Structures Subjected to Repeated Loading: Stability and strength (Stability and Strength)

STRUCTURES SUBJECTED TO REPEATED LOADING Stability and Strength STRUCTURES SUBJECTED TO REPEATED LOADING Stability a...

Author: R. Narayanan

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STRUCTURES SUBJECTED TO REPEATED LOADING

Stability and Strength

STRUCTURES SUBJECTED TO REPEATED LOADING Stability and Strength STABILITY AND STRENGTH A Series of Books Reviewing Advances in Structural Engineering Edited by R.Narayanan, The Steel Construction Institute, Silwood Park, Ascot, Berkshire SL5 7QN, UK The inspiration for this series comes from the recognition of the significant advances made in our understanding of the behaviour of structures as a result of research during the past decade. These advances have, in turn, set new trends and caused major changes in the design codes in many countries. Even the philosophy of design has seen a major shift from the permissible stress basis to the concepts of limit state. Much research effort continues to be directed towards a better understanding of the complex behaviour of structures in the post-elastic, post-buckling and ultimate stages. Nevertheless, the ultimate benefit to be derived from the substantially improved knowledge depends on its effective implementation. Much needs to be done to bridge the gap between the results of research and their effective use by practitioners of the art. The purpose of these books is to explain the current theories and to present material which has been (or will be) influential in the generation of design specifications. Each volume in the series is devoted to a central theme and contains a number of papers of the ‘state-of-the-art’ type on various topics within the selected theme. The aim is to present articles concerned with current developments along with sufficient introductory material, so that a graduate engineer with some basic analytical background and familiarity with structural stability concepts can follow it without having to undertake any substantial background reading. An effort has been made to limit the treatment to advances of practical significance and avoid lengthy theoretical discussions. Throughout the series emphasis has been given to the international aspects of structural steel research. In terms of both design codes and safety parameters, the experiences and practices in Europe, North America and Japan are widely referenced. All the authors are well known experts who are internationally recognised for their contributions in the relevant fields. Titles of published volumes: (1) Axially Compressed Structures (2) Plated Structures (3) Beams and Beam Columns

iii

(4) Shell Structures (5) Steel Framed Structures (6) Concrete Framed Structures (7) Steel-Concrete Composite Structures (8) Structural Connections

STRUCTURES SUBJECTED TO REPEATED LOADING Stability and Strength Edited by

R.NARAYANAN M.Sc. (Eng.), Ph.D., D.I.C., C.Eng., F.I.C.E., F.I.Struct.E., Manager (Education and Publications), The Steel Construction Institute, Ascot, United Kingdom. and T.M.ROBERTS B.Sc., Ph.D., C.Eng., M.I.C.E, M.I.Struct.E School of Engineering, University of Wales College of Cardiff, United Kingdom.

ELSEVIER APPLIED SCIENCE LONDON and NEW YORK

ELSEVIER SCIENCE PUBLISHERS LTD Crown House, Linton Road, Barking, Essex IG11 8JU, England This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Sole Distributor in the USA and Canada ELSEVIER SCIENCE PUBLISHING CO., INC. 655 Avenue of the Americas, New York, NY 10010, USA WITH 18 TABLES AND 153 ILLUSTRATIONS © 1991 ELSEVIER SCIENCE PUBLISHERS LTD British Library Cataloguing in Publication Data Structures subjected to repeated loading. 1. Structures. Loads. Effects I. Narayanan, R. II. Roberts, T.M. III. Series 624.172 ISBN 0-203-97504-9 Master e-book ISBN

ISBN 1-85166-567-6 (Print Edition) Library of Congress Cataloging in Publication Data Structures subjected to repeated loading: stability and strength edited by R.Narayanan and T.M.Roberts. p. cm. Includes bibliographical references and index. 1. Structural stability. 2. Strength of materials. I. Narayanan. R. II. Roberts. T.M., 1946TA656.S78 1991 624.1′7–dc20 90–14075 CIP No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Special regulations for readers in the USA This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside the USA, should be referred to the publisher. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission of the publisher.

PREFACE

It gives us great pleasure to write a short preface to the book on Structures Subjected to Repeated Loading, the ninth in the series of volumes on Stability and Strength of Structures. Understanding fatigue damage consequent on repeated loading of structures, becomes extremely important in view of the relatively low margins of safety implicit in present day designs using Limit State Codes. Fatigue is a complex phenomenon, influenced by a large number of independent and often unquantifiable variables. Much research has been carried out in recent years, with a view to developing satisfactory techniques of quantifying fatigue effects, which would aid in safe and acceptable designs. The book begins with an introduction to fatigue assessment and fracture mechanics applicable to structures; this is followed by state-of-the-art reports on several facets of the effect of repeated loading. As with other books in the series, the book is aimed at the graduate student and the designer; it can be followed by anyone with a basic knowledge of structural analysis. We have continued the practice of inviting internationally acclaimed experts to make contributions on several selected areas. As Editors, we wish to express our gratitude to all the contributors for their willing cooperation in producing this volume. We sincerely hope that the book will be stimulating and useful to its readers. R.NARAYANAN T.M.ROBERTS

CONTENTS

Preface List of Contributors

vi viii

1.

Fatigue Assessment and Fracture Mechanics in Structural Engineering I.F.C.SMITH

1

2.

Fatigue Damage Accumulation under Varying-Amplitude Loads F.J.Z.WERNEMAN

22

3.

Effect of Residual Stresses on the Fatigue of Welds S.SARKANI

48

4.

Fatigue Cracking in Plate and Box Girders Y.MAEDAY.KAWAI & I.OKURA

63

5.

Fatigue Strength of Adhesively Bonded Cover Plates P.ALBRECHT

90

6.

Fatigue of Tubular Joints in Offshore Structures J.J.A.TOLLOCZKO

111

7.

Case Studies and Repair of Fatigue-Damaged Bridge Structures J.W.FISHER & C.C.MENZEMER

158

8.

Reinforced-Concrete Frames Subjected to Cyclic Load C.MEYER

182

9.

Unstiffened Steel Plate Shear Walls G.L.KULAK

210

Index

245

LIST OF CONTRIBUTORS

P.ALBRECHT Department of Civil Engineering, University of Maryland, College Park, Maryland 20742, USA J.W.FISHER ATLSS Engineering Research Center, Lehigh University, Pennsylvania 18075, USA Y.KAWAI Research and Development Center, Kawasaki Steel Corporation, Chiba, Japan G.L.KULAK Department of Civil Engineering, University of Alberta, Edmonton, Canada, T6G 2G7 Y.MAEDA, Emeritus Professor, Osaka University, 2–1, Yamadaoka, Suita, Osaka 565, Japan C.C.MENZEMER ATLSS Scholar, Lehigh University, Pennsylvania 18015, USA C.MEYER Department of Civil Engineering and Engineering Mechanics, Columbia University, New York 10027, USA I.OKURA Department of Civil Engineering, Osaka University, 2–1, Yamadaoka, Suita, Osaka 565, Japan S.SARKANI Department of Civil, Mechanical and Environmental Engineering, The George Washington University, Washington, DC 20052, USA

ix

I.F.C.SMITH ICOM, Federal Institute of Technology, Lausanne, Switzerland J.J.A.TOLLOCZKO Offshore Engineering Division, The Steel Construction Institute, Silwood Park, Ascot, Berkshire SL5 7QN, UK F.J.ZWERNEMAN School of Civil Engineering, Oklahoma State University, Stillwater, Oklahoma 74078–0327, USA

Chapter 1 FATIGUE ASSESSMENT AND FRACTURE MECHANICS IN STRUCTURAL ENGINEERING I.F.C.SMITH ICOM, Federal Institute of Technology, Lausanne, Switzerland

SUMMARY Although fatigue is a complicated phenomenon, special conditions in structural engineering create opportunities for simplification. Variations in material properties of steels and environmental conditions are usually of secondary importance owing to built-in stresses, discontinuities and stress gradients. Thus, the two most important factors affecting the fatigue life of a structure are geometry and stress range. Also, structures which are subject to fatigue loading need to be designed, fabricated, erected, inspected and maintained according to special criteria. A summary of basic fracture-mechanics principles is presented and some applications in structural engineering are examined. Although fracturemechanics concepts are gaining acceptance in structural engineering, they should be used with caution when assessing large structures. Often, the complexity of the design problem precludes the use of sophisticated approaches which are employed in other fields. Aside from their obvious utility for strength calculations, fracture-mechanics principles increase the qualitative understanding of structures containing cracks. Also, fracture-mechanics methods enable quantitative analyses to be undertaken for associated problems which were previously solved more subjectively—such as decisions regarding inspection and repair. NOTATION a ai

Crack length Initial crack length

2 I.F.C.SMITH

acr a0 ′a A CTOD d E J K ′K ′ Kth l m n N r t W Y ′ v σ σy σ yy ′σ

Critical crack length Detectable crack length Crack length increment Crack-growth constant Crack-tip-opening displacement Crack opening Young’s modulus J integral Stress-intensity factor Stress-intensity factor range Threshold stress intensity factor range Length of attachment Crack growth constant Number of inspections Number of cycles Polar coordinate from crack tip Plate thickness Correction for stress field Correction for crack shape and plate edges Polar coordinate from crack tip Poissons’s ratio Stress remote from a crack Effective yield strength Local stress in y-direction Stress range 1.1 INTRODUCTION

Fatigue cracking may threaten structures and consequently fatigue has long been recognised as a design constraint. Failures may occur in bridges, cranes, gantry girders, chimneys, transmission towers and marine platforms. The cost of repairing a fatigue-damaged structure may exceed the initial capital investment. Therefore, fatigue assessments should be based upon a rational evaluation of all factors which are relevant to the intended use. Although standard fatigue-assessment methods provide sufficient information for good design under most cases of repeated loading, their simplicity precludes explicit consideration of all factors. For example, standard methods cannot evaluate the effects of cracks in structural elements. Fracture-mechanics (henceforth FM) principles enable a more detailed assessment of such elements to be carried out. This chapter reviews fatigue assessments for cases of large structures and discusses considerations necessary for good fatigue design and effective

FATIGUE ASSESSMENT AND FRACTURE MECHANICS 3

management during post-design stages. Also, basic fracture mechanics concepts are described. Some problems which are particular to structural engineering are outlined, and examples of FM applications are presented. Although the occurrence of fatigue cracking in concrete structures is increasing, discussions in this chapter are limited to steel structures for two reasons. Firstly, there is general agreement in the engineering community regarding fatigue assessment of steel structures, while corresponding procedures for concrete structures vary greatly. Secondly, FM analyses for steel structures are better verified and probably more versatile than for concrete structures. 1.2 FACTORS AFFECTING FATIGUE LIFE OF STRUCTURES 1.2.1 Four Categories of Factors The factors which influence the fatigue life of structures can be placed into the following four categories: – – – –

geometry, stresses, environment, material properties.

Geometrical factors determine where and how quickly fatigue failure occurs. These factors incude span, spacing between elements, orientation of elements, joint geometry, stress concentrations, as well as small discontinuities such as scratches, surface pitting, grinding marks, non-fatigue cracks, and weldingprocess discontinuities. Factors in the stress category influence whether or not fatigue cracking occurs. Unfortunately, stress parameters often possess the greatest statistical uncertainty in a fatigue analysis. Stress range, mean stress, residual stresses, stress sequence, impact stresses, lack-of-fit stresses, stresses due to thermal gradients, and movement-induced stresses are important factors. Environmental factors can induce fatigue cracking in an otherwise safe structure and can accelerate fatigue cracking through corrosion and creep mechanisms near the crack tip. Environmental factors include the effects of corrosive liquids or gases, temperature, humidity, hydrogen and irradiation. Fortunately in most cases, environmental factors can be neglected if protection is provided. Material properties determine how the structure reacts to factors in the other three categories. Material properties include stress-strain behaviour, grain size

4 I.F.C.SMITH

and shape, hardness, energy absorption, chemical composition, homogeneity and microstructural discontinuities. Different combinations of these factors create a wide range of fatiguecracking conditions. Design strategies for one case may not be appropriate for other cases if the most dominant factors are not the same. For example, it would be unwise to employ a given fatigue-design strategy for bridge design because it is known that the strategy suffices for aircraft. Fortunately, the factors associated with fatigue cracking in large structures create a situation where simplifications are possible. These factors are discussed next. 1.2.2 Simplifications for Large Welded Steel Structures In large welded steel structures, it is very difficult to avoid built-in stresses at crack locations. Residual stresses due to welding, stresses caused by unwanted deformations during fabrication, and lack-of-fit problems during erection can create situations where stresses near possible crack locations approach the tensile yield strength of the material. In addition, the configuration of the structure may contribute to high stresses. Fig. 1.1 shows three magnitudes of geometrical effects. The highest magnitude, the overall structural configuration, can be such that differential settlement of supports, thermal gradients in the structure, and the behaviour of expansion joints contribute to these built-in stresses. Considering all these factors leads to the conclusion that even without applied loading, real levels of stress are unknown in all but very simple structures. Furthermore, it should be assumed that at possible crack locations, stresses near the tensile yield strength of the steel are present. Applied mean stress levels and stress sequence have little importance under such conditions. Therefore, applied stress range becomes the only significant factor in the stress category. Many international and national design documents already incorporate this simplification, e.g. ECCS (1985), Fisher (1977), SIA (1979), BSI (1980a). Other simplications are possible owing to the special geometrical factors for the middle and smallest magnitudes of Fig. 1.1. For the middle magnitude, local stress concentrations are caused by details such as welded attachments. Generally, effects of such attachments are very important; they may magnify the applied stress by more than five times. Fortunately, such magnification is extremely localised in an area that, for typical stress ranges, is only tens of grains large. Elsewhere, the stress ranges are not magnified to similar degrees. This means that the structure behaves elastically during fatigue loading. For the smallest magnitude (Fig 1.1) small weld-induced discontinuities are unavoidable. Even rigorous quality requirements have been unsuccessful in removing crack-like discontinuities less than 0·1 mm deep (Smith & Smith, 1982). Such discontinuities are not easily detectable (their depth equals the thickness of an average piece of paper) but their presence eliminates any significant period of crack nucleation and short-crack growth (sometimes called crack initiation). This


FIG. 1.1. Three magnitudes of geometric effects.

is especially true when built-in stresses are high; as discussed above, this is generally the case. The combination of these factors has two more important consequences. Firstly, the fatigue process in the presence of crack-like discontinuities subject to elastic stress ranges, sometimes called long-crack propagation, leads to propagation rates which are known to be indepen dent of steel quality and yield strength for steels employed in large welded structures (Rolfe & Barsom, 1977). Therefore, changes in material properties do not affect fatigue life—provided that the steel is of standard structural quality. This provides an important simplification as well as a justification for international harmonisation of design guidelines. Secondly, the presence of crack-like discontinuities and predominantly elastic conditions reduces the effect of changes in environmental conditions. Exposure to a mildly corrosive environment, such as the humidity under a bridge over a river, does not usually mean that the fatigue life is reduced. For the same reason, the usual initial corrosion which is expected of weathering steels does not reduce fatigue lives of unpainted structures made from this type of steel. 1.2.3 The Two Most Important Factors From the above discussion, it can be concluded that only two factors affect the fatigue life of large welded steel structures: stress range and geometry. This has been verified experimentally (e.g. Hirt et al., 1971). Figure 1.2 summarises the

6 I.F.C.SMITH

FIG. 1.2. Special conditions leading to simplifications of fatigue assessments.

reasons behind this conclusion. Note that such simplifications are possible only under certain conditions; generalisation to all cases of fatigue cracking is not justified. For example, assessment of offshore structures requires a more detailed consideration of environmental effects. Since many factors have a reduced influence upon fatigue lives of large structures, the relative importance of stress range and geometry is increased. Moreover, applied stress ranges are increasing in structures owing to the trend toward use of higher-strength steels. Also, an unanticipated increase in use generally increases the number and magnitude of stress ranges during the life of the structure. Therefore, geometrical factors are becoming even more important to the integrity of structures subject to fatigue loading. In large structures, the severity of geometrical effects can be minimized by judicious selection of dimensions and configurations for the largest and middle magnitudes of Fig. 1.1. Such selection can be called ‘fatigue-resistant design’ and it is discussed further in the next section. 1.3 FATIGUE-RESISTANT DESIGN Good design practice involves special considerations in three areas: structural systems, detailing, and factors affecting fabrication and erection. The following discussion provides recommendations for good fatigue design according to these areas. These recommendations provide examples for bridges only; a complete documentation for all types of structures is beyond the scope of this chapter.


1.3.1 Fatigue-Resistant Structural Systems Structures need to be designed so that they are not subject to large numbers of high stress ranges. For example, a railway-bridge structure should not be sensitive to wheel loads and dynamic effects. Wheel-load sensitivity can be minimised through avoiding short members, employing composite construction and using ballast. On highway bridges, dynamic effects are reduced if smooth surfaces over joints can be maintained throughout the life of the structure. This may require a review of foundation-design criteria and specification of reliable expansion joints. Also, the number of expansion joints should be kept to a minimum; they should be employed only if analysis demonstrates a need. Continuousspan bridges need fewer expansion joints and, therefore, they experience less dynamic stress magnification. Components in structures should be analysed correctly for their service behaviour. For example, partial fixity provided at the connection between two members should be included in the assessment. Although such fixity, such as that between a beam and a diaphragm in a bridge, is ignored for ultimate-limitstate assessments, service stresses can induce fatigue cracking at such connections. The model used during analysis should also be able to cope with deformation-induced stresses. For example, simplified force models do not usually provide enough information for analysis of skew bridges. It is important that structures subjected to fatigue loading are designed so that they are easy to inspect and to maintain. Fatigue critical details should be identified and remain accessible for regular inspection. The structure should be protected from corrosive effects of rain water, de-icing salts and drainage which does not feed directly into the ground. Decks which are correctly sealed and drainage pipes which avoid contact of the structure with rain water prevent corrosion-induced fatigue cracking. Finally, structures should be designed to be tolerant of potential fatigue cracks. Certain structural systems can redistribute loads; in the event of local fatigue cracking, this redistribution may provide an early warning of a potential overall failure. Other engineering fields have termed this ‘fail-safe design’. Several damage scenarios should be investigated, and the remaining redundancy of the structure after fatigue cracking has occurred should be evaluated. Furthermore, local failure may lead to unacceptable service behaviour; for example, fatigue cracking of a railway-bridge element supporting a rail may lead to derailment. 1.3.2 Fatigue-Resistant Detailing In general terms, details should be designed so that they have low stress concentrations and small discontinuities. This is achieved through avoiding

8 I.F.C.SMITH

details such as: web gaps at the end of stiffeners, sudden changes in stiffness, partial-penetration welds, load-carrying fillet welds, and intermittent welding. Note that non-load-carrying attachments may in fact carry enough load to induce cracking. In such cases, built-in defects caused by lack of penetration are potential fatigue-crack sites. In all cases, the key to fatigue-resistant detailing is a rational consideration of factors affecting fatigue life early in the design process. A fatigue check performed after details have been designed to other criteria may result in a costly and inadequate structure. 1.3.3 Factors Affecting Fabrication and Erection At the design stage, fabricators and contractors should be consulted. Such consultation identifies the most appropriate specifications for criteria such as quality assurance. Also, this contact provides an opportunity to point out areas which are most sensitive to fatigue cracking, and to discuss special precautions. Finally, and probably most important, establishing communication links at an early stage encourages the transfer of new information concerning the structure during post-design stages. Such information may originate from modifications during erection, or from last-minute decisions regarding a change in use of the structure. In all cases, the designer should monitor closely fabrication and erection procedures. No modifications to designs should be made without approval of all parties. 1.4 FRACTURE MECHANICS IN STRUCTURAL ENGINEERING Originally, most structural-engineering applications of FM were limited to inservice assessments of structures. Sizes and shapes of discontinuities have been correlated to fabrication quality levels (BSI, 1980b). Consequently, code-drafting committees have employed FM calculations in order to evaluate specifications for quality assurance. Furthermore, when no specifications were applicable, FM has been used directly by the engineer for evaluating non-destructive inspection (henceforth NDI) results. Currently, FM analyses are used to examine a range of other structuralengineering problems. For example, recommendations have been made for FM calculations of inspection intervals (ECCS, 1985) and researchers are performing parametric studies in order to identify important geometric effects. Also, application of FM to failure of concrete structures is being investigated (Wittman, 1984). In spite of all this activity, FM remains a method of analysis which is used primarily by mechanical engineers. Originally used to explain the rupture of


glass specimens (Griffith, 1921), the development of FM began in the 1940s after several catastrophic failures of ships and pressure vessels. Today, FM is used to assess elements in SPacecraft, robotic structures, turbine blades, pipelines, automotive parts, and many other components. Academic journals, conference proceedings, trade magazines, and several organizations propose a multitude of design approaches. Unfortunately, much of this information cannot be used by structural engineers. In order to decide what portion of the information is relevant to the particular application, design engineers are often confused by complicated mathematics, an array of formulae, many empirical constants, and different boundary conditions. In addition, a traditional structural-engineering education does not provide the competence to choose the most appropriate approach. Therefore, there is a risk that insufficient information and inappropriate use may result in unsafe and costly decisions. 1.5 BASIC FRACTURE-MECHANICS CONCEPTS For simplicity, discussions are limited to cases where the load is applied remote from crack locations, and normal to crack surfaces (mode 1). Review articles provide more detailed information (e.g. Rolfe, 1977) and several mechanicalengineering books dealing with FM principles and applications have been published recently (e.g. Broek, 1986). 1.5.1 How to Account for a Crack Five cases of a plate containing a crack are shown in Fig. 1.3. It requires no knowledge of FM to agree that cases 1 to 5 are placed in order of increasing severity. Considering case 1 to be the basis for comparison, the following important FM parameters can be identified: (i) a longer crack (case 2); (ii) the crack location at a plate edge (case 3); (iii) the effect of bending (case 4), all of which weaken the plate. Case 5 is of greatest practical importance. A sharp stress concentration, such as an abrupt change in section, in combination with a crack may weaken a plate to a fraction of its uncracked strength. These cases illustrate fundamental parameters considered in all FM analyses. Note also that for these cases, allowable stress criteria used with net-section stresses are unreliable and often dangerous. A magnification of the area around a crack tip in an infinitely wide plate is given in Fig. 1.4. This resembles closely the conditions of case 1 when the crack length is small compared to the plate width. When a remote stress σ is applied, the crack opens a certain distance d. The separated crack surfaces cannot carry any stress. The stress which this cross-sectional area would have carried is diverted to the uncracked area of the plate. This diversion creates a high concentration of stress

10 I.F.C.SMITH

FIG. 1.3. Five different cases of a plate containing a crack.

FIG. 1.4. A crack in an infinitely wide plate.

in the vicinity of the crack tip. Theoretically, this stress is infinite at the crack tip but, in real materials, plastic zones are formed because the strain exceeds the material’s ability to behave elastically. This process, whereby an applied load: (i) causes a crack to open, (ii) relieves crack surfaces of stresses, and (iii) creates crack-tip plastic straining is the fundamental mechanism which weakens structures containing cracks or crack-like discontinuities. How can one describe the stress field near the crack tip? Ignoring plasticity, a mathematical development employing special stress functions provides a solution using the coordinates r and ′ . When ′=0, the stress in the y-direction is (1.1)


provided that the crack length a is much larger than the distance r from the crack tip. The numerator in eqn (1.1) determines the gradient of the (theoretical) stresses as they rise to infinity when r approaches zero. This numerator is called the stress intensity factor K: (1.2) The advantage of this model is that any combination of stress and crack length can be characterised by a single parameter. Other analytical solutions exist for particular geometrical configurations and loading conditions; these are summarised in handbooks (Tada et al., 1985) and compendia (Rooke & Cartwright, 1976). However, many practical cases cannot be solved analytically. In such instances, the following expression is used to approximate K: (1.3) where Y corrects for the presence of plate edges and curved crack fronts, while W corrects for non-uniform local-stress fields caused by the presence of residual stresses, stress concentrations, stress gradients due to thermal effects, etc. Usually, such correction factors are determined using numerical methods. Equation (1.1) is based on linear-elastic material properties and cannot account for the presence of plasticity at the crack tip. Furthermore, stress redistribution due to plasticity alters the stress field outside the crack-tip plastic zone. Nevertheless, if this zone is much smaller, say less than 2% of plate thickness, crack length, and size of uncracked ligament, the stress intensity factor K remains an acceptable model. These limitations are violated in many practical situations. For example, when stress concentrations cause localised plasticity, elasticplastic analyses may be required. The most common analyses use either expressions such as J in order to model the change in potential energy with respect to crack length, or the cracktip opening displacement value, CTOD. Further description of elastic-plastic analyses is available elsewhere (Broek, 1986). Occasionally, an equivalent K is calculated (Irwin, 1983): (1.4) where E is Young’s modulus and σ y is the effective yield strength. When plate dimensions are large enough to restrict behaviour to essentially two-dimensional straining (plane strain), the material constant E is replaced by E/(1−v2) where v is Poisson’s ratio. A further restriction on K exists for small crack lengths where all of the approaches explained above lose their validity. When the size of the crack or initial discontinuity is of the order of the grain size, microstructural properties, such as grain orientation, may influence crack growth (Lankford, 1985). Microstructural FM models may then become necessary. These models are not yet well defined and no generally accepted design rules are available. Usually, if the crack length is greater than about five grain diameters, the models which

12 I.F.C.SMITH

assume an isotropic continuum, e.g. those employing K or J, are sufficiently accurate. 1.5.2 Fatigue Crack Growth Fatigue cracking results from repeated loading and occurs most commonly in structures such as bridges, towers and cranes. Although today’s steel structures use higher-toughness materials and thus are more fracture resistant than ever before, many structural elements remain equally susceptible to fatigue-crack growth. Consequently, if the fracture limit state is attained, it is most likely due to f atigue-crack growth after many years of trouble-free service. Such an occurrence is covered by the definition of the fatigue limit state. In addition to higher-toughness properties, developments of low-alloy and fine-grain microstructures have increased the yield strengths of steels available for construction. Consequently, higher service stresses have been allowed in recently built structures. Furthermore, previous practices of bolting and riveting have been replaced by welding, thereby lowering fatigue strengths of connections. Therefore, modern structures can be more susceptible to fatigue cracking than older structures. Nevertheless, the number of old structures which have exceeded their design life is growing exponentially. The combined effect of these two trends is increasing the importance of fatigue assessments of new and existing structures. All structural elements contain metallurgical or fabrication-related discontinuities, and most include severe stress concentrators such as weld toes. Therefore, fatigue failure is often the result of slow crack growth from a discontinuity at a stress concentration. This growth may begin before the structure is put in service. The stress intensity factor can also represent fatiguecrack growth by adapting eqn (1.3) to account for repeated loading. For a constant-amplitude stress range, ′ σ this becomes (1.5) Equation (1.5) is related empirically to the crack-growth rate, da/dN, which is obtained from the slope of the curve of crack-growth measurements—see Fig. 1.5(a). This slope becomes the ordinate when plotted against ′ K in a doublelogarithmic representation—see Fig. 1.5(b). The values of ′ K are calculated using eqn (1.5) for particular magnitudes of crack length, a. At very low growth rates, the curve for crack growth (in steel) becomes vertical, indicating a crackgrowth threshold at ′ Kth, the threshold stress intensity factor. At higher values of ′ K, the curve straightens to a near-constant slope and becomes vertical again when the fracture toughness is approached at the maximum stress in the cycle. Material properties, stress level, and environment have a greater influence on the vertical ends of the curve shown in Fig. 1.5(b) than in the middle. In this centre portion, the Paris law is useful (Paris & Erdogan, 1963):


FIG. 1.5. The stress-intensity factor and fatigue-crack growth. (a) Crack length vs number of cycles. (b) Crack-growth rate vs the stress-intensity factor range. (c) Magnification of the lower portion of the curve in (b).

(1.6) where A and m are constants which are determined by means of a regression analysis of test data. These constants are reliable when similar materials, loadings, and environments are studied. Also, the regression analysis is dependent upon the domain of crack-growth rates considered since the centre portion of the curve in Fig.l.5(b) is not perfectly straight. Many structural applications involve repeated loading of over one million cycles and this requires a precise knowledge of slow crack-growth rates near ′ Kth—see Fig. 1.5(c). Conservative assessments result when eqn (1.6) is extrapolated into this region. The error resulting from the extrapolation is dependent upon the magnitude of the stress ratio R. This ratio is often used to examine the effects of mean stress on crack growth. When ′ K is much larger than ′ Kth, eqn (1.6) can be integrated to calculate the crack-propagation fatigue life N: (1.7) where ai and acr are the initial and critical crack lengths.

14 I.F.C.SMITH

Variable-amplitude loading may be assessed in a cycle-by-cycle integration. This integration may follow a procedure comparable to the calculation of the equivalent constant-amplitude stress range which is recommended in some design codes. In doing so, interaction effects, called either crack acceleration or crack retardation, are ignored. Many models have been proposed when these effects are important. Further work is needed to identify those models which are most useful for structural-engineering applications. Small crack sizes and excessive plasticity may invalidate models which employ the stress intensity factor. Non-conservative calculations may result if socalled short-crack behaviour occurs (Suresh & Ritchie, 1984). Fortunately, such situations are not common when assessing large structures. Usually, the stress intensity factor remains a useful characterising parameter for conditions of fatigue-crack growth since discontinuities are large and a high percentage of crack growth occurs under conditions where K is valid. Also, structuralengineering applications have particular characteristics which reduce the occurrence of this anom alous behaviour. Some of these characteristics are presented in the following section. 1.6 STRUCTURAL-ENGINEERING APPLICATIONS Structural-engineering applications of FM must consider many variables which are difficult to determine accurately. This difficulty influences the degree of sophistication possible in FM analyses. The following paragraphs outline particular loading and fabrication factors, and discuss the consequences for design assessments. An accurate definition of the loading on a structure can be the greatest challenge in design. Often, the structure is not analysed in detail at the design stage. Usually, impact and other dynamic loadings are poorly defined and out-ofplane movements as well as other so-called secondary effects are neglected. While this may be adequate for traditional static assessments, such omissions are not recommended for FM evaluations. Cracks are very sensitive to local stresses. At typical structural details, local stresses are high in a very small region around potential crack sites. Stresses due to imposed deformations must be considered. Temperature effects due to the sun and other heat sources can transform nominally compressive stresses into tensile stresses. Furthermore, support settlement, frost, and erosion can lead to unexpected tensile stresses in some structures. Discontinuities exist in every structure. Some discontinuities are very sharp, having root radii which were measured by Smith & Smith (1982) to be less than 5μ m. Even if a connection appears to be free of any discontinuity, tiny crack-like imperfections may exist—see Fig. 1.6(a). Post-weld treatments which remove these discontinuities, such as grinding, appear effective (Fig. 1.6(b)). However,


FIG. 1.6. Weld-toe discontinuities. (a) As-welded. (b) After burr grinding. (c) Magnification of (b).

usual quality-assurance procedures may not be able to guarantee that new discontinuities are not introduced (Fig. 1.6(c)). Generally, non-destructive detection of surface discontinuities is impossible if they are less than 0·5 mm deep and difficult if they are less than 5 mm deep. Furthermore, access to the connection can be so restricted that detection of even larger discontinuities is not practicable using the equipment available. Therefore, it is important that designs include detail configurations which avoid inspection difficulties. In any case, it is prohibitively expensive to prove that a large structure is completely free of discontinuities. Most steel structures contain welded connections and attachments. The welding process introduces tensile residual stresses in areas where crack-like discontinuities may exist. Attempts to reduce residual stresses by stress-relieving procedures are not always effective. Furthermore, plate deformations are unavoidable and attempts to correct them may not be successful. Unwanted deformations may lead to lack of fit and additional tensile residual stresses. The previous discussion demonstrated that an accurate FM analysis is complicated in structural engineering. Determination of all stresses is very important; this includes stresses due to all loading events, residual stresses due to fabrication, erection, and service loads, as well as stresses due to imposed deformations. Discontinuities are always present and, consequently, their sizes

16 I.F.C.SMITH

must be measured accurately. In many designs, a poor definition of such important information precludes a high level of analytical sophistication. Fatigue-strength assessments in structural engineering must remain simple for similar reasons. Unknown static-load levels have two conse quences for FM assessments of fatigue-crack growth. First, the effect of compressive loading cannot be excluded from the stress range unless it can be proved that this loading never causes any crack opening. Secondly, the intrinsic threshold stress intensity should be used regardless of the applied mean stress. The possibility of sharp discontinuities prohibits consideration of a crackinitiation stage. Also, steep stress gradients at stress concentrations preclude short-crack-growth effects during typical service loading. Stress concentrations may be high but steep gradients cause their effects to diminish as the cracks increase in length. Thus, simple linear-elastic fracture-mechanics models are sufficient for most structural-engineering designs. 1.6.1 Fracture Mechanics Used as a Qualitative Design Tool Engineering designers rarely use FM as a design tool. Older concepts, such as stress-range (hereafter SN) models, remain the most practical design tools for evaluating many structures. Nevertheless, the concepts of FM enable the designer to increase his qualitative understanding of structures containing cracklike discontinuities. Since more parameters are explicit in FM analyses, designers can identify more easily those parameters which influence the strength of the structure. SN curves do not exist for every possible structural shape and detail. In design guidelines, it is often possible to identify details which are similar to the one under consideration for a given structure. A parametric study using FM is able to determine whether the identification is correct or what extra effects are to be considered. For example, approximately ten years ago, FM analyses showed that increasing the plate thicknesses of connections employing transverse fillet welds (cruciform joints) reduced the fatigue life (Gurney, 1979). After testing verified this trend (e.g. Berge, 1983) new code provisions were proposed (HMSO, 1984). A comparison of theoretical and experimental studies is shown in Fig. 1.7. Relative fatigue lives were calculated by dividing the fatigue life by the value corresponding to a thickness of t=12·5 mm for each case. Similarly, a FM study revealed an increase in fatigue life with increasing plate thickness for longitudinal fillet-welded attachments (Smith & Gurney, 1986). Again, some testing has supported the trend predicted by FM (Hirt, 1979, unpublished data). Initially, a typical fatigue crack propagates very slowly into the plate thickness from the surface, taking the form of a so-called thumbnail crack. The stress concentration, modelled by the parameter W in eqn (1.5), affects the crackgrowth rate. Connections classified in different detail categories (ECCS, 1985) may have different crack-propagation characteristics for the same fatigue life-see


FIG. 1.7. The effect of thickness on fatigue life: comparison of theoretical and experimental studies.

FIG. 1.8. Comparison of fatigue crack-growth behaviour for two different details having the same fatigue life.

Fig. 1.8. The detail with a lower-severity stress concentration (butt-welded plate) consumes a greater percentage of the total fatigue life during the propagation of a crack to a size a than the detail with a higher-severity stress concentra tion (attachment). Thus, the same size crack may be identified earlier at the attachment than at the butt weld. Other factors, such as differences in inspection feasibility, influence the exact timing of crack identification. Nevertheless, such crack-growth behaviour should be considered when fixing inspection intervals. 1.6.2 Quantitative Design Using Fracture Mechanics Many documents provide guidelines for employing FM during structural assessments. The damage-tolerant-design concept is the major justification for

18 I.F.C.SMITH

the use of FM. This concept accepts the possibility that an element remains useful when it has been subject to damage upon fabrication, transportation, or even after several years in service. Damage tolerance is now a standard design concept in some engineering fields; acceptance of certain aspects is apparent in structural engineering. For example, the following terms imply that structural engineers recognise that structures may contain cracks or crack-like discontinuities: (a) Fitness for purpose (acceptance levels) (b) Engineering critical assessment (c) In-service inspection for cracks (d) Remaining fatigue life (e) Strengthening and repair of cracked elements (f) Local vs general fatigue failure (g) Fail-safe fatigue and fracture design Each term implies considerations and calculation which FM can rationalize. A few examples are provided below. The concepts of fitness for purpose and engineering critical assessment imply that discontinuities exist in structures after fabrication. Discontinuities are often rejected during standard fabrication controls although, in reality, the presence of many types of discontinuites does not influence the performance of the structure. The economic consequences of remedial measures can be severe. Qualityassurance clauses in many codes are based more upon accepted standard practice, experience, and detection capability than upon scientific accuracy. FM provides a more analytical method. Often, determination of intervals between in-service inspections does not consider the behaviour of the structure if it were to contain a crack. Crackgrowth curves can be predicted for all critical details in a structure, as shown schematically in Fig. 1.9. Between the crack lengths when first detection is


FIG. 1.9. Inspection intervals which shorten according to the time in service.

possible, a0, and the critical crack size acr, a set number of inspections, n, should be performed. A constant crack-length increment ′ a should be fixed rather than a constant time interval. The number of inspections should be determined using reliability considerations which include crack-detection probabilities and the consequences of failure. More frequent inspections are needed as the structure ages; current practice in some engineering applications reflects this. Therefore, FM calculations offer a more rational approach to an empirical field. As the number of structures which approach or exceed their design life grows exponentially each year, so does the number of structures which have a high probability of containing fatigue cracks. FM analysis will become an essential part of the assessment of these structures and, consequently, the decisions concerning their strengthening, repair and replacement. Recently, cracking in several NorthAmerican steel bridges was reviewed in a collection of case studies (Fisher, 1984). Fatigue-life improvement methods give the designer the possibility of using an otherwise unacceptable structural detail (instead of reducing the stresses). The fatigue life of such a detail can be increased by more than ten times when using an appropriate improvement method. However, the exact amount of improvement is not always known, as it depends upon the type of detail, the steel strength, and fabrication conditions. Standard SN curves found in codes cannot accommodate such parameters. FM analysis may provide an alternative in many situations. Also, FM can be used to examine the performance of some improvement methods in order to develop quality-assurance technology for these applications. Finally, FM analyses may assist in the assessment of complex structures not covered by test results. SN test data of small specimens may be inapplicable to large structures and to complex loading. This is particularly true for offshore structures, some of which have been analysed thoroughly using FM.

20 I.F.C.SMITH

1.7 CONCLUSIONS 1. The two most important parameters in fatigue assessments of large structures are geometry and stress range. 2. Achieving fatigue-resistant design requires careful and deliberate consideration of all factors which affect fatigue life. A concentrated effort is needed to design fatigue-resistant structural systems, to produce good detail configurations, and to ensure adequate consideration of fatigue during postdesign stages. 3. Fracture-mechanics concepts are primarily mechanical-engineering design tools, and therefore, special considerations are needed for civil-engineering structures. 4. In terms of fracture-mechanics applications, large structures need particular attention. In steel structures, there may be poorly defined loading, imposed deformations, high built-in tensile stresses, and many sharp discontinuities. Often, these circumstances cannot justify the use of concepts more sophisticated than simple linear-elastic approaches. 5. Fracture mechanics permits a greater qualitative understanding of parameters which influence fatigue, such as size effects and crack growth. 6. Calculations using fracture-mechanics models help to determine inspection intervals, evaluate structures containing crack-like discontinuities, develop improvement methods, perform remaining-life calculations, assess complex design problems. ACKNOWLEDGEMENTS The author is grateful to the Swiss National Science Foundation which sponsors research at ICOM in the area of fatigue, as well as Professor M.A.Hirt who coauthored the following articles from which this chapter is derived: ‘Fatigue design concepts’, IABSE Surveys S-29,1984; ‘Fracture mechanics in structural engineering’, Steel Structures: Recent Research Advances and Their Applications to Design, Elsevier, 1986; and ‘Fatigue resistant steel bridges’, J. Construct. Steel Research, 12, 1989. Also, the staff at ICOM, particularly G.J.Kimberley, are thanked for their help in preparing this chapter. REFERENCES BERGE, S. (1983) Effect of Plate Thickness in Fatigue of Cruciform Welded Joints. Report, MK/R 67, Norwegian Institute of Technology, Trondheim. BROEK, D. (1986) Elementary Engineering Fracture Mechanics. Martinus Nijhoff, The Hague. BSI (1980a) BS 5400: Steel Concrete and Composite bridges. Part 10: Code of Practice for Fatigue, British Standards Institution, London.


BSI (1980b) BSI PD 6493: Guidance on Some Methods for the Derivation of Acceptance Levels for Defects in Fusion Welded Joints. British Standards Institution, London. Eccs (1985) Recommendations for the fatigue design of structures. Publication No. 43. European Convention for Constructional Steelwork, Brussels. FISHER, J.W. (1977) Bridge Fatigue Guide. American Institute of Steel Construction, New York. FISHER, J.W. (1984) Fatigue and Fracture in Steel Bridges. Wiley, New York. GRIFFITH, A.A. (1921) The phenomena of rupture and flow in solids. Phil. Trans. Royal Soc., London, A221, 163–98. GURNEY, T.R. (1979) Theoretical Analysis of the Influence of Attachment Size on the Fatigue Strength of Transverse Non-Load-Carrying Fillet Welds. Members Report 91/1979, Welding Institute, Cambridge. HIRT, M.A, YEN, T.Y. & FISHER, J.W. (1971) Fatigue strength of rolled and welded steel beams. J. Structural Div., ASCE, 97(ST7), 1897–911. HMSO (1984) Offshore Installations: Guidance on Design and Construction. HMSO, London. IRWIN, G.R. (1983) A summary of fracture mechanics concepts. J. Testing & Evaluation, 11, 56–65. LANKFORD J. (1985) The influence of microstructure on the growth of small fatigue cracks. Fatigue Fracture Eng. Materials Structures, 8, 161–75. PARIS, P. & ERDOGAN, F. (1963) A critical analysis of crack propagation laws. Trans. ASME (Series D), 85, 528–34. ROLFE, S.T. & BARSOM, D.J.M. (1977) Fracture and fatigue control in steel structures. Eng. J. Am. Inst. Steel Constr., 14, 2–15. ROOKE, D.P. & CARTWRIGHT, D.J. (1976) A Compendium of Stress Intensity Factors. HMSO, London. SIA (1979) Steel Structures. SIA 161. Swiss Society of Engineers and Architects. SMITH, I.F.C. & GURNEY, T.R. (1986) Changes in the fatigue life of plates with attachments due to geometrical effects. Welding J. 65,244s-250s. SMITH, I.F.C. & SMITH, R.A. (1982) Defects and crack-shape development in filletwelded joints. Fatigue of Engineering Materials and Structures, 5, 151–65. SURESH, S. & RITCHIE, R.O. (1984) Propagation of short fatigue cracks. Int. Metals Rev., 29, 445–76. TADA, H., PARIS, P. & IRWIN, G. (1985) The Stress Analysis of Cracks Handbook. Paris Productions Inc., St. Louis. WITTMANN, F.H. (ed.) (1984) Fracture Mechanics of Concrete. Elsevier Applied Science, Amsterdam.

Chapter 2 FATIGUE DAMAGE ACCUMULATION UNDER VARYING-AMPLITUDE LOADS F.J.ZWERNEMAN School of Civil Engineering, Oklahoma State University, USA

SUMMARY Fatigue tests conducted in the laboratory have traditionally been performed using loads which vary sinusoidally with time at a constant amplitude. Loads carried by a structure in service also vary with time, but amplitudes of load cycles generally are not constant. To estimate the service life of a structure, an engineer must convert the variable-amplitude service loads to a form compatible with experimental data. In this chapter, methods are described for assessing, on the basis of constant-amplitude data, fatigue damage caused by variable amplitude load-time histories. NOTATION a ac an ap ao ao1 A C Cp

Crack length Current crack length Total crack length after n cycles Crack length plus width of overload plastic zone Original crack length Crack length at end of overload application Empirical constant equal to the N-axis intercept on a plot of the logarithm of stress range versus the logarithm of number of cycles to failure Empirical constant equal to the da/dN axis intercept on crack-growth-rate plots Retardation parameter

C1 da/dN

Constant dependent on plane stress or plane strain conditions Crack growth rate

FATIGUE DAMAGE ACCUMULATION 23

Ei K σK ′ Keff m

m′ n NB NC Ni ni Nt Pi Reff Ry Ryap Ryo1 Sreff Sri σ ap (σ c)max (σ c)max,eff (σ c)min,eff σ res σy (′ σ c)eff

Total number of cycles per block equal to or exceeding pi times the maximum stress in the cycle Stress-intensity factor Stress-intensity range Effective stress-intensity range Empirical constant equal to the slope on a plot of the logarithm of stress range versus the logarithm of number of cycles to failure; also the slope on a crack-growth-rate plot Shaping exponent Number of sizes of minor cycles or total number of cycles Number of complex cycles to failure Number of constant amplitude cycles to failure at the major cycle stress range Number of cycles to failure at stress range Sri Number of cycles applied at stress range Sri Total number of cycles to failure Ratio of minor cycle range to major cycle range Effective stress ratio Extent of current yield zone Plastic zone size needed to expand the current plastic zone outside the overload plastic zone Overload plastic zone size Effective stress range Stress range of the ith cycle Stress required to produce Ryap Current maximum stress Maximum effective stress Minimum effective stress Residual stress Yield stress Effective stress range 2.1 INTRODUCTION

Fatigue damage is produced by loads which vary cyclically with time, such as the loads applied to a bridge as a vehicle crosses or to an aircraft in flight. A record of service load versus time for such structures contains cycles of widely varying amplitudes and mean levels. Load-time histories used in laboratory tests, in most cases, bear only a vague resemblance to service load-time histories. In the laboratory, fatigue tests are generally conducted using loads which vary sinusoidally with time at some constant amplitude. To estimate the service life of a bridge or an aircraft, the engineer must either conduct fatigue tests using the

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FIG. 2.1 Conversion of service load-time history to constant-amplitude history.

applicable load-time history or convert the applicable history to a form compatible with available constant-amplitude data. Since conducting fatigue tests is expensive and time consuming, it is preferable to make use of available constant-amplitude data. A variety of techniques have been developed to make the transition from variable-amplitude service loads to constant-amplitude laboratory data. Generally, the variable-amplitude cycles are analytically reduced to equivalent constant-amplitude cycles. Empirical models are then used to assess and sum the damage produced by the equivalent constantamplitude cycles. The process is illustrated in Fig. 2.1. In this chapter, a number of the more popular cycle-counting techniques and damage models will be described. A discussion of the strengths and weaknesses of the various techniques will be presented to assist the designer in making the best choice for a specific application. Recommendations for design and suggestions for additional work are also presented.


