Thin-Walled Structures - Advances and Developments

THIN-WALLED STRUCTURES ADVANCES AND DEVELOPMENTS Elsevier Science Internet Homepage http://www.elsevier.nl (Europe) h...

Author: J. Zaras | K. Kowal-Michalska | J. Rhodes

221 downloads 2176 Views 45MB Size Report

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

DOWNLOAD PDF

THIN-WALLED STRUCTURES ADVANCES AND DEVELOPMENTS

Elsevier Science Internet Homepage http://www.elsevier.nl (Europe) http://www.eisevier.com (America) http://www.elsevier.co.jp (Asia) Consult the Elsevier homepage for full catalogue information on all books, journals and electronic products and services. Elsevier Titles of Related Interest

BAI Pipelines & Risers. ISBN: 008-043 712-5

MAKELAINEN & HASSINEN Light-Weight Steel and Aluminium Structures. (ICSAS '99). ISBN: 008-043014-7

CHAN & CHUI Non-Linear Static and Cyclic Analysis of Steel frames with Semi-Rigid Connections. ISBN: 008-042998-X

OHTSUBO & SUMI Proceedings of the 14th International Ship and Offshore Structures Congress. (3 Volume Set) ISBN: 008-043602-1

CHAN & TENG ICASS '99, Advances in Steel Structures. (2 Volume Set). ISBN: 008-043015-5

OWENS Steel in Construction (CD-ROM Proceedings with Printed Abstracts Volume, 268 papers).

DUBINA SDSS '99 - Stability and Ductility of Steel Structures.

ISBN: 008-042997-1

ISBN: 008-043016-3

SHANMUGAM, LIEW & THEVENDRAN Thin-Walled Structures: Research and Development.

FRANGOPOL, COROTIS & RACKWlTZ Reliability and Optimization of Structural Systems.

ISBN: 008-043003-1

ISBN: 008-042826-6

USAMI & ITOH Stability and Ductility of Steel Structures.

FUKUMOTO Structural Stability Design.

ISBN: 008-043320-0

ISBN: 008-042263-2

GODOY Thin-Walled Structures with Structural Imperfections: Analysis and Behavior. ISBN: 008-042266-7

KELLY & ZWEBEN Comprehensive Composite Materials

VASSALOS Contemporary Ideas on Ship Stability. ISBN: 008-043652-8

WATSON Practical Ship Design. ISBN: 008-042999-8

YOUNG Wind Generated Ocean Waves.

ISBN: 008-042993-9

ISBN: 008-043317-0

KO&XU Advances in Structural Dynamics (2 Volume Set) (ASD 2000).

ZHAO & GRZEBIETA Structural Failure and Plasticity (IMPLAST2000)

ISBN: 008-043792-3

ISBN: 008-043875-X

Related Journals Free specimen copy gladly sent on request. Elsevier Science Ltd, The Boulevard, Lang]'ord Lane, Kidlington, Oxford, OX5 1GB, UK

Applied Ocean Research Advances in Engineering Software CAD Coastal Engineering Composite Structures Computers and Structures Construction and Building Materials Engineering Failure Analysis Engineering Fracture Mechanics

Engineering Structures Finite Elements in Analysis and Design International Journal of Solids and Structures Journal of Constructional Steel Research Marine Structures NDT & E International Ocean Engineering Structural Safety Thin-Walled Structures

To Contact the Publisher Elsevier Science welcomes enquiries concerning publishing proposals: books, journal special issues, conference proceedings, etc. All formats and media can be considered. Should you have a publishing proposal you wish to discuss, please contact, without obligation, the publisher responsible for Elsevier's civil and structural engineering publishing programme:

Ian Salusbury Phone: +44 1865 843425 Senior Publishing Editor Fax: +44 1865 843920 Elsevier Science Ltd E.mail: [email protected] The Boulevard, Langford Lane Kidlington, Oxford OX5 IGB, UK General enquiries, including placing orders, should be directed to Elsevier's Regional Sales Offices - please access the Elsevier homepage for full contact details (homepage details at the top of this page).

THIN-WALLED STRUCTURES ADVANCES AND DEVELOPMENTS Third International Conference on Thin-Walled Structures

Edited by

J. Zara~ Technical University of L6d~ Poland

K. Kowal-Michalska Technical University of L6d~ Poland

J. Rhodes University of Strathclyde, UK

e

2001

ELSEVIER Amsterdam

- London

- New

York-

Oxford

- Paris - Shannon

- Tokyo

ELSEVIER SCIENCE Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 IGB, UK

9 2001 Elsevier Science Ltd. All rights reserved.

This work and the individual contributions contained in it are protected under copyright by EL~'-vierScience Ltd, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Rights & Permissions Departm~t, PO Box 800, Oxford OX5 IDX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also contact Rights & Permissions directly through Elsevier's home page (http-3/www.eisevier.nl), selecting fn~t 'Customer Support', then 'General Information', then 'Permissions Query Form'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham C..ounRoad, London WIP OLP, UK; phone: (+44) 171 436 5931; fax: (+44) 171 436 3986. Other countries may have a local reprographic rights agency for payments. Derivative Works Subscribers may reproduce tables of contents for internal circulation within their institutions. Permission of the publisher is required for resale or distribution of such material outside the institution. Permission of the publisher is required for all other derivative works, including compilations and translatiom. Electronic Stmage or Usage Permission of the publisher is required to stme or use electronically any material contained in this work, including any chapter or part of a chapter. Contact the publisher at the address indicated. Except as outlined above, no part of this work may be ~:r stored in a reirieval system or trammitted in any form or by any means, electronic, mechanical, photocopying, reconting or otherwise, without prior written permission of the publisher. Address pmuissiom ~ to: Elsevier Science Rights & Pmnim'om Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assmned by the i~blisher for any injury and/or damage to p e n o u or ~ as a maUe of pnxhlcts liability, neglil~ce or otherwise, or from any use or opmmion of any methods, pmducm, instmcfiem or ideas contained in the material herein. llecmme of rapid advances in the medical ~ , in particular, bJdePesldemvedficmion of diagnmes and drag dmages sh~ be made"

First edition 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

ISBN:

0-08- 043955-1

Transferred to digital printing 2005 Printed and b o u n d by A n t o n y R o w e Ltd, E a s t b o u m e

ORGANISING COMMITTEE Chairman

M. Kr61ak,

Co-Chairmen

J. Rhodes, J. Loughlan, K.P. Chong,

Director Secretary

J. Zara~ M. Marciniak

Members

R. Gr~dzki M. Jaroniek, Z. Kotakowski M. Kote|ko K. Kowal-Michalska T. Kubiak M. Macdonald J. Swiniarski A. 2:eligowski

INTERNATIONAL SCIENTIFIC COMMITTEE Chairman N. E. Shanmugam, Singapore Members W. Abramowicz, Poland M. A. Bradford, Australia Y. K. Cheung, Hong Kong China C. K. Choi, Korea J. M. Davies, United Kingdom P. J. Dowling, United Kingdom D. Dubina, Romania M. Eisenberger, Israel R. Evans, United Kingdom Y. Fukumoto, Japan K. Ghavami, Brazil N. K. Gupta, India G. J. Hancock, Australia J. E. Harding, United Kingdom T. H6glund, Sweden N. Jones, United Kingdom S. Kitipornchai, Australia R. A. LaBoube, USA L. Librescu, USA M. Mahendran, Australia A. Manevich, Ukraine

J. Murzewski, Poland R. Narayanan, India T. Pekfz, USA W. Pietraszkiewicz, Poland K. J. R. Rasmussen, Australia J. Rondal, Belgium J. Spence, United Kingdom J. Stupnicki, Poland C. Szymczak, Poland A. S. Tooth, United Kingdom V. Tvergaard, Denmark S. Ujihashi, Japan T. Usami, Japan G. I. van den Berg, South Africa R. Vaziri, Canada A. C. Walker, United Kingdom F. Werner, Germany T. Wierzbicki, USA H. D. Wright, United Kingdom W. W. Yu, USA J. Zielnica, Poland

This Page Intentionally Left Blank

vii

PREFACE

There has been a substantial growth in knowledge in the field of Thin-Walled Structures over the past few decades. Lightweight structures are in widespread use in the Civil Engineering, Mechanical Engineering, Aeronautical, Automobile, Chemical and Offshore Engineering fields. The development of new processes, new methods of connections, new materials has gone hand-in-hand with the evolution of advanced analytical methods suitable for dealing with the increasing complexity of the design work involved in ensuring safety and confidence in the finished products. Of particular importance with regard to the analytical process is the growth in use of the finite element method. This method, about 40 years ago, was confined to rather specialist use, mainly in the aeronautical field, because of its requirements for substantial calculation capacity. The development over recent years of extremely powerful microcomputers has ensured that the application of the finite element method is now possible for problems in all fields of engineering, and a variety of finite element packages have been developed to enhance the ease of use and the availability of the method in the engineering design process. This volume contains the papers presented at the Third International Conference on Thin-Walled Structures, Cracow, Poland on June 5-7, 2001. This is the third conference in the "Thin-Walled Structures" series, and is a sequel to the first two, which were held in Glasgow in 1996 and in Singapore in 1998. There are 83 papers of which 5 are Keynote papers. They are arranged in 12 sections as follows: Keynote Papers Analysis, Design and Manufacture Bridge Structures Cold-Formed Sections Composites Dynamic Loading (Cyclic, Impact and Vibration)

Finite Element Analysis Laminate and Sandwich Structures Optimization and Sensitivity Analysis Plate Structures Shell Structures Ultimate Load Capacity

The Conference is organised by the Department of Strength of Materials and Structures of the Technical University of L6d~., Poland jointly with the Department of Mechanical Engineering, University of Strathclyde and the College of Aeronautics, Cranfield University, UK and is supported by the State Committee for Scientific Research of Poland. The Editors should like to express their appreciation of the role played by the International Journal "Thin-Walled Structures" in the propagation of research in this field and in providing the impetus for this Conference.

Jan Zara~ Katarzyna Kowal-Michalska Jim Rhodes


CONTENTS

Preface

vii

Section I: KEYNOTE PAPERS

Shear Strength of Empty and Infilled Cassettes J.M. Davies and A.S. Fragos Stability and Ductility of Thin High Strength G550 Steel Members and Connections C.A. Rogers, D. Yang and G.J. Hancock

19

Thin-Walled Structural Elements Containing Openings N.E. Shanmugam

37

Sensitivity Analysis of Thin-Walled Members, Problems and Applications C. Szymczak

53

Some Observations on the Post-Buckling Behaviour of Thin Plates and Thin-Walled Members 3. Rhodes

69

Section II" ANALYSIS, DESIGN AND MANUFACTURE

Residual Stresses in Unstiffened Plate Specimens M.R. Bambach and K.J.R. Rasmussen

87

Behaviour and Design of Valley-Fixed Steel Cladding Systems M. Mahendran and D. Mahaarachchi

95

Behaviour of Flexible Thin-Walled Steel Structures for Road and Railway Applications J. Vaslestad, B. Bednarek and L. Janusz

103

A Simplified Computation Model for Arch-Shaped Corrugated Shell Roof Fan Xuewei

109

Fabrication Accuracy and Cost of Thin-Walled Steel Plate Girders in Shop Assembling K. Yoshikawa and S. Shimizu

119

Section III: BRIDGE STRUCTURES

Web Breathing as a Fatigue Problem in Bridge Design U. Kuhlmann and H.-P. Giinther

129

Erosion of the Post-Buckled Reserve of Strength of Thin-Walled Structures due to Cumulative Damage M. ~kaloud and M. Z6rnerov6

137

Evaluation of Strength and Ductility Capacities for Steel Plates in Cyclic Shear T. Usami, P. Chusilp, A. Kasai and T. Watanabe

145

A Rational Model for the Elastic Restrained Distortional Buckling of Half-Through Girder Bridges Z. VrceO"and M.A. Bradford

153

Instability Testing of Steel Plate Girders with Folded Webs T. Yabuki, Y. Arizumi, J.M. Aribert and S. Guezouli

161

Section IV: COLD-FORMED SECTIONS

Modelling of the Behaviour of a Thin-Walled Channel Section Using Beam Finite Elements H. Deg~e

171

Codification of Imperfections for Advanced Finite Analysis of Cold-Formed Steel Members D. Dubina, V. Ungureanu and 1. Szabo

179

Innovative Cold-Formed Steel Structure for Restructuring of Existing RC or Masonry Buildings by Vertical Addition of Supplementary Storey D. Dubina, V. Ungureanu, M. Georgescu and L. Fiilb'p

187

EC.3-Annex Z Based Method for the Calibration of (a) Generalized Imperfection Factor in Case of Thin-Walled Cold-Formed Steel Members in Pure Bending M. Georgescu and D. Dubina

195

Design Aspects of Cold-Formed Portal Frames P. Frti and L. Dunai

203

Local Buckling and Effective Width of Thin-Walled Stainless Steel Members H. Kuwamura, Y. lnaba and A. lsozaki

209

Bifurcation Experiments on Locally Buckled Z-section Columns K.J.R. Rasmussen

217

Buckling Load Capacity of Stainless Steel Columns Subject to Concentric and Eccentric Loading J. Rhodes, M. Macdonald, M. Kotetko and W. McNiff

225

Experimental Behavior of Pallet Racks and Components K. S. (Siva) Sivakumaran

233

Structural Behaviour of Cold-Formed Steel Header Beams S.F. Stephens and R.A. LaBoube

241

Compression Tests of Thin-Walled Lipped Channels with Return Lips J. Yan and B. Young

249

Experimental Investigation of Stainless Steel Circular Hollow Section Columns B. Young and W. Hartono

257

Section V: COMPOSITES

A Model for Ferrocement Thin Walled Sructures D. Abruzzese

269

Effects of Manufacturing Variables on the Service Reliability of Composite Structures A.R.A. Arafath, R. Vaziri, H. Li, R.O. Foschi and A. Poursartip

277

Post-Failure Analysis of Thin-Walled Orthotropic Structural Members M. Kotetko

285

Modal Coupled Instabilities of Thin-Walled Composite Plate and Shell Structures M. Kr6lak, Z. Kotakowski and M. Kotetko

293

Induced Strain Actuation and its Application to Buckling Control in Smart Composite Structures J. Loughlan and S.P. Thompson

301

Failure Analysis of FRP Panels with a Cut-out Under Static and Cyclic Load A. Muc, P. Kcdziora, P. Krawczyk and M. Sikoh

313

Some New Applications of the Theory of Thin-Walled Bars J.B. Obrqbski

321

Buckling Behaviour of Thin-Walled Composite Columns Using Generalised Beam Theory N. Silvestre, D. Camotim, E. Batista and K. Nagahama

329

Shear Connection Between Concrete and Thin Steel Plates in Double Skin Composite Construction H.D. Wright, A. Elbadawy and R. Cairns

339

Section Vl: DYNAMIC LOADING (CYCLIC, IMPACT AND VIBRATION)

Regular and Chaotic Behaviour of Flexible Plates J. Awrejcewicz, V. A. Krysko and A. V. Krysko

349

Seismic Performance of Arc-Spot Weld Deck-to-Frame Connections W.F. Bond, C.A. Rogers and R. Tremblay

357

Dynamic Buckling and Collapse of Rectangular Plates Under Intermediate Velocity Impact S. Cui, H. Hao and H.K. Cheong

365

Structural Optimization of Thin Shells Under Dynamic Loads S.A. Falco, Luiz E. Vaz and S.M.B. Afonso

373

Vibrations of Compressed Sandwich Bars J. Hahkkowiak and F. Roman6w

381

Numerical Simulation of Damaged Steel Pier Using Hybrid Dynamic Response Analysis T. Ikeuchi and N. Nishimura

391

xii Cyclic Response of Metal-Clad Wood-Framed Shear Walls W. Pan and K.S. (Siva) Sivakumaran

399

Vibration of Imperfect Structures J. Ravinger and P. Kleiman

407

Section VII: FINITE ELEMENT ANALYSIS

The Finite Element Method for Thin-Walled Members - Basic Principles M.C.M. Bakker and T. Pek6z

417

Nonlinear Analysis of Locally Buckled I-Section Steel Beam-Columns A.S. Hasham and K.J.R. Rasmussen

427

The Finite Element Method for Thin-Walled Members - Applications A.T. Sarawit, Y. Kim, M.C.M. Bakker and T. PekOz

437

Finite Element Methods for the Analysis of Thin-Walled Tubular Sections Undergoing Plastic Rotation T. Wilkinson and G. Hancock

449

On Finite Element Mesh for Buckling Analysis of Steel Bridge Pier E. Yamaguchi, Y. Nanno, H. Nagamatsu and Y. Kubo

459

Section Vlll: LAMINATE AND SANDWICH STRUCTURES

The Elasto-Plastic Postbuckling Behaviour of Laminated Plates Subjected to Combined Loading R. Grqdzki and K. Kowal-Michalska

469

Axial Post-Buckling of Thin Orthotropic Cylindrical Shells with Foam Core X. Huang and G. Lu

477

Nonlinear Stability Problem of an Elastic-Plastic Sandwich Cylindrical Shell Under Combined Load L. Jaskuta and J. Zielnica

483

Buckling Analysis of Multilayered Angle-Ply Composite Plates H. Matsunaga

491

Optimum Design for Laminated Panel with Cutout: The Genetic Algorithm Approach Z Li and P. W. Khong

499

The Effect of Membrane-Flexural Coupling on the Compressive Stability of Anti-Synunetric Angle-Ply Laminated Plates J. Loughlan

507

Homogeneous and Sandwich Elastic and Viscoelastic Annular Plates Under Lateral Variable Loads D. Pawlus

515

xiii Local Buckling Behaviour of Sandwich Panels N. Pokharel and M. Ma;:endran

523

Section IX: OPTIMIZATION AND SENSITIVITY ANALYSIS

Optimal Design of Steel Telecommunication Towers by Interior Point Algorithms for Non-Linear Programming N.A. Cerqueira, G.S.A. Falcon, J.G.S. da Silva and F.J. da C.P. Soeiro

533

Design Optimisation of Shell Structures with Dimensional Analysis Resources M.P.R. C. Gomes

541

Vector Optimization of Stiffened Plates Subjected to Axial Compression Load Using the Canadian Norm S.A. Falco and K. Ghavami

549

Bicriteria Optimization of Sandwich Cylindrical Panels Aided by Expert System R. Kasperska and M. Ostwald

559

Shaping of Open Cross Section of the Thin-Walled Beam with Flat Web and Multiplate Flange E. Magnucka-Blandzi, R. Krupa and K. Magnucki

567

Two-Criteria Optimization of Thin-Walled Beams-Columns Under Compression and Bending A.1. Manevich and S. V. Raksha

575

Optimization of Volume for Composite Plated and Shell Structures A. Muc and W. Gurba

585

Sensitivity Analysis of Structures Made of Thin-Walled E-Profiles K. Rzeszut, W. Kqkol and A. Garstecki

593

Section X: PLATE STRUCTURES

Buckling Loads of Variable Thickness Plates A. Alexandrov and M. Eisenberger

603

Buckling and Post-Buckling Behavior of Plates on a Tensionless Elastic Foundation A.S. Holanda and P.B. Gonfalves

611

Degree of Wall Joint Work Together with Stiffening Rib in Steel Bunker M.1. Kazakevitch and D.O. Bannikov

619

Pure Distortional Buckling of Closed Cross-Section Columns K. Takahashi, H. Nakamura and K. lmamura

623

Elasto-Plastic Large Deflection of Uniformly Loaded Sector Plates G.J. Turvey and M. Salehi

631

xiv Section XI: SHELL S T R U C T U R E S

Validation of Analytical Lower Bounds for the Imperfection Sensitive Buckling of Axially Loaded Rotationally Symmetric Shells G.D. Gavrylenko and J.G.A. Croll

643

On the Analysis of Cylindrical Tubes Under Flexure F. Guarracino and M. Fraldi

653

Buckling of Aboveground Storage Tanks with Conical Roof L.A. Godoy and J. C. Mendez-Degr6

661

On the Collapse of a Reinforced Concrete Digester Tank L.A. Godoy and S. Lopez-Bobonis

669

Closed Cylindrical Shell Under Longitudinal Self-Balanced Loading V.L. Krasovsky and G. K Morozov

677

Coupled Instability of Cylindrical Shells Stiffened with Thin Ribs (Theoretical Models and Experimental Results) A.1. Manevich

683

Instability Modes of Stiffened Cylindrical Shells J. Murzewski

693

Refined Strain Energy of the Shell R.A. Walentyhski

701

Numerical and Experimental Studies on Generalised Elliptical Barrelled Shells Subjected to Hydrostatic Pressure P. Wang

709

Section XII: ULTIMATE LOAD CAPACITY

Experimental Techniques for Testing Unstiffened Plates in Compression and Bending M.R. Bambach and K.J.R. Rasmussen

719

Effects of Anchoring Tensile Stresses in Axially Loaded Plates and Sections M.R. Bambach and K. J.R. Rasmussen

729

A Probabilistic Approach to the Limit State of Centrally Loaded Thin-Walled Columns Z. Kala, J. Kala, B. Teplf4 M. ,qkaloud

739

Rotational Capacity of I-Shaped Aluminium Beams: A Numerical Study G. De Matteis , V. De Rosa and R. Landolfo

747

Author Index

757

Keyword Index

759

Section I KEYNOTE PAPERS


Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

SHEAR STRENGTH OF EMPTY AND INFILLED CASSETTES J M Davies and A S Fragos Manchester School of Engineering, University of Manchester Manchester M13 9PL, UK

ABSTRACT This paper considers the shear buckling of light gauge steel cassette sections both with and without an infilling of relatively rigid thermal insulation. In cassette construction, In-plane shear stresses usually arise as a consequence of stressed skin (diaphragm) action and, in this context, it is local buckling of the wide flange that usually governs the design. Although there are some rudimentary equations for the shear strength of plain cassettes in Part 1.3 of Eurocode 3, this is a subject that is not fully understood and which raises some fundamental questions such as: What are the boundary conditions for plate buckling - is it sufficient to consider individual plate elements or is it necessary to consider the whole section? Is it sufficient to neglect the post-buckling strength or should this be incorporated in the design equations, as it is for plate elements in compression? How can the favourable interaction with a relatively rigid, thermally-insulating infill be incorporated in the design equations? Thus, this Keynote paper addresses some fundamental questions using the title topic to provide an illustration of the principles involved.

KEYWORDS Buckling, cassettes, finite element analysis, insulation, polystyrene, post-buckling, shear, steel, testing.

INTRODUCTION Typical cassette wall construction is illustrated in Figure 1. The first author has been interested in this form of construction for many years and has written a number of papers (eg Davies 1998a, Davies 1998b, Davies 2000a) advocating this method for houses and other low-rise commercial buildings. He has also been responsible for the structural design of a number of projects incorporating this form of load-bearing wall construction. This has led to a detailed study of the design of these structural elements. The design of light gauge steel cassettes (sometimes also known as structural liner trays) subject to axial load, bending or shear, is covered in Part 1.3 of Eurocode 3 (1996) (EC3). Figure 2 shows the most general form of a cassette section. The range of validity of the design procedures in EC3 is stated to be as follows:

0.75 mm 30 mm 60 mm 300 mm

< < <
15 : C = 2 . 0

0.5

0.0 0

5

!

i

10

15

,

t

I

20

25

30

dlt Figure 3 Existing and Proposed Bearing Coefficients for Bolted Connections SCREWED CONNECTION STUDIES

Background Investigations of single overlap connections in G550 steel concentrically loaded in shear have been reported by Rogers and Hancock (1999). The steel ranged in thickness from 0.42 mm to 1.00 mm, and the type, number and placement of screws were varied.

Proposed Design Provisionsfor Screwed Connections A proposed method to accommodate for the change in bearing behaviour with sheet steel thickness, which relies on the ratio of screw diameter to sheet thickness, d/t, was presented in Rogers and Hancock (1999). Unconservative predictions of connection bearing capacity have demonstrated a need for a gradated bearing coefficient which is dependent on the stability of the edge of the screw hole in a similar manner to the bolted connections described above. These unsatisfactory results have been recorded for test specimens where two different thickness sheet steels are connected and loaded in shear. The screwed connection test specimens that have two elements of the same thickness all failed in a combined bearing/tilting mode and have acceptable test-to-predicted ratios. Macindoe and Pham (1996) tested a number of screwed connections where bearing failure was forced to occur because of a large differential in the thickness of the connected sheets. This behaviour differs from that exhibited for the majority of the connections that were tested for by Rogers and Hancock, and by Macindoe and Pham, where failure was caused by a combination of beating and tilting. The connection resistance that was calculated for the screwed connection tests where bearing/tilting failure occurred is

25 reasonably accurate. Hence, the proposed method includes the tilting formulation that is specified in both the AS/NZS 4600 (1996) and AISI (1997) design standards. It also includes the gross yielding (1) and net section fracture (2) failure provisions described for bolts in the preceding sections i.e. no stress reduction factor is used. The material properties for thin G550 sheet steels are not reduced by the 0.75 factor. At present, the bearing coefficient that is contained in the AS/NZS-4600 (1996) and AISI (1997a) design standards is a constant 2.7 for screw connections as shown in Fig. 4. The CAN/CSA-S136 design standard (1994) requires that the bearing coefficient vary depending on the ratio of d/t, as shown in Fig. 4. The proposed method contains a gradated bearing coefficient which is also dependent on d/t, however, the maximum allowed value is lowered to 2.7 and the rate of change of the bearing coefficient is modified accordingly.

,~- C S A - S 1 3 6 //

4.0

ro 3.5 .~ 3.0 ~

2.5

~

2.0

//

d/t~: 1 0 : C - 3 . 0 10 < d/t < 1 5 : C = 3 0 t / d d/t 2 15 : C = 2.0

C=2.7

f

=P r o p o s e d d/t < 6 : C = 2.7 6 < d/t < 13 : C = 3.3 - O. l d/t d/t ~ 1 3 : C = 2.0

o~ 1.5 1.0 ~

- A S / N Z S 4600, A I S I

(1.5 0.0 0

3

6

9

12

15

18

d/t Figure 4 Proposed bearing coefficient for screwed connections

FRACTURE STUDIES

Background Test results from Rogers and Hancock (1997) reveal that the ability of G550 sheet steels to undergo deformation is dependent on the direction of load within the plane of the sheet, where transverse specimens exhibit the least amount of overall, local and uniform elongation. Typically, transverse specimens fail immediately after exiting the linear elastic phase of deformation with minimal plastification of the net section or the gauge length. The fracture resistance of the material, which in reality contains microscopic cracks, may have influenced this behaviour. The G550 sheet steels that were tested do not meet the Dhalla and Winter (1974b) elongation and ultimate strength to yield stress ratio requirements regardless of direction, except for the uniform elongation of longitudinal coupon specimens. The paper on fracture toughness of G550 sheet steels subject to tension by Rogers and Hancock (2001) reports on the fracture resistance properties of G550 sheet steels that were tested in tension. The fracture resistance of the G550 sheet steels was measured for a range of temperatures, and a numerical study of the effect of cracks on structural performance in the elastic deformation range was completed using the FRANC2D 1995) finite element computer program. A summary of the results from this paper are included here.

26

Measurement of Critical Stress Intensity Factors Stress distribution in a loaded member is greatly affected by the presence of cracks or discontinuities. The classical structural mechanics approach deals with these matters by a numerical multiplier referred to as a stress concentration factor, which can be thought of as the increase in stress caused by a change in geometry such as a notch. Fracture mechanics, however, recognises that the stress intensity at the tip of the crack can be expressed as a stress intensity factor, K, as follows, K = o',pp~

(4)

where travp is the nominal stress applied to the member and a is the size of the crack. Thus K can grow as crsop or a or simply the product of these grows. However, K cannot grow to a value larger than the fracture toughness of the material, Kc, which is the critical stress intensity factor at the tip of a crack and is def'med as follows,

IL = 4Eao

(5)

where E is the elastic constant of the material (Young's Modulus) and Gc is the toughness of the material. As the stress intensity factor at the tip of the crack, K, increases with increased loading, it may reach the value of Kc, when the balance of elastic energy release from the loaded body exceeds the energy requirement for crack extension. At this point a running crack that is known as unstable fracture takes place. The stress intensity factor, K, at the tip of the crack should be kept at a value less than the characteristic Kc of the material under investigation if unstable fracture of the structure is to be avoided. This is analogous to the requirement that the cross-sectional stress, f, must lie below fy if one does not want yielding to occur. A total of 30 notch specimens were tested using the same steels as for the ductility, bolted connection and screwed connection tests described above. The main objective of this experimental testing phase was to determine the critical Mode I, i.e. crack opening as opposed to crack sliding (Mode II) or crack tearing (Mode III), stress intensity factors, Ke, for 0.42 and 0.60 mm G550 sheet steels. Tests were completed to measure the magnitude of the crack tip stress field where ultimate failure was caused by unstable fracture of the notch specimens. All of the test pieces within a material and thickness type were cut from the same sheet, although similar specimens were cut from various locations to avoid localised material properties. The material properties of cold reduced sheet steels have been shown to be anisotropic, hence, specimens were cut from three directions within the sheet; longitudinal, transverse and diagonal with respect to the rolling direction. Critical stress intensity factors were calculated for all of the notch test specimens. Of the 30 notch tests, 18 were completed at a temperature of 21.5~ (room temperature) and the remaining at temperatures which varied from I~ to -21~ The room temperature mean value test results are provided in Table 1, and detailed information for each individual notch specimen can be found in Rogers and Hancock (1998b). All of the sheet steel types that were tested have critical stress intensity factors that exceed 3000 MNm "3r2. A significant decrease in the fracture toughness of the G550 sheet steels is not evident for the transverse direction in comparison to the longitudinal and diagonal directions. However, the transverse Kc values do fall below the longitudinal and diagonal values for both the 0.42 mm and 0.60 mm G550 sheet steels. The measured Kr values are atypically high partially because of the thinness of the G550 sheet steels, which did not allow for plane strain conditions to occur during testing. This does not mean that the measured Kc values are incorrect, but that these values are only valid for the thicknesses tested. Ashby (1981) associates the failure of materials that are found to have Ke values in the range measured with the plastic rupture ductile fracture failure mode.

27 No significant variation in the measured critical stress intensity factors of the G550 sheet steels was observed for the range of temperatures used in testing, i.e. 21.5~ to -21 ~ (see Rogers and Hancock (1998b)). This is an indication that the transition temperature from ductile to brittle fracture behaviour of the G550 sheet steels that were tested lies below the range of temperatures used. Table 1: Mean Measured Kc Values at Room Temperature (21.5~

Material Type & Direction

K~ 3/2

(MNm")

0.42ram G550 Long. 0.42ram G550 Tran. 0.42mm G550 Diag. 0.60ram G550 Long. 0.60mm G550 Tran. 0.60ram G550 Diag. IIIII

3767 3182 3748 3551 3260 3743 I

Evaluation of the Critical Stress Intensity Factorsfor Perforated Coupon Specimens An analytical study of cracked specimens fabricated from G550 sheet steels was completed to determine the design implications of possible failure by unstable fracture in the elastic deformation range. Critical stress intensity factors were computed using a finite element model, and then were compared with the measured critical stress intensity factors obtained from tests. The FRANC2D (1995) finite element computer program, distributed and written by the Comell University Fracture Group, was used because it had been specifically developed for the analysis of crack behaviour. The FRANC2D program allows the user to easily define element meshes at crack tips, extend cracks in any direction and remesh the crack area, as well as calculate stress intensity factors at the crack tip. In this analytical study, it was assumed that for the modelled test specimens rapid unstable fracture resulting in ultimate failure would occur when the stress intensity at the prescribed crack tip reached the critical measured I~ value. This is a conservative assumption that is dependent on the following; 1) once the specimen has reached its ultimate load carrying capacity the maximum load does not decrease as the crack size increases, hence rapid fracture of the specimen is not abated, and 2) that loading occurs over a short period of time so that stable crack growth does not occur prior to ultimate failure, i.e. the length of the fatigue crack is not extended by any further crack growth except at ultimate failure. All of the elements used in the finite element models had elastic-isotropic material properties that were defined using the results obtained from the coupon tests of G550 sheet steels (Rogers and Hancock, 1997). Numerical analyses of representative coupon models, with 1, 2, 5 and 7 mm circular, as well as 5 mm diamond and square perforations, for the two G550 sheet steel types were completed using the FRANC2D (1995) computer program. The nominal size of the previously tested perforated coupons was used (see Rogers and Hancock (1997)). Additional 0.25 mm long non-cohesive cracks were placed on either side of the perforation at the position of the highest stress concentration, perpendicular to the direction of load. The applied ultimate edge stress was calculated based on the assumption that the average stress over the net section, which was calculated using only the perforation dimension and not the crack length, would equal the yield stress of the material (see Rogers and Hancock' 1997). The loaded and boundary edges of the coupon models were defined as found for the notch specimen finite element model.

28 Numerical analysis, using the FRANC2D computer program, of the perforated coupons with additional edge cracks revealed that the applied stress intensity factors, Kapp, did not reach a critical level, Kc, as defined by the tested notch specimens (see Figures 5 and 6 for 0.42 mm G550 steel). These results indicate that failure of the real perforated G550 sheet steel coupon specimens (Rogers and Hancock, 1997) can be attributed to yielding and ultimate rupture of the material at the net section, and not unstable fracture in the elastic deformation range. The results also show that the sharp geometry of a diamond perforation and the presence of a small crack will cause the largest applied stress intensity factors.

Figure 5 Perforated Coupons Kapp versus K~ (0.42 mm G550 Longitudinal)

Figure 6 Perforated Coupons Kapp versus I~ (0.42 mm G550 Transverse)

Evaluation of the Critical Stress Intensity Factorsfor Triple Bolted Specimens

The longitudinal and transverse triple bolt G550 sheet steel connections that are documented in Rogers and Hancock (1998), and which failed through rupture of the net section, were also analysed using the FRANC2D (1995) computer program. For each bolted connection, additional 0.5 mm long noncohesive cracks were placed on either side of the perforation of the innermost bolt hole at the position of the highest stress concentration, perpendicular to the direction of load. Distributed loads were

29 applied to all of the bolt holes along the edge where the bolt and sheet steel were in contact. The boundary edge was defined as previously noted for the notch Specimen finite element model. Analysis of the bolted connection specimens using the FRANC2D computer program revealed that the applied stress intensity factors, Kapp, did not reach a critical level, Kr as defined by the notch specimens that were tested (see Figure 7). These results indicate that failure of the triple bolt connection G550 sheet steel specimens can be attributed to yielding and ultimate rupture of the material at the net section and not unstable fracture in the elastic deformation range. However, the applied stress intensity factors have increased in comparison with the values obtained in the analysis of the perforated coupons. This increase can be attributed to the greater width of the test piece, greater crack length and more localised load distribution. It is most noticeable for the transverse 0.42 mm G550 test specimens where the applied stress intensity factors fall just short of reaching the critical level. Further increases in the crack length would result in elevated applied stress intensity factors and ultimately, unstable fracture of the test specimens in the elastic deformation range.

Figure 7 Triple Bolt Connections Kapp versus Kr

Conclusions on Fracture o f G550 Sheet Steels The Mode I critical stress intensity factor, Kc, for the 0.42 mm G550 and 0.60 mm G550 sheet steels was determined for three directions in the plane of the sheet. Single notch test specimens with fatigue cracks were loaded in tension to determine the resistance of G550 sheet steels to failure by unstable fracture in the elastic deformation range. The critical stress intensity factors obtained by testing were then used in a finite element study to determine the risk of unstable fracture in the elastic deformation zone for a range of different G550 sheet steel structural models. It was determined that the perforated coupon and bolted connection specimens that were previously tested for this research project were not at risk of failure by unstable fracture in the elastic zone. Notch specimens that were tested at different temperatures (1 to -21~ did not provide crack resistance values that significantly varied from those tested at room temperature (21.5~ These results are an indication that the ductile-brittle transition temperature lies below the range of temperatures used in testing.

30 COMPRESSION STUDIES

Background and Geometry of Sections Tested The tests by Wu, Yu and LaBoube (1996a, 1996b) discussed in the introduction to this paper were performed on decking sections in bending and so the stiffened flange elements in compression were supported by 2 webs in bending. In order to produce a section with all elements in compression, box sections were formed from 0.6 mm G550 sheet steel at the University of Sydney. The same steel as used for the earlier ductility tests was used. It was not possible to form a complete box without some type of connection at at least one of the comers. Consequently, boxes were formed from two pieces brake-pressed then connected at two of the comers as shown in Fig. 8. Epoxy was used between the sheet steel sections at the comers. These sections were designed to be as close to a true square hollow section as possible without any welding which would have produced considerable distortion of the very thin sections. The sections ranged from 20 mm to 100 mm fiat widths. The b/t ratios therefore varied from 33.3 to 167. The unstiffened lip elements were kept at 7.5 mm long for all section sizes. The local buckling stresses varied from approximately 30 MPa for the 100 mm section to 710 MPa for the 20 mrn section. The effect of the double thickness comer (lip) elements was to increase the theoretical local buckling coefficients to approximately 4.5 due to slight torsional restraint at 2 of the comers.

Figure 8 G550 Box Column Section- Type 1

Test Results and Comparisons with Design Standards The sections were tested in an MTS Sintech 300 kN testing machine with machined end plates. Pattemstone was used between the end plate located against the top of the specimen and the testing machine top platen to ensure even bearing on the specimens. The first few specimens were found to fracture in the epoxy at loads very close to the ultimate load. It was therefore decided to drill 3 mm holes in the comer lips and to place small diameter bolts and nuts to ensure that the comers did not come apart. Initially, these bolts were only located at the ends, but subsequent testing resulted in increased numbers of bolts with the bolt spacing at approximately 20 mm then 10 ram. A photograph of a box section with the small bolts after test and showing the local buckles is shown in Fig. 9. The measured yield stress of the steel was 711 MPa.

Figure 9 G550 Box Section after Test

The test results have been plotted in Fig. 10 versus the theoretical stub column test strengths computed using AS/NZS 4600 (his) and the AISI Specification (Pn). In Fig. 10, the areas of the holes have not been allowed for. The test results with a square are those for the holes at approximately 20 mm spacing, and those with the crosses are for the tests with the 10 mm hole spacing. It is clear that decreasing the hole spacing decreases the load. Obviously the holes are having an effect. All results lie below the theoretical strength based on R~fy as included in Section A3.3.2 of the latest addendum to the AISI Specification (2000). The results of the test have been plotted in Fig. 11 with the areas of the holes removed from the effective area on the assumption that the lips are fully effective and the holes can simply be removed. In this case, the majority of the test results lie above the revised AISI Section A3.3.2 curve. It is also interesting to note that for more slender sections, the test results are closer to the theoretical strengths assuming no reductions in the yield stress. This is to be expected since the effect of no strain hardening is likely to have a lesser effect for sections which are slender and fail principally by post-local instability and then yielding rather than inelastic local buckling. For the more stocky sections, the results are closer to the revised AISI strengths. At the time of writing this paper, further tests were underway for sections without holes and bolts.

32

1.20

I

1.15

-

1.10

-

1.05

-

I

I

I

I

I

I

I

'1

+ Holes at 10mm spacing o Holes at 20mm spacing

_

1.00

Ntes---! 0 . 9 5 Ns 0.90

. . . . . . . . . . . . . . . . . . . . .~,,, AISI supplement No. 1, 1999 -oo

O.ooo. . . . . .[] .

0.85

9 9 ~::~o. o. o

[] 0.80 -

~:

+

m

~

~176o o o . ~

t:]

[]

+

.p

0.75 0.70

,

0

I

,,,

20

],

I

I

I

I

I

I

I

40

60

80

100 b/t

120

140

160

180

200

Figure 10 Comparison of G550 Stub Box Compression Tests with Design Standards (holes excluded from calculations) 1.20

I

II

Ill

I

III I

I

I'

I

1.15

+ Holes at 10mm spacing D Holes at 20mm spacing

1.10

_

1.05 1.00

Ntest Ns 0.95

[] '''~176176176

....

9

0.90

B O

[]

....

~176

o

0.85

i ....... .§ ..... .~

AISI supplement No. 1, 1999

0.80 0.75 0.70

0

I

I=

20

40

I ....

60

I

80

I

I

~

i

100 b/t

120

140

160

180

200

Figure 11 Comparison of G550 Stub Box Compression Tests with Design Standards (holes included in calculations)

33 CONCLUSIONS The use of cold reduced G550 coated sheet steel is becoming more common in the construction industry for structural members, particularly in residential construction. A wide range of research projects has been performed mainly at the University of Sydney and the University of Missouri-Rolla to ascertain the strength of these steels when used in load beating applications. The areas of members in compression, bending and tension, and screwed and bolted connections have been investigated. Fracture toughness has been investigated where it has been shown that all specimens and connections tested to date have failed by ductile failure rather than sudden fracture. Further research is required in the area of compression members to ascertain whether sections composed of G550 sheet steels can carry their design capacity without significant reductions in the yield stress being required in the design equations. Ongoing work is being performed in this area at the University of Sydney.

ACKNOWLEDGEMENTS The authors would like to thank the Australian Research Council and BHP Coated Steel Division for their financial support for these projects performed at the University of Sydney. The advise of Associate Professor Kim Rasmussen on the compression tests is gratefully acknowledged. REFERENCES

American Iron and Steel Institute. (1997). "1996 Edition of the Specification for the Design of ColdFormed Steel Structural Members", Washington, DC, USA. American Iron and Steel Institute. (2000). "1996 Edition of the Specification for the Design of ColdFormed Steel Structural Members, Supplement 1, July 1999", Washington, DC, USA.

American Society for Testing and Materials A 611. (1997). "Standard Specification for Steel, Sheet, Carbon, Cold-Rolled, Structural Quality", Philadelphia, PA, USA American Society for Testing and Materials A 653. (1997). "Standard Specification for Steel Sheet, ZincCoated (Galvanized) or Zinc-iron Alloy-Coated (Galvannealed) by the Hot-Dip Process", Philadelphia, PA, USA. American Society for Testing and Materials A 792. (1997). "Standard Specification for Steel Sheet, 55% Aluminum-Zinc Alloy-Coated by the Hot-Dip Process", Philadelphia, PA, USA. American Society for Testing and Materials A 875. (1997). "Standard Specification for Steel Sheet, Zinc5% Aluminium Alloy-Coated by the Hot-Dip Process", Philadelphia, PA, USA. Ashby, M.F.. (1981). Prog. Mat. Sci., Chalmers Anniversary Volume, pp. 1-25. Canadian Standards Association, S 136. (1994). "Cold Formed Steel Structural Members", Etobicoke, Ont, Canada.

34 Dhalla, A.K., Winter, G.. (1974a). "Steel Ductility Measurements", Journal of the Structural Division, ASCE, Vol. 100, No. ST2, pp. 427-444. DhaUa, A.K., Winter, G.. (1974b). "Suggested Steel Ductility Requirements", Journal of the Structural Division, ASCE, Vol. 100, No. ST2, pp. 445-462. Daudet,R., and Klippstein,K.H. (1994), "Stub Column Study using Welded, Cold-Reduced Steel", 12th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1994, pp 285 - 302. FRANC2D. (1995). "Tutorial and User's Guide", Version 2.7, ComeU University Fracture Group, Ithaca NY, USA. Hancock, G.J. and Murray, T.M (1996), "Residential Applications of Cold-Formed Stmcaaal Members in Australia, 13th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1996, pp 505 - 511. Macindoe, L. and Pham, L., (1995), Test Data from Screwed and Blind Rivetted Connections", CSIRO Division of Building, Construction and Engineering, Document 96/22(M), Higher Victoria, Australia Mahendran, M, (1990), "Static Behaviour of Corrugated Roofing under Simulated Wind Loading",Civil Engg Transactions, Inst. Engrs. Aust., Vo132, No. 4, pp 211-218. Mahendran, M, (1994), "Behaviour and Design of Crest Fixed Profiled Steel Roof Claddings under High Wind Forces",Engg. Struct., Vol. 16 No. 5. McAdam, J.N, Brockenbrough, R.A., LaBoube, R.A., Pekoz, T, and Schneider, E.J., "Low Strain Hardening Ductile Steel Cold-Formed Members", 9~ International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Nov 1988. Rogers, C.A., Hancock, G.J.. (1997). "Ductility of G550 Sheet Steels in Tension", Journal of Structural Engineering, ASCE, Vol. 123, No. 12, 1586-1594. Rogers, C.A., Hancock, G.J.. (1998a). "Bolted Connection Tests of Thin G550 and G300 Sheet Steels", Journal of Structural Engineering, ASCE, Vol. 124, No. 7, pp. 798-808. Rogers, C.A., Hancock, G.J.. (1998b). "Tensile Fracture Behaviour of Thin G550 Sheet Steels", Research Report No. R773, Centre for Advanced Structural Engineering, University of Sydney, Sydney, NSW, Australia Rogers, C.A., Hancock, G.J. (1999). "Screwed Connection Tests of Thin G550 and G300 Sheet Steels", Journal of Structural Engineering, ASCE, Vol. 125, No. 2, pp. 128-136. Rogers, C.A. and Hancock, G.J. (2001). "Fracture Toughness of G550 Sheet Steels subjected to Tension", Journal of Constructional Steel Research, Vo157, pp 71-89. Standards Australia. ( 1 9 9 3 ) . "Steel sheet and strip - Hot-dipped zinc-coated or aluminium/zinc coated - AS 1397", Sydney,NSW, AusUalia Standards Australia / Standards New Zealand. (1996). "Cold-formed steel structures - AS/NZS 4600", Sydney, NSW, Australia

35 Tang, and Mahendmn, M. (1998),"Local Failures in Trapezoidal Steel Claddings", Thin-Walled Stractures Research and Developments, Eds Shanmugam, Liew and Thevendran, Elsevier

-

Wu, S., Yu, W.W, and LaBoube, R.A. (1996a), "Strength of Flexural Members using Structural Grade 80 of A653 Steel (Deck Panel Tests)", Second Progress Report, Department of Civil Engineering, University of Missouri-Rolla, November. Wu, S., Yu, W.W, and LaBoube, R.A. (1996b), "Flexural Members using Structural Grade 80 of A653 Steel (Deck Panel Tests)", 13th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1996, pp 255-274. Wu, S., Yu, W.W, and LaBoube, R.A. (1997a), "Strength of Flexural Members using Structural Grade 80 of A653 Steel (Web Crippling Tests)", Third Progress Report, Depamnent of Civil Engineering, University of Missouri-Rolla, November. Wu, S., Yu, W.W, and LaBoube, tLA. (1997b), "Web Crippling Strength of Members using High Strength Steels", 14th International Specialty Conference on Cold-Formed Steel Structures, St Louis, Missouri, Oct 1998, pp 193-208.


Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michaiskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

37

THIN-WALLED STRUCTURAL ELEMENTS CONTAINING OPENINGS N.E.Shanmugam Department of Civil Engineering The National University of Singapore 10, Kent Ridge Crescent, Singapore-119260

ABSTRACT This paper is concerned with thin-walled structural elements that contain openings provided to allow for locating services, to facilitate fabrication process and to reduce the self-weight of the structures. Extensive analytical and experimental investigations have been carried out in the past to examine the behavior of thin-walled structures containing openings. However, due to lack of an organized database the results have not been fully utilized by other researchers. As part of this review an endeavor is made to cluster the experimental and analytical results with particular importance on plates, beams and plate girders and compression members and cellular structures containing perforations as an elucidative database using a handy spreadsheet program written in Visual Basic. KEYWORDS Thin-walled, structural elements, local buckling, in-plane loading, shear loading, review, experiments, analyses INTRODUCTION Openings are often introduced in thin-walled structures such as automobiles, aircrafts, bridges, ships and storage racking structures to facilitate access for services and inspection. Presence of these openings also help to lighten the structure and to enhance the constructability of thin-walled sections by allowing easy consolidation of building services, piping, electric-wiring, plumbing etc., within the section depth. Access for repair and maintenance works are provided through openings in webs of plate girders, box girders and ship grillage. In aircraft industry openings are often provided for air passage to the interior. Thin-walled members for some specific tasks are generally manufactured with a regular pattern of multiple holes in order to meet the requisites. Typical openings in thin-walled structure are shown in Figure 1. Presence of openings results in redistribution of the membrane stresses, change in buckling and strength characteristics leading to drop in load-carrying capacity. Geometry of the thin-walled structural element (e.g. angle, channel - plain or lipped, etc.), type of applied stress (e.g. compressive, tensile, shear etc) and shape, size, location and number of the openings have greater influence on the

38

Figure: 1 Some examples of Thin-Walled Structures Containing Openings behaviour of structural members containing openings. Thus, the precise analysis and design of structural steel members with perforated elements are imperative. A considerable amount of research has been directed towards this problem during the last three or four decades and approximate design methods for evaluating the ultimate capacity of thin plate elements containing openings and subjected to in-plane axial or shear loading have been proposed by researchers. A review on the research work to account for the effects of openings on the behaviour of thin-walled elements in steel structures has been presented by Shanmugam (1997). Attention is directed, in the review, to the analytical and experimental work carried out on centrally or eccentrically placed, reinforced or unreinforced circular or rectangular openings in perforated plate elements, stiffened flanges, shear webs and cold-formed steel structural elements. Also, design methods of thin-walled steel structural elements that account for the presence of openings are highlighted. However, detailed information from each of the works is not presented in this paper and, one has to dig out the references listed for such information. The object of this paper is, therefore, to summarize the works and present a data bank in a systematic manner so that future researchers could have one source from which they would be able to get any details of the past work done for their reference. An effort has been made in this paper to provide a comprehensive review and to compile the experimental and analytical results on perforated thin-walled structures in a user-friendly format. The review is presented, for convenience of the readers, under four major topics viz. plates, thin-plated beams, thin-walled columns and cellular structures.

PLATES A number of researchers studied initially perforated flange and web plates subjected to different types of loadings such as uniaxial compression, biaxial compression and shear and investigated the stress concentration around openings. Finite element method was applied to determine the elastic buckling of plate elements with cutouts. The method was extended further to include inelastic and largedeflection behaviour of such plate elements.

Uniaxiai and Biaxial Loadings Pennington Vann (1971) employed finite element method for elastic buckling analysis of uniaxially loaded plates. The theoretical and experimental results presented indicate that unless a central nonflange,d hole is rather large, it will have a small effect on the elastic buckling load of a plate, and that a flanged hole can be expected to make the elastic buckling load greater than that of the corresponding non-perforated plate. Tentative results concerning the ultimate strength of pierced plates indicate that a small non-flanged hole has essentially no effect on the ultimate strength, and a stiffened hole may or may not affect the ultimate strength. The finite element method was extended later (Shanmugam and

39 Narayanan, 1982; Sabir and Chow, 1983)to encompass other forms of loading and support conditions for both square and circular openings. The knowledge of post-buckling behaviour is necessary to determine the ultimate load-carrying capacity. Yu and Davies (1971) and Ritchie and Rhodes (1975) proposed that the effective width concept with slight modification could be extended to the postbuckling analysis of axially compressed perforated plates, taking size and shape of openings into consideration. The proposed method was validated with the experimental results on the buckling and post-buckling behaviour of thin-walled structural elements containing centrally located hole. The finite element formulation, which incorporated the material and geometric non-linearity and the initial plate imperfections, based on variational principles (Azizian and Roberts, 1983 and 1984) could predict the non-linear behaviour of perforated plates, right up to failure. Narayanan and Chan (1985) proposed an approximate method based on energy approach to predict the ultimate load capacity of uniaxially compressed plates under linearly varying load across the width. Plate elements subjected to biaxial loading commonly occur in ship double bottoms, dock gates and multi-cell girders. Only limited information is available on the behaviour of such members in the literature. Elastic buckling of such plate elements containing openings was investigated by employing finite element method (Shanmugam and Narayanan, 1982; Sabir and Chow, 1983). A simplified procedure was proposed to estimate the peak load of biaxially loaded plates containing openings was proposed by Narayanan and Chow (1984a). The proposed procedure has been found to give reasonable predictions compared to the corresponding experimental values. Taking into account of the time and cost factor involved in such methods, Narayanan and Chow (1984b, 1984c, 1987)suggested an approximate design method for evaluating the ultimate capacity of plates containing openings and subjected to uni-axial or biaxial compression. All these methods require a proper understanding of the plate behaviour and the effective width concept. Shanmugam et al. (1999) proposed a design formula using finite element method that does not require any rigorous mathematical computations. The designers need to compute different sets of coefficients and substitute into the general equation to predict the ultimate capacity of the circular and square perforated plates for different shapes of openings, different types of axial loading and different combinations of boundary conditions. Shear Loading

Finite element method was employed by Rockey et al. (1967) to study the buckling of a square plate having a centrally located circular hole when subjected to edge shear load. The introduction of openings was found to result in drop of critical shear stress for square plate panel. Studies by other researchers (Shanmugam and Narayanan, 1982; Sabir and Chow, 1983; Narayanan and Chow, 1984c; Naryanan and Der Avenesian, 1984) of the elastic buckling behaviour of perforated plates under shear loading include different shapes of openings and boundary conditions. It may be necessary to place openings eccentrically in the plate. The effects of such eccentricity on elastic buckling behaviour under shear loading of plates were extensively analysed by Narayanan and Der Avenesian (1984). Reinforcements around rectangular and circular openings were also considered in these studies. For plates with central circular and rectangular cutout, an approximate relationship for the shear-buckling coefficient was suggested based on the finite element results. Narayanan and Chow (1984), based on experimental investigations, concluded that the effect of opening size on buckling is more significant than that of location. Stiffened Plates

It becomes essential for flange plates employed in ship double bottoms, dock gates and box girder bridges to be stiffened longitudinally in order to enhance the efficiency of the structure to resist loads. Such stiffened plates often contain openings in bed-plates in which case the failure load may be determined using a method proposed by Mahendran et al. (1994, 1996). This method has been found

40 to predict the failure load to an acceptable degree of accuracy compared with a number of experimental failure loads (Shanmugam et al. 1985, 1986).

PERFORATED BEAMS Bending Hoglund (1971) concluded that the reduction in bending strength of a thin web I-girder having web openings located at the centre is small since the flanges carry most of the bending moment. Experimental and analytical studies were carried out by Shan (1994) to determine the local buckling characteristics of a flexural member containing web opening and proposed an equation to determine the flexural capacity of such sections. Yu and Davis (1973) studied the behaviour of thin-walled steel members with perforated web elements and concluded that the presence of a circular hole results in reduction of shear capacity; they have proposed an empirical reduction factor to quantify the reduction. An approximate method was proposed by Narayanan and Rockey (1981) for the analysis of perforated plate girders. The method was based on experimental observations on plate girders containing web openings. Equilibrium solutions by Narayanan and Der Avanesian (1983 a-d, 1984 a, b, 1985, 1986) based on further assumptions on tension field action has covered shapes and locations of openings in plate girder webs. Though there is no theoretical basis to confirm the hypothetical tension band around openings, the observed experimental pattern of failure substantiated the concept. The predictions obtained by the equilibrium solutions, compared with test results, have invariably been conservative, however, there has been no systematic study to verify if this would generally be the case. Shan (1994) developed both linear and non-linear reduction factors to determine the nominal shear strength, based on the findings from experimental and analytical studies of the behaviour of web elements of cross sections with web openings subjected to uniform shear force. Schuster (1999)carried out experiments to establish an analytical method for calculating the shear resistance of perforated cold-formed steel Csections subjected to constant shear at the University of Waterloo. Combined effect of bending moment and shear force was investigated at the University of MissouriRolla (Shan, 1994; Shan et al., 1996) on standard C- shaped members containing web openings. The current interaction equation in the AISI Specifications, which adequately predicts the web capacity of the nominal shear and bending strengths, are appropriately modified to account for the presence of web openings based on the test results. In the case of beams and columns subjected to concentrated loads in the steel framing structures provision must be made for the load transfer into the web from the flange. Web crippling occurs when web flange intersection subjected to a large compressive force and, this problem becomes more critical in the presence of openings. This problem of web crippling in coldformed steel members has been addressed in the recent research findings (Sivakumaran, 1988; Sivakumaran et al., 1989; Shan et al., 1993 and 1994). Yu and Davies (1973) have also reported on the reduction in web crippling strength due to presence of web openings. This is based on the experimental results on cold-formed steel members. Recent experimental and theoretical studies by Chung (1995) have shown that for perforated sections with practical shapes and significant sizes, the resistance of the sections to web crippling load resistances is often not significant.

Lateral Buckling The problem of lateral buckling in deep slender beams is an important issue that needs adequate consideration in the design of such beams. However, only limited information could be traced in the published literature on the effect of openings on the buckling capacity of beams (Bower, 1968; ASCE Task Committee Report, 1971; Redwood and Uenoya, 1979). Redwood and Uenoya (1979) have treated the problem of webs as a stability problem of a perforated plate with simplified edge loadings and support conditions. The finite element method has been used to solve the resulting problem. Coull and Alvarez (1980) based on their experimental studies have proposed an empirical method for

41 determining the lateral buckling capacity of beams with a number of openings, either circular or rectangular. Their formulae do not seem to allow for openings Other than those considered by them. Thevendran and Shanmugam (1991, 1992a,b) proposed using the principle of minimum total potential energy a numerical method to predict the elastic lateral buckling load of narrow rectangular and I beams containing web openings and subjected to single concentrated load applied at the centroid of the cross section. The accuracy of the method has been verified with results obtained from experiments similar to those shown in Figure 2. Simply supported and cantilever beams were considered and the method is capable of predicting the effects of size, location and shape of openings on the elastic critical load.

Figure 2 : Experiments on Lateral Buckling of Beams with Openings

Reinforced Web Openings It is an expensive operation to provide reinforcements around openings in order to minimize the strength reduction. However, if the loss of strength implicit in cutting a web hole is unacceptable, the web opening will need to be reinforced around its periphery so that the opening can still be introduced. The method of designing an appropriate circular ring reinforcement to restore the strength lost due to the cutout serves as a useful tool in the design office. Based on their investigation, Narayanan and Der Avenesian (1984 a, b) proposed an equilibrium solution for prediction of the ultimate capacity of girders containing a reinforced circular or rectangular hole. Since the tension field in a non-perforated web is developed predominantly along a diagonal band, it is wise to locate openings away from this band, so that the girder does not suffer any significant drop in strength. The effect of eccentrically placed openings on the behaviour of plate girder webs was investigated by Narayanan et al. (1984,1985). THIN-WALLED COLUMNS Local buckling and post-buckling strength for stiffened and unstiffened doubly symmetric perforated cold-formed steel compression members were first considered by Davis and Yu (1972) and their findings based on experimental investigations formed the basis for further developments in the evaluation of the effective design width of perforated plate elements. Yu and Davis proposed an effective design width equation for plates with either central or square perforation. Because of the intricacy involved in the computation of various constants, this equation is not popular with designers. The effective width equation, in the same form as that of Winter's formula (1947), recommended by Ortiz-Colberg (1981) is now currently practiced in the AISI Specifications (1996). But this equation is only applicable for stiffened plates under uniform compression with certain limitation on slenderness ratio and circular perforation to width ratio. Sivakumaran and Banwait (1987 a, b, c) recommended reassessment of effective design width equations in the design codes CAN3-S 136-M84 and AISI-1986,

42 in which the strength of sections is overestimated. They emphasized on the need for experimental data on the sections with manufacturer's hole. Miller and Pekoz (1994) attempted to modify the unified effective width approach to local buckling in the Specification for Design of Cold-Formed Steel Structural Members (AISI 1986) to model the perforated section. It was concluded that the effect of perforations is negligible unless the perforation is extended to the effective portion of the compressed element. An effective design width equations to determine the ultimate strength of non-perforated and perforated cold-formed steel compression members was proposed (Rahman, 1997; Rahman and Sivakumaran, 1998) based on a proven finite element model (Sivakumaran, Rahman, 1998). It has been found that this equation could predict well for manufacturer's hole.

Multiple Openings In the field of storage racking, an ample proportion of cold-formed steel construction, the upright is generally having many perforations, otten in the form of a repeating pattern to a significant extent as shown in Figure 3. The strut capacity of such member is generally assessed on the basis of tests. A more general, but relatively conservative design approach for perforated wall stud assemblies is given in the Canadian standard S136-94 (CSA 1994), which is not applicable for single members in compression. The CSA $136-1974 equations for non-stiffened plates are not suitable for analyzing perforated studs with short holes similar to those tested. Loov (1984)developed an equation to

Figure 3 : Typical Examples of Multiple Openings determine the effective width of non-stiffened portion of the web beside the holes based on the experiments. The test results support the present equation for the average yield stress in Canadian Standards Association Standard S136-1974 but the present code equations for non-stiffened plates are unduly conservative when applied to the design of the web adjacent to openings of the size considered. Rhodes and Schneider (1996)studied experimentally the effects of multiple circular perforations on the ultimate compressive strength of cold-formed steel channel sections. The application of existing coldformed steel design codes to perforated members is examined on the basis of comparison with the tests, and various modifications to the design codes are considered to take perforations into account. Rhodes and Macdonald (1996) extended the investigation to study the effects of perforation length on the stub column capacity. The study concluded that consideration should be given to the perforation geometry along the member as well as the perforation geometry across the member. Wall structures, especially used in the Nordic Countries, include web-perforated C- or sigma-sections as studs and U-sections as tracks and, e.g. gypsum wallboards attached to the stud flanges to provide

43 quality thermal performance. Jyrki Kesti and Pentti Makelainen (1999)proposed design curves to determine the distortional buckling strength of the section based on the experimental study. The comparison showed that the method used gives reasonable results for C-sections but overestimates the compression capacity of sigma sections. CELLULAR STRUCTURES Multi-cell structures, otten used in bridges, storage tanks, dock gates and ship double bottoms, usually contain large openings in webs (Figure 4). The influence on shear deflection component of web openings resulted in large deflection of this form of structures. Studies (Shanmugam and Evans, 1979 a,b) have also shown that openings have influence on longitudinal stresses too. Empirical curves, based on suitable reduction coefficients for shear and torsional stiffness of the structure, can be used adequately to represent the effect of the web openings in grillage analyses of multi-cellular structures. The analysis was also applied to investigate the elastic behaviour and vibrational characteristics of thin-walled multi-cell structures containing web openings (Shanmugam and Balendra, 1985). Evans and Shanmugam (1981,1985) observed tension field action in the webs of cellular structures also.

Figure 4 : Celluar Structure Models with Web Perforations COMPILATION Extensive research materials on the behavior of perforated thin-walled structures have been reported in the published literature. However, there is no collection of data organized especially for rapid search and retrieval. Review is mandatory to prevent consuming valuable time in replicating the previous work. Compilation is essential to avoid unnecessary time in extracting important information from the other researchers work. An effort is, therefore, made to cluster the experimental and analytical results with particular emphasis on plates, beams, compression members and cellular structures containing perforations as an elucidative database using a handy spreadsheet program. The database in visual basic programming contains main record and sub records preceded by the user-friendly instruction sheet. Main record is carefully organized in such a way that a user can easily view through the sheet and select the research work he needed to retrieve from the sub records. The main record is presented in a table format in Appendix I. The sub records include research group, material properties, experimental and analytical results, design equations critical comments and final results for 59 different sets, categorized under different structures, Plate, Beam and Column. An attempt is made to provide all the necessary data needed in a comprehensible format. So the future researcher can refer this database to get all the information needed about perforated thin-walled members. Sample set is given for each category of plate, beam and column in Appendix II.

