7

99.4 49.7

67.6

1

88.4

BOLTED CONSTRUCTION

8 - 25

sum of rut, the factored force per bolt due to the tensile force, and qu, the additional tension per bolt resulting from prying action produced by deformation of the connected parts. While the effect of prying action is considered in the design of the bolts, it is primarily a function of the connected elements; thus, the connected elements must possess adequate flexural strength and it is their stiffness which is the key to satisfactory performance. Refer to “Hanger Connections” in Part 11 for treatment of prying action. Slip Resistance

In slip-critical connections, the fully tensioned bolt creates resistance to slip through friction on the faying surface between two connected parts. This slip resistance is a function of the slip coefficient µ of the faying surface. Clean mill scale with no coating is defined as a Class A surface with µ = 0.33. Blast-cleaned surfaces with no coatings are defined as Class B surfaces with µ = 0.50. Hot-dip galvanized and roughened surfaces are defined as Class C surfaces with µ = 0.40.

Ru

Ru

Splitting of plate

(a) Tear fracture for smaller end distance

Ru

Ru

Tearout of Plate

(b) Tear fracture for longer end distance

Ru

Ru

Deformation

(c) Deformation of material at bolt hole Fig. 8-5. Bearing strength at bolt holes. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 26


Table 8-13. Design Bearing Strength at Bolt Holes, kips/in. thickness Two or more holes in line of force with Le ≥ 1.5d and s ≥ 3d; hole deformation considered* Nominal Bolt Diameter d, in. 5⁄ 8

3⁄ 4

7⁄ 8

1

15⁄ 16

11⁄8

15⁄16

11⁄2

Fu, ksi

17⁄8

21⁄4

25⁄8

3

STD, OVS SSL, LSLP

58 65 70

65.3 73.1 78.8

78.3 87.8 94.5

91.4 102 110

LSLT

58 65 70

54.4 60.9 65.6

65.3 73.1 78.8

76.1 85.3 91.9

11⁄8

11⁄4

13⁄8

11⁄2

111⁄16

17⁄8

21⁄16

21⁄4

33⁄8

33⁄4

41⁄8

41⁄2

1.5d

Hole Type

3d 104 117 126

117 132 142

131 146 158

144 161 173

157 176 189

87.0 97.5 105

97.9 110 118

109 122 131

120 134 144

131 146 158

STD = Standard Hole OVS = Oversized Hole SSL = Short-Slotted Hole LSLP = Long-Slotted Hole parallel to line of force LSLT = Long-Slotted Hole transverse to line of force *When s < 3d, or when hole deformation is not a design consideration, refer to LRFD Specification Section J3.10. When Le < 1.5d or for one hole in the line of force, refer to Table 8-14.

Table 8-14. Design Bearing Strength at Bolt Holes, kips/in. thickness One hole in line of force or top bolt with Le < 1.5d* Nominal Bolt Diameter d, in.

Fu, ksi

1

11⁄8

11⁄4

13⁄8

11⁄2

15⁄8

13⁄4

17⁄8

58 65 70

43.5 48.8 52.5

48.9 54.8 59.1

54.4 60.9 65.6

59.8 67.0 72.2

65.3 73.1 78.8

70.7 79.2 85.3

76.1 85.3 91.9

81.6 91.4 98.4

*Design strength from Table 8-14 shall not exceed tabular value from Table 8-13. For remaining bolts, when s − d / 2 > 2.4d, refer to Table 8-13; otherwise refer to LRFD Specification Section J3.10.

Slip coefficients for all other coated blast-cleaned surfaces must be determined by the Testing Method to Determine the Slip Coefficient Used in Bolted Joints; refer to Appendix A of the RCSC Specification. When the tests results in 0.33 ≤ µ < 0.50, the coating is a Class A coating and the design slip coefficient is µ = 0.33. If the test results in µ ≥ 0.50, the coating is a Class B coating and the design slip coefficient is µ = 0.50. The surface requirements for slip-critical connections apply only to the faying surfaces and do not include the surfaces under the bolt, washer, or nut. Bolts in slip-critical connections may be designed at either service loads or factored loads with the provisions of LRFD Specification Section J3.8. From LRFD Specification Section J3.8a, when subjected to shear only, the resistance to slip for comparison with service loads is φRn, where Rn = (Fv Ab)n AMERICAN INSTITUTE OF STEEL CONSTRUCTION

BOLTED CONSTRUCTION

8 - 27

Table 8-15. Design Tensile Strength of Bolts, kips Nominal Bolt Diameter d, in. 5⁄ 8

3⁄ 4

7⁄ 8

1

11⁄8

11⁄4

13⁄8

11⁄2

1.227

1.485

1.767

2

Nominal Bolt Area, in.

ASTM Desig.

φFt, ksi

A325

67.5

A490 A307

84.8 33.8

0.3068

0.4418

0.6013

20.7

29.8

40.6

41.4

59.6

81.2

26.0

37.4

52.0

74.9

51.0 102

0.7854 53.0 106 66.6 133

0.9940 67.1

82.8

134

166

84.2 169

100

119

200

239

104

126

150

208

252

300

10.4

14.9

20.3

26.5

33.5

41.4

20.7

29.8

40.6

53.0

67.1

82.8

50.1 100

59.6 119

and φ = 1.0 for standard holes, oversized holes, short-slotted holes, and long-slotted holes perpendicular to the direction of the load; φ = 0.85 for long-slotted holes parallel to the direction of the load. In the above equation, n is the number of bolts in the connection. In general, slip is likely to occur at 1.4 to 1.5 times the service loads. Note that the values of Fv tabulated in LRFD Specification Table J3.6 for bolts in slip-critical connections assume Class A surfaces with µ = 0.33. As stated in LRFD Specification Section J3.8a, it is permissible to increase Fv to the applicable value in the RCSC Specification for other surfaces. When subjected to combined shear and tension, the slip capacity for comparison with service loads must be reduced by the factor:  T 1 −  Tb   as specified in LRFD Specification Section J3.9a, where T is the unfactored force on the connection and Tb is the minimum bolt tension from LRFD Specification Table J3.1. From LRFD Specification Appendix J3.8a, the design slip resistance for comparison with factored loads is φRstr, Effective thickness in bearing db 2

db 4

Ru Ru db

Fig. 8-6. Effective thickness for bearing of countersunk bolts. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 28


where Rstr = 1.13µTm Nb Ns and φ is equal to 1.0 for standard holes, 0.85 for oversized and short-slotted holes, 0.70 for long-slotted holes perpendicular to the direction of the load, and 0.60 for long-slotted holes parallel to the direction of the load. When subjected to combined tension and shear, the design slip resistance for comparison with factored loads must be reduced by the factor:   Tu 1 −  1.13Tm Nb   as specified in LRFD Specification Appendix J3.8b. In the above equations, Tu is the factored force on the connection, Tm is the minimum bolt tension from LRFD Specification Table J3.1, and Nb is the number of bolts in the connection. For convenience, slip capacities for comparison with service loads and design slip resistances for comparison with factored loads are tabulated in Tables 8-16 and 8-17, respectively. ECCENTRICALLY LOADED BOLT GROUPS

When the line of action of an applied load does not pass through the center of gravity (CG) of a bolt group, the load is eccentric and results in a moment which must be considered in the design of the connection. Eccentricity in the Plane of the Faying Surface

Eccentricity in the plane of the faying surface produces additional shear. The bolts must be designed to resist the combined effect of the direct shear from the applied load Pu and the additional shear from the induced moment Pu e. Two methods of analysis for this type of eccentricity will be discussed: (1) the instantaneous center of rotation method; and, (2) the elastic method. Instantaneous Center of Rotation Method

Also known as the ultimate strength method (Crawford, 1968), this method considers the load-deformation relationship of each bolt and, thus, more accurately predicts the ultimate strength of the eccentrically loaded connection. Eccentricity produces both a rotation about the centroid of the bolt group and a translation of one connected element with respect to the other. The combined effect of this rotation and translation is equivalent to a rotation about a point defined as the instantaneous center of rotation (IC) as illustrated in Figure 8-7a. The location of the IC depends on the geometry of the bolt group as well as the direction and point of application of the load. The individual resistance of each bolt is assumed to act on a line perpendicular to a ray passing through the IC and the centroid of that bolt as illustrated in Figure 8-7b. The load-deformation relationship of one bolt is illustrated in Figure 8-8, where R = Rult(1 − e−10∆)0.55 In the above equation, AMERICAN INSTITUTE OF STEEL CONSTRUCTION

ECCENTRICALLY LOADED BOLT GROUPS

8 - 29

Table 8-16. Slip-Critical Connections Design Resistance to Shear at Service Loads,* kips (Class A faying surface, µ = 0.33) Nominal Bolt Diameter, in. 5⁄ 8

ASTM Desig.

Hole Type

A325

STD

A490

3⁄ 4

7⁄ 8

1

11⁄8

11⁄4

13⁄8

11⁄2

Nominal Bolt Area, in.2 Loading

0.3068 0.4418 0.6013 0.7854 0.9940

1.227

1.485

1.767

S D

5.22 10.4

7.51 15.0

10.2 20.4

13.4 26.7

16.9 33.8

20.9 41.7

25.2 50.5

30.0 60.1

OVS SSL

S D

4.60 9.20

6.63 13.3

9.02 18.0

11.8 23.6

14.9 29.8

18.4 36.8

22.3 44.5

26.5 53.0

LSLP

S D

3.13 6.26

4.51 9.01

6.13 12.3

8.01 16.0

10.1 20.3

12.5 25.0

15.1 30.3

18.0 36.0

LSLT

S D

3.68 7.36

5.30 10.6

7.22 14.4

9.42 18.8

11.9 23.9

14.7 29.5

17.8 35.6

21.2 42.4

STD

S D

6.44 12.9

9.28 18.6

12.6 25.3

16.5 33.0

20.9 41.7

25.8 51.5

31.2 62.4

37.1 74.2

OVS SSL

S D

5.52 11.0

7.95 15.9

10.8 21.6

14.1 28.3

17.9 35.8

22.1 44.2

26.7 53.5

31.8 63.6

LSLP

S D

3.93 7.85

5.65 11.3

7.70 15.4

10.1 20.1

12.7 25.4

15.7 31.4

19.0 38.0

22.6 45.2

LSLT

S D

4.60 9.20

6.63 13.3

9.02 18.0

11.8 23.6

14.9 29.8

18.4 36.8

22.3 44.5

26.5 53.0

STD = Standard Hole OVS = Oversized Hole SSL = Short-Slotted Hole LSLP = Long-Slotted Hole parallel to line of force LSLT = Long-Slotted Hole transverse to line of force S = Single Shear D = Double Shear *For design slip resistance at factored loads, refer to Table 8-17.

R = shear force in one bolt at a deformation ∆, kips. Rult = ultimate shear strength of one bolt, kips. ∆ = total deformation of a bolt, including shearing, bearing, and bending deformation, plus local bearing deformation of the plate, in. e = 2.718…, base of the natural logarithm. Applying a maximum deformation ∆max to the bolt most remote from the IC, the maximum shear strength of that bolt may be determined. For other bolts, deformations are assumed to vary linearly with distance from the IC, and shear strengths can be obtained from this relationship. The strength of the bolt group is, then, the sum of the


8 - 30


Table 8-17. Slip-Critical Connections Design Slip Resistance at Factored Loads, kips (Class A faying surface, µ = 0.33) Nominal Bolt Area, in.2 5⁄ 8

ASTM Desig.

Hole Type

A325

STD

3⁄ 4

7⁄ 8

1

11⁄8

11⁄4

13⁄8

11⁄2

Minimum ASTM A325 Bolt Tension, kips Loading

19.0

28.0

39.0

51.0

56.0

71.0

85.0

103

S D

7.09 14.2

10.4 20.9

14.5 29.1

19.0 38.0

20.9 41.8

26.5 53.0

31.7 63.4

38.4 76.8

OVS SSL

S D

6.02 12.0

8.88 17.8

12.4 24.7

16.2 32.3

17.8 35.5

22.5 45.0

26.9 53.9

32.6 65.3

LSLP

S D

4.25 8.50

6.26 12.5

8.73 17.5

11.4 22.8

12.5 25.1

15.9 31.8

19.0 38.0

23.0 46.1

LSLT

S D

4.96 9.92

7.31 14.6

10.2 20.4

13.3 26.6

14.6 29.2

18.5 37.1

22.2 44.4

26.9 53.8

Minimum ASTM A490 Bolt Tension, kips

A490

24.0

35.0

49.0

64.0

80.0

102

121

148

STD

S D

8.95 17.9

13.1 26.1

18.3 36.5

23.9 47.7

29.8 59.7

38.0 76.1

45.1 90.2

55.2 110

OVS SSL

S D

7.61 15.2

11.1 22.2

15.5 31.1

20.3 40.6

25.4 50.7

32.3 64.7

38.4 76.7

46.9 93.8

LSLP

S D

5.37 10.7

7.83 15.7

11.0 21.9

14.3 28.6

17.9 35.8

22.8 45.6

27.1 54.1

33.1 66.2

LSLT

S D

6.26 12.5

9.14 18.3

12.8 25.6

16.7 33.4

20.9 41.8

26.6 53.3

31.6 63.2

38.6 77.3

STD = Standard Hole OVS = Oversized Hole SSL = Short-Slotted Hole LSLP = Long-Slotted Hole parallel to line of force LSLT = Long-Slotted Hole transverse to line of force S = Single Shear D = Double Shear

individual strengths of all bolts. If the correct location of the IC has been selected, the three equations of in-plane statics will be satisfied; i.e., ΣFx = 0, ΣFy = 0, and ΣM = 0. Tables 8-18 through 8-25 employ the instantaneous center of rotation method for the bolt patterns and eccentric conditions indicated and inclined loads at 0°, 15°, 30°, 45°, 60°, and 75°. The load-deformation relationship is based on data obtained experimentally for 3⁄4-in. diameter ASTM A325 bolts, where Rult = 74 kips, and ∆max = 0.34 in. The non-dimensional coefficient C is obtained by dividing the factored eccentric force Pu by Rult.



8 - 31

For any of the bolt group geometries shown, the design strength of the eccentrically loaded bolt group is φRn, where φRn = C × φrn

lo e

Pu

IC

CG

(a) Instantaneous center of rotation (IC)

e

lo

Pu CG

IC lrm

ax

ru max

(b) Forces on bolts in group Fig. 8-7. Instantaneous center of rotation method. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 32


In the above equation, φrn is the least design strength of one bolt determined from the limit states of bolt shear strength, bearing strength at bolt holes, and slip resistance (if the connection is to be slip-critical). The design strength φRn must be greater than or equal to the required strength Pu. Thus, by dividing Pu by φrn, the minimum coefficient C is obtained, and a bolt group can be selected for which the coefficient is of that magnitude or greater. These tables may be used with any bolt diameter and are conservative when used with ASTM A490 bolts. Linear interpolation within a given table between adjacent values of ex is permitted. Design strengths determined with these tables provide a factor of safety equivalent to that for bolts in connections less than 50 inches long, subjected to shear produced by a concentric load in either bearing-type or slip-critical connections. Although this procedure is based on connections which may experience slip under load, both load tests and analytical studies (Kulak, 1975) indicate that it may be conservatively extended to slip-critical connections. A convergence criterion of one percent was employed for the tabulated iterative solutions. Straight line interpolation between values for loads at different angles may be significantly unconservative. Therefore, unless a direct analysis is performed, use only the values for the next lower angle for design. For bolt group patterns not treated by these tables, a special ultimate strength analysis is required if the instantaneous center of rotation method is to be used.

Example 8-1

Given:

Refer to Figure 8-9. Determine the largest eccentric force Pu for which the design shear strength of the bolts in the connection is adequate using the instantaneous center of rotation method. Use 7⁄8-in. diameter A325-N bolts, φrn = 21.6 kips/bolt. A. Assume the load is vertical as illustrated in Figure 8-9 (θ = 0°°) B. Assume the load acts at an angle of 75°° with respect to vertical (θ = 75°°)

Solution A:

From Table 8-20 with θ = 0°, with s = 3 in., e = 16 in., and n = 6: C = 3.55 Design Shear Strength φRn = C × φrn = 3.55 × 21.6 kips/bolt = 76.7 kips Thus, Pu must be less than or equal to 76.7 kips.

Comment:

Note that this eccentricity has effectively reduced the shear strength of this bolt group by about 70 percent when compared with the concentrically loaded case.

Solution B:

From Table 8-20 with θ = 75°°, s = 3 in., e = 16 in., and n = 6: AMERICAN INSTITUTE OF STEEL CONSTRUCTION


8 - 33

C = 7.90 Design shear strength φRn = C × φrn = 7.90 × 21.6 kips/bolt = 171 kips Thus, Pu must be less than or equal to 171 kips. Comment:

In Solution B, the vertical component of the design strength is φRn sin75°° = (171 kips)(0.966) = 165 kips and the horizontal component of the design strength is φRn cos75°° = (171 kips)(0.259) = 44.3 kips

Elastic Method

Alternatively, the elastic method may be used to analyze eccentrically loaded bolt groups. It offers a simplified, conservative approach but does not render a consistent factor of safety and, in some cases, provides excessively conservative results. Furthermore, the elastic method ignores both the ductility of the bolt group and the load redistribution which occurs. Refer to Higgins (1971). In the elastic method, for a force applied parallel to the Y principal axis of the bolt group as illustrated in Figure 8-10, the eccentric force Pu is resolved into a force Pu acting through the center of gravity (CG) of the bolt group and a moment Pu e where e is the eccentricity. Each bolt is then assumed to support an equal share of the concentric force Pu, and a share of the eccentric moment Pu e which is proportional to its distance from the

80

R, kips

60 R = Rult (1 – e

–10 ∆ 0.55

)

40

20

0

0.10

0.20

0.30

∆, in.

Fig. 8-8. Load-deformation relationship for bolts. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 34


CG. The bolt most remote from the CG, then, is the most highly stressed. The resultant vectorial sum of these forces ru is the required strength for the bolt. The direct shear force per bolt due to the concentric force Pu is r1, where r1 =

Pu n

and n is the number of bolts. The shear force in each bolt due to the moment Pu e varies with distance from the CG and will be maximum in the bolt which is must remote from the CG. The maximum shear due to the moment Pu e is rm, where Y e = 16 in. Pu =60 kips

X

CG

1½

5@3=1 ′-3

W14x82 A572 gr. 50

X PL 7/8, A36

2¾

2¾

5½ Y

Fig. 8-9. Bolted bracket plate for Examples 8-1 and 8-2. Y

Pu

e CG

X



rm =

8 - 35

Pu ec Ip

In the above equation, c = distance from CG to center of bolt most remote from CG, in. Ip = polar moment of inertia of the bolt group, in.4 per in.2 (see any text on statics). To determine the resultant force on the most highly stressed bolt, rm must be resolved into vertical component r2 and horizontal component r3, where Pu ecx Ip Pu ecy r3 = Ip

r2 =

In the above equation, cx and cy are the horizontal and vertical components of the diagonal distance c. Thus, the resultant factored force is ru, where r2 r1

ru = √  (r1 + r2)2 + (r3)2

r3 rm ru

and the bolts must be chosen such that the design strength φrn exceeds the required strength ru. For the more general case of an inclined eccentric force, i.e., not parallel to the Y principal axes of the bolt group, the effect of the X-direction component of the direct shear must also be included. Refer to Iwankiw (1987).

Example 8-2

Given:

Refer to Example 8-1. Recalculate the largest eccentric force Pu for which the design shear strength of the bolts in the connection is adequate using the elastic method. Compare the result with that of Example 8-1. Use 7⁄8-in. diameter A325-N bolts, φrn = 21.6 kips. Ip = 406 in.4 per in.2

Solution:

Direct shear force per bolt: r1 =

Pu n

=

Pu 12

Additional shear force on bolt due to eccentricity: r2 =

Pu ecx Ip AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 36


 51⁄2−in.   Pu (16 in.)   2  = 406 in.4 per in.2 = 0.108 Pu Pu ecy r3 = Ip Pu (16 in.) (71⁄2− in.) = 406 in.4 per in.2 = 0.296 Pu Resultant shear force:  (r1 + r2)2 + (r3)2 ru = √

 √ 2

  Pu 2  12 + 0.108Pu  + (0.296Pu )   = 0.352 Pu =

Since ru must be less than or equal to φrn, φrn 0.352 21.6 kips ≤ 0.352 ≤ 61.3 kips

Pu ≤

This 20 percent reduction in the strength predicted by the instantaneous center of rotation method in Example 8-1a is indicative of the conservatism of the elastic method. Eccentricity Normal to the Plane of the Faying Surface

Eccentricity normal to the plane of the faying surface produces tension above and compression below the neutral axis of the bracket connection illustrated in Figure 8-11. The eccentric load Pu can be resolved into a concentric force Pu acting at the faying surface of the connection and a moment Pu e normal to the plane of the faying surface where e is the eccentricity. Each bolt is then assumed to support an equal share of the concentric force Pu, and the moment is resisted by tension in the bolts above the neutral axis and compression between the lower part of the bracket and the column flange. The forces for which the bolts in this connection must be designed must be determined by balancing the tensile forces in the bolts above the neutral axis with the resultant compressive force below the neutral axis. The analysis of such a connection is straightforward and usually begins with one of two assumptions: Case I assumes the neutral axis is not at the center of gravity (CG) while Case II assumes the neutral axis is at the CG. For a bearing-type connection, the limit state of bolt tension, including the effect of prying action and the shear stress present, must still be checked as specified in LRFD Specification Section J3.7. For a slip-critical connection, the bolts above the neutral axis subject to tension would lose a portion of their clamping force. The overall connection, however, would experience no reduction in total clamping force because the clamping AMERICAN INSTITUTE OF STEEL CONSTRUCTION


8 - 37

force below the neutral axis is increased by an equivalent amount. Therefore, it would be unnecessary to reduce the strength of this connection for the interaction of tension and shear above the neutral axis. However, the limit state of bolt tension, including the effect of prying action and the shear stress present, must still be checked as specified in LRFD Specification Section J3.9. Case I—Neutral Axis Not at Center of Gravity

The shear force per bolt due to the concentric force Pu is ruv, where ruv =

Pu n

and n is the number of bolts in the connection. To determine the location of the neutral axis, assume a trial position of the neutral axis at one-sixth of the total bracket depth, measured upward from the bottom. In Figure 8-12a, this is indicated by the line X-X. To provide for reasonable proportions and to recognize that the effective bearing area will depend upon the bracket flange or support flange bending stiffness, the effective width of the compression block Weff should be taken as: Weff = 8tf ≤ bf where tf = lesser of bracket flange and support flange thicknesses, in. bf = bracket flange width, in. This effective width is valid for bracket flanges made from W or S shapes, welded plates, and angles. Where the bracket flange thickness is not constant, the average flange thickness should be used. Having assumed the width of the compression block, it is possible to check an assumed location of the neutral axis by checking static equilibrium assuming an elastic stress

e

Pu

Tee Bracket

Fig. 8-11. Bolts subjected to eccentricity normal to the plane of the faying surface. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 38


distribution. Equating the moment of the bolt area above the neutral axis with the moment of the compression block area below the neutral axis, ΣAb × y = Weff × d ×

d 2

In the above equation, ΣAb = sum of the areas of all bolts above the neutral axis, in.2 y = distance from line X-X to CG of of the bolt group above neutral axis, in. d = depth of compression block, in. The value of d may then be adjusted until a reasonable equality exists. Once the neutral axis has been located, the tensile force per bolt rut, as illustrated in Figure 8-12b may be determined as: rut =

Pu ec × Ab Ix

where c = distance from neutral axis to most remote bolt in group, in. Ix = combined moment of inertia of bolt group and compression block about neutral axis, in.4 Bolts above the neutral axis are subjected to the shear force ruv, the tensile force rut, and the effect of prying action; bolts below the neutral axis are subjected to the shear force ruv only. Case II—Neutral Axis at Center of Gravity

This method provides a more direct, but also a more conservative result. As for Case I, the shear force per bolt due to the concentric force Pu is ruv, where

tf

Pu n

CG (tension group)

2rut

y

d = Depth/6

Depth

ruv =

X

NA X

Weff (a) Initial approximation of location of NA

(b) Force diagram with final location of NA

Fig. 8-12. Case I—Neutral axis (NA) not at center of gravity (CG). AMERICAN INSTITUTE OF STEEL CONSTRUCTION


8 - 39

and n is the number of bolts in the connection. The neutral axis is assumed to be located at the CG of the bolt group as illustrated in Figure 8-13. The bolts above the neutral axis are in tension and the bolts below the neutral axis are said to be in “compression.” To obtain a more accurate result, a plastic stress distribution is assumed; this assumption is justified because this method is still more conservative than Case I. Accordingly, the tensile force rut in each bolt above the neutral axis due to the moment Pue is: rut =

Pu e n′dm

where n′ = number of bolts above the neutral axis dm = moment arm between resultant tensile force and resultant compressive force, in.

tf

Bolts above the neutral axis are subjected to the shear force ruv, the tensile force rut, and the effect of prying action; bolts below the neutral axis are subjected to the shear force ruv only.

2rut

NA

Fig. 8-13. Case II—Neutral axis (NA) at center of gravity (CG). AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 40


Table 8-18. Coefficients C for Eccentrically Loaded Bolt Groups Angle = 0°° C req =

Pu

φrn

or φR n = C × φ rn

where

ex = e

s

Pu = factored force, kips φ rn = design strength per bolt, kips φ R n = design strength of bolt group, kips e = eccentricity of Pu with respect to centroid of bolt group, in. (not tabulated, may be determined by geometry.) e x = horizontal component of e, in. s = bolt spacing, in. C = coefficient tabulated below.

s

s

s

Pu

Number of bolts in one vertical row, n

s, in. ex, in.

3

6

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.18 0.88 0.69 0.56 0.48

2.23 1.75 1.40 1.15 0.97

3.32 2.81 2.36 2.01 1.73

4.39 3.90 3.40 2.96 2.59

5.45 4.98 4.47 3.98 3.55

6.48 6.06 5.56 5.05 4.57

7.51 7.12 6.64 6.13 5.63

8.52 8.17 7.72 7.22 6.70

9.53 9.21 8.78 8.30 7.79

10.5 10.2 9.84 9.38 8.87

11.5 11.3 10.9 10.4 9.96

7 8 9 10 12

0.41 0.36 0.32 0.29 0.24

0.83 0.73 0.65 0.59 0.49

1.51 1.34 1.21 1.09 0.92

2.28 2.04 1.83 1.66 1.40

3.17 2.85 2.59 2.36 2.00

4.13 3.75 3.42 3.14 2.68

5.15 4.72 4.34 4.00 3.44

6.20 5.73 5.31 4.92 4.27

7.28 6.78 6.32 5.89 5.15

8.36 7.85 7.36 6.90 6.09

9.44 8.93 8.42 7.94 7.06

14 16 18 20 24

0.21 0.18 0.16 0.15 0.12

0.42 0.37 0.33 0.29 0.25

0.79 0.70 0.62 0.56 0.47

1.21 1.06 0.95 0.85 0.71

1.74 1.53 1.37 1.24 1.03

2.33 2.06 1.84 1.67 1.40

3.01 2.67 2.39 2.16 1.82

3.75 3.33 3.00 2.72 2.29

4.55 4.06 3.66 3.33 2.81

5.41 4.85 4.38 3.99 3.37

6.31 5.68 5.15 4.70 3.99

28 32 36

0.11 0.09 0.08

0.21 0.18 0.16

0.40 0.35 0.31

0.61 0.54 0.48

0.89 0.78 0.69

1.20 1.05 0.94

1.57 1.37 1.22

1.97 1.73 1.54

2.42 2.13 1.90

2.92 2.57 2.29

3.45 3.04 2.72

2 3 4 5 6

1.63 1.39 1.18 1.01 0.88

2.71 2.48 2.23 1.98 1.75

3.75 3.56 3.32 3.07 2.81

4.77 4.60 4.39 4.15 3.90

5.77 5.63 5.45 5.23 4.98

6.77 6.65 6.48 6.28 6.06

7.76 7.65 7.51 7.33 7.12

8.75 8.66 8.52 8.36 8.17

9.74 9.66 9.53 9.38 9.21

10.7 10.7 10.5 10.4 10.2

11.7 11.6 11.5 11.4 11.2

7 8 9 10 12

0.77 0.69 0.62 0.56 0.48

1.56 1.40 1.26 1.15 0.97

2.58 2.36 2.17 2.01 1.73

3.64 3.40 3.17 2.96 2.59

4.73 4.47 4.22 3.98 3.55

5.81 5.56 5.30 5.05 4.57

6.89 6.64 6.39 6.13 5.63

7.95 7.72 7.47 7.22 6.70

9.00 8.78 8.55 8.30 7.79

10.1 9.84 9.61 9.38 8.87

11.1 10.9 10.7 10.4 9.96

14 16 18 20 24

0.41 0.36 0.32 0.29 0.24

0.83 0.73 0.65 0.59 0.49

1.51 1.34 1.21 1.09 0.92

2.28 2.04 1.83 1.66 1.40

3.17 2.85 2.59 2.36 2.00

4.13 3.75 3.42 3.14 2.68

5.15 4.72 4.34 4.00 3.44

6.20 5.73 5.31 4.92 4.27

7.28 6.78 6.32 5.89 5.15

8.36 7.85 7.36 6.90 6.09

9.44 8.93 8.42 7.94 7.06

28 32 36

0.21 0.18 0.16

0.42 0.37 0.33

0.79 0.70 0.62

1.21 1.06 0.95

1.74 1.53 1.37

2.33 2.06 1.84

3.01 2.67 2.39

3.75 3.33 3.00

4.55 4.06 3.66

5.41 4.85 4.38

6.31 5.68 5.15



8 - 41

Table 8-18 (cont.). Coefficients C for Eccentrically Loaded Bolt Groups Angle = 15°° C req =

Pu

φ rn

or φ R n = C × φrn

where

ex

Pu 15°

s

s


s

s

e


s, in. ex, in.

3

6

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.15 0.86 0.67 0.55 0.47

2.20 1.76 1.42 1.17 0.99

3.28 2.78 2.35 2.00 1.73

4.34 3.85 3.36 2.94 2.58

5.39 4.92 4.41 3.94 3.52

6.42 5.98 5.48 4.98 4.52

7.45 7.03 6.55 6.04 5.55

8.46 8.08 7.61 7.11 6.61

9.47 9.11 8.67 8.18 7.67

10.5 10.1 9.72 9.24 8.74

11.5 11.2 10.8 10.3 9.81

7 8 9 10 12

0.41 0.36 0.32 0.29 0.24

0.86 0.75 0.67 0.61 0.51

1.52 1.35 1.22 1.10 0.93

2.30 2.06 1.86 1.69 1.43

3.16 2.86 2.60 2.38 2.03

4.11 3.74 3.43 3.16 2.71

5.10 4.69 4.32 4.00 3.46

6.13 5.68 5.27 4.90 4.28

7.18 6.70 6.26 5.85 5.15

8.24 7.74 7.28 6.84 6.06

9.30 8.80 8.31 7.85 7.01

14 16 18 20 24

0.21 0.19 0.17 0.15 0.12

0.43 0.38 0.34 0.30 0.25

0.81 0.71 0.63 0.57 0.48

1.24 1.09 0.97 0.88 0.73

1.76 1.56 1.39 1.26 1.06

2.37 2.10 1.88 1.70 1.43

3.04 2.70 2.43 2.20 1.86

3.78 3.37 3.04 2.76 2.33

4.57 4.09 3.70 3.37 2.86

5.41 4.87 4.42 4.03 3.43

6.30 5.69 5.18 4.74 4.04

28 32 36

0.11 0.09 0.08

0.22 0.19 0.17

0.41 0.36 0.32

0.63 0.55 0.49

0.91 0.80 0.71

1.23 1.08 0.96

1.60 1.41 1.26

2.02 1.77 1.58

2.47 2.18 1.95

2.97 2.62 2.34

3.51 3.10 2.78

2 3 4 5 6

1.61 1.36 1.15 0.98 0.86

2.69 2.45 2.20 1.96 1.76

3.72 3.52 3.28 3.03 2.78

4.74 4.56 4.34 4.10 3.85

5.74 5.59 5.39 5.16 4.92

6.74 6.60 6.42 6.21 5.98

7.73 7.61 7.45 7.25 7.03

8.73 8.61 8.46 8.28 8.08

9.71 9.61 9.47 9.30 9.11

7 8 9 10 12

0.75 0.67 0.61 0.55 0.47

1.57 1.42 1.29 1.17 0.99

2.55 2.35 2.16 2.00 1.73

3.60 3.36 3.14 2.94 2.58

4.66 4.41 4.17 3.94 3.52

5.73 5.48 5.23 4.98 4.52

6.80 6.55 6.30 6.04 5.55

7.85 7.61 7.36 7.11 6.61

8.90 8.67 8.43 8.18 7.67

9.94 9.72 9.49 9.24 8.74

11.0 10.8 10.5 10.3 9.81

14 16 18 20 24

0.41 0.36 0.32 0.29 0.24

0.86 0.75 0.67 0.61 0.51

1.52 1.35 1.22 1.10 0.93

2.30 2.06 1.86 1.69 1.43

3.16 2.86 2.60 2.38 2.03

4.11 3.74 3.43 3.16 2.71

5.10 4.69 4.32 4.00 3.46

6.13 5.68 5.27 4.90 4.28

7.18 6.70 6.26 5.85 5.15

8.24 7.74 7.28 6.84 6.06

9.30 8.80 8.31 7.85 7.01

28 32 36

0.21 0.19 0.17

0.43 0.38 0.34

0.81 0.71 0.63

1.24 1.09 0.97

1.76 1.56 1.39

2.37 2.10 1.88

3.04 2.70 2.43

3.78 3.37 3.04

4.57 4.09 3.70

5.41 4.87 4.42

6.30 5.69 5.18


10.7 10.6 10.5 10.3 10.1

11.7 11.6 11.5 11.3 11.1

8 - 42



Pu

φrn

or φ R n = C × φ rn

where

ex

Pu

s


s

30°

s

s

e


s, in. ex, in.

3

6

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.14 0.86 0.69 0.57 0.49

2.20 1.80 1.50 1.27 1.09

3.25 2.79 2.40 2.08 1.82

4.30 3.83 3.39 3.00 2.68

5.33 4.87 4.41 3.98 3.60

6.36 5.92 5.45 4.99 4.57

7.38 6.96 6.49 6.02 5.58

8.39 7.99 7.53 7.06 6.60

9.40 9.02 8.57 8.11 7.64

10.4 10.0 9.61 9.15 8.68

11.4 11.1 10.6 10.2 9.72

7 8 9 10 12

0.43 0.38 0.34 0.31 0.26

0.95 0.83 0.75 0.67 0.56

1.61 1.44 1.30 1.19 1.01

2.40 2.17 1.98 1.82 1.55

3.27 2.98 2.74 2.52 2.17

4.20 3.86 3.57 3.31 2.87

5.17 4.79 4.46 4.15 3.64

6.17 5.76 5.39 5.05 4.46

7.18 6.75 6.35 5.98 5.33

8.21 7.77 7.34 6.95 6.24

9.25 8.79 8.35 7.93 7.17

14 16 18 20 24

0.23 0.20 0.18 0.16 0.14

0.48 0.42 0.38 0.34 0.28

0.87 0.77 0.69 0.62 0.52

1.35 1.20 1.07 0.97 0.81

1.90 1.69 1.52 1.37 1.16

2.53 2.26 2.04 1.85 1.57

3.23 2.89 2.62 2.38 2.02

3.98 3.58 3.25 2.97 2.53

4.78 4.33 3.94 3.61 3.09

5.63 5.11 4.67 4.30 3.69

6.51 5.94 5.45 5.02 4.33

28 32 36

0.12 0.10 0.09

0.24 0.21 0.19

0.45 0.40 0.35

0.70 0.61 0.55

1.00 0.88 0.78

1.36 1.19 1.07

1.75 1.54 1.38

2.20 1.94 1.74

2.69 2.38 2.13

3.22 2.85 2.56

3.79 3.37 3.03

2 3 4 5 6

1.59 1.34 1.14 0.98 0.86

2.66 2.43 2.20 1.99 1.80

3.69 3.48 3.25 3.02 2.79

4.70 4.52 4.30 4.06 3.83

5.71 5.54 5.33 5.11 4.87

6.70 6.55 6.36 6.14 5.92

7.70 7.55 7.38 7.17 6.96

8.69 8.56 8.39 8.20 7.99

9.68 9.55 9.40 9.22 9.02

7 8 9 10 12

0.77 0.69 0.63 0.57 0.49

1.64 1.50 1.37 1.27 1.09

2.59 2.40 2.23 2.08 1.82

3.60 3.39 3.19 3.00 2.68

4.64 4.41 4.19 3.98 3.60

5.68 5.45 5.22 4.99 4.57

6.73 6.49 6.26 6.02 5.58

7.77 7.53 7.30 7.06 6.60

8.80 8.57 8.34 8.11 7.64

9.83 9.61 9.38 9.15 8.68

10.9 10.6 10.4 10.2 9.72

14 16 18 20 24

0.43 0.38 0.34 0.31 0.26

0.95 0.83 0.75 0.67 0.56

1.61 1.44 1.30 1.19 1.01

2.40 2.17 1.98 1.82 1.55

3.27 2.98 2.74 2.52 2.17

4.20 3.86 3.57 3.31 2.87

5.17 4.79 4.46 4.15 3.64

6.17 5.76 5.39 5.05 4.46

7.18 6.75 6.35 5.98 5.33

8.21 7.77 7.34 6.95 6.24

9.25 8.79 8.35 7.93 7.17

28 32 36

0.23 0.20 0.18

0.48 0.42 0.38

0.87 0.77 0.69

1.35 1.20 1.07

1.90 1.69 1.52

2.53 2.26 2.04

3.23 2.89 2.62

3.98 3.58 3.25

4.78 4.33 3.94

5.63 5.11 4.67

6.51 5.94 5.45


10.7 10.6 10.4 10.2 10.0

11.7 11.5 11.4 11.2 11.1


8 - 43


Pu

φrn


where

ex 45°

Pu

s

s


s

s

e


s, in. ex, in.

3

6

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.17 0.92 0.75 0.64 0.55

2.23 1.89 1.63 1.42 1.25

3.26 2.87 2.54 2.25 2.01

4.28 3.87 3.50 3.17 2.88

5.29 4.88 4.49 4.13 3.80

6.30 5.90 5.49 5.11 4.76

7.31 6.91 6.51 6.11 5.73

8.32 7.93 7.52 7.11 6.73

9.32 8.94 8.53 8.12 7.73

10.3 9.95 9.55 9.14 8.73

11.3 11.0 10.6 10.2 9.74

7 8 9 10 12

0.49 0.44 0.40 0.36 0.31

1.11 0.99 0.90 0.81 0.68

1.81 1.64 1.49 1.37 1.17

2.63 2.41 2.22 2.06 1.79

3.51 3.25 3.02 2.82 2.47

4.43 4.14 3.87 3.63 3.22

5.38 5.06 4.77 4.50 4.02

6.36 6.01 5.69 5.39 4.87

7.34 6.98 6.64 6.32 5.74

8.34 7.96 7.61 7.27 6.65

9.34 8.96 8.58 8.23 7.58

14 16 18 20 24

0.27 0.24 0.21 0.19 0.16

0.59 0.52 0.46 0.41 0.35

1.03 0.91 0.82 0.74 0.63

1.58 1.41 1.27 1.16 0.98

2.20 1.97 1.78 1.62 1.38

2.88 2.60 2.36 2.16 1.85

3.62 3.29 3.00 2.76 2.37

4.41 4.03 3.70 3.41 2.94

5.24 4.81 4.43 4.10 3.56

6.11 5.63 5.21 4.84 4.22

6.99 6.48 6.02 5.61 4.92

28 32 36

0.14 0.12 0.11

0.30 0.26 0.23

0.54 0.48 0.43

0.85 0.75 0.67

1.19 1.05 0.94

1.61 1.43 1.28

2.08 1.84 1.65

2.58 2.30 2.07

3.14 2.80 2.53

3.73 3.34 3.02

4.37 3.92 3.55

2 3 4 5 6

1.57 1.35 1.17 1.03 0.92

2.64 2.43 2.23 2.05 1.89

3.66 3.46 3.26 3.06 2.87

4.67 4.48 4.28 4.07 3.87

5.67 5.49 5.29 5.09 4.88

6.66 6.49 6.30 6.10 5.90

7.66 7.50 7.31 7.12 6.91

8.65 8.49 8.32 8.13 7.93

9.64 9.49 9.32 9.13 8.94

10.6 10.5 10.3 10.1 9.95

11.6 11.5 11.3 11.1 11.0

7 8 9 10 12

0.83 0.75 0.69 0.64 0.55

1.75 1.63 1.52 1.42 1.25

2.70 2.54 2.39 2.25 2.01

3.68 3.50 3.33 3.17 2.88

4.68 4.49 4.30 4.13 3.80

5.69 5.49 5.30 5.11 4.76

6.71 6.51 6.30 6.11 5.73

7.72 7.52 7.31 7.11 6.73

8.74 8.53 8.33 8.12 7.73

9.75 9.55 9.34 9.14 8.73

10.8 10.6 10.4 10.2 9.74

14 16 18 20 24

0.49 0.44 0.40 0.36 0.31

1.11 0.99 0.90 0.81 0.68

1.81 1.64 1.49 1.37 1.17

2.63 2.41 2.22 2.06 1.79

3.51 3.25 3.02 2.82 2.47

4.43 4.14 3.87 3.63 3.22

5.38 5.06 4.77 4.50 4.02

6.36 6.01 5.69 5.39 4.87

7.34 6.98 6.64 6.32 5.74

8.34 7.96 7.61 7.27 6.65

9.34 8.96 8.58 8.23 7.58

28 32 36

0.27 0.24 0.21

0.59 0.52 0.46

1.03 0.91 0.82

1.58 1.41 1.27

2.20 1.97 1.78

2.88 2.60 2.36

3.62 3.29 3.00

4.41 4.03 3.70

5.24 4.81 4.43

6.11 5.63 5.21

6.99 6.48 6.02


8 - 44



Pu

φrn


where

ex 60°

Pu

s

e

s

s

s



s, in. ex, in.

3

6

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.27 1.05 0.89 0.77 0.68

2.32 2.05 1.83 1.65 1.49

3.32 3.02 2.77 2.54 2.34

4.31 4.00 3.72 3.47 3.24

5.30 4.98 4.69 4.41 4.16

6.30 5.97 5.66 5.37 5.10

7.29 6.96 6.64 6.34 6.06

8.27 7.94 7.62 7.32 7.02

9.27 8.94 8.61 8.29 7.99

10.3 9.93 9.60 9.28 8.97

11.3 10.9 10.6 10.3 9.95

7 8 9 10 12

0.61 0.56 0.51 0.47 0.40

1.37 1.26 1.16 1.07 0.93

2.17 2.01 1.87 1.74 1.52

3.03 2.83 2.66 2.50 2.22

3.93 3.71 3.51 3.32 3.00

4.85 4.61 4.39 4.19 3.82

5.79 5.54 5.30 5.08 4.67

6.74 6.48 6.23 5.99 5.55

7.71 7.43 7.17 6.92 6.45

8.67 8.39 8.12 7.86 7.37

9.64 9.35 9.07 8.81 8.30

14 16 18 20 24

0.35 0.32 0.29 0.26 0.22

0.81 0.72 0.65 0.58 0.49

1.35 1.21 1.09 1.00 0.85

2.00 1.81 1.66 1.53 1.32

2.73 2.49 2.30 2.12 1.84

3.50 3.23 2.98 2.77 2.41

4.32 4.00 3.72 3.47 3.05

5.16 4.81 4.50 4.21 3.73

6.03 5.65 5.31 4.99 4.45

6.92 6.51 6.14 5.80 5.21

7.83 7.40 7.00 6.63 5.99

28 32 36

0.19 0.17 0.15

0.42 0.37 0.33

0.74 0.65 0.59

1.15 1.02 0.92

1.61 1.43 1.29

2.13 1.91 1.72

2.71 2.44 2.21

3.34 3.02 2.74

4.00 3.63 3.31

4.70 4.28 3.92

5.44 4.97 4.57

2 3 4 5 6

1.60 1.42 1.27 1.15 1.05

2.65 2.48 2.32 2.18 2.05

3.65 3.48 3.32 3.17 3.02

4.64 4.48 4.31 4.15 4.00

5.64 5.47 5.30 5.14 4.98

6.63 6.46 6.30 6.13 5.97

7.62 7.45 7.29 7.12 6.96

8.61 8.44 8.27 8.11 7.94

9.60 9.44 9.27 9.10 8.94

10.6 10.4 10.3 10.1 9.93

11.6 11.4 11.3 11.1 10.9

7 8 9 10 12

0.96 0.89 0.83 0.77 0.68

1.93 1.83 1.73 1.65 1.49

2.89 2.77 2.65 2.54 2.34

3.86 3.72 3.59 3.47 3.24

4.83 4.69 4.55 4.41 4.16

5.81 5.66 5.51 5.37 5.10

6.80 6.64 6.49 6.34 6.06

7.78 7.62 7.47 7.32 7.02

8.77 8.61 8.45 8.29 7.99

9.76 9.60 9.43 9.28 8.97

10.8 10.6 10.4 10.3 9.95

14 16 18 20 24

0.61 0.56 0.51 0.47 0.40

1.37 1.26 1.16 1.07 0.93

2.17 2.01 1.87 1.74 1.52

3.03 2.83 2.66 2.50 2.22

3.93 3.71 3.51 3.32 3.00

4.85 4.61 4.39 4.19 3.82

5.79 5.54 5.30 5.08 4.67

6.74 6.48 6.23 5.99 5.55

7.71 7.43 7.17 6.92 6.45

8.67 8.39 8.12 7.86 7.37

9.64 9.35 9.07 8.81 8.30

28 32 36

0.35 0.32 0.29

0.81 0.72 0.65

1.35 1.21 1.09

2.00 1.81 1.66

2.73 2.49 2.30

3.50 3.23 2.98

4.32 4.00 3.72

5.16 4.81 4.50

6.03 5.65 5.31

6.92 6.51 6.14

7.83 7.40 7.00



8 - 45


Pu

φ rn


where

ex 75°

s


s

Pu

s

s

e


s, in. ex, in.

3

6

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.49 1.32 1.18 1.07 0.98

2.51 2.33 2.18 2.04 1.92

3.49 3.30 3.14 2.99 2.85

4.46 4.27 4.09 3.93 3.79

5.44 5.24 5.05 4.88 4.73

6.42 6.21 6.01 5.84 5.67

7.40 7.18 6.98 6.79 6.62

8.38 8.15 7.95 7.75 7.57

9.36 9.13 8.92 8.72 8.53

10.3 10.1 9.89 9.68 9.49

11.3 11.1 10.9 10.7 10.5

7 8 9 10 12

0.90 0.84 0.78 0.73 0.65

1.82 1.72 1.63 1.55 1.41

2.73 2.62 2.51 2.41 2.23

3.65 3.52 3.40 3.29 3.08

4.58 4.44 4.31 4.19 3.95

5.52 5.37 5.23 5.10 4.84

6.46 6.30 6.16 6.02 5.75

7.40 7.24 7.09 6.94 6.66

8.36 8.19 8.03 7.88 7.59

9.31 9.14 8.97 8.81 8.51

10.3 10.1 9.92 9.76 9.45

14 16 18 20 24

0.58 0.53 0.48 0.44 0.38

1.30 1.20 1.11 1.03 0.89

2.06 1.92 1.78 1.66 1.46

2.88 2.70 2.53 2.38 2.12

3.73 3.52 3.33 3.16 2.85

4.60 4.38 4.17 3.97 3.63

5.50 5.26 5.03 4.82 4.44

6.40 6.15 5.91 5.69 5.27

7.31 7.05 6.80 6.56 6.13

8.23 7.96 7.70 7.45 6.99

9.16 8.88 8.61 8.35 7.87

28 32 36

0.34 0.30 0.27

0.79 0.70 0.62

1.29 1.16 1.05

1.90 1.73 1.58

2.59 2.38 2.19

3.33 3.08 2.85

4.11 3.81 3.55

4.91 4.58 4.28

5.73 5.37 5.05

6.57 6.19 5.84

7.43 7.02 6.65

2 3 4 5 6

1.71 1.60 1.49 1.40 1.32

2.72 2.61 2.51 2.42 2.33

3.70 3.59 3.49 3.39 3.30

4.69 4.57 4.46 4.37 4.27

5.67 5.55 5.44 5.34 5.24

6.66 6.53 6.42 6.31 6.21

7.64 7.52 7.40 7.29 7.18

8.79 8.50 8.38 8.26 8.15

9.78 9.48 9.36 9.24 9.13

10.8 10.5 10.3 10.2 10.1

11.7 11.5 11.3 11.2 11.1

7 8 9 10 12

1.25 1.18 1.13 1.07 0.98

2.25 2.18 2.11 2.04 1.92

3.22 3.14 3.06 2.99 2.85

4.18 4.09 4.01 3.93 3.79

5.14 5.05 4.97 4.88 4.73

6.11 6.01 5.92 5.84 5.67

7.07 6.98 6.88 6.79 6.62

8.05 7.95 7.85 7.75 7.57

9.01 8.92 8.81 8.72 8.53

10.0 9.89 9.78 9.68 9.49

11.0 10.9 10.8 10.7 10.5

14 16 18 20 24

0.90 0.84 0.78 0.73 0.65

1.82 1.72 1.63 1.55 1.41

2.73 2.62 2.51 2.41 2.23

3.65 3.52 3.40 3.29 3.08

4.58 4.44 4.31 4.19 3.95

5.52 5.37 5.23 5.10 4.84

6.46 6.30 6.16 6.02 5.75

7.40 7.24 7.09 6.94 6.66

8.36 8.19 8.03 7.88 7.59

9.31 9.14 8.97 8.81 8.51

10.3 10.1 9.92 9.76 9.45

28 32 36

0.58 0.53 0.48

1.30 1.20 1.11

2.06 1.92 1.78

2.88 2.70 2.53

3.73 3.52 3.33

4.60 4.38 4.17

5.50 5.26 5.03

6.40 6.15 5.91

7.31 7.05 6.80

8.23 7.96 7.70

9.16 8.88 8.61


8 - 46



Pu

φrn

or φR n = C × φ rn ex = e

where

s, in.

3

6

Pu

s

s

s

s


3


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

0.84 0.65 0.54 0.45 0.39

2.54 2.03 1.67 1.42 1.22

4.48 3.68 3.06 2.59 2.25

6.59 5.67 4.86 4.21 3.69

8.72 7.77 6.84 6.01 5.32

10.8 9.91 8.93 8.00 7.17

12.9 12.1 11.1 10.1 9.16

15.0 14.2 13.2 12.2 11.2

17.0 16.3 15.4 14.4 13.4

19.0 18.3 17.5 16.5 15.5

21.0 20.4 19.6 18.7 17.7

23.0 22.5 21.7 20.8 19.8

7 8 9 10 12

0.35 0.31 0.28 0.26 0.22

1.08 0.96 0.86 0.78 0.66

1.99 1.78 1.60 1.46 1.24

3.27 2.93 2.65 2.42 2.06

4.74 4.27 3.87 3.53 3.01

6.46 5.86 5.34 4.90 4.19

8.33 10.3 12.4 7.60 9.50 11.5 6.97 8.75 10.7 6.42 8.10 9.91 5.51 7.01 8.63

14.5 13.6 12.7 11.8 10.4

16.7 15.7 14.7 13.8 12.2

18.8 17.8 16.8 15.9 14.2

14 16 18 20 24

0.19 0.17 0.15 0.14 0.12

0.57 0.51 0.45 0.41 0.34

1.08 0.95 0.85 0.77 0.65

1.78 1.57 1.41 1.27 1.07

2.62 2.32 2.07 1.88 1.58

3.66 3.24 2.90 2.63 2.21

4.82 4.27 3.83 3.48 2.93

6.15 5.47 4.92 4.47 3.77

7.61 6.79 6.11 5.55 4.69

9.19 10.9 12.7 8.23 9.78 11.4 7.43 8.85 10.4 6.76 8.07 9.48 5.72 6.85 8.06

28 32 36

0.10 0.09 0.08

0.29 0.26 0.23

0.56 0.49 0.43

0.92 0.80 0.72

1.36 1.19 1.06

1.90 1.67 1.49

2.53 2.22 1.98

3.25 2.86 2.55

4.05 3.57 3.18

4.95 4.36 3.90

2 3 4 5 6

0.84 0.65 0.54 0.45 0.39

3.24 2.79 2.41 2.10 1.85

5.39 4.93 4.44 3.97 3.55

7.47 7.08 6.60 6.11 5.62

9.51 9.17 8.75 8.27 7.77

11.5 11.2 10.9 10.4 9.93

13.5 13.3 12.4 12.5 12.1

15.5 15.3 15.0 14.6 14.2

17.5 17.3 17.0 16.7 16.3

19.5 19.3 19.1 18.7 18.4

21.5 21.3 21.1 20.8 20.4

23.4 23.3 23.1 22.8 22.5

7 8 9 10 12

0.35 0.31 0.28 0.26 0.22

1.64 1.47 1.34 1.22 1.04

3.18 2.87 2.61 2.39 2.04

5.17 4.75 4.39 4.06 3.52

7.27 6.79 6.34 5.92 5.20

9.43 8.92 8.43 7.96 7.10

11.6 11.1 10.6 10.1 9.12

13.7 13.3 12.7 12.2 11.2

15.9 15.4 14.9 14.4 13.4

18.0 17.5 17.1 16.6 15.6

20.1 19.6 19.2 18.7 17.7

22.1 21.7 21.3 20.9 19.9

14 16 18 20 24

0.19 0.17 0.15 0.14 0.12

0.90 0.80 0.71 0.64 0.54

1.77 1.57 1.41 1.28 1.07

3.09 2.75 2.48 2.25 1.90

4.61 4.12 3.72 3.38 2.86

6.36 5.74 5.21 4.77 4.06

8.27 10.3 12.4 7.52 9.44 11.7 6.87 8.68 10.6 6.31 8.02 9.85 5.40 6.91 8.55

14.5 13.5 12.6 11.8 10.3

16.7 15.7 14.7 13.8 12.2

18.9 17.8 16.8 15.9 14.1

28 32 36

0.10 0.09 0.08

0.46 0.41 0.36

0.93 0.81 0.73

1.64 1.44 1.29

2.47 2.18 1.94

3.52 3.11 2.78

4.70 4.16 3.72

6.05 5.37 4.81


7.52 6.69 6.02

5.93 5.23 4.67

7.00 6.18

5.52

9.12 10.8 12.6 8.15 9.71 11.4 7.34 8.78 10.3


8 - 47


Pu

φrn

or φR n = C × φ rn ex

where

s, in.

3

6

15°

s

s


Pu

s

s

e

3


ex, in.

1

2

3

4

5

2 3 4 5 6

0.87 0.68 0.55 0.47 0.41

2.54 2.04 1.69 1.44 1.25

4.47 3.71 3.11 2.66 2.31

6.54 5.63 4.85 4.21 3.70

8.63 7.69 6.79 6.00 5.34

7 8 9 10 12

0.36 0.32 0.29 0.27 0.23

1.10 0.98 0.88 0.81 0.68

2.04 1.83 1.65 1.51 1.28

3.29 2.96 2.68 2.45 2.09

4.79 4.32 3.94 3.61 3.08

6.46 5.87 5.37 4.93 4.24

8.30 10.2 12.3 7.60 9.45 11.4 6.99 8.74 10.6 6.45 8.11 9.88 5.58 7.05 8.66

14 16 18 20 24

0.20 0.17 0.16 0.14 0.12

0.59 0.52 0.47 0.42 0.35

1.11 0.98 0.88 0.79 0.67

1.82 1.61 1.44 1.31 1.10

2.69 2.38 2.13 1.93 1.62

3.71 3.29 2.96 2.68 2.26

4.90 4.36 3.92 3.56 3.00

6.21 5.54 4.99 4.54 3.84

7.67 6.86 6.20 5.65 4.79

9.23 10.9 12.7 8.29 9.83 11.5 7.51 8.93 10.4 6.85 8.17 9.57 5.82 6.96 8.17

28 32 36

0.10 0.09 0.08

0.30 0.27 0.24

0.57 0.50 0.45

0.94 0.83 0.74

1.40 1.23 1.10

1.95 1.72 1.53

2.60 2.28 2.04

3.32 2.93 2.61

4.15 3.66 3.27

5.05 4.46 3.98

2 3 4 5 6

0.87 0.68 0.55 0.47 0.41

3.21 2.76 2.38 2.07 1.83

5.35 4.88 4.40 3.96 3.56

7.42 7.00 6.53 6.04 5.56

9.45 9.09 8.65 8.17 7.67

11.5 11.1 10.7 10.3 9.80

7 8 9 10 12

0.36 0.32 0.29 0.27 0.23

1.63 1.47 1.34 1.23 1.05

3.22 2.92 2.66 2.45 2.09

5.12 4.73 4.37 4.05 3.53

7.19 6.72 6.29 5.90 5.21

9.30 11.4 8.81 10.9 8.33 10.4 7.88 9.95 7.06 9.04

14 16 18 20 24

0.20 0.17 0.16 0.14 0.12

0.91 0.81 0.72 0.66 0.55

1.83 1.62 1.45 1.32 1.11

3.11 2.78 2.50 2.28 1.93

4.64 4.17 3.77 3.45 2.93

6.35 5.75 5.24 4.80 4.10

8.22 10.2 12.2 7.51 9.38 11.4 6.88 8.66 10.5 6.34 8.02 9.82 5.46 6.95 8.57

28 32 36

0.10 0.09 0.08

0.48 0.42 0.37

0.96 0.84 0.75

1.67 1.47 1.32

2.54 2.24 2.00

3.57 3.16 2.83

4.78 4.24 3.80

6

7

10.7 12.8 9.80 11.9 8.84 10.9 7.94 9.98 7.15 9.09

13.5 13.2 12.8 12.4 11.9

8

9

10

11

12

14.8 14.0 13.0 12.1 11.1

16.9 16.1 15.2 14.2 13.2

18.9 18.2 17.3 16.3 15.3

20.9 20.2 19.4 18.4 17.4

22.9 22.3 21.5 20.5 19.6

14.3 13.4 12.6 11.8 10.4

16.4 15.5 14.6 13.7 12.2

18.6 17.6 16.6 15.7 14.1

6.05 5.34 4.78

7.12 6.29 5.64

15.5 15.2 14.9 14.5 14.0

17.4 17.2 16.9 16.5 16.1

19.4 19.2 18.9 18.6 18.2

21.4 21.2 20.9 20.6 20.3

23.4 23.2 22.9 22.6 22.3

13.6 13.1 12.6 12.1 11.1

15.7 15.2 14.7 14.2 13.2

17.8 17.3 16.8 16.3 15.3

19.9 19.4 18.9 18.5 17.5

21.9 21.5 21.0 20.6 19.6

14.3 13.4 12.5 11.7 10.3

16.5 15.5 14.5 13.7 12.1

18.6 17.6 16.6 15.7 14.0

6.11 5.44 4.89

7.58 6.77 6.10


9.15 10.8 12.6 8.21 9.75 11.4 7.42 8.85 10.4

8 - 48



Pu

φrn


where

Pu

s, in.

3

6

s


s

30°

s

s

e

3


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

0.97 0.75 0.62 0.52 0.45

2.60 2.12 1.78 1.53 1.34

4.52 3.83 3.29 2.85 2.51

6.54 5.71 4.99 4.39 3.89

8.59 7.71 6.88 6.16 5.54

10.6 9.75 8.87 8.06 7.33

12.9 11.8 10.9 10.0 9.23

14.7 13.9 13.0 12.1 11.2

16.7 15.9 15.1 14.1 13.2

18.8 18.0 17.1 16.2 15.3

20.8 20.0 19.2 18.3 17.3

22.8 22.1 21.3 20.4 19.4

7 8 9 10 12

0.40 0.36 0.32 0.30 0.25

1.19 1.07 0.97 0.88 0.75

2.23 2.00 1.81 1.66 1.41

3.48 3.15 2.87 2.64 2.27

5.01 4.57 4.19 3.87 3.34

6.70 6.14 5.66 5.24 4.54

8.51 10.4 7.86 9.68 7.28 9.02 6.77 8.43 5.92 7.43

12.4 11.6 10.9 10.2 9.04

14.4 13.6 12.8 12.0 10.8

16.4 15.6 14.7 13.9 12.5

18.5 17.6 16.7 15.9 14.4

14 16 18 20 24

0.22 0.19 0.17 0.16 0.13

0.65 0.58 0.52 0.47 0.39

1.23 1.08 0.97 0.88 0.74

1.98 1.76 1.58 1.43 1.21

2.93 2.60 2.34 2.12 1.79

3.99 3.56 3.21 2.92 2.48

5.24 4.69 4.24 3.87 3.29

6.61 5.94 5.38 4.92 4.18

8.09 7.30 6.64 6.08 5.19

9.67 11.4 8.77 10.3 8.00 9.45 7.34 8.70 6.29 7.48

28 32 36

0.12 0.10 0.09

0.34 0.30 0.26

0.64 0.56 0.50

1.04 0.92 0.82

1.55 1.36 1.21

2.14 1.89 1.69

2.85 2.51 2.25

3.63 3.21 2.87

4.52 4.00 3.59

5.49 4.87 4.37

2 3 4 5 6

0.97 0.75 0.62 0.52 0.45

3.20 2.75 2.39 2.10 1.87

5.31 4.86 4.42 4.02 3.67

7.37 6.95 6.49 6.04 5.61

9.39 9.01 8.57 8.11 7.66

11.4 11.1 10.6 10.2 9.73

7 8 9 10 12

0.40 0.36 0.32 0.30 0.25

1.69 1.53 1.40 1.29 1.12

3.36 3.08 2.84 2.63 2.28

5.21 4.84 4.51 4.21 3.70

7.21 6.79 6.40 6.04 5.39

9.27 11.4 8.82 10.9 8.39 10.4 7.98 9.99 7.23 9.16

14 16 18 20 24

0.22 0.19 0.17 0.16 0.13

0.98 0.87 0.79 0.71 0.60

2.00 1.78 1.60 1.45 1.23

3.29 2.95 2.68 2.45 2.08

4.86 4.40 4.02 3.70 3.17

6.57 6.01 5.52 5.09 4.39

8.41 10.3 7.75 9.60 7.17 8.93 6.65 8.33 5.79 7.32

28 32 36

0.12 0.10 0.09

0.52 0.46 0.41

1.06 0.93 0.83

1.82 1.61 1.44

2.77 2.45 2.20

3.85 3.42 3.08

5.11 4.56 4.12

13.4 13.1 12.7 12.3 11.8

6.54 5.81 5.22

13.1 12.0 11.0 10.1 8.75 7.68 6.83 6.15

15.4 15.1 14.7 14.3 13.9

17.4 17.1 16.8 16.4 16.0

19.4 19.1 18.8 18.4 18.0

21.3 21.1 20.8 20.4 20.1

23.3 23.1 22.8 22.5 22.1

13.4 13.0 12.5 12.0 11.2

15.5 15.1 14.6 14.1 13.2

17.6 17.1 16.7 16.2 15.3

19.6 19.2 18.7 18.3 17.3

21.7 21.3 20.8 20.4 19.4

12.3 11.5 10.8 10.1 8.95

14.4 13.5 12.7 12.0 10.7

16.4 15.5 14.7 13.9 12.5

18.5 17.6 16.7 15.9 14.4

6.49 5.82 5.27


7.99 7.20 6.53

9.59 11.3 13.0 8.68 10.3 11.9 7.91 9.37 10.9


8 - 49


Pu

φ rn

or φ R n = C × φrn ex

where

Pu

3

6

s

s

45°

s

e

s, in.

s


3


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.17 0.92 0.75 0.64 0.55

2.79 2.32 1.99 1.74 1.54

4.67 4.06 3.57 3.17 2.84

6.62 5.92 5.31 4.78 4.33

8.61 7.86 7.16 6.53 5.98

10.6 9.83 9.09 8.39 7.76

12.6 11.8 11.1 10.3 9.63

14.6 13.9 13.1 12.3 11.6

16.6 15.9 15.1 14.3 13.5

18.6 17.9 17.1 16.3 15.5

20.6 19.9 19.1 18.3 17.5

22.6 21.9 21.1 20.3 19.5

7 8 9 10 12

0.49 0.44 0.40 0.36 0.31

1.38 1.25 1.14 1.05 0.90

2.57 2.33 2.13 1.96 1.68

3.93 3.60 3.31 3.06 2.65

5.49 5.06 4.69 4.36 3.83

7.20 6.70 6.25 5.85 5.17

9.00 10.9 8.43 10.3 7.91 9.67 7.44 9.14 6.63 8.20

12.8 12.1 11.5 10.9 9.86

14.8 14.0 13.4 12.7 11.6

16.8 16.0 15.3 14.6 13.4

18.7 18.0 17.2 16.5 15.2

14 16 18 20 24

0.27 0.24 0.21 0.19 0.16

0.78 0.69 0.62 0.56 0.48

1.47 1.31 1.17 1.06 0.90

2.33 2.08 1.88 1.71 1.45

3.40 3.05 2.76 2.52 2.14

4.61 4.16 3.77 3.45 2.94

5.95 5.38 4.91 4.51 3.87

7.41 6.74 6.18 5.69 4.91

8.97 10.6 12.3 8.20 9.75 11.4 7.55 9.00 10.5 6.97 8.34 9.80 6.04 7.26 8.57

28 32 36

0.14 0.12 0.11

0.41 0.36 0.32

0.77 0.68 0.61

1.26 1.11 0.99

1.86 1.64 1.47

2.56 2.27 2.03

3.38 3.00 2.70

4.30 3.82 3.44

5.30 4.73 4.26

2 3 4 5 6

1.17 0.92 0.75 0.64 0.55

3.24 2.84 2.51 2.24 2.03

5.30 4.90 4.52 4.17 3.86

7.32 6.93 6.53 6.15 5.78

9.33 8.96 8.56 8.15 7.76

11.3 11.0 10.6 10.2 9.77

13.3 13.0 12.6 12.2 11.8

15.3 15.0 14.6 14.2 13.8

17.3 17.0 16.6 16.2 15.8

19.3 19.0 18.6 18.3 17.9

21.3 21.0 20.6 20.3 19.9

23.2 23.0 22.6 22.3 21.9

7 8 9 10 12

0.49 0.44 0.40 0.36 0.31

1.85 1.70 1.57 1.46 1.28

3.59 3.35 3.13 2.94 2.60

5.45 5.13 4.85 4.58 4.11

7.39 7.03 6.70 6.38 5.81

9.38 9.00 8.63 8.28 7.64

11.4 11.0 10.6 10.2 9.54

13.4 13.0 12.6 12.2 11.5

15.4 15.0 14.6 14.2 13.5

17.5 17.1 16.7 16.3 15.6

19.5 19.1 18.7 18.3 17.5

21.5 21.1 20.7 20.3 19.5

14 16 18 20 24

0.27 0.24 0.21 0.19 0.16

1.13 1.01 0.92 0.84 0.72

2.32 2.09 1.90 1.73 1.47

3.71 3.36 3.07 2.83 2.43

5.31 4.88 4.50 4.18 3.64

7.06 6.55 6.09 5.69 5.00

8.89 10.8 8.31 10.2 7.78 9.56 7.31 9.02 6.48 8.08

12.7 12.0 11.4 10.8 9.76

14.7 14.0 13.3 12.7 11.5

16.7 15.9 15.2 14.6 13.3

18.7 17.9 17.2 16.5 15.2

28 32 36

0.14 0.12 0.11

0.62 0.55 0.49

1.28 1.13 1.01

2.13 1.90 1.71

3.22 2.88 2.61

4.45 3.99 3.62

5.80 5.24 4.77

8.86 10.5 12.2 8.09 9.65 11.3 7.43 8.90 10.4

14.0 13.0 12.0

7.28 6.62 6.05


6.41 5.73 5.17

7.59 6.80 6.15

14.1 13.1 12.1 11.3 9.95 8.85 7.94 7.20

8 - 50



Pu

φrn


where

3

6

s s

60°

s

e

s, in.

Pu

s


3


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.51 1.24 1.04 0.89 0.77

3.17 2.76 2.43 2.16 1.95

4.97 4.47 4.04 3.70 3.40

6.85 6.30 5.81 5.39 5.01

8.77 8.19 7.65 7.17 6.73

10.7 10.1 9.53 9.01 8.52

12.7 12.0 11.5 10.9 10.4

14.6 14.0 13.4 12.8 12.3

16.6 16.0 15.3 14.7 14.2

18.6 17.9 17.3 16.7 16.1

20.6 19.9 19.3 18.6 18.0

22.5 21.9 21.2 20.6 20.0

7 8 9 10 12

0.68 0.61 0.56 0.51 0.43

1.77 1.62 1.49 1.38 1.20

3.13 2.90 2.70 2.52 2.21

4.67 4.37 4.09 3.84 3.40

6.33 5.96 5.62 5.31 4.76

8.07 7.65 7.26 6.89 6.25

9.88 9.42 8.98 8.58 7.85

11.7 11.2 10.8 10.3 9.53

13.6 13.1 12.6 12.1 11.3

15.5 15.0 14.5 14.0 13.0

17.4 16.9 16.3 15.8 14.9

19.4 18.8 18.2 17.7 16.7

14 16 18 20 24

0.38 0.34 0.30 0.27 0.23

1.06 0.95 0.85 0.78 0.66

1.96 1.76 1.60 1.46 1.24

3.05 2.75 2.51 2.30 1.97

4.30 3.92 3.59 3.32 2.87

5.71 5.24 4.84 4.48 3.90

7.23 6.68 6.19 5.76 5.04

8.83 10.5 8.20 9.79 7.64 9.16 7.14 8.60 6.29 7.64

12.2 11.5 10.8 10.1 9.06

14.0 13.2 12.4 11.7 10.6

15.8 14.9 14.1 13.4 12.1

28 32 36

0.20 0.18 0.16

0.57 0.50 0.45

1.07 0.95 0.85

1.72 1.52 1.37

2.52 2.24 2.02

3.44 3.07 2.77

4.47 4.01 3.63

5.61 5.06 4.59

2 3 4 5 6

1.51 1.24 1.04 0.89 0.77

3.39 3.08 2.80 2.57 2.37

5.36 5.04 4.73 4.45 4.20

7.33 7.01 6.69 6.39 6.11

9.31 8.98 8.66 8.35 8.05

7 8 9 10 12

0.68 0.61 0.56 0.51 0.43

2.19 2.04 1.91 1.80 1.60

3.98 3.77 3.59 3.42 3.11

5.85 5.61 5.38 5.17 4.78

7.76 7.49 7.24 7.00 6.54

9.70 9.41 9.13 8.87 8.37

14 16 18 20 24

0.38 0.34 0.30 0.27 0.23

1.44 1.31 1.20 1.10 0.95

2.85 2.63 2.43 2.26 1.97

4.43 4.12 3.84 3.58 3.15

6.13 5.74 5.40 5.08 4.53

7.91 7.48 7.08 6.71 6.06

9.74 9.27 8.84 8.43 7.69

28 32 36

0.20 0.18 0.16

0.84 0.74 0.67

1.73 1.54 1.39

2.80 2.52 2.28

4.08 3.71 3.39

5.52 5.05 4.65

7.06 6.51 6.02

11.3 11.0 10.6 10.3 10.0

6.85 6.20 5.65

8.17 7.41 6.77

9.55 11.0 8.70 10.1 7.98 9.26

13.3 12.9 12.6 12.3 12.0

15.2 14.9 14.6 14.3 13.9

17.2 16.9 16.6 16.2 15.9

19.2 18.9 18.6 18.2 17.9

21.2 20.9 20.5 20.2 19.9

23.2 22.8 22.5 22.2 21.8

11.7 11.6 11.1 10.8 10.2

13.6 13.3 13.0 12.7 12.1

15.6 15.3 15.0 14.7 14.1

17.6 17.2 16.9 16.6 16.0

19.5 19.2 18.9 18.6 18.0

21.5 21.2 20.9 20.5 19.9

11.6 11.1 10.7 10.2 9.39

13.5 13.0 12.5 12.0 11.2

15.4 14.9 14.4 13.9 12.9

17.4 16.8 16.3 15.7 14.8

19.3 18.7 18.2 17.6 16.6

8.68 10.4 12.1 8.05 9.66 11.3 7.49 9.03 10.7

13.9 13.1 12.3

15.7 14.8 14.0



8 - 51


Pu

φ rn


where

75°

3

6

s

Pu

s

s

e

s, in.

s


3


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.84 1.71 1.57 1.44 1.31

3.63 3.41 3.19 2.98 2.79

5.44 5.17 4.90 4.65 4.41

7.29 6.97 6.67 6.39 6.12

9.17 8.82 8.50 8.19 7.90

11.1 10.7 10.4 10.0 9.71

13.0 12.6 12.2 11.9 11.6

14.9 14.5 14.1 13.8 13.4

16.9 16.4 16.0 15.7 15.3

18.8 18.4 18.0 17.6 17.2

20.8 20.3 19.9 19.5 19.1

22.7 22.3 21.8 21.4 21.0

7 8 9 10 12

1.20 1.10 1.01 0.93 0.81

2.61 2.45 2.31 2.18 1.95

4.19 3.99 3.81 3.63 3.33

5.88 5.65 5.43 5.23 4.86

7.62 7.37 7.14 6.91 6.49

9.42 9.14 8.89 8.65 8.19

11.3 11.0 10.7 10.4 9.94

13.1 12.8 12.5 12.2 11.7

15.0 14.7 14.3 14.1 13.5

16.9 16.5 16.2 15.9 15.3

18.8 18.4 18.1 17.8 17.2

20.7 20.3 20.0 19.6 19.0

14 16 18 20 24

0.71 0.63 0.57 0.52 0.44

1.77 1.61 1.48 1.36 1.18

3.06 2.83 2.63 2.45 2.15

4.53 4.23 3.96 3.72 3.30

6.11 5.75 5.42 5.12 4.60

7.76 7.36 6.98 6.63 6.02

9.47 11.2 9.03 10.8 8.61 10.3 8.23 9.88 7.53 9.12

13.0 12.5 12.0 11.6 10.8

14.8 14.3 13.8 13.3 12.4

16.6 16.1 15.6 15.1 14.2

18.4 17.9 17.4 16.9 15.9

28 32 36

0.38 0.34 0.30

1.04 0.92 0.83

1.91 1.71 1.55

2.95 2.67 2.43

4.16 3.78 3.47

5.49 5.04 4.65

6.93 6.41 5.94

11.7 10.9 10.3

13.3 12.6 11.9

15.0 14.2 13.5

2 3 4 5 6

1.84 1.71 1.57 1.44 1.31

3.66 3.49 3.32 3.16 3.02

5.55 5.36 5.18 5.01 4.84

7.48 7.27 7.08 6.89 6.72

9.42 9.20 9.00 8.81 8.62

11.4 11.2 10.9 10.7 10.5

13.3 13.1 12.9 12.7 12.5

15.3 15.1 14.8 14.6 14.4

17.6 17.0 16.8 16.6 16.3

19.6 19.0 18.7 18.5 18.3

21.5 21.0 20.7 20.5 20.2

23.5 22.9 22.7 22.4 22.2

7 8 9 10 12

1.20 1.10 1.01 0.93 0.81

2.88 2.75 2.63 2.52 2.32

4.69 4.54 4.40 4.27 4.03

6.55 6.39 6.24 6.09 5.82

8.44 8.27 8.11 7.95 7.66

10.4 10.2 10.0 9.83 9.52

12.3 12.1 11.9 11.7 11.4

14.2 14.0 13.8 13.6 13.3

16.1 15.9 15.7 15.6 15.2

18.1 17.9 17.7 17.5 17.1

20.0 19.8 19.6 19.4 19.0

22.0 21.8 21.5 21.3 20.9

14 16 18 20 24

0.71 0.63 0.57 0.52 0.44

2.15 2.00 1.87 1.75 1.55

3.82 3.62 3.44 3.28 2.98

5.57 5.35 5.14 4.94 4.57

7.38 7.13 6.90 6.67 6.24

9.22 8.95 8.69 8.45 7.98

11.1 10.8 10.5 10.3 9.75

13.0 12.7 12.4 12.1 11.6

14.9 14.5 14.2 13.9 13.4

16.7 16.4 16.1 15.8 15.2

18.7 18.3 18.0 17.7 17.1

20.6 20.2 19.9 19.5 18.9

28 32 36

0.38 0.34 0.30

1.40 1.27 1.16

2.74 2.52 2.33

4.24 3.95 3.68

5.85 5.49 5.16

7.54 7.13 6.75

9.28 11.1 8.83 10.6 8.41 10.1

12.9 12.4 11.9

14.7 14.1 13.7

16.5 16.0 15.4

18.3 17.8 17.3

8.45 10.0 7.86 9.37 7.32 8.78


8 - 52



Pu

φrn


where

s, in.

3

6

Pu

s

s

s

s


5½


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.14 0.94 0.80 0.70 0.62

2.75 2.32 1.99 1.74 1.54

4.59 3.92 3.39 2.96 2.62

6.61 5.80 5.10 4.51 4.03

8.69 7.82 6.98 6.24 5.60

10.8 9.90 9.00 8.15 7.39

12.9 12.0 11.1 10.2 9.30

14.9 14.1 13.2 12.3 11.3

17.0 16.2 15.3 14.4 13.4

19.0 18.3 17.4 16.5 15.5

21.0 20.4 19.6 18.6 17.7

23.0 22.4 21.7 20.8 19.8

7 8 9 10 12

0.55 0.50 0.46 0.42 0.37

1.38 1.25 1.14 1.04 0.90

2.36 2.14 1.96 1.80 1.55

3.63 3.30 3.01 2.78 2.39

5.07 4.61 4.22 3.89 3.36

6.72 6.15 5.66 5.23 4.53

8.53 10.5 7.84 9.67 7.23 8.97 6.70 8.34 5.82 7.28

12.5 11.6 10.8 10.1 8.87

14.6 13.6 12.8 12.0 10.6

16.7 15.7 14.8 13.9 12.4

18.8 17.8 16.9 15.9 14.2

14 16 18 20 24

0.32 0.29 0.26 0.24 0.20

0.79 0.70 0.63 0.57 0.48

1.36 1.21 1.09 0.99 0.84

2.10 1.87 1.68 1.53 1.29

2.96 2.64 2.37 2.16 1.83

3.99 3.55 3.20 2.91 2.46

5.13 4.58 4.14 3.77 3.19

6.44 5.76 5.21 4.75 4.03

7.87 7.05 6.38 5.82 4.94

9.42 11.1 12.8 8.47 9.99 11.6 7.68 9.08 10.6 7.02 8.30 9.69 5.97 7.07 8.28

28 32 36

0.18 0.16 0.14

0.42 0.37 0.33

0.73 0.64 0.57

1.11 0.98 0.88

1.58 1.39 1.24

2.13 1.88 1.68

2.77 2.44 2.18

3.49 3.08 2.75

4.29 3.79 3.39

5.19 4.58 4.10

2 3 4 5 6

1.14 0.94 0.80 0.70 0.62

3.25 2.86 2.52 2.24 2.00

5.37 4.93 4.47 4.04 3.65

7.45 7.05 6.59 6.12 5.66

9.49 9.14 8.72 8.25 7.77

11.5 11.2 10.8 10.4 9.91

13.5 13.2 12.9 12.5 12.1

15.5 15.3 15.0 14.6 14.2

17.5 17.3 17.0 16.7 16.3

19.5 19.3 19.0 18.7 18.4

21.4 21.3 21.0 20.8 20.4

23.4 23.3 23.0 22.8 22.5

7 8 9 10 12

0.55 0.50 0.46 0.42 0.37

1.80 1.64 1.50 1.38 1.19

3.31 3.02 2.77 2.56 2.21

5.23 4.84 4.49 4.18 3.65

7.29 6.83 6.39 5.99 5.29

9.42 8.93 8.45 7.99 7.16

11.6 11.1 10.6 10.1 9.15

13.7 13.2 12.7 12.2 11.2

15.8 15.4 14.9 14.4 13.4

17.9 17.5 17.0 16.5 15.5

20.0 19.6 19.2 18.7 17.7

22.1 21.7 21.3 20.8 19.8

14 16 18 20 24

0.32 0.29 0.26 0.24 0.20

1.04 0.93 0.84 0.76 0.64

1.95 1.74 1.57 1.43 1.21

3.24 2.90 2.62 2.39 2.02

4.72 4.24 3.84 3.50 2.98

6.44 5.83 5.31 4.87 4.16

8.32 10.3 12.4 7.59 9.48 11.5 6.95 8.74 10.7 6.39 8.08 9.89 5.49 6.99 8.61

14.5 13.6 12.6 11.8 10.4

16.7 15.7 14.7 13.8 12.2

18.8 17.8 16.8 15.9 14.1

28 32 36

0.18 0.16 0.14

0.55 0.49 0.43

1.05 0.93 0.83

1.76 1.55 1.38

2.59 2.29 2.05

3.63 3.21 2.88

4.80 4.25 3.81

6.13 5.45 4.90


7.59 6.77 6.09

6.15 5.44 4.87

7.21 6.38 5.72

9.18 10.9 12.7 8.21 9.76 11.4 7.41 8.83 10.4


8 - 53


Pu

φ rn


where

Pu

s, in.

3

6

s


s

15°

s

s

e

1 5½


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.18 0.97 0.83 0.72 0.64

2.78 2.34 2.02 1.77 1.57

4.61 3.97 3.45 3.03 2.70

6.59 5.80 5.11 4.54 4.06

8.64 7.78 6.97 6.26 5.65

10.7 9.83 8.94 8.12 7.39

12.8 11.9 11.0 10.1 9.27

14.8 14.0 13.1 12.1 11.2

16.8 16.1 15.2 14.2 13.3

18.9 18.1 17.3 16.3 15.4

20.9 20.2 19.3 18.4 17.5

22.9 22.2 21.4 20.5 19.6

7 8 9 10 12

0.57 0.52 0.48 0.44 0.38

1.41 1.28 1.17 1.07 0.93

2.43 2.20 2.01 1.85 1.60

3.66 3.34 3.06 2.82 2.44

5.13 4.68 4.30 3.98 3.44

6.74 6.18 5.70 5.27 4.58

8.52 10.4 7.86 9.65 7.27 8.97 6.76 8.36 5.90 7.34

12.4 11.6 10.8 10.1 8.91

14.4 13.5 12.7 11.9 10.6

16.5 15.6 14.7 13.8 12.4

18.6 17.6 16.7 15.8 14.2

14 16 18 20 24

0.33 0.30 0.27 0.25 0.21

0.81 0.72 0.65 0.59 0.50

1.40 1.25 1.13 1.02 0.87

2.15 1.91 1.72 1.57 1.33

3.03 2.70 2.44 2.22 1.88

4.05 3.62 3.27 2.98 2.53

5.22 4.68 4.23 3.86 3.27

6.51 5.84 5.28 4.83 4.11

7.94 7.14 6.48 5.93 5.05

9.47 11.1 12.8 8.54 10.1 11.7 7.77 9.16 10.7 7.11 8.40 9.78 6.07 7.19 8.39

28 32 36

0.18 0.16 0.14

0.43 0.38 0.34

0.75 0.66 0.59

1.15 1.01 0.90

1.63 1.43 1.28

2.19 1.93 1.73

2.84 2.50 2.24

3.57 3.15 2.82

4.39 3.88 3.48

5.29 4.68 4.19

2 3 4 5 6

1.18 0.97 0.83 0.72 0.64

3.24 2.85 2.51 2.23 2.00

5.34 4.90 4.45 4.05 3.68

7.40 6.99 6.53 6.07 5.62

9.43 9.07 8.63 8.16 7.69

11.5 11.1 10.7 10.3 9.80

7 8 9 10 12

0.57 0.52 0.48 0.44 0.38

1.81 1.65 1.52 1.40 1.21

3.36 3.08 2.83 2.62 2.27

5.20 4.82 4.48 4.18 3.66

7.22 6.78 6.36 5.98 5.31

9.31 11.4 8.83 10.9 8.37 10.5 7.93 9.97 7.13 9.08

14 16 18 20 24

0.33 0.30 0.27 0.25 0.21

1.07 0.95 0.86 0.78 0.66

2.00 1.79 1.62 1.47 1.25

3.25 2.92 2.65 2.42 2.06

4.76 4.29 3.90 3.58 3.05

6.44 5.85 5.34 4.91 4.21

8.28 10.2 12.3 7.58 9.43 11.4 6.97 8.72 10.6 6.43 8.09 9.87 5.55 7.03 8.64

28 32 36

0.18 0.16 0.14

0.57 0.50 0.45

1.08 0.95 0.85

1.79 1.58 1.42

2.66 2.35 2.11

3.68 3.26 2.93

4.87 4.33 3.90

13.5 13.2 12.8 12.4 11.9

6.28 5.56 4.99

7.33 6.50 5.84

15.4 15.2 14.8 14.5 14.0

17.4 17.2 16.87 16.5 16.1

19.4 19.2 18.9 18.6 18.2

21.4 21.2 20.9 20.6 20.2

23.4 23.1 23.0 22.6 22.3

13.5 13.1 12.6 12.1 11.1

15.7 15.2 14.7 14.2 13.2

17.7 17.3 16.8 16.3 15.3

19.8 19.4 18.9 18.4 17.4

21.9 21.4 21.0 20.6 19.6

14.3 13.4 12.5 11.7 10.4

16.4 15.5 14.6 13.7 12.2

18.6 17.6 16.6 15.7 14.1

6.19 5.52 4.97

7.65 6.84 6.18


9.22 10.9 12.6 8.27 9.81 11.4 7.49 8.91 10.4

8 - 54



Pu

φrn


where

Pu

s, in.

3

6

s


s

30°

s

s

e

5½


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.30 1.08 0.92 0.80 0.71

2.90 2.47 2.14 1.89 1.69

4.72 4.13 3.64 3.24 2.91

6.66 5.94 5.30 4.76 4.29

8.65 7.86 7.12 6.46 5.88

10.7 9.85 9.04 8.29 7.61

12.7 11.9 11.0 10.2 9.45

14.7 13.9 13.0 12.2 11.4

16.7 16.0 15.1 14.2 13.4

18.7 18.0 17.1 16.3 15.4

20.8 20.0 19.2 18.3 17.4

22.8 22.1 21.2 20.4 19.5

7 8 9 10 12

0.64 0.58 0.53 0.49 0.42

1.53 1.39 1.28 1.18 1.02

2.63 2.40 2.20 2.03 1.76

3.90 3.57 3.29 3.04 2.65

5.38 4.95 4.58 4.26 3.72

7.01 6.49 6.02 5.61 4.92

8.76 10.6 8.14 9.92 7.59 9.29 7.09 8.72 6.25 7.73

12.5 11.8 11.1 10.4 9.31

14.5 13.7 12.9 12.2 11.0

16.5 15.7 14.9 14.1 12.8

18.6 17.7 16.8 16.0 14.6

14 16 18 20 24

0.37 0.33 0.30 0.27 0.23

0.90 0.80 0.72 0.66 0.56

1.55 1.38 1.25 1.13 0.96

2.34 2.09 1.89 1.73 1.46

3.29 2.95 2.67 2.43 2.07

4.37 3.92 3.55 3.25 2.77

5.58 5.03 4.57 4.19 3.57

6.93 6.26 5.70 5.23 4.47

8.38 7.59 6.93 6.36 5.47

9.93 11.6 9.03 10.6 8.27 9.70 7.62 8.95 6.56 7.73

28 32 36

0.20 0.18 0.16

0.48 0.43 0.38

0.83 0.73 0.66

1.27 1.12 1.00

1.79 1.58 1.42

2.41 2.13 1.91

3.11 2.76 2.47

3.90 3.46 3.10

4.78 4.25 3.81

5.75 5.11 4.59

2 3 4 5 6

1.30 1.08 0.92 0.80 0.71

3.27 2.89 2.56 2.29 2.08

5.33 4.91 4.50 4.13 3.80

7.36 6.96 6.53 6.10 5.69

9.38 9.01 8.58 8.14 7.70

11.4 11.0 10.6 10.2 9.75

13.4 13.1 12.7 12.3 11.8

15.4 15.1 14.7 14.3 13.9

17.4 17.1 16.8 16.4 15.9

19.3 19.1 18.8 18.4 18.0

21.3 21.1 20.8 20.4 20.0

23.3 23.0 22.8 22.5 22.1

7 8 9 10 12

0.64 0.58 0.53 0.49 0.42

1.89 1.74 1.61 1.49 1.30

3.51 3.25 3.02 2.81 2.47

5.31 4.96 4.64 4.35 3.85

7.27 6.86 6.49 6.13 5.51

9.30 8.86 8.44 8.04 7.31

11.4 10.9 10.5 10.0 9.22

13.4 13.0 12.5 12.1 11.2

15.5 15.0 14.6 14.1 13.2

17.6 17.1 16.7 16.2 15.3

19.6 19.2 18.7 18.3 17.3

21.7 21.3 20.8 20.4 19.4

14 16 18 20 24

0.37 0.33 0.30 0.27 0.23

1.15 1.03 0.93 0.85 0.72

2.19 1.96 1.78 1.62 1.38

3.44 3.11 2.83 2.60 2.23

4.98 4.54 4.16 3.83 3.30

6.67 6.12 5.63 5.21 4.51

8.49 10.4 7.83 9.66 7.26 9.00 6.74 8.41 5.89 7.40

12.4 11.6 10.8 10.2 9.02

14.4 13.5 12.8 12.0 10.7

16.4 15.6 14.7 13.9 12.5

18.5 17.6 16.7 15.9 14.4

28 32 36

0.20 0.18 0.16

0.63 0.55 0.50

1.20 1.06 0.95

1.95 1.73 1.55

2.89 2.57 2.31

3.96 3.53 3.18

5.21 4.67 4.22

6.59 5.92 5.36


8.07 7.28 6.61

6.78 6.04 5.44

13.3 12.2 11.2 10.4 8.99 7.91 7.06 6.36

9.66 11.3 13.1 8.75 10.3 12.0 7.98 9.43 11.0


8 - 55


Pu

φ rn


where

Pu

3

6

s

s

45°

s

e

s, in.

s


5½


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.53 1.30 1.11 0.98 0.87

3.18 2.76 2.43 2.17 1.95

4.96 4.42 3.97 3.60 3.28

6.84 6.22 5.67 5.19 4.77

8.77 8.09 7.46 6.89 6.37

10.7 10.0 9.32 8.68 8.09

12.7 12.0 11.2 10.6 9.90

14.7 14.0 13.2 12.5 11.8

16.7 15.9 15.2 14.4 13.7

18.7 17.9 17.2 16.4 15.6

20.7 19.9 19.2 18.4 17.6

22.6 21.9 21.2 20.4 19.6

7 8 9 10 12

0.78 0.71 0.65 0.60 0.52

1.78 1.63 1.50 1.39 1.22

3.01 2.77 2.57 2.39 2.08

4.40 4.07 3.78 3.52 3.09

5.91 5.50 5.13 4.81 4.26

7.56 7.07 6.64 6.25 5.58

9.31 11.1 8.76 10.5 8.26 9.97 7.81 9.45 7.01 8.54

13.0 12.4 11.8 11.2 10.2

14.9 14.2 13.6 13.0 11.9

16.9 16.2 15.5 14.8 13.6

18.8 18.1 17.4 16.7 15.4

14 16 18 20 24

0.45 0.41 0.37 0.33 0.28

1.08 0.96 0.87 0.79 0.68

1.85 1.65 1.50 1.37 1.16

2.75 2.48 2.25 2.06 1.76

3.82 3.45 3.14 2.88 2.47

5.02 4.55 4.16 3.82 3.28

6.34 5.77 5.29 4.87 4.21

7.76 7.09 6.53 6.04 5.23

9.28 10.9 8.53 10.1 7.87 9.30 7.30 8.65 6.35 7.55

12.6 11.6 10.8 10.1 8.85

14.3 13.3 12.4 11.6 10.2

28 32 36

0.25 0.22 0.20

0.59 0.52 0.46

1.01 0.89 0.80

1.53 1.35 1.21

2.15 1.91 1.71

2.87 2.55 2.29

3.69 3.29 2.96

4.61 4.11 3.70

5.61 5.01 4.53

2 3 4 5 6

1.53 1.30 1.11 0.98 0.87

3.39 3.04 2.74 2.49 2.28

5.36 4.99 4.64 4.31 4.02

7.35 6.98 6.60 6.24 5.89

9.35 8.98 8.60 8.21 7.84

11.3 11.0 10.6 10.2 9.82

13.3 13.0 12.6 12.2 11.8

15.3 15.0 14.6 14.2 13.8

17.3 17.0 16.6 16.3 15.9

19.3 19.0 18.6 18.3 17.9

21.3 21.0 20.6 20.3 19.9

23.2 22.9 22.6 22.3 21.9

7 8 9 10 12

0.78 0.71 0.65 0.60 0.52

2.10 1.94 1.81 1.69 1.50

3.76 3.53 3.32 3.13 2.80

5.57 5.28 5.00 4.74 4.29

7.48 7.13 6.81 6.50 5.94

9.44 9.07 8.71 8.37 7.74

11.4 11.0 10.7 10.3 9.61

13.4 13.0 12.7 12.3 11.5

15.5 15.1 14.7 14.3 13.5

17.5 17.1 16.7 16.3 15.5

19.5 19.1 18.7 18.3 17.5

21.5 21.1 20.7 20.3 19.5

14 16 18 20 24

0.45 0.41 0.37 0.33 0.28

1.34 1.21 1.10 1.01 0.86

2.52 2.29 2.09 1.92 1.64

3.89 3.55 3.26 3.01 2.61

5.45 5.02 4.65 4.33 3.79

7.17 6.67 6.22 5.82 5.13

8.98 10.9 8.41 10.2 7.89 9.65 7.42 9.11 6.60 8.17

12.8 12.1 11.5 10.9 9.84

14.7 14.0 13.4 12.7 11.6

16.7 16.0 15.3 14.6 13.4

18.7 17.9 17.2 16.5 15.2

28 32 36

0.25 0.22 0.20

0.75 0.67 0.60

1.44 1.27 1.14

2.30 2.05 1.85

3.36 3.02 2.73

4.58 4.12 3.74

5.92 5.35 4.88

8.95 10.6 12.3 8.18 9.73 11.4 7.52 8.98 10.5

14.1 13.0 12.1

7.38 6.72 6.15


6.69 6.00 5.43

7.87 7.07 6.40

9.11 8.20 7.44

8 - 56



Pu

φrn


where

3

6

s

s

60°

s

e

s, in.