2.2 CYCLE COUNTING Fatigue-life estimates for structures in service are based on stress range versus number of cycles to failure curves or crack-growth-rate curves (see Fig. 2.2). In most cases these curves are generated in experimental laboratories using constant-amplitude load-time histories. Unfortunately, service loads seldom vary sinusoidally with time at a constant amplitude. An analytical method must be employed to reduce the variable-amplitude service load-time history to a combination of constant-amplitude cycles or to a single equivalent constantamplitude cycle. These constant-amplitude cycles should produce the same fatigue life as the service history they are replacing. Many such analytical techniques have been proposed. For the current discussion, the techniques will be categorized as peak counting, level crossing, and range counting (Schijve, 1963). Peak-counting techniques involve the counting of all maximum and minimum peaks in the load-time history. Minimum peaks on the positive side of the datum and maximum peaks on the negative side of the datum are ignored. A load range is defined as the difference between the peak and the datum. The technique is illustrated in Fig. 2.3. One weakness of this technique is that it artificially increases the range of some small load cycles. A variation to the basic peakcounting technique involves counting only absolute maximum and minimum peaks between consecutive datum crossings. This variation does away with the problem of increasing the size of small cycles, but creates the opposite problem of excluding some cycles from the count. The level-crossing technique involves counting each time the varying load crosses predetermined load levels. The restriction may be imposed that only crossings with a positive slope are counted on the positive side of the datum and only crossings with a negative slope are counted on the negative side of the datum. The technique is illustrated in Fig. 2.4. The number of peaks and range of cycles cannot be determined on the basis of level-crossing counts. Range pair, rainflow (Endo et al., 1974), and reservoir counting are three essentially identical range-counting techniques (Dowling, 1972; Hoadley, 1982). The rainflow technique is illustrated in Fig. 2.5. The load-time history is rotated so that the abscissa is vertical, with time increasing in the downward direction. An imaginary raindrop starts at the inside tip of each peak or valley and flows down the sloped ‘roof’. The raindrop falls until it comes opposite a peak or valley greater than the one from which it started, or until it contacts the path of another raindrop started from a higher peak or valley. A half-cycle is counted for each raindrop. The load range for a half-cycle is the difference between the load at the starting point of a raindrop and the load at the termination point. Halfcycles are counted between points 1 and 2, 2 and 5, 3 and 4, 4 and 3′, 5 and 8, 6 and 7, and 7 and 6′. Full cycles are formed by combining half-cycles of the same load range, but opposite in flow direction.

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FIG. 2.2 Fatigue-life and crack-growth-rate curves. (a) Stress range versus number of cycles to failure (S–N curve). (b) Crack growth-rate curve.

Range-counting techniques are generally preferred over peak-counting and level-crossing techniques. Range-counting techniques count each cycle without altering the range of any cycles and allow a mean load level to be assigned to each cycle. 2.3 DAMAGE MODELS Once the variable-amplitude history has been reduced to a combination of constant-amplitude cycles at known mean load levels, an empirical model can be used to assign a degree of damage to each cycle. This damage takes the form of either an increment of crack growth or a percentage reduction in remaining fatigue life. Damage produced by individual cycles is summed to arrive at an


FIG. 2.3 Peak counting. All maximum peaks above the datum and all minimum peaks below the datum are counted.

FIG. 2.4 Level crossing. Positive-slope crossings are counted on the positive side of the datum and negative-slope crossings are counted on the negative side of the datum.

estimate of fatigue life. Models discussed here are grouped into categories of linear damage models, non-linear damage models, and interaction models.

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FIG. 2.5 Rainflow counting.

2.3.1 Linear Damage Models In the context of fatigue-damage models, the word ‘linear’ refers to the amount of damage attributed to individual load cycles in variableamplitude load-time histories relative to the amount of damage attributed to these individual cycles in constant-amplitude load-time histories. When a linear damage model is used, the user is assuming that the damage produced by individual cycles in a variableamplitude history can be calculated directly from the fatigue-life equation (N=A Sr−m) or the crack-growth-rate equation (da/dN=C ′ Km). To determine total damage produced by the history, damage produced by individual cycles making up that history is added linearly. It is assumed that all cycles are damaging and that cycles of different sizes do not interact to retard or accelerate growth.


The best-known and most widely used fatigue-damage model is the PalmgrenMiner (Palmgren, 1924; Miner, 1945) linear damage model. According to this model, failure occurs when (2.1) where ni is the number of cycles applied at stress range Sri, and Ni is the number of cycles to failure at Sri. Use of the model is illustrated in Fig. 2.6. To improve clarity in the figure, the numbers of cycles used in this illustration are much lower than would be encountered in practice. The Palmgren-Miner model can be combined with the standard fatigue-life equation to produce an expression for effective stress range. Effective stress range is the stress range of a constant-amplitude load-time history which will produce the same fatigue damage as a variableamplitude history, for the same number of cycles (Schilling et al., 1978). The method is demonstrated below:

(2.2)

where Nt is the total number of cycles to failure. Setting where Sreff is the effective stress range: (2.3) The concept is illustrated in Fig. 2.7. If the mean stress level of the cycles has a significant effect on fatigue life, this can be accounted for by allowing the N-axis intercept, A, to vary with the mean. Barsom (1973) and Schilling et al. (1978) conducted an extensive series of fatigue and crack-growth rate tests and concluded that a good fit to their data could be achieved by using m=2·0. If m is taken to be 2·0, the model is referred to as the root mean square (RMS) model. The drawback to using m equal to 2·0 is that, from the derivation, m should equal the slope of the S–N curve. The slope of this curve for most structural details is 3·0 (Keating & Fisher, 1987). If m is taken to be 3·0, the model is referred to as the root mean cube (RMC) model. Fisher et al. (1983) used the RMC model to accurately estimate crack-growth rates for a series of variable amplitude tests. Fatigue-life estimates based on the RMC model are shorter than estimates based on the RMS model, but accuracy relative to experimental data is not substantially different for the two models. The effective stress range calculated with either the RMS or RMC model should not be confused with the RMS of the stress history. The relevant quantity in eqn (2.3) is the range of stress under varying load, not the stress magnitude. To apply a linear damage model, the question of how to deal with cycles whose stress ranges are below the fatigue limit must be addressed. Fisher et al. (1983), Swensson (1984), Schijve et al. (1985), Heuler & Seeger (1986) and

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FIG. 2.6 Palmgren-Miner linear damage model.

Dowling (1988) conducted tests with a large percentage of the cycles below the fatigue limit and concluded that these cycles are damaging when they are part of a variable-amplitude history containing some cycles above the fatigue limit. Fisher & Swensson have recommended that all cycles in a variable-amplitude history be linearly added in the damage summation. In accordance with this recommendation, the variable Sri in eqn (2.3) includes all cycles in the history, not just cycles above the fatigue limit. The linear damage models described above are relatively easy to use and understand, which probably accounts for much of the popularity and widespread


FIG. 2.7 Effective stress range. (a) Variable-amplitude history. (b) Equivalent constantamplitude history.

use of these models. The major weakness of linear models is that they do not account for the effects of cycle interaction. Damage produced by individual cycles in variable-amplitude load-time histories is strongly affected by prior loading history. Linear damage models used indiscriminately can produce fatigue-life estimates significantly in error. Whether this error is conservative or unconservative depends on the load-time history. 2.3.2 Non-linear Damage Models Non-linear damage models are based on the assumption that all cycles in a variable-amplitude history do not cause the same damage they would if they were part of a constant-amplitude load-time history. Damage produced by a cycle may be assumed to depend on such things as the range of that cycle relative to the range of other cycles in the history, the mean level of cycles within the history, or the crack length at the time that cycle is applied. This type of model is still not an interaction model, because damage is not treated as a function of prior load history on a cycle-by-cycle basis. A non-linear model proposed by Albrecht & Friedland (1979) essentially ignores damage produced by cycles below the constant-amplitude fatigue limit. Albrecht calculates a Palmgren-Miner effective stress range based only on cycles above the fatigue limit. The number of cycles to failure at this effective stress range is then determined from a constantamplitude S–N curve. The total number

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FIG. 2.8 Number of cycles to failure for a variable-amplitude stress-time history. (a) S–N curve for constant-amplitude loading. (b) Variable amplitude load-time history.

of variable amplitude cycles to failure, including cycles below the fatigue limit, is calculated by dividing the number of cycles to failure at the effective stress range by the percentage of cycles above the fatigue limit. One drawback to this technique is that an accurate knowledge of the fatigue limit is required. There is a relatively small amount of data available in the vicinity of the limit, and there tends to be a great deal of scatter in the available data. If the S–N curve can be defined as in Fig. 2.8(a), Albrecht’s model can be used to calculate fatigue life for the stress-time history shown in Fig.2.8(b). Notice that 1/5 of the cycles in the variable amplitude history are above the fatigue limit and 4/5 are below. Assuming cycles below the fatigue limit are nondamaging, as proposed by Albrecht: N=(4×1012)(200)−3=5×105 Sreff=[(1·0)(200)3]1/3=200 N=(5×105)/(1/5)=2·5×106 (cycles above fatigue MPa (all cycles) limit) If it is assumed that all cycles are damaging and the Palmgren-Miner model is applied: Sreff=[(4/5)(50)3+(1/5)(200)3]1/3=119 N=(4×1012)(119)−3=2·35×106(all MPa cycles) In this example, Albrecht’s model estimates fatigue life to be 6% longer than the Palmgren-Miner model with the assumption that all cycles are damaging. The effect of Albrecht’s model on the S–N curve is shown in Fig. 2.9. When all cycles in the variable-amplitude history are above the fatigue limit, the S–N curve plots as a straight line. When all cycles are below the limit, the eifective


FIG. 2.9 Effective stress range versus number of cycles to failure based on non-linear damage models.

stress range is zero and the fatigue life is infinite. Between these two extremes, the S–N curve is non-linear on a log-log plot. Albrecht has used this technique to accurately predict fatigue life for a limited number of tests. An approach supported by Haibach (1970) and by Tilly & Nunn (1980) involves a fracture-mechanics-based modification to the Palmgren-Miner approach. The amount of crack growth produced by a load cycle depends on load range and crack length. For a constant load range, the amount of crack growth per cycle increases as the crack length increases. If a small cycle is applied to a sufficiently small crack, there will be no growth. This corresponds to the crackgrowth threshold in growth rate tests and to the fatigue limit in S–N tests. If the same size cycle is applied to a larger crack, a growth increment may occur. The modification to the Palmgren-Miner approach proposed by Haibach and by Tilly & Nunn involves predicting, on the basis of a fracture-mechanics analysis, which cycles are damaging and including only those cycles in the damage summation. In a variable-amplitude history, it is assumed that only the large cycles drive the crack when the crack is small; as the crack increases in length the smaller cycles become damaging. A small cycle is omitted from the damage summation when the fatigue crack is small and included when the crack has grown to near failure. The rationale behind this technique is not difficult to understand, but application is considerably more complex than application of the linear models. To use the technique directly, it is necessary to calculate the stress intensity range for changing crack lengths and to use this information to determine whether a cycle is damaging. Calculation of the stress intensity range for components encountered in normal engineering practice can be extremely difficult and stress intensity range thresholds are not well-defined. To overcome these difficulties, proponents of the technique recommend using a bilinear S–N curve to predict number of cycles to failure for inclusion in damage summations. As illustrated in Fig. 2.8(a), the slope of the S–N curve is changed from m to some value greater than m (Haibach proposes (2m−1) and Tilly & Nunn propose (m +2), both of which reduce to 5·0 if m is taken to be 3·0) for cycles below the

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constant-amplitude fatigue limit. The technique still suffers from the need for accurate knowledge of the fatigue limit. If the fatigue limit can be defined as in Fig. 2.8(a), Haibach’s model can be used to calculate number of cycles to failure for the stress-time history shown in Fig. 2.8(b): (2.4) where NB is the number of complex cycles to failure. A complex cycle is composed of the major cycle and any number of minor cycles. Complex cycles in Fig. 2.8(b) are composed of one major and four minor cycles. Considering only cycles below the fatigue limit: N1=4×1016(50)−5=128×106 Considering only cycles above the fatigue limit: N2=4×1012(200)−3=0·5×106 Combining cycles above and below the limit: NB(4/128×106+1/0·5×106) NB=492308(complex N=(5)(492 308)=2·46×106 =1·0 cycles) (all cycles) It was shown earlier that for the same stress-time history, if all cycles are assumed damaging, fatigue life is estimated at 2·35×106 cycles or 5% less than the estimate based on Haibach’s model. As seen in Fig. 2.9, changing the slope of the S–N curve for long fatigue lives produces an effect similar in appearance to that produced by the Albrecht model. When the stress range is high, all cycles are above the fatigue limit and are included in the damage summation. When the stress range is low, some of the small cycles are initially below the crack growth threshold and are not included in the summation. Omitting some of the cycles from the summation for some portion of the fatigue life increases the predicted fatigue life, moving the lower portion of the curve in Fig. 2.9 to the right. The curve is not shifted as far to the right for this model as for the Albrecht model because the Albrecht model assumes that cycles below the fatigue limit are never damaging, while this model assumes that the damage produced by small cycles is reduced. Accuracy of the technique has been demonstrated for a limited series of tests. A model completely independent from the Palmgren-Miner approach has been proposed by Gurney (1981, 1983). Gurney performed fatigue tests using the loadtime histories shown in Fig. 2.10. Each of the different histories in the figure is treated as one complex cycle. In Gurney’s fatigue tests, a complex cycle was applied repeatedly until the specimen failed. The Palmgren-Miner model significantly underestimated fatigue lives for these tests. Gurney noted from his data that the logarithm of the number of complex cycles to failure varied linearly with the ratio of minor cycle stress range to major cycle stress range. Based on this observation, Gurney showed that the number of complex cycles to failure can be calculated as:


FIG. 2.10 Load-time histories used by Gurney.

(2.5) where NB is the number of complex cycles to failure, NC is the number of constant-amplitude cycles to failure at the major cycle stress range, n is the number of sizes of minor cycles, Ei is the total number of cycles per block equal to or exceeding pi times the maximum stress in the cycle, and Pi is the ratio of minor cycle range to major cycle range. Fatigue lives for Gurney’s data are in better agreement with eqn (2.5) than with the Palmgren-Miner damage model. The models proposed by Albrecht, Haibach, and Tilly & Nunn produce fatiguelife estimates greater than or equal to the estimate produced by the PalmgrenMiner model, so could be more unconservative for Gurney’s data than is the Palmgren-Miner model. Tests performed by Joehnk (1982) and Zwerneman (1983, 1985) confirm Gurney’s finding that the Palmgren-Miner model can be unconser vative for some load-time histories. Zwerneman has proposed that crack growth caused by minor cycles in a variable-amplitude load-time history is dependent on the mean level of that cycle relative to the mean of the major cycle and on the number of minor

36 F.J.ZWERNEMAN

cycles between major cycles. Fatigue life is unconservatively estimated by the Palmgren-Miner model if the mean level of the minor cycles is high and there are few minor cycles between major cycles. This dependence on mean load level is much greater than observed by conducting multiple constant-amplitude tests with each test at a different mean. The dependence has been observed even in tests on steel weldments, where it has been demonstrated (Fisher et al., 1970) that the mean level of cycles in constant-amplitude tests has no effect on fatigue life. This ‘mean stress effect’ has not been formally incorporated in a damage model. Non-linear damage models described in this section are capable of improving fatigue-life estimates for some load-time histories. The models tend to work very well for experimental data generated by the researcher proposing the model, but can be seriously inaccurate when applied to data from other researchers using different load-time histories. Use of these models is justified only if the loadtime history under consideration matches the history used in developing the model. 2.3.3 Interaction Models An infinitely sharp crack and fully elastic behavior are assumed in deriving expressions for the stress intensity factor. As shown in Fig. 2.11, tbese assumptions result in elastic normal stresses at the crack tip approaching infinity. This obviously cannot occur in a real material with a finite yield stress. High stresses near the crack tip must cause the material in this area to yield. The stress distribution that would exist for a finite-yield material is labeled as elasticperfectly plastic in the figure. The plastic zone shown ahead of the crack tip is the zone of material which has yielded under the applied tensile stress. To simplify this discussion, the plastic zone is assumed to be circular. Notice that the plastic zone extends beyond the point where the elastic curve intersects the yield stress line. In order to satisfy statics, the lost elastic stress near the crack tip must be replaced by this extension of the plastic stress zone. The stress distribution near the crack tip becomes more complex when cyclic loads are applied (Rice, 1967). In cyclic loading, the first half-cycle involves application of a force, while the second half-cycle involves removal of that force. The removal of a tensile force is equivalent to the concurrent application of a compressive force of the same magnitude as the tensile force. The fictitious compressive force completes the unloading half of the load cycle by cancelling the tensile force. The application of the compressive force will theoretically produce an infinite negative stress at the crack tip, in the same way as the tensile force produced an infinite positive stress at the tip. Of course, the compressive stress cannot actually reach infinity. The material at the crack tip must yield in compression. Compression yielding due to unloading is illustrated in Fig. 2.12.


FIG. 2.11 Stress distribution in an elastic-perfectly plastic material.

The unloading plastic zone shown in Fig. 2.12(a) is much smaller than the loading plastic zone. The size of the plastic zone depends on the amount of stress the material can withstand before yielding. Going from tensile yield to compressive yield requires twice as much applied stress as going from zero stress to tensile yield stress. Therefore, the unloading plastic zone is smaller than the loading plastic zone. If the cyclic loading continues with the reapplication of a tensile half-cycle, stress in the small plastic zone must go from compressive yield to tensile yield. This small plastic zone is referred to as the reverse plastic zone, since the stress direction in this zone reverses as load is applied and removed. The larger plastic zone which forms ahead of the crack tip with the initial application of load is referred to as the forward plastic zone. Both the forward and reverse plastic zones move ahead of the crack tip as a crack grows across a plate. The plastic zones ahead of the crack tip are generally small relative to the overall plate width. The bulk of the plate, which has not experienced stresses of a sufficient magnitude to cause yielding, responds elastically to the application and removal of load. When external load is removed, the elastic material seeks to return to its original undeformed position. In the plastic zones, however, zero

38 F.J.ZWERNEMAN

FIG. 2.12 Stresses and strains produced by cyclic loading. (a) Stress distribution for loading and unloading. (b) Cyclic stress-strain curve for material near crack tip.

stress does not mean zero deformation. As can be seen in Fig. 2.12(b), when crack tip unloading stress is zero, crack tip strain is greater than zero. A compressive force is required to reduce crack tip strains back to zero. This compressive force is supplied by the elastic material surrounding the plastic zone. The plastic and elastic portions of the plate must exist in a state of equilibrium. The plastic zone seeks to remain in its expanded form by spreading apart the surrounding elastic material. The elastic material seeks to return to its original form by compressing the expanded plastic zone. The important end result is that for an externally unloaded specimen, the material immediately ahead of the crack tip exists in residual compression. Residual stresses and strains ahead of the crack tip are believed to be the cause of a crack-growth phenomenon known as retardation. Retardation is a reduction in growth rate following the application of an isolated overload. The phenomenon is illustrated in Fig. 2.13. The application of an overload causes


FIG. 2.13 Crack growth retardation caused by an overload. (a) Load-time history. (b) Crack length versus number of cycles. (c) Crack-growth rate versus stress-intensity range.

plastic deformation ahead of the crack tip in excess of plastic deformations produced by constant-amplitude cycling. This, in turn, results in higher than normal compressive stresses ahead of the tip. Three popular retardation models incorporate this idea of plastic deformation at the crack tip as the cause of retardation. The residual stress model attributes retardation to the high compressive residual stresses which develop ahead of the crack tip following the application of an overload (Hudson & Hardrath, 1961, 1963). The crack tip blunting model attributes crack growth retardation to blunting of the crack tip by an overload (Rice, 1967). The stress concentration produced by a blunt crack is less than that produced by a sharp crack, resulting in a retardation of growth rate. The crack closure model (Elber, 1971), like the residual stress model, attributes retardation to residual deformations ahead of the crack tip. The difference in the two models is the way in which these residual deformations are assumed to act as growth retarders. The residual stress model maintains that residual deformations retard crack growth by producing residual

40 F.J.ZWERNEMAN

compressive stresses ahead of the crack tip. The crack closure model maintains that residual deformations retard growth by closing the crack behind the tip. In the closure model, retardation occurs after the crack has advanced into the overload plastic zone. The deformed material behind the tip causes the crack to close at a stress greater than zero. An analytical model developed by Wheeler (1972) employs the residual stress model to account quantitatively for retardation due to cycle interaction. Development of the Wheeler model begins with the basic crack-growth-rate equation: (2.6) When applying this equation to a variable amplitude load-time history, consideration must be given to the fact that load is a discontinuous variable. Since load is a discontinuous variable, it is not possible to rearrange and integrate the crack-growth-rate equation to find number of cycles to failure (Paris & Erdogan, 1963). Wheeler solves this problem by summing crack growth cycle by cycle: (2.7) where an is the total crack length after n cycles, a0 is the original crack length, and ′ Ki is the stress intensity range for cycle i. To account for crack-growth retardation, Wheeler introduces a retardation parameter Cpi: (2.8) This retardation parameter varies from zero immediately following an overload to 1·0 when no overload is influencing crack growth. The retardation parameter is calculated as shown below: Cp=(Ry/(ap−a))m′ for (a+Ry)ap (2.9) where Ry is the extent of the current yield zone, ap is the crack length plus the width of the overload plastic zone, a is crack length, (ap−a) is the distance from crack tip to elastic-plastic interface, and m′ is the shaping exponent. The physical meaning of these terms is illustrated in Fig. 2.14. It can be seen that Cpi is a minimum immediately after the application of the overload when (ap−a) has its maximum value. As a approaches ap, Cp increases. The influence of the overload vanishes when the current yield zone moves through the overload yield zone. Wheeler was able to fit his model to crack-growth-rate data by adjusting the value of the shaping exponent m′. The major weakness of the Wheeler model is in this shaping exponent. The exponent must be determined empirically for every material used in fatigue applications. Also, there is some variation of m′ with


load spectra. The need for this empirical shaping exponent in the Wheeler model seriously damages the model’s usefulness for ordinary analysis purposes. A model developed by Willenborg (Engle & Rudd, 1974) is conceptually similar to the Wheeler model, but the Willenborg model avoids the shaping exponent problem. Development of the Willenborg model begins with the expression for the radius of the plastic zone at the tip of a fatigue crack: (2.10) where C1 is a constant dependent on plane stress or plane strain conditions, K is the stress intensity, and σ y is the yield stress. Referring to Fig. 2.14: ap=ao1+Ryo1 (2.11) where ao1 is the crack length immediately after the overload application, and Ryo1 is the overload plastic zone radius. The overload is effective in retarding crack growth until the current crack length, ac, plus the current plastic zone is greater than ap. To this point, the Willenborg and Wheeler models are conceptually identical. Both models retard crack growth with a compressive residual stress zone. The retardation is maximum immediately following the overload. The effect of the overload decays as the crack moves through the overload plastic zone. The models differ in the way the overload effect is forced to decay. Wheeler employs an empirical reduction factor while Willenborg bases the decay on the difference between the stress needed to expand the current plastic zone outside the overload plastic zone and the stress currently being applied. The technique is demonstrated below:

(2.12)

where Ryap is the plastic zone size needed to expand the current plastic zone outside the overload plastic zone, and σ ap is the stress required to produce Ryap. Willenborg then makes the assumption that residual stress due to the overload is equal to ′ ap minus the current maximum stress, (σ c)max: σres=σap−(σc)max (2.13) The residual stress is subtracted from the current maximum and minimum stresses to calculate effective applied stresses:

42 F.J.ZWERNEMAN

(2.14)

Notice that the effective stress range is reduced below the applied stress range only if the minimum and/or the maximum effective current stress falls below zero. The effective stress intensity range shown above is used in the standard crackgrowth-rate equation: (2.15) In addition to a possible reduction in stress range, the compressive residual stresses could reduce growth rate by lowering the mean level of the stress cycle. This effect would be accounted for through the constant C in the crack-growthrate equation. This constant varies with the ratio of minimum stress to maximum stress in a cycle for some materials. The Willenborg model has been used as the basis for a program developed by Engle & Rudd (1974) to estimate fatigue life of aircraft. Engle’s program has been used to estimate fatigue lives for specimens tested under isolated overload, block load, and service stress-time histories. Specimens used in these tests were made from aluminum and steel alloys. The Willenborg model predicted fatigue lives within 200% for the isolated overload and service stress-time histories. This level of accuracy corresponds to the scatter band generated in constant-amplitude tests on similar samples. The model was less accurate for the block load stresstime histories. The crack closure model has been analytically reproduced by Newman (1977, 1981) using two different approaches. The first approach, based on twodimensional elastic-plastic finite-element methods, shows that compressive yielding occurs in material behind an advancing crack tip at load levels above the minimum. Significant strain does not occur ahead of the tip until sufficient load is applied to remove these compressive stresses. The second approach is based on stress superposition and strip-yield approximations. The finite-element model more accurately matches observed behavior, but computation time required to solve problems of interest prohibits the use of the method for the general case. The second approach is less computationally sophisticated and has been used to predict crack growth under aircraft spectrum loading. Unfortunately, the model’s reliance on several empirical constants limits its usefulness.


FIG. 2.14 Wheeler residual stress model.

2.4 RECOMMENDATIONS FOR DESIGN To count cycles, the range pair method described in Section 2.2 is preferred over other methods. This method counts the entire load-time history and allows mean load levels to be calculated for each cycle. The disadvantage to this technique is that it obscures sequence effects. The choice of damage models is dependent on the load-time history being considered. If the load-time history contains few large cycles separated by many small cycles and the minimum load level for the small cycles is near the bottom of the load-time history, the non-linear models proposed by Albrecht, Haibach, or Tilly & Nunn could be appropriate. If a higher level of computational sophistication is desired, the Willenborg model as described by Engle could be used. The Willenborg model provides especially good correlation with experimental data when crackgrowth retardation is significant. All of these models provide fatigue-life estimates greater than or equal to the estimate that would be provided by the Palmgren-Miner model.

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If the load-time history is made up of large cycles separated by few small cycles and the mean level of the small cycles is at or above the mean of the large cycle, the non-linear model proposed by Gurney is appropriate. Gurney’s model provides fatigue-life estimates that are less than or equal to estimates provided by the Palmgren-Miner model. Unfortunately, current understanding of fatigue-crack growth is not sufficient to specify how many cycles make up ‘many’ or ‘few’ so that precise criteria can be established for selecting between the models proposed by Albrecht, Haibach, and Tilly & Nunn, and the model proposed by Gurney. For the general case, the non-linear and interaction models increase the level of sophistication of calculations without providing a parallel improvement in accuracy of fatigue-life estimates. In general, the Palmgren-Miner model is the preferred fatigue-damage model. The model is very simple to understand and apply, and the accuracy of fatiguelife estimates is comparable to those of the more sophisticated models. If the model is applied to a history containing an isolated overload, the fatigue life estimate will be conservative; the same is true for the non-linear models. If the model is applied to a history containing large cycles separated by few small cycles, and the mean of the small cycles is high relative to the mean of the large cycles, the fatigue life estimate will be unconservative; the same is true for the interaction models and the non-linear models proposed by Albrecht, Haibach, and Tilly & Nunn. If, however, the model is applied to a general service loadtime history such as could be measured on a highway bridge and all cycles in the history are assumed to be damaging, the fatigue life estimate will be acceptably accurate. 2.5 SUGGESTIONS FOR ADDITIONAL RESEARCH To improve our present ability to estimate fatigue life under variable amplitude loads, additional crack-growth studies are needed. These studies should be conducted on both the macroscopic and the microscopic level. On the macroscopic level, conventional crack-growth-rate tests can be used to better define the transition between the accelerated growth rates observed by Gurney, Joehnk, and Zwerneman, and the retarded growth rates observed by Albrecht, Haibach, and Tilly & Nunn. On the microscopic level, cycle-by-cycle monitoring of damage produced by variable-amplitude loading could provide information relevant to the influence of the mean level of cycles on growth produced by those cycles. Cycle-by-cycle monitoring could also help to resolve the question of whether cycles below the crack growth threshold are damaging when they are part of a variable-amplitude load-time history. Such cycle-by-cycle monitoring would be difficult for tests conducted at or near threshold growth rates, but may be possible with the aid of scanning electron microscopy or through the study of acoustic emissions emitted during cycling.


REFERENCES ALBRECHT, P. & I.M.FRIEDLAND. (1979) Fatigue-limit effect on variable-amplitude fatigue of stiffeners. J. Structural Div. ASCE, 105(ST12), 2657–75. BARSOM, J.M., (1973) Fatigue-crack growth under variable-amplitude loading in ASTM A514-B steel. In Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536. American Society for Testing and Materials, pp. 147–67. DOWLING, N.E., (1972) Fatigue failure predictions for complicated stress-strain histories. Journal of Materials. 7(1), 71–87. DOWLING, N.E., (1988) Estimation and correlation of fatigue lives for random loading. Int. J. Fatigue.10(13), 179–85. ELBER, W., (1971) The significance of fatigue crack closure. Damage Tolerance in Aircraft Structures. ASTM STP 486. American Society for Testing and Materials.pp. 230–42. ENDO, T.,MITSUNAGA, K., TAKAHASHI, K.,KOBAYISHI, K., & MATSUISHI, M. (1974) Damage evaluation of metals for random or varying load—three aspects of rainflow method. Mechanical Behavior of Materials, Proceedings of the 1974 Symposium on the Mechanical Behaviour of Materials, The Society of Materials Science, Japan. ENGLE, R.M. & RUDD, J.L., (1974). Analysis of crack propagation under variable amplitude loading using the Willenborg retardation model. AIAA/ASME/ SAE 15th Structures, Structural Dynamics and Materials Conference, Las Vegas, Nevada, April, AIAA Paper No. 74–369. FISHER, J.W., FRANK, K.H, HIRT, M.A. & MCNAMEE, B.M., (1970) Effect of Weldments on the Fatigue Strength of Steel Beams. National Cooperative Highway Research Program Report 102, Highway Research Board, National Research Council, Washington, D.C. FISHER, J.W., MERTZ, D.R. & ZHONG, A., (1983). Steel Bridge Members Under Variable Amplitude Long Life Fatigue Loading. National Cooperative Highway Research Program Report 267. Transportation Research Board, National Research Council, Washington, D.C. GURNEY, T.R. (1981). Some Fatigue Tests on Fillet Welded Joints Under Simple Variable Amplitude Loading. Research Report, The Welding Institute. GURNEY, T.R. (1983). Fatigue Tests on Fillet Welded Joints to Assess the Validity of Miner’s Cumulative Damage Rule. Proc. Roy. Soc. London, A386, 393– 408. HAIBACH, E. (1970) (Contribution to Discussion) In Fatigue of Welded Structures, Proceedings of the Conference. The Welding Institute, Brighton, England, Vol. 2, pp. xx–xxii. HEULER, P. & SEEGER, T. (1986) A criterion for omission of variable amplitude loading histories. Int. J. Fatigue. 8(4), 225–30. HOADLEY, P.W. (1982) Estimation of the Fatigue Life of a Welded Steel Highway Bridge from Traffic Data. Master’s Thesis, Department of Civil Engineering, The University of Texas at Austin. HUDSON, C.M. & HARDRATH, H.F. (1961) Effects of Changing Stress Amplitude on the Rate of Fatigue Crack Propagation in Two Aluminum Alloys. NASA Technical Note D-960, National Aeronautics and Space Administration, Washington, D.C.

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HUDSON, C.M. & HARDRATH, H.F. (1963) Investigation of the Effects of VariableAmplitude Loadings on Fatigue Crack Propagation Patterns. NASA Technical Note D-1803, National Aeronautics and Space Administration, Washington, D.C. JOEHNK, J.M. (1982) Fatigue Behavior of Welded Joints Subjected to Variable Amplitude Stresses. Master’s Thesis, Department of Civil Engineering, The University of Texas at Austin. KEATING, P.B. & FISHER, J.W., (1987) Fatigue behavior of variable loaded bridge details near the fatigue limit. In Bridge Needs, Design, and Performance. Transportation Research Record 1118, Transportation Research Board, National Research Council, Washington, D.C. MINER, M.A. (1945) Cumulative damage in fatigue. J. Appl. Mech., Trans. ASME, 67, pp. A159-A164. NEWMAN, J.C., Jr. (1977) Finite element analysis of crack growth under mono tonic and cyclic loading. Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Growth, ASTM STP 637, American Society for Testing and Materials, pp. 56–80. NEWMAN, J.C., Jr. (1981) A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. Methods and Models for Predicting Fatigue Crack Growth Under Random Loading. ASTM STP 748. (J.B.Chang & C.M.Hudson, eds). American Society for Testing and Materials, pp. 53–84. PALMGREN, A. (1924) Bertschrift des Vereines Ingenieure, 58, 339. PARIS, P.C. & ERDOGAN, F.E., (1963) A critical analysis of crack propagation laws. J. Basic Eng. Series D. Trans. ASME, 85, 528–34. RICE, J.R. (1967) Mechanics of crack tip deformation and extension by fatigue. Fatigue Crack Propagation. ASTM STP 415. American Society for Testing and Materials, pp. 247–311. SCHIJVE, J. (1963) The analysis of random load-time histories with relation to fatigue tests and life calculations. Fatigue of Aircraft Structures, Symposium on Fatigue of Aircraft Structures, Paris, 1961. (W.Barrois & E.L.Ripley, eds). MacMillan. SCHIJVE, J., VLUTTERS, A.M., ICHSAN, & PROVO KLUIT, J.C. (1985) Crack growth in aluminium alloy sheet material under flight-simulation loading. Int. J. Fatigue, 7(3), 127–36. SCHILLING, C.G., KLIPPSTEIN, K.H., BARSOM, J.M. & BLAKE, G.T., (1978) Fatigue of Welded Steel Bridge Members Under Variable-Amplitude Loadings. National Cooperative Highway Research Program Report 188. Transportation Research Board, National Research Council, Washington, D.C. SWENSSON, K.D. (1984) The Application of Cumulative Damage Theory to Highway Bridge Fatigue Design. Master’s Thesis, Department of Civil Engineering, The University of Texas at Austin. TILLY, G.P. & NUNN, D.E., (1980) Variable amplitude fatigue in relation to highway bridges. Proc. Inst. Mech. Eng. Appl. Mech. Group. 194(27), 259–67. WHEELER, O.E. (1972) Spectrum loading and crack growth. J. Basic Eng. Trans. ASME, March, pp. 181–6. ZWERNEMAN, F.J. (1983) Influence of the Stress Level of Minor Cycles on Fatigue Life of Steel Weldments. Master’s Thesis, Department of Civil Engineering, The University of Texas at Austin.


ZWERNEMAN, F.J. (1985) Fatigue Crack Growth in Steel Under Variable Amplitude Load-Time Histories. Ph.D. Dissertation, Department of Civil Engineering, The University of Texas at Austin.

Chapter 3 EFFECT OF RESIDUAL STRESSES ON THE FATIGUE OF WELDS S.SARKANI Department of Civil, Mechanical and Environmental Engineering, The George Washington University, USA

SUMMARY This chapter deals with the effect of residual stresses due to welding on fatigue life under both constant- and variable-amplitude loadings. It is shown that residual stresses can significantly alter the S–N curve for welded joints. In particular, it is shown that, under constant-amplitude loading, residual stress affects the mean stress of small-amplitude cycles. Using simple hypotheses for the initial stress state and simple stress-strain behavior, the effect of mean stress for small amplitudes, using both the modified Goodman and the Gerber formulae, is investigated. Finally it is shown that consideration of residual stresses changes the S–N curve enough to cause substantial modification of variable-amplitude fatigue-life predictions by the commonly used Rayleigh approximation method. NOTATION A* Ay Dj K m N Ny

S S′

Largest stochastic amplitude Applied stochastic stress amplitude equal to the yield stress Fatigue damage due to cycle j Constant of the S–N curve Inverse slope of the S–N curve Number of cycles to fatigue failure Number of cycles to fatigue failure when the applied stress range is twice the yield stress Constant-amplitude stress range having tensile mean Equivalent constant-amplitude stress range having zero mean

EFFECT OF RESIDUAL STRESSES ON THE FATIGUE OF WELDS 49

Se Sr Sv S′v T t X(t) v0 σ σ

Equivalent stress range having zero mean Applied stress range having tensile mean stress Variable-amplitude stress range having tensile mean Equivalent variable-amplitude stress range having zero mean Time at which fatigue failure is expected Time Stochastic stress Rate of zero crossing of stochastic stress Standard deviation (SD) Stress (subscripts: a, applied; m, mean; u, ultimate; y, yield) 3.1 INTRODUCTION

Fatigue is a common mode of failure for a broad range of civil engineering structures, particularly when such structures are made up of members that are connected by welds. Highway bridges and offshore structures are two examples which are generally made up of welded members and for which fatigue failure can be disastrous. Unfortunately, there are many uncertainties involved in predicting fatigue failure for such structures under service load conditions. Welded joints provide a particular challenge since they contain one of the more common locations for fatigue-crack growth, and there also exists uncertainty regarding the initial stress distribution and the physical properties of the weld material. In order to predict fatigue life under service load, use is generally made of the results of laboratory tests using constant-amplitude loadings. Such constantamplitude tests are run at each of two or more amplitude levels, and the results are plotted as the so-called S–N curve. The service fatigue life is predicted by dividing the variable-amplitude service loadtime history into cycles of constant amplitude and then using the S–N data to predict the damage done by each constant-amplitude cycle. Despite a large volume of work carried out to date, there are still questions as to the best way to do each of these steps in variableamplitude fatigue prediction, but they are secondary to the present discussion. This chapter deals with the determination of the S–N curve and with the extent to which variable-amplitude fatigue-life predictions can be affected by the procedures used in getting that curve. More specifically, it deals with the nature and magnitude of the effects of residual stresses which may be introduced by welding. Using simple hypotheses about the residual stress distribution and stress-strain behavior of the material, it is shown that residual stresses can have a significant effect on the mean stress at critical locations in a joint when that joint is subsequently subjected to small-amplitude cyclic loading. The change in mean stress then alters the resulting S–N curve significantly. Finally, using the Rayleigh approximation, it is shown that the variable-amplitude fatiguelife

50 S.SARKANI

predictions can be substantially altered for loadings that have small standard deviations. 3.2 BACKGROUND In order to predict the fatigue life of a structure due to any complicated loading history, one needs to know the fatigue behavior of that structure under constantamplitude loadings. The constant-amplitude results are usually designated the S– N curve, where S denotes the range (double amplitude) of either the loading process or the response and N denotes the number of cycles until failure. Empirical data of this type can usually be approximated quite well by an equation of the form N=KS−m (3.1) where K and m are constants related to the material properties. Equation (3.1) ignores the existence of any fatigue limit (or endurance limit) which gives a minimum range necessary to cause damage. This shortcoming is considered to be relatively unimportant in predicting fatigue life for realistic variable-amplitude loadings, provided that there remain many cycles which are above the fatigue limit. Knowledge of the S–N relationship enables one to compute the average incremental damage Dj caused by a single cycle with stress range Sj: (3.2) Note that this gives the accumulated damage to be unity at the time of failure in the constant-amplitude tests. The Palmgren-Miner hypothesis (Palmgren, 1924; Miner, 1945) is that variable-amplitude failure also occurs when the accumulated damage is unity when eqn (3.2) is used to find the damage in each cycle. This implies that the damage due to any loading cycle is a function of the stress range in that cycle only. Under variable-amplitude laoding, in order to use eqn (3.2) one must first identify the cycles and determine their ranges. This is quite easy for a very narrowband process, but is much less obvious in other situations. The probabilistic form of eqn (3.2) is (3.3) in which E(Dj) is the expected or mean value of the incremental damage per cycle and fs(s) is the probability density function of the stress ranges. The corresponding approximation of the number of cycles to failure is (3.4) Finding the time t=T at which failure is expected requires an estimate of the number of cycles per unit time in the variable-amplitude process. One of the simplest and most widely used analytical techniques for estimating fatigue life under variable-amplitude loading is the so-called Rayleigh


approximation method (Miles, 1954). This technique is based on the assumption that the damage caused by any stochastic stress X(t) is the same as the damage caused by a very narrowband normal stress with the values of standard deviation, σ x, and rate of zero-crossing, v0, equal to those of the original process. Since the peaks of a very narrow-band normal process are Rayleigh distributed, one has (3.5) in which fA(a) is the probability density function of the peaks. Finally, the magnitude of a stress range is taken to be twice the value of a. Based on the above assumptions, the expected value of the damage is (3.6) where ′ (•) denotes the gamma function, and the estimated failure time is (3.7)

3.3 INFLUENCE OF RESIDAL STRESS The above discussion has said nothing about the effects of mean stresses. When applied loading is symmetric about zero, it is natural to begin with the assumption that mean stresses are also zero. It is logical to expect, though, that the welding may have left considerable residual stress, so that the material at some particular point in a specimen might have a significant mean stress. In particular, it seems likely that the thermal shrinkage of the weld material caused very large tensile residual stresses in the base metal immediately adjacent to the weld. This is particularly significant for fatigue, since the fatigue cracks are typically initiated at the weld toe, which is part of this postulated region of tensile residual stress. Even though it is very difficult to accurately determine the magnitude and distribution of residual stresses around the weld toe, recent experimental measurements (Sarkani & Lutes, 1988; Berge & Eide, 1982) indicate that the magnitude of these weld-toe residual stresses is close to the yield stress of the base metal, and decreases rapidly as the distance from the weld-toe is increased. Thus, for a joint geometry in which the cracks initiate around the weld toe, it is important to determine the combined effect of residual stresses and the applied loading on the fatigue of the joint. Figure 3.1 shows the effect of the residual stresses on two different load cycles, under the simplifying assumptions that the base metal has an elastic, perfectly plastic stress-strain curve and that the yield stress has not been affected by the welding. Two cases are considered: (a) when the amplitude of the applied

52 S.SARKANI

FIG. 3.1 Hypothesized effect of residual stresses.

nominal stress at the weld-toe is near the yield stress, and (b) when the applied nominal stress at the weld-toe is less than the yield stress of the base metal. It is assumed that the gross behavior of the specimen is linear, even though localized yielding occurs. Thus, the changes in strains (both nominal and local) are proportional to the load and the nominal stress. Consider Fig. 3.1(a), where the applied cyclic nominal stress has amplitude close to the yield stress. Upon the application of the first quarter-cycle of the loading, i.e. from A to B, even though the nominal stress is varied from zero to yield, the local weld-toe stress remains the same as before while the local strain is increased from point A to point B. As the second and third quarter-cycles of the loading are applied, i.e. from B to D, the weld-toe local stress and nominal stress both change from tensile yield stress to compressive yield stress. Note that it is being assumed that unloading is linear, both nominally and at the weld-toe of the specimen. Application of further cycles of the loading would cause the weld-toe stress to vary between points D and B. Thus, the weld-toe mean stress is zero. Figure 3.1.(b) illustrates the effect of application of a loading cycle with amplitude less than the yield stress of the base metal. Similar to the previous case, upon application of the first quarter-cycle of loading, i.e. from A to B, the local weld-toe stress remains at yield as the local strain is increased from A to B. However, in this case, upon application of the second and third quarter-cycles of the loading, the local weld-toe stress varies from yield to point D, which is equal to the yield stress minus the stress range of the applied loading cycle. Again, application of further cycles of loading will continue to vary the local weld-toe


stress between points D and B. In this case the mean weld-toe stress remains equal to σm=σy−σa (3.8) where σ m=weld-toe mean stress; σ y=yield stress of base metal; σ a=amplitude of applied nominal stress cycles. Based on Fig. 3.1(a), it is reasonable to assume that weld residual stresses should not effect the fatigue life of a specimen subjected to a symmetric loading containing large tensile nominal stresses. This generally applies to the large constant-amplitude loadings used to define the low-cycle end of the S–N curve. However, for stresses on the high-cycle end of the S–N curve (nominal stress amplitude less than yield), the presence of tensile residual stress causes a tensile mean stress level, as shown in Fig. 3.1(b). Such tensile mean stresses can significantly reduce the fatigue life. There are a number of methods available to account approximately for the effect of mean stress on fatigue life of a specimen subjected to cyclic loading. These methods are based on experimental observations. For a given stress range and mean stress, such methods provide the methodologies for calculating an equivalent stress range with zero mean stress. The most widely used correction methods are the modified Goodman (Goodman, 1899; Muvdi & McNabb, 1980) and the Gerber parabola (Gerber, 1874; Muvdi & McNabb, 1980). The modified Goodman correction is based on the following equation: (3.9) The Gerber equation is (3.10) where Sr=applied stress range having tensile mean stress; Se=equivalent stress range having zero mean; σ m=mean stress; σ u=ultimate stress of the material; σ y=yield stress of the material. In order to examine the magnitude of the change for typical welded joints, the S–N curve of eqn (3.1) is rewritten as (3.11) in which Ny is the fatigue life when the applied stress range is Sy i.e. twice the yield stress of the specimen material. Comparison of eqns (3.1) and (3.11) indicates that Using the models previously presented for taking into account the effect of residual stresses, eqn (3.11) can be converted to an equivalent zeromean S–N curve as follows:

54 S.SARKANI

FIG. 3.2 Influence of mean stresses on S–N curve.

(3.12) Setting f=1 and 2 would represent the equivalent zero-mean S–N curve based on the modified Goodman and the Gerber corrections, respectively. Note that when f is set equal to a large positive number, eqn (3.12) reduces to the original S–N curve of eqn (3.11). Close examination indicates that eqn (3.12) is no longer a straight line on the log-log plot. However eqn (3.12) is attractive because it can easily be implemented for variable-amplitude fatigue-life predictions, as will be demonstrated later. Equation (3.12) is plotted in Fig. 3.2 for three different f values which would correspond to corrections based on the modified Goodman, the Gerber and no correction for mean stresses. In these plots, σ y=50 ksi, σ u=77·5 ksi, Ny=20000 and m is set equal to 4. These are typical values for welded joints. (Note: 1 Ksi=6·9 N/mm2) In order to determine which one of the above correction procedures more closely predicts the effect of mean stresses on the fatigue of welded joints, Sarkani and Lutes (1988) carried out an experimental investigation. The specimens they used were welded plate tees. Their study began by obtaining the S–N curve as is done customarily in fatigue studies. However, in addition to the usual S–N curve tests, they tested several specimens under constant-amplitude loading which had previously been subjected to a few large cycles, so that residual stresses were


FIG. 3.3 Influence of residual stresses on S–N curve.

redistributed or ‘wiped out’ at the point of crack initiation. The few large cycles had the nominal stress amplitude equal to the yield stress of the specimen material, whereas all the other cycles until failure had a smaller nominal stress amplitude equal to 40% of the yield stress. Because of the effect of the few large-amplitude cycles, the new data points were considered to represent fatigue with no mean stress effect. They then corrected their original S–N for the effect of residual stresses by employing the simplified model presented above in conjunction with modified Goodman method and the Gerber parabola. The original and corrected S–N curves are shown in Fig. 3.3. Also shown in Fig. 3.3 are the data points which are considered to be fatigue lives with no mean stress effect. It was concluded that the Gerber correction adequately predicts the effect of residual stresses on the constant-amplitude fatigue lives in terms of elevated mean stresses. 3.4 VARIABLE-AMPLITUDE FATIGUE-LIFE PREDICTIONS From the preceding discussion, it is clear that residual stresses present in welded specimens alter the S–N curve predicted for specimens without residual stresses. Futhermore, this change is particulary significant for the small-stress portion of the S–N curve. The next step is to determine whether the alteration in the S–N curve is large enough to significantly effect the fatigue lives predicted under variable-amplitude situations.