44 CONCLUSIONS Significant progress has been made in the investigation of thin walled structures containing openings. Several analytical models have been proposed to predict the effects of cutouts on the behaviour of thinwalled members, particularly on singly and doubly symmetric sections with single and multiple perforations subjected to different types of loadings. A number of experiments have been carried out to verify the proposed models. The investigations have resulted in simplified design methods of thinwalled elements containing openings. An attempt has been made in this paper to consolidate all the research works carried out on perforated thin-walled structural elements and to give a comprehensive review and a data base which could be of use to the future researchers in this field. REFERENCES

Abdel - Rahman, N. (1997). Cold-Formed Steel Compression Members with Perforations, PhD Thesis, Mc Master University, Hamilton, Ont. Azizian, Z. G. and Roberts, T. M. (1983). Buckling and Elasto-Plastic Collapse of Structures,

Proceedings of the International Conference on Instability and Plastic Collapse of Structures, Manchester, 322-328. Balendra, T. and Shanmugam, N.E. (1985). Vibrational Characteristics of Multi-Cellular Structures, Journal of Structural Engineering, ASCE, 111:7, 1449-1459. Banwait, A. S. (1987). Axial Load Behaviour of Thin-Walled Steel Sections with Openings, M.Eng Thesis, McMaster University, Hamilton, Ontario, Canada. Bower, J.E., (1968). Design of Beams with Web Openings, 3'. Struct. Div., ASCE 94, 783-807. Sub-committee on Beams with Web Openings of the Task Committee on Flexural Members of the Structural Division. (1971). Suggested Design Guides for Beams with Web Holes, J. Struct. Div., ASCE 97, 2707-2727. Chung, K. F. (1995). Structural Performance of Cold-Formed Sections with Single and Multiple Web Openings- Part 1" Experimental Investigation, The Structural Engineer, 73:9, 141-149. Chung, K. F. (1995). Structural Performance of Cold-Formed Sections with Single and Multiple Web Openings- Part 2: Design Rules, The Structural Engineer, 73:9, 141-149. Coull, A. and Alvarez, M.C. (1980). Effect of Openings on Lateral Buckling of Beams, J. Struct. Div., ASCE 106, 2553-2560. Davis, C. S. and Yu. W. (1972). The structural Behaviour of Cold-Formed Steel Members with Perforated Elements, Department of Civil Engineering, University of Missouri Rolla, Rolla, Mo. Evans, H. R. and Shanmugam, N. E. (1979). The Elastic Analysis of Cellular Structures containing Web Openings, Proceedings, The Institution of Civil Engineers, 67:2, 035-1063. Evans, H. R. and Shanmugam, N. E. (1981). An Experimental study of the Ultimate Load Behaviour of Small-scale Box Girder Models with Web Openings, Journal of Strain Analysis, 16:4, 251-259. Hoglund, T. (1971). Strength of Thin Plate Girders with Circular or Rectangular Web Holes without Web Stiffeners, Proceedings, Colloquium of the International Association of Bridge and Structural Engineering, London. Jyrki Kesti and Penti Makelainen. (1999). Compression Behaviour of Perforated Steel Wall Studs, Lightweight Steel and Aluminum Structures. Loov, R. (1984). Local Buckling Capacity of C-Shaped Cold-Formed Steel Sections with Punched Webs, Canadian Journal of Civil Engineering, 11, 1-7. Mahendran, M., Shanmugam, N. E. and Richard Liew, J. Y. (1994). Strength of Stiffened Plates with Openings, Proceedings, Twelfth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, 29-40. Mahendran, M., Shanmugam, N. E. and Liew, J. Y. (1996). Design of Stiffened Plates with Openings, Journal of Institution of Engineers, Singapore, 36:2, 15-21.

45 Miller, T. H. and Pekoz, T. (1994). Unstiffened Strip approach for Perforated Wall Studs, ASCE Journal of Structural Engineering, 120:2, 410-421. Nabil Abdel-Rahman and Sivakumaran, K. S. (1998). Effective Design Width for Perforated Cold-Formed Steel Compression Members, Canadian Journal of Civil Engineering, 25, 319-330. Narayanan, R. and Rockey, K. C. (1981). Ultimate Load Capacity of Plate Girders with Webs containing Circular Cut-outs, Proceedings, Institution of Civil Engineers, 71:2, 845-862. Narayanan, R. and Der Avenesian, N. G. V. (1983a). Strength of Webs Containing Circular Cutouts, IABSE Proceedings P-64/83, 141-152. Narayanan, R. and Der Avenesian, N. G. V. (1983b). Equilibrium Solution for Predicting the Strength of Webs with Rectangular Holes, Proceedings of the Institution of Civil Engineers, 75:2, 265-282. Narayanan, R. and Der Avenesian, N. G. V. and Ghannam, M. M. (1983). Small-scale Model Tests on Perforated Webs, The Structural Engineer, 61 (3), 47-53. Narayanan, R. (1983). Ultimate Shear Capacity of Pate Girders with Openings in Webs, Plated Structures-Stability and Strength, ed. R. Narayanan, Applied Science Publishers, London, pp. 3976. Narayanan, R. and Der Avenesian, N. G. V. (1984a). Design of Slender Webs Containing Circular Holes, IABSE Proceedings P-72/84, pp. 25-32. Narayanan, R. and Der Avenesian, N. G. V. (1984b). An Equilibrium Method for assessing the Strength of Plate Girders with Reinforced Web Openings, Proceedings, Institution of Civil Engineers, 77:2, pp. 107-137. Narayanan, R. and Der Avenesian, N. G. V. (1984c). Elastic Buckling of Perforated Plates under Shear, Thin-Walled Structures, 2, 51-73. Narayanan, R. and Der Avenesian, N. G. V. (1984d). Strength of Webs with Comer Openings, The Structural Engineer, 62B, 6-11. Narayanan, R. and Chow, F. Y. (1984a). Strength of Biaxially Compressed Perforated Plates, Proceedings, Seventh International Specialty Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, 55-73. Narayanan, R and Chow, F.Y. (1984b). Ultimate Capacity of Uniaxially Compressed Perforated Plates, Thin-Walled Structures, 2, 241-264,. Narayanan, R and Chow, F.Y. (1984c). Buckling of Plates Containing Openings, Proceedings, Seventh International Specialty Conference on Cold-Formed Steel Structures, University of Missouri Rolla, 39-53. Narayanan, R. and Chow, F. Y. (1984d). Experiments on Perforated Plates Subjected to Shear, Thin-Walled Structures, 2, 51-73. Narayanan, R. and Der Avenesian, N. G. V. (1985). Design of Slender Webs having Rectangular Holes, dournal of Structural Engineering, ASCE, 111:4, 777-787. Narayanan, R. and Darwish, I. Y. S. (1985). Strength of Slender Webs having Non-Central Holes, The Structural Engineer, 63B, 57-62,. Narayanan, R. and Chan, S.L. (1985). Ultimate Capacity of Plates Containing Holes under Linearly Varying Edge Displacements, Computers and Structures, 21:4, 841-849. Narayanan, R. and Der Avenesian, N. G. V. (1986). Analysis of Plate Girders with Perforated Webs, Thin-Walled Structures, 4, 363-380. Narayanan, R. (1987). Simplified Procedures for the Strength Assessment of Axially Compressed Plates with or without Openings, Proceedings, International Conference on Steel and Aluminum Structures, Cardiff, 592-606. Ortiz-Colberg, R. (1981). The Load Carrying Capacity of Perforated Cold-Formed Steel Columns, M.Sc. Thesis, Cornell University, Ithaca, N.Y. Pennington Vann, W. (1971 ) Compressive Buckling of Perforated Plate Elements, Proceedings of the first Specialty Conference on Cold-Formed Structures, University of Missouri Rolla, 52-57. Redwood, R.G. and Uenoya, M. (1979). Critical Loads for Webs with Holes, d. Struct. Div., ASCE 105, 2053-2067.

46 Ritchie, D. and Rhodes, J. (1975). Buckling and Post-Buckling Behaviour of Plates with Holes, Aeronautical Quarterly, 281-296. Roberts, T. M. and Azizian, Z. G. (1984). Strength of Perforated Plates subjected to in-plane Loading, Thin-Walled Structures, 2:2, 153-164. Rockey, K. C., Anderson, R. G. and Cheung, Y. K. (1967). The behaviour of square Shear Webs having Circular Hole, Proceedings of the Swansea Conference on Thin-Walled Structures, Crosby Lockwood and Sons, London, 148-169. Rhodes, J., and Macdonald, M. (1996). The effects of Perforation Length on the behaviour of Perforated Elements in Compression, Proceedings, 13th International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Mo., 91-101. Rhodes, J. and Schneider, F. D. (1996). The Compressional behaviour of Perforated Elements, Proceedings, 12th International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Mo., 11-28. Sabir, A.B. and Chow, F.Y. (1983). Elastic Buckling of Flat Panels Containing Circular and Square Holes, Proceedings, International Conference on Instability and Plastic Collapse of Structures, Manchester, pp. 311-321. Said, A. N. (1995). The behaviour of Thin-Walled Perforated Elements under Compression, Final Year Project Thesis, University of Strathclyde, Glascow. Schuster, R. M. (1999). Perforated Cold-Formed Steel C-Sections subjected to shear (Experimental Results), Light-Weight Steel and Alumnium Structures, 779-788. Shan, M.Y., LaBoube, R.A. and Yu W.W. (1993). Shear behaviour of web elements with openings, Proceedings, Structural Stability Research Council Annual Technical Session. 103-113. Shan, M. Y., Batson, K. D., LaBoube, R. A. and Yu, W. W. (1994). Local Buckling flexural Strength of Webs with Openings, Engineering Structures, 16:5, 317-323. Shan, M. Y. (1994). Behaviour of Web Elements with Openings subjected to Bending, Shear and the combination of Bending and Shear, PhD dissertation, Dept. of Civil Engg., University of Missouri Rolla, Roila, Mo.. Shan, M. Y., LaBoube, R. A. and Yu. W. W. (1966). Bending and Shear Behaviour of Web Elements with Openings, Journal of Structural Engineering, ASCE. Shanmugam, N. E. and Evans, H. R. (1979). An Experimental and Theoretical Study of the effects of Web Openings on the Elastic Behaviour of Cellular Structures, Proceedings, The Institution of Civil Engineers, 67:2, 653-676. Shanmugam, N.E and Narayanan, R. (1982). Elastic Buckling of Perforated Square Plates for Various Loading and Edge Conditions, Proc., Proceedings, International Conference on finite element methods, Shanghai, China, 241-245. Shanmugam, N. E. and Evans, H. R. (1985). Structural Response and Ultimate Strength of Cellular Structures with Perforated Webs, Thin-Walled Structures, 3,255-271. Shanmugam, N.E. and Balendra, T. (1985). Model Studies on Multi-Cell Structures, Proceedings, Institution of Civil Engineers, Part 2: Research and Theory, 79, 55-71. Shanmugam, N. E. and Paramasivam and Lee, S. L. (1985). Ultimate Strength of Axially Compressed Stiffened Plates Containing Openings, Proceedings of the International Conference on Metal Structures, Melbourne, Australia, 48-52. Shanmugam, N. E. and Paramasivam and Lee, S. L. (1986). Stiffened Flanges Containing Openings, Journal of Structural Engineering, ASCE, 112:10, 2234-2246. Shanmugam, N. E. and Thevendran, V. (1992). Critical Loads of Thin-Walled Beams Containing Web Openings, Thin-Walled Structures, 14, 291-305. Shanmugam, N.E. (1997). Openings in Thin-Walled Structures. Thin-Walled Structures, 28:3-4, 355-372. Shanmugam, N. E., Thevendran. V. and Y. H. Tan. (1999). Design Formula for Axially Compressed Perforated Plates, Thin-Walled Structures, 34, 1-20.

47 Sivakumaran, K. S., Banwait, A. S. (1987). Effect of Perforation on the Strength of Axially Loaded Cold-Formed Steel Section, Proceedings, International conference on Steel and Aluminum Structures, Cardiff, UK, 8-10 July 1987, pp. 428-437. Sivakumaran, K. S. (1987). Load Capacity of Uniformly Compressed Cold-Formed Steel Section with Punched Web, Canadian Journal of Civil Engineering, 14, 550-558. Sivakumaran, K. S. and Nabil Abdel-Rahman. (1998). A Finite Element Analysis Model for the Behaviour of Cold-Formed Steel Members, Thin-Walled Structures, 31,305-324. Sivakumaran, K.S. (1988). Some Studies on Cold-Formed Steel Sections with Web Openings, Proceedings, Ninth International Specially Conference on Cold-Formed Steel Structures, St. Louis, 513-527. Sivakumaran, K.S. and Zielonka, K. M. (1989). Web Crippling Strength of Thin-Walled Steel Members with Web Openings, Thin-Walled Structures, 8, 295-319. Thevendran, V and Shanmugam, N. E. (1991). Lateral Buckling of Doubly Symmetric Beams Containing Openings, Jr. Eng. Mech., ASCE, 117:7, 1427-41. Thevendran, V and Shanmugam, N. E. (1992). Lateral Buckling of Narrow Rectangular Beams Containing Openings, Computers and Structures, 43(2), 247-254. Winter, G. (1947). Strength of Thin Steel Compression Flanges, Transactions of the American Society of Civil Engineers, 112, 527-554. Yu, W.W. and Davies, C.S. (1971). Bucking Behaviour and Post-Buckling Strength of Perforated Stiffened Compression Elements, Proceedings, The first Specially Conference on Cold-Formed Structures, University of Missouri Rolla, 58-64. Yu, W. W. and Davies, C. S. (1973). Cold-Formed Steel Members with Perforated Elements, ASCE Journal of Structural Engineering Division, 99(ST 10), 2061-2077. APPENDIX I: SAMPLE OF THE MAIN R E C O R D OF THE DATA B A S E STRUCTURE : WORKSHEET:

No

Year

PLATE PLATE

Title

Source

Author

Status

1971

CompressiveBuckling of Perforated Plate Elements

The first SpecialityConference PenningtonVann, on Cold-FormedStructures W

Platel

1971

BuckingBehaviourand PostBuckling Strength of Perforated Stiffened CompressionElements

The first SpecialityConference Yu, W.W., on Cold-FormedStructures Davies, C.S

Plate2

1975

Bucklingand Post-Buckling Behaviour of Plates with Holes

Aeronautical Quarterly

1982

ElasticBuckling of Perforated Square Plates for Various Loading and Edge Conditions

Proc., International Conference Shanmugam,N.E, on finite element methods Narayanan,R

Plate4

1983

Elastic Buckling of Flat Panels Containing Circular and Square Holes

Proc., International Conference Sabir,A.B., on Instability and Plastic Chow, F.Y Collapse of Structures

Plate5

1983

Buckling and Elasto-Plastic Collapse of Perforated Plates

Proc., International Conference Azizian,Z.G, on Instability and Plastic Roberts, T.M Collapse of Structures

Plate6

1984

Ultimate Capacity of Uniaxially Compressed Perforated Plates

Thin-Walled Structures, (Vol.2) Narayanan,R, Chow, F.Y

Plate7

Ritchie, D., Rhodes, Plate3 J

48

STRUCTURE : BEAM WORKSHEET:

BEAM

1

1967

The Behaviour of Square Shear Webs having Circular Hole

Proc., The Swansea Conference on Thin-Walled Structures

Rockey,K.C., Anderson, R.G., Cheung, Y.K

Beaml

2

1971

Bucking Behaviour and PostBuckling Strength of Perforated Stiffened Compression Elements

The first Speciality Conference on Cold-Formed Structures

Yu, W.W., Davies, C.S

Beam2

3

1971

Strength of Thin Plate Girders with Circular or Rectangular Web Holes without Web Stiffeners

Proc., Coloquium, International Association of Bridge and Structural Engineering, London

Hoglund, T.

Beam3

4

1981

Ultimate Load Capacity of Plate Girders with Webs Containing Circular Cut-Outs

Proc., Institution of Civil Engineers, Part2, Vol.71, pp 845-862

Narayanan, R., Rockey, K.C.

Beam4

5

1983

Design of I beams with Web Perforations

Beams and Beam ColumnsStability Strength

Redwood, R.G.

Beam5

6

1983

Equilibrium Solution for Predicting the Strength of webs with Rectangular Holes

Proc., Institution of Civil Engineers, Part2, Vol.75, pp 265-282

Narayanan, R., Der Avenessian, N.G.V.

Beam6

7

1983

Strength of Webs Containing Circular Cut-Outs

IABSE Proceedings P-64/83, pp 141-152

Narayanan, R., Der Avenessian, N.G.V

Beam7

8

1983

Small-Scale Model Tests on Perforated Webs

The Structural Engineer, Vol. 61B, No.3, pp. 47-53

Narayanan, R., Der Avenessian, N.G.V Ghannam, M.M.

Beam8

STRUCTURE

:

WORKSHEET :

COLUMN COLUMN

1

1971

Compressive Buckling of Perforated Plate Elements


Pennington Vann, W

Columnl

2

1971

Bucking Behaviour and PostBuckling Strength of Perforated Stiffened Compression Elements


Yu, W.W., Davies, C.S.

Column2

3

1973

Cold-Formed Steel Members with Perforated Elements

ASCE Journal of Structural Engineering Division, Vol.99(STI 0), pp. 2061-2077

Yu, W.W., Davies, C.S.

Column3

4

1984

Local Buckling Capacity of CShaped Cold-Formed Steel Sections with Punched Webs

Canadian Journal of Civil Engineering, Vol. 11, pp. 1-7

Loov, R.

Column4

5

1984

Effect of Perforation on the Strength of Axially Loaded ColdFormed Steel Section

Proc. Int. natl. Conf. On Steel Aluminum Structures, Cardiff, UK, 1987, pp. 428-437

Sivakumaran, K.S., Banwait, A.S.

Column5

6

1987

Load Capacity of Uniformly Compressed Cold-Formed Steel Section with Punched Web

Canadian Journal of Civil Engineering, Vol. 14, pp. 550558

Sivakumaran, K.S.

Column6

49

APPENDIX

II: SAMPLE

SET OF DATA BASE

PLATE 7 Year Subject Title Author Structure Boundary Condition

: : : : :

1956 Ultimate Capacity of Uniaxially Compressed Perforated Plates Thin-Walled Structures, (Vol.2) Narayanan, R and Chow, F.Y Plate

:

Simply Supported

An approximate method of evaluating the ultimate capacity has been suggested using simple elastic-plastic concepts. Design Curves for centrally perforated plates have been suggested for designers. The analysis is based on small deflection theory, hence the method is not valid for wide plates with slenderness values in excess of 80 or so

Table I : DETAILS OF TEST SPECIMENS CONTAINING CENTRAL CUTOUTS Group Circular

a (mm)

t (mm)

a/t = b/t

d or a' (mm)

d/a or a'/a

Imper (mm)

lmper/t

YldStress N/mm2

CIR2a CIR2b CIR3a CIR4a CIR4b CIR5a CIR6 CIR7 CIR8 CIR9 CIRI 0 CIR11 CIR12

125.0 125.0 125.0 125.0 125.0 125.0 86.0 86.0 86.0 86.0 86.0 86.0 86.0

1.615 1.615 1.615 1.615 1.615 1.615 2.032 1.615 0.972 0.693 2.032 1.615 0.972

77.40 77.40 77.40 77.40 77.40 77.40 42.30 53.23 88.48 124.10 42.30 53.25 88.48

25.0 25.0 37.5 50.0 50.0 62.5 25.0 25.0 25.0 25.0 40.0 40.0 40.0

0.2 0.2 0.3 0.4 0.4 0.5 0.291 0.291 0.291 0.291 0.465 0.465 0.465

0.229 0.097 0.136 0.304 0.127 0.279 0.254 0.229 0.102 0.051 0.102 0.279 0.152

0.142 0.060 0.084 0.188 0.079 0.173 0.143 0.142 0.105 0.074 0.050 0.173 0.156

323.3 323.3 323.3 323.3 323.3 323.3 334.7 323.3 317.6 322.8 334.7 323.3 317.6

Square SQ2 SQ3 SQ4 SQ5

125.0 125.0 125.0 125.0

1.615 1.615 1.615 1.615

77.40 77.40 77.40 77.40

25.0 37.5 50.0 62.5

0.2 0.3 0.4 0.5

0.097 0.141 0.113 0.209

0.060 0.087 0.070 0.129

323.3 323.3 323.3 323.3

Table 2 : DETAILS OF TEST SPECIMENS CONTAINING ECCENTRICAL CUTOUTS Yield Stress = 317.6 N/mm2 Specimen Circular

a t (mm) (mm)

aJt = b/t

d or a' (mm)

d/a or a'/a

e (mm)

e/a

lmper (mm)

Imper/t

UEC 1 UEC 2

125.0 0.972 125.0 0.972

128.60 128.60

37.5 62.5

0.3 0.5

12.5 12.5

0.1 0.1

0.254 0.102

0.261 0.105

Square UES 1 UES 2 UES 3 UES4

125.0 125.0 125.0 125.0

128.60 128.60 128.60 128.60

37.5 62.5 37.5 62.5

0.3 0.5 0.3 0.5

12.5 12.5 25.0 25.0

0.1 0.1 0.2 0.2

0.127 0.229 0.078 0.132

0.131 0.236 0.080 0.136

0.972 0.972 0.972 0.972

50

Table 3: E X P E R I M E N T A L RESULTS FOR UNIAXIALLY LOADED PLATES HAVING CENTRAL HOLES Specimen

a/t

d/a or a'/a

Observed Values Ku Avg. Failure Load (Pf) (kN)

Circulzr

(mm)

(mm)

Pcr Avg. (kN)

PL CIR2a CIR2b CIIL3a CIR4a CIR4b CIR5a CIR6 CIR7 CIR8 CIR9 CIRI 0 CIR11 CIR12 Square Hole SQ2 SQ3 SQ4 SQ5

77.40 77.40 77.40 77.40 77.40 77.40 77.40 42.30 53.23 88.48 124.10 42.30 53.25 88.48

0.0 0.2 0.2 0.3 0.4 0.4 0.5 0.291 0.291 0.291 0.291 0.465 0.465 0.465

25.064 22.504 23.228 21.311 19.706 18.358 19.482 6.341 2.320 5.926

4.013 3.604 3.720 3.413 3.156 2.940 3.120 3.205 3.235 2.995

77.40 77.40 77.40 77.40

0.2 0.3 0.4 0.5

22.600 20.290 18.230 19.170

3.620 3.250 2.920 3.070

Pf/Psq

Pxh/Psq

Pxh/Pf

39.32 37.46 38.70 33.94 29.57 28.39 27.35 42.17 26.18 12.35 7.33 33.64 22.14 10.89

0.603 0.574 0.593 0.520 0.453 0.435 0.419 0.721 0.583 0.465 0.381 0.575 0.493 0.410

0.610 0.560 0.560 0.510 0.470 0.470 0.420 0.700 0.615 0.480 0.410 0.560 0.510 0.410

1.012 0.976 0.944 0.981 1.038 1.080 1.002 0.971 1.055 1.032 1.076 0.974 1.034 1.000

33.48 28.85 25.52 21.86

0.525 0.460 0.400 0.340

0.525 0.460 0.400 0.340

1.024 1.041 1.023 1.015

BEAM 12

Year : 1984 Subject : Strength of Webs x~ith Con~er Openings Title : The Structural Engineer, Vol. 62B, pp. 6-11 Author : Narayanan, R and Der-Avanessian, N.G Structure: Plate Girder Material: Steel An equilibrium method of predicting the ultiamte capacity of plate girder webs containing comer web openings and loaded in shear is presented and based on the post-critical behaviour of such webs. Ultimate load = elastic critical load on the web + load carried by the tensile membrane stresses developed in the post=critical stages + load carried by the flanges. TABLE 2: E X P E R I M E N T A L VALUES OF U L I T M A T E LOADS C O M P A R E D WITH PREDICTED VALUES Girder

NCP13 NCPI3 NCPl4 NCPI4 NCP 15 NCP15 NCPl 6 NCPl6

Diameter of Cutout (mm) 125 175 250 325 180 270 360 480

h/t (nominal)

b/h (nominal)

d/h (nominal)

250 250 250 250 360 360 360 360

1.5 1.5 1.5 1.5 1 l l l

0.25 0.35 0.5 0.65 0.25 0.375 0.5 0.67

Observed Ultimate load (kN) 176.4 168.6 125.4 85 234.4 218.8 170.2 112

Equilibrium Solution Predicted load Pre/obs (kN) (kN) 162.9 0.923 157.2 0.932 121.7 0.97 79.7 0.937 221.5 0.945 202.1 0.924 158.8 0.933 87.4 0.78

The theory and experiments confirm that it is advantageous to locate the holes in the comers of the web panels in the compression diagonal i.e far away from the tension field.

51

COLUMN 6 Year : Subject: Title : Author : Structure: Material: Internal Radius: Loading : Section : Opening :

1984 Local Buckling Capacity of C-Shaped Cold-Formed Steel Sections with Punched Webs Canadian Journal of Civil Engineering, Vol. 11, pp. 1-7 Loov, R Column Steel 3.2 Stub-column test Lipped Channel Section Each of the studs had a 38.1 x 44.5 mm rectangular hole centrally punched in the web. Additionally, a 22.2mm diameter hole was centered 41.3ram from one edge of the rectangular opening & a 27.7mm diameter hole was centered in 41.3mm on the opposite side along the centerline of the web.

Specimen

Web

Flange

Lip

Hole Width

Fy

Test

1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

(mm) 63.8 63.8 63.8 63.7 63.7 63.5 63.5 63.5 92.3 92.3 92.3 92.1 92.1 92.1 92.4 92.4 92.4 152.8 152.8 152.8 152.5 152.5 152.5 152.6 152.6 152.6 203.7 203.7 203.7 203.0 203.0 203.0 204.0 204.0 204.0

(mm) 42.6 42.6 42.6 42.2 42.2 42.4 42.4 42.4 42.2 42.2 42.2 42.2 42.2 42.2 42.4 42.4 42.4 42.1 42.1 42.1 41.9 41.9 41.9 42.2 42.2 42.2 42.1 42.1 42.1 41.9 41.9 41.9 42.7 42.7 42.7

(mm) 12.5 12.5 12.5 13.0 13.0 12.1 12.1 12.1 12.5 12.5 12.5 13.0 13.0 13.0 12.8 12.8 12.8 12.7 12.7 12.7 12.8 12.8 12.8 12.9 12.9 12.9 12.5 12.5 12.5 12.9 12.9 12.9 12.9 12.9 12.9

(mm) 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38. I 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1

MPa 384.0 384.0 384.0 382.9 382.9 268.5 268.5 268.5 268.5 268.5 268.5 382.9 382.9 382.9 384.0 384.0 384.0 268.5 268.5 268.5 382.9 382.9 382.9 384.0 384.0 384.0 268.5 268.5 268.5 382.9 382.9 382.9 384.0 384.0 384.0

(kN) 103.62 103.92 102.40 78.65 76.00 44.76 43.87 42.80 46.92 48.11 48.22 72.00 76.40 78.90 119.20 118.00 118.20 51.52 52.32 52.46 84.10 83.40 86.70 128.30 130.00 125.20 51.33 48.70 46.02 87.20 81.70 86.70 133.00 133.50 135.40


Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

53

SENSITIVITY ANALYSIS OF THIN-WALLED MEMBERS, PROBLEMS AND APPLICATIONS C. Szymczak Department of Civil Engineering, Technical University of Gdafisk, U1. G. Narutowicza 11, 80-340 Gdafisk, Poland

ABSTRACT A review of problems related to sensitivity analysis of thin walled members with open monosymmetric or bisymmetric cross-section is presented. Three different kinds of restraint imposed on angle of crosssection rotation, transverse displacement and cross-section warping are taken into account. The consideration is based upon the classical theory of thin-walled beams with nondeformable cross-section for linear elastic range of the member material. Attention is focused on the members undergoing torsion, because this topic is not so popular as bending and/or compression. The first variations of state variables due to a change of the design variable are investigated. Arbitrary displacement, internal force or reaction of the member subject to static load, critical buckling load, frequency and mode of torsional vibration are assumed to be the state variables. The dimensions of the cross-section, the material constants, the restraints stiffness, and their locations, position of the member ends are taken as the design variables. All problems under consideration are illustrated by numerical examples. Accuracy of the approximation changes of the state variables achieved by sensitivity analysis is also discussed. Finally some problems dealing with generalization of the sensitivity analysis elaborated upon the space frames and grids assembled with thin-walled members are described.

KEYWORDS Thin-walled structures, statics, stability, vibrations, restraints, sensitivity analysis.

INTRODUCTION Behavior of the thin-walled members is described by means of so-called state variable s such as: displacements, internal forces, reactions, critical buckling loads and frequencies and modes of free vibrations. The values of these state variables depend on many parameters of the members, known as design variables d. In many problems of engineering practice it is very useful to know a direct relation between the state variable variation 6s and the design variable variation of 5d. Sensitivity analysis (Haug, Choi & Komkov (1986)) enables to derive such relations. Since the sensitivity analysis of structures undergoing bending and compression or tension is well developed, the present paper deals with the sensitivity analysis of members subjected to torsion. From the mathematical point of view, one can distinguish two kinds of design variables:

54 Continuous variables, for example, the cross-section dimensions and the member material constants, Discrete variables, for instance, the restraints stiffness and their location and the support position. In case of the variation of the continuous design variable the first order variation of the state variable sought can be expressed as follows -

-

1

~Ss= IFsa(z)Sd dz

(1)

0

where the underintegral function Fsd(z)can be considered as the influence line of the state variable variation due to the unit point variation of the design variable. If the discrete design variables are taken into account, then a similar relation between the state variable variation and the vector of the design variable variation 5d is 5s = WsT &l

(2)

where vector Wsd consists of the first order sensitivity coefficient corresponding to the design variable and (...)T denotes transposition of the vector. The usual assumptions of the classical theory of thin-walled members with nondeformable crosssection (Vlasov (1959)) adopted in this paper are: (1) The thin-walled member is prismatic, (2) The static loads are conservative, (3) The member cross-section is not deformed in its plane but it is subject to warping in the longitudinal direction, (4) The shear deformation in the middle surface vanishes, (5) The deformations and the strains are small, (6) The member material is homogeneous, isotropic and obeys Hooke's law. Because a lack of a general theory of thin-walled members with arbitrary variable cross-section, the sensitivity analysis is restricted to the member cross-section with one or double axis of symmetry. It is well known that for the bisymmetric cross-section, torsion of the member can be considered independently of bending, as in case of monosymmetric cross-section bending with respect of the symmetry axis independently of torsion and bending with respect to the second axis. Three types of elastic restraints are considered in this paper: the flexural restraint against the lateral displacement of the member axis, the torsional restraint against the member cross-section rotation, and the warping restraint against the cross-section warping. The behavior of the restraints is modelled by suitable linear elastic supports. The sensitivity analysis problems are investigated only for linearly elastic range of the member material behavior.

SENSITIVITY ANALYSIS OF MEMBERS SUBJECT TO STATIC LOADS

Continuous design variable Consider a thin-walled beam with bisymmetric open cross-section shown in Figure 1. The rectangular coordinate system x,y,z is so chosen that x and y coincide with the principal axis of the cross-section and z coincide with the longitudinal axis of the beam. The beam of length I is subjected to a distributed torque m, and axial end loads P. Moreover, two types of restraint, torsional and warping, are imposed on the member. Following Szewczak, Smith & DeWolf (1983) continuously distributed restraints under consideration are modelled by linear elastic foundation with corresponding stiffness k0, kw. In this case the torsion of the member can be considered independently of bending. A detailed treatment of the member behavior is presented by Trahair and Bild (1990). The first variation of the state variable 5s determining the behavior of the member due to some variations of the design variable 5d is sought.

55

m

P._l'.z/r

kw, k0

:-i

'

k0

\

kw

Figure 1" Thin-walled member with bisymmetric cross-section The angle of cross-section rotation and its first derivative, the internal force at a specified cross-section or reactions at the supports and restraints are assumed to be the state variable. Our attention in this section is concentrated on the continuous design variables, for example,: the cross-section dimensions, the material moduli, and the stiffness of the continuously distributed restraints. Taking advantage of the concept of adjoint structure (Dems & Mr6z (1983)), the first variation of the design variable can be written as (compare Budkowska & Szymczak (1992a)) 1

5s=

),d

+

+kw,d

+k0,d

0 0 where E and G are Young's and shear modulus, respectively; Iw stands for the warping constant, Id denotes the St. Venant torsion constant, r represents the radius of gyration of the cross-section, 0 m

and 0 are the angle of the cross-section rotation for the external loads and adjoint member, respectively. The derivative with respect to z is denoted by prime and ("'),d is the partial derivative with respect to the design variable d. The underintegral function Fsd(Z) is considered to be the influence line of the state variable variation due to point unit variation of the design variable. The static analysis of the member necessary to determine the influence line can be carried out by the FEM. The first numerical example deals with continuous thin-walled I beam presented in Figure 2. The computer program elaborated on the basis of the sensitivity analysis presented enables to obtain the influence line of the torsional moment on the right side of support B due to point torsional restraint with unit stiffness K0 = 1. m = 2 kNm/m for 0< z 12.5 mm. The Eurocode formula does not have this shortcoming, but it is more conservative than the AISI formula for dw 5.106

For rod welded girders (which always exhibit larger irregularities in the welds and elsewhere): log Nloo = 12.301 - 3 log Ats p log Nloo = 16.036 - 5 log Ats p

N < 5.106 N > 5.106

AGp is the principal surface stress range in the crack-prone areas for which formulae were established.

References Skaloud M. and Roberts T.M. (1998). Fatigue Crack Initiation and Propagation in Slender Web Breathing Under Repeated Loading. Proc. of the 2nd World Conf. on Steel in Construction, 417419. ~kaloud M., Z6rnerovfi M., Kuhlman U. and Spiegelhalder U. (1999). Prague and Stuttgart Experimental Research on Web Breathing. Proc. of the Int. Conf. Eurosteel'99 1, 75-78. Skaloud M., ZSrnerovfi M. and Roberts T.M. (1997). Fatigue Assessment of the ,,Breathing" Webs of Steel Plate Girders. Proc. of the 18th Czech-Slovak International Conference on Steel Structures and Bridges 4, 59-66.

Third InternationalConferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

145

EVALUATION OF STRENGTH AND DUCTILITY CAPACITIES FOR STEEL PLATES IN CYCLIC SHEAR T. Usami i, p. Chusilpl, A. Kasai l, and T. Watanabe I ~Department of Civil Engineering, Nagoya University, Nagoya, 464-8603, Japan

ABSTRACT This study attempts to develop a simple method for evaluation of maximum shear strength and ductility capacity for unstiffened steel plates subjected to predominant shear loading. The following two cases of plates are investigated: web plates without flanges and web-flange assemblages. To model the material nonlinearity, a modified two-surface model is employed. Both residual stresses and initial out-of-plane deflection are taken into account. Elasto-plastic large displacement analyses are conducted with the consideration of monotonic and cyclic loading patterns. From the results, the effects of slenderness parameter of the plate, flange thickness to web thickness ratio, and loading pattern are discussed. The maximum shear strength and ductility capacity obtained are compared with some approximate formulas available in the literature, and design equations for predicting maximum shear strength and ductility capacity are suggested for the web plates and web-flange assemblages. KEYWORDS Unstiffened plate, stability design, shear strength, ductility, shear buckling, cyclic loading INTRODUCTION Steel flat plates subjected to predominant shear loading are commonly found in beams and beam-tocolumn joints in thin-walled steel bridge piers. These plates are characterized by high width-thickness ratio and, thus, susceptible to severe buckling under a catastrophic earthquake. A pictorial view of shear panels buckling at the mid-span of the beam in a bridge pier damaged in the 1995 Kobe Earthquake is presented in Fig. 1. As shown, stability is an important problem that governs the performance (strength and ductility) of slender web plates. In evaluation or design of these plates, maximum shear strength and ductility capacity are ones of major indices that need to be checked with the seismic demands. Prediction methods of the maximum shear strength have been developed so far. A pilot study was carded out by Basler (1961). Since then, various maximum shear models based on the concept of diagonal tension band were proposed. Among them, the model of Porter et al. (1975) seems to be the most comprehensive because of good agreement between predicted and experimental results. Most studies mentioned above were conducted on plate girders with widely spaced transverse stiffeners

146

TABLE 1 BOUNDARYCONDITIONSOF WEB MODEL u

,,

w

o~

o~

o~

x=O

1

0

1

1

0

1

x=a

0

0

1

1

0

1

~y=O

0

0

1

0

1

0

y=b

0

0

1

0

1

0

free = O, fixed = 1 Figure 1. Shear buckling in web plates of a bridge pier

Figure 2: Finite element model of web model

Figure 3" Distribution of residual stresses of web model

under monotonic, predominant shear loading. The maximum shear strength can alternatively be computed from an empirical formula (Nara et al., 1988), which takes an advantage of simpler design method. For the ductility capacity, very few attempts have been made. Experimental investigations of unstiffened webs under cyclic shear by Krawinkler & Popov (1982) and Fukumoto et al. (1999) are examples. However, their work is limited to few cases of plate parameters. In the present study, maximum shear strength and ductility capacity of unstiffened web plates are investigated. Web plates with and without flanges are considered to gain insight into the contribution of the flanges. Key geometrical parameters of the plates are varied over the practical range, and both monotonic and cyclic loading patterns are adopted. A commercial computer program, ABAQUS (1998), including a constitutive law developed at Nagoya University is employed to carry out nonlinear finite element analyses. Finally, the effects of the geometrical parameters are discussed and design equations for the evaluation of shear strength and ductility capacity are suggested. ANALYTICAL METHOD Unstiffened web plates and web-flange assemblages (referred as web model and web-flange model, respectively) considered are those found in practical steel bridge piers having thin-walled box section. All webs have the width, b, of 2000 mm. The length of the web, a, being the spacing of diaphragm, is kept equal to the width b, since preliminary analyses show insignificant effect of the web aspect ratio, ot (equal to a/b), on the maximum shear strength and ductility capacity. These plates are fabricated from the steel type SS400 (Japanese specification) having the modulus of elasticity, E, of 206 GPa, the yield stress in tension, Cy, of 235 MPa, and the Poisson's ratio, v, of 0.3.

147

Figure 4: Web-flange model

Figure 5: Loading Patterns

Web Model The web model represents the case of web-flange assemblage whose flanges are extremely flexible. Its finite element model is shown in Fig. 2. The boundary conditions are assumed in accordance with Table 1, where u, v, and w denote x-, y-, and z-displacement degree-of-freedoms (d.o.f.'s), respectively, and 0~, Oy, and 0., represent rotational d.o.f.'s, respectively. The node (0,0) is hinged, and shear stress on the edges x = 0 and x = a is generated by means of the displacement 8 at the node (a,0) imposing the constraint that y-displacements at nodes on each loaded edge are linearly interpolated from those at the two ends of the edge. The other two edges unloaded are modeled as infinitely rigid bars and, as a result, the in-plane displacements at nodes along each unloaded edge are linearly interpolated from those at the two ends. According to the assumptions mentioned above, the stress state of the plate before initiation of buckling is close to pure shear condition. The web is divided into 14x14 mesh. Such element division is found to be sufficient to obtain accurate solution. A 4-node doubly curved shell element with five integration points along the plate thickness is employed. In the analyses, material and geometrical imperfections are taken into account. The distribution of residual stresses is illustrated in Fig.3. The tensile residual stress, art, and the compressive one, crc, are applied in x-direction with the magnitudes of ay and 0.3ay, respectively. The initial out-of-plate deflection, w, is assumed in Eqn. 1. The deflection amplitude ww is set as b/150. w= w w sin( a~-~)sin(~ )

(1)

Web-flange Model For the web-flange assemblages (box section), the width of the flange is assumed equal to that of the web. Only a half of the box section is modeled due to the symmetry about the plane z = -b/2 (Fig. 4). As a consequence, Ox, Oy, and w of nodes on the symmetric plane are vanished. Boundary conditions and constraints adopted are similar to those of the web model, except at the edges where the flanges are connected. In addition, all displacement d.o.f.'s on the edge x = 0, y = 0 are constrained to represent hinge supports. At all nodes on the plane x = a, the rotation 0x is not allowed and the displacement u is assumed identical throughout the plane. Vertical shear displacement 8 is applied uniformly along the edge x = a, y = 0.

148 Finite element division of the web-flange model is 14x14 mesh for the web and 7x14 mesh for the half of the flange considered. The distribution of the residual stresses similar to that of the web model is applied to the web and the flanges (Fig. 4). The initial deflection is introduced in the form of Eqn. 2, where ww = wf= b/150. - wwsin(nx / a) sin(ny / b) w = ~- w/sin(nx / a) sin(m / b) [w: sin(~ / a) sin(m / b)

for z = 0 for y = 0

(2)

for y = b

The web and web-flange models are characterized by the web's slenderness parameter

R~=

k

=twV

k . n 2E

= I4.00+ 5.34/t~ 2

for ct__l

(3)

(4)

Here, tw = web thickness, xy = yield stress in shear (equal to C~y/4~according to the von Mises yield criterion), Xcr = buckling shear stress, and k = elastic buckling coefficient of web plate in shear approximated by Eqn. 4 (DIN 4114, 1953) in which cc = 1.0. Elasto-plastic large displacement analyses are carded out by employing ABAQUS computer program (1998). A modified two-surface model (2SM) developed at Nagoya University (Shcn et al., 1995) is incorporated to trace the material nonlinearity. Two loading patterns: monotonic and cyclic cases shown in Fig. 5 are applied to the web and web-flange models by means of the displacement ~5, in order to investigate the effect of loading history on shear strength and ductility capacity. The magnitude of ~5is increased step-by-step as a multiple of the yield displacement of the plate in pure shear, ~Sy.The internal force and deformation of the two models are determined by means of average shear stress, ~, and average engineering shear strain,~, as follows:

~=P/(btw) and ~ = 8 / a

(5)

where P is the summation of shear stressesalong the loaded edge. All models are loaded up to failure defined at the stateat which the strength drops by 5 % aRer shear buckling. However, there exists some case where shear buckling is not dominant. The average shear stress increases until yielding, yield plateau forms, and strain hardening develops. In this case, the failure is encountered when the shear strength drops to 95% of the yield value.

EVALUATION O F WEB MODEL Parametric study is carried out on the web model with the parameter R~ ranging from 0.3 to 1.3 (b/t = 34 to 147). In this paper, some results are discussed. More detail of the investigation can be found elsewhere (Usami et al., 1999). Analysis results show distinguished difference in the ultimate behavior of thin and thick web models. Relatively thin plates (R~w> 0.3) quickly reach the failure aRer shear buckling, while thick plate (R~ = 0.3) develops yield plateau and strain hardening without any sign of shear buckling. In Fig. 6,

149 1.4

1.2

~

~ ' 1---9 9 I .... ........... ~ ..........i " ' ] - ' -

i'~i ~,

,

Euler curve G u i d e l i n e (1987) N a r a r al. (1988)

i

i

15

i :

,

!

v t~"9

0 n

.............. " ....... " ~ " ! ......

i

1

o

Monotonic One-side

V

Two-side

-

...."-V--

~.~]o lib-

~ 0.8

I o' M..o,o.,. I

0.6

0.4

"I

12

I v ,

,!,

One-side

I".N"

t .............. "~'" .......'""~"~

r,.o-,~e I ,

I

,

,!,

,

0.5

,,

,

I

5

0 0.5

0

Figure 6: Maximum shear strength of web model

I

Figure 7: Ductility capacity of web model

normalized maximum shear strength, Xm/Xy, obtained from monotonic (circular marks) and cyclic (triangular marks) loading are presented with the design curve suggested by the Japanese guideline for stability design (published in 1987 and adopted in Japan Road Association, 1996) and the Euler curve. An empirical formula (Eqn. 6) proposed by Nara et al. (1988) is chosen for comparison, due to its simplicity. The computed results agree very well with Nara et al.'s curve. It is also observed that the effect of the loading history on the shear strength is insignificant. x_..~= [ 0.486 / 0.333 "Cy

R~ )

for 0.486 < R~ _ 1,0 bi (i=l ..... n) Correction terms Mean value correction Correlation coefficient Variation coefficient

b p Vr

b > 1.1 P > 0.9 Vr < O.1O0

Safety factor

?g

YM = 1,1

As ((XLT) values increase, more and more (bi) values (initially less than unity) will comply to the relation: b~ = re--L/___1.0

(12)

These values are in fact individual safety factors for each specimen and when ((XLT) reaches such a value that every bi > 1.0 this means: rei > rti,Vi = 1,...,n In other words, for this value of the generalized imperfection factor, the theoretical model is on the safe side compared to the available experimental values. Therefore the calibration criterion proposed by the authors is the following: CR/TERION: The calibrated (aLr) value is corresponding to the obtention of all (bJ values greather than unity and simultaneously of their average (b) greather-equal than 1.1 (this last value being in fact the safety factor prescribed by Annex Z in case of stability phenomena) i.e.: m

199

{~

~>1.0

V i = l ..... n

(13)

>1.1

PROCEDURE APPLICATION ON ' T ' H.R. AND WELDED STEEL MEMBERS The calibration procedure focuses on the model for members in bending adopted in EUROCODE 3ENV-1993-1-1, where the generalized imperfection is taken according to eqn (2) instead of eqn. (1) The experimental results supplied in the frame of EC.3-Background Documentation, Chapter 5 / Document 5.03/ October 1989, (Eds. Sedlacek G. et. al.), have been used to apply the proposed procedure. a) In case of hot-rolled steel profiles a number of 144 test results, from a total of 243 tests (selected by European experts as representative for lateral-torsional buckling of beams) have been available. For what regards the structural shapes used for the tests, the studied profiles are representative for most of the hot-rolled sections used arround the world: I or H sections produced in Western Europe, North America and Japan. It must however be noticed that the depth of the tested beams never exceeded 305 mm so that the representativity of the tested beam population is restricted to this depth range ! Because the 144 tests were carried out by several researchers in different laboratories all over the world, it was accepted that they are well representative of the testing conditions. In order to check for the influence of steel strength on hot rolled profiles classification, the authors have divided the initial 144 hot rolled specimens in two sub-sets i.e. 123 specimens with fyOm < 355 N / m m 2 and 21 specimens with f;om > 355 N / m m 2 The influence of thermal treatment on analyzed profiles framing was also in view, by selecting annealed profiles and calibrating separately on subsequent subset. TABLE 2 - CALIBRATION RESULTS OBTAINED ON H.R. AND WELDED STEEL PROFILES Subset name:

Dibley Suzuki Lindner Trahair LindnerSchmidt U.N. SUB-1 SUB-2 SUB-3 Reunited specimens SUB-4 SUB-5 Reunited specimens

Total nr. of specim.

21 54 11 29 15 90 33 16 144

35 18 53

Spec. within coupling range

Correlation coefficient

Variation coefficient

Safety factor"

(p)

(Vr)

(YM)

Calibrated (GLLT) value

1.2268 1.1640 1.2667 1.1827 1.1579

0.152 0.214 0.181 0.115 0.197

0.096 0.101 0.122 0.098 0.106

1.2505 1.1524 1.2606 1.2110 1.1607

0.181 0.197 0.174 0.044 0.197

WELDED "I" PROFILES 0.962 0.126 0.967 0.095 29 0.991 0.127

1.2461 1.1805 1.2217

0.444 0.232 0.350

HOT ROLLED "I" PROFILES 0.994 0.098 17 0.859 0.094 0.946 0.092 0.926 0.088 19 0.996 0.096

38 13 57

18

0.953 0.993 0.991 0.994 0.994

200 b) In case of welded beams, a number of 71 test results (selected as representative by European experts from a total of 96 tests) have been available. All data concerning the specimens are listed in EC.3-Backgrounds. For every test specimen, all actual properties (mechanical and geometrical) were measured. All the beams were submitted to moment loading. In order to check for the influence of steel strength on welded profiles classification, the initial set of 71 welded specimens was divided in two subsets i.e. 51 specimens with fyOm< 355 N / m m 2 and 20 specimens with f~om > 355 N / mm 2 All specimens having a reduced slenderness less than 2Lr = 0.4 have been eliminated from the application set as irrelevant for the proposed model (out of model definition range). The last proposal of prEN-1993-1-1, to use YM0 = 1.0 on this range of member reduced slenderness (instead of YM0 = 1.1 ) was not yet analyzed by the authors, being part of a future research. A thorough analysis has been performed by subdividing the available test results into various subsets in order to observe the possible differences in terms of (aLT) calibrated values. The results are presented in Table 2. NOTE: The meaning of SUB-1 to SUB-5 denominations are the following: 9 SUB-1 is containing all H.R. specimens with f~o., = 235 N / m m 2 , for which no thermal treatment was applied; 9 SUB-2 is containing all H.R. specimens with f,om = 235 N / m m 2 , annealed; 9 SUB-3 is containing all H.R. specimens with f,o,, = 450 N / m m 2 , for which no thermal treatment was applied. 9 SUB-4 is containing all welded specimens with f;om = 235 N / m m 2 and jyf.... = 314_

N/mm 2

9 SUB-5 is containing all welded specimens with fyom = 450 N / m m 2 and f~om = 690 N / m m 2 OBSERVATIONS ON THE OBTAINED RESULTS (TABLE 1): a) The short member range of ~ ~ [0-0.4] proposed by the model has been confirmed by the excellent corelation values and percent deviations obtained; b) Framing of hot-rolled I members on curve "a" is confirmed; c) Framing of welded I members on curve "c" is confirmed; d) Members made of high strength steel seem to require framing on the previous higher buckling curve ("a0" instead of"a" for hot rolled profiles and "b" instead of"c" for welded members); e) Thermal treatment (annealing) of hot rolled profiles also seems to lead to profile framing on proximum higher buckling curve. According to the second draft of prEN 1993-1-1, a new model was recently proposed by TC.8 Committee of ECCS, i.e. for bending member of constant cross section, the value of XLT for the m

appropriate non-dimensional slenderness ~,LT shall be determined from:

~LT ----

(I)

+

1 4- r ~T -- ~" ~2L'-T but ~ LT < 1,1

(14)

LT

where (I)LT :0,5"[13+(XLT(;~LT--0,2)+~LT ] and ~=0,87. New calibrations are thus required on this model, using upper sets of experimental results and possibly other results, in order to test its validity.

201 PROCEDURE EXTENSION ON T.W.C.F. STEEL MEMBERS The proposed procedure may be also extended on TWCF members in pure bending, starting from eqn. (4). On this base, the non-dimensional form of the resistance moment may be written as:

m MLTMb'Rd _ ZLr 9Weff,y 9fy =~LT" Weff,y -- ZLT" QLT Mp, Wpl,y 9fy Wpl,y

(15)

where Wpl,y denotes the plastic modulus of member gross cross section and Weft the elastic modulus of the effective cross-section In uper relation, the reduction factor (ZLT) is given by the following equation: 1

ZLT --

--2 0.5 < 1.0 (I)LT "}"[(I)2T--~LT]

(16)

where:

(I)LT-- 0.5 " I1 + (XLTCLT -- 0.4)q- ~2LT]

(17)

The relative reduced slenderness of eqn. (17) results from the following equation:

-

/woffly.JMp,

~LT-'VWpI,Y

VM- : Q~-LT''/~p'-v Mr

(18)

! One should observe that for short members (with ~,LT ~ 0,4 ) where ZLT "- 1 and the local instability mode is present only, equation (15) becomes: W~

MLT =

y

' =QLT Wpl,y

(19)

In the frame of the ECBL Theory proposed by Dubina (1993), the following relation exists in the coupling point of local and global instability mode (located on the buckling curve by abscissa -

A,tr = ~

1

)

MLT = (1-- tl/LT)-QLT

(20)

where WLTis called "erosion coefficient" Furthermore, a link relation of the type described in eqn. (3-b) exists in this point, between the generalized imperfection factor "OtLT" and "WET": Thus, by determining the position of the coupling point (i.e. through a calibration of ~dLTusing the procedure proposed by the authors) one can control the buckling curve defined by eqns. (16) to (18) The upper described procedure was applied on the available sets of TWCF profiles in pure bending (lipped channel sections called "C" and channel sections called "U" tested by Lovell in 1985) in order to calibrate realistic "O~LT"values required by test evidence. The obtained results are hereby presented:

202 TABLE 3 - RESULTS OBTAINED ON T.W.C.F. MEMBERS IN PURE BENDING Subset name:

Total hr. of specim.

Spec. within coupling range

Correlation coefficient

Variation coefficient

Safety factor

Calibrated

(p)

fVr)

(~/M)

value

27 9

7 6

0.980 0.956

0.120 0.103

1.2172 1.2119

0.07 11.14

C-Lovell U-Lovell

(~LT)

According to these results, the "U" sections in bending would be framed on buckling curve "a" while "C" sections on buckling curve "a0". This remark must be, of course, limited to the analyzed set of specimens. For a general conclusion, further studies are necessary.

CONCLUDING REMARKS A unified procedure built to calibrate (or) generalized imperfection factor used in the Ayrton-Perry equation of the European buckling curves was presented by the authors. This procedure is valid for hot rolled, welded or thin-walled cold-formed steel members, either in compression or in pure bending. A review of subsequent design models and present cross section classifications prescribed by EC.3 is thus possible. An application of the procedure for members in bending in connection with the present preocupations of ECCS-TC.8 Committee aiming to issue Euronorm EN-1993-1-1 is presented.

REFERENCES Byfield M.P., Nethercot D.A.- "An improved method for calculating partial safety factors" Seventh International Conference on Applications of Statistics and Probability, Paris, 1995 Dubina D. - "Coupled Instabilities in Thin-walled Structures: Erosion Coefficient Approach in Overall-Local Buckling Interaction" - Comission of the European Comunities, cooperation in Science and Technology with Central and Eastern European Countries, Research Report, Ref. ERB 3510PL922443, Liege, October 1993. Georgescu M., Dubina D . - "'ECBL and Eurocode Annex Z based Calibration Proc. for Buckling Curves of Compression Steel Members"- SDSS '99 Proceedings of the International Colloquium On Stability & Ductility of Steel Structures, Eds. Dubina D. & Ivanyi M., Elsevier ,1999, pp 501508 Lovell M . H . - "Lateral buckling of light gauge steel b e a m s " - Research Report, Dept. of Civil engineering, University of salford, 1985 Rondal J. and Maquoi R . - "Formulation d'Ayrton-Perry pour le flambement des barres metalliques" - Construction Metallique, hr. 4, 1979 Dubina D . - "The ECBL Approach for interactive Buckling of Thin-walled Steel Members"- Steel and Composite Structures, vol. 1, Nr. 1,2001-01-05 Georgescu M. - "Coupled Instabilities in case of Thin-walled Cold-formed Members"-Ph.D. Thesis, The "Politehnica" University of Timisoara / Romania, 1999

Third International Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

203

DESIGN ASPECTS OF COLD-FORMED PORTAL FRAMES P. Frti and L. Dunai Department of Structural Engineering, Budapest University of Technology and Economics, Budapest, Hungary

ABSTRACT The paper presents a design method and its background for portal frames built-up from cold-formed profiles. It has a main focus on the most critical point of this type of cold-formed frame that is the comer. To follow the complicate behaviour of the new type semi-rigid connection experimental tests are performed. The results and their application in the design is presented. An other important part of the design is the interactive behaviour modes of the structural members. The interaction of the buckling phenomena is studied by finite strip analysis. Finally, the practical application of the system is discussed.

KEYWORDS cold-formed C-profile, portal frame, semi-rigid connection, test based design, finite strip method

INTRODUCTION General

Thin-walled cold-formed structural elements are widely used as secondary structural systems of steel building structures (e.g. purlins, wall beams). Recently the product range is grown due to the developments in the manufacturing technology. Cold-formed beams and trapezoidal sheeting are used in floors; new kind of light-weight trusses are developed for residential houses and for smaller industrial buildings. The application of portal frames built-up from cold-formed profiles, however is not typical. It is due to the difficulties in the forming of moment resistant joints by the application of uni-symmetrical profiles. In the paper a recently developed portal frame system is introduced, emphasising its typical and most important design aspects.

Structural System The Lindab Small Building System (SBS) is small-span (3-10 meter) complex, light-weight, steel building system, developed in the co-operation of the Budapest University of Technology and Lindab Ltd, Hungary. In the development the most important aspects are the application of cold-formed profiles, the simplicity of the site connections, the simple- do-it-yourself type- erection technology.

204 The skeleton of the structure is shown on Figure 1. The primary load-bearing structure (two hinged portal frame) is built up from single cold-formed C-profiles. The section height is between 150 and 300 mm and the thickness is between 1.2 and 3.0 mm. The moment resistant semi-rigid connections are solved by selfdrilling screws. The screws join the webs of the C-profiles that meet back-to-back. The distance between the frames is 1.0 meter. The secondary load-bearing elements are built-up from cold-formed hat profiles. The distance of the puffins is 400-800 ram, depending on the applied cladding. The wind and side bracings are in the end-bays. It is to be noted, that the sheeting elements of the roof and wall cladding are also considered in carrying the horizontal forces.

Figure 1: Structural system Structural Problems

The above solution of the structural systems arises the following problems: 9 Structural elements: - the single C-profiles connected to each other eccentrically, one of the flanges are usually not braced laterally; distorsional and global buckling modes and also their interaction with local buckling is possible. 9 Structural joints: - structural elements are connected in the joint eccentrically, - forces and moments are transferred only through the webs of the connecting profiles, - the application of self-drilling screws in moment resistant connections brings many uncertainties; yet there is no reliable design method nor practical experience. There is no design standard which can follow directly the above behaviour characteristics. The base of the applied design method is the related part of Eurocode 3 (1996). The complicated joint behaviour is analysed in laboratory and the verification is done by design assisted by testing method. The design values of moment resistance, the stiffness and the ductility of the frame comers are obtained from the tests. The buckling modes and the critical forces of the structural elements are determined with the finite strip method. From the results the slendemess ratios are calculated. The static analysis- using the properties of the structural elements, joints and sheeting plates - is realised with plane- and space flame models, taking into account the stiffening effect of the trapezoidal sheeting. In the following the paper deals with the two most important aspects of the development, namely the test based joint design and the stability design of the members by the finite strip method. -

205 JOINT BEHAVIOUR AND DESIGN

ExperimentalProgram One of the basic requirement of the development that the structure have to be quick an easy to be erected. That is why it is decided to use single C-profiles connected each other back-to-back through their webs, without the application any additional elements (e.g. gusset plates). This simplification in the structural solution, however, results in a very complicated behaviour mode. To create a reliable design model of the joints laboratory test program is performed (Dunai, F6ti, Kaltenbach & K~ill6, 1999-2000) to follow the difficult structural behaviour. During the development, in three steps, a total of 26 full-scale tests are completed. Figure 2 shows a typical test setup with self-drilling screws in the frame comer connection of the specimen.

\ 125125 125 125 125 125,,I

~

,

I-I.5 m

_L~

..'/y/x,'/,"

:: -- ___

Figure 2: Test setup and a typical joint with self-drilling screws To keep the frame in plane hat profile purlins and a C-profile under the joint is used, simulating the lateral supports of a real structure. This longitudinal C-profile is also used to provide aid under the erection of the frames. The concentrated force is applied on the structure with 1.0-1.5 m arm through the beam, adjusting the proper ratio of bending moment and shear force. Table 1 contains the range of varying parameters of the experimental program. It is noted, that beside the self-drilling screws, metric bolts and high tolerance bolts are also used in tests. During the design of the test setup and the loading history the recommendations of Eurocode 3 is followed. TABLE 1. TEST PARAMETERS

Number of experiments 26

Type of fasteners

Number of fasteners 4-22

Thickness of the[ Height of the section ,, I section 1,0-3,0 [mm] I 15.0-300 [mm]

b/t ratio of section 100-200

In the tests the following parameters are measured: concentrated force, displacements of the joints, rotation and stresses (in 2 points on the flanges of the beam, near to the joint). Test Results

The results are evaluated basically by the moment-rotation diagrams. Three different modes of failure are separated: 1) local buckling in the C-profile, 2) pull-out of fasteners and 3) sheafing of fasteners.

206 In the behaviour the interaction of the above characteristics could be observed, as it is detailed in F6ti & Dunai (2000). In most cases, however, one of them is dominant in the failure mode. Due to the type and arrangement of the fasteners local buckling could develop in two main different ways. Figure 3 shows the difference in the two types of failure modes by the moment-rotation diagrams.

~

a"

g

3

~2 oE

,,b"

4

.:,:.3

1

1

0

0

0s

Os

0s

0s

OD8

0.10

0,12

0

0,14

0s

i)s

0s

0s

0s

OD8

I : ~ t i ~ [rad]

Rota-tion Ira d]

Figure 3" Local buckling in a) metric bolted and b) screwed specimen On Figure 3a the behaviour of a classical bolted connection can be seen. There is a large slip about in the beginning of the load application due to the shift between the bolt and the hole diameter. To avoid this slip the gap had to be eliminated. High tolerance bolts or self drilling screws can solve this problem. The previous possibility results in good structural behaviour but requires very accurate fabrication and erection. The usage of self-drilling screws much simpler from practical point of view, and the behaviour is favourable, as it can be seen in Figure 3b. 7

.b"

15

,,

8

~4 Eo3 ~2

~. 10

~ 5

1

0

,

0

0s

0s

0.12

0.18

0

,

ODO

OD1

0s

Rotation [rad]

OD3

0 04

0s

R otation [rad]

Figure 4: Shearing of screws a) one by one b) all at once When the failure mode is shearing, two different types of behaviour are experienced, as it is shown in Figure 4. On Figure 4 a - after the initial nonlinear part- the screw failure happened one by one, while on Figure 4.b all the screws sheared once and caused sudden failure.

