LUI 39: Lehr- und Uandbiicher der Ingenieurwissenschaften Band 39

Birkhauser Verlag Basel· Boston· Stuttgart

J. Djubek, R. Kodnar and M. Skaloud Limit State of the Plate Elements of Steel Structures

1983 Birkhauser Verlag Basel· Boston· Stuttgart

Authors' addresses

Jozef Djubek Assoc. Prof., D. Sc., Civ. Eng. Head of Department at the . Institute of Structures and Architecture Slovak Academy of Sciences Bratislava (Czechoslovakia)

Miroslav Skaloud Assoc. Prof., D. Sc., Civ. Eng. Head of Department at the . Institute of Theoretical and Applied Mechanics Czechoslovak Academy of Sciences Prague (Czechoslovakia)

Rudolf Kodnar RNDr., Ph. D. Head of the Section of Mathematical

Modelling at the Institute of Applied Mathematics Comenius University Bratislava (Czechoslovakia)

Library of Congress Cataloging in Publication Data

Djubek, Jozef. Limit state of the plate elements of steel

structures. (Lehr- und Handbiicher der Ingenieurwissenschaften;

Bd.39) «Published in co-edition with VEDA, Publishing House

of the Slovak Academy of Sciences, Bratislava» - Verso t.p.

I. Building, Iron and steel. 2. Plastic analysis (Theory of structures). 3. Plates, Iron and steel. I. Kodnar, Rudolf. II. Skaloud, Miroslav. III. Title. IV. Series. TA684.D597 1983 624.1821 83-2764

CIP-KurztiteIaufnahme cler Deutschen Bibliothek

Djubek, Jozef: Limit state of the plate elements of steel structures / J. Djubek, R. Kodnar and M. Skaloud. - Basel; Boston; Stuttgart: Birkhauser, 1983.

(Lehr- und Handbiicher der Ingenieurwissen­schaften ; Bd.39)

NE: Kodnar, Rudolf:; Skaloud, Miroslav:; GT

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

© 1983 by J. Djubek, R. Kodnar, M. Skaloud (© of the translation: M.Skaloud)

Softcover reprint of the hardcover 1 st edition 1983

Licensed edition for the distribution in all non-socialist countries by Birkhauser Verlag, Basel 1983 Published in co-edition with VEDA, Publishing House of the Slovak Academy of Sciences, Bratislava

ISBN 978-3-0348-7499-1 ISBN 978-3-0348-7497-7 (eBook)

DOl 10.1 007/978-3-0348-7497-7

Contents

Prefal·e. . . . . . . . . . . . . . 11

1 Basic Assumptions of Theory of Slender Webs. . . . . . . . . . . . .. 15 1.1 Basic Notions and Formulation of Basic Assumptions . . . . . . . .. 15 1.2 Relationships between Web Deformations and Displacements in Its Middle

Plane . . . . . . . . . . . . . . . 17 1.3 Relationships between Deformations and Stresses 21

1.31 Hook's Law . . . . . . . . . . 21 1.32 Incremental Theory of Plasticity . . . 22 1.33 Ilyushin's Deformation Plasticity Theory 24

References

2 Mathematical Problems of the Fundamental Equations. . . . . . . . . 2.1 Introductory Remarks. . . . . . . . . . . . . . . . . . 2.2 Some Types of Equations from Considerations about the Flexure of Plates 2.3 Basic Boundary Value Problems

References . . . . . . . .

3 Approximate Methods of Solution 3.1 Compactness Method . 3.2 Topological Method . 3.3 Variational Methods . 3.4 Uniqueness of Solution 3.5 Two-Sided Estimations of Approximate Solutions.

References . . . . . . . . . .

4 Bifurcation Problems of Basic Equations 4.1 Properties of the Homogeneous Problem 4.2 Problem of Bifurcation Points . . . . 4.3 One Property of Problem (4.10, 4.11) .

References . . . . . . . . . . . .

5 Problems of Solution of a System of Non-Linear Algebraic Equations 5.1 Simple Iteration Methods 5.2 The Newton-Raphson Method 5.3 TheBoothMethod. . . . . 5.4 An Improvement of the Initial Approximation by Means of Extrapolation. 5.5 Method of Prolongation Df Solution with Respect to Parameter 5.6 The Perturbation Method . . . . . . . . . . . . 5.7 Perturbation Method for an Ideally Plane Web. . . . . .

