Thin-Walled Structures with Structural Imperfections

THIN-WALLED STRUCTURES WITH STRUCTURAL IMPERFECTIONS

Analysis and Behavior

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TANAKA & CRUSE Boundary Element Methods in Applied Mechanics CHRYSSOSTOMIDIS BOSS '94, Behaviour of Offshore Structures OEHLERS & BRADFORD Composite Steel and Concrete Structural Members: Fundamental Behaviour SHANMUGAM & CHOO 4th Pacific Structural Steel Conference- Structural Steel Related Journals (free specimen copy gladly sent on request)

Advances in Engineering Software Composite Structures Computers and Structures Construction and Building Materials Engineering Analysis with Boundary Elements Engineering Failure Analysis Engineering Structures Finite Elements in Analysis and Design International Journal of Solids and Structures Journal of Constructional Steel Research Marine Structures Mathematical and Computer Modelling Ocean Engineering Reliability Engineering and System Safety Soil Dynamics and Earthquake Engineering Structural Engineering Review Thin-Walled Structures

THIN-WALLED STRUCTURES WITH STRUCTURAL IMPERFECTIONS Analysis and Behavior

by Luis A. Godoy University of Puerto Rico at Mayaguez Puerto Rico, USA

PERGAMON

UK

Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK

USA

Elsevier Science Inc., 660 White Plains Road, Tarrytown, New York 10591-5153, USA

JAPAN

Elsevier Science Japan, Tsunashima Building Annex, 3-20-12 Yushima, Bunkyo-ku, Tokyo 113, Japan Copyright 9 1996 Elsevier Science Ltd All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publisher. First edition 1996

Library of Congress Cataloging-in-Publication Data Godoy, Luis A. (Luis Augusto) Thin-walled structures with structural imperfections : analysis and behavior /by Luis A. Godoy.-- 1st ed. p. cm. Includes index. 1. Thin-walled structures. 2. Structural analysis (Engineering) I. Title. TA660. T5G63 1996 624. 1'71--dc20 95-53855

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 08 042266 7

Printed and bound in Great Britain by BPC Wheatons Ltd, Exeter

Para mis padres Matilde y Juan Carlos, que siempre me apoyaron

This Page Intentionally Left Blank

Preface Thin-walled structures are designed with sophisticated numerical analysis techniques and constructed using fabrication processes requiring highly skilled workmanship. There are, however, a number of factors that may result in a structure that is not exactly coincident with what was considered during the design calculations. These features may be associated with changes in the properties of the structure (such as non-uniformity or local variations in the physical properties of the materials), in the geometry (deviations in the shape, eccentricities, local indentations), and many others. But even small changes in the structure may sometimes produce significant changes in its response. There are numerous excellent books on shell theory, analysis, and behavior that can be very useful to researchers and engineers. Unfortunately, little information is available in such books about the influence of structural imperfections. The present work is intended to introduce professionals and researchers to the effects of imperfections on the stresses in thin-walled structures. The main idea behind the presentation is that small imperfections may introduce changes in the stresses that are nearly equal to the stresses due to the loads. This was realized during the last 25 years, thanks to the pioneering contributions by M. Soare, C. Calladine, J. G. A. Croll, K. O. Kemp, and several others. The book is organized into two main parts. The first part (Chapters 2 to 6) covers the techniques for analyzing imperfections. Chapters 3 and 4 are the most important because they introduce the direct, equivalent load and perturbation techniques in intrinsic and in geometric imperfections. The techniques of analysis are simple, and some mechanical models are analyzed in detail. In the second part the emphasis is on applications which at present may be found scattered in scientific and professional journals. The vii

viii

Thin-Walled Structures with Structural Imperfections

core of part two is contained in Chapters 9 to 11 about spherical and cylindrical shells and cooling towers. The behavior of such shells is simple, but not obvious. More practical aspects of imperfections are discussed in Chapter 12. It is assumed that the reader is familiar with finite element techniques, and with the basics of thin-walled structures. Two appendices are included, to cover basic aspects of perturbation techniques and the main equations of shells. In the selection of topics there is a certain bias towards my main fields of interest; for example, the book is primarily concerned with elastic structures. The book is not intended as a comprehensive treatise on the subject, nor is it exhaustive, but it can be a valuable preparatory reading to the work of other authors, so many references are included at the end of each chapter. I hope that those engineers with a different orientation and interests from mine will nevertheless find this book useful. Because the book is not written for any specific course I have not included suggested problems at the end of each chapter. However, it can be used as supplementary material in courses on Theory of Plates and Shells. It was very difficult to have complete uniformity in the notation, mainly because of the many sources used. The reader will notice minor changes in notation between chapters, and I hope that he or she forgives me for that. My interest in imperfect thin-walled structures began during my doctoral studies at University College London, and increased when I had to work as a consultant in several cases of large reinforced concrete shells and spherical nuclear containment that had damage or imperfections. Many people have contributed to my own understanding in this field, and to the developments presented here. Certainly, the most important influence has been Prof. J. G. A. Croll, who introduced me to this area and with whom I worked for several years. I also thank Prof. K. O. Kemp for his guidance, and Dr. C. P. Ellinas, all of them at University College in London. The book is largely based on investigations carried out at the University of Cordoba in Argentina (19851993) and at the University of Puerto Rico at Mayaguez (1994-1995). Special thanks are due to Dr. C. Prato, and to my former students and colleagues in Cordoba Dr. F. Flores, S. Raichman, and N. Novillo.

Preface

ix

During short visits to Mexico and Barcelona, I worked in this field with Prof. P. Ballesteros, Prof. E. Onate and Prof. B. Suarez. I discussed many topics of spherical shells with Dr. G. Sanchez-Sarmiento and R. Carnicer at the Nuclear Agency for Electricity Plants of Argentina. The discussions and work that I did with all those people led to the publication of papers, and those form the basis of this book. As a member of the scientific staff of the Science and Technology Research Council of Argentina (CONICET), I had the opportunity to carry out research without other academic duties. I also thank the adequate environment provided by the Civil Infrastructure Research Center of the University of Puerto Rico. The support and encouragement of the director of the Department of Civil Engineering at the University of Puerto Rico, Prof. I. Pagan Trinidad, has been particularly important. Generous support for my research on shells from several funding agencies over the years has also helped to make this book possible. Grants from CONICET, from the Science and Technology Research Council of the Province of Cordoba in Argentina (CONICOR), and from Puerto Rico EPSCoR-National Science Foundation have been particularly important. I should thank several other people: J. Gambino and C. Olgado prepared the drawings, M. Gotay and D. Dayton, both of the University of Puerto Rico, read the manuscript and improved the style. Dr. J. Milne was the publishing editor of Elsevier, and he encouraged me to continue with this work that I had interrupted for some time. Most of all, I thank my wife Nora for her love, patience, and encouragement.

Luis A. Godoy Mayaguez, November 1995

This Page Intentionally Left Blank

Contents 1

INTRODUCTION 1.1 T H I N - W A L L E D S T R U C T U R E S AND I M P E R F E C T I O N S ................ 1.1.1 Thin-Wa, lled S t r u c t u r e s . . . . . . . . . . . . . . 1.1.2 Imperfect T h i n - W a l l e d S t r u c t u r e s . . . . . . . . 1.1.3 S t r u c t u r a l Consequences of I m p e r f e c t i o n s . . . . 1.2 C L A S S I F I C A T I O N O F I M P E R F E C T I O N S IN T H I N WALLED STRUCTURES ................ 1.2.1 A Criterion Based on t h e R e l a t i o n Between an Imperfection and t h e Loading Process . . . . . . . . . . . . . . . . . . 1.2.2 A Criterion Based on the Space D i s t r i b u t i o n of I m p e r f e c t i o n s . . . . . . . . . . 1.2.3 Imperfections in Intrinsic a n d in Geometric Parameters .............. 1.3 S O M E S H E L L S T R U C T U R E S IN W H I C H I M P E R F E C T I O N S MAY B E I M P O R T A N T . . . . . . . . . . . . . . . . . 1.3.1 Reinforced C o n c r e t e Cooling Towers . . . . . . 1.3.2 Reinforced C o n c r e t e Silos a n d Tanks . . . . . . 1.3.3 Spherica,1 Steel C o n t a i n e r s . . . . . . . . . . . . 1.3.4 Shallow C o n c r e t e Shells . . . . . . . . . . . . . 1.3.5 T u b u l a r M e m b e r s in Off-Shore S t r u c t u r e s . . . 1.3.6 C o m p o s i t e T h i n - W a l l e d S t r u c t u r e s . . . . . . . 1.4 A F E W S I T U A T I O N S IN W H I C H IMPERFECTIONS ARE CONSIDERED . . . . . . . . . . . . . . . . . . . . . . xi

1 1 1 2 4

5

5 6 7

9 9 11 12 14 14 17

17

Thin-Walled Structures with Structural Imperfections

,o

Xll

1.5

2

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

1.5.1

Scope and O b j e c t i v e of this Book

1.5.2

A b o u t the C o n t e n t s . . . . . . . . . . . . . . . .

SINGLE DEGREE-OF-FREEDOM IMPERFECTIONS 2.1

INTRODUCTION

2.2

I M P E R F E C T I O N S IN GEOMETRIC PARAMETERS

2.3

2.4

2.5

2.6

3

O U T L I N E OF THE B O O K

21 ........

SYSTEMS

21 22

WITH 26

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

26

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

28

2.2.1

Initial Strain Model of I m p e r f e c t i o n . . . . . . .

29

2.2.2

Equilibrium Condition

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

31

2.2.3

P e r t u r b a t i o n Analysis

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

31

2.2.4

Simplified P e r t u r b a t i o n Analysis . . . . . . . . .

34

2.2.5

E q u i v a l e n t Load Analysis

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

35

2.2.6

Simplified E q u i v a l e n t Load . . . . . . . . . . . .

36

2.2.7 N u m e r i c a l Results ................ I M P E R F E C T I O N S IN I N T R I N S I C PARAMETERS . . . . . . . . . . . . . . . . . . . . . . 2.3.1 P e r t u r b a t i o n Analysis . . . . . . . . . . . . . .

37 38 40

2.3.2

42

E q u i v a l e n t Load Analysis

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

2.3.3 N u m e r i c a l Results . . . . . . . . . . . . . . . . I M P E R F E C T I O N S IN B O T H G E O M E T R I C AND I N T R I N S I C PARAMETERS . . . . . . . . . . . . . . . . . . . . . . 2.4.1 P e r t u r b a t i o n Analysis . . . . . . . . . . . . . . 2.4.2 E q u i v a l e n t Load Analysis . . . . . . . . . . . .

44 45 46

MAXIMUM VALUES OF T H E RESPONSE . . . . . . . . . . . . . . . . . . . . . . . .

47

2.5.1

General One Degree-of-Freedom Systems . . . .

47

2.5.2

S t r u c t u r a l Systems

48

CONCLUDING REMARKS

IMPERFECTIONS

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

IN I N T R I N S I C

PARAMETERS

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

43

49

52

3.1

INTRODUCTION

3.2

BASIC E Q U A T I O N S . . . . . . . . . . . . . . . . . . .

54

52

3.2.1

Stiffness M a t r i x . . . . . . . . . . . . . . . . . .

54

3.2.2

Models of Degeneracies in Intrinsic Parameters ....................

56

Contents

xiii

3.2.3

3.3

3.4

E x a m p l e s of Functions for Intrinsic ~imperfections . . . . . . . . . . . . . . . . . . . D I R E C T ANALYSIS . . . . . . . . . . . . . . . . . . 3.3.1 Finite E l e m e n t E q u a t i o n s . . . . . . . . . . . .

56 58 58

3.3.2 R e m a r k s on the Direct Analysis . . . . . . . . . E Q U I V A L E N T L O A D ANALYSIS . . . . . . . . . . . 3.4.1 Equivalent Load . . . . . . . . . . . . . . . . . .

59 61 61

3.4.2 3.4.3 3.5

3.7

3.8

4

R e m a r k s on the Equivalent Load Analysis

P E R T U R B A T I O N ANALYSIS . . . . . . . . . . . . . 3.6.1 E x p a n s i o n of the Thickness . . . . . . . . . . . 3.6.2 P l a t e with Thickness C h a n g e . . . . . . . . . . 3.6.3 Perturbation Equations . . . . . . . . . . . . . . 3.6.4 Stress R e s u l t a n t s . . . . . . . . . . . . . . . . . 3.6.5 Relation Between Equivalent Load and P e r t u r b a t i o n Techniques . . . . . . . . . . . 3.6.6 R e m a r k s on P e r t u r b a t i o n Analysis . . . . . . . TWO DEGREE-OF-FREEDOM SYSTEM WITH INTRINSIC IMPERFECTIONS .................... 3.7.1 Basic f o r m u l a t i o n . . . . . . . . . . . . . . . . . 3.7.2 Intrinsic P a r a m e t e r s . . . . . . . . . . . . . . . 3.7.3 Solution via P e r t u r b a t i o n Analysis . . . . . . . 3.7.4 Equivalent Load Analysis . . . . . . . . . . . . FINAL R E M A R K S . . . . . . . . . . . . . . . . . . . .

SYSTEMS WITH GEOMETRICAL IMPERFECTIONS 4.1 4.2

....

3.4.4 Errors in the Iterative Scheme . . . . . . . . . . EXPLICIT F O R M OF THE EQUIVALENT LOAD . . . . . . . . . . . . . . . . . . 3.5.1 Mindlin P l a t e with Thickness Changes . . . . . 3.5.2 Mindlin Plate with Step C h a n g e in Thickness . 3.5.3 Shell of Revolution with Thickness C h a n g e . . . 3.5.4

3.6

Kirchhoff Plate with Thickness Changes Iterative Solution Scheme . . . . . . . . . . . .

INTRODUCTION .................... M O D E L S OF G E O M E T R I C IMPERFECTIONS .................... 4.2.1 Cosine Imperfection Profile . . . . . . . . . . .

. . .

62 64 66 67 68 69 70 72 73 73 74 75 77 78 78

79 79 83 84 86 89

95 95 97 98

Thin-Walled Structures with Structural Imperfections

xiv 4.2.2 4.3

4.4 4.5

4.6

4.7

P o l y n o m i a l I m p e r f e c t i o n Profile . . . . . . . . .

99

DIRECT ANALYSIS . . . . . . . . . . . . . . . . . . .

101

4.3.1

102

Some p r o b l e m s in t h e direct analysis

......

4.3.2 A few Solutions . . . . . . . . . . . . . . . . . . INITIAL STRAIN MODEL OF GEOMETRIC IMPERFECTIONS ...........

105

EQUIVALENT LOAD ANALYSIS ........... 4.5.1 The Equivalent Load ............... 4.5.2 Itera, tive Solution S c h e m e . . . . . . . . . . . . 4.5.3 C o n v e r g e n c e of t h e I t e r a t i v e Solution . . . . . . EXPLICIT EQUIVALENT LOAD ........... 4.6.1 Explicit E q u i v a l e n t L o a d in Shallow Shells . . . 4.6.2 Simplified E q u i v a l e n t Loads . . . . . . . . . . . 4.6.3 Discussion on t h e E q u i v a l e n t L o a d Analysis . .

109 109 110 111 111 112 114 I15

PERTURBATION 4.7.1 4.7.2

ANALYSIS

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

Perturbation Equations .............. Stress R e s u l t a n t s . . . . . . . . . . . . . . . . .

116 117 119

4.7.3

4.8

4.9

5

Relation Between Perturbation and Equivalent Load Techniques . . . . . . . . . . . 4.7.4 Discussion on t h e P e r t u r b a t i o n Analysis . . . . . TWO DEGREE-OF-FREEDOM SYSTEM WITH GEOMETRIC IMPERFECTION . . . . . . . . . . . . . . . . . . . . 4.8.1 The Problem Considered ............. 4.8.2 Basic F o r m u l a t i o n . . . . . . . . . . . . . . . . . 4.8.3 E q u i v a l e n t L o a d AnaIysis . . . . . . . . . . . . 4.8.4 Simplified E q u i v a l e n t Loa, d . . . . . . . . . . . . 4.8.5 P e r t u r b a t i o n Analysis . . . . . . . . . . . . . . 4.8.6 Numerical Results ................

103

4.8.7 Remarks . . . . . . . . . . . . . . . . . . . . . . IMPERFECTIONS DEFINED IN T E R M S O F T W O PARAMETERS . . . . . . . . . . . . . . . . . . . . .

119 20

120 120 121 124 126 127 128 129

130

SYSTEMS WITH INTERACTING IMPERFECTIONS

135

5.1 5.2 5.3

135 136 139

INTRODUCTION . . . . . . . . . . . . . . . . . . . . DIRECT ANALYSIS . . . . . . . . . . . . . . . . . . . PERTURBATION ANALYSIS . ............

Contents 5.3.1

6

Solution of the P e r t u r b a t i o n E q u a t i o n s

NON-LINEAR STRUCTURES

ANALYSIS

.....

144

INTRODUCTION

6.2 6.3

D I R E C T ANALYSIS . . . . . . . . . . . . . . . . . . . P E R T U R B A T I O N ANALYSIS O F GEOMETRIC IMPERFECTIONS ........... 6.3.1 Non-Linear E q u a t i o n s of E q u i l i b r i u m . . . . . . 6.3.2 P e r t u r b a t i o n E q u a t i o n s and Solution . . . . . . 6.3.3 An A l g o r i t h m for the Solution of P e r t u r b a t i o n E q u a t i o n s . . . . . . . . . . . . P E R T U R B A T I O N ANALYSIS O F INTRINSIC IMPERFECTIONS .............

6.5

6.6

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

7.3

144 145 146 146 147 149 150

6.4.1 Non-linear E q u a t i o n s of E q u i l i b r i u m . . . . . . 6.4.2 P e r t u r b a t i o n E q u a t i o n s and Solution . . . . . . E X A M P L E OF NON-LINEAR ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Direct Analysis . . . . . . . . . . . . . . . . . . 6.5.2 P e r t u r b a t i o n Analysis . . . . . . . . . . . . . .

153 153 156

FINAL REMARKS . . . . . . . . . . . . . . . . . . . .

159

PLATES AND PLATE ASSEMBLIES IMPERFECTIONS 7.1 7.2

142

OF IMPERFECT

6.1

6.4

7

xv

150 151

WITH

INTRODUCTION .................... FINITE STRIPS FOR PLATE BENDING . . . . . . . . . . . . . . . . . . . . . . . . . PLATES WITH THICKNESS CHANGES . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Equivalent Load Analysis . . . . . . . . . . . . 7.3.2 P e r t u r b a t i o n Analysis . . . . . . . . . . . . . .

162 162 163 164 164 168

7.4

PLATE ASSEMBLIES WITH THICKNESS CHANGES ................

171

7.5

PLATES WITH IMPERFECTIONS IN T H E M O D U L U S . . . . . . . . . . . . . . . . . . . 7.5.1 Basic F o r m u l a t i o n . . . . . . . . . . . . . . . . . 7.5.2 Equivalent Load M e t h o d . . . . . . . . . . . . . 7.5.3 P e r t u r b a t i o n Analysis . . . . . . . . . . . . . .

172 176 177 182

FINAL R E M A R K S . . . . . . . . . . . . . . . . . . . .

183

7.6

Thin-Walled Structures with Structural Imperfections

xvi

8

IMPERFECT 8.1

INTRODUCTION

8.2

E Q U I V A L E N T L O A D IN GEOMETRIC IMPERFECTIONS 8.2.1

8.3

8.4

9

SHALLOW SHELLS

185

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

185 ...........

186

Simplified Imperfection Analysis . . . . . . . . .

186

ELLIPTICAL PARABOLOIDAL SHELL . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

8.3.1

Equivalent Load . . . . . . . . . . . . . . . . . .

189

8.3.2

Equivalent Load R e p r e s e n t a t i o n . . . . . . . . .

190

8.3.3

Results for a Specific Imperfection . . . . . . . .

190

8.3.4

Influence of Imperfection P a r a m e t e r s

191

......

FINAL REMARKS . . . . . . . . . . . . . . . . . . . .

197

IMPERFECT SPHERICAL SHELLS 9.1 I N T R O D U C T I O N .................... 9.1.1 Short L i t e r a t u r e Review . . . . . . . . . . . . . 9.2 T O O L S O F ANALYSIS . . . . . . . . . . . . . . . . . 9.2.1 Direct Finite Element Model of the Imperfect Shell . . . . . . . . . . . . . . . . . . 9.2.2 Equivalent Load Analysis . . . . . . . . . . . . 9.3 A X I S Y M M E T R I C G E O M E T R I C A L IMPERFECTIONS .................... 9.3.1 Stresses in an Imperfect Shell . . . . . . . . . . 9.3.2 Influence of Shell and Imperfection P ar ameters . . . . . . . . . . . . . . . . . . . . 9.3.3 Simplified Analysis . . . . . . . . . . . . . . . . 9.3.4 9.4

200 200 203 204 204 206 211 211 215 221

Conclusions about Axisymmetric Imperfections . . . . . . . . . . . . . . . . . . .

223

LOCAL GEOMETRICAL IMPERFECTIONS ....................

223

9.4.1

Stresses in an imperfect shell . . . . . . . . . . .

223

9.4.2

Influence of Shell and Imperfection Parameters ....................

224

9.4.3

Simplified Analysis

230

9.4.4

Comparison Between Axisymmetric and Local Imperfections ...............

9.4.5

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

Conclusions about Local Imperfections

230 .....

232

Contents 9.5

9.6

xvii

NON LINEAR BEHAVIOR OF GEOMETRICALLY IMPERFECT SPHERES . . . . . . . . . . . . . . . . . . . . . . . . .

235

9.5.1

Stresses in an Imperfect Shell

235

9.5.2

Influence of Imperfection P a r a m e t e r s

9.5.3

D e n t e d Imperfections . . . . . . . . . . . . . . .

242

9.5.4

Conclusions Concerning Non-Linear Behavior . . . . . . . . . . . . . . . . . . . . . .

244

T H I C K N E S S V A R I A T I O N S IN SPHERES . . . . . . . . . . . . . . . . . . . . . . . . .

246

9.6.1

Stresses in a Shell with Thickness Changes . . .

246

9.6.2

Linear versus Non-linear Analysis . . . . . . . .

250

9.6.3

Conclusions about Thickness Changes

250

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

.....

10 C Y L I N D R I C A L S H E L L S WITH IMPERFECTIONS 10.1 I N T R O D U C T I O N

240

256

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

10.1.1 Short L i t e r a t u r e Review

256

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

10.2 I N F L U E N C E O F M E R I D I O N A L IMPERFECTIONS ....................

257 257

10.2.1 Equivalent Load . . . . . . . . . . . . . . . . . .

258

10.2.2 Ring Solution

259

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

10.2.3 Bending Solution 10.2.4 Case S t u d y

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

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

260 260

10.2.5 O t h e r Imperfection Shapes . . . . . . . . . . . .

262

10.3 I N F L U E N C E O F L O C A L DAMAGE . . . . . . . . . . . . . . . . . . . . . . . . .

263

10.4 L O C A L I M P E R F E C T I O N S IN LARGE VERTICAL CYLINDERS

268

...........

10.4.1 Bulge Imperfections . . . . . . . . . . . . . . . . 10.5 I N T E R A C T I O N B E T W E E N G E O M E T R I C A L AND MATERIAL IMPERFECTIONS

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

269

274

10.5.1 A Simplified Model . . . . . . . . . . . . . . . .

274

10.5.2 Reinforced Concrete Shell

276

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

1 0 . 5 . 3 0 r t h o t r o p i c Model of an Imperfect Cylinder . . . . . . . . . . . . . . . . . . . . . .

277

xviii

Thin-Walled Structures with Structural Imperfections

10.6 I N T E R A C T I O N B E T W E E N GEOMETRICAL IMPERFECTIONS 10.6.1 Behavior of with Cracks 10.6.2 Influence of 10.6.3 Influence of

AND C R A C K S . . . . . . . . . . . an I m p e r f e c t Shell . . . . . . . . . . . . . . . . . . . . Shell F o r m . . . . . . . . . . . . . . Imperfection Form . . . . . . . . .

10.6.4 Influence of Crack P a r a m e t e r s . . . . . . . . . . 10.6.5 Discussion . . . . . . . . . . . . . . . . . . . . . 10.7 E X P E R I M E N T A L S T U D I E S . . . . . . . . . . . . . .

278 278 286 288 290 294 295

10.8 I N F L U E N C E O F C H A N G E S IN T H I C K N E S S . . . . . . . . . . . . . . . . . . . . .

297

10.9 C O N C L U S I O N S

298

11 I M P E R F E C T

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

HYPERBOLOIDS

OF REVOLUTION

11.1 I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . 11.1.1 Short L i t e r a t u r e Review . . . . . . . . . . . . . 11.2 S T R E S S E S IN A P E R F E C T COOLING TOWER . . . . . . . . . . . . . . . . . . . 11.3 I N F L U E N C E O F M E R I D I O N A L IMPERFECTIONS . . . . . . . . . . . . . . . . . . . . 11.3.1 Mechanics of Stress R e d i s t r i b u t i o n . . . . . . . 11.3.2 Influence of t h e P a r a m e t e r s t h a t Define the Shell and Imperfection . . . . . . . . . . . . . . . . . . 11.4 I N F L U E N C E O F CIRCUMFERENTIAL IMPERFECTIONS . . . . . . . . . . . . . . . . . . . . 11.4.1 Mechanics of Stress R e d i s t r i b u t i o n . . . . . . . 11.4.2 P a r a m e t e r s Influencing Behavior . . . . . . . . . 11.5 I N F L U E N C E O F L O C A L IMPERFECTIONS . . . . . . . . . . . . . . . . . . . . 11.5.1 Case S t u d y

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

11.5.2 A n t i s y m m e t r i c Imperfections

..........

11.5.3 Bulge I m p e r f e c t i o n s . . . . . . . . . . . . . . . . 11.6 I N T E R A C T I O N B E T W E E N GEOMETRIC IMPERFECTIONS AND CRACKS . . . . . . . . . . . . . . . . . . . . . . 11.6.1 M o t i v a t i o n . . . . . . . . . . . . . . . . . . . . . 11.6.2 Case S t u d y

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

304 304 306 308 310 310 313

316 316 319 321 321 323 324

325 325 327

Contents

xix

11.7 C R E E P O F I M P E R F E C T COOLING TOWERS ................... 11.7.1 Creep Models . . . . . . . . . . . . . . . . . . . . 11.7.2 Creep in an Imperfect Cooling Tower Shell . . . 11.7.3 Combined Creep and Changes in Stiffness . . . 11.8 F I N A L R E M A R K S . . . . . . . . . . . . . . . . . . . .

12 I M P E R F E C T I O N S

IN PRACTICE

12.1 I N T R O D U C T I O N .................... 12.2 S U R V E Y I N G G E O M E T R I C A L IMP E R F E C T I O N S . . . . . . . . . . . . . . . . . . . . 12.2.1 Surveying Cooling Towers, Silos, and O t h e r Large Reinforced Concrete Shells . . . . . 12.2.2 Surveying Large Stiffened Metal Shells . . . . . 12.2.3 Scanning Small Scale Metal Shells . . . . . . . . 12.3 D E T E C T I O N O F I M P E R F E C T I O N S IN T H E MATERIAL . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Material Imperfections in Reinforced Concrete Shells . . . . . . . . . . . . . . . . . . 12.3.2 Masonry Structures . . . . . . . . . . . . . . . . 12.4 S O M E R E C O R D S O F STRUCTURAL IMPERFECTIONS .................... 12.4.1 Examples of Imperfections in the Aerospace I n d u s t r y . . . . . . . . . . . . . . . . 12.4.2 Examples of Imperfections in the Off-Shore I n d u s t r y . . . . . . . . . . . . . . . . 12.4.3 Examples of Imperfections in Cooling Towers . . . . . . . . . . . . . . . . . . 12.4.4 Examples of Imperfections in Spherical Shells . . . . . . . . . . . . . . . . . . 12.5 C O L L A P S E O F S T R U C T U R E S WITH IMPERFECTIONS ................ 12.5.1 Failure as a Guide to Success . . . . . . . . . . 12.5.2 Ferrybridge, England, 1965 . . . . . . . . . . . . 12.5.3 Ardeer, Scotland, 1973 . . . . . . . . . . . . . . 12.5.4 Fiddlers Ferry, England, 1984 . . . . . . . . . . 12.5.5 O t h e r Collapses . . . . . . . . . . . . . . . . . .

330 330 332 333 334

342 342 343 343 345 348

348 348 351

352 353 355 356 359 361 361 364 364 365 366

Thin-Walled Structures with Structural Imperfections

XX

12.6 P O S S I B L E E X P L A N A T I O N S FOR COLLAPSES . . . . . . . . . . . . . . . . . . . . 12.6.1 Collapse M e c h a n i s m for A x i s y m m e t r i c Imperfections . . . . . . . . . . . . . . . . . . . 12.6.2 An e s t i m a t e of t h e collapse load . . . . . . . . . 12.6.3 Discussion . . . . . . . . . . . . . . . . . . 12.6.4 Collapse M e c h a n i s m for C i r c u m f e r e n t i a l fections . . . . . . . . . . . . . . . . . . . 12.7 G E O M E T R I C T O L E R A N C E LIMITS . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Design R e c o m m e n d a t i o n s . . . . . . . . . . . . 12.7.2 Reinforced C o n c r e t e Silos . . . . . . . . . . . . 12.7.3 Cooling Towers . . . . . . . . . . . . . . . . . . 12.8 F I N A L R E M A R K S . . . . . . . . . . . . . . . . .

371

. . .

372 372 373 373 374

. . .

371

381

13.1 I N T R O D U C T I O N . . . . . . . . . . . . . . . 13.2 P L A T O ON T H E I M P E R F E C T WORLD, AND OTHER CONTRIBUTIONS . . . . . . . . . . . . . . . 13:2.1 P l a t o . . . . . . . . . . . . . . . . . . . 13.2.2 al-Qabisi . . . . . . . . . . . . . . . . . 13.2.3 Galileo . . . . . . . . . . . . . . . . . . 13.2.4 Brunel . . . . . . . . . . . . . . . . . . 13.2.5 Buckling enters t h e Field . . . . . . . . . . . . . 13.2.6 Flugge . . . . . . . . . . . . . . . . . . 13.2.7 New P a r a d i g m . . . . . . . . . . . . . . . . . . 13.2.8 Stress R e d i s t r i b u t i o n s .............. 13.2.9 N e o - P l a t o n i s m ..................

. . . . .

381

. . . . .

382 382 382 383 384 384 385 386 387 387

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

13.2.10 S u m m a r y . . . . . . . . . . . . . . . . . . . . . 13.3 O T H E R A P P R O A C H E S . . . . . . . . . . . . . . . . .

A.1 A.2 A.3

366 369

. . . Imper. . .

13 C L O S U R E

A B A S I C S OF P E R T U R B A T I O N

366

TECHNIQUES

388 388

392

INTRODUCTION . . . . . . . . . . . . . . . . . . . . TAYLOR EXPANSIONS ................. EXPLICIT AND IMPLICIT

392 393

PERTURBATION TECHNIQUES ........... A.3.1 P e r t u r b a t i o n via Implicit Differentiation . . . . A.3.2 P e r t u r b a t i o n via Explicit S u b s t i t u t i o n .....

394 394 396

Contents A.4 A.5 A.6

DEGENERATE AND NON-DEGENERATE PROBLEMS ........... REGULAR AND SINGULAR PERTURBATIONS . . . . . . . . . . . . . . . . . . . . FINAL REMARKS . . . . . . . . . . . . . . . . . . . .

B SHELL E Q U A T I O N S B.1 B.2

SUMMARY OF SHELL THEORY ........... APPLICATIONS . . . . . . . . . . . . . . . . . . . . .

SUBJECT INDEX

xxi

396 397 398

400 400 404

407

This Page Intentionally Left Blank

Chapter 1 INTRODUCTION This book is about the influence of structural imperfections on the static response of thin-walled shells and plates. This introductory chapter explains wha.t we understand by structural imperfections (Section 1.1) and how they arise in real thin-walled structures. There are several ways to classify imperfections, some of which are shown in Section 1.2. This is followed by examples in some common shell structures, where imperfections have been found to be important (Section 1.3). The reasons why we may need to understand the influence of imperfections are discussed in Section 1.4, with reference to practical situations in which an interested party needs to pursue this kind of investigation. Finally, Section 1.5 contains an outline of the book, stating the scope, objectives, contents, and possible uses of the work. ,

1.1

1.1.1

THIN-WALLED STRUCTURES AND IMPERFECTIONS Thin-Walled Structures

Thin-walled structures are used as structures or structural components in many engineering applications, including civil, naval, aeronautical, mechanical, chemical, and nuclear engineering. Usual geometries of such structures are thin-walled cylinders, spheres, assemblies of flat plates, and shells of revolution. One of the main advantages of such structural forms is that they are very efficient in carrying loa.ds acting

Thin-Walled Structures with Structural Imperfections perpendicular to the mid-surface by membrane action, with only minor contribution of bending stresses to satisfy equilibrium. The analysis of thin-walled structures can often be more involved than the analysis of thicker structures, and the mathematical modeling of shells has been one of the most challenging problems in solid mechanics. The fundamental equations that govern the mechanical behavior of thin shells were first formulated by Love towards the end of the last century [14], and since then the contributions to the stress analysis of shells have mainly concentrated on studies of the consistency of the governing equations, numerical and analytical techniques to obtain solutions, new applications in problems of design, optimization, and imperfection sensitivity. During the last 25 years, the trend has been to construct ever thinner and more slender structures. There are many reasons to explain this trend, such as the desire for lighter structures that are easier to construct, transport, and install; higher stiffness to load performance; and others. "There is a tendency to build taller and span wider; in other words, to exploit the high strength by doing more with less. This is not always accompanied by appropriate analysis and understanding of all conditions present and the material properties" [15]. There are many books dealing with the stress analysis of shells and thin-walled structures. In this book, we try to accompany the appropriate analysis and understanding of some conditions related to the influence of structural imperfections in thin-walled shells.

1.1.2

Imperfect Thin-Walled Structures

Attempting a non-trivial definition of structural imperfections is a difficult task. Rather than having a formal definition, we will explain what we mean by that. The design of a structure involves an idealization of what should be built or constructed. This idealization is reflected in specifications regarding the geometry, the material, and the processes to be followed. The structure that is defined in this way will be called the "as-designed" or "perfect" structure.

Introduction

3

But there are many differences between this as-designed structure and a "real" structure constructed according to such design. The real structure may also be called the "as-constructed" or "imperfect" structure. Why are there differences between the as-designed and the asconstructed structures? A first cause of differences may be summarized with the word "errors". A second cause of differences is associated with the word "damage". Error-induced imperfections arise, for example, due to the following 9 Defects of fabrication 9 Errors in the construction 9 Defects in the materials of construction 9 Changes introduced during the fabrication Damage-induced imperfections, on the other hand, may have various sources" 9 Damage sustained during the early stages of the structure 9 Damage sustained during the life-time of the structure 9 Damage sustained due to effects that were initially considered to be secondary In this book we use the term structural imperfections to refer to deviations from the as-designed situation. Alternative words, often employed in the literature, are defects, flaws, and degeneracies. This book is concerned with structural imperfections, but only with some aspects of imperfections and with some types. First, we are not here interested in the process by which an imperfection is produced. In a few specific cases, there will be some discussion about the causes of certain imperfections, but this is not the central topic of the book. Second, we are not concerned with the singularities associated with some imperfections. For example, cracks are considered from the point of view of the overall redistribution of stresses caused by them. We will not study stress concentrations at the tip of a crack that are best considered in texts on fracture mechanics. Finally, severe changes in the constitutive relations that are studied in texts on damage mechanics and plasticity are beyond the scope of this book.

Thin-Walled Structures with Structural Imperfections

1.1.3

Structural Consequences of Imperfections

Interest in the influence of geometric and load imperfections in structures was stimulated in the 1960s by the discovery that they had a paramount importance on the computation of buckling loads [12]. Structures became classified as "imperfection sensitive" and "imperfection insensitive", according to the changes obtained in the buckling loads with and without imperfections. The study of stresses in imperfect structures emerged when a large shell collapsed in the early 1970s, and its failure was attributed to the construction errors coupled with cracks [11]. A further step was that such changes in stresses occur and are important in other shell forms, such as silo structures, spherical pressure vessels, metal cylindrical shells, and shallow shell roofs. New questions can now be formulated regarding the structural importance of imperfections"

9 How important is the stress redistribution for a class of imperfection occurring in a specific structure? 9 How does the stress redistribution depend on the specific features of the imperfection? 9 How can one produce tolerance specifications that are rationally based on the effects of the imperfection? 9 Is the response linear with the characteristics of the imperfection, or is it non-linear? 9 Will a structure resist the loads in the new, deteriorated situation produced by the imperfection?

This book is written with these questions in mind and attempts to provide tools of analysis and to show examples of the behavior of imperfect thin-walled structures.

Introduction

5

C L A S S I F I C A T I O N OF IMPERFECTIONS IN THINWALLED STRUCTURES

1.2

When we deal with imperfections, we refer to some difference between what was designed and what we have standing at a certain stage of the life of a structure. At first, one should think that every imperfection is different from others, and that one should model each one as an individual case. However, it is convenient to classify structural imperfections using some criteria. This provides some basis to perform systematic studies of imperfections. Three possible classifications are given in this section. These are based on the features shown by the imperfections themselves rather than by their effects on the structural behavior.

1.2.1

A C r i t e r i o n B a s e d on t h e R e l a t i o n B e t w e e n an I m p e r f e c t i o n a n d t h e Loading Process

Hayman [9] [10] distinguished between imperfections induced prior to, under, or by the loads.

Imperfections induced prior to loading occur during construct, ion, erection, or installation of the structure, and before it stands against the major loading condition for which it has been designed. Examples of this kind are cracking of concrete shells due to shrinkage, thermal effects during construction, etc. They also occur during the early stages in the life of the structure. Imperfections in the shape produced as a consequence of errors during the construction also belong to this group. Other examples are problems of non-uniformity of the thickness and lack of homogeneity of the constitutive materia,1. Imperfections of this kind usually occur at random, and their exact definition is not known a priori. However, there may be some imperfections that are more likely to occur than others, or have certain preferred directions or localizations.

Thin-Walled Structures with Structural Imperfections

Imperfections induced under load by an external actionIn this case, an imperfection is induced by an agent that is independent of the main loading system. Examples are the damage and even the loss of part of a structure during its service life due to some external agency. Such imperfections may be associated with a collision, or explosion, or with deterioration of part of the structure due to chemical action, corrosion, or any other factor.

Imperfections induced by the load" In many cases, the main loading system produces cracking, fracture, or plasticity in the structure, and the structure has to satisfy equilibrium for further increments of the same main loading. The loads, in such cases, play a double role, both generating the change in the properties and acting as the control parameter. This is the subject of the fields of fracture mechanics, damage mechanics, and plasticity. In this work we consider imperfections induced prior to loading, and also some imperfections induced under load by an external action.

1.2.2

A Criterion B a s e d on t h e Space D i s t r i b u t i o n of I m p e r f e c t i o n s

We say that a structure is prismatic if the properties of geometry and material are constant in at least one direction. Most of the structures considered in this book are prismatic, and have the geometry of a shell of revolution or a thin-walled folded plate. In those cases the analysis is conveniently simplified by the use of semi-analytical methods, such as finite strips or finite rings. However, the presence of an imperfection may affect the prismatic property of the structure depending on the geometry of the imperfection itself. Thus, a second possible classification of imperfections refers to its space distribution. Such a classification has been very useful in shells of revolution and is generalized in this section: 9 I m p e r f e c t i o n s w i t h prismatic properties: In thin-walled prismatic structures, an imperfection may affect the properties in one direction, but still preserve the prismatic characteristics of the body. Examples are axisymmetric imperfections in shells of revolution. This class may be modeled conveniently by semianalytical methods.

Introduction

7

9 I m p e r f e c t i o n s w i t h some repeated p a t t e r n " There are imperfections that destroy the prismatic properties of a structure, but still show some repeated symmetry. Examples are circumferential imperfections in shells of revolution (i.e., when the parallels are not circular). They may be investigated by isolating a part of the shell in which the pattern is present. Localized imperfections" In this case, an imperfection is a single, isolated defect of the complete structure. Examples are bulge-type imperfections, with curvature errors in two principal directions of a shell. In localized imperfections, one cannot introduce simplifications of prismatic or repeated properties. In principle, the whole structure with the localized imperfection should be modeled; however, if the stress changes associated with the imperfection are expected to affect only a region in the neighborhood, the analysis may be simplified by assuming some fictitious repeated pattern. . Global imperfections" There are structures in which the part affected by the imperfection is almost the complete area. Thus, most of the structure shows the influence of the defect. If there is no repeated pattern in the space distribution, a model of the complete structure should be produced. Examples of the areas affected by the above imperfections for a circular cylindrical shell are presented in Fig. 1.1.

1.2.3

I m p e r f e c t i o n s in Intrinsic and in Geometric Parameters

This is a very useful distinction between the sources of imperfections, which for many years has been employed in the field of sensitivity analysis (see, for example Ref.[2]). 9 Intrinsic parameters are parameters present in the constitutive relations. Depending on the constitutive material considered, such parameters may be the modulus of elasticity, Poisson's ratio, parameters of orthotropy, yield stress, etc. In thin-walled components, the thickness is also included in the constitutive equations and may be considered in this group. A distinctive

Thin-Walled Structures with Structural Imperfections

Figure 1.1" Area of a shell affected by (a) imperfection with prismatic properties; (b) imperfection with a repeated pattern; (c) localized imperfection; and (d) global imperfection. feature of intrinsic parameters is that the equilibrium equations of the system are an explicit function of them. Examples of imperfections in intrinsic parameters are local changes in the elastic parameters, a local inhomogeneity, etc. A local reduction in the thickness (grooves) in metal structures may also be interpreted as imperfections of this class, although sometimes they are not imperfections but rather part of a design. Partial cracks in reinforced concrete shells, in which the concrete shows cracking but the steel has not reached yield, may be viewed as an imperfection in an intrinsic parameter and modeled through thickness changes. In the limit, a locally reduced value of thickness may be zero, and one should refer to it as a "full" or "passing-through" crack. In fact, a full crack represents a new boundary condition (stress-free), and as such it produces a new definition of the domain. However, it may be convenient to think of a full crack as a special case of a groove.

Introduction

9

Geometric p a r a m e t e r s define the mid-surface and the boundaries of a thin-walled structure. Imperfections in geometric parameters may affect the size, the shape, the position of the middle surface, etc. If the analysis is carried out by a finite element procedure, these parameters affect the strain-displacement matrix (matrix B in the finite element notation), and the limits of integration of the volume or surface considered. They may also be reflected in the volume of the complete body, and in the boundary conditions. An important feature of these imperfections is that the stiffness matrix is not an explicit flmction of them. Examples of imperfections in geometric parameters are out-of-roundness in cylindrical shells a,nd in shells of revolution, deviations in the shape of the meridional curve in shells of revolution, bulges due to construction defects, and denting damage in metal structures among many others.

1.3

SOME SHELL STRUCTURES IN WHICH IMPERFECTIONS MAY BE IMP ORTANT

This section discusses some practical engineering applications in which the stress redistribution associated with structural imperfections has been identified to be important. Only a, few examples a,re presented here, to illustrate cases of special concern in the studies of this book: shells made with reinforced concrete, steel, and fiber-reinforced composites. The first two are considered in more detail in this book.

1.3.1

Reinforced Concrete Cooling Towers

Cracking and imperfections in the geometry of cooling towers and other reinforced concrete shells of large dimensions have been known to occur for a number of years. At; least two major collapses of cooling towers in the United Kingdom ha,ve been influenced by the presence of imperfections in the shell [11] [7]. It is now generally accepted that cracks in cooling towers have their origins in a combination of factors, of which thermally induced stresses and shrinkage of the concrete are the most frequent ([11] [13]).

10

Thin-Walled Structures with Structural Imperfections

Figure 1.2" Geometrical imperfection in a cooling tower shell (Photograph by the author)

Figure 1.3" Concrete spalling in a cooling tower shell (Photograph by the author)

11

Introduction

Overall heating of the tower does not cause cracking, except in zones near the boundaries of the shell, but high temperature gradients across the thickness produce significant moments in the section that can sometimes result in cracking. The differences in temperature between the faces of the shell needed to produce these gradients could occur in several ways: for example, when the tower is working, a gradient would be produced due to low outside temperatures; when the tower is not in use, a high outside temperature produces a similar effect. It seems that the former is more important. Reinforced concrete cooling towers are very prone to geometric imperfections, mainly because their large dimensions cause difficulties in the setting-out and construction of the shell. Figure 1.2 shows the imperfect meridian of a cooling tower in Argentina, in which the deviations of the mid-surface are on the order of the thickness (or even larger). Other ca,uses of geometrical imperfections are the influence of dead and temporary loads during construction, deformations of the supporting frames and panel works employed to cast concrete, etc. In other cases, geometrical imperfections occur due to damage of the structure, as reported in Ref. [8] for a cooling tower affected by a tornado. Concrete spalling, another structural problem encountered in cooling towers, is illustrated in Fig. 1.3. Here, part of the concrete cover is lost, and the reinforcement is exposed to the environment.

1.3.2

Reinforced

Concrete

Silos and Tanks

It seems that cracking has been of more concern to practicing engineers than imperfections in the geometry. If only because of all the pathological states of reinforced concrete, it is probably the one most easily observed imperfection. In reinforced concrete shells, geometrical imperfections on the order of the thickness of the shell are difficult to detect visually. Cracks, on the other hand, are often seen at considerable distance from the structure, in many cases simply because the lines of the cracks are emphasized by stains. Cracks are illustrated in the cylindrical silo structures of Fig. 1.4. The loss of strength produced by discrete, well-defined cracks is of particula.r importance if it occurs in the region and in the direction in which the shell has to provide important membrane or bending contributions to reach equilibrium. This situation is not uncommon in

12

Thin-Walled Structures with Structural Imperfections

Figure 1.4: Cracking of a silo structure (Photograph by the author) silos, and it could eventually lead to a collapse. Structural imperfections are often found in storage tanks made of reinforced concrete. A case of deterioration of a circular cylindrical tank is illustrated in Fig. 1.5.

1.3.3

Spherical Steel Containers

Interest in the structural behavior of very thin closed spherical shells has been largely stimulated during the past decades due to their use as main structural components in pressurized water reactor, or pressurized heavy water reactor nuclear power plants. For example, in a typical plant design, the spherical steel containment shell may have a radius to thickness ratio of 1000. Figure 1.6 shows a steel sphere used

13

Introduction

Figure 1.5" Reinforced concrete tank with imperfections in the material (Photograph by the author)

Figure 1.6" A spherical steel gas container

14

Thin-Walled Structures with Structural Imperfections

to store gas. Although the tolerances in geometry adopted for the fabrication of such shells are stringent, deviations from the as-designed geometry, with amplitudes larger than the shell thickness, may occur as a consequence of damage of the shell. In that case, it is necessary to evaluate the level of stresses and the nature of the stress redistribution that takes place in the shell with the modified geometry.

1.3.4

S h a l l o w C o n c r e t e Shells

Figure 1.7 shows the geometry of an elliptical paraboloidal shell employed as a roofing system in industrial and sports buildings. The origin of imperfections in the geometry in this case may be due to imperfections of the formwork itself; moreover, for these very flexible structures, removal of the concrete formwork is accompanied by significant deflections. Finally, creep of concrete may also induce additional deflections under self weight. Ballesteros [1] investigated the collapse of an elliptical paraboloidal shell in which important geometric imperfections had been detected. The amplitude of the deviation from the as-designed mid-surface was between 1 and 1.5 times the thickness, covering an area which could be modeled by an elliptical shape in plan.

1.3.5

T u b u l a r M e m b e r s in O f f - S h o r e S t r u c t u r e s

The economic importance of off-shore platforms for drilling and oilproduction cannot be over-stressed. There may be hundreds of them in an oil-producing area. Structural damage in off-shore structures occurs in a large number of cases" Ellinas and Valsgaard [6], in a study of damage due to accidents in off-shore platforms involving different countries, found that about 17% were produced by the impact of a marine vessel with the platform, while about 12% were due to impacts from dropped objects. About 14% of the accidents led to major problems for the structure. A typical geometrical damage in a tubular member is illustrated in Fig. 1.8. In their normal operation, supply ships are very close to fixed and compliant off-shore structures, and ship collision incidents have to be taken into account in the analysis and design of both. The collision itself is a complex dynamic problem involving the mass and the stiffness

~176

~

~

o

i

9

9 c~

9

im-A

~.~o

9 9

~mi~

Gr~

~

16

Thin-Walled Structures with Structural Imperfections

Figure 1.8" Denting of a tubular member in off-shore platforms

of the two colliding structures, and the outcome may induce severe permanent deformations in the off-shore structure. For example, localized deformations may occur in cylindrical components, with a subsequent behavior that is similar to the response of the same shell with a static bulge imperfection. From the point of view of the safety of the structure, it is usually accepted that some damage can occur in the event of such collisions, but this damage should not lead to the failure of the component, or should not produce unacceptable flooding of compartments that are otherwise dry. More frequent but less serious incidents are usually termed "operating incidents", but those should not induce significant damage to the structure. Damaged off-shore structures should be carefully investigated to assess their strength and stability capacities, and this can be done using the same procedures as in imperfect structures. Sometimes related with the previous example, the evaluation of the stresses developed at the hull of a ship following damage may be of great importance, as established by Ref. [5], and many others.

In troduction

1.3.6

17

C o m p o s i t e Thin-Walled Structures

The development of composite technologies ha,s now extended from aeronautical to mechanical and civil engineering. Fiber-reinforced composite shells and thin-walled structures are also prone to geometrical as well as to material imperfections. For example, imperfections in cylindrical shells, manufactured by lay-up, are reported in Ref. [4], in an attempt to assess the influence of different methods of manufa,cturing on the slope imperfections. Other forms of imperfections in structures of composite materials a,rise a,s a, consequence of damage, such a,s impa, ct damage. Reductions in strength on the order of 40% have been reported from experiments on laminated composites initially subjected to low velocity impact a,nd subsequently tested under static and fatigue conditions. The degradation caused by impact damage is an important topic in structures made of composite materials.

1.4

A FEW SITUATIONS IN WHICH IMPERFECTIONS ARE CONSIDERED

We discuss here examples of situa,tions in which imperfections play a,n importa, nt role in the design or in the evaluation of the sa,fety of a structure. Throughout this section, when we refer to "an interested party", we have in mind the owner of the structure, a firm operating the facility, a government agency interested in the safety of the community, an insurance company, etc. Let us summa,rize six possibilities involving imperfect structures. Situation 1

9 A structure has been completed some (let us say five) years ago. 9 It is expected that the structure should have a reasonable service life (say, thirty years). 9 Imperfections ha,ve been discovered in the structure. 9 Field measurements have allowed a more precise identification of the imperfections.

18

Thin-Walled Structures with Structural Imperfections 9 The actual source that gave rise to the imperfection is not clearly known. 9 An interested party has the following questions. - Is the safety of the structure significantly affected by the imperfections? - Will the imperfections grow with time? - What will be the safety of the structure by the end of the original service life? - Has the service life decreased? - Is it possible to increase the strength or the stiffness to compensate for the effects of the imperfection? - What would be a likely collapse mode? - Should the structure be demolished?

This situation is exemplified by cases the of the cooling towers in England, reported in Ref. [16]. Situation 2

9 A structure suffered damage prior to being completed, or during operation. 9 Assessments of damage have been made, and show significant changes with respect to the original design. 9 It is necessary to estimate the stresses in the shell during the event tha,t led to damage. 9 Repair and completion of the shell should be investigated to evaluate the cost in time and budget,. Reports of this kind of study may be found in Ref. [8] for a cooling tower damaged during a tornado, and in Ref. [5] for ship collisions. Situation 3

9 A structure was designed for a certain life span (say, 20-years).

Introduction

19

9 At the end of the life span, the structure is still in operation (say, after 30-40 years). 9 The structures shows structural imperfections. 9 The cost to replace the structure is prohibited. There is a strong incentive to assess the safety of the existing structure, with the damage and aging that it has. This situa, tion is described in Ref. [17] for dented tubular members in off-shore structures. Situation 4

9 A structure has collapsed. 9 Imperfections were detected in the structure prior to the collapse. 9 A committee is called to investigate the causes of the collapse. 9 Some of the questions posed to the committee are: - Did the structural collapse have any relation with the existence of the imperfections? - Is there a mechanism of events that may explain the collapse based on the stress redistributions due to imperfections? - Is the mode of collapse and the position of the debris consistent with the failure mode assumed based on the influence of the imperfections? A situation similar to this is described in the reports of the collapse of cooling towers at Ardeer [11] and Fiddlers Ferry [3] in Great Britain. Situation 5

9 A given structural component is produced in large numbers for specific applications. 9 The firm that produces these components wants to know what imperfections would have a significant influence on the structural response.

20

Thin-Walled Structures with Structural Imperfections 9 From the identification of these imperfections, a quality control program is to be established. 9 If it is found that the process of fabrication is associated with a critical imperfection that is difficult to remove, an improved process may be necessary.

Situation 6 9 A code of practice for a class of structures will establish tolerance specifications for imperfections. 9 Some questions that need to be answered are:

How should imperfections be measured? What parameters should be used to characterize the imperfections? How often should imperfections be measured?

Conclusions In all the previous situations, we presented problems of shell structures that can be large or expensive, and in which imperfections are of concern. The questions that we formulate in each case may have important practical consequences in terms of cost and safety. However, these questions are related to more basic knowledge that appears to be theoretical in principle. The main question that has to be considered seems to be:

What is the mechanics of redistribution of stresses that the imperfect shell structure is likely to exhibit? This book addresses this question. Based on the answers, one can proceed towards design recommendations for specific structures.

21

Introduction

1.5 1.5.1

OUTLINE

OF THE BOOK

Scope and Objective of this Book

This book focuses on the influence of imperfections in thin-walled structures and components. By structural imperfection we mean some relatively small change in the structural system with respect to the as-designed situation. These changes may affect the parameters that define the stiffness matrix (with which intrinsic parameters are associated) or those that define the geometry (in which case geometric parameters are identified). A variety of imperfections may be modeled in this way, but special attention is given to changes in the position of the mid-surface of a shell, defined in terms of geometric parameters; and to changes in the thickness or in the constitutive parameters defined in terms of intrinsic parameters. The imperfections studied in this book may have occurred as a consequence of unknown agents, but in the new, deteriorated situation, the structure still has to withstand the loading conditions. The main purposes of the book are: 9 To explain how imperfections change the stress response of a structure; and 9 To provide techniques of analysis adequate to model problems with imperfections. The models studied in the book are described in terms of imperfection parameters, and the resulting problem becomes non-linear in these new parameters. Considerable effort is placed throughout the book to present a unified treatment of the techniques for modeling imperfections, with discussion of their relative merits. The book considers discrete conservative systems, for which a total potential energy can be defined. The structural systems are holonomic in the sense that there are no constraints on the displacements. Furthermore, we assume them to be scleronomic, since time does not appear explicitly in the formulation. Each topic is introduced with reference to simple mechanical models (either one or two degree-of-freedom systems), and the presentation progresses toward the more general multi degree-of-freedom systems that arise from finite element discretizations.

22

1.5.2

Thin-Walled Structures with Structural Imperfections

About the Contents

Chapter 1 is a general introduction to the field of imperfect shells and other thin-walled structures. Classifications of imperfections are introduced here, based on different criteria: the relation of imperfections with the loading process; with the space distribution; and with the parameters that are changed. Practical problems of structures with imperfections leading to significant stress redistributions are also shown. Part. I, including Chapters 2 to 6, refers to the stress analysis of structures with imperfections. Chapter 2 gives an introductory account of linear one degree-offreedom systems, for which simple solutions are possible. The material in this chapter is also used to introduce the three techniques of analysis employed in the text: the direct, the equivalent load, and the perturbation technique. The presentation progresses through examples of geometric imperfections, intrinsic imperfections, and cases with both classes of imperfections. Chapters 3 to 6 refer to multi degree-of-freedom systems, in which the finite element method is employed as a technique of discretization. Intrinsic imperfections are the subject of Chapter 3; while geometric imperfections are discussed in Chapter 4. Problems with both types of imperfections are considered in Chapter 5. The influence of kinematic non-linearity is discussed in Chapter 6. Part II (Chapters 7 to 11) refers to the behavior of thin-walled structures with imperfections. These chapters may be viewed as examples to illustrate the importance of the changes in stresses due to imperfections. We believe that the designer of a thin-walled structure must have a certain intuition about stresses due to imperfections. The objective of Part II is to convey to the designer a "feel" for the role of imperfections, whether they are due to geometrical errors, to cracks or grooves, or to anisotropy or inhomogeneity of the constitutive material. An intuitive understanding can be gained by studying examples from practical shell structures. With enhanced intuitive knowledge about imperfections in shells, the designer should have an improved ability to foresee situations in which problems due to imperfections might occur. He or she will be able to set up more appropriate models for analytical predictions and to interpret the results.

Introduction

23

Applications of the techniques to the analysis of plates and plate assemblies are presented in Chapter 7. Several cases of imperfections are considered: geometrical deviations from the flat surface; thickness changes with respect to a uniform value; and changes in the constitutive properties. Chapter 8 is devoted to shallow shells; it is short and serves as a bridge between flat plates and more general shells. The applications here are to shell roofs. Spherical shells are studied in Chapter 9. These are important shell forms and they have been studied extensively by the author and other researchers. Geometrical as well as thickness imperfections are modeled, and both linear and non-linear kinematic relations are assumed. Chapter 10 is the study of geometrical and thickness imperfections in cylindrical shells. Analytical and finite element tools are used to model geometrical imperfections, with special reference to axisymmetric imperfections. Thickness changes in the form of grooves are also studied coupled with geometric deviations. Finally, there is a section on the influence of creep on the deformations of the shell. Chapter 11 deals with cooling tower shells that have imperfections in geometry and in elastic properties. A review of results obtained by different researchers is presented, together with some results that attempt to explain possible collapse mechanisms of hyperboloids of revolution. Practical aspects of imperfections are the subject of Chapter 12, which explains how imperfections are assessed in real situations, according to the type of imperfection and to the scale of the structure. Levels of real imperfections detected at some locations are also discussed, including amplitude of imperfections and their shape. In this chapter we also discuss the role of imperfections in the collapse of several large and expensive shells. The closing chapter reviews some philosophical and historical aspects of imperfections and summarizes the main conclusions derived in previous chapters. Appendix A explains basic topics of perturbation techniques, and some aspects of shell theory are reviewed in Appendix B.

24

Thin-Walled Structures with Structural Imperfections

References [1] Ballesteros, P., Non-linear dynamic and creep buckling of elliptical paraboloidal shells, Bulletin of the Int. Association for Shell and Space Structures, No. 66, 1978, 39-60. [2] Brockman, R. A. and Lung, F. Y., Sensitivity analysis with plate and shell elements, Int. J. Numerical Methods in Engineering, 26, 1988, 1129-1143. [3] Central Electricity Generating Board, Report On Fiddlers Ferry Power Station Cooling Tower Collapse 13 January 1984, CEGB, London, 1985. [4] Chryssanthopoulus, M. K., Gravotto, V., and Poggi, C., Statistical imperfection models for buckling analysis of composite shells, in: Buckling of Shells on Land, in the Sea and in the Air, Ed: J. F. Jullien, Elsevier Applied Science, Barking, England, 1991, 43-52. [5] Davies, I. L., A method for the determination of the reaction forces and structural damage arising in ship collisions, J. of Petroleum Technology, 1981, 2006-2014. [6] Ellinas, C. P. and Valsgard, S., Collision and damage of off-shore structures: A state of the art, J. Energy Resources Technology, ASME, 107(9), 1985, 297-314. [7] Fullalove, S. and Greeman, A., Cooling tower collapse exposes calculated gamble, New Civil Engineer, The Institution of Civil Engineers, UK, 19, 1984, 4-5.

Introduction

25

[8] Gould, P. L. and Guedelhoefer, O. C., Repair and completion of damaged cooling towers, J. Structural Engineering, ASCE, 115(3), 1989, 576-593. [9] Hayman, B., The Stability of Degenerate Structures, Ph.D. Thesis, University College, University of London, London, 1970. [10] Hayman, B. and Chilver, A. H., The Effect of Structural Degeneracy on the Stability of Cooling Towers, Report 71-17, Engineering Department, University of Leicester, Leicester, England, 1971. [11] Imperial Chemical Industries, Report of the Committee of Inquiry into the Collapse of the Cooling Tower at A rdeer Nylon Works, Ayrshire on Thursday 27th September 1973, I. C. I. Ltd., Petrochemicals Division, London, 1973. [12] Koiter, W. T., Over the Stabiliteit van het Elastich Evenwicht, Delft Institute of Technology, Delft, 1945. English translation, NASA Report TTF-10, 1967. [13] Larrabee, R. D., Billington, D. P., and Abel, J. F., Thermal loading of thin-shell concrete cooling towers, J. of Structural Division, ASCE, 100 (ST12), 1974, 2367-2383. [14] Love, A. E. H., A Treatise of the Mathematical Theory of Elasticity, 1st Edition, 1988; 4th Edition, 1927. Also available in Dover Publications, New York, 1944. [15] Pfrang, E. O. et al., Building structural failures" their cause and prevention, IABSE Periodica 3, 1986, 17-28. [16] Pope, R. A., Grubb, K. L., and Blackhall, J. D., Structural deficiencies of natural draught cooling towers at U.K. power stations" Part 2, Surveying and structural appraisal, Proc. Inst. Civil Engineering, Structures and Buildings, 104, 1994, 11-23. [17] Ricles, J. M., Gillum, T. E., and Lamport, W., Grout repair of dented offshore tubular bracing- Experimental behavior, J. of Structural Engineering, ASCE, 120(7), 1994, 2086-2107.