Pu

s


5½


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.78 1.62 1.45 1.31 1.18

3.55 3.26 2.97 2.71 2.48

5.34 4.95 4.57 4.23 3.93

7.17 6.71 6.27 5.86 5.50

9.04 8.53 8.04 7.58 7.16

10.9 10.4 9.86 9.36 8.90

12.9 12.3 11.7 11.2 10.7

14.8 14.2 13.6 13.1 12.5

16.7 16.1 15.5 15.0 14.4

18.7 18.1 17.5 16.9 16.3

20.6 20.0 19.4 18.8 18.2

22.6 22.0 21.4 20.7 20.1

7 8 9 10 12

1.07 0.98 0.90 0.83 0.72

2.28 2.11 1.97 1.84 1.62

3.66 3.43 3.22 3.03 2.70

5.18 4.88 4.61 4.37 3.93

6.79 6.45 6.12 5.82 5.28

8.48 10.2 8.09 9.80 7.72 9.39 7.37 9.00 6.73 8.28

12.0 11.6 11.1 10.7 9.91

13.9 13.4 12.9 12.5 11.6

15.7 15.2 14.7 14.2 13.4

17.6 17.1 16.6 16.1 15.1

19.5 19.0 18.4 17.9 16.9

14 16 18 20 24

0.64 0.57 0.52 0.47 0.40

1.45 1.31 1.19 1.09 0.93

2.43 2.21 2.02 1.85 1.59

3.56 3.24 2.98 2.75 2.37

4.81 4.42 4.07 3.77 3.28

6.19 5.71 5.29 4.93 4.32

7.66 7.11 6.63 6.19 5.46

9.22 10.9 8.60 10.2 8.05 9.55 7.55 8.98 6.69 8.01

12.5 11.8 11.1 10.5 9.41

14.3 13.5 12.7 12.1 10.9

16.0 15.2 14.4 13.7 12.4

28 32 36

0.35 0.31 0.28

0.82 0.72 0.65

1.39 1.24 1.11

2.08 1.86 1.67

2.90 2.59 2.34

3.83 3.43 3.11

4.86 4.37 3.97

5.99 5.41 4.93

2 3 4 5 6

1.78 1.62 1.45 1.31 1.18

3.59 3.35 3.11 2.89 2.70

5.48 5.20 4.93 4.66 4.42

7.41 7.12 6.82 6.53 6.26

9.36 9.06 8.75 8.45 8.16

7 8 9 10 12

1.07 0.98 0.90 0.83 0.72

2.52 2.36 2.23 2.10 1.89

4.19 3.99 3.81 3.64 3.34

6.01 5.77 5.55 5.35 4.97

7.88 7.62 7.37 7.13 6.70

9.79 9.51 9.24 8.98 8.49

14 16 18 20 24

0.64 0.57 0.52 0.47 0.40

1.71 1.57 1.44 1.33 1.16

3.08 2.85 2.65 2.47 2.17

4.63 4.32 4.04 3.79 3.36

6.29 5.92 5.58 5.26 4.71

8.04 7.62 7.22 6.86 6.21

9.85 9.39 8.95 8.55 7.82

28 32 36

0.35 0.31 0.28

1.02 0.91 0.82

1.92 1.72 1.56

3.00 2.71 2.46

4.26 3.88 3.55

5.67 5.20 4.80

7.19 6.64 6.16

11.3 11.0 10.7 10.4 10.1

7.21 6.54 5.98

8.51 7.75 7.10

9.88 11.3 9.02 10.4 8.29 9.55

13.3 13.0 12.7 12.3 12.0

15.3 15.0 14.6 14.3 14.0

17.2 16.9 16.6 16.3 15.9

19.2 18.9 18.6 18.2 17.9

21.2 20.9 20.6 20.2 19.9

23.2 22.9 22.5 22.2 21.9

11.7 11.4 11.1 10.9 10.3

13.7 13.4 13.1 12.8 12.2

15.6 15.3 15.0 14.7 14.1

17.6 17.3 17.0 16.7 16.1

19.6 19.2 18.9 18.6 18.0

21.5 21.2 20.9 20.6 19.9

11.7 11.2 10.7 10.3 9.50

13.6 13.1 12.6 12.1 11.2

15.5 15.0 14.4 13.9 13.0

17.4 16.9 16.3 15.8 14.8

19.3 18.8 18.2 17.7 16.7

8.80 10.5 12.2 8.17 9.77 11.4 7.61 9.14 10.7

14.0 13.1 12.4

15.8 14.9 14.1



8 - 57


Pu

φ rn


where

3

6

s s s

e

s, in.

75° P u

s


5½


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.92 1.87 1.82 1.75 1.68

3.82 3.72 3.60 3.47 3.33

5.70 5.54 5.37 5.18 5.00

7.57 7.36 7.14 6.92 6.69

9.45 9.19 8.94 8.68 8.42

11.3 11.1 10.8 10.5 10.2

13.2 12.9 12.6 12.3 12.0

15.2 14.8 14.5 14.1 13.8

17.1 16.7 16.3 16.0 15.7

19.0 18.6 18.2 17.9 17.5

20.9 20.5 20.1 19.8 19.4

22.9 22.5 22.1 21.7 21.3

7 8 9 10 12

1.60 1.52 1.45 1.38 1.25

3.19 3.06 2.93 2.80 2.57

4.81 4.63 4.46 4.29 3.98

6.47 6.26 6.05 5.85 5.48

8.17 7.93 7.70 7.48 7.07

9.92 9.66 9.41 9.16 8.71

11.7 11.4 11.2 10.9 10.4

13.5 13.2 12.9 12.6 12.1

15.3 15.0 14.7 14.4 13.9

17.2 16.9 16.5 16.2 15.7

19.1 18.7 18.4 18.1 17.5

20.9 20.6 20.3 19.9 19.3

14 16 18 20 24

1.13 1.03 0.95 0.87 0.75

2.36 2.18 2.02 1.88 1.65

3.70 3.45 3.23 3.03 2.69

5.15 4.85 4.57 4.32 3.87

6.69 6.34 6.01 5.71 5.17

8.29 7.90 7.54 7.19 6.57

9.96 9.53 9.13 8.75 8.05

11.7 11.2 10.8 10.4 9.60

13.4 12.9 12.5 12.0 11.2

15.2 14.7 14.2 13.7 12.9

16.9 16.4 15.9 15.4 14.5

18.7 18.2 17.7 17.2 16.2

28 32 36

0.66 0.59 0.53

1.46 1.31 1.19

2.42 2.18 1.99

3.50 3.19 2.92

4.71 4.32 3.98

6.03 5.56 5.15

7.44 6.90 6.42

12.1 11.4 10.7

13.7 12.9 12.2

15.4 14.6 13.8

2 3 4 5 6

1.92 1.87 1.82 1.75 1.68

3.80 3.70 3.59 3.48 3.36

5.69 5.55 5.40 5.26 5.11

7.59 7.42 7.25 7.09 6.93

9.51 9.32 9.14 8.96 8.78

11.5 11.2 11.1 10.9 10.7

13.4 13.2 13.0 12.8 12.6

15.4 15.1 14.9 14.7 14.5

17.6 17.1 16.9 16.6 16.4

19.6 19.0 18.8 18.6 18.4

21.5 21.0 20.8 20.5 20.3

23.5 23.0 22.7 22.5 22.2

7 8 9 10 12

1.60 1.52 1.45 1.38 1.25

3.24 3.13 3.02 2.91 2.72

4.97 4.84 4.71 4.58 4.34

6.77 6.62 6.47 6.33 6.07

8.62 8.45 8.29 8.14 7.85

10.5 10.3 10.2 9.98 9.67

12.4 12.2 12.0 11.9 11.5

14.3 14.1 13.9 13.7 13.4

16.2 16.0 15.8 15.6 15.3

18.1 17.9 17.7 17.6 17.2

20.1 19.9 19.7 19.5 19.1

22.0 21.8 21.6 21.4 21.0

14 16 18 20 24

1.13 1.03 0.95 0.87 0.75

2.54 2.38 2.24 2.11 1.88

4.13 3.92 3.74 3.57 3.27

5.82 5.59 5.38 5.17 4.80

7.57 7.32 7.09 6.87 6.44

9.38 9.10 8.85 8.61 8.15

11.2 10.9 10.7 10.4 9.90

13.1 12.8 12.5 12.2 11.7

15.0 14.6 14.3 14.0 13.5

16.8 16.5 16.2 15.9 15.3

18.7 18.4 18.1 17.7 17.1

20.6 20.3 19.9 19.6 19.0

28 32 36

0.66 0.59 0.53

1.70 1.55 1.42

3.00 2.77 2.57

4.47 4.17 3.90

6.06 5.70 5.37

7.72 7.31 6.93

9.43 11.2 8.99 10.7 8.57 10.3

13.0 12.5 12.0

14.8 14.3 13.8

16.6 16.1 15.5

18.4 17.9 17.3

8.93 10.5 8.32 9.81 7.78 9.21


8 - 58



Pu

φrn


where

s, in.

3

6

Pu

s

s

s

s


8


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.31 1.12 0.98 0.87 0.79

2.91 2.54 2.24 1.99 1.80

4.71 4.14 3.66 3.27 2.95

6.66 5.95 5.33 4.80 4.35

8.69 7.90 7.15 6.48 5.90

10.8 9.93 9.10 8.33 7.63

12.8 12.0 11.1 10.3 9.49

14.9 14.1 13.2 12.3 11.5

16.9 16.2 15.3 14.4 13.5

18.9 18.2 17.4 16.5 15.6

21.0 20.3 19.5 18.6 17.7

23.0 22.4 21.6 20.7 19.8

7 8 9 10 12

0.71 0.65 0.60 0.56 0.49

1.63 1.49 1.38 1.28 1.11

2.68 2.46 2.27 2.11 1.84

3.97 3.65 3.37 3.13 2.73

5.40 4.97 4.59 4.27 3.73

7.02 6.48 6.01 5.59 4.90

8.77 10.7 8.13 9.91 7.55 9.24 7.04 8.64 6.19 7.63

12.6 11.8 11.1 10.4 9.18

14.6 13.8 13.0 12.2 10.9

16.7 15.8 14.9 14.1 12.6

18.8 17.9 17.0 16.1 14.5

14 16 18 20 24

0.44 0.39 0.36 0.33 0.28

0.99 0.89 0.80 0.73 0.63

1.64 1.47 1.33 1.22 1.04

2.42 2.17 1.97 1.80 1.53

3.31 2.98 2.70 2.47 2.10

4.36 3.91 3.55 3.25 2.77

5.50 4.95 4.50 4.12 3.51

6.80 6.13 5.57 5.10 4.35

8.20 7.40 6.73 6.17 5.28

9.73 11.4 13.1 8.80 10.3 11.9 8.02 9.39 10.9 7.35 8.62 9.99 6.30 7.39 8.59

28 32 36

0.25 0.22 0.20

0.55 0.48 0.43

0.91 0.80 0.72

1.33 1.18 1.06

1.83 1.62 1.45

2.41 2.13 1.91

3.06 2.71 2.43

3.79 3.36 3.01

4.60 4.08 3.66

5.50 4.87 4.37

2 3 4 5 6

1.31 1.12 0.98 0.87 0.79

3.28 2.93 2.63 2.37 2.15

5.35 4.94 4.52 4.13 3.78

7.42 7.03 6.59 6.15 5.72

9.47 9.12 8.70 8.25 7.78

11.5 11.2 10.8 10.4 9.90

13.5 13.2 12.9 12.5 12.0

15.5 15.3 14.9 14.6 14.1

17.5 17.3 17.0 16.6 16.2

19.5 19.3 19.0 18.69 18.3

21.4 21.3 21.0 20.7 20.4

23.4 23.3 23.0 22.8 22.4

7 8 9 10 12

0.71 0.65 0.60 0.56 0.49

1.97 1.81 1.67 1.55 1.35

3.47 3.19 2.95 2.75 2.40

5.32 4.95 4.62 4.33 3.82

7.33 6.89 6.48 6.10 5.43

9.43 8.95 8.49 8.05 7.25

11.6 11.1 10.6 10.1 9.21

13.7 13.2 12.7 12.2 11.3

15.8 15.4 14.9 14.4 13.4

17.9 17.5 17.0 16.5 15.5

20.0 19.6 19.1 18.7 17.7

22.1 21.7 21.3 20.8 19.8

14 16 18 20 24

0.44 0.39 0.36 0.33 0.28

1.20 1.08 0.97 0.89 0.76

2.14 1.92 1.75 1.60 1.37

3.41 3.07 2.79 2.56 2.18

4.86 4.40 4.00 3.67 3.14

6.56 5.96 5.46 5.02 4.32

8.40 10.4 12.4 7.69 9.56 11.5 7.06 8.83 10.7 6.52 8.18 9.97 5.62 7.11 8.71

14.5 13.6 12.7 11.9 10.4

16.7 15.7 14.7 13.9 12.3

18.8 17.8 16.8 15.9 14.2

28 32 36

0.25 0.22 0.20

0.66 0.58 0.52

1.19 1.05 0.95

1.90 1.68 1.51

2.75 2.44 2.19

3.78 3.35 3.01

4.93 4.38 3.94

6.26 5.58 5.02


7.70 6.88 6.21

6.46 5.73 5.15

7.51 6.67 5.99

9.27 11.0 12.7 8.31 9.85 11.5 7.52 8.93 10.4


8 - 59


Pu

φ rn


where

Pu

s, in.

3

6

15°

s

s


s

s

e

8


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.35 1.16 1.02 0.90 0.81

2.96 2.58 2.28 2.03 1.84

4.75 4.20 3.73 3.35 3.03

6.67 5.98 5.37 4.85 4.40

8.67 7.90 7.17 6.53 5.96

10.7 9.89 9.08 8.34 7.66

12.7 11.9 11.1 10.3 9.48

14.8 14.0 13.1 12.2 11.4

16.8 16.0 15.2 14.3 13.4

18.8 18.1 17.3 16.3 15.4

20.9 20.2 19.3 18.4 17.5

22.9 22.2 21.4 20.5 19.6

7 8 9 10 12

0.74 0.68 0.63 0.58 0.51

1.67 1.53 1.42 1.31 1.15

2.76 2.53 2.34 2.17 1.90

4.02 3.70 3.43 3.19 2.79

5.48 5.05 4.68 4.36 3.82

7.06 6.53 6.07 5.66 4.97

8.79 10.6 8.17 9.91 7.61 9.27 7.12 8.69 6.28 7.69

12.6 11.8 11.0 10.4 9.23

14.5 13.7 12.9 12.2 10.9

16.6 15.7 14.8 14.0 12.6

18.6 17.7 16.8 16.0 14.4

14 16 18 20 24

0.45 0.41 0.37 0.34 0.29

1.02 0.91 0.83 0.76 0.65

1.69 1.51 1.37 1.26 1.07

2.48 2.23 2.02 1.85 1.58

3.40 3.05 2.77 2.54 2.16

4.43 3.99 3.63 3.32 2.84

5.61 5.05 4.60 4.21 3.60

6.88 6.21 5.66 5.19 4.45

8.29 7.50 6.84 6.28 5.39

9.79 11.4 8.88 10.4 8.11 9.48 7.45 8.73 6.40 7.52

28 32 36

0.25 0.23 0.20

0.56 0.50 0.45

0.93 0.83 0.74

1.37 1.22 1.09

1.89 1.67 1.50

2.47 2.19 1.96

3.14 2.78 2.49

3.88 3.44 3.09

4.71 4.18 3.75

5.61 4.98 4.47

2 3 4 5 6

1.35 1.16 1.02 0.90 0.81

3.29 2.94 2.64 2.38 2.17

5.33 4.93 4.52 4.15 3.82

7.39 6.99 6.55 6.12 5.70

9.42 9.05 8.63 8.18 7.72

11.4 11.1 10.7 10.3 9.80

13.4 13.1 12.8 12.4 11.9

15.4 15.2 14.8 14.4 14.0

17.4 17.2 16.9 16.5 16.1

19.4 19.2 18.9 18.5 18.2

21.4 21.2 20.9 20.6 20.2

23.4 23.2 22.9 22.6 22.3

7 8 9 10 12

0.74 0.68 0.63 0.58 0.51

1.99 1.83 1.69 1.58 1.38

3.52 3.25 3.02 2.81 2.47

5.31 4.95 4.63 4.34 3.84

7.28 6.86 6.46 6.10 5.45

9.33 8.87 8.43 8.00 7.23

11.4 11.0 10.5 10.0 9.15

13.5 13.1 12.6 12.1 11.2

15.6 15.2 14.7 14.2 13.2

17.7 17.3 16.8 16.3 15.3

19.8 19.4 18.9 18.4 17.4

21.9 21.5 21.0 20.5 19.6

14 16 18 20 24

0.45 0.41 0.37 0.34 0.29

1.23 1.10 1.00 0.92 0.78

2.20 1.98 1.80 1.65 1.41

3.44 3.11 2.83 2.60 2.23

4.91 4.46 4.08 3.75 3.22

6.56 5.99 5.49 5.06 4.36

8.38 10.3 12.3 7.69 9.52 11.5 7.09 8.82 10.7 6.56 8.20 9.96 5.70 7.15 8.74

14.4 13.5 12.6 11.8 10.4

16.5 15.5 14.6 13.8 12.2

18.6 17.6 16.6 15.7 14.1

28 32 36

0.25 0.23 0.20

0.68 0.60 0.54

1.23 1.09 0.97

1.95 1.73 1.55

2.82 2.50 2.25

3.83 3.41 3.07

5.02 4.47 4.03

6.32 5.64 5.09

7.76 6.96 6.30


6.59 5.86 5.27

13.1 11.9 11.0 10.1 8.71 7.64 6.80 6.12

9.31 11.0 12.7 8.38 9.90 11.5 7.60 9.01 10.5

8 - 60



Pu

φrn


where

s, in.

3

6

s


Pu

s

30°

s

s

e

8


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.49 1.29 1.13 1.00 0.90

3.12 2.74 2.43 2.18 1.98

4.91 4.39 3.95 3.58 3.26

6.80 6.16 5.60 5.10 4.67

8.75 8.04 7.37 6.77 6.23

10.7 9.98 9.24 8.55 7.93

12.7 12.0 11.2 10.4 9.72

14.7 14.0 13.2 12.4 11.6

16.7 16.0 15.2 14.3 13.5

18.7 18.0 17.2 16.3 15.5

20.8 20.0 19.2 18.4 17.5

22.7 22.1 21.3 20.4 19.5

7 8 9 10 12

0.82 0.75 0.70 0.65 0.57

1.81 1.67 1.55 1.44 1.26

2.99 2.76 2.56 2.38 2.09

4.30 3.97 3.69 3.44 3.03

5.76 5.35 4.98 4.66 4.13

7.37 6.87 6.42 6.02 5.34

9.08 10.9 8.49 10.2 7.96 9.62 7.49 9.07 6.66 8.12

12.8 12.0 11.4 10.8 9.67

14.7 13.9 13.2 12.5 11.3

16.7 15.9 15.1 14.4 13.0

18.7 17.8 17.0 16.2 14.8

14 16 18 20 24

0.50 0.45 0.41 0.38 0.32

1.12 1.01 0.92 0.84 0.72

1.86 1.67 1.52 1.39 1.19

2.71 2.44 2.22 2.03 1.74

3.69 3.33 3.03 2.78 2.38

4.78 4.33 3.95 3.62 3.11

5.99 5.44 4.97 4.57 3.93

7.33 6.66 6.10 5.62 4.84

8.75 10.3 11.9 7.98 9.39 10.9 7.32 8.64 10.1 6.75 7.98 9.30 5.83 6.92 8.08

28 32 36

0.28 0.25 0.23

0.63 0.56 0.50

1.04 0.92 0.83

1.52 1.35 1.21

2.08 1.84 1.66

2.72 2.41 2.17

3.44 3.06 2.75

4.24 3.77 3.40

5.13 4.57 4.11

2 3 4 5 6

1.49 1.29 1.13 1.00 0.90

3.36 3.02 2.73 2.48 2.27

5.36 4.97 4.60 4.26 3.96

7.37 6.99 6.58 6.18 5.80

9.38 9.01 8.61 8.18 7.76

11.4 11.0 10.7 10.2 9.79

13.4 13.1 12.7 12.3 11.8

15.4 15.1 14.7 14.3 13.9

17.4 17.1 16.7 16.4 15.9

19.3 19.1 18.8 18.4 18.0

21.3 21.1 20.8 20.4 20.0

23.3 23.1 22.8 22.4 22.1

7 8 9 10 12

0.82 0.75 0.70 0.65 0.57

2.09 1.93 1.80 1.68 1.49

3.68 3.43 3.21 3.01 2.67

5.44 5.11 4.81 4.53 4.05

7.36 6.97 6.61 6.27 5.67

9.35 8.93 8.53 8.14 7.43

11.4 11.0 10.5 10.1 9.31

13.5 13.0 12.6 12.1 11.3

15.5 15.1 14.6 14.2 13.3

17.6 17.1 16.7 16.2 15.3

19.6 19.2 18.7 18.3 17.4

21.7 21.2 20.8 20.4 19.4

14 16 18 20 24

0.50 0.45 0.41 0.38 0.32

1.33 1.20 1.09 1.00 0.86

2.39 2.16 1.97 1.81 1.55

3.65 3.31 3.03 2.80 2.41

5.15 4.71 4.34 4.01 3.48

6.81 6.27 5.79 5.37 4.68

8.60 10.5 7.96 9.76 7.39 9.12 6.89 8.53 6.04 7.53

12.4 11.7 10.9 10.3 9.14

14.4 13.6 12.8 12.1 10.8

16.5 15.6 14.8 14.0 12.6

18.5 17.6 16.8 15.9 14.5

28 32 36

0.28 0.25 0.23

0.75 0.67 0.60

1.35 1.20 1.08

2.12 1.89 1.70

3.06 2.73 2.46

4.13 3.69 3.34

5.36 4.81 4.36

6.72 6.05 5.50


8.19 7.40 6.74

6.09 5.43 4.89

7.12 6.36 5.74

13.6 12.5 11.5 10.7 9.32 8.24 7.37 6.66

9.76 11.4 13.2 8.86 10.4 12.0 8.09 9.53 11.1


8 - 61


Pu

φ rn


where

Pu

3

6

s

s

45°

s

e

s, in.

s


8


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.70 1.51 1.35 1.21 1.10

3.43 3.09 2.78 2.52 2.30

5.22 4.76 4.34 3.97 3.67

7.06 6.52 6.01 5.57 5.17

8.95 8.35 7.78 7.25 6.78

10.9 10.2 9.60 9.01 8.47

12.8 12.2 11.5 10.8 10.2

14.8 14.1 13.4 12.7 12.1

16.8 16.1 15.3 14.6 13.9

18.7 18.0 17.3 16.6 15.9

20.7 20.0 19.3 18.5 17.8

22.7 22.0 21.3 20.5 19.8

7 8 9 10 12

1.00 0.92 0.85 0.79 0.69

2.12 1.96 1.82 1.70 1.50

3.40 3.17 2.96 2.78 2.46

4.82 4.51 4.23 3.97 3.54

6.35 5.96 5.60 5.28 4.73

7.97 7.51 7.08 6.70 6.04

9.67 11.5 9.15 10.9 8.68 10.4 8.24 9.86 7.46 8.97

13.3 12.7 12.1 11.5 10.6

15.2 14.5 13.9 13.3 12.2

17.1 16.4 15.7 15.1 14.0

19.0 18.3 17.6 17.0 15.7

14 16 18 20 24

0.61 0.55 0.50 0.46 0.40

1.34 1.21 1.11 1.02 0.87

2.21 2.00 1.82 1.67 1.43

3.18 2.88 2.64 2.42 2.09

4.27 3.89 3.56 3.29 2.84

5.48 5.01 4.60 4.25 3.68

6.80 6.23 5.74 5.31 4.62

8.21 7.54 6.97 6.47 5.65

9.70 11.3 8.95 10.4 8.30 9.71 7.73 9.06 6.77 7.96

12.9 12.0 11.2 10.5 9.23

14.6 13.6 12.7 11.9 10.6

28 32 36

0.35 0.31 0.28

0.76 0.68 0.61

1.26 1.12 1.00

1.83 1.63 1.46

2.49 2.22 2.00

3.24 2.89 2.60

4.07 3.64 3.29

5.00 4.47 4.04

6.00 5.38 4.87

2 3 4 5 6

1.70 1.51 1.35 1.21 1.10

3.52 3.23 2.96 2.72 2.51

5.44 5.11 4.79 4.48 4.20

7.40 7.06 6.70 6.36 6.03

9.37 9.03 8.67 8.30 7.94

11.4 11.0 10.7 10.3 9.90

13.3 13.0 12.7 12.3 11.9

15.3 15.0 14.6 14.3 13.9

17.3 17.0 16.6 16.3 15.9

19.3 19.0 18.6 18.3 17.9

21.3 21.0 20.6 20.3 19.9

23.2 22.9 22.6 22.3 21.9

7 8 9 10 12

1.00 0.92 0.85 0.79 0.69

2.33 2.18 2.04 1.92 1.71

3.96 3.73 3.53 3.35 3.02

5.73 5.45 5.19 4.94 4.50

7.60 7.27 6.96 6.67 6.13

9.53 9.17 8.83 8.50 7.88

11.5 11.1 10.8 10.4 9.73

13.5 13.1 12.7 12.4 11.6

15.5 15.1 14.7 14.3 13.6

17.5 17.1 16.7 16.3 15.5

19.5 19.1 18.7 18.3 17.5

21.5 21.1 20.7 20.3 19.5

14 16 18 20 24

0.61 0.55 0.50 0.46 0.40

1.55 1.41 1.29 1.19 1.03

2.75 2.51 2.31 2.13 1.84

4.12 3.78 3.49 3.24 2.82

5.65 5.22 4.85 4.53 3.99

7.33 6.83 6.39 6.00 5.32

9.11 11.0 8.55 10.3 8.04 9.77 7.57 9.25 6.76 8.32

12.9 12.2 11.6 11.0 9.97

14.8 14.1 13.4 12.8 11.7

16.8 16.0 15.3 14.7 13.5

19.8 18.0 17.3 16.6 15.3

28 32 36

0.35 0.31 0.28

0.90 0.80 0.72

1.62 1.44 1.30

2.50 2.24 2.02

3.56 3.20 2.90

4.76 4.30 3.92

6.09 5.52 5.04

9.08 10.7 12.4 8.32 9.85 11.5 7.66 9.10 10.6

14.2 13.1 12.2

7.53 6.86 6.30


7.08 6.37 5.78

8.24 7.43 6.75

9.47 8.56 7.79

8 - 62



Pu

φrn


where

3

6

s

s

60°

s

e

s, in.

Pu

s


8


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.86 1.77 1.66 1.54 1.43

3.71 3.52 3.31 3.10 2.90

5.56 5.29 4.99 4.70 4.41

7.41 7.07 6.70 6.34 6.00

9.28 8.88 8.45 8.04 7.64

11.2 10.7 10.3 9.79 9.35

13.1 12.6 12.1 11.6 11.1

15.0 14.5 13.9 13.4 12.9

16.9 16.4 15.8 15.3 14.7

18.8 18.3 17.7 17.1 16.6

20.8 20.2 19.6 19.0 18.5

22.7 22.1 21.6 21.0 20.4

7 8 9 10 12

1.33 1.24 1.16 1.08 0.96

2.71 2.54 2.38 2.24 2.00

4.15 3.92 3.70 3.51 3.17

5.68 5.39 5.12 4.88 4.44

7.27 6.94 6.63 6.34 5.82

8.94 10.7 8.56 10.3 8.22 9.86 7.89 9.49 7.28 8.81

12.4 12.0 11.6 11.2 10.4

14.2 13.8 13.3 12.9 12.1

16.1 15.6 15.1 14.6 13.8

17.9 17.4 16.9 16.4 15.5

19.8 19.3 18.7 18.2 17.3

14 16 18 20 24

0.86 0.77 0.70 0.65 0.56

1.81 1.64 1.51 1.39 1.20

2.88 2.64 2.43 2.25 1.95

4.07 3.74 3.46 3.21 2.80

5.36 4.95 4.59 4.28 3.76

6.73 6.25 5.83 5.45 4.81

8.19 7.64 7.15 6.71 5.96

9.72 11.3 9.11 10.7 8.56 10.0 8.06 9.48 7.19 8.50

13.0 12.2 11.6 11.0 9.88

14.7 13.9 13.2 12.5 11.3

16.4 15.6 14.8 14.1 12.8

28 32 36

0.49 0.43 0.39

1.06 0.94 0.85

1.72 1.54 1.39

2.48 2.22 2.01

3.34 3.00 2.72

4.29 3.87 3.52

5.34 4.83 4.40

6.47 5.87 5.36

2 3 4 5 6

1.86 1.77 1.66 1.54 1.43

3.72 3.55 3.36 3.17 2.99

5.59 5.37 5.14 4.90 4.67

7.50 7.25 6.98 6.72 6.46

9.43 9.16 8.88 8.59 8.31

7 8 9 10 12

1.33 1.24 1.16 1.08 0.96

2.82 2.67 2.52 2.40 2.17

4.46 4.26 4.08 3.91 3.61

6.21 5.98 5.76 5.56 5.20

8.05 7.79 7.55 7.32 6.90

9.92 9.65 9.39 9.14 8.66

14 16 18 20 24

0.86 0.77 0.70 0.65 0.56

1.98 1.82 1.69 1.57 1.37

3.35 3.11 2.91 2.72 2.41

4.87 4.57 4.30 4.05 3.61

6.51 6.15 5.81 5.50 4.96

8.23 10.0 7.81 9.56 7.43 9.13 7.07 8.73 6.43 8.00

28 32 36

0.49 0.43 0.39

1.22 1.09 0.99

2.15 1.94 1.76

3.25 2.94 2.69

4.49 4.10 3.77

5.88 5.41 5.00

11.4 11.1 10.8 10.5 10.2

7.68 6.99 6.41

8.97 10.3 11.7 8.19 9.46 10.8 7.53 8.71 9.96

13.3 13.0 12.7 12.4 12.1

15.3 15.0 14.7 14.4 14.1

17.3 17.0 16.7 16.3 16.0

19.2 18.9 18.6 18.3 18.0

21.2 20.9 20.6 20.3 19.9

23.2 22.9 22.6 22.2 21.9

11.8 11.5 11.3 11.0 10.5

13.8 13.5 13.2 12.9 12.4

15.7 15.4 15.1 14.8 14.2

17.7 17.3 17.0 16.7 16.1

19.6 19.3 19.0 18.7 18.1

21.6 21.3 20.9 20.6 20.0

11.8 11.4 10.9 10.5 9.67

13.7 13.2 12.7 12.2 11.4

15.6 15.1 14.5 14.1 13.2

17.5 16.9 16.4 15.9 15.0

19.4 18.9 18.3 17.8 16.8

8.97 10.6 12.3 8.34 9.92 11.6 7.78 9.30 10.9

14.1 13.3 12.5

15.9 15.0 14.2

7.38 6.83 6.35



8 - 63


Pu

φr n


where

3

6

s s

s

e

s, in.

75° P u

s


8


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.94 1.92 1.89 1.85 1.81

3.87 3.82 3.75 3.67 3.59

5.79 5.70 5.60 5.48 5.35

7.70 7.58 7.43 7.28 7.11

9.61 9.45 9.26 9.07 8.87

11.5 11.3 11.1 10.9 10.6

13.4 13.2 12.9 12.7 12.4

15.3 15.1 14.8 14.5 14.2

17.3 17.0 16.7 16.4 16.1

19.2 18.9 18.5 18.2 17.9

21.1 20.8 20.4 20.1 19.8

23.0 22.7 22.3 22.0 21.6

7 8 9 10 12

1.76 1.71 1.66 1.61 1.51

3.50 3.40 3.30 3.20 3.01

5.22 5.08 4.94 4.80 4.53

6.94 6.76 6.59 6.42 6.08

8.67 8.46 8.26 8.06 7.67

10.4 10.2 9.96 9.73 9.30

12.2 11.9 11.7 11.4 11.0

14.0 13.7 13.4 13.2 12.7

15.8 15.5 15.2 14.9 14.4

17.6 17.3 17.0 16.7 16.2

19.4 19.1 18.8 18.5 17.9

21.3 21.0 20.6 20.3 19.7

14 16 18 20 24

1.41 1.31 1.23 1.15 1.01

2.82 2.65 2.48 2.34 2.08

4.27 4.03 3.80 3.60 3.23

5.76 5.47 5.19 4.93 4.48

7.31 6.96 6.64 6.34 5.80

8.90 10.5 8.52 10.1 8.16 9.73 7.82 9.36 7.20 8.67

12.2 11.8 11.3 10.9 10.2

13.9 13.4 13.0 12.6 11.8

15.6 15.2 14.7 14.2 13.4

17.4 16.9 16.4 15.9 15.0

19.2 18.6 18.1 17.7 16.7

28 32 36

0.90 0.81 0.73

1.87 1.69 1.54

2.93 2.67 2.45

4.08 3.75 3.45

5.33 4.91 4.55

6.65 6.17 5.74

12.6 11.9 11.2

14.2 13.5 12.8

15.9 15.1 14.3

2 3 4 5 6

1.94 1.92 1.89 1.85 1.81

3.86 3.80 3.74 3.66 3.58

5.77 5.68 5.57 5.46 5.35

7.68 7.55 7.42 7.29 7.15

9.60 9.45 9.29 9.14 8.98

11.5 11.4 11.2 11.0 10.8

13.5 13.3 13.1 12.9 12.7

15.4 15.2 15.0 14.8 14.6

17.6 17.2 16.9 16.7 16.5

19.6 19.1 18.9 18.7 18.5

21.5 21.1 20.8 20.6 20.4

23.5 23.0 22.8 22.6 22.3

7 8 9 10 12

1.76 1.71 1.66 1.61 1.51

3.49 3.40 3.31 3.22 3.05

5.23 5.12 5.00 4.89 4.67

7.01 6.88 6.74 6.61 6.36

8.83 8.68 8.53 8.38 8.10

10.7 10.5 10.4 10.2 9.89

12.5 12.4 12.2 12.0 11.7

14.4 14.3 14.1 13.9 13.6

16.3 16.2 16.0 15.8 15.4

18.3 18.1 17.9 17.7 17.3

20.2 20.0 19.8 19.6 19.2

22.1 21.9 21.7 21.5 21.1

14 16 18 20 24

1.41 1.31 1.23 1.15 1.01

2.88 2.73 2.58 2.45 2.21

4.46 4.26 4.08 3.90 3.59

6.12 5.89 5.68 5.47 5.10

7.84 7.59 7.35 7.13 6.71

9.61 9.33 9.08 8.84 8.38

11.4 11.1 10.8 10.6 10.1

13.3 12.9 12.7 12.4 11.9

15.1 14.8 14.5 14.2 13.6

17.0 16.6 16.3 16.0 15.5

18.9 18.5 18.2 17.9 17.3

20.8 20.4 20.1 19.7 19.1

28 32 36

0.90 0.81 0.73

2.01 1.84 1.70

3.32 3.08 2.87

4.77 4.47 4.19

6.32 5.97 5.64

7.96 7.56 7.19

9.65 11.4 9.21 10.9 8.80 10.5

13.1 12.7 12.2

14.9 14.4 13.9

16.7 16.2 15.7

18.5 18.0 17.5

8.06 7.51 7.01

9.52 11.0 8.91 10.4 8.36 9.77


8 - 64



Pu

φrn


ex = e

where

s, in.

3

6

s

Pu

s

s


3

3 6


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.71 1.42 1.21 1.05 0.92

4.07 3.40 2.90 2.51 2.21

6.81 5.79 4.97 4.34 3.85

9.86 8.61 7.53 6.64 5.91

13.0 11.7 10.4 9.24 8.27

16.1 14.8 13.4 12.1 11.0

19.3 18.0 16.6 15.2 13.9

22.3 21.1 19.8 18.3 16.9

25.4 24.3 23.0 21.5 20.0

28.5 27.4 26.1 24.7 23.2

31.5 30.5 29.3 27.9 26.4

34.5 33.6 32.5 31.1 29.7

7 8 9 10 12

0.81 0.72 0.64 0.58 0.49

1.96 1.76 1.60 1.46 1.24

3.44 3.11 2.83 2.59 2.21

5.31 4.80 4.38 4.02 3.44

7.46 6.78 6.20 5.71 4.91

9.95 12.7 9.09 11.6 8.34 10.7 7.70 9.91 6.65 8.59

15.6 14.4 13.3 12.4 10.8

18.6 17.3 16.1 15.0 13.2

21.8 20.4 19.1 17.9 15.7

25.0 23.5 22.1 20.8 18.5

28.2 26.7 25.2 23.8 21.3

14 16 18 20 24

0.42 0.37 0.33 0.29 0.24

1.08 0.95 0.85 0.77 0.64

1.92 1.70 1.52 1.37 1.15

3.00 2.66 2.39 2.16 1.82

4.30 3.82 3.43 3.11 2.62

5.83 5.19 4.67 4.24 3.57

7.57 6.75 6.08 5.53 4.67

9.53 11.7 8.51 10.5 7.68 9.45 6.99 8.61 5.92 7.30

14.0 12.6 11.4 10.4 8.8

16.5 14.9 13.5 12.3 10.5

19.2 17.3 15.8 14.4 12.3

28 32 36

0.21 0.18 0.16

0.55 0.49 0.43

0.99 0.87 0.77

1.57 1.38 1.23

2.26 1.98 1.77

3.08 2.71 2.42

4.04 3.55 3.17

5.12 4.51 4.03

2 3 4 5 6

1.71 1.42 1.21 1.05 0.92

4.85 4.24 3.72 3.29 2.93

8.04 7.36 6.66 6.00 5.41

11.2 10.6 9.86 9.14 8.44

7 8 9 10 12

0.81 0.72 0.64 0.58 0.49

2.63 2.38 2.17 2.00 1.71

4.90 4.46 4.09 3.78 3.27

7.79 10.9 7.20 10.2 6.67 9.54 6.20 8.94 5.41 7.88

14 16 18 20 24

0.42 0.37 0.33 0.29 0.24

1.49 1.32 1.19 1.08 0.91

2.87 2.55 2.30 2.09 1.76

4.78 4.28 3.86 3.51 2.97

7.01 6.29 5.70 5.20 4.42

9.61 12.4 8.69 11.3 7.91 10.4 7.25 9.54 6.19 8.19

28 32 36

0.21 0.18 0.16

0.78 0.69 0.61

1.52 1.33 1.19

2.57 2.27 2.03

3.84 3.39 3.03

5.39 4.77 4.27

14.2 13.7 13.1 12.4 11.6

6.33 5.58 4.99

7.67 6.77 6.05

9.13 10.7 8.06 9.47 7.21 8.48

17.3 16.8 16.2 15.6 14.9

20.3 19.9 19.4 18.7 18.1

23.2 22.9 22.4 21.9 21.2

26.2 25.9 25.5 25.0 24.4

29.2 28.9 28.5 28.1 27.5

32.2 31.9 31.6 31.1 30.6

35.1 34.9 34.6 34.2 33.7

14.1 13.4 12.6 12.0 10.7

17.3 16.6 15.8 15.1 13.7

20.6 19.8 19.1 18.3 16.8

23.7 23.0 22.3 21.6 20.0

26.9 26.2 25.5 24.8 23.3

30.0 29.4 28.7 28.0 26.5

33.2 32.6 31.9 31.2 29.8

15.4 14.2 13.1 12.1 10.4

18.6 17.2 15.9 14.8 12.9

21.8 20.3 18.9 17.7 15.5

25.0 23.5 22.0 20.7 18.3

28.2 26.7 25.2 23.8 21.2

9.15 11.4 13.7 8.13 10.1 12.3 7.30 9.10 11.1

16.3 14.6 13.2

19.0 17.1 15.5

7.14 6.33 5.67



8 - 65


Pu

φ rn


ex Pu

where

s, in.

3

6

s

15° e

s

s


3

3 6


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.77 1.47 1.25 1.08 0.94

4.10 3.45 2.95 2.57 2.26

6.84 5.86 5.07 4.44 3.93

9.82 8.61 7.55 6.67 5.96

12.9 11.6 10.4 9.26 8.33

16.0 14.7 13.3 12.1 11.0

19.1 17.8 16.4 15.1 13.8

22.2 20.9 19.5 18.1 16.8

25.2 24.1 22.7 21.3 19.8

28.3 27.2 25.8 24.4 23.0

31.3 30.3 29.0 27.6 26.1

34.3 33.3 32.1 30.7 29.3

7 8 9 10 12

0.83 0.74 0.66 0.60 0.50

2.01 1.81 1.64 1.50 1.28

3.52 3.18 2.90 2.65 2.27

5.37 4.87 4.45 4.10 3.52

7.55 6.88 6.31 5.81 5.01

9.97 12.7 9.13 11.7 8.40 10.8 7.77 9.99 6.74 8.71

15.5 14.4 13.3 12.4 10.9

18.5 17.2 16.1 15.0 13.2

21.5 20.2 18.9 17.8 15.8

24.7 23.2 21.9 20.7 18.4

27.8 26.4 25.0 23.6 21.2

14 16 18 20 24

0.43 0.38 0.34 0.30 0.25

1.11 0.98 0.88 0.79 0.67

1.98 1.75 1.57 1.42 1.19

3.08 2.73 2.45 2.22 1.87

4.40 3.91 3.52 3.19 2.69

5.93 5.29 4.77 4.33 3.66

7.69 6.87 6.20 5.65 4.78

9.62 11.8 8.62 10.6 7.80 9.59 7.12 8.76 6.04 7.45

14.1 12.7 11.5 10.5 8.99

16.5 15.0 13.6 12.5 10.7

19.1 17.4 15.9 14.6 12.5

28 32 36

0.22 0.19 0.17

0.57 0.50 0.45

1.02 0.90 0.80

1.61 1.42 1.26

2.32 2.04 1.82

3.17 2.79 2.49

4.14 3.65 3.26

5.24 4.62 4.13

2 3 4 5 6

1.77 1.47 1.25 1.08 0.94

4.83 4.22 3.71 3.28 2.94

7.98 7.31 6.64 6.01 5.45

11.1 10.5 9.77 9.06 8.38

7 8 9 10 12

0.83 0.74 0.66 0.60 0.50

2.65 2.40 2.20 2.02 1.74

4.97 4.55 4.18 3.86 3.34

7.75 10.8 7.17 10.1 6.66 9.49 6.20 8.92 5.43 7.91

14 16 18 20 24

0.43 0.38 0.34 0.30 0.25

1.52 1.35 1.22 1.10 0.93

2.94 2.62 2.36 2.14 1.81

4.82 4.32 3.91 3.57 3.03

7.07 6.38 5.79 5.30 4.52

9.60 12.4 8.71 11.3 7.95 10.4 7.31 9.60 6.26 8.28

28 32 36

0.22 0.19 0.17

0.80 0.71 0.63

1.56 1.37 1.23

2.63 2.32 2.08

3.93 3.47 3.11

5.47 4.85 4.35

14.1 13.6 12.9 12.2 11.5

6.47 5.72 5.11

7.82 6.92 6.20

9.31 10.9 8.24 9.66 7.38 8.66

17.2 16.7 16.1 15.4 14.7

20.2 19.7 19.2 18.5 17.8

23.2 22.8 22.3 21.7 21.0

26.1 25.8 25.3 24.8 24.1

29.1 28.8 28.3 27.8 27.2

32.1 31.8 31.4 30.9 30.3

35.0 34.8 34.4 33.9 33.4

13.9 13.2 12.5 11.9 10.6

17.1 16.4 15.6 14.9 13.6

20.3 19.6 18.8 18.1 16.6

23.5 22.7 22.0 21.3 19.8

26.6 25.9 25.2 24.5 23.0

29.7 29.1 28.4 27.6 26.1

32.8 32.2 31.5 30.8 29.3

15.3 14.1 13.0 12.1 10.5

18.4 17.0 15.8 14.8 12.9

21.5 20.1 18.8 17.6 15.5

24.6 23.2 21.8 20.5 18.2

27.3 26.3 24.9 23.5 21.1

9.24 11.4 13.8 8.23 10.2 12.4 7.41 9.23 11.2

16.3 14.7 13.3

18.9 17.1 15.6

7.26 6.45 5.80


8 - 66



Pu

φrn


ex Pu

where

s, in.

3

6

s

30°

s


s

e

3

3 6


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

1.94 1.61 1.37 1.19 1.04

4.26 3.63 3.15 2.77 2.45

6.99 6.09 5.35 4.74 4.23

9.90 8.80 7.83 7.00 6.30

12.9 11.7 10.6 9.54 8.67

16.0 14.7 13.5 12.3 11.3

19.0 17.7 16.5 15.2 14.1

22.0 20.8 19.5 18.2 17.0

25.1 23.9 22.6 21.2 19.9

28.1 27.0 25.7 24.3 23.0

31.1 30.0 28.7 27.4 26.0

34.1 33.1 31.8 30.5 29.1

7 8 9 10 12

0.92 0.82 0.74 0.67 0.56

2.19 1.98 1.80 1.65 1.41

3.81 3.45 3.16 2.90 2.49

5.71 5.22 4.79 4.42 3.82

7.92 10.4 7.27 9.58 6.71 8.88 6.22 8.26 5.41 7.22

13.0 12.1 11.2 10.5 9.23

15.8 14.8 13.8 12.9 11.5

18.7 17.6 16.5 15.5 13.8

21.7 20.5 19.3 18.2 16.4

24.7 23.4 22.2 21.1 19.0

27.8 26.4 25.2 24.0 21.8

14 16 18 20 24

0.48 0.42 0.38 0.34 0.28

1.23 1.08 0.97 0.88 0.74

2.18 1.93 1.73 1.57 1.32

3.36 2.99 2.69 2.44 2.06

4.78 4.26 3.85 3.50 2.96

6.40 5.73 5.18 4.73 4.01

8.22 10.3 12.4 7.40 9.25 11.3 6.71 8.41 10.3 6.14 7.70 9.42 5.22 6.58 8.08

14.8 13.4 12.3 11.3 9.72

17.2 15.7 14.4 13.3 11.5

19.8 18.2 16.7 15.4 13.4

28 32 36

0.24 0.21 0.19

0.64 0.56 0.50

1.14 1.00 0.89

1.78 1.57 1.40

2.56 2.26 2.02

3.48 3.07 2.75

4.54 4.01 3.59

2 3 4 5 6

1.94 1.61 1.37 1.19 1.04

4.86 4.27 3.78 3.39 3.06

7.96 7.32 6.70 6.14 5.64

11.0 10.4 9.75 9.10 8.48

7 8 9 10 12

0.92 0.82 0.74 0.67 0.56

2.78 2.54 2.34 2.16 1.87

5.19 4.80 4.45 4.14 3.61

7.91 10.9 7.38 10.3 6.90 9.67 6.46 9.14 5.71 8.20

14 16 18 20 24

0.48 0.42 0.38 0.34 0.28

1.65 1.47 1.33 1.21 1.02

3.20 2.86 2.58 2.35 2.00

5.10 4.60 4.19 3.84 3.29

7.41 6.74 6.17 5.68 4.89

28 32 36

0.24 0.21 0.19

0.88 0.78 0.70

1.73 1.52 1.36

2.86 2.54 2.27

4.28 3.80 3.41

14.1 13.5 12.9 12.2 11.5

5.73 5.07 4.54

7.05 6.25 5.61

8.51 10.1 11.8 7.55 8.96 10.5 6.78 8.06 9.44

17.1 16.6 15.9 15.3 14.6

20.1 19.6 19.0 18.4 17.7

23.1 22.6 22.1 21.5 20.8

26.0 25.6 25.1 24.5 23.9

29.0 28.6 28.1 27.6 27.0

32.0 31.6 31.1 30.6 30.1

35.0 34.6 34.2 33.7 33.1

13.9 13.3 12.6 12.0 10.9

17.0 16.3 15.7 15.0 13.8

20.1 19.4 18.7 18.1 16.8

23.2 22.6 21.9 21.2 19.8

26.3 25.7 25.0 24.3 22.9

29.4 28.8 28.1 27.4 26.0

32.5 31.9 31.2 30.5 29.1

9.95 9.12 8.39 7.75 6.71

12.7 11.7 10.8 10.1 8.78

15.6 14.5 13.5 12.6 11.1

18.5 17.3 16.2 15.2 13.5

21.5 20.3 19.1 18.0 16.1

24.6 23.3 22.0 20.9 18.8

27.7 26.4 25.0 23.8 21.6

5.90 5.25 4.73

7.77 6.95 6.28

9.83 12.1 14.5 8.83 10.9 13.1 8.00 9.88 11.9

17.0 15.4 14.1

19.6 17.9 16.4



8 - 67


Pu

φrn


ex Pu

s

where

3

6

s

e

s, in.

45°

s


3

3 6


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.23 1.89 1.63 1.42 1.25

4.67 4.06 3.57 3.17 2.84

7.33 6.50 5.84 5.27 4.78

10.2 9.19 8.36 7.63 6.99

13.1 12.0 11.1 10.2 9.40

16.0 14.9 13.9 12.9 12.0

19.0 17.9 16.8 15.7 14.7

22.0 20.9 19.7 18.6 17.6

25.0 23.9 22.7 21.5 20.4

28.0 26.9 25.7 24.5 23.4

31.0 29.9 28.7 27.5 26.3

33.9 32.9 31.7 30.5 29.3

7 8 9 10 12

1.11 0.99 0.90 0.81 0.68

2.57 2.33 2.13 1.96 1.68

4.36 3.99 3.68 3.40 2.95

6.42 5.92 5.49 5.10 4.46

8.70 11.2 8.09 10.5 7.54 9.80 7.05 9.21 6.22 8.19

13.8 13.0 12.2 11.6 10.4

16.6 15.7 14.8 14.0 12.7

19.4 18.4 17.5 16.6 15.1

22.3 21.2 20.3 19.3 17.7

25.2 24.1 23.1 22.1 20.3

28.2 27.0 26.0 24.9 23.0

14 16 18 20 24

0.59 0.52 0.46 0.41 0.35

1.47 1.31 1.17 1.06 0.90

2.59 2.31 2.08 1.89 1.60

3.95 3.54 3.20 2.92 2.48

5.55 4.99 4.54 4.15 3.54

7.35 6.65 6.06 5.56 4.76

9.34 11.5 8.49 10.5 7.77 9.64 7.15 8.90 6.15 7.70

13.8 12.7 11.7 10.8 9.39

16.2 14.9 13.8 12.8 11.2

18.7 17.3 16.1 15.0 13.1

21.3 19.8 18.5 17.2 15.2

28 32 36

0.30 0.26 0.23

0.77 0.68 0.61

1.38 1.22 1.08

2.15 1.90 1.69

3.08 2.72 2.44

4.16 3.68 3.30

5.39 4.79 4.30

2 3 4 5 6

2.23 1.89 1.63 1.42 1.25

5.02 4.50 4.05 3.68 3.36

8.01 7.44 6.89 6.40 5.96

11.0 10.4 9.86 9.30 8.78

7 8 9 10 12

1.11 0.99 0.90 0.81 0.68

3.09 2.86 2.65 2.47 2.16

5.57 5.22 4.90 4.61 4.11

8.29 11.2 7.84 10.6 7.43 10.2 7.04 9.69 6.35 8.85

14 16 18 20 24

0.59 0.52 0.46 0.41 0.35

1.92 1.72 1.56 1.43 1.22

3.69 3.34 3.04 2.79 2.38

5.76 5.25 4.82 4.44 3.84

8.11 10.7 7.47 9.94 6.91 9.26 6.43 8.66 5.62 7.64

28 32 36

0.30 0.26 0.23

1.06 0.94 0.84

2.08 1.84 1.65

3.37 3.00 2.71

4.98 4.46 4.04

14.0 13.5 12.9 12.3 11.7

6.77 6.03 5.42

8.28 7.39 6.66

9.91 11.7 8.87 10.5 8.02 9.49

13.5 12.2 11.1

17.0 16.5 15.9 15.3 14.7

20.0 19.5 18.9 18.3 17.7

23.0 22.5 21.9 21.3 20.7

25.9 25.5 24.9 24.4 23.8

28.9 28.4 27.9 27.4 26.8

31.9 31.4 30.9 30.4 29.8

34.8 34.4 33.9 33.4 32.8

14.1 13.6 13.0 12.5 11.6

17.1 16.5 16.0 15.4 14.4

20.1 19.5 19.0 18.4 17.3

23.2 22.6 22.0 21.4 20.2

26.2 25.6 25.0 24.4 23.2

29.2 28.6 28.0 27.4 26.2

32.3 31.7 31.1 30.4 29.2

13.4 12.6 11.8 11.1 9.84

16.2 15.3 14.4 13.6 12.2

19.1 18.1 17.2 16.3 14.7

22.1 21.0 20.0 19.0 17.3

25.0 23.9 22.9 21.9 20.0

28.0 26.9 25.8 24.7 22.8

8.82 11.0 13.4 7.97 10.0 12.2 7.27 9.18 11.2

15.8 14.6 13.4

18.4 17.0 15.7

21.1 19.5 18.1

6.81 6.12 5.56


8 - 68



Pu

φrn


ex

where

3

6

s

60°

s

e

s, in.

Pu

s


3

3 6


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.59 2.32 2.07 1.84 1.65

5.21 4.73 4.29 3.90 3.56

7.88 7.27 6.69 6.18 5.73

10.6 9.91 9.23 8.63 8.08

13.4 12.7 11.9 11.2 10.6

16.3 15.5 14.6 13.9 13.2

19.2 18.3 17.5 16.6 15.9

22.1 21.2 20.3 19.5 18.7

25.0 24.1 23.2 22.3 21.5

28.0 27.0 26.1 25.2 24.3

30.9 30.0 29.0 28.1 27.2

33.9 32.9 32.0 31.0 30.1

7 8 9 10 12

1.49 1.35 1.23 1.12 0.95

3.27 3.01 2.78 2.58 2.25

5.32 4.95 4.63 4.34 3.84

7.59 10.0 7.13 9.48 6.71 8.98 6.33 8.52 5.67 7.70

12.6 12.0 11.4 10.9 9.91

15.2 14.5 13.9 13.3 12.3

17.9 17.2 16.5 15.9 14.7

20.7 19.9 19.2 18.5 17.3

23.5 22.7 22.0 21.2 19.9

26.3 25.5 24.7 24.0 22.6

29.2 28.4 27.6 26.8 25.3

14 16 18 20 24

0.83 0.73 0.65 0.59 0.49

1.98 1.77 1.60 1.46 1.24

3.43 3.09 2.81 2.57 2.20

5.11 4.64 4.24 3.90 3.35

7.00 6.40 5.89 5.44 4.72

9.08 11.3 8.36 10.5 7.73 9.74 7.19 9.09 6.27 7.99

13.7 12.7 11.9 11.1 9.85

16.1 15.1 14.2 13.3 11.9

18.7 17.5 16.5 15.6 14.0

21.3 20.1 19.0 17.9 16.2

23.9 22.6 21.5 20.4 18.5

28 32 36

0.42 0.37 0.33

1.07 0.95 0.85

1.91 1.69 1.51

2.93 2.60 2.34

4.15 3.70 3.34

5.55 4.97 4.49

12.6 11.5 10.5

14.7 13.4 12.3

16.8 15.4 14.2

2 3 4 5 6

2.59 2.32 2.07 1.84 1.65

5.32 4.94 4.57 4.25 3.95

8.17 7.73 7.31 6.91 6.55

11.1 10.6 10.2 9.73 9.32

14.0 13.5 13.1 12.6 12.2

17.0 16.5 16.0 15.5 15.1

19.9 19.4 19.0 18.5 18.0

22.9 22.4 21.9 21.4 20.9

25.8 25.4 24.9 24.4 23.9

28.8 28.3 27.8 27.4 26.9

31.8 31.3 30.8 30.3 29.8

34.7 34.3 33.8 33.3 32.8

7 8 9 10 12

1.49 1.35 1.23 1.12 0.95

3.69 3.46 3.25 3.06 2.73

6.22 5.92 5.64 5.39 4.92

8.94 8.58 8.25 7.94 7.37

11.8 11.4 11.0 10.6 9.97

14.6 14.2 13.8 13.4 12.7

17.5 17.1 16.7 16.3 15.5

20.5 20.0 19.6 19.1 18.3

23.4 22.9 22.5 22.0 21.2

26.4 25.9 25.4 24.9 24.1

29.3 28.8 28.4 27.9 27.0

32.3 31.8 31.3 30.8 29.9

14 16 18 20 24

0.83 0.73 0.65 0.59 0.49

2.46 2.23 2.04 1.88 1.63

4.52 4.18 3.87 3.60 3.15

6.85 6.39 5.97 5.59 4.94

9.36 8.80 8.28 7.81 6.99

12.0 11.4 10.8 10.2 9.25

14.7 14.0 13.4 12.8 11.7

17.5 16.8 16.1 15.4 14.2

20.3 19.6 18.8 18.1 16.8

23.2 22.4 21.6 20.9 19.5

26.1 25.3 24.4 23.7 22.2

29.0 28.1 27.3 26.5 25.0

28 32 36

0.42 0.37 0.33

1.43 1.27 1.15

2.79 2.49 2.25

4.41 3.97 3.61

6.31 5.74 5.26

8.44 10.7 13.1 7.74 9.90 12.2 7.13 9.17 11.4

15.7 14.6 13.7

18.2 17.1 16.1

20.9 19.7 18.6

23.6 22.3 21.1

7.10 6.38 5.79

8.81 10.7 7.95 9.65 7.23 8.81



8 - 69


Pu

φ rn


ex

where

3

6

s s

e

s, in.

75° P u

s


3

3 6


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.86 2.77 2.66 2.53 2.40

5.68 5.49 5.27 5.04 4.81

8.47 8.19 7.89 7.58 7.27

11.3 10.9 10.5 10.2 9.81

14.1 13.7 13.2 12.8 12.4

16.9 16.4 16.0 15.5 15.1

19.8 19.2 18.8 18.3 17.8

22.6 22.1 21.6 21.0 20.6

25.5 24.9 24.4 23.9 23.3

28.4 27.8 27.2 26.7 26.2

31.3 30.7 30.1 29.5 29.0

34.2 33.6 33.0 32.4 31.8

7 8 9 10 12

2.26 2.13 2.00 1.89 1.67

4.57 4.35 4.13 3.93 3.57

6.97 6.69 6.41 6.15 5.67

9.47 9.13 8.82 8.51 7.95

12.0 11.7 11.3 11.0 10.4

14.7 14.3 13.9 13.5 12.9

17.4 16.9 16.5 16.1 15.4

20.1 19.6 19.2 18.8 18.0

22.9 22.4 21.9 21.5 20.7

25.6 25.1 24.7 24.2 23.4

28.4 27.9 27.4 27.0 26.1

31.3 30.7 30.2 29.8 28.8

14 16 18 20 24

1.49 1.34 1.21 1.10 0.93

3.25 2.97 2.73 2.53 2.19

5.25 4.87 4.54 4.24 3.75

7.44 6.98 6.56 6.18 5.52

9.77 9.23 8.74 8.28 7.48

12.2 11.6 11.1 10.5 9.59

14.7 14.1 13.5 12.9 11.8

17.3 16.6 16.0 15.3 14.2

19.9 19.2 18.5 17.8 16.6

22.6 21.8 21.1 20.4 19.1

25.3 24.5 23.7 23.0 21.6

28.0 27.2 26.4 25.6 24.2

28 32 36

0.80 0.71 0.63

1.93 1.72 1.55

3.34 3.01 2.74

4.97 4.51 4.12

6.79 6.20 5.70

13.2 12.3 11.5

15.5 14.5 13.6

17.9 16.8 15.9

20.4 19.2 18.2

22.9 21.7 20.6

2 3 4 5 6

2.86 2.77 2.66 2.53 2.40

5.66 5.49 5.30 5.10 4.91

8.48 8.25 8.02 7.79 7.56

11.3 11.1 10.8 10.6 10.3

14.2 13.9 13.6 13.4 13.1

17.1 16.8 16.5 16.2 15.9

20.1 19.7 19.4 19.1 18.8

23.0 22.7 22.3 22.0 21.7

26.4 25.6 25.2 24.9 24.6

29.3 28.5 28.2 27.8 27.5

32.3 31.5 31.1 30.8 30.4

35.2 34.4 34.0 33.7 33.3

7 8 9 10 12

2.26 2.13 2.00 1.89 1.67

4.72 4.54 4.37 4.21 3.90

7.34 7.14 6.94 6.75 6.39

10.1 9.83 9.61 9.40 9.00

12.9 12.6 12.4 12.1 11.7

15.7 15.4 15.2 14.9 14.4

18.5 18.3 18.0 17.7 17.2

21.4 21.1 20.8 20.6 20.0

24.3 24.0 23.7 23.4 22.9

27.2 26.9 26.6 26.3 25.7

30.1 29.8 29.5 29.2 28.6

33.0 32.7 32.4 32.1 31.5

14 16 18 20 24

1.49 1.34 1.21 1.10 0.93

3.63 3.39 3.17 2.98 2.65

6.06 5.75 5.47 5.22 4.76

8.63 8.29 7.96 7.66 7.10

11.3 10.9 10.6 10.2 9.57

14.0 13.6 13.2 12.9 12.2

16.8 16.3 15.9 15.5 14.8

19.6 19.1 18.7 18.2 17.5

22.4 21.9 21.4 21.0 20.2

25.2 24.7 24.2 23.8 22.9

28.1 27.5 27.0 26.6 25.7

30.9 30.4 29.9 29.4 28.5

28 32 36

0.80 0.71 0.63

2.38 2.16 1.97

4.37 4.03 3.73

6.60 6.15 5.75

8.99 11.5 8.45 10.9 7.96 10.3

14.1 13.4 12.8

16.7 16.0 15.3

19.4 18.7 17.9

22.1 21.3 20.6

24.8 24.0 23.3

27.6 26.8 26.0

8.78 10.9 8.08 10.1 7.47 9.40


8 - 70



Pu

φrn


ex= e

where

s, in.

3

6

s

Pu

s

s


6

6 12


ex, in.

1

2

3

2 3 4 5 6

2.15 1.91 1.71 1.55 1.42

4.55 4.06 3.65 3.31 3.02

7.17 6.43 5.80 5.27 4.82

7 8 9 10 12

1.31 1.21 1.12 1.05 0.92

2.77 2.56 2.38 2.21 1.94

4.44 4.10 3.81 3.55 3.12

6.34 5.87 5.46 5.09 4.48

8.46 10.8 7.85 10.1 7.31 9.39 6.84 8.79 6.03 7.78

14 16 18 20 24

0.81 0.72 0.64 0.58 0.49

1.72 1.53 1.38 1.26 1.06

2.77 2.48 2.25 2.05 1.73

3.99 3.58 3.25 2.96 2.52

5.38 4.84 4.40 4.02 3.42

6.95 6.27 5.70 5.21 4.45

8.69 10.6 12.7 7.85 9.60 11.5 7.15 8.75 10.5 6.55 8.03 9.65 5.60 6.88 8.29

28 32 36

0.42 0.37 0.33

0.92 0.81 0.72

1.50 1.32 1.18

2.19 1.93 1.72

2.97 2.63 2.35

3.87 3.42 3.06

4.88 4.32 3.87

2 3 4 5 6

2.15 1.91 1.71 1.55 1.42

4.94 4.48 4.07 3.71 3.40

7.98 7.39 6.81 6.27 5.79

11.1 10.5 9.86 9.22 8.61

7 8 9 10 12

1.31 1.21 1.12 1.05 0.92

3.13 2.90 2.69 2.51 2.21

5.35 4.97 4.64 4.34 3.85

8.05 11.0 7.53 10.4 7.07 9.78 6.64 9.24 5.91 8.27

14 16 18 20 24

0.81 0.72 0.64 0.58 0.49

1.96 1.76 1.60 1.46 1.24

3.44 3.11 2.83 2.59 2.21

5.31 4.80 4.38 4.02 3.44

7.46 6.78 6.20 5.71 4.91

9.95 12.7 9.09 11.6 8.34 10.7 7.70 9.91 6.65 8.59

28 32 36

0.42 0.37 0.33

1.08 0.95 0.85

1.92 1.70 1.52

3.00 2.66 2.39

4.30 3.82 3.43

5.83 5.19 4.67

4

5

10.0 13.0 9.06 11.9 8.23 10.9 7.51 9.97 6.88 9.16

14.2 13.6 13.0 12.3 11.7

6

7

8

9

10

11

12

16.0 14.9 13.7 12.7 11.7

19.1 17.9 16.7 15.5 14.4

22.2 21.0 19.8 18.5 17.3

25.3 24.1 22.9 21.5 20.3

28.3 27.2 26.0 24.7 23.3

31.4 30.3 29.1 27.8 26.4

34.4 33.4 32.3 31.0 29.6

13.4 12.5 11.7 10.9 9.70

16.1 15.1 14.1 13.3 11.8

19.0 17.9 16.8 15.8 14.1

22.0 20.7 19.6 18.5 16.6

25.1 23.7 22.5 21.3 19.1

28.2 26.8 25.5 24.2 21.9

14.9 13.6 12.4 11.4 9.82

17.3 15.8 14.4 13.3 11.5

19.9 18.1 16.6 15.3 13.2

6.00 5.32 4.77

7.24 6.42 5.76

8.59 10.1 11.6 7.62 8.93 10.3 6.84 8.02 9.29

17.2 16.7 16.1 15.5 14.8

20.2 19.8 19.3 18.6 18.0

23.2 22.8 22.3 21.8 21.1

26.2 25.8 25.4 24.9 24.3

29.2 28.9 28.5 28.0 27.4

32.1 31.9 31.5 31.0 30.5

35.1 34.8 34.5 34.1 33.6

14.1 13.4 12.8 12.1 11.0

17.3 16.6 15.9 15.2 13.9

20.5 19.8 19.0 18.3 16.9

23.6 23.0 22.2 21.5 20.0

26.8 26.1 25.4 24.7 23.2

29.9 29.3 28.6 27.9 26.4

33.1 32.5 31.8 31.1 29.7

15.6 14.4 13.3 12.4 10.8

18.6 17.3 16.1 15.0 13.2

21.8 20.4 19.1 17.9 15.7

25.0 23.5 22.1 20.8 18.5

28.2 26.7 25.2 23.8 21.3

9.53 11.7 14.0 8.51 10.5 12.6 7.68 9.45 11.4

16.5 14.9 13.5

19.2 17.3 15.8

7.57 6.75 6.08



8 - 71


Pu

φ rn


ex

Pu

where s

15°

s, in.

3

6

e

s

s


6

6 12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.22 1.97 1.77 1.61 1.47

4.62 4.13 3.72 3.38 3.10

7.25 6.53 5.91 5.39 4.93

10.1 9.13 8.31 7.60 6.98

13.0 11.9 10.9 10.1 9.28

16.0 14.9 13.7 12.7 11.8

19.0 17.9 16.7 15.5 14.4

22.1 20.9 19.7 18.4 17.2

25.1 24.0 22.7 21.4 20.2

28.2 27.1 25.8 24.5 23.2

31.2 30.1 28.9 27.6 26.2

34.2 33.2 32.0 30.7 29.3

7 8 9 10 12

1.35 1.25 1.16 1.08 0.94

2.85 2.63 2.44 2.28 2.00

4.54 4.21 3.91 3.65 3.20

6.45 5.98 5.57 5.21 4.59

8.59 10.9 7.98 10.2 7.45 9.51 6.97 8.92 6.16 7.91

13.5 12.6 11.8 11.1 9.84

16.1 15.1 14.2 13.4 11.9

19.0 17.8 16.8 15.9 14.2

21.9 20.7 19.5 18.5 16.6

24.9 23.6 22.4 21.2 19.2

27.9 26.6 25.3 24.1 21.9

14 16 18 20 24

0.83 0.74 0.66 0.60 0.50

1.77 1.58 1.43 1.30 1.10

2.85 2.56 2.31 2.11 1.79

4.09 3.68 3.34 3.05 2.59

5.50 4.96 4.51 4.13 3.52

7.08 6.40 5.83 5.34 4.56

8.84 10.8 12.8 8.00 9.75 11.7 7.30 8.91 10.7 6.70 8.19 9.82 5.74 7.03 8.45

15.0 13.7 12.6 11.6 10.0

17.4 15.9 14.6 13.5 11.7

19.9 18.2 16.8 15.5 13.4

28 32 36

0.43 0.38 0.34

0.95 0.84 0.75

1.55 1.37 1.22

2.25 1.99 1.78

3.06 2.70 2.42

3.98 3.52 3.15

5.01 4.43 3.98

2 3 4 5 6

2.22 1.97 1.77 1.61 1.47

4.97 4.50 4.10 3.75 3.45

7.97 7.40 6.84 6.32 5.86

11.0 10.5 9.82 9.20 8.61

7 8 9 10 12

1.35 1.25 1.16 1.08 0.94

3.18 2.95 2.75 2.57 2.26

5.44 5.07 4.73 4.44 3.93

8.06 11.0 7.55 10.4 7.09 9.78 6.67 9.26 5.96 8.33

14 16 18 20 24

0.83 0.74 0.66 0.60 0.50

2.01 1.81 1.64 1.50 1.28

3.52 3.18 2.90 2.65 2.27

5.37 4.87 4.45 4.10 3.52

7.55 6.88 6.31 5.81 5.01

9.97 12.7 9.13 11.7 8.40 10.8 7.77 9.99 6.74 8.71

28 32 36

0.43 0.38 0.34

1.11 0.98 0.88

1.98 1.75 1.57

3.08 2.73 2.45

4.40 3.91 3.52

5.93 5.29 4.77

14.1 13.5 12.9 12.3 11.6

6.15 5.45 4.89

7.40 6.57 5.90

8.77 10.2 11.8 7.79 9.12 10.5 7.01 8.20 9.49

17.1 16.6 16.0 15.4 14.7

20.1 19.7 19.1 18.5 17.8

23.1 22.7 22.2 21.6 20.9

26.1 25.7 25.2 24.7 24.1

29.1 28.7 28.3 27.8 27.2

32.1 31.7 31.3 30.8 30.3

35.0 34.7 34.3 33.9 33.3

14.0 13.3 12.7 12.1 11.0

17.1 16.4 15.7 15.1 13.8

20.3 19.5 18.8 18.1 16.8

23.4 22.7 22.0 21.3 19.8

26.5 25.8 25.1 24.4 23.0

29.6 29.0 28.3 27.6 26.1

32.7 32.1 31.4 30.7 29.3

15.5 14.4 13.3 12.4 10.9

18.5 17.2 16.1 15.0 13.2

21.5 20.2 18.9 17.8 15.8

24.7 23.2 21.9 20.7 18.4

27.8 26.4 25.0 23.6 21.2

9.62 11.8 14.1 8.62 10.6 12.7 7.80 9.59 11.5

16.5 15.0 13.6

19.1 17.4 15.9

7.69 6.87 6.20


8 - 72



Pu

φrn


ex Pu

where

s, in.

3

6

s

30°

s


s

e

6

6 12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.40 2.15 1.94 1.76 1.61

4.89 4.40 3.99 3.65 3.35

7.53 6.84 6.24 5.74 5.29

10.3 9.45 8.69 8.02 7.42

13.2 12.2 11.3 10.5 9.72

16.1 15.1 14.0 13.1 12.2

19.1 18.0 16.9 15.8 14.8

22.1 21.0 19.8 18.7 17.6

25.1 24.0 22.8 21.6 20.4

28.1 27.0 25.8 24.6 23.4

31.1 30.0 28.8 27.6 26.3

34.1 33.0 31.9 30.6 29.3

7 8 9 10 12

1.49 1.37 1.28 1.19 1.04

3.10 2.87 2.67 2.49 2.19

4.90 4.55 4.24 3.97 3.50

6.89 6.42 6.00 5.63 4.98

9.06 11.4 8.47 10.7 7.94 10.1 7.47 9.49 6.64 8.48

13.9 13.1 12.4 11.7 10.5

16.6 15.6 14.8 14.0 12.6

19.3 18.3 17.4 16.5 14.9

22.2 21.1 20.0 19.1 17.3

25.1 23.9 22.8 21.8 19.9

28.1 26.9 25.7 24.6 22.5

14 16 18 20 24

0.92 0.82 0.74 0.67 0.56

1.95 1.75 1.58 1.44 1.22

3.12 2.81 2.55 2.33 1.98

4.46 4.03 3.66 3.35 2.86

5.97 5.40 4.92 4.52 3.87

7.64 6.93 6.33 5.82 5.00

9.46 11.4 8.61 10.4 7.89 9.59 7.27 8.85 6.26 7.65

13.6 12.4 11.4 10.6 9.16

15.8 14.5 13.4 12.4 10.8

18.2 16.7 15.5 14.4 12.5

20.7 19.1 17.7 16.4 14.4

28 32 36

0.48 0.42 0.38

1.06 0.93 0.83

1.72 1.52 1.36

2.49 2.20 1.97

3.37 2.99 2.68

4.37 3.88 3.48

5.48 4.87 4.38

2 3 4 5 6

2.40 2.15 1.94 1.76 1.61

5.11 4.66 4.26 3.92 3.63

8.05 7.51 6.99 6.52 6.09

11.1 10.5 9.90 9.34 8.80

7 8 9 10 12

1.49 1.37 1.28 1.19 1.04

3.38 3.15 2.95 2.77 2.45

5.70 5.35 5.03 4.74 4.23

8.30 11.1 7.83 10.6 7.40 10.0 7.00 9.54 6.30 8.67

14 16 18 20 24

0.92 0.82 0.74 0.67 0.56

2.19 1.98 1.80 1.65 1.41

3.81 3.45 3.16 2.90 2.49

5.71 5.22 4.79 4.42 3.82

7.92 10.4 7.27 9.58 6.71 8.88 6.22 8.26 5.41 7.22

28 32 36

0.48 0.42 0.38

1.23 1.08 0.97

2.18 1.93 1.73

3.43 2.99 2.69

4.78 4.26 3.85

14.1 13.5 12.9 12.3 11.7

6.71 5.97 5.38

8.06 7.18 6.47

9.51 11.1 8.49 9.91 7.66 8.95

12.8 11.4 10.3

17.1 16.5 16.0 15.3 14.7

20.1 19.6 19.0 18.4 17.7

23.0 22.6 22.0 21.5 20.8

26.0 25.6 25.1 24.5 23.9

29.0 28.6 28.1 27.6 27.0

32.0 31.6 31.1 30.6 30.0

34.9 34.6 34.1 33.6 33.1

14.1 13.5 12.9 12.3 11.3

17.1 16.5 15.8 15.2 14.1

20.2 19.5 18.8 18.2 17.0

23.2 22.6 21.9 21.2 19.9

26.3 25.7 25.0 24.3 23.0

29.4 28.7 28.1 27.4 26.0

32.5 31.8 31.2 30.5 29.1

13.0 12.1 11.2 10.5 9.23

15.8 14.8 13.8 12.9 11.5

18.7 17.6 16.5 15.5 13.8

21.7 20.5 19.3 18.2 16.4

24.7 23.4 22.2 21.1 19.0

27.8 26.4 25.2 24.0 21.8

8.22 10.3 12.4 7.40 9.25 11.3 6.71 8.41 10.3

14.8 13.4 12.3

17.2 15.7 14.4

19.8 18.2 16.7

6.40 5.73 5.18



8 - 73


Pu

φ rn


ex Pu

where

3

6

s s

e

s, in.

45°

s


6

6 12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.64 2.43 2.23 2.05 1.89

5.30 4.90 4.52 4.17 3.86

8.01 7.44 6.89 6.40 5.96

10.8 10.1 9.38 8.75 8.20

13.6 12.8 12.0 11.2 10.6

16.4 15.6 14.7 13.9 13.1

19.3 18.4 17.5 16.6 15.7

22.3 21.3 20.3 19.3 18.4

25.2 24.2 23.2 22.2 21.2

28.1 27.2 26.1 25.0 23.99

31.1 30.1 29.0 27.9 26.9

34.0 33.1 32.0 30.9 29.8

7 8 9 10 12

1.75 1.63 1.52 1.42 1.25

3.59 3.35 3.13 2.94 2.60

5.57 5.22 4.90 4.61 4.11

7.70 7.25 6.83 6.45 5.78

9.99 9.43 8.91 8.44 7.60

12.4 11.7 11.1 10.6 9.58

14.9 14.2 13.5 12.8 11.7

17.5 16.7 15.9 15.2 14.0

20.2 19.3 18.5 17.7 16.3

23.0 22.1 21.2 20.3 18.8

25.8 24.8 23.9 23.0 21.3

28.7 27.7 26.7 25.7 23.9

14 16 18 20 24

1.11 0.99 0.90 0.81 0.68

2.32 2.09 1.90 1.73 1.47

3.69 3.34 3.04 2.79 2.38

5.21 4.74 4.33 3.98 3.42

6.90 6.29 5.77 5.33 4.60

8.73 10.7 8.00 9.85 7.36 9.10 6.81 8.44 5.91 7.35

12.8 11.8 10.96 10.2 8.91

15.0 13.9 12.9 12.1 10.6

17.4 16.1 15.0 14.1 12.4

19.8 18.5 17.3 16.2 14.3

22.3 20.9 19.5 18.4 16.3

28 32 36

0.59 0.52 0.46

1.28 1.13 1.01

2.08 1.84 1.65

2.99 2.65 2.38

4.03 3.59 3.23

5.20 4.63 4.17

12.8 11.6 10.5

14.6 13.3 12.1

2 3 4 5 6

2.64 2.43 2.23 2.05 1.89

5.38 5.02 4.67 4.34 4.06

8.22 7.78 7.33 6.90 6.50

11.1 10.7 10.2 9.66 9.19

14.1 13.6 13.1 12.5 12.0

17.0 16.6 16.0 15.5 14.9

20.0 19.5 19.0 18.4 17.9

22.97 22.5 22.0 21.4 20.9

25.9 25.5 25.0 24.4 23.9

28.9 28.5 28.0 27.4 26.9

31.9 31.4 31.0 30.4 29.9

34.8 34.4 34.0 33.4 32.9

7 8 9 10 12

1.75 1.63 1.52 1.42 1.25

3.80 3.57 3.36 3.17 2.84

6.16 5.84 5.54 5.27 4.78

8.76 8.36 7.99 7.63 6.99

11.5 11.1 10.6 10.2 9.40

14.4 13.9 13.4 12.9 12.0

17.3 16.8 16.2 15.7 14.7

20.3 19.7 19.2 18.6 17.6

23.3 22.7 22.1 21.5 20.4

26.3 25.7 25.1 24.5 23.4

29.3 28.7 28.1 27.5 26.3

32.3 31.7 31.1 30.5 29.3

14 16 18 20 24

1.11 0.99 0.90 0.81 0.68

2.57 2.33 2.13 1.96 1.68

4.36 3.99 3.68 3.40 2.95

6.42 5.92 5.49 5.10 4.46

8.70 11.2 8.09 10.5 7.54 9.80 7.05 9.21 6.22 8.19

13.8 13.0 12.2 11.6 10.4

16.6 15.7 14.8 14.0 12.7

19.4 18.4 17.5 16.6 15.1

22.3 21.2 20.3 19.3 17.7

25.2 24.1 23.1 22.1 20.3

28.2 27.1 26.0 24.9 23.0

28 32 36

0.59 0.52 0.46

1.47 1.31 1.17

2.59 2.31 2.08

3.95 3.54 3.20

5.55 4.99 4.54

9.34 11.5 13.8 8.49 10.5 12.7 7.77 9.64 11.7

16.2 14.9 13.8

18.7 17.3 16.1

21.3 19.8 18.5

7.35 6.65 6.06

6.49 5.80 5.23

7.90 7.07 6.40

9.42 11.1 8.46 9.95 7.67 9.04


8 - 74



Pu

φrn


ex

where

3

6

s

60°

s

e

s, in.

Pu

s


6

6 12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.83 2.72 2.59 2.46 2.32

5.64 5.43 5.18 4.92 4.66

8.45 8.13 7.77 7.40 7.03

11.3 10.8 10.4 9.92 9.46

14.1 13.6 13.0 12.5 12.0

16.9 16.3 15.7 15.1 14.5

19.8 19.1 18.5 17.8 17.1

22.6 21.9 21.2 20.5 19.8

25.5 24.8 24.0 23.2 22.5

28.4 27.6 26.8 26.0 25.2

31.3 30.5 29.7 28.9 28.0

34.2 33.4 32.5 31.7 30.8

7 8 9 10 12

2.19 2.07 1.95 1.84 1.65

4.41 4.17 3.95 3.74 3.38

6.68 6.35 6.04 5.75 5.22

9.02 8.61 8.22 7.86 7.19

11.4 11.0 10.5 10.1 9.28

13.9 13.4 12.9 12.4 11.5

16.5 15.9 15.3 14.8 13.8

19.1 18.4 17.8 17.3 16.2

21.8 21.1 20.4 19.8 18.6

24.5 23.7 23.0 22.4 21.1

27.2 26.5 25.7 25.0 23.7

30.0 29.2 28.5 27.7 26.3

14 16 18 20 24

1.49 1.35 1.23 1.12 0.95

3.06 2.79 2.55 2.35 2.02

4.76 4.37 4.02 3.72 3.22

6.61 6.09 5.64 5.24 4.57

8.58 10.7 7.95 9.93 7.39 9.28 6.90 8.69 6.06 7.68

12.9 12.0 11.3 10.6 9.43

15.2 14.2 13.4 12.6 11.3

17.5 16.5 15.6 14.8 13.3

20.0 18.9 17.9 17.0 15.4

22.5 21.3 20.3 19.3 17.5

25.0 23.8 22.7 21.7 19.8

28 32 36

0.83 0.73 0.65

1.76 1.56 1.40

2.84 2.53 2.27

4.04 3.61 3.26

5.39 4.84 4.38

12.0 11.0 10.1

14.0 12.8 11.7

16.0 14.7 13.5

18.1 16.7 15.4

2 3 4 5 6

2.83 2.72 2.59 2.46 2.32

5.64 5.44 5.21 4.97 4.73

8.47 8.19 7.88 7.57 7.27

11.3 11.0 10.6 10.3 9.91

14.2 13.8 13.4 13.1 12.7

17.1 16.7 16.3 15.9 15.5

20.0 19.6 19.2 18.8 18.3

23.0 22.6 22.1 21.7 21.2

25.9 25.5 25.0 24.6 24.1

28.9 28.4 28.0 27.5 27.0

31.8 31.4 30.9 30.4 30.0

34.8 34.3 33.9 33.4 33.0

7 8 9 10 12

2.19 2.07 1.95 1.84 1.65

4.51 4.29 4.09 3.90 3.56

6.97 6.69 6.43 6.18 5.73

9.56 9.23 8.92 8.63 8.08

12.3 11.9 11.5 11.2 10.6

15.0 14.6 14.3 13.9 13.2

17.9 17.5 17.0 16.6 15.9

20.8 20.3 19.9 19.5 18.7

23.7 23.2 22.8 22.3 21.5

26.6 26.1 25.6 25.2 24.3

29.5 29.0 28.6 28.1 27.2

32.4 32.0 31.5 31.0 30.1

14 16 18 20 24

1.49 1.35 1.23 1.12 0.95

3.27 3.01 2.78 2.58 2.25

5.32 4.95 4.63 4.34 3.84

7.59 10.0 7.13 9.48 6.71 8.98 6.33 8.52 5.67 7.70

12.6 12.0 11.4 10.9 9.91

15.2 14.5 13.9 13.3 12.3

17.9 17.2 16.5 15.9 14.7

20.7 19.9 19.2 18.5 17.3

23.5 22.7 22.0 21.2 19.9

26.3 25.5 24.7 24.0 22.6

29.2 28.4 27.6 27.0 25.3

28 32 36

0.83 0.73 0.65

1.98 1.77 1.60

3.43 3.09 2.81

5.11 4.64 4.24

9.08 11.3 13.7 8.36 10.5 12.7 7.73 9.74 11.9

16.1 15.1 14.2

18.7 17.5 16.5

21.3 20.1 19.0

23.9 22.6 21.5

7.00 6.40 5.89

6.86 6.19 5.62

8.47 10.2 7.66 9.26 6.98 8.46



8 - 75


Pu

φ rn


ex

where

3

6

s s

e

s, in.

75° P u

s


6

6 12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.92 2.89 2.86 2.82 2.77

5.83 5.77 5.70 5.61 5.51

8.73 8.63 8.51 8.38 8.23

11.6 11.5 11.3 11.1 10.9

14.5 14.3 14.1 13.9 13.6

17.4 17.2 16.9 16.6 16.3

20.3 20.0 19.7 19.4 19.0

23.1 22.8 22.5 22.1 21.8

26.0 25.7 25.3 24.9 24.5

28.9 28.5 28.1 27.7 27.2

31.8 31.4 30.9 30.5 30.0

34.7 34.2 33.7 33.3 32.8

7 8 9 10 12

2.72 2.66 2.60 2.53 2.40

5.40 5.29 5.16 5.04 4.78

8.06 7.89 7.71 7.53 7.16

10.7 10.5 10.3 10.1 9.57

13.4 13.1 12.8 12.6 12.0

16.0 15.7 15.4 15.1 14.5

18.7 18.3 18.0 17.7 17.0

21.4 21.0 20.6 20.3 19.6

24.1 23.7 23.3 22.9 22.1

26.8 26.4 26.0 25.6 24.8

29.6 29.1 28.7 28.3 27.4

32.3 31.9 31.4 31.0 30.1

14 16 18 20 24

2.26 2.13 2.00 1.89 1.67

4.52 4.27 4.03 3.81 3.41

6.80 6.45 6.12 5.80 5.24

9.12 8.68 8.27 7.88 7.18

11.5 11.0 10.5 10.1 9.22

13.9 13.3 12.8 12.3 11.4

16.4 15.8 15.2 14.6 13.6

18.9 18.2 17.6 17.0 15.9

21.4 20.7 20.1 19.4 18.2

24.0 23.3 22.6 21.9 20.7

26.6 25.9 25.1 24.4 23.1

29.3 28.5 27.7 27.0 25.6

28 32 36

1.49 1.34 1.21

3.06 2.77 2.52

4.75 4.33 3.97

6.56 6.02 5.56

12.6 11.8 11.1

14.9 13.9 13.1

17.1 16.1 15.2

19.5 18.4 17.4

21.9 20.7 19.7

24.3 23.1 22.0

2 3 4 5 6

2.92 2.89 2.86 2.82 2.77

5.82 5.76 5.68 5.59 5.49

8.71 8.60 8.47 8.34 8.19

11.6 11.4 11.3 11.1 10.9

14.5 14.3 14.1 13.9 13.7

17.4 17.1 16.9 16.7 16.4

20.3 20.0 19.8 19.5 19.2

23.5 22.9 22.6 22.4 22.1

26.4 25.8 25.5 25.2 24.9

29.3 28.7 28.4 28.1 27.8

32.3 31.7 31.3 31.0 30.7

35.2 34.6 34.2 33.9 33.6

7 8 9 10 12

2.72 2.66 2.60 2.53 2.40

5.39 5.27 5.16 5.04 4.81

8.04 7.89 7.74 7.58 7.27

10.7 10.5 10.4 10.2 9.81

13.4 13.2 13.0 12.8 12.4

16.2 16.0 15.8 15.5 15.1

19.0 18.8 18.5 18.3 17.8

21.8 21.6 21.3 21.0 20.6

24.6 24.4 24.1 23.9 23.3

27.5 27.2 27.0 26.7 26.2

30.4 30.1 29.8 29.5 29.0

33.3 33.0 32.7 32.4 31.8

14 16 18 20 24

2.26 2.13 2.00 1.89 1.67

4.57 4.35 4.13 3.93 3.57

6.97 6.69 6.41 6.15 5.67

9.47 9.13 8.82 8.51 7.95

12.0 11.7 11.3 11.0 10.4

14.7 14.3 13.9 13.5 12.9

17.4 16.9 16.5 16.1 15.4

20.1 19.6 19.2 18.8 18.0

22.9 22.4 21.9 21.5 20.7

25.6 25.1 24.7 24.2 23.4

28.4 27.9 27.4 27.0 26.1

31.3 30.7 30.2 29.8 28.8

28 32 36

1.49 1.34 1.21

3.25 2.97 2.73

5.25 4.87 4.54

7.44 6.98 6.56

9.77 12.2 9.23 11.6 8.74 11.1

14.7 14.1 13.5

17.3 16.6 16.0

19.9 19.2 18.5

22.6 21.8 21.1

25.3 24.5 23.7

28.0 27.2 26.4

8.49 10.5 7.84 9.77 7.27 9.10


8 - 76



Pu

φrn


ex = e

where

s, in.

3

6

s

Pu

s

s


3

3

3

9


ex, in.

1

2

2 3 4 5 6

2.60 2.23 1.94 1.69 1.49

5.70 4.92 4.30 3.79 3.37

9.24 13.2 8.05 11.7 7.09 10.4 6.30 9.29 5.65 8.37

7 8 9 10 12

1.32 1.18 1.07 0.98 0.83

3.03 2.74 2.50 2.29 1.96

5.10 4.63 4.24 3.89 3.34

7.59 10.4 6.92 9.56 6.35 8.81 5.86 8.15 5.06 7.06

14 16 18 20 24

0.73 0.65 0.58 0.53 0.44

1.72 1.52 1.37 1.24 1.04

2.92 2.59 2.33 2.11 1.78

4.44 3.95 3.55 3.23 2.72

6.21 5.54 4.99 4.53 3.83

8.27 10.6 13.2 7.39 9.48 11.9 6.67 8.57 10.7 6.07 7.81 9.77 5.14 6.62 8.30

28 32 36

0.38 0.34 0.30

0.90 0.79 0.71

1.54 1.36 1.21

2.35 2.07 1.85

3.31 2.91 2.60

4.45 3.92 3.50

2 3 4 5 6

2.60 2.23 1.94 1.69 1.49

6.48 5.75 5.12 4.58 4.13

10.7 9.79 8.91 8.10 7.37

7 8 9 10 12

1.32 1.18 1.07 0.98 0.83

3.74 3.41 3.13 2.89 2.50

6.74 10.5 6.20 9.73 5.73 9.05 5.31 8.45 4.63 7.43

14 16 18 20 24

0.73 0.65 0.58 0.53 0.44

2.19 1.95 1.76 1.60 1.35

4.09 3.65 3.29 2.99 2.53

6.60 5.93 5.37 4.90 4.16

9.53 12.9 8.59 11.7 7.81 10.8 7.15 9.85 6.10 8.44

28 32 36

0.38 0.34 0.30

1.17 1.03 0.92

2.19 1.93 1.72

3.61 3.19 2.85

5.31 4.69 4.20

3

4

14.9 14.0 13.1 12.2 11.3

5

6

7

8

9

10

11

12

17.3 15.6 14.0 12.6 11.5

21.4 19.7 18.0 16.4 14.9

25.6 23.9 22.1 20.3 18.7

29.7 28.1 26.3 24.4 22.7

33.8 32.3 30.5 28.6 26.7

37.9 36.5 34.7 32.9 30.9

41.9 40.6 38.9 37.1 35.2

45.9 44.7 43.1 41.4 39.4

13.7 12.6 11.6 10.8 9.37

17.2 15.9 14.7 13.7 12.0

21.0 19.5 18.1 16.9 14.9

24.9 23.3 21.7 20.3 17.9

29.0 27.3 25.6 24.1 21.3

33.2 31.4 29.6 27.9 24.9

37.5 35.5 33.7 31.9 28.6

16.0 14.4 13.1 11.9 10.2

19.1 17.2 15.7 14.3 12.2

22.4 20.2 18.4 16.9 14.4

25.8 23.4 21.4 19.6 16.8

5.73 5.05 4.51

7.20 6.35 5.68

8.82 10.6 12.6 7.79 9.38 11.1 6.96 8.39 9.95

14.7 13.0 11.6

18.9 18.2 17.4 16.5 15.5

23.0 22.3 21.6 20.7 19.7

27.0 26.4 25.7 24.9 24.0

31.0 30.5 29.9 29.1 28.3

34.9 34.5 33.9 33.2 32.5

38.9 38.5 38.0 37.4 36.7

42.9 42.5 42.0 41.4 40.8

46.8 46.5 46.1 45.5 44.9

14.5 13.6 12.8 12.1 10.7

18.8 17.8 16.9 16.0 14.3

23.1 22.1 21.1 20.1 18.3

27.4 26.4 25.4 24.4 22.5

31.6 30.6 29.7 28.7 26.7

35.8 34.9 34.0 33.0 31.0

40.0 39.1 38.3 37.3 35.3

44.1 43.3 42.5 41.5 39.6

16.7 15.2 14.0 12.9 11.1

20.6 19.0 17.5 16.2 14.0

24.7 22.9 21.3 19.8 17.3

29.0 27.1 25.3 23.6 20.8

33.3 31.3 29.4 27.6 24.5

37.6 35.5 33.6 31.7 28.3

9.69 12.3 15.2 8.61 11.0 13.6 7.73 9.89 12.3

18.4 16.5 14.9

21.8 19.6 17.8

25.3 22.9 20.8

7.37 6.53 5.85



8 - 77


Pu φ rn


ex Pu

where s

15°

s, in.

3

6

e

s

s


3

3

3 9


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.68 2.30 1.99 1.74 1.53

5.77 5.00 4.38 3.88 3.45

9.31 8.17 7.22 6.43 5.77

13.2 11.7 10.4 9.37 8.47

17.2 15.6 14.1 12.8 11.6

21.3 19.6 17.9 16.4 15.0

25.4 23.7 22.0 20.2 18.6

29.5 27.9 26.0 24.2 22.5

33.6 32.0 30.2 28.3 26.6

37.7 36.2 34.4 32.5 30.6

41.7 40.2 38.5 36.7 34.8

45.7 44.3 42.7 40.9 39.0

7 8 9 10 12

1.36 1.22 1.11 1.01 0.86

3.10 2.81 2.57 2.36 2.02

5.21 4.74 4.34 4.00 3.44

7.71 10.6 7.05 9.70 6.48 8.95 5.98 8.29 5.18 7.21

13.7 12.7 11.8 10.9 9.52

17.2 15.9 14.8 13.8 12.2

20.9 19.5 18.1 17.0 15.0

24.8 23.2 21.7 20.4 18.1

28.8 27.1 25.5 24.0 21.4

32.9 31.1 29.4 27.7 24.9

37.1 35.2 33.4 31.6 28.5

14 16 18 20 24

0.75 0.67 0.60 0.54 0.46

1.77 1.57 1.41 1.28 1.08

3.01 2.68 2.40 2.18 1.84

4.55 4.05 3.65 3.32 2.80

6.36 5.67 5.12 4.66 3.94

8.43 10.8 13.3 7.54 9.66 12.0 6.81 8.74 10.9 6.21 7.98 9.95 5.26 6.78 8.47

16.1 14.6 13.3 12.1 10.4

19.2 17.3 15.8 14.5 12.4

22.4 20.3 18.6 17.1 14.6

25.8 23.5 21.5 19.8 17.0

28 32 36

0.40 0.35 0.31

0.93 0.82 0.73

1.59 1.40 1.25

2.43 2.14 1.91

3.41 3.00 2.68

4.56 4.03 3.60

12.8 11.3 10.2

14.9 13.2 11.9

2 3 4 5 6

2.68 2.30 1.99 1.74 1.53

6.48 5.75 5.13 4.61 4.17

10.6 9.75 8.91 8.14 7.45

7 8 9 10 12

1.36 1.22 1.11 1.01 0.86

3.79 3.46 3.19 2.94 2.55

6.84 10.4 6.30 9.71 5.83 9.05 5.42 8.47 4.73 7.47

14 16 18 20 24

0.75 0.67 0.60 0.54 0.46

2.24 2.00 1.80 1.64 1.39

4.18 3.74 3.38 3.08 2.60

6.66 6.00 5.45 4.98 4.25

9.62 12.9 8.71 11.8 7.94 10.8 7.28 9.92 6.23 8.54

28 32 36

0.40 0.35 0.31

1.20 1.06 0.94

2.26 1.99 1.78

3.69 3.26 2.92

5.43 4.81 4.31

14.7 13.9 13.0 12.1 11.2

5.89 5.19 4.65

7.37 6.51 5.83

9.02 10.9 7.98 9.59 7.15 8.59

18.8 18.1 17.2 16.3 15.3

22.9 22.2 21.4 20.5 19.5

26.9 26.3 25.5 24.7 23.7

30.9 30.3 29.7 28.9 27.9

34.9 34.4 33.7 33.0 32.2

38.8 38.3 37.7 37.1 36.3

42.8 42.4 41.8 41.2 40.4

46.7 46.3 45.8 45.2 44.5

14.5 13.6 12.8 12.1 10.7

18.6 17.6 16.8 15.9 14.3

22.8 21.8 20.9 20.0 18.2

27.0 26.0 25.1 24.1 22.2

31.3 30.3 29.3 28.3 26.4

35.4 34.5 33.5 32.6 30.6

39.6 38.7 37.8 36.8 34.8

43.7 42.9 42.0 41.0 39.1

16.6 15.2 14.0 13.0 11.2

20.5 18.9 17.5 16.2 14.1

24.5 22.8 21.2 19.8 17.3

28.6 26.8 25.1 23.5 20.8

32.9 30.9 29.1 27.4 24.4

37.1 35.1 33.2 31.4 28.1

9.85 12.5 8.77 11.1 7.89 10.0

15.4 13.8 12.5

18.5 16.6 15.1

21.8 19.7 17.9

25.3 22.9 20.9

7.48 6.65 5.97


8 - 78



Pu

φrn


ex Pu

s

where

s, in.

