asymptotic methods in the buckling theory of elastic shells

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SERIES ON STABILITY,

V IBRATION AND CO NTROL OF SYSTEM S

Ser ies A

Asymptotic Methods

in the Buckling Theory

of Elastic Shells

Petr

E.

Tovstik & A nd rei L. Sm irnov

Edited by

Peter R. Fr ise & A rde sh i r G u ra n

World Scient i f ic



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.

Th e material is ideal for a gene ral scientific and en gineering 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. T eo, and T. Vincent

Selected Topics in Structronics and Mechatronic Systems

Editors: A. Belyayev and A. Guran

Selected Volumes in Series A

Vol. 1 Stability The ory of Elastic Rods

Author: T. Atanackovic

Vol. 2 Stability of Gyroscopic System s

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 Asym ptotic Methods in Buckling The ory of Elastic Shells

Authors: P . E. Tovstik and A. L Sm irinov

Vol. 5 Gen eralized Point Models in Structural Mechanics

Author: I. V. Andronov

Vol. 6 Mathem atical Problems of the Control Theory

Author: G. A. Leonov

Vol.

7 Vibrational Mech anics: Theory and Applications to the Problems of

Nonlinear Dynamics

Author: llya I. Blekhmam

SERIES ON STABILITY, VIBRATION AND CO NTR OL OF SYSTEMS

<^Q >

S e r i e s

A

Volume

4

Series Editors:

A.

Guran,

A.

Belyayev,

H.

Bremer,

C.

Christov

& G.

Stavroulakis

Asymptotic Methods

in the

Buckling Theory

of

Elastic Shells

Pe tr E. Tovstik

Andrei L. Smirnov

St.

Petersburg State University, Russia

Edited

by

Peter

R.

Frise

Windsor University, Canada

Ardeshir G uran

Institute for Structronics, Canada

V f e World Scientific

wfc Singapore'New Jersey •London*ingapore • New Jersey • London • Hong Kong



Published by

W orld Scientific Pu blishing Co . Pte. Ltd.

P O Box 128, Farrer Road, Singapore 912805

USA o ffice: Suite I B , 1060 Main Street, River Edge, NJ 07661

UK o ffice:

57 Shelton Street, Covent Garden, London WC2H 9H E

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 © 200 1 by W orld Scientific Pub lishing Co . Pte. Ltd.

All rights reserved. This book, or parts

thereof,

may not be reproduced in any form or by any means,

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ISBN 981-02-4726-5

Printed in Singapore.



Pre face

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

the buckling of thin-walled s tructures , and the solutions of many important

problem s have been obtain ed. In spite of th e variety of boo ks th a t 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

part ial ly f i l led by the monograph "Asymptotic Methods in Buckling Theory of

Elast ic Shells", which is a revised and exten ded ed it ion of th e m ono graph "T he

Stabil i ty 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 applicat ion of asymptotic methods to the problem of

shell buckling. Previou sly these m etho ds have been applied to problem s of

shell statics and vibrations.

Th ere are m any new results in this rathe r comp act book. Th e s tat ic s tabil i

ty p roblem for th e case of small mem brane deform ations an d angles of rota t ion

is stud ied comp letely. In th e genera l form ulation, th e principal questions on

th e developm ent of local buckling m odes are stud ied. T he results pre sented

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

modern numerica l methods .

One of the unique features of the book is the large bibliography of litera

ture on the applicat ion 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 general ly been translated

into English bu t is s ti ll not widely know n, despite the high s ta nd ard s p revalent

in this body of research.

The book is directed both to researchers working on the analysis and con

s t ruc t ion of th in-wal led s t ruc tures and cont inuous media and a lso to math

ematicians who are interested in the applicat ion of asymptotic methods to

v



VI

Preface

problems of thin shell buckling. The book may also be useful for g radua te and

post-graduate s tudents of Mathe ma tics , Engineering and Physics.

The present book

is a

result

of the

scientific cooperation

of the

Depar t

men t s of Theoret ical and Applied Mechanics of the Faculty of Mathemat ics

and Mechanics at St. Petersburg State University in Russia and Depar tmen t

of Mechanical and Aerospace Engineering at Carleton University in Ottawa,

C an ad a . 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 Ins t i tu te 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. Pe te r sburg and Carleton

University made

it

possible

to

prepare th is manuscr ip t .

This work was suppor ted in par t by Russian Foundation for Fundamen ta l

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 par t of the

book

and

drew most

of the

pictures.

We

would also like

to

thank

our

students

N . Kolysheva, Yu. Dyld ina, Yu. Balanina, N. Vasilieva, E. Kreis, A. Yam-

barshev,

V.

Braulov,

and M.

Antonov

for

their help

in

preparat ion

of the

manuscr ip t .

Peter Tovstik and Andrei Smirnov



C o n t e n t s

P r e f ac e v

I n t r o d u c t i o n 1

B a s i c n o t a t i o n 5

1 Eq u a t ion s o f T h in E las t i c S h e l l T h e or y 7

1.1 Ele m en ts of Surface Th eo ry 7

1.2 Equ il ibrium Equ ations and Bo und ary Con dit ions 10

1.3 Er ror s of 2D Shell Th eor y of Kirchhoff-Love Ty pe 16

1.4 M em bran e Stress St ate 23

1.5 Technical Shell Th eory Eq uatio ns 26

1.6 Technical The ory Eq uatio ns in the Oth er Cases 29

1.7 Sha llow Shells 30

1.8 In itia l Im perfe ctions 31

1.9 C ylin dric al Shells 31

1.10 T he Po ten tial Energy of Shell Deform ation 33

1.11 Pr ob lem s an d Exercises 34

2 B as i c Eq u a t ion s o f S h e l l Bu c k l in g 37

2.1 Ty pe s of Elastic Shell Buck ling 37

2.2 Th e Buckling Equ ations 42

2.3 Th e Buckling Eq uations for a M emb rane State 42

2.4 Buckling Eq uation s of the General Stress Sta te 46

2.5 Pro blem s and Exercises 48

3 S i m p l e B u c k l i n g P r o b l e m s 4 9

3.1 Bu ckling of a Shallow Con vex Shell 49

3.2 Shallow Shell Bu ckling M odes 53

3.3 T he Non -Uniqu eness of Buck ling M odes 55

3.4 A Circu lar Cy lindrical Shell Un der Ax ial Com pression 57

vn



viii Contents

3.5 A Circular Cylindrical Shell Un der Ex tern al Pressu re 60

3.6 Es tim ate s of Critical Load 62

3.7 Problem s and Exam ples 67

4 B u c k l i n g M o d e s L o c a l i z e d ne a r P a r a l l e l s 6 9

4.1 Local Shell Buc kling M odes 69

4.2 Co nstru ction Alg orithm of Buckling M odes 70

4.3 Buc kling Mo des of Conv ex Shells of Re volu tion 77

4.4 Buc kling of Shells of Re volu tion W it ho ut Torsion 80

4.5 Buck ling of Shells of Re volu tion Un der Torsion 86

4.6 Pro ble m s and Exercises 89

5 N o n - h o m o g e n e o u s A x i a l C o m p r e s s i o n o f C y l i n d r i c a l S h e l l s 9 1

5.1 Buckling Modes Localized near G en erat rix 92

5.2 Recon struction of the As ym ptotic Expan sions 95

5.3 Axial Com pression and Ben ding of Cylin drical Shell 98

5.4 T he Influence of Inte rna l Pre ssure 102

5.5 Buc kling of a N on-C ircula r Cy lindric al Shell 103

5.6 Cylind rical Shell with C urv atu re of Variable Sign 104

5.7 Pr ob lem s and Exercises 108

6 B u c k l i ng M o d e s L o c a l i z e d a t a P o i n t 1 1 1

6.1 Local Buckling of Convex Shells I l l

6.2 Co nstru ction of the Buckling M ode 114

6.3 Ellipsoid of Rev olution Und er Co m bined Load 119

6.4 Cylind rical Shell Un der Axial Com pression 122

6.5 Co nstru ction of the Buckling M odes 127

6.6 Pro ble m s an d Exercises 130

7 S e m i - m o m e n t l e s s B u c k l i n g M o d e s 1 3 3

7.1 Basic Eq uatio ns and Bo und ary Co nditio ns 133

7.2 Buc kling M odes for a Conic Shell 135

7.3 Effect of Ini tia l M em bra ne Stress R es ul tan ts 141

7.4 Semi-M om entless Buckling Modes of Cylind rical Shells 143

7.5 Pro ble m s and Exercises 148

8 E ff ec t o f B o u n d a r y C o n d i t i o n s o n S e m i - m o m e n t l e s s M o d e s 151

8.1 Construct ion Algor i thm for Semi-Mom entless Solu tions . . . . 151

8.2 Sem i-M om entless Solu tions 154

8.3 Ed ge Effect So lutio ns 158

8.4 Sep aration of Bo und ary Co ndition s 161

8.5 Th e Effect of Boun dary Con ditions on the Crit ical Load . . . . 170

8.6 B ou nd ary Co nd ition s and Buc kling of a Cy lindric al Shell . . . 175



Contents

ix

8 .7 Co n i c S h e l ls U n d e r E x t e r n a l P r e s s u r e 1 79

8 .8 P r ob le m s an d Exe rc i ses 184

9 T o r s i o n a n d B e n d i n g o f C y l i n d r i c a l a n d C o n i c S h e l l s 1 8 5

9 .1 To r s io n o f C yl in dr i c a l She l l s 185

9 .2 Cy l i n d r i c a l S h e l l u n d e r C o m b i n e d L o a d i n g 1 88

9 .3 A S h e ll w i t h N o n - C o n s t a n t P a r a m e t e r s U n d e r T o r s i o n 1 91

9 .4 B en din g o f a C yl in dr i c a l She l l 195

9 .5 T h e To r s ion an d B en din g o f a Co nic She l l 196

9 .6 P r o b l e m s a n d E x e r c i s e s 1 99

1 0 N e a r l y C y l i n d r i c a l a n d C o n i c S h e l l s 2 0 1

1 0 .1 Ba s i c R e l a t i o n s 2 0 1

1 0 .2 B o u n d a r y P r o b l e m i n t h e Z e r o t h A p p r o x i m a t i o n 2 0 4

10 .3 Bu ck l in g o f a Ne ar ly C yl in dr i c a l She l l 206

10 .4 To r s ion o f a Ne ar ly C yl in dr i c a l She l l 210

1 0 .5 P r o b l e m s a n d E x e r c i s e s 2 1 1

1 1 S h e l l s o f R e v o l u t i o n o f N e g a t i v e G a u s s i a n C u r v a t u r e 2 1 3

1 1 .1 I n i t i a l E q u a t i o n s a n d T h e i r S o l u t i o n s 2 1 4

1 1 .2 S e p a r a t i o n o f t h e B o u n d a r y C o n d i t i o n s 2 1 5

1 1 .3 B o u n d a r y P r o b l e m i n t h e Z e r o t h A p p r o x i m a t i o n 2 1 9

1 1 .4 B u c k l i n g M o d e s W i t h o u t T o r s i o n 2 2 3

1 1 .5 T h e Ca s e o f t h e N e u t r a l S u r f a c e Be n d i n g 2 2 5

1 1 .6 T h e Bu c k l i n g o f a T o r u s S e c t o r 2 2 8

1 1 .7 S h e l l w i t h G a u s s i a n C u r v a t u r e o f V a r i a b l e S i g n 2 3 2


1 2 S u r f a c e B e n d i n g a n d S h e l l B u c k l i n g 2 3 9

1 2 .1 T h e T r a n s f o r m a t i o n o f P o t e n t i a l E n e r g y 2 3 9

1 2 .2 P u r e Be n d i n g Bu c k l i n g M o d e o f S h e l ls o f Re v o l u t i o n 2 4 2

1 2 .3 T h e B u c k l i n g o f a W e a k l y S u p p o r t e d S he ll of R e v o l u t i o n . . . . 2 4 5

1 2 .4 W e a k l y S u p p o r t e d Cy l i n d r i c a l a n d Co n i c a l S h e l ls 2 5 2

1 2 .5 W e a k l y S u p p o r t e d S h e ll s o f N e g a t i v e G a u s s i a n C u r v a t u r e . . . 2 5 9


1 3 B u c k l i n g M o d e s L o c a l i z e d a t a n E d g e 2 6 7

1 3.1 R e c t a n g u l a r P l a t e s U n d e r C o m p r e s s i o n 2 6 7

1 3.2 C y l i n d r i c a l S h el ls a n d P a n e l s U n d e r A x i a l C o m p r e s s io n . . . . 2 70

1 3 .3 Cy l i n d r i c a l P a n e l w i t h a W e a k l y S u p p o r t e d E d g e 2 7 5

1 3 .4 S h a l l o w S h e l l w i t h a W e a k E d g e S u p p o r t 2 7 9

13 .5 M od es o f She l l s o f R ev o lu t ion Loca l i zed ne ar an Ed ge 284

1 3 .6 Bu c k l i n g M o d e s w i t h T u r n i n g P o i n t s 2 8 7



x

Contents

13.7 M odes Localized nea r the W eakest Po int on an Edg e 290

13.8 Pro blem s and Exercises 294

1 4 S h e l l s o f R e v o l u t i o n u n d e r G e n e r a l S t r e s s S t a t e 2 9 7

14.1 T he Basic Eq ua tio ns an d Edge Effect So lution s 297

14.2 Buckling with Pseu do-b end ing Modes 303

14.3 T he Cases of Significant Effect of Pre -bu cklin g stra ins 308

14.4 Th e W eakest Parallel Coinciding W ith an Edge 316

14.5 Pr ob lem s and Exercises 319

B i b l i o g r a p h y 3 2 1

L i s t o f F i g ur e s 3 3 5

Lis t o f Ta bles 33 9

I n d e x 3 4 1

A b o u t t h e A u t h o r s a n d E d i t o r s 3 4 5



2

Introduction

l inear problems of shallow shell buckling will not be considered in this book.

Likewise, the effect of initial imperfections is not taken into account.

Development of approximate asymptotic formulae for expected buckling

modes and corresponding critical loads is the basic subject of the book. Shells

with various neutral surface geometry and load and support conditions are

analyzed.

Significant attention is given to construction of localized buckling modes,

in which many small pi ts appear.

In some cases these pits cover the entire neutral surface while in other

cases the localization of buckling modes near some weak lines or points on the

neutral surface occurs. This localization is related to non-homogeneity of the

initial stress-strain state, or to variability of shell Gaussian curvature and/or of

shell thickness. Localization in the neighbourhood of an edge may be related

to specific features of the edge support.

In publications on thin shell buckling (see [21, 22, 59, 61, 149, 175]) one

can find numerous examples of buckling modes which are localized in neigh

bourhood of certain lines or points (for example, the buckling problem for a

stretched spheroid under external pressure when the pits form along the equa

t o r ) . However, the analy tical description of these localized buckling mode s

that result from the asymptotic solution of the buckling equations is practi

cal ly absent in shell theory monographs. This fact prompted the authors to

write this book.

The methods of solution of the buckling problems used below are based on

the methods of the asymptotic solution and quali tat ive analysis developed by

A .L. Goldenv eizer a nd his pu pils [51, 52, 57, 70, 88] in connectio n w ith the

problems of shell static, vibration and buckling.

From the point of view of asymptotic solutions, shell free vibration and

buckling have much in common. This is because, in both cases, we come to

the boundary value problem with a small parameter in the higher derivatives.

The main difference is that in the buckling problems we are interested only in

the smallest (or, perhaps only in the smallest few) eigenvalues.

The typical s i tuation for an asymptotic integrat ion of equations with non-

constant coefficients is that when in the integration domain, transition lines

appear [15, 57, 88, 125, 180], (they are called caustics in the field of acoustics

[174]) the region is sep ara ted into p ar ts with qu alita tive ly different solution

behaviour. In shell buckling problems the transi t ion l ines separate the part of

the neutral surface which contains the pits which appear under buckling.

The lowest eigenvalue in which we are interested corresponds to the buck

ling m ode which has pits only on a sm all po rtion of the n eut ral surface (i.e. lo

calised in the neighbourhood of the weakest line or point). The peculiarities

of asymptotic methods used below are connected with this circumstance.

In the case when the weakest point does not coincide with a shell edge, we



4

Introduction

co inc ide wi th a she l l edge a re p resen ted . In Chap te r 5 , t he buck l ing modes fo r

a c y l i n d r i c a l s h e l l u n d e r n o n - h o m o g e n e o u s a x i a l p r e s s u r e a r e d e v e l o p e d .

In Chap te r 6 , t he buck l ing modes fo r convex and cy l indr i ca l she l l s loca l i zed

i n t h e n e i g h b o u r h o o d o f t h e w e a k e s t p o i n t w h i c h d o e s n o t c o i n c i d e w i t h a s h e l l

e d g e a r e c o n s t r u c t e d .

T h e ne x t 4 ch ap te r s ( f rom 7 to 10) a r e de vo ted to the con s id era t ion o f s em i -

moment l es s buck l ing modes fo r cy l indr i ca l and con ic she l l s . For such she l l s the

p i t s f o r m e d u n d e r b u c k l i n g s t r e t c h a l o n g t h e g e n e r a t r i x . If t h e s t r e s s - s t r a i n

s t a t e v a r i e s i n t h e c i r c u m f e r e n t i a l d i r e c t i o n t h e n t h e l o c a l i s a t i o n o f b u c k l i n g

m o d e n e a r t h e w e a k e s t g e n e r a t r i x w il l o c c u r . T h e e x t e r n a l n o r m a l p r e s s u r e ,

t o r s i o n , a n d b e n d i n g a r e t y p i c a l l o a d s w h i c h c a n c a u s e t h i s b u c k l i n g m o d e . T h e

effect o f bo u n da ry c on d i t io ns on th e c r i t i ca l bu ck l in g load i s a l so ex am in ed .

I n Ch a p t e r 1 1 t h e b u c k l i n g of s h e ll s o f r e v o l u t i o n w i t h n e g a t i v e G a u s s i a n

c u r v a t u r e i s i n v e s t i g a t e d . U n d e r b u c k l i n g , t h i s t y p e of s h e l ls g e n e r a l l y d i s p l a y s

p i t s which cover the en t i r e neu t r a l su r f ace .

C h a p t e r 1 2 p r e s e n t s a d i s c u s s i o n of t h e b u c k l i n g o f w e a k l y s u p p o r t e d s h e l ls .

T h e r e l a t i o n s h i p b e t w e e n t h e b e n d i n g of t h e n e u t r a l su r f a c e a n d t h e b u c k l i n g

of the she l l i s examined .

T h e p r o b l e m s f o r t h e s h e l l s w i t h i n i t i a l l y m e m b r a n e s t r e s s - s t r a i n s t a t e t h a t

r e s u l t s

T h e b u c k l i n g m o d e s l o c a l iz e d n e a r a s h e ll e d g e a r e c o n s i d e r e d i n C h a p t e r 1 3 .

A g a i n t h e i n i t i a l s t r e s s - s t r a i n s t a t e o f t h e s h e l l i s a s s u m e d t o b e m e m b r a n e .

In some cases , buck l ing i s connec ted wi th a weak edge suppor t whi l e in o ther

cases ,

the weakes t l i ne o r po in t co inc ides wi th the edge .

C h a p t e r 14 c o n c l u d e s t h e b o o k a n d p r o v i d e s a t r e a t m e n t o f b u c k l i n g p r o b

l e m s w h e r e t h e i n i t i a l s t r e s s - s t r a i n s t a t e is a c o m b i n a t i o n o f m e m b r a n e , b e n d

ing and ed ge e ff ect s. T h i s typ e of s t r es s - s t r a in s t a t e i s r e f e rr ed to as a gen era l

s t r e s s - s t r a i n s t a t e i n t h i s b o o k . T h e i n i t i a l s t r e s s - s t r a i n s t a t e is r e p r e s e n t e d a s

t h e s u m o f a m e m b r a n e s t r e s s - s t r a i n s t a t e a n d a n e d g e e ffe ct. P r o b l e m s w h e n

t h e t r a n s i t i o n t o a g e n e r a l f o r m u l a t i o n s l i g h t l y c h a n g e s t h e b u c k l i n g l o a d , a n d

p r o b l e m s w h e n t h e m e m b r a n e f o r m u l a t i o n g iv e s i n c o r r e c t v a l u e s o f t h e b u c k

l ing load a re cons idered .

In r ea d in g se lec ted c ha p te r s it i s use fu l to no te th a t in C h ap te r s 1 -3 wel l -

k n o w n i n t r o d u c t o r y i n f o r m a t i o n i s p r e s e n t e d , C h a p t e r s 4 - 1 1 a r e d i v i d e d i n t o

f o u r p a r t s ( w e a k l y c o n n e c t e d t o e a c h o t h e r ) a c c o r d i n g t o t h e m e t h o d o f a s y m p

tot ic in tegrat ion used: ( i ) Sect ions 4 and 5 , ( i i ) Sect ion 6 , ( i i i ) Sect ions 7 , 8 , 9

an d 10 an d ( iv ) Sec t ion 11 . F in a l ly C h ap te r s 12 , 13 an d 14 co n ta in in fo rm at io n

o f g e n e r a l n a t u r e a n d a r e la r g e l y i n d e p e n d e n t o f t h e e a r l ie r p a r t s o f t h e b o o k .

B a s i c n o t a t i o n

Th ree digit indexing of equ ations is used in the book. Each equ ation index

number includes the number of the chapter, the number of the section within

the chapter, and finally the number of the equation. For brevity the chapter

number is omitted if we refer to equations from within the same chapter, and

the chapter and section numbers are omitted if we refer to equations from the

same section.

Two digit indexing of figures, tables and remarks is used, comprising the

number of the chapter and the number of the figure, table or remark.

To designa te th e order of a function, ip(h*), so as to com pare it with ano ther

function, \l>{h

t

), as h* —> 0 (or fi,e —> 0) the symbols O, o, ~ are used.

The no ta t ion ip = 0(tp) indicates the existence of a constant A > 0 which

is independent of h* and \ip\ < A\4>\. Sometimes instead of 

—>

0 as

h*

—»• 0. Th e symbol ~

is used for functions of the same asymptotic order, that is (p ~

ip

means tha t

ip

= 0(VO and

ip

= 0{<p) simultaneously.

B a s i c S y m b o l i c N o t a t i o n

a, 0 ( a i , a

2

) — orthogonal curvilinear coordinates of a point on the neutral

surface;

A, B (A\, A2) — Lame's coefficients on the neutral surface;

E — Young 's modulus ;

e i , e

2

, n — unit vectors of the coordinate system connected with a point

on the neutral surface before deformation;

e*, e\, n* — unit vectors of the coordinate system connected with a point

on the neutral surface after deformation;

Sij

— membrane strains in the neutral surface;

£i, u> — membrane strains in the l inear approximation;

5

6

Basic notation

e — a s m a l l p a r a m e t e r , e

8

= / i * / ( 1 2 ( l — v$)) e > 0;

£ o — a s m a l l p a r a m e t e r , e

6

= / i ^ / ( 1 2 ( l — UQ)), S

0

> 0;

i

— a n i m a g i n a r y u n i t ,

i = \f—\,

h — t h e s h e l l t h i c k n e s s ;

ho

— the charac te r i s t i c she l l t h i ckness in the case o f a she l l o f non-cons tan t

t h i c k n e s s ;

h„

— t h e d i m e n s i o n l e s s s h el l t h i c k n e s s , t h e p r i n c i p a l s m a l l p a r a m e t e r ,

h* —

h/R>0;

Eh Eh

3

K = 7T, a n d D = -—— ^- — th e she ll s t i f fnesses;

1 — i/

z

12(1 — v

1

)

&ij &2 — th e p r inc ipa l cu rva tu r es o f the ne u t r a l su r f ace ;

L\,

L i — s izes of a sha l low sh el l ;

A — l o a d p a r a m e t e r (A > 0 ) ;

H

— a s m a l l p a r a m e t e r ,

u\

A

— h%(l2(l — VQ)),

/i > 0;

Mi, Hi, H — p r o j e c t i o n s o f i n t e r n a l s t r e s s - c o u p l e s i n t o t h e u n i t v e c t o r s

e*, n* r e l a t e d t o t h e u n i t l e n g t h b e f o re t h e d e f o r m a t i o n ;

P i , P, M\, M

— ex te rn a l fo rces an d m o m en ts on an edge o f a she l l o f

rev ol ut i on (see Fi gu re 1 .3);

v — P o i s s o n ' s r a t i o ;

p ,

q — w a v e n u m b e r s ;

I I — po te n t i a l energ y o f de for m at io n o f a she l l ;

9 i , 92 , 9 (9 i , 92, 9*) — pro jec t ion s o f th e vec to r o f ex te r na l loa d in te ns i ty

i n t o u n i t v e c t o r s

e\, e^, n, (e*, e*

2

, n*)

r e l a t e d t o t h e u n i t a r e a o f t h e n e u t r a l

s u r f a c e b e f o r e t h e d e f o r m a t i o n ;

•Ri) R-2 — t h e p r i n c i p a l r a d i i o f c u r v a t u r e o f t h e n e u t r a l s u r f a c e ;

R

— th e ch ara c te r i s t i c l in ear s ize o f th e ne u t r a l su r f ace ;

Qi, Qi

— s h e a r s t r e s s - r e s u l t a n t s r e l a t e d t o t h e u n i t l e n g t h b e f o r e t h e d e

f o r m a t i o n ;

T{, S{, S — p r o j e c t i o n s o f i n t e r n a l s t r e s s - r e s u l t a n t s i n t o u n i t v e c t o r s e* , e*

2

r e l a t e d t o t h e u n i t l e n g t h b e f o r e t h e d e f o r m a t i o n ;

ti — m e m b r a n e d i m e n s i o n l e s s i n i t i a l s t r e s s - r e s u l t a n t s ;

t — th e ind ex o f va r i a t i on o f a s t r es s - s t r a in s t a t e ( see Sec t ion 1 .3 );

s

— the l eng th o f an a rc genera t r ix ( fo r she l l s o f r evo lu t ion and she l l s o f

z e r o c u r v a t u r e ) ;

<fi — a n g l e i n t h e c i r c u m f e r e n t i a l d i r e c t i o n ;

9

— ang le be tw een t he ax i s o f ro ta t io n a nd th e no rm al ( fo r a she l l o f

r e v o l u t i o n ) ;

U{, w (u, v, w) — p r o j e c t i o n s o f d i s p l a c e m e n t ;

Wi,

7f — ang le s o f ro ta t io n o f un i t ve c to r s e ; ,

n

i n t h e l i n e a r a p p r o x i m a t i o n ;

Xi, T

— b e n d i n g a n d t o r s i o n a l s t r a i n s of t h e n e u t r a l s u r f a c e i n t h e l i n e a r

a p p r o x i m a t i o n .



C h a p t e r 1

E q u a t i o n s of T h i n E l a s t ic

S h e l l T h e o r y

In this chapter we will consider the relations of the theory of surfaces and

surface deform ations, th e equilibrium equa tions of shell theory, the con stitu

t ive relations and some approximations of these equations and relations. The

reader can become acquainted with the derivation and detailed discussion of

these equations in monographs [12, 24, 28, 33, 51, 52, 87, 111, 119] and others.

1.1 E lem en ts of Sur face T h e o ry

Consider the curvilinear coordinate system

a, f3

on th e un deformed shell

ne utra l surface. Th e system axes coincide with the lines of curv atur e. Th e

position of a point M on the neutral surface is defined by its radius vector

r = r(a, /?).

We introduce local orthogonal coordinate system by means of unit vectors

e i , e-2, and n, where

I dr I dr dr 9r

C1

= W

e2 =

B~W

n

=

e

^ ^

A

- ^ '

B=

d-p-

( h L 1 )

The first and second quadratic forms of the surface are:

I = ds

2

=A

2

da

2

+

B

2

df3

2

,

1 1 =

^-da

2

+

^d/3

2

,

(1.1.2)

where

ds

is the arc length on the surface,

R\, R?

are the principal radii of

curvature. We also use the notation ki = R^~ .

7

8 Chapter 1. Equations of Thin Elastic Shell Theory

Here A, B, Ri, and R

2

are the functions of the coordinates a and /? satis

fying the Codazzi-Gauss relations.

R

2

dp'

1 dB

(1.1.3)

1

R1R2

The following formulae for the differentiation of unit vectors are valid:

dei

_

_}_d_A

A_ de

x

_

1

dB

da

~

Blp

e2 +

Ri™' ~W~Ada

e2

'

de

2

1 dA de

2

I dB B

da B dp dp A da R

2

dn

A dn _ B

a»7

~ ~R~i

ei

~dp~~~R

2

e2

'

After deformation, a point M occupies positio n

M*,

which is defined by the

radius vector

r*.

We assume that

T*=T

+ U,

U = uie\ + u

2

e

2

+ wn, (1.1.5)

where U

is

the displacem ent vector and u

x

, u

2

and w are

its

projections

on

the unit vectors before deformation.

Assuming

1

d r

*

, 1

-7-*— =

I

1

+

£

i )

e

i +

w

i

e

2

-

7i™,

-p -^-3-

=

(1

+

£2) e

2

+ oj

2

e

x

-

f

2

n.

we get

_ 1 dui I dA w _ 1 du

2

1 dB w

£ l =

A~da~

+

A~B'dp

U2

~R'

1

'

£2=

B~dJ

+

~AB~l)a~

Ul

~ R~

2

'

1 du

2

1 &4 1 dux 1 8B

Wl =

A-fo-ABW

1

'

U

'

2=

BW~AB^

U2

'

(1

-

L7)

_ 1 dw

«i _ 1

<9w

«2

7 1

~

~A~da~~~R

1

~ '

7 2 _ _

B ^ " ^ '

After deformation the coordinates a and /? become non-orthogo nal. We

then introduce unit vectors

1 9r* 1 dr* „ el x e$

e i

=

A:aa7'

e2 =

5:^-'

n =

- 5 s r -

(1

-

L8)



1.1.

Elemen ts of Surface Theory

9

where 6 is the angle between e\ and e*

2

( e ' e l = cos# ) .

In the deformed state the first quadratic form of the surface is

I , =

Al da

2

+ 2 i » f l , c o s

6dad/3 + B

2

d/3

2

,

(1.1.9)

where

1 1

£11 = £l + ^ (

£

1 +

W

l + 7 l ) , £22 = £2 + ^ (

£

2 +

w

2 + ll) >

COS0 = £12 = U + £ i W

2

+ £ 2 ^ 1 + 7 1 7 2 , ( 1 .1 .1 0 )

A* = A

2

(l + 2en), B

2

= B

2

(1 + 2 e

2 2

) ,

W

=

W l

+

W 2

.

Th e values e n , £12 and £22 are the m em bran e strains of the ne utra l surface

and we assum e tha t the y are sma ll com pare to unity. Th e variables

e\

,

to

an d £2

are the membrane strains in the l inear approximation. For small deformations

wi and w

2

are the rotation angles of the unit vectors B\, e

2

in the tangential

plane, and 71 and 72 are the rotation angles for the normal n.

After deformation the second quadratic form of the surface may be repre

sented as

I I ,

=

L

u

da

2

+ 2L

12

da

d/3

+ L

22

dp

2

,

(1.1.11)

where

d

2

r* d

2

r* d

2

r*

The expressions for L ij can be found by formulae (4), (6) and (8) which

are complicated non-linear functions of the displacements. We can write the

approximate formulae in which the terms of the second order with respect to

the displacements are neglected

10

Chapter 1. Equa tions of Thin Elastic Shell Theory

W ith th e same precision the derivatives of the unit vectors after deform ation

have the form [61]

de\ 1 dA

„ /

1 d(Bu)

1

dA,

\

, /

1

\

,

d e *

2

IdB

,

/ 1 0 ( J 4 W )

1

9 B

t

\

, /

1

\ ,

da \B

t

d/3 B da J

l

B d/3

de\

( 1 8B, u8A\

,

wdB

t

_ , ,,

, , ^

-

T J

)

e

2--7 -<

+

Brn

*,

(1.1.14)

5/? V-4*

da A

dp

J

* Ada

d/3 \R

2

' V V R2)

In the linear approximation the relationships for the unit vectors both be

fore and after the deformation are the following

e*

=

e i + w

1

e

2

- 7 i n ,

e\ = e

2

+ w

2

e i - ~ f

2

n, (1.1.15)

n* = n+ 7 i d + 7 2 e

2

.

The discussion below is limited to the small displacement case since the

ap plic atio n of the ap pro xim ate formu lae (13)— (15) does not allow us to consider

problem s with finite (but no t small) displacem ents and rot atio n angles of a shell

element. The case of finite displacements and rotation angles is discussed in

[87,

26, 131].

1.2 E q u i l i b r i u m E q u a t i o n s a n d B o u n d a r y C o n

d i t ions

The equilibrium equations of a deformed element of the shell are

—

(BFi) +

—

(A F

2

) + ABq = 0,

d a dp ,

12

jx

-(BG

1

)+-(iG

2

) +

^5e;xF

1

+4fl,e^xf

2

= 0.



1.2. Equilibrium Equations

and

Boundary Conditions

11

Here F\ and G\ (F

2

and G

2

) are the stress-resultant and the stress-couple

vectors related

per the

unit length before

the

deformation acting

on the

line

a

=

const

(/? =

const)

and q is the

intensity

of the

external surface loads

per

the unit area before the deformation (see Figure 1.1). If the intensity of the

external surface loads q* per the unit area after the deformation is introduced

(for example, the hydrostatic pressure) then the last term in the first rela tion

(1)

is to be

replaced

by

A*B*q*

sin

6.

Figure 1.1: The stress-resultants and stress-couples acting on a shell element.

We in troduce the projections of these vectors on the basis vectors after

deformation

F i = T

t

e\ + S

ie

*

2

+ Q

in

*,

F

2

= T

2

e*

2

+ S

2

e\ +

Q

2

n*

,

(1.2.2)

d = ff

ie

*-M

ie

£,

G

2

=

M

2

el

-

H

2

e*

2

,

q

= q\e\ + qle*

2

+

q*n*.

The scalar equilibr ium equations

are

obtained

by

projecting equation

(1)

onto the basis vectors after deformation and using formulae (1.14). Then we



12

Chapter 1. Equations of Thin Elastic Shell Theory

can write [61]

djBTj) AdB* d(AS

2

) B 8A. „ OA

-A B (T - ^ ) Q

2

+ ABq\

= 0,

d(AT

2

) B dA. d{BSi) A dB. dB

-AB (T

- | 0

Qx

+ ABq*

2

=

0,

d{B

Ql

X

+

8SAQl

+AB

(±

+

\

Ti+

da 80 \Ri

+AB (Si + S

2

) T + AB

Q - +

x

2

\ T

2

+ AB q* -

0,

d(BHi)

+

d(AM

2

)

+

(AQB+_ _ d{Aw)\

H

^

+

da dp \A. da dp

dA

__

dB

„ ,

dp da

d[AH

2

)

+

d(BMi)

+

(_B_0A. _

d(BuY

R

^

(1.2.3)

dp da \B* dp da

+

BA,Q

1 +

AB.Q

2 +

{.-)M

+

dB

TT

dA , ,

+ JT-UH2- ^ w M i = 0 ,

da dp

A.BSi - AB.S

2

+ AB f

1

- + xA Hi~

-AB (^- + x

2

\ H

2

-

{Mi

-

M

2

) ABr =

0.

We consider shells of absolutely elastic isotropic Hookeian material, with

Young 's modulus

E

and Poisson's ratio

v.

To derive two-dim ensional shell

theory equations we use the Kirchhoff-Love hypotheses. Let us take the con-



16

Cha pter 1. Equa tions of Thin Elastic Shell Theory

(1 .2 .15)

on edge a = a

0

the va lues

M i .

f o r m u l a e ( 1 2 ) m a y b e t r a n s f o r m e d i n t h e s a m e w a y .

I n a d d i t i o n t o t h e a b o v e a s s u m p t i o n s w e l i m i t t h i s d e v e l o p m e n t t o b o u n d

ary cond i t ions , desc r ibed as e i the r f ixed o r f r ee cond i t ions in the genera l i zed

d i r e c t io ns (9 ) o r (11) . W e wi l l cons id er on ly the s im ple case w he n the gener

a l i zed d i s p la ce m en t o r gen era l i zed fo rce cor resp on din g to i t i s eq ua l to ze ro a t

th e edg e . In th e o th er word s , i t i s a s s um ed th a t the bo un d ar y s ti ffness in each

d i r e c t i o n c o i n c i d i n g w i t h t h e a x e s o f t h e t r i h e d r o n ( m ,

t,

« ) , i s e q u a l e i t h e r

to inf in i ty or to zero .

T h e r e f o r e a t e a c h e d g e f o u r b o u n d a r y c o n d i t i o n s s h o u l d b e i n t r o d u c e d

u

m

= 0 o r T

m

= 0 ,

v

t

=

0 o r

S

t

= 0,

(1 .2 .16)

ID

= 0 or Q

mt

, = 0,

y

m

= 0 o r M

m

= 0

Also we wi l l no t cons ider the cases o f e l as t i c suppor t o f the type u

m

+ /3 T

m

= 0

o r s u p p o r t b y a r i b .

We d e n o t e v a r i a n t s o f t h e b o u n d a r y c o n d i t i o n s b y f o u r - f i g u r e n u m b e r s o f

O 's a n d l ' s w h e r e 1 i n d i c a t e s t h e s u p p o r t i n t h e c o r r e s p o n d i n g d i r e c t i o n a n d

p l a c i n g t h e d i g i t s i n t h e s a m e o r d e r s a s t h e c o n d i t i o n s i n ( 1 6 ) . F o r e x a m

p le ,

1 1 11 m e a n s a c l a m p e d e d g e , 0 0 00 m e a n s a f re e e d g e a n d 0 1 1 0 m e a n s a

s i m p l e s u p p o r t . T h e b o u n d a r y c o n d i t i o n s a r e o f te n d e n o t e d b y C , a n d 5,-

(i = 1,2,..., 8) [21] (see Ta b le 1.1).

1.3 E rr o rs of 2D Shel l T h e o ry of

Kirchhoff-

Love Type

T h e en t i r e sys tem of the equ a t i on s o f 2D she l l t he ory o f th e Ki rch hof f -Lov e

t y p e c o n s i s t s o f fiv e e q u i l i b r i u m e q u a t i o n s ( 2 . 6 ) , s e v e n c o n s t i t u t i v e r e l a t i o n s



1.3. Errors of 2D Shell Theory of Kirchhoff-L ove Type 17

T a b l e 1 .1 : T h e b o u n d a r y c o n d i t i o n s

u

m

= 0

T

m

=0

0

0

0

0

0

0

0

0

w»

= 0

S

t

=Q

1

1

1

1

0

0

0

0

1

I

1

1

0

0

0

0

w =

0

O n , = 0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

7m = 0

M

m

=0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

C i

Si

c

5

s

5

c

3

s

3

c

7

s

7

c

2

s

2

Ce

s

6

C

A

SA

c

8

Ss

clamped edge

simply supported edge

(Navier 's conditions)

free edge

(2 .4 ) ,

n i n e r e l a t i o n s fo r s m a l l s t r a i n s , a n g l e s o f r o t a t i o n a n d d i s p l a c e m e n t s

( 1 . 7 ) a n d ( 1 . 1 3 ) a n d t h r e e f o r m u l a e f o r t h e n o n l i n e a r m e m b r a n e d e f o r m a t i o n s

( 1 . 1 0 ) . T h e r e f o r e t h e e n t i r e s y s t e m c o n s i s t s o f 2 4 a l g e b r o - d i ff e r e n t ia l e q u a t i o n s

i n 15 c i n e m a t i c a n d 9 f o rc e u n k n o w n s

« i , u

2

, w, e i , e

2

, wi , w

2

, 71 , 72 , xi, x

2

, r, e n , e

2

2, £12,

T

U

T

2

,S

U

S

2

,M

1

,M

2

,H,Q

1

,Q

2

T h e s o l u t i o n o f t h i s s y s t e m w i t h f o u r b o u n d a r y c o n d i t i o n s o n e a c h e d g e

is cal led

the general stress-strain state

of a shel l . In the cases when the de

f o r m a t i o n s m a y b e i g n o r e d t h e s t r e s s - s t r a i n s t a t e is c a l l e d s h o r t l y the stress

state.

I n t h i s s e c t i o n t h e e r r o r a r o s e u n d e r t r a n s i t i o n f r o m 3 D n o n l i n e a r e q u a t i o n s

of the th eo ry o f e l as t i c i ty to 2D eq ua t io ns o f th e the ory o f she l l s i s d i s cu s sed .

W e a l so g ive th e c l as s i f ica t ion o f th e pa r t i a l s t r es s - s t r a in s t a t e s o f a she l l ,

wh ich no t nece s sa r i ly s a t is fy the b o u nd a ry co nd i t ion s . As a ru le for l inea r

p r o b l e m s t h e g e n e r a l s t r e s s - s t r a i n s t a t e i s a s u m o f t h e p a r t i a l s t r e s s - s t r a i n

s t a t e s .

18

Cha pter 1. Equations of Thin Elastic Shell Theory

To make the asymptotic estimates and to compare the orders of the specific

terms we introduce, for convenience, a small thickness parameter

h*

= ^ , (1.3.2)

where

R

is a characteristic linear size of the neutral surface, for example, the

radius of the curvature, and

ho

is the characteristic shell thickness. In the case

of a shell of a constant thickness we assume ho = h.

Errors in shell theory equations are caused, in the most cases, by the Kirch-

hoff-Love hyp othese s, which leads to one or ano ther v arian t of the cons titutive

relation s. At the sam e tim e, the expressions for strai ns and the equilibrium

equations generally can be written exactly.

Firstly we discuss the errors of the linear shell theory. The equations of the

shell theory contain linear and quadratic terms in unknowns (1). The linear

theory is obtained assuming all quadratic terms be equal to zero.

In [121] it was shown that for the linear theory of shells the error A of the

Kirchhoff-Love hypotheses has the order of the relative shell thickness, i.e.

A ~ A * . (1 .3 .3 )

In (see [52, 77]) the similar estimates for the shell stress-strain state unessen

tially differed from the above one have been obtained.

Let us introduce the index of variation t of stress-strain state as

~A-*z, (1 .3 .4)

where z is any unknown function which determines this state. We will deter

m ine , thro ugh ~ , the values of the same asym pto tic order. A m ore general

error estimation than (3) has the form

A ~ m a x { / i „ ,

hl~

2t

) ,

(1.3.5)

and it follows from here that estimates (3) and (5) coincide as t = 1/2. In [52]

it was shown that, by choosing more complicated constitutive relations than

(2.4) the error A can be reduced to

A~hl~

2t

, Q t<l,

(1.3.6)

as

t

< 1/2. The advantages and disadvantages of this approach are discussed

in [52].

R e m a r k 1 . 1 . T he error of the shell theo ry equ atio ns is den oted in form ulae

(3), (5) and (6) by A. The error of the fixed equation is equal to a fraction of

max

dz

da

i

dz

dp

1.3. Errors of 2D Shell Theory of KirchhofF-Love Type

19

t h e m a x i m u m o f t h e o m i t t e d t e r m s a n d t h e m a i n t e r m of t h i s e q u a t i o n . T h e

e r r o r of t h e s y s t e m i s e q u a l t o t h e m a x i m u m of i t s e q u a t i o n e r r o r s . T h e e r r o r o f

t h e s y s t e m d e p e n d s o n t h e t y p e o f s o l u t i o n ( m a i n l y o n i t s i n d e x t o f v a r i a t i o n ,

see (4 ) ) . I t may occur tha t the o rder , A

s

, of the er ror of the solut ion i s not

eq ua l to th e o rder , A , of th e e r ro r of th e sys tem . For ex am pl e in [56] fo r t he

pr ob le m of f ree v ib ra t io ns o f a she l l o f r ev o lu t ion i t wa s sho w n th a t th e s o lu t ion

e r r o r i s l a r g e r t h a n t h e s y s t e m e r r o r . I n s t e a d o f ( 6) t h e e s t i m a t e A

s

~

hl~

3t

i s v a l i d . A s i m i l a r g r o w t h of e r r o r m a y a l so o c c u r w h e n b u c k l i n g m o d e s a r e

a n a l y z e d . H e r e w e d o n o t c o n s i d e r t h i s p r o b l e m . F o r t y p i c a l b u c k l i n g m o d e s

11"?

t <J 1 /2 an d we m ay expe c t th a t A

s

< hj as A ~ h*. W e n o t e a l s o t h a t

f o r t h e i n t e g r a l s o l u t i o n c h a r a c t e r i s t i c s , s u c h a s t h e f r e q u e n c y o f v i b r a t i o n s o r

the c r i t i ca l load , the o rder o f the i r e r ro r s A \ does no t exceed the o rder o f the

s y s t e m e r r o r A .