Chapter 2 SINGLE DEGREE-OF-FREEDOM SYSTEMS WITH IMPERFECTIONS 2.1

INTRODUCTION

The degrees-of-freedom of a structure are amplitudes of deformation patterns that the structure can have. Real engineering structures are usually modeled using many degrees-of-freedom (sometimes ten, other times one hundred, often one thousand, and on occasion ten thousand and more). In this chapter we restrict our attention to systems with one degree of freedom. There are two types of such systems: 9 In some cases, a single degree-of-freedom model is associated with an assemblage of rigid bodies, in which the deformations can occur only in localized discrete elements (such as a spring, or a moment spring) constrained by supports and hinges. The displacements are permitted following only one shape pattern. In this case, the system itself is discrete, and there may be just one type of displacement allowed by the hinges and the supports. 9 Continuum systems with elastic properties allow for an infinite number of displacement patterns; however, for the purpose of 26

Single Degree-of-Freedom Systems with Imperfections

27

the analysis, they are sometimes modeled using one deformation pattern. In this case, the system becomes discrete by a process of analysis, and is assumed to have just one deflection mode. In continuum cases like this, the mathematics involved reduces the model to a single degree-of-freedom. The restrictions to one shape of deformation arises as an engineering assumption rather than as a, property of the system. The studies in this chapter are based on models with one degreeof-freedom. The presentation here has three main purposes: 9 To illustrate how different techniques of analysis may be used to model an imperfect system; 9 To introduce a first estimate of errors involved in the different approximations; and 9 To establish general aspects of the mechanics of behavior of imperfect structures. It is expected that these studies will serve as an adequate introduction to the more complex, multi degree-of-freedom problems to be tackled in the following four chapters, in which some aspects of the discretization procedures and the numerical solutions may somehow obscure the most elementary aspects. We want to emphasize that the simple mechanical models studied are not considered here to describe precisely the response of any particular structural component under a given loading condition. In general, the response of a structure cannot be modeled by just one generalized coordinate; on the contrary, it requires a large number of them. Let us consider a system governed by a linear equilibrium condition of the form

k O - f -O

(2.1)

in which k is some stiffness of the model, and is represented by a single physical discrete element, which could be a spring or a moment spring; f is the applied force; and 0 is the response (the displacement or rotation at some point of the structure). It will be shown in this chapter that for imperfections characterized by amplitude {, the imperfect model leads to an equilibrium equation of the form:

28

Thin-Walled Structures with Structural Imperfections: Analysis

k (1 - a ( + b~2 _ c~3 + ...) O - f - 0

(2.2)

where a, b, and c are coefficients associated with the specific nature of the imperfection. In the first part of this chapter, the imperfections considered are "geometric", while in the second part we study "intrinsic" imperfections. For a geometrical imperfection, Section 2.2 employs a model based on initial strains; equilibrium conditions that depend linearly on the displacements but that are a non-linear function of the imperfection parameter are obtained. The perturbation and equivalent load techniques are introduced for the first time in this book, and the results are compared using a specific example. Simplified models of analysis are also considered, because they are often employed in the literature. A similar sequence of analysis is followed in Section 2.3 for systems with imperfections in intrinsic parameters. Section 2.4 considers imperfect systems with two independent parameters; for the purpose of generality, we choose one geometric and one intrinsic parameter and investigate the response. In Section 2.5 we study the conditions that produce a maximum in the displacement response, as a function of the amplitude of the imperfection. Some general conclusions are presented in Section 2.6.

2.2

IMPERFECTIONS IN GEOMETRIC PARAMETERS

Let us consider a simple mechanical model of an arch under a point load, illustrated in Figure 2.1. The initial angle of the bars is a; the displacement ~ in this case is the rotation of the bars with respect to their original position; the load P is the force applied at the center; and the only stiffness is provided by a linear spring. The total potential energy of this system, denoted as V, is given by V-

1

~N A - P 5

(2.3)

in which N, the stress in the spring, is related to the displacement of the spring, A, by the constitutive equation

Single Degree-of-Freedom Systems with Imperfections

N-

K A

29

(2.4)

For a detailed discussion on the total potential energy and its properties, the reader is referred to the various texts on the subject, for example, Refs. [9], [7], and [3]. Approximate kinematic expressions for this model are [2] A -

(2L sin a)0

-

...

ti - ( L cos c~)0 + ...

(2.5)

The boundary conditions in this problem are a pin-support and a roller.

2.2.1

Initial Strain M o d e l of I m p e r f e c t i o n

To introduce the influence of a geometrical imperfection, the displacement 0 in the imperfect model is written as the difference between the current and the initial configurations, i.e. 0 -

0p -

(

(2.6)

where 0p is measured with respect to the perfect or idealized geometry; and ~ is the amplitude of the imperfection, represented in Fig. 2.1 as an initial angle. The deformation of the spring is now A - A p - A~

(2.7)

For small strains and small rotations, the following approximations are used Ap - [2L sin a - ( 0 + ~) L cos c~ ](0 + ~) A~ -- (2L sin a - ( L cos a)~

(2.8)

Replacing Eq.(2.8)into Eq.(2.7)leads to A - 2L(sin a - ~ cos a)0

(2.9)

30

Thin-Walled Structures with Structural Imperfections: Analysis

Figure 2.1: Model considered for geometrically imperfect system The internal force N can now be obtained as N - 2 L K (sin a - ~ cos a)O

(2.10)

Finally, the energy V of the imperfect system results in V - 2L2K [sin a - ~ cos a]20 ~

- P (L cos a)0

(2.11)

The strain energy is quadratic in 0, while the load term is linear in both 0 and P.

Single Degree-of-Freedom Systems with Imperfections

2.2.2

31

Equilibrium Condition

The equilibrium states of the imperfect system can be obtained from the condition of stationary total potential energy; they yield dV

d6

- [/to + ~IQ + ~2K2] 9 - P (L cos a) - 0

(2.12)

with KoK1

-

[K (2L sin a)2] [-8L 2 sin a cos a]

K2- [4(Lcosa) 2]

(2.13)

A more convenient form of writing Eq. (2.12) for a one degree-offreedom system is

k(1 -

a~ + b ( 2 ) O - f - 0

(2.14)

in which a-~;

2 tana

k - K(2L sin a)2;

1 (tan a) 2

b-

f - P L cos a

(2.15)

Notice that the problem of Eq. (2.14) is non-linear in terms of the amplitude of the imperfection (. In most studies in the literature, a linearized version of Eq. (2.14) is considered.

2.2.3

Perturbation Analysis

The perturbation technique is a convenient way to investigate the response of a non-linear system. There are several books on this technique [1], [5], [6], [4], and [8], and a short presentation of perturbation techniques is given in Appendix A. The unknown 0 in Eq. (2.12) is expanded in terms of a perturbation parameter. For this case of regular, non-degenerate perturbations, we choose this parameter as the amplitude ( of the imperfection, so as to obtain an explicit relation between 0 and (. The expansion becomes

32

Thin-Walled Structures with Structural Imperfections: Analysis

6} -- ~90 + ~ 1

+ ~"2 6}2 +

(2.16)

~3"~-'"

=C6. for n in the range of 1 to the number of terms to be included in the expansion. A generic coefficient 0n is defined in the form of the nth order derivative of 0 with respect to the perturbation parameter, evaluated at ~ = 0:

= n!d~,~ I~=o

(2.17)

Eq. (2.16) and (2.17) represent a Taylor expansion of a variable 8 in the neighborhood of a given value ~90. The value ~90is usually known by simple calculations, and represents the solution of the system without any imperfection, for which ~ = 0. The second term in the series, ~91, is linear in the amplitude of the imperfection, and depends on the first derivative of t9 with respect to ~, evaluated at ~ = 0. The third term is quadratic in ~, and contains the second derivative. The number of terms to be included in the analysis depends on the accuracy required. Substitution of Eq. (2.16) into Eq. (2.14) yields the equilibrium condition

[kSo -

f] +

~ [k (01

-

-

aSo)] + ~2 [k (02 - a01 + bSo)]

+ ~3 [k (03 - aO: + bO1)] +

.

.

.

-

0

-

(2.18)

Next, we perform successive differentiation of the equilibrium condition with respect to (, and obtain

- [k~90- f] + ~ [k (01

[k

d

-

--

at?0)] + ~2 [k (02 -

+ boa)]

+

...

-

a~}l

+ boo)]

0

-- [k (01 -- a00)] + ~ [k (02 -- aO1 + b0o)]

Single Degree-of-Freedom Systems with Imperfections

-4-~ 2 [/i; (03 -- a02 +

d2(dV)

b01)]-~-...

-- [1~ (02 --

33

- 0

aO1 -4- boo)]

+ ~ k (03 - a02 + b01) + ... - 0

d~ 3

-~

-

/~ ( 0 3 -

a02 -~- bO1)-~-...- 0

(2.19)

Finally, we evaluate the above expressions at ( - 0, and obtain a set of linear perturbation equations

kOo - f - 0 k (01 - aOo) -

0

k (02 --a01 + boo) - 0 k (03 --

a02 + bO1) - 0

(2.20)

Notice that the coefficient of the unknown 00 in the first of Eq. (2.20) is k. The next equation is called the first order perturbation equation, in which 00 is known and 01 is unknown. The coefficient of 01 is again k. The following equation is the second order perturbation equation, with 02 as unknown and k as its coefficient. What changes from one equation to the next is only the constant term, which is computed from previous results of perturbation systems. The first equation has just one unknown, and since it is a linear system, it can be easily solved for 00. The second equation is a function of 00 and 01, but only 01 is unknown at this stage. Again, the equations are linear and can be solved for 01. The third equation depends on 02, 01, and 00, so that it can be solved to obtain 02. Thus, the solution of the set of Eq.(2.20) leads to O0 - f- - P cos o k 4L (sin c~)2

34

Thin-Walled Structures with Structural Imperfections: Analysis

O1 ~

aOo

(2.21) Replacing Eq. (2.21)into Eq. (2.16), one obtains

o-f

[ l + a ~ + ( a 2 _ b)~2_+_ (a 3 - 2ab)~3_~_ ...]

(2.22)

From Eq. (2.10), the stress N results in N

-4-(a 2 - b - a c o s

__.

P cos a [1 + (a - cos a) 2 (sina) 2

a)~2-t-(a 3 - a 2 cos a - 2ab-4- b cos a)~3_~_...] (2.23)

The displacement variable 0 can also be written in terms of the original parameters of the problem, and is 02.2.4

P cosa [ 1 + 2 ( + 3 ( 2 + 4KL(sina)2

] "'"

(2.24)

Simplified Perturbation Analysis

It is possible to assume that the quadratic term in Eq.(2.14) is negligible for imperfections of very small amplitude, i.e. values of ~ << 1. In this case the solution reduces to 0 ~ f [1 + a~ + a2~2+ a3~3+ ...]

(2.25)

P cos a 0 ,~ 4KL (sin a) 2 [1 + 2~ + 4~2 + ...]

(2.26)

or else

]

Eq. (2.25) is the simplified version of Eq. (2.22), and Eq. (2.26) is the simplified Eq. (2.24). In the present example, there is no need to introduce such simplifications, but the need will be present in systems with many degrees of freedom.

Single Degree-of-Freedom Systems with Imperfections

2.2.5

35

Equivalent Load Analysis

The equivalent load is another technique to solve non-linear equations in structural mechanics. In this case, the non-linear terms are treated as a new load acting on the system, and a set of linear problems is solved. In the equivalent load analysis, Eq. (2.14) is written in the form

kO - f + f*({, O)

(2.27)

f*((, O) - - k ( - a ~ + b~2)O

(2.2s)

where

The value of f* can be seen as a load equivalent to the effect that the imperfection produces on the structure, and because of that it is called the equivalent load. Notice that f* depends on the displacement 0. The load f* is defined with a negative sign, so that it can be treated in the same way as the original load f. The solution of Eq. (2.27) can be obtained in an approximate form by using a direct iteration procedure, described below. S t e p 1:

f*=0 The problem reduces to

kOo- f;

f Oo- ~ k

S t e p 2: ]r

-- f*(~, 00) -

k(a~ - b,~2)Oo

and the solution is 01 -- ( a ~ -- b~2)O0

S t e p 3: ~02 -- f * * ( ~ , 01) -- ~ ( a ~ -- b~2)O1

leading to

Thin-Walled Structures with Structural Imperfections: Analysis

36

02 - ( a ~ - b~2)O1 Step 4:

k03- f***(~, 0 2 ) - k(a~- b~r from which 03

-

-

(a~- b~2)02

The complete approximate solution takes the form 0 --- 0 0 -~- 01 -~- 0 2 "[- 0 3 -~-...

(2.29)

Notice that one can also write

0o01

--

f(a~- b~2)

02 - f(a~ - b~2)2 f

03 - k ( a ~ - b~2)3 and the complete approximate solution becomes

o - f [l + (a~- b~2) + (a~ -- b~2)2 + ( a ~ - b~2)3 -~ ...]

(2.30)

It is important to notice that this result is different from Eq.(2.22) for the perturbation method.

2.2.6

Simplified Equivalent Load

Again, let us investigate the influence of neglecting the quadratic term in Eq. (2.14) by setting b - 0. The solution reduces to O-- f [1 + a~ + (a~)2 + (a~)3 + ...]

(2.31)

Single Degree-of-Freedom Systems with Imperfections

/ /t

1.4

0~

.,=.

/.

1.0j~

...2'

.,,.,. Gm~ p~

0.6

0.2

_

j

/

I

.

37

/

.

~/

/ /

I

-

0.0

I

I

I .... I

-0.1

I 0.0

! .I

I 0.1

Geometric Imperfection ei [rad] Figure 2.2" Perturbation solution of a one degree-of-freedom system with geometric imperfection Eq. (2.31) is the simplified version of Eq. (2.30). Again, the computational advantage of such simplifications is more evident in multiple degree-of-freedom systems.

2.2.7

Numerical Results

Let us consider a structure in which c~ - 0.1257r and investigate the response of the imperfect model at a load P - 0.01(4KL). The perturbation solution in this problem yields a sensitivity plot of displacements versus amplitude of imperfection as in Fig. 2.2. The first order perturbation analysis in this case leads to a very good approximation, which is only marginally corrected by higher order perturbation terms. For values o f - 0 . 0 5 a _< ~ _< 0.05a, the changes in rotations are less than 10%. The results for an imperfection amplitude ( - 0.05a are presented in Table 2.1

Thin-Walled Structures with Structural Imperfections" Analysis

38

FIRST

SECOND

THIRD

Perturbation

6.8972

6.9397

6.9424

Perturbation Si mplified

6.897

6.9538

6.9592

EquivalentLoad

6.8831

6.9371

6.942

EquivalentLoad Si mplified

6.8972

6.95382

6.95318

Exact

6.942

Table 2.1 Comparison of rotations for different techniques of analysis for the one degree-of-freedom model of Fig. 2.1.

The values in Table 2.1 for perturbation and equivalent load methods are different at each level of approximation, although both converge to the exact solution. For this specific problem, a first order approximation using perturbations has an error of 0.6%, while the first order equivalent load leads to 0.8% error. The simplified solutions lead to higher values of rotations.

2.3

IMPERFECTIONS PARAMETERS

IN INTRINSIC

For this example we have chosen a simple model in bending, with two bars and a central rotational spring. This model is shown in Fig. 2.3. The moment M at the joint satisfies the elastic constitutive equation MThe total potential energy is

C(20)

(2.32)

Single Degree-of-Freedom Systems with Imperfections

39

Figure 2.3" Model considered for imperfection in intrinsic parameter

1

V (0, P) - -~C(20) 2 - PLO

(2.33)

Let us investigate the influence of imperfections in the stiffness of the spring, C, and write it in the form C --

~ t3

(2.34)

where 7 is a constant, and t is the variable to be investigated. This situation is typical of the bending of a beam, in which the bending stiffness depends on the cube of the thickness of the beam. A degradation in the intrinsic property t is next introduced by t-

to(1 - r);

0 _< r <__ 1

(2.35)

where to is a reference value, that is, a value of t from which deviations are to be studied. If we substitute Eq. (2.35) into Eq. (2.34), we get (2.36) where Co - 7 t3The energy V results in

40

Thin-Walled Structures with Structural Imperfections: Analysis

V (O, P, T) -- ~1C 0 (1 -- 3r -~- 3r2 -- T 3) (20 ) 2

PLO

The condition of equilibrium becomes dV

- 4Co (1 - 3r + 3r ~ -

k d6 Let us introduce the notation

f = PL;

k = 4C0;

r3~/ ~ 9 -

a = 3;

PL

-

b = 3;

(2.37)

0

c= 1

Then, Eq. (2.37) can be written in the form k ( 1 - aT +

br 2 -

CT3) e -

f --

0

(2.38)

This equation has the same form as Eq. (2.2), and can be solved using the same techniques that we employed in geometrically imperfect models.

2.3.1

Perturbation Analysis

The displacement 0 is again expanded in terms of the imperfection parameter, called r in this case: 0 = 6o q- tO1 q- r202 -t- T303 -k ...

(2.39)

where On-- ldnOl~.=o n! dr ~ Substitution of Eq. (2.39)into Eq. (2.38)leads to [k8o - f] + r [k (9~ - a9o)] + r 2 [k (92 - aO~ + boo)]

+ T3 [k (93 - a92 4 b01 - O9o)] + . . . - 0

(2.40}

We can obtain the derivatives of Eq. (2.40) with respect to r, and evaluate them at r = 0. This leads to a set of equations that are linear perturbation equations in the form

Single Degree-of-Freedom Systems with Imperfections

41

kOo- f = 0 k (01 - aOo) -- 0

k (02 -- a01 Jr- boo) - 0

k (03 - a02 +

bO1 -

coo)

-

0

(2.41)

The above equations can be solved in sequential order, and yield the solutions Oo = f-

k

01 ~

aOo

02 = a01 + bOo

03 = a02 -- b01 4- coo

(2.42)

From Eq. (2.42) and (2.39), it follows that

0 -- f -t- T (aOo) -t- T 2 (aOa -4- boo) -t- T 3 (a02 -- bO1 Jr coo) -~ ...

(2.43)

This approximation of 0 can also be written in terms of 00 from Eq. (2.42)

0 - ~ [1 + aT + (a 2 - b)T2-[ - (a 3 - 2ab + c)T3+ ...]

(2.44)

Bending moments may now be obtained from Eq. (2.32), and lead to M - C0(200) + 0 + 0 + ...

(2.45)

In a one degree-of-freedom system, a change in the stiffness cannot produce a redistribution of stresses, and since equilibrium needs to be

Thin-Walled Structures with Structural Imperfections: Analysis

42

satisfied, there are no changes in the moments, M. This result is not valid in systems with many degrees-of-freedom, for which stresses can and do occur. Notice that Eq. (2.22) is a particular case of nq. (2.44) with c - 0. This, however, only affects the cubic term in the asymptotic expansion.

2.3.2

Equivalent Load Analysis

We consider again Eq. (2.38), but in the form

kO - f + f*(T, O)

(2.46)

f*(T, O ) - k (aT -- bT2 + CT3) 0

(2.47)

where

This is an equivalent load, and depends on the displacement field, 0. Let us employ a direct iteration procedure: Step 1:

oo__f Step 2: f'(T, 8 ) - k (aT -- b7"2+ cT 3) eo o1-

+

Oo

Step 3:

f**(r,e)-k(ar-br2+cra)el Step 4:

/

O3

/a~ - b~ ~ + ~ )

\

0~

The complete solution can now be written as

Single Degree-of-Freedom Systems with Imperfections

9

43

9

sS/

0k/f-

J/

SS

~~

1.0

f ~

s"~

~

J

/

J

0.6

0.2

.,...

0.0

I

I

I

-0.1

I I I

!

!

0.0

0.1

Figure 2.4" Perturbation solution of a one degree-of-freedom system with intrinsic imperfection

0 - Oo + 01 + 02 + 03 +

...

(2.48)

Eq. (2.48) can also be written in terms of 00 as O-- f[1 + (aT- b72+ CT3)

+ (aT- bT2+ CT3)2 + (aT- bT2+ CT3) 3 + ...1 2.3.3

Numerical

(2.49)

Results

Let us consider an imperfection amplitude T -- 0.1, (or else f - 2(70). A perturbation solution of the rotation is shown range of variation o f - 0 . 1 _< v _< 0.1. Changes in found to be up to 30% with respect to the perfect

with P - 2Co/L in Fig. 2.4, for a the rotations are system. For the

44

Thin-Walled Structures with Structural Imperfections: Analysis

range considered, a linear solution provides a good approximation, but for larger values of v it is necessary to include higher order terms. The results of this problem are shown in Table 2.2.

FIRST

SECOND

THIRD

Perturbation

0.65

0.68

0.685

Equivalent Load

0.6355

0.6722

0.6821

Exact

0.6858

Table 2.2 Comparison of different techniques of analysis for the one degree-of-freedom model of Fig. 2.3.

A perturbation analysis with just one term yields only 5% error, while the equivalent load technique with one term has an error of 7%.

2.4

IMPERFECTIONS IN BOTH GEOMETRIC AND INTRINSIC PARAMETERS

We can also model problems with imperfections in the geometry and in some intrinsic parameter acting simultaneously. An extension of our previous ideas would lead to a mathematical model with two parameters, r and (:

k (1

-

a17" +

517 .2 -

Cl T3) (1 -- a2~ + b2~2)0 - f - 0

(2.50)

In the following, we apply the two techniques of analysis to this system.

Single Degree-of-Freedom Systems with Imperfections 2.4.1

Perturbation

45

Analysis

The response 0 of this system is now expanded in terms of two parameters, rather than just one. Thus, 0 -- 000 "~- 7"010 "4- ~001 -~- 7"2020 + r 2 002 + 7"~011 -4- T 3 030 -~-..-

= r~(mo~m

(2.51)

where 1 1 0 (n+m)O O Tt,m

~-

n! m! OTni)~~

(2.52)

with m , n - 0, 1, 2, .... The perturbation of Eq. (2.50), up to cubic terms, is

(kOoo - f) -I- 7-[k (0~o - al0oo)] --[-~ fig (001 - - a20oo)]

ArT 2

fig (020 -- al010 -4- hi000)] -[- ~2 [k (002 - a2001 At- b2000)]

"-[--T~ [k (011 - a149Ol - a201o - ala2Ooo)]

--[--T3 fig (030 -- a~02o.+ b1~910 - c1000)]--[:- ~3 [/~ (003 - a2~02 ~- b2001)]

-[--T~ 2 [k (012 -- a2011 '[ 52010 -- a l a 2 0 0 1 -- alb2000)]

-4- T2~ [~ (~21 -- alOll -4- blO01 - ala2OlO - a2blOo0)] -~-... - 0

The solution of the perturbation equations is

eoo 010-

al0o0;

f

~01 --

a2000

(2.53)

46

Thin-Walled Structures with Structural Imperfections: Analysis

020- (al2

-bl)000;

030--(a3-2albl 012-

002- (a 2 --b2)0oo; + Cl) 000;

2ala~Ooo;

011 ~ ala20oo

0o3-- (a3-2a2b2) Ooo

021 --

2a2a~Ooo

(2.54)

The solution that we wanted in Eq. (2.51) is now

0-- f[1 +

alY + a2~ + ala2T~ + (a21-

bl)T2+ (a22--b2)~2

+ (a 3 --2albl-+-Cl) T3+ (2al a2) Y2~ -4- (2a2al2) ~27" + (a 3 -- 2a2b2)~3 + ...]

2.4.2

(2.55)

Equivalent Load Analysis

From Eq. (2.50) it is possible to write (2.56)

kO - f + f*(~, T, O)

where f*((, T, O) -- kO(alT - bl T2 "-{-ClT3 -4-a2~ -4- ala2T~ + bla2T2~ -- Cla2T3~

-- b2~2 +

alb2T~ 2 --

blb2-r2~2 +

clb2T3~ 2)

or else

0) - k0r (,, where r (v, ~) is the term in parenthesis in Eq. (2.57). Iterative solution of the equilibrium equation (2.56) leads to Step 1"

(2.57)

Single Degree-of-Freedom Systems with Imperfections

47

Oo= f Step 2: f*(~, r, 0) - kr 01 = r

r

--

Step 3: f**(~, T, 0) - ]~r 02-

r

-- r

Step 4: f***(~, T, O) -- kr 03-

r

-- r

The solution using the equivalent load method is O - f (1 + r 1 6 2

2.5

2.5.1

MAXIMUM RESPONSE

(2.5s)

+Ca+...)

VALUES

OF THE

General One Degree-of-Freedom

Systems

We have considered systems for which linear equilibrium takes the form of Eq. (2.1), while an associated imperfect system is of the form of Eq. (2.38). For such general systems, it is important to investigate if there is an imperfection for which the response of the system attains a maximum value. Such a situation would be extremely useful in design, as it would be associated with the kind of imperfection producing higher values of displacements or stresses in the system.

48

Thin-Walled Structures with Structural Imperfections" Analysis

A maximum in the space of ( 0 - 00) versus the amplitude of imperfection z is characterized by dO dT

= 0

(2.59)

To investigate the maximum, the perturbation procedure presented in this chapter is especially convenient, since it provides an analytical relation between 0 and r, so it will be adopted in this section. Let us consider the general case, for which the solution is of the form of Eq. (2.44). Its substitution into Eq. (2.59)leads to dO

d---T= a + 2 (a 2 - b) 7 -4-3 (a 3 - 2ab-4-c)7"2+ . . . - 0

(2.60)

If one retains up to quadratic terms in 0, Eq. (2.60) results in 1

a

Tmax -- --~a 2 - b

(2.61)

But since the range of values of r that actually reduce the values of the properties of the system are r < 0, then the condition of Eq. (2.61) reduces to a2 - b < 0

(2.62)

One may now state the following conclusion: For a general one degree-of-freedom system k O - f with an associated imperfect system of the form k ( 1 - ar bT2 - - C T 3 - J r - . . . ) 0 - f - O, in which r is the amplitude the imperfection, a necessary condition for the existence a maximum of 0 is that a 2 - b < O.

2.5.2

0, + of of

Structural Systems

Next, we will see if the above conclusion is of relevance to the kind of structural systems considered in this chapter. For the geometrically imperfect model, we obtained the following coefficients c - O;

2 a - ~ ; tanc~

b-

1 (tan c~)2

(2 63) "

Single Degree-of-Freedom Systems with Imperfections

49

Thus, 2 a2-b-->0 3 and a maximum does not exist. In the problem with intrinsic imperfection, we have a-

3;

b - 3;

c-

1

(2.64)

(2.65)

and a2-b-6>O

(2.66)

Again, there is no maximum associated with that problem. This leads to the following new conclusion: One degree-of-freedom structural systems with one imperfection parameter cannot present a maximum in the response with respect to the amplitude of the imperfection. This conclusion is only valid for one degree-of-freedom systems, and we shall see that there are maximum values of displacements and stresses in systems with many generalized coordinates.

2.6

CONCLUDING

REMARKS

In this chapter we considered simple structural systems, formed by rigid bars and springs, in which the energy depends on one degree of freedom. Geometrical imperfections are introduced by small modifications in the initial geometry, leading to initial strains but no stresses. Intrinsic imperfections are modeled as changes in the properties of the springs. We look for results showing the changes in the response as a function of the amplitude of the imperfection. The analysis introduced is based on equivalent load and on perturbation techniques. In the first, the imperfection is studied as a new load acting on the perfect structure. In the perturbation analysis, the solution is expanded in terms of the amplitude of the imperfection. For linear kinematic and constitutive relations, the equilibrium conditions are a non-linear function of the amplitude of the imperfection. The examples show that the two techniques require the solution of

50

Thin-Walled Structures with Structural Imperfections: Analysis

several linear systems to obtain the final solution; however, the solutions obtained by perturbation and equivalent load techniques are not identical. Systems that have both imperfections in the geometry and in constitutive parameters contain two amplitudes of imperfection and one degree of freedom. The analysis is more involved, but it follows the same lines described previously. The analysis of systems that have many degrees of freedom is more involved, and requires the use of matrix notation. This is presented in the next four chapters.

Single Degree-of-Freedom Systems with Imperfections

51

References [1] Bellman, R. E., Perturbation Techniques in Mathematics, Physics and Engineering, Holt Rinehart and Winston, New York, 1964. [2] Croll, J. G. A. and Walker, A. C., Elements of Structural Stability, Macmillan, London, 1972. [3] E1 Naschie, M. S., Stress, Stability and Chaos in Structural Engineering: An Energy Approach, McGraw-Hill, London, 1990. [4] Keller, L. B., Perturbation theory. In: Teoria de Bifurcacoes e suas Aplicacoes, vol. 2, Laboratorio Nacional de Computacao Cientifica, LNCC/CNPq, Rio de Janeiro, 1985, 83-152. [5] Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973. [6] Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981. [7] Reddy, J. N., Energy and Variational Methods in Applied Mechanics, Wiley, New York, 1984. [8] Simmonds, J. G. and Mann, J. E., A First Look at Perturbation Theory, R. E. Krieger, Florida, 1986. [9] Washizu, K., Variational Methods in Elasticity and Plasticity, 3rd. ed., Pergamon, New York, 1982.

Chapter 3 IMPERFECTIONS INTRINSIC PARAMETERS 3.1

IN

INTRODUCTION

In the last chapter, we were exclusively concerned with simple onedimensional problems in which the influence of geometrical and intrinsic imperfections was studied. This led to an initia,1 understanding of the analysis and behavior of very simple imperfect systems. However, in most situations of practical interest, a one degree-of-freedom model is inadequate to describe the behavior of a structure. Furthermore, new features in the analysis and in the behavior are present in more complex structuralsystems. Finally, we want some way of modeling not only the behavioral characteristics, but also an accurate description of the local stress redistribution in the area of a,n imperfection. In this and the following chapters we consider more general multiple degree-of-freedom systems, to model imperfect thin-walled structures or structural components. The structural systems considered here are investigated for a fixed level of the load; however, being basically linear systems in terms of the relation between loads and displacements, the response for other load levels would simply be proportional. Non-linear problems are considered in Chapter 6. It is assumed that the structure is characterized by multiple de52

Imperfections in Intrinsic Parameters

53

grees of freedom (or generalized coordinates), and they arise as a consequence of a discretization of the continuous body. Special attention is given to discretizations in terms of finite elements, in which the degrees of freedom are displacements or rotations at the nodes of an element. Other main assumptions adopted are related to linear kinematic relations and linear elastic constitutive material. As stated in Chapters 1 and 2, we consider that imperfections may affect either intrinsic or geometric parameters, and distinguish between them as follows: 9 Intrinsic parameters are those that define the constitutive equations of the problem, and include the modulus of the material and some sectional properties, such as thickness, area, inertia, etc. The element stiffness matrix of the problem is usually an explicit function of these parameters. 9 Geometric parameters are those that define the mid-surface of the structure, the boundaries, or the domain. The element stiffness matrix is not an explicit function of geometric parameters, which enter into the analysis through the integration process.

This chapter deals with the techniques of analysis of imperfections that can be modeled as changes affecting intrinsic parameters of a thin-walled structure. In the next chapter we focus our attention on imperfections in geometric parameters, also called geometrical imperfections. Many imperfections can be modeled by modifications of intrinsic parameters. Some examples tha, t involve significant stress redistributions are described below. 9 In structures that have closely spaced cracking, the reduction in the stiffness may be reflected by modifications of the stiffness parameters. Typical smeared-out approaches to cracking include the re-definition of the stiffness or modulus of the material in given directions. This may even lea,d to consideration of a new material with orthotropic properties. 9 In other structures, a sharp reduction in the thickness may be seen as an imperfection, and modeled by a, suitable function that follows the va,ria,tion in t.

Thin-Walled Structures with Structural Imperfections: Analysis

54

9 In many concrete structures, the partial loss of the thickness of a thin-walled member in some areas may be caused by spalling of the material. A model of this problem should include the change in the thickness, and this is done as an intrinsic parameter. 9 A variation in one of the moduli of the material may be a. consequence of a degradation of some of the properties. This degradation can be caused by environmental action, or by aging. In Section 3.2, we present the basic equations that define the imperfection in terms of parameters. An imperfection is assumed to have a maximum amplitude, a space distribution, and a position in the domain of the structure. A direct method of analysis, which only requires large computational capabilities is briefly discussed in Section 3.3; and advantages and problems in its use are outlined. A matrix formulation of the equivalent load is presented in Section 3.4. Solution of the equivalent load analysis is achieved by direct iteration; and errors in the iterative procedure are discussed in Section 3.4. Some explicit forms of equivalent loads for thickness changes based on the differential equations of the problem are presented in Section 3.5. In Section 3.6, intrinsic imperfections are modeled by a perturbation technique, following the presentation of Ref. [7]. The equations are derived with emphasis on imperfections that can be modeled as thickness changes, because this is one of the most complex problems (the bending stiffness is a cubic function of the thickness). Comparison between perturbation and equivalent load techniques are made. Finally, an example of a two degree-of-freedom system is presented in Section 3.7, where perturbation and equivalent load techniques are applied to illustrate their uses and differences. This is a simple problem with three rigid links and moments and line springs, but it is complex enough as to require the computation of the matrices defined in this chapter.

3.2 3.2.1

BASIC EQUATIONS Stiffness M a t r i x

We employ here the usual notation of finite element analysis, in which a symmetric stress tensor with six different components is represented

Imperfections in Intrinsic Parameters

55

as a vector a. Similarly, a strain vector r is considered to represent the six different components of a symmetric strain tensor. A simple linear constitutive equation may be written in the form - D~

(3.1)

where D is the constitutive matrix and ~r and e have to be measures of stress and strain consistent between them. For a thin-walled structure, the computation of D depends on the modulus of the material (modulus of elasticity, E; Poisson ratio, v; etc.) and on the thickness t.

The kinematic relations are written as - B a

(3.2)

where B is the linear strain-displacement matrix, which depends on derivatives of shape functions N and on geometric variables. In the context of a finite element analysis, Eq.(3.1) may be interpreted as the constitutive relation at an element level. The stiffness matrix K ~ of an element e may be written as K ~ - f S T D B dv

(3.3)

The element load vector p~ is given by pr - f N T f dv

(3.4)

v

where f is the vector of load intensities; and N contains the interpolation or shape functions. Assembly of the element contributions K ~ , a ~ and p~ yields a linear set of equations in the matrix form K a-

p - 0

(3.5)

subject to boundary conditions. In this chapter we consider forms of imperfection that can be modeled by modifications of the intrinsic parameters that define K. Problems in which the boundaries, the domain, or the position of the midsurface of a thin-walled structure change with respect to a design or ideal situation are considered in the next chapter. Familiarity with finite element techniques is assumed in this chapter, and the reader is referred to the various books on the subject. It

Thin-Walled Structures with Structural Imperfections: Analysis

56

is impossible to give credit to all available books on the finite element method, covering different aspects of the method in solid mechanics; however the fundamentals of our current understanding of the method may be found in Ref. [31, [18], [12], [13], and [1].

3.2.2

Models of Degeneracies in Intrinsic Parameters

A distinctive feature of the modeling of intrinsic parameters is that the variational equation, or the equivalent equilibrium condition, is an explicit function of them. If an intrinsic parameter is denoted by b, it is convenient to write an explicit dependence of b on three parameters: 9 on b0, indicating the value of b in an ideal, reference, or perfect system; 9 on an amplitude parameter of the imperfection T; and 9 on the space distribution of the imperfection.

Thus, b - b (b0, T, xi)

(3.6)

For example, Eq. (3.6) may take the form b - b0[1 + r F(x~)]

(3.7)

where F(xi) represents the space distribution of the imperfection. A decrease in the value of bo is associated with negative values of T, whereas a positive ~" denotes an increment in b with respect to bo.

3.2.3

Examples of Functions for Intrinsic Imperfections

A local imperfection in b may be modeled by simple analytical functions F(xi). A step change in the value of b in the domain x I < x < x F results in (Fig. 3.1.a) b - b0[1- T F(x)] with

(3.8)

57

Imperfections in Intrinsic Parameters

F(x)-

0

for

xI > x > xF

F(x) - 1

for

X I <__ X <__ X F

A smooth change in b may be modeled as (Fig. 3.1.b) f(x)-

COS

-~

(X--X

C)

for

XI < X < XF

where h -

X F~

X C -- X C -- X I

The extent of the degeneracy, modeled by a parameter q, may also be introduced a,s a variable in b, leading to (3.9)

b - b 0 [1 + r F (q,x~,x~)] An example of the function F in Eq. (3.9) is

(,, x,, x~) -cos ~ (x_ x~)

]

so~

(x c - ~) _< x _< (x c + ~)

A linear dependence of the value of b on the amplitude of change r has been assumed. The dependence on the extent 7/, on the other hand, is non-linear, and in the form

1(~-~) l( x x)4 l( x x)0 b - b0{1 + r [ 1 - _

~

~

_

7/

+...])

Many simple degeneracies can be characterized in terms of three parameters" the amplitude r; the extent q ; and the position x/c. More complex degeneracies require more parameters included in the model. This is a significant difference with respect to one degree-of-freedom systems, which depend on just one imperfection parameter T.

58

Thin-Walled Structures with Structural Imperfections: Analysis

Figure 3.1" Examples of thickness reductions

3.3

DIRECT

ANALYSIS

Attention is now given to the techniques that can be employed in the analysis of a structural system with imperfections in intrinsic parameters.

3.3.1

Finite Element Equations

The simplest idea is to employ a finite element (or an equivalent discrete) formulation to follow the local variation in b. Thus, a change in b is included in the data of the problem. The direct approach is somehow inherent to the finite element method, in which details of variations in the properties of a structure can be taken into account through the local properties of the elements. In the direct analysis, the constitutive relation is internally computed as a - D (b) e (3.10) In this approach, matrix D does not require any modifications with respect to a standard formulation. The element stiffness matrix results

Imperfections in Intrinsic Parameters

59

Figure 3.2: Semi-analytical and finite element discretizations of a cooling tower in

K ~ ( b ) - f B T D (b) B dv

(3.11)

13

The assembled system of equations is K (b) a -

p - 0

(3.12)

which must be solved for a. The direct approach involves a, simple use of the finite element method, in which details of variations in the properties of a structure are taken into account through the local properties of elements. In the direct approach there is no formulation a,s such, other than the standard finite element formulation.

3.3.2

R e m a r k s on t h e D i r e c t A n a l y s i s

A direct approach to model intrinsic imperfections is attractive because: . It can be performed using a general finite element code. . It does not require any modifica, tions in the program. However, there are some drawbacks in its use:

60

Thin-Walled Structures with Structural Imperfections: Analysis

Figure 3.3" Semi-analytical and finite element discretizations of a folded plate assembly 9 It may be expensive to perform a direct analysis, since it requires a full two (or three) dimensional discretization of the thin-walled structure. 9 It does not take advantage of the fact that K ~ is an explicit function of b. 9 For each specific imperfection studied, a new finite element analysis has to be carried out. 9 Only computational experimentation is possible to isolate relevant ranges of variables, or to understand the main features of the behavior. A typical example to illustrate some difficulties with the direct approach is found in the analysis of shells of revolution and plate assemblies shown in Fig. 3.2 and 3.3. Significant computational economies are obtained in shells of revolution with a semi-analytical element; but if a degeneracy modifies the properties in the prismatic direction (the circumference), then a direct analysis would require a new discretization using a two-dimensiona.1 shell element. Something similar occurs with prismatic plate assemblies, in which the finite strip method is the most common computational tool employed. An imperfection affecting the longitudinal direction destroys

Imperfections in Intrinsic Parameters

61

the prismatic nature of a property of the structure, and the direct analysis of the imperfect plate assembly has to be carried out by means of rectangular elements.

3.4

EQUIVALENT

LOAD ANALYSIS

In the equivalent load analysis, a degeneracy is modeled by means of a new loading system acting on the perfect or ideal structure. Such technique has the obvious advantage that, the degeneracy is confined to the load vector and does not affect the stiffness matrix. Equivalent load techniques to account for local changes in intrinsic parameters were perhaps first employed in Ref. [15]. This work considered thickness changes in plates using a finite strip code based on Mindlin's theory of plates. The changes in thickness were expanded in Fourier series in the prismatic direction to model arbitrary (but symmetric with respect, to the mid-surface) changes. Details of the computation of strip matrices leading to the equivalent load were given for a two-node finite strip with linear interpolation of nodal rotations. This work is discussed in Chapter 7. For shells of revolution, Ref. [14] implemented the equivalent load using a semi-analytical element that is a conical-frustra based on the Mindlin-Reissner hypothesis. This work also contains an explicit definition of the equivalent load, which is not just a pressure but a load vector with five components: three pressure and two moment components. This work is discussed in Chapter 10.

3.4.1

Equivalent Load

To present the equivalent load technique, we start by rewriting the constitutive equation (3.10) as a - [Do + D~]e

(3.13)

where Do is associa, ted to the perfect system; and D~ is due to the degeneracy. Replacement of Eq. (3.13) into Eq. (3.3) results in K~(r) - K~) + K~

(3.14)

62

Thin-Walled Structures with Structural Imperfections: Analysis

where K~) - j" B T Do B dv

(3.15)

v

K~ - f B T D . B dv

(3.16)

v

Assembling the element contributions in Eq. (3.14) into a global system yields [K0 + K . ] a -

p - 0

(3.17)

Notice that numerical approximations other than those associated with the discretization technique have not been introduced in Eq. (3.17). Let us now define a vector p* as p* - - K ~

a

(3.18)

and rewrite Eq. (3.17) in the form K0 a - p + p*

(3.19)

The vector p* is called the equivalent load because it can be interpreted as a load acting on the perfect system and whose influence is equivalent to the effect of the degeneracy. Notice that p* is a displacement-dependent load, since the displacement vector a needs to be computed before p* is evaluated. Fig. 3.4 illustrates the equivalent load system for a shallow beam with a change in the thickness. The resulting force p* is normal to the beam axis and localized in the zone in which the thickness is reduced. As a consequence of p*, an additional displacement is introduced (due to the larger flexibility of the area affected), which has to be added to those occurring in the beam of uniform properties.

3.4.2

Kirchhoff Plate with Thickness Changes

To illustrate the computation of Do and D~ , let us consider a change in the thickness of a flat plate under Kirchhoff theory. For a thinwalled structure, the generalized stress and strain vectors are defined as

Imperfections in Intrinsic Parameters

63

Figure 3.4: A loa,d equivalent to a change in the thickness

o'T--{ Nll --

s

N22 N12 Mll

s

s

Xll

M22

M12 }

X22 X12

where Nij are the membrane stress resultants; Mij are the moment resultants, both per unit length in one direction; eij are the membrane strains; a,nd Xij are the changes in curvature. The elasticity matrix D under the Kirchhoff hypothesis is of dimension 6x6, and takes the form

o [om 0] 0

Db

(3.20)

Thin-Walled Structures with Structural Imperfections: Analysis

64

The membrane contribution in Eq. (3.20) is

om

1

--

v2

v

1

1

0 0

o 1

0

(3.21)

while the bending part of Eq. (3.21) may be written as

ob

-12(1

Iiv

- v 2)

0 1

v 1 1 0 0 0 ~(1-v)

(3.22)

As an example of change in an intrinsic parameter, let us consider a step reduction in thickness t. Previously we saw that the thickness variation in the zone of imperfection, takes the form t-

to (1 + r)

(3.23)

as illustrated in Fig. 3.1. For this particular case, matrix Do is of the same form as D in Eq. (3.20), but with t - to; whereas D , in Eq. (3.13) is D~

rD~

-

(3.24)

Db,r -- ( 3 T --[- 3T2-~ - T 3) D ob

3.4.3

Iterative

Solution

(3.25)

Scheme

A convenient way to solve Eq. (3.19) is by means of an iterative technique, in which the equivalent load is computed from a previous solution of the displacement vector a. The first step is the solution of a perfect system K0a0 - p -

0

(3.26)

leading to a0 -

(3.27)

Kolp

Next, a first order equivalent load p* is computed as p* -

- K,

for which equilibrium is satisfied by

a0

(3.28)

Imperfections in Intrinsic Parameters

al

-- Kolp

65

*

(3.29)

The n-order equivalent load p**"* results in p*** = - K~ an-1

(3.30)

where the n-order approximation in displacements is given in the form an -- K o l p * ' *

(3.31)

The complete solution can now be written by adding the response to the original load and the response to the equivalent loads, i.e. a-

a0 + al + a2 + ... an

(3.32)

It is convenient to look at the contributions of equivalent load in terms of ao, i.e. p* -- -- Kr ao

,.,- -~ ,,,~=-~

~-

a=- - ~

~ [~o1~.] a. [~o1~] [~o1~] a.

This is not an efficient way to compute the equivalent load, but it shows that everything depends on the solution of the perfect system, a0.

The stress field is next evaluated at an element level using the constitutive equations c r - [D0 q- D~] B a ~ or else a - [ D o + Dr] B (a~ + a~ -+- a~ -t-...) = ao + 31 + ~2 + ... where ao - D o B a~

(3.33)

Thin-Walled Structures with Structural Imperfections: Analysis

66

(71

--

Do B a~ + D . B a~

~2 - Do B a~ + D , B a~ Since r has not been retained as an explicit parameter, the original problem of equilibrium is linear in a. In the equivalent load technique, we solve a series of linear problems. Has the computational effort increased with respect to a direct approach? The main difference is that each linear system has the same stiffness matrix Ko of the perfect structure, while the direct approach needs the solution using (K0 + K,). In many structures, such as shells of revolution or a plate assemblies, a semi-analytical finite element method is convenient for the analysis of the perfect structure. If the imperfect structure is studied by means of the equivalent load, a semi-analytical solution can still be used in the discretization, and this may save a large amount of computational effort.

3.4.4

Errors in t h e I t e r a t i v e S c h e m e

In all iterative procedures it is important to know about the existence of bounds of the errors. The error e~ in the n-order approximation is

en

II a. II- II a._l II li a.-1 il

--"

=

Ilanll-1 If an-1 it

(3.34)

in which If ali represents a suitable norm of vector a. But since

an -- - K o l K ,

an-1

then,

Ila.l,- II ol . a--lll It may be shown that [8]

(3.35)

Imperfections in Intrinsic Parameters

IIi, o 1K~ an-ill <-

67

o d(Ko)IlK.

I liar_ill

IIKo I

(3.36)

where cond (K0)is the condition number of matrix K0; and IIKII is a suitable norm of K. Substitution of Eq. (3.36) into Eq. (3.34) leads to en <

cond(Ko) IlK" II

-

]lK0ll

1

(3.37)

Eq. (3.37) shows some important features of the equivalent load solution: 9 The error is bounded for each iteration. 9 This bound is the same for all iterations. 9 We do not expect to have a chaotic solution.

3.5

EXPLICIT EQUIVALENT

FORM

OF

THE

LOAD

The equivalent load p* has been obtained in the la,st section as the product of a matrix K , times the displacement vector of the last iteration, a n - 1 . This requires the computation of K , , and carrying the product K , an-1. In some problems it is possible to introduce some further simplifications at a physical level, on the linear differential equilibrium equations, and obtain an explicit form of the equivalent load. Such equivalent load is usually an approximation of the exact one, but the procedure has two main advantages: 9 It assigns some physical meaning to the effect of the structural imperfection (that is not present in K,). 9 It may lead to a significant economy in the computations.

Thin-Walled Structures with Structural Imperfections: Analysis

68

3.5.1

Mindlin P l a t e with T h i c k n e s s Changes

Let us consider a plate structure for which a thickness change may be written in the form [15]" t - to [: - ~ - ( ~ ) ]

(3.3s)

The equilibrium equation is (~2Mll

(~2M12

(~2M22

= p(xl x2) (3.39) Ox~ + 2 (~Xl(~X2 (~X22 For Mindlin's plate theory, the constitutive equations result in Mll

+

Et 3 ( 00: 092 ) 12(1 - v 2) \~Xl + v~x2

-

Et 3 ( (~1 (~2) M22 = 12(1 - v 2) vO~x~+ ~

Et 3 1 - v M12= 1 2 ( 1 - v 2) 2

(3.40)

(aOl aO2) ~ +

where 8i are rotations of the normal to the mid-surface. Substitution of Eq. (3.38) and (3.40)into Eq. (3.39) leads to an equivalent load in the form

Et3o

P* (Xl' X2' ~1' ~2) -- 12(1 - v 2) 3~1

•

(0

o~

(~3~1 +

(~3~2

03~2 )

OxiO~ + o~ox~ + Ox~ (926}1

+ 6 ( 1 - - 2 T + T 2) (l+v2

OX l C~X2

1

1 -- ?2 0202

2

Ox~

a20:

0r Ox~ Ox2

- 6 ( 1 - r) v ~ x 1 + ~-~x2] + 3(1 - 27 + T

001

002~ 02r

(3.41)

Imperfections in Intrinsic Parameters

69

Thus, for a thickness variation in one coordinate direction, the equivalent load p* is a function of the first, second, and third order derivatives of the rotations 01 and 02; and of the function that approximates the changes in thickness r, and its first and second derivatives.

3.5.2

M i n d l i n P l a t e w i t h Step C h a n g e in Thickness

For the same problem of the previous section, we next assume a step change in the thickness in the direction of x2. The function ~- is piecewise constant in the form

t-to

for

xI2 ~ x2 ~ x F

t-0

for

x~ >_ x2 >_ x F

(3.42)

where x / and x f indicate the initial and final coordinates of the area affected by the change in thickness, and r is a parameter that represents the amplitude of the change. The equivalent load reduces to [15]

Et3o P* (01' 02) - 12(1

0301 X

0301

-

v 2)

( 3 7 - 372

+ T3)

0302

0302 )

OX31 + OXlOX2 + OX20X2 + .OX3

(3.43)

for x / < x2 < xF; and

p* (01, 02) -- 0 for

_> x2 _> x f

Even for this simple case, it is clear that the equivalent load depends on displacement variables that are not present in a typical finite element discretization, such as the third derivative of rotations. In this case, the explicit calculation of the equivalent load can only be implemented in an analytical solution; or else, in a semi-analytical finite element approximation, in which the direction of the thickness change in x2 is coincident with the prismatic direction of the problem. In the

70

Thin-Walled Structures with Structural Imperfections: Analysis

latter case, because trigonometric functions are employed in the prismatic direction, it is possible to obtain (a392/ax~) and (0391/OXlaX~). An approximation to Eq. (3.43) is

Et3 P* (01' 02) -- 12(1 - v 2)

3.5.3

(37"- 37"2+ v3) ( (~301

0X 10X22

+ 0302)

(3.44)

Shell of R e v o l u t i o n with Thickness C h a n g e

Let us consider a simplified form for an equivalent load in shells of revolution. We start by assuming Kirchhoff-Love hypothesis, and considering that Xl is the coordinate along the meridional curve; x2 = rg, is the circumferential coordinate; and R1 and R2 are the principal radii of curvature. Following Ref. [10], equilibrium of the shell may be written as 0Nil (~Xl (~N22 0X2

f

0N12 0X2

1 Or f- (Nll - N22) + Pl -- 0 r~Xl

1 Or 0N21 {- -(N12 ~- N21) -4- P2 -- 0 0XI r~Xl

Nil N22 0N13 0N23 +P3 - 0 R I + -~2 + (9Xl "~ OX 2 N13 __

-N23 -

(3.45)

1 Or (M11- M22)

0Mll (~Xl

0M21 fOX2

r C~Xl

c3M22 0x 2

0M12 0x I

1 Or (M12 -+- M21) r Ox I

with the usual definitions in the KirchhoffoLove theory for twisting moments M and shear S [10] M - ~ 1 (M12 + M21 )

S ' - N12

M21 R2 -- N21

M12 R1

(3.46)

We assume here a thickness variation in the circumferential direction

Imperfections in Intrinsic Parameters

t-

71

t0[1 - r(x~)]

(3.47)

and the thickness remains constant in the meridional direction. It is expected that the major contribution to the equivalent load will be given by a pressure normal to the mid-surface of the shell, and special attention should be given to the third of Eq. (3.45). Replacement of the shear resultants in terms of bending moment resultants and introducing the transverse displacement u3 lead to a simplified equilibrium condition

N22

Et 3 12(1 - v 2)

Nil "lR1 R2

~74tt3 +

P3 - 0

(3.48)

where V4u3

-

04U3

"-""'--"~,,04U3 04u3

+2OxOx

(3.49)

To obtain Eq. (3.48), it has been assumed that the thickness is piecewise constant. The membrane contributions are linear in t, whereas the bending contribution is cubic. Replacement of Eq. (3.47) into (3.48) leads to

R1

R2

N~ V4tt3 -- --P3 -- P~

(3.50)

where

P3----

-~1 + - ~ 2

T + 12(1--v 2)

and

Eto N~ : (1 - v 2) (s

+ ?J s

Eto N~ = (1 - v 2) (e22 + v s

(3.52)

To compute the equivalent load, Eq. (3.51), the in-plane stress resultants for constant thickness are readily obtained from a previous

Thin-Walled Structures with Structural Imperfections" Analysis

72

iteration. However, the bending term depends on fourth-order derivatives. For a typical element, that employs cubic interpolation of u3 in zl and trigonometric functions in z2, the first term in Eq. (3.49) is automatically neglected.

3.5.4

R e m a r k s on the Equivalent Load Analysis

An equivalent load approach is attractive when it is used in conjunction with semi-analytical methods of discretization (finite strips or finite ring elements for shells of revolution), in which the displacement field is approximated by finite elements in one direction and trigonometric functions in the other (prismatic) direction. It is also very convenient when the solution is obtained in an analytical form. Advantages of the equivalent load technique must be mentioned here: 9 There is a more physical interpretation of the effects of the imperfection. 9 One can deal with non-prismatic imperfections using a formulation for prismatic thin-walled structures. 9 If the equivalent load is obtained from the equilibrium equations of the system, it does not require any modifications to standard semi-analytical programs. Some drawbacks of the technique should also be mentioned" 9 For each specific imperfection studied, a new finite element analysis has to be carried out. For example, if a decrease in thickness of 30% has been studied, a new analysis is required to consider the influence of a reduction of 40%. 9 Only computational experimentation is possible to isolate the most critical imperfection parameters (as in the direct analysis). 9 Several matrices need to be computed, which are not present in standard finite element codes.

Imperfections in Intrinsic Parameters

3.6

PERTURBATION

73

ANALYSIS

Perturbation techniques have been employed in mechanics to follow a non-linear path, starting at an equilibrium condition that is known, and using a perturbation parameter to advance along the path. They have played a central role in many problems, including the theories of elastic stability and non-linear dynamics. Perturbation approaches have been recently developed for sensitivity analysis (see, for example, Ref. [2], [9]). We have included an appendix on the basic aspects of perturbation techniques, since they constitute an important part of the development of this book. One of the first applications of perturbation techniques to expand material properties is due to Westergaard [17], who considered changes in Poisson's ratio. Vinson [16] expanded the stiffness coefficients in a composite plate with reference to an isotropic plate. The perturbation technique was formulated via, a variational approach in Ref. [7] and implemented in Ref. [11] for a finite strip model of a Kirchhoff plate assembly. The strip matrices used in the perturbation analysis were given for a three-node strip. Results have been compared with the equivalent load and direct analyses. Fertis and Mijatov [6], using some earlier work by Fertis, have developed a technique that they call equivalent system for variable thickness plates. The equivalent system proposed there may be viewed as a perturbation technique applied to the differential equation governing the problem. What they call the equivalent system is the set of perturbation equations presented here. The form of the thickness variation considered was a step change [6].

3.6.1

Expansion of the Thickness

In the following, it is assumed that the change in the thickness (or other intrinsic parameter) of a thin-walled structure is characterized by its amplitude, and that the zone that is affected by the change is fixed. This assumption eliminates the extent of the degeneracy as a parameter in the perturbation procedure. A more complete definition of the degeneracy in terms of several parameters would be appropriate for analytical studies; however, if a finite element is employed to model the structure, the associated space discretization is not amenable to leaving the extent of the degeneracy as a variable.

74

Thin-Walled Structures with Structural Imperfections: Analysis

Before examining the application of perturbation techniques to the analysis of degeneracies, it is convenient to stress that we are dealing with a non-linear problem in terms of the amplitude of degeneracy r and the solution vector a. For a cubic dependence of the constitutive matrix on the intrinsic variable b, Eq. (3.13) may be written as

(3.53)

D(b) - Do + TD1 + 7-2D2 -+-r3D3

where Do is computed using bo; and the last three matrices Dk are associated to the change in constitutive properties, which are linear, quadratic, and cubic in the amplitude T. Replacement of Eq. (3.53) into Eq. (3.3) results in K ~ - K~ + rK~ + r 2 K~ + r3K~

(3.54)

where K~

-

f BTDkB dv

(3.55)

for k = 1, 2, 3. For simplicity in the presentation, it may be assumed that each element affected by the degeneracy has the same amplitude v; that is, a single imperfection parameter is considered in the analysis. However, more than one imperfection parameter could be taken into account without any further conceptual difficulties. Assembling the element contributions, Eq. (3.55) into a global system yields Ko + TK1 + T2K2 -+"r3K3] a - p - 0

3.6.2

(3.56)

Plate with Thickness Change

For the same plate problem considered in the equivalent load, the matrices D are:

0

o]

Db~ +T

0

o]

D~

+ r2

0

where D~ - D~;

D~ - 3Dbo

o]

D~

+ ra

0

o]

D~

75

I m p e r f e c t i o n s in I n t r i n s i c P a r a m e t e r s

D ~ - 0;

D~ - 3Dbo

D ~ - 0;

Db3 - Dbo

Thus, the construction of matrices Dk, K~ and Kk is made with only minor modifications of an existing code.

3.6.3

Perturbation Equations

The variable b associated to the structural degeneracy has been expanded in terms of an amplitude parameter T, leading to the global equilibrium condition, Eq. (3.56). We shall now consider that the displacement vector a may be expanded in terms of a perturbation parameter s in the form a-

ao -f- s a l -+- s 2 a 2 -4-...

in which 1 dna an = n! ds n I~=0

A convenient choice of the perturbation parameter is to adopt s T; that is, to expand a in terms of the same amplitude parameter 7" chosen to identify the degeneracy. Thus, a-

ao + T al + T 2 a 2 -1- ...

(3.57)

with 1 dna an =

n! dT n

I~=0

(3.58)

The Taylor expansion in Eq. (3.57) contains a first term a 0 , which is the solution of the problem for s = 0. Since s now represents the amplitude of imperfection, the displacement vector a0 is the solution for the perfect case, with b = b0 . We say that a0 is the reference solution. The first-order perturbation term contains the derivative of a with respect to the imperfection, and we assume that such derivative exists. Similar assumptions are implicit in higher order terms.

76

Thin-Walled Structures with Structural Imperfections: Analysis Replacement of Eq. (3.57) into Eq. (3.56) leads to N 2 ( TnK0 an "1"-Tn+lK1 an + rn+2K2 an + Tn+3K3 an) - p -- 0 n--0

(3.59) in which N is the number of perturbation terms included in the expansion. Eq. (3.59) may also be written as (Koao - p) + v (Koal + i l a o ) -4-...

+ Tn (Koan + Klan-1 + K2an-2 + K3an-3) - 0

(3.60)

The equilibrium Eq. (3.60) may be differentiated as many times as desired with respect to the perturbation parameter T, and evaluated for r - 0. This produces a set of perturbation equations: Koa0 - p

Koal -- -- K1ao Koa2 - - ( K l a l -t-K2ao) K0a3 = - ( K l a 2 + K2al + K3a0) Koan - - ( K l a n - 1

+ K2an-2 + Kzan-a)

(3.61)

As in all perturbation analysis, Eq. (3.61) is a set of equations that can be solved sequentially. The matrix associated to the vector of unknowns, K0, is the same in all cases, and for linear problems in displacements, it is non-singular. The terms on the right hand side of Eq. (3.61) are all known, since a n - l , a n - 2 , a n - 3 have been solved in previous perturbation equations. The solution of Eq. (3.61) is next replaced into Eq. (3.57) to obtain a parametric solution of a in terms of 7. Thus, not just the result for

Imperfections in Intrinsic Parameters

77

a specific amplitude of imperfection, but a whole range of variation of a is obtained as an explicit function of r. From Eq. (3.61) it is clear that matrix K3 is only present in the third-order perturbation equation; thus, this should be the minimum number of perturbation equations to be included in the analysis if the solution of the complete Eq. (3.56) is to be obtained.

3.6.4

Stress Resultants

Since b is a parameter in the domain of the problem, the elasticity matrix D is obtained from Eq. (3.53), whereas the vector of nodal unknowns is derived from the series expansion (Eq. 3.57), with the coefficients obtained from each perturbation system. At an element level, the displacement field is written as ae - a0 e + r a e1 + r2a~ -4- ...

(3.62)

Next, the constitutive relations within each element are applied in the form

a-

[Do+TDI+T2D+TZD]

B { a o~+ r a 1~+ r 2 %~ + . . . }

Expanding the above expressions, leads to O" - - (70 -+- 7"0" 1 -4- 7"20"2 - ~ - . . . T n O ' n

(3.63)

where o'o-

DoBa~)

is the generalized stress vector for the reference situation; and the other stress components are ~rl - D o B a ~ + D I B a ~ )

or2 - D o B a ~ + D I B a ~ + D2Ba~)

an - D o B a ~ + D I B a ~ _ 1 + D 2 B a ~ _ 2 + D 3 B a ~ _ 3

(3.64)

Thin-Walled Structures with Structural Imperfections: Analysis

78

3.6.5

Relation B e t w e e n Equivalent Load and Perturbation Techniques

A similar notation has been used to present both techniques, but it is important to state that the systems of equations to be solved are different. To show that, the equivalent load p*, Eq. (3.18), may be written as

p*-- [TK 1 -~-T2K2-~- T3K3] a

(3.65)

The iterative solution may now be written as the sequence of equations Koao = p

Koal - - [ K o + TK1 + r2K2 + 73K3] K0a2 - - [ K 0 + Koan -

ao

TK1 ~-T2K2 + T3K3] al

- [Ko + TK1 + T2K2 + T3K3] an-1

Comparison between Eq. (3.66) and Eq. (3.61) shows that matrices K0, K1, K2, and K3 have the same meaning in both sets, but the solution in Eq. (3.57) is different from Eq. (3.32). The iterative scheme may not be obtained from the perturbation analysis: this means that the vectors an in Eq. (3.66) do not represent derivatives of the solution with respect to the amplitude parameter. Furthermore, Eq. (3.32) is associated to a specific value of ~', whereas Eq. (3.57) is valid for any amplitude r.

3.6.6

Remarks on Perturbation Analysis

Some advantages of this approach may be summarized as follows: 9 It takes advantage of the fact that K ~ is an explicit function of B.

9 The solution provides a range of values of displacements and stresses in terms of the amplitude of imperfection.

Imperfections in Intrinsic Parameters

79

9 A search of extreme values of the response in terms of the imperfection amplitude is now possible. The main drawback is similar to one discussed in the equivalent load: 9 Several matrices need to be computed, which are not present in standard finite element codes.

3.7

TWO DEGREE-OF-FREEDOM SYSTEM WITH INTRINSIC IMPERFECTIONS

The example in this section is presented in order to clarify the application of the techniques discussed, using a problem that is simple and yet sufficiently complexity to require the computation of all the matrices involved. This simple two degree-of-freedom system has been employed by Croll and Walker [5] to illustrate buckling. Let us emphasize here that this model is not meant to describe precisely the response of any particular structure to the application of loads.

