Asymptotic methods in the buckling theory of elastic shells

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS

Series Editors: A. Guran, A. Belyayev, H. Bremer, C. Christov &

G. Stavroulakis

About the Series

Rapid developments in system dynamics and control, areas related to many other

topics in applied mathematics, call for comprehensive presentations of current

topics. This series contains textbooks, monographs, treatises, conference proceed￾ings and a collection of thematically organized research or pedagogical articles

addressing dynamical systems and control.

The material is ideal for a general scientific and engineering readership, and is

also mathematically precise enough to be a useful reference for research specialists

in mechanics and control, nonlinear dynamics, and in applied mathematics and

physics.

Selected Volumes in Series B

Proceedings of the First International Congress on Dynamics and Control of Systems,

Chateau Laurier, Ottawa, Canada, 5-7 August 1999

Editors: A. Guran, S. Biswas, L. Cacetta, C. Robach, K. Teo, and T. Vincent

Selected Topics in Structronics and Mechatronic Systems

Editors: A. Belyayev and A. Guran

Selected Volumes in Series A

Vol. 1 Stability Theory of Elastic Rods

Author T. Atanackovic

Vol. 2 Stability of Gyroscopic Systems

Authors: A. Guran, A. Bajaj, Y. Ishida, G. D'Eleuterio, N. Perkins,

and C. Pierre

Vol. 3 Vibration Analysis of Plates by the Superposition Method

Author: Daniel J. Gorman

Vol. 4 Asymptotic Methods in Buckling Theory of Elastic Shells

Authors: P. E. Tovstik and A. L. Smirinov

Vol. 5 Generalized Point Models in Structural Mechanics

Author: I. V. Andronov

Vol. 6 Mathematical Problems of the Control Theory

Author G. A. Leonov

Vol. 7 Vibrational Mechanics: Theory and Applications to the Problems of

Nonlinear Dynamics

Author: llya I. Blekhmam

SERIES O N STABILITY, VIBRATION AN D CONTROL OF SYSTEM!

Volume '

Series Editors: A. Guran, A. Belyayev, H . Bremer, C. Christov & G. Stavroulak

Asymptoti c Method s i n th e

Bucklin g Theor y o f Elasti c Shell ;

Petr E. Tovsti k

Andre i L . Smirno v

St. Petersburg State University, Russia

Edited by

Peter R. Fris e

Windsor University, Canada

Ardeshi r Gura n

Institute for Structronics, Canada

D A I HOC THAIjjGUYE N

TRUNG TA M HOCLlI u

Worl d Scientifi c

m Singapore • New Jersey • London • Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

P O Box 128, Farrer Road, Singapore 912805

USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ASYMPTOTIC METHODS IN THE BUCKLING THEORY OF ELASTIC SHELLS

Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,

electronic or mechanical, including photocopying, recording or any information storage and retrieval

system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to

photocopy is not required from the publisher.

ISBN 981-02-4726-5

Printed in Singapore.

Prefac e

Many publications both in Russia and abroad are devoted to the analysis of

the buckling of thin-walled structures, and the solutions of many important

problems have been obtained. In spite of the variety of books that are avail￾able on this subject however, there is no one book which contains a general

overview of buckling problems and sound base of methodology for the qualita￾tive study of a large number of different types of problems. This gap may be

partially filled by the monograph "Asymptotic Methods in Buckling Theory of

Elastic Shells", which is a revised and extended edition of the monograph "The

Stability of Thin Shells. Asymptotic Methods" by P.E. Tovstik published in

1995 in Russian.

The book contains numerous results developed by the authors and their

pupils by means of the application of asymptotic methods to the problem of

shell buckling. Previously these methods have been applied to problems of

shell statics and vibrations.

There are many new results in this rather compact book. The static stabili￾ty problem for the case of small membrane deformations and angles of rotation

is studied completely. In the general formulation, the principal questions on

the development of local buckling modes are studied. The results presented

give a clear, qualitative picture which is necessary for further development of

modern numerical methods.

One of the unique features of the book is the large bibliography of litera￾ture on the application of asymptotic methods to the problems of thin shell

buckling. The reference list includes Western as well as the Russian literature

on asymptotic methods. The Russian material has generally been translated

into English but is still not widely known, despite the high standards prevalent

in this body of research.

The book is directed both to researchers working on the analysis and con￾struction of thin-walled structures and continuous media and also to math￾ematicians who are interested in the application of asymptotic methods to

v

vi Preface

problems of thin shell buckling. The book may also be useful for graduate and

post-graduate students of Mathematics, Engineering and Physics.

The present book is a result of the scientific cooperation of the Depart￾ments of Theoretical and Applied Mechanics of the Faculty of Mathematics

and Mechanics at St. Petersburg State University in Russia and Department

of Mechanical and Aerospace Engineering at Carleton University in Ottawa,

Canada. The authors would like to express their special thanks to the editors

of the book, Prof. Peter R. Frise from the University of Windsor and Prof.

Ardeshir Guran from the Institute of Structronics and also to Prof. John

Goldak amd Prof. D.R.F. Taylor of Carleton University whose support of

the program of scientific cooperation between St. Petersburg and Carleton

University made it possible to prepare this manuscript.

This work was supported in part by Russian Foundation for Fundamental

Research (grants 9801.01010, 0101.00327) and Soros International Foundation

(grant # 54000) and the Natural Sciences and Engineering Research Council

of Canada under Dr. Frise's individual research grant.

Our special thanks to Mrs. V. Sergeeva who typeset the main part of the

book and drew most of the pictures. We would also like to thank our students

N. Kolysheva, Yu. Dyldina, Yu. Balanina, N. Vasilieva, E. Kreis, A. Yam￾barshev, V. Braulov, and M. Antonov for their help in preparation of the

manuscript.

Peter Tovstik and Andrei Smirnov

Content s

Preface v

Introduction 1

Basic notation 5

1 Equations of Thin Elastic Shell Theory 7

1.1 Elements of Surface Theory 7

1.2 Equilibrium Equations and Boundary Conditions 10

1.3 Errors of 2D Shell Theory of Kirchhoff-Love Type 16

1.4 Membrane Stress State 23

1.5 Technical Shell Theory Equations 26

1.6 Technical Theory Equations in the Other Cases 29

1.7 Shallow Shells 30

1.8 Initial Imperfections 31

1.9 Cylindrical Shells 31

1.10 The Potential Energy of Shell Deformation 33

1.11 Problems and Exercises 34

2 Basic Equations of Shell Buckling 37

2.1 Types of Elastic Shell Buckling 37

2.2 The Buckling Equations 42

2.3 The Buckling Equations for a Membrane State 42

2.4 Buckling Equations of the General Stress State 46

2.5 Problems and Exercises 48

3 Simple Buckling Problems 49

3.1 Buckling of a Shallow Convex Shell 49

3.2 Shallow Shell Buckling Modes 53

3.3 The Non-Uniqueness of Buckling Modes 55

3.4 A Circular Cylindrical Shell Under Axial Compression 57

vii