5.71 Boundary Conditions, System of Algebraic Equations . 5.72 Numerical Results

References

27

29 29 30 35 40

42 42 47 51 54 55 59

61 61 65 67 69

70 70 71 72 74 75 76 81 81 84 86

6 Contents

6 Large Deflections of Eiasticisotropic Webs . . . . . . . . . . . .. 88 6.1 Introduction. . . . . . . . . . . . . . . . . . . . ., 88

. 6.2 A Slender Rectangular Web with Flanges Flexible in the Web Plane and Subject to Compression. . . . . . . 88

6.21 Formulation of the Problem 88 . 6.22 Boundary Conditions 91

. 6.23 Solution to the Equilibrium Equation 93 6.24 Numerical Results . . . . . . 95

6.3 A Slender Web Subjected to Compression and to Combined Compression and Bending, with Boundary Members Flexible in the Web Plane. Unsymmetrical Cross-Section . . . . . . . . . . . . . . . . . . . . .. 98

6.31 Formulation of the Problem, Equations and Boundary Conditions of the Web. . . . . . . . . . . . . . . . . . . . . .. 98

6.32 Numerical Results . . . . . . . . . . . . . . . . .. 104 6.4 Slender Webs Loaded in Shear with Boundary Members Flexible or Inflexible in

the Web Plane . . . . . . . 107 6.41 Theoretical Solution. . . . . . . . . . . . 107 6.42 BoundaryValueProblems. . . . . . . . . . 107 6.43 Numerical Solution to the Boundary Value Problem A . 11 0 6.44 Numerical Solution to the Boundary Value Problem B . 110 6.45 Comparison of the Theoretical Solution with Experimental Results 110

References 113

7 Large Deflections of Orthotropic Webs . . . . . . 115 A. Orthotropic Webs in Compression with Imperfections 115 7.1 Equations of Equilibrium . . 115

7.11 Rigidities . . . . 116 7.12 Cross-Sectional Forces 118

7.2 The Basic Differential Equations of Large Deflections of a Web with Structural Orthotropy . . . . . . . . . . . . . . . 120

7.21 Compatibility Equations (Membrane Equations) 120 7.22 Equilibrium Equations (Slab Equations) 121 7.23 Solution to the Compatibility Equation. 125 7.24 Boundary Conditions . . . . . 126 7.25 Solution to the Equilibrium Equation . 128

7.3 AnIsotropicWeb . . . . . . . . . . 131 B. Design of Longitudinally Stiffened Compression Flanges of Steel Box-Girder

Bridges . . . . . . . . . . . . . . . . . . . . . . 134 7.4 The Limit Load of a Compression Flange Plate with Structural Orthotropy 134

7.41 Reduction Factor mIN 135 7.42 Reduction Factor m2N 135

7.5 Numerical Results. . . . 140 7.51 A Parametric Study ofthe Problem 144

7.6 Approximate Relationship for the Determination of the Reduction Factor mIN 153 7.7 Comparison of Theoretical and Experimental Values 155

References . . . . . . . . . . . . . . . . . . . . . . .. 162

Contents 7

8 Large Deflections of Elasto-Plastic Webs 164 8.1 Introduction. . . . . . . . 164 8.2 Cyclic Plasticity. . . . . . . 164 8.3 Equations Equivalent to the Prandtl-Reuss Equations. 166 8.4 Relationships between Forces on Unit Length and Deformations. 169

8.41 Incremental Theory of Plasticity . 169 8.42 Parameters ib, . . . . . . . 170 8.43 Deformation Theory of Plasticity. 172 8.44 Parameters Cj • • • • • • 174

8.5 Residual Stresses . . . . . . . 176 8.6 Basic System of Differential Equations 178 8.7 Solution to the Equilibrium Equations 181 8.8 NumericalResults 181

References . . . . . . . 186

9 Ultimate Load Theories of Webs . . . . . . 187 9.1 Introduction. . . . . . . . . . . 187 9.2 Ultimate Load Behaviour of Webs in Shear . 187 9.3 The Rockey and Skaloud Theory for the Ultimate Load of Webs in Shear 189