3.7.1

Basic formulation

The model is a three-link arch, with concentrated bending stiffness at the joints B and C and a spring at D. The load system is a force on the link BC, and boundary conditions prevent displacements at A and D (see Fig. 3.5). This problem has been solved using the symbolic manipulator Maple V [4]. With reference to Fig. 3.5, the total potential energy of the model may be written as V -

1

~ ( / 1 / ~ 1 -~- M2/~2 -~- F A 1 )

-

PA2

The kinematic variables are taken as in Ref. [5], i.e. /~1 -- L (cos 01 -~- cos 0 2 -~- cos (~ - 1 - 2 cos a)

(3.67)

80

Thin-Walled Structures with Structural Imperfections" Analysis

Figure 3.5" Two degree-of-freedom model studied

1

~-

~(Ux + u~)

~1 -- 0 / - - 0 1 - ~ - ' r

f12 -

a -

02 -

r

(3.68)

It is convenient to write the vertical components of displacements in the non-dimensional form

u1

u2

II,2 - - ~

721----L;

(3.69)

The non-dimensional displacements are (51 --

A1

--(lZ 1 -it.- ?22)sin a ~2 =

A2

_

(u~ + u 2 - uxu2)

1

L - 2 (~ + ~) /31 - 2Ul - u2

(3.70)

Imperfections in Intrinsic Parameters

~2 -

81

2u2 - Ul

(3.71)

The constitutive relations for an elastic material are F-

K/k1;

M1

-- C1/~l;

M2 - C2/~2

(3.72)

where C1 and C2 are the stiffness of the moment springs at joints B and C; and K is the stiffness of the linear spring at joint D. Substitution of Eq. (3.72)into Eq. (3.67) leads to

1 (C1/~I2 + C2,/~ + I{,A~) - P A 2 This can be conveniently written in terms of non-dimensional variables aS

1 (C,/3~ + C2/3~ + C2k~) - C2p~2 where

KL: k=~;

p=

C2

PL

C2

Next, the above equations are written in matrix form, adequate for the formulation presented in this chapter. The variables take the form:

(3.73)

92

a -

{Ul}

(3.74)

U2

~-

/r / M1

(3.75)

M2

and are related by the kinematic equations ~- Ba with

(3.76)

82

Thin-WMled Structures with Structural Imperfections: Analysis

I sina B 2 -1

sinal -1 2

(3.77)

The constitutive equations are a=De

(3.78)

where k

0

0

0 0

C1/C2

0 l

I

D=C2

0

(3.79)

The total potential energy now results in the form 1

V - :-cTDe- PA2

2

-

~laT

[BTDB]a -

aTf

(3.8o)

where f-pC2

1/2 } 1/2

(3.81)

Equilibrium states for this two degree-of-freedom problem are obtained from the condition OV 5V - 0---~-Sa- 0

(3.82)

where OV Oa

-

[B'rDB] a - f

(3.83)

or else Ka- f - 0

(3.84)

Imperfections in Intrinsic Parameters

83

Intrinsic P a r a m e t e r s

3.7.2

We start by assuming that the moment stiffness at the joint B, indicated by C1 in Fig. 3.5, is a function of a parameter r, with a form similar to Eq. (3.53), and may be written as

C1 - O1 (1 + 3r + 3r 2 + r a)

(3.85)

where C1 is a reference value of the moment stiffness considered in the analysis, i.e. C1 - C1 for r - 0. The constitutive matrix takes the form

I

k _ 0 0 C1 (1 + 3r + 3r 2 + r a)/C2 0 0

D - C2

0 1 0 1

(3.86)

Eq. (3.87) can also be written as the sum of four contributions, each one being multiplied by the parameter r to a given exponent

(a.sT)

D - Do + rD1 + r2D2 + raDa where

Do - C2

0

C1/C2

0

0

0

1

0

D2-

~ ~

0 0

D1 -

0

0 3(~1 0 0

;

0

3C1

0

0

0

0

I~ ~ ~ [0

;

0

0

D 3 - [ 0 01 0

0

0

0

(3.88)

0

The stiffness matrix takes the form of Eq. (3.54), with the equilibrium equation as in Eq. (3.56). The explicit forms of matrices K~

are

Ko- BTDoB - C2

[ k sin 23 + 4C1/C2 - --[-1 k sin 23 - 2C~ / C2 - 2

K1 - BTDIB -

_

k sin 23 - 2C1 / C2 - 2 k sin 23 + 01/C2 + 4

]

I 12C1 -6C1 -6C'1 3&1

] (3.89)

(3.90)

Thin-Walled Structures with Structural Imperfections: Analysis

84

K2 - BTD2B -

1201

-6C1

]

--6C1 3C1

(3.91)

K 3 - B TDaB - 3 [ 12C1 -6C'1 ]

-601

(3 92)

36'1

Notice that gl

-

K2 - 3K3

(3.93)

All the matrices in the example are computed for the complete structure, so there is no need to consider the element contributions.

3.7.3

Solution via Perturbation Analysis

If the vector of unknowns a is written as the perturbation expansion in Eq. (3.57), the solution of the first set of perturbation equation is the reference problem, not affected by the imperfection: ao-

Kolf

The inverse of Ko is K o l - ~1 [ ( C 1 ~ - C 2 ) k sin 2 o~-1!-C1] -1

-

k sin 2a + ClJ C2 + 4 - k sin 2a + 2C1/C2 q- 2 - k sin 2a + 2C1/C2 ~- 2 k sin 2c~ + 4C1 / C2 ~- 1 a0 =

P 6 [(C1 + C2)k sin2 a _F C1]

{

C1 + 2C2 -

2C1 q- C2

}

] (3.94)

The first order perturbation coefficient of the solution al results in

al --

KoXKI ao

where

K~

[(C1-[- C2) C1

k sin 2 Or -[- C1]

85

Imperfections in Intrinsic Parameters

2 ( k s i n 2 a + 2) -(ksin2a+2) ] - 2 (k sin2a - 1) (ksin 2a - 1) This leads to -(ksin2a+2) } (k sin 2a - 1)

PCIC2 a l --

2 [(~'~1 "+ C2)ksin 2 C~"-~-~'r ] 2

(3.95)

The second order perturbation solution is a2 -- - K o

-al-

I (Klal

+

K2a0)

[ o1 1] al

or else PC1C 2 [(2C1-

a2-

2

62)k sin2 a + 2C1]

[(el-t-62) ksin2 c~ + C'1]3

(ksin2a+2) } 3.96) - ( k s i n 2a - 1) (

The third order perturbation analysis leads to a3 -- - K o i ( K l a 2 + K2al + K3ao)

1

= --[KolK1] (~a0 + al + a2)

(3.97)

The generalized stresses may now be computed from Eq. (3.63) and Eq. (3.64). The results of sensitivity of displacements with respect to the perturbation parameter 7 are shown in Fig. 3.6, using first and second order perturbation approximations. Positive values of 7 mean C1 > C2. This produces a decrease in the vertical displacement at the joint B (called ui in the graph), and an increase in the displacement at joint C (called u: in the graph). For this particular problem, a linear solution including only first order perturbation equations shows the general trend of the sensitivity, but is only valid for Iv[ _< 0.5. The second order perturbation analysis shows an improvement in the overall picture of sensitivity, and the values of displacements are close to the exact

Thin-Walled Structures with Structural Imperfections" Analysis

86

solution for r < 1.5. In this particular problem, the exact solution for negative r shows large changes and this can only be approximated by higher order perturbation analysis. Sensitivity of stress resultants is shown in Fig. 3.7.

3.7.4

Equivalent Load Analysis

The matrix K~ can be constructed as

K~ -

K1 + K2 -4- K3

but on account of Eq. (3.93), we may write

K,.r-

(T'-t'-T2-~ " 3 T3) U 1

(3.98)

where K1 has the same meaning as in Eq. (3.90). The reference solution leads to the same value as in the perturbation approach, Eq. (3.94). The first order equivalent load is p* - -K~ao

(3.99)

or p.

-P

-~

X

54 [(C1 ~- C2)ksin2 c~+ C1] 2 c~ 1] ~(2~'~1_.~._C2)} { (4 +ksin 23 + 2~_~)(C, + 2C2) + ( 2 - ksin2a + 2C' ( 2 - k sin2c~ + ~2 ) ( C ' + 2C2) +(l+ksin2(~+ 4C1~ c2] (2~'~1+ C2) This leads to the solution (3.100) The second order equivalent load is

p** -K~al

(3.101)

-

and the second order solution is al

(3.102)

87

Imperfections in Intrinsic Parameters

Ul

0.002 2nd

0.0016

~ ~xact

1st

0.0012

~.

Exact

~

2

n

d 1st

0.0008 O.OOO4 0

U2

I

-0.4

I

-0.2

I

0

0.0016 ~

"

1st.

o :2nd

4

1st

I

0.4

~ . . . ~

0.0012 -

,,

0.2

Exact

0.0008

0.0004

I

-0.4

I

-0.2

0

I

0.2

I

0.4

Figure 3.6" Displacements due to perturbation analysis of the model of Fig. 3.5, for values ofC1/C2 - 1; k - 1,000 and c~ - 0.2 rad; C1 - C1 (1 + 3T + 3r 2 + r 3)

88

Thin-Walled Structures with Structural Imperfections: Analysis

Mt,l 0.003 1st

0.002 0.001

Exact l s t ~ l .~,., o

-0.4

F

~

2nd Exact

o'~176~

-0.2

I

0

0.2

0

0.2

i

0.4

0.6

0.5 0.4 0.3 0.2 0.1 I

-0.4

I

-0.2

I

I

0.4

Figure 3.7" Stresses due to perturbation analysis of the model of Fig. 0.2 rad; 6'1 = 3.5, for values of C1/C2 - 1; k - 1,000 and a C~ (1 + 3r + 3r 2 + r 3)

Imperfections in Intrinsic Parameters

89

The third order equivalent load results in p*** -- - K r a 2 and

a 2 - - r 1LKoiKj ai The complete solution in the equivalent load method is given by Eq. (3.32). Notice that in the equivalent load method, all matrices K1, K2, K3 are used from the beginning, even for the first order load. In the perturbation approach, on the other hand, the matrix K1 is employed in the first order perturbation; K1 and K2 in the second order perturbation analysis; and so on. The displacements have been computed for a set of values of v, using Maple V, and are plotted in Figure 3.8. For positive values of T the equivalent load results are close to the exact solution. The values of a[1] for negative r tend to diverge from the exact results, although convergence in a[2] is good. Results for stresses are presented in Fig. 3.9. Notice that the sensitivity of the solution with respect to the imperfection parameter depends on the actual values of original stiffness of the model considered. This particular example does not model any specific real structure and is employed to highlight the techniques of analysis presented. Solutions to real structural components are presented in the second part of the book.

3.8

FINAL

REMARKS

In this chapter we have considered imperfections that can be modeled through changes in intrinsic parameters. The methods developed for the analysis were a direct technique, an equivalent load, and a perturbation technique. One of the advantages of the perturbation technique is that it provides sensitivity results of the solution with respect to a parameter employed to define the imperfection. The order of the perturbation solution is associated with linear, quadratic, cubic, or other, dependence of the solution with the chosen parameter. In the equivalent load analysis, the solution is found for a specific value of amplitude

90

Thin-Walled Structures with Structural Imperfections" Analysis

U2 0.0016

. -..'.ft..xact

.

0.0012 2nd

0.0008

~:xact

I

0.0004

I

-0.4

Ul 0.0016

I

-0.2

0

I

I

0.2

0.4

Exact "

Oo,~

2n

9 "~"

0.0012 0.0008

"

~xact

-

1

0.0004

I

-0.4

I

-0.2

0

I

0.2

I

0.4

Figure 3.8" Displacements from equivalent load analysis, of the model of Fig. 3.5, for values of k - 1,000 and a - 0.2 tad; C1 = C1 (1 + 3r + 3r 2 + r 3)

91

Imperfections in Intrinsic Parameters

M[~] 0.003

SS,

0.002

xact

0.001 !

-0.4

F

1

-0.2

I

0

0.2

0

0.2

I

0.4

1;

0.6 0.5 0.4

0.3 0.2 0.1 I

-0.4

I

-0.2

I

I

0.4

Figure 3.9" Stresses from equivalent load analysis, of the model of Fig. 3.5, for values of k - 1,000 and a - 0.2 rad; Ca = C1 (1 + 3T + 372 + 73 )

92

Thin-Walled Structures with Structural Imperfections: Analysis

of imperfection, and parametric studies have to be carried out to investigate sensitivity of the response with respect to the imperfection parameter. Application of these techniques in practical situations is presented in Part II of this book, and results are discussed in order to understand the behavior of specific structures. In the next chapter we will study geometric imperfections, with special reference to imperfections that are deviations of the mid-surface of a thin-walled structure. The same techniques discussed in this chapter are also applicable for geometric imperfections, and they will be presented using a similar notation.

Imperfections in Intrinsic parameters

93

References [1] Bathe, K. J., Finite Element Procedures, Prentice Hall, Englewood Cliffs, N J, 1996. [2] Benedettini, F. and Capecchi, D. , A perturbation technique in sensitivity analysis of elastic structures, Meccanica, 23, 1988, 5-10. [3] Brebbia, C. A. and Connor, J. J., Fundamentals of Finite Element Techniques for Structural Engineers, Butterworth, London, 1973. [4] Char, B. W. et at., Maple V Language Reference Manual, Springer-Verlag, New York, 1991. [5] Croll, J. G. A. and Walker, A. C., Elements of Structural Stability, Macmillan, London, 1972. [6] Fertis, D. G. and Mijatov, M. M., Equivalent systems for variable thickness plates, J. of the Structural Division, ASCE, 115 (ST10), 1989, 2287- 2300. [7] Godoy, L. A., A perturbation formulation for sensitivity and imperfection analysis of thin-walled structures, Latin American Applied Research, 20, 1990, 147- 153. [8] Leon, S. J., Linear Algebra, with Applications, Macmillan, New York, 1990. [9] Mroz, Z. and Haftka, R., Design sensitivity analysis of non-linear structures in regular and critical states, Int. J. Solids and Structures, 31(15), 1994, 2071-2098. [10] Novozhilov, V. V., The Theory of Thin Elastic Shells, P. Noorhoff, Groningen, 1959.

94

Thin-WMled Structures with Structural Imperfections: Analysis

[11] Raichman, S. R. and Godoy, L. A., A perturbation/finite strip approach for static analysis of non-prismatic plate assemblies, Int. J. Computers and Structures, 40(3), 1991, 629-637. [12] Reddy, J. N., An Introduction to the Finite Element Method, 2nd Edition, McGraw-Hill, New York, 1993. [13] Spyrakos, C., Finite Element Modeling in Engineering Practice, West Virginia University Press, Morgantown, WV, 1994. [14] Suarez, B., Metodos Semianaliticos para el Calculo de Estructuras Prismaticas, Centro Internacional de Metodos Numericos en Ingenieria, Barcelona, 1991. [15] Suarez, B., Godoy, L. A. and Onate, E, Analisis de estructuras prismaticas de espesor variable pot el metodo de la banda finita, In: Mecanica Computacional, vol. 7 (Ed." L. Godoy, F. Flores and C. Prato), Asociacion Argentina de Mecanica Computacional, Santa Fe (Argentina), 1988, 73-88. [16] Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials, Kluwer, Dordrecht, 1987. [17] Westergaard, H. M., Theory of Elasticity and Plasticity, Harvard University Press, Cambridge, MA, 1952. [18] Zienkiewicz, O. C. and Taylor, R., The Finite Element Method, Volume 1, 4th Edition, McGraw-Hill, New York, 1990.

Chapter 4 SYSTEMS WITH GEOMETRICAL IMPERFECTIONS 4.1

INTRODUCTION

In many built engineering structures, the final constructed shape has significant deviations with respect to an ideal, or as-designed geometry. For example, plates are not so fla,t; circular cylinders are not quite circular; and spheres are not perfectly spherical. Furthermore, the size may also ha,ve defects, and the curves that define the boundaries may be constructed with errors. All these defects will be enclosed in the category of "geometrical imperfections", and they so named because they somehow affect the definition of the geometry of the thin-walled structure. The most common causes of these imperfections are 9 problems arising during the construction process, since sometimes it may be very difficult to build something with the exact shape specified in the design; or 9 damage that occurred after the structure was completed. To understand the structural significance of geometric imperfections, one must remember that smooth shells carry loads by means of a membrane mecha, nism, and only a, marginal part of the load is 95

96

Thin-Walled Structures with Structural Imperfections: Analysis

equilibrated by bending action. But in zones of high changes in the geometric curvature (as would occur due to local changes in the position of the mid-surface), a local redistribution of stresses induces high membrane as well as bending stresses.

This important redistribution of stresses in geometrically imperfect shells has not been evident until recently. Engineers carried out the analysis of shells totally neglecting considerations about the influence of geometrical imperfections. Research in the 1960s acknowledged that buckling loads were modified by initial imperfections, but only following the collapse of a large cooling tower [17] did engineers reflect on the importance of imperfections as stress-concentrators.

In this chapter we attempt to show different ways to analyze geometrically imperfect structures. The behavior of shell structures with imperfections in the shape is discussed in Chapters 7 to 11, for different shell forms. Specific forms of imperfections found in practice are presented in Chapter 12, while a historical review of approaches to the world of imperfections is discussed in Chapter 13.

Section 4.2 is concerned with the geometric description of an imperfection, and two such geometries are discussed: a cosine-shaped imperfection and an imperfection modeled as a polynomial function. The direct analysis is presented in Section 4.3; this leads to some problems, so possibilities to overcome them are discussed. In Section 4.4 we present a model of geometric imperfections that assumes initial strains and performs the analysis with respect to a perfect geometry. Equivalent load techniques are the subject of Section 4.5, and a direct iteration procedure is followed to solve the resulting equations. But instead of computing the equivalent load by means of products of matrices ~nd vectors, it is sometimes possible to compute them from the differential equations of the problem, and this is the subject of Section 4.6. Perturbation techniques are applied to compute displacements and stresses in geometrically imperfect structures in Section 4.7. Finally, the techniques are used in Section 4.8 to solve a two degree-of-freedom structure.

Systems with Geometrical Imperfections

4.2

97

MODELS OF GEOMETRIC IMPERFECTIONS

Records of geometrical imperfections obtained in large shells are discussed in Chapter 12, but we can anticipate here that they show large variations from one type of shell to another, depending on the construction procedure and on the details of fabrication. But for one specific shell, it is clear that there is no simple shape that can be employed to describe the imperfection. The amplitude varies from one part of the shell to another, and the wavy pattern is far from being uniform in different zones. Recalling the categories listed in Section 1.2.2, most geometrical imperfections recorded in real structures would be examples of global imperfections. This can be seen from the examples discussed in Chapter 12, and from many other references listed there. But any attempt to study the stress redistributions in geometrically imperfect shells exactly as in the survey records would constitute an almost impossible task from the computational point of view. It is clear then that simplifications need to be introduced in the analysis, in order to deal with some rather simpler imperfections. Referring again to the categories of spatial distributions of imperfections, one would like to find prismatic properties or repeated patterns in actual recorded imperfections. For example, the committee on the Ardeer collapse [17] identified some tendency in the imperfections to remain constant over a considerable distance in the circumferential direction. The theoretical studies were thus restricted to axisymmetric imperfections. This assumption apparently constitutes an over simplification of the problem, but it allows some insight into the nature of the resulting stress redistribution to be gained. Ellinas et al. [8] gave special attention to the zone of maximum imperfection amplitude and modeled a local imperfection. Parametric studies were conducted to understand how the imperfection parameters influenced the behavior. In all the cases discussed above, the imperfect shell was not modeled in every detail, nor was it exactly modeled at a specific zone. What is usually done is to identify a zone where the most severe stress concentrations are expected to occur. This depends on two factors: high stresses in that zone, even in the perfect structure, and high changes in the geometric curvature due to the imperfection. The reasons why these two factors are important will be more clear in Section 4.6. For

98

T h i n - W a l l e d Structures with Structural Imperfections" Analysis

the zone selected, the parameters that may define the imperfection should be identified, and these parameters are employed to generate some idealized imperfection profile. As stated in Chapter 1, the stiffness matrix does not exhibit an explicit dependence on geometric variables. They are the limits of the volume integrals, the coordinates of the mid-surface, or are included in the kinematic boundary conditions. In describing the geometric configuration of a thin-walled structure, the mid-surface will be identified by the coordinates xj.P If deviations from the perfect geometry (which will be here denoted by xj) are considered, the geometry needs to be modified. Similar to intrinsic variables, the geometry of the imperfect shell will be written with reference to the perfect shell in the form xj - x~ + ~F(x~)

(4.1)

where F(xj) is a function that models the space distribution of the imperfection, and ~ is amplitude. Let us consider two possible shapes of imperfections: one that follows a cosine curve, and a second one that is represented by a polynomial. Once again, we stress that these are examples of curves that can be employed to model a more complex, real, geometrical imperfection.

4.2.1

Cosine I m p e r f e c t i o n Profile

Let us consider a shell of revolution as in Fig. horizontal radius is represented as

4.1, for which the (4.2)

r - rp + ri

A cosine variation of r about a level z - Zc may be written in the form ri - ( [1 + c~

- h zc )

(4.3)

in which ( is the maximum amplitude of the imperfection, occurring at the middle of the imperfection band, and h is the extent of the band. In order to compute the radius of curvature of the imperfect zone, the derivatives are obtained as follows dr

dr p

dr i

d z = dz + dz

(4.4)

99

S y s t e m s with Geometrical I m p e r f e c t i o n s

Figure 4.1" Notation for perfect and imperfect geometries

d2r

d2r p

d2r i

dz 2

dz 2

dz 2

where dr i dz d2r i dz 2

-~- sin

- _2

(z - Zo)

]

(z_zo)]

(4.5)

With Fig. 4.2 showing the variation of r i and its derivatives (Eq. 4.5), it is seen that the second derivative (and hence the curvature of the imperfect meridian) has discontinuities at the top and at the bottom of the imperfection. The effect of these discontinuities on the numerical solution requires some attention (see, for example, Ref [13]). For the cosine imperfection described above, the distribution of positive curvature errors is equal to the distribution of negative curvature errors.

4.2.2

Polynomial Imperfection Profile

There are many functions that could be used to describe geometric imperfections; for example, r i may be expressed as a polynomial in z so that

100

T h i n - W a l l e d S t r u c t u r e s with S t r u c t u r a l Imperfections: Analysis

m

h m

Zo

d /dz

. = = . . - .

d2 /s /

Figure 4.2: A cosine imperfection profile and its geometrical consequences

ri-- Z

aj

z (j-l)

dr i dz - E ( J - 1 )

cUz (/-2)

J d2r i dz 2 = ~ - ~ ( j - 1 ) ( j - 2)

a j z (j-3) (4.6) J with the coefficients aj being determined from the conditions of geometric continuity with the perfect shell. If continuity of geometric curvature is required at all points in the imperfect shell, an eighthorder polynomial may be constructed using eight conditions. Six of them can be ri=

at z -

zc

dr i dz

=

d2r i dz 2

=0

(4.7)

h / 2 and z - zo + h/2; and the remaining two can be

dr i ri - ~;

dz = 0

(4.8)

at z - z c. The resulting imperfection profile is illustrated in Fig. 4.3. Now 43% of the imperfection length has positive curvature error and 57% has negative ones.

Systems with Geometrical Imperfections

101

h

2

Figure 4.3" A polynomial imperfection profile and its geometrical consequences Other polynomial profiles could be constructed in a similar way, with different levels of continuity or different ratios of negative to positives curvature errors.

4.3

DIRECT

ANALYSIS

A deviation in the geometry of a shell can be modeled as a more complex shell, for example by means of finite difference or finite element methods, and without any special reference to the "ideal" or "desired" shape of the structure. In this way, the imperfection enters into the analysis through the coordinates of the mid-surface of the thin-walled component, to follow some assumed model such as those described in the previous section. This would be the approached followed by an analyst with extensive computer facilities available. For a linear problem of equilibrium, the global system results in K(() a + p - 0

(4.9)

where the stiffness matrix K(() is an implicit function of the imperfection characteristics. The first studies in this field were performed by means of a direct analysis: Flugge [9]; Soare [24]; Croll and co-workers [5] [6] [8]; and others who wanted to check the equivalent load results. Direct analysis

102

Thin-Walled Structures with Structural Imperfections: Analysis

Imperfection Measured

N

Discretization Figure 4.4" Errors in the discretization of a geometrical imperfection of a part of a cooling tower and of the complete tower including measured geometrical imperfections has also been done in specific cases [22] assuming elastic response.

4..3.1

Some problems in the direct analysis

Two comments should be made on the modeling of imperfections by direct analysis: First, if parametric elements are employed, in which the geometry is interpolated by some shape function within an element, the interpolation should not introduce geometric errors larger than the ones we intend to model. Fig. 4.4 illustrates a smooth imperfection profile, which is modeled by few elements with linear interpolation of the coordinates in each one of them. Although, the position of the shell has been preserved at some points, a sharp change in slope has been "artificially" introduced at the center and at the ends of the imperfection, and severe stress concentrations will arise there on top of the ones associated with the original shape of the imperfection. We know that this problem is always present in the modeling of curved shells by flat elements, but it is even more severe in this application, since imperfections may occur in small areas of a shell and with significant changes in curvature. Ref. [18] shows an illustration of this problem. Second, the use of curved shell elements based on the Kirchhoff hypothesis may present some problems when used to model geometric imperfections. It was shown in Section 4.1 that if the imperfection profile is represented by a cosine function (and this is the case in many

Systems with Geometrical Imperfections

103

studies in the literature), then the slope of the shell is continuous at the junction between imperfect and perfect segments, but the geometric curvature is discontinuous. In most descriptions of an imperfection some form of discontinuity arises, either in the geometric curvature or even in the slope. The influence that such discontinuities have on the behavior of the numerical solution has been mentioned in Ref. [1], among others, and it has been shown that compatibility conditions are not satisfied. In particular, curved element formulations that impose continuity of three displacements and their first derivatives between elements present difficulties where the geometric curvature exhibits discontinuities between adjacent elements. The numerical results of Fig. 4.5 (from Ref. [13]), for the variation of meridional membrane resultants Nil in the meridional direction, exhibit important disturbances at the ends of the imperfection. Also shown in the figure is a correct solution, and it is possible to see that the deviations of the numerical solutions from these values are of the same magnitude as the stress resultants that would occur in a perfect cylinder. But there are other cases in which not even the slope is continuous between segments of a meridian in a shell of revolution. For the shell theory of Kirchhoff-Love, this problem is illustrated in Ref. [5].

4.3.2

A few S o l u t i o n s

The first problem mentioned may be avoided by the use of parametric elements with quadratic interpolation of the geometry. In any case, a reasonable mesh in the zone of imperfection would include, at least, eight elements. The second problem is not present in elements based on the MindlinReissner hypothesis, but arises whenever the shell equations use the assumptions of Love-Kirchhoff. Several solutions that have been presented for this problem are reviewed here.

Substructuring Technique One may define separate nodes at the level of the discontinuity, with different generalized coordinates for each node. Continuity of the shell at that level would be restored by enforcing equilibrium and compatibility to the global stiffness matrix. This procedure has been implemented in finite differences [5], leading to the explicit satisfaction of

104

Thin-Walled Structures with Structural Imperfections: Analysis

1

1 m

m

I 9

9 |

!

9

9

I

!

~ I

I

-85

I I

I

t I

,-ff

~

E

Z

.

-

-

.

.

-

-

.

.

.

.

l

.

9

"._______

I I

I

-80 I

Z

t

I t

I I .

I

I

I

I

I

I

I

-75 : '

! i

!

! .

-21 N m m " -0.5

I

X I

Imperfection length

h

,': -1.0

-1.5

9

Figure 4.5: Comparison of meridionaI stress resultant N i l for a cyhnder with axisymmetric cosine imperfection, using exact and approximate satisfaction of continuity conditions. - - - -Perfect cylinder; Imperfect cylinder, exact representation of continuity; - . . - Imperfect cylinder, approximate representation of continuity; ........ Imperfect cylinder, exact membrane solution

Systems with Geometrical Imperfections

105

all continuity conditions. The method increases the size of the global system of equations due to the introduction of additional degrees of freedom; moreover, unless a special algorithm for sub-structuring is available, its application to finite element analysis has the drawback that the system of equations becomes non-symmetric. Notice that in finite differences this does not represent a disadvantage, since the system of equations is, in any case, non-symmetric on account of the boundary conditions.

Use of Lagrange Multipliers A different treatment was presented in Ref. [2] for an energy-based finite difference formulation dealing with the buckling and vibration analysis of axisymmetric shells. For segmented and branched shells, the method of Lagrange multipliers was used to join segments of different geometries, and only the compatibility condition was explicitly satisfied. Such formulation preserves the symmetry of the stiffness matrix.

Transformation of Generalized Coordinates A consequence of the continuity conditions of equilibrium and compatibility at a junction where discontinuity in geometric parameters exists is that some generalized coordinates of one element should be different from those at the same node in the adjoining element. In particular, compatibility of rotations and equilibrium of membrane stress resultants provide linear transformations between the generalized coordinates of both elements. In Ref. [13] it is proposed to modify the stiffness matrix of one of the elements adjacent to a discontinuity in slope or curvature using this linear transformation. This procedure preserves the symmetry and band width of the global stiffness matrix.

4.4

INITIAL STRAIN MODEL OF GEOMETRIC IMPERFECTIONS

The problem of the mechanical behavior of an imperfect thin-walled structure has the special feature that one of the variables that is present

106

Thin-Walled Structures with Structural Imperfections" Analysis

in the analysis is geometric (the displacement field): thus, displacements may be directly employed to model a geometric imperfection. This is not possible, for example, in heath conduction problems, in which the temperature field does not represent any geometric characteristic. Neither would it be possible if a force method was employed. First, we consider a reference configuration, in which the structure is assumed to have a "perfect" or "ideal" shape. Displacements of a discrete system from the reference configuration will be denoted by a vector ap. At an element level in a finite element discretization, such vector is a~o. Second, a geometrical imperfection will next be modeled in terms of the same geometric degrees of freedom employed in a;, and represented by a~. Third, the displacements a ~ associated to the loads and measured from the imperfect geometry, that produce strains in the body, result in a ~ - ap~ - a i~ (4 10) Following Donnell [7], "the net strains occurring under load can then be calculated as those which would be produced by a lateral deflection of (w0 + w) minus those which would be produced by a lateral deflection of w0 alone". In the present work, w0 is represented by ai and w by a. The usual finite element notation in problems with non-linear straindisplacement equations is as follows [25] - [B + BL(a~)] a ~

(4.11)

where B a ~ is the linear part of the strains, and BL(a~)a ~ is the quadratic part. The matrix BL(a ~) is often called "large displacements matrix". Each one of the three displacement components (a ~, a~0, and a~) in Eq. (4.10) can be thought to induce strains (e, ep, and e~) via the kinematic relations in Eq. (4.11). The strains measured with respect to the perfect geometry may be written as (4.12) r - [B + BL(a~ 4- a~)] (a~ + a ~) and the initial strains due to the imperfection are r

[B + BL(a~)] a~

(4.13)

The net strains e, due to the loads, result from the difference between ep and ei

Systems with Geometrical Imperfections

107

- ep - 5 ~ - [B + 2BL(a~) + BL(a~)] a ~ in which the property

BL(a~)a ~ - BL(a~)a~ has been employed. In this chapter only linear problems in terms of displacements a ~ are considered, and thus BL (a~)a ~ may be neglected. This non-linearity will be reconsidered in Chapter 6. Thus (4.14)

e - [B + 2BL(a~)] a ~ The stresses are computed from Eq. (4.14) as

(4.15)

a - D [B + 2BL(a~)] a ~

At an element level, the total potential energy V ~ may be written aS

V~ _ 1 f aT [B + 2BL(ai)] T D [B + 2BL(ai)] a 2

dv-

aTf

which, upon expansion, yields

V ~ -- laT 2

f[BrDB+ 2BTDBL(ai)

+ 2BLT(ai)DB + 4 B T ( a i ) D B T ( a i ) ] a - aTf

(4.16)

The first matrix contribution is the linear stiffness matrix

K~

-

f BTDBdv

(4.17)

which corresponds to the perfect structure. The next two terms in the integral (4.16) are linearly dependent on the geometric deviation, a~. and can be grouped together as

K~ ( a ~ ) -

f~

[BTDBL(a~)+ BLT(a~)DB]

dv

(4.18)

108

Thin-Wafted Structures with Structural Imperfections: Analysis

The last term is quadratic in ai, and is here denoted as

K~ (a~, a ~ ) - f~ [BLT(a:)DB/(a:)] dv

(4.19)

The expression for V ~ in Eq. (4.16) results

V r - laT [K~) + 2K~ + 4K~] a r 2

a Crfr

(4.20)

The element contributions are assembled into a global system, V - laT [K0 + 2K1 + 4K2] a - aTf 2 The first variation of V with respect to the net displacements yields the equilibrium condition

OV =0 0a leading to

[Ko + 2K1 + 4K~] a - f - 0

(4.21)

or else

[Ko + K e ] a - f -

0

(4.22)

where

Kr

[2K1 + 4K21

(4.23)

Equation (4.22) is the starting point of the equivalent load method. In the perturbation technique, we write the global displacement vector for the complete structure in the form ap -- (ai + a

(4.24)

in which ~ is an amplitude parameter, which allows the imperfection shape ai to be scaled. The total potential energy of the structure results in V - ~ a l T [K0 + 2~KI + 4~2K2] a - aTf (4.25) where Kx and K2 are obtained by assembly of the element contributions K~ and K~, calculated for ( - 1.

Systems with Geometrical Imperfections

109

Equilibrium is now Ko + 2~K1 + 4(2K2] a -

f - 0

(4.26)

This is the starting point of the perturbation technique. Notice that the system of equations (4.26) is linear in the unknown displacements and quadratic in the imperfection parameter ~r The original problem has been rewritten, using the initial strain representation of the imperfection, in an apparently more complex form; but it will be shown that this form is very convenient to carry out indirect analysis. The matrix K0 is computed from the geometry in the perfect configuration, and the influence of the imperfection has been confined to K~. It is interesting to observe that K~ has a linear and a quadratic contribution in terms of ~; this is because the geometric non-linearity has been assumed as quadratic. Finally, it should be pointed out that only one amplitude parameter has been employed in the preceding discussion to model the imperfect geometry of the structure. Should two or more parameters be necessary to represent a geometric deviation in the complete structure, a similar procedure to the one presented here could be used, but the expansion should be made in terms of alI parameters.

4.5 4.5.1

EQUIVALENT The

Equivalent

LOAD ANALYSIS

Load

The initial strain model presented led to an equilibrium equation of a form that is similar to what was discussed in Chapter 2. We may then proceed a,s before by defining a vector p* as p* - - K ~ a

(4.27)

The equilibrium condition for the geometrically imperfect structure becomes now K0a - p + p*(a) (4.28) which may be interpreted as a new problem defined on the perfect structure, but under a loading condition that depends on the displacements. Such displacement-dependent load will be called the equivalent load.

110

4.5.2

Thin-Walled Structures with Structural Imperfections: Analysis

Iterative Solution Scheme

The solution of the matrix equation (4.28) may be obtained by means of an iterative scheme, in which an initial value is assumed for p* and it is later updated using the last computed value of the displacement vector a. The first stage may be to consider p* - 0; thus K0a0 - p

(4.29)

Kolp

(4.30)

from which a0 -

The first order equivalent load p , can be computed from ao, leading to p*(ao) -

-Kr

(4.31)

This additional load is equilibrated by al - Kolp *

(4.32)

The second order equivalent load p** results in -K~al

(4.33)

a2 - K o l p **

(4.34)

p**(al) -

from which we find that

A similar procedure is followed for higher order equivalent loads, but they are seldom required. The solution of Eq.(4.28) is approximated by a - a0 + al + a2 + ... + a~ (4.35) The stress resultant can be computed as a - DBa ~

(4.36)

at an element level. Matrix D is the conventional constitutive matrix, because the intrinsic parameters are assumed to remain unchanged by the geometrical imperfection. Notice that Eq. (4.32) and (4.34) can also be written as

ax- -[KoIK ]

Systems with Geometrical Imperfections

111

This is not an efficient way to carry out the computations; however, it shows that the solution of the imperfect system depends on the solution of the perfect system. The original linear problem of the imperfect structure has been solved as a series of linear problems, the difference being that the former would require the construction of the stiffness matrix of the imperfect system, whereas the latter only works with the stiffness of the perfect configuration. An important practical consequence for semi-analytical finite element solutions, in which the imperfection changes the geometry in the prismatic direction, is that it would still be possible to obtain the solution from a series of linear systems discretized by semi-analytical elements. This may represent a large saving in computational effort, or may help in the study of several imperfections with little extra cost.

4.5.3

C o n v e r g e n c e of the Iterative Solution

Since the equivalent load system for geometric imperfection is similar to that obtained for degeneracies in intrinsic variables, in which K~ is substituted by Kr the same convergence characteristics apply so they will not be repeated here. The reader is referred to Section 3.4.4 for more details.

4.6

EXPLICIT

EQUIVALENT

LOAD

In intrinsic degeneracies, a general procedure for the calculation of the equivalent load was formulated, and then explicit forms for particular problems were obtained. However, the development of equivalent load analysis of geometric imperfections started from a simple explicit expression. The equivalent load was obtained as the product of membrane stress resultants Nij by the errors in geometric curvature of the shell [6]; i.e.

pj -0;

9

o

p~- N~ ~j

i , j - 1, 2

(4.a7)

where N~j are the stress resultants in the perfect shell; and Xi~ are the errors in geometric curvature, introduced by the imperfection. But how

112

Thin-Walled Structures with Structural Imperfections: Analysis

can one show that this is a reasonable representation of the influence of the imperfection? To show that, we start from the differential equations of equilibrium, and include the imperfection as initial displacements, but linear as well as non-linear terms need to be retained, according to Section 4.4. Non-linear terms which only involve net displacements could be neglected in a linearized analysis. These non-linear terms should be retained in eigenvalue or non-linear analysis, as in Chapter 6. Terms involving the imperfection could be grouped into an "equivalent load"; they also depend on net displacements and the solution should be achieved following the same iterative scheme presented above.

4.6.1

Explicit Equivalent Load in Shallow Shells

In this section, we consider a shallow shell, for which the basic equations are summarized in Chapter 8, and follow Ref. [12]. Displacements from the imperfect configuration are denoted as ua; the initial imperfection as u~ and displacements measured with respect to the perfect geometry as u~. Thus, from Eq. (4.10)

We have here assumed that the deviation from the ideal geometry involves only displacements in the out-of-plane direction xa, so that u~ - u~ - O. The strains are 0

C i j - - Cip

-- Cij

where kij is the curvature tensor of the perfect shallow shell; /3i are rotations; and/3 ~ denote the slope of the imperfections, i.e.

Oxi The first three terms do not involve initial displacements and are calculated from the perfect geometry k~j. The last term is due to the presence of the imperfection. In linear analysis we may neglect the third term, which is non-linear in rotations.

Systems with Geometrical Imperfections

113

Next, consider the changes in curvature:

o

l(Ofl ~

Off~

(440)

x~j--~ -~xj +-~x~

[

Xijp - -~ 1 ~0 (flo + fli) + ~

0

(flo + flj)

]

(4.41)

from which p

0

l(Ofli

Oflj)

(4.42)

Changes in curvature have been assumed to be linearly dependent on displacements; thus, a geometric imperfection does not introduce further changes at this level. The complete set of strain-displacement equations (4.39) and (4.42) may be written in the compact form ~,j - ~ , j ( ~ ) + ~,j(~, ~0)

(4.43)

Xij - Xij(uk)

(4.44)

with

1 (flifl o + floflj) in which the first term of each equation is the kinematic relation without imperfection and the second term contains u~. Only the membrane stress resultants are modified by the imperfection, leading to Nij - Nij(uk) + Nij(uk, u~ (4.45) where

Nij(uk, u~ -- 1 + v

[1

+

v

1 - v fl, fl?]

(4.46)

Equilibrium of the shell is written as

ONij Oxi

+ pj - 0

(4.47)

and

02M~j OxiOxj

+ (k~ + x~)lv~ + ;3 - o

(4.48)

Thin-Walled Structures with Structural Imperfections: Analysis

114

Replacement of Eq. (4.45) into (4.47) yields

ONij

P; -

P39 - Xij0 N~j + (kij + Xij0 + Xij)Nij

(4.49)

(4.50)

Equations (4.49) and (4.50) are the complete set of equivalent loads, as they are obtained from the differential equations of equilibrium. However, they are never used in their complete versions, and some simplifications are considered next.

4.6.2

Simplified Equivalent Loads

The in-plane components of the equivalent load are usually neglected. Let us consider the normal pressure component p; and several levels of approximation.

o p) is clearly the same as the "intuitive" . The first term ( xijNij equivalent pressure of Eq. (4.37) and results linear in both u3 and u~. 9 The term (x,jN~j) is linear in u ~ but non-linear in u3 and may be neglected in a linear analysis such as the ones considered in this chapter, leading to

P39 - X 0~jN~j + ( kij + X oij )N j

(4.51)

9 The other two terms are linear in u3, but while (kijNij) is also linear in u~a, the other contribution (xi~ is non-linear in U 0 . For imperfections of small amplitude, it may be acceptable to neglect(xi~ in comparison with the linear terms in u3~ and this reduces the equivalent load to 9

0

p

P3 -- x ijNij -'~ k ijNij

(4.52)

9 For a fiat plate, on the other hand, in which kij - O, only the remains traditional equivalent pressure (Xi~

S y s t e m s with Geometrical Imperfections

115

0 p P39- xijNij

(4.53)

In the work of Ref. [19], the influence of the curvature of the shell itself is neglected; that is, in order to compute the equivalent load, an approximation of a flat inclined plate is assumed.

4.6.3

Discussion on the Equivalent Load Analysis

The equivalent load technique was perhaps first suggested by Munro in an unpublished report in 1967 [20]. Munro considered an arbitrary imperfect bulge and employed the technique for some design checks. A theoretical justification of the explicit equivalent load in axisymmetric shells was given in the fundamental paper by Calladine [3]. The technique was applied by Croll and Kemp [4], and validated (for some problems of small amplitude of imperfections) by Oyekan [21], Godoy [10], Croll et al. [6], Gould et al. [15], Hang and Tong [16] and Moy and Niku [19]. A lucid account of the technique is found in Croll et al. [6]. The equilibrium equations were again the starting point in the work of Moy and Niku [19] for shells of revolution and local imperfections. A more consistent derivation of the equations for shells was presented by Godoy [12]. A complete equivalent load analysis of geometric imperfections in linear systems is quadratic; however, a linear representation has been shown to be a good approximation for certain imperfections of small amplitude. For such cases in which the equivalent load may be calculated directly, it has been employed with significant advantages over direct techniques. When the discretization is carried out by means of semi-analytical finite elements or finite differences, the explicit equivalent load can be used even if the geometric deviation destroys the symmetry in the prismatic direction. For the specific application to the analysis of cooling tower shells, a first order equivalent load has often been employed, especially for axisymmetric imperfections. For localized deviations with amplitude ~ larger than the shell thickness, on the other hand, the technique underestimates the stresses in the structure. The advantage of the explicit equivalent load is not only a reduction in the computational effort, but makes a physical interpretation of the influence of the imperfections possible. Thus, in its simplest form,

116

Thin-Walled Structures with Structural Imperfections: Analysis

the imperfection affects the structure due to the errors in geometric curvature that are introduced, and the membrane stress field present in the zone of imperfection is also directly relevant. The question of what errors in curvature most affect the behavior of a shell should thus be answered by also looking at the stress state in the different directions. If the technique is limited to a first-order linearized analysis, then a new analysis is not necessary for different amplitudes of imperfection. However, if more than one iteration is required, some care must be taken to add the solution from different orders of equivalent load, if the solution for an amplitude is employed to calculate another. On the other hand, if the complete equivalent load is employed, a new analysis needs to be produced for each new imperfection amplitude considered. In some problems, especially when localized geometric imperfections are considered, it may be convenient to employ a combination of two of the techniques explained above. One situation in which such combination of techniques of analysis has been implemented is in semianalytical finite element discretization of a shell of revolution, in which the imperfect meridian is represented by a direct model, whereas the variation in the circumferential direction is followed by an equivalent load approach. This has been employed by Han and Tong [16] for cooling tower shells.

4.7

PERTURBATION

ANALYSIS

We shall next retain the amplitude of the geometric deviation as a variable, and fix the function that determines the area affected by the imperfection. This limitation is associated with the use of a finite element discretization of the structure; but if semi-analytical elements or completely analytical solutions were employed, the extent of the imperfection could also be treated as a variable. Consider again Eq. (4.26), which represents equilibrium of the whole structure in global coordinates. The system is quadratic in the amplitude ~, but linear in the unknown displacement vector a.

Systems with Geometrical Imperfections

117

Perturbation Equations

4.7.1

The solution of Eq. (4.26) is written as a-

ao + s al q- s2a2 + ...

in which

1 dna an

--"

n[ ds n

and s is a suitable perturbation parameter. A convenient choice of the parameter s is the imperfection amplitude itself, ~, leading to a - ao + ~al + ~2a2 + ...

with an

=

1 dna n! d~~

(4.54)

(4.55)

Substitution of Eq. (4.54)into Eq. (4.26) results in OV = [Koao + p] + ( [ K o a l + 2Klao] Oa + ~2 [Koa2 + 2 K l a l + 4K2ao] + ... - 0

(4.56)

or else Ko (aj ~J) +

2K 1

(aj ~j+l)

+

4K2 (aj~ j+2) + p - 0

(4.57)

Differentiating with respect to the perturbation parameter ~, we get -[Koal

+ 2Klao] +

[Koa2 + 2K1a1 + 4K2ao] + . . . - 0

d (OV) -[Koa2

de 2 ~ a

-+- 2Klal + 4K2ao]-t-...- 0

(4.58)

(4.59)

Next, we evaluate the above expressions at ~ - 0, and obtain the following perturbation equations of order zero

118

Thin-Wafted Structures with Structural Imperfections" Analysis

Koao + p - 0

(4.60)

First order perturbation equations" K o a l q- 2K~ao - 0

(4.61)

Second order perturbation equations: Koa2 + 2Klal + 4K2ao - 0

(4.62)

Perturbation equations of order n: Koa~ + 2 K l a n - 1 + 4K2an-2 - 0

(4.63)

Notice that matrix K2 is only present in the second order (and higher) perturbation equation, and as such this would be the minimum number of perturbation equations to be included in the analysis if the complete Eq. (4.26) is solved. The system of Eq. (4.60-4.63) is independent of the amplitude parameter ~. The solution may be obtained in sequential order in the form a0 -- -KoXp

(4.64)

al -- - 2 K o l K l a o

(4.65)

a2 -- - K o I [2Kla1 + 4K2a0]

(4.66)

a~ - - K o ~ [2Klan-1 4- 4K2an-2]

(4.67)

The matrix K0 that is multiplied by the vector of unknowns is computed on the assumption of a perfect geometry and is the same in all systems of perturbation equations. It is convenient to use an efficient solver for Eq. (4.60) and only change the vector on the righthand side of Eq. (4.61-4.63) each time. The solution of (4.64-4.67) is replaced in (4.54) to obtain a parametric solution of a as a function of a -

ao + ~ax + ~2a2 + ...

Systems with Geometrical Imperfections

119

Thus, the displacement field a is not obtained for a specific value of ~, as in the direct or in the equivalent load approaches, but for a range of values of ~.

4.7.2

Stress Resultants

Calculation of stress resultants here are simpler than for degeneracies in intrinsic variables, since at an element level, the constitutive matrix D is not a function of ~. Thus, (7" -- (7"0 -~- ~0" 1 ~- ~20" 2 -~-... ~no" n

(4.68)

where a0 = DBa~

O" 1

--

DBa~

~ - DBa~

(4.69)

The first of Eq. (4.69) is the stress field in the perfect structure, whereas the other contributions are associated with the solution of each perturbation equation.

4.7.3

Relation B e t w e e n Perturbation and Equivalent Load Techniques

The present perturbation approach may be related to the equivalent load technique, and it may be shown that the two are different approaches. In order to do so, a load contribution p may be written from Eq. (4.63) as Pn -- --(2Klan-1 + 4K2a~_2) (4.70) with the perturbation equation (4.63) now given by K0a,~ - pn

Notice that the load equivalent load (4.33), equivalent load depends bation analysis uses the

(4.71)

contribution in Eq.(4.70) is different from the and their solutions are also different. Each on the previous solution, whereas the perturlast two solutions.

120

4.7.4

Thin-Walled Structures with Structural Imperfections: Analysis

D i s c u s s i o n on t h e P e r t u r b a t i o n A n a l y s i s

The method of analysis for shape deviations in shells of revolution used in Ref. [23] is essentially a perturbation technique applied to the differential equation of the problem. However, only first order perturbation equations were solved. A variational approach based on the total potential energy was presented in Ref. [11]. Unlike the equivalent load analysis, in the perturbation approach an explicit form is not possible so two matrices K1 and K2 need to be constructed. If the analysis is restricted to imperfections of small amplitude, it could be carried out solely on the basis of K1, and this simplifies the computational effort. This type of linearized analysis is bound to be useful for approximately the same conditions as in the equivalent load. For finite element analysis of imperfect structures, the perturbation approach thus requires some modifications in the computer code to include the construction of K~ and K~ at an element level, and then assemble them into a global matrix. The advantages of the perturbation analysis are that the parametric solution is obtained in terms of the amplitude parameter (.

4.8

4.8.1

TWO DEGREE-OF-FREEDOM SYSTEM WITH GEOMETRIC IMPERFECTION The Problem Considered

Let us consider the two degree-of-freedom system represented in Fig. 4.6, for which the influence of geometric imperfections will be investigated. The displacement field is described by the two variables 0 and A, and the strain measures will be denoted by /3 (the rotation of the moment spring), and u (the shortening associated to the axial spring K). The related moment and force are M and F, and P is the apphed load.

Systems with Geometrical Imperfections

121

Figure 4.6" (a) Unloaded system; (b) Deformed configuration of the imperfect system

4.8.2

Basic F o r m u l a t i o n

The kinematics of the system may be written as A - 2 [L (1 - cos 0) + u cos 0]

0-

1

~/~

(4.72)

(4.73)

A positive displacement of the linear spring, u, is assumed to shorten the spring. A positive displacement A is assumed to shorten the distance between the two supports, acting in the same direction as the load P. The inverse form of Eq. (4.72) and (4.73) are - c-

L

1A ~COS 0

(4.74)

- 20

(4.75)

An approximate form of the kinematic equations may be obtained as /Z,~

...

(4.76)

122

Thin-Walled Structures with Structural Imperfections: Analysis

if only quadratic terms are retained in the expansion. In matrix form, the above expressions are written as - B0 a

or else

+

BL(a) a

~ o o

[1/2o o

(4.77)

The non-linear terms in the kinematic equations are due to the presence of the rotations 0 in the axial deformation u. The constitutive relations for linear elasticity are:

a-

{r} M

M-

C fl

(4.78)

F-

K u

(4.79)

-

o]{u}

0

C

fl

-Da

(4.80)

Notice that C and K are dimensional quantities. For the perfect system, the total potential energy V takes the form 1

1

Y - -~M fl + -~2Fu- P A

(4.81)

In the finite element notation the quadratic approximation of Eq. (4.81) results in -'1

v

~~o~K~-~;p

-

(4.82)

where K

Ko -

---o------nTnno-

v

0

0] 4C

p-{o

(4.83)

(4.84)

The linearized perfect system may be written as ~- 0 0 4C

]{ } { } A 0

_

P 0

(4.85)

123

Systems with Geometrical Imperfections with t,he solution

4P

A, = -

K

0" = 0

(4.86)

Notlice t,ha.t,Eqs. (4.86) a,re independent, of L a.nd C. The "st,ra,i~i"va.ria,bles a,re

( .1.87 ) and the "stress" field is give11 hy

.w= 0

(3.88)

('learly. this is a **nieiiiI)rane** soliitioii. The iniperfec-t syst,eni is dssiiiiie(1 t o have ari initial ivtatioii 19,. t)ilt no initial displac-ernent A , , a s iiitlic-atecl in Fig. 4.6. The dispiac-emeiits uieasiired from t lie imperfect geoinet ry (see Eq. (4.76)), a r e

(4.89) with A, = 0. Thus, the nun-lineai stiain matrix associated with the initial displacements. from Eq. (4.77) and (4.89), a r e

(4.90)

The stiffness Imtrix components required by the initial strain model are

2K1 = LK0;/2

[ -' ] [ ]

4 K 2 = Ii'L28:

-1

0 0

0

(4.91) (4.92)

124

Thin- Walled Structures with Structural Imperfections: Analysis

The global system for the imperfect case result,s as

(4.93) which is quadratic in the initial displacement 8;.

4.8.3

Equivalent Load Analysis

The terms due to 8; are included in an equivalent load vector

[ “b’“ ,“c] { f } = - L11‘8a

1

-112

0 -112

(4.94)

LO;

Equation (4.86) is the solution for the perfect system. The firsborder equivalent, load is represented in Fig. 4.10, and result,s in

{ $}

= -LKO,

=

[ {

0 -1/2 2POLOi

-112

]{

4PiK

}

LOi

}

(4.95)

The solution of equilibrium under the loads of Eq. (4.95) is

{:} { =

0 PLO;/2C

}

(4.96)

This first-order correction term is linear in the amplitude of the imperfection, 8;. The sec,ond-order equivalent, loa,d is

(4.97)

~~

9

0~

.J

0

o

~,.~~

0

o

C~

+

o~

I

~

I

.4,

N

N

~

I

~

0

~. ~ ~

~J

~.,~~

~

c~

0

~

0

~--

0

~

rar

i,,~~

~-~

~

o

~. ~

9

~~'

oo

~

~~

~.~ ~

,-'.co

~~

~

~" ~" ~. ~

~'

~

oo

~

0

~

r

o

L'q

i

o

:~

o

_~

o ~9

o

rar~

o~

~J

0

9

9a

.~

i.,~~

c~

bO q.,vl

Ca~

0

"1

"U

0

Car~

126 Thin- Walled Structures with Structural Imperfections: Analysis

:I[: }:{I;

{:}=[: 4.8.4

={ g }

(4.102)

Simplified Equivalent Load

This solution is obtained if the term associated with in the analysis. Eq. (4.94) is now written as

[IY4 -

:c]{

[

LK8; -192

:}={ ;} -10/2] { t }

K 2

is neglected

(4.103)

In this particular example, (A1,Ol) result as previously. The secondorder equivalent loads are

{ $ } = - T P HK1L{2

-114

}

(4.104)

from which one obtains

(4.105) The approximate solution may be now written as

4P

{ :}=F{ PL 2c

1 0 )

L3K

(4.106)

It is interest,ing to notice that the simplified and complete equivalent loads would produce identical first order solution in 8, and second order solution in A. But higher order corrections are very different from the complete equivalent load.

127

Systems with Geometrical Imperfect ions

4.8.5

Perturbation Analysis

In this case, the reference solution, {AO,Oo}, results as in Eq. (4.86). From Fig. 4.6 we write

8 = 8p - 8; where t is the amplit,ude parameter employed to consider different imperfections, and 8; is a specific imperfection. The complete solution in terms of perturbations is

The first, set, of perturbation equations leads to

{:

=

[

4/ Ii' 0 l/L

={

][

0 -1/2

0 --PLBa/2C

-1/2] 0

1

{

4 P y

}

(4.107)

The second set) takes the form

(4.108) Finally, the third set results in 0

-I{PL383/8C2

[ :] { }

128

Thin-Walled Structures with Structural Imperfections: Analysis

0.08

0 0.0

/

i

I

-0.08

,

|

-0.05

0.0

ii 0.05

Imperfection 0i [rad] Figure 4.8" Changes in displacements due to initial lack of alignment

0 The complete perturbation solution, up to third-order approximation, is

+~

PLOi/2C

+

0

+

0

+""

(4.110)

which is coincident with the exact solution. The solution of the imperfect system depends on L and C, unlike its counterpart for the perfect structure.

4.8.6

Numerical Results

To illustrate the solution of this problem using equivalent load and perturbation techniques, we use data given by Oi - 0.05 under a load P / A E - 0.001. The results using perturbation techniques are shown

Systems with Geometrical Imperfections

ZX/L 4x10 -3

129

s

% % wS

%%

m

m

9 m

,

~

i ~ l l l l . ~ o

-

-

~

m

I

!

-0.05

-

0.0

0.05

Imperfection Oi [rad] Figure 4.9" Changes in rotations due to initial lack of alignment

in Fig. 4.8 and 4.9 and indicate that the displacements are not affected by the sign of the initial angle between the bars. The rotation at the hinge, on the other hand, represents a. growth of the initial angle of imperfection.

4.8.7

Remarks

In the example considered, the perturbation analysis leads to the exact solution of the problem using a second-order perturbation expansion, and higher-order corrections are zero. This is the result one would expect to obtain. If we look at the equivalent load analysis, on the other hand, the first-order approximation is the same as in the perturbation technique, but the second-order approximation includes a value of 02 -7(= 0, and higher-order corrections exist.

130

4.9

Thin-Walled Structures with Structural Imperfections: Analysis

IMPERFECTIONS DEFINED IN TERMS OF TWO PARAMETERS

In some cases, we may be interested in the sensitivity of the response with respect to more than one imperfection parameter. For example, let us consider the imperfection in the form 1

ai - ~( (1 + cos ( ~ ) ) for values of x in the range [-h _< x _< h]. We seek sensitivity with respect to the amplitude ( and the extent h of the imperfection. To achieve this, it is convenient to expand the cosine in power series to give 1 1 7rx 2 1 ? ) ai - ~ [ ~ . ( - ~ ) - ~ ( - -

+~1 (~;)-6 ~.1 ( h )~ +...1 or else ( 7t2z2)1 a~- ~[ 2!

(~4x4) 1

1

-

4!

h--i

( oxo) 1 ( sxs) 1 +

6!

~-

81

~ + "'']

The displacements in the mid-surface may be explicited as ap - ai + a The equilibrium condition for such imperfection is

0 The solution a is written in terms of two independent parameters, and ~b - 1/h 2, in the form a-

a00 -I- al0~ -I- a01

Systems with Geometrical Imperfections

131

-[-a11~ -~- a20~ 2 -[- a02~ 2 -~-...anm~n~ m We shall not pursue this analysis here, but a similar case will be studied in the next chapter on interacting imperfections.

132

Thin-Walled Structures with Structural Imperfections: Analysis

References [1] Brebbia, C. A. and Connor, J. J., Fundamentals of Finite Element Techniques for Structural Engineers, Butterworth, London, 1973. [2] Bushnell, D., Analysis of buckling and vibration of ring-stiffened, segmented shells of revolution, Int. J. Solids and Structures, 6, 1970, 157-181. [3] Calladine, C. A., Structural consequences of small imperfections in elastic thin shells of revolution, Int. J. Solids and Structures, 8, 1972, 679-697. [4] Croll, J. G. A. and Kemp, K. O., Specifying tolerance limits for meridional imperfections in cooling towers, J. of the American Concrete Institute, 76, 1979, 139-158. [5] Croll, J. G. A., Kaleli, F., and Kemp, K. O., Meridionally imperfect cooling towers, J. of the Engineering Mechanics Division, ASCE, 105(5), 1979, 761-777. [6] Croll, J. G. A., Kaleli, F., Kemp, K. O. and Munro, J., A simplified a.pproach to the analysis of geometrically imperfect cooling tower shells, J. Engineering Structures, 1, 1979, 92-98. [7] Donnell, L. H., Beams, Plates and Shells, McGraw-Hill, New York, 1976. [8] Ellinas, C. P., Croll, J. G. A. and Kemp, K. O., Cooling towers with circumferential imperfections, J. of the Structural Division, ASCE, 106(12), 1980, 2405-2423. [9] Flugge, W., Stresses in Shells, Springer-Verlag, Berlin, 1960.

Systems with Geometrical Imperfections

133

[10] Godoy, L. A., Stresses in Shells of Revolution with Geometrical Imperfections and Cracks, Ph.D. Thesis, University College, University of London, London, 1979. [11] Godoy, L. A., A perturbation formulation for sensitivity and imperfection analysis of thin-walled structures, Latin American Applied Research, 20, 1990, 147- 153. [12] Godoy, L. A., On loads equivalent to geometrical imperfections in shells, J. of Engineering Mechanics, ASCE, 119(1), 1993, 186-190. [13] Godoy, L. A. and Croll, J. G. A., Geometric discontinuities in thin shell finite element formulations, Int. J. Computers and Structures, 14(1-2), 1981, 37-41. [14] Gould, P. L. and Guedelhoefer, O. C., Repair and completion of damaged cooling tower, Journal of Structural Engineering, ASCE, 115(3), 1989, 576-593. [15] Gould, P. L., Han, K. J. and Tong, G. S., Analysis of hyperbolic cooling towers with local imperfections, H Int. Symposium on Natural Draught Cooling Towers, Bochum, 1984, 397-411. [16] Han, K. J. and Tong, G. S., Analysis of hyperbolic cooling towers with local imperfections, J. Engineering Structures, 7, 1985, 273279. [17] Imperial Chemical Industries, Report of the Committee of Inquiry into the Collapse of the Cooling Tower at A rdeer Nylon Works, Ayrshire on Thursday 27th September 1973, Imperial Chemical Industries Ltd., Petrochemicals Division, London, 1973. [18] King, D., Cooling tower analysis, Int. Symposium on Very Tall Reinforced Concrete Cooling Towers, IASS, Paris, 1978. [19] Moy, S. S. J. and Niku, S. M., Finite element techniques for the analysis of cooling tower shells with geometric imperfections, Thin Walled Structures, 1, 1983, 239-263. [20] Munro, J., Unpublished Report, Imperial Chemical Industries Ltd., London, 1967.

134

Thin-Walled Structures with Structural Imperfections: Analysis

[21] Oyekan, G. L., Analysis of Cooling Towers with Structural Imperfections, Ph.D. Thesis, The University of Southampton, Southampton, 1978. [22] Pope, R. A., Grubb, K. P., and Blackhall, J. D., Structural deficiencies of natural draught cooling towers at U.K. power stations, Part 2: surveying and structural appraisal, Proc. Inst. Civil Engineers, Structures and Buildings, 104, 1994, 11-23. [23] Popov, A. Iu. and Chernyshev, G. N. (1984), Effect of a small deviation in the form of the shells of revolution from axial symmetry on their state of stress, Prikl. Matem. Mekhan., 48(1), 154-160. [24] Soare, M., Cooling towers with constructional imperfections, Concrete, November, 1967, 367-379. [25] Zienkiewicz, O. C. and Taylor, R., The Finite Element Method, vol. 2, 4th Edition, McGraw-Hill, New York, 1991.

Chapter 5 SYSTEMS WITH INTERACTING IMPERFECTIONS 5.1

INTRODUCTION

In Chapter 3 we presented the analysis of imperfections in intrinsic pa,rameters, such a,s devia.tions in the va,lues of modulus of the ma,terial or the thickness. The ana,lysis of imperfections in the geometry was discussed in Chapter 4. This chapter is about the combined action of imperfection in intrinsic and in geometric parameters, leading to an intera, ction between them. The ma,in problem tha,t a,rises with coupling of imperfections is that one imperfection may try to induce high stresses in a given direction in order to equilibrate the external loads, while the other imperfection may reduce the stiffness in that direction. In other words, the thin-walled structure ha.s a preference to redistribute stresses due to one imperfection, but finds that the other imperfection has already weakened the much needed resistance capacity. This may seem an unlikely combination of imperfections, but in fact it is very common in practical structures with structural deterioration. An interesting example of this interaction is the occurrence of imperfections in the geometry of a reinforced concrete shell, causing a stress redistribution from membrane to higher membrane and bending a,ctions. A second imperfection in the form of a partial or through-the135

136

Thin-Walled Structures with Structural Imperfections: Analysis

thickness crack, inhibits the development of these required membrane a,nd bending. The outcome is that a new redistribution takes place, often involving perma,nent deformations and pla,sticity, which may even lead to the collapse of the structure. A second example occurs in thin-walled metal structures with sharp changes in the thickness or with grooves; on the other hand, deviations in the geometry of such shells may occur due to accidents. The combined action of the two effects may require unexpected redistributions of stresses. Hairline cracking in the regions of weld has been observed in the la.rge steel cylinders a,dopted for buoyancy in compliant off-shore structures. This a,lso occurs in corlbination with geometrical distortions of the shell, sometimes due ;o fabrication problems but often ca,used by denting due to collision with external objects. Therefore, it is of great importallce to understand under what conditions geometrical and intrinsic imperfections combine their effects in such a way as to undermine the safety of the structure. The literature in this topic is scarce, and the usual approach has been to uncouple the imperfections and superimpose the results. The experimental work of Ref. [4] on a, shell of revolution with grooves and carefully controlled deviations from the cylindrical geometry, provided clear evidence that this is a coupled problem leading to a, complex stress redistribution. These experiments and the numerical models investigated to understand the beha,vior [3] will be discussed in Chapter 10. In Section 5.2 we present a brief description of direct models of combined imperfections, their problems and achievements. Perturbation techniques are discussed in Section 5.3. We believe that this is important because it highlights the phenomenon of interaction that is present in the model.