3

6

30°

s


s

e

3

3

3

9


ex, in.

1

2

2 3 4 5 6

2.90 2.50 2.18 1.91 1.69

6.06 5.31 4.70 4.18 3.75

9.59 13.4 8.52 12.1 7.62 10.9 6.85 9.86 6.19 8.98

7 8 9 10 12

1.51 1.36 1.23 1.13 0.96

3.38 3.07 2.81 2.59 2.23

5.63 5.14 4.73 4.37 3.78

8.21 11.2 7.55 10.3 6.97 9.54 6.46 8.88 5.62 7.78

14 16 18 20 24

0.84 0.74 0.67 0.61 0.51

1.95 1.73 1.56 1.42 1.20

3.32 2.96 2.66 2.42 2.04

4.96 4.43 4.00 3.65 3.09

6.90 6.19 5.60 5.11 4.34

9.08 11.6 8.17 10.4 7.41 9.46 6.77 8.67 5.77 7.41

28 32 36

0.44 0.39 0.35

1.03 0.91 0.81

1.77 1.56 1.39

2.68 2.36 2.11

3.77 3.32 2.97

5.01 4.43 3.97

2 3 4 5 6

2.90 2.50 2.18 1.91 1.69

6.59 5.88 5.30 4.81 4.38

10.6 9.83 9.05 8.35 7.72

7 8 9 10 12

1.51 1.36 1.23 1.13 0.96

4.01 3.69 3.41 3.16 2.76

7.15 10.7 6.64 10.0 6.19 9.41 5.79 8.85 5.09 7.88

14 16 18 20 24

0.84 0.74 0.67 0.61 0.51

2.44 2.18 1.97 1.80 1.53

4.54 4.08 3.70 3.38 2.87

7.08 10.1 6.41 9.21 5.85 8.45 5.37 7.80 4.61 6.74

28 32 36

0.44 0.39 0.35

1.32 1.17 1.05

2.49 2.20 1.97

4.02 3.57 3.21

3

4

14.7 13.9 13.0 12.3 11.4

5

6

7

8

9

10

11

12

17.3 15.8 14.4 13.2 12.1

21.3 19.8 18.2 16.8 15.5

25.3 23.8 22.1 20.6 19.1

29.4 27.8 26.1 24.5 22.9

33.4 31.9 30.1 28.4 26.8

37.5 35.9 34.2 32.5 30.7

41.4 40.0 38.3 36.6 34.8

45.4 44.0 42.4 40.7 38.9

14.4 13.3 12.4 11.6 10.2

17.8 16.6 15.5 14.6 12.9

21.4 20.0 18.8 17.7 15.8

25.2 23.7 22.3 21.1 18.9

29.1 27.5 26.1 24.7 22.2

33.1 31.4 29.9 28.3 25.7

37.1 35.4 33.7 32.2 29.3

14.2 12.9 11.8 10.8 9.22

17.1 15.5 14.2 13.1 11.2

20.2 18.4 16.9 15.5 13.4

23.4 21.4 19.7 18.2 15.7

26.8 24.6 22.7 21.0 18.2

13.9 12.3 11.1

16.1 14.4 13.0

6.46 5.71 5.12

8.05 7.14 6.40

9.83 11.8 8.72 10.5 7.84 9.41

18.7 18.0 17.1 16.3 15.4

22.7 22.0 21.2 20.4 19.5

26.7 26.1 25.4 24.5 23.6

30.8 30.1 29.4 28.6 27.8

34.7 34.1 33.5 32.7 31.9

38.7 38.2 37.5 36.8 35.9

42.6 42.2 41.5 40.8 40.0

46.6 46.1 45.5 44.9 44.1

14.6 13.8 13.0 12.4 11.2

18.6 17.7 16.9 16.2 14.7

22.7 21.8 20.9 20.1 18.5

26.9 25.9 25.0 24.1 22.4

31.0 30.0 29.1 28.2 26.4

35.1 34.2 33.3 32.4 30.6

39.2 38.3 37.4 36.5 34.6

43.3 42.4 41.6 40.6 38.8

13.4 12.3 11.4 10.5 9.16

17.0 15.7 14.6 13.6 11.9

20.9 19.4 18.1 16.9 14.9

24.7 23.2 21.8 20.4 18.1

28.8 27.1 25.6 24.1 21.5

32.9 31.1 29.4 27.9 25.1

36.9 35.1 33.4 31.8 28.8

8.07 10.5 13.3 7.20 9.45 11.9 6.49 8.55 10.9

16.3 14.6 13.4

19.4 17.6 16.0

22.7 20.7 18.9

26.2 23.9 22.0

5.91 5.26 4.73



8 - 79


Pu φ rn


ex Pu

s

where

3

6

s

e

s, in.

45°

s


3

3

3

9


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.26 2.87 2.54 2.25 2.01

6.62 5.92 5.31 4.78 4.33

10.2 9.19 8.36 7.63 6.99

13.9 12.7 11.7 10.8 9.94

17.7 16.4 15.2 14.1 13.1

21.6 20.2 18.9 17.6 16.5

25.5 24.0 22.6 21.3 20.1

29.4 27.96 26.5 25.1 23.8

33.4 32.0 30.5 29.0 27.5

37.4 35.9 34.4 32.9 31.4

41.3 39.9 38.4 36.9 35.3

45.3 43.9 42.4 40.8 39.3

7 8 9 10 12

1.81 1.64 1.49 1.37 1.17

3.93 3.60 3.31 3.06 2.65

6.42 5.92 5.49 5.10 4.46

9.20 8.55 7.96 7.44 6.55

12.2 11.4 10.7 10.1 8.93

15.5 14.6 13.7 13.0 11.6

18.9 17.9 16.9 16.0 14.4

22.5 21.3 20.3 19.2 17.5

26.2 24.9 23.8 22.7 20.7

30.0 28.6 27.4 26.2 24.0

33.9 32.4 31.1 29.9 27.5

37.7 36.3 34.9 33.6 31.1

14 16 18 20 24

1.03 0.91 0.82 0.74 0.63

2.33 2.08 1.88 1.71 1.45

3.95 3.54 3.20 2.92 2.48

5.83 5.24 4.75 4.35 3.71

8.00 10.5 7.23 9.47 6.59 8.66 6.04 7.96 5.18 6.84

13.1 12.0 10.9 10.1 8.71

15.9 14.6 13.5 12.5 10.8

19.0 17.5 16.1 15.0 13.0

22.1 20.4 18.9 17.6 15.4

25.4 23.6 21.9 20.5 18.0

28.8 26.8 25.0 23.5 20.7

28 32 36

0.54 0.48 0.43

1.26 1.11 0.99

2.15 1.90 1.69

3.23 2.86 2.56

4.52 4.00 3.59

13.7 12.3 11.2

16.0 14.4 13.1

18.5 16.7 15.2

2 3 4 5 6

3.26 2.87 2.54 2.25 2.01

6.89 6.28 5.74 5.27 4.85

10.8 10.1 9.38 8.75 8.20

7 8 9 10 12

1.81 1.64 1.49 1.37 1.17

4.49 4.16 3.87 3.62 3.19

7.70 11.3 7.25 10.7 6.83 10.2 6.45 9.65 5.78 8.75

14 16 18 20 24

1.03 0.91 0.82 0.74 0.63

2.84 2.56 2.33 2.13 1.82

5.21 4.74 4.33 3.98 3.42

7.97 11.1 7.30 10.3 6.72 9.48 6.21 8.83 5.38 7.74

28 32 36

0.54 0.48 0.43

1.59 1.41 1.26

2.99 2.65 2.38

4.74 4.22 3.81

14.8 14.0 13.3 12.6 11.9

5.99 5.31 4.77

7.65 6.81 6.13

9.50 11.5 8.48 10.3 7.64 9.30

18.7 18.0 17.3 16.5 15.7

22.7 22.0 21.2 20.4 19.7

26.6 26.0 25.3 24.5 23.7

30.6 30.0 29.2 28.5 27.7

34.6 33.9 33.2 32.5 31.7

38.5 37.9 37.2 36.5 35.7

42.5 41.9 41.2 40.5 39.7

46.5 45.9 45.2 44.5 43.8

15.1 14.4 13.8 13.1 12.0

19.0 18.2 17.5 16.9 15.6

22.9 22.2 21.4 20.7 19.3

26.9 26.1 25.4 24.6 23.2

30.9 30.1 29.4 28.5 27.0

34.9 34.1 33.3 32.5 31.0

39.0 38.2 37.4 36.6 35.0

43.0 42.2 41.4 40.6 39.0

14.5 13.5 12.6 11.8 10.4

18.1 16.9 15.9 15.0 13.4

21.8 20.6 19.4 18.4 16.5

25.6 24.3 23.0 21.9 19.8

29.5 28.1 26.7 25.5 23.2

33.4 32.0 30.6 29.2 26.8

37.4 35.9 34.4 33.1 30.5

9.30 12.0 14.9 8.38 10.9 13.6 7.62 9.89 12.5

18.0 16.5 15.2

21.3 19.5 18.0

24.7 22.8 21.1

28.2 26.1 24.3

6.87 6.17 5.59


8 - 80



Pu

φrn


ex

where

3

6

s

60°

s

e

s, in.

Pu

s


3

3

3

9


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.63 3.38 3.10 2.84 2.60

7.25 6.77 6.27 5.80 5.36

10.9 10.3 9.55 8.92 8.33

14.6 13.8 13.0 12.2 11.5

18.3 17.4 16.5 15.6 14.8

22.1 21.1 20.1 19.1 18.2

25.9 24.8 23.7 22.7 21.7

29.7 28.6 27.5 26.4 25.4

33.6 32.4 31.3 30.1 29.1

37.5 36.3 35.1 33.9 32.8

41.4 40.2 38.9 37.8 36.6

45.3 44.1 42.8 41.6 40.4

7 8 9 10 12

2.38 2.19 2.02 1.87 1.62

4.96 4.60 4.28 3.99 3.51

7.79 10.9 7.30 10.2 6.85 9.68 6.45 9.17 5.75 8.27

14.1 13.4 12.7 12.1 11.0

17.4 16.7 15.9 15.2 13.9

20.9 20.0 19.2 18.4 17.0

24.4 23.5 22.6 21.8 20.3

28.0 27.1 26.1 25.3 23.6

31.8 30.7 29.7 28.8 27.0

35.5 34.4 33.4 32.4 30.6

39.3 38.2 37.1 36.1 34.1

14 16 18 20 24

1.43 1.27 1.15 1.04 0.88

3.12 2.81 2.56 2.34 2.00

5.18 4.70 4.29 3.95 3.39

7.50 10.1 6.85 9.23 6.28 8.52 5.80 7.89 5.01 6.87

12.9 11.9 11.0 10.2 8.98

15.8 14.7 13.7 12.8 11.3

18.9 17.6 16.5 15.5 13.8

22.1 20.7 19.5 18.4 16.4

25.4 24.0 22.6 21.4 19.2

28.9 27.3 25.9 24.5 22.1

32.4 30.7 29.1 27.7 25.2

28 32 36

0.76 0.67 0.60

1.74 1.54 1.38

2.96 2.63 2.36

4.39 3.91 3.52

12.3 11.2 10.2

14.8 13.5 12.3

17.4 15.9 14.5

20.1 18.4 16.9

23.0 21.1 19.4

2 3 4 5 6

3.63 3.38 3.10 2.84 2.60

7.29 6.88 6.46 6.06 5.69

11.1 10.6 10.0 9.55 9.09

14.9 14.3 13.8 13.2 12.7

18.8 18.2 17.6 17.0 16.4

22.7 22.1 21.5 20.9 20.3

26.6 26.0 25.4 24.7 24.2

30.5 29.9 29.3 28.7 28.1

34.5 33.9 33.3 32.6 31.9

38.4 37.8 37.2 36.5 35.9

42.4 41.8 41.1 40.4 39.8

46.3 45.7 45.1 44.4 43.8

7 8 9 10 12

2.38 2.19 2.02 1.87 1.62

5.34 5.03 4.74 4.47 4.01

8.66 8.27 7.90 7.55 6.93

12.2 11.7 11.3 10.9 10.1

15.9 15.4 14.9 14.5 13.6

19.7 19.1 18.6 18.1 17.2

23.6 22.9 22.4 21.9 20.8

27.4 26.8 26.2 25.7 24.5

31.3 30.7 30.1 29.5 28.3

35.2 34.6 34.0 33.4 32.2

39.2 38.5 37.9 37.3 36.0

43.1 42.4 41.8 41.2 39.9

14 16 18 20 24

1.43 1.27 1.15 1.04 0.88

3.63 3.31 3.04 2.81 2.44

6.38 5.91 5.49 5.12 4.49

9.46 8.84 8.28 7.77 6.90

12.8 12.1 11.3 10.8 9.62

16.2 15.4 14.6 13.9 12.6

19.9 18.9 18.0 17.2 15.8

23.5 22.6 21.6 20.8 19.1

27.3 26.3 25.2 24.3 22.6

31.0 30.0 28.9 28.0 26.1

34.9 33.8 32.7 31.7 29.8

38.7 37.6 36.5 35.4 33.4

28 32 36

0.76 0.67 0.60

2.15 1.91 1.73

3.99 3.58 3.24

6.18 5.58 5.08

8.70 11.5 14.5 7.93 10.6 13.4 7.27 9.76 12.5

17.7 16.5 15.4

21.1 19.7 18.4

24.5 23.0 21.6

28.0 26.4 24.9

31.6 29.9 28.3

6.07 5.43 4.91

7.97 10.1 7.15 9.06 6.48 8.22



8 - 81


Pu

φ rn


ex

where

3

6

s

Pu

s

e

s, in.

75°

s


3

3

3

9


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.86 3.79 3.70 3.59 3.47

7.69 7.53 7.34 7.13 6.89

11.5 11.2 11.0 10.6 10.3

15.3 14.9 14.6 14.2 13.8

19.1 18.6 18.2 17.7 17.2

22.9 22.4 21.9 21.3 20.8

26.7 26.1 25.5 24.9 24.4

30.5 29.9 29.2 28.6 28.0

34.3 33.6 33.0 32.3 31.7

38.2 37.5 36.7 36.1 35.4

42.1 41.3 40.6 39.8 39.1

45.9 45.1 44.3 43.6 42.9

7 8 9 10 12

3.34 3.20 3.07 2.94 2.68

6.65 6.40 6.16 5.91 5.45

9.98 9.64 9.31 8.98 8.36

13.4 12.9 12.6 12.2 11.5

16.8 16.4 15.9 15.4 14.6

20.3 19.8 19.3 18.8 17.9

23.8 23.3 22.8 22.2 21.3

27.4 26.9 26.3 25.7 24.8

31.1 30.4 29.9 29.3 28.3

34.8 34.1 33.5 32.9 31.8

38.5 37.8 37.1 36.6 35.4

42.2 41.5 40.8 40.2 39.0

14 16 18 20 24

2.45 2.24 2.06 1.90 1.63

5.03 4.65 4.31 4.01 3.51

7.79 10.7 7.28 10.1 6.81 9.55 6.40 9.03 5.69 8.13

13.9 13.2 12.5 11.9 10.8

17.1 16.3 15.5 14.9 13.6

20.4 19.6 18.8 18.0 16.6

23.8 22.9 22.0 21.2 19.7

27.3 26.3 25.4 24.5 22.9

30.8 29.8 28.8 27.9 26.2

34.3 33.2 32.2 31.3 29.5

37.9 36.8 35.8 34.8 32.9

28 32 36

1.43 1.27 1.14

3.11 2.79 2.53

5.11 4.62 4.22

12.5 11.5 10.7

15.4 14.3 13.3

18.3 17.1 16.0

21.4 20.0 18.9

24.6 23.2 21.8

27.8 26.3 24.9

31.1 29.5 28.0

2 3 4 5 6

3.86 3.79 3.70 3.59 3.47

7.67 7.51 7.32 7.12 6.92

11.5 11.2 11.0 10.7 10.4

15.3 15.0 14.7 14.4 14.1

19.1 18.8 18.4 18.1 17.7

23.0 22.6 22.2 21.9 21.5

26.9 26.4 26.0 25.6 25.3

30.8 30.4 29.9 29.5 29.1

35.2 34.3 33.8 33.3 32.9

39.1 38.1 37.7 37.3 36.8

43.0 42.1 41.6 41.1 40.7

47.0 46.0 45.5 45.0 44.6

7 8 9 10 12

3.34 3.20 3.07 2.94 2.68

6.70 6.49 6.28 6.08 5.69

10.2 9.92 9.66 9.42 8.95

13.8 13.5 13.2 12.9 12.4

17.4 17.1 16.8 16.5 15.9

21.1 20.8 20.5 20.2 19.5

24.9 24.5 24.2 23.9 23.2

28.7 28.3 28.0 27.6 26.9

32.6 32.1 31.8 31.4 30.7

36.4 36.0 35.6 35.2 34.5

40.2 39.8 39.5 39.0 38.3

44.1 43.7 43.3 42.9 42.1

14 16 18 20 24

2.45 2.24 2.06 1.90 1.63

5.33 4.99 4.69 4.42 3.95

8.51 8.10 7.72 7.36 6.74

11.9 11.5 11.0 10.6 9.83

15.4 14.9 14.4 13.9 13.1

19.0 18.5 17.9 17.4 16.5

22.6 22.1 21.5 21.0 20.0

26.3 25.7 25.1 24.6 23.6

30.1 29.4 28.8 28.2 27.1

33.8 33.1 32.5 31.9 30.7

37.6 36.9 36.2 35.6 34.4

41.4 40.7 40.0 39.3 38.1

28 32 36

1.43 1.27 1.14

3.57 3.25 2.98

6.21 5.74 5.33

9.16 12.4 8.56 11.6 8.02 11.0

15.7 14.9 14.1

19.0 18.2 17.3

22.5 21.6 20.7

26.1 25.1 24.1

29.7 28.6 27.6

33.3 32.2 31.2

36.9 35.9 34.8

7.36 6.71 6.15

9.83 9.02 8.31


8 - 82



Pu φ rn


ex= e

where

s, in.

3

6

s

Pu

s

s


4

4

4

12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.82 2.50 2.23 2.01 1.81

5.98 5.31 4.74 4.27 3.86

9.46 8.43 7.58 6.86 6.24

13.3 12.0 10.8 9.82 8.96

17.3 15.7 14.3 13.1 12.0

21.3 19.7 18.2 16.7 15.4

25.5 23.8 22.2 20.5 19.0

29.6 28.0 26.3 24.5 22.9

33.7 32.2 30.4 28.6 26.9

37.7 36.3 34.6 32.8 31.0

41.8 40.4 38.8 37.0 35.2

45.8 44.6 43.0 41.3 39.4

7 8 9 10 12

1.64 1.49 1.36 1.25 1.07

3.52 3.22 2.96 2.73 2.37

5.70 5.24 4.83 4.47 3.89

8.22 11.1 7.57 10.2 7.01 9.48 6.51 8.83 5.68 7.74

14.2 13.2 12.3 11.4 10.1

17.6 16.4 15.3 14.3 12.6

21.3 19.9 18.6 17.5 15.5

25.2 23.6 22.1 20.8 18.5

29.2 27.5 25.9 24.4 21.8

33.3 31.5 29.8 28.2 25.3

37.5 35.6 33.8 32.1 29.0

14 16 18 20 24

0.94 0.83 0.75 0.68 0.58

2.08 1.86 1.67 1.52 1.29

3.42 3.05 2.75 2.50 2.12

5.02 4.49 4.06 3.70 3.14

6.86 6.15 5.56 5.07 4.30

8.95 11.3 8.04 10.2 7.29 9.22 6.65 8.43 5.66 7.18

13.8 12.5 11.4 10.4 8.88

16.6 15.0 13.7 12.6 10.8

19.6 17.8 16.3 14.9 12.8

22.8 20.7 19.0 17.5 15.0

26.2 23.9 21.9 20.2 17.4

28 32 36

0.50 0.44 0.40

1.12 0.98 0.88

1.84 1.62 1.45

2.72 2.40 2.15

3.73 3.30 2.95

4.92 4.34 3.89

13.1 11.6 10.4

15.2 13.5 12.1

2 3 4 5 6

2.82 2.50 2.23 2.01 1.81

6.54 5.90 5.33 4.84 4.42

10.6 9.81 9.01 8.27 7.60

7 8 9 10 12

1.64 1.49 1.36 1.25 1.07

4.05 3.73 3.45 3.20 2.80

7.02 10.6 6.51 9.94 6.06 9.30 5.66 8.72 4.98 7.73

14 16 18 20 24

0.94 0.83 0.75 0.68 0.58

2.47 2.21 2.00 1.82 1.55

4.43 3.98 3.60 3.29 2.79

6.92 6.25 5.68 5.21 4.45

28 32 36

0.50 0.44 0.40

1.34 1.18 1.06

2.42 2.14 1.92

3.87 3.43 3.07

14.8 14.0 13.1 12.2 11.4

6.24 5.51 4.94

7.73 6.84 6.13

9.37 11.2 8.29 9.90 7.43 8.88

18.9 18.1 17.3 16.4 15.5

22.9 22.3 21.5 20.6 19.7

26.9 26.4 25.7 24.8 24.0

30.9 30.4 29.8 29.0 28.2

34.9 34.5 33.9 33.2 32.4

38.9 38.5 37.9 37.3 36.6

42.8 42.5 42.0 41.4 40.7

46.8 46.5 46.0 45.5 44.8

14.6 13.7 13.0 12.2 10.9

18.8 17.8 16.9 16.1 14.5

23.0 22.0 21.1 20.2 18.4

27.3 26.3 25.3 24.4 22.5

31.5 30.6 29.6 28.6 26.7

35.7 34.8 33.9 32.9 30.9

39.9 39.1 38.2 37.2 35.2

44.1 43.3 42.4 41.5 39.5

9.81 8.90 8.13 7.47 6.40

13.2 12.0 11.0 10.1 8.72

16.8 15.4 14.2 13.1 11.3

20.7 19.1 17.7 16.4 14.3

24.8 23.0 21.4 20.0 17.5

29.0 27.1 25.3 23.7 20.9

33.2 31.3 29.4 27.7 24.5

37.5 35.5 33.6 31.7 28.3

5.59 4.95 4.44

7.64 6.79 6.10

9.96 12.6 8.87 11.2 7.98 10.1

15.5 13.8 12.5

18.6 16.7 15.1

21.9 19.7 17.9

25.5 23.0 20.9



8 - 83


Pu

φ rn


ex Pu

where

s, in.

3

6

s

15° e

s

s


4

4

4

12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

2.91 2.57 2.30 2.06 1.86

6.06 5.40 4.84 4.37 3.96

9.56 8.57 7.72 6.99 6.37

13.3 12.0 10.9 9.93 9.09

17.2 15.8 14.4 13.2 12.1

21.3 19.7 18.2 16.7 15.5

25.3 23.7 22.1 20.5 19.0

29.4 27.8 26.1 24.4 22.8

33.5 31.9 30.2 28.5 26.7

37.5 36.1 34.3 32.6 30.8

41.6 40.2 38.5 36.7 34.9

45.6 44.3 42.6 40.9 39.0

7 8 9 10 12

1.69 1.53 1.40 1.29 1.11

3.61 3.31 3.04 2.81 2.44

5.83 5.36 4.95 4.59 4.00

8.36 11.2 7.72 10.4 7.15 9.64 6.65 9.00 5.82 7.90

14.3 13.3 12.4 11.6 10.2

17.7 16.5 15.4 14.5 12.8

21.3 19.9 18.7 17.6 15.6

25.1 23.6 22.2 20.9 18.7

29.0 27.4 25.8 24.4 21.9

33.1 31.3 29.7 28.1 25.3

37.2 35.3 33.6 31.9 28.9

14 16 18 20 24

0.97 0.86 0.78 0.71 0.60

2.15 1.92 1.73 1.57 1.33

3.52 3.15 2.84 2.59 2.19

5.15 4.61 4.17 3.80 3.23

7.02 6.30 5.71 5.21 4.43

9.12 11.5 8.21 10.3 7.45 9.41 6.81 8.61 5.80 7.36

14.0 12.7 11.6 10.6 9.07

16.8 15.2 13.9 12.8 11.0

19.8 18.0 16.5 15.2 13.0

22.9 20.9 19.2 17.7 15.3

26.3 24.0 22.1 20.4 17.6

28 32 36

0.52 0.46 0.41

1.15 1.02 0.91

1.90 1.68 1.50

2.80 2.48 2.22

3.85 3.40 3.04

5.05 4.46 4.00

13.4 11.9 10.7

15.5 13.8 12.4

2 3 4 5 6

2.91 2.57 2.30 2.06 1.86

6.57 5.93 5.37 4.89 4.48

10.6 9.81 9.04 8.33 7.70

7 8 9 10 12

1.69 1.53 1.40 1.29 1.11

4.12 3.80 3.52 3.27 2.86

7.13 10.6 6.62 9.95 6.17 9.32 5.77 8.76 5.09 7.80

14 16 18 20 24

0.97 0.86 0.78 0.71 0.60

2.54 2.27 2.06 1.88 1.59

4.53 4.08 3.70 3.38 2.88

7.00 6.34 5.78 5.30 4.54

9.92 9.02 8.26 7.60 6.54

28 32 36

0.52 0.46 0.41

1.38 1.22 1.09

2.50 2.21 1.98

3.96 3.51 3.15

5.72 5.08 4.56

14.7 13.9 13.0 12.2 11.4

6.41 5.67 5.08

7.91 7.01 6.29

9.59 11.4 8.50 10.1 7.63 9.09

18.8 18.0 17.2 16.3 15.4

22.8 22.1 21.3 20.5 19.5

26.8 26.2 25.5 24.6 23.7

30.8 30.3 29.6 28.8 27.9

34.8 34.3 33.6 32.9 32.1

38.8 38.3 37.7 37.0 36.2

42.7 42.3 41.7 41.1 40.3

46.7 46.3 45.8 45.1 44.4

14.5 13.7 12.9 12.2 11.0

18.6 17.7 16.8 16.0 14.5

22.8 21.8 20.9 20.0 18.3

27.0 26.0 25.1 24.1 22.3

31.2 30.2 29.3 28.3 26.4

35.4 34.4 33.5 32.5 30.6

39.5 38.6 37.7 36.8 34.8

43.7 42.8 41.9 41.0 39.0

13.2 12.0 11.1 10.2 8.84

16.8 15.4 14.2 13.2 11.5

20.6 19.0 17.7 16.4 14.4

24.6 22.9 21.3 19.9 17.5

28.7 26.9 25.2 23.6 20.9

32.8 30.9 29.1 27.5 24.5

37.1 35.1 33.2 31.4 28.2

7.77 10.1 12.7 6.92 9.03 11.4 6.23 8.15 10.3

15.6 14.0 12.7

18.7 16.8 15.3

22.0 19.9 18.1

25.4 23.1 21.1


8 - 84



Pu

φrn


ex Pu

where

s, in.

3

6

s

30°

s


s

e

4

4

4

12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.14 2.79 2.50 2.25 2.04

6.41 5.75 5.19 4.71 4.29

9.91 8.95 8.16 7.45 6.83

13.6 12.4 11.4 10.5 9.65

17.5 16.1 14.9 13.7 12.7

21.4 20.0 18.5 17.2 16.0

25.4 23.9 22.4 20.9 19.6

29.4 27.9 26.3 24.7 23.3

33.4 31.9 30.3 28.6 27.1

37.4 35.9 34.3 32.6 31.0

41.4 40.0 38.4 36.7 35.0

45.4 44.0 42.4 40.7 39.0

7 8 9 10 12

1.85 1.69 1.55 1.43 1.23

3.93 3.61 3.33 3.08 2.68

6.28 5.80 5.38 5.00 4.37

8.92 11.8 8.27 11.0 7.70 10.3 7.19 9.64 6.32 8.52

15.0 14.0 13.1 12.3 11.0

18.3 17.2 16.2 15.3 13.6

21.9 20.6 19.4 18.4 16.5

25.6 24.2 22.9 21.7 19.6

29.4 27.9 26.5 25.2 22.8

33.3 31.7 30.2 28.8 26.2

37.3 35.6 34.0 32.5 29.8

14 16 18 20 24

1.08 0.96 0.87 0.79 0.67

2.36 2.11 1.91 1.74 1.48

3.88 3.47 3.14 2.86 2.43

5.62 5.05 4.57 4.18 3.56

7.61 6.86 6.24 5.71 4.88

9.83 12.3 8.89 11.1 8.10 10.2 7.43 9.35 6.36 8.03

14.9 13.6 12.4 11.5 9.87

17.8 16.2 14.9 13.8 11.9

20.8 19.0 17.5 16.2 14.1

24.0 22.0 20.3 18.9 16.4

27.3 25.2 23.3 21.6 18.9

28 32 36

0.58 0.51 0.46

1.28 1.13 1.01

2.11 1.87 1.67

3.10 2.74 2.45

4.25 3.76 3.37

5.55 4.92 4.41

14.5 12.9 11.7

16.7 14.9 13.5

2 3 4 5 6

3.14 2.79 2.50 2.25 2.04

6.75 6.12 5.58 5.13 4.73

10.7 9.94 9.23 8.58 8.00

7 8 9 10 12

1.85 1.69 1.55 1.43 1.23

4.38 4.06 3.78 3.53 3.10

7.47 10.9 6.98 10.3 6.55 9.72 6.15 9.18 5.47 8.25

14 16 18 20 24

1.08 0.96 0.87 0.79 0.67

2.76 2.48 2.25 2.06 1.76

4.90 4.43 4.04 3.70 3.17

7.46 10.4 6.79 9.55 6.22 8.79 5.72 8.14 4.93 7.06

28 32 36

0.58 0.51 0.46

1.53 1.35 1.21

2.76 2.45 2.19

4.32 3.84 3.46

14.7 13.9 13.1 12.4 11.6

7.02 6.23 5.60

8.65 10.4 12.4 7.69 9.29 11.0 6.91 8.36 9.95

18.7 18.0 17.2 16.3 15.5

22.7 22.0 21.2 20.4 19.5

26.7 26.1 25.3 24.5 23.6

30.7 30.1 29.4 28.6 27.7

34.7 34.1 33.4 32.7 31.8

38.6 38.1 37.5 36.7 35.9

42.6 42.1 41.5 40.8 40.0

46.6 46.1 45.5 44.8 44.1

14.7 14.0 13.3 12.6 11.4

18.7 17.9 17.1 16.3 14.9

22.7 21.9 21.0 20.2 18.6

26.8 25.9 25.1 24.2 22.5

31.0 30.1 29.2 28.3 26.5

35.1 34.2 33.3 32.4 30.6

39.2 38.3 37.4 36.5 34.7

43.3 42.4 41.5 40.6 38.8

13.7 12.6 11.7 10.9 9.48

17.2 16.0 14.9 13.9 12.2

21.0 19.6 18.3 17.1 15.2

24.9 23.3 21.9 20.6 18.3

28.8 27.2 25.7 24.2 21.7

32.9 31.2 29.5 28.0 25.3

37.0 35.2 33.5 31.9 28.9

8.38 10.8 13.5 7.50 9.73 12.2 6.77 8.82 11.1

16.5 14.9 13.6

19.6 17.8 16.3

22.9 20.9 19.1

26.3 24.1 22.2

6.22 5.54 5.00



8 - 85


Pu

φ rn


ex Pu

s

where

3

6

s

e

s, in.

45°

s


4

4

4

12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.46 3.15 2.87 2.61 2.39

6.96 6.38 5.84 5.36 4.93

10.5 9.73 8.97 8.30 7.69

14.2 13.2 12.3 11.4 10.7

18.0 16.8 15.7 14.7 13.9

21.8 20.6 19.3 18.2 17.2

25.7 24.4 23.1 21.8 20.7

29.6 28.2 26.9 25.5 24.3

33.5 32.1 30.7 29.3 28.0

37.4 36.1 34.6 33.2 31.8

41.4 40.0 38.6 37.1 35.6

45.3 44.0 42.5 41.0 39.5

7 8 9 10 12

2.19 2.01 1.86 1.72 1.49

4.55 4.21 3.90 3.63 3.18

7.15 6.66 6.21 5.82 5.14

9.98 9.34 8.76 8.24 7.33

13.0 12.2 11.5 10.9 9.76

16.2 15.3 14.5 13.8 12.4

19.6 18.6 17.7 16.8 15.2

23.1 22.0 21.0 20.0 18.3

26.7 25.5 24.4 23.3 21.4

30.4 29.2 27.9 26.8 24.7

34.2 32.9 31.6 30.4 28.1

38.1 36.7 35.3 34.0 31.6

14 16 18 20 24

1.32 1.17 1.06 0.96 0.82

2.82 2.53 2.29 2.10 1.79

4.59 4.14 3.76 3.44 2.94

6.58 5.95 5.43 4.98 4.26

8.81 11.3 8.00 10.3 7.32 9.44 6.74 8.71 5.81 7.53

13.9 12.7 11.7 10.9 9.43

16.7 15.4 14.2 13.2 11.5

19.7 18.2 16.9 15.7 13.8

22.8 21.2 19.7 18.4 16.2

26.1 24.3 22.7 21.2 18.7

29.5 27.5 25.7 24.2 21.4

28 32 36

0.71 0.63 0.56

1.56 1.38 1.23

2.56 2.26 2.03

3.73 3.31 2.97

5.09 4.52 4.06

14.4 12.9 11.7

16.7 15.1 13.7

19.2 17.3 15.8

2 3 4 5 6

3.46 3.15 2.87 2.61 2.39

7.09 6.58 6.09 5.66 5.26

10.9 10.3 9.65 9.07 8.54

14.8 14.1 13.4 12.8 12.1

18.7 18.1 17.3 16.6 15.9

22.7 22.0 21.3 20.6 19.8

26.7 26.0 25.3 24.5 23.8

30.6 30.0 29.3 28.5 27.8

34.6 33.9 33.3 32.5 31.8

38.5 37.9 37.3 36.5 35.8

42.5 41.9 41.2 40.5 39.8

46.5 45.9 45.2 44.5 43.8

7 8 9 10 12

2.19 2.01 1.86 1.72 1.49

4.91 4.59 4.30 4.04 3.59

8.07 7.63 7.23 6.85 6.19

11.6 11.0 10.5 10.0 9.14

15.3 14.6 14.0 13.4 12.4

19.1 18.4 17.7 17.1 15.9

23.0 22.3 21.5 20.8 19.5

27.0 26.2 25.5 24.7 23.3

31.0 30.2 29.4 28.6 27.2

35.0 34.2 33.4 32.6 31.1

39.0 38.2 37.4 36.6 35.1

43.0 42.2 41.4 40.6 39.1

14 16 18 20 24

1.32 1.17 1.06 0.96 0.82

3.22 2.91 2.66 2.44 2.10

5.62 5.13 4.71 4.35 3.76

8.38 11.4 7.71 10.6 7.12 9.87 6.61 9.22 5.76 8.11

14.8 13.8 12.9 12.1 10.8

18.3 17.2 16.2 15.3 13.7

22.0 20.8 19.6 18.6 16.7

25.8 24.4 23.2 22.1 20.0

29.6 28.2 26.9 25.7 23.4

33.5 32.1 30.7 29.4 27.0

37.5 36.0 34.6 33.2 30.6

28 32 36

0.71 0.63 0.56

1.83 1.63 1.46

3.30 2.94 2.64

5.08 4.54 4.11

9.64 12.3 8.71 11.2 7.93 10.2

15.2 13.9 12.7

18.3 16.7 15.4

21.5 19.8 18.3

24.9 23.0 21.3

28.4 26.3 24.5

7.22 6.50 5.90

6.61 5.89 5.30

8.31 10.2 12.2 7.42 9.11 11.0 6.69 8.23 9.91


8 - 86



Pu

φrn


ex

where s

Pu


3

6

s s

e

s, in.

60°

4

4

4

12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.74 3.57 3.38 3.17 2.97

7.46 7.12 6.75 6.36 5.99

11.2 10.7 10.2 9.61 9.09

14.9 14.3 13.6 12.9 12.3

18.6 17.9 17.1 16.4 15.6

22.4 21.6 20.7 19.8 19.0

26.2 25.3 24.3 23.4 22.5

30.0 29.0 28.0 27.0 26.1

33.9 32.8 31.8 30.7 29.7

37.7 36.7 35.6 34.5 33.4

41.6 40.5 39.4 38.2 37.1

45.5 44.4 43.2 42.0 40.9

7 8 9 10 12

2.78 2.60 2.44 2.28 2.02

5.63 5.29 4.98 4.69 4.18

8.59 8.13 7.69 7.28 6.56

11.7 11.1 10.6 10.1 9.16

14.9 14.2 13.6 13.0 11.9

18.2 17.5 16.8 16.1 14.9

21.6 20.8 20.1 19.3 18.0

25.1 24.3 23.4 22.7 21.2

28.7 27.8 26.9 26.1 24.5

32.3 31.4 30.4 29.5 27.8

36.0 35.0 34.0 33.1 31.3

39.8 38.7 37.7 36.7 34.8

14 16 18 20 24

1.80 1.62 1.47 1.34 1.15

3.76 3.40 3.10 2.85 2.45

5.95 5.43 4.99 4.61 3.99

8.38 11.0 7.70 10.2 7.11 9.42 6.59 8.76 5.73 7.67

13.8 12.8 11.9 11.1 9.82

16.7 15.6 14.6 13.7 12.2

19.8 18.6 17.4 16.4 14.6

23.0 21.6 20.4 19.3 17.3

26.3 24.8 23.5 22.2 20.1

29.6 28.1 26.7 25.3 23.0

33.1 31.4 29.9 28.5 26.0

28 32 36

1.00 0.88 0.79

2.15 1.91 1.72

3.51 3.13 2.81

5.06 4.52 4.08

13.2 11.9 10.9

15.6 14.2 13.0

18.2 16.6 15.3

20.9 19.2 17.7

23.8 21.8 20.2

2 3 4 5 6

3.74 3.57 3.38 3.17 2.97

7.47 7.16 6.82 6.47 6.14

11.2 10.8 10.4 9.94 9.52

15.0 14.6 14.1 13.6 13.1

18.9 18.4 17.8 17.3 16.7

22.8 22.2 21.7 21.1 20.5

26.7 26.1 25.5 24.9 24.3

30.6 30.0 29.4 28.8 28.2

34.5 33.9 33.3 32.7 32.1

38.5 37.9 37.3 36.6 36.0

42.4 41.8 41.2 40.5 39.9

46.4 45.8 45.1 44.5 43.8

7 8 9 10 12

2.78 2.60 2.44 2.28 2.02

5.82 5.52 5.24 4.98 4.51

9.11 8.73 8.37 8.03 7.41

12.6 12.1 11.7 11.3 10.6

16.2 15.7 15.2 14.8 14.0

19.9 19.4 18.9 18.4 17.5

23.7 23.2 22.6 22.1 21.1

27.6 27.0 26.4 25.8 24.8

31.5 30.8 30.2 29.7 28.5

35.3 34.7 34.1 33.5 32.3

39.3 38.6 38.0 37.4 36.2

43.2 42.5 41.9 41.3 40.1

14 16 18 20 24

1.80 1.62 1.47 1.34 1.15

4.10 3.76 3.46 3.21 2.79

6.86 6.37 5.94 5.56 4.91

9.91 9.29 8.74 8.23 7.34

13.2 12.4 11.8 11.2 10.1

16.6 15.8 15.0 14.3 13.0

20.1 19.2 18.4 17.6 16.2

23.8 22.8 21.9 21.0 19.5

27.5 26.5 25.5 24.6 22.9

31.2 30.2 29.2 28.2 26.4

35.0 33.9 32.9 31.9 30.0

38.9 37.7 36.6 35.6 33.6

28 32 36

1.00 0.88 0.79

2.47 2.21 2.00

4.38 3.95 3.58

6.61 5.99 5.46

9.13 11.9 8.33 11.0 7.65 10.1

14.9 13.8 12.8

18.1 16.8 15.7

21.4 20.0 18.7

24.7 23.2 21.9

28.2 26.6 25.1

31.8 30.1 28.5

6.80 6.11 5.53

8.76 10.9 7.89 9.83 7.16 8.95



8 - 87


Pu φ rn


ex

where

3

6

s s

e

s, in.

75° P u

s


4

4

4

12


ex, in.

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6

3.89 3.84 3.79 3.72 3.65

7.75 7.66 7.54 7.40 7.25

11.6 11.5 11.3 11.1 10.8

15.5 15.2 15.0 14.7 14.4

19.3 19.0 18.7 18.3 17.9

23.1 22.7 22.4 21.9 21.5

26.9 26.5 26.1 25.6 25.1

30.8 30.3 29.8 29.3 28.7

34.6 34.1 33.5 32.9 32.4

38.5 37.9 37.3 36.7 36.1

42.3 41.7 41.0 40.4 39.8

46.2 45.5 44.8 44.1 43.5

7 8 9 10 12

3.56 3.47 3.37 3.27 3.07

7.08 6.90 6.71 6.52 6.14

10.6 10.3 10.0 9.77 9.23

14.1 13.7 13.4 13.1 12.4

17.6 17.2 16.8 16.4 15.6

21.1 20.6 20.2 19.8 18.9

24.6 24.1 23.7 23.2 22.3

28.2 27.7 27.2 26.7 25.7

31.8 31.3 30.7 30.2 29.1

35.5 34.9 34.3 33.7 32.6

39.1 38.5 37.9 37.3 36.2

42.8 42.2 41.6 41.0 39.8

14 16 18 20 24

2.87 2.68 2.50 2.34 2.06

5.76 5.40 5.07 4.76 4.23

8.71 8.22 7.76 7.33 6.57

11.8 11.1 10.6 10.0 9.10

14.9 14.2 13.5 12.9 11.8

18.1 17.3 16.6 15.9 14.7

21.4 20.5 19.7 19.0 17.6

24.7 23.8 23.0 22.2 20.7

28.1 27.2 26.3 25.5 23.9

31.6 30.6 29.7 28.8 27.1

35.1 34.1 33.1 32.2 30.4

38.7 37.6 36.6 35.6 33.8

28 32 36

1.82 1.63 1.48

3.78 3.41 3.11

5.94 5.41 4.95

13.5 12.6 11.7

16.4 15.3 14.3

19.3 18.1 17.0

22.4 21.0 19.8

25.5 24.1 22.8

28.7 27.2 25.8

32.0 30.4 28.9

2 3 4 5 6

3.89 3.84 3.79 3.72 3.65

7.74 7.64 7.52 7.38 7.23

11.6 11.4 11.2 11.0 10.8

15.4 15.2 14.9 14.7 14.4

19.3 19.0 18.7 18.4 18.1

23.1 22.8 22.5 22.1 21.8

27.0 26.6 26.3 25.9 25.6

30.9 30.5 30.1 29.7 29.3

35.2 34.4 34.0 33.6 33.2

39.1 38.3 37.8 37.4 37.0

43.0 42.2 41.7 41.3 40.8

47.0 46.1 45.6 45.2 44.7

7 8 9 10 12

3.56 3.47 3.37 3.27 3.07

7.07 6.90 6.73 6.56 6.21

10.6 10.4 10.1 9.92 9.48

14.2 13.9 13.6 13.4 12.9

17.8 17.5 17.2 16.9 16.4

21.5 21.2 20.8 20.5 19.9

25.2 24.9 24.5 24.2 23.6

29.0 28.6 28.3 27.9 27.3

32.8 32.4 32.0 31.7 31.0

36.6 36.2 35.8 35.5 34.7

40.4 40.0 39.6 39.3 38.5

44.3 43.9 43.5 43.1 42.3

14 16 18 20 24

2.87 2.68 2.50 2.34 2.06

5.88 5.57 5.27 4.99 4.50

9.07 8.67 8.29 7.94 7.29

12.4 11.9 11.5 11.1 10.3

15.9 15.4 14.9 14.4 13.6

19.4 18.8 18.3 17.8 16.9

23.0 22.4 21.9 21.3 20.4

26.6 26.0 25.5 24.9 23.9

30.3 29.7 29.1 28.5 27.4

34.1 33.4 32.8 32.2 31.0

37.8 37.1 36.5 35.8 34.7

41.6 40.9 40.2 39.6 38.3

28 32 36

1.82 1.63 1.48

4.08 3.73 3.43

6.73 6.25 5.82

9.67 12.8 9.06 12.1 8.51 11.4

16.1 15.3 14.5

19.4 18.6 17.8

22.9 22.0 21.1

26.4 25.4 24.5

30.0 29.0 28.0

33.6 32.5 31.5

37.2 36.1 35.1

8.30 10.9 7.61 10.0 7.01 9.26


8 - 88


ANCHOR RODS OR THREADED RODS

Cast-in-place anchor rods, illustrated in Figure 8-14, are generally made from unheaded rod material or headed bolt material. Drilled-in anchor rods, illustrated in Figure 8-15, are not normally used; their design is governed by manufacturer’s specifications. Refer also to Cannon, Godfrey, and Moreadith (1981). LRFD Specification Section A3.4 permits the use of unheaded rod material from the following ASTM specifications as anchor rods or threaded rods: A36, A193, A354, A449, A572, A588, and A687. Additionally, LRFD Specification Section A3.4 permits the use of headed bolts conforming to the provisions of LRFD Specification Section A3.3 for use as anchor rods. Headed bolts, however, are generally available only in lengths up to about eight inches. Furthermore, designations such as ASTM A325, A490, and A307 apply only to bolts manufactured with a head and it is, therefore, improper to specify unheaded anchor rods or other similar threaded devices as ASTM A325, A490, or A307. The availability and strength of the aforementioned ASTM specifications for unheaded rod material and headed bolt material are summarized in Table 8-26. Suitable nuts may be selected from ASTM A563 or ASTM A194 grade 7. Because base plates typically have holes larger than oversized holes to allow for tolerances on the location of the anchor rod, washers are usually furnished from ASTM A36 steel plate; they may be round, square, or rectangular, are generally about 1⁄2-in. thick, and generally have holes which are 1⁄16-in. larger than the anchor rod diameter. Minimum Edge Distance and Embedment Length

The recommendations of Shipp and Haninger (1983) for minimum anchor-rod (concrete) edge distance and embedment length for tensile forces, adopted from ACI 349, are summarized in Table 8-26. The edge distance requirement is intended to prevent blow-out of the side of the concrete foundation and is based on concrete with fc′ = 3,000 psi. For edge distance requirements for shear, refer to Shipp and Haninger (1983). In addition to providing the recommended minimum embedment length, anchor rods must extend a distance above the foundation that is sufficient to permit full thread engagement of the nut; from RCSC Specification Section 2(b), “…the end of the [anchor rod] will be flush with or outside the face of the nut when properly installed.”

Lh

(a) Hooked

(b) Headed

(c) Threaded with Nut

Fig. 8-14. Typical cast-in-place anchor rods. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


8 - 89

Note that it is seldom possible to fully tension anchor rods since the concrete usually cannot provide the necessary anchorage. Welding to Anchor Rods

Though not typical, welds must sometimes be used in lieu of nuts to attach anchor rods to base plates. The use of weldable steels such as ASTM A36 or A572 is recommended for this purpose; anchor-rod material which is quenched and tempered should not be welded. Hooked Anchor Rods

Hooked anchor rods should be used only for axially loaded columns to locate and prevent the displacement or overturning of columns due to erection loads or accidental collisions during erection. Additionally, high-strength steels are not recommended for use in hooked rods since bending with heat may materially affect their strength. For the hooked rod of Figure 8-14a, the tensile force is resisted through bond development along the length and the mechanical anchorage of the hook. However, because smooth rods do not always form a reliable bond (due to oil used in threading among other things), the design of such anchor rods should be based upon the anchorage provided by the hook only. To prevent the anchor rod from pulling out and straightening, the hook should be designed to resist one-half the design tensile strength of the anchor rod φRn, where φ = 0.75 Rn = φtFu Ag In the above equation, φt = 0.75. From Fisher (1981), the bearing strength of the concrete is: 0.7fc′dLh

Grout

Fig. 8-15. Drilled-in anchor rods. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 90


Table 8-26. Anchor Rod Material Availability and Strength Availability

Headed Bolt Mat. (Only)

Headed Bolt or Unheaded Rod Material

Unheaded Rod Material (Only)

Type

ASTM Design.

Material Typeb

Strength

Grade

Diameter, d, in.

Proof Load

Min. Yield, ksi

Minimum Min. Embdmt. Minimum Tensile, Length, Edge Dist., in.e ksi in.

A36

C

—

to 8

—

36

58

12d

5d

A572

HSLA

42

to 2

—

42

60

12d

5d

50

to 6

—

50

65

17d

7d

—

to 4

—

50

70

17d

7d

over 4 to 5

—

46

67

17d

7d

over 5 to 8

—

42

63

17d

7d

to 3

—

105

150c

19d

7d

to 21 ⁄2

120

130

150

19d

7d

105

115

140

19d

7d

105

109

125

17d

7d

95

99

115

17d

7d

85

92

120

17d

7d

11 ⁄8 to 11 ⁄2

74

81

105

17d

7d

13 ⁄4 to 3

55

58

90

17d

7d

to 4

—

—

60

12d

5d

85

92

120

17d

7d

74

81

105

17d

7d

120

—

150

19d

7d

A588

HSLA, ACR

A687

A, QT, NT

—

A354

A, QT

BD

5⁄ 1⁄

4

8

over 21 ⁄2 to 4 BC

1⁄

4

to 21 ⁄2

over 21 ⁄2 to 4 A449d

C, QT

—

A307

C

—

A325a,d

C, QT

—

1⁄

1⁄

4

2

to 1

to 1

11 ⁄8 to 11 ⁄2 A490a,d

A, QT

—

1⁄

2

to 11 ⁄2

aAvailable with weathering (atmospheric corrosion resistance) characteristics comparable to ASTM A242 and

A588 steels. bA = Alloy Steel bACR = Atmospheric-Corrosion-Resistant Steel bC = Carbon Steel bHSLA = High-Strength Low-Alloy Steel bNT = Notch-Tough Steel (CVN 15 @ −20°°F) bQT = Quenched and Tempered Steel cMaximum (ultimate tensile strength) dThreaded rod material with properties meeting ASTM A325, A490, and A449 specifications may be obtained

with the use of an appropriate steel (such as ASTM A193, grade B7), quenched and tempered after fabrication. eNot less than 4 in.

Thus, the minimum hook length Lh min is: φRn 2 Lh min = 0.7fc′d where fc′ is the specified strength of the concrete, ksi. The total embedded anchor rod length is then the hook length Lh plus the minimum embedment length from Table 8-26. Headed Anchor Rods

When anchor rods are required for a calculated tensile force Tu, a more positive anchorage is formed when headed anchor rods, illustrated in Figure 8-14b, are used. With adequate embedment and edge distance, the limit state is either a tensile failure of the anchor rod or the pull-out of a cone of concrete radiating outward from the head (Marsh and Burdette, 1985a) as illustrated in Figure 8-16. The design tensile strength of the anchor rod is φRn, AMERICAN INSTITUTE OF STEEL CONSTRUCTION


8 - 91

where φ = 0.75 Rn = φtFu Ag In the above equation, φt = 0.75. Using the projected surface area of the concrete cone and a limiting average stress on this area of 4√ fc′ , the minimum anchor rod length Lmin is Lmin =

√ 

Acp 3.14

where Acp =

Tu φt√ fc′

fc′ = specified strength of the concrete, psi

Tu

45° L

Failure Plane

Projected Surface

Fig. 8-16. Concrete cone subject to pull-out. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 92


Table 8-27. Dimensions and Weights of Clevises

D

p

w

b

a

n

t Grip Grip = plate thickness + ¼ in.

t

Thread: UNC Class 2B

Dimensions, in. Clevis Number

Max. D

2

5⁄

2 1 ⁄2

3

3 1 ⁄2

4 5 6 7 8

7⁄

8

8 13 ⁄8 11 ⁄2 13 ⁄4

2 21 ⁄2 3 4

Max. p 3⁄

4

11 ⁄2 13 ⁄4 2 21 ⁄4 21 ⁄2 3 33 ⁄4 4

b

n

17 ⁄16 5⁄8 21 ⁄2 11⁄8 3 15 ⁄16 31 ⁄2 15⁄8 4 13⁄4 21⁄4 5 6 23⁄4 3 7 4 8

a

w

37 ⁄8 4 5 6 6 7 8 9 10

11 ⁄16 11 ⁄4 11 ⁄2 13 ⁄4 2 21 ⁄2 3 31 ⁄2 4

t

Weight, pounds

Design Strength φR n*, kips

(+ 1 ⁄32, −0) 1 16 (+ ⁄32, −0) 1 ⁄ (+ 1 ⁄ 2 32, −0) 1 ⁄ (+ 1 ⁄ 2 32, −0) 1 ⁄ (+ 1 ⁄ 2 32, −0) 5 ⁄ (+ 1 ⁄ 8 16, −0) 3 ⁄ (+ 3 ⁄ 4 32, −0) 7 ⁄ (+ 1 ⁄ −0) 8 8, 1 1 1 ⁄2 (+ ⁄8, −0)

1 2 4 6 8 16 26 36 80

5.25 11.3 22.5 27.0 31.5 56.4 81.0 103 203

5⁄ 5⁄

16

Notes: Weights and dimensions of clevises are typical; products of all suppliers are essentially similar. User shall verify with the manufacturer that product meets design-strength specifications above. *Tabulated design strengths for comparison with factored loads are based on φ=0.3. To determine safe working load (kips) for comparison with service loads, divide tabular design strength by 1.5. Safe working load, then, corresponds to a 5:1 factor of safety using maximum pin diameter.

Tu = tensile force in the anchor rod, kips When the concrete cone intersects an edge of the pedestal or the cone from another anchor rod, the effective area of concrete is reduced; refer to the AISC Design Guide Column Base Plates (DeWolf and Ricker, 1990) and Marsh and Burdette (1985). Marsh and Burdette (1985) showed that the head of the anchor rod usually provides sufficient anchorage and the use of an additional washer or plate does not add significantly to the anchorage. The nut and threading shown in Figure 8-14c is acceptable in lieu of a bolt head. The nut should be welded to the rod to prevent the rod from turning out when the top nut is tightened. For the design of anchor rods for shear or a combination of tension and shear, see AISC Design Guide Column Base Plates (DeWolf and Ricker, 1990), Fisher (1981), Shipp and Haninger (1983), and ACI 349. OTHER MECHANICAL FASTENERS Clevises

Dimensions, weights, and design strengths of clevises are listed in Table 8-27. Compatability of clevises with various rods and pins is given in Table 8-28. Turnbuckles

Dimensions, weights, and design strengths of turnbuckles are listed in Table 8-29. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

OTHER MECHANICAL FASTENERS

8 - 93

Table 8-28. Clevis Numbers Compatible with Various Rods and Pins Diameter of Pin, in.

Dia. of Tap, in.

5⁄ 8

5⁄ 8 3⁄ 4 7⁄ 8

2 — —

2 21⁄2 21⁄2 21⁄2 21⁄2 21⁄2 21⁄2 21⁄2 21⁄2 21⁄2 — 21⁄2 21⁄2 21⁄2 21⁄2

1 11⁄4 13⁄8

— — —

— — —

— — —

11⁄2 13⁄4

— —

— —

— —

31⁄2 31⁄2 31⁄2 — 4 4

4 5

4 5

5 5

5

2 21⁄4

— —

— —

— —

— —

— —

5 —

5 —

5 6

5 6

5 6

6 6

6 6

7

7

21⁄2 23⁄4

— —

— —

— —

— —

— —

— —

— —

6 —

6 —

6 7

7 7

7 7

7 7

7 8

7 8

3 31⁄4

— —

— —

— —

— —

— —

— —

— —

— —

— —

7 —

8 8

8 8

8 8

8 8

8 8

8 8

31⁄2 33⁄4

— —

— —

— —

— —

— —

— —

— —

— —

— —

— —

8 8

8 8

8 8

8 8

8 8

8 8

4

—

—

—

—

—

—

—

—

—

—

8

8

8

8

8

8

3⁄ 4

7⁄ 8

1

3 3 3

11⁄4 11⁄2 13⁄4

3 3 3

3 3 3

2

3 3 31⁄2 31⁄2 31⁄2

21⁄4 21⁄2 23⁄4

3

31⁄4 31⁄2 33⁄4

4

4

Notes: Tabular values assume that the net area of the clevis through the pin hole is greater than or equal to 125 percent of the net area of the rod, and is applicable to round rods without upset ends. For other net area ratios, the required clevis size may be calculated by reference to the dimensions tabulated in Tables 8-7 and 8-27.

Sleeve Nuts

Dimensions and weights of sleeve nuts are listed in Table 8-30. Recessed-Pin Nuts

Dimensions and weights of recessed-pin nuts are listed in Table 8-31. Cotter Pins

Dimensions and weights of cotter pins are listed in Table 8-32.


8 - 94


Table 8-29. Dimensions and Weights of Turnbuckles c a

n

n

g

e

D

Threads: UNC and 4UN Class 2B

Weight (pounds) for Length a, in.

Dimensions, in. Diameter D, in.

a

3⁄ 8

6

9⁄

1⁄ 2 5⁄ 8 3⁄ 4 7⁄ 8

6 6 6 6

3⁄

32 11⁄16 7 1 ⁄32

1 1 ⁄8 1 1 ⁄4 1 3 ⁄8

1

6 6 6 6

13 ⁄8 19⁄16 13 ⁄4 115 ⁄16

1 1 ⁄2 1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

6 6 6 6

2 1 ⁄4

2

n 16

29 ⁄

4

c

g

6

11 ⁄32

0.41

71 ⁄2 11⁄16 15 ⁄16 713 ⁄16 13 ⁄16 11 ⁄2 81 ⁄8 15 ⁄16 123 ⁄32 87 ⁄16 13 ⁄32 17 ⁄8

0.75 1.00 1.45 1.85

71 ⁄8

83 ⁄4 91 ⁄8 91 ⁄2 97 ⁄8

e 9⁄

16

9

12

18

24

26

Design Strength, φR n *, kips 1.80

0.80 1.38 1.63

1.00 1.50 2.13 2.83

2.43 3.06 4.20

3.30 5.25 7.80 10.8

4.25 5.43

19 ⁄32 21 ⁄32 113 ⁄32 29 ⁄32 19 ⁄16 217 ⁄32 111⁄16 23 ⁄4

2.60 2.72 3.58 4.50

3.20 4.70 4.70

4.40 6.10 7.13

6.85

10.0

11.3

13.1

21 ⁄8 21 ⁄4 21 ⁄2 23 ⁄4

101 ⁄4 127 ⁄32 31 ⁄32 101 ⁄2 131 ⁄32 39 ⁄32 11 21 ⁄8 39 ⁄16 111 ⁄2 23 ⁄8 4

5.50 7.50 9.50 11.5

8.00

9.13

16.8

19.4

15.3

16.0

19.5

6 6

23 ⁄4 33 ⁄8

111 ⁄2 23 ⁄8 123 ⁄4 211⁄16

4 45 ⁄8

11.5 18.0

15.3 35.3

27.5 43.5

55.8 72.0

2 1 ⁄2 2 3 ⁄4

6 6

33 ⁄4 41 ⁄8

131 ⁄2 141 ⁄4

3 31 ⁄4

5 55 ⁄8

23.3 31.5

33.6

42.4 54.0

90.0 113

3 1 ⁄4

3

6 6

41 ⁄2 51 ⁄4

15 161 ⁄2

35 ⁄8 37 ⁄8

61 ⁄8 63 ⁄4

39.5 60.5

145 183

3 1 ⁄2 3 3 ⁄4

6 6

51 ⁄4 6

161 ⁄2 18

37 ⁄8 45 ⁄8

63 ⁄4 81 ⁄2

60.5 95.0

183 252

4 1 ⁄4

4

6 9

6 63 ⁄4

18 221 ⁄2

45 ⁄8 51 ⁄4

81 ⁄2 93 ⁄4

95.0

4 1 ⁄2 4 3 ⁄4

9 9

63 ⁄4 63 ⁄4

221 ⁄2 221 ⁄2

51 ⁄4 51 ⁄4

5

9

71 ⁄2

24

6

14.0 17.4 22.8 26.1 31.5 36.8 42.5 55.8

152

252 351

93 ⁄4 93 ⁄4

152 152

351 351

10

200

442

Notes: Weights and dimensions of turnbuckles are typical; products of all suppliers are essentially similar. User shall verify with the manufacturer that product meets design strength specifications above. *Tabulated design strengths for comparison with factored loads are based on φ = 0.3. To determine safe working load (kips) for comparison with service loads, divide tabular design strength by 1.5. Safe working load, then, corresponds to a 5:1 factor of safety using maximum pin diameter.



8 - 95

Table 8-30. Dimensions and Weights of Sleeve Nuts l

4

½

D

n

Long Dia.

½

Short Dia.

n

c

Inspection hole (optional)

Thread: UNC and 4 UN Class 2B Screw Dia. D, in. 3⁄ 8 7⁄ 16 1⁄ 2 9⁄ 16 5⁄ 8 3⁄ 4 7⁄ 8

1 11⁄8 11⁄4 13⁄8 11⁄2 15⁄8 13⁄4 17⁄8 2 21⁄4 21⁄2 23⁄4 3 31⁄4 31⁄2 33⁄4 4 41⁄4 41⁄2 43⁄4 5 51⁄4 51⁄2 53⁄4 6

Dimensions, in. Short Dia.

Long Dia.

Length l

Nut n

Clear c

Weight, pounds

11⁄ 16 25⁄ 32 7⁄ 8 15⁄ 16 11⁄16 11⁄4 17⁄16 15⁄8 113⁄16

25⁄ 32 7⁄ 8

4 4 4 5 5 5 7 7 71⁄2 71⁄2 8 8 81⁄2 81⁄2 9 9 91⁄2 10 101⁄2 11 111⁄2 12 121⁄2 13 131⁄2 14 141⁄2 15 151⁄2 16 161⁄2 17

— — — — — — 17⁄16 17⁄16 15⁄8 15⁄8 17⁄8 17⁄8 21⁄16 21⁄16 25⁄16 25⁄16 21⁄2 23⁄4 215⁄16 33⁄16 33⁄8 35⁄8 313⁄16 41⁄16 43⁄4 5 51⁄4 51⁄2 53⁄4 6 61⁄4 61⁄2

— — — — — — 1 11⁄8 11⁄4 13⁄8 11⁄2 15⁄8 13⁄4 17⁄8 2 21⁄8 23⁄8 25⁄8 27⁄8 31⁄8 33⁄8 35⁄8 37⁄8 41⁄8 43⁄8 43⁄4 5 51⁄4 51⁄2 53⁄4 6 61⁄4

0.27 0.34 0.43 0.64 0.93 1.12 1.75 2.46 3.10 4.04 4.97 6.16 7.36 8.87 10.4 12.2 16.2 21.1 26.7 33.2 40.6 49.1 58.6 69.2 75.0 90.0 98.0 110 122 142 157 176

2 23⁄16 23⁄8 29⁄16 23⁄4 215⁄16 31⁄8 31⁄2 37⁄8 41⁄4 45⁄8 5 53⁄8 53⁄4 61⁄8 61⁄2 67⁄8 71⁄4 75⁄8 8 83⁄8 83⁄4 91⁄8

1 11⁄16 17⁄32 17⁄16 15⁄8 113⁄16 21⁄16 21⁄4 21⁄2 211⁄16 215⁄16 31⁄8 35⁄16 31⁄2 315⁄16 43⁄8 413⁄16 51⁄4 55⁄8 6 63⁄8 67⁄8 71⁄2 715⁄16 83⁄8 87⁄8 91⁄4 93⁄4 101⁄8 105⁄8

Notes: Weights and dimensions of sleeve nuts are typical; products of all suppliers are essentially similar. User shall verify with the manufacturer that strengths of sleeve nut are greater than the corresponding connecting rod when the same material is used.


8 - 96


Table 8-31. Dimensions and Weights of Recessed-Pin Nuts

Short Dia.

Long Dia.

Tc

Grip

d

D s t

Material: Steel

Thread: 6 UN Class 2A/2B

Pin Dimensions, in. Thread

Pin Dia. d, in.

D

T

2, 2 1 ⁄4 2 1 ⁄2 , 2 3 ⁄4 3, 3 1 ⁄4 , 3 1 ⁄2 3 3 ⁄4 , 4 4 1 ⁄4 , 4 1 ⁄2 , 4 3 ⁄4 5, 5 1 ⁄4 5 1 ⁄2 , 5 3 ⁄4 , 6 6 1 ⁄4 , 6 1 ⁄2 6 3 ⁄4 , 7 7 1 ⁄4 , 7 1 ⁄2 7 3 ⁄4 , 8, 81 ⁄4 8 1 ⁄2 , 8 3 ⁄4 , 9 9 1 ⁄4 , 9 1 ⁄2 9 3 ⁄4 , 10

11 ⁄2 2 21 ⁄2 3 31 ⁄2 4 41 ⁄2 5 51 ⁄2 51 ⁄2 6 6 6 6

1 11 ⁄8 11 ⁄4 13 ⁄8 11 ⁄2 15 ⁄8 13 ⁄4 17 ⁄8 2 2 21 ⁄4 21 ⁄4 23 ⁄8 23 ⁄8

c

Grip

½

d+1

d Bolt ¾

1⁄ 1⁄ 1⁄ 1⁄ 1⁄ 1⁄ 1⁄ 3⁄

3⁄ 3⁄ 3⁄ 3⁄ 3⁄ 3⁄

8 8 8 4 4 4 4 8 8 8 8 8 8 8

Nut Dimensions, in. Thickness t 7⁄

8

1 11 ⁄8 11 ⁄4 13 ⁄8 11 ⁄2 15 ⁄8 13 ⁄4 17 ⁄8 17 ⁄8 21 ⁄8 21 ⁄8 21 ⁄4 21 ⁄4

Diameter Short Dia. 3 35 ⁄8 43 ⁄8 47 ⁄8 53 ⁄4 61 ⁄4 7 75 ⁄8 81 ⁄8 85 ⁄8 93 ⁄8 101 ⁄4 111 ⁄4 111 ⁄4

Recess

Long Dia. Rough Dia. 33 ⁄8 41 ⁄8 5 55 ⁄8 65 ⁄8 71 ⁄4 81 ⁄8 87 ⁄8 93 ⁄8 10 107 ⁄8 117 ⁄8 13 13

25 ⁄ 8 31 ⁄ 8 37 ⁄ 8 43 ⁄ 8 51 ⁄ 4 53 ⁄ 4 61 ⁄ 2 7 71 ⁄ 2 8 83 ⁄ 4 95 ⁄ 8 105⁄8 105⁄8

s 1⁄ 1⁄ 3⁄ 3⁄ 1⁄ 1⁄ 5⁄

5⁄ 3⁄

3⁄ 3⁄ 3⁄ 3⁄ 3⁄

4 4 8 8 2 2 8 8 4 4 4 4 4 4

Weight, pounds 1 2 3 4 5 6 8 10 12 14 19 24 32 32

Notes: Although nuts may be used on all sizes of pins as shown above, a detail similar to that shown at the left is preferrable for pin diameters over 10 inches. In this detail, the pin is held in place by a recessed cap at each end and secured by a bolt passing completely through the caps and pin. Suitable provisions must be made for attaching pilots and driving nuts.

¾

Typical Pin Cap Detail for Pins over 10 in. in dia. Dimensions shown are approximate



8 - 97

Table 8-32. Dimensions and Weights of Cotter Pins HORIZONTAL OR VERTICAL PIN 3/8

HORIZONTAL PIN 1″

1

GRIP + ½

p

d

h

GRIP + 1 ″

1″

d

c

l l = Length of pin, in.

Pins with Heads Head Diameter h, Pin Diameter d, in. in.

11⁄4 11⁄2 13⁄4 2 21⁄4 21⁄2 23⁄4 3 31⁄4 31⁄2 33⁄4

11⁄2 13⁄4 2 23⁄8 25⁄8 27⁄8 31⁄8 31⁄2 33⁄4 4 41⁄4

Weight of One, pounds

0.19 + 0.35l 0.26 + 0.50l 0.33 + 0.68l 0.47 + 0.89l 0.58 + 1.13l 0.70 + 1.39l 0.82 + 1.68l 1.02 + 2.00l 1.17 + 2.35l 1.34 + 2.73l 1.51 + 3.13l

Cotter Length c, in.

Diameter p, in.

Weight per 100, pounds

7⁄

1⁄ 4 1⁄ 4 1⁄ 4 3⁄ 8 3⁄ 8 3⁄ 8 3⁄ 8 1⁄ 2 1⁄ 2 1⁄ 2 1⁄ 2

2.64 3.10 3.50 9.00 9.40 10.9 11.4 28.5 28.5 33.8 33.8

8

1 11⁄8 11⁄4 13⁄8 11⁄2 15⁄8 13⁄4 17⁄8 17⁄8 21⁄4


8 - 98


WELDED CONSTRUCTION

While AWS D1.1 is the traditional design specification for weld stresses in both buildings and bridges, AASHTO/AWS D1.5 also exists for dynamically loaded structures. There are significant differences between the two codes and, in the case of building structures, AWS D1.1 is normally used unless contract documents state otherwise. Welds in building structures are predominantly designed for static loading. Some parts, however, such as crane runways and machinery supports, are subjected to dynamic loading. When this is the case, additional requirements and special joint details may be necessary. This may include reinforcing fillet welds at tee and corner joints, radius cuts on terminations of gusset type connections, radiographic or ultrasonic testing for quality control, or joint details in accordance with LRFD Specification Appendix K3. The contract documents should specifically enumerate these additional requirements when they are determined to be necessary. Weldability of Steel

AWS has defined weldability as the capacity of a metal to be welded, under the fabrication conditions imposed, into a specific, suitably designed structure, and to perform satisfactorily in the intended service. AWS D1.1 is based on certain weldable grades of steel as listed therein by ASTM designation. It contains all of the steels permitted by LRFD Specification Section A3.1a. The effect a steel’s properties have upon its weldability relates to the reaction of the steel to the drastic heating and cooling cycle of welding. This weld quench can range from the practically instantaneous cooling of an accidental arc strike to the 10 minutes required to cool a high-heat-input electroslag weld. Due to the rapid cooling of the arc strike, the full-quench hardness for the carbon equivalent of the steel may be realized, resulting in brittleness and the potential for cracking. In contrast, the slower cooling rate of the electroslag weld may produce a more ductile and lower-strength metallurgical structure in the heat-affected zone (HAZ) of the base metal. As they cool, welds develop residual shrinkage strains that can approach the yield strain as a limit; ductility and notch resistance are needed to accommodate these strains. Since chemical composition, grain size, and thickness affect ductility and notch resistance, they are the most important properties for weldability. These factors, discussed below, assume greater significance as the structure becomes large and must store greater elastic energy. Table 8-33 summarizes several ASTM specifications and their requirements for the aforementioned properties. Note that there is a greater flexibility in grain size and carbon equivalents in these specifications for shapes, plates, and bars. Also, maximum tensile strength requirements are listed to exclude steels from the upper end of the chemical composition range which might require special welding procedures or weld repairs. In contrast, the requirements for structural tubing, pipe, sheet, and strip do not limit grain size or maximum tensile strength, but generally impose smaller limits on thickness. Chemical analysis of a heat of steel is usually made during the processing as a control and upon completion after it has been tapped into a ladle. This heat analysis is used to compile a mill test report which also lists the customer’s order number, steel grade, quantity and dimension of pieces shipped, and the results of any mechanical testing (tensile, flexural, Charpy impact, or other). This information may be obtained by request from the steel supplier when placing an order and is essential for good control of welded fabrication. It is imperative that the grade of steel to be welded is known since the proper welding procedure depends upon this information. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 99

Table 8-33. ASTM Requirements for Properties Affecting Weldability of Steels Max. Carbon content, % by weight (heat analysis)

Max. tensile strength, ksi

Grain Size

Max. thickness, in.

80

—*

none

ASTM Specification

Products Covered

A36

shapes

0.26

plates

0.25–0.29

A242 A514

bars

0.26–0.29

shapes, plates, bars

type 1, 0.15

plates—quenched varies among 13 and tempered grades, 0.14–0.21

none

—

4

130

fully killed, fine grain

6 1⁄

A529


0.27

85

—

A572

shapes, plates, bars, sheet piling

varies among grades, 0.21–0.26

none

—*

Gr. 42: 6 Gr. 50: 4 Gr. 60, 65: 11 ⁄4

A588


varies among 5 grades, 0.15–0.20

none

fine grain

F y = 50: 4 F y = 42: 8

2

A852

plates

0.19

110

fine grain

4

A53 Grade B

tubing, pipe

0.30

none

—

2.344, 24 dia.

A500

tubing, pipe

Gr. A, B: 0.26 Gr. C: 0.23

none

—

A501

tubing, pipe

0.26

none

—

1

A618

tubing, pipe

Gr. Ia: 0.15 Gr. Ib: 0.20 Gr. II: 0.22 Gr. III: 0.23

none

—

11 ⁄2

A570, Gr. 36, 50

sheet, strip

0.25

none

—

0.23

A606

sheet, strip

0.22

none

—

none

A607

sheet, strip

0.22–0.26

none

—

none

1⁄

2

*Supplemental requirements can specify killed fine grain.

Chemical Composition

The most important element affecting weldability is carbon, however, the effect of other elements on weldability is related through a carbon equivalent formula. Weldability is enhanced as carbon equivalent decreases because the maximum hardness and consequent brittleness that a steel may reach after rapid liquid quenching from high temperature is directly proportional to the carbon equivalent. This relationship is illustrated in Figure 8-17 and is applicable to the surface in contact with the quench liquid where the quench rate is greatest. Although no liquid is present in welding, the HAZ is subject to rapid cooling and consequent hardening by conduction of weld heat into the base metal. As the thickness of the section increases, so does the cooling rate, producing progressively harder and less ductile metallurgical constituents. Alloys such as Ni, Cr, and Mo in the steel permit hardening at slower cooling rates and at depths below the surface where the cooling rate is slower; pre-heat is the common remedy for reducing the cooling rate and hardness. As the carbon content increases from 0.10 percent to 0.20 percent by weight, the maximum as-quenched hardness increases from 40 to 50 Rockwell C. Using the known hardness-strength relationship, it can be shown that the maximum as-quenched tensile AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 100


strength increases from 180 to 260 ksi. Welding procedures are designed to keep weld quench rates far below these maximum rates. Also, electrodes are usually designed to deposit weld metal containing about 0.008 to 0.12 percent carbon to avoid cracking. Grain Size

In general, weldability will be enhanced by steel with a finer grain size. As illustrated in Figure 8-18, grain size is a prime variable affecting the ductility and impact resistance for a wide variety of steel compositions. The grain size of weld metal also varies and has a similar effect. Because they experience a slower cooling rate, high-heat-input welds show a larger grain size than the same process and electrode at a lower heat input. This is one reason the AWS D1.1 limits multi-pass SAW groove weld layers to a maximum size of 1⁄4-in. Also, a subsequent pass will refine the grain of a previous pass. Thickness

In general, as the thickness to be welded decreases, the weldability of the material is enhanced. Because of their greater mass, thick plates extract heat from and quench the weld more rapidly than thin plates with the identical weld. As a partial remedy, the plates may be pre-heated and held at temperatures of a few hundred degrees Fahrenheit for the welding operation. This pre-heat appreciably slows the quench rate and reduces weld hardness, as does post-heating. As plate thickness increases, the notch impact resistance decreases as shown in Figure 8-19. This test was conducted on American Bureau of Shipping (ABS) class C ship plate in 3⁄4-in., 1-in., 2-in., and 3-in. thicknesses using a severe crack-like notch in the ASTM A208 drop-weight test. The use of fine-grain steelmaking practice as specified by ASTM can improve notch toughness where required by the service of a particular structure.

60 255

50

Maximum hardness for carbon and alloy steels

40

180 140

30 20

Equivalent tensile strength, ksi

Maximum hardness, Rockwell, C

70

10

0

0.20

0.40

0.60

0.80

1.0

Carbon, per cent

Fig. 8-17. Influence of carbon content on the maximum hardness of steel as quenched (Stout and Doty, 1978), courtesy Welding Research Council. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 101

Structural Welding Materials and Processes

Filler metal and flux specifications are exclusively AWS specifications, having been removed from ASTM specifications. Additionally, AWS uses a coding system for consumable electrodes to designate the tensile strength and coating or flux combination. Since the coding for the several filler/flux combinations are consistent only with respect to the types of electrode used, it is very important that the applicable specifications be reviewed when specifying such welding requirements. The welding processes discussed in this text are: shielded metal arc welding (SMAW), submerged arc welding (SAW), gas-metal arc welding (GMAW), flux-cored arc welding (FCAW), electroslag welding (ESW) and electrogas welding (EGW). Except for electroslag welding, each of these processes use electrical energy from an arc discharge between a steel-wire electrode and the base metal to provide heat for fusion. Electroslag welding uses a high-electrical-resistance molten-slag bath which occupies the entire joint. This slag melts both the electrode and the base metal.

40

CVN Transition Temperature, °F

0

– 40

– 80

c c c

–120

–160

— Plain Carbon c — Nickel — Manganese — Molybdenum — Chromium

c

–200

(After Kottcamp and Stout) 9

8

7

6

5

4

ASTM Ferrite-grain Size Number

Fig. 8-18. Effect of ferrite-grain size on CVN transitional temperature (Stout and Doty, 1978), courtesy Welding Research Council. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3

8 - 102


Each of the aforementioned processes will be summarized here; a full description may be found in AWS (1978). Additionally, thermal cutting and air arc gouging will be discussed. SMAW

There are two AWS Specifications for SMAW electrodes: AWS A5.1 and AWS A5.5. A condensation of the provisions of these specifications is given in Table 8-34. AWS notation for SMAW electrodes is illustrated in Figure 8-20. This has also been extended to other processes. The welding positions noted in Figure 8-20 (flat, horizontal, vertical, and overhead) are illustrated in Figure 8-21. SMAW (stick) electrodes are made in a variety of low-carbon compositions. The extruded coatings contain aluminum, silicon, and other deoxidizers; the deposited weld is a mini-electric-furnace-killed steel with excellent ductility and resistance to cracking from weld shrinkage strains. In the arc stream, moisture breaks down and liberates atomic hydrogen which is readily soluble in molten iron (Stout and Doty, 1978); see Figure 8-22. As the weld solidifies,

70

Increase in NDT Transition Temperature, °F

60

50

40

30

20

ABS-C, Drop-Weight NDT

10

0 0

1

2 Plate Thickness, in.

3

Fig. 8-19. Effect of plate thickness on the drop-weight NDT ductility transition temperature (Stout and Doty, 1978), courtesy Welding Research Council. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 103

Table 8-34. Condensed AWS Specifications for SMAW Electrodes Electrode

Type

AWS Spec.

Carbon Steel

A5.1

Low Alloy

A5.5

Impact Test Criteria Criteria for Min. Criteria for Tensile Composition Strength, of Deposited Charpy V- Weld Metal Radiographic Grades Weld Metal Notch Test Condition Soundness ksi 60

62

70

72

70 80 90 100 110 120

70 80 90 100 110 120

Not stipulated Required for As-welded some grades Stipulated only Stipulated (all grades)

Required for Some assome grades welded, only some stressrelieved

Stipulated for all but E6012, E6022 Stipulated for all grades

Note: A particular production welding condition may be more severe than the test conditions specified for the above.

hydrogen becomes much less soluble and the atoms are rejected into voids where pairs combine to form a much less mobile molecular H2. This molecular hydrogen can then exert pressure in lattice imperfections which is sufficient, when combined with weld shrinkage strains, to cause “fisheyes” or cracking in the weld material. This can be prevented by maintaining the moisture content of consumable electrodes below specified levels and through proper pre-heat. E7015, E7016, E7018, and E7028 low-hydrogen electrodes have specially compounded and baked extruded coatings containing a limited moisture (hydrogen) content by weight. Coatings for the E70 electrode series can contain a maximum of 0.04 percent moisture, while the E120 electrode series is limited to only 0.015 percent. As the tensile strength of the base metal increases, electrodes with lower moisture content must be selected to avoid weld cracking. Since the electrode coating will absorb moisture when stored in damp or humid conditions, drying ovens near points of use in the shop are necessary for low-hydrogen electrodes.

6

—

A1 Weld metal composition

1

Coating characteristics**

70 70,000 psi min. tensile

Electrode

E

Position code*

ELECTRODE PROPERTIES

**1 = All flat, vertical, overhead, and horizontal **2 = Flat and horizontal only **5, 6, 8 = Low hydrogen

Fig. 8-20. AWS classification system for SMAW electrodes. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 104


Fillet Welds

Groove Welds

(a) Flat

(b) Horizontal

(c) Vertical

(d) Overhead Fig. 8-21. Welding positions. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 105

The electrodes to be used with various base metals are shown in AWS D1.1 Table 4.1. Low-hydrogen electrodes are used with ASTM A572 and A588 steels among others. Filler metal matching the color of ASTM A588 steel is listed in AWS D1.1 Table 4.2. SAW

The automatic and semi-automatic SAW processes provide consistent, high quality, and economical deposits which are particularly suitable for long welds. Their major limitation is that the work must be positioned to allow for near flat or horizontal welding. In the SAW process, fluxes may be fused or agglomerated (finely powdered constituents bonded together with silicates), but are classified in AWS specifications only according to the weld metal properties produced in the standard specified weld tests. The applicable specifications are: AWS A5.17 and AWS A5.23. AWS notation for SAW electrodes and fluxes is illustrated in Figure 8-23. Fluxes must be kept dry in storage to avoid an increase in moisture content and subsequent chance of hydrogen cracking in steels with higher yield strengths or highly restrained joints in thick members.

Hydrogen, cm.3 per 100 g. of iron 5

10

15

20

25

30

2100

2500 Austenite

Delta Iron

1 atmos.

1100

1500

600 Ferrite

Temperature, °C.

1600

500

100 0

0.00075

0.0015 Hydrogen, percent

0.00225

Fig. 8-22. Solubility of hydrogen in iron (Stout and Doty, 1978), courtesy Welding Research Council. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

0.003

Temperature, °F.

Liquid

3500

8 - 106


GMAW

The GMAW process can be used with mixtures of argon and two percent oxygen, argon and carbon dioxide, or pure carbon dioxide. While argon is inert, carbon dioxide can react with the weld metal and result in a reduction in ductility and impact properties at low temperatures. Despite this, 70 ksi electrodes have commonly been used with carbon dioxide gas with good results; a CVN 20 (20 ft-lb Charpy V-notch impact value) at −20°F is specified in the AWS tests. Alloy electrodes producing up to 120 ksi minimum tensile strength with CVN 20 at −60°F, and three percent nickel electrodes producing 80 ksi minimum tensile strength with CVN 20 at −100°F are available. There are two AWS Specifications for GMAW electrodes: A5.18 and A5.28. Identification of these electrodes is illustrated in Figure 8-24. FCAW

FCAW electrodes are made by forming a thin sheet strip into a U-shape and filling it with flux. After closing the tube, it is drawn to size as a continuous coil. AWS classifies these

M

12

K

Silicon killed

Medium Mn (1.00% ± )

E

Nominal carbon (0.12%)

6

Electrode

A


CVN 20 @ − 60° F

7

Tested as welded

Flux

F

70,000 psi min. tensile

FLUX CAPABILITY

Fig. 8-23. AWS classification system for SAW materials.

ELECTRODE PROPERTIES B2

L Low carbon (0.05% max.)

—

Cr (11/ 4 %); Mo (1/ 2 %)

S

Solid electrode


R

Rod*

Electrode

E

*Can be used as feed rod with independent heat source (e.g., tungsten arc)

Fig. 8-24. AWS classification system for GMAW electrodes. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 107

electrodes according to: (1) whether or not carbon dioxide is used as a separate shielding gas; (2) suitability for either single or multiple pass applications; (3) the type of current; (4) the welding position; and, (5) the as-welded mechanical properties of the weld metal. High weld-production rates may be attained with semi-automatic equipment which may be used in any position with the appropriate electrode. Where required by service conditions, flux-cored electrode grades can provide weld metal with CVN 20 impact values at temperatures in steps from 20°F to −100°F. Some of the deposits of the carbon steel electrodes will develop CVN 20 at −20°F, while the low alloy electrodes will develop CVN 20 at −100°F. The applicable specifications are AWS A5.20 and AWS 5.29 (symbols are similar to AWS 5.20 with the addition of an alloy composition at the extreme right). The AWS classification system is illustrated in Figure 8-25. ESW and EGW

With the ESW and EGW processes, 18-in. and greater thicknesses may be welded in one pass, using multiple electrodes, with the joint in a vertical plane. A single-electrode, semi-portable welding machine can join plates up to five inches thick. Furthermore, using either of these processes, it is possible to make girder flanges by welding mill-width plates and subsequently longitudinally cutting out three or more flange widths. Note that AWS prohibits the use of these welding processes on quenched and tempered steels. The composition of cored electrodes is based on weld-metal analysis, and the composition of solid electrodes is based on wire analysis. The coarse grains in the slow-cooled electroslag weld may make it difficult to test ultrasonically and the minimum size flaw detectable by RT is about 11⁄2 percent of the thickness. This creates difficulty in the inspection of electroslag welding. AWS A5.25 requires electrodes which contain nickel to provide CVN 15 impact values at either 0°F or −20°F. This specification is patterned after AWS A5.17 and A5.18 insofar as the electrodes are concerned; refer to Figure 8-26.

1

—

2 Usability code**

T

Position code*


Electrode

E

Tubular (flux cored)


**2 = Flat and horizontal only **1 = All position **2 = Single pass CO2 shielded only

Fig. 8-25. AWS classification system for FCAW electrodes. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 108


Thermal Cutting and Air-Arc Gouging

Thermally cut welding bevels are required to be smooth and free of notches or grooves in which weld slag may be trapped. Two cutting systems, oxy-fuel gas and plasma arc, are available. Oxy-fuel gas cutting may be used to cut almost any plate thickness commercially available except in stainless steel which must be plasma cut. Plasma arc cutting will cut thicknesses only up to about 11⁄2-in., but is much faster than oxy-fuel gas cutting. This speed advantage increases as the plate thickness decreases; at a thickness of one inch, the cutting speed is over 300 percent faster with a water-injection plasma torch. The plasma arc cutting process, however, also leaves a slight taper in the cut as it descends. If the plate being cut contains large discontinuities or non-metallic inclusions, turbulence may be created in the oxy-fuel cutting stream. As result, this may cause notches or gouges in the edge of the cut. The plasma arc stream is less susceptible to this as it moves with a higher velocity. Within the depth limits of the specifications, it is usually better practice to remove these by grinding than to weld repair and grind. Additionally, re-entrant thermal cuts should provide a smooth transition. Carbon-air-arc gouging is a convenient method for removing weld defects, gouging the weld root to sound metal, or forming a U-groove on one side of a square butt joint. The carbon arc travels over the work and melts a weld-nugget-shaped area of the metal. This molten material is then blown away by a jet of compressed air, directed from the holder, parallel to the carbon electrode. Thus, air-arc gouging may be considered the opposite of welding in that each pass removes approximately one weld pass. Because the arc quench is similar in both air-arc gouging and welding, any pre-heat required for welding should also be used for air-arc gouging. Inspection

The five most commonly used testing methods for welding inspection are: visual (VT), dye penetrant (DPT), magnetic particle (MT), radiographic (RT), and ultrasonic (UT). These methods are discussed in the following sections; refer also to AWS B1.0. Visual inspection is the most commonly specified procedure. Other, more stringent methods can

Electrode

W

Fig. 8-26. AWS classification system for ESW materials. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

T

1

CNV 15 @ 20°F

E


2

Weld metal tested as deposited

7

CNV 15 @ 20°F

ES

Electroslag flux

Flux

F



FLUX CAPABILITY

WELDED CONSTRUCTION

8 - 109

add significant cost to the project and, therefore, should be specified only when essential to the integrity of the structure. The engineer of record (EOR) must specify in the contract documents which type of weld inspection is required as well as the extent and application of each type. In the absence of instruction, AWS D1.1, paragraph 6.6.5 states that the fabricator or erector is responsible only for those weld discontinuities found by visual inspection. If additional inspection more stringent than visual is later required, the owner is normally responsible for the cost of weld repairs other than those identified by the visual inspection. VT

Visual testing provides the most economical approach to checking weld quality. It is particularly good for inspecting single-pass welds, but is limited in that only surface imperfections may be detected. This type of inspection is especially effective when it includes both a check of the joint for accuracy and cleanliness before welding and an observation of the welding procedure. Acceptance criteria are specified in the AISC Code of Standard Practice and Quality Criteria and Inspection Standards (AISC, 1988), as well as AWS D1.1. DPT

A red dye penetrant is applied to the work and penetrates any crack or crevice open to the surface. After removing excess dye, a white developer is applied. Where cracks are present, the red dye seeps through the developer, producing a visible red image. This process is summarized in Figure 8-27. DPT may be used to detect tight cracks as long as they are open to the surface. Like VT, however, only surface cracks are detectable. Furthermore, deep weld ripples and scratches may give a false indication when DPT is used. MT

A magnetizing current is introduced into the weldment to be inspected as shown in Figure 8-28. The magnetic field induced in the work will be distorted by any cracks, seams, inclusions, etc., located on or within approximately 1⁄10-in. of the surface. A dry magnetic powder spread lightly on the surface will gather at such discontinuities, leaving a distinct mark. These magnetically held particles then show the size, location, and shape of the discontinuity. This method will detect surface cracks filled with slag or contaminants which dye in DPT could not enter. Additionally, the powder may be picked up and preserved with clear

Visible Indication

Subvisible Crack

Cleaned Surface

Penetrant Applied

Excess Removed

Fig. 8-27. Schematic diagram of DPT. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Developer Applied

8 - 110


tape, providing accurate and detailed records of inspection results. However, this method requires relatively smooth surfaces and while cleanup is easy, demagnetization, when necessary, may not be. RT

This method uses a radioactive source and an X-ray film process. RT can detect porosity, slag, voids, cracks, irregularities, and lack of fusion. To be detected, the imperfection must be oriented roughly parallel to the impinging radiation beam and occupy about 11⁄2 percent of the metal thickness along that beam. The film negative provides a permanent record of the inspection. Defects smaller than about 11⁄2 percent of the metal thickness and defects not parallel to the beam may not register. RT of closed, inaccessible pipe joints is difficult to obtain and interpret and should be discouraged. Additionally, when the particle beam must penetrate varying thicknesses, as at fillets and tee or corner joints, RT is not readily interpreted and the resulting inspection may be less consistent. When this is the case, other inspection methods should be used. Other limitations of RT are that the required exposure time increases with material thickness and there is a worker hazard due to the radiation used in the method. The precautions for avoiding these hazards and the equipment and film costs make this method the most expensive inspection method. UT

This process, illustrated in Figure 8-29, is analogous to radar and operates on a principle called pulse-echo. A short pulse of high-frequency sound is introduced into the metal. The reflection of this sound wave from the far end of the member and any voids encountered along the way may then be detected. Any reflections are displayed as pips on a display in which the horizontal grid represents the distance through the metal, and the vertical scale represents the area, and therefore the strength, of the reflecting surface. The point of origin of the sound wave can be readily moved around to check many orientations and can project the wave into the metal at angles of 90°, 70°, 60°, and 45°. While UT can detect favorably oriented, flat discontinuities smaller than 1⁄64-in. in carbon and low-alloy structural steels, austenitic stainless steels and extremely coarse-

Current

Part

Magnetic Field

Fig. 8-28. Schematic diagram of MT. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 111

grained steels such as electroslag weld metal are difficult to inspect. Also, certain joint geometry limits the use of UT and it is difficult to inspect members less than 5⁄16-in. thick because there is a “dead area” at the origin of the sound wave. The accuracy of UT depends upon the skill and training of the operator and frequent calibration of the instrument. ASNT has set training standards for UT operators. Despite the fact that UT is a more versatile, expedient, and economical inspection method than RT, it does not provide a permanent record like the X-ray negative in MT. Instead the operator must make a written record of discontinuity indications. For more information, see Krautkramer (1977) and Institute of Welding (1972). Economical Considerations

On a weight basis, the cost of weld metal far exceeds the cost of any other material in a structure. Therefore, in addition to designing joints for the best welding position, significant economy can be achieved by selecting the proper weld type and an arrangement for the welds which requires a minimum amount of weld metal and the least amount of deposit time. Acceptance of prior qualification of welding procedures can also result in a more economical structure.