La te r in th i s boo k , th e co ns t i tu t iv e r e l a t ion s o f ty pe (2 .4 ) wi l l be use d .

T h a t i s w h y i n e a c h e q u a t i o n , t e r m s o f o r d e r / i* a n d s m a l l e r i n c o m p a r i s o n

w i t h t h e m a i n t e r m m a y b e n e g l e c t e d w i t h o u t l o s i n g a c c u r a c y ( se e [1 2 0 ]) .

I n l i n e a r s h e l l t h e o r y t h e p r o b l e m o f c o m p a r i n g t h e o r d e r s o f t e r m s f o r c h a r

a c t e r i s t ic p a r t i a l s t r e s s - st r a i n s t a t e s su c h a s m e m b r a n e , p u r e b e n d i n g , s e m i -

m o m e n t l e s s , s i m p l e e d g e e f f e c t , o r a g e n e r a l s t r e s s - s t r a i n s t a t e w h i c h i s a c o m

b i na t io n of so m e of these s t r es s - s t r a in s t a t e s and th e def in i t ion o f wh ich a re

g iven be low h as bee n s tu d ie d ex ten s ive ly [24 , 52 , 57] .

The o rder o f each o f the non- l inear t e rms depends on the l eve l o f the ex

t e r n a l s u rf a c e a n d b o u n d a r y l o a d i n g . E a c h b u c k l i n g p r o b l e m is c h a r a c t e r i z e d

by i t s own l eve l o f load and p re -buck l ing s t r a ins .

W e d e n o t e t h e m a x i m u m v a l u e o f t h e m e m b r a n e p r e - b u c k l i n g s t r a i n s b y e

e

= m a x { | e u | , | e i

2

| , |£22|} • (1 .3 .7)

Fur ther , i n r e l a t ions s imi la r to (7 ) we wi l l omi t the module s ign in es t i

m a t i n g t h e a b s o l u t e v a l u e s o f t h e v a r i a b l e s . I n t h e b o o k w e o n l y c o n s i d e r t h e

p r e - b u c k l i n g s t r e s s - s t r a i n s t a t e s f o r w h i c h t h e f o l l o w i n g e s t i m a t e i s v a l i d

e<K, (1 .3 .8)

w h i c h m e a n s t h a t t h e o r d e r o f

e

i s no t l a rger than the o rder o f / i* . In par t i cu la r

t h e e s t i m a t e

e ~ / i , (1 .3 .9 )

i s va l id fo r the buck l ing o f a membrane s t r es s - s t r a in s t a t e fo r a convex she l l

( see Sec t ion 3 .1 ) an d a l so for a m od er a t e l y long cy l in dr i ca l she l l un de r ax ia l

compres s ion ( s ee Sec t ion 3 .4 ) .

20


For the buckling of a cylindrical shell under external pressure (see Sec

tion 3.5) or torsion (see Section 3.6) one can use the estimate

e«h*. (1.3.10)

Let us now consider one formula of (1.10)

£ n

£i + ^{4+^>l + J

2

i)-

(1.3.11)

Generally, a small value of

e\\

does not necessarily lead to sma ll values of

S\,

u\,

and 7 J . Indeed for pure plate bending, under which the plate turns into

a cylindrical shell, e n = 0, bu t £ i, 71 ~ 1. T h at is why we assum e tha t both

the linear membrane strains and the squares of the angles of rotation are also

small variables

m a x { e i , u,u>i, 7,

2

} < h*. (1.3.12)

A ssum ption (12) lim its the class of buckling problem s considered below. In

fact, this assumption has already been used in Sections 1.1 and 1.2 to simplify

the geometr ic relations and equilibr ium equations.

In some buckling problems characterized by a large bending of the neutral

surface estimates (8) does not hold. In particular, in the case of the axisym-

metric deformation of a shell of revolution under axial compression the part of

the neutral surface has the form close to the mirror reflection from the plane

orthogonal to the axis of symmetry

00.

In Figu re 1.3 the ge ne ratr ix of the

shell of revolution before the deformation (solid line) and after the deformation

(dashed line) is shown.

o

Figure 1.3: The large bending of the neutral surface

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

21

In the area of the large bend ing (near the p oint A in Figure 1.3) the fol

lowing estimates are valid

{e

u

e

2

}~hl<

2

, x

x

~R-

l

h-;

112

.

(1.3.13)

The error A increases and for its order instead of (3) we have

A ~ / i i

/ 2

. (1.3.14)

In [167, 168] the consti tutive relations for such type of problem s have been

obtained by mea ns of the as ym ptotic integration of the 3D nonlinear eq uation s.

These relations generalizing equations (2.4) have the relative error of order /i».

In particular, expression for T\ has the form

Ti = K(ei + ue

2

+ M

2

+ Ehh^), (1.3.15)

where the additional terms with the multipliers b\ and 62 take into account the

quadrat ic dependence of stresses on strains and in their tur n the m ultipliers

61 and 62 depend on the nonlinear elastic properties of shell material.

Errors of 2D shell eq uatio ns and of their simplified forms are discussed in

[13].

In the book the problems involving the es t imates (13) and (14) are not

1/2

analyzed. In Section 14.1 one pro ble m in which £ ~

hj

is pointed out.

The special assumptions on the character of the stress-strain state and the

form of the ne utr al surface lead to further simplification of equ ation s (2.6),

(2.4),

(1.7), (1.10), (1.13). Below we give the classification of the characteristic

stress-strain states of a shell proposed by Go ldenv eiser [52] and Novozhilov

[119]. This classification is based upon the relations between tange ntial and

bending strains (stresses) and the index of variation of the stress-strain sta te,

t. The maximal tensile-compressive stresses, a

s

and bending stresses a

b

are

equal to

a

s

= -

max-fTl,

T

2

, S

u

S

2

], <r

b

= m a x { M i , M

2

,

H).

(1.3.16)

Fur ther in this section the relation s between the orders of cr

s

and at, ob

tained

in

[52] are given.

1) For

the mem brane (mom entless)

stress-strain state

a

s

~^>

a

b

and

t

= 0.

The shear stress-resultants Q\ and Q2 may be omit in the first three relatio ns

(2.6).

In the next ch apte rs we consider me m bran e pre-buckling stress-s train

s ta te . In more details the m em brane stress-strain state is discussed in Sec

tion 1.4. For the m em bran e stress-strain s tate with the index of variation

22


t

= 0 the following order relation

a

b

~

h\a

s

is valid. If we supp ose th at

0 < t < 1/2 then

<r

b

~

h

2

t

~

4t

a

s

as 0 < t < 1/2. (1.3.17)

2) For the pure bendings stress-strain state , the shell neutral surface de

forms w ithou t tensile and shear, i .e. e n = e

1 2

= £22 = 0. If th e nonlin

ear tangential s trains are replaced with their l inear approximations, then the

stress-s train sta te with £1 =

u

= £2 = 0 is called

infinitesimal pure bendings.

In accordance with assumption (12) we do not make a difference between pure

bending and infinitesimal pure bendings. The buckling modes for the weakly

sup port ed shell of revolution are close to the pure ben ding s.

3) If the tensile deform ations of the shell ne utra l surface are small c om pared

to its bending deformations, i.e.

maxJEi ,

u,

£

2

} -C i?max{x i , r ,

x

2

},

(1.3.18)

then the corresponding stress-strain state is called the pseudo-bendings. Such

buckling modes are characteristic for shells of zero and negative Gaussian cur

vature.

4) The bending of the shell neutral surface generates

the pure mom ent

stress-strain state, for which the bending deformations and stresses are

signif

icantly larger than the tangential deformations. For 0 < t < 1/2 the relation

opposite to (17)

<r

s

~hl-

4t

(T

b

as 0 <

<

< 1/2. (1.3.19)

is valid.

5) Under buckling of shells of zero Gaussian curvature under external pres

sure and/or torsion the pseudo-bending leads to the semi-momentless stress-

strain state, for which one of the bending moments may be neglected Mi <C

M

2

. For such stress-strain state the indices of variations in axial and circum

ferential directions (so-called partial indices of variation),

t\ —

0 and

t?

= 1/4

are different

dz dz

1/4

da op

where z denotes any of variables (1).

6) For

the combined

stress-strain state the tangential and bending stresses

have equal orders

cr

s

~ "V As it follows from re latio ns (17) and (19) for th is

s ta te t — 1/2. T he deta iled d escrip tion of this st at e is given in Section 1.5 (see

formulae (5.1) and (5.2)). The buckling of convex shells and cylindrical shells

under axial compression is accompanied with the combined stress-strain state.

7) The above mentioned stress-strain states do not always satisfy the bound

ary conditions and the stress-strain state called

the edge effect

may form in the

1.4. Membrane Stress State

23

neighbourhood of the shell edges. Consider the edge a = c*o. All variables (1)

describing the edge effect decrease exponentially away from the shell edge and

have different indices of variations in the edge direction,

t\

< 1/2, and in the

direction that is orthogonal to the edge, t, = 1/2:

^~h:

1/2

z, ^ - h ^ ' z , < 2 < l / 2 . (1 .3 .2 0)

oa op

If i

2

= 0, then the edge effect is called

the simple edge effect.

If the edge effect

depends on the pre-buckling stresses it is called the non-linear edge effect (see

Section 14.1).

8) The partial case of the general stress-strain state is called

the mom ent

stress-strain state if it is a combination of the membrane (or semi-momentless)

stre ss-s train st at e and the edge effect. In pa rtic ula r, es tim at e (3) is valid for

moment stress-strain states.

Next we write the simplified equations for the characteristic stress-strain

states of a shell.

1.4 M em b r a n e S t r e s s S t a t e

This state is characterized by slow variation of unknown functions (t = 0).

Displacements u,- and

w

are small variables of equa l ord ers. In the gene ral

case, the strains e,-,

u,

7,-, and

Rxi

are also sm all variab les of equ al orders .

The bending stresses are less than the membrane stresses

{MiR-^HR-^Qi} ~ h

2

,{Ti,S}.

(1.4.1)

The membrane stresses satisfy the non-linear system of equations

da oa A

0/3

HA T,) 8A d(B*S)

-bT~ ~W

1+

~^da~

+ ABq

- - °'

( L 4

.

2 )

(*1 + xi) Ti + (*

2

+ *

2

)

T

2 +

1ST + q*

= 0 ,

Ti = K {en + VE21), T-2 = K (e

2

2 + ve

n

), S = —^ Ke

12

-

If the curvatures k{ are not equal or close to zero in the entire domain

being considered, then in the third equation of system (2) the non-linear terms

XjTi, rS

m ay be neglected. Furth er, if the surface deforma tion in the en tire

24


do m ain is far from the bend ing e* = w = 0, then in cons titutive relation s (2)

the non-linear membrane strains may be l inearized

dJBT,) 8B d(A*S) _

~ d -

a

— - ^

T 2 +

^dF

+ ABqi

- °-

d(AT

2

) dA d{B*S) _

-dp--w

1 +

~B^T

+ q2

"

0>

kiTi +

k

2

T

2

+ q = 0,

71 = ff(ei+i/e

2

), T

2

= K(e

2

+ ve

1

), S

l - v

•Ku,

(1.4.3)

i.e. we get linear membrane equations [52, 119].

Let us write the explicit formulae for the membrane stresses for some par

ticular load cases.

Figure 1.4: Shell of Revolution.

Consider a shell of revolution where as the curvilinear coordinates a,/? we

introduce the meridian arc length s and the circular angle <p (see Figure 1.4).

Then

„ , „ „ • „

d9

1

dB

A=l, B =

R

2

smV, — = —-—, -3— = —cos

I

as R\ as

(1.4.4)

where

6

is the angle between the axis of rotation and the normal on the neutral

surface. Let the shell be limited by two para llels (si ^

s

^

s

2

)

and be under

st at e is a m em bra ne stress sta te with the stress-re sultan ts (see [52], C ha pte r 13)

1

Ti =

S =

B

/

(

d

A_

T2

^_

\d<p ds

B

2

fi+ I

I

^ - T

2

^ + B

gi

)

ds

-R

2

q,

B

2

= B (s

2

)

52

1

5 2

B

2

f

2

+

I [ B ^ +

B

2

q

2

) ds

(1.4.7)

(1.4.8)

Assuming

s = s\

in (7), we find stresses TJ and

S

at the other shell edge.

1.5 Tec hn ica l She ll T h eo ry E q u a t i o n s

Here we will examine the mixed stress-strain state for which the membrane

and bending stresses are of the same orders. We need to make some assump

tions to characterize this sta te . Let the index of varia tion

t

be

t

= 1/2 and

let

w

~

h*R, ki

~

R

1

, U j < t o .

(1.5.1)

Then from the equilibr ium equations, the consti tutive relations and the geo

metrical relations we find the orders of all unknown functions

max{u ,} ~

hj w;

max { £ j , u , , £ j j } ~ R

_1

w ~ h

t

;

max{7,} ~ / i* R~

x

w ~ hj ;

m a x { x ; , r } ~ h~

1

R~

2

w ~ R~

lm

,

max{Ti , 5} ~ KR~

l

w ~ Kh*;

m a x j Q i } ~ hl^KR-

1

™ ~ Kh

Z

J

2

;

max{Mi,H}

~

KKw

~

KRh

2

,.

(1.5.2)

Relations (2) give the orders of the maximum values of variables from one

group.

We can use the orders of the relations above to simplify equations (2.6),



1.5. Technical Shell Theory Equations

27

(2.4),

(1.13), (1.10), (1.7). We then obtain

da da Aop

d(AT

2

) dA d(B

2

S)

_

~W

"

lp

Tx +

~Bda-

+ ABq

> '

° '

1 (d{BQ

x

) d(AQ

2

)\

AB {-Jo-

+

—dT)

{l X) X

+2TS

+ (k

2

+

x

2

)T

2

+

q*

= 0,

^ - >

^ + ^ = ••

where

T

x

= K{e

u

+

ve

22

),

T

2

= K{e

22

+ ve

n

),

Mi = D ( x i + i /x

2

) , M

2

= D(x

2

+ uxi),

y

= LjLKen, H = {\-V)DT,

(1.5.3)

1

2

1 3« i 1

dA w

en =

e i + 2 7 i , *i = J " ^ " + ^ 0 0 "

2

~ T V

1

2

1 du

2

1 3 5 u>

£22 = £ 2 + ^ 7 2 , £2 = B-dJ

+

AB^

Ul

-R -

2

'

£12 = w + 7 i7 2 ,

_

B d /u

2

\ A d

/ « r

W

~ A~d^\~B)

Jr

B~dJi\A,

I dw I dw

X l

~

fa \A~fa)

+

IB

2

"dp 1)0'

1 ( \ dw\ 1 dB dw

3/?

\B dp) BA

2

da da'

1 3 / 1 3 u A 1 dB dw

Bdp\Ada) BA

2

da dp

1 d (}_dw\ 1 dAdw

Ada \BdpJ AB

2

dp da'

http://bdpj/

http://bdpj/

http://bdpj/




System (3) is known as the system of equations of the technical shell theory

[32, 61, 109, 171, 175]. All of the simplifications made in defining this system

introduce an error of order

h*

due to (1) and (2) . Th us, this system m ay

be considered to be exact within the limits of the Kirchhoff-Love hypotheses

under the assumptions described above (see (1) and t = 1/2).

W e note some pecu liarities of this syste m . Th ere are no shear stress-

res ulta nts in the first two equ atio ns. In the third e qua tion all of the term s

are equa lly significant. In the e xpressions for £;J , both terms are of the same

orde rs an d th ere are no tang en tia l disp lace m en ts in the expression s for 7,-, x,-

and r. System (3) was used as the sta rtin g poin t for m an y researchers in this

field. We will also consider this system further in the book.

Without surface tangential loading (gj = g

2

= 0) system (3) can be written

in the compact form ([61])

-DAAw +

(* _+ xi) Ti +

2TS + (k

2

+

x

2

) T

2

+

q*

= 0

A A $ + A r

2

x

2

+ Ar

2

xi + x

x

x

2

- r

2

= 0

A A

„ , , . . , , _ 2 _

n

(1-5-4)

Eh

where

. 1

Aw —

——

AB

d (Bdw\ d (Adw

fa \~A~do~)

+

~dp

V ^ 5 / ?

(1.5.5)

is the L aplace operato r in the curvil inear coordinates and the m em bra ne stress-

resultants may be expressed through the stress function $ as

1 0 / 1

d$\

1

dBd$

Bd/3 \Bdl3J A

2

B da da'

r,

=

IifI

+

- L M ^ ( 1 . 5 . 6 )

2

Ada \A da J

B*

A dp dp*

1

(

5

2

$ 1 3 , 4 5 $

\dBd§\

~ ~AB \da~dp ~ A Jp ~da~ ~ ~B ~da~~dd~) '

For these

T{

,

S

the first two equilibrium equations are valid with the relative

errors of order h

t

, bu t for AriA;2 = 0 sy stem s (3) an d (4) are eq uiva lent .

R e m a r k 1 .2 . Here, equat ions (4) are obta ined assuming tha t

D

= const,

Eh =

const. For shells with variable thickness and elastic material properties

with an error of order

h*,

the first terms in (4) must be replaced by

-A(DAw), A Q ^ A * ) . (1.5.7)

It often occurs that the shell stress-strain state is a sum of the main stress-

str ai n sta te a nd an edge effect. W e consider an edge effect which has th e

http://bdl3j/

http://bdl3j/

http://bdl3j/

1.6. Technical Theory Equ ations in the Other Cases

29

large index of variation

(t

= 1/2) in the neighborhood of line

a

= ao in the

direction orthogonal to the edge and small index of variation

(t '

= 0) in the

edge direction. Such an edge effect spreads in a band, the width of which is of

1/2

order RhJ at the line a = ao . T o describe th e edge effect we m ay use syste m

(3).

1/2

With an error of order hj we can also use th e following eq ua tio n [61]

„

d

A

w

m

8

2

w Eh

/ 1 i/ \ ,

n

„

r n

.

where

dx

= Ada, ^L +

q

*

1=

0, T

2

=

v T

x

- ^ . (1.5.9)

ox tii

d

2

W

The term T\ „ is non-linearly depe nden t on the loading and th at is why

eq ua tio n (8) usua lly is called th e non-line ar edge effect eq ua tio n.

If

dB

5 = 0 , ^ - = 0 a t

a = a

0

,

(1.5.10)

da

the accuracy of equation (8) increases, and the error is of order h*.

We note that if the load level satisfies estimate (3.9), the first four terms

in (8) are of the same orders. For smaller loads, we arrive at the simple edge

effect equation

-"£- -=••

<"•»>

For shells of revolution equation (8) is discussed in Section 14.1.

1.6 Tec hn ica l T h eo ry E q ua t ion s in t h e O th e r

Cases

We obtained equations (5.3) using three important assumptions

< = - , i « ~ i ? -

1

/ i

H

. = h

0

, ki~R-\ (1.6.1)

but the equations have wider applicability. Let us discuss the cases when some

of assumptions (1) are not fulfilled.

30

Cha pter 1. Equa tions of Thin Elastic Shell Theory

T h e a s s u m p t i o n t h a t

w

~

h

0

a f f ec t s on ly the r e l a t ive va lues o f the non

l ine ar t e rm s in th e th i rd eq ua t io n (5 .3 ) an d in th e exp res s ion s for e , j . For

w <C ho s y s t e m ( 5 . 3 ) b e c o m e s l i n e a r a f t e r n e g l e c t i n g t h e n o n - l i n e a r t e r m s .

Let 0 < t < 1 /2 , i . e . t he s t r e s s - s t r a in s t a t e ha s sm al l e r ind ex o f va r i a t ion

th a n th e s imp le edge ef fec t. In th i s case , sy s t e m s (5 .3 ) an d (5 .4 ) ca n a l so be

used bu t the i r acc urac y decrea ses and th e e r ro r A i i s o f o rd er

A i ~ / i f . (1 .6 .2)

T h i s e r r o r i s m a i n l y c a u s e d b y n e g l e c t i n g t h e t a n g e n t i a l d i s p l a c e m e n t s i n e x

pre ss io ns (1 .7) for 7,- an d by the fact th a t fun ct io ns (5 .6) do no t ac cu ra te l y

s a t i s f y t h e f i r s t e q u i l i b r i u m e q u a t i o n s .

Ca s e s w h e n t h e n e u t r a l s u rf a c e d e f o r m a t i o n i s c l os e t o b e n d i n g , i. e . t h e

va lues o f £ , - ,u> ,£ i j a r e smal l due to the r educ t ion o f the main t e rms in the

expres s ions fo r these va lues , r equ i r e spec ia l cons idera t ion . In th i s case

£,-, w, Sij <^R~

1

w. (1 .6 .3)

and th i s case wi l l be d i s cus sed in Sec t ion 12 .1 .

1.7 Sh allow Sh ells

L e t u s s u p p o s e t h a t \Rk{\ <C 1 (here R i s the charac te r i s t i c s i ze o f a she l l ) .

In th i s case the e r ro r in equa t ions (5 .3 ) and (5 .4 ) decreases and becomes equa l

t o

A i ~ max{Rki} • h

2

J . (1.7 .1)

A s s u m i n g t h a t

A, B, k\, ki

a r e a p p r o x i m a t e l y c o n s t a n t , o n e c a n s i m p l if y

t h e f o r m o f t h e s e e q u a t i o n s . F o r A = B = 1 eq ua t io ns (5 .4 ) tu rn to [171 , 173 ,

175].

<9

2

$ / , 8

2

w\

-DAA

w+w

(k

+

^

+

d

2

f, d

2

w\

o

s

2

$

d

2

w

t n ni9

s

+

^{

k

^w)-

2

do-op o^

+ q

=

° '

(L 7

'

2)

1

A

A

^ ,

9

2

w

,

a

2

w

d

2

w d

2

w

(

o

2

w

\

2

F o r k\ = &2 = 0 e q u a t i o n s ( 2) b e c o m e t h e p l a t e n o n - l i n e a r b e n d i n g e q u a t i o n s

[29,

46, 72] , the er rors of which are of order

e

( se e ( 3 . 7 ) ) . T h e m a t h e m a t i c a l

a s p e c t s o f e x i s t e n c e a n d u n i q u e n e s s of s o l u t i o n s o f b o u n d a r y v a l u e p r o b l e m s

in th e non l ine ar the ory o f p la t es and sh a l low she l l s a r e dusc usse d in [107 , 108 ,

178].

1.8.

Initial Imperfections 31

.

1 9

£ u = £ i + 7 i 7 i + - 7 i ,

. 1 9

£22

= £2 +

7275

+ 2 72.

£12 = w + 7 j 7

2

+ 7 * 7 i + 7 i 7 2 -

7?

=

72* =

i

dw*

A da

1 dw*

B

dp

1.8 In i t i a l Im pe r f ec t io ns

Let

the

unperturbed neutral surface

of the

shell

be

defined

by the

p a r ame

ters

A, B, R\

and

R

2

.

Function w'(a,j3) defines the init ial normal deviations

of

the

real surface from

the

unper tu rbed

one. The

projections

of

displacement

which was caused by the deformation of the shell are denoted by u\, u

2

, and

w. Th en

u\, u

2

, w' + w

are p rojections of the total displacement. We assume

t h a t

w* < hg.

Equat ions

(5.3) and (5.4)

mus t

be

modified

to

account

for the

init ial

im

perfections. The following relations are valid

(1.8.1)

In

the

third equilibr ium equation

(5.3) and in

equations

(5.4) x, r are

replaced

by

X{

+ x*,

T

+ T', where x* =

Xi(w'),

r* = r(w'). Then equat ions (5.4) are

represented

as

- D A A w + (h

+ x

x

+ x{)

T

1

+ 2{T + T')S+

+ {k

2

+ x

2

+ xl)T

2

+

q*

=0,

1 (1.8.2)

— - A A $ +

fci><2

+

k

2

x\

+

x\x

2

-

T

2

+

Eh

x\x* + x

2

x\ — 2TT* =

0.

The above relations

are

valid with relative errors

of

order

/i*, if the

index

of

variation w* is not larger than 1/2. We will not consider shells with initial

imperfections beyond this point.

Buckling of shells with imperfections of the neutral surface and post-buck

ling behaviour of the imperfect shells are considered in [9, 10, 11, 19, 48, 61,

64, 76, 148, 149]

1.9 C yl in dr ic a l Shell s

For a cylindrical shell we assume that

A

= B = R,

Jfci

= 0,

k

2

(/3)=R~

1

k(l3), (1.9.1)

where, for a non-circular cylindrical shell,

R

is the character ist ic radius of

curvature . For a circular shell k(f3) = 1.



32

Chapter 1. Equations

of

Thin Elastic Shell Theory

Many buckling problems involving cylindrical shells can be solved on the ba

sis of systems (5.3) or (5.4) after simplification by means of formulae (1). Some

problems however, especially for long cylindrical shells, cannot be described by

these equations. An example is t h a t of semi-mom entless stress-strain states,

for which

d

2

w

d

2

w

and s ta tes for which the contour of th e shell-cross section is not deformed

(or only slightly deformed). For such stress-stra in sta tes , the shear stress-

resu l tan ts

in

the first e quilibrium equ ations and the ta ng en tial displace m ents

in the expressions for the bending deformations must not be omitted.

We can write

a

system of equations

for a

non -circula r cy lindrica l shell

9Ti

dS

n

.

dT

2

0S_

dp

+

da

f+^

z

-R{k

2

+ x

2

)Q2

+ Rq*

2

= 0,

^ -

+

^ +

i J [ x

1

r

1

+ 2 r 5 + ( *

2

+

x

2

) T

2

+

«*]

= 0,

8H 8M

2

- « - +

-57T

+

AQ2

= 0,

da

dp

dH

dp

£ i

da

1 du

x

R~~da~'

1 (du

x

du

2

R~\ )p~

+

~da~

1 d

2

w

(1.9.3)

1

(

du

* u

£

2

= —

~K7, kw

\dp

1 d fdw

"

l

~ R

2

da

2

' *

2

~ R?dp\dp

+kU2

1

d fdw

,

The expressions for e,j and the constitutive relations we take in the form

of (5.3).

There exists

a

number

of

different vers ions

in

nota t ion

for the

resolving

equations of a circular cylindrical shell [27, 33, 47, 61, 135, 141, 173, 175] and



1.10. The Potential Energy of Shell Deformation 33

othe rs. Below we will present the eq uation s which are valid descriptions of

both the fast varying and the semi-momentless states mentioned above

(1.9.4)

-DAAw+-^~+T

1

x

1

+2Sr + T

2

(± + x

2

)+q* =

0,

— A A $ + - X

1

+ K

1

X

2

-T

2

= 0,

where the stress function $ is introduced by formulae

l

~ B?dp' B?dadf3'

2

~R

2

da*

+

R-

l

'

Here and in (5.4) we assume that q\ = q\ — 0.

With an error of order e (see (3.7)) we can surmise that in (3) and (4)

"»=£(£

+

-)-

• • • • »

System (4) is valid for almost all of the stress-strain states considered below.

The various stress-strain states are characterized by different main terms of the

system. Buckling problems of long cylindrical shells under axial compression

are not, however, described by system (4) (see Sections 2.3 and 2.4).

1.10 T h e P o te n t i a l E ne rg y of She ll D eform a

t ion

We can write an expression for the potential energy of shell deformation as

n = n, + n* =

- / /

K (e

2

u

+

2z/e

11

e

22

+

e\

2

+ — ^ 1 2 )

dn

+

n

\ If D (x

2

+ 2vx

x

x

2

+ K\

+ 2 (1 -

v) r

2

) dQ,

n

dQ. = AB dad/3, (1.10.1)

where the potential energy of shell extension and bending are denoted by II

£

and

Yl

H

respectively. As in Section 1.2 we neglect the difference in metrics of

the neutral surface before and after deformation.

34


The first variation of the potential energy is equal to

fll

=

ff(T

1

Sen +

T

2

Se

2

2

+ SSe

1

2 +

a

+M

1

8x

1

+ 2H8T + M

2

8x

2

\dQ,

(1.10.2)

where

S =

S

1

--=S

2

-§-, (1-10.3)

tt

2

tl\

and the other stress-resultants and stress-couples are given by formulae (2.4).

We can also write the elemental work

8 A

of the external surface load

q

and

the edge stress-resultant vectors F/, and the edge stress-couples vectors Gk

as

8A= [J q8U<m + I(F

k

SU + G

k

6u\ds, (1.10.4)

where 8U and 8u> are vectors of the elemental displacement and rotation.

The equilibr ium equations and static boundary conditions can be obtained

from the relation

8U = 8A

(1.10.5)

for any virtual displacement 8 U of the neutral surface that satisfied the geo

metr ic boundary conditions.

1.11 P ro b le m s an d Exe rc i ses

1.1. Find the index of variation for the function

F(x, h)

=

g(x)

sinh

(z)

for

z ~ h~

t

f(x) as h —> 0 assuming f'(x)

^

0 and t> 0.

Answer The index of variation is equal to t.

1.2. Find th e general index of varia tion, and the direction in which the

partial index of variation is minimal as

h

—>• 0 for the following functions:

a)

F(x, y, h) = g(x)

sin

(z(x — ay)),

where

z — h~

t

,

b) F(x, y, h) = exp (z~ J, where z = h~

f

,

assuming t > 0.

Answer For a) and b) the index of variation is equal to —t.

For a) in the direction x = ay the partial index of variation is equal to 0.



1.11.

Problems and Exercises

35

For b) in the direction x = 0 the partial index of variation is equal to 0.

1.3. Derive relations (1.13) and (1.14) using formulae (1.4-1.10)

1.4. Derive relation s (4.6) from t he equ ilibrium equatio ns for the pa rt of a

shell of revolution between the parallels

s

and s = s

2

•

Prove th at the o btained

stress-resultants satisfy (4.3).



C h a p t e r 2

B a s i c E q u a t i o n s o f S h e l l

B u c k l i n g

The various research methods of the buckling of shells under static loading are

briefly discussed in this Chapter (see also [12, 15, 21, 33, 45, 48, 61, 62, 95,

123,

166, 149, 170, 175, 178] although only one of these will be employed in the

following chap ters. T he app roach presented in this book will pe rm it the ex

amination of the stability of equilibrium states under conservative surface and

edge loading by determination of the critical loads using linearized equilibrium

equations which will be presented below.

2.1 T y p e s of E las t ic Shel l Bu ckl ing

The equil ibrium state or motion of a mechanical system is termed

stable

if

the deviat ion from this s tate at the moment

t

is as small as desired for any

sufficiently small perturbations at the initial moment of time

t —

0.

The general method of research in s tabil i ty is to s tudy perturbed motion

in the ne ighbourhood of unper turbed mot ion. The method ( the dynamic cr i

terion of stability) for conservative mechanical systems was first proposed by

Lagrange. A.M. Liapunov later developed an accurate mathematical theory of

motion stability [86].

The dynamic criterion can be used for any problem of shell stability but

the study of perturbed shell motion is a much more difficult problem than the

study of its equilibrium states. That is why, unless it is necessary, the dynamic

criterion is rarely used for in the stability analysis of shell equilibrium states.

Later on in the book only the problems with the conservative loads are studied,

for which the bifurcation (Euler) or the lim it poin t citeria m ay be used. For

37



38 Cha pter 2. Basic Equa tions of Shell Buck ling

t h e s e c l a s s o f p r o b l e m s th e s e c r i t e r i a a r e e q u i v a l e n t t o t h e d y n a m i c c r i t e r i o n .

I t i s i m p o r t a n t t o n o t e , h o w e v e r , t h a t t h e u s e o f t h e d y n a m i c c r i te r i o n i s

t h e o n l y p o s s i b i l i t y in a n u m b e r of c a s e s . E x a m p l e s w o u l d b e p r o b l e m s o f s h e ll

m o t i o n s t a b i l i t y u n d e r d y n a m i c [ 8 5 , 1 4 6 , 1 4 7 , 1 7 6 ] a n d n o n - c o n s e r v a t i v e l o a d s ,

such as the mot ion o f a she l l i n a gas f lux [124 , 176 , 177] , and paramet r i c she l l

ins t ab i l i ty [17 , 48 , 126] . In th i s wo rk the se p ro b le m s a re no t cons ide red an d

t h e d y n a m i c c r i t e r io n o f s t a b i l i t y is n o t u s e d .

W e wi l l cons ide r a she l l l oad ed w i th s t a t i c con serv a t ive su r f ace an d edge

loa ds . Lo ad s a re r e fe r red to as con serv a t ive , if t h e work do ne by th em de

p e n d o n l y o n t h e e n d s t a t e s a n d d o n o t d e p e n d o n t h e w a y o f d e f o r m a t i o n . I n

p a r t i c u l a r , l o a d s w h i c h d o n o t c h a n g e t h e i r v a l u e s a n d d i r e c t i o n s , a r e c o n s e r

va t ive . However , t hese loads do no t compr i se the en t i r e c l as s o f conserva t ive

l o a d i n g . H y d r o s t a t i c p r e s s u r e l o a d s , t h e d i r e c t i o n of w h i c h d e p e n d o n t h e

s t r e s s - s t r a i n s t a t e , a r e a l so c o n s e r v a t i v e . N o t e t h a t o n l y t h e d y n a m i c c r i t e r i o n

g ives ac cu ra te r es u l t s fo r cases o f s t a t i c no n-c on serv a t iv e loa ds [17 , 126 , 182] .

In c l as s i ca l p r ob lem s o f l inea r e l as t i c i ty w here in fin i t e ly sm al l de for m at io ns

a r e a s s u m e d , t h e e q u i l i b r i u m c o n d i t i o n s a r e a s s u m e d t o b e s a t i s f i e d b y t h e

f o rc e s a c t i n g o n t h e u n d e f o r m e d e l a s t ic s y s t e m . T h i s a s s u m p t i o n w h i c h is e s

sen t i a l for Ki rchhof f ' s gen era l un iq uen ess t he or em [75] l ead s to un iq ue so lu t ion s

f or s u c h l i n e a r p r o b l e m s . O n t h e o t h e r h a n d , i n f o r m u l a t i n g b u c k l i n g p r o b

l e m s t h i s a s s u m p t i o n is d r o p p e d a n d t h e e q u i l i b r i u m c o n d i t i o n s a r e s a t is fi e d

by the fo rces ac t ing on the deformed e las t i c sys tem. Th i s l eads to an es sen

t i a l l y n o n l i n e a r f o r m u l a t i o n o f s u c h p r o b l e m s i n t h e se n s e t h a t d i s p l a c e m e n t s

a r e n o t l i n e a r l y p r o p o r t i o n a l t o t h e e x t e r n a l l y a p p l i e d l o a d s , a n d , i n f a c t , o f te n

t h e d e f o r m a t i o n o f a s t r u c t u r e w i l l n o t b e u n i q u e l y d e t e r m i n e d b y t h e a p p l i e d

l o a d i n g .

A cco rd in g to Ki rch hof f ' s t h eo re m th ere i s on ly one se t of so lu t ion s o f

s t r e s s e s , s t r a i n s , a n d d i s p l a c e m e n t s f o r a n e l a s t i c b o d y i n e q u i l i b r i u m , s a t

i s fy ing a l l bas i c equa t ions o f l inear e l as t i c i ty fo r a g iven body load and bound

ary con d i t ion s . In f ac t , any two se t s o f so lu t ion fo r th e s a m e bo dy load an d

b o u n d a r y c o n d i t i o n s , a t m o s t m a y d i f f e r o n l y b y t h e r i g i d b o d y d i s p l a c e m e n t

of the sys tem, i . e . , the di f ference in any two sets of solut ion descr ibes the r ig id

b o d y m o t i o n o f t h e s y s t e m . T h e r e f o r e , t h e s o l u t i o n of t h e s u c h l i n e a r l y f o r m u

la t ed p r ob lem is a lw ays s t a b le . Th e su ff ic i en t c on d i t i on r eq u i r e d for s a t i s fy ing

K i r c h h o f f ' s t h e o r e m i s t h a t t h e p o t e n t i a l e n e r g y o f t h e e l a s t i c s y s t e m s h o u l d

be a po s i t iv e def in i te fun ct io n. T h is co nd i t i on i s fu lf il led for shel l s ( see Sec t ion

1.10).

T h e p h y s i c a l l y n o n - l i n e a r f o r m u l a t i o n o f t h e p r o b l e m ( i . e ., t h e n o n - l i n e a r

d e p e n d e n c i e s o f t h e st r e s s e s o n t h e s t r a i n s ) is n o t c o n s i d e r e d b e l o w . O n l y

b u c k l i n g , w h i c h i s c a u s e d b y t h e g e o m e t r i c n o n - l i n e a r i t y o f t h e p r o b l e m , i s

s t u d i e d . T h e g e o m e t r i c n o n - l i n e a r i t y is t h e n o n - l i n e a r d e p e n d e n c e of t h e s t r a i n s

on the d i sp lacement s and on the non- l inear t e rms , due to the d i f f e rence o f the



2.1. Types of Elastic Shell Buckling 39

coordinate systems before and after deformation. On the physically non-linear

pr ob lem of shell buc klin g refer to [61, 79, 119, 167, 168].

As with any elastic system it is possible to consider two types of shell

buckling from the equilibrium state [1]. The first type is connected with the

bifurcation (or bran ching ) of the equilibrium states , and th e second is accom

panied with the appearance of the l imit point .

C

O w O w O w

(a) (b) (c)

Figure 2.1: Relationship between load and deflection for an ideal shell.

The relationships between the load (P) and the characteristic deflection

(w)

are plotted schematically in Figure 2.1. Considering Figures 2.1 (a) and (b) let

the load increase smoothly from zero, remaining less then P*. The equilibrium

state is then stable, the deflection is uniquely determined (the section

OB

in Figure 2.1). For P ^. P* one or more adjacent equil ibrium states BE, in

addition to the basic one

BD,

appea r in the neighbourhood of point

B.

Point

B is called the bifurcation poin t of the equilibrium state s. Th e e quilibrium

states in the neighbourhood of point

B

can be both s table and unstable. In

Figure 2.1 the unstable equil ibrium states are plotted by the dotted l ines . The

basic equil ibrium state

CD

is , as a rule , unsta ble. Th e s table equilibrium s tate

BE after bifurcation is plo tted in Figu re 2.1 (a). T he case when there are no

stable equilibrium states close to point

B,

is shown in Figure 2.1 (b).

Point B in Figure 2.1 (b) is the buckling point. For P ~ P

t

the adjacent

equil ibrium states are unsta ble.

Rigorously speaking, poin t

B

in Figu re 2.1 (a) can no t be called the b uckling

point. But for P « P» a modification of the deflection shape occurs, and the

deformation abruptly increases with the small increase in the load. It is clear

that the evaluation of P* is important in engineering analysis. Taking this into

account it is reasonable to call P* the critical load. The same term is used in

the cases of Figure 2.1 (b) and 2.1 (c).

Figure 2.1 (c) shows the case when the deflections grow abruptly as the



40

Chapter 2. Basic Equa tions of Shell Buckling

loading increases smoothly to P„, i .e.

jp^°o

a s P - > P » . (2 .1 .1 )

Buckling occurs at point

C.

Th e arc of curve

CD

consists of the unstable

equilibrium states. There are no close equilibrium states for

P

> P» and there

is no restructuring of the buckling mode, only the growth of its amplitude as

P approaches P*. Point C is called the limit point.

We can now show some characteristic examples. The buckling of a circu

lar cylindrical shell under homogeneous external pressure follows the scheme

shown in Figure 2.1 (a); the buckling of a circular cylindrical shell under ax

ial compression corresponds to Figure 2.1 (b) and the buckling of the convex

shallow shell under normal loading is illustrated in Figure 2.1 (c).

The deformation of a thin shell under large displacements is much more

complicated than is shown in Figure 2.1. Arc

BD

includes other bifurcation

points besides point B. Arcs BE and CD may contain points of the secondary

bifurcation and so on. A m ong these equil ibrium states there are bo th s table

and unstable s tates and the translat ion from one s table equil ibrium state to

another can be accomplished continuously (see point

B

in Figure 2.1 (a)) or

by an abrupt jump (points

B

and

C

in Figures 2.1 (b) and (c)).

At the present time an understanding of the deformation of thin shells un

der large deflections is not fully developed even for shells of simple geometry.

The problem of shallow shell deformation and the axisymmetric deformation

of shells of revolution has been studied in some details [29, 38, 107, 108, 138,

145, 149, 156, 170, 178]. The bifurcation points of the axisymmetric equilib

riu m stat e of shells of revolution into non -axisy m m etric sta te have also been

obtained mainly by numerical methods [22, 149, 156, 170].

The non-axisymmetric deformation of shells of revolution is considered in

[95].

We note also works [132, 133], in which th e geo me tric research m eth od s

for studying buckling are used.

It is not normally necessary for the analysis of real structures to obtain a

detailed picture of post-buckling shell behaviour because in this case the shell

ha s failed a nd is no longer of inte rest . Ex cep tions t o this are shells which work

in the large displacement regime.

For engineering analysis it is necessary to determine the critical load P,

accurately. But in many cases the comparison of theoretical and experimental

values of P, reveals their essential divergence. For example, in the problem of

a cylindrical shell under compression by an axial force, the theoretical value of

P , exceeds the ex perim enta l one by a factor of two or thre e an d thi s divergence

has been systematically observed in many experiments [33, 59, 61].

This flaw in the theory of shell stability was resolved by taking into account

sm all impe rfections, mainly of the neutr al surface. It becomes clear tha t the



2.1. Types of Elastic Shell Buck ling

41

critical load is very sensitive to these imperfections [9, 10, 11, 19, 64, 148].

Figure 2.2: The relationship between the load and deflection for an ideal shell

and for a shell with imperfections.

The comparison of the "load-deflection" curves for an ideal shell and a

shell w ith imperfections is shown in Figure 2.2. Curve OBNE corresponds

to the ideal shell and is similar to the curve plotted in Figure 2.1 (b), the

curve

OCQG

correspond s to a shell with imperfections. Arc

OC

of this curve

corresponds to the pre-buckling stress-strain state. Arcs EN and QG show

the stable post-buckling equilibrium states under large deflections.

We will introd uce t he following no tati on to assist in the an alysis. T he

ordinates P " and P " of poin ts

B

and

C

are called the upper critical loads

for the ideal shell and th e shell with imp erfections. T he sm allest load under

which the post-buckling stable equilibrium state is possible, is called the lower

critical load. In Figure 2.2, points N and Q correspond to the lower critical

loads Pj and Pj;.

Calc ulation of P " a nd P " is com paratively simple problem . Explicit an a

lytical expressions for P " have been obtain ed for ma ny prob lem s of ideal shell

buckling. In a num be r of cases it is possible to evalua te P " by mea ns of one

of the numerical methods [62, 95, 115, 170]. The evaluation of P^ and Pj is,

however, a complicated non-linear problem, which is not completely solved at

the present t ime.

A very important question is which load should be taken as the designed

load in an actua l design. Th e obvious answer might be to take P " , but i ts

evaluation is complicated because, as a rule, the actual imperfections present

in the shell are unknow n. At the same time it occurs th at the app roxim ate

values of PJ; obtained by approximation of the deflection by a series with a few

term s (one or two) retain ed, agrees very well with exper im enta l results. T he

following studies reveal, however, that

Pi

essentially depends on the accuracy



42

Chap ter 2. Basic Equations of Shell Buckling

of the deflection approximation and sometimes for a large number of terms it

is equal to 1% of P " or it m ay be neg ative. T h at is why at the present time

the upper critical load of a shell with imperfections

P"

is taken as the correct

value in a ctu al designs [1, 48, 77].

2 .2 T h e Buck l ing E qu a t io ns

Buckling equations are obtained by considering variations of the nonlinear

equ ation s, introd uced in Ch ap ter 1 (for exam ple, equation s (1.2.6) or (1.5.3)

or (1.5.4) or (1.9.4)).