~

1,5

~0,5 0 0

0,05

0.1 Rocmion [tad]

0.15

s

Figure 5" Tilting and pull-out of screws

207 Beside the above typical modes of failure a special one is experienced when relatively small number of self-drilling screws are used in thin profiles: the tilting and pull-out of screws. The thin web plate of the profile suffer local bending in the surroundings of the screw instead of shearing the fasteners. The fasteners tilting and as the moment increasing, starting to pull-out. The typical moment-rotation diagram can be seen on Figure 5. Conclusions on the Test Results

After performing 26 tests it is found that for practical design purposes the most suitable connections are the self-drilling screwed joints with spread arrangement. They perfectly eliminate the slip in the initial range. By using suitable number and arrangement of fastener shearing of screw can be avoid and local buckling in the member can be reached instead. They also represent ductile behaviour after the top of the load-bearing capacity is reached. Derivation of Design Values

On the bases of the test results the design values of the joint characteristics are derived. The joint behaviour is characterised by the initial stiffness, moment resistance and ductility. The design values of these parameters are defined as follows: The initial stiffness is defined as the secant stiffness at the design moment level. The characteristic value of the stiffness is taken as a mean value of at least two tests (Eurocode 3 recommendation). The moment resistance is calculated from the maximum moment, measured in the tests. The characteristic moment resistance Mk corresponding to this test is obtained from the test result Madj as it is given in Eqn. 1, according to the recommendation of Eurocode 3. In the equation rlk is taken 0.7. Mk = 0.9"qk"Madj

(1)

When the failure is in the members (local buckling), there is a close relation between the failure moment measured in the tests (Mr) and the cross-sectiorial moment-resistance of the member section (Mc.rd). Due to the test conditions (force transfer through the web, eccentricities) this ratio is about 60%. The ductility is defined as an amount of rotation and it is measured from the zero point until the descending branch reaches the level of the ultimate moment.

ANALYSIS AND DESIGN OF STRUCTURAL ELEMENTS Thin-walled cold-formed profiles under moment and/or axial force lose load-bearing capacity by buckling failure. Due to the geometric properties the following buckling modes must be considered: 9 local buckling, 9 distorsional buckling and 9 global buckling (flexural-torsional or lateral-torsional buckling). It is very important, that the above buckling modes have usually interaction in the behaviour of the structural elements. The interaction phenomena between them have to be taken into account. In this design process the finite strip method is used to obtain the linear critical loads. The structural elements with same cross-sections but different lengths are checked under moment and axial force. It can be handled as one bar buckled in different lengths. The virtual lengths of the elements are equal to buckling half-wavelengths of the examined bar. The results are elastic critical forces for these different half-wavelengths as shown in Figure 6. The diagram is based on a 150 mm high and 1.5 mm thick C-profile subjected to pure bending moment. The curves for local, distorsional and overall buckling modes can be clearly separated.

208

ZE f...

3lJ

0 E m

211

0

9

~,~

uckling

,

,] I1)

I(H}

I~XX)

Half-wavelength [mm] Figure 6: Critical buckling moment for different lengths m

From these critical forces the relative slenderness ()~) for a bar in a given length can be calculated. Below there is an example for the calculation in case of lateral-torsional buckling of members subject to bending by the Eurocode 3 (1996). To calculate the design buckling resistance moment (Mb,Rd)the reduction factor (~x) of Eurocode 3 is used, in the function of )~LTwhich can be calculated as shown in Eqn. 2.

~LT ._ /L "Weft V Mcr

(2)

In the equation fy is the yield strength, Weffis the section modulus of the effective cross-section, and Mcr is the elastic critical moment for lateral-torsional buckling, obtained from the finite strip analysis.

APPLICATION OF THE DESIGN METHOD The above design procedures of joints and structural elements are built into the global structural design method of cold-formed portal frames. On this bases the Lindab SBS system is developed. Its architectural and structural design, fabrication and sale management is supported by an integrated Lindab-SBS SOFT computer program system. The development of the system began in January 1999, the selling of the new product started in the spring of 2000. In the practical application more than 100 Lindab SBS buildings are built by the end of 2000, serving different functions. Among them the smallest is 3x5 meter and the biggest is 10x22 meter floorspace.

Acknowledgement The research work has been conducted under the financial support of OTKA T035147 and Lindab Ltd.

References ENV 1993 Eurocode 3. (1996). Design of steel structures - Part 1.3. General rules - Supplementary rules for cold formed thin gauge members and sheeting Dunai L., F6ti P., Kaltenbach L. and K~l16 M. (1999-2000). Experimental study on frame corner joints built-up from cold-formed C-profiles, Research Reports (1-3) (in Hungarian), Technical University of Budapest, Department of Steel Structures, Hungary P. F6ti and L. Dunai: (2000). Interaction phenomena in the cold-formed frame comer behaviour. The Third Int. Conf. on Coupled Instabilities in Metal Structures (CIMS 2000, Lisbon, Portugal), 459-466.


209

LOCAL BUCKLING AND EFFECTIVE WIDTH OF THIN-WALLED STAINLESS STEEL MEMBERS

H. Kuwamura, Y. Inaba, and A. Isozaki Department of Architecture, School of Engineering, the University of Tokyo, 7-3-1Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

ABSTRACT Local buckling behaviors of thin-walled stainless steel stub-columns were experimentally investigated. Six types of sections, i.e., angle, channel, lipped channel, H-shaped, square box, and circular cylindrical sections were tested. These specimens were formed from two grades of austenitic stainless steels designated SUS304 and SUS301L 3/4H, whose specified yield strengths for design are 235 and 440 N/mm2, respectively. Effective width-to-thickness ratios of unstiffened and stiffened plate elements and limit diameter-to-thickness ratios of circular cylinders were established from the test data. It was found that the effective width-to-thickness ratios must be expressed by different equations for the two strength grades because of the difference in strain hardening.

KEYWORDS stainless steel, local buckling, thin-walled section, effective width, stub-column

INTRODUCTION Stainless steel has a significance in its corrosion resistance, while it has been scarcely used in structural skeletons of buildings due to high price. However, recent change of social mind from mass production and abundant consumption to ecological coexistence with natural environment, sustaining a long life of buildings is of much concern in construction engineering. In that context, stainless steel is expected to be promising material for architectural construction. The research for utilizing stainless steel in building structures has been conformed in the advancement

210

of cold-formed steel, especially in the USA, by G. Winter and his followers, which is resulted in a design manual such as "Design of Cold-Formed Stainless Steel Structural Members -Proposed Allowable Stress Design Specification with Commentary (Lin, Yu, & Galambos 1988)." Japanese history of research on structural stainless steel for building use is back to only a decade. At that period of virtual growth of economy, researchers and engineers intended to use stainless steel in heavy steel construction and then a specification of design and construction was published by SSBA (Stainless Steel Building Association of Japan 1995). The most promising use of stainless steel in buildings is obviously in the form of light-weight members relying on its corrosion resistance, which may compensate for its high cost in fabrication as well as material. Thus, we need a design method of thin-walled sections of stainless steel with a higher strength in order to reduce weight. In this study, SUS301L 3/4H which has about twice strength of SUS304 is studied. The engineering concern in applying such thin-walled members is focused on local buckling of plate elements. In this study, stub-column test is performed to establish effective width-to-thickness ratios on the basis of post-buckling strength.

MATERIAL Stainless steels investigated in this study are SUS304 and SUS301L 3/4H, whose specified yield strengths for design (F-values) are 235 N/mm 2 and 440 N/mm2, respectively. Both stainless steels are austenitic types which are solution heat-treated after cold rolling, but SUS301L 3/4H is further thermal-refining rolled to increase strength. Exceptionally, SUS304 circular hollow sections of 76.3-mm and 48.6-mm diameters are formed from hot-rolled strips. Their chemical compositions and mechanical properties described in mill sheet are summarized in Table 1. It is observed that SUS301L 3/4H strips have higher yield ratios and less ductility than SUS304 strips. TABLE 1 MATERIAL PROPERTIES IN MILL SHEET stainless steel

thickspecification

chemical composition(%)

mechanical properties

ness C

Si

Mn 0.78

P

S

Ni

Cr

{rnrrl) 3.0

0.07

0.52

cold-rolledstrip

1.0

JIS G 3459

1.5

0.05 0.05 0.05 0.04

0.38 1.00 0.26 1.04 0 . 5 3 0.98 0.59 0.96

0.023

0.36

iJIS G 4305 SUS304

~,,

pipe JIS G 4305

SUS301L 3/4H cold-rolled strip ~hermalrefmin$)

3.0 1.5

0.2%-offset tensile yield elongastrength strength ratio tion (~l/mm2~ fN/mm2~ L'%)

0.040! 0.005

8.06

18.30

0.036 0.032 0.034 0.028

0.004: ! 0.007, 0.005 0.006

8.10 8.11 8.33 8.29

18.26 18.45 18.12 18.18

279 .. --312 261

641

0.44

57

633 649 583 612

--0.54 0.43

55 52 53 59

1 . 4 0 0.030 0.005

6.62

17.45

508

829

0.61

43

511

832

0.61

41

stub-column specimen angle, channel, lipped channel,

H-shape,dr square tube lipped channel circular tube(except'bellows) ' circular tube #76. 3 ..... circular tube ~46.8 angle, channel, lipl~d channel, H-shaped, square tube circular tube

Mechanical properties of coupons are investigated also in laboratory as shown in Table 2. It is noted that laboratory test coupons of circular sections are taken from pipes. It is known that stainless steels have following distinguishable mechanical properties in comparison with carbon steels : (1) proportional limit of stainless steel is fairly low and non-linearity appears at a low stress level, (2) strain hardening of stainless steel is considerable and thus yield ratio (yield strength / tensile strength ) is very low, and (3) initial Young's modulus of stainless steel is slightly less than carbon steel ( nominal value of Young's modulus of stainless steel is 193,000 N/mm 2, while that of carbon steel is 205,000 N/mm2).

211 From the first item, yield strength of stainless steel for design is defined as 0.1%-offset yield strength of coupon test (Stainless Steel Building Association of Japan 1995), while 0.2%-offset yield strength is generally used in material specification as in Japanese Industrial Standards. This study follows the def'mition of 0.1%-offset yield strength. TABLE 2 MECHANICAL PROPERTIES OBTAINED FROM LABORATORY COUPON TEST nominal 0.1%uniform ~rupture initail thick- measureJ tensile yield elonga-elonga- Young's thicknes~ offset strength stub-columnspecimen hess (ram) strength (N/ram2) ratio tion tion modulus . . . . (•m) ,, (N/~Eq:~) , , (%) (%~ (]N/mm2) angle, channel, lipped channel, cold-roiled strip 3.0 . . 29.2 . . 249 . . . . . . 203,000 H.shapedr.squaretube 1.0 0.94 257 . . . . 203,000 lippexl' channel SUS304 pipe ~48.6 'i.42 239 600 0.40 57 63 202,000 076.3 1.40 239 694 0.34 69 73 202~000 iPress-formed ~101.6 1.5 1.38. 33 ! 737 0.45 55 60 222~000 circular tube pipe from cold- ~139~8 1.38 318 750 0.42 58 ..... 62 222t000 rolled strip ~165.2 1.36 301 742 0.41 , 57 61 2221000 angle, channel, lipped channel, cold-rolled strip 3.0 3.01 497 845 0.59 40 42 206,000 H.shapedTsquaretube ~48.6 1.51 496..... 932 0.53 45 51 ! 50r01)0 SUS301L 3/4H press-formed d~76.3 1.50 458 928 0.49 46 53 1771000 pipe from cold- d~!01.6 1.5 1.51 451 904 0.50 .45 51 185~000circular tube roiled strip ~!39.8 ! 1.49 420 902 0.47 48 .. 54 1571000 , ~165.2 1.49 420 889 0.47 53 59 188,000 stainless

shape

steel

STUB-COLUMN SPECIMENS

Sections of stub-columns are angle with equal legs, channel, lipped channel, H-shaped, square hollow, and circular hollow as shown in Figure 1. Angle, channel, and lipped channel are cold-press-formed from 3-mm thick cold-rolled stainless steel strips, partly 1-mm thick only for lipped channel. H-shaped sections are built-up from 3-mm thick cold-rolled stainless steel strips by laser beam welding or partly by TIG welding. Square hollow sections are built up from two cold-press-formed channels by laser beam welding. Circular hollow sections are cold-press-formed from 1.5-mm thick cold-rolled stainless steel strips by means of TIG welding except 76.3-mm and 48.6-mm diameters of SUS304 which are formed from hot-rolled strips by automatic arc welding.

r__~BJ ~ L

I

,B - ~ ~ "~rBs

~ B t, - ~

~ D r__~_._d___~ ]j,r

arc weld _

B

~? rL?

p~ess b" .~1Dlt.~es s D['?a~' ~l~li,_t l-/~' bt,press s~

B

b=B-r

~

.

O]? ~dl~ a st__,JF or T I G er laser weld w e l d~ t

t

t

~

!

_

b=B-r d=D-2r

b=B-2r d=D-2r bs=Bs-r

b=(B-t)/2 d=D-2t

d=D-2r

Figure 1 9Shape of stub-column section and method of forming Width in this paper is defined as the width of a flat plate element excluding comer as shown in Figure 2. For example, the width b of an angle is equal to B - r, in which B and rare the whole width and

212

the outer radius of the comer, respectively, and the width d of a channel web is equal to D - 2r, in which D is the whole depth. For an H-shaped section, flange width b and web width d are determined by neglecting weld, because the fillet size by laser or TIG welding is very small. For circular hollow sections, outer diameter D is adopted. Length of each stub-column specimen is three times the whole width of the section. For channel, lipped channel, and H-shaped sections, the length is larger of 3B and 3D. Seventy three specimens of stub-columns are scheduled as listed in Table 3. The section sizes in the table are nominal, while measured sizes, which are much more important for thin plates, are used in analysis. TABLE 3 SCHEDULE OF STUB-COLUMN SPECIMENS section of stub-column angle channel

stainless steel SUS304 SUS301L 3/4H

specified yield stress for design

nominal thickness (mm)

F ~/mm2~ 235 440

SUS304

235

SUS301L 3/4H

440

3

3 SUS304

235

lipped channel

H-shaped

square hollow circular hollow

1 ,, SUS3OIL 3/4H

440

3

SUS304 "' SUS301L 3/4H

235

3

440

SUS304 SUS301L 3/4H

235 440

SUS304 SUS301L 3/4H

235 440

, 3 1.5

nominal width (mm~

B 25"~60 B 25"--~ B 25 ~-50 D 50--- !50 B 25---50 D 50---150 B 50---75, Bs 20---25 D 100"200 B 17"-25, Bs 7--'8 D 33"67 B 50"-75, Bs 2 0 " 2 5 D 100"200 B 50--" ! 50 D 50"--200 B 50"--150 D 50~'200 D 50---200 D 50"-200 D 4 8 . 6 " 165.2 D 48.6 "-~ 165.2

number nominal width-to-thickness of snecimens ratio 6 b/t 6"-- ! 8 .... 6 b/t 6"-- 18 b/t 6--- 15 d/t. 13-,-46 b/t 6--- 15 d/t 13"-,46 ' bit 13"--21, b~t 5",~6 d/t 2 9 " 6 3 b/t 1 3 " 2 1 , bs/t 5"~6 d/t 2 9 " 6 3 b/t 13"-'21, bs/t 5"~'-6 d/t 29"-63 b/t 8--~25 d/t 13"--65 b/t 8"--25 d/t 13"--65 d/t 13---63 d/t 13"-63 D/t 33 "~ 110 D/t 33"-110

6 5 4 4 4

8 8 6 6 5 5

EFFECTIVE WIDTH Application of Karman Equation

Onset of local buckling of a flat plate does not mean the failure, because post-buckling stability with an elevation of strength is usually expected. This post-buckling strength is owing primarily to redistribution of stress in a buckled plate in which higher stress beyond buckling stress is distributed in a section adjacent to its support edge, which is enough to compensate the release of axial stress at the middle of the bent plate. Since the out-of-plane deformation at the maximum strength is not serious, a thin plate can be economically designed on the basis of post-buckling strength in which a concept of effective width is applied. According to Karman, following formula is commonly used in the allowable stress design of light-gage sections of steel, in which C is determined from experiment.

be= C

,

(1)

,N,

The effective width denoted by be or d e, as shown in Figure 2, is a virtual width adjacent to its edge,

213 over which edge stress equal to yield stress equal to the maximum compressive force.

//Ar

O'y is

uniformly distributed and the stress resultant is

e/2~Ar be de/,2~Ar be/~" '-be~2de/2~r bebe

~_~ d~

de/'2~_~XA r

a:zI/ar b:; ~b/2 Figure 2 :Effective width

Effective Width o f Angle Sections

An angle section with equal legs is composed of two unstiffened plate elements each of which is pin-supported at one edge and free at the other. Now, we define an experimental effective width be as follows:

(~,bet+~,Ar)'tYy=emax

(2)

where the first Y. means the summation of effective sectional areas of flat plates and the second is the summation of the comer areas Ar. For an angle section two fiat plates and one corner are involved. O'y is the 0.1%-offset yield stress obtained from tension test, and Pmax is the maximum compressive force of the stub-column. Experimentally determined effective width-to-thickness ratios be /t which are calculated from Eqn. 2 are plotted against actual width-to-thickness ratios b/t in Figure 3. It is noted that be /t-values of angles with small width-to-thickness ratios are greater than b/t-values, because strain hardening beyond full yielding of entire section can be attained. The observed limit value of b / t to assure full yielding is 14.7 for SUS304 and 9.6 for SUS301L 3/4H, beyond which experimental be /t-values tend to keep constant. The yield stress of these plates are 249 and 497 N/mm 2, respectively. Substituting these values into Eqn. 2, the values of C of Eqn. 1 can be obtained with the following results: be be

230 215

for unstiffened plates of SUS304

(3a)

for unstiffened plates of SUS301L 3/4H

(3b)

It is noted that the same number is not assigned to C for the two grades of stainless steels. This indicates that the patterns of stress distribution at the maximum strength are not the same for the two grades of stainless steels. The reason why SUS304 has a higher C-value than SUS301L 3/4H may be attributed to the fact that the former material has larger strain-hardening than the latter.

214 -- .....I 0

20

ii

I

8US304

9

i ..........

I

SUS3OILa/aH

~i

15

................. .. ................. 10

i

I}

0

.

6

5

10

-

14.7 15

20

b/t

Figure 3 9Effective width-to-thickness ratio vs. actual width-to-thickness ratio of angles

Effective Width of Square Hollow Sections A square hollow section is composed of four stiffened plate elements each of which is pin-supported at both edges. As is the case of an angle, Eqn. 2 is applied to square hollow sections with a change of be by d e, from which d e / t vs. d / t is plotted in Figure 4. The observed limit value of d / t to assure full yielding is 39.0 for SUS304 and 23.8 for SUS301L 3/4H which have yield stress of 249 and 497 N/mm 2, respectively. Substituting these values into Eqn. 1, C is calculated and the following equation is established. In this case, too, different C-values are given to SUS304 and SUS301L 3/4H. d e = 615 t ~y de 530

-;-=7,

.-.

>

(4a)

for stiffened plates of SUS301L 3/4H

(4b)

L /- sus ol, ,4, r

40

for stiffened plates of SUS304

...... ["o

sus3o4

"

i ....... .............. 9 '-..............

:

, "........... ".............. ! .............. "......... = ~ .....O ............... : ! ............

i

3o

o

............ i ............ O ...........

i

:

i

i~i

i .............. i .............. i .............. i

i-"i

!o, i

lO

el/i 0

2~.~ i ~-~ 10

20

30

40

i ..................... ~ 50

60

70

d/t

Figure 4 9 Effective width-to-thickness ratio vs. actual width-to-thickness ratio of square tubes

Effective Width of Channel, Lipped-channel, and H-shaped Sections

215 The plate elements constituting channel, lipped channel, and H-shaped sections restrain each other at the state of post-buckling. Here, however, we assume that no interaction acts between the adjacent plates. For example, a channel is assumed to be composed of three independent plates such that a web plate is pin-supported at both edges and flange plates are pin-supported at one edge and free at the other. From this simplification, effective width can be calculated from Eqn. 3 for flange plates and Eqn. 4 for web plates. Effective width-to-thickness ratios predicted by this method are summed up over the entire section, which is denoted by ( ~ be / t)pred. On the other hand, stub-column test gives the experimental effective width as follows:

(~,bet+ZAr)'Cry=mJn{Acry,Pmax}

(5)

From this equation, (~,be/t)exp can be calculated. Both are compared ir~ Figure 5. The predicted width-to-thickness ratios on the assumption of no interaction tend to underestimate the actual ones in experiment. However, the error is less than 20% and conservative, which indicates that effective width of channel, lipped channel, and H-shaped sections can be calculated from Eqns. 3 and 4 on the assumption of no interactions between plate elements.

6oS~ 9 iiiiiiiiii,i!!!!!!!! oo12O ii 84ii 84

lOo12O, ........ .. .............. . . .,......... . . . ............... ............,

~oo

,o

o

~ooi

o

0

20

40

(Zb~ t

60

)pred

(a) channel

80

0

40

80

(Zb~ t )pred

120

(b) lipped channel

..................!

......i.........~........i

o ~ ...... i ........ i ......... i......... i ......... i ........ i 0 40 80 120

(Zbe/t )pred

(c) H-shaped

Figure 5 : Prediction vs. experiment of effective width-to-thickness ratios

Limit Diameter-to-thickness Ratio of Circular Hollow Sections Limit diameter-to-thickness ratio to assure full yielding of a thin circular cylinder for design is generally represented by following equation and the coefficient C2 is determined from experiment.

o) _c: T lim

O'y

(6)

All of the cylinders in this study can sustain full yield strength as shown in Figure 6. Thus, the limit value cannot be obtained. However, from a conservative consideration, C2-values are tentatively assign as follows which are the maximum values verified in the test:

216 C2 = 32,500 for circular cylinder of SUS304 C2 = 46,800 for circular cylinder of SUS301L 3/4H

(7a) (7b)

1.5 ..................................i..................................................... ~I.O 0.5

~

~I

03 ! 32.5xl .................................. i-;.;;............................... - 4 6 . 8 x l

.... ~

0.0 0

;I

03

9 SUS301L a/ H

, , ........... !................. 10 20 30 40 50 ( D / t)cy ( X 103N/rn~)

Figure 6 : Maximum strength of circular cylinders

SUMMARY Thin-walled stub-columns formed from two grades of stainless steels designated by SUS304 and SUS301L 3/4H are tested. Based on post-buckling strength, effective width-to-thickness ratios be /t and d e /t are determined. The be /t of unstiffened plates is derived from angle specimens and is given by Eqn. 3. The d e /t of stiffened plates is derived from square tube specimens and is given by Eqn. 4. The effective width of channel, lipped channel, and H-shaped sections can be calculated on the assumption of no interaction between adjacent plates. The limit diameter-to-thickness ratios of circular cylinders are conservatively given by Eqns. 6 and 7.

REFERENCES Lin S-H., Yu W-W., & Galambos T.V. (1988). Design of Cold-Formed Stainless Steel Structural Members -Proposed Allowable Stress Design Specification with Commentary, A Research Project Sponsored by ASCE, Univ. of Missouri-Rolla, Rolla, Missouri, USA Stainless Steel Building Association of Japan (1995). Design and Construction Standards of Stainless Steel Buildings, SSBA, Tokyo, Japan

Third InternationalConferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 200I ElsevierScience Ltd. All rights reserved

217

BIFURCATION EXPERIMENTS ON LOCALLY BUCKLED Z-SECTION COLUMNS K.J.R. Rasmussen Department of Civil Engineering, University of Sydney, Australia

ABSTRACT The paper describes a series of tests on thin-walled plain Z-sections compressed between fixed ends. Two tests each were performed on seven lengths covering the range from short stub columns to long columns. Most columns failed by interaction of local and overall flexural buckling. The test results are compared with theoretical solutions for the bifurcation load of a locally buckled point-symmetric column. The theoretical buckling loads require calculation of the instantaneous stiffnesses against minor and major axis flexure. These stiffnesses are obtained using a geometric and material nonlinear finite strip analysis. The theoretical results predict that while overall bifurcation of a non-locally buckled column will occur by pure flexure about the minor axis, the locally buckled column will bifurcate in a flexural mode involving coupling between minor and major axis flexure. This result is verified using a geometric nonlinear finite element analysis.

KEYWORDS Z-sections, Steel structures, Tests, Local buckling, Flexural buckling, Interaction buckling, Finite strip analysis, FEM.

INTRODUCTION

It is well-known that local buckling may influence the overall buckling behaviour of thin-walled sections. The influence depends on the end support conditions (pinned of fixed) and the symmetry characteristics. For a doubly symmetric section, such as an I-section, local buckling reduces the flexural rigidity and precipitates overall buckling at a reduced load but does not induce overall displacements. For a singly symmetric cross-section, such as a channel section, local buckling induces overall bending when the column is compressed between pinned ends but not when compressed between fixed ends (Young and Rasmussen 1997). The present paper focuses on point-symmetric columns, such as Z-section columns. Theoretical results (Rasmussen 2000) have shown that local buckling of point-symmetric columns does not induce overall displacements, as it does in pin-ended singly symmetric columns, but causes a coupling between the minor and major axis buckling displacements. In physical terms, this implies that the direction of overall buckling occurs about an axis rotated from the minor principal axis. According to the theory, torsional and flexural overall buckling are uncoupled. However, the torsional buckling

218 mode may become critical in the case of Z-sections with very slender flanges because local buckling reduces the warping rigidity (El~o) more severely than the minor axis flexural rigidity (E/y). The purpose of this paper is to present tests and finite element analyses of fixed-ended Z-section columns to verify experimentally and numerically the behaviour predicted by the theory (Rasmussen 2000).

TEST P R O G R A M

The test specimens were brake-pressed into section from nominally 1.5 mm thick G500 sheet steel. G500 is an Australian produced steel to AS1397 (1993) with galvanized coating and nominal yield stress of 500 MPa. It has low tensile strength to yield stress ratio and limited ductility of the order of 10-15%. The average measured cross-section dimensions are shown in Table 1 using the nomenclature defined in Fig. la. The coefficients of variation of the measured widths of the flanges (bf) and web (bw) were 0.022 and 0.007 respectively, indicating that a tight tolerance was achieved on the cross-section dimensions. The average measured thickness (T) was 1.58 mm. After removing the galvanizing layer by etching, the base metal thickness (t) was measured as 1.495 mm. The b/t-ratios for the flanges and web were 17.1 and 80.5 respectively. The elastic local buckling stress (trl) and half-wavelength (/) were determined from a finite strip analysis (Hancock 1978), as also shown in Table 1. The local buckling mode is shown in Fig. lb. Local buckling was precipitated by instability of the web. y,v

y

V-

-rwf "Ct

bw

w

.

r_.. ~

.

o

........ :

I bf I (a) Section geometry

(b) Local buckling mode

(c) Overall buckling mode

Fig. 1: Symbol definitions and buckling modes

bf

bw

(n~..4)

(ram) 121.8

t* (mm) 1.495

ro (mm) 3.0

A (ram~) 256.5

ly, (mm) 9573

Ix (mm') 5.01•

a 6.95"

trl (MPa) 137.2

l (mm) 120

* base metal thickness

Table 1: Average Measured Cross-section Dimensions and Section Constants Five tensile coupons were cut from the steel sheets in the same direction as the longitudinal axis of the test specimens. Figures 2a and 2b show a typical stress-strain curve obtained from one of the coupon tests. The average values of initial Young's modulus (E0), yield stress (try) and ultimate tensile strength (o,) are shown in Table 2. The COV of E0, Oy and Ou based on the five tests were 0.039, 0.0039 and 0.003 respectively indicating close resemblance of the mechanical properties. The sheets probably pertained to the same batch. In the absence of a sharp yield point, the yield stress was obtained as the 0.2% proof stress. Significant softening was observed in the vicinity of yield, as shown in Fig. 2a. Based on an average value of

219

0.01% proof stress of ~0.01=383 MPa, as determined from the stress-strain curves, the RambergOsgood n-parameter was calculated as n=9.1.

,~

700

700 '

600

600

500

500

400

~ 400

t~

r~

200

,

,

--

r

200

100

100 00

,

~ 30o

300 r~

,

0.001

0.002 0.003 Strain

0.004

0

0.005

0

I

I

0.02

0.04

0.()6 0.08' 0.10 Strain

0.12

0.14

a) initial curve b) complete curve Fig. 2: Typical Stress-strain Curve E0

O'y

(GPa) 216

....

Cu

(MPa) 533 ,,

(MPa) 576

Table 2" Average Mechanical Properties The test specimens were cut in lengths varying from 250 mm to 1600 mm. Subsequently, the ends were milled fiat to ensure even loading. For each length, two nominally identical specimens were prepared. Local and overall geometric imperfections were measured on all specimens prior to testing. The overall geometric imperfections are shown as u0 in Table 3 corresponding to the measured out-of straightness at midlength in the direction of the major x-axis, as shown in Fig. 1c. The out-of-flatness of the web (Ww0) measured at the centre of the web at mid-length is also shown in Table 3. The averages of the overall imperfection relative to the length (uo/L) and the local imperfection of the web (Wwo) were 1/4050 and 0.19 mm respectively. Specimen zf250a zf250b zf600a zf600b zf600a zf600b zf800a zfS00b zf1000a zfl000b ....zfl200a zfl200b zfl400a zfl400b zfl600a zfl600b Average ,

,

Length (L)

uo

(mm)

(mm)

250 249.5 399.5 399 600.5 60O 799 800.5 1000 1000 1199 1198 1400.5 1400 1597 1599

0.075 0.121 0.121 0.036 0.241 0.188 0.445 0.214 0.401 0.455 0.250 0.314 0.205 0.367 0.359 0.374

uo /L 1/3330 1/2060 1/3300 1/11080 1/2490 1/3190 1/1800 1/3740 1/2500 1/2200 1/4800 1/3810 1/6830 1/3820 1/4450 1/5350 1/4050

Wwo

ME

(mm)

(leq)

(~-,~) 65.3

36.0 35.0 38.0 35.5 36.0 38.0 36.5 33.0 34.0 33.5 32.6

64.5 61.2 61.2 60.2 60.1 57.3 57.2 54.5 51.8 42.5 44.6 35.5 29.5 29.0 29.8

0.28 0.00 0.22 0.'14 0.15 0.14 0.25 0.28 0.30 0.i 1 0.30 0.18 0.23 0.20 0.08 0.18 0.19

35.3

Table 3" Geometric Imperfections and Ultimate Loads

N~

N./N1 1.95 1.83 1.74 1.74 1.71 1.71 1.63 1.63 1.55 1.47 1.21 1.27 1.01

0.824 X:~J

220 The specimens were loaded between fixed ends in a vertical position. The top end platen was rigidly connected to the cross-head, thus preventing flexural and torsional rotations. At the base, a lockable spherical seat was used to ensure full contact between the end platen and the specimen during setup. Once contact was achieved, the seat was locked by tightening a bolt at each comer such that flexural rotations could no longer occur. The specimens were uniformly compressed until failure using a 250 kN capacity MTS Sintec testing machine. The ultimate load (Nu) was recorded, as shown in Table 3, and the test then continued into the post-ultimate range. Readings were taken at regular intervals of local and overall deformations. Local deformations were measured using transducers mounted on an aluminium frame, which was attached to the comers of the specimen at midlength, as shown in Fig. 3. The frame followed the specimen during overall buckling and ensured that the local buckling deformations were measured at the same points in the cross-section throughout loading. Readings were taken at the centre of the web and near the free edge of the flanges. Two transducers were used at each of these location, spaced approximately a quarter-wave longitudinally to ensure non-zero local buckling readings from at least one of the two transducers. In addition, three transducers were used to record overall displacements at midlength. Local buckling of the web and flanges can be clearly seen in Fig. 3. The experimental local buckling load was estimated using the N vs w 2 method, according to which the local buckling load is the intersection of the load axis with the line fitted through the graph of the load (N) versus the square of the plate buckling deformation in the initial post buckling range. The experimental local buckling loads (NlE) are shown in Table 3. They are generally close with an average of 35.3 kN and a COV of 0.053. The average experimental local buckling load (Nm=35.3 kN) was nearly equal to the theoretical value (N1=35.2 kN).

Figure 3: Specimen zf1000a during Test

221

BIFURCATION

ANALYSIS

The general theory (Rasmussen 1997) for calculating overall buckling loads of locally buckled members was applied to fixed-ended point-symmetric columns, such as Z-sections, in Rasmussen (2000). It was shown that the buckling displacements Ub and Vb in the principal x- and y-axis directions respectively are determined from the differential equations, ?t

tr

t

[(Eiy)t u o,t] + [(EIxy)t v ott] + (N cruo ,) = 0 tt tt ? I(Eixy)t UbttI "1-[(EIy)t ]2bt,] + (N crVbt) -" 0

(1) (2)

where Nor is the overall buckling load and (E/x)t, (Ely)t, and (Elxy)t are tangent flexural rigidities calculated at the buckling load, as described in detail in Rasmussen (1997). The tangent rigidities can be found by subjecting a length of section equal to the local buckle half-wavelength to increasing levels of compression and then superimposing small curvatures about the major and minor principal axes at each compression level. The tangent rigidities are the ratios of the resulting moments to the applied curvatures, eg

- A-----~"

(4)

The nonlinear inelastic finite strip local buckling analysis described and applied by Key and Hancock (1993) has been used in this paper to calculate the tangent rigidities. For fixed-ended columns, the governing equations (1,2) and the boundary conditions are satisfied by the displacement field, u b _ v~ = 1 - c o s (.2xz ~)

(5)

L

- c--;

which, upon substitution into Eqns (1,2), leads to the eigenvalue problem, 2

cr (6)

Non-trivial solutions are obtained for,

NCr ~ -

B + ~/B 2 - 4AC

where A=1

(8)

B -- - ( N x + N y )

N x = 4n'2(EI~),

(11)

(7)

2A (9)

N =

L~.

(12)

222 Figure 4 compares buckling curves determined from Eqns (7-12) with the test strengths shown in Table 3. The buckling curves are those corresponding to elastic material behaviour and elastic perfectly-plastic material behaviour with values of Young's modulus (E0) and yield stress (Oy) as shown in Table 2. The Euler curve assuming no local buckling is also included. The base metal thickness and average measured values of flange and web widths were used in the numerical calculations, as shown in Table 1. A local geometric imperfection was incorporated in the nonlinear finite strip analysis, assumed to be in the shape of the critical local buckling mode with a magnitude at the centre of the web of Wwo=0.19mm. This value was the average measured local geometric imperfection, as shown in Table 3, and was chosen according to the recommendation made in Hasham and Rasmussen (2001). Comer radii were ignored in the numerical analyses. One and two harmonics were used to model out-of-plane (flexural) and in-plane (membrane) buckling displacements respectively. The overall buckling loads and test strengths are n0ndimensionalised with respect to the local buckling load (N1=35.2 kN) in Fig. 4. 3.0

,,.

I

~

"

I

I

I

\ . . . . . . :

2.5

Flexural-(u),undistorted ---

Flexural-(uv), distorted (elastic)

99

\ \

,

I

Flexural-(uv), distorted (el-pl)

"

9

Test

\~"..

2.0

1.5 o

2: 1.0

0.5 1

0

500

I

I

I

1000 1500 2000 Column length, L(mm)

/

2500

3000

Figure 4: Test Strengths and Bifurcation Curves It follows from Fig. 4 that the elastic and inelastic bifurcation curves are somewhat higher than the test strengths. It is also noted that the plateau reached by the inelastic bifurcation curve at a length of about 750 mm is a measure of the stub column strength predicted by the finite strip analysis, and that the plateau is 15 % higher than the experimental stub column strength obtained from the tests on 250 mm long columns. This result suggests that the displacement field used in the finite strip analysis may not have been adequate for analysing the localised deformations developing near the ultimate load, as observed in the tests. The discrepancy between the bifurcation curves and the test strengths may also be explained by the presence of overall geometric imperfections and the fact that the material was not elastic perfectlyplastic as assumed in the finite strip analysis. As shown in Table 3, the average overall geometric imperfection in the major principal direction was L/4050, which was not negligible, and likewise, significant softening of the material was observed in the tension coupon tests, as shown in Figure 2a. The twist rotation was calculated from the three overall transducers located at midlength. It was found to be negligible at ultimate for all specimens, thus confirming that local buckling does not produce coupling between torsion and flexure in the bifurcation of point-symmetric sections.

223 DIRECTION OF OVERALL BUCKLING According to the classical theory for non-locally buckled point-symmetric sections; overall flexural buckling occurs about the minor y-axis and is uncoupled from flexural buckling about the major xaxis. However, it follows from Equations (1,2) that for a locally bucked section, flexural buckling about the minor principal axis (Ub) is coupled with flexural buckling about the major principal axis. From a physical viewpoint, this simply means that the direction of overall buckling changes from that of the major x-axis. Elastically, the change occurs because the section loses stiffness near the tips of the flanges which causes the axis of buckling to align itself more with the axis through the web. The change in rotation (Aa) of the axis of buckling can be determined from Eqn. (6),

Aa = tan -1( vb ) = tan-' (Cv)= tan-'

2

9

(13)

Equation (13) is plotted against the nondimensionalised load (N/Nt) in Fig. 5 for both the elastic and inelastic cases. In the inelastic case, the axis of buckling first rotates towards the web, then starts rotating back toward the principal y-axis direction at a load of about N/N~=I.9. The latter change in axis of rotation is a result of the fact that when membrane yield occurs near the comers, the material loses stiffness in this region. 2.5

1 i I l e Abaqus (geometricnonlinear elasticanalysis) Elastic 2.0 bifurcation _

1.5

L= 800mm~ . r~

1.0 -

0.5 -

L=1000mm~

L= 1 2 0 0 m m ~ 0

1

]/'

Inelastic

-

analysis i

i I~ 2 N/N I

I 3

4

Figure 5: Change in direction of overall buckling (Aot), (positive counter-clockwise) It was sought to verify the predicted change in direction of overall buckling using the experimental plots of major (u) and minor (v) axis overall buckling displacements. However, as shown by the inelastic curve in Fig. 5, the change of angle (Aot) was less than about 1~ which could not be detected experimentally because overall geometric imperfections produced displacements in both principal directions from the onset of loading. While overall buckling principally occurred in the major x-axis direction, there was significant random variation in the calculation of Aot--tan-l(v/u). A geometric nonlinear finite element analysis (Hibbert et al. 1995) was therefore performed on three lengths of column, L=800 mm, L=1000 mm and L=1200 mm, using SR4 shell elements for the discretisation of web and flanges. A local geometric imperfection was introduced with the same magnitude (Ww0=0.19mm) as that used in the nonlinear finite strip analysis. A small overall flexural geometric imperfection (u0) of L/10,000 was also introduced to avoid numerical instability and/or possible skipping of the overall bifurcation point. The overall geometric imperfection was purely in

224 the direction of the major x-axis. As for the nonlinear finite strip analysis, the width and thickness of the web and flanges, as well as the Young's modulus, were the average measured values given in Tables 1 and 2. The material was assumed to be linear-elastic. The change in direction of overall buckling from the major principal direction (Aa=tanl(v/u)) was calculated using the values of u and v recorded at the ultimate load, as obtained from the nonlinear analysis. The points are labeled "Abaqus" in Figure 5 and seen to agree with the curve predicted by the bifurcation theory, particularly for ultimate loads less than twice the local buckling load. CONCLUSIONS

Tests have been presented on thin-walled Z-sections uniformly compressed between fixed ends. Fourteen tests were conducted at seven lengths ranging from short to long columns, featuring failure by interaction of local and overall buckling. The test specimens were brake-pressed from high strength steel with a nominal yield stress of 500 MPa. Coupon tests demonstrated significant material softening at stresses well below the yield stress. The tests strengths have been compared with elastic and inelastic buckling loads predicted for a locally buckled section bifurcating in an overall mode. The bifurcation loads were higher than the test strengths with increasing discrepancy in the short to intermediate length range. The discrepancy has been attributed to overall geometric imperfections present in the test specimens, gradual material softening and inability of the nonlinear finite strip analysis to model localised curvature for the number of harmonics chosen. The tests confirmed that local buckling does not produce coupling between torsion and flexure in Z-section columns. According to the bifurcation analysis, the direction of overall buckling changes from the major principal axis in a locally buckled section. This result was verified using a geometric nonlinear finite element analysis. However, the change in direction was less than 1~ for the Z-sections tested and could not be demonstrated experimentally because of random variations in the direction of overall buckling resulting from overall geometric imperfections. ACKNOWLEDGMENTS The tests were conducted by Ms Anh Ton and Ms Ruth Tirtaatmadja as part of their BE honours thesis project. Their substantial contribution to this paper is gratefully acknowledged. REFERENCES AS1397, 1993. Steel Sheet and Strip- Hot-dipped Zinc-coated or Aluminium/zinc-coated, Standards Association of Australia, Sydney. Hancock, GJ, 1978. Local, distortional and lateral buckling of I-beams, Journal of the Structural Division, ASCE, 104(ST11), 1787-1798. Hasham, AS and Rasmussen, KJR, 2001. Nonlinear Analysis of Locally Buckled I-section Steel Beam-columns, Proceedings, 3rd International Conference on Thin-walled Structures, Cracow, Poland. Hibbitt, Karlsson and Sorensen, Inc., 1995, 'ABAQUS Standard, Users Manual', Vols 1 and 2, Ver. 5.5, USA Key, PW and Hancock, GJ, 1993. A Finite Strip Method for the Elastic-plastic Large Displacement Analysis of Thin-Walled and Cold-Formed Steel Sections. Thin-walled Structures 16, 3-29. Rasmussen, KJR, 1997. Bifurcation of Locally Buckled Columns, Thin-walled Structures, 28(2), 117154. Rasmussen, KJR, 2000. Overall Bifurcation of Locally Buckled Point-symmetric Columns, Proceedings, Coupled Instabilities in Metal Structures, C1MS'2000, 163-170. Young, B and Rasmussen, KJR, 1997. Bifurcation of Singly Symmetric Columns, Thin-walled

Structures, 28(2), 155-177.

Third International Conferenceon Thin-WalledStructures J. Zara~,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

225

BUCKLING LOAD CAPACITY OF STAINLESS STEEL COLUMNS SUBJECT TO CONCENTRIC AND ECCENTRIC LOADING J.Rhodes 1, M.Macdonald 2, M. Kotelko 3 and W.McNiff 1 1.Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK 2.Department of Engineering, Glasgow Caledonian University, Glasgow, UK 3. Department of Strength of Materials and Structures. Technical University ofLodz, Lodz, Poland

ABSTRACT This paper describes the results obtained from a series of compression tests performed on cold formed stainless steel Type 304 columns of lipped channel cross-section. The cross-section dimensions, the column length and the eccentricity of the applied compressive load are varied to examine the effects on the buckling load capacity of the columns. The results obtained from the tests and from a finite element analysis are compared to those obtained from the relevant design specifications in America and in Europe. Conclusions are drawn on the basis of the comparisons.

KEYWORDS Stainless steel, cold formed, columns, lipped channels, eccentric loading.

INTRODUCTION

Design code specifications for cold-formed carbon steel members have been published in many countries, e.g. (1), (2), (3). Stainless steel members have fewer design specifications available, the main design code for stainless steel members being the ASCE code in the USA (Ref. 4). In Europe, Eurocode 3, Part 1.4 (Ref 5) has been introduced recently and is currently under examination in the member countries of the EEC. The Eurocode has taken a substantially different viewpoint on some aspects of design than that adopted by the ASCE, e.g. the evaluation of column capacity. The non-linearity of the stress-strain law is taken into account in the ASCE code whilst the Eurocode uses the initial elastic modulus and assumes a larger imperfection parameter to take care of the degradation of the elastic modulus with increase in stress. In dealing with combined bending and axial loading, the differences between the Eurocode and the ASCE code are compounded by virtue of the different interaction formulae used. In view of these differences it was felt that an examination of the effects of concentric and eccentric compressive loading on the buckling behaviour and load capacity of cold formed stainless steel lipped channel section columns and comparison of the predictions of the ASCE Codes and Eurocode 3, Part 1.4 with experiments would be informative. The interaction formulae incorporate the member moment capacity. Since in a parallel series of tests on the same sections (Ref. 6), the experimental moment section capacity was determined, this is also used in the

226 interaction formulae (for comparison with the theoretical moment capacity) to investigate if any improvements to the code predictions are obtained by using the known true moment capacity.

DESIGN CODE RECOMMENDATIONS

Short to medium length cold formed stainless steel columns of lipped channel cross-section are investigated under two different load conditions. In condition (i) load is compressive and applied through the section centroid (concentric loading), and in condition (ii) the load is applied through a point at a fixed distance from the centroid (eccentric loading). In this paper, design code predictions based on full section properties are presented (reported by Macdonald et al (Ref. 7)). The cross-section dimensions were such that it could be considered to be fully effective, with details of the experimental investigation described later. The design rules given in the ASCE code and in Eurocode 3: Part 1.4, set in a form directly applicable to the particular loading conditions examined experimentally, are given below. In dealing with the Eurocode, Part 1.4 does not directly give details of bending/axial load interaction, but instead refers to Part 1.3 or Part 1.1.

ASCE:

Under concentric loading the design axial strength is given by Pn = 0-85AFn

where A and

= gross cross-sectional area

(N)

(1)

(mm 2)

n2E'

Fn

= flexural buckling stress =

with

K = buckling coefficient = 1 for pinned ends L = column length (ram) r = cross-section radius of gyration (ram)

and

E t = tangent modulus =

(KL/r) 2

(N/mm 2)

EoF.

~/mm2)

F~ + O.OOZnEo(o / Z ) "-~ in which

n Eo

Fv and

t~

= plasticity factor = 6.216 (Ref. 7) = initial elastic modulus (N/mm 2) = virgin or full section 0.2% proof stress = F n (N/ram 2)

(N/mm 2)

In the case of a column subjected to a load of constant eccentricity, the maximum eccentric load P, applied at eccentricity e from the neutral axis, is given by equation (2)

P ~+ P,, where M. = and

9_ Y

Pe

p~

= moment capacity of the cross section

P, = /r 2E,I~,. = Euler buckling capacity L2

(2)

l.5h

1.5h Span Length = L

Single Point Load Configuration

L1

L2

m N=38mm

L3

m N=38mm

N=76mm

R

1.5h

1.5h

> 1.5h

Span Length = L

Two Point Load Configuration

Figure 1" Typical loading configurations.

1.

N=76

247 Top Track Section

FTop Track x / / Section

Screw Locations

Bottom Track ~ . ~ Section

-'-

'-

----_____

Screw Locations

Back-to-Back Channel Sections

1

~-Channel Sections

Bottom Track Section

Figure 2: Typical I-beam and box-beam sections

1.6

......

1.4

1.2

1.0

9

=E ~

9

.~

ff

4'

0.8

9

"~

""

9

, -

0.4

. . .k.P, ,k,

,

,

1.2

1.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.6

1.8

Pt/Pn

Figure 3" I-Beam data with P. for built-up I-sectiom.

2.0

2.2

248 1.8 1.6 1.4 1.2 ee :

9

1.0

~

0.8

0.6

=1

9

] . 0 7 r P' ~ + [ M ' ~ = 1.42 - - - - - - ' ~ 0.4

L,,,)

0.2

.,

iD

i

t

""l :...- ..'" .............

0.0 , ~ 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

P~Pn

Figure 4: Box beam data with P, for single web C-sections.

1.8

1.6 1.4 1.2 0

9

= 1.0 :S

:S

0.8

0.6 0.4 0.2 :: ......

0.0 0.0

0.2

0.4

''

........

0.6

0.8

::::',::::I 1.0

1.2

......... 1.4

1.6

' .... 1.8

' .... 2.0

' ........ 2.2

2.4

P~1.4xPn

Figure 5: Box beam data with P. for single web C-sections.

1, 2.6

Third International Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

249

COMPRESSION TESTS OF THIN-WALLED LIPPED CHANNELS WITH RETURN LIPS J. Yan and B. Young School of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798

ABSTRACT Longitudinal edge stiffeners have been used to enhance the local buckling stress of thin-walled steel channel sections. This in turn leads to the increase of the column strength. Apart from conventional simple lips in channel sections, the use of return lips can further improve the buckling behaviour of the sections and hence improve the column strength. This paper presents the results of an experimental investigation into the strengths and behaviour of cold-formed lipped channels with return lips compressed between fixed ends. A series of tests were performed on channel sections brake-pressed from high strength structural steel sheets. Two different cross-section geometry were tested over a range of lengths which involved local buckling, distortional buckling and flexuraltorsional buckling. The experimental column strengths are compared with the design strengths predicted using the American Specification and the Australian/New Zealand Standard for cold-formed steel structures. Design column curves are also plotted. It is shown that the American Specification generally overestimated the test strengths of the specimens. The Australian/New Zealand Standard is generally conservative, except that it overestimated the test strengths of the specimens having more slender flanges for intermediate and long columns.

KEYWORDS Buckling, Channel columns, Cold-formed steel, Design strengths, Effective length, Experimental investigation, Fixed-ended, Return lips, Structural design, Steel structures, Test strengths.

INTRODUCTION The buckling stress of thin-walled channels can be enhanced by the use of edge stiffeners which provide a continuous support along the longitudinal edge of the flange. Hence, thin-walled channels with edge stiffeners can lead to an economic design. Apart from conventional simple lips used in channel sections, the use of return lips can further improve the column strength of the channels. The current design rules for uniformly compressed elements with an edge stiffener in the American Iron and Steel Institute (AISI, 1996) Specification for the Design of Cold-Formed Steel Structural Members and the Australian/New Zealand Standard (AS/NZS 4600, 1996) for Cold-Formed Steel Structures allow for stiffeners other than simple lips. However, these design provisions were based on

250 the experimental investigations dealing only with simple lip stiffeners on adequately stiffened and partially stiffened elements conducted by Desmond, Pekoz and Winter (1981 ). The extension of these provisions to other types of stiffeners was purely intuitive. Therefore, the design rules for other types of stiffeners such as return lips are not supported by test data. Hence, it is important and necessary to obtain test data for sections with other types of stiffeners. In addition, it is essential to assess the ability of the current design rules for the prediction of column strengths of such sections. This paper presents a series of fixed-ended column tests of thin-walled lipped channels with return lips. The test strengths are compared with the design strengths predicted using the AISI Specification and the AS/NZS Standard for cold-formed steel structures. Design column curves are also plotted. The purpose of this paper is firstly to provide test data for fixed-ended lipped channel columns with return lips for use in international steel design specifications and secondly to assess the ability of the current design rules in predicting the column strengths of lipped channels with return lips.

EXPERIMENTAL INVESTIGATION

Test Specimens Two series of tests were performed on lipped channels with return lips subjected to pure compression between fixed ends. The specimens were brake-pressed from high strength zinc-coated Grade G450 structural steel sheets having a nominal yield stress of 450 MPa and specified according to the Australian Standard AS1397 (1993). The specimens were supplied from the manufacturer in uncut lengths of 4000 mm. Each specimen was cut to a specified length ranging from 500 mm to 3500 ram. Both ends of the specimens were welded to 25 mm thick mild steel end plates to ensure full contact between the specimen and end bearings. The shortest specimen lengths complied with the Structural Stability Research Council (SSRC) guidelines (Galambos, 1988) for stub column lengths. The test specimens had a nominal thickness of 1.5 mm, a nominal lip width of 25 mm, a nominal return lip width of 15 mm and a nominal web width of 150 mm. The measured inside comer radius of the specimens is 2.0 mm. The nominal flange width was either 80 mm or 120 mm and was the only variable in the cross-section geometry. The two series are labeled as T1.5F80 and TI.5F120, where "T" refers to thickness and "F" refers to flange. The numbers following "T" and "F" are the nominal thickness and nominal flange width respectively. The nominal flange width to thickness ratios are 53.3 and 80.0 for Series T1.5F80 and T1.5F120 respectively. Tables 1 and 2 show the measured cross-section dimensions of the test specimens for Series TI.5F80 and T1.5F120 respectively, using the nomenclature defined in Fig. 1. The cross-section dimensions shown in Tables 1 and 2 are the averages of measured values at both ends for each test specimen.

Bl

j

-I

~,-~..... .i",~ T. -, -~-I,,T Yi

ri W

. . . . . .

.._...~

tt

~Drl tl

II ii ii |1 |, ;I |l

i ,r i -. ,~.~. ~_.~.

BW Figure 1" Definition of symbols

251

TABLE 1 MEASURED SPECIMEN DIMENSIONSAND EXPERIMENTALULTIMATE LOADS FOR SERIES T1.5F80 Return Lips Flanges Lips s, sr Brt (mm) (mm) (mm) 17.7 28.2 83.4 T1.SF80L0500 17.6 27.6 83.4 T1.5F80L1000 28.0 83.3 T1.5F80L1500 17.6 T1.5F80L2000 17.3 28.0 83.2 28.1 83.4 T1.5F80L2500 17.6 T~.SF80L3000 17.7 28.3 84.4 'T1.5F80L3500 17.4 28.0 83.3 Mean 17.6 28.0 83.5 0.009 COV 0.008 0.005 Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN * Base metal thickness Specimen

Web

B..

Thickness

t

t"

Radius r,

(ram) (mm) (mm) (mm) 153.4 1.547 152.7 1.564 153.9 1.519 154.3 1.535 153.7 1.532 153.7 1.549 153.9 1.534 153.6 1.540 0.003 0.009 .

1.489 1.506 1.461 1.477 1.474 1.491 1.476 1.482 0.010

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 0.000

Column Length L

Area

Exp.Ult. Load

A

imm~)

P~e

(mm) 499.5 1000.2 1498.9 1998.2 2498.6 3001.2 3501.2 9

. . . .

,

588 592 578 583 583 593 583 586 "0.010

172.0 166.9 163.4 161.7 158.8 154.8 124.4 ,

TABLE 2 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTALULTIMATE LOADS FOR SERIES T1.5F120 Specimen

Return Lips

Lips

Flanges

Web

Br!

Bz

Bf

B,,.

Thickness

t

t"

Radius

ri

(mm) (mm) (mm) (mm) (mm) (mm) (mm) T1.5F 120L0500 17.2 27.6 1 2 2 . 9 154.2 T1.5F 120L1000 17.5 28.2 1 2 3 . 7 152.7 T1.5F120L1500 17.5 28.4 1 2 4 . 0 152.7 T1.5FI20L2000 17.2 28.4 124.1 153.5 T1.5F120L3000 17.3 28.3 123.3 153.3 T1.5F 120L3500 17.4 27.9 1 2 3 . 2 153.7 Mean 17.4 28.1 123.5 15314 COV 0.009 0 . 0 1 1 0 . 0 0 4 0 . 0 0 4 ..Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN * Base metal thickness

1.521 1.527 1.524 1.529 1.525 1.530 i.526 0.002

1.473 1.479 1.476 1.481 1.477 1.482 1.478 0.002

2.0 2.0 2.0 2.0 2.0 2.0 2.0 0.000

Column Length

Area

Exp.Ult. Load P~/,

L

A

(mm)

(mm~)

(le,o

497.5 1002.5 1495.6 2001.8 3002.0 3502.2

697 703 703 705 701 703 702 "0.004

166.9 159.3 145.7 139.5 131.3 127.4 , . .

.

The base metal thickness is the effective thickness o f the specimens without zinc coating. In Tables 1 and 2, the base metal thickness (t*) was measured by removing the zinc coating b y acid etching. The thickness of the zinc coating was measured as 29 ~un for Series T1.5F80 and 24 lxrn for Series T1.5F120.

Specimen Labeling The test specimens were labeled such that the series and specimen length could be identified from the label. The specimen label is basically the label o f the test series followed by a letter "L" and the nominal length o f the specimen, where "L" refers to the length o f the specimen. For example, the label "T1.5F80L1500" defines the specimen belonging to Series T1.5F80 having a nominal length of 1500 mm.

Material Properties Tensile coupon tests were performed to determine the material properties o f the test specimens. Longitudinal coupon was taken from the center o f the web plate o f the untested specimen belonging to

252 the same batch of the column test specimens for each series. The coupons were prepared and tested according to the Australian Standard AS1391 (1991), having a gauge length of 50 mm and a width of 12.5 mm. The coupons were tested in a 300 kN capacity INSTRON displacement controlled testing machine using friction grips to apply the loading. The longitudinal strain was measured by using a calibrated extensometer of 50 mm gauge length as well as two strain gauges attached to the coupon at the center of each face. The strain gauges readings were used to determine the Young's modulus. The load, gauge length extensions and strain gauges readings were recorded at regular intervals by using a data acquisition system. The static load was obtained by pausing the applied straining for one and a half minutes near the 0.2% tensile proof stress and the ultimate tensile strength. This allowed the stress relaxation associated with plastic straining to take place. The material properties obtained from the coupon tests are summarized in Table 3. The table contains the nominal and measured static 0.2% tensile proof stress (o0.2), the Young's modulus (E), the static tensile strength (ou) and the elongation after fracture (ca) based on a gauge length of 50 mm. The measured static 0.2% proof stresses were used as the corresponding yield stresses in calculating the design column strengths. TABLE 3 NOMINALANDMEASUREDMATERIALPROPERTIES "i'est Series

! Nominal 00.2 (MPa) T1.5F80 450 T1.5F120 . 450 . . . .

E (GPa) 204 208

Measur'ed 00.2.... ou (MPa) (MPa) 522 554 .,507 550

eu (%) 10 10

_

Test Operation The test setup is shown in Fig. 2. Compressive axial force was applied to the specimen by using a servo-controlled hydraulic machine. The upper end support was moveable to allow tests to be conducted at various specimen lengths. The top end plate of the specimen was bolted to a rigid flat bearing plate connected to the upper end support. The rigid fiat bearing plate was restrained from the minor and major axes rotations as well as twist rotations and warping. The load was then applied at the lower end through a spherical bearing. Initially, the spherical bearing was free to rotate in any directions. The ram of the actuator was moved slowly towards the lower end of the specimen until the spherical bearing was in full contact with the bottom end plate of the specimen with an initial load of approximately 2 kN applied on the specimen. This procedure eliminated any possible gaps between the spherical bearing and the bottom end plate of the specimen. The bottom end plate of the specimen was bolted to the spherical bearing which was then restrained from rotations and twisting by using vertical and horizontal bolts respectively. The vertical and horizontal bolts were used to lock the spherical bearing in position. Hence, the spherical bearing became a fixed-ended bearing which was considered to be restrained against both minor and major axes rotations as well as twist rotations and warping. Three displacement transducers were positioned on the fixed-ended bearing to measure the axial shortening of the specimen. Displacement control was used to drive the hydraulic actuator at a constant speed of 0.2 mm/min during the test. The use of displacement control allowed the tests to be continued in the post-ultimate range. The applied load and displacement transducers readings were recorded at regular intervals by using a data acquisition system. The static load was obtained by pausing the applied straining for one and a half minutes near the ultimate load.

253

Figure 2: Test setup

Geometric Imperfections Initial overall flexural geometric imperfections about the minor axis of the specimens were measured prior to testing. A theodolite was used to obtain readings at mid-length and near both ends of the specimens. The maximum flexural imperfections at mid-length were 1/1540 and 1/1670 of the specimen length for Series T1.5F80 and T1.5F120 respectively.

Test Results The experimental ultimate loads (PExP)obtained from the tests are shown in Tables 1 and 2 for Series T1.5F80 and T1.5F120 respectively. The test results are also plotted against the effective length for minor axis flexural buckling (ley) in Figs. 3 and 4 for the respective series. The failure modes observed at ultimate load of each test specimen are also shown in Figs. 3 and 4, where "L" refers to local buckling, "D" refers to distortional buckling and "FT" refers to flexural-torsional buckling.

254 COMPARISON OF TEST STRENGTHS WITH DESIGN STRENGTHS The fixed-ended column test strengths (PF.xp) are compared in Figs. 3 and 4 with the unfactored design strengths predicted using the American Specification (1996) and the Australian/New Zealand Standard (1996) for cold-formed steel structures. The AS/NZS Standard was adopted from the AISI Specification. The design rules for compression members in the AS/NZS Standard are identical to those in the AISI Specification, except that the AS/NZS Standard has a separate check for distortional buckling of singly-symmetric sections as specified in Clause 3.4.6. The ultimate loads of the test specimens are plotted against effective length in Figs. 3 and 4. The effective lengths for major (lex) and minor (Icy) axes flexural buckling as well as torsional buckling (let) are assumed equal to one-half of the column length for the fixed-ended columns (lex = ley = let = L / 2), where L is the actual column length. This is because the fixed-ended bearings are restrained against the major and minor axes rotations as well as twist rotations and warping. The experimental local buckling loads are also indicated in Figs. 3 and 4. In addition, the theoretical elastic minor axis flexural and flexural-torsional buckling loads of the fixed~ columns are plotted against the effective length for minor axis flexural buckling (Icy) in Figs. 3 and 4. The equations of the theoretical buckling loads are detailed in Young and Rasmussen (1995 and 1998). The design strengths and the theoretical elastic flexural and flexural-torsional buckling loads were calculated using the average measured cross-section dimensions and the measured material properties summarized in Tables 1-3. The base metal thickness was used in the calculation. The fixed-ended columns were designed as a concentrically loaded members, that is, the load is assumed to act at the centroid of the effective cross section as recommended by Young and Rasmussen (1998). For the AS/NZS Standard, the distortional buckling loads were calculated according to Clause 3.4.6 of the Standard. The elastic distortional buckling stresses (foal) were obtained from a rational elastic buckling analysis (Papangelis and Hancock, 1995). For Series T1.5F80, the AISI Specification overestimated the test strengths of the short (ley 1000 mm), except that the test strength at an effective length of 1750 mm was slightly overestimated. The design strengths predicted by the AS/NZS Standard were conservative for all the columns, except that the test strength at an effective length of 1750 mm was slightly overestimated. The failure modes observed in the tests were combined local and distortional buckling for short and intermediate columns, except that distortional buckling was not observed at an effective length of 250 mm. Combinations of local and flexural-torsional buckling modes were observed for the long columns. The AISI Specification predicted flexural-torsional buckling for all the test specimens, which was in agreement with the failure modes observed in the tests for long columns, but not for short and intermediate columns. The AS/NZS Standard predicted distortional buckling for short and intermediate columns and flexuraltorsional buckling for long columns, which were in agreement with the experimental observations, except for the shortest column. For Series T1.5F120, the AISI Specification overestimated the test strengths of all the test specimens, as shown in Fig. 4. The AS/NZS Standard overestimated the test strengths of the intermediate and long columns, but it underestimated the test strengths of the short columns. The failure modes observed in the tests were combined local and distortional buckling for short and intermediate columns, except that distortional buckling was not observed at the shortest column. Combinations of local and flexural-torsional buckling modes were observed for the long columns. In addition, distortional buckling was observed at an effective length of 1500 ram. The failure modes predicted by the AISI Specification and the AS/NZS Standard follow a similar trend as Series T1.5F80. The AISI Specification predicted flexural-torsional buckling for all the test specimens, which was in agreement with the failure modes observed in the tests for long columns, but not for short and intermediate

255 400

9

Tests AISI - - AS/NZS

350

y 15o

300

Flexural-torsional

a, 250 =

O I,,i t~ =

@

L

200

150

Flexural Buckling

so

9

9 --

=

L,D I

I

,

L,D

.