5.2

DIRECT

ANAI

YSIS

In this section we follow the work of Ref. [1] in which the shell is studied by means of two-dimensional finite elements ba,sed on Kirchhoff-Love theory. The elements employed have 36 degrees-of-freedom, and are capable of modeling doubly curved shells. This is an essential feature required in geometricMly imperfect structures, in which the geometry changes rapidly.

Systems with Interacting Imperfections

137

Figure 5.1" A cylindrical shell with a,xisymmetric imperfection and grooves

The problem under consideration has an axisymmetric imperfection with a, cosine shape, and vertical grooves at equally spaced locations. Thus, the domain of the analysis is reduced to a panel of the shell, using the center line of groove, the meridia.n between two cracks, and the mid-height of the shell as lines of symmetry. The effects of the end-boundary conditions of the shell were loca,lized and could not be detected in the region of the geometric imperfection and grooves. The geometry of the problem is shown in Figure 5.1. Finite element meshes employed for convergence studies are presented in Fig. 5.2, ranging from 45 elements to aa6 elements. The grooves were modeled using a technique known as the "working boundary" method [2]. No attempt was made to investigate the singularity of the stress field at the tip of the groove, but the interest was focused on the overall stress redistributions that occur in the shell due to the groove.

!s~uama[a 9I[; (a)!s,~ua~a[a 08[ (q)~s~,uomaio ~lz ('e) [-~ -~'tam. u,,aoqs maiqo.ld aq,J,jo s!sfI~eU~e ~,:)a.I!p aqJ, .toj saqsam ,J,uatuaIa a,~,!u!d "g'~ a.m/~.d immmmmnmmmmm inmmmmmmmmmm

lnmmmmmmmmmm lmmmmmmnmmmm

mlll m m m m m m m [ili

d!l ~oeJo

!!!]

nmmmmmm mmmmmmm mmmmmmm m m m m i N m m m m m m m

m i m m

m m i m

m m a l m m m m n

mnmmmm mmmmmn

mmmmmm immmmmm

INN Lmml m=m==-.

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mm n

m

I i

i

llllmmm Plllmmm

!

lnnnnmm

i

inmmmmm ~mm,~, ] '--~mm--

IIII

J

m m

-illl

-II!!

!!!1 II lll l I 1111 I !!!! I

i

(o)

(p)

mmmmmmmmmm

mmmmmmmmmm mmmnmmmmmm

---mmmmmmm , mmmmmmn

mmm

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m

mmmmmmm

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s'fs.4I~uV :su~176

[q)

(e}

[~.m~an.z~S rt~f"~ s~

POll~At-ur. rtJ,

8~ I

139

S y s t e m s with I n t e r a c t i n g I m p e r f e c t i o n s

Different models are possible, depending on the degree of approximation that is feasible and required. In Ref. [5], a geometrical imperfection is modeled by means of axisymmetric finite elements (elements of shell of revolution), while the influence of cracks was modeled using orthotropic properties of the shell. In such model, cracks are not discrete structural imperfections, but are smeared in a much larger region of the shell. In all direct analyses, extensive convergence studies should be carried out in order to obtain reliable meshes to compute stresses.

5.3

PERTURBATION

ANALYSIS

The constitutive ma,trix should be modified due to the presence of an imperfection in intrinsic parameters. If we denote the amplitude of the intrinsic imperfection by r, then the constitutive matrix D becomes as in Chapter 3" D - Do + 7D1 + r2D2 + 73D3

(5.1)

The geometrical imperfection is modeled using the initial strain approach presented in Chapter 4. The strains become e - [B + 2~BL(ai)] a ~

(5.2)

We combine Eqs. (5.1) and (5.2) to compute the stresses ~r as

cr- D c-

[Do + rD1 + r2D2 + ~-3D3] [B + 2~BL(ai)] a ~

(5.3)

Next, we substitute ~r in the total potential energy, " _ 1_ / cr

d v _ a~Tf ~

2

and obtain V ~ - lasT f [B + 2~Bg(ai)] T 2

• [Do + wD1 d- r2D2 + r3D,] [B + 2(Bg(a~)] d v a ~ - a~Tff

(5.4)

140

Thin-Walled Structures with Structural Imperfections: Analysis

Expanding Eq. (5.4) we get, ve

/([BTDB]

= 3 I

+ [B*DlB] + [BTD2B]T~+ [BTD3B]T~ T

+

+

+2 [BTD2B~(a,)B;(a;)DzB] T ~ [ 4 [Bz(a,)DIB~(ai)] t2r

+ 4 [BE(ai)D3B~(a;)] c2r3))dz>ae - aeTP

(5-5)

The element contribut#ionsto t,he st,iffness matrix can be displayed as follows:

1 2

V e = ,aeT[K&,+ K ; O+~ &,r2

+ K;,T~

+ K;p3( + K;272[2 + K ; , ~ ~ [ ~ j aaeTfe ~ -

where

K& = K;o =

/ [BTDB]dv

1

[BTDIB]dv

(5.6)

Systems with Interacting Imperfections

141

K;,- j 2 [BrDBL(a~)+ BT(ai)DB] dv K;. - / 4 [BT(ai)DBL(ai)] dv

- j 2 [n~eln~(a~)+

nLZ(ai)D1n]dv

Z,~l -/2[BTD,BL(ai) + BDr(ai)D.B]

dv

Z~2- S 4 [BTL(ai)DIBL(ai)] dv

i 2 [BTD.BD(ai)+ B~(ai)DaB] dv

U;l-

K:, - i 4 [BTL(ai)D,BL(ai)] dv

K~.- S

4 [BLT(ai)DaBD(ai)] dv

(5.7)

In the first line of Eq. (5.6) we have grouped the uncoupled terms, which axe identical to those presented in Chapters 3 and 4. The second line contains the coupled terms involving the interaction between geometric and intrinsic imperfections. The element matrices are assembled to obtain the energy V of the complete structure. The condition of stationary total potential energy leads to

OV

0a

= [Koo + Klo7- -t- K2or 2 + K3or 3

+Kol~ -t- Ko2f 2 + Kll T~ -i- K21T2~ + K12w~2 -+-K3ar3( + K22w2~2 + K32~-3(2]a- f

(5.8)

142

5.3.1

Thin-Walled Structures with Structural Imperfections: Analysis

Solution of the P e r t u r b a t i o n E q u a t i o n s

We write the solution of the set of perturbation equations in the form a -- a00 4-

7a10 -4- (a01 -[- "/~all -[- r2a20 + ~2a02

_.[_T 3 a30-~- T 2 ~a21 -~- T~2a12 -~- ~3 a03+...

(5.9)

Eq. (5.9) is replaced into Eq. (5.8) to obtain a, polynomial in terms of the imperfections amplitudes r a,nd ~. Use of the perturbation technique in this case leads to the following set of evaluated perturbation equation

Kooaoo = f

Kooa11 -

Kooalo -

- (Kolaoo)

Kooaol -

-(Kloaoo)

-(KolalO

~- K l o a o l ~- K11aoo)

(5.10)

Higher order terms ca,n be obtained following the same procedure. Whenever two imperfections are studied simultaneously, simple examples become more complex. Numerical examples of cylinders and hyperboloids of revolution with combination of imperfections are shown in Chapters 10 and 11.

Systems with Interacting Imperfections

143

References [1] Godoy, L. A., Stresses in Shells of Revolution with Geometrical Imperfections and Cracks, Ph.D. Thesis, University College, University of London, London, 1979. [2] Godoy, L. A., Croll, J. G. A., and Kemp, K. O., The numerical modeling of cra.cks in shells, In: Applied Numerical Modeling, Ed. E. Alarcon and C. A. Brebbia, Pentech Press, London, 1978. [3] Godoy, L. A., Croll, J. G. A., a.nd Kemp, K. O., Stresses in axially loa.ded cylinders with imperfections and cracks, J. Strain Analysis, 20(1), 1985, 15-22. [4] Godoy, L. A., Croll, J. G. A., Kemp, K. O., and Jackson, a. F., An experimental study of the stresses in a shell of revolution with geometrical imperfections and cracks, J. Strain Analysis, 16(1), 1981, 59-65. [5] Jullien, J. F., Aflak, W., and L'Huby, Y., Cause of deformed shapes in cooling towers, J. Structural Engineering, ASCE, 120(5), 1994, 1471-1488.

Chapter 6 NON-LINEAR ANALYSIS OF I M P E R F E C T STRUCTURES 6.1

INTRODUCTION

All the studies of Chapters 3 to 5 dealt with small rotation theories of strain-displacement relations. We assumed there that the errors caused by neglecting quadratic terms in the kinematic relations were small; however, there is only a range of loads and geometries for which this is correct. For very thin shells, and for load levels that induce large rotations, it is necessary to refine the analysis. Such geometrically non-linear analysis is the subject of this chapter. As we mentioned before, the importance of non-linearity in the results depends on the specific problem considered. The studies of Refs. [8] and [10] on cooling towers show only minor differences with respect to a linear analysis. Nevertheless, the results of Ref. [3] on spherical shells show that if non-linearity is neglected, the stresses can be significantly overestimated. In Section 6.2 we review some basic aspects of geometrically nonlinear analysis, with reference to studies of imperfect shells in which this has been applied. Imperfections in the geometry are considered in Section 6.3, using perturbation techniques. Imperfections in intrinsic parameters are studied in Section 6.4, again using perturbation techniques. The reader will notice that perturbation techniques become 144

Non-Linear Analysis of Imperfect Structures

145

cumbersome for this class of problems; however, it is an interesting exercise to go through the formulation, since it throws some light regarding the nature of the non-linear problem considered. An example from Ref. [5] is presented in Section 6.5, in which there is an imperfection in intrinsic parameters, and non-linearity of the strain-displacement relations is included. Both direct and perturbation techniques are employed and sensitivity plots are discussed. Although the example is simple, it shows some features that should be considered in every nonlinear analysis if the results are to be meaningful.

6.2

DIRECT

ANALYSIS

We take here the elements of geometrically non-linear finite element analysis from Ref.[ll], [1], [9], [7], [2], and [13]; but the notation that we follow is from Ref.[4]. Let us write the stra.in-displacement equations as

-[B

+ BL(a)] a

(6.1)

The constitutive equations are assumed as in linearly elastic solids, i.e.

~r- D c

(6.2)

The total potential energy of an element of the structure is obtained in the form V~ - -~ 1 f c T (7 dv - a T f

(6.3)

The element contributions are assembled into the energy of the complete structure, from which equilibrium results as 00 aV = [ K 0 +

3 ~

K 1 (a)+2K 2 (a,a)]a-0

(6.4)

where K~ - ] BTDBdv

(6.5)

is the part of the stiffness matrix that is constant. The linear part of the stiffness is given by

146

Thin-Walled Structures with Structural Imperfections: Analysis

K~ ( a ) - i [BTDBL ( a ) + BLT (a)DB] dv

(6.6)

Finally, the quadratic part is

K~ (a~ a ) -

i

[BLT(a)DBL (a)] dv

(6.7)

Equation (6.4) is non-linear in terms of the displacement vector a, and to solve it one must employ numerical techniques, such as NewtonRaphson, perturbation, or any other technique suitable for non-linear analysis. Examples of non-linear studies of geometrically imperfect thinwalled structures are presented in Refs. [8], [10], [3], [4], and [12]. Nonlinear studies of imperfections in intrinsic parameters may be found in Ref. [6]. A direct analysis is the only practical way to carry out the analysis at present; however, it is interesting to consider the formulation that is obtained by use of perturbation techniques. This is done in the following sections.

6.3

6.3.1

PERTURBATION ANALYSIS OF GEOMETRIC IMPERFECTIONS Non-Linear Equations of Equilibrium

The analysis here is carried out in a way similar to Chapter 4, but we now include non-linear terms in a that were previously neglected. The strain-displacement relations are

e -- [B + 2~BL(ai)+ nL(a)] a

(6.8)

in which the influence of the imperfection has been included as an initial strain field. The stresses become

cr - D [B + 2 ( B / ( a ~ ) + B/(a)] a

(6.9)

The potential energy of an element is written as usual in the form V~ - -~ 1 i eta dv-aTf

(6.10)

Non-Linear Analysis of Imperfect Structures

147

The equation of equilibrium of the complete system is obtained upon assembly of the element contributions, and may be written as OV = [K0 + 3 K (a) + 2K (a, a)]a 0a ~ 1 2

+ 2~[K1 (ai)+ 3K2 (ai, a)]a +4~2[K~ (ai, a i ) ] a - 0

(6.11)

where Ko, K1 (a), and K2 (a, a ) a are assembled from the element matrices given in Eqs. (6.5-6.7), and the other matrices are constructed in a similar way, i.e. K 1 ( a i ) - f [BTDBL (ai)+ BLT (ai)DB] dv K~ (ai, a i ) -

i [BLT (ai) DBL (ai)] dv

K~ (a, a i ) -

j [ . ~ (a)DBL (ai)] dv

(6.12) (6.13)

(6.14)

The first line in Eq. (6.11) is identical to the non-linear problem in Eq. (6.4), and the influence of the imperfection is reflected in the second line of Eq. (6.11). One can see that the equilibrium equations result non-linear in terms of both ~ and a. The corresponding equations in Chapter 4 were linear in a.

6.3.2

P e r t u r b a t i o n E q u a t i o n s and Solution

The global solution in terms of displacements is constructed using a perturbation technique as a-

a0 + ~al + ~2a2 + ~3a3 + ...

(6.15)

The perturbation equations of order zero are a non-linear system in terms of a, which is the reference state given by

[K0 aK + ~

1

]

(a0) + 2K2 (ao, a0) ao - p - 0

(6.16)

Any technique to solve a set of non-linear equations can be employed to obtain a0 in Eq. (6.16), such as Newton-Raphson or perturbation methods.

148

Thin-Walled Structures with Structural Imperfections: Analysis The first-order perturbation equations result in [Ko + 3K1 (ao)+ 6K2 (ao, ao) ] al = - 2 [K1 (ai) -t- 3K2 (ao, ai)] ao

(6.17)

Notice that now the left-hand side of Eq. (6.17) is linear in al, since the value of ao is already known. Unlike what was found in Eq. (6.16), the first and the higher-order perturbation systems are linear in the current unknown. The second-order perturbation equation becomes [Ko + 3K1 (ao) -+- 6K2 (ao, ao) ] a2

-- --2 [K1 (ai) -k- 3K2 (ao, ai)] al -

- [6K2 (ao, ai) ] a l

--

[~3K1 (al) -+-6K2 (ao, al) ] al 4 [K2 (ai, ai) ] ao

(6.18)

Finally, the third-order perturbation equations become [Ko + 3K1 (ao) -k- 6K2 (ao, ao) ] a3 -- --2 [K1 (ai) -k- 3K2 (ao, ai)] a2

_2[

K1 (al) -+- 6K2 (ao, al) ] a2 - [6K2 (ao, ai)] a2

- 4 [K2 (ai, ai)]

al

--

2 [K2 (al, al) -t- 3K2 (al, ai) ] al

(6.19)

Notice that Eqs. (6.17-6.19) have the same matrix associated to the unknowns; but this matrix is different from that in Eq. (6.16). The right-hand side of each equation can be constructed using information that was available in the previous equations, and thus the number of computations needed is significantly reduced.

Non-Linear Analysis of Imperfect Structures

6.3.3

149

An A l g o r i t h m for the Solution of P e r t u r b a t i o n Equations

The equations of the previous section can be organized in a more efficient way as follows Zero-order analysis: Solve the non-linear problem Ko + ~3K 1 (ao) + 2K2 (ao ao) a o = p First-order perturbation analysis" Compute KA -- [Ko + 3K1 (ao) + 6K2 (ao, ao) ] Ks

-[K1 (ai) + K2 (ao, ai) ]

Solve the linear system KAal ----KBao Second-order perturbation analysis" Compute K1 (al) + 6K2 (ao, al) ] KD -- 4 [K2 (ai, ai) ] KE -- 6 [K2 (ao, ai)]

Solve the linear problem KAa: -- --[Ks + Kc + KE] al -- KDa0 Third-order perturbation analysis: Compute KF -- 6 [K2 (al, ai)] + 2 [K2(al, al)] Solve the linear problem

Thin-Wedled Structures with Structural Imperfections: Analysis

150

KAa3 -- - - [ K , + 2Kc + KE] a2 --[KF] al Solution

a - - a o -+-~al -[--~2a2 +~ 3 a3q-...

6.4

6.4.1

PERTURBATION ANALYSIS INTRINSIC IMPERFECTIONS

OF

Non-linear Equations of Equilibrium

We start from the non-linear kinematic relations of Eq. (6.1) e -[B

+ BL(a)] a

but now the constitutive matrix is expanded to reflect changes in the intrinsic parameter, i.e. D - Do + TD1 -[- r2D2 + T3D3

(6.20)

The stresses at the level of an element take the form a -

[Do + TD1 + T2D2 + T3D3] ~

(6.21)

The total potential energy of each element is obtained as in Eq. (6.3), that is

V~ - -~ 1 f eTa d v - aT f leading to the global equilibrium condition

OV 0a

.2

= [K0 + ~K1 (a)-k 2K2 (a, a)]a 3

+ r [K01 -4- ~K 11 ( a ) + 2K21 (a, a)]a +T2[K02 + ~3 K 12 ( a ) + 2K

(a, a)]a

Non-Linear Analysis of Imperfect Structures

151

+ 73[K~ + 32 K 13 (a) + 2K23 (a, a)]a - 0

(6.22)

where K0, K1 (a), and K2 (a, a ) a axe assembled from the element matrices given in Eqs. (6.5-6.7), and the new matrices are assembled from the following element contributions K~I - f BTD1Bdv K~2 - f BTD2Bdv (6.23)

K~3 -- f BTD3Bdv K;l(a

f [BT(a)DIB + BTDIBT(a)]

dv

f

[BT(a)D2B

+ BTD:BT(a)]

dv

- f

[BLT(a)D3B

+ BTD3BLT(a)] dv

) -

K~2(a ) K;3(a )

U~l(a )

--

j

(6.24)

[nT(a)elnT(a)] dv

K,~2(a) - ] [BT(a)D2BLT(a)] dv K~3(a ) - f [BLT(a)D3BT(a)] dv

(6.25)

The problem in ha,nd, Eq. (6.22), is non-linear in both the displacements a and the imperfection parameter 7. The solution will next be found using perturbation techniques.

6.4.2

Perturbation Equations and Solution

Let us assume the solution of Eq. (6.22) in the form a(7) - a0 + val + 72a2 + T3a3 + ...

(6.26)

The perturbation equation is as follows: the zero-order is the nonlinear problem taken as a reference situation, and is given by Eq. (6.16)

152

Thin-Walled Structures with Structural Imperfections: Analysis

[Ko + ~K1 3 (ao)+ 2K2 (ao, ao) ] ao -

p - 0

The solution of this non-linear system leads to the values of ao. The perturbation equation of order one is [Ko + 3K1 (ao)+ 6K2 (ao, ao)] a l

[

3

]

-- -- Kol + ~Kll (ao)+ 2K21 (ao, ao) ao

(6.27)

This is a linear problem in terms of al, and can be solved without difficulties. The second-order perturbation equations are [Ko + 3K1 (ao)+ 6K2 (ao, ao) ] a2 = -[Kol + 3Kll (ao)+ 6K21 (ao, ao) ] al 3 6K2o (ao, al) + ~K lO (al)] al

--

~

-

-

[Ko2] ao

K 12 (ao) ] ao - [2K22 (ao ao) ] ao

Finally, the third-order perturbation equations are [Ko + 3K1 (ao) + 6K2 (ao, ao) ] a3 - -[Kol -4- 3Kll (ao)+ 6K21 (ao, ao) ] a2

[

3

]

- 2 6K2o (ao, al) -}- ~Klo (al) a2 - [Ko2] al --2 [~K12 (ao)] al - 3 [2K22 (ao, ao)] al 3 Ko3 + ~K 13 (ao) + 2K23 (ao, ao) ] ao

(6.28)

Non-Linear Analysis of Imperfect Structures

153

Figure 6.1" Two degree-of-freedom non-linear model investigated

[3K11 (al)+ 2K2o (al,al) ] al

--[6K21 (ao, al)] al

(6.29)

Equation (6.26) collects the solution of the perturba, tion systems, and is the approximate solution of the non-linear problem.

6.5

EXAMPLE ANALYSIS

OF NON-LINEAR

As an application of the formulation presented in this chapter, we investigate the non-linear behavior of a simple, two degree-of-freedom system. We choose the same system on intrinsic imperfections considered in Chapter 3, but now include kinematic non-linearities. The structure is shown again in Fig. 6.1. The example is taken from Ref. [5]. 6.5.1

Direct

Analysis

The non-linear kinematic rela.tions are reflected in the B matrices, i.e.

B -

and

sina 2 -1

sina] -1 ] 2

(6.30)

154

Thin-Walled Structures with Structural Imperfections: Analysis ii

300

11 1/I /

- -

P 200

i I ;i/ i//

-

,11" V

I

./"

"~.

~"" ""

\ xXx.

\/2

.,

.Zl / - /'-

//;" i/" /,g ."

100

/[././t f

-

.17r ft," /

Ii/

0.00

\'\

/

.t(./

Z

X

f

i

,,,,

[

0.02

I

0.06

I 0.10

I

\

Ul

Figure 6.2" Non-linear paths for different values of imperfection parameter, for the model of Fig. 6.1. From Ref. [5]

u2/2 - ?~1 7 2 1 / 2 - u2

B L --

I

0

0

0

0

(6.31)

The matrices of the non-linear analysis Kl(a) and K2(a, a) have been computed in Ref. [5]. The stiffness of the linear spring, denoted by k - K L 2 / C , is expanded in the form k - k(1 + T)

(6.32)

where k is a reference value obtained for the perfect system, and r shows the influence of increasing the amplitude of the imperfection. In this simple example, the imperfection is reflected in a reduction in the value of k that is linear in r. In the direct analysis, the non-linear path in the load versus displacement space is computed, for different values of the parameter r. Results for a specific case are presented in Fig. 6.2, in which the path is drawn for different values of 7 ranging from r = - 0 . 4 to r = 0.4. The displacements are Ul = u2, so that only one displacement compo-

Non-Linear Analysis of Imperfect Structures 300

155

m

P

/,4,/i/

200

~

100

ol/

000

o

I 0.02

I

I 0.06

0.10

Ul Figure 6.3" Linear paths for different values of imperfection parameter, for the model of Fig. 6.1. From Ref. [5] nent needs to be plotted. Because each path is computed separately, there is no restriction on the value of T in the sense that it does not have to be a small value. For this arch-like structure, the path reaches a limit point and then drops with increasing displacements. We re, strict our attention to the ascending part of the path. Curve (1) in Fig. 6.2 is for r = 0; curve (2) is for r = 0.2; curve (a) corresponds to r - -0.2; curve (4) to r = 0.4; and curve (5) to r - -0.4. For linear kinematic relations, as those considered in Chapter 3, the paths are presented in Fig. 6.3, in which the curves have the same meaning as in Fig. 6.2, and are drawn in the same scale. The sensitivity of the displacements is represented in Fig. 6.4 for linear and non-linear equilibrium paths for a load p = 174. For this load level, the linear solution represents a reasonable approximation to the sensitivity of the model, and for positive values of r the displacements are overestimated, while they are underestimated for negative values of 7".

In the linear solution the displacements are proportional to the loads, so that sensitivity can be computed for any load level. For example, it can be computed for a higher load level p = 228, and we

156

Thin-WMled Structures with Structural Imperfections: Analysis

Ul

0.04 ~

o

linear path

~

o~

" ' ~ " ~ :..= . . . . . . .

0.02

non-linear path

0.00 -0.2

0.0

T

0.2

Figure 6.4" Sensitivity using direct methods, for the model of Fig. 6.1. From Ref. [5] would find a response sensitivity similar to that presented in Fig. 6.4. But for negative values of r the solution would be wrong, because it does not satisfy equilibrium. This will be discussed again in the next section.

6.5.2

P e r t u r b a t i o n Analysis

The matrices required to perform the perturbation analysis of the nonlinear system are defined in Section 6.4 as

KOl

-

It 2 - - 2

K11

--

]~ [

K21 = ~-

k

sin 2 a

[11]

k sin 2 a tt 1

(--UI -- U2)/2

(2 tt I --'/t2) 2

(U1 __ 2u2)(2 U 1 -- U2)

(6.33)

1 1

(--It 1 -- t t 2 ) / 2 ] U 1 -- 2 tt 2

]

(~t I -- 21t2)(2 U 1 -- U2) ] (ltl -- 2 U2) 2

(6.34)

(6.35)

Non-Linear Analysis of Imperfect Structures

157

0.04 _

Ul

0.0261

0.02

0.00 -0.2

0.0

0.2

T Figure 6.5: Sensitivity using perturbation methods, for the model of Fig. 6.1 at a load p=174. From Ref. [5]

Based on these matrices, we have computed the matrices required for the first-order perturbation equation, and limited our analysis to that. For the same case solved using the direct method, we have obtained resultsusing perturbation techniques. Let us consider two load levels indicated in Fig. 6.2: for a load p = 174 the results are in Fig. 6.5; while for p = 228 the results are plotted in Fig. 6.6. In the first case, a first-order perturbation analysis produces the changes indicated in the figure, and the exact values, computed by the direct method are also shown. Perturhation in this case underestimates the displacements by about 15%, for positive and negative values of the parameter r. In the case of a higher load, the solution for positive r is good when compared with the exact solution, but it is not satisfactory for negative values of r in the order of 20%. We see that there are no exact values of sensitivity for 7 - -0.2. The reason for the deficiency of this latter result is more clearly seen in Fig. 6.2, in which for the load p = 228, the structure with

158

Thin-Walled Structures with Structural Imperfections: Analysis

Ul 0.06

O.O4

0.02

o.oo -0.2

I

I 0.0

T

0.2

Figure 6.6" Sensitivity using perturbation methods, for the model of Fig. 6.1 at a load p=228. From Ref. [5]

T = --0.2 does not have an equilibrium state. A limit point has been reached along the curve (3), although curve (1) can still take more load before reaching a limit point. The perturbation analysis has no way to identify this limit point, and would thus compute the sensitivity as if there was equilibrium for the load level required. A second feature can be observed in Fig. 6.2: for the load p = 174, a perturbation of T -- --0.4 would face a similar problem. For this larger perturbation, there is no equilibrium state in the path (5). The simple example illustrates a point that was not present in the linear analysis considered in Chapters 3 and 4: that because the present perturbation analysis does not check equilibrium of the perturbed solution, it is possible to compute values of sensitivity of the response that do not satisfy equilibrium. A similar situation occurs in the direct analysis with linear kinematics, in which satisfaction of equilibrium along a linear path is no guarantee that it will be satisfied along the true non-linear path. Great care must be exercised in non-linear problems, and to be on a safe side, a direct non-linear analysis is recommended.

Non-Linear Analysis of Imperfect Structures

6.6

FINAL

159

REMARKS

In this chapter we considered the influence of non-linearity on the response of a structure to imperfections. Two techniques were discussed to perform the analysis: a direct method, and perturbation methods. In the latter case, the formulation differs depending on the type of imperfection, but always one has to compute first the equilibrium state using any technique of non-linear analysis, and then produce perturbation from there. This perturbation process requires the solution of linear problems. Thus, perturbations require the solution of a nonlinear problem to get the reference state, and then linear problems to obtain sensitivity. In the direct method, on the other hand, the non-linear path has to be computed for several values of the imperfection parameter. Although accurate, this procedure is computationally expensive. A warning is given in this chapter regarding the existence of the equilibrium states computed using linear or perturbation schemes. It may be so that for a given load level there is an equilibrium state; but that due to the effects of imperfections, the path is modified in such a way that there is a limit point along an imperfect path occurring before the load level considered. It is then important to have a means of checking if the solution obtained satisfies equilibrium. Because of this, some of the computational advantages of perturbation techniques are now lost, and it may be more efficient to use the direct method if we expect a limit point in the vicinity of our study. A similar situation occurs in the neighborhood of a bifurcation point.

160

Thin-Walled Structures with Structural Imperfections: Analysis

References [1] Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, NJ, 1981. [2] Criesfield, M. A., Non Linear Finite Element Analysis of Solids and Structures, Wiley, Chichester, 1991. [3] Flores, F. G. and Godoy, L. A. (1988), Linear versus non-linear analysis of imperfect spherical pressure vessels, Int. J. Pressure Vessels and Piping, 33, 95-109. [4] Flores, F. G. and Godoy, L. A. (1990), Finite element applications to the internally pressure loadings on spherical and other shells of revolution, Chapter 9 in Finite Element Applications to Thin-Walled Structures, Ed. J. W. Bull, Elsevier Applied Science, London, 1990. [5] Godoy, L. A., Sensitivity of non-linear equilibrium paths using perturbation techniques, To appear in Applied Mechanics in the Americas, vol. 4, American Academy of Mechanics, Blacksburg, VA. [6] Godoy. L. A. and Flores, F. G., Thickness changes in pressurized shells, Int. J. Pressure Vessels and Piping, 55, 1993, 451-459. [7] Hinton, E., (Ed.), Introduction to Non-linear Finite Elements, National Agency for Finite Elements, NAFEMS, UK, 1990. [8] Kato, S. and Yokoo, Y. (1980), Effects of geometric imperfections on stress redistributions in cooling towers, J. Engineering Structures, 2, 150-156. [9] Kleiber, M., Incremental Finite Element Modeling in Non-linear Solid Mechanics, Ellis Horwood, 1989.

Non-Linear Analysis of Imperfect Structures

161

[10] Niku, S.M., Finite Element Analysis of Hyperbolic Cooling Towers, Springer-Verlag, Berlin, 1986. [11] Oden, J. T., Finite Elements of Non Linear Continua, McGrawHill, New York, 1972. [12] Ohtani, Y., Koguchi, H. and Yada, T., Non-linear stress analysis for thin spherical vessels with local non-axisymmetric imperfections, Int. J. Pressure Vessels and Piping, 45, 1991, 289-299. [13] Zienkiewicz, O. C. and Taylor, R., The Finite Element Method, 4th edition, vol. 2, McGraw-Hill, New York, 1991.

Chapter 7 PLATES AND PLATE ASSEMBLIES WITH IMPERFECTIONS 7.1

INTRODUCTION

This second part of the book (Chapters 7 to 11) is dedicated to the behavior of specific structural forms, or structural components, and the influence that imperfections have on them. We start by investigating flat plates, which are the simplest geometrical forms; and pla,te assemblies, which are a,n intermediate situation between flat pla,tes and curved shells. Under transverse load, a plate develops a bending mechanism of equilibrium. This can be done without membrane stresses provided the rotations are small. The linear equilibrium path of this problem is not very sensitive to the presence of imperfections, and that it is not important if the specific imperfections considered induce changes in the geometry, in the material, or in the thickness. On the other hand, plates under in-plane loa,d require a membra, ne state of stresses to satisfy equilibrium. Such membrane mechanism is more sensitive to imperfections, and it may need the development of a new bending stress state. Finally, plate assemblies usually work by coupled bending and membrane actions and may be sensitive to imperfections. In Appendix B we review some basic elements of bending a,nd membra, ne a,ctions in plates. The numerical techniques of finite strips to 162

Plates and Plate Assemblies

163

model them are discussed in Section 7.2. Thickness changes in plates are explored in Section 7.3 using equivalent load and perturbation techniques. Plate assemblies are considered in Section 7.4 by means of perturbation techniques and finite strips. In Section 7.5 the origin of the imperfection is a change in the modulus of the material, and explicit equivalent loads and perturbations are developed and employed. Final remarks are presented in Section 7.6.

7.2

FINITE STRIPS FOR PLATE BENDING

Semi-analytical elements in which the displacement field is approximated by a combination of polynomial and trigonometric functions often present significant computational advantages over more classical two or three-dimensional finite element formulations. However, this gain in efficiency is accompanied by limitations regarding possible variations of the parameters that define the system. Perhaps the most widely employed of such semi-analytical procedures is the finite strip method [1] for the analysis of folded plate structures. In the finite strip method, the displacement field is taken in the form tt 1 - -

Z

rNTan / sin 1

2a

n

u2--~

NTa

cos

2a

N a a~ sin

2a

(7.1)

Tt

tt 3

-n

where N T are the polynomial interpolation functions for the ui displacement used in the x2 direction, and a n is the vector of unknown displacements associated to the n-th trigonometric function in the Xl direction. The trigonometric expressions chosen satisfy the following boundary conditions" 02U3

Ul

723

OX 22

= 0

(7.2)

164 Thin-Walled Structures with Structural Imperfections: Behavior at the ends X 1 : 0 and X 1 "-- 2a. In its classical formulation, the finite strip method is restricted to the analysis of prismatic structures in which the material and the geometry of the transverse section remain constant in the longitudinal direction. However, most imperfections induce changes in the properties that depart from the prismatic situation. To deal with changes in the properties, Ref. [7] presents a finite strip formulation based on cubic B-spline functions for the displacement interpolation in the longitudinal direction, and third order polynomials for transverse displacements. Numerical integration is used to take into account the longitudinal changes in the thickness of a strip. The procedure developed in Ref. [7] can handle rectangular slab structures with variable thickness in the longitudinal direction. Two additional formulations have been presented in the literature: In Ref. [5], a Mindlin plate theory has been used with an equivalent load procedure to model thickness changes. A similar purpose was achieved using perturbation techniques in Ref. [4]. Results obtained with both formulations are presented in this chapter.

7.3

PLATES WITH CHANGES

THICKNESS

Local variations in the thickness may be useful models of some more complex imperfections in plates or plate assemblies. The influence of step changes is discussed in this section for flat plates, and is extended to plate assemblies in the next section. The simplest of such problems are perhaps those of plates with central grooves. They are studied in the following.

7.3.1

Equivalent Load Analysis

This equivalent load study is based on work reported in Ref. [5], and deals with the application of finite strip techniques and equivalent load analysis to rectangular (Mindlin) pla,tes. Due to limitations of the finite strip technique employed, the results are restricted to simply supported plates. The kind of problems tha, t can be modeled are shown in Fig. 7.1. Other boundary conditions require the use of two-dimensional finite elements or other discretization techniques.

Plates and Plate Assemblies

165

Figure 7.1" A plate with changes in the thickness

Figure 7.2" Square plate with a, step change in the thickness, and finite strip discretization

166 Thin-Walled Structures with Structural Imperfections: Behavior

U3 -0.08 -0.16 -0.24 -0.32

\

-0.40

\.

-0.48 -0.56

,

,

,

,

0.00

,

"X. ,

0.25

0.50 X1

N13 -0.02

-0.06

-0.10 -0.14

-0.18 -0.22

:

/

!

tl

f ././'"

-0.26

, 0.00

,

,

,

, , , , 0.25

=

= 0.50

X1 Figure 7.3" Square plate with step change in the thickness. Equivalent load analysis" Displacement and shear

167

Plates and Plate Assemblies

Mll

(x lo')[ 0.00

-\ -tl

-0.08 -0.16

-0.24

- f\ -

%

m

,\ 9

-0.32

.\

"

-0.40

X~4p..

..

II

It%

O,,.

It

I -0.48 0.00

I

t

t

I

0.25

t

a

I

I

I

0.50

Xl

M22 (x 10") -0.04

-o.12 -0.20 -0.28

~\~\ 0\~ -..

-0.36

-0.44 0.00 0.10 0.20 0.30 0.40 0.50 Xl

Figure 7.4" Square plate with step change in the thickness. Equivalent load analysis: Bending moments

168 Thin-Walled Structures with Structural Imperfections: Behavior Typical results for a square plate with simple supports and a step variation in the thickness are illustrated in Fig. 7.2. For values of 0.4 _< x2/2a <_ 0.6, the thickness t is reduced locally to 75% of its original value to, that is t = 0.75 to. Under transverse uniform load p3 = 1, the out-of-plane displacements at the position x2/L = 0.5, and shear, are illustrated in Fig. 7.3. The bending moments and the shear for a location x 2 / L - 0.475 are also shown in Fig. 7.4. The presence of the imperfection in the thickness leads to an increase in the maximum displacements of 25% with respect to a plate of constant thickness. The bending moments Mll, on the other hand, are decreased along the weakened zone by 25%; while M22 decreases by 44%. Since the results are obtained using a semi-analytical procedure (finite strips), it is important to consider the discretization employed. The results show that an increase from 9 to 19 trigonometric terms in the approximation does not substantially improve the results. The number of iterations in equivalent loads employed have been 3 and 5; but, certainly, 3 is a.dequate for practical purposes.

7.3.2

Perturbation Analysis

A similar model, but based on perturbation techniques, has been presented in Ref. [4] and is discussed in this section. The formulation follows exactly the one described in Chapter 3 for perturbation analysis of thickness changes, so it will not, be repeated here. Matrices K0, K1, K2 and Kz need to be constructed. In the first example, a thin square plate with changes in the thickness is studied under uniform transverse pressure. The plate slenderness is t/L - 0.01, with a groove of width a - 0.3L, and is studied by means of a Kirchhoff theory and a, finite strip analysis. The results for the deflection in the central section and bending moments, employing 15 trigonometric terms, are shown in Fig. 7.5 and 7.6. An exact solution is available in this case. The perturbation approximation leads to errors less than 1%. Sensitivity of the response with a parameter r, defined as r

t-- to =

to

(7.3)

is presented in Fig. 7.7. This also illustrates the influence of taking

169

Plates and Plate Assemblies

3

6 5

x

4

2 1 0

1

2

3

4

5

Xl

Figure 7.5" Square plate with step change in thickness. Perturbation analysis: Displacements at the center x2=0.5

different number of perturbation systems (abbreviated a,s NPS in the figure) in the solution, so they a,re compa, red with the deflections in the associated plate of constant thickness. For values of the perturbation parameter larger than r = 0.4, it is advisable to obtain the solution of the reference state and three perturbation systems. This is consistent with our findings for a simple two degree-of-freedom system in Chapter 3. The change in displacements is clearly non-linear with the depth of the groove, modeled by r. For example, a value of r = 0.2 produces a change in displacements of about 30% the displacement in the perfect plate; whereas for r = 0.4 the change in displacements reaches almost

80%. In the second example, we take the rather thicker square plate previously investigated by Mindlin theory and equivalent load and study it again using Kirchhoff plate theory. The new results are in Fig. 7.8 and 7.9, using 19 trigonometric terms and perturbation analysis of third order. The curves for the plate of constant thickness are also included in the drawing. This example shows the good behavior of both perturbation and equivalent load solutions, and the numerical differences observed are related to the different plate hypothesis and to the

170 Thin-Walled Structures with Structural Imperfections: Behavior

Mll B

prismatic ==

perturbation

1

o

1

I

2

I

3

'

I

4

I

5

Xl

M22 D

prisma

N,

/perturbation

-

-

1

I

i

I

I

Ii

2

3

4

5 X1

Figure 7.6: Square plate with step change in thickness. Perturbation analysis" Bending moments at the center x2=0.5

P l a t e s and P l a t e A s s e m b l i e s

171

1.2 9N P S = 3 1.0

NPS = 2

0.8 0

...0

03 v

2~ !

o3

0.6

9N P S = 1

r

2~ 0.4 0.2

0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

Figure 7.7" Square plate with thickness change. Perturbation technique: Sensitivity analysis numerical errors of each procedure.

7.4

PLATE ASSEMBLIES WITH THICKNESS CHANGES

The perturbation procedure of Section 7.3.2 has also been employed in Ref. [4] to model thickness changes in thin plate assemblies. A thin-walled angle beam with changes in the thickness is illustrated in Fig. 7.10; this is an interesting example because it shows a typical membrane-bending interaction under its own weight. The ratio t / L 0.003; a l L - 0.005 for a plate with L - 2,000; b - 100; a - 60~ E = 210,000; u = 0.15; and unit load is acting as self weight. Displacements and generalized stresses at the vertex of the midsection are presented in Fig. 7.11 and 7.12 for different values of the amplitude parameter r. The results were obtained employing 11 trigonometric terms and third order perturbation systems.

172 Thin-Walled Structures with Structural Imperfections: Behavior

U3

I"

perturbation~

0.5

\ \/~

0.4

.,~

equivalent load

0.3

0.2 0.1 0

0.1

0,2 0.3 0.4 0.5 Xl

Figure 7.8: Plate with changes in the thickness. Comparison between equivalent load and perturbation techniques: Displacements The displacements at the vertex of the mid-section are increased by a factor of 6 for a perturbation parameter r = 0.3, that is, a local decrease in the thickness of 30%. At the vertex, the longitudinal stress resultants change by 30% for v = 0.3, while bending actions are increased by almost 65%. The changes observed in this example are much larger than in the plates studied in Section 7.3, the main difference being that there is a coupling between membrane and bending in plate assemblies, and changes in the system introduced by imperfections modify the way in which the structure sa.tisfies equilibrium.

7.5

PLATES WITH IMPERFECTIONS IN THE MODULUS

Some imperfections in plates can be modeled by means of modifications of the constitutive parameters. For that, we need to introduce constitutive equations for orthotropic plates; these are:

173

Plates and Plate Assemblies

Mll

0 x

prismatic ! ~'-

0.5 "

e quivalent

0.4 0.3 0.2 0.1

/ / t ~ perturbation I

0.1

I

0.2

I

0.3

I

I

0.4 0.5 Xl

M22 "7

Q vx

0.5 0.4

perturLbation"

0.3 0.2

/ /"

equivalent

/t~

load

',

0.1 0

0.1

0.2

0.3

0.4

0.5

Xl Figure 7.9: Square plate with thickness changes. Comparison between equivalent load and perturbation techniques" Bending moments

174 Thin-Walled Structures with Structural Imperfections: Behavior

Figure 7.10: Thin-walled beam with changes in the thickness

Figure 7.11: Thin-walled beam with step change in thickness. Perturbation analysis: Displacements

Plates and Plate Assemblies

175

-0.5

-0.4 o o

*"

v

Z!

/

-0.3

eq c~l

Z

Z

-0.2

-0.1

.

/

/

/ I

0

-0.1 -0.2

o,,I o,,I

o

./"

-0.6

o,I r oll

/

-0.4

-0.2

0

=

-0.5

0

-0.8 0

I

-0.3 -0.4

l

-0.'1

-0.2"

-0.3'

- 0 . 4'

-0.5I

Figure 7.12: Thin-walled beam with step change in thickness. Perturbation analysis" Stress resultants

176 Thin-Walled Structures with Structural Imperfections" Behavior

/ 1 1 -- Dl1~11 -~- D12X22 q- D16~12

(7.4)

M22 - D12Xll q- D22X22 + D26X12

M12 -- D16X11 + D26X22 q- D66X12

with the coefficients Dll, D22 being the stiffness in each direction. Constitutive models like this one are currently employed in the analysis of composite laminates [3]. An interesting investigation into the analysis of composite plates by means of perturbation techniques, in which the properties of the composite were expanded with respect to an isotropic state, was carried out by Vinson [8]. Such approach has been also followed in Ref. [2] to model changes in the properties of the material induced by imperfections, and this latter work is summarized in this section.

7.5.1

Basic Formulation

As usual, to get an equivalent load in explicit form, we start from the differential equation. Consider here an orthotropic plate [6]. The outof-plane displacement u3 results from the solution of the differential equation

04U3

D12 q- 2D66 04u3

Ox----T -1- 2

D22 04lz3

Ox20x 2 ~ Dll Ox 4

Dll

p3 Dll = 0

(7.5)

Next, we introduce a cha,nge of variables defined by [8] 1

1

r

Y --

Dll ]

_

x2

or else

x2

_

b-

\Dll ]

b

or else

b-

\ Dll ]

,/.==

VDll

We substitute 9, b in the derivatives and obtain

(7.6)

Plates and Plate Assemblies

02(

177

(D22~-1 02( ) \Dll] 0~2

) _

Oqx2

7 Dll 02( ) ~ 0~2

_

04 ( ) D22 Oy4

04( )

Dll

Ox 4

(7.7)

The differential equation in terms of the new variables becomes

04U3

ox~l -~ 2

D12 + 2D66,/D11 04U3 04lt3 Dll VD22 0Xl20y2 ~ 094

P3

Dll = 0

(7.8)

It is possible to define a parameter ~- that contains the coefficients of orthotropy in the form

D12 + 2D66) ~- -- 2 1 -

v/D11D22

(7.9)

With this new definition, the differential equation can be written

&S 047/,3 Ox---T1 + ( 2 - r or else

( 04tt3 )

04U3 p3

Ox1~O9~

094

Oll

=0

04U3 DaaV4u3 - p3 - TD110x2109 2 = 0

(7.10)

where V4u3

-

04u3 04u3 + 2 ~ + Ox 4 0x21092

04u3 094

(7.11)

The solution of Eq. (7.10) will be obtained by equivalent load and by perturbation techniques following Ref. [2].

7.5.2

Equivalent Load M e t h o d

For r > 0, the new term in Eq. (7.10), i.e.

04U3

P3 -- -TD110x~O~/2 has the same effect as the original load p3. This is a problem which can be solved by means of a,n equivalent load technique.

178 Thin-Walled Structures with Structural Imperfections: Behavior

Figure 7.13" Plate with changes in the modulus: Reference solution The reference problem in this case is an isotropic plate with Dll = D~2 - D12 + 2D66, leading to r - 0. The set of equivalent loads is p* -- 7"011

p** -- T D l l

(x l , )

Ox~Of;2

Ox~Of;~

The complete solution of plate with changes in the modulus of the material is obtained by summation of the individual contributions ua-u ~

a+u 3 +...

The plate problem considered for a numerical example is simply supported, and is examined under a uniformly distributed load. A Fourier analysis of equivalent load can be carried out for this simple problem, with the details given in Ref. [2]. The steps in the analysis, detailing the different equivalent loads are illustrated in Fig. 7.13. The solutions of different approximations, are presented in Fig. 7.14. The solution has been computed for a number of values of the parameter r, so that sensitivity can be obtained numerically. This is shown in Fig. 7.15, where in each curve the number indicates the order of equivalent load employed.

Plates and Plate Assemblies

179

Figure 7.14: Plate with changes in the modulus. Equivalent load of first, second and third-order

180 Thin-Walled Structures with Structural Imperfections: Behavior

Figure 7.15: Plate with cha,nges in the modulus. First, second and third-order corrections in displacements

Plates and Plate Assemblies

U3

181

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 r

0.2 -3

-2

-1

0

1

2

3

Figure 7.16" Displacement sensitivity using equiva,lent load techniques. The numbers indicate the order of equivalent load a,na,lysis

U3

1.2 1.0 0.8 0.6 0.4 0.2 0.3

-2

-1

I

0

[

1

|

2

,,,

3

Figure 7.17: Displacement sensitivity using perturbation technique

182 Thin-Wafted Structures with Structural Imperfections: Behavior

7.5.3 Perturbation Analysis This problem of changes in the modulus of the material can also be modeled by perturbation techniques applied to the differential formulation. The reference problem is the same as in the equivalent load method; and the perturbation expansion is of the form

~ - ~ + ~ ~ + ~ .~ + . . . We substitute u3 in the differential equation, and obtain

04~ ] [V11~z4u~ - p] + T D11~z4tL 1 -- V i i 0x12 00-~----

+~'r

201104ul]

Dll

OX2Ofl2 -'[-...-- 0

But since r =fi 0, then the first order perturbation solution is obtained from 04U 0 ~ 7 4 ~ _ Ox~09~ = o

The second order perturbation solution results from ~747232 --

04ul

--" 0

Ox~092 The reference problem is the same as in the equivalent load analysis. First order perturbation solution ~ - 0.137

The second order perturbation solution u~ - 0.0344 The complete solution becomes 1 u3 - 0.539 + 0.137T + =0.0688T 2 2 The sensitivity of this problem with respect to changes in the parameter r is shown in Fig. 7.16, for different levels of approximation.

Plates and Plate Assemblies

183

Curve (1)is the first order, and curve (2)is the second order perturbation solution. From Eq. (7.9) we observe that positive values of r indicate In11D22 > D12 + 2D66

7.6

FINAL REMARKS

In this chapter we have presented some applications of the techniques introduced in the first part of the book for the analysis of plates and plate assemblies. In the case of plates, we illustrated the problems of imperfections by changes in thickness and changes in the modulus of the material; however, it is clear that the corresponding modifications in the stress field were not significant. Under transverse load, plates only develop bending, and with just one mechanics of resistance there is no redistribution. This means that a decrease in bending in one area of the plate is compensated by an increase in other areas. In plate assemblies, on the other hand, there are both bending and membrane contributions to equilibrium, and for the example with changes in the thickness we noticed some level of redistribution of stresses from one mechanism to the other. Only linear kinematics have been used here, in view of the limited practical value of the results in plates; but in the following chapters the influence of nonlinearity on the response will be investigated.

184 Thin-Walled Structures with Structural Imperfections: Behavior

References [1] Cheung, Y. K., Finite Strip Method in Structural Analysis, First edition, Pergamon Press, Oxford, 1976. [2] Godoy, L. A., Almanzar, L., Despradel, S. and Guzman, A., Numerical techniques to model structural changes in plate structures, Proc. VII Puerto Rico EPSCOR Annual Conference, Dorado (Puerto Rico), 1995.

[3] Jones, R. M., Mechanics of Composite Materials, Hemisphere, New York, 1975. [4] Raichman, S. R. and Godoy, L. A., A perturbation/finite strip approach for static analysis of non-prismatic plate assemblies, Int. J. Computers and Structures, 40(3), 1991, 629-637. [5] Suarez, B., Godoy, L. A., and Onate, E., Analisis de estructuras prismaticas de espesor variable por el metodo de la banda finita, In" Mecanica Computacional, 7, 1989, 73-88.

[6] Ugural, A. C., Stresses in Plates and Shells, McGraw Hill, New York, 1981. [7] Uko, C. E. A. and Cusens, A. R., Application of spline finite strip analysis to variable depth bridges, Communications in Applied Numerical Methods, 4, 1988, 273-278. [8] Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials, Kluwer, Dordrecht, 1987.

Chapter 8 IMPERFECT SHELLS 8.1

SHALLOW

INTRODUCTION

Structural problems in thin reinforced concrete shallow shells with ~ocal imperfections have been reported by Ballesteros [2] in relation to the collapse of several elliptical paraboloidal shells. The dimensions of those particular shells were 27x27 m in plan, with thickness of t = 0.06 m; and the maximum deviation in shape at the apex was estimated to be about twice the thickness. The imperfection was localized at the apex of the shell, and was mainly due to initial deflections following gravity after the form work of the shell was removed. A finite element code was used by Ballesteros to model the imperfect geometry of the shell. Other imperfections in concrete shallow shells were reported in Ref. [3]. This chapter contains some simple solutions and numerical results for shallow shells commonly employed as roofs in industrial or sports buildings. The equivalent load equations are presented in Section 8.2, in which the solution of the bending equations is made using a Ritz technique. The advantage of this solution is that it yields explicit equations for displacements and stress resultants of the imperfect shell. The solution presented is linear in the amplitude of imperfection, and is also restricted to the first order equivalent load (that is, one iteration in the solution). This approximation may however be enough to have an estimate of the stress levels in an imperfect shallow shell. Section 185

186 Thin-Walled Structures with Structural Imperfections: Behavior

Figure 8.1: Notation and positive convention for a shallow shell with local imperfection. 8.3 contains results for a,n elliptical paraboloidal shell, to illustrate how important the stress are in a practical situation. This is taken from Ref. [4].

8.2

8,2.1

EQUIVALENT LOAD IN GEOMETRIC IMPERFECTIONS

Simplified Imperfection Analysis

The bending equations that govern the behavior of a shallow shell are given in Appendix B. A main difference between a flat plate and a shallow shell is the geometric curvature of the shell. The curvature parameters of the perfect mid-surface, kij , are given by O2X3

k~j = Ox~Oxj

(8.1)

Shallow Shells

187

where xa defines the geometry of the shell. For the sake of simplicity, the geometric deviations ~ from the perfect shape considered in this work are restricted to the analytically convenient form u3 - ~

1+cos

l+cos

a

b

(8.2)

for a ----

a <

b

X1 <

b <

X2 <

--

and u~ - 0 elsewhere, with ~ being the maximum amplitude of imperfection, assumed to occur at Xl - x2 - 0; and (k, l) the parameters defining the wavelength of the imperfection. A specific choice of k, l and ~ defines an imperfection in a shell with plan dimensions 2a and 2b. This imperfection is represented as an initial displacement in Eq. (8.2). 0 may be calculated from u 8i The error in geometric curvature, Xij, aS 0

X i j - OXiOXj

(8.3)

The shell equations should be solved for a mid-surface defined by X 3 - - X~ -~- X 3i

(8.4)

where x~ is the coordinate in the perfect shell, and x/z is the coordinate of the initial imperfection. An approximate method to study the influence of localized imperfections in shells is the equivalent load method, described in Chapter 4. In this method the shell equations are written for the perfect geometry, but a pressure is applied, which approximates the effects of the imperfections. The first order load representation of an imperfection is obtained from 9

p3 -

0

(8.5)

Notice that Xij~is computed from Eq. (8.3), while Nij are obtained from an analysis with the actual loads acting on the perfect shell. For

188

Thin-Walled Structures with Structural Imperfections: Behavior

the case of uniform stress fields in the X 1 and x2 directions, and with N12 - 0 , the pressure intensity p~ results in

P39 __ _4

1+

2 {Nil

+ N22

1 + cos

cos

cos

b

cos

a

a

b

}

(8.6)

The load component p~ is next represented by its amplitudes in a double Fourier series

*

Pa - p~m cos

( ngxl ) cos ( m7f'x2 ) 28 2b

(87)

This has two advantages: first, the load can be better understood by means of its Fourier components; and second, a Rayleigh-Ritz analysis using the Fourier series to represent the displacements can be carried out. The load components Pmn result in pm~ - - ab

Pa cos

2a

cos

dxl dx2

" 2b

(8.8)

Replacement of Eq. (8.6) into Eq. (8.8) leads to t: p*~ - "--kbr2 (N~IQ~,~ + N22Rm~) (8.9) 4 The coefficients Qm~ and R,,~ depend only on the wavelength defining the imperfection (k, g), and on the particular harmonic (n, m) being considered. They can be written as Q~n - ~ (k n) ,~ (l, m) -+- - - sin

[

4

Rm, - ,$ (1, m) r ( k , n ) - + - - - s i n mrr

~

(8.10)

where r ' for n -J: 2k; and

n)_4_ n sin (~-~) rr 4k 2 - n 2

(8.11)

189

Shallow Shells

0.15 if"--"1

E E E

Z

//

0.10 0.05

0

,/ I/

L__._J

n

i'

2r~,,,

t

-0.05 I

!

I

1

I

1

-2 -1 0 1 2

-4

I

I

I

4

• Ira] Figure 8.2" Equivalent load for an elliptical paraboloidal shell. Contributions for m = l , a,nd (1) n=27; (2) n=15; and (3) n=7

l

( k, n) - --s

(8.12)

for n - 2k. The load factors p~. are large for n < 4k, m < 41, and decrease for large values of n, and m.

8.3

8.3.1

ELLIPTICAL PARABOLOIDAL SHELL Equivalent Load

To illustrate the behavior of a concrete shallow shell with local imperfections, an elliptical paraboloidal shell is considered in this section. The mid-surface is defined by the equation

x3-

(8.13t

190

Thin-Walled Structures with Structural Imperfections: Behavior

Let us consider the following dimensions to compute stresses" fl = f2 - 2 m; a - b - 15 m; and t - 0.08 m. These shell dimensions are somehow similar to those reported in Ref. [2].

8.3.2

Equivalent Load Representation

First, an imperfection with wavelength k - l - 3 and amplitude = t - 80 mm has been considered, under the action of a uniform stress field Nll - N22 - 52.65 N / m m in the perfect shell. The reason to choose this specific value of stresses in the perfect shell is that they are m a x i m u m values under self weight. A profile of the equivalent load p3 m a y be obtained for a given number of harmonics included in the summation. Fig. 8.2 shows the equivalent pressure for summations including up to n - m - 7, 15 and 27. From the results, it seems that a load that includes terms up to n - m - 5k is a reasonable approximation. Similar conclusions were obtained for the spherical shell, in Chapter 9. The contribution of each harmonic n in the definition of the equivalent pressure for a fixed value m - 1 is presented in Fig. 8.3. The m a x i m u m values occur for n < 4k, reaching a m a x i m u m at n - 7, and then decreases. Contributions are positive and negative in alternate bands. For large values of n, the contribution pm~ tends to zero. From a load spectra as shown in Fig. 8.3, it is possible to identify what terms should be included in the analysis.

8.3.3

Results for a Specific Imperfection

For the imperfect shell described in the previous section, the changes in stress and moment resultants with respect to the perfect situation have been indicated by ni*j and mi*j in Fig. 8.4 and 8.5. The computations have been obtained by means of a Ritz solution that employs trigonometric functions, as explained in Appendix B. Convergence was achieved for n = m = 27. It may be seen that even a rather crude representation with n = m = 7 produces results which follow the general stress behavior of the shell. The wavy pattern of stress changes in Fig. 8.4 and 8.5 is similar to those found in hyperboloids of revolution and spherical shells with local imperfections. This mechanism of redistributions of stress resultants t h a t occur in the imperfect shells are discussed in Chapters 10 and 11,

Shallow Shells

191

8

;2

4

7=o -2

0

5

10

15

20

25

30

35

40

45

n Figure 8.3: Equivalent load for an elliptical paraboloidal shell. Harmonic components for m = l

simply because they were first discovered in the context of cylindrical shells and cooling towers. Notice that for an imperfection amplitude equal to the thickness, the stress changes ni~ are about 60% of the stress resultants Nij in the perfect shell. Although imperfections are usually considered as secondary effects in terms of the stresses in the structure, the results in this chapter show that they induce stress changes of the same order of magnitude as the stresses due to the main loads. Since in the present model the load is directly proportional to (, the stress changes for other amplitudes of imperfection should be multiplied by the factor (/t.

8.3.4

Influence of Imperfection Parameters

A specific imperfection was considered in the previous studies, so it would be important to show how the wavelength of imperfection affects the stress resultants. Fig. 8.6 shows variations in n~l and m~2 for increasingly shorter imperfections, and it may be seen that the bending action is increased by a, factor of 3.6 when the imperfection wavelength

192 Thin-Walled Structures with Structural Imperfections: Behavior

~

I/

e-. 20

30 i

0

I

1

I

I

I

5

i

• _3

,,/-E

,z,

10

B

2.

I'ff"

"-,,'7 20 F ..-')/\ 30

,;/ F '

0

1

5

x2 [m] Figure 8.4: Membra,ne stress changes in the elliptical paraboloidal shell. (1)n=m=27; (2)n=m=7

193

Shallow Shells

E E

E E

Z

L____J

o

0.2

"1r

E

o o J~

0.4

0.6

0

1

x, [m]

0.2

0 0.2

0.4

0.6

0

1

5

x2 [m] Figure 8.5" Moment changes in the elliptical paraboloidal shell (1) n=m=27; (2) n=m= 7

194 T h i n - W a l l e d S t r u c t u r e s with S t r u c t u r a l Imperfections: B e h a v i o r is reduced by a factor of 2.33. A more general picture of the influence of the imperfection wavelength is presented in Fig. 8.7, in which maximum changes in stresses are plotted for different imperfections identified by k, 1. The results are very similar in nature to those presented for cooling towers in Chapter 11. Maximum values of stress resultants n~l occur for k, l approximately equal to 5, and decrease for short imperfections. The bending contribution rn22, on the other hand, increase in short imperfection, but reaches a maximum value. Notice that now we are dealing with two parameters to describe the imperfection: the amplitude ~ and the wavelength, as reflected by the number of waves k and 1. Very short imperfections are characterized by large numbers k and l; while long imperfections have small values of k and 1. Long imperfections produce small membrane stress resultants and negligible bending because the errors in curvature are smooth. As the imperfection extends over a smaller region of the shell, and for the same amplitude, the membrane action increases dramatically. In the present case, a maximum in n~l and n~2 is reached for k, l - 5, that is, 1/5 of the dimensions of the shell. Notice that the bending action, although high, is not at a maximum. For very short imperfections, the curvature errors are high and most of the imperfection is taken by bending, with membrane stresses being identical to those in the perfect shell. If the thickness of the shell is increased for the same amplitude of imperfection, the results have been plotted in Fig. 8.7. A reduction in t from 80 mm to 60 mm reduces the bending stiffness of the shell, and also the membrane stiffness, although less. The membrane stress resultants tend to increase by more than 30%. An increase in the thickness reduces the membrane stress resultants and increases the bending moments mi~. From the point of view of the designer, it is important to have upper bounds to stress and moment resultants for a given imperfection amplitude, without having to specify the wavelength of the imperfection. Both the membrane and bending resultants in Fig. 8.7 are seen to reach maximum values (n~l for long, while m~2 for short imperfections). Those maximum values are plotted in Fig. 8.8 for shells of different slenderness and imperfection amplitude. The stress changes are approximately linear with the radio r / t , in which r - 1 / k l l - 1/k22. The results of Fig. 8.8 could be used to estimate maximum changes in

195

Shallow Shells

\ E E Z

0.2

1

0

\\ \

,It

t-

0.2

0.4 0.6

m

I

o

I

I

I

I

1

i

i

5

x, [ml '\

1.5

\

E E E Z

E

I

;! t 0.5

?'

-"" 1

1

o

~\I~~

-0.5 i

0 1

t

I

I

I

I

I

5

X2 [m] Figure 8.6" Influence of imperfection wavelength. k=l-5; (3/k=l-7

(1) k=l=3; (2)

Cb

~C~

...j~

oo

II

.~~

II

DO

ED

r

i--,~

9

OO

m

0

tI

22

m* [Nmm/mm] 0

X

0

0

!

i

I !

/

0

x

f

0

0

n 11[N/mm] 0

~176

t~

ra~

e-,N

I

i.,,,i ~

Shallow Shells

197

stress resultants for a given imperfection amplitude.

8.4

FINAL

REMARKS

The stress resultants in shallow shells are sensitive to the presence of small deviations in the geometry. For reinforced concrete shells constructed with the shape of an elliptical paraboloid, there is clear evidence of short wave imperfections in the geometry, with amplitude larger than the shell thickness. For such thin shells, the results in this chapter show that the level of stress changes is of the same order as those produced by self weight. Bending stresses are also increased. The changes due to imperfections depend on several factors, such as the area covered by the imperfection, the maximum deviation, and the geometry of the shell. These factors also affect the relative contributions between bending and membrane actions, to equilibrate the imperfect shell. All this indicates that shallow shells should be carefully checked to ensure that the geometry follows tolerance limits based on stresses in imperfect shells. Detail studies should be done whenever imperfections of the order of the thickness are found, and this may lead to the need to provide strengthening, as presented in Ref. [2]. We have not considered non-linear kinematics, plasticity, and creep effects in this chapter. This could increase the amplitude of the imperfections in time; which leads to the need to implement periodic controls of imperfections to record how they grow in time.

198 Thin-Walled Structures with Structural Imperfections: Behavior

k=l=3

XlO 3

,-~

=,.=

..

Z

e.-

,1r

30

qr=.. ,1=1,

~ ~

~m,= -,~~~'`~'`~'`-~

20

10

I O. 1

I 0.08

. r,

vt

I 0.06

xlO 3

rE. E

2

=.,

I

I

O. 1

0.08

I

..