To this end, each unkn ow n function u,-, w, Ti, Mi,. .. , in thes e eq ua tion s is

replaced by u? +

u,-, w° + w, T?+Ti, M° + Mi,....

Here, functions «?;

w°,...

describe the initial stress state. The stability of this initial state which satisfies

the no nlinear system of equ ation s is to be investig ated. Fu nction s u,-, w, . . .

describe the adjacent infinitesimally close equilibrium state. They satisfy the

linear homogeneous equations (the buckling equations) and the homogeneous

boundary conditions, that are obtained as a result of linearization by w,, w, . . .

of the initial non-linear equations.

The existence condition for the non-trivial solution of the buckling equation

is used for the evalu ation of the critical load. In dealing with buckling prob lem s

it is convenient to assume that the load varies proportionally to a loading

para m eter A > 0. The n the pre-buckling s tate functions (u°, w°, Tf, M,-, . . .)

and the coefficients of the buckling equations depend on A. In this way, the

buckling problem is reduced to an eigenvalue pro blem . Th e least (positive)

eigenvalue is tak en as the first critica l value A = A* leading to the correspon ding

buckling mo de. Such an appro ach is called

equilibrium or Euler analysis of

stability

due to L. Euler who in 1744 used this appro ach to study the stability

of axially compressed bars. His paper [36] is considered to be the first work on

structural s tabil i ty.

In the general case the buckling equations are rather complicated (see [60,

61]) an d are no t writ ten h ere. La ter in this boo k simplified forms of the

buckling equations, applicable under specific cases depending on the character

of the initial stress state and buckling mode are presented.

2 .3 T h e B u c k l in g E q u a t i o n s for a M e m b ra n e

S ta t e

The membrane stress state of a shell is described by equations (1.4.3). Let

the displacement

u°, w°

and the s tress-resultants

T°

,

S°

characterising this

state, be weakly varying functions of

a

an d /? (i.e. the inde x of variatio n

44

Chapter 2. Basic Equations of Shell Buckling

where A is the Laplace operator (1.5.5)

1

AkW

=

ki><2

+ fc

2

xi =

AB

d fk

2

Bdw\ d fkiAdw

da

V

A da J dp \ B dp

(2.3.5)

A

C

T

= T f x + 2S°T +

T

2

°xr

2

.

Substituting the expression for xt, and r from (1.5.3) into (5) and using

the membrane equations for 7}°, and

S°

we get

where

A

TW

= —

A?w = ATW + TW,

8 (BT?dw\ d (A7$dw *

(2.3.6)

da

V

A da J dp \ B dp

d ( dw\ d

+

da-{^W)

+

dp

(*£)]•

(2.3.7)

- 7i9i - 7292

Substituting (6) into the first equation of (4), we obtain the equilibrium

equation in the projection on the normal to the neutral surface before defor

mat ion .

The operators

DAAw, AkW,

A j w in (4) an d (7) are form ally self-adjoint.

This means that integrating by parts several t imes the integral

ff vAkU

dQ. in

the domain fi occupied by a shell, we arrive at the integral

ff uAkV

dQ. and

an integral by the boundary contour 7.

The term Tw in (6) m akes system (4) not self-adjoint. At the sam e tim e

the follower surface load with a projection on the plane which is tangent to

the neu tral surface, is non-con servative. It should be note d th at th e non-

conservative term Tw is relatively sm all. Indeed, let t > 0 be the index of

variation of the buckling mode. This term then has order h^ compared to the

main terms in (4) .

Since the method used here is, strictly speaking, not valid for non-conser

vative loads we suppose further that Tw = 0, i.e. con side ring the follower

load we restrict ourselves to a nor m al follower load (no rm al press ure). Th en

system (4) may be rewritten in the form

-DAAw + A

T

w + A

k

$ -

0,

— - A A $ + A

k

w = 0.

hifi

(2.3.8)

In the case of a "dead" surface load (i.e. one which does not follow the

direction of the shell deformation) the terms Xi in equations (1) differ from



46

Chapter 2. Basic Equa tions of Shell Buckling

F o r a n o n - c i r c u l a r c y l i n d r i c a l s h e l l , i n n o t a t i o n ( 1 . 9 . 1 ) , t h e b u c k l i n g e q u a

t i o n s m a y b e w r i t t e n a s

w h e r e x , - , r m a y b e e v a l u a t e d b y f o r m u l a e ( 1 . 9 . 3 ) a n d ( 1 . 9 . 6 ) , a n d t h e s t r e s s -

r e s u l t a n t s

Ti,S

a r e c o n n e c t e d w i t h $ b y f o r m u l a e ( 1 . 9 . 5 ) .

In the p rob lem of ax ia l compres s ion o f a long cy l indr i ca l she l l , sys t em

(12) g ives an inco r rec t r esu l t . To cor rec t i t one m us t t ake in to ac co un t th e

d i ff e rence in m et r i cs o f th e ne u t r a l su r f ace before an d a ft e r de fo rm at i on , w hich

w a s n e g l e c t e d a b o v e . T h e v a l u e o f

T

2

i n ( 1 2 ) m u s t b e c a l c u l a t e d b y f o r m u l a

[61].

Now, accord ing to (1 .9 .3 ) and (1 .9 .6 ) sys tem (12) may be wr i t t en in the

f o r m

/

A

2

d

2

w w\

1 3

2

$

T? (0

2

w OuA

T

2

° fd

2

w

+

d{ku

2

) _ duA

+

25°

/ d

2

w

+

d(ku

2

y

+z

^

0

(2 .3 .14)

R

2

\d(3

2

d/3 da J B? \dad(3 da

Eh R

3

da

2

S y s t e m ( 1 4 ) c o n t a i n s t e r m s , d e p e n d i n g o n

u\

a n d

u

2

,

which a re es sen t i a l

o n l y if i n e q u a l i t y ( 1 .9 . 2 ) h o l d s . F o r su c h s tr e s s s t a t e s w e m a k e a n a p p r o x i m a

t i o n w i t h t h e a s s u m p t i o n t h a t

e

2

= w =

0 and ge t the fo l lowing r e l a t ions

du

x

du

2

du

2

, „ „ . , >

^ T + ^ — = 0,

—--kw = 0

2 .3 .15

dp da op

c los ing sys tem (14) .

2 .4 Bu ck l ing E qu a t io ns o f t h e G en er a l S t r es s

S t a t e

Co n s i d e r t w o t y p e s o f t h e i n i t i a l s t r e s s s t a t e : ( i ) t h e m o m e n t s t r e s s s t a t e ,

wh ich is a su m of th e me m br a ne s t r es s s t a t e an d the edge eff ec t a nd ( ii ) t he

2.4. Buckling Equations of the General Stress State 47

general stress state, which is a s um of the m em brane s t ress s ta te and the stress

s ta te (may be fast varying) due to local imperfections of the ne utra l surface.

In both cases

the the

ne ut ra l surfaces before

and

after deform ation c ann ot

be

identified. Indeed,

let the

initial deflection

to

0

~

h, then

for the

edge effect

the

initial changes of curvature x ? , r° ~ . f t

- 1

, i.e. they are not small .

Let the conditions of applicability of the system of equat ions of the technical

theory (1.5.3)

be

fulfilled

and

assume t ha t

the

initial stress state

is the sum of

a membrane s t ress s ta te

and an

edge effect.

The

buckling equations

we

take

in the form

d_BT

L

_d_B_ lO(A'S)

=

da

da

2

A dp

dAJh_8A 18(B

2

S)

dp

dp B da

-DAAw+k

1

T

1

+ k

2

T

2

-

(

2 A 1

)

~ J B - L [

5 ( T l

°

7 1 + T l 7 l

°

5

°

7 2 +

"

-

- - [MT +T

+ S^t+Sj?)]

=

0.

This system

may be

writ ten also through

the

stress function

<3> in the

form

-DAAw+ A

T

w + A

k

$ + A?$ = 0,

1 (2-4-2)

— - A A $

+

A

k

w +

A°

k

w

=

0,

fcih

where

A°$ = x\T

x

+ x\T

2

+

2T°S,

AJV = H\H

2

+ >$x\ - 2r°r,

(2.4.3)

and Ti

and

S

are

expre ssed th ro ugh <3>

by

formulae (1.5.6).

The

operators

A , Afc and Ay are the same, as in (3.8).

The boundary conditions which

are to be

introduced

to

solve systems (3.8),

(3.9)

and (2) are

obta ined

by

l inearization

of

variables (1.2.9)-(1.2.15). We con

sider edge a = a

0

and assume tha t the external b oun dary load is conservative.

Then, with

an

error

of

order

h* we

will equate

to

zero

the

generalized

additional displacements u\, u

2

, u, 7x or stress-resultants Ti, S, Q\

t

, Mi,

where

1 r) M

Qu

= Qi~ g^J - Thx - S°J

2

- 7?Ti

-

J°

2

S. (2.4.4)



48

Chapter 2. Basic Equations of Shell Buckling

If the initia l stress sta te is m em bra ne the n the last two term s in (4) m ay be

I In

neglected (with an error of order hj ).

Let there exist initial imperfections of the form

w*.

We can represent the

entire deflection in the form

w* + w° + w,

where the term

w* + w°

describes

the initial stress state, the stability of which is being considered and

w

is the

additional deflection. Equations (1.8.2) give the buckling equations

-DAAw + A

T

w + A

f c

$ + A £ $ + A J $ = 0 ,

—

- A A $ + A

k

w + A°

k

w + A'

k

w

= 0. ^ ' '

'

Eh

K K

The init ial imperfections,

w*

, are included in the operator AJ, which is

obtained from (3) replacing

w°

by

w*,

and in the term

ATW,

in which the

init ial s tress-resultants T°, S° depend on w' and may be found from system

(1.8.2).

2 .5 P r o b l em s an d E xe r c i s e s

2.1 Using dynamic criterion investigate the stability of the equilibrium

sta te w = 0 for the simply sup port ed b ar unde r axial com pression. T he bar

has a mater ial density of

p.

(Note th at , unlike in Euler criterion, using the

dynamic criterion one needs to know the mass distribution of the system.)

2.2 The horizontal pipe carries a fluid. The pipe has a length

L ,

modulus

of elasticity of

E,

and the m om ent of ine rtia / . Th e velocity of the flow is

V

with a mass of m per second f lowing through the pipe. Determine the stabil i ty

of the tube if:

1) both ends of the pipe are simply supported;

2) one end of the pipe is fixed with the other end being free.

2.3 Use the kinetic approach to investigate the stability of the equilibrium

sta te w = 0 of a massless bar when subjected to the axial follower load. The

bar is clamped at the bottom but i t carr ies a mass

m

at i ts top.

2. 4 Derive buckling equ ations (2.3.9) based on varia tiona l equ ation (2.3.10).

Hint. First ly represent the variations

8T1

£

and

STl*

in the form

SU

e

= / / ( T i f c i + T

2

Se

2

+ SSu>)dn,

a

SIl* = [[(Mrfx + M

2

Sx

2

+ HSr)dn.



C h a p t e r 3

S i m p l e B u c k l in g P r o b l e m s

In this chap ter we will analyze buckling problems for a me m bran e homog eneous

stress state. Consider a shallow shell and circular cylindrical shell, for which

the problem leads to equations with the constant coefficients. The shell edges

are assumed to be simply supported which allows us to write the explicit form

of the solution. We will also consider the buckling modes for which a shell is

covered by a regular pattern of small pits.

In addition, for the case of a non-homogeneous membrane stress state we

will determine an estimate for the critical load by means of the energy method

and obtain the expected buckling modes (see Section 3.6)

3 . 1 B u c k l i n g o f a S h a l l o w C o n v e x S h e l l

Consider the buckling of a thin, shallow shell under a membrane stress

stat e as determ ined by the init ial s tress-resultants 7}° and 5 ° . T he m etr ic

coefficients A and B, the curvatures kj and stress-resultants T® and S° are

assumed to be constant.

As buckling equations we take system (2.3.8), which, in this case has con

sta nt coefficients. We can write the syste m in the dimensionless form

H

2

AAw + XA

t

w

- A

f c

$' = 0,

(3.1.1)

/ i

2

A A $ '

+ A

fe

ti> = 0,

49

50

Chapter 3. Simple Buckling Problems

where

_

d

2

w d

2

w

dx

2

dy

2

'

,

d

2

w

,

d

2

w

A

kW

= k

2

—

+ kl

— , (3.1.2)

d

2

w

n

d

2

w d

2

w

ox* oxoy dy

2,

In (1) and (2) the following notation is used:

{T°,T°,S°} = -\Ehn

2

{h,t

2

,t

3

},

- -

R

^ -

h

"

I _

Ri' ^ ~ 12(1-u

2

)R

2

~

PB

y =

JT'

hi

1 2 ( 1 - I /

2

)

(3.1.3)

Here

fi

> 0 is a small param eter , A > 0 is an unknown load param e

ter. It is assum ed, th at the loading is one -pa ram etric , i .e. th at th e initial

stress-resultants increase proportional to a para m ete r A, under loading. This

assumption doesn't violate the generality since, under such loading, any stress

sta te m ay be obt aine d. T he symb ol "min us" in the first g roup of formulae (3)

is introduced for convenience, since buckling is characterised by compressive

(negative) stress-resultants

Tf.

Below, the prim e sym bol by $ is om itte d.

The solution of system (1) we seek in the form

K $ } =

{w

Q

,$

0

}

exp

li

PX + q y

\ ,

(3.1.4)

where

WQ, <&Q, p,

and

q

are unknown constants.

After s ub stitu tion in (1) we find the equ ation w hich wave num be rs

p

and

q

satisfy. We write this e quation in a form, solved w ith respect to A

\-t< \ -

(P

2

+ <?

2

)

4

+ (*2p

2

+ fcl<?

2

)

2

„ , , ,

"

/ ( p

'

q )

~ (pi+mhP

2

+2tsp

q

+t

2

q

2

)

{ 3 L 5 )

While not considering the question of satisfying the boundary conditions,

we assume, that

p

and

q

m ay at ta in any real values. T he critical value Ao of

the pa ram ete r A is given by formula:

(+)

A

0

= m i n / =

f(po,qo)-

(3.1.6)

P,9

3.1. Buckling

of a.

Shallow Convex Shell

51

Let

p

=

rcos<p,

q = r

sirup,

(3.1.7)

then the function / has the form:

f

=

r

2

a(ip)

+

r-

2

b(ip)

(3.1.8)

and its m in imu m by

r

may be found imm ediately

(+) 2(k

2

cos

2

<p+kisin

2

<p) _

m i n / = —

:

'••

— c(y>). (3 .1.9)

r

t\ cos

2

 0 (the (+) symbol

in

(10) and

also

in (6) and (9)

r e m i n d

us of

thi s) . Since

the

shell

is

convex

* i , k

2

> 0, t h e

n u m e r a t o r

of c((p) is

posi t ive

for a ll <p.

I f s imul taneous ly

t i

< 0, <

2

< 0, tl-tit

2

<0,

(3.1.11)

t h e n

the

d e n o m i n a t o r

of

c(ip)

for all <p is

non-pos i t ive

and the

shell will

no t

buckle since there

a re no

compress ive ini t ia l s tresses act ing

in an y

direct ion.

If inequali t ies (11) are not fulfilled si mu ltane ousl y, th en

4&iAr

2

,

tiki

- t

2

k

2

+ r

3

s g n t

3

*

0

=

T~Z

—

m.

—i

>

cotipo-

—— ,

' i * i + '2«2 +

T

3

2k

2

t

3

(3.1.12)

r

3

=

({tiki

-

t

2

k

2

)

2

+ 4 * i M )

1 / 2

> o.

D ue to (3) the critical values of the stress-resultants T° and S° satisfy the

equality

((T°R

2

-

T°Ri)

2

+

4RiR

2

(S°)

2

y

-

T»R

2

-

T°Ri

= J f

f t

. (3.1.13)

V 3 ( l - v

2

)

The above solution was obta ined in [139]. Earlier, the problem witho ut

shear stress (S° — 0) was solved in [134]. Similar solutions for some partial

cases of loading have been found by geometr ic m ethods in [132, 133].

The upper cr i t ical loads, given by A.V . Pogorelov in [132, 133] differ from

those found by other auth ors by the factor

y/\

—

v

2

.

The difference may be

5 2

Chapter

3.

Simple Buckling Problems

e x p l a i n e d as fo l lows . The b a s i s is the s e a r c h for the p o t e n t i a l e n e r g y of shell

d e f o r m a t i o n u n d e r b e n d i n g . One of the p o t e n t i a l e n e r gy c o m p o n e n t s is the

p o t e n t i a l e n e r g y

of

t a n g e n t i a l d e f o r m a t i o n

(II

£

)

w h i c h

is

c o n c e n t r a t e d

inthe

n e i g h b o u r h o o d

of the

f ol d a p p e a r i n g d u r i n g

the

is o m e t r i c t r a n s f o r m a t i o n

of

t he su r f ace (see (2 .3 .12) )

n

-4//(

'(

+

2veie

2

+e\+

—^-u'

1

I

dil

T o c a l c u l a t e t h i s e n e r g y

the

a u t h o r a s s u m e d t h a t

£i 0 and

e

2

— u>

= 0.

A c t u a l l y the e q u a l i t y £2 =0 i n t r o d u c e s a n o n - e x i s t e n t tie on the sur face defor

m a t i o n .

If in

e v a l u a t i n g

II

£

we

a s s u m e t h a t

£2 =

—v£i ( w h i ch c o r r e s p o n d s

to

t h e m i n i m u m of the II

£

by £

2

) t h e n the r e s u l t s g i v e n by A.V. P o g o r e l o v ' s g e o

m e t r i c a l m e t h o d [ 1 3 3 ] for the u p p e r c r i t i c a l l o a d s c o i n c i d e e x a c t l y w i t h t h o s e

o b t a i n e d

by the

b u c k l i n g e q u a t i o n s .

L e t us con s ider so m e spec ia l cases .

Pure shear.

L e t

T°

—

T

2

=0 and

S° ^

0. T h e n the c r i t i c a l v a l u e of t h e s t r e s s - r e s u l t a n t s

a n d the p a r a m e t e r s of the b u c k l in g m o d e are the fo l lowing

Eh

2

±

fki

2

o ^1^2

S°=

.

cot<p

0

= \-±, rg =2, , . (3 .1 .14)

/Z{\-u

2

)RiR

2

V*2 ° *i +*2

V ;

The absence

of

external norm al pressure.

D u e to (1 .4 .3 ) tiki

+ t

2

k

2

=0 and f o r m u l a (13) can be s impl i f ied as

((T?fRl + (SyRiR

2

)> = f*

(3 .1 .15)

The absence

of

shear stress.

In th i s case d i r ec t ly f rom (9) we find

. 2*2

n

^0

Eh

2

Ao =- — ,

fo

=0,

Ti —

h

'

L

R

2

y/3(1

- v

2

)

for

tiki

> t

2

k

2

;

(3 .1 .16)

2ki

ir

0

_ Eh

2

A

0 — ——, ¥>0 — 77 >

1

2 — ~

h '

^

2' - Riy/Hl-U

2

)

for tiki<t

2

k

2

. (3 .1 .17)

T h e c a s e w h e n tiki =

t

2

k

2

is s p e c i a l and is c o n s i d e r e d in S e c t i o n 3.3.

54


w h e r e t h e m i n i m u m w e s e e k t o t h o s e m ,

n

for which A

m n

> 0.

Le t us cons ider the case when t\k\ >

<2&2J

^3 = 0 , for which the pi t s

a r e e l o n g a t e d a l o n g a x i s Y. I t w a s s t a t e d a b o v e t h a t f or a r b i t r a r y p a n d q

t h e m i n i m u m i s a t t a i n e d a t p

0

= y/k\, q

0

= 0 . T h a t is w h y m i n i m u m ( 6) is

a t t a i n e d a t n = 1 an d p

m

wh ich i s c losest to po . W e ca n es t im at e the d i ff e rence

be tw een Ao an d Ao- By t ak in g in to acco un t th a t | p

m

— p o |< 7T/i /(2/ i) we f ind

t h a t

I Ap - A

0

| 7T

2

Ao - ^ 1 2 ( 1 - j /

2

)

There fore i f Ri, Li ~ R e r ro r (7 ) has o rder h*.

Whi le i t i s more d i f f i cu l t t o cons t ruc t the buck l ing modes fo r

S°

^ 0 an d

for the o th er bo u n d ar y co nd i t ion s i t i s r e l a t iv e ly easy to so lve the p r ob lem for

5 ° = 0 in the case w hen tw o op po s i t e edges of a r e c t an gu la r p l an e she l l a r e

s i m p l y s u p p o r t e d a n d a t t h e o t h e r tw o e d g e s a r b i t r a r y b o u n d a r y c o n d i t i o n s

a r e i n t r o d u c e d .

T h e e x a c t c o n s t r u c t i o n o f a b u c k l i n g m o d e w a s i n t r o d u c e d a b o v e . N o w l e t

u s c o n s i d e r t h e e s t i m a t e o f t h e c r i t i c a l l o a d u n d e r d i f f e r e n t b o u n d a r y c o n d i

t i o n s . H e r e w e c a n m a k e t w o s t a t e m e n t s .

F i r s t , s ince th e bu ck l in g m o d e ha s a loca l ch ara c te r , i nc re as in g the s ti ffnes s

o f t h e b o u n d a r y ( u p t o a c l a m p e d e d g e s u p p o r t c o n d i t i o n u\ — u

2

= w =

71 = 0 ) may l ead to an inc rease o f the c r i t i ca l va lue o f the load paramete r A

compared to (1 .12) on ly by a smal l va lue o f o rder

h*.

T h i s s t a t e m e n t w il l b e

proved a t the end o f Sec t ion 3 .6 .

S e c o n d , t h e r e e x i s t 8 v a r i a n t s o f w e a k s h el l e d g e s u p p o r t ( w i t h a r b i t r a r y

s u p p o r t o f t h e o t h e r e d g e s ) f or w h i c h t h e p a r a m e t e r A d e c r e a s e d . T h e b u c k l i n g

m od e does no t cover th e en t i r e ne u t r a l su r f ace bu t r a th er i t i s l oca l i zed in the

n e i g h b o u r h o o d o f t h e w e a k l y s u p p o r t e d e d g e . T h i s c a s e w i ll b e c o n s i d e r e d i n

Sec t ion 13 .4 .

R e m a r k 3 . 1 . G e n e r a l l y s p e a k i n g , t h e m e t h o d of s t u d y d e s c r i b e d a b o v e

is not val id for shel l s of negat ive

{k\k

2

< 0) an d zero

{k\k

2

= 0 ) G a u s s i a n

c u r v a t u r e .

I n d e e d , l e t

k

2

>

0 an d fci < 0 . Th e n w e f ind f rom (1 .9) an d (1 .10) th a t

Ao = 0, r

0

= 0, cotipo =±J-'j^-,

f rom wh ich i t fo llows th a t th e o rde r of th e c r i t i ca l loa d dec rease s an d the

d i s t a n c e b e t w e e n t h e p i t s , d u e t o ( 1 ) , i n c r e a s e s . T h e a n g l e o f i n c l i n a t i o n o f t h e

p i t s ip , d e p e n d i n g o n t h e i n i t i a l s t r e s s e s , c o u l d o n l y b e f o u n d . A n e x c e p t i o n t o

th i s i s the case when

k\

=

T%

= 5 ° = 0 an d th i s wil l be ex am in ed in S ec t ion

3.4.

R

2

h 2R

z

2

h

LI

RiLl

T?R

2

(3.2.7)

3.3. The Non-Uniqueness of Buckling Mo des 55

3 .3 T h e N on - U n i qu en es s of Bu ck l ing M od es

For the condition

T?R

2

=

T%R

X

< 0,

S°

= 0

the critical load could be found simply from

2k

2

2Ari

(3.3.1)

ti

1

K 2 - i

2

n i -

Eh

2

vW ^

2 (3.3.2)

but the buckling mode is s t i l l indeterminate.

Let

p

and

q

in (1.4) be arbit rary values. W e can w rite (1.5) in the form

z

=

k

2

p

2

+ kiq

2

(p

2

+ q

2

)

^2 "

(3.3.3)

Figure 3.1: Wave numbers for the special case of a convex shell.

It is clear that the critical load is realized for z — \, i.e. for any p and q,

lying on the curve

(see Figure 3.1 or in polar coordinates (1.7))

r

2

= fc

2

cos

2

<p

+ k\

sin

2

<p.

(3.3.4)

(3.3.5)



56


If p a n d q a r e f ixed a nd sa t i sfy e qu a t i on (4 ) , t h en t he p i t s ( as in Se c t ion

3 .2 ) a r e h i g h l y e l o n g a t e d i n t h e s a m e d i r e c t i o n . H o w e v e r , n o w b o t h ± p a n d

±q

s a ti s fy e q u a t i o n ( 4 ) . C o m p o s i n g t h e l i n e a r c o m b i n a t i o n o f t h e f u n c t i o n s

exp I - (±

px

± qy) I (3.3.6)

w e o b t a i n t h e p o s s i b l e b u c k l i n g m o d e

{w,

$ } =

{

W Q

,

* o } s in fc^l s in ^ M , (3 .3 .7 )

w h e r e p a n d q s a ti sf y e q u a t i o n ( 4 ) , a n d XQ an d j /o a r e a r b i t r a ry .

R e m a r k 3 . 2 .

S i m i l a r t o (1 . 4) b u c k l i n g m o d e s (7 ) a r e o b t a i n e d n o t d e p e n d

i n g o n a n y b o u n d a r y c o n d i t i o n . T h e i m p o r t a n c e of c o n s t r u c t i n g s u c h m o d e s is

t h a t t h e y r ef le c t t h e p r i n c i p a l q u a l i t a t i v e p e c u l i a r i t i e s o f t h e b u c k l i n g m o d e s

f or r e a l b o u n d a r y c o n d i t i o n s in t h i s , a n d o t h e r m o r e d if fi cu lt p r o b l e m s ( s u c h

as tho se w i th var i a b le in i t i a l s t r es ses , cu rva tu r es , e t c . ) . A s fo r th e c r i t i ca l load ,

th e va lu e o f Ao sou gh t he re is a go od f ir st a pp ro x i m at io n o f th e c r i t i ca l load .

T h e f o l l o w i n g d i s c u s s i o n c o n f i r m s t h i s s t a t e m e n t .

I f t h e s h e ll h a s a r e c t a n g u l a r p l a n a r f o r m a n d i t s e d g e s a r e s i m p l y s u p p o r t e d ,

t h e n m o d e ( 2 . 5 ) s a t i s f i e s b o t h e q u a t i o n s ( 1 . 1 ) a n d b o u n d a r y c o n d i t i o n s ( 2 . 4 ) .

D u e t o ( 3 ) , r e l a t i o n ( 2 . 6 ) m a y b e r e p l a c e d b y s e a r c h i n g f o r t h e m i n i m u m

k

2

p

2

m

+k

1

ql

" " " I ~ i w i •*-17 "Tim ~— / n . *> \ •}

(Pm + r

n

y

m i n | z

m

„ — 1 | ,

z

mn

=

-j-£

— |—

2\2~- \6.i-o)

F o r s m a l l

h*

p o i n t s

p

m

a n d

q

n

a r e s i t u a t e d r a t h e r d e n s e l y o n (p ,

q)

p l a n e

a n d t h e m a j o r i t y o f t h e m li e i n t h e n e i g h b o u r h o o d o f c u r v e ( 4 ) . W e n o t e t h a t

for

z

mn

- 1 ~

h

l

J

2

(3.3.9)

^ f ^ ~ h. (3 .3 .10)

Ao

T h e s p e c i a l c a se c o n s i d e r e d h e r e i s i m p o r t a n t i n t h e p r o b l e m o f t h e b u c k l i n g

o f a s p h e r i c a l sh e l l u n d e r e x t e r n a l p r e s s u r e . F o r t h a t T° = T® — — — , R\ —

i?2 = R an d by fo rm ula (2 ) we can f ind the c r i t i ca l ex te r na l p res su re ( see

[61, 183])

qo

= ,

2Eh2

(3 .3 .11)

y

^ 3 ( 1 -

v?)R

2 V

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3.4. A Circular Cylindrical Shell Under Axial Com pression

57

Although a complete sphere is not a shallow shell, within each pit, the shell

may be considered as shallow and that justifies the use of equations (1.1).

R e m a r k 3 . 3 .

It should not be supposed that the buckling of a real shell

occurs by a mode for which the difference \z

mn

— 1 | is a minimum. Init ial im

perfections and other factors not taken into account tend to distort the picture.

On the other hand, the fact that eigenvalues A

m n

con cen trate n ear Ao ha s a

large effect on the sens itivity of the critica l load to any initia l im perfe ction s. It

is known that a spherical shell under external pressure (mentioned above) and

a cylindrical shell under axial compression are the most sensitive (see Section

3.4). For both of these problems, buckling with many modes can occur.

The increase in sensitivity is caused by two factors. The more likely is that

an occasional imperfection is close to one of the buckling modes and the other

is that multiple and adjacent eigenvalues are more sensitive to perturbations

(imperfections) than a simple one. This problem is considered in more detail

in [14, 78, 145, 150].

3.4 A C i rcu lar C yl ind r ica l Shel l U n d e r A xia l

C o m p r e s s i o n

Let us consider the buckling problem of circular cylindrical shell of radius

R and length L \ under a m em brane stress sta te T° < 0, T | = 5° = 0. For the

case of a shell of moderate length

{L \

~

R)

we can use equations (1.1) as the

buckling equations, in which

ki = 0, k

2

= l, 0<x< l

l

= -±,

0 < y < 2 7 r . (3.4.1)

If edges

x —

0, and

l\

are simply supported (2.4), then the specific case

considered in Section 3.3 takes place and the buckling mode is defined by

formula (3.7), in which

p

and

q

satisfy e qu atio n (3.4)

(p

2

+ q

2

f= P

2

,

( 3 A 2 )

and the critical load, by virtue of (3.2) is equal to

A

0

= - , T ° = , ^ (3.4.3)

tx

l

v / 3 ( l - z ,

2

) i ?

V ;

Formula (3) has been obtained by R. Lorenz [94] and S.P. Timoshenko

[154].

The set of values of p and q satisfying (2) which is a couple of circles of

radius 1/2 is shown in Figure 3.2.

58

Chap ter 3. Simple Buckling Problem s

F i g u re 3 .2 : Wa v e n u m b e r s f o r a x i a l c o m p re s s i o n o f a c y l i n d r i c a l s h e l l .

S a t i s fy i n g t h e b o u n d a ry c o n d i t i o n s a n d t h e p e r i o d i c c o n d i t i o n s b y

y

we find

t h a t

p

a n d

q

m a y t a k e o n o n ly t h e d i s c r e t e v a l u e s

firmr

, q

n

= pn, x

0

~0

(3.4.4)

(m =

1,2,..., n

= 0 , 1 , 2 , . . . ) . T h e e x p e c t e d b u c k l i n g m o d e is p o s s i b le fo r a

coup le o f va lues (m,

n)

for which the va lue of

i s c lo s e t o u n i t y . A m o n g t h e m a r e t h e v a lu e s p

m

= 1 a n d q

n

= 0, i .e.

n

= 0,

m

L i I I < / 1 2 ( 1 - I / 2 ) _ ,

— ?7 — F=^= — 'Oi

npR

WRh

(3 .4 .5)

(3.4.6)

c o r r e s p o n d i n g t o a n a x i a ll y s y m m e t r i c b u c k l in g m o d e .

F o r m ul a (3) is i na pp l i ca b l e fo r ve ry long and ve ry sh o r t she l l s an d so we

wi l l n o w c o n s i d e r t h e m s e p a ra t e l y .

Consider suff ic ien t ly long she l l s where /

0

> 1 ( see (6 ) ) , t h en we can ap p ly

fo rm u l a (3 ) . I f

IQ

< 1 t h e n we a s s u m e

m —

1 , and

n —

0 an d t he c r i t i ca l l oad

i s t he n equ a l t o

T° = -

Eh

2

y/l2(l-v

2

)R

(l

2

o+lo

2

)-

(3.4.7)

I f , bes ides t ha t ,

IQ

<§; 1 , th en ne gle c t in g t h e f irst t e rm in (7) w e co m e to th e

E u l e r f o rm u l a

T° =

Eh

3

7T

2

\2{l-u

2

)L\

(3 .4 .8)



3.4. A Circular Cylindrical Shell Under Axial Com pression 59

for the buckling of a beam-strip cut out from the shell by generators.

Now consider long shells. It follows from F igur e 3.2 th a t a m on g the ex

pected buckling modes there are modes for which

p

m

and

q

n

are simultaneously

close to zero (i.e.

m

and

n

are small) . This case corresponds to the smaller

variation of the stress state and system of equations (1.1) is not applicable.

For small p

m

and q

n

p^ -C q„ (see Figure 3 .2), i.e. ineq ualit y (1.9.2) is

fulfilled and we can use systems (2.3.12) and (2.3.13). We seek the solution in

the form

{w,

(f)}

= {w

Q

,

cj>o} s in

-^—

s in

ny,

PmWQ p

m

X .

u\ = — cos s in ny, «2 =

lin

z

\i

As a resul t we get

T? • f 4 (A2, +

n

' )

2

- 2"

2

+

1

,

Eh= {

A»,(l

+

„-»)

+

w

0

. p

m

x

sin cos ny.

n (j,

A

2

(A 2

l

+ n 2 ) 2

( 1 + n

- 2

)

(3.4.9)

(3.4.10)

where A

m

=

rmv/li.

By me an s of this formu la we can also get formulae (3)

and (8). In the case A

2

^ -C

n

2

formula (10) may be simplified

^ = m i n ( / ( "

2

-

1

^

2

+

A

- X (3 4 11)

Eh ^ f \l(ni + l)

+

n2(n2+ l)J-

(6AAl

>

Minimizing it by A

m

we get the formula developed by P. Southwell and S .P. Ti m -

oshenko [61], generalizing (3) for the case of small

n

^ - V S & ^ T T ' ^ = " » ^ T , (3.4,2)

For very large

l\

the case when

m = n

= 1 tak es place. In thi s case we find

from (11)

Eh /TI -

X

2

T

=

- ^ U J '

(3

-

413)

and the buckling of the shell occurs in a manner similar to a rod of circular

cross-section.

Co m paring th e values of T ° , given by formula (12) for n — 2 and by form ula

(13) we find that buckling occurs similar to a rod with

£ > 3 .7 (f) "

2

. (3.4,4)

More accurately, the effect of the critical loads

T°

on the characteristics of

the shell are st ud ied in [45, 61].

60 Chapter 3. Simple Buckling Problems

3 .5 A C i r cu l a r C y l indr i ca l She ll U n d e r E x te r

n a l P r e s s u r e

We will now consider the buckling problem of a membrane stress state

T? = S° = 0, T$ = -qR (3.5.1)

of circular cylind rical shell described in Section 3.4. In th e case of sim ply

supported edges the buckling mode is

7TX

w = WQ sin — sin ny. (3.5.2)

« i

After substitution into equations (2.3.14) we find

T° = - min

- ^ - [ f

((A

2

+ n

2

)

2

- In

2

+

l) + A?(A

2

+

n

2

)~

2

]

, (3.5.3)

where Ai = 7r//i, l\ = L \/R.

Fi rst, we will consider a m od era tely long th in shell (/( < 1, Ai ~ 1).

Minimizing by

n

and assuming that we can neglect

\\

and 1 com pared to

n

2

in (3) (i.e.

(n

~

hZ '

)) , we approximately get

T

2

° = - m in Eh U

A

n

2

+ A ^ n

- 6

) . (3.5.4)

n

We could also obtain this formula by starting with system (1.1).

Minimizing (4) by n we get the Southwell-Papkovich formula [61, 104, 127,

144]

T

2

° _ 4Ehfi

4

n

2

_ 0.856 E ( h

5

\

1/2

q

° - ~ l l -

3R ~

( 1 - 1 /2 ) 3 /4

\L\lp) '

(ibb

>

where

/ 2

, ,

0

/ n 3 \

1/4

no

(

\ \

l

/

2

/ n 3 \

i/i

Buckling occurs for an integer

n

which is closest to

UQ.

For these

n,

the

critical value qo may be obtained more precisely by means of formula (3). The

1

/ 9

error of formula (5) is of order

hj

and is a result of the inac cura cy of form ula

(4) and with necessity of transition to an integer value of n.

From formulae (5) and (6) it follows that both q

0

and n

0

decrease with the

growth of shell length

L \.

Th e dependence of

q

0

on the length of the shell

may be also used to explain why buckling mode (2), has one half-wave in the

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3.5. A Circular Cylindrical Shell Und er External Pressure

61

longitudinal direction. The case of a few half-waves is similar to the case of a

shorter shell.

The minimum value of n is equal to 2. For sufficiently long shells (Ai < /i)

we can neglect the second term in (4), and for

n

= 2 we get

Eh

3

90

=

4 ( 1 - „ » ) # •

( 3 5

-

7 )

This is the Grashof-Bress formula [18]. The shell length is not introduced

in formula (7). This formula can be derived by considering the buckling of a

ring made from a shell under an external follower pressure [1].

The dependence of no and #0 on the shell parameters for small no = 2-6 is

studied in [61].

Let us now consider the combined loading of a cylindrical shell under an

axial force and a normal pressure

(Tf,

T

2

° ^ 0,

S°

= 0). We will restrict the

discussion to the case of a moderately long shell with simply supported edges

an d consider sy stem (1.1). It occurs [61], th at for

T°

< 0 (as in the case when

T°

= 0) that the buckling mode has only one wave in the longitudinal direction

and a few waves in the circum ferential dire ction (m = 1, n ~ /i* ).

With an error of order

hj

, the critical values of th e stresses

T°

and

T%

can be found from the relation

„ 4

( A

2

+

n

2

}

4

+

g

A

2

+

ll

n

^

( A

2

+

n

2

) 2

+

A

4

= 0 j

( 3 5 8 )

which, with an error of the same order may be written as

2 r

1 +

c £ T

2

= £

2

+ r

2

, (3.5.9)

where

T" -0 T- iO

J

i

J

2 (3.5.10)

Tl = - 2Ehfi

2

, T

2

* = - cEhXm

3

, c = 4 • 3 ~

3

/

4

=

1.755.

Here TI and r

2

are the nondimensional stresses related to the critical values

obtained in (4.3) and (5). The critical value of T-I for fixed

T\

we get from (9)

minimizing it by £. This dependence is represented in Figure 3.3.

For r

2

< 0, (the internal pressure) the value of T\ does not d epend on T? in

the l inear approximation and so the buckling mode is axially symmetric. For

T\, T2 > 0 the curve is close to straight line T\ + T? = 1. The tangents to this

curve drawn through points (1,0) and (0,1) are

• n + 0 . 8 7 7 r

2

= 1, 0.866 n + r

2

= 1. (3.5.11)

This case of combined loading is discussed in more detail in [45, 61, 155].



62

Chapter 3. Simple Buckling Problem s

Figure 3.3: Dimensionless stress-resultants for a cylindrical shell under com

bined loading.

3 .6 E s t im a te s of C r i t ica l Lo ad

The problems considered in this chapter comprise the class of shell buck

ling problems which have exact closed form solutions. In subsequent chapters

different ap pro xim ate solution s will be develope d. T he order of the critical

loads for membrane buckling are obtained by energy methods. Generally, the

results given below are contained in [54, 55]. A similar approach was used in

[57, 158] in which shell free vib ratio ns were stu die d. A ppl icat ion s of various

variational principles to shell theory are discussed in

[169].

Membrane stress state buckling may be studied starting from the variation

problem which comes from (2.3.10)

A

= m i n

( + )

J ,

J =

n

2

n '

T

(3.6.1)

where

e\ + 2z/£ie

2

+ el H — w

2

£

E

0

h

0

2JJ E

0

n

*=w

=

V/*£?

[ x l+2

^

2+

^

+2(1

-

^^

(3.6.2)

Here, A > 0 is an unknown load para m ete r. Th e m inim um in (1) is taken

over all possible complementary displacements

u\,

«2,

w,

for which the value

3.6. Estimates

of

Critical L oad

63

of

J in

(1)

is

po s i t ive ( the (+ ) s ign r em ind s us o f th i s ) an d w hich sa t i sf i es

t h e g e o m e t r i c b o u n d a r y c o n d i t io n s . W e a s s u m e t h a t t h e s m a l l p a r a m e t e r

/u

(or /i+)

in

(1)

is

t h e o n l y e s s e n t i a l p a r a m e t e r . I n o t h e r w o r d s , w e a n a l y z e

a

m o d e r a t e l y lo n g s h e ll w i t h s m o o t h l y v a r y i n g m e t r i c s , c u r v a t u r e a n d m e m b r a n e

in i t i a l s t r es ses .

T h e v a l u e o f

J

i n ( 1 ) d o e s n o t d e p e n d o n t h e r a n g e o f t h e c o m p l e m e n t a r y

d i s p l a c e m e n t s a n d t h e r e f o r e , w e w i ll a s s u m e f or c o n v e n i e n c e t h a t

w ~ R

a n d

a l l geomet r i c she l l s i zes a r e r e l a t ed to R. T h e n , a l l t h e v a r i a b l e s i n (1 ) b e c o m e

n o n - d i m e n s i o n a l .

T h e r e s e a r c h m e t h o d u s e d b e l o w c o n s i s t s o f t h e s u b s t i t u t i o n i n t o ( 1)

of

v a r i o u s d i s p l a c e m e n t s w h i c h s a t i s f y t h e g e o m e t r i c b o u n d a r y c o n d i t i o n s . For

t h e l o a d p a r a m e t e r , A, w e o b t a i n a n e s t i m a t e of t h e u p p e r b o u n d . T h e s y m b o l

O

i s used ins t ead o f ~i n the case when the exac t o rder wi th r espec t to

h*

of

t h e e s t i m a t e d v a r i a b l e , i s n o t f o u n d .

I t i s c l ea r tha t Tl'

e

> 0,

H'^

> 0

f o r a n y o f t h e d i s p l a c e m e n t s U{,

w.

T h e

va lue o f

H'

T

m ay have a ny s ign dep en d i ng on T^° an d

S°

, b u t w e a r e i n t e r e s t e d

o n l y i n d i s p l a c e m e n t s f o r w h i c h H'

T

>

0 . C o n s i d e r a t i o n o f t h e q u a d r a t i c f o r m

in 71 and 72 int h e expres s ion fo r I I^ , show s th a t such d i s p la ce m en t s e x i s t

on ly in cases when inequa l i t i e s (1 .11) a r e no t fu l f i l l ed s imul t aneous ly . In the

o t h e r w o r d s

it

i s neces sa ry fo r bu ck l in g th a t a t l eas t one of th e ineq ua l i t i e s be

fulfilled

T ° < 0 or T

2

° < 0 or ( S

0

)

2

-

T?T$

>0. (3.6.3)

T h e s e i n e q u a l i t i e s p r o v i d e t h e e x i s t e n c e o f t h e d i r e c t i o n s a l o n g w h i c h t h e

c o m p r e s s i v e i n i t i a l s t r e ss e s a c t . C o n d i t i o n (3 ) is n o t n e c e s s a r y int h e e n t i r e

d o m a i n Q. o c c u p i e d b y t h e s h e l l .

It

i s suf fic ient th a t th is co nd i t i on b e fulf il led

on ly in a c e r t a i n p o r t i o n

£li

o f t h i s d o m a i n .

L e t u s c o n s i d e r t h e s y s t e m o f e q u a t i o n s

e i

= e

2

=

w

= 0,

(3 .6 .4)

w h e r e t h e t a n g e n t i a l d e f o r m a t i o n s £ , - , uare def ined

in

( 1 . 1 . 7 ) , ( 1 . 1 . 1 0 ) . T h e

n o n - t r i v i a l s o l u t i o n s o f t h i s s y s t e m w i t h r e s p e c t to

U{,

w, s a t i s fy ing th e ge

o m e t r i c b o u n d a r y c o n d i t i o n s a r e c a l l e d i n f i n i t e s i m a l s u r f a c e b e n d i n g . If, for

them, the va lues o f x , - ,

r

d if fe r f rom zero th en we ha ve no n- t r iv i a l be nd in g .

W e a s s u m e t h a t t h e r e e x i st s n o n - t r i v i a l b e n d i n g fo r w h i c h t h e d i s p l a c e m e n t s

Ui,

w

are s lowly vary ing func t ions ( index o f va r i a t ion

t =

0 ) . T h e n w e o b t a i n

f rom (1) the fo l lowing es t imates

X = 0(K), T° = o( pi,

T

0

=max{7;°,S

0

}.