Good Bond

Slag Inclusion

Crack or Incomplete Fusion Fig. 8-29. Variations in UT reflections due to differences in acoustic properties caused by defects at the boundary. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 112


Welding Position

When weld metal is deposited in the flat position, it can be deposited more quickly since gravity does not adversely affect the deposit. As a result, large electrodes and high currents may be used. In the vertical and overhead positions, electrode diameters above 5⁄ -in. produce weld pools with surface tensions and arc forces which are unable to 32 overcome the pull of gravity, causing the weld metal to run. Since the deposition rate in the flat position and in the horizontal position for single-pass fillet welds (not greater than 5⁄ -in.) is approximately four times faster than that in the vertical or overhead position, 16 there is strong economic incentive to design and position work for welding in the flat or horizontal position. Weld Type

In general, in the flat position, the SAW, GMAW, or FCAW processes will be more economical than the SMAW process. However, the selection of the welding process should be left to the fabricator since the equipment and training of personnel will vary from one shop to another. It is appropriate, though, for the designer to specify the type of weld to be used, e.g., fillet, groove, etc. The fillet weld will be most economical and should generally be selected instead of the groove weld in applications for which groove welds are not required. Additionally, fillet welds result in lesser distortion of the connected material. There are, however, situations, such as joints subjected to fatigue loading, in which the performance of the groove weld is superior. Complete-joint-penetration groove welds may incur the additional costs of non-destructive testing, backgouging, or backing bars; refer to Alexander (1991). Fillet welds around the inside of a hole or slot require less weld metal than plug or slot welds of the same size. It should be noted, however, that the diameters of holes and widths of slots for fillet welds should be somewhat larger than those for plug and slot welds in metal of the same thickness to accommodate the necessary tilt of the electrode. Weld Metal Volume

Welds which are oversized waste weld metal and labor time, resulting in an unnecessary increase in the cost of the connection. Thus, it is important to use the proper weld size required for strength or based upon the minimum weld size from the LRFD Specification and to not over-specify weld size. While the strength of a fillet weld is in direct proportion to its size, the volume of the weld metal increases as the square of the weld size. Thus, a 5⁄8-in. fillet weld is twice as strong as a 5⁄16-in. fillet weld but also four times more costly. For this reason, it is more desirable to specify a smaller-sized and longer weld than a larger-sized and shorter weld. In groove welds, double-bevel, double-V, double-J, and double-U welds are typically more economical than single welds of the same type since they use less weld metal. As an added benefit, the resulting symmetry results in less rotational distortion strain. Double welds, however, require more labor in edge preparation and proper cleaning of the weld root prior to commencing the weld on the second side. There may also be added cost if the piece must be repositioned to perform the weld on the second side. For this reason, many fabricators prefer a single weld in thicknesses up to about one inch. Where single- or double-groove welds are to be used, bevel- and V-groove welds are usually less expensive since they may be flame cut; J- and U-groove welds are more expensive since they must be planed or air-arc gouged. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 113

Deposit Time

Fillet welds sizes up to 5⁄16-in. may be deposited in a single pass when deposited in the flat or horizontal position. Larger-size welds must be deposited in multiple passes which will require appreciably more time and weld metal. Thus, fillet welds sized not greater than 5⁄16-in., where possible, will result in a significant savings in deposit time, weld material, and cost. Prior Qualification of Procedures

Evidence of prior qualification of welding procedures, welders, welding operators, or tackers may be accepted at the discretion of the engineer of record (EOR). Fabricators certified in the AISC Quality Certification Program have the experience and documentation necessary to assure that the EOR could accept such prior qualifications (refer to Part 6 for a description of the AISC Quality Certification Program). Significant economic savings may be achieved by accepting such prior qualifications. Minimizing Weld Repairs

Added cost in the form of weld repairs or replacement may be minimized if the designer considers the possibilities of lamellar tearing, fatigue cracking, notch development, and reduced impact toughness when designing welded connections. Lamellar Tearing

A lamellar tear is a separation or crack in the base metal caused by through-thickness weld shrinkage strains. When steel is hot-rolled, sulphides or other inclusions are elongated to form microscopic platelets in the plane of the steel plate. These inclusions reduce the strength of the steel in the through-thickness direction below that in the longitudinal or transverse direction. While special practices are available to produce low-sulphur steel which is resistant to lamellar tearing and ASTM A770 provides a testing method by which the throughthickness strength of the base metal may be measured, it is difficult to assure freedom from the possibility of lamellar tearing. Lamellar tearing is a phenomenon which can occur even in material with superior mechanical properties. Instead, the joint detail is most important in preventing lamellar tearing. Some joint designs are inherently susceptible to lamellar tearing (AISC, 1973). For example, the complete-joint-penetration groove-welded tee joints in thick sections shown in Figure 8-30 can develop lamellar tears in the crossbar of the tee flange. Such tears can be detected with UT. Other susceptible joints are shown with improved details in Figures 8-31 and 8-32. The probability of lamellar tearing may be minimized through good joint design and proper welding procedures. The joint design should minimize the weld size and, therefore, the resulting shrinkage strains. Additionally, the design should reduce the restraint which intensifies the local strains. The welding procedure should then establish a sequence to minimize component and internal restraint. Welding with low-hydrogen processes and effective pre-heat has also been shown to minimize lamellar tearing (Kaufmann, Pense, and Stout, 1981). Fatigue Cracking

Because of their inherent rigidity, welded members are subjected to severe restrictions at service loads if subjected to the repeated variations in stress (fatigue loading). In a AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 114


dynamically loaded structure, fatigue cracks at notches progress at a rate proportional to the stress range and to the number of stress cycles. Gradual transitions of sections will help to alleviate these concentrations. The fatigue resistance of a butt weld in a tension member, for example, can be improved approximately 25 percent by grinding the weld reinforcement flush. Thus, any notches in the tension areas should be ground out. Additionally, all grinding should be done in the direction of the stress. Refer to LRFD Specification Appendix K3 for further information. Notch Development

When subjected to lateral movement, a severe notch can result at locations of one-sided welds. For the fillet-welded joint subjected to lateral loading in Figure 8-33, the unwelded side has no strength in tension and a notch may form from the unwelded side. Using one fillet weld on each side will eliminate this condition. This is also true with partial-jointpenetration groove welds. In the case of the backing bar of Figure 8-34a, the location of the tack welds may cause fatigue notches. An improved detail would be as shown in Figure 8-34b, where the backing bar is tack welded inside the groove. Any undercut would then be filled, or at least backed up, by the final weld joint. This is also applicable in the case of box members with corner backup. Note that backing bars should also be continuous throughout the length to avoid discontinuities at the base of the weld profile. Impact Toughness

Different classifications of alloy electrodes and fluxes can produce welds with CVN 20 at selected temperatures between 0°F and −150°F. Arc Strikes

Arc strikes may occur during welding procedures if the welding rod is lifted from the work while the current is on, or during magnetic particle testing if the magnetizing prod is lifted from the work while the current is on. As stated in Quality Criteria and Inspection Standards (AISC, 1988), arc strikes need not be removed in statically loaded structures.

Fig. 8-30. Lamellar tear resulting from shrinkage of large welds in thick material under high restraint. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 115

Other Considerations in Welded Construction

Matching Electrodes

AWS D1.1 Table 4.1 lists matching electrodes for various steels by ASTM Specification and is referenced in LRFD Specification Table J2.5. Use of electrodes one strength-level higher than matching is permitted. Typical structural steel grades with Fy equal to 36 ksi and 50 ksi are normally welded with electrode material of 70 ksi nominal strength, indicated as E70XX for SMAW or its equivalent. Welding Shapes from ASTM A6 Groups 4 and 5

When heavy shapes are spliced, extremely high shrinkage strains may develop in the base metal, inhibiting ductile deformation in the material and increasing the possibility of brittle fracture. Additionally, interior portions of heavy hot-rolled shapes and plates may contain a coarser grain structure and/or lower notch-toughness properties than other areas of the product.

(a)

Susceptible Detail

Improved Detail

Susceptible Detail

Improved Detail

Susceptible Detail

Improved Detail

(b)

(c)

Fig. 8-31. Susceptibility to lamellar tearing can be reduced by careful detailing of welded connections. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 116


LRFD Specification Sections A3.1c, J1.5, J1.6, J2.8, and M2.2 contain special material and fabrication requirements for ASTM A6 Groups 4 and 5 rolled shapes, shapes built-up from plates more than two inches thick, welded together to form the cross section, and shapes where the cross section is to be spliced by welding and subjected to primary tensile stress due to tension or flexure. These special requirements address notch toughness, access hole profiles, welding procedures, pre-heat, thermal cutting, grinding, and inspection requirements and are intended to minimize the possibility of cracking. The corresponding sections of the Commentary on the LRFD Specification provide further information, including alternative splice details and details for weld-access holes and beam copes. Intersecting Welds and Triaxial Stresses

If a stiffener were to be welded into and around the corner as it meets two elements of a shape (i.e., the flange and web of a column), the welding arc would take the path of least resistance to the three plates meeting at the corner and a lack of fusion or slag pocket would result in that corner. In addition to creating a discontinuity, this would add to the weld shrinkage strains in that corner. Corners of stiffeners, then, should be clipped generously to preclude this problem.

Susceptible Detail

Improved Detail Figure 8-32. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 117

In general, a 3⁄4-in. clip will be adequate. In small stiffeners, where such a clip would remove a large portion of the effective area of the stiffener, and in shapes, the radii of which require a clip in excess of 3⁄4-in., the clip dimension may be adjusted to suit conditions. For further information, see Butler, Pal, and Kulak (1972) and Blodgett (1980). Painting Welded Connections

Paint is normally omitted in areas to be field welded. LRFD Specification Section M3.5 requires that, unless otherwise provided in the plans and specifications, surfaces within two inches of any field weld shall be free of materials that would prevent proper welding or produce objectionable fumes during welding. Since little is gained by an exhaustive identification of the small areas involved, most fabricators prefer to use the general note, “No paint on OSL of connection angles,” where OSL stands for outstanding leg. This

Weak

Strong

Notch

Fig. 8-33. One-sided fillet weld results in a severe notch. A similar effect exists with a one-sided partial-penetration groove weld.

Fillet weld tacks can result in notches that reduce fatigue resistance.

tacks are incorporated in weld

(a) Susceptible Detail

(b) Improved Detail

Fig. 8-34. Backing bar tack welds. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 118


“no paint” requirement does not apply to shop welding where painting is normally done after the welds are made. Clearances for Welding

Clearances are required to allow the welder to make proper welds. In the SMAW process, for example, the welder must hold an electrode, about 3⁄8-in. in diameter and 14 inches to 18 inches long, in full control, and in such a position that the far end of the rod is in near contact with the base metal. This welder must observe the weld through a protective window of very dark glass in a bulky protective hood. Furthermore, the welder must keep control of the stiff electrical cable which powers the welding process. These conditions make welding difficult and it is imperative that other factors do not further hamper the welder. Ample room must be provided so that the welder or welding operator may manipulate the electrode and observe the weld as it is being deposited. The preferred position of the electrode when welding in the horizontal position is in a plane forming 30° with the vertical side of the fillet weld being made. However, this angle, shown as angle x in Figure 8-35, may be varied somewhat to avoid contact with some projecting part of the work. A simple rule which may be used to provide adequate clearance for the electrode in horizontal fillet welding is that the clear distance to a projecting element should be at least one-half its height; distance y / 2 in Figure 8-35b. A special case of minimum clearance for welding with a straight electrode is illustrated in Figure 8-36. The 20° angle is the minimum which will allow satisfactory welding along the bottom of the angle and therefore governs the setback with respect to the end of the beam. If a 1⁄2-in. setback and 3⁄8-in. electrode diameter were used, the clearance between the angle and the beam flange could be no less than 11⁄4-in. for an angle with a leg dimension w of three inches, nor less than 15⁄8-in. with a w of four inches. When it is not possible to provide this clearance, the end of the angle may be cut as noted by the optional cut in Figure 8-36 to allow the necessary angle. However, this secondary cut will increase the cost of fabricating the connection. Fillet Welds

In Figure 8-37a, fillet welds A are loaded in longitudinal shear and fillet weld B is loaded in transverse shear. If the force Ru is increased to exceed the strength of the welds, rupture will occur on the planes of least resistance. As shown in Figure 8-37b, this is assumed to take place in the weld throat where the least cross-sectional area is present. Tests of fillet welds using matching electrodes have demonstrated that the weld will fail through its effective throat before the material will fail along the weld leg. Fillet welds are approximately one-third stronger in the transverse direction than in the longitudinal direction. While this increased strength is ignored in LRFD Specification Section J2.4, the provisions of LRFD Specification Appendix J2.4 may be used to take advantage of this increased strength. Effective Area

The effective area of a fillet weld Aw is the product of the effective length of the fillet weld times the effective throat thickness of the fillet weld. The effective length l of the fillet weld is the overall length of the full-sized fillet weld. Except for fillet welds made with the SAW process, the effective throat thickness of the fillet weld is 0.707w, where w is the weld size. The deep penetration of fillet welds made by the SAW process is recognized in the LRFD Specification Section J2.2a wherein the effective throat thickAMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 119

ness is considered to be equal to the weld size for 3⁄8-in. and smaller welds, and equal to the effective throat thickness plus 0.11 in. for fillet welds sizes over 3⁄8-in. Minimum Effective Length

The minimum effective length of a fillet weld when used alone and not as a part of a continuing joint boundary (i.e., an end return or corner) must be greater than or equal to four times the nominal weld size. Thus, the shortest length of 5⁄16-in. fillet weld which is permitted to be considered to transmit load is 11⁄4-in. Conversely, regardless of the fillet-weld size used, the maximum effective size is limited to one-fourth the weld length. Intermittent fillet welds likewise are subject to this provision with the added requirement that the incremental length of weld must not be less than 11⁄2-in; refer to LRFD Specification Section J2.2b. Minimum Fillet Weld Size

When very small fillet-weld sizes are used, rapid cooling after welding creates internal stresses which, in turn, may lead to cracking of the weld. To preclude this, the minimum fillet-weld size is established in LRFD Specification Section J2.2b as a function of the thickness of the thicker of the parts joined. From this, if two 7⁄8-in. plates are joined, the minimum permissible fillet-weld size is 5⁄16-in., even if a 1⁄4-in. weld might provide adequate strength. Where different thicknesses are joined, the weld size need not exceed the thickness of the thinner part, unless a larger size is required for strength. If this is the case, adequate pre-heat must be provided to assure soundness of the weld. Maximum Fillet-Weld Size

The maximum fillet-weld size on the edge of the material is limited in LRFD Specification Section J2.2b to the thickness of the element for material less than 1⁄4-in. thick and 1⁄ -in. less than the thickness of the element for material greater than or equal to 1⁄ -in. 16 4 thick, unless the drawing is specially noted to build up the weld to achieve full throat size. This limitation recognizes that the exposed corner of the welded edge tends to melt Electrode x

END VIEW

PLAN VIEW

(a) y

Electrode A x

y /2

A PLAN VIEW

SECTION A-A (b)

Fig. 8-35. Clearances for welding. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 120


into the weld as illustrated in Figure 8-38, thereby reducing the leg dimension and the weld throat. Additionally, the toes of most rolled shapes do not have an ideal 90° corner. Thus the actual thickness of material at the weld is less than the nominal thickness t of the member. While the LRFD Specification permits the use of a larger weld size if the weld is built up to the full throat size, this is difficult to achieve. End Returns

LRFD Specification Section J2.2b gives requirements on when fillet weld terminations must be returned around ends or sides. This is illustrated in Figure 8-39. Weld returns reinforce the effective weld where it is most highly stressed and, thus, inhibit cracking and progressive tearing throughout the length of the weld. Thus, they are required in fatigue applications and for connections which assume flexibility exists in the connected part or parts (e.g., the support legs of a double angle connection). If welds are not returned, they must terminate not less than two times the nominal weld size from the sides or ends. Also, based upon LRFD Specification Section J2.2b, Figures 8-40 and 8-41 indicate examples where welds must be interrupted or should not be returned. In these instances, the welds, while in the same plane, lie on opposite sides of the contact surfaces. An attempt to weld around the corner will melt the corner material, creating a reduced thickness and notch. Furthermore, such welds cannot be made with a fully effective throat. Welding around such a corner should be avoided. It is not recommended that weld be applied in the gap at the end of the beam web between the heels of the angles, as this reduces the flexibility of the connection angles. Furthermore, the setback of the beam web is not a controlled dimension as it may be used to account for the tolerance in length of the beam and may vary from zero in. to 1⁄2-in. or

Setback

w

Optional cut

20° PLAN VIEW Electrode

45° to 50°

ELEVATION

END VIEW

Fig. 8-36. Clearances for welding. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 121

more. In any case, most beam webs are too thin for an effective minimum weld size to be applied along such an edge. Fillet Welds in Holes or Slots

The recommended minimum hole diameters or slot widths for fillet welding are shown in Table 8-35. It is important to distinguish between plug or slot welds and fillet welds placed around the inside of a hole or slot. In the case of such fillet welds, the shear strength is the product of the effective throat thickness and the weld length measured along the line bisecting the throat area. If this effective area should exceed the area of the hole or slot, it cannot be considered to be a fillet weld and must be designed as a plug or slot weld. Other Limitations on Fillet Welds

In concentrically loaded welded joints, the stresses are assumed to be uniformly distributed throughout the length of the welds. The design strength of a concentrically loaded fillet-weld group, then, is the sum of the design strengths of each weld in the group. LRFD Specification Section J1.8 provides that the center of gravity of a weld group should coincide with the gravity axis of an axially loaded member, or provision must be made for the resulting eccentricity. Certain welded members not subject to fatigue loading are excluded from this provision: “Eccentricity between gravity axes of such members…may be neglected in statically loaded members, but shall be considered in members subject to fatigue loading.” This provision permits very significant cost savings in weld material

Ru

Ru

Welds A Weld B

Ru

Ru (a)

(b)

t

1/16

t

1/16

Fig. 8-37. Fillet welds.

(a)

(b)

Figure 8-38. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 122


and labor in the fabrication and erection of such statically loaded members as roof and floor trusses, bracing, etc. Additionally, LRFD Specification Section J2.2b imposes other limitations on proportions of lap joints. Minimum Shelf Dimensions

In Figure 8-42, the recommended minimum shelf dimensions for normal size SMAW fillet welds are summarized. This dimension is critical to the deposition of the weld. SAW fillet welds would require a greater shelf dimension to contain the flux, although this is sometimes provided by clamping auxiliary material to the member. In Figure 8-43, the distance b must be large enough so that a full-size weld may be deposited on it. Select a gage that will permit enough clearance b to deposit an effective weld. The dimension b should be sufficient to accommodate the combined tolerances of the framing-angle length, the cope depth, and the beam mill over/underrun as well as the specified weld size. Complete-Joint-Penetration Groove Welds

Assuming compliance with LRFD Specification Section J2, the design strength of complete-joint-penetration groove welds is equal to that of the base metal in all respects. Therefore, no allowance for the presence of such welds need be made in proportioning the connections of structural members for any type of static loading. Where members are of unequal cross section or different material strength, the strength of the complete-jointpenetration groove weld is limited to the strength of the weaker member. Extension, Runoff, Backing, and Spacer Bars

When groove welds are used to splice plate girders and beams, LRFD Specification Section J7 requires that the splice be capable of developing the full strength of the smaller spliced section or 100 percent of the full section if the spliced sections are of the same size. To obtain a fully welded cross section, the termination at either end of the joint must Provide end returns having length = twice nominal weld size if subjected to cyclical (fatigue) loading or = four times nominal weld size if needed for connection flexibility. Otherwise, terminate welds not less than nominal weld size from ends with no end returns.

1

2 x nominal weld size

/4 for Pts. 1 1 for Pts. 2 2

5 /16

Note: Locations of Pts. 1 and Pts. 2 are shown on the erection diagram (not included).

Fig. 8-39. Weld returns. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 123

be of sound weld metal. Extension or runoff bars are usually used to assure the soundness of the end of the weld. Frequently, the joint will require a backing or spacer bar which can be extended to serve as the extension or runoff bar. Figure 8-44 demonstrates the application of extension, backing, and spacer bars in a splice or moment connection. Extension and backing bars should be of approved weldable material as specified in AWS D1.1, Section 8.2.4; spacer bars must be of the same material specification as the base metal. This can create a procurement problem since small tonnage requirements may make them difficult to obtain in the specified ASTM designation. Also indicated in Figure 8-44 is the use of a cover plate or seat angle for backing the weld.

Do not tie welds together here terminate welds 2 x nominal weld size from end.

Do not tie welds together here

Fig. 8-40. Fillet welds on opposite sides of a common plane should not be continuous. Do not return welds here terminate welds 2 x nominal weld size from end.

Fig. 8-41. Fillet welds should not be returned across thickness of material. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 124


Table 8-35. Recommended Minimum Hole Diameters or Slot Widths for Fillet Welding, in. Plate Thickness, in. 3⁄

16

Min. Diameter or Width, in.

and 1⁄4

11⁄ 16 13⁄ 16 15⁄ 16 11⁄16 13⁄16 15⁄16

5⁄ 16 3⁄ 8 7⁄ 16 1⁄ 2 5⁄ 8

Shown in Figure 8-45 are flat-type extension bars, normally used with beveled grooves, and contour-type extension bars, normally used with J-grooves or U-grooves and shaped to follow the contour of the joint geometry. While the contour-type extension bar is shown as though it were comprised of two pieces, some fabricators might elect to mill the full contour in one piece and subsequently cut it to suit job requirements. AWS D1.1, Section 3.12 states that runoff and extension bars need not be removed in statically loaded structures unless required by the engineer of record (EOR). Such might be the case where these bars would create an interference with other work. In dynamically and cyclically loaded structures, however, they must be removed and the welds made smooth and flush to the base metal abutting edges.

Vertical or horizontal section

Fillet Weld Size (in.)

Min. Shelf Dim. (in.)

3

/16

7

1

1

5

9

3

5

7

11

1

3

/4

/16 /8 /16 /2

/16 /2

/16

/8 /16

/4

Fig. 8-42. Recommended minimum shelf dimensions for SMAW fillet welds. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 125

According to AWS D1.1, Section 3.13, backing bars on groove-welded joints must be fully spliced to avoid stress concentrations or discontinuities and should be thoroughly fused with the weld metal. It is further required on dynamically loaded structures that the backing bars be removed and the surfaces finished smooth when they are transverse to the direction of stress. If this were the case for the flange splice of Figure 8-44, removal of the backing bars would be required and, therefore, the splice might be made more economically with another joint profile. Weld Access Holes

The beam web is provided with an access hole or “rathole”, as illustrated in Figure 8-44, to permit down-hand welding to the backing bars located below both the top and bottom flanges. The weld-access hole also provides increased relief from concentrated weld shrinkage strains and prevents the intersection or close juncture of welds in orthogonal directions. Weld-access holes should not be filled with weld metal since it is difficult to provide sound weld metal to fill such a void and doing so may introduce a state of triaxial stress under loading. Partial-Joint-Penetration Groove Welds

b

b

b

Gage

Gage

Partial-joint-penetration groove welds are used primarily for welded compression splices, the connection of elements in heavy box sections and pedestals, and, in general, for joints where the stress to be transferred is substantially less than that which would require complete-joint-penetration groove welds. This type of weld is not, however, recom-



1

t

Extension bars

/8 ″ Min.

w

L

W

8 - 126

Extension bars Backing bar Backing bar

Col.

Beam access hole

Access Hole

Seat angle

Spacer bar (when req’d)

Beam flange Overlapping cover plate Note: Extension bars should be at least ¼ ″ thick to reduce hazard of weld “blow through.” Figure 8-44 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 127

mended in joints subject to dynamic or cyclical loading, except for joining the components of built-up members. Effective Area

The effective area of a partial-joint-penetration groove weld Aw is the product of the effective length of the weld times the effective throat thickness of the fillet weld. These quantities are determined as follows. The effective length is the width of the part joined. The effective throat thickness E is as determined from LRFD Specification Table J2.1, but not less than specified in LRFD Specification Table J2.3. Nomenclature of partial-joint-penetration welds is shown in Figure 8-46. Note that the effective throat thickness shown is less than the dimensioned groove-weld size. AWS prequalified partial-joint-penetration welds establish for each joint an effective throat E as a function of the material thickness, weld-preparation size, or depth S. Thus, the design drawings should specify the effective weld length and the required effective throat. The shop drawings should then show the groove depth S and geometry which will provide for the specified effective throat E. Some fabricators may indicate both the weld size and the effective throat on the shop drawings to eliminate confusion. The comments on “Extension, Runoff, Backing, and Spacer Bars” and “Weld Access Holes” for complete-joint-penetration groove welds also apply to partial-joint-penetration groove welds. Intermittent Welds

In preparing the joint profile for intermittent partial-joint-penetration groove welds, a transition or “faring-in” of the joint at beginning and termination must be provided to ensure proper fusion with the base metal. The nominal angular value of this transition should be 45°as shown in Figure 8-46. Flare Welds

A flare weld is a special case of the partial-joint-penetration groove weld wherein the convex surface of the connected part creates the joint preparation. This convexity may be the result of an edge preparation, but more often results when one (or both) joint component consists of a round rod or a shape with a rounded bend or corner radius created by bending or rolling as shown in Figure 8-47.

Extension bars Runout plate or backing bar extension

Figure 8-45. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 128


Effective Area

The effective area of a flare weld Aw is the product of the effective length of the weld times the effective throat thickness of the flare weld; the effective length is the width of the part joined and the effective throat thickness E is as determined from LRFD Specification Table J2.2. Limitations

The deposition of effective weld metal to the bottom of the flare groove is very difficult because the welding arc short-circuits across the surfaces due to the sharp angular slopes. Thus, the quality of this weld is difficult to control; LRFD Specification Section J2.1a permits examination and adjustment of the weld strength based on random testing and special qualification. Note that weldability of concrete reinforcing bars is not a part of ASTM specifications. In past experience, improperly welded concrete reinforcing bars have cracked and separated under no-load conditions. Typical deformed-type concrete reinforcing bars, such as ASTM A615, A616, and A617, are not produced to a controlled chemistry and their weldability must be carefully evaluated; refer to AWS D1.4. Plug and Slot Welds

The use of plug and slot welds for stress transfer is limited to resisting shear loads in joint planes parallel to the faying surfaces. These welds should not be subjected to tensile stresses and are limited when subjected to stress reversal. Furthermore, some specificaEffective A

Transition

length

T

S

E

45° Min.

A

SECTION A-A

Fig. 8-46. Partial-joint-penetration groove weld nomenclature.

Flare-V-groove effective throat=R/2

R

R Flare-bevel groove effective throat = 5R/16

Fig. 8-47. Flare weld nomenclature. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 129

tions do not permit their use as load-carrying welds. Because of these limitations, plug and slot welds are more frequently employed as stitch welds rather than as a means of primary stress transfer. The effective area of a plug or slot weld Aw is the nominal cross-sectional area of the hole or slot. The proportions and spacing of holes and slots and the depth of weld are stipulated in LRFD Specification Section J2.3b and illustrated in Figure 8-48. Design Strength of Welds

The design strength of welds is determined in accordance with LRFD Specification Sections J2.2 and J2.4. LRFD Specification requirements are based upon the provisions of AWS D1.1, except as noted in LRFD Specification Section J2. For welds, the limit states of the weld-metal strength and the base-metal strength must be checked as applicable in LRFD Specification Table J2.5. These limit states assume that the matching electrode requirements of LRFD Specification Section J2.6 and Table J2.1 are met. Weld Metal Design Strength

From LRFD Specification Section J2.4, the weld metal design strength is φRn, where φ is a resistance factor from LRFD Specification Table J2.5 and: Rn = Fw Aw In the above equation, Fw = 0.60FEXX Aw = effective area of the weld, in.2 and φ is determined as follows: For a fillet weld loaded in shear on its effective area, φ = 0.75; For a complete-joint-penetration groove weld loaded in shear on its effective area, φ = 0.80; For a partial-joint-penetration groove weld loaded in shear parallel to the axis of the weld, φ = 0.75; For a partial-joint-penetration groove weld loaded in tension normal to the effective area, φ = 0.80; For a plug or slot weld loaded in shear on its effective area, φ = 0.75. Base Metal Design Strength

From LRFD Specification Section J2.4, the base metal design strength is φRn, where φ is a resistance factor from LRFD Specification Table J2.5 and: Rn = FBM ABM AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 130


In the above equation, ABM is the cross-sectional area of the base metal. For a fillet weld loaded in tension or compression parallel to the axis of the weld, φ = 0.90 FBM = Fy For a complete-joint-penetration groove weld loaded in tension or compression normal to its effective area, φ = 0.90 FBM = Fy For a complete-joint-penetration groove weld loaded in shear on its effective area, φ = 0.90 FBM = 0.60Fy d

l

l

R′

d

d

S′

t

W

W

W

S

W

S

R

Plate thickness, in. 3/16 & 1/4

Min. hole dia. or slot width, d, in. 9/16

5/16 & 3/8

11/16

7/16 & 1/2

13/16

9/16 & 5/8

15/16

Hole and slot proportions, spacing and depth of weld d ≥ (t + 5/16), round to next higher odd 1/16; also d ≤ 2 1/4W S ≥ 4d S ′≥ 2 l l ≤ 10W R = d/2 R≥t

Where t ≤ 5/8, W = t Where t > 5/8, W = t/2 but, not less than 5/8

Fig. 8-48. Plug and slot welds. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

WELDED CONSTRUCTION

8 - 131

For a partial-joint-penetration groove weld loaded in tension or compression normal to its effective area or tension or compression parallel to the axis of the weld, φ = 0.90 FBM = Fy Prequalified Welded Joints

AWS D1.1 contains provisions for prequalified welded joints which provide joint geometries, such as root openings, angles, and clearances, as illustrated in Figures 8-49 and 8-50, that will permit a qualified welder to deposit sound weld material. Thus, prequalified joints are concerned almost exclusively with the welding process as a method of joining metal and deal with welded joints only from fusion boundary to fusion boundary. The designer must satisfy all provisions of AWS D1.1 Sections 2, 3, and 4 before a joint is considered prequalified. Prequalified welded joints are not, in themselves, adequate consideration of welded design details. To emphasize this, the AWS D1.1 Section 1.1 states: “…The use of prequalified joints is not intended as a substitute for engineering judgment with respect to the suitability of application of these joints to a weld assembly.” The design and detailing for successful welded construction requires consideration of factors which include, but are not limited to, the magnitude, type, and distribution of forces to be transmitted, access, restraint against weld shrinkage, thickness of connected materials, residual stress, and distortion. Accordingly, the design and detailing must also satisfy the requirements of LRFD Specification Section J2. The prequalified welded joints in Table 8-36 meet the requirements of the 1992 version of AWS D1.1 as well as the 1993 LRFD Specification. Because AWS D1.1 is revised every other year, designers and fabricators should verify this information with the latest issue of AWS D1.1. The designations such as B-L1a, B-U2, and B-P3 are those used in AWS standards. Note that lowercase letters, e.g., a, b, c, etc., are often used to differentiate between joints that would otherwise have the same joint designation. These prequalified welded joints are limited to those made by the SMAW, SAW, GMAW (except short circuit transfer), and FCAW procedures. Small deviations from dimensions, angles of grooves, and variation in depth of groove joints are permissible within the tolerances given. In general, all fillet welds, whether illustrated or not, are prequalified, provided they conform to the requirements of AWS D1.1. Groove welds are classified using the conventions indicated in the tables. Welded joints other than those prequalified by AWS may be qualified, provided they are tested and qualified in accordance with AWS D1.1.


8 - 132


ctive Effe ength dL Wel

Weld face Throat area (shaded)

45°

45°

t line

e

tiv

Roo

c ffe

E Leg size

h

gt

d

el

n le

/12 N o Th rma r o Siz at l e

W al rm at o N ro Th ze Si

Root

Size 135° max.

60° min.

Penetration

t

roa

Th

t

oa

r Th

t oa

r

Th

Normal Throat Size

p

e De

CONVEX

n tio e tra iz ne t S Pe hroa T

al e rm Siz No at ro Th Fig. 8-49. Fillet weld nomenclature. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

CONCAVE

WELDED CONSTRUCTION

Groove face

8 - 133

Groove (and bevel) angle

Groove angle Bevel angle

Groove radius

Root

Root opening

Spacer bar

Root Root Root face opening

Backing bar

Root opening

PREPARATION Penetration (fusion zone)

Weld face

Root Backing bead

Weld size

Reinforcement

Weld throat= Weld size

Root opening

0

Root face

PARTIAL-JOINT-PENETRATION

COMPLETE-JOINT-PENETRATION

Groove size Root face

Groove angle Fillet size

1/8

Effective throat

Eff. at thro

PARTIAL-JOINT-PENETRATION (When Reinforcing Fillet is Specified) Fig. 8-50. Groove weld nomenclature. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 134


Table 8-36. Prequalified Welded Joints Symbols for Joint Types B C T

butt joint corner joint T-joint

L U P

limited thickness, complete-joint-penetration unlimited thickness, complete-joint-penetration partial-joint-penetration

1 2 3 4 5

square-groove single-V-groove double-V-groove single-bevel-groove double-bevel-groove

S G F

submerged arc welding SAW gas metal arc welding GMAW flux cored arc welding FCAW

BC butt or corner joint TC T- or corner joint BTC butt, T-, or corner joint

Symbols for Base Metal Thickness and Penetration

Symbols for Weld Types 6 7 8 9 10

single-U-groove double-U-groove single-J-groove double-J-groove Flare-bevel-groove

Symbols for Welding Processes if not Shielded Metal Arc welding (SMAW):

Symbols for Welding Positions F flat H horizontal V vertical OH overhead The lower case letters, e.g., a, b, c, d, etc., are used to differentiate between joints that would otherwise have the same joint designation.

Notes to Prequalified Welded Joints A B Br C E J

J2 L M Mp N

Q Q2 R

V

Not prequalified for GMAW using short circuiting transfer. Refer to AWS D1.1 Appendix A. Joints welded from one side only. Bridge applications limit the use of these joints to the horizontal position. Refer to AWS D1.1 Section 9.12.5. Back gouge root to sound metal before welding second side. Minimum effective throat (E) as shown in LRFD Specification Table J2.3; S as specified on drawings. If fillet welds are used in buildings to reinforce groove welds in corner and T-joints, they shall be equal to 1 ⁄4 T 1 , but need not exceed 3 ⁄8 -in. Groove welds in corner and T-joints in bridges shall be reinforced with fillet welds equal to 1 ⁄4 T 1 , but not more than 3 ⁄8 -in. If fillet welds are used in buildings to reinforce groove welds in corner and T-joints, they shall be equal to 1 ⁄4 T 1 , but not more than 3 ⁄8 -in. Butt and T-joints are not prequalified for bridges. Double-groove welds may have grooves of unequal depth, but the depth of the shallower groove shall be not less than one-fourth of the thickness of the thinner part joined. Double-groove welds may have grooves of unequal depth, provided they conform to the limitations of Note E. Also, the effective throat (E), less any reduction, applies individually to each groove. The orientation of the two members in the joints may vary from 135° to 180°, provided the basic joint configuration (groove angle, root face, root opening) remains the same and the design throat thickness is maintained. For corner and T-joints, the member orientation may be changed provided the groove dimensions are maintained as specified. The member orientation may be changed provided the groove dimensions are maintained as specified. The orientation of two members in the joint may vary from 45° to 135° for corner joints and from 45° to 90° for T-joints, provided the basic joint configuration (groove angle, root face, root opening) remains the same and the design throat thickness is maintained. For corner joints, the ouside groove preparation may be in either or both members, provided the basic groove configuration is not changed and adequate edge distance is maintained to support the welding operations without excessive edge melting.


WELDED CONSTRUCTION

8 - 135

Table 8-36 (cont.). Prequalified Welded Joints Basic Weld Symbols

Back

Groove or Butt

Plug or Slot

Fillet

Square

V

Bevel

U

J

Flare Bevel

Flare V

Supplementary Weld Symbols

Backing

Weld All Around

Spacer

Contour Field Weld

Flush

Convex

For other basic and supplementary weld symbols, see AWS A2.4

Standard Location of Elements of a Welding Symbol Finish symbol

Groove angle or included angle or countersink for plug welds

Contour symbol Root opening, depth of filling for plug and slot welds

Length of weld in inches

Effective throat

Pitch (c. to c. spacing) of welds in inches

F

Depth of preparation or size in inches

A

S(E)

sides)

R

Specification, process, or other reference

(Both

T

Tail (omitted when reference is not used)

Field weld symbol

(Arrow (Other side ) side )

Reference line

Weld-all-around symbol

@P

Elements in this area remain as shown when tail and arrow are reversed.

Basic weld symbol or detail reference

A

B

Arrow connects reference line to arrow side of joint. Use break as at A or B to signify that arrow is pointing to the grooved member in bevel or J-grooved joints.

Note: Size, weld symbol, length of weld, and spacing must read in that order, from left to right, along the reference line. Neither orientation of reference nor location of the arrow alters this rule. The perpendicular leg of

,

,

,

, weld symbols must be at left.

Arrow and other side welds are of the same size unless otherwise shown. Dimensions of fillet welds must be shown on both the arrow side and the other side symbol. The point of the field weld symbol must point toward the tail. Symbols apply between abrupt changes in direction of welding unless governed by the ‘‘all around’’ symbol or otherwise dimensioned. These symbols do not explicitly provide for the case that frequently occurs in structural work, where duplicate material (such as stiffeners) occurs on the far side of a web or gusset plate. The fabricating industry has adopted this convention: that when the billing of the detail material discloses the existence of a member on the far side as well as on the near side, the welding shown for the near side shall be duplicated on the far side.


8 - 136


Table 8-36 (cont.). Prequalified Welded Joints Fillet Welds

Notes: 1. En, En′ = effective throats dependent on magnitude of root opening Rn. See AWS D1.1 Section 3.3.1 Subscript n represents 1, 2, 3, or 4. 2. t = thickness of thinner part. 3. Not prequalified for gas metal arc welding using short circuitry transfer. Refer to AWS D1.1. 4. Part (f), apply Z loss factor of AWS D1.1 Table 2.4 to determine effective thrust. 5. Part (f), not prequalified for angles under 30°°. For welder qualfications, see AWS D1.1 Table 10.5, Column 10. *Angles smaller than 60°° are permissible, however, if the weld is considered to be a partial-joint-penetration groove weld.


WELDED CONSTRUCTION

8 - 137

Table 8-36 (cont.). Prequalified Welded Joints Complete-Joint-Penetration Groove Welds Square-groove weld (1) Butt joint (B) Corner Joint (C)

Welding Process

Joint Designation

SMAW

B-L1a

GMAW FCAW

Base Metal Thickness (U = unlimited)

Groove Preparation Tolerances

Gas Shielding for (FCAW)

T1

T2

Root Opening

As Detailed

As Fit Up

Permitted Welding Positions

1 ⁄4

max

—

R = T1

+ 1 ⁄16 , −0

+ 1 ⁄4 , − 1 ⁄16

All

—

N

C-L1a

1 ⁄4

max

U

R = T1

+ 1 ⁄16 , −0

+ 1 ⁄4 , − 1 ⁄16

All

—

—

B-L1a-GF

3 ⁄8

max

—

R = T1

+ 1 ⁄16 , −0

+ 1 ⁄4 , − 1 ⁄16

All

Not Required

A, N

Notes

Square-groove weld (1) Butt joint (B)


Welding Process

Joint Designation

SMAW

B-L1b

1 ⁄4

GMAW FCAW

B-L1b-GF

SAW SAW



T1

T2

Root Opening

As Detailed

As Fit Up


max

—

R = T1 / 2

+ 1 ⁄16 , −0

+ 1 ⁄6 , − 1 ⁄8

All

—

C, N

3 ⁄8

max

—

R = 0 to 1 ⁄8

+ 1 ⁄16 ,

+ 1 ⁄6 ,

− 1 ⁄8

All

Not Required

A, C, N

B-L1-S

3 ⁄8

max

—

R=0

±0

+ 1 ⁄16 , −0

F

—

N

B-L1a-S

5 ⁄8

max

—

R=0

±0

+ 1 ⁄16 , −0

F

—

C, N


Notes

−0

Notes

Square-groove weld (1) T-joint (T) Corner joint (C)

Welding Process

Joint Designation

SMAW

TC-L1b

GMAW FCAW SAW



T1

T2

Root Opening

As Detailed

As Fit Up


1 ⁄4

max

U

R = T1 / 2

+ 1 ⁄16 , −0

+ 1 ⁄16 , − 1 ⁄8

All

—

C, J

TC-L1-GF

3 ⁄8

max

U

R = 0 to 1 ⁄8

+ 1 ⁄16 , −0

+ 1 ⁄16 , − 1 ⁄8

All

Not Required

A, C, J

TC-L1-S

3 ⁄8

max

U

R=0

±0

+ 1 ⁄16 , −0

F

—

J, C


8 - 138


Table 8-36 (cont.). Prequalified Welded Joints Complete-Joint-Penetration Groove Welds Single-V-groove weld (2) Butt joint (B)

Welding Process

Joint Designation

SMAW

B-U2a

GMAW FCAW

B-U2a-GF

Tolerances


As Detailed

As Fit Up

R = +1 ⁄16 , −0

+ 1 ⁄4 , − 1⁄16

α = +10°°, − 0°°

+10°°, − 5°°

Groove Preparation

Gas Shielding for FCAW

Notes

T1

T2

Root Opening

Groove Angle


U

—

R = 1 ⁄4

α = 45°°

All

—

N

R = 3 ⁄8

α = 30°°

F, V, OH

—

N

R = 1 ⁄2

α = 20°°

F, V, OH

—

N

R = 3 ⁄16

α = 30°°

F, V, OH

Required

A, N

R = 3 ⁄8

α = 30°°

F, V, OH

Not req.

A, N

R = 1 ⁄4

α = 45°°

F, V, OH

Not req.

A, N

U

—

SAW

B-L2a-S

2 max

—

R = 1 ⁄4

α = 30°°

F

—

N

SAW

B-U2-S

U

—

R = 5 ⁄8

α = 20°°

F

—

N

Tolerances

Single-V-groove weld (2) Corner joint (C)

Welding Process

Joint Designation

SMAW

C-U2a


As Detailed

As Fit Up

R = +1 ⁄16 , −0

+ 1 ⁄4 , − 1⁄16

α = +10°°, − 0°°

+10°°, − 5°°

Groove Preparation


Notes

T1

T2

Root Opening

Groove Angle


U

U

R = 1 ⁄4

α = 45°°

All

—

Q

R = 3 ⁄8

α = 30°°

F, V, OH

—

Q

R = 1 ⁄2

α = 20°°

F, V, OH

—

Q

R = 3 ⁄16

α = 30°°

F, V, OH

Required

A

R = 3 ⁄8

α = 30°°

F, V, OH

Not req.

A, Q

GMAW FCAW

C-U2a-GF

U

U

R = 1 ⁄4

α = 45°°

F, V, OH

Not req.

A, Q

SAW

C-L2a-S

2 max

U

R = 1 ⁄4

α = 30°°

F

—

Q

SAW

C-U2-S

U

U

R = 5 ⁄8

α = 20°°

F

—

Q


WELDED CONSTRUCTION

8 - 139

Table 8-36 (cont.). Prequalified Welded Joints Complete-Joint-Penetration Groove Welds Single-V-groove weld (2) Butt joint (B)

Welding Process

Joint Designation

SMAW



Root Opening Root Face Groove Angle

As Detailed

As Fit Up

Gas Permitted Shielding Welding for Positions Notes FCAW

T1

T2

B-U2

U

—

R = 0 to 1⁄8 f = 0 to 1 ⁄8 α = 60°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°°, − 0°°

+1 ⁄16 , − 1 ⁄8 Not limited +10°°, − 5°°

All

—

C, N

GMAW FCAW

B-U2-GF

U

—

R = 0 to 1⁄8 f = 0 to 1 ⁄8 α = 60°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°, − 0°°

+1 ⁄16 , −1 ⁄8 Not limited +10°, − 5°°

All

Not required

A, C, N

SAW

B-L2c-S

Over 1 ⁄2 to 1

—

R = 0, α = 60°° f = 1 ⁄4 max

R = ±0 f = +0, − f α = +10°°, − 0°°

+1 ⁄16 , − 0

F

—

C, N

Over 1 to 11 ⁄2

—

R = 0, α = 60°° f = 1 ⁄2 max

Over 11 ⁄2 to 2

—

R = 0, α = 60°° f = 5 ⁄8 max

± 1 ⁄16 +10°°, − 5°°

Single-V-groove weld (2) Corner joint (C)

Welding Process

Joint Designation

SMAW




As Detailed

As Fit Up

Gas Permitted Shielding Welding for Positions FCAW Notes

T1

T2

C-U2

U

U

R = 0 to 1 ⁄8 f = 0 to 1 ⁄8 α = 60°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°°, − 0°°

+1 ⁄16 , − 1 ⁄8 Not limited +10°°, − 5°°

All

—

C, J, R

GMAW FCAW

C-U2-GF

U

U

R = 0 to 1 ⁄8 f = 0 to 1 ⁄8 α = 60°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°, − 0°°

+1 ⁄16 , −1⁄8 Not limited +10°°, − 5°°

All

Not required

A, C, J, R

SAW

C-L2b-S

U

U

R=0 f = 1 ⁄4 max α = 60°°

±0 +0, − 1 ⁄4 +10° − 0°

+1 ⁄16 , − 0

F

—

C, J, R

± 1 ⁄16 +10°°, − 5°°


8 - 140


Table 8-36 (cont.). Prequalified Welded Joints Complete-Joint-Penetration Groove Welds Tolerances

Double-V-groove weld (3) Butt joint (B)

Welding Process SMAW

B-U3a

SAW

B-U3a-S


As Fit Up

R = ±0

+ 1 ⁄4 , −0

f = ±0

+1 ⁄16 , − 0

α = +10°°, − 0°°

+10°°, − 5°°

SAW

±0

+1 ⁄16 , − 0

SMAW

±0

+1 ⁄8 , − 0

Spacer

Joint Designation

As Detailed

Groove Preparation

Gas Permitted Shielding Welding for Positions (FCAW) Notes

T1

T2

Root Opening

Root Face

Groove Angle

U Spacer = 1 ⁄8 × R

—

R = 1 ⁄4

f = 0 to 1 ⁄8

α = 45°°

All

—

R = 3 ⁄8

f = 0 to 1 ⁄8

α = 30°°

F, V, OH

—

R = 1 ⁄2

f = 0 to 1 ⁄8

α = 20°°

F, V, OH

—

U Spacer = 1 ⁄4 × R

—

R = 5 ⁄8

f = 0 to 1 ⁄4

α = 20°°

F

—

C, M, N

C, M, N

Double-V-groove weld (3) Butt joint (B) For B-U3c-S only T1

Over 2 21 ⁄ 2 3 35 ⁄ 8 4 43 ⁄ 4 51 ⁄ 2

S1

to 21 ⁄2 3 35 ⁄8 4 43 ⁄4 51 ⁄2 61 ⁄4

13 ⁄8 13 ⁄4 21 ⁄8 23 ⁄8 23 ⁄4 31 ⁄4 33 ⁄4

For T 1 > 61⁄4, or T 1 ≤ 2 S 1 = 2 ⁄3 ( T 1 − 1 ⁄4 )

Welding Process

Joint Designation

SMAW

B-U3b

GMAW FCAW

B-U3-GF

SAW

B-U3c-S

Base Metal Thickness (U = unlimited) T1

T2

U

—

U

—


Gas Permitted Shielding Welding for Positions FCAW


As Detailed

As Fit Up

R = 0 to 1 ⁄8 f = 0 to 1 ⁄8 α = β = 60°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°°, − 0°°

+1 ⁄16 , − 1 ⁄8 Not limited +10°°, − 5°°

All

—

C, M, N

All

Not required

A, C, M, N

R=0 f = 1 ⁄4 min α = β = 60°°

+1 ⁄16 , − 0 +1 ⁄4 , − 0 +10°, − 0°°

+1 ⁄16 , − 0 +1 ⁄4 , − 0 +10°, − 5°°

F

—

C, M, N

To find S 1 see table above; S 2 = T 1 − (S 1 + f )


Notes

WELDED CONSTRUCTION

8 - 141


Single-bevel-groove weld (4) Butt joint (B)

Welding Process

Joint Designation

SMAW

B-U4a

GMAW FCAW


B-U4a-GF


GMAW FCAW

SAW

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄4 , − 1⁄16

α = +10°°, − 0°°

+10°°, − 5°°

Groove Preparation


Notes Br, N

T1

T2

Root Opening

Groove Angle


U

—

R = 1 ⁄4

α = 45°°

All

—

R = 3 ⁄8

α = 30°°

All

—

Br, N

R = 3 ⁄16

α = 30°°

All

Required

A, Br, N

R = 1 ⁄4

α = 45°°

All

Not req.

A, Br, N

R = 3 ⁄8

α = 30°°

F

Not req.

A, Br, N

U

—

Tolerances

Single-bevel-groove weld (4) T-joint (T) Corner joint (C)

Joint Designation

As Detailed


As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄4 , − 1⁄16

α = +10°°, − 0°°

+10°°, − 5°°

Groove Preparation


Notes

—

J, Q, V

T1

T2

Root Opening

Groove Angle


TC-U4a

U

U

R = 1 ⁄4

α = 45°°

All

R = 3 ⁄8

α = 30°°

F, V, OH

—

J, Q, V

TC-U4a-GF

U

U

R = 3 ⁄16

α = 30°°

All

Required

A, J, Q, V

R = 3 ⁄8

α = 30°°

F

Not req.

A, J, Q, V

R = 1 ⁄4

α = 45°°

All

Not req.

A, J, Q, V

R = 3 ⁄8

α = 30°°

F

—

J, Q, V

R = 1 ⁄4

α = 45°°

TC-U4a-S

U

U


8 - 142


Table 8-36 (cont.). Prequalified Welded Joints Complete-Joint-Penetration Groove Welds Single-bevel-groove weld (4) Butt joint (B)

Base Groove Preparation Metal Thickness Tolerances (U = unlimited) Root Opening As As Fit Root Face T1 T2 Groove Angle Detailed Up

Welding Process

Joint Designation

SMAW

B-U4b

U

—

GMAW FCAW

B-U4b-GF

U

—

R = 0 to 1 ⁄8 f = 0 to 1 ⁄8 α = 45°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°°, − 0°°

+1 ⁄16 , − 1⁄8 Not limited +10°°, − 5°°



Notes

All

—

Br, C, N

All

Not required

A, Br, C, N

Single-bevel-groove weld (4) T-joint (T) Corner joint (C)

Base Groove Preparation Metal Thickness Tolerances (U = unlimited) Root Opening As As Fit Root Face T1 T2 Up Groove Angle Detailed

Welding Process

Joint Designation

SMAW

TC-U4b

U

U

GMAW FCAW

TC-U4b-GF

U

U

SAW

TC-U4b-S

U

U


Notes

R = 0 to 1 ⁄8 f = 0 to 1 ⁄8 α = 45°°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°°, − 0°°

+1 ⁄16 , − 1 ⁄8 Not limited +10°°, − 5°°

All

—

C, J, R, V

All

Not required

A, C, J, R, V

R=0 f = 1 ⁄4 max α = 60°

±0 +0, − 1 ⁄8 +10°, − 0°

+1 ⁄4 , − 0

F

—

C, J, R, V

± 1⁄16 +10°, − 5°


WELDED CONSTRUCTION

8 - 143


Double-bevel-groove weld (5) Butt joint (B) T-joint (T) Corner joint (C)



Groove Preparation

As Fit Up

R = ±0

+ 1 ⁄4 , −0

f = 1 ⁄16 , − 0

± 1 ⁄16

α = +10°°, − 0°°

+10°, − 5°°

+1⁄16 , − 0

+1 ⁄8 , − 0

Spacer

Joint Designation

As Detailed


T1

T2

Root Opening

Root Face

Groove Angle

B-U5b

U Spacer =1⁄8 × R

U

R = 1 ⁄4

f = 0 to 1 ⁄8

α = 45°°

All

—

Br, C, M, N

TC-U5a

U Spacer =1⁄4 × R

U

R = 1 ⁄4

f = 0 to 1 ⁄8

α = 45°°

All

—

C, J, M, R, V

R = 3 ⁄8

f = 0 to 1 ⁄8

α = 30°°

F, OH

—

C, J, M, R, V

Notes

Double-bevel-groove weld (5) Butt joint (B)

Base Groove Preparation Metal Thickness (U = unlimited) Root Opening Tolerances Joint As As Fit Welding DesigRoot Face T1 T2 Groove Angle Detailed Up Process nation SMAW

B-U5a

GMAW B-U5-GF FCAW

Gas Permitted Shielding Welding for Positions FCAW Notes

U

—

R = 0 to 1 ⁄8 f = 0 to 1⁄8 α = 45°° β = 0°° to 15°°

+1 ⁄16 , − 0 +1 ⁄16 , − 1⁄8 Not limited +1 ⁄16 , − 0 α + β , +10°°, − 0° α + β , +10°°, − 5°

All

—

Br, C, M, N

U

—

R = 0 to 1 ⁄8 f = 0 to 1⁄8 α = 45°° β = 0°° to 15°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 α + β = +10°°, − 0°

All

Not req.

A, Br, C, M, N

+1 ⁄16 , − 1⁄8 Not limited α + β = +10°°, − 5°


8 - 144


Table 8-36 (cont.). Prequalified Welded Joints Complete-Joint-Penetration Groove Welds Double-bevel-groove weld (5) T-joint (T) Corner joint (C)

Welding Process

Joint Designation

SMAW

TC-U5b

GMAW TC-U5-GF FCAW SAW

TC-U5-S

Base Groove Preparation Metal Thickness Tolerances (U = unlimited) Root Opening As As Fit Root Face T1 T2 Groove Angle Detailed Up U

U

U

U

U

U



Notes

R = 0 to 1⁄8 f = 0 to 1 ⁄8 α = 45°

+1 ⁄16 , − 0 +1 ⁄16 , − 0 +10°°, − 0°

+1 ⁄16 , − 1 ⁄8 Not limited +10°°, − 5°

All

—

C, J, M, R, V

All

Not req.

A, C, J, M, R, V

R=0 f = 3 ⁄16 max α = 60°

±0 +0, − 3 ⁄16 +10°°, − 0°

+1 ⁄16 , − 0

F

—

C, J, M, R, V

± 1 ⁄16 +10°°, − 5°


WELDED CONSTRUCTION

8 - 145


Single-U-groove weld (6) Butt joint (B) Corner joint (C)

Welding Process

Joint Designation

SMAW

B-U6

GMAW FCAW


As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄16 , − 1⁄8

α = +10°°, − 0°°

+10°°, − 5°°

f = ± 1 ⁄16

Not limited

r = +1 ⁄8 , − 0

+1 ⁄8 , − 0

Groove Preparation Gas Shielding for FCAW

Notes C, N

T1

T2

Root Opening

Groove Angle

Root Face

Groove Radius


U

U

R = 0 to 1 ⁄8

α = 45°°

f = 1 ⁄8

r = 1 ⁄4

All

—

R = 0 to 1 ⁄8

α = 20°°

f = 1 ⁄8

r = 1 ⁄4

F, OH

—

C, N

R = 0 to 1 ⁄8

α = 45°°

f = 1 ⁄8

r = 1 ⁄4

All

—

C, J, R

C-U6

U

U

R = 0 to 1 ⁄8

α = 20°°

f = 1 ⁄8

r = 1 ⁄4

F, OH

—

C, J, R

B-U6-GF

U

U

R = 0 to 1 ⁄8

α = 20°°

f = 1 ⁄8

r = 1 ⁄4

All

Not req.

A, C, N

C-U6-GF

U

U

R = 0 to 1 ⁄8

α = 20°°

f = 1 ⁄8

r = 1 ⁄4

All

Not req.

A, C, J, R

Double-U-groove weld (7) Butt joint (B)

Welding Process

Joint Designation

SMAW


T2

B-U7

U

—

GMAW FCAW

B-U7-GF

U

—

SAW

B-U7-S

U

—

Tolerances

Tolerances

For B-U7 and B-U7-GF

For B-U7-S

As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄16 , − 1⁄8

R = ±0

+ 1 ⁄16 , − 0

+10°°, − 5°°

α = +0°°, − 1 ⁄4 °

± 1 ⁄16

f = ± 1 ⁄16 , − 0

Not limited

r = +1 ⁄4 , − 0

± 1 ⁄16

Gas Permitted Shielding Groove Welding for Radius Positions FCAW

Groove Angle

Root Face

R = 0 to 1 ⁄8 α = 45°°

f = 1 ⁄8

r = 1 ⁄4

All

R = 0 to 1 ⁄8 α = 20°°

f = 1 ⁄8

r = 1 ⁄4

R = 0 to 1 ⁄8 α = 20°°

f = 1 ⁄8

r = 1 ⁄4 r = 1 ⁄4

R=0

As Fit Up

α = +10°°, − 0°°

Groove Preparation Root Opening

As Detailed

α = 20°° f = 1 ⁄4 max


Notes

—

C, M, N

F, OH

—

C, M, N

All

Not required

A, C, M, N

F

—

C, M, N

8 - 146



Single-J-groove weld (8) Butt joint (B)

Welding Process

Joint Designation

SMAW GMAW FCAW


As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄16 , − 1⁄8

α = +10°°, − 0°°

+10°°, − 5°°

f = ± 1 ⁄16 , − 0

Not limited

r = +1 ⁄4 , − 0

± 1 ⁄16

Groove Preparation Permitted Groove Welding Radius Positions

Notes

T1

T2

Root Opening

Groove Angle

Root Face

B-U8

U

—

R = 0 to 1 ⁄8

α = 45°°

f = 1 ⁄8

r = 3 ⁄8

All

—

Br, C, N

B-U8-GF

U

—

R = 0 to 1 ⁄8

α = 30°°

f = 1 ⁄8

r = 3 ⁄8

All

Not required

A, Br, C, N

Tolerances

Single-J-groove weld (8) T-joint (T) Corner joint (C)

Welding Process

Joint Designation

SMAW

TC-U8a

GMAW FCAW


TC-U8a-GF


Groove Preparation

As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄16 , − 1⁄8

α = +10°°, − 0°°

+10°°, − 5°°

f = ± 1 ⁄16 , − 0

Not limited

r = +1 ⁄4 , − 0

± 1 ⁄16


T1

T2

Root Opening

Groove Angle

Root Face

U

U

R = 0 to 1 ⁄8

α = 45°°

f = 1 ⁄8

r = 3 ⁄8

All

—

C, J, R, V

R = 0 to 1 ⁄8

α = 30°°

f = 1 ⁄8

r = 3 ⁄8

F, OH

—

C, J, R, V

R = 0 to 1 ⁄8

α = 30°°

f = 1 ⁄8

r = 3 ⁄8

All

Not required

A, C, J, R, V

U

U


Notes

WELDED CONSTRUCTION

8 - 147


Double-J-groove weld (9) Butt joint (B)

Welding Process

Joint Designation

SMAW GMAW FCAW


As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄16 , − 1⁄8

α = +10°°, − 0°°

+10°°, − 5°°

f = ± 1 ⁄16 , − 0

Not limited

r = +1 ⁄8 , − 0

± 1 ⁄16

Groove Preparation Permitted Groove Welding Radius Positions

T1

T2

Root Opening

Groove Angle

Root Face

B-U9

U

—

R = 0 to 1 ⁄8

α = 45°°

f = 1 ⁄8

r = 3 ⁄8

All

—

Br, C, M, N

B-U9-GF

U

—

R = 0 to 1 ⁄8

α = 30°°

f = 1 ⁄8

r = 3 ⁄8

All

Not required

A, Br, C, M, N

Welding Process

Joint Designation

SMAW

TC-U9a

TC-U9a-GF

Notes

Tolerances

Double-J-groove weld (9) T-joint (T) Corner joint (C)

GMAW FCAW



Groove Preparation

As Detailed

As Fit Up

R = +1 ⁄16, − 0

+ 1 ⁄16 , − 1⁄8

α = +10°°, − 0°°

+10°°, − 5°°

f = +1 ⁄16 , − 0

Not limited

r = +1 ⁄8 , − 0

± 1 ⁄16


T1

T2

Root Opening

Groove Angle

Root Face

U

U

R = 0 to 1⁄8

α = 45°°

f = 1 ⁄8

r = 3 ⁄8

All

—

C, J, M, R, V

R = 0 to 1⁄8

α = 30°°

f = 1 ⁄8

r = 3 ⁄8

F, OH

—

C, J, M, R, V

R = 0 to 1⁄8

α = 30°°

f = 1 ⁄8

r = 3 ⁄8

All

Not required

A, C, J, M, R, V

U

U


Notes

8 - 148


Table 8-36 (cont.). Prequalified Welded Joints Partial-Joint-Penetration Groove Welds Square-groove weld (1) Butt joint (B)


Welding Process

Joint Designation

T1

SMAW

B-P1a

1 ⁄8

max

B-P1c

1 ⁄4

max


T2

Root Opening

As Detailed

As Fit Up


Effective Throat (E)

Notes

—

R = 0 to 1⁄16

+1 ⁄16 , − 0

± 1 ⁄16

All

T 1 − 1 ⁄32

B

—

T1 R= min 2

+1 ⁄16 , − 0

± 1 ⁄16

All

T1 2

B

Square-groove weld (1) Butt joint (B)

E 1 + E 2 must not exceed

Welding Process

Joint Designation

SMAW

B-P1b

3T1 4


T2

max

—

1 ⁄4

Groove Preparation Tolerances Root Opening

As Detailed

As Fit Up



T1 R= 2

± 1 ⁄16 , − 0

± 1 ⁄16

All

3T1

Notes

4

Single-V-groove weld (2) Butt joint (B) Corner joint (C)

Welding Process

Joint Designation

SMAW

BC-P2

GMAW FCAW

BC-P2-GF

SAW

BC-P2-S

Groove Preparation Base Metal Thickness Tolerances (U = unlimited) Root Opening Root Face As As Fit T1 T2 Up Groove Angle Detailed 1 ⁄4

1 ⁄4

7 ⁄16

min

min

min

U

U

U

R=0 f = 1⁄32 min α - 60°°

0, +1 ⁄16 +u, −0 +10°°, − 0°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄8 min α - 60°°

0, +1 ⁄16 +u, −0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄4 min α - 60°°

±0 +u, −0 +10°°, − 0°°

+1 ⁄16 , − 0*



All

S

B, E, Q2

All

S

A, B, E, Q2

F

S

B, E, Q2

± 1 ⁄16 +10°°, − 5° ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 +10°°, − 5°°


Notes

WELDED CONSTRUCTION

8 - 149

Table 8-36 (cont.). Prequalified Welded Joints Partial-Joint-Penetration Groove Welds Double-V-groove weld (3) Butt joint (B)

Welding Process

Joint Designation

SMAW

B-P3

GMAW FCAW

B-P3-GF

SAW

B-P3-S

Base Groove Preparation Metal Thickness Tolerances (U = unlimited) Root Opening Root Face As As Fit T1 T2 Groove Angle Detailed Up 1 ⁄2

1 ⁄2

3 ⁄4

min

min

min

—

—

—

R=0 f = 1 ⁄8 min α = 60°°

+1⁄16 , − 0 +u, −0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄8 min α = 60°°

+1⁄16 , − 0 +u, −0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄4 min α = 60°°

±0 +u, −0 +10°°, − 0°°

+1 ⁄16 , − 0*



All

S

E, Mp, Q2

All

S

A, E, Mp, Q2

F

S

E, Mp, Q2

± 1 ⁄16

Notes

+10°°, − 5°° ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 +10°°, − 5°°


8 - 150


Table 8-36 (cont.). Prequalified Welded Joints Partial-Joint-Penetration Groove Welds Single-bevel-groove weld (4) Butt joint (B) T-joint (T) Corner joint (C)

Welding Process

Joint Designation

SMAW

BTC-P4

GMAW FCAW

BTC-P4-GF

SAW

TC-P4-S

Base Groove Preparation Metal Thickness Tolerances (U = unlimited) Root Opening Root Face As As Fit T1 T2 Groove Angle Detailed Up U

1 ⁄4

7 ⁄16

U

U

min

min

U

R=0 f = 1 ⁄8 min α = 45°°

+1 ⁄16 , − 0 unlimited +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄8 min α = 45°°

+1 ⁄16 , − 0 unlimited* +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄4 min α = 60°°

±0 +u, −0 +10°°, − 0°°

+1 ⁄16 , − 0



All

S − 1⁄8

B, E, J2, Q2, V

F, H V, OH

S

S − 1 ⁄8

A, B, E, J2, Q2, V

F

S



All

(S − 1 ⁄8 ) − 1 ⁄4

E, J2, L, Mp, Q2, V

All

(S 1 + S 2 )

A, E, J2, L, Mp, Q2, V

± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 +10°°, − 5°°

Notes

B, E, J2, Q2, V

Double-bevel-groove weld (5) Butt joint (B) T-joint (T) Corner joint (C)

Welding Process

Joint Designation

SMAW

BTC-P5

GMAW FCAW

BTC-P5-GF

SAW

TC-P5-S

Base Groove Preparation Metal Thickness Tolerances (U = unlimited) Root Opening Root Face As As Fit T1 T2 Groove Angle Detailed Up 5 ⁄16

1 ⁄2

3 ⁄4

min

min

min

U

U

U

R=0 f = 1 ⁄8 min α = 45°°

+1 ⁄16 , − 0 unlimited +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄8 min α = 45°°

+1 ⁄16 , − 0 unlimited +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄4 min α = 60°°

±0 unlimited +10°°, − 0°°

+1 ⁄16 , − 0*

± 1 ⁄16

+10°°, − 5°° ± 1 ⁄16 +10°°, − 5°°

± 1 ⁄16 +10°°, − 5°°

*For flat and horizontal postiions f = + u , −0


− 1 ⁄4

F

S1 + S2

Notes

E, J2, L, Mp, Q2, V

WELDED CONSTRUCTION

8 - 151

Table 8-36 (cont.). Prequalified Welded Joints Partial-Joint-Penetration Groove Welds Single-U-groove weld (6) Butt joint (B) Corner joint (C)

Welding Process

Joint Designation

SMAW

BC-P6

GMAW FCAW

BC-P6-GF

SAW

BC-P6-S

Base Groove Preparation Metal Thickness Root Opening Tolerances (U = unlimited) Root Face Groove Radius As As Fit T1 T2 Groove Angle Detailed Up 1 ⁄4

1 ⁄4

7 ⁄16

U

min

U

min

min

U

R=0 f = 1 ⁄32 min r = 1 ⁄4 α = 45°°

+1 ⁄16 , − 0 +u, −0 +1 ⁄4 , − 0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄8 min r = 1 ⁄4 α = 20°°

+1 ⁄16 , − 0 +u, −0 +1 ⁄4 , − 0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄4 min r = 1 ⁄4 α = 20°°

±0 +u, −0 +1 ⁄4 , − 0 +10°°, − 0°°

+1 ⁄16 , − 0



All

S

B, E, Q2

All

S

A, B, E, Q2

F

S

B, E, Q2



All

S1 + S2

E, Mp, Q2

All

S1 + S2

A, E, Mp, Q2

F

S1 + S2

E, Mp, Q2

± 1 ⁄16 ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 ± 1 ⁄16 +10°°, − 5°°

Notes

Double-U-groove weld (7) Butt joint (B)

Welding Process

Joint Designation

SMAW

B-P7

GMAW FCAW

B-P7-GF

SAW

B-P7-S

Base Groove Preparation Metal Thickness Root Opening Tolerances (U = unlimited) Root Face Groove Radius As As Fit T1 T2 Groove Angle Detailed Up 1 ⁄2

1 ⁄2

3 ⁄4

min

min

min

—

—

—

R=0 f = 1 ⁄8 min r = 1 ⁄4 α = 45°°

+1 ⁄16 , − 0 +u, −0 +1 ⁄4 , − 0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄8 min r = 1 ⁄4 α = 20°°

+1 ⁄16 , − 0 +u, −0 +1 ⁄4 , − 0 +10°°, − 0°°

+1 ⁄8 , − 1 ⁄16

R=0 f = 1 ⁄4 min r = 1 ⁄4 α = 20°°

±0 +u, −0 +1 ⁄4 , − 0 +10°°, − 0°°

+1 ⁄16 , − 0

± 1 ⁄16 ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 ± 1 ⁄16 +10°°, − 5°° ± 1 ⁄16 ± 1 ⁄16 +10°°, − 5°°


Notes

8 - 152


Table 8-36 (cont.). Prequalified Welded Joints Partial-Joint-Penetration Groove Welds Single-J-groove weld (8) Butt joint (B) T-joint (T) Corner joint (C)


Joint Designation TC-P8*

SMAW


T2

1 ⁄4

min

U

BC-P8**

1 ⁄4

min

U

GMAW FCAW

TC-P8-GF*

1 ⁄4

min

U

GMAW FCAW

BC-P8-GF**

1 ⁄4

min

U

SAW

TC-P8-S*

7 ⁄16

min

U

SAW

C-P8-S**

7 ⁄16

min

U

Groove Preparation Root Opening Tolerances Root Face As As Fit Groove Radius Up Groove Angle Detailed R=0 +1 ⁄16 , − 0 +1 ⁄8 , − 1 ⁄16 1 f = ⁄8 min ± 1 ⁄16 +u, −0 r = 3 ⁄8 +1 ⁄4 , − 0 ± 1 ⁄16 α = 45°° +10°°, − 0°° +10°°, − 5°° R=0 +1 ⁄16 , − 0 +1 ⁄8 , − 1 ⁄16 f = 1 ⁄8 min ± 1 ⁄16 +u, −0 3 r = ⁄8 +1 ⁄4 , − 0 ± 1 ⁄16 α = 30°° +10°°, − 0°° +10°°, − 5°° R=0 +1 ⁄16 , − 0 +1 ⁄8 , − 1 ⁄16 f = 1 ⁄8 min ± 1 ⁄16 +u, −0 3 r = ⁄8 +1 ⁄4 , − 0 ± 1 ⁄16 α = 45°° +10°°, − 0°° +10°°, − 5°° 1 1 R=0 + ⁄16 , − 0 + ⁄8 , − 1 ⁄16 f = 1 ⁄8 min ± 1 ⁄16 +u, −0 r = 3 ⁄8 +1 ⁄4 , − 0 ± 1 ⁄16 α = 30°° +10°°, − 0°° +10°°, − 5°° 1 R=0 ±0 + ⁄16 , − 0 f = 1 ⁄4 min ± 1 ⁄16 +u, −0 r = 1 ⁄2 +1 ⁄4 , − 0 ± 1 ⁄16 α = 45°° +10°°, − 0°° +10°°, − 5°° R=0 ±0 +1 ⁄16 , 0 f = 1 ⁄4 min ± 1 ⁄16 +u, −0 r = 1 ⁄2 +1 ⁄4 , − 0 ± 1 ⁄16 α = 20°° +10°°, − 0°° +10°°, − 5°°

*Applies to inside corner joints. **Applies to outside corner joints.


Permitted Effective Welding Throat Positions (E) All S

Notes E, J2, Q2, V

All

S

E, J2, Q2, V

All

S

A, E, J2, Q2, V

All

S

A, E, J2, Q2, V

F

S

E, J2, Q2, V

F

S

E, J2, Q2, V

WELDED CONSTRUCTION

8 - 153

Table 8-36 (cont.). Prequalified Welded Joints Flare Welds Double-J-groove weld (9) Butt joint (B) T-joint ( T) Corner joint (C)


Joint Designation BTC-P9*

GMAW FCAW


T2

1 ⁄2

min

U

BTC-P9-GF**

1 ⁄2

min

U

SAW

C-P9-S*

3 ⁄4

min

U

SAW

C-P9-S**

3 ⁄4

min

U

SAW

T-P9-S

3 ⁄4

min

U

Groove Preparation Root Opening Tolerances Root Face As As Fit Groove Radius Up Groove Angle Detailed 1 1 R=0 + ⁄16 , − 0 + ⁄8 , − 1 ⁄16 f = 1 ⁄8 min ± 1 ⁄16 +u, −0 r = 3 ⁄8 ± 1 ⁄16 +1 ⁄4 , − 0 α = 45°° +10°°, − 0°° +10°°, − 5°° 1 1 R=0 + ⁄16 , − 0 + ⁄8 , − 1 ⁄16 f = 1 ⁄8 min ± 1 ⁄16 +u, −0 r = 3 ⁄8 +1 ⁄4 , − 0 ± 1 ⁄16 α = 30°° +10°°, − 0°° +10°°, − 5°° R=0 ±0 +1 ⁄16 , − 0 f = 1 ⁄4 min ± 1 ⁄16 +u, −0 r = 1 ⁄2 +1 ⁄4 , − 0 ± 1 ⁄16 α = 45°° +10°°, − 0°° +10°°, − 5°° R=0 ±0 − 1⁄16 , 0 f = 1 ⁄4 min ± 1 ⁄16 +u, −0 r = 1 ⁄2 +1 ⁄4 , − 0 ± 1 ⁄16 α = 20°° +10°°, − 0°° +10°°, − 5°° R=0 ±0 +1 ⁄16 , 0 f = 1 ⁄4 min ± 1 ⁄16 +u, −0 1 r = 1 ⁄2 + ⁄4 , − 0 ± 1 ⁄16 α = 45°° +10°°, − 0°° +10°°, − 5°°

Permitted Effective Welding Throat Positions (E) All S1 + S2

Notes E, J2, Mp, Q2, V

All

S1 + S2

A, J2, Mp, Q2, V

F

S1 + S2

E, J2, Mp, Q2, V

F

S1 + S2

E, J2, Mp, Q2, V

F

S1 + S2

E, J2, Mp, Q2

Single-J-groove weld (B) Butt joint (B) T-joint (T) Corner joint (C)


Joint Designation BTC-P10


T2

3 ⁄16

U

min GMAW FCAW

BTC-P10-GF

SAW

T-P10-S

3 ⁄16

U

min 1 ⁄2

1 ⁄2

min

min

Groove Preparation

Tolerances Root Opening As As Fit Root Face T3 Bend Radius Detailed Up 1 1 T 1 min R=0 + ⁄16 , − 0 + ⁄8 , − 1 ⁄16 f = 3⁄16 min +U, − 1 ⁄16 +U, −0 3T1 − 0, +Not- − 0, +NotC= min Limited Limited 2 T 1 min R=0 +1⁄16 , − 0 + 1 ⁄8, − 1⁄16 f = 3⁄16 min +U, − 1 ⁄16 +U, −0 3T 1 − 0, +Not- − 0, +NotC= min Limited Limited 2 N/A R=0 ±0 + 1 ⁄16, −0 1 f = ⁄2 min +U, − 1 ⁄16 +U, −0 3T1 − 0, +Not- − 0, +NotC= min Limited Limited 2

*Applies to inside corner joints. **Applies to outside corner joints.


Permitted Effective Welding Throat Positions (E) Notes 5 ⁄8 T 1 J2, All Q2, Z

All

5 ⁄8 T 1

A, J2, Q2, Z

F

5 ⁄8 T 1

J2, Q2, Z

8 - 154


ECCENTRICALLY LOADED WELD GROUPS

When the line of action of an applied load does not pass through the center of gravity (CG) of a weld group, the load is eccentric and results in a moment which must be considered in the design of the connection. Eccentricity in the Plane of the Faying Surface

Eccentricity in the plane of the faying surface produces additional shear and the welds must then be designed to resist the combined effect of the direct shear from the applied load Pu and the additional shear from the induced moment Pu e. Two methods of analysis for this type of eccentricity will be discussed: (1) the instantaneous center of rotation method; and, (2) the elastic method. Instantaneous Center of Rotation Method

Also known as the ultimate strength method (Crawford, 1968), this method considers the load-deformation relationship of each weld element as well as the variation in weld strength with respect to the direction of the applied force and, thus, more accurately predicts the ultimate strength of the eccentrically loaded connection (Butler, Pal, and Kulak, 1972). Eccentricity produces both a rotation about the centroid of the weld group and a translation of one connected element with respect to the other. The combined effect of this rotation and translation is equivalent to a rotation about a point defined as the instantaneous center of rotation (IC) as illustrated in Figure 8-51a. The location of the IC depends on the geometry of the weld group as well as the direction and point of application of the load. The individual resistance of each unit weld element is assumed to act on a line perpendicular to a ray passing through the instantaneous center and the centroid of that element, as illustrated in Figure 8-51b. The load-deformation relationship of a single unit-weld element was originally given by Butler, Pal, and Kulak (1972) for E60 electrodes. New strength curves for E70 electrodes (Lesik and Kennedy, 1990) are illustrated in Figure 8-52, where: R = 0.60FEXX(1.0 + 0.50 sin1.5θ) [p (1.9 − 0.9p)]0.3 In the above equation, R FEXX θ p

= shear force per unit area in a single unit-weld element at a deformation ∆, kips = weld electrode strength, ksi = angle of loading measured from the weld longitudinal axis, degrees = ratio of element deformation to its deformation at maximum stress

Unlike the load-deformation relationship for bolts, strength and deformation of welds are dependent on the angle θ that the resultant elemental force makes with the axis of the weld element. The critical weld element is usually the weld element farthest from the IC. While this may not always be the case, for the purpose of explanation, this will be assumed. The maximum deformation ∆max may be determined as ∆max = 1.087w (θ + 6)−0.65 ≤ 0.17w where w is the leg size of the weld and θ is expressed in degrees. The deformation of other weld elements is assumed to vary linearly with distance from the IC as, AMERICAN INSTITUTE OF STEEL CONSTRUCTION


∆=

lr lr max

8 - 155

∆max

More discussion of this method is contained in LRFD Specification Appendix J2.4 and its Commentary. These new provisions permit, for the first time, weld strength to exceed the 0.6FEXX nominal value, which is the least strength applicable to longitudinally loaded (θ = 0°) elements. Load-deformation curves in Figure 8-52 for values of θ = 0°, 30°, 45°, Pu

e

lo

CG

IC

(a) Instantaneous center of rotation (IC)

lo e

IC

CG

l ax

rm

ru max

(b) Forces on weld elements Fig. 8-51. Instantaneous center of rotation method. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Pu

8 - 156


60°, 75°, and 90° are shown relative to Ro = 0.6FEXX. The ductility of the weld group is governed by ∆max of the element that first reaches its limit. The total strength of all weld elements is the sum of the individual resistances of all welds in the group. If the correct location of the instantaneous center has been selected, the three equations of statics will be satisfied, i.e., ΣFx = 0, ΣFy = 0, ΣM = 0. Because of the non-linear nature of the requisite iterative solution, a minimum of twenty weld elements for the longest line segment is generally recommended for sufficient accuracy. Tables 8-38 through 8-45 employ the instantaneous center of rotation method in accordance with LRFD Specification Appendix J2.4 for the weld patterns and eccentric conditions indicated and inclined loads at 0°, 15°, 30°, 45°, 60°, and 75°. Thus, unlike the First Edition LRFD Manual, tabulated values are not limited to a maximum weld nominal strength of 0.6FEXX. For some cases, significant increases of up to 50 percent of values tabulated previously are possible; many values reflect more moderate but, nevertheless, substantial increases on the order of 10 to 30 percent. The traditional and more conservative designs based upon a constant fillet weld nominal strength of 0.6FEXX is also permitted, refer to AISC (1986). For any of the weld group geometrics shown, the design strength of the eccentrically loaded weld group is φRn, where In the above equation, 1.6

θ = 90° θ = 75°

1.4

θ = 60° θ = 45°

1.2 θ = 30°

θ = 0°

1.0

( RR ) o

0.8

0.6

0.4 Ro = 0.6Fexx

0.2

0 0.000

0.050

∆ w

0.100

Fig. 8-52. Fillet weld strength as a function of force angle, θ. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

0.150


8 - 157

C = tabular value (which includes φ = 0.75) φRn = CC1Dl C1 = electrode coefficient from Table 8-37 which adjusts tabular value, which is based on E70XX electrodes, for other electrodes. Note that this coefficient includes an additional reduction factor of 0.90 for E80 and E90 electrodes and 0.85 for E100 and E110; this accounts for the uncertainty of extrapolation to the higher strength electrodes. D = number of sixteenths-of-an-inch in the weld size l = length of the reference weld, in. The first line in each table (a = 0) gives the design strength of a concentrically loaded weld group in accordance with LRFD Specification Appendix J2.2a. Linear interpolation within a given table between adjacent a and k values is permitted. Figure C-J2.5 from LRFD Specification Commentary Section J2 indicates that, for equal-leg fillet welds, the area of the fusion surface is always larger than the leg dimension times the weld length. Therefore, the tabulated values are based upon the strength through the throat of the weld of (0.75 × 0.6 × FEXX × 0.707 × 1⁄16) Tabulated values are valid for weld metal with a strength level equal to or matching the base material. A convergence criterion of less than 0.5 percent unbalanced force was employed for the tabulated iterative solutions. Straight line interpolation between these angles may be significantly unconservative. Therefore, unless a direct analysis is performed, use only the values tabulated for the next lower angle. Since the coefficients in these tables were derived from physical tests with loading at ultimate strength levels, they should be used only for the weld patterns indicated and not in combination with any additional loading. In cases not treated by these tables, a special ultimate strength analysis is required if the instantaneous center of rotation method is to be used.

Example 8-3

Given:

Refer to Figure 8-53. Determine the largest eccentric force Pu for which the design shear strength of the welds in the connection is adequate using the instantaneous center of rotation method. Use 3⁄8-in. fillet weld and 70 ksi electrode weld size. A. Assume the load is vertical as illustrated in Figure 8-53 (θ = 0°°) B. Assume the load acts at an angle of 75°° with respect to vertical (θ = 75°°)

Solution A:

l = 10 in. kl = 5 in. k = 0.5 From Table 8-42 with θ = 0°, x = 0.125 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 158


Table 8-37. Electrode Strength Coefficients Electrode

FEXX (ksi)

C1

E60 E70 E80 E90 E100 E110

60 70 80 90 100 110

0.857 1.00 1.03 1.16 1.21 1.34

xl + al = 10 in. 0.125(10 in.) + a (10 in.) = 10 in. a = 0.875 By interpolation from Table 8-42 with θ = 0°, C = 1.41 Design shear strength φRn = CC1Dl = 1.41(1.0)(6 sixteenths)(10 in.) = 84.6 kips Comment:

Note that this eccentricity has effectively reduced the shear strength of this weld group by 60 percent when compared with the eccentrically loaded case.

Solution B:

From Solution A, k = 0.5 a = 0.875 By interpolation from Table 8-42 with θ = 75°°, C = 2.59 Design shear strength φRn = CC1Dl = 2.59(1.0)(6 sixteenths)(10 in.) = 155 kips

Comment:

In Solution B, the vertical component of the design strength is φRn sin75°° = (155 kips)(0.966) = 150 kips and the horizontal component of the design strength is AMERICAN INSTITUTE OF STEEL CONSTRUCTION


8 - 159

φRn cos75°° = (155 kips)(0.259) = 40.1 kips Elastic Method

Alternatively, the elastic method may be used to analyze eccentrically loaded weld groups. It offers a simplified, conservative approach but does not render a consistent factor of safety and, in some cases, provides excessively conservative results. Furthermore, the elastic method ignores both the ductility of the weld group and the load redistribution which occurs. Refer to Higgins (1971). In the elastic method, for a force applied parallel to the Y principle axis of the weld group, the eccentric force Pu is resolved into a force Pu acting through the center of gravity (CG) of the weld group and a moment Pu e where e is the eccentricity. Each weld element is then assumed to support an equal share of the concentric force Pu, and a share of the eccentric moment Pu e which is proportional to its distance from the CG. The weld most remote from the CG, then, is the most highly stressed. The resultant vectorial sum of these forces ru is the required strength for the weld element. The shear force per linear inch of weld due to the concentric force Pu is r1, where r1 =

Pu l

and l is the total length of the weld measured along the axis of each element. The shear force per linear inch of weld due to the moment Pu e varies with distance from the CG and will be maximum in the weld element which is most remote from the CG. The maximum shear due to the moment Pu e is rm, where rm =

Pu ec Ip

In the above equation,

10 in.

Pu

k l = 5 in.

l = 10 in.

1.25 in.

Figure 8-53. Illustration for Example 8-3 and 8-4. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 160


c = distance from CG to point on weld most remote from CG, in. Ip = polar moment of inertia of the weld group, in.4 per in.2 (Ip = Ix + Iy). Refer to Figure 8-54. For section moduli and torsional constants of various welds treated as line elements, refer to Table 5 (page 7.4–7) of Blodgett (1966). To determine the resultant force on the most highly stressed weld element, rm must be resolved into vertical component r2 and horizontal component r3, where Pu ecx Ip Pu ecy r3 = Ip

r2 =

In the above equations, cx and cy are the horizontal and vertical components of the diagonal distance c. Thus, the resultant force is ru, where r2 r1

ru = √  (r1 + r2)2 + (r3)2

r3 rm ru

and the weld size must be chosen such that the design strength of the weld exceeds the required strength ru. For the more general case of an inclined eccentric force, i.e., not parallel to the Y principle axis of the bolt group, the effect of the X-direction component of the direct shear must also be included. Refer to Iwankiw (1987).

Example 8-4

Given:

Refer to Example 8-3a. Recalculate the largest eccentric force Pu for which the design shear strength of the welds in the connection is adequate using the elastic method. Compare the result with that of Example 8-3a. Use 3⁄8-in. weld size, E70XX electrodes Ip = 385 in.4 per in.2

Solution:

Direct shear force per inch of weld Pu l Pu = 20 in.

r1 =

Additional shear force on weld due to eccentricity Pu ecx Ip Pu (8.75 in.) (3.75 in.) = 385 in.4 per in.2 = 0.0852Pu

r2 =



r3 =

8 - 161

Pu ecy

Ip

Pu (8.75 in.) (5 in.) = 385 in.4 per in.2 = 0.114Pu Resultant shear force per inch of weld  (r1 + r2)2 + (r3)2 ru = √

y

(p) x

xo cg (po )

x

yo

y

xo

xo

x

yo

y

l = 6.283R

x (p)

x

n

y

x (p)

ln2 12

ln2 + l(dy)2 12 lm2 Iyo = 12 lm2 Iy = + l(dx)2 12 Ix =

yo a

xo

cg (po ) xo

dx

y

xo

R

dy

yo

xo

m

Ixo =

y

a = 0.637R l = 3.142R

Ixo = πR 3 Ix = πR 3 + l(dy)2 Iyo = πR 3 Iy = πR 3 + l(dx)2

(p) x

yo

R

a

R

xo cg (po ) x

yo d y x

cg (po )

dy

xo

dx

xo

Ixo = 0 Ix = l(dy)2 l3 Iyo = 12 l3 Iy = + l(dx)2 12

12 l3 Ix = + l(dy)2 12 Iyo = 0 Iy = l(dx)2

cg (po )

y

yo

l3

l /2

/2

dy

xo

y

dx

dy

Ixo =

l l

a

yo

yo

y

dx

l

dy

xo x

yo

y

dy

dx cg (po )

l/ 2

l

yo

x (p)

x yo

y

x (p)

a = 0.637R l = 1.571R

Ix =

π 3 R + l(dy)2 2 π 4  Iyo =  −  R 3  2 π

π 2  Ixo =  −  R 3  4 π π 2  3 Ix =  −  R + l(dy)2  4 π π 2  Iyo =  −  R 3  4 π

π 4  Iy =  −  R 3 2 π

π 2  Iy =  −  R 3 + l(dx)2  4 π

Ixo =

π 3 R 2

Fig. 8-54. Moments of inertia of various weld segments. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 162


 √

  Pu 2  20 + 0.0852Pu  + (0.114Pu )   = 0.177Pu 2

=

Since ru must be less than or equal to φrn, φrn 0.177 1.392D ≤ 0.177 1.392 (6 sixteenths) ≤ 0.177 ≤ 47.2 kips

Pu ≤

This is a 44 percent reduction in the strength predicted by the instantaneous center of rotation method in Example 8-3a.