56 S.SARKANI

The three S–N curves to be considered are that corrected by the modified Goodman method, that by the Gerber method and the one obtained by assuming that residual stresses have no effect (eqn (3.11)). Recall that these three S–N curves are supposed to give the fatigue lifetime without means stress, in spite of the different assumptions made in obtaining the curves. When considering the effect of residual stresses, it is reasonable to use one of the corrected S–N curves and consider mean stresses in any variable-amplitude loading for which fatigue life is to be predicted. If one uses the uncorrected S–N curve, then it is consistent to ignore mean stresses in the variable-amplitude loadings as well. The common Rayleigh approximation will now be used to predict variableamplitude fatigue lives using the three S–N curves. Comparison of the results will give an indication of the consequences of the assumptions about residual stress. Even though this ignores questions about the accuracy of the Rayleigh approximation for certain service loads, the comparison of the S–N curves and the relative effects of residual stresses is nonetheless valid. Probabilistic fatigue-life prediction can be considerably more complicated if one uses any of the correction methods to obtain an equivalent stress range for each range in a variable-amplitude time history, since Rayleigh distribution of stress ranges will be lost. However, one can still use the S–N curve of eqn (3.11) to obtain variable-amplitude fatigue-life estimates. Also, if the mean stress at any specific time is taken to be a function of the largest cycle prior to that time, then the damage will be non-stationary. These complications can be simplified if the residual stresses at the crack-initiation point are assumed to have been removed. Since variable-amplitude loadings contain large stress cycles, then this large cycle, if applied early during the loading process, can remove the residual stresses and subsequently remove the mean stress on the following variableamplitude loading cycles. Let Ay denote the applied stress amplitude that brings the nominal stress in the specimen to the yield level of the material. Recall that the assumed stress-strain behavior gives removal of the residual stress at the location of crack initiation upon application of a stress of amplitude Ay. It is pertinent to consider how frequently one should expect the stress amplitude to exceed Ay or any specified fraction of Ay. These numbers for the recurrence interval of Ay, 0·75Ay, 0·50Ay and 0·25Ay, have been computed from the assumed Rayleigh distribution of amplitudes with various standard deviation (Sd) values and are given in Table 3.1. From Table 3.1 it is clear that when Sd is small it is much less likely that an amplitude of Ay will occur early in the stochastic loading. For Sd/Ay=0·15, for example, on average an amplitude equal to Ay will occur once in every 447×107 cycles. Now for a given variable-amplitude stress range Sv, assuming that the largest stress amplitude that occurs throughout the loading process is A* and furthermore assuming that A* occurs early during the loading process, allows one to obtain the equivalent zero-mean stress range Sσv as follows:


(3.13) Now the Palmgren-Miner hypothesis is employed to calculate the fatigue damage caused by Sσv. To accomplish this, use is made of the zero-mean TABLE 3.1 RECURRENCE INTERVAL FOR STRESS EQUAL TO A*

0·05 0·10 0·15 0·20 0·25 0·30

A*=0·25Ay

A*=0·5Ay

A*=0·75Ay

A*=Ay

268×103 23 4 2 2 1

518×1019 268×103 259 23 7 4

721×1046 164×1010 268×103 1131 90 23

723×1084 518×1019 447×107 268×103 2981 259

S–N curve of eqn (3.12). In particular, a constant-amplitude stress range S can be converted to an equivalent zero-mean constant amplitude stress range S′ given by (3.14) Now setting the right-hand side of eqns (3.13) and (3.14) equal to each other and solving for S in terms of Sv allows one to obtain the damage Dv due to a cycle Sv in terms of the S–N curve of eqn (3.12). Setting f = 1 and 2, one can obtain S in terms of Sv for both the modified Goodman and the Gerber correction methods as follows: (modified Goodman) (3.15a) (Gerber) (3.15b) (3.15C) Using eqns (3.15a) or (3.15b) for equivalent constant-amplitude stress range, one can obtain the damage Dv due to a variable amplitude cycle Sv. Assuming that the original stress range, Sv i.e. not corrected for effect of residual stress, is twice the amplitude, and that the amplitude is taken to have a truncated Rayleigh probability distribution (since the largest amplitude that can occur is A*), results in the following expression for the expected number of cycles to failure:

58 S.SARKANI

(3.16) in which for the modified Goodman correction the mth moment of the stress ranges can be calculated as (3.17a) for the Gerber correction, one gets (3.17b) in which

Note that fA(a) is the truncated Rayleigh distribution (see eqn (3.5)): (3.18) In the above formulation, it is explicity assumed that the largest amplitude throughout the random loading does not exceed A* and furthermore that A* occurs early during the loading process so that it results in some removal of the residual stresses. Note that when A* reaches the yield stress of the specimen material, then it is assumed that all the residual stresses are removed under the variable-amplitude loading. Let N0, N1 and N2 denote lifetimes predicted from eqn (3.16) using the uncorrected stress ranges, i.e. neglecting the effect of residual stress, and those corrected for the effect of residual stresses by the modified Goodman and the Gerber methods, i.e. eqns (3.15a) and (3.15b). Tables3.2 and 3.3 represent values of N0, N1 and N2 corresponding to various A* values (the largest amplitude of the stochastic stress process) for several values of the standard deviation of the stochastic stress. Also shown in Tables3.2 and 3.3 are the recurrence intervals of A* which are given in the final column of these tables. Note that the results in these tables are based on σ y=50 ksi, σ u=77·5 ksi and Ny=20000. From Tables 3.2 and 3.3 it is clear that the effect of correction for residual stress becomes much more significant when the SD of the stress process is smaller and when A* approaches the yield stress of the base metal. The effect of the modified Goodman correction is more severe that that of the Gerber correction, particularly when the SD of the stochastic stress process is smaller. On the other hand, one must note that when the SD is small, it also becomes much less likely that a large value of A* will occur early in the stochastic loading. For example, for m=3, SD/Ay=0·05 and assuming A*=Ay, ignoring the residual stress in the variableamplitude fatigue-life calculations would cause a conservative error of 283% if one used the Gerber correction. However, for this situation, it is very unlikely that an amplitude equal to or larger than Ay would


actually occur. On the other hand, for SD/Ay=0·25 it is very likely that the stochastic stress would exceed Ay long before fatigue failure. For this situation, ignoring the residual stress would cause a conservative error of 48% based on the Gerber correction. One may also note that introduction of an endurance limit into S–N curves, and subsequently into variable-amplitude fatigue-life calculations, would have a significant effect on the damage predictions for small SD TABLE 3.2 VARIABLE-AMPLITUDE FATIGUE- LIFE PREDICTIONS N0 (cycles) N1 (cycles) (neglecting (corrected for residual stress) residual stress, modified Goodman) (a) m=3, A*=0·25σ y 0·05 42·6×106 0·10 7·1×106 0·15 4·4×106 0·20 3·8×106 0·25 3·5×106 0·30 3·4×106 (b) m=3, A*=0·50σ y 0·05 42·6×106 0·10 5·3×106 0·15 1·6×106 0·20 0·9×106 0·25 0·6×106 0·30 0·6×106 (c) m=3, A*=0·75σ y 0·05 42·6×106 0·10 5·3×106 0·15 1·6×106 0·20 0·7×106 0·25 0·4×106 0·30 0·3×106 (d) m=3, A*=σ y 0·05 42·6×106 0·10 5·3×106 0·15 1·6×106 0·20 0·7×106 0·25 0·3×106

N2 (cycles) N1/No N2/N0 A* recurrence (corrected for interval residual stress, Gerber)

84·1×106 9·8×106 5·7×106 4·7×106 4·4×106 4·2×106

67·2×106 8·8×106 5·2×106 4·4×106 4.1×106 3·9×106

1·98 1·39 1·28 1·25 1·24 1·23

1·58 1·24 1·17 1·15 1·15 1·14

268×103 23 4 2 2 1

213·1×106 18·0×106 3·7×106 1·6×106 1·1×106 0·9×106

113·5×106 10·4×106 2·5×106 1·2×106 0·8×106 0·7×106

5·00 3·38 2·25 1·80 1·64 1·57

2·67 1·96 1·52 1·34 1·27 1·24

518×109 268×103 259 23 7 4

423·8×106 40·1×106 8.3×106 2·4×106 1·0×106 0·6×106

149·4×106 14·0×106 3·2×106 1·1×106 0·5×106 0·3×106

10·17 7·53 5·27 3·54 2·64 2·23

3·51 2·64 2·02 1·62 1·41 1·31

721×1046 164×1010 268×103 1131 90 23

766·8×106 75·2×106 16·9×106 5·1×106 1·8×106

162·9×106 15·4×106 3·5×106 1·2×106 0·5×106

18·02 14·14 10·71 7·70 5·29

3·83 2·90 2·23 1·77 1·48

723×1084 518×107 447×109 268×103 2981

60 S.SARKANI

0·30

N0 (cycles) N1 (cycles) (neglecting (corrected for residual stress) residual stress, modified Goodman)

N2 (cycles) N1/No N2/N0 A* recurrence (corrected for interval residual stress, Gerber)

0·2×106

0·3×106

0·8×106

3·84

1·32

259

TABLE 3.3 VARIABLE-AMPLITUDE FATIGUE-LIFE PREDICTIONS N0 (cycles) N1 (cycles) (neglecting (corrected for residual stress) residual stress, modified Goodman) (a) m=4, A*=0·25σ y 0·05 400·1×106 0·10 39·3×106 0·15 22·4×106 0·20 18·7×106 0·25 17·3×106 0·30 16·6×106 (b) m=4, A*=0·5σ y 0·05 400·0×106 0·10 25·0×106 0·15 5·4×106 0·20 2·5×106 0·25 1·7×106 0·30 1·4×106 (c) m=4, A*=0·75σ y 0·05 400·0×106 0·10 25·0×106 0·15 4·9×106 0·20 1·6×106 0·25 0·8×106 0·30 0·5×106 (d) m=4, A*=σ y 0·05 400·0×106 0·10 25·0×106 0·15 4·9×106 0·20 1·6×106 0·25 0·6×106

N2 (cycles) N1/N0 N2/N0 A* recurrence (corrected for interval residual stress Gerber)

924·1×106 56·9×106 29·6×106 24·1×106 22·0×106 21·0×106

699·2×106 49·9×106 26·7×106 21·9×106 20·1×106 19·2×106

2·31 1·45 1·32 1·29 1·27 1·27

·75 1·27 1·20 1·17 1·16 1·16

268×103 23 4 2 2 1

3258·8×106 111·5 13·3×106 4·7×106 2·9×106 2·3×106

1415·1×106 56·1×106 8·5×106 3·4×106 2·2×106 1·8×106

8·15 4·46 2·49 1·92 1·73 1·64

3·54 2·24 1·58 1·37 1·3 1·26

518×1019 268×103 259 23 7 4

8472·7×106 336·0×106 38·0×106 6·8×106 2·2×106 1·2×106

2050.5×l06 84·1×106 11·3 2·7×106 1·1×106 0·6×106

21·18 13·44 7·68 4·24 2·92 2·40

5·13 3·36 2·20 1·69 1·44 1·32

721×1046 164×1010 268×103 1131 90 23

18298.5×106 793·6×106 102·1×106 19·1×106 4·4×106

2303·7×106 95·5×106 12·8×106 3·0×106 1·0×106

45·75 31·74 20·68 12·22 6·86

5·76 3·82 2·60 1·89 1·51

723×1084 518×109 447×107 268×103 2981


0·30

N0 (cycles) N1 (cycles) (neglecting (corrected for residual stress) residual stress, modified Goodman)

N2 (cycles) N1/N0 N2/N0 A* recurrence (corrected for interval residual stress Gerber)

0·3×106

0·4×106

1·5×106

4·44

1·26

259

values. In effect, the endurance limit would remove the damage caused by the small-amplitude cycles and it is in this region that the correction for residual stresses is most significant. Comparison of results in Tables 3.2 and 3.3 also indicates that, as the value of m increases, the effect of residual stress becomes more significant. Overall, it appears that there exists a range of stochastic loads for which it is likely that the mean stresses will be near zero throughout most of the variableamplitude fatigue life, and for which neglecting the effect of residual stresses gives significant errors. On the other hand, for some other stochastic loadings, the effect of residual stresses is not necessarily significant compared to other uncertainties in practical fatigue-life prediction, but it could very well mask the effect of other factors which could be of considerable research interest. For instance, the bandwidth effects predicted by rainflow analysis (Lutes et al., 1984; Wirsching & Light, 1980) and the experimentally measured effect of weathering (Albrecht and Sidani, 1986) are generally of a smaller order of magnitude than the effect of residual stresses as presented here. It is not clear at this point whether the residual stress effect should be considered in design codes. Even though the model presented here indicates that the occasional application of yielding stress does extend fatigue life, it is not clear that one should rely on this possible benefit in order to increase the allowable level of design stress within a structure. Careful consideration must be given to the probability of occurrence of large loads in the intended service conditions and initial distribution and magnitude of residual stresses. 3.5 SUMMARY AND CONCLUSIONS Several recent experimental investigations have concluded that the magnitude of residual stress present at the crack-initiation location of welded joints is close to yield stress of the base metal. Using simple hypotheses, the effect of residual stress on the constant-amplitude fatigue S–N curve was investigated. It was concluded that residual stress aifects the mean stress of applied loading on the joint. Futhermore, the Gerber formula for correcting the resulting mean stress gives results agreeing with experimental findings. Finally, the effect of residual stress on variable-amplitude fatigue life was investigated. It was shown that the effect of residual stress is particularly

62 S.SARKANI

significant for variable-amplitude loadings that have small SD values typical of service conditions. ACKNOWLEDGMENT The author would like to express his sincere thanks to Professor Loren D.Lutes of Texas A&M University for initially introducing him to the topic of stochastic fatigue. Some of the material presented here was drawn from a paper by the author and Professor Lutes (Sarkani & Lutes, 1988). Thanks are also extended to Mr. David P. Kihl, a doctoral student at George Washington University, for his assistance throughout the preparation of this chapter. REFERENCES ALBERCHT, P. & SIDANI, M. (1986). Fatigue Strength of 8-Year Weathered Stiffeners in Air and Salt Water. Civil Engineering Report, University of Maryland, College Park, Maryland. BERGE, S. & EIDE, O.I. (1982). Residual Stresses and Stress Interaction in Fatigue Testing of Welded Joints. ASTM STP 776, pp. 115–131. GERBER, W.Z.(1874). BAYER Archit. Ing. Ver., 6 101. GOODMAN, J. (1899). Mechanics Applied to Engineering. Longman, Green and Company, London. LUTES, L.D., CORAZAO, M. HU, S-L.J. & ZIMMERMAN, J.J. (1984). Stochastic fatigue damage accumulation. J. Structural Div. ASCE, Paper 19823, 110(ST11), 2585–601. MILES, J.W. (1954). On structural fatigue under random loading. J. Aeronautical Sci. 753–62. MINER, M.A. (1945). Cumulative damage in fatigue. J. Appl. Mech. 12 A159– A164. MUVDI, B.B. & McNABB, J.W. (1980). Engineering Mechanics of Materials. MacMillan, New York. PALMGREN, A. (1924). Die Lebensdauer vo Kugallagern. Ver. Deut. Ingr., 68, 339–41. SARKANI, S. & LUTES, L.D. (1988). Residual stress effects on fatigue of welded joints. J. Structural Div. ASCE, Paper 22231, 114(ST.2), 462–74. WIRSCHING, P.H. & LIGHT, M.C. (1980). Fatigue under wide band random stresses. J. Structural Div. ASCE, Paper 15574, 106(ST7), 1593–607.

Chapter 4 FATIGUE CRACKING IN PLATE AND BOX GIRDERS Y.MAEDA Emeritus Professor, Osaka University, Osaka, Japan Y.KAWAI Research and Development Center, Kawasaki Steel Corp., Chiba, Japan & I.OKURA Department of Civil Engineering, Osaka University, Osaka, Japan SUMMARY The fatigue behavior of plate and box girder structures subjected to repeated loading is presented, based on the results of analyses, laboratory tests and observations of model or actual structures carried out by the authors. First of all, fundamental crack patterns occurring in thin-walled plate girder bridges are discussed, including cracking at the connections of cross-beams to the main girders of highway bridges. Secondly, fatigue cracking of two types of box girder monorail guideways is considered. Finally, the fatigue cracking observed in heavy-duty crane girders, such as welded crane girders in workshops and apron girders of unloaders in harbors, is discussed. NOTATION a Width of a rectangular plate or interval between adjacent vertical stiffeners b bfr bfs bie boe Cv

Height of a rectangular plate or web depth Width of an upper flange of a runway rail Width of an inner flange of a stiffening frame Effective width of an inner flange of a stiffening frame Effective width of skin plate as an outer flange of a stiffening frame Coefficient related to contact pressure distribution of a pneumatic tire

64 Y.MAEDA, Y.KAWAI AND I.OKURA

d Dc e E If Ig IQ IR J JR, Jf, Jw k kml, km3, kml23, kbl, kb3, kbl23 kv l m, n Nc po P ′P R Ri s tfr tfs tw tT w0′, w0 max wp x, y Z σ σ σ ′σ σ s0′σ g σ

Distance between a vertical loading point and a webto-flange welded joint Flexural rigidity of a concrete slab Magnitude of rail misalignment Young’s modulus Moment of inertia of an upper flange Moment of inertia of a main girder Moment of inertia of a cross-beam Moment of inertia of a rail JR+Jf+Jw St. Venant’s torsional rigidities of a rail, upper flange and web, respectively Factor ′ σ /′ P Constants Maximum running speed of a monorail vehicle Span length of a plate girder Positive integers Number of cycles when fatigue cracks are observed Air pressure of a pneumatic tire Magnitude of a wheel load Load range Ratio of in-plane bending stresses σ o min/σ o max Radius-of-curvature of an inner flange of a stiffening frame Spacing between main girders Thickness of an upper flange of a runway rail Thickness of an inner flange of a stiffening frame Web thickness Thickness of a tie-pad Initial deflection and its maximum of a plate, respectively Width of a pneumatic tire

Abscissa and ordinate, respectively (IQ/Ig)[l/(2s)]3 Coefficient for adjusting the thickness of a tie-pad to its effective thickness Web slenderness ratio b/tw Coefficient Local strain range Rotations of concrete slab and of cross-beam, respectively

FATIGUE CRACKING IN PLATE AND BOX GIRDERS 65

v Poisson’s ratio σb Plate-bending stress at web panel boundaries σ b max′ σ b min Maximum and minimum plate-bending stresses which are caused by σ o max and σ o min, respectively σd Local bearing stress at a flange-to-web weld σe σ 2E/[12(1–v2)σ 2] σ fb Plate-bending stress in an upper flange of a runway rail σl Longitudinal stress in a lower flange σm Membrane stress produced in a connection plate at a cross-beam connection σo In-plane bending stress at the junction of flange and web σ o max′ σ o min Maximum and minimum of in-plane bending stress σ o, respectively σr Radial stress in a curved flange σσ Circumferential stress in a curved flange ′ σ bf Fatigue strength of fillet welds subjected to platebending stress ′ σo Range of in-plane bending stress (σ o max–σ o min) σ cr Shear buckling strength of a rectangular plate simply supported along four edges σo In-plane shearing stress σ o max Maximum of in-plane shearing stress σ o

4.1 INTRODUCTION During the past decade, a great deal of research has been carried out on the effects of repeated loading on welded structures such as bridges, monorail guideways, crane runways, etc. As well as knowledge obtained from the observations of actual structures, these studies have led to a better understanding of fatigue behavior of plate and box girders, and particularly to improvement of fatigue provisions in bridge design codes like AASHTO (1989), BS5400 (1980) and EUROCODE 3 (1990). Traditionally, fatigue cracking and fatigue resistance of girders have been considered in relation to design details and initial defects or cracks due to geometrical discontinuities or poor quality welds during fabrication. A lowfatigue-strength detail or built-in defects may produce only a single significant crack. However, the behavior of secondary members has not always been as obvious. These members interact with the main members and are subjected to more cycles of a higher stress range than anticipated. The most remarkable fatigue problem of girder structures in recent years is cracking from secondary stresses due to the repetition of web deflection. Interaction between the longitudinal girder and the transverse framing will


produce secondary and displacement-induced stresses at the connections to the main members, which may cause many fatigue cracks simultaneously in the overall structure, depending upon structural conditions. The fatigue cracks of the secondary members and the displacement-induced fatigue cracks are discussed and some recommendations are provided as to how the problems may be avoided or minimized to insure the intended performance. 4.2 THIN-WALLED PLATE GIRDER As pointed out by Goodpasture & Stallmeyer (1967), Maeda (1971), Mueller & Yen (1968), Patterson et al. (1970), Toprac & Natarajan (1971), and Yen & Mueller (1966), when a thin-walled plate girder is subjected to repeated loading, there are possibilities of the initiation and propagation of fatigue cracks along the fillet welds around the web panel boundaries. As shown in Fig. 4.1, the fatigue cracks are classified as follows, depending on loading conditions. (a) In girders under bending, the following three types of fatigue cracks occur. – Type 1 crack. This crack is initiated at the toe on the web side of the fillet weld connecting the web to the compression flange. It grows

FIG. 4.1. Fatigue cracks in thin-walled plate girders. (a) Girder in bending. (b) Girder in shear.

gradually along the weld toe with loading cycles. As shown in Fig. 4.2, the crack is caused by the plate-bending stress at the weld toe due to out-of-plane deformation of the web under in-plane bending. – Type 2 crack. This crack is observed at the toe on the web side of the fillet weld connecting the vertical stiffener to the web. It propagates towards the tension flange. Penetration of the crack into the tension flange leads to the


FIG. 4.2. Plate-bending stress due to web deflection.

collapse of the girder. As the crack is produced on the tension side of the neutral axis of the girder, its propagation speed is faster than that of a Type 1 crack. – Type 3 crack. This crack occurs at the fillet weld connecting the web to the tension flange. Its initiation is due to incomplete penetration of the fillet weld or to discontinuities on the weld surface. (b) In girders under shear, Type 4 cracks are initiated at the toe, on the web side of the fillet welds, near to the corners where a diagonal tension field is expected to be anchored. They propagate along the weld toe, and then branch out into the web in directions approximately perpendicular to the tension field. Then the girder will lose its load-carrying capability because of the diminished tension field action. The cause of cracking is, as shown in Fig. 4.2, the plate-bending stress at the weld toe, due to out-of-plane deformation of the web under in-plane shear. Of the cracks mentioned above, Types 2 and 3 cracks are the most common cracks in rolled beams or built-up girders. The allowable stress ranges for them are specified in the fatigue design of various codes (AASHTO, 1989; BS5400, 1980; EUROCODE 3, 1990). Thus, it is possible to predict their initiation by comparing the range of applied in-plane bending stress, estimated by beam theory at each cracking point, with the corresponding allowable stress range. However, it is still difficult to predict the initiation of Type 1 cracks, because the relation between applied load and plate-bending stress is not clearly defined. The characteristics of Types 1 and 4 cracks are peculiar to thin-walled plate girders and the design parameters are described in subsequent sections. 4.2.1 Bending The finite deflection analysis of a rectangular plate under in-plane bending was carried out by the finite-element method (FEM) to show the effects of an initial web deflection on the plate-bending stresses along the web panel boundaries (Maeda & Okura, 1981, 1983). Referring to Fig. 4.3, the plate is simply supported at x=0 and a, fixed at y=0 and b against out-of-plane deflection, and


FIG. 4.3. Rectangular plate under inplane bending.

free to move in the in-plane direction. It is assumed that the plate has an initial deflection in the following form: w0=w0 max sin(mπx/a) sin(nπy/b) (4.1) where w0 and w0 max are the initial deflection and its maximum value respectively, and m and n are positive integers. Figure 4.4 shows the relation between the in-plane bending stress σ o and the plate-bending stress (σ b, at the edge y=b of a square plate. The following notation is used in the figure: σ e=σ 2E/{12(1−v2)/σ 2}, E= Young’s modulus, v=Poisson’s ratio, σ =b/tw=web slenderness ratio, and tw=web thickness. It can be seen that the initial deflection shape for m=1 restrains the increase in plate-bending stress, while the initial deflection shapes for m=2 and 3 increase the plate-bending stress greatly. The condition to prevent Type 1 cracks is as follows: (4.2) where σ b max and σ b min are the maximum and minimum plate-bending stresses which are caused by the maximum and mininum in-plane bending stresses σ o max and σ o min, respectively, and ′ σ fb is the fatigue strength of fillet welds subjected to plate-bending stress. The fatigue strength ′ σ fb was obtained by Maeda (1978) and Mueller & Yen (1968). The relation between the in-plane bending stress σ o and the plate-bending stress ′ b was formulated for the initial deflection shapes defined by m=2 and 3 in eqn (4.1) by Maeda & Okura (1984). Therefore, solving the σ o versus σ b relation subject to eqn (4.2), the relation between the maximum in-plane bending stress σ o max and the web slenderness ratio σ satisfying eqn (4.2), can be obtained. Since the σ o versus σ b relation is non-linear, the σ o max versus σ relation varies with the ratio R of the in-plane bending stresses defined by R=σo min/σo max (4.3) Therefore the range of in-plane bending stress ′ σ o to satisfy enq (4.2) is expressed as


FIG. 4.4. Relation between in-plane bending stress and plate-bending stress.

∆σo=σo max−σo min=σo max(1−R) (4.4) The variation of ′ σ 0 with R is smaller than that of σ o max for test girders (Okura et al. 1991). Figure 4.5 shows the relation between ′ σ o and σ for the girders tested in bending by Maeda (1971), Mueller & Yen (1968), Patterson et al (1970), Toprac & Natarajan (1971), and Yen & Mueller (1966). Figures 4.5(a) and (b) give the classification of cracking at 5×105 cycles and at 2×106 cycles, respectively. The allowable stress ranges for Types 2 and 3 cracks, specified as Detail Classes E and C respectively, in BS5400 (1980), are also given in the figures. For in each of Figs 4.5(a) and (b), Type 1 cracks do not occur below the allowable stress ranges for Types 2 and 3 cracks. However, Type 1 cracks occur below the allowable stress ranges for Types 2 and 3 cracks for σ >200, as shown in Fig. 4.5(a), and below the allowable stress range for Type 3 cracks for σ >200, as shown in Fig. 4.5(b). Therefore, it is necessary to limit the web slenderness ratio to 200 for stiffened plate girders to prevent Type 1 cracks. 4.2.2 Shear Actual plate girders which are subjected to shear are necessarily subjected to coexistent bending. The finite out-of-plane deformation of a square plate under combined in-plane shear and in-plane bending was analyzed by FEM (Okura & Maeda, 1985). Figure 4.6 shows the relation between the in-plane shearing stress σ o and the plate-bending stress σ b for the initial deflection shapes defined by (m, n)=(1, 1), (2, 2), (2, 1) and (1, 2) in eqn (4.1). It is seen that the plate-bending stresses become closer to one another with increase in the in-plane shearing


FIG. 4.5.Relation between ′ σ o and σ . (a) Classification for 5×105 cycles. (b) Classification for 2×106 cycles.

stress. Hence, in plate girders under shear, the influence of the shape of initial web deflection on the increase in the plate-bending stress, and on the cracking, is small. The relation between the in-plane shearing stress σ o and the plate-bending stress σ b was formulated for the initial deflection defined by (m, n)=(1, 1) in eqn (4.1) (Maeda et al., 1985; Okura & Maeda, 1985). From the equation for the σ o versus σ b relation, Type 4 cracks may be prevented by keeping the applied maximum in-plane shearing stress below the shear buckling strength of a rectangular plate simply-supported along four edges (Okura et al., 1991).


FIG. 4.6. Relation between in-plane shearing stress and plate-bending stress.

Figure 4.7 shows the relation between σ o max/σ cr and σ for the girders tested by Goodpasture & Stallmeyer (1967), Mueller & Yen (1968), Patterson et al. (1970), Toprac & Natarajan (1971), and Yen & Mueller (1966). Here, ′ o max is the applied maximum in-plane shear stress, and ′ cr is the shear buckling strength of a rectangular plate simply-supported along four edges. In Fig. 4.7, Type 4 cracks were initiated at filled circles before 2×106 cycles, while they were not at open circles. There is one filled circle below the horizontal line σ o max/σ cr=1·0. At this filled circle (Toprac and Natarajan, 1971), Type 4 cracks were detected at 1·626×106 cycles, which is close to two million cycles. Moreover, there are several open circles above the horizontal line and the shear buckling strength σ cr generally represents a conservative estimate. 4.3 CONNECTIONS IN PLATE GIRDER HIGHWAY BRIDGES Fatigue cracks occurs at the connections of cross-beams to main girders in many plate girder highway bridges in the urban area of Japan. The fatigue cracks observed in the plate girder bridges of the Hanshin Expressway in Osaka are classified as shown in Fig. 4.8. – Type 1 crack. This crack is initiated at the end of a fillet weld between the connection plate and the top flange of the main girder. – Type 2 crack. This crack is initiated at the upper scallop of the connection plate, and grows diagonally through the connection plate itself. – Type 3 crack. This crack is initiated at the end of the fillet weld connecting the connection plate to the main girder web, and grows downwards along the weld toe on the connection plate side.


FIG. 4.7.Relation between σ o max/σ cr and σ .

FIG. 4.8. Fatigue cracks at a cross-beam connection in a plate girder highway bridge.

– Type 4 crack. This crack is initiated and grows along the toe on the web side of the fillet weld between the top flange and the web of the main girder. Field stress measurements were carried out for an existing plate girder bridge of the Hanshin Expressway to determine the stress states at the cross-beam connections (Okura et al., 1987; Okura & Fukumoto, 1988).


FIG. 4.9. Local stresses σ m and σ b.

FIG. 4.10. Rotations σ s0 and σ g.

It revealed that the membrane stress σ m and the plate-bending stress σ b, as illustrated in Fig. 4.9, were the stresses governing the initiation of Types 1 and 4 cracks, respectively. Then, the relationship between the local stresses and the rotations of the concrete slab and cross-beam was formulated as follows (Okura et al., 1988);

where σ s0 is the rotation of the concrete slab due to the slab-deformation caused by wheel loads (see Fig. 4.10), σ g is the rotation of the cross-beam due to the vertical displacements of main girders (see Fig. 4.10), σ is a coefficient depending on the position of a vehicle in the direction of the roadway width, and Km1, Km3, Kml23, Kbl, Kb3 and Kbl23 are constants which relate the local stresses to the rotations. The structural parameters governing the cracking at the cross-beam connection were deduced from eqn (4.5), and are as follows (Okura et al., 1989a,b): (a) For the concrete-slab rotation: Dc/s (b) For the cross-beam rotation,


FIG. 4.11. Relationship between IQ/s2 and the number of bridges in which Type 1 cracks were observed.

where Dc is the flexural rigidity of the concrete slab, s is the spacing between main girders, IQ is the moment of inertia of a cross-beam, Ig is the moment of inertia of a main girder, l is the span length of a main girder, and Z=(IQ/Ig){l/(2s)}3. The plate girder bridges with smaller values of the above structural parameters are more susceptible to cracking, since a decrease of the parameters increases σ s0 and σ g, resulting in an increase in the local stresses ′ m and σ b. Figure 4.11 shows the relationship between IQ/s2 and the number of bridges in which Type 1 cracks were observed. The influence of σ g on the local stress σ m, which causes Type 1 cracks, is small in plate girder bridges with IQ/s2=3·1 cm2 (Okura et al., 1989a,b). In Fig. 4.11, however, Type 11 cracks occur in the bridges with This indicates that Type 1 cracks can be initiated by the concreteslab rotation σ s0 alone. Figure 4.12 shows the relationship between IQ/s2 and the number of bridges in which Type 4 cracks were observed. With the increase in IQ/s2, the number of bridges suffering from cracking gradually decreases, since the influence of σ g on the local stress σ b, which causes Type 4 cracks, becomes small. No cracks occur in the bridges with


FIG. 4.12. Relationship between IQ/s2 and the number of bridges in which Type 4 cracks were observed.

4.4 BOX GIRDERS FOR MONORAIL GUIDEWAYS 4.4.1 Straddle-type Guideway girders for straddle-type monorails are now under construction in Osaka, Japan. Prestressed concrete girders with a span length of 22 m are usually used for the guideway girders. Steel girders are, however, built at the intersection with an existing road, or at a crossing above a river. As shown in Fig. 4.13, they are narrow box girders 660 mm wide and 2·2–3·3 m in height. The parallel girders are connected by cross beams of I section. Wheels of monorail vehicles run on the top flange of the girders. Figure 4.14 shows schematically the bending moment and shearing force acting on a cross-beam. As shown in Fig. 4.14(a), when the load is applied to girder G1, the bending moment has the same magnitude but opposite sign at the girders G1 and G2, and is zero at the middle of the cross-beam. When, as shown in Fig. 4.14(b), the load is applied to the girder G2, the bending moment and shear force have opposite signs to those in Fig. 4.14(a). Hence, alternating


FIG. 4.13. Guideway girders for straddle-type monorail.

FIG. 4.14. Bending moment and shearing force acting on a cross-beam.

stresses are induced at the cross-beam connections when monorail vehicles run on the G1 and G2 girders alternately. To investigate the fatigue strength of the cross-beam connections, as shown in Fig. 4.15, three types of specimens were designed, based on the results of a FEM analysis (Maeda et al., 1989). A part of the girder below the middle of the plate for running of stabilizing wheels, and a part of the cross-beam with a length of half the spacing between two girders, were included in the specimens. Models CP and TP were proposed as connection details to improve the fatigue strength at the cross-beam connection in Model ST. In Model CP specimens, a corner plate, as shown in the figure, was provided at the intersection of the top flange of


FIG. 4.15. Three types of specimens.

the cross-beam with the track girder web. In Model TP specimens, it was possible to conduct fatigue tests on the right and left sides of the same specimen. The size of the corner plate of Model CP specimens, and the increased depth of the cross-beams of Model TP specimens, were both limited by the clearance of monorail vehicles. The top of the girder of the test specimens was fixed to a rigid floor, and alternating loads were applied vertically to the end of the cross beam.


FIG. 4.16. ′ σ versus Nc relation.

In Model CP specimens, fatigue cracks were initiated at the upper edge of the corner plate. Their initiation was very sensitive to the shape of the upper edge of the corner plate. On the other hand, in Model ST and TP specimens, fatigue cracks were initiated at the intersection of the top flange of the cross-beam with the girder web. Figure 4.16 shows the relation between ′ σ and Nc. Nc is the number of cycles at which fatigue cracks were detected. ′ σ is the range of the local strain measured at the cross-beam connection, which was obtained as follows: Figure 4.17 shows the distributions of the strains measured near the intersection of the top flange of the cross-beam with the girder web. The distributions of the strains are linear from about 10 mm ahead of the weld toe. The strain values at the weld toe, obtained by extrapolation of the straight lines, were converted to the strain range ′ σ The value of the factor k, defined as k=′ σ /′ P, is a measure of the improvement of the connection detail. Here, ′ P is the range of load. A crossbeam connection with larger values of k is more susceptible to cracking. The k values for Models ST and TP are shown in Table 4.1. Except for Model TP2L, the k values of Model TP specimens are about half of that for Model ST. When a Model TP connection detail is used at the cross-beam connection, a hole is required in the diaphragm for fabrication work. The influence of such a hole on cracking was investigated in Model TP2L. The results showed that, as illustrated in Fig. 4.18, fatigue cracks were TABLE 4.1 VALUES OF k Model

k(μ /kN)

ST

5·0


FIG. 4.17. Measured strains at cross-beam connection. Model

k(μ /kN)

TPIR TP1L TP2R TP2L Note: k=′ σ /′ P

2·9 2·4 2·8 4·2

initiated at the corners of the hole in the diaphragm and along the fillet welds at one corner of the diaphragm above the rib plate, after the fatigue cracks occurred at the intersection of the top flange of the cross-beam with the girder web. Thus, the improvement of the connection detail by Model TP is very effective, but the hole for fabrication work must be covered by a steel plate. 4.4.2 Suspended-type Two types of guideway girders for a suspended-type monorail are shown in Fig. 4.19, with stiffening frames inside or outside the skin plate. Both are thin-


FIG. 4.18. Fatigue cracks observed in Model TP2L specimen.

walled open box girders with a longitudinal slit at the lower flange to let suspension links of monorail vehicles pass through. Wheels of monorail vehicles run on the runway rails. Lateral loads due to wind or meandering of the vehicles act on the guideway rails. The skin plate acts as flanges and webs of the box girder. The stiffening frames restrain the cross-sectional deformation of the box girder and also support the runway rails. It was pointed out by Hikosaka et al. (1982) that the stiffening frames and runway rails were susceptible to cracking. As shown in Fig. 4.20, two types of cracks may be initiated at the corner of the stiffening frame, produced by the circumferential stress σ σ and radial stress σ r. Such local stresses can be estimated by in-plane curved beam theory, using the effective widths boe and bie, given by the following equations for the skin plate and for the curved inner flange of the stiffening frame, respectively: boe=0·6b (4.6) (4.7) where and Ri, tfs and bfs are the radius of curvature, thickness and width of the curved inner flange of the stiffening frame, respectively. Equation (4.7) was obtained by Anderson (1950). In the fatigue design of the stiffening frame, such


FIG. 4.19. Stiffening of a suspended monorail guideway.

FIG. 4.20. Two types of fatigue cracks in a stiffening frame.

dimensions as the radius of curvature, thickness and width of the curved inner flange must be determined, so that both the circumferential stress σ σ and the radial stress σ r can satisfy the corresponding allowable fatigue strengths given in Fig. 4.21 (Yamasaki & Kawai, 1983). As shown in Fig. 4.22, cracking may occur at two locations of the runway rail. The cracking in the lower flange at the intermediate ribs, which are provided to prevent local deformation of the upper flange of the rail, is caused by the longitudinal stress σ l in the lower flange. This longitudinal stress is given by superposing the local stress due to individual wheel loads on the overall bending stress and warping torsional bending stress due to the total vehicle loads. The local stress due to the individual wheel loads can be estimated by treating the rail as a three-span continuous beam supported by the stiffening frames. The cracking in the upper flange of the runway rail in Fig. 4.22 is caused by the plate-bending stress σ fb which is produced every time the individual wheels run through between the adjacent intermediate ribs. The thickness of the upper flange may be designed as a T-shaped welded joint subjected to repeated platebending stress. Introducing the effects of contact pressure distribution of a pneumatic tire into the equation by Senior & Gurney (1963), the following equation is obtained for the plate-bending stress σ fb:


FIG. 4.21. S–N curves for fatigue design of a stiffening frame.

FIG. 4.22. Two types of fatigue cracks in a runway rail.

(4.8) where tfr and bfr are the thickness and width of a runway rail respectively, wp and po are the width and air pressure of a pneumatic tire respectively, Cv is a


coefficient related to contact pressure distribution of a pneumatic tire, and Kv is the maximum running speed of a monorail vehicle in kilometers/hour. Sadamasu (1969) proposed 0·017h/km for Cv. 4.5 HEAVY-DUTY CRANE GIRDERS 4.5.1 Welded Crane Runway Girders In recent decades, many studies have been carried out on fatigue behavior of welded crane runway girders (Senior & Gurney, 1963; Maas, 1972; Demo & Fisher, 1976; Nishiyama et al., 1978, 1980, 1983; Reemsnyder & Demo, 1978; Umino & Mimura, 1982). Referring to Fig. 4.23, fatigue cracks that have been observed in welded crane runway girders are summarized as follows: (1) Runway girder Type a crack: upper flange-to-web welded joint Type b crack: upper flange-to-transverse stiffener welded joint Type c crack: welded joint in lower flange Type d crack: lower flange-to-secondary member welded joint (2) Back girder (wall girder and lateral bracings) Type e crack: lateral bracing member Type f crack: welded joint of gusset plate (3) Supporting members Type g crack: connection bolts between runway girder and post Type h crack: bearing plate on building post The characteristics of Types a and b cracks in the runway girder and of Types e and f cracks in the back girder are presented in more detail below. Types a and b cracks are mainly caused by repeated local stresses due to wheel loads of a travelling crane. Referring to Fig. 4.24, the local stress at the flange-to-web welded joint for Type a cracks, is given by summing the bearing stress σ d and plate-bending stress σ b. The former stress is due to the direct force of wheels, and the latter is due to eccentricity between the centers of wheel loading and the web plane. By modifying the formulae proposed by Parks (1952) and Oxfort (1963) with the results of experimental studies by Nishiyama et al. (1980), eqns (4.9) and (4.11) are obtained for σ d and σ b, respectively: (4.9) (note: σ d in MPa, P in N, and tw, d and e in mm) where P is the magnitude of a wheel load, e is the rail misalignment, and d is the distance between a virtual loading point and a web-to-flange welded joint, which is given by


FIG. 4.23. Typical fatigue cracks observed in crane runway girders.

FIG. 4.24. Local stress at flange-to-web weld.

(4.10) (note: d, tw and tT in mm; IR and If in mm4) where Ir and If are the moments of inertia of the rail and the upper flange respectively, tT is the thickness of a tiepad, and σ is the coefficient for adjusting the thickness of the tie-pad to its effective thickness. (4.11) (note: σ b in MPa, P in N, tw, a, b and e in mm, J in mm4, and σ in mm−1) where J=JR+Jf+Jw, and JR, Jf and Jw are St. Venant’s torsional rigidities of the rail,


FIG. 4.25.General view of unloader.

upper flange and web, respectively, and ′ = (note: σ in mm−1, tw and b in mm, and J in mm4). Equations (4.9) and (4.11) are effective for e between 0 and 5tw. The structural details of the upper part of the girder have to be designed so that the total stresses σ d and σ b can meet the allowable fatigue strength given by Maeda (1978) for fillet welds subjected to plate-bending stress. A study of fatigue damage of crane runway girders carried out by the Society of Steel Construction of Japan JSSC, 1976, revealed that Types e and f cracks were initiated more often than any other types of cracks. They are caused by locally induced stresses not generally considered in the design practice of crane runway girders. It is very difficult to predict such local stresses, since they are produced by the three-dimensional behavior of the whole of a crane runway girder. There are some studies on the relationship between the local stresses and structural behavior of a crane runway girder (Nishiyama et al., 1978). However, satisfactory conclusions are not yet available and further investigation is required. 4.5.2 Apron Girders in Unloaders An unloader for removing cargo from a ship is shown in Fig. 4.25. The apron girder and runway girder in the unloader are susceptible to cracking, since they are subjected to severe repeated loading. As shown in Fig. 4.26, the apron and runway girders are either a twin box-girder or a mono box-girder. As shown in Fig. 4.27, they suffer from cracking at the following locations: Type a crack: flange-to-web welded joint Type b crack: connection of a rail to a flange


FIG. 4.26. Two types of apron and runway girders. (a) Twin box-girder type. (b) Mono box-girder type.

FIG. 4.27. Typical fatigue cracks in unloader girders.

Type c crack: weld to connect a diaphragm to a skin plate Type a cracks have the same features as those of Type a cracks in welded crane runway girders. The local stress for cracking can be estimated by eqns (4.9) to (4.11). To prevent Type b cracks, it is recommended that the gap between the flange surface and the bottom of the rail should be kept small. 4.6 CONCLUSIONS Recent findings concerning fatigue cracking in plate and box girders have been presented in relation to stress analysis in design. General patterns and characteristics of deformation-induced fatigue cracking in thinwalled plate


girders subjected to repeated bending and shear, have been considered in relation to the interaction between overall and local effects. Furthermore, cracking observed at the connections of cross-beams to the main girders of highway bridges has been identified as one of the major fatigue problems in bridge maintenance. Cracking in box girders has been illustrated by reference to straddle- and suspended-type monorail guideways. The effects of detail design and fabrication on the fatigue cracking of two types of welded girder in a crane runway and an unloader have also been discussed. REFERENCES AASHTO. (1989) Standard Spe, cations for Highway Bridges. The American Association of State Highway and Transportation Officials, Washington, D.C., 14th edn. ANDERSON, C.G. (1950) Flexural stress in curved beams of I- and box section. Proc. Inst. Mech., London, 162, 295–306. BSI(1980) 5400:Part 10. Steel, Concrete and Composite Bridges. British Standards Institution, London. DEMO, D.A. & FISHER, J.W. (1976) Analysis of fatigue of welded crane runway girders. J. Structal. Div., ASCE, 102(5), 919–33. EUROCODE 3 (1990) Chapter 9. Design of Steel Structures. Edited draft issue 3. Commission of the European Communities. GOODPASTURE, D.W. & STALLMEYER, J.E. (1967) Fatigue Behavior of Welded Thin Web Girders as Influenced by Web Distortion and Boundary Rigidity. C.E. Studies SRS No. 328, University of Illinois. HIKOSAKA, H., KAWAI, Y.&TAKEMI, K. (1982)An experimental and theoretical study on the static behavior of curved guideways for suspended monorail system. Memoirs of Faculty of Engineering, Kyushu University, Japan, 42(2), 1–8. JSSC (1976) Report of the investigation on fatigue damage of crane runway girders. Society of Steel Construction of Japan, 12(128), 9–22 (in Japanese). MAAS, G. (1972) Investigation concerning craneway girders. Iron and Steel Engineering, London, March, pp. 49–58. MAEDA, Y. (1971) Ultimate static strength and fatigue behavior of longitudinally stiffened plate girders in bending. IABSE, London Colloquium, March, pp. 269–82. MAEDA, Y. (1978) Fatigue cracks of deep thin-walled plate girders. Bridge Engineering. Transpoftation Research Board, Washington, D.C., pp. 120–8. MAEDA, Y. & OKURA, I. (1981) Interaction between initial web deflection and fatigue crack initiation in thin-walled plate girders. The Design of Steel Bridges-Conference Discussion, (H.R.Evans ed.). University College, Cardiff, Wales, pp. 12.9–12.16. MAEDA, Y. & OKURA, I. (1983) Influence of initial deflection of plate girder webs on fatigue crack initiation. Eng. Struct., (England), 5 January, pp. 58–66. MAEDA, Y. & OKURA, I. (1984) Fatigue strength of plate girder in bending considering out-of-plane deformation of web. Proc. JSCE, Structural Eng./Earthquake Eng., 1 (2), 149s-159s. MAEDA, Y., OKURA, I. & HIRANO, H. (1985) Formulation of finite out-of-plane deformation of rectangular plate in shear. Technology Reports of Osaka University, 35, 91–100.