9.31 Three Stages of the Behaviourofa Shear Girder . 189 9.32 Stage 1: Web Operates in Pure Shear 190 9.33 Stage 2: Web Operatesasa Tension Band. 191 9.34 Stage3:FailureMechanism . . . . . 195 9.35 Ultimate Load ......... 199 9.36 Extension of the Theory to Unsymmetrical Girders, to Webs Fitted with

Both Transverse and Longitudinal Stiffeners and to Webs Subject to Combined Shear and Bending. . . . . . 200

9.4 Other Ultimate Load Theories for Webs in Shear 201 References . . . . . . . . . . . . . 202

10 Large Deflections of Slender Webs Fitted with Ribs 203 10.1 Introduction. . . . . . . . . . . 203 10.2 Solution to the Problem . . . . . . . 204

10.21 Differential Equations and Boundary Conditions 204 10.22 Assumption for the Deflection Surface of the Web 205 10.23 Stress Function and Membrane Stresses 206 10.24 Relative Displacements of the Opposite Web Edges, Displacements

u and v, and Shear Deformation e . . . . . . . . 207 10.25 Evaluation of the Parameters W;j by the Energy Method . 210 10.26 Critical Stress of the Stiffened Web . . . . . . . . . 213 10.27 Deformation ofthe Stiffened Web in the Post-Buckled Range 215 10.28 Efficiency of the Stiffener in the Post-Buckled Range 218

References . . . . . . . . . . . . . . . . . . 219

11 Buckling of the Compression Flanges of Steel Box-Girder Bridges 220 11.1 Linear Buckling Theory of Compression Flanges . . . 220

11.11 Definition of the Optimum Rigidity of Stiffeners. . 220

8 Contents

11.12 Solution to the Stability Problems of Thin-Walled Plated Structures 223 11.13 Application of Folded Plate Theory to the Solution to the Stability

Problems of Thin-Walled Plated Structures . . . . . . . . . . 228 11.14 Stability Problem of a Compression Flange Panel Stiffened by Numerous

Longitudinal Ribs. . . . . . . . . . . . . . . . . . . 238 11.15 Stability Problem of a Compression Flange Panel Stiffened by a Small

Number of Longitudinal Ribs. . . . . 246 11.2 Post-Buckled Behaviour of Compression Flanges . . . . . . . . . .. 249

11.21 GeneralSolution. . . . . . . . . . . . . . . . . . . 249 11.22 Numerical Solution for a Compression Flange Stiffened by Two Lon-

gitudinal Closed-Section Ribs . 252 References . . . . . . . . . . . . . . . . . . . . . . 255

12 Interaction of the Buckling of Thin-Walled Bars with the Buckling of Their Plate Elements. . . . . . . . . . . . . . . . . . . . . . . 256

12.1 Critique of the Classical Concept of the Design of the Plate Elements of Compressed Bars . . . . . . . . . . . . . . . . . . . .. 256 12.11 Classical Concept of the Design of the Plate Elements of Compressed

Bars . . . . . . . . . . . . . . . . . . . . . . . 256 12.12 Critique of the Classical Concept . . . . . . . . . . . . . 256

12.2 Theoretical Investigation into the Interaction between Overall and Local Bucklingof"Actual"Thin-WalledBars. . 259 12.21 StabilitySolutiontotheProblem. . . . . . . . . . . . .. 259 12.22 Mathematical Solution to the Problem . . . . . . . . . . . . 265 12.23 Critical Loads of "Ideal" Thin-Walled Columns Calculated for an

Effective Cross-Section. . . . . 268 12.24 Results ofthe Investigation. . . .

12.3 Eccentrically Loaded Thin-Walled Columns 12.31 Stability Solution to the Problem . 12.32 Results ofthe Investigation.

References . Subject Index . . . . . . . . .