8 - 163

Table 8-38. Coefficients C for Eccentrically Loaded Weld Groups Angle = 0°° φ R n = CC1Dl

C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D ex = a l Pu

l

where Pu = factored force, kips

D = number of sixteenths-of-an-inch in the fillet weld size l = characteristic length of weld group, in. a = e x / l, in. e x = horizontal component of eccentricity of Pu with respect to centroid of weld group, in.

kl ex = a l

C = coefficient tabulated below which includes φ = 0.75 C 1 = electode strength coefficient from Table 8-37 (1.0 for E70XX electrodes

P

Special Case

u

(Load not in plane of weld group) Use C-values for k = 0 Any equal distances

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.78 2.78 2.75 2.64 2.48

2.78 2.78 2.75 2.63 2.48

2.78 2.78 2.74 2.63 2.48

2.78 2.78 2.73 2.62 2.47

2.78 2.78 2.71 2.60 2.47

2.78 2.77 2.70 2.59 2.46

2.78 2.75 2.69 2.58 2.46

2.78 2.74 2.67 2.57 2.45

2.78 2.73 2.66 2.56 2.45

2.78 2.71 2.64 2.55 2.44

2.78 2.70 2.63 2.54 2.44

2.78 2.67 2.60 2.52 2.43

2.78 2.64 2.58 2.50 2.41

2.78 2.61 2.55 2.48 2.40

2.78 2.59 2.53 2.46 2.39

2.78 2.78 2.50 2.44 2.38

0.30 0.40 0.50 0.60 0.70

2.32 2.00 1.72 1.50 1.32

2.32 2.00 1.72 1.50 1.32

2.32 2.01 1.74 1.52 1.34

2.32 2.03 1.77 1.55 1.38

2.33 2.05 1.80 1.59 1.42

2.33 2.07 1.83 1.63 1.47

2.33 2.08 1.86 1.67 1.51

2.33 2.10 1.89 1.71 1.55

2.33 2.11 1.91 1.74 1.59

2.33 2.12 1.93 1.77 1.62

2.33 2.14 1.95 1.79 1.65

2.33 2.15 1.99 1.84 1.71

2.33 2.16 2.01 1.87 1.75

2.32 2.17 2.03 1.90 1.79

2.32 2.18 2.05 1.92 1.81

2.31 2.18 2.06 1.94 1.84

0.80 0.90 1.00 1.20 1.40

1.17 1.05 0.957 0.806 0.695

1.18 1.06 0.963 0.812 0.701

1.20 1.08 0.986 0.835 0.724

1.24 1.12 1.02 0.872 0.758

1.28 1.17 1.07 0.916 0.799

1.33 1.22 1.12 0.963 0.844

1.38 1.27 1.17 1.01 0.889

1.42 1.31 1.21 1.06 0.932

1.46 1.35 1.26 1.10 0.973

1.50 1.39 1.29 1.14 1.01

1.53 1.43 1.33 1.17 1.05

1.59 1.49 1.40 1.24 1.12

1.64 1.54 1.45 1.30 1.18

1.68 1.59 1.50 1.35 1.23

1.71 1.62 1.54 1.40 1.28

1.74 1.66 1.58 1.44 1.32

1.60 1.80 2.00 2.20 2.40

0.611 0.544 0.491 0.447 0.410

0.616 0.550 0.496 0.452 0.415

0.638 0.570 0.515 0.470 0.431

0.670 0.600 0.542 0.495 0.455

0.708 0.635 0.576 0.526 0.484

0.750 0.674 0.612 0.560 0.516

0.792 0.714 0.650 0.596 0.550

0.833 0.753 0.687 0.631 0.583

0.873 0.791 0.723 0.665 0.616

0.911 0.828 0.758 0.699 0.648

0.947 0.863 0.792 0.731 0.679

1.01 0.928 0.855 0.792 0.738

1.07 0.987 0.912 0.848 0.792

1.13 1.04 0.964 0.899 0.842

1.17 1.09 1.01 0.945 0.887

1.22 1.13 1.05 0.988 0.929

2.60 0.379 0.384 0.399 0.421 0.448 0.478 0.510 0.542 0.573 0.604 0.634 0.691 0.743 0.791 0.836 0.877 2.80 0.352 0.357 0.371 0.392 0.417 0.446 0.476 0.506 0.536 0.565 0.594 0.649 0.699 0.746 0.790 0.830 3.00 0.329 0.333 0.347 0.366 0.390 0.417 0.446 0.474 0.503 0.531 0.559 0.611 0.661 0.706 0.748 0.788


8 - 164


Table 8-38 (cont.). Coefficients C for Eccentrically Loaded Weld Groups Angle = 15°° φ R n = CC1Dl

C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D ex = a l Pu

15° l



kl ex = a l Pu

C = coefficient tabulated below which includes φ = 0.75 C 1 = electode strength coefficient from Table 8-37 (1.0 for E70XX electrodes)

15°

Special Case


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.97 2.84 2.76 2.63 2.48

2.97 2.84 2.76 2.63 2.48

2.97 2.84 2.75 2.63 2.48

2.97 2.83 2.75 2.62 2.48

2.97 2.82 2.74 2.62 2.49

2.97 2.82 2.73 2.62 2.49

2.97 2.81 2.72 2.61 2.49

2.97 2.80 2.72 2.61 2.49

2.97 2.80 2.71 2.61 2.49

2.97 2.79 2.70 2.61 2.50

2.97 2.78 2.70 2.60 2.50

2.97 2.77 2.69 2.60 2.50

2.97 2.75 2.68 2.59 2.51

2.97 2.74 2.67 2.59 2.51

2.97 2.73 2.66 2.58 2.51

2.97 2.72 2.65 2.58 2.51

0.30 0.40 0.50 0.60 0.70

2.32 2.01 1.74 1.52 1.34

2.32 2.01 1.74 1.52 1.35

2.32 2.02 1.76 1.54 1.37

2.33 2.04 1.78 1.57 1.40

2.34 2.06 1.82 1.62 1.45

2.35 2.09 1.86 1.66 1.50

2.36 2.12 1.89 1.70 1.54

2.37 2.14 1.93 1.75 1.59

2.38 2.16 1.96 1.78 1.63

2.39 2.18 1.99 1.82 1.67

2.39 2.19 2.01 1.85 1.70

2.41 2.22 2.05 1.90 1.77

2.42 2.25 2.09 1.95 1.82

2.43 2.27 2.12 1.99 1.87

2.43 2.28 2.15 2.02 1.90

2.43 2.30 2.17 2.05 1.94

0.80 0.90 1.00 1.20 1.40

1.20 1.08 0.979 0.826 0.714

1.20 1.08 0.985 0.832 0.719

1.22 1.11 1.01 0.856 0.743

1.26 1.15 1.05 0.893 0.778

1.31 1.19 1.09 0.938 0.820

1.36 1.24 1.15 0.987 0.866

1.41 1.30 1.20 1.04 0.913

1.46 1.34 1.25 1.09 0.960

1.50 1.39 1.29 1.13 1.00

1.54 1.43 1.33 1.17 1.05

1.58 1.47 1.37 1.21 1.09

1.65 1.54 1.45 1.29 1.16

1.71 1.60 1.51 1.35 1.23

1.76 1.66 1.57 1.41 1.28

1.80 1.70 1.62 1.46 1.34

1.84 1.74 1.66 1.51 1.39

1.60 1.80 2.00 2.20 2.40

0.628 0.560 0.506 0.461 0.423

0.633 0.566 0.511 0.466 0.428

0.656 0.587 0.530 0.484 0.445

0.688 0.617 0.558 0.510 0.469

0.727 0.653 0.592 0.541 0.499

0.770 0.693 0.630 0.577 0.532

0.815 0.735 0.669 0.613 0.566

0.859 0.777 0.708 0.650 0.601

0.901 0.817 0.746 0.687 0.636

0.941 0.855 0.783 0.722 0.670

0.980 0.893 0.819 0.757 0.703

1.05 0.963 0.887 0.822 0.765

1.12 1.03 0.949 0.882 0.824

1.18 1.09 1.01 0.938 0.878

1.23 1.14 1.06 0.989 0.928

1.28 1.19 1.11 1.04 0.974

2.60 0.391 0.396 0.412 0.434 0.462 0.493 0.526 0.559 0.591 0.624 0.656 0.716 0.772 0.825 0.873 0.918 2.80 0.363 0.368 0.383 0.404 0.430 0.460 0.491 0.522 0.553 0.584 0.614 0.672 0.727 0.778 0.825 0.869 3.00 0.339 0.344 0.358 0.378 0.403 0.430 0.460 0.489 0.519 0.549 0.578 0.634 0.686 0.736 0.781 0.824



8 - 165


C min =

Pu

D min =

C1Dl

Pu

Pu

lmin =

CC 1 l

CC 1 D

ex = a l

Pu 30° l



kl ex = a l

Pu


30°

Special Case


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.28 3.03 2.87 2.72 2.57

3.28 3.03 2.87 2.73 2.57

3.28 3.04 2.87 2.73 2.57

3.28 3.04 2.87 2.73 2.58

3.28 3.04 2.88 2.74 2.59

3.28 3.05 2.88 2.74 2.61

3.28 3.05 2.88 2.75 2.62

3.28 3.06 2.89 2.76 2.63

3.28 3.06 2.90 2.76 2.64

3.28 3.06 2.90 2.77 2.66

3.28 3.07 2.91 2.78 2.67

3.28 3.07 2.92 2.79 2.68

3.28 3.07 2.94 2.81 2.70

3.28 3.06 2.94 2.82 2.72

3.28 3.06 2.95 2.83 2.73

3.28 3.05 2.95 2.84 2.74

0.30 0.40 0.50 0.60 0.70

2.41 2.11 1.84 1.62 1.44

2.41 2.11 1.85 1.63 1.45

2.42 2.12 1.86 1.65 1.47

2.43 2.14 1.89 1.68 1.51

2.45 2.17 1.93 1.73 1.56

2.47 2.20 1.98 1.78 1.61

2.49 2.24 2.02 1.83 1.66

2.51 2.28 2.06 1.88 1.72

2.53 2.30 2.10 1.92 1.77

2.54 2.33 2.13 1.96 1.81

2.56 2.35 2.17 2.00 1.85

2.59 2.40 2.22 2.07 1.92

2.61 2.43 2.27 2.12 1.99

2.63 2.46 2.31 2.17 2.04

2.64 2.49 2.34 2.21 2.09

2.66 2.51 2.37 2.25 2.13

0.80 0.90 1.00 1.20 1.40

1.30 1.17 1.07 0.907 0.786

1.30 1.18 1.08 0.913 0.792

1.33 1.20 1.10 0.939 0.817

1.37 1.24 1.14 0.979 0.855

1.42 1.29 1.19 1.03 0.901

1.47 1.35 1.25 1.08 0.951

1.52 1.41 1.30 1.14 1.00

1.58 1.46 1.36 1.19 1.05

1.63 1.51 1.41 1.24 1.10

1.68 1.56 1.46 1.29 1.15

1.72 1.61 1.51 1.33 1.20

1.80 1.69 1.59 1.42 1.28

1.87 1.76 1.66 1.49 1.35

1.93 1.82 1.73 1.56 1.42

1.98 1.88 1.78 1.62 1.49

2.02 1.92 1.84 1.68 1.54

1.60 1.80 2.00 2.20 2.40

0.693 0.619 0.559 0.510 0.469

0.699 0.625 0.565 0.516 0.474

0.723 0.648 0.587 0.536 0.493

0.759 0.681 0.617 0.564 0.520

0.801 0.721 0.655 0.599 0.552

0.848 0.765 0.696 0.638 0.589

0.897 0.811 0.740 0.679 0.628

0.946 0.857 0.783 0.720 0.667

0.993 0.901 0.825 0.761 0.705

1.04 0.945 0.866 0.800 0.743

1.08 0.987 0.907 0.838 0.779

1.16 1.07 0.984 0.912 0.850

1.24 1.14 1.05 0.981 0.917

1.31 1.21 1.12 1.04 0.978

1.37 1.27 1.18 1.10 1.04

1.42 1.32 1.24 1.16 1.09

2.60 0.433 0.439 0.456 0.481 0.512 0.546 0.583 0.620 0.657 0.693 0.728 0.795 0.860 0.920 0.975 1.03 2.80 0.403 0.408 0.424 0.448 0.477 0.509 0.544 0.579 0.614 0.649 0.683 0.748 0.809 0.867 0.922 0.972 3.00 0.376 0.382 0.397 0.419 0.446 0.477 0.510 0.543 0.577 0.610 0.642 0.705 0.764 0.821 0.873 0.923


8 - 166



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

ex = a l 45°

Pu

l



kl ex = a l


45° Pu

Special Case


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.61 3.37 3.13 2.94 2.77

3.61 3.37 3.13 2.94 2.77

3.61 3.38 3.15 2.95 2.78

3.61 3.38 3.17 2.97 2.80

3.61 3.40 3.20 2.99 2.83

3.61 3.42 3.23 3.03 2.86

3.61 3.43 3.25 3.06 2.89

3.61 3.44 3.28 3.10 2.93

3.61 3.46 3.30 3.13 2.97

3.61 3.47 3.33 3.17 3.01

3.61 3.48 3.35 3.20 3.04

3.61 3.50 3.38 3.25 3.11

3.61 3.51 3.41 3.29 3.16

3.61 3.52 3.43 3.33 3.21

3.61 3.52 3.45 3.35 3.25

3.61 3.52 3.46 3.38 3.28

0.30 0.40 0.50 0.60 0.70

2.61 2.32 2.06 1.84 1.66

2.61 2.32 2.07 1.85 1.66

2.63 2.34 2.09 1.87 1.69

2.65 2.37 2.12 1.91 1.73

2.68 2.41 2.17 1.96 1.79

2.71 2.45 2.22 2.02 1.85

2.75 2.50 2.27 2.08 1.91

2.79 2.54 2.33 2.14 1.97

2.83 2.59 2.38 2.19 2.03

2.86 2.63 2.42 2.25 2.09

2.90 2.66 2.47 2.30 2.14

2.97 2.73 2.54 2.38 2.23

3.04 2.80 2.61 2.45 2.31

3.09 2.87 2.68 2.52 2.38

3.14 2.92 2.74 2.58 2.44

3.18 2.97 2.79 2.63 2.50

0.80 0.90 1.00 1.20 1.40

1.50 1.37 1.26 1.08 0.938

1.51 1.38 1.26 1.08 0.946

1.54 1.40 1.29 1.11 0.975

1.58 1.45 1.34 1.16 1.02

1.64 1.51 1.40 1.21 1.07

1.70 1.57 1.46 1.28 1.13

1.76 1.64 1.53 1.34 1.19

1.83 1.70 1.59 1.40 1.25

1.89 1.76 1.65 1.46 1.31

1.95 1.82 1.71 1.52 1.37

2.00 1.88 1.77 1.58 1.42

2.10 1.98 1.87 1.68 1.52

2.18 2.07 1.96 1.77 1.62

2.26 2.14 2.04 1.85 1.70

2.32 2.21 2.11 1.93 1.77

2.38 2.27 2.17 2.00 1.84

1.60 1.80 2.00 2.20 2.40

0.831 0.745 0.675 0.617 0.568

0.838 0.752 0.682 0.624 0.574

0.866 0.779 0.707 0.647 0.596

0.908 0.817 0.743 0.681 0.628

0.958 0.864 0.787 0.722 0.667

1.01 0.917 0.836 0.768 0.710

1.07 0.972 0.888 0.818 0.756

1.13 1.03 0.941 0.868 0.804

1.19 1.08 0.992 0.917 0.851

1.24 1.13 1.04 0.964 0.897

1.29 1.19 1.09 1.01 0.941

1.39 1.28 1.18 1.10 1.03

1.48 1.37 1.27 1.18 1.11

1.56 1.45 1.35 1.26 1.18

1.64 1.52 1.42 1.33 1.25

1.71 1.59 1.49 1.40 1.32

2.60 0.526 0.532 0.553 0.583 0.619 0.660 0.703 0.749 0.794 0.838 0.881 0.963 1.04 1.12 1.18 2.80 0.489 0.496 0.515 0.544 0.578 0.617 0.658 0.701 0.743 0.786 0.827 0.906 0.982 1.05 1.12 3.00 0.458 0.464 0.482 0.509 0.542 0.579 0.618 0.658 0.699 0.739 0.779 0.856 0.929 0.998 1.06

1.25 1.18 1.12



8 - 167


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D ex = a l 60° P

u

l



kl ex = a l


60° Pu

Special Case (Load not in plane of weld group) Use C-values for k = 0 Any equal distances

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.91 3.65 3.46 3.27 3.10

3.91 3.66 3.47 3.28 3.10

3.91 3.68 3.49 3.30 3.13

3.91 3.71 3.53 3.35 3.17

3.91 3.74 3.58 3.40 3.23

3.91 3.78 3.63 3.47 3.30

3.91 3.80 3.68 3.54 3.38

3.91 3.83 3.73 3.60 3.46

3.91 3.84 3.76 3.65 3.53

3.91 3.85 3.79 3.69 3.58

3.91 3.86 3.80 3.73 3.63

3.91 3.86 3.83 3.78 3.70

3.91 3.87 3.84 3.80 3.75

3.91 3.86 3.85 3.82 3.78

3.91 3.86 3.85 3.83 3.80

3.91 3.86 3.86 3.84 3.82

0.30 0.40 0.50 0.60 0.70

2.95 2.68 2.44 2.24 2.05

2.95 2.69 2.45 2.24 2.06

2.98 2.71 2.48 2.27 2.09

3.02 2.76 2.53 2.32 2.15

3.07 2.81 2.59 2.39 2.21

3.14 2.88 2.66 2.46 2.29

3.22 2.95 2.73 2.54 2.37

3.31 3.03 2.80 2.62 2.45

3.39 3.12 2.88 2.69 2.53

3.46 3.21 2.96 2.77 2.60

3.53 3.29 3.05 2.84 2.67

3.62 3.43 3.22 3.01 2.82

3.68 3.53 3.36 3.17 2.98

3.73 3.61 3.46 3.29 3.12

3.76 3.66 3.54 3.39 3.24

3.78 3.70 3.60 3.47 3.34

0.80 0.90 1.00 1.20 1.40

1.89 1.75 1.62 1.42 1.25

1.90 1.76 1.63 1.43 1.26

1.93 1.79 1.67 1.46 1.30

1.99 1.85 1.73 1.52 1.36

2.06 1.92 1.80 1.59 1.42

2.14 2.00 1.88 1.67 1.50

2.22 2.08 1.96 1.75 1.57

2.30 2.16 2.04 1.83 1.65

2.38 2.24 2.12 1.91 1.73

2.45 2.32 2.20 1.98 1.81

2.52 2.39 2.27 2.06 1.88

2.66 2.53 2.41 2.19 2.01

2.81 2.65 2.53 2.32 2.13

2.95 2.80 2.66 2.43 2.25

3.08 2.93 2.79 2.54 2.35

3.20 3.05 2.91 2.66 2.45

1.60 1.80 2.00 2.20 2.40

1.12 1.01 0.922 0.847 0.782

1.13 1.02 0.932 0.856 0.791

1.17 1.06 0.964 0.887 0.820

1.22 1.11 1.01 0.932 0.863

1.28 1.17 1.07 0.986 0.914

1.35 1.24 1.13 1.05 0.971

1.43 1.31 1.20 1.11 1.03

1.50 1.38 1.27 1.18 1.10

1.58 1.45 1.34 1.25 1.16

1.65 1.52 1.41 1.31 1.22

1.72 1.59 1.47 1.37 1.28

1.85 1.72 1.60 1.49 1.40

1.98 1.84 1.72 1.61 1.51

2.09 1.95 1.82 1.71 1.62

2.19 2.05 1.92 1.81 1.71

2.29 2.14 2.02 1.90 1.80

2.60 0.726 0.734 0.762 0.803 0.852 0.907 0.964 1.03 1.09 1.15 2.80 0.677 0.686 0.712 0.750 0.797 0.849 0.905 0.963 1.02 1.08 3.00 0.635 0.643 0.668 0.704 0.749 0.799 0.852 0.906 0.963 1.02

1.21 1.14 1.07

1.32 1.24 1.18

1.43 1.35 1.28

1.53 1.45 1.37

1.62 1.54 1.46

1.71 1.62 1.55


8 - 168



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

ex = a l

75°

Pu l



kl ex = a l


75° Pu

Special Case (Load not in plane of weld group) Use C-values for k = 0

Any equal distances

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

4.11 3.88 3.76 3.64 3.53

4.11 3.90 3.77 3.65 3.54

4.11 3.95 3.83 3.71 3.60

4.11 4.00 3.90 3.79 3.69

4.11 4.04 3.96 3.88 3.78

4.11 4.07 4.01 3.94 3.87

4.11 4.08 4.04 3.99 3.93

4.11 4.09 4.06 4.03 3.98

4.11 4.09 4.08 4.05 4.01

4.11 4.09 4.08 4.06 4.03

4.11 4.09 4.09 4.07 4.05

4.11 4.09 4.09 4.08 4.07

4.11 4.09 4.09 4.09 4.08

4.11 4.09 4.09 4.09 4.08

4.11 4.09 4.09 4.09 4.09

4.11 4.09 4.09 4.09 4.09

0.30 0.40 0.50 0.60 0.70

3.43 3.24 3.07 2.91 2.77

3.44 3.25 3.08 2.92 2.78

3.49 3.29 3.12 2.97 2.82

3.58 3.38 3.20 3.05 2.90

3.69 3.50 3.32 3.15 3.00

3.78 3.62 3.45 3.29 3.13

3.86 3.72 3.57 3.42 3.27

3.92 3.80 3.67 3.54 3.40

3.97 3.86 3.75 3.63 3.51

4.00 3.91 3.82 3.71 3.60

4.02 3.95 3.87 3.78 3.68

4.05 4.00 3.94 3.88 3.80

4.07 4.03 3.99 3.94 3.88

4.07 4.05 4.02 3.98 3.93

4.08 4.06 4.04 4.01 3.97

4.08 4.07 4.05 4.03 4.00

0.80 0.90 1.00 1.20 1.40

2.63 2.50 2.38 2.17 1.99

2.64 2.52 2.40 2.18 2.00

2.69 2.57 2.45 2.24 2.05

2.77 2.64 2.53 2.32 2.13

2.87 2.74 2.63 2.41 2.23

2.99 2.86 2.74 2.52 2.33

3.13 3.00 2.87 2.64 2.45

3.26 3.13 3.01 2.78 2.57

3.39 3.26 3.14 2.91 2.71

3.49 3.38 3.26 3.04 2.84

3.58 3.48 3.37 3.16 2.96

3.72 3.64 3.55 3.37 3.18

3.81 3.75 3.68 3.52 3.36

3.88 3.83 3.77 3.64 3.50

3.93 3.88 3.83 3.73 3.61

3.96 3.93 3.88 3.80 3.69

1.60 1.80 2.00 2.20 2.40

1.83 1.69 1.57 1.46 1.37

1.84 1.70 1.58 1.48 1.38

1.89 1.75 1.63 1.52 1.43

1.97 1.83 1.70 1.59 1.49

2.06 1.92 1.79 1.68 1.57

2.17 2.02 1.89 1.77 1.67

2.28 2.13 1.99 1.87 1.76

2.39 2.24 2.10 1.97 1.86

2.52 2.35 2.21 2.08 1.97

2.65 2.48 2.33 2.19 2.07

2.78 2.61 2.45 2.30 2.18

3.01 2.84 2.68 2.54 2.40

3.20 3.05 2.89 2.75 2.61

3.36 3.22 3.08 2.94 2.81

3.49 3.36 3.23 3.10 2.98

3.59 3.48 3.36 3.24 3.13

2.60 2.80 3.00

1.28 1.21 1.14

1.30 1.22 1.15

1.34 1.27 1.20

1.40 1.33 1.26

1.48 1.40 1.33

1.57 1.49 1.41

1.66 1.58 1.49

1.76 1.67 1.59

1.86 1.77 1.68

1.96 1.86 1.77

2.07 1.96 1.87

2.28 2.16 2.06

2.49 2.37 2.26

2.68 2.56 2.45

2.85 2.73 2.62

3.01 2.89 2.78



8 - 169


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

4.18 3.24 2.92 2.65 2.41

4.18 3.27 2.95 2.68 2.44

4.18 3.36 3.03 2.75 2.50

4.18 3.48 3.15 2.85 2.60

4.18 3.61 3.29 2.99 2.73

4.18 3.73 3.43 3.15 2.88

4.18 3.83 3.56 3.30 3.04

4.18 3.91 3.68 3.43 3.19

4.18 3.97 3.77 3.55 3.33

4.18 4.01 3.84 3.65 3.44

4.18 4.05 3.90 3.73 3.54

4.18 4.09 3.98 3.85 3.70

4.18 4.12 4.04 3.93 3.81

4.18 4.13 4.07 3.99 3.89

4.18 4.14 4.09 4.03 3.95

4.18 4.15 4.11 4.06 3.99

0.30 0.40 0.50 0.60 0.70

2.20 1.86 1.60 1.40 1.24

2.23 1.88 1.62 1.42 1.26

2.29 1.95 1.68 1.47 1.30

2.39 2.03 1.76 1.54 1.37

2.50 2.14 1.85 1.63 1.45

2.64 2.26 1.96 1.73 1.54

2.81 2.39 2.08 1.84 1.64

2.96 2.55 2.21 1.96 1.75

3.11 2.71 2.36 2.08 1.86

3.24 2.86 2.51 2.22 1.98

3.36 2.99 2.66 2.36 2.11

3.54 3.22 2.91 2.63 2.38

3.68 3.40 3.12 2.86 2.61

3.79 3.55 3.29 3.05 2.81

3.86 3.66 3.43 3.21 2.99

3.92 3.74 3.55 3.34 3.14

0.80 0.90 1.00 1.20 1.40

1.11 1.00 0.914 0.777 0.674

1.13 1.02 0.929 0.789 0.685

1.17 1.06 0.965 0.821 0.713

1.23 1.11 1.02 0.866 0.753

1.30 1.18 1.08 0.920 0.802

1.39 1.26 1.15 0.984 0.856

1.48 1.35 1.23 1.05 0.918

1.58 1.44 1.31 1.13 0.982

1.68 1.53 1.40 1.20 1.05

1.79 1.63 1.49 1.28 1.12

1.90 1.73 1.59 1.36 1.19

2.15 1.96 1.79 1.53 1.34

2.39 2.19 2.02 1.72 1.50

2.60 2.40 2.22 1.92 1.68

2.78 2.59 2.41 2.11 1.85

2.95 2.76 2.59 2.28 2.02

1.60 1.80 2.00 2.20 2.40

0.594 0.531 0.481 0.439 0.404

0.604 0.541 0.489 0.446 0.410

0.629 0.563 0.509 0.465 0.427

0.665 0.595 0.538 0.492 0.452

0.709 0.635 0.574 0.524 0.483

0.759 0.680 0.616 0.562 0.517

0.812 0.729 0.661 0.604 0.555

0.871 0.780 0.708 0.647 0.596

0.931 0.836 0.757 0.693 0.638

0.993 0.892 0.810 0.740 0.681

1.06 0.950 0.862 0.789 0.727

1.19 1.07 0.971 0.890 0.820

1.33 1.20 1.09 0.994 0.917

1.48 1.33 1.21 1.10 1.02

1.64 1.47 1.33 1.22 1.12

1.80 1.62 1.47 1.34 1.23

2.60 0.374 0.379 0.396 0.419 0.447 0.479 0.514 0.552 0.592 0.632 0.674 0.760 0.850 0.941 1.04 1.14 2.80 0.348 0.353 0.368 0.390 0.416 0.446 0.479 0.514 0.551 0.589 0.628 0.709 0.793 0.878 0.966 1.06 3.00 0.325 0.330 0.344 0.364 0.389 0.417 0.448 0.481 0.516 0.552 0.588 0.664 0.742 0.822 0.904 0.989


8 - 170



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


Pu

ex = a l


15° kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

4.11 3.29 2.97 2.70 2.46

4.11 3.30 2.99 2.71 2.48

4.11 3.34 3.03 2.76 2.53

4.11 3.43 3.11 2.84 2.61

4.11 3.55 3.22 2.94 2.71

4.11 3.66 3.35 3.07 2.82

4.11 3.76 3.48 3.20 2.95

4.11 3.83 3.59 3.34 3.09

4.11 3.89 3.68 3.45 3.22

4.11 3.94 3.76 3.55 3.34

4.11 3.97 3.82 3.64 3.44

4.11 4.02 3.90 3.76 3.61

4.11 4.04 3.96 3.85 3.72

4.11 4.06 4.00 3.91 3.81

4.11 4.07 4.02 3.95 3.87

4.11 4.07 4.04 3.98 3.91

0.30 0.40 0.50 0.60 0.70

2.25 1.91 1.65 1.44 1.28

2.27 1.93 1.66 1.46 1.29

2.32 1.98 1.71 1.50 1.34

2.40 2.05 1.79 1.57 1.40

2.50 2.15 1.87 1.65 1.48

2.61 2.25 1.97 1.75 1.56

2.73 2.37 2.08 1.85 1.66

2.87 2.49 2.19 1.95 1.76

3.00 2.62 2.30 2.06 1.85

3.13 2.75 2.42 2.17 1.96

3.25 2.87 2.55 2.28 2.06

3.44 3.10 2.79 2.51 2.27

3.58 3.29 2.99 2.72 2.48

3.69 3.44 3.17 2.91 2.67

3.77 3.56 3.32 3.08 2.85

3.83 3.65 3.44 3.22 3.00

0.80 0.90 1.00 1.20 1.40

1.15 1.04 0.945 0.803 0.697

1.16 1.05 0.958 0.814 0.707

1.20 1.09 0.993 0.846 0.735

1.26 1.14 1.05 0.892 0.776

1.33 1.21 1.11 0.946 0.825

1.41 1.29 1.18 1.01 0.881

1.50 1.37 1.26 1.08 0.942

1.59 1.45 1.34 1.15 1.01

1.69 1.54 1.42 1.22 1.07

1.78 1.63 1.50 1.30 1.14

1.88 1.72 1.59 1.37 1.21

2.07 1.90 1.75 1.52 1.34

2.27 2.08 1.92 1.67 1.47

2.45 2.26 2.10 1.82 1.61

2.63 2.43 2.26 1.97 1.74

2.79 2.59 2.42 2.12 1.87

1.60 1.80 2.00 2.20 2.40

0.615 0.550 0.497 0.454 0.418

0.624 0.559 0.505 0.461 0.424

0.649 0.581 0.526 0.480 0.442

0.686 0.614 0.556 0.508 0.467

0.731 0.655 0.593 0.542 0.499

0.781 0.701 0.635 0.580 0.534

0.835 0.751 0.681 0.623 0.573

0.894 0.802 0.729 0.667 0.615

0.954 0.858 0.778 0.713 0.657

1.01 0.913 0.830 0.760 0.700

1.08 0.968 0.881 0.808 0.745

1.20 1.08 0.985 0.904 0.834

1.32 1.19 1.09 0.999 0.924

1.44 1.30 1.19 1.10 1.01

1.56 1.41 1.29 1.19 1.10

1.68 1.52 1.39 1.28 1.19

2.60 0.387 0.392 0.409 0.433 0.462 0.495 0.531 0.570 0.609 0.650 0.691 0.775 0.859 0.942 1.03 1.11 2.80 0.360 0.365 0.381 0.403 0.430 0.461 0.494 0.530 0.568 0.606 0.644 0.724 0.802 0.881 0.959 1.03 3.00 0.336 0.341 0.356 0.377 0.402 0.431 0.463 0.496 0.532 0.568 0.604 0.678 0.753 0.827 0.900 0.972



8 - 171


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


Pu

30° kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.91 3.37 3.07 2.82 2.60

3.91 3.38 3.07 2.83 2.61

3.91 3.40 3.10 2.85 2.63

3.91 3.45 3.14 2.89 2.68

3.91 3.50 3.20 2.94 2.73

3.91 3.56 3.27 3.01 2.80

3.91 3.62 3.35 3.09 2.87

3.91 3.67 3.43 3.17 2.94

3.91 3.71 3.50 3.26 3.03

3.91 3.75 3.56 3.35 3.12

3.91 3.77 3.61 3.42 3.21

3.91 3.81 3.69 3.54 3.36

3.91 3.83 3.74 3.62 3.48

3.91 3.84 3.77 3.68 3.56

3.91 3.85 3.79 3.72 3.62

3.91 3.85 3.81 3.75 3.67

0.30 0.40 0.50 0.60 0.70

2.40 2.06 1.80 1.58 1.41

2.41 2.07 1.80 1.59 1.42

2.44 2.10 1.84 1.62 1.45

2.49 2.16 1.89 1.68 1.51

2.55 2.22 1.97 1.75 1.58

2.62 2.30 2.04 1.83 1.66

2.69 2.38 2.13 1.91 1.74

2.76 2.46 2.21 2.00 1.82

2.84 2.54 2.29 2.07 1.90

2.92 2.61 2.36 2.15 1.97

3.01 2.69 2.44 2.23 2.05

3.18 2.83 2.57 2.36 2.18

3.32 2.99 2.70 2.49 2.30

3.43 3.14 2.85 2.61 2.42

3.51 3.25 2.98 2.74 2.53

3.57 3.35 3.10 2.86 2.64

0.80 0.90 1.00 1.20 1.40

1.26 1.15 1.05 0.891 0.774

1.27 1.15 1.06 0.900 0.783

1.31 1.19 1.09 0.932 0.812

1.37 1.25 1.14 0.979 0.855

1.43 1.31 1.20 1.03 0.906

1.51 1.38 1.27 1.10 0.963

1.59 1.46 1.35 1.17 1.02

1.66 1.53 1.42 1.23 1.09

1.74 1.61 1.49 1.30 1.15

1.82 1.68 1.57 1.37 1.21

1.89 1.75 1.63 1.43 1.27

2.03 1.88 1.76 1.56 1.39

2.15 2.01 1.88 1.67 1.50

2.26 2.12 1.99 1.78 1.60

2.36 2.22 2.09 1.88 1.70

2.46 2.32 2.19 1.97 1.79

1.60 1.80 2.00 2.20 2.40

0.684 0.612 0.554 0.506 0.465

0.692 0.620 0.562 0.513 0.472

0.719 0.644 0.584 0.534 0.491

0.757 0.679 0.616 0.563 0.519

0.805 0.723 0.655 0.600 0.553

0.857 0.771 0.700 0.641 0.591

0.913 0.823 0.749 0.686 0.633

0.972 0.876 0.799 0.733 0.676

1.03 0.931 0.848 0.779 0.720

1.09 0.985 0.899 0.826 0.764

1.14 1.04 0.949 0.874 0.808

1.26 1.14 1.05 0.965 0.896

1.36 1.24 1.14 1.05 0.979

1.46 1.33 1.23 1.14 1.06

1.55 1.42 1.31 1.22 1.14

1.64 1.51 1.39 1.30 1.21

2.60 0.431 0.437 0.455 0.481 0.512 0.548 0.587 0.628 0.669 0.711 0.752 0.835 0.915 0.992 1.07 1.14 2.80 0.401 0.406 0.423 0.448 0.477 0.511 0.547 0.585 0.625 0.664 0.704 0.782 0.858 0.931 1.00 1.07 3.00 0.375 0.380 0.396 0.419 0.447 0.478 0.513 0.548 0.586 0.623 0.661 0.734 0.808 0.878 0.946 1.01


8 - 172



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


45° Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.61 3.37 3.13 2.94 2.77

3.61 3.37 3.13 2.94 2.77

3.61 3.38 3.15 2.95 2.78

3.61 3.38 3.17 2.97 2.80

3.61 3.40 3.20 2.99 2.83

3.61 3.42 3.23 3.03 2.86

3.61 3.43 3.25 3.06 2.89

3.61 3.44 3.28 3.10 2.93

3.61 3.46 3.30 3.13 2.97

3.61 3.47 3.33 3.17 3.01

3.61 3.48 3.35 3.20 3.04

3.61 3.50 3.38 3.25 3.11

3.61 3.51 3.41 3.29 3.16

3.61 3.52 3.43 3.33 3.21

3.61 3.52 3.45 3.35 3.25

3.61 3.52 3.47 3.38 3.28

0.30 0.40 0.50 0.60 0.70

2.61 2.32 2.06 1.84 1.66

2.61 2.32 2.07 1.85 1.66

2.63 2.34 2.09 1.87 1.69

2.65 2.37 2.12 1.91 1.73

2.68 2.41 2.17 1.96 1.79

2.71 2.45 2.22 2.02 1.85

2.75 2.50 2.27 2.08 1.91

2.79 2.54 2.33 2.14 1.97

2.83 2.59 2.38 2.19 2.03

2.86 2.63 2.42 2.25 2.09

2.90 2.66 2.47 2.30 2.14

2.97 2.73 2.54 2.38 2.23

3.04 2.80 2.61 2.45 2.31

3.09 2.87 2.68 2.52 2.38

3.14 2.92 2.74 2.58 2.44

3.18 2.97 2.79 2.63 2.50

0.80 0.90 1.00 1.20 1.40

1.50 1.37 1.26 1.08 0.938

1.51 1.38 1.26 1.08 0.946

1.54 1.40 1.29 1.11 0.975

1.58 1.45 1.34 1.16 1.02

1.64 1.51 1.40 1.21 1.07

1.70 1.57 1.46 1.28 1.13

1.76 1.64 1.53 1.34 1.19

1.83 1.70 1.59 1.40 1.25

1.89 1.76 1.65 1.46 1.31

1.95 1.82 1.71 1.52 1.37

2.00 1.88 1.77 1.58 1.42

2.10 1.98 1.87 1.68 1.52

2.18 2.07 1.96 1.77 1.62

2.26 2.14 2.04 1.85 1.70

2.32 2.21 2.11 1.93 1.77

2.38 2.27 2.17 2.00 1.84

1.60 1.80 2.00 2.20 2.40

0.831 0.745 0.675 0.617 0.568

0.838 0.752 0.682 0.624 0.574

0.866 0.779 0.707 0.647 0.596

0.908 0.817 0.743 0.681 0.628

0.958 0.864 0.787 0.722 0.667

1.01 0.917 0.836 0.768 0.710

1.07 0.972 0.888 0.818 0.756

1.13 1.03 0.941 0.868 0.804

1.19 1.08 0.992 0.917 0.851

1.24 1.13 1.04 0.964 0.897

1.29 1.19 1.09 1.01 0.941

1.39 1.28 1.18 1.10 1.03

1.48 1.37 1.27 1.18 1.11

1.56 1.45 1.35 1.26 1.18

1.64 1.52 1.42 1.33 1.25

1.71 1.59 1.49 1.40 1.32

2.60 0.526 0.532 0.553 0.583 0.619 0.660 0.703 0.749 0.794 0.838 0.881 0.963 1.04 1.12 1.18 2.80 0.489 0.496 0.515 0.544 0.578 0.617 0.658 0.701 0.743 0.786 0.827 0.906 0.982 1.05 1.12 3.00 0.458 0.464 0.482 0.509 0.542 0.579 0.618 0.658 0.699 0.739 0.779 0.856 0.929 0.998 1.06

1.25 1.18 1.12



8 - 173


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


60°

Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.28 3.20 3.10 2.98 2.89

3.28 3.19 3.09 2.98 2.89

3.28 3.19 3.10 2.98 2.89

3.28 3.19 3.10 2.98 2.89

3.28 3.19 3.10 2.98 2.89

3.28 3.19 3.10 2.99 2.90

3.28 3.19 3.10 2.99 2.90

3.28 3.19 3.10 3.00 2.91

3.28 3.18 3.10 3.01 2.91

3.28 3.18 3.10 3.01 2.92

3.28 3.18 3.10 3.02 2.92

3.28 3.17 3.10 3.02 2.94

3.28 3.16 3.10 3.03 2.95

3.28 3.15 3.09 3.03 2.96

3.28 3.14 3.09 3.02 2.96

3.28 3.13 3.08 3.02 2.96

0.30 0.40 0.50 0.60 0.70

2.80 2.63 2.44 2.27 2.09

2.80 2.63 2.44 2.27 2.10

2.80 2.63 2.45 2.28 2.11

2.81 2.64 2.47 2.29 2.13

2.81 2.65 2.48 2.32 2.16

2.82 2.66 2.50 2.35 2.19

2.82 2.67 2.52 2.37 2.23

2.83 2.68 2.54 2.40 2.27

2.83 2.69 2.55 2.42 2.29

2.84 2.70 2.57 2.44 2.32

2.85 2.71 2.58 2.46 2.34

2.86 2.73 2.61 2.49 2.39

2.87 2.74 2.63 2.52 2.42

2.89 2.76 2.65 2.55 2.45

2.89 2.77 2.66 2.57 2.48

2.90 2.78 2.68 2.59 2.50

0.80 0.90 1.00 1.20 1.40

1.94 1.80 1.67 1.46 1.28

1.94 1.80 1.67 1.46 1.29

1.96 1.82 1.69 1.48 1.31

1.98 1.85 1.73 1.52 1.35

2.02 1.89 1.77 1.57 1.40

2.06 1.93 1.82 1.62 1.46

2.10 1.98 1.87 1.67 1.51

2.14 2.02 1.92 1.73 1.57

2.18 2.06 1.96 1.78 1.62

2.21 2.10 2.00 1.82 1.67

2.23 2.13 2.04 1.86 1.71

2.29 2.19 2.10 1.93 1.79

2.33 2.24 2.15 2.00 1.86

2.37 2.28 2.20 2.05 1.92

2.40 2.32 2.24 2.10 1.97

2.42 2.34 2.27 2.14 2.01

1.60 1.80 2.00 2.20 2.40

1.15 1.03 0.940 0.861 0.794

1.15 1.04 0.946 0.867 0.800

1.18 1.06 0.971 0.892 0.825

1.22 1.11 1.01 0.932 0.863

1.27 1.16 1.06 0.979 0.909

1.32 1.21 1.11 1.03 0.959

1.38 1.27 1.17 1.09 1.01

1.44 1.32 1.22 1.14 1.06

1.49 1.37 1.28 1.19 1.11

1.54 1.42 1.32 1.24 1.16

1.58 1.47 1.37 1.28 1.21

1.66 1.55 1.45 1.37 1.29

1.73 1.63 1.53 1.44 1.36

1.80 1.69 1.60 1.51 1.43

1.85 1.75 1.66 1.57 1.49

1.90 1.80 1.71 1.63 1.55

2.60 0.736 0.743 0.767 0.804 0.848 0.896 0.946 0.997 1.05 1.09 1.14 2.80 0.686 0.693 0.716 0.752 0.794 0.841 0.889 0.937 0.985 1.03 1.07 3.00 0.643 0.649 0.672 0.706 0.746 0.792 0.838 0.885 0.931 0.975 1.02

1.22 1.16 1.10

1.29 1.23 1.17

1.36 1.30 1.24

1.42 1.36 1.30

1.48 1.42 1.36


8 - 174



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l 75°


Pu kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.97 2.86 2.89 2.88 2.87

2.97 2.86 2.89 2.88 2.87

2.97 2.87 2.89 2.88 2.87

2.97 2.87 2.89 2.88 2.87

2.97 2.88 2.89 2.88 2.86

2.97 2.88 2.89 2.87 2.86

2.97 2.88 2.89 2.87 2.85

2.97 2.89 2.89 2.86 2.85

2.97 2.89 2.88 2.86 2.84

2.97 2.89 2.88 2.85 2.84

2.97 2.89 2.87 2.85 2.83

2.97 2.88 2.86 2.84 2.81

2.97 2.87 2.85 2.82 2.80

2.97 2.85 2.83 2.81 2.79

2.97 2.84 2.82 2.80 2.78

2.97 2.55 2.81 2.78 2.76

0.30 0.40 0.50 0.60 0.70

2.86 2.84 2.79 2.74 2.66

2.86 2.83 2.79 2.73 2.66

2.86 2.83 2.78 2.73 2.66

2.85 2.82 2.77 2.72 2.66

2.85 2.82 2.77 2.72 2.65

2.84 2.81 2.76 2.71 2.65

2.84 2.80 2.76 2.70 2.64

2.83 2.79 2.75 2.70 2.64

2.82 2.79 2.74 2.69 2.64

2.82 2.78 2.73 2.68 2.63

2.81 2.77 2.73 2.68 2.63

2.80 2.76 2.71 2.67 2.62

2.78 2.74 2.70 2.66 2.61

2.77 2.73 2.69 2.65 2.61

2.76 2.72 2.68 2.64 2.60

2.74 2.71 2.67 2.63 2.59

0.80 0.90 1.00 1.20 1.40

2.59 2.51 2.42 2.25 2.08

2.59 2.51 2.42 2.25 2.08

2.59 2.51 2.43 2.25 2.09

2.58 2.51 2.43 2.27 2.11

2.58 2.51 2.43 2.28 2.13

2.58 2.51 2.44 2.30 2.15

2.58 2.51 2.44 2.31 2.18

2.58 2.51 2.45 2.32 2.19

2.58 2.52 2.45 2.33 2.21

2.58 2.52 2.46 2.34 2.23

2.58 2.52 2.46 2.35 2.24

2.57 2.52 2.47 2.37 2.27

2.57 2.52 2.48 2.38 2.29

2.56 2.52 2.48 2.39 2.31

2.56 2.52 2.48 2.40 2.32

2.56 2.52 2.48 2.40 2.33

1.60 1.80 2.00 2.20 2.40

1.92 1.78 1.65 1.54 1.44

1.93 1.78 1.66 1.54 1.44

1.94 1.80 1.67 1.56 1.46

1.96 1.83 1.70 1.60 1.50

1.99 1.86 1.74 1.64 1.54

2.02 1.89 1.78 1.68 1.59

2.05 1.93 1.82 1.72 1.63

2.08 1.96 1.86 1.76 1.68

2.10 1.99 1.89 1.80 1.71

2.12 2.02 1.92 1.83 1.75

2.14 2.04 1.95 1.86 1.78

2.17 2.08 2.00 1.92 1.84

2.20 2.12 2.04 1.96 1.89

2.22 2.15 2.07 2.00 1.93

2.24 2.17 2.10 2.03 1.97

2.26 2.19 2.12 2.06 2.00

2.60 2.80 3.00

1.35 1.26 1.19

1.35 1.27 1.20

1.37 1.29 1.22

1.41 1.33 1.26

1.45 1.37 1.30

1.50 1.42 1.35

1.55 1.47 1.40

1.60 1.52 1.45

1.64 1.56 1.50

1.67 1.60 1.54

1.71 1.64 1.58

1.77 1.71 1.64

1.82 1.76 1.70

1.87 1.81 1.75

1.91 1.85 1.79

1.94 1.88 1.83



8 - 175


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


l

Pu


kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

0.00 0.10 0.15 0.20 0.25

2.78 2.78 2.75 2.64 2.48

3.20 3.07 3.05 2.95 2.79

3.62 3.42 3.37 3.25 3.10

4.04 3.78 3.71 3.57 3.40

4.45 4.15 4.06 3.91 3.72

4.87 4.53 4.42 4.25 4.04

5.29 4.91 4.78 4.59 4.38

5.71 5.30 5.15 4.94 4.71

6.12 5.69 5.52 5.30 5.06

6.54 6.08 5.89 5.66 5.40

6.96 6.47 6.27 6.02 5.75

7.80 7.25 7.02 6.75 6.46

8.63 8.03 7.78 7.49 7.18

9.47 10.3 11.1 8.82 9.61 10.4 8.54 9.31 10.1 8.23 8.98 9.74 7.91 8.65 9.39

0.30 0.40 0.50 0.60 0.70

2.32 2.00 1.72 1.50 1.32

2.61 2.26 1.95 1.70 1.50

2.92 2.54 2.20 1.92 1.70

3.22 2.83 2.47 2.17 1.93

3.52 3.12 2.75 2.44 2.17

3.83 3.41 3.03 2.70 2.43

4.15 3.71 3.31 2.97 2.68

4.47 4.01 3.59 3.24 2.93

4.80 4.32 3.89 3.52 3.20

5.14 4.64 4.19 3.80 3.47

5.48 4.96 4.50 4.09 3.75

6.17 5.62 5.13 4.70 4.33

6.88 6.30 5.78 5.33 4.93

7.59 6.99 6.46 5.98 5.56

8.32 7.70 7.14 6.64 6.20

9.05 8.42 7.84 7.32 6.86

0.80 0.90 1.00 1.20 1.40

1.17 1.05 0.957 0.806 0.695

1.33 1.20 1.09 0.916 0.790

1.52 1.37 1.24 1.05 0.908

1.73 1.56 1.42 1.21 1.05

1.95 1.77 1.62 1.38 1.20

2.20 2.00 1.83 1.57 1.37

2.43 2.23 2.05 1.76 1.54

2.67 2.45 2.26 1.95 1.72

2.92 2.69 2.49 2.15 1.89

3.18 2.93 2.72 2.36 2.08

3.45 3.18 2.96 2.57 2.27

4.00 3.71 3.45 3.02 2.68

4.58 4.26 3.98 3.50 3.12

5.18 4.84 4.53 4.00 3.58

5.80 5.44 5.10 4.53 4.06

6.44 6.05 5.69 5.08 4.57

1.60 1.80 2.00 2.20 2.40

0.611 0.544 0.491 0.447 0.410

0.694 0.619 0.558 0.509 0.467

0.800 0.714 0.645 0.588 0.540

0.923 0.825 0.746 0.680 0.625

1.06 0.950 0.860 0.785 0.721

1.21 1.09 0.984 0.898 0.827

1.37 1.23 1.12 1.02 0.939

1.53 1.37 1.25 1.14 1.05

1.69 1.52 1.38 1.27 1.17

1.86 1.68 1.53 1.40 1.29

2.03 1.84 1.67 1.54 1.42

2.40 2.18 1.99 1.83 1.69

2.80 2.54 2.33 2.14 1.99

3.23 2.93 2.69 2.48 2.30

3.67 3.35 3.08 2.84 2.64

4.15 3.79 3.49 3.22 3.00

2.60 0.379 0.431 0.499 0.578 0.667 0.765 0.869 0.977 1.09 1.20 2.80 0.352 0.401 0.464 0.538 0.621 0.712 0.809 0.911 1.01 1.12 3.00 0.329 0.375 0.434 0.503 0.580 0.666 0.757 0.853 0.949 1.05

1.32 1.23 1.16

1.57 1.47 1.38

1.85 1.73 1.62

2.15 2.01 1.89

2.46 2.31 2.17

2.80 2.63 2.47


1.8

2.0

8 - 176



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


15°

l

Pu


kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

0.00 0.10 0.15 0.20 0.25

2.97 2.84 2.76 2.63 2.48

3.38 3.16 3.10 2.96 2.79

3.79 3.52 3.44 3.30 3.12

4.20 3.89 3.79 3.64 3.45

4.61 4.28 4.14 3.98 3.78

5.02 4.66 4.51 4.32 4.11

5.43 5.05 4.87 4.67 4.45

5.84 5.43 5.24 5.02 4.78

6.25 5.83 5.61 5.37 5.13

6.66 6.22 5.98 5.73 5.47

7.07 6.62 6.35 6.09 5.82

7.89 7.41 7.11 6.82 6.53

8.71 8.20 7.86 7.55 7.25

9.54 10.4 11.2 8.99 9.79 10.6 8.63 9.41 10.2 8.30 9.05 9.81 7.98 8.72 9.46

0.30 0.40 0.50 0.60 0.70

2.32 2.01 1.74 1.52 1.34

2.61 2.26 1.96 1.72 1.52

2.92 2.54 2.21 1.94 1.72

3.24 2.84 2.48 2.19 1.95

3.57 3.15 2.77 2.46 2.19

3.89 3.46 3.07 2.74 2.46

4.22 3.77 3.37 3.03 2.73

4.54 4.08 3.66 3.30 3.00

4.88 4.39 3.96 3.59 3.27

5.21 4.71 4.27 3.88 3.55

5.55 5.04 4.58 4.18 3.83

6.24 5.70 5.21 4.79 4.42

6.95 6.38 5.87 5.42 5.03

7.66 7.07 6.54 6.07 5.66

8.39 7.78 7.23 6.74 6.31

9.13 8.50 7.93 7.43 6.97

0.80 0.90 1.00 1.20 1.40

1.20 1.08 0.979 0.826 0.714

1.36 1.22 1.11 0.938 0.810

1.54 1.39 1.27 1.07 0.930

1.75 1.58 1.45 1.23 1.07

1.98 1.80 1.65 1.41 1.23

2.22 2.03 1.87 1.60 1.40

2.48 2.27 2.09 1.81 1.58

2.74 2.52 2.32 2.01 1.77

3.00 2.76 2.55 2.21 1.95

3.26 3.01 2.79 2.43 2.14

3.53 3.26 3.03 2.65 2.34

4.09 3.80 3.54 3.10 2.76

4.67 4.36 4.07 3.59 3.20

5.28 4.94 4.63 4.10 3.67

5.91 5.54 5.21 4.64 4.16

6.55 6.16 5.81 5.19 4.68

1.60 1.80 2.00 2.20 2.40

0.628 0.560 0.506 0.461 0.423

0.713 0.636 0.575 0.524 0.481

0.820 0.733 0.663 0.605 0.556

0.946 0.846 0.766 0.699 0.643

1.09 0.974 0.882 0.805 0.741

1.24 1.11 1.01 0.922 0.849

1.41 1.26 1.15 1.05 0.965

1.57 1.42 1.29 1.18 1.09

1.74 1.57 1.43 1.31 1.21

1.91 1.73 1.57 1.45 1.33

2.09 1.89 1.73 1.59 1.47

2.48 2.24 2.05 1.89 1.75

2.89 2.62 2.40 2.21 2.05

3.32 3.02 2.77 2.56 2.38

3.78 3.45 3.17 2.93 2.72

4.26 3.89 3.59 3.32 3.09

2.60 0.391 0.445 0.514 0.595 0.686 0.786 0.894 1.01 1.12 1.24 2.80 0.363 0.414 0.478 0.554 0.638 0.732 0.832 0.938 1.05 1.16 3.00 0.339 0.386 0.447 0.518 0.597 0.684 0.779 0.878 0.981 1.09

1.36 1.27 1.19

1.63 1.52 1.43

1.91 1.79 1.68

2.22 2.07 1.95

2.54 2.38 2.24

2.89 2.71 2.55


1.8

2.0


8 - 177


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


30°

l

Pu


kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.28 3.03 2.87 2.72 2.57

3.67 3.45 3.25 3.07 2.88

4.06 3.85 3.63 3.43 3.22

4.45 4.24 4.02 3.80 3.57

4.84 4.62 4.40 4.17 3.93

5.23 5.00 4.77 4.53 4.29

5.62 5.38 5.15 4.89 4.65

6.01 5.76 5.52 5.25 4.99

6.40 6.14 5.89 5.61 5.34

6.79 6.52 6.26 5.97 5.69

7.18 6.90 6.64 6.34 6.05

7.96 7.67 7.39 7.09 6.77

8.74 8.44 8.15 7.84 7.52

9.53 10.3 9.22 10.0 8.93 9.70 8.60 9.37 8.27 9.04

11.1 10.8 10.5 10.2 9.81

0.30 0.40 0.50 0.60 0.70

2.41 2.11 1.84 1.62 1.44

2.70 2.36 2.07 1.83 1.63

3.02 2.64 2.32 2.06 1.84

3.35 2.95 2.60 2.31 2.07

3.70 3.27 2.90 2.59 2.33

4.05 3.60 3.21 2.88 2.60

4.40 3.94 3.53 3.18 2.89

4.75 4.28 3.86 3.50 3.19

5.09 4.61 4.19 3.82 3.50

5.43 4.94 4.50 4.12 3.79

5.77 5.27 4.83 4.43 4.09

6.48 5.95 5.48 5.07 4.70

7.21 6.65 6.16 5.73 5.33

7.95 7.36 6.85 6.40 5.99

8.71 8.09 7.56 7.09 6.66

9.48 8.85 8.28 7.79 7.35

0.80 0.90 1.00 1.20 1.40

1.30 1.17 1.07 0.907 0.786

1.46 1.32 1.21 1.03 0.890

1.65 1.50 1.37 1.17 1.02

1.87 1.70 1.56 1.34 1.17

2.11 1.93 1.78 1.53 1.34

2.37 2.18 2.01 1.73 1.52

2.64 2.43 2.25 1.95 1.72

2.93 2.70 2.51 2.18 1.93

3.22 2.98 2.77 2.41 2.14

3.50 3.25 3.02 2.65 2.34

3.79 3.52 3.28 2.88 2.56

4.38 4.08 3.82 3.37 3.01

4.98 4.67 4.38 3.89 3.49

5.61 5.28 4.97 4.44 3.99

6.26 5.90 5.57 5.00 4.52

6.93 6.55 6.20 5.59 5.07

1.60 1.80 2.00 2.20 2.40

0.693 0.619 0.559 0.510 0.469

0.785 0.703 0.635 0.579 0.533

0.901 0.808 0.731 0.668 0.614

1.04 0.931 0.844 0.771 0.710

1.19 1.07 0.970 0.887 0.818

1.36 1.22 1.11 1.01 0.935

1.53 1.38 1.26 1.15 1.06

1.72 1.55 1.41 1.30 1.20

1.91 1.73 1.57 1.45 1.34

2.10 1.90 1.74 1.60 1.48

2.30 2.08 1.91 1.75 1.62

2.72 2.47 2.26 2.08 1.93

3.16 2.88 2.64 2.44 2.27

3.62 3.31 3.05 2.82 2.62

4.12 3.77 3.48 3.22 3.00

4.63 4.25 3.93 3.64 3.40

2.60 0.433 0.493 0.569 0.658 0.758 0.867 0.984 1.11 1.24 2.80 0.403 0.458 0.529 0.613 0.706 0.808 0.918 1.03 1.16 3.00 0.376 0.428 0.495 0.573 0.661 0.757 0.860 0.969 1.08

1.37 1.28 1.20

1.51 1.41 1.32

1.80 1.68 1.58

2.11 1.98 1.86

2.45 2.29 2.16

2.80 2.63 2.48

3.18 2.99 2.82


8 - 178



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


45°

l

Pu


kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.61 3.37 3.13 2.94 2.77

3.97 3.74 3.52 3.29 3.10

4.33 4.11 3.89 3.65 3.44

4.70 4.48 4.26 4.02 3.79

5.06 4.84 4.62 4.38 4.14

5.42 5.21 4.99 4.75 4.51

5.78 5.58 5.36 5.12 4.87

6.14 5.94 5.74 5.50 5.25

6.50 6.31 6.11 5.88 5.63

6.86 6.68 6.49 6.26 6.01

7.22 7.04 6.86 6.64 6.40

7.95 7.78 7.61 7.39 7.16

8.67 8.52 8.36 8.15 7.92

9.39 10.1 9.25 9.99 9.11 9.85 8.91 9.66 8.68 9.45

10.8 10.7 10.6 10.4 10.2

0.30 0.40 0.50 0.60 0.70

2.61 2.32 2.06 1.84 1.66

2.91 2.59 2.30 2.06 1.86

3.24 2.87 2.57 2.30 2.08

3.57 3.18 2.86 2.58 2.34

3.92 3.51 3.16 2.87 2.61

4.27 3.85 3.49 3.17 2.91

4.63 4.20 3.82 3.50 3.22

5.00 4.55 4.17 3.83 3.54

5.38 4.92 4.52 4.18 3.88

5.76 5.29 4.89 4.53 4.22

6.15 5.67 5.26 4.89 4.56

6.91 6.42 5.97 5.57 5.22

7.67 7.18 6.71 6.28 5.90

8.44 7.95 7.47 7.02 6.62

9.21 8.72 8.24 7.78 7.35

9.98 9.50 9.01 8.54 8.10

0.80 0.90 1.00 1.20 1.40

1.50 1.37 1.26 1.08 0.938

1.69 1.54 1.41 1.21 1.06

1.90 1.74 1.60 1.38 1.21

2.13 1.96 1.81 1.57 1.38

2.40 2.21 2.05 1.78 1.57

2.68 2.48 2.30 2.01 1.78

2.98 2.76 2.58 2.26 2.01

3.29 3.06 2.86 2.52 2.24

3.61 3.37 3.15 2.78 2.49

3.93 3.68 3.45 3.06 2.74

4.27 4.00 3.76 3.34 3.00

4.91 4.62 4.35 3.90 3.52

5.56 5.25 4.97 4.48 4.06

6.25 5.91 5.61 5.07 4.62

6.97 6.61 6.28 5.70 5.21

7.70 7.32 6.97 6.35 5.81

1.60 1.80 2.00 2.20 2.40

0.831 0.745 0.675 0.617 0.568

0.939 0.843 0.764 0.699 0.644

1.07 0.966 0.877 0.804 0.741

1.23 1.11 1.01 0.925 0.854

1.41 1.27 1.16 1.06 0.981

1.60 1.45 1.32 1.21 1.12

1.80 1.63 1.49 1.37 1.27

2.02 1.83 1.67 1.54 1.43

2.24 2.04 1.87 1.72 1.59

2.48 2.25 2.06 1.90 1.77

2.71 2.48 2.27 2.10 1.95

3.20 2.93 2.69 2.49 2.32

3.70 3.40 3.14 2.91 2.71

4.23 3.90 3.60 3.35 3.13

4.78 4.42 4.09 3.81 3.57

5.36 4.96 4.61 4.30 4.03

2.60 0.526 0.597 0.687 0.792 0.911 1.04 1.18 2.80 0.489 0.556 0.641 0.739 0.850 0.972 1.10 3.00 0.458 0.520 0.600 0.693 0.796 0.911 1.03

1.33 1.24 1.17

1.48 1.39 1.30

1.65 1.54 1.45

1.82 1.70 1.60

2.16 2.03 1.91

2.54 2.38 2.24

2.93 2.75 2.60

3.35 3.15 2.97

3.78 3.57 3.37



8 - 179


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


60°

l

Pu


kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.91 3.65 3.46 3.27 3.10

4.23 3.97 3.78 3.60 3.42

4.56 4.30 4.11 3.92 3.74

4.89 4.64 4.45 4.26 4.07

5.22 4.99 4.80 4.61 4.41

5.54 5.35 5.17 4.97 4.78

5.87 5.70 5.53 5.35 5.15

6.20 6.05 5.90 5.72 5.53

6.53 6.40 6.26 6.09 5.92

6.85 6.74 6.61 6.46 6.29

7.18 7.08 6.97 6.82 6.67

7.84 7.75 7.66 7.54 7.40

8.49 8.42 8.34 8.24 8.12

9.15 9.08 9.02 8.93 8.83

9.80 9.74 9.69 9.61 9.52

10.5 10.4 10.4 10.3 10.2

0.30 0.40 0.50 0.60 0.70

2.95 2.68 2.44 2.24 2.05

3.25 2.96 2.70 2.47 2.28

3.57 3.26 2.98 2.74 2.53

3.89 3.57 3.29 3.04 2.81

4.23 3.90 3.61 3.35 3.12

4.59 4.24 3.95 3.69 3.45

4.97 4.61 4.31 4.04 3.79

5.35 4.99 4.68 4.40 4.14

5.74 5.38 5.06 4.77 4.50

6.12 5.77 5.45 5.15 4.87

6.50 6.16 5.84 5.54 5.25

7.25 6.93 6.61 6.31 6.02

7.98 7.69 7.37 7.08 6.79

8.70 8.43 8.12 7.83 7.55

9.41 10.1 9.16 9.87 8.87 9.60 8.58 9.31 8.30 9.04

0.80 0.90 1.00 1.20 1.40

1.89 1.75 1.62 1.42 1.25

2.10 1.95 1.81 1.59 1.41

2.34 2.18 2.03 1.79 1.59

2.62 2.44 2.29 2.02 1.81

2.92 2.73 2.56 2.28 2.04

3.23 3.04 2.86 2.55 2.30

3.56 3.36 3.17 2.84 2.57

3.91 3.69 3.49 3.15 2.85

4.26 4.03 3.82 3.46 3.14

4.62 4.38 4.17 3.78 3.45

4.99 4.74 4.52 4.11 3.76

5.75 5.49 5.24 4.80 4.41

6.51 6.24 5.98 5.50 5.07

7.27 6.99 6.72 6.21 5.75

8.02 7.74 7.46 6.93 6.44

8.76 8.48 8.20 7.66 7.16

1.60 1.80 2.00 2.20 2.40

1.12 1.01 0.922 0.847 0.782

1.26 1.14 1.04 0.956 0.884

1.43 1.30 1.19 1.09 1.01

1.63 1.48 1.36 1.25 1.16

1.85 1.69 1.55 1.43 1.33

2.09 1.91 1.75 1.62 1.51

2.34 2.14 1.97 1.83 1.70

2.60 2.39 2.20 2.04 1.91

2.88 2.65 2.45 2.27 2.12

3.16 2.92 2.70 2.51 2.35

3.46 3.20 2.97 2.76 2.58

4.08 3.78 3.52 3.28 3.08

4.69 4.36 4.08 3.82 3.59

5.34 4.97 4.65 4.37 4.11

6.01 5.61 5.26 4.94 4.66

6.70 6.27 5.90 5.55 5.24

2.60 0.726 0.821 0.943 1.08 1.24 2.80 0.677 0.767 0.881 1.01 1.16 3.00 0.635 0.719 0.827 0.952 1.09

1.41 1.32 1.24

1.59 1.49 1.40

1.78 1.67 1.58

1.99 1.87 1.76

2.20 2.07 1.95

2.42 2.28 2.15

2.89 2.73 2.58

3.38 3.20 3.03

3.88 3.68 3.49

4.40 4.18 3.97

4.96 4.70 4.47


8 - 180



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


75°

l

Pu


kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

0.00 0.10 0.15 0.20 0.25

4.11 3.88 3.76 3.64 3.53

4.40 4.17 4.04 3.92 3.80

4.70 4.49 4.36 4.23 4.11

5.00 4.81 4.69 4.57 4.45

5.29 5.14 5.03 4.92 4.81

5.59 5.45 5.36 5.26 5.16

5.89 5.76 5.69 5.60 5.50

6.18 6.07 6.00 5.92 5.84

6.48 6.36 6.31 6.24 6.16

6.78 6.66 6.61 6.55 6.48

7.07 6.96 6.91 6.86 6.80

7.67 7.53 7.50 7.46 7.41

8.26 8.11 8.09 8.05 8.01

8.85 8.69 8.67 8.64 8.60

9.45 10.0 9.27 9.84 9.25 9.83 9.23 9.81 9.19 9.78

0.30 0.40 0.50 0.60 0.70

3.43 3.24 3.07 2.91 2.77

3.70 3.51 3.34 3.17 3.02

4.00 3.80 3.63 3.46 3.31

4.34 4.14 3.95 3.78 3.63

4.70 4.49 4.31 4.13 3.97

5.05 4.86 4.67 4.49 4.33

5.40 5.22 5.04 4.86 4.70

5.75 5.57 5.40 5.23 5.07

6.08 5.91 5.75 5.60 5.44

6.41 6.25 6.10 5.95 5.80

6.72 6.57 6.43 6.30 6.16

7.35 7.22 7.08 6.96 6.84

7.96 7.85 7.71 7.60 7.50

8.56 8.46 8.34 8.23 8.13

9.16 9.07 8.96 8.84 8.75

9.75 9.67 9.57 9.46 9.36

0.80 0.90 1.00 1.20 1.40

2.63 2.50 2.38 2.17 1.99

2.87 2.74 2.62 2.39 2.20

3.16 3.02 2.89 2.66 2.45

3.48 3.34 3.20 2.96 2.74

3.81 3.67 3.53 3.28 3.05

4.17 4.02 3.87 3.61 3.38

4.53 4.38 4.23 3.96 3.71

4.91 4.75 4.60 4.32 4.06

5.28 5.13 4.98 4.69 4.42

5.65 5.50 5.35 5.06 4.79

6.01 5.87 5.72 5.44 5.16

6.72 6.58 6.45 6.18 5.91

7.39 7.27 7.15 6.90 6.64

8.04 7.93 7.83 7.60 7.36

8.66 8.57 8.48 8.28 8.06

9.27 9.20 9.11 8.93 8.74

1.60 1.80 2.00 2.20 2.40

1.83 1.69 1.57 1.46 1.37

2.03 1.88 1.75 1.63 1.53

2.27 2.11 1.97 1.84 1.73

2.54 2.37 2.22 2.08 1.96

2.84 2.66 2.49 2.34 2.21

3.16 2.96 2.79 2.63 2.48

3.49 3.28 3.10 2.92 2.77

3.82 3.61 3.41 3.23 3.07

4.17 3.95 3.74 3.55 3.37

4.53 4.29 4.08 3.88 3.69

4.90 4.65 4.43 4.22 4.02

5.64 5.39 5.15 4.92 4.71

6.39 6.13 5.88 5.64 5.42

7.12 6.87 6.62 6.37 6.14

7.83 7.59 7.34 7.10 6.87

8.53 8.30 8.07 7.83 7.59

2.60 2.80 3.00

1.28 1.21 1.14

1.44 1.36 1.28

1.63 1.54 1.46

1.85 1.75 1.66

2.09 1.98 1.88

2.35 2.23 2.12

2.62 2.50 2.38

2.91 2.78 2.65

3.21 3.07 2.92

3.52 3.36 3.22

3.84 3.67 3.52

4.51 4.32 4.14

5.20 5.00 4.80

5.91 5.70 5.49

6.63 6.41 6.19

7.36 7.13 6.91


2.0


8 - 181


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

4.18 3.24 2.92 2.65 2.41

4.45 3.51 3.18 2.89 2.64

4.73 3.81 3.49 3.19 2.93

5.01 4.15 3.81 3.52 3.25

5.29 4.51 4.16 3.85 3.58

5.57 4.87 4.53 4.21 3.92

5.85 5.22 4.89 4.58 4.28

6.12 5.55 5.25 4.95 4.65

6.40 5.87 5.60 5.31 5.02

6.68 6.17 5.94 5.67 5.39

6.96 6.47 6.27 6.02 5.75

7.52 7.03 6.89 6.69 6.45

8.07 7.56 7.47 7.32 7.12

8.63 8.08 8.03 7.92 7.76

9.19 8.59 8.58 8.50 8.37

9.74 9.09 9.11 9.06 8.96

0.30 0.40 0.50 0.60 0.70

2.20 1.86 1.60 1.40 1.24

2.43 2.07 1.79 1.57 1.39

2.70 2.31 2.01 1.77 1.58

3.00 2.59 2.27 2.01 1.79

3.33 2.90 2.55 2.26 2.03

3.66 3.22 2.85 2.54 2.28

4.01 3.54 3.16 2.83 2.56

4.37 3.88 3.48 3.13 2.84

4.74 4.23 3.80 3.44 3.13

5.11 4.60 4.15 3.76 3.44

5.48 4.96 4.50 4.09 3.75

6.20 5.70 5.22 4.79 4.40

6.90 6.43 5.95 5.50 5.09

7.57 7.14 6.68 6.22 5.80

8.21 7.83 7.39 6.95 6.51

8.83 8.49 8.09 7.66 7.23

0.80 0.90 1.00 1.20 1.40

1.11 1.00 0.914 0.777 0.674

1.25 1.13 1.03 0.880 0.764

1.42 1.29 1.18 1.01 0.879

1.62 1.48 1.35 1.16 1.01

1.84 1.68 1.54 1.32 1.16

2.07 1.89 1.74 1.50 1.31

2.32 2.13 1.96 1.69 1.48

2.59 2.38 2.19 1.89 1.67

2.87 2.64 2.44 2.11 1.86

3.15 2.91 2.69 2.34 2.06

3.45 3.18 2.96 2.57 2.27

4.07 3.77 3.51 3.08 2.73

4.72 4.39 4.10 3.61 3.21

5.41 5.05 4.73 4.18 3.73

6.11 5.73 5.38 4.78 4.28

6.82 6.42 6.06 5.40 4.86

1.60 1.80 2.00 2.20 2.40

0.594 0.531 0.481 0.439 0.404

0.675 0.604 0.547 0.499 0.459

0.777 0.696 0.630 0.576 0.530

0.895 0.803 0.728 0.665 0.612

1.03 0.921 0.836 0.765 0.705

1.17 1.05 0.953 0.873 0.805

1.32 1.19 1.08 0.990 0.913

1.48 1.34 1.22 1.11 1.03

1.66 1.49 1.36 1.25 1.15

1.84 1.66 1.51 1.39 1.28

2.03 1.84 1.67 1.54 1.42

2.44 2.21 2.02 1.86 1.72

2.89 2.62 2.39 2.20 2.04

3.36 3.06 2.80 2.58 2.39

3.87 3.52 3.23 2.97 2.76

4.39 4.00 3.67 3.39 3.14

2.60 0.374 0.425 0.491 0.567 0.653 0.747 0.847 0.955 1.07 1.19 2.80 0.348 0.396 0.457 0.529 0.608 0.696 0.790 0.890 0.998 1.11 3.00 0.325 0.370 0.428 0.495 0.569 0.651 0.740 0.834 0.935 1.04

1.32 1.23 1.16

1.60 1.49 1.40

1.90 1.77 1.66

2.22 2.08 1.95

2.57 2.40 2.25

2.93 2.74 2.57


8 - 182



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


Pu

ex = a l

15°


kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

0.00 0.10 0.15 0.20 0.25

4.11 3.29 2.97 2.70 2.46

4.40 3.56 3.24 2.95 2.70

4.70 3.85 3.53 3.24 2.98

5.00 4.19 3.85 3.56 3.29

5.29 4.55 4.20 3.89 3.63

5.59 4.91 4.56 4.25 3.97

5.89 5.26 4.93 4.62 4.33

6.18 5.61 5.30 4.99 4.70

6.48 5.95 5.66 5.37 5.08

6.78 6.29 6.01 5.73 5.45

7.07 6.62 6.35 6.09 5.82

7.67 7.26 7.01 6.78 6.54

8.26 7.88 7.65 7.45 7.23

8.85 8.49 8.27 8.08 7.90

9.45 10.0 9.09 9.69 8.89 9.51 8.71 9.31 8.54 9.17

0.30 0.40 0.50 0.60 0.70

2.25 1.91 1.65 1.44 1.28

2.48 2.12 1.83 1.61 1.43

2.75 2.36 2.06 1.82 1.62

3.05 2.65 2.32 2.06 1.84

3.38 2.95 2.60 2.32 2.08

3.71 3.27 2.90 2.60 2.34

4.06 3.61 3.22 2.89 2.62

4.43 3.95 3.54 3.20 2.91

4.80 4.30 3.88 3.52 3.21

5.17 4.67 4.22 3.84 3.51

5.55 5.04 4.58 4.18 3.83

6.29 5.78 5.31 4.88 4.49

7.01 6.52 6.04 5.60 5.19

7.70 7.25 6.78 6.33 5.90

8.36 7.95 7.51 7.06 6.62

9.01 8.64 8.23 7.78 7.34

0.80 0.90 1.00 1.20 1.40

1.15 1.04 0.945 0.803 0.697

1.29 1.17 1.07 0.908 0.789

1.46 1.33 1.22 1.04 0.907

1.67 1.52 1.39 1.19 1.04

1.89 1.72 1.59 1.36 1.19

2.13 1.95 1.79 1.54 1.35

2.38 2.19 2.02 1.74 1.53

2.65 2.44 2.25 1.95 1.72

2.93 2.70 2.50 2.17 1.91

3.23 2.98 2.76 2.40 2.12

3.53 3.26 3.03 2.65 2.34

4.15 3.86 3.60 3.16 2.80

4.82 4.49 4.19 3.70 3.30

5.50 5.15 4.83 4.27 3.82

6.21 5.83 5.48 4.86 4.35

6.92 6.52 6.15 5.48 4.92

1.60 1.80 2.00 2.20 2.40

0.615 0.550 0.497 0.454 0.418

0.698 0.625 0.565 0.516 0.475

0.803 0.719 0.651 0.595 0.548

0.924 0.829 0.752 0.687 0.633

1.06 0.951 0.863 0.790 0.728

1.21 1.08 0.984 0.902 0.832

1.36 1.23 1.12 1.02 0.943

1.53 1.38 1.26 1.15 1.06

1.71 1.54 1.40 1.29 1.19

1.90 1.71 1.56 1.43 1.32

2.09 1.89 1.73 1.59 1.47

2.51 2.28 2.08 1.91 1.77

2.97 2.69 2.47 2.27 2.10

3.45 3.13 2.87 2.65 2.45

3.93 3.58 3.29 3.04 2.82

4.45 4.06 3.73 3.45 3.20

2.60 0.387 0.440 0.508 0.586 0.675 0.772 0.875 0.987 1.11 1.23 2.80 0.360 0.409 0.473 0.546 0.629 0.719 0.817 0.921 1.03 1.15 3.00 0.336 0.383 0.442 0.511 0.589 0.674 0.765 0.863 0.967 1.08

1.36 1.27 1.19

1.65 1.54 1.44

1.96 1.83 1.72

2.28 2.14 2.01

2.63 2.46 2.31

2.99 2.80 2.63


2.0


8 - 183


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


Pu

30° kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.91 3.37 3.07 2.82 2.60

4.23 3.70 3.38 3.11 2.87

4.56 4.02 3.71 3.42 3.16

4.89 4.36 4.03 3.74 3.47

5.22 4.71 4.38 4.07 3.80

5.54 5.08 4.74 4.42 4.14

5.87 5.45 5.12 4.80 4.51

6.20 5.81 5.50 5.18 4.88

6.53 6.18 5.88 5.57 5.27

6.85 6.55 6.26 5.96 5.66

7.18 6.90 6.64 6.34 6.05

7.84 7.61 7.37 7.10 6.82

8.49 8.30 8.10 7.85 7.58

9.15 8.98 8.80 8.58 8.33

9.80 9.65 9.50 9.30 9.06

10.5 10.3 10.2 10.0 9.79

0.30 0.40 0.50 0.60 0.70

2.40 2.06 1.80 1.58 1.41

2.65 2.29 2.00 1.76 1.57

2.93 2.55 2.23 1.98 1.77

3.24 2.83 2.50 2.23 2.01

3.56 3.14 2.79 2.50 2.26

3.89 3.46 3.10 2.79 2.54

4.25 3.81 3.42 3.10 2.82

4.62 4.16 3.76 3.42 3.13

5.00 4.52 4.11 3.75 3.44

5.39 4.89 4.46 4.09 3.76

5.77 5.27 4.83 4.43 4.09

6.55 6.04 5.57 5.15 4.77

7.31 6.81 6.33 5.88 5.47

8.06 7.57 7.09 6.61 6.18

8.81 8.32 7.84 7.35 6.90

9.55 9.06 8.58 8.09 7.63

0.80 0.90 1.00 1.20 1.40

1.26 1.15 1.05 0.891 0.774

1.42 1.29 1.18 1.01 0.875

1.60 1.46 1.34 1.15 1.00

1.82 1.66 1.53 1.32 1.15

2.06 1.89 1.74 1.50 1.32

2.31 2.13 1.96 1.70 1.49

2.59 2.38 2.20 1.91 1.69

2.87 2.65 2.46 2.14 1.89

3.17 2.93 2.72 2.38 2.10

3.47 3.22 3.00 2.62 2.33

3.79 3.52 3.28 2.88 2.56

4.44 4.14 3.87 3.42 3.05

5.10 4.77 4.47 3.97 3.56

5.78 5.42 5.09 4.54 4.08

6.48 6.09 5.74 5.13 4.63

7.19 6.79 6.41 5.76 5.20

1.60 1.80 2.00 2.20 2.40

0.684 0.612 0.554 0.506 0.465

0.774 0.694 0.629 0.574 0.528

0.889 0.798 0.723 0.662 0.609

1.02 0.919 0.834 0.763 0.703

1.17 1.05 0.957 0.877 0.809

1.33 1.20 1.09 1.00 0.924

1.51 1.36 1.24 1.14 1.05

1.69 1.53 1.39 1.28 1.18

1.88 1.70 1.55 1.43 1.32

2.09 1.89 1.73 1.59 1.47

2.30 2.08 1.91 1.75 1.62

2.75 2.50 2.29 2.11 1.95

3.22 2.94 2.69 2.48 2.30

3.70 3.38 3.11 2.87 2.67

4.21 3.86 3.55 3.29 3.06

4.74 4.35 4.01 3.72 3.47

2.60 0.431 0.489 0.565 0.652 0.750 0.857 0.972 1.10 1.23 2.80 0.401 0.456 0.526 0.608 0.699 0.800 0.908 1.02 1.15 3.00 0.375 0.426 0.492 0.569 0.654 0.749 0.851 0.959 1.07

1.36 1.27 1.20

1.51 1.41 1.32

1.82 1.70 1.60

2.15 2.01 1.89

2.49 2.34 2.20

2.86 2.68 2.52

3.24 3.05 2.87


8 - 184



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


45° Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.61 3.37 3.13 2.94 2.77

3.97 3.74 3.52 3.29 3.10

4.33 4.11 3.89 3.65 3.44

4.70 4.48 4.26 4.02 3.79

5.06 4.84 4.62 4.38 4.14

5.42 5.21 4.99 4.75 4.51

5.78 5.58 5.36 5.12 4.87

6.14 5.94 5.74 5.50 5.25

6.50 6.31 6.11 5.88 5.63

6.86 6.68 6.49 6.26 6.01

7.22 7.04 6.86 6.64 6.40

7.95 7.78 7.61 7.39 7.16

8.67 8.52 8.36 8.15 7.92

9.39 10.1 9.25 9.98 9.11 9.85 8.91 9.67 8.68 9.45

10.8 10.7 10.6 10.4 10.2

0.30 0.40 0.50 0.60 0.70

2.61 2.32 2.06 1.84 1.66

2.91 2.59 2.30 2.06 1.86

3.24 2.87 2.57 2.30 2.08

3.57 3.18 2.86 2.58 2.34

3.92 3.51 3.16 2.87 2.61

4.27 3.85 3.49 3.17 2.91

4.63 4.20 3.82 3.50 3.22

5.00 4.55 4.17 3.83 3.54

5.38 4.92 4.52 4.18 3.88

5.76 5.29 4.89 4.53 4.22

6.15 5.67 5.26 4.89 4.56

6.91 6.42 5.97 5.57 5.22

7.67 7.18 6.71 6.28 5.90

8.44 7.95 7.47 7.02 6.62

9.21 8.72 8.24 7.78 7.35

9.98 9.50 9.01 8.54 8.10

0.80 0.90 1.00 1.20 1.40

1.50 1.37 1.26 1.08 0.938

1.69 1.54 1.41 1.21 1.06

1.90 1.74 1.60 1.38 1.21

2.13 1.96 1.81 1.57 1.38

2.40 2.21 2.05 1.78 1.57

2.68 2.48 2.30 2.01 1.78

2.98 2.76 2.58 2.26 2.01

3.29 3.06 2.86 2.52 2.24

3.61 3.37 3.15 2.78 2.49

3.93 3.68 3.45 3.06 2.74

4.27 4.00 3.76 3.34 3.00

4.91 4.62 4.35 3.90 3.52

5.56 5.25 4.97 4.48 4.06

6.25 5.91 5.61 5.07 4.62

6.97 6.61 6.28 5.70 5.21

7.70 7.32 6.97 6.35 5.81

1.60 1.80 2.00 2.20 2.40

0.831 0.745 0.675 0.617 0.568

0.939 0.843 0.764 0.699 0.644

1.07 0.966 0.877 0.804 0.741

1.23 1.11 1.01 0.925 0.854

1.41 1.27 1.16 1.06 0.981

1.60 1.45 1.32 1.21 1.12

1.80 1.63 1.49 1.37 1.27

2.02 1.83 1.67 1.54 1.43

2.24 2.04 1.87 1.72 1.59

2.48 2.25 2.06 1.90 1.77

2.71 2.48 2.27 2.10 1.95

3.20 2.93 2.69 2.49 2.32

3.70 3.40 3.14 2.91 2.71

4.23 3.90 3.60 3.35 3.13

4.78 4.42 4.09 3.81 3.57

5.36 4.96 4.61 4.30 4.03

2.60 0.526 0.597 0.687 0.792 0.911 1.04 1.18 2.80 0.489 0.556 0.641 0.739 0.850 0.972 1.10 3.00 0.458 0.520 0.600 0.693 0.796 0.911 1.03

1.33 1.24 1.17

1.48 1.39 1.30

1.65 1.54 1.45

1.82 1.70 1.60

2.16 2.03 1.91

2.54 2.38 2.24

2.93 2.75 2.60

3.35 3.15 2.97

3.78 3.57 3.37



8 - 185


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l 60°


Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

3.28 3.20 3.10 2.98 2.89

3.67 3.59 3.50 3.38 3.27

4.06 3.98 3.90 3.79 3.66

4.45 4.37 4.29 4.17 4.04

4.84 4.76 4.67 4.56 4.43

5.23 5.14 5.05 4.94 4.80

5.62 5.53 5.43 5.31 5.17

6.01 5.92 5.82 5.69 5.55

6.40 6.30 6.20 6.07 5.92

6.79 6.69 6.58 6.45 6.29

7.18 7.08 6.96 6.83 6.67

7.96 7.86 7.74 7.59 7.43

8.74 8.63 8.51 8.36 8.19

9.53 10.3 9.41 10.2 9.29 10.1 9.13 9.91 8.96 9.74

11.1 11.0 10.9 10.7 10.5

0.30 0.40 0.50 0.60 0.70

2.80 2.63 2.44 2.27 2.09

3.16 2.96 2.74 2.54 2.35

3.54 3.30 3.06 2.84 2.63

3.92 3.66 3.40 3.16 2.93

4.29 4.02 3.75 3.49 3.25

4.66 4.38 4.10 3.83 3.58

5.02 4.74 4.46 4.18 3.92

5.39 5.09 4.80 4.53 4.26

5.76 5.44 5.14 4.86 4.59

6.13 5.80 5.49 5.20 4.92

6.50 6.16 5.84 5.54 5.25

7.25 6.89 6.54 6.23 5.93

8.01 7.64 7.28 6.93 6.63

8.77 8.40 8.02 7.66 7.34

9.55 10.3 9.17 9.94 8.78 9.56 8.42 9.18 8.07 8.82

0.80 0.90 1.00 1.20 1.40

1.94 1.80 1.67 1.46 1.28

2.18 2.02 1.88 1.64 1.45

2.44 2.27 2.11 1.85 1.64

2.73 2.54 2.37 2.09 1.86

3.03 2.83 2.65 2.35 2.10

3.35 3.14 2.95 2.62 2.35

3.68 3.46 3.26 2.91 2.63

4.01 3.78 3.58 3.21 2.91

4.34 4.11 3.90 3.52 3.20

4.66 4.43 4.21 3.82 3.48

4.99 4.74 4.52 4.11 3.76

5.65 5.40 5.16 4.73 4.35

6.34 6.07 5.82 5.36 4.96

7.04 6.76 6.50 6.02 5.58

7.75 7.46 7.19 6.69 6.23

8.49 8.18 7.90 7.38 6.89

1.60 1.80 2.00 2.20 2.40

1.15 1.03 0.940 0.861 0.794

1.30 1.17 1.06 0.974 0.899

1.47 1.33 1.21 1.11 1.03

1.67 1.51 1.39 1.27 1.18

1.89 1.72 1.58 1.46 1.35

2.13 1.95 1.79 1.65 1.54

2.39 2.19 2.01 1.86 1.73

2.66 2.44 2.25 2.09 1.94

2.93 2.69 2.49 2.31 2.15

3.19 2.94 2.72 2.53 2.36

3.46 3.20 2.97 2.76 2.58

4.01 3.72 3.47 3.24 3.03

4.60 4.28 3.99 3.74 3.52

5.20 4.85 4.55 4.27 4.02

5.82 5.45 5.12 4.82 4.55

6.47 6.07 5.72 5.40 5.11

2.60 0.736 0.834 0.956 1.10 1.26 2.80 0.686 0.778 0.893 1.03 1.18 3.00 0.643 0.729 0.837 0.964 1.11

1.43 1.34 1.26

1.62 1.52 1.43

1.82 1.70 1.61

2.02 1.89 1.79

2.22 2.08 1.97

2.42 2.28 2.15

2.86 2.69 2.55

3.32 3.13 2.97

3.80 3.60 3.41

4.31 4.08 3.88

4.84 4.59 4.37


8 - 186


Table 8-41 (cont.). Coefficients C for Eccentrically Loaded Weld Groups Angle = 75°° φ R n = CC 1 D l

C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


75° Pu

kl

l


k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.97 2.86 2.89 2.88 2.87

3.38 3.27 3.24 3.19 3.17

3.79 3.67 3.65 3.61 3.57

4.20 4.08 4.06 4.02 3.98

4.61 4.49 4.47 4.43 4.38

5.02 4.90 4.87 4.83 4.79

5.43 5.31 5.28 5.24 5.19

5.84 5.72 5.69 5.64 5.59

6.25 6.14 6.10 6.05 5.99

6.66 6.55 6.50 6.45 6.39

7.07 6.96 6.91 6.86 6.80

7.89 7.78 7.72 7.67 7.60

8.71 8.60 8.54 8.48 8.40

9.54 9.42 9.36 9.29 9.21

10.4 10.2 10.2 10.1 10.0

11.2 11.1 11.0 10.9 10.8

0.30 0.40 0.50 0.60 0.70

2.86 2.84 2.79 2.74 2.66

3.17 3.16 3.13 3.07 3.00

3.54 3.51 3.47 3.42 3.34

3.93 3.88 3.83 3.76 3.68

4.33 4.26 4.19 4.11 4.02

4.73 4.64 4.56 4.47 4.37

5.13 5.02 4.93 4.83 4.72

5.53 5.41 5.30 5.19 5.07

5.93 5.79 5.68 5.56 5.43

6.33 6.18 6.05 5.93 5.79

6.73 6.57 6.43 6.30 6.16

7.52 7.36 7.20 7.04 6.89

8.32 8.15 7.96 7.80 7.63

9.12 8.94 8.75 8.56 8.38

9.92 9.73 9.53 9.33 9.14

10.7 10.5 10.3 10.1 9.90

0.80 0.90 1.00 1.20 1.40

2.59 2.51 2.42 2.25 2.08

2.91 2.82 2.73 2.53 2.35

3.25 3.15 3.05 2.84 2.63

3.59 3.49 3.38 3.15 2.94

3.93 3.82 3.71 3.48 3.25

4.26 4.15 4.03 3.80 3.57

4.61 4.49 4.37 4.12 3.88

4.95 4.83 4.70 4.44 4.19

5.30 5.17 5.03 4.77 4.51

5.66 5.52 5.38 5.10 4.83

6.01 5.87 5.72 5.44 5.16

6.74 6.58 6.42 6.12 5.83

7.47 7.30 7.14 6.82 6.52

8.21 8.04 7.86 7.53 7.22

8.96 8.78 8.60 8.26 7.93

9.71 9.52 9.34 8.99 8.65

1.60 1.80 2.00 2.20 2.40

1.92 1.78 1.65 1.54 1.44

2.17 2.01 1.87 1.74 1.63

2.44 2.26 2.11 1.97 1.84

2.73 2.54 2.37 2.22 2.08

3.04 2.84 2.65 2.49 2.34

3.35 3.14 2.95 2.77 2.61

3.65 3.43 3.24 3.06 2.89

3.95 3.73 3.53 3.34 3.17

4.26 4.04 3.82 3.63 3.44

4.58 4.34 4.12 3.92 3.73

4.90 4.65 4.43 4.22 4.02

5.55 5.30 5.05 4.83 4.62

6.23 5.96 5.70 5.47 5.24

6.92 6.63 6.37 6.12 5.89

7.62 7.33 7.05 6.79 6.55

8.33 8.03 7.75 7.48 7.23

2.60 2.80 3.00

1.35 1.26 1.19

1.52 1.43 1.35

1.73 1.63 1.53

1.95 1.84 1.74

2.20 2.08 1.97

2.47 2.34 2.22

2.74 2.60 2.48

3.01 2.86 2.73

3.28 3.13 2.99

3.56 3.40 3.25

3.84 3.67 3.52

4.43 4.25 4.07

5.04 4.84 4.66

5.67 5.46 5.26

6.32 6.10 5.89

6.98 6.75 6.53



8 - 187


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


Pu

l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.39 1.39 1.37 1.32 1.24

1.81 1.71 1.69 1.63 1.56

2.28 2.09 2.05 1.98 1.88

2.65 2.48 2.43 2.33 2.22

3.06 2.88 2.81 2.70 2.57

3.48 3.28 3.20 3.08 2.93

3.90 3.69 3.60 3.46 3.29

4.32 4.10 4.00 3.84 3.65

4.73 4.51 4.40 4.23 4.03

5.15 4.92 4.80 4.62 4.40

5.57 5.33 5.21 5.01 4.77

6.40 6.16 6.02 5.80 5.53

7.24 6.99 6.84 6.58 6.30

8.07 7.82 7.65 7.38 7.07

8.91 8.65 8.47 8.17 7.84

9.74 9.48 9.28 8.97 8.62

0.30 0.40 0.50 0.60 0.70

1.16 0.998 0.860 0.748 0.659

1.46 1.27 1.09 0.952 0.838

1.77 1.55 1.35 1.17 1.04

2.09 1.84 1.61 1.41 1.25

2.42 2.13 1.87 1.65 1.46

2.76 2.43 2.14 1.89 1.68

3.10 2.74 2.41 2.14 1.91

3.45 3.06 2.70 2.40 2.15

3.81 3.38 3.00 2.67 2.40

4.16 3.71 3.30 2.95 2.66

4.53 4.04 3.61 3.24 2.93

5.26 4.73 4.25 3.84 3.50

6.00 5.43 4.92 4.47 4.09

6.75 6.14 5.60 5.13 4.72

7.51 6.87 6.30 5.80 5.36

8.27 7.61 7.01 6.49 6.03

0.80 0.90 1.00 1.20 1.40

0.586 0.527 0.478 0.403 0.348

0.746 0.671 0.609 0.512 0.441

0.922 0.829 0.752 0.633 0.546

1.11 1.00 0.909 0.766 0.661

1.31 1.18 1.08 0.910 0.787

1.51 1.37 1.25 1.06 0.922

1.72 1.56 1.43 1.22 1.06

1.94 1.77 1.62 1.39 1.21

2.17 1.98 1.82 1.56 1.36

2.42 2.21 2.03 1.75 1.53

2.67 2.44 2.25 1.94 1.70

3.20 2.95 2.73 2.36 2.08

3.77 3.48 3.23 2.81 2.48

4.36 4.05 3.77 3.29 2.92

4.98 4.64 4.33 3.80 3.38

5.62 5.25 4.92 4.34 3.86

1.60 1.80 2.00 2.20 2.40

0.305 0.272 0.245 0.223 0.205

0.387 0.345 0.311 0.283 0.260

0.479 0.427 0.385 0.350 0.321

0.581 0.518 0.467 0.425 0.390

0.692 0.618 0.558 0.508 0.467

0.813 0.727 0.657 0.599 0.551

0.938 0.840 0.760 0.694 0.639

1.07 0.958 0.868 0.793 0.729

1.21 1.09 0.983 0.897 0.826

1.36 1.22 1.10 1.01 0.929

1.51 1.36 1.23 1.13 1.04

1.85 1.66 1.51 1.38 1.27

2.21 1.99 1.81 1.66 1.53

2.61 2.35 2.14 1.97 1.82

3.03 2.74 2.50 2.30 2.12

3.48 3.15 2.88 2.65 2.46

2.60 0.189 0.240 0.297 0.360 0.431 0.509 0.591 0.675 0.765 0.860 0.961 1.18 2.80 0.176 0.223 0.276 0.335 0.401 0.474 0.550 0.628 0.712 0.801 0.895 1.10 3.00 0.164 0.208 0.257 0.313 0.375 0.443 0.514 0.588 0.666 0.749 0.838 1.03

1.42 1.33 1.24

1.69 1.57 1.48

1.98 1.85 1.73

2.29 2.14 2.01

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800


8 - 188



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l

Pu = factored force, kips


15°

Pu l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.48 1.42 1.38 1.32 1.24

1.89 1.77 1.73 1.66 1.56

2.31 2.15 2.09 2.01 1.91

2.72 2.55 2.47 2.37 2.25

3.13 2.96 2.86 2.74 2.60

3.54 3.38 3.25 3.11 2.96

3.95 3.79 3.64 3.49 3.32

4.36 4.20 4.04 3.87 3.68

4.77 4.61 4.44 4.25 4.05

5.18 5.02 4.84 4.64 4.41

5.59 5.43 5.24 5.02 4.79

6.41 6.25 6.05 5.80 5.53

7.23 7.08 6.85 6.57 6.29

8.05 7.90 7.66 7.36 7.06

8.87 8.73 8.46 8.15 7.82

9.69 9.55 9.28 8.94 8.60

0.30 0.40 0.50 0.60 0.70

1.16 1.00 0.869 0.759 0.670

1.46 1.27 1.10 0.961 0.849

1.79 1.56 1.35 1.18 1.05

2.12 1.85 1.62 1.42 1.26

2.45 2.16 1.89 1.67 1.48

2.79 2.46 2.17 1.92 1.72

3.13 2.77 2.45 2.18 1.95

3.48 3.09 2.74 2.44 2.20

3.83 3.41 3.04 2.72 2.45

4.19 3.74 3.34 3.00 2.71

4.54 4.07 3.65 3.29 2.98

5.27 4.76 4.30 3.90 3.55

6.01 5.46 4.96 4.53 4.15

6.75 6.17 5.64 5.18 4.78

7.51 6.90 6.34 5.86 5.43

8.27 7.63 7.06 6.55 6.10

0.80 0.90 1.00 1.20 1.40

0.598 0.539 0.490 0.413 0.357

0.758 0.683 0.621 0.524 0.452

0.934 0.842 0.766 0.646 0.558

1.12 1.02 0.924 0.781 0.675

1.33 1.20 1.10 0.928 0.804

1.54 1.40 1.28 1.09 0.943

1.76 1.60 1.47 1.25 1.09

1.99 1.81 1.66 1.42 1.24

2.22 2.03 1.87 1.61 1.40

2.47 2.26 2.08 1.80 1.57

2.72 2.50 2.31 1.99 1.75

3.26 3.01 2.79 2.42 2.13

3.83 3.55 3.30 2.88 2.54

4.43 4.12 3.84 3.36 2.99

5.06 4.71 4.41 3.88 3.45

5.70 5.33 5.00 4.42 3.95

1.60 1.80 2.00 2.20 2.40

0.314 0.280 0.253 0.230 0.211

0.398 0.355 0.320 0.291 0.267

0.491 0.438 0.395 0.360 0.330

0.595 0.531 0.479 0.437 0.401

0.708 0.633 0.572 0.522 0.479

0.833 0.746 0.675 0.616 0.566

0.967 0.867 0.784 0.717 0.659

1.10 0.988 0.896 0.819 0.753

1.25 1.12 1.01 0.926 0.853

1.40 1.26 1.14 1.04 0.959

1.56 1.40 1.27 1.16 1.07

1.90 1.71 1.55 1.43 1.31

2.27 2.05 1.87 1.71 1.58

2.67 2.42 2.20 2.03 1.87

3.11 2.81 2.57 2.37 2.19

3.56 3.24 2.96 2.73 2.53

2.60 0.195 0.247 0.305 0.370 0.444 0.524 0.610 0.697 0.790 0.889 0.993 1.22 2.80 0.182 0.230 0.284 0.344 0.412 0.487 0.568 0.649 0.736 0.827 0.925 1.14 3.00 0.170 0.214 0.265 0.321 0.386 0.456 0.531 0.607 0.688 0.774 0.866 1.06

1.47 1.37 1.28

1.74 1.62 1.52

2.04 1.90 1.79

2.36 2.20 2.07

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800



8 - 189


C min =

Pu

D min =

C1Dl

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


30° Pu

l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.64 1.52 1.44 1.36 1.28

2.03 1.93 1.82 1.72 1.61

2.42 2.32 2.21 2.09 1.97

2.81 2.72 2.60 2.47 2.33

3.20 3.11 2.99 2.84 2.69

3.59 3.50 3.37 3.21 3.04

3.98 3.89 3.75 3.58 3.40

4.37 4.28 4.14 3.95 3.76

4.76 4.68 4.52 4.33 4.11

5.15 5.07 4.91 4.70 4.48

5.54 5.46 5.29 5.08 4.84

6.33 6.24 6.07 5.84 5.59

7.11 7.03 6.84 6.60 6.34

7.89 7.81 7.62 7.38 7.10

8.67 8.60 8.41 8.15 7.87

9.45 9.39 9.19 8.93 8.65

0.30 0.40 0.50 0.60 0.70

1.20 1.05 0.921 0.812 0.722

1.51 1.32 1.16 1.02 0.908

1.84 1.61 1.41 1.25 1.11

2.18 1.91 1.68 1.49 1.33

2.53 2.23 1.96 1.74 1.56

2.88 2.55 2.26 2.01 1.81

3.22 2.87 2.56 2.29 2.07

3.56 3.19 2.86 2.57 2.33

3.91 3.52 3.16 2.86 2.60

4.26 3.85 3.47 3.15 2.87

4.62 4.18 3.79 3.45 3.15

5.33 4.86 4.44 4.07 3.74

6.07 5.57 5.11 4.71 4.36

6.82 6.28 5.80 5.38 5.00

7.58 7.02 6.51 6.06 5.67

8.35 7.77 7.24 6.77 6.35

0.80 0.90 1.00 1.20 1.40

0.647 0.586 0.535 0.454 0.393

0.816 0.739 0.674 0.572 0.496

0.998 0.905 0.827 0.703 0.610

1.20 1.09 0.994 0.847 0.736

1.41 1.28 1.18 1.00 0.874

1.64 1.49 1.37 1.17 1.02

1.88 1.71 1.58 1.36 1.19

2.12 1.95 1.80 1.55 1.36

2.37 2.18 2.02 1.75 1.53

2.63 2.42 2.25 1.95 1.72

2.90 2.68 2.48 2.16 1.91

3.46 3.21 2.99 2.61 2.32

4.05 3.77 3.52 3.10 2.75

4.66 4.36 4.08 3.61 3.22

5.30 4.97 4.67 4.15 3.72

5.96 5.60 5.28 4.71 4.24

1.60 1.80 2.00 2.20 2.40

0.347 0.310 0.280 0.255 0.234

0.437 0.391 0.353 0.322 0.296

0.538 0.481 0.435 0.397 0.365

0.650 0.582 0.526 0.481 0.442

0.773 0.693 0.627 0.573 0.528

0.909 0.816 0.740 0.677 0.623

1.06 0.949 0.861 0.789 0.727

1.21 1.09 0.988 0.904 0.833

1.37 1.23 1.12 1.02 0.944

1.53 1.38 1.26 1.15 1.06

1.70 1.54 1.40 1.28 1.18

2.07 1.87 1.71 1.57 1.45

2.47 2.24 2.05 1.88 1.74

2.90 2.64 2.41 2.22 2.06

3.36 3.06 2.81 2.59 2.40

3.85 3.52 3.23 2.98 2.77

2.60 0.217 0.273 0.337 0.409 0.489 0.577 0.674 0.772 0.875 0.983 1.10 1.34 2.80 0.201 0.254 0.314 0.381 0.455 0.538 0.628 0.719 0.815 0.916 1.02 1.26 3.00 0.188 0.238 0.293 0.356 0.426 0.504 0.588 0.673 0.763 0.858 0.959 1.18

1.62 1.51 1.42

1.92 1.79 1.68

2.24 2.10 1.97

2.59 2.43 2.28

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800


8 - 190



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



45°

l


Pu

c.g.

xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.81 1.68 1.57 1.47 1.39

2.17 2.06 1.95 1.83 1.72

2.53 2.43 2.32 2.19 2.06

2.89 2.80 2.68 2.55 2.41

3.25 3.17 3.05 2.91 2.76

3.61 3.55 3.42 3.28 3.12

3.97 3.92 3.80 3.64 3.48

4.33 4.29 4.17 4.02 3.85

4.70 4.66 4.54 4.39 4.22

5.06 5.03 4.92 4.76 4.58

5.42 5.40 5.29 5.13 4.95

6.14 6.14 6.03 5.88 5.69

6.86 6.86 6.77 6.62 6.44

7.58 7.58 7.51 7.37 7.19

8.31 8.31 8.25 8.12 7.94

9.03 9.03 8.99 8.86 8.69

0.30 0.40 0.50 0.60 0.70

1.31 1.16 1.03 0.921 0.829

1.62 1.43 1.28 1.14 1.03

1.94 1.72 1.54 1.38 1.25

2.27 2.03 1.81 1.63 1.48

2.61 2.34 2.10 1.90 1.73

2.96 2.66 2.40 2.18 1.99

3.31 2.99 2.71 2.47 2.26

3.67 3.33 3.03 2.77 2.54

4.03 3.68 3.36 3.08 2.84

4.39 4.03 3.70 3.41 3.15

4.75 4.37 4.03 3.72 3.46

5.49 5.08 4.70 4.37 4.08

6.23 5.82 5.41 5.04 4.72

6.99 6.57 6.15 5.76 5.40

7.75 7.32 6.90 6.49 6.11

8.50 8.09 7.66 7.23 6.84

0.80 0.90 1.00 1.20 1.40

0.751 0.685 0.629 0.538 0.469

0.935 0.854 0.785 0.674 0.589

1.13 1.04 0.956 0.822 0.720

1.35 1.24 1.14 0.985 0.864

1.58 1.45 1.34 1.16 1.02

1.82 1.68 1.56 1.35 1.19

2.08 1.92 1.78 1.56 1.38

2.35 2.18 2.03 1.78 1.58

2.63 2.45 2.29 2.01 1.79

2.93 2.73 2.55 2.25 2.00

3.22 3.00 2.81 2.49 2.22

3.81 3.57 3.35 2.98 2.67

4.43 4.17 3.93 3.51 3.16

5.08 4.79 4.53 4.07 3.68

5.77 5.45 5.16 4.66 4.23

6.47 6.14 5.83 5.28 4.81

1.60 1.80 2.00 2.20 2.40

0.416 0.373 0.338 0.308 0.284

0.522 0.468 0.424 0.388 0.357

0.639 0.574 0.521 0.477 0.439

0.769 0.692 0.628 0.575 0.531

0.911 0.821 0.746 0.685 0.632

1.07 0.964 0.879 0.806 0.745

1.24 1.12 1.02 0.939 0.868

1.42 1.29 1.17 1.08 0.999

1.61 1.46 1.33 1.22 1.13

1.80 1.63 1.49 1.37 1.27

2.00 1.82 1.66 1.53 1.42

2.42 2.20 2.02 1.86 1.73

2.87 2.62 2.41 2.23 2.07

3.35 3.07 2.84 2.63 2.44

3.87 3.56 3.29 3.05 2.84

4.41 4.06 3.76 3.50 3.27

2.60 0.263 0.331 0.407 0.492 0.587 0.692 0.807 0.928 1.05 1.18 2.80 0.245 0.308 0.379 0.458 0.548 0.646 0.753 0.867 0.983 1.10 3.00 0.229 0.288 0.355 0.429 0.513 0.606 0.707 0.814 0.922 1.04