'/t

0.06

Figure 8.8" Influence of shell slenderness r/t on the maximum values of stress changes

Shallow Shdls

199

References [1] Apeland, K., Analysis of bending stresses in translational shells, including anisotropic and non-homogeneous properties, A cta Politecnica Scandinavia, Civil Engineering Series, 22, 1963. [2] Ballesteros, P., Non-linear dynamic and creep buckling of elliptical paraboloidal shells, Bulletin of the Int. Association for Shell and Spatial Structures, 66, 1978, 39-60. [3] Ballesteros, P., Nonlinear dynamic buckling of an imperfect spherical concrete shell, In: Non-Linear Behavior of Reinforced Concrete Spatial Structures, IASS Symposium, Damstadt, Germany, 1978, 205-218. [4] Godoy, L. A., Quinones, D. and Wagner, R., Shallow shells with local deviations in shape, Bulletin of the Int. Association for Shell and Spatial Structures, 97, 1988, 55-62. [5] Reissner, E., Stresses and small displacements for shallow spherical shells, J. Applied Mechanics, 25, 1946, 279-300.

Chapter 9 IMPERFECT SHELLS 9.1

SPHERICAL

INTRODUCTION

Spherical shells have a number of applications as structural components, specially as pressure vessels. Very large thin-walled complete spherical shells are used as main structural components in pressurized water reactors (PWR) or pressurized heavy water reactors (PHWR) in nuclear power plants. In a typical plant design, the spherical steel containment m a y have a diameter of 50 m (r/t - 9 0 0 - 1000); for example, Fig. 9.I shows the sphere of a nuclear power plant with such dimensions [15] during the construction. Large cranes are employed to assemble the parts into the sphere. Liquefied natural gas is transported by sea by large carriers that may have five spherical shells in line. Each shell is supported at the equator on a cylinder. Typical dimensions are 15 to 20 m in diameter and r / t - 3 5 0 - 800.[14] High-pressure gas containers are another important application of large spherical shells, with 200 < r/t < 400. One shell of this type is shown in Fig. 1.6. Spherical shells are also used as end-closures of pressure vessels, as shown in Fig. 9.2. In this case, the pressure vessel contains gas, and some imperfections were detected in the spherical part. Thin-walled spheres are also used in liquid storage, chemical plants, and ship and aerospace structures. Although the tolerances in geometry adopted for the fabrication of 200

Spherical Shells

201

Figure 9.1" Steel spherical shell of a nuclear power plant such shells are stringent, it may so happen that deviations from the asdesigned geometry, with amplitudes of the order of the shell thickness, occur as a consequence of damage of the shell. Changes or defects in the shape of a metal shell can also be due to welding of panels that form the complete shell. The assembly of elements and the forming process may induce residual stresses, too. Other problems found in the construction of large metal shells are the lack of alignment of the mid-surface when panels are joined and the joints of sectors of the shell with different thicknesses. In those cases, it is necessary to evaluate the level of stresses and the nature of the stress redistribution that takes place in the modified shell. Stress concentrations associated with an imperfection are important due to the stress levels induced, which may produce yielding of the material, and when they are coupled with other effects. Changes in the thickness of a pressurized shell may occur because of a defect in fabrication, or because of the requirements of design. In both cases, a step change in the thickness induces elastic stresses which may be significant when they are compared with stresses due to other design conditions. For example, a spherical shell containment of a nuclear power plant has been analyzed in Ref. [15] considering the

202

Thin-Walled Structures with StructurM Imperfections: Behavior

Figure 9.2" Steel pressure vessel with spherical end-closure (photograph by the author) local stresses induced by a change in the thickness. The governing equations for spherical shells are summarized in Appendix B, so that the reader can follow the developments of the following sections. Section 9.2 refers to the tools of analysis that have been employed by the author and his co-workers to compute the subsequent results in this chapter. These tools include a finite element formulation for shells of revolution as the basis of a direct analysis of geometric and intrinsic imperfections, and a simplified equivalent load technique for geometric imperfections. In Sections 9.3 to 9.5, we present some results on the elastic behavior of thin spherical shells having geometrical imperfections with amplitude equal to or larger than the shell thickness. Emphasis is placed on the mechanics of behavior of the shell, ra,ther than on design rules. The results presented in Section 9.5 are restricted to the influence of meridional (or a,xisymmetric) imperfections. Local imperfections are considered in Section 9.4. At the end of this section, a comparison between stress resultants due to axisymmetric and local imperfections is given. The nonlinear behavior of imperfect thin spheres is discussed in Section 9.5, for local and axisymmetric imperfections. Finally, in Section 9.6 we discuss

Spherical Shells

203

thickness changes in pressurized shells as a possible form of intrinsic imperfections that induce a concentration of stresses.

9.1.1

Short L i t e r a t u r e R e v i e w

Perhaps Flugge was the first to evaluate stresses in a geometrically imperfect spherical shells. In Ref. [5], Flugge considered a spherical shell ( r / t - 200) under uniform dead load, and an axisymmetric imperfection with maximum amplitude equal to half the thickness and meridional shape defined by three circular arcs. Using a membrane solution, Flugge computed significant changes in the circumferential stress resultant N22 but found no variations in the meridional stress resultant NIl with respect to the membrane stresses in the perfect shell. To account for bending, he used the force method and restored compatibility at the junctions between segments of the imperfection. As a result, the values of N22 were reduced, but still showed large variations with Nil and the meridional bending moment Mix following similar banded patterns. Heyman [10] confirmed that small variations in the shape of the meridian of a masonry dome produce large changes in N22 without affecting Nll. Calladine performed a systematic study on the influence of axisymmetric imperfections in cylindrical shells [1], and extended them to spherical shells by using the assumptions of Geckeler. The model provided good estimates of stresses, except for a limited region near the crown of the sphere, for which he noticed that the results should not be applied. More recent interest in this field ha,s been inspired by the achievements of detailed studies on imperfect cooling towers, drawn by the needs of the nuclear power industry. This is reflected in Ref. [7], [8], [6], [3], and [13]. Thickness reductions have been studied in Ref. [15], [4], and [9]. Other forms of defects, including stress concentrations due to welding, have been investigated in Ref. [17], [11], and [12].

204 Thin-Walled Structures with Structural Imperfections: Behavior

t

I

u ~,/~~,~ P W

1 Figure 9.3" Spherical shell with imperfection

9.2

TOOLS OF ANALYSIS

9.2.1

Direct Finite E l e m e n t M o d e l of the I m p e r f e c t Shell

In this chapter, imperfect shells are studied using a formulation for shells of revolution in which stress resultants rather than local stresses across the thickness are computed. The Love-Kirchhoff hypotheses are imposed a priori, with the consequence that stress profiles are necessarily linear with the coordinate normal to the mid-surface. A finite element formulation has been developed to model such shells [4], in which the meridian is discretized by one-dimensional elements and the circumferential direction is expanded in terms of Fourier series. The load and the response can be non-axisymmetric, but the geometry is a shell of revolution. Details of the element and the formulation are given in Ref. [4]. Geometric imperfections have been represented by a direct model of the imperfect meridian, with the amplitude of the imperfection being added to the radius r of the sphere. With reference to Fig. 9.3, the radius r/ of the shell having an

Spherical Shells

205

imperfection of amplitude ~ is given by ri - r + ~

(9.1)

The studies of axisymmetric imperfections have been restricted to the following geometry"

[

- ~~0 1 + cos

-~0 (r - r

)]

for

r

r

< - r -< r

r176 (9.2)

- 0

for

r

r

fi(~

> 4) > r + :-x~ 2 2

where sx0 is the maximum amplitude; r is the angular extent; and r is the angular position of the center of the imperfection. Of course, this is an idealization of real imperfections that may occur in practice. The effects of different profiles of imperfections on the stresses in thin shells of revolution have been investigated by several authors, and it has been shown that, provided no discontinuities in slope occur, the stress distributions due to very different imperfections do not differ so much from each other. Thus, although the present studies are restricted to one particular imperfection profile, they may be seen as indicative of the behavior of the shell with other meridional imperfections but similar values of (o, r and r In the study of local imperfections, some restrictions on the load and imperfection have been considered to simplify the studies: First, the loads are assumed to produce uniform pressure normal to the shell; and second, the local imperfection considered has the same curvature errors in all directions. For the class of load and imperfection described, the shell can still be analyzed as a shell of revolution, with the advantage that the finite element code used in axisymmetric imperfections can also be applied to the stress analysis of locally imperfect spherical shells. The type of imperfection studied is representative of local damage that is likely to occur in the shell after the construction is completed. For the localized imperfections of Fig. 9.4, the amplitude is now given by

206 Thin-Walled Structures with Structural Imperfections: Behavior

I I

i i i I Figure 9.4" Localized imperfection in a spherical shell

1 [1 +cos (2zrr r ]

(-~(0

27r0)] [l+cos(~-o

for

r 2

0


(9.3) and

--0

for

00 < 0 < 00 2-2 r >r r 2 _--~

and

Oo > 0 > Oo 2 -

in which 00 is the angle in the circumferential direction. Typical meshes include at least 18 elements along the meridian, of which 9 cover the imperfect area. Finally, thickness changes are modeled as in Fig. 9.5. Again, the change in the thickness is located at the crown of the sphere to take advantage of the conditions of symmetry. It is assumed that a step reduction in the thickness occurs in a small circular area of the spherical shell. Furthermore, to simplify the computations, this local reduction in thickness is symmetric with respect to the middle surface of the shell. Typical meshes have 12 finite elements in the meridian of the shell.

9.2.2

Equivalent

Load Analysis

The present section is concerned with the modeling of a geometric imperfection with tools simpler than those mentioned in the previous sec-

Spherical Shells

207

Figure 9.5" Thickness change in spherical shell tion. To carry out the analysis, the equivalent load method of Chapter 4 is used together with Fourier expansions of the load and displacement fields. This is a convenient way to obtain simple explicit expressions for some typical forms of imperfection, in which the accuracy of the results depends on the number of modes considered. Following Chapter 4, the first-order load representation of an imperfection is obtained from

P* -- ~101 /11 -~" ~02 /22 ~- 2 Xl~ N12

(9.4)

in which Nl1, N22 and N12 are the stress resultants in the perfect shell; and X1~ )/o2 and )/o2 are the curvature errors. Let us assume an imperfection ~(xi, x2), in which the radius of the imperfect shell, ri, is written as in Eq. (9.1). The curvature error associated with the imperfection is given in terms of the amplitude ~ by

02~

02~

02(

(9 5)

To compute the equivalent load p* in Eq. (9.4), the stress resultants in the perfect spherical shell should be known. For the case of uniform pressure q, the membrane stresses result in

r Nil - N~2 - - ~ q ;

N~2 - 0

(9.6)

208 Thin-Walled Structures with Structural Imperfections: Behavior and Eq. (9.4) reduces to

(9.7) Axisymmetric Imperfection For an axisymmetric imperfection with

~ - 2(0 [1 + cos ( - ~ ) ]

7rr<-x2<-n 7rrn

for

(9.8)

where n defines the wavelength of the imperfection, it may be shown [6] that the equivalent load is of the form

t9 t Fourier components can now be obtained for this load distribution, and they are given in explicit form in Ref. [6]. At this stage, it is interesting to consider the contribution of each Fourier component p*(0, j) in the definition of the equivalent load. For a typical imperfection with central angle r = 30o, and amplitude ~0 = - 3 t , on a shell with r/t = 993, the values of Fourier components are plotted in Fig. 9.6. This is similar to Fig. 8.3 for shallow shells. The maximum contributions are seen to be those with j < 2n, and the values of p*(0, j) are significantly smaller for higher harmonic numbers j. This shows that the analysis should at least include 2n harmonics. The membrane components are given by

ix1

n22 (0, j ) cos ~

n22 i

roll

r

j

-- Y~Emll(0,j)c~176 i j

cos

jx2 r

ix1

jx2

r

r

m 2 2 ~ ?A r o l l

where [6]

jn 2 2/zr sin jTr n~2(O,j) - -(oq (n 2 _ j2) [4 + (j2/cR)2] n

for

j#n (9.10)

Spherical Shells

209

5 4 n

:E

3

I'

2

o

,qp..

X

1

13.

0

0

20

40

60

80

100

J Figure 9.6: Pressure components of the equivalent load, in different wavelength numbers j

n

n~2(O,j) - (oq [4 + (j2/cR)2]

for

j - n

and

m~l(0, j ) --

~oq jan~ t sin jTr 2~r (n 2 - j2) c2R [4 + (j2/cR)2] n

for

jsCn (9.11)

re;l(0, j ) _

~oq t 4 c 2R [4 + (j2/cR)2]

for

j~rt

The values of c and R are defined as usual c 2 - 3(1 - v2);

R - r/t

(9.12)

Local Imperfections For a local imperfection, the amplitude ~r is written as

~ -- 4~0 [1-t-COS(m:l/] [1-~-COS(-~}]

(9.13)

210 Thin-Walled Structures with Structural Imperfections" Behavior leading to the equivalent load

p,- ____~0

8r qn2{m

2[

l+cos

[

(_~)]

(taxi)] r

+ n 2 1 + cos

cos

cos

(mxl) r

(-~)

}

(9.14)

Again, the Fourier components of the load are given in explicit form in Ref. [6]. Use of the equivalent load technique and a simplified solution of the spherical shell based on a trigonometric representation of displacements, the stress field is similar to that of an axisymmetric imperfection, but now [6]

,

1 j2 [

n22(i'J) - 27r2 i 2 + j2

4+

cR

~oqQ~j

(9.15)

and

-1 ,

1 (vi2+j2)

mll(i' J) -- 871.2

t

c-~oqQij

cR

(9.16)

where 2 Q~j - m2~a(m, i) ~(n, j) + _ sin - n 3

[

2

+ n2~a(n, j) ~(m, i) + _ sin

$

(9.17) and r

2i ~ri i) - - m 2 _ i2 sin --rn ~(m,/)-

71" rn

--

for

for

i~ m

(9.18)

i-rn

Equations (9.15) and (9.16) are very similar to Eq. (9.10) and (9.11) for the banded imperfection: n~2 also depends on ~0q, and m~l is related to ~oqt/R. Thus, two shells with the same value of R have the same changes in stress resultants; but in the thicker shell the moment resultants will be larger. Values of n~l and m~2 can be obtained by interchanging the modal and imperfections' numbers in Eq. (9.15) and (9.16).

Spherical Shells

211

1.00 .= .

.

.

.

.

.

.

m . i

0.50

Nll.

N~,

(internal presure),

E E

N~, (self weight)

Z

30mm

-0.51 -60

!

L

I

I

.

-30

0

30

60

90

Figure 9.7" Stress resultants due to self-weight and internal pressure

9.3

9.3.1

AXISYMMETRIC IMPERFECTIONS

GEOMETRICAL

S t r e s s e s in an I m p e r f e c t S h e l l

To illustrate the mechanics of behavior of a thin sphere with an axisymmetric imperfection, calculations have been performed on one particular shell geometry. The case studied has radius r = 28000 ram, thickness t = 30 ram, and a ratio r/t = 933; the modulus of elasticity is assumed as E - 2.1 x 105 N/mm 2, with Poisson's ratio v - 0.3. Self weight and internal pressure have been considered to show redistributions of stresses. It is not our intention to establish limits to the load for t h e m at this stage; thus, an arbitrary value of internal pressure p - 0.05 N/mm has been assumed, together with a specific weight 3' - 7.64 x 10 .5 N/mm 3. A shell with similar characteristics has been studied in a different context in Ref. [16].

212

Thin-WMledStructures with Structural Imperfections: Behavior

Before examining the behavior of an imperfect shell, let us consider the level of stresses in a shell with perfect spherical geometry. Fig. 9.7 shows the stress resultants Nix and N22, for dead weight and for internal pressure, at different meridional positions. For dead weight, the stresses are not significant in the upper hemisphere, but increase near to the bottom of the shell. For the level of internal pressure considered, the associated stresses are significantly higher than those due to dead load, and the trend is only reversed for values of r near to -62.50. The influence of an inward imperfection with its shape described by Eq. (9.1) and with amplitude ~0 - - t , extending over r - 300 and centered at r = 0o, will next be studied. This particular imperfect shell has slope continuity at all points, but shows discontinuities in meridional curvature at the intersections between perfect and imperfect segments of the shell. The results have been computed using a finite element direct model of the imperfect shell. The changes in hoop stress resultants, n~2, and in meridional bending moments, m~l , are plotted in Fig. 9.8 for self-weight. As in the case of cylinders and hyperboloids of revolution, n~2 follows a banded pattern, with alternate bands of tension and compression. The role of these bands of n~2 in equilibrating the out-of-balance moment produced by Nix acting on the imperfect shell has been discussed in Ref. [2]. In Fig. 9.8, the moments m~l show several changes in sign; similar results have been found for other shells. For this rather long imperfection, the actual changes in stress resultants are not significant: for instance, at the center of the imperfection, N22 = 64 N/mm in the perfect shell, while the changes n~2 are of 4.7 N/mm. If the amplitude of the imperfection ~0 is increased, the stresses increase directly proportional to the amplitude. For example, at r = 0o , for ~0 - - t the changes in n~2 are of 4.7 N/mm; while for ~0 - - 3 t the changes in n~2 are of 14.5 N/mm. Similar distributions of n~2 and m~l are found to occur for the case of internal pressure acting on the same shell and imperfection, but now the signs are reversed. And again, the changes in stresses are proportional to the amplitude of the imperfection. Notice that the changes in n~2 and m~l are directly proportional to the Nil stress resultants in the perfect shell.

I

II

I

II

..,,,

O

""

0

O~

i=,=~ 9

Or3

i===~

.o

oo

II ~

"8-

I~ 0

__L

O

O"

-

-

0

0

0

/

0

b

0

0

'

0

O

I

. ~ ~

0

m*~, [Nmm/mm]

mm i

I

-

i

0 I

-

PO 0 -

0

-

0 f~s""-

0

~ , - -

"

01 -

9

O

/ t" "

I

t~1

s.,

~~

-'.-"

I

01

o "~

I

0

i

.--L

I

01 I

|i

i

~

. SSO00 mm.

i

[N/mm]

G.I

214

Thin-Walled Structures with StructurM Imperfections: Behavior

'E'

.1

0.3

E

,

0.2 -

Z

o.1

1

2

, ....

I

I

S .o'

0

~'"

~1:;:~ -0.1 -0.2 -0.3

,E

E

1

"E ,~

-20

,

0

10

20

~

0.3

,~~, 2

0.2 0.1

, ~,

o

~'-

-10

~J;""

',lP,-

-o.~

-

-0.2

-

-0.3 ~

I -20

S.J. ~

iv I

-10

0

I

I

10

'

"1

I

20

Figure 9.9" Imperfect shell under internal pressure (1) ~o=-3t; (2) ~o=2t; (3)~o=-t

Spherical Shells

9.3.2

215

Influence of Shell and I m p e r f e c t i o n Parameters

The particular example presented in the previous section was indicative of the pattern of stress redistribution that occurs in a thin spherical shell with a rather long axisymmetric imperfection. But to understand how each chosen parameter affects the elastic response of the structure, parametric studies will be carried out in which imperfection and shell parameters are modified. First, the influence of the extent of the imperfection along the meridian, as represented by the angle r is studied in a shell with r/t - 933 under internal pressure. For an amplitude ~0 - - 3 t , the results for imperfection with r from 20 to 3o are plotted in Fig. 9.10; and it may be seen that n~2 and m~l not only change in values with r but also the actual shape of the banded diagrams changes. In general, it can be said that the changes are inversely proportional to the angle r at least for r _> 4o. For short imperfections, i.e. r - 5o, the maximum changes occur at the center of the imperfection; but for longer ones, say r - 20o, the highest values of m{1 are computed near the ends of the imperfection. The importance of these changes in stresses and moment is summarized in Fig. 9.11, in which n~2 and m~l are plotted as a function of r Extreme values of m~l , positive and negative, have been included in Fig. 9.11, since negative moments are higher than positive ones for long imperfections. This does not occur with n~2 , which is always maximum for positive values in inward imperfections. For the shell with ~o - - 3 t , n~2 increases from 0.36 k N / m m to 1.19 k N / m m (an increase of 330%) when the angle r is reduced from 200 to 10o. For the same reduction in r m{1 increases from 0.35 to 4.73 kN/mm, that is by 1300%. But from Fig. 9.11 it may be seen that the increase in the membrane hoop contribution to equilibrium, even in the purely elastic case, is not unlimited, and for r < 50, n~2 decreases, with the consequence that m~i is severely increased. This change in the relative membrane to bending contribution was previously pointed out in Ref. [2] for cooling towers. From the results of Fig. 9.11, it may be seen that for short imperfections (say, r - 5o), with ~0 - - t , the values of n~2 are of the same order a,s the stress resultants Nll and N22 in the perfect shell; whereas they are about 3 times the perfect stresses for ~0 - - 3 t .

216 Thin-Walled Structures with Structural Imperfections: Behavior 2

9

-I

0

I

I

I

I

i

2

4

6

8

10

lb

'" e l

E

mm

56000

E

, I",

'~//,L5 -7"" I

il ~ I?,L..' "

-10 0

2

4

6

8

I0

Figure 9.10" Influence of imperfection extent (1) r (3) r (4) r (5)r

(2) r

Spherical Shells

217

m

1

E E z

2 1

-

u__._a

,Ic

\ '\

Cq

l-

\

3

r'-~,.

X

X \ '\, ,,,...\ ,,

\

~"~"

0

'

'

4

8

I

12

"=" I

I

16

I

20

,o 9

, 40

n

1

20

2,

3,, ,,,\ "~ ~ o~

-20

I

I

4

I

8

q..... ,.

~.=~-:~

I

12

I

16

'

20

Figure 9.11" Summary of changes in stress resultants (1) ~o - -3t; (2) ~o--2t;

(3)~o--t

218 Thin-Walled Structures with Structural Imperfections" Behavior

m

40

1

20

o

-20 -20

E

'

F::

8o

E E z

60 4o

.Ic ""

E

'

-10

0

10

20

1

/2 t

'

20

3

J

=.. L

o

--;J

ki

-20 -40

-60 I

-20

I

-10

o

I

I

~o

I

I

20

Figure 9.12" Influence of the position of the imperfection, for a shell with an inward imperfection (1) r (2) r (3) r

Spherical Shells

219

So far only imperfections centered at 4~i = 0 have been considered. It would be of importance to understand what is the influence of the actual position of the axisymmetric imperfection on the stresses. The shell with imperfection defined by ~0 = - 3 t , q~0 = 30o has been studied, with q~i taking values between 0o and -500. The stresses due to internal pressure show insignificant variation with 4~i, and only the results for self-weight are plotted in Fig. 9.12. Since the meridional stress resultant Nil increa, ses near the bottom of the shell, it was expected that n~2 would also increase in that zone. This is confirmed by the numerical results, and it is seen that for the same imperfection parameters, an imperfection centered at 4~i - - 3 0 o produces maximum n~2 which is more than twice the corresponding stresses for q~i = 0. But if those changes are normalized with respect to Nll in the perfect shell at different levels, it may be concluded that n~2 is almost proportional to Nix, and only marginally affected by small variations in horizontal radius of the shell. Thus, from the point of view of practical analysis, to account for the case of self-weight of the shell it would be acceptable to consider the solution due to internal pressure that produces the same level of Nil at the center of the imperfection, as in the spherical shell under self-weight. The influence of the shell parameters, as reflected by the ratio r/t, on the stress redistribution, will next be examined for shells under internal pressure. An imperfection with parameters ~0 = - 9 0 ram, ~b0 = 20o and q~i = 0, acting on a shell with r = 28000 m m has been considered, together with thickness values t = 30, 40 and 60 rnm, leading to ratios r/t = 933,700 and 467 respectively. The results show only small variations in n~2 for these very different shells, but as the thickness increases, the bending contribution also increases. Because of the larger contribution of membrane rather than bending stresses to equilibrium, a small decrease in n~2 is compensated by a large change in m~x. The results are summarized in Fig 9.13 for different values of 4~0 in the same form as in Fig. 9.11. A reduction in the angular extent of imperfection, q~0 , produces an increase in n~2 and m~x , since the shell now has a smaller zone over which bands of hoop tension and compression can develop. For a shell with r/t = 983 and stress resultants Nil - N22 - 0.7 kN/mm in the perfect shell, values of n~2 - 1.94 kN/mm are computed for an imperfection with ~0 - -3t and q~0 - 5o; but if the r/t ratio is taken as 467, n~2 is reduced to 0.96 kN/mm.

220 T h i n - W a l l e d S t r u c t u r e s w i t h S t r u c t u r a l I m p e r f e c t i o n s : B e h a v i o r

_

i%

\

2= "~\ \

E E

1

-

3....~~,\.

z .Ic

e-

I

-1

4

I

8

I

12

I

16

i

20

,o 30

3

20

1 \i ' \

..~

"=...:.. , , ~ 9-

0

-20

0

i3 4

i

8

t

12

i

16

i

20

,o Figure 9.13" Influence of shell parameter r/t for a shell with inward imperfection; r=28000 mm, (1) t=30 mm; (2) t=40 mm; (3) t - 6 0 mm

Spherical Shells

9.3.3

221

Simplified Analysis

It may be seen from Eq. (9.10) and (9.11) that the stress and moment resultants are directly proportional to the amplitude ~0 of the imperfection and to the applied pressure q. The membrane stress resultant n~2 does not depend on the dimensions of the shell, but increase with the ratio R - r/t. However, for imperfections not smaller than 5~ the values of n~2 become almost independent of R. On the other hand, the moments m~l are directly proportional to the thickness t and to 1/R. This confirms numerical results obtained from the finite element model for a similar shape of imperfection. The influence of the imperfection wavelength, as reflected by n, cannot be explained in terms of Eq. (9.10) and (9.11), and numerical evaluations ha,ve to be performed in order to understand how they affect the stress field. Convergence of the results to the exact solution of the model should be obtained for large number of Fourier components; however, it is important to investigate the number of terms necessary to obtain good estimates of the solution. To illustrate the convergence characteristics of the solutions derived here, the example of Fig. 9.9 is again solved. The imperfection is initially modeled as axisymmetric, with n = 12. The load coefficients for the equivalent load are those represented in Fig. 9.6, and it may be seen that the largest contributions are obtained for values of j near to 12. The contributions follow a wavy pattern of alternating positive and negative values, with the extreme coefficients decreasing as j is increased. The results of n~2 and m~l for different number of terms have been plotted in Fig. 9.14, together with the results from the finite element model of the same shell and imperfection. It may be observed that the stress resultant n~2 for j - 54 shows only minor differences with the results for j - 102. However, the moment resultant m~l for j 54 has 25~ error in maximum values with respect to the converged solution j = 102. Thus, 102 modes should be used in the case for convergence of moment resultants, although convergence of membrane stress resultants can be obtained with only 54 modes.

222 Thin-Walled Structures with Structural Imperfections: Behavior

300 E

200

-

/

100-

.//-

/ 71

2

3

. \1 _ ~

-'~

4

A-

0

-200 -300 0

5~

10"

imperfection

E

15 ~

20 ~

25 ~

r

-": 1

1,--"

100

Z

\

,1r

e-100

-200

0

5"

10 ~

15 ~

20 ~

25 ~

r Figure 9.14" Changes in stress resultants for diferent number of harmonics n included in the analysis (1) I=126; (2) I=54; (3) 1-30; (4) I=102

Spherical Shells

9.3.4

223

Conclusions about A x i s y m m e t r i c Imperfections

Our conclusions on the influence of axisymmetric geometric imperfections in thin spherical shells are as follows" 9 The changes in hoop membrane and meridional bending stress resultants are directly proportional to the amplitude of the imperfection. 9 The imperfections that produce the largest changes in stress resultants are short, with angular extent of about 50. In imperfections shorter than 5~ the bending contribution becomes dominant; while in longer ones it is the membrane contribution that becomes responsible for providing equilibrium. 9 The changes in stress resultants depend on the meridional stress resultant acting in the perfect shell, but they are not significantly affected by the actual position of the imperfection along the meridian. 9 An increase in the thickness of the shell produces only small reductions in the hoop membrane stresses. From the results presented, it.follows that a short imperfection with amplitude equal to the thickness and approximately axisymmetric shape produces changes in hoop stresses of the same order as the meridional stresses in the perfect spherical shell. The significance of these high stresses depends on the particular design considered and the factors of safety adopted; but in any case, the results suggest that if imperfections or damage produce deviations from the spherical geometry, their influence on the stresses should be of concern to the designer.

9.4

LOCAL GEOMETRICAL IMPERFECTIONS

9.4.1

Stresses in an imperfect shell

Although important for the conceptual understanding of the mechanical behavior, a purely axisymmetric deviation from the perfect geome-

224

Thin-Walled Structures with Structural Imperfections: Behavior

try is unlikely to occur in practice, and in most engineering situations an imperfection introduces local curvature errors in two principal directions. But in what respect will the stress fields due to axisymmetric and local imperfections differ from each other? Flugge's studies on imperfect spheres concentrated on a banded imperfection. For local rather than axisymmetric imperfection it was assumed that %he essential features of the stress disturbance will be similar" [5]. However, it will be shown here that some essential features are very different, i.e. the linear relationship between amplitude of imperfection and stress resultants does not hold true for local imperfections. To highlight the main features of the behavior of a spherical shell with local imperfection, a shell with r = 28000 m m and t = 30 rum has been analyzed, with an imperfection of amplitude (0 - - t and central angle r - 30~ In Fig. 9.15, membrane and bending stress resultants, n~2 and m~l , are plotted. Because of the symmetric in load and geometry, the other resultants n~l and m~2 yield the same values as n~2 and m~l in Fig. 9.15. For this rather long imperfections with (0 = - t , the membrane stresses and moment resultants follow a banded pattern, much in the same way as in the axisymmetric imperfection, but with higher values. For example, the maximum values in n~2 are 57 and 53 N / m m ; and in m~lare 73 and 97 k N / m m for the local imperfection and axisymmetric imperfection, representing differences of 7.5% in n~2 and of 33% in m~l .

9.4.2

Influence of Shell and I m p e r f e c t i o n Parameters

If the amplitude (o of the local imperfection is increased from t to 3t, Fig 9.15 shows that the stress and moment changes are not linear with ~0. For example, this increase by a factor of 3 in the amplitude of the imperfection produces increments in the maximum values of n~2 of 258~ and of 4040-/; for m~l. This non linear dependence of stress changes on the amplitude of imperfection will be seen to be even more severe in short imperfections. Notice that an equivalent load model would be unable to detect this kind of non-linearity. The stress changes are strongly dependent on the wavelength of the local imperfection. This is illustrated by the studies of Fig 9.16, in

Spherical Shells

225

ii

3O mm E 0.2

1

0.1

119 o~

t't

i

k.... 9

c:

i ,,..

i

.o/

i

_ =,=nlm=~=~

.;,"

-0.1

-0.2

I

I

10"

ii

Illl

I,

I

20*

I

r

n I

30 mm ~ A

E

E

0.2

"E

0.2

.Z ~

o.1

~-J

o

E

-0.1

1 2

-.

-0.2 -0.3 0

10"

20*

r

Figure 9.15" Changes in stresses for an inward imperfection, for different amplitudes of imperfection (1) ~o - - 3 t (2) ~o - - 2 t (3) ~o - - t

226 Thin-Walled Structures with Structural Imperfections" Behavior

-

'

'

I

iAl ~

i~

4

iIi~. \ ','.~~ \\-"

~1/:

L f,:

0

4

,," /

-.",'

8

12

20

16

20 9 E

F:

!

3O

P

9

~o

1o

E E

Z

,-~,

0 I_/~i

E

'~.

i/

'4" -10

-

4fl'~ ' 4

i i

'r-" 8

i

I

I 12

i 16

20

Figure 9.16: Influence of angular extent of local imperfection (1) $0 =3; (2)r -5; (3)r =7; (4) r -10; (5) r -15

Spherical Shells

227

which the central angle is increased from 3 to 15~ The distribution of n~2 and m~l along a meridian is represented in Fig. 9.16, and it can be seen that the shape of the diagram changes with 4)o. The banded pattern in n~2 illustrated in Fig. 9.15 is seen to be representative of long imperfections and is similar to that found to occur in axisymmetric imperfection. But in short imperfections the sign at the center is reversed, with an inward local imperfection producing compressive rather than tensile stresses. This effect is quite different from what was observed in axisymmetric imperfection. Changes in m~l are also seen to occur in short local imperfection. It is important to study how this maximum values of n~2 and m~l change with 4)o. The variations are plotted in Fig. 9.17. A shell with ~0 - - 3 t shows maximum values of n~2 and m~l at approximately q~o - 10~ and they show a strong decrease for q~o < 10. It must be noticed that although a similar behavior was detected in n~2 for axisymmetric imperfection, in the local imperfection m~l also shows a maximum depending on 4)o. To understand the existence of such maximum values, it must be considered that a local imperfection which extends over a very small area would remain unnoticed by the shell, and would not change the stress distribution. On the other hand, if the imperfection affects a very large area, the curvature errors are small and small values of stress resultants integrated over a large area would provide the necessary contribution to equilibrium. A maximum value is reached between both situations, that is, for imperfections of intermediate length. Fig 9.18 shows the stress changes for shells with different R = r/t ratios. An increase in t reduces the parameter R, and affects the moments m~l. The results indicate that the membrane stress resultant remains almost unaffected by a reduction in R from 933 to 467. However, the small modifications in n~2 are accompanied by large changes in m{1 , which is inversely proportional to R. From the results, the relationship between m~land R is non-linear. The dependence of maximum stress changes on the angular extent of the local imperfection has also been investigated, and the results are summarized in Fig. 9.18. As shown previously in Fig. 9.17, maximum values of n~2 and m~l are identified for certain values of qS0.

228

Thin-Walled Structures with Structural Imperfections: Behavior

E E

1.0

1

Z

2 '("~

Od O4

0.5

%

Jr---.. ~

3A,

,, 9

0

I

i

;

I

4

I

8

12

".~ %.

~.-. I

I

16

2O

30

'E'E2o-

;~

,..~,

f

\

,,.9 0

0

, 4

I 8

= 12

"'*"f .... z ~ 16 20

0o Figure 9.17" Influence of angular extent of local imperfection on maximum stress resultants (1)~o=-3t; (2) ~o--2t; (3) ~o=-t

Spherical Shells

229

/

.,i-.--.,~1

.

3

E E Z

~ \

. . . . . . . .

|

..

--

I

,.~."

OJ

l...~i.~ .\

s

y"

..),r

, ,,|.

8

4

]"

, 12

, 16

, 20

_

30 "

'E'

''

"

3

20

E E

E

f]

lO

J"

Z IL

9

4r

E

0

+J 9

-10 0

I

4

I

8

/S

~"I

12

I

16

i

20 0

Figure 9.18: Influence of shell parameter r/t for the shell with inward local imperfection (1) t=30 mm; (2) t - 4 0 mm; (3) t=60 mm

230

Thin-Walled Structures with Structural Imperfections: Behavior

9.4.3

Simplified Analysis

Next, convergence of the equivalent load solution for a local imperfection is studied. For the shell studied in Fig. 9.15, with an imperfection defined by n = m = 12, the results are plotted in Fig. 9.19. Again, convergence is obtained with 54 modes for n~2 and 102 modes for m~l. Studies conducted for other imperfection and shell characteristics indicate that the recommended number of harmonics in each direction should be 9n/2 for membrane stress resultants, and 17n/2 for moments resultants, where n defines the wavelength in each direction. Because the present solution is based on a first-order equivalent load model of the imperfection, it has some limitations associated with the model considered. The main limitation is that the model yields a solution in terms of stress resultants that is directly proportional to the amplitude of the imperfection. It has been shown that this is reasonable for banded imperfections, even for ~0 = 3t; but it is only valid for long-wave local imperfections. For local imperfections in which the curvature errors are severe, that is short imperfections (r < 10~ with amplitude larger than the shell thickness, the response is non-linear with the amplitude of imperfection, and the equivalent load solution is only a first approximation.

9.4.4

Comparison Between Axisymmetric and Local Imperfections

The results in Section 9.3 for axisymmetric imperfection, and in Section 9.4 for local imperfection show different stress changes for two possible families of imperfections in spherical shells. One of the main features highlighted by the results is the dependence of maximum stress resultants on the imperfection characteristics, as reflected by the amplitude ~0 and the angular extent r Results are plotted in Fig. 9.20, to compare maximum values as a function of imperfection parameters for internal pressure q - 0.05 N/mm 2. The membrane stress changes indicate that for short wavelengths, the axisymmetric imperfection yields higher values than the corresponding local imperfection. For approximately r - 1 0 - 13~ the curves intersect, and for r > 140 it is the local imperfection that produces higher values. If the bending moments are compared, again the axisymmetric imperfection yields higher values for short wavelengths. For r = 5 - 7o the maximum due to a local

Spherical Shells

231

i

imperfection

~:

50

~'~ ~

1

25 -

iJ ~ ~ " ~ ~

3

10 ~

20"

0 -25 -50

0

15 ~

25 ~

N

5O E

5"

_

1

2

25

o -25

-50

_j 0

l 5~

10"

15 ~

20 ~

25 ~

Figure 9.19" Changes in stress resultants using simplified solution (1) I=126; (2)I=54; (3)I=30

232

Thin-Walled Structures with Structural Imperfections" Behavior

imperfection is equal to that given by the axisymmetric imperfection. A peak value of m~l is obtained in local imperfection, which is not present in axisymmetric imperfection. The solid curves in Fig. 9.20, for values of ~0 - - 3 t , - 2 t and - t , show the maximum values of n~2 and m~l for axisymmetric imperfection and local imperfection. For the design engineer, it would be important to have some more general picture of maximum stresses that can be expected for a given shell, without having to specify what particular imperfection produces it. For a shell with a maximum deviation of 3t, one could thus obtain maximum values of n~2 and m~l that can occur. If the wavelength is restricted to r > 5o (that is, if very short imperfections are excluded), it can be seen in Fig. 9.20 that n~2 and m~l show clear maximum values. For n~2 the maximum is associated to the axisymmetric imperfection, whereas for bending moments the maximum is due to the local imperfection. Although both values do not occur for the same imperfection, they represent bounds on stress redistributions for the cosine shape imperfection. The results of Fig. 9.20 have been obtained for a shell with r/t = 933. A similar picture can be found for thicker shells. Results of maximum membrane and bending stress resultants for a given value of amplitude and for different spherical shells have been plotted in Fig. 9.21. For example, a shell with r/t = 500 under uniform internal pressure q - 0.05 N / m m 2 having an unspecified imperfection with maximum amplitude equal to the shell thickness, would yield maximum values of n~2 - 0 . 3 7 k N / m m and m ~ l - - 15 kN/mm. Notice that for the range of shells studied, linear relations exist between maximum stress and moment resultants and the slenderness of the shell, R. The dependence on ~o/t, on the other hand, is non-linear.

9.4.5

Conclusions about Local Imperfections,

In Section 9.4, the influence of localized imperfections on the stresses in thin spherical shells has been investigated. Although the behavior for this kind of imperfection is similar to the banded imperfection studied in Section 9.3, in the sense that both produce concentrations of stresses that only affect the zone of imperfection, there are some distinctive features which are summarized as follows:

SphericM Shells

233

~o = "3 t ~o = -2 t r

9

E E Z

i

n O4 O4

,1r

r

0

I

5

0

I

10

,

I

n

15

20

,o r ~

E E E E

t t I t

t t I

25.0

\

4 I

d

PJ

9

Z a

9

,1r

E

12.5

0

5

10

15

20

(~0

Figure 9.20" Comparison between local and axisymmetric imperfection

Thin-Walled Structures with Structural Imperfections" Behavior

234

q = 0.05 N/mm

M

E E

2

-

If

.E +

30

E E E

==

Z ,

Z

" 20

J

Od t~

|

c-

,

4~

E

1"*'l~l*2Z ~ 0

~

~

I'.'.

10

"tl

~

~

,=

I""

0 400

I

I

I

I

I

500

600

700

800

900

1000

R-r/t Figure 9.21" Maximum values of changes in stress resultants for shells with cosine shape imperfection 9 The changes in membrane and bending stress resultants are related to the amplitude of the imperfection in a non-linear way. 9 For a given amplitude of local imperfection, the largest changes in stress resultants occur for 4)o - 5 - 10~ 9 Contrary to the case of banded imperfections, in the local imperfection a maximum value is identified as a function of the wavelength. 9 An increase in the thickness of the shell produces only small reductions in the hoop membrane stresses. 9 For a given amplitude and extent of imperfection, a local imperfection yields higher values of stress resultant than the banded imperfection in the range of long wavelengths; whereas the reverse holds true for short imperfections.

Spherical Shells

235

On the basis of the results, it seems reasonably to establish bounds on stress resultants for a given shell and maximum amplitude of imperfection. These bounds have been defined in this chapter by straight lines that approximate the relations between maximum stresses and shell slenderness. It must be noticed that the model used in the present analysis is restricted to linear strain-displacement relations, and as such the results represent upper bounds to stresses in imperfect shells. The influence of kinematic non-linearity on the response of imperfect spherical shells will be considered in Section 9.5.

9.5

9.5.1

N O N L I N E A R B E H A V I O R OF GEOMETRICALLY IMPERFECT SPHERES S t r e s s e s in an I m p e r f e c t Shell

Very thin shells are known to have geometrically non linear behavior, and this is also expected to occur in very thin imperfect spherical shells. The influence of geometric non-linearity depends on the load level considered; for low values of pressure the response should be almost linear in a load-displacement diagram, and highly non-linear for pressures approaching the yield condition of the material. A finite element model of shells of revolution, accounting for nonlinear strain-displacement relations, has been used to evaluate level of stresses in imperfect spheres, and the results are presented in this section following Ref. [3]. A particular case is first investigated to understand the differences in the displacements and stresses when linear and non-linear assumptions are used. The imperfection considered is defined by r / ~ o - 200 and 80 - 5o; this is an imperfection of intermediate extent and large amplitude. Results are presented for two slenderness ratios: r / t 1,000 and 200, which may be considered as representative of the nuclear power and the gas containment applications. Fig. 9.22 shows the variations of the out-of-plane displacement component w (normalized with respect to the linear solution w0) when the pressure is increased, and each curve corresponds to a different pressure level which is char-

236 Thin-Walled Structures with Structural Imperfections: Behavior

U3

r ~-

U30

1000

20 ~% ,,

-% %%

%%

%% %

0

U3 U30

0.5

1

,I ;- oo

0

0

!

0.5

I

1

r162 Figure 9.22" Linear and non-linear displacements for local imperfection (1) linear solution; (2) Cro - 20 N/mm2; (3) Cro - 50 N/mm2; (4) ao 100 N/mm2; (5)ao - 150 N/mm2; (6) ao= 200 N/mm2

Spherical Shells

237

acterized by cr0. For the thinner shell, the ratio w/wo at the center of the imperfection is reduced from 20.7 in the linear analysis to 7.18 for a0 = 200 N / m m 2. Thus, for this pressure level, the linear displacements are overestimated by a factor of 2.9. On the other hand, for a shell with r/t - 200, the nonlinearity has the effect of reducing w/wo from 3.66 to 2.89, with an error of 22% in the linear solution. The results presented show a non-linear dependence of displacements and stresses on the pressure considered, and as such, it is necessary to limit the values of pressure to practical values. A maximum stress of cr0 - 200 N / m m 2 (q - 400t/r in N / m m 2) has been used to obtain the values of this section. Such a stress level corresponds to about 40% of the yield stress for the type of steel that is usually employed in pressure vessels. The errors in displacements are also reflected in the stress distributions plotted in Fig. 9.23. The membrane stresses crm have been computed as the equivalent von Mises stress ~--~

1

[O"121-~-0"222-~-(fill-0"22)2]}"

(9.19)

at the mid-surface of the shell, where the bending stresses are zero. This membrane stresses are normalized with respect to ~r0, the linear solution. A strong nonlinearity is shown in Fig. 9.23 for r/t = 1000, in which not only the values but also the profile of the stress field is seen to change at the higher levels of pressure. These changes are evidence of a redistribution of stresses that takes place in the shell from the initial linear response, and show the trend that the imperfect shell has to behave as a perfect shell at high pressure values. If the bending stiffness of the shell is increased to r / t = 400, as in Fig. 9.23, the behavior follows the same general picture as in the thin shell, but the changes are not so important. The influence of the bending effects is more clearly appreciated in Fig. 9.24, in which the equivalent von Mises stress crm+b is plotted at the inner or outer surface of the shell, depending on which is the most highly stressed. The curves again show a trend towards stresses in the perfect shell, but it may be seen that the reduction in the bending stresses is not as important as with the purely membrane stresses.

238 Thin-Walled Structures with Structural Imperfections: Behavior

~m

= 1000

~o

..~1

-

:.-.,. /i ~

eeoellle.q..- I ~ dpeJ

i ~,p# } ''' '" ''B ~ 0

~,m

"~ ",,.\

I 0.5

r _ 200 t

(3' o

0

0

r 1

I 1

4

.

I

I

0.5

1

r162 Figure 9.23: Membrane equivalent stress ratio. Key as in Fig. 9.22

Spherical Shells

239

{~'m + b

"---.,.

Oo ~ ",, 4 I"

7

....... -...

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

1

r=

"'.,,,

,,.. " * * ,

2

- ......

".,, ~ 99

"........

,, . . . . o%.q,Do ~ o

,,,.~,q,

.,...~2

..

.,.

%

~..13 _ _

.,.,,,.~,,,~

...:.~..?:.~

s

/

..........:::::~::.?. .....

1000

"%.

I

...."

I,

0

,,

0

i

0.5

.

I

1

,/r

i

(~m+b

Go

r_

4

t

200

3

1

2

,..,:..

6

1

0

I 0

0.5

.

I 1

r162

Figure 9.24" Tota,1 equivalent stress ratio. Key as in Fig. 9.22

240 Thin-Walled Structures with Structural Imperfections: Behavior

9.5.2

Influence of I m p e r f e c t i o n P a r a m e t e r s

In the numerical examples of the previous section, an inward imperfection was assumed. In the linearized solution, the sign of the imperfection affects only the sign of the changes in displacements and stresses. However, for the higher values of pressure considered in this section, it is expected that a change in the sign may considerably affect the non-linear shell response. To understand the differences between inner and outer imperfections, results have been obtained for the same absolute values of r and r/~o of Fig. 9.23 and 9.24, but for which the imperfection has the effect of increasing the radius of the shell. Curves of ~rm/ao and a,~+b/~ro as a function of the angle r are plotted in Fig. 9.25, and it may be observed that the changes in stresses with respect to the perfect shell are significantly smaller than the corresponding stresses associated to the inner imperfection of Fig. 9.23 and 9.24. Furthermore, the behavior of the shell is not so much affected by the non-linear terms in the kinematic relations, and the bending stresses in Fig. 9.25.b are almost linear. An important parameter that should be considered, is the extent of the imperfection, represented in this case by the angle r Maximum stresses am+b have been plotted in Fig. 9.26 as a function of r for three different slendernesses, and for a pressure given by a0 - 200 N / m m 2. For the very thin shell, r/t = 1000, the linear results are strongly dependent on r while r does not influence the values to a great extent when the non-linear terms are included in the analysis. For that particular shell, the stresses are limited between

max 1.5 < ~rm+b/ao < 2.5

for values of 80 < 10~ Finally, the slenderness adopted corresponds to two types of applications of spherical shells. The results show that a non-linear analysis should be employed for high levels of stresses in very thin applications; whereas linear results would be a good approximation for r/t - 200. Extensive parametric studies on non-linear response of spherical shells with imperfections may be found in Ref. [3].

Spherical Shells

0 13 0

241

lr

- = 1 000 t

0.5

0 O' o

-

r_ t

1

r

200 1

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

4

0

I

I

0.5

1

r162

Figure 9.25: Equivalent stress ratio for outer imperfection (1) am+b linear; (2) cr.~ linear; (3) ~rm+b non-linear; (4) ~.~ non-linear, for a o 20ON/ram 2

242 Thin-Walled Structures with Structural Imperfections: Behavior

max

C~'m+ b O' o

o ~ ==Ibm% /

9

.

I

%

I

0

,

i

1

2

I

5

%

I

7.5

I

1

,0

Figure 9.26: Maximum total equiva.lent stress ratio for different shell slenderness. (1) r / t - 1000, linea.r; (2) r / t - 500, linear; (3) r / t = 200, linear; ( 4 ) r / t - 1000, non-linear; (5) r / t = 500, non-linea, r; ( 6 ) r / t = 200. Non-linear is computed for a0 - 200 N / m m 2

9.5.3

D e n t e d Imperfections

Just one imperfection profile has been investigated in the previous sections, and this takes the form of a cosine shape in one direction (axisymmetric imperfections) or in two directions (local imperfections). A cosine shape is rather smooth, in the sense that the slope is continuous at all points and the changes in geometric curvature are also continuous except at the end of the imperfection. In this section we report results from Ohtani, Koguchi and Yada [13], for local imperfections with a dented shape, that have more severe changes in slope and curvature. The geometry considered in Ref. [13] is described by the form

1- or )

(9 20)

for the zone near to the center of the imperfection, i.e. 0 _< 0 _< 00; and by

Spherical

243

Shells

0

-t

-(Oo+Ol) -Oo O:O Oo (0o+01) Circumferential angle 0 0

0=0 -t

I

!

I

-r 0 r Meridionai angle 4)

Figure 9.27: Errors introduced by a dented local imperfection

-~'o(1-0-0~ for 00 _< 0 _< 00 + 01. The parameters required to identify a specific imperfection are the amplitude, ~o; the angle of the zone where the imperfection is constant in the circumferential direction, 00; the angle where the change in position is linear, 01; and the angle in the meridional direction where the change in position is linear, r The actual changes in the geometry introduced by such imperfection are illustrated in Fig. 9.27. Notice that this imperfection shape has to be added to the geometry of the perfect shell, as indicated in Eq. (9.1). The analysis was performed by a direct approach, and for a shell with r / t - 280; r - 9850 m m ; t - 35 m m ; ~o - 20 m m and 0o - 01. N/mm2; The material wa,s modeled using modulus E - 2.058 •

244 Thin-Walled Structures with Structural Imperfections: Behavior u - 0.3; yield stress ay - 6.86 x 105 N / m m 2, with hardening slope Ep - 20.58 x 105 N / m m 2. The pressure applied is 2.9 N / m m 2, leading to primary stresses of 414 N / m m 2 (about 60% of the yield stress); and self weight was also applied. Notice that the level of stresses considered in Ref. [13] is more than twice that considered in Ref. [3], and this means that the influence of non-linearity will now be greater. Furthermore, the dented imperfection with slope discontinuity should induce higher levels of stress concentrations. Typical results are presented in Fig. 9.28, in terms of pressure versus strain curves. Outer and inner surfaces of the shell are identified by super-index (o) and (i) respectively, and are given for strains in the meridional and in the circumferential directions. Other results in Ref. [13] tend to support that the angular extent of the imperfection, 00, does not affect the changes in non-linear strain due to the imperfection. Comparisons with banded imperfections indicate that "... if the angular extent of 00, 01 of local imperfection widens to a certain magnitude, then the difference in the stress-strain relation between the local and the banded one can be neglected... Thus, it is inferred that the tendency of the strain concentration for these kinds of imperfection can be characterized by the initial shape w(r of the imperfect meridian and the maximum amplitude of imperfection" [13].

9.5.4

Conclusions Concerning Non-Linear Behavior

The main conclusion concerning the influence of geometric non-linearity on the response of imperfect spherical shells are: 9 For very thin shells (r/t > 500), a linear response yields extremely conservative values of stresses and displacements. 9 For thin shells with r/t - 200, a linear response seems to be reasonably accurate for engineering purposes, provided the errors introduced by them do not present sharp changes in slope. 9 For imperfections with changes in slope at some points, there are significant differences between linear and non-linear behavior in shells with r/t >_ 200.

9

Pt

<1) 0~q~ 9

i.-=.

O

E/3

rj ~ E/3

o?

i-.a.

~

r.j3

r

.

~

Crl

O

II

II CD

G)

O

C) (1)

=o c-

9 ~

X

Membrane stress (3m [Mpa]

I

w

l.-,. :3

r.~

r

=

!

i

-

=.

Crl

b

O

II

II G)

<:D O

C) tl)

Membrane stress O m [Mpa]

t'~

C3

~

Thin-Walled Structures with Structural Imperfections: Behavior

246

9 Internal imperfections produce higher levels of stress changes than outer ones; and the former show a stronger geometric nonlinearity than the latter. 9 The maximum non-linear stresses are not so sensitive to the wavelength of the imperfection. 9 The non-linear terms have a stronger influence on the behavior as the amplitude of the imperfection increases.

9.6

9.6.1

THICKNESS SPHERES

VARIATIONS

IN

S t r e s s e s in a Shell w i t h T h i c k n e s s C h a n g e s

In this section we study the stress changes in a metal spherical shell with elastic properties and dimensions given in Fig. 9.7. The maximum internal pressure considered is q - 0.5 N / m m 2, for which a membrane field of N22 = 7 k N / m m occurs in the shell of constant thickness. With reference to Fig. 9.5, the parameters that define the change in the thickness are r = 2, 3 and 5~ with reductions in thickness from t = 30 m m to 20, 15 and 10 mm. The results from the geometrically non-linear analysis are plotted in Fig. 9.29 to 9.31, and show values of membrane stress resultants in the circumferential direction N22, and bending moment resultants in the meridional direction, Mll. The curves in Fig. 9.29 are computed for central angles of r = 2~ and for three different thickness reductions. There is a concentration of N22 at the section where the change in thickness occurs, with alternate bands of tension (outer region) and compression (inner region of reduced thickness). For to - t / 3 , N22 increases by 45%, while for to - t/2 changes in N22 are of 27%; and for to - 2t/3, N22 increases by 18%. These changes in N22 are not severely affected by the area of the thickness reduction, as reflected by angle r This may be seen from Fig. 9.29.a, 9.30.a, and 9.31.a. For to = t/3, the moments Mll increase from zero to 2 k N m m / m m . This change in moment resultant produces stresses that are about 10~ of those associated with N22. This seems to be a reasonable result, since the perfect shell has a very low bending stiffness, and this is further reduced

247

S p h e r i c a l Shefls

11 10 9

==

"

3

8 -

2

1

7 Z

.~..~..'-r'~...

~n-~

r-,-

.

.

.

.

.

6 5 I

I

!

I

!

1

2

3

4

5

6

10

,-ff

E E E

Z

3

4

-2 -4

I

I

I

I

I

1

2

3

4

5

6

Figure 9.29: Non-linear behavior of spherical shell with step variation in thickness, for 4)o - 20 (1) to - t / 3 ; (2) to - t / 2 ; (3) to - 2t/3; (4) Perfect shell

248

Thin-Walled Structures with Structural Imperfections: Behavior

11 10

,-ff

B

3

m

E n

Z Z

0

t

I

I

I

I

1

2

3

4

5

6

I0

E E E

Z

4

-

3

2 o -2 -4

0

t

I

I

t

t

1

2

3

4

5

6

Figure 9.30" Non-linear behavior of spherical shell with step variation in thickness, for r - 3~ Key as in Fig. 9.27

9

o-q

i-,.,~ ~

O

II

r.~

O

I,--,~ ~

9

i,,,.,~ ~

i.,.,,~ ~

i-,,.~ ~

i,,,.~

dl)

=r"

i.-,,,~ ~

m-'

~

=

I,.,.,i ~

~176

I,~

~'L'q

9

-e-

O'J

01

0

-

_

"%!A

~iI

ii

/

M,, [kNmm/mm]

-e-

t31

O~

PO

0

!

i

i

!

t

i !

|

,

|

--~

I~_

,!

I,

i

,

.

sl

/

li =,J

i!

I t

o

a i

i

i

/

i /

:/"L. i; ! I

9

/ #

.".,. .1~

/

N22 [kN/mm]

ch

250

Thin-Walled Structures with Structural Imperfections" Behavior

by the local reduction in t. Thus, equilibrium to the internal pressure is basically provided by changes in the membrane stress resultants. A summary of the above results is presented in Fig. 9.32, in the form of maximum values of changes in N22 and Mix as a function of the angle r for the case in which t0 - t/3.

9.6.2

Linear versus N o n - l i n e a r Analysis

The results presented in the last section were obtained using a nonlinear shell theory. But the first type of analysis that one would attempt would be a linear one, and it is important to see how the assumptions of non-linearity may affect the results for these very thin shell. Figure 9.33 shows results for a shell with thickness reduction of to - t/3 and for several values of r These curves should be compared with those in Fig. 9.29 to 9.31. It may be seen that the changes in hoop membrane stress resultants are slightly underestimated, but the moments Mll are overestimated in the linear solution by as much as 100%. Fig. 9.32 also includes linear results for comparison with non-linear ones. The linear model does not produce bounds to the solution in N22, since for small angles r < 4~ it overestimates the stresses, while for r > 4~ the model underestimates them. These errors are, however, of the order of 10%. The bending moments, on the other hand, are consistently overestimated by the linear solution by approximately 100%. Linearized bending stresses are thus not reliable in this model.

9.6.3

C o n c l u s i o n s about T h i c k n e s s C h a n g e s

From the results presented for shells with sudden reductions in thickness, it is possible to draw some conclusions" 9 A redistribution of stresses occurs as a consequence of the reduction in thickness, with a decrease in the carrying capacity of the weakened zone that is compensated by an increase in stresses in the vicinity of the stress change. 9 Tensile stresses are induced at the edges of the area affected, with a stress concentration that affects about the same radius as the reduction in thickness.

Spherical Shells

251

0.66 i

E

t

- - - 7 - - -

0.66 t

Z Z

':;

04 04

,,,,

1

0

0.33 t

-

0

' 1

I 2

I 3

I 4

I 5

6 0

5 -

4-

//

3-

,;~

2 t

/f

///

066t

1t0

0

t 1

2

3

4

5

Figure 9.32: Maximum changes in stress resultants

6

252 Thin-Walled Structures with Structural Imperfections" Behavior

11

~o= 3~

-~o=Z \

10

to= 0.33t

0

E

Z Z

Od Od

,4 "4 1

0

I

1

I

2

I

3

I

4

10

E E E

4

.~

2

=_-

o

Z

5

,

6

to= 0.33t i ~ O ~- 5 e

E

L.___U

I

-2 -4

0

1

2

3

4

5

6

Figure 9.33" Kinematic linear results for different thickness changes

Spherical Shells

253

9 This increase in tensile stresses may be as high as 45% if the thickness is reduced to one third of its original value. 9 Changes in stresses that are compressions arise at the inner circular zone; however, for the internally pressurized shell, the net stresses in this zone are tensile. 9 The reduced thickness is also accompanied by an increase in bending moments. However, the stresses associated with them are only 25% of those changes due to the membrane action. 9 A linear shell theory predicts unrealistically high levels of bending action, coupled with membrane stress resultants which do not bound the correct solutions. 9 Some care must be exercised when using linear solutions of spherical shells with thickness changes.

254 Thin-Walled Structures with Structural Imperfections: Behavior

References [1] Calladine, C. A., Structural consequences of small imperfections in elastic thin shells of revolution, Int. J. Solids and Structures, 8, 1972, 679-697. [2] Croll, J. G. A., Kaleli, F., and Kemp, K. O., Meridionally imperfect cooling towers, J. of the Engineering Mechanics Division, ASCE, 105 (EM5), 1979, 761-777. [3] Flores, F. G. and Godoy, L. A., Linear versus non-linear analysis of imperfect spherical pressure vessels, Int. J. Pressure Vessels and Piping, 33, 1988, 95-109. [4] Flores, F. G. and Godoy, L. A., Finite element applications to the internal pressure loadings on spherical and other shells of revolution, Chapter 9 in: Finite Element Applications to Thin-Walled Structures, Ed. J. W. Bull, Elsevier, London, 1990, 259-296. [5] Flugge, W., Stresses in Shells, Springer Verlag, Berlin, 1963. [6] Godoy, L. A., A simplified bending analysis of imperfect spherical pressure vessels, Int. J. Pressure Vessels and Piping, 27, 1987, 385-399. [7] Godoy, L. A. and Flores, F. G., Stresses in thin spherical shells with imperfections. Part I" Influence of axisymmetric imperfections, Int. J. Thin-Walled Structures, 5, 1987, 5-20. [8] Godoy, L. A. and Flores, F. G., Stresses in thin spherical shells with imperfections. Part II" Influence of local imperfections, Int. J. Thin-Walled Structures, 5, 1987, 145-155. [9] Godoy, L. A. and Flores, F. G., Thickness changes in pressurized shells, Int. J. Pressure Vessels and Piping, 55, 1993, 451-459.

Spherical Shells

255

[10] Heyman, J., On shell solutions for masonry domes, Int. J. Solids and Structures, 3, 1967, 227-241. [11] Ohtani, Y., Koguchi, H., and Yada, T., Theoretical analysis of strain concentration for various shapes of distortion at weld in a pressure vessel, Trans. of the Japan Soc. of Mechanical Engineers, 55-520, 1989, 2516-2520. [12] Ohtani, Y., Koguchi, H., and Yada, T., Nonlinear analysis of strain concentration occurred at welded joint with initial distortion in a spherical pressure vessel, Int. J. Pressure Vessels and Piping, 45, 1991, 3-21. [13] Ohtani, Y., Koguchi, H., and Yada, T., Nonlinear stress analysis of thin spherical vessels with local non-axisymmetric imperfections, Int. J. Pressure Vessels and Piping, 45, 1991, 289-299. [14] Pedersen, P. T. and Jensen, J. J., Buckling behavior of imperfect spherical shells subjected to different load conditions, Thin- Walled Structures, 23, 1995, 41-55. [15] Sanchez Sarmiento, G., Bergman, A., Lotorto, A., and Luxwurm, H., Tecnicas computacionales para el analisis tensional de cascaras de contention esfericas, Revista Argentina de Ingenieria Estructural, 1(3), 1983, 1-19. [16] Sanchez Sarmiento, G., Idelsohn, S. R., Cardona, A., and Sonzogni, V., Failure internal pressure of spherical steel containment, Nuclear Engineering and Design, 90, 1985, 209-222. [17] Yada, T., Koguchi, H., and Ohtani, Y., A consideration of strain concentration at a welded joint with angular distortion in a thin spherical pressure vessel (Influence of the position of the distorted joint), Trans. of the Japan Soc. of Mechanical Engineers, 55-510, 1989, 326-332.

C h a p t e r 10 C Y L I N D R I C A L SHELLS WITH IMPERFECTIONS 10.1

INTRODUCTION

Thin-walled cylinders are used in many engineering applications, including silos, tanks and containers. They are also widely used in pressure vessels and many other structures in marine, mechanical, and aeronautical engineering. A cylindrical shell is a highly optimized shape, in the sense that under axial load or lateral pressure it carries loads exclusively by membrane action. With imperfections in the geometry or in other properties, the mechanism of resisting loads is modified so bending as well as membrane actions are required. In this chapter we investigate changes in stress resultants due to a variety of imperfections affecting cylindrical shells. In Appendix B, we review some basic aspects of the equations of cylindrical shells, which will be of value to interpret the following formulations and results. Models and results for constructional errors that induce geometrical imperfections in cylindrical shells are presented in Section 10.2, reviewing mainly equivalent load methods that lead to explicit solutions. The influence of denting damage in metal cylinders is discussed in Section 10.3. Silos are the subject of Section 10.4, in which out-of-circularity and bulge-type imperfections are considered. Section 10.5 contains studies in cylinders with combined imperfections in modulus and in the geometry. Such studies are useful for the simplified modeling of reinforced concrete problems with smeared cracking. 256

Cylindrical Shells

257

Section 10.6 and 10.7 are concerned with combined geometric deviations and discrete cracks. In the former, numerical studies are presented, while experiments are reviewed in the latter. The influence of thickness changes is studied in Section 10.8. Conclusions are presented in Section 10.9.