271 282 283 285 292 293

List of the Most Important Notations

Et3

~ = 12(1 ~ V 2 )

E

to h h* h,

L M x , My, Mxy N" Ny, Nxy

p

w

web panel length area of full cross-section of column area of effective cross-section of column stiffener cross-section area flange panel width (or web panel width) partial flange panel width (or partial web panel width) effective width of plate element distance of inner plastic hinge in flange from web corner

flexural rigidity of unit width of flanges eccentricity of compressive load Young's modulus of elasticity of flange (or web) mate­rial amplitude of initial curvature of column depth of longitudinal stiffener optimum depth of longitudinal stiffener depth of transverse stiffener radius of inertia of full cross-section of column moment of inertia of effective cross-section of column moment of inertia of flange moment of inertia of stiffener optimum moment of inertia of longitudinal stiffener buckling length of column, flange panel length (i.e. spacing of transverse stiffeners) length of whole flange internal moment (per unit width) in flange (or web) internal in-plane forces (per unit width) in flange (or web) compressive load acting on column critical load of column calculated for its full cross-section critical load of column calculated for its effective cross­section limit load of column shear forces (per unit width) in flange (or web) thickness of web sheet thickness of flange sheet displacements in direction of axes x, Y buckling of flange (or web) initial curvature of flange (or web)

10

x, y, z Y Ymax Yo Z,

E1, Y=Db

y*

'T/ V

A =l/i a

a:".mu

a' a

a. = y a~ + a~ - apy + 3 'f.y

at a;

List of Most Important Notations

coordinate axes deflection of stiffener (or of column) maximum deflection of stiffener (or of column) initial curvature of column, initial deflection of stiffener plastic modulus of flange

ratio of flexural rigidity of stiffener to flexural rigidity of flange (or web) optimum rigidity of stiffener resulting from linear buck­ling theory

non-dimensional parameter of plastic behaviour of flange efficiency factor of stiffeners Poisson's ratio of flange (or web) material slenderness ratio of column compressive load acting on flange critical load of stiffened flange critical load of stiffened flange related to nodal-line buckling mode maximum value of critical load of stiffened flange related to nodal-line buckling mode critical load of corresponding unstiffened flange critical load of stiffened flange related to symmetrical buckling mode critical load of partial flange panel of stiffened flange determined on assumption that partial flange panel is plate simply supported on all boundaries

effective (comparison) stress

. tension band stress value of tension band stress at which the tension band subjected to a combination of shear stress 'fa and tension band stress at starts to yield stresses in flange (or web) shear stress in web critical shear stress in web ultimate shear stress in web inclination of tension band inclination of geometrical diagonal

Preface

The necessity to save steel leads to a marked tendency towards thin-walled structures. Such structures are made of thin plating, the behaviour - and, of course, design - of which is very significantly affected by stability phenomena. In fact, with up-to-date thin-walled steel plated structures, it is very frequently the point of view of stability that governs the design. So it is not astonishing that the attention of a great number of research teams in various parts of the world has been for a good many years directed to investigations into numerous aspects of the buckling behaviour of steel plated structures. However, the current problems of buckling research, which require to account for the effect of initial imperfections, post-buckled behaviour and plastic reserve of strength (this leading in theoretical research to the necessity to solve boundary value problems of geometrically and physically non-linear partial differential equations, and in experimental studies to conduct experiments on full-size test girders) are very complex and time-consuming. Then it is beyond the means of one investigator, or even of one research team, to deal successfully with such problems and, conse­quently, effective cooperation is indispensable. This was also the reason for the initiation of a fruitful collaboration between the first author of this book (Assoc. Prof. J. Djubek, D.Sc.) and the third author (Assoc. Prof. M. Skaloud, D.Sc.) and, of course, also between their respective Departments, namely, (i) the Department of Mechanics of Struc­tural Systems of Homogeneous Materials at the Institute of Structures and Architecture of the Slovak Academy of Sciences in Bratislava, headed by the first author, and (ii) the Stability Department at the Institute of Theoretical and Applied Mechanics of the Czechoslovak Academy of Sciences in Prague, the Head of which is the third author. These working contacts have been in operation for about twenty years now, and its fruits are numerous. About twenty-five papers and other contributions, dealing with various aspects of the post-critical behaviour and limit state of plate elements of steel structu­res, have so far resulted from the cooperation. Both authors also played an important part in the preparation of a number of Czechoslovak Design Standards. For example, the chapter "Strength of Webs and Flanges" in the Czechoslovak Design Specifications CSN 73 1401 "Design of Steel Struc­tures" (1976) and in the new edition of CSN 73 6205 "Design of Steel Bridge Structures" is based on the work of both writers and their associates. The first and third authors were also awarded, for their achievements in