MAEDA, Y., FUKUOKA, T., OKURA, I. & ISOZAKI, K. (1989) Fatigue test of cross beam connections in steel track girders for straddle-type monorail. Proc. Japan Society of Civil Engineers, 404(1–11), 425–34 (in Japanese). MUELLER, J.A. & YEN, B.T. (1968) Girder Web Boundary Stresses and Fatigue. Welding Research Council, Bull. No. 127. NISHIYAMA, R., TAKEMOTO, H., KAWAI, Y. & HONGO, K. (1978) Static loading tests on crane runway girders. Annual Conference of Architectural Institute of Japan, pp. 1335–9 (in Japanese). NISHIYAMA, R., TAKEMOTO, H., MASUDA, H., HONGO, K. & KAWAI, Y. (1980) Experimental study on local stresses in upper part of crane runway girders due to rail misalignment. Annual Conference of Architectural Institute of Japan, pp. 1207–10 (in Japanese). NISHIYAMA, R., TAKEMOTO, H., HONGO, K.&KAWAI, Y. (1983) Fatigue damage in bearing supports of crane runway girders. Symposium on Structural Engineering, Japan, 29, 51–8 (in Japanese). OKURA, I. & FUKUMOTO, Y. (1988) Fatigue of cross beam connections in steel bridges. IABSE, 13th Congress, Helsinki, pp. 741–6. OKURA, I. & MAEDA, Y. (1985) Analysis of deformation-induced fatigue of thinwalled plate girder in shear. Proc. JSCE, Structural Eng./Earthquake Eng., 2(2), 377s-384s. OKURA, I., HIRANO, H. & YUBISUI, M. (1987) Stress measurement at cross beam connections of plate girder bridge. Technology Reports of Osaka University, 37, 151–60. OKURA, I., YUBISUI, M., HIRANO, H. & FUKUMOTO, Y. (1988) Local stresses at cross beam connections of plate girder bridges. Proc. JSCE, Structural Eng./ Earthquake Eng., 5(1), 89s-97s. OKURA, I., TAKIGAWA, H. & FUKUMOTO, Y. (1989a) Structural parameters governing fatigue cracking at cross-beam connections in plate girder highway bridges. Technology Reports of Osaka University, 39 289–96. OKURA, I., TAKIGAWA, H. & FUKUMOTO, Y. (1989b) Structural parameters governing fatigue cracking in highway bridges. Proc. JSCE, Structural Eng./ Earthquake Eng. 6(2), 423s–426s. OKURA, I., YEN, B.T. and FISHER, J.W. (1991) Fatigue of thin-walled plate girders. Proc. JSCE, Structural Eng./Earthquake Eng. (Submitted). OXFORT, J.K. (1963) Zur Beanspruchung der Obergurte Vollwandiger Kranbahnträger durch Torsionsmomente und durch Querkraftbiegung unter dem örtlichen Radlastangriff. Der Stahlbau, Berlin, 32(12), 360–7. PARKS, E. (1952) The stress distribution near a loading point in a uniform flanges beam. Phil. Trans. Royal Soc., London, Series A, 244(886). PATTERSON, P.J., CORRADO, J.A., HUANG, J.S. & YEN, B.T. (1970) Fatigue and Static Tests of Two Welded Plate Girders. Welding Research Council, Bull. No.155. REEMSNYDER, H.S. & DEMO, D.A. (1978) Fatigue cracking in welded crane runway girders: causes and repair procedures. Iron and Steel Engineering, U.S.A., April, 52–6. SADAMASU, F. (1969) Contact pressure of a running pneumatic tire. Technical Journal of Public Works Research Institute, Ministry of Construction, Japan, 11(8), 3–5 (in Japanese).


SENIOR, A.G. & GURNEY, T.R. (1963) The design and service life of upper part of welded crane girders. Structural Engineer, England, 141(10), 301–12. TOPRAC, A.A. & NATARAJAN, M. (1971) Fatigue strength of hybrid plate girders. J. Structural Div., ASCE, 97(4), 1203–25. UMINO, S. & MIMURA, H. (1982) Fatigue research on welded crane runway girders. IABSE Lausanne Colloquium, pp. 577–84. YAMASAKI, T. & KAWAI, Y. (1983) Fatigue strength of stiffening frame in guideway beams for suspended monorail system. Trans. of Japan Society of Civil Engineers, 15, 78–81. YEN, B.T. & MUELLER, J.A. (1966) Fatigue Tests of Large-size Welded Plate Girders. Welding Research Council, Bull. No. 118.

Chapter 5 FATIGUE STRENGTH OF ADHESIVELY BONDED COVER PLATES PEDRO ALBRECHT Department of Civil Engineering, University of Maryland, USA

SUMMARY Adhesively bonding a cover plate to the tension flange of a steel girder and high-strength bolting the ends to prevent debonding has been shown experimentally to increase the fatigue life by a factor of 20 over that of welded cover plates, from Category E to Category B. The number of end bolts should be that needed to develop the cover plate’s portion of the maximum moment at the theoretical cut-off point. Parametric analyses of two-span-continuous, multigirder bridges have shown that improvements in cover plate design, which raise the fatigue strength to that of Category B, make them virtually fatigue-proof for highway truck loading. NOTATION Ac Ab b fr fr,X fr,E fr,A Fb Fsr Fv Kf m

Gross cross-sectional area of cover plate Gross cross-sectional area of bolt Intercept of S–N line Calculated stress range Fatigue strength of Detail X Fatigue strength of Category E details Fatigue strength of Category A details Allowable bending stress Allowable stress range Allowable shear stress for slip critical joints Fatigue notch factor Slope of S–N line

FATIGUE STRENGTH OF ADHESIVELY BONDED COVER PLATES 91

Mr n N Nd NE Nx Pb Pc s S Sc

Bending moment range Number of bolts Number of cycles Number of design cycles Mean number of cycles to failure of Category E details Mean number of cycles to failure of Detail X Shear force in bolt Tensile force in cover plate Standard deviation Section modulus of beam without cover plate Section modulus of beam with cover plate 5.1 INTRODUCTION

The aerospace industry is extensively bonding components and members fabricated from metals and composites for two main reasons. First, replacing mechanical fasteners with adhesives prolongs the fatigue life. Secondly, bonding lightweight composites to underlying structures greatly reduces the weight of components. Although epoxy resin adhesives were introduced to the construction industry at about the same time as to the aerospace industry, their application so far has been limited. To date, epoxy resins have been applied to bridge construction and rehabilitation in the following areas: (1) epoxy/aggregate mortar; (2) coating and surfacing; (3) bonding of secondary members; (4) corrosion protection of reinforcing bars for concrete structures; and (5) strengthening of concrete members with externally bonded steel plates. The writer and his coworkers have explored ways of extending the application of adhesives by testing bonded structural details that are commonly found on steel bridge construction (Albrecht et al., 1984). They reported elsewhere the results of tests on the fatigue strength (Albrecht & Sahli, 1986) and static strength (Albrecht & Sahli, 1988) of bolted and adhesively bonded splices of tension members and beams. The present study examines the fatigue strength of steel beams with adhesively bonded cover plates. 5.2 TEST PROGRAM 5.2.1 Specimens The 17 beams Bl through B18 (beam B16 had no cover plates), shown in Fig. 5.1, consisted of 355-mm (14-in) deep rolled beams, US standard

92 PEDRO ALBRECHT

FIG. 5.1. Coverplated beam specimens.

designation W14×30, weighing 440 N/m (30 lb/ft). The W14×30 section has 10mm×171-mm flanges and a 7-mm thick web. They were tested on a 4570-mm span under 2-point symmetrical loading. Two 13-mm×171mm×1143-mm long cover plates were adhesively bonded to the tension flange, one on each side of the center line of symmetry. The initial plan was to bond only the cover plates to the flange. After the tests of beams Bl to B3 had shown that the cover plate ends were gradually debonding, the cover plate ends on beam B4 closest to mid-span and both cover plate ends on beams B5 to B18 were clamped to the flange with two 19-mm diameter ASTM A325 High-Strength Bolts for Structural Steel Joints. If the bolts alone had to develop the cover plate’s portion of the maximum moment at the theoretical cut-off point, the required number of bolts would be (5.1) in which Pc is the cover plate force; Pb is the bolt shear force; Fb is the allowable bending stress; S is the section modulus of beam without cover plate; Sc is the section modulus of beam with cover plate; Ac is the gross area of cover plate; Fv is the allowable shear stress for slip critical joints of blast-cleaned low-alloy steel; and Ab is the gross area of bolt. Substituting in eqn (5.1) the applicable allowable stresses from the Standard Specifications for Highway Bridges adopted by the American Association of State Highway and Transportation Officials (AASHTO), and the cross-sectional properties of the test beam, gives n=4·4. Accordingly, the two high-strength bolts, with which the cover plate ends


on beams B5 to B18 were clamped, represent about one-half of the bolts needed to develop the cover plate’s portion of the maximum moment. The cover plates on beams B1 to B13 terminated in the shear span, 152 mm away from the loading points, and were cycled at stress ranges of 145 and 189 MPa (lMPa=1 N/mm2). Those on beams B14 to B18 were extended 108 mm into the constant bending moment region and were cycled at stress ranges of 159 and 207 MPa. The stress ranges were calculated from (5.2) in which Mr is the bending moment range at the cross section through cover plate end, and S is the gross section modulus of W14×30 beam without cover plates. The minimum stress was 7 MPa in all tests. 5.2.2 Materials The steel for the beams and cover plates conformed to the requirements of the ASTM A588 Specification for High-Strength Low-alloy Steel with 345 MPa Minimum Yield Point to 100 mm Thick. The measured tensile properties and chemical composition of the A588 Grade B steel beams and cover plates are given in Tables 5.1 and 5.2, respectively. The cover plates were bonded with Versilok 201, an acrylic structural adhesive for bonding metals, composites and engineering thermoplastics. The adhesive consisted of two parts, the acrylic and the accelerator No. 4. The adhesive cured on contact with the accelerator. The bond became TABLE 5.1 MECHANICAL PROPERTIES OF A588 GRADE B STEEL SPECIMENS Component

Yield point (MPa)

Tensile strength Elongation 200- Charpy energy (MPa) mm gage (%) at −4°C (J)

W14×30 beam 390 535 Cover plate 462 615 aCharpy specimen size: 10 mm×7·5 mm.

144a 106

23·2 20·0

TABLE 5.2 CHEMICAL COMPOSITION OF A588 GRADE B STEEL SPECIMENS Component

Composition (%)

C

Mn

P

S

Si

Cu

Ni

Cr

V

Al

W14×30 Coverplate

0·16 0·12

0·94 1·05

0·009 0·019

0·020 0·011

0·23 0·25

0·30 0·28

0·29 0·25

0·54 0·47

0·27 0·06

0·02 —

94 PEDRO ALBRECHT

handleable after 8–16 minutes of curing at 24°C and developed full strength after 24 hours. The mean shear strength of the adhesive, measured as part of this study with 13 quad-lap specimens under rapid loading, was 27·3 MPa with 2·2MPa standard deviation. The Versilok 201 adhesive had been recommended by Nara & Gasparini (1981), for its good shear and tension strength under rapid loading, rapid cure at room temperature and minimal requirements for surface preparation. In subsequent tests, Versilok 201 was found to have low creep strength under longterm sustained loading in severe environments (Albrecht et al., 1984; Mecklenburg, et al., 1985), but this behavior did not adversely affect the fatigue strength of the cover plates in the short-term fatigue tests under cyclic loading. 5.2.3 Bonding The contact surfaces were prepared, and the cover plates were bonded to the tension flange as follows: 1. Shot-blast (wheelabrate) the surfaces with No. 330 cast steel shot to a nearwhite condition. 2. Wipe the surfaces with a cloth. When needed, remove oil and crayon marks with trichloroethylene. 3. Apply accelerator No. 4 on both surfaces. 4. Apply Versilok 201 adhesive on one surface, after the accelerator has dried. 5. Sprinkle 0·25-mm diameter glass beads with a salt shaker on the surfaces to obtain the desired 0·25-mm bond line thickness. 6. Place the cover plate on the flange. 7. Install the bolts at the cover plate ends (Beams B5 to B18) and tension them to 70% of the ultimate tensile strength. 8. Clamp the cover plates to the flange with steel blocks, and firmly handtighten the C-clamps (Fig. 5.2). 9. Cure the adhesive at room temperature for at least 48 hours, but not longer than 2 weeks, before the fatigue test begins. 5.3 RESULTS 5.3.1 Test Program The six cover plates on beams Bl to B3 were bonded but not bolted to the tension flange. They gradually debonded during stress cycling. For example, an


FIG. 5.2. Adhesively bonded cover plate clamped to tension flange during curing.

ultrasonic compression wave scan of cover plate B2–1 showed that its midspan end had debonded over a 125-mm length after 1986000 cycles of 145 MPa stress range. The debonded areas are shown dark in Fig. 5.3, and the bonded areas clear. The midspan ends of the bonded cover plates on beam B4 were bolted, but not the support ends. After the cover plate B4–1 had debonded from the support end, the cover plate B4–2 was secured at the support end with two high-strength C clamps. Thereafter, all cover plates bonded to beams B5 to B18 were bolted at both ends prior to stress cycling. Beam B16 had no cover plates. 5.3.2 Crack Initiation and Propagation There were 28 bonded and bolted midspan ends of cover plates on Beams B4 to B18. Of those, 9 failed from cracks that initiated at the bolt holes in the flange, 2 from fretting cracks in the flange, and 16 did not fail. The data for the cover plate B4–1 was discarded, because the cover plate had debonded from the support end (Table 5.3). The bolt-hole cracks initiated along the bore or at the intersection of the bore with the flange surfaces. The cracks initiated on either side of

96 PEDRO ALBRECHT

FIG. 5.3. Compression wave scan of cover plate end B2–1. Bonded areas are light, debonded areas are dark. TABLE 5.3 FATIGUE TEST DATA FOR BONDED COVER PLATES Beam no.

Gross area stress range, fr (MPa)

Fatiguea life, N Log-mean life life Mean fatigue life (×103) (×103)

Category E Category B (×103) (×103) With non-bolted ends B1-1 145 B1-2 145 B2–1 145 B2–2 145 B3–1 145 B3–2 145 With bolted ends B7–1 145 B7–2 145 B8–1 145 B8–2 145 B9–1 145 B9–2 145 B13–1 145 B13–2 145 B14–1 159 B14–2 159 B15–1 159 B15–2 159

4156a 4156a 1986a 1986a 1282a 1282a

>2200

160

2580

>5560

160

2580

>2220

121

1900

2900 4471c 10012c 2568 9006c 5502c 5502c 2920 3394c 150 2130


Beam no.

Gross area stress range, fr (MPa)

Fatiguea life, N Log-mean life life Mean fatigue life (×103) (×103)

Category E Category B (×103) (×103) B4–1 189 _b B4–2 189 2239c B5–1 189 1302 B5–2 189 2130c B6–1 189 684 B6–2 189 1251 B10–1 189 2277 B10–2 189 2505c B11-1 189 1311c B11–2 189 1311c B12–1 189 842 B12–2 189 1752c B17–1 207 905 B17–2 207 905c B18–1 207 1029 B18–2 207 1168c aCover plate debonded from midspan end. bCover plate debonded from support end. cDetail did not fail, or test was discontinued.

>1480

70

1050

>970

53

780

the line formed by the intersection of the bore with the normal plane through the center of the bolt. The initiation points were offset from 3 mm ahead of to 8 mm behind the normal plane. The cracks initiated in about equal numbers from the bolt-hole sides facing either the web or the edge of the flange. Cracks in some beams grew simultaneously from both bolt holes. The bolt-hole cracks propagated, at first, as part-through or corner cracks until their length along the bore was equal to the flange thickness. Thereafter, they propagated as through cracks across the flange and up into the web. The two fretting cracks initiated on the flange surface, as a result of friction between the bolt head and the cover plate. They initiated in cover plate B14–1 14 mm ahead of, and in cover plate B15–2 11 mm behind the bolt center line. They propagated as part-through cracks until they reached the inside flange surface and then as two-ended through cracks across the flange. No fatigue cracks initiated in the cover plate, nor did flange cracks propagate across the bond line and into the cover plate. Figure 5.4 shows a typical bolt-hole crack. After a cover plate end had failed, the flange was temporarily spliced with plates and high-strength C-clamps. The

98 PEDRO ALBRECHT

FIG. 5.4. Bolt-hole crack at end of cover plate B9–1. Bolt installed at crack tip in web is part of temporary repair.

crack tip in the web was arrested by drilling a hole and installing a high-srength bolt. Then, cycling was continued until the other cover plate on the same beam failed. 5.3.3 Fatigue Test Data Each beam provided two data points, one per cover plate end nearest the midspan, for a total of 34 data points. The fatigue lives of the cover plates are listed in Table 5.3, and the data points are plotted in Fig. 5.5. The fatigue lives of the non-bolted cover plates on beams B1 to B3 are shown with circular symbols. No cracks were observed in the tension flange or in the cover plate of these beams, but the gradual debonding would eventually have separated the entire cover plate from the flange and limited the load-carrying capacity of the beam to that of the W14×30 section alone. The number of cycles to fatigue failure is ambiguous in this case. If, as is commonly done, fatigue failure is defined as the presence of a large crack in the tension flange or cover plate, these beams cannot be said to have failed in fatigue. However, in an actual bridge girder, the gradual debonding of the cover plate from the ends towards mid-length in the direction of increasing bending moment is unacceptable, because the cover plate would eventually separate from the girder, causing the girder to fail at maximum load. Such gradual debonding must be prevented.


FIG. 5.5. S–N plot for bonded cover plates.

The 27 data points for the bonded and bolted cover plates on beams B4 to B18 are shown with triangular symbols. The 11 solid triangles are for cover plate ends that fatigue-cracked to failure, and the 16 open triangles are for cover plates that did not fail. In the latter case, the tests were discontinued for two reasons: (1) the detail had reached 10 million cycles or the upper confidence limit for Category B details without cracking; or (2) a fatigue crack reinitiated at another previously cracked and temporarily repaired detail on the same beam. The data points for the two cover plate ends that failed from fretting cracks are identified with the letter F. The mean S–N line log N=b−m logfr (5.3) for the 27 data points was found by regression analysis to have an intercept b=16·571 5 for stress range in units of MPa (N/mm2), slope m=4·574, and standard deviation s=0·1983. The data for the non-bolted ends were excluded from the regression analysis. Equation (5.3) is shown in Fig. 5.5. The fatigue test data were compared against the AASHTO Category B rhomboid, herein defined as the space containing 95·5% of the data points for plain welded beams from which Fisher et al. (1970) derived the Category B allowable S–N line. The rhomboid, shown in Fig. 5.5, is delineated by the following four lines: 1. Lower confidence limit, equal to the AASHTO allowable S-N line for Category B details: log Nd=(b−2s)−m log Fsr (5.4) in which b=intercept=13·698 m=slope=3·372

100 PEDRO ALBRECHT

s=standard deviation=0·147 Nd=number of design cycles Fsr=allowable stress range 2. Upper confidence limit: log N=(b+2s)−m log fr (5.5) 3. Lower cut-off, equal to the AASHTO fatigue limit of 110 MPa for Category B plain welded beams subjected to over 2000000 cycles of loading. 4. Upper cut-off, equal to the highest stress range, fr=290 MPa, at which welded beams were tested. Category B is one of seven categories of details for which the AASHTO specifications provide allowable S–N lines. These lines are equal to the mean minus two standard deviations of S–N data for the following members that were tested in the laboratory: A, rolled beams; B, beams built up of plates that are connected with continuous filled welds; B′, members built up of plates that are connected with groove welds; C, transverse stiffeners and welded attachments 50-mm long or less; D, welded attachments 100-mm long or less; E, cover plates welded to flanges 20-mm thick or less; and E′, cover plates welded to flanges thicker than 20-mm. Details other than those listed above are assigned to the category for which most data points just exceed the allowable S–N line for that category. As Fig. 5.5 shows, all data points fell inside or to the right of the Category B rhomboid, indicating that the fatigue strength of bonded cover plates with bolted ends exceeds that of Category B details. Table 5.3 compares at each stress range the log-mean life of the bonded cover plates with the means of Category E welded cover plates and Category B plain welded beams. The latter were calculated from eqn (5.3) using the intercepts and slopes listed in Table 5.1 of Albrecht and Simon (1981). The mean fatigue life of the bonded cover plates exceeded the mean for Category B welded beams at all stress range levels. The factor increase in life over the mean for Category E welded cover plates was at least 20 for the data at the three highest stress ranges, and 35 at the lowest stress range near the fatigue limit. Six of eight bonded and bolted cover plates tested at the lowest stress range (145 MPa) did not fail, suggesting that the fatigue limit may lie about half way between the AASHTO fatigue limits for Category A rolled beams (165 MPa) and Category B welded beams (110 MPa). Clearly, bonding and end bolting has the potential of fatigueproofing cover plates in typical bridge girders (Al-brecht et al, 1983). 5.3.4 Comparison with Previous Work Since 1969, several investigators have reported results of 755 fatigue tests of cover plates on steel beams. One group of studies established the fatigue strength of various end details (Figs 5.6 (a) to (e); Fisher et al., 1970), cover plates welded to


FIG. 5.6. Welded cover plates.

thick flanges (Fig. 5.6(a); Fisher et al., 1979a; Roberts et al., 1977), and the behavior under variable amplitude loading (Figs 5.6(a) and (b); Schilling et al., 1978). The other group of studies on cover plates focused on improving the fatigue strength by grinding, shot peening, and TIG remelting (Fig. 5.6 (a); Fisher et al., 1979b), end-welding and grinding (Fig. 5.6 (f); Yamada & Albrecht, 1977), end bolting (Figs. 5.7(a) to 5.7(c); Wattar et al., 1985), and retrofitting (Fig. 5.7(d); Sahli et al., 1984). The fatigue test data for each aforementioned series of tests are summarized in Table 5.4. The factor increase in mean fatigue life of a Detail X over that of a Category E detail is defined in Fig. 5.8 as (5.6) The stress ranges for Detail X and Category E at 500 000 cycles were calculated from the corresponding mean S–N lines (see eqn (5.3)). The exponent, m=3·2 is the average slope of the S–N lines that formed the basis of the AASHTO fatigue specifications. The fatigue notch factor (5.7)

102 PEDRO ALBRECHT

FIG. 5.7. End-bolted and adhesively bonded cover plates.

gives the factor on stress range by which the mean S–N line of a Detail X falls below the mean S–N line for Category A rolled beams. The results of the calculations, listed in Table 5.4 and plotted in Fig. 5.9, show that the various techniques improved the mean fatigue life over that of Category E by the following factors: 1·5, 2·1 and 4·4 for ground, shot-peened, and TIG remelted toes of end welds, respectively; 6·5 for end-welding and grinding the cover plate to a 1:3 taper; 19 or more for bolting the non-welded ends with enough bolts to develop the cover plate’s portion of the maximum moment at the theoretical cut-off point; and 18 and 13, respectively, for retrofitting with a bolted splice the non-cracked and cracked ends of welded cover plates. In comparison, bonding and end bolting cover plates (Fig. 5.7(f) increased the fatigue life by a factor of 20 in the finite-life regime and by a factor of 35 near the fatigue limit. Evidently, the combined use of adhesive and bolts can greatly increase the fatigue life. In this case, the adhesive transfers the longitudinal shear stress from the flange to the TABLE 5.4


COMPARISON OF PREVIOUS AND PRESENT FATIGUE TEST DATA FOR COVER PLATE ENDS Symbol in Fig. 5.9

Series

No. of details tested

Stress range at 500000 cycles fr,x (MPa)

Fatigue notch factor, Kf

Square-ended, prismatic cover plates (Fisher, et al., 1970) CR, CW 103 98 3·57

Fatigue life increase, Nx/NE

Comments

0·94

Welded end Unwelded end Welded end Welded end Unwelded end Welded end Welded end

CR, CW 100

109

3·22

1·31

CT

30

101

3·49

1·01

CB

30

102

3·44

1·06

CB

30

79

4·41

0·48

CM

30

102

3·45

1·05

All

193

100

3·51

1·00

End-welded and ground cover plates (Yamada & Albrecht, 1977) CG 26 180 1·94 6·54

Toe cracks

Ground, shot-peened, and TIG remelted end welds (Fisher et al., 1979b) GA 8 115 3·06 1·53 PA

16

127

2·77

2·12

TA

16

159

2·20

4·40

Cover plates subjected to variable amplitude loading (Schilling et al., 1975) B&C 27 98 3·58 0·93 B&C

45

103

3·40

1·10

Var. ampl.

Full-size coverplated beams (Fisher et al., 1979a; Roberts, et al., 1977) B 38 84 4·19 0·56 End-bolted cover plates (Wattar et al., 1985) 6-bolt 4 …

1·35

21

6-bolt

16

…

1·40

19

4-bolt

12

…

1·30

24

2-bolt

12

…

1·56

13

Retrofitted cover plates (Sahli et al., 1984)

Cleaned

104 PEDRO ALBRECHT

Symbol in Fig. 5.9

Series

No. of details tested

Stress range at 500000 cycles fr,x (MPa)

Fatigue notch factor, Kf

Fatigue life increase, Nx/NE

Comments

B

14

…

1·42

18

A

15

…

1·58

13

Noncracked Precracke d

…

20

fr159 MPa

…

35

fr=145 MPa

Adhesively bonded cover plates (present study) B 19 … B

8

…

FIG. 5.8. Calculation of fatigue notch factor and increase in fatigue life.

cover plate, while the clamping effect of the bolts prevents debonding of the cover plate ends. End-bolting bonded cover plates is about equally effective in increasing the fatigue life as end-bolting welded cover plates (Wattar et al., 1985). In both cases, coverplated highway bridge girders would eventually fail from bolt-hole cracks or fretting cracks, but the increase in fatigue strength virtually fatigueproofs the cover plate ends for truck loading as will be shown in Section 5.5. The only other known fatigue tests of long attachments adhesively bonded to steel beams are those reported by Nara and Gasparini (1981). In that study 14mm×229-mm×1220-mm long cover plates were bonded to the tension flange of two W14×30 steel beams (Fig. 5.7(e)), with the same adhesive as was used in the present study. The two beams were subjected to 6000000 cycles and 3 500 000 cycles of 138 MPa stress range. Measured curves of load versus midspan deflection had indicated a progressive decrease in beam stiffness with stress


FIG. 5.9. Comparison of mean fatigue strength of all data sets with mean of Categories A through E′ data. See Table 5.4.

cycling caused by debonding from the ends. The writer calculated the change in stiffness as a function of debonding length and compared the results against the measured changes in stiffness reported by Nara & Gasparini (1981). According to the calculations, the reported changes in stiffness indicated that the cover plate ends on the beam cycled 6000000 times had debonded over a length of 320 mm. This problem was solved in the present study by clamping the ends with highstrength bolts. 5.4 COVER PLATE DEVELOPMENT Two important observations were made in a companion study on the static strength of bolted beam splices with adhesively bonded contact surfaces (Albrecht & Sahli, 1988). First, the adhesive failed at a load equal to or less than the load needed to cause the steel beam to yield in bending at the edge of the bond line. Epoxy adhesives typically have an ultimate strain less than about 1/20 of the ultimate strain of structural steel. Unlike steels, adhesives cannot deform plastically. Second, the resistance of the splice was controlled by either the ultimate shear strength of the adhesive or the ultimate shear strength of the bolts, whichever was higher. The strengths of the two connecting elements were not additive. The above findings suggest that, under static loading, a beam with an adhesively bonded cover plate can fail suddenly when the flange near the theoretical cut-off

106 PEDRO ALBRECHT

point begins to yield if the ends are clamped with less than the number of bolts needed to develop the cover plate’s portion of the bending moment. In this case, yielding of the flange could trigger the following sequence of events: (1) shear failure of the adhesive at the cover plate end; (2) shift of the longitudinal shear transfer from the adhesive to the bolts, which are not designed to develop the cover plate’s portion of the bending moment; (3) shear failure of the bolts; (4) progressive yielding of the steel beam towards midspan; (5) unzipping of the central portion of the bonded cover plate; and (6) failure of the steel beam at the point of maximum moment, which is much greater than the moment at the theoretical cut-off point. To prevent the occurrence of a progressive static failure, the number of bolts should be increased from one-half (as tested herein) to the full number of bolts needed to develop the cover plate’s portion of the bending moment at the theoretical cut-off point, given by eqn (5.1). Full cover plate development by the bolts reduces the bending stresses in the tension flange, enables the adhesive to transfer the reduced longitudinal shear stresses away from the regions of stress concentration at the cover plate ends, and may even allow the formation of a plastic hinge in the steel beam at the theoretical cut-off point. 5.5 APPLICATION TO HIGHWAY BRIDGES For reasons of economy, rolled wide flange sections are frequently used for girders in short-span highway bridges. The range of spans for which rolled girders are suitable can be extended by welding cover plates to the flanges in regions of high bending moment, thus increasing the moment-carrying capacity of the girder. The disadvantage of welded coverplated girders is that the cover plate ends have very low fatigue strength. As a result, the cover plates must be extended beyond the theoretical cut-off points for maximum stress to points where the stress range induced by the live load at the extreme fiber of the rolled girder is equal to or smaller than the allowable stress range for Category E. To demonstrate the benefits of adhesive bonding cover plates in lieu of welding, four two-span continuous highway bridges, with the following details, were analyzed: 1. Span length: 21·3–21·3 m, 24·4–24·4 m, 27·4–27·4 m, and 30·5– 30·5 m. 2. 13·4 m wide roadway. 3. Composite construction (positive moment region only); rolled beam with cover plate; allowable stress design. 4. Six girders equally spaced 2·39 m apart. 5. 20-cm thick reinforced concrete slab. 6. AASHTO HS-20 loading consisting of a 3-axle truck with axle loads of 35·6, 142·4, and 142·4 kN.


7. AASHTO Fatigue Specifications, Article 1.7.2; assume Category B for bonded and end-bolted cover plate, and Category E′ for welded cover plate. 8. Steel: Fy=250 MPa. 9. Concrete: f’c=27 MPa, modular ratio n=ES/EC=8. 10. Cover plate cut-off point: (1) for maximum moment, -two times cover plate width plus 75 mm beyond theoretical cut-off point (see AASHTO Art. 1.17. 12); and (2) for fatigue, where computed stress range equals allowable stress range. Envelopes were constructed for the maximum and minimum moments at each tenth point along the length of the girders. The sizes of the 915-mm deep girders (W36 sections) and the cover plates needed to resist the maximum positive moment in the span and the maximum negative moment over the interior support are given in Table 5.5 for all four span lengths. The stress ranges in the beam flange at the cover plate ends were calculated using eqn (5.2) in which Mr is the algebraic difference between maximum and minimum moment and 5 is the section modulus of beam without cover plates. The fatigue analysis was performed for loading case I, that is ‘over 2000000 cycles of loading’ producing the maximum stress range at the cover plate end (AASHTO Art. 10.3.1). Accordingly, the allowable stress ranges are 18 MPa for Category E′ and 110 MPa for Category B. These stress ranges correspond to the safe fatigue limit below which the details are not expected to fail irrespective of the applied number of cycles. TABLE 5.5 SIZE OF GIRDER AND COVER PLATES FOR TWO-SPAN CONTINUOUS BRIDGES Span length (m)

Rolled section

Positive moment: bottom

Negative moment: top and bottom

21·3–21·3 24·4–24·4 27·4–27·4 30·5–30·5

W36×l35 W36×160 W36×l82 W36×245

Cover plate thickness and width (mm)

10×250 13×250 19×250 16×275

19×250 32×250 44×275 32×350

Figure 5.10 shows the arrangement of the (a) bonded and end-bolted and (b) welded cover plates in the 21·3–21·3 m long bridge. The bonded and end-bolted cover plates were terminated at the cut-off points for maximum stress, since fatigue did not govern the design in this case (Fig. 5.10(a)). The low fatigue strength of the welded cover plates required extensions of the cover plates to points where the calculated stress range was equal to the allowable stress range of 18 MPa. The right end of the 10-mm×250-mm cover plate for maximum positive moment on the bottom flange was extended to point

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FIG. 5.10.Cover plate arrangement for 21·3–21·3-m two-span continuous bridge.

No. 2 in Fig. 5.10(b), where it was butt-welded to the 19-mm×250-mm cover plate that was needed to resist the maximum negative moment. This design was chosen because it was the most economical. The ground and tapered butt weld, made prior to welding the combined plate to the flange, has Category B fatigue strength and was not fatigue critical. The left end of the 10-mm×250-mm cover plate was extended to point No. 1. The top flange cover plate for maximum negative moment also had to be extended, in this case to point No. 3 (Fig. 5.10(b)). For reasons of economy, this extension was done with a lighter 10-mm×250-mm cover plate, which was buttwelded to the 19-mm×250-mm cover plate. Both the top and bottom flange splices of the negative-moment cover plates were located at the theoretical cutoff point for the thinner plate. The designs of the bridge girders with 24·4–24·4-m, 27·4–27·4-m, and 30·5– 30·5-m spans were similar to that for the 21·3–21·3 spans shown in Fig. 5.10. For all spans, the lengths of the bonded and end-bolted cover plates were governed by maximum stress. The stress range at the cut-off points of the cover plates for maximum moment varied from 65 to 105 MPa, depending on the cover plate end and span length. All values were lower than the allowable stress range for Category B details. The length of the welded cover plates was always governed by fatigue. In addition to being fatigue proof, girders with bonded and end-bolted cover plates are, on average, 5% lighter than girders with welded cover plates. 5.6 CONCLUSIONS The results of the 28 tests performed in the present study showed that cover plates adhesively bonded to the tension flanges of steel beams have Category B fatigue strength under cyclic loading if the cover plate ends are high-strength


bolted to the flange to prevent gradual debonding. When this was done, the fatigue life was 20 times longer than that of welded Category E cover plates. Analysis of four two-span-continuous, multigirder bridges has shown that improvements in cover plate design which raise the fatigue strength to that of Category B also make them fatigue-proof to truck loading. From the viewpoint of fatigue strength, it is sufficient to clamp the cover plate ends with one-half of the number of bolts needed to develop the cover plate’s portion of the maximum bending moment at the theoretical cut-off point. However, to permit yielding of the flange at the theoretical cut-off point under static loading without triggering a progressive debonding failure, the ends should be clamped with the number of bolts required to fully develop the cover plate. It is emphasized that the Versilok 201 adhesive, like many other adhesives, has low creep strength at temperatures and relative humidities that are high but still within the range of those found in service environments (Albrecht et al., 1984, 1987). Although end bolting bonded cover plates may prevent creep, adhesives with better durability over a 50-year service life are generally needed. As the aerospace industry’s success with bonding components suggests, the widespread application of adhesives to bonding civil engineering structures is only a matter of time. REFERENCES ALBRECHT, P. & SAHLI, A.H. (1986). Fatigue strength of bolted and adhesive bonded structural steel joints. Symposium on Fatigue in Mechanically Fastened Composite and Metallic Joints, ASTM STP 927. American Society for Testing and Materials, Philadelphia, Pennsylvania, 72–94. ALBRECHT, P. & SAHLI, A.H. (1988). Static strength of bolted and adhesively bonded joints. Symposium on Adhesively Bonded Joints—Testing, Analysis and Design, ASTM STP 981, American Society for Testing and Materials, Philadelphia, Pennsylvania, pp. 229–51. ALBRECHT, P. & SIMON, S. (1981). Fatigue notch factors for structural details. J. Structural Div., ASCE, 107(7), 1279–96. ALBRECHT, P., WATTAR, F. & SAHLI, A. (1983). Towards fatigue-proofing cover plate ends. Proceedings of W.H. Munse Symposium on Metal Structures, Convention, American Society of Civil Engineers, New York, pp. 24–44. ALBRECHT, P. & SAHLI, A.H., CRUTE, D., ALBRECHT, Ph. & EVANS, B. (1984). Application of Adhesives to Steel Bridges. Report No. FHWA/RD-84/037, Federal Highway Administration, Washington, D.C. ALBRECHT, P., MECKLENBURG, M.F., WANG, J.R. & HONG, W.S. (1987). Use of Adhesives to Replace Welded Connections in Bridges. Report No.87/029 Federal Highway Administration, Washington, D.C. FISHER, J.W., FRANK, K.H., HIRT, M.A. & MCNAMEE, B.M. (1970). Effect of Weldments on the Fatigue Strength of Steel Beams. NCHRP Report No. 102, Transportation Research Board, National Research Council, Washington, D.C.

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FISHER, J.W., HAUSAMMANN, H. & PENSE, A.W. (1979a). Retrofitting Procedures for Fatigue Damaged Full-Scale Welded Bridge Beams. Fritz Engineering Laboratory Report No. 417–3(79), Lehigh University, Bethlehem, Pennsylvania. FISHER, J.W., HAUSAMMANN, H., SULLIVAN, M.D. & PENSE, A.W. (1979b). Detection and Repair of Fatigue Damage on Welded Highway Bridges. NCHRP Report No. 206, Transportation Research Board, National Research Council, Washington, D.C. MECKLENBURG, M.F., ALBRECHT, P. & EVANS, B. (1985) Screening of Structural Adhesives for Application to Steel Bridges. Interim Report, Federal Highway Administration, NTIS Report No. PB 86 142 601, Washington, D.C. NARA, H. & GASPARINI, D. (1981). Fatigue Resistance of Adhesively Bonded Structural Connections. Report No. 45K1–114, Department of Civil Engineering, Case Institute of Technology, Cleveland, Ohio. ROBERTS et al. (1977). Determination of Tolerable Flaw Sizes in Full-Size Welded Bridge Details. Report No. FHWA-RD-77–170, Federal Highway Administration, Washington, D.C. SAHLI, A.H., ALBRECHT, P. & VANNOY, D.W. (1984). Fatigue strength of retrofitted cover plate ends. J. Structural Eng., ASCE, 110(6), 1374–88. SCHILLING, C.G., KLIPPSTEIN, K.H., BARSOM, J.M. & BLAKE, G.T. (1978). Fatigue of Welded Steel Bridge Members Under Variable-Amplitude Loading. NCHRP Report No. 188, Transportation Research Board, National Research Council, Washington, D.C. WATTAR, F., ALBRECHT, P. & SAHLI, A.H. (1985). End-bolted cover plates. J. Structural Eng., ASCE, 111(6), 1235–49. YAMADA, K. & ALBRECHT, P. (1977). Fatigue behavior of two flange details. J. Structural Div., ASCE, 103(4), 781–91.

Chapter 6 FATIGUE OF TUBULAR JOINTS IN OFFSHORE STRUCTURES J.J.A.TOLLOCZKO Offshore Engineering Division, The Steel Construction Institute, Ascot, UK

SUMMARY This chapter introduces the reader to the very complex problem of offshore tubular joint behaviour and fatigue design. A review of available knowledge on the fatigue behaviour of tubular joints is beyond the scope of any single document. However, this chapter addresses the general parameters which influence fatigue and also discusses a number of these parameters with particular reference to experimental techniques and recent data. Tubular joint design is an area which has attracted substantial research on a worldwide basis. This has been necessary so that the offshore industry can, with reasonable confidence, design and install structures to resist the continuous cyclic loads generated by winds and waves. The rationalisation and refinement of tubular joint fatigue design will continue for many years to come but it is unlikely that all the factors influencing the fatigue behaviour of tubular joints will be fully understood, in the near future. 6.1 INTRODUCTION The need for an offshore platform is usually established by a requirement to provide an offshore facility which consists of an operational deck, supported above sea level, having a specified working area and capable of carrying a predefined load. The design of the supporting structure parallels in many ways that of a land-based structure but with two additional requirements. The first is that the supporting structure is fabricated onshore and then transported and installed

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offshore. The second is the requirement to design the structure to resist cyclic wave loading so as to prevent structural fatigue failure during its life. By far the most common type of offshore platform in current use is the steel jacket structure, illustrated in Fig. 6.1. This type of platform consists of a prefabricated steel support structure (jacket), that extends from the seabed to above the water surface, with a prefabricated steel deck located on top of the support structure (topside). The structure is fixed to the seabed by piles, which are designed to provide adequate resistance against lateral loading from wind, waves and currents. Jackets and topsides have to be designed to be self-supporting during fabrication, transportation and installation. The entire assembled installation has to withstand loading due to self-weight, operational requirements and environmental loads. The main criteria considered in the design of a jacket and topside are: (a) Fabrication (b) Loadout (c) Transportation (d) Lifting and/or Launching (e) Temporary Stability (f) In-place (g) Damaged The in-place loading condition, which generally governs the design of jacket structures, is usually due to waves. This therefore makes the assessment of fatigue life a primary design criterion. It is usual in offshore structural design to analyse primary units such as jackets, decks, modules, flare booms/towers, bridges, and the completed platform structure, as three-dimensional space frames for all the applicable structural states and loading conditions. The effect of the interaction between the topside and jacket and between the jacket and foundations is also included in the design of the assembled structure. A moderately large topside module may have 120 members and a medium-size jacket may have 2000 members. In view of the large number of structural states and loading conditions that have to be considered in a design, extensive use of computers becomes necessary to analyse the structures and check the strength of the members and joints. Most steel offshore support structures are three-dimensional frames fabricated from cylindrical steel members. These give the best compromise in satisfying the requirements of low drag coefficients, high buoyancy and high strength-to-weight ratio. Tubular joints, which are formed by welding the contoured end of one tubular member (brace member) onto the undisturbed outside of the other (chord member), are a major source of difficulty and high cost in the design, construction and maintenance of steel offshore structures. The resulting joint

J.J.A.TOLLOCZKO 113

FIG. 6.1. Typical North Sea steel-piled self-contained drilling/production platform. 1. Jacket substructure; 2. Module support frame; 3. Piles; 4. Drilling derrick; 5. Pile sleeves; 6. Launch runners.

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FIG. 6.2. Typical jacket framing configurations.

configurations can get very congested, and it is not unusual to have up to eight brace members intersecting onto one chord member. Because of the harsh environment to which North Sea structures are subjected, the area of design which has undoubtedly received the most attention is that of fatigue. A jacket concept study which determines the form the structure will take, is as likely to be influenced by fatigue requirements as static strength considerations. Every welded connection which sees cyclic stresses due to environmental loading is assessed and designed such that a satisfactory fatigue life is attained. Some connections are checked by implication, as it is normal to reduce computation time by checking the worst of a group. 6.2 TUBULAR JOINTS IN OFFSHORE STRUCTURES 6.2.1 Joint Types and Geometric Notations The number, size and orientation of members meeting at a joint vary significantly according to the configuration and size of the structure. Jacket configurations include X-braced, K-braced and diagonally braced structures. A two-dimensional example of each structure is shown in Fig. 6.2. The configuration is normally chosen to provide the best horizontal and torsional resistance to the particular environmental forces under consideration. Associated with the main legs of a large jacket structure there are likely to be many K, KT and YT joints (see Fig. 6.3) and, depending on the position and number of legs, some of these will be double joints and have members in more than one plane. At the other extreme, in the horizontal conductor framing system, most joints will be of the T or DT configuration because of the regular orthogonal grid of conductors. For ease of reference, joints are generally classified by the system of notation shown in Fig. 6.3. Many of the codes, and in particular the American Welding Society Code D1.1(1986), adopt this reference system. It should be noted that this classification refers only to geometry. Several design codes, and in particular

J.J.A.TOLLOCZKO 115

FIG. 6.3. Preferred notation for joint configurations.

the American Petroleum Institute Code API RP2A (1987), require consideration of both the applied loads and the joint configuration to determine the design classification. Joints are placed in the following four categories for design purposes: (1) (2) (3) (4)

Simple welded joints Complex welded joints Cast steel joints Composite joints

Each joint type presents different design and fabrication problems. As well as describing the detailing of the joints, the first two classes indicate the relative

116 J.J.A.TOLLOCZKO

ease of design. The remainder of this section details joint types and nondimensional geometric ratios for joints classified as ‘simple joints’. To be considered ‘simple’, a joint must be formed by welding two or more tubular members in a single plane without overlap of brace members and without the use of gussets, diaphragms, stiffeners or grout. To prevent excessively high local stresses and to provide adequate static strength, simple joints may (and generally do) contain a short length of thicker-walled tube, sometimes of higherstrength material, in the connection area. This higher-strength is termed the ‘joint can’. The geometric notation and non-dimensional parameters describing simple joints are given in Fig. 6.4. Definitions of eccentricity and gap for KT joints are given in Fig. 6.5. The basic dimensions which describe a simple joint are: (1) (2) (3) (4) (5) (6) (7)

Chord outside diameter D Brace outside diameter d Chord wall thickness T Brace wall thickness t Included angle between chord and brace σ Gap g between brace toes (for K and KT joints) Chord length L, defined as the distance between end restraints or points of contraflexure of the chord

To facilitate the design and assessment of simple joints the above joint parameters must be rendered dimensionless. In general the following nondimensional geometric parameters are used for design and assessment purposes: Chord length parameter σ , is defined as the ratio of chord length L to chord radius D/2 (i.e. 2L/D) and gives an indication of chord beam bending characteristics. Diameter ratio σ , is the ratio of the brace diameter to the chord diameter (i.e. d/ D) and describes the compactness of the joint. Chord thinness ratio σ , is the ratio of the chord radius to the chord wall thickness (i.e. D/2T) and gives an indication of the thinness and radial stiffness of the chord. Wall thickness ratio σ , is the ratio of the wall thickness of the brace to that of the chord. It is a measure of the likelihood that the chord wall will fail before the brace cross section fractures. Gap parameter σ , is defined as the ratio of the gap to chord diameter. This parameter describes the proximity of other brace members to the subject brace member for joints with more than one brace. Another parameter which is sometimes used is σ , defined as σ /sin σ . This gives an indication of the principal length of the chord plug. Further, the terms ‘crown’ and ‘saddle’ are introduced to facilitate the description of position on the intersection periphery. These terms are illustrated in Fig. 6.6.