L,D

~

100 -

L,FT L,FT 9

, ~/L~

L,FT

Buckling

50 I

,

500

I

I

I

I

1000

1500

2000

2500

3000

Effective length, l ~ (mm) Figure 3: Comparison of test strengths with design strengths for Series T1.5F80

400 350

9 Tests . . . . AISI " " AS/NZS

Ix y ~ 1 5 1.~5lZ0 -- ! " |

~

Flexural Buckling

300 250 =

Flexural-torsional Buckling ~

200

t,1

150

L ...~

L,D . .,,

L,D

~

9

..-.

D

,

II

F

T

= 100 @

L,FT

/ / L o c a l Buckling .

.

.

.

.

.

.

.

.

~

.

.

.

.

50 I

500

9

I

1000

.,

I

I

I

1500

2000

2500

Effective length, 1~ (mm) Figure 4: Comparison of test strengths with design strengths for Series T1.5F120

3000

256 columns. The AS/NZS Standard predicted distortional buckling for short and intermediate columns and flexural-torsional buckling for long columns, which were in agreement with the failure modes observed in the tests, except for the shortest column. At an effective length of 1500 mm, distortional buckling was observed in the test but not predicted by the AS/NZS Standard.

CONCLUSIONS This paper has presented the experimental results of fixed-ended column tests on thin-walled lipped channels with return lips. The test strengths were compared with the design strengths obtained using the American Iron and Steel Institute (AISI, 1996) Specification for the Design of Cold-Formed Steel Structural Members and the Australian/New Zealand Standard for Cold-Formed Steel Structures (AS/NZS 4600, 1996). It is concluded that the design strengths predicted by the AISI Specification were generally unconservative for the lipped channels with return lips. The design strengths predicted by the AS/NZS Standard were generally conservative, except for the specimens having more slender flanges. The failure modes predicted by the AISI Specification were generally in agreement with the failure modes observed in the tests for long columns, but not for short and intermediate columns. However, the failure modes predicted by the AS/NZS Standard were generally in agreement with the failure modes observed in the tests for all columns.

ACKNOWLEDGMENTS

The test specimens were provided by BHP Steel Building Products. The authors would like to express their sincere thanks to Miss Siew Ping YEO and Mr. Sun Ping YEN who are undergraduate students in the School of Civil and Structural Engineering of the Nanyang Technological University for their assistance in the experimental program.

REFERENCES

American Iron and Steel Institute, (1996). Specificationfor the Design of Cold-formed Steel Structural Members, AISI, Washington DC. Australian Standard, (1991 ). Methodsfor Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Australian Standard, (1993). Steel Sheet and Strip- Hot-dipped Zinc-coated or Aluminium/Zinccoated, AS 1397, Standards Association of Australia, Sydney, Australia. Australian/New Zealand Standard, (1996). Cold Formed Steel Structures, AS/NZS 4600:1996, Standards Australia, Sydney, Australia. Desmond T.P., Pekoz T. and Winter G. (1981). Edge Stiffeners for Thin-walled Members. Journal of Structural Engineering, ASCE, 107:2, 329-353. Galambos T.V. (1988). Ed. Guide to Stability Design Criteriafor Metal Structures, 4th Edition, Wiley Inc., 708-710. Papangelis J.P. and Hancock G.J. (1995). Computer Analysis of Thin-Walled Structural Members. Computers and Structures, 56:1, 157-176. Young B. and Rasmussen K.J.R. (1995). Compression Tests of Fixed-ended and Pin-ended Coldformed Lipped Channels. Research Report R715, School of Civil and Mining Engineering, University of Sydney. Young B. and Rasmussen K.J.R. (1998). Design of Lipped Channel Columns. Journal of Structural Engineering, ASCE, 124:2, 140-148.

Third International Conferenceon Thin-Walled Structures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

257

EXPERIMENTAL INVESTIGATION OF STAINLESS STEEL CIRCULAR HOLLOW SECTION COLUMNS B. Young

and

W. Hartono

School of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798

ABSTRACT The paper describes a test program on cold-formed stainless steel circular hollow section (CHS) columns compressed between fixed ends. A series of tests consisting of three cross-section geometries was performed. The specimens were cold-rolled from stainless steel sheets. The tests were performed over a range of column lengths, which involved local buckling and overall flexural buckling. Measurements of overall geometric imperfections and material properties were conducted. The test strengths are compared with the design strengths predicted using the American, Australian/New Zealand and European specifications for cold-formed stainless steel structures. Furthermore, the test strengths are compared with the column strengths obtained from the design rules proposed by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997). Generally, it is shown that the specifications unconservatively predict the strengths of the tested CHS columns, whereas Rasmussen & Hancock and Rasmussen & Rondal conservatively predict the column strengths.

KEYWORDS Buckling, Cold-formed steel, Design strengths, Experimental investigation, Fixed-ended columns, Stainless steel, Structural design, Test strengths, Tubular members.

INTRODUCTION Cold-formed stainless steel tubular members are used increasingly for structural applications. Members subjected to compression force are one of the major components in a structural design. There are several design specifications available for the design of cold-formed stainless steel tubular columns, such as the American Society of Civil Engineers (ASCE, 1991) Specification for the Design of ColdFormed Stainless Steel Structural Members, the Australian/New Zealand Standard (Aust/NZS, 2000) for Cold-formed Stainless Steel Structures and the European (Euro Inox, 1994) Design Manual for Structural Stainless Steel. In addition, design rules for such columns are also proposed by other researchers, such as Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997). The design rules in the specifications as well as the design rules proposed by the aforementioned researchers are mainly based on the investigations of pin-ended columns. Little test data are available on the strength offtxed-

258

ended cold-formed stainless steel tubular columns. The pin-ended support conditions are rarely realised in practice. In most cases, some degree of rotational restraint is offered at the end supports, and the column is somewhere between fixed and pinned. Therefore, it is also important to obtain test data for fixed-ended columns. The objective of this paper is firstly to present a series of tests of fixed-ended cold-formed stainless steel circular hollow section (CHS) columns, and secondly to compare the test strengths with the design strengths predicted using the American (1991), Australian/New Zealand (2000) and European (1994) specifications for cold-formed stainless steel structures. In addition, the test strengths are compared with the design strengths predicted by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997). Weld

t

ITM

J"l

Figure 1" Definition of symbols TABLE 1 MEASURED SPECIMEN DIMENSIONS FOR SERIES C1

Specimen

Diameter

Thickness

Length L

Area

(mm)

(mm)

(mm)

(ram2)

2.88 2.71 2.83 2.78 2.78 2.68 2.78 0.027

549 1000 1500 2001 2499 3001

781 735 768 753 752 726 752 0.027

89.2 89.0 89.2 89.0 88.9 88.9 89.0 Mean 0.002 COV Note: 1 in. = 25.4 mm. COV = coefficient of variation. C1L0550 C1L1000 CIL1500 C1L2000 C1L2500 C1L3000

A

EXPERIMENTAL INVESTIGATION

Test Specimens The tests were performed on stainless steel circular hollow sections (CHS). The test specimens were cold-rolled from annealed flat strips of type 304 stainless steel. The specimens were supplied from the manufacturer in uncut lengths of 6000 ram. Each specimen was cut to a specified length ranging from 550 mm to 3000 mm, and both ends were welded to stainless steel end plates to ensure full contact between specimen and end bearings. Three series of CHS were tested, having an average measured thickness of 2.78 ram, 3.34 ram, 4.32 mm and the average outer diameter of 89.0 ram, 168.7 ram, 322.8 mm for Series C 1, C2, C3 respectively. The average measured outer diameter to thickness (D/t) ratios are 32.0, 50.5 and 74.7 for Series C1, C2 and C3 respectively. Tables 1-3 show the measured cross-section dimensions of the test specimens using the nomenclature defined in Fig. 1. The cross-

259 section dimensions shown in Tables 1-3 are the averages of measured values at both ends for each test specimen. The specimens were tested between fixed ends at various column lengths. The test specimens were labeled such that the series and specimen length could be identified from the label. For example, the label "C2L0550R" defines the following specimen: 9 The first two letters indicate that the specimen belongs to test Series C2. 9 The third letter "L" indicates the length of the specimen. 9 The last four digits are the nominal length of the specimen in mm (550 mm). 9 If a test was repeated, then the letter "R" indicates the repeated test. TABLE 2 MEASUREDSPECIMENDIMENSIONSFOR SERIES C2 Specimen

Diameter

(mm) C2L0550 168.8 C2L0550R 168.8 C2L1000 168.5 C2L1500 168.8 C2L2000 168.7 Mean 168.7 COV 0.001 Note: 1 in. = 25.4 ram. COV = coefficient of variation. ,,

Thickness

Length L

Area A

(ram)

(mm)

~mm~

3.39 3.36 3.31 3.37 3.26 3.34 0.016

550 548 999 1500 2004

1762 1746 1718 1751 1694 1734 0.016

...

,11

TABLE 3 MEASUREDSPECIMENDIMENSIONSFOR SERIES C3 Specimen

C3LIO00 C3L 1500 C3L2000 C3L2500 C3L3000

Diameter D

Thickness t

Len~h L

(mm)

(mm)

(mm)

(ram~)

4.25 4.25 4.48 4.24 4.38 4.32 0.025

1000 1499 1999 2498 2998

4249 4249 4446 4261 4402 4322 0.022

.

.

.

.

322.5 322.5 320.4 324.1 324.3 322.8 COV 0.005 Note: 1 in. = 25.4 ram. COV = coefficient of variation. M e a n

Area A

.,

Material Properties The material properties of each series of specimens were determined by tensile coupon tests. Six longitudinal coupons (two from each series of specimens) were tested. The coupons were taken from the finished specimens at the location 90 ~ angle from the weld. The coupon dimensions were conformed to the Australian Standard AS 1391 (1991) for the tensile testing of metals using 12.5 mm wide coupons of gauge length 50 mm. The coupons were also tested according to AS 1391 (1991) in a 300 kN capacity Instron UTM displacement controlled testing machine using friction grips. A calibrated extensometer of 50 mm gauge length was used to measure the longitudinal strain. In addition, two linear strain gauges were attached to each coupon at the center of each face. The strain gauges readings were used to determinate the initial Young's modulus. A data acquisition system was used to record the load and the readings of strain at regular intervals during the tests. The static load was obtained by pausing the applied straining for 1.5 minutes near the 0.2% tensile proof stress and the

260 ultimate tensile strength. This allowed the stress relaxation associated with plastic straining to take place. The material properties obtained from the coupon tests are summarized in Table 4, namely the measured static 0.2% (00.2) and 0.5% (o0.s) tensile proof stresses, the static tensile strength (ou), as well as the initial Young's modulus (Eo) and the elongation after fracture (c,,) based on a gauge length of 50 mm. The 0.2% proof stresses were used as the corresponding yield stresses in calculating the design strength of the columns. The measured stress-strain curves were used to determine the parameter n using the Ramberg-Osgood expression (Ramberg and Osgood, 1943), e = m + 0.002

(1)

Eo where e is the strain, o is the stress and n is a parameter that describes the shape of the curve. The parameter n was obtained from the measured 0.01% (o0.01)and 0.2% (00.2)proof stresses using n = In(0.01/0.2) / ln(o0.01/o0.2). This expression provided the values of n = 4, 7 and 5 for Series C1, C2 and C3 respectively, as shown in Table 4. TABLE 4 MEASUREDMATERIALPROPERTIESFROMTENSILECOUPONTESTS Series (GPa) (MPa) CI 191 272 CI 188 268 Mean 190 270 C2 190 291 C2 200 285 Mean 195 288 C3 200 266 C3 203 255 Mean 202 261 Note:l ksi= 6.89MPa.

(MPa) 301 291 296 309 310 310 295 279 287

(MPa) 706 673 690 707 672 690 629 603 616

(%) 62 58 60 61 56 59 68 62 65

7 7 7 5 5 5

Test Rig and Operation The test rig and a test set-up are shown in Fig. 2. A servo-controlled hydraulic testing machine was used to apply compressive axial force to the specimen. Two stainless steel end plates were welded to the ends of the specimen. A moveable upper end support allowed tests to be conducted at various specimen lengths. A rigid flat bearing plate was connected to the upper end support, and the top end plate of the specimen was bolted to the rigid flat bearing plate, which was restrained against the minor and major axes rotations as well as twist rotations and warping. The load was then applied at the lower end through a spherical beating. Initially, the spherical bearing was free to rotate in any directions. The ram of the actuator was moved slowly toward the specimen until the spherical beating was in full contact with the bottom end plate of the specimen having a small initial load of approximately 2 kN. This procedure eliminated any possible gaps between the spherical bearing and the bottom end plate of the specimen. The bottom end plate of the specimen was bolted to the spherical beating. The spherical beating was then retrained from rotations and twisting by using vertical and horizontal bolts respectively. The vertical and horizontal bolts of the spherical bearing were used to lock the bearing in position after full contact was achieved. Hence, the spherical bearing became a fixed-ended bearing. The fixed-ended bearing was considered to restrain both minor and major axes rotations as well as

261 twist rotations and warping. Three displacement transducers were positioned at the loading end to measure the axial shortening of the specimen. Displacement control was used to drive the hydraulic actuator at a constant speed of 0.5 mm/min. The use of displacement control allowed the tests to be continued into the post-ultimate range. A data acquisition system was used to record the applied load and the readings of displacement transducers at regular intervals during the tests. The static load was recorded by pausing the applied straining for 1.5 minutes near the ultimate load.

Figure 2: Test set-up Stub Column Tests

The shortest specimen lengths complied with the Structural Stability Research Council (SSRC) guidelines (Galambos, 1988) for stub column lengths. The measured cross-section dimensions and the measured specimen length of the stub columns are given in Tables 1-3. The stub columns C1L0550, C2L0550, C2L0550R and C3L1000 were tested for Series C1, C2 and C3. Four longitudinal strain gauges were attached at mid-length of the stub columns. The material properties of the complete crosssection in the cold-worked state were obtained for the stub columns. Table 5 shows the measured initial Young's modulus (Eo), the static 0.2% tensile proof stress (or0.2) and the parameter n for the stub columns. TABLE 5 MEASUREDMATERIALPROPERTIESFROMSTUBCOLUMNTESTS

Eo

Series

CI 190 C2 195 C3 202 Note: 1 ksi = 6.89 MPa. i

ao.2 242 247 248

262

Measured Geometric Imperfections Initial overall geometric imperfections o f the specimens were measured prior to testing. The geometric imperfections were measured along the weld o f the specimens. A theodolite was used to obtain readings at mid-length and near both ends of the specimens. The maximum overall flexural imperfections at mid-length were 1/630, 1/2200 and 1/2000 of the specimen length for Series C1, C2 and C3 respectively.

Test Results The experimental ultimate loads (PE~) o f the test specimens are shown in Tables 6, 7 and 8 for Series C 1, C2 and C3 respectively. A test was repeated for Series C2. The test result for the repeated test is very close to the first test value, with a difference o f 0.7%. The small difference between the repeated test demonstrated the reliability o f the test results. Failure modes at ultimate load o f the columns involved local buckling, overall flexural buckling and combined local and overall flexural buckling. TABLE 6 COMPARISON OF TEST STRENGTHSWITH DESIGN STRENGTHS FOR SERIES C 1 Specimen

Test

P~

. _

'"

Pexp PaStE

1.16 C1L0550 235.2 0.98 C1LI000 198.4 0.87 C1L1500 177.4 0.82 C 1L2000 165.1 0.85 C1L2500 151.6 0.84 C1L3000 133.4 0.92 Mean -0.140 COV -Note: 1 kip = 4.45 kN. COV = coefficient of variation. i

Comparison

Paust/m s

PEu,o

PEp PR&H

P~, PR&R

1.1( 0.98 0.87 0.93 0.98 0.96 0.98 0.098

1.16 0.98 0.87 0.81 0.79 0.74 0.89 0.171

1.29 1.09 0.99 0.99 0.96 0.89 1.04 0.135

1.29 1.09 1.02 1.05 1.03 0.95 1.07 0.109

.,

,.

i

TABLE 7 COMPARISON OF TEST STRENGTHSWITH DESIGN STRENGTHSFOR SERIES C2 Corn )arison

Specimen

P~P

PASCE

(k~9 495.6 0.99 492.2 0.99 474.9 0.95 461.0 0.92 431.6 0.86 -0.94 Mean -0.055 COV Note: 1 kip = 4.45 kN. COV = coefficient of variation. C2L0550 C2L0550R C2L1000 C2L 1500 C2L2000

i

....

0.99 0.99 0.95 0.92 0.86 0.94

0.99 0.99 0.95 0.92 0.86 0.94

1.16 1.15 1.11 1.08 1.01 1.10

1.16 1.15 1.11 1.08 1.01 1.10

0.055

0.055

0.055

0.055

263 TABLE 8 COMPARISONOF TESTSTRENGTHSWITHDESIGNSTRENGTHSFOR SERIESC3 Specimen

Test

P~P

Comparison

P~,.

P.+,,

P~,,

P~,,

P~,,

PaSCE

PAust/ ms

PEuro

PRa,H

PR&R

1.00 0.99 0.96 0.93 0.90 0.96 0.046

1.00 0.99 0.96 0.93 0.90 0.96 0.046

1.05 1.04 1.02 0.98 0.94 1.01 0.046

1'05 1.04 1.02 0.98 0.94 1.01 0.046

Oa9 1123.9 1.00 1119.7 0.99 1087.8 0.96 1045.7 0.93 1009.5 0.90 Mean -0.96 COV -0.046 Note: 1 kip = 4.45 kN. COV = coefficient of variation. C3L1000 C3L1500 C3L2000 C3L2500 C3L3000

DESIGN RULES

The design strengths of the columns were predicted using the American (ASCE, 1991), Australian/New Zealand (Aust/NZS, 2000) and European (Euro Inox, 1994) specifications for coldformed stainless steel structures. In addition, design rules proposed by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997) were also used. The design rules to calculate design strengths for cold-formed stainless steel columns in the American, Australian/New Zealand and European specifications as well as those proposed by Rasmussen & Hancock and Rasmussen & Rondal are either based on Euler column strength or Perry curve. Euler column strength requires the calculation of tangent modulus (Et) and the Ramberg-Osgood parameter n to determine the design stress, which involve an iterative design procedure. On the other hand, the design rules based on Perry curve needs only the initial modulus (Eo) and a number of parameters to calculate the design stress. In both design rules, the design stress should be less than or equal to the 0.2% proof stress of the material. Euler column strength is adopted in the ASCE Specification while in the Aust/NZS Standard one can choose either Euler column strength (identical to that of ASCE Specification) or Perry curve. For the purpose of comparison, this paper uses the latter. The European Specification adopts the Perry curve for the column design strength. The design rules proposed by Rasmussen & Hancock adopts the Euler column strength while Rasmussen & Rondal adopts the Perry curve. In calculating the design strengths, effective length (le) was assumed equal to one-half of the column length (L) for the fixed-ended columns (le = L/2). The design strengths were calculated using the average measured cross-section dimensions and the average measured material properties as detailed in Tables 1-5. The material properties obtained from the tensile coupon tests were used to calculate the design strengths for the three specifications. On the other hand, the material properties obtained from the stub column tests were used to calculate the design strengths predicted by Rasmussen & Hancock and Rasmussen & Rondal as required. The three specifications require the determination of effective cross-section area (Ae) of the column, whereas Rasmussen & Hancock and Rasmussen & Rondal use the gross cross-section area (A). In ASCE Specification and A u s t ~ Z S Standard, the effective area was found to be equal to the gross area of cross-section for Series C1, C2 and C3. In European Specification, the effective area was taken as the gross area of the cross-section for all test series. For the ASCE Specification, the tangent modulus (Et) was determined using Equation (B-2) in Appendix B of the Specification. For the Aust/NZS Standard, the values of the required parameters t~, [3 ko and ~,l obtained from Table 3.4.2 of the Standard are shown in Table 9. These parameters depend

264 on the type of stainless steel used in the column. For the European Specification, the values of imperfection factor and limiting slenderness were taken as 0.49 and 0.4 respectively, which were obtained from Table 5.1 of the Specification. These values depend on how the column is made. The test specimens were cold-formed and seam welded. The design rules proposed by Rasmussen & Hancock (1993) are identical to those in the ASCE Specification, except that the tangent modulus and 0.2% proof stress were determined from the stub column tests rather than the tensile coupon tests. As mentioned earlier, the design rules proposed by Rasmussen & Rondal (1997) use the Perry curve as basic strength curve, and specify the imperfection parameter in terms of the non-dimensional proof stress (t~o.2/Eo) and the Ramberg-Osgood parameter n. These values were determined from the stub column tests, as shown in Table 5. Rasmussen & Rondal use the same parameters as in the Aust/NZS Standard, and equations were proposed for determination of these parameters. The values of these parameters are given in Table 9 for Series C 1, C2 and C3. The values of ot and 13are smaller than the values obtained from the Aust~ZS Standard, while the values of ~o and ~,i are close to those of the Standard. TABLE 9 VALUESOF PARAMETERS ct, [3, ~o, ~l Parameter

Rasmussen & Rondal Series C1 SeriesC2 SeriesC3 0.788 0.998 0.792 0.157 0.152 0.161 0.550 0.566 0.545 0.235 0.285 0.229

Aust/NZS All series 1.590 0.280 0.550 0.200

,

_

COMPARISON OF TEST STRENGTHS W I T H DESIGN STRENGTHS The fixed-ended test strengths (PExp) are compared with the unfactored design strengths predicted using the American (PAscE), Australian/New Zealand (PA~tmzs) and European (PEuro) specifications for coldformed stainless steel structures. The test strengths are also compared with the unfactored design strengths predicted by Rasmussen & Hancock (PR~n) and Rasmussen & Rondal (PR,~). Tables 6, 7 and 8 show the comparison of the test strengths with the design strengths for Series C1, C2 and C3 respectively. The design strengths predicted by the three specifications are generally unconservative for Series C1, C2 and C3. However, the design strengths predicted by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997) are generally conservative. For Series C1, the mean values of the test strength to design strength ratios are 0.92, 0.98 and 0.89 with the coefficients of variation (COV) of 0.140, 0.098 and 0.171 for the American, Australian/New Zealand and European specifications respectively. However, the mean values of the test strength to design strength ratios are 1.04 and 1.07 with the COV of 0.135 and 0.109 for the predictions by Rasmussen & Hancock and Rasmussen & Rondal respectively, as shown in Table 6. Similar results were obtained for Series C2 and C3, as shown in Tables 7 and 8 respectively.

CONCLUSIONS An experimental investigation of cold-formed stainless steel circular hollow section (CHS) columns has been described. Three series of CHS having diameter to thickness ratios ranging from 32.0 to 74.7 were tested between fixed ends at various column lengths. The test strengths were compared with the design strengths predicted using the American (1991), Australian/New Zealand (2000) and European (1994) specifications for cold-formed stainless steel structures, and the design strengths were calculated based on the material properties obtained from tensile coupon tests. Furthermore, the test

265 strengths are compared with the design strengths predicted by Rasmussen & Hancock (1993) and Rasmussen & Rondal (1997), and the design strengths were calculated based on the material properties obtained from stub column tests. It is shown that the design strengths predicted by the three specifications are generally unconservative for the tested fixed-ended cold-formed stainless steel CHS columns. However, the design strengths predicted by Rasmussen & Hancock and Rasmussen & Rondal are generally conservative. The design strengths were calculated based on an effective length of onehalf of the column length.

ACKNOWLEDGMENTS The authors are grateful to the Ministry of Education of Singapore for the support through an AcRF research grant. The authors are thankful to Mr Choong Kiat TAN and Miss Yah Hwee YEO for their assistance in the experimental program as part of their final year undergraduate research project at the Nanyang Technological University, Singapore.

REFERENCES

ASCE. (1991). Specification for the Design of Cold-formed Stainless Steel Structural Members, American Society of Civil Engineers, ANSI/ASCE-8-90, New York. Aust/NZS. (2000). Cold-formed Stainless Steel Structures, Draft Australian/New Zealand Standard, Document DR 00011, Standards Australia, Sydney, Australia. Australian Standard. (1991 ). Methodsfor Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Euro Inox. (1994). Design Manual for Structural Stainless Steel, European Stainless Steel Development & Information Group (Euro Inox), Nickel Development Institute, Toronto, Canada. Galambos T.V. (1988). Ed. Guide to Stability Design Criteria for Metal Structures, 4 th Edition, John Wiley & Sons, Inc., New York, 708-710. Ramberg W. and Osgood W.R. (1943). Description of Stress Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, Washington, D.C. Rasmussen K.J.R. and Hancock G.J. (1993). Design of Cold-formed Stainless Steel Tubular Members. I: Columns. Journal of Structural Engineering, ASCE, 119:8, 2349-2367. Rasmussen K.J.R. and Rondal J. (1997). Strength Curves for Metal Columns. Journal of Structural

Engineering, ASCE, 123:6, 721-728.


Section V COMPOSITES


Third InternationalConferenceon Thin-Walled Structures J. Zara.4,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

269

A MODEL FOR FERROCEMENT THIN WALLED STRUCTURES Donato Abruzzese Department of Civil Engineering, University of Rome "Tor Vergata" Via di Tor Vergata, 00133 Rome, ITALY

ABSTRACT The ferrocement material has been widely adopted in the past in Italy, mostly by famous Nervi, for several kinds of structures. Most of the structures are thin walled, and this kind of structures seem to have a good resistance and as well as a good duration. In the paper is presented a simple and reliable model for thin structural elements made by this composite material, taking into account the information obtained from some experimental results and the construction technology. The model considers the structural element composed by several layers of reinforcing net in a homogeneous matrix of concrete. A large "waved" ttmnel roof designed by Nervi and built in the '50s has been evaluated with the proposed method, and the results were compared with those obtained by Nervi who used less sophisticated mathematical tools.

KEYWORDS Ferrocement, fiber-reinforced concrete, composite material, thin structures, Nervi. THE WIDESPREAD REINFORCEMENT IN THE CONCRETE The limited tensile resistance of the concrete needs to be improved by adding reinforcement, such as steel, fiber glass, carbon fiber or others. The most challenge way to improve in all the directions the resistance of the concrete is to distribute, in a homogeneous way, short fibers in the cement matrix. In a composite material this will result with a behaviour almost equal for tensile and compressive stress. A large literature on the fiberreinforced concrete is available, and also many experimental results can be utilized to investigate better the composite. The ACI Committee guide 549, 1996, states that the ferrocement is a composite material of hydraulic cement and filler reinforced by multiple steel net layers. The characteristic resistance of the matrix should not be less than 35 N/ram 2, while the characteristic of the steel used for the nets should not be lower than the values indicated in the Table 1 (referred to different type of nets, see Fig.l), and the yield stress not greater than 690 N/mm 2.

270

Veld stress fv (N/mmz) E, effective longitudinal

Woven square mesh 450

Welded Square mesh 450

Hexagonal mesh 310

Expanded mesh 310

Wire

138

200

104

138

200

165

200

69

69

(kN/mm~) Er effective transversal

(kN/mm2)

414

Tab. 1 - Minimum recommanded values stress/elasticity modulus of the steel in the net According to the recommendations of the ACI Committee, the structures made by ferrocement should be calculated in the same way as for reinforced concrete. This seems the simplest way to approach the problem, since several things are different between the two composite material. For instance, the way to evaluate the elasticity modulus of the reinforcing wires will differ from type to type of net. Sometime it is required to perform specific test to evaluate the global elasticity modulus for the composite. The ultimate stress analysis of the structure, with this assumptions, will appear as the following:

b----d

__'___ Fig. 1 - Stress distribution for the flexural behaviour of a rectangular beam (limit analysis) THE FERROCEMENT AND PIER LUIGI NERVI

In the 1948, P. L. Nervi patented the ferrocement technique, and started to use several layers of steel net with very thin wires kept together in almost any kind of shape by a mortar composed of cement and sand. Nervi used the ferrocement technique and material for many structures, taking advantage of the moderate thickness, and consequently less weight, relying on the greater ductility of the composite material steel-cement, and assuming the bidimensional behaviour of the new material, which is able to support compressive or tension loads. In fact, the first experimental tensile tests on samples with different number of layers of net, performed in 1949 at the University of Milano by Oberti and Grandori, produced encouraging results which showed well the increasing of the deformation of the material at the first crack. In Tab. 2 the values of the stress (~r and the strain er corresponding to the first crack are presented, versus the number and type of net adopted. The percentage of the steel takes into account only the area of the parallel wires in the longitudinal directions, neglecting the transversal ones. As it is possible to see in Tab. 2, samples with greater steel quantities resulted in greater values of the deformation at the first crack. This behaviour of the ferrocement can be explained considering that the cement, with an high percentage of steel, corresponding to a large frictional surface, can move into an ultra-elastic state, allowing large deformation and ductility.

271 Sample N. layers l 1 2 3 4 5 6 7 8

3 6 4 20 10 10 10 10

P kg/m2 0.4 0.4 0.4 0.4 1 1 1 1

d ,m,m 0.57 0.57 0.57 0.57 0.9 0.9 0.9 0.9

L mm 10 10 10 ..... 10 10 10 10 10

s mm 17 16 17 18 17 17 16 16

g % 0.45 0.95 1.35 2.83 3.74 3.74 3.98 , 3.98

~r E-05 12.5 11.5 11.5 60 66 67 64 64

or orf k g / c m 2 kg/cm2 25 270 20 250 20 250 42 1290 60 1420 60 1440 60 1380 60 1380

Tab. 2 - The experimental results by Oberti & Grandori (1949) The typical tensile response of a ferrocement sample is shown in Fig.2. The stress-strain function can be divided in three phase. In the first phase the composite behaves as linear elastic material, in the second one, starting at the first crack, several cracks appear in the matrix (concrete). This second phase can be considered as the most probably normal life of the structure. In the last phase, starting from the yielding point of the steel, the number of the cracks remains almost the same, but their dimension increases. // /

/

(c~-

/

"'!'-

8

COMI~ITE 9 $TI~IJN

Fig. 2 - Typical tensile behaviour of ferrocement Therefore we can state that the ultimate tensile load of the ferrocement depends neither on the thickness of the concrete nor its quality, since the matrix, once cracked, does not contribute to the overall resistance of the sample, granted only by the reinforcement. In the primary elastic phase the stress-strain relation is defined with the elastic modulus Ec of the composite. The initial value can be calculated: Ec =Em ( 1 - rl Vr) + Er rl Vr

(1)

While the value of the stress in the composite at the first crack is given by the following experimental function (Naaman e Shah): (~cr = (~mu + 25 1"1Sr

1"1Sr< 0.2 mln2/mm 3.

(2)

where Era,= matrix elasticity; ~mu = tensile resistance of the matrix, r I = net effective factor (in the stress direction) The net effective factor indicates the available contribute of the reinforcing net in the stress direction. In Tab. 3 the net effective load factor has been shown depending on the different available net shapes.

272 Woven square mesh 0.50 0.50 0.35

Stress direction . ,

Longitudinal rh Trasversal qt Angle 45 ~ rl4s

Welded Square mesh 0.50 0.50 0.35

. . . .

Hexagonal mesh 0.45 0.30 0.30

Expanded mesh 0.65 0.20 0.30

,=

Tab.3 - Net effective factors for different net shapes As the number of the cracks increases, the contribute of the matrix decreases progressively, and in this phase the value of the elastic modulus Er of the reinforcement can be assumed: Ec= nVrEr

(3)

The maximum dimension of the cracks ~i~max in this phase can be easily calculated with the experimental function proposed by Naaman e Shah: per O'r < 345 11 Sr per Or > 345 1"1Sr

~max = 3500 / Er Wm~ = 20 / Er(175 + 3.69 ( t~r- 345 11 Sr ))

(4) (5)

where ar "- stress in the net when the load is applied Once the steel reaches the yield point, the load remains unchanged and is carried entirely by the net. The collapse load can be calculated: t~cu = fy 1"1Vr (6) THE MODELS FOR THE FERROCEMENT In Fig. 3 is shown the experimental function stress-strain corresponding to a sample of ferrocement of dimensions 25.4 x 9.5 mm and 150 mm long, reinforced with a quarter-inch squared nets. N=

1

5

8

10

14

16

10-

_

-o15

Wov~I Steel Mesh Vr = 0.023 Sr = 0.27 (~ Phase /

t

yeld point

Phase II

10

20

Composite strain, ram*

30

40 I0

50

4

Fig. 3 - Result from tensile test of ferrocement element

Eo'

E1

Fig. 4 - Experimental results of a ferrocement sample

273 The first tensile test required by Nervi in 1949 and performed at the Politecnico of Milano on ferrocement samples produced the graph stress-strain in Fig. 4, in which the strain is limited to the limit elastic deformation e = 0.001 of the steel. The gradient of the function decreases until the value el for the more reinforced samples, while in the range e > el the function is linear, because the contribution of the matrix, already cracked, remain constant until the collapse of the sample. The deformation limit, corresponding to el, does not depend on the reinforcing percentage. In the graph it is possible recognize two different phases. The first one for 0<e<el, in which the concrete is acting the main role, with the secant elastic modulus El, corresponding to the deformation e=Cl; a second one for Cl<e<e r, when the contribute to the resistance is given by the reinforcement, and the elastic modulus can be assumed equal to E'. Let us consider the line from the origin of the coordinates, corresponding to an elastic modulus value: E'o = Eo ( 1 + ~/ n )

(7)

that is corresponding to the starting tangent modulus of the experimental graphs, since the reinforcement steel does not yet reach the yield point. The modulus Eo' can be assumed, instead of the secant modulus, in the first phase. If we write the contribute of the concrete to the resistance of the composite: S t~ic = S - Nd t~c (8) this is the effective contribute, since the load will be distributed considering the reinforcing wires (number N and diameter d). We obtain, for e<el, that the behaviour does not depend on the reinforcement, since we are still in the elastic phase. Also, the modulus in the second phase seems depend on the reinforcement. If we assume eric = el Eo, for e=el, since eric is not depending on the reinforcement, we can then calculate E1 in the following way. Let SN be the thickness corresponding to the nets in the total S thickness, and So the thickness corresponding to the concrete: SN = N d,

So = S - N d

(9) (10)

the elastic modulus corresponding to the thickness of the nets, neglecting the contribute of the concrete in that part, can be calculated: N.n.d 2 EN = E f ~ 4.1.S N

(11)

where Ef = Eo n. Then the value of the effective total elastic modulus E 1 is: EI=E0"S0/S+EN-S~t/S

= E 0 ( 1 . + y . n ) . N 'sd

(12)

The value obtained is very close to that one of the secant modulus of the experimental tests. In the starting elastic phase, the elastic modulus and the crack stress are given by the (1) and (2), after that point we can assume the medium value of the global elastic modulus Ec, given by the (3). The values of the elastic modulus can be always evaluated experimentally. The reinforcing net behaviour is elasto-plastic, with an elastic modulus equal to the effective global elastic modulus of the material matrix-net Er, taking into account for each layer of net the effective resistant area Asi in the stress direction, calculated by means of the global efficiency factor. The

274 behaviour of the concrete can be assumed non linear for compressive stresses, given by the Sargin's formula, and linear elastic for tensile stress, until the first crack. A higher resistance of the concrete can be evaluated considering the amount of reinforcing in the matrix, as calculated in (2). A simplified tensile model of the behaviour of the ferrocement can be obtained by linearizing the described phases and assuming the value for E and o" corresponding at the specified (Fig.5).

~r

--

E2

o.0o4Compression It

I

' i .,

Tension

., 0.9

(~c

Fig.5 - Ferrocement mechanical model

THE PALACE FOR T H E F E R R O C E M E N T ROOF

EXHIBITION

IN TORINO.

THE MODELING

OF THE

The Palace for the exhibition in Torino, Italy, has a rectangular shape with dimension of 96 m x 156 m. The main roof is a cylindrical vault, with the span of 80 m, and internal high 18.10m, with a waved shaped showing a thickness of the cross section ranging from 1 m to 1.60 m. The roof is composed by several precasted elements in ferrocement, with width 2.50 m and length 4.50 m., with some openings for the windows"

Fig.6- The cross section of the waved roof

Fig. 7 - Wires and bars for the reinforcement in the precast ferrocement elements The idea to use precast structural elements in ferrocement allowed to build a very light and not very expensive structure in a short time. Using the traditional concrete would result in a heavy structure, need of casting, and probably the architectural effect will be changed.

275 The variable thickness of the roof is related to the structural assumption of the two hinges arch. If a static calculation of the roof, as it is, is performed, a pressure curve can be observed, not so far from the that one assuming two hinges at the end. Also, the pressure curve due to the dead load only, is very close to the center of the section of the 'beam'. The static calculation has been done considering a distribute load on the roof equal to 150 Kg/m2, an asymmetric load (half roof loaded) and a thermal load of 20~ Our aim has been to evaluate the static behaviour of the ferrocement roof of the Palace of the Exhibition of Turin. The simple constitutive model, obtained by linearizing the experimental curves, offers a reliable instrument to investigate existing structures made by ferrocement. The results of the calculation have been compared with those reported by Nervi in his report. To investigate the flexural behaviour of the ferrocement structural elements the simplified model utilized is based on the concept of homogenization of composite materials. Since the reinforcement is widespread and well distributed, the composite is modeled as an equivalent ideal homogenous material with a different behaviour in compression and in tension. In compression an elastic perfectly - plastic model has been applied, similar to those adopted for the fiberreinforced. The compressive elastic modulus has been calculated by the following homogenization law: Ecr =

Ecru V m +

E,. V r 11

(13)

where Vm, Vr and Ecru, Er are, respectively, the percentage and the Young's modulus of the matrix and the reinforcement. The coefficient rl is the efficient factor of the reinforcement in the stress direction. For the tensile behaviour of the material, considering the high percentage of steel, a simplified model taking into account the different phases of the composite, has been assumed, as described previously. The simple constitutive model, obtained by experimental curves, offers a reliable instrument to investigate existing structures made by ferrocement. The results of the calculation (maximum stress in the concrete ere = 58 Kg/cm2) have been compared with those reported by Nervi in his report (co = 53 Kg/cm2). For each different structural element the ultimate domain M-N has been calculated. The medium safety factor of the roof of the Palace of the Exhibition is slight less then 4. Of course, during the regular activities, when the loads are represented only by the self weight, the safety factor is very large.

ANALYSIS RESULTS AND CONCLUSIONS The aim of the study has been to find a reliable model to evaluate the static behaviour of ferrocement elements, and to assess the safety factor of the structures. As case test, the roof of the Palace of the Exhibition in Torino, designed by P. L. Nervi in 1949, has been analyzed, and the results seem comparable with those calculated by Nervi. The constitutive model assumed seems reliable, and able to describe the behaviour of the bidimensional structural elements in the elastic and ultra-elastic phases. A further experimental investigation on flexural behaviour of ferrocement elements seems useful to validate and refine the parameters of the proposed model.

276 REFERENCES

ACI Committee 549.(1996). Guide for the design, Construction and Repair of Ferrocement. ACI Manual of Concrete Practice 1996. Part 5. ACI Committeee (1996). ACI Manual of concrete practice. Part 5. Balaguru, Naamaan and Shah. (1978) Analysis and behavior of ferroeement in flexure. ASCE Journal of the Structural Division. Vol. 103 Barberio, Goffi and Mattone (1981). Comparison of the flexural behaviour of thin ferrocement and fibre reinforced concrete slabs. International symposium on ferrocement. Rilem. edit by Oberti and Shah Colin, Johnston and Mattar (1976). Ferrocemem behavior in tension and compression. ASCE Journal of the structural division. Vol. 102 Lim, Parasivam, and Lee. (1987). Bending behavior of steel fiber concrete beams. ACI Structural Journal. V01.84. Mansur and Parasivam (1990). Ferroeement short columns under axial and eccentric compression. A CI Structural Journal. Nervi P. L. (1956). I1 ferroeemento: sue r e possibilitY. L'ingegnere. Oberti G. and G. Grandori (1949). Prime esperienze sulla deformabilit~ e resistenza a trazione di provini in ferro eementato. Report of the Laboratory at Politecnico of Milano. Parasivam, Ong and Lee (1988). Ferrocement structures and structural elements. Material Composite. Somayaji and Shah (1981). Prediction of tensile response or ferrocement International symposium on ferrocement. Rilem. edit by Oberti and Shah Wamg Kai Ming (1981). Calculation of strength for ferrocement. International symposium on ferrocement, Rilem, edit by Oberti and Shah

Third Intemational Conference on Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

277

EFFECTS OF MANUFACTURING VARIABLES ON THE SERVICE RELIABILITY OF COMPOSITE STRUCTURES A.R.A. Arafath, R. Vaziri, H. Li, R.O. Foschi and A. Poursartip Departments of Civil Engineering and Metals and Materials Engineering The University of British Columbia 2324 Main Mall, Vancouver, B.C., V6T 1ZA, Canada

ABSTRACT A combined deterministic-stochastic simulation approach is presented to demonstrate the manner in which manufacturing-induced variabilities in composite structures can be controlled to achieve targeted reliability levels in structural performance. A finite element code, which deterministically simulates the various physical phenomena during manufacturing of composite structures, is integrated with non-linear structural analysis and reliability analysis to compute the statistics of the parameters that must be controlled at the manufacturing level in order to result in an optimum or reliable structural performance during service. The methodology is demonstrated through a case study that examines the buckling behaviour of a composite plate in the presence of manufacturing-induced imperfections. The reliabilitybased approach adopted here which uses the as-manufactured, rather than the as-designed conditions as the basis for structural analysis can potentially lead to the reduction of uncertainties and costs associated with the use of empirical safety factors currently employed in design of composite structures.

KEYWORDS Fibre-Reinforced Laminated Composite Structures, Manufacturing Imperfections, Process Modelling, Reliability, Buckling, Residual Stresses, Variability.

INTRODUCTION

Background Fibre-reinforced plastic composite materials are replacing traditional ones in many of today's structural engineering applications. When using composites, unlike their metallic counterparts, the complete (largescale) structure is manufactured from the raw materials in one step. Therefore, the manufacturing process and its consequences on the material and geometrical properties of the final structure must be taken into account when considering the response to, and reliability under, the applied loads. Otherwise, the designer

278 of composite structures is forced to use empirical safety factors, which may result in a conservatism that, in many cases, nullifies the benefits that are offered by composite materials. There are many input process parameters that influence, to various degrees, the outcome of the manufacturing, influencing, in turn, the structural performance. Many of these parameters have inherent stochastic variability that result in uncertainty in the final part geometry and material properties. By coupling stochastic and deterministic computational models of the physical events one can study the effect of those variabilities as they propagate through the various stages of the analyses, and eventually influence the response of the composite structure to in-service loading. This would ultimately allow process variables to be controlled or manipulated at the outset (manufacturing level) to result in an optimum or reliable structural performance during service. The objective of this paper is to present a computational framework for analysing the complete sequence of events from the manufacturing stage to the in-service structural performance of a composite structure, while taking into account the stochastic nature of the parameters involved. Specifically, we combine a deterministic model for the manufacturing process and structural analysis with probability-based reliability models to assess the reliability of the composite structure under service loads. To illustrate the approach, we consider the buckling behaviour of a slender laminated composite plate in the presence of imperfections arising from the manufacturing.

Deterministic Process Model Our focus is a high-performance structural component made of advanced thermoset matrix composites typically employed in the aerospace industry. An autoclave process is commonly used to manufacture such structures. Broadly speaking, this process involves stacking of pre-impregnated sheets of unidirectional fibres (commonly called prepreg) at various orientations over a tool of desired shape and then subjecting the whole assembly to a controlled cycle of temperature and pressure inside an autoclave. The process results in the compaction and curing of the composite part. However, because of the residual stresses that build up during the process, the precise shape and dimensions of the final part after tool removal are often difficult to control. In thin-walled composite structures spring-back and warpage are commonly observed process-induced imperfections. A comprehensive, multi-physics, 2D finite element code, COMPRO, has been developed to analyze industrial autoclave processing of composite structures of intermediate size and complexity. The model caters for a number of important processing parameters and the development of residual stress and deformation. This model advances previous work by other researchers and, in addition, accounts for the effects of tool/part interactions, which have been neglected by other investigators. The model assumptions, theoretical background, solution strategies and case studies demonstrating its predictive capabilities have been documented in recent articles by Hubert et al (1999) and Johnston et al (2001). Few investigators have explicitly considered the effect of manufacturing imperfections on the structural performance. Those who have incorporated the as-manufactured conditions in their analyses (e.g. Li et al, 1995) limited them to measurable quantities such as thickness variations. However, using a deterministic numerical process model, such as COMPRO, the final state of a manufactured component with all its spatial variations in geometry, material properties and residual stresses, which do not lend themselves to simple experimental measurements, are readily available as initial conditions for structural analyses.

Probability-Based Modelling Uncertainties in composite structural behaviour arise at different levels (Shiao and Chamis, 1999) such as the material level (uncertainties in the raw material properties), the manufacturing level (e.g. uncertainties

279 in ply orientation, ply thickness, fibre volume fraction, cure cycle) and the structural level (e.g. uncertainties in loading, boundary conditions, geometry). To account for these uncertainties in the analysis and design of composite structures require a stochastic modelling approach. Among several available approaches, sampling techniques (e.g. the Monte Carlo simulation) that choose samples purely randomly from the range of input parameters are theoretically simple. However, since the deterministic models for each input parameter set has to be invoked, the total computational time is a strong function of the number of samples chosen. Typically, a huge amount of computer time and memory is required especially when finite element methods are used for the deterministic computations. An alternative approach is to use analytical-based reliability models, which are theoretically more complex but in return provide powerful and efficient tools for predicting the probability distributions of output parameters of interest. Of particular interest in performance-based design is the inverse reliability problem, whereby design parameters must be found so that different performance criteria are satisfied with target reliabilities. Li and Foschi (1998) have recently developed an inverse reliability software (IRELAN) to study the inverse problem. In this paper, we present how process-induced uncertainties can be included in the analysis of composite structures and how the processing parameters can be controlled to achieve a certain reliability level in structural performance. For illustrative purposes, the buckling behaviour of a composite plate in the presence of process-induced residual stresses and deformations is examined using an integrated deterministic-probabilistic analysis approach. The deterministic process simulation software, COMPRO, the commercial structural analysis software, ABAQUS, and the inverse reliability software, IRELAN, are used here to determine the process parameters that must be controlled to meet structural performance standards with target reliabilities.

M E T H O D O L O G Y AND RESULTS

Statement of the Problem A unidirectional, carbon fibre-reinforced plastic (CFRP) composite laminate with a span of 1200mm, a width of 100mm and a thickness of 1.6mm (consisting of 8 plies of thickness 0.2 mm) is considered for this study. As part of an extensive experimental study (Twigg, 2001), various samples were manufactured on flat aluminum tooling inside an autoclave and the resulting warpages were measured to provide a benchmark for comparison with the process modelling predictions of COMPRO. Initial imperfections resulting from the manufacturing process were computed using COMPRO in terms of three main variables: the fibre orientation angle 0, the fibre volume fraction ratio Vf and the autoclave hold temperature T. The virtually manufactured structures were then considered to be simply-supported and subjected to progressively increasing compressive axial load P as shown in Figure I.

i(w),..7

,1,~,

,,.

"~

-x (u)

t.. IT M

K//

y (v)

Fibre Orientation ~.6mm ~

1200 mm '

t

p

~~,

-5

SIO0

mm

,~1 "7

Figure 1" Schematic of the composite plate loading and boundary conditions. The corresponding non-linear structural analysis was carried out using ABAQUS. The material properties used in these calculations, which are listed in Table 1, were fully-cured properties of the laminate computed using the micro-mechanical models in COMPRO (Johnston et al, 2001). The non-linear relation

280

between the axial load, P and the out-of-plane displacement, w, was obtained by increasing the load P incrementally up to the limiting buckling load Per and computing the corresponding displacement. Using the eigenvalue analysis option in ABAQUS, the critical buckling load for a flat plate, free of imperfections, was found to be Pcr = 29.5 N. Of interest to our subsequent reliability study was the deflection A75~ at a load level of P = 0.75Pcr. TABLE 1 LAMINATEMATERIALPROPERTIES Longitudinal elastic modulus, Eml Transverse elastic modulus, E22 =

126 GPa E33

Major Poisson's ratio in-plane, vl2 Longitudinal shear modulus, GI2 = Transverse shear modulus, G23

10.9 GPa 0.264

Gl3

5.04 GPa 3.28 GPa

Process Modelling The finite element mesh used in COMPRO consisting of 4-noded isoparametric quadrilateral elements under plane strain conditions, is shown in Figure 2. Because of the symmetry, only half of the geometry is modelled. Our experimental and numerical research has shown that the operative mechanism at the interface between the composite part and the tool plays a significant role in the final distortion of parts with various geometrical shapes. At present, our process model in COMPRO assumes a simple, linear elastic shear layer at the tool/part interface as shown in Figure 2. The properties of the shear layer material depend on many parameters, which at present are not fully characterized. These properties are currently adjusted in order to match the experimental measurements of the distorted shape. In our case, using a shear layer modulus of 5.52 kPa resulted in an accurate prediction of the deflection profile as shown in Figure 3.

Figure 2: Finite element mesh used in COMPRO

Figure 3: Comparison of COMPRO predictions of warpage (for half-length) with experiment

281

Structural Modelling Being a 2D model, COMPRO can only provide information on a 2D plane strain section of the composite part. Therefore, to obtain the 3D spatial distribution of residual stresses and deformations necessary for the structural analyses, the output from COMPRO had to be mapped onto the structural elements (plates and shells) in ABAQUS as initial conditions. To accomplish this, all six components of the process-induced residual stresses (3 stresses in x-z plane and 3 stresses in the x-y plane arising from the plane strain constraints) evaluated at the midpoint of the 2D solid elements in COMPRO were initially assigned to corresponding layers of shell elements in ABAQUS (Figure 4a,b). The initial configuration of the composite plate in ABAQUS was taken to be cylindrical with its curved geometry in the x-z plane conforming to the deformed shape predicted by COMPRO (Figure 4b). The shell elements were then allowed to deform freely (i.e. with the plane strain constraints removed) in order to equilibrate under the action of these initial stresses (Figure 4c). Independent, full-blown 3D analyses of a similar problem, a bimetallic strip subjected to thermal loading, were carried out to verify the validity of this approach.

i ~

""~f"-!

L t ' 4-4-d -'

; ~TI"4

i ~.----r-,-t"7-.. , , , t-7---'-

(a)

""

r

'

t /i

(b)

....1 ,",

(c)

Figure 4: Schematic of the procedure used to map the information from COMPRO to ABAQUS

Parametric Study A parametric study was performed to identify the most important process variables that have an effect on the warpage of the plate. The sensitivity factor, which is defined as the ratio of the percentage increase in warpage to the percentage increase in the parameter, was calculated and plotted for each parameter as shown in Figure 5. According to the chart, only 3 parameters (T, Vf, and 0) have much effect on warpage. Hence, only these 3 parameters are considered as random variables in the reliability analysis. 1.6 1.4

T - Temperature Vf - Fibre Volume Fraction 0 - Ply Misalignment HR - Heating Rate p - Pressure HT - Hold Time CR - Coolin~ Rate

t..

o

1.2 1

~ o.8 9- 0.6 9~ 0.4 0.2

T

Vf

0

HR

p

HT

CR

Figure 5: Sensitivity of warpage to changes in selected processing variables

Reliability.Based Determination of Process Control Parameters The distorted shape of the plate after processing, w, is a function of T, Vf, and 8, while the critical load, Pc,

282 is only a function of 0and Ve, which influence the material properties. Since the variables T, Vf, and 19are subject to uncertainty due to possible lack of tight process control, the outcomes w and Per are also random. As a first step, the range in these outcomes is determined by constructing a database for w = A75~ (i.e. displacement at 0.75Per) and Pcr, using COMPRO and ABAQUS in a deterministic manner, allowing T, Vf, and 0 to take values within their respective, likely ranges. These two deterministic databases can be expanded at will by increasing the number of variable combinations studied. For the purposes of this study the performance standards for the plate were defined as follows: 1) The deflection w should be within a range, from 20mm to 40mm, with high confidence (90%). The lower limit will ensure that buckling will not result in a sudden departure from the flat configuration (i.e. there would be some warning before collapse). The upper limit will ensure that the plate will not be excessively deformed. These two conditions can be described by a set of two "performance functions", Gl and G2 as follows

G~ = w(O,V/,T)- 20 G 2 = 4 0 - w(O,V/,T) 2) The applied load P should result in a sufficient reliability against buckling. This condition is expressed as a performance function G3,

o~ =5.,(o,v:)-P Given that the variables are random, the failure or non-performance probabilities correspond to the events G~ < 0,G2 < 0 and G3 < 0. Provided that the statistics of the variables are defined, these probabilities can be readily evaluated by forward reliability procedures. In the software RELAN (Foschi et al., 1999), the iterative evaluation algorithm requires the calculation of the functions G~. G2 and G3 for different combinations of the variables. This calculation is carried out by local interpolation of the deterministic database (Foschi and Li, 2000). The probability of failure in each case is obtained by importance sampling simulation around an anchor point, which is determined by using an approximate response surface and the First Order Reliability Method (FORM). In each case, the probability of failure can be expressed by the associated reliability index ,8.

Example I m

For the inverse problem, let us assume that the task is to find the design parameters V:, T and P , mean values of the corresponding variables, assuming that the remaining relevant statistics are as shown in Table 2. That is, knowing the accuracy with which the angle 0 and the variables Ve and T can be controlled, their mean values are to be found (except for O = 0.0~ Along with these, the mean of the load P that can be applied is to be determined in such a way that the plate response has the following target reliabilities for each of the three performance criteria: 1) For Gt, a failure probability of 0.05, or a target reliability index fl~ = 1.645 2) For G2, a failure probability of 0.05, or a target reliability index ,82 = 1.645 3) For G3, a failure probability of 6.2x10 3, or a target reliability index ~ = 2.500 The first two conditions imply a 90% confidence in the interval 20mm < w 0 - to the flange deflected towards the outside of the column contour. The value of the critical load corresponding to the global flexural-torsional buckling decreases with an increase in the curvature b~/R by nearly 1.5 times. The value of the global flexural critical load undergoes slight changes. A similar situation occurs for the local antisymmetric buckling. The shell element exerts the strongest influence on the local symmetric buckling. Even a slight flange curvature results in a rapid increase in the critical load. The lowest value of the critical load of the beam-column under analysis occurs for: local symmetric buckling mode for -0.5 < b~/R ___0.25; local antisymmetric buckling mode for b~/R = 0.5. The further analysis has concerned the interactive buckling and has been aimed at determination of the sensitivity of the columns under discussion to global and local geometrical imperfections. It has been assumed that the beam-column has the following initial deflections ~g~ = I1.01, ~-10.21. In Table 1 the theoretical load-carrying capacity N, [Nm], the ratios of the theoretical load-carrying capacity, determined within the first order non-linear approximation, to the minimum critical load value, that is to say, NdNm, where Nm = min(Ng, N0, are presented. In each case the sign of the imperfection has been chosen in the most unfavourable value, i.e. so that N, would have its minimum value (for more a detailed analysis see Refs. by Kolakowski and Krolak (1995), Krolak and Kolakowski (1995)). In Table 1 the ratios of the global critical load to the local one, for the same conditions along the symmetry axis of the cross-section, i.e. the ratios of the critical loads: global flexural-torsional (antisymmetric) to local antisymmetric; global flexural (symmetric) to local symmetric, are included as well. TABLE 1 LOAD-CARRYING CAPACITY WITHIN THE FIRST ORDER APPROXIMATION

bl/R -0.5 0.0 0.5

Antisymmetric modes N, [Nm] NdNm Ng/Nl 0.139 0.784 1.949 0.115 0.766 1.626 0.115 0.762 1.344

N, [him],, 0.138 0.0621 0.133

Symmetric modes N~qm 0.858 0.797 0.858

Ng/Nl 4.077 9.085 4.423

300 The interaction of the global flexural buckling mode with the local symmetric one has turned out to be the most dangerous for -0.5 _.Ex'.'-.x.x.. olfxxl (:Sss = V,,sEs/(l-VsxV,,s)

l~xs

Es/(1-VsxVxs )

o

0

0

,

1ess~

(1)

Gxs l~,~ J

The objectives of this work are (i) to extend the existing GBT, in order to make it applicable to orthotropic thin-walled structural members and (ii) to use the derived equations to analyse the local and global buckling behaviour of composite FRP columns with commonly used cross-sections. The GBT results are validated through a comparison with numerical ones, yielded by finite strip analyses. Finally, a parametric study is carried out to investigate the variation of the critical bifurcation stress and buckling mode nature with both (i) the column length and cross-section dimensions and (ii) the composite material properties.

GBT

FOR ORTHOTROPIC MATERIALS

The simplifying assumptions adopted in the derivation of GBT for isotropic materials (Schardt, 1989), namely neglecting the membrane shearing strain and transversal extension, with the respect to the longitudinal extension values, remain perfectly valid in the context of the linear stability analysis of orthotropic columns. According with such assumptions, the relevant strain-displacement (kinematic) relations employed are defined by exF• = --ZW,xx

esFs= -- ZW,ss

7xFs= -- 2ZW,xs

M

1

9

Cxx= U.x + 5 (V,'x+

X

W2

,x)

x(u) t

~

ds / l / z(w)

Figure 1: Column geometry, coordinate system and displacement components.

, (2)

331 where x and s are coordinates along the length and mid line of the cross-section wall (see figure 1) and the superscripts ( )" and ( )v stand for the origin of the strain components (membrane orflexural). In order to obtain a displacement representation compatible with the classical beam theory, Schardt (1989) prescribes that each displacement component (u, v or w) at any given point of the mid line of a crosssection comprised of n wall segments (i.e., containing n+l nodes and/or degrees of freedom) must be expressed as a combination of de n+l orthogonal functions (u,), all of which vary linearly between consecutive nodes. One has, therefore, U(X,S) -- UkCk, x

V(X,S) -- Vkr k

W(X,S) -- Wkr k

,

(3)

where the summation convention applies to the subscript k (1 G2)induces a similar reduction in the oJbe and ~ values LP J LP D (yb.r 7/59"~t~bDt;2/~b.C~l=43/69=O. 625). Next, it is intended to investigate the influence of the lipped channel column cross-section dimensions on the nature of the local critical buckling mode and corresponding orb value. Taking into consideration the cross-section proportions of the commonly used pultruded columns, the following dimension ratios are considered: (i) b,,/t=20 and 40, (ii) b/b,,,=0.25, 0.50 and O.75 and (iii) bl/bw=O.05 to 0. 40. Figure 6 shows the variation of the orb with bdbw, for the LPM (dotted line), DM (solid-dashed lines).

120

Gb (MPa)

40

~b (MPa) beqgw=0.75

b~/bw=0.50

............................................................................

90

bet,w=0.50 b~ow=0.75 bc'bw=0.25

6o

30

bt4bw=0.25

20

30 bl/bw

0, 0

0.1

0.2

(a)

0.3

0.4

0 -! 0

, 0.1

. 0.2

0.3

bl/bw 0.4

(b)

Figure 6: Variation of orb with b/bw and bl/bw. (a) bw/t=20 (b) bwlt=40

337 From this limited parametric study, it is possible to conclude that: (i) As expected, the cr~1' values are practically independent of the lips width (the dotted lines are practically horizontal). They only depend on the web (mainly) and flanges widths. (ii) The r vs. bdbw curves display maximum values for 0il) bdbwzO.16 (b/b,,,=0.25), (ii2) bdb,,,zO.24 (by~b,,,=0.50) and (ii3) bdb,~=0.38 (b/b,,=O. 75). Notice also that, for (unrealistically) large bl/bw and when bfbw < 0.75, cd~is not associated with a local minimum (dashed portions of the curves). (iii) For the lower plate slendemess value (bJt=20), the local critical stress is always associated to the DM (O'crt~OJ~). For the upper plate slenderness value (bw/t=40), on the other hand, it may be associated to either the DM (o'crt~6~) or the LPM (O'cr~(Tlbl'),depending on the b fib,,, value. (iv) The lip width values associated to critical DM-LPM transitions (white circles in figure 6(b) have an obvious practical interest, as they lead to the definition of "optimally efficient" lips (stiffeners). Therefore, design formulae to estimate such width values would be rather useful.

CONCLUDING REMARKS (i) (ii)

(iii)

(iv)

(v)

The existing GBT, developed in the context of isotropic materials, was extended in order to enable its application to orthotropic thin-walled structural members. The derived GBT equations were first validated, by means of a comparison with numerical results obtained through finite strip analyses, and, then, applied to study the local and global buckling behaviour of lipped channel FRP columns. In particular, a mixed flexural-distortional buckling mode was identified, which does not appear in isotropic (e.g., cold-formed steel) columns. Such mode was shown to be critical for intermediate length columns. A limited parametric study was carried out to investigate the influence of (iii0 the composite material properties and (iii2) the column length and cross-section dimensions on the critical bifurcation stress and buckling mode nature. The buckling behaviour of five geometrically identical lipped channel columns made of different fiber reinforced plastics, all with an epoxy matrix and unidirectionally aligned glass, kevlar or carbon fibers, was investigated. The analysis unveiled markedly different behaviours and the results obtained showed the carbon fibers to be the ones providing the highest local and global buckling resistance. Design formulae to evaluate the widths of"optimally efficient" lips (i.e., widths associated to critical DM-LPM transitions) would be very useful in practice and, therefore, should be sought.

References Datoo MH (1991), Mechanics of Fibrous Composites, Elsevier Science, London. Davies JM, (1999), Modeling, Analysis and Design of Thin-Walled Steel Structures, Light-Weight

Steel and Aluminium Structures, Elsevier Science, London, 3-18. Davies JM, Leach P and Heinz D (1994), Second-Order Generalised Beam Theory, Journal of Constructional Steel Research, 31,221-241. Nagahama K, Batista E (2000), Stability Analysis of Glass Fiber Composite Pultmded Members, CDROM Proceedings of the XXIX Jornadas Sudamericanas de Ingenieria Estrutural, Punta Del Este, Uruguay, paper # 2.7.5 (20 pages). (in portuguese) Schardt R (1989), Verallgemeinerte Technische Biegetheorie, SpringerVerlag, Berlin. (in german)


Third International Conference on Thin-Walled Structures J. Zara~, K. KowaI-Michalska and J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

339

SHEAR CONNECTION BETWEEN CONCRETE AND THIN STEEL PLATES IN DOUBLE SKIN COMPOSITE CONSTRUCTION Howard D Wright 1 Anwar Elbadawy 2 & Roy Cairns 3 1Professor, University of Strathclyde, Glasgow, UK 2 Research student, University of Strathclyde, Glasgow, UK 3 Lecturer, University of Strathclyde, Glasgow, UK

ABSTRACT Double skin composite (DSC) systems are formed from steel-concrete-steel sandwich elements that consist of a layer of un-reinforced concrete, sandwiched between two layers of thin steel plates. These in turn are connected to the concrete by welded shear connectors. Shear studs or bars are used to transfer slip shear between the outer steel skins and concrete core. To investigate the behaviour of the shear connectors when welded to the normally thin steel plates an experimental study, on nine model push-out tests, has been carried out and is presented in this paper. The connectors were 6mm diameter bars welded between each plate providing a connector to plate thickness ratio of 3. Micro-concrete with maximum aggregate size smaller than 2.41 mm was used. The micro-concrete core was pushed through the plates in direct shear. Failure modes are defined for each of three series of tests and more detailed observations regarding the structural action of this form of construction are given. The studies show that failure occurred by yielding for all bar connectors and that the spacing of connectors influenced the extent of cracking in the concrete core.