(3 .6 .5)

64

Chapter 3. Sim ple Buckling Problem s

T he es tim at e A ~ /i» ta kes place in the buckling p robl em for a long or free

cylindrical shell under external pressure (see (5.7)), and also in the buckling of

a weakly supported shell of negative Gaussian curvature (see Section 12.2).

Later it will be shown (see Chapter 12) that in the case when non-trivial

bending exists, functional (1) does not give the asymptotically exact value of

th e critical load A as ft*

—>

0, although estimate (5) is valid.

To explain this, we represent A in the form A = a

0

h" + o(h^). Th en, if

bending exists, function (1) gives the correct value of a and in the general case,

an incorrect value of do. At the same time for the other cases considered in

this section, the accuracy of (1) is sufficient for the evaluation of both a and

a

0

.

Now, we assum e tha t no non-trivial solution satisfying the geom etric b ound

ary conditions of system (4) exists. As the test functions for the substitution

into (1) take first the displacement field

m =

M

2

= 0,

w

=

<p(a,P)sinz, z = h~ *f (a,{3).

(3.6.6)

The geometric boundary conditions are fulfilled by a special choice of func

tion ip th a t is assum ed to vanish toge the r with the few first deriva tives on th e

boundary of domain 12.

•7T- 1 + ( T77T I > 0 ha s ind ex of va ria tio n t. Subst i tu t ion

da) \d/3J

of (6) into (1) gives 11^ = 0 ( 1 ) ,

W„

~

h~

4 t

.

T he integrand in 11^ takes the

form

L — 2t 2

2

h

t

\p

cos

z

dfV „ dfdf

,

fdf

da~)

+2t3

dp~da-

+ f2

\dp

+ 0(h:

f

) (3.6.7)

and if conditions (3) are fulfilled it is possible to find functions

(p

an d / in (6)

for which 0 < 11^ ~ h~

2t

. Th e sub stitu tion of these estim ates into (1) gives

J =

Oik*-

1

+ hl~

2t

)

an d for

t =

1/2 we obtain

J =

O ( l ) .

Hence, indepe nden t of the shap e of the shell and th e me tho d of shell su ppo rt

we obtain under the above assumptions

A = 0 ( 1 ) , T

0

= o ( ^ Y (3.6.8)

For "well" supported convex shells (&i&2 > 0) estimate (8) is exact i.e.

A ~ 1 (see Sections 3 .1, 12.3, 13.4). Here th e shell is called "well s up po rte d"

if restrictions on the displacements prevent neutral surface bending.

In the case of weakly supported shells, estimate (8) may be improved,

however we will not consider this case here (see Chapter 12).

We will, however, try to improve estimate (8) for arbitrarily supported

shells of zero and negative Gaussian curvature. Let us consider a shell of zero

3.6. Estimates of Critical L oad

65

dA

curvature k\ = 0. By virt ue of th e Co da zz i-G au ss re latio n (1.1.3) -77-7 = 0.

The displacement field we take in the form [158]

«i = / i»Vi sin z , u

2

— hl<f2 cos z + h^ips sin z,

w = tpsmz, z =

K

t

f(/3), / ' ( /? ) ^ 0 , (3.6 .9)

B

2

d( VS

=

-JJ-^-

By direct substitution we can verify that (ei ,£

2

,w) ~ h

2

*. Hence H'

£

~ h

4t

n ^ ~

h~

4t

.

Fu rth er 71 ~ 1, 72 ~

h~*

and by virtue of (2)

*

J

0 < n^, ~

h~

2t

for

T°

< 0,

0 < n^, ~

h-

f

for T

2

° = 0,

S° +

0,

0 < n^ , ~ 1 for T

2

° = 5 ° = 0, 7\

0

< 0 and

for T

2

° > 0,

{S

0

)

2

-

T?T$

> 0.

(3.6.10)

Substituting the above estimates into (1) and minimizing by

t

in the three

cases of (10) we obtain respectively

t

= 1/4 and

A = o(/

l

y

4

),

T

° = °(

Eh

°(it)

5 /

\

( 3 6 1 1 )

A = O(I),

T

° = ° ( ^ r ) -

For the particular case of a circular cylindrical shell, the first and third

estimates have been given in Sections 3.5 and 3.4. We note that in the case of

A

~ 1, displacements (6) for t = 1/2 lead to the same estimate. Therefore, the

improvement of the general estimate (8) takes place only in the first two cases.

R e m a r k 3 . 4 . Es tim ates (11) rem ain valid in the case when the inequalit ies

in (10) are fulfilled only in a certain (not too small) portion fli of domain fi.

To obtain estimate (11) we take the function in (9) which is non-zero only in

fii. W e dea l in a similar way in obtain ing estim ate (8), if con dition s (3) are

fulfilled in a portion of domain Q..

Now we consider a shell of negative Gaussian curvature, i.e.

k\k

2

< 0. We

6 6


t a k e

u i =K<pi cos z, <pi=Aki[—\ ip

- l

u

2

= ht>

( 3

'

6

'

1 2 )

w = <psinz, z = h~

t

f(a,/3),

w here func t ion / s a t i s fi es the eq ua t io n

For such displacements we f ind e , - , u> ~ hi. L e t u s t u r n t o t h e e s t i m a t i o n o f

Tl'

T

a n d a s s u m e t h a t

k\

< 0,

k

2

> 0.

T h e m e t h o d of t h e c o n s t r u c t i o n o f d i s p l a c e m e n t s ( 1 2 ) i m p o s e s c o m p l e m e n

t a r y r e s t r i c t i o n ( 1 3) f or 7— a n d — c o n t a i n e d i n ( 7 ) . T h e r e q u i r e m e n t t h a t

( 7 ) b e p o s i t i v e l e a d s t o t h e c o n d i t i o n

T°k

2

-T%k

1

- S°\/-kik < 0. (3.6.1 4)

I f th is condi t ion i s fu l f i l led a t leas t in a par t of domain Cl, t h e r e a r e d i s p l a c e

m en t s for wh ich 0 < 1 1^ ~

h~

2t

.

A f t e r s u b s t i t u t i o n i n t o ( 1 ) a n d m i n i m i z a t i o n b y

t

we find

* = 1/3, A = 0 ( / t i

/ 3

) ,

T°

=

otEh

0

( \

J . (3.6.15)

Co n d i t i o n ( 1 4 ) i m p o s e s s t r o n g e r r e s t r i c t i o n s o n t h e i n i t i a l s t r e s s - r e s u l t a n t s

T f , S° th a n do co nd i t io ns (3) . I f co nd i t io ns (3) are fu lf i lled , bu t (14) are

no t fu lf il led , the n buck l ing occ ur s . How ever i t i s pos s ib le to gu ar an te e on ly

es t imate (8 ) fo r the c r i t i ca l load .

R e m a r k 3 . 5 . T h e i m p r o v e m e n t o f e s t i m a t e ( 8 ) i s c o n n e c t e d w i t h t h e

c o n s t r u c t i o n o f s u r fa c e p s e u d o - b e n d i n g i .e . s u c h s u r fa c e d i s p l a c e m e n t s ( 9 ) a n d

( 1 2 ) f o r w h i c h t h e t a n g e n t i a l d e f o r m a t i o n s a r e s m a l l .

So f a r we cons id ered ne i th e r th e cases o f weak she l l su pp or t no r the cases

w h e n t h e G a u s s i a n c u r v a t u r e o f t h e s u rf a c e c h a n g e s i t s si g n .

T h e e s t i m a t e s o f t h e c r i t i c a l l o a d o b t a i n e d i n t h i s s e c t i o n a n d t h e e x p e c t e d

b u c k l i n g m o d e s a r e u s e d b e lo w fo r c o n s t r u c t i o n o f t h e a s y m p t o t i c s o l u t i o n s .

Le t us r e tu rn to the case cons idered in Sec t ions 3 .1 -3 .2 (ki,k

2

>

0) an d

pro ve th a t the inc rea se o f s ti ffnes s in th e bo un d ar y su pp or t doe s no t g rea t ly

3.7. Problems and Examples

67

increase the critical value compared to the value given by (1.12). We write the

displacements in the form

ui

=

p a wo

cos

z, U2 = fibwo

cos

z,

w = w

0

sinz, z = fi~

1

(px + qy) + z

0

,

_ (q

2

- vp

2

)(k

2

p

2

+ k

iq

2

) ki

(3.6.16)

p{p

2

+ q

2

)

2

P '

b=

(p

2

- yq

2

)(k

2

p

2

+ k

iq

2

) k

2

q{p

2

+ q

2

)

2

q '

where the values of p and q have been found in Section 3 .1. Su bst ituti on of

(16) into (1) for too = 1 gives A eq ual to (1.1 2).

Now we take WQ

= WQ(X, y)

and require that

w

0

= = 0 ,

x,y€T, (3.6.17)

where T is a conto ur of dom ain Q. Th en functions (16) satisfy the condition

of a clamped support. For each integral 11^, 11^.,

H'

T

in (1) we have

K'j^rfj f[w

2

0

dn +

O(n

2

),

/ i - > 0 ,

j =

e,x,T,

(3.6.18)

n

where n*- are such that it is possible to write formula (1.5) in the form A

0

=

(11° + n ^ ) ^ ) -

1

. It follows from above th a t A < A

0

+ 0{n

2

).

3 .7 P r o b l e m s a n d E x a m p l e s

3 . 1 .

Based on system (2.3.8) derive the system of equa tions describing

the membrane pre-buckling axisymmetric stress-strain state of the shell of

revolution. Use the curvilinear coordinates (s,(p), where

s

is the length of the

genetatr ix and ip is the angle in the circumferential direction.

A n s w e r The system has the form (3.1.1), where

<

fc

= - A W *

= 1

'

2

where

T£(s)

are the m em bra ne pre-buckling stresses,

A

is a loading parameter ,

4

h

2

B(s) , ,

p

=

pirn, p

is a smal l parameter

p =

—

T

„, 6 = — — -,

w(s, <p) =

12(1 — l/^jK H

w(s)

cos(m<p), an d $ ( s ,

<p) —

$( s ) cos(ray>),

68

Chapter 3. Sim ple Buckling Problem s

3. 2. Th e mem brane pre-buckling axisymm etric stress-strain sta te of the

shell of revolution is described by system of equations (3.1.1) (see Problem 3.1)

Construct the asymptotic expansions of the solutions of the system

A n s w e r

1

f

u>(s,fi) ~ "^2fi

h

w

k

(s)exp- p(s)ds

k=o V

Js

°

$ ( s , n) ~ ^2 ^^k (s) exp - / p(s)ds

k=0

where p(s) is a root of equation

,2 r V

n

/ , hr'Vf, r>\ , f,_ _

2

k^

P'-Tl +*[tiP

t

--Tr) P'-T, + *1P

6

2

7 V P V b

2

J V 6

2

«. - ( • -£ ) (»"

and

3 . 3 .

Derive from (3.6.4) relations for the tangential displacements U\ and

«2 for the shells of zero and negative Gaussian curvature (3.6.9) and (3.6.12).

Derive the estimates (3.6.11) and (3.6.15).

H i n t

Represent the deflection in the form

w

= <p sin

z.



C h a p t e r 4

B u c k l i n g M o d e s L o c a l i z e d

n e a r P a r a l l e l s

The buckling modes of convex shells of revolution under axisymmetric mem

brane initial stress states are considered in this chapter. Only buckling prob

lems in which the deflections are localized near some parallel (henceforth called

the weakest parallel) are studie d. It is assum ed tha t th e weakest parallel is

sufficiently far from the shell edges to avoid edge effects.

4.1 L oca l Shel l B uck l ing M o d es

The buckling modes of thin elastic shells under initial membrane stress

states may be divided into two classes. In the first class the buckling concavity

occupies all or the greater part of the shell surface. A typical example of this

class is th e buckling of a shallow convex shell und er ex tern al press ure. T he

second class is when the buckling mode consists of many small pits formed on

the surface of the shell.

The size and position of the pits and the value of the buckling load essen

tially depend on some determining functions such as the radii of the curvature

of the neutral surface, the shell thickness, the initial membrane stresses, etc. In

simple cases when these functions may be assumed to be approximately con

stant, the buckling pits occupy the entire shell surface (see Section 3.1). This

case is important, for example, for the buckling of a circular cylindrical shell

under axial compression (see Section 3.4) or under external lateral pressure

(Section 3.5) or torsion (Section 9.1). Shells with negative Gaussian curvature

also, as a rule, lose their stability by modes for which the pits occupy the entire

shell surface (see Chapter 11).

69



70

Chapter 4. Buckling Modes L ocalized near Parallels

If t h e d e t e r m i n i n g f u n c t i o n s v a r y f ro m p o i n t t o p o i n t o f t h e n e u t r a l s u r fa c e

t h e n l o c a l i z a t i o n o f t h e b u c k l i n g m o d e w i l l o c c u r .

For the f i r s t l oca l i za t ion type , the p i t s a r e loca l i zed near the weakes t l i ne

on the she ll su r f ace . T h e buck l ing mo de s o f conv ex she l l s o f r ev o lu t ion un de r

a x i s y m m e t r i c l o a d i n g ( d i s c u ss e d i n t h i s C h a p t e r ) a n d o f c y l i n d r i c a l s h e l ls u n d e r

c o m b i n e d a c t i o n of a n a x i a l f o rc e a n d a b e n d i n g m o m e n t ( C h a p t e r 5 ) a r e

p r o b l e m s o f t h i s t y p e .

T h e second lo ca l i z a t ion t yp e is pos s ib le fo r conv ex she l ls fo r wh ich th e s m al l

b u c k l i n g p i t s a r e c o n c e n t r a t e d n e a r t h e w e a k e s t p o i n t o n t h e n e u t r a l s u r f a c e .

T h e b u c k l i n g m o d e o f a c o n v e x s h e l l o f r e v o l u t i o n u n d e r c o m b i n e d l o a d i n g m a y

be o f th i s type ( s ee Chap te r 6 ) .

Let us n ot e th a t th e for the se loc al iz at io n ty pe s i t i s m o re di f ficul t to f ind

th e buck l ing mo de s if t h e weake s t po in t o r l ine i s c lose to an edge of the

she l l . I n th i s case i t i s nece s sa ry to s a t i sfy th e bo un da ry c on d i t ion s an d to

t a k e i n t o a c c o u n t t h e i n i t i a l b e n d i n g s t r e s se s a n d p r e - b u c k l i n g d e f o r m a t i o n s

( see Chap te r 14) .

T h e th i rd lo ca l i z a t ion ty pe i s pos s ib le fo r the she l l s o f ze ro Ga uss ian cur

va tu re ( i . e . for cy l in dr i ca l an d con ic she l l s ) . Bu ck l in g i s ac co m pa ni ed by the

a p p e a r a n c e o f c o n c a v i t i e s w h i c h a r e s t r e t c h e d a l o n g t h e s h e l l g e n e r a t r i x f r o m

o n e s h e ll e d g e t o a n o t h e r . T h e d e p t h of t h e c o n c a v i t i e s is m a x i m u m n e a r t h e

w e a k e s t g e n e r a t r i x a n d d e c r e a s e s f a s t a w a y f r o m t h e g e n e r a t r i x . T h e b u c k l i n g

m od es of no n-c i r cu la r cy l in dr i ca l an d con ic she l l s ( an d of c i r cu la r she l ls w i th

s l a n t e d e d g e s) u n d e r e x t e r n a l p r e s s u r e o r t o r s i o n h a v e t h e s e f o r m s . T h i s t y p e

o f b u c k l i n g o c c u r s f o r c i r c u l a r c y l i n d r i c a l s h e l l s i n b e n d i n g ( s e e Ch a p t e r s 7 a n d

9).

We n o t e t h a t f o r t h e s e c o n d a n d t h i r d l o c a l i z a t i o n t y p e s t h e r e e x i s t t w o

b u c k l i n g m o d e s c h a r a c t e r i z e d b y v e r y c l o s e c r i t i c a l l o a d s .

For th e case wh en th e we ake s t l ine or po in t i s far f rom a shel l edg e th e

a p p r o x i m a t e a n a l y t i c a l f o r m u l a e f o r b u c k l i n g m o d e s a n d u p p e r c r i t i c a l l o a d s

h a v e b e e n o b t a i n e d i n a s s u m p t i o n t h a t t h e r e l a t i v e s h e l l t h i c k n e s s i s s m a l l .

Unfor tuna te ly , i f weakes t po in t o r l ine i s c lose to a she l l edge , the r esu l t s a r e

no t so s imple .

4 .2 C on s t r u c t i o n A l go r i t hm of Buck l i ng M od es

T h i s a l g o r i t h m i s a n o n e - d i m e n s i o n a l v e r s i o n o f M a s l o v ' s a l g o r i t h m [ 9 9 ] . I n

she l l buck l ing p rob lems th i s a lgor i thm was used in [160 , 163] ( s ee a l so [15 , 88] ) .

W e d e s c r i b e t h i s a l g o r i t h m u s i n g , a s a n e x a m p l e , t h e s e l f- a d j o in t o r d i n a r y

4.2. Construction Algorithm of Buckling Mo des

7 1

di f f e ren t i a l eq ua t io n o f o rd er In

k= 0

X 7

w h e r e fi > 0 i s a sm al l pa ra m et e r an d the coef fi c ien ts a^ a re r ea l and de pe nd

l i n e a rl y u p o n t h e p a r a m e t e r A > 0

a

k

=a'

k

-Xa'i, a'

k

, a'

k

'

€ C °° (4 .2 .2)

a n d t h e f u n c t i o n s a'

k

an d a j.' a re inf in i te ly di f ferent iab le w i th resp ec t to x.

At each end o f the in t e rva l

(x\, x?)

t h e

n

s e l f- a d jo i n t b o u n d a r y c o n d i t i o n s

a r e i n t r o d u c e d , b u t w e d o n o t s a ti s fy t h e m e x a c t l y . W e s e ek e i g e n v a l u e s , A '

m

' ,

f o r th e p ar am et e r , A , for w hich th e re ex i s t l oca l i zed so lu t ion s o f eq ua t io n (1 )

w h i c h e x p o n e n t i a l l y d e c a y w h e n

x

a p p r o a c h e s t h e e n d s o f t h e i n t e r v a l

(x\, x^).

We w i l l s t u d y o n l y t h e c a s e w h e n t h e r e e x i s t s a p o i n t

XQ (X\

<

XQ

<

X2)

s u c h t h a t t h e s e s o l u t i o n s e x p o n e n t i a l l y d e c a y a s

\x — XQ\

i n c r e a s e . P o i n t

xo

is

r e fe r r ed to as th e we akes t po in t .

S o m e s h e ll b u c k l i n g p r o b l e m s m a y b e r e d u c e d t o t h i s p r o b l e m a ft e r s e p a

ra t io n o f th e var i ab les . A s a ru le we ge t a sy s te m of eq ua t io ns o f ty pe (1 ) w i th

c o e f f i c i e n t s r e g u l a r l y d e p e n d i n g o n t h e p a r a m e t e r

\i.

T h e a l g o r i t h m d e s c r i b e d

b e l o w m a y b e a l s o u s e d i n t h i s c a s e w i t h o u t m a j o r c h a n g e s .

I f we seek the solut ion of equat ion (1) as a formal ser ies

w(x,fi)

=

y^ii-'wj(x)

e x p

\

—

I p(x)dx>,

(4 .2 .3)

3= 0 ^

J

>

t h e n p(x) s a ti sf ie s t h e a l g e b r a i c e q u a t i o n

n

5 >

f c

( x ) p

2 f c

= 0. (4.2.4)

fc=0

F u n c t i o n ( 3 ) s a ti sf ie s t h e d e c a y i n g c o n d i t i o n m e n t i o n e d a b o v e if t h e r e e x i s t

p o i n t s

x\, x\

s u c h t h a t

9 p ( a ; ) > 0 fo r

x

>

x\,

Qp(x) = 0

for z | <$ a: < z 5 , (4 .2 .5 )

9 p ( a ; ) < 0 for

x

<

x\.

T h e s im ul t a ne ou s fu l fi lment o f co nd i t io ns (5 ) i s pos s ib le on ly in th e case

w h e n t h e r o o t s o f e q u a t i o n ( 4 ) a r e m u l t i p l e f o r

x

=

x\

a n d

x = x\.

In th e

n e i g h b o u r h o o d of p o i n t s

x = x*

a n d

x = x\

( w h i c h a r e c a l l e d t h e t u r n i n g

p o i n t s ) s o l u t i o n s ( 3 ) w i t h l i m i t e d Wj(x) do no t ex i s t .

72


T o c o n s t r u c t l i m i t e d a s y m p t o t i c s o l u t i o n s i n t h i s c a se , t h e s t a n d a r d e q u a

t io n m e th o d [84] ha s bee n used in [57 , 157] . Fu r th er we p ro po se a s imp le r

m e t h o d , w h i c h is b a s e d o n t h e a s s u m p t i o n t h a t t h e t u r n i n g p o i n t s x\ a n d x*

2

are c lose to each o th er a nd th a t th e coeff ic i en t s o f eq ua t io n (1 ) a r e r ea l . W e

seek a so lu t ion sa t i s fy ing the decay ing cond i t ion as a fo rmal s e r i es

w{x, n) = w

t

e x p { i (fi-

1/2

p

0

li + ( 1 / 2 ) <

2

) } , (4.2 .6)

w h e r e

X >

f c / 2

M O ,

t = »~

1/2

{' -

*o)

-

(4-2.7)

fc=0

a n d

Wk(i)

a r e p o l y n o m i a l s i n £ . P a r a m e t e r A is a l so e x p a n d e d i n t o a s e ri es i n

A*

A = A

0

+ / x A i + ^

2

A

2

+ . . . (4 .2 .8)

Here uifc(£), \k , XQ, po a n d a m a y b e f o u n d a f t e r s u b s t i t u t i o n i n t o e q u a t i o n

( 1 ) . Sa t i s fy ing th e decay co nd i t ion fo r so lu t ion (6 ) , we f ind th a t po mus t be r ea l

and 9 ( a ) > 0 . For tha t

XQ

is t h e w e a k e s t p o i n t , p a r a m e t e r

po

c h a r a c t e r i z e s

th e osc i l l a t ion f r equenc y o f func t ion w, a n d 9 ( a ) g i v e s t h e r a t e o f d e c r e a s e of

t h e f u n c t i o n a m p l i t u d e w h e n

x

d e v i a t e s f r o m

XQ.

T o f i n d t h e p a r a m e t e r s xo an d po we r eso lve equ a t io n (4 ) in A

I X

OOP

2

*

A = ^

= / ( p , * ) . (4.2.9)

£ < ( * ) p

2 f c

fc=0

L e t t h e m i n i m u m

A

0

= m i n { / ( p ,

a

: ) } = / ( p o , x o ) ( 4.2 .1 0 )

exis ts for real p a n d x\ <Z. x

<^.

X2, a n d x\ < xo < x^ , an d the s eco nd d i f f e ren t i a l

o f func t ion / a t po in t (po , *o) i s the pos i t iv e def in i t e q ua d ra t i c fo rm

d

2

f = f„ dp

2

+

2f

px

dp dx + f

xx

dx

2

>0.

(4 .2 .11)

H e r e d e r i v a t i v e s f° , / ° a n d f

xx

a r e e v a l u a t e d a t p o i n t ( p o, XQ).

W e r e w r i t e e q u a t i o n ( 1) i n t h e f o r m

H ( - i H ^ , x, \,v) w = 0, (4 .2 .12)

4.2. Con struction Algorithm of Buckling Modes

73

where

k-0 j=0

are polynomials of order In in p. After th e sub stitu tion of solution (6) in

equation (12) we obtain

H (po + /i

1/2

« - « 4 ) - *o + H

ll2

L Ao + Mi + • • • , A*) £ f

k/2wk

(0 = °-

(4.2.13)

The expansion into a power series of fi

1

'

2

leads to equ ations

k

Y,HU

)

xv

k

.

j

= 0, * = 0 , 1 , 2 , . . . , (4.2 .14)

i=o

where the first differential operators H^> are

HW

= # (

P o

, x

0

, A

0

, 0 ) ,

* < » = i l l i ^ - 2 i . { i - , - . ^ )

+

(4.2.15)

+

H I (ae

- a | ) + \ H°„e+*i # °+#°;

S # „

d

2

H

-tip — <-j , ripx

p

dp '

p:E

dp

<9x'

The superscript ° on the derivatives indicates that they are calculated for

p = Po, x

=

x

0

,

A = A

0

,

ft = 0.

Since, due to (10)

#(°) = ff

p

° = tf ° = 0, (4.2.16)

we exam ine the equ ation

H^w

0

= 0. Th is equ ation has a solution in the form

of a polyn om ial in £, only if th e coefficient a t £

2

is equal to zero, from which

we get

H°

p

a

2

+

2H°

px

a + H°

xx

= 0.

(4.2.17)



4.2. Construction Algorithm of Buckling Mo des

75

is satisfied.

We note the following property of operators (15),

H^"\

If the poly nom ial

P(£) is even then the polynomial

H^P(^)

is even for even

n

and odd for

odd

n.

If the polynomial P(£) is odd then the polynomial H^P(^) is od d for even

n

and even for odd

n.

It follows th a t con diti on (24) is alw ays fulfilled. W e

chose the solution of equation (23), satisfying the condition

{w\, H

m

)

= 0.

The following approximation leads to the equation

# ( 4 )

wo = 0,

which may be rewritten in the form

M

m

[w

2

(0)] = P

m 2

(0). (4.2.25)

Polynomial P

m

2{0) contains the term 2r~

1

\2H

m

(0), whe re the value of

A

2

may be found from the condition (P

m

2 ,

H

m

)

= 0. T his process m ay be

contin ued infinitely. At the even ste p 2

k

th e value of co rrec tion A& for th e

eigenvalue of A m ay be found.

T he critical value A* m ay be ob taine d for

m

= 0 in formula (20) and

K = X

0

+ fi~ + O{fi

2

), w

0

(t) = \.

(

4

-

2

-26)

Above we also stu die d th e case when m ^ 0, since it does not re quire ad di

t ional computations. The information about the distance between the cr i t ical

value and the other eigenvalues may be used to estimate the sensitivity of the

critical value to pertu rba tion s (imperfections) and also for the develop m ent of

numerical algorithms for the calculation of A*.

When we seek the minimum of (10), two essentially different cases are

possible

Po

=

0 (case

A)

P o f 0 (case

B).

y

'

R e m a r k 4 . 1 . Case

A

may only take place if

f^

x

= 0. In this case eigen-

functions (6) are real and eigenvalues (8) are simple. They may be expressed

by formulae

*->(„

= »

P

(-*^1) [*. ( ^ p l )

+

o( , - / ,

A = A

0

+ ^ ( m + - ) r +

0(ii

2

),

(4.2.28)

76

Chapter 4. Buckling M odes L ocalized near Parallels

where

A

0

0

_ 2 ( a K - « o « i )

« )

Jxx

dx

2

\a%J'

r

~ \Jxx Jpp)

>

c —

\Jxx) \Jpp)

'

(4.2.29)

All of the functions in formulae (29) are calculated for x = XQ. The eigenfunc-

tion for m = 0 is plotted in Figure 4.1.

Figure 4.1: The eigenfunction when p

0

= 0.

Re m a r k 4 . 2 . In

case

B

eigenfunctions (6) are com plex. Since equatio n

(1) is real, both the real part of function (6) and the imaginary part are eigen

functions. It m ay seem th a t the correspond ing eigenvalue (8) is doub le, bu t

this is incorrect, since series (6) and (8) are asymptotic (and not convergent)

series and they do not exactly represent the eigenvalues and eigenfunctions.

Two real eigenfunctions correspond to each eigenvalue (8), the asymptotic

representations of which are the following

„(i)

(3tu)*

COSZJ

—

9u>* sinzj) exp

< ^— — >

, (4.2.30)

where

p

0

(x -x

0

) 5Ra(x

*o)

2

+ i

]'

i = i,2.

The init ial phase,

9j,

has quite definite values

6\

and

02

(0 ^

0\, 62 < 2n).

Eigenfunctions (30) for m = 0 are shown in Figure 4.2. Moreover, the exact

eigenvalues corresponding to eigenfunctions (30) are different. We denote the m

as A'™'

1

) and

\^

m

'

2

\

T he following es tim ate is valid for any

N

AA = A ^

1

) - A(

m

'

2

) =

O (»

N

)

Henceforth, we will call such eigenvalues asymptotically double which refers

to the fact that they are nearly (but not exactly) the same and approach each

other .

4.3. Buckling Mod es of Convex Shells of Revolution

77

Figure 4.2: The even and odd eigenfunctions when po ^ 0.

The method of the asymptotic integration used here does not permit the

determination of the difference AA and the values of the initial phases 9\ and

02 • For one particular case these values will be found by numerical integration

in Section 5.3.

4 .3 B uck l ing M o de s of Co nve x She lls o f R ev o

lu t ion

We will now introduce curvilinear coordinates on the neutral surface of a

shell of revo lution . As the first coo rdina te we chose the g ene ratrix leng th

s'

(s'j <;

s'

^

s'

2

)

and as the second, the circumferential angle ip. We sta r t analysis

with system (2.3.8) rewritten in a dimensionless form as in (3.1.1)

u.

2

A(dAw ) + \A

t

w- A

f c

$ = 0,

^ A ^ A ^ + A f c w = 0,

(4.3.1)

where, for a shell of revolution

Aw —

AfcW =

A,w

=

id_

b ds

ld_

b ds

10_

b ds

OS

1

d

2

w

b

2

dip

2

'

ki d

2

w

+

b

2

lhp~

2

'

t

2

d

2

w

+

i

d

b

2

d(p

2

b ds

< 3

dw

d(p

+

t

3

d

2

w

b dsdtp'

(4.3.2)

78


H e r e

,

4

_

hi

Eh

3

(l-v

2

)

_

1 =

Eoh

1

B_

12(l-i/

0

2

)i?

2

'

E

0

hl(l-viy

9

Eh' R'

ki

= W

i

s

i>

s

'}

=

R

{

s

i>

s

}> {T?,S°}=-\T{t

i

,t

3

},

( 4 3

'

3 )

T

=

E oh

of

i

2

,

» = 1 , 2 .

T h e f u n c t i o n s

b,

d, g, &,• a n d <,• do n o t d e p e n d on th e an gl e <p. Sys tems (1 )

i s w r i t t e n f o r t h e c a s e w h e n p a r a m e t e r s

E, v

a n d

h

d e p e n d o n

s

( se e R e m a r k

1 .1 ). T y p i c a l v a l u e o f t h e s e p a r a m e t e r s a r e d e n o t e d b y EQ, VQ a n d ho.

If E, v

a n d

h

a r e c o n s t a n t t h e n

d

=

g

= 1.

We seek a so lu t ion o f sys tem (1) w i th nw a v e s i n t h e c i r c u m f e r e n t i a l d i r e c

t i o n

in

t h e f o r m

of

(2.6)

w(s,tp,n)

=

i o , e x p

j i (//-

1/ 2

p

0

£ +

l / 2 a £

2

)

+ in{<p -

<p

0

)}

,

(4.3.4)

w h e r e £ =

p~

l

l

2

(s

— So),

a n d

w*

h a s t h e f o r m of (2 .7) .

We seek the func t ion

$ in

t h e s a m e f o r m ( 4 ) .

T h e f u n c t i o n / in fo rm ula (2 .9 ) is

U ^

d(

P

2

+ q

2

)

4

+ g(k

2

p

2

+

k

iq

2

)

2

f (P . P,

S

) =

— *

9^ 1

(4.3.5)

( p

2

+<7

2

)* (tlP

2

+

2t

3P

q

+

t

2

qi)

w h e r e q = pb~

l

, p =

pin.

Fu nc t io n (5) d i ffers f rom fun ct io n (3 .1 .5) only by

m u l t i p l i e r s

d

a n d

g.

As inS e c t i o n 3 . 1 , wea s s u m e t h a t

k\k

2

> 0and also

tha t cond i t ions (3 .6 .3 ) a r e s a t i s f i ed . Ind o m a i n — o o o , s o ) , ( 4 . 3 . 6 )

P,P,s

a s s u m i n g t h a t p a r a m e t e r

p

c h a n g e s a p p r o x i m a t e l y c o n t i n u o u s l y . W e t h e n o b

t a i n

A

0

=min

5

( + )

{ T (

S

) } = 7 ( * O ) ,

i(s)

=

, ,

J ; : '

,

TS

=

( ( t

1

* i - <

2

*

2

) + 4 * 1 * 2 * 1 )

•

tiki+t

2

k

2

+

T3

\

K

)

(4

3

7)

6 ( f c

2

r

0

+& i)

2

(9 \

l

l

A

Poro

K

' ' '

5 ) -

r

0

= ( 2 *

2

< 3 ) ~

{tiki

-t

2

k

2

+r

3

) .

P0=

r

2

+ l ^

'

P0 =

4.3. Buckling Mo des of Con vex Shells of Revolution

79

Here, the values po, po and ro are evaluated for s = so. The parallel s =s

0

in which the function

j(s)

at ta ins its m inim um is called the weakest parallel.

The buckling pits are localized near parallel

s

= s

0

. We assum e th at th e shell

edge supports do not cause buckling. In Chapter 13 we will list the cases of

weak edge support for which buckling occurs near the shell edge.

We can write the expressions for initial membrane stress-resultants T° and

S°

for

a

shell u nde r axial force

P

and torsiona l m om ent M , app lied to edge

s = S2,

and und er e xtern al surface loadings g,(s) and

q(s)

applied to the shell

surface (see (1.4.6))

^ o

PR* R*'[ ( &

B

dB\

»i

** = - ^ k - ^

+

« r k / ( " S

+

'

f i

^ )

J S

' <«•">

The stress resultants may be transformed to the dimensionless stress-resul

tan ts

ti

due to (3). The extraction of the multiplier

A

(the load parameter) from

the initial stress resultants Tf and S° is arbitrary and the choice of parameter

A depe nds on the problem at han d. T he finite results do not dep end on th at

choice.

Let us study some particular cases of shell loading. We are going to use

the algorithm described in Section 4.2 which is the reason that the discussion

is limited to cases when the weakest parallel is far from a shell edge.

Let the initial shear be absent (£3 =0). In this case formulae (7) are not

convenient and may be replaced by

l{s)=

2

- {gdfl\

P o

=

0,

P

{s) =

b( pj ,

po =

P

(s

0

) (4.3.9)

for <2&2 >

tiki

and

l{s) =

2

-h

{gd

fl\

po=

(iRy\

po

= Q

( 4 . 3 . 1 0 )

for tiki > ^2^2- Here, as in (7), eva luat ing po and po we assume that s = so-

If tiki =

<2^2 for

s = SQ

then as in Section 3.3 nume rous buckling m ode s

are possible.

80

Chapter 4. Buckling Modes Localized near Parallels

R e m a r k 4 . 3 .

It follows from the results of this Section that the often used

m e th o d of coefficient "freezing" allows us to corre ctly find th e valu e of Ao in

the expansion of (2.8) and parameters

p

0

and

n

in buckling mode (4). To find

the correction Ai and the parameter

a

in (4) determ ining the buckling mo de

localization it is necessary to use the more detailed study given below.

4.4 Buckl ing

of

Shells of R evo l u t i on W i t h o u t

Tors ion

Let us first study the case <2&2

>

^l^i in which buckling is mainly deter

mined by the hoop stress-resultants ,

T°.

T hi s case is desc ribed by formulae

(3.9) and was called case

A

in Section 4.2. The critical load and buckling mode

are given by formulae (2.28) and (2.29).

For m

=

0 we get

A =Xo+fiXi +0(fi

2

),

, y )

=

e x p | -

c ( 8

^

o ) 2

} c o 8 n ( y - y > o ) [ l +

O

(/i

1

'

2

)]

, ^ ' ^

[s

where

A

0

= m i n ,

7

(

S

) , X, = 1/2 (/»„ / ° . )

1 / 2

,

c =

{tfj fj"

2

,

n - p o / i "

1

,

l(s)= (

9

d)

1

, P(')=

b

(

i

j)

1

'

A

> (4-4.2)

ds

2

' \p dsj '

Jpp

k

x

q

'

The values of f°

s

and f° in form ulae (2) are ev alua ted for s = SQ- The

ini tia l pha se y7o in (1) is ar bi tra ry and the refore th e critica l load is do ubl e. Due

to inequali t ies /°

5

>

0 and

f° >

0 inequ ality (2.11) is satisfied. T he trans ition

from

a

non -intege r value of

n —

po/*

_ 1

to the closest integer number of waves

in the circumferential direction

n

does not change the terms given by (1) for

A.

The buckling mode shown in Figures 4.1 and 4.3 illustrates the fact that

the buckling pits are elongated in the meridional direction. T he tange ntial

displacements

u\

and

u-i

are of order

ft

compared with the normal deflection

w.

Now let us study some particular loading cases.

Let a shell which has the form of a cupola or spheroid be under the normal

4.4. Buckling of Shells of Revolution Withou t Torsion

8 1

F i g u r e 4.3: The b u c k l i n g m o d e of a convex she l l of r e v o l u t i o n at t

2

k

2

>

tiki.

h o m o g e n e o u s p r e s s u r e q. T h e n , due to f o r m u l a e (3.8) we get

T h e c a s e

T

2

Ri < T°R

2

w h i c h we are e x a m i n i n g c o r r e s p o n d s to e x t e r n a l

p r e s s u r e (q > 0) for R

2

< Ri and to i n t e r n a l p r e s s u r e for R

2

> 2R\. If we

a s s u m e t h a t

\q\ =

\TR~

l

,

T

=

E

0

h

0f

i

2

t h e n

in

f o r m u l a e

(2)

h =

go

2kn

2k

2

- h

h =

—2p—

q

°'

1o

= sgnq.

(4.4.4)

(4.4.5)

A s s u m i n g t h a t the w e a k e s t p a r a l l e l s =SQ is far f rom a she l l edge , one ma y

u s e f o r m u l a e (1) to e v a l u a t e the c r i t i ca l load and b u c k l in g m o d e . If p a r a m e t e r s

E, v and h are c o n s t a n t t h e n d = g = 1 and the e x p r e s s i o n for the c r i t i ca l lo ad

m a y be r e w r i t t e n in the f o r m

1

=

2E h

2

f / k

2

(ki - k

2

)

^ 3 ( l - i /

2

)

{2RiR

2

-R

2

2

)

I

fX

\k

2

(ki-2k

2

)

2

1

1/2

ki — 2k

2

d

2

( kikl \

1 /

d

, , , ,

N

1/ 2

kik

2

ds

2

\ki-2k

2

J k\b* \ds

+

0(P

2

)}.

(4.4.6)

82


He re a l l o f the func t ions a re eva lua ted a t th e po in t wh ere s = SQ, w h i c h m a y

b e f o u n d f r o m t h e m i n i m u m c o n d i t i o n f o r f u n c t i o n

7(«)

4 * i A |

(4.4.7)

2* 2 - * i

A s a n e x a m p l e w e w i l l s t u d y a n e l l i p s o i d o f r e v o l u t i o n g e n e r a t e d b y t h e r o t a t i o n

of an e l l ipse wi th s emi -axes

O,Q

a n d 6Q a r o u n d a x is 6Q ( see F ig ur e 4 .4 ) .

F i g u r e 4 . 4 : A n o b l a t e (S < 1 ) and a p ro la t e (S > 1) e l l ipso id of rev ol ut i on .

A s a c h a r a c t e r i s t i c s i z e w e t a k e R = OQ . T h e n

*

2

= (s in

2

6 + S

2

c o s

2

6>)

1 / 2

6 = *

2

s in(

S =

b

0

a

0

,

*

1

= < J -

2

* ^ ,

d

- h —

(4.4.8)

w h e r e 9 i s an an g le be t we en th e ax i s o f r ev o lu t ion an d a n o rm al to the she l l

su r f ace .

F i r s t we wi l l cons ider an p ro la t e e l l ipso id (S > 1) u n d e r e x t e r n a l p r e s s u r e .

T h e w e a k e s t p a r a l l e l i s t h e e q u a t o r ( p o i n t B i n F i g u r e 4 . 4 ) . By u s i n g f o r m u l a e

(6) an d (2 ) , we o b t a i n

2 Eh

2

1+

V 3 ( l - i /

2

K ( 2 « P - l )

h_\

1 / 2

( J

2

- 1 ) ( 4 J

2

- 1 )

1

/

2

, < W y 3 ( l - i /

2

) ( 2 *

2

- l )

+

o (M

(4.4.9)

n

^m^)^yf\

c =

,-4

( M a

_

1/2)

1/ 2

Under in t e rna l p res sure , e l l ipso id buck ing may occur fo r 2<S

2

< 1. In th is

c a s e t h e c r i t i c a l v a l u e s o f p a r a m e t e r s q a n d n may be found by di f ferent for

m u l a e a c c o r d i n g t o t h e s i g n o f 28

—

1.

4.4. Buckling of Shells of Revolution Without Torsion 83

Indeed, function (7) attains its minimum at the weakest parallel. For 1 <

26 < V2 this minimum is attained at the equator and the cr i t ical pressure q

and wave number

n

may be found by the same formulae (9) as for external

pressu re. For 25 < 1 function (7) at ta in s its local m ax im um at the equ ato r

and minimum at

/ 3<J

2

01 = a rc sin W — — ,

6

2

= n

-

$i

(4.4.10)

(point C in Figure 4.4), i .e. the pits move away from the equator.

The critical values of q and n are equal to

lQEh

2

6

2

? = -

a

2

^ 3 ( 1 - 1 / 2 ) L

1 +

* V ' V 193(1-4*) V

/ 2

+ 0 ( M

,a

0

J \W6 ^12(1 - v

2

)

(4.4.11)

For

26

= 1 we have

f° ,

= 0. In this case, the cond ition

f°

s

> 0 is no t

satisfied and formula (1) for the buckling mode must be made more precise,

since it does not provide for the decrease of the deflection away from parallel

s = So- However, for estimating q and n, one may use formulae (9) and (11).

The problem of ellipsoid buckling under homogeneous pressure is also con

sidered in [30, 110, 175] .

We will now study the buckling of a shell of revolution under an axial tensile

force

(P

> 0). In this case, by virtue of (3.8)

T,° -

PR 2

TP -

PR 2

< ? ° - 0 (A

4

1

2)

J l -

2 7 r £ 2 ' •

2

~ 2nB

2

R

1

'

b

~ " ' ^

AU)

Assuming that

XT = P/(2nR)

we get

tl =

~bk~

2

'

t2 =

& k' 7 ( s ) = 2 ( < 7 < O

1 / 2

s i n

2

0, (4.4.13)

where the angle

9

is the same as in formula (8) (see Figure 4.4).

I f parameters E, v and h are constant (d = g — 1) then function ~/(s)

attains its minimum at one of the edges (we recall that the shell is convex).

Let the shell thickness now depend on

s.

Then function

j(s) =

—-^— sin

2

6

• "o .

and cases when this function attains its minimum inside the shell are possible.

84


We consider

an

ellipsoid

of

revolution

(8)

which

has its

minim um w all thickness

at

its

d iameter

h (a) = h

Q

[1 + a (<? -

T T / 2 )

2

]

, a > 0, (4.4.14)

For a >

1 the

d iameter

is the

weakest parallel

and the

critical values

of

P

and

n are

equal

to

2irEhl

>/3(l-^

a

)

h

0

1+ - 7^

a

0

1/ 2

(«-l)

1/ 2

b

2

o (3(1-./'))

1/ 4

+ 0(ft„)

(4.4.15)

</\2{\-u

2

)

/^oo^

1/2

The buckling mode described

by

formula

(1) is

shown

in

Figure

4.3.

Now

we

consider

the

combined loading

by an

axial force P

and a

homoge

neous pressure q

and

find

the

cases

in

which

the

buckling modes

are the

same

as

in

Figure 4.3.

For an

ellipsoid

of

revolution

of

con stan t thickn ess described

above

we

assume

P = 2ira

0

TP

0

\,

. <?o Po

t\ =

q = Ta

Q

1

q

0

X, q

0

= sgnq,

q

0

{2k

2

- ki)

P

0

ki

(4.4.16)

U

=

2fc

2

k

2

b

2

k

2

'

Then buckling occurs near

the

equator if

I: i _ S

2

< ^ <

(1

- <$

2

) (l - 4<5

2

) ,

<7o

II: ( I _ * 2 ) ( I _ 4 J 2 ) < H ^ < i _2*2,

9o

+

b

2

k

2

•

go

1o

- 1 .