12 Preface

stability and other research, a number of Czechoslovak distinctions, among them being also the Czechoslovak Klement Gottwald State Prize, presented to them in 1975. Both authors also had the opportunity to spend extended periods of time in well-known universities abroad; for example, the first author carried out his three years' post-graduate study at the Leningrad Technical University (USI) and spent one year in Professor K.C. Rockey's Department at University College, Cardiff, and the third author (i) one year in working with Prof. Ch. Massonnet at University of Liege and (ii) three years in working with Professor K.C. Rockey at Swansea and Cardiff. These stays gave them inspiration and encouragement for their following research activities. They also profited from scientific contacts with numerous Czechoslovak and foreign colleagues and research teams, whether this be through personal contact or through the membership of the third author in a number of international working commissions and task groups. During the last years the authors were frequently asked to sum up their to-date research results in a monograph, which would provide some guidance for a modern design of webs and other plate elements with regard to their post-buckled behaviour; so they eventually directed their activities towards this aim. That was the beginning of work on this publication. Design of steel structures (both buildings and bridges) in Czechoslovakia and a good many other countries is currently based on Limit State Design philosophy. Two limit states are considered in the corresponding analysis, viz.

a) limit state ultimate, which works with factored loads, b) limit state of serviceability, operating with non-factored (working)

loads. So it is understandable that the authors' research, and also the objective of the envisaged book, was bound to focus on providing a reliable scientific background for the determination of the limit states of the plate elements of up-to-date constructional steelwork. As the number of pages available for the prospective monograph was rather limited, it was indispensable to restrict the scope of the book so that only the main results of the authors' most important theoretical studies could be included in the publication. When deciding to write a monograph about their theoretical studies, the first and third writers took the view that it was desirable to explain in the publication at least some basic parts of the mathematical background needed to solve those boundary value problems of non-linear partial differential equations to which an investigation into the post-buckled behaviour of thin webs and other plate elements leads. So they invited the second author (Dr.

Preface 13

R. Kodnar, Ph.D.), Head of the Section of Mathematical Modelling at the Institute of Applied Mathematics of Comenius University in Bratislava, who had intensively cooperated with the first writer in his research, to join them as a member of the writers' team for the contemplated book. The authorship of the individual chapters then developed as follows. The first author (1. Djubek) wrote Chapters 1,6, 7 and 8, the second author (R. Kodnar) Chapters 2, 3, 4 and 5 and the third author (M. Skaloud) Chapters 9, 10, 11, 12. The preface was written jointly by the first and third authors. Nevertheless, all three writers cooperated closely in the course of the whole work, discussing all important points of the pUblication. The English translation of the text was carried out by M. Skaloud. The authors wrote this monograph in the hope that it will contribute, jointly with the striving of other research teams, to further progress in the know­ledge about the limit states of thin-walled steel plates structures. To conclude this preface, the writers would like to thank Academician J. Nemec, Director of the Institute of Theoretical and Applied Mechanics of the Czechoslovak Academy of Sciences, Prague, and Academician R. Sknicany, Director of the Institute of Construction and Architecture of the Slovak Academy of Sciences, Bratislava, for their continuous support and encouragement, which substantially contributed to the success of the authors research activities. Further, the writers are indepted to Professor A. Schindler, D.Sc., the scientific editor of the book, and Professor V. Krupka, D.Sc. and Professor A. Tesar, D.Sc., who as reviewers of the manuscript helped them - by numerous valuable comments and constant encouragement - to improve many parts of the text, figures and tables. The authors are also pleased to be able to take this opportunity to express their thanks to their numerous co-workers, whether these be members of their respective Departments in Prague or Bratislava, or work in other institutes, for their prolonged cooperation. The names of some of them are several times quoted in the publication, this highlighting the value of their work. Last but not least, the authors want to mark their gratitude to Mrs. Kostanjevcova for typing the manuscript and to Mr. Both for drawing the figures. Their assistance made it possible to bring the work to a successful end.

Bratislava-Prague November, 1981