J.J.A.TOLLOCZKO 117

FIG. 6.4. Simple joint: geometric notation.

6.2.2 Local Joint Behaviour As described previously, tubular joints are formed by the intersection of brace and chord members. There are almost an infinite number of combinations of joint geometries, sizes and types for both simple and complex joints. Because of the circular member cross sections, the intersection details and the associated weld are of a complicated geometry even for the simplest type of tubular joint. This results in major distortions of the nominal stresses adjacent to the intersection. The stress distribution around the joint and the maximum ‘hot-spot’ stress play a critical role in fatigue design. The stress field around a tubular joint arises from three main effects: • The basic structural response of the joint applied load. This is termed the ‘nominal’ stress. • To maintain continuity at the intersection, the tubular walls deform giving rise to the ‘deformation’ stresses. • The weld introduces a geometrical discontinuity, giving rise to ‘notch’ stresses.

118 J.J.A.TOLLOCZKO

FIG. 6.5. Simple joint: definition of eccentricity and gap for KT joint.

Nominal stresses arise due to the framing action of the jacket structure under applied external loads. These stresses are calculated by carrying out a global analysis of the structure. Deformation stresses are best illustrated by the example of a simple T-joint in Fig. 6.7. Under tension loading in the brace, Points 1 and 2 displace along the brace axis by similar amounts owing to the constant axial stiffness of the brace. To maintain compatibility at the intersection, the chord will deform and introduce bending and membrane stresses in the chord wall. Since the stiffness of the chord at the saddle (Point 2) is greater than at the crown (Point 1), a larger force will be required at the saddle than at the crown. This results in a maldistribution of the nominal stress near the intersection. The bending and membrane stresses in the chord wall and the maldistribution of the nominal stress give rise to the deformation stresses. Notch stresses are the result of the geometric discontinuity of the tubular walls at the weld toes where an abrupt change of section occurs. The notch stresses decay through the wall thickness and there is, therefore, a local region with stresses varying rapidly in three dimensions. The weld also stiffens the walls of the tubular locally, and thus affects the local stresses in the tubulars up to and beyond the position of the weld toe. The presence of the weld has an effect on the deformation stresses owing to its local stiffening effect. This leads to the notion of ‘gross’ and ‘local’ deformation stresses. ‘Gross’ deformation stresses are due to the overall geometry of the joint, disregarding the weld and weld profile (e.g. as obtained from thin-shell finite-element analysis). ‘Local’ deformation stresses are due to the effect of the geometry of the joint, as well as the local stiffening offered by the weld (e.g. as measured by strain gauges on a steel model). The stresses at the intersection can vary greatly owing to the stress ‘raisers’ identified above. The maximum stress, known as the ‘hot-spot’ stress, can

J.J.A.TOLLOCZKO 119

FIG. 6.6. Typical tubular joints.

greatly exceed the nominal stress. The ratio of ‘hot-spot’ stress to the nominal stress is known as the stress concentration factor (SCF), which is generally used to describe the relationship between the stress at any point at the intersection to the nominal brace stress. The calculation of SCFs is discussed in Section 6.3.3. 6.3 FATIGUE RESISTANCE DESIGN OF TUBULAR JOINTS 6.3.1 General Two basic approaches are available for the fatigue-life assessment of structural components. The first method, which is currently in general use in design, relies on empirically derived relationships between the applied stress ranges and fatigue life (S−N approach). The second method is based on linear elastic fracture mechanics, which considers the growth rate of an existing defect at each stage of its propagation. The current practice limits the use of the fracture

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FIG. 6.7. Deformation stresses in a T joint under tension load.

mechanics approach for appraisal of existing joints with known defects and the specification of tolerable defect sizes during the fabrication stage. The following sections are primarily concerned with the S–N approach and address the fatigue limit state from the resistance standpoint. S–N curves and the associated design approaches recommended by design codes and guidance documents such as the American Petroleum Institute Code RP2A (1987) and the Department of Energy Guidance Notes (1984) have been derived from an examination of available test data. These recommendations are updated regularly to reflect any changes that may be warranted as new data become available. Examination of design codes and other published guidance documents such as UEG UR33 (1985) give the following parameters which affect fatigue strength: (a) The geometry of the tubular joint and weld. This geometry determines the SCF and governs the hot-spot stress location where fatigue cracking would be expected to initiate for a particular type of loading. The influence of

J.J.A.TOLLOCZKO 121

chord wall thickness on fatigue life is also an area which is currently receiving a lot of interest (thickness/size effect). (b) The type, amplitude, mean level and distribution of applied loads. The applied nominal stress history is increased at the hot-spot and these hot-spot stresses promote fatigue cracking at or near the welds. (c) The fabrication process. The procedures adopted during fabrication will determine the local material properties, the residual stress pattern and the distribution of defects. These factors will control the initiation period of the fatigue-failure process. (d) Post-fabrication processes applied to the tubular joint in order to improve fatigue life or some other aspect of performance. These include – PWHT (post-weld heat treatment) – weld toe grinding, hammer peening, TIG dressing, overall profiling, etc. – corrosion protection (e) The environment in which fatigue cracks initiate and grow. Three areas are relevant: – Air-well above the splash zone, although such air could have a high water vapour and salt content – Mixed air and seawater—splash zone where immersion is con trolled by wave height and tide – Seawater—complete immersion (f) Static chord load. The static load present in a chord member of a tubular joint owing to dead load or buoyancy, for example, can influence fatigue performance. In addition to the above parameters, the fatigue strength of a tubular joint may be influenced by the ratio R (ratio of trough applied stress to peak applied stress), seawater temperature and frequency of testing compared with wave frequency. While the discussion of all these aspects is beyond the scope of this chapter, a number of these parameters are introduced, discussed and assessed in Sections 6. 3.3 and 6.3.4 within the context of identifying their influence on fatigue life from recently published data. 6.3.2 Applied Loads Before carrying out a fatigue analysis at a tubular joint, it is necessary to determine the stress response over the range of sea conditions which the structure can expect to experience during its life. Random sea conditions are usually described in the short term, i.e. over a period of a few hours, by one of the many

122 J.J.A.TOLLOCZKO

direction wave spectrum formulae. These give the amplitude of a component wave at each frequency and direction in terms of parameters such as significant wave height, mean zero-crossing period, mean wave direction, etc. The proportion of time for which each seastate persists (ideally measured at the site over a number of years) completes the description of the sea. Hydrodynamic loading on tubular jacket structures is calculated using Morison’s (1950) equation. This expresses the wave loading as a function of local water particle velocity and acceleration. Various wave theories exist for calculating particle motion in regular waves; the most commonly used are the Stokes, Airy and stream-function theories. Analytical techniques currently used for calculating long-term stress distributions fall into the following categories: (1) Deterministic methods (2) Spectral methods A detailed review of these methods is beyond the scope of this chapter. 6.3.3 Local Joint Behaviour 6.3.3.1 Hot-Spot Stress Concept as Determined by Experimental Models Various methods are available for determining hot-spot stresses/stress concentration factors (HSS/SCF) to be used in fatigue design. These include the use of design formulae, finite-element (FE) analysis and testing. The methods vary not only in their accuracy and reliability, but also in their acceptance by designers. The generally preferred method is the use of parametric equations to calculate the SCF/maximum stress range at a joint. The method has the obvious advantage of simplicity and does not incur the significant cost and time penalties associated with FE analysis and testing. However, the use of parametric formulae is mainly applicable only to simple joints, and even then the confidence with which they can be used is questionable. For joints not covered by the current formulae, FE analysis and testing need to be considered. Since the use of parametric formulae is generally preferred, their status and reliability are discussed in Section 6.3.3.2. Section 6.3.3.3 addresses the areas of uncertainties associated with FE analysis. The remainder of this section discusses the measurement of SCFs from experimental models. Model testing is used primarily to obtain the strain-stress distributions at an intersection, which is then extrapolated to obtain the strain-stress at the weld toe.

J.J.A.TOLLOCZKO 123

FIG. 6.8. Typical strain gauge layout.

The interpretation of this data provides the HSS/SCF values to be used in design. Strain gauges are used to obtain the strain and stress distribution around the tubular joint intersection. It is usually good practice to locate the positions of maximum stress around the intersection prior to the placing of extrapolation gauges to determine the stresses at the weld toe. For joints where the position of the maximum stress is known, the location of the extrapolation gauges does not present a problem. However, the actual position of the maximum stress under a unique loading mode is not usually known. Ideally, in this case, a set of rosette gauges are placed around the intersection in the first instance. Once the location of peak stress is identified, extrapolation gauges can then be mounted. Frequently, owing to cost and time restraints, the maximum stress is calculated from measured values at adjacent points. A typical layout of gauges is shown in Fig. 6.8. The interpretation of measured strain data is open to debate. The approach to interpretation should rest on the primary argument that the resulting ‘hot-spot’ stress should be compatible with an appropriate S–N curve. It is generally accepted that the S–N curve should itself take into consideration any ‘notch’ effects so that the design ‘hot-spot’ stress need only reflect the deformation stress. This is a sensible approach since the ‘notch’ stresses are not easily measurable, owing to the rapid three-dimensional variation of ‘notch’ stresses over a few millimetres near the weld toe. Stresses within the ‘notch’ region may also be influenced by weld geometry and weld bead size. Although the ‘notch’ stresses are undoubtably important in describing fatigue-crack initiation and early growth at the ‘hot-spot’, these phenomena form only a small part of the total fatigue life of a tubular joint. Such stresses decay rapidly as crack growth takes place along the weld toe.

124 J.J.A.TOLLOCZKO

It is generally agreed that, although not taking into consideration ‘notch’ effects, some form of stress extrapolation to the weld toe is necessary. The methods used for extrapolation are yet another subject for debate. The position of gauges and type of extrapolation will determine the magnitude of ‘hot-spot’ stress and which stress raisers are included in the measured value. It is apparent from the above that certain factors need to be considered when considering experimental test data. These fall into the following categories: • The stress distribution around a tubular joint, taking into consideration the various wave directions and the resulting principal stress directions at the ‘hotspot’. • The positioning of strain gauges relative to the weld toe. This is particularly relevant to the influence of the weld on the ‘hot-spot’ stress definition. • The methods of extrapolation of strain gauge readings to the weld toe. These fall under the categories of linear and curvilinear extrapolation. The method of curvilinear extrapolation also needs to be clearly defined. • Consideration needs to be given to whether maximum principal stresses, maximum principal strains, individual strain components or strains perpendicular to the weld toe should be extrapolated to obtain ‘hot-spot’ stress values. Stresses perpendicular to the weld toe may be a favoured option in view of the fact that the crack driving forces are also in this direction. If the maximum principal stress philosophy is used, then the shear stress component should be considered in deriving SCFs. When addressing the above factors, it is necessary to consider their implications on current design S-N curves. The UK Department of Energy Guidance Notes (1984) T-curve, for example, has been derived from test results based on linear extrapolation of the maximum principal stress to the weld toes. This would be appropriate in the use of orthogonal joints (e.g. T and DT joints), where the principal planes are essentially perpendicular to the weld toe. However, for joints with inclined braces where linear extrapolation may be unconservative in determining ‘hot-spot’ stress, the use of the current T-curve may be inappropriate. It may therefore be necessary either to derive new S–N curves for particular joints, or more logically to determine the relationships between the various methods available for ‘hot-spot’ stress analysis and the current S–N curves. A sensitivity study of the influence of design methods on fatigue life would obviously be appropriate. Care should be exercised when using acrylic instead of steel models. The use of acrylic allows weld details to be either included or excluded. Models without any weld fillets yield the ‘gross’ deformation strains at the intersection, whereas the S–N curves used in design are based on ‘local’ deformation and include ‘notch’ stresses. For this reason the effect of weld profile must be ascertained if realistic results are to be obtained. Variation of weld profile and weld tolerances cannot be satisfactorily simulated on acrylic models. A further disadvantage in

J.J.A.TOLLOCZKO 125

using acrylic is that it creeps under load (and variable temperature). The rate of creep is approximately 3% per minute after 1 minute of application of load. Any strain gauge reading must therefore be taken after a set time, usually 1 minute. The Young’s modulus must be determined at the time the gauges are read. Acrylic tubing can contain residual stresses, which may result in crazing of the surface if solvents are brought into contact with it. This may cause difficulty during assembly of the tubes after machining of brace ends, application of brittle lacquer, or application of resistance foil strain gauges. However, these problems can be overcome and acrylic models have been used extensively in recent years. 6.3.3.2 SCF Parametric Equations The designer relies heavily on available formulae for the calculation of SCFs. These have been developed by various authors and are based on either acrylic model tests of simple joints or on theoretical work, such as finite-element analysis. The published formulae cover the following simple joint types: T ′ ′ ′

Kuang (1977)* Gibstein (1978)* Wordsworth(1981)+ UEG UR33 (1985) Buitrago (1984)* Efthymiou (1988)*

′ ′ ′

′ ′ ′

Y ′

K/YT ′

KT ′

X/DT

′

′

′

′

′ ′ ′

′

′

′

′

* Formulae derived by FE methods +Formulae derived from acrylic model tests (Note: The reader should refer to the individual references for actual formulae.)

Since the formulae are derived from the methods discussed in Sections 6.3.3.1 and 6.3.3.3, it is apparent that the variations in SCF/‘hot-spot’ stress definitions and derivations are also passed on to these various formulae. It is, therefore, not surprising that the use of these formulae results in variations in calculated values of SCF for the same joint. The existing formulae show considerable differences in the influence of individual joint parameters on SCFs. There are also significant differences in the validity ranges of these formulae. These differences can be shown by considering a simple T-joint, which is covered by all six formulae, and comparing the variation in SCF at the chord saddle with σ ratio under axial loading (Fig. 6.9). It can be seen that the validity ranges vary considerably and, more significantly, the

126 J.J.A.TOLLOCZKO

predicted SCFs range from 6.5 to 10 at a σ equal to 0·6 depending on which formula is used. These significant differences are typical and can be shown to exist for other joint parameters and types. The implications for fatigue design of using these formulae is that calculated SCFs may be unconservative in some cases. It is necessary to determine the method of derivation and the reliability of these formulae in predicting SCFs and also in the prediction of fatigue life. Work has been undertaken to look at the basic reliability of current formulae. This has been reported by Lalani (1986) and Tolloczko and Lalani (1988). The method adopted in both references was to compare available SCF test data from realistically constructed steel tubular joints with predicted values using the appropriate current formulae. Although there are still certain areas of uncertainty regarding steel model tests, their use provides the most realistic method of HSS/SCF measurement. Recently reported SCF data extracted from the UKOSRP II (1987) project and the SIMS 87 Conference (1987) are presented in Tables 6.1 to 6.4, and have been collated following the restrictive constraints adopted in UEG UR33 (1985). Tables 6.1 to 6.4 present data from a total of 24 T/Y/H joint tests,

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FIG. 6.9. Chord saddle SCF variation with σ ratio.

32 non-overlapping K joint tests, 8 stiffened T joint tests and 25 overlapping K joint tests. A total of 352 different SCF values are available for various load cases and locations. A comparison between measured and predicted SCFs, as predicted by the UEG UR33 (1985) equations and Efthymiou (1988) equations, is given in Figs 6.10 to 6.14. These equations have been selected for comparison since they were shown by Lalani (1986) to be potentially the most reliable. The following points are worthy of note from examination of the figures: • The recent data are superimposed on the database of SCF results reported in UEG UR33 (1985).

(1) s= saddle; c=crown. (2) Two values denote readings at the two saddle or crown locations for each specimen.

NEW TUBULAR JOINT SCF DATA FOR T, Y AND H JOINTS

TABLE 6.1

128 J.J.A.TOLLOCZKO

(1) Both braces have same included angle.

NEW TUBULAR JOINT SCF DATA FOR NON-OVERL

TABLE 6.2 APPING K JOINTS

J.J.A.TOLLOCZKO 129

(1) s=saddle; c=crown. (2) Specimen T223CV unstiffened. (3) Specimen numbers ending with Gl provided with two ‘light’ internal ring stiffeners. (4) Specimen numbers ending with G2 provided with two ‘heavy’ T section ring stiffeners.

NEW TUBULAR JOINT SCF DATA FOR RING-STIFFENED T JOINTS

TABLE 6.3

130 J.J.A.TOLLOCZKO

(1) The two brace values reflect SCFs on the 90° and 45° brace respectively. (2) Load type is balanced axial unless otherwise stated. (3) All SCF values normalised to nominal stress in 90° brace for balanced axial load tests.

NEW TUBULAR JOINT SCF DATA FOR OVERLAPPING K JOINTS

TABLE 6.4

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FIG. 6.10.Reliability of UEG equations for T/Y/H joints.

• Only the UEG and Efthymiou equations are considered, for the reason described above. • Both sets of equations exhibit scatter in their prediction ability. • The new data do not significantly change the reliability trends described by Lalani (1986) for simple joints, i.e. both sets of equations exhibit good reliability. It follows that the main conclusion drawn by Lalani still holds true following inclusion of the new SCF data for simple joints, i.e. the UEG UR33 equations predict SCFs which can be expected to be exceeded in 16% of cases; in contrast, the corresponding figure for the Efthymiou equations is 24%.

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FIG. 6.11.Reliability of Efthymiou equations for T/Y/H joints.

• The UEG equations do not cater for overlapping joints. In Fig. 6.14 the accuracy of the Efthymiou equations for overlapping joints is shown. It can be seen that the Efthymiou equations predict safe SCFs for overlapping joints; the low ratios of test result to prediction are primarily a reflection of the low SCFs that occur in overlapping joints (see Table 6.3).

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FIG. 6.12.Reliability of UEG equations for non-overlapping K/YT joints.

6.3.3.3 SCF Finite-Element Analysis Stress analyses of tubular joints can be performed by FE methods and the results used to calculate SCFs. A number of computer programs of an acceptable standard have been developed, and these have been used to produce parametric SCF formulae and to analyse problem joints in offshore structures. There are, however, still a number of uncertainties with this type of analytical work. These concern: • Modelling of the welds

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FIG. 6.13.Reliability of Efthymiou equations for non-overlapping K/YT joints.

• Representation of the boundary conditions • Techniques for the determination of SCFs from the computed stress distribution • Load interactions Most finite-element models of tubular joints consist entirely of shell elements. At tubular intersections, there is considerable approximation since the weld is not modelled. Spurious local bending of the member walls may be induced. At the present time, it is not certain how localised the effect of the approximation is, although the calculated stress levels are likely to be higher. To overcome these

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FIG. 6.14. Reliability of Efthymiou SCF equations for overlapping K/YT joints.

problems, the intersection region can be modelled using three-dimensional elements. This introduces the effect of local deformation stresses (but not ‘notch’ stresses) into the model. Depending on the choice of S–N curve for the subsequent fatigue analysis, this may not be appropriate. This too requires clarification, since the behaviour of the tubular joint represented by the FE model must be consistent with the assumptions of the fatigue analysis. Finite-element models usually have short chords, with the chord length parameter a typically having a value of about 11. Diaphragms may be added at the free ends of the chord to provide pinned supports. Depending on the

J.J.A.TOLLOCZKO 137

configuration of elements and/or boundary conditions, these may artificially prevent ovalisation of the ends of the chord. Diaphragms are introduced to facilitate the imposition of boundary conditions for the FE analysis and often a restricted length of chord is introduced, based on engineering judgement. Both these aspects are known to influence the SCFs at the tubular intersections, and further assessment is required on the influence of a and end conditions. Prescribed displacements rather than applied loads can be applied to the free ends of members, so that the deformed shape is realistic. End diaphragms in short chords reduce deformations. In T/Y joints, for example, this results in lower saddle SCFs on both the chord and brace. The length of the chord influences beam bending. Consequently, crown SCFs in T/Y joints increase steadily with ′ . The definition of ‘hot-spot’ stress should be consistent with the S–N curve used. The method of stress extrapolation should also be consistent, but this may not always be possible because physical models (which are used to determine S– N curves) and analytical models differ, particularly at the weld intersection. At present time, there are different ways of determining SCFs from the calculated stress distributions. The outer surface stresses at nodes along the tubular intersections are sometimes used. However, these lines of nodes do not necessarily coincide with the weld toe, sometimes lying outside the weld and sometimes lying inside it. Alternatively, the stresses at the weld toe can be used. The stresses at the weld toe can be obtained by two different techniques, which will invariably lead to slightly different SCFs. The stress at the weld toe can be obtained by extrapolation of the stress distribution in the region of stress linearity, in accordance with the procedure used in the UKOSRP tests (1987). Linear or curvilinear extrapolations can be used, although other methods may be more suitable. The second technique involves plotting the stress distribution along a line perpendicular to the weld toe and obtaining the stress there by interpolation. In view of the approximations in the modelling of the intersection regions with shell elements, this procedure may be hard to justify, and better approximations can be obtained using brick elements local to the intersection. A study in which the different techniques of extracting SCFs from the results of an FE analysis are compared would clearly be useful and would demonstrate whether the differences are significant. SCFs are determined for loads applied to individual members only. The simultaneous application of loads to other members of a joint will influence the SCF of that member due to the interaction of the stress fields of neighbouring connections. By allowing for load interactions, fatigue lives may be altered appreciably. However, such an approach requires a revision of the manner in which SCFs are presented, perhaps following the procedure described by Buitrago (1984).

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FIG. 6.15. Typical S–N curve.

6.3.4 Fatigue Life Assessment (S−N Method) 6.3.4.1 General There are two basic approaches used in the fatigue-life assessment of tubular joints. The first method relies on the use of empirically derived S–N curves which relate the stress range at a point under consideration to the number of cycles to failure. The second method is based on linear elastic fracture mechanics and considers an analysis of crack propagation at the point under consideration. A typical S–N curve is given in Fig. 6.15. The ordinate describes the stress range and the abscissa describes the number of cycles to failure. Logarithmic scales are conventionally used for both axes, and the S–N curves are based on test data. To carry out fatigue-life predictions, a linear fatigue damage model is used in conjunction with the relevant S–N curve. One such fatigue damage model is that postulated by Miner (1945). In essence, the model assumes that irreversible damage occurs with each stress cycle and that damage accumulates linearly to a fixed level at which failure occurs. The long-term distribution of relevant stress ranges is established for each potential crack location around the periphery of the intersection of a tubular joint. Each stress cycle experienced by the joint will have an associated stress range ′ f. For the ith stress cycle with stress range ′ fi, an increment of damage equal to 1/Ni occurs, where Ni is the corresponding number of stress cycles to failure under constant-amplitude loading of ′ fi Fatigue failure occurs as soon as the linear cumulative damage of the cycles in the variable-amplitude loading sequence has reached a fixed level, i.e.

J.J.A.TOLLOCZKO 139

(6.1) where ni is the expected number of cycles of the various stress ranges ′ fi in the design spectrum which are assumed to occur in the design life of the structure, Ni is the corresponding number of cycles to failure under constant amplitude loading obtained from the relevant S-N curve, and Ds is the damage summation failure limit. The value of Ds is often taken as unity. However, this value is sometimes chosen to reflect the safety levels required for the particular joint under consideration. 6.3.4.2 Stress Range The fatigue design is normally based on the ‘hot-spot’ stress range concept. The nominal brace stress ranges are increased at the intersection by appropriate stress concentration factors (SCFs). SCFs and their definition are discussed in detail in previous sections. In essence, the relevant stress range to be used in design is defined as the maximum principal stress range obtained by linear or curvilinear extrapolation to the weld toe of the maximum principal stress distribution near the weld toe, but far enough away from it not to be influenced by the ‘notch’ stress. The recommended S–N curve is based on this definition; the effects of ‘notch’ stresses and residual welding stresses are appropriately reflected in the choice of curve. 6.3.4.3 Failure Criteria During the fatigue testing of tubular joints, fatigue crack propogation/failure is classified under three primary headings. These are, in chronological order: (a) N1: First discernible surface cracking, usually detected from visual inspection or strain gauge monitoring adjacent to that portion of the weld toe where the ‘hot-spot’ stress is measured. (b) N2: First through-thickness cracking, detected by (i) visual means; (ii) internally-applied air pressure; (iii) monitoring strain gauges. (c) N3: End of test, defined as (i) extensive cracking; (ii) test rig limitations;

140 J.J.A.TOLLOCZKO

(iii) member pull-out; (iv) surface cracking around about 80% of the periphery of the joint. N1 is liable to considerable error and, in jacket structures in service, is likely to be well in excess of that observed in the laboratory. N3 is also liable to considerable differences since the definition of this failure criterion changes with each specimen and the place of testing. In service conditions, joints are usually repaired before this condition arises. In general the relevant life of a joint is defined as N2. Apart from the fact that the degree of scatter observed from tests is less, a number of other justifications can be stated: (i) Such a crack should be both detectable and repairable. (ii) The structure should be capable of tolerating such a crack without the intervention of catastrophic fracture. (iii) Such a crack should be small enough for the jacket structure not to have to shed load, leading to damage of other joints. Therefore, providing that crack instability does not occur first (i.e. the joint does not reach the fracture limit state), failure within the context of fatigue design is defined as the number of cycles to first through-thickness cracking. 6.3.4.4 Available Design S–N Curves Figure 6.16 presents the family of S–N curves derived from experimental data and recommended by both the American Petroleum Institute (API) Code RP2A (1987) and the Department of Energy Guidance Notes (1984). Examination of this figure and the recommendations in the two documents reveal the following. (a) Basic S–N curves. API RP2A presents two curves, the X and X′ curve. The X curve is recommended for joints in which the welds merge smoothly with the adjoining base metal. For welds without such profile control, the X′ curve is recommended. The so-called ‘thickness and size effect’ is taken by API to be appropriately reflected in these two curves, i.e. it is assumed that the curves are sufficiently devalued that the effect of wall thickness is largely negated if profiled welds are specified. An endurance limit at 2×108 cycles is specified, and it is assumed that all stress cycles below 35 N/mm2 or 23 N/mm2 for the X and X ′ type weld profiles, respectively, are non-damaging. The Guidance Notes, on the other hand, recommend specific size/thickness correction factors, as shown in Fig. 6.16. The size correction takes the form: f=fb(32/T)1/4 (6.2) where f=fatigue strength of joint under consideration (N/mm2); fb=fatigue strength using the basic T curve derived from tubular joints with chord wall thickness of 32 mm (N/mm2);

J.J.A.TOLLOCZKO 141

FIG. 6.16. S–N curves recommended by API RP2A and the UK Guidance Notes.

T=wall thickness of the joint under consideration (mm). Figure 6.16 illustrates S–N curves for various wall thicknesses; a lower limiting value of T=22 mm is imposed for calculating fatigue lives of joints with lesser thicknesses. No upper limit is imposed. The curves are bilinear in format, and do not encompass a threshold endurance limit, on the argument that whilst a threshold may exist under constant amplitude load conditions, this trend may not be valid under variable-amplitude load conditions, when damage caused by high stress cycles is likely to render the low stress cycles damaging in subsequent load cycles. (b) Influence of environment. The API RP2A curves presume effective cathodic protection. No quantitative guidance is presented for other conditions, apart from a comment that ‘For splash zone, free corrosion, or excessive cathodic protection conditions, appropriate reductions to the allowable cycles should be considered’. The Guidance Notes curves are valid for adequate corrosion protection conditions. For unprotected joints exposed to seawater, a penalty factor of 2.0 is imposed, and the beneficial bilinear correction applied at 107 cycles is not allowed. (c) Weld improvement technique. The API curves explicitly allow the designer to take benefit for properly profiled welds. No other benefit is recommended. The Guidance Notes do not provide any quantitative recommendations regarding weld profile benefit. However, the document recommends an increase in fatigue life by a factor of 2.2 if controlled local machining or grinding of the weld toe is carried out.

142 J.J.A.TOLLOCZKO

(d) Damage summation model. Both documents recommend the same damage model, i.e. (6.3) where Ds=cumulative fatigue damage ratio; n=number of cycles applied at a given stress range; N=number of cycles for which the given stress range would be allowed by the appropriate S–N curve. This model is in the same form as given in eqn (6.1). 6.3.5 New Fatigue Data New fatigue data extracted from the UKOSRP II (1987) and SIMS 87 (1987) are presented in Tables 6.5 to 6.9. The new data encompass: • • • • • • • • •

4 axially loaded H joints 3 axially loaded T joints 3 balanced axially loaded simple K joints 6 axially loaded stiffened T joints 15 balanced axially loaded overlapping K joints 2 balanced axial plus moment loaded overlapping K joints 4 axially loaded toe ground H joints 4 axially loaded PWHT T joints 6 in-plane moment loaded X joints in air or seawater and 10 in the as-welded, TIG dressed or shot-peened condition • 4 axially loaded T joints in seawater and cathodically protected. The hot-spot stress range noted in Tables 6.5 to 6.9 is defined as the maximum stress obtained by linear extrapolation to the weld toe of the stress distribution near the weld toe, but far enough away from it not to be influenced by the notch stress. For the fatigue lives, three levels of failure have been recorded. These have been defined in Section 6.3.4.3. The assessment of this data, together with that presented in UEG UR33 (1985), is presented below with particular reference to the available design S–N curves and the influence of size, post-weld heat treatment (PWHT) and environment on fatigue life.

J.J.A.TOLLOCZKO 143

6.3.6. Analysis of New Data 6.3.6.1 Constant-Amplitude, As-Welded, in Air, Tubular Joint Data Figures 6.17 and 6.18 present the new as-welded data in the standard S–N format (hot-spot stress range against life to through-thickness cracking). Superimposed on the graphs are the data from the UEG UR33 (1985) design guide. Data are separately identified according to the specimen wall thickness (16 mm and 32 mm in this case) to enable the size corrections recommended by the Guidance Notes to be assessed. The relevant design curves from both API RP2A and the Guidance Notes are also shown in these figures. It can be seen that all the data fall on the safe side of the design curves. The new data fall within the scatterband of the data collated within the UEG UR33. It is of interest to note the safety margin provided by the API X′ curve; at the high stress cycle range, for example, the two curves deviate appreciably. For 32-mm thick joints, at a stress range of say 200 N/mm2, the allowable cycles using the API X curve is approximately 50000 under constant-amplitude conditions; the corresponding value using the Guidance Notes 32-mm curve gives 200000 cycles, a fourfold increase. Similar comparisons using data for the 16-mm joints, give an even greater difference. This is due to the differences in slopes adopted in the two documents and the size correction factors introduced in the Guidance Notes (see Fig. 6.16). The evidence presented in Fig. 6.17 and 6.18 indicates that the data are approximately parallel to the slope adopted in the Guidance Notes. 6.3.6.2 Influence of Size Figure 6.19 shows the new tubular joint as-welded data plotted in the S–N format. The data cover joints with wall thicknesses of 16 mm, 32 mm, 40 mm and 76 mm. The tubular diameters also increase with increasing thickness. Examination of the figure reveals the following: • There is a distinct indication that fatigue strength reduces with increasing size. • All the data lie above the relevant API X′curve, including the 76-mm data. Similar conclusions can be drawn from Fig. 6.20, which shows a plot of the new PWHT tubular joint data. The size effect is present, even after application of post-weld heat treatment. Of interest is the comparison of potential size effects for as-welded (Fig. 6.19) and PWHT (Fig. 6.20) joints. The figures appear to indicate that the effects are different, which lends support to the case that stress

NEW TUBULAR JOINT FATIGUE DATA FOR AS-WELDED, AXIALLY LOADED H, T AND K JOINTS UNDER CONSTANT-AMPLITUDE LOADING

TABLE 6.5

144 J.J.A.TOLLOCZKO

levels local to the weld toe and through the wall thickness are, at least in part, responsible for the size effect. Insufficient tubular joint data are available to quantify the size effect throughout the range of thicknesses which occur in practice. However, reference

(1) For Gl and G2 type stiffeners, see Table 6.5. (2) *=cracks for G2 stiffener specimens ran under the weld between the root and chord toe. (3) **=hot-spot stress range relates to appropriate inter-section value; loading relate to tension in 90° brace and compression in 45° brace. (4) For specimen K5, bracket brace stress range value relate to maximum value at entire intersection. (5) For specimens K4-K6, t=8 mm. (6) For specimens in LA and LB series, t is given in brackets after T. (7) For specimens LA2–3 and LA2–4, IPB loads also applied.

NEW TUBULAR JOINT FATIGUE DATA FOR AS-WELDED, AXIALLY LOADED, OVERLAPPING AND STIFFENED JOINTS UNDER CONSTANT-AMPLITUDE LOADING

TABLE 6.6

J.J.A.TOLLOCZKO 145

(1) *=specimens relate to ground end of H joints. (2) **=post-weld heat treated.

NEW TUBULAR JOINT FATIGUE DATA FOR POST-WELD TREATED H AND T JOINTS UNDER CONSTANT-AMPLITUDE LOADING

TABLE 6.7

146 J.J.A.TOLLOCZKO

(a) Brace 1 in air. (b) Brace 2 in sea water. (1) ′ HSS=′ ′HS×l·2× E. (2) As-welded. (3) Tig-dressed. (4) Shot-peened. (5) Fy=540 N/mm2.

NEW TUBULAR JOINT FATIGUE DATA FOR AS-WELDED AND POST-WELD TREATED X JOINTS UNDER CONSTANT-AMPLITUDE LOADING

TABLE 6.8

J.J.A.TOLLOCZKO 147

(1) *=tests in seawater, CP level 850 mV. with respect to Ag/Ag=l electrode, at 0.167 Hz.

NEW TUBULAR JOINT FATIGUE DATA FOR AS-WELDED T JOINTS TESTED IN SEA WATER UNDER CONSTANT-AMPLITUDE LOADING

TABLE 6.9

148 J.J.A.TOLLOCZKO

J.J.A.TOLLOCZKO 149

FIG. 6.17. 16 mm Tubular joint fatigue data (in air, constant-amplitude, aswelded).

FIG. 6.18. 32 mm Tubular joint fatigue data (in air, constant-amplitude, as-welded).

can be made to the wealth of weldment data which are available, many of which are new and were generated recently. In order to present all the available data (both tubular joint and weldments) on a common basis, the following approach has been adopted:

150 J.J.A.TOLLOCZKO

FIG. 6.19. Influence of thickness using the new as-welded tubular joint data.

(a) For each set of points relating to a specific thickness, a eastsquares analysis was undertaken. For tubular joints, the following sets were encompassed: • • • • •

16-mm wall thickness, as-welded data-sample size 39 (see Fig. 6.17) 32-mm wall thickness, as-welded data-sample size 22 (see Fig. 6.18). 76-mm wall thickness, as-welded data-sample size 4 (see Fig. 6.19). 32-mm wall thickness, PWHT data-sample size 4 (see Fig. 6.20). 76-mm wall thickness, PWHT data-sample size 3 (see Fig. 6.20).

For weldment data, the following sets were encompassed: • 32 sets of new data covering various wall thicknesses, R ratios, as-welded and PWHT conditions, reported in UKOSRP II (1987) and SIMS (87) (1987)—sample size 188 in total. • 15 sets of data reported in UEG UR33 (1985) covering various wall thicknesses, R ratios and load/non-load carrying attachments. (b) For each set, the fatigue strength at 0·5×106 cycles, 1×106 cycles and 2×106 cycles was calculated. (c) All the fatigue strengths were normalised to 32 mm to reflect the value adopted in the Guidance Notes. This resulted in the identification of relative fatigue strengths.

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FIG. 6.20. Influence of thickness using the new PWHT tubular joint data.

The relative fatigue strengths thus derived are presented in Fig. 6.21 for 2×106 cycles. Similar trends can be demonstrated for 0.5×106 cycles and 1×106 cycles and are not reported here. Examination of this figure reveals the following: • The tubular joint data and the weldment data fall within the same scatterband. • The size effect is adequately represented by the Guidance Notes recommendation of an inverse slope of 0.25 for relative fatigue strengths normalised to 32-mm wall thickness. • The data fall within the scatterband noted for 16-mm wall thickness tubular joints to achieve a 98% probability of survival. This observation lends support to the continued use of the (32/T)1/4 derating function. • None of the individual test results (totalling more than 300 datapoints) fall below the relevant design S–N curves, provided the size correction is taken into account. The size effect issue has been the subject of considerable research and debate over the past decade. Significant advances have been made in the understanding of the mechanisms which govern this apparent reduction in fatigue strength with increasing size. Figure 6.21 demonstrates categorically that a size effect exists. The API RP2A curves recognise and quantify weld profile effects; the Guidance Notes recognise and quantify size effects. Neither document properly accounts for both size and profile effects; it is now generally accepted that weld profiling does not completely negate size effect, and, therefore, both size and profile effects should ideally be reflected in the design process. The American

152 J.J.A.TOLLOCZKO

FIG. 6.21. Influence of thickness—relative fatigue strength at 2 million cycles for all data.

Welding Society Code D1.1 (1986) provisions go some way in reconciling and recognising both effects. 6.3.6.3 Influence of Post-Weld Heat Treatment Figure 6.22 and 6.23 present the new as-welded and PWHT tubular joint data in conventional S–N format for the 32-mm and 76-mm wall thickness specimens. The figures indicate that, for both sizes of tubular joints, the PWHT process has a significant fatigue benefit when compared with the as-welded condition. This observation is in line with UEG UR33 (1985), which demonstrated that residual stresses play a part in the development of fatigue cracks. The extent of their influence is dependent on the applied load cycles and the R ratio. Exploitation of the benefits of PWHT has, to date, not been possible because of difficulties in the following: • determining that the residual stresses, following PWHT, have been significantly relieved; • identifying the level of reduction of residual stresses for the specimens tested; • demonstrating that a major part of the expected stress history in service would be compressive in action; • demonstrating that the lack-of-fit stresses introduced during jacket fabrication are insignificant.

J.J.A.TOLLOCZKO 153

FIG. 6.22. Influence of PWHT using the new 32 mm tubular joint data.

Notwithstanding the above, it seems prudent that efforts should be directed towards overcoming the noted difficulties so that all the potential benefits of PWHT can be exploited in the design of those joints in which fatigue life is a limiting criterion. 6.3.6.4 Influence of Environment Figure 6.24 shows the new tubular joint as-welded data plotted in the standard S– N format. The data covers three joints tested in air and four joints tested in seawater with cathodic protection at a level of −850 mV (Ag/AgCl). Although the data are limited, tests with cathodic protection clearly indicate that the fatigue life to through-thickness cracking is reduced when compared to joints tested in air. Included in Fig. 6.24 is the relevant Guidance Notes curve recommended for joints with cathodic protection. Whilst both the air and protected data points lie above the design curve, a reduced degree of safety for the joints with cathodic protection can be noted. Also included in Fig. 6.24 is the Guidance Notes curve for unprotected joints. Use of this design curve for protected joints would reinstate the desired level of safety for these joints. It is therefore apparent that joints with cathodic protection may require treatment in the same manner as freely corroding joints. This behaviour is in marked contrast to that expected from an analysis of welded plate specimens, and calls into question the applicability of plate data to identify environmental effects. It should be noted that the previously held belief in industry that adequate protection restores the in-air behaviour was essentially based on an assessment of welded plate data.

154 J.J.A.TOLLOCZKO

FIG. 6.23. Influence of PWHT using the new 76 mm tubular joint data.

FIG. 6.24. Influence of environment using the new 32 mm tubular joint data.

From the evidence noted in Fig. 6.24, it would appear that further research work on the effects of cathodic protection should concentrate on tubular joints and not on welded specimens, although it is recognised that this would have a major impact on the cost of tests.

J.J.A.TOLLOCZKO 155

6.3.7 Fatigue-Life Assessment (Fracture-Mechanics Method) Fracture mechanics (FM) is applied to the assessment of crack-like defects. For members and joints under fatigue loading, it provides an alternative means of calculating fatigue lives. In this application, FM techniques are consistent with conventional S–N methods and Miner’s rule. However, FM techniques are much more powerful than S–N methods for a number of reasons. Firstly, whilst an S–N analysis gives a single number—the fatigue life—the corresponding FM analysis gives the crack growth characteristic, which is vital when planning an inspection programme. Furthermore, FM techniques provide the means for assessing crack stability, that is, whether the fatigue life will be shortened by intervening fracture. Under closely controlled conditions, it may be considered that FM techniques represent an essentially ‘exact’ science. However, under actual conditions, particularly for welded structures in an offshore environment, there are many unknowns and analytical results become less exact. In this application, FM techniques are best suited to performing comparative studies. In the role of determining an underwater inspection programme, FM analysis is able to rank the joints in order of their fatigue lives and defect tolerance, permitting an optimised inspection plan to be formulated. To date, fracture mechanics studies in the offshore industry have tended to lie at extreme ends of the spectrum in terms of the complexity of the analyses. For quick and simple defect assessments, design curve methods have been widely used. Whilst having the advantage of being economical to perform, the gross simplifications in the assessment procedure greatly undermine confidence in the results. At the other extreme, very detailed studies of individual components, frequently involving finite-element analyses, offer greater accuracy. The improvement in accuracy, however, seldom justifies the order-of-magnitude increase in the cost which is involved, particularly when the uncertainties of defect characteristics and loading are considered. A review of crack-growth mechanics, FM crack-growth laws, crack-growth constants and stress-intensity factor solutions is beyond the scope of this chapter. Initial reference to the overview of FM given in UEG UR33 (1985) shows that FM analysis methods, once fully developed and accepted by industry, will provide a powerful fatigue-life assessment tool. Although unlikely to replace the S–N fatigue design method, FM will be used in the design of more complex joints and in the assessment of defects in offshore structures. 6.4 RECOMMENDATIONS FOR FUTURE RESEARCH A review and assessment of the parameters influencing fatigue and also new data recently generated have been presented in the previous sections. Whilst advances

156 J.J.A.TOLLOCZKO

have been made in the area of fatigue, many areas of uncertainty still warrant further investigation. Some of these are listed below: (i) Identification of experimental errors inherent in SCF measurements. (ii) Hindcast forecasting studies, correlating predicted hot-spot stress ranges using the available parametric SCF equations with measured fatigue lives. (iii) Identification of weld profile and thickness/size effects through studies of notch stress and through-thickness stress levels. (iv) Further investigations into corrosion fatigue, particularly the influence of cathodic protection. (v) Generation of further tubular joint data on complex joints. (vi) Development of non-destructive techniques to determine residual stress levels in tubular joints. (vii The fatigue-life assessment of tubular joints using fracture mechanics ) techniques with particular reference to developing techniques and methodologies suitable to engineering applications. 6.5 CONCLUDING REMARKS The subject of offshore tubular joint fatigue design is one which has attracted substantial reseach on a worldwide basis, costing many millions of pounds. This has been necessary so that the offshore industry, particularly with North Sea developments, can, with reasonable confidence, design and install structures to resist the continuous cyclic loadings generated by winds and waves. The consequences of fatigue failure can indeed be catastrophic. However, the more common consequence of fatigue failure is the requirement to repair cracking at great expense to the operating company. It is hoped that the preceding sections of this chapter have given some indication to the reader of the complexity of the fatigue behaviour and design of tubular joints. The influence of geometry, manufacturing methods, loading and in-service environment results in no two joints ever being the same and hence the design methods developed are generally and necessarily conservative. The rationalisation and refinement of tubular joint fatigue design will continue. This will be accelerated by the operating companies seeking more confidence in designs so as to reduce the requirements for costly underwater inspection and repairs. It is therefore apparent that research and experimentation will continue in this area for many years to come, but it is unlikely that all the factors influencing the fatigue behaviour of tubular joints will be understood for a long time.

J.J.A.TOLLOCZKO 157

REFERENCES AMERICAN PETROLEUM INSTITUTE (1987) Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, API RP2A, 17th edn. AMERICAN WELDING SOCIETY (1986) Structural Welding Code—Steel, ANSI/ AWS D1.1–86. BUITRAGO, J. (1984) Combined hot-spot stress procedure for tubular joints. Paper OTC 4775, Offshore Technology Conference, Texas. DEPARTMENT OF ENERGY (UK) (1984) Offshore Installations: Guidance on Design and Construction, HMSO, London. EFTHYMIOU, M. (1988) Development of SCF formulae for use in fatigue analysis. Offshore Tubular Joint Conference, OTJ (88), UEG, CIRIA. GIBSTEIN, M.B. (1978) Parametric stress analysis of T joints. Paper 26, European Offshore Steels Research Seminar, Cambridge. KUANG, J.G. (1977) Stress concentrations in tubular joints. Paper OTC 2205, Offshore Technology Conference, Texas. (Modifications in SPE Journal, August 1977, 287–99. LALANI, M. (1985) Improved fatigue life estimation of tubular joints. Paper OTC 5306, Offshore Technology Conference, Texas. LALANI, M. (1987) The fatigue behaviour of overlapping joints. Paper TS13, Proceedings of the 3rd International ECSC Offshore Conference on Steel in Marine Structures (SIMS 87), Delft, the Netherlands. MINER, M.A. (1945) Cumulative damage in fatigue. J. Appl Mech., Trans. ASME, 12, 159–64. MORISON, J.A. (1950) The forces exerted by surface waves on piles. Petroleum Trans., AIME, 189, 149–54. SIMS 87 (1987) Proceedings of the 3rd International ECSC Offshore Conference on Steel in Marine Structures, Delft, the Netherlands. TOLLOCZKO, J.J.A & LALANI, M. (1988) The implication of new data on the fatigue life assessment of tubular joints. Paper OTC 5662, Offshore Technology Conference, Texas. UEG UR33 (1985) Design of Tubular Joints for Offshore Structures. UEG Publication UR33, CIRIA, London. UKOSRP II (1987) Summary and Task Reports. HMSO, London. WORDSWORTH P. (1981) Stress Concentration Factors at K and KT Tubular Joints, Fatigue in Offshore Structural Steels. Institute of Civil Engineers, Westminster, London.