(4.4.17)

The domains determined

by

these inequalities

are

shown in Figure

4.5.

D ue to formulae

(1),

(2) and

(3.3) the

critical values of P

and

q

may

be

determined from the relation

- ^ +

9

( 2 *

2

- l ) =

2Eh

2

ira

n

v i- K

i +

'h((S

2

- 1) go + 2 Pp) ((4*

2

- 1) (S

2

- 1) go - 2 Pp)

v

/ 3 ( l - z /

2

) a o ( ( 2 J

2

- l ) g

0

+ 2 P o )

2

1/ 2

+ (4.4.18)

+

0 M

4.4 . Buckling of Shells of Revolution

W i th o u t

Torsion

85

2P/q

0

Figure 4.5: Critical value domains for the buckling of an ellipsoid near an

equator .

which generalizes formula (9) for the case when

P

^ 0.

In all of the cases considered above the hoop stress-resultant

T®

was the

main stress determining buckling. Now we come to the case when

<iAri > t

2

k

2

,

(4.4.19)

and the principal compressive stress-resultant T° is th at on the meridian . In

this case, according to formula (3.10), the load parameter is equal to

A

0

= min(+)|^ (W)

1/2

}

(4.4.20)

Let the parameters E, v and h be constant (d = g = 1). Then, in the case

considered of combined loading or in cases when the shell is under either an

axial force P or a normal pressure q, function (20) att ai ns its m inim um at one

of the edges.

Let us now consider a shell of non-constant thickness, h = h(s). Let min

imum (20) be attained at

s

= so (*i < *o < ^2)- Then, due to (3.10) the

buckling mode is axisymmetric and is described by function (3.4) for n = 0.

The buckling mode is the set of axisymmetric concavities the depth of which

decreases away from parallel

s =

SQ

w = exp{i(

f

i-

1/2

Pot+l/2ae)}

(l +

O

(p

1

'

2

)) ,

A

=

Ao

+ ^ A i + O ^

2

) .

(4.4.21)

86


Ins tea d of (3.5) we can tak e function (3.5) in the form

dp

4

+gk%

tip

For the evaluation of the parameters in (21) we then obtain

1/4

f=

«_L_Zp

1

.

( 4 A 2 2 )

Po

(

—r

, Ai - 1/2 I /°

p

f°

s

- (f°

3

)

) ,

f°

p

- —

,

\ a J

v

' ti (4.4.23)

.n , d fd\ ,

n

, d

2

fd\ 1 d

2

fgkl

n

(if

fp

S

= 0

then

f°

s

=

7° ,

).

P a r ame te r

a

m ay be calculate d by formula

(2.18). Here all of the variables are evaluated at s =so- The critical load is

asymptotically double (see Remark 4.2) and the corresponding buckling modes

which are nearly identical for close values of load are shown in Figure 4.2.

As an example consider the same ellipsoid of revolution (14) with variable

thickness under an axial compressive force (P <0) . Assuming tha t P =

—2naoT\, we obta in

7 ( s ) =

2 ^ M ^ )

=

2^)

sm2

,

( 4 4 2 4 )

ho

ho

Buckling near the equator occurs for a > 1 and the asym ptotically double

critical load is equal to

p =

2nEh

2

3(1- )

ho\

1/2

a

2

{a-iyl

2

ao) b

2

( 3 ( i _

z /

2 ) )

1

+ 1 - ) %._\. L i / 4 + 0 ( M

(4.4.25)

We note t ha t the absolu te value of the critical force found above coincides w ith

the absolute value of the critical load in the case of tensile loading of the shell

(see (15)), but that the buckling modes are essentially different.

R e m a r k 4 . 4 .

The formulae for the critical load (see (1), (6), (9), (11), (18)

and (25)) all have the same structure. They consist of a main term which may

be found by the method of freezing coefficients and of small corrections of the

1 / 2

relative order hj . Due to (2.28) the next eigenvalue of the load parameter

may be obtained by increasing this correction by a factor of three.

4 . 5 B u c k l i n g o f S h e l ls o f R e v o l u t i o n U n d e r T o r

s ion

Let us now study the buckling of shells of revolution under homogeneous

pressure

q,

axial force P and torsional m om ent

M

>0. We introdu ce the



4.5. Buckling of Shells of Revolution Under Torsion

87

load param eter as

A

M

2nR

2

T'

Th en

due to (3.3) and (3.8) we get

q

0

P

0

q

0

(2k

2

- ki) Pokt

H

=

~i 7 T T ~ J *2 = ^To r

2k

2

b

2

k

2

'

1

2/5

2

t3

-~v>

Po =

PR

9o

b

2

k

2

'

2irR

3

q

(4.5.1)

(4.5.2)

M '

3U

M

Assuming that conditions (3.6.3)

are

satisfied,

we

find

by

formulae

(3.7)

A

0

= m i n , (

+

) 7 ( s ) =j(s

0

),

y{s)

=2

6

2

^ ( W )

1 / 2

g o

6

'

|

(

A

2

|

k

i

2k

2

+

\

A +

k

2

1/2'

- 1

r

0

=A-JA* +

* I

*=TT

+ £ »

, t a n *

0

= - r o ,

(4.5.3)

Figure

4.6: The

buckling m ode

of a

convex shell

of

revolution und er torsion .

In the case w hen the weakest p arallel so is far from a shell edges (see Figure

4.6)

the

buckling mode

is

given

by (3.4)

which

we can

write

in the

form

%+

a

-f-

+ n(r-^))+0(^)\, (4.5.4)

ce/2

cos

88 Chapter 4. Buckling Mo des L ocalized near Parallels

where £ = p~

1

'

2

(s~so) and a = a\ +ic. T he values of

po

and n were found in

Section 4.3 and

a

and Ai are evaluated for

m

= 0 by means of formulae (2.18)

and (2.20), in which the function / m us t be take n in th e form of (3.5).

As in (4.1), concavities (4) are elongated (their sizes are of order RhJ x

RhJ

) and they are inclined to the m eridia n by the angle \P

0

(tan\Po = —

r

o) -

Let us now study some particular cases. First, we assume that the torsional

mo men t

M

> 0 and

P = q

= 0 and that the parameters

E, v

and

h

are

con stant. We also assume tha t function

b(s)

is an even function in

s

— so.

Then, for «i < s

0

< s

2

we find from (3), (3.7), (2.18), and (2.20) that the load

pa ra m et er A and the coefficients in (4) are the following

A = Ao + p A i + 0 ( / i

2

) , A

0

= 7 ( s o ) ,

7 (s) = 2 6

2

v

/

M l , A: = 1 /2 ( /° . /

p

°

p

)

1 / 2

,

2 6

2

iff (4.5.5)

f

pp

= - ^ = ( 5 At?-2

* * , +

* » ) , t a n t f

0

=

m

Jpp

~ VW h

y

U

V *

2

'

7 ° , \

1 / 2

V2hbk

2

v

_ . „

7o~ '

n

- " 7 T T T T ' Po = - , , , , a i = 0.

J p p /

^ ( * i + * 2 ) fc + Ar

2

The calculations show that for an ellipsoid of revolution, function

j(s)

is

convex upwards and therefore i t can not atta in th e minim um within th e interval

(

s

i

i

s

2

) (see form ulae (7) below ). H owever, for some convex shells of revolu tion

this is possible. Indeed, by using the Codazzi-Gauss relations (1.1.3) we can

rewrite function f(s) in the form

7

( s ) = 2 ( - 6

3

6 " )

1 / 2

. (4.5.6)

Taking

b(s) =

1 -

as

2

- /3s

4

,

we find that for

a

> 0 and 2/? > a

2

function

"f(s)

at ta ins i ts minimum at

s

= 0.

Assuming that the parameters

E, v

and

h

are constant, we try to find in

which cases the ellipsoid under combined simu ltane ous loading of M, P and

q

has the weakest parallel at the equator. By using designations (1), (2), and

(4.8) we rewrite the conditions (7 > 0, 7" > 0) in the form

A

=

q

0

S

2

+ Ai > 0,

(2P

0

+

(S

2

-

1) go) + 4<J

2

>0,

-2g

0

Ai<J

2

(S

2

- 1) +

+ ( 2 P

0

+

(S

2

-

1) go) (2P o +

(S

2

- 1) (1 - 2<J

2

) go) + 4J

2

< 0.

(4.5.7)

4.6. Problems and Exercises

89

For buckling at the equator these three inequalities must be satisfied simulta

neously. For that, the critical values of dimensional parameters

M, P

and

q

satisfy relations

nqa

0

b

o + J (a

0

P + n q a

0

(b%

- a

2

0

)} +

b

2

a M

2

4 7ra2£

0

/ i o A <

2

( l + 0 ( / i ) ) .

(4.5.8)

Let us consider ine qua lities (7) in m ore det ail . It is clear th a t for go = 0

th e last of ine qua lities (7) is no t satisfied. Let now g

0

^ 0 and PQ = 0. T he

domain of parameters go and

S

in which buckling at the equator is possible is

shown by the cross-hatching in Figure 4.7.

?o

3

O

-3

m

A B

Figure 4.7: The domains of the critical values in the buckling of an ellipsoid

near an equator .

The transition of the point (go,

6)

through th e bound of the stabil i ty dom ain

I (</o > 0) and through the part

AC

of the bound of the domain II (go < 0)

means that the weakest parallel deviates away from the equator. In approach

ing the domain II boundary BC the critical values of q and M increase (as

in formula (4.9) as

28

2

—>

1) , and af ter the transit ion through this boundary

shell buckling does not occur (at the equator) since inequalities (3.6.3) are not

satisfied.


4.1 Consider the boundary value problem

fj,

4

w""+ 2fi

2

Xw"+ a

0

(x)w = 0, a

0

(x) = 2 - c o s z , n < 1

with the boundary conditions

w(±\)

= u / ( ± l ) = 0 .



C h a p t e r 5

N o n - h o m o g e n e o u s A x i a l

C o m p r e s s i o n o f C y l i n d r i c a l

She l l s

This Chapter will present a treatment of buckling problems for a cylindrical

shell of m ode rate length under a mem brane stress state an d a non-hom ogeneous

axial pressure w hich can be solved by sep aratin g the variables. T he edges

of the shell are assumed to be simply sup po rted . It is also assum ed th at

the determining functions (the axial compression, shell thickness and elastic

character ist ics of the mater ial) do not depend on the longitudinal coordinate

but may depend on the circular coordinate. Buckling modes localized in the

neighbourhood of the weakest generatrix are considered, and are developed by

means of the algorithm described in Section 4.2.

The buckling of a cylindrical shell under non-homogeneous axial pressure,

particular ly under a bending moment, has been considered in many previous

studies (see reviews [59, 61, 149]). In [68], the Bubnov-Galerkin method was

applied and the deflection was approximated by a double trigonometrical series.

In [151, 153] the method of asymptotic integration described below was used.

Long cylindrical shells subjected to a bending moment can experience flat

tening of the cross-section - this is called the D ub ja ga -K ar m an -B raz ie r effect

and it must be taken into account (see [12, 61]) in most analyses. The flat

tening effect on the critical load for moderately long simply supported shells

examined below is, however, not significant and will not be considered here.

91



92 Chapter 5. Non-homogeneous Axial Com pression of Cylindrical Shells

5 .1 Bu ck l ing M od es Loca li zed ne a r G en e r a t r i x

U n d e r t h e a s s u m p t i o n s m a d e a b o v e w e c a n w r i t e t h e s y s t e m o f b u c k l i n g

equa t ions in the fo rm ( see (2 .3 .8 ) , (4 .3 .1 ) )

d

2

w 3

2

$

Ox Ox ^ _

L 1

j

2

A / i * *x , 9

2

w

n

A

d

2

d

2

w h e r e d i m e n s i o n l e s s c o o r d i n a t e s x a n d y, s imi la r to those o f Sec t ion 3 .4 a re

i n t r o d u c e d ,

T? = -XTt

lt

T= Eohofi

2

, (5.1.2)

a n d t h e o t h e r n o t a t i o n (fi, d, g and £2) i s the s ame as in (4 .3 .3 ) . Here , t\, k

2

,

d

a n d

g

d e p e n d o n

y

an d a re as s um ed to be in f in i te ly d i f f e ren t i ab le .

The she l l occup ies the r eg ion 0 s j x ^ / = L/R, y\ ^ y ^ y

2

, w h e r e L is

t h e s h e l l l e n g t h , a n d

R

i s t h e c h a r a c t e r i s t i c r a d i u s , t a k e n a s t h e l e n g t h u n i t .

T h e s i m p l e s u p p o r t c o n d i t i o n s ( 3 .2 . 4 ) a r e i n t r o d u c e d a t t h e c u r v i l i n e a r e d g e s

x — 0,1.

T h e b o u n d a r y c o n d i t i o n s a t e d g e s

y

=

yk

m a y b e a r b i t r a r y . F o r a

s h e l l w h i c h i s c l o se d in t h e c i r c u m f e r e n t i a l d i r e c t i o n , t h e p e r i o d i c i t y c o n d i t i o n s

b y

y

mus t be ful f i l led .

S e p a r a t i n g t h e v a r i a b l e s a s

w (x, y) = w

m

(y) s in - ^ — , $ (x, y) =

$

m

(j/)

s i n - ^ — ,

^ ** (5 .1 .3 )

miv fi

Pm =

—

- ,

— , m = 1 ,2 , . . . ,

w e g e t t h e s y s t e m o f o r d i n a r y d i ff e r en t ia l e q u a t i o n s f or u n k n o w n f u n c t i o n s

w

m

{y ) a n d &

m

(y)

A

m

(dA

m

w

m

) - Xtip^Wm + k

2

p

2

n

$

m

= 0,

d

2

(5-1-4)

A

m

(g -

1

A

m

$

m

) - k

2

p

m

w

m

= 0 ,

A

m

=p

2

n

+ H ' / * )

2

^ -

As in Sec t ion 4 .2 , we seek a so lu t ion o f sy s te m (4) in the fo rm (no te th a t

t h e s u b s c r i p t

m

in

w

m

a n d $

m

is o m i t t e d )

w(y,fi) = iy» e x p j j ^

_ 1 / 2

g o C +

l /

2 a

C

2

) } ,

OO

« , . = ^ / / / ^ ( C ) , C = »-

1/2

(y-yo), (5.1.5)

fc=0

A = A

0

+ ^ 1

H

5.1. Buckling Mo des L ocalized near Generatr ix

93

W e seek function $ in the same form as (5). For 3 a > 0 solution (5)

approaches zero away from the weakest generatrix y — y

0

. To find function

$ instead of the boundary conditions on the edges

y = yk

or the periodicity

conditions the assumptions

S>

o > 0 and 2/1 < 2/o < 2/2 should be used.

For the evaluation of Ao we have

A

0

= m i n ,

i y > m

{ / } = / ( g o , 2/o),

d(p

2

m

+ q

2

)

4

+ 9k

2

2

p:

2

i „ 2 ^ 4 , „ J . 2 ^ 4 (5.1.6)

/

trfM + q*)

^2

Here the minimum is evaluated for 0 ^

q

2

< oo,

y\

^ j / i j 2/2,

rn =

1,2,...

Let the m inim um of (6) be atta ine d for g = go, y = yo

a n

d m = m

0

. Fir st, we

assume tha t m (and hence p

m

) is fixed and also th a t k

2

> 0-

Now we rewrite (6) to the form

/ (

, „

) =

M ( ,

+

I ) „

=

M^ (;)-,

7

=

^ .

(

„

7 )

As in Section 4.2, there are possible two cases here: go = 0 (case

A)

and

q

0

^ 0 (case

B).

The minimum of / is at tained for

z =

1 in which case it is equ al to

A

0

= m i n {7(2/)} = 7(2/0) (5.1.8)

y

or for go = 0 and it then is equal to

A'

0

= m i n { 7 ' } = 7 ' U / o ) , y

=

d

^ + / ^ . (5.1.9)

Gase >1 occurs if Ag < Ao, otherwise if Ao < A

0

, then the case B occurs.

If Ao = A'

0

, then we have a special case which does not fit the scheme of

Section 4.2 and which will be considered in Section 5.2. We will consider all of

these cases, since each of them may occur under specific conditions depending

on the relationship of the various parameters to each other.

After the weakest generatrix 2/0 (or 2/0) is found, it is useful to choose the

values of E

0

, v

0

, h

0}

R and t\ such, that

d

= g = k

2

= l,

t

1 =

= 2

for

y = y

0

{y = y'

0

).

(5.1.10)

For that, due to (8) and (9)

Ao = l,

K = \{p

2

m+p-

m

2

)-

(5-1.11)

94 Chap ter 5. Non-ho mog eneous Axial Com pression of Cylindrical Shells

Hence , case A takes place i f p

m

> 1 , an d case B t akes p lace i f p

m

< 1. If th e

shel l i s sh or t an d

L < n n R = n

2*>2 \ l / 4

R

2

h

1 2 ( 1 - i /

2

)

(5 .1 .12)

th en du e to (3) p i > 1 for

m— \

an d case

A

t a k e s p l a c e .

Le t ine qu a l i ty (12) be fu lf il led , th en d ue to (4 .2 .28) , for the load pa ra m et e r

e i g e n v a l u e s a n d b u c k l i n g m o d e s w e h a v e

w

(m,p)

<y-y '

a

fl(^)

H

p

^(y-j/

0

)j+O(^

s i n

•

1/2

PmX

^ = \(p

2

m+ Pm') + H (p + \ ) U

( 1 -

Pm") j £ \ + Ofr

w h e r e m = 1, 2 , . . . , p = 0, 1, . . . an d 7 ' is giv en by (9 ) ,

1/2

(5 .1 .13)

c =

1 d

2

j

2 ( 1 - P m

4

) r f y ( ,

2

(5 .1 .14)

N o t e t h a t t h e e i g e n v a l u e s A (

m , p

) a r e s imple .

P a r a m e t e r

\(

m

'

p

>

a t t a i n s i t s m i n i m u m fo r

m =

1,

p —

0 . F o r t h a t ,

HQ =

1,

t h e b u c k l i n g m o d e i s a s i n g l e p i t , a n d A '

1

'

0

) g ives a cr i t ica l load c lose to (3 .4 .7)

for a shor t c i rcular cyl indr ical shel l .

Now we pas s to case

B.

L e t

p

m

< 1 an d as s um e th a t r e l a t ion s (10) a r e

va l id . T h en Ao = 1 an d con d i t io n z = 1 gives

90 =Pm -P

2

m

-

(5 .1 .15)

If we diff ere ntia te (7) , we f ind

f°

=

Jqq

J

qy

f°

--

yy

16(1 —

Pm

= 0,

fl +

dyo

Pm)

(-

\ayo

( W 5 )

(5 .1 .16)

N o w w e c a n d e t e r m i n e t h e b u c k l i n g m o d e a n d a s y m p t o t i c a l l y d o u b l e c r i ti c a l

5.2. Reconstruction of the Asymptotic Expansions

95

value of parameter

A

by using the formulae of Section 4.2

wW(x, y) =

e

- ^ - ^ " )

2

/ (2 . ) L

s

fio(y-y

0

)

+ e

. \

+ 0

(

^ i /

2 )

sin

A = l + ^

4 ( 1

p m

P m )

/ °

y

y

/ 2

+ 0 ( p

2

) , (5.1.17)

/

P

f°

\

1 / 2

1 6 ( l -

P r o

)

Here, as in (4.2.30), the p hases

0\

and

9

2

are not found. The buckling mode

is a system of pits, whose depth decreases away from the weakest generatrix.

5.2 R e c o n s t r u c t i o n of t h e A s y m p t o t i c E x p a n

sions

Consider the case when m varies. Then due to (1.3), p

m

takes on a sequence

of discrete values. We will examine the dependence of

A

on

p

m

.

Let

p

m

< 1,

the n d ue to (1.11) and (1.17) Ao = 1 and does no t dep end on p

m

, and the term

of order

p.

decreases as

p

m

increases and vanishes for

p

m

= 1.

Nevertheless, for

p

m

~ 1, (1.17) are not applicable, since

f°

= 0 desp ite

assumption (4.2.11). If p

m

= 1 then c = oo and the te rm p A2, which is

not written out in (1.17) but for one particular case is given in (5.3.3), also

approaches infinity.

For

p

m

> 1, pa ram ete r A shows a stronger dep endence on

p

m

.

Due to

(1.11) the value of Ao increases with p

m

. For p

m

~ 1 (1.13) are not applic able

for the same reason as (1.17) (see also (3.6)).

From the above development i t follows tha t p aram eter A reaches i ts m ini

mum for

p

m

~ 1, hence the case whe n

p

m

~ 1 deserves special cons ideratio n.

Let us assume that

p

m

= l+ep'

m

,

A = l + e

2

A',

y

-

yo

=

£

ri

h

2 x i / 6 (5-2.1)

. 1 2 ( 1 - ^ )

and seek a solution of system (1.4) in the form

^ V w / % ) ,

$ = ^e

f c

$ (

f c

) ( r

?

). (5.2.2)

=

fc=o

k=o

96

Chapter 5. Non-hom ogeneous Axial Com pression of Cylindrical Shells

In the zeroth approximation after substituting (2) into (1.4) we obtain the

fourth order equation

A

d

4

w° .

o

,

x

d

2

w° dw°

+ (a7/

2

+ a

3

»7 + 4 p £ - 2 A ' ) w ° = 0, (5.2.3)

where coefficients a,- and a depend on the first and second derivatives of the

functions

t\, ki, d

and

g

for

y

=

yo

and do not depend on

s.

In the case when

the first derivatives of these functions are equal to zero for y = yo, we have

arj = 0 and equation (3) is simplified

d

4

= £ ?

> o-

(5.2.4)

Here 7 is given by (1.7).

We seek the values of A', for which the non-trivial solutions of equations

(3) or ( 4), con verg ing t o zero as 77 —> ±0 0, exist . These equations are not

integrable in the known functions, however, the fact that the coefficients of

these equations do not depend on p, provides the opportunity to make the

following conclusions.

For p

m

~ 1 the critica l load differs from Ao = 1 by a qua nt ity of order p

A

l

3

(for

p

m

< 1 th is difference ha s orde r

p.

due t o (1.17)). For

p

m

~ 1, th e inde x

of variation of the solution in the y direction is equal to t = 1/3 (see (1)), for

p

m

< 1 it is equ al to

t =

1/2, and for

p

m

> 1 it is equ al to

t

= 1/4. In other

words, in the neighbourhood of

p

m

—

1 buckling m ode reconstruction takes

place.

Let us consider equation (4). Applying Fourier transform

0 0

ti;

0

(ij) = - T L

f w

F

{u)e

iu

"idLj

(5.2.5)

V

2

7T J

— 0 0

we come to a second order equation for function w

F

(u>).

a ^ f + [ 2 A ' - 4 ( ^ +

W

2

)

2

] ^ = 0. (5.2.6)

We transform this equation to a form with only two para m eters A and

k,

^

+

(A-(

x

*+ kf)w

F

= 0, (5.2.7)

5.2. R econstruction of the Asymptotic Expansions

97

where

=

(?)'"-

- 4 ; f

•

A

4G

)

2 / 3

(5.2.8)

For each A; (—oo < k < oo) there exists the countable set A,(&) of values

of A, for which there exist non-trivial solutions of equation (7) converging to

zero as

x

—>• ±oo. Funct ions

A

0

(k )

an d Ai(Ar) are given in Fig ure 5. 1.

A

6

5

4,-

" ^ . . - ' 3

X ^ 2

1

i i i

A i

t

/

/

/

y

t

t

t

t >

/

'

A

0

'

- 3 - 2 - 1 0 1 2 k

Figure 5.1: Functions Ao(fc) and Ai(Ar).

By virtue of (1) and (8) we get

' a \ 2 / 3

1/3

\W = l + 2^^y

/3

A

i

(k) + 0(u.*),

k

=

H-Wfrm - 1)

(5.2.9)

Fu nc tion Ao(Ar) att ai ns its m ini m um Ao = 0, 905 for

k

= —0.44. Hence, the

critical load is minimal for

P m

= 1 - 0.44 /z

2

/

3

( ^ )

a \ i / 3

4 /

and is equal to

A

m i n

=

1

+ 1.81 (J)

fl*

+

0{f).

(5.2.10)

(5.2.11)

For other

p

m

< 1 the critic al load differs from this value by an am ou nt of order

li (see (1.17)).

It should be noted that for

k

< 0, the curves of Ao and Ai approach each

oth er as |A;| increases (see Figu re 5.1). T hi s m ean s th a t the eigenva lues are

asymptotically double for p

m

< 1.

98

Chapter 5. Non-hom ogeneous Axial Compression of Cylindrical Shells

5 .3 A xia l C om pre s s ion an d B en d in g o f Cyl in

dr ical Shel l

Let us consider a circular cylindrical shell with constant parameters h, E

and

v

under axial force

P

and bending moment

M\

(see Figure 5.2). The curvi

linear shell edges are simply supported and the initial stress state is assumed

to be membrane and defined by the stress

T°

2nR

+

M\

cos

y

nR

2

'

(5.3.1)

M i

Figure 5.2: The non-homogeneous axial compression of a cylindrical shell.

For Mi < 0, PR + 2Mi < 0, the generatrix j /

0

= 0 is the we akest one . Let

us assume that in (1.2)

2{

a

+ cosy) PR

ti = — —: -, a = „ , , > - 1 .

a + l

2 Mi

(5.3.2)

Assuming that

d = g

= &2 = 1

m

th e formulae of Section 5.1 we now

find that for

p

m

<

1 the buck ling mo de is given by expression (1.17). In th is

case due to symmetry it is possible to define angles #i and 02, namely (?i = 0,

6-2

= 7r/2, that correspond to the even and odd buckling modes by y. The

5.3. Axial Com pression and Bending of Cylindrical S hell

99

method used here gives the same critical load for these modes [151]

A

1 + 2 M

j ' / ( Q ) ( i - ^ y

/ 2

+

+ ^

2p

m

( 1 - P m ) < {

v

( 0 ) < y ( 0 ) ( - 6 p g , + 2 1 p

r o

- 1 4 )

4Pm<i'(0) 6 4 p

m

( l - p

m

)

+

Mi =

<i'(0)

=

+ 0 ( / *

3

) ,

2nR

2

XT

(5.3.3)

2 M a

1 + a '

2

* {

v

( o ) =

2

T= Ehfi

2

,

1 + a '

1 + a

In [151] to analyze the differences in critical load corresponding to even and

odd buckling modes, system of equations (1.4) has been integrated numerically.

The following boundary conditions correspond to the even mode

w' = w'"

= $ ' = $ " ' = 0 a t

y =

0,

n,

and to the odd mode

w = w"

= $ = $" = 0 at

y =

0,

n.

(5.3.4)

(5.3.5)

For p roblems wi th parameter s — = 0 .01 , / = — = 2 , a = l and

v

= 0.3,

R H

the values of A are given in Table 5.1 for various values of p

m

.

Table 5.1: Critical buckling loads and wave numbers for a cylindrical shell

under combined loading

/

Pm

0.173

0.518

0.691

0.864

0.951

II

^even

1.180667

1.076492

1.051833

1.024585

1.010245

III

A

odd

1.180671

1.076494

1.051847

1.031065

1.031077

IV

A

a

1.1706

1.0734

1.0486

1.0257

1.0177

In columns

II

and 7 7 / the values of A

e v e n

and A

odd

corresponding to the

even and odd buckling modes found by integrating of system (1.4) are given.

The values of A

a

found by formula (3) are presented in column

IV .

It follows

from Table 5.1 that for p

m

< 1 the v alues of A

even

and A

o dd

are rather close,

hence the treatment of A

a

, as if it is asy m ptotic ally doub le is justified. T he

values A

even

and A

o d d

diverge as

p

m

appro aches unity. In each case the sm aller

critical load corresponds to the even buckling mode.

Now let

p

m

> 1. T he buck ling m od e is given by expression (1.13) for

p

= 0,

and para m eter A is equal to

A =

\ (P

2

m

+P

m

2

)+Vp

m

3

(-\t'{(0)

(P

m

~

I ) ) ' + P

2

3 *i

v

(o )0 4 - i )

< foi

,

A

A \ _

n

( 9 5 p g , - 6 4 p 6

l

+ 1 7 2p 2, + 6 4 p 2 , + 4 5 )

(5.3.6)

Here

P

and

M\

are dete rm ined by the sam e formulae (3). If

m

= 1 for a shor t

shell (see (1.12)) the buckling mode is in the form of a single pit. By virtue of

(1.13)

u>'

1,0) = e X P

{ - ^ }

S i n

v (

1 + O ( / i l / 2 )

) <

5

-

3

-

7

)

where

c = [ 2 ( i -

P

r

4

) ( i

+

) ]

1 / 2

,

pi

=

^ -

(

5

-

3

-

8

)

For the next (odd according to y) buckling mode

U J '

1 , 1

)

~ j / u /

1

'

0

) corre

sponds to

A =

\ (P

m

+

P -

2

) + 3 p p -

3

(-\t»(0) (P

8

m

~l))

12

+ 0

(/ ? ). (5.3.9)

For p

m

~ 1 we can use th e results of Section 5.2. We repre sen t the critical

load in the form

A = l + 2 ^ / 3

A

.

( / : )

( 4

( 1 + a )

) -

2

/

3

+

o (^

)

k =

^ - 2 / 3 4 1 / 3 ^ _

1 } )

where functions Ao(fc) and A\(k) are shown in Figu re 5.1.

Consider an example from

[152].

Let — =

0.001,

I

= 2,

a

= 0 and

v

= 0.3.

H

In Figure 5.3 parameters A

even

and A

o dd

obtained numerically for even and

odd buckling modes respectively are shown.

For p

m

< 1, the values A

even

and A

odd

approach each other . A comparison

of num erical and asy m pto tic results is presented in Ta ble 5.2. For even and

5.3. Axial Compression and Bending of Cylindrical Shell

101

Figure

5.3:

Load parameters

for

even

and odd

buckling modes

of a

cylindrical

shell under non-homogeneous axial compression.

Table

5.2:

Critical buckling loads

and

wave numbers

for a

cylindrical shell

under combined loading

Pm

1

0.50

0.90

0.95

1.00

1.05

1.15

1.30

\even

2

1.0516

1.0138

1.0055

1.0041

1.0108

1.0480

1.1523

A

odd

3

1.0519

1.0165

1.0111

1.0143

1.0240

1.0660

1.1756

\even

4

1.0348

1.0126

1.0078

CO

1.0168

1.0495

1.1523

A

o d d

5

1.0348

1.0126

1.0078

CO

1.0203

1.0640

1.1726

\

even

6

-

1.0093

1.0046

1.0034

1.0113

-

-

A

o d d

7

-

1.0104

1.0093

1.0136

1.0222

-

-

odd buckling modes

the

numerical values

of A are

given

in

co lumns

2 and 3.

Columns

4 and 5

give

the

values

for

p

m

<

1 found

by

formula

(4)

(they

are the

same for even and odd modes since the eigenvalues are asymptotically double)

and

the

values

for

p

m

> 1

obta ined

by

formulae

(6) and (9). For

p

m

tn

1

formulae

(4), (6) and (9) are not

applicable,

and

t h a t

is why in the

co lumns

6

and

7 the

values

of

A found

by

formula

(10) are

given.

The results

in

Table

5.2

i l lustrate

the

fact that

the

asymptot ic formulae

derived

for

p

m

< 1,

p

m

m

1 and

p

m

> 1,

cover

all

possible values

of

p

m

and

so the

areas

of

their applicability intersect.

The

asymptotically double

eigenvalues

for

p

m

< 1

transforms

for

p

m

> 1 to

eigenvalues which differ

by

order \i (see (6) and (9)).

http://even/

http://even/

http://even/

http://even/

102 Chapter 5. Non-hom ogeneous Axial Com pression of Cylindrical Shells

5 .4 T h e In fluence o f In t e rn a l P re s s u re

It was noted in Section 3.5 that for a shell under axial compression, the

tensile stress-resultant T%, gen erat ed by the in tern al press ure d oes no t affect

the critical load and also that the buckling mode is axisymmetric. In the cases

of non-homogeneous compression considered in this Section the buckling mode

is no t axially sy m m etr ic. The refore, a mo re significant effect of the ten sile

stress T

2

° on the value of the critical load should be expec ted. T his question

will be considered below (see also [97]).

Let the initial membrane stress state be defined by stress-resultants

T®(y)

and

T$ (y).

W e ad d to the left side of th e first eq ua tion (1.4) th e following

term

- * < - ' * > • £ ( < > < » >

^ ) .

<

5

- « >

where t

2

is related to T by an expression similar to (1.2). We assume that

t

2

< 0 .

Instead of (1.6), in this case function / has the form

1

(hp

m

+t

2

ql)(pl +

q

^-

{

-->

The value of A

0

m ay be found by the m inim izatio n of / by q, m and y. As in

Section 5.1 we assume that relations (1.10) apply at the weakest generatrix.

For

p

m

;> 1, the m ini m um of A

0

= m i n / is reached for

q

0

—

0 and it is

equal to A

0

= i ( p ^ + p -

2

) .

For p

m

< 1 the m in im um is reached for go ^ 0, an d Ao > 1. To verify thi s,

let us represent / in the form

f=hh,

2f

1=

z

+

K

/

2

= ( l ¥ r > * = * £ + £ . (5.4.3)

z

\ Pm / Pm

We have /1 ^ 1, f

2

> 1, an d /1 = 1 for gi = s/p

m

- P

m

,

a n d

h - 1 for g

2

= 0.

T he ine qu alit y Ao > 1 is fulfilled since

q\

^

q

2

.

For different values of

t

2

th e dep end enc e of Ao on

p

m

is shown in Figure

5.4.

Now consider a shell of m od era te leng th. By virtu e of (1.3) we m ay assum e,

that there exists a number

m

such that

p

m

~ 1. Unlike the case considered in

Section 5.2, the case where

p

m

=

1 and g

0

= 0 is not special here for

t

2

<

0.

Inde ed, from (2) we find th at

f

o

__

f

o , 0 _

fl f

0 _ j£_ (d + gk\\

/ « - *2.

hy ~

u

>

hy ~ dy

2

0

\ h ) '

[

'

Let £2(0) > 0, then for the construction of the critical load and buckling

mode, the results of Sections 5.1 and 5.2 may be applied and we can write the

formulae for the load parameter A. Let buckling occur with

m

half-waves at

the generatrix (see (1.3)). Then for

p

m

< 1, due to formu la (1.17) we find

A = l + 2/W

— k%(0)) +0(n

2

), (5.5.2)

V

Pm J

For p

m

~ 1 accord ing to (2.9) we have

k=

p

"

(5.5.3)

p2 /3

\k%(0),

where Ao(fc)

is

ob ta ined f rom Figu re

5 . 1 . A t

las t ,

for p

m

> 1,

for mu la (1.13)

gives

A

= \{p

m

+ P

m

2

) + »

(2-\

Pm

2

-

Pm

6

)

* £ ( 0 ) )

1 / 2

+ O {n

2

).

(5.5.4)

We consider

for

e x a m p l e ,

a

shell with

a

cross-sect ion

i n t h e

form

of an

el l ipse with semi-axes

a an d b (a > b).

T h e n ,

in

( 2 ) - (4 )

w e

should assu me th a t

b

2

a

2

h

2

b

2

- a

2

, / ,

N

1/2 (5 .5.5 )

Pm =

</12(l -

^

2

)

I U ,

By virtue of (1.2) the stress-resultant, T°, is related to

A

by th e expression

=

Eh

2

aX

1

^ V

12

1

-^

2

)

To evaluate the critical value Tf we should minimize

A

from formulae (2)-(4)

by m .

5 .6 C yl in dr ica l She ll w i th C u rv a tu re o f V ar i

able Sign

For

&2

= 0 in the case when p

m

^ 1, the solu tion co nstru cte d in Section 5.1

is not applicable. At the same time, the weakest generatrix is where fc

2

= 0 .

It is assumed below that

M O) = 0 , M O ) =

fe

20 > 0. (5.6.1)



5.6. Cylindrical Shell with Curvature of Variable Sign

105

These conditions are fulfilled for a shell which has a generatrix with a point

of inflection as shown in Figure 5.5. The weakest generatrix

yo

= 0 is marked

with point

A.

Figure 5.5: The generatrix of a cylindrical shell with the curvature of variable

sign.

We now return to system of equations (1.4) and assume that

d

—

g = t\ =

1

for

y

= 0. As before, we seek solutions that exponentially decay as |y| increases.

Assuming that k'

20

~ 1, we find the minimum value of parameter A, for which

such a solution exists.

Let us consider a thin shell of moderate length. Then in (1.3) / ~ 1 and

parameter p

m

has values which are very close to each other. We minimize A

by

m,

assuming that parameter

p

m

is continuous.

We assume in system (1.4) that

y = H

t

a-

l

r],

p

m

=n

a

a,

A = //A', a ,A '~l (5.6.2)

and equating the orders of the main terms we find that

i = /? = 2/3,

a

=1/3 . (5.6.3)

Now, the solution of system (1.4) we seek in the form

oo

oo

k-0 k=0

(5.6.4)

A

= eAo + e

3

A + ---,

e

= p

2

'

3

,

and we require that all functions

Wk{rj), k(v)

converge to zero as

r)

—>• ±oo.

In the zeroth approximation we have the system of equations

( W / T - I )

2

- A ) ™

0

+ ^ O = 0,

&

3

( ^ 2 - l )

2

*

0

- ^ o = 0, (5.6.5)

where

106

Chapter 5. Non-hom ogeneous Axial Com pression of Cylindrical Shells

The next approximations for

Wk

an d $fc satisfy n on-h om oge neo us equa

tions, the left sides of which coincide with the left sides of equations (5), and

the right sides of which include functions which depend on the earlier approx

imat ions .

As in Section 4.2, due to the alternation of parity of functions Wk(i]), the

solutions of equations in the even approximations converge to zero as

77 —>

±00

always exist. T he values Ai, A2 and . . . in (4) are eva luate d from the existence

condition for functions w^, W4, and . . . which decrease to zero as 77 —» ±00 .

Consider the zeroth approximation in more detail. As in Section 5.2, after

applying the Fourier transform (2.5) we arrive at the second order system of

equations

= & V + 1 )

2

^ ,

d&

= b

3

{u

2

+ l)

2

w

F

-A w

F

.

(5.6.7)

By means of numerical integration we find the first eigenvalue, Ao(fe), for

which a non-trivial solution of system (7) converging to zero for u> —>• ±0 0

exists. We obtain from (6)

A

0

= ( ^

0

)

2

/

3

6 -

1

A

0

( 6 ) .

(5.6.8)

Figure 5.6 shows a plot of function b

1

Ao(6). It corresponds to the buckling

mode which is even by w and odd by <J>.

1.5

b

Figure 5.6: The function

b

1

Ao(b).

The minimum value of A by p

m

is reached for

6 = 0.466, Pm = A«

1/3

(*2o)

1/3

&

(5.6.9)

5.6. Cylindrica l Shell with Curvature of Variable Sign

107

and it is equal to

A = 1.577 (nk'

20

)

2

/

3

+ O{u.

2

).

(5.6.10)

For a discrete change in the parameter p

m

, buck ling occurs for one value

of p

m

de ter m ine d by (1.3) which is closest to value (9).

Let us give the expression for the characteristic deformation half-wave

length L

x

in the direction of the x axis

L x

= E £ i * =

7 r

i ? 6 - V

2 / 3

( ^ o ) ~

1 / 3

(5-6.11)

Pm

or in dimensional variables

If the shell length is

L

<

L

x

, then it should be assumed that

m = l ,

Pm

=

Pl

= ——.

(5.6.13)

As before, the critical load is defined by formula (8), where Ao(6) is taken

from Fig ure 5.6, and for 6 S> 1

A

0

(6) = 6

3

+ V 2 + O ( 6 -

3

). (5.6.14)

The curve

b

2

+ b~

1

y/2

is shown in Figure 5.6 as a dotted line.

To derive formula (14) in system (7) we make the substitution u> =

b~

3

'

2

u>i.

Then in the zeroth approximation we come to the Weber equation [69]

d

2

w

F

du>

2

+ ( A - 6

3

- 2 w

2

) w ^ = 0. (5.6.15)

For A — 6

3

= y/2 (1 + 2n ) , n = 0 ,1 , 2 . . . this equation has solutions which

converge to zero as wi —>• ±oo. Formula (14) is obtained for

n =

0.

Since in this problem it is inconvenient to use the radius of the curvature

of the directrix as the characteristic shell size

R,

we can write the form ula

for the critical compressive stress-resultant

T°

in the dimensional variables.

Accounting for (1.2) the main term of (4) gives

If

=

-Eh

(5.6.16)

b

3

= b

3

(

h

2

d f

V 1 2 ( l - * /

2

) d s V

n

3

hm

3

L

3

^/I2(l

- i /

2

)

"

(~)

1

s

V R 2 / 0 .

-'Aoib)

- 1

108 Chapter 5. Non-homogeneous Axial Compression of Cylindrical Shells

The problem considered here is, to some degree, similar to the buckling

problem

of a

simply supported plate under compression

in one

direction.

The

buckling

of a

short plate occurs with

one

half-wave

in the

longitudinal direction

while the buckling of a long plate displays several half-waves [16].

5.7 Problems and Exercises

5.1 Examine the buckling of a non-circular cylindrical shell under homoge

neous axial pressure, when k^y) = y

2n

+ 1. Find the critical pressure.

5.2 Find the critical compressive stress-resultant T® for a short cylindrical

shell with curvature

of

variable sign where

6 =

d i > l .

A n s w e r

EhW

( y ^

1

1 2 ( l - t /

2

) L

2

I

+

6?

where the m ain term coincides with the Euler formula (3.4.8), while the second

term in the parentheses takes into account the bending of the directrix.

5.3

Obtain critical load under axial compression

of the

non-circular cylin

drical shell of m oderate length and constant thickness

h.

The shell cross-section

is an ellipse with the semi-axes do and

&o

(ao <

bo)-

H i n t

Find

the

ellipse curvature

and

apply relation (2.11).

A n s w e r

r

1 =

,.

Eh

l

a

„,fi

+

i . 8 i ^

2

-

a 2

)

N 2 / 3

&V12(1-^

2

) V

' V 4a

2

h

2

a

2

1 2 ( l - i /

2

) 6

4

'

To calculate the axial force multiply the stress-resultant T\ by the length

of

the

ellipse

arc.

5.4 Obtain the critical load for the circular simply supported cylindrical

shell of radius

R,

moderate length

L

~

R

and constant thickness

h

under

bending moment Mi .

H i n t Use

relation (2.11).

A n s w e r

M

1

= IvtfEhi? (l + 0 . 7 2 / /

3

) , / =

1 2 ( 1

^

2 ) J

^

2

.

109

5.5 Obtain the critical load for the simply supported short

(L < ir/iR)

circular cylindrical shell of radius Rand constant thickness h under bendin g

mo men t M i

.

H i n t

Use relation (1.13).

A n s w e r

2*R*EV (h(P

2

i + Pi*) + /i O f e

1

)

V 2

) .

h

2

_

fnvR

12(1

- u

2

)R ? '

Pl =

~T~'

5.6 Obtain the critical load for the cylindrical panel with the curvature

of variable sign of the moderate length L

~

{a, 6}, and constant thickness h

under axial compression. The panel generatrix has a sine form Y =

a sin(X/b),

—7T&/2 < X <7r6/2. The panel is simply suppo rted at the curvilinear edges

x

=0,

L

and c lamped at the re ctilinear edges X =±7:6/2.

H i n t Use relation (6.10).

A n s w e r

T

1 =

1.577 (

12(1

_;

)

2

;

(a2 + 6 2 )

)

2 / 3

.

.A

C h a p t e r 6


a t a Po in t

In th i s Ch ap te r w e wi l l con s ider th e buck l ing of a th in she l l for which th e

d e t e r m i n i n g f u n c t i o n s ( s e e S e c t i o n s 4 . 1 a n d 6 . 1 ) a r e v a r i a b l e i n b o t h d i r e c t i o n s

u n d e r a m e m b r a n e s t r e s s s t a t e . W e s ee k t h e b u c k l i n g m o d e s w h i c h a r e l o c a l iz e d

i n t h e n e i g h b o u r h o o d o f w e a k p o i n t s w h i c h a r e a s s u m e d t o b e l o c a t e d fa r f r o m

the edges of the shel l .

