Thin-Walled Structures with Structural Imperfections

THIN-WALLED STRUCTURES WITH STRUCTURAL IMPERFECTIONS Analysis and Behavior Elsevier Titles of Related Interest TANA...

Author: Luis A. Godoy

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THIN-WALLED STRUCTURES WITH STRUCTURAL IMPERFECTIONS

Analysis and Behavior

Elsevier Titles of Related Interest

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


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


,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


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


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


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


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


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.


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.


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


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


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


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


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


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


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


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


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


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.


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


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)]


-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


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 ~ 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


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.


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.


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


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


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


F(x)-

0

for

xI > x > xF

F(x) - 1

for

X I 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


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


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)


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.


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


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


[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)


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)


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


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.


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


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.


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.


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


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, 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.


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.


179

Figure 7.14: Plate with changes in the modulus. Equivalent load of first, second and third-order


Figure 7.15: Plate with cha,nges in the modulus. First, second and third-order corrections in displacements


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.


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.


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


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
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


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-


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.


~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


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


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

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


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


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


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.


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


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".


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


,,

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]

[]7I 'II] ~uv~lnsaa ssaals [vuo!p.taalAI "9I'OI aan$!d

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


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


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


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

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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

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gmmm. m

~

-

-

m

n

.,

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Z

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~

(mmmmbm

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0.017 I n m ' W

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I=~

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~

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m m

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0.017

m

l

0.004

m

0

l

w~

m

,

WBm q, m l

am

~m,.~

10 ~

20 ~

2t

'0

Z-0

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= 0.048 -

9 .....

0.034

Z

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0

. m

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"-

m m m m

d~mmm m m m m m

0 0 0 4 - - " -~.

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m

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10 ~

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~

m

m

--- - - , , m

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mud

20 ~

O Figure 10.22" Effect of imperfection amplitude [11,14]

290


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

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gL .

.

.

.

.

0

,,,,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

9o'~

g'O

O'L

g'L

z' 0"~

3 3 3 3

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g~'O

0

Z IX3 r

9

Z

3 3 3 3

oO -

g].'O -

q/I

~6~;

294


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


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


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].


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].


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


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


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


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


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]


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

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