KEYWORDS Composite construction, Push-Out tests, Shear connectors, Thin-Walled Structures, Micro-Concrete

INTRODUCTION Double skin composite elements (DSC) are basically steel-concrete-steel sandwich elements that consist of a layer of un-reinforced concrete, sandwiched between two layers of thin steel plate. These in turn are connected to the concrete by welded shear connectors.

340 Several experimental and analytical studies have been carried out to understand the behaviour of DSC elements (Oduyemi and Wright 1989, Wright, Oduyemi & Evans 1991, Wright, Oduyemi & Evans 1991, Roberts et al 1996). The main conclusion from these studies was that DSC elements could generally be designed in accordance with normal reinforced concrete practice satisfying the following criteria; (a) Yielding of the tension steel plate. (b) Yielding or buckling of the compression plate. (c) Shear failure of the connectors. (d) Crushing of the concrete in compression. (e) Shear failure of the concrete. (f) Pull out failure of connectors. Of these, (b), (c), (e) and (f) are specific to DSC. Those criteria specific to DSC are influenced by the thickness of the steel plate that is typically only 1/20th of the total element thickness and consequently DSC easily falls within the category of a Thin-Walled Structure. An experimental study on DSC elements will be presented in the paper. The main aim is to investigate the behaviour of bar shear connectors when welded to thin steel plates. The experimental study involves push tests consisting of two thin steel plates connected together through a core of concrete by bar shear connectors. The connector diameter to plate thickness ratio was 3. The concrete core was pushed through the plates in direct shear allowing the behaviour of the bar to plate connection to be observed without the need for full panel bending tests. Current design guidance for concrete slab to steel beam flanges in composite beams merely places a limit on steel plate thickness in relation to stud diameter. The limit is often inappropriate for DSC and the paper will comment on this.

TEST PROGRAM

Nine model push-out tests were fabricated. Micro-concrete was used the mix of which was established by Hossain (1995). Table 1 show the properties of micro-concrete of control mix. TABLE 1 MICRO-CONCRETE PROPERTIES Cube Strength fo~ (N/mm 2) 28.0

Cylinder Strength fo

(N/mm 2) 19.55

Splitting Strength fs (N/ram 2)

Density (Kg/m s)

1.13

2400

Modulus of Rupturefb (N/mm 2)

(KN/mm 2)

4.51

14.55

Ec

,,, _.

The nine models are classified in three series and identified in the text as POT 1 to POT9. A typical pushout test model is shown in figure 1 and full details of each series are given in table 2. In summary: Three test series were carried out as follows: l-Three models with three connectors in one column (1 x3) with spacing 100ram. 2-Three models with six connectors in two-columns (2x3) with spacing 150ram in two directions. 3-Three models with six connectors in two-columns (2x3) with spacing 200mm in two directions. The perimeter of the plates was stiffened with additional steel frame members attached to the plates using a sufficient number of bolts. The models were tested by applying uniformly compressive force over the breadth of the top surface of micro-concrete core to push it through the plates in direct shear.

341 p.. Test

Frame

20 mm DIe. Corner Pin with $poeer~/

Members

?irection

i

o?s o,o. c. . . . . ,,o . , ,

[--,f>'7-

t !tU 6 m m Shear Stud

i

J

I

-]~-4,~ . It. ~ _

11

'.

I ! i

~

I !1

'

I!1

:! -I Te,~ .... ,ember,

I

Ill;

.l.h

!

I i-_-_-_-_-_-_-_-_-~

I

:I ! I

of Loading

I (

'

i

If

I,/"

I ~f:

[

!

I i I :Test

I

=

i

I i I:~

Frame

Members

1" Geometry

'

'

.!

\i

!

\1:1o mm Die. Intermediate

;l

ll

Elevation Figure

m' |-'.

6 ,,n, Shear Stud

..,..I."'.J.,. Connectors

!

I

i-tll

:

;'

I i I : ",.1o mm m I I~-F----//Dio.lntermediofe Coifs !LI

I

:"~ """ ~~.....

Section of

Push-Out

Test Model

TABLE 2 DETAIL MODELS OF PUSH-OUT TEST No

Series

Specimen

1

2

~

POT1 POT2 POT3 POT4 POT5 POT6 ,, POT7 POT8 ......... POT9

Of Studs lx3 lx3 lx3 2x3 2x3 2x3 2x3 i ,, 2x3 2x3

Stud Spacing Horizontal Vertical mm

111171

150 150 150 200 200 200

100 100 100 150 150 150 200 200 200

End spacing mm Top & Left & Bottom Right 100 50 100 50 100 50 75 75 75 75 , 75 75 160 loo 100 100 100 100

Material properties The properties of the steel plates were determined from tensile tests on random samples taken from each batch of steel. A summary of the steel plates tensile test results shown in the table 3. The properties of the bar connectors were determined from tensile tests on three specimens cut at random from the bar material. A summary of the bars' tensile test results shown in the table 4. The micro-concrete consisted of Ordinary Portland Cement, sea-dredged sand of 2.41-mm maximum size. A summary of the results is given in table 1. The properties of micro-concrete in each individual series of models were determined from at lest three tests on 100-mm cubes and three tests of 200 mm long by 100-mm diameter cylinders

342 (for split cylinder tensile and compression cylinder tests). A summary of these test results is shown in the table 5. The models were cast vertically and in stages. Following casting the models were covered with polythene and the micro-concrete was then allowed to cure in air until testing commenced. Figure 2 show the models as cast. TABLE 3 STEEL PLATE PROPERTIES Thickness (mm) 1.93

0.2%Proof stress (N/mm2) . 315

Ultimate stress (N/mm2) 393

E$

(r,~/mm~-) 195

TABLE 4 BAR CONNECTOR PROPERTIES Diameter (mm) 6.22

0.2%Proof stress (N/mm2) , 360

Ultimate stress (N/mm2) 517

ES

(KN/mm2) 196 ,,

TABLE 5 PROPERTIES OF MICRO-CONCRETE IN PUSH-OUT TESTS

Series

Cube Strength (N/ram 2)

Cylinder strength (N/mm 2)

Splitting strength (N/mm 2)

Series 1 Series 2 Series 3

25.33 24.33 29.75

20.31 18.72 22.73

2.53 1.69 1.85

Figure 2: Micro-Concrete casting

Remark

7 days 7 days 12-28 days.

343

Test procedure and instrumentation The compressive force was applied to the top of micro-concrete core of the model by means of a 250 KN actuator using deflection control mode. Model instrumentation is shown in Figure 3. The movement of the micro-concrete core relative to the steel plate was measured by two dial gauges, which were attached to concrete core at 5 cm from the bottom level of concrete core. One was attached on each face. The load slip values were simultaneously recorded and printed.

Figure 3: Model instrumentation

Loading and test observation At the start of each test the initial dial gauge reading was recorded. The compressive force was applied to the model by increasing increment loads gradually until the failure load. The slip between steel plates and micro-concrete core was recorded at each increment load until end of the test. Figure 4 shows the typical load-slip relationship of the push-out tests for series 1, 2 and 3. The observations on each test series is as follows:

Series 1 (Specimen POT1, POT2 and POT3) During the test cracking noises were heard at loads between 10-14kN, 22-28kN and 37-38kN. All specimens cracked vertical in the middle of concrete core along the line of connectors. The specimens failed at a compressive load of 37kN, 40kN and 37kN for specimen POT1, POT2 and POT3 respectively. In specimen POT 1 it was noted that a part of concrete core disturbed one of the dial gauges following lateral movement. In specimen POT3 the micro-concrete core started cracking in the middle of concrete core from the top and separated through the connectors in an inclined crack above the dial gauges. Bar yielding occurred in all the specimens.

344 Series 2 (Specimen POT4, POT5 and POT6) During the test cracking noises were heard at loads between 15-18kN, 44-56kN and 95-107kN. All specimens cracked vertically in two lines through the connectors. The specimens failed at a compressive load of 95kN, 107kN and 101kN for specimen POT4, POT5 and POT6 respectively. Bar yielding failure occurred in all the specimens.

i

./

j

i

i

! !

i ! t .

-1

.....

:

.~

--,r , .

'

,

0

,

.

1

,

.

,

. . . . . . .

2

,. . . .

~. . . . . .

3

4

I,

s.rm3

i ....

I i

w

7

,

5

"

i,

Blip mm

Figure 4: Typical load-slip of the push-out tests

Series 3 (Specimen POT7, POT8 and POT9) During the test low noises were heard throughout and a high cracking noise at failure. On dismantling none of the specimens were found to have a cracked concrete core. The specimens failed at a compressive load of 108kN, 77kN and 86kN for specimen POT7, POT8 and POT9 respectively. Yielding failure of the bars occurred in all the specimens. This led to separation of the connectors from the steel plates.

DESIGN STUDY The behaviour of the DSC systems in general has been reported in many studies. This paper concentrates on the behaviour of the bar shear connectors when welded to thin steel plates. The arrangement and the properties of bar shear connectors could be designed using BS 5950 pt 3 (1990) EC 4 (1994) or studies by Wright et al (1991) Roberts (1994) and Obeid (1986).

Stud~plate ratio BS 5950 limits the ratio between shear stud diameter to steel plate thickness as not greater than 2.5. EC4 uses the same ratio, with an additional criterion that the connector diameter must be less than steel plate thickness. Obeid showed that an appropriate limit to the ratio between stud diameter to steel plate thickness was 3, a figure he derived from a study of 3, 4, 5, 6 and 8 mm shear studs connectors welded to

345 thin steel plates. The tests confirmed that a figure of at least 3 was suitable as there was no indication of plate failure or stud pull-out in any of the tests.

Steel plate buckling and shear connector spacing The ratio of the centre to centre distance between stud shear connectors Sc to plate thickness ts~ is limited in BS 5950. The maximum spacing between studs being 600 mm or 4 times the concrete core thickness and the minimum spacing being not less than the 5 times the stud diameter. The limitation in EC4 is stated as maximum stud spacing to plate thickness ratio of 40. Wright (1993) confirmed that this ratio should not be greater than 40 for stud layouts where a column-buckling mode is likely. Column buckling occurs where the studs are spaced regularly allowing a line buckle to form between rows of connectors. Wright evaluated this ratio when the compression plate was in contact with a rigid medium (as in the case of DSC elements) and found that the limit increased to 52. In this paper this ratio was 50 in series 1, 75 in series 2 and 100 in series 3. During the tests local buckling was not noted however this was due to the fact that these tests are in direct shear with little compression developing in the plates. It should be noted that the direction of the load might not always be perpendicular to the stud layout. In two-way spanning panels the yield lines may be diagonal to the edges. In this case the ratio must be checked with the diagonal spacing between shear connectors. TABLE 6 COMPARISON WITH VARIOUS CODES OF PRACTICE AND RESEARCH STUDIES Data Connector diam/plate ratio Connector spacing

. . . .

Connector spacing/plate thickness Connector capacity

BS 5950 pt.3 30 mm 11.41 mm 30 Sc tu,(t) + 0 ( ~ u,.u ~t 2-/.t '2

~) (16)

and t

c I~F(t)dt + ~ E Il a(~u,e) = kt 2 +-~ , : u'~u kt - / u

2

+o(~)

(17) The symbols I~> and in the Dirac notation. ~t in this formula is given by (23). To determine constants c and d, (24) is introduced to Eq.(2) after which Eq.(2) is multiplied by functions t~(t) and t.,(t). The procedure described can be expressed in vector language as follows §

I-/2 >)-- 0

< -/21{02 + [8 + s . ~v]}(c I2 > +d I-/2 >) = 0

(25) where the second derivative of Eq.(2) is denoted by D z . These homogeneous equations can be solved only when condition 8 = A ( ~ , 6 ) = (~) ~ +El< ~ I~' I- ~ >1-~ < ~ I ~ l a >

(26) exists for any periodic perturbation W and It satisfying (23). Eqs (26) mean that 8 considered as a function of the characteristic exponent It and the perturbation parameter e experiences jumps of magnitude

2 ~ ~ ~b~ ztn

7/'n

(27) a possible mechanism of which is explained in Fig.3 /x

Figure 3" Jumps of functions 8(bt)

387 We interpret Eqs (26) as equations defining stability regions in F. It is interesting that matrix elements appearing in these equations are expressed by the Fourier coefficients of the perturbation W: zcn zcn 1L 2feint) < ~ I ~ I - ~ > - < ~a I~U l- ~a > - - -L! ~ ( t ) e x p ( - ~ a dt=~,

(28) and ~n 1~ iron ~ > = 1 i ~ ( t ) d t =hu0 -- rt~ = f = 190 x 10.6

Increasing the material resistance to o0 - 30.0 M N / m 2, the limit load for the same thin-wall structure is: q = f . a0 = 190 x 10.6 x 30 = 5700 x 10 .6= 5.7 x 103 M N / m 2 = 5.7 k N / m 2 R e d u c i n g the thickness, increasing the material resistance, and c o m p a r i n g the limit load, it is possible to c h o o s e the best solution for the thin-walled shell in wind pressure problems.

R u n way shell cover analysis D e t e r m i n a t e the limit load for a reinforced c o n c r e t e shell to cover a run way, using the following data: L = 12.00m 8 = 2.00 cm h = 8.00 cm rt3=L/h = 1200 / 8 = 150

graphics

rt2=6/h = 2 / 8 = 2.5 x l 0 .2

>rtl =f= 325 x 10 .6

I f c0 = 20.0 M N / m 2 is the material resistance, then the limit load o f the structure is: q = f . g0 = 325 x 10.6 x 20 = 6500 x 10 .6 = 6.50 x 10 .3 M N / m 2 = 6.5 k N / m 2 g = 0.08 x 25 = 2.0 k N / m 2 p = q - g = (6.5 - 2.0) = 4.5 k N / m 2 I f the thickness is r e d u c e d to h = 6 cm and the material resistance is increased to Oo = 30.0 M N / m 2, then the limit l o a d is: 7t3=L/h = 1200 / 6 = 200 rtz=6/h = 2 / 6 = 3.3 x l 0 "2

graphics ) ~ 1 ~

=

225 x 10.6

q = f . a0 = 225 x 10.6 x 30 = 6750 x 10.6 = 6.75 x 10"3 M N / m 2 =

546 6.75 kN/m 2 g = 0 . 0 6 x 2 5 = 1.5kN/m z p = q - g = (6.75 - 1.5) = 5.25 kN/m 2 By reducing the thickness and increasing the reinforced concrete resistance, it is possible to control the allowable overload on the shell. CONCLUSIONS With the aid of dimensional analysis resources, it is possible to analyse any type of structure, including shells with complex shapes. For shells, the relationship between the thickness and other parameters, for example the length of the shell projection, the deflection and the limit load, is very important. The graphics were prepared for a particular shell shape, but the same process can be generalised for any shell shape. The objective, when preparing these graphics, is to solve the shell thickness problem for a particular material quickly. This idea is very important, because shell structures without suitable thickness might not supports their own dead load. ACKNOWLEDGEMENTS The author wishes to thank Prof. Fernando Luiz Lobo Barboza Carneiro, who has taught her how to use dimensional analysis, a very powerful tool for solving many shell structure problems, including the application presented in this work.

References Carneiro F.L.L.B. (1993). Dimensional Analysis, UFRJ, Rio de Janeiro, Brazil Gomes M.P.R.C. (1993). Elasto-plastic and Geometrically Non-linear Analysis of Shells by Finite Element Method, Coppe/UFRJ, Rio de Janeiro, Brazil Gomes M.P.RC. (1996), Analysis of Shell Structures with Similar Shapes, In Proc. of Joint

Conference of Italian Group of ComputationalMechanics and Ibero-Latin American Association of Computational Methods in Engineering, XVII CILAMCE, Padova, Italy Gomes M.P.R.C. (1996), Dimensional Analysis for Hyperbolical-Paraboloid Shells, In Proc. of Joint Conference of Italian Group of ComputationalMechanics and lbero-Latm American Association of Computational Methods in Engineering, XVII CILAMCE, Padova, Italy Gomes M.P.RC. (1996), Stability Verification of Prestressed Concrete Shell Structures, In Proc. of Joint Conference of ltalian Group of Computational Mechanics and Ibero-Latin American Association of ComputationalMethods in Engineering, XVII CILAMCE, Padova, Italy Gomes M.P.R.C. (1997), Limit Load of Structural Shells with Similar Shapes, In Proc. oflCCBE-VII, Seoul, Korea

547 Gomes M.P.R.C. (1997), Abacos para determinagao da Carga Limite de Cascas Semelhantes, In Proc. of XXVIll Jornadas Sul-Americanas de Engenharia Estrutural, S~o Paulo, Brazil Gomes M.P.R.C. (1997), Carga Limite de Cascas Estruturais com Formas Reversas, In Proc. of XENIEF, Bariloche, Argentina Gomes M.P.R.C. (1998), Design Optimization of Structural Shells with Reversal Shapes, In Proc. of Joint Conference of IV WCCM and XIX CILAMCE, Buenos Aires, Argentina Gomes M.P.R.C. (1998), Otimizagao do Projeto de Cascas Estruturais com Formas Reversas, In Proc. oflII SIMMEC, Ouro Preto, Minas Gerais, Brasil Gomes M.P.R.C. (1999), Optimizaci6n del disefio de bovedas y estructuras similares, In Proc. of IV COMNI, SeviUa, Spain Gomes M.P.R.C. (1999), Abacos para projeto de cascas estruturais, In Proc. of XX CILAMCE, Sao Paulo, Brazil Gomes M.P.R.C. (2000), Optimizag~o do Projecto de Cascas Estruturais, In Proc. of V1 CNMAC, Aveiro, Portugal Gomes M.P.R.C. (2000), Graphics to determinate the limit load of shells with similar shapes, In Proc. of Fourth International Colloquium on Computation of Shell & Spatial Structures, IASS-IACM-2000, Chania, Greece Gomes M.P.R.C. (2000), Design Optimization of Conoidal Shell Structures, In Proc. of XXI CILAMCE, Rio de Janeiro, Brazil


Third International Conferenceon Thin-Wailed Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScienceLtd. All rights reserved

549

VECTOR OPTIMIZATION OF STIFFENED PLATES SUBJECTED TO AXIAL COMPRESSION LOAD USING THE CANADIAN NORM Susana Ang61ica Falco and Khosrow Ghavami Department of Civil Engineering Pontificia Universidade Cat61ica do Rio de Janeiro - PUC-Rio Rua Marquis de S~o Vicente 225/301L, G/tvea CEP 22453-900, Rio de Janeiro, RJ, Brazil

ABSTRACT This paper is concerned with the vector optimization problem of longitudinally stiffened plates subjected to axial compression load. This leads to a non-linear vector optimization problem with a finite number of sideconditions. As an application it is considered the optimal design of such plates where the ultimate buckling load should be maximal and the weight minimal. Among the different formulae and recommendations for the prediction of the ultimate buckling load, the Canadian norm CAN-S 136-M89 (1989) was chosen for its simplicity and for producing one of the best results. The chosen design variables are the number, the thickness and the height of the stiffeners. Several constraints are considered, Brosowski and Conci (1983), and constrains avoiding the failure due to the lateral buckling for torsion of the longitudinal stiffeners are introduced considering different norms Falco (1996). These side-conditions created a bounded feasible set of points. The compromise solution of this multi-objective optimization problem is established by an interactive procedure using the Efficient Points Method, developed by Brosowski (1985), which leads the problem, with n objective-functions, to a problem with (n- 1) ones. In the particular case of the optimization of stiffened plates, it is used a simplification procedure ofscalarization, which consists of the minimization of only one objective-function with only one variable within an established interval. An example is considered illustrating the potentiality of this method for practical cases. KEYWORDS Vector optimization, Stiffened plates, Ultimate loading, Buckling, Efficient points, Scalarization, Simplified method, Non-linear problems.

550 INTRODUCTION Stiffened plates are very efficient structures, as a large increment of the strength created by a small addition of weight used as stiffeners. The collapse mechanism of stiffened plates subjected to axial compression load is complex because it depends on several geometricvariables beside the mechanical properties of the materials used and buckling failure modes. Predicting the buckling behavior is possible both analytically and through numerical methods, such as finite element method, which need to use a large number of elements and is not considered in this work. A semi-analytical method based on the Canadian norm CAN-S 136-M89 (1989) is considered in this paper for the prediction of the ultimate buckling load. This method has been chosen based on an extensive experimental investigation realized in the Civil Engineering Department ofPontificia Universidade Catolica of Rio de Janeiro, Ghavami (1994). In this program a total of seventeen tests on simply supported steel plates were executed. The main variables considered were the type of stiffeners cross-section and the spacing between them. Plates with three types of longitudinal stiffeners i.e. L, T and rectangular (R) crosssections with and without transversal stiffeners o f t cross-section have been studied. The comparison of the calculated loads showed that the Canadian norm, the Murray's method, Murray (1975), and the AISC, American Institute of Steel Construction (1978), gave the response very close to the experimental ones, Falco (1996) and Brosowski, B. and Ghavami k. (1997). Based on these results, the Canadian norm was chosen for the simplicity of its formulation comparing with others methods and it is given in detail. In this paper, the design problem is carried out for an isotropic plate stiffened by rectangular stiffeners parallel in equal distance along the axis of the load application. The geometrical characteristics of this stiffened plate are shown in Figure 1, where L, B and t are the length, width and thickness of the plate, respectively, and tl, d and b are the thickness, height and distance between the stiffeners, respectively. The axial load is denominated as Px.

Figure 1. Geometrical characteristics of the stiffened plate under axial load. Otten the design of such structure has several requirements such the case of minimization of the weight and maximization of the buckling collapsing load. Thus, the designer of this structure is confronted with the problem of satisfying two conflicting objectives; such problems are called multi-objective or vector optimization problems. In general, the objective-functions do not attain their optimum in a common point of the set of feasible points. Intuitively, it is clear that one can expect only compromises in the sense that an improvement of any of the objective functions may presumably lead to a worsening of the value of some of the other objectives. Compared with ordinary optimization one has to develop a new concept of a solution, which is called an efficient compromise. In this work, this compromise solution of this multi-objective optimization problem is established by an interactive procedure using the Efficient Points Method, developed by Brosowski (1985).

551 ULTIMATE BUCKLING LOAD USING THE CANADIAN STANDARD The Canadian Standards is based on the concept of the effective width, including the effects of local buckling. The project stress ap, or reduced stress of buckling is calculated multiplying the stress of the global elastic buckling by the reduction coefficient 0.833, which takes into account the initial imperfections of the plate and the welding residual stress, Eqn. 1. The effects of the inelastic buckling are considered from the equations of the curve of CRC (Column Research Council), Figure 2. Therefore, after calculating the project stress, the obtained stress is adjusted using the curve of CRC. Thus, the critical stress of buckling a~ is obtained comparing the project stress with the proportionality stress, based on the limit of the proportionality ~y/2, Table 1. Op= 0.833 OE = 0.833 rc2E

O'a :

/~ ~y

~y

(1)

2 t~ Y

4~p

~ TABLE 1 CRITICALSTRESSOF BUCKLING

OE =~2E/(kL/r)2

2

For % > o r / 2 9 o~ = O'y/2

Oy-4--~n I

For~p"

.,..a

0,31MN/m

-'~

"'vr

/ -

m

==

,

.z

Px=0, 2 MNIm .~I

u.

0.4 --

F

O.4

T

i

T

I

VECTOR OPTIMIZATION ! {Preferred solutione ) }

l

0--

t

i

[

0--

0.005 External pressure q [MPa]

0 External pressure q [MPa]

Figure 5 The change in the face thickness (scalar optimization, WAGA=0)

,,

i T

T

J-

32 --[

Px=O 3 MNIm

1

0.007

Figure 6 The change in the face thickness (vector optimization) t

I

I

,,1

I

I VECTOR OPTIMIZATIONI(Pre, . . . . d s o l u t i o n . ) l

_:.

0.006

r

....

l

,/

)

E

r.

==

10

Px=0, 2 MNIm

28--

'~

._o -=_

z~/

"~' Px=0,2 MNIm

0 24

Px=0,1 MNIm

8

I 0

0.001

0.002 0.003 0.004 0.005 External pressure q 0VlPa]

0.006

PX=0,1 MNIm

J

2O 0.007

Figure 7 The change in the core thickness (scalar optimization, WAGA=0)

0

0.001

0.002 0.003 0.004 0.005 External pressure q [MPa]

0.006

0.007

Figure 8 The change in the core thickness (vector optimization)



567

SHAPING OF OPEN CROSS SECTION OF THE THIN-WALLED BEAM WITH FLAT WEB AND MULTIPLATE FLANGE E. Magnucka-Blandzil, R. Krupa 2 and K. Magnucki 3 1Institute of Mathematics, Poznafi University of Technology ul. Piotrowo 3, 60-965 Poznafi, Poland 2Metalplast Invest plac Wolno~ci 6, 61-738 Poznafi, Poland 3 Institute of Technology, University of Zielona G6ra al. Wojska Polskiego 69, 65-625 Zielona Grra, Poland

ABSTRACT The subject of the analysis is a thin-walled prismatic beam with flat web and multiplate flange. The beam is loaded at both ends and submitted to pure bending. Open cross section of the beam is described by five dimensionless parameters controlling its shape. Basic values of cross section area resulting from classical theory of beam bending, warping function and warping moment of inertia resulting from general theory of warping torsion have been determined. The parametric shaping of the cross section is reduced into the problem of maximization of bending moment for fixed cross section area. Constraints of the solution result from geometrical, strength, and stability conditions. Numerical analysis has been performed.

KEYWORDS Thin-walled beam, monosymmetrical open cross section, warping function, lateral buckling.

INTRODUCTION Thin-walled structures are endangered with destruction mainly due to insufficient strength of material or loss of stability. The strength and stability of thin-walled beams, particularly coldformed, are the subject of present-day research. JOnson (1999) developed a classical theory of thin-walled beams, taking into account the change in their cross-section shape. Kotakowski, Kowal-Michalska and K~dziora (1997) numerically analyzed stability of thin-walled beams of open and closed cross-section profiles. Kubo and Kitahori (1998) performed experimental investigations of the stability of a monosymmetrical I-bar of various flange widths. Pi, Put and Trahair (1999) discussed the results of stability analysis of thin-walled channel and Z-bars. Optimization of the thin-walled structure is their further improvement. Zyczkowski (1990) made a basic contribution into development of shell optimization taking into consideration their

568 stability. He discussed in his monograph the results of research carried out in the domain. Bochenek and Zyczkowski (1990) have optimized an arc of I-section under stability constraints. Magnucki and Magnucka-Blandzi (1999) have determined an optimal shape of thin-walled beam section of variable thickness. Magnucki and Monczak (2000) have optimized a cross-section profile of a beam subject to bending. Manevich and Raksha (2000) have optimized a channel cross section of a thin-walled beam. Rao (1995) discussed mathematical essentials of optimization applied to technical problems. The topic of the work is a thin-walled beam subject to pure bending. The optimization is performed with a view to defining the cross-section profile of the beam of maximal safe bending moment.

ANALYTICAL DESCRIPTION OF THE OPEN CROSS SECTION The curvilinear shape of the cross-section for many purposes is unpractical, therefore it is often replaced by a piecewise straight line (Fig. 1). Total depth of the beam is H = 2a + t. The web of the beam is located in the middle of the cross-section width.

b&Y

0

H

z ~''

Figure 1: The open cross-section of the thin-walled beam The cross-section is characterized by the following dimensionless parameters b x1 = - , a

c x2 = - , a

d x 3 = --, a

and the angle 13.

(1)

The beam cross section area (2)

A l = 2 a t . fi

where 1 + 2cosl3- sin 13 A -- A ( X l ' X 2 ' X 3 ' ~ ) -- 1 +

2 cos 13

X1 + X2X3 .

The cross-section geometrical stiffness of torsion 2 Jt = -~tS a" f l 9

(3)

569 The coordinate of the centroid of cross-section

1 f~ z0--~a~,

(4)

where

f2 "- f2(Xl)X2)X3~)--Xl

I

X1 4" X2 -- X2X3 / ' 4cosl3

The moment of inertia of the cross-section area with respect to the y and z axes 1

dy =-~a3t" f3,

(5)

dz =2a3t" f4,

where 6cosl3 x~ + x 2 + X2X 3

A

x, ( 1 + ~1- t g 2 ) f4 = f4(xl,x2,x3,~5)= 2cos ~ fl~ +~ fll tgl3xl 13x~ +2cosl3 + 1 .-(I,~§ 3

3

)3

)'

tg~ +x2]-

fll = f l l ( X l , X 2 , ~ ) = l - - f l x l

The sectorial coordinate with respect of the point B shown in Fig. 2.

I

..... ~

3

i

2

i

-~ ff--'~ i

z-"

Figure 2: Plot of the sectorial coordinate-point B. Values of these coordinates are as follows:

toB~ = 0 '

1

, coB2 =--a2x~f~l 2

co~ - ~1 a ~x, (2+x2 - fll + x ~ ) .

1

c~ =-~a2x~(x2 -f~l),

f.OB4 --- -~1 a 2xi (2 + x 2 -f~,),

570 The distance from the shear center A of the cross-section to point B (Fig. 2)

zA-zs=-~1 AI03sydA=laf' 6 74,

(6)

where

f5 -- f5(Xl,X2,X3,~)- ~3Xl f 03B3 "+03B4 -'1-[2(1 -- X2).-F fll X2X 3

+~

a

] 03Bl }+-~[(3--X2)f-OB3+(3--2X2)O3B2 6cosl3

{[2(1- X2X3 )+ l]f.0B5 + (3 -- X2X3 )03B4 }"

The main warping function (the main sectorial coordinate) 03 for the open cross-section is shown in Fig.3.

i 3 5 2

60" 1

B

A

Z

Figure 3" Plot of the main sectorial coordinate-point A. Values of these coordinates are as follows: o3i = a 2 "~i

for

i = 1,2 ..... 5,

where ~.

1

f5

031=--6f11~4 ' 1 c~ =-2 xl

...

1 (x2 _ 1)~_ "

('t)2 ='6

x,

,14 --"~-fll,

...

1

lf5 (x z - fl, ) - - - 6f4

f5 (2+X2-fl, ) 6l f4 '

.~ =1-~ x, ( 2 + x 2 - f ` , + x~x~ )- ~. ~ ( l 1_ x z x 3 ) o~ The warping moment of inertia of the cross-section (the sectorial moment of inertia) 2

J,o =-~aSt"f6, where

(7)

571

"~ +2cOs~ X1 f6 = f6(Xl'X2'X3'~) = IfllO~

(~ + ~ + ~1~2)+ X2(~ + ~ + ~2~3)+

+Xl(~ +~42 +~3~4)+ X2X3(~2+~2 +~4~5)]. Parameters x 1,x2,x3,[3 determine the above geometric quantities of the open cross-section and they are parameters in the optimization problem.

FORMULATION OF PARAMETRIC OPTIMIZATION P R O B L E M A thin-walled beam with open cross section is in pure bending state - the bending moment M is constant. The optimization problem is defined as follows: the area A1 of the cross section of the beam is constant. For this value the parameters xt, x 2, x 3 , fS, the thickness t and the depth H are sought for which bending moment M will be will maximum. The strength and stability conditions are constraints of the solution. The depth of the beam (Fig. 1) n = 2a + t,

from which

a = 0.5(H - t).

(8)

Substituting the size a into the cross section area (2) one determines the parameter x 1 controlling the width of the beam x 1 = 2 (H

-

A-----!---~ t)t 0 +x2x3 -

)I

cos13 1 - sin [3+ 2 cos [3

(9)

The strength condition of the beam M H
> projects

With the aim of finding a ~.,,.8.,,,=

0.0 - P ~ - ~ , q:3.o! I 0.00 0.04 0.08 0.12 0.16 0.20 0.24 W ho

(a)

100.0

/

7""

f""

80.0 4 60.0 q ro

,0/,, . ,,(__

4

o

o ' y

E~17640.0

20.0 0

(b)

0.0

.

0 0.3

~ 0.6

0.9

1.2

W ho

Figure 2: Pressure versus centre deflection - (a) 13= 0.4 and (b) 13= 3.

638

10.0

8.0 m.,==:=:==~ 9

6.0

4

qro

Eoh~

4.0

P ---0.3 9 SS-F!I q.--'2.7 E] SS-FR,q=2.75 CL-FI, q =3.03 0 GL-FI,qy:=~.95.8.5x12 A CL-FR~qv=3.03

I /

I 2.0 --if 9

I

0.0 - 9 0.00

I

i

I

0.02

0.04 2 Nr

I

0.08

0.06

r o

3

Eoho

(a)

100.0 .

.

.

.

,,,;,li,

80.0

4

qro

le"

tO'

lO/~ m

i

m" 9

60.0

.I 9

9

m"

Eoho 40.0_t7;~ ~ . l ' 9 /~-'~,ff.~~ , . .m ,~j~

2O.O -ifvI.4/"

u=o.~

9 $S-FI, q=20.3 12 $ S - ~ , (~ =20.3

"

, CL-n., ~ ,

511~"/"/ ii

0.0 "! 0.0

CL-FR,qT=22-7

0

i

0.4

i

0.8

i

2

1.2

i

1.6

2.0

Nr r o

3

Co)

Eoho

Figure 3: Pressure versus central radial stress resultant - (a) [3 = 0.4 and (b) [3 = 3.

639 0.12 0.10 -

/,i

0.08 Mr

2

r o 4

Eoho

0.06

-

0.04 -

0.02 -

,,/"

/ / ps

o

,E~

0.00 -i 0.0

CL_.~_nL~,--~_~03 ....

A CL-FR, q =3.03

'I

2.0

' 'I' "

I

4.0

6.0

qro

'i

8.0

10.0

4 4

Eoho

(a)

0.8 0 (20o

0.62 Mrro 4

0.4-

Eoho

0.2-

//

C S$-FR. qT=2q. CL-Flt.g, ==22"6

/9

0.04 0.0

Ill

II

20.0

J

40.0

J

60.0

nl II liB ill ii

80.0

i

i

100.0

4

qro

4

(b)

Eoho

Figure 4: Central stress couple versus pressure - (a) 13 = 0.4 and (b) [3 = 3.

640 CONCLUDING REMARKS The governing system of equations for the nonlinear response of pressure loaded sector plates, based on the Von Mises layered yield plasticity model, have been outlined. In comparison to the Ilyushin full-section yield model, it has been shown that first yield pressures predicted with the present elastoplastic constitutive model are about one-third lower. The centre deflection, stress resultant and stress couple responses with increasing pressure have also been presented for stocky and slender 60~ sector plates with simply supported and clamped edges. It is shown that the in-plane edge restraint has a significant effect on the response of simply supported slender plates, especially as the elasto-plastic state in the plate becomes well-developed. The in-plane edge restraint is shown to have little influence on the response of stocky plates, except for the radial stress resultant and then only over part of its response range, when the plate edges are simply supported. The results presented quantify extremes of response for 60~ sector plates and provide benchmark data for checking similar analyses, based on alternative numerical techniques. ACKNOWLEDGEMENTS The second author wishes to acknowledge financial support for this research provided by the Iranian Ministry of Higher Education. Both authors wish to record their appreciation to colleagues in the Engineering Department for encouragement and support. REFERENCES

Otter J.R.H., Cassell A.C. and Hobbs R.E. (1966). Dynamic Relaxation. Proceedings of the Institution of Civil Engineers (Research and Theory) 3:2, 633-656. Timoshenko S.P. and Woinowsky-Krieger S. (1959). Theory of Plates and Shells, McGraw-Hill, New York, USA. Turvey G. J. (1978) Large Deflection of Tapered Annular Plates by Dynamic Relaxation. Journal of

the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers 104:EM2, 351-366. Turvey G.J. and Salehi M. (1990). DR Large Deflection Analysis of Sector Plates. Computers and Structures 34:1, 101-112. Turvey G.J. and Salehi M. (1997). Full-Section Yield Analysis of Uniformly Loaded Sector Plates. In Trends in Structural Mechanics: Theory, Practice & Education, Roorda J. and Srivastava N.K. (eds.) Kluwer Academic Publishers, Dordrecht, Holland, 289-298. Turvey G.J. and Salehi M. (1999). Elasto-Plastic Response of Uniformly Loaded Sector Plates: FullSection Yield Model Predictions and Spread of Plasticity. In Proceedings of CIVIL-COMP 99 F, 157169.

Section XI SHELL STRUCTURES


Third Intemational Conferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

643

VALIDATION OF ANALYTICAL LOWER BOUNDS FOR THE IMPERFECTION SENSITIVE BUCKLING OF AXIALLY LOADED ROTATIONALLY SYMMETRIC SHELLS G.D.Gavrylenko 1 and J.G.A.Croll 2 1Professor, Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev, Ukraine 2 Professor, Head of Department, Department of Civil Engineering, University College London, London WC1E 6BT, UK

ABSTRACT Doubly curved shells of revolution, with cylindrical and near cylindrical form continue to provide 0.

650

9

Analysis of Fig.3 allows the following conclusions to be drawn: Non-dimensional minimum lower bound critical stress, decreases with growth of r/t;

9 the growth of r/p leads to growth of O'cm/t~cl and especially in comparison with a smooth cylindrical shell (r/p=0); 9 when r/t trends to 2000 and more values acrn /C~elstabilize and almost do not exchange. METHODS ESTIMATION OF CARRYING CAPACITY OF THIN-WALLED RIBBED STRUCTURES.

3.

Two methods of defining carrying capacities for thin-walled ribbed structures, having initial imperfections of geometry, have been examined and compared: 9 numerical method based on using of nonlinear theory of shell together with method of finite differences, [8,9] created by Prof. G. Gavrylenko (NAS Ukraine, Kiev); 9 method of an analytical calculation suggested by Prof. J. Croll (UCL, London). Numerical method [6-9] uses the procedure of defining of minimal critical loads in compressive ribbed cylindrical shells having given initial imperfections of form with take in account an eccentricity of ribs and their discrete placing. As local so regular axisymmetrical and nonaxisymmetrical imperfections are investigated. The parameters of minimal critical load Pl = Per / P~ is determined (p~,~-criticalload of nonideal shell, pcl=0,605EtF/r, where F- area of cross section of skin ). The theoretical basis of analytical estimation of carrying capacity is expounded in articles [ 1-3] and named by authors reduced stiffness method - RSM. In Fig. 4 a representative shell [ 10] is examined; this has the following characteristics: 1/r=l; r/t=400; ds/ts=3; ts/t=3; k=40, where d s and t s are height and width of longitudinal stiffening elements, k- number of stringers. Plmin is found for indicated shell having axisymmetrical imperfections w 0 = B ~ sin mwx/1. Others type of imperfections give larger values pl 9In Fig.4 are represented: 9 two curve (m = 3 & m = 5, using a finite difference mesh having IxJ = 61 x31, continuous lines - solution of nonlinear problem (O - m=3, 13 - m=5) and dash lines- solution of linear problem ( O - m=3, II-m=5); 9 straight line- result received by RSM. Values p t m founded by numerical and analytical methods draw together when 1,2 < Bin, / t < 2,5. P1 1,5 1,25

J

. . ~

9

..

,Qo

o

"'t) ....

0...

0,75 0,6

i

0

~

1

i

i

2

i

i

lOmnl/t

Fig. 4 Comparison of minimum lower bound non-dimensional critical stress pl founded by RSM and numerical method.

651 4. CONCLUSIONS. The reduced stiffness method [ 1- 3] is extended to shells near to cylindrical form ( r/o ~ 0 ). Suggested and realized is an approach which allows definition of an analytical solution for lower bounds for the buckling of these shells. In result of calculations has been found: 9the values of classical critical stress of shells with positive Gaussian curvature are close to equal that for cylindrical shells; 9in contrast shells with negative Gaussian curvature have classical critical stresses that tend to zero with growth Ir / p J.

Despite of above mentioned deduction the following main result was received: o non-dimensional minimum lower bound critical stress found by RSM predict lower knockdown with growth of r / p for shells with positive Gaussian curvature;

9when r / t ratio tends to 2000 - 3000, values of the minimum lower bound stabilize at a constant asymptote;

9 for a longitudinally stiffened cylinders, lower bounds

of non-linear imperfection sensitivity studies are shown to be accurately bounded by the analytical predictions from the reduced stiffness method.

REFERENCES 1. Batista, R.C., (1979)Lower bound estimates for cylindrical shell buckling. P h . D . thesis, University College, London. 2. Croll J.G.A., and Batista R.C., (1981) Explicit lower bounds for the buckling of axially loaded cylinders. Int. J. Mech. Sci., 23, 331-343.. 3.Croll J.G.A., (1995) Towards a Rationally Based Elastic-Plastic Shell Buckling Design Methodology. Thin-Walled Structures, 23, 67-84. 4. Croll J.G.A., Gavrylenko G.D. (1998) Substantiation of Reduced Stiffness Method//International Journal ~Strength of Materials)), N5. 39-58. 5. Croll J.G.A., and Gavrylenko G.D., (1999) Reduced Stiffness Method in the Theory of Smooth Shells and the Classical Analysis of Stability (review). Strength of Materials, 31, N2, 138-154. 6. G.D. Gavrylenko, (1995) Stability of Smooth and Ribbed Shells of Revolution in a Nonuniform Stress-Strain State, Int. Appl. Mech. 31, N7, 501-520. 7. G.D. Gavrylenko, (1983) Basic Nonlinear and Linearized Equations of Imperfect Ribbed Shells of Revolution. Int. Appl. Mech. 19, N7, 610-614. 8. Gavrylenko G.D. Stability Ribbed Cylindrical Shells with Nonuniform Stress-Strain State. Kiev, Nauk. Dumka. (1989), 176. 9. Gavrylenko G.D. (1999) Stability of Imperfect Ribbed Shells, Kiev, Printed in Institute of Mathematics of NAS Ukraine.. 190. 10. Ellinas C.P., Croll J.G.A., & Batista R.C. (1981) Overall buckling of stringer stiffened cylinders //Proc. Inst. Civ. Engrs., Part II,, 71, June, 479-512.



653

ON THE ANALYSIS OF CYLINDRICAL TUBES UNDER FLEXURE F. Guarracino and M. Fraldi Dipartimento di Scienza delle Costruzioni, Universit~ di Napoli "Federico II", via Claudio, 21 - 80125 Napoli, ITALY

ABSTRACT In the present paper the collapse behaviour of infinitely long, cylindrical, elastic tubes under pure bending is investigated. This behaviour is characterised by a smooth global maximum on the load-deflection curve which is due to the well-known yon K~rm~in effect, that is to the progressive flattening of the cross section of the tube under bending moment. Even if this phenomenon has been extensively investigated in many classical works, nevertheless the comparison of the predictions from these approaches with experimental data shows that there are still some discrepancies which seem to be worth of further considerations. To this purpose a straightforward solution is presented which takes fully into account some effects in the deformation of the cross section which appear to have been neglected in the past and which, on the contrary, can give reason for several experimental findings.

KEYWORDS Cylindrical tubes, ovalization, bending strength, limit moment. INTRODUCTION The present work is concerned with the analysis of the bending strength and with the determination of detbrmation, stiffness and stresses of originally straight tubes. In fact, it is known that the cross section of thin walled beams deform during bending and that this deformation affects the bending strength giving occurrence to a flattening instability phenomenon. This phenomenon was firstly described and modelled by von Kmanfin (von K~rm~n, 1911) in the case of curved tubes and, successively, by Brazier (Brazier, 1927) and Chwalla (Chwalla, 1933) in the case of originally straight tubes. A large number of works have been published on this subject, both from a theoretical (e.g. Ades, 1957, Reissner, 1961, Fabian, 1977, Gellin, 1980) and an experimental point of view (e.g. Ellinas and Walker, 1985, Corona and Kyriakides, 1988) and several quantitative results for the non-linear behaviour of stresses and deformations in the tube have been made available, both in the purely elastic and in the elastic-plastic range. However, some of these results happen to differ quite significantly from each other (as it is the case, for example, of the limit moment derived from Brazier's paper and that derived from Chwalla's one) and many of them are not able to predict or give full reason of some experimental findings, so that, in the opinion of the authors of the present work, it is still worth to attempt to formulate a more precise formulation of this classical problem. Therefore, in this paper a straightforward modelling of the phenomenon is presented, and the relative solution is

654

obtained in a closed form. The analysis is carried out in the purely elastic range, but the findings can be usefully extended to the elastic-plastic range and this extension will be the object of a forthcoming work. The developed formulation leads to a clear explanation of the asymmetry of the deformed shape of the crosssection of the tube with respect to the neutral axis of flexure, which has been pointed out in some experimental works (Ellinas and Walker, 1985, Corona and Kyriakides, 1988). This asymmetric behaviour seems to have been overlooked in practically all the previous solutions of the problem, with the noticeable exception of the Brazier's work in which, however, the asymmetry deriving from the contribution of the St. Venant's solution of the flexure problem is first taken into account in the formulation of the variational problem and then discarded in the final solution. In the present paper it is shown that there are also other terms, apart from the St. Venant's ones, that contribute to the asymmetry of the problem and that the characteristics of inertia of the deformed cross-section are affected by this deformation mode. Finally, two numeric examples, one of which relative to a tube extensively tested at the University of Surrey (Ellinas and Walker, 1985), are presented. FORMULATION

OF THE PROBLEM AND RELATIVE SOLUTION

Let us start by taking into consideration the initial shape of the circular cross-section of a cylindrical tube, as shown in Figure 1. r is the mean radius and t denotes the thickness of the wall. /

y

M

/ M

/

M

Figure 1 Under the Bernouilli's hypothesis that the cross-sections of the tube remain plane on bending, we assume that the displacement field of the points of the cross section in its own plane results (see Figure 2) u=~'+~=

VCr 2

sin2~:+t7

2

v= V+ ~=-

(1)

VCr 2

2

cos 2~: +

where ~" and V are the linear components of displacement, yielded by the St. Venant's solution to the flexure problem, and t~ e F represent the non-linear part of the displacement field, due to the yon K ~ m ~ effect, that is to the ovalising pressure produced by the longitudinal stresses which give origin to a resultant directed towards the centre of curvature of the deformed element of tube, as shown in Figure 1. The usual linear bending moment-curvature relationship can be assumed to hold true

1 R

dip ds

c. . . . .

M El(c)

(2)

where R is the radius of curvature of the initially straight longitudinal axis of the tube, dip is the slope between

655

two the

sections

at

distance

t

ds

after

deformation

has occurred,

Y

M

is

the

bending

moment,

E

is

s

~

u

,

, "; 8

,

'

udr ~ "' --~fl

Figure 2 Young's modulus and I is the second moment of area of the section about its neutral axis. On account of the cross-section deformation in its own plane, I results a function of the curvature, that is I= I(c). The ovalising pressure p acting on the unit of area of the tube wall in the normal direction to the neutral axis, can be expressed as

p . dR.tc . . cr~ctrd~ . . . c2tEy c2trEsin~

ds

rd~

(3)

where dR.z is the force acting on the element of circumference ds = r sin~:, and o2 is the stress normal to the shell thickness, which can be evaluated by the classic Navier's formula. To the purpose of determining the expression of the non-linear displacement field, t7 e ~, we take into consideration the initially circular cross-section subject to the ovalising pressure p. As it is usual in problems of this type, we make the hypothesis that the system a e ~ is inextensional and that the shear deformation through the shell thickness is negligible. With reference to Figure 2, let us impose the equilibrium of a generic element of the shell in the deformed configuration, and write the compatibility relationships between the components of displacement and the slope fl(~) in the plane of the cross section, as well as between the slope fl(~) and the curvature Z(~:). We are thus led to the following set of differential equations

a=a,(g) rot(g) r ~ ag = + ' ~ ' ( g ) = - E J

ar (4) + :,, (r - - r fl(r

rz(4)- ~fl(4)

656 where tT,. and 12. are, respectively, the components of the non-linear displacement field in the normal and tangential direction to the undeformed cross-section, fiB(C:) is the bending moment acting through the shell thickness and J is the flexural stiffness of the shell, that is tadl

J =

We now define the load Q(r

(5)

12(1-v 2)

see Figure 3, as (6)

Q ( ~ ) = ds~c~ p " r " d v

Y P

Q(O

I

,@x

x

Q Figure 3 Quite straightforwardly, the expression for the shell bending moment, 9Yt(~:), results OB(~)

= Q.[r.(1-cos~)]-a(~).[r.(1

-

cos~:)-d]+X

(7)

~r

where X is an unknown parameter, Q = ds~j p- r - d ~ is the resultant of the vertical pressures, which equates the vertical reaction of the support, and d is given by ~r

d s . ~ j p . [r . ( 1 - c o s ~ ) ] d= ~ Q(r

. r . d~

= r'sin 2 2

(8)

The unknown parameter X can be obtained by means of the virtual work theorem, that is I ~ 9"A ( ~ ) . r " d~=O .IO

EJ

~

X

--- ~ ! c 2 t

. E . r3ds

(9)

4

Thus, the expression (7) of the shell bending moment, 9B(~), takes the form 93~(~) . ds ~: cos ~:) + sin2 "~] . .c2t. E. r 3 [ - ~1 + c~ ~" (c~ --2

(10)

We are now able to pursue the solution of the problem represented by the set of differential equations (4). From the first of the equations of the system (4), provided we take into consideration the following additional conditions,

657

\9, r

....

0

(symmetry of the displacements with respect to the y-axis)

(11)

(the displacement field is inextensional) we obtain

c2 r 5 _ V 2 t2r = - - - 7 ( 1 )- coseC:

(12)

and by integrating the second equation of the system (4), under the condition fi, (~: =0) = 0 , we have c2r 5 fi.~(~) = ,10 t~, (~)d~: = -2.-7(1- 1,'2 ) sin 2~

(13)

Given that we can express the displacement field in the reference frame x-y as fi = -fi, cos ~: - fi.~sin ~: = -ti r sin ~ + iT, cos ~

(14)

on account of the formulae (12) and (13), the equations (1), which fully describe the deformation of the genetic cross section in its own plane, take the form

u =

VCF2

c2r5

2 sin2~'--7(1-v~)/-[c~176

2

sin2~']sin~:}

Vcr2 t-c2r5 1 v = - - - - - - - c o s 2~:----7-(1-v2){-[cos2~:]sin~:-[ sin 2~:]cos~:} 2

(15)

Figure 4 Figure 4 shows the deformed shape of the cross section in the x-y plane. It is worth underlining that it results by no means symmetrical, contrary to what happens to practically all the solutions proposed in literature. As the value of the external bending moment grows, the square terms in the longitudinal axis curvature, c, tend to play a more significant role in the deformation of the cross section. Moreover, as the curvature c increases, the thickness t of the tube wall and the length ds of a generic element of its circumference become t ' = x][t +(G' -u~)] 2 + [u.~'-u~] 2 (16) ds '= x/[d(r cos ~:+ u)] 2 + [d(r sin ~ + v)] 2

658

where u~' and u~ are, respectively, the displacement components in the direction normal to the mid-surface of the shell of two points which, along the same normal, represent the intersections with the external and the internal boundary of the shell, u]' and u~ are the displacement components of the same points in the direction tangent to the mid-surface of the shell. After some algebraic manipulation, we can write

t'= t41 + c2v2r 2 + 2 c r v s i n ~ = V(c,~)-t ds '= ds~/1 + c2v 2r 2 + 2 c r v s i n ~ = V(c,r

(17)

ds

so that the shell circumferential strain, e,, results e.~ = - - - - ds

=v-i

(18)

This strain, according to the hypothesis made that the system fir and tT. is inextensional, is due to the linear part of the displacement field only. As a direct consequence of the asymmetry of the deformed shape of the cross section with respect to the x-axis, which can be described by means of the strain factor ~(c,~:), the position of centre of the mass, G, of the deformed section results displaced with respect to the origin of the axes x-y. The actual y-coordinate of G can be easily obtained by means of the formulae expressing the actual area of the cross section, A ', and of its first moment, S'. We have

A'=~ t'ds'=fci'~r.t.~2(~).d~ (19)

S '= ~st'. (y + v)ds '= ~ci'r.t. ~r2(~). [r sin r + v(r so that it results

, y~ . S A'

ft21r r . t . ~ 2 ( ~ ) . [ r s i n ~ + v ( ~ ) ] . d ~

. ~

.

. . ~,i=r.t.~2(r

.

= ( c r 2 v)

[1- 3 " c Z ' r 4 " ( 1 - v z ) ] 4t2 [l+(c.r.v)2l

(20)

The actual value of the second moment of the area, l(c), which appears in the moment-curvature relationship (2), is l(c) = I,, - A ' y 6 =

r.t.

(~).[rsin~+

-

Y6

9

,

r t

(~:) -d~:

(21)

that is, in a closed form,

l(c) =

[I,, + ~ f i ( r , t , v ) . c 2k ] ~=' [1 + c 2- v 2- r 2]

(22)

where

f ~ ( r , t , v ) = ~~. .{/5. 5. t 2 . v 2 - 6

9r 2 -(1- V 2 )}

f 2 ( r , t , v ) = 8.---~3.{20 ~'r7 . t 4. v. 4 . r 2 (1-v2)-[8 - t 2 91,,2 - 5 - ( 1 - v 2)]} (23)

L - (r,t,v) = x" 8./3 r 9 . V.~ 2 "{2"t4

.V 4

_ r z . ( 1 - v 2 ) . [ 2 0 . t 2 .v 2 - ( 1 - v2)] }

/~'. r 15 . V 4

f , ( r , t , v ) = t----.--.--y~- 5 - (1 - v2) 8. Finally, equation (2) and equation (22) allow us to write the relationship between the external bending moment, M, and the longitudinal axis curvature, c, in the following form

659

[l,,+~fi(r,t,v).c

zk ]

k=i [1 + C2- V2- r 2]

M(c)=E.c.I(c)=E.c

(24)

This is a compact and quite handy expression, formally similar to that proposed by one of the present authors under the assumption that the deformed shape of the cross-section could be approximated to an ellipse (Guarracino and Minutolo, 1996). It allows the direct evaluation of the limit bending strength of the tube as a function of the curvature of its longitudinal axis, as well as of its actual stiffness. Its simple expression can turn useful in the analysis of more complicated problems as, for example, pipelaying in deep waters (Guarracino and Mallardo, 1999). NUMERICAL EXAMPLES AND CONCLUSIONS We consider the cases of two different tubes, with D / t ratios equal to 500 and 14, respectively. Both tubes are high grade steel made, with E=2.07 • d a N / c m ~"and v=0.3. The first case is that of a very thin tube, characterised by an overall diameter D= 100.2 c m and a wall thickness t=0.2 c m , whose collapse can be obtained experimentally for suitable lengths in the purely elastic range. The bending moments vs. curvatures plots, see Figure 5, show the results from the present formulation as a continuous line and the results from the Brazier's solution as a dashed line. The attainment of the limit moment results evident in both contexts, with a difference of about 5% in the value of the maximum bending moment (Mlim=4.60425X106 d a N c m according to the present formulation and MHm=4.3854x106 d a N c m according to the Brazier's solution) and about 9% in the value of the critical curvature (c=4.39443x10 -~ c m -~ according to the present formulation and c=4.03871 x l 0 -5 c m 1 according to the Brazier's solution). M

5xlO ~

l

4xlO ~ 3•

I

/

\ '\\

6

j

\

'\, \

2x10 6

"'\

\

I

\\\

\

IxlO ~

! \ \

/ ,

. . . .

L

0.~2

,

,

.

|

0.~4

.

,

,

~

0.00006

,

,

.

/ ~

0.00008

J

,

.

i

C

0.0001

Figure 5 Similar findings were obtained in the second case, which is the case of a pipeline extensively tested at the University of Surrey-UK (Ellinas and Walker, 1985), characterised by an overall diameter D= 16.2 c m and a wall thickness t= 1.2 cm. It must be noticed that, as it is the case for the vast majority of tubes of practical interest, the actual collapse of this tube cannot be obtained in the purely elastic range. Nevertheless, in order to perform a comparison in the elastic range, we plotted the results from the present formulation as a continuous line and the results from the Brazier's solution as a dashed line, see Figure 6. After the proposed solution, the limit moment results M~im=2.60323• d a n c m in correspondence of a critical curvature c=1.070205x10 2 c m -I. After the Brazier's solution, we have Mlim=2.47863• d a N c m and c=0.983094x10 2 c m ~ It can be concluded that, according to the present formulation, the values of the limit bending moments, as well as those of the critical curvatures, do not differ significantly from those deriving from the Brazier's theory in the elastic range. This is meaningful, because Brazier's results have received a significant experimental validation in the elastic range in the past seventy years. However, it is noticeable that the shape of the deformed cross section deriving from the present analysis differ

660 significantly, to the best of the author's knowledge, from those proposed in other works. The asymmetry of the deformed shape gives origin to a difference of about 25% between the maximum value of the longitudinal compression and the maximum value of the longitudinal tension in the vicinity of the limit bending load in the first of the cases considered above (tf=-3305 daN/cm 2, ot=2648 daN/cm2). This fact, apart from being in accordance with the experimental findings, gives also reason for the different locations of the plastic zones under bending, as well as for the onset of the bifurcation buckling of the ovalised tube (Fabian, 1977).

I

4xlO 7

f / I

3xi0 7

2xlO

I

7

I/

\\\\\

IxlO ~

O.005

O.01

O.015

I O.02

O.025

Figure 6

References Ades, C.S. (1957). Bending strength of tubing in the plastic range. J. Aero. Sci., 24, 605-610. Brazier L.G. (1927). On the flexure of thin cylindrical shells and other thin sections. Proc. Roy. Soc. A, 116, 104-114. Chwalla, E. (1933). Reine Biegung schlanker, diinnwandinger Rohre mit gerader Achse. ZAMM, 13, 48-53. Corona, E. and Kyriakides, S. (1988). On the collapse of inelastic tubes under combined bending and pressure. Int. J. Solids Structures, 24, 505-535. Ellinas, C.P., Walker, A.C. et al. (1985). A development in the reeling method for laying subsea pipelines. Proc. 1'~tPetr. Tech .Austr. Conf., Perth, Australia, 26-29 November. Fabian, O. (1977). Collapse of cylindrical elastic tubes under combined bending, pressure and axial loads. Int. J. Solids Structures, 13, 1257-1270. Gellin, S. (1980). The plastic buckling of long cylindrical shells under pure bending. Int. J. Solids Structures, 16, 397-407. Guarracino, F and Minutolo, V. (1996). Analisi della ovalizzazione di condotte circolari in regime di spostamenti finiti. Scritti in onore di M.Ippolito, Ass. Idrot. It., Napoli (in italian). Guarracino, F. and Mallardo, V. (1999). A refined analysis of submerged pipelines in seabed laying. Applied Ocean Research, 21,281-293. Reissner, E. (1961). On finite pure bending of cylindrical tubes. Osterr. Ing. Arch., 15, 165-172. yon K~rm~in, Th. (1911). Ueber die Form~inderung dtinnwandinger Rohre, insbesondere federnder Ausgleichrohre. Zeitschrifi des Vereines deutscher Ingenieur,. 45, 1889-1895.

Third InternationalConferenceon Thin-WalledStructures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

661

BUCKLING OF ABOVEGROUND STORAGE TANKS WITH CONICAL ROOF Luis A. Godoy and Julio C. Mendez-Degr6 Department of Civil Engineering, University of Puerto Rico, Mayagiiez, PR 00681-9041, Puerto Rico

ABSTRACT The buckling of aboveground circular steel tanks with conical roof is considered in this paper. The specific source of loads investigated is wind action during hurricane storms in the Caribbean islands. The structure is modeled using a finite element discretization with the computer package ALGOR. Bifurcation buckling of the shell is computed for a given static wind pressure distribution. Then the bifurcation loads and buckling modes are compared with the evidence of real tanks that failed during hurricane Georges in Puerto Rico in 1998. Several pressure distributions are assumed for the roof of the tank, and it is shown that the results are highly sensitive to the choice of pressures.

KEYWORDS ALGOR, Bifurcation analysis, buckling, finite elements, hurricane winds, metal tanks, mode shape, shells, wind pressures. INTRODUCTION The failure of tanks employed to store water and oil in the Caribbean Islands has been studied in recent years by the first author. Buckling of aboveground circular steel tanks with and without a roof was observed in St. Croix in 1990 (hurricane Hugo), St. Thomas in 1995 (hurricane Marilyn), and in Puerto Rico in 1998 (hurricane Georges). For tanks without a roof, or for those that lost the roof before the cylindrical part buckled, it was possible to reproduce the expected behavior using computer modeling. For example, a tank without a roof that failed in St. Thomas was modeled using standard wind pressured distributions around the circumference (Flores & Godoy 1997). For such a pressure, the computer model displays buckling for the wind speeds usually found during a hurricane. However, a far more difficult job is faced in an attempt to model the failure of the cylindrical shell in a tank with a conical roof or a shallow dome. The main questions regarding the pressure distributions on the roof are not answered within the current state of the art. In this paper we employ computer modeling to identify adequate pressure distributions that are compatible with the structural evidence showing buckling due to hurricane winds.