U n d e r t h e s e a s s u m p t i o n s w e w i l l c o n s t r u c t t h e b u c k l i n g m o d e s o f a c o n

vex she l l [101 , 160 , 163] an d a cy l in dr i ca l she l l un de r no n-h om og en eo us ax ia l

c o m p r e s s i o n [164].

6 . 1 L o c a l B u c k l i n g o f C o n v e x S h e l ls

Co n s i d e r m e m b r a n e s t r e s s s t a t e b u c k l i n g t h a t is d e fi n ed b y t h e i n i t i a l s t r e s s -

r e s u l t a n t s T° a n d S°. W e a s s u m e t h a t t h e d e t e r m i n i n g f u n c t i o n s ( s tr e s s-

r e s u l t a n t s 7 ] ° a n d S°, m et r i c coef f ic ients A\ a n d A2, c u r v a t u r e s ki and &2,

t h i c k n e s s h a n d t h e e l a s t i c c h a r a c t e r i s t i c s o f t h e m a t e r i a l E a n d v) d e p e n d

on the curv i l inear coord ina tes <* i and a? an d a re in f in i t e ly d i f f e ren t i ab le w i th

r e s p e c t t o t h e m . H e r e t h e v a l u e s A, B, a a n d (3 a r e d e n o t e d b y A\, A2, a\

a n d a 2 -

T h e b u c k l i n g e q u a t i o n s w e t a k e i n t h e f o r m o f ( 2 . 3 . 8 ) , ( 4 . 3 . 1 )

p

2

A ( d A w ) + A A

t

u ) - A

f c

$ = 0,

^ A f o r ^ A f c J + AfctD = 0 ,

111

112

Chapter 6. Buckling Mod es Localized at a Point

AiA

2

/

-r'dai \Ai

3a,-

1 Y ^ 9 (Ajkj dw

where

. 1 v-^ d (' At dw

AkW

= — Y

J i - (

^l ^L )

,

( 6 . 1 . 2 )

A

1

A

2

^da

i

\ Ai daij'

A

1 v ^ 9 (Ajti dw , dw \ . .

1

„ . . .

AtW

=A^22da-

i

{'A-d^

+ t3

d^)'

J,J = l

'

2

'

3+l

-

Here, all of the linear variables are referred to the characteristic size of the

neutral surface,

R,

and retain their original designations. The dimensionless

stress-resultants ti are related to the stress-resultants Tf and S° by expressions

(4.3.3)

(T°,T°,S°) = -XE

0

h

Q

^(t

l

,t

2

,t

3

). (6.1.3)

The smal l parameter fi and functions d and g are also defined by formulae

(4.3.3).

We assume that kik

2

> 0. Let point a* =

a*

be fixed and change the

dete rm inin g functions in system (l) by their values at this po int . T hen we

come to system (3.1.1), with constant coefficients, the solution of which has

been constructed in Section 3.1 in the form of (3.1.4)

( to ,*) = ( IBO.SO) c' "

- 1

( P I « > + P ^ ) , (6.1.4)

where A satisfies t he rela tion

A =

d

(ll + <1

2

2

)

4

+ 9 {k2qj + k

iq

2

2

)

2

q

.

=

EL (6 1 5)

(?? + ?1)

2

(*i9?

+ 2*39192 + <29l) ' '

A

''

Now we vary the point a* = a* . The right side in (5) is a function of the four

arg um ent s p,- and a* w hich we deno te by

f

(p,,a*). Let

A

0

= m i n { / } = / ( p ? , a ° ) , (6.1.6)

Pi,a'

where we seek the minimum for all a* on the neutral surface and for real values

of

pi.

To evaluate this minimum it is convenient first to find the minimum by

Pi

for fixed

a*

using the results of Section 3.1.

Let the minim um of (6) be attain ed at

p,-

= p° ,

a*

= a° and let the weakest

poin t

a°

be far from the edge of the shell. Then

6.1.

L ocal Buckling of Convex Shells

113

The solution of system (1) which decreases exponentially away from point

a ? inall directions, is constructed below. Here we will not consider the ques

tion of satisfying the boundary conditions but rather we will assume that the

mentioned decay replaces the satisfaction of the boundary conditions.

The case when point a° coincides with

a

shell edge is m ore c om plex , since

then wemust satisfy thebou nda ry conditions. Th is case is considered in

Section 13.7.

Th e me th o d

of

construction

of

the solution

of

system (1)

is a

generaliza

tion of the algorithm described inSection 4.2 for the case of two independent

variables

ai

and «2-

We also assume that the function /(p,-, a,) has a s t rong minim um at poin t

(p°,a°) and also that the quadratic form

d

2

f

is positive definite

d 2

/ = £ ^ : ^ ; >

0

> (s-i-8)

i , i = i

'

3

where

(x\,

X2, £3 , £4) =

(pi,P2, <*i,

012), and the derivatives are evalu ated at

point x®.

We are looking for the formal asymptotic solutions

of

system (1)

in the

neighbourhood of point a° in th e form [99]

(«;,*) =

(w(a

k

,fi),

*(«*,/*)) e ~

ls

\ (6.1.9)

and we require that

<3S(a°

k

) = 0, 9 5 H > 0 ( a * # a 2 ) . (6.1.10)

The last inequali ty leads to the fast dam ping of functions (9) and as a

consequence provides the possibility of not satisfying the boundary conditions.

We can find functions

S(a

k

),

w(a

k

,fi)

and $((**/*) by su bst itu ting th em

into system (1) . Let us make a subs ti tution for the ind ependen t variables

a

k

=

a°

k

+n^

2

C

k

,

* = 1 , 2 , (6.1 .1 1)

and expand S into a series in

£

k

k

K

k,l

K

where

S

R

=

O

(fj,

3

'

2

).

Substi tuting (12) into (9) we note that

H~

1

SR = O

(/i

1

/

2

) and this term

may be included into

w

and $. Therefore, we seek

S(a

k

)

inthe form

S(a

k

) = ix

x

>

2

(p»Ci +P2C2)

+

£ 5 > « C * G , (6-1.13)

114

Chapter 6. Buckling Mod es L ocalized at a Point

where

p°

k

provides a minim um value of (6) and the sym m etric m atr i x with

constant elements So =

{ski}

is still unknown. Due to (10) 5

S o

is a positive

definite matrix. We seek functions

w

and $ in the form

oo oo

«5

= 5>* /V (c , - ),

<i

= £ y / ^ ( c o , (6.1.14)

fc=0 k=0

where

w

k

and $

fc

are polynomials in

Q.

Sim ultane ously we find th e expan sion

of the load parameter from the existence condition for this solution

X = X

0

+ nXi+n

2

X

2

+-- (6.1.15)

In Section 6.2 the algorithm for constructing the polynomials

w

k

(Q )

and

4>

fc

(£;) a nd unk now n p ar am et er s A,- an d Ski is given. Here we note t ha t, as

in Section 4.2 (case

B)

each eigenvalue (15) is proven to be asymptotically

double. Due to (9) the buckling mode is given by the relation

{ -^4

?Rw cosz - 9 w s in z) exp < — C

T

S2C \ > (6.1.16)

where

C = (G,C

2

)

T

,

So =

S + i 5

2

.

As in Section 4.2, the initial phase

9

takes on one of two possible values (0 ^

6\, 02 < 27r), which can not be dete rm ined by the propose d m eth od . For

the smallest eigenvalue corresponding to the critical value A, in the zeroth

approximat ion

w

~ 1 (see Section 6.2).

The size of the buckled area is of order

R hj

an d is covered by a series

of stretched pits the sizes of which are of order

RhJ

x

RhJ .

These pits

are inclined in the a

2

direc tion by angle y>o such t h a t ta n <po

—

—P2/P1 (

s e e

Figure 6.1).

The main difference between this mode and that described in Section 3.2

is the decay in the depth of the pits away from point a° (see Figure 6.1).

6 .2 C on s t r u c t i o n of t h e Buck l ing M o d e

We assume that

w

=

w\

and $ = u>2 and substitute (1.9) into (1.1). The

result we write as

Y,H

nk

(-ip JL,

aj

, A u>

fc

e"«"

l5

<"'-> =0, n = 1,2,

(6.2.1)

6.2. Construction of the Buckling Mo de

115

In «^«°

Figure 6.1: The buckling mode of a convex shell localized near the weakest

point.

where

H(pj, aj,p)

are polyno mia ls in

pj

such that

# n ( — in T;—> oij, p, ) z — n

4

A(dAz) + / i

2

A A

(

z ,

V d<Xj J

H

2X

z = -H

l2

z = fi

2

A

k

z, H

2

2Z = p

4

A ( j i

_ 1

A z ) .

We introduce the expansions

Hnk (Pj,(*j,n) = H°

k

(pj,aj) - inHl

k

{j>j,aj) + ••• ,

then, by virtue of (1.2) and (1.15)

H°

n

= d(ql + q\f - X

0

H°, H°

12

= -H & = k

2

q\ + k

iq

2

2

,

H°

22

=

9-Hll + ll)\ H? = t

iq

\ + 2t

3q i

q

2

+

t

2

ql

TT O

„

P"

LA\A

2

m

OOL

m

dp

m

A„

(6.2.2)

(6.2.3)

(6.2.4)

where S

nk

is the Kronecker de lta.

We transform (1) to the variables Q using formulae (1.11) and taking into

account (1.13) we obtain

£ H

nk

(p° + »

1/2

(J2

s

JiO " '" 7 ) >

a

°j + /^

1/2

0- A m = 0. (6.2.5)

If we expand the left side in a power series of /i

1

'

2

we get

(6.2.6)

116 Chapter 6. Buckling Mo des L ocalized at a Point

where the first operators P

nk

are

l

nk - 2 ^

J

L

dc

i

2

l

nk

24

3°

0Cr+

3a? 3p° I

2

V '

0 0

j ^ o V Z -

2 i

0 ^ - - * i r ) +

5

3 f t ( E

8

i ' « " » ^ -

a c

5p?<?P°

vf

d

2

- ^ ^ - ^ " - d c T d c ; )

Q

dCr

ih

nk-

(6.2.7)

Here h

J

nk

are the values of functions

H

3

nk

at a, = a ° , and p,- = p °.

Substituting series (1.14) into (6) we get the sequence of systems of equa

tions

Yy

nk

<

= 0- " = 1-2 (6.2.8)

E(W+w+w) = o,..

(6.2.9)

(6.2.10)

The determinant of system (8) we denote as

h

0

= H°(a°,p°)=h<l

1

h°

2

+(h

0

12

y

(6.2.11)

and note that the equality

h°

— 0 corresponds to equality (1.6).

We next introduce

w\

=

/ ^ ( / i ^ )

- 1

^ .

System (9) is com patib le by virtue

of (1.7) and (7). We can write

^ ( ^ - E M W )

- 1

and the compatibili ty condition for system (10) is then

(6.2.12)

(6.2.13)



6.2. Construction of the Buckling Mod e

117

where

L

0

Z

= -^ti{AZ

cc

) -iZ^FC - (^tiF + XidA Z,

C =

(Ci,C

a

)

T

, z

c

= ( ^ , ^ ) , z« = {^k}>

D=^[SoASo + B

T

So + S

0

B + C], F = AS

0

+ B, di =/i?/i§

2

>

2 J , 0 1 ( a 2 i, 0 1 r ^ 2 L 0

d

2

h

A

n

-[

d2h

°

\

r =

f ^ ° 1

dp°

m

dp°

n

(6.2.14)

Here C and Zf are two-dim ensional vectors,

Z^, A, B

and

C

are second

order square matr ices, tr

X

denotes the sum of the diagonal matr ix elements

X, and

T

is the matr ix transpose operation.

Equation (13) has a solution in the form of a polynomial in Ci and £2 only

if

2D =

S

0

AS

0

+ B

T

So + SoB + C = 0,

(6.2.15)

From t his equ ation one can find

So.

To solve equ ation (15) we use the alg orith m

developed in [99]. We note that the fourth order matrix

A

T 2 ) (6.2.16)

B

1

C

r

is positive definite due to inequality (1.8), since

d

2

h° d

2

f

dx

m

dx

n

dx

m

dx

n

'

Consider the system of vector equations

-B

T

u-Cv = /3u, Au + Bv=/3v, (6.2.18)

where u and v are two-dimensional vectors.

Let /? = /?j be th e eigenvalues of/?, for which syste m (16) ha s n on -triv ial

solutions Uj and Vj, where j = 1,2,3,4. Since matrix (17) is positive definite

it follows that

f3j

are pure imaginary numbers.

Let

Pk=iu

k

, /3

k+2

= -iu>k, w

f c

> 0 , k = 1 , 2 . (6 .2 .19)

118

Chapter 6. Buckling Modes Localized

a t a

Point

The following orthogonality condition is valid

u

T

m

v

n

- u

T

n

v

m

= 0 , p

m

+Pn± 0. (6.2.20)

We introduce the second order matr ices

U

and

V

the columns of which

coincide with the vectors

u\,

u<i and

v\, v^.

Then

S

0

= UV~

1

,

(6.2.21)

and So does not depend on the normalization of u

k

and v

k

. We can verify

directly the satisfaction of equation (15). The relation

u\v2

=

u^vi

leads to

the symmetry of matr ix

So.

We will now prove tha t m atr ix S2 is positive definite. Consider th e q ua dra tic

form T = x

T

Sox. We m ust prove th at SJT > 0 for any x ^ 0. After substitu

tion y = V~

x

x we get

T = y

T

Ty, T =

{

7

m„} =

V

T

U

= {v

T

m

u

n

}. (6.2.22)

From (18) we find

v

T

k

u

k

= ~ (ulAu

k

+ 1u

T

k

Bv

k

+ v

T

k

Cv

k

) .

(6.2.23)

Due to the fact that matrix (16) is positive definite, the expression in the

parentheses is also positive. Then we note that u\, t>\ is the eigenvector that

corre spo nds to /?3 = — iu>i and use equality (20) ujv2 = vju?. We then obtain

712 =7 21 and i t fo llows tha t 9 T > 0 .

Now we come to the solution of equation (13). We make a substitution for

the independent variables £ =

Vrj.

As a result , matr ix

F

becomes

F' =

K

_ 1

F l / = d i a g ( i o ;

1

, iw

2

). (6.2.24)

In its turn, equation (13) may be written as

fc.m

^

dw\ (wi

+ w

2

, A

0

+

2_JJ

k

r\

k

-

— + I Aidi I

w\

= 0,

(6.2.25)

where

A> = {a

km

} = V-iA(V-y.

We introduce arbitrary integers m,

n

;> 0 and seek the solution of equation

(25) in the form of a polynomial with the main term n™^. T he n for

A

(

r"

)

= rfr

1

i

w

i(

m

+ 2 J

+ W 2

i

n +

2

(6.2.26)

6.3. Ellipsoid of Revolution Under Com bined L oad

119

t h e u n k n o w n p o l y n o m i a l h a s t h e f o r m

wl

=

P

mn

= n?r%+

Y, P^iti, ( 6 - 2 - 2 7 )

k<m,l<n

where the coef f ic ients pki a r e un ique ly def ined and the sums o f the ind ices k + l

a t each t e rm a re odd i f

m + n

i s odd and even i f m +

n

i s even.

T h e c r i ti c a l l o a d is d e t e r m i n e d b y t h e e x p r e s s i o n

, (0 ,0 ) W i + W 2 o _ i (c o 9S \

A

v

i — ^ , w

x

= 1. (6.2 .28 )

W e a r e n o t c o n s i d e r i n g h e r e t h e d e v e l o p m e n t of h i g h e r o r d e r a p p r o x i m a

t i o n s w h i c h m a y b e m a d e b y m e t h o d s s i m i l a r t o t h a t p r e s e n t e d i n S e c t i o n 4 . 2

(see also [160]) .

6 .3 E l lip s o id of R evo l u t i on U n d e r C om bi n ed

Load

W e w i l l n o w c o n s i d e r a n e x a m p l e [ 10 1] t h a t i l l u s t r a t e s t h e m a t e r i a l of t h e

t w o p r e v i o u s S e c t i o n s . We w i l l s t u d y a n p r o l a t e e l l i p s o i d o f r e v o l u t i o n u n d e r

a h o m o g e n e o u s e x t e r n a l p r e s s u r e q a n d a b e n d i n g m o m e n t M\. T h e v e c t o r

My i s a s s u m e d t o b e o r t h o g o n a l t o t h e a x i s o f s y m m e t r y ( s ee F i g u r e 1 .3 ).

P a r a m e t e r s E, v a n d h a r e c o n s t a n t .

L e t u s i n t r o d u c e a s y s t e m o f c u r v i l i n e a r c o o r d i n a t e s

9,

<p (0 <

9Q

^ # ^

f ~ #0) 0 ^ <p <

2n)

on the ne u t r a l su r f ace o f th e e l lipso id . He re we wi l l use

t h e s a m e n o t a t i o n a s i n S e c t i o n 4 . 4 ( s e e F i g u r e 4 . 4 ) a n d a s t h e c h a r a c t e r i s t i c

s ize ,

R,

w e w i l l t a k e t h e s m a l l e r s e m i - a x i s

R

= ao-

U n d e r o n l y p r e s s u r e q, b u c k l i n g o f t h e e l l i p s o i d o c c u r s i n t h e n e i g h b o u r

h o o d o f t h e w e a k e s t p a r a l l e l ( e q u a t o r ) #o = T / 2 ( se e S e c t i o n 4 . 4 ) . U n d e r t h e

c o m b i n e d l o a d i n g o f t h e p r e s s u r e a n d a m o m e n t

Mi

, t h e p o i n t s o f t h e e q u a t o r

a re un de r d i f fe ren t load con d i t i on s an d we can expec t th a t buck l ing wi l l occ ur

i n t h e n e i g h b o u r h o o d of t h e w e a k e s t p o i n t .

I n t h e p r o b l e m u n d e r c o n s i d e r a t i o n t h e i n i t i a l s t r e s s - r e s u l t a n t s a r e t h e f o l

lowing (1 .4 .6 )

0

_ _ 1 Mi cos <p

0

_ M i cos 9 s in ip

l

~ 2

2<?

7 r B

2

s i n 0 ' ~ 7 r 5

2

s i n 0 '

(6 .3 .1)

Let

2R

X

J * TTB

2

R

lS

me

XEhu

2

a

0

Mi = ira%qm. (6 .3 .2)

120


at a

Point

Then by vir tue of (1.3) and (4.4.8) we get

_

1

mk\ cosip

1

k

2

mk\ cos (p

1

=

W

2

~ sin

3

0 '

t2

= V

2

~2P

+

S

2

sm

3

9

'

m kn cos 0sin 

/

>\i/2

r

6

0

t

3

= —=TT -, ^2 =(sin

2

9 + S

2

c os

2

6 ) ' ,

<J

= — > 1.

sin 6 a

0

(6.3.3)

Let m

>

0.

If

then

0° =

7r/2, y>°

= 0 is

the weakest point. Assum ing tha t conditions

(4)

are fulfilled, we can construct the buckling mode and find the critical load by

means of the method discussed in Sections 6.1 and 6.2

.

Under the above assumptions we have in (1.5)

d = g=l, A

1 =

S

, A

2

=

S

- , (6.3.5)

the values of £,• and

k

2

are defined by formulae (3). In(1.6) the minimum

A

0

=2 U i +J

2

- i j (6 .3 .6)

is a t ta ined at

p° =0,

P

°

2

=

S-\

0°

= n/2, <p° =

0.

(6.3.7)

By means of formulae (2.14), (2.17) and (1.5) we can find the elements of

matr ices A, Band C. In the case under consideration, matr ices Aand C are

diagonal and

B

is a tr ivial ma tr ix . Th e non-zero elements of m atr ices

A

and

C are the following

a n =

4(6

2

+ 2m-l)S-

s

(6

2

+ m-l/2)-

1

,

a

22

= 8 8~

6

,

c

u

= 2

[(S

2

- 1)

{A S

2

-l)-m(S

2

+

2)]

S~

8

(8

2

+m -

1/2)-

1

,

K

' ' '

c

2 2

= 2m<J-

8

((J

2

+m - l / 2 ) -

1

.

By directly solving equa tion (2.15) we find th at

.

1/2

- 1 / 2

. 1/2

- 1 / 2

n

/a o n\

«11

=

iC[[

a

n

,

s

22

= ic

22

a

22

,

S\

2

= s

2

i

=

U (0.J.9J

122 Chapter 6. Buckling Modes Localized a t a Point

Figure 6.2: The buckling mode of a elliptic shell localized near the weakest

point.

6 .4 C yl indr i ca l She ll U n d er A xia l C om pre s s ion

Let us now consider th e buckling of a cylind rical shell un de r axia l com

pression. The init ial s tress state is assumed to be membrane.

In the case of a circular cylindrical shell when th e dete rm inin g functions are

constant and the shell edges are simply supported the waves due to buckling

cover the entire neutral surface (see Section 3.4). If the compression is non-

homogeneous in the circumferential direction, the buckling pits are localized

in the neighbourhood of the weakest generatrix (see Chapter 5).

T he gene ral case of a non-circular cylindrical shell with va riable dete rm ining

functions is considered below. On the shell surface there may exist a weak point

in the neighbourhood of which the buckling mode is localized. Assuming that

this point exists and that it is far from the shell edges, approximate expressions

for the critical load and the buckling mode are found.

We will use the system of equations (1.1) which in this case has the form

fi

2

A(dAW) + \

dx

( < • £ ) -

A

x

2

A ( ^ -

1

A$) + fe

2

d

2

$

dx

2

8

2

W

dx

2

o ,

o ,

(6.4.1)

where

d =

Eh

3

(l-v$)

E

0

h

3

0

(l-^)'

— \Tt\,

i?2

9 =

Rk;

Eh

Eoho

h

2

0

12

(1

-

v

2

) B? '

d

2

T=E

0

h

ol

>',

A=—

+ w

.

(6.4.2)

The functions

E(x,y), h(x,y), v(x,y), h(x,y)

and

k

2

(y)

are assum ed to be

infinitely differentiable.

6.4. Cylindrical Shell Under Axial Com pression

123

Relation (1.5) may be rewritten as

* = f(x,y,p,q) = f°(x,y)

C

-

±

^— ,

(6.4.3)

where

£,«,>•», c =

*->.

* = ( * ) . . = - £ , ,

Evaluating the minimum of function / by all of its arguments we get

Ao

=

m i n / °

=

/ V

) 2

/

0

) ,

C=l-

(6-4-5)

Let function f° a t ta in i ts minim um at poin t

(x°, y°),

the second differential

d

2

f

=

Y,f

kl

dx

k

dx

u

f

kl

= J ^ ,

(x,y)

= (x

u

x

2

) (6.4.6)

OXfc OXi

at this point is the p ositive definite qu ad rati c form and the poin t (a;

0

, y° ) is

situated far from the edges of the shell.

We choose

R,

ho, EQ and

t\

in such a way th at

<i =2, k

2

= d = g=l (6.4.7)

at point (x°,y°). T he n Ao = 1, which agrees with th e Loren z-Tim oshenk o

formula (3.4.3).

As in Section 6.1 we seek a deflection sha pe th a t decreases away from th e

weakest point

(x°,y°).

How ever, since th e values of

p°

and

q°

could not be

uniquely determined from the minimum condition for

/,

we can not use the

method employed in Section 6.1 for the present problem.

We apply Fourier transform

oo oo

W{x,y)= w{p,q)e

iz

'dpdq, $ (x,y) = jj<p (p,q) e

iz

' dpdq, (6.4.8)

— oo

—oo

where

z*

=/ i

_ 1

[p (x

-

x°) + q (y

-

y

0

)] .

T he n we ex pan d the coefficients of

system (1) into

a

power series

in x

— x°

and

y

— y°.

As

a

resu lt, after th e

elimination of <p for w (p , q) we get the equation

124


at

a Point

where

h{p

k

,n)

=

H(p

k

,x°

k

,n),

h

k

= —

T

, h

kl

dx

k

dx°

k

dx°'

••

(Pi,Pa)

=

(p ,«) , H= H°-iuH

1

+--

H

=

d i P +q)

~

XhP

+

( W'

H=

2{lb8t

+

m)-

(6.4.10)

We transform equation (9) to the variables z and

r

by mea ns of the formulae

P

Q

z

=

P

2

+q

2

'

T

=

,2

_i_

„ 2

p

2

+

g

(6.4.11)

an d we seek a solutio n w hich is localized at

z$ —

1,

r =

ro in the form (ZQ

=

— 1 ,

T = — TQ leads to the complex conjugate solution)

0 0

W

( p ,

g i /

i )

=

u)e-(«C +»" ) /

2

, u)

=

5 ; / i* /

4

u ,

f c

(C, '?) ,

^ - ,

d > = >

.

/ l

- ^

( C l f |

, ,

k

=o (6-4.12)

z = H V

/ 2

C ,

r = r

0

+ / J

1

/

4

^ ,

A =1 +u

\

t

+

/j

3

/

2

A

2

+

.

• • ,

where w

k

are unknow n polynom ials in C and

n,

a,

b,

TQ and \

k

are unknown

constants (5ft a

>

0,

U

b

>

0).

Substitution (12) into equation (9) gives

where

(

y

L

0

+

U

1

'

4

L

1

+^

2

L

2

+

---)w

= Q,

L

0

w

= (lot

2

-Xi)w+ ih U-QZ ~«C

2

+ 2)

w

~

(6.4.13)

hi

f

d

T{dC~

a<:]

w

L iW

=

T] L '

0

w +

ih<i-h2\-Q£-a(i

dr)

— bn I w,

1

7 / ( 9

\

L 2W

=

2

V

2

L'oW -

A

2

*

- -y f

-7; bnj w + L *w,

E

dtp

dz _ 1

. .

k

'dxJWu ~

aij +

4

ih

'

(6.4.14)



6.4. Cylindrical Shell Under Axial Com pression

125

Here f° and ip are the same functions as in (4), lj and Uj are functions of

To, operators

L '

0

and

L '

0

'

are ob taine d from

L o

af ter the substi tutions instead

of the coefficients

l\

and

In

their first a nd second derivativ es by r

0

(the first

term in

L o

becomes zero). We denote the terms that do not influence

wo(C,»?)

an d A2 du e to (2.1) by

L*w.

Let us consider the equation

L

0

w

0

= 0. Th is equ ation has a solution in the

form of a polynomial in C if th e coefficient of C

2

is equal to zero. This yields

a = {r-ih)l

n

\

r = ( 4 / i i - Z 2 )

1 / 2

= ( 4 a u )

1 / 2

> 0 . (6 .4 .15)

For

Ax = Aj

n )

= r

(n +

± V n = 0 , l , 2 , . . . (6 .4 .1 6)

the equat ion

L0W0 —

0 has the solution

w

0

(C,r,)=v

0

(v)H

n

(9), e = aC, a = (j-J

, (6.4.17)

where t>o(»7) is still an unknown function, and

H

n

(0 )

is the He rm ite polynom ial

of the n-th order that satisfies the equation

d^ii

du

M

n

u=-^-29 — + 2nu = 0, H

0

(6) = 1.

(6.4.18)

The equation of the f irst approximation

L0W1 + L^wo

=0 after tran sitio n

to the variable

0

takes the form

M

n

w

1

= F

1

(9)r

1

vo(v) + F

2

(9 ) f^--bti\v

0

= F(e,ri), (6.4.19)

where

Fi(9)

= -

l

k H'

r

i-2aeH -aH

n

+

-H

n

\

+

+

i (eH'

n

- H

n

+

1

-H

n

), (6.4.20)

r \ a

2 /

ft

W

= (£ ,„-,„(„*.._£,,.))

are known polynomials.

Equation (19) has a solution in the form of a polynomial if

e

2

f F(9,r,)H

n

{0)e~^de =

O. (6.4.21)



126

Chapter 6. Buckling Mod es L ocalized at

a

Point

Condition (21) gives the equation dr/dro — 0 for th e eva luat ion of TQ and

equation (19) has the solution

«>i(0, r,) = «ifo) H

n

(9 ) + W ? ) G i(0 ) + (~ -br,v

0

\ G

2

(0), (6.4.22)

where v\

(TJ)

is an unknown function,

d = Atf H'

n

+ A

3

H

n+2

, G

2

= A

3

H'

n

+A

4

eH

n

,

A

1=

-A

2

, A

2

= ( 4 a

2

r ) -

1

( a

2

/ '

1 1

+ 2 i a / '

1

) , (

6

-

4

-

23

)

A

3

= -ar~

1

l

l2

-2A

4

, A

4

- - (ar)-

1

(a l

12

+ il

2

).

Now we come to the second approximation

L

0

w

2

+ Liw

1

+ L

2

w

0

= 0. (6.4.24)

Condition (21) of the existence of a solution of equation (24) in the form of a

polynomial in £ leads to

-Q\§--

br

lj v

0

+ (Rr]

2

-X

2

)v

0

= 0, (6.4.25)

where

Q = -xh

2

-r \A

4

\

2

= - a

lx

(a

u

a

22

- a\

2

) > 0,

R -

2U+l

\l'{

1

\a \

2

+ il'{(a

2

-2a)-4

| a / '

n

+ iZ'j

|

2

1

= (6.4.26)

16 r

2n+l d

2

r _ 1 d

2

X

1

4

drj'^'drf'

If

b = R

1

'

2

Q-

1

'

2

>0,

A

2

= A

2

m , n )

= ( 2 m + l ) ( Q i ? )

1

/

2

, m = 0 , l , . . .

(6.4.27)

then equation (25) has a solution in the form of the polynomial v

0

(rj) =

H

m

(b-^

V

).

The parameter

r

in (12) could be found from the minimum condition for

Ai, thus R> 0 in (26) .



6.5. Construction of the Buckling Mo des

127

We give the final expression for A

A

=

A

(m,„)

_ i

+

( 2 n + l ) / j a

1

1

{

2

+

+ / i

3

'

2

( 2 m +1) (2n +l )

1

/

2

( 4 a

1 1

) -

3 / 4

K '

1

( a

1 1

a

2 2

- a

2

2

) ]

1 / 2

+ 0 (/ i

2

)

m , n

=

0 , 1 , 2 , . . .

(6.4.28)

As in Section 6.1 each eigenvalue A(

m ,n

) is asymptotically double. Buckling

corresponds to the smallest eigenvalue, which is obtained from (28) for m

=

n = 0. The neighbouring eigenvalues allow us to conclude that the critical load

is sensitive to any initial imperfections.

6 .5 C o ns t r u c t i o n of t h e Buck l i ng M od es

Let us find the buckling mode which corresponds to the critical load ob

tained previously.

In

order

to

do this we subs titu te expression (4.12) into

integral (4.8) assuming that w =1 +O (/i

1

^

4

)- We get

+oo

W(x,y) = Jj{l

+

0{^

4

))expl-(px'

+

qt/)-

— OO

-\

a

C

2

-\b

V

2

\dpdq,

(6.5.1)

where we denote x' =x - x° and y' = y — y°. In (1) we come to the in teg rati ng

variables C and n. By virtue of (4.11) and (4.12) we find

p

=

po +

/ i

1 / 2

p

2

C

+

f*

1/4

Pif)

+

M

1/2

P3'?

2

+

• • •

q = go +M

1/2

g

2

C +

1 /4

gi77 +

J

u

1

/

2

g

3

'7

2

+---

where

Po =

2

, go =ToPo, pi =q

2

=- 2 Topi,

Qi

=

~P2

=

(1

-

r

0

2

) pjj, p

3

=

(3 r

2

-

l) pg,

q

3

= r

0

( r

2

-

3) p§ .

Evaluating integral (1), we get

W =

C e - « e x p { - ( p

0

z '

+

g o 2

/ ) } ( l + 0 ( ^

1 / 4

) ) ,

f

_ {Pix' + qiy'f (p

2

x' + q

2

y')

2

(6.5.2)

(6.5.3)

(6.5.4)

2fi

3

P(b-2i (p

3

x' + q

3

y')) 2/xa

128 Chapter 6. Buckling Modes Localized at a Point

It follows from (4) that buckling is accompanied by the formation of a series

of elongated pits inclined by angle 7 to the cylinder generatrix (tan 7 = —

r^

1

).

The distance between the centres of the pits is

2-KU\

(1 +

T Q ) .

T he factor e

-

^ in

(4) describes slowly varying pit amplitudes (depths) and phases.

We fix c > 0 and find the domain in which 3?£ < c. Outside of this domain,

the deflections are smaller than e

_ c

. The domain bound is given by the curve

jpix' + qiy') b (P2x' + q2y')

^ « _

2 ^

3

/

2

[fc

2

+ 4/i"

1

fax' + q

3

y'f]

2

1 *

l

a

(6.5.5)

and we obtain the inequali t ies

12.. . A

1

/

2

\P2x'

+ q

2

y'\

<

2 | a |

2

c ^

IPix' + q^l^^Ho,

= H

1/2

a

0

,

60 =

l e a

— U

2

b \"

+

( l + r

2

) S R a

1 / 2

(6.5.6)

The domain obtained above is reminiscent of the form of an the ellipse with

semi-axes

S'

2

a

0

(l + T*),

VL

3

'%(1

+ TS).

The large semi-axis of the "ellipse" is inclined to the cylinder generatrix by

the angle

S

( tan

S

= 2 r

0

(l - r

0

2

)

- 1

) .

T he buckling m ode is shown schem atically in Figure 6.3. T he series of

strongly elongated pits inclined to axis

x

by angle 7 covers an "ellipse" whose

size is of order

u

1

2

x / /

3

/

4

that is inclined to axis

x

by angle

S.

Two linear combinations of the real and imaginary parts of (4) are the

eigen functio ns. As was found in Section 6 .1 , we ca nn ot find th e coefficients of

this linear combination by use of the present method.

Let us return to the evaluation of TQ. By virtue of (4.14)

011 = /i°i(l -

T

2

)

2

-

4 /

1

°

2

ro(l -

r

2

)

+ 4/

2

°

2

r

0

2

-

(6.5.7)

The value of r

0

m ay be found by minim izing the function a n . According

to the above assumption on

f°(x, y),

the value of TQ is unique for /°

2

^ 0 and

A V o X ) , | r

0

| < \ / l +

/ ?

2

- / ? ,

Let

fi

2

-

0. Th en for /f t < 2 /

2

°

2

we get r

0

= 0.

0

Jr.

2

f°

Jn

(6.5.8)

We are no t considering the case where /f t = 2 /

2 2

since

an =

2 /

2 2

(1 + r^)

and

d

2

a

n

/dT

2

=

0 for r

0

= 0. For /ft >

2/

2

°

2

we get r

0

= ± (l - /

2

°

2

/ / f t )

1 / 2

-

.5 . Con struction of the Buckling Modes

y

129

p

x

x '

+ q

x

y = Q

Figure 6.3: The localized buckling mode of a cylindrical shell near the weakest

point.

In the last case, due to the existence of two values of

To,

the critical load is

asymptotically tetra-multiple and the buckling mode is a l inear combination

of the forms described above for ±ro. The expected buckling mode is shown

schem atically in Figure 6.4. T he do m ain filled with pits is com prised of a

combination of the two elongated

ellipses".

Figure 6.4: The localized buckling mode of a cylindrical shell at a/?

2

>

(1

— a )

2

.

As an example, we will consider the bending of a circular cylindrical shell

with variable thickness. We take

t\ = 2 cos y, h = ho

1

— a cos fix

0 < a < 1.

(6.5.9)

130

Chapter 6. Buckling Modes Localized at

a

Point

The weakest point is then

x° = y°

= 0. We get

f°(x,y)

1<x0-

KQ

cos

y'

/ i i = 7 T 3 f e > ^ = °- /22 = 1- (6-5.10)

For / ? < 2 /

2

°

2

, i.e. a/?

2

< (1 - a )

2

we get r

0

= 0,

A

=

1

+ A* (A ° i)

1 /2

+ / /

3 / 2

( 4 /

1

°

1

) -

1

/

4

( 2 - / ? , ) / ' + O (p

2

). (6.5.11)

In this case the pits are elongated in the circumferential direction and the

"ellipse" filled by them is elongated along generatrix

y =

0 in contrast with

the previous case (see Figure 6.5).

y

4$MM§

Figure 6.5: The localized buckling mode of a cylindrical shell at a/?

2

<

(1

— a)

2

For a/?

2

> ( 1 - a )

2

we find

70

A

= ± 1 -

( l - « )

;

a/3

2

2 \

1/2

1 + 2fl[1

~7f,

1/ 2

+

(6.5.12)

/ 1 \

5 / 4

/ 4 \

1 / 2

and the buckling mode is shown in Figure 6.4.

In conclusion, we note that the buckling modes constructed in Sections 6.4

and 6.5 (especially the m ode shown in Figure 6.4) look rat he r un usu al. Perh aps

this is related to the ass um ption s m ade in Section 4. 1, the m ajor one of which

states that the determining functions (for example, the thickness) are variable,

but change slowly, while at the same time the neutral surface imperfections,

which are characterized by faster variability, are not taken into account.

6 .6 P r o b l e m s an d E xe r c i s e s

6 . 1 . Prove that the parameter of the critical load

A

has expansion (1.15) in




131

integer pow ers of// wherea s mo de (1.14) is exp and ed in the series in fra ctiona l

powers of p. In particular, prove that the term A

3

/

2

^

3 /

'

2

in (1.15) vanishes.

6.2. Consider a prolate truncated ellipsoid of revolution under the external

pressure q an d edge force P i (see Fig ure 1.3). For P i = 0 the equa tor is th e

weakest line. Find the position of the weakest point for Pi ^ 0 and the critical

load in the zeroth approximation.

H i n t U se relations (1.4.6) to calculate the stress-resu ltants, form ulae (4.4.8)

for the parameters of ellipsoid and (1.5) to determine the weakest point and

the critical load.

6 . 3 .

Consider prol ate trun cat ed ellipsoid of revolution under the extern al

pressure q an d edge force Pi (see Fig ure 1.3). W it h the help of form ulae (5.3)

and (5.4) for /i = 0.02, TQ = ± 0 .5 , 6 = 1 constru ct the level lines for four real

buckling modes which are odd or even in

x'

and

y'

respectively.

H i n t

One of this modes is schematically plotted in Figure 6.2.

C h a p t e r 7

S e m i - m o m e n t l e s s B u c k l i n g

M o d e s

In this Chapter we will study the class of buckling problems for shells with

zero Gaussian curvatu re under a m em bran e init ial s tress sta te. For this class

of problems the buckling concavities are stretched along the asymptotic lines

of the neutral surface and may be localized near one (the weakest) line. The

additional stress state appearing under buckling is a semi-momentless state

[119]. T he m eth od ma y be applied to cylindrical and conic shells of mo der ate

length, the cross-section of which need not necessarily be circular and the edges

of which are not plane curves. The two-dimensional problem may be reduced

to a sequence of fourth order one-dimension al prob lem s. In some pa rtic ula r

cases for cylindrical shells, approximate solutions have been obtained.

In this Chapter we will use the results obtained in [100, 152, 161, 162, 163].

Buckling problems for non-circular cylindrical shells and shells with variable

thickness are also studied in [6, 7, 8, 23, 34, 42, 83, 122, 149, 181] and others.

7 .1 B a s i c E q u a t i o n s a n d B o u n d a r y C o n d i t i o n s

We will consider th e buckling of a conic shell under a m em bra ne stress sta te .

In the neutral shell surface we introduce an orthogonal system of curvilinear

coordinates

s,

<p, where s = R~

1

s',

s'

is the distance between the point of a

surface and the cone vertex,

R

is the characteristic size of the neutral surface,

tp is the coo rdina te in a directrix which is so chosen th a t the first qu ad ra tic

surface form is equal to

da

2

=

R

2

(ds

2

+

s

2

dip

2

). Th e radius of curvatu re,

R

2

,

is

equal to R

2

= .Rs&

_1

(<£>). The shell is closed in the circumferential direction,

133

7 .2. Buckling Modes for

a

Conic Shell

135

In Section 7.2 we will study only one case of simply supported edges (S

2

in

Table 1.1)

T

m

=

v

t

= w = M

m

= 0 at s = si ,

s = s

2

,

(7-1-4)

where T

m

and M

TO

are the stress-resultant and the stress-couple respectively in

the direction orthogonal to the edge, and vt is the displacement in the direction

tangential to the edge (see (1.2.11) and (1.2.12)). To construct the main stress

state with an error of order e

2

we must satisfy the conditions

w = $ = 0 at s = s\, s = s

2

. (7.1.5)

If the shell edge is a plane curve supported by a diaphragm which is rigid in-

plane but flexible out-of-plane, then we must reformulate boundary conditions

(4) but conditions (5) are valid with an error of the same order e

2

. Some other

variants of the boundary conditions may also be reduced to conditions (5) (see

Section 8.4).

7 .2 B uck l ing M o d es for a C onic She l l

Let

ti,t2,t3

= 0 ( 1 ) . Th en, due to formulae (3.6.11) and (1.3) we have

estimates

A

~ 1 at

t

2

> 0, (7-2.1)

A ~ A ; r

1 / 4

a t <

2

= 0 , i

3

^ 0 , (7.2 .2)

and it is sufficient that conditions in estimates (1) and (2) be fulfilled, at least

in some subdomain of the neutral surface G.

We sta rt with the case when

t

2

> 0 in some sub dom ain of

G.

Then, buckling

occurs such that concavities are elongated in the longitudinal direction from

one shell edge to the other. This is the situation where a conic or cylindrical

shell is buck ling under e xte rna l pressure (see Section 3.5). In this Section

we assume additionally that the shell generatrices are under different loading

conditions (the meaning of this assumption will be clarified below) and the

buckling mode does not spread over the entire neutral surface but rather is

localized near the weakest generatrix.

Under these assumptions we can seek the solution of system (1.2) in the

form

w (a, 0 ,

136

Chapter

7.

Semi-momentless Buckling Modes

where w

n

(C , s) are polynom ials in £.

We seek

the

function

$ in the

same form

as (3). The

value

of

pa ramete r

q

which determines

the

buckling mode oscillation

in the

direction <p

is

real.

The weakest generatrix

is

ip

=

ipo

and

p a ra me t e r

a

characterizes

the

ra te

of

decrease of the dep th of the buckling con cavities as we go away from the weak

est generatrix.

We

no te tha t mode

(3)

generalizes solution (4.2.6)

for

ordinary

differential equation (4.2.1). Here solution

(3)

depends also

on

variable s.

To determine unknown functions w

n

, $„ and the values of q , a, <p

0

and A„,

we sub sti tute solution

(3)

into sy stem

(1.2) and

equalize

the

coefficients

by the

same powers

of

e

1

/

2

.

We

expand

the

coefficients

of

system

(1.2)

depending

on

<p in a power series of <p — (po = e

1

'

2

C-

Firs t ,

it is

convenient

to

express function

$«

throu gh w*

by

v i r tue

of

second

equation (1.2).

We get

$ *

4 6

1

/

2

/ ,

.dw*\

aQw* -i -ZT I +

w*

„4

gq

5

dip

— taw*

d

2

w,

gs^

ds

2

'

+

Now, the first equation of (1.2) gives the sequence of equations

m ining functions w

n

,

which

may be

w ri t ten

in the

form

H

0

w

Q

= 0 H

0

u>i + Hiw

0

= 0 H

0

w

2

+ H\W\ H

2

w

0

= 0 ..

Here

(7.2.4)

for deter-

(7.2.5)

H

0

z

=

H

t

z =

H

2

z =

H.z

k

2

da

4

^A

2

z

+

-^-z-X

0

Nz,

Nz

q

2

t

2

gs

3

q

•z,

dH

0

dH

0

\ , .8H

0

dz

a — 1— -— 1 Qz — i

dq

„ 2

dip

d

2

H

0

+ 2a

dq dC

d

2

H

0

d

2

H

0

Cz-

— i \ a

dq

2

' dq dip d(p

2

d

2

H

0

d

2

H

0

\ dz

1

d

2

H

0

• +

c

q

2

dqdip)^ d( 2 dq

2

i d

2

H

0

2 dqd(p

z + H

t

z-XiNz,

. .

fdt

3

_ _

dz

i z +

5C

2

(7.2.6)

7 .2. Buckling M odes for a Conic Shell

137

The substi tution of solution (3) in the boundary condition

w =

0 for

s = s

k

{ip)

gives

w

o

((,s)=0, w

1

((,s) + Cs'-^ = 0,

ds

(7.2.7)

u»

2

(C.