Chapter 7 CASE STUDIES AND REPAIR OF FATIGUE-DAMAGED BRIDGE STRUCTURES J.W.FISHER Joseph T. Stuart Professor of Civil Engineering, Director, ATLSS Engineering Research Center, Lehigh University, USA & C.C.MENZEMER ATLSS Scholar, Lehigh University, USA SUMMARY This chapter briefly reviews the major factors governing the fatigue strength of welded details. Case studies which illustrate some of the causes off fatigue cracking being experienced in highway bridge structures and the methods used to repair and retrofit the welded details are examined. Examples of severe corrosion and the inability to detect corrosion cell activity are summarized. NOTATION a b c C da/dn E(k) Fe Fg Fs Fw H K KT ′K r

Crack size Width of hanger Crack width Crack-growth constant Crack-propagation rate Elliptic integral of the second form

Crack shape correction Stress gradient correction factor Correction for free surface Correction for finite width Weld leg size Stress-intensity factor Stress-concentration factor Stress-intensity range Radius of rivet hole

REPAIR OF FATIGUE-DAMAGED BRIDGE STRUCTURES 159

Sr Sre tcp tf tp σi ′ σ σ

Stress range Effective stress range Cover plate thickness Flange thickness Plate thickness Frequency of occurrence of stress range cycles Sri as a fraction of the total cycles Angle for calculated stress intensity Radius of retrofit hole Stress 7.1 INTRODUCTION

The two primary factors governing the fatigue strength of welded steel details are the applied stress range and detail type (Fisher et al., 1970, 1974). Welded structural joints behave differently because of the variation in the local stress fields around the weldments, as well as the inherent variability in the distribution of initial discontinuities. All welding processes introduce small defects in or around the weldment, although proper fabrication will minimize both the number and size of the initial flaws. A large number of experimental programs have generated fatigue data which were used in the development of design guidelines (Keating & Fisher, 1986). Figure 7.1 is a typical plot of fatigue data for short web attachments. Stress range versus life behavior of welded details can be adequately described by straightline relationships on log-log scale plots. The lower bound is normally established such that it is at least two standard deviations below the mean. As a result, lowerbound design curves are associated with a specific probability of survival at a given confidence level. A typical set of fatigue design curves is shown in Fig. 7.2. Each curve corresponds to a type of detail. Structural joints with similar stress concentration and discontinuity conditions are grouped into a single category. Details which provide the lowest fatigue resistance usually involve connections which experience fatigue-crack growth from weld toes and terminations where there is a large stress concentration. Details which fail from internal discontinuities will generally have higher allowable stress range values as there is no geometrical stress concentration more severe than the discontinuity itself. Inspection of Fig. 7.2 reveals horizontal cut-offs for each of the design curves, which represent the constant-amplitude fatigue limit. For stress range cycles below this limit, no fatigue crack propagation would be expected to occur. Constant-amplitude fatigue limit values decrease with the severity of the detail. As bridges are subjected to variable-amplitude loadings, several studies were conducted to examine member behavior under this type of loading (Schilling et al.,

160 J.W.FISHER AND C.C.MENZEMER

FIG. 7.1. Laboratory fatigue data for web attachments.

FIG. 7.2. Fatigue design curves.

1978; Fisher et al., 1983). An important conclusion from these studies was that if any of the stress range cycles in a loading spectrum exceeded the constantamplitude fatigue limit, life could be estimated by employing a cumulative damage law such as that originally proposed by Miner’s.


FIG. 7.3. Examples of distortion-induced cracking.

7.2 FATIGUE CRACKING OF WELDED BRIDGE DETAILS Since the early 1960s, a number of localized failures have developed in steel bridge members owing to fatigue-crack propagation. In some instances, fatiguecrack propagation has resulted in brittle fracture (Fisher, 1984, 1986). A majority of the fatigue cracks can be placed into one of two broad categories. The largest category is a result of outof-plane distortion in a small, unstiffened segment of girder web. When distortion-induced cracking develops in a bridge component, large numbers of cracks develop nearly simultaneously because the cyclic stress is high and the number of stress cycles needed to produce cracking is relatively small. Displacement-induced cracking has developed in a wide variety of structures, including suspension, two-girder floorbeam, tiedarch, and box-girder bridges. In general, the cracks form parallel to the primary stress field and are not detrimental to the performance of the structure provided they are discovered and retrofitted before turning perpendicular to the applied stresses. A typical example of distortion-induced cracking is depicted in Fig. 7.3. An unstiffened segment of girder web is created at the end of the floorbeam connection plate when a positive attachment between the girder flange and connection plate is not provided. As the floorbeam rotates under traffic loading, the segment of web is pulled out-of-plane, producing a large stress gradient in the small portion of web. Figure 7.4 shows a typical stress gradient which was measured in one structure (Fisher, 1978). The large cyclic stress near the web flange weld and the weld termination at the end of the connection plate produces fatigue cracking in a relatively small number of stress cycles. The second largest category of fatigue-damage to bridge members and components comprises large initial defects or cracks. Defects in this category have resulted from poor-quality welds that were produced before non-destructive test methods were well established. In addition, a number of the failures in this category have developed because the groove-welded component was considered a secondary member or attachment. As a result, weld quality criteria were not established and non-destructive test methods were not employed.


FIG. 7.4. Measured stress gradient.

A majority of the remaining failures resulted from the use of low-strength details that were not anticipated to have such a low strength at the time of the original design. 7.3 LOW FATIGUE STRENGTH DETAILS— YELLOW MILL POND BRIDGE 7.3.1 Structure Description and History of Cracking The Yellow Mill Pond Bridge is located on Interstate-95 (I-95) Bridgeport, Connecticut. Constructed in 1956–1957, the bridge was opened to traffic in 1958 and consists of 28 simple span cover-plated steel beam bridges. Three lanes of roadway traffic are carried in each direction. A typical framing plan is shown in Fig. 7.5. Fatigue cracking was first observed in 1970 (Fisher et al., 1970; Fisher, 1986). Fatigue-crack growth resulted in fracture of a beam tension flange as shown in Fig. 7.6. Inspection of several other adjacent beams revealed cracks which had extended halfway through the tension flange thickness. Other small cracks were also observed. Follow-up inspections conducted in 1973, 1976, 1979 and 1981 revealed additional crack growth from the cover-plated details. The extensive numbers of cracks that developed at the ends of the welded cover plates were primarily due to the large volume of truck traffic and the unanticipatedly low fatigue resistance of the details. Both eastbound and westbound bridges of span 10 were selected for major stress history studies during this time. Beams of span 10 are rolled sections of A242 steel (σ y=310 MPa (45ksi)). All beams, except the interior facia beam of the eastbound roadway, are fitted with


FIG. 7.5. Span 10 framing plan.

single cover plates on the compression flange and two cover plates on the tension flange (Fig. 7.5). Primary cover plates for the exteror facia beams of both roadways are full length. 7.3.2 Stress History Measurements Both eastbound and westbound bridges of span 10 were selected for two stress history studies by the State of Connecticut and the US Federal Highway Administration (FHWA). An initial investigation was conducted in 1971, with a follow-up study continuing from April 1973 to April 1974. Limited additional measurements were acquired in 1976. Figure 7.7 shows a typical stress-range histogram for cover plate details from the westbound structure. In addition to strain measurements, vehicle distributions, lane positions, and truck weights were recorded. From recorded information on truck traffic, coupled with an FHWA Loadometer survey, it was estimated that 35 million trucks had crossed the eastbound and westbound structures between 1958 and 1976. Composite stress-range response spectra were constructed using the various sets of strain measurements. Root mean cubed or Miner’s effective stress-range values were determined for all strain gages which had been placed directly under the web on the tension flange, adjacent to the cover plate terminations: (7.1) Measurements indicated that the effective stress range varied from 7·6 MPa to 13·6 MPa (1·1 ksi to 1·98 ksi) for the gages close to coverplate ends. Beams located


FIG. 7.6. Cracked girder at cover-plate termination.

FIG. 7.7. Stress-range histograms.

under the outside roadway lanes tended to have larger recorded stress range values. Careful examination of individual stress response records for single truck passages showed that more than one stress cycle event occurred for each truck crossing. Approximately 1.8 stress cycle events per truck passage resulted, corresponding to a total of 63 000 000 cycles after 18 years of service.


FIG. 7.8. Crack-size relationships.

7.3.3 Fatigue-Crack Growth Analysis Studies on welded steel details (Fisher et al., 1974) have shown that the secondstage fatigue-crack propagation behavior can be adequately described by a thirdorder power law of the form. (7.2) The presence of initial cracks or defects in or around the weldment is presumed. Time spent in crack initiation is assumed to be negligible, as all welding processes give rise to sharp discontinuities in the form of slag inclusions, porosity, etc. Discontinuities, in the zone of high stress concentration at the weld toe, act as potential crack sites. Usually, a number of small defects grow along the weld toe and coalesce into a primary crack front. Empirical relationships for both the coalescence of cracks and minor-to-major semidiameter axis ratios have been determined for different details. A set of such relationships is shown in Fig. 7.8. Calculation of stress-intensity factors for cracks in complex structural joints can be formulated by application of correction factors to the solution of a through crack in an infinite plate, subjected to remote tension loading (Albrecht & Yamada, 1977). (7.3) Values for the correction factors can be found in a number of handbooks and other publications (Paris & Sih, 1965; Broek, 1982; Tada et al., 1987). Cracks which formed at the cover-plate weld toes were modeled as semielliptical surface


FIG. 7.9. S–N curve for cover-plate details.

cracks in the flange under remote tension. Correction factors and crack size relationships were taken as (7.4) (7.5) (7.6) (7.7) (7.8) (7.9) c=5.46 a1·133 (inch) (7.10) With an initial crack size equal to 0·75 mm (0·03 in) and the effective stress range as 13 MPa (1·9 ksi), the number of cycles to grow a crack 25 mm (1 in) deep was estimated as (7.11) This is consistent with the larger fatigue cracks which formed between 1958 and 1976. Figure 7.9 is an S–N curve derived from laboratory data on thick cover-plated details. A composite stress-range spectrum for the worst stress conditions


monitored in 1971, 1973, and 1976, yielded an effective Miner’s stress range of 13·1 MPa (1·9 ksi). Plotted as an open circle, and assuming 1 stress cycle event/ truck, the data point falls below the lower-bound fatigue resistance. Application of the observed 1.8 stress cycle events/truck, results in the filled circle plotted in Fig. 7.9. Larger fatigue cracks observed in the structure are consistent with data developed on a number of cover-plated beams tested in the laboratory. Many smaller cracks observed in other beams at lower effective stress range levels are compatible with tests conducted under variable-amplitude loading. This suggests that the crack-growth threshold is lowered by stress cycles exceeding the constantamplitude fatigue limit and that a proportionally larger number of smaller stress cycles contribute to damage propagation. 7.3.4 Retrofit Procedures In 1981, the entire structure was retrofitted by hammer peening the weld toes at smaller cracks and installing bolted splices at larger cracks. Weld-toe peening introduces local compressive stresses around the weldment surface, thereby reducing the applied stress range and prolonging the life of the detail. 7.4 LACK OF FUSION AND DISTORTION— I-93 CENTRAL ARTERY 7.4.1 Structure Description and Inspection Results The Central Artery carries I-93 traffic through Boston, Massachusetts and surrounding communities. A viaduct structure, located in the northern area of the Artery, consists of bilevel rigid steel frame bents which support two girder floorbeam structures on each level (Demers & Fisher, 1989). Columns of the rigid bent supports are box sections, while transverse beams are either box or I sections (Fig. 7.10). Figure 7.11 shows the I-beam-to-column connections which contain closure plates fillet welded to the edges of the I section and column flanges. Webs of the I beams are groove-welded to the columns, while beam flanges (both box and I) and box beam webs are connected via full-penetration groove welds with back-up bars. Beam flanges of the upper level of the bents are continuous into the column. The cope openings in the box-beam webs at the beam-column connection points are covered with small plates which are filletwelded to the box beam webs. For each of the two girder floorbeam superstructures, longitudinal girders form composite action with the reinforced-concrete decks. Both top and bottom flanges of the longitudinal girders are coped by flame cutting to permit a bolted, double


FIG. 7.10. Typical bent in viaduct.

angle web connection with transverse beams of the bent. The connection angles are fitted to both the top and bottom flanges of the transverse beams. Flanges of the floorbeams are also coped to permit bolting to connection plates which are fillet-welded to the longitudinal girder webs. Inspection of the structure revealed several areas which contained fatigue cracks. In addition, lateral displacement of several longitudinal girder bottom flanges at transverse beam connections was observed. Fatigue cracks were found along the longitudinal girder web to flange weld at the flange terminations and vertical cracks were observed in the girder web at the re-entrant corner of the bottom copes, as illustrated in Fig. 7.12. Cracking was also observed along the bolt fixity line of a few end connection angles. These cracks typically extended along the angle fillet of the girder-transverse bearn end connection. The cracks shown in Fig. 7.12 were found only at locations where the longitudinal girder web was bolted full-depth to the connection angles. Other locations, where the girder web was not bolted full-depth to the connection angles, as shown in Fig. 7.13, exhibited no evidence of cracking. Inspection of the rigid frame bent revealed cracks in the fillet welds at several corner joints where the transverse beam closure plates were joined to the bent column flange. Grinding into the fillet weld throat in the cracked corner region of the closure plates exposed the crack which had extended into the groove weld connecting the top flange of the transverse beam to the column, as shown in Fig. 7.14. Further grinding exposed the crack which extended from the back-up bar through the groove weld thickness. Other bents revealed cracks at the web cope closure plates of the transverse box beams. Grinding of the fillet weld


FIG. 7.11. Typical I-beam-to-column connection.

throat into the beam flange exposed cracks extending from the back-up bar into the groove weld thickness. 7.4.2 Field Measurements and Probable Causes of Failure Coping by flame cutting the longitudinal girder webs resulted in high tensile residual stresses along the cut edge as well as a re-entrant corner. In addition, coping greatly reduced the section modules for in-plane bending and left an unstiffened segment of girder web between the flange terminations and the bolt


FIG. 7.12. Fatigue cracks in girder cope.

fixity line of the longitudinal girders. At locations where the longitudinal girders were not bolted full-depth to the connection angles, the girder web could rotate in-plane without develop ing significant compression in the lower portion of the girder. This resulted in no lateral displacement, no out-of-plane web distortion, and no fatigue cracking in the cope or connection angles. In locations where a full-depth bolted connection for the longitudinal girder was provided, the bottom flanges had an initial lateral displacement. As the end of the girder rotated under traffic loading, connection restraint resulted in additional lateral displacement of the bottom flange, which caused large cyclic stresses to develop in the cope region. Hence, fatigue cracks in the web copes of the longitudinal girders, as well as cracks along the bottom edges of the connection angles were all distortion induced. A maximum stress range of 90 MPa (12·9 ksi) was measured for the web cope locations. As such, crack extension would continue unless sufficient retrofit measures were taken. Electrical resistance strain gages were installed in four bents during the first set of field tests. ages were placed on the transverse beam top flanges, and on the box webs or closure plates in cross sections adjacent to the bent. Significant differences in stress-range values for corner gages on the same transverse beam flange were attributed to axial bending caused by longitudinal traction/braking forces. On one bent, the axi-mum stress range on one corner gage was 93 MPa (13·3 ksi), while the other corner gage recorded a value of 45 MPa (6·4 ksi). A majority of the current design guidelines do not account for the effect of longitudinal forces when evaluating the fatigue resistance of bridge members. Partial penetration fillet welds at the corner joints of the bent transverse beams produced a fabricated lack-of-fusion defect perpendicular to the beam primary


FIG. 7.13. Bolted connection of girder to transverse beam with no detectable cracking.

FIG. 7.14. Crack in closure plate weld at rigid frame connection.

stress field. Large initial defects and cracks are low-fatigue-resistance details, and with maximum measured stress-range values in the variable spectrum exceeding the crack growth threshold of 31 MPa (4·5 ksi), propagation would be expected. In addition, at least one corner gage exceeded the threshold for each


FIG. 7.15. Model for lack-of-fusion crack.

cross section measured. This suggests that crack extension would occur in at least one location for every transverse beam-to-column joint. 7.4.3 Crack-Growth Analysis Measured stress-range spectra were used in the evaluation of the fatigue resistance and crack development at the box corner connections of the steel bent. Effective stress-range spectra were determined using Miner’s rule for several truncated minimum stress-range values. In addition, the recorded strain data were used to develop a cycle/hour frequency for each gage so that estimates of past service life could be made. The lack-of-fusion defect in the closure plate detail was modeled as a loadcarrying fillet weld, with crack growth from the weld root (Fig. 7.15). A crack growth rate which follows a third-order power relationship was assumed, and the applied stress intensity was estimated from (Frank & Fisher 1979) (7.12) where A1 and A2 are functions of H/tp, and w is equal to H+tp/2. The resulting stress range versus cyclic life is plotted in Fig. 7.16, along with an S–N curve derived from laboratory data on thick cover plates and web attachment. In addition, effective stress-range values with the corresponding results of the cumulative service life cycles are plotted for gages adjacent to the corner joints on the instrumented bents. Observed cracking is consistent with the model. Several gages plot to the right of both the resistance curves. As the maximum stress range in the spectrum exceeds the threshold, cracks would be expected to develop at these and other box corner connections. At other gage locations, the effective stress range is less, as a number plot around the 14 MPa (2 ksi) level


FIG. 7.16. Predicted curve for the lack-of-fusion crack.

for 2–6 million cycles. Damage accumulation is much slower at these locations, so fatigue cracking would not be expected for some time. 7.4.4 Retrofit Procedures Retrofits recommended for fatigue cracking in the longitudinal girders included the drilling of holes at the end of the cracks to blunt the tips. Further retrofit was required to prevent the lateral displacement of the bottom flange and minimize the out-of-plane distortion in the web cope. Othewise, fatigue-crack reinitiation from the retrofit holes and continued propagation along the bolt fixity line would result. Several rows of bolts were removed from the girder connection to reduce the compression in the web and move the flange towards normal alignment. Angles were bolted to the inside longitudinal girder web and flange and to the connection angles (Fig. 7.17) to prevent lateral motion of the bottom flange and reduce the bending stress on the girder web. Transfer of the live load forces from the beam top flange to the column had to be ensured for an effective retrofit of the beam-column connections of the bent. Short sections of wide flange shapes were modified by partial removal of the web and were bolted to the flange of the transverse beams and webs of the columns (Fig. 7.18). If fatigue cracks in the groove welds continued to propagate and eventually led to fracture of the top flange-to-column connection, the live load forces would be transferred by the bolted splices. To verify the effectiveness of the retrofit procedures, a second set of


FIG. 7.17. Retrofit angles bolted to both girder and transverse beam.

field tests was made. Stress-range spectra were observed on one corner joint without the retrofit collar and after the bolted splices were installed. These measurements demonstrated that the bolted retrofit collars reduced the cyclic stress at the corner connection by 75%. In addition, they also demonstrated that the bolted collar was effective in resisting the live load. 7.5 CORROSION—I-95 OVER THE SUSQUEHANNA RIVER 7.5.1 Structure Description I-95 crosses the Susquehanna River in Maryland. The structure is a multiplespan, deck truss bridge whose members are either built-up riveted sections or rolled structural shapes. Transverse floorbeams and longitudinal stringers make up the floor system, and both form composite action with the reinforced-concrete deck. All of the floorbeams are supported by the top chord of the main deck trusses as shown in Fig. 7.19. Suspended truss spans contain pin and hanger assemblies. Hangers are riveted box sections (Fig. 7.20).


FIG. 7.18. Retrofit collars for the transverse-beam-to-column connection.

7.5.2 Inspection Results and Field Measurements Inspection of the structure revealed fatigue cracks emanating from the rivet holes in several box sections, as shown in Fig. 7.21. In one hanger web, a crack had reinitiated from a prior retrofit hole which had been placed to arrest crack growth. Significant amounts of corrosion product were found between the corner angles of the hanger sections and both web and flange plates. Environmental corrosion was also found between the gusset and hanger plate connection, as well as on the gusset at the pin elevation (Fig. 7.22). Several sets of field measurements were made on the structure between 1985 and 1987. Initial strain measurements were acquired on a single cracked hanger. Small-tension stress cycles developed in all the strain gages mounted on the hanger, with typical cyclic stress values of 10·5 MPa (1·5 ksi). In addition to the tension cycles, both low-level vibration of the hanger and bending of the hanger web were observed. Large dynamic responses occurred periodically in all gages mounted on the hanger and were on the order of the yield point (Fig. 7.23).


FIG. 7.19. Susquehanna River bridge.

A subsequent set of field measurements was made after pins of the originally damaged hanger were lubricated. For comparison, several other hangers were instrumented. At locations where pins had been lubricated, both the bending of the hanger webs and the large dynamic responses were minimized. Cracked hanger locations with lubricated assemblies experienced a maximum stress range of 35 MPa (5 ksi). In contrast, unlubricated pin assemblies had hangers which experienced significant axial and bending stresses. Typical values varied up to 76 MPa (11 ksi). Final field measurements were conducted after all pin assemblies had been lubricated. A maximum stress range of 30 MPa (4.3 ksi) was observed for the original cracked hanger web. A cyclic stress range of 55 MPa (7.9 ksi) was recorded for a cracked hanger flange plate in a second location where only one of the two pins was lubricated. 7.5.3 Probable Causes of Failure and Retrofit Procedures Field inspections and strain measurements demonstrated that the development of corrosion caused pin fixity which, in turn, led to the introduction of bending in the hangers. The hanger connections are located directly below open roadway expansion joints which allow salt, water and debris to fall and collect on the pin and hangers, thereby assisting and accelerating the corrosion process. The constant-amplitude fatigue limit for riveted members is approximately 48 MPa (7 ksi). Unlubricated pin assemblies showed stress range values up to 76 MPa (11

FIG. 7.20. Pin and hanger assembly.


ksi) which is consistent with the observed cracks. Lubrication of the pins was effective in minimizing hanger bending and dynamic effects by reducing pin fixity and as such, was recommended as a routine maintenance practice. Measured stress-range values for lubricated assemblies were less than 35 MPa (5


FIG. 7.21. Fatigue crack from rivet hole.

FIG. 7.22. Corrosion at the pin connection.

ksi). This level of stress range would not produce fatigue cracking in a virgin hanger. However, continued propagation of smaller cracks which might exist under rivet heads would occur. A longer-term correction would require a redesign of the pin connection as well as effective control of water and debris by the elimination of open expansion joints.


FIG. 7.23. Strain-time response for dynamic excursion.

Cracked hanger plates were retrofitted by the drilling of holes to blunt the crack tips. For a crack to reinitiate, the relationship given in eqn. 7.13 must be exceeded. (7.13) For a 19·1-mm (3/4-in) arrest hole and a 152-mm (6-in) crack, the stress intensity factor range becomes (7.14) With a ligament width of 622 mm (24.5 in) and a root radius of 9.5 mm (0·375 in), (7.15) or Sr>29 ksi (7.16) This stress-range level was exceeded prior to lubrication of the pins owing to the observed large dynamic stress excursions. Hence, a crack would have been expected to form at the prior arrest holes. With the significant reduction in stress range for lubricated joints, fatigue cracking will not be expected to occur at any of the arrest holes. If small cracks exist under the rivet heads, the cracks will propagate at a slower rate owing to the reduction of applied stress range. For a crack tip near the edge of a rivet hole, the stress-intensity range is given by (7.17) For a hole radius of 11·9 mm (0·47 in) and an existing crack of 10·2 mm (0·4 in), this becomes (7.18)


When the stress intensity range becomes larger than the crack-growth threshold, crack propagation will develop. For a riveted section with tension from dead load, the crack growth threshold is approximately 4·5 MPa (4 ksi-in). As such, an applied stress range of 17 MPa (2·4 ksi) or greater will propagate the cracks at a slow rate. 7.6 CONCLUSIONS As replacement costs for highway bridges are often prohibitive, it is often desirable to repair and retrofit existing structures to maximize the benefit from limited funds. A majority of fatigue-damaged details can be repaired by drilling holes, weld-toe peening, or bolting splices over the damaged areas to strengthen the connection. In some instances, welding can be used, although care should be exercised so that low-fatigue-resistance details do not result. Once corrosion is identified as a problem, extensive repairs are often reqired. Examination of the causes of fatigue cracking and corrosion, coupled with implementation of new design tools, code revisions and effective technology transfer will help to minimize future structural deficiency problems. Simple, realistic models and analysis procedures for member interaction and connection behavior are needed to reduce the occurrence of distortion-induced fatigue cracking. Identification of low-fatigue-resistant details and effective methods for detail classification would ease the burden on designers and limit the incidence of fatigue cracking. REFERENCES ALBRECHT, P. & YAMADA, K. (1977) Rapid calculation of stress intensity factors. J. Structural Div., ASCE., 103(ST2), 377–89. BROEK, D. (1982). Elementary Engineering Fracture Mechanics. Martinus Nijhoff, Boston, Massachusetts. DEMERS, C. & FISHER, J.W. (1989) A Survey of Localized Cracking in Steel Bridges. ATLSS Report #89–01, Lehigh University, Bethlehem. FISHER, J.W. (1977) Bridge Fatigue Guide-Design and Details. American Institute of Steel Construction, New York. FISHER, J.W. (1978) Fatigue cracking in bridges from out-of-plane displacements. Canadian J. Civil Eng., 5(4), 542–6. FISHER, J.W. (1984) Fatigue and Fracture in Steel Bridges. Wiley Interscience, New York. FISHER, J.W. (1986) Failures of bridge components. Metals Handbook, Vol. 11. American Society for Metals, Metals Park, Ohio. FISHER, J.W. & MERTZ, D.R. (1985) Hundreds of bridges and thousands of cracks. Civil Eng., ASCE, 55, 64–7.


FISHER, J.W., FRANK, K.H., HIRT, M.A., & MCNAMEE, B.M. (1970) Effect of Weldments on the Fatigue Strength of Steel Beams. National Cooperative Highway Research Program Report 102, Washington, D.C. FISHER, J.W., ALBRECHT, P., YEN, B.T., KLINGERMAN, D.J. & MCNAMEE, B.M. (1974) Fatigue Strength of Steel Beams with Welded Stiffeners and Attachments. National Cooperative Highway Research Program Report 147, Washington, D.C. FISHER, J.W., MERTZ, D.R. & ZHONG, A. (1983) Steel Bridge Members Under Variable Amplitude Long life Fatigue Loading. National Cooperative Highway Research Program Report 267, Washington, D.C. FRANK, K.H. & FISHER, J.W. (1979) Fatigue strength of fillet welded cruciform joints. J. Structural Div., ASCE, 105(ST9) , 1727–40. KEATING, P.B. & FISHER, J.W. (1986) Evaluation of Fatigue Tests and Design Criteria on Welded Details. National Cooperative Highway Research Program, Report 286, Washington, D.C. PARIS, P. & SIH, G.C. (1965) Stress Analysis of Cracks. ASTM STP 391. American Society of Testing Materials, Philadelphia, Pennsylvania. SCHILLING, C.G., KLIPPSTEIN, K.H., BARSOM, J.M. & BLAKE, G.T. (1978) Fatigue of Welded Steel Bridge Members Under Variable Amplitude Loading. National Cooperative Highway Research Program Report 186. TADA, H., PARIS, P. & IRWIN, G. (1987) The Stress Analysis of Cracks Handbook. Del Research Corporation, Paris Productions, St. Louis, Missouri.

Chapter 8 REINFORCED-CONCRETE FRAMES SUBJECTED TO CYCLIC LOAD C.MEYER Department of Civil Engineering and Engineering Mechanics, Columbia University, USA

SUMMARY This chapter summarizes important facts known about the response of reinforced concrete frames to cyclic loads. Since the most severe of such loads result from destructive earthquakes, much information on the behavior characteristics, analysis techniques, and design considerations, is drawn from the earthquake-engineering literature. The response of concrete frames to severe cyclic load is accompanied by a gradual process of strength and stiffness degradation, which can be considered a low-cycle fatigue phenomenon. Recent advances in the fields of modeling and analyzing concrete frames for such loading are reviewed. 8.1 INTRODUCTION Structural design is based typically on the requirement that the structural strength or resistance S exceed the load effect Q, S>Q (8.1) As structures are rarely subjected only to static loads, that is, exclusively to loads that do not change in time, the condition of eqn (8.1) poses two separate problems. The first one involves the determination of the load effect, i.e. the demand placed on a structure’s capacity to resist load needs to be determined, while properly accounting for the complications caused by the repeated application of load. The other task consists of determin ing the structure’s capacity or resistance, which is different for multiple-load applications from that for single, static loads. This is especially true for a highly non-linear material such as reinforced concrete, which suffers progressive damage as it is being exposed to a history of repeated

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FIG. 8.1. Cyclic load categories. (a) Static load. (b) Load fluctuation. (c) Load swell. (d) Load reversal.

load applications. This damage manifests itself in a gradually advancing stiffness and strength degradation, which may reach a point where the structure cannot any longer resist the imposed load. When considering realistic loads on structures, it is important to distinguish whether only moderate stress fluctuations or complete stress reversals are the result. In the fatigue analysis of structures it is common to introduce a parameter R defining the ratio of minimum to maximum stress or load amplitude (see Fig. 8.1): (8.2) Common sense and experience tell us that a material subjected to a load history with R=−1 (complete load reversal) is likely to suffer more damage than one subjected to a load history with only moderate or no amplitude fluctuations, everything else being equal. Most structures which are subjected to gravity-type loads experience load histories of the type shown in Fig. 8.1(b). The dead weight, say of a highway bridge, establishes a base load level, and any temporary traffic or live load constitutes a more or less significant deviation therefrom. Stress reversals are not very common in this case, except within certain regions of continuous multispan bridges. If we consider building structures which are subjected to lateral loads, the situation may be different. There are virtually no steady-state horizontal

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loads, so that temporary loads are more likely to involve complete reversals. But also here we have to distinguish between different cases. Gravity loads produce stresses both in the girders (shear forces and bending moments) and in the columns (mostly axial forces). Horizontal loads applied to either side of a building frame cause bending moments in both the beams and the columns. Since in the beams these moments are superimposed on the gravity load moments, it may very well be that no moment reversals will occur (R>0). In the columns, complete moment reversal is more likely to take place (R=−1) yet, because of the precompression caused by the gravity load, the actual stresses in the steel and concrete do not necessarily change signs, unless the lateral loads are large compared with the gravity loads. The most important sources of lateral loads on buildings are wind, blast, and earthquakes. Wind loads are typically characterized by a steady-state pressure, with superimposed pressure fluctuations due to gusting (Simiu & Scanlan, 1978). The result is not unlike the load history of Fig. 8.1(b), i.e. with load reversals being unlikely. The important exception is the case of across-wind oscillations caused by vortex-shedding (see Fig. 8.2). Blast loads consist typically of a single impulse, which will cause structural vibrations, but the load itself is not of cyclic nature and therefore is not considered here. Earthquake ground motions are basically of a cyclic nature and the loads they cause are among the most destructive known. It is the combination of repeated load application, potential dynamic amplification, and the duration of a seismic disturbance, which creates one of the most challenging tasks for the structural engineer. Structures are generally designed to respond linear elastically to wind loads. In earthquake-resistant design, however, entirely elastic response is often neither feasible nor desirable, because the energy dissipation through inelastic action can


FIG. 8.2. Across-wind oscillations.

be utilized to limit both the seismic loads and important structural response quantities. Balancing these two design objectives, namely reducing the seismic loads through inelastic behavior on the one hand and keeping overall displacements, often expressed in terms of ductility ratios, within acceptable bounds on the other hand, constitutes one of the main challenges of earthquakeresistant design. A second important problem is caused by the degradation of strength and stiffness that concrete experiences under repeated load applications, especially if complete load reversals and inelastic action are involved. By paying careful attention to reinforcing details it is possible to turn the essentially brittle concrete into a rather ductile material. The resulting ‘ductile concrete’ has become an increasingly popular building material in seismic regions of the world. It is the objective of this chapter to present a concise overview of what is known about the response of reinforced concrete frames to cyclic loads. Findings from experimental investigations will be covered as well as important lessons learned from post-earthquake investigations. Next, a summary of the current state of the art in numerical simulation of concrete frame response to cyclic loads will be given. This analysis task is of extraordinary difficulty and requires careful calibration of the mathematical models against experimentally obtained data, but our capabilities of modeling reinforced concrete for such analyses have now reached the point where they can be utilized in selected situations in lieu of expensive experimental investigations. The final section will discuss the implications of this new technology for the design of reinforced-concrete frames against lateral loads.

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8.2 OBSERVED RESPONSE OF R-C FRAMES TO CYCLIC LOAD Mother Nature is a stern but very effective teacher. An actual earthquake will decide with incorruptible impartiality and often deadly accuracy which building types and construction details are appropriate and which ones are not. Lessons learned from post-earthquake investigations are invaluable and form the basis of much of our knowledge. But from a modern engineering standpoint we can do and should do better than wait for the next earthquake to happen as a full-scale test of new design concepts. In fact, a considerable number of laboratory investigations have been performed to systematically study the factors that determine reinforced-concrete response to cyclic load and to translate the observations and conclusions into practical guidelines for design. These investigations range from quasi-static tests of single members to pseudo-dynamic tests of full-scale buildings. 8.2.1 Quasi-Static Tests of R–C Members In quasi-static experiments, loads or deformations are applied to the test specimens relatively slowly, thereby ignoring both the inertia effects and any influence that the strain rates of actual loadings may have on the material response. The neglect of inertia effects is acceptable because these can readily be transformed into equivalent static forces. Ignoring the strain-rate effect is more problematic, because both concrete and steel have been observed to experience strength and stiffness increases up to 10% and even 20% under strain rates common for earthquake-type loads. This fact has to be recalled when data obtained from quasi-static tests are interpreted and utilized. Significant test series involving reinforced-concrete members subjected to cyclic loads have been conducted at numerous institutions, e.g. the University of California at Berkeley (Bertero et al., 1974; Atalay & Penzien, 1975; Ma et al., 1976 and Bertero & Popov, 1977), the University of Illinois (Hwang, 1982), and the University of Michigan (Scribner & Wight, 1978), to name a few. These investigations have led to a wealth of knowledge. A brief overview of some of the important points will suffice here. Figure 8.3 shows the load-deflection curve for a cantilever beam tested at UC Berkeley under cyclic loading of increasing amplitudes (Ma et al., 1976). The following observations can be made from this and similar experimental results: (1) The load-deflection curve is fairly linear up to the yield moment. Thereafter, the yielding of the steel causes a considerable drop in stiffness. (2) The stiffness associated with unloading is initially not much different from the stiffness under initial loading. However, the farther the load excursion


into the inelastic range, the more noticeable is the stiffness decay upon unloading. Much of this stiffness degradation can be attributed to the Bauschinger effect and bond deterioration. (3) The stiffness associated with reloading is even more affected by the extent of previous non-linear load excursions than that for unloading. (4) For small numbers of load cycles with constant amplitude, the beam behavior is remarkably stable. In the example of Fig. 8.3, no strength decay is noticeable prior to the final load amplitude of over 2 in (5 cm); that is until then approximately the same load is required in successive cycles to produce the same deflection. (5) If more load cycles were applied for each load amplitude, strength degradation would be more noticeable; this is a low-cycle fatigue phenomenon and results from the gradual accumulation of damage (concrete cracking and bond slip) until failure. (6) Failure is generally due to buckling of the flexural reinforcing bars, after they have undergone considerable plastic deformations. Such buckling failures can be delayed by reducing the spacing of the lateral reinforcement. (7) A considerable increase of a beam’s energy dissipation capacity is possible if equal amounts of steel are provided for the positive and negative reinforcement in the critical regions, as compared to members with less reinforcement in the bottom layer of the beam. (8) In spite of special precautions, large deformations due to bond deterioration (bar pull-out) were observed in most experiments after several load reversals. The hysteresis loops of Fig. 8.3 are rather stable and convex, because the beam behavior was controlled largely by flexure. The contrast with the load-deflection curve of Fig. 8.4 is striking, because in this example the influence of shear was much more significant. This can be explained as follows. Under combined bending and shear, diagonal cracking develops, at an angle of approximately 45° with the beam axis. Upon load reversal, a second set of diagonal cracks forms, roughly orthogonal to the first one. After unloading, a crack is only partially closed, and relatively little load is needed to close it completely. The associated part of the load-deflection curve has very small slope (stiffness). After crack closing, the stiffness increases considerably, until renewed yielding of the steel again causes a stiffness decrease. As a result, the energy dissipation capacity of flexural members is greatly reduced in the presence of high shear forces. This can be restored only partially by providing closely spaced lateral reinforcement, because vertical flexural cracks may eventually extend over the entire cross section, especially after extensive yielding of both top and bottom reinforcement, at which point the only significant source of shear resistance is dowel action. Diagonal shear reinforcement was found to overcome this problem and to permit large plastic hinge rotations. In fact, the ductilities achieved with such

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FIG. 8.3. Load-deflection diagram for cantilever beam with low shear (Ma et al., 1976).

FIG. 8.4. Load-deflection diagram for cantilever beam with high shear (Ma et al., 1976).

special web reinforcement, as shown in Fig. 8.5, were comparable with those of compact steel beams (Bertero et al., 1974). The presence of an axial force in addition to shear and bending, affects the member behavior in several ways, as shown in Fig. 8.6 (Atalay & Penzien, 1975). First, it impacts on the member’s moment capacity, as is best illustrated in


conventional interaction diagrams. Next, the axial force causes an additional bending moment (P−σ effect), which, under increasing displacement amplitudes, leads to instability. In columns, the axial force varies as a function of time, owing to the overturning moment associated with lateral load. The result is a momentrotation curve such as the one shown in Fig. 8.7 (Abrams, 1987). Whereas previously opened cracks are slower to close when the axial force decreases (or even turns tensile), an increase in axial compression causes the cracks to close sooner, thus leading to a noticeable stiffness increase. Such an asymmetric response may lead to a large accumulation of plastic strain in the reinforcement, accompanied by large crack widths and a reduction of shear strength. Similar effects can be observed in beams with large differences between negative and positive reinforcement, which can reduce the member’s energy absorption capacity. A significant effect of axial force application is the increase in concrete strain, which translates into a decrease of residual strain capacity available for further loading. This reduction in ductility, together with the P–σ effect, is the main rationale for the ‘strong column-weak beam’ design criterion, which seeks to avoid inelastic column behavior during seismic disturbances in the first place, and assign the task of energy dissipation to inelastic deformation of the beams only. All of the above observations were made regardless of the confinement of the concrete that may be provided by lateral reinforcement such as continuous spirals or closely spaced stirrups. Such confinement reinforcement affects the behavior of reinforced-concrete members in three ways. (1) It causes an increase in the concrete’s strength. The restraint of lateral strains creates a three-dimensional state of compression, under which the strength of concrete is known to increase considerably. (2) It increases the concrete’s deformability. The strain at peak stress increases moderately, but the strain-softening branch of the stress-strain curve has a much smaller negative slope than that for an unconfined concrete specimen. As a result, the energy dissipated by the specimen, as measured by the area under the stress-strain curve, even in a monotonic load test, is increased enormously. The reason for this behavior is the delay of large-scale cracking following the peak stress. (3) Under cyclic loading, confinement plays a similar role to that it plays for monotonic loading. It delays the strength and stiffness degradation mentioned earlier, thereby giving the material a ductility which the normally brittle concrete does not have. Hysteresis loops for well-confined concrete members are much more stable, and as a result, the energy dissipation capacity of such members is increased by large amounts. In Fig. 8.8 the hysteresis loops of two column specimens are juxtaposed (Ozcebe & Saatcioglu, 1987), one with a confinement steel ratio of 1.69%, and one with 2. 54%. The effect of the confinement steel is dramatic.

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FIG. 8.5. Load-deflection diagram for beam with special web reinforcement (Bertero et al., 1974).

8.2.2 Frame Subassemblies The question of whether conclusions derived from individual member tests can be used directly to predict the response of entire buildings prompted the investigation of so-called frame subassemblies (see Fig. 8.9). The most


FIG. 8.6. Load–deformation diagram for column (Atalay & Penzien, 1975).

significant feature of such tests, not present when working with single members, is the beam–column joint, which is known to influence greatly the overall response of R–C frame buildings. The single most critical factor was found to be bond deterioration within the joint panel zone. It generally does not require many load cycles, with beam reinforcement yielding in tension at one face of the column and in compression at the other column face, to pull through. Such bond deterioration, coupled with high shear stresses, causes an early strength and stiffness degradation of beam– column joints, visible in severely pinched hysteresis loops, which means greatly reduced energy dissipation capacity. 8.2.3 Other Tests A very instructive testing method was made possible with the introduction of shaking tables. In the United States, the earthquake research laboratories at the Universities of Berkeley–California, Urbana–Illinois, and Buffalo–New York have provided valuable insight into the response of structures to ground motion. Weight restrictions limit the scope of such studies to scale models. For example, at the University of Illinois a series of 10-story building frames (approximately one-tenth scale) have been tested on the shaking table (Healey & Sozen, 1978), and at UC Berkeley a two-story building model of approximately half scale was studied (Clough & Gidwani, 1976). The importance of these investigations derives from the fact that they permit detailed monitoring of structural response to earthquake-type loads under closely controlled conditions and thus can serve

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FIG. 8.7. Moment-rotation relation for column axial force–deflection variation (Abrams, 1987).

as benchmark cases for the verification of numerical models and analysis procedures as well as novel design concepts. The most comprehensive single investigation was sponsored by the United States and Japan as a joint research effort and involved a series of quasi-static component tests, scale-model tests, shaking table tests, and a pseudo-dynamic test of a full-scale seven-story reinforced concrete building in Tsukuba (Wight, 1985). This well-coordinated research effort resulted in a better understanding of concrete behavior under cyclic load and raised the confidence level for mathematical models to simulate such behavior, both for scale and full-scale buildings. 8.2.4 Post-Earthquake Observations Each earthquake which subjects concrete structures to severe ground shaking has the potential to expand our knowledge of their response to such loads (Moehle & Mahin, 1991). Whereas public attention generally focuses on those structures which suffer severe damage or even collapse, the fact is that most structures behave rather well. For example, after the 1985 Mexico City earthquake, much has been written and said about the more than 200 multistory buildings that


FIG. 8.8. Effect of confinement reinforcement on column degradation (Ozcebe and Saatcioglu, 1987). (a) 1.69% confinement steel ratio. (b) 2.54% confinement steel ratio.

collapsed, yet hundreds of thousands of buildings were subjected to the earthquake, and most of them responded very satisfactorily. Each earthquake teaches us lessons on mistakes made in the past and to be avoided in the future. One of the most significant examples was the San Fernando Earthquake of 1971, because it was the first full-scale test of a densely populated area with many modern reinforced-concrete buildings engineered to resist seismic loads. The Olive View Hospital, for example, barely completed, fared so poorly during that earthquake that it had to be torn down. Much of what is now known about proper reinforced-concrete design was learned as a result of the investigations prompted by that earthquake. The Loma Prieta Earthquake of 1989 promises to be comparably consequential. At the time of this writing the in-depth investigations

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FIG. 8.9. Frame subassemblies (Bertero & Popov, 1977).

of the effects of this earthquake have barely begun, but a wealth of information is expected to result from these. Among the fundamental rules for aseismic construction are the requirements that the structural system have a simple, regular, and compact layout and offer redundant load paths in case of local distress. Discontinuities tend to increase loads or deformations often with severe concentrations of stress or strain at the discontinuities. For example, discontinuities of vertical structural elements have frequently led to failure. A common type of this discontinuity is a shear wall that extends over the entire height of a building except for the first story. As a result, damage is likely to concentrate in this first story. Similarly, irregularities of any kind can cause problems, which can be solved with proper detailing, but which it is wiser to avoid. Examples are setbacks, changes in story height and heavy concentrated masses, for example on mechanical floors. Also the role of nonstructural components is important, especially if they do in fact influence the structural response. For example, flexible frames with masonry infill, if properly designed and detailed, can improve the building response by contributing to both stiffness and strength, but if improperly designed they may fail prematurely or otherwise impair the structural efficiency. A common problem is associated with infill walls rising only up to the window sills and thus effectively reducing the column heights. Such short columns may not be able to develop their flexural strengths before failing in shear. A most essential ingredient of the continuity requirement is that both structural and non-structural elements be properly tied together so as to permit a continuous load path from the position of large masses such as floor slabs to the vertical load-resisting elements and from there to the foundations. Numerous


instances of failure or heavy damage have been observed, in which cladding elements or unreinforced brick walls were torn loose, thereby endangering building occupants or pedestrians on sidewalks. Part of this requirement is that adequate diaphragm action can develop both in the roof and the floor slabs, to ensure that all inertia loads are safely carried into the walls and columns. The value of non-structural components and building contents typically exceeds that of the structure proper, and their damage can constitute a considerable economic loss. To limit the potential of damage, lateral drift deformations have to be limited by providing sufficient stiffness. Excessive drift deformations not only cause large damage to building contents and endanger occupants, but also endanger non-ductile structural elements as well. Finally, drift control is important in preserving the vertical stability of a frame. The P–σ effect can give rise to instability, particularly for tall and slender buildings. 8.3 FRAME ANALYSIS 8.3.1 General The purpose of structural analysis is to predict, in a rational way, the response of a given structure to specified loads. The analysis problem considered here is made particularly difficult by two factors: (1) the loading (due to earthquake ground shaking) is of a highly random, unpredictable nature; (2) the load– deformation relationships of the struc ture and its components are strongly nonlinear. For practical purposes, a determination has to be made as to what degree of accuracy of the calculations is warranted. For most applications, the uncertainty of seismic ground motions alone precludes detailed and potentially costly analyses. This is true for preliminary designs and most routine designs of conventional buildings. But in the case of tall buildings and other important structures, it may be preferable to predict analytically the margin of safety against serious damage or collapse. When ground acceleration histories are available, which are representative of the seismicity of the building site, it is appropriate to employ accurate models and analysis techniques which realistically simulate the stiffness and strength degradation of concrete members during strong inelastic load cycles. Otherwise, the response predictions, especially if based on over-simplified models and analysis methods, cannot constitute much more than educated guesses. Much has been written to justify the use of such simplified analysis methods, with the principal argument that buildings designed on such a basis have behaved rather satisfactorily in earthquakes. The fact is that buildings behave well only if they have been properly designed, well detailed, and well constructed. If these three conditions are met, then the sophistication of the analysis method employed is indeed of

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secondary importance. But it is the essence of the engineering approach that structural response be predicted beforehand as accurately as possible so that any subsequent design decisions can be based on a rational foundation. It is true, though, that in practical design situations the use of advanced analysis techniques is generally limited to special or important structures or for evaluation of a completed or existing design. 8.3.2 Time History Analysis If a sufficiently accurate mathematical model of a reinforced-concrete building is available and the building response is linear, then the computation of this response to load is a relatively routine process and involves the application of wellestablished computer programs. The methods employed by these programs are described in texts on structural dynamics (e.g Newmark & Rosenblueth, 1971; Clough & Penzien, 1975; Bathe, 1982). Time history analyses require that the loading, i.e. the earthquake ground motion, be represented by deterministic acceleration histories such as may have been recorded during actual earthquakes, or generated synthetically using random vibration techniques (Shinozuka & Tan, 1983). Equations of motion are established for as many degrees of freedom as are needed to describe the stiffness and mass characteristics of the building. These are of the well-known form, (8.3) where M, C, K are, respectively, the mass, damping, and stiffness matrices of the structure; x, are, respectively, the displacement, velocity, and acceleration histories of the structure’s degrees of freedom; I is the identity matrix and is the ground acceleration history. The equations 8.3 assume that the ground motion has only translational and no rotational components. They represent a set of n coupled ordinary differential equations, where n is the number of degrees of freedom of the structure. To solve these equations, one may employ a direct integration scheme, in which all n equations are integrated simultaneously in the time domain. If the structural response is linear (i.e. K remains constant), it is also possible to first solve the associated eigenvalue problem and then use the modal matrix to transform eqns (8.3) to so-called natural coordinates (Clough & Penzien, 1975). It is these new equations then that are integrated in the time domain. This method has the advantage that the transformation results in diagonal coefficient matrices, i.e. it uncouples the differential equations so that they can be integrated independently of each other. Moreover, it is known that the structural response is controlled only by a relatively small number of significant modes, i.e. only m of the n equations need to be solved, where The drawback of this method is that the m eigenvalues and their corresponding mode shapes need to be determined first in an eigenvalue analysis. But the


determination of these eigenvalues, i.e. the structure’s natural frequencies, is often desirable anyway. Again, this method is applicable only if the structure remains linear-elastic. If the structure response is non-linear, as is generally the case for reinforcedconcrete frames responding to strong cyclic load, then the constant structure stiffness K in eqns (8.3) has to be replaced by the tangent stiffness Kt, which in general will change at each time step of integration. In this case, it is appropriate to rewrite eqns (8.3) in incremental form to reflect the requirement that the incremental dynamic forces acting on a structure during some small time step ′ t be in equilibrium, (8.4) For earthquake-type loading functions, it is common to integrate these equations by some implicit numerical algorithm, such as the Newmark method. These consist of approximations for incremental accelerations and velocities with which eqns (8.4) are transformed into a set of quasistatic equilibrium equations: K* Δx=∆F* (8.5) Equations (8.5) can be solved for the displacement increments ∆x But this task involves factorization of the effective stiffness matrix K*, which is a computationally expensive task, especially if K* continually changes. This happens whenever Kt changes, as in the case of inelastic behavior, or when ′ t is varied. The piecewise linearization of response histories causes unbalanced forces at the end of each time step which can be resolved by using numerical analysis techniques that are beyond the scope of this chapter (Bathe, 1982). Implicit algorithms such as the Newmark method are often unconditionally stable, i.e. the solution remains stable irrespective of the choice for the integration time step ′ t. By selecting a value for ′ t which is much larger than the natural periods of most of the structure’s modes of vibration, the numerical integration scheme effectively damps out the contributions of all such modes. Only the significant modes with periods well above ′ t continue to contribute to the computed structural response. 8.3.3 Modeling of Building Components for Cyclic Load Analysis The computation of the dynamic response of a reinforced-concrete building to cyclic loading such as an earthquake can only be as accurate as the model used to represent the structure. Although it is possible to model for each member the concrete, the steel and the bond interface between the two in great detail (ASCE, 1982; Meyer & Okamura, 1986), the enormous effort required for realistic applications precludes this approach for most practical applications. The alternative is the use of so-called semi-empirical ‘macro-elements’, for which the

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FIG. 8.10. Hysteresis rules (Chung et al., 1989).

stiffness matrix of a single frame member is calculated on the basis of certain hysteresis rules, which have been calibrated against experimental test results. In most of these member-size models, all inelastic action is assumed to take place within concentrated plastic hinges at the member ends, while the member proper is assumed to remain linear elastic (Clough et al, 1965; Otani & Sozen, 1972). In more recent models (Roufaiel & Meyer, 1987; Chung et al, 1989), the finite sizes of the plastic regions are accounted for explicitly. The momentcurvature relationship of an R–C section is simulated using a set of rules first proposed by Takeda et al. (1970). It may be represented by linear branches of the five different types identified in Fig. 8.10; (1) one for elastic loading and unloading, valid as long as the moment does not exceed the section’s yield capacity; (2) one for inelastic loading, used when the moment exceeds the yield moment and is still increasing; (3) one for inelastic unloading, for moments decreasing after the yield moment has been exceeded; (4) one for inelastic reloading during closing of previously opened cracks; and (5) one for inelastic reloading after closing of previously opened cracks. The transition from branch 4 to 5 is determined by a ‘crack-closing’ moment, which is a function of the member’s shear span-to-depth ratio. The inclusion of this detail in the model makes it possible to reproduce the pronounced pinching of the hysteresis loops that can be observed in the presence of high shear forces. A member’s stiffness degradation progresses as a function of the degree to which the yield point has been exceeded. Both loading and unloading stiffnesses are affected.