662 The search for adequate modeling of loads due to natural hazards is of great importance for the prediction of the safety of structures. There are various ways in which this is done at present. On the experimental side one can perform a full-scale test on a real structure or instrument it until an event occurs. The use of small-scale models is another possibility. Computer modeling of the environmental action on the structure is now possible, specially thanks to advances in Computational Fluid Dynamics. And there is the possibility of linking the failure of uninstnmaented structures to the loads that led to their failure. This work explores the last possibility within the context of tanks exposed to hurricanes winds. For tanks with a roof (either conical, spherical, or flat), the literature on wind load buckling reduces to a few contributions. Some extensive books on the design of tanks (Myers 1997, Ghali 1979) do not consider buckling under wind load. Early studies in the 1960s on the wind pressures on cylinders with shallow cap roofs were reported by Maher (1966), while an extension for fiat roof was published by Purdy, Maher & Frederick (1967). The literature on silos may also be relevant for short tanks; however the aspect ratio of tanks (height to diameter) of interest in this work is of the order of 1/5, while for silos this is higher than 1. Esslinger, Ahmed & Schroeder (1971) investigated silos with a dome roof under wind. This work is also discussed by Greiner (1998). Other wind tunnel tests were performed by Resinger & Greiner (1982), but there is no information about pressure distributions on the roof. The influence of group effects in silos under wind was reported by Rotter, Trahair & Ansourian (1980) from buckling experiments in a wind tunnel. The silos tested were aligned in a single row perpendicular to the direction of the wind, and closely spaced. Esslinger, Ahmed & Schroeder (1971) tested two cylinders with spherical cap roof separated by a distance of the order of a diameter, and for various directions of wind incidence. For two tanks aligned in the direction of the wind, the first tank shelters the second one, and develops pressures on the windward side, suction close to 90 degrees from the wind direction, and small pressures on the leeward side. The roof has suction on a small part of the windward side, and pressure on the leeward side. Such non-uniform pressure distribution on the roof is also found for other orientations of wind. The influence of an internal operating vacuum may further modify the wind pressures. Flores and Godoy (1997) studied the nonlinear dynamic response of short tanks and found that inertia effects were not significant in this class of shells, so that static analysis could well be carried out to estimate instability under wind load. In the following sections we describe the computational finite element model employed and consider a specific structure to investigate its buckling failure. Several pressure distributions are assumed for the roof and the results of buckling pressure and mode shape are compared with the evidence from real cases. COMPUTATIONAL MODEL A metal circular tank with a conical roof under a static pressure distribution (Figure 1) is modeled in this paper in order to evaluate bifurcation buckling loads and modes. Because there is evidence that the real tank failed during hurricane Georges in Puerto Rico, then we have an upper limit to the wind velocity at the time of buckling. The buckling mode observed in the real tank includes plasticity effects associated to an advanced post bucking behavior; however, the localization of the mode at the windward meridian and the mode shape are considered similar to the initiating elastic buckling mode. Preliminary studies using geometrically nonlinear analysis indicate that instability in this case has small displacements in the pre buckling equilibrium states and that the maximum load and mode attained by the tank are well represented by a bifurcation study. This is by no means a general conclusion and applies only for the geometry of the tanks considered, which is rather short. At least for open tanks, it has been shown (Godoy & Flores 2000) that the aspect ratio and thickness slenderness are crucial to determine the type of instability that may be found in the structure.

663 The general purpose finite element package ALGOR (1999) was employed to build the computational model using 2250 shell elements, including 1250 quadrilateral elements for the cylinder and 1000 elements (either quadrilateral or triangular) for the roof. The boundary conditions at the bottom of the shell are assumed as clamped. Only one-half of the tanks is modeled, with symmetry in the plane of incidence of wind. The mesh of elements is shown in Figure 2. A series of bifurcation buckling analysis were made in order to identify adequate pressure distributions on the roof, which are compatible with the evidence observed in real structures. This was accomplished by calculating the critical bifurcation buckling mode and pressure and comparing them with the maximum load expected to occur during a hurricane. CASE STUDIED The theme structure in this paper is a tank with variable thickness, as shown in Figure 1. The tank has 30.5 m of diameter and a height of 12.2 m. It is made of steel with the assumed properties listed in Table 1. The tank is located in Pefiuelas, an industrial area in the south of Puerto Rico, where a large number of tanks were build in order to store petrochemical products for the many industrial companies that developed several years ago. TABLE I DATA ASSUMEDIN THE ANALYSIS

Figure 1. Dimensions of the tank.

Figure 2. Finite element mesh.

This tank was damaged by winds during hurricane Georges in 1998, as shown in Figure 3. The tank was empty at the time when the hurricane hit the area; it belongs to a group of tanks that are separated by a distance of approximately 50 m, and its location is close to the coast (about 300 m from the coast).

664

Figure 3. Tank investigated that buckled during hurricane Georges in 1998. ASSUMED PRESSURE DISTRIBUTIONS

It was assumed that the pressure distribution around the circumference has positive values on the windward meridian, and negative pressure (suction) on the rest of the cylinder, and was modeled as a constant unit pressure in the vertical direction. This pressure pattern was used by many authors before (ACI-ASCE 1991, Flores & Godoy 1998, 1999). In order to propose a pressure distribution for the roof of the tank we used the photographs as a guide to how the tank buckled under wind load. For this purpose we used an inverse technique of cause and effect, in which the photographs showed the effect of the hurricane winds. Once the buckling mode shape was available, we attempted to find the pressures that caused this failure. Several bifurcation buckling analyses were made using ALGOR for different pressure distributions on the roof. Then the buckling load was calculated for each case. Every load assumption used caused a different buckling load and mode shape in the structure. Finally, we searched for a buckling load similar to the load expected during a hurricane that caused a buckling shape similar to the one photographed just after the hurricane occurred. RESULTS FOR DIFFERENT PRESSURE DISTRIBUTIONS ON THE ROOF

In Load Case 1 the tank was modeled by taking the influence of the roof with upper boundary conditions instead of the roof itself. This class of models is attractive because one does not model the roof with finite elements, but it neglects the pressures that may act on the roof surface. First, the cylindrical tank clamped at the top was considered. This lead to a bucklin~ mode consistent with what was observed in the structure, but for a high load factor o f k c = 3.35 kN/m" (or wind speed of v = 66.9 m/s). For a simply supported condition on top, the values changed to ~c = 3.33 kN/m 2. This model shows a buckling mode shape similar to the real mode. For a free condition at the top it is not expected that the model simulates a roof, and is only considered here as a reference case. The load factor changed to ~c = 1.3605 kN/m 2, but the mode shape from such computations was very different to the one expected. Load Case 2: In order to include the influence of the roof into the model we assumed different patterns of wind pressures acting on the roof. As a first option, we included a downward constant pressure as a percentage of the maximum pressure (1 kN/m 2 ) applied on the walls. The distribution of pressure (ACI-ASCE, 1991) applied around the circumference has a maximum value of 1 kN/m 2. For this load

665 case we obtained the results plotted in Figure 4. The buckling mode of each of these cases was very similar to the real one, as shown in Figure 5.

Figure 4. Load Case 2

Figure 5. Load Case 2.

Load Case 3 was a variable pressure load acting on the roof. It was assumed that the meridian of the shell has pressures with the same sign, and following a circumferential distribution similar to a tank without a roof. This pressure load on the roof has the same orientation as the pressures acting on the walls, and we call this a "variable pressure" because it varies in the circumferential direction. The magnitude of the pressures varies according to the direction with respect to the incidence of the winds. The positive pressures produce a downward pressure on the roof, while negative magnitudes of pressure produce an upward pressure (suction). The scope of this analysis was the same as in Load Case 2, varying the percentage of the pressure load to study the variation of the critical load, Figure 6. In this case the critical buckling loads were very high, exceeding the expected range.



Load Case 4: Consultations with wind-experts led to the recommendation that an upward pressure should be present, following the experience with pressures on buildings with rectangular plan. For this case we applied a constant upward (negative) pressure to the roof. In this model only suction occurs on the roof. With full maximum constant pressure (lkN/m 2) acting upward on the roof the buckling load was -2.98 kN/m 2. With 90% of the maximum constant pressure acting the buckling load was-1.83 kN/m 2. Negative values of critical buckling load mean that the pressure distribution should be applied in the opposite direction. The buckling mode for this case was very interesting: The roof suffered large buckling deflections instead of the walls, and the buckling mode shows a totally different shape, Figure 7. This shape is far from what is observed in the real situation.

666 Load Case 5: Wind-tunnel experiments in Germany (Esslinger, Ahmed & Schroeder 1971) have shown that for silos, the pressure on a conical roof is negative on the windward part of the roof and positive on the leeward part. At the center of the conical roof the pressures are zero, and they take non-zero values on a ring which spans half way between the center and the edge of cone. Our cases 5 and 6 take that into account. For Load Case 5 we applied a variable pressure but with the "inverse orientation" of the pressure acting on the walls (Greiner, 1998). That means that the roof is modeled with positive pressure in the places where the wind is negative on the walls, and negative (suction) pressure in the places where the walls have positive pressures. The critical buckling loads are similar to the expected load corresponding to the wind speed measured for the area in which the tanks are built. The variations of the critical load due to the increase of variable pressure load are plotted in Figure 8. Notice that this behavior is similar to what was obtained for Load Case 2. Furthermore, the modes of buckling computed were also similar to those expected in comparison with the photographs (Figure 9).

Figure 10. Load Case 6. Load Case 6: We assumed that the pressure acting on the wall only affect some part of the roof. In this case the pressure was applied over the circumference of the roof to a distance of one fourth of the diameter of the roof measured from the comer (the junction between the roof and the wall). This distribution left the middle section of the roof without any pressure. The orientation of the load was the same assumed for the Load Case 5. The critical load computed by ALGOR was 3.03 kN/m 2. The buckling mode shape for this case is also similar to those obtained from Load Cases 2 and 5. The buckling mode shape corresponding to this distribution is shown in Figure 10.

667 DISCUSSION For the first load case (Load Case 1) the tank was modeled using three different boundary conditions instead of the roof. For the tank clamped at the top the buckling load was larger than the expected range of values (wind speed larger than that registered in the area in which the tank is built), although the buckling mode shape was similar to the one expected. The same situation occurred with the tank modeled with a simply supported condition at the top. For a free condition at the top the buckling load was smaller than the other two cases but the buckling mode shape was totally different of the one expected. In Load Case 2, a downward constant pressure was applied on the roof. The results obtained vary depending on the magnitude of the pressure used. As discussed before, the magnitudes considered vary in percentages of the maximum constant pressure (lkN/m2). For 50% up to 120% of the maximum constant pressure the critical buckling load resulted in values very close to the one expected. These values decrease as the percent of full pressure acting on the roof increases. The estimated range of the velocity pressure for an open area is between 2.40 kN/m 2 and 3.00 kN/m 2, corresponding to wind velocities from 56 m/s up to 67 m/s (125 mph up to 150 mph). The values between 10% and 40% lead to high wind speeds. In terms of the buckling mode shape, the results obtained for the different magnitudes of pressure were similar to the shape expected according to the photographic evidence. For Load Case 3, in which the applied load was a variable pressure acting on the roof, the results were not satisfactory in terms of the buckling load. That means that if we applied on the roof a distribution that varies according to the direction of winds (positive pressures on the cylinder produce a downward pressure on the roof, while negative magnitudes of pressure on the cylinder produce an upward pressure on the roof) we obtain very large values of the critical buckling load. These values also changed the sign if we increase the pressure to more than 50% of the value of the pressure on the cylinder. Such pressure distribution is not recommended in this context. For Load Case 4 (constant upward pressure on the roof) we obtained negative values of buckling load, which means that the distribution used in this case was not satisfactory. This case considers also the distribution used on the walls, and for this reason this is not a good distribution to model the wind. The buckling mode shape obtained was very different from the one expected, i.e. the tank model failed in the roof instead of the walls. In Load Case 5 we applied a variable pressure distribution but with the inverse orientation of the pressure acting on the walls (Esslinger, Ahmed & Schroeder 1971, Greiner 1998). The critical buckling load computed in this case was similar to that computed in Load Case 2 and the results are also satisfactory. The modes of buckling were also similar to those obtained for Load Case 2. For Load Case 6, which is a variation of Load Case 5, we obtained a critical buckling load just larger than the one obtained for the 100% of the load in Load Case 5. The value was over the expected range, but very close to the maximum load expected. The buckling mode shape was similar to those modes obtained for load cases 2 and 5. CONCLUSIONS The pressures distributions obtained for silo structures with shallow cap roof by Esslinger and coworkers (1971) seem to be an adequate representation for short tanks with conical roof. In this case the pressure distribution on the roof is not uniform, but has suction close to the meridian of incidence of wind, and positive pressure on the leeward side of the shell, while the central part of the roof does not have pressures. For such pressure distributions the buckling pressure (and associated wind velocities)

668 and the buckling mode shape were consistent with those found in the field following hurricane Georges in 1998. This preliminary study illustrates the sensitivity of the buckling response with the pressure distribution assumed on the roof. More detailed experimental evidence of pressure distributions for short tanks is required, and this could be obtained from wind-tunnel experiments. Such studies are now being performed at the University of Puerto Rico at Mayagtiez. ACKNOWLEDGEMENTS

This work was sponsored by the US National Science Foundation (NSF) under grant CMS-9907440, and by the US Federal Emergency Management Administration (FEMA) under grant PR-0060-A. The support of both institutions is greatly appreciated. REFERENCES

ACI-ASCE Committee 334 (1991) Reinforced concrete cooling tower shells: Practice and Commentaries. American Concrete Institute, New York. ALGOR (1999) Linear and nonlinear static and dynamic finite element stress analysis. ALGOR lnc., Pittsburgh, PA, USA. Esslinger M., Ahmed S. & Schroeder H. (1971). Stationary wind loads of open-topped and roof-topped cylindrical silos (in German). Der Stalbau, 1-8. Flores F. G. & Godoy L. A. (1998) Buckling of short tanks due to hurricanes, Engineering Structures, 20(8), 752-760. Flores F. G. & Godoy L. A. (1999) Forced vibrations of silos leading to buckling, Journal of Sound and Vibration, 224(3), 431-454. Ghali A. (1979), Circular Storage Tanks and Silos, EFN Spon, London. Godoy L. A. & Flores F. G. (2000). Imperfection-sensitivity of wind-loaded tanks. Submitted for possible publication. Greiner R. (1998). Cylindrical shells: wind loading. Chapter 17 in: Silos (Ed. C. J. Brown & L. Nilssen), EFN Spon, London, 378-399. Maher F. J. (1966) Wind loads on dome-cylinder and dome-cone shapes. ASCE Journal of the Structural Division, 92, 79-96. Myers P. E. (1997), Aboveground Storage Tanks, McGraw-Hill, New York. Purdy D. M., Maher P. E. & Frederick D. (1967). Model studies of wind loads on fiat-top cylinders. ASCE Journal of the Structural Division, 93,379-395. Resinger F. & Greiner R. (1982). Buckling of wind loaded cylindrical shells: Application to unstiffened and ring-stiffened tanks. In Buckling of Shells, E. Ramm (Ed.), Springer-Verlag, Berlin, 305331. Rotter J. M., Trahair N. S. & Ansourian P. (1980). Stability of plate structures, Proceedings, Symposium on Steel Bins for Bulk Solids, Australian Institute of Steel Construction / Australian Welding Research Association, Sydney, 36-42.

Third International Conference on Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 Elsevier Science Ltd. All rights reserved

669

ON THE COLLAPSE OF A REINFORCED CONCRETE DIGESTER TANK Luis A. Godoy and Sandra Lopez-Bobonis Department of Civil Engineering, University of Puerto Rico, Mayagtiez, PR 00681-9041, Puerto Rico

ABSTRACT The investigation following the collapse of a large reinforced concrete dome, which was part of a digester tank, is presented. The shell was constructed in 1987, and had construction errors related to the location of the single layer of reinforcement, which were discovered as a consequence of the collapse. The shell collapsed without the occurrence of any natural hazard. It is believed that a high internal pressure developed on the day of the collapse because of a problem with a valve, which allowed the discharge of large quantities of sewage inside the tank and filled the structure completely. A finite element analysis of the structure shows the stress levels in the structure, and support the hypothesis of a failure mechanism which coupled the construction errors with the internal pressure. Finally, the strengthening of the shell with externally bonded fiber composite sheets is described as a possibility to improve the safety of other tanks in similar situations.

KEYWORDS Collapse, composite materials, construction errors, digester tanks, reinforced concrete, shells. INTRODUCTION The failure of reinforced concrete shells has been reported in a number of cases, as mentioned in the texts by Billington (1990), Godoy (1996), Gould (1999), and others. For example, Ballesteros (1978) reported the collapse of an elliptical paraboloidal shell during the process of removing the formwork; the structure showed clear imperfections in the geometry and had construction defects. Many cases of reinforced concrete tanks (or similarly shaped structures) that fail due to structural or construction problems are not reported in the open literature; however, this lack of publication does not help other researchers to learn lessons from failures. This paper reports on the collapse of a large reinforced concrete shell dome, which was part of a digester tank, under an internal pressure produced by the accidental filling of the tank. The change in the operational conditions resulted in a limit state for which the tank was not adequately reinforced. Possible ways to repair existing tanks in the same plant with composite sheets are also considered.

670 THE SHELL STRUCTURE INVESTIGATED The digester studied here is sufficiently far from other digester tanks in the same facility so that it can be considered as an isolated structure. The structure is a closed reinforced concrete containment structure formed by a cylindrical shell, a sector of a spherical shell roof, and a conical shell at the bottom, as shown in Figure 1. Part of the structure was built underground. The horizontal radius of the cylinder is r0 = 14.5m, the height is 10m, and the wall thickness is 0.5m. According to the plans of the design, the cylinder was reinforced with two layers of steel (#6 bars at 250mm in the vertical direction and #9 bars at 38mm in the horizontal direction).

Figure 1. A typical digester tank considered in this study.

CL

11 0.228m #,1@20

0.30rnT ~ .1" .

#4Q20cm

0.50m

3 m

!

11.5

,

Figure 2. Reinforcement of the dome of the digester tank. The dome is a spherical cap (Figure 2) with radius of curvature R = 30.8m and thickness t = 228mm, so that R / t = 135. This is considered a thin shell for reinforced concrete, and should have been designed according to the ACI provisions already published at the time of design (ACI Committee 334, 1986). As a reference value, large reinforced concrete cooling towers have R/t of the order of 150. The dome is connected to a cylinder by means of a ring. The maximum elevation of the dome with respect to its

671 supports on the ring is approximately 2.90m. A PVC liner was attached to the bottom surface of the dome. The steel section in both directions was found to be #4 at 200mm, leading to As = 645mm2/m. For the area of concrete Ac, the ratio As / Ac -- 0.28% is a low amount of reinforcement for this class of shells. The structure was constructed in 1987. The dome should have been designed to resist primary compressive forces due to self-weight and accidental loads, but sufficient bending capacity should have been provided for situations other than gravity load. For this shell, which is exposed to an aggressive environment, it would be expected to have a 38mm concrete cover.

Figure 3. Partial view of the dome after the collapse.

Figure 5. Details of damage of the shell.

Figure 4. Location of the reinforcement.

Figure 6. Details of cracks in the shell.

CONDITIONS OF THE STRUCTURE PRIOR TO THE COLLAPSE Figure 3 shows the structure following the collapse of a large part of the dome. Because part of the structure did not collapse, it was possible to observe some details of the construction: (a) The single layer of the reinforcement was not placed at the center of the thickness, as indicated in the drawings, but it was displaced to the bottom surface of the shell. This has important consequences for the membrane and bending resistance of the shell.

672 (b) The concrete cover on the bottom surface of the shell was not sufficient. For this shell, which was constructed to operate in an aggressive environment, the concrete cover was found to be only 12.7mm and perhaps less in some zones. This is illustrated in the photograph of Figure 4. (c) Furthermore, the concrete cover in the zone close to the supports of the dome on the top surface of the dome was not adequate: in some parts of the shell it was possible to see the steel bars in the meridional direction. (d) Corrosion of the steel bars had occurred for some time. This could be seen at many places on the external surface of the dome. Corrosion has the consequence of reducing the effective diameter of a steel bar in a localized way. (e) Meridional cracks are clearly visible on the external surface of the dome (Figures 5 and 6), even in parts of the shell sufficiently far from the area that collapsed. Those are not new cracks formed as a consequence of the collapse, but were formed some time ago. Cracks larger than 3mm in the meridional direction occur at a spacing of about 2m. This reduces the bending capacity of the shell for negative hoop moments. (f) Circumferential cracks on the external surface are visible. Again, this reduces the bending capacity of the shell for negative meridional moments. Both meridional and circumferential cracks reduce the tensile capacity of the shell. (g) There are some fiat parts of the shell between meridional cracks, with the consequence that the shell had rotated taking the cracks as hinges. SEQUENCE OF EVENTS LEADING TO THE COLLAPSE OF THE DOME There was no report of high winds, earthquake, or small amplitude ground motion on the day of the collapse. However, a problem was reported on a valve, which caused the filling of the digester up to the top, with large internal pressures acting on the dome. A worker observed that material stored in the digester was being discharged at the top of the roof. A large crack (l.5m) formed on the external surface in the meridional direction, and large quantities of sewage material started flowing through the crack, coming from inside the digester. Next, a bulge formed close to the crack, and extended for at least 2m in the circumferential direction. The amplitude of the bulge (the elevation with respect to the external surface of the shell) was estimated to be at least 0.10m. The 3m part of the shell that had double reinforcement remained in the structure and the central part of the dome collapsed towards the inside of the digester. No explosion was reported. All debris were found inside the digester. STRUCTURAL CONSIDERATIONS The main loading condition of the dome shell is self-weight, leading to compression in two directions. The self-weight of the structure produces in-plane stresses of the order of 363KPa. Such stresses are small compared to the compressive strength of the concrete, f'c = 20.7MPa. Even if the weight is increased by accidental loads, the stresses are still small. Failure of concrete under compression is ruled out as a main cause of the collapse. A buckling load of the dome was estimated (Billington 1990, pp. 320) using E = 21.2GPa, leading to a critical pressure pC = 360KPa. This pressure is much higher than any pressure associated to gravity action. Thus, it is not likely that buckling under the main compressive (membrane) action led to a limit state in the shell. It has been shown that for thin-walled shells geometric distortions produced by various causes may induce bending stresses of the same order as the primary stresses. The collapse of several large reinforced concrete shells has been attributed to this effect. A review of several cases and their causes is reported in Godoy (1996). This dome clearly had significant geometric distortions, as reported by engineers previous to the collapse. Since no measurements of the actual shape of the dome were performed, it is difficult to assess the amplitude and extent of such distortions. At the time the shell

673 showed signs of a critical condition, an engineer reported a bulge with amplitude of about 0.10m and extending with a diameter of at least 2m. This may be a significant source of stress concentrations in the shell. But for the reinforcement present in the shell, an internal pressure may be the triggering cause of the collapse. Notice that a reinforced concrete spherical cap is extremely efficient to resist self-weight because it can develop compression; however, if the load is reversed the dome becomes a most inefficient structure under tension. Had the reinforcement been placed on a single layer at the center of the thickness, a tensile force T could develop due to the contribution of the steel section As. The concrete could have taken a compressive zone at the lower part of the thickness, to produce the required bending and thus equilibrate the internal pressure p'. Cracks were clearly present in the dome, in both directions. Such cracks were not new, and may have been produced by a variety of reasons, including the early life of concrete, thermal action, and others. A factor that must have played a role in the crack formation is the position of the reinforcement, which was displaced towards the bottom of the cross section for some unknown reason. In the real situation, the as-built shell has the steel reinforcement on the inner side. Under bending produced by the reversed load, the tensile force T had to be very large since the distance between the tensile and the compressive forces is small. Furthermore, the compressive force C developed by the concrete small section (the concrete cover) must be extremely high. For a steel with yield stress a y = 450MPa, a limit state could be reached with an internal pressure higher than the self-weight of the dome. FINITE ELEMENT STRESS ANALYSIS A shell with dimensions similar to the central part of the dome, for which a single layer of reinforcement was present, has been studied using a finite element model. The structure was considered as an axisymmetrie solid with quadrilateral elements, as illustrated in Figure 7. The data assumed is shown in Table I.

6 ,L2:_:: ~.,-._3__

~ ...........

1

Figure 7. Dimensions of the dome investigated in the analysis. The stresses and displacements of the shell have been computed under an internal pressure to simulate the influence of the sewage at the time a valve permitted the filling of the tank under pressure. It is not known the value of the pressure that was induced by the sewage, so that a reference value of 0.30 times the self-weight is adopted for the computations. Since this is an elastic analysis, the pressure should be scaled to evaluate a limit state.

674

TABLE I PROPERTIES FOR CONCRETE AND STEEL

Properties

Concrete

Steel

E, Elasticity Modulus

20.68 GPa

200 GPa

Mass density

2402.7 kg/m3

7861.4 kg/m3

v, Poisson ratio

0.15

0.29

Thermal Coefficient

.0000108

.0000117

Shear modulus

9.0 GPa

77.2 GPa

v. ";5

,,,

!

[

i ,

0.1

~zr

0-

- oooo

!

.

i

]

15~00

200 ~ 0

!

!1 oloo '

]

i

I

!

:

2 5 1 ~ ~

350'000

; -0.05 ~ !

-v.'

!

1 i

: i - vr ~. 4|~,

t

'

,

t

Stresses (N/m2) r

E=Econcrete/10--e--E=Econcrete/3

-"

E=Econcrete(2/3)

•

E=Econcrete

Figure 8. Stresses in the dome with and without deterioration o f concrete. o.0008

i Econcrete*l/10

0.0007 ~....,,~.~-~.~

g

le

~. 0.0005

Ec~

E

,

i f

" ~ ~ ~ , ~§ Econaete'2/3

Econ~me 2.07E§

a

!

I

i

'

0.0000 2.07E*08

5.21E*09

I 1.02E+10

!

I 1.52E+10

, 2.02.E+10

2.52E+10

Elasticity M o d u l u s (NIm'L2)

Figure 9. Displacements in the shell for various conditions o f the concrete.

675 The stresses in the meridional direction are shown in Figure 8 for E = Ec. This is a linear response including membrane and bending action. For the perfect shell to crack under internal pressure p', the value of pi should be 3.65 times the self-weight of the shell. However, there are clear signs that the shell had serious deterioration and cracking prior to the occurrence of internal pressure. To investigate the elastic stresses in the shell including damage of the concrete, parametric studies are shown in Figure 8 for several values of the modulus of elasticity. The actual area affected by damage was assumed over the central part of the dome, extending one-third of the total arc in the meridional direction and on the top half of the thickness. The results depart from classical shell theory assumptions, i.e. a linear distribution of stresses through the thickness is lost, with a significant reduction of stresses on the top part. To compensate for that, the stresses in the lower part of the thickness are largely increased, in order to maintain the net tensile force on the overall section (membrane contribution). In the deteriorated section with E = 0.1 Ec the tensile stresses increase by 40%. The bending moment, however, now acts in the opposite direction. The influence of the deterioration of concrete on the maximum displacements is shown in Figure 9, leading to a 100% increase for the case with E = 0.1 Ec. STRENGTHENING THE DOMES In view of the catastrophic consequences of the construction errors for some loading conditions, the safety of other digester tanks in the same plant and with similar characteristics to the tank that collapsed had to be evaluated. Possible ways to strengthen a dome were considered, and the most convenient way evaluated was the use of externally bonded fiber composite sheets.

V

-

_

,o

T

/

,

/

00 Stresses

ro00-0o (N/m21

-~-One layer of CFRP -e-Without strengthening Figure 10. Stress redistributions in the shell with an externally bonded carbon fiber sheet. Composite sheets have been employed to strengthen reinforced concrete bridges, columns, and beams, and this is an area of great interest in terms of research and civil engineering practice. A carbon fiber composite (CFRP) could be laid on the external surface of the thickness, to restore the tensile capacity of the dome in case of a reversed bending situation. Due to limitations of space it is not possible to describe in full the behavior of the dome with such reinforcement, and only one plot of the stress redistribution in the meridional direction is shown in Figure 10.

676 The main problems to be considered in the design of this reinforcement are the shear transfer between the composite and the concrete, and how the behavior of the shell is modified under normal (selfweight) conditions. CONCLUSIONS The preliminary conclusions of this work may be summarized as follows: (a) The shell could resist normal operation under self-weight and accidental loads due to gravity. This was done even taking cracking and the real location of the reinforcement into account. (b) Effects due to buckling of the shell have not been a crucial factor that could explain the collapse. Seismic or wind load were not identified on the day of the collapse so as to study a dynamic action as a possible explanation. (c) Because the digester was accidentally pressure-filled on the day of the collapse, it is expected that high internal pressures developed on the dome. If the shell had been constructed as designed (with a central layer of reinforcement in two directions) then it is expected that it would have resisted an internal pressure. But for the actual shell with the reinforcement at the bottom of the thickness, with cracking on the top part of the concrete, and with geometric distortions, it seems that the structure would not be able to take the internal pressure. (d) Strengthening the dome on the top side with carbon fiber composite sheets may be a convenient way to improve the safety of a structure in such conditions. The results show that the stress levels are reduced in the concrete, and that the bending capacity is restored thanks to the tensile contribution o f the composite. REFERENCES ACI Committee 334 (1986), Concrete Shell Structures: Practice and Commentaries, American Concrete Institute, pp. 14. Ballesteros P. (1978). Nonlinear dynamic and creep buckling of elliptical paraboloidal shell. Bulletin of the Int. Association for Shell and Spatial Structures, 66, pp. 39-60. Billington D. P. (1990), Thin Shell Concrete Structures, 2 nd Ed., McGraw-Hill, New York. Godoy L. A. (1996), Thin-Walled Structures with Structural Imperfections: Analysis and Behavior, Pergamon Press, Oxford, UK. Gould P. L. (1999), Analysis of Plates and Shells, Prentice Hall, New Jersey.

Third Intemational Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalska and J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

677

CLOSED CYLINDRICAL SHELL UNDER LONGITUDINAL SELF-BALANCED LOADING V. L. Krasovsky and G. V. Morozov Prydnieprovsk State Academy of Civil Engineering and Architecture, Chernyshevsky Street, Dnepropetrovsk, 49600, Ukraine

ABSTRACT One of factors, influencing stress-strain state (SSS) of large-sized vertical cylindrical tanks for oil storage, is an subsidence of foundations. The magnitudes of this subsidence are defined as vertical displacements of the lower edge of the tank shell. The constituents of these displacements that correspond to the second and subsequent harmonics of trigonometric series may lead to considerable stresses in the tank shell. The problem formulated in this paper may be reduced to solving the SSS problem for a closed cylindrical shell at non-uniform kinematic loading applied to one of the edges. The problem is solved using a specially developed automatic calculation package in finite differences. The investigation of the problem has shown a number of effects in the shell behavior at specified kind of loading. Particularly, it was found that in case of free edge at unloaded edge of the shell considerable radial displacements occur in its vicinity that lead to increase in the bending stresses. It is also found that when the circular displacements at this edge are restricted (v-0), the SSS in the shell is close to the momentless one.

KEYWORDS Cylindrical shell, kinematic loading, storage tank, foundation subsidence, finite differences method.

PROBLEM F O R M U L A T I O N The problem of SSS definition in a closed cylindrical shell (see Figure 1) is considered. The shell is loaded at one edge by the longitudinal kinematic load: *

ny

u = Asin--k-- ,

(1)

where A is the amplitude of displacement, n is the number of harmonics that defines the alternation of longitudinal displacements of the shell edge in the direction of circular co-ordinate y.

678

S=

i

....

d

N

M21

l~

Figure 1: Cylindrical shell

MI

MI2

& ~ N!

d Q2

M1

Figure 2. Internal forces and displacements in shell

The solution is built on the base of equations of the linear theory of cylindrical shells written in displacements, Vlasov (1949): # OW

02U 1 - # 02U ~ + ~ ~ + Or,2 2 Oy2

l+bt 02V

1+#

1-,u 02v 10w 3-,u

02u

20xOy

+

+--~-c~ OxOy R Ox

2

02v

+ ~ ~ + -~ 20x 2

--m-C # OU (03U 1-# R Ox

Ox3

2

(03W

1-#

&3

2

~ c R Ok, 2

03w 1-#2 Ox20y

+

3+ 1 OV --C3-# . 03V

03U

Ox@2

R Oy

03W ) 1 - # 2 )+ X =0 OxOy2 Et '

2

Et

Y =0,

(2)

(3)

+

Ox20y

(4) + c R V2V2w + ------:---+ R

+

R2

Et

Z = 0.

where x, y are the unknown variables (longitudinal and circular, respectively); u, v, w are the displacements in longitudinal x, circular y and radial z directions (figure 2); R, t are the radius and thickness of the shell, E, # are Young's modulus and Poisson's ratio of the material; X, Y, Z are load components in the directions x, y, z; ~ . _ .

t2 12R

V2

9

__

02 02 ~ +~

_

Laplace's operator.

The expressions for components of the internal forces (see Figure 2) that correspond to the equilibrium equations (2-4) can be expressed as follows: N1

1_#2

= ~

+#

-c

+---C

,

s~ 20+#) Oy Ox

MI:D

R --~+#

OxOy)

-

-'~+#7

, N2= 1- #2

S2 =

w ,

+~-+,u-~- +c OY2 ),

+--+c~

2(1+#) Oy ox

OxOy

,

(5)

(6)

(7)

679

M12 = -D (1-/.t) /

R 0x +OxOy)'

Q1 =D

M21 = - O

0x2

2

2 . -~-

-

0y2 + 2R 0x0y

+2-~j

0x

+

,

(8)

w ,

(9)

Q2:D

(1-P) R Ox2

R 2 Oy

03, ~-T +

w .

(10) where D is the cylindrical stiffness. The expressions for the generalized shear ( S 1 ) and transverse (Q1) forces, according to Kolkunov (1987), are as follows: S1 =S 1+M12 R

D(l-p)

6 ~ 6 1 ~ ~ + +

0v 0y

3 02w 2R 0xOy'

(ll) Q1 = Q1 +-

=D

0y

0x2

-

~

2

0y2 + - 2

3 - ,u O2v

03w

0x0y - ( 2 - / t ) 0X0y2

03w

0x 3 .

(12) Boundary conditions on the lower edge of the cylinder were taken in the form: u = u*, v = 0, w = 0, M~=0. Different boundary conditions were considered on the upper edge. The solution and the analysis of presented problem have been performed within the framework of an automated calculation package using the finite differences method (FDM). This package was developed for solving various shells problems, connected with calculation of vertical cylindrical tanks. All calculations have been performed on base of FDM models for cylindrical shells. The difference templates of extra-high accuracy have been used for presentation of initial differential equations.

RESULTS AND ANALYSIS In the case of flee support of shell on the upper edge (NI "-" 0, S l * - 0, Q1*= o, MI = 0) it was found that even slight longitudinal displacement of the unloaded edge may cause considerable radial displacements on the upper edge. The nature of these displacements corresponds to the longitudinal kinematic loading of the lower edge. In this case the bending stresses or(M2) prevail but the membrane stresses are negligible. The deformed shell at n = 4 is shown in Figure 3 (a, b). In Figure 3 (c) the maximum radial displacements w are shown with solid line (marked "o") along the length of the shell with parameters R = H = 10 m, R/t = 1000, A = t, n = 2. It has been found that the increase of amplitude of displacements A of the lower edge of the shell leads to proportional increase of the radial displacements and stresses on the opposite free edge. The proportional increase of deformations and stresses can be also seen when the shell height H increased. The increase of number of harmonics n for given longitudinal displacements at the constant values of these parameters results in an increase of radial displacements proportional to n 2.

680 To check the possibility of such an effect in real shells the experiments have been performed on samples made of solid paper. One of models is presented in figure 4. The shell 1 is installed on a cross-piece 3. The dead load 4 is applied to the lower edge. Such loading causes the state close to that described by (1) with n= 4. The radial displacements of the shell on the lower edge are restricted with a rigid plate 2 installed inside the shell. The results of the experiments confirm the qualitative peculiarities of the shell deformation obtained in the calculations. As it is seen on the photograph, the shape of sample is similar

to the configuration obtained in calculations. To examine the behavior of the shell at different boundary conditions and to reveal the conditions that considerably influence the SSS at given kind of loading the following test has been performed. The boundary conditions were varied at constant level of load for the shell (R = H = 10 m, R/t = 1000, A = t , n = 2) on the upper (loaded) and lower (unloaded) edges. The results are presented in Table 1. TABLE 1 DISPLACEMENTS, FORCES AND STRESSES IN THE SHELL WITH DIFFERENT BOUNDARY CONDITIONS (R = H = 10 M, R/T = 1000, .4 - T, N = 2) Displacements, forces and stresses Boundary conditions (upper/lower)

w/t

N~ (kN) 0.77 0 (1)

N2 (kN)

S~ (kN)

M~ (kNm)

M2 (kNm)

(MPa)

2.92 (u)

0.435 (1)

6.41e-3 (u)

21.3e-3 (u)

1.53 (u)

O'IV

N! = 0, $1 = 0, Q1 = 0, M1 = 0; u=u*, v=O, w=O,M~=O

4.0 (u)

N1 = 0, $1 = 0, Ql = 0, M1 = 0; u= u, v= 0, w= 0, w ' = 0

3.99 (u)

3.30 (1)

58.9 (u)

5.13 (1)

0.614 (1)

0.184 (1)

32.9 (u)

u - 0 , S~=0, Q I = 0 , M 1 = 0 ; u=u', v=O, w=O,M~=O

1.70 (u)

2060 (1,u)

615 (1)

26.8 (1)

1.27 (1)

0.383 (1)

250 (l)

N l = 0 , v=0, Q 1 = 0 , M I = 0 ; u=u*,v=O,w=O,M~=O

0.234 (1)

2090 (1)

625 (1)

1180 (1)

0.556 (1)

0.168 (1)

225 (l)

N~ = 0, v= 0, w= O, w'= 0; u=u , v=O, w=O, w ' = 0

0.209 (1)

2090 (l)

625 (1)

1050 (1)

1.67 (1)

0.501 (1)

274 (l)

Momentless solution (upper edge: N1 = O, v = O)

0.305 (1)

2090 (1)

,

.,

,

,,.

1050 (1)

182 (1)

681 Here the maximum values of the corresponding characteristics are given that occurred in the area of either unloaded lower (1) or unloaded upper (u) edge of the shell, in this table $1 is the shear force. The maximum equivalent stresses o~v are defined according to the energy theory of strength:

O'IV =

~/ 2

2 2 O"x + Cry + 3"r x - O ' x C r y

,

N1 6M1 where crx = ~' + ---~--,

N2

S1

6M2

O'y = ~ t + / 2

'

Z"X - - ~

t

As it is seen from the Table 1 the rigid fixation (in comparison with hinge) does not considerable influence the behavior and the general SSS. At the same time the rigid fixation of the upper edge does not change the SSS too much. This SSS passes on from the bending (moment) one in the case of free edge to the state close to the momentless one. It becomes apparent from the considerable reduction in the radial displacements and bending stresses and in the increase (by one or two orders) in the membrane stresses. It is found that the most important factor among boundary conditions that defines the SSS is the restriction of the circular displacements of the unloaded edge (v=0). To show this effect in Figure 3(c) together with maximum radial displacements of the shell with free upper edge (curve "o"), the displacements obtained for fixed upper edge are presented in circular (v=0; curve ~v))) and longitudinal directions (u=0; curve (~u))). The solution of the momentless SSS problem (v=0) has been found also on the base of the momentless theory of shells [6]. The results of this solution (line "Momentless solution" in Table 1) show practically complete coincidence of the basic forces N~ and S~ obtained in the momentless and moment solutions at the corresponding boundary conditions. In practice the canonical boundary conditions (free edge, rigid fixation and others) are realized rarely. Usually the edges of shell are elastically supported. The walls of large-sized tanks are supported at the upper edge with a stiffness ring. To study the influence of reinforcement of unloaded edges on the shell behavior the problem of thinwalled cylinder SSS has been solved. The cylinder is stiffened with a ring of rectangular cross-section which is fastened to the butt-end symmetrically with respect to the medial surface. The equilibrium equations for the ring are accepted as Karmishin et al (1975). At given geometry of the cylinder and loading (R = H = 10 m, R/t = 1000, A = t, n = 2) the dimensions of transversal cross-section of the ring were varied (a is the ring width, b is the ring height ). The results of calculations are presented in Table 2. As in Table 1, here the maximum values of obtained displacements and forces are shown and the points where they were defined are marked: (1) - at the lower edge, (u) - upper edge of shell. The values Jr and .L are the moments of inertia of the ring in its plane and out-of-plane, respectively. The first line of the table contains the results of calculations when the ring is absent (free edge). It is seen from the table that the increase of the ring rigidity leads to the decrease of maximum radial deflections in the area of junction of the ring and the shell. At the same time it makes the normal and shear stresses rise. When the rigidity increases, the maximum values of these forces move from the point of junction with the shell toward the lower edge to which the self-balanced kinematic load is applied. Thus, when the ring rigidity increases the smooth transformation occurs from the bending stress state to the momentless one. It was also found that in real storage tanks it is almost impossible to restrict the radial displacements in the area of coupling of the shell with a ring. The stiffness of the ring has to be so high that it is hard to make such ring with accepted shape of cross-section. This can be explained by high level of shear forces emerging in the shell that are transferred to the ring and causes the bending. To stiffen the free edge of the cylindrical shell at the self-balanced longitudinal loading it is recommended to use roofs, rigid membrane coverings. If it is impossible, and if one has to use the rings of ri-

682 gidity (as for example in large-sized vertical tanks with a floating roof) the emergence of self-balanced loads on a shell should be restricted. Speaking about restrictions it should be pointed that these effects of non-homogeneity of radial displacements in the radial direction have been observed in storage tanks of large volume, reinforced with a ring of rigidity, in the process of filling up, i.e. when the foundations settle. TABLE 2 INFLUENCEOF THE RING STIFFNESSON DISPLACEMENTSAND FORCESIN THE SHELL Displacements and forces

Geometry of ring b

a

Jr" 102

N1

N2

S12

ml

M2 (~-M)

(u)

0.770 (1)

2.92 (u)

0.435 (1)

0.00641 (u)

0.0213

Jx" 104

~) 4.00

(u)

'0.1

0.5

10.4

0.417

39.3 (u)

29.8 (1)

46.6 (u)

17.2 (1)

0.603 (u)

0.201 (u)

0.2

1.0

1.67

6.670

3.2]. (u)

395 (1)

119 (1)

206 (1)

0.483 (u)

0.161 (u)

'0.3

1.5

8.47

33.80

1.81

1120 (1)

336 (1)

565 (1)

0.611 (1)

0.183

0.4

2.0

(u)

1630 (l)

488 (1)

816 (l)

1.164 (l)

0.344

0.40 (u)

1890 (l)

565 (l)

985 (u)

1.440 (1)

0.433

0.5

(u)

2.5

26.0 65.1

107.0 260.0

0.87

(1) (1) (1)

References Krasovsky V.L., Morozov G.V.(1997) The Stress-Strain State of Closed Cylindrical Shell Under Longitudinal Self-Balanced Loading of its Edge. Theoretical Foundations of Civil Engineering, Warsaw, 133-138 (in Russian). Krasovsky V.L., Morozov G.V. (1998) Stresses in a Wall of Steel Vertical Cylindrical Tanks under Foundation Subsidence. Lightweight Structures in Civil Engineering. Proceedings of International Colloquium, Warsaw, 131-136. Vlasov V.Z. (1949) General shell theory and it application in technique, Moscow (in Russian). Kolkunov N.V. (1987) Bases of elastic shell calculation, Moscow (in Russian). Karmishin A.V., Laskovets V.A., Machenkov V.I., Frolov A.N. (1975) Static and Dynamic of Thin Walled Shell, Structures, Moscow (in Russian).

Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

683

COUPLED INSTABILITY OF CYLINDRICAL SHELLS STIFFENED WITH THIN RIBS (theoretical models and experimental results)

A. I. Manevich Department of Theoretical Mechanics and Strength of Materials, Ukrainian State University of Chemical Engineering, Dniepropetrovsk, 49005, Ukraine

ABSTRACT Coupled instability of longitudinally stiffened cylindrical shells under axial compression is considered. The attention is focused on features of theoretical models for shells with thin ribs of large bending stiffness for which the buckling and postbuckling behaviour can be strongly affected by local displacements of stiffeners. The interaction of overall modes with two types of local modes (> and ) is taken into account. The exact analytical solution of the linear buckling problem with a plate model for stiffeners, the Koiter's asymptotic method and the amplitude modulation approach (in a specific form) are employed in the analysis of the coupled buckling. Comparison of the theoretical solution with the experimental data is carried out, and limitations on the use of the asymptotic theory for prediction of the failure loads are discussed. KEYWORDS Stiffened cylindrical shells, buckling, coupled instability, experimental investigations.

INTRODUCTION The first theoretical and experimental investigations of coupled instability of stiffened shells have been carried out in 70-s (Koiter (1976), Byskov .& Hutchinson (1977), Manevich A.I. et al. (1971), Manevich (1977, 1979) and were refined in 80-s (Byskov & Hansen (1980), . Hue et al (1981), Manevich (1983, 1985). In this paper we focus our attention on cylindrical shells stiffened with relatively small number of thin-walled stringers of large bending stiffness (welded, riveted), behavior of which may essentially differ from that of shells stiffened with a large number of weak fibs. Due to the ribs large stiffness such shells can carry out considerable load after the skin local buckling. But for these shells it is principally important to account for effects of the ribs local buckling.

684 In the paper are briefly summarized results of the theoretical investigations of stability of cylindrical shells stiffened with longitudinal thin ribs (Figure 1) which were carried out by the author in the framework of the Koiter's theory of initial postbuckling behavior. The exact analytical solution for the linear local buckling problem allowing for the local deformations of stringers is used. The theoretical conclusions are compared with qualitative and quantitative results of experiments which had been carried out by the author and his collaborators. The experiments confirm the main conclusions of the coupling buckling theory, but the asymptotic approach is found to be inadequate for prediction of ultimate loads of shells with preliminary local ~skin buckling)) since the theory of initial postbuckling behavior does not account for some important features of buckling process in these shells.

Tu7

Figure 1: Stiffened cylindrical shell and cross-sections of ribs THE L I N E A R P R O B L E M

In the first investigations of the coupled instability in stiffened shells rather simplified theoretical models for analysis of buckling modes, especially, local modes were employed; later they were gradually refined. Byskov & Hutchinson (1977) used the sinusoidal approximations of local modes, i.e. the stiffeners only determined the periodicity of the local mode in the circumferential direction, their stiffness was not taken into account. Byskov & Hansen (1980) took into account the ribs torsional stiffness. Hue et al (1981) accounted for the axial rigidity and the eccentricity of the fibs, in order to describe the redistribution of stresses between the skin and the fibs. But the ribs were considered as bars which do not lose their stability (parametrical terms in the equations of equilibrium were neglected). An exact solution of the linear local buckling problem for a stiffened shell as a skin-fib system allowing for the local buckling of the ribs themselves has been obtained in papers (Manevich (1983), Manevich (1985)). Side by side with the bar model for ribs (with account for the parametrical terms in stringers equilibrium equations) the plate model for stringers of rectangularand T- cross section was employed. For shells with thin ribs one should discern two types of local modes (Figure 2): 1) ~skin local buckling, modes initiated by instability of the skin between stiffeners (with wavelength of order of distance between stiffeners), 2) ~rib local buckling)) modes initiated by buckling of the stiffeners walls (wavelength of order of the rib height).

685 The bar-beam model of ribs is adequate for the local modes of type 1 (critical stresses and modes obtained with using the plate model and bar model differ slightly, see upper drawing at Fig. 2). But for the local modes of type 2 only the plate model is applicable. The lower drawing in Fig. 2 illustrates large differences in the buckling modes obtained with the bar model (displacements of the skin between the ribs are large) and the plate model ( displacements of the skin are negligible). The bar-beam model can not describe real modes and therefore considerably overestimates the local critical stresses crx. In Fig. 3 the dimensionless local critical stresses cr~ = O-xl0 3 / E are presented for the shell R/L=0.5, R/h=300, ks=60, co=td/(ha)=0.2 (R, L, h = radius, length and thickness of the shell, t, d = rib thickness and height, ks=the number of fibs, a = distance between the fibs) via the number of longitudinal half-waves m for various values of ribs ((slenderness parameter)) t / d = 0.10; 0.08; 0.06; 0.05; dark points (solid lines) correspond to the plate model, light points (dashed lines) to the bar model (with account of the parametrical terms). G

J

E 10

. . . . beam model plate model

4.s

~

./0.08

\

,,/'/,-" ' ',

~

1-"skin local buckling"

,

3.s

d ~

~

010

0.06

,~" " ",'

~

,,,

..y: 2 -"rib local buckling" ~, ~ ~, ~,r ..P ~- '" ''d ~" ~,-,b ~ ~ .~ ~ ~

2.6 3

Figure 2: Two types of local modes - - - - - the plate model, ..............the . bar-beam model.

5

7

O

ill

Figure.3: Comparison of local critical stresses for beam model and plate model of ribs. L/R=0.5, R/h=300, ks=60, co=tdl(ha)=0.2, t / d = 0.10; 0.08; 0.06; 0.05.

When ribs are relatively thick ( t / d > 0.07) both the models result in nearly coincident results; for thinner ribs the bar model overestimates cr~ and this overestimation increases with increasing m. THE NON-LINEAR COUPLED BUCKLING P R O B L E M The Koiter's theory of initial postbuckling behavior results in the following expression for the potential energy of a shell in the case of two interactive modes - overall (i=l) and local (i=2): 1 22 1 ~2_) 1 2 P=~ao + ~ a 1 ~ ' 2 ( 1 - 2 1 +sazr"~(1-~-2)+a122G,~'22 + 1 + 4 allll

1

, 2

~': + 4 a 2 2 2 2 ~"~ - a l ~"1 ffl ~

, 2

(1)

+ 0 2 ~'2 ~'2 "~2

where ~ is the amplitude of ith mode (normalized, in given case, by the condition of equality of the maximal deflection to the shell thickness h), ~i is the amplitude of initial imperfection in ith mode, 2

686 is the load factor, ,;1,i is the critical value of 2 for ith mode (the periodicity and symmetry conditions are accounted for in Eqn.1). Coefficients a~,,, a2222 govern the postbuckling behavior at one-modal overall and local buckling and are determined by solving the nonlinear problems for one-modal buckling, coefficient a m (or all22, if am=0) governs the coupled buckling. Equilibrium paths are determined by the equations OP/cT~i =0. These equations together with the condition of vanishing jacobian I=0 determine the limit load ';[,(~'1,~'2 )as a function of amplitudes of the initial imperfections. Here we consider in detail only the coupling buckling problem. There exist two principal approaches to the analysis of the interaction between the overall and local modes in stiffened shells, both based on W. Koiter's works: 1) the asymptotic approach, 2) the method of modulation of the local mode amplitude. When the asymptotic method is employed (Byskov & Hutchinson (1977)) the cubic terms in the potential energy vanish due to conditions of periodicity and symmetry ( a~::=0), and the mode interaction can be accounted for only in the second asymptotic approach (by coefficient az~:2) through solving a boundary problem for the mixed second order mode. But the presence of a cluster of short-wave local modes with close critical stresses makes this boundary problem badly conditioned (Koiter (1976)). The amplitude modulation method takes into account that overall deflections either promote or suppress the local buckling depending on the direction of the overall deflection. This method deals with the amplitude of the local mode as a slowly varying function, thereby allowing for the mode interaction already in the first nonlinear approach ( a~22s0). It can be expected (and this is confirmed by experiments (Manevich A.I. et al. (1971), Manevich (1979)), that in the case of very short wavelengths (in particular, for the ~ribs local buckling>> modes) the tendency to the coupled buckling exhibits itself only on portions of a shell with certain direction of the overall deflection. When the overall buckling is oscillatory then zones on the shell surface with different character of buckling - coupled and uncoupled - alternate. So the following simple approach can be proposed (Manevich (1983)) which may be considered as a limit variant of the amplitude modulation method. We assume that the coupled buckling occurs only on portions of shells with certain direction of deflection in the overall mode, i.e. on the half of the shell surface (Fig.4). We remain the procedure of the asymptotic method in calculating the coefficient a m , only the domain of integration in corresponding integrals is to be changed. If the sign of a122is found to be negative then the mode interaction occurs namely in the chosen zones; but when a~22 is positive we have to change the domain of integration (to take other half of the shell surface) that leads only to the change of the sign of a~22(in this case the coupled buckling takes place on the shell portions with the opposite direction of the overall deflection). Numerical analysis of the solution (Manevich (1983)) shows that in the case of external stringers the unstable postbuckling behavior is pertinent to zones with radial overall deflections directed inward of the shell; in the case of internal stringers it remains valid for relatively thick fibs, but for thin ribs the coupled instability is pertinent to zones where the overall deflection is directed along the external normal. In intermediate t/d range (including the domain of optimal parameters) values of a~22for internally stiffened shells are small. So the conclusion can be drawn about much more pronounced mode interaction and imperfection sensitivity of shells with externally spaced stringers. This effect is similar to the effect which has been revealed by Hutchinson & Amazigo (1967), but it has another nature (in the later paper the scheme was employed and the mode interaction was not considered).

687 The second important feature of the nonlinear behavior of shells with thin ribs is the presence of two local minima in dependence of the limit load on m (if to account for the interaction between the overall mode and each local mode separately). The first minimum is determined by the minimum of the linear critical stresses cr~ and usually corresponds to a > mode, the second by the maximum of the coefficient iau~t and corresponds to a mode. The lam! values turn out to be much greater for modes with wave-lengths of order of stringer height than for those with wave-lengths of order of the distance between stringers. At small t/d the second minimum is found to be global, even in the cases of higher o-~ values. In Fig. 5 the dimensionless limit stresses of coupled buckling versus modes. Thus the local modes with prevailing displacements of ribs can govern the carrying capacity even in the cases when the linear critical stresses for them are higher than for >.

I 3,fi

3,6

3

3

2

2

0,04 0,06 0,08 0,1 t/d a

.-"

0,04 0,06 0,08 0,1 t/d b

Figure 4: Alternation of zones with 0,. Later the value 0, for a cylindrical shell was refined with account for the stringer torsional stiffness (Stephens (1971)) and finite longitudinal stiffness and eccentricity of the ribs (Hue et al (1981)). The upper bound of the range of stable postbuckling is found to be higher (in the examples considered - up to 0.9). For the shells with stiff thin-walled stringers at construction of the second order local modes the possibility of"rib local buckling" is to be accounted for. A solution of the problem with account of the parametrical terms in the equations of equilibrium of stringers was obtained in (Manevich A.I. (1989)). In the examples considered the 0o value was found to be slightly lesser in comparison with results (Hue et al (1981)), but quantitatively the effect is small (0.=0.85-0.89). Shells stiffened with heavy fibs, as a rule, have a small number of them, and the Koiter's parameter usually is large: 0. >1. So for these shells coefficient a2222 Cro); B - the shells with much lower local critical stress: crL ) the discrepancies between the experimental and theoretical limit loads remain considerable. The values of Koiter's parameter /9 for these shells lay in the range /9=0.87 - 1.59. As we noted above, this values correspond to negative values of the postbuckling coefficient a2222,i.e. to the unstable local postbuckling behavior. So the theoretical limit load 2, accordingly to the Koiter's theory is lower than the local critical load 22. However for many shells the experimental limit stresses were higher than the linear local critical stresses crL. As a rule, they lay between o"L and linear overall critical stresses Cro: crL < O'lim < o"o . The buckling process in all these shells was rather complicated. In Fig. 7 typical diagrams of deforming are presented for the shells with internal (a) and external (b) stringers. At first the local buckling mode appeared (the > mode - the stringers were only twisted). After a small decrease the load increased again until the overall buckling occurred. In the process of increasing load the local mode usually underwent transformations, and new local modes could appear, with small falls and subsequent rises of the load. Moreover, after the appearance of the overall mode the load can again rise ( at internal stringers) by 10-20%, in spite of a partial loss of rigidity, until the failure load was reached. The theory of initial postcritical behavior can not describe such a buckling process. It can only pretend to predict the first critical point ( local skin buckling) in dependence on imperfections and the initial slope of the postbuckling path. If one would constrain himself with this two items then the Koiter's theory turns out to be adequate. Local buckling always happened with a snap and a fall of the load, in concordance with the theory prediction. However after reaching certain local deflection this mode became stable. The theory of initial postbuckling in the first and second approximations is not able to predict such changes in character of the postbuckling. It follows that this theory, basically, can not predict the limit loads of shells, if the first critical point does not coincide with the failure load or is not a stable point. Thus the Koiter's asymptotic theory which correctly describes the initial postbuckling, can not be used for prediction of failure loads of rather wide class of stiffened cylindrical shells in which the buckling process includes several stages (transformations of an initial mode, the appearance of new modes with unstable initial and stable following behavior and so on).

L

kN

7,

kN

o retail buckl~ng~,,~

/buckhng ! r 0

10

~

~ ~

overall

~ckl~ng

\

2R=143ram, h=O.19 ram, L=130 ram, 24 internal stringers 2.3x3.0x0.234 0,1

0.2

0.3

0.4

0.5

EL, mm

0

0,1

0.2

0.3

0.4

0.5

,A ./.,~ mm

b Figure 7: Typical diagrams of deforming stringer-shells with internal (a) and external (b) stringers

691

References Brush D.O. (1968). Imperfection Sensitivity of Stringer Stiffened Cylinders. AIAA 9'. 6:12,. 24452447. Byskov E. and Hutchinson J.W. (1977). Mode Interaction in Axially Stiffened Cylindrical Shells. A/AA J. 15: 7, 941-948. Byskov E. and Hansen J.C. (1980). Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction. J. Struct. Mech. 8:2, 205-224. Hue D., Tennyson R.C. and Hansen J.C. (1981). Mode Interaction of Axially Stiffened Cylindrical Shells: Effects of Stringer Axial Stiffness, Torsional Rigidity and Eccentricity. 3'. Appl. Mechs, Trans. of ASME, Ser. E 48:4,. 915-922. Hutchinson J.W.and Amazigo J.C. (1967). Imperfection sensitivity of Eccentrically Stiffened Cylindrical Shells. AIAA 3". 5, 3,. 392-401. Koiter W.T. (1956). Buckling and Postbuckling Behaviour of a Cylindrical Panel under Axial Compression. NLR Report N 476. Amsterdam. Koiter W.T. (1976). WTHD Report N 590 (Delft Univ. of Technology, 41 p. Kostyrko V.V., Krasovsky V.L. and Manevich A.I. (1983). About the Effect of Stiffeners Parameters on the Carrying Capacity of Compressed Shells. Civil Engineering and Design of Buildings 3, 5255. Moscow (in Russian). Manevich A.I. (1977). Coupled Loss of Stability of a Longitudinally Stiffened Cylindrical Shell. Hydroaeromechanics and Theory of Elasticity 22, Dniepropetrovsk, DGU, 104-114 (in Russian). Manevich A.I. (1979). Stability and Optimal Design of Stiffened Shells. Kiev-Donetsk, Vyshcha shkola, 152 p.(in Russian). Manevich A.I. (1983). Loss of Stability of Compressed Longitudinally Stiffened Cylindrical Shells at Finite Displacements with Account for Local Buckling of Ribs - Plates. Mechanics of Solids, Izv. of USSR AS 2, 136-145. Moscow (in Russian). Manevich A.I. (1985). Stability of Shells and Plates with T-cross-section Ribs. Civil Engineering and Design of Buildings 2, 34-38. Moscow (in Russian). Manevich A.I. (1989). Nonlinear Theory of Stability of Stiffened Plates and Shells with Accounting for Mode Interaction. Thesis on Dr. Sci. degree. Dnepropetrovsk (in Russian). Manevich A.I. et al. (1971). An Experimental Investigation of Stability of Longitudinally Stiffened Cylindrical Shells under Axial Compression. In: Design of Spatial Structures 14, Moscow, Stroyizdat, 87-102 (in Russian). Stephens W.B. (1971). Imperfection Sensitivity of Axially Compressed Stringer Reinforced Cylindrical Panels under Internal Pressure. AIAA J.. 9: 9, 1713-1719.


Third International Conferenceon Thin-Walled Structures J. Zara~, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

693

INSTABILITY MODES OF STIFFENED CYLINDRICAL SHELLS

J. Murzewski Faculty of Civil Engineering, Politechnika Krakowska, 31-155 Krakrw, Poland

ABSTRACT

Circular cylindrical shell with circumferemial stiffening frames (tings) is subject to axial force Ns. Two-dimensional state of stress occurs in the shell because of stiffening action of the frames. Thanks to this fact a small enhancement ~g of resistance of the elastic cylinder may be taken into account; however, much stronger reduction 7~of the resistance may happen if circumferential plastic hinges arise along the stiffening rings and in the middle of the cylindrical segment (Figure 1). Quite different equations have been proposed for local elastic instability of thin walled cylindrical shells. Coupled plastic and elastic instability modes are taken into consideration. Probability principles are applied to assess interaction of plastic and elastic instability limits and a coupled reduction factor cp is derived as a function of shell slenderness r/t and stiffener spacing a/t (Figure 2). Severe reductions of resistance of transversally stiffened shells with not too thick walls indicate that design and construction of large diameter cylindrical shafts of towers, guyed masts etc. without longitudinal stiffeners rather shall be avoided..

KEYWORDS

critical strength, cylindrical shell, elastic-plastic buckling, partial safety factor, plastic hinges.

694 INTRODUCTION

Certain of the background material relating to behaviour of stiffened cylinders under axial compression and/or bending have been published more than 40 years ago (Becker, 1958). Perfect elastic behaviour of the stiffened circular cylinder is a classical problem presented by Timoshenko (1962), Flugge (1967) etc. In the 1960-ies some novel concepts about plastic instability were derived by the author from theoretical considerations and experimental tests as results of research projects on steel shell galleries (Murzewski, 1964, 1967) and tubular guyed masts (Murzewski, 1968,1969). Coupled instability ideas came before the proper time to attract interest and discussions of the specialists. Code-writers and designers were not disposed to study non-classical plastic problems and the new rules has not been implemented. There is some progress in recent publications and standards on structural design, but a new Polish draft on design of steel towers and masts (prPN-B-03204) is still conservative. No advantage of plastic behaviour is allowed; however, reduction of buckling resistance with respect to both the ideal Euler's critical stress and the yield strength has been always taken into account. Now, computer facilities and a better knowledge of coupled instabilities may help to reassess instability modes of cylindrical shells. A very simple problem is taken into consideration in this paper: a transversally stiffened cylindrical shell (tube) subject to axial compression (Figure 1).

Figure 1: Stiffened cylindrical shell before (a) and after plastic hinge formation (b)

695 STABLE ELASTIC BEHAVIOUR

First, two-dimensional state of stress is assumed in a perfectly elastic shell under axial compression. Normal stresses ox, fly of the cylindrical shell in axial (x) and circumferential (y) directions (Figure 1) are balanced by axial tension

fff

of the stiffening frame. The stresses are derived from:

a) the Hooke's law for the shell and the frame

gx =(t~x -Vt~y)/E,

E:y =(Oy-VOx)/E , 8f --fff/E;

(1)

b) compatibility condition between the shell and the frame

9

gv = g f ,

o

(2)

c) integral equilibrium equations in a cross-section and an axial section of the cylinder atOy +Aft~f =0 ;

Ao x =N s,

where E = 205 GPa,

(3)

v=0,3 - elastic constants for structural steel,

Af, a - cross-section area of stiffening frames (rings) and their spacing (length of the segment), A = r~ r t, r, t - cross-section area, middle surface diameter and thickness of the cylinder, Ns [kN] - the compressive axial force, Ns > 0.

Solution of the set of linear equations (1), (2), (3) is as follows

~

-

N 2rtrt

-o

o

'

o

o o l+ot(1-v

gx = where

E

l+ct

Y

va l+a

= - ~ o

2)

o o '

ct = Af/a t - non-dimensional frame spacing,

v 9 l+ot ~176

o,

o ~ = ~

(4)

v

~ Y = ~ ~E = ~l+0t f

;

(5)

o0 = Ns/A >0 - the compressive stress.