«) +

C

« ' - 5 - + ^ r ( a ' r - o - + * " ^ r = 0, . . .

and from the condition $ = 0 we obtain, taking into account formula (4),

d

2

w

0

d

2

wi d

3

w

0

OS* OS

1

OS-* (n 2 g \

5 s

2

^

s

3 s

3

^ 2 V

ds

4

^ 8s

3

J q ds

3

Let us consider the boundary value problem in the zeroth approximation

k

2

(fo) d

2

(

a

d

2

w

0

\ q

4

d .

AT

wo —

3 , = 0 at s = si(v?

0

), s = «2(v

:,

o)-

(7.2.9)

In addition to the principal para m ete r Ao this problem also contain s tw o

parameter s ,

q

and <p

0

. For fixed values of

q

and ipo the m inim um eigenvalue

Ao is a function of these pa ra m et er s, i.e. Ao =

f(q,fo)-

Here , for pro ble m (9)

we consider only positive eigenvalues. We denote

\

0

0

=mm{f(q,<p

0

)} = f(q

0

,<p°

0

). (7.2.10)

Below we will omit the superscript (°) on A° and (p°.

The conditions

<9A

0

9A

0

/

7 0 1 1

i

- 5 — = — = 0 a t

q-qa, <P = fo

(7.2.11)

must be fulfilled.

As in the previous Sections (see formulae (4.2.11) and (6.1.8)) we assume

th at t he second differential of function / at poin t (qo, <fo) is a positive definite

quadratic form, i.e.

d

2

f = \

qq

dq

2

+ 2 \

w

dq dip + A

w

(fy>

2

> 0, (7.2.12)

7.2. Buckling Modes for

a

Con ic Shell

139

w h e r e t h e

w°

i s a n e i g e n f u n c t i o n f or p r o b l e m ( 9) u n d e r c o n d i t i o n s ( 1 1 ) , a n d

P

0

i s a n u n k n o w n f u n c t i o n t o b e d e t e r m i n e d f r o m t h e f o l l o w i n g a p p r o x i m a t i o n s .

In the f i r s t approx imat ion we ge t

H

0

w

i +

r

P

I

dH

° _L

dH

°

Po[a — h - —

dq 0<po

t » n

o,

Wi+CPos'

,dwl _d

2

w

33...0

i + C V ^ ^

(7.2.19)

a t

Sfc.

Due to fo rmulae (13)—(16) , t he ex i s t ence cond i t ion fo r the so lu t ion o f p rob

l e m ( 1 9 ) i s a n a l o g o u s t o e q u a l i t i e s ( 1 1 ) f r o m w h i c h t h e p a r a m e t e r s go a n d ip°

h a v e b e e n f o u n d . I n p a r t i c u l a r , it f o ll o w s t h a t a n o n - a r b i t r a r y g e n e r a t r i x m a y

b e i n t r o d u c e d a s t h e w e a k e s t o n e i n f o r m u l a ( 3 ) . F r o m ( 19 ) w e t h e n o b t a i n

«>i(C, «) = Pitt) ™°o +<Po{Q(aw

q

+ w

v

) - iPo(C)w

q

(7.2.20)

w h e r e

w

q

a n d

w^,

a r e so lu t ions o f p rob lems (13) fo r WQ

= w®

a n d b o t h f u n c t i o n s

Pa a n d P\ a r e u n k n o w n .

I n t h e s e c o n d a p p r o x i m a t i o n w e h a v e t h e e q u a t i o n fr o m w h i c h t a k i n g i n t o

a c c o u n t b o u n d a r y c o n d i t i o n s ( 7 ) , ( 8) a n d d u e t o ( 1 5 ) , w e g e t t h e e x i s t e n c e

c o n d i t i o n f o r t h e s o l u t i o n W2

L P

0

=

- 1

\

qq

P'

Q

' +

&CP

0

' +

(v -

Ai + ^

6

+

ctA P

0

=

0, (7.2.21)

w h e r e

0 — l \Q, \qq -\- Xqtpj , 2C — d Xqq + ZdAq^ + A < ^ ,

'

=

£ / ( <

Hy

dq

o _ -o

dHo

-Wu, — W,

p

w

0

dq

UV ds+

+

4k

2

s

3

s' fdw°

0

d

3

w°

0

dw°

0

d

3

w°

0

ds ds

3

ds ds

3

«2

-2i w^H+w^ds

«2

f w°

0

Nw°

0

ds.

(7.2.22)

C on di t io n c = 0 i s nece s sa ry for the ex i s t ence o f a po ly no m ia l fo rm so l u t io n

of eq ua t io n (2 1) . F r om th e sq ua re equ a t i on c = 0 acc ord ing to (12) we find

t h e u n i q u e v a l u e o f

a

such that 9 a > 0 .

140

Chapter 7 . Semi-mom entless Buckling M odes

For

c = 0, Ai =

(n+-jb +

T),

n =

0 ,1 ,2 , . . . (7 .2 .23)

equation (21) has the solution

Po — H

n

(Q,

where

H

n

are n- th degree Hermite

polynomials. The most interesting case

is

when

n = 0

and

H

n

=

1 since

in

this case the value of Ai

is a

m inim um . Indeed, due

to

(22) an d (12)

b > 0.

The following approximations may be constructed in a sim ilar fashion. We

note only that Wk(C, s) are either even or odd po lynom ials in £ and the existence

condition for w

2

k+

2

gives

L P

2k

+ A

fc

Po + F

2k

(C ) = 0, AT > 0, (7.2.24)

where the

L

is the operator in th e left side of eq ua tion (21) for c = 0.

The value of Afc m ay be found from the existence cond ition for a polynomial

form solution of (24). If the polynom ials

Wk, Pk

an d

Fk

are even, then the

polynomials Wk+i, Pk+i and Fk+\ are odd and vice-versa. That is w hy the

equat ion L P

2

k+i + F

2

k+i = 0 always ha s a polynom ial solution.

In fact, the values of

Afc (k

> 2) are not found below. First, because of the

difficulty of the calculations and, second, since the value of

A2

depends on the

terms which were omitted in the derivation of sys tem (1 .2) . T ha t is why the

above discussion is important only to justify the expansion by powers of e (3)

of the parameter A. In [42] th e corre ctio n of th e seco nd ord er A2 is found for a

cylindrical shell with a slanted edge under uniform exte rnal pressu re. For this

aim instead of system (1.2) the more exact system (1.2.6) was used.

Separating the real and the imaginary parts in (3) we find that each eigen

value A is asymptotically double. The buckling mode has the form

w = (3? w

t

cos

z —

9

w*

s in

z)

exp < — (

2

3 a > ,

I

2

J (7.2.25)

Z =

e

- l / 2

g o C +

_

C

2 s

Ka +

^

As in Section 4.2 the m etho d used here does not perm it th e de term inatio n of

the init ial phase

0

— const which is equ al to 0 <

9\

,

9

2

< 2n,

nor does it allow

one to distinguish the c orresponding eigenvalues.

R e m a r k 7.2. It follows from the me th o d of so lu tion tha t the results

obtained are also valid for a shell which is no t closed in the circumferential

direction. It is only necessary th at the weakest gen eratr ix b e far from the shell

rectilinear edges.

7 .3. Effect of Initial Mem brane Stress Resultants 141

7.3 Effect of I n i ti a l M e m b r a n e S t r e s s R e s u l

t a n t s

In Section 7.2 we assumed in developing the solution that the initial stress-

resu l tan ts T?, T

2

° and S° had the same orders and that the stress-resultant TS

was compressive (see (2.1), and also (1.3), which represents the dimensional

stress resultants T®, T$ and S° through the dimensionless ones t\, t% and £3).

Under this assumption, the terms Ao + eAi in expansion (2.3) for the critical

load depen d only on the stres s-resu ltant <2 and not on t\ and £3.

Indeed,

in

this case the function

WQ(S)

is real an d 77

= 0 in

(2.23).

The

value of A2 depends on <i and £3 (see (2.3)) and to derive A2

it

is ne cessary

to

assemble the next two approximations.

In order to estimate the effect of the stress-resultants t\ and t

3

o n A we u se

the following method. We assume temporarily that the variables £2, £ti, and

etz are of the same order, i.e. the order of variables t\ and £3 are greater th an

the order of <2- This assumption causes the operators Nand H* in(2.5) to

change

Nz

=SJl

z

-

q£

( .

z +

2t

3

] =

N

0

z +

ieNiz,

S

\OS S

J

H,z =

£

\—(sh—)---^-z

[ds \ ds J 2 dsdipo

The rest of the formulae obtained above are still valid. Taking Nz in the form

of (1), we represent the boundary value problem (2.9) as following

H

0

w

0

= Hoowo -ieX

0

Niw

0

=0 (7.3.2)

and expand its solution in a pow er series in e

w

0

= woo + iswoi + £

2

u>02

+•••

,

A

0

=

A Q

+

e

2

A

0

2

+

. . . (7.3.3)

Here WQQ and AQ are the same as in (2.9).

The function WQI(S) is the solution of the non-hom ogeneo us pro blem

rj2

#oow>oi -

X Q N I W O O =

0,

w

01

=

2

=

0,

s = si,s

2

,

(7.3.4)

which is always solvable due to (2.15) because

/ wooNiWoods =0. (7.3.5)

7.3.1

7 .4. Semi-Mom entless Buckling Modes of Cylindrical Sh ells

143

Let

t

2

< 0 and stres s-resultan t £3 be such th at

et

3

has order not less than

the order of t

2

. No te th a t we do not exclude the case when t

2

= 0. Concerning

the stress-resultant

t\

, we assum e th at its order is not larger tha n th e order of

The boundary value problem in the zeroth approximation has the form

k

2

d

2

( ^d

2

w^\ q

4

w

0

"T n T

, . / q

2

t

2

w

0

. / dt'

3

dw

0

\\ , - „ , „ ,

+

H——

+ t q

r

o

js-

+ 2 t 3

-d7))

= 0

( 7 3 1 3 )

q* ds

2

\ 8s

2

J s

3

, . 2 J „ „ _ .

o.i

where

t'

3

— et

3

and the boundary conditions are of the same form as in (2.9).

This problem has a real discrete spectrum but the eigenfunctions are complex.

Indeed, let WQ be an eigenfunction. We mul tiply (13) by WQ and integrate

it from si to s

2

. Th en we get

« 2

f/k s

O w

0

2

s

3

8

2

^

2

«

4

<f

ds

2

1 I | 2 \ ,

+ ~3 \

W

°\ )

d S

8 2

. 2

(7.3.14)

/ (

1

t2

I I

2

, W

9V dU

\\j

S '

'

v

OS OS )

where we denote WQ =

u + iv.

It is possible to show th a t the re exists an infinite

set of both positive and negative eigenvalues. As in Section 7.2 we denote the

m inim um posit ive eigenvalue

Ao

by

f(q,

ipo). We m ake the following e valua tions

in a manner similar to that in Section 7.2.

7 .4 Se m i - M om en t l e s s Buck l ing M o de s of Cy l in

dr ical Shel ls

Let the shape and size of a cylindrical shell be determined by the following

relation

Si(lf) < S < S

2

(ip), 0<<p<ip

2

da

2

= R

2

(ds

2

+ d )

where the

R

is the characteristic radius of curvature. For a circular cylindrical

shell, R is its radius and • 0. But we still may use all of the relations derived

144

Chapter 7. Semi-momentless Buckling Modes

in Sections 7.1-7.3 for the case of a cylindrical shell by replacing all of the

factors s" in them by unity.

For a cylindrical shell the boundary value problem in the zeroth approxi

mation (2.9) has the form

e

o

2

( « 9

2

u , o \

,

4

.

2

,

9 - 5 - 5 - + dq*w

0

- \oq't

2

w

0

- 0,

I"

ds2

\

ds

" / (7.4.2)

d

2

w

0

o=

Q

, = 0 at s = si, s = s

2

.

All of the formulae in this Chapter are given for non-constant variables

E,

v

and

h

through which the variables

d

and

g

are expressed according to (1.3).

If

E, v

and

h

are constant then we assume that

d

—

g =

1.

If the variables g, d and t

2

do not depend on s then problem (2) has an

explicit solution

.

T T ( S - S I ) dq

2

TT

A

k

2

g

wo

= sin

;

, Ao = — + ,

6

, l(<p)

= s

2

-si,

(7.4.3)

/ t

2

12' 9

from which we find

Ao = i m n

\TU\-TT) '

%

= -& r

lfi =

ip

a

(7.4.4)

and the value of Ai in (2.3), by virtue of (2.22) and (2.23), is equal to

A

1 =

l / 2 ( A ,

g

A ^ - A ^ )

1 / 2

, (7.4.5)

where

16d . „

d fd

Xqq

-i7'

Xw

-

8qo

d^{r

2

\tptp

— ? o

d\ l

4

d 3

2

(k

2

g

(7.4.6)

d<p

2

0

\t

2

J Wgdyl

\l

A

h)\'

Here, all of the functions are evaluated at the weakest generatrix <p = <po-

We note that the zeroth approximation for the critical load, Ao, coin

cides with the value obtained by the Southwell-Papkovich formula for a cir

cular cylindrical shell of con stan t length

L \ — Rio

and radius

Rk$

(here

'0 = l{<Po), k

0

= k(<p

0

)) (see (3.5.5)).

In the following examples we evaluate both the zeroth approximation and

the first correction for a critical load. In Examples 1, 2 and 4 we assume that

parameter s E, v and h are constant.

http://tu/-tt

http://tu/-tt

http://tptp/

http://tptp/

http://tptp/

http://tu/-tt

7 .4. Semi-M omentless Buckling Modes of Cylindrical Shells 145

E x a m p l e

7.1. Consider the buckling of a circular cylindrical shell of mod

erate length with a sloped edge under homogeneous external pressure

p.

Let

si{<p)=0,

s

2

(<p)

= l (<p)=l

0

+(cos<p-l) ta.n/3,

(7.4.7)

where /? is the edge slope angle.

In (3), (4) and (6) we assume that

ti =

1,

k =

1. Then

4? r

8

_

37T

4

^°-°'

Ao

- sWv

q

° ~ If'

\ - ifi \ - n \ -

4 7 r t a n

P

Aqq — 1 0 , A

qv

— U, A

vlp

— o 3 / 4 / 2

(7.4.8)

and the critical value of pressure is the following

For fc* = 1 we get the Sou thw ell-Pap kovic h form ula which corresp onds to

the maximum shell length. The second term in

k„

takes into acc ount the slope

edge influence. For ex am ple , for /?

=

45° ,

v —

0.3 and

R/h =

400 the slope

edge increases the critical load by 14%.

E x a m p l e 7 . 2 .

Consider th e buckling of a non-circu lar cylindrical shell

of moderate length

L \

under homogeneous external pressure

p.

The weakest

generatr ix is <po with minim um curvature

k

(<p). As a characteristic linear size,

R,

we take the maximum radius of curvature of the generatr ix .

Th en

k

(ipo)

= 1 and we assume

p = Ehe

6

XR-\ (7.4.10)

2

-k^y

Then due to (4)-(6) we get

'

= £

A - J ? L \ - i f i \ - n \ -

2 7 r 3 l / 4 dH

0 —

- 3 / 4 1 ' 11 ~ ' 9¥>

—

'

A

f

^° (7.4.11)

4

7

r e

6

£ / i , ,

1

3

7

/

8

e f,d

2

k\

1/2

„ . . .

For A;» = 1, as in the pre vious ex am ple , we get the So uth w ell- Pa pk ov ich

formula for a circular cylindrical shell of radius

R.

146

Chapter 7 . Semi-momentless Buckling Modes

E x a m p l e 7 . 3 . N o w w e will consider t h e buck l ing of a c i rcu lar cyl indr ic al

shel l of cons t an t l eng th a n d var iab le th ickness i n t h e c i rcum feren t ia l d i rec t ion

unde r homogeneous ex t e rna l p r e s su re

p .

The weakes t genera t r ix is t ha t w i th t h e m i n i m u m t h i c k n e ss . I n t h e neigh

b o u r h o o d of th is g en er at ri x (y>o = 0 ) l e t th e shel l th ick ness hav e t h e expans ion

h(

v

)=h

0

(l+y

(7.4.12)

Suppose t ha t

t h e

p a r a m e t e r s

E a n d v a r e

c o n s t a n t ,

w e

f ind t h a t

- l

(7.4.13)

ow

p

'

(4)-(6)

d

=

give

47r

Eh

0

e

6

~ 3

3

/

4

Li

( ' •

K*

,

aip

2

^

2

* . =

) " •

1 +

9

3

7 / 8 J

• ( '

+

' 2 a L A

,3TVRJ

ap

1

2

1/2

e

E x a m p l e 7 . 4 .

L e t u s look a t t h e buck l ing of a cant i lev er c i rcular cyl ind r i

cal shell

of

cons t an t l eng th

L \

= /

R

u n d e r n o n - h o m o g e n e o u s

( i n t h e

c i rcum

fe ren t ia l d i rec t ion) ex te rna l p ressure . L e t t h e l o a d p a r a m e t e r A descr ibe t h e

pressure p b y t h e express ion p — po<2, where po =

A

E h s

6

R

_1

.

We in t roduce t h e d imens ion less in i t i a l s t ress - resu l tan t s J,- as

t\ = — — s

2

( « i

c os

) , 0:1,0:2 > 0 .

T h i s p r o b l e m w a s solved in [61] b y m e a n s o f t h e B u b n o v - G a l e r k i n m e t h o d

us ing a l a rge number of coo rd in a te func t ions a n d be low, w e wi l l compare ou r

resu l t s wi th those ob ta ined in [61].

Let

a\

+

012 >

0 . T h e n t h e weakes t gener a t r ix is

(po

= 0 . K e e p i n g t h e t w o

first terms in expans ion (2 .3 ) we ge t th e cr i t ica l value of p a r a m e t e r

p

0

in th e

form

4 T T

Ehe

6

k*

eA , - , , „ >

P=&n

j

M

.

n

,

n

y

k

*

=

1

+

r-

(7.4.16)

3

J

'

4

L \

(1

+ at\ +

or

2

)

A

o

Using

t h e

re la t ions der ived

i n

Sec t ion

7 .2

yields

i

r w

, . yi 1/2 . 4 7 T 3 -

3

/

4

-Sai

a= -A

0

( a i + 4 a

2

) ,

A

o = 7 7 7 - - r,A j = — — — 7.4.17

4

L

J

< (1+0:1+02) 1 + ai + 02

7 .4. Semi-Mom entless Buckling Modes of Cylindrical Shells

147

and the expression for the parameter /t* which takes into account the variable

load is

We introduce the length of a half-wave concavity in the circumferential

direction A<p and parameter

p

which takes into account the decrease in de pth of

the concavity in the circum ferential direction away from t he weakest gen eratrix .

We obtain

A „ = ^ =

1.132„,

, = i ^ f ^

=

0.513 f - ^ ± ^ )

1 / 2

, . (7.4 .19)

The depths of the concavities are proportional to 1,

e~

p

,

e ~

4 p

, . . .

We may use expansions (2.3) under the assumption that the number of

waves in the circumferential direction is large. Taking into account the expres

sions for Aip and y we conclude that the proposed method is valid for shells of

m ode rate length. As length

L \

increases, the value of

Aip

also increases and

the accuracy of the method decreases.

From a qualitative point of view, formula (19) fully agrees with the results

of [61]. Quantitatively, the error in formula (19) essentially depends on the

problem para m ete rs. Th e mo st unfavourable combination of para m eters is

R/h = 100 and

L\/R

— 10 for w hich the error at ta in s 15 % for som e values of

c*i and a

2

. W h er e h/R and L\jR decrease , th e error also decreases and for

R/h = 400,

L i/R =

2.5 it is less than 5

% .

In Figure 17.4 of [61] the buckling mode for R/h = 1000, L

x

/R - 10,

OL\ = 0.5 an d

a.

2

= 0 is shown. T he localization of the concavities at the

weakest generatr ix (po = 0 is clearly seen. In this case, form ulae (20) give

Aip

= 36.4°,

p

= 0.186. Th is m ean s th at th e dept hs of the concavities are

pro po rtio na l to 1, 0.83, 0.48, 0.19, 0.05. One can see in Figu re 17.4 th a t

A<p ca 30 °. W e no te t h a t in this case the error in form ula (19) is less tha n 1 %.

Formula (19) does not account for the initial stress-resultants

T°

and

S°.

Their contribution may be estimated by use of (3.8) and (3.12). As a result

one mu st add the te rm 0.181 (c*i + 4 a

2

) ( l + « i + a

2

)~

1

y

2

, to th e value of k*.

This term partly takes into account terms of order e

2

.

In [61] the even functions in <p were used as the coordinate functions for the

Bubnov-Galerk in method. That is why the buckl ing mode obtained , w(s,<p),

is also an even function in <p. For a load which is very close to the critical load,

the odd buckling mode can also occur.

The method of this chapter is generalized in [40] for buckling of cylindrical

shells (pipes) joined with a mitred joint under homogeneous external pressure

and in [102] for wave propagation in a non-circular cylindrical shell.

148

Chapter 7 . Semi-mom entless Buckling M odes

7 .5 P r o b l e m s an d E xe r c i se s

7 . 1 . Consider t he buck ling of a circular cylindrical shell of m od era te length

with a sloped edge under homogeneous external pressure

p.

Plot the graph of

the function of critical pressure vs edge slope angle when the other parameters

are constant.

7.2 . Consider the buckling of a non-circular cylindrical shell with th e sim ple

supported edges

s

= 0,

I

of moderate length

L\ = Rl

under homogeneous

external pressure

p,

w hen ge ne ratr ix is an ellipse with se m i-axes ao an d &o

(ao > bo). Find the critical pressure. Plot the graph for critical pressure vs.

eccentricity.

A n s w e r

The critical pressure may be found by formula (4.11) in which

S = 3 (^ - l ),

difl

yhere

R

= ^ ,

k(ip) = (cos

2

e + Sfsin

2

e)

3/2

,

do

-—= kl, d l = — > 1 .

dip

OQ

For

R/h

— 400,

v

— 0.3, / = 1 an d

Si

= 2 the critical pressure for an elliptic

shell is 52 % large r t ha n th e critical pressure for a circula r shell of rad ius

R.

7 .3 . Consider the buckling of a non-circular cylindrical shell with the sim

ple supported edges s = 0, / of moderate length L\ — Rl under homogeneous

external pressure

p,

w hen ge ne ratr ix is an ellipse w ith sem i-axes ao an d &o

(ao > bo). Note that for an elliptic shell the critical load obtained is asymptot

ically tetra-multiple since in the neighbourhood of each generatrices #o — 0,7r

two buckling modes corresponding to nearly identical loads m ay exist. Th is

problem permits the exact separation of the variables

{w,

$ } ( s , ip) =

{w ,

$}{<p) sin —

and may be reduced to the system of ordinary differential equations of the

eights order.

Construct even and/or odd with respect to the ellipse diameters modes

solving the problem numerically on the quarter of ellipse 0 < 9 < TT/2.

Similar calculations for the problem of shell free vibrations has been made

in [81].




149

H i n t .

The evenness and oddness conditions have correspondingly the fol

lowing forms

w' = w'" = $ ' = $ '" = 0 , w = w" = $ = $ " = 0.

7. 4. Prove th at the par am ete r of the critical load A ha s expansion (2.3)

in integer powers of e whereas mode

w„

is expanded in the series in fractional

powers of e. In particular , prove that the term A3/2£

3

^

2

in (1.15) vanishes.



C h a p t e r 8

E f f e c t o f B o u n d a r y

C o n d i t i o n s o n

S e m i - m o m e n t l e s s M o d e s

In this chapter we will study the same problems as in Chapter 7 but we will

present the solutions in ano ther analy tical form. Here we will seek the unkn ow n

functions as an integer power series of the small parameter

e.

Various boundary conditions in addition to simply supported are studied

and the problem of the separation of the boundary conditions (i.e. of extracting

two linear com bina tions of four bou nd ary cond itions), which mu st be satisfied

under the development of the main stress state is solved.

The dependence of the critical load on the boundary conditions is also

examined.

8.1 C o n s t r u c t i o n A l g o r i t h m for S e m i - M o m e n t -

less Solut ions

Consider system (7.1.2)

e

4

A (d

A

w) + Xe

2

A

t

w

- A

f c

$ = 0,

(8.1.1)

e

4

A{g-

1

A$) + A

k

w = 0

under the same assumptions as in Section 7.1.

In Section 7.2 the solution of this system in the form of a power series

(7.2.3) of e

1

/

2

was developed. Here we will construct the solution in the form

151

152

Chap ter 8. Effect of Boundary Cond itions on Sem i-momentless Mo des

of a formal asymptotic integer powers series of e.

oo

e) = w ° e x p | -

q(f)d<pj,

A = A

0

+ eAi + eX

2

H

w°

=

£ V

n = 0

(8.1.2)

The function $ is represented in the same form. As in the previous Sections

we seek the solution which exponentially decreases away from the weakest

generatrix o) = 0 , 9 a > 0 ,

a=-p~-

(8.1.3)

dip

0

(compare with the conditions in (7.2.3)).

The substitution of solutions (2) into equation (1) leads to a sequence of

equ ations for the unknow n functions

q (p), w

n

(s , <p)

a n d $ „

( s , <p)

and the values

of A

n

. For simply supported edges (see conditions (7.1.4) and (7.1.5)) in the

zeroth approximation we come to the boundary value problem

. 3

2

$o q

A

d d

2

w

0

g

4

K-5-5

r

w

0

+ X

0

Nwo =

0, K -5 -5 -H r $ o = 0 ,

os

z

s*

5

as

1

gs

A

a

2

t

N=

;

w

0

— $0 = 0 at

s

—

s

x

Up), s = s

2

(<p),

s

(8.1.4)

which after the elimination of $0 is comparable to problem (7.2.9). In the

solution of problem (7.2.9) we assumed that

q

and

ip

are the independen t

pa ram ete rs and the n th e eigenvalue Ao =

f (q,<p)

was found as a function of

these para m ete rs. Th en , due to (7.2.10) we searched for the m inim um value

of Ao by q and ip and the correspon ding values of X°, go and ip

0

.

Here we assume that the value of Ao is fixed and is equal to A°, the angle

ip

is the v ariable para m ete r and the function

q — q (<p)

is unkn ow n. T hen the

equality

Ag = /(«(?>)>¥>) (8-1-5)

may be considered as an equation for evaluating

q (<p).

If relations (7.2.11) and

(7.2.12) are fulfilled then equation (5) has a unique solution in the neighbour

hood of ip — (po satisfying conditions (3).

It follows from the expansion of /

(q , ip)

in form ula (5) in a power series in

q —

go and ip —

po

tha t

X

qq

(q

- go)

2

+ 2 A

w

(g - g

0

) {<p - <p

0

) + X

vv

{<p - <Po)

2

+ • • • = 0. (8.1.6)

To find a we get the same quadratic equation c = 0 (see (7.2.22)).

154 Chapter 8. Effect of Boundary Conditions on Sem i-momentless Modes

We will not consider higher order approximations because these formulae are

very complicated and also because system (1) is not sufficiently accurate since

it does not contain some terms which effect the second approximation.

8 .2 S e m i - M o m e n t l e s s S o l u t i o n s

In Section 8.1 we developed only the functions

w

and $ and studied simply

supported boundary conditions. In order to examine other types of boundary

conditions it is necessary to have expressions for the other functions which

describe the stress-strain state of the shell.

In this Section we will find the displacements («i = u,

U2

= v), the normal

rotation angles (7;) , the tangential s tress-resultants

(Tj, S),

the stress-couples

(M,, H)

and the shear stress-resultants

(Qi).

Let us introduce the dimensionless values (marked with prime symbols)

with the relations

(«{,«/) = ±(ui,w), (T;,

S > ,

Q;.) = - i - (Ti,s, Qi),

K rj

0

hQ

(8.2.1)

Later, we will omit the prime symbols for simplicity.

Any of mentioned functions we denote by

x

and seek them in the form

similar to (1.2),

. *

x(s,v,e) =

x°exp{-

q{v)d<p), x° =

e

«°(-> £ s

n

x°

n

(s , <p), (8.2.2)

where the a

0

is the index of intensity depending on the choice of the function

x

and

ao(w)

= ao($) = 0. As a rule, we will find two first terms of series (2).

T he second eq uat ion in (1.1) gives the relatio n between t he functions $ °

and w°

0

s^k_g_

5 V _

4iegs

3

d_ f

9 V \ _

q

4

8s

2

q

5

0 ^

(823)

Due to formulae (1.5.6) we get

4

n

a

2

<t

, 1

9$\

c

_

4

a / i 0 * \

_

4

s

2

$

r

'

= ( 4

? ¥

+

^ '

s =

-

e

T,{-.a;)>

T2

= £

V >

^

8.2. Sem i-Mom entless Solutions

155

and a f t e r the subs t i tu t ion o f (2 ) and (3 ) we ob ta in

s

2

s

2

\ dip

4

/ | W 1 3<E

0

'

\ s

2

dip

2

s ds

us

\ s J as \s

o<p

< 9

2

$ °

12

~

£

ds

2

•

T o e v a l u a t e t h e d i s p l a c e m e n t s w e u s e f o r m u l a e

-f L-kw + u] = e

2

= I(r

2

-i/Ti).

1 du 9 /v\ 2(1

+J/)

„

- O -

* Q - ( -

= w = —i ^5

s dip os \sJ g

and accord ing to fo rmula (2 ) we f ind

n

e

2

s

2

k

d fw

0

^

q

2

ds\s

T )

+

i£3s2

dS

, 4

\ < r s o<p\q)

qs oip\ q

l

/ /

/

2

a

/ t / $ ° \

2 ( l

+

z/ )s

d

/ < E ° \

ks

2

d (

d_(u^\

I 3 s

V

s

2

# / 5 ds\sJ

q

4

ds\ ds\ s J

2

fl / 1 9 p 9 /Arw°

3 s \ s g

2

3ip \g 3y

(f))

(8 .2 .5)

(8.2.6)

+ (8.2.7)

gs 3y> \ g

2

d<p\

q ) q dip\ q

2

I) J

J

.

kw° e

2

d /kw° ,

i£ + — ^ +

- ) •

d<p\

q

?

/kw°\\

q dip

\q d < p \ q I) q

3

ds\ s J gs

^

3

( -—(-—(—W

ks 2

d

(

w

°)

u q

® °

156

Chapter 8. E ffect of Boundary Conditions on Semi-mom entless Mo des

Then, by using formulae

_

dw

1

dw

OS S dip

_ d

2

w 1 d

2

w

a s

2

s

2

ay?

2

d (\ aw

N

T —

—

ds \s

dip

M i =

e*vdx

u

M

2

= £

8

dx

2

,

(

8

-

2

-

8

)

H = £

8

{\-u)dT,

1 d .

M

, . M

2

1 <9#

W j

,

fi

s w

s s dip

Q

2

= F ^ +

A

<37i+<272 e

6

,

s ay?

we find

(8.2.9)

n

dw°

„ io „ 1

dw°

7 i = - - 5 — . 72 = «> *— >

ds es s d<p

M° = i/M$,

M ° =

J-

w

°

+

—- [2q—— + q'w°

s

2

s

z

\ dip

«•

= « v ( - ^

+

i|(^)

+

^ | ( ^ ) )

+

+ A e

6

( < i 7 i + < 3 7 2 ) ,

The expressions for the generalized displacements u

m

, v

t

, w and 7

m

and for

the corresponding generalized stresses T

m

, St, Qm and M

m

at edge s = so(y)

which does not coincide with the line of curv ature we find according t o formulae

8.2. Semi-Mo mentless Solutions

157

(1.2.11)

- „o

-r

rpO

s?

o°

M°

=

u

co sx + u s inX;

v

t

=

7 ? c o s x + 7 2

s m

Xi 7?

— u°

sin x +

v°

cos x,

= T

1

0

c o s

2

x + 2 5 ° s i n x c o s x + T

2

°s in

2

x ,

= (T ° - T °) sin x cos x + 5 ° cos 2

X

,

+ ( ^ - ^ | f ) s i n x + A

e

6

( f „

j 7

^

+

<

T 7 t

0

) , (8-2.10)

(M

2

° - Mj

0

) sin x

cos

x +

H°

cos 2x,

dw° 1

d /i/dw°\\

s

3

s ds\ s 11

V

s°

s ds \ s

= < i c o s

2

x + 2 t

3

s i n x c o s x + i 2 s i n

2

x ,

=

{h

- < i ) s i n x c o s x + ^ 3 C o s 2 x ,

= M

1

°co s

2

x + 2 / f ° s i n x c o s x + M j s in

2

x -

Here, m and t are the geodesic normal and the tangent to curve s = sa(ip),

the angle x is shown in Figure 8.1. The values of Q\a and Q^* are given for

the case when the edge load maintains its original direction.

Figure 8.1: The parallel coordinates near a slanted edge.

One must substi tute the l inear combinations of these solutions and the

edge effect solutions into the boundary conditions. The edge effect solutions

are developed in the next section.

158

Chapter 8. E ffect of Boundary Conditions on Semi-momentless Modes

8.3 E d g e Effect S olu t ion s

Consider a conic shell where the edge does not coincide with the line of

curvature and is given by the equation s — so(<p) in the coordinate system s,

(p

introduced in Section 7.1.

In the neighbourhood of an edge we introduce the orthogonal system of

parallel coordinates s

m

, ip

t

([24]), where the curve s

m

= 0 coincides with the

shell edge, the lines tp

t

= const are the geodesic normals to the edge, (ft is the

value of

ip

at the edge and R s

m

is the distance between the point an d the edge

as measu red along the geodesic no rm al (see Figu re 8.1). L et |x | < —, i.e. th e

generatrices are nowhere tangent to the edge.

The coordinates s, ip and the coordinates s

m

, ipt are relate d by formulae

s = s(s

m

,

I

\ (s

m

s\nx\ / e o n

f

=

f(s

m

,ft) = ft

+ arcsin I l = (8.3.1)

= ft +

— s in x 5"

s l n

X

c o s

X + 0 ( s „ ) ,

so *o

s

i

so

= s

0

(f

t

),

x = X(ft),

t a n x = - — ,

s

o

where the derivative with respect to

f

t

is m arked by a prim e sym bol.

The first surface quadratic form is

da

2

= R

2

(ds

2

m

+ B

2

{s

m

,f

t

) df

2

)

,

B

t

=

B

t0

(1 +

x

t

s

m

) + O

(«^ ) ,

B

tQ

= (s

2

0

+

s'

0

2

)

1

2

= - f 2 _ ,

Ht

= 5 -

0

3

(«§ +

2s'

0

2

-

SQ

s'A ,

V / cos x

^ '

where R~

l

x

t

is the geodesic curv atu re of an edge.

For a cylindrical shell, formulae (1) and (2) take the form

s= s

0

+ s

m

cosx, f = ft + s

m

s'mx, t a n x = - s '

0

>

/ 2N

1

/

2

1 (8.3.3)

Bto=[l + s

0

) = , x

t

= - s '

0

' c o s

3

x ,

V / cos x

and t he rest of formu lae (2) are th e sam e.

The edge effect solutions in the case when the curvilinear coordinates do

not coincide with th e lines of cu rva ture are co nstru cted in [24, 52].

8.3. Edge Effect Solutions

159

Let

y

be one of the functions describing the stress-strain state of a shell.

We seek these solutions as the formal series

v>

y{s

m

,ip

t

,e) =y

e

e

mr

^exp{

1

- fqd<p},

%o (8.3.4)

O O

n = 0 ^

where

y

e

n

are polynomials in

m.

The variability of solutions (4) with respect to

s

m

is greater tha n w ith

respect to <p

t

(the indices of va riati on are equ al to 1/2 an d 1/4 resp ectiv ely) .

The functions WQ and w\ do not depend on m and they are arbitrary functions

in

tpt-

The functions

w„, (n

^ 2) are polynomials in

m

and they contain

arbitrary terms depending on

ip

t

.

The number

a

e

(y)

describe s th e edge effect

intensity for function y.

The operators included in equations (1.1) in the coordinate system s

m

, ipt

have the form

1

d ( dw\

1

d (

1

dw\ d

2

w

A w = - - —

B

t

-z

— +

A

k

w

B

t

ds

m

\ ds

m

J B

t

dipt \B

t

d<p

t

J ds

2

^

1 r

d (B

t

dw \

3 / 1

di

. ,

B

t

[ds

m

\R

t

ds

m

J ds

m

\R

mt

d<p

t

d f_\_ dw_\ _d_ f

1

dw

dipt \Rmt ds

m

J dipt \R

m

B

t

dipt

l t d Rt d

x 2

~ — I 1

R

t

\ds

m

B

t

R

m

t dipt

where

1 _ k cos

2

x 1 _ fe sin x cos x 1 _ k sin

2

\

Rt SQ R

m

t

so

R

m

s

o

(8.3.5)

(8.3.6)

The approximate expressions (5) are given with an error of order e

2

. All

of the solutions found later in this section will be determined with the same

accuracy.

After the substitution of (4) into (1.1) we get the equation for

r

4 , 4 (•> ,

e 6

V

n

4 9 k

2

cos

4

x , i q s'p (»-}

7

\

\ r J

dsg

s

0

B

t

o

from which we get r

r = r

j

=r

0

l3

j

+ eb +

O(e

2

),

$ +

1 = 0,

j = 1,2,3,4

(8.3.8)

160

Chap ter 8. Effect of Bound ary Conditions on Semi-mo mentless Modes

a n d t h e r e l a t i o n s

Pi

9k

c o s

2

x

/ 1

2eb\

s

0

[r] r* )

W

(8.3.9)

w

e

=

WQ

+ e w\

, $

e

= $ o +

e

^ i -

We f i n d t h e t a n g e n t i a l s t r e s s - r e s u l t a n t s

T

m

a n d

St

by fo r m ula e (see (1 .5 .6 ) )

T

m

=e

A

j _

_a_

/ j _ j 9 ^ \ j _ aB

t

5$

5

t

9 y t \ -Bt * ty>

t

/

B

t

ds

m

ds

d ( 1 <9$

<9s

m

V

# t d p t / '

(8 .3 .10)

f ro m wh i c h we o b t a i n

it

T

m

= e

2

I r x , - ^ r ) $

e

+ ^ ( 2q ^ - + B t o ^ - I T M ^ I ,

B

2

S? = -

e ~

$

e

- e

2

B

to

n

2

r d$

e

B

t

o dipt '

^ l

+

B

t

o—( A

dft dip

t

\B

t0

T o d e v e l o p t h e d i s p l a c e m e n t s

u

m

, v

t

we u s e t h e r e l a t i o n s

( l - j /

2

) T

m

du

m

w

= £m+V£t = TT - +

US

rn

f^m

^

1

dvt

1 3 5 ,

ID

2 ( l + i / ) 5

t

and f ina l ly we ob ta in

B

t

d<pt ' B

t

ds

m

m

R

t/

~

_ J_ du

m

d f^t_\ 2w

mt

B

t

d^t

*

d s

m

\B

t

) Rmt

(8 .3 .11)

(8 .3 .12)

v.* = e

2

k

( s in

2

x

+

v

c o s

2

x)

•> 2?

k vq

s in x co s

2

x .

— w

e

— e —7 i w

rs

0

2

2k

sin x cos x

rs

0

r

2

s

2

w

e

+

,

3

[2i(l + v)q , , i q k

( s in

2

x +

v

c o s

2

x )

e

+ e — — $ - 7 . w

e

\ g B

t0

r

2

s

0

Bto

(8 .3 .13)

The exp ress ions fo r t he va lues o f

f

m

, M

m

a n d

Q

m

+,

wh i c h m a y b e c o n t a i n e d

i n s o m e b o u n d a ry c o n d i t i o n s a r e t h e f o l l o wi n g

7m =

£ 2

TW

e

,

Q ,

M

e

n _ —

dHmt

n

e

B

t

dipt

s

A

dr

2

w

e

,

e dr

3

w

e

.

(8 .3 .14)

8.4. Separation of Boundary Conditions

161

W e n o t e t h a t i n t h e c a s e w h e n t a n

x =

s

'o

= 0 ( i .e . t h e edge co inc ide s w i th

the l ine o f curva tu re o r i s t angen t to i t ) t he o rder o f the va lue

v\

d e c r e a s e s .

8.4 Se pa r a t i on of B o u n da r y C on d i t i ons

I n t h i s S e c t i o n w e w i l l d i s c u s s t h e p r o b l e m o f t h e d e t e r m i n a t i o n o f t h e t w o

m a i n b o u n d a r y c o n d i t i o n s f r o m f o ur b o u n d a r y c o n d i t i o n s ( 1 .2 .1 6 )

u

m

= 0 or T

m

= 0,

v

t

= 0 or

S

t

—

0,

w = 0 or Q

mt

= 0,

f

m

= 0 or

M

m

= 0 a t

s = s

0

,

w h i c h m u s t b e s a ti sf ie d i n d e v e l o p i n g t h e s e m i - m o m e n t l e s s s o l u t i o n s . H e r e ,

w e w i l l c o n s i d e r o n l y t h e s i m p l e ca s e w h e n t h e g e n e r a l i z e d d i s p l a c e m e n t o r

gen era l i z ed fo rce co r re spo nd ing to it i s eq ua l to ze ro a t the edge . In th e o t he r

w o r d s , i t i s a s s u m e d t h a t t h e b o u n d a r y s t i f f n e s s i n e a c h d i r e c t i o n c o i n c i d i n g

w i t h t h e a x e s o f t h e t r i h e d r o n ( m , t, n), i s eq ua l e i t he r to inf in i ty or to ze ro .

N o t e t h a t e l a s t i c s u p p o r t o f t h e t y p e u

m

+ /3T

m

— 0 i s no t con s ide red .

I n T a b l e 8 .1 t h e i n t e n s i t y i n d i c e s o f t h e s e m i - m o m e n t l e s s s o l u t i o n s a o w h i c h

are fou nd in Sec t ion 8 .2 a n d of th e edge effect so lut io ns

ct

e

( see Sect ion 8 .3)

a re g iven fo r the func t ions inc luded in cond i t ions (1 ) .

W e m ak e a d i ff e rence be tw een th e case o f s in% = 0 an d s in% ^ 0 . T h e

fi rs t case tak es place if a shel l edg e coin cide s w i th th e l ine of c ur v at u re . In

pa r t i cu la r , i f a she l l o f r ev o lu t ion i s l im i t ed by a pa ra l l e l . W e hav e the s a m e

case wh en s in x = 0 on ly a t the po in t o f th e in t e r sec t ion o f th e edge an d th e

w e a k e s t p a r a l l e l

<p

= <po.

T h e d e t e r m i n a t i o n o f t h e m a i n b o u n d a r y c o n d i t i o n s f o r s h e l l s o f r e v o l u t i o n

i s d i s cus sed in [57]. For th e case o f a f ree edge , the ma in b o un da ry co nd i t io ns

are found in [24, 25, 123] .

T h e r e l a t i v e i n t e n s i t y of s e m i - m o m e n t l e s s s o l u t i o n s ( c o m p a r e d w i t h e d g e

effect s o lu t ion s ) i s ch ara c te r i ze d b y th e va lue o f A a = ao — a

e

( see T ab le 8 .1) .

F o r t h e m a i n b o u n d a r y c o n d i t i o n s , t h e v a l u e o f A a m u s t b e s t r i c t ly l e ss t h a n

f o r t h e a d d i t i o n a l b o u n d a r y c o n d i t i o n s . F o r t h a t t o b e t r u e t h e f o l lo w i n g

condition, A, m u st be fulfi l led (see [57]) :

in the zeroth approxim ation both the semi-m omentless solutions included

in the ma in bo undary conditions and the edge effect solutions included in the

additional boundary conditions must be independ ent.