FIG. 8.11. Frame member model.

Once the moment–curvature relationship has been derived from the stress– strain curves for steel and concrete at any member section and for any possible loading branch, it is relatively straightforward to compute the member deformations (flexibility coefficients) by integrating the curvatures over the member length, thereby explicitly accounting for the finite sizes of the plastic regions shown in Fig. 8.11 (Roufaiel & Meyer, 1987). The tangent member stiffness is then determined by using standard methods of structural analysis. In addition to stiffness degradation, R-C members experience strength deterioration under cyclic loading beyond the yield level. Atalay & Penzien (1975) had noticed some correlation between commencement of strength deterioration and the spalling of the concrete cover. But Hwang’s experiments (Hwang, 1982) showed that strength deterioration can start at considerably lower load levels. Even for loads only slightly above the yield level, damage and strength deterioration can be observed, provided a sufficiently large number of load cycles is applied. It is suggested therefore that strength starts to deteriorate as soon as the yield load level is exceeded, and this deterioration accelerates as the critical load level is approached. This phenomenon can be reproduced with a strength drop index Sd, illustrated in Fig. 8.12 (Chung et al., 1989): (8.6) where ′ M is the moment capacity reduction in a single load cycle up to curvature ′ Mf is the fictitious moment capacity reduction in a single load cycle up to failure curvature is the yield curvature, and is a parameter which depends on material properties and reinforcing details. When modeling reinforced-concrete columns, it is important to include the effect the axial force has on the strength and deformation capacity. Since the

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FIG. 8.12. Strength drop index.

yield moment is a function of the time-dependent axial force, an accurate model should recompute the yield moment and monotonic load–deformation curve at each time step (Keshavarzian & Schnobrich, 1984), which is a computationally expensive undertaking. For practical purposes, it is often acceptable to base the yield moment capacity on the axial force caused by gravity loads alone. The application of such models to structural walls has to be done with great care. Because the cross section of a shear wall tends to be narrow, it may be in danger of buckling, either globally or between stories. This danger is increased after extensive yielding of the reinforcement, when only the bars temporarily resist the reversed load. Since it is impractical to confine the concrete over the entire section depth, confinement is usually limited to the boundary members. In any concrete member, the neutral axis changes its position in the cross section as the member cracks and crushes. In a slender column or girder, this effect tends to be small, but not in a wide shear wall. During cyclic loading, the neutral axis tends to be close to the compression face, so that the wall tilts first about one edge, then the other, accompanied by significant vertical movements at the wall centerline. The large moment capacity of a wall, often further increased by flanges or wing walls, increases the potential for shear failure. When shear stresses are relatively low, flexural cracks near the base tend to be horizontal, which reduces the wall’s shear capacity, as only little effective truss action can develop and a sliding shear failure becomes a possibility. Under high nominal shear stresses, characteristic diagonal cracks control the wall behavior, so that an efficient shear-resisting truss can develop, whose capacity is controlled mostly by the web’s compression capacity (Oesterle et al., 1976, 1979). All of these factors, among others, complicate the behavior of walls under cyclic load and make the mathematical modeling task extraordinarily difficult (Meyer, 1984). When assembling individual component models to produce a model of a complete building and to analyze how the structural members interact to resist load, a number of additional factors need to be considered.


Beam-column joints in reinforced-concrete frames are not rigid. In particular, bond slip is an important cause of joint deformation. A standard procedure is to assume rigid connections between girders and columns, and to account for joint deformations in the girder and/or column elements. It is common to assume that floors are rigid in-plane. This assumption may not always be reasonable, and some computer programs consider the in-plane floor deformations (Meyer, 1991). Analyses of three-dimensional response are complex and expensive, so that two-dimensional models are normally used. Such models may, however, neglect some important interactions among the different frames of a building. Most concrete buildings are quite stiff, and the P–σ effect tends to be small unless the lateral loads are very large or damage is growing severely. The boundary conditions associated with the foundations influence the frame response. More important, however, is the dynamic soil-structure interaction. At present most seismic analyses of detailed reinforced-concrete building models have been carried out for research purposes, and there have been relatively few applications in practical design. Hence, experience is limited, and there is a great deal of uncertainty about ‘correct’ pro cedures, both for creating mathematical models and for interpreting analysis results. ‘Guidelines’ therefore cannot be given beyond general discussions. It is up to the engineer to make the various decisions when modeling the structure and interpreting the analysis results for design purposes. 8.3.4 Modeling of Damage The response of concrete buildings to cyclic loads is inextricably tied to the concept of damage. Each load cycle is likely to increase the amount of concrete cracking, if not yielding of flexural reinforcement or even crushing of concrete in compression. Most damage indices proposed in the literature (Reitherman, 1985; Chung et al., 1987) are of an empirical nature and are tied to more or less subjective post-earthquake inspections for assessing damage. Because of the important influence damage has on the safety and reliability of a building in regard to some future load, it is highly desirable to quantify damage in a rational way such that it can be incorporated into the mathematical model of the building. Damage of a reinforced concrete member will therefore signify a specific degree of physical deterioration with clearly defined consequences regarding the member’s capacity to resist further load. Similarly, ‘failure’ of the member signifies a specific level of damage, associated with a negligible residual capacity to resist further load (Chung et al., 1989). A damage index can then be defined as the damage value normalized with respect to the failure level, so that a value of 1 corresponds to failure. Consider, for example, a simple cube of plain concrete subjected to repeated application of load up to a given strain level (Fig. 8.13) (Bang & Meyer, 1989). It is advantageous to define damage at time t as

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FIG. 8.13. Strength degradation of plain concrete under cyclic loading (Bang & Meyer, 1989).

(8.7) where E(t) is the energy dissipated by time t, and σ the total energy dissipation capacity of the material. The determination of σ requires the definition of failure. To eliminate arbitrariness as far as possible, failure may be defined to take place when all strength reserves for further loading have been exhausted. This will be the case when the stress-strain curve for plain concrete or the moment-curvature curve for a reinforced concrete member reaches a horizontal slope. The total energy dissipation capacity σ depends not only on the strain level but also on the details of the load history if variable-amplitude loading is applied. In the case of reinforced concrete, the confinement steel and reinforcing details have a major effect on σ . As damage accumulates (or the energy dissipation capacity is exhausted), strength deteriorates. This process can be simulated by incorporating the damage parameter into the hysteresis law for the member through the strength drop index Sd of eqn (8.6). Figure 8.14 illustrates the typical agreement between an experimental load-deflection curve and the analytically generated counterpart (Chung et al., 1987). For variable-amplitude loading, damage does not accumulate linearly as the widely used Palmgren-Miner hypothesis suggests, (8.8) Here, ni is the number of actually applied cycles of load with amplitude i, while Ni is the number of cycles with load amplitude i that leads to failure. It has been


FIG. 8.14. Numerical simulation of degrading R-C beam. (a) Experiment. (b) Analysis. (Chung et al., 1987).

shown that eqn (8.8) does not correctly represent damage accumulation in metal structures either (ASCE, 1982), and a power law representation has been suggested (Kutt & Bieniek, 1988). Our own preliminary investigation indicates that damage in plain concrete accumulates faster in the earlier load cycles, (Fig. 8.15) (Bang & Meyer, 1989), and may likewise be represented for loading with constant amplitude i by a power law; (8.9) where σ i is a material parameter which depends on the load amplitude, i.e. the strain level σ i. For variable-amplitude loading, it is useful to convert the number of cycles ni with some strain level σ i to an equivalent

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FIG. 8.15. Damage accumulation in plain concrete (Bang & Meyer, 1989).

number of cycles with a different strain level σ j, on the premise that ni and cause the same amount of damage, i.e. (8.10) from which (8.11) By generalizing eqn (8.11) it is possible to trace the accumulation of damage of the material up to failure. The incorporation of a damage model into a general frame analysis program gives the design engineer a tool for design options that would be difficult to pursue otherwise. Chung et al. have incorporated their damage model into SARCF (Chung et al., 1988a), a program for the seismic analysis of reinforcedconcrete frames, which in conjunction with a set of built-in design rules permits the execution of an automatic damage-controlled design option (Chung et al., 1988b, 1990). In order to keep damage in the building to a minimum, the program attempts to achieve a uniform level of damage (or energy dissipation) throughout the frame. The program user specifies acceptable values for the mean and standard or maximum deviation from the mean damage, and the program will automatically perform iterations towards those target design values.


8.4 DESIGN FOR LATERAL LOADS It is the ultimate objective of all structural design to devise structures that fulfill their intended purposes with a minimum risk of failure, given all statistical uncertainties of loads and resistances, and economic constraints. A comprehensive review of current lateral load design approaches is beyond the scope of this chapter and is addressed elsewhere (Park & Paulay, 1975; Wakabayashi, 1986). Here, only a brief overview of current thinking on the subject will suffice. The loads associated with major earthquakes are so severe that it is seldom feasible to provide structures with sufficient strength to remain elastic. In fact, it is all but universally accepted practice to permit local inelastic actions to take place. Even though the resulting inelastic deformations may be larger than the corresponding elastic deformations would be, the design can be safe provided that the inelastic (plastic) regions are detailed properly to preclude failure under cyclic loading and that instability does not become an issue. In principle, there are several design approaches (ACI-ASCE, 1988). The standard choice for most structures subjected to static loads is strength design. Governed by codes such as the ACI Code (ACI, 1989), member strengths are assigned on the basis of forces and moments determined in a linear elastic analysis. Any deviations between real and assumed forces are easily accomodated through inelastic moment redistribution, provided sufficient ductility is available. In capacity design (Park, 1986), a special hierarchy of member strengths is devised that assures that plastic hinges form in a prespecified sequence. The simplest and most common example is the strong column-weak beam concept, which assures that hinges form only in the beams and not in the columns. Earthquake-resistant design deals with imposed deformations rather than loads. Thus the internal forces and moments are controlled by the member strength capacities. At the same time, sufficient ductility capacities have to be ensured, and care has to be taken that the lateral drift associated with large inelastic deformations does not become excessive, lest instability ensues. There are basically three different approaches to dissipating the large amounts of energy that a damaging earthquake imparts on a reinforced-concrete building. In the first approach, a number of structural members are deliberately selected to act as ‘fuses’, i.e. weak spots assigned to develop plastic hinges and to dissipate energy under tightly controlled conditions. The designer has to detail the selected structural elements very carefully to ensure that the energy dissipation demand can indeed be met without premature failure. Examples of this design philosophy are Park and Paulay’s solution for coupled shear wall buildings (Park & Paulay, 1975) and Popov’s eccentrically braced steel frames. In a second approach, all or most structural elements are called upon to share equally in the task of energy dissipation, with the result that damage is uniformly distributed over the entire frame and therefore can be kept down to a minimum average value. This line of

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thinking has been widespread among practitioners and is the basis for the damage-controlled automatic design procedure mentioned earlier. This design concept is easily combined with the strong column-weak beam principle, so that only beams are permitted to develop plastic hinges; but all of them are expected to dissipate similar amounts of energy. A third design approach attempts to avoid altogether the energy dissipation issue within the structure proper by relying on base isolation, frictional dampers, and active or passive control mechanisms. Current design practice recognizes three different levels of design earthquakes. A building should resist minor but frequent earthquakes without any damage. It should be sufficiently stiff so that deformations remain small and damage to non-structural components is insignificant. For moderately strong earthquakes of less frequency, some non-structural damage is acceptable, but the structure itself should not be damaged. This implies that for such load levels, the structure’s response remains essentially elastic. For strong earthquakes with very low probability of occurrence, both structural and non-structural damage is acceptable. It is desirable that such damage be repairable, but life safety remains the overriding design principle, i.e. under no circumstances should the structure lose its integrity or suffer collapse. To achieve this goal, it is permissible to activate the entire energy-dissipation capacity of the building short of reaching failure. An important key to design success or failure is the attention paid to design details, especially for beam–column joint regions and all other regions in which inelastic action is expected. Beam–column joints are subject to high shear forces, axial forces, bending moments and bond stresses and are particularly susceptible to failure under cyclic load. Design guidelines are given in the report of ACI Committee 352 (ACIA- SCE, 1985) and can be summarized as follows. (1) The compression forces from the columns require adequate lateral confinement of the concrete. This can be achieved by lateral reinforcement or transverse members framing into the joint, or both. Bar offsets should be avoided in joint regions. (2) When detailing the joint for shear, the degradation of the concrete shear capacity under cyclic load should be recognized, which increases the share of the load to be carried by shear reinforcement. (3) Since it is preferable to have plastic hinges form in the beams rather than in the columns, the columns should have flexural strengths 1.4 times those of the beams framing into the same joints if these joints are part of the primary system that resists seismic lateral load. (4) The bending moments acting on a joint zone are such that any bar, vertical or horizontal, is simultaneously pushed and pulled on opposite sides. Thus, especially high demands are placed on the bar’s capacity to transfer bond stresses within the limited joint size. This demand calls for large joint dimensions relative to the selected bar sizes.


Concrete will invariably deteriorate under strong load reversals. But if it is sufficiently confined, the degradation process can be slowed appreciably, thereby increasing the structure’s reliability. It is the inherent toughness of ductile concrete that assures the life safety of reinforced-concrete buildings. 8.5 CONCLUSIONS The response of reinforced-concrete frames to cyclic load is characterized by gradual accumulation of damage in the concrete. Whereas in metals the length of a single crack serves as a useful damage indicator, in concrete there is a complex system of microcracks, which passes through various phases until one or more predominant cracks develop, which eventually cause failure. Failure of metal structures is typically a sudden phenomenon, once a crack becomes unstable. In reinforced concrete, the failure is more gradual, provided sufficient confinement reinforcement is present. It is also possible that the flexural reinforcement itself suffers low-cycle fatigue failure. This failure mode, observed occasionally in laboratory experiments, is difficult to predict numerically because of the lack of experimental data. Because of this failure mode, the random and unpredictable nature of earthquake motions, and uncertainty concerning the properties of the materials subjected to strong cyclic loads, the survival of a structure cannot be fully assured. But the present state of the art of reinforced-concrete design has advanced to the point where such survival can reasonably be expected if proper design, detailing, and construction practices are followed. REFERENCES ABRAMS, D.P. (1987) Influence of axial force variations of flexural behavior of reinforced concrete columns. ACI Structural J., 84(3), 246–54. ACI COMMITTEE 318 (1989) Building Code Requirements for Reinforced Concrete. ACI Standard 318–89. ACI-ASCE COMMITTEE 352 (1985) Recommendations for Design of Beam-Column Joints in Monolithic Reinforced Concrete Structures. ACI Report 352R-85. ACI-ASCE COMMITTEE 442 (1988) Response of Concrete Buildings to Lateral Forces. ACI Report 442R-88. ASCE TASK COMMITTEE ON FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE STRUCTURES, A.H. NILSON, CHMN. (1982) Finite Element Analysis of Reinforced Concrete. ASCE, Special Publication. ASCE COMMITTEE on Fatigue and Fracture Reliability (1982) Fatigue Reliability. J. Structural Div. ASCE, 108(ST). ATALAY, M.B. & PENZIEN, J. (1975) The Seismic Behavior of Critical Regions of Reinforced Concrete Components as Influenced by Moment, Shear and Axial Force. Earthquake Engineering Research Center, Report No. EERC 75–19, University of California, Berkeley.

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BANG, M.S. & MEYER, C. (1989) Damage of Plain Concrete as a Low-Cycle Fatigue Phenomenon. Dept. of Civil Engineering and Engineering Mechanics, Columbia University, New York. BATHE, K.J. (1982) Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, New Jersey. BERTERO, V.V. & POPOV, E.P. (1977) Seismic behavior of ductile moment-resisting reinforced concrete frames. In Reinforced Concrete Structures in Seismic Zones. ACI Special Publication SP-53. BERTERO, V.V., POPOV, E.P. & WANG, T.Y. (1974) Hysteretic Behavior of Reinforced Concrete Flexural Members with Special Web Reinforcement. Earthquake Engineering Research Center, Report No. EERC 74–9, University of California, Berkeley. CHUNG, Y.S., MEYER, C. & SHINOZUKA, M. (1987) Seismic Damage Assessment of Reinforced Concrete Members. Technical Report NCEER-87–22, National Center for Earthquake Engineering Research, State University of New York at Buffalo. CHUNG, Y.S., MEYER, C. & SHINOZUKA, M. (1988a) Automated Seismic Design of Reinforced Concrete Buildings. Technical Report NCEER-88–24, National Center for Earthquake Engineering Research, State University of New York at Buffalo. CHUNG, Y.S., MEYER, C. & SHINOZUKA, M. (1988b) SARCF—Seismic Analysis of Reinforced Concrete Frames. Technical Report NCEER-88–44, National Center for Earthquake Engineering Research, State University of New York at Buffalo. CHUNG, Y.S., MEYER, C. & SHINOZUKA, M. (1989) Modeling of concrete damage. ACI Structural J. 86(3), 259–71. CHUNG, Y.S., MEYER, C. & SHINOZUKA, M. (1990) Automatic seismic design of RC building frames. ACI Structural J. 87(3), 326–40. CLOUGH, R.W. & GIDWANI, J. (1976) Reinforced Concrete Frame 2: Seismic Testing and Analytical Correlation. Earthquake Engineering Research Center, Report No. EERC 76–15, University of California, Berkeley. CLOUGH, R.W. & PENZIEN, J. (1975) Dynamics of Structures. McGraw-Hill, New York. CLOUGH, R.W. BENUSKA, K.L. & WILSON, E.L. (1965) Inelastic earthquake response of tall buildings. Third World Conference of Earthquake Engineering, New Zealand. HEALEY, T.J. & SOZEN, M.A. (1978) Experimental Study of the Dynamic Response of a Ten-Story Reinforced Concrete Frame with a Tall First Story. Structural Research Series No. 450, University of Illinois, Urbana. HWANG, T.H. (1982) Effects of Variation of Load History on Cyclic Response of Concrete Flexural Members, Ph.D. Thesis, Dept. of Civil Engineering, University of Illinois, Urbana. KESHAVARZIAN, M. & SCHNOBRICH, W.C. (1984) Computed Seismic Response of R/C Wall-Frame Structures. Structural Research Series No. 515, University of Illinois, Urbana. KUTT, T.V. & BIENIEK, M.P. (1988) Cumulative damage and fatigue life prediction. AIAA Journal, 26, 213–19. MA, S.M., BERTERO, V.V. & POPOV, E.P. (1976) Experimental and Analytical Studies on the Hysteretic Behavior of Reinforced Concrete Rectangular and T-Beams. Earthquake Engineering Research Center, Report No. EERC 76–2, University of California, Berkeley.


MEYER, C. (1984) Earthquake analysis of structural walls. In Proc. of the International Conference on Computer-Aided Analysis and Design of Concrete Structures, Vol. 2, ed., F.Damjanic et al. Pineridge Press, Swansea. MEYER, C. & OKAMURA, H. (eds) (1986) Finite element analysis of reinforced concrete structures. Proceedings of U.S.-Japan Seminar, Tokyo, ASCE Special Publication, ASCE, New York. MEYER, C. (1991) Computation of inelastic response. In Inelastic Design Procedures for Reinforced Concrete Structures Subjected to Earthquakes. ACI Special Publication. MOEHLE, J.P. & MAHIN, S.A. (1991) Observations on the behavior of reinforced concrete buildings during earthquakes. In Inelastic Design Procedures for Reinforced Concrete Structures Subjected to Earthquakes. ACI Special Publication. NEWMARK, N.M. & ROSENBLUETH, E. (1971) Fundamentals of Earthquake Engineering. Prentice-Hall, Englewood Cliffs, New Jersey. OESTERLE, R.G. et al. (1976) Earthquake Resistant Structural Walls—Tests of Isolated Walls. Portland Cement Association, Skokie, Illinois. OESTERLE, R.G. et al. (1979) Earthquake Resistant Structural Walls—Tests of Isolated Walls, Phase II. Portland Cement Association,Skokie, Illinois. OTANI, S. & SOZEN, M.A. (1972) Behavior of Multistory Reinforced Concrete Frames During Earthquakes. Structural Research Series No. 392, University of Illinois, Urbana. OZCEBE, G. & SAATCIOGLU, M. (1987) Confinement of concrete columns for seismic loading. ACI Structural J., 84(4), 308–15. PARK, R. (1986) Ductile design approach for reinforced concrete frames. Earthquake Spectra, EERI, 2(3), 565–619. PARK, R. & PAULAY, T. (1975) Reinforced Concrete Structures. Wiley, New York. REITHERMAN, R. (1985) A review of earthquake damage estimation methods. Earthquake Spectra, EERI, 1(4). ROUFAIEL, M.S.L. & MEYER, C. (1987) Analytical modeling of hysteretic behavior of R/C frames. J. Structural Eng. ASCE, 113(3). SCRIBNER, C.F. & WIGHT, J.K. (1978) Delaying Shear Strength Decay in Reinforced Concrete Flexural Members Under Large Load Reversals. Report No. UMEE 78R2, Department of Civil Engineering, University of Michigan, Ann Arbor. SHINOZUKA, M. & TAN, R.Y. (1983) Seismic Reliability of Damaged Reinforced Concrete Beams. J. Structural Eng., ASCE, 109(7), 1617–34. SIMIU, E. & SCANLAN, R.H. (1978) Wind Effects on Structures. Wiley, New York. TAKEDA, T., SOZEN, M.A. & NIELSEN, N.N. (1970) Reinforced concrete response to simulated earthquakes. J. Structural Div. ASCE, 96(ST12), 2557–73. WAKABAYASHI, M. (1986) Design of Earthquake Resistant Buildings. McGraw-Hill, New York. WlGHT, J.K., (ed.) (1985) Earthquake Effects on Reinforced Concrete Structures, U.S.Japan Research. ACI Special Publication SP-84.

Chapter 9 UNSTIFFENED STEEL PLATE SHEAR WALLS G.L.KULAK Department of Civil Engineering, University of Alberta, Canada

SUMMARY A number of buildings have been constructed in recent years using steel plate shear walls, most notably in the United States and Japan. Design practice has been to calculate the capacity of the shear wall either on the basis of attainment of shear yield or using the stress that will produce shear buckling of the plate. Any additional strength that might be present after the web has buckled is neglected. The consequence of the conservative approach used is that either relatively thick plates must be used or the plates must be heavily stiffened. An obvious analogy exists between a steel plate shear wall core and a plate girder. This analogy has been used to develop a simple method of analysis which utilizes the post-buckling strength. The analysis indicates that very thin, unstiffened webs can meet normal strength and stiffness requirements. An extensive experimental programme has been used to verify the analytical results. This included examination of stiffness (drift), load excursions under wind loading levels, ultimate strength, and the seismic behaviour of steel plate shear wall assemblies. NOTATION Ab Ac fQu h Ic

Cross-sectional area of beam Cross-sectional area of column Ultimate frame load Panel height, storey height Second moment of area of column

UNSTIFFENED STEEL PLATE SHEAR WALLS 211

L P Py Qy QULT w w Qb w Qy σ σ ′ σ

Panel length Axial load in column Yield load of column System yield load System ultimate load Panel thickness Ultimate wall load Yield wall load Angle of inclination of tension field Deflection of column, deformation Deflection required to develop tension field Stress 9.1 INTRODUCTION

In the design of high-rise buildings, it is usual to use one of four sytems to resist lateral forces. These are moment-resisting frames, braced frames, shear walls, and so-called tubular structures. In general, shear walls have proved to be an effective and economical bracing system for buildings in the range of 15–40 storeys. Until recently, shear cores have been constructed almost exclusively of reinforced concrete, regardless of whether the main structural frame was concrete or steel. A number of steel-framed buildings have been built throughout the world using steel plate shear walls, most notably in the United States and in Japan. One of the earliest examples, however, is a structure in West Germany. This is the Bayer-Hochhaus in Leverkusen. It is a 32-storey building and the lateral load resistance is provided by steel plate shear walls in the lower 18 storeys, with Kbracing used in the upper 14 floors. In general, a steel plate shear wall core will be much lighter than reinforced concrete, yielding reduced foundation costs. It is usually quicker to erect, reduces the number of different trades working at any one time, and should result in a useful increase in usable floor area. Two features have inhibited more widespread use of steel plate shear walls. One is a lack of understanding of how to design the system and the other is a lack of knowledge as to the seismic behaviour of the system. Current design practice in North America is to calculate the shear resistance on the basis of either the shear yield strength of the plate or the shear buckling capacity. For the usual dimensions involved, the latter will probably govern whether the plate is unstiffened. Resulting plate thicknesses are relatively large or, alternatively, stiffeners must be used at frequent intervals. Both of these have the effect of reducing the economic attractiveness of the system, particularly if the amount of fabrication necessary is high. The approach used by the Japanese has been strongly influenced by seismic requirements. Generally, the steel plate

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FIG. 9.1. Web plate in shear.

shear walls used contain many stiffeners, both vertical and horizontal. The goal is that shear buckling be precluded prior to attainment of shear yield in the plate. A major difference between the approach used in the United States and that taken by the Japanese has to do with the treatment of the gravity loads. The American approach, in general, is to include the gravity loads in the steel plate shear wall analysis. The Japanese approach is to assign only lateral loads to the shear wall system. An obvious analogy exists between a steel plate shear wall stack and a vertically oriented plate girder. In the shear wall stack, the building columns act as the girder flanges, the steel shear wall is the web, and the floor beams are the transverse stiffeners of the girder. Since civil engineering practice recognizes the considerable post-buckling strength that may be present in a plate girder web due to tension field action, it seemed reasonable to examine this approach for the case of steel plate shear walls. The analytical method developed on this basis to describe the strength and deformation characteristics of a steel plate shear wall will be described. The results of physical testing of large-size specimens under static loading, repetitive displacement under wind load, and quasi-seismic loading will be presented. In all cases, the steel plate was unstiffened in the panel region formed by the intersection of the columns and beams bounding the core. 9.2 BASIS OF THE ANALYTICAL METHOD Prior to buckling, a web plate subjected to pure shear is considered to behave as shown in Fig. 9.1. The shear stresses shown on the element located orthogonally with respect to the girder (Fig. 9.1(a)) can be replaced by an equivalent case, the element located at 45°, as shown in Fig. 9.1(b). The stresses on this element are the principal stresses, one tensile and one compressive. The shear force applied can be increased until the shear buckling stress of the panel is exceeded. At this point, the panel buckles (moves out-of-plane). The resisting mechanism developed in the plate is changed by the buckling action. After buckling, the compressive principal stress cannot increase any further. The tensile principal stress, however, is limited only by the yield strength of the material, and it will continue to increase in response to the load after the web


FIG. 9.2. Steel plate shear wall core.

has buckled, up to the yield stress level (Fig. 9.1(c)). Saying the same thing in another way, the buckles that form in the plate after the buckling stress has been exceeded do not inhibit the strength of the material in the direction of the diagonal tension field. This tension field action can provide significant postbuckling strength for the panel. Wagner (1931) was the first to present a theory describing the postbuckling strength which develops in thin webs subjected to shearing forces. Aluminium alloys, used almost exclusively for aircraft membranes, have a low modulus of elasticity and, hence, a low buckling strength; in practice, web buckling often occurs at loads less than the design load. This led Wagner to assume that the shear buckling resistance of the web was negligible, and his formulation considered only the contribution resulting from the diagonal tension field (‘pure diagonal tension’). Kuhn et al. (1952), expanded upon Wagner’s work and developed a theory of ‘incomplete diagonal tension’. They studied the intermediate case of webs falling between the two extremes of shear-resistant webs, that is, those webs whose strength is considered to be limited by shear buckling, and pure diagonal tension webs. Their work involved some empiricism, based on tests of aluminium alloy plate girder webs, and reverted to a trial-and-error solution if the flanges were not infinitely stiff. The method of Wagner and Kuhn is the basis for the analytical method described below. 9.3 METHOD OF ANALYSIS The basis of the proposed method of analysis for the static strength of an unstiffened steel plate shear wall has already been presented. To summar ize, the steel plate shear wall core (Fig. 9.2) consisting of the building columns, beams, and web plate in the core will be treated as a vertical plate girder. The method of analysis will take the flexural stiffness of the columns (the plate girder flanges) into account. A number of simplifying assumptions are made:

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FIG. 9.3. Strip model.

(1) The shear buckling capacity of the web will be neglected. The plate thickness of the web will generally be so small relative to the plate width and height that the load required to buckle the plate will be very low. Moreover, it can usually be expected that the plate will already be non-planar following completion of fabrication, either as a result of handling or due to welding distortions. (2) The tension field, assumed to act as shown in Fig. 9.3, will be modelled as a series of inclined strips. (3) The limit of action of a single strip will be that corresponding to the tension yield strength of the strip. This neglects any beneficial effects of strainhardening and ignores the effect of the compressive stresses acting on the strip (Fig. 9.1(c)). (4) Bending of the floor beam due to the action of the tension field is assumed to be nil for a typical interior panel. This assumption is justified on the basis that the difference in tension field intensity from floor to floor will, in general, not be large.

As the final step in modelling the typical panel, the angle of inclination of the tension field must be determined. The procedure used herein was based on that developed by Wagner (1931), namely, the application of the principle of least work. As detailed by Thorburn et al. (1983) and Timler & Kulak (1983), expressions are written for the work in one panel due to the tension field and due to the associated forces in the two boundary columns and one beam. The expression for total work is then minimized by differentiating with respect to the angle of inclination, σ . The resulting expression is (9.1) where σ is the angle of inclination of the tension field L and h are the panel length and height, respectively, w is the panel thickness, Ab and Ac are the crosssectional areas of beam and column, respectively, and Ic is the second moment of area of the column. Fig. 9.4 shows the assembly of the model. The tension field is represen ted by the series of bars inclined at angle σ , each bar having an area equal to the product


FIG. 9.4. Strip model for typical storey.

of the bar width and the web plate thickness. The bar is assumed to be pinconnected to the surrounding frame and capable of transmitting only axial force. The bars are obtained by dividing the plate into a series of strips of equal width. The number of strips required for satisfactory modelling depends on the particular circumstances, but a typical 10-storey building showed that 10 strips per panel was an adequate representation (Thorburn et al., 1983). In the structure shown in Fig. 9.4, the beams are pin-connected to continuous columns. It should be noted that the angle of inclination of the tension field, σ , given by eqn (9.1), is uniquely related to the boundar conditions shown in Fig. 9.4. Other beam or column boundaries can, of course, be modelled. Using the inclined bar model, a plane frame program can be employed to analyse the response of a panel to an applied shearing force. By means of this strip representation of the plate, the distribution of forces in a given shear panel can be established. These include the axial force, shear, and moment at various locations in the boundary members and the tension forces in the web plate. The program also calculates the lateral deflection of the system. The forces imposed on the boundary members are used to check the capacities of those members. The forces in the web plate are used to establish a strength Innit for the plate and to design the connection between the plate and the frame. The deflection of the panel is checked against the drift limit, and the sum of all storey deflections and the effect of column shortening are combined to calculate the overall deflection of the entire structure. The stiffness characteristics of a given wall can be expressed in an alternative way by replacing the tension zone of the steel plate with an equivalent truss element having the same storey stiffness. (The truss diagonal would, of course, run between alternate column-beam intersections; it would not be inclined at the same angle ′ as the tension field.) A similar approach was taken earlier by Japanese researchers. Using the inclined bar model, the deflection of the steel plate shear wall system can be calculated. The deflection of the equivalent truss

216 G.L.KULAK

system is set equal to this and a virtual-work analysis of the truss is used to compute the brace area required to give it the same lateral stiffness. To calculate the area of an equivalent brace, it is still necessary to model the structure as described above and use the plane frame analysis. Therefore, there would appear to be no advantage to the equivalent truss approach. Nevertheless, it would still be advantageous to consider use of the equivalent brace system in multistorey structures in which there is frequent repetition of sizes. If geometry, column, beam and plate sizes are common within a given zone, the equivalent brace area could be calculated for a typical panel in this zone. A similar procedure would be used for other zones and then the various equivalent braces applied within their appropriate zones. The deflection of the entire assembled equivalent braced structure can then be calculated using any of the common frame analysis programs. This would result in considerably less work than an analysis of the total structure using the inclined bar model. 9.4 SOME ANALYTICAL RESULTS The extent to which a tension field forms in a given panel, the panel stiffness, and the distribution of forces within the panel are influenced by panel geometry (bay width and storey height), column stiffness, web thickness, and the angle of inclination of the tension field. This last factor, defined in eqn (9.1), is itself dependent upon the panel dimensions, column and beam areas, column stiffness, and web plate thickness. As an illustration of the forces within the tension field, the frame of Fig. 9.5 can be considered. Here the frame dimensions, beam size, and web

FIG. 9.5. Distribution of tension field forces.


plate thickness are held constant as the column moment of inertia is varied. The stiffest column, Fig. 9.5(a), results in a fairly uniform and well-developed stress field in the web. Figure 9.5(b) shows that as the column stiffness is reduced, the web is less effective in developing a tension field. The use of the very flexible column, Fig. 9.5(c), results in portions of the ‘tension’ field actually in compression. In order to produce an acceptable design, both strength and stiffness requirements must be met. (It must also be noted that the former relates to factored loads, and the latter to specified loads if a limit states design format is being used.) To achieve this, one approach is to satisfy the stiffness requirements and then to simply accept the resulting stress field in the web. The stress in the most highly loaded strip would be compared to the factored yield strength of the material to ensure that the stresses are within the acceptable range. Alternatively, the designer’s main objective could be to make optimum use of the web material; that is, stress as much of the web as possible to the permissible limit. As a final step in this approach, the resulting structure would have to be examined to see whether it satisfies the drift limitations. Although there is no general way of deciding which of these two sequences is preferable, experience does indicate that the design of cores in high-rise buildings is usually controlled by drift. As an illustrative example of building design under static loading only, Thorburn, et al. (1983) considered a 25-storey building located in Edmonton, Canada. The storey height is 3·66 m and the bay width in the core area is 9·00 m. The width of the building perpendicular to the direction of the wind is 45 m. For these conditions and for the first storey, the storey shear due to wind is 5760 kN and the factored shear load is 8640 kN. The former value is used for the deflection calculations and the latter for strength calculations. The first storey has two identical shear walls, each resisting one-half of the total applied force. The drift restriction is h/500 per storey. The yield strength of the steel was taken as 300 MPa (N/mm2). The design aids provided by Thorburn et al. show that for these panel dimensions, column stiffnesses (I/L) greater than about 500×103 mm3 give stresses through the tension field that are approximately uniform. The corresponding moment of inertia for the storey height of 3·66 m is 1850×106 mm4. Based on axial loads only, the columns in the first storey of a 25-storey office building would be expected to be in this size range. Selecting a trial column section for which Ic=225×106 mm4 and Ac=48 600 mm2, the Thorburn et al. design aids suggest that a web plate 3.1 mm thick would provide adequate strength and meet the drift limitation. To check the preliminary design, a panel with a 3·5 mm thick web plate was modelled as a series of strips and analysed. The resulting storey drift (including the effect of column shortening) was 6·80 mm and the maximum stress in the tension field due to the factored shear force was 220 MPa. Since the permissible stiffness and strength limits are 7·32 mm and 270 MPa, respectively, the panel

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design is satisfactory. It should be noted that handling considerations would probably preclude use of any plate thinner than about 4·5 mm. 9.5 RELATED DESIGN CONSIDERATIONS When designing the beams and columns that form the boundaries of a shear wall web, it must be recognized that additional forces are imposed upon these members by the action of the tension field. The column must be designed for the additional axial compression and bending which result from the vertical and horizontal components of the web forces. The beam design must now accommodate an additional axial load (the member now becomes a beam– column), and the design should include a recognition of possible instability problems if sufficient lateral support is not present. If the structure has been modelled using the inclined bar model described above and analysed using a plane frame program, all of the forces in the boundary members will be available as part of the output. At the extreme top and bottom panels in a multistorey shear wall system, the vertical components of the tension field forces which act on the beams are not balanced by equal and opposite forces in an adjacent panel. The vertical components of the tension field at the ends of a shear wall stack must therefore either be taken out of the core or resisted internally. The former can be accomplished by providing a rigid element at the extreme top and bottom of the core in order to anchor the inclined stresses in the adjacent panels. At the bottom level of a steel shear core this rigid element could be large girder, and for the top panel the resisting element could be provided in the form of a truss or a deep girder. Alternatively, the web in end panels can be proportioned in such a way that the shear stresses will not exceed the critical buckling stress for the panel. To ensure that buckling will not occur in these ‘anchor’ panels, a restriction must be imposed on the panel dimensions, such as is commonly done in the design of a plate girder. Connection forces can be obtained directly from the member forces in the strip model used for analysis. The conventional methods used for the design of highstrength bolts or welds can then be applied. Direct connection of a web plate to its beam and column boundaries can be done with either high-strength bolts in a slip-resistant connection or by using welds. In the former case, a fish plate or angles would be attached in the fabrication shop to the boundary elements and the shear wall plate bolted onto this in the field. The second arrangement also uses a fish plate shop-welded to the boundary members. In the field, the web plate is lapped over the fish plate, aligned and held in place with a few erection bolts or tack welds, and then finally welded.


9.6 EXPERIMENTAL VERIFICATION The experimental verification of the analytical method included both static loading and quasi-earthquake loading cases. The first of these also included the case of quasi-wind cyclic loading, and the second also included the ultimate load test of a specimen that had been already cycled under quasi-earthquake loading. Following a description of these tests and presentation of the test results, an analytical model for use under earthquake loading is presented. 9.6.1 Static Loading Case, Including Quasi-Wind Cyclic Loading In order to substantiate the proposed analytical method, a single largescale specimen was fabricated and tested. The major areas of interest of the testing programme were the examination of the tension field development within the web plate, the out-of-plane behaviour of the plate under service load reversals (quasi-wind cyclic loading), and the ultimate load behaviour of the system. A second test of the validity of the analytical method was obtained when the specimen used for quasi-earthquake loading was loaded finally to failure. (see Section 9.6.2). Figure 9.6 shows the specimen tested. It represents two single-storey, one-bay steel shear wall elements. The members oriented vertically in the test specimen correspond to the beams in the prototype; the horizontal members in the test specimen represent the columns. The use of a symmetric model provided a condition of infinite stiffness at the interior beam located vertically at the centre of the specimen. By having equal and opposing transverse (horizontal) components of the tensile field acting on either side of the interior beam, a situation of nil bending exists for this member. This is consistent with the design assumption for interior beams located in shear wall stacks (Thorburn et al., 1983). In the prototype structure, beam-to-column connections would probably be made using web framing angles. They would be assumed to act as a simple connection. True pin connections were used in this test specimen; the arrangement was chosen in order to provide a severe test of the effect of the beam-to-column rotations on the web plate at this corner. The sizes of the various components in the specimen and the panel geometry are shown in Fig. 9.6. The bay width was 3·75 m and the storey height was 2·50 m. The columns, oriented horizontally in Fig. 9.6, were built-up sections approximately equivalent to a Canadian section W310×129 (AISC designation: W12×87), i.e. a section approximately 310 mm deep and with a mass of 129 kg/ m. The vertically oriented members, the beams in the prototype, were also builtup sections. They were approximately equivalent to W460×144 (AISC designation: W18×97) sections. These dimensions and framing sizes represent

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FIG. 9.6. Test specimen: static loading case.

reasonable structural proportions. Although the beam size chosen is somewhat larger than would be expected in actual construction practice with respect to the column size, this cross-sectional properties were necessary in order that the tension field in the web plate of this test specimen have a satisfactory boundary. Because the purpose of the experimental programme was to test a thin-webbed unstiffened steel shear wall, the thinnest hot-rolled plate readily available was used. A 5-mm web plate was selected. It was connected to the adjacent framing members by means of the fish plate arrangement shown in Section A–A of Fig. 9.6. The web plate was aligned such that its plane coincided with the planes of the webs of the beams and columns. All steel except that used for fittings was to have a yield strength of 300 MPa. (The measured static yield strength of the plate used for the web was 271 MPa.) The fabricated structure was tested as a simply supported deep beam. Because of the symmetry, two shear panels were thereby tested. The overall test set-up is shown in Fig. 9.7. The loading was applied vertically at mid-span and the reactions to the floor were transferred at the ends. A pin-connected clevis delivered either tensile or compressive loads from the crosshead of the 6200-kN capacity testing machine to the top of the interior beam. (The terms ‘tensile’ or ‘compressive’ are used here only to describe the sense of the load delivered by the testing machine. In so far as the behaviour of the specimen is concerned, these loads simply correspond to the shear load on the panel delivered first in one direction and then in the other.)