1.32 1.23 1.16

1.61 1.51 1.42

1.93 1.81 1.70

2.28 2.14 2.02

2.66 2.50 2.36

3.06 2.88 2.72

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800



8 - 191


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



60°

l

Pu


c.g.

xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.95 1.82 1.73 1.63 1.55

2.28 2.15 2.05 1.96 1.87

2.61 2.48 2.38 2.28 2.18

2.94 2.82 2.71 2.61 2.50

3.26 3.16 3.06 2.94 2.83

3.59 3.51 3.41 3.29 3.17

3.92 3.86 3.76 3.65 3.53

4.25 4.21 4.12 4.01 3.89

4.57 4.56 4.47 4.37 4.25

4.90 4.90 4.83 4.73 4.62

5.23 5.24 5.18 5.09 4.98

5.88 5.88 5.87 5.80 5.70

6.54 6.54 6.54 6.49 6.41

7.19 7.19 7.19 7.18 7.11

7.85 7.85 7.85 7.85 7.79

8.51 8.51 8.51 8.51 8.47

0.30 0.40 0.50 0.60 0.70

1.47 1.34 1.22 1.12 1.03

1.78 1.62 1.48 1.36 1.25

2.09 1.91 1.75 1.61 1.49

2.40 2.20 2.03 1.88 1.74

2.72 2.51 2.32 2.15 2.00

3.05 2.83 2.63 2.44 2.28

3.40 3.16 2.95 2.75 2.58

3.76 3.52 3.28 3.07 2.89

4.13 3.88 3.63 3.41 3.22

4.50 4.24 4.00 3.76 3.56

4.87 4.61 4.36 4.13 3.91

5.60 5.36 5.10 4.86 4.62

6.32 6.09 5.84 5.59 5.33

7.02 6.82 6.57 6.32 6.05

7.72 7.53 7.30 7.03 6.78

8.40 8.23 8.01 7.75 7.50

0.80 0.90 1.00 1.20 1.40

0.945 0.874 0.812 0.709 0.626

1.16 1.07 0.999 0.875 0.776

1.38 1.28 1.20 1.06 0.939

1.62 1.51 1.41 1.25 1.12

1.87 1.75 1.64 1.46 1.31

2.14 2.01 1.89 1.69 1.52

2.42 2.28 2.15 1.93 1.75

2.72 2.57 2.43 2.19 1.98

3.04 2.87 2.72 2.46 2.23

3.36 3.18 3.02 2.74 2.49

3.70 3.51 3.34 3.03 2.76

4.39 4.17 3.97 3.62 3.31

5.09 4.85 4.63 4.23 3.89

5.80 5.55 5.32 4.89 4.50

6.52 6.26 6.02 5.56 5.15

7.24 6.98 6.73 6.26 5.82

1.60 1.80 2.00 2.20 2.40

0.560 0.506 0.461 0.423 0.391

0.696 0.630 0.575 0.528 0.489

0.845 0.766 0.701 0.645 0.597

1.01 0.916 0.839 0.774 0.718

1.18 1.08 0.993 0.917 0.851

1.38 1.26 1.16 1.07 0.999

1.59 1.46 1.34 1.24 1.16

1.81 1.66 1.53 1.42 1.33

2.04 1.88 1.74 1.61 1.51

2.28 2.11 1.95 1.81 1.69

2.54 2.34 2.17 2.02 1.89

3.05 2.82 2.62 2.44 2.29

3.59 3.33 3.11 2.90 2.72

4.17 3.88 3.62 3.39 3.19

4.79 4.46 4.17 3.91 3.68

5.43 5.07 4.75 4.47 4.21

2.60 0.363 0.454 0.556 0.669 0.794 0.932 1.08 1.24 2.80 0.339 0.424 0.519 0.625 0.744 0.875 1.02 1.17 3.00 0.317 0.398 0.488 0.587 0.700 0.824 0.956 1.10

1.41 1.33 1.25

1.59 1.49 1.41

1.77 1.66 1.57

2.15 2.02 1.91

2.56 2.42 2.29

3.01 2.84 2.69

3.48 3.29 3.12

3.98 3.77 3.58

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800


8 - 192



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



75°

Pu l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.05 1.94 1.88 1.82 1.76

2.35 2.22 2.15 2.09 2.04

2.65 2.51 2.44 2.38 2.32

2.94 2.81 2.75 2.69 2.63

3.24 3.13 3.07 3.01 2.95

3.54 3.44 3.39 3.33 3.27

3.83 3.75 3.70 3.65 3.60

4.13 4.05 4.01 3.97 3.92

4.43 4.35 4.32 4.29 4.24

4.72 4.65 4.63 4.60 4.56

5.02 4.94 4.93 4.90 4.87

5.61 5.53 5.52 5.50 5.47

6.21 6.11 6.10 6.09 6.07

6.80 6.68 6.68 6.67 6.66

7.39 7.26 7.26 7.25 7.24

7.99 7.83 7.83 7.83 7.82

0.30 0.40 0.50 0.60 0.70

1.71 1.62 1.53 1.46 1.38

1.99 1.89 1.80 1.71 1.63

2.27 2.16 2.07 1.97 1.89

2.57 2.46 2.35 2.25 2.16

2.89 2.77 2.65 2.55 2.45

3.21 3.09 2.98 2.86 2.76

3.54 3.42 3.31 3.19 3.08

3.87 3.76 3.64 3.53 3.42

4.19 4.09 3.98 3.87 3.76

4.51 4.41 4.31 4.21 4.10

4.83 4.73 4.64 4.54 4.44

5.44 5.36 5.27 5.19 5.11

6.05 5.98 5.90 5.83 5.75

6.64 6.58 6.51 6.44 6.38

7.23 7.18 7.11 7.04 6.99

7.81 7.77 7.71 7.64 7.59

0.80 0.90 1.00 1.20 1.40

1.31 1.25 1.19 1.09 0.994

1.56 1.49 1.42 1.30 1.20

1.81 1.73 1.66 1.53 1.41

2.07 1.99 1.91 1.77 1.65

2.35 2.27 2.18 2.03 1.90

2.66 2.56 2.47 2.31 2.16

2.98 2.88 2.78 2.60 2.44

3.31 3.21 3.10 2.91 2.74

3.65 3.54 3.44 3.24 3.05

3.99 3.89 3.78 3.57 3.38

4.34 4.23 4.13 3.92 3.71

5.01 4.92 4.82 4.61 4.40

5.67 5.59 5.49 5.30 5.09

6.31 6.24 6.16 5.98 5.78

6.93 6.87 6.80 6.64 6.46

7.54 7.48 7.43 7.29 7.13

1.60 1.80 2.00 2.20 2.40

0.914 0.845 0.784 0.730 0.683

1.11 1.03 0.956 0.894 0.838

1.31 1.22 1.14 1.07 1.01

1.53 1.43 1.34 1.26 1.19

1.77 1.66 1.56 1.47 1.39

2.03 1.91 1.80 1.70 1.61

2.30 2.16 2.04 1.94 1.84

2.58 2.44 2.31 2.19 2.08

2.88 2.73 2.58 2.45 2.33

3.20 3.03 2.88 2.73 2.60

3.52 3.35 3.18 3.03 2.89

4.20 4.00 3.82 3.65 3.49

4.89 4.68 4.49 4.30 4.12

5.58 5.38 5.18 4.97 4.77

6.27 6.07 5.86 5.65 5.43

6.94 6.76 6.55 6.33 6.11

2.60 0.641 0.788 0.949 1.13 2.80 0.604 0.744 0.897 1.07 3.00 0.570 0.704 0.851 1.01

1.32 1.25 1.19

1.53 1.45 1.38

1.75 1.66 1.59

1.98 1.89 1.80

2.22 2.12 2.03

2.48 2.37 2.27

2.76 2.63 2.52

3.34 3.20 3.07

3.95 3.78 3.63

4.58 4.39 4.23

5.23 5.04 4.85

5.90 5.70 5.50

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800



8 - 193


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



Pu

l

c.g.


kl xl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.39 1.39 1.38 1.32 1.24

1.81 1.72 1.70 1.63 1.55

2.23 2.10 2.05 1.96 1.86

2.65 2.48 2.41 2.31 2.18

3.06 2.86 2.77 2.65 2.51

3.48 3.24 3.13 2.99 2.84

3.90 3.62 3.50 3.34 3.17

4.32 4.00 3.86 3.68 3.50

4.73 4.38 4.22 4.03 3.83

5.15 4.76 4.58 4.38 4.16

5.57 5.13 4.94 4.72 4.50

6.40 5.88 5.66 5.42 5.17

7.24 6.63 6.38 6.12 5.86

8.07 7.37 7.11 6.83 6.55

8.91 8.12 7.84 7.55 7.25

9.74 8.87 8.57 8.27 7.96

0.30 0.40 0.50 0.60 0.70

1.16 0.998 0.860 0.748 0.659

1.45 1.26 1.08 0.942 0.828

1.75 1.52 1.31 1.14 1.00

2.05 1.79 1.56 1.35 1.20

2.36 2.08 1.81 1.58 1.41

2.68 2.36 2.07 1.83 1.63

2.99 2.65 2.34 2.07 1.86

3.31 2.94 2.60 2.32 2.10

3.63 3.24 2.88 2.58 2.34

3.95 3.54 3.15 2.84 2.58

4.27 3.84 3.44 3.11 2.83

4.93 4.46 4.02 3.66 3.35

5.60 5.09 4.63 4.23 3.90

6.27 5.74 5.25 4.83 4.47

6.96 6.41 5.89 5.45 5.06

7.66 7.08 6.55 6.08 5.67

0.80 0.90 1.00 1.20 1.40

0.586 0.527 0.478 0.403 0.348

0.735 0.661 0.599 0.505 0.436

0.895 0.807 0.734 0.621 0.537

1.07 0.971 0.885 0.751 0.651

1.27 1.15 1.05 0.893 0.776

1.47 1.34 1.23 1.05 0.912

1.69 1.54 1.41 1.21 1.06

1.91 1.75 1.61 1.38 1.21

2.13 1.96 1.81 1.56 1.37

2.36 2.18 2.01 1.75 1.54

2.60 2.40 2.22 1.94 1.71

3.09 2.86 2.66 2.33 2.07

3.61 3.35 3.13 2.75 2.45

4.15 3.87 3.62 3.20 2.86

4.72 4.41 4.14 3.68 3.30

5.30 4.97 4.68 4.17 3.75

1.60 1.80 2.00 2.20 2.40

0.305 0.272 0.245 0.223 0.205

0.383 0.341 0.308 0.280 0.257

0.472 0.422 0.381 0.347 0.318

0.574 0.512 0.463 0.422 0.388

0.685 0.613 0.554 0.506 0.465

0.806 0.722 0.654 0.597 0.548

0.936 0.839 0.760 0.694 0.639

1.07 0.963 0.873 0.798 0.735

1.22 1.10 0.993 0.909 0.836

1.37 1.23 1.12 1.02 0.942

1.53 1.38 1.25 1.15 1.06

1.86 1.68 1.53 1.41 1.30

2.21 2.00 1.84 1.69 1.56

2.58 2.35 2.15 1.99 1.85

2.98 2.72 2.50 2.31 2.15

3.41 3.11 2.87 2.65 2.47

2.60 0.189 0.238 0.294 0.358 0.430 0.508 0.592 0.681 0.773 0.872 0.976 1.20 2.80 0.176 0.221 0.274 0.333 0.400 0.472 0.550 0.633 0.720 0.811 0.908 1.12 3.00 0.164 0.207 0.256 0.311 0.374 0.442 0.515 0.593 0.673 0.758 0.850 1.05

1.46 1.36 1.27

1.72 1.61 1.51

2.00 1.88 1.76

2.30 2.16 2.03

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800


8 - 194



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



15°

Pu l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.48 1.42 1.38 1.32 1.24

1.89 1.77 1.73 1.65 1.55

2.31 2.15 2.09 1.99 1.87

2.72 2.53 2.45 2.33 2.20

3.13 2.91 2.80 2.67 2.53

3.54 3.29 3.16 3.01 2.85

3.95 3.66 3.52 3.35 3.17

4.36 4.04 3.87 3.69 3.50

4.77 4.41 4.22 4.03 3.82

5.18 4.78 4.58 4.36 4.15

5.59 5.14 4.93 4.70 4.47

6.41 5.89 5.64 5.38 5.13

7.23 6.63 6.36 6.08 5.80

8.05 7.38 7.08 6.78 6.49

8.87 8.14 7.81 7.49 7.18

9.69 8.90 8.54 8.21 7.89

0.30 0.40 0.50 0.60 0.70

1.16 1.00 0.869 0.759 0.670

1.45 1.25 1.09 0.950 0.838

1.75 1.51 1.31 1.14 1.01

2.06 1.78 1.55 1.35 1.20

2.37 2.06 1.80 1.58 1.41

2.68 2.35 2.05 1.81 1.62

3.00 2.64 2.32 2.06 1.86

3.31 2.94 2.60 2.33 2.10

3.62 3.23 2.88 2.59 2.35

3.93 3.53 3.17 2.87 2.61

4.25 3.82 3.45 3.14 2.87

4.89 4.43 4.03 3.69 3.40

5.54 5.05 4.63 4.27 3.95

6.21 5.69 5.25 4.86 4.52

6.89 6.35 5.89 5.47 5.10

7.58 7.03 6.54 6.11 5.71

0.80 0.90 1.00 1.20 1.40

0.598 0.539 0.490 0.413 0.357

0.747 0.672 0.611 0.516 0.446

0.905 0.818 0.746 0.633 0.548

1.08 0.980 0.896 0.763 0.663

1.27 1.15 1.06 0.905 0.789

1.47 1.34 1.23 1.06 0.925

1.68 1.54 1.42 1.22 1.07

1.91 1.75 1.61 1.39 1.22

2.15 1.97 1.82 1.58 1.39

2.39 2.20 2.04 1.77 1.56

2.64 2.44 2.26 1.97 1.74

3.14 2.92 2.72 2.39 2.12

3.66 3.41 3.19 2.82 2.52

4.21 3.94 3.69 3.27 2.93

4.77 4.48 4.21 3.75 3.38

5.36 5.04 4.75 4.25 3.84

1.60 1.80 2.00 2.20 2.40

0.314 0.280 0.253 0.230 0.211

0.393 0.351 0.317 0.289 0.265

0.484 0.432 0.391 0.356 0.327

0.586 0.524 0.474 0.433 0.398

0.698 0.626 0.567 0.518 0.476

0.820 0.737 0.668 0.611 0.562

0.951 0.854 0.776 0.710 0.654

1.09 0.981 0.891 0.816 0.753

1.24 1.11 1.01 0.929 0.857

1.39 1.26 1.14 1.05 0.967

1.56 1.41 1.28 1.18 1.09

1.91 1.73 1.58 1.45 1.34

2.27 2.06 1.89 1.74 1.61

2.65 2.42 2.22 2.05 1.90

3.06 2.80 2.57 2.38 2.21

3.49 3.19 2.95 2.73 2.54

2.60 0.195 0.245 0.303 0.368 0.441 0.521 0.606 0.698 0.795 0.898 1.01 1.24 2.80 0.182 0.228 0.282 0.343 0.410 0.485 0.565 0.650 0.741 0.837 0.937 1.16 3.00 0.170 0.213 0.263 0.320 0.384 0.453 0.528 0.609 0.694 0.782 0.877 1.08

1.50 1.40 1.31

1.77 1.66 1.56

2.06 1.93 1.82

2.37 2.23 2.10

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800



8 - 195


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


30°

Pu l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.64 1.52 1.44 1.36 1.28

2.03 1.92 1.81 1.70 1.60

2.42 2.30 2.17 2.04 1.91

2.81 2.66 2.53 2.37 2.22

3.20 3.02 2.87 2.70 2.53

3.59 3.38 3.21 3.02 2.83

3.98 3.74 3.55 3.35 3.14

4.37 4.10 3.90 3.67 3.45

4.76 4.46 4.24 4.00 3.77

5.15 4.83 4.59 4.34 4.09

5.54 5.19 4.94 4.68 4.42

6.33 5.93 5.66 5.37 5.10

7.11 6.67 6.39 6.09 5.79

7.89 7.43 7.13 6.82 6.51

8.67 8.19 7.88 7.56 7.24

9.45 8.96 8.64 8.32 8.00

0.30 0.40 0.50 0.60 0.70

1.20 1.05 0.921 0.812 0.722

1.49 1.30 1.14 1.01 0.895

1.79 1.56 1.36 1.20 1.07

2.08 1.82 1.59 1.41 1.26

2.37 2.08 1.83 1.63 1.47

2.66 2.35 2.09 1.87 1.69

2.96 2.63 2.35 2.12 1.93

3.25 2.92 2.63 2.39 2.17

3.56 3.22 2.92 2.66 2.43

3.89 3.52 3.21 2.94 2.70

4.21 3.84 3.51 3.23 2.98

4.86 4.46 4.12 3.82 3.55

5.53 5.10 4.73 4.41 4.11

6.23 5.77 5.37 5.01 4.69

6.95 6.45 6.03 5.64 5.30

7.68 7.15 6.70 6.30 5.93

0.80 0.90 1.00 1.20 1.40

0.647 0.586 0.535 0.454 0.393

0.803 0.726 0.663 0.563 0.489

0.966 0.878 0.805 0.687 0.599

1.15 1.05 0.960 0.825 0.721

1.34 1.23 1.13 0.973 0.855

1.54 1.42 1.31 1.13 0.997

1.76 1.62 1.50 1.30 1.15

2.00 1.84 1.71 1.48 1.31

2.24 2.07 1.92 1.68 1.49

2.49 2.31 2.15 1.88 1.67

2.75 2.56 2.39 2.10 1.86

3.30 3.08 2.88 2.55 2.28

3.85 3.61 3.39 3.02 2.72

4.41 4.15 3.91 3.50 3.16

4.99 4.71 4.45 4.01 3.63

5.60 5.30 5.02 4.53 4.12

1.60 1.80 2.00 2.20 2.40

0.347 0.310 0.280 0.255 0.234

0.432 0.386 0.349 0.319 0.293

0.530 0.475 0.430 0.393 0.361

0.639 0.574 0.521 0.476 0.439

0.760 0.684 0.620 0.568 0.524

0.890 0.802 0.729 0.668 0.616

1.03 0.928 0.846 0.776 0.716

1.18 1.06 0.969 0.890 0.823

1.33 1.21 1.10 1.01 0.936

1.50 1.36 1.24 1.14 1.06

1.68 1.52 1.39 1.28 1.19

2.06 1.87 1.71 1.58 1.46

2.46 2.25 2.06 1.91 1.77

2.88 2.63 2.42 2.25 2.09

3.31 3.04 2.80 2.60 2.42

3.77 3.47 3.21 2.98 2.78

2.60 0.217 0.271 0.335 0.406 0.486 0.572 0.665 0.765 0.870 0.983 1.10 1.36 2.80 0.201 0.252 0.311 0.378 0.453 0.534 0.621 0.714 0.813 0.919 1.03 1.28 3.00 0.188 0.236 0.291 0.354 0.424 0.500 0.582 0.669 0.763 0.862 0.967 1.20

1.65 1.55 1.45

1.95 1.83 1.72

2.27 2.13 2.00

2.60 2.45 2.31

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800


8 - 196



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



45°

Pu l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.81 1.68 1.57 1.47 1.39

2.17 2.04 1.93 1.81 1.70

2.53 2.38 2.25 2.12 1.99

2.89 2.71 2.56 2.41 2.27

3.25 3.04 2.86 2.69 2.53

3.61 3.37 3.18 2.99 2.82

3.97 3.71 3.51 3.30 3.12

4.33 4.06 3.85 3.64 3.44

4.70 4.41 4.19 3.98 3.77

5.06 4.77 4.55 4.33 4.12

5.42 5.12 4.91 4.69 4.47

6.14 5.84 5.64 5.41 5.18

6.86 6.56 6.37 6.14 5.92

7.58 7.30 7.10 6.88 6.66

8.31 8.03 7.85 7.63 7.40

9.03 8.76 8.59 8.38 8.15

0.30 0.40 0.50 0.60 0.70

1.31 1.16 1.03 0.921 0.829

1.60 1.41 1.26 1.13 1.02

1.88 1.67 1.48 1.33 1.20

2.14 1.91 1.71 1.55 1.41

2.40 2.16 1.96 1.78 1.63

2.68 2.43 2.22 2.03 1.86

2.97 2.71 2.49 2.29 2.11

3.28 3.01 2.77 2.55 2.37

3.60 3.31 3.06 2.84 2.63

3.94 3.63 3.37 3.13 2.91

4.28 3.96 3.68 3.43 3.21

4.98 4.64 4.34 4.06 3.82

5.70 5.33 5.02 4.72 4.45

6.42 6.03 5.70 5.38 5.09

7.17 6.74 6.38 6.06 5.75

7.92 7.46 7.09 6.75 6.42

0.80 0.90 1.00 1.20 1.40

0.751 0.685 0.629 0.538 0.469

0.920 0.840 0.772 0.664 0.581

1.10 1.01 0.930 0.804 0.706

1.29 1.19 1.10 0.957 0.845

1.50 1.38 1.28 1.12 0.995

1.72 1.59 1.48 1.30 1.16

1.95 1.81 1.69 1.49 1.33

2.20 2.05 1.92 1.69 1.51

2.46 2.30 2.15 1.91 1.71

2.72 2.55 2.40 2.13 1.91

3.00 2.82 2.65 2.37 2.13

3.59 3.38 3.19 2.87 2.59

4.21 3.98 3.77 3.40 3.09

4.82 4.58 4.35 3.95 3.60

5.46 5.20 4.95 4.50 4.12

6.12 5.83 5.57 5.09 4.67

1.60 1.80 2.00 2.20 2.40

0.416 0.373 0.338 0.308 0.284

0.515 0.463 0.420 0.384 0.354

0.629 0.566 0.515 0.471 0.435

0.754 0.681 0.621 0.569 0.526

0.892 0.807 0.736 0.677 0.626

1.04 0.943 0.862 0.793 0.734

1.20 1.09 0.995 0.918 0.850

1.36 1.24 1.14 1.05 0.974

1.54 1.41 1.29 1.19 1.11

1.73 1.58 1.45 1.34 1.25

1.93 1.77 1.62 1.50 1.40

2.36 2.17 2.00 1.85 1.72

2.82 2.60 2.40 2.23 2.08

3.31 3.05 2.83 2.64 2.46

3.80 3.52 3.27 3.05 2.85

4.31 4.00 3.73 3.48 3.27

2.60 0.263 0.328 0.403 0.488 0.582 0.683 0.792 0.909 1.03 1.16 2.80 0.245 0.306 0.376 0.455 0.543 0.639 0.741 0.851 0.966 1.09 3.00 0.229 0.286 0.352 0.427 0.510 0.600 0.697 0.799 0.910 1.03

1.31 1.22 1.15

1.61 1.51 1.43

1.95 1.83 1.73

2.31 2.17 2.05

2.68 2.53 2.39

3.07 2.90 2.74

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800



8 - 197


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



60° Pu

l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.95 1.83 1.73 1.63 1.55

2.28 2.13 2.03 1.93 1.84

2.61 2.42 2.30 2.19 2.09

2.94 2.72 2.58 2.45 2.34

3.26 3.03 2.88 2.74 2.62

3.59 3.36 3.19 3.04 2.92

3.92 3.69 3.52 3.36 3.23

4.25 4.03 3.86 3.69 3.55

4.57 4.37 4.20 4.03 3.88

4.90 4.71 4.55 4.38 4.22

5.23 5.05 4.89 4.73 4.56

5.88 5.73 5.59 5.43 5.26

6.54 6.41 6.27 6.13 5.97

7.19 7.08 6.96 6.82 6.67

7.85 7.74 7.63 7.51 7.37

8.51 8.40 8.31 8.19 8.06

0.30 0.40 0.50 0.60 0.70

1.47 1.34 1.22 1.12 1.03

1.76 1.60 1.46 1.34 1.23

2.00 1.84 1.69 1.56 1.44

2.25 2.08 1.93 1.79 1.67

2.52 2.34 2.18 2.04 1.91

2.81 2.62 2.45 2.30 2.16

3.12 2.91 2.73 2.57 2.42

3.43 3.22 3.04 2.86 2.70

3.76 3.54 3.35 3.16 3.00

4.09 3.87 3.67 3.48 3.30

4.42 4.20 3.99 3.80 3.62

5.11 4.87 4.66 4.46 4.27

5.80 5.54 5.33 5.14 4.94

6.51 6.23 6.01 5.81 5.62

7.22 6.91 6.69 6.49 6.30

7.93 7.61 7.37 7.16 6.98

0.80 0.90 1.00 1.20 1.40

0.945 0.874 0.812 0.709 0.626

1.14 1.06 0.983 0.862 0.766

1.34 1.25 1.17 1.03 0.922

1.55 1.45 1.37 1.21 1.09

1.79 1.68 1.58 1.41 1.27

2.03 1.91 1.81 1.62 1.46

2.29 2.16 2.05 1.84 1.67

2.56 2.42 2.30 2.08 1.89

2.84 2.70 2.57 2.33 2.13

3.14 2.98 2.84 2.59 2.37

3.45 3.28 3.13 2.86 2.63

4.08 3.91 3.75 3.44 3.18

4.75 4.56 4.39 4.06 3.76

5.42 5.24 5.05 4.70 4.38

6.10 5.91 5.73 5.36 5.02

6.79 6.60 6.41 6.03 5.66

1.60 1.80 2.00 2.20 2.40

0.560 0.506 0.461 0.423 0.391

0.688 0.623 0.569 0.524 0.484

0.831 0.756 0.692 0.638 0.591

0.988 0.901 0.828 0.765 0.710

1.16 1.06 0.975 0.902 0.840

1.34 1.23 1.13 1.05 0.979

1.53 1.41 1.30 1.21 1.13

1.74 1.60 1.48 1.38 1.29

1.95 1.80 1.67 1.56 1.46

2.18 2.02 1.88 1.75 1.64

2.43 2.25 2.09 1.95 1.83

2.94 2.73 2.55 2.39 2.24

3.50 3.26 3.05 2.86 2.69

4.08 3.82 3.58 3.37 3.17

4.70 4.41 4.14 3.90 3.69

5.32 5.00 4.71 4.45 4.21

2.60 0.363 0.451 0.551 0.662 0.785 0.916 1.06 1.21 2.80 0.339 0.421 0.515 0.620 0.736 0.861 0.994 1.14 3.00 0.317 0.395 0.484 0.584 0.693 0.812 0.939 1.07

1.37 1.29 1.22

1.54 1.45 1.37

1.72 1.62 1.53

2.11 2.00 1.89

2.54 2.40 2.28

3.00 2.85 2.70

3.49 3.31 3.15

3.99 3.79 3.60

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800


8 - 198



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



75° Pu l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.05 1.94 1.88 1.82 1.76

2.35 2.21 2.14 2.07 2.01

2.65 2.48 2.40 2.32 2.26

2.94 2.77 2.67 2.59 2.53

3.24 3.06 2.96 2.88 2.82

3.54 3.37 3.26 3.18 3.11

3.83 3.68 3.58 3.48 3.41

4.13 3.99 3.89 3.78 3.71

4.43 4.31 4.20 4.10 4.02

4.72 4.60 4.51 4.42 4.32

5.02 4.88 4.82 4.73 4.63

5.61 5.46 5.42 5.35 5.26

6.21 6.04 6.00 5.96 5.88

6.80 6.61 6.58 6.54 6.49

7.39 7.19 7.16 7.12 7.08

7.99 7.77 7.74 7.70 7.67

0.30 0.40 0.50 0.60 0.70

1.71 1.62 1.53 1.46 1.38

1.95 1.86 1.77 1.69 1.61

2.20 2.10 2.01 1.93 1.85

2.47 2.36 2.27 2.18 2.09

2.76 2.65 2.55 2.45 2.36

3.05 2.95 2.84 2.74 2.65

3.36 3.25 3.15 3.05 2.95

3.66 3.56 3.46 3.36 3.26

3.96 3.86 3.77 3.67 3.57

4.26 4.16 4.07 3.98 3.89

4.56 4.46 4.38 4.29 4.21

5.17 5.06 4.98 4.91 4.83

5.79 5.66 5.58 5.51 5.44

6.42 6.26 6.17 6.10 6.05

7.03 6.88 6.76 6.69 6.64

7.62 7.49 7.35 7.28 7.22

0.80 0.90 1.00 1.20 1.40

1.31 1.25 1.19 1.09 0.994

1.54 1.47 1.40 1.29 1.18

1.77 1.70 1.63 1.50 1.39

2.01 1.94 1.87 1.73 1.61

2.28 2.19 2.12 1.98 1.85

2.56 2.47 2.39 2.24 2.10

2.85 2.76 2.68 2.51 2.36

3.16 3.07 2.98 2.80 2.64

3.48 3.38 3.29 3.11 2.94

3.79 3.70 3.60 3.42 3.24

4.11 4.02 3.93 3.74 3.56

4.75 4.66 4.57 4.39 4.21

5.37 5.29 5.21 5.04 4.86

5.98 5.92 5.84 5.69 5.52

6.58 6.52 6.46 6.32 6.17

7.17 7.13 7.07 6.95 6.80

1.60 1.80 2.00 2.20 2.40

0.914 0.845 0.784 0.730 0.683

1.10 1.02 0.947 0.886 0.832

1.29 1.21 1.13 1.06 0.996

1.50 1.41 1.32 1.25 1.18

1.73 1.63 1.53 1.45 1.37

1.97 1.86 1.75 1.66 1.57

2.23 2.10 1.99 1.89 1.79

2.50 2.36 2.24 2.13 2.03

2.78 2.64 2.50 2.38 2.27

3.08 2.93 2.78 2.65 2.53

3.39 3.23 3.07 2.93 2.80

4.02 3.85 3.68 3.53 3.38

4.68 4.50 4.33 4.16 3.99

5.34 5.16 4.98 4.81 4.64

6.00 5.82 5.65 5.47 5.29

6.65 6.49 6.31 6.13 5.96

2.60 0.641 0.782 0.940 1.11 2.80 0.604 0.738 0.890 1.06 3.00 0.570 0.699 0.844 1.00

1.30 1.23 1.17

1.49 1.42 1.36

1.71 1.62 1.55

1.93 1.84 1.76

2.17 2.07 1.98

2.42 2.31 2.22

2.68 2.57 2.46

3.24 3.11 2.99

3.84 3.70 3.56

4.47 4.31 4.16

5.12 4.95 4.79

5.78 5.61 5.44

x

0.000 0.008 0.029 0.056 0.089 0.125 0.164 0.204 0.246 0.289 0.333 0.424 0.516 0.610 0.704 0.800



8 - 199


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l Pu

yl


l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.39 1.39 1.38 1.32 1.24

1.60 1.53 1.52 1.47 1.39

1.81 1.71 1.69 1.63 1.55

2.02 1.90 1.87 1.80 1.71

2.23 2.09 2.05 1.98 1.88

2.44 2.29 2.24 2.15 2.04

2.65 2.48 2.43 2.33 2.21

2.85 2.68 2.62 2.52 2.39

3.06 2.88 2.82 2.70 2.57

3.27 3.08 3.01 2.89 2.74

3.48 3.29 3.21 3.08 2.93

3.90 3.70 3.61 3.46 3.30

4.32 4.11 4.01 3.85 3.68

4.73 4.52 4.42 4.25 4.06

5.15 4.94 4.83 4.65 4.46

5.57 5.35 5.26 5.05 4.85

0.30 0.40 0.50 0.60 0.70

1.16 0.998 0.860 0.748 0.659

1.30 1.12 0.965 0.840 0.739

1.45 1.25 1.08 0.935 0.822

1.61 1.39 1.20 1.04 0.913

1.77 1.54 1.32 1.15 1.01

1.92 1.68 1.46 1.27 1.12

2.08 1.82 1.60 1.40 1.24

2.25 1.97 1.73 1.53 1.36

2.42 2.13 1.87 1.66 1.48

2.59 2.29 2.02 1.80 1.61

2.77 2.45 2.17 1.94 1.74

3.13 2.79 2.50 2.24 2.03

3.50 3.15 2.84 2.57 2.34

3.88 3.51 3.19 2.90 2.66

4.26 3.89 3.55 3.26 3.00

4.65 4.27 3.92 3.62 3.35

0.80 0.90 1.00 1.20 1.40

0.586 0.527 0.478 0.403 0.348

0.658 0.591 0.536 0.452 0.389

0.732 0.658 0.597 0.503 0.433

0.813 0.731 0.663 0.558 0.481

0.901 0.811 0.736 0.620 0.535

1.00 0.900 0.818 0.690 0.596

1.11 0.999 0.909 0.769 0.665

1.22 1.11 1.01 0.856 0.743

1.34 1.21 1.11 0.947 0.824

1.45 1.32 1.21 1.04 0.904

1.58 1.44 1.32 1.13 0.990

1.85 1.69 1.56 1.35 1.18

2.14 1.97 1.82 1.58 1.39

2.45 2.26 2.10 1.83 1.62

2.77 2.57 2.40 2.10 1.86

3.11 2.90 2.71 2.38 2.12

1.60 1.80 2.00 2.20 2.40

0.305 0.272 0.245 0.223 0.205

0.342 0.305 0.275 0.250 0.229

0.381 0.339 0.306 0.278 0.255

0.423 0.377 0.340 0.309 0.284

0.470 0.419 0.378 0.344 0.316

0.525 0.468 0.423 0.385 0.354

0.586 0.524 0.473 0.431 0.396

0.655 0.585 0.529 0.483 0.443

0.728 0.652 0.590 0.539 0.495

0.800 0.717 0.649 0.593 0.546

0.877 0.787 0.713 0.652 0.600

1.05 0.941 0.854 0.780 0.719

1.24 1.11 1.01 0.927 0.855

1.44 1.30 1.19 1.09 1.00

1.67 1.51 1.38 1.26 1.17

1.91 1.73 1.58 1.45 1.34

2.60 0.189 0.212 0.236 0.262 0.292 0.327 0.366 0.410 0.458 0.505 0.555 0.667 0.792 0.933 1.09 1.25 2.80 0.176 0.197 0.219 0.244 0.271 0.304 0.340 0.381 0.426 0.470 0.517 0.621 0.739 0.870 1.01 1.17 3.00 0.164 0.184 0.204 0.228 0.254 0.284 0.318 0.356 0.398 0.440 0.483 0.581 0.692 0.815 0.950 1.10

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


8 - 200


Table 8-44 (cont.). Coefficients C for Eccentrically Loaded Weld Groups Angle = ±15°° φ R n = CC1Dl

C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


15° yl

15°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.48 1.42 1.38 1.32 1.24

1.69 1.56 1.53 1.48 1.39

1.89 1.73 1.69 1.63 1.55

2.10 1.90 1.85 1.78 1.69

2.31 2.09 2.03 1.94 1.85

2.51 2.28 2.20 2.11 2.00

2.72 2.47 2.39 2.28 2.16

2.92 2.67 2.57 2.46 2.33

3.13 2.88 2.76 2.64 2.50

3.33 3.09 2.95 2.82 2.67

3.54 3.30 3.15 3.00 2.85

3.95 3.73 3.54 3.38 3.21

4.36 4.17 3.95 3.77 3.59

4.77 4.59 4.37 4.16 3.97

5.18 5.02 4.80 4.57 4.36

5.59 5.45 5.23 4.97 4.76

0.30 0.40 0.50 0.60 0.70

1.16 1.00 0.869 0.759 0.670

1.31 1.13 0.978 0.854 0.753

1.46 1.27 1.09 0.953 0.840

1.60 1.41 1.22 1.06 0.936

1.74 1.53 1.34 1.18 1.04

1.89 1.66 1.46 1.29 1.15

2.04 1.80 1.59 1.41 1.26

2.20 1.94 1.72 1.53 1.37

2.36 2.09 1.85 1.65 1.48

2.53 2.25 2.00 1.79 1.61

2.70 2.41 2.15 1.93 1.74

3.05 2.74 2.47 2.23 2.03

3.42 3.09 2.80 2.55 2.33

3.79 3.45 3.15 2.89 2.65

4.18 3.83 3.51 3.24 2.99

4.57 4.21 3.88 3.60 3.34

0.80 0.90 1.00 1.20 1.40

0.598 0.539 0.490 0.413 0.357

0.672 0.605 0.550 0.464 0.401

0.749 0.675 0.613 0.517 0.446

0.834 0.750 0.681 0.574 0.496

0.927 0.834 0.758 0.639 0.552

1.03 0.925 0.842 0.712 0.616

1.13 1.02 0.933 0.791 0.686

1.23 1.12 1.03 0.878 0.764

1.34 1.23 1.12 0.964 0.841

1.46 1.33 1.23 1.06 0.921

1.58 1.45 1.34 1.15 1.01

1.85 1.71 1.58 1.36 1.20

2.14 1.98 1.84 1.60 1.41

2.45 2.28 2.12 1.86 1.64

2.78 2.59 2.42 2.13 1.89

3.12 2.91 2.73 2.41 2.15

1.60 1.80 2.00 2.20 2.40

0.314 0.280 0.253 0.230 0.211

0.352 0.314 0.283 0.258 0.237

0.392 0.350 0.315 0.287 0.263

0.436 0.389 0.351 0.319 0.293

0.485 0.433 0.391 0.356 0.327

0.542 0.484 0.437 0.398 0.366

0.605 0.541 0.489 0.446 0.410

0.675 0.604 0.546 0.499 0.458

0.745 0.668 0.606 0.554 0.510

0.818 0.735 0.666 0.609 0.561

0.897 0.806 0.731 0.669 0.616

1.07 0.962 0.875 0.801 0.738

1.26 1.14 1.04 0.950 0.877

1.47 1.33 1.21 1.11 1.03

1.70 1.54 1.41 1.29 1.20

1.94 1.77 1.62 1.49 1.38

2.60 0.195 0.219 0.243 0.271 0.302 0.338 0.379 0.424 0.472 0.520 0.571 0.685 0.814 0.957 1.12 1.28 2.80 0.182 0.203 0.226 0.252 0.281 0.314 0.352 0.394 0.439 0.484 0.532 0.639 0.759 0.894 1.04 1.20 3.00 0.170 0.190 0.211 0.235 0.262 0.294 0.329 0.368 0.411 0.453 0.498 0.598 0.711 0.838 0.977 1.13

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167



8 - 201


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



30°

yl

30°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.64 1.52 1.44 1.36 1.28

1.83 1.68 1.60 1.51 1.43

2.03 1.85 1.76 1.67 1.58

2.22 2.03 1.91 1.83 1.73

2.42 2.21 2.08 1.98 1.88

2.61 2.39 2.25 2.14 2.03

2.81 2.58 2.42 2.30 2.18

3.01 2.77 2.60 2.47 2.34

3.20 2.97 2.78 2.64 2.51

3.40 3.16 2.97 2.82 2.68

3.59 3.36 3.16 3.00 2.85

3.98 3.77 3.55 3.37 3.21

4.37 4.18 3.95 3.76 3.59

4.76 4.59 4.37 4.16 3.98

5.15 5.00 4.79 4.57 4.37

5.54 5.41 5.20 4.98 4.78

0.30 0.40 0.50 0.60 0.70

1.20 1.05 0.921 0.812 0.722

1.34 1.18 1.03 0.910 0.811

1.49 1.31 1.15 1.02 0.908

1.63 1.44 1.27 1.13 1.01

1.78 1.58 1.40 1.25 1.12

1.92 1.71 1.52 1.36 1.22

2.06 1.84 1.64 1.47 1.33

2.22 1.98 1.77 1.59 1.44

2.37 2.13 1.91 1.72 1.56

2.54 2.28 2.06 1.86 1.69

2.71 2.44 2.21 2.00 1.83

3.06 2.78 2.53 2.31 2.12

3.43 3.13 2.87 2.64 2.43

3.81 3.50 3.23 2.98 2.77

4.20 3.88 3.60 3.34 3.12

4.60 4.27 3.98 3.71 3.48

0.80 0.90 1.00 1.20 1.40

0.647 0.586 0.535 0.454 0.393

0.729 0.660 0.601 0.510 0.441

0.815 0.736 0.671 0.568 0.492

0.908 0.820 0.747 0.632 0.547

1.01 0.912 0.831 0.704 0.610

1.11 1.01 0.920 0.783 0.680

1.21 1.10 1.01 0.868 0.756

1.31 1.20 1.11 0.952 0.835

1.43 1.31 1.21 1.04 0.916

1.55 1.42 1.31 1.14 1.00

1.67 1.54 1.43 1.24 1.10

1.95 1.81 1.68 1.47 1.30

2.25 2.10 1.96 1.72 1.53

2.58 2.41 2.25 1.99 1.77

2.91 2.73 2.56 2.28 2.04

3.26 3.07 2.89 2.58 2.32

1.60 1.80 2.00 2.20 2.40

0.347 0.310 0.280 0.255 0.234

0.389 0.347 0.314 0.286 0.263

0.433 0.387 0.349 0.318 0.292

0.482 0.430 0.389 0.354 0.325

0.537 0.480 0.433 0.395 0.363

0.600 0.536 0.485 0.442 0.406

0.668 0.599 0.542 0.495 0.455

0.742 0.666 0.604 0.552 0.508

0.814 0.733 0.666 0.610 0.562

0.893 0.805 0.732 0.670 0.618

0.978 0.882 0.802 0.735 0.679

1.16 1.05 0.958 0.880 0.813

1.37 1.24 1.13 1.04 0.963

1.60 1.45 1.33 1.22 1.13

1.84 1.67 1.54 1.42 1.31

2.10 1.92 1.76 1.63 1.51

2.60 0.217 0.243 0.270 0.301 0.335 0.376 0.421 0.471 0.521 0.573 0.630 0.754 0.896 1.05 1.22 2.80 0.201 0.226 0.251 0.280 0.312 0.349 0.392 0.438 0.486 0.535 0.588 0.704 0.836 0.984 1.15 3.00 0.188 0.211 0.235 0.261 0.292 0.327 0.366 0.410 0.455 0.501 0.550 0.660 0.784 0.924 1.08

1.41 1.32 1.24

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


8 - 202



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



45° yl

45°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.81 1.68 1.57 1.47 1.39

1.99 1.83 1.71 1.61 1.52

2.17 1.99 1.86 1.75 1.65

2.35 2.14 2.01 1.90 1.80

2.53 2.30 2.17 2.06 1.96

2.71 2.47 2.33 2.22 2.11

2.89 2.64 2.50 2.37 2.26

3.07 2.82 2.67 2.54 2.42

3.25 3.00 2.84 2.71 2.58

3.43 3.18 3.02 2.88 2.75

3.61 3.37 3.21 3.06 2.93

3.97 3.76 3.59 3.43 3.29

4.33 4.15 3.99 3.83 3.68

4.70 4.55 4.39 4.23 4.07

5.06 4.95 4.80 4.64 4.48

5.42 5.35 5.21 5.06 4.90

0.30 0.40 0.50 0.60 0.70

1.31 1.16 1.03 0.921 0.829

1.43 1.27 1.14 1.02 0.919

1.56 1.40 1.25 1.13 1.02

1.71 1.53 1.38 1.24 1.13

1.86 1.67 1.51 1.36 1.24

2.01 1.81 1.64 1.49 1.36

2.15 1.95 1.77 1.62 1.48

2.31 2.10 1.91 1.74 1.60

2.46 2.25 2.05 1.88 1.73

2.63 2.40 2.20 2.02 1.87

2.80 2.57 2.36 2.18 2.01

3.16 2.92 2.70 2.50 2.32

3.54 3.28 3.05 2.85 2.66

3.93 3.66 3.43 3.21 3.02

4.33 4.06 3.81 3.59 3.39

4.75 4.47 4.21 3.98 3.77

0.80 0.90 1.00 1.20 1.40

0.751 0.685 0.629 0.538 0.469

0.835 0.764 0.702 0.603 0.527

0.928 0.849 0.782 0.674 0.589

1.03 0.943 0.870 0.751 0.655

1.14 1.05 0.966 0.836 0.730

1.25 1.15 1.07 0.926 0.811

1.36 1.26 1.16 1.01 0.894

1.47 1.36 1.27 1.11 0.979

1.60 1.48 1.38 1.21 1.07

1.73 1.61 1.50 1.32 1.17

1.87 1.74 1.63 1.43 1.28

2.17 2.03 1.90 1.69 1.51

2.49 2.34 2.21 1.97 1.77

2.84 2.68 2.53 2.27 2.05

3.20 3.03 2.87 2.59 2.34

3.58 3.39 3.22 2.92 2.65

1.60 1.80 2.00 2.20 2.40

0.416 0.373 0.338 0.308 0.284

0.467 0.419 0.379 0.346 0.318

0.521 0.466 0.422 0.385 0.355

0.580 0.519 0.470 0.429 0.395

0.646 0.579 0.524 0.479 0.441

0.721 0.647 0.587 0.536 0.493

0.799 0.720 0.654 0.599 0.552

0.877 0.793 0.722 0.663 0.613

0.961 0.869 0.794 0.730 0.675

1.05 0.953 0.871 0.801 0.741

1.15 1.04 0.954 0.878 0.812

1.36 1.24 1.14 1.05 0.971

1.60 1.46 1.34 1.24 1.15

1.86 1.70 1.56 1.45 1.34

2.14 1.96 1.80 1.67 1.56

2.43 2.23 2.06 1.92 1.78

2.60 0.263 0.295 0.328 0.365 0.408 0.457 0.511 0.570 0.627 0.689 0.756 0.904 1.07 1.26 2.80 0.245 0.274 0.305 0.340 0.380 0.425 0.476 0.532 0.585 0.643 0.707 0.845 1.00 1.18 3.00 0.229 0.256 0.286 0.318 0.355 0.398 0.446 0.498 0.549 0.604 0.663 0.794 0.942 1.11

1.46 1.37 1.29

1.67 1.57 1.48

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167



8 - 203


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



60° yl

60°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.95 1.83 1.73 1.63 1.55

2.12 1.94 1.84 1.74 1.66

2.28 2.07 1.96 1.87 1.78

2.44 2.20 2.10 2.00 1.92

2.61 2.35 2.25 2.15 2.06

2.77 2.51 2.40 2.31 2.22

2.94 2.68 2.57 2.47 2.38

3.10 2.85 2.74 2.64 2.55

3.26 3.03 2.92 2.81 2.72

3.43 3.21 3.10 2.99 2.89

3.59 3.40 3.28 3.18 3.07

3.92 3.78 3.67 3.56 3.45

4.25 4.17 4.07 3.96 3.85

4.57 4.55 4.47 4.37 4.27

4.90 4.90 4.86 4.78 4.68

5.23 5.23 5.23 5.16 5.08

0.30 0.40 0.50 0.60 0.70

1.47 1.34 1.22 1.12 1.03

1.58 1.44 1.32 1.21 1.12

1.70 1.56 1.43 1.32 1.22

1.83 1.69 1.56 1.44 1.33

1.98 1.83 1.69 1.57 1.46

2.13 1.98 1.83 1.71 1.59

2.29 2.13 1.99 1.85 1.73

2.46 2.29 2.14 2.00 1.88

2.63 2.45 2.30 2.15 2.02

2.80 2.62 2.46 2.31 2.18

2.98 2.80 2.63 2.48 2.34

3.35 3.16 2.99 2.83 2.68

3.75 3.55 3.37 3.21 3.05

4.16 3.96 3.77 3.60 3.44

4.58 4.38 4.19 4.01 3.85

5.00 4.81 4.61 4.43 4.26

0.80 0.90 1.00 1.20 1.40

0.945 0.874 0.812 0.709 0.626

1.03 0.958 0.893 0.783 0.695

1.13 1.05 0.983 0.865 0.771

1.24 1.16 1.08 0.957 0.855

1.36 1.27 1.19 1.06 0.948

1.49 1.40 1.31 1.17 1.05

1.63 1.53 1.44 1.29 1.16

1.76 1.66 1.56 1.40 1.26

1.90 1.79 1.69 1.52 1.38

2.05 1.94 1.83 1.65 1.50

2.21 2.09 1.98 1.79 1.63

2.55 2.42 2.30 2.09 1.91

2.91 2.77 2.65 2.42 2.22

3.29 3.15 3.01 2.76 2.55

3.69 3.53 3.39 3.13 2.89

4.09 3.93 3.79 3.51 3.25

1.60 1.80 2.00 2.20 2.40

0.560 0.506 0.461 0.423 0.391

0.623 0.564 0.515 0.473 0.438

0.693 0.629 0.575 0.530 0.489

0.771 0.701 0.642 0.590 0.545

0.857 0.781 0.716 0.659 0.608

0.952 0.868 0.798 0.735 0.680

1.05 0.957 0.880 0.813 0.755

1.15 1.05 0.964 0.891 0.829

1.25 1.15 1.06 0.978 0.910

1.36 1.25 1.16 1.07 0.997

1.49 1.37 1.26 1.17 1.09

1.75 1.61 1.49 1.39 1.30

2.04 1.89 1.75 1.63 1.53

2.35 2.18 2.03 1.90 1.78

2.68 2.50 2.33 2.18 2.05

3.03 2.83 2.64 2.48 2.33

2.60 0.363 0.407 0.454 0.506 0.565 0.632 0.704 0.774 0.850 0.932 1.02 1.22 2.80 0.339 0.380 0.424 0.472 0.527 0.590 0.658 0.726 0.797 0.875 0.958 1.14 3.00 0.317 0.356 0.397 0.442 0.494 0.553 0.618 0.683 0.750 0.824 0.903 1.08

1.43 1.35 1.27

1.67 1.57 1.49

1.93 1.82 1.72

2.20 2.08 1.97

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


8 - 204



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



75°

75°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.05 1.94 1.88 1.82 1.76

2.20 2.01 1.95 1.90 1.84

2.35 2.11 2.05 2.00 1.95

2.50 2.23 2.17 2.12 2.07

2.65 2.37 2.31 2.26 2.21

2.79 2.52 2.47 2.41 2.36

2.94 2.69 2.63 2.58 2.53

3.09 2.87 2.81 2.76 2.71

3.24 3.05 3.00 2.95 2.89

3.39 3.24 3.19 3.14 3.09

3.54 3.44 3.39 3.34 3.29

3.83 3.80 3.77 3.73 3.69

4.13 4.09 4.09 4.09 4.06

4.43 4.38 4.38 4.38 4.38

4.72 4.67 4.67 4.67 4.67

5.02 4.96 4.96 4.96 4.96

0.30 0.40 0.50 0.60 0.70

1.71 1.62 1.53 1.46 1.38

1.79 1.70 1.62 1.54 1.47

1.90 1.81 1.72 1.64 1.57

2.02 1.93 1.84 1.76 1.69

2.16 2.07 1.98 1.90 1.82

2.31 2.22 2.13 2.05 1.97

2.48 2.39 2.30 2.21 2.13

2.66 2.56 2.47 2.38 2.30

2.84 2.75 2.66 2.57 2.48

3.04 2.94 2.85 2.76 2.67

3.24 3.14 3.04 2.95 2.86

3.65 3.54 3.44 3.34 3.25

4.03 3.95 3.85 3.75 3.65

4.38 4.32 4.25 4.16 4.07

4.67 4.66 4.61 4.54 4.46

4.96 4.96 4.94 4.90 4.84

0.80 0.90 1.00 1.20 1.40

1.31 1.25 1.19 1.09 0.994

1.40 1.34 1.28 1.17 1.07

1.50 1.44 1.38 1.27 1.17

1.62 1.55 1.49 1.38 1.28

1.75 1.68 1.62 1.50 1.40

1.90 1.83 1.76 1.64 1.53

2.06 1.98 1.91 1.79 1.67

2.22 2.15 2.08 1.95 1.83

2.40 2.33 2.25 2.11 1.98

2.59 2.51 2.43 2.28 2.14

2.77 2.69 2.60 2.45 2.31

3.16 3.07 2.98 2.82 2.67

3.56 3.47 3.38 3.21 3.04

3.97 3.88 3.79 3.61 3.44

4.38 4.29 4.20 4.02 3.85

4.76 4.68 4.60 4.43 4.26

1.60 1.80 2.00 2.20 2.40

0.914 0.845 0.784 0.730 0.683

0.992 0.920 0.857 0.801 0.751

1.08 1.01 0.941 0.881 0.828

1.19 1.11 1.04 0.973 0.916

1.30 1.22 1.14 1.08 1.01

1.43 1.34 1.26 1.19 1.12

1.57 1.47 1.39 1.31 1.24

1.72 1.62 1.52 1.44 1.36

1.86 1.76 1.66 1.57 1.49

2.02 1.90 1.80 1.70 1.62

2.18 2.06 1.95 1.85 1.76

2.53 2.39 2.27 2.16 2.06

2.89 2.75 2.62 2.50 2.39

3.28 3.13 2.99 2.86 2.73

3.68 3.52 3.37 3.23 3.10

4.10 3.93 3.77 3.62 3.48

2.60 0.641 0.706 0.781 0.865 0.960 1.07 1.18 2.80 0.604 0.666 0.738 0.819 0.910 1.01 1.12 3.00 0.570 0.631 0.700 0.777 0.863 0.961 1.07

1.29 1.23 1.17

1.41 1.34 1.28

1.54 1.46 1.39

1.67 1.59 1.52

1.96 1.87 1.79

2.28 2.18 2.09

2.62 2.51 2.41

2.97 2.86 2.74

3.35 3.22 3.10

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167



8 - 205


C min =

Pu

D min =

C 1D l

Pu

Pu

lmin =

CC 1 l

CC 1 D

where ex = a l



yl

Pu

l

c.g.


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.39 1.39 1.38 1.32 1.24

1.60 1.55 1.53 1.47 1.38

1.81 1.74 1.71 1.63 1.52

2.02 1.93 1.88 1.79 1.66

2.23 2.12 2.06 1.94 1.80

2.44 2.31 2.22 2.09 1.94

2.65 2.49 2.38 2.23 2.07

2.85 2.66 2.54 2.37 2.21

3.06 2.83 2.69 2.52 2.35

3.27 2.99 2.83 2.67 2.51

3.48 3.14 2.98 2.81 2.66

3.90 3.45 3.28 3.12 2.96

4.32 3.76 3.59 3.42 3.27

4.73 4.08 3.91 3.75 3.59

5.15 4.42 4.24 4.08 3.91

5.57 4.76 4.58 4.41 4.25

0.30 0.40 0.50 0.60 0.70

1.16 0.998 0.860 0.748 0.659

1.29 1.11 0.958 0.833 0.733

1.42 1.22 1.05 0.913 0.805

1.54 1.32 1.14 0.998 0.884

1.67 1.42 1.24 1.09 0.967

1.79 1.54 1.34 1.18 1.06

1.92 1.66 1.46 1.29 1.15

2.06 1.80 1.58 1.41 1.26

2.20 1.94 1.72 1.54 1.38

2.35 2.09 1.86 1.67 1.51

2.51 2.24 2.01 1.81 1.64

2.82 2.55 2.31 2.10 1.92

3.12 2.85 2.61 2.40 2.21

3.44 3.16 2.91 2.69 2.50

3.76 3.48 3.22 2.99 2.79

4.10 3.81 3.54 3.31 3.09

0.80 0.90 1.00 1.20 1.40

0.586 0.527 0.478 0.403 0.348

0.652 0.586 0.532 0.448 0.387

0.719 0.648 0.589 0.497 0.430

0.791 0.715 0.651 0.551 0.476

0.868 0.786 0.717 0.609 0.528

0.951 0.864 0.791 0.674 0.586

1.04 0.949 0.870 0.744 0.648

1.14 1.04 0.957 0.820 0.716

1.25 1.15 1.05 0.904 0.790

1.37 1.26 1.16 0.997 0.874

1.50 1.38 1.27 1.10 0.963

1.77 1.63 1.51 1.31 1.16

2.05 1.90 1.77 1.55 1.37

2.32 2.17 2.03 1.80 1.60

2.61 2.44 2.29 2.04 1.83

2.90 2.73 2.57 2.30 2.07

1.60 1.80 2.00 2.20 2.40

0.305 0.272 0.245 0.223 0.205

0.340 0.303 0.273 0.249 0.228

0.378 0.337 0.304 0.277 0.254

0.419 0.374 0.338 0.308 0.283

0.466 0.416 0.376 0.343 0.315

0.518 0.463 0.419 0.382 0.351

0.573 0.514 0.465 0.425 0.391

0.634 0.569 0.516 0.472 0.434

0.702 0.630 0.572 0.523 0.482

0.776 0.697 0.633 0.579 0.534

0.856 0.771 0.700 0.641 0.591

1.03 0.932 0.848 0.778 0.718

1.23 1.11 1.02 0.932 0.861

1.44 1.31 1.20 1.10 1.02

1.66 1.51 1.39 1.28 1.18

1.88 1.72 1.58 1.47 1.36

2.60 0.189 0.211 0.235 0.261 0.291 0.325 0.362 0.402 0.446 0.495 0.548 0.667 0.800 0.945 1.10 1.27 2.80 0.176 0.196 0.218 0.243 0.271 0.302 0.337 0.375 0.416 0.461 0.511 0.622 0.747 0.882 1.03 1.18 3.00 0.164 0.183 0.204 0.227 0.253 0.282 0.315 0.350 0.389 0.431 0.478 0.582 0.700 0.825 0.962 1.11

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


8 - 206



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



yl

15° 15°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.48 1.42 1.38 1.32 1.24

1.69 1.57 1.54 1.47 1.39

1.89 1.74 1.70 1.63 1.54

2.10 1.92 1.86 1.78 1.68

2.31 2.09 2.02 1.94 1.81

2.51 2.27 2.19 2.09 1.94

2.72 2.44 2.35 2.22 2.07

2.92 2.61 2.51 2.36 2.21

3.13 2.78 2.67 2.51 2.35

3.33 2.95 2.83 2.66 2.50

3.54 3.12 2.98 2.82 2.66

3.95 3.45 3.30 3.13 2.98

4.36 3.78 3.62 3.45 3.30

4.77 4.12 3.95 3.78 3.63

5.18 4.46 4.29 4.12 3.96

5.59 4.81 4.64 4.46 4.30

0.30 0.40 0.50 0.60 0.70

1.16 1.00 0.869 0.759 0.670

1.30 1.12 0.970 0.847 0.746

1.43 1.23 1.06 0.926 0.820

1.55 1.33 1.15 1.01 0.899

1.68 1.44 1.25 1.10 0.983

1.80 1.55 1.36 1.20 1.07

1.93 1.68 1.48 1.31 1.18

2.06 1.81 1.60 1.43 1.29

2.21 1.95 1.74 1.56 1.41

2.36 2.10 1.88 1.69 1.53

2.51 2.25 2.02 1.83 1.67

2.83 2.57 2.33 2.12 1.95

3.15 2.88 2.64 2.43 2.25

3.48 3.20 2.95 2.74 2.54

3.81 3.53 3.28 3.05 2.84

4.15 3.87 3.61 3.37 3.15

0.80 0.90 1.00 1.20 1.40

0.598 0.539 0.490 0.413 0.357

0.666 0.600 0.545 0.461 0.398

0.734 0.663 0.604 0.511 0.442

0.807 0.730 0.666 0.565 0.490

0.885 0.803 0.734 0.625 0.543

0.969 0.881 0.807 0.690 0.601

1.06 0.968 0.888 0.761 0.664

1.17 1.06 0.979 0.840 0.734

1.28 1.17 1.08 0.927 0.812

1.40 1.28 1.18 1.02 0.897

1.52 1.40 1.30 1.12 0.988

1.79 1.66 1.54 1.34 1.19

2.08 1.93 1.80 1.58 1.40

2.37 2.21 2.08 1.84 1.64

2.66 2.49 2.34 2.09 1.88

2.96 2.78 2.63 2.35 2.12

1.60 1.80 2.00 2.20 2.40

0.314 0.280 0.253 0.230 0.211

0.350 0.312 0.282 0.257 0.236

0.389 0.347 0.313 0.286 0.262

0.432 0.386 0.348 0.318 0.292

0.479 0.429 0.388 0.354 0.325

0.532 0.476 0.431 0.393 0.362

0.589 0.528 0.479 0.438 0.403

0.651 0.585 0.531 0.486 0.447

0.721 0.649 0.589 0.538 0.496

0.798 0.718 0.652 0.597 0.550

0.879 0.793 0.721 0.660 0.609

1.06 0.958 0.873 0.801 0.740

1.26 1.14 1.04 0.958 0.886

1.48 1.34 1.23 1.13 1.05

1.70 1.55 1.43 1.32 1.22

1.93 1.77 1.63 1.51 1.40

2.60 0.195 0.218 0.242 0.270 0.301 0.335 0.373 0.415 0.460 0.510 0.565 0.687 0.824 0.975 1.13 1.31 2.80 0.182 0.203 0.225 0.251 0.280 0.312 0.348 0.386 0.428 0.476 0.527 0.641 0.770 0.910 1.06 1.22 3.00 0.170 0.189 0.211 0.234 0.261 0.291 0.325 0.361 0.401 0.445 0.494 0.601 0.722 0.852 0.993 1.15

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167



8 - 207


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



30°

yl

30°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.64 1.52 1.44 1.36 1.28

1.83 1.68 1.59 1.51 1.42

2.03 1.85 1.75 1.66 1.56

2.22 2.03 1.91 1.81 1.71

2.42 2.20 2.07 1.96 1.85

2.61 2.38 2.23 2.12 1.99

2.81 2.55 2.40 2.26 2.12

3.01 2.73 2.56 2.41 2.26

3.20 2.90 2.72 2.56 2.41

3.40 3.06 2.88 2.72 2.57

3.59 3.23 3.05 2.89 2.74

3.98 3.59 3.41 3.24 3.09

4.37 3.95 3.77 3.59 3.43

4.76 4.32 4.13 3.94 3.77

5.15 4.70 4.51 4.31 4.13

5.54 5.08 4.89 4.68 4.49

0.30 0.40 0.50 0.60 0.70

1.20 1.05 0.921 0.812 0.722

1.34 1.17 1.03 0.904 0.804

1.47 1.29 1.13 0.993 0.884

1.60 1.40 1.22 1.08 0.968

1.74 1.51 1.33 1.18 1.06

1.87 1.64 1.45 1.29 1.16

1.99 1.76 1.57 1.41 1.27

2.13 1.90 1.70 1.53 1.38

2.28 2.04 1.83 1.66 1.51

2.43 2.19 1.98 1.80 1.64

2.60 2.35 2.13 1.94 1.78

2.95 2.68 2.45 2.25 2.08

3.28 3.03 2.80 2.59 2.40

3.62 3.35 3.13 2.93 2.73

3.97 3.69 3.46 3.25 3.06

4.32 4.04 3.80 3.58 3.37

0.80 0.90 1.00 1.20 1.40

0.647 0.586 0.535 0.454 0.393

0.722 0.654 0.596 0.506 0.438

0.796 0.722 0.659 0.560 0.487

0.874 0.794 0.727 0.620 0.540

0.957 0.872 0.799 0.685 0.597

1.05 0.957 0.879 0.755 0.660

1.15 1.05 0.968 0.833 0.729

1.26 1.16 1.07 0.920 0.806

1.38 1.27 1.17 1.02 0.891

1.51 1.39 1.29 1.12 0.984

1.64 1.51 1.40 1.22 1.08

1.92 1.78 1.66 1.46 1.29

2.23 2.08 1.94 1.72 1.53

2.55 2.39 2.24 1.99 1.79

2.87 2.70 2.55 2.28 2.05

3.19 3.02 2.85 2.56 2.32

1.60 1.80 2.00 2.20 2.40

0.347 0.310 0.280 0.255 0.234

0.386 0.345 0.312 0.285 0.262

0.430 0.384 0.347 0.317 0.291

0.477 0.427 0.386 0.352 0.324

0.529 0.474 0.429 0.392 0.360

0.585 0.526 0.477 0.436 0.401

0.648 0.583 0.529 0.484 0.446

0.717 0.645 0.586 0.537 0.495

0.794 0.715 0.650 0.595 0.549

0.878 0.791 0.720 0.660 0.609

0.967 0.874 0.796 0.730 0.674

1.16 1.05 0.961 0.884 0.817

1.38 1.25 1.15 1.06 0.978

1.61 1.47 1.35 1.24 1.15

1.87 1.70 1.57 1.45 1.34

2.12 1.94 1.79 1.66 1.55

2.60 0.217 0.242 0.269 0.299 0.334 0.372 0.413 0.459 0.510 0.566 0.626 0.760 0.910 1.08 1.25 2.80 0.201 0.225 0.250 0.278 0.311 0.346 0.385 0.428 0.475 0.527 0.584 0.710 0.851 1.01 1.17 3.00 0.188 0.210 0.234 0.260 0.290 0.324 0.360 0.401 0.445 0.494 0.547 0.666 0.799 0.944 1.10

1.44 1.35 1.27

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


8 - 208



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


45° yl

45°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.81 1.68 1.57 1.47 1.39

1.99 1.83 1.71 1.60 1.51

2.17 1.99 1.86 1.74 1.64

2.35 2.15 2.02 1.88 1.77

2.53 2.32 2.18 2.04 1.92

2.71 2.50 2.36 2.21 2.07

2.89 2.67 2.53 2.38 2.24

3.07 2.85 2.69 2.54 2.41

3.25 3.02 2.86 2.71 2.58

3.43 3.19 3.03 2.88 2.75

3.61 3.37 3.21 3.06 2.93

3.97 3.74 3.58 3.42 3.29

4.33 4.10 3.95 3.80 3.66

4.70 4.48 4.33 4.18 4.04

5.06 4.85 4.71 4.56 4.42

5.42 5.21 5.08 4.94 4.80

0.30 0.40 0.50 0.60 0.70

1.31 1.16 1.03 0.921 0.829

1.43 1.27 1.13 1.01 0.911

1.55 1.38 1.23 1.11 1.00

1.67 1.49 1.35 1.22 1.11

1.81 1.63 1.48 1.34 1.21

1.95 1.77 1.61 1.46 1.33

2.11 1.92 1.74 1.58 1.45

2.28 2.07 1.88 1.72 1.58

2.45 2.23 2.03 1.86 1.71

2.62 2.39 2.19 2.01 1.86

2.80 2.57 2.36 2.18 2.01

3.16 2.92 2.72 2.52 2.35

3.54 3.27 3.06 2.87 2.70

3.91 3.64 3.40 3.21 3.04

4.29 4.03 3.76 3.55 3.37

4.66 4.41 4.14 3.91 3.71

0.80 0.90 1.00 1.20 1.40

0.751 0.685 0.629 0.538 0.469

0.828 0.757 0.696 0.598 0.523

0.915 0.839 0.773 0.666 0.582

1.01 0.927 0.854 0.735 0.644

1.11 1.02 0.938 0.810 0.712

1.22 1.12 1.03 0.892 0.786

1.33 1.22 1.14 0.985 0.868

1.45 1.34 1.25 1.09 0.960

1.58 1.47 1.36 1.20 1.06

1.72 1.60 1.49 1.31 1.16

1.87 1.74 1.63 1.43 1.28

2.19 2.05 1.92 1.70 1.52

2.53 2.38 2.24 2.00 1.80

2.88 2.73 2.58 2.31 2.09

3.21 3.05 2.91 2.64 2.40

3.54 3.38 3.23 2.96 2.71

1.60 1.80 2.00 2.20 2.40

0.416 0.373 0.338 0.308 0.284

0.464 0.416 0.377 0.344 0.317

0.516 0.463 0.419 0.383 0.353

0.572 0.513 0.466 0.426 0.392

0.633 0.570 0.518 0.474 0.436

0.700 0.631 0.574 0.526 0.485

0.774 0.699 0.636 0.583 0.538

0.857 0.775 0.705 0.647 0.598

0.948 0.858 0.782 0.718 0.664

1.05 0.947 0.865 0.795 0.736

1.15 1.04 0.954 0.878 0.812

1.38 1.25 1.15 1.06 0.983

1.63 1.49 1.37 1.26 1.17

1.90 1.74 1.60 1.49 1.38

2.19 2.02 1.86 1.73 1.61

2.50 2.31 2.14 1.99 1.85

2.60 0.263 0.293 0.327 0.363 0.405 0.450 0.500 0.555 0.617 0.684 0.756 0.916 1.09 1.29 2.80 0.245 0.273 0.304 0.338 0.377 0.420 0.467 0.518 0.576 0.638 0.707 0.857 1.03 1.21 3.00 0.229 0.256 0.285 0.317 0.353 0.393 0.437 0.486 0.540 0.599 0.663 0.805 0.964 1.14

1.50 1.41 1.33

1.73 1.62 1.53

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167



8 - 209


C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D

where ex = a l



60° yl

60°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

1.95 1.83 1.73 1.63 1.55

2.12 1.94 1.84 1.74 1.65

2.28 2.07 1.96 1.86 1.76

2.44 2.21 2.09 1.98 1.88

2.61 2.36 2.24 2.12 2.02

2.77 2.53 2.40 2.28 2.18

2.94 2.70 2.57 2.45 2.35

3.10 2.88 2.75 2.63 2.53

3.26 3.06 2.94 2.82 2.72

3.43 3.24 3.13 3.00 2.90

3.59 3.41 3.31 3.20 3.08

3.92 3.76 3.67 3.57 3.46

4.25 4.11 4.03 3.94 3.84

4.57 4.45 4.38 4.30 4.21

4.90 4.78 4.72 4.65 4.56

5.23 5.12 5.06 4.99 4.91

0.30 0.40 0.50 0.60 0.70

1.47 1.34 1.22 1.12 1.03

1.58 1.44 1.31 1.20 1.11

1.68 1.54 1.41 1.30 1.20

1.80 1.66 1.53 1.42 1.32

1.94 1.79 1.67 1.55 1.45

2.09 1.94 1.81 1.69 1.59

2.26 2.10 1.97 1.85 1.74

2.44 2.27 2.13 2.01 1.90

2.62 2.45 2.30 2.17 2.06

2.81 2.64 2.48 2.35 2.23

2.99 2.83 2.66 2.52 2.40

3.36 3.20 3.04 2.89 2.75

3.73 3.56 3.42 3.27 3.12

4.11 3.93 3.78 3.64 3.50

4.48 4.29 4.14 4.00 3.87

4.83 4.65 4.49 4.36 4.23

0.80 0.90 1.00 1.20 1.40

0.945 0.874 0.812 0.709 0.626

1.02 0.950 0.886 0.777 0.690

1.12 1.04 0.973 0.858 0.765

1.23 1.15 1.08 0.952 0.852

1.35 1.27 1.19 1.06 0.945

1.49 1.40 1.32 1.17 1.04

1.64 1.54 1.45 1.28 1.15

1.79 1.68 1.58 1.41 1.26

1.95 1.83 1.73 1.54 1.39

2.11 1.99 1.88 1.68 1.52

2.28 2.16 2.05 1.84 1.66

2.63 2.51 2.40 2.17 1.97

2.99 2.87 2.75 2.52 2.31

3.36 3.23 3.10 2.87 2.66

3.73 3.60 3.46 3.21 3.00

4.10 3.96 3.83 3.56 3.34

1.60 1.80 2.00 2.20 2.40

0.560 0.506 0.461 0.423 0.391

0.619 0.561 0.512 0.471 0.436

0.689 0.626 0.573 0.527 0.487

0.769 0.697 0.636 0.585 0.541

0.850 0.771 0.705 0.649 0.600

0.938 0.853 0.780 0.719 0.666

1.04 0.943 0.864 0.797 0.738

1.14 1.04 0.957 0.883 0.819

1.26 1.15 1.06 0.977 0.907

1.38 1.27 1.17 1.08 1.00

1.52 1.39 1.28 1.19 1.10

1.81 1.66 1.53 1.43 1.33

2.12 1.96 1.82 1.69 1.58

2.46 2.28 2.12 1.98 1.85

2.80 2.61 2.44 2.28 2.14

3.13 2.94 2.77 2.60 2.44

2.60 0.363 0.405 0.452 0.502 0.558 0.620 0.687 0.763 0.847 0.937 1.03 1.25 2.80 0.339 0.379 0.422 0.469 0.521 0.580 0.644 0.715 0.793 0.878 0.969 1.17 3.00 0.317 0.355 0.395 0.440 0.489 0.545 0.605 0.672 0.746 0.826 0.913 1.10

1.48 1.39 1.31

1.74 1.63 1.54

2.01 1.90 1.79

2.30 2.17 2.06

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


8 - 210



C min =

Pu

D min =

C 1D l

Pu

lmin =

CC 1 l

Pu

CC 1 D


ex = a l


75° yl

75°

c.g.

l

Pu


xl kl

k a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.10 0.15 0.20 0.25

2.05 1.94 1.88 1.82 1.76

2.20 2.01 1.94 1.89 1.83

2.35 2.08 2.03 1.97 1.92

2.50 2.20 2.15 2.09 2.04

2.65 2.34 2.29 2.24 2.19

2.79 2.49 2.44 2.39 2.35

2.94 2.65 2.61 2.56 2.52

3.09 2.81 2.77 2.73 2.69

3.24 2.97 2.94 2.90 2.86

3.39 3.13 3.10 3.07 3.03

3.54 3.28 3.25 3.23 3.20

3.83 3.59 3.56 3.53 3.51

4.13 3.93 3.87 3.84 3.81

4.43 4.25 4.20 4.15 4.11

4.72 4.57 4.52 4.48 4.42

5.02 4.88 4.84 4.80 4.75

0.30 0.40 0.50 0.60 0.70

1.71 1.62 1.53 1.46 1.38

1.78 1.69 1.61 1.53 1.46

1.87 1.79 1.70 1.63 1.56

2.00 1.91 1.83 1.75 1.68

2.14 2.05 1.97 1.89 1.82

2.30 2.21 2.12 2.05 1.97

2.47 2.38 2.29 2.21 2.14

2.65 2.56 2.47 2.39 2.31

2.82 2.74 2.66 2.57 2.49

3.00 2.92 2.84 2.76 2.68

3.17 3.10 3.03 2.95 2.87

3.49 3.43 3.38 3.31 3.25

3.79 3.75 3.70 3.65 3.60

4.09 4.05 4.02 3.97 3.93

4.39 4.35 4.32 4.28 4.24

4.70 4.65 4.61 4.58 4.55

0.80 0.90 1.00 1.20 1.40

1.31 1.25 1.19 1.09 0.994

1.39 1.33 1.27 1.16 1.07

1.49 1.43 1.37 1.26 1.16

1.61 1.54 1.48 1.37 1.27

1.75 1.68 1.62 1.50 1.40

1.90 1.83 1.77 1.64 1.54

2.06 1.99 1.93 1.80 1.69

2.24 2.16 2.10 1.97 1.85

2.41 2.34 2.27 2.14 2.01

2.60 2.52 2.45 2.31 2.19

2.79 2.71 2.64 2.50 2.36

3.17 3.10 3.03 2.87 2.73

3.54 3.47 3.41 3.26 3.12

3.88 3.82 3.76 3.64 3.50

4.20 4.16 4.10 3.99 3.87

4.51 4.47 4.43 4.34 4.23

1.60 1.80 2.00 2.20 2.40

0.914 0.845 0.784 0.730 0.683

0.987 0.915 0.852 0.797 0.748

1.08 1.00 0.937 0.878 0.825

1.18 1.11 1.04 0.972 0.915

1.30 1.22 1.15 1.08 1.02

1.44 1.35 1.27 1.20 1.13

1.58 1.49 1.40 1.33 1.26

1.74 1.64 1.55 1.46 1.39

1.90 1.80 1.70 1.60 1.52

2.07 1.95 1.84 1.74 1.65

2.24 2.11 2.00 1.89 1.80

2.59 2.45 2.33 2.21 2.10

2.97 2.82 2.69 2.56 2.44

3.36 3.21 3.06 2.92 2.80

3.74 3.61 3.46 3.31 3.17

4.11 3.99 3.85 3.71 3.56

2.60 0.641 0.704 0.778 0.865 0.963 1.07 1.19 2.80 0.604 0.664 0.736 0.819 0.913 1.02 1.13 3.00 0.570 0.628 0.698 0.777 0.867 0.966 1.07

1.32 1.25 1.18

1.44 1.37 1.30

1.57 1.49 1.42

1.71 1.62 1.55

2.00 1.91 1.82

2.33 2.23 2.13

2.68 2.57 2.46

3.05 2.93 2.81

3.42 3.30 3.18

x y

0.000 0.005 0.017 0.035 0.057 0.083 0.113 0.144 0.178 0.213 0.250 0.327 0.408 0.492 0.579 0.667 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250 0.227 0.208 0.192 0.179 0.167


CONSTRUCTION COMBINING BOLTS AND WELDS

8 - 211

Eccentricity Normal to the Plane of the Faying Surface

Figure 8-55 shows a bracket welded to a column flange. The eccentric load Pu can be resolved into a concentric force Pu at the faying surface of the connection and a moment Pu e normal to the plane of the faying surface where e is the eccentricity. Each weld element is then assumed to support an equal share of the concentric force Pu, and the moment Pu e is resisted by tension in the welds above the neutral axis and compression below the neutral axis. In contrast to bolts, where the interaction of shear and tension must be considered, for welds, shear and tension may be combined vectorially for welds into a resultant shear. Thus, the solution of a weld loaded eccentrically normal to the plane of the faying surface is parallel to that discussed previously for welds loaded eccentrically in the plane of the faying surface; with the neutral axis assumed to be located at the CG of the weld group, this case is identical to that described previously for the elastic method. CONSTRUCTION COMBINING BOLTS AND WELDS

In bearing-type connections in new construction, the rigidity of the welds prevents the initial joint slippage necessary to develop the strength of all the bolts in a connection that might combine both welds and bolts. Thus, bearing-type connections combining welds and bolts are permissible only if the design strength of the welds φRn alone exceeds the required strength of the connection Ru. However, in situations where it can safely be assumed that joint slippage has occurred before welding is performed, welds may be used to reinforce existing bolted or riveted joints. Such is the case with structures previously in service. In this case, the design strength of the original bolt group may be used to carry the existing dead loads and the design strength of the welds need be adequate only to carry additional loads. Refer to LRFD Specification Section J1.9. In slip-critical connections, since connection slip is neither expected nor required for the bolts to develop their strength, the design strengths of welds and high-strength bolts are additive. When high-strength bolts and welds are used together in a slip-critical connection, the bolts should preferably be fully tensioned before welding is performed. The design drawings should clearly indicate where this type of connection occurs.

Pu e

Fig. 8-55. Welds subjected to eccentricity normal to the plane of the faying surface. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 212


CONNECTED ELEMENTS

Connected elements are the angles, plates, tees, gussets, and other connecting elements used in connections to transfer load from one structural member to another as well as the affected elements of the connected members. Economical Considerations

Cost effective steel fabrication requires close cooperation between the designer, detailer, and fabricator. Effective communication and planning will allow the project to take full advantage of the strengths of all parties involved. Often, potential problems can be avoided through early consultation and good communication during the full life of a project. Designs and details should be suited to the shop practices and standards of the fabricator. The resulting similarity throughout the project will further lend itself to the minimization of errors. For example, once gage lines conforming to standard machine set-ups are determined, they should be utilized as much as possible throughout any one job. Furthermore, it is desirable to keep the same bolt spacing throughout a project. Longitudinal spacing should preferably be three inches or a multiple of three inches, since most shops consider this to be standard. At a minimum, gages and hole sizes on any one member should not be varied throughout the length of that member. This prevents unnecessary material re-handling and the need for multiple punching or drilling. Design Strength of Connected Elements

The design strength of connecting elements is determined in accordance with the provisions of LRFD Specification Sections J4 and J5; the applicable limit states are shear yielding, shear rupture, block shear rupture, tension yielding, and tension rupture. Shear Yielding

This limit state applies to the gross section of the connected element. From LRFD Specification Section J5.3, the design shear yielding strength is φRn, where φ = 0.90 Rn = 0.60Fy Ag Shear Rupture

This limit state applies to the net section From LRFD Specification Section J4.1, the design shear rupture strength is φRn, where φ = 0.75 Rn = 0.60Fu Anv Table 8-46 gives the reduction in area for standard, oversized, short-slotted, and long-slotted holes in material thicknesses from 3⁄16-in. to 1 in.; for other material thicknesses, multiply the tabular value for 1-in. thickness by the actual thickness. Block Shear Rupture

The term block shear rupture describes a material tearing limit state which occurs in a combination of shear and tension. This phenomenon can occur at the end of a coped beam, shown in Figure 8-56, or at the end of a tension connection, shown in Figure 8-57. This AMERICAN INSTITUTE OF STEEL CONSTRUCTION

CONNECTED ELEMENTS

8 - 213

Table 8-46. Reduction in Area for Holes, in.2 C

Thckns. t, in.

A

B

STD Standard Hole

OVS Oversized Hole

3⁄ 4

7⁄ 8

D

A

A

SSL Short-Slotted Hole

LSL Long-Slotted Hole

A ×t

B ×t

Bolt Diameter d b , in.


1

1 1 ⁄8

1 1 ⁄4

1 3 ⁄8

1 1 ⁄2

3⁄ 4

7⁄ 8

1

1 1 ⁄8

1 1 ⁄4

1 3 ⁄8

1 1 ⁄2

3⁄ 16 1⁄ 4

0.164 0.188 0.211 0.234 0.258 0.281 0.305 0.188 0.211 0.246 0.281 0.305 0.328 0.352 0.219 0.250 0.281 0.313 0.344 0.375 0.406 0.250 0.281 0.328 0.375 0.406 0.438 0.469

5⁄ 16 3⁄ 8 7⁄ 16 1⁄ 2

0.273 0.328 0.383 0.438

0.313 0.375 0.438 0.500

0.352 0.422 0.492 0.563

0.391 0.469 0.547 0.625

0.430 0.516 0.602 0.688

0.469 0.563 0.656 0.750

0.508 0.609 0.711 0.813

0.313 0.375 0.438 0.500

0.352 0.422 0.492 0.563

0.410 0.492 0.574 0.656

0.469 0.563 0.656 0.750

0.508 0.609 0.711 0.813

0.547 0.656 0.766 0.875

0.586 0.703 0.820 0.938

9⁄ 16 5⁄ 8 11⁄ 16 3⁄ 4

0.492 0.547 0.602 0.656

0.563 0.625 0.688 0.750

0.633 0.703 0.773 0.844

0.703 0.781 0.859 0.938

0.773 0.859 0.945 1.03

0.844 0.938 1.03 1.13

0.914 1.02 1.12 1.22

0.563 0.625 0.688 0.750

0.633 0.703 0.773 0.844

0.738 0.820 0.902 0.984

0.844 0.938 1.03 1.13

0.914 1.02 1.12 1.22

0.984 1.09 1.20 1.31

1.05 1.17 1.29 1.41

13⁄ 16 7⁄ 8 15⁄ 16

0.711 0.766 0.820 0.875

0.813 0.875 0.938 1.00

0.914 0.984 1.05 1.13

1.02 1.09 1.17 1.25

1.12 1.20 1.29 1.38

1.22 1.31 1.41 1.50

1.32 1.42 1.52 1.63

0.813 0.875 0.938 1.00

0.914 0.984 1.05 1.13

1.07 1.15 1.23 1.31

1.22 1.31 1.41 1.50

1.32 1.42 1.52 1.63

1.42 1.53 1.64 1.75

1.52 1.64 1.76 1.88

1 3 ⁄8

1 1 ⁄2

1

Thckns. t, in.

3⁄ 4

7⁄ 8

C ×t

D ×t



1

1 1 ⁄8

1 1 ⁄4

1 3 ⁄8

1 1 ⁄2

3⁄ 4

7⁄ 8

1

1 1 ⁄8

1 1 ⁄4

3⁄ 16 1⁄ 4

0.199 0.223 0.258 0.293 0.316 0.340 0.363 0.363 0.422 0.480 0.539 0.598 0.656 0.715 0.266 0.297 0.344 0.391 0.422 0.453 0.484 0.484 0.563 0.641 0.719 0.797 0.875 0.953

5⁄ 16 3⁄ 8 7⁄ 16 1⁄ 2

0.332 0.398 0.465 0.531

0.371 0.445 0.520 0.594

0.430 0.516 0.602 0.688

0.488 0.586 0.684 0.781

0.527 0.633 0.738 0.844

0.566 0.680 0.793 0.906

0.605 0.727 0.848 0.969

0.605 0.727 0.848 0.969

0.703 0.844 0.984 1.13

0.801 0.961 1.12 1.28

0.898 1.08 1.26 1.44

0.996 1.20 1.39 1.59

1.09 1.31 1.53 1.75

1.19 1.43 1.67 1.91

9⁄ 16 5⁄ 8 11⁄ 16 3⁄ 4

0.598 0.664 0.730 0.797

0.668 0.742 0.816 0.891

0.773 0.859 0.945 1.03

0.879 0.977 1.07 1.17

0.949 1.05 1.16 1.27

1.02 1.13 1.25 1.36

1.09 1.21 1.33 1.45

1.09 1.21 1.33 1.45

1.27 1.41 1.55 1.69

1.44 1.60 1.76 1.92

1.62 1.80 1.98 2.16

1.79 1.99 2.19 2.39

1.97 2.19 2.41 2.63

2.14 2.38 2.62 2.86

13⁄ 16 7⁄ 8 15⁄ 16

0.863 0.930 0.996 1.06

0.965 1.04 1.11 1.19

1.12 1.20 1.29 1.38

1.27 1.37 1.46 1.56

1.37 1.48 1.58 1.69

1.47 1.59 1.70 1.81

1.57 1.70 1.82 1.94

1.57 1.70 1.82 1.94

1.83 1.97 2.11 2.25

2.08 2.24 2.40 2.56

2.34 2.52 2.70 2.88

2.59 2.79 2.99 3.19

2.84 3.06 3.28 3.50

3.10 3.34 3.57 3.81

1

failure is usually the result of high reactions imposed on relatively thin material through a short connection. The design block shear rupture strength is φRn, where φ = 0.75 and Rn is determined as follows. For bolted connections, from LRFD Specification Section J4.3, when Fu Ant ≥ 0.6Fu Anv, shear yielding occurs in combination with tension rupture and, Rn = 0.6Fy Agv + Fu Ant AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 214


This case is the basis of Tables 8-47, where φFu Ant is tabulated per inch of material thickness in Table 8-47a and φ(0.6Fy Agv) is tabulated per inch of material thickness in Table 8-47b. When 0.6Fu Anv > Fu Ant , shear rupture occurs in combination with tension yielding and, Rn = 0.6Fu Anv + Fy Agt This case is the basis of Tables 8-48, where φ(0.6Fu Anv) is tabulated per inch of material thickness in Table 8-48a and φFy Agt is tabulated per inch of material thickness in Table 8-48b. For welded connections, block shear rupture is treated as for bolted connections; the only difference is that, in the absence of bolt holes, Anv = Agv and Ant = Agt. L eh

L ev n bolts @ s spacing

Shear Area

Shear Area

Tension Area

Tension Area

(a) Bolted Connections

(b) Welded Connections

Fig. 8-56. Block shear rupture in coped beams.

L eh

L ev Shear Area

n bolts @ s spacing

Tension Area

Fig. 8-57. Block shear rupture in ends of tension members. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

CONNECTED ELEMENTS

8 - 215

Table 8-47a. Block Shear Rupture Tension Rupture Component per inch of thickness, φ[FuAnt] / t, kips/in. L eh

Fu, ksi 58

65

70




3⁄ 4

7⁄ 8

3⁄ 4

7⁄ 8

3⁄ 4

7⁄ 8

1

24.5

21.8

19.0

27.4

24.4

21.3

29.5

26.3

23.0

1 1 ⁄8 1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

29.9 35.3 40.8 46.2

27.2 32.6 38.1 43.5

24.5 29.9 35.3 40.8

33.5 39.6 45.7 51.8

30.5 36.6 42.7 48.8

27.4 33.5 39.6 45.7

36.1 42.7 49.2 55.8

32.8 39.4 45.9 52.5

29.5 36.1 42.7 49.2

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

51.7 57.1 62.5 68.0

48.9 54.4 59.8 65.3

46.2 51.7 57.1 62.5

57.9 64.0 70.1 76.2

54.8 60.9 67.0 73.1

51.8 57.9 64.0 70.1

62.3 68.9 75.5 82.0

59.1 65.6 72.2 78.8

55.8 62.3 68.9 75.5

78.8 89.7 101 111

76.1 87.0 97.9 109

73.4 84.3 95.2 106

88.4 101 113 125

85.3 97.5 110 122

82.3 94.5 107 119

95.2 108 121 135

91.9 105 118 131

88.6 102 115 128

Leh, in.

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3

1

1


1

8 - 216


Table 8-47b. Block Shear Rupture Shear Yielding Component per inch of thickness, φ[0.6FyAgv] / t, kips/in.

L ev n bolts @ 3 ″ spacing

Fy, ksi 36

50



n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

12

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

562 564 566

563 565 567

564 566 568

780 783 786

782 785 788

783 786 789

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

568 570 572 574

569 571 573 575

570 572 574 576

789 792 795 797

790 793 796 799

792 795 797 800

3

578 582 586 590

579 583 587 591

580 584 588 592

803 809 814 820

804 810 816 821

806 811 817 823

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

513 515 517

514 516 518

515 517 519

713 716 719

714 717 720

716 719 721

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

519 521 523 525

520 522 524 527

521 523 525 528

721 724 727 730

723 726 728 731

724 727 730 733

3

530 534 538 542

531 535 539 543

532 536 540 544

735 741 747 752

737 743 748 754

738 744 750 755

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

465 467 469

466 468 470

467 469 471

645 648 651

647 650 653

648 651 654

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

471 473 475 477

472 474 476 478

473 475 477 479

654 657 660 662

655 658 661 664

657 660 662 665

481 485 489 493

482 486 490 494

483 487 491 495

668 674 679 685

669 675 681 686

671 676 682 688

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

11

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

10

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


CONNECTED ELEMENTS

8 - 217

Table 8-47b (cont.). Block Shear Rupture Shear Yielding Component per inch of thickness, φ[0.6FyAgv] / t, kips/in.


Fy, ksi 36

50



n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

9

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

416 418 420

417 419 421

418 420 422

578 581 584

579 582 585

581 584 586

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

422 424 426 428

423 425 427 429

424 426 428 430

586 589 592 595

588 591 593 596

589 592 595 598

3

432 436 440 444

433 437 441 446

434 438 442 447

600 606 612 617

602 608 613 619

603 609 615 620

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

368 370 372

369 371 373

370 372 374

510 513 516

512 515 518

513 516 519

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

374 376 378 380

375 377 379 381

376 378 380 382

519 522 525 527

520 523 526 529

522 525 527 530

3

384 388 392 396

385 389 393 397

386 390 394 398

533 539 544 550

534 540 546 551

536 541 547 553

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

319 321 323

320 322 324

321 323 325

443 446 449

444 447 450

446 449 451

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

325 327 329 331

326 328 330 332

327 329 331 333

451 454 457 460

453 456 458 461

454 457 460 463

335 339 343 347

336 340 344 348

337 341 345 349

465 471 477 482

467 473 478 484

468 474 480 485

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

8

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

7

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


8 - 218




Fy, ksi 36

50



n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

6

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

270 272 274

271 273 275

272 274 276

375 378 381

377 380 383

378 381 384

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

276 278 280 282

277 279 281 284

278 280 282 285

384 387 390 392

385 388 391 394

387 390 392 395

3

287 291 295 299

288 292 296 300

289 293 297 301

398 404 409 415

399 405 411 416

401 406 412 418

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

222 224 226

223 225 227

224 226 228

308 311 314

309 312 315

311 314 316

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

228 230 232 234

229 231 233 235

230 232 234 236

316 319 322 325

318 321 323 326

319 322 325 328

3

238 242 246 250

239 243 247 251

240 244 248 252

330 336 342 347

332 338 343 349

333 339 345 350

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

173 175 177

174 176 178

175 177 179

240 243 246

242 245 248

243 246 249

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

179 181 183 185

180 182 184 186

181 183 185 187

249 252 255 257

250 253 256 259

252 255 257 260

189 193 197 201

190 194 198 203

191 195 199 204

263 269 274 280

264 270 276 281

266 271 277 283

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

5

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

4

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


CONNECTED ELEMENTS

8 - 219



Fy, ksi 36

50



n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

125 127 129

126 128 130

127 129 131

173 176 179

174 177 180

176 179 181

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

131 133 135 137

132 134 136 138

133 135 137 139

181 184 187 190

183 186 188 191

184 187 190 193

3

141 145 149 153

142 146 150 154

143 147 151 155

195 201 207 212

197 203 208 214

198 204 210 215

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

76 78 80

77 79 81

78 80 82

105 108 111

107 110 113

108 111 114

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

82 84 86 88

83 85 87 89

84 86 88 90

114 117 120 122

115 118 121 124

117 120 122 125

92 96 100 104

93 97 101 105

94 98 102 106

128 134 139 145

129 135 141 146

131 136 142 148

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

2

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


8 - 220


Table 8-48a. Block Shear Rupture Shear Rupture Component per inch of thickness, φ[0.6FuAnv] / t, kips/in.


Fu, ksi 58

65

70




n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

12

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

631 635 638

594 597 600

556 560 563

707 711 715

665 669 673

623 627 631

762 766 770

717 721 725

671 675 679

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

641 644 648 651

604 607 610 613

566 569 573 576

718 722 726 729

676 680 684 687

634 638 642 645

774 778 782 786

728 732 736 740

683 687 691 695

3

657 664 670 677

620 626 633 639

582 589 595 602

737 744 751 759

695 702 709 717

653 660 667 675

793 801 809 817

748 756 764 772

703 711 719 726

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

576 579 582

542 545 548

507 511 514

645 649 653

607 611 614

569 572 576

695 699 703

654 658 662

612 616 620

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

586 589 592 595

551 555 558 561

517 520 524 527

656 660 664 667

618 622 625 629

580 583 587 590

707 711 715 719

665 669 673 677

624 628 632 636

3

602 608 615 622

568 574 581 587

533 540 546 553

675 682 689 697

636 644 651 658

598 605 612 620

726 734 742 750

685 693 701 709

644 652 660 667

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

520 524 527

489 493 496

458 462 465

583 587 590

548 552 556

514 517 521

628 632 636

591 595 599

553 557 561

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

530 533 537 540

499 502 506 509

468 471 475 478

594 598 601 605

559 563 567 570

525 528 532 536

640 644 648 652

602 606 610 614

565 569 573 577

546 553 560 566

515 522 529 535

484 491 498 504

612 620 627 634

578 585 592 600

543 550 558 565

660 667 675 683

622 630 638 646

585 593 600 608

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

11

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

10

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


CONNECTED ELEMENTS

8 - 221

Table 8-48a (cont.). Block Shear Rupture Shear Rupture Component per inch of thickness, φ[0.6FuAnv ] / t, kips/in.