Length of the cylinder decreases and radius of the shell increases Aa = - a z ~ ,

Ar = ~ r

(6)

696 The effective stress, so called according to the Huber-Mises yield criterion, is reduced thanks to the two-dimensional state of stress VOt,

VO~

oefr = o o 1 - 1 - ~ a +

=WOo 1.0, the meridional profile is actually part of the generalised ellipse determined by Eqn. (1). When variables nl or n2 approach relatively large values, for example, nl = n2= 5.0, the meridional curvature will approach to a form of straight line and thus the geometry of the elliptical shell will approach to the geometry of cylindrical shell. Since elliptical shells possess same truncation radius R0 and axial length Lo as that of the cylinder, the wall-thickness t of the elliptical shells must be thinner than cylinder wallthickness to due to the surface' s bowing-out. For example, for cylinder of IxJR0 = 1.0, R0/t0 = 33.33

711 and b/Lo = 1.0, the derived wall-thickness ratio t/t0 is 0.67 for (nl,n2) = (1.0,2.0) and 0.98 for (nl,n2) = (10.0,2.0). .-200 .

~

i

i ,

~

,

J.

,.

_,.

-

~

!

at'ha 9 at 9 nt - I.S

-lS0

"

n, 9nt 92.0 nt 9

-t00

i I

-50 0

It~

R

1110 150

Figure 1: Geometry of cylindrical (Fig. 1a) and elliptical barrelled (Fig. l b) shells.

Figure 2: Different meridional profiles of elliptical shells (Lo/Ro =2.0, b/Lo = 1.0, nl = n2)

In all pertaining analyses, the boundary conditions (BC) are applied to the top and bottom openings of the shell structure and the only non-zero BC is the axial movement of the shell's top and bottom edges. All shell structures are subjected to external hydrostatic pressure and therefor, the loading type is actually combination load of pure external pressure P plus axial compressive load with edge loading density of RoP/2. Mild steel material properties are assumed and elastic-perfectly plastic material with modules of elasticity E = 207 KN/mm 2, yield point Ovv = 300 N/mm 2 and Poisson's ratio v = 0.3 are employed throughout numerical calculations.

3 NUMERICAL RESULTS BOSOR5 [10] software has been adopted as the main numerical tool in this paper to perform collapse and bifurcation analyses. ABAQUS [11] software is also used for some selected points. Since collapse and non-axisymmetric bifurcation are the two possible failure modes of cylindrical and elliptical shells, both non-linear axisymmetric stress (collapse) analysis and bifurcation analysis from both software were performed to predict the failure modes and loads. The meridional curve was modelled by 91 small-segments in BOSOR5 and both 3-node axisymmetric line element SAX2 and doubly-curved 8-node shell element $8R5 were used for FE modelling in ABAQUS.

3.1 Buckling Pressure of Cylinder (Lo/Ro =1.0 and Ro/to --33.33) Numerical results of cylindrical shell from both analytical codes are presented in Table I. It is seen that the two methods predicted almost same collapse pressures. As for bifurcation failure, ABAQUS can not predict bifurcation failure mode while BOSOR5 predicted that the Lo/Ro =1.0 and R0/to =33.33 cylinder will lose its load carrying capacity by bifurcation with 7 circumferential waves in eigenmode. TABLE 1 PREDICTIONSOF BIFURCATION[ COLLAPSEPRESSURES Pbif (MPa)

Peon (MPa)

HEImH ABAQUS

i0.52

10.38

* Value in bracks denotes the circumferentialwave numbers in eigenmode

712

3.2 Buckling Pressure of Elliptical Barrelled Shells Due to the fact that the geometry profile of the elliptical barrel-shaped shells could only be determined once all three variables n~, n2 and b have been decided, enormous calculation efforts are required to draw the whole picture of the performance of the elliptical shells. Subjected to the limitation of the calculation resources, it is unpractical to fulfil this target. Thus, additional restrains were introduced to transform the 3-variable-involved problem to 1-variable-involved (by setting variable b unchanged and nl = n2) and 2-variable-involved (by setting variable b unchanged) problem.

3.2.1 One-Dimensional Analysis of Elliptical Shells (b/Lo = 1.0 and nl

=

n2)

For cylindrical shell of Lo/Ro =1.0 and R0/to =33.33, by setting pararheter b unchanged, the only variable nl(= n2) is needed to determine the geometry of the elliptical barrel-shaped shells in onedimensional analysis. Numerical results obtained from BOSOR5 are shown in Fig. 3 and the point of wall-thickness t = 3 mm corresponds to the cylindrical shell structure. It is seen from Fig. 3 that, without adding additional material and without losing the cylinder's axial length, the load carrying capacity of the cylindrical shell could be improved simply by re-arranging the material distribution. The peak point (nl = n2 = 2.1) fails at pressure 14.30 MPa which is 41% higher than the ultimate failure pressure of mass-equivalent cylinder. It is worth to point out that, with the continuous changing of variable ni = n2, the failure modes of the elliptical barrelled shells also change. Cylindrical shell and elliptical shells of n~ = n2 > 2.5 fail by bifurcation; elliptical barrels of nl = n2 < 2.5 fail by collapse, including the optimal (peak) point. 15.0

4.0.'

~.)

-

x

~s

10.0

3.0

i j

~ m

112 2.5

5.0

2.0 Collapse . . . . . .

0.0

i

2.0

2.2

i

B ifurcation

i

2.4 2.6 Wall-thickness t (ram)

1.5

, i

2.8

3.0

Figure 3: Variation of bifurcation/collapse load for elliptical shells of b/L0 = 1.0, n~ = n2

1.6 i

1.0

i 1.5

i 2.0

!

2.5

i 3.0

! 3.5

4.0

Ill

Figure 4: Contour plot of elliptical barrelled shell's performance (b/Lo = 1.0)

3.2.2 Two-Dimensional Analysis of Elliptical Shells (b/Lo = 1.0) In this section, the imposed restrain in Section 3.2.1of setting n~ = n2 is freed and the aim is to find the optimal structural geometry in the whole searching domain of (nl, n2). The optimal point is defined as such that the load carrying capacity at this point is higher than all its surrounding points. The searching process was performed like this: after a coarse study at some even-spaced points in (nl, n2) domain, finer computations will only concentrate on the region where optimal point could possibly occur based on the coarse study results. In this paper, four values of variable b were evaluated, i.e., b/Lo= 0.75, 1.0, 1.25 and 1.5. Only results of b/Lo= 1.0 are discussed in detail. 2-D contour plot was achieved to reveal the elliptical shells' performance of b/L0 = 1.0, shown in Fig. 4. The optimal point happens at (nt, n2) = (2.2, 2.0) and the magnitude of failure pressure is 14.71 MPa, which is 46% higher than that of the cylinder. Also, the failure mode at this optimal point is collapse. Fig. 5 illustrates the squashing process of the optimal barrel by external

713 hydrostatic pressure. The horizontal axis represents the axial displacement of the top (bottom) edge and the vertical axis represents the applied external pressure. The yield pressure of the optimal elliptical barrel at which the structural stress level at some parts of the structure exceeds the yield strength of the material is Py = 9.17 MPa. Deformed profiles of the meridional surface also show the spread of bending in the process of increasingly applied external pressure. Three other values of variable b were checked to find their respective optimal elliptical barrelshaped shells. The results are listed in Table 2 with information as where these optimal failure pressures were obtained and Fig. 6 illustrates how optimal pressures various versus variable b. TABLE 2 FAILURE PRESSURES OF OPTIMALBARREL-SHAPEDSHELLS b/Lo 0.75 1.00 1.25 1.50 .

.

.

.

.

.

.

.

.

.

.

.

.

.

....

Popt (MPa) 14.53 14.71 14.68 14.33 .

.

.

(nbn2) (4. 6 , 1.9) (2.2, 2.0) (1.29, 2.09) (1.005, 2.05)

.

.,,

..

n3 (mm) 114.5 115.0 113.3 111.7

1.48 16.0 It

1.46 ~, 12.0

== o.

Q. 9~

8.0

1.44

n

ir m

X 1.42'

4.0

0.0

0

0.01

0.02 0.03 0.04 Axial Disp. (ram)

0.05

0.06

Figure 5: Pressure versus end-edge axial displacement of optimal elliptical shell (b/L0 = 1.0). Magnitudes of yield and collapse pressures are marked and meridional deformed profiles are given

1.40 0.70

0.90

1.10 blLo

1.30

1.50

Figure 6: Normalised optimal pressures (Popt/Pcylinder) versus variable value b/Lo

4 E X P E R I M E N T A L DETAILS Since the load carrying capacity of cylindrical shells under external hydrostatic pressure can be improved by bowing out through generalised elliptical profiles, it is decided to perform several evaluation experimental tests to prove the numerical predictions.

4.1 General Information Four experimental specimens were manufactured, two of them are identical cylindrical shells and two are identical optimal elliptical barrel-shaped shells of b/L0 = 1.0. All specimens were CNC machined with top and bottom internal flanges and stress relief was performed at 650~ for 1 hour under vacuum conditions. The nominal dimension of the cylindrical specimen is L0 = R0 = 100mm and to = 3mm. Fig. 7 shows the pictures of finished specimen S 1-1 (cylinder) and El-1 (optimal elliptical barrel).

714

Figure 7" CNC machined specimens. (a) cylinder; (b) optimal elliptical barrel (b/Lo = 1.0) Wall thickness of all testing specimens was measured along 20 equally spaced circumferentials at 18 ~ intervals and 10 equally spaced meridians of arc-length by ultrasonic probe. Table 3 shows the average, minimum and maximum values of the wall thickness of all specimens. The shape measurement of testing specimens' external surface was conducted by laying each testing specimen on a rotating table and taking relative readings from two gauges. Measurement results revealed that the maximum initial radial variation was 4.1%. TABLE 3 WALL THICKNESS MEASUREMENT tave (mm) 3.02

Specimen SI-1 Sl-2

El-1 El-2

.

.

.

.

3.05

2.57 2.56

tmin (mm) 2.96 2.99 2.50 2.46 ,

tmax (mm) 3.07 3.10 2.67 2.65

. .

,

.

.

.

.

.

Tensile tests on round bars were carried out to determine the mechanical properties of the material. Four coupons were cut from longitudinal direction of the steel tube and test results showed that the average upper and lower yield point of material was 348.13 MPa and 329.40 MPa respectively. During evaluation test, each specimen was mounted in a pressure chamber and external pressure was supplied by hydraulic pump and measured by pressure transducer. Copper pipe connecting inside of the full-oil-filled specimen to an oil container placed on an electronical scale was used to measure the specimen's inside volume change during load applying process. At each pressure increment, the weight of oil being pressurised out from inside of each specimen was recorded when no more oil drops from the copper pipe was observed. In case material plasticity happens, the continuous material deformation would cause continuously oil dropping. It was decided that the reading of the weight should be taken 20s after each incremental pressure was met. Specimen failure (bifurcation or collapse) was considered to have taken place when such following happens: (1) sudden decrease in pressure of the chamber; (2) an audible sound or, (3) sudden large amount of oil being pressurised out. 4.2 Results and Discussion

Experimental data of all testing specimens are given in Table 4 together with numerical predictions from BOSOR5 and ABAQUS. Average lower yield strength, average wall-thickness, average radial diameter and average shell axial length measurement values were used in numerical calculations. Photographs of failed specimens S 1-1 and E 1-1 are shown in Fig. 8. First, let's look at the failure modes of testing specimens. Specimen S 1-1 and S 1-2 failed by bifurcation with 4 wave numbers across the circumferential direction while BOSOR5 predicts 7 waves. By measuring the dimension of each wave of the failed testing specimen, it revealed that the circumferential length of the 4 waves of specimens S 1-1 and S 1-2 exactly accounts for 4/7 of

715 TABLE 4 COMPARISON OF EXPERIMENTAL & ANALYTICAL RESULTS

Sample

BOSOR5 Results

Experimental Results Pexpt (MPa)

PBOSOR5 (MPa)

11.66 11.58

11.17 (7) 11.23 (7)

16.96 16.82

15.80 15.66

SI-1 S1-2 El-1 El-2

Pexpt-PBosOR5 Pexpt 4.2%

ABAQUS Results

PABAQUS (MPa)

3.1%

11.60 11.71

6.8% 6.9%

15.87 15.78

PexP:-PABAQUS Pexpt 0.5% - 1.1%

6.4% 6.2%

Figure 8: View of failed testing specimens. (a) cylinder S 1-1; (b) optimal barrel E 1-1 whole circumferential length. In other words, the above cylindrical testing specimens just failed by bifurcation mode BOSOR5 predicted. The reason why perfect bifurcation mode (full-wave numbers) did not happen is due to the unevenly distributed wall-thickness. As for specimens E 1-I and E 1-2, they all failed by collapse and it could be seen from Fig. 8 that, apart from inward radial deflection due to the applied external pressure, both specimens also deformed along the meridional direction and absorbed large amount of strain energy for the bending deformation. Surface cracks even happened at the outside surface of specimen E 1-1 although it was not a wall-through one. Next, let us consider the difference of the experimental results and the numerical results. It is seen from Table 4 that the percentage errors between experimental results and numerical results vary from -1.1% to 4.2% for bifurcation failure specimens (cylindrical shells) and from 6.2% to 6.9% for collapse failure specimens (optimal elliptical barrelled shells) respectively. Although the overall numerical predictions are a bit conservative compared with experimental results, all predictions from both analytical methods (BOSOR5 and ABAQUS) exhibit direct engineering-usevalue.

5 CONCLUSION Numerical and experimental studies are presented for the bucking analysis of metallic cylindrical and mass equivalent elliptical barrel-shaped shells subjected to external hydrostatic pressure. In the study, numerical software BOSOR5 and FE software ABAQUS were employed to perform bifurcation buckling and collapse analyses and experimental studies were conducted according to numerical results by CNC machined testing specimens. Main results obtained may be summarised as follows: (1) Buckling strength of mild steel cylindrical shells under external hydrostatic pressure can be improved by bowing-out without increasing the material volume (weight) while at the same time keep the original axial length unchanged. For the cylinder geometry of Lo/R0 =1.0, Ro/t0 =33.33,

716 the buckling pressure of the optimal elliptical barrel-shaped shell can be 46% higher than that of the cylindrical shell. (2) The finding procedure of the optimal elliptical shell geometry presented in this paper involved only 2 variables although 3 variables are actually associated to determine the elliptical surface profile. Due to the huge amount of calculation efforts needed to cover all possibilities, appropriate optimisation methods must be introduced and, at this time, optimisation technique called TABU search method is now being tested. (3) Four experimental specimens made of mild steel were CNC machined according to numerical computation result. Experimental result proved that BOSOR5 predicts failure mode more accurate than ABAQUS and agreed well with numerical predictions of bifurcation buckling pressure; for structural collapse, both software predict nearly the same collapse pressures and the predictions are a bit conservative.

ACKNOWLEDGEMENTS The author wishes to thank ORS Award from Committee of Vice-Chancellors and Principles of the Universities (UK) and Hsiang Su Coppin Memorial Scholarship (The University of Liverpool) for their financial supports.

REFERENCES

1. Kruzelecki J. (1997). On optimal barrel-shaped shells subjected to combined axial and radial compression. In: Gutkowski W, Mroz Z, editors. Structural and Multidisciplinary optimisation. IFTR P A N - WE Lublin 1,467-472. 2. Blachut J. (1987). Optimal barrel-shaped shells under buckling constraints. AIAA 25, 186-188. 3. Blachut J. (1987). Combined axial and pressure buckling of shells having optimal positive gaussian curvature. Computers & Structures 26(3), 513-519. 4. Blachut J. and Wang P. (2000). Buckling of barrelled shells subjected to external hydrostatic pressure. ASME PVP Codes and Standards 407,107-114. 5. Zintilis G. M. and Croll J. G. A. (1983). Combined axial and pressure buckling of end supported shells of revolution. Eng. Struct. 5,199-206. 6. Lukasiewicz S. and Wawrzyniak A. (1976). Stability of shells revolution with slightly curved generator under complex load. Mech Teor Stos 14, 535-545. 7. Odland J. (1981). Theoretical and experimental buckling loads of imperfect spherical shell segment. J. of Ship Research 25(3),201-218. 8. Schmidt H. and Krysik R. (1991). Towards recommendations for shell stability design by means of numerically determined buckling loads. In: Jullien J. F., editors. Buckling of Shell Structures, on Land, in the Sea and in the Air, Elsevier Appl. Sci., London NY, 508-519. 9. Blachut J. and Wang P. (1999). On the performance of barrel-shaped composite shells subjected to hydrostatic pressure. In: Toropov V. V., editors. Engineering Design Optimisation, Proc. Of I st ASMO UK/ISSMO Cofer., 59-65. 10. Bushnell D. (1976). BOSOR5-program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Computers & Structures 6, 221-239. 11. Hibbit H. D., Karlsson B. I. and Sorensen E. P. (1998). ABAQUS User's Manual, Version 5.8, Pawtucket, R102860-4847, USA.

Section XII ULTIMATE LOAD CAPACITY



719

EXPERIMENTAL TECHNIQUES FOR TESTING UNSTIFFENED PLATES IN COMPRESSION AND BENDING M. R. Bambach 1 and K. J. R. Rasmussen I 1Department of Civil Engineering, Sydney University, Australia

ABSTRACT

Details of a dual-actuator rig developed for testing rectangular plates supported on three sides, with the remaining (longitudinal) edge free, under combined uni-axial compression and in-plane bending are presented. Particular attention is given to ensuring a constant strain gradient at the loaded ends, as opposed to a constant load eccentricity, in order to determine the post-buckling behavior and ultimate load and moment capacities of unstiffened thinwalled elements. Strain gradients varying from pure compression to pure bending are facilitated. Attention is also given to ensuring simply supported boundary conditions, and the methods used for anchoring the tensile stresses that develop at the loaded edges as a result of large plate deflections. Details of the methods for controlling the applied displacements are given, for which a system of four laser displacement devices were employed in order to achieve the required strain gradient. The operation of the rig is verified against established theoretical solutions. KEYWORDS

Plate testing rig, unstiffened plate elements, stress gradients, dual-actuator control INTRODUCTION

Open thin-walled sections consist of stiffened and unstiffened component plates. If these elements are sufficiently slender, the section will locally buckle at a load significantly less than the ultimate load carrying capacity. The section will continue to resist load after local buckling due to the redistribution of longitudinal stress from the most flexible regions. This redistribution may be simplified for design purposes by assuming that certain regions of the cross section remain effective up to the yield point of the material, whilst the remainder is ineffective in resisting load. Current specifications for the design of open thin-walled sections provide equations for determining the effective width of stiffened and unstiffened elements, and the ultimate capacity of the section is calculated from the effective section properties.

720

Extensive experimental and analytical studies on stiffened elements (supported along both longitudinal edges) have been carried out by many researchers and have led to well established equations for the estimation of the effective width of such elements in uniform compression and under stress gradients. In comparison, experimental investigations on unstiffened elements (supported along one longitudinal edge) are quite limited. In the 1970s the applicability of the effective width concept to unstiffened elements under uniform compression was studied in detail by Kalyanaraman et al. (1977), who tested a large number of beams and short columns that contained web elements that were fully effective. Tests of a similar nature previously reported by Winter (1946, 1970), and the experimental and analytical research by Kalyanaraman et al. (1977), led to the adoption of the effective width approach for uniformly compressed unstiffened elements in the 1986 edition of the AISI Specification. Beam tests on sections that contain fully effective webs and simple edge stiffeners subjected to stress gradients have been reported by Desmond et al. (1981) and Winter (1947), and on open channels with inclined flanges under stress gradients by Rhodes (2000). Single plate test data for unstiffened elements with stress gradients have only been reported by Rhodes et al. (1975), where results of 4 tests on individual plates simply supported on three sides are given for varying values of load eccentricity. Due to the lack of test data, current AISI (1996) and Australian specifications (AS/NZS 4600 1996) treat unstiffened elements with stress gradients as if they were uniformly compressed for effective width calculations. Recent studies however, of cold-formed plain channels by Rasmussen (1994) and fabricated Isections by Chick and Rasmussen (1999) have demonstrated a marked conservatism in this method. Experimental and analytical results show that beam-column capacities were typically 30% or more higher than those estimated by the current specifications. The conservatism could be traced directly back to the bending capacities, which were of the order of 50% of the actual strengths, due to excessive conservatism inherent in the design procedures for unstiffened elements with stress gradients. This paper outlines the experimental investigation initiated as a means to establishing more accurate design methods for unstiffened elements under stress gradients. In order to determine the fundamental behaviour of these elements, the tests are performed on rectangular plates simply supported on three sides, as opposed to testing complete sections, so as to avoid unquantified restraints from adjoining elements. LOADING CONDITION- CONSTANT STRAIN ECCENTRICITY

The method for the application of the load is critical for unstiffened plate specimens. Researchers testing stiffened plates under stress gradients commonly use rigs that apply a load eccentrically to the ends of the plate, allowing the ends to rotate in-plane to maintain constant loading eccentricity. This method however, does not facilitate accurate results for unstiffened plates. For example, if a load is applied at the centre of an unstiffened plate (zero load eccentricity), at elastic local buckling the longitudinal stresses in the plate will redistribute towards the supported edge, and the resistance offered by the plate becomes eccentric. If the load point remains at the centre of the plate severe in-plane bending is induced, and consequently little post-buckling strength is achieved. This condition is also not congruent to that of plane sections remaining plane in assemblies of plates, which must be satisfied if accurate design methods are to be produced from the tests, since this is the condition of the nodal lines in a member with a number of locally buckled half-wavelengths.

721

To accurately capture the post-buckling behaviour of an unstiffened plate, and maintain a condition of plane sections remaining plane, a rig has been developed that can apply a constant strain eccentricity to the ends of the plate. El

~

j

Freeedge

El

El

El

El

El

.................. tf Z f !il 1t Figure 1" Strain Gradients for Test Series

DUAL-ACTUATOR RIG

Figure 2: Dual Actuator Rig - Plan and Elevation The rig incorporates the use of two hydraulic actuators, a (primary) 200 ton Dartec actuator applying compressive strains, and a (secondary) 25 ton MTS actuator that is connected to lever arms as shown in Figure 2, applying bending strains to the specimen. The dual action can apply strain gradients varying from pure compression to pure bending as required.

Primary Actuator- Compression Rig The primary actuator is mounted as a compression rig, with high strength Maccalloy tension bars providing the reaction force to the actuator, as shown in Figure 2. The rig is quite flexible compared with those having a fixed reaction frame. As the actuator applies load the tension bars extend elastically, such that the specimen will axially shorten and shift in the longitudinal direction.

722

J11

IIIII

,,5

ilL,

Figure 3: Detailed Schematic of the Rig - Side View

Secondary Actuator- Bending Rig The secondary actuator is seated on support beams and connected to a long SHS as shown in Figure 2. The actuator and the SHS are connected to lever arms that extend to the end platens of the compression rig, where they are rigidy connected. Extension or contraction of the actuator causes the lever arms to rotate about the pin joints shown in Figure 2. The inplane rotation of the end platens can thus be controlled directly. Since the actuator must move relative to its supports, friction is a concem. To minimise this the actuator is seated on nine ball bearings constrained in a grid, denoted J15 in Figure 2, and the SHS is seated on four ball bearings, such that the lever arm rig is essentially floating.

Boundary Conditions Establishing simply supported boundary conditions is essential, particularly for the loaded ends of the plate. The elastic critical buckling stress for slender plates is significantly higher if rotational restraint exists at the simply supported loaded ends. In order to maintain free rotation, the loaded ends of the plate are seated in machined key-ways cut in segments of circular rods, which are fitted into split spherical needle bearings housed in solid bearing blocks, as shown in Figure 4. The rod segments are cut in 20mm lengths, such that the loaded ends of the plate may rotate by varying degrees across the width.

Figure 4: Loaded Edge of a Plate, Rod Segment and End Bearing Block

723

In a slender unstiffened element, the longitudinal stress may redistribute to such an extent after local buckling that tensile stresses may develop at the unsupported edge. If these tensile stresses are not anchored, the loaded edge will deform and the assumption of plane sections remaining plane will no longer be valid. It is important then to restrain the loaded edge, and for this purpose holes are drilled in the rod segments and in the plate specimens where tensile stresses are likely to occur as shown in Figure 4, and pins are inserted such that the end bearings may resist tension while allowing rotation. This condition is important to the operation of the rig, particularly for the case shown in Figure 1 where compressive strain is applied to the unsupported edge and varies linearly to zero at the supported edge. Slender plates under this load condition have large out-of-plane deflections of the free edge in the post buckling range, and as a result the loaded edges are prone to deformation. The pure bending load case in Figure 1, whereby compressive strain is applied to the free edge and tensile strain to the supported edge, presents a unique problem. As the plate is loaded in the post buckling range, the region of the specimen in compression becomes less effective, and the neutral axis shifts. As a result, a large portion of the cross-section yields in tension in order to attain ultimate capacity. Drilling holes through the plate to insert pins for resisting tension is not possible, as the plate will fracture through the line of holes and the ultimate condition cannot be captured. For specimens tested under this load condition end plates are welded to the specimen, such that the boundary conditions of the loaded edges are now fixed, not simply supported. Another effect of the neutral axis shift for this load condition is that the plates develop net tension after local buckling. Stiff compression struts are required between the northern and southem headstocks of the compression rig to facilitate the net tension.

Figure 5: Discreet Finger Support The simple support condition along the unloaded supported edge is achieved by the use of discreet 'finger' supports, as shown in Figure 5, which were originally developed at Cambridge University (Moxham 1971). The plate edge is inserted between the first set of pins only and clamped by tightening the top bolt, thus restraining out-of-plane displacements of the edge while allowing rotation. The use of fingers requires no edge preparation to the plate. The fingers are supported on a rod by means of a spherical bearing, such that the fingers may rotate to allow small in-plane transverse displacements of the edge. The rod has a secondary rod beneath it to assist in minimising bending, as shown in Figure 3, and both rods are seated in linear bearings. The linear bearings allow the entire finger assembly to shift in the longitudinal direction, in accordance with the shift of the specimen as the tension bars of the compression rig extend. The fingers contain a slender section near the base such that they are flexible in the longitudinal direction, thus negligible load is transferred through the fingers to the support. The test setup in Figure 6 shows the use of the end bearings and fingers to provide three sided simple support.

724

Figure 6: End Bearings and Fingers Provide 3-Sided Simple Support CONTROL OF THE APPLIED STRAIN GRADIENT The flexibility of the rig requires that an independent measurement system be used to control the applied strain gradient. Initially LVDTs were mounted on the end bearings on each side of the specimen, as shown in Figure 7b, to measure the longitudinal displacement between the end bearings on the east and west sides. However internal displacements within the end bearings preclude accurate measurements of the axial shortening of the plate. The displacements are twofold; firstly there is slack in the bearing between the rod segments and the spherical bearing, and within the spherical needle bearing itself, and secondly the bearing block elastically compresses. For the case of uniform compression the slack can be excluded from the results, and experiments have shown this to be of the order of 0.2mm. The compression of the bearing blocks causes a change in slope of the load-displacement curve, or apparent change in stiffness, and this too could be removed from the results with careful processing of the test data. The difficulty arises however with the cases where strain gradients are applied. For these cases one longitudinal edge is compressed while the other edge remains unloaded. The individual rod segments settle by different amounts as the slack is only taken up when load is applied, therefore differential settlement within the bearings occurs across the width of the plate. A measurement system is thus required that can measure the displacement of the loaded ends of the plate directly.

Figure 7: Displacment Measurement Systems Various displacement measurement systems have been investigated, and their accuracy determined by comparison with readings from strain gauges that am fixed to both sides of the plates on each longitudinal edge at mid-length, which can be seen in Figure 6. The gauges are accurate to within 5 microstrain. The most reliable system is to rigidly attach a rod across the width of the specimen with epoxy glue, as close to the ends as is practical, as

725

shown in Figures 6 and 7c. LVDTs are mounted on each side of the plate running longitudinally, such that the displacement of the east and west sides is known. The ratio of the values prescribes the strain gradient on the plate (Figure 7a). This method was found to be inaccurate for the more slender specimens however, due to large out-of-plane displacements of the plate causing the rod ends to displace vertically. For slender specimens, targets are mounted on the rods and laser measurement devices are positioned at each rod end (Figure 7d). The mounting arrangement uses bearings to ensure verticality of the target at all stages of plate deflections. The difference between the two readings on each side of the plate gives the displacements of the east and west sides respectively, and the strain gradient may be deduced. If the targets are set carefully only small inaccuracies occur when the rod ends displace vertically (i.e. when large lateral displacements occur). In all tests, the strain gauges are used to determine the applied strain gradient until out-of-plane deflections occur. The gauge readings are not applicable in determining the (average) applied strain after local buckling, since they record the Iocalised strain at the centre and do not account for shortening resulting from plate deflections. The tests are performed by setting stroke speeds on the digital controllers. The controller for each actuator is set to a prescribed control-slave system according to the strain gradient required, whereby one actuator runs at a set stroke speed, and the other actuator runs as a slave at a ratio of that stroke speed. This is achieved by sending the voltage signal from the intemal LVDT of the master actuator to the controller for the slave actuator, and amplifying or reducing the signal accordingly. Due to the flexibilty of the rig, the stroke ratio is determined experimentally and needs to be manually adjusted continuously throughout the test. Continuous monitoring of the displacement measurement system allows the strain gradient to be known at all times during the test, and the stroke ratio is adjusted to produce the required gradient.

OUT-OF-PLANE DISPLACEMENT MEASUREMENT A measurement frame is situated above the specimen as shown in Figure 3, and has two high-prescision rails running longitudinally. A trolley is mounted by means of linear bearings on the rails, and is attached to a timing belt and pre-programmed stepper motor. Transducers are mounted from the trolley and positioned in a line across the width of the specimen. The line of transducers is run along the length of the specimen before and during the test, producing a fine representation of the initial imperfections and buckled shape.

TEST SPECIMENS The test series consists of 80 mild steel specimens, all of nominal thickness 5mm. Plate width to thickness ratios vary from 12 to 35, representing slendemess ratios (Z = ~fi/for ) of 0.75 to 2.17. The aspect ratio is 5 for all specimens, such that the buckling coefficient is close to the asymptotic value of 0.425 for unstiffened elements (Bulson 1970), and such that end effects are minimised. Half the specimens have been heat treated to induce residual stresses approximating those that exist in fabricated sections. Details of the processes and magnitudes of stress induced are given in Bambach and Rasmussen (2001). For each of the four load cases shown in Figure 1, 20 specimens are tested. All specimens have 4 strain gauges attached, one on each side of the plate near the longitudinal edges at mid-length. The 20 specimens that are to be tested under the pure bending case shown in Figure 1 have welded end plates.

726

VERIFICATION

Since the behaviour of unstiffened elements under uniform compression has been more widely reported on than for those under stress gradients, the compression case will be used to verify the rig. The primary concern in establishing that the rig is performing correctly is in the boundary conditions of the loaded edge. A number of pure compression tests were performed on specimens of nominal dimensions 200x600x5mm, for which established theoretical solutions by Bulson (1970) estimate that the elastic critical buckling load will be 60kN if the loaded edges are simply supported, and 97kN if they are fixed. Tests without the pins in the end bearings showed the bearings to be performing well, freely allowing rotation at the loaded edges and the specimens buckled at loads close to 60kN. The tangent to the Load vs. Lateral Displacement Squared method, as described by Venkataramaiah and Roorda (1982), is used to deduce the critical load. It was found that when the pins were used in the end bearings, the bearings had a tendancy to lock and not freely allow rotations. To avoid this, the end bearings must be parallel to the ends of the plates to within a very fine tolerance. To achieve this tolerance, the strain gauges were introduced. During setup, the primary actuator applies a small compressive load and the secondary actuator is adjusted (i.e. the end bearings are rotated in-plane) until the strain gauges read equal compressive strain on each longitudinal edge. Following this procedure ensures that the end bearings are free to rotate, and critical buckling loads show that the plates are effectively simply supported at the loaded ends.

Figure 8: Typical Load - Lateral Displacement Squared Curve

Figure 9: Specimen at Ultimate Load

Figure 8 shows a test result for a three sided simply supported plate under uniform compression, with the pins inserted. The elastic buckling load is very close to that for a plate with simply supported ends (60kN). The photo in Figure 9 shows the plate at the ultimate load carrying capacity, where the buckle has Iocalised at the loaded end shown. Figure 8 verifies that the application of a constant strain eccentricity creates a testing condition whereby the full post-buckling behaviour of unstiffened plates may be captured. The specimen reaches an ultimate capacity of about 2.7 times its elastic buckling load. Some test results for ultimate load carrying capacites of specimens under uniform compression are compared with tests on beams and short columns (Kalyanaraman et al. 1977) in Figure 10, and with the Winter curve for unstiffened elements in pure compression, given by:

727

1.2

I

I

,

I

I

,-, ~ " "~. .

1.0

O

~

0.6

SS

SS

Kalyanaraman et al. (1977) Tests:

. 9

9 Stub column + Beam

o

"'. ~~'~".,.,.;.... x

0.4

I

SS

"'..

0.8

,,

buckling Local

~

"'..~-. +

....... +

-

~ 1 7 6 1 7 6 1 7 6 1 7o6 O O , o o o . , O o o

0.2

- ...... 0 x

0

0.5

~ Winter for u n s t i f f e n e d ~ e l e m e n t s Test results (Bambach and Rasmussen, 2001) - Unwelded specimens Test results (Bambach and Rasmussen, 2001) - Welded specimens I I I I I

1.0

-,

i

1.5

2.0 2.5 3.0 3.5 Slenderness X Figure 10: Comparison of Test Results for Unstiffened Plates in Uniform Compression

CONCLUSIONS Experimental techniques for testing rectangular plates simply supported on three sides, in combined compression and in-plane bending have been presented. Results for uniform compression tests have been verified against established theoretical solutions, and shown to be in general agreement with existing test data of sections. The paper details methods for ensuring simply supported longitudinal and loaded edges, anchoring tensile stresses at the loaded edges, and measuring the applied strains.

REFERENCES

American iron and Steel Institute. (1996) Specification for the Design of Cold-Formed Steel Structural Members. 1996 Edition, Washington D.C. AS/NZS 4600 (1996). Australian/NewZealand Standard. Cold-Formed Steel Structures. 1996 Edition, Sydney Bambach M.R. and Rasmussen K.J.R. (2001). Residual Stresses in Unstiffened Plate Specimens. Proceedings of the Third International Conference on Thin-Walled Structures,Cracow, Poland. Bulson P.S. (1970) The Stability of Flat Plates. Chatto and Windus, London Desmond T.P., Pekoz T. and Winter G. (1981). Edge Stiffeners for Thin-Walled Members. Journal of the Structural Division, ASCE, 107:ST2, pp 329-353 Chick C.G. and Rasmussen K.J.R. (1999). Thin-Walled Beam-Columns.2:Proportional Loading Tests. Journal of Structural Engineering, ASCE, 125:11, pp 1267-1276 Kalyanaraman V., Pekoz T. and Winter G. (1977). Unstiffened Compression Elements. Journal of the Structural Division, ASCE 103:ST9, pp 1833-1848 Moxham K.E. (1971) Buckling Tests on Individual Welded Steel Plates in Compression. University of Cambridge, Report CUED/C-Struct/TR.3 Rasmussen K.J.R. (1994). Design of Thin-Walled Columns with Unstiffened Flanges. Engineering Structures, 16:5, 1994, pp 307-316 Rhodes J., Harvey J.M. and Fok W.C. (1975). The Load-Carrying Capacity of Initially Imperfect Eccentrically Loaded Plates. International Journal of Mechanical Sciences, Vol. 17, pp161-175 Rhodes J. (2000). Buckling of Thin Plates and Thin-Plate Members. Structural Failure and Plasticity, IMPLAST 2000, pp 21-42 Venkataramaiah K.R. and Roorda J. (1982). Analysis of Local Plate Buckling Data. Proceedings of the Sixth International Specialty Conference on Cold-Formed Steel Structures. 1982. Winter G. (1947). Strength of Thin Steel Compression Flanges. Transactions, ASCE. Vol. 112, pp 527-576 Winter G. (1970). Commentary on the 1968 Edition of the Specification for the Design of Cold-Formed Steel Structural Members, American Iron and Steel Institute, New York



729

EFFECTS OF ANCHORING TENSILE STRESSES IN AXIALLY LOADED PLATES AND SECTIONS M. R. Bambach 1 and K. J. R. Rasmussen 1 1Department of Civil Engineering, Sydney University, Australia

ABSTRACT Thin-walled compression members are commonly designed on the assumption that the loaded edges remain straight. Under this assumption, tensile stresses develop in the most flexible parts of the component plates at advanced local buckling deformation, and thus are assumed to be 'anchored' at the ends. However, current design rules for plate elements, such as the Winter formulae, are based partly on tests in which the load was applied by use of rigid platens that did not permit tensile stresses to develop. Recent literature has pointed to the apparent inconsistency between the assumption of straight loaded edges and the use of a design curve calibrated from tests in which the loaded edges of component plates may not have remained straight. The paper addresses this apparent inconsistency by comparing finite element solutions for the conditions of straight loaded edges and loading by use of a contact surface between the plate edge and a non-deformable rigid body end platen, where there is no constraint for the plate edge to remain in contact with the rigid body. Solutions are provided for a single halfwavelength of unstiffened and stiffened plate elements simply supported along three and four edges respectively. The effect of multiple half-wavelengths is also investigated, as is the effect of interaction between elements in practical sections comprising stiffened and unstiffened elements. KEYWORDS Local buckling, stiffened and unstiffened plate elements, FEM, ultimate capacity, stub columns, tests INTRODUCTION Compression members such as the unlipped channel shown in Figure 1, will locally buckle in a number of half-wavelengths and will retain this mode shape into the post-local buckling range, until Iocalisation occurs in one of the half-waves, propagating failure. It is well-known that loads at failure may be considerably higher than those at which local buckling occurs,

730

due to the redistribution of longitudinal stresses from the flexible to the stiff parts of the component plates. In the stiffened element, supported along both longitudinal edges, the longitudinal stress pattem across the width of the loaded edges changes from a condition of uniform stress prior to local buckling, to that shown in Figure 2a in the post-buckled range. The stresses redistribute from the deformed centre of the plate to the supported edges. Similarly for unstiffened elements, supported on one longitudinal edge, the stresses redistribute from the deformed edge of the plate to the supported edge, as shown in Figure 2b. If the component plates are sufficiently slender, the large plate deflections may cause redistribution of the longitudinal stresses to such an extent that tensile stresses develop at the nodal lines in the regions of maximum deformation. Half-wavelength

(a)

dal planes

Locally

(b) Figure 1: Locally Buckled Member In a uniformly compressed long column, consisting of many local buckle half-waves (cells), the condition of continuity at the nodal lines allows the tensile stresses in each cell to be anchored in its neighbouring cells. In comparison, if a single cell, such as in Figure l a, is uniformly compressed, the nodal lines are now the loaded ends of the short column, and tensile stresses may only develop if the ends are anchored. In a testing situation, anchorage could be achieved by welding stiff end plates to the short column. In the absence of welded end plates, which is the most common situation, the loaded edges will pull away from the end platens at advanced stages of local buckling in the most flexible regions of the cross-section, and as a result the loaded edges will not remain straight. This is shown later in the paper to be the case for unstiffened elements, and to a lesser extent for stiffened elements, (Figures 6 and 7). (a) Stiffened Element (b) Unstiffened Element

~"-Lor~tudnal stresses/ ~:ends Figure 2: Postbuckled Plates

731

Winter(1970) proposed and verified experimentally the following effective width formulae for stiffened (Eqn. 1) and unstiffened (Eqn. 2) compression elements, based on experimental programs (Winter 1947,1970 and Kalyanaraman et al. 1977). b~='/~'II-O'22-/f'~l/b ~l,,t, ~./~

(1)

Many of the tests were conducted on beams where the compression elements locally buckled with several half-wavelengths, and anchorage of tensile stresses could be expected due to continuity of the member at its nodal lines and 'compressed ends' (being the points of application of the beam loads). However, a number of the tests were on stub columns, particularly those tests on unstiffened elemGnts, for which the ends were free to pull away from the end platens. Since the Winter formulation is the basis of the effective width formulae used in the current American specification AISI (1996) and the AS/NZS 4600 (1996) specification, there is an apparent inconsistency between the common practical condition of long columns where tensile stresses can be expected to develop in the area of failure near the centre, and the method used to design it, being calibrated from tests in which loaded edges may not have remained straight. The objective of this paper is to quantify the effect of this inconsistency. From a practical viewpoint, the anchorage of tensile stresses is likely to be important mainly for short members, notably columns forming a single half-wave. For such members, tensile stresses can only develop if the member is welded to stiff adjoining elements, as mentioned above. If a short member is connected through a few points only, such as a stud screwed to the track of a wall frame, tensile stresses cannot develop at the ends and the ultimate load is likely to be affected. However, for long members forming a large number of local buckles and failing near the centre, the ultimate load is unlikely to be significantly affected by the end support conditions. This is because the nodal lines remain essentially straight in the central region, thus allowing tensile stresses to develop. It is noted that in the case of an even number of half-waves, the central nodal line remains exactly straight by symmetry, and evidently, the larger the number of half-waves, the closer the adjoining nodal lines will be to remaining straight. FEM ANALYSIS

In order to examine the effects of the loaded edge conditions on the post-buckling strength of sections, the stiffened and unstiffened components are analysed separately, as four sided and three sided simply supported plates respectively. Finite element analyses using the FEM program Abaqus v5.7 are carried out, using 4-node isoparametric shell elements with five integration points through the thickness. Initially a linear elastic analysis is performed and verified against well established theoretical solutions for elastic buckling stresses (Bulson, 1970), and the buckled shape is then scaled to generate geometric imperfections in the model. A maximum out-of-plane imperfection of 10% of the thickness is used for all models. Non-linear material properties are introduced and a geometric and material non-linear analysis is performed, using the Riks Arc Length method to determine the ultimate load carrying capacity. Two conditions are analysed in the FEM analysis. The 'Straight' condition, whereby the loaded edges are constrained to remain straight by using the geometric loading condition of

732

Figure 3: Contact Models Loaded by Rigid Bodies prescribing all nodes at the loaded ends to displace an equal amount in the longitudinal direction. The second, 'Contact' condition, uses a contact surface between the loaded ends and non-deformable rigid bodies, where there is no constraint for the plate ends to remain in contact with the rigid bodies. In the analysis, the rigid bodies are prescribed to move in the longitudinal direction only (no in-plane rotation is permitted), and bear on the contact surface of the plate. The advantage of this model is that the loaded ends of the plate are allowed to deform, since the contact elements allow the loaded ends to pull away from the rigid bodies. These two conditions reflect the loading of short columns between rigid end platens, with and without welded end plates respectively. Figure 3 shows the Contact models for (a) unstiffened plates and (b) stiffened plates. VERIFICATION WITH TESTS

The non-linear models are compared with experimental data by the authors, and with the Winter Equations 1 and 2. The ongoing experimental program by the authors includes tests on individual steel plates, simply supported on three sides, under various combinations of compression and in-plane bending. The test results from those specimens under uniform compression are discussed here. Further details of the experimental method are given in Bambach and Rasmussen (2001). Uniform compression tests were performed on unstiffened mild steel plates of nominal thickness 5mm, widths of 60 to 125mm, and lengths corresponding to an aspect ratio of five for all specimens. Coupon data showed the plates to have a yield stress of 272MPa, and a Young's modulus of 2.02x105 MPa. Imperfection measurements showed the average out-ofplane imperfection of the free edge to be 0.37mm, which is close to the assumed value of 0.5mm in the model. Two tests were conducted for each specimen dimension. For comparison with the tests, the material properties of the above specimens were included in the FEM model, assuming an elastic perfectly-plastic material stress-strain curve. The FEM results for stiffened elements are compared with Winter Eqn. 1 in Figure 4, and for unstiffened elements are compared with the tests and Winter Eqn. 2 in Figure 5. These results are for a single half-wavelength, as detailed in the next section. The FEM analyses are extended to include all width to thickness (b/t) ratios within the limits as set out in AS/NZS 4600 (1996), being 500 for a stiffened compression element (a slenderness of

733

k=10), and 60 for an unstiffened element (a slenderness of X=3.6).The critical buckling stress

(Iol) and slenderness (~.) are given by:

J;~ = 12(1_o2------~

2_1 = [ f i t

i

o

......

I

I

'

1

'"'

0.75 I~ ~ .

-. , . . ~ "-

Winter for stiffened elements Abaqus-straight loaded edge Abaqus-contact loaded edge

0.50 r~

0.25

0

,

I

0

,.

I

2

I

I

4

6 Slenderness k

8

10

Figure 4" Strength Curve for Stiffened Elements It is noted here that the comparison of the FEM results for unstiffened elements with the Winter Eqn. 2 for unstiffened elements shows a slight optimism in the Winter equation in the slenderness range 0 . 7 5 - 1.55, and a slight conservatism for slenderness values exceeding 1.75. This trend is verified experimentally in the results obtained by Kalyanaraman et al. (1977) and by those of the authors (noting that the authors experimental program has yet to be extended to slenderness values exceeding 1.75). 1.2

r~

i

'

t

I

9

I

'

[

i

O

0.8

Kalyanaraman et ai. (1977) Tests:

9

9 Stub column + Beam

oo~..

.,..t

'1 t~

....

1.0 "'.oo

r.f]

i

0.6

+

~

"

r~

0.4

0.2

-o..

......

W i n t e r for unstiffened elements

---. o

Abaqus-straight loaded edge Abaqus-contact loaded edge Testresults (Bambach and Rasmussen, 2001)

........

I

I

I

I

I

1.0

1.5

2.0

2.5

3.0

I

3.5

Slendemess k

Figure 5: Strength Curve for Unstiffened Elements

734

FEM RESULTS FOR A SINGLE HALF-WAVELENGTH

Figures 4 and 5 show the strength curves based on the section dimensions and ultimate loads given in Table 1. They represent plate specimens which buckle as a single halfwavelength, for which a stiffened element will have an aspect ratio of one. Unstiffened plate elements have one longitudinal edge simply supported, and if this boundary condition is satisfied and no rotational restraint is applied to the edge, the plate will form a single buckle with a half-wavelength equal to the length of the plate. Bulson (1970) has shown that the buckling coefficient for unstiffened elements asymptotes to a value of 0.425 at large aspect ratios, and in order to achieve a similar value for the coefficient in the tests, an aspect ratio of five was used. This aspect ratio was also used in the FEM analysis in Table 1. The influence of the aspect ratio will be discussed further in this paper. STIFFENED W i d t h . Length (mm) (mm) 115 115 155 155 195 195 240 240 335 F 335 465 465 620 620 775 775 1550 1550 UNSTIFFENED Width Length (mm) ( m m ) 60 300 80 400 100 500 115 575 125 ,' 625 150 750 175 i 875 210 ~ 1050 250 1250 300 1500 ,

,

,

,,

,,,

,

,

,

|

EM~MENTS Slenderness

Thickness 3mm Ultimate Load (kN) Difference (~) Straight Contact 0.74 90.5 90.5 0.0% 1.00 103.1 0.0% 103.1 1.26 104.8 0.0% 104.8 0.0% 1.55 109.3 109.3 118.8 0.0% 2.17 118.8 126.5 3.01 126.4 0.1% 1.2% 130.9 129.3 4.01 133.3 - 0.2% 5.01 133.6 141.5 - 2.2% 138.5 10.02 Nominal Thickness 5mm Thickness 5mm "TESTS ELEMENTS Slenderness, Ultimate Load (kN) Difference Width Slendemess Ultimate Load (kN) (~,) , Straight Contact (mm) (;L) 0.0% 61.16 0.74 79.1 0.71 80.7 80.7 0.0% 60.62 0.73 82.5 1.03 110.8 110.8 0.0% 80 1.07 95.7 1.19 101.5 101.5 0.0% 79.92 1.06 103.4 1.37 101.6 101.6 0.1% 100.36 1.21 113.3 1.49 ~ 104.5 104.4 2.1% _- 100.3 1.21 94.5 1.79 , 117.7 115.2 4.0% 125.64 1.52 94.8 2.08 , 132.1 126.8 8.2% = 125.75 1.52 94.3 2.50 i 150.1 137.8 10.6% 2.98 171.3 153.2 3.57 176.2 160.4 .

,

.

.,.

..

....

i

j

.....

=

|

Table 1" FEM and Test Results for 1 Half-Wavelength Stiffened Elements

The strength curves in Figure 4 for the two loaded edge conditions, Straight and Contact, are virtually coincident. Only at high slendemess values, greater than 3, do tensile stresses at the loaded ends have any effect. Values of tensile stress and end deformation for the specimen of slenderness ;k=5, corresponding to a large but still practical bit ratio of 260, are detailed in Figures 6a and 6b. Figure 6a shows the longitudinal stress variation across the width of the loaded ends at ultimate. The stress decreases at the supported edges since these edges are free to pull in transversely. In the central region of the loaded edge of the stiffened plate, for the Straight loaded edge condition, small values of tension are present. The maximum value of tensile stress at ultimate is 2 MPa. For the Contact condition, where

735 there is no restraint to keep the edge straight, the central region pulls away from the rigid body. The longitudinal stress is zero in this region. The pull-in of the loaded edge is very small at ultimate, with a maximum of 4xlO3mm, as shown in Figure 6b. The distribution of longitudinal stress for the two cases are virtually coincident, and as a result the ultimate loads are nearly the same. The simplified stress distribution using the Winter effective width Eqn. 1 is also shown for comparison, and gives a good approximation of the ultimate load capacity for this specimen. It is noted that while the longitudinal stress magnitude is not at yield at the supported edges, the element von Mises stresses are, due to shear stress components. 50 0

15

-50 -100

-200

(][01 .........

a

-150 .~

=::I,E= ;::] ~

~

~'/

!

~-"

Winter Abaqus-co for tnact-ultimate stiffened element s

0 = 0

,,J

0,

,

,

0 1553104656;Z) 775 Width of the Loaded Edges (mm)

(b)

-250 -300

(a)

Width of the Loaded Edges (mm)

Figure 6: Mid-Surface Longitudinal Stress and Displacement at Ultimate Load for Stiffened Element of Slenderness ~.=5 Unstiffened Elements Once the slenderness exceeds a value of ~=1.5 (b/t ratio of 25), the ultimate load carrying capacity is reduced by the Contact condition, and the strength curves for the two cases diverge, as shown in Figure 5. This corresponds to the slendemess after which the specimens in the Straight condition will attain tensile stresses at the unsupported edge, and in the Contact condition this edge will pull away from the rigid loading body. The maximum reduction in strength is 10.6% for a slenderness of ~,=3 (b/t ratio of 50). For this slendemess value, the maximum value of tensile stress at ultimate is 158MPa in the Straight condition, and the pull-in of the free edge in the Contact condition is O.3mm. Figure 7a shows the distribution of longitudinal stress across the width of the loaded edges at ultimate, for a slenderness value of ~.=2.1 (b/t ratio of 35). For this slenderness value, the Contact condition causes a 4% drop in ultimate load carrying capacity. The difference in the stress distributions at ultimate between the two loaded edge conditions is noticeable. The Winter Eqn. 2 for unstiffened elements is conservative in this slenderness range, as seen in Figure 5, and the difference in the assumption for the stress distribution is noticeable also for the Straight condition. The deformation of the loaded edge for the Contact condition is plotted in Figure 7b. For both stiffened and unstiffened elements, the values in Table 1 show that the percentage difference in the ultimate loads between the two loaded edge conditions decreases at very

736

large slendemess values. This is due to a mode change of the deflected shape for very slender elements. i

300

i "

t 200

~ z

' i i Abaqus-straight-utimate Abaqus-contact-ultimate Winter for unstiffened elements

1oo r~

i

a

o

0.5 .... Q4 Q3

-lOO

2-:i0 ~- 02 _=" 0.1

-200

_~ g

50

r~

1o

~

1(

-

111 -!

,.I

-300

-400

/

0

/

/

I

, 0 125 250 Widthof the LoadedEdges (mm) (bl

(a) Width of the Loaded Edges (mm)

Figure 7: Mid-Surface Longitudinal Stress and Displacement at Ultimate Load for Unstiffened Element of Slendemess ~=2.1 FEM RESULTS FOR MULTIPLE H A L F - W A V E L E N G T H S

The FEM results discussed thus far have been for specimens that buckle in a single halfwavelength. For practical purposes, tests for determining section strength will typically be conducted on stub columns, and will generally ensure the formation of 2-3 local buckles. Accordingly, the FEM analyses are extended here to determine the ultimate load carrying capacity of stiffened and unstiffened elements for the two conditions of Straight and Contact loaded edges, when the plates buckle in three half-waves. This mode is achieved for stiffened elements by changing the aspect ratio to three. For unstiffened elements it is achieved by restraining the out-of-plane deflections (only), across the entire width of the plate, at the points corresponding to 1/3 of the length and 2/3 of the length. The length is tripled in order that each half-wave retains the aspect ratio of five. UNSTIFFENED Width (mm) 60 100 125 150 175 250

ELEMENTS

Thickness 5ram Ultimate Load ( k N ) Difference Length Slenderness Straight Contact (mm) (;~) 900 0.71 81.0 81.0 0.0% 1500 1.19 101.6 101.6 0.0% 1875 1.49 104.6 104.6 0.0% 2250 1.79 118.0 116.3 1.4% 2625 2.08 132.3 127.9 3.3% 3750 2.98 171.9 154.6 10.1% 3.57 163.9 6.6%

. . . ........ /

i

I

/J

]

.-]

A .r

~ ~..."

/[

Table 2: FEM Results for Unstiffened Elements with 3 Half-Wavelengths The results for three half-wavelengths are shown in Table 2 for unstiffened elements. As one would expect, the effect of not constraining the loaded edges to remain straight (Contact)

737

diminishes compared to the results shown in Table 1, since there are now two nodal lines along the length at which the tensile stresses of one buckled cell may anchor into the next. APPLICATION TO SECTIONS CONTAINING BOTH STIFFENED AND UNSTIFFENED ELEMENTS

It is well documented that slender sections containing stiffened and unstiffened elements, such as the unlipped channel in Figure 1, will locally buckle at a half-wavelength different to that if the components were treated separately, due to interaction between the flanges and the web. As a result, it is likely that the unstiffened element of the section will buckle at a halfwavelength less than the previously assumed aspect ratio of five. Table 3 shows the ultimate load carrying capacities of unstiffened elements for the Straight and Contact loaded edge conditions, when the aspect ratio is reduced from five to one. The study is for elements buckling in three half-waves. The effect of end loading condition is seen to be strongly dependent on the aspect ratio, eg for a b/t-ratio of 50, the ultimate loads are equal for an aspect ratio of one, while they differ by 10.1% for an aspect ratio of 5. The percentage differences shown in Table 3 are plotted against b/t in Figure 8 for each of the five aspect ratios. UNSTIFFENED ELEMENTS - 3 Half-Wavelen~lths I Aspect ratio of I | Ultimate Load Diff. Width!b/t Length Straight Conlact Length 175 35 525 150.6 15 c.6 0.0% 1050 250 50 750 171.4 171.4 0.0% 1500 Aspect ratio of 3 Ultimate Load Diff, Width b/t Length Straight Contact! Length 150 30 1350 120.8 120.3 0.4% 1800 175 35! 1575 133.4 131.3 1.6% 2100 i

. . . . . . . .

Thickness 5mm Aspect ratio of 2 -Ultimate Straight 135.2 159.3

Load Contact 134.9 156.8 Aspect ratio, of 4 Ultimate Load Straight Contact 119.3 118.0 133.1 129.4

Diff.

0.2% 1.5%

i

Aspect ratio of 5 I Ultimate " .oad Diff. Length Straight C :intact 1.1% 2250 118.0 16.3 1.4% 2.8% 2625 132.3 127.9 3.3% Diff.

Table 3: FEM Results for 3 Half-Wavelengths and Varying Aspect Ratios In practical sections comprising stiffened and unstiffened elements, the stiffness of the web with respect to that of the flanges will cause the section to locally buckle in half-wavelengths producing aspect ratios of the buckles in the web between 1 and 2. In this case, the aspect ratio of the unstiffened element will be in the range from b~/bf to 2b~bf. Given that the b~/bfratio of practical sections typically varies between 2 and 4, the aspect ratios for unstiffened elements in practice are likely to fall in the range from 2 to 8. CONCLUSIONS

The influence of the two limiting conditions of the loaded edges, Straight and Contact, on the ultimate load carrying capacity of stiffened and unstiffened elements has been investigated. These two conditions are analogous to short columns with and without stiff welded end plates respectively. The influence of multiple half-wavelengths, aspect ratio and interaction between stiffened and unstiffened elements comprising a section are investigated. It is shown that the ultimate load carrying capacity of stiffened elements is virtually unaffected by the loaded edge condition. However, for unstiffened elements, the ultimate load can be

738

10.0~ Aspect Ratio of flange local buckle:

Open Sections Locally Buckling In 3 Half-Wavelengths

9.0%

5

8.0~ 7.0% w

ca &O%

0 ,.I

o w Q.

5.0% A s p e c t ratio =

m 4.0% tO 3.0% 2.0% 1.0% 0.(7'/0 25

30

35

40

45

50

Range W i ~ a n g e 1Ndmess

Figure 8: Section Capacity loss if the Loaded Edges are allowed to Deform Three Half-Wavelengths assumed reduced by about 10% when the loaded edges are allowed to pull in (Contact condition). This percentage corresponds to a plate slenderness value (b/t) of 50 and an aspect ratio of 5. The reduction is shown to depend strongly on the aspect ratio and less so on the number of halfwaves. It is concluded then that the Winter formulae provide a satisfactory estimation of the effective width of stiffened compression elements. The Winter formula for unstiffened elements is conservative at high slenderness ratios (;~>2), which may be partly because it was calibrated on the basis of stub column tests which allowed the ends to pull away from the loading platens. REFERENCES

American Iron and Steel Institute. (1996) Specification for the Design of Cold-Formed Steel Structural Members. 1996 Edition, Washington D.C. AS/NZS 4600 (1996). Australian/New Zealand Standard. Cold-Formed Steel Structures. 1996 Edition, Sydney Bambach M.R. and Rasmussen K.J.R. (2001a). Experimental Techniques for Testing Unstiffened Elements in Compression and Bending. Proceedings of the Third International Conference on Thin-Walled Structures. /CTWS (2001), Cracow, Poland. Bulson P.S. (1970). The Stability of Flat Plates, Chatto and Windus, London Kalyanaraman V., Pekoz T. and Winter G. (1977). Unstiffened Compression Elements. Journal of the Structural Division, ASCE 103:ST9, pp 1833-1848 Winter G. (1947). Strength of Thin Steel Compression Flanges. Transactions, ASCE, Vol. 112, pp 527-576 Winter G. (1970). Commentary on the 1968 Edition of the Specification for the Design of Cold-Formed Steel Structural Members, American Iron and Steel Institute, New York.

Third International Conferenceon Thin-WalledStructures J. Zarag, K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

739

A PROBABILISTIC APPROACH TO THE LIMIT STATE OF CENTRALLY LOADED THIN-WALLED COLUMNS Z. KALA, J. KALA, B. TEPL~" Faculty of Civil Engineering, Bmo University of Technology, Veveri 95, Bmo M. SKALOUD

Czech Acad. of Sciences, Inst. of Theor. & Appl. Mechanics, Prosecka 76, Prague

ABSTRACT The problem of the interaction of local buckling and the loss of stability as a whole is studied taking into account different imperfections. Thin-walled closed section columns in compression are analysed for several models of boundary conditions utilizing the probabilistic approach and by means of a detailed FEA model. A comparison to the load-bearing capacity according to Eurocode is then made.

KEYWORDS Imperfection, limit states, thin-walled cross-sections, finite elements methods, non-linear solution, probabilistic simulation methods. 1. INTRODUCTION It is generally known that in the strength calculations of thin-walled compressed columns it is necessary to include the influence of the interaction of global and local buckling, see for example, [3], [6], [7]. The conception of the normative buckling strength of the compressed columns is based on the analysis of a real structural element that, in contrast with the model of an ideal column, shows a number of initial imperfections affecting its load-carrying capacity. These imperfections are divided into the following three categories: 1. Geometrical deviations: The initial curvature of the bar axis, the eccentricity of the load point, the violation of the theoretical arrangement of the cross-section (the size and shape tolerance of the cross section), etc. 2. Structural imperfections: Dispersion of mechanical properties of a material (the nonhomogeneity of a material that is manifesting itself by dispersion of yield point values, breaking strength values, elasticity modulus values, etc.), the initial state of stress (residual stress as a result of technological production processes), etc.

740 3. Construction imperfections: Imperfections in the execution of joints, connections, bearing and other construction details that manifest themselves in deviations of the action of the real load-carrying system, compared with the theoretical assumptions that are being introduced during the solution of the idealized system. By their nature, the imperfections of the first and the second types are random quantities showing a larger or smaller variability. The analysis of their influence on response may be studied using a number of probability methods (Monte Carlo, Latin Hypercube SamplingLHS, Importance Sampling, etc.)- see, e.g., [1 ]. The influence of the third type of imperfections should be taken into consideration in a more accurate solution of the system. In a number of practical cases when a bar model was used, the construction imperfection cannot always be expressed in greater detail. In this context, let us now concentrate our attention on the thin-walled box-type cross- section where the load-carrying capacity may be considerably affected by the buckling of its slender walls. The most buckled soft centre of the wall transfers relatively smaller loads than the rigid edges (comers) as a result of the irregular rigidity of the thin-walled cross-section. This is also in relation with the modelling of boundary conditions when, for example, by introducing end forces to relatively more rigid comers, higher values of load-carrying capacities may be obtained than in cases when the model mediates the transfer of reactive forces (external forces) uniformly to the whole cross-section or to the soft centres of the walls. The subject of our investigation will be how the change of the type of boundary conditions will affect the load-carrying capacity of the bar. Aiming to make a provision for the effect of the local buckling on the loss of stability as accurately as possible we modelled the problem by means of the shell finite elements. The load-carrying capacity of the bar was evaluated by a geometrically and physically non-linear analysis. The realizations of input imperfections were simulated using the LHS [7] method that makes it possible to simulate the realization of the data from the known statistics of the input values in a similar way as they could be obtained by direct measurements. Based upon the simulated input data, we evaluated three series of load tests for different structure types by introducing the load to the ends of the bar. Thus, the materialisation of the real boundary condition was determined when, due to various structural adaptations (end-stiffening plate, etc.), the load may be irregularly redistributed. The problems of the simplifying relations of EC3 [9] where the load-carrying capacity of the thin-walled column may be evaluated based on the internal forces determined by means of the column model, and where the boundary conditions are considered in a very simple way (the hinge in the cross-section centre of gravity) are also discussed in the paper. 2. CLASSIFICATION OF CLASS 4 CROSS-SECTION Eurocode EC3 [9] takes into consideration the effect of the local buckling with the aid of the coefficient flA = Aeff/A which is determined as the ratio of the so-called effective to the real surface of the cross-section, see Fig. lb. According to [9], the cross-section where the ratios of the breadth and depth of the compressed parts of the cross-section may be put in a class 4 classification at the web or flange plate respectively are

(h-20/t > 42 e,

(b-20/t> 42

c,

(1)

while particular parts of the cross-section may generally be in a different class. Parameter r for steel of S 235 series may be considered according to [9] by value 6 = 1. Then, the influence of local buckling in these types of cross-sections is considered only approximately by means of effective zones- see relation (2) presented hereinafter.