10.1.1

Short L i t e r a t u r e R e v i e w

Studies on the stresses associated to the out-of-circularity in cylinders may be found as early as 1908 in a paper by Marbec [21]. "A tube that is not quite circular" was investigated by Haigh in 1936 [17]. This was followed some twenty years later by studies of Carlson and McKean [6]. The influence of imperfections in thin-walled cylindrical shells was discussed by Burmistrov [3] in 1949, and by Clark and Reissner in 1956 [7]. Most of the interest in this field was stimulated by the design and safety assessment of pressure vessels. There was a good awareness of the importance of sudden changes in the slope of the meridian, which usually happened at the junction between cylindrical segments and other shell components [6] [26] [27]. These are not geometrical imperfections as the ones considered in this book, because they are part of the geometric design of a. pressure vessel; however, they are illustrations of how stress concentrations arise due to the lack of smoothness in the meridian of a shell of revolution. Stresses in cylindrical shells that are grooved internally in the circumferentia,1 direction were investigated in Ref. [19]. Recent interest in the behavior of cylinders with imperfections is reflected in Ref. [11], [15], and [14], which investigate the interaction between geometric and intrinsic imperfections. Damage in tubular members, in the context of off-shore platforms, has been studied in Refs. [30] and [31]. Other fields of interest include silo structures with imperfections, as reported in Refs. [24] and [13].

10.2

INFLUENCE OF MERIDIONAL IMPERFECTIONS

The governing equations of cylindrical shells under static loads are given in Appendix B. In this section we consider the equivalent load

258 Thin-Walled Structures with Structural Imperfections" Behavior

Figure 10.1" Cylinder with axisymmetric imperfection for simple imperfections. In the case of circular imperfections, the slope is continuous at all points, and different levels of analysis are presented: a ring analysis and a cylinder analysis.

10.2.1

Equivalent Load

Consider a cylindrical shell with imperfections in the meridian, as illustrated in Fig. 10.1. Under the action of external pressure and axial load, the shell deflects and increases the amplitude of the deviations with respect to the perfect cylinder. A simplified equivalent load model has been found to be a convenient way to model these types of problems, in which the equivalent load is given by the pressure

p : - Nl1~~

(10.1)

We should consider specific shapes of imperfections to compute the errors in the geometric curvature X~ Let us study an imperfection with the shape formed by four circular segments, as shown in Fig. 10.2. The changes in curvature introduced by such imperfection were studied in Ref. [8] and are

259

Cylindrical Shells

Figure 10.2: Equivalent load for an imperfection modeled by four circular arcs

(10.2)

1 --(h/4)2 for the part of the imperfection where h/4 <

~01

--

for-h/4 < x~ < - h / 2 and h/4 <

10.2.2

--

~---'~'~

X1 <

-h/4; and

(10.3)

X 1 < h/2.

Ring Solution

Croll and Kemp [8] modeled the shell in a simple way as if the behavior was similar to a ring, with the same radius as the local radius of the shell considered. This led to the following stress resultants r

M~-0

(10.4) (10.5)

This model was specifically developed for the analysis of cooling tower shells, and the emphasis is on the membrane action, and as such, it provides upper bounds to changes in N22.

260

T h i n - W a l l e d Structures with Structural Imperfections: B e h a v i o r

10.2.3

Bending

Solution

Again, let us consider an imperfection in the form of four circular arcs. Use of the equivalent load into the bending equations of a cylinder leads to the following solution [12] 9

Ua - -8

~--~N1112 "~ n (e/3xl - e -/3xl) sin (/~Xl) + g (~Xl + {~-~xl) COS(~Xl) ]

(10.6)

where _

_

3 ( 1 - v 2)

(10.7)

(rt) A - - 2 e -zh/4 sin (/3h/4) + e -zh/2 sin (/~h/2) B - - 2 e -zh/4 cos (flh/4) + e -zh/2 cos (/3h/2)

(10.8)

The hoop stress resultants become t N2*2 - E - u 3

F

(10.9)

and the meridional moments can be computed as

02U3 M;a--D

Ox--~

(10.10)

A similar procedure was followed in Ref. [16] but it is reviewed in Chapter 11 because it was specifically oriented to cooling towers.

10.2.4

Case Study

The formulation based on an equivalent load and a direct finite element, solution have been employed to model a cylindrical shell with a geometric imperfection. The shell has also been tested in a laboratory, as explained later in this chapter, and the dimensions for the analysis are r -- 117.5 m m ; r / t -- 38.3; r / ~ - 21; r / h - 1.5; u - 0.35; and N~I - 1 N / m m . In the direct approach, a cosine imperfection profile was assumed; while in the equivalent load analysis, a series of circular arcs were considered.

Cylindrical Shells

261

1.25 1.00 ,---

i2]i

0.75 0.50 0.25 0 -0.25 -0.50 -0.75 i

0

i

i

0.25

I

I

0.50

_x, h

1.00 i E

0.75

E

0.50

Z

0.25 0 -0.25 -0.50 -0.75 -

I

I

0.25

I

i

0.50

i

h Figure 10.3: Metal cylinder with geometrical imperfections

262 Thin-WMled Structures with Structural Imperfections: Behavior The presence of the imperfection in the meridian produces an outof-balance moment M0 given by Mo - N l l ~

(10.11)

The results in terms of changes in hoop membrane stress resultants and meridional bending are shown in Fig. 10.3, and it may be seen that the differences between direct and equivalent load analysis are very small, more so if we consider that the geometry of the imperfections in each case are not identical. The pattern of changes in hoop membrane stress resultants, N~2 , and in bending moment resultants, M~I , are very similar to those already obtained in previous chapters for shallow and for spherical shells. The results show bands of tension and compression that provide an overall resistance to the out-of-balance moment. The out-of-balance moment is equilibrated by hoop bands of N~2 , as explained in Chapter 12, and by bending M~I. In the present case, in which there is symmetry of revolution in the geometry and in the response, the equilibrium condition can be established at each meridian. The proportion of M0 carried by flexural action in this rather thicker shell is about 20%.

10.2.5

Other Imperfection Shapes

In Ref. [16] the equivalent load method is used so that the pressure depends on a shape parameter. With this parametric representation it is possible to model several imperfection profiles using a single equation. The changes N~2 are highest in the imperfection formed by circular arcs in short imperfections (h/x/~ < 6), and in the imperfection formed by straight lines in the longer ones (h/x/~ > 6). Maximum values occur for h / v ~ - 4.2,5. The bending contribution M1"1, on the other hand, is maximum in imperfections formed by straight lines, with peak values occurring for h/v"-~ - 2.43. Again, as we found in spherical shells, there is a maximum in the stress contributions as a function of the extent h of the imperfection. According to the studies of Ref. [16] the product N2*2M~l is independent of the thickness of the shell:

Cylindrical Shells

263

"For a given value of the product, therefore, any increase in Mll can be compensated for by a proportional decrease or increase i'n N22. However, it would not be desirable to depend heavily on this stress redistribution because that would mean excessive cracking and yielding, which may lead to a mode of failure not considered..." [16]. The following estimates of changes in stress resultants are derived, useful "when the exact shape of the imperfection is not known" [16] N2*2 -- 1.3~-Nll t

MI*1 -- 0.375~Nll

(10.12)

for h/v/-~ < 2; and N2"2- 1.32V~~~ N l l

M ; 1 --

for h/47

> 2.

10.3

INFLUENCE DAMAGE

0.38x/~~Nll

(10.13)

OF LOCAL

Damage of tubular members produced under the impa, ct of an object or collision with another body is a common situation in off-shore platforms. In a survey paper, Ellinas [10] demonstrated this to be the case. Because of the operation of the platform, ships have to be very close during several hours, and under adverse environmental conditions there are impa.cts leading to damage of tubular members. Local damage induces changes in the geometry of tubular members, with many possible shapes of distortion. According to Refs. [10] and [29], there are three main modes of damage-induced deformations" 9 Overall bending without local denting; 9 Local denting without overall bending; and

264

Thin-Walled Structures with Structural Imperfections: Behavior

A ! I

' 9 ,,

~x

,L

i

~

~_

r

,

,'A.

e

=

sectionA-A

Figure 10.4: Damaged tubular members, from Ref. [30] 9 Local denting with overall bending. The shapes of damage-induced distortions in the geometry of a circular tubular member is quite different from fabrication-induced distortions" damage introduces more severe changes in the curvature of the shell and is localized in a relatively small area (short wave lengths). Under axial load, there may be significant levels of elastic stress concentration in a damaged member. This may reduce the load carrying capacity of the components in a number of ways, i. e. [30]" . Under the cyclic loading found in the ocean, fatigue cracks may be initiated in zones of stress concentration. 9 Short-term increases in the stress levels may lead to the occurrence of plasticity, and this, in turn, to a gradual increase in the amplitude of the geometric distortion. . Reduction in the buckling capacity. Furthermore, the maximum deviations of the damaged surface with respect to the perfect shape can be as high as 20 to 30 times the wall thickness.

Cylindrical Shells

265

To model the elastic stress concentration in an elastic axially loaded cylinder, Tam and Croll [30] adopted the following meridional shape

ri--~rr

-bz#

where b is an axial decay parameter, and ri is the radial deviation. In the circumferential direction, the distorted shape is modeled by a flat region and circular arcs. Such distortion is localized to a small area, and can be investigated without interactions between the imperfection and the boundary conditions. Fig. 10.4 shows the shape of the damaged zone in a cylindrical shell. Results for a specific damage model, with ~ = 0.05, b = 20 in a shell with r / - 100 and L / r - 2 are computed in Ref. [30], and presented in Figs. 10.5 and 10.6. This deviation is a 5% of the horizontal radius of the shell, a value that many codes would accept. The main changes occur in the meridiona,1 moment resultants, M~I , and in the circumferential membrane resultant, N2*2. The stress variation of M{a and N~2 in the circumferential direction are seen to closely follow the variation in the geometry itself, with positive and negative stresses extending over 2~. Stress changes in the meridional direction, on the other hand, are restricted to the zone of damage and do not affect the boundaries of the shell. For unit axial load Nax = 1, the changes in N22 reach a value of 3.0 in tension and above 1.0 in compression. Bending moment changes are in the order of M~I = 0.5 NII~ r and are even more localized than N~2. The mechanism of resistance in the case of dented distortions is similar to that found in imperfections with more gradual changes in curvature that, model defects in the fabrication of the shell. However, the values of stress changes are now higher. The analysis employed an equivalent load of first order, together with a Fourier solution in two principal directions. The number of terms required to compute converged results varied with the specific dent profile; but about 60 Fourier terms were required in the axial direction and 25 in the circumferential direction. Parametric studies reported in Ref. [30] show that the ratio of the thickness to the radius of the shell is very important: in thinner shells, the out-of-balance moment is mainly equilibrated by hoop stress

266 Thin-Walled Structures with Structural Imperfections" Behavior

0.05

L

Circumferential direction ,

,

0

0.025 0.00 -0.025 --/1:

0

/1:

0

/I;

0

/1:

11r

o.5 0.25 0.0

-0.25 -/I:

ill

N11

3.0

1.5 0.0

-1.5

--/1:

Figure 10.5" Damaged tubular members" Stresses in circumferential direction, from Ref. [30]

267

Cylindrical Shells

0.05

L

Meridional direction I

>

X

0.025

0.00 -0.025

-I/2

0

I/2

12

0

112

0

112

0.5 0.25 0.0

-0.25

N,,

3.0

1.5 0.0

-1.5

-I/2

Figure 10.6: Damaged tubular members" Stresses in meridional direction, from Ref. [30]

268

Thin-Walled Structures with Structural Imperfections: Behavior

resultants, and high stress concentrations occur. The extent of the damaged zone in the meridional direction is also important: so that for increasing values of b, the circumferential membrane action decreases. It has been shown that the circumferential membrane stress resultants N2*2 reach a maximum when they account for about 50% of the out-ofbalance moment. Finally, the shape adopted to model the distortion in the meridional direction also affects the results. Tam and Croll looked at several damage profiles, ranging from that represented in Fig. 10.4 to others with smooth transitions between segments, and found results of surfa,ce stress concentrations varying from 2 to 14, for a unit compressive axial load. The particular shape of a zone affected by damage is then extremely important in the evaluation of stress concentration factors. These shapes of damaged zones are idealizations of more irregular shapes that occur in tubular members in off-shore platforms; thus, careful monitoring of damage and subsequent evaluation of stress levels should be done in this class of structures.

10.4

LOCAL IMPERFECTIONS IN LARGE VERTICAL CYLINDERS

Problems due to constructional imperfections in vertical silos were detected experimentally in Ref. [2]. Slip-form construction tends to reduce the geometric deviations in silos, but still they have been observed in both isolated cylinders and also in silos formed by groups of cylinders. We have already shown a case of a reinforced concrete silo structure with severe cracking in Fig. 1.2. Studies of cra,cked silos were reported in Ref. [20], and in many other references in the literature, showing that this is a relatively common situation. Engineers, on the other hand, tend to ignore the importance of these structural imperfections. But it seems that imperfections act as stress concentrators in silos in much the same way as we studied them in other shell forms and application, and that we should be more concerned about their influence on the stress beha.vior in thin-walled silos. This is the subject of the present section.

269

Cylindrical Sh ells

Tz/D [

I P

D~ IB 9

I

!

,

I

H h ffs'fff

f/fff

D

B

Figure 10.7: Imperfect silo under pressure of stored material, from Ref. [24]. Reproduced with permission of Trans Tech Publications

10.4.1

B ulge Imp erfections

Here we report results obtained by E1 Rahim for bulge imperfections in vertical reinforced concrete cylindrical silos of moderate thickness, under the pressure due to cement stored inside the shell [24]. Direct analysis based on finite element modeling of the imperfect geometry was carried out, using elastic properties. The geometry of the shell studied is characterized by r / t - 23.8, and H / r - 5.4, with material properties E - 2.7 x 104 N / m m ~ and u - 0.167. The height of the silo, H, is 2.7 m and the diameter is D - 1 m. The pressure variation is shown in Fig. 10.7, together with the dimensions of the area affected by the imperfection. The imperfection has a maximum amplitude ~ / t - 2.4, with total height h / r - 1.4 in the meridional direction, and width 0.52 rn in the

270 Thin-Walled Structures with Structural Imperfections" Behavior

Figure 10.8" Pressure changes in stored material due to imperfection, from Ref. [24]. Reproduced with permission of Trans Tech Publications circumferential direction. The center of the imperfections is located at an elevation of 0.3H. This is a rather narrow imperfection, with an extent of 1/4 of the height and 1/12 of the circumference. To understand the significance of this imperfection, we have to look at the stresses in the perfect shell, and at the equivalent loads that they induce. The average hoop stress resultant in the perfect shell is N~2 = 0.48, leading to a circumferential out-of-balance moment Mo - N~'2~

(10.14)

equal to 0.024. In the meridional direction, on the other hand, the stress resultants N~I are very small, and the extent is 2.7 times larger than in the vertical direction. This means that circumferential effects are dominant, with only minor influence from the meridional errors. The changes in N22 due to the imperfection are of the order of 20%, while the disturbance in Nll is small in comparison, being about 25% of the changes in hoop membrane action. This is accompanied by circumferential bending/142"2, that shows a wavy pattern following the zone of imperfection. Bending changes in the meridional direction are about 25% of those in the circumferential direction. For the thick shell considered, the imperfection does not dramatically change the stress field, as it does in thinner shells. But according to Ref.[24] the pressure itself is modified in the zone of the imperfection, with local increment in magnitude of 5. The situation is illustrated in Fig. 10.8, for the same imperfection and shell data, in which there is a local increase at the top of the imperfection to p = 5, and a decrease to p = 0 at the bottom of the imperfection. Hoop membrane action is shown in Fig. 10.9 to display bands

271

Cylin drical Shells

,i

N22

,

\

D-D

'J

I y , 0e ec""e"

0.8

tJ

0.0

/

!

I

t

,,--- ~ , _ ~-J- . . . . \

I

. = ~ ==.-...,, ,, f

9

I

9

I

B-B -0.8

[

2.0

1.0

0.0 N22 0.0

- 5.27

/'1

I ! I

1 I

z/D

1

I I I V

I ,,

-271;

0

271:

Figure 10.9: Hoop stress resultant in imperfect silo under pressure of stored material, from Ref. [24]. Reproduced with permission of Trans Tech Publications

272 Thin-Walled Structures with Structural Imperfections: Behavior

0.012 ./

M22

/

\

B-B

\.

/ .

/

"~.

0.0

6 ~

%

/ -0.012

D-D

!

_

J

1

P

f

I

1.0

0.0

0.011

I

!

2.0

z/D

i l

! I I

M22

0.006

0 ~

I I

%

" ~S

~s

LI

II

-2/~ 0 27t Figure 10.10" Hoop moment resulta, nt in imperfect silo under pressure of stored material, from Ref. [24]. Reproduced with permission of Trans Tech Publications

of tension and compression, a,nd with values that are several times the stresses in the perfect shell. Variation in M22 (Fig. 10.10) are also very important, with maximum values of 0.012. The stress changes in the meridional direction are smaller, as illustrated by Figs. 10.11 and 10.12, with maximum bending M~I being about 1/3 the values in M2*2. The shell studied in Ref. [24] is rather thick, with r / t - 23.8, and the redistributions of stresses are not typical of thinner shells. However, the redistribution of pressure due to the imperfection, coupled with the redistribution of stress resultants, may lead to large changes with respect to the ideal or perfect situation. Cylindrical shells can have imperfections in which the circumfer-

Cylindrical Shells

273

2.0

h

Nll 0.0

-2.0

._

i-

q"

M

\

\

I

I

.~

/

J

I

!

I

2.0

1.0

0.0

N11

1.33 0.0 0.52

-2

71;

I

z/D

s

I

-

B-B

~wJ

I

1

27~

Figure 10.11" Vertical stress resultant in imperfect silo under pressure of stored material, from Ref. [24]. Reproduced with permission of Trans Tech Publications

ential effects are much more important than the meridional ones. In such cases, the imperfection can be modeled as a purely circumferential deviation from the perfect geometry. The analysis of this problem can then be reduced to a horizontal section of the shell, in which M0 is equilibrated by M22. Computations for this problem have been carried out by means of a plane frame analysis, and are fully reported in Ref. [13]. We shall see in Chapter 11 that there is a mechanism of resisting M0 by bands of meridional Nix, but this is only possible in shells with meridional curvature.

Thin-Walled Structures with Structural Imperfections: Behavior

274

Mll

/"

"l

i

0.0002

-

I

B-B

I

0.0

~..--

q

f

I f % ,

.

/

/

-0.0002

P

/

D-D I

!

I

2.0

1.0

0.0

0.0045 1

Mll

0.0

]

t 9 -2 g

,

z/D

"~

. ....I

-- -- ....

I

~

Q

L

f

' I

I 0

2/~

Figure 10.12" Meridional moment stress resultant in imperfect silo under pressure of stored material, from Ref. [24]. Reproduced with permission of Trans Tech Publications

10.5

INTERACTION

BETWEEN

GEOMETRICAL MATERIAL 10.5.1

A Simplified

AND

IMPERFECTIONS Model

One of the applications of modulus imperfections in the literature has been a way to model closely spaced cracks that reduce the stiffness in one direction of the shell. This case is different from well defined and isolated cracks, which will be studied in the next section. Let us assume that the membrane stiffness in the circumferential

Cylindrical Shells

275

0.6 0.4

E E

0.2

Z Z

r

-0.2

-0.4

/

-0.6

(1=

1.0

0

0.25

0.50

X

h .

0.6

o.,/o~ \

0.4 0.2

I

~~

1.0

F

-0,2 ~

-0.4 -0.6

0

0

I

i

0.25

I

n

0.50

X

Figure 10.13" Reinforced concrete cylinder with geometrical imperfection and reductions in the modulus of the material [12]

276

Thin-Walled Structures with Structural Imperfections: Behavior

direction is reduced by a factor T, but the bending stiffness in the meridional direction is not affected. In this case, the differential equilibrium equation becomes [12]

04U3 Et D-~x~oi + (1 + r ) ~ T u a - pa

(10.15)

where (1 + r) is the factor that modifies the membrane stiffness. The solution of the imperfect shell with the modified stiffness, is the same as Eq. (10.6), but now /34 - (1 + T) 3(1 -- u 2)

(rt)

(10.16)

A similar approach, based on a different model of imperfection, has been followed in Ref. [16].

10.5.2

Reinforced Concrete Shell

The influence of r on the solution of a cylindrical shell with an imperfection in the geometry is next studied for a specific example of a reinforced concrete shell. The imperfection is defined as in Fig. 10.2, and the data for the analysis is r = 22.5 m; r/t = 150; r/( = 60; r/h - 1.5; u - 0.15; and N~'I - 1 N/mm. The dimensions of this shell may seem somehow large for a cylinder; however, they are only an approximation of an hyperbolic cooling tower. More examples like this will be studied in the next chapter. The results are plotted in Fig. 10.13. The hoop membrane action is reduced with a decrease of the stiffness, as represented by increases in negative values of r. From r -- 0 (geometric imperfection but no stiffness reduction) to r - 0.6, the maximum stress resultants are reduced from 0.67 to 0.28 N/mm, while the maximum bending moments are increased from 8.5 to 36.5 Nmm/mm. Thus, a 60% reduction in the hoop capacity of the shell is reflected in a 60% decrease in the hoop membrane action (a linear relation), and a 330% increase in the meridional bending. Although this example has been solved by means of a simplified tool, it is clear that a decrease in the vital circumferential stiffness produces a severe redistribution of stresses, from membrane to bending. To compensate for the loss of membrane action, the shell has to provide

Cylindrical Shells

277

very high bending stresses. Depending on the reinforcement of the shell, it may or may not be able to provide this required bending action.

10.5.3

Orthotropic Model of an Imperfect Cylinder

A model based on orthotropy for the constitutive material was employed in Ref. [1] oriented towards the analysis of cooling towers. The geometry of the imperfection considered in this case is as follows" 1 4 (_~) [g + aN Cos

11

+ 14----~cos

h

+ 15---~cos

h

]

(10.17)

where r i is the horizontal radius of the imperfect cylinder, and rP is the radius in the perfect cylinder. The change in hoop stress resultants and meridional moments are r

N2"2 - ~,~-ffNfl

(10.18)

MI*~ - ~NI~

(10.19)

The values of fl and 7 are functions of t, r, h, and of the parameters of orthotropy Kll, E22, Ell, which can be found in Ref. [1]. Because of the similarities in the behavior of a, cylindrical shell and the hyperboloid of revolution, the above results were applied to the analysis of cooling towers. A conclusion that highlights the importance of the loss of hoop capacity is written in Ref. [1] as follows: "A 50% reduction in the membrane stiffness due to vertical cracking would decrease the maximum circumferential force by 28% for a cooling tower shell having twice the radius and thickness dimensions of the Ardeer tower, and the maximum meridional moment would increase by 580-/; for a shell having the radius and thickness dimensions of the base of the Ardeer tower. These changes are judged sufficiently large to justify an orthotropic analysis".

278 Thin-Walled Structures with Structural Imperfections" Behavior

10.6

INTERACTION BETWEEN GEOMETRICAL IMPERFECTIONS AND CRACKS

Another important structural imperfection occurs when discrete cracks are present in the structure in combination with imperfections in the geometry. We shall see here that cracks annihilate the stiffnesses that are required by the imperfections in shape, thus leading to further stress redistributions. This section is based on the results of Refs. [11] and [14], which were obtained using a direct finite element modeling of the shell.

10.6.1

B e h a v i o r of an I m p e r f e c t Shell w i t h Cracks

Detailed analysis of one crack configuration on the cylinder of Fig. 10.14 is presented here in order to establish the main features of the behavior of shells with cracks and shape imperfections. The geometry was chosen so as to be compatible with the test model described in Ref. [11] and [14]; in this case the crack configuration studied has six full cracks evenly spaced around the circumference, with crack length/? equal to the height of the banded imperfection, h. This is a very severe cracked condition, since the out-of-balance moment due the meridional stress acting on the geometrically imperfect shell must be completely equilibrated by flexural action in the vicinity of the crack. The situation would be typical of that occurring in a reinforced concrete shell after yielding of the horizontal reinforcement across a crack has taken place. The geometric imperfection is assumed as axisymmetric. Comparisons between the stress state in the uncracked and cracked imperfect shells are summarized in Figs. 10.15 to 10.17. It can be seen that a complete loss of hoop stress capability at the crack has significant implications for the stress distribution. In the uncracked shell, shown by the dotted curves in Fig. 10.15, the resistance to the out-of-balance moments arising from the eccentricity of the meridional stress Nl1 acting on the imperfection is dominated by the development of alternating bands of hoop tension and compression N2*2. The meridional moments MI*1 provide a relatively smaller contribution. This mechanism has been discussed in this and previous chapters.

Cylindrical Shells

Figure 10.14" Geometry and cracks of the shell investigated

279

280 Thin-Walled Structures with Structural Imperfections: Behavior

,,

rE~

-- 0 ~

------

m

E E E

4-

Uncracked Cracked

=.

2-

Z

l

Z

04

0 -2

0

I

I

I

I

0.25

0.5

0.75

1.0

Z

h

E E

3

~~ 21I| _._z/j_o ....... 0 -1

-2~ 0

z/h=0.5 I

10'

I

20'

I

30'

Figure 10.15" Behavior of the imperfect shell of Fig. 10.18 with 6 full cracks [11, 14]

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oOt~ I

oO~ I

oO~

o

I

..

S'O-qlz

'~-,,

Z Z

9

3 3

0

q

.,....

Z

I

l

c3l'O

0"I.

c3"0

S~'O

l

0

I

-! 0 7 3 3

tl II

rz +

II

o0-

-0

f

sHat/s

Igg

I~or.apm.l*D

282 Thin-Walled Structures with Structural Imperfections" Behavior

1.~ - - 0 ~

,.

E E E

|

Z

4-

9

f

-2 0

0.25

-•,,,,•h

rE

E E E

Z

-

0.5

0.75

1.0

=0

0

0.25 -1 -2

_f---

0.5 0.5

-3 I

0

100

I

20 ~

I

30 ~

Figure 10.17" Meridional moments [11, 14]

Z

Cylindrical Sh ells

283

Ill 9'5,

Ill

,"

J,

I x

I

~

,,

/

xx

I

/

/

',

\

ii 1 3

,

I #

I

/

!

%

/

t

/ /

\\

\

I ~I tl.I/

I/

./ %

/

/

i Figure 10.18: Axial stresses around the crack [11, 14]

Loss of these important hoop stresses N2*2 in the vicinity of a vertical crack can be seen to have the anticipated result of greatly increasing the role of meridional moments. At the location of the vertical crack, where the circumferential coordinate 0 = 0, Fig. 10.17 shows that M~I is increased by some 300% at the mid-height. The rather smaller increase occurring at the bottom of the imperfection means that the combined contribution to the resistance of out-of-balance moments from meridional moment is increased to roughly 250% of its value at the same location in the untracked shell. So that while moments contributed some 26% to the equilibration of out-of-balance moments in the uncracked shell, they provide around 66% at the cracked meridian for an equivalent cracked shell. The reason why an even greater increase in meridional moments does not occur is due to a redistribution in

284

Thin-Walled Structures with Structural Imperfections: Behavior

vertical loads that accompanies cracking. Loss of the vertical load carrying capacity provided by N l l near the crack is not directly evident in Fig. 10.16 as a result of a superposition of bands of vertical tension and compression that are developed along the line of the crack boundary. These bands are the result of the same phenomenon experienced in plates in which compressions are developed along the lines of cracks which are transverse to an uniform, uniaxial tension field in a flat plate [25]. This is illustrated in Fig. 10.18 for the axially loaded shell with cracks. For the present situation, this produces the meridional compression at mid-height where the hoop stress being relieved by the crack is tensile. At the top and bottom of the imperfection, the loss of the hoop compressions at the crack induce, in the same way, the bands of vertical tension. An average vertical stress, drawn along the ~ = 0 meridian, would show the above reduction to be around the 66% indicated by the meridional moments. Associated with this redistribution of vertical load resistance away from the cracked region, is a generally increased level of hoop stress N~2 and meridional moments MI*1 among adjacent cracks. The out-of-balance moment is not constant in the circumferential direction because of the local reduction in meridional stress resultants. This variation is represented in Fig. 10.19, for the cracked shell. In the neighborhood of the full crack, there is no contribution from the hoop membrane mechanism; thus, the value of M0 has to be taken by M{1. This is shown in Fig. 10.20, where the bending contribution is maximum at the edge of the crack, and decreases away from it. This decrease is due to the contribution from hoop membrane stresses, that can develop away from the crack. This means that at an angle ~ = 15o the bending contribution is reduced from 1 to 0.4, and if the cracks are far away from each other, then the full membrane contribution can be restored. To obtain a clearer insight into the way in which the shell, imperfection, and crack characteristics affect the behavior of the shell, a number of parametric studies have been performed. The studies examine the distributions of the stresses and the nature of the stress redistributions around a crack and their dependence on a number of non-dimensional parameters. The numerical model employs a finite element mesh of 216 elements, that was found to give good accuracy in stress resultants [11]. The boundary conditions assumed were simple support at both ends, with again a uniform meridional compression of

285

Cylin drical Shells

1.0

b 0

"10 0

0.5

r L_

o

V

0

0.0

I 30

15

0

0 Figure 10.19" Circumferential variation of the out-of-balance moment [11] "0 (1.} 0

1.0

0

V

C)

t" "ID r 0 ..Q em

V

0

0.0

I 30

=|

15

0

0 Figure 10.20" Proportion of the average out-of-balance moment carried by bending [11]

286

Thin-Walled Structures with Structural Imperfections: Behavior

unit value.

10.6.2

I n f l u e n c e of Shell F o r m

In order to have a more general understanding of the behavior, several parameters concerning the length, thickness and meridional curvature of the shell have been studied for a given imperfection and crack configurations [11]. All the cases studied have eight full cracks in which f / h - 1, and a cosine imperfection profile with h/r - 2/3.

Effects of Shell Thickness As Fig. 10.21 makes clear, for a thin, uncracked shell, the hoop stress carries a much larger proportion of the out-of-balance moment at the meridional imperfection. Accordingly, the meridional moment accounts for a much smaller contribution to the out-of-balance moment near a potential crack than it does for the thicker example discussed in relation to Figs. 10.15 to 10.17. A meridional crack across a banded imperfection, in the absence of any redistribution of vertical stress away from the crack, would have the effect of forcing the shell to reach equilibrium by developing meridional flexural moments of the same order as the out-of-balance moment, Nax~ = 4 N m m / m m . The loss of hoop capacity at a crack in a very thin shell could therefore be anticipated to result in a much more severe increase in the meridional moments at meridians near to the crack than in relatively thicker shells. This is seen to be the case in Fig. 10.21, where although an increase in r/t results in Mi*1 being sma,ller in the untracked shell, it also results in the proportional increase of moments for the cracked shell being considerably greater relative to their untracked values. These increased moments, acting on a thinner shell, would result in increasingly severe meridional flexural stresses as r/t becomes greater. Variations of N~2 caused by cracking are also significantly dependent upon r/t. For thick shells (r/t - 40 in Fig. 10.21), the zone of interference arising from the crack is such that it extends over the entire inter-crack spacing. This is a, reflection of the preference of the shell to generate the resistance lost by cracking in the form of bending rather than membrane action. As the thickness is reduced, so the circumferential extent of the crack interference zone is decreased, until for extremely thin shells the hoop stress N2*2 at 0 - 0~/2 would be little

287

Cylindrical Shells

Z=O E E E E

qlBllm ~

80

=,=== ,=,=~ aallD ~ l l B 9mlB I l l l l

GIIII a U J Ulllm IIIMI i m l l ~ U l l ~ i i i m i n i i i I I t a m p

Z

Z

150

, ,Iil~J ~

,mmmmm

m

~

40

m

150/

m

r

/

_y ~

Immmll m

~

~

i

m

m

m

R/t=

mind m m m

~mmJ ~ m m

~

r

40

04

0

0

10 ~

20 ~

Z-O r

R/t=40

E E cE

1"

m

0

0

--

~

80

~

.

m

- -. . T. s. ; - - -i _' - -_

.

m

m

_

m

_

m

_

m

_

m

_

10 ~

Figure 10.21" Effect of shell thickness [11, 14]

m

m

-,,-1 20 ~

288 Thin-Walled Structures with Structural Imperfections: Behavior affected by the crack. On the other hand, the stresses generated by the moments in the vicinity of the crack would be almost certain to induce material failure. From the above it is apparent that the bending stiffness of the shell, as expressed through the r / t ratio, plays a crucial role in the redistributions from membrane to bending resistance.

10.6.3

Influence of I m p e r f e c t i o n Form

Studies on hyperboloids of revolution with geometric errors in Refs. [8], [9], and [23] have shown that several parameters related to the geometry of meridional imperfections affect the behavior of the shell, but that the dominant ones are the amplitude and the length of the imperfection. Both parameters have been studied for a cylinder with r / t - 80, having eight equally spaced cracks for which g/h - 1.

Effect of imperfection amplitude The imperfection amplitude has a direct effect on the out-of-balance moments, ~r For an uncracked shell, the disturbances in stress are linearly related to the imperfection amplitude ~/r, which confirms studies on cooling towers [9]. In contrast, when cracks are present, this linear relationship no longer applies. Different amplitudes of imperfection, as shown in Fig. 10.22, exhibit very different redistribution characteristics. The meridional stress resultant Nil not shown, is significantly affected by the parameter ~/r. The values of N2*2 at the center of the imperfection increase away from the crack, but for large amplitude of imperfections, N2*2 does not reach between cracks the level of the corresponding stresses in the uncracked shell. So that even between cracks, the hoop contribution in the cracked shell is less than in the uncracked shell, provided the cracks are close to each other. Associated with these, the moments MI*1 are increased between cracks to several times their values in an uncracked shell. If the amplitude of the imperfection was very small (i.e. ~/r - 0.004 in Fig. 10.22) the shell would not have to develop high stresses to resist the small out-of-balance moments, even for the case of full-depth cracking.

Cylindrical Shells

289

i,

2 r

E E

.

02 ~

m

i... m

q m m m m

0.034

m

W

- -

m

i

m

~mmm, m

~mmb

~mmm

z = o a ~

m i n d ,romp

~,/R=o.o48 - -

gmmm. m

~

-

-

m

n

.,

Z

Z

im,~ m

~

(mmmmbm

m l

0.034 _...-...--

04 04

w~

~,mD

0.017 I n m ' W

m.m

~

I=~

m~tlD

~

t

0.004

m m

,rim:_

0.017

m

l

0.004

m

0

l

w~

m

,

WBm q, m l

am

~m,.~

10 ~

20 ~

2t

'0

Z-0

rE

E E E

= 0.048 -

9 .....

0.034

Z

0.017 m m

"''-" ~

0

. m

~mD

"-

m m m m

d~mmm m m m m m

0 0 0 4 - - " -~.

"

m

m l m m m m

10 ~

.ram. m m

~

m

m

--- - - , , m

I m m m

mud

20 ~

O Figure 10.22" Effect of imperfection amplitude [11,14]

290

Thin-Walled Structures with Structural Imperfections" Behavior

Effect of imperfection height The effect of the height of the imperfection has been studied for a configuration with g/h = 1 of which some selected results are presented in Fig. 10.23. The out-of-balance moment, defined by ~Nll is independent of the imperfection wavelength; however, in short imperfections, the shell has a shorter length over which to develop bands of hoop tension and compression, and the corresponding higher membrane stress gradients in the untracked shell require higher levels of moments in the cracked shell. The study shows that short imperfections have a more harmful effect than longer ones in cracked shells; moreover, the higher meridional curvatures of the short imperfections tend to make the shell resist more by developing flexural moments. For example, if h/r is halved from 2/3 to 1/3, the moments Ma*~ increase about 80%, while the hoop stresses N2*2 increase by only a 30% away from the cracks.

10.6.4

I n f l u e n c e of Crack P a r a m e t e r s

In the previous parametric studies, the spacing and length of the cracks related to the imperfection height were assumed to be the same in all cases. In the following, the effects of the length and spacing of full thickness cracks are examined for a shell having L/r - 2.7, (/r = 0.048, r / t - 40, and h / r - 2/3.

Effect of crack spacing It could be expected that if the cracks were a sufficient distance apart in the circumferential direction, there would be no interaction among them; but that as the separation is reduced, the interaction among cracks would become stronger. Numerical studies were carried out on shells having 3, 4, 5, 10, and 20 equally spaced cracks (that is, central angles 0c - 120, 90, 60, 36, and 18o, respectively) with g / h - 1. With Fig. 10.24 showing M~I at mid-height of the imperfection, it is seen that it is only for the case in which 0~ - 120 ~ that MI*1 is not affected by the cracks at 0 - 0~/2. In this case, the effects of individual cracks are non-interacting. Up to six cracks (0~ - 60 ~ the values of M~*~in the vicinity of the cracks are not dependent on their separation, and only away from the cracks occurs some increase. This could be considered as

291

Cylindrical Sh ells

rE1 E

Z-O

1.5

1

0.5

0

10"

20 ~

0

'E' E E E

Z-O

1.5

Z

, .9. , ~ ~- . , . ~ ~

0.5

0

0.66

Inmmo ~

________

~

0.5 ~

. 0.33 . . . . . .

ammmo ~

~

,n,m,~ ~ Q i a l U U ~

10 ~

~

U

~n'

~,iUl~

~

lume,~

GUNI

20 ~

O Figure 10.23" Effect of imperfection height [11, 14]

292 Thin-WMled Structures with Structural Imperfections: Behavior

E

Z-O 3 1.~r

Z.

==

180

o

2

,

36

0 0

60 ~

90 ~

I

I

I

I

9

18

30

45

120 ~

60 ~

Figure 10.24" Effect of crack spacing [11, 14] a case of limited interaction. With 10 or 20 cracks, there are variations in M~I at all distances from the crack. It may be anticipated that for central angles smaller than 10o, MI*1 should be almost constant in the circumferential direction, indicating that the shell is unable to develop the requisite N~2 action to alleviate the high meridional moment even away from the crack. Effect of crack l e n g t h

The effect that the length of the crack has upon M~I is shown in Fig. 10.25, where M~I is at the center of the crack increasing almost linearly with g/h; Nll shows little variation at the same location. A more dramatic effect that the crack length has on the redistribution of stresses can be observed in N2*2. For f / h _< 0.3 the shell still develops an important membrane hoop contribution to the out-of-balance moment with moderate increase in N2"2, even at 0 - 0. However, as the crack length approaches f / h - 0.5, the shell is forced to develop very large hoop stresses (i.e., for a unit value of applied meridional stress Nll there would be values of N2*2 larger than five), and material yield may occur at the ends of the cracks, thus tending to increase the effective crack length. It is at this stage that hoop action cannot be relied upon

[~I 'II] q ~ u ~ ~t~

008

jo ~oj~t "~g'OI ~

gL .

.

.

.

.

0

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.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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g].'O -

q/I

~6~;

294

Thin-Walled Structures with Structural Imperfections" Behavior

to offer substantial reserves for resisting out-of-balance moments.

10.6.5

Discussion

The above parameter studies have shown that two basic mechanisms accompany the growth of a discrete vertical crack across the stress concentration caused when vertical loads act on a meridionally distorted cylinder. First, there is a redistribution of vertical load away from the cracked zone, with an associated general increase in the level of stresses away from this cracked zone. But secondly, and even more significant for the integrity of the shell, once cracks are present, there is a redistribution from the hoop membrane action to meridional bending action to equilibrate those vertical loads that continue to be transmitted down the cracked meridian. The relative importance of these two forms of redistribution, as well as the extent of redistribution that is actually necessary, have been demonstrated to be strongly dependent upon those shell, imperfection, and crack parameters that most significantly influence the relative bending and membrane stiffness of the shell. Most significant in controlling the relative importance of these bending and membrane effects are the r/t of the shell and the imperfection wavelength. The thinner the shell, the higher is the redistribution of vertical load away from the cracked zone, and consequently the higher are the hoop stresses at some distance from the crack. Even so, the redistribution in the vicinity of the crack produces flexural stress which, although accounting for a lower proportion of the out-of-balance moment carrying capacity than in thick shells, becomes increasingly likely to cause material failure as the shell thickness is reduced. Conversely, for very thick shells, the major part of the out-of-balance moment, caused by the vertical stress acting on the meridional imperfection, would be resisted by bending action. These are much less affected by cracking since the hoop membrane actions that would otherwise exist at the cracked locations are relatively less important. The increased significance of bending action which accompanies a decreased meridional wavelength of imperfection likewise reduces the need for redistribution when cracking occurs. Increasing thickness and decreasing imperfection wavelength for that reason produce very similar effects. However, there are other parameters that quantitatively and qualitatively affect stress redistributions which accompany cracking. The

Cylindrical Sh ells

295

spacing between cracks and the length of the crack have significant influences. It has been shown that the crack length must exceed a certain threshold proportion of the imperfection height before a substantial loss of the hoop membrane resistance takes place. If the cracks are short, around 30~ of the imperfection height, the shell retains its capability for developing the moment of resistance through membrane hoop action. Lengthening the crack beyond this values has been seen to produce a rapid degeneration of membrane hoop capacity at the crack which is compensated by a development of large meridional bending moments. What the above parameter studies confirm, is that those parameters that have little influence on the stresses in the uncracked imperfect shell are also found to have little influence on the stresses accompanying cracking. Shell length falls into this category, provided it does not come within the region affected by the imperfection generated stress concentrations.

10.7

EXPERIMENTAL

STUDIES

An experimental program on the stress redistributions in metal cylindrical shells with various structural imperfections was carried out by the author between 1977 and 1979 [11] [15]. Only one shell and one imperfection shape were investigated. Grooves and cracks in the shell were progressively extended to study a total of seven imperfect situations. A model was fabricated of duraluminium, largely because of its machining advantages over other metals. Due to machining convenience, the ratio t/r (thickness to radius of curvature of the shell) was fixed to a relatively high value of 1/40. The geometry of the model is shown in Fig. 10.14, with the amplitude of the controlled imperfection being a cosine function of the vertical coordinate z. The final shape was obtained by machining a thick tube in a center lathe. Manufacture of the specimen was made with the help of a copying attachment. A plate with the same shape as the desired outer meridian was fixed to the copying attachment, which guided the cutting tool of the lathe following the shape of this plate. Similar opera, tions were performed on the inside surface. The thickness and shape of the shell were monitored after the lab-

296

Thin-Wafted Structures with Structural Imperfections: Behavior

rication, and the mean value of the recorded thickness was found to be 3.07 mm. The results for the recorded meridians showed variations in the wall thickness of + 2%. Grooves were cut along a meridian producing a controlled discrete loss of stiffness in the circumferential direction. Some tentative results obtained from numerical studies indicated the approximate depths of groove that would produce experimentally measurable strains. The width of the cracks was fixed between 1 and 2 mm, determined by the width of the cutters commercially available. Since there were serious problems in trying to cut the grooves on the inside of the shell, they were cut on the outside only. The grooves, therefore, had an'eccentricity with respect to the middle surface of the shell. A special cutting tool was designed for the fabrication of the cracks, which is fully described in Ref. [11] [15]. The cracks were carefully measured after fabrication: the total length of the cracks had maximum variations of

+3%. For any one crack configuration, all the cracks were designed to have the same characteristics in order to maintain cyclic symmetry with respect to the center of each crack. First, the shell was tested with a geometrical imperfection. The first crack configuration had four grooves equally spaced around the circumference, with length g equal to the height h of the banded imperfection. The depth of the grooves was about 2t/3 and the eccentricity of the center of the remaining wall at the crack was t/3. A slot of nominally 1/4 of the imperfection length was symmetrically cut in the existing grooves through the thickness of the shell to produce a second crack configuration. The slots were further extended to 1/2, 3/4 and 1 times the imperfection length. This allowed the observation of the effect of the crack length on the redistribution of stresses. Notice that t and h are different in several of the crack configurations studied. The model was tested under uniform axial compression, which for meridional imperfections produces the highest meridional stresses, and consequently the highest out-of-balance moment (the product of the meridional stress resultant times the maximum amplitude of the geometric deviation). The strains were recorded at a number of positions near the crack, where the highest values were expected to occur. Electric resistance foil strain gauges were placed on the top half of the cylinder, with additional gauges on the bottom half to check overall symmetry. Gauges

Cylindrical Shells

297

were also placed near the top and the bottom of the shell in order to check that the load was uniformly applied in the vertical direction at both ends. In all cases, the gauges were fixed on both inner and outer surfaces so that average direct strains and changes in curvature could be obtained. The values of loads an strains recorded were processed to obtain strains and changes in curvature of the mid-surface in the principal curvilinear directions, for a normalized uniform compression. Full details of the results may be found in Ref. [11]. In Section 10.6 we presented theoretical results based on a direct model of imperfections and grooves. That model was solved by means of finite elements for two-dimensional, double curved shells [11]. The good news are that the differences between theoretical and experimental results are in the range of 5 to 7%, which is reasonable in view of the inevitable errors in the experimental results, and the approximations involved in the theoretical formulation.

10.8

INFLUENCE OF CHANGES IN THICKNESS

Stresses in cylindrical shells with internal circumferential grooves were investigated in Ref. [19]. This problem is of interest to the paper industry, in which a rotating pressure vessel with such grooves is stressed and deformed by the pressure of cylindrical pressure rolls. The rolls are modeled by means of longitudinal line loads. The reasons to have grooved walls are related to the need to promote heath flow through the shell wall. The cylinders are made of cast iron, with 3.5 m diameter. Again, the sudden changes in the properties arise as a consequence of design, but they alert us of the importance of the stress redistributions that are associated to such local weaknesses. An interesting similar research is the influence of sudden changes in the thickness of a cylindrical shell of Ref. [28], in which the change in the thickness occurs in the circumferential direction, so that there are two parts of a shell joined by a meridian. Fig. 10.26 and 10.27 are obtained from Ref. [28], where a numerical study was carried out. The model developed was an equivalent load of various orders using semi-analytical techniques to model the shell.

298

Thin-Walled Structures with Structural Imperfections: Behavior

Figure 10.26" Cylinder with sudden change in the thickness, from Ref. [28] Mindlin shell theory was employed to account for rather thick cylindrical shells. The height of the cylinder is 20, with radius r and thickness to = 1; material properties considered are E = 1 and ~ = 0.3. This shell is rather thick, with an r/t ratio of 10. The thickness is reduced from to to 0.74 to on half of the cylinder. The results of a solid line in Fig. 10.27 are obtained with 3 iterations of equivalent load and 9 circumferential harmonic components in the element of a shell of revolution; while the dotted line was obtained by means of 5 circumferential harmonics. They showed a decrease in the out-of-plane displacements in the thicker part, with respect to a shell of constant thickness, and an increase in the thinner part. Such changes oscillate between a 10% decrease and almost 40% increase in displacements with respect to the "perfect" shell.

10.9

CONCLUSIONS

In this chapter we tackled the cylindrical shell and investigated the response with various structural imperfections. With geometrical imperfections, it was clear that the behavior was similar to that obtained for other shell geometries, such as shallow shells and spherical shells. This similarity in the behavior will be extended to hyperboloids of

Cylindrical Shells

299

0.45

.2 t'O

0 V r

0

I

I

I

I

I

30

60

90

120

150

180

Figure 10.27: Changes in displa.cements due to sudden change in the thickness of a cylindrical shell, from Ref. [28] revolution in the next chapter. With a need to provide high hoop membrane action to equilibrate the out of balance moments in the imperfect cylinder, it is importa,nt to investigate what happens when the hoop stiffness is decreased by some material imperfection. This was illustrated by changes in the modulus of the material, leading to further redistributions from hoop membra, ne to meridional bending effects. The next step in this chapter was the reduction of hoop stiffness a,t discrete locations, i.e. at grooves of passing-through cracks in the meridional direction of the shell. This produced major redistributions of stresses, and led to high bending contributions. Experimental evidence confirmed the theoretical results of a direct analysis of cylinders with combined cracks and geometrical errors. Redistributions of stress accompanying the growth of cracking across geometric imperfections, demonstrated to be consistent with those observed in other shells with defects. Limiting the consequences of both imperfection and crack must, therefore, represent an important constraint in the design of reinforced concrete and metal shells.

300 Thin-Wailed Structures with Structural Imperfections: Behavior

References [1] Alexandridis, A. and Gardner, N. J., Tolerance limits for geometric imperfections in hyperbolic cooling towers, J. Structural Engineering, ASCE, 118(8), 1992, 2082-2100. [2] Askegaard, V. and Nielsen, J., Probleme bei der Messung des Silodruckes mit Hilfe yon Druckzellen, Die Bautechnik, 3, 1972. [3] Burmistrov, E. F., Symmetrical deformations of nearly cylindrical shells, Prikladnaya Matematika I Mekhanika, 13, 1949, 401-412 (in Russian). [4] Calladine, C. R., Structural consequences of small imperfections in elastic thin shells of revolution, Int. J. Solids and Structures, 8, 1972, 679-697. [5] Calladine, C. R., Theory of Shell Structures, Cambridge University Press, Cambridge, 1983, 374-387. [6] Carlson, W. B. and Mc Kean, J. D., Cylindrical pressure vessels" Stress systems in plane cylindrical shells and in plane and fierced daumheads, Proc. Inst. Mechanical Engineers, 169, 1955, 269-293. [7] Clark, R. A. and Reissner, E., On axially symmetric bending of nearly cylindrical shells of revolution, J. Applied Mechanics, 23(1), 1956, 59-67. [8] Croll, J. G. A. a.nd Kemp, K. O., Specifying tolerance limits for meridional imperfections in cooling towers, J. American Concrete Institute, 1979, 139-158. [9] Croll, J. G. A., Kaleli, F. and Kemp, K. O., Meridionally imperfect cooling towers, J. Engineering Mechanics, ASCE, 105(5), 1979, 761-777.

Cylindrical Shells

301

[10] Ellinas, C. P., Ultimate strength of damaged tubular bracing members, J. Structural Division, ASCE, 110(2), 1984, 245-254. [11] Godoy, L. A., Stresses in Shells of Revolution with Geometrical Imperfections and Cracks, Ph.D. Thesis, University of London, London, 1979. [12] Godoy. L. A., Un modelo simplificado de imperfecciones geometricas en cascaras de hormigon, Revista Brasileira de Engenharia, Caderno de Estruturas, Rio de Janeiro, 1985. [13] Godoy, L. A. and Despradel, S., Stress redistributions due to circumferential imperfections and cracks in vertical silos, Bulk Solids Handling, 1996. [14] Godoy, L. A., Croll, J. G. A. and Kemp, K. O., Stresses in axially loaded cylinders with imperfections and cracks, J. Strain Analysis, 20(1), 1985, 15-22. [15] Godoy, L. A., Croll, J. G. A., Kemp, K. O. and Jackson, J. F., An experimental study of the stresses in a shell of revolution with geometrical imperfections and cracks, J. Strain Analysis, 16, 1981, 59-65. [16.] Gupta, A. K. and Al-dabbagh, A., Meridiona.1 imperfection in cooling tower design: Update, J. Structural Division, ASCE, 108(8), 1982, 1697-1708. [17] Haigh, B. P., An estimate of the bending stresses induced by pressure in a tube that is not quite circular, Appendix IX in Welding Research Committee, 2nd Report, Proc. Institution of Mechanical Engineers, 133, 1936, 96-98. [18] Kemp, K. O. and Croll, J. G. A., The role of geometric imperfections in the collapse of a cooling tower, The Structural Engineer, 54, 1976, 33-37. [19] Kinsbury, H. B. and Elderkin, R. L., A line load analysis of cylinders with grooved or reinforced walls, Thin Walled Structures, 3, 1985, 231-254.

302 Thin-Walled Structures with Structural Imperfections: Behavior [20] Maj, M. and Trochanowski, A., Stress State Analysis of a Cracked Silo Container for Cement Powder, Ph.D. Thesis, Wroclaw Polytechnic Institute, Wroclaw (Poland), 1983 (in Polish). [21] Marbec, M., Theorie de l'equilibre d'ume lame elastique soumise a une pression uniforme, Bulletin de l'Association Technologic Maritime, 19, 1908, 181. [22] Novillo, N. and Godoy, L. A., Deformaciones diferidas en cilindros de hormigon con imperfecciones, In: Colloquia 87, Porto Allegre (Brazil), 1987, 119-131. [23] Oyekan, G. L., Analysis of Hyperbolic Cooling Towers with Structural Imperfections, Ph.D. Thesis, University of Southampton, Southampton, 1978. [24] Rahim, H. A., Effect of constructional imperfections on the internal forces in silo walls, Bulk Solids Handling, 9(2), 1989. [25] Rolls, M., The Elastic Stability of Discontinuous Structural Systems, Ph.D. Thesis, University of London, London, 1969. [26] Steel, C. R., Juncture of shells of revolution, J. of Spacecraft and Rockets, 3, 1966, 881-884. [27] Steel, C. R. and Skogh, J., Slope discontinuities in pressure vessels, J. Applied Mechanics, 37, 1970, 587-595. [28] Suarez, B., Metodos Semi-Analiticos para el Calculo de Estructuras Prismaticas, Centro Internacional de Metodos Numericos en Ingenieria, Barcelona, 1991. [29] Taby, J., Moan, T. and Rashed, S. M. H., Theoretical and experimental study of the behavior of damaged tubular members in offshore structures, Norwegian Maritime Research, 2, 1981, 26-33. [30] Tam, C. K. W. and Croll, J. G. A., Elastic stress concentrations in cylindrical shells containing local damage, In: Applied Solid Mechanics, 2, Ed. A. S. Tooth and J. Spence, Elsevier, Essex, 1988.

Cylindrical Shells

303

[31] Tam, C. K. W. and Croll, J. G. A., Stress concentrations in circular tubular members containing local damage, Proc. VII Int. Conf. on Offshore Mechanics and Artic Engineering (OMAE), Houston, Texas, ASME Paper OMAE-88-672, 1988. [32] Timoshenko, S. and Woinowski-Krieger, S., Theory of Plates and Shells, Second Edition, McGra,w-Hill Kogakusha, Tokyo, 1959, 471-475. [33] Vandepitte, D. and Lagae, G., Buckling of spherical domes made of micro concrete and creep buckling of such domes under longterm loading, IUTAM Symposium on Non-Linear Analysis of Plates and Shells, PUC, Rio de Janeiro, 1985.

C h a p t e r 11 IMPERFECT HYPERBOLOIDS REVOLUTION 11.1

OF

INTRODUCTION

Hyperboloids of revolution here refer to just one type of structure: reinforced concrete cooling towers. These towers axe formed by a thin shell and supported on columns. A typical geometry is illustra, ted in the photograph of Fig. 11.1, taken at the construction of a cooling tower in Dampierre, France. Towers like these are employed in nuclear power and other electricity plants, in petrochemical and in metallurgical plants. However restricted this structural type may seem, the number and cost of cooling towers make it an important field to explore the influence of imperfections. But this is even more relevant when we consider tha, t there ha,ve been a few major collapses of cooling towers due to, or influenced by, imperfections of some kind. The height of cooling towers has increased from 60 m in the early 1960s, to 100 m in the late sixties and early 1970s, and to 180 m from the late 1970s onwards. This is an increase of three times in the height of a structure in only 20-30 yea,rs. Some impression of this increase may be gained from Ta,ble 12.1. One of the rea,sons for this increase in size has been a, shift, in conceptual design, from a number of small towers to the construction of fewer but larger towers. Not all such cooling towers are designed with the shape of a hyper304

Hyperboloids of Revolution

305

Figure 11.1" A hyperbolic cooling tower shell during construction at Dampierre, France (Photograph by the author)

boloid of revolution: in the sixties, m a n y cooling towers were designed with cone-toroidal shape. At present, they are almost exclusively built as hyperboloids of revolution. This chapter continues with a short literature review covering the period 1967-1995. In Section 11.2, a cooling tower is considered for the evaluation of the stresses in a perfect shell. The influence of meridional imperfections is studied in Section 11.3 by looking a,t the mechanics of stress redistribution and the influence of the parameters that affect the behavior. Circumferential imperfections are the subject of Section 11.4, while a combination of errors in the meridional and circumferential direction are investigated in Section 11.5. In Section 11.6 we consider the interaction between geometrical imperfections and discrete vertical cracks in cooling towers. Creep effects are studied in Section 11.7. Some final remarks about the influence of imperfections in cooling towers are made in Section 11.8.

306

Thin-Walled Structures with Structural Imperfections: Behavior

11.1.1

Short L i t e r a t u r e R e v i e w

The first work known to the author about the influence of shape imperfections on the stresses in cooling tower shells was published by Soare in 1967 [58]. This pa,per showed that under gravity loa,d, constructional imperfections induced tensile hoop stresses even larger than the meridional compression. This research employed a membrane theory, and as such, the stresses represented an upper bound with respect to a more accurate bending solution. But even within these limitations, the paper showed that important stress levels should be expected in imperfect hyperboloids. Unfortunately, the work of Soare did not make an impact in engineering practice. It remained ignored for severa,1 years, waiting for the first major collapse in this field to occur. The standard practice and code recommenda,tions did not take into a.ccount the stresses due to imperfections" this may be seen from in US recommendations [2], and the British standards [9]. A state-of-the-art in cooling tower design and construction, published in 1976 [20], practically ignored the issue. Tolerances in the geometry of the shell were not related to the stress field, but to engineering judgment. Cracking, on the other hand, had been of some concern with regard to vibrations of the tower [50] [1]. In 1973, a 106 m high cooling tower, which was known to have severe meridiona,1 shape imperfections a,t the time of construction (with errors in the ra,dius of 305 mm), colla,psed at Ardeer, Scotland. The committee of inquiry into the colla,pse showed [42] that the imperfections induced tensile hoop stresses of the same order of magnitude as the meridional compression. This case is further discussed in Chapter 12. Several investigations followed the Ardeer collapse, to clarify the effects of geometrical imperfections as stress concentrators. As stated in Ref. [19], it was necessary a "damaging collapse to first awaken the recognition of the importance of geometric tolerance control in shells of this type". Croll and Kemp, who produced calculations for the committee of inquiry, explained the mecha, nism of stress redistributions in a shell with meridiona,1 imperfections [47] [15] [17], and also developed guida, nce to tolerance specifications in cooling towers [16]. This work wa,s later extended to consider imperfections in the circumference [21] [22]. Further studies in this direction were reported in Ref. [26] and

[27].

Hyperboloids of Revolution

307

Independent research in England confirmed the Ardeer results [54] [51] [52]. In the [IS, the results of Refs. [34], [3] and [37] were based on calculations using a simplified model of a meridionally imperfect shell, and emphasized the importance of bending stresses. Out-of-roundness effects were investigated by Boresi [48]. Gould and co-workers studied imperfect towers in Refs. [38] [40]. The influence of geometrical imperfections was a,lso studied in Japan, with results for non-linear analysis being presented in Refs. [45] and [46]. In France, where some very large cooling towers were built in the late seventies, there has also been concern with structural imperfections [44]. The techniques of analysis played a very important role in this field. The equivalent load technique for cooling tower-type shells was established by Croll and co-workers [18]. Further studies may be found in Ref. [51]; and a, derivation of the complete equivalent load is in Ref.

[29]. But it may ta,ke more than a, model of geometrica,1 imperfections to investigate a deteriorated cooling tower. Cracks were detected at the Ardeer tower, and have been monitored in a number of other cooling towers [55] [56]. The combination of geometrical imperfections and cracks was mentioned in Ref. [42] as the most probable cause of the collapse; however, the first studies on this combination were only ready in 1978 [32]. Several studies tackled the mechanism of stress redistributions around discrete cracks in shells with imperfections in the meridian [2,5] [27] [33], and an experimental confirmation was presented in Ref.[31], for an isotropic shell with grooves and manufactured imperfections. The influence of smeared-out cra,cks was considered in Ref. [37]. Thin-walled, reinforced concrete cooling tower shells represent a unique case for which structural imperfections proved to be of major importance in eroding the safety of the shell. Initial distortions that were regarded to induce negligible effects resulted in major consequences. Thus, the collapse at Ardeer was followed by a failure at Bouchain, in Fra,nce, in 1979, and a.nother one a.t Fiddlers Ferry, in England, in 1984 [12]. Because of the evidence of major collapses, other towers with severe distortions were investigated. In France, some towers were demolished because of the risk of collapse [43]; and the safety of almost 100 cooling towers with structural imperfections was evMuated in Great Britain during the last decade [56].

308 Thin-Walled Structures with Structural Imperfections: Behavior

t t

T

Figure 11.2" Geometry of a cooling tower shell Cases of damage of cooling towers leading to geometric distortions and loss of integrity have also been reported. A cooling tower in the US was under construction when it was hit by the centrally mounted tower crane that collapsed during a tornado [35]. The event induced a bulge with maximum amplitude of about 20 mm, and a V-notch in the upper region of the tower.

11.2

STRESSES IN A PERFECT COOLING TOWER

Unlike our previous studies on cylindrical, shallow a.nd spherical shells, in the hyperboloid it is not possible to obtain analytical solutions, and one must use numerical methods. Two numerical techniques have been employed in the literature: finite differences and finite elements, mainly in their semi-analytical versions, in which trigonometric functions are assumed in the circumferentia,1 direction. An account of semi-analytical and two dimensional finite elements for cooling towers may be found in Gould's book [34]. Before examining the behavior of an imperfect shell, it is importa,nt to consider the stresses in a cooling tower with perfect hyperboloidal

Hyperboloids of Revolution

309

....

O

500

O

Gravity , Wind + Gravity

i

E Z

0

z -500

z0 I

0

I

0.5

I

I

1

Z

R

Gravity Wind + Gravity 50

E

70 o

Z

Z

O4 O4

~ e O ~

-50

OOO

-100

0

1

Z

R

Figure 11.3" Stress resultants for wind and gravity load according to BS4485, for the shell of Fig. 11.2, f,'om Ref. [22]. Reproduced with permission of the American Society of Civil Engineers

310

Thin-Walled Structures with Structural Imperfections: Behavior

geometry. The specific example studied is shown in Fig. 11.2. This shell is representative of the British designs in the sixties. It has a maximum height of H - 106.45 m, and thickness t - 0.127 m, with top radius of 24.07 m, bottom ra,dius 39.4 m, throa,t ra,dius of 22.3 m, and the throat is at an elevation of 83.3 m. The material properties are E = 2.59 x l07 kN/m2; u - 0.2; and specific weight is ~ , - 22.5 k N / m 3. The elastic stress resultants under wind and gravity loadings have been obtained from a bending analysis using finite differences [22]. For the purpose of the analysis, the wind pressures were calculated for a design wind speed of 51.5 re~sac, and were taken as constant in elevation, and with a circumferential distribution as in the British Standard [9]. Load factors were 1.0 for dead load and 1.4 for wind load. The most significant stress resultants a,re Nll and N22, and their values are shown in Fig. 11.3 for the most stressed meridians. It may be seen that the values of Nll are approximately five times larger tha, n N22 at most elevations. The bending moments were computed, and were not significant away from the zone where the shell is supported on columns. Let us now look at the influence of different kinds of imperfections in the geometry, and how they affect the stresses.

11.3

INFLUENCE OF MERIDIONAL IMPERFECTIONS

The analysis of a cooling tower with a, meridional imperfection is the simplest one because the shell still has rotational symmetry, and at present it is the geometrical imperfection that has received more attention. We consider here the mechanics of stress redistributions and introduce the concept of out-of-balance moment. Then we investigate the parameters that define the stress field.