Fu, ksi 58

65

70




n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

9

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

465 468 471

437 440 444

409 413 416

521 525 528

490 494 497

459 463 466

561 565 569

528 532 536

494 498 502

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

475 478 481 484

447 450 453 457

419 422 426 429

532 536 539 543

501 505 508 512

470 473 477 481

573 577 581 585

539 543 547 551

506 510 514 518

3

491 498 504 511

463 470 476 483

436 442 449 455

550 558 565 572

519 527 534 541

488 495 503 510

593 600 608 616

559 567 575 583

526 534 541 549

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

409 413 416

385 388 392

361 364 367

459 463 466

431 435 439

404 408 411

494 498 502

465 469 473

435 439 443

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

419 422 426 429

395 398 401 405

370 374 377 380

470 473 477 481

442 446 450 453

415 419 422 426

506 510 514 518

476 480 484 488

447 451 455 459

3

436 442 449 455

411 418 424 431

387 393 400 406

488 495 503 510

461 468 475 483

433 441 448 455

526 534 541 549

496 504 512 520

467 474 482 490

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

354 357 361

333 336 339

312 315 318

397 400 404

373 377 380

349 353 356

427 431 435

402 406 410

376 380 384

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

364 367 370 374

343 346 349 352

321 325 328 331

408 411 415 419

384 388 391 395

360 364 367 371

439 443 447 451

413 417 421 425

388 392 396 400

380 387 393 400

359 365 372 378

338 344 351 357

426 433 441 448

402 410 417 424

378 386 393 400

459 467 474 482

433 441 449 457

408 415 423 431

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

8

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

7

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


8 - 222




Fu, ksi 58

65

70




n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

6

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

299 302 305

281 284 287

263 266 269

335 338 342

314 318 322

294 298 302

360 364 368

339 343 347

317 321 325

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

308 312 315 318

290 294 297 300

272 276 279 282

346 349 353 356

325 329 333 336

305 309 313 316

372 376 380 384

350 354 358 362

329 333 337 341

3

325 331 338 344

307 313 320 326

289 295 302 308

364 371 378 386

344 351 358 366

324 331 338 346

392 400 408 415

370 378 386 394

348 356 364 372

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

243 246 250

228 232 235

214 217 220

272 276 280

256 260 263

239 243 247

293 297 301

276 280 284

258 262 266

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

253 256 259 263

238 241 245 248

223 227 230 233

283 287 291 294

267 271 274 278

250 254 258 261

305 309 313 317

287 291 295 299

270 274 278 282

3

269 276 282 289

254 261 268 274

240 246 253 259

302 309 316 324

285 293 300 307

269 276 283 291

325 333 341 348

307 315 323 331

289 297 305 313

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

188 191 194

176 179 183

165 168 171

210 214 218

197 201 205

185 188 192

226 230 234

213 217 221

199 203 207

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

197 201 204 207

186 189 192 196

175 178 181 184

221 225 229 232

208 212 216 219

196 199 203 207

238 242 246 250

224 228 232 236

211 215 219 222

214 220 227 233

202 209 215 222

191 197 204 210

239 247 254 261

227 234 241 249

214 221 229 236

258 266 274 282

244 252 260 268

230 238 246 254

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

5

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

4

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


CONNECTED ELEMENTS

8 - 223



Fu, ksi 58

65

70




n

Lev, in.

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3⁄ 4

7⁄ 8

1

3

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

132 135 139

124 127 131

116 119 122

148 152 155

139 143 146

130 133 137

159 163 167

150 154 158

140 144 148

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

142 145 148 152

134 137 140 144

126 129 132 135

159 163 166 170

150 154 157 161

141 144 148 152

171 175 179 183

161 165 169 173

152 156 159 163

3

158 165 171 178

150 157 163 170

142 148 155 161

177 185 192 199

168 176 183 190

159 166 174 181

191 199 207 215

181 189 197 205

171 179 187 195

1 1 ⁄4 1 3 ⁄8 1 1 ⁄2

77 80 83

72 75 78

67 70 73

86 90 93

80 84 88

75 79 82

93 96 100

87 91 95

81 85 89

1 5 ⁄8 1 3 ⁄4 1 7 ⁄8

86 90 93 96

82 85 88 91

77 80 83 86

97 101 104 108

91 95 99 102

86 90 93 97

104 108 112 116

98 102 106 110

93 96 100 104

103 109 116 122

98 104 111 117

93 100 106 113

115 122 130 137

110 117 124 132

104 112 119 126

124 132 140 148

118 126 134 142

112 120 128 136

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

2

2 2 1 ⁄4 2 1 ⁄2 2 3 ⁄4

3


8 - 224


Table 8-48b. Block Shear Rupture Tension Yielding Component per inch of thickness φ[FyAgt] / t, kips/in. L eh

Fy, ksi Leh, in.

36

50

1

27.0

37.5

11⁄8 11⁄4 13⁄8 11⁄2

30.4 33.8 37.1 40.5

42.2 46.9 51.6 56.3

15⁄8 13⁄4 17⁄8 2

43.9 47.3 50.6 54.0

60.9 65.6 70.3 75.0

21⁄4 21⁄2 23⁄4 3

60.8 67.5 74.3 81.0

84.4 93.8 103 113


CONNECTED ELEMENTS

8 - 225

Tension Yielding

From LRFD Specification Section J5.2, the design tension yielding strength is φRn, where φ = 0.90 Rn = Fy Ag Tension Rupture

From LRFD Specification Section J5.2, the design tension rupture strength is φRn, where φ = 0.75 Rn = Fu An In the above equation, An is the net area not to exceed 0.85Ag. Table 8-46 gives the reduction in area for standard, oversized, short-slotted, and long-slotted holes in material thicknesses from 3⁄16-in. to 1 in.; for material thicknesses not listed, multiply the tabular value for 1-in. thickness by the actual thickness. Members with Copes, Blocks, or Cuts

When structural members frame together, a minimum clearance of 1⁄2-in. should be provided, when possible. In cases where material removal is necessary to provide such a clearance, material may be removed by coping, blocking, or cutting as illustrated in Figures 8-58. Note the recommended practices for coping illustrated in Figure 8-59; the potential notch left by the first cut will occur in waste material which will subsequently be removed by the second cut. All re-entrant corners must be shaped notch-free per AWS D1.1 to a radius. An approximate minimum radius to which this corner must be shaped is 1⁄2-in. Material removal is costly, and should be avoided when possible. For example, the elevations of the tops of infill beams could be established at a sufficient distance below the tops of girders to clear the girder fillet. Alternatively, coping could be eliminated with a connection as illustrated in Figure 8-60; this detail also allows the use of a shorter beam length. When necessary, coping is usually the most economical method to remove material. Copes, blocks, and cuts can significantly reduce the design strengths of members and may require web reinforcement; it may be more economical to use a heavier member than to provide such reinforcement. The design strength of the unreinforced coped member is determined from the limit states of flexural yielding, local buckling, and lateral torsional buckling, if applicable. Web reinforcement of coped beams is discussed in Part 9. Flexural Yielding

The flexural yielding strength of a supported beam which is coped at the top and/or bottom is φbMn, where φb = 0.90 Mn = Fy Snet AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 226


In the above equation, Snet is the net elastic section modulus, in.3 Values of Snet are tabulated in Table 8-49. The beam-end reaction Ru must be such that: Ru ≤

φbMn e

where e is the distance from the face of the cope to the point of inflection of the beam, in. It is usually assumed that the point of inflection is located at the face of the supporting member and e is as shown in Figure 8-61. However, depending upon the connection type and stiffness and support condition, the point of inflection may move away from the face of the supporting member; when this is the case, a lesser value of e may be justified. In any case, the choice of e shown in Figure 8-61 will be conservative. Local Web Buckling

For short copes no greater than the length of the connection angle(s), plate, or tee, local web buckling will generally not occur. If, however, the depth of the cope were such that

c dc

c

Cut not grid preferred Cut and grid if surface must be flush with web

c dc

(a) Cope

(b) Blocks

(c) Cut

Fig. 8-58. Copes, blocks, and cuts.

first cut

resulting notch occurs in waste

0 to 15° as required second cut

second cut potential notch

AVOID

first cut

RECOMMENDED

Fig. 8-59. Recommended coping practice. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

CONNECTED ELEMENTS

8 - 227

dc > 0.2d, the unreinforced web could buckle between the top of the cope and the beam flange if the beam web were thin. In a reduced section, the design strength in local web buckling may be more critical than the design strength in flexural yielding. This design strength is critical at the compression zone of the web near the cope and is dependent on three parameters: (1) cope depth dc; (2) cope length c; and (3) web thickness tw. It should be noted that, for convenience, the dimension h0 in Figure 8-61 is used instead of the more correct dimension h1; this eliminates the detailed calculation required to locate the neutral axis of the coped beam. Alternatively, the dimension h1 may be substituted for h0 in the following local buckling calculations. The beam end reaction Ru must be such that: Ru ≤

φFbc Snet e

where Snet = elastic section modulus of the net section, in.3 from Table 8-49 e = distance from the end reaction to the face of the cope, in. and φFbc is determined as follows. When a beam is coped at the top flange only, the design recommendations are based on the classical plate buckling formula with a k-factor based on three edges simply supported and one free edge. An additional factor f, which generally accounts for stress concentration at the cope, was developed to correlate with the coped beam buckling solutions (Cheng, et. al., 1984). From Figure 8-61, when the c ≤ 2d and dc ≤ d / 2, 2

π2E  tw    fk Fcr = 12(1 − v2)  ho  where E = 29,000 ksi, modulus of elasticity of steel ν = 0.3, Poisson’s ratio f = plate buckling model adjustment factor

(a) Coping Required

(b) Coping Eliminated

Fig. 8-60. Minimizing coping requirements. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 228


k = plate buckling coefficient ho = d − dc, reduced beam depth, in. Thus, the design buckling stress φFbc for a beam coped at the top flange only is, 2

 tw  φFbc = 23,590   fk  ho  where f and k are determined from the following equations: c c f = 2   for ≤ 1.0 d d   c c f = 1 +   for > 1.0 d  d  ho  c k = 2.2   for ≤ 1.0 ho c  ho  c k = 2.2   for > 1.0 ho c When a beam is coped at both flanges, the design recommendations are based on the lateral buckling model with an adjustment factor fd (Cheng, et al., 1984). From Figure 8-62, when at both flanges c ≤ 2d and dc ≤ 0.2d, Fcr = 0.62πE

t2w f cho d

Thus, the design buckling stress φFbc for a beam coped at both flanges is, φFbc = 50,840

t2w f cho d

and e c

Buckling checked here dc

Setback

ho

Ru

d

h1

tw

Simple shear connection

Fig. 8-61. Local buckling of beam web coped at top flange only. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

N.A.

CONNECTED ELEMENTS

8 - 229

d  d

c fd = 3.5 − 7.5 

where dc is the larger of the top cope depth dct and the bottom cope depth dcb. Lateral Torsional Buckling

In laterally unbraced beams, copes, blocks, and cuts further reduce the out-of-plane rotational restraint. Cheng, et al. (1984) discusses the design strength of laterally unbraced coped beams. For laterally unbraced beams coped at the top only, this design strength may be determined with this information and the provisions of LRFD Specification Section F1.2. For laterally unbraced beams coped at the top and bottom, this design strength may be determined with this information and the provisions of LRFD Specification Appendix F1. A detailed discussion of this topic is beyond the scope of this text.

e c

Buckling checked here

Simple shear connection

d

Ru

tw

d cb

d – d ct – dcb

d ct

Setback

Fig. 8-62. Local buckling of beam web coped at both flanges. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

N.A.

8 - 230


So

Snet

d

Sx

d

d

dc

Table 8-49. Section Modulus of Coped W Shapes

Snet, in.3 dc, in.

Designation

d in.

tf in.

Sx in.3

So in.3

2

3

4

5

6

7

8

9

10

W44×335 W ×290 W ×262 W ×230

44.0 43.6 43.3 42.9

1.77 1.58 1.42 1.22

1410 1240 1120 969

492 417 374 330

451 382 342 301

431 365 327 288

411 348 312 274

392 332 297 261

373 316 283 249

355 300 269 236

337 285 255 224

320 270 241 212

303 255 228 200

W40×593 W ×503 W ×431 W ×372 W ×321 W ×297 W ×277 W ×249 W ×215 W ×199 W ×174

43.0 42.1 41.3 40.6 40.1 39.8 39.7 39.4 39.0 38.7 38.2

3.23 2.76 2.36 2.05 1.77 1.65 1.58 1.42 1.22 1.07 0.830

2340 1980 1690 1460 1250 1170 1100 992 858 769 639

810 673 567 480 406 374 335 299 256 247 234

— — — — 368 339 304 271 231 224 211

— 584 491 415 350 323 289 258 220 213 201

671 556 467 394 332 306 274 245 208 202 190

639 528 444 374 315 290 260 232 197 191 180

607 501 421 354 298 275 246 219 186 180 170

575 475 398 335 282 259 232 207 176 170 160

545 449 376 316 266 245 219 195 166 160 151

515 424 355 298 250 230 206 183 156 150 142

486 399 334 280 235 216 193 172 146 141 133

W40×466 W ×392 W ×331 W ×278 W ×264 W ×235 W ×211 W ×183 W ×167 W ×149

42.4 41.6 40.8 40.2 40.0 39.7 39.4 39.0 38.6 38.2

2.95 2.52 2.13 1.81 1.73 1.58 1.42 1.22 1.03 0.830

1710 1440 1210 1020 971 874 785 682 599 512

705 581 483 396 371 320 286 244 234 217

— — — 360 337 291 259 221 212 196

613 504 419 342 321 276 246 210 201 186

584 480 398 325 305 262 234 199 191 177

556 456 378 308 289 249 221 189 181 167

528 432 358 292 274 235 209 179 171 158

500 409 339 276 259 222 198 168 161 149

474 387 320 261 244 210 186 159 152 140

448 365 302 245 230 197 175 149 143 132

422 344 284 231 216 185 165 140 134 123

W36×848 W ×798 W ×650 W ×527 W ×439 W ×393 W ×359 W ×328 W ×300 W ×280 W ×260 W ×245 W ×230

42.5 42.0 40.5 39.3 38.3 37.8 37.4 37.1 36.7 36.5 36.3 36.1 35.9

4.53 4.29 3.54 2.91 2.44 2.20 2.01 1.85 1.68 1.57 1.44 1.35 1.26

3170 2980 2420 1950 1620 1450 1320 1210 1110 1030 953 895 837

1094 1016 794 618 503 443 400 360 328 305 285 269 253

— — — — — — — 324 295 274 256 241 227

— — — 531 430 378 341 307 279 259 242 228 214

903 836 649 503 407 358 322 290 264 245 228 215 202

858 794 615 476 384 338 304 273 249 230 215 203 190

813 752 582 449 362 318 286 257 234 217 202 190 179

770 712 550 423 341 299 269 242 220 203 190 178 168

728 673 518 398 320 281 252 226 206 190 177 167 157

687 634 487 374 300 263 236 212 192 178 166 156 146

647 597 457 350 280 246 220 197 179 165 154 145 136


CONNECTED ELEMENTS

8 - 231

So

Snet

d

Sx

d

d

dc

Table 8-49 (cont.). Section Modulus of Coped W Shapes

Snet, in.3 Designation

d in.

tf in.

Sx in.3

W36×256 W ×232 W ×210 W ×194 W ×182 W ×170 W ×160 W ×150 W ×135

37.4 37.1 36.7 36.5 36.3 36.2 36.0 35.9 35.6

1.73 1.57 1.36 1.26 1.18 1.10 1.02 0.940 0.790

895 809 719 664 623 580 542 504 439

W33×354 W ×318 W ×291 W ×263 W ×241 W ×221 W ×201

35.6 35.2 34.8 34.5 34.2 33.9 33.7

2.09 1.89 1.73 1.57 1.40 1.28 1.15

W33×169 W ×152 W ×141 W ×130 W ×118

33.8 33.5 33.3 33.1 32.9

W30×477 W ×391 W ×326 W ×292 W ×261 W ×235 W ×211 W ×191 W ×173 W30×148 W ×132 W ×124 W ×116 W ×108 W ×99 W ×90

So in.3

dc, in. 2

3

4

5

6

7

8

9

329 295 272 249 234 218 206 195 181

297 266 245 224 211 196 185 176 163

281 251 232 212 199 185 175 166 154

266 238 219 201 188 175 165 157 145

251 224 207 189 178 165 156 148 137

237 211 195 178 167 155 147 139 129

223 199 183 167 157 146 138 130 121

209 186 172 157 147 137 129 122 113

196 174 161 146 137 128 120 114 105

183 163 150 137 128 119 112 106 98.1

1230 1110 1010 917 829 757 684

373 330 300 268 250 230 209

— 295 268 239 223 205 186

315 278 253 226 210 193 175

297 262 238 212 197 181 165

279 246 223 199 185 170 154

262 230 209 186 173 159 144

245 216 195 174 162 148 135

229 201 182 162 150 138 125

213 187 169 151 140 128 116

198 173 157 139 129 118 107

1.22 1.06 0.960 0.855 0.740

549 487 448 406 359

191 176 165 155 143

170 157 147 138 128

161 148 139 130 120

151 139 130 122 113

141 130 122 114 106

132 122 114 107 98.6

124 114 106 100 91.9

115 106 98.8 92.5 85.4

107 97.9 91.6 85.7 79.1

98.6 90.5 84.6 79.2 73.0

34.2 33.2 32.4 32.0 31.6 31.3 30.9 30.7 30.4

2.95 2.44 2.05 1.85 1.65 1.50 1.32 1.19 1.07

1530 1250 1030 928 827 746 663 598 539

475 378 305 269 240 211 192 174 158

— — — 238 212 186 170 153 139

398 315 254 223 198 174 159 143 130

374 295 237 208 185 163 148 133 121

350 276 221 194 172 152 138 124 112

327 257 206 180 160 141 128 115 104

305 239 191 167 148 130 118 106 96.1

283 222 177 155 137 120 109 97.7 88.4

262 205 163 142 126 110 100 89.6 81.0

242 188 150 130 115 101 91.2 81.8 73.9

30.7 30.3 30.2 30.0 29.8 29.7 29.5

1.18 1.00 0.930 0.850 0.760 0.670 0.610

436 380 355 329 299 269 245

152 139 131 124 118 110 98.7

134 123 115 109 103 96.4 86.7

125 115 108 102 96.5 90.0 80.9

117 107 100 95.3 89.9 83.9 75.4

109 99.3 93.4 88.6 83.6 77.9 70.0

101 92.1 86.5 82.1 77.4 72.1 64.8

93.3 85.1 79.9 75.8 71.4 66.5 59.7

86.0 78.3 73.6 69.7 65.7 61.1 54.9

78.9 71.8 67.4 63.9 60.1 56.0 50.2

72.1 65.5 61.5 58.2 54.8 51.0 45.7


10

8 - 232


So

Snet

d

Sx

d

d

dc



d in.

tf in.

Sx in.3

So in.3

W27×539 W ×448 W ×368 W ×307 W ×281 W ×258 W ×235 W ×217 W ×194 W ×178 W ×161 W ×146

32.5 31.4 30.4 29.6 29.3 29.0 28.7 28.4 28.1 27.8 27.6 27.4

3.54 2.99 2.48 2.09 1.93 1.77 1.61 1.50 1.34 1.19 1.08 0.975

1570 1300 1060 884 811 742 674 624 556 502 455 411

W27×129 W ×114 W ×102 W ×94 W ×84

27.6 27.3 27.1 26.9 26.7

1.10 0.930 0.830 0.745 0.640

W24×492 W ×408 W ×335 W ×279 W ×250 W ×229 W ×207 W ×192 W ×176 W ×162 W ×146 W ×131 W ×117 W ×104

29.7 28.5 27.5 26.7 26.3 26.0 25.7 25.5 25.2 25.0 24.7 24.5 24.3 24.1

W24×103 W ×94 W ×84 W ×76 W ×68 W24×62 W ×55

dc, in. 2

3

4

5

6

7

8

9

10

509 404 321 259 233 212 193 174 155 145 131 118

— — — — 203 185 168 152 134 126 113 102

— — 262 211 189 172 156 141 125 117 105 95.0

394 310 244 196 176 159 145 130 115 108 97.2 87.7

367 288 226 181 162 147 134 120 106 100 89.5 80.7

341 267 209 167 150 136 123 111 97.6 91.5 82.0 74.0

316 247 193 154 137 124 113 101 89.3 83.6 74.9 67.5

292 227 177 141 126 114 103 92.3 81.3 76.1 68.1 61.3

269 209 162 128 114 103 93.2 83.7 73.6 68.8 61.5 55.3

247 191 147 116 104 93.3 84.2 75.5 66.3 61.9 55.3 49.7

345 299 267 243 213

117 106 94.2 88.0 80.5

101 91.6 81.6 76.2 69.6

94.0 84.9 75.6 70.6 64.5

86.9 78.4 69.8 65.1 59.5

80.1 72.2 64.3 59.9 54.7

73.5 66.2 58.9 54.9 50.1

67.2 60.5 53.7 50.1 45.7

61.1 54.9 48.8 45.4 41.4

55.3 49.6 44.0 41.0 37.3

49.7 44.6 39.5 36.8 33.5

3.54 2.99 2.48 2.09 1.89 1.73 1.57 1.46 1.34 1.22 1.09 0.960 0.850 0.750

1290 1060 864 718 644 588 531 491 450 414 371 329 291 258

420 331 261 210 184 167 149 136 124 115 104 94.4 84.4 75.4

— — — — 158 143 127 117 106 98.0 88.5 80.3 71.7 64.1

— — 209 167 146 132 117 107 97.6 90.0 81.2 73.7 65.7 58.7

316 247 193 154 134 121 107 98.2 89.4 82.3 74.2 67.3 60.0 53.5

292 227 177 141 123 111 98.0 89.5 81.4 74.9 67.5 61.1 54.5 48.6

269 209 162 128 112 101 89.0 81.2 73.8 67.9 61.1 55.3 49.2 43.8

247 191 147 116 101 91.0 80.4 73.3 66.5 61.1 54.9 49.7 44.2 39.3

226 173 133 105 91.2 81.8 72.2 65.8 59.6 54.7 49.1 44.3 39.4 35.0

205 157 120 94.3 81.7 73.1 64.4 58.6 53.0 48.6 43.6 39.3 34.8 30.9

186 141 108 84.0 72.6 64.9 57.0 51.8 46.8 42.8 38.3 34.5 30.5 27.1

24.5 24.3 24.1 23.9 23.7

0.980 0.875 0.770 0.680 0.585

245 222 196 176 154

82.9 76.2 68.3 62.6 57.5

70.7 64.9 58.0 53.2 48.8

64.9 59.5 53.2 48.7 44.7

59.3 54.3 48.6 44.5 40.8

53.9 49.4 44.1 40.4 37.0

48.8 44.6 39.8 36.4 33.4

43.9 40.1 35.8 32.7 29.9

39.2 35.8 31.9 29.1 26.6

34.8 31.7 28.2 25.8 23.5

30.6 27.9 24.8 22.6 20.6

23.7 23.6

0.590 0.505

131 114

56.9 51.1

48.3 43.4

44.3 39.7

40.4 36.2

36.7 32.9

33.1 29.7

29.7 26.6

26.5 23.7

23.4 20.9

20.5 18.3


CONNECTED ELEMENTS

8 - 233

So

Snet

d

Sx

d

d

dc



d in.

tf in.

W21×201 W ×182 W ×166 W ×147 W ×132 W ×122 W ×111 W ×101

23.0 22.7 22.5 22.1 21.8 21.7 21.5 21.4

W21×93 W ×83 W ×73 W ×68 W ×62

dc, in.

Sx in.3

So in.3

2

3

4

5

6

7

8

9

1.63 1.48 1.36 1.15 1.04 0.960 0.875 0.800

461 417 380 329 295 273 249 227

125 111 99.3 91.2 81.0 74.1 67.1 60.4

105 93.3 83.0 76.1 67.5 61.6 55.7 50.1

95.2 84.8 75.3 68.9 61.1 55.7 50.4 45.3

86.2 76.6 68.0 62.1 55.0 50.2 45.3 40.7

77.6 68.8 61.0 55.7 49.2 44.8 40.4 36.3

69.4 61.4 54.4 49.5 43.7 39.8 35.9 32.1

61.6 54.4 48.1 43.7 38.5 35.0 31.5 28.2

54.2 47.8 42.2 38.2 33.6 30.5 27.4 24.5

47.3 41.6 36.6 33.1 29.0 26.3 23.6 21.1

21.6 21.4 21.2 21.1 21.0

0.930 0.835 0.740 0.685 0.615

192 171 151 140 127

67.2 59.0 51.5 48.1 44.1

56.0 49.1 42.7 39.9 36.5

50.7 44.4 38.7 36.1 33.0

45.7 40.0 34.8 32.4 29.7

40.9 35.7 31.0 29.0 26.5

36.3 31.7 27.5 25.6 23.4

32.0 27.9 24.2 22.5 20.5

27.9 24.3 21.0 19.6 17.8

24.1 20.9 18.1 16.8 15.3

W21×57 W ×50 W ×44

21.1 20.8 20.7

0.650 0.535 0.450

111 94.5 81.6

43.4 39.2 35.2

36.1 32.5 29.1

32.6 29.4 26.3

29.3 26.4 23.6

26.2 23.6 21.0

23.2 20.8 18.6

20.4 18.3 16.3

17.7 15.9 14.1

15.2 13.6 12.1

W18×311 W ×283 W ×258 W ×234 W ×211 W ×192 W ×175 W ×158 W ×143 W ×130

22.3 21.9 21.5 21.1 20.7 20.4 20.0 19.7 19.5 19.3

2.74 2.50 2.30 2.11 1.91 1.75 1.59 1.44 1.32 1.20

624 564 514 466 419 380 344 310 282 256

186 166 148 130 115 102 92.1 81.7 72.5 65.2

— — — — 94.5 83.4 75.1 66.4 58.8 52.8

140 124 110 96.1 84.8 74.7 67.2 59.3 52.4 47.0

126 111 98.3 85.9 75.6 66.5 59.7 52.6 46.4 41.5

113 99.3 87.4 76.2 66.9 58.7 52.6 46.2 40.7 36.4

100 87.8 77.2 67.1 58.7 51.4 45.9 40.2 35.4 31.5

88.2 77.1 67.5 58.5 51.0 44.5 39.6 34.6 30.4 27.0

77.0 67.0 58.5 50.4 43.8 38.1 33.8 29.4 25.7 22.8

66.5 57.6 50.0 43.0 37.1 32.1 28.4 24.6 21.5 19.0

W18×119 W ×106 W ×97 W ×86 W ×76

19.0 18.7 18.6 18.4 18.2

1.06 0.940 0.870 0.770 0.680

231 204 188 166 146

61.7 54.4 48.9 43.1 37.6

49.8 43.8 39.3 34.6 30.1

44.3 38.9 34.9 30.6 26.7

39.1 34.3 30.7 26.9 23.4

34.2 29.9 26.8 23.4 20.3

29.5 25.8 23.1 20.2 17.5

25.2 22.0 19.6 17.1 14.8

21.2 18.5 16.4 14.3 12.3

W18×71 W ×65 W ×60 W ×55 W ×50

18.5 18.4 18.2 18.1 18.0

0.810 0.750 0.695 0.630 0.570

127 117 108 98.3 88.9

42.4 38.3 35.0 32.4 29.1

34.1 30.8 28.1 26.0 23.4

30.3 27.3 24.9 23.0 20.7

26.7 24.0 21.9 20.2 18.2

23.3 20.9 19.1 17.6 15.8

20.1 18.0 16.4 15.1 13.5

17.1 15.3 13.9 12.8 11.5

14.3 12.8 11.6 10.7 9.54

W18×46 W ×40 W ×35

18.1 17.9 17.7

0.605 0.525 0.425

78.8 68.4 57.6

28.9 24.9 22.7

23.2 20.0 18.2

20.6 17.7 16.1

18.1 15.5 14.1

15.7 13.5 12.3

13.5 11.6 10.5

11.5 9.80 8.88

9.56 8.16 7.37


10

8 - 234


So

Snet

d

Sx

d

d

dc


Snet, in.3 dc, in.

Designation

d in.

tf in.

Sx in.3

So in.3

2

3

4

5

6

7

W16×100 W ×89 W ×77 W ×67

17.0 16.8 16.5 16.3

0.985 0.875 0.760 0.665

175 155 134 117

44.4 39.0 33.1 28.3

34.9 30.6 25.9 22.1

30.5 26.7 22.6 19.2

26.4 23.1 19.4 16.5

22.6 19.7 16.5 14.0

19.0 16.5 13.8 11.7

15.7 13.6 11.4 9.58

W16×57 W ×50 W ×45 W ×40 W ×36

16.4 16.3 16.1 16.0 15.9

0.715 0.630 0.565 0.505 0.430

92.2 81.0 72.7 64.7 56.5

29.4 25.6 22.9 20.1 18.8

23.0 20.0 17.9 15.6 14.6

20.1 17.4 15.5 13.6 12.7

17.3 15.0 13.4 11.7 10.9

14.8 12.7 11.3 9.89 9.21

12.4 10.7 9.47 8.24 7.67

10.2 8.74 7.75 6.73 6.25

W16×31 W ×26

15.9 15.7

0.440 0.345

47.2 38.4

17.1 14.9

13.3 11.6

11.6 10.1

10.0 8.64

8.44 7.31

7.03 6.08

5.73 4.95

W14×808 W ×730 W ×665 W ×605 W ×550 W ×500 W ×455

22.8 22.4 21.6 20.9 20.2 19.6 19.0

5.12 4.91 4.52 4.16 3.82 3.50 3.21

1400 1280 1150 1040 931 838 756

— — — — — — —

— — — — — — —

W14×426 W ×398 W ×370 W ×342 W ×311 W ×283 W ×257 W ×233 W ×211 W ×193 W ×176 W ×159 W ×145

18.7 18.3 17.9 17.5 17.1 16.7 16.4 16.0 15.7 15.5 15.2 15.0 14.8

3.04 2.85 2.66 2.47 2.26 2.07 1.89 1.72 1.56 1.44 1.31 1.19 1.09

707 656 607 559 506 459 415 375 338 310 281 254 232

164 150 135 122 107 94.4 83.1 73.2 64.9 57.6 52.2 45.7 40.9

— — — — — — 64.1 56.1 49.5 43.8 39.5 34.5 30.7

W14×132 W ×120 W ×109 W ×99 W ×90

14.7 14.5 14.3 14.2 14.0

1.03 0.940 0.860 0.780 0.710

209 190 173 157 143

38.1 34.2 30.0 27.2 24.3

28.6 25.5 22.3 20.2 18.0

24.3 21.7 18.9 17.0 15.2

W14×82 W ×74 W ×68 W ×61

14.3 14.2 14.0 13.9

0.855 0.785 0.720 0.645

123 112 103 92.2

28.0 24.4 22.2 19.7

20.9 18.2 16.5 14.6

17.7 15.4 13.9 12.3

451 365 317 275 238 208 182

— — — — 153 131 113

— 220 187 158 134 115 98.2

244 195 165 139 117 99.4 84.6

216 172 144 121 101 85.3 72.1

87.6 78.7 70.1 61.9 53.5 46.3 40.0 34.6 30.2 26.4 23.6 20.4 18.0

75.2 67.2 59.6 52.3 44.9 38.7 33.3 28.6 24.8 21.6 19.2 16.5 14.5

63.8 56.7 50.0 43.6 37.2 31.8 27.1 23.2 19.9 17.3 15.2 13.0 11.4

20.3 18.1 15.7 14.2 12.6

16.7 14.8 12.8 11.5 10.2

13.4 11.8 10.2 9.15 8.07

14.8 12.8 11.6 10.2

12.1 10.4 9.42 8.28

9.64 8.31 7.46 6.54

— 101 104 91.1 93.7 81.4 83.4 72.3 72.7 62.7 63.6 54.6 55.5 47.4 48.4 41.3 42.6 36.1 37.5 31.7 33.8 28.5 29.4 24.7 26.1 21.9


8

9

10

CONNECTED ELEMENTS

8 - 235

So

Snet

d

Sx

d

d

dc


Snet, in.3 dc, in.

Designation

d in.

tf in.

Sx in.3

So in.3

2

3

4

5

6

W14×53 W ×48 W ×43

13.9 13.8 13.7

0.660 0.595 0.530

77.8 70.3 62.7

19.1 17.3 15.3

14.2 12.8 11.3

12.0 10.8 9.50

9.93 8.93 7.84

8.07 7.23 6.34

6.39 5.71 4.99

W14×38 W ×34 W ×30

14.1 14.0 13.8

0.515 0.455 0.385

54.6 48.6 42.0

16.0 14.4 13.2

12.0 10.8 9.88

10.2 9.14 8.37

8.48 7.62 6.96

6.94 6.22 5.68

5.54 4.95 4.51

W14×26 W ×22

13.9 13.7

0.420 0.335

35.3 29.0

12.3 10.7

9.20 7.97

7.80 6.75

6.50 5.62

5.31 4.58

4.23 3.64

W12×336 W ×305 W ×279 W ×252 W ×230 W ×210 W ×190 W ×170 W ×152 W ×136 W ×120 W ×106 W ×96 W ×87 W ×79 W ×72 W ×65

16.8 16.3 15.9 15.4 15.1 14.7 14.4 14.0 13.7 13.4 13.1 12.9 12.7 12.5 12.4 12.3 12.1

3.00 2.71 2.47 2.25 2.07 1.90 1.74 1.56 1.40 1.25 1.105 0.990 0.900 0.810 0.735 0.670 0.605

— — — — — 49.0 42.3 36.5 31.6 27.5 23.7 19.8 17.4 15.8 14.1 12.6 11.2

83.1 71.4 63.1 54.2 47.5 41.6 35.7 30.7 26.5 22.9 19.7 16.3 14.3 13.0 11.5 10.3 9.16

71.4 61.0 53.5 45.7 39.9 34.7 29.7 25.3 21.7 18.7 16.0 13.2 11.5 10.4 9.23 8.24 7.28

60.6 51.4 44.8 38.0 32.9 28.5 24.2 20.5 17.5 14.9 12.6 10.4 9.03 8.11 7.16 6.37 5.61

50.8 42.7 36.9 31.0 26.7 22.9 19.3 16.2 13.7 11.6 9.70

W12×58 W ×53

12.2 12.1

0.640 0.575

78.0 70.6

14.8 13.9

10.4 9.74

8.52 7.94

6.79 6.31

5.24 4.85

W12×50 W ×45 W ×40

12.2 12.1 11.9

0.640 0.575 0.515

64.7 58.1 51.9

14.8 13.1 11.4

10.4 9.27 8.03

8.54 7.56 6.54

6.82 6.02 5.19

5.27 4.63 3.98

W12×35 W ×30 W ×26

12.5 12.3 12.2

0.52 0.44 0.38

45.6 38.6 33.4

12.3 10.5 9.08

8.85 7.47 6.47

7.30 6.15 5.32

5.89 4.94 4.27

4.61 3.86 3.32

W12×22 W ×19 W ×16 W ×14

12.3 12.2 12.0 11.9

0.425 0.350 0.265 0.225

25.4 21.3 17.1 14.9

9.60 8.39 7.43 6.61

6.89 6.01 5.30 4.71

5.69 4.95 4.36 3.86

4.59 3.98 3.50 3.10

3.59 3.11 2.72 2.41

483 123 435 108 393 96.1 353 83.7 321 74.2 292 65.6 263 57.0 235 49.6 209 43.3 186 37.9 163 32.8 145 27.6 131 24.3 118 22.2 107 19.9 97.4 17.9 87.9 16.0


7

8

9

10

8 - 236


So

Snet

d

Sx

d

d

dc



d in.

tf in.

dc, in.

Sx in.3

So in.3

2

3

4

25.7 22.3 19.1 16.2 13.9 12.1 10.5 9.46

17.5 15.0 12.8 10.7 9.13 7.88 6.79 6.10

13.9 11.9 10.0 8.35 7.10 6.09 5.22 4.68

10.8 9.12 7.62 6.29 5.30 4.52 3.86 3.44

W10×112 W ×100 W ×88 W ×77 W ×68 W ×60 W ×54 W ×49

11.4 11.1 10.8 10.6 10.4 10.2 10.1 8.00

1.25 126 1.12 112 0.990 98.5 0.870 85.9 0.770 75.7 0.680 66.7 0.615 60.0 0.560 54.6

W10×45 W ×39 W ×33

10.1 2.00 3.00

0.620 0.530 0.435

49.1 42.1 35.0

9.75 8.49 7.49

6.33 5.48 4.80

4.88 4.20 3.67

3.61 3.08 2.67

W10×30 W ×26 W ×22

10.5 10.3 10.2

0.510 0.440 0.360

32.4 27.9 23.2

8.64 7.33 6.51

5.75 4.86 4.29

4.51 3.80 3.34

3.41 2.85 2.50

W10×19 W ×17 W ×15 W ×12

10.2 10.1 9.00 7.00

0.395 0.330 0.270 0.210

18.8 16.2 13.8 10.9

6.52 6.01 5.52 4.43

4.33 3.98 3.64 2.91

3.39 3.10 2.83 2.26

2.55 2.33 2.12 1.68

W8×67 W ×58 W ×48 W ×40 W ×35 W ×31

0.00 5.00 0.00 5.00 2.00

0.935 0.810 0.685 0.560 0.495 0.435

60.4 52.0 43.3 35.5 31.2 27.5

12.2 10.4 7.89 6.71 5.66 5.06

7.42 6.24 4.63 3.89 3.24 2.88

5.44 4.52 3.32 2.74 2.28 2.01

W8×28 W ×24

6.00 3.00

0.465 0.400

24.3 20.9

5.04 4.23

2.89 2.40

2.02 1.67

W8×21 W ×18

8.00 4.00

0.400 0.330

18.2 15.2

4.55 4.02

2.67 2.35

1.91 1.66

W8×15 W ×13 W ×10

1.00 9.00 9.00

0.315 0.255 0.205

11.8 9.91 7.81

4.03 3.61 2.65

2.36 2.10 1.54

1.68 1.49 1.08

5


6

7

8

9

10

CONNECTED ELEMENTS

8 - 237

Other Elements in Connections

Shims

Shims are furnished to the erector for use in filling the spaces allowed for field clearance which might be present at connections such as simple shear connections, PR and FR moment connections, column base plates, and column splices. These shims, illustrated in Figure 8-63, may be either strip shims, with round punched holes, or finger shims, with slots cut through the edge. Whereas strip shims are less expensive to fabricate, finger shims may be laterally inserted and eliminate the need to remove erection bolts or pins already in place. Finger shims, when inserted fully against the bolt shank, are acceptable for slip-critical connections and are not to be considered as an internal ply with the slotted hole determining the design strength of the connection. This is because less than 25 percent of the contact surface is lost and this is not enough to affect the performance of the joint. Fillers

A filler is furnished to occupy spaces which will be present because of dimensional separations between elements of a connection across which load transfer occurs. Examples where fillers might be used are beams framing off center on a column and raised beams. From LRFD Specification Section J6, fillers in welded connections and fillers thicker than 3⁄4-in. in bolted bearing-type connections must be fully developed. In bolted bearing-type connections, fillers between 1⁄4-in. and 3⁄4-in. thick, inclusive, need not be developed, provided the design shear strength of the bolts is reduced by the factor 0.4(t − 0.25) where t is the total thickness of the fillers up to 3⁄4-in. In bolted slip-critical connections, fillers need not be fully developed.

Strip

Finger

Fig. 8-63. Shims. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

8 - 238


REFERENCES

Alexander, W. G., 1991, “Designing Longitudinal Welds for Bridge Members,” Engineering Journal, Vol. 28, No. 1, (1st Qtr.), pp. 29–36, AISC, Chicago, IL. American Concrete Institute, 1985, ACI 349 Code Requirements for Nuclear Safety Related Concrete Structures, Appendix B, ACI, Detroit, MI. American Institute of Steel Construction, Inc., 1993, Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, IL. American Institute of Steel Construction, Inc., 1989, Manual of Steel Construction— Allowable Stress Design, 9th ed., AISC, Chicago, IL. American Institute of Steel Construction, Inc., 1988, Quality Criteria and Inspection Standards, 3rd ed., AISC, Chicago, IL. American Institute of Steel Construction, Inc., 1973, “Commentary on Highly Restrained Welded Connections,” Engineering Journal, Vol. 10, No. 3, (3rd Qtr.), pp. 61–73, AISC, Chicago, IL. American Welding Society, 1978, Welding Handbook—Volume 2, 7th ed., AWS, Miami, FL. American Welding Society, 1977, Guide for the Non-Destructive Inspection of Welds, (AWS B1.0-77), AWS, Miami, FL. Astaneh, A., 1985, “Procedure for Design and Analysis of Hanger-Type Connections,” Engineering Journal, Vol. 22, No. 2, (2nd Qtr.), pp. 63–66, AISC, Chicago, IL. Blodgett, O. W., 1966, Design of Welded Structures, James F. Lincoln Arc Welding Foundation, Cleveland, OH. Blodgett, O. W., 1980, “Detailing to Achieve Practical Welded Fabrication,” Engineering Journal, Vol. 17, No. 4, (4th Qtr.), pp. 106–119, AISC, Chicago, IL. Bowman, M. D. and M. Betancourt, 1991, “Reuse of A325 and A490 High-Strength Bolts,” Engineering Journal, Vol. 28, No. 3, (3rd Qtr.), pp. 110–118, AISC, Chicago, IL. Butler, L. J., S. Pal, and G. L. Kulak, 1972, “Eccentrically Loaded Welded Connections,” Journal of the Structural Division, Vol. 98, No. ST5, (May), pp. 989–1005, ASCE, New York, N.Y. Cannon, R. W., D. A. Godfrey, and F. L. Moreadith, 1981, “Guide to the Design of Anchor Bolts and Other Steel Embedments,” Concrete International, Vol. 3, No. 7, (July 1981), pp. 28–41, ACI, Detroit, MI.. Cheng, J. J., J. A. Yura, and C. P. Johnson, 1984, “Design and Behaviour of Coped Beams,” Department of Civil Engineering, The University of Texas at Austin, Austin, TX. Crawford, S. F. and G. L. Kulak, 1968, “Behavior of Eccentrically Loaded Bolted Connections,” Studies in Structural Engineering, (No. 4), Department of Civil Engineering, Nova Scotia Technical College, Halifax, Nova Scotia. DeWolf, J. T. and D. T. Ricker, 1990, Column Base Plates, AISC, Chicago, IL Fisher, J. M., 1981, “Structural Details in Industrial Buildings,” Engineering Journal, Vol. 18, No. 3, (3rd Qtr.), pp. 83–89, AISC, Chicago, IL. Fisher, J. W. and J. H. A. Struik, 1974, Guide to Design Criteria for Bolted and Riveted Joints, John Wiley & Sons, Inc., New York, NY. Grover, L., 1946, Manual of Design for Arc Welded Steel Structures, Air Reduction Sales Co., New York, NY. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

REFERENCES

8 - 239

Higgins, T. R., 1971, “Treatment of Eccentrically Loaded Connections in the AISC Manual,” Engineering Journal, Vol. 8, No. 2, (April), pp. 52–54, AISC, Chicago, IL. Institute of Welding, 1972, Procedures and Recommendations for the Ultrasonic Testing of Butt Welds, London, England. Iwankiw, N. R., 1987, “Design for Eccentric and Inclined Loads on Bolt and Weld Groups,” Engineering Journal, Vol. 24, No. 4, (4th Qtr.), pp. 164–171, AISC, Chicago, IL. Kaufmann, J., A. W. Pense, and R. D. Stout, 1981, “An Evaluation of Factors Significant to Lamellar Tearing,” Welding Journal Research Supplement, Vol. 60, No. 3, (March), AWS, Miami, FL. Krautkramer, J., 1977, Ultrasonic Testing of Materials, 2nd. ed., Springer-Verlag, Berlin, West Germany. Kulak, G. L., 1975, “Eccentrically Loaded Slip-Resistant Connections,” Engineering Journal, Vol. 12, No. 2, (2nd Qtr.), pp. 52–55, AISC, Chicago, IL. Lesik, D. F. and D. J. L. Kennedy, 1990, “Ultimate Strength of Fillet-Welded Connections Loaded in Plane,” Canadian Journal of Civil Engineering, Vol. 17, No. 1, National Research Council of Canada, Ottawa, Canada. Kulak, G. L. and Timler, 1984, “Tests on Eccentrically Loaded Fillet Welds,” Department of Civil Engineering, University of Alberta, Edmonton, Canada. Kulak, G. L., J. W. Fisher, and J. H. A. Struik, 1987, Guide to Design Criteria for Bolted and Riveted Joints, 2nd ed., John Wiley & Sons, New York, NY. Marsh, M. L., and E. G. Burdette, 1985a, “Anchorage of Steel Building Components to Concrete,” Engineering Journal, Vol. 15, No. 4, (4th Qtr.), pp. 33–39, AISC, Chicago, IL. Marsh, M. L., and E. G. Burdette, 1985b, “Multiple Bolt Anchorages: Method for Determining the Effective Projected Area of Overlapping Stress Cones,” Engineering Journal, Vol. 15, No. 4, (4th Qtr.), pp. 29–32, AISC, Chicago, IL. Research Council on Structural Connections, 1988, Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts, AISC, Chicago, IL. Shipp, J. G. and E. R. Haninger, 1983, “Design of Headed Anchor Bolts,” Engineering Journal, Vol. 20, No. 2, (2nd Qtr.), pp. 58–69, AISC, Chicago. IL. Stout, R. D. and W. D. Doty, 1978, Weldability of Steels, 3rd. ed., Welding Research Council, New York, NY Thornton, W. A., 1985, “Prying Action—A General Treatment,” Engineering Journal, Vol. 22, No. 2, (2nd Qtr.), pp. 67–75, AISC, Chicago, IL. Tide, R. H. R., 1980, “Eccentrically Loaded Weld Groups—AISC Design Tables,” Engineering Journal, Vol. 17, No. 4, (4th Qtr.), pp. 90–95, AISC, Chicago, IL.


9-1

PART 9 SIMPLE SHEAR AND PR MOMENT CONNECTIONS SIMPLE SHEAR CONNECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7 Double-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11 Shear End-Plate Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-91 Unstiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-128 Stiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-138 Single-Plate Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-147 Single-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-161 Tee Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-170 SHEAR SPLICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-176 SPECIAL CONSIDERATIONS FOR SIMPLE SHEAR CONNECTIONS . . . . . . . . . 9-185 Web Reinforcement of Coped Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-185 Simple Shear Connections at Stiffened Column-Web Locations . . . . . . . . . . . . . 9-190 Eccentric Effect of Larger-Than-Normal Gages . . . . . . . . . . . . . . . . . . . . . . 9-192 Simple Shear Connections for Large End Reactions . . . . . . . . . . . . . . . . . . . . 9-196 Double Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-196 Beams Offset from Column Centerline . . . . . . . . . . . . . . . . . . . . . . . . . . 9-202 Connections for Raised Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-211 Connections for Tubular and Pipe Members . . . . . . . . . . . . . . . . . . . . . . . . 9-215 Non-Rectangular Simple Shear Connections

. . . . . . . . . . . . . . . . . . . . . . . 9-215

PR MOMENT CONNECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-234 Flange-Plated PR Moment Connections . . . . . . . . . . . . . . . . . . . . . . . . . . 9-246 Flexible Wind Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-253 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-263


9-2

SIMPLE SHEAR AND PR MOMENT CONNECTIONS


OVERVIEW

9-3

OVERVIEW Part 9 contains general information, design considerations, examples, and design aids for the design of simple shear connections, shear splices, PR moment connections, and special considerations in the aforementioned topics. It is based upon the provisions of the 1993 LRFD Specification. Supplementary information may also be found in the Commentary on the LRFD Specification. Following are the general topics addressed. SIMPLE SHEAR CONNECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7 Considerations for Economical Simple Shear Connections . . . . . . . . . . . . . . . . . 9-7 Comparing Two-Sided, Seated, and One-Sided Connections . . . . . . . . . . . . . . . . 9-8 Erectability Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9 Computer Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-10 Double-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12 Recommended Angle Length and Thickness . . . . . . . . . . . . . . . . . . . . . . 9-12 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12 All-Bolted Double-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . 9-13 Bolted/Welded Double-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . 9-15 All-Welded Double-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . 9-16 Shear End-Plate Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-91 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-91 Recommended End-Plate Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 9-91 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-91 Bolted/Welded Shear End-Plate Connections . . . . . . . . . . . . . . . . . . . . . . 9-92 Unstiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-128 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-129 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-130 All-Bolted Unstiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . . 9-130 Bolted/Welded Unstiffened Seated Connections . . . . . . . . . . . . . . . . . . . . 9-132 All-Welded Unstiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . 9-132 Stiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-138 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-139 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-140 All-Bolted Stiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . . . 9-140 Bolted/Welded Stiffened Seated Connections . . . . . . . . . . . . . . . . . . . . . . 9-140 Single-Plate Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-147 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-147


9-4


Recommended Plate Length and Thickness . . . . . . . . . . . . . . . . . . . . . . 9-148 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-149 Bolted/Welded Single-Plate Connections . . . . . . . . . . . . . . . . . . . . . . . 9-149 Single-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-161 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-161 Recommended Angle Length and Thickness . . . . . . . . . . . . . . . . . . . . . . 9-161 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-161 All-Bolted Single-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . . . 9-162 Bolted/Welded Single-Angle Connections . . . . . . . . . . . . . . . . . . . . . . . 9-163 Tee Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-170 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-170 Recommended Tee Length and Flange and Web Thicknesses . . . . . . . . . . . . . 9-171 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-171 SHEAR SPLICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-176 SPECIAL CONSIDERATIONS FOR SIMPLE SHEAR CONNECTIONS . . . . . . . . 9-185 Web Reinforcement of Coped Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 9-185 Doubler Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-185 Longitudinal Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-185 Combination Longitudinal and Transverse Stiffening . . . . . . . . . . . . . . . . . 9-185 Simple Shear Connections at Stiffened Column-Web Locations . . . . . . . . . . . . . 9-190 Eccentric Effect of Larger-Than-Normal Gages . . . . . . . . . . . . . . . . . . . . . 9-192 Column-Web Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-192 Girder-Web Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-194 Alternative Treatment of Eccentric Moment . . . . . . . . . . . . . . . . . . . . . . 9-195 Simple Shear Connections for Large End Reactions . . . . . . . . . . . . . . . . . . . 9-196 Double Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-196 Supported Beams of Different Nominal Depths . . . . . . . . . . . . . . . . . . . . 9-196 Supported Beams Offset Laterally . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-200 Beams Offset from Column Centerline . . . . . . . . . . . . . . . . . . . . . . . . . . 9-202 Framing to the Column Flange from the Strong Axis . . . . . . . . . . . . . . . . . 9-202 Framing to Column Flange from the Weak Axis . . . . . . . . . . . . . . . . . . . . 9-204 Framing to the Column Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-209 Connections for Raised Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-211 Connections for Tubular and Pipe Members . . . . . . . . . . . . . . . . . . . . . . . 9-215 Non-Rectangular Simple Shear Connections . . . . . . . . . . . . . . . . . . . . . . . 9-215 AMERICAN INSTITUTE OF STEEL CONSTRUCTION

OVERVIEW

9-5

Skewed Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-215 Sloped Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-224 Canted Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-225 Inclines in Two or More Directions (Hip and Valley Framing) . . . . . . . . . . . . . 9-228 PR MOMENT CONNECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-234 Modeling PR Moment Connections for Gravity Loads . . . . . . . . . . . . . . . . . . 9-234 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-237 The Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-239 Elastic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-240 Non-Rigid Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-242 Plastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-244 Real Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-244 Flange Plated PR Moment Connections . . . . . . . . . . . . . . . . . . . . . . . . . . 9-246 Force Transfer in PR Moment Connections . . . . . . . . . . . . . . . . . . . . . . . 9-248 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-248 Shop and Field Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-248 Flexible Wind Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-253 Design Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-254 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-263


9-6



SIMPLE SHEAR CONNECTIONS

9-7


The ends of members with simple shear connections are assumed to be unrestrained or free to rotate under load as illustrated in Figure 9-1. While simple shear connections do actually possess some rotational restraint, as illustrated by curve A in Figure 9-2, this small amount is usually neglected and the connection is idealized to be completely flexible. Accordingly, simple shear connections are sized only for the end reaction or shear Ru of the supported beam. Note that simple shear connections must provide flexibility to accommodate the required end rotation of the supported beam. When members are designed with simple shear connections, provision must be made to stabilize the frame for gravity loads and also to resist lateral loads. A positive steel bracing system, such as X- or K-bracing, PR or FR construction, and concrete or masonry shear walls are three commonly used methods. PR moment connections (including flexible wind connections) are treated in this Part. FR moment connections are treated in Part 10. Bracing systems and connections are treated in Part 11. For the design of concrete or masonry shear walls, refer to ACI 318. Considerations for Economical Simple Shear Connections

The AISC Code of Standard Practice states that, after the engineer of record (EOR) designs the structural members, the EOR may design and detail the connections or the EOR may have the fabricator develop the detailed configuration of the simple shear connections. In both cases, the fabricator must submit shop drawings for approval and verification that the EOR’s design criteria and intent have been satisfied. Regardless of which approach is taken, the AISC Code of Standard Practice states that the EOR is responsible for the adequacy of these connections. The fabricator is responsible for the accuracy of the detail dimensions, clearances, and general fit-up of the structural steel members and connecting materials for field assembly (refer to the AISC Code of Standard Practice Section 2 for definition of which items are and are not considered structural steel). The latter approach is usually taken since there are economies inherent in allowing the fabricator to choose the most efficient connections for the fabricator’s shop and erection processes. Whenever possible, the designer should give the fabricator and erector the flexibility to choose the connection types which offer the most economical shop fabrication and safest and most economical erection. In taking this approach, however, some engineers of record specify general design criteria (e.g., one-half the total factored uniform load) from which the connections are to be developed without regard to the actual reactions. Thornton (1992) describes several of these practices and provides examples of the uneconomical and/or unsafe connections which can result from their use. Because of this, when the fabricator or detailer is to θ

θ No restraint Ends free to rotate

Note: top angle not shown for clarity.

Figure 9.1. Illustration of simple shear connection. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9-8


develop the detailed configuration of the connections, the EOR must indicate the actual design reactions on the contract drawings or provide the fabricator with a method to accurately determine the required strength. In the absence of such information, connections will be selected to support one-half the total factored uniform load for the given beam, span, and grade of steel specified; no consideration will be given for the effects of any other loads unless specified on the contract drawings. Comparing Two-Sided, Seated, and One-Sided Connections

Following is a general discussion of the advantages of two-sided, seated, and one-sided connections. Two-sided connections, such as double-angle and shear end-plate connections, offer the following advantages: (1) suitability for use when the end reaction is large; (2) compactness (usually, the entire connection is contained within the flanges of the supported beam); and, (3) eccentricity perpendicular to the beam axis need not be considered for usual gages. Unstiffened and stiffened seated connections offer the following advantages: (1) seats may be shop attached to the support, simplifying erection; (2) ample erection clearance is provided; (3) erection is fast and safe; and, (4) the bay length of the structure is easily maintained (seated connections may be preferable when maintaining bay length is a concern for repetitive bays of framing). Note that seated connections can cause erection

FR moment connections

Fixed end moment PR moment connections

End moment

Beam line

Simple shear connections A Simple beam rotation

Rotation

A

Figure 9-2. Simple shear connection behavior. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9-9

interference when floors are close, beams are deep, or seats protrude excessively from the column face; the practice of leaning or tilting the columns to erect a column-web connection is difficult, unsafe, and should always be avoided. One-sided connections such as single-plate, single-angle, and tee connections offer the following advantages: (1) shop attachment of connecting materials to the support, simplifying shop fabrication and erection; (2) reduced material and shop labor requirements; and, (3) excellent safety during erection since double connections may be eliminated. Erectability Considerations

In field-bolted connections, when beams or girders frame opposite each other and take the same open holes in the web of a column, as illustrated in Figure 9-3, the first member to be erected must be supported while the second member to be erected is brought into its final position. Note that hanging the beam on a partially inserted bolt or drift pin is dangerous; such a makeshift practice should not be attempted. A temporary erection seat, usually an angle, is sometimes provided in the column web and located to clear the bottom flange of the supported member by approximately 3⁄8-in. to accommodate mill, fabrication, and erection tolerances. The erection seat is sized and attached to the column web with sufficient bolts or welds to support the dead weight of the member, unless additional loading is indicated. The sequence of erection is most important in determining the need for erection seats. If the erection sequence is known, the erection seat is provided on the side needing the support. If the erection sequence is not known, a seat can be provided on both sides of the column web. Erection seats may be reused at other locations, but are not generally required to be removed unless they create an interference, detract from the architectural appearance, or such removal is required in the contract documents. In field-welded connections in which some means of temporary support must be provided until final welding is performed, temporary erection bolts are usually provided.

Column First beam to be erected

Clearance

Second beam to be erected

Temporary erection seat

Figure 9-3. Erection seat. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 10


Note that it is not necessary that these bolts be removed subsequent to final welding. Subject to the provisions of LRFD Specification Section J1.9, erection bolts may also serve as permanent attachment; refer to “Construction Combining Bolts and Welds” in Part 8. Safety laws require that two bolts be placed for erection safety. As a general rule, then, two erection bolts are used for framing angles or similar connecting elements up to 12 inches long, four bolts are used for connecting elements up to 18 inches long, and six bolts are used for longer connecting elements. Additional erection bolts may be provided and serve two purposes: (1) they provide for the contingency of large temporary loads during erection; and, (2) they assist in pulling the connection angles up tightly against the web of the supporting beam prior to welding. Some engineers prefer to locate erection bolts below the mid-depth of the connection; theoretically, this provides the greatest possible flexibility near the top of the connection, where the angles are expected to flex away from the supporting member. However, this practice does not ensure a close fit-up of the angle before welding. Other engineers prefer the more general practice of spacing the bolts equally along the length of the angles. In this latter case, the bolts are placed as closely as practical to the toes of the outstanding leg to provide greater flexibility. Computer Software

CONXPRT is fully automated connection design software which provides for rapid design of economical simple shear connections. Based upon the AISC Manual of Steel Construction, Volume II—Connections and the engineering knowledge and experience of respected fabricators and design engineers, CONXPRT comes with preset guidelines, but can be modified to meet individual standards. It is menu-driven with a built-in shapes database and provides complete documentation of all design checks.



9 - 11

Double-Angle Connections

A double-angle connection is made with two angles, one on each side of the web of the beam to be supported, as illustrated in Figure 9-4. These angles may be bolted or welded to the supported beam as well as to the supporting member. When the angles are welded to the support, adequate flexibility must be provided in the connection. As illustrated in Figure 9-4c, line welds are placed along the toes of the angles with a return at the top per LRFD Specification Section J2.2b. Note that welding across the entire top of the angles must be avoided as it would inhibit the flexibility and, therefore, the necessary end rotation of the connection; the performance of the resulting connection is unpredictable.

(a) All-bolted

w

(b) Bolted/welded, angles welded to supported beam

w

2w Note: weld returns on top of angles per LRFD Specification Section J2.2b.

w

(c) Bolted/welded, angles welded to support Figure 9-4. Double-angle connections. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 12


Table 9-1. Fillet Encroachment Chart k − tf, in. 5⁄

k

tf encr.

1

/32 in.

tw

encr., in.

16

1⁄ 8

3⁄ 8

3⁄ 16

7⁄

16

3⁄ 16

1⁄ 2

3⁄ 16

9⁄

16

1⁄ 4

5⁄ 8

1⁄ 4

11⁄ 16

1⁄ 4

3⁄ 4

1⁄ 4

13⁄ 16

1⁄ 4

7⁄ 8

5⁄ 16

1

5⁄ 16

Design Checks

The design strengths of the bolts and/or welds and connected elements must be determined in accordance with the LRFD Specification; the applicable limit states are discussed in Part 8. In all cases, the design strength φRn must equal or exceed the required strength Ru. For usual gages of three inches and standard or short-slotted holes, eccentricity in double-angle connections may be neglected, except in the case of a double vertical row of bolts through the web of the supported beam, as illustrated in Figure 9-5. Eccentricity should always be considered in the design of welds for double-angle connections. Recommended Angle Length and Thickness

To provide for stability during erection, it is recommended that the minimum angle length be one-half the T-dimension of the beam to be supported. The maximum length of the connection angles must be compatible with the T-dimension of an uncoped beam and the remaining web depth, exclusive of fillets, of a coped beam. Note that the angle may encroach on the fillet or fillets by 1⁄8-in. to 5⁄16-in., depending upon the radius of the fillets; refer to Table 9-1. To provide for flexibility, the maximum angle thickness for use with usual gages should be limited to 5⁄8-in. Shop and Field Practices

Double-angle connections may be made to the webs of supporting girders and to the flanges of supporting columns. Because of bolting and welding clearances, double-angle connections may not be suitable for connections to the webs of W8 columns, unless gages are reduced or bolts are staggered, and may be impossible for W6 columns. When framing to a girder web, both angles are usually shop attached to the web of the supported beam. When framing to a column web, both angles may be shop attached to AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9 - 13

the supported beam or to the column web. In the latter case, the bottom flange of the supported beam is coped to allow knifed erection (the beam web is lowered into place between the angles from above). Knifed erection requires that a total erection clearance of about 1⁄8-in. be provided between the angles as illustrated in Figure 9-6a. For bolted construction, this clearance may vary as gages will occur in minimum increments of 1⁄ -in. Shims must be furnished whenever measured clearances exceed 1⁄ -in. 16 8 When framing to a column flange, provision must be made for possible mill variation in the depth of the columns. If both angles are shop attached to the beam web, the beam length could be shortened to provide for mill overrun and shims could be furnished at the appropriate intervals to fill the resulting gaps or to provide for mill underrun; in general, shims are not required except for fairly long runs (i.e., six or more bays of framing). If both angles are shop attached to the column flange, the erected beam is knifed into place and play in the open holes usually furnishes the necessary adjustment to compensate for the mill variation in the columns; short slots can also be used. Alternatively, in any of the aforementioned cases, one angle could be shop attached to the support and the other shipped loose. In this case, the spread between the outstanding legs should equal the decimal beam web thickness plus a clearance which will produce an opening to the next higher 1⁄16-in. increment, as illustrated in Figure 9-6b; short slots in the support-leg of the angle eliminate the need to provide for variations in web thickness. However, shipping one angle loose is not a desirable practice since it requires additional material handling as well as added erection costs and difficulty. All-Bolted Double-Angle Connections

Tables 9-2 are design aids for all-bolted double-angle connections. Design strengths are tabulated for supported and supporting member material, as well as angle material with Supporting member

g1

g3

E

g2

E

E indicates that eccentricity must be considered in this leg. Gages g1, g2, g 3 are usual gages as shown below Usual gages* in angle legs, in. Leg

8

7

6

5

4

31⁄2

3

21⁄2

2

13⁄4

11⁄2

13⁄8

11⁄4

1

g1 g2 g3

41⁄2 3 3

4 21⁄2 3

31⁄2 21⁄4 21⁄2

3 2 13⁄4

21⁄2

2

13⁄4

13⁄8

11⁄8

1

7⁄ 8

7⁄ 8

3⁄ 4

5⁄ 8

*Other gages are permitted to suit specific requirements subject to clearances and edge distance limitations.

Figure 9-5. Eccentricity in double-angle connections. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 14


Fy = 36 ksi and Fu = 58 ksi and with Fy = 50 ksi and Fu = 65 ksi. All values, including slip-critical bolt design strengths, are for comparison with factored loads. Tabulated bolt and angle design strengths consider the limit states of bolt shear, bolt bearing on the angles, shear yielding of the angles, shear rupture of the angles, and block shear rupture of the angles. Values are tabulated for 2 through 12 rows of 3⁄4-in., 7⁄8-in, and 1 in. diameter A325 and A490 bolts at 3 in. spacing. For calculation purposes, angle edge distances Lev and Leh are assumed to be 11⁄4-in. Tabulated beam web design strengths, per inch of web thickness, consider the limit state of bolt bearing on the beam web. For beams coped at the top flange only, the limit state of block shear rupture is also considered. Additionally, for beams coped at both the top and bottom flanges, the tabulated values consider the limit states of shear yielding gage

Provide approximately 1/8 in. erection clearance between angles; spread should be a multiple of 1/16 in.

(a) Both angles shop attached to the column flange (beam knifed into place)

gage

Provide erection clearance so that spread is the next larger multiple of 1/16 in. greater than the beam web thickness.

(b) One shop attached to the column flange, other shipped loose Figure 9-6. Double-angle connection erection clearances. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9 - 15

and shear rupture of the beam web. Values are tabulated for beam web edge distances Leh from 11⁄4-in. to 3 in. and for beam end distances Leh of 11⁄2-in. and 13⁄4-in.; for calculation purposes, these end distances have been reduced to 11⁄4-in. and 11⁄2-in., respectively, to account for possible underrun in beam length. For coped members, the limit states of flexural yielding and local buckling must be checked independently. These limit states are discussed in Part 8; web reinforcement of coped members is treated in this Part under “Special Considerations”. Tabulated supporting member design strengths, per inch of flange or web thickness, consider the limit state of bolt bearing on the support. Bolted/Welded Double-Angle Connections

Table 9-3 (see page 9-88) is a design aid arranged to permit substitution of welds for bolts in connections designed with Tables 9-2. Electrode strength is assumed to be 70 ksi. All values are for comparison with factored loads. Holes for erection bolts may be placed as required in angle legs that are to be field welded. Welds A may be used in place of bolts through the supported-beam-web legs of the double angles or welds B may be used in place of bolts through the support legs of the double angles. Although it is permissible to use welds A and B from Table 9-3 in combination to obtain all-welded connections, it is recommended that such connections be chosen from Table 9-4. This table will allow increased flexibility in selection of angle lengths and connection strengths since Table 9-3 conforms to the bolt spacing and edge distance requirements for the bolted double-angle connections of Tables 9-2. Weld design strengths are tabulated for the limit state of weld shear. Design strengths for welds A are determined by the instantaneous center of rotation method using Table 8-42 with θ = 0°. Design strengths for welds B are determined by the elastic method. With the neutral axis assumed at one-sixth the depth of the angles measured downward and the tops of the angles in compression against each other through the beam web, the design strength of these welds is φRn, where φRn = 2 ×

1.392DL

 √ 1+

12.96e2 L2

In the above equation, D is the number of sixteenths-of-an-inch in the weld size, L is the length of the connection angles, and e is the width of the leg of the connection angle attached to the support. The tabulated minimum thicknesses of the supported beam web for welds A and the support for welds B match the shear yielding strength of these elements with the strength of the weld metal. Given the design shear yielding strength per unit length from LRFD Specification Section J5.3 as 0.9(0.60Fy t) and the weld strength constant (unit length design strength per 1⁄16-in. weld size for 70 ksi electrodes) as 1.392 kips/in., the minimum supported beam web thickness for welds A (two lines of weld) is tmin =

D × 1.392 × 2 5.16D = Fy 0.9 × 0.60Fy

where D is the number of sixteenths in the weld size. Similarly for welds B (one line of weld) the minimum supporting flange or web thickness is AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 16

tmin =


2.58D

Fy

When welds line up on opposite sides of the support, the minimum thickness is the sum of the thicknesses required for each weld. In either case, when less than the minimum material thickness is present, the tabulated weld design strength must be reduced by the ratio of the thickness provided to the minimum thickness. The minimum angle thickness when Table 9-3 is used is the weld size plus 1⁄16-in. but not less than the angle thickness determined from Table 9-2. The angle length L must be as tabulated in Table 9-3. In general, 2L4×31⁄2 will accommodate usual gages, with the 4 in. leg attached to the supporting member. Width of web legs in Case I may be optionally reduced from 31⁄2-in. to 3 in. Width of outstanding legs in Case II may be optionally reduced from 4 in. to 3 in. for values of L from 51⁄2 through 171⁄2-in. All-Welded Double-Angle Connections

Table 9-4 (see page 9-89) is a design aid for all-welded double-angle connections. Electrode strength is assumed to be 70 ksi. All values are for comparison with factored loads. Holes for erection bolts may be placed as required in angle legs that are to be field welded. Weld design strengths are tabulated for the limit state of weld shear. Design strengths for welds A are determined by the instantaneous center of rotation method using Table 8-42 with θ =0°. Design strengths for welds B are determined by the elastic method as discussed previously for bolted/welded double-angle connections. The tabulated minimum thicknesses of the supported beam web for welds A and the support for welds B match the shear yielding strength of these elements with the strength of the weld metal and are determined as discussed previously for bolted/welded double angle connections. When welds line up on opposite sides of the support, the minimum thickness is the sum of the thicknesses required for each weld. When less than the minimum material thickness is present, the tabulated weld design strength must be reduced by the ratio of the thickness provided to the minimum thickness. The minimum angle thickness when Table 9-4 is used must be equal to the weld size plus 1⁄16-in. The angle length L must be as tabulated in Table 9-4. Use 2L4×3 for angle lengths greater than or equal to 18 in.; use 2L3×3 otherwise.

Example 9-1

Given:

Refer to Figure 9-7. Use Table 9-2 to design an all-bolted double-angle connection for the W18×50 beam to W21×62 girder web connection. Ru = 60 kips W18×50 tw = 0.355 in. d = 17.99 in. Fy = 50 ksi, Fu = 65 ksi top flange coped 2 in. deep by 4 in. long, Lev = 11⁄4-in., Leh = 13⁄4-in. (Assumed to be 11⁄2-in. for calculation purposes to account for possible underrun in beam lengths) AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9 - 17

W21×62 tw = 0.400 in. Fy = 50 ksi, Fu = 65 ksi Use 3⁄4-in. diameter A325-N bolts in standard holes. Assume angle material with Fy = 36 ksi and Fu = 58 ksi. Solution:

Design bolts and angles (refer to Part 8) From Table 9-2, for 3⁄4-in. diameter A325-N bolts and angle material with Fy = 36 ksi and Fu = 58 ksi, select three rows of bolts and 1⁄4-in. angle thickness. φRn = 76.7 kips > 60 kips o.k. Check supported beam web From Table 9-2, for three rows of bolts, beam material with Fy = 50 ksi and Fu = 65 ksi, and Lev = 11⁄4-in. and Leh = 13⁄4-in. (Assumed to be 11⁄2-in. for calculation purposes to account for possible underrun in beam lengths) φRn = (204 kips/in.)(0.355 in.) = 72.4 kips > 60 kips o.k. Check flexural yielding on the coped section (refer to Part 8) From Table 8-49, Snet = 23.4 in.3 φRn =

φFy Snet

e 0.9 (50 ksi) (23.4 in.3) = (4 in. + 1⁄2jin.) = 234 kips > 60 kips o.k.

Check local web buckling at the cope (refer to Part 8) c 4 in. = = 0.222 d 17.99 in. 4 in. c = = 0.250 ho (17.99 in. − 2 in.) c Since ≤ 1.0, d c f =2  d = 2(0.222) = 0.444 c Since ≤ 1.0, ho AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 18


1.65

 ho  k = 2.2   c

 1   = 2.2   0.250 

1.65

= 21.7 2

 tw  φFbc = 23,590   fk  ho  2

 0.355 in.  = 23,590   (0.444) (21.7)  17.99 in. − 2 in. = 112 ksi φFbc Snet φRn =

e (112 ksi) (23.4 in.3) = (4 in. + 1⁄2jin.) = 582 kips > 60 kips o.k.

Check supporting girder web From Table 9-2, for three rows of bolts and girder material with Fu = 65 ksi, φRn = (527 kips/in.)(0.400 in.) = 211 kips > 60 kips o.k. The connection, as summarized in Figure 9-7, is adequate.

Example 9-2

Given:

Refer to Figure 9-8. Use Table 9-2 to design an all-bolted double-angle connection for the W36×230 beam to W14×90 column-flange connection. Ru = 225 kips W36×230 tw = 0.760 in. Fy = 50 ksi, Fu = 65 ksi W14×90 tf = 0.710 in. Fy = 50 ksi, Fu = 65 ksi Use 3⁄4-in. diameter A325-N bolts in standard holes. Assume angle material with Fy = 36 ksi and Fu = 58 ksi.

Solution:

Design bolts and angles AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9 - 19

From Table 9-2, for 3⁄4-in. diameter A325-N bolts and angle material with Fy = 36 ksi and Fy = 58 ksi, select eight rows of bolts and 5⁄16-in. angle thickness. φRn = 254 kips > 225 kips o.k. Check supported beam web From Table 9-2, for eight rows of bolts, beam material with Fy = 50 ksi and Fu = 65 ksi, and Leh = 13⁄4-in., φRn = (702 kips/in.)(0.760 in.) = 534 kips > 225 kips o.k. Check supporting column flange From Table 9-2, for eight rows of bolts and column material with Fy = 50 ksi and Fu = 65 ksi, φRn = (1,404 kips/in.)(0.710 in.) = 997 kips 225 kips o.k.

Example 9-3

Given:

Refer to Example 9-1. Use Table 9-3 to substitute welds for bolts in the supported-beam-web legs of the double-angle connection (welds A).

Solution:

From Table 9-3, for three rows of bolts (an angle length of 81⁄2-in.), a 3⁄ -in. weld size provides φR = 110 kips. For beam web material with 16 n Fy = 50 ksi, the minimum web thickness is 0.31 in. Since tw = 0.355 in. > 0.31 in., no reduction in the tabulated value is required. φRn = 110 kips > 60 kips o.k. Check minimum angle thickness The minimum angle thickness for Table 9-3 is the weld size plus 1⁄16-in., but not less than the thickness determined from Table 9-2. tmin = 3⁄16-in. + 1⁄16-in. = 1⁄4-in. This thickness is equal to the thickness chosen previously from Table 9-2.

Example 9-4

Given:

Refer to Example 9-2. Use Table 9-3 to substitute welds for bolts in the support legs of the double-angle connection (welds B).

Solution:

From Table 9-3, for eight rows of bolts (an angle length of 231⁄2-in.), a 5⁄ -in. weld size provides φR = 279 kips. For beam web material with 16 n AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 20


Fy = 50 ksi, the minimum column flange thickness is 0.26 in. Since tf = 0.710 in. > 0.26 in., no reduction of the tabulated value is required. φRn = 279 kips > 225 kips o.k. Check minimum angle thickness The minimum angle thickness for Table 9-3 is the weld size plus 1⁄16-in., but not less than the thickness determined from Table 9-2. tmin = 5⁄16-in. + 1⁄16-in. = 3⁄8-in. Thus, the angle thickness must be increased to 3⁄8-in. to accommodate the welded legs of the double-angle connection. Example 9-5

Given:

Refer to Example 9-2. Use Table 9-4 to design an all-welded doubleangle connection for the W36×230 beam to W14×90 column-flange connection.

Solution:

Design supported-beam-web angle leg welds (welds A) From Table 9-4, for L = 24 in., a 3⁄16-in. weld A size provides φRn = 259 kips. For beam web material with Fy = 50 ksi, the minimum supported beam web thickness is 0.31 in. Since tw = 0.760 in. > 0.31 in., no reduction of the tabulated value is required. φRn = 259 kips > 225 kips o.k. Design support angle leg welds (welds B) From Table 9-4, for L = 24 in., a 1⁄4-in. weld B size provides φRn = 229 kips. For column flange material with Fy = 50 ksi, the minimum column flange thickness is 0.21 in. Since tf = 0.710 in. > 0.21 in., no reduction of the tabulated value is required. Check minimum angle thickness The minimum angle thickness for Table 9-4 is the weld size plus 1⁄16-in. tmin = 1⁄4-in. + 1⁄16-in. = 5⁄16-in. Use 2L4×3×5⁄16.



9 - 21

¾φ A325-N

/16*

¼ 1 3/8 max.

11/4 min.

W18×50

11/16

2 15/16

A

3

61/8 min. (use 6¼)

L ev = 1¼

3 3 3¼ 2

2¼ L eh = 1¾

2L 4×3½×¼×8½ (SLBB)

Section at A * This dimension (see sketch, section at A) is determined to be one-half of the decimal web thickness rounded to the next higher 1/16 in. Example: 0.355/2 = 0.1775; use 3/16 in. This will produce spacing of holes in the supporting beam slightly larger than detailed in the angles to per mit spreading of angles (angles can be spread but not closed) at time of erection to supporting member. Alternatively, consider using horizontal slots in the support legs of the angles. Fig. 9-7.

2½

¾φ A325-N /16 *

/16

7

2L 5×3× 5/16 ×1″-1½

Section at B * This dimension is one-half decimal web thickness rounded to the next higher 1/16 in., as in example 9-1. Fig. 9-8. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

1 3/8 max.

1¼ min.

W36×230

3 9/16

5

B

1 7/16

7@3 = 1 ″-9

1¼

6¾ min. (use 8)

1¾

9 - 22


Fy = 36 ksi Fu = 58 ksi

Table 9-2. All-Bolted Double-Angle Connections

3⁄ -in. 4

Bolts

Bolt and Angle Design Strength, kips

12 Rows

ASTM Thread Desig. Cond.

W44

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

326

382

382

382

X

—

326

408

477

477

SC

STD

251

251

251

251

OVS

213

213

213

213

SSLT

213

213

213

213

STD

326

380

380

380

OVS

307

323

323

323

SSLT

323

323

323

323

N

—

326

408

477

477

X

—

326

408

489

596

SC Class A

STD

313

313

313

313

OVS

266

266

266

266

SSLT

266

266

266

266

STD

326

408

475

475

OVS

307

383

403

403

SSLT

326

403

403

403

A325 Varies

t

11@3 = 33

Class A

SC Class B 2 1/ 4

L eh

Angle Thickness, in.

Hole Type

11@3 = 33

Lev

A490

Lev

SC Class B

Beam Web Design Strength per Inch Thickness, kips/in. Coped at Top Flange Only Hole Leh,* UnType in. coped 11⁄4 STD OVS SSLT

Coped at Both Flanges

Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

940

665

668

672

675

685

711

653

659

666

672

685

711

13⁄4

940

672

675

678

682

691

717

653

659

666

672

691

717

11⁄2

940

628

631

634

637

647

673

613

620

626

633

647

673

13⁄4

940

634

638

641

644

654

680

613

620

626

633

653

680

11⁄2

940

665

668

672

675

685

711

653

659

666

672

685

711

13⁄4

940

672

675

678

682

691

717

653

659

666

672

691

717

Notes: Support Design STD = Standard holes Strength per Inch OVS = Oversized holes Thickness, kips/in. SSLT = Short-slotted holes transverse to direction of load

1879

N = Threads included X = Threads excluded SC = Slip critical

*Tabulated values include 1⁄4-in. reduction in end distance Leh to account f or possible underrun in beam length



9 - 23


Table 9-2 (cont.). All-Bolted Double-Angle Connections 3⁄ -in 4

Bolts


12 Rows


W44

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

366

382

382

382

X

—

366

457

477

477

SC

STD

251

251

251

251

OVS

213

213

213

213

SSLT

213

213

213

213

STD

366

380

380

380

OVS

323

323

323

323

SSLT

323

323

323

323

N

—

366

457

477

477

X

—

366

457

548

596

SC

STD

313

313

313

313

OVS

266

266

266

266

SSLT

266

266

266

266

STD

366

457

475

475

OVS

344

403

403

403

SSLT

366

403

403

403

11@3 = 33

Class A

SC Class B 2 1/ 4

L eh Lev

A490


Hole Type

11@3 = 33

Class A

SC Lev

Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

1053

754

758

762

765

776

806

731

739

746

753

775

806

13⁄4

1053

764

767

771

775

786

815

731

739

746

753

775

815

11⁄

2

1053

712

716

720

723

734

764

687

695

702

709

731

764

13⁄4

1053

722

725

729

733

744

773

687

695

702

709

731

773

11⁄2

1053

754

758

762

765

776

806

731

739

746

753

775

806

13⁄

1053

764

767

771

775

786

815

731

739

746

753

775

815

4


2106


*Tabulated values include 1⁄4-in. reduction in end distance Leh to account for possible underrun in beam length.


9 - 24



Table 9-2 (cont.). All-Bolted Double-Angle Connections

3⁄ -in. 4

Bolts


11 Rows


W44, 40

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

299

350

350

350

X

—

299

373

437

437

SC

STD

230

230

230

230

OVS

195

195

195

195

SSLT

195

195

195

195

STD

299

348

348

348

OVS

281

296

296

296

SSLT

296

296

296

296

N

—

299

373

437

437

X

—

299

373

448

547

SC Class A

STD

287

287

287

287

OVS

244

244

244

244

SSLT

244

244

244

244

STD

299

373

435

435

OVS

281

351

370

370

SSLT

299

370

370

370

A325 Varies

t

Class A 10@3 = 30


Hole Type

SC Class B 2 1/ 4

L eh

10@3 = 30

Lev

A490

Lev

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

861

610

613

616

619

629

655

597

604

610

617

629

655

13⁄4

861

616

620

623

626

636

662

597

604

610

617

636

662

11⁄2

861

575

579

582

585

595

621

561

568

574

581

595

621

13⁄4

861

582

585

589

592

602

628

561

568

574

581

600

628

11⁄2

861

610

613

616

619

629

655

597

604

610

617

629

655

13⁄4

861

616

620

623

626

636

662

597

604

610

617

636

662


1723





9 - 25


Table 9-2 (cont.). All-Bolted Double-Angle Connections 3⁄ -in. 4

Bolts


11 Rows


W44, 40

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

335

350

350

350

X

—

335

418

437

437

SC

STD

230

230

230

230

OVS

195

195

195

195

SSLT

195

195

195

195

STD

335

348

348

348

OVS

296

296

296

296

SSLT

296

296

296

296

N

—

335

418

437

437

X

—

335

418

502

547

SC

STD

287

287

287

287

OVS

244

244

244

244

SSLT

244

244

244

244

STD

335

418

435

435

OVS

314

370

370

370

SSLT

335

370

370

370

Class A 10@3 = 30


Hole Type

SC Class B 2 1/ 4

L eh

10@3 = 30

Lev

A490

Class A

Lev

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

965

692

696

700

703

714

743

669

676

684

691

713

743

13⁄4

965

702

705

709

713

724

753

669

676

684

691

713

753

11⁄

2

965

654

657

661

665

676

705

629

636

644

651

673

705

13⁄4

965

663

667

671

674

685

714

629

636

644

651

673

714

11⁄2

965

692

696

700

703

714

743

669

676

684

691

713

743

13⁄

965

702

705

709

713

724

753

669

676

684

691

713

753

4


1931




9 - 26




3⁄ -in. 4

Bolt


10 Rows


W44, 40, 36

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

271

318

318

318

X

—

271

338

398

398

SC

STD

209

209

209

209

OVS

178

178

178

178

SSLT

178

178

178

178

STD

271

316

316

316

OVS

254

269

269

269

SSLT

269

269

269

269

N

—

271

338

398

398

X

—

271

338

406

497

SC Class A

STD

261

261

261

261

OVS

222

222

222

222

SSLT

222

222

222

222

STD

271

338

396

396

OVS

254

318

336

336

SSLT

271

336

336

336

A325 Varies

t

Class A 9@3 = 27


Hole Type

SC Class B 2 1/ 4

Lev

L eh

Lev

9@3 = 27

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

783

554

557

561

564

574

600

542

548

555

561

574

600

13⁄4

783

561

564

567

571

580

607

542

548

555

561

580

607

11⁄2

783

523

526

530

533

543

569

509

515

522

529

543

569

13⁄4

783

530

533

536

540

549

576

509

515

522

529

548

576

11⁄2

783

554

557

561

564

574

600

542

548

555

561

574

600

13⁄4

783

561

564

567

571

580

607

542

548

555

561

580

607


1566





9 - 27



Bolts


10 Rows


W44, 40, 36

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

303

318

318

318

X

—

303

379

398

398

SC

STD

209

209

209

209

OVS

178

178

178

178

SSLT

178

178

178

178

STD

303

316

316

316

OVS

269

269

269

269

SSLT

269

269

269

269

N

—

303

379

398

398

X

—

303

379

455

497

SC

STD

261

261

261

261

OVS

222

222

222

222

SSLT

222

222

222

222

STD

303

379

396

396

OVS

285

336

336

336

SSLT

303

336

336

336

Class A 9@3 = 27


Hole Type

SC Class B 2 1/ 4

Lev

L eh

9@3 = 27

A490

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

878

630

634

637

641

652

681

607

614

622

629

651

681

13⁄4

878

639

643

647

650

661

691

607

614

622

629

651

691

11⁄

2

878

595

599

603

606

617

647

570

578

585

592

614

647

13⁄4

878

605

608

612

616

627

656

570

578

585

592

614

656

11⁄2

878

630

634

637

641

652

681

607

614

622

629

651

681

13⁄

878

639

643

647

650

661

691

607

614

622

629

651

691

4


1755




9 - 28




3⁄ -in. 4

Bolts


9 Rows


W44, 40, 36, 33

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

243

286

286

286

X

—

243

304

358

358

SC

STD

188

188

188

188

OVS

160

160

160

160

SSLT

160

160

160

160

STD

243

285

285

285

OVS

228

242

242

242

SSLT

242

242

242

242

N

—

243

304

358

358

X

—

243

304

365

447

SC Class A

STD

235

235

235

235

OVS

200

200

200

200

SSLT

200

200

200

200

STD

243

304

356

356

OVS

228

285

303

303

SSLT

243

303

303

303

A325 Varies

t

8@3 = 24

Class A

SC Class B

2 1/ 4

Lev

L eh

A490

Lev

8@3 = 24


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

705

499

502

505

508

518

544

486

493

499

506

518

544

13⁄4

705

505

509

512

515

525

551

486

493

499

506

525

551

11⁄2

705

471

474

477

481

491

517

457

463

470

476

491

517

13⁄4

705

478

481

484

487

497

523

457

463

470

476

496

523

11⁄2

705

499

502

505

508

518

544

486

493

499

506

518

544

13⁄4

705

505

509

512

515

525

551

486

493

499

506

525

551


1409





9 - 29



Bolts


9 Rows


W44, 40, 36, 33

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

272

286

286

286

X

—

272

340

358

358

SC

STD

188

188

188

188

OVS

160

160

160

160

SSLT

160

160

160

160

STD

272

285

285

285

OVS

242

242

242

242

SSLT

242

242

242

242

N

—

272

340

358

358

X

—

272

340

409

447

SC

STD

235

235

235

235

OVS

200

200

200

200

SSLT

200

200

200

200

STD

272

340

356

356

OVS

256

303

303

303

SSLT

272

303

303

303

8@3 = 24

Class A

SC Class B

2 1/ 4

Lev

L eh

8@3 = 24

A490


Hole Type

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

790

568

572

575

579

590

619

545

552

559

567

589

619

13⁄4

790

577

581

585

588

599

628

545

552

559

567

589

628

11⁄

2

790

537

540

544

548

559

588

512

519

527

534

556

588

13⁄4

790

546

550

554

557

568

597

512

519

527

534

556

597

11⁄2

790

568

572

575

579

590

619

545

552

559

567

589

619

13⁄

790

577

581

585

588

599

628

545

552

559

567

589

628

4


1580




9 - 30




3⁄ -in. 4

Bolts


8 Rows


W44, 40, 36, 33, 30

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

215

254

254

254

X

—

215

269

318

318

SC

STD

167

167

167

167

OVS

142

142

142

142

SSLT

142

142

142

142

STD

215

253

253

253

OVS

202

215

215

215

SSLT

215

215

215

215

N

—

215

269

318

318

X

—

215

269

323

398

SC Class A

STD

209

209

209

209

OVS

178

178

178

178

SSLT

178

178

178

178

STD

215

269

316

316

OVS

202

253

269

269

SSLT

215

269

269

269

A325 Varies

t

7@3 = 21

Class A

SC Class B

2 1/ 4

Lev

L eh

7@3 = 21

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

626

443

446

450

453

463

489

431

437

444

450

463

489

13⁄4

626

450

453

456

460

470

496

431

437

444

450

470

496

11⁄2

626

419

422

425

429

438

464

405

411

418

424

438

464

13⁄4

626

425

429

432

435

445

471

405

411

418

424

444

471

11⁄2

626

443

446

450

453

463

489

431

437

444

450

463

489

13⁄4

626

450

453

456

460

470

496

431

437

444

450

470

496


1253





9 - 31



Bolts


8 Rows


W44, 40, 36, 33, 30

A325 Varies


Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

241

254

254

254

X

—

241

302

318

318

SC

STD

167

167

167

167

OVS

142

142

142

142

SSLT

142

142

142

142

STD

241

253

253

253

OVS

215

215

215

215

SSLT

215

215

215

215

N

—

241

302

318

318

X

—

241

302

362

398

SC

STD

209

209

209

209

OVS

178

178

178

178

SSLT

178

178

178

178

STD

241

302

316

316

OVS

227

269

269

269

SSLT

241

269

269

269

t

7@3 = 21

Class A

SC 2 1/ 4

Class B

Lev

L eh

7@3 = 21

A490

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

702

506

509

513

517

528

557

483

490

497

505

527

557

13⁄4

702

515

519

522

526

537

566

483

490

497

505

527

566

11⁄

2

702

478

482

486

489

500

530

453

461

468

475

497

530

13⁄4

702

488

491

495

499

510

539

453

461

468

475

497

539

11⁄2

702

506

509

513

517

528

557

483

490

497

505

527

557

13⁄

702

515

519

522

526

537

566

483

490

497

505

527

566

4


1404




9 - 32




3⁄ -in. 4

Bolts


7 Rows W44, 40, 36, 33, 30, 27, 24 S24

Varies


1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

188

223

223

223

X

—

188

234

278

278

SC

STD

146

146

146

146

OVS

124

124

124

124

SSLT

124

124

124

124

STD

188

221

221

221

OVS

176

188

188

188

SSLT

188

188

188

188

N

—

188

234

278

278

X

—

188

234

281

348

SC Class A

STD

183

183

183

183

OVS

155

155

155

155

SSLT

155

155

155

155

STD

188

234

277

277

OVS

176

220

235

235

SSLT

188

234

235

235

A325

t

6@3 = 18

Class A

SC Class B

2 1/ 4


Hole Type

Lev

L eh

Lev

6@3 = 18

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

548

388

391

394

398

407

433

375

382

388

395

407

433

13⁄4

548

394

398

401

404

414

440

375

382

388

395

414

440

11⁄2

548

367

370

373

376

386

412

352

359

365

372

386

412

13⁄4

548

373

377

380

383

393

419

352

359

365

372

392

419

11⁄2

548

388

391

394

398

407

433

375

382

388

395

407

433

13⁄4

548

394

398

401

404

414

440

375

382

388

395

414

440


1096





9 - 33



Bolts


7 Rows W44, 40, 36, 33, 30, 27, 24 S24

Varies

ASTM Thread Desig. Cond. A325

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

210

223

223

223

X

—

210

263

278

278

SC

STD

146

146

146

146

OVS

124

124

124

124

SSLT

124

124

124

124

STD

210

221

221

221

OVS

188

188

188

188

SSLT

188

188

188

188

N

—

210

263

278

278

X

—

210

263

315

348

SC

STD

183

183

183

183

OVS

155

155

155

155

SSLT

155

155

155

155

STD

210

263

277

277

OVS

197

235

235

235

SSLT

210

235

235

235

6@3 = 18

Class A

SC Class B

2 1/ 4


Hole Type

Lev

L eh

6@3 = 18

A490

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

614

444

447

451

455

466

495

420

428

435

442

464

495

13⁄4

614

453

457

460

464

475

504

420

428

435

442

464

504

11⁄

2

614

420

423

427

431

442

471

395

402

410

417

439

471

13⁄4

614

429

433

437

440

451

480

395

402

410

417

439

480

11⁄2

614

444

447

451

455

466

495

420

428

435

442

464

495

13⁄

614

453

457

460

464

475

504

420

428

435

442

464

504

4


1229




9 - 34




3⁄ -in. 4

Bolts


6 Rows W44, 40, 36, 33, 30, 27, 24, 21 S24

Varies


1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

160

191

191

191

X

—

160

200

239

239

SC

STD

125

125

125

125

OVS

107

107

107

107

SSLT

107

107

107

107

STD

160

190

190

190

OVS

150

161

161

161

SSLT

160

161

161

161

N

—

160

200

239

239

X

—

160

200

240

298

SC Class A

STD

157

157

157

157

OVS

133

133

133

133

SSLT

133

133

133

133

STD

160

200

237

237

OVS

150

188

202

202

SSLT

160

200

202

202

A325

t

5@3 = 15

Class A

SC Class B

2 1/ 4

5@3 = 15

Lev

L eh

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

470

332

336

339

342

352

378

320

326

333

339

352

378

13⁄4

470

339

342

346

349

359

385

320

326

333

339

359

385

11⁄2

470

314

318

321

324

334

360

300

307

313

320

334

360

13⁄4

470

321

324

328

331

341

367

300

307

313

320

339

367

11⁄2

470

332

336

339

342

352

378

320

326

333

339

352

378

13⁄4

470

339

342

346

349

359

385

320

326

333

339

359

385


940





9 - 35



Bolts


6 Rows W44, 40, 36, 33, 30, 27, 24, 21 S24

Varies


t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

179

191

191

191

X

—

179

224

239

239

SC

STD

125

125

125

125

OVS

107

107

107

107

SSLT

107

107

107

107

STD

179

190

190

190

OVS

161

161

161

161

SSLT

161

161

161

161

N

—

179

224

239

239

X

—

179

224

269

298

SC

STD

157

157

157

157

OVS

133

133

133

133

SSLT

133

133

133

133

STD

179

224

237

237

OVS

168

202

202

202

SSLT

179

202

202

202

5@3 = 15

Class A

SC 2 1/ 4

Class B

5@3 = 15

Lev

L eh

Lev

A490


Hole Type

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

527

381

385

389

392

403

433

358

366

373

380

402

433

13⁄4

527

391

394

398

402

413

442

358

366

373

380

402

442

11⁄

2

527

361

365

369

372

383

413

336

344

351

358

380

413

13⁄4

527

371

374

378

382

393

422

336

344

351

358

380

422

11⁄2

527

381

385

389

392

403

433

358

366

373

380

402

433

13⁄

527

391

394

398

402

413

442

358

366

373

380

402

442

4


1053




9 - 36




3⁄ -in. 4

Bolts


5 Rows


W30, 27, 24, 21, 18 S24, 20, 18 MC18

Varies

A325

4@3 = 12

1⁄ 2

159

159

159

X

—

132

165

198

199

SC

STD

104

104

104

104

OVS

88.8

88.8

88.8

88.8

SSLT

88.8

88.8

88.8

88.8

STD

132

158

158

158

OVS

124

134

134

134

SSLT

132

134

134

134

N

—

132

165

198

199

X

—

132

165

198

249

SC Class A

STD

131

131

131

131

OVS

111

111

111

111

SSLT

111

111

111

111

STD

132

165

198

198

OVS

124

155

168

168

SSLT

132

165

168

168

Lev

A490

4@3 = 12

3⁄ 8

132

L eh

L ev

5⁄ 16

—

SC Class B

2 1/ 4

1⁄ 4

N

Class A

t


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

392

277

280

283

287

296

322

264

271

277

284

296

322

13⁄4

392

284

287

290

293

303

329

264

271

277

284

303

329

11⁄2

392

262

265

269

272

282

308

248

254

261

268

282

308

13⁄4

392

269

272

275

279

288

315

248

254

261

268

287

315

11⁄2

392

277

280

283

287

296

322

264

271

277

284

296

322

13⁄4

392

284

287

290

293

303

329

264

271

277

284

303

329


783





9 - 37



Bolts


5 Rows


W30, 27, 24, 21, 18 S24, 20, 18 MC18

Varies

A325

159

159

X

—

148

185

199

199

SC

STD

104

104

104

104

4@3 = 12

OVS

88.8

88.8

88.8

88.8

SSLT

88.8

88.8

88.8

88.8

STD

148

158

158

158

OVS

134

134

134

134

SSLT

134

134

134

134

N

—

148

185

199

199

X

—

148

185

222

249

SC

STD

131

131

131

131

OVS

111

111

111

111

SSLT

111

111

111

111

STD

148

185

198

198

OVS

139

168

168

168

SSLT

148

168

168

168

Lev 4@3 = 12

1⁄ 2

159

L eh

L ev

3⁄ 8

148

SC

A490

5⁄ 16

—

Class B

2 1/ 4

1⁄ 4

N

Class A

t


Hole Type

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

439

319

323

327

330

341

370

296

303

311

318

340

370

13⁄4

439

329

332

336

340

351

380

296

303

311

318

340

380

11⁄

2

439

303

306

310

314

325

354

278

285

293

300

322

354

13⁄4

439

312

316

320

323

334

363

278

285

293

300

322

363

11⁄2

439

319

323

327

330

341

370

296

303

311

318

340

370

13⁄

439

329

332

336

340

351

380

296

303

311

318

340

380

4


878




9 - 38




3⁄ -in. 4

Bolts


4 Rows


W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18

A325

3@3 = 9

1⁄ 2

127

127

127

X

—

104

131

157

159

SC

STD

83.5

83.5

83.5

83.5

OVS

71.0

71.0

71.0

71.0

SSLT

71.0

71.0

71.0

71.0

L eh

STD OVS

104 97.9

127

127

127

108

108

108

104

108

108

108

N

—

104

131

157

159

X

—

104

131

157

199

SC Class A

STD

104

104

104

104

L ev

SSLT A490

3@3 = 9

3⁄ 8

104

SC Class B

L ev

5⁄ 16

—

t

2 1/ 4

1⁄ 4

N

Class A Varies


Hole Type

SC Class B

OVS

88.8

88.8

88.8

88.8

SSLT

88.8

88.8

88.8

88.8

STD OVS SSLT

104 97.9 104

131

157

158

122

134

134

131

134

134



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

313

221

225

228

231

241

267

209

215

222

228

241

267

13⁄4

313

228

231

235

238

248

274

209

215

222

228

248

274

11⁄2

313

210

213

216

220

230

256

196

202

209

215

230

256

13⁄4

313

217

220

223

226

236

262

196

202

209

215

235

262

11⁄2

313

221

225

228

231

241

267

209

215

222

228

241

267

13⁄4

313

228

231

235

238

248

274

209

215

222

228

248

274


626





9 - 39



Bolts


4 Rows


W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

117

127

127

127

X

—

117

146

159

159

SC

STD

83.5

83.5

83.5

83.5

OVS

71.0

71.0

71.0

71.0

SSLT

71.0

71.0

71.0

71.0

Class A Varies


Hole Type

3@3 = 9

t

SC

STD

117

127

127

127

OVS

108

108

108

108

SSLT

108

108

108

108

N

—

117

146

159

159

X

—

117

146

176

199

SC

STD

104

104

104

104

Class B

2 1/ 4

L ev

L eh

L ev

3@3 = 9

A490

Class A

SC Class B

OVS

88.8

88.8

88.8

88.8

SSLT

88.8

88.8

88.8

88.8

STD

117

146

158

158

OVS

110

134

134

134

SSLT

117

134

134

134


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

351

257

261

264

268

279

308

234

241

249

256

278

308

13⁄4

351

266

270

274

277

288

318

234

241

249

256

278

318

11⁄

2

351

244

248

252

255

266

296

219

227

234

241

263

296

13⁄4

351

254

257

261

265

276

305

219

227

234

241

263

305

11⁄2

351

257

261

264

268

279

308

234

241

249

256

278

308

13⁄

351

266

270

274

277

288

318

234

241

249

256

278

318

4


702




9 - 40




3⁄ -in. 4

Bolts


3 Rows


W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 MC18, 13, 12

A325

*Limited to W10×12, 15, 17, 19, 22, 26, 30.