741

@

z

(~)

~ i i

Z

# i

Non-effective zone

.... ~y

.2

Gross

t~

cross-section b

~t

"

Effective cross-section with effective z o n e A.ff

Fig. 1" Class 4 cross-section, axial force In cases when the relation (1) is not fulfilled the column is put in class 3, and the loadcarrying capacity is determined irrespective of the influence of local buckling. Therefore, in our numerical study we focused our attention on the column, which although put in class 4, was analysed by means of a detailed FEA model. 3. CALCULATION MODEL was studied using a column of length L = 12 m, a rectangular shape of the cross-section according to Fig. la, of nominal value of breadth b = 0.5 m, height h = 0.3 m and sheet thickness t = 6 mm. The relative slenderness of the column is 2, = 1. Especially as far as the wall thickness is concerned, this is a very slender column. The reason why this column was selected was to emphasize the effect of the boundary arrangements. To solve the problem by means of FEA, the program system ANSYS was used. With respect to the symmetry of the column solved, and to the exacting character of the numerical calculation, only half of the column length was considered. The symmetry plane in x = 6 was prescribed by preventing the displacement in axis x and by the rotation around the axes y and z (UX, ROTY, ROTZ). The centrically compressed column was hinged in x = 0 m. The hinge was modelled by preventing the displacements UY, UZ at two points in the centre of shorter walls, which made the angular displacement of the end cross-section in plane xz possible. The column geometry was modelled by a four-node shell element provided in [2] as SHELL 43 which makes it possible to solve physical and geometrically non-linear problems. Non-linear material behaviour was included in the calculation by means of a bi-linear kinematics model without stiffening. The density of the elements network had to be selected high enough to enable the description of the local buckling of walls. The model was formed by 2000 finite elements with 2040 nodes. The solution of the problem results in the system of 12 116 equations. Aiming to obtain the information about the contingent error in the determination of the loadcarrying capacity due to different boundary conditions, we modelled the column in four different ways: Variant A) Parabolic distribution of the uniform load at wall edges, the stiffening of the end part- see Fig 2a. Variant B) Constant distribution of the uniform load at wall edges, the stiffening of the end part- see Fig 2b. Variant C)Local transmission of the reaction to the centre of the wall, the stiffening of the end part. Variant D) The same as in C, but the stiffening ring is of a higher rigidity (50 times).

742

9

B

Fig. 2 a, b: Variants of boundary conditions In Variant A, we assumed a redistribution of the end load that approximately corresponds to the stress distribution in remote cross-sections which are not influenced by the boundary condition model, e.g., at the column axis of symmetry. Besides this, the end cross-section was stiffened, which in a way, substitutes the stiffeners or another detail of the column attachment. Variant B is the same as Variant A, the difference rests in the fact that the load was distributed uniformly along the cross-section circumference. Therefore, we may expect that redistribution of internal forces towards the stiffer comers of the cross-section will occur 9 Variant C differs from the previous ones in fact that the load (reaction) was introduced to the centres of shorter walls (in the place where displacements UY, UZ are prevented). As a result of the poor utilization of the material for the transmission of the given load, the load-carrying capacity of this model may be expected to be lower compared with models in Variants A, B. Variant D roughly models a marked stiffening of the end attachment (only a deterministic analysis was carried out). Note that Variant C is probably out of the range of practical reality and serves here (as well the Variant D does) for comparative purposes only showing a certain limiting cases. The loadcarrying capacity was determined as the load under which the determinant of the structure stiffness matrix will equal, with the selected accuracy, to zero. 4. STOCHASTIC CALCULATION MODEL The question is, to what extent the standard relations are truthful. A large number of tests made on real columns (i.e. also with respect to imperfections) were difficult to produce and also from the economical point of view, and would not offer sufficient data for the evaluation of results with respect to the assumed dispersion and heterogeneity of imperfections. Since we know a number of statistically usable data for some columns obtained by measuring or taken from the tolerance standards, the evaluation of the load-carrying capacity based upon numerical simulations carried out by means of the computer is less troublesome [5]. The Yield limit was the first random quantity that was considered in the calculation. The yield point of strength class S 235 steel was expressed by a histogram (according to statistical assessment steel manufacturing in Czech Republic). The mean value of the histogram is 285.74 MPa, the standard deviation is 23.57 MPa and its skewness is practically negligible. Another imperfection that may substantially affect the load-carrying capacity is the initial curvature of the column. The shape of the curvature was introduced in a form of the sine function in interval O - re, both for the initial deflection in the plane of the original bending in direction of axis z (cot) and for the lateral deflection in direction of axis y (co2). With the amplitudes of the maximum initial deflection eol, eo2,, we considered a Gaussion distribution of probability with the mean value 0 (straight bar). The standard deviation for both deflection directions was considered Seol = Seol = 6 mm according to the rule 2 Sx.

743 The initial buckling of the profile walls was introduced to the model by the first proper shape of buckling standardized to the maximum deviation. Its value is assumed as a random quantity eo7, with the mean value meoz = O, and the standard deviation Seoz = 1.67 mm. The initial rotation of the walls of the box- type cross- section is another imperfection affecting, in a way, the load-carrying capacity, see Fig. 2. In random quantities eo3, eo4, we assumed the mean value of the normal distribution to equal to the value of 0 mm and the standard deviation being 1.25 mm. Similarly, in the second direction (eos, eo6) of the shorter wall, we also assumed the mean value of the normal distribution to be 0 mm and the standard deviation to be 0.626 mm. The nominal value of the wall thickness from Table 1 was considered to be the mean value of the normal distribution. The coefficient of variation equalled to the value of 0.07 in all cases, similarly as in ref. [5]. 5. THE STATISTIC ANALYSIS OF THE LOAD-CARRYING CAPACITY By the statistical analysis we understand the problem of determining the main statistical parameters of the construction response, in this case the limit load-carrying capacities of the centrically compressed thin-walled columns of the box-type cross-section. The analysis was made using the Latin Hypercube Sampling method, which in contrast to the Crude Monte Carlo method, offers sufficient accuracy even with a considerably smaller number of runs. This is a crucial and necessary requirement in our task. Here, each run is a non-linear solution of the shell by means of FE! 100 simulation runs were always used. Table 1 summarizes the results of this numerical study. As expected, Variant C leads to an unrealistically low value of the load-carrying capacity. Variant A (in our view the most realistic) leads to the highest load-carrying capacity. This has been verified by the values of statistical moments. However, the design value of the load-carrying capacity that is evaluated in the Table in harmony with Annex A [10] as 0.1% of the quantile for several types of probability distribution is decisive for design. The last of the above types has been selected as the most fitting one (Hermite distribution). The Table also presents detailed values for two distributions determined by means of the relations from Table A3 [10] that only represent the simplification based upon certain weight coefficients. The load-carrying capacity is determined according to EC3 is presented in the last part of Table 1. The difference is remarkable. Table 1 The values of statistical moments of the load-carryinl capacity Variant A Variant B Variant C Arithmetic mean 1304.2kN 1230.2 i~q 924~91 kN Statistical moments Stand. deviation 183.49 kN 176.36 kN 154.42 kN Stand. skewness 0.0130 0.1819 0.3627 Stand. kurtosis 3.0210 2.9575 2.8137 Normal iGauss) 737.191 kN 685.215 kN 447.709 kN 0,1% Lognormal 837.925 kN 783.683 kN 546.508 kN quantile Hermite 737.330 kN 731.956 kN 540.674 kN 0,1% quantile accord Normal (Gauss) 746.39 kN 694.0656 455.4732 to EC 1 table. A3, p. 6,~ Lognormal 850,35 kN 795.6197 556.7675 EC3 938.82 kN ....

In all variants, the hinge at the end cross-section was modelled by preventing the displacements on axes y, z at two points in the centre of the shorter walls. The hinged simple

744 beam whose load-carrying capacity is given in the last line of the Table was determined according to EC3, using relation 5.45, page 105: (2)

Nb.Ra = Z flA A f j, / YM, '

where flA = Aeff/A for the class of cross-section 4 holds, Z is the buckling coefficient, fy is the characteristic yield point and Yu~ is partial reliability coefficient. In relation (2) nothing is said about the detailed location of the external load and its consequent re-distribution along the bar length. The results of the statistical analysis are presented once again in Fig. 3 from which it is obvious that the standard result Nu=938 kN is close to the mean value rated for Variant C, i.e., the variant describing probably in the worst way the arrangement of the hinged support of the thin-wall column. The mean value, however, is not relevant with respect to the required reliability of the design. It is necessary to say that one solution of the imperfect column according to Variant D offered the load-carrying capacity that was identical with the corresponding imperfect column according to Variant B. The necessity of stiffening is pronounced. 0.0027 Relative Frequency 0.0024

Variant C

Variant B Variant A

0.0021 0.0019 0.0016 0.0013 0.0011 0.0008 0.0005 0.0003 0.0000 . 524

.

. 661

. . 798

. 935

. 1073 1210 1347 1484 Load-carrying capacity [kN]

1622

1759

1896

Fig.3 Probability distribution functions of variants A, B, C. Fig. 4 presents the correlation dependence between the results for Variants A and B, i.e., their considerable "uniformity" describes a certain information about the quality of statistical results. Each realization of the random load-carrying capacity plotted on the horizontal and vertical axes represents here a result of two "loading tests" of one column simulated by the computer. This is a great advantage of this procedure since after completing this "loading test", the real destruction of the column does not occur, and the test may be repeated for other boundary conditions as well. With a real bar loaded in the laboratory, the repeated measuring of the load-carrying capacity could not be accomplished on one bar, since after loading the bar to collapse (the maximum load-carrying capacity), a permanent deformation is introduced to the bar, and therefore its initial imperfections are modified.

745 A strong correlation between the results for Variants A and B suggests that the effect of the variability of input quantities upon the variability of the load-carrying capacity is practically the same for both variants. However, the load-carrying capacity of Variant B is lower on average. The extent of the sensitivity of the load-carrying capacity to particular input quantities (especially imperfections) and their statistics are presented in [4]. 31.5

Variant A (parabola)

[kN]

29.9 28.3 26.7

q," .',

9

mm 9

9

9

25.1 :" ,,,u

23.5

.~., ,~g.

21.9

.#,

20.3 18.7 17.1 15.5 15.5

[~] I

'

17.1

I

18.7

I

20.3

I

'1

I

I

21.9 23.5 25.1 26.7 Variant B (constant)

'

I

28.3

'

'

I

29.9

31.5

Fig. 4: Correlation between Variants A, B.

6. THE DISCUSSION It is evident from Table 1 that practically the highest values of the load-carrying capacity were obtained in Variant A for the parabolic distribution of the load where the maximum utilization of the material for the transfer of a given load may be assumed. The cross-section is loaded by such a distribution of the external load that corresponds to the load-carrying capacity of its particular portions. On the contrary, the lowest values of the load-carrying capacity were obtained in Variant C where the introduction of load in the form of a concentrated force in the middle of the shorter walls highly affected the total reduction of the load-carrying capacity compared with Variant A. The reduction of the load-carrying capacity in Variant C is caused by a different rigidity of the cross-section in comers and in the centre of the slender walls. In this way the validity the Bernoulli hypothesis is limited, which in its consequence, would highly affect the state of stress in the welded or bolted joints. Constructional adaptations and additional stiffening of the end cross-section in place of actions of forces are necessary, of course. The above-described results of the statistical analysis suggest that such an analysis may offer interesting data on the consequences of imperfections, on the importance of the truthful modelling of constructions and their details, and may, in this way, reveal the procedure that does not always comply with normative documents. However, these conclusions cannot be generalized since only one element was investigated here.

746 In conclusion we may say that when determining the load-carrying capacity according to the standard relation (2) on the implementation of boundary conditions should be considered with a high degree of caution. It is also necessary to focus our attention on welded and bolted joints through which the element is included in the system of bars. The irregular distribution of rigidity of the welded backing may influence not only the load-carrying capacity of the welded joint itself, but consequently even the distribution of reactive forces, and thus the stress of the basic material. The above studies should be enlarged by more detailed analyses of other boundary conditions and structural arrangements. It would be appropriate to analyse the bars of other cross-sections and other slenderness ratios too. The question is, what would be the effect on the load-carrying capacity of the perfectly rigid distribution plate, which would secure the coaction of all parts of the cross-section, and this way would limit the warping of the end edge.

This study was written during the solution of Project No 103/00/0603 of the Grant Agency of the Czech Republic References

1. O. Ditlevsen, H. O. Madsen: Structural Reliability Methods, Wiley, 1999. 2. ANSYS Element reference, Release 5.5, ANSYS, Inc., 1998. 3. Johnson, C. P., Will, K. M., Beam Buckling by FE Procedure, in Journ. Struct. Div. ASCE, ST3, 1974, pp. 6 6 9 - 685. 4. Kala, J., Kala, Z., Tepl3~, B., Skaloud, M." Probabilistic aspects in the interactive global and local buckling of thin-walled columns, Proceedings of the Third International Conference on Coupled Instabilities in Metal Structures CIMS '2000, Lisabon, 21th23rd.September 2000, pp. 121 - 128, ISBN 1-86094-252-0. 5. Kala, Z., Kala, J., Tepl)~, B." Non-linear analysis of a thin-walled steel beam computed with the random influence of imperfections, Proceedings of the Conference, Bratislava, 2 "d - 3ra October 2000, pp. 117 - 122 (in Czech). 6. Rondal, J., Batista, E., Stsability Problem of Thin-Walled Cold-Formed Steel Columns. in Stavebnicky ?asopis, No. 7, Bratislava, 1988. 7. ~kaloud, M.: The limit state of thin-walled columns with regard to the interaction of the deformation of the columns as a whole with the buckling of its plate elements, Acta technica (~SAV, No 1 1975. 8. Mc Kay,M.D., Beckman,R.J. and Conover, W.J., A comparison of three methods for selecting values of input variables, Technometrics, 2, pp 239-245, 1979. 9. ENV 1993-1-1, EUROCODE 3, Design of steel structures, PART 1-1: General rules and rules for buildings. 10. Eurocode 1 1994. Basis of Design and Actions on Structures, Part 1: Basis of Design. Brussels, CEN

Third InternationalConferenceon Thin-Walled Structures J. Zarag,K. Kowal-Michalskaand J. Rhodes 9 2001 ElsevierScience Ltd. All rights reserved

747

R O T A T I O N A L CAPACITY OF I-SHAPED A L U M I N I U M BEAMS: A N U M E R I C A L STUDY G. De Matteis 1, V. De Rosa I and R. Landolfo 2 University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy 2 University of Chieti G. D'Annunzio, Viale Pindaro 42, 65127 Pescara, Italy

ABSTRACT In the current study a non-linear finite element model aiming at obtaining a reliable evaluation of the rotational capacity of I-shaped aluminium alloy beams subjected to a moment gradient loading and prone to both local and global buckling phenomena is established. Such a numerical model is then used for a parametric analysis, so allowing determining an upper and a lower bound to the rotational capacity of the considered aluminium members as a function of the main influencing factors (local slenderness of the cross-section and global slenderness of the member) as well as of some other important factors (moment gradient and web stiffness). The results obtained show the importance, in some circumstances, of taking into account the steepness of the moment gradient and the torsional restraint provided by the web. But, above all, the paramount influence of interactions between flange and lateral-torsional buckling on the values of both the global rotational capacity and its stable part is unquestionably recognized. Therefore, a radical change in the cross-sectional classification approach provided by major structural codes seems to be necessary. KEYWORDS Aluminium Beams, Cross-Sectional Classification, FEM Model, I-Beams, Imperfections, LateralTorsional Buckling, Local Buckling, Rotational Capacity. INTRODUCTION For metal structures, rotational capacity is one of the most significant behavioural parameters of a member under monotonic actions. In fact, a correct evaluation of the member available ductility plays a fundamental role in assessing the available ductility of the whole structure, i.e. its capacity to redistribute plastic strains locally induced by particularly severe loading events, such as earthquake, explosions or structural impacts. Since many decades, in the field of rotational capacity, several works have been carried out for steel. In particular, the remarkable research on simply supported I-shaped steel beams under moment gradient has led to theoretical (Kemp and Dekker, 1991), semi-empirical (Kato, 1990; Mazzolani and Piluso, 1992) and empirical (Akiyama, 1985; Sedlacek and Spangemacher, 1992) models, allowing for the prediction of the rotational capacity of steel I beams by a simplified way. On the other hand, more accurate models based on the finite-element method have been proposed by Greshick et al. (1989), but their results did not fully fit the experimental ones, showing the difficulty of such an approach. Nonetheless, existing modem structural codes (namely, Eurocode 3 and 9, AISC LRFD, CSA S 16-2001) allow for a qualitative estimation of the member inelastic behaviour (with respect to both resistance and ductility) by means of the cross-sectional classification. Such a classification is based upon fixing suitable slenderness limits for each one of the individual structural elements composing the cross-section, so that

748 the whole section is classified according to the less favourable class (i.e. the highest normalised slenderness) among these elements. According to this approach, several geometrical and mechanical factors are generally taken into account as influencing parameters, namely: conventional elastic material strength, stress state on the section and on the considered element (axial/flexural compression), width-tothickness ratio of the element, manufacturing process of the member (welded/unwelded, hot rolled/cold formed), position of the element in the cross section (internal/outstand). On the other side, the simplicity of the approach does not allow considering several other important factors, which are partly common to all metal structures, and partly peculiar to the aluminium ones. Generally, lacks relevant to all metal structures spring from ignoring that the interaction between lateral-torsional buckling and local buckling may produce a relevant decreasing of the ductility of the whole (Earls, 1999). As far as aluminium beams are concerned, the available knowledge is still lacking (Landolfo, 2000). Several experimental analyses have been carried out by Mazzolani et al. (1996) and by Faella et al. (2000), in order to evaluate the ultimate resistance of heat-treated aluminium alloy sections failing in local buckling under uniform compression. Actually, these studies constitute the basis of the cross-sectional classification presently adopted by the Eurocode 9, in a similar approach as for steel. But these results do not consider the effect of stress gradient through the section and moment gradient through the member. Besides, selected specimens do not allow the effect of strain hardening to be suitably investigated (De Matteis et al., 1999). Finally, they do not take into consideration aluminium plastic anisotropy and the possibility of premature tensile failure due to reduced material ductility (De Matteis et al., 2000). Both Opheim (1996) and Moen (1999) have afterwards established numerical models for aluminium beam, in order to: (i) predict the load-deflection curve and so the rotational capacity of experimentally tested aluminium beams; (ii) conduct parametric studies on aluminium beams, aimed at improving the comprehension of their inelastic behaviour and the cross-sectional classification provided by structural design codes. In particular, the numerical model established by Moen et al. (1999b), using the non-linear implicit code ABAQUS (1997), was demonstrated to be capable of accurately predicting both the increasing and the decreasing branches of the behavioural curve, as far as the only local buckling of ductile and compact aluminium beams is concerned; on the contrary, it has shown to be not able to accurately predict the influence of interactions between local (flange and web) and global (lateraltorsional) buckling modes (I beams), so giving rise to considerable discrepancies between the decreasing branch of the numerical behavioural curve and the experimental one. Starting from the above considerations, the aim of the current study is twofold. Firstly, based on the numerical model established by Moen et al. (1999b), an investigation is carried out into the feasibility of establishing a non-linear finite element model, aiming at obtaining a reliable evaluation of the rotational capacity of I-shaped aluminium alloy beams subjected to a moment gradient loading and prone to both local and global buckling phenomena. Secondly, the proposed numerical model, once calibrated, is used: i) to conduct a parametric analysis so to establish both an upper and a lower bound to rotational capacity of the considered members as a function of the applied boundary conditions in the mid-span and of geometrical imperfections; ii) to determine and evaluate any influence of several parameters (moment gradient, web stiffness, local slenderness, global slenderness) on the lower bound to the rotational capacity, in order to assess their impact on the cross-sectional classification presently provided by relevant structural codes. ADOPTED NUMERICAL PROCEDURE

Analysed Scheme Aluminium beams investigated in this study are extruded profiles with I-shaped cross-section, made of AA 6082 alloy Temper T6, characterised by different values of both local and global slenderness. The analysed scheme is the same used for full-scale tests carried out by Moen et al. (1999a). Therefore, beams are simply supported and vertically loaded at the mid-span, by means of an actuator (a compact steel block with a total width of 150 mm) imposing displacements on the top of the upper flange of the beam.

749 At the supports, vertical and lateral displacements of the bottom flange are prevented, while no restraint is provided at the upper flange. As far as lateral restraints in the mid-span section are concerned, they have been experimentally provided by two rigid plates, adjacent to the central region of the beam. In the numerical model, for the sake of simplicity and computational efficiency, such a complex boundary condition is replaced by bilateral restraints set at the web-to-flange intersections. Furthermore, in order to avoid web crippling due to concentrated forces the beam length is extended beyond the end supports.

Basis of the Numerical Model The numerical model has been established by using ABAQUS/Standard (1997) non-linear FEM code. A 4-node shell element with reduced integration (S4R) has been used. A mesh refinement in the central region has been adopted to well interpret local contact and plastic buckling phenomena (Figure 1). The loading device has been modelled using a rigid surface, having the same geometry as the experimental one and imposing vertical displacements to the deformable surface at the upper flange of the beam. The ABAQUS/Standard default algorithm has been used to represent the contact between the master and the slave surfaces. The incremental loading process has been governed through the default automatic arclength control (Riks) procedure provided by the code (Abaqus, 1997).

- ~ 2 elts. ? 1

i

--- 8 elts.

z..

"

~ 50 mm

10 mm ~

q~--~~~

6 elts.

--r

[

1

1I

['i',l',',

l

~....

!

!

1'

i

1 O0 mm

------- 200 mm

Figure 1" The FEM model The multi-component strain-hardening model proposed by Hopperstad (1993) is adopted to describe the uniaxial stress-plastic strain behaviour of the material. The Hill yield criterion (1950) is used to take account for the experimentally observed plastic anisotropy (Moen et al., 1999b). Because of the lateral displacement restraint provided at mid-span, symmetry has been assumed about the transversal vertical plane (xy), so that only one half of the beam has been modelled. However, some numerical simulations have been performed in order to find out possible differences between the halfbeam model and the entire-beam one: the perfect coincidence of results, also in case of beams endowed with geometrical imperfections, has allowed using the simplified, less time-consuming model (see Figure 2a, which is related to the numerical simulation of the beam 'I1-2m' referred in Moen et al., 1999a,b). Some numerical simulations have been also carried out so as to investigate any possible effect of the coarseness of the mesh on obtained results (see Figure 2b, which is still related to the beam 'I 1-2m', but without using any geometrical imperfection in the model). Accordingly to Opheim (1996) and Moen et al. (1999b), it has been found that a very slight influence may appear in the decreasing branch of the behavioural curve of initially perfect systems. On the other hand, it has been noticed that, in case of imperfect systems, coarser meshes often prevent the numerical model from converging on the solution, this letting to prefer the fine-mesh model for performing the following numerical program.

Modelling of Geometrical Imperfections Generally speaking, extruded aluminium beams are characterised by small initial imperfections, really hardly detectable basing on visual inspections (Mazzolani 1995). Recent experimental studies on hollow rectangular thin-walled members (Opheim 1996, Moen et al. 1999a) have indicated that geometrical imperfections are very slight, even though measured data do not allow for general conclusions to be drawn

750 on their exact distribution and amplitude. On the other hand, as far as aluminium 1-section beams are concerned, it has been pointed out that imperfections are no more negligible. In particular, Moen (1999) recognized that beam out-of-straightness (i.e. deflection in the longitudinal planes) is relatively large, if compared to flange straightness (i.e. sinusoidal sweeps of the flange in the longitudinal vertical plane yz) and flange flatness (i.e. deviation of the mid-flange compared to the flange comer in a section). This sort of imperfection is very significant, since it gives rise to a consistent twist between the web and the flange and, as a consequence, to the possibility of influencing the potential buckling mode of the member. Opheim (1996) showed that an initial imperfection of the compression flange of I-sections, involving both flange flatness and beam straightness deviations, reduces the load carrying capacity of the beam obtained from numerical simulations, and the extent of such a decrease is also influenced by the imperfection magnitude. 90 80

,._, 80

70

~ 70

oo 60

~ 6o ~ 50

~o 50

~9 40

~ 40

g 30 ~

20

: ,7.

N 10

Entire-beam model

~ 30

Half-beam model

~9 20

odd [

rt

Coarse-mesh model

10

/ ! i

0 0

a)

50

100

150

M idsp an vertical disp lacement [mm]

200

b)

O

50

100

150

200

M idspan vertical displacernent [mm]

Figure 2: Effect of 'half-beam' modelling (a) and mesh coarseness sensitivity (b) It has been found by several researchers that introducing geometrical imperfection patterns in numerical models of beams, which are not prone to global buckling, has generally limited influence on their global behaviour. As an example, the numerical model established by Moen et al. (1999b) to simulate the inelastic behaviour of RHS and SHS aluminium beams showed a good agreement with the experimental results both in the initial model (where no geometrical imperfections had been introduced) and in the sensitivity study model (where geometrical imperfections at the compressed flange of the box section throughout the beam were applied). Therefore, it may be concluded that imperfections, either absent or characterised by a small magnitude, do not influence significantly the behaviour of hollow section beams in terms of both load carrying capacity and post-buckling behaviour. On the other hand, when dealing with members prone to global buckling phenomena, introducing a relevant imperfection pattern is deemed essential in order to get a reasonable failure mode. In such a case, the global behaviour of the structural member generally depends on both shape and magnitude of the imperfection pattern introduced in the numerical model (Galambos, 1998). Considering all the suggestions from the research works quoted above, it appears clear that a suitable representation of imperfections in I-shaped aluminium beams is generally advisable when dealing with local buckling only, and is mandatory when global failure modes and their possible interactions with the local ones are taken into account. It is generally assumed that an initial imperfection pattern is defined through an assigned shape and a scalar parameter, governing its magnitude. In order to establish such a function, in the current work, as also suggested by Espiga (1996), a linear elastic buckling analysis is firstly performed, aiming at evaluating the eulerian eigenmodes. Bifurcation analyses have shown that, within the range of both local and global slenderness parameters covered by tested beams, the first eigenmode is mainly "global", involving a horizontal translation and a rotation of the beam cross sections. Such eigenmode features can be easily recognised not only for high values of global slenderness, but also for those beam configurations highly prone to local buckling. Therefore, it is here deemed that global

751 imperfection patterns should be appropriately adopted in every numerical simulation. Once ascertained the global nature of the imperfection pattern, a standard shape, proposed by Earls (1999) for I-section high-strength steel profiles, has been adopted. The chosen imperfection (see Figure 1) is characterised by the rigid rotation, which is kept constant throughout the beam length, of both flanges (the same amount both for the upper and the lower one) as respect to the web, which is kept straight. In this way, a tendency for the beam to experience out-of-plane deformations is indirectly created, so giving rise to inelastic flexural modes characterised by coupling of local and lateral-torsional buckling. It has been undeniably pointed out that this sort of imperfection is both effective in order to obtain a suitable numerical model of collapse and post-collapse behaviour of high-strength steel beams under moment gradient, but it has also been shown that this imperfection pattern is the one favouring the most detrimental mode, it being related to the lowest rotational capacity (Earls 1999). Finally, it has been here verified that the imperfection magnitude (i.e. the flange rigid rotation q~),has a negligible influence on the overall response of the beam (Figure 3). Therefore, it does not represent a relevant factor to be taken into account in the performing numerical program. In particular, as imperfection magnitude, a value of q~= b/500 has been assumed for all the analysed cases. 90

90

(

80

q't

70

60

60

~

50

50

"~ 40

"~ 40 Experimental

= 30 .......

~ 20

~9

N lO

9

o

a)

/

.......

~ 20

~ = b/500

50

100

150

M idspan vertical displacement [mm]

200

b)

- - . . . .

-------q~b/lO00

N I0

q~= b / l ~ _ _ _ ~

. . . . .

Experimental "Perfect" mo(

= 30

"Perfect" model

~

"

rl 9

0 0

(

80

;~ 70

~o = b/500 ~ - b/100 w

0

50

100

150

200

M idspan vertical displacement [mm]

Figure 3: Effects of imperfection magnitude and mid-span restraint condition Model Calibration

The numerical model has been calibrated against available full-scale tests concerning 1-section profiles made of different aluminium alloys (Moen et al., 1999a). The FEM inelastic model originally established by Moen et al. (1999b), which did not consider any global geometrical imperfection ("perfect" model), provided an effective simulation for three of the four available experimental results, giving in one case (beam 'I1-2m') numerical results which were remarkably on the unsafe side (Figure 3). In such a case, simulations performed with the numerical model proposed here, endowed with the above initial geometrical imperfection pattern, have shown a very significant influence of the mid-span restraint condition on the slope of the decreasing branch of the behavioural curve. In particular, numerical analyses have been carried out for bilateral restraint preventing lateral displacements located both at both the upper and lower flange (Figure 3a) and at upper flange only (Figure 3b). Conservative result has been provided for the latter mid-span restraint condition. Therefore, for the following numerical test program, an "upper and lower bound criterion" has been applied in order to evaluate the global rotational capacity. In particular, the upper bound is given by the inelastic model adopted by Moen et al. (1999b), which is related to a beam without any geometrical imperfection so that lateral buckling modes are not activated in the numerical model, while the lower bound is yielded by the numerical model defined in the current paper, which is characterised by the above geometrical imperfection and the lateral restraint provided at the upper flange only (Figure 3b).

752 NUMERICAL TEST PROGRAM A parametrical study has been carried out in order to investigate the effect of several parameters (local and global slenderness, web restraining action and moment gradient) on the non-linear response of aluminium beams in bending. Since the analysis is mainly dealing with the effect of lateral-torsional buckling on the rotational capacity of the member experiencing plastic excursions, only class 1 and class 2 cross-sections (plastic and compact, according to the limits provided by present Eurocode 9) have been considered, as the other ones are assumed not to be able to develop conventional plastic moment. Furthermore, all the analysed cases have been devised in such a way the class of the cross-section is always determined by flange slenderness rather than by web slenderness.

2,oc=2eq/20has been accounted for by normalising the equivalent v2)/(7 9p).d/t with respect to 2 o = n'. where d is the plate element

Local (cross-sectional) slenderness slenderness 2eq = ~/12. (1-

width and t its thickness, while coefficients 7 and p depend on the relative boundary conditions and load shape, respectively (Mazzolani, 1995). Global slenderness has been expressed through the parameter

A~r = ~/a i We, "fo.2/Mcr, provided by Eurocode 9, where Mcr is the critical elastic buckling moment, We, is the elastic resistance modulus of the cross-section and a is the generalised shape factor of the crosssection, corresponding to an ultimate value of the curvature (it is equal to 10 times the elastic limit curvature and equal to the ratio Wv/Wet, for class 1 and class 2 cross-sections, respectively). Finally, according to De Matteis et al. (2001), the web restraining action has been controlled by the parameter kw=twS/h, where tw is the thickness of the web and h is the distance between mid-planes of the flanges, while the steepness of the moment gradient has been characterised through the beam compactness ratio L/b, where L is the half-span of the beam and b is the width of its cross-section. As far as the material is concerned, according to tensile tests reported in Moen et al. (1999a) relative to specimen I 1, the AA 6082 alloy Temper T6 considered in this study has been characterised through a conventional elastic strength f0.2=283 Nmm 2, a Young's modulus E=66.7 kNmm 2, a Poisson's ratio v=0.33, hardening parameters (the Hopperstad multicomponent stress-strain curve is considered) C~=2.559, Qt=31.20 Nmm 2, C2 =5.32, Q2=143 Nmm ~. It is worthy noticing that the first results obtained by numerical tests presented here are to be intended as dependent on the material characteristics, since normalisation is carried out with respect to E andfo.2 only, and not with respect to Ci and Qi. On the other hand, previous studies have shown the remarkable effect provided by the material strain hardening (De Matteis et al., 1999, 2001), which is being investigated by further analyses. Study cases have been selected in order to allow for the investigation of all the above parameters in a realistic and relevant range of variation. In particular, three different values have been considered for both local and global slenderness; besides, a specified value of web restraining action and beam compactness ratio parameters has been associated to the former and the latter, respectively. Nevertheless, for the sake of simplicity, for the highest value of the global slenderness (2Lr = 0.90), only the two values of 2tochave m

been considered. All main data of the examined cases (which, according to the above "upper and lower bound criterion", correspond to 16 numerical analyses) are given in Table 1 (tI = flange thickness). TABLE 1 MAIN DATAFOR BEAMSEXAMINEDIN THE NUMERICALTEST PROGRAM Beam No. 1 2 3 "'4" .........

5 6 -'~.......... 8

L

b

h

tw

ty

kw

L/b

Class

~oc

2~r

[mm] 1150 1250 1200

[mm] 105 120 126

[mm] 120 120 125

[mm] 12.0 10.0 15.0

[mm] 22.0 18.0 15.0

[mm2] 14 8 27

11 10 10

(EC9) 1 2 2

0.25 0.35 0.44

0.50" 0.50 0.50

~'6~ .......

i~6 .......

i2:15 .....

"2:~.6 ......

f4" ........

"1"f~ ......

f ..........

"67:~3- .... &:7"6""

"f966.....

2000 120 120 10.0 18.0 8 1950 126 125 15.0 15.0 27 3"6f3"6..... i-6~....... i]Z-6....... i~.15..... -;i2.6 ...... l'/l......... 3050 120 120 10.0 18.0 8

17 2 15 2 ~ ...... i .......... 25 2

0.35 0.70 0.44 0.70 "6~3".... /3.!i6"'0.35 0.90

753 OBTAINED RESULTS The structural response of analysed beams is presented in terms of normalised moment (M/Mo.2)-rotation (0/00.2) curves, where: M = P (L-D~2)~4, Mo.2=fo.2 We, 0 is the rotation at the beam support, 00.2=f0.z/E (L+D/2)/h, D being the extension of the load zone. Furthermore, according to Mazzolani and Piluso (1997), the obtained behavioural curves have been evaluated in terms of both the stable part of rotational capacity (130) and the global rotational capacity (13), defined as follows:

0u

0c-1

130 = 7 0 2 - 1 ;

(1)

13=00.2

where: 0u is the rotation corresponding to the maximum moment resistance, 0c is the rotation corresponding to the conventional collapse moment, which is attained when the decreasing branch of the moment-rotation curve reaches the conventional elastic moment resistance. In Figure 4, for each analysed case, both the behavioural curve relative to the perfect beam (upper bound model) and the one relative to the imperfect beam (lower bound model) are presented. From the examination ofbehavioural curves, it has to be noted that, when analysing beam 8, no solution has been obtained for the imperfect model. Such an event is not uncommon when performing numerical studies dealing with highly non-linear problems (Pekoz and Yogwook, 1999). Secondly, it may be observed that, generally speaking, the interaction between local and global buckling phenomena, while practically unaffecting the peak load-bearing capacity, may have a very great influence on the beam ductility. Nonetheless, the influence of interaction between local and global phenomena is certainly much more significant and extensive when considering the global rotational capacity 13. In fact, the stable part of the rotational capacity (130)often shows no substantial variations when comparing the case where only local buckling phenomena are allowed and the case where also lateral-torsional buckling takes place, while the characteristics of the unstable branch, and therefore the numerical value of 13, always show non negligible reductions when dealing with beams characterized by a relatively high value of the global slenderness. In particular, for the analysed cases, the rotational capacity 13is reduced up to one quarter (and, however, at least to the hal0 of the value provided by the upper-bound numerical model (Table 2). It is worthy noticing that the reduction shown by 130 and 13 as a consequence of the local-global interactions, and the extent of such a reduction, besides of the values of local and global slenderness, depends to some extent also on the other aforementioned parameters (i.e. moment gradient and web stiffness). As far as the steepness of the moment gradient is concerned, its influence can be obtained by examining the cases related to the model without any geometrical imperfection. It may be observed that the stable part of the rotational capacity is not significantly influenced, so as the beam loading bearing capacity, while the post-collapse behavioural branch (and then the global rotational capacity) is fairly sensitive to the L/b ratio (see Figure 5). Also, the effect of the L/b variation on 130 and 13 shows no dependence on the cross-sectional geometry (i.e. kw and Atoc ). m

TABLE 2 COMPARISONBETWEENROTATIONALCAPACITIESFOR THE TWO ADOPTEDMODELS

.

.

.

.

Beam

Modelwithoutimpe~ections

No.

~Q ....

~

~Q

1

4.0

14

2 3 4 5 6 7 8

2.2 2.7 3.2 2.0 2.3 2.6 1.6

14 12 9.0 8.4 7.2 6.1 6.3

3.2 2.2 2.7 2.0 1.8 1.9 1.2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Modelwithimperfections

.

.

~ = . =

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

7.8 4.7 6.6 3.6 2.3 2.3 1.5

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

754 1.6

1.6

...........

~am

1.4 1.4 l 1.2

1.2

1.0

1.0

0.6

~

i

0.4

-"

0.2 ] ~

~ : model with imperfections

0.0 I. i 0

.

model without imperfections

.

.

3

. 6

9

12

0.8 0.6

;,

0.4

!

0.2

i

0.0

15

- -"

model without imperfections model with imperfections

0

3

6

O/00.2

9

12

O/00.2

1.6

1.6

Baam 4

Beam 3

o

2

1.4

1.4

1.2

1.2

1.0

1.0

0.8

~

0.6

0.8 0.6

0.4 0.2

i

n

0.0

.

0

model with imperfections .

3

.

i

.

6

12

~.


0.2

[]

model with imperfections

0.0

i

9

0.4

0

15

3

6

9

12

0/00~

O/0o.2 1.6

1.6

14 t

,4

1.2

Beam 6

1.2

1.0 0.8 0.6

~

1.0

~

0.8 0.6

0.4 I~

-

modelwithout imperfections

0.4

_ _ _

0.2 ~

i

0.0

9 modelwithout imperfections

0.2

[]


0.0 0

3

6

9

12

15

0

3

6

0/00.2 1.6

Beam 7

1.4 1.2

1.6

Beam 8

1.4 1.2

~. 1.0

1.0

~9 0.8

-.. 0.8

0.6

0.6

o

A

0.4 i 0.2 0.0 .' 0

9

0 / 00.2


0.4


0.2

,

,

,

,

3

6

9

12

0 1 0o.~

hout imperfections

0.0 15

0

,

,

,

,

3

6

9

12

15

0 1 00.2

Figure 4: Comparison between behavioural curves yielded by the two adopted numerical models

755

--Ik

w=14 mm2;"~toc=O'25f

k w=27 mm 2" ~toc=0.44 [

9

9

~6

~6

r~ {3o

5

l0

15

L/b

20

25

~0

5

,

,

10

15

,

,

20

25

30

L/b

Figure 5' Influence of moment gradient on rotational capacity As far as the web stiffness is concerned, the selected study cases do not allow for significant conclusions to be drawn. Anyway, it has been recognized the slight influence of kw on the peak load-bearing capacity, unless very low values of this parameter are considered. On the contrary, higher values of the web stiffness are able to provide a somewhat decreasing of the slope of the descending branch of the beam behavioural curve, producing a corresponding slight increment of the global rotational capacity. On the other hand, local slenderness has certainly a paramount role in determining the rotational capacity of the beam, but the numericalm tests which have been performed clearly show that, once given the local slenderness parameters (i.e..2~oc and kw), the global rotational capacity is strongly affected by the global slenderness value as well. Figure 6 shows this effect on both 130and 13.For the sake of comparison, in such a figure local slenderness limits for class 1 and class 2 cross-sectional classification according to Eurocode 9 have also been reported. As far as {30is concerned, it may be easily noticed that, up to ALr - 0.50, the analysed case endowed with class 1 cross-section provides a rotational capacity higher than 3.0, which could be considered enough for classification purposes. On the contrary, when the global slenderness parameter exceeds this limit, the stable part of rotational capacity is affected by a relevant reduction, definitely higher that the one concerned with local slenderness (2~oc), emphasising the need to account for the member global slenderness in the definition of cross-section ductility classes. Such an effect is even more evident for the global rotational capacity {3, but it is to be mentioned that the present codified approaches do not account for the unstable part of rotational capacity in defining cross- section ductility classes. 8

8

7 6 5 e~. 4 3 2 1 0 0.20

,

7

O "~Lr = 0.50

6 5

r'l "~t,r = 0.70

9

2Lr

9x'--Lr=

0.70

o

,2Lr = 0.90

~.

9 "~r

4

= 0.90

3

|

D

2

Class 1 0.25

Class 2 0.30

0.35 m

~loc

t

1 0 0.40

0.45

0.50

i

Class 1

Class 2

i

0.20

0.25

0.30

,

|

0.35

0.40

0.45

0.50

B

~loc

Figure 6: Influence of global buckling phenomena on rotational capacity CONCLUSIVE REMARKS The results obtained by the performed numerical analyses show that the rotational capacity of aluminium 1-section beams is affected by several influential factors which are not only related to the geometry of the cross-section and to the slenderness of the plate element susceptible to local buckling. In particular, in some circumstances, taking into account the steepness of the moment gradient may be significantly important, since it affects the slope of the decreasing branch of the member behavioural curve. On the other hand, taking into account the effect of the restraint provided by the web on the compressed flange may be significant only in the case of very slender webs. But, above all, the performed numerical tests

756 unquestionably demonstrate the paramount influence of the interaction between local and global lateraltorsional buckling phenomena on both the global rotational capacity and its stable part. Eurocode 9 and also the major current structural codes for steel structures deal separately with these two phenomena, and cross-section ductility classes are deemed not to depend on their combined effect, which, on the contrary, may be noticeably important. Therefore, a radical change in the codified cross-sectional classification approach of metal structures seems to be required. References

ABAQUS/Standard (1997), Theory Manual- version 5. 7, Hibbit, Karlsson & Sorensen Inc., Pawtucket, Rhode Island. Akiyama H. (1985), Earthquake Resistant Limit State Design for Buildings, University of Tokyo Press, Tokyo, Japan. De Matteis G., Moen L., Hopperstad O.S, Landolfo R., Langseth M., Mazzolani F.M (1999). A parametric study on the rotation capacity of aluminium beams using non-linear FEM. Light-Weight Steel and Aluminium Structures (ICSAS'99), Elsevier (P. Makelainen and P. Hassinen, eds.), Oxford (UK), 637-646. De Matteis G., Landolfo R., Mazzolani F.M. (2000). Inelastic Behaviour of Hollow Rectangular Shaped Aluminium Beams. Finite element: Techniques and Developments (CST 2000), Civil Comp Press (B.H.V. Topping ed.), Edinburgh (UK), 373-379. De Matteis G., Moen L.A., Langseth M., Landolfo R., Hopperstad O.S., Mazzolani F.M. (2001). Cross-Sectional Classification for Aluminium Beams - A Parametric Study. Journal of Structural Engineering (ASCE), (under press). Earls C.J. (1999). On the Inelastic Failure of High-Strength Steel I-shaped Beams. Journal of Constructional Steel Research 49, 1-24. Espiga F. (1996). Numerical Investigations on Local and Global Inelastic Buckling Modes of I-Beams, 2"a Int. Conf. on Coupled Instability in Metal Structures, Liege, 85-92. Faella, C., Mazzolani, F.M., Piluso, V., Rizzano, G. (2000). Local Buckling of Aluminium Members: Testing and Classification. Journal of Structural Engineering, ASCE, 126(3), 353-360. Galambos, T. (editor) (1998). Guide to Stability Design Criteria for Metal Structures, 5th Edition, John Wiley & Sons, Inc. Greschik G., White D.W., McGuire W. (1989). Evaluation of the Rotational Capacity of Wide-Flange Beams Using Shell Finite Elements. Proc. Structures Congress (Vol. 3 ), Steel Structures, ASCE ed., New York, 590-599. Kato B. (1990). Deformation Capacity of Steel Structures. Journal of Constructional Steel Research 17:1, 33-94. Kemp A.R., Dekker N.W. (1991). Available Rotation Capacity in Steel and Composite Beams. Structural Engineer 69:5, 88-97. Landolfo, R. (2000). Coupled Instabilities in non-linear materials, 3rd Int. Conf. on Coupled Instability in Metal Structures, Lisbon, 263-272. Lay M.G., Galambos T.V. (1967). Inelastic Beams Under Moment Gradient. Journal of the Structural Division (ASCE) 93:ST1, 381-399. Lukey A.F., Adams P.F. (1969). Rotation Capacity of Wide-Flange Beams Under Moment Gradient. Journal of the Structural Division (ASCE) 96:ST6, 1173-1188. Mazzolani F.M., Piluso V. (1992). Evaluation of the Rotation Capacity of Steel Beams and Beam-Colunms, I s' Cost C1 Workshop, Strasbourg, France. Mazzolani F.M. (1995), Aluminium Alloy Structures (2"dEdition), Chapman & Hall, London, United Kingdom. Mazzolani F.M., Faella C., Piluso V., Rizzano G. (1996). Experimental Analysis of Aluminium Alloy SHS-members Subjected to Local Buckling Under Uniform Compression. 5'h International Colloquium on Structural Stability (SSRC), Brazilian Session, Rio de Janeiro, Brazil. Mazzolani F.M., Piluso V. (1997). Prediction of the Rotation Capacity of Aluminium Alloy Beams. Thin-Walled Structures 27:1, 103-116. Moen L.A. (1999). Rotational Capacity of Aluminium Alloy Beams. Doctoral Thesis, Norwegian University of Science and Technology, Trondheim, Norway. Moen L.A., Hopperstad O.S., Langseth M. (1999a). Rotational Capacity of Aluminium Beams under Moment Gradient. I: Experiments. Journal of Structural Engineering (ASCE) 125:8, 910-920. Moen L.A., De Matteis G., Hopperstad O.S., Langseth M., Landolfo R., Mazzolani F.M. (1999b). Rotational Capacity of Aluminium Beams under Moment Gradient. II: Numerical Simulations. Journal of Structural Engineering (ASCE) 125:8, 921-929. Opheim B.S. (1996). Bending of Thin Walled Aluminium Extrusions. Doctoral Thesis, Norwegian University of Science and Technology, Trondheim, Norway. Pekoz, P., Yogwook, K. (1999). Integrated Numerical Method and Design Provisions for Aluminium Structures, 2"d Aluminium Structures Workshop, Ithaca, New York. Sedlacek R., Spangemacher G. (1992). On the Development of a Computer Simulator for Tests of Steel Structures, Proceedings of the I s' World Conference on Constructional Steel Design, Acapulco, Mexico.

757

AUTHOR

Abruzzese, D. 269 Afonso, S.M.B. 373 Alexandrov, A. 603 Arafath, A.R.A. 277 Aribert, J.M. 161 Arizumi, Y. 161 Awrejcewicz, J. 349 Bakker, M.C.M. 417, 437 Bambach, M.R. 87, 719, 729 Bannikov, D.O. 619 Batista, E. 329 Bednarek, B. 103 Bond, W.F. 357 Bradford, M.A. 153 Cairns, R. 339 "Camotim, D. 329 Cerqueira, N.A. 533 Cheong, H.K. 365 Chusilp, P. 145 Croll, J.G.A. 643 Cui, S. 365 da C. P. Soeiro, F.J. 533 da Silva, J.G.S. 533 Davies, J.M. 3 De Matteis, G. 747 De Rosa, V. 747 Deg6e, H. 171 Dubina, D. 179, 187 Dunai, L. 203 Eisenberger, M. 603 Elbadawy, A. 339 Falco, S.A. 373,549 Falcon, G.S.A. 533 Fan Xuewei, 109 Foschi, R.O. 277 F6ti, P. 203 Fragos, A.S. 3 Fraldi, M. 653 Ftil6p, L. 187

Garstecki, A. 593 Gavrylenko, G.D. 643 Georgescu, M. 187, 195 Ghavami, K. 549

INDEX

Godoy, L.A. 661,669 Gomes, M.P.R.C. 541 Gongalves, P.B. 611 Gr~dzki, R. 469 Guarracmo, F. 653 Guezouli, S. 161 Gtinther, H.-P. 129 Gurba, W. 585 Hafidkowiak, J. 381 Hancock, G.J. 19, 449 Hao, H. 365 Hartono, W. 257 Hasham, A.S. 427 Holanda, A.S. 611 Huang, X. 477 Ikeuchi, T. 391 Imamura, K. 623 Inaba, Y. 209 Isozaki, A. 209 Janusz, L. 103 Jaskuta, L. 483 K~kol, W. 593 Kala, J. 739 Kala, Z. 739 Kasai, A. 145 Kasperska, R. 559 Kazakevitch, M.I. 619 K~dziora, P. 313 Khong, P.W. 499 Kim, Y. 437 Kleiman, P. 407 Kotakowski, Z. 293 Kotetko, M. 225,285, 293 Kowal-Michalska, K. 469 Krasovsky, V.L. 677 Krawczyk, P. 313 Kr61ak, M. 293 Krysko, A.V. 349 Krysko, V.A. 349 Krupa, R. 567 Kubo, Y. 459 Kuhlmann, U. 129 Kuwamura, H. 209 LaBoube, R.A. 241 Landolfo, R. 747

Li, H. 277 Li, Z. 499 Lopez-Bobonis, S. 669 Loughlan, J. 301,507 Lu, G. 477 Macdonald, M. 225 Magnucka-Blandzi, E. 567 Magnucki, K. 567 Mahaarachchi, D. 95 Mahendran, M. 95,523 Manevich, A.I. 575,683 Matsunaga, H. 491 McNiff, W. 225 Mendez-Degr6, J.C. 661 Morozov, G.V. 677 Muc, A. 313,585 Murzewski, J. 693 Nagahama, K. 329 Nagamatsu, H. 459 Nakamura, H. 623 Nanno, Y. 459 Nishimura, N. 391 Obr~bski, J.B. 321 Ostwald, M. 559 Pan, W. 399 Pawlus, D. 515 Pek6z, T. 417,437 Poldaarel, N. 523 Poursartip, A. 277 Raksha, S.V. 575 Rasmussen, K.J.R. 87, 217, 427, 719, 729 Ravinger, J. 407 Rhodes, J. 69, 225 Rogers, C.A. 19, 357 Roman6w, F. 381 Rzeszut, K. 593

Salehi, M. 631 Sarawit, A.T. 437 Shanmugam, N.E. 37 Shimizu, S. 119 Sikofi, M. 313 Silvestre, N. 329

758 Sivakumaran, K.S. (Siva) 233, 399 Skaloud, M. 137, 739 Stephens, S.F. 241 Szabo, I. 179 Szymczak, C. 53

Ungureanu, V. 179, 187 Usami, T. 145

Xuewei Fan 109

Vaslestad, J. 103 Vaz, L.E. 373 Vaziri, R. 277 Vrcelj, Z. 153

Takahashi, K. 623 Tepl3~, B. 739 Thompson, S.P. 301 Tremblay, R. 357 Turvey, G.J. 631

Walentyfiski, R.A. 701 Wang, P. 709 Watanabe, T. 145 Wilkinson, T. 449 Wright, H.D. 339

Yabuki, T. 161 Yamaguchi, E. 459 Yan, J. 249 Yang, D. 19 Yoshikawa, K. 119 Young, B. 249, 257

Zielniea, J. 483 Z~merovfi, M. 137

759 KEYWORDS INDEX actuation forces, 301 adaptive control, 301 ALGOR, 661 aluminum, 437 aluminium beams, 747 analysis, 321 analytical lower bounds, 643 analytical-numerical methods, 469 angle-ply, 491,500 annular, 515 antisymmetric, 507 any cross-sections, 321 arch-shaped corrugated shell roof, 109 arc-spot welds, 357 axial load, 643 beam capacity, 233 beam columns, 427 beam finite elements, 171 beams, 69, 449 bearing capacity, 103 bending, 69, 179 bending strength, 653 bifurcation, 709 bifurcation analysis, 661 bifurcation stresses, 329 bolted connections, 187 box cross-section, 623 box-beams, 241 breathing, 137 bridges, 129, 153 buckling, 3, 69, 249, 257, 277, 293, 301,313,407, 459, 477, 491,515, 549, 603,623,643,661,683 buckling analysis, 437 buckling and post-buckling, 129 buckling behaviour, 329 buckling curves, 195 buckling of composite panel, 499 built-up headers, 241 built-up sections, 187 bunker, 619

calculations exactness, 321 cassettes, 3 channel columns, 249 chaos, 349 characteristic exponent, 381 closed cross-section, 623 cold formed, 225 cold-formed C-profile, 203 cold-formed sections, 179 cold-formed steel, 187, 233,241,249, 257, 437, 449 collapse, 365,669, 709 columns, 69, 225 composite, 491 composite bars, 321 composite construction, 339 composite laminated plates, 469 composite materials, 269, 285,669 composite plates, 301, 313 composite structures, 293 composites, 585 compression, 69, 179 compressive stability, 507 computer algebra, 701 connection flexibility, 233 connections, 357 constitutive relations, 483,701 construction errors, 669 corrugated steel structure, 103 corrugation effect, 109 cost, 119 coupled instabilities, 195,293 coupled instability, 683 critical strength, 693 critical stress, 507 cross-sectional classification, 747 C-sections, 241 cumulative damage, 137 cut-outs, 313 cyclic loading, 145,459 cyclic tests, 399 cylinder, 709 cylindrical shell, 677, 693

760 cylindrical tubes, 653 damage, 313 damping, 365 degree of joint work, 619 design optimization, 499 design strengths, 249, 257 detail category, 129 digester tanks, 669 dimensional analysis, 541 dimensioning, 321 distortion, 171,623 distortional buckling, 153 distortional mode, 329 dual-actuator control, 719 ductility, 19, 145 dynamic buckling criterion, 365 dynamic relaxation, 631 dynamic stability, 407, 515 dynamics, 373 EC.3-Annex Z Procedure, 195 earthquake damage, 391 eccentric loading, 225 effective length, 249 effective width, 209, 523 efficient, 187 efficient points, 549 eigenproblem, 381 elastic, 153,701 elasticity, 631 elastic-plastic behavior, 457 elastic-plastic buckling, 693 elastic-plastic shells, 483 elasto-plastic postbuckling behaviour, 469 elephant-foot buckling, 391 elliptical barrel-shaped shell, 709 energy absorption, 357 energy procedure, 153 erosion, 137, 179 errors, 417 exact stiffness matrix, 603 experimental, 233 experimental investigation, 249, 257 experimental investigations, 683

experimental results, 195 experiments, 37, 95 expert system, 559 extended Kantorovich method, 603

fabricated thin-walled sections, 87 fabrication, 119 farm buildings, 399 fatigue, 129, 313 fatigue cracks, 137 FE analysis, 313, 585 FEA, 709 FEM, 217, 427, 619, 727 FEM model, 747 Ferrocement, 269 fiber reinforced plastics (FRP), 329 fiber-reinforced concrete, 269 fibre-reinforced laminated composite structures, 277 finite difference method, 349 finite differences, 631 finite differences method, 677 finite element analysis, 3,449 finite element mesh, 459 finite element method, 109, 203, 417, 437, 459, 499, 515, 611,739 finite elements, 373,661 finite strip analysis, 217 finite strip method, 507 fixed-ended, 249 fixed-ended columns, 257 flexible plates, 349 flexural buckling, 217 flexural-torsional buckling, 427 foam core, 477 folded plate, 161 foundation subsidence, 677 frames, 321 generalised beam theory (GBT), 329 generalized imperfection factor, 195 genetic algorithm, 499 genetic algorithms, 585 geometric imperfections, 427 geometric non-linear theory, 407

761 high strength steel, 19 higher-order theory, 491 homogeneous, 515 hurricane winds, 661 hybrid dynamic response analysis, 391 I-beams, 241,747 impact load duration, 365 imperfection, 179, 437, 643, 739 imperfections, 195,593,747 Ince-Strutt diagram, 381 initial imperfection, 119 initial imperfections, 407 m-plane loading, 37 Interaction buckling, 217 interactive buckling, 179 intermediate velocity impact, 365 IOF loading, 241 kinematic loading, 677 labour-saving, 119 laminated plate, 491 laminated plates, 507 large deflection, 631 lateral buckling, 567 lateral-torsional buckling, 195, 747 light, 187 limit moment, 653 limit state, 137 limit states, 739 linear complementarity problem, 611 linear-damage accumulation, 129 lipped channels, 225 load-capacity, 285 local buckling, 37, 161,171,209, 217, 427, 449, 523,729, 747 local plate mode, 329 manufacturing imperfections, 277 material model, 437 MA THEMA TICA, 701 MathTensor, 701

membrane-flexural coupling, 293, 507 metal cladding, 399

metal tanks, 661 micro-concrete, 339 mode shape, 661 monosymmetrical open cross-section, 567 multi-criteria minimization, 575 multicriteria optimisation, 559 natural half-wavelength, 507 Nervi, 269 Nickel-Titanium alloy, 301 nonlinear, 701 nonlinear analysis, 417 non-linear problems, 549 non-linear solution, 739 numerical methods, 643 numerical models, 171 optimization of volume, 585 orthotropic cylindrical shell, 477 orthotropic material, 329, 491 ovalization, 653 pallet racks, 233 pareto-optima, 575 pareto-optimum, 559 partial and ordinary differential equations, 349 partial safety factor, 693 penthouse, 187 periodic axial force, 381 perturbation theory, 381 photoelasticity, 313 pipe arch, 103 pipe section, 459 plastic hinges, 693 plasticity, 631 plate, 515 plate elements, 87 plate girders, 119 plate instability, 611 plate testing rig, 719 plates, 69, 585,603,611 polystyrene, 3 polystyrene foam core, 523 portal frame, 203

762 post-buckled reserve of strength, 137 postbuckling, 301 post-buckling, 3, 69, 477 post-buckling analysis, 437 post-buckling of plates, 611 post-failure behaviour, 285 potential energy, 477 pressure, 631 probabilistic simulation methods, 739 process modelling, 277 pull-through failures, 95 push-out tests, 339

railway bridges, 129 rectangular plate, 3685 reinforced concrete, 669 reliability, 277 repair of defects, 119 residual stress, 437 residual stresses, 87, 277 restraints, 153 return lips, 249 reversal cover shell structures, 541 reversal shell structures analysis, 541 reversal thin-walled shell structures, 541 review, 37 RHS, 449 ribbed wall, 619 Ritz method, 483 roof diaphragm, 357 rotation capacity, 449 rotational capacity, 747

sandwich, 515 sandwich panels, 523 sandwich shell, 559 sandwich shells, 483 scalarization, 549 secondary bending stresses, 129 sector, 631 seismic, 357 semi-rigid connection, 203,233 sensitivity, 593

sensitivity analysis, 53 sequential quadratic programming, 373 shape memory alloy, 301 shear, 3 shear buckling, 145 shear center, 623 shear connectors, 339 shear loading, 37 shear stiffness, 399 shear strength, 137, 399 shear wall, 399 sheet steel, 357 sheeting, 187 shell design optimisation, 541 shell design with graphics, 541 shell theory, 483 shells, 585,661,669 simplified method, 549 smart-structures, 301 soil-structure interaction, 103 spatial structure, 109 stability,19, 153, 161,407, 483,593 stability design, 145 stainless steel, 209, 225,257 statics, 53 steel, 3 steel bridge pier, 459 steel pier, 391 steel cladding systems, 95 steel light structure, 109 steel plate girders, 161 steel plates, 523 steel structures, 217, 249, 427, 533 steel telecommunication towers, 533 stiffened and unstiffened plate elements, 729 stiffened cylindrical shells, 683 stiffened plates, 549 storage tank, 677 strain energy, 483,701 strength, 119 stress concentration, 313 stress gradients, 719 stress range spectra, 129 structural analysis, 533 structural connections, 19

763 structural design, 249, 257 structural elements, 37 structural members, 19 structural optimisation, 533 structural optimization, 373,575 struts, 69 stub columns, 729 stub-column, 209, 233 symmetric shells, 643 tapered, 603 tendon force, 87 tensionless foundation, 611 tensor analysis, 701 test based design, 203 test strengths, 249, 257 testing, 3 tests, 217, 729 theory of plasticity, 483 theory of shells, 701 thin shells, 373 thin structures, 269 thin-walled, 37 thin-walled beam, 567 thin-walled column, 623 thin-walled columns, 329 thin-walled cross-sections, 739 thin-walled members, 69, 161,575 thin-walled orthotropic beams and columns, 285 thin-walled section, 209 thin-walled structure, 109

thin-walled structures, 53, 129, 339, 407, 593 through-girders, 153 tubular members, 257 ultimate capacity, 427, 729 ultimate loading, 49 unilateral contact, 611 unstiffened plate, 145 unstiffened plate elements, 719 upright frame tests, 233

valley-fixed, 95 variability, 277 variable thickness, 603 vector optimization, 549 vibration, 407 vibrations, 53 viscoelasticity, 515 von K/lrmgn equations, 349 warping function, 567 washers, 357 web breathing, 129 web crippling, 241 weld shrinkage, 87 wind pressures, 661 wind uplift, 95 wood frame, 399 Z-sections, 217


Thin-Walled Structures - Advances and Developments

Thin-Walled Structures: Advances and Developments

Advances in Steel Structures

Boiling Heat Transfer: Modern Developments and Advances

Advances in Tourism Economics: New Developments

Advances in Tourism Economics: New Developments

Multiple Sensorial Media Advances and Applications: New Developments in Mulsemedia

Essentials of Autopsy Practice: New Advances, Trends and Developments

Palgrave Advances in Peacebuilding: Critical Developments and Approaches

Essentials of Autopsy Practice. Topical developments, trends and advances

Jellyfish Blooms: Causes, Consequences and Recent Advances (Developments in Hydrobiology)

Ribonucleases: Structures and Functions

Crystals and Crystal Structures

Ecdysone: Structures and Functions

Structures and Strategies

Ribonucleases: Structures and Functions

Foliations and Geometric Structures

Carbon Bonding and Structures

Fullerenes and Related Structures

Physics and fractal structures

Ecdysone: Structures and Functions

Categories, types and structures

Fullerenes and Related Structures

Ecdysone, Structures and Functions

Structures and Structure Determination

Structures And Structure Determination

Physics and Fractal Structures

Data Structures and Algorithms

Specialist Yarn and Fabric Structures: Developments and Applications (Woodhead Publishing Series in Textiles)

Petroleum Biotechnology - Developments and Perspectives

Raman Microscopy: Developments and Applications

Thin-Walled Structures - Advances and Developments