11.3.1

M e c h a n i c s of Stress R e d i s t r i b u t i o n

The mechanics of resistance of a, shell with imperfections in the meridia,n are first discussed for the cylinder of Fig. 11.4. In the perfect cylinder, the meridional forces are equilibrated by meridional stresses Nil. If the shell has an axisymmetric imperfection, the meridional stresses Nll at the top and center produce a couple M0 of value

Hyperboloids of Revolution

l'ltl

311

i

1" T ' - -

) N,,T

Figure 11.4: Membrane mechanism in a cylinder with meridional imperfection

(11.1)

M0 - ~ N l l

This couple is known as the out-of-balance moment [15]. Since N i l are usually large vaJues in the perfect shell, even small deviations ( may produce large out-of-baJance moments. Let us see how M0 is equilibrated: partial equilibrium is provided by changes in the bending moments, M~'I, and their overall contribution is MObending

9

---- / 1 1

Itop -Mll9 [c e n t e r

(11.2)

There is a second mechanism of resistance, which is associated to the membrane action in the circumferential direction. There are changes in hoop membrane stress resultants, N2".2, that develop bands of tension and compression, and the resulting forces of ea,ch one of them are identified here a,s T and C. They produce a, couple T.d, which is illustrated in Fig. 11.4. The meridional component of T.d contributes a resisting moment M y *~b~n*, of value equal to Mgembrane

__

f

z=zo+h/~ z N ~ 2 d z

J z--'zo

r

312 Thin-Walled Structures with Structural Imperfections: Behavior

I

II

-10

,g,

-20

Z

-30 O4 04

Z

-40

0.3

0.4

0.5

0.6

Z 0.8

H 2OO IO0

-100 -200 0.3

0.5

Z

0.8

H Figure 11.5: Stresses in the cooling tower of Fig. 11.2 with meridional imperfection, under gravity load [28]

Hyperboloids of Revolution

313

=

T.d r

(11.3)

In a thin shell, the bending stiffness is small, so that most of M0 is equilibrated by membrane action. The significance of the above mechanism in a cooling tower is discussed for the shell of Fig. 11.2 with an imperfection of cosine shape, centered at z0 - 0.5H, with ~ - -0.314 m and h - 30.4 rn. For gravity loading, the stress resultant N22 and moment resultant Mll are shown in Fig. 11.5, where it can be seen that modifications in the stress field are confined to the zone of the imperfection itself. At z - 0 . 5 H , the N22 stress resultants increase from 32 to 57 k N / m . For this particular example, the membrane action accounts for 98% of M0, and the rest is taken by bending action. This behavior of meridionally imperfect cooling tower has been confirmed by a number of authors [47] [15] [17] [16] [54] [3] [38] [45] [26] [37] [34] [4] [52] [5].

11.3.2

Influence of the P a r a m e t e r s that Define the Shell and I m p e r f e c t i o n

Several parameters have been used in the definition of the shell and the imperfection, and their influence on the behavior is discussed in this section. Such studies were first performed by Croll and co-workers using a direct finite difference model of the imperfect shell [17].

Profile of the Imperfection In the definition of an imperfection in the meridian of a cooling tower, different profiles of imperfection lead to different distributions of curvature errors along the extent of the imperfect zone. For cooling towers with different profiles of imperfections, and for a unit axial load, the maximum changes in hoop membrane stress resultants, N~2 , computed in Ref. [17] are in the range 0.54 __ N2*2 _< 1.13

(11.4)

For the most important types of imperfection models employed in practice, the va.lues given by Croll have been confirmed in Ref. [37], using a simplified solution for a cylinder. As indicated in Ref. [37], it is only for the long wave imperfections that the actual profile matters

314 Thin-Walled Structures with Structural Imperfections: Behavior on N2*2. The bending moments, on the other hand, are more heavily dependent on the imperfection shape, but in any case their contribution to equilibrium is small.

Imperfection Amplitude and Height The changes in N2*2 and in MI*1 are approximately proportional to the maximum amplitude ~ of the imperfection, at least for values ~/t <_ 2 [22] [37] [25]. If the sign of the imperfection is reversed, i.e. inner imperfection, as opposed to outer imperfection, the same linear relation between ~ and the changes in stress resultants is obtained, but with a different sign in the stresses. The wavelength, on the other hand, is inversely proportional to N2*2 and MI*1 for imperfections of intermediate height. More precisely, there seems to be a relation between the inverse of the square of the imperfection height, and the changes in the hoop stress resultants. In short imperfections, the hoop contribution decreases and the bending contribution increases as a consequence of the high curvature of the meridian.

Shell Thickness A change in t modifies the shell flexural stiffness relative to its membrane stiffness. As the thickness is reduced, N2*2 increases until it approaches a constant value, and the solution tends to a membrane sta,te [17] [25]. An interesting physical interpretation of the reduction of t rela.tive to the other dimensions of the shell has been presented in Ref. [17], as a, way to investigate" "... the consequences of a loss of vertical flexural stiffness in the shell. In the present context, such a loss of vertical flexural stiffness could result from a flexural yielding of the vertical steel, so that if the imperfection wavelength is not too short and the moment is large enough to cause flexural yielding, the shell would still appear to have the capability of accommodating such an imperfection so long as it is provided with sufficient, hoop strength".

Hyperboloids of Revolution

315

Shell Radius The changes in stress resultants have an almost linear relation with the radius r of the shell at the center of the imperfection [17]. This effect is importa, nt because of two reasons: First, for the same tower, there seems to be an influence of the specific location where the imperfection occurs (i.e., at different locations, the horizontal radius changes). Second, in larger towers, the stress resultants increase with respect to those found in smaller towers.

Curvature of the Meridian The actual curvature stress redistribution. parameters A and B, where the equation of

of the perfect shell may have an effect on the The geometry of a shell is a function of the the two axes of the hyperboloid of revolution, the meridian is given by r --

+ (z/A)

(11.5)

The magnitude of the meridional curva, tures of perfect shells are small in comparison with the curvature errors associated with the imperfections. For example, a shell with B / A - 0.41 has maximum values of curvature k l l - - 0.0014 r n m - 1 at the throat while the curvature error of the imperfection at that level may be X ~ I - - 0.0172 mm -1, that is, about 12 times larger. The shape of the geometric imperfections were kept the same for all these parametric studies, and given by H/h = 2.25; r/t = 40; ~ / r - 0.047 and h / r - 2/3. The results of Ref. [25] show that the changes in the response are not so dramatically changed by the differences in the shape of the meridian. For this reason, many authors model a cooling tower shell by an equivalent cylinder, which will be used in Section 11.6 and 11.7.

Nature of the External Loading The influence of different kinds of loading (i.e., gravity, wind, etc.), is only reflected in the value of the meridional stress resultant Nll in the associated "perfect" shell. This means that how this value of Nll is induced is not a primary factor, but it is the actual value that matters. This is related to the out-of-balance moment, given by Eq. (11.1).

316

Thin-WalledStructures with Structural Imperfections: Behavior

11.4

INFLUENCE OF CIRCUMFERENTIAL IMPERFECTIONS

In a circumferential imperfection, the error in curvature remains constant along the meridian. Particular attention is given in this section to imperfections with periodic variations around the circumference. The analysis of cooling towers with such imperfections is far more involved than cooling towers with banded imperfections, since it is not possible to assume a prismatic model for the imperfect shell. Finite difference and finite element studies are reported in Refs. [22], [26], and [48], and they axe concerned with the mechanics of stress redistributions.

11.4.1

M e c h a n i c s of Stress R e d i s t r i b u t i o n

The stress changes produced by a specific circumferential imperfection with eight full waves around the circumference are considered here. The amplitude of the imperfection is ( = -0.5 m, and the dimensions of the tower are given in Fig. 11.2. The results in this section were previously published in Ref. [26], a,nd were obtained using doubly curved shell elements with 36 degreesof-freedom per element. Symmetry was used to model a vertical sector of the shell under dead load. Wind load would require a mesh covering a. much la,rger a,rea. of the shell (from 0 to 27r in the circumferential direction). The changes in stresses due to purely circumferential imperfections affect the complete height of the shell, so that it is necessary to model the complete tower in elevation. The most important variations in the stress field occur for M22 and Nll. Values at critical meridians are shown in Fig. 11.6 for gravity load. The hoop moments, M2~, which were almost zero in the perfect shell, reach values of approximately 6 kNm/m in the imperfect shell; whereas the meridional stresses Nil a.t mid-height are increased from 150 to 300 kN/m, a.nd from 240 to 410 kN/m at the bottom of the shell. The maximum increments in Mll, on the other hand, are from 0 to 1.5 kNm/m; and in N~2 from 45 to 65 kN/m at the lower part of the shell. To understand the reason for the above increases in M22 and Nll, the simple static mechanisms of Fig. 11.7 are considered following the

Hyperboloids of Revolution

317

2

E

Z

0

f

ii

L____l

-2 -4 -6

t

t

I

I

I

0

I

I

I

0.5

Z

R

1

t~=O ~

Z

-loo IX

/

Z

f

f

I

f

~~ 1.~= IT, 8

-300 I

0

.t

t

I

I

I

I

I

I

0.5

1

Z

R Figure 11.6: Stresses in the cooling tower of Fig. 11.2, with circumferential imperfection under gravity load, from Ref. [28]

318 Thin-Walled Structures with Structural Imperfections" Behavior

[

r

r

TRId

N22 Figure 11.7: Membra.ne mechanism in a shell with circumferentia,1 imperfection arguments of Ref. [22]. The hoop stress resultant in the perfect shell, N22, acts on different parallel circles, thus producing an out-of-balance moment M0 given by (~ N22) at each eleva,tion. Contrary to the case of meridional imperfections, this value of (~ N22) is not constant in the meridional direction. A useful measure of the out-of-balance moments may be obtained by integration of (~ N22) along the meridian Mo - ~z z=H (~ N22)dz

(11.6)

"-0

Notice that now what is really important is the level of hoop stresses existing in the perfect structure, and the actual source is not entirely relevant. The results in Fig. 11.3 show that the combination of gravity and wind loads produces high values of compressive hoop stresses in the windward meridia,n, so that it is in this area that the out-of-balance moment will be highest. One obvious way in which the shell partially equilibrates M0 is by the development of circumferential bending moments M22, with a contribution to equilibrium given by [22] a,=o

*2 10=o-M2"2 [o=2"/j dz

(11.7)

in which j is the number of circumferential waves of the imperfection,

Hyperboloids of Revolution

319

and M2*2 are the changes in moments. A membra, ne mechanism of resistance is provided by alternate bands of tensile and compressive Nll, which produce a, couple (T.d), the circumferential component of which is M~ ~'~b'~n~ -

T.d R1

(11.8)

Eq. (11.3) is similar to Eq. (11.8), but since r is usually smaller than the radius of curvature of the meridian, R1, then the NI*1 stress in the circumferential imperfection should be much larger than the N2*2 stresses in the meridional imperfection to produce the same contribution. Thus, in a circumferentially imperfect tower, most of the out-of-balance moment is equilibra, ted by circumferential bending and a smaller part is ta,ken by membrane meridional action. The mechanism of equilibrium explained can thus equilibrate the out-of-balance moments, first, by increasing circumferential moments, and, second, by developing bands of tensile and compressive meridional stress resultants. But many reinforced concrete cooling towers show signs of vertical cracking, perhaps developed shortly after construction. The M22 capacity required is largely undermined by cracking, and this means that the first choice in the resistance of the shell would not be available. It is also possible tha, t the high circumferential bending exceeds the capa, city of the shell, producing cracking. The meridional stresses, already increased if the full bending capacity is present, would require to take the redistribution from M22 that cannot be ta,ken due to cracking. With designs of cooling towers not providing tensile and compressive strength to account for such large redistributions, the shell does not have the capacity to withstand normal design wind loads. It is believed that this mechanism of redistribution played an important pa,rt in the colla.pse of the cooling tower a.t Fiddlers Ferry [12].

11.4.2

Parameters Influencing Behavior

Let us consider which geometric parameters change the stresses associated to the imperfection. In this section we study the influence of the wavelength of the imperfection, the amplitude, and the thickness of the shell. Other parameters axe also considered, such as the meridional curvature of the perfect shell.

320 Thin-Walled Structures with Structural Imperfections: Behavior

Wavelength of the Imperfection In the example chosen for parametric studies, an imperfection with eight full waves ( j - 8) was assumed. Results from Ref. [22] show that for this shell, the stresses N~I reach a maximum, while the bending contribution is 50 % of the out-of-balance moment. For j > 8 (short wave imperfections), the stresses NI*1 decrease, but the M2*2 contribution increases, until all M0 is equilibrated by pure bending. On the other hand, for j < 8 (long wa,ve imperfections), M~ and NI*~ decrease, and most of M0 is taken by the membrane mechanism. The conclusions from Ref. [22] are that" "For very short wavelengths, in which the wavelength is shorter than one twentieth part of the circumference, the resistance arises principally from the development of circumferential bending action. For longer wavelengths, alterna,ting bands of meridiona,1 tension and compression contribute a significant part to the resistance of the effects of circumferential imperfections" [22].

Imperfection Amplitude Again, as with the meridional imperfections, the changes in stress resultants are approximately proportional to the amplitude of the imperfection ~, provided the amplitude is small. But it is expected that non-linearity will show even more than in banded imperfections, so that some caution must be exercised with linear results.

Thickness of the Shell Changes in the thickness affect the bending stiffness of the shell, and as such they modify the ratio of membrane to bending contributions. A thinner shell ha.s a. reduced bending capacity (depending on t 3) but the membrane stiffness is less affected (linea,rly with t). But the influence of vertical era,eking would also be to selectively reduce the bending stiffness ra,ther than the membrane stiffness, and some insight into this can be gained by reducing the shell thickness. The results of Ref. [22] show that with a reduction of 40% in the thickness the ]l/[bendin9 decreases by 20% and NI*a is increased by 50% .~

~vz 0

~

9

Hyperboloids of Revolution

321

In the limit, as t ~ 0, the shell tends to have a membrane behavior, in which N~'1 increases by a factor of 5. This shows that for circumferential imperfections it is crucial to provide both hoop bending and meridional membrane resistance to take the additional stresses induced by a,n imperfection.

11.5

INFLUENCE OF LOCAL IMPERFECTIONS

We say that an imperfection is local if the magnitudes of the curvature error in the meridian and in the circumference are approximately the same. The direct analysis of such large imperfection requires the use of general two-dimensional shell elements, even for gravity loads. A more economic approach to do the computations is presented in Ref. [39] which combines general two-dimensiona.1 shell elements in the zone of the imperfection with a,xisymmetric elements in the rest of the shell. The equiva, lent loa,d method has also been employed, including three terms in the first order and in the second order equivalent loads. Notice that to simulate interactions between different curvature errors occurring in bulge imperfections, second order equivalent loads cannot be neglected. The most significant second order contribution is due to the meridional stresses, so that "four separate analysis, three for the first order and one for the second order load, are necessary for a bulge imperfection in a typical cooling tower shell" [36]. Even though, this equivalent loa,d model can only represent imperfections of small amplitude.

11.5.1

Case S t u d y

One particular type of imperfections is discussed here, namely local imperfections of periodic nature around the circumference. The numerical model becomes manageable by introducing the assumption of circumferential periodicity, and only a sector of the entire shell needs to be considered in a finite element discretization. For the shell of Fig. 11.2, the influence of a local imperfection with eight full waves, ( = 0.5 rn, h = 60.8 m and centered at the midheight of the tower has been a,na,lyzed. Such imperfection introduces

322 Thin-Walled Structures with Structural Imperfections: Behavior

E

2

E Z

0

r

"2

-4

0

0.5

Z

1

R

0 -100

Z -300

h 0

I

I

I

I

I

0.5

I

I

I

I

Z

R Figure 11.8" Stresses in the cooling tower of Fig. 11.2, with local imperfection under gra,vity load, from Ref. [28]

Hyperboloids of Revolution

323

maximum errors in curvature of X~a - 1.3 x 10 -a xm and X~2 - 2.5 x i0-3 1. 77~

The numerical results in terms of stress resultants shown in Fig. 11.8, show large variations in Nix and M22, as in the circumferential imperfection. Also, N ~ a,nd Mix have high changes, as in meridional imperfections, but in this case, the effects associated to circumferential errors dominate the solution. But even for the imperfect case with X~I - X'~2, the stresses from the circumferential errors are larger than those due to the errors in the meridian [26]. In order to classify an imperfection, it seems that a better indicator would be to consider the products (NIl ~1)and (N22 X~2), rather than the dimensions of the imperfection. One importa.nt parameter studied in Ref. [26] is the height of the imperfection, and it has been shown tha, t it is the purely circumferential imperfection tha.t produces the largest changes in NIl and M22. From Ref. [22], which studied a bulge imperfection with changes in the vertical extent h, the bending action M22 showed not to be very sensitive to h, but the meridional stress contribution Nl1 increased with h. Thus, for a given circumferential wavelength, short meridional imperfections produce smaller stress concentrations than purely circumferential ones. It seems that a conservative a,ssumption would be to model imperfection in the meridian only and imperfections in the circumference only, as extreme cases for design.

11.5.2

Antisymmetric Imperfections

The stress redistributions in cooling towers with antisymmetric imperfections were considered in Ref. [45]. The imperfections studied had a meridional deviation and with circumferential variation in the form of the sum of harmonic numbers 0, 1, and 2. Static stresses were computed under self-weight, seismic and wind loading, using a non-linear analysis. For the imperfections of Ref. [45], the errors in the circumferential direction are small in comparison with the meridional direction, so that the maximum changes in stress resultants are very similar for harmonic imperfections 0, 1, and 2. However, under dead load, antisymmetric imperfections show both positive and negative values of stress resultants at different locations a,round the circumference, so that this is a major difference with banded or axisymmetric imperfections with the

324 T h i n - W a l l e d S t r u c t u r e s with S t r u c t u r a l Imperfections" B e h a v i o r same meridional wavelength and amplitude. From the non-linear studies, the changes in hoop stress resultants are found to be proportional to the amplitude { of the imperfection, for a given imperfection height h. The bending action is summarized in Ref. [45] as follows

M e , - MI', -

k U~

(11.9)

where M~ and N~ the stress resultants in the perfect shell; Mal is the bending resultant in the imperfect shell; and kb is a coefficient given by the authors. This is a non-linear relation. The results are not significantly affected by the type of imperfection or loads studied in Ref. [45].

11.5.3

Bulge Imperfections

Isolated bulge imperfections have been studied in Ref. [36], for a 106 m height cooling tower under gravity and wind loads. The bulge has a vertical height of 12.6 m and a circumferential angular extent of 300, with amplitude of ~ = 0.304 m. Changes in the membrane and bending stress resultants a,re very similar to those presented in the previous section, with the stress disturbances being confined to the zone of the imperfection. This means that bulge imperfections may be approximated by models with periodic nature around the circumference, provided the perfect stress field in the area of the imperfection is employed. Finally, if a bulge imperfection has a large amplitude (i.e. even for ~ / t - 4) but extends over a very large part of the shell with very smooth changes in curvature, the resulting stress changes are not very importa,nt. This is reported in Ref. [48] using a membrane theory of shells.

Hyperboloids of Revolution

11.6

INTERACTION BETWEEN GEOMETRIC IMPERFECTIONS AND CRACKS

11.6.1

Motivation

325

A most illustrative case of interaction between geometrical imperfections and cra,cks is the tower a,t Ardeer, mentioned in the introduction of this chapter. The Ardeer cooling tower had severe shape imperfections even at the time of construction. After the collapse, a committee of inquiry investigated the effects of the imperfections by theoretical methods, and showed that imperfections induced tensile hoop membrane stresses of the same order of magnitude as the meridional compression. In an untracked shell, these high tensile stress resultants could have been carried by the concrete without major problems. However, existing cra,cks (proba,bly of therma,1 origin) grew into the zone of imperfection and eventually left, the steel reinforcing bars to provide the necessary hoop action. With a low percentage of circumferential steel, as in the case of the Ardeer shell, the cracked sections would not have been able to develop the necessary hoop tensions because of yielding of the reinforcement. Failure of the shell would thus occur at lower wind speeds than would be the case for the uncracked but imperfect shell. The committee concluded that the most probable cause of the collapse was the imperfections in the shape together with meridional cracking that developed during its service life. But theoretica,1 calculations carried out by the committee of inquiry were based on an homogeneous and elastic idealization of the tower. In contrast, the Ardeer shell showed severe meridional cracking before collapse, so that in the calculations it wa,s "only possible to speculate on the redistribution of stress which would take place following era,eking a,nd/or local failure of the hoop reinforcement" [42]. The numerical study of interaction between geometrical imperfections and discrete meridiona,1 cracks reported in this section was carried out using a doubly curved shell finite element described in Ref. [25].

326

Thin-Walled Structures with Structural Imperfections: Behavior

E E

,.%

1

-- i

."- %~

s~S

Z

i.

(',4 O4

i

Z

0=0

.,~. I

~

/

\

/~-~ " ~

/

..

,~_..s

"

T

-1

-0.5

0.0

0.5

z/H

0 I".~/2 H=O I "~. 1 N22_1~--" - -- yL---.-~. ~-: 1

0 11.9"

0

15

Figure Hoop stress resultants in the shell of imperfect shell (1) Vncracked; (2) Eight cracks

Fig.

10.14, for

The geometric imperfections were modeled by defining the actual geometry of the imperfect meridians, with an axisymmetric cosine shape for the imperfection assumed in all cases. The effects of discrete cracks were also incorporated in the finite element analysis. It has been shown that in a shell of revolution a crack produces significant redistributions of stresses only when there is no membrane stiffness present [32] [25]. In the context of reinforced concrete shells, very.thin shells, such as cooling towers, are less sensitive to concrete cracking than thicker ones, provided that the reinforcement remains elastic at the crack. Full cracks, in which there is no membrane stiffness present, correspond to cracked concrete and steel yielding for an increment of the load, after yield of the horizontal reinforcing bars has occurred, and they can be considered as stress-free boundary conditions.

Hyperboloids of Revolution

E E

327

100

E

~'~.

.//" \'(

o

E

.

_

Z

.

.

2

1

.

J T==

-loo

k./

o=o

I

I

-0.5

0.0

100 -

9

2

1

zlH

0.5

z/H=O

0 0

0

15

Figure 11.10" Meridional moments in the shell of Fig. 10.14, for imperfect shell (1) Uncracked; (2) Eight cracks

11.6.2

Case Study

To highlight the main features of the behavior of this class of shell, one example with similar material properties and local geometry to the Ardeer cooling tower in the region of a major imperfection (that is, between 55 and 67 m above soffit), has been studied in Ref. [27]. It, is shown that the meridional curvature of the perfect shell does not significantly affect the concentration of stresses in shells with axisymmetric imperfections or with such imperfections and cracks. Because of this, here we concentrate on a cylindrical shell form, with the same radius in the region of imperfection as in a cooling tower. However, if the region of imperfection were positioned lower down the tower, where slope effects may be significant, it, would be advisable to take into account the actual geometry of the shell. The data, for the problem studied is shown in Fig. 10.14, with horizontal radius r = 22.5 m; shell thickness t = 0.15 m; and an

328 Thin-Walled Structures with Structural Imperfections: Behavior

0=0 I

9

X %

9

E E

X

t

I

Z

I

9

k

t

, I

I I

I o

Ib

Z

-2

%

--

I

x

%

! %

,

1 .~

i,,

0

X X

.

.

,

.

S

I

9

to

%

~

2

I

9

#

0

e I

%

~..- I

..--.,

9

t

'

I

-0.5

0.0

Nll

z/H

I_.,'

0

0.5

I I

I

-2

--!

I # f

0

f

1

t

2 z/H=O

0

20

Figure 11.11" Meridiona,1 stress resultants in the shell of Fig. 10.14, for imperfect shell (1) Uncracked; (2) Eight cracks

Hyperboloids of Revolution

329

inward imperfection with maximum amplitude .s = 0.305 rn and height h = 15 m. Convergence studies were carried out, and a finite element mesh with 216 elements, modeling half of the shell length and half of the spacing between cracks, was selected to perform the calculations. As the circumferential membrane contribution is crucial in equilibrating the out-of-balance moment, it is very important to understand what would happen if the membrane circumferential resistance of the shell were lost at some section for further load increments, as a consequence of yielding of the horizontal bars. The behavior of the shell with eight full meridional cra.cks has been studied for a unit vertical axisymmetric load, and the stress resultants are shown in Fig. 11.9 to 11.11. The cracks are assumed to occur at a uniform spacing, with length equal to the height of the banded imperfection. In the cracked shell, the hoop stresses N22 become very high at the extremes of the crack, but are zero along the crack. Thus, for a unit axial load, the hoop stresses contribute a much smaller part to the equilibration of vertical force in the vicinity of the crack than in the uncracked shell. This reduction in the hoop stress mechanism at the cracked meridian has two important consequences" First, there is an increase in bending moments Mll near the crack (in the example of Fig. 11.10, Mll is increased by a factor of 10); and second, there is a redistribution of the vertical stresses Nll to meridians away from the crack (Fig. 11.11). Thus, associated with the redistribution of membrane stress resultant there is a redistribution from the dominantly membrane resistance to a bending resistance in the region of the crack. Away from the crack, the bending action decreases as a consequence of the increasing capacity of the bands of hoop tension and compression N22 to provide the resistance to the out-of-balance moment. Such a mechanism has been discussed in Ref. [33]. In the present example, a compressive axisymmetric load and an inward imperfection are considered. Different combinations of axial force (tensile or compressive) and of axisymmetric imperfection (inward or outward) produce different concentrations of stresses. However, for the analysis of cooling tower shells, mainly compressive forces are of interest, because the maximum tensile forces occurring on the windward meridian(for wind and gravity load) may be about 10 times smaller than the maximum compressive forces at the meridian 72~ from the wind direction (see, for example, Ref. [42], Appendix J). For an inward imperfection under compressive load, the maximum

330

Thin-Walled Structures with Structural Imperfections: Behavior

tensile hoop stress resultant N22 occurs near the ends of the imperfections in the uncracked shell; whereas the worst combination of Nll and Mla in the cracked shell is at the center of the imperfection. For an outward imperfection, the situation would reverse, with the maximum tensile N22 at the center of the imperfection in the uncracked shell, and the worst combination of Nll and Mll near the ends of the imperfection in the cracked shell. The results of Ref. [25] for curvatures ranging from a cylinder to a typical cooling tower show small variations in the stresses, but in any case, it is in the cylinder that the stress changes are highest.

11.7

C R E E P OF I M P E R F E C T COOLING TOWERS

The survey of dimensional errors in a cooling tower at the Rathcliffe Power Station [56] showed that there was an increase in the size of the geometrical deviations of the order of 100%, in a, period of about 12 years. The purpose of this section is to investigate if this kind of imperfection growth could be accounted for by viscoelastic effects. We follow here the work of Ref. [53] in which the simplified model of axisymmetric geometric imperfection is combined with simple creep models.

11.7.1

Creep Models

There are several models employed in the literature to approximate the effects due to creep in concrete structures. Some of them may be seen in Refs. [23] and [13]. Let us restrict our attention to problems in which linear constitutive equations are a, good approximation to the behavior. This may be relevant in problems of concrete shells under self weight, in which case the stresses at mid-height of the shell may be of the order of 1 / 10 of the ultimate compression stress. For such linear problems, the principle of correspondence is valid for constant stresses, and the effect of creep can be replaced by an elastic solution in which the modulus are adapted at each instant in time. Three visco-elastic models are considered here:

Hyperboloids of Revolution

331

Solid M o d e l of Kelvin In this model creep is restricted to the deviatoric strains, and volumetric effects are considered to be elastic. A Kelvin solid with three parameters may be adapted to compute Poisson's ratio u and modulus of elasticity E as p-

fie -~(t-t') q- 1

(11.10)

E - 3K(t')[(qo- ql/~)e -;~(t-t')

q-

ql]

(11.11)

where t' is the time at which the loading started; t is the time at which strains are computed; K(t') is the volumetric modulus of the material; and p, r/, )~, q0, and ql are non-dimensiona,1 parameters evaluated in laboratory tests. Coefficients like these have been obtained by Vandepitte [59].

Solid M o d e l of Kelvin with C o n s t a n t Poisson's Ratio According to Bazant [7], a good estimate can be obta,ined by using a consta, nt value of u and adjusting the value of E due to creep. This is not a, drastic simplification for concrete, in which variations in u due to creep may be between 0.15 and 0.25. Equation (11.10) is now simplified to ~ = 0.18

(11.12)

and (11.11) is of the same form but with new set of non-dimensional parameters.

Fluid M o d e l For short times, say 10 days, fluid models may be a better representation of the behavior of a, material like concrete. Ballesteros [6] used Eq. (11.12) a,nd a modulus of elasticity given by 1

E

_

1 [1 + r Eo

~]

(11.13)

332

Thin-Walled Structures with Structural Imperfections: Behavior

. . . . .

Solid model Fluid model

= 0.25

& 0

o3

= 0.5

a--0.5 = 0.75

0~=1

0

20

40

60

80 100 120 140 160

t-(days) Figure 11.12: Displacements in reinforced concrete cylinder with geometrical imperfection and creep [53]

11.7.2

Creep in an Imperfect Cooling Tower Shell

In the following we combine creep models for reinforced concrete structures with the simplified model of a geometrical imperfection tha, t has rotational symmetry. We consider a shell with r = 24.5 m; r/t = 161; r/(80; r / h - 1.95; u - 0.15; and self weight N~~ - 187.3 N/mm. The da.ta, of the different creep models has been obtained from the literature, and a,re K(t') - 15, 196 N/ram2; p - -0.84; r/ - 0.0001133; A = 0.0385; q0 = 0.505; and ql = 0.182, for the solid model of Kelvin. In the model with constant u we adopt K ( / ' ) - 16,146 N/mm2; q0 - 0.476; and qa - 0.171. The fluid model is based on n - 1/8; r 1.103;and E0 - 31,000 N/'mm 2. The stress level was obtained at 0.7 of the total height of a cooling tower shell of 105 m. At the center of the imperfection, the displacements were to grow in Fig. 11.12 from one (instantaneous elastic deflection) to a,lmost three times in 130-150 days, and then they remain a,lmost constant. Virtua, lly, the same results were obtained for the two solid models. The fluid model was observed to depend on the value of the pa,rameter r For solid models, the changes in displacements with

Hyperboloids of Revolution

333

10

4

~

~11_3 O0days " 1 0 0 days 50 aays

2 -

0

"

0

0.25

~

0.5

'

~

0.75

....... 0 days

10

(X

Figure 11.13" Displacements in reinforced concrete cylinder with geometrical imperfection, change in hoop stiffness, and creep [53] time ra,nge from 0 to 100 days.

11.7.3

Combined Creep and Changes in Stiffness

Next, let us combine the creep actions with a reduction in the hoop stiffness, as it would happen in a shell with closely spaced fine cracking. Results for this case are presented for displacements and a,s a function of a, reduction in the hoop stiffness, as expressed by the parameter c~ = 1 - T. The influence of a reduction in hoop stiffness is important, and can change the level of displacements. For example, a reduction of hoop stiffness of 50% duplicates the displacements. A more complete picture of the maximum displacements can be obtained from Fig. 11.13, in which we ha,ve plotted maximum displacements versus reduction in hoop stiffness for different times. However, looking only at changes in displacements may be misleading, because the elastic deflections are themselves very small. Thus, one has to consider hoop stresses and meridional moments. This is illustrated in Fig. 11.14. Linear changes occur in time, but even for 300 days and high reductions in stiffness, the changes in stress levels

334

Thin-Walled Structures with Structural Imperfections: Behavior

E

Z Z

ol

Solid model, 1.1Variable, G = 1

158.00 t 157.98 R 157.96 157.94 157.92 157.90

t"

0

I

50

I

I

100 150

I

200

I

250

r"

300

t-(days) Figure 11.14: Hoop stress resultant in a reinforced concrete cylinder with geometrical imperfection, change in hoop stiffness, and creep [53] are not significant. Moment changes are plotted in Fig. 11.15. The overall conclusion of this section seems to be that creep itself does not induce a significant growth in the amplitude of a geometrical imperfection, and consequently, the increase in stresses due to creep is not very high. Cases with higher stresses due to creep may occur in structures in which the elastic instantaneous deflections are of the order of the thickness of the shell. For such studies linear viscoelasticity would not be an accurate representation of the phenomenon, and nonlinear viscoela, stic properties should be taken into account.

11.8

FINAL

REMARKS

In this chapter we have considered various aspects of structural imperfections in cooling tower shells made of reinforced concrete. There is a large number of research papers published about, imperfect cooling towers, comparatively to other shell forms, including the spherical and shallow shells. Cooling towers are among the largest structures ever built by men, and there may be practical difficulties in the construction of these structures that lead to structural imperfections. In ma,ny towers there is a combination of shape imperfections with

Hyperboloids of Revolution

335

Solid model, 15 Variable, (X = 1

E -1.36

-1.34

~r

I

I

I

I

I

I

0

50

100

150

200

250

300

t-(days) Figure 11.15" Moments in a, reinforced concrete cylinder with geometrical imperfection, change in hoop stiffness, and meridiona,1 bending with creep [53] cracking. The studies in this chapter have shown the important changes in stress resultants in cooling towers with imperfections in the meridian and in the circumference. In a homogeneous shell, this may be severe but perhaps can be taken by the shell; however, in a cracked shell some of the required resistance is not available. In Section 11.6 we have shown possible redistributions due to the interaction between cracks and shape imperfections. Finally, viscoelastic effects have been included in the models to investigate changes with time. The results show that this is not in itself a plausible explanation for the growth of imperfections observed in real cooling towers, and more complex models should perhaps be used to account for that. The relevance of this information in practical situations is discussed in the next chapter, where the possible collapse mechanisms in real cooling towers are outlined. The role of tolerances in design guidelines is also discussed in Chapter 12.

336 Thin-WMled Structures with Structural Imperfections: Behavior

References [1] Abel, J. F. and Billington, D. P., Effect of shell cracking on dynamic response of concrete cooling towers, Proc. IASS World Congress on Space Enclosures, Building Research Center, Concordia University, Montreal, 1976. [2] ACI-ASCE Committee 334, Reinforced Concrete Cooling Tower Shells, Practice and Commentary, Report A CI 334.2R-91, America,n Concrete Institute, Detroit, 1991. [3] A1-Dabbagh, A. and Gupta, A. K., Meridional imperfection in cooling tower design, J. of the Structural Division, ASCE, 105 (ST6), 1979, 1089- 1102. [4] Alexandridis, A. and Gardner, N. J., Direct evaluation of meridional imperfection forces in cooling tower shells, Civil Engineerin 9 for Practicing and Design Engineers, 4, 1985, 847-881. [5] Alexandridis, A. and Gardner, N. J., Tolerance limits for geometric imperfections in hyperbolic cooling towers, J. Structural Engineering, ASCE, 118(8), 1992, 2082-2100. [6] Ballesteros, P., Nonlinear dynamic and creep buckling of elliptical paraboloidal shell, Bulletin of the Int. Association for Shell and Space Structures, 66, 1978, 39-60. [7] Ba,zant, Z. and Wu, S. T., Dirichlet series creep function for aging concrete, J. Engineering Mechanics Division, ASCE, 99(2), 1973, 367. [8] Billington, D. P., Thin Shell Concrete Structures, Second Ed., McGraw-Hill, New York, 1990.

Hyperboloids of Revolution

337

[9] British Standards Institution, BS~,.~85, Part ~" Structural Design of Cooling Towers, BSI, London, 1975. [10] British Sta,ndards Institution, Dra.ft revision of BS4485, Part 4" Structural Design of Cooling Towers, BSI, Document 92/17117 DC, London, 1992. [11] Central Electricity Generating Board, Report of the Committee of Inquiry into the Collapse of Cooling Towers at Ferrybridge, CEGB, London, 1965. [12] Centra,1 Electricity Generating Board, Report on Fiddlers Ferry Power Station Cooling Tower Collapse 13 January 1984, CEGB, London, 1975. [13] Creuss, G., Viscoelasticity: Basic Theory and Application to Concrete Structures, Springer-Verlag, Berlin, 1986. [14] Croll, J. G. A., Discussion on" Meridional imperfection in cooling tower design, J. of the Structural Division, ASCE, 106 (ST3), 1980, 744-746. [15] Croll, J. G. A. and Kemp, K. O., Design implications of geometric imperfections in cooling towers, Proc. IASS World Congress on Space Structures, Montreal, 1976. [16] Croll, J. G. A. and Kemp, K. O., Specifying tolerance limits for meridional imperfections in cooling towers, J. of the American Concrete Institute, 76, 1979, 139-158. [17] Croll, J. G. A., Kaleli, F. a.nd Kemp, K. O., Meridionally imperfect cooling towers, J. of the Engineering Mechanics Division, ASCE, 105 (EM5), 1979, 761-777. [18] Croll, J. G. A., Kaleli, F., Kemp, K. O. and Munro, J., A simplified approach to the analysis of geometrically imperfect cooling tower shells, Engineering Structures, 1, 1979, 92-98. [19] Croll, J. G. A., Ellina.s, C. P. and Kemp, K. O., Collapse as a stimulus for research to improve design, Proc. Conf. Institution of Mechanical Engineers, U.K., 1981.

338 Thin-Walled Structures with Structural Imperfections" Behavior [20] Diver, M., and Patterson, A. C., Large cooling tower shells" The present trend, The. Structural Engineer, 55(10), 1977, 431-445. [21] Ellinas, C. P., Croll, J. G. A. and Kemp, K. O., Towards tolerance specifications for circumferential geometric imperfections, Proc. IASS Conf. on Very Tall Reinforced Concrete Cooling Towers, Paris, 1978. [22] Ellinas, C. P., Croll, J. G. A. and Kemp, K. O., Cooling towers with circumferential imperfections, J. of the Structural Division, ASCE, 106 (ST12), 1980, 24O5-2423. [23] Flugge, W., Viscoelasticity, Blaisdel, Walthom, MA, 1967. [24] Fullalove, S. and Greeman, A., Cooling tower collapse exposes calculated gamble, New Civil Engineer, The Institution of Civil Engineers, U.K., 19 January 1984, 4-5. [25] Godoy, L. A., Stresses in Shells of Revolution with Geometrical Imperfections and Cracks, Ph. D. Thesis, University College, University of London, London, 1979. [26] Godoy, L. A., Tensiones en torres de enfriamiento con imperfecciones geometricas: Evaluation de dos tecnicas numericas, Proc. XXI South America.n Conference on Structural Engineering, Rio de Janeiro, 1981. [27] Godoy, L. A., On the collapse of cooling tower shells with structural imperfections, Proc. Institution of Civil Engineers, Part 2, 77, 1984, 419-427. [28] Godoy, L. A., Influence of geometric imperfections on cooling towers, Proc. Conf. on Structural Analysis and Design of Nuclear Power Plants, Porto Allegre (Brazil), vol. 1, 1984, 67-85. [29] Godoy, L. A., On loads equivalent to geometrical imperfections in shells, J. Engineering Mechanics, ASCE, 119(1), 1993, 186-190. [30] Godoy, L. A., Discussion to" Cause of deformed shapes in cooling towers, J. Structural Engineering, ASCE, 121, 1995.

Hyperboloids of Revolution

339

[31] Godoy, L. A., Croll, J. G. A., Kemp, K. O. and Jackson, J. F., An experimental study of the stresses in a shell of revolution with geometrical imperfections and cracks, J. of Strain A nalysis, 16(1), 1981, 59-65. [32] Godoy, L. A., Croll, J. G. A., and Kemp, K. O., Stress redistribution in vertically cracked cooling tower type shells, Proc. IASS Conf. on Very Tall Reinforced Concrete Cooling Towers, Paris, 1978. [33] Godoy, L. A., Croll, J. G. A. and Kemp, K. O., Stresses in axia.lly loaded cylinders with imperfections and cracks, J. of Strain Analysis, 20(1), 1985, 15- 22. [34] Gould, P., Finite Element Analysis of Shells of Revolution, Pitman Advanced Publishing Program, Boston, MA, 1985. [35] Gould, P. L. and Guedelhoefer, O. C., Repair and completion of damaged cooling tower, J. Structural Division, ASCE, 115(3), 1989, 576-593. [36] Gould, P. L., Han, K. J. and Tong, G. S., Analysis of hyperbolic cooling towers with local imperfections, II Int. Symposium on Natural Draught Cooling Towers; Bochum, 1984, 397-411. [37] Gupta, A. K. and A1-Dabbagh, A., Meridional imperfection in cooling tower design: Update, J. of the Structural Division, ASCE, 108 (STS), 1982, 1697-1708. [38] Han, K. J. a.nd Gould, P. L., The effect of geometric imperfection on cooling towers, Bulletin of the Int. Association for Shell and Space Structures, 20, 1979, 63-68. [39] Han, K. J. and Gould, P. L., Shells of revolution with local deviations, Bulletin of the Int. Association for Shell and Space Structures, 20, 1984, 305-313. [40] Han, K. J. and Tong, G. S., Analysis of hyperbolic cooling towers with local imperfections, Engineering Structures, 7, 1985, 273-279. [41] IASS Working Group, Recommendations for the Design of Hyperbolic and other Similarly Shaped Cooling Towers, Int. Association for Shell and Spatial Structures, I.A.S.S., Brussels, 1977.

340 Thin-Walled Structures with Structural Imperfections: Behavior [42] Imperial Chemical Industries, Report of the Committee of Inquiry into the Collapse of the Cooling Tower at A rdeer Nylon Works, Ayrshire on Thursday 27th September 1973, ICI Ltd., Petrochemicals Division, London, 1973. [43] Jullien, J. F. and Reynouard, J. M., Reflections on the origin of deflections observed on the cooling towers of Pont sur Sambre a.nd Anserevilles, IASS 3rd Int. Symposium on Natural Draught Cooling Towers, Paris, 1989, 595-604. [44] Jullien, J. F., Aflack, W. and L'Huby, Y., Cause of deformed shapes in cooling towers, J. Structural Engineering, ASCE, 120(5), 1994, 1471-1488. [45] Kato, S. and Yokoo, Y., Effects of geometric imperfections on stress redistributions in cooling towers, Engineering Structures, 2, 1980, 150-156. [46] Kato, S., Miyamura, A. and Murata, M., Geometric imperfections and buckling in cooling tower shells, II Int. Symposium on Natural Draught Cooling Towers, Bochum, (Ed." P. L. Gould et. al.), Springer-Verlag, New York, 1984, 264-278. [47] Kemp, K. O. and Croll, J. G. A., The role of geometric imperfections in the colla.pse of a cooling tower, The Structural Engineer , 54, 1976, 33-37. [48] Langha, ar, H. L. and Boresi, A. P., Effect of out-of-roundness on stresses in a cooling tower, Thin,-Walled Structures, 1, 1983, 31-54. [49] Larrabee, R. D., Billington, D. P. and Abel, J. F., Thermal loading of thin- shell concrete cooling towers, J. of Structural Division, ASCE, 100 (ST12), 1974, 2a67-23S3. [50] Leckie, F. A., Palacios, O. F. and Williams, J. J., The influence of meridional cracking on the vibration of cylindrical shells, Proc. Institution of Civil Engineers, U.K., 51, 1972, 91-106. [51] Moy, S. S. and Niku, S.M., Finite element techniques for the analysis of cooling tower shells with geometric imperfections, Thin Walled Structures, 1, 1983, 239-263.

Hyperboloids of Revolution

341

[52] Niku, S. M., Finite Element Analysis of Hyperbolic Cooling Towers, Springer-Verlag, Berlin, 1986. [53] Novillo, N. and Godoy, L. A., Deformaciones diferidas en cilindros de hormigon con imperfecciones, In: Colloquia 87, Porto Allegre (Brazil), 1987, 119-131. [54] Oyekan, G. L., Analysis of cooling towers with structural imperfections, Ph. D. Thesis, The University of Southampton, Southampton, 1978. [55] Pope, R. A., Structural deficiencies of natural draught cooling towers at U.K. power stations. Part 1" failures at Ferrybridge and Fiddlers Ferry, Proc. Inst. Civil Engineers, Structures and Buildings, 104, 1994, 1-10. [56] Pope, R. A., Grubb, K. P. and Blackhall, J. D., Structural deficiencies of natural draught cooling towers at U.K. power stations. Part surveying and structural appraisal, Proc. Inst. Civil Engineers, Structures and Buildings, 104, 1994, 11-23. [57] Rascon, O. A. and Mendoza, C. J., Design Manual for Civil Engineering Works, Section C, Topic 2, Chapter ~," Cooling Towers, Federal Electricity Board, Mexico, 1981. [58] Soare, M., Cooling towers with constructional imperfections, Concrete, 1967, 367-379. [59] Vandepitte, D. and Laga,e, G., Buckling of spherical domes ma.de of microconcrete a.nd creep buckling of such domes under longterm loading, IUTAM Symposium on Non-Linear Analysis of Plates and Shells, Springer-Verlag, Berlin, 1985. [60] Woolley, G. R. and Van der Cruyssen, D., Structural deficiencies of natural draught cooling towers at U.K. power stations. Part 3" strengthening of natura.1 draught cooling tower shells, Proc. Inst. Civil Engineers, Structures and Buildings, 104, 1994, 25-37.

C h a p t e r 12 IMPERFECTIONS PRACTICE 12.1

IN

INTRODUCTION

Chapters 2 to 6 of this book were devoted to techniques of analysis used to account for the influence of stress cha,nges in thin-walled structures with imperfections. In Chapters 7 to 11, we looked at different shells and introduced simplified forms of imperfections to illustrate the stress changes associated with certain imperfections. Such studies are relevant for the understanding of the mechanics of the behavior of imperfect shells, but are based on idealiza,tions of some imperfections. In this chapter we review some more practical aspects related to imperfections, and attempt to answer questions about 9 What imperfections do occur in real structures? 9 How are imperfections detected a,nd measured? 9 Is there a relation between patterns of imperfections and fabrication processes? 9 Have actual structures failed because of the stresses induced by imperfections? Of the hundreds of applications of shells in different fields of engineering, only a few ca,n be considered here, certainly those more closely 342

Imperfections in Practice

343

related to the interest of the author, and those that have received important coverage in the technical literature. But even a few specific examples should serve as a guide to problems that may be present in other shell structures.

12.2

SURVEYING GEOMETRICAL IMPERFECTIONS

How do we actually measure an imperfection in the geometry? What tools do we need? Can one use the same technique in any shell, or do we have to consider specific techniques adequate for certain structural types? In this section we review techniques to measure amplitudes of imperfections in the shape of large shells, of the size of cooling towers, and then move to smaller ones, used in the off-shore and astronautics industries.

12.2.1

Surveying Cooling Towers, Silos, and Other Large Reinforced Concrete Shells

Practical techniques to survey the actual shape of a large reinforced concrete shell were stimulated by the need to assess the safety of existing cooling towers. For shells with heights between 60 and 160 m, and thickness between 0.10 and 0.25 m, one needs to record deviations in the geometry with amplitudes that can reach two or three times the thickness. This is a. specialized field, and it requires some expertise to obtain meaningful records. There is a balance that should be satisfied between accuracy and cost of the survey. A comprehensive study of adequate techniques was presented in Ref. [31], and is followed in this section. A practical way of recording the shape of a large shell of revolution is by means of triangulation using theodolites. This technique requires the use of several survey stations, of which two are used simultaneously to acquire information on a part of the shell. The dimensions of the shell are obtained with respect to a polygonal line ABCD...H, that joins the survey stations. A specific point T on the shell surface is identified by a single laser beam generated at L, as illustrated in Fig.

344

Thin-Walled Structures with Structural Imperfections

Figure 12.1" Geometric imperfections in a large shell of revolution, obtained by triangulation 12.1. The angular position of T is obtained from stations A and B, which include horizontal and vertical angles. The length LT can be computed from the recorded angles and the length AB. To map points along a meridian of the shell, the laser beam is rotated to hit points at different elevations in the shell. From the same two stations A and B, a number of meridians can be surveyed. For cooling tower shells, the number of points measured to obtain an accurate surface is of the order of one thousand [31]. In the method of theodolite triangulation the observations have to be recorded and stored in a computer to speed the surveying process. One advantage of this method is that redundant information is generated for each point T, thus, it is possible to check the accuracy of the readings and discard those that have higher errors than the ones admitted for the readings. Other techniques can also be employed, such as the use of a single laser range-finder device. Just one observation is here necessary to

Imperfections in Practice

345

obtain the required distance LT: it can be obtained by one person using one instrument. Problems with this technique are reported in Ref. [31]. A third alternative uses photogrammetry, and also allows to obtain the irregular surface of the imperfect shell. The photographs themselves can be kept afterwards for future inspection. Other factors may also influence the adoption of one technique or a,nother: tria, ngula, tion should be done by night, so that the laser spot is visible from the theodolites; wherea,s single laser and photogrammetry are done during the da,y. Evalua, tions of accuracy indicate better results from triangulation. This technique proved to be accurate a,nd economic for the survey of about 100 cooling tower shells, operated by the Central Electricity Generating Board of the United Kingdom, between 1985 and 1987.

12.2.2

Surveying Large Stiffened Metal Shells

Our second report on geometrical records of shells is from a study of tubular members for the off-shore industry, carried out at Cambridge University in England [36] [16]. The studies comprised large tubular members with diameters of 7 and 9 m, with skin thickness of 0.025 to 0.04 m in the smaller diameters. The ratios r/t are in the range of 115 to 215. The tubular members axe used to provide buoya, ncy in three structures that were later installed off-shore. Mainly out-of-circularity was recorded, and this was initially done by taking readings of the internal diameter of the shell at different positions. An estimate of the degree of out-of-circularity can be obtained by considering the extreme values of diameters recorded at one level, that is Dmax and Dmi~. The error would thus be given by

e

--

Dmax- Drain D

A problem may be anticipated with such readings in cases where the diameter has positive and negative initial displacements. For example, in Fig. 12.2, one would find that the error e is a smaller value than what is real. Another problem is that "the out-of-roundness extracted only the magnitude of the geometric imperfection and disregarded its shape" [35].

346

Thin-Walled Structures with Structural Imperfections

Figure 12.2: Problems with measuring diameters to obtain out-ofroundness

Figure 12.3: Profile gauge

Imperfections in Practice

347

A profile gauge can also be used to measure out-of-roundness in a sector of the shell. In this case, it is the gap between the shell and the profile gauge that should be measured, as indicated in Fig. 12.3. A third technique takes internal readings of radius, rather than diameters. This is done with the help of a piano-wire that measures from the center of the shell to the inner surface. The circle that best fits the readings is identified by means of a least square approximation. This radial deviation dr with respect to the best fit is normalized by the diameter, i.e.

f=

dr D

and the out-of-roundness is given by the maximum value of f. This short review based on Ref. [16] indicates wha, t the oil companies and fabricators use to accept or reject cylindrical shells in the off-shore industry. More accurate laboratory techniques have been developed in the aerospace industry. Full-scale shell structures were surveyed since 1975, and the technique is described as follows:

"The first shell measured was a 3.05 m diameter aluminum alloy integral stiffened shell . . . The technique employed measured the deviation of the cylinder outer surface relative to an imaginary cylinder reference surface. Here this was accomplished with a 3.05 m long aluminum guide rail supported on the outside diameter of the two steel end rings and a direct current differential transformer instrument on a trolley car. The trolley car, which was spring loaded to roll with continuous contact along the guide rail, was slowly driven along the guide rail by an electric motor. The position of the car was electronically measured using a, potentiometer that rotated as the car moved along the track. The accuracy of the displacement measured was to within +0.05 mm." [35].

Other techniques are reviewed in Ref. [35].

Thin-Walled Structures with Structural Imperfections

348

12.2.3

Scanning Small Scale M e t a l Shells

Shells are also used in small scale models that attempt to represent the behavior of larger shells but in a laboratory environment. Many such shells are made of steel or aluminum, with diameters between 0.07 and 0.40 m. Information about shapes of imperfections in such small shells is vital mainly beca,use they are used to evaluate buckling capacities. The techniques for measuring geometric, imperfections in small shells of revolution are very different from those described previously, a,nd are based on scanning the inside surfa.ce with contact dial gauges. An example of the use of those measurements is reported in Ref. [6].

12.3

DETECTION OF IMPERFECTIONS MATERIAL

IN THE

The evaluation of imperfections that affect the physical, rather than geometrical, properties of shells is difficult in most cases. In the context of this book, such imperfections are modeled by imperfection parameters, but they arise from different sources according to the constitutive material of the shell, fabrication details, etc.

12.3.1

Material I m p e r f e c t i o n s in Reinforced Concrete Shells

In most reinforced concrete shells, damage is a localized consequence of 9 gradual deterioration, leading to long-term changes in structural properties 9 natural disaster, which suddenly changes the structural properties Visual inspection of reinforced concrete shells often reveals significant levels of damage. Fig. 12.4 shows damage of the concrete and exposed reinforcement in a shell of revolution of large dimensions. Concrete spalling is another form of structural damage to be expected in reinforced concrete structures, and is illustra, ted in Fig. 12.5.

Imperfections in Practice

349

Figure 12.4" Exposure of reinforcement due to poor cover (Photograph by the author)

Whenever shells like these are part of a power plant or an industrial site, the workers and engineers tend to keep a record of such structural imperfections. More careful records are sometimes done to assess the integrity of the structure. In the systematic evaluation of about one hundred cooling towers in Great Britain [31], cradles were suspended from the top of the tower, in order to inspect the outer surface of the shell. Each damage found has to be recorded and identified in the shell, to follow any possible evolution. Core samples are important to identify the state of the reinforcement in parts of a shell. Off course, extracting material in a shell that already has damage may be dangerous, as it tends to weaken the strut-

350

Thin-Wailed Structures with Structural Imperfections

Figure 12.5: Exposure of reinforcement due to spalling (Photograph by the author) ture. Usual core samples are cubes of 0.1 m. per side. At a cooling tower in Zapla, Argentina, core samples revealed high corrosion in the reinforcing bars, and the diameter of some bars was between 1/4 and 1/3 of their original values [21]. Non-destructive techniques should be preferred whenever possible. There are many such techniques available nowadays, and only a couple are mentioned in the following: 9 Low a m p l i t u d e v i b r a t i o n tests" The structure is subject to low amplitude excitation, and the dynamic response is recorded. Then, a system identification technique is employed to estimate the actual stiffness of the structure [1]. C h a n g e s in m o d a l p a r a m e t e r s " "During the past decade, many research studies have focused on the possibility of using the vibration characteristics of structures as an indication of structural damage ... More recently, attempts have been made to monitor structural integrity of bridges ... and to investigate the feasibility of damage detection in large space structures using changes in modal parameters ... Also, a few studies have investigated modal uncertainty in structural systems ... and assessed

Imperfections in Practice

351

Figure 12.6" Cracks in the dome of the church at Hormigueros, Puerto Rico (Photograph by the author) the effect of model uncertainty on damage detection accuracy ... [28] 9 R a d a r t e c h n i q u e s " They are based on surface scanning impulse radar systems. The equipment for pulse radar is formed by a "pulse generator connected to a, transmitting antenna, which is usually combined with a similar receiving antenna to form a single transducer, and is connected to a signal processor, data storage and display system" [8].

12.3.2

Masonry Structures

The maintenance and repair of historic masonry buildings is vital to preserve the cultural heritage of a society. Unlike modern structures,

352

Thin-Walled Structures with Structural Imperfections

for which we (should) have enough documented information, in most historic buildings we do not even know the exact year of completion of the work. Cracks, as well as geometric imperfections, are often found in church domes made of masonry. An example is shown in Fig. 12.6. But not only damage parameters are unknown for such studies: the masonry itself needs to be identified. Non-destructive techniques provide valuable information about the real situation of the structure [33]. In cases like this, radar techniques may be of great help, from the estimation of wall thickness to actual damage inside the wall. Such techniques are valuable when only one surface of the shell is accessible, as is often the case of church domes, where the outer shell can be reached but not the inner surface.

12.4

SOME RECORDS STRUCTURAL IMPERFECTIONS

OF

In previous sections of this chapter we have mentioned ways employed to measure imperfections, specially those of geometric nature. When the structure has been completed, the imperfections can be recorded and the measurements can be introduced in the analysis; which could be done using the techniques presented in Chapters 3 and 4. In modeling a, problem like this, one could use the complete information of the imperfection or some idealization concerning the features of interest. But often an engineer or researcher needs information about realistic imperfections expected to occur in certain structural forms, from the previous experience of other engineers and researchers. This information has great value at the design stage, if one wants to look at, stress changes due to a typical imperfection that may be induced during fabrication or construction. The importance of collecting information about imperfections in thin shells has been recognized for some time, but only one attempt has been made to create a data bank. This is discussed next.

Imperfections in Practice

353

Ref. [16]

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12.4.1

Examples of Imperfections in the Aerospace Industry

The International Initial Imperfection Data, Bank is the effort of researchers at the University of Delft and the University of Haifa to record systematic information about imperfections in metal cylindrical shells [5]. The motivation is the study of buckling of shells used in the field of astronautics. Such very thin shells exhibit buckling characteristics that are very sensitive to small imperfections. Although this information has been collected with aerospace erientation in mind, we can still take advantage of it as input in studies of stress redistributions. An example of the imperfections recorded is presented in Fig. 12.7, taken from Ref. [5]. To compare data from different sources, it is important to have the information available in a systematic way. A convenient representation is by writing the geometric deviations as a doubly Fourier series, i.e.

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354

T h i n - W a l l e d S t r u c t u r e s with S t r u c t u r a l I m p e r f e c t i o n s

circumferential direction (summation in k), and usually 5 half waves in the longitudinal direction (summation in 1). All deviations are measured with respect to the so-called best fit cylinder. A Fourier characterization of imperfections has the additional advantage that it can be more easily incorporated into finite element data, and specially so in semi-analytical elements. Furthermore, it is easy to compare different imperfections, and classify them. Finally, it is possible to correlate Fourier contents of imperfections with different fabrica,tion techniques. Typical plots of imperfection data are presented in the form of an amplitude ( versus the circumferential wave number l, for specific values of meridional half waves, k. The amplitude ~ is defined as ^

- v/C~z + D~,

(12.2)

and is normalized with respect to the thickness of the shell. Several shells are considered in Ref. [5], where it is interesting to distinguish those made by machining and those assembled from panels. 9 Shells made in a laboratory by machining seamless tubes are frequent in research. The shells considered in [5] have stringer stiffeners, and the data shows that the largest amplitude of occurs for k - 1 and 1 < l < 5. In one specific example, the maximum amplitude found was d - 0.34, with 1 - 2 (out-ofroundness) and k - 1. For k - 2, the maximum was d - 0.21; while for k - 3, the maximum was ~ - 0.11. 9 When shells are assembled from identical panels, rather than machined, the imperfections show two maximum values" one due to out of roundness (1 - 2), and a second in coincidence with the number of panels which form the shell, or multiples of it. For example, for six identical longitudinal panels with riveted joints, in a shell with r - 945 ram, r / t - 1488 and L / r - 2.9, and closely spaced Z-shape stringers, the second maximum occurs for l - 6, for which ( - 2.4. ^

9 The relative importance of the out-of-roundness errors depends on the fabrication process" for example, carefully machined rigid endings may considerably reduce the amplitudes of l - 2 . This also occurs in pressure vessels, whenever end-domes are forcedfitted into the cylindrical part.

Imperfections

in P r a c t i c e

355

9 The imperfection pattern changes if the joints are welded, rather than riveted. In welded joints, the maximum amplitudes ~ are associated with one half wave in the longitudinal direction and with multiples of the number of welds in the circumferential direction. The information of the International Data Bank [5] clearly shows a correlation between the shape of the imperfection and the technique of fabrication employed, at least for shells that are currently used in the aerospace industry.

12.4.2

Examples of Imperfections in the Off-Shore Industry

Our second excursion is in the field of compliant off-shore structures, again dealing with metal cylindrical shells. Measurements of imperfections were reported in Refs. [36] and [16], for shells with radius r -- 4,600 r a m , r / t - 184 to 115; and r - 3,650 r a m , r / t - 228 to 152. The shells were formed by an assembly of panels welded at three joints. They had ring stiffeners located closer than a radius apart, and stringer stiffeners in some parts. Those large tubular members were used to provide buoyancy. Some characteristics of the imperfections, of relevance to the presentations in this book are as follows: The measured circumferential imperfections showed a clear influence of the number of panels used in the fabrication. This is reflected in the example of Fig. 12.7. At the seams, the shell moved inwards, and the number of circumferential waves was clearly dominated by l = 3. It is reported that special care was taken to avoid out-of-roundness errors by controlling the dimensions of the ring stiffeners. 9 More localized imperfections occurred in the vicinity of the welds. The circumferential welds induced local distortions in the longitudinal direction reached amplitudes up to 6 r a m in a shell with r - 4,600 r a m . 9 Welds in the longitudinal direction induced imperfections that were clearly visible, leading to cusps towards the outside of the shell. Typical angles of these cusps were 3.5 to 4.5o.

Thin-Walled Structures with Structural Imperfections

356

9 Misalignement at welds are also detected, and led to values of the order of 6 m m in amplitude [36]. The limited examples in this section, taken from the off-shore industry, tend to confirm the general results of measurements of the aerospace industry, although specific amplitudes are very much influenced by the fabrication process and overall dimension of the shells.

12.4.3

Examples of Imperfections in Cooling Towers

There are many records of imperfections in hyperbolic cooling tower shells in the technical literature, but there has not been any attempt to produce a data bank comparable to that available in the aerospace industry. Another example is provided by a tower surveyed in 1975 and reported in Ref. [17]. Horizontal deviations are plotted in Fig. 12.8 for this case. The figure shows imperfections at three elevations in the tower, namely at 20, 40 and 60 m above soffit level. The most significant outward imperfections reach values of ~/t > 3, with very small amplitudes of inward imperfections. If we consider the level z = 20 m, the dominant number of circumferential harmonics is approximately eight. But the imperfect shell cannot be described simply in geometric terms, so cracks (denoted by triangles in Fig. 12.8) have to be investigated. The state of the concrete and steel in the zones of cracks is assessed by taking cores from the shell. The shell has one single layer of reinforcement, and core samples reveal flexural yielding of the circumferential reinforcement. This tower was constructed in the early 1960s, and it is clear that imperfections have grown with time during the service life of the tower. Due to the cracks, the shell behaved as if it had panels among themi and the joints at the cracks acted as concentrators of deformation, with reduced moment stiffness. In a way, the shell is similar to those metal cylinders considered before, for which panels welded at the joints induced imperfections. Further measurements of imperfections and cracking have been reported in Ref. [31]. In 1984, the Central Electricity Generating Board

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(CEGB) of Great Britain operated 139 large reinforced concrete cooling towers, of which 74 were 114 m height and 65 towers were smaller in size but with minimum height of 65 m. Following the failure at Fiddlers Ferry, reported in Section 12.5, the CEGB investigated the imperfections in the towers and produced perhaps the most important assessments available in this field. Out of a number of towers investigated, and for which field surveys were conducted, most appeared to have imperfections in the geometry and cracks. Two cases are discussed in Ref. [31]" tower B2 at West Burton Power Station, and tower 2A at Radcliffe Power Station. Radial imperfections at West Burton are shown in Fig. 12.9, as recorded in 1986 at 33 m above soffit level. The maximum outward errors detected in the tower were close to 600 mm. For a shell of nominal thickness of 127 mm, this is an error of over four times the thickness. The situation is even worse if one considers that the actual thickness measured in the tower was reduced by 10 to 15% at certain places. The imperfections showed a circumferential pattern with nine waves. Cracks were detected in coincidence with such high peaks in the imperfect surface on both sides of the shell. Corrosion reduced the diameter of the bars of the single layer reinforcement from 10 mm to values of

Imperfections in Practice

359

6 mm at some locations. Other imperfections were also found concerning displacements of the single layer of reinforcement towards the inside of the shell, leaving sections with about 50% of the cover originally designed. Imperfections recorded at Radcliffe are shown in Fig. 12.10, at elevations 24 and 31 m above soffit. This is a very interesting case, because it shows the evolution of radial errors between 1975 and 1986, and between 1986 a,nd 1989. Errors in 1975 were of the order of 280 mm, and increased to 330 mm in 1986. Crack widths of 15mm were also recorded in this tower. According to Ref. [31], maximum errors of 650 mm were measured at Radcliffe. The cause of deformed shapes in cooling towers has been the subject of some debate recently. Factors influencing the growth of imperfections can be identified as: 9 Non uniform settlement of the foundation, as reported in Ref. [15] 9 Stresses due to thermal effects, arising from the normal operation of the tower, as well as from solar action 9 Cracks, caused by shrinkage of the concrete and by thermal effects, are often present since the construction of the tower is completed. At such weakened sections, the growth in amplitude of imperfections may be accelerated 9 There is also the possibility that the reverse can occur: "it is likely that many meridional; cracks below the throat (in the body) of cooling tower shells a,re induced by as-constructed errors in radial dimensions" [31] 9 Other authors have related the growth of imperfections to buckling [27], but at present there seems to be limited supporting evidence for that [20]

12.4.4

Examples of Imperfections in Spherical Shells

Large steel spherical shells are used in the nuclear industry. The spheres are assembled by welding carefully controlled panels; however,

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Imperfections in Practice

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there are cases of damage that occur during the final construction of the shell. Records of such damage for a shell with r/t = 933 are shown in Fig. 12.11. Here, damage-induced imperfections are localized at an elevation of 45o above the equator of the sphere and affecting less than 1/4 of the total circumference. They reach maximum errors of the order of four times the thickness of the shell.