The predicted angle of inclination of the tension field of the specimen was 51·0°. (Equation (9.1) cannot be applied directly. That expression was developed for the boundary conditions commonly present in multistorey buildings and as pictured in Fig. 9.4. The boundary conditions present in the test specimen, as shown in Fig. 9.6, were somewhat different.) Two loading sequences were used during the testing of this specimen; a cyclic loading to the allowable serviceability deflection limit, and a final loading excursion until failure of the structural system was reached. During the load reversals carried out within the first loading procedure, both tensile and compressive forces were applied by the testing machine. The specimen was cycled three times such that a drift limit of h/400 was reached during each excursion. This drift limit corresponded to a deflection at the centreline of the test specimen of 6·25 mm, and it was attained by application of a load of 2104 kN. Following completion of the service load excursions, the specimen was loaded to failure by application of a compressive load from the testing machine. A full description of the test results is available elsewhere (Timler & Kulak, 1983) and only a summary of selected results will be given here. Generally, satisfactory agreement between observed and predicted behaviour was obtained for strains in the columns, beams, and web plate, deflection of the system, and ultimate load. Of course, no prediction could be made for the out-of-plane deflections of the web. A plot of load versus frame deflection for the excursions corresponding to the drift limit of h/400 is shown in Fig. 9.8. Test No. 2 was a compression loading cycle, from 0 to 2145 kN. It was followed by Test No. 3, where the loading sequence was 0 to 2145 to 0 to 2063 kN. (The discrepancy between maximum load levels is accounted for by the dead load of the specimen.) The slopes of Test 2 and 3 agree well enough, and the hysteresis behaviour of Test 3 shows that the slopes between identical load excursions are essentially the same. The larger frame displacement during the tension cycles is probably the result of elongation of tie-down bolts in the reaction fixtures. For all load applications, linear elastic behaviour is observed. The very small residual deflections noted upon unloading may have been a result of yielding within the frame system, but more likely they resulted from local yielding in the various parts of the loading apparatus. The web, as fabricated, was not planar under no-load; the measured initial outof-flatness was approximately 9 mm over a length of 1275 mm. Loading of the specimen either increased or decreased the amplitude of the initial deflections, depending upon the direction of the applied load. The maximum out-of-plane deflection under service load was about 20 mm, measured with respect to a theoretical, perfectly flat plate. This meant that the increase in amplitude over the initial profile of this plate was only about 11 mm. The profile of the plate under load was nearly identical under successive load excursions and the plate returned to its no-load profile, within measurement tolerances, after each cycle. The

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FIG. 9.7. Overall test set-up.

buckles which formed at maximum load were barely discernible to an observer standing only a short distance away from the specimen. The measured value of the angle of inclination of the tension field, as obtained from strain gauge readings, was between 47° and 53° in the lower portion of the panel; the predicted value of the angle is 51·0°. A plot of load versus the lateral deflection of the frame is given in Fig. 9.9. Both the actual response and the predicted response are shown. The analytical model used corresponded exactly to that suggested by Thornburn et al. (1983) for the


FIG. 9.8. Specified load-versus-deflection.

FIG. 9.9. Predicted-versus-experimental frame deflection: Static load case.

behaviour up to the load at which the first inclined bar in the model reached yield. In this case, the predicted value of the load causing first yield is 3450 kN. As closely as can be determined from the plot of the experimental values, this was the load at which non-linearity did indeed start. The deflection predicted for

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the frame at the time of first yield of a bar is about 10% less than that measured in the test. An analytical model limited by the first yield load in one bar is obviously conservative; the frame will have considerable reserve strength beyond this level. Timler and Kulak (1983) used a model which accounted both for sequential yielding of inclined bars and for yielding of boundary members. The model, as modified, did not take strain-hardening into account, nor did it account for the presence of residual stresses in the framing members. Nevertheless, as can be seen in Fig. 9.9, the predicted curve provides an acceptable estimate of the true behaviour in the region between first yield and ultimate. The overall behaviour of the frame shows that linear behaviour extended well beyond the serviceability load limit (3450 kN vs 2063 kN). This was followed by a range of gradual softening of the frame, that is, ductile behaviour, until the ultimate load of 5395 kN was attained. The ultimate load did not occur as a result of behaviour of the principal elements, but was a connection failure. A fillet weld tear at the web plate to fish plate connection adjacent to one of the pins was followed by a local lateral instability around the pin in a region where a flange had been cut back. The eccentricity of load through the fish plate appeared to contribute to both of these effects. A standard web framing angle connection would provide much more lateral stability locally than that present around the pin in this test. 9.6.2 Quasi-Seismic Loading Case Although it is possible to develop analytical descriptions of the load-versusdeformation behaviour of structural elements or assemblages acting under simulated earthquake loading, it is usually preferable to use physical testing as the primary method of obtaining this description. Thus, the physical tests described in this section were conducted first, and then the analytical model described in Section 9.7 was developed. In order to examine the seismic characteristics of the unstiffened steel plate shear wall arrangement, another large-scale specimen was tested. The general arrangement, configuration, and test set-up were similar to those used for the specimen described earlier for the static loading case. The two-panel assembly had a bay width of 2·75 m, a storey height of 2·20 m, and used a 3·25 mm thick unstiffened steel plate web. There were two significant differences between this specimen and the one used for the static load case, however. One was that bolted connections using web framing angles connected the beams to the columns. The connection was designed as a slip-resistant joint. Thus, this connection was expected to be very stiff as compared with the pinned connections used in the earlier test. The second major difference was that, in order to simulate the erection sequence that might be expected to occur in the field, an initial preload of 0·09Py


was introduced into the columns before the web plate was inserted and fastened to the boundary members. After the web had been installed, an additional column load of 0·12Py was applied. These column preloads were obtained using prestressing bars of the type normally used in concrete structures. The bars ran from one end of the specimen to the other, over the two-storey height, and remained horizontal as the test piece deflected vertically. Testing was done in two phases. In the first phase, a slowly applied cyclic load was applied, with gradually increasing displacements. This provided information about the response of the system to earthquake loading. The second phase involved a monotonic load applied so as to attain the ultimate static capacity of the specimen, including, of course, any deleterious effect introduced by the previous quasi-earthquake loading. A photograph of the test specimen after testing is shown in Fig. 9.10. Full details of the test have been reported by Tromposch & Kulak (1987). Figure 9.11 shows the complete history of the load versus displacement response for the 28 cycles of load that were applied. At the higher load levels, the response is that characteristic of any steel framing system which contains elements that buckle: the curve can be described as S-shaped or pinched. An examination of individual loops would show that, for this test arrangement, the response is essentially linear for about 13 excursions, corresponding to a storey displacement of about 1/400 times the storey height. Thereafter, the effect of the buckling of the web plate as the tension field forms and reforms in opposite directions is to produce the flatter portions of the curve in the central region. It should be noted that the amount of energy absorption continued to increase throughout the 28 cycles of load applied in this test. The application of alternating load had to be terminated because the capacity of the testing machine had been reached. Although several small tears in the web plate had occurred, these did not seem to be affecting the strength of the system and it was continuing to carry load. The central, flatter portion of the load-versus-deformation response curve is primarily the response of the frame alone. The web is in an intermediate stage as load is alternately applied. In one direction the web has buckled, but in the other direction the diagonal tension field has not yet formed. Because of the particular arrangement tested, in this case this slope represents (approximately) the behaviour of a single-storey frame, fixed at its bottom end and with pinned beamto-column connections at the top. In any multistorey building, the situation could be significantly more favourable than this. The columns would be continuous and the beam-to-column connections would offer somewhat more restraint. Thus, the response curve in this region could be substantially stiffer than that demonstrated in this test. The response curve and energy absorption capability of this thin, unstiffened steel plate shear wall under quasi-seismic loading can be compared with other structural systems for resisting lateral load. Both reinforced-concrete shearwalls and K- or X-braced steel frames also exhibit the characteristic S-shaped

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FIG. 9.10. Test specimen: quasi-earthquake case.

FIG. 9.11. Seismic response of thin, unstiffened steel plate shear wall.

hysteresis loops. Of course, steps can be taken to improve the situation if a larger absorption of energy and a stiffer structure are considered essential. The eccentrically braced steel frame is one example of this type of improvement (Popov, 1980). It has also been demonstrated that steel plate shear walls which


have intermediate stiffeners can produce a spindle-shaped hysteresis loop (Takahashi et al., 1973). Following completion of the alternating load portion of the test of this shearwall, it was loaded monotonically in order that another test result for the ultimate strength of an unstiffened shearwall might thereby be generated. It was also important to establish whether or not the frame would be able to attain its theoretical load following the simulated earthquake loading. The boundary conditions for this specimen were not well-defined (slipresistant connections were used, not real pins as had been used in the specimen for the static test), and both welding residual stresses and stresses due to column preloading were present in the web plate. The initial stress in the web plate (the net of welding tensile residual stress and compressive stress due to column preload) was estimated to be about 50 MPa (Tromposch & Kulak, 1987). This is about 20% of the measured yield strength of the web plate material. When the shearwall was being loaded in the direction described as ‘compression’, an analysis that assumed that the beam-to-column joints were fixed gave an excellent prediction of the actual monotonic load-response curve. When the load was applied in the other direction (the ‘tension’ case), the best representation was obtained by assuming that the joints were fixed until their theoretical slip load was reached, after which it was assumed that they were pinned. However, in both cases, a model that assumes pinned connections provides a satisfactory estimate of the actual response and is conservative. Figure 9.12 is a plot of the monotonic load-versus-deflection response. The initial portion was obtained by plotting the peak compressive loads obtained during the cyclic loading test. The column axial loads had to be removed prior to failure of the specimen and there appears to be a slight increase in strength at the end of the cyclic loading and the commencement of the continuation of the load to failure. The relatively small increase in strength seems to indicate that the effect of the column load acting through the member displacement (P−σ ) had only a minor influence on the structural behaviour during the cyclic loading phase. In the monotonic loading phase, the specimen was loaded to the capacity of the testing machine, sightly over 6000 kN. A maximum midspan deflection in excess of 70 mm was reached. Tears in the welds and fish-plates that had developed during the cyclic tests lengthened only slightly under the monotonic loading. New tears in the welds and fish-plates occurred in the top outside corners of the test specimen. These corner fractures appear to have been caused by the very severe buckles that took place in the corners as a result of joint slip and joint rotation. It is clear that the ultimate strength of this specimen was unaffected by its previous quasi-earthquake loading.

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FIG. 9.12. Monotonic load-versus-deflection response.

9.7 ANALYTICAL TREATMENT OF EARTHQUAKE LOADING 9.7.1 General Requirements The hysteresis curve, developed experimentally by applying a quasi-static cyclic load in alternate directions, provides the member or frame behaviour information needed when conducting a dynamic analysis and also provides a general idea of the suitability of the system for a structure that must be designed to resist earthquakes. The area enclosed by the hysteresis loop is equal to the energy absorbed by the system. Systems that absorb a large amount of energy and exhibit sound and stable hysteresis loops have generally performed well in earthquakes. Structural systems that perform well in seismic events generally also exhibit good ductility. A measure of the ductility of a system is defined as the ductility factor. This factor can be defined as the maximum strain, rotation, or displacement divided by the yield strain, rotation, or displacement, respectively (Popov, 1980). The amount of ductility required for a given structure is difficult to determine. For structures subjected to strong siesmic activity, Popov (1980) has suggested that a displacement ductility factor in the order of 6 may be necessary. Good ductility, however, cannot compensate for poor energy absorption; it merely changes the failure mode.


FIG. 9.13. Hysteresis behaviour of a moment-resisting frame (after Popov, 1980).

9.7.2 Cyclic Loading Behaviour of Common Structural Systems When examining the cyclic loading response of a steel plate shear wall, a comparison with other commonly used systems will be useful. These too must all be compared with what is considered desirable behaviour. For frames, three types of structural systems are commonly considered for lateral load resistance. These are a moment-resisting frame, a simply supported frame (continuous columns but simple beam-to-column connections) containing a core in which steel bracing (usually K- or X-bracing) is present, or a simply supported steel frame containing a vertical shear wall. Moment-resisting frames generally exhibit good hysteresis behaviour. Figure 9.13 is the hysteresis curve developed by a structural assemblage comprised of two half-columns and two beams (Popov, 1980). The load-versus-displacement loops are sound and fully developed, showing excellent ductility. A displacement ductility factor in excess of 10 is achieved. Because the columns are generally carrying large axial loads, the effect of the axial load acting through the displaced distance (P−σ ) causes the stiffness to decrease at large deflections. It is generally the contribution of the P–σ effects that causes the failure of moment-resisting frames subjected to cyclic loading. The second type of lateral load-resisting system commonly used in diagonal bracing. It comes in many forms, but the two most typical configurations are Kand X-bracing. Both of these systems exhibit a degenerating pinched-loop behaviour when subjected to cyclic loads. Figure 9.14 is the hysteresis curve for the particular X-braced frame illustrated in the upper left corner of the figure (Popov, 1980). The pinched loops are the result of buckling of the yielded members before they can be recompressed. A recent innovation that improves the hysteresis performance of braced frames is to attach the diagonal brace to the beam a short distance away from the beam-to-column connection. The

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FIG. 9.14. Hysteresis behaviour of a diagonally braced frame (after Popov, 1980).

FIG. 9.15. Hysteresis behaviour of a reinforced concrete shear wall (after Oesterle et al., 1978).

eccentrically braced frame is designed so that the short link beam yields prior to the yielding or the buckling of the diagonal members (Popov, 1980). The hysteresis curves produced by cyclically loading a eccentrically braced frame are fully developed and stable, as would be expected since only the short link beams are yielding. Shear walls are the third type of commonly used lateral load-resisting system. Until recently, shear walls were exclusively made of reinforced concrete. The hysteresis performance of reinforced-concrete shear walls varies greatly, depending on the configuration and the details used. Figure 9.15 illustrates the shear force versus shear distortion hysteresis loops for a simple reinforced concrete shear wall (Oesterle et al., 1978). The pinched loops in this case are caused by the yielding of the reinforcement. As load is applied in one direction, a permanent deformation remains in the reinforcement after unloading. When the load is then applied in the other direction, only the reinforcement is effective in resisting the applied moment prior to crack closure. This in turn results in a reduced stiffness prior to crack closure and causes the pinched loops. In this case, the shear distortion of 0·01 radians corresponds to a lateral deflection of about 20 mm. The rotational ductility factor in this case is in excess of 10. For shear walls, the rotational ductility factor is nearly equivalent to the


FIG. 9.16. Hysteresis behaviour of an unstiffened steel plate shear wall panel (after Takahashi et al., 1973).

displacement ductility factor. The hysteresis performance of concrete shear walls can be improved by linking slender shear walls with heavily reinforced coupling beams which act as the main energy-absorbing units in the structure (Park and Paulay, 1975). 9.7.3 Previous Research In Japan, the stiffened steel plate shear wall has been used exclusively because of its superior hysteresis performance when compared to unstiffened panels. The high seismic risk in Japan makes this a key factor in design. The first steel plate shear wall structures were designed on the basis that stresses in the panels are limited to the elastic range and that buckling of the panels does not occur. The shear walls were assumed to carry only lateral loads and no vertical loads. The lateral resistance was assumed to derive totally from the shear resistance of the panels. The structures were analysed using a form of the Wagner model, with the stiffness characteristics then equated to an equivalent pair of diagonal braces (Canadian Institute of Steel Construction, 1980). It is unclear whether the resulting analytical approximation was ever verified by physical tests. Extensive research by Takahashi et al., (1973) on the hysteresis properties of stiffened steel plate panels demonstrated the large ductility available and the superior hysteresis properties of stiffened shear wall panels as compared to unstiffened panels. Figure 9.16 shows the hyster esis shear stress-versus-shear strain curve for an unstiffened steel shear wall panel 2·10 m by 0·90 m by 2·3 mm thick, mounted on very stiff boundary members, and with idealized pinned connections. This arrangement produces hysteresis loop information which reflects the lateral strength of the panel only if it is assumed that the very stiff boundary members did not yield. Note that the web thickness used (2·3 mm) constitutes a very thin web panel. Figure 9.17 shows the hysteresis curve for a

232 G.L.KULAK

FIG. 9.17. Hysteresis behaviour of a heavily stiffened steel plate shear wall panel (after Takahashi et al., 1973).

heavily stiffened panel of the same geometrical configuration as that illustrated in Figure 9.16. Note that the heavily stiffened panel displays a superior hysteresis curve, enclosing considerably more area and therefore absorbing more energy. The ductility of these specimens was very large. The maximum shear deformation was of the order of 0·1 radians for some of the specimens tested. The principal recommendations of these researchers were that steel plate shear wall panels be designed so that elastic buckling does not occur, and so that when inelastic buckling occurs it does not extend across the entire panel. If these design recommendations are followed, the resulting shear wall panel will display sound, stable hysteresis loops. Mimura & Akiyana (1980) followed this work by developing general expressions for predicting the monotonic and the hysteresis behaviour of steel plate shear wall panels. They assumed in their derivation that the steel panels developed a tension field to resist the applied loads. The assumed inclination of the tension field was that developed by Wagner (1931). The monotonic loadversus-deformation curve they developed is an elastic–plastic model that superimposes the frame and the plate stiffnesses. Figure 9.18 illustrates the concept. The notation fQu, wQb, wQy, Qy, and QULT represents the ultimate frame load, the ultimate wall load, the wall yield load, the idealized system yield load, and the idealized system ultimate load, respectively. Mimura and Akiyana then developed a theoretical hysteresis curve for a steel plate shear wall panel in the following way. If the shear buckling strength is greater than the shear yield strength (Von Mises), then the behaviour of the structure will be assumed to be similar to a conventional steel beam in bending. For cases where the shear buckling load is less than the shear yield load, the theoretical hysteresis curve shown in Fig. 9.19 was proposed. Line O–A–H represents the monotonic load-versus-deflection curve for the panel. If the panel loading is taken to a load such as point B and then unloaded, the unloading path will be parallel to the elastic curve to point C′. Applying load in the negative sense, the


FIG. 9.18. Load-versus-deflection behaviour.

FIG. 9.19. Theoretical hysteresis curve proposed by Mimura & Akiyana (1980).

deflections would continue to be parallel to the elastic curve until shear buckling of the panel occurs at a point C. The deflection C–D is that required to develop the tension field. This distance is approximately one-half the distance O– C′. From point D, the curve follows a linear transition back to the point of negative yield at A′. The curve then repeats itself, with yielding to point E, unloading to point F, and redevelopment of the tension field at point G. The distance F′–G′, which is equal to F–G, is taken as the average of the distances O– F′ and O–D′. From point G, a linear transition back to the point of last maximum

234 G.L.KULAK

load, B, is assumed. For further cycles, the hysteresis curves would follow the same form. The reasons for assuming that it would require a deflection of one-half the permanent plastic deformation to redevelop the tension field are based on the assumptions of a tension field angle of 45°, Poisson’s ratio effectively equal to 0·5, and an initially unbuckled panel. If a panel is displaced a distance σ and the material reaches yield, a plastic tensile principal stress σ will develop in one direction and a perpendicular compressive stress of magnitude 0·5σ will develop in the other (Mimura & Akiyana, 1980). Thus, a deformation of 0·5σ , the result of the difference between σ and 0·5σ , can be considered to correspond to the buckle deformation. When the load is applied in the opposite sense, a deformation of 0·5σ beyond the zero-load permanent deformation, C′, will result in the buckles cancelling each other and a tension field developing in the opposite sense. Mimura & Akiyana (1980) conducted a series of four tests on plate girders in order to verify their proposed analytical model. The panel width ranged from 549 mm to 599 mm, the panel height ranged from 264 mm to 599 mm and the web thickness was 1·0 mm for three of the specimens and 1·6 mm for the fourth. Other details, including the material properties and stiffnesses of the panel boundaries have been summarized by Tromposch & Kulak (1987). Test behaviour correlated reasonably well with the predicted behaviour. However, the number of loading cycles was small in each of the tests and therefore the stability of the hysteresis loop was not established in these tests. 9.7.4 Analytical Model for Cyclic Loading The analytical model proposed by Mimura & Akiyana (1980) was used as the basis for predicting the test results of Tromposch & Kulak (1987). However, expanding the theoretical hysteresis curve developed by Mimura and Akiyana by several cycles shows that the theoretical curve does not degenerate at a rate similar to that observed in the test. The deflection theoretically required to redevelop the tension field, CD in Fig. 9.19, is much less than the deflection measured in the test. Furthermore, this portion of the theoretical hysteresis curve has a zero slope. This results from the assumption that the boundary members provide no lateral resistance, an assumption consistent with the Japanese tests. However, in the Tromposch & Kulak test (and in any real structure) the shear wall frame did contribute to the lateral stiffness of the shear wall and the observed slope in this region of the hysteresis curve was not zero. This lateral stiffness, however, would still be small when compared to that provided by the tension field of the panel. The influence of the frame stiffness on the overall stiffness of the structure can be seen in Fig. 9.20, the final hysteresis cycle of the test frame. The portion of the curve covering the displacement required to redevelop the tension field has a slope of approximately 93 kN/mm. Also plotted in Fig. 9.20 is the stiffness (70·6


FIG. 9.20. Hysteresis cycle 28.

kN/mm) of two pin-ended columns alone (assuming the steel panel is absent). Because by the final cycle the beam-to-column connections had slipped several times, they can be assumed to be pinned. The slopes of the two lines are reasonably similar, with the difference attributable to the uncertainty in the restraint provided by the bolted connections. In the early loading cycles, these bolted joints were behaving more like fixed connections and the slope of the curve in the redevelopment phase was greater in the earlier cycles than in the final cycles. The theoretical hysteresis curve development proposed by Mimura & Akiyana (1980) can be modified to incorporate the effects of the frame stiffness and the effects of the low panel buckling strength. Four assum ptions are made: (1) The stiffness of the structure during the tension field redevelopment phase can be represented by the elastic stiffness of the framing members alone. (2) When a sufficient number of plastic hinges are produced in the boundary members to form a mechanism, the slope during the tension field redevelopment phase will be zero. (3) Given that the buckling load for a thin unstiffened panel is very low, then the deflection necessary to produce buckling is very small and can be neglected. (4) The deflection required to redevelop the tension field is based solely on the amount of yielding experienced by the panels in each direction. Figure 9.21 illustrates the development of a theoretical hysteresis curve for an unstiffened steel plate shear wall using these assumptions. When load is applied to the structure, the response will follow the monotonic load-versus-deflection curve past first yield to an arbitrary maximum load at point B. When the structure is unloaded from this point, the load-versus-deflection curve will

236 G.L.KULAK

FIG. 9.21. Proposed theoretical hysteresis curve.

unload elastically to point C. Applying load in the other sense, the panel must develop a tension field in the other direction. The deflection required for the tension field to develop is a distance CD′ (derivation of this distance to follow). During this rebuckling deflection, the stiffness of the structure is equal to the elastic stiffness of the frame only. Once the tension field has reformed (point D), a linear transition is assumed to apply up to the point of negative yield, E. The formation of the hysteresis loops is then a continuation of the previous processes; yielding to point F, unloading to point G, reformation of the tension field at point H, and a linear transition back to the point of the previous maximum tension, point B. The deflection required to redevelop the tension field in any given cycle (distances CD′ and GH′ in the example of Fig. 9.21) can be derived using the same assumptions made by Mimura & Akiyana, namely, that Poisson’s ratio is effectively equal to 0·5 when yielding occurs and that the angle of inclination of the tension field is equal to 45°. (The actual angle of inclination is generally very close to 45°.) This deflection is given by the following expression: ∆r=∆1+∆2−0·5∆2 (9.2a) or ∆r=∆1+0·5∆2 (9.2b) where ′ r is the deflection required to redevelop tension field, ′ 1 is the yielded deflection from previous cycle in the direction of loading, and ′ 2 is the yielded deflection just completed in the opposite loading direction from that under consideration. As shown in eqn (9.2a), the deflection required to redevelop the tension field is made up of two deflection components, ′ 1 and ′ 2, but is also influenced by the Poisson effect, in this case 0·5′ 2. The statement is then simplified as given by eqn (9.2b). For the first hysteresis cycle in Fig. 9.21, the deflections required to redevelop the tension fields are given by CD′= 0·5OC (9.3) GH′=OC+0·50G (9.4)


FIG. 9.22. Theoretical and actual hysteresis curves for cycle 16.

Note that in eqn (9.3) there is no ′ 1 term: for the first loading cycle the yielded deflection from the previous cycle is equal to zero. Using the relationships developed above, theoretical hysteresis curves were developed for two of the observed loading cycles obtained for the test panel. The theoretical hysteresis curves were developed using the above relationships, the load-versus-deflection curves generated by the plane frame inclined bar model, and the peak cycle deflections observed in the test. Figure 9.22 provides a comparison between the theoretical hysteresis curve and the actual hysteresis curve for cycle 16. The two curves are reasonably similar, but the theoretical curve encloses about 36% less area than does the actual curve. This is to be expected, since the joint restraint of the actual frame is greater than the assumed restraint (pinned joints) used in the analytical model. Thus, the predicted behaviour is less stiff than the actual behaviour during the tension field redevelopment phase. Figure 9.23 compares the theoretical and the observed curves for the ast hysteresis cycle in this test, cycle 28. Again, the two curves are reasonably similar. The slopes of the curves in the tension field redevelopment region correlate better than they did for cycle 16, with the theoretical hysteresis curve now enclosing about 20% more area than the actual hysteresis curve. The discrepancy is due in part to the differences in the actual monotonic behaviour and the predicted monotonic behaviour. Because the relationships developed above result in a reasonable prediction of the experimentally obtained hysteresis loops, it is practical to develop predictions of the hysteresis behaviour of other shear walls with different configurations,

238 G.L.KULAK

FIG. 9.23. Theoretical and actual hysteresis curves for cycle 28.

member section properties, or different connection characteristics. Three such examples follow. Figure 9.24 is a hysteresis loop developed for a structure identical to the specimen tested by Tromposch & Kulak (1987) except that the web thickness is increased from 3·25 mm to 5 mm. (A 5-mm thick panel is probably the practical minimum web plate thickness, as governed by handling considerations.) All material properties and member section properties were similar to those used in the analysis of the test specimen. The beam-to-column joints and the support locations were all assumed to be pinned throughout the load history. Using the new panel thickness, the angle of inclination of the tension field was calculated to be 43·7°. The hysteresis loop was developed based on the deflections taken in cycle 28 of the conducted test and the theoretical load-versus-deflection curve developed by the plane frame model. As can be seen from Fig. 9.24, the increase in panel thickness increases the ultimate strength of the structure but does not change the basic shape of the hysteresis loop. The stiffness of the frame, 134·2 kN/mm, was comparable to the frame stiffness of the test specimen. As a consequence, the amount of pinching and the enclosed area are similar in the two cases. Figure 9.25 shows the theoretical hysteresis loop for a structure identical to the previous case (Fig. 9.24) except that the beam-to-column joints are fixed throughout the load history. The change in the beam-to-column connection results in a stiffer and stronger structure, with a considerable increase of area within the hysteresis loop (150%). This is mainly due to the near doubling of the frame stiffness (from 134·2 to 265·2 kN/mm). The horizontal portions of the loop, AB and CD, reflect the formation of a mechanism in the frame. This


FIG. 9.24. Theoretical hysteresis loop for a 5-mm thick panel and pinned joints.

FIG. 9.25. Theoretical hysteresis loop for a 5-mm thick panel and fixed joints.

mechanism will result in an increase in the amount of pinching in the hysteresis loops at higher load levels. Finally, the hysteresis behaviour of a panel in a configuration typical of a multistorey building is examined. Figure 9.26 illustrates the plane frame model used to evaluate the hypothetical shear wall structure. The column and beam sections used are the same as used in all previous cases. The panel thickness is set equal to 3·25 mm with a yield strength of 242·5 MPa. The total vertical load

240 G.L.KULAK

FIG. 9.26. Example three-storey structure.

applied to the columns, 350 kN per floor, is approximately equal to the column load applied to the test specimen. The applied lateral loads are of equal magnitude for the first and second floors and the lateral load applied to the top floor is one-half the magnitude of the load applied to the lower floors. At any level, two shear walls are assumed to resist the total lateral force Q. All beam-tocolumn joints are assumed to be pinned joints but all member nodes along line AB (Fig. 9.26) are assumed to be fixed joints. (The lower storey of a typical shear wall would normally consist of a very stiff anchor panel. This is provided so that the tension fields in the panels above can develop fully.) The panels were divided into only seven tension members in order to reduce the computations involved. The hysteresis loop developed for the lower storey of this structure, using the maximum deflections recorded during cycle 28 of the reported test, is shown in Fig. 9.27. Note that the loop is severely pinched, with a small enclosed area of about 60% of the area enclosed by actual hysteresis loop (Fig. 9.23). This is due to the very low frame stiffness (32·11 kN/mm). An identical structure, except for the use of rigid beam-to-column connections, was analysed and the resulting theoretical hysteresis loop produced is shown in Fig. 9.28. The area enclosed under this curve is 3·3 times greater than the area under the previous hysteresis


FIG. 9.27. Theoretical hysteresis loop for the bottom storey of the three-storey example structure—pinned joints.

FIG. 9.28. Theoretical hysteresis loop for the bottom storey of the three-storey example structure—fixed joints.

loop, Fig. 9.27. This is due to the large increase in the frame stiffness achieved by fixing the beam-to-column connections. Initially, the concept of using fixed beam-to-column connections may seem undesirable; however, fixing only two connections per shear wall stack per floor results in an increase of 230% in the amount of absorbed energy for this arrangement of beams and columns.

242 G.L.KULAK

9.7.5 Comparison of Hysteresis Behaviours When examining the hysteresis behaviour of any structural system, it should be compared with what is considered ideal behaviour. Ideal hysteresis behaviour is one that maximizes the enclosed area while minimizing the final deflection. Ideal hysteresis behaviour also displays loops that are stable and nondegenerating, or, more simply, these hysteresis loops continue to follow approximately the same load-versus-deflection path for each subsequent cycle. The hysteresis behaviour of a moment-resisting frame (Fig. 9.13) is very close to the ideal behaviour described. The amount of area enclosed by the hysteresis loops is very large and the loops are very stable, following the same path for each of the higher loading cycles. This type of hysteresis behaviour is also displayed by the eccentrically braced frame and by the heavily stiffened steel plate shear wall. The hysteresis behaviour of a steel plate shear wall is dependent on the resistance of the panels to buckling. If the panels buckle before yielding occurs, some pinching of the loops will occur. Pinched hysteresis loops are the typical behaviour of structures that have members that buckle when subjected to compressive loads. Conventional steel cross bracing (Fig. 9.14) displays this type of behaviour; the members yielded in tension cannot be recompressed without buckling. This structural system must deflect by a larger amount to absorb the same amount of energy (enclosed area) as a system which displays the same monotonic load-versus-deflection response but has fully developed hysteresis loops. Pinched hysteresis loops are also a characteristic of reinforced-concrete shear walls. The amount of pinching and the stability of the loops is dependent on the structural details and reinforcement details used. The pinched loops are not due to buckling of members but are the result of the change in stiffness due to the cracking of the concrete and the yielding of the reinforcement. The observed hysteresis behaviour of the unstiffened steel plate shear wall can be compared with the structural systems described above. It is apparent that the behaviour of an unstiffened steel plate shear wall panel is unlikely to achieve the ideal behaviour shown by the eccentrically braced frame or by the momentresisting frame. However, the behaviour is similar to that displayed by conventional steel cross bracing or by conventional reinforced-concrete shear walls. 9.10 CONCLUDING REMARKS An obvious analogy exists between a steel plate shear wall core and a plate girder. This analogy has been used to develop a simple method of analysis which utilizes the post-buckling strength of the system. The analysis indicates that very thin, unstiffened webs can meet the normal strength and stiffness requirements


for the core of a high-rise building. The experimental programme reported in this chapter used representative sizes and fabrication techniques to verify the model. The examination has included stiffness (drift), load excursions at wind loading levels, ultimate strength, and the seismic behaviour of steel plate wall assemblies. The steel shear wall system is not difficult to analyse and it appears to offer the potential for economies in both material and construction. It should be considered for use both in new structures and in the retrofitting of existing structures. ACKNOWLEDGEMENT The work described in this chapter is based upon the theoretical and experimental studies carried out by the author’s students, Jane Thorburn, Peter Timler and Eric Tromposch. Some of the original ideas in the development of the shear wall model are the contribution of the author’s former colleague, Dr C.J.Montgomery. The author expresses his appreciation to all of these people. REFERENCES CANADIAN INSTITUTE of STEEL CONSTRUCTION (1980) A Guide to the Structure of B-class Japanese Buildings, (English translation). Willowdale, Ontario. KUHN, P., PETERSON, J.P. & LEVIN, L.R. (1952) A Summary of Diagonal Tension. Part I—Methods of Analysis. NACA Technical Note 2662. Washington, D.C. MIMURA, H. & AKIYANA, H. (1980) Load Deflection Relationship on Earthquake Resistant Steel Shearwalls Developed Diagonal Tension Field. (English translation) Canadian Institute of Steel Construction, Willowdale, Ontario. OESTERLE, R.G., FIORATA, A.E., ARITIZABAL-OCHOA, J.D. & CORLEY, W.G. (1978) Hysteretic Response of Reinforced Concrete Structural Walls. PCA Research and Development Construction Laboratories, Skokie, Illinois. PARK, R. & PAULAY, T. (1975) Reinforced Concrete Structures. Wiley, New York. POPOV, E.P. (1980) Seismic behaviour of structural subassemblages. J. Structural Eng., ASCE, 106(7) 1451–74. TAKAHASHI, Y., TAKEDA, T., TAKEMOTO, Y. & TAKAGAI, M. (1973) Experimental Study on Thin Steel Shear Walls and Particular Steel Bracings Under Alternative Horizontal Loading. Preliminary Report IABSE Symposium on Resistance and Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads, Lisbon, pp. 185–191. THORBURN, L.J., KULAK, G.L. & MONTGOMERY, C.J. (1983) Analysis and Design of Steel Shear Wall Systems. Structural Engineering Report 107, Department of Civil Engineering, University of Alberta, Edmonton. TIMLER, P.A. & KULAK, G.L. (1983) Experimental Study of Steel Plate Shear Walls. Structural Engineering Report 114, Department of Civil Engineering, University of Alberta, Edmonton.

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TROMPOSCH, E.W. & KULAK, G.L. (1987) Cyclic and Static Behaviour of Steel Plate Shear Walls. Structural Engineering Report No. 145, Department of Civil Engineering, University of Alberta, Edmonton. WAGNER, H. (1931) Flat Sheet Metal Girders with Very Thin Metal Webs. Part I— General Theories and Assumptions. NACA Technical Memo. 604. Washington, DC.

INDEX

Across-wind oscillation, 207–9 Acrylic adhesives, 106–7 Acrylic tubing, 142 Adhesively-bonded cover plates, 103–25 bonding, 107–8 crack initiation, 108–12 crack propagation, 108–12 fatigue testing, 112–14 materials for, 106–7 Albrecht’s model, 36–7, 50–1 Amplitude, and fatigue life prediction, 64– 70 Apron girders, 98–100 mono box, 99 twin box, 99

Constant-amplitude history, 26–8 Corrosion, bridges, 196–202 Coverplated beam specimens, 105–6 Cover plates, adhesives and, 103–25 bonding, 107–8 development, 119–20 highway bridges, 120–3 materials for, 106–7 welded, 114–19 Crack closure model, and Newman, 49 Crack growth, 13–16 plate/box girders, 73–101 Crack growth analysis, of bridges, 185–7, 193–4 Crack growth retardation, 44–5 Crack initiation, 5 adhesively bonded cover plates, in, 108–12 Cracks, accounting for, 10–13 Crack tip stress, and interaction model, 41– 9 Crane fatigue cracking, 95–100 Crane girders, 95–100 apron, in unloaders, 98–100 welded runways, 95–8 Cross-beam rotation, 85 Cycle-by-cycle crack growth summation, 46 Cycle counting, 28–32 level crossing technique for, 29–30

Bending, with plate girder fatigue, 78–82 shear, 80–2 Bolt-hole cracks, 108, 111–12 Bonded cover plates, 103–25 Box girder fatigue cracking, 86–95 Box girders, monorails, 86–95 Bridges, 7, 82–6, 120–3 repair of, 177–203 Buckling, plate shear walls, 239–41 Chord saddle, 143–4 Chord thickness, tubular joints, 132 Concrete, damage to, 228–30 Concrete-slab rotation, 85 245

246 INDEX

peak, 28, 30 rainflow, 29, 31 Cycle counting—contd. range pair, 29, 30, 32 reservoir, 29 Cyclic load, 205–35 Cyclic loading analytical model, 265–73 common structural systems, 258–61 quasi-wind, 247–53 Damage modelling, R-C frames, 227–31 Damage models, 32–49 interaction, 41–9 linear, 32–5 non-linear, 35–41 See Fatigue damage models Design fracture damage accumulation, and, 50– 1 lateral loads, 231 Detailing, fatigue-resistant, 8 Diagonal bracing, 259 Discontinuities, 5, 14–18 Distortion-induced cracking, 180–1 Earthquake loading, 258–74 cyclic loading behaviour, 258–61 hysteresis behaviour comparisons, 274 model for, 265–73 Earthquakes, 218–19 Efthymiou equation, 144, 149–53 Elastic-perfectly plastic stresses, 41–2 Engle-Rudd program, 48 Environment, and fatigue life, 3–4 Erection, and fracture mechanics, 8–9 Fabrication, and fracture mechanics, 8–9 ‘fail-safe design’, 8 Fatigue assessment, 1–23 Fatigue crack growth, 13–16 see also Crack growth Fatigue cracking plate/box girders, 73–101 welded bridge details, 180–2 Fatigue damage accumulation, 25–53 design for, 50–1

models for, 32–49 research into, 51 Fatigue-damage bridge structures, 177–203 Fatigue, and fracture mechanics, 1–23 Fatigue life, 3–7 factors affecting, 3–4 geometry and, 6–7 large welded steel structures, 4–6 stress range, 6–7 Fatigue life assessment, of tubular joints, 155–60 Fatigue life prediction, variable amplitude, 64–70 Fatigue notch factor, 118 Fatigue-resistant design, 7–9 detailing, 8 erection, 8–9 fabrication, 8–9 structural systems, 7–8 Fatigue strength, adhesively-bonded cover plates, 103–25 Finite analysis, and SCF, 152–5 FM. See Fracture mechanics Fracture mechanics, 10–16 applications, 16–22 crack accounting, 10–13 crack growth, 13–16 fatigue and, 1–23 qualitative tool design, 18–20 quantitative design, 20–2 structural engineering, 9–10 structural engineering applications, 16– 22 Frames. See Reinforced-concrete frames Gauges, 140–1, 192–3 Geometric notations, for offshore structures, 130–4 Geometrical effect magnitudes, 4–6 Geometry, and fatigue life, 3, 6–7 Gerber formula, 55, 62, 64, 66–7, 70 Girders, 190–1 apron, unloaders, 98–100 bridges, 82–6 Girders—contd. cranes, 95–100 monorails, 86–95 thin-walled, 76–83

INDEX 247

Goodman formula, 55, 61–2, 64 Gravity loads, 207 ‘Gross’ deformation, 136–7 Guideway girders, 86–95 Gurney load-time histories, 39–41 Gurney model, 39–40, 50–1 Haibach model, 38–9, 50–1 Heavy-duty crane fatigue cracking, 95–100 Highway bridge cover plates, 120–3 Hot-spot stress concept, 139–42 Hysteresis behaviour, steel plate shear walls, 258–74 comparative, 274 Hysteresis rules, 223–4 I-93 Central Artery Bridge (Mass.), 188–96 crack growth analysis, 193–4 failure causes, 191–3 field measurements, 191–3 retrofitting, 194–6 I-95 Susquehanna River Bridge (Md.), 196–202 field measurements, 197–200 retrofitting, 200–2 ‘Incomplete diagonal tension’ theory, 240 Infinitely wide plate cracking, 11 Interaction models, fatigue damage, 41–9 Joints, in offshore structure, 130–4 cast steel, 132–3 complex, 132–3 composite, 132–3 local behaviour, 134–7, 139–55 K-braced load resistance, 259 Kuhn theory, 240 Lack-of-fusion cracks, 193–4 curve prediction, 194 Large welded steel structures, 4–6 Lateral load designs, 231–3 Lateral loads, 207 Level crossing counting, 29–30 Linear damage model, 32–5 applications, 35 Palmgren-Miner, 32–4

root mean cube, 35 root mean square, 35 Load-deflection curves, 210–14 Loads, varying amplitude, and fatigue, 25– 53 ‘Local’ deformation, 136–7 Low fatigue bridge strength details, 182–7 Mono box girder, 98–100 Monorail box girders, 86–95 Newman crack closure model, 49 Newmark method, 222 Non-linear damage model, 35–41 Albrecht’s model, 36–7 Gurney model, 39–40 Haibach model, 38–9 Palmgren-Miner model, 36, 38–41 Tilly-Nunn approach, 37–8 Zwerneman model, 40–1 Offshore structures, 127–75 Oil platform. See Offshore structures Palmgren-Miner linear damage model, 32– 5, 50, 57–8, 65, 228 linear damage, 32–4 non-linear damage, 36–41 Paris law, 15 Park-Paylay solution, 232 Peak counting technique, 28–32 Plastic zones, 41–5 Plate cracks, 10–13 Plate girder bridge connection fatigue, 82– 6 concrete-slab rotation, 85 crack types, 82–4 cross-beam rotation, 85–6 Plate girder fatigue cracking, 73–86 Plate girders highway bridge, 82–6 thin-walled, 76–82 Plate shear walls, 237–76 PWHT tubular joints, 166–8 post-weld heat treatment, 170–1 Qualitative design tools, 18–20

248 INDEX

Quantitative design, and fracture mechanics, 20–2 Quasi-seismic loading, 253–8 Quasi-static R-C member testing, 209–16 Quasi-wind cycling loading, 247–53 Rails. See Monorail box girders Rainflow counting, 29, 31 Range pair counting, 29, 30, 32 Rayleigh approximation method, 55, 57, 58, 64, 67 Reinforced-concrete frames, 205–35 analysis, 220–31 building components analysis, 222–7 damage modelling, 227 earthquakes and, 218–19 quasi-static tests, 209–16 subassemblies, 216–17 time history analysis, 220–1 Reservoir, 29 Residual stress, 59–64 hypothesized, 59–61 weld fatigue, 55–71 Residual stress model, 44–5 Wheeler model, and, 45–9 Retardation models, 44–5 Retardation parameter, 46 Retrofitting, 187, 194–6, 200–2 Root mean cube model, 35 Root mean square model, 35 Runway girders, 95–8 SCF finite-element analysis, 152–5 SCF parametric equation, 142–52 Service load-time history, 26–8 Shear, plate girders, 80–2 bending, and, 81–2 Short-crack behaviour, 15 Simplification, fatigue assessment, 4–7 Small weld-induced discontinuities, 5 S–N curves, 57–8, 61–2, 64 coverplates, 112–14 Palmgren-Miner model, and, 36–8 S–N methods, and tubular joints, 155–60 available curves, 158–60 Steel plate shear walls, 237–76 Straddle-type monorail box girders, 86–92

Strain gauge layout, 140–1 Strength-drop index, 225 Stress concentration factor, See SCF Stress intensity factor, 12–16 Stress-range models, 18–22 Stress superposition, 49 Strip models, 241–3 Strip yield approximations, 49 Structural engineering, and fracture mechanics, 1–23 applications of, 16–22 fatigue life, 3–7 fatigue-resistant design, 7–9 Susquehanna River Bridge, 196–202 Suspended-type monorail box girders, 92–5 Thin-walled plate girders, 76–82 crack types, 76–8 bending, 78–80 shear, 80–2 Tilly-Nunn model, 37–8, 50–1 T-joint deformation, in tubular joints, 135– 6 Truncated Rayleigh distribution, 67 Tubular joint fatigue, offshore structures, 127–75 applied load, 138–9 environment and, 171–2 failure criteria, 157–8 fatigue resistance design, 137–73 Tubular joint fatigue—contd. fatigue strength parameters, 137–8 fracture-mechanics method, 172–3 geometric notation, 130–4 hot-spot stress, 139–42 joint behaviour, 134–7, 139–55 joint types, 130–4 new data, 160–72 new data, specific joint types, 145–8 research directions in, 173–4 SCF finite-element analysis, 152–5 SCF parametric equation, 142–51 size effects, 166–71 S–N method, 155–60 stress range, 157 Twin box girders, 98–100

INDEX 249

Two-dimensional elastic-plastic finiteelement method, 49 Two-span continuous highway-bridge cover plates, 120–3 UEG UR33 equations, 144, 149–53 Unloader girders, 98–100 Unloading plastic zone, 42–3 Unstiffened steel plate shear walls, 237–76 Variable-amplitude fatigue-life prediction, 64–70 Yariable-amplitude fatigue-life prediction tables, 68–9 Variable-amplitude stress range, 65 Varying-amplitude loads, 26–53 Versilok-201, 106–7 Viaduct bents, 188 Wagner theory, 240 Web attachment, 178–9 Web deflection, 77 Web plate, and shear, 239–40 Weld fatigue, and residual stress, 55–71 Weld-induced discontinuities (small), 5 Weld-toe discontinuities, 16–18 Welded bridge fatigue cracking, 180–2 Welded crane runway girders, 95–8 Welded steel structures, 4–6 Wheeler model, 45–9 Willenborg model, and, 47–8 Willenborg model, 47–50 Wheeler model, and, 47–8 X-braced load resistance, 259 Yellow Mill Pond Bridge (Conn.), 182–7 fatigue-crack growth analysis, 185–7 retrofitting, 187 stress history, 183–5 Zwerneman model, 40–1, 50–1

Structures Subjected to Repeated Loading: Stability and strength (Stability and Strength)

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Principles of Naval Architecture: Stability and Strength

Strength of Materials and Structures

Strength of materials and structures

Strength of Materials and Structures, Fourth Edition

Strength of Ships and Ocean Structures 2008

Stability

Stability of Structures: Principles and Applications

Stability

Stability

Strength and Honor

Stability

Stability

Strength and Honor

Structural Stability And Morphogenesis

Energy stability and convection

Aircraft stability and control

Aircraft Stability and Control

Stochastic stability and control

Stochastic Stability and Control

Ship Hydrostatics and Stability

Garnet Strength

Muscle Strength

Hydrodynamic and hydromagnetic stability

Swarm Stability and Optimization

Mechanics and Strength of Materials

Garnet Strength

Brute Strength

Starting Strength

Brute Strength

Structures Subjected to Repeated Loading: Stability and strength (Stability and Strength)

Theory of Arched Structures. Strength, Stability, Vibration

Principles of Naval Architecture: Stability and Strength

Strength of Materials and Structures

Strength of materials and structures

Strength of Materials and Structures, Fourth Edition

Strength of Ships and Ocean Structures 2008

Stability

Stability of Structures: Principles and Applications

Stability

Stability

Strength and Honor

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Structural Stability And Morphogenesis

Energy stability and convection

Aircraft stability and control

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