Varies

Lev

95.4

95.4

X

—

76.7

95.8

SC

STD

62.7

62.6

62.6

62.6

OVS

53.3

53.3

53.3

53.3

SSLT

53.3

53.3

53.3

53.3

STD

76.7

94.9

94.9

94.9

OVS

71.8

80.7

80.7

80.7

SSLT

76.7

80.7

80.7

80.7

N

—

76.7

95.8

115

119

X

—

76.7

95.8

115

149

SC Class A

STD

76.7

78.3

78.3

78.3

OVS

66.6

66.6

66.6

66.6

SSLT

66.6

66.6

66.6

66.6

STD

76.7

95.8

115

119

OVS

71.8

89.7

101

101

SSLT

76.7

95.8

101

101

A490

Lev

1⁄ 2

95.4

2 /4

3 3

3⁄ 8

76.7

1

L eh

5⁄ 16

—

SC Class B

3 3

1⁄ 4

N

Class A

t


Hole Type

SC Class B

115

119



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

235

166

169

172

176

185

212

153

160

166

173

185

212

13⁄4

235

173

176

179

182

192

218

153

160

166

173

192

218

11⁄2

235

158

161

164

168

177

203

144

150

157

163

177

203

13⁄4

235

164

168

171

174

184

210

144

150

157

163

183

210

11⁄2

235

166

169

172

176

185

212

153

160

166

173

185

212

13⁄4

235

173

176

179

182

192

218

153

160

166

173

192

218


470





9 - 41



Bolts


3 Rows


W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 MC18, 13, 12

A325

*Limited to W10×12, 15, 17, 19, 22, 26, 30.

Varies

Lev Lev

95.4

95.4

95.4

—

85.9

SC

STD

62.6

62.6

62.6

62.6

OVS

53.3

53.3

53.3

53.3

SSLT

53.3

53.3

53.3

53.3

STD

85.9

94.9

94.9

94.9

OVS

80.4

80.7

80.7

80.7

SSLT

80.7

80.7

80.7

80.7

N

—

85.9

107

119

119

X

—

85.9

107

129

149

SC

STD

78.3

78.3

78.3

78.3

OVS

66.6

66.6

66.6

66.6

SSLT

66.6

66.6

66.6

66.6

STD

85.9

107

119

119

OVS

80.4

101

101

101

SSLT

85.9

101

101

101

2 /4

3 3

1⁄ 2

X

1

A490

3⁄ 8

85.9

SC

L eh

5⁄ 16

—

Class B

3 3

1⁄ 4

N

Class A

t


Hole Type

Class A

SC Class B

107

119

119


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

263

195

199

202

206

217

246

172

179

186

194

216

246

13⁄4

263

204

208

212

215

226

256

172

179

186

194

216

256

11⁄

2

263

186

189

193

197

208

237

161

168

176

183

205

237

13⁄4

263

195

199

203

206

217

246

161

168

176

183

205

246

11⁄2

263

195

199

202

206

217

246

172

179

186

194

216

246

13⁄

263

204

208

212

215

226

256

172

179

186

194

216

256

4


527




9 - 42




3⁄ -in. 4

Bolts


2 Rows


W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

48.9

61.2

63.6

63.6

X

—

48.9

61.2

73.4

79.5

SC

STD

41.8

41.8

41.8

41.8

OVS

35.5

35.5

35.5

35.5

SSLT

35.5

35.5

35.5

35.5

STD

48.9

61.2

63.3

63.3

OVS

45.7

53.8

53.8

53.8

SSLT

48.9

53.8

53.8

53.8

N

—

48.9

61.2

73.4

79.5

X

—

48.9

61.2

73.4

97.9

SC Class A

STD

48.9

52.2

52.2

52.2

OVS

44.4

44.4

44.4

44.4

SSLT

44.4

44.4

44.4

44.4

STD

48.9

61.2

73.4

79.1

OVS

45.7

57.1

67.2

67.2

SSLT

48.9

61.2

67.2

67.2

A325

Class A Varies


Hole Type

t

3

SC Class B

2 1/ 4

Lev

L eh

3

Lev

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

157

110

114

117

120

130

156

97.9 104

111

117

130

156

13⁄4

157

117

120

124

127

137

157

97.9 104

111

117

136

157

11⁄2

157

106

109

112

115

125

151

91.4

97.9 104

111

125

151

13⁄4

157

112

116

119

122

132

157

91.4

97.9 104

111

131

157

11⁄2

157

110

114

117

120

130

156

97.9 104

111

117

130

156

13⁄4

157

117

120

124

127

137

157

97.9 104

111

117

136

157


313





9 - 43



Bolts


2 Rows


W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

54.8

63.6

63.6

63.6

X

—

54.8

68.6

79.5

79.5

SC

STD

41.8

41.8

41.8

41.8

OVS

35.5

35.5

35.5

35.5

SSLT

35.5

35.5

35.5

35.5

STD

54.8

63.3

63.3

63.3

OVS

51.2

53.8

53.8

53.8

SSLT

53.8

53.8

53.8

53.8

N

—

54.8

68.6

79.5

79.5

X

—

54.8

68.6

82.3

99.4

SC

STD

52.2

52.2

52.2

52.2

OVS

44.4

44.4

44.4

44.4

SSLT

44.4

44.4

44.4

44.4

STD

54.8

68.6

79.1

79.1

OVS

51.2

64.0

67.2

67.2

SSLT

54.8

67.2

67.2

67.2

Class A Varies


Hole Type

t

3

SC

2 1/ 4

Class B

Lev

L eh

3

Lev

A490

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

176

133

136

140

144

155

176

110

117

124

132

154

176

13⁄4

176

142

146

149

153

164

176

110

117

124

132

154

176

11⁄

2

176

127

131

135

138

149

176

102

110

117

124

146

176

13⁄4

176

137

140

144

148

159

176

102

110

117

124

146

176

11⁄2

176

133

136

140

144

155

176

110

117

124

132

154

176

13⁄

176

142

146

149

153

164

176

110

117

124

132

154

176

4


351




9 - 44




7⁄ -in. 8

Bolts


12 Rows


W44

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

307

383

460

520

X

—

307

383

460

613

SC

STD

307

349

349

349

OVS

286

297

297

297

SSLT

297

297

297

297

STD

307

383

460

520

OVS

286

358

429

450

SSLT

307

383

450

450

N

—

307

383

460

613

X

—

307

383

460

613

SC Class A

STD

307

383

439

439

OVS

286

358

373

373

SSLT

307

373

373

373

STD

307

383

460

613

OVS

286

358

429

565

SSLT

307

383

460

565

A325 Varies

t

11@3 = 33

Class A

SC Class B 2 1/ 4

L eh


Hole Type

11@3 = 33

Lev

A490

Lev

SC Class B

Beam Web Design Strength per Inch Thickness, kips/in. Coped at Top Flange Only


Lev, in.

Lev, in.

Hole Type

Leh, Unin. coped 11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

STD

11⁄2

1096

628

631

634

637

647

673

613

620

626

633

647

673

13⁄4

1096

634

638

641

644

654

680

613

620

626

633

653

680

11⁄2

1096

589

592

595

598

608

634

573

579

586

592

608

634

13⁄4

1096

595

599

602

605

615

641

573

579

586

592

612

641

11⁄2

1096

628

631

634

637

647

673

613

620

626

633

647

673

13⁄4

1096

634

638

641

644

654

680

613

620

626

633

653

680

OVS SSLT


2192





9 - 45



Bolts


12 Rows


W44

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

344

430

516

520

X

—

344

430

516

649

SC

STD

344

349

349

349

OVS

297

297

297

297

SSLT

297

297

297

297

STD

344

430

516

520

OVS

321

401

450

450

SSLT

344

430

450

450

N

—

344

430

516

649

X

—

344

430

516

687

SC

STD

344

430

439

439

OVS

321

373

373

373

SSLT

344

373

373

373

STD

344

430

516

649

OVS

321

401

481

565

SSLT

344

430

516

565

11@3 = 33

Class A

SC Class B 2 1/ 4

L eh Lev

A490


Hole Type

11@3 = 33

Class A

SC Lev

Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

1229

712

716

720

723

734

764

687

695

702

709

731

764

13⁄4

1229

722

725

729

733

744

773

687

695

702

709

731

773

11⁄

2

1229

669

672

676

680

691

720

642

649

656

664

686

720

13⁄4

1229

678

682

685

689

700

729

642

649

656

664

686

729

11⁄2

1229

712

716

720

723

734

764

687

695

702

709

731

764

13⁄

1229

722

725

729

733

744

773

687

695

702

709

731

773

4


2457




9 - 46




7⁄ -in. 8

Bolts


11 Rows


W44, 40

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

281

351

421

476

X

—

281

351

421

561

SC

STD

281

320

320

320

OVS

262

272

272

272

SSLT

272

272

272

272

STD

281

351

421

476

OVS

262

327

393

412

SSLT

281

351

412

412

N

—

281

351

421

561

X

—

281

351

421

561

SC Class A

STD

281

351

402

402

OVS

262

327

342

342

SSLT

281

342

342

342

STD

281

351

421

561

OVS

262

327

393

518

SSLT

281

351

421

518

A325 Varies

t

Class A 10@3 = 30


Hole Type

SC Class B 2 1/ 4

L eh

10@3 = 30

Lev

A490

Lev

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1005

575

579

582

585

595

621

561

568

574

581

595

621

13⁄4

1005

582

585

589

592

602

628

561

568

574

581

600

628

11⁄2

1005

540

543

546

549

559

585

524

530

537

543

559

585

13⁄4

1005

546

550

553

556

566

592

524

530

537

543

563

592

11⁄2

1005

575

579

582

585

595

621

561

568

574

581

595

621

13⁄4

1005

582

585

589

592

602

628

561

568

574

581

600

628


2010





9 - 47



Bolts


11 Rows


W44, 40

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

314

393

472

476

X

—

314

393

472

595

SC

STD

314

320

320

320

OVS

272

272

272

272

SSLT

272

272

272

272

STD

314

393

472

476

OVS

294

367

412

412

SSLT

314

393

412

412

N

—

314

393

472

595

X

—

314

393

472

629

SC

STD

314

393

402

402

OVS

294

342

342

342

SSLT

314

342

342

342

STD

314

393

472

595

OVS

294

367

440

518

SSLT

314

393

472

518

Class A 10@3 = 30


Hole Type

SC Class B 2 1/ 4

L eh

10@3 = 30

Lev

A490

Class A

Lev

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

1126

654

657

661

665

676

705

629

636

644

651

673

705

13⁄4

1126

663

667

671

674

685

714

629

636

644

651

673

714

11⁄

2

1126

614

618

621

625

636

665

587

594

602

609

631

665

13⁄4

1126

623

627

631

634

645

674

587

594

602

609

631

674

11⁄2

1126

654

657

661

665

676

705

629

636

644

651

673

705

13⁄

1126

663

667

671

674

685

714

629

636

644

651

673

714

4


2252




9 - 48




7⁄ -in. 8

Bolts


10 Rows


W44, 40, 36

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

254

318

382

433

X

—

254

318

382

509

SC

STD

254

291

291

291

OVS

238

247

247

247

SSLT

247

247

247

247

STD

254

318

382

433

OVS

238

297

356

375

SSLT

254

318

375

375

N

—

254

318

382

509

X

—

254

318

382

509

SC Class A

STD

254

318

365

365

OVS

238

297

311

311

SSLT

254

311

311

311

STD

254

318

382

509

OVS

238

297

356

471

SSLT

254

318

382

471

A325 Varies

t

Class A 9@3 = 27


Hole Type

SC Class B 2 1/ 4

Lev

L eh

Lev

9@3 = 27

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

914

523

526

530

533

543

569

509

515

522

529

543

569

13⁄4

914

530

533

536

540

549

576

509

515

522

529

548

576

11⁄2

914

491

494

497

501

510

537

475

482

488

495

510

537

13⁄4

914

498

501

504

507

517

543

475

482

488

495

514

543

11⁄2

914

523

526

530

533

543

569

509

515

522

529

543

569

13⁄4

914

530

533

536

540

549

576

509

515

522

529

548

576


1827





9 - 49



Bolts


10 Rows


W44, 40, 36

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

285

356

428

433

X

—

285

356

428

541

SC

STD

285

291

291

291

OVS

247

247

247

247

SSLT

247

247

247

247

STD

285

356

428

433

OVS

266

333

375

375

SSLT

285

356

375

375

N

—

285

356

428

541

X

—

285

356

428

570

SC

STD

285

356

365

365

OVS

266

311

311

311

SSLT

285

311

311

311

STD

285

356

428

541

OVS

266

333

399

471

SSLT

285

356

428

471

Class A 9@3 = 27


Hole Type

SC Class B 2 1/ 4

Lev

L eh

9@3 = 27

A490

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

1024

595

599

603

606

617

647

570

578

585

592

614

647

13⁄4

1024

605

608

612

616

627

656

570

578

585

592

614

656

11⁄

2

1024

559

563

567

570

581

610

532

540

547

554

576

610

13⁄4

1024

569

572

576

580

591

620

532

540

547

554

576

620

11⁄2

1024

595

599

603

606

617

647

570

578

585

592

614

647

13⁄

1024

605

608

612

616

627

656

570

578

585

592

614

656

4


2048




9 - 50




7⁄ -in. 8

Bolts


9 Rows


W44, 40, 36, 33

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

228

285

343

390

X

—

228

285

343

457

SC

STD

228

262

262

262

OVS

213

223

223

223

SSLT

223

223

223

223

STD

228

285

343

390

OVS

213

266

320

337

SSLT

228

285

337

337

N

—

228

285

343

457

X

—

228

285

343

457

SC Class A

STD

228

285

329

329

OVS

213

266

280

280

SSLT

228

280

280

280

STD

228

285

343

457

OVS

213

266

320

424

SSLT

228

285

343

424

A325 Varies

t

8@3 = 24

Class A

SC Class B

2 1/ 4

Lev

L eh

A490

Lev

8@3 = 24


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

822

471

474

477

481

491

517

457

463

470

476

491

517

13⁄4

822

478

481

484

487

497

523

457

463

470

476

496

523

11⁄2

822

442

445

449

452

462

488

426

433

439

446

462

488

13⁄4

822

449

452

455

459

468

495

426

433

439

446

465

495

11⁄2

822

471

474

477

481

491

517

457

463

470

476

491

517

13⁄4

822

478

481

484

487

497

523

457

463

470

476

496

523


1644





9 - 51



Bolts


9 Rows


W44, 40, 36, 33

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

256

320

384

390

X

—

256

320

384

487

SC

STD

256

262

262

262

OVS

223

223

223

223

SSLT

223

223

223

223

STD

256

320

384

390

OVS

239

299

337

337

SSLT

256

320

337

337

N

—

256

320

384

487

X

—

256

320

384

512

SC

STD

256

320

329

329

OVS

239

280

280

280

SSLT

256

280

280

280

STD

256

320

384

487

OVS

239

299

358

424

SSLT

256

320

384

424

8@3 = 24

Class A

SC Class B

2 1/ 4

Lev

L eh

8@3 = 24

A490


Hole Type

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

921

537

540

544

548

559

588

512

519

527

534

556

588

13⁄4

921

546

550

554

557

568

597

512

519

527

534

556

597

11⁄

2

921

504

508

512

515

526

556

478

485

492

500

522

556

13⁄4

921

514

518

521

525

536

565

478

485

492

500

522

565

11⁄2

921

537

540

544

548

559

588

512

519

527

534

556

588

13⁄

921

546

550

554

557

568

597

512

519

527

534

556

597

4


1843




9 - 52




7⁄ -in. 8

Bolts


8 Rows


W44, 40, 36, 33, 30

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

202

253

303

346

X

—

202

253

303

405

SC

STD

202

233

233

233

OVS

189

198

198

198

SSLT

198

198

198

198

STD

202

253

303

346

OVS

189

236

283

300

SSLT

202

253

300

300

N

—

202

253

303

405

X

—

202

253

303

405

SC Class A

STD

202

253

292

292

OVS

189

236

249

249

SSLT

202

249

249

249

STD

202

253

303

405

OVS

189

236

283

377

SSLT

202

253

303

377

A325 Varies

t

7@3 = 21

Class A

SC Class B

2 1/ 4

Lev

L eh

7@3 = 21

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

731

419

422

425

429

438

464

405

411

418

424

438

464

13⁄4

731

425

429

432

435

445

471

405

411

418

424

444

471

11⁄2

731

393

397

400

403

413

439

377

384

390

397

413

439

13⁄4

731

400

403

407

410

420

446

377

384

390

397

417

446

11⁄2

731

419

422

425

429

438

464

405

411

418

424

438

464

13⁄4

731

425

429

432

435

445

471

405

411

418

424

444

471


1462





9 - 53



Bolts


8 Rows


W44, 40, 36, 33, 30

A325 Varies


Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

227

283

340

346

X

—

227

283

340

433

SC

STD

227

233

233

233

OVS

198

198

198

198

SSLT

198

198

198

198

STD

227

283

340

346

OVS

211

264

300

300

SSLT

227

283

300

300

N

—

227

283

340

433

X

—

227

283

340

453

SC

STD

227

283

292

292

OVS

211

249

249

249

SSLT

227

249

249

249

STD

227

283

340

433

OVS

211

264

317

377

SSLT

227

283

340

377

t

7@3 = 21

Class A

SC 2 1/ 4

Class B

Lev

L eh

7@3 = 21

A490

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

819

478

482

486

489

500

530

453

461

468

475

497

530

13⁄4

819

488

491

495

499

510

539

453

461

468

475

497

539

11⁄

2

819

450

453

457

461

472

501

423

430

438

445

467

501

13⁄4

819

459

463

466

470

481

510

423

430

438

445

467

510

11⁄2

819

478

482

486

489

500

530

453

461

468

475

497

530

13⁄

819

488

491

495

499

510

539

453

461

468

475

497

539

4


1638




9 - 54




7⁄ -in. 8

Bolts


7 Rows W44, 40, 36, 33, 30, 27, 24 S24

Varies


1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

176

220

264

303

X

—

176

220

264

352

SC

STD

176

204

204

204

OVS

164

173

173

173

SSLT

173

173

173

173

STD

176

220

264

303

OVS

164

205

246

262

SSLT

176

220

262

262

N

—

176

220

264

352

X

—

176

220

264

352

SC Class A

STD

176

220

256

256

OVS

164

205

217

217

SSLT

176

217

217

217

STD

176

220

264

352

OVS

164

205

246

329

SSLT

176

220

264

329

A325

t

6@3 = 18

Class A

SC Class B

2 1/ 4


Hole Type

Lev

L eh

Lev

6@3 = 18

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

639

367

370

373

376

386

412

352

359

365

372

386

412

13⁄4

639

373

377

380

383

393

419

352

359

365

372

392

419

11⁄2

639

344

348

351

354

364

390

329

335

342

348

364

390

13⁄4

639

351

354

358

361

371

397

329

335

342

348

368

397

11⁄2

639

367

370

373

376

386

412

352

359

365

372

386

412

13⁄4

639

373

377

380

383

393

419

352

359

365

372

392

419


1279





9 - 55



Bolts


7 Rows W44, 40, 36, 33, 30, 27, 24 S24

Varies


t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

197

247

296

303

X

—

197

247

296

379

SC

STD

197

204

204

204

OVS

173

173

173

173

SSLT

173

173

173

173

STD

197

247

296

303

OVS

184

230

262

262

SSLT

197

247

262

262

N

—

197

247

296

379

X

—

197

247

296

395

SC

STD

197

247

256

256

OVS

184

217

217

217

SSLT

197

217

217

217

STD

197

247

296

379

OVS

184

230

276

329

SSLT

197

247

296

329

6@3 = 18

Class A

SC Class B

2 1/ 4


Hole Type

Lev

L eh

6@3 = 18

A490

Lev

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

717

420

423

427

431

442

471

395

402

410

417

439

471

13⁄4

717

429

433

437

440

451

480

395

402

410

417

439

480

11⁄

2

717

395

399

402

406

417

446

368

376

383

390

412

446

13⁄4

717

404

408

412

415

426

456

368

376

383

390

412

456

11⁄2

717

420

423

427

431

442

471

395

402

410

417

439

471

13⁄

717

429

433

437

440

451

480

395

402

410

417

439

480

4


1433




9 - 56




7⁄ -in. 8

Bolts


6 Rows W44, 40, 36, 33, 30, 27, 24, 21 S24

Varies


1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

150

188

225

260

X

—

150

188

225

300

SC

STD

150

175

175

175

OVS

140

148

148

148

SSLT

148

148

148

148

STD

150

188

225

260

OVS

140

175

210

225

SSLT

150

188

225

225

N

—

150

188

225

300

X

—

150

188

225

300

SC Class A

STD

150

188

219

219

OVS

140

175

186

186

SSLT

150

186

186

186

STD

150

188

225

300

OVS

140

175

210

280

SSLT

150

188

225

282

A325

t

5@3 = 15

Class A

SC Class B

2 1/ 4

5@3 = 15

Lev

L eh

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

548

314

318

321

324

334

360

300

307

313

320

334

360

13⁄4

548

321

324

328

331

341

367

300

307

313

320

339

367

11⁄2

548

296

299

302

305

315

341

280

286

293

299

315

341

13⁄4

548

302

306

309

312

322

348

280

286

293

299

319

348

11⁄2

548

314

318

321

324

334

360

300

307

313

320

334

360

13⁄4

548

321

324

328

331

341

367

300

307

313

320

339

367


1096





9 - 57



Bolts


6 Rows W44, 40, 36, 33, 30, 27, 24, 21 S24

Varies


t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

168

210

252

260

X

—

168

210

252

325

SC

STD

168

175

175

175

OVS

148

148

148

148

SSLT

148

148

148

148

STD

168

210

252

260

OVS

157

196

225

225

SSLT

168

210

225

225

N

—

168

210

252

325

X

—

168

210

252

336

SC

STD

168

210

219

219

OVS

157

186

186

186

SSLT

168

186

186

186

STD

168

210

252

325

OVS

157

196

235

282

SSLT

168

210

252

282

5@3 = 15

Class A

SC 2 1/ 4

Class B

5@3 = 15

Lev

L eh

Lev

A490


Hole Type

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

614

361

365

369

372

383

413

336

344

351

358

380

413

13⁄4

614

371

374

378

382

393

422

336

344

351

358

380

422

11⁄

2

614

340

344

348

351

362

392

314

321

328

335

357

392

13⁄4

614

350

353

357

361

372

401

314

321

328

335

357

401

11⁄2

614

361

365

369

372

383

413

336

344

351

358

380

413

13⁄

614

371

374

378

382

393

422

336

344

351

358

380

422

4


1229




9 - 58




7⁄ -in. 8

Bolts


5 Rows


W30, 27, 24, 21, 18 S24, 20, 18 MC18

Varies

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

124

155

186

216

X

—

124

155

186

248

SC

STD

124

145

145

145

OVS

115

124

124

124

SSLT

124

124

124

124

STD

124

155

186

216

OVS

115

144

173

187

SSLT

124

155

186

187

N

—

124

155

186

248

X

—

124

155

186

248

SC Class A

STD

124

155

183

183

OVS

115

144

155

155

SSLT

124

155

155

155

STD

124

155

186

248

OVS

115

144

173

231

SSLT

124

155

186

235

A325

Class A

4@3 = 12

t

SC Class B

2 1/ 4

Lev

L eh

4@3 = 12

A490

L ev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

457

262

265

269

272

282

308

248

254

261

268

282

308

13⁄4

457

269

272

275

279

288

315

248

254

261

268

287

315

11⁄2

457

247

250

253

257

266

293

231

238

244

251

266

293

13⁄4

457

254

257

260

263

273

299

231

238

244

251

270

299

11⁄2

457

262

265

269

272

282

308

248

254

261

268

282

308

13⁄4

457

269

272

275

279

288

315

248

254

261

268

287

315


914





9 - 59



Bolts


5 Rows


W30, 27, 24, 21, 18 S24, 20, 18 MC18

Varies

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

139

174

208

216

X

—

139

174

208

271

SC

STD

139

145

145

145

OVS

124

124

124

124

SSLT

124

124

124

124

STD

139

174

208

216

OVS

129

162

187

187

SSLT

139

174

187

187

N

—

139

174

208

271

X

—

139

174

208

278

SC

STD

139

174

183

183

OVS

129

155

155

155

SSLT

139

155

155

155

STD

139

174

208

271

OVS

129

162

194

235

SSLT

139

174

208

235

Class A

4@3 = 12

t

SC Class B

2 1/ 4

Lev

L eh

L ev

4@3 = 12

A490


Hole Type

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

512

303

306

310

314

325

354

278

285

293

300

322

354

13⁄4

512

312

316

320

323

334

363

278

285

293

300

322

363

11⁄

2

512

286

289

293

297

308

337

259

266

273

281

303

337

13⁄4

512

295

299

302

306

317

346

259

266

273

281

303

346

11⁄2

512

303

306

310

314

325

354

278

285

293

300

322

354

13⁄

512

312

316

320

323

334

363

278

285

293

300

322

363

4


1024




9 - 60




7⁄ -in. 8

Bolts


4 Rows


W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18

1⁄ 4

N

—

97.9

X

—

SC

3⁄ 8

1⁄ 2

122

147

173

97.9

122

147

196

STD

97.9

116

116

116

OVS

91.1

98.9

98.9

98.9

SSLT

97.9

98.9

98.9

98.9

STD

97.9

122

147

173

OVS

91.1

114

137

150

SSLT

97.9

122

147

150

N

—

97.9

122

147

196

X

—

97.9

122

147

196

SC Class A

STD

97.9

122

146

146

OVS

91.1

114

124

124

SSLT

97.9

122

124

124

STD

97.9

122

147

196

OVS

91.1

114

137

182

SSLT

97.9

122

147

188

A325

Class A Varies

3@3 = 9

t

SC Class B

2 1/ 4

L ev

L eh

3@3 = 9

A490

L ev


Hole Type

SC Class B

5⁄ 16



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

365

210

213

216

220

230

256

196

202

209

215

230

256

13⁄4

365

217

220

223

226

236

262

196

202

209

215

235

262

11⁄2

365

198

201

205

208

218

244

182

189

195

202

218

244

13⁄4

365

205

208

211

215

224

250

182

189

195

202

221

250

11⁄2

365

210

213

216

220

230

256

196

202

209

215

230

256

13⁄4

365

217

220

223

226

236

262

196

202

209

215

235

262


731





9 - 61



Bolts


4 Rows


W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

110

137

165

173

X

—

110

137

165

216

SC

STD

110

116

116

116

Class A Varies


Hole Type

OVS

98.9

98.9

98.9

98.9

SSLT

98.9

98.9

98.9

98.9

3@3 = 9

t

SC

STD

110

137

165

173

OVS

102

128

150

150

SSLT

110

137

150

150

N

—

110

137

165

216

X

—

110

137

165

219

SC

STD

110

137

146

146

OVS

102

124

124

124

SSLT

110

124

124

124

STD

110

137

165

216

OVS

102

128

153

188

SSLT

110

137

165

188

Class B

2 1/ 4

L ev

L eh

L ev

3@3 = 9

A490

Class A

SC Class B


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

410

244

248

252

255

266

296

219

227

234

241

263

296

13⁄4

410

254

257

261

265

276

305

219

227

234

241

263

305

11⁄

2

410

231

235

238

242

253

282

204

211

219

226

248

282

13⁄4

410

240

244

248

251

262

292

204

211

219

226

248

292

11⁄2

410

244

248

252

255

266

296

219

227

234

241

263

296

13⁄

410

254

257

261

265

276

305

219

227

234

241

263

305

4


819




9 - 62




7⁄ -in. 8

Bolts


3 Rows


W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 MC18, 13, 12

Varies

1⁄ 4

N

—

71.8

X

—

SC

3⁄ 8

1⁄ 2

89.7

108

130

71.8

89.7

108

144

STD

71.8

87.3

87.3

87.3

OVS

66.7

74.2

74.2

74.2

SSLT

71.8

74.2

74.2

74.2

STD

71.8

89.7

108

130

OVS

66.7

83.4

100

112

SSLT

71.8

89.7

108

112

N

—

71.8

89.7

108

144

X

—

71.8

89.7

108

144

SC Class A

STD

71.8

89.7

108

110

OVS

66.7

83.4

93.2

93.2

SSLT

71.8

89.7

93.2

93.2

STD

71.8

89.7

108

144

OVS

66.7

83.4

100

133

SSLT

71.8

89.7

108

141

A325

*Limited to W10×12, 15, 17, 19, 22, 26, 30

Class A

t

SC Class B

3 3 1

2 /4 L eh Lev

A490

Lev

3 3


Hole Type

SC Class B

5⁄ 16



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

274

158

161

164

168

177

203

144

150

157

163

177

203

13⁄4

274

164

168

171

174

184

210

144

150

157

163

183

210

11⁄2

274

149

153

156

159

169

195

133

140

146

153

169

195

13⁄4

274

156

159

163

166

176

202

133

140

146

153

173

202

11⁄2

274

158

161

164

168

177

203

144

150

157

163

177

203

13⁄4

274

164

168

171

174

184

210

144

150

157

163

183

210


548





9 - 63



Bolts


3 Rows


W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 MC18, 13, 12

A325

*Limited to W10×12, 15, 17, 19, 22, 26, 30

Varies

1⁄ 4

N

—

80.4

X

—

80.4

SC

STD

80.4

87.3

87.3

87.3

OVS

74.2

74.2

74.2

74.2

SSLT

74.2

74.2

74.2

74.2

STD

80.4

OVS

74.7

SSLT

80.4

N

—

X SC

Class A

t

SC Class B

3 3 1

2 /4 L eh Lev

A490

Lev

3 3


Hole Type

Class A

SC Class B

5⁄ 16

3⁄ 8

1⁄ 2

101

121

130

101

121

161

101

121

130

112

112

101

112

112

80.4

101

121

161

—

80.4

101

121

161

STD

80.4

101

110

110

OVS

74.7

93.2

93.2

93.2

SSLT

80.4

93.2

93.2

93.2

STD

80.4

OVS

74.7

SSLT

80.4

93.4

101 93.4 101

121

161

112

141

121

141


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

307

186

189

193

197

208

237

161

168

176

183

205

237

13⁄4

307

195

199

203

206

217

246

161

168

176

183

205

246

11⁄

2

307

176

180

184

187

198

227

149

157

164

171

193

227

13⁄4

307

186

189

193

197

208

237

149

157

164

171

193

237

11⁄2

307

186

189

193

197

208

237

161

168

176

183

205

237

13⁄

307

195

199

203

206

217

246

161

168

176

183

205

246

4


614




9 - 64




7⁄ -in. 8

Bolts


2 Rows


W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

45.7

57.1

68.5

86.6

X

—

45.7

57.1

68.5

91.4

SC

STD

45.7

57.1

58.2

58.2

OVS

42.3

49.4

49.4

49.4

SSLT

45.7

49.4

49.4

49.4

STD

45.7

57.1

68.5

86.6

OVS

42.3

52.9

63.4

74.9

SSLT

45.7

57.1

68.5

74.9

N

—

45.7

57.1

68.5

91.4

X

—

45.7

57.1

68.5

91.4

SC Class A

STD

45.7

57.1

68.5

73.1

OVS

42.3

52.9

62.1

62.1

SSLT

45.7

57.1

62.1

62.1

STD

45.7

57.1

68.5

91.4

OVS

42.3

52.9

63.4

84.6

SSLT

45.7

57.1

68.5

91.4

A325

Class A Varies


Hole Type

t

3

SC Class B

2 1/ 4

Lev

L eh

3

Lev

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

183

106

109

112

115

125

151

91.4 97.9 104

111

125

151

13⁄4

183

112

116

119

122

132

158

91.4 97.9 104

111

131

158

11⁄2

183

100

104

107

110

120

146

84.6 91.1

97.6 104

120

146

13⁄4

183

107

110

114

117

127

153

84.6 91.1

97.6 104

124

153

11⁄2

183

106

109

112

115

125

151

91.4 97.9 104

111

125

151

13⁄4

183

112

116

119

122

132

158

91.4 97.9 104

111

131

158


365





9 - 65



Bolts


2 Rows


W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8

A325

1⁄ 4

5⁄ 16

3⁄ 8

N

—

51.2

64.0

76.8

X

—

51.2

64.0

76.8

SC

STD

51.2

58.2

58.2

58.2

OVS

47.4

49.4

49.4

49.4

SSLT

49.4

49.4

49.4

49.4

STD

51.2

64.0

76.8

86.6

OVS

47.4

59.2

71.1

74.9

SSLT

51.2

64.0

74.9

74.9

N

—

51.2

64.0

76.8

102

X

—

51.2

64.0

76.8

102

SC

STD

51.2

64.0

73.1

73.1

OVS

47.4

59.2

62.1

62.1

SSLT

51.2

62.1

62.1

62.1

STD

51.2

64.0

76.8

OVS

47.4

59.2

71.1

94.1

SSLT

51.2

64.0

76.8

94.1

Class A Varies


Hole Type

1⁄ 2

86.6 102

t

3

SC

2 1/ 4

Class B

Lev

L eh

3

Lev

A490

Class A

SC Class B

102


11⁄


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

2

205

127

131

135

138

149

179 102

110

117

124

146

179

13⁄4

205

137

140

144

148

159

188 102

110

117

124

146

188

11⁄

173

2

205

122

125

129

133

144

173

94.8 102

109

117

139

13⁄4

205

131

135

138

142

153

182

94.8 102

109

117

139

182

11⁄2

205

127

131

135

138

149

179 102

110

117

124

146

179

13⁄4

205

137

140

144

148

159

188 102

110

117

124

146

188


410




9 - 66




1-in. Bolts


12 Rows


W44

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

286

358

429

573

X

—

286

358

429

573

SC

STD

286

358

429

456

OVS

258

323

387

388

SSLT

286

358

388

388

STD

286

358

429

573

OVS

258

323

387

516

SSLT

286

358

429

573

N

—

286

358

429

573

X

—

286

358

429

573

SC Class A

STD

286

358

429

573

OVS

258

323

387

487

SSLT

286

358

429

487

STD

286

358

429

573

OVS

258

323

387

516

SSLT

286

358

429

573

A325 Varies

t

11@3 = 33

Class A

SC Class B 2 1/ 2

L eh


Hole Type

11@3 = 33

Lev

A490

Lev

SC Class B

Beam Web Design Strength per Inch Thickness, kips/in. Coped at Top Flange Only


Lev, in.

Lev, in.

Hole Type

Leh, Unin. coped 11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

STD

11⁄2

1253

589

592

595

598

608

634

573

579

586

592

608

634

13⁄4

1253

595

599

602

605

615

641

573

579

586

592

612

641

11⁄2

1253

534

538

541

544

554

580

516

523

529

536

554

580

13⁄4

1253

541

544

548

551

561

587

516

523

529

536

555

587

11⁄2

1253

589

592

595

598

608

634

573

579

586

592

608

634

13⁄4

1253

595

599

602

605

615

641

573

579

586

592

612

641

OVS SSLT


2506





9 - 67


Table 9-2 (cont.). All-Bolted Double-Angle Connections 1-in. Bolts


12 Rows


W44

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

321

401

481

642

X

—

321

401

481

642

SC

STD

321

401

456

456

OVS

289

362

388

388

SSLT

321

388

388

388

STD

321

401

481

642

OVS

289

362

434

579

SSLT

321

401

481

588

N

—

321

401

481

642

X

—

321

401

481

642

SC

STD

321

401

481

573

OVS

289

362

434

487

SSLT

321

401

481

487

STD

321

401

481

642

OVS

289

362

434

579

SSLT

321

401

481

642

11@3 = 33

Class A

SC Class B 2 1/ 2

L eh Lev

A490


Hole Type

11@3 = 33

Class A

SC Lev

Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1404

669

672

676

680

691

720

642

649

656

664

686

720

13⁄4

1404

678

682

685

689

700

729

642

649

656

664

686

729

11⁄2

1404

608

612

615

619

630

659

579

586

593

601

622

659

13⁄4

1404

617

621

625

628

639

669

579

586

593

601

622

669

11⁄2

1404

669

672

676

680

691

720

642

649

656

664

686

720

13⁄4

1404

678

682

685

689

700

729

642

649

656

664

686

729


2808




9 - 68




1-in. Bolts


11 Rows


W44, 40

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

262

327

393

524

X

—

262

327

393

524

SC

STD

262

327

393

418

OVS

236

295

354

356

SSLT

262

327

356

356

STD

262

327

393

524

OVS

236

295

354

472

SSLT

262

327

393

524

N

—

262

327

393

524

X

—

262

327

393

524

SC Class A

STD

262

327

393

524

OVS

236

295

354

446

SSLT

262

327

393

446

STD

262

327

393

524

OVS

236

295

354

472

SSLT

262

327

393

524

A325 Varies

t

Class A 10@3 = 30


Hole Type

SC Class B 2 1/ 2

L eh

10@3 = 30

Lev

A490

Lev

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1148

540

543

546

549

559

585

524

530

537

543

559

585

13⁄4

1148

546

550

553

556

566

592

524

530

537

543

563

592

11⁄2

1148

490

494

497

500

510

536

472

479

485

492

510

536

13⁄4

1148

497

500

504

507

517

543

472

479

485

492

511

543

11⁄2

1148

540

543

546

549

559

585

524

530

537

543

559

585

13⁄4

1148

546

550

553

556

566

592

524

530

537

543

563

592


2297





9 - 69




11 Rows


W44, 40

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

294

367

440

587

X

—

294

367

440

587

SC

STD

294

367

418

418

OVS

265

331

356

356

SSLT

294

356

356

356

STD

294

367

440

587

OVS

265

331

397

529

SSLT

294

367

440

539

N

—

294

367

440

587

X

—

294

367

440

587

SC

STD

294

367

440

525

OVS

265

331

397

446

SSLT

294

367

440

446

STD

294

367

440

587

OVS

265

331

397

529

SSLT

294

367

440

587

Class A 10@3 = 30


Hole Type

SC Class B 2 1/ 2

L eh

10@3 = 30

Lev

A490

Class A

Lev

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1287

614

618

621

625

636

665

587

594

602

609

631

665

13⁄4

1287

623

627

631

634

645

674

587

594

602

609

631

674

11⁄2

1287

559

562

566

570

581

610

529

536

544

551

573

610

13⁄4

1287

568

572

575

579

590

619

529

536

544

551

573

619

11⁄2

1287

614

618

621

625

636

665

587

594

602

609

631

665

13⁄4

1287

623

627

631

634

645

674

587

594

602

609

631

674


2574




9 - 70




1-in. Bolts


10 Rows


W44, 40, 36

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

238

297

356

475

X

—

238

297

356

475

SC

STD

238

297

356

380

OVS

214

268

321

323

SSLT

238

297

323

323

STD

238

297

356

475

OVS

214

268

321

428

SSLT

238

297

356

475

N

—

238

297

356

475

X

—

238

297

356

475

SC Class A

STD

238

297

356

475

OVS

214

268

321

406

SSLT

238

297

356

406

STD

238

297

356

475

OVS

214

268

321

428

SSLT

238

297

356

475

A325 Varies

t

9@3 = 27

Class A

SC Class B 2 1/ 2

Lev

L eh

9@3 = 27

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1044

491

494

497

501

510

537

475

482

488

495

510

537

13⁄4

1044

498

501

504

507

517

543

475

482

488

495

514

543

11⁄2

1044

446

450

453

456

466

492

428

435

441

448

466

492

13⁄4

1044

453

456

460

463

473

499

428

435

441

448

467

499

11⁄2

1044

491

494

497

501

510

537

475

482

488

495

510

537

13⁄4

1044

498

501

504

507

517

543

475

482

488

495

514

543


2088





9 - 71




10 Rows


W44, 40, 36

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

266

333

399

532

X

—

266

333

399

532

SC

STD

266

333

380

380

OVS

240

300

323

323

SSLT

266

323

323

323

STD

266

333

399

532

OVS

240

300

360

480

SSLT

266

333

399

490

N

—

266

333

399

532

X

—

266

333

399

532

SC

STD

266

333

399

477

OVS

240

300

360

406

SSLT

266

333

399

406

STD

266

333

399

532

OVS

240

300

360

480

SSLT

266

333

399

532

Class A 9@3 = 27


Hole Type

SC Class B 2 1/ 2

Lev

L eh

9@3 = 27

A490

Lev

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1170

559

563

567

570

581

610

532

540

547

554

576

610

13⁄4

1170

569

572

576

580

591

620

532

540

547

554

576

620

11⁄2

1170

509

513

516

520

531

560

480

487

494

502

524

560

13⁄4

1170

519

522

526

530

540

570

480

487

494

502

524

570

11⁄2

1170

559

563

567

570

581

610

532

540

547

554

576

610

13⁄4

1170

569

572

576

580

591

620

532

540

547

554

576

620


2340




9 - 72




1-in. Bolts


9 Rows


W44, 40, 36, 33

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

213

266

320

426

X

—

213

266

320

426

SC

STD

213

266

320

342

OVS

192

240

288

291

SSLT

213

266

291

291

STD

213

266

320

426

OVS

192

240

288

384

SSLT

213

266

320

426

N

—

213

266

320

426

X

—

213

266

320

426

SC Class A

STD

213

266

320

426

OVS

192

240

288

365

SSLT

213

266

320

365

STD

213

266

320

426

OVS

192

240

288

384

SSLT

213

266

320

426

A325 Varies

t

8@3 = 24

Class A

SC Class B

2 1/ 2

Lev

L eh

A490

Lev

8@3 = 24


Hole Type

SC Class B

Beam Web Design Strength per Inch Thickness, kips/in. Coped at Top Flange Only Hole Leh,* UnType in. coped 11⁄4 STD OVS SS:T


Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

940

442

445

449

452

462

488

426

433

439

446

462

488

13⁄4

940

449

452

455

459

468

495

426

433

439

446

465

495

11⁄2

940

402

405

409

412

422

448

384

390

397

404

422

448

13⁄4

940

409

412

415

419

428

455

384

390

397

404

423

455

11⁄2

940

442

445

449

452

462

488

426

433

439

446

462

488

13⁄4

940

449

452

455

459

468

495

426

433

439

446

465

495


1879





9 - 73




9 Rows


W44, 40, 36, 33

A325 Varies

t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

239

299

358

478

X

—

239

299

358

478

SC

STD

239

299

342

342

OVS

215

269

291

291

SSLT

239

291

291

291

STD

239

299

358

478

OVS

215

269

323

430

SSLT

239

299

358

441

N

—

239

299

358

478

X

—

239

299

358

478

SC

STD

239

299

358

430

OVS

215

269

323

365

SSLT

239

299

358

365

STD

239

299

358

478

OVS

215

269

323

430

SSLT

239

299

358

478

8@3 = 24

Class A

SC Class B

2 1/ 2

Lev

L eh

8@3 = 24

A490


Hole Type

Lev

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

1053

504

508

512

515

526

556

478

485

492

500

522

556

13⁄4

1053

514

518

521

525

536

565

478

485

492

500

522

565

11⁄2

1053

460

463

467

471

482

511

430

438

445

452

474

511

13⁄4

1053

469

473

476

480

491

520

430

438

445

452

474

520

11⁄2

1053

504

508

512

515

526

556

478

485

492

500

522

556

13⁄4

1053

514

518

521

525

536

565

478

485

492

500

522

565


2106




9 - 74




1-in. Bolts


8 Rows


W44, 40, 36, 33, 30

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

189

236

283

377

X

—

189

236

283

377

SC

STD

189

236

283

304

OVS

170

212

255

259

SSLT

189

236

259

259

STD

189

236

283

377

OVS

170

212

255

340

SSLT

189

236

283

377

N

—

189

236

283

377

X

—

189

236

283

377

SC Class A

STD

189

236

283

377

OVS

170

212

255

325

SSLT

189

236

283

325

STD

189

236

283

377

OVS

170

212

255

340

SSLT

189

236

283

377

A325 Varies

t

7@3 = 21

Class A

SC Class B

2 1/ 2 Lev

L eh

7@3 = 21

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

835

393

397

400

403

413

439

377

384

390

397

413

439

13⁄4

835

400

403

407

410

420

446

377

384

390

397

417

446

11⁄2

835

358

361

365

368

378

404

340

346

353

359

378

404

13⁄4

835

365

368

371

375

384

410

340

346

353

359

379

410

11⁄2

835

393

397

400

403

413

439

377

384

390

397

413

439

13⁄4

835

400

403

407

410

420

446

377

384

390

397

417

446


1670





9 - 75




8 Rows


W44, 40, 36, 33, 30

A325 Varies


Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

211

264

317

423

X

—

211

264

317

423

SC

STD

211

264

304

304

OVS

190

238

259

259

SSLT

211

259

259

259

STD

211

264

317

423

OVS

190

238

286

381

SSLT

211

264

317

392

N

—

211

264

317

423

X

—

211

264

317

423

SC

STD

211

264

317

382

OVS

190

238

286

325

SSLT

211

264

317

325

STD

211

264

317

423

OVS

190

238

286

381

SSLT

211

264

317

423

t

7@3 = 21

Class A

SC Class B

2 1/ 2 Lev

L eh

7@3 = 21

A490

Lev

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

936

450

453

457

461

472

501

423

430

438

445

467

501

13⁄4

936

459

463

466

470

481

510

423

430

438

445

467

510

11⁄2

936

410

414

418

421

432

461

381

388

395

403

425

461

13⁄4

936

420

423

427

431

442

471

381

388

395

403

425

471

11⁄2

936

450

453

457

461

472

501

423

430

438

445

467

501

13⁄4

936

459

463

466

470

481

510

423

430

438

445

467

510


1872




9 - 76




1-in. Bolts


7 Rows W44, 40, 36, 33, 30, 27, 24 S24

Varies


1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

164

205

246

329

X

—

164

205

246

329

SC

STD

164

205

246

266

OVS

148

185

222

226

SSLT

164

205

226

226

STD

164

205

246

329

OVS

148

185

222

296

SSLT

164

205

246

329

N

—

164

205

246

329

X

—

164

205

246

329

SC Class A

STD

164

205

246

329

OVS

148

185

222

284

SSLT

164

205

246

284

STD

164

205

246

329

OVS

148

185

222

296

SSLT

164

205

246

329

A325

t

6@3 = 18

Class A

SC Class B

2 1/ 2


Hole Type

Lev

L eh

Lev

6@3 = 18

A490

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

731

344

348

351

354

364

390

329

335

342

348

364

390

13⁄4

731

351

354

358

361

371

397

329

335

342

348

368

397

11⁄2

731

314

317

320

324

334

360

296

302

309

315

334

360

13⁄4

731

321

324

327

330

340

366

296

302

309

315

335

366

11⁄2

731

344

348

351

354

364

390

329

335

342

348

364

390

13⁄4

731

351

354

358

361

371

397

329

335

342

348

368

397


1462





9 - 77




7 Rows W44, 40, 36, 33, 30, 27, 24 S24

Varies


t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

184

230

276

368

X

—

184

230

276

368

SC

STD

184

230

266

266

OVS

166

207

226

226

SSLT

184

226

226

226

STD

184

230

276

368

OVS

166

207

249

331

SSLT

184

230

276

343

N

—

184

230

276

368

X

—

184

230

276

368

SC

STD

184

230

276

334

OVS

166

207

249

284

SSLT

184

230

276

284

STD

184

230

276

368

OVS

166

207

249

331

SSLT

184

230

276

368

6@3 = 18

Class A

SC Class B

2 1/ 2


Hole Type

Lev

L eh

6@3 = 18

A490

Lev

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

819

395

399

402

406

417

446

368

376

383

390

412

446

13⁄4

819

404

408

412

415

426

456

368

376

383

390

412

456

11⁄2

819

361

365

368

372

383

412

331

339

346

353

375

412

13⁄4

819

370

374

378

381

392

421

331

339

346

353

375

421

11⁄2

819

395

399

402

406

417

446

368

376

383

390

412

446

13⁄4

819

404

408

412

415

426

456

368

376

383

390

412

456


1638




9 - 78




1-in. Bolts


6 Rows W44, 40, 36, 33, 30, 27, 24, 21 S24

Varies


1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

140

175

210

280

X

—

140

175

210

280

SC

STD

140

175

210

228

OVS

126

157

189

194

SSLT

140

175

194

194

STD

140

175

210

280

OVS

126

157

189

252

SSLT

140

175

210

280

N

—

140

175

210

280

X

—

140

175

210

280

SC Class A

STD

140

175

210

280

OVS

126

157

189

243

SSLT

140

175

210

243

STD

140

175

210

280

OVS

126

157

189

252

SSLT

140

175

210

280

A325

t

5@3 = 15

Class A

SC Class B

2 1/ 2

5@3 = 15

Lev

L eh

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

626

296

299

302

305

315

341

280

286

293

299

315

341

13⁄4

626

302

306

309

312

322

348

280

286

293

299

319

348

11⁄2

626

270

273

276

280

289

315

252

258

265

271

289

315

13⁄4

626

277

280

283

286

296

322

252

258

265

271

291

322

11⁄2

626

296

299

302

305

315

341

280

286

293

299

315

341

13⁄4

626

302

306

309

312

322

348

280

286

293

299

319

348


1253





9 - 79




6 Rows W44, 40, 36, 33, 30,27, 24, 21 S24

Varies


t

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

157

196

235

314

X

—

157

196

235

314

SC

STD

157

196

228

228

OVS

141

176

194

194

SSLT

157

194

194

194

STD

157

196

235

314

OVS

141

176

211

282

SSLT

157

196

235

294

N

—

157

196

235

314

X

—

157

196

235

314

SC

STD

157

196

235

286

OVS

141

176

211

243

SSLT

157

196

235

243

STD

157

196

235

314

OVS

141

176

211

282

SSLT

157

196

235

314

5@3 = 15

Class A

SC 2 1/ 2

Class B

5@3 = 15

Lev

L eh

Lev

A490


Hole Type

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

702

340

344

348

351

362

392

314

321

328

335

357

392

13⁄4

702

350

353

357

361

372

401

314

321

328

335

357

401

11⁄2

702

311

315

319

322

333

363

282

289

297

304

326

363

13⁄4

702

321

324

328

332

343

372

282

289

297

304

326

372

11⁄2

702

340

344

348

351

362

392

314

321

328

335

357

392

13⁄4

702

350

353

357

361

372

401

314

321

328

335

357

401


1404




9 - 80




1-in. Bolts


5 rows


W30, 27, 24, 21, 18 S24, 20, 18 MC18

Varies

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

115

144

173

231

X

—

115

144

173

231

SC

STD

115

144

173

190

OVS

104

130

156

162

SSLT

115

144

162

162

STD

115

144

173

231

OVS

104

130

156

207

SSLT

115

144

173

231

N

—

115

144

173

231

X

—

115

144

173

231

SC Class A

STD

115

144

173

231

OVS

104

130

156

203

SSLT

115

144

173

203

STD

115

144

173

231

OVS

104

130

156

207

SSLT

115

144

173

231

A325

Class A

4@3 = 12

t

SC Class B

2 1/ 2

Lev

L eh

4@3 = 12

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

522

247

250

253

257

266

293

231

238

244

251

266

293

13⁄4

522

254

257

260

263

273

299

231

238

244

251

270

299

11⁄2

522

226

229

232

236

245

271

207

214

221

227

245

271

13⁄4

522

232

236

239

242

252

278

207

214

221

227

247

278

11⁄2

522

247

250

253

257

266

293

231

238

244

251

266

293

13⁄4

522

254

257

260

263

273

299

231

238

244

251

270

299


1044





9 - 81




5 Rows


W30, 27, 24, 21, 18 S24, 20, 18 MC18

Varies

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

129

162

194

259

X

—

129

162

194

259

SC

STD

129

162

190

190

OVS

116

145

162

162

SSLT

129

162

162

162

STD

129

162

194

259

OVS

116

145

174

233

SSLT

129

162

194

245

N

—

129

162

194

259

X

—

129

162

194

259

SC

STD

129

162

194

239

OVS

116

145

174

203

SSLT

129

162

194

203

STD

129

162

194

259

OVS

116

145

174

233

SSLT

129

162

194

259

Class A

4@3 = 12

t

SC Class B

2 1/ 2

Lev

L eh

Lev

4@3 = 12

A490


Hole Type

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

585

286

289

293

297

308

337

259

266

273

281

303

337

13⁄4

585

295

299

302

306

317

346

259

266

273

281

303

346

11⁄2

585

262

266

269

273

284

313

233

240

247

254

276

313

13⁄4

585

271

275

279

282

293

323

233

240

247

254

276

323

11⁄2

585

286

289

293

297

308

337

259

266

273

281

303

337

13⁄4

585

295

299

302

306

317

346

259

266

273

281

303

346


1170




9 - 82




1-in. Bolts


4 Rows


W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

91.1

114

137

182

X

—

91.1

114

137

182

SC

STD

91.1

114

137

152

OVS

81.7

102

123

129

SSLT

91.1

114

129

129

STD

91.1

114

137

182

OVS

81.7

102

123

163

SSLT

91.1

114

137

182

N

—

91.1

114

137

182

X

—

91.1

114

137

182

SC Class A

STD

91.1

114

137

182

OVS

81.7

102

123

162

SSLT

91.1

114

137

162

STD

91.1

114

137

182

OVS

81.7

102

123

163

SSLT

91.1

114

137

182

A325

Class A Varies

3@3 = 9

t

SC Class B

2 1/ 2

Lev

L eh

3@3 = 9

A490

Lev


Hole Type

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

418

198

201

205

208

218

244

182

189

195

202

218

244

13⁄4

418

205

208

211

215

224

250

182

189

195

202

221

250

11⁄2

418

182

185

188

191

201

227

163

170

176

183

201

227

13⁄4

418

188

192

195

198

208

234

163

170

176

183

203

234

11⁄2

418

198

201

205

208

218

244

182

189

195

202

218

244

13⁄4

418

205

208

211

215

224

250

182

189

195

202

221

250


835





9 - 83




4 Rows


W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

102

128

153

204

X

—

102

128

153

204

SC

STD

102

128

152

152

114

129

129

Class A Varies


Hole Type

OVS

91.6

3@3 = 9

t

SC Class B

2 1/ 2 L eh

102

128

129

129

STD

102

128

153

204

114

137

183

OVS

91.6

SSLT

102

128

153

196

N

—

102

128

153

204

X

—

102

128

153

204

SC

STD

102

128

153

191

114

137

162

Lev Lev

3@3 = 9

A490

SSLT

Class A

SC Class B

OVS

91.6

SSLT

102

128

153

162

STD

102

128

153

204

114

137

183

128

153

204

OVS SSLT

91.6 102



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

468

231

235

238

242

253

282

204

211

219

226

248

282

13⁄4

468

240

244

248

251

262

292

204

211

219

226

248

292

11⁄2

468

213

216

220

224

235

264

183

190

198

205

227

264

13⁄4

468

222

226

229

233

244

273

183

190

198

205

227

273

11⁄2

468

231

235

238

242

253

282

204

211

219

226

248

282

13⁄4

468

240

244

248

251

262

292

204

211

219

226

248

292


936




9 - 84




1-in. Bolts


3 Rows


W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 MC18, 13, 12

A325

*Limited to W10×12, 15, 17, 19, 22, 26, 30

Varies

Lev Lev

1⁄ 2

83.4

100

133

X

—

66.7

83.4

100

133

SC

STD

66.7

83.4

100

114

OVS

59.6

74.5

89.5

97.0

SSLT

66.7

83.4

97.0

97.0

STD

66.7

83.4

OVS

59.6

74.5

SSLT

66.7

83.4

100

133

N

—

66.7

83.4

100

133

X

—

66.7

83.4

100

133

SC Class A

STD

66.7

83.4

100

133

OVS

59.6

74.5

SSLT

66.7

83.4

100

122

STD

66.7

83.4

100

133

OVS

59.6

74.5

SSLT

66.7

83.4

A490

3 3

3⁄ 8

66.7

2 1/ 2 L eh

5⁄ 16

—

SC Class B

3 3

1⁄ 4

N

Class A

t


Hole Type

SC Class B

100

133

89.5

119

89.5

119

89.5

119

100

133



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

313

149

153

156

159

169

195

133

140

146

153

169

195

13⁄4

313

156

159

163

166

176

202

133

140

146

153

173

202

11⁄2

313

137

141

144

147

157

183

119

126

132

139

157

183

13⁄4

313

144

148

151

154

164

190

119

126

132

139

158

190

11⁄2

313

149

153

156

159

169

195

133

140

146

153

169

195

13⁄4

313

156

159

163

166

176

202

133

140

146

153

173

202


626





9 - 85




3 Rows


W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 C18, 13, 12

A325

*Limited to W10×12, 15, 17, 19, 22, 26, 30

Varies

Lev Lev

3 3

1⁄ 2

93.4

112

149

X

—

74.7

93.4

112

149

SC

STD

74.7

93.4

112

114

OVS

66.8

83.5

97.0

97.0

SSLT

74.7

93.4

97.0

97.0

STD

74.7

93.4

112

149

OVS

66.8

83.5

100

134

SSLT

74.7

93.4

112

147

N

—

74.7

93.4

112

149

X

—

74.7

93.4

112

149

SC

STD

74.7

93.4

112

143

OVS

66.8

83.5

100

122

SSLT

74.7

93.4

112

122

STD

74.7

93.4

112

149

OVS

66.8

83.5

100

134

SSLT

74.7

93.4

112

149

2 1/ 2

A490

3⁄ 8

74.7

SC

L eh

5⁄ 16

—

Class B

3 3

1⁄ 4

N

Class A

t


Hole Type

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

351

176

180

184

187

198

227

149

157

164

171

193

227

13⁄4

351

186

189

193

197

208

237

149

157

164

171

193

237

11⁄2

351

163

167

170

174

185

214

134

141

148

156

178

214

13⁄4

351

173

176

180

183

194

224

134

141

148

156

178

224

11⁄2

351

176

180

184

187

198

227

149

157

164

171

193

227

13⁄4

351

186

189

193

197

208

237

149

157

164

171

193

237


702




9 - 86




1-in. Bolts


2 Rows


W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

42.3

52.9

63.4

84.6

X

—

42.3

52.9

63.4

84.6

SC

STD

42.3

52.9

63.4

76.1

OVS

37.6

47.0

56.4

64.7

SSLT

42.3

52.9

63.4

64.7

STD

42.3

52.9

63.4

84.6

OVS

37.6

47.0

56.4

75.2

SSLT

42.3

52.9

63.4

84.6

N

—

42.3

52.9

63.4

84.6

X

—

42.3

52.9

63.4

84.6

SC Class A

STD

42.3

52.9

63.4

84.6

OVS

37.6

47.0

56.4

75.2

SSLT

42.3

52.9

63.4

81.1

STD

42.3

52.9

63.4

84.6

OVS

37.6

47.0

56.4

75.2

SSLT

42.3

52.9

63.4

84.6

A325

Class A Varies


Hole Type

t

3

SC Class B

2 1/ 2

Lev

L eh

3

Lev

A490

SC Class B

Beam Web Design Strength per Inch Thickness, kips/in. Coped at Top Flange Only Hole Type STD OVS SSLT

Leh,* Unin. coped


Lev, in. 11⁄4

13⁄8

11⁄2

15⁄8

Lev, in. 2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

209

100

104

107

110 120 146 84.6 91.1 97.6 104

120 146

13⁄4

209

107

110

114

117 127 153 84.6 91.1 97.6 104

124 153

11⁄2

209

13⁄4

209

100

103

107

110 120 146 75.2 81.7 88.2

11⁄2

209

100

104

107

110 120 146 84.6 91.1 97.6 104

120 146

13⁄4

209

107

110

114

117 127 153 84.6 91.1 97.6 104

124 153

93.4

96.7

99.9 103 113 139 75.2 81.7 88.2


418

94.7 113 139 94.7 114 146





9 - 87




2 Rows


W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8

A325

1⁄ 4

5⁄ 16

3⁄ 8

1⁄ 2

N

—

47.4

59.2

71.1

94.8

X

—

47.4

59.2

71.1

94.8

SC

STD

47.4

59.2

71.1

76.1

OVS

42.1

52.7

63.2

64.7

SSLT

47.4

59.2

64.7

64.7

STD

47.4

59.2

71.1

94.8

OVS

42.1

52.7

63.2

84.2

SSLT

47.4

59.2

71.1

94.8

N

—

47.4

59.2

71.1

94.8

X

—

47.4

59.2

71.1

94.8

SC

STD

47.4

59.2

71.1

94.8

OVS

42.1

52.7

63.2

81.1

SSLT

47.4

59.2

71.1

81.1

STD

47.4

59.2

71.1

94.8

OVS

42.1

52.7

63.2

84.2

SSLT

47.4

59.2

71.1

94.8

Class A Varies


Hole Type

t

3

SC

2 1/ 2

Class B

Lev

L eh

3

Lev

A490

Class A

SC Class B



Lev, in.

Lev, in.

13⁄8

11⁄2

15⁄8

2

3

11⁄4

13⁄8

11⁄2

15⁄8

2

3

11⁄2

234

122

125

129

133

144

173

94.8

102 109

117

139

173

13⁄4

234

131

135

138

142

153

182

94.8

102 109

117

139

182

11⁄2

234

114

117

121

125

136

165

84.2 91.6

98.9 106

128

165

13⁄4

234

123

127

130

134

145

174

84.2 91.6

98.9 106

128

174

11⁄2

234

122

125

129

133

144

173

94.8

102 109

117

139

173

13⁄4

234

131

135

138

142

153

182

94.8

102 109

117

139

182


468




9 - 88


Length of return 2 x weld size

½

Welds B

L

L

3 spa.

3 spa.

Web thickness

k (min.)

Table 9-3. Combination Bolted/Welded Double-Angle Connections

3½

3½

k (min.)

Varies 2¼

Welds A

Welds A (70 ksi) Weld Size, in.

n

L

12

351 ⁄2

5⁄

16

1⁄ 3⁄

11

321 ⁄2

10

291 ⁄2

9

261 ⁄2

16

5⁄

16 1⁄ 4 3⁄ 16 5⁄

16 1⁄ 4

3⁄

16

5⁄

16

1⁄ 3⁄

8

7

231 ⁄

2

201 ⁄2

6

2

16

16 1⁄ 4 3⁄ 16 5⁄ 16

16

5⁄

16

3⁄

141 ⁄2

4

111 ⁄2

3

81 ⁄2

16

16 1⁄ 4 3⁄ 16 5⁄ 16 1⁄ 4 3⁄

16

5⁄

16

3⁄

51 ⁄2

4

5⁄

1⁄

2

4

3⁄

1⁄

5

4

5⁄

1⁄

171 ⁄

4

5⁄

4

16

16 1⁄ 4 3⁄ 16

Welds B (70 ksi)

Min. Web Thickness, in.

φRn, kips


593 475 356 548 439 329 506 405 304 464 371 278 423 338 254 379 304 228 334 267 200 287 230 172 237 190 142 184 147 110 125 100 75.2

0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43 0.72 0.57 0.43

0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31 0.52 0.41 0.31

Weld Size, in. 3⁄

8

5⁄

16 1⁄ 4 3⁄

8

5⁄

16 1⁄ 4 3⁄ 8 5⁄ 16 1⁄

4

3⁄

8

5⁄

16

1⁄

4

3⁄

8

5⁄

16 1⁄ 4 3⁄

5⁄

8

16

1⁄

4

3⁄

8

5⁄

16 1⁄ 4 3⁄ 8 5⁄ 16 1⁄ 4 3⁄ 8 5⁄ 16 1⁄

4

3⁄

8

5⁄

16 1⁄ 4 3⁄

5⁄

8

16 1⁄ 4

φRn, kips 550 458 366 496 414 331 443 369 295 389 324 259 335 279 223 280 234 187 226 188 150 172 143 115 120 100 79.9 72.2 60.1 48.1 32.8 27.3 21.9


Min. Support Thickness, in.

Fy = 36 ksi Fy = 50 ksi 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29 0.43 0.36 0.29

0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21 0.31 0.26 0.21


9 - 89


k (min.)

Table 9-4. All-Welded Double-Angle Connections

½

L

Weld B

3

k (min.)

Web thickness 4 in. for L > 18 in. (typ.) 3 in. for L < 18 in. (typ.)

Weld A

Welds A (70 ksi)

L 36

Weld Size, in. 5⁄

0.52

3⁄

4

483

0.57

0.41

5⁄

3⁄

16

362

0.43

0.31

1⁄

5⁄

16

574

0.72

0.52

3⁄

4

459

0.57

0.41

5⁄

3⁄

16

345

0.43

0.31

1⁄

5⁄

16

546

0.72

0.52

3⁄

4

437

0.57

0.41

5⁄

3⁄

16

328

0.43

0.31

1⁄

5⁄

16

516

0.72

0.52

3⁄

4

413

0.57

0.41

5⁄

3⁄

16

310

0.43

0.31

1⁄

5⁄

16

487

0.72

0.52

3⁄

4

390

0.57

0.41

5⁄

3⁄

16

292

0.43

0.31

1⁄

5⁄

16

459

0.72

0.52

3⁄

4

367

0.57

0.41

5⁄

3⁄

16

275

0.43

0.31

1⁄

5⁄

16

432

0.72

0.52

3⁄

4

346

0.57

0.41

5⁄

3⁄

16

259

0.43

0.31

1⁄

5⁄

16

404

0.72

0.52

3⁄

4

323

0.57

0.41

5⁄

3⁄

16

242

0.43

0.31

1⁄

5⁄

16

376

0.72

0.52

3⁄

4

301

0.57

0.41

5⁄

3⁄

16

226

0.43

0.31

1⁄

5⁄

16

348

0.72

0.52

3⁄

4

278

0.57

0.41

5⁄

16

209

0.43

0.31

1⁄

1⁄

30

1⁄

28

1⁄

26

1⁄

24

1⁄

22

1⁄

20

1⁄

18


Weld Size, in.

0.72

1⁄

32


603

16

1⁄

34

φRn, kips

Welds B (70 ksi)

1⁄ 3⁄

φRn, kips



8

558

0.43

0.31

16

465

0.36

0.26

4

372

0.29

0.21

8

523

0.43

0.31

16

436

0.36

0.26

4

349

0.29

0.21

8

487

0.43

0.31

16

406

0.36

0.26

4

325

0.29

0.21

8

452

0.43

0.31

16

376

0.36

0.26

4

301

0.29

0.21

8

416

0.43

0.31

16

347

0.36

0.26

4

277

0.29

0.21

8

380

0.43

0.31

16

317

0.36

0.26

4

253

0.29

0.21

8

344

0.43

0.31

16

286

0.36

0.26

4

229

0.29

0.21

8

307

0.43

0.31

16

256

0.36

0.26

4

205

0.29

0.21

8

271

0.43

0.31

16

226

0.36

0.26

4

181

0.29

0.21

8

235

0.43

0.31

16

196

0.36

0.26

157

0.29

0.21

4


9 - 90



k (min.)

Table 9-4 (cont.). All-Welded Double-Angle Connections

½

L

Weld B

3

k (min.)

Web thickness 4 in. for L > 18 in. (typ.) 3 in. for L < 18 in. (typ.)

Weld A

Welds A (70 ksi)

L 16

Weld Size, in. 5⁄

0.52

3⁄

4

255

0.57

0.41

5⁄

3⁄

16

191

0.43

0.31

1⁄

5⁄

16

546

0.72

0.52

3⁄

4

437

0.57

0.41

5⁄

3⁄

16

328

0.43

0.31

1⁄

5⁄

16

516

0.72

0.52

3⁄

4

413

0.57

0.41

5⁄

3⁄

16

310

0.43

0.31

1⁄

4

5⁄

16

8

1⁄

10

8

7

6

0.26

4

148

0.29

0.21

8

185

0.43

0.31

16

154

0.36

0.26

4

123

0.29

0.21

8

149

0.43

0.31

16

124

0.36

0.26

0.29

0.21

0.52

3⁄

0.41

5⁄

3⁄

16

292

0.43

0.31

1⁄

5⁄

16

459

0.72

0.52

3⁄

4

367

0.57

0.41

5⁄

3⁄

16

275

0.43

0.31

1⁄

5⁄

16

432

0.72

0.52

3⁄

4

346

0.57

0.41

5⁄

3⁄

16

259

0.43

0.31

1⁄

5⁄

16

404

0.72

0.52

3⁄

4

323

0.57

0.41

5⁄

3⁄

16

242

0.43

0.31

1⁄

5⁄

16

376

0.72

0.52

3⁄

4

301

0.57

0.41

5⁄

3⁄

16

226

0.43

0.31

1⁄

5⁄

16

348

0.72

0.52

3⁄

4

278

0.57

0.41

5⁄

3⁄

16

209

0.43

0.31

1⁄

5⁄

16

318

0.72

0.52

3⁄

4

255

0.57

0.41

5⁄

16

191

0.43

0.31

1⁄

1⁄

4

0.31

0.36

0.57

1⁄

5

0.43

185

0.72

1⁄

1⁄ 3⁄


16

390

1⁄


222

487

1⁄

φRn, kips

8

4

1⁄

9


Weld Size, in.

0.72

1⁄

12


318

16

1⁄

14

φRn, kips

Welds B (70 ksi)

99.3

0.43

0.31

94.6

0.36

0.26

4

75.7

0.29

0.21

8

96.2

0.43

0.31

16

80.2

0.36

0.26

4

64.2

0.29

0.21

8

79.5

0.43

0.31

16

66.3

0.36

0.26

4

53.0

0.29

0.21

8

63.6

0.43

0.31

16

53.0

0.36

0.26

4

42.4

0.29

0.21

8

48.7

0.43

0.31

16

40.6

0.36

0.26

4

32.4

0.29

0.21

8

35.1

0.43

0.31

16

29.2

0.36

0.26

4

23.4

0.29

0.21

8

23.2

0.43

0.31

16

19.3

0.36

0.26

15.5

0.29

0.21

16

4

113



9 - 91

Shear End-Plate Connections

A shear end-plate connection is made with a plate length less than the supported beam depth as illustrated in Figure 9-9. The end plate is always shop welded to the beam web with fillet welds on each side, but may be field bolted or welded to the supporting member. Welds connecting the end plate to the beam web should not be returned across the thickness of the beam web at the top or bottom of the end plate because of the danger of creating a notch in the beam web. When the plate is welded to the support, adequate flexibility must be provided in the connection. Line welds are placed along the vertical edges of the plate with a return at the top per LRFD Specification Section J2.2b. Note that welding across the entire top of the plate must be avoided as it would inhibit the flexibility and, therefore, the necessary end rotation of the connection; the performance of the resulting connection is unpredictable. The use of steels with Fy greater than 36 ksi for the end plate should be based on an engineering investigation that confirms that adequate flexibility will be provided. The strength and end-rotation characteristics of the shear end-plate connection will closely approximate that of the double-angle connection for similar thicknesses, gage lines, and length of connection. Design Checks

The design strengths of the bolts and/or welds and connected elements must be determined in accordance with the LRFD Specification; the applicable limit states are discussed in Part 8. Note that the limit state of shear yielding of the beam web must be checked along the length of weld connecting the end plate to the beam web. In all cases, the design strength φRn must equal or exceed the required strength Ru. Recommended End-Plate Dimensions

End plates should be designed with a plate thickness between 1⁄4-in. and 3⁄8-in., inclusive. The gage g should be between 31⁄2-in. and 51⁄2-in., inclusive, with top and bottom edge distances of 11⁄4-in.; lesser values of edge distance should be avoided. Shop and Field Practices

Shear end-plate connections may be made to the flanges of supporting columns and to the webs of supporting girders. Because of bolting and welding clearances, shear end-plate connections may not be suitable for connections to the webs of W8 columns, unless gages are reduced, and may be impossible for W6 columns.

Figure 9-9. Shear end-plate connections. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 92


When framing to a column flange, provision must be made for possible mill variation in the depth of the columns. The beam length could be shortened to provide for mill overrun and shims could be furnished at the appropriate intervals to fill the resulting gaps or to provide for mill underrun; in general shims are not required except for fairly long runs (i.e., six or more bays of framing). Shear end-plate connections require close control in cutting the beam to the proper length and in squaring the beam ends such that both end plates are parallel. Additionally, any beam camber must not result in out-of-square end plates which make erection and field fit-up difficult. Bolted/Welded Shear End-Plate Connections

Tables 9-5 are design aids for shear end-plate connections bolted to the supporting member and welded to the supported beam. Design strengths are tabulated for supported and supporting member material with Fy = 36 ksi and Fu = 58 ksi and with Fy = 50 ksi and Fu = 65 ksi. End-plate material is assumed to have Fy = 36 ksi and Fu = 58 ksi. Electrode strength is assumed to be 70 ksi. All values, including slip-critical bolt design strengths, are for comparison with factored loads. Tabulated bolt and end-plate design strengths consider the limit states of bolt shear, bolt bearing on the end plate, shear yielding of the end plate, shear rupture of the end plate, and block shear rupture of the end plate. Values are included for 2 through 12 rows of 3⁄4-in., 7⁄8-in., and 1 in. diameter A325 and A490 bolts at 3 in. spacing. End-plate edge distances Lev and Leh are assumed to be 11⁄4-in. Tabulated weld design strengths consider the limit state of weld shear assuming an effective weld length equal to the plate length minus twice the weld size. The tabulated minimum beam web thickness matches the shear yielding strength of the web material with the strength of the weld metal. As developed previously for double-angle connections, tmin =

5.16D Fy

where D is the number of sixteenths-of-an-inch in the weld size. When less than the minimum material thickness is present, the tabulated weld design strength must be reduced by the ratio of the thickness provided to the minimum thickness. Tabulated supporting member design strengths, per inch of flange or web thickness, consider the limit state of bolt bearing.

Example 9-6

Refer to Figure 9-10. Design a shear end-plate connection for the W18×50 beam to W21×62 girder web connection. Ru = 60 kips W18×50 tw = 0.355 in.

d = 17.99 in.

Fy = 50 ksi, Fu = 65 ksi top flange coped 2 in. deep by 41⁄2-in. long W21×62 AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9 - 93

tw = 0.400 in. Fy = 50 ksi, Fu = 65 ksi Use 3⁄4-in. diameter A325-N bolts in standard holes and 70 ksi electrodes. Assume plate material with Fy = 36 ksi and Fu = 58 ksi. Design bolts and end-plate From Table 9-5, for 3⁄4-in. diameter A325-N bolts and end-plate material with Fy = 36 ksi and Fu = 58 ksi, select three rows of bolts and 1⁄4-in. plate thickness φRn = 76.7 kips > 60 kips o.k. Check weld and beam web From Table 9-5, for a 1⁄4-in. weld size and three rows of bolts (an end-plate length of 81⁄2-in.), a 1⁄4-in. weld size provides φRn = 89.1 kips. For beam web material with Fy = 50 ksi, the minimum web thickness is 0.41 in. Since tw = 0.355 in. < 0.41 in. the tabular value must be reduced. Thus,  0.355 in.  φRn = 89.1 kips   0.41 in.  = 77.1 kips > 60 kips o.k. Check flexural yielding on the coped section From Table 8-49, Snet = 23.4 in.3 0.9Fy Snet e

2

φRn =

3 3 3¼

A

W18×50

PL ¼×6×8½

3½

Solution:

Section at A Fig. 9-10. AMERICAN INSTITUTE OF STEEL CONSTRUCTION

9 - 94


0.9 (50 ksi) (23.4 in.4) (41⁄2−in. + 1⁄4−in.) = 222 kips > 60 kips o.k. =

Check local web buckling at the cope 41⁄2−in. c = = 0.250 d 17.99 in. 41⁄2−in. c = = 0.281 ho (17.99 in. − 2 in.) c Since ≤ 1.0, d c f =2  d = 2(0.250) = 0.500 c Since ≤ 1.0, ho 1.65

 ho  k = 2.2   c

 1   = 2.2   0.281 

1.65

= 17.9 2

 tw  φFbc = 23,590   fk  ho  2

 0.355 in.  = 23,590   (0.500) (17.9)  17.99 in. − 2 in. = 104 ksi φFbc Snet φRn =

e

(104 ksi) (23.4 in.3) = (41⁄2−in. + 1⁄4−in.) = 512 kips > 60 kips o.k. Check supporting girder web: From Table 9-5, for three rows of bolts and girder material with Fu = 65 ksi, φRn = (527 kips/in.)(0.400 in.) = 211 kips > 60 kips o.k. The connection, as summarized in Figure 9-10, is adequate. AMERICAN INSTITUTE OF STEEL CONSTRUCTION


9 - 95 3⁄ -in. 4


Diameter Bolts 12 Rows W44

Table 9-5. Bolted/Welded Shear End-Plate Connections Bolt and End-Plate Design Strength, kips End-Plate Thickness, in.

ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

326

382

382

X

—

326

408

477

SC

STD

251

251

251

OVS

213

213

213

SSLT

213

213

213

STD

326

380

380

OVS

307

323

323

SSLT

323

323

323

N

—

326

408

477

X

—

326

408

489

SC Class A

STD

313

313

313

OVS

266

266

266

SSLT

266

266

266

STD

326

408

475

OVS

307

383

403

SSLT

326

403

403

Class A

SC Class B

A490

SC Class B

Weld (70 ksi) and Beam Web Design Strength, kips Minimum Beam Web Thickness, in. 70 ksi Weld Size, in.

φRn, kips

36

50

3⁄ 16

293

0.43

0.31

Support Design Strength per Inch Thickness, kips/in.

1⁄ 4

390

0.57

0.41

Fu, ksi

5⁄ 16

485

0.72

0.52

58

65

3⁄ 8

580

0.86

0.62

1879

2106

Fy, ksi

STD = Standard holes OVS = Oversized holes SSLT = Short slotted holes transverse to direction of load



9 - 96

SIMPLE SHEAR AND PR MOMENT CONNECTIONS 3⁄ -in. 4

Diameter Bolts

Fy = 36 ksi

11 Rows

Fy = 58 ksi

W44, 40

Table 9-5 (cont.). Bolted/Welded Shear End-Plate Connections Bolt and End-Plate Design Strength, kips End-Plate Thickness, in.

ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

299

350

350

X

—

299

373

437

SC

STD

230

230

230

OVS

195

195

195

SSLT

195

195

195

STD

299

348

348

OVS

281

296

296

SSLT

296

296

296

N

—

299

373

437

X

—

299

373

448

SC Class A

STD

287

287

287

OVS

244

244

244

SSLT

244

244

244

STD

299

373

435

OVS

281

351

370

SSLT

299

370

370

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

268

0.43

0.31


1⁄ 4

356

0.57

0.41

Fu, ksi

5⁄ 16

444

0.72

0.52

58

65

3⁄ 8

530

0.86

0.62

1723

1931

Fy, ksi





9 - 97 3⁄ -in. 4


Diameter Bolts 10 Rows

W44, 40, 36


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

271

318

318

X

—

271

338

398

SC

STD

209

209

209

OVS

178

178

178

SSLT

178

178

178

STD

271

316

316

OVS

254

269

269

SSLT

269

269

269

N

—

271

338

398

X

—

271

338

406

SC Class A

STD

261

261

261

OVS

222

222

222

SSLT

222

222

222

STD

271

338

396

OVS

254

318

336

SSLT

271

336

336

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

243

0.43

0.31


1⁄ 4

323

0.57

0.41

Fu, ksi

5⁄ 16

402

0.72

0.52

58

65

3⁄ 8

480

0.86

0.62

1566

1755

Fy, ksi




9 - 98

SIMPLE SHEAR AND PR MOMENT CONNECTIONS 3⁄ -in. 4

Fy = 36 ksi

Diameter Bolts

Fy = 58 ksi

9 Rows W44, 40, 36, 33


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

243

286

286

X

—

243

304

358

SC

STD

188

188

188

OVS

160

160

160

SSLT

160

160

160

STD

243

285

285

OVS

228

242

242

SSLT

242

242

242

N

—

243

304

358

X

—

243

304

365

SC Class A

STD

235

235

235

OVS

200

200

200

SSLT

200

200

200

STD

243

304

356

OVS

228

285

303

SSLT

243

303

303

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

218

0.43

0.31


1⁄ 4

290

0.57

0.41

Fu, ksi

5⁄ 16

360

0.72

0.52

58

65

3⁄ 8

430

0.86

0.62

1409

1580

Fy, ksi





9 - 99 3⁄ -in. 4



W44, 40, 36, 33, 30


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

215

254

254

X

—

215

269

318

SC

STD

167

167

167

OVS

142

142

142

SSLT

142

142

142

STD

215

253

253

OVS

202

215

215

SSLT

215

215

215

N

—

215

269

318

X

—

215

269

323

SC Class A

STD

209

209

209

OVS

178

178

178

SSLT

178

178

178

STD

215

269

316

OVS

202

253

269

SSLT

215

269

269

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

193

0.43

0.31


1⁄ 4

256

0.57

0.41

Fu, ksi

5⁄ 16

318

0.72

0.52

58

65

3⁄ 8

380

0.86

0.62

1253

1404

Fy, ksi




9 - 100


3⁄ -in. 4

Fy = 36 ksi

Diameter Bolts

Fy = 58 ksi

7 Rows W44, 40, 36, 33, 30, 27, 24 S24


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

188

223

223

X

—

188

234

278

SC

STD

146

146

146

OVS

124

124

124

SSLT

124

124

124

STD

188

221

221

OVS

176

188

188

SSLT

188

188

188

N

—

188

234

278

X

—

188

234

281

SC

STD

183

183

183

OVS

155

155

155

SSLT

155

155

155

STD

188

234

277

OVS

176

220

235

SSLT

188

234

235

Class A

SC Class B

A490

Class A

SC Class B


φRn, kips

36

50

3⁄ 16

168

0.43

0.31


1⁄ 4

223

0.57

0.41

Fu, ksi

5⁄ 16

277

0.72

0.52

58

65

3⁄ 8

330

0.86

0.62

1096

1229

Fy, ksi





9 - 101 3⁄ -in. 4



W44, 40, 36, 33, 30, 27, 24, 21 S24


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

160

191

191

X

—

160

200

239

SC

STD

125

125

125

OVS

107

107

107

SSLT

107

107

107

STD

160

190

190

OVS

150

161

161

SSLT

160

161

161

N

—

160

200

239

X

—

160

200

240

SC

STD

157

157

157

OVS

133

133

133

SSLT

133

133

133

STD

160

200

237

OVS

150

188

202

SSLT

160

200

202

Class A

SC Class B

A490

Class A

SC Class B


φRn, kips

36

50

3⁄ 16

143

0.43

0.31


1⁄ 4

189

0.57

0.41

Fu, ksi

5⁄ 16

235

0.72

0.52

58

65

3⁄ 8

280

0.86

0.62

940

1053

Fy, ksi




9 - 102


3⁄ -in. 4

Fy = 36 ksi

Diameter Bolts

Fy = 58 ksi

5 Rows W30, 27, 24, 21, 18 S24, 20, 18 MC18


ASTM Desig.

Thread Cond.

Hole Type

A325

N

—

132

159

159

X

—

132

165

198

SC Class A

STD

104

104

104

SC Class B

A490

1⁄ 4

5⁄ 16

3⁄ 8

OVS

88.8

88.8

88.8

SSLT

88.8

88.8

88.8

STD

132

158

158

OVS

124

134

134

SSLT

132

134

134

N

—

132

165

198

X

—

132

165

198

SC Class A

STD

131

131

131

OVS

111

111

111

SSLT

111

111

111

STD

132

165

198

OVS

124

155

168

SSLT

132

165

168

SC Class B


φRn, kips

36

50

3⁄ 16

118

0.43

0.31


1⁄ 4

156

0.57

0.41

Fu, ksi

5⁄ 16

193

0.72

0.52

58

65

3⁄ 8

230

0.86

0.62

783

878

Fy, ksi





9 - 103 3⁄ -in. 4



W24, 21, 18, 16 S24, 20, 18, 15 C15 MC18


ASTM Desig.

Thread Cond.

Hole Type

A325

N

—

104

127

127

X

—

104

131

157

SC Class A

STD

83.5

83.5

83.5

OVS

71.0

71.0

71.0

SSLT

71.0

71.0

71.0

SC Class B

A490

1⁄ 4

STD

104

OVS

97.9

5⁄ 16

3⁄ 8

127

127

108

108

SSLT

104

108

108

N

—

104

131

157

X

—

104

131

157

SC Class A

STD

104

104

104

SC Class B

OVS

88.8

88.8

88.8

SSLT

88.8

88.8

88.8

STD

104

OVS

97.9

SSLT

104

131

157

122

134

131

134

Weld (70 ksi) and Beam Web Design Strength, kips Minimum Beam Web Thickness, in. 70 ksi Weld Size, in. 3⁄ 16

Fy, ksi

φRn, kips

36

50

92.9

0.43

0.31


Fu, ksi

1⁄ 4

122

0.57

0.41

5⁄ 16

151

0.72

0.52

58

65

3⁄ 8

180

0.86

0.62

626

702




9 - 104


3⁄ -in. 4

Fy = 36 ksi

Diameter Bolts

Fy = 58 ksi

3 Rows W18, 16, 14, 12, 10* S18, 15, 12 C15, 12 MC18, 13, 12 *Limited to W10×12, 15, 17, 19, 22, 26, 30.


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

A325

N

—

76.7

95.4

95.4

X

—

76.7

95.8

SC Class A

STD

62.7

62.7

62.7

OVS

53.3

53.3

53.3

SSLT

53.3

53.3

53.3

SC Class B

A490

3⁄ 8

N

115

STD

76.7

94.9

94.9

OVS

71.8

80.7

80.7

SSLT

76.7

80.7

80.7

—

76.7

95.8

115 115

X

—

76.7

95.8

SC Class A

STD

76.7

78.3

78.3

OVS

66.6

66.6

66.6

SSLT

66.6

66.6

66.6

SC Class B

STD

76.7

95.8

115

OVS

71.8

89.7

101

SSLT

76.7

95.8

101


Fy, ksi

φRn, kips

36

50

3⁄ 16

67.9

0.43

0.31


1⁄ 4

89.1

0.57

0.41

Fu, ksi

5⁄ 16

110

0.72

0.52

58

65

3⁄ 8

129

0.86

0.62

470

527





9 - 105 3⁄ -in. 4



W12, 10, 8 S12, 10, 8 C12, 10, 9, 8 MC13, 12, 10, 9, 8


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

48.9

61.2

63.6

X

—

48.9

61.2

73.4

SC Class A

STD

41.8

41.8

41.8

OVS

35.5

35.5

35.5

SSLT

35.5

35.5

35.5

STD

48.9

61.2

63.3

OVS

45.7

53.8

53.8

SSLT

48.9

53.8

53.8

N

—

48.9

61.2

73.4

X

—

48.9

61.2

73.4

SC Class A

STD

48.9

52.2

52.2

OVS

44.4

44.4

44.4

SSLT

44.4

44.4

44.4

STD

48.9

61.2

73.4

OVS

45.7

57.1

67.2

SSLT

48.9

61.2

67.2

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

42.8

0.43

0.31


1⁄ 4

55.7

0.57

0.41

Fu, ksi

5⁄ 16

67.9

0.72

0.52

58

65

3⁄ 8

79.3

0.86

0.62

313

351

Fy, ksi




9 - 106


7⁄ -in. 8

Diameter Bolts

Fy = 36 ksi

12 Rows

Fy = 58 ksi

W44


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

307

383

460

X

—

307

383

460

SC

STD

307

349

349

OVS

286

297

297

SSLT

297

297

297

STD

307

383

460

OVS

286

358

429

SSLT

307

383

450

N

—

307

383

460

X

—

307

383

460

SC Class A

STD

307

383

439

OVS

286

358

373

SSLT

307

373

373

STD

307

383

460

OVS

286

358

429

SSLT

307

383

460

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

293

0.43

0.31


1⁄ 4

390

0.57

0.41

Fu, ksi

5⁄ 16

485

0.72

0.52

58

65

3⁄ 8

580

0.86

0.62

2192

2457

Fy, ksi





9 - 107 7⁄ -in. 8


Diameter Bolts 11 Rows W44, 40


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

281

351

421

X

—

281

351

421

SC

STD

281

320

320

OVS

262

272

272

SSLT

272

272

272

STD

281

351

421

OVS

262

327

393

SSLT

281

351

412

N

—

281

351

421

X

—

281

351

421

SC Class A

STD

281

351

402

OVS

262

327

342

SSLT

281

342

342

STD

281

351

421

OVS

262

327

393

SSLT

281

351

421

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

268

0.43

0.31


1⁄ 4

356

0.57

0.41

Fu, ksi

5⁄ 16

444

0.72

0.52

58

65

3⁄ 8

530

0.86

0.62

2010

2252

Fy, ksi




9 - 108


7⁄ -in. 8

Diameter Bolts

Fy = 36 ksi

10 Rows

Fy = 58 ksi

W44, 40, 36


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

254

318

382

X

—

254

318

382

SC

STD

254

291

291

OVS

238

247

247

SSLT

247

247

247

STD

254

318

382

OVS

238

297

356

SSLT

254

318

375

N

—

254

318

382

X

—

254

318

382

SC Class A

STD

254

318

365

OVS

238

297

311

SSLT

254

311

311

STD

254

318

382

OVS

238

297

356

SSLT

254

318

382

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

243

0.43

0.31


1⁄ 4

323

0.57

0.41

Fu, ksi

5⁄ 16

402

0.72

0.52

58

65

3⁄ 8

480

0.86

0.62

1827

2048

Fy, ksi





9 - 109 7⁄ -in. 8



W44, 40, 36, 33


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

228

285

343

X

—

228

285

343

SC

STD

228

262

262

OVS

213

223

223

SSLT

223

223

223

STD

228

285

343

OVS

213

266

320

SSLT

228

285

337

N

—

228

285

343

X

—

228

285

343

SC Class A

STD

228

285

329

OVS

213

266

280

SSLT

228

280

280

STD

228

285

343

OVS

213

266

320

SSLT

228

285

343

Class A

SC Class B

A490

SC Class B


φRn, kips

36

50

3⁄ 16

218

0.43

0.31


1⁄ 4

290

0.57

0.41

Fu, ksi

5⁄ 16

360

0.72

0.52

58

65

3⁄ 8

430

0.86

0.62

1644

1843

Fy, ksi




9 - 110


7⁄ -in. 8

Fy = 36 ksi

Diameter Bolts

Fy = 58 ksi

8 Rows W44, 40, 36, 33, 30


ASTM Desig.

Thread Cond.

Hole Type

1⁄ 4

5⁄ 16

3⁄ 8

A325

N

—

202

253

303

X

—

202

253

303

SC

STD

202

233

233

OVS

189

198

198

SSLT

198

198

198

STD

Load & Resistance Factor Design: Manual of Steel Construction

Read more

Load & Resistance Factor Design: Manual of Steel Construction, Volume-I & Volume II, Connections

Read more

Steel Manual

Read more

Detailing for Steel Construction

Read more

Design of steel structures

Read more

Construction manual

Read more

Steel Design

Read more

Analysis, Design and Construction of Steel Space Frames

Read more

White Resistance Manual

Read more

Electrical Load Forecasting: Modeling and Model Construction

Read more

Steel Detailers' Manual

Read more

Steel Detailers' Manual

Read more

Steel Designers Manual

Read more

Practical Design of Steel Structures

Read more

steel structures - design of joints

Read more

Glass Construction Manual

Read more

API RP 2A-LRFD Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Load and Resistance Factor Design

Read more

Construction Superintendent Operations Manual

Read more

Brick and Block Construction: Masonry Construction Manual

Read more

Masonry Construction Manual

Read more

Cold formed Steel Design

Read more

Instructor Solutions - Steel Design

Read more

Architectural Design in Steel

Read more

Steel Bridge Construction, Myths & Realities D432-07 (2008)

Read more

Design Manual

Read more

Aluminium design and construction

Read more

Aluminium design and construction

Read more

Support & Resistance Simplified

Read more

Steel Forgings: Design, Production, Selection, Testing and Application, Manual 53

Read more

Aluminium Design and Construction

Read more

Load & Resistance Factor Design: Manual of Steel Construction

Load & Resistance Factor Design: Manual of Steel Construction

Load & Resistance Factor Design: Manual of Steel Construction, Volume-I & Volume II, Connections

Steel Manual

Detailing for Steel Construction

Design of steel structures

Construction manual

Steel Design

Analysis, Design and Construction of Steel Space Frames

White Resistance Manual

Electrical Load Forecasting: Modeling and Model Construction

Steel Detailers' Manual

Steel Detailers' Manual

Steel Designers Manual

Practical Design of Steel Structures

steel structures - design of joints

Glass Construction Manual

API RP 2A-LRFD Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Load and Resistance Factor Design

Construction Superintendent Operations Manual

Brick and Block Construction: Masonry Construction Manual

Masonry Construction Manual

Cold formed Steel Design

Instructor Solutions - Steel Design

Architectural Design in Steel

Steel Bridge Construction, Myths & Realities D432-07 (2008)

Design Manual

Aluminium design and construction

Aluminium design and construction

Support & Resistance Simplified

Steel Forgings: Design, Production, Selection, Testing and Application, Manual 53

Aluminium Design and Construction

Recommend Documents