In o rder t o s a t is fy th i s co nd i t io n , as a ru le , i t i s nec es sa ry to use l in ear

c o m b i n a t i o n s o f b o u n d a r y c o n d i t io n s . L e t

x\ = X2

= 0 b e t h e i n d e p e n d e n t

(8 .4 .1)

162

Chapter 8. Effect of Boundary Conditions on Semi-momentless Modes

Table 8.1: The intensity indices

/

X

u

V

w

7i

7i

S

Qi*

M i

u

m

v

t

w

im

T

m

St

M

m

U

V

II

a

0

2

1

0

0

2

3

6

6

1

1

0

- 1

2

2

5

6

2

1

3

/ / /

x

0

0

+ exl

*o = 0,

ds \ s J

w°

w°

dw°

ds

$ 0

ds v s )

o

9w°

os

w°

* o # 0 ,

as

w°

w°

o

dw

°

w + e a

3

- j —

a s

$o

w

u

+ £ a

5

- r -

w

u

+

ea

6

— —

d-s{-v)

+£a7

d^

o

9w°

„ds

d /*°\

ds\ s )

Remark.

If t

3

^ 0

a

0

= 5,

IV

<*e

sin

x — 0

2

3

0

- 2

2

1

2

4

sin

x 0

(?)

2

2

0

- 2

2

1

2

4

2

2

1

for <5i*

we

Ac*

= 3.

V

x

0

+ e Xf.

r~

l

w

e

r

2

w

e

w

e

r w

e

(r~

l

+ b

x

r~

2

) w

e

r~

l

w

e

r

3

w

e

r

2

w

e

(r'

1

+ eb

2

r-

2

)w

e

(r-

1

+ sb

3

r-

2

)w

e

w

e

rw

e

( r -

1

+

eb

4

r-

2

)w

e

( r

_ 1

+ eb

5

r~

2

) w

e

r

3

w

e

r

2

w

e

( r

_ 1

+ eb

6

r~

2

) w

e

( r

_ 1

+eb

7

r~

2

) w

e

(r~

l

+

e

b

8

r~

2

)

w

e

assume that

VI

Ac*

0

- 2

0

2

0

2

4

2

- 1

- 1

0

1

0

1

3

2

0

- 1

2

163

linear com binations forming the m ain bou nda ry c onditions and £3 = X4 = 0

provide the additional boundary conditions. We consider the value of

S

= m i n { A a

3

, A a

4

} - max{Ac*i , A a

2

} , Aa,- = A a ( x ;) . (8.4 .2)

We conclude that the problem of separation of the boundary conditions is

solved if

6^

2.

In order determine whether or not the solutions are linearly dependent in

Table 8.1, we represent the functions which, with an error of order e

2

, are

proportional to the semi-momentless solutions and to the edge effect solutions

respec tively. T he coefficients of pr op ort ion ali ty and t he coefficients a; an d &,•

may be found from the formulae given in Sections 8.2 and 8.3.

In calculating the entries of column III for the sem i-mom entless solution s,

the terms containing derivatives of unknown functions with respect to

ip

have

been eliminated. For example, consider the value of v° a t s'

0

^ 0. Re taining

the first two terms in relation (2.7) we obtain

v° = C

1

w° + C

2

^-, Ci = Ci{ip,e), C

2

= 0 ( e C i ) , (8 .4 .3 )

If the boundary condition

v° =

0 is introduc ed at edge

s

=

s°

then

w°(so(<p),

ip) = 0{e). Differentiating this relation with respect to <p we

obtain

dw° . dw° ,_ , , ,. , , ,

_

+ s

'

0

_

= O { £

) , (8.4.4)

and then

v° =

C

x

w°

-

C

2

s'

0

^ - + 0 (e

2

C

lW

°) .

(8.4.5)

os

It follows from Table 8.1 that the semi-momentless solutions v, v

t

, w and

Mi are proportional to each other with an error of order e

2

and solutions

u

m

,

v

t, w,

7

m

,

Q

mt

and

M

m

are prop ortio nal to each othe r with an error of order

e. The edge effect solutions are proportional if they contain the multiplier r

(see Column

V)

in equal powers or in powers which differ by four.

Below, the determination of the main boundary conditions is made for all

16 variations of boundary conditions that follow from (1), though not all of

these variants are equally interesting for the practical problems.

First, let us study a simple case when the edge tangent at the weakest point

is or thogonal to the generatr ix

(s '

0

= 0). In this case due t o (3.7) and (3.8),

6 = 0 and the values r in (3.4) are proportional to the roots of equation (3.8)

$ +

1

= 0.



164 Chapter 8. Effect of Boundary Conditions on Semi-momentless Modes

Assuming at the beginning that the edge under consideration

s =

So((p) is

s

= S2(tp) (s2 >

Si).

Th en we can write

A . =

^ ± ^ - (» .« )

The edge effect integrals which decreases away from the edge s = S2 correspond

to these roots.

Let us examine the case of excluding the edge effect solutions from the main

boundary condition

x° + x

e

( C i r f +

C

2

r() =

0,

r

5

=

r

0

^ ,

j = 1,2

(8.4.7)

by the use of two additional boundary conditions

a? + a | ( C

1

r f ' + C

2

r «

i

) = 0,

i = 1,2.

(8.4.8)

Here, two cases are possible. In the first f3 is equal to one of the values a ; or

differs from it by 4. For exa m ple , let /? = ot\. Then the l inear combination

x

o _ £ _ ^ i

= 0 ( 8 4 9 )

a

i

does not contain any edge effect solutions.

In the second case f3 ^ oc\,ot2 and exclud ing th e edge effect solutio ns we

obtain the formula

*° = x

e

E ( - l ) " ' s g n

(/?,-,-/?,)

2

C

-

^4~

a

',

(8.4.10)

i = r l , 2

where

/? -

at =

4 Hi

+ a°, oij

— a,- = 4

riji

+

atji,

Oi=l\^-2\(-l)

a

°,

{ « ? , « « } =

{ 1 ,2 ,3 } ,

(8.4.11)

and ri i and riji are integers.

For example, at

0

= — 1 , a i = 0 , a% = 1 formu la (10) ha s the form

x

w M 4 _ ^ 4 V

(8

.

4

,

2)

165

Naturally, by means of this method we exclude only the main terms of

the edge effect sol utio ns. If we wish to exclude th e higher order t erm s it is

necessary to repeat the procedure.

We also neglect the mutual effect of the edge effect solutions at edges

s\

and

s

2

• That is why below we will only discuss the conditions at one of the edges

without dependence on the boundary conditions at the other edge. Below it is

assumed by default that

so =

s

2

> «i

•

R e m a r k 8 . 1 . If

we consider edge

s

0

= si <

s

2

then in (10) we must

change the value r

0

by - r

0

. If

s'

Q

^ 0 then in (3.7) 6 ^ 0 and formula (10)

must be corrected. In the case when

s

0

= s

2

, the value of

r

0

must be changed

by

r

a

+eb\/2

and in the case when

s

0

= si it must be changed by —

r

0

+ eby/2.

The 16 boundary condition variants considered here may be split into four

groups depending on which boundary conditions are to be satisfied in the

integration of system (1.4)

1) the

c l a m p e d s u p p o r t

group consists of 6 variants (1111, 1110, 1101,

1100, 1011, 1010) and in the zeroth approximation leads to the conditions

^0 = ^ = 0; (8.4.13)

OS

2) the s i m p l e s u p p o r t group also consists of

6

variants (0111, 0110, 0101 ,

0100, 0011,0010)

W o

= $

0

= 0; (8.4.14)

3) the w e a k s u p p o r t group contains 2 variants

(1001,

1000)

(=0

= S ( T ) = *

< » • » >

fw

0

\ d ($>

o~s

4) and the f r e e e dg e group also contains 2 variants (0001, 0000)

$ o = ^ = 0. (8.4.16)

OS

We can write (1.2) for the loading parameter

A

= Ao + e A i + e

2

A

2

+ •••

The groups differ from each other by the main terms A

0

. But it occurs that

within one group the values of

Ao

and Aj coincide and in all cases the value of

Ai is determined by (1.11).

The difference between the boundary conditions within one group are re

vealed in the terms e

2

A

2

. Below, for the clamped support and simple support



166 Chapter

8.

Effect

of

Boundary Conditions

on

Semi-mom entless Mod es

groups,

we can

write

the

main boundary conditions more accurately

by

keeping

the terms

of

order

e

2

which allow

us to

evaluate

the

value

of

\

2

-

For the

weak

suppor t

and

free edge support groups this cannot

be

done since

the

accuracy

of the edge effect solutions developed in Section 8.3 is not sufficient. Als o, such

boundary conditions

are

quite rare

in

real buckling problems.

It should also

be

noted that

in

most cases

we

specify

the

group

by the

tangential boundary conditions. The variants 1011, 1010, 0011 and 0010, in

which

the

tangential condition S —

0 is the

additional boundary condition

are

clearly exceptions.

We will now s tudy the c lamped supp or t group in deta i l . For var ian ts 1111,

1110,

1101 and 1100 the

main boundary conditions

are the

tangential condi

tions

u°

= v° = 0,

(8.4.17)

which in the zeroth approximation by vir tue of (2.7) lead to conditions (13).

According

to (2.7) we get

w°

= 0{e

2

), — = 0(e

2

)

at

s = s

0

. (8.4.18)

From this point onward

we

will normalize

the

solutions assuming that away

from

the

edges

the

deflection w°

is of

order unity.

We begin with

the

case

of

c lamped suppor t

1111 (in

other words u

=

v =

w

= 71 =0) .

According

to

es t imate

(18), the

edge effect intensity

is e

2

t imes

less than what is shown in Table 8.1 and

A a ( u )

= 0,

A « ( D )

=

- 2 ,

A a (10) = 2, A a (71) =4,

(8.4.19)

i.e.

due to (2)

S

= 2 for

conditions

(17). In

order

to

make conditions

(17)

more accurate we e l iminate the edge effect in condition u = 0 by mean s of the

relations

U° + e

2

—

(r^C

1

+r^C

2

)=Q,

S

°

(8.4.20)

w°

+ C

1

+C

2

= 0, r

1

C

1

+ r

2

C

2

= 0.

According

to

formula

(12) we can

more precisely determine

the

main boundary

conditions which take into account terms of order e

2

u°-e

2

^ w

0

= 0, v° = 0. (8.4.21)

soro

It follows from Table

8.1

t h a t

the

edge effect intensity

of

function

v is two

orders

of

magnitude smaller (with respect

to e)

than

the

intensity

of

function

u

and the

condition v°

= 0

does

not

change.



8.4. Separation of Boundary Cond itions 167

For the variant 1110 (ie.

u = v = w

=

M\

= 0) similar calculations give

u

0 _

£

2 J ^ _

w

0

Q j M

0

= 0 ( 8 4 2 2 )

s0r0V2

For the remaining two variants, 1101 and 1100, (i .e. u = v =Q i* =0,

ji =

0 or Mi

=

0), conditions (17) may be used with an error of order

e

5

.

To estimate the error, we take into account the previously noted decay of the

intensity of the edge effect by virtue of (18) and note that the edge effect

solutions for the functions

u

and Qi* are prop ort iona l to each other. Th e

pa ra m et er s A differ in cases 1101 and 1100 by the s am e order of e

5

In cases 1011 and 1010 (ie.

u = S\ = w =

71 = 0 or

M\ =

0) the conditions

u

0 _„„o

0, (8.4.23)

are main ones and S — 2 due to (2) and Table 8.1.

In case 1011, cond itions (23) after being m ad e mo re precise are the following

u

o

+£

ij

s

o

=Qj w

o

+

i*lr

g 0 =0 > (g 42 4 )

91 y/2egqk

and in case 1010, the second term in the left hand side of the second condition

(24) increases by a factor of two.

Now let us s tudy the s imple support group and write the modified main

boundary condit ions for al l 6 variants

V2x

t

s

0

) w°-

-.0, v° = 0 (0111);

y o

+ g 2

_ £ A i * o

=

0 | v

o

=0

( o n o )

. (8.4.25)

s

0

r

0

V2

T f + £ 4

^ ^

= 0

' °

= 0 (0101);

Tj

0

= 0 ,

v° =

0

(0100);

s

3

0

r

3

Q

V2

ds q \

8lr

0

y/2j

w

o _

£ 2

1

d_

+

j4ro

S 0

=

Q

r

0

V2

ds

egkqV2

T

o_

£

i

Jox

ls

o

=Qt w0 +

is

lroV2

s0 =Q ( 0 0 1 ( ) )

q egqk

168

Chapter 8. Effect of Boundary Conditions on Semi-momentless Mod es

It follows from these formulae that for the four first variants, the tangential

boundary conditions are the main ones and for the remaining two variants

that the non-tangential condition,

w°

= 0, becomes the most important. The

correction terms in conditions (25) are of order e

2

compared to the main ones.

Conditions T° = v° — 0 in case 0100 pro vide an erro r of orde r e

4

.

Now we will briefly discuss the problem of separation of the boundary

conditions at a slanting edge (s '

0

^ 0) . As in the case when s'

0

= 0, the

boundary conditions split into four groups

1) the clamped support group (1111, 1110, 1101, 1100),

2) the special group (1011, 1010),

3) the simplesupport group (0111, 0110, 0101 , 0100, 001 1, 0010, 1001,1000),

4) the free edge group

(0001,

0000).

In the zeroth approximation for the clamped support group, the simple

support group and the free edge group, the main boundary conditions (13),

(14) and (16) are the same as in the case when

s'

0

= 0, but the redistribu

tion of the boun dar y co nditions between th e groups mu st be not ed. Va riants

1011 and 1010, which previously belonged to the clamped support group, now

form a special group for which the main boundary conditions in the zeroth

approximation have the form

u,° = ^ - a , - * ° = 0. (8.4.26)

os

The variants 1001 and 1000 which earlier formed the weak support group

are now contained in the simple support group. Here we keep only the correc

tion terms in the main boundary conditions that are of order e compared to

the main terms, unlike the case S'Q — 0 when we keep the corrections terms up

to order e

2

.

Let us first s tud y th e clam ped su pp ort g rou p. It follows from T able 8.1

at s'

0

^ 0 th at the m ain term s of solutions u^ and

v®

are linearly dependent.

That is why we introduced their linear combinations u° and v° (see Table 8.1)

for which we can find the edge effect solutions by means of formulae

u

e

= M ^ c o s x - < s i n x , v

e

= u

e

m

s i n * + v

e

t

c o s* . (8.4.27)

In all of the variants the conditions u° = v° — 0 dom inate and according to

(2.7) they give with an error of order

e

2

as q os

2

In order to derive the main boundary conditions (26) for the special group

it is necessary to exclude the edge effect solutions. This may be done by means

169

of formula (10). As a result we can write for variant 1011

i\/2k g

3 .

u°

m

= S°-

£t

"

/ 9<

l

COS X

w° = 0 (1011) (8.4.29)

r

0

So

from which we get conditions (26) with

<y

2

r

0

sin

2

x

, ,

a i

=- = - . (8.4.30)

V 2 kg so cosx

In a similar way we obtain a

2

=2 a

x

in formula (26) for variant 1010. Here we

are neglecting terms of order

e.

For the sim ple supp ort group we find th at , w ith an error of order e

2

, for

variants 0111, 0110, 0101, 0100, 0011, and 0010, the conditions T£

=

v?

=

0

(or

w°

=0) are dominant and that leads to conditions (14). For variants 1001

and 1000 the main conditions are u„

=

5 °

=

0 which due to (2.10), (2.7) and

(2.5) take on the form

w0 +

iesp g ^

=

$ 0

_ iesp

g ^

= ( )

q

sin

x

cos x ds

q

sin

x

cos

x

os

Finally we study the free edge conditions 0000 (i.e. T =5 ° =

Q^*

=

M °, =0). According to formulae (2.5) and (2.10) solutions T„ and

S°

are

linearly dependent (the main terms) and that is why instead of 5° we use the

linear combination

S° = S°

cosx +

T°

sin x- (8.4.32)

According to Table 8.1, the difference between the orders of the semi-

momentless solutions and the orders of the edge effect solution are equal to

each other

A « ( T

m

) =0, A

a (S*)

=2 , A a ( Q

m

* ) = 3 , A a ( M

m

) =2, (8.4.33)

from which it follows that the condition

T^

=0 is the dominant one.

To derive the second main condition we eliminate from function 5* the

edge effect solutions (in the zeroth and first approximations) by using the rest

of the boundary conditions. As a result we obtain the l inear combination

for which A a =0. After the transformations, the main boundary conditions,

with an error of order e

2

, take on the form

g O

=

y_2i

g

W

m0

_i£«W _^W

= 0

.

(8

.

4

.

35)

OS

K

Sg

k \ S

0

J

OS

We note that, under the construction of conditions (34) it is necessary to use

the refined edge effect solutions (see Remark 8.1).

170 Chapter 8. E ffect of Boundary Conditions on Sem i-mom entless Mo des

8 .5 T h e Effect o f B o u n d ar y C on d i t io ns on th e

Cr i t i ca l Load

In Section 8.4 some variants of the main boundary conditions are intro

duced. We can write them in the general form

'.

?

* + c'i*H = 0 at s = s

k

(ip), (8.5.1)

where

lt

k

=l

k

(

w

° ,& )=Ai

kW

° + B l

k

^ + Cl

k

*° + D{

k

^, ^ = 1 , 2 ,

(8.5.2)

and the coefficients A

J

ik

, B\

k

, ... may depend on o,*o) = 0 a t

s =

s

k

(ip). (8.5.3)

Conditions (3) may be of one of the types (4.13)-(4.16) and (4.25) and in all the

cases the boundary value problem in the zeroth approximation is self-adjoint.

In the first approximation we come to the boundary value problem consist

ing of equations (1.7) and the boundary conditions

& ( u > i , * i ) + & ( i i > o , * o ) = 0 a t

8 = s

k

{<p).

(8.5.4)

As in Section 8.1 we introduce the function V(<p) and write the compati

bility condition for the first approximation

l

^

q

-l^

+

2-

Z

d^{

Z

^a-)

V+{Xl

-

r,)V = 0

-

(8

-

5

'

5)

The value of dXo/dq = 0 at ip —

ipo-

Th e solution of this equatio n of type

(1.9) which is regular at tp = ip

0

exists if

X

1

=

r,

+

(1 /2 +

n)

( A , , A

W

- A ^ )

1 / 2

. (8.5.6)

Here, the value of r\ depends on the type of boundary conditions in the zeroth

approximation. For boundary conditions (4.13)—(4.16) it is equal to

1 f2A

3

*7l*o|

2

,,

| 2

V • . , ,

2

ft2s' , \

"

=

--

z

{-^{-

9

—

d

KI

2

J+*gAoKI

2

(^— -h)

+

+ ^ _

W o

_ *

1

_

+

_ 4

0

_

1 1

,

1

_ j j ^.

(8.5.7)



8.5. The Effect of Boundary Conditions on the Critical L oad

171

In this Section we will not study boundary conditions (4.25). From the physical

sense of the problem r) is real. Direct calculations show that in all of the cases

considered abov e (except for the spec ial grou p 1011 an d 1010 at s' ^ 0 which

have not been considered) we get

7

= 0, (8.5.8)

i.e. the critical load parameter may be found by the formula

\ = \

0

+ e\

1

+

O(s

2

),

A

1 =

i ( A

g g

A

w

- A ^ )

1 / 2

. (8.5.9)

If we take the boundary conditions contained in one group both for

s' —

0 and

for s' ^ 0 and evaluate A, then the differences in the results is of order e

2

.

The complete development of the term e

2

A2 in the expansion for

A

in e is a

complex problem the solution of which one must start by using a more precise

initial system th an syste m (1.1). We will not solve this problem h ere and

the discussion will be limited to the development of the £

2

A

2

portion of this

term which inside one group of boundary conditions depends on the particular

choice of boundary conditions. Here we will assume that s' = 0, i.e. the edge

is straight.

We can write the regularity condition for the second approximation (which

is similar to conditions (6) and (7)) and keep only the terms which depend on

the boundary conditions. Then we get

f

2

A'

2

= e

2

( r /

2

- r

? 1

) ,

k fd$>2 -

,

dw

0

dw

2

= 5 $

0

»72,1 = - £ —

WO

- $ 2

~ X 1"

- 5 — $ 0 - » 2 - 5 —

Z \ OS OS OS OS

wh ere 772 an d 771, which are ind ep en de nt of each ot he r, t ak e in to accou nt th e

boundary condition effect at

s = s

2

and at

s = s\.

T h e form ulae for 7/2 are

given below.

R e m a r k 8.2 . For specific bou nda ry condition we represent the loading

parameter in the form

A = A

0

+ s Ai + e

2

(A '

2

+ A'

2

' ) + 0(e

3

) ,

where the term A'

2

is determ ined above and the term A

2

depending on the error

of system (1.1) is unknown. Since the terms Ao, Ai, and A

2

are the same for the

boundary conditions from one group the differences A

2i

—

A'

2

•

m ay be used to

analyze the effect of the boundary conditions within one group. Here indexes

i

and

j

are the numbers of boundary conditions within the group. As it follows

from relation (10) effects A'

2

,- are to be summarized for the both edges.

V

u . u . i u ;



172 Chapter 8. Effect of Boundary Conditions on Semi-momentless Modes

If we do not know X'2 then the term \'

2

may be omitted in the expression

for A. For a cylindrical shell the value A

2

' is found in [42]. Numerical results

for the various shell parameters and for various boundary conditions (identical

for the both edges) are ptrsented in [22]

Consider the clamped support group (4.13). The modified main boundary

conditions for all 6 variants contained in the group have the form of (4.17),

(4.21),

(4.22), (4.24) or generally

w

° =

e

>C

lt

^ = — + £

2

C

2

, (8.5.11)

os s

where

C\

and

C2

linearly depend on $0, -7;—• According to formulae (10) we

OS

find that

* = - I (

C A

-

C ,

* E ( T ) ) - (••»•»)

By means of formulae (2.7) and (2.5) we can obtain the expressions for C\

and

C

2

for all 6 variants

Ci = ai, C

2

= a

2

+ a

3

(1111);

C i = a i ,

C

2

= a

2

+-a

3

(1110);

Ci = ai, C

2

= a

2

(1101,1100); (8.5.13)

Ci = a

4

,

C

2

= a

2

+ a

5

(1011);

Ci=2a

4

, C

2

= a

2

+ a

5

(1010),

where

ai

=

r®o,

a

3

=

, 7 — $ 0 ,

(8.5.14)

gk\pl as V s J '

5

gk ds \ s

In order to simplify (12) and (13) we use Remark 8.2 and change (13) by

Ci = 0,

C

2

= a

3

(1111);

C* = 0, C

2

= ^a

3

(1110);

Ci = 0, C

2

= 0 (1101,1100); (8.5.15)

Ci =04 — 01, C

2

= a

5

(1011);

Ci = 2a

4

- a i , C

2

= a

5

(1010),

8.5. The Effect of Boundary Conditions on the Critical L oad 173

which lead s t o th e following exp ressio ns for 772

(

l ) _

V

2

q

4

V2

. ,2

.

,

h ~ _ _

3

_ „ 1*01 ( H l l ) i

(1110); r/

2

3)

= 0 (1101, 1100);

%

<7 S

3

T o ^

4

2 )

= l / 2 ^

%

W _ "<?

2

/ i 9 *

0

^ . 9 *o \ «

3

ro

* 0 - 5 — + * 0 a

5 2 V o s a s

a / *

0

(

5

)

- l i

f

*_

^*£L

a. * -

^ * ° \

_

g 3 r

° ^

5 2

' = — ^ I *o ^ — + *o „5 2 V a s

os

ds \ s

d

/ *

ds V

s

(1011);

(1010).

(8.5.16)

Here ro and z are evaluated by (3.7) and (1.8). By calculating 771 (for edge

s = si), the value of ro m ust be changed by — r

0

.

It is simple to verify that the larger value of 7/2 corresponds to the more

stiff support namely

n^ > ^ > nf\

>>\

^>^K

(8.5.17)

We will now study the simple support group. By differentiating the equality

w (.* (ip) ,<p) = 0 at s' = 0 twice with respect to (p we obtain

8

2

w

dr.

dip

2

ds'

and then according to (2.7), condition v° — 0 takes on the form

n

,

x

t

s

2

dw°

w

=

e

(8.5.18)

(8.5.19)

q

2

ds

By using relations (2.5) and (19) we write conditions (4.25) in the form

<D°

=

e

2

a

( i )

dw°

( i )

5$°

d7

+ a2

~ds~

w

o

=

£

, (

a

( 0 * £

+

a

( . ) ^ ,

f

(8

.

5

.

20)

and then due to (10) we obtain

dw

0

^

= ~ l ^

ds

2

(,) <9*o dw

0

, (i) 9w)

0

9* o , (i)

ds ds ds ds

9*o

a s

(8.5.21)

174 Chapter 8. Effect of Boundary Conditions on Semi-mom entless Mo des

Here the index

i (i

— 1,2,.. . ,6) means the number of the boundary condition

variant (using the same sequence of variants as in (4.25)). For simplicity we

use Remark 8.2 and give only the non-zero coefficients

a£'

(l) _

gks fr

0

xfs

2

V2

,

q

2

V2^

ai

-7%{~F~~^

+

, w _ £ * ^ L

(0110

);

<

,(5) _ (3) (5) _ (5) _ J _ _ *t^_ (8.5.22)

As with the clamped support group, the inequali t ies

W> W>$\ $>>W >$\ W > vi

5)

>

4

9)

8 . 5 . 2 3 )

are fulfilled for arbitrary values of —— and ——.

a s os

formulae (10), (16) and (21) may be used to study of the effect of boundary

conditions on the critical load. The main disadvantage of this approach is that

the initial stress state is assumed to be membranous up to the edge while,

in reality, the initial stress state is not truly membrane at the edge and the

magnitude of the initial bending stresses (strains) depends on the particular

type of boundary condition. This question will be discussed in more detail in

Chapter 14.

The results obtained in this Chapter are valid if the shell has an initial

tensile strain such that under the external load the stress state becomes truly

membrane .

R e m a r k 8 . 3 . Here we obtain the formula for the critical load in the form

A = A o + eA i+

e

2

A'

2

+ - • •, where the main term, Ao, is found from the solution of

the boundary value problem in the zeroth approximation. The term eAi takes

into account the localization of the buckling mode at the weakest generatrix.

For boundary conditions which are contained in one group, the first two

terms coincide and the third summand,

e

2

X'

2

,

ta ke s into accoun t t he effect of the

boundary condition within the group and does not depend on the localization

effect. Th ese results ma y be used also for ax isym m etric ally loade d cylin drical

8.6. Boun dary Con ditions and Buckling of a Cylindrical Shell 175

and conic shells of revolution. In these cases

A =

Ao

+£

2

X'

2

,

the buckling pits

occupy the entire shell surface and the term e

2

A^ (as in the general case) takes

into account the boundary condition effect.

8.6 B o u n d a ry C on di t i on s an d Bu ck l ing of a Cy

l indr ical Shel l

As

it

was no ted above in Section 7.4, the form ulae for

a

cylindrical shell

may be obtained from the formulae for

a

conic shell given abov e by th e form al

substi tution of 1 instead of s

a

. In this case the ne utr al surface me tric is defined

by (7.4.1).

The boundary value problem in the zeroth approximation consists of the

equations

k

TT+-*o = 0, k - -q

4

dw

0

+

Xq

2

t

2

wo

= 0 (8.6.1)

os'

1

g os

and the boundary conditions at edges

s

= s\(ip) and

s

= S2(<p) which, accord

ing to the type of support, may have the form of (4.13)-(4.16) and condition

(4.15) is to be replaced by ——

=

——

=

0.

Os os

In cases when the values of d,

g

and

t

2

do not depend on

s,

prob lem (1)

ma y be resolved in the eleme ntary functions. We will consider three vari ants

of the main boundary conditions at two edges, namely

( i ) c lamped suppor t—clamped suppor t ,

( i i) clamped support—simple support ,

( ii i) simple support— simp le supp ort .

In the following formulae the value of a corresponds to the minimum eigen

values.

(i) If at both edges, clamped support conditions (4.13) are introduced then

cos aC cosh aC , Ida ( cos aC cosh aC

cos a /2 cosh a / 2 ' V 3 \ cos a / 2 cosh a / 2 / (S 6 21

(ii) If at edge s — s

2

clamped support conditions (4.13) and at edge s = si

simple support conditions (4.14) are introduced then

sinaC sinh a( . Ida / s i n a C sin h aC

sinh

a V 3 \

sin

a

sinh

a y

(8 6 31

a

=

3.9266, (=^-^-.

176

Chapter 8. Effect of Boundary Conditions on Semi-mom entless Mo des

(iii) If at both edges simple support conditions (4.13) apply then

w

0

= \/2sma£,

$

0

= W - — s i n a C ,

a = ir,

C = —

j ~ ^ -

(8.6.4)

To simplify the following formulae we will assume that the normalization

in (7.1.3) is such that

d = g = k = t2 = la,t

the weakest gen eratrix

ip =

<po.

Then, due to (3.7) r

0

= 1.

As in Section 7.4 we find

4 a 3

1/,4

a

A

°

=

3374/'

q

°

=

~

— '

l(f)-

s

2-si, z =

3

1/4

a, (8.6.5)

where z is the same as in (1.8) and (5.10).

We assume that edge

s — S2

is considered an d

s'

2

(<po)

= 0 . In o rder to

evaluate the value of

772,

which takes into account the effect of the support

(see (5.10)), we use formulae (5.16), (5.21) and (5.22). We can then write the

values of functions u>o and $0 an d th eir d eriva tives which are con taine d in

these formulae

. . , ,

3 $

0

5.366

<Po = 1 .1 5 5 , —— = —

-

— (c lamped—clamped) ,

os I

$0 = 1.155, - 5 — = —

'-r—

( simple suppo r t— clam ped) ,

ds I

dw

0

_

5.71

d$

0

_

3.12

ds I ' ds I

dw

Q

_ 4 .44 <9£o _ 2.565

~ds~ ~

F '

h~~ V~

(c lamped—simple suppor t) ,

( s imple suppor t—simple suppor t) .

(8.6.6)

The effect of the support at edge s =

s%

on the critical load A can be

described by the parameter

77*

introdu ced by

\~ \

0

+ e

2

m

= \

0

(l +

£

-n

t

), n* =

1

-^- (8-6.7)

/ " ' " X

The term eAi is not included in (7) since it does not depend on the boundary

condition variant within a given group.

If we consider the buckling of an axisymmetrically loaded circular cylin

drical shell with straight edges and with constant values of

E, v

and

h

then

Ai = 0 (see Remark 8.3). Formula (7) takes into account only the effect of the

support on edge s = s

2

. For edge

s = s\,

a similar term must be introduced.

If edges

s = s\

and

s = s

2

are supported identically then the effect of the type

of support is doubled.



8.6. Boundary Conditions and Buckling of a Cylindrical Shell

177

Table 8.2: Parameter

r)„

corresponding to the boundary conditions

/

m i

1110

1101, 1100

1011

1010

c lamped—clamped

II

1.414i/

2

0.707

u

2

0

1.494

v

- 0.788

1.494

v

- 1.576

III

0.29

0.14

0

- 0 . 7 7

- 2 . 5 6

simple suppor t—clamped

IV

\A\Au

2

0.707

v

2

0

1.524 v - 0.816

1.524i/ - 1.631

V

0.29

0.14

0

- 0 . 8 1

- 2 . 6 6

Table 8.3: The simply supported edge

/

0111

0110

0101

0100

0011

0010

clamped—simple suppor t

II

1.295 - 0.354 * , +

+ 0 .070x?

0 .024x?

0.647

0

1.161 - 0 .1 3 9 * ,

-0 . 38 6 - 0 .193**

III

2.27

0.87

1.47

0

0.88

- 2 . 6 3

simple suppor t— simple suppor t

IV

1.223 - 0 .4 1 9 * ,+

+ 0.071*2

0.036 **

2

0.612

0

1.125 -

0 . 2 4 1 * ,

-0 .4 0 8 - 0 .241 * ,

V

1.55

1.31

1.39

0

0.36

- 3 . 1 1

In columns

II

and

IV

of Ta bles 8.2 and 8.3, the values of 77, corresponding

to the bo un da ry cond itions of colum n / are given. T able 8.2 is for the clam ped

edge, s = S2, and Table 8.3 corresponds to the simply supported edge, s = sz-

Columns II in bo th Tables correspond to the clam ped edge s = s\ and columns

IV

are for the simp ly sup po rted edge,

s = s\.

It follows from (7) and Tables 8.2 and 8.3 that the boundary condition

effect within one group is determined by the value of e

2

l~

l

r]

9

and depends on

http://a/Au2

http://a/Au2

http://a/Au2



178

Chapter 8. Effect of Boundary Conditions on Semi-mom entless Mo des

Table 8.4: Main boundary conditions

1

2

3

4

5

6

7

8

9

10

S = S i

clamped

clamped

simple support

c lamped

clamped

simple support

simple support

weak support

weak support

free

s =

s

2

clamped

simple support

simple support

weak support

free

weak support

free

weak support

free

free

A

0

/

8.30

6.89

5.51

4.15

3.29

2.75

0

0

0

0

three dimensionless parameters

y = — £ — ( 1 2 (1 - 1 / ' ) ) , v, x * = i x

t

= — , (8.6 .8)

and for the straight edge x* = 0.

As an example, consider the same shell as in Example 7.1. We take R/h =

400, L

0

/R = 4,/3 = 45°, and v = 0.3. Then x, = 4, and e

2

/ "

1

= 0.0227.

In co lumns 77/ and

V

of Tables 8.2 and 8.3 the value e

2

/

- 1

?;* for the above

mentioned parameters is given ( in percentages) .

One can see that , within the clamped support group, the boundary con

dition effect at edge

s = S2

does not exceed 3 % , and for the sim ple su pp ort

grou p it does not exceed 5 % . Th is influence increases (as

^/Tu)

with the shell

thickness.

In general the boundary condition effect within a given group is small. The

effect of the b ou nd ary cond itions on the pa ram ete r A when going from one

bou nda ry condition group to anothe r is s tronger . Th e reader may com pare

the results.

Table 8.4 includes all possible combinations of main boundary conditions

in the zeroth approximation and the corresponding values of Ao. For the first

three lines in Table 8.4, the dependence of the parameter A on the full boundary

conditions was discussed above. The method of studying buckling proposed in

Chapters 7 and 8 is not valid for the variants shown in the last four lines in

Table 8.4 since in these cases the buckling mode is not localized. The cases of

shells of revolution are discussed in Chapter 12.

8.7. Conic Shells Under External Pressure

179

The boundary condition effect on the critical normal pressure for the buck

ling of a circular cylindrical shell has been extensively studied in [22, 61, 113,

136,

143, 149] and others. In [143] the numerical results are given as a function

of the para m eter z

m

= L

{Rh)~

l

l

2

{\

— v

2

)

l

l

4

which is related to our par am ete r

e

2

l~

l

(see (7) an d (8 )). Th e effect of th e bo un da ry con dition s on the shell pa r

allel on the upper critical load of the shell reinforced with the ring is examined

in [41, 42].

In [39] buckling of a mitred pipe joint consisting of two connected circular

cylindrical thin shells with slope edges under homogeneous external pressure

is analyzed.

8 .7 Co n ic She lls U n d er E x t e r n a l P re ss u re

Consider a conic shell with constant parameters E, v and h under a con stant

external pressure

p.

We assume th a t

s

*2 = r7^T >

P =

XEhe

6

. / / i \

5 / 2

= A t f ( £ ) ( l 2 ( l - ^ ) ) ~

3 / 4

(8.7.1)

k { 

-w"o~=^yr

+a

bwo

=

0

(8J

-

2)

which is convenient for comparison with a cylindrical shell.

Here introduce the designations

s =(l-xy)s

2

, l = S2 — si, H = IS^

1

,

q

4

= ai>, ^ = kl-

2

s

3

2

, X

0

= br), rj = A :

3

/

2

^

3 7 4

/ -

1

.

(8.7.3)

Shell edges si(y>) and S2(<p) due to (3) are transformed into

y

= 1 and

y

= 0

at which the two main boundary condit ions must be introduced. Here we wil l

only consider shells with clamped (4.13) or simply supported (4.14) edges.

For x = 0 w e obta in a cylindrica l she ll and the case of >c = 1 corresponds

to a conic shell closed in a vertex. In the last case of system (2) for y = 1 there

is a s ingular point and the boundary condit ions at

y

= 1 are replaced by th e

limiting conditions for the solution.

System (2) is not integ rable analytically in the known functions. By nu

merical solution of boundary value problem (2) with corresponding boundary

conditions at edges y — 0,1 for the fixed values of a and n one can obtain

function 6 =

b (a, K)

and then find its minimum with respect to

a

and

x.

180

Chapter 8. E ffect of Boundary Conditions on Sem i-mom entless Modes

Conditions (7.2.11) determining A§ and the values of go, and

ipo

ma y be

written in the form

— = — = 0 ?h. = b—+—— =0 (8 7 4)

dq da 'dip dip dx dip

The first of these equations determines the functions

a = a(x), b°{x) = b(a{x),x),

(8.7.5)

And from the second of equations (4) we find the weakest generatrix <p

= <p

0

and the corresponding values of

^o =

mK , it =

aoV'o, (8.7 .6)

where

a

0

= a(x

0

), b% = b°(x

0

), (8 7 7)

x

0

=

x(tp

Q

),

Vo = tp

(<Po),

Vo =

,

n(

l

Po)-

To use formula (1.11) to evaluate the correction e\\ of th e value A° one

must find the values \

qq

, \

qv

, and \

w

. Due to (3), (4), and (5) we get

where

\ q

=

i]ob

aa

a

q

,

\ip = 11obaa<lq (< V ~

a

xX<p) ,

\ip —

b rftpq,

+ b„ {qoHw + 2n

lfi

x

v

)

+

+

rjobaaa^

{a y — 2a

it

x

v>

) + ^ob^x^,

4 go aoYv

a

a

= " —

(8.7.8)

V ' o ' V'o

(8.7.9)

With subscripts we denote the derivatives with respect to the corresponding

variables calculated at g = go and tp =

<po.

To derive (8) we use the relation

b

a> e

= —b

aa

a,x

obtained by means of differentiating the first of equations (4)

by

x.

In order to use formulae (8) for calculations it is necessary to know the

functions b°(x), a(x), b^x), a^x), b

aa

{x), and b^^x), which are determ ined

in the numerical solution of boundary value problem (2).

Figure 8.2 shows plots of the functions

b°(x), a

0

{x), b

aa

(x),

and b

itit

(x) for

four boundary condition variants: s imple support—simple support

(SS),

sim

ple suppor t—clamped

{SC),

c lamped— simple suppor t

(CS),

and c lamped—

clamped

(CC).

In each case, the wider shell edge is m ent ion ed first.

8.7. Conic Shells Under External Pressure

181

O

0.5

\ x O

0.5

Figure 8.2: The functions

b°(x), a

0

(x), b

aa

(x),

and

b

Ki{

(x)

1 x

The funct ions b* and a* may be found approximately by means of numer

ically differentiating curves 6° and

a

which are shown in Figure 8.2.

As it is expected for small values of x, the curves SC and CS for functions

b°(x)

and

ao(x)

in Figu re 8.2 converge since for shells which are nea rly c ylin

drical i t is not important which of the edges is clamped or simply supported.

On the contrary, for

x

~ 1 th e pair of curves labelled

SS, SC

and

CS, CC

converge since then the shell stiffness increases as the shell edge approaches the

cone vertex and the type of support for the narrow edge becomes unimportant .

System (2) at

y — x~

l

ha s an essen tial sing ular po int [142, 180]. If

x

is

close to 1, then the singular point is close to edge point y = 1 th a t influences

the behaviour of the curves as shown in Figure 8.2. It is especially notable for

functions

b

aa

(x )

and

b

itH

(x )

since, after differentiating, the oscillation am pli

tude increases.

Under these assumptions the existence of the weakest generatrix

ip

=

tpo

is

connected with the variabili ty of functions

n (<£>)

a nd

x(<p)

(see (3)) which is

dete rm ined by the dependen ce of functions

s\, S2,

and

k

on

<p.

In particular, we are interested in a circular conic shell with a slanted edge

and vertex angle 2a (see Figure 8.3). The upper edge is inclined by angle

(3

to the lower edge. As a characteristic size,

R,

we take the lower edge radius.

182

Chapter 8. E ffect of Boundary Conditions on Semi-mom entless Mod es

Figure 8.3: The slanted conic shell.

Then

k = cot a, «2 = — , ' (</>) = *2

(1 + c) (1

—

/o sin a )

1 + c cos

 ^ 27rs ina , x = / s i n a , c = tan a tan j3 < 1, (8.7.10 )

I

1

s in a I

Ao = sin a (cos a )

3

l

2

b* (x),

where the IQ

= I

(0) corresponds to the longest gen eratr ix.

From the last formula it follows due to (4) that function 6* attains its min

imum value at the weakest generatr ix . Function

b*(x)

is shown in Figure 8.4.

1 *

Figure 8.4: The functions

b*{x)

and

d{x).

8.7. Conic Shells Under External Pressure 183

In the buckling problem of a circular cylindrical shell with a slanted edge

under a homogeneous external pressure, the longest generatrix is also the weak

est (see Example 7.1). It then follows from (10) and Figure 8.4 for a conic shell

with

x

m

< 0.5 th a t the longest ge ne ratr ix also is the weakest one. Here, we

denote the value of x corresponding to the longest generatrix by x

m

.

If

x

> 0.5 the dependence of

b*

on

x

is mo re complex.

If the narrow shell edge is clamped (see the curves CC and SC in Fig

ure 8.4), then function

b*(x)

is monotonic and at

x

—>• 1 (the full cone) it

converges to

b*(l)

= 26.75 for clam ped su pp ort of the wide edge

(CC)

and to

6*(1) = 18.30 for simple support of the wide edge

(SC).

The difference between b*(x) and b* (1) is less than 1 % for x > x\, where

xi

= 0.55 in case

CC

and

x\

= 0.63 in case

SC.

T h a t is why, if

x

m

>

xi,

we recommend the use of

A

= Ao at

b* — b*(l)

in the analysis. The correction

term eAi (see (1.2)) may be neglected since it takes into account the local

character of the buckling mode, but in our case

b*(x)

is close to constant. This

contradicts the assumption about the local character of the buckling mode.

If the narrow shell edge is simply supported then the function

b*(x)

has an

infinite number of minimums at

x —>•

1, only one of which at

x

=

XQ

can be

seen in Figure 8.4 (see curves CS and SS).

We have

x

0

=

0.53,

b*(x

0

)

=

26.27 in case

CS

and

x

0

= 0.59,

b*(x

0

)

=

17.85 in case SS. For x

m

> XQ we recom m end t ha t one use in the analysis

A = Ao at b* — b*(xo) and that the term eAi be neglected for the same reason

as before.

Let us now return to the case when x

m

< 0.5 and by m ea ns of (1), (1.2),

(1.11) and (8) represent the critical pressure in the form

>~<T(»»-*rHH<$&T^

/2

+ (

(8.7.11

where

6°

xo = /os ina , AQ = — (cos a )

3

/

2

,

n 0

a a

1 - x

0

) I . XQby, \

a

0

' 6

aa

(l

-x

0

) ( x

0

b,

6 ° V

b

°

For the boundary condition variants considered, the function

d(x)

is show n

in Figure 8.4. For x > 0.5 this function approaches zero which agrees with the

earlier proposal about neglecting the term eAi. For

a = x

0

—

0, (11) is valid

for a cylindrical shell with a slanted edge and in the case when both edges are

simply supported it coincides with (7.4.9).

184

Chapter 8. E ffect of Boundary Conditions on Semi-mom entless Mod es

For a circular conic shell with straight edges we assume that d = 0 in (11),

and that the number of waves

n

in the circumferential direction is equal to

_ q sin a _ I a\{>t) R

2

h

2

cos

2

a

U

~ ~~T~

=

V 12 (l-v

2

)

L

4

where

a

0

{x )

is show n in Figure 8.2.

8 .8 P ro b le m s an d Ex erc i ses

8.1 Derive formula (4.10).

8.2 . Solve num erically the buckling proble m for circular cylindrical shell

under external normal pressure that is a linear function in coordinate s for

different main boundary conditions at edges

s

= 0,1. The results represent in

the form of a table similar to Table 8.4.

H i n t .

Integrate system (6.1) for k — g = q = 1, <2 = s/l.

8.3 Find the weakest generatrix of the truncated non-straight conic shell

unde r extern al homo geneous pressure. Th e shell edges are