12.5

COLLAPSE OF STRUCTURES WITH IMPERFECTIONS

12.5.1

Failure as a Guide

to Success

In a recent book about error and judgment in engineering, Petrovski argues that failure is central to the process of design, since successful designs are achieved by thinking about how to avoid failure. "It has long been practically a truism among practicing engineers a.nd designers that we learn much more from failures than from successes" [29]. Other authors have also discussed the growth of knowledge as a consequence of the failure of structures [7]. As engineers, we tend to be confident whenever a certain design proves to be safe. Sibly and Walker [34] investigated the failures of large metal bridges in Great Britain and North America, and showed that designs that were considered successful at some stage, contained "a certain factor that was of secondary importance with regard to stability or strength. With increasing scale, however, this factor became of prima,ry importance and led to failure. The accidents h a p p e n e d . . , because of the unwitting introduction of a new type of b e h a v i o r . . . Following a period of successful construction a, designer . . . simply extended the design method once too often" [34]. Relevant to the subject of this book, we find that the history of the construction of shell structures contains several dramatic failures that have been attributed to the influence of imperfections becoming a primary factor, rather than a secondary one. A major change in a design, that may lead to a collapse of this kind, is the change in the scale of the construction. Nowhere has the author

Thin-Walled Structures with Structural Imperfections

362

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Figure 12.11" Geometric imperfections in a steel spherical shell

Imperfections in Practice

363

seen such important change in scale as in the case of increasing size and slenderness of cooling towers. Table 12.1 shows some towers that were designed at different places since the beginning of the century.

TOWER

COUNTRY

YEAR

H[m]

Heerlen Hams Hall Ferrybridge Niederaufiem Troja.n Neur at h Dampierre Mulheim K ernkr aft werk s Pampat Tihange Kendal Dry

Netherlands UK UK Germany USA Germany France Germany Germany India Be lg ium USA

1914 1931 1965 1966 1971 1972 1976 1976 1980 1980 1980 1984

35 68 114 117 151 126 160 162 160 124 159 165

Table 12.1 The remarkable increase has reached heights over 160 m. These are some of the largest constructions in the world. Small imperfections were not significant in the small towers, with large r/t ratios; but became important with the increasing slenderness of the shell. But a similar trend is found in other structures, such as off-shore platforms, nuclear containments, industrial and sports buildings, and many others. During this century we have also witnessed tremendous advances in techniques of analysis using computers, a,nd the power to compute often gives us a, sense of control over the design. However, we can only compute what we model and can only model by including approximations about effects that we understand, and neglect other effects that we do not understand. This means tha, t every design involves a number of approximations, some of which are simply not true. "Virtually all design is conducted in a state of relative ignorance of the full behavior of the system being designed" [29]

Thin-Walled Structures with Structural Imperfections

364

"Calculations are not made in the abstract; someone must be motivated to make them and must realize their relevance and significance for a possible new or altered failure mode" [29]. In this section we report a few dramatic failures of large cooling tower shells. We should try to look at such failures as full scale experiments, carried out under loads that are not fully known, and with properties that can only be approximately evaluated.

12.5.2

Ferrybridge, England, 1965

This failure was not due to imperfections in the shell, but it was considered to be so influential in future practice that we start our review here. The eight towers were cone-toroids, a shape not used nowadays, with a height of 114 m. The collapse of three towers at Ferrybridge occurred on November 1st, 1965. The investigations following the collapse pointed towards high meridional tensile stresses developed on the windward side. The committee of inquiry identified "tensile failure within the shell fabric to have been the dominant initial mode of failure" [11]. As a consequence of this failure, wind loads assumed in the analysis, buckling loads, and vibration, were investigated in several research centers.

12..5.3

Ardeer, Scotland, 1973

The Ardeer cooling tower at Ayrshire, Scotland, was constructed following the collapses at Ferrybridge, and it was known to comply with the findings of the committee of inquiry. The tower was completed in 1966. The construction a,t Ardeer presented imperfections in the geometry below the throat, with maximum amplitude of the order of 305 mm at 55 m above soffit. Meridiona! cracks were also found during inspections, and one crack was reported to extend into the zone of large geometrical imperfections. The Ardeer tower colla,psed at wind speeds estimated between 70 mph (31m/s) and 80 mph (36 m / s ) o n September 27th, 1973. The debris of the tower were found in a different position than in the Ferrybridge case: now the top ring and most of the remains of the shell

Imperfections in Practice

365

were found inside the base diameter of the shell. Another difference was that the columns and the bottom part of the shell remained standing at Ardeer. The committee of inquiry investigated several possible causes, and found that geometrical imperfections played a major role in the "tension failure of the circumferential reinforcement"; and this was considered as "the most probable initiating cause of the collapse of the Ardeer tower" [25]. Imperfections must have induced tensile hoop membrane stresses of the same order a.s the vertical compressive stresses. In an untracked shell, these high tensile stresses could have been carried by the concrete without major problems, and would not have caused cracking for the loads that the tower had to withstand between 1966 and 1973. However, the existing cracks that grew into the zone of major imperfections left these sections with only the reinforcement to provide the necessary hoop action. With only 0.15% of reinforcement placed in a single layer, the cracked sections would ha.ve not been able to develop the circumferential contribution required for equilibrium. Failure of the reinforcement would thus occur for lower wind speeds than in the untracked shell.

12.5.4

Fiddlers Ferry, England, 1984

The cooling tower B2, at Fiddlers Ferry Power Station, in England, was designed to be 115 rn high, with shell thickness of 127 ram. The tower was under construction when three towers collapsed at Ferrybridge, in November 1965. Because of the concern created by the event, the construction at Fiddlers Ferry stopped until the calculations were verified; this led to a change in the shell, which was redesigned with two layers of reinforcement instea.d of one, and a stronger top rim was provided. The wind loa,d considered in the design was high for the standard practice in the sixties. Errors in the geometry were recorded during the construction of the tower, with a bulge developing at the bottom of the shell, mainly affecting a horizontal band between 6 and 13 m above soffit. The amplitude of the circumferential errors (between 101 and 116 ram) did not exceed the thickness of the shell, and it was thought that such changes in the tower would not be relevant for its safety. The tower collapsed under high winds, with gusts over 80 mph

Thin-Walled Structures with Structural Imperfections

366

(36 m/s), on January 13th, 1984 [18]. The position of the debris at Fiddlers Ferry was similar to what was found at Ardeer [30], and the investigation considered the influence of structural imperfections as a possible cause of the collapse. Calculations of stresses for gravity loads indicated high hoop tensile stresses, close to what would be required for cracking of concrete. With a low (0.15%) percentage of steel, the high hoop stresses would initiate a sequence of events leading to the collapse. Thus, the committee of inquiry identified the presence of the outward circumferential bulge above the ring beam as the most important factor contributing to the collapse of the tower at Fiddlers Ferry [12] [30].

12.5.5

Other Collapses

Other failures of cooling towers in France have been reported:

"In 1979 at the Bouchain power station in France, a cooling tower, which had been in service for only ten years, collapsed when the wind load was apparently minimal, the shell of this structure was known to have serious dimensional errors: these could have been the cause of stresses which may have been instrumental in the failure. Other towers with severe distortion of their shells were identified at the Pont sur Chambre and Ansereuilles stations, and were demolished because of the risk of failure" [31]. An account of the problems in Pont sur Chambre and Ansereuilles may be found in Ref. [26].

12.6

POSSIBLE EXPLANATIONS FOR COLLAPSES

12.6.1

Collapse Mechanism for Axisymmetric Imperfections

Following the work in the previous chapter, here we concentrate on a mechanism leading to the collapse of a cooling tower with similar reinforcement characteristics to the Ardeer shell. Simple approximations

Imperfections in Practice

367

to the behavior of the shell are used, and it is argued that the combination of horizontal steel yield under tension and meridional steel yield under flexural action at some local areas could lead to useful estimates of the collapse load. The Ardeer tower was constructed with a low ratio of horizontal steel (0.15% of the cross-sectional area), but with higher values of vertical steel (0.44% in the levels studied) on account of the uplift forces [25] and following the teachings of a previous collapse at Ferrybridge [11]. In the region of the tower where major imperfections were later detected, the reinforcement was placed on a single layer. For a shell with such reinforcement characteristics, and based on the studies of Ref. [25] and those of Chapter 11, it is possible to envisage a mechanism leading to collapse. Thus, the following sequence of events (associated with increments in compressive axial load for an inward imperfection) leading to a form of collapse, may be suggested: (a) Initially, the shell develops high hoop membrane stresses and meridional bending to equilibrate the out-of-balance moments. (b) As the load is increased, yield of the horizontal reinforcement occurs in the regions of maximum tensile stresses. For any further increments of the load there would be a redistributions of stresses from hoop membrane to meridional bending. (c) At the ends of the zone of yield there are high hoop tensile stresses, so that an extension in the length of the zone of yield is produced. A situation is reached where the propagation of yield is accompanied by an insignificant increase in the contribution of N22 to the equilibration of the out-of-balance forces. (d) As no additional membrane action can be produced by the shell in the vicinity of the crack, high values of moments Mll are developed there. (e) With an increase in the load, the meridional reinforcement reaches yield locally. (f) As a result of the low bending stiffness of the shell, yield of the meridional reinforcement rapidly propagates along the imperfection and next to the crack.

Thin-Walled Structures with Structural Imperfections

368

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Imperfections in Practice

369

(g) The area, near the crack becomes unable to resist any increase in the load. At this stage, there is a significant circumferential redistribution of the vertical load, with the meridians between cracks developing high hoop membrane stresses. (h) For further increments in the load, the whole area along the imperfection and between the cracks is unable to provide global equilibrium. In the proposed mechanism, it has been assumed that yield of the horizontal steel under tensile stresses propagates along the crack as the load increases. Even if the high compressive stresses were able locally to close the crack, the shell would still reach a situation in which it would be unable to provide the bands of tensile hoop stresses, so that there would be no increment in the membrane contribution to equilibrate the out-of- balance moment. A numerical investigations to follow the growth of plasticity and the associated redistributions of stresses outlined above would require a non-linear material finite element model of the shell. However, some insight into the problem may be obtained by means of a simplified model and this is discussed next.

12.6.2

A n e s t i m a t e of t h e collapse load

With an appropriate finite element code, it is possible to follow some important stages in the mechanism leading to the collapse of a cooling tower. To do so, some simplifications have been made in Ref. [19]: The concentrations of stresses that a,ffect the whole area of the imperfection need large load increments to produce local yield, whereas very local concentrations of stresses(singularities in the stress field) propagate with small load increments. A simplified model based on these assumptions would only require the computations of steps (a), (b), (d) and (e) in the mechanism of the collapse previously described. It has also been assumed that once the zone in the vicinity of the crack has reached local yield of the horizontal and vertical reinforcement, the shell would not have enough ductility to redistribute stresses away from the crack. Thus, steps (g) and (h) will not be computed. The above ideas were applied to the shell of Fig. 10.14 to estimate a collapse load under axial loading condition. The values for

370

Thin-Walled Structures with Structural Imperfections

reinforced concrete considered are f' - 31 N/mm2; f~ - 2.1 N/mm2; and fy - 350 N/mm 2. From the values of Fig. 12.12, in the uncracked shell, a meridional load of 185 N~mm produces a maximum tensile N22 of 81 N/mm and a maximum moment Mix of 0.89 kNmm/mm. This value of N22 is still below the concrete tensile strength of the section, but if the shell has meridional cracks (i.e. due to thermal effects), the ultimate strength of the section would be reached when the circumferential stress yields. For a shell with 0.15% horizontal reinforcement, this would occur when N22 = 81 N/mm, and any increase in the load would not produce an increase in N22 (as shown in Fig. 12.12), but the extra out-of-balance moment would be resisted by flexural action. On account of the assumptions of the model, for increments in the load after yield of the horizontal reinforcement, the shell would behave as one with full vertical cracks. For example, a load of 250 N/mm would produce a moment of 5 kNmm/mm; and for a load of 295 N/mm, Mix would reach its limit. In this case, the meridional bending capacity has been calculated taking into account the interaction between Mll and Nll, with a tensile Nll decreasing the Mll capacity of the section. The load versus Mll path is shown in Fig. 12.12. For this simplified model, it seems that the shell would still have some 60% of extra load capacity after yield of the circumferential reinforcement has occurred (p = 185 N/mm) and before the meridional flexural capacity is exceeded ( p - 295 N/mm). Stresses in the Ardeer tower for wind and gravity load at the meridian 72 ~ from the wind direction (where maximum compressions occur) and at 63 m above soffit (where major imperfections were detected) are next considered. For a 128 Km/h wind velocity, the maximum compressions computed were 384 N/mm (see Ref. [25], Appendix J, Table 2). As it is believed that the collapse of the tower occurred for a wind speed near to 128 Km/h (35.6 re~s), the simplified model of Ref. [19] produces a conservative estimate of the collapse load, with the actual collapse load being 30% higher than the estimate computed. It is important to notice that the shell with 0.3% horizontal steel and the same imperfection and cracking condition would have a much larger hoop membrane capacity, with the estimated collapse load being 475 N/mm (Fig. 12.12). This is well above the stresses at the time of collapse, and the results suggest that a shell with simila.r imperfections to the Ardeer tower, but with a 0.3% hoop reinforcement, would not have collapsed under the same wind conditions.

Imperfections in Practice

371

In the previous studies, the interaction between Mll and Nll was taken into account to specify a limit to the meridional flexural capacity of the shell. If such a limit is fixed on just flexural strength of the section, the estimated collapse load becomes p - 413 N/mm, which is slightly above the stresses at the time of collapse.

12.6.3

Discussion

In the comparison between the present simplified model and the Ardeer cooling tower at the time of collapse, the nature of the simplifications made must be considered. These are: first, the cylindrical shape, as opposed to the hyperboloidal shape of the real tower; second, the axisymmetric nature of the imperfection assumed, which is an approximation of the more complex and localized imperfection presented in the tower; third, the axisymmetric nature of the load in the finite element model, while the load varied in the circumferential direction in the real structure under gravity and wind load; and fourth, the assumptions about the plastic behavior of the shell. However, the simple model for the shell described above is consistent with the mechanism of collapse which was believed to occur in the tower, and produces a reasonable estimate of the collapse load. Such a simplified model may contribute to the evaluation of the safety of an existing tower with structural imperfections. The practical conclusions that could be drawn for designers seem to be that if the horizontal reinforcement is designed in such a way as to carry the membrane hoop stresses in an imperfect shell, then even if the cooling tower develops vertical cracks in zones of geometric imperfections, the horizontal steel will be able to develop hoop action. However, if imperfections are not considered in the design of the horizontal steel, but do occur during construction, there is danger of the horizontal reinforcement reaching yield, with the consequence that the severe redistribution of stresses described in this section may take place.

12.6.4

Collapse M e c h a n i s m for Circumferential Imperfections

In the previous considerations, only imperfections with nearly axisymmetric imperfections were mentioned. We saw in Chapter 11 that there

Thin-Walled Structures with Structural Imperfections

372

are significant differences between the mechanism of stress redistributions due to circumferential imperfections and due to axisymmetric imperfections, so this should be reflected in the mechanism of failure. The importance of circumferential imperfections is considered in this section, following the studies of Ref. [31]. The sequence of events leading to the collapse of the structure with circumferential imperfections may be suggested to occur as follows: (a) Initially, the shell develops high circumferential bending and meridional membrane stresses to equilibrate the outof-balance moments. (b) Vertical cracking of the concrete may occur from some other actions. It is speculated in Ref. [31] that tensile stresses due to geometrical imperfections could be the cause of vertical cracks below the throat. (c) In a cracked section, yield of the horizontal reinforcement occurs due to circumferential bending. (d) As no additional bending action can be produced by the shell in the vicinity of the cracks, high values of Nll are developed there. (e) Under increasing load, the meridional reinforcement is strained until it reaches yield. (f) For further increments of the load, the shell is unable to provide global equilibrium.

12.7

GEOMETRIC LIMITS

TOLERANCE

12.7.1

Design Recommendations

The information about the mechanics of stress redistribution due to structural imperfections should be reflected in the codes of practice and in the design recommendations available to the engineer. But sometimes the recommendations provided are too general to be of any direct use. An example of the lack of specific guidance is the following sentence, taken from a code for nuclear plants:

Imperfections in Practice

373

"The engineer shall specify the tolerances for the shape of the shell. If construction results in deviations from the shape greater than the specified tolerances, an analysis of the effect of the deviations shall be made". [4]

12.7.2

Reinforced Concrete Silos

Construction tolerances for reinforced concrete silos specify the errors in the geometry and thickness variations admitted. "The maximum horizonta,1 deviation .. of any point on the structure relative to a corresponding point at the base of the structure shall not be over 80 mm at any location in any 30 m of height, nor over 100 mm total for silos over 30 m high" [3]. Out-of-roundness is limited to 25 mm or 4 mm per m of diameter, whichever is larger; and should not exceed 75 mm. It is also specified that the changes in the thickness should be in the range [-10 mm, 25 mm][3].

12.7.3

Cooling Towers

Among the first to introduce tolerance limits to geometric deviations were the British Standard and the IASS recommendations of 1977. The British Standard for cooling towers [9] fixed a maximum error in the slope of the meridian of +1%, and a maximum error in the horizontal radius of +5 cm. The minimum specified reinforcement was 0.20% of the concrete cross sectional area, in both directions. The IASS working group on cooling towers [24] fixed a maximum errors in the slope of the meridian of 4-1.5%, with the error in the horizontal radius not exceeding +0 10 m or the value of 1 x / ~ . The minimum reinforcement recommended is 0.25% high tensile steel or 0.35% mild steel in both directions. The reasons for having such tolerance specifications is not clear, and may be the "result of an intuitive feeling of what has been possible in past practice" [14]. After the collapse of the Ardeer tower, Croll and Kemp proposed to specify tolerances suing the membrane mechanism of the shell, leading to 9

374

Thin-Walled Structures with Structural Imperfections

<

---~h

-

N::

24 r Nll

(12.3)

in which N~2 is the hoop stress at yield of the horizontal reinforcement. Eq. (12.3) is based on the philosophy of providing sufficient hoop reinforcement as to resist the hoop membrane action N22 even in the absence of any contribution from the concrete. It is assumed that "flexural failure due to the imperfection generated meridional moments will not precipitate collapse, but that the shell has sufficient ductility to enable redistribution from bending to membrane action to occur" [13]. Thus, hoop reinforcement should be provided as if only N22 were to equilibrate the out-of-balance moment. For typical cooling towers with short imperfections, both BS4485 and IASS recommendations are not conservative, and would require less hoop reinforcement than Eq. (12.3). This equation was introduced in the recommendations of Ref.

[32]. For very short imperfections, as reflected by the ratio h/r, the tolerance limits should be "based upon the ultimate vertical flexural capacity of the shell. Such a tolerance could be based upon the assumption that the out-of-balance moment is resisted by flexural action, and would be applicable to those meridional; deviations that might occur over a height of two to three construction lifts" [14]. The subject of tolerance specifications in cooling tower design is still under discussion. The reader can find some useful contributions in Refs. [22] and [2].

12.8

FINAL

REMARKS

Fig. 12.13 summarizes the steps in the evaluation of an existing structure. Experimental techniques, like the ones described in this chapter, are required to assess the structure. Sometimes, because of the nature of the structure and the safety levels required, it is necessary to perform measurements and monitoring on a regular basis. In other cases, it is only because damage has been detected that a study like this one is required.

Imperfections in Practice

375

J Existing Structure J Measurement and Monitoring

.~

Experimental Techniques

.~

Engineering Judgment and Knowledge

Computational work

~

Analysis Techniques

Response Interpretation

.~

and Knowledge

Representation of the Existing Structure Material and Geometric Hypothesis Model of Imperfect

Structure

Comparisonwithcodes and recommendations

Decision regarding the existing structure

Figure 12.13" Steps in the evaluation of an imperfect structure

What we then have is a representation of the structure: the best we know about the structure with the techniques employed. Further hypotheses are introduced, regarding the geometry of the structure and its constitutive material. For example, one may decide to simplify the recorded geometric imperfections because the data obtained by monitoring is too complex. This is often made, and a complex surface is acknowledged, but a, simpler one is modeled to continue the studies. Something similar occurs regarding material imperfections, in which engineering judgment is required to model the imperfect situation. This leads to a model of the imperfect structure. The techniques of analysis of Cha,pters 3 to 6 in this book are now of use, and computational work is carried out to evaluate the response of the imperfect structure. In many cases this step has been

376

Thin-Walled Structures with Structural Imperfections

eliminated, and engineers have replaced the calculations with their judgment. But the complexity of the stress redistributions in imperfect structures, as we have studied here, make it clear that we need to use our judgment, and knowledge only after having some computed response of the imperfect structure. The decision about the future of the imperfect structure can lead to basically three outputs: the structure can be demolished, it can be repaired, or it can be left as it is. In the latter case it may be that there is no need to take any flarther strengthening action, or that its operation is limited but continues to take place. The decision is made based on a number of factors, including the cost of strengthening and repair, the time needed to repair, the obsolescence of the structure and the purpose it serves, the social and economic cost of a possible collapse, and several others. Whenever there are codes of practice with indications about tolerance specifications, they have to be taken into account, but they are not a substitute for a thorough investigation about the safety of the structure. We must remember that codes are also enriched by cases of failure that were not adequately covered by them, so that we have to be careful and avoid producing a new cause for modification of the existing recommendations.

Imperfections in Practice

377

References [1] Agbabian, M. S., Masri, S. F., Miller, R. K. and Caughey, T. K., System identification approach to detection of structural changes, J. Engineering Mechanics, ASCE, 117(2), 1991, 370-390. [2] Alexandridis, A. and Gardner, N. J., Tolerance limits for geometric imperfections in hyperbolic cooling towers, J. of Structural Engineering, ASCE, 118(8), 1992, 2082-2100. [3] ACI Committee 313-77, Recommended Practice for Design and Construction of Concrete Bins, Silos, and Bunkers for Storing Granular Materials, American Concrete Institute, Detroit, MI, 1983. [4] ACI Committee 349-85, Code Requirements for Nuclear Safety Related Concrete Structures, Chapter 19, American Concrete Institute, Detroit, MI, 1985. [5] Arbocz, J., Shell stability analysis" theory and practice, In" Collapse, The Buckling of Structures in Theory and Practice, Ed. J. M. Thompson and G. W. Hunt, Cambridge University Press, Cambridge, 1983. [6] Batista, R. C., Lower Bound Estimates.for Cylindrical Shell Buckling, Ph.D. Thesis, University College, University of London, London, 1979. [7] Blockley, D. I. and Henderson, J. R., Structural failures and the growth of engineering knowledge, Proc. Inst. Civil Engineers, 68, 1980, 719-728.

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Thin-Walled Structures with Structural Imperfections

[8] Bungey, J. H. and Millard, S. G., Radar inspection of structures, Proc. Inst. Civil Engineers, Structures and Buildings, 99, 1993, 173'186.

[9] British Standards Institution, BS~485, Part ~" Structural Design of Cooling Towers, BSI, London, 1975.

[10] British Standards Institution, Draft, revision of BS4485, Part 4" Structural Design of Cooling Towers, BSI, Document 92/17117 DC, London, 1992. [11] Central Electricity Generating Board, Report of the Committee of Inquiry into the Collapse of Cooling Towers at Ferrybridge, Monday 1 November 1965, CEGB, London, 1966. [12] Central Electricity Generating Board, Report on Fiddlers Ferry Power Station Cooling Tower Collapse 13 January 198~, CEGB, London, 1985. [13] Croll, J. G. A. and Kemp, K. O., Specifying tolerance limits for meridional imperfections in cooling towers, J. of the American Concrete Institute, 76, 1979, 139-158. [14] Croll, J. G. A., Kaleli, F. and Kemp, K. O., Meridionally imperfect cooling towers, J. of the Engineering Mechanics Division, ASCE, 105 (EM5), 1979, 761-777. [15] Cooling Tower Settlement Evaluation, Paul C. Rizzo Associates, Monroeville, PA, 1993. [16] Dwight, J. B., Imperfection levels in large stiffened tubulars, In" Buckling of Shells in Offshore Structures, Ed. J. E. Harding, P. J. DoMing and N. Agelidis, Granada Publishing, London, 1982, 393-412. [!7] Ellinas, C. P., Croll, J. G. A., and Kemp, K. O., Cooling towers with circumferential imperfections, J. Structural Division, ASCE, 106, 1980, 2405-2423. [18] Fullalove, S. and Greeman, A., Cooling tower collapse exposes calculated gamble, New Civil Engineer, The Institution of Civil Engineers, UK, 19 January 1984, 4-5.

Imperfections in Practice

379

[19] Godoy, L. A., On the collapse of cooling tower shells with structural imperfections, Proc. Inst. of Civil Engineers, Part 2, 77, 1984, 419-427. [20] Godoy, L. A., Discussion of: Cause of deformed shapes in cooling towers, J. Structural Engineering, ASCE, 1995. [21] Godoy, L. A., Prato, C. A. and Decanini, L. D., Report on the Safety of a Cooling Tower at Zapla, Jujuy, unpublished report to SINDINSA, Buenos Aires. 1979. [22] Gupta, A. K. and A1-Dabbagh, A., Meridional imperfection in cooling tower design: Update, J. of the Structural Division, ASCE, 108 (8), 1982, 1697-1708. [23] Hayman, B., FlexuraJ properties of thin reinforced concrete members, Proc. Inst. of Civil Engineers, Part 2, 1978, 253-269. [24] IASS Working Group, Recommendations for the Design of Hyperbolic and other Similarly Shaped Cooling Towers, Int. Association for Shell and Spatial Structures, IASS, Brussels, 1977. [25] Imperial Chemical Industries, Report of the Committee of Inquiry into the Collapse of the Cooling Tower at A rdeer Nylon Works, Ayrshire on Thursday 27th September 1973, ICI Ltd., Petrochemicals Division, London, 1973. [26] Jullien, J. F. and Reynouard, J. M., Reflections on the origin of deflections observed on the cooling towers of Pont sur Chambre and Ansereuilles, Proc. III IASS Int. Symposium on Natural Draught Cooling Towers, Paris, 1989, 595-604. [27] Jullien, J. F., Aflack, W., and L'Huby, Y., Cause of deformed shapes in cooling towers, J. Structural Engineering, ASCE, 120(5), 1994, 1471-1488. [28] Kim, J. T. and Stubbs, N., Model-uncertainty impact and damage-detection accuracy in plate girder, J. of Structural Engineering, ASCE, 121(10), 1995, 1409-1417. [29] Petrovski, H., Design Paradigms: Case Studies of Error and Judgment in Engineering, Cambridge University Press, Cambridge, 1994.

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Thin-Walled Structures with Structural Imperfections

[30] Pope, R. A., Structural deficiencies of natural draught cooling towers at U. K. power stations: Part 1" failures at Ferrybridge and Fiddlers Ferry, Proc. Inst. of Civil Engineers, Structures and Buildings, 104, 1994, 1-10. [31] Pope, R. A., Grubb, K. P. and Blackhall, J. D., Structural deficiencies of natural draught cooling towers at U.K. power stations, Part 2" surveying and structural appraisal, Proc. Inst. Civil Engineers, Structures and Buildings, 104, 1994, 11-23. [32] Rascon, O. A. and Mendoza, C. J., Design Manual for Civil Engineering Works, Section C, Topic 2, Chapter ~" Cooling Towers, Federal Electricity Board, Mexico, 1981. [33] Schuller, M. P., Atkinson, R. H. and Noland, J. L., Evaluation of historic masonry using non destructive test methods, Association for Preservation Technology, XXV Anniversary Conf., Ottawa, 1993. [34] Sibly, P. and Walker, A. C., Structural accidents and their causes, Proc. Inst. of Civil Engineers, 62, 1977, 191-208. [35] Singer, J. and Abramovich, H., The development of shell imperfection measurement techniques, Thin-Walled Structures, 23, 1995, 379-398. [36] White, J. D. and Dwight, J. B., Residual Stresses and Geometrical Imperfections in Stiffened Tubulars, CUED Tech. Report CUED/C-Struct/TR.64, Cambridge University, Cambridge, 1977.

Chapter 13 CLOSURE

13.1

INTRODUCTION

In a book about "imperfections", as is our case, it is important to give the reader some reference about the framework in which this concept developed. Although the specific equations that describe the mechanical behavior of thin-walled shells are very recent (they were formulated about 100 years ago), the philosophical and social context in which we employ them to obtain conclusions are much older. Often, this means that we use old ideas to understand what we obtain with new tools. Even more, we tend to think the "new" problems using old categories of thought. In Section 13.2, we try to highlight some of the relations between the new and the old in imperfections. Any study is restricted to a certain focus, and leaves outside a number of other relevant questions and topics. In the last section we mention some limitations of the studies of this book, some of which are developed elsewhere and some should be seen as topics for further research. 381

Thin-Walled Structures with Structural Imperfections

382

13.2

PLATO ON THE IMPERFECT WORLD, AND OTHER CONTRIBUTIONS

13.2.1

Plato

A most appropriate starting point may be the work of Plato, who was born in Athens about the year 428 B.C. Of specific interest here is his doctrine of Forms, a doctrine running through his earlier dialogues (Parmenides), in which he shows a clear separation between the perfect and the imperfect. Plato argues that beyond the world of physical things, there is a higher realm of Forms, or Ideas, only apprehensible by the mind. Each Form is a pattern of a particular category of things in this world, and is eternal and changeless, such as the Forms of shape, of color, ... Nothing is perfect in the world in which we live, but there is a sense of perfection about Forms. Yet, the things in this world are only imperfect copies of these perfect forms [10]. These ideas not only shaped the times of Plato, but they were incorporated in one way or another in different cultures, such as those emerging with Christianity and in the Islamic world between 800 and 1000 AD. Far from the topic of this book, as they seem to be, we may identify that here we discuss Forms like The Sphere, The Cylinder, or The Flat Plate. And yet, in the real world, these Forms become the imperfect sphere, the imperfect cylinder and the imperfect plate.

13.2.2

al-Qabisi

A further historical topic worth of consideration is in the work of an Arab astronomer, known in the West as al-Qabisi, or Alchabitius, born around 950 AD. We are here interested in his writings on the insignificance of the irregularity of the surface of the Earth. From a translation of his work, one reads the statement of the problem as" "One might ask oneself how the area of the Earth's surface can be obtained despite its having high mountains and deep valleys. In reply to this, we say that we regard them as huge [only] in comparison with us, and not with the totality of the Earth" [11].

Closure

383

He then employs arithmetic series to investigate the influence of an object on the surface of a sphere, and concludes" ..."Then, this [object] will be on the surface of the sphere like a roughness imperceptible by virtue of its smallness. And similarly, mountains and valleys altogether will be, compared with the entire Earth, imperceptible, and the Earth will appear as a smooth surface" [11]. The whole idea about the lack of importance of irregularities in an otherwise sphere, with reference to the computation of the surface of the sphere so clearly explained by al-Qabisi, is also present in more recent authors in mechanics.

13.2.3

Galileo

The influence of imperfections is discussed by Galileo in his book on Two New Sciences [3]. The presentation that follows takes the form of a dialogue between Sagredo and Salviati concerning effects of scale. In this book, Salviati represents the mature Galileo, while Segredo is the young Galileo. "Sagredo remarks that 'one cannot argue from the small to the large, because many devices which succeed on a small scale do not work on a large scale', but he confesses that he cannot explain why from geometrical principles alone. Salviati acknowledges that the scale effect was commonly attributed to 'imperfections and variation of the material', but dismisses this explanation and asserts that 'imperfections in the material, even those which are great enough to invalidate the clearest mathematical proof, are not sufficient to explain the deviations observed between machines in the concrete and in the abstract"' [9]. From the writings of Galileo we may draw some conclusions relevant for the subject of this book" 9 At the time of Galileo, imperfections and variations in the material were thought to have important consequences on the behavior of constructions

Thin-Walled Structures with Structural Imperfections

384

9 Galileo states that large imperfections can invalidate mathematical proofs; thus, imperfections play a role in the mathematical formulation, even if only at the level of assumptions 9 Galileo states that imperfections are not enough to explain differences that were observed between as-designed and real machines

13.2.4

Brunel

Isamabad Kingdom Brunel, a famous British engineer of the Great Western Rail, refers to the influence of imperfections in the context of cast-iron girder bridges. Following the collapse of the Dee bridge in 1847, he argued that "with the proper care of eliminating inhomogeneities and other imperfections, reliable iron castings of almost any form and of twenty or thirty tons weight could be ensured"

[9]. In the explanation of Brunel, imperfections become responsible for the problems observed between small and larger structures. This seems to be a similar way of though as tha, t prevalent at the time of Galileo.

13.2.5

B u c k l i n g enters the Field

The first studies on the influence of structural imperfections in shells concentrated on imperfections in the geometry, and were a consequence of the lack of agreement between theoretical predictions and experimental results in the buckling of thin-walled shells. Koiter [6] and his followers, in developing a general theory of elastic stability, reached a general conclusion around 1970, that could be summarized as follows:

Small deviations in the geometry of the shell (of the order of the shell thickness) cause a reduction in buckling loads. To highlight the importance of this behavior, a recent conference gathered around the topic of buckling of imperfection-sensitive structures.

Closure

13.2.6

385

Flugge

For loads smaller than those required for buckling, Flugge investigated the influence of imperfections on the stresses in a shell. Here we recall the work of Flugge regarding the lack of importance of irregularities in an otherwise perfect sphere, with reference to the computation of the stress field" "Because of the limits of accuracy of all workmanship, the middle surface of an actual shell always deviate a. little from its intended shape. Although such deviations should be small compared with the over-all dimensions of the shell, they may easily be of the order of the wall thickness, and the curvature of the middle surface may locally be rather different from that used in computing the membrane forces. The problem is to determine what influence such deviations from the true form have on the stresses" [2]. Flugge found that the hoop membrane stresses for a specific case "look rather wild"; however, he argued that when bending was ineluded, it would modify the stresses so that "the curve makes only a feeble attempt to follow the two high peaks of the membrane force". Furthermore, Flugge generalized this in the form: "The deformity of the shell which we have investigated here is of a rather special kind, since it has axial symmetry, but it may be assumed that also in shells with locally restricted deviations the essential features of the stress disturbance will be similar" (pp. 364-368 in Ref. [2]). This led to a final generalization: "Such thin shells, whether test specimens or parts of real structures, can hardly be expected to be very perfectly cylindrical. Deviations from the exact shape may easily be of the order of the wall thickness or even more. On p. 367 when studying the bending stresses of an almost spherical shell, we have seen that such deviations are not a matter of great concern" (pp.463 in Ref. [2]).

Thin-Walled Structures with Structural Imperfections

386

The conclusions from the above work represented a state of the art at the time, that could be worded as follows:

Small deviations in the geometry of the shell (of the order of the shell thickness) do not significantly affect the stresses. Let us make it clear that the book of Flugge represented a huge contribution to the stress analysis of shells in his time, and that here we only want to highlight the presence of a paradigm certainly broader than its application in mechanics.

13.2.7

New Paradigm

The paradigm of Flugge was challenged during the early seventies, with the collapse of a large reinforced concrete cooling tower, in which imperfections and cracks had been monitored after construction. This was the first major evidence that the stress concentrations associated with a structural imperfection may be of the same order of magnitude as the membrane stresses in the perfect shell. Research into the stress redistributions in cooling tower shells with deviations from the as-designed shape developed since 1975, and the problem has been considered by several authors during the last decade, as reported in this book. Other structural forms with problems due to imperfections have been detected: for example, shallow shell roofs, spherical steel containments, and pressure vessels. New forms of structural imperfections have also been recently modeled, including imperfections in the constitutive properties of the structure. The new paradigm regarding the stresses in imperfect shells could be now stated as:

Small deviations in the geometry of the shell (of the order of the shell thickness) cause additional stresses that can be of the same order of magnitude as the membrane stresses in the perfect shell. This book is written within the context of this new paradigm.

Closure

13.2.8

387

Stress Redistributions

To consider the stress redistributions due to imperfections, one has to be clear about the stress distributions in thin-walled structures. Basically, a thin-walled structure has two mechanisms to equilibrate external loads: membrane and bending. One mechanism is preferential (membrane action), and the other one is complementary (bending action). Loads may act in-plane (and they are resisted by membrane action); or out-of-plane (and in curved shells they are taken by membrane and bending). How does the preference for membrane action arise? It arises as a consequence of being thin. In thicker shells, this preference is not so notorious, and eventually it may reverse. An imperfection introduces new conditions into a system, and may modify this relative importance between membrane and bending contributions to equilibrium. An imperfection is detrimental whenever it produces a jump from one mechanism to another. Furthermore, there may be coupling between imperfections and loads, to create more severe conditions than those that were originally present. A load may induce membrane plus flexural effects. The shell develops membrane action (to account for the transverse loads) and bending (with the same purpose). If the first mechanism is not possible, the shell goes to bending only to equilibrate the external loads, but increases bending stresses far too much. The problem of sensitivity of the fundamental path to imperfections occurs whenever the main mechanism of resistance in the perfect structure is membrane. If it was bending (like in beams or plates), there is not too much to be re-distributed under small displacements. Redistributions require here large displacements, so that membrane stresses begin to count.

13.2.9

Neo-Platonism

The reader may have noticed that there is a specific view point adopted in this work, and could be summarized as follows: There is an ideal, or as-designed, structural shape, which is taken as a reference condition. This is characterized as a "perfect" state. Then, there are perturbations from this condition into other configurations that incorporate imperfections. What do these imperfections attempt to model?

Thin-Walled Structures with Structural Imperfections

388

They try to model real-world, real engineering situations, that arise as a consequence of constructional problems, early misbehaviors or damage suffered by the structures. All these expressions are employed in this work, and have been previously used by many authors. Certainly, there is no clear intention on our side to ascribe to the theory of Plato; however, it seems that we think our problems in much the same way as he did over two thousand years ago.

13.2.10

Summary

Some of the lessons from this section may be summarized as follows: 9 For a long time there has been a recognition that imperfections exist in geometry and in mechanics 9 In many cases, imperfections were loosely pointed as the main cause of discrepancy between an expected behavior and what was really observed. Often this was done without proof and was not based on sound knowledge 9 There have been cases in which imperfect structures were evaluated by means of mathematical models, with the idea of dismissing the influence of imperfections, even if the calculations showed otherwise 9 There is a need to assess the influence that imperfections have in our constructions This book is an attempt to put together what we know about stresses in imperfect shells. It includes the mathematical methods to model the problem; the behavior of simple structural components with idealized imperfections; and data and lessons obtained from real structures.

13.3

OTHER

APPROACHES

A deterministic approach has been followed here, in the sense that specific imperfections are investigated. An alternative is to consider

Closure

389

a non-deterministic analysis, so that probabilistic displacements and stresses are obtained. This requires information about real imperfections in many structures, i.e. a databank for each specific class of application, which is desirable. We have seen in Chapter 12 that there are already moves in this direction in the field of astronautics, and that there is information available as to develop something similar in the field of cooling tower shells. However, the author believes that there will be sometime before such information becomes available. In the meantime deterministic analysis will play a dominant role in the evaluation of stresses in imperfect thin-walled structures. A second limitation of this presentation regards the static nature of the loads applied. Information about imperfections in the dynamic response of structures is not provided here, and is seen as an important but separate topic. Buckling loads and the influence of imperfections on the stability of structures is not considered here. There are other books that deal with the subject, such as Refs. [7], [1], [5], [8], and several others. Finally, there is a close relation between the topics discussed in this book and a field known as sensitivity analysis. In sensitivity analysis, the changes in a given response are investigated when there are changes in selected parameters that define the system. This is a well established field by now, with applications in optimization and in reanalysis of structures, both linear and non-linear. An account is in the book of Ref. [4], and a recent review may be found in Ref. [7]. There are clear similarities between the techniques developed in this text and those of sensitivity analysis. But they differ more in the emphasis on the applications rather than in the essence of the analysis that is done. Thus, the first part of this book could be seen as the sensitivity of a structural system with respect to changes in the parameters that define an imperfection, and this is called imperfection sensitivity. In design sensitivity analysis, on the other hand, one investigates sensitivity of the structural system with respect to changes in the parameters that define the system itself.

390

Thin-Walled Structures with Structural Imperfections

References [1] Croll, J. G. A. and Walker, A. C., Elements of Structural Stability, Macmillan, London, 1972. [2] Flugge, W., Stresses in Shells, Springer-Verlag, Berlin, 1963. [3] Galileo Galilei, Discorsi e Dimostrazioni Matematiche, Intorno a Due Nuoue Scienze A ttenenti alla Mecanica ~ i Movimenti Locall, Elsevier, 1638. Translated as Dialogues Concerning Two New Sciences, Dover, New York, 1954. [4] Haug, E. J., Choi K. K. and Komkov, V., Design Sensitivity Analysis of Structural Systems, Academic Press, London, 1985. [5] Huseyin, K., Nonlinear Theory of Elastic Stability, Noordhoff, Leyden, 1975. [6] Koiter, W. T., On the Stability of Elastic Equilibrium, Ph.D. Thesis, Delft, 1945 (in Dutch) [7] Mroz, A. and Haftka, R. T., Design sensitivity analysis of nonlinear structures in regular and critical states, Int. J. Solids and Structures, 31(15), 1994, 2071-2098. [8] E1 Naschie, M. S., Stress, Stability and Chaos in Structural Engineering: An Energy Approach, McGraw Hill, London, 1990. [9] Petrovski, H., Design Paradigms: Case Histories of Error and Judgment in Engineering, Cambridge University Press, Cambridge, 1994. [10] Russell, B., History of Western Philosophy, George Allen & Unwin, London, 1961.

Closure

391

[11] Sesiano, J., A treatise by al-Qabisi (Alchabitius) on arithmetical series, in: From Deferrent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E. S. Kennedy, Ed. D. A. King and G. Saliba, Annals of the New York Academy of Sciences, New York, vol. 500, 1987, 483-500. [12] Thompson, J. M. T. and Hunt, G. W., A General Theory of Elastic Stability, Wiley, London, 1973.

Appendix A B A S I C S OF P E R T U R B ATI O N TECHNIQUES A.1

INTRODUCTION

Perturbation theory is the theory of perturbation methods that are used to solve non-linear problems. The specific form of solution that is employed in perturbation methods is based on the study of the effects that small modifications produce on the response of a system. To carry out the study, a parameter known as perturbation parameter is adopted to follow a non-linear path. Both control and response parameters are expanded using this parameter. We shall distinguish between regular and singular perturbations. Problems leading to regular perturbations are those in which small changes in the system lead to small changes in the response; thus, if the changes tend to zero, the response also tends to zero. On the contrary, if a decrease in the changes the response remains constant, or does not decrease in amplitude, then we say that perturbations are

singular. Several fundamental aspects of perturbation methods are presented in this Appendix, as a guide to the use that is made of them in Part I on the analysis of imperfect structural systems. For a more detailed treatment of the subject, the reader is referred to the various texts and works in this subject, such as Refs. [1] [5] [4] [6] [3]. 392

Appendix A: Perturbation Techniques

A.2

393

TAYLOR EXPANSIONS

Let us consider a problem modeled by the condition F (a, A ) - 0

(A.1)

where a is the response vector (displacements), and A is the control parameter. In this book we couple finite elements and perturbation techniques, so that Eq. A.1 arises from the discretization of a continuum by finite elements; thus, a is a set of nodal degrees of freedom. Cases in which F is a function of continuous variables are out of the scope of this presentation. We seek to obtain the response a in the form a-

a(s)

(A.2)

where s is the perturbation parameter, chosen in such way that it can highlight the behavior of interest in the system. The perturbation parameter is usually present in the definition of the problem, and is a scalar parameter. Sometimes it may be useful to adopt a new parameter not present in the definition of the problem, but, this need did not arise in the book. More than one perturbation parameter is needed in some cases, such as in the studies on interacting imperfections in Chapter 5. Substitution of Eq. (A.2)into Eq. (A.1) leads to F [a (s), A (s), s] - 0

(A.3)

Let us assume that the response of Eq. (A.1) is known for a particular value of s, such as s = so; this will be termed the reference solution, and denoted by ao = a0(s0). The value of so chosen is usually associated with a particularly simple solution of the system. Furthermore, let us assume that the properties of a(s) are known at s = so, such as continuity, differentiability, or others required. Then, an analytica~ approximation can be obtained for the solution Eq. (A.2) in the form of a Taylor series

a ( s ) -- ao -t-(s - S o ) a l + (s - So) 2 a2 + (s - So) 3 a3 + ...

(A.4)

where a~ represents the n-order derivatives of the components of a with respect to the perturbation parameter s, and are evaluated at s - So,

394

Thin-Walled Structures with Structural Imperfections

dna a~ = ds n I~=~o

(A.5)

An alternative approach is to employ power series to expand a. The control parameter A is next expanded in the form

A(s) - Ao + (s - So)11 q- (s

-- So) 2

A2 + (s - So) 313 + ...

(A.6)

To illustrate the approximation that may be obtained with Taylor series, let us consider the expansion of the function F - cos(0) for 0 < 0 < ~r. Each new term added to the series alternatively represents an upper and a lower bound to the exact solution, which is thus enclosed by two successive approximations. This is an extremely useful property in terms of convergence of the results, but cannot be guaranteed in multiple degree of freedom systems.

A.3

EXPLICIT AND IMPLICIT PERTURBATION TECHNIQUES

There are several ways to obtain perturbation equations, and two of them are described in this section.

A.3.1

Perturbation via Implicit Differentiation

We start by differentia,tion of Eq. (A.3) with respect to the perturbation parameter s; thus, we obtain the first order perturbation equation OF

OF

OF

0---~-alq- ~-~-A1 q- 0s

= G(a, A, s) - 0

(A.7)

The second order perturbation equation may be obtained by differentiation of the first one, leading to OG

OG

OG

0---~al q- ~--~A1 q- Os = H(a,A,s) - 0

(A.8)

395

Appendix A: Perturbation Techniques

Expanding Eq. (A.8) in terms of F, one gets

[o r o(0 ) o(or)] 0a0aal+oa

-~- ~

-~a

+

~

al

Al-t- ~a

(02 )

0(

11

al -~- 0A '~ A1-+-~-~

-~aa2+

(or)02 ] ~

A2+0s2

(A.9)

-0

The third order perturbation equation becomes OH

OH

OH

0---~-al -~--~A1 + Os = I(a, A, s) - 0

(A.10)

which can be written in terms of F. In some cases, it may happen that F is not a direct function of s, but only of the generalized coordinates a and parameter A; in this case, we have the simplifications OF 02F = = =0 Os Os 2 "'"

(A.11)

To obtain the values of al, a2, ... and A1, A2, ..., we evaluate the perturbation equations in s - So, G(a, A, So) = 0 H(a, A, s o ) - 0

(A.12)

I(a, A, so) - 0 The set of Eqs. (A.12) can be written in terms of F in the form OF --al Oa OF

OF + - ~ A 1 Is=so +G(ao, Ao, so) - 0

OF

O----~a2+ ~--~A2 1,=,o +H(al, ao, nl, no,

so)

-

0

(A.13)

Thin-Walled Structures with Structural Imperfections

396

OF

+

OF

A3 Is=so +/(a2, al, ao, A2, A1, Ao, so) - 0

In the first perturbation equation, evaluated at So, there are two unknowns, namely al and A1. Once these unknowns are solved from the linear system, the second perturbation equation only contains a2 a,nd A2. Similarly, a3 and A3 are the unknowns in the third set of perturbation equations. Thus, this is a system of equations that can be solved sequentially. Each system has the same linear operators associated with the unknowns, and they are [-~] and [OF bX]" Each system is linear, but has one unknown more than the number of equations; however, the choice of the perturbation parameter is the additional condition required.

A.3.2

Perturbation via Explicit Substitution

An alternative way to present the perturbation technique is by expliciting the dependence of Eq. (A.1) with s, as in Eq. (A.3). In Eq. (A.3) we group terms that contain s with similar powers, and obtain a polynomial in the form Ao + sA1 + s2A2 + ... + s~A~ - 0

(A.14)

We can now follow the fundamental theorem of perturbations [6] and say that if the coefficients Ao, A1, A2, ... s~A~ are independent of s, then A o - A1 - A2 = ... = A~ - 0

(A.15)

Notice that Eqs. (A.15) are the evaluated perturbation equations written in Eq. (A.13), but using a different notation.

A.4

DEGENERATE AND NON-DEGENERATE PROBLEMS

It was previously mentioned that F may be a set of equations that take a matrix form. Its derivatives are also matrices. If the linear operator

Appendix A: Perturbation Techniques

397

of the perturbation systems has an inverse, then we say that the system is non-degenerate, and each evaluated perturbation equation has a solution. There are cases in which there is no inverse of the linear operators, so the technique of solution outlined above is not directly applicable. Such cases are said to be degenerate. The linear operator is singular, and the order of singularity may be one, or perhaps higher than one. To solve degenerate perturbation problems, it is necessary to provide additional equations, one for each order of singularity found. One way to obtain these equations is by means of the mechanism of contraction [7], by which a scalar equation is obtained by multiplication of a vectorial equation by a vector, so that part of this equation is eliminated due to orthogonality. oF is singular at s -- So, and tha.t it is symmetLet us assume that 5-~ ric. Thus, a vector x will exist, so that

OF

0---~x =0

(A.16)

Premultiplication of Eq. (A.13) by x T leads to the scalar equation

x rOF -~aal

+ x rOF ~-AA1l~=~0 +xTG(a0,

A0, so) - 0

(A.17)

But since [OF] 5-~ is symmetric, the first term is zero and we obtain one equation with one unknown. This additional condition eliminates the singularity of the system. The same procedure can be used to solve higher order perturbation equations. There are other techniques to solve degenerate problems, for example that presented in Ref. [4]. L

A.5

J

REGULAR AND SINGULAR PERTURBATIONS

A function a(s) is regular in so if it has all required derivatives with respect to s at s - so. In this case, the function a(s) can be investigated in the vicinity of so by means of regular perturbations. But if not all derivatives are defined in s = so, then, the function a(s) must be studied using singular perturbations. There are several ways to solve singular perturbations, and one of them is based on the

Thin-Wailed Structures with Structural Imperfections

398

least degenerate expansion. In this case, the expansion is made in terms of unknown exponents and both the coefficients and the exponents have to be obtained. This is explained in Ref. [2]. Singularities at s = so may arise from features of the domain or from the model itself. There are many problems in mechanics in which a singularity is present in the domain under study, such as crack tips, semi-infinite domains, and many others. Singularities in the model are only observable in the response, such as in boundary layer problems, in which no matter how small the perturbation parameter is, the response never vanishes. The bending behavior of a shell near to the boundaries is an example of a boundary layer problem in solid mechanics. All problems considered in this book fall within the category of regular perturbations.

A.6

FINAL REMARKS

Perturbation techniques lead to approximate solutions (and not to exact solutions); however, the solutions take the form of analytical functions. This is why perturbation techniques are analytical approximate solutions, in contrast to numerical methods of approximation. Some advantages of perturbation techniques are: 9 Perturbation techniques reveal the explicit dependence of the solution with the governing parameters in a more explicit way than numerical solutions. Sensitivity expressions are obtained immediately from the solution. 9 Perturbation techniques provide information about derivatives at a certain state, and thus they allow to produce a classification of behavior based on the values of the derivatives. On the other hand, there are disadvantages" 9 Numerical methods tend to be more efficient, and provide a solution that is more accurate far away from s - so. 9 New matrices have to be computed to obtain the set of perturbation equations.

Appendix A" Perturbation Techniques

399

References [1] Bellman, R. E., Perturbation Techniques in Mathematics, Physics and Engineering, Holt Rinehart and Winston, New York, 1964. [2] Godoy, L. A. and Mook, D. T., Higher order sensitivity to imperfections in bifurcation buckling analysis, Int. J. Solids and Structures, 1995. [3] Godoy, L., Flores, F., Raichman, S. and Mirasso, A., Tecnicas de Perturbacion en el A nalisis No linear mediante Elementos Finitos,

Asociacion Argentina, de Mecanica Computacional, Cordoba (Argentina), 1990. [4] Keller, L. B., Perturbation Theory, In" Teoria de Bifurcacoes e suas Aplicacoes, vol 2, Laboratorio Nacional de Computacao Cientifica, LNCC/CNPq, Rio de Janeiro, 1985, 83-152. [5] Nayfeh, A., Perturbation Methods, John Wiley, New York, 1973. [6] Simmonds, J. G. and Mann, J. E., A First Look at Perturbation Theory, R. E. Krieger Publishing Co., Florida, 1986. [7] Thompson, J. M. T. and Hunt, G. W., A General Theory of Elastic Stability, Wiley, London, 1973.

Appendix B SHELL EQUATIONS B.1

SUMMARY

OF SHELL THEORY

In this appendix we review some of the main results of a theory of shells, which is used in several chapters of the book. There is a number of formulations for the analysis of shells, and here we present a summary of those equations developed by Novozhilov [4]. A shell, in the theory of elasticity, is a curved surface in which the transverse characteristic dimension is small in comparison with the lateral characteristic dimensions. The geometry of the shell may be defined by specifying the position of its middle surface and its thickness at every point. Let us define the principal coordinates of the surface by al and a2, and the principal radius of curvature at the point considered by R1 and R2. The middle surface of the shell may be written in terms of Cartesian coordinates xi as Xi - - f i ( o L 1 ,

~2)

(B.1)

where i = 1, 2. A segment dsi along the curvilinear coordinates of the surface is given by

d s i - Aidai

(B.2)

where in Eqs. (B.7) and (B.2) there is no summation in i. The parameters of Lame of the surface are here represented by Ai. They satisfy the conditions of Codazzi 400

Appendix B: Shell Equations

401

l e3 e2 el

o~2

Figure B.I: Principal coordinates of the middle surface of a shell

0 ( A 2 ) _ 10A2 0012 ~

0

t~1 00[.1

A1

0Ctl ( R l l ) -

10A1

(B.3)

R2 0C~2

and the condition of Gauss

o (1 oA2) o (1 oA1) +

Oal A10al

~

A20a2

AI A2 R1 R2

(B.4)

Using the Kirchhoff-Love hypothesis, the displacement field of a point located on the normal to the shell at a distance x3 from the middle surface may be expressed in terms of the mid-surface displacements in the form ui(x~) = u~ + x~gi

(B.5)

u3(x3) - u3

(B.6)

in which the rotations/3i are /3i -

10u 3

r

ui

Ai Oc~i Ri

The linear strain-displacement equations can be obtained as

(B.7)

Thin-Walled Structures with Structural Imperfections

402

10U 1 e l l "--

A1 0Ctl

1 OA1 tL3 ] AIA20a2 u2 -[- R1

10u2

1 OA2 ~

e22 - A2 0c~2 { AI A2 0al

l[A20 (u2

C12 -- 2

A1 00L1

~11 --

~22

u3 ll I -~-

R2

+ A2 0 ~

~11

(B.8)

1 0~1 1 OA1 A1 0OL1 -'~ AIA20a2/32

1 0/32 1 OA2 X22 = A20a2 + AI A2 0al /~1

~12

1 ( 02u3 AI A2 OOZlOOZ

10A1 Du3 A1 00~2001.1

10A20u3'~ A20al Oa2J

+-~1 A2 0c~2

10A1 ) AI A2 0c~2ul

1 ( 1 0u2 + ~ A~ 0G~1

AIA2 0Clgl ?22

1 ( 1 0Ul

10A2

) (B.9)

Non-linear terms are sometimes required. For the purpose of modeling imperfections, the simplest terms added to Eq. (B.9) are .

1

c i j - ~3i/3j

(B.10)

The stress resultants Nll and moment resultant Mll are defined by integration of stresses Nil

N22 -

J-t/2

O"11

1+ ~

dx3

~,j2r~/2(1§

X3

~22

t/2

N12 - J-t/~ O"12

-a-

(1

dx3

(B.11 /

403

Appendix B: Shell Equations

N21 -Mll

it~2 X3) dx3 J-t~2O"21 ( 1 + R22 1+ ~

J-t/2 ~

dx3

(B.12)

with similar definitions for the other stress components. The constitutive equations for elastic and isotropic material are given in the form N l l - K (s

-~-//s

N22 - K (s

-~-/2s

N12 - K (1 - v) s

(B.13)

and M l l -- D (~11 -~- U~22)

M22 - D (X22 + U~11) M12 - D (1 - u) X12

(B.14)

In Eq. (B.13) and (B.14), the membrane and bending stiffnesses are given by Et K

__

1 -- U2

Et 3 (B.15) 1 2 ( 1 - u 2) where E is the modulus of elasticity, t is the thickness of the shell, and u is Poisson's ratio. Finally, the total potential energy of the shell can be written as

D -

:j'/

V - -~

(Nijcij + M i j x i j ) t d x l d x 2 -

//

(piui + p3u3)tdxldx

(e.16) In the theory of Mindlin-Reissner, one preserves the rotations/3i as independent variables instead of employing Eqs. (B.5-B.6).

404

B.2

T h i n - W a l l e d S t r u c t u r e s with S t r u c t u r a l I m p e r f e c t i o n s

APPLICATIONS

In a fiat plate, the parameters that define the geometry are 1/R1 - 1/R2 - 0

A1

~1

--

--

A2 - 1

Xl

(B.17)

C~2 - -

X2

In a shallow shell, the parameters of the geometry are determined with the assumptions

Oxi]

<<1

(B.18)

and the contributions of the displacement components Ul and u2 are neglected in the computation of changes in curvature Xij. The curvature parameters of the perfect mid-surface, kij, are given by (~2X 3

(B.19)

kij ---- OxiOxj

The geometry of the cylinder is represented using the following parameters 1/R1 - 0

A1

-

1

OlI

--

X 1

R2 - r

A2 - r

(B.20)

OL2 - -

In the sphere, the Lame parameters are R1 - r

R2 - r

A1- 1

A2- 1

~1

~

Xl

~2

~

X2

(B.21)

Appendix B: Shell Equations

405

References [1] Novozhilov, V. V., The Theory of Thin Elastic Shells, P. Noorhoff, Groningen, 1959.

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Subject Index corrosion, 350,358 crack, 3, 11, 12, 53, 290,307, 352, 356 creep, 330

aerospace, 353 algorithm, 149 arch, 28 buckling, 4, 96, 384 concrete spalling, 10 constitutive relation, 55, 58, 61 collapse, 96, 361 load, 368,369 mechanism, 366,371 collision, 14, 18 composite materials, 17 condition number, 67 containers, 12, 13, 200 cooling towers, 9, 18, 19, 59, 144, 304, 332,344, 356, 363,373 at Ardeer, 97, 306,307, 325,327, 364, 367 at Bouchain, 366 at Dampierre, 304 at Ferrybridge, 364 at Fiddlers Ferry, 307, 365 at Radcliffe, 330,358 at West Burton, 358 at Zapla, 350 407

damage, 3, 14, 95 dented, 242 local, 263 mechanics, 3 defects, 3,201 degeneracies, 3 direct analysis, 58, 101,136, 145, 153,204 problems, 59, 102 equilibrium, 31, 68, 109 equivalent load analysis, 35, 42, 46, 61, 72, 86, 109, 124, 164 and perturbations, 78, 119 errors, 66, 111 itera, tion, 64, 110 simplified, 36, 114, 126, 186, 189, 206, 221,230,258 experiments, 295

408

Thin-Walled Structures with Structural Imperfections

finite elements, 54, 58, 102,137, finite strips, 60, 163 geometric imperfection, 9, 28, 95, 137, 146 and cracks, 278,325 and creep, 332 and intrinsic imperfection, 44, 135,274 antisymmetric, 323 axisymmetric, 205,208, 211,258, 310 bulge, 269,324 causes of, 11 circumferential, 316 Fourier components, 354 height, 290, 314 initial strain model, 29, 105 local, 205, 209,223,268, 321 profile, 98, 99, 262,277, 288, 313 surveying, 97, 343 groove, 137, 295 intrinsic imperfection, 7, 38, 53, 83, 150 and geometric imperfection, 44, 135,274 examples of, 56 in the modulus, 54, 172 imperfection classification, 5 consequences, 4 importance, 9, 17 space distribution, 6 two-parameter, 130

Lagrange multipliers, 105 load, 4, 315 vector, 55 masonry, 351 mechanical model one degree-of-freedom, 26 two degree-of-freedom, 79, 120, 153 non-linear analysis, 144, 146 behavior, 235,246 path, 155 off-shore structures, 14, 16, 19, 355 orthotropy, 277 out-of-balance moment, 262, 270, 285, 311,318 perturbation analysis, 31, 40, 45, 73, 75, 84, 116, 127, 139, 146, 150, 156, 168, 182 degenerate, 396 explicit, 396 implicit, 394 regular, 397 simplified, 34 singular, 397 plasticity, 3 plate assemblies, 60, 73, 171 plates, 74, 114, 162 Kirchhoff, 62, 169 Mindlin, 68, 69, 169 pressure vessels, 200 prismatic, 6 probabilistic analysis, 389

409

Subject Index

profile gauge, 347 radar techniques, 351 ring, 259 sensitivity, 37, 157, 168, 178, 181,389 shell applications, 404 cylindrical, 137, 256,385 elliptical paraboloida,1, 15, 189 equations, 400 form, 286, 315 hyperboloidal, 304 of revolution, 70 shallow, 14, 112, 185 slenderness, 198,219 spherical, 12, 200,359,385 steel, 345,348 theory, 70, 400 silos, 11,373 stiffness matrix, 54, 107, 145 strain, 55 -displacement relations, 55, 106, 139, 145, 146 stress resultants, 55, 77, 119 redistribution, 4, 20, 310, 316, 387 -strain relations, 55 substructures, 103 tanks, 11 Taylor series, 393 thickness, 196, 286,314, 320 change, 53, 62, 69, 164, 206,246, 297 expansion, 73

tolerance limits, 372 total potential energy, 28, 107, 139, 145 welds, 355 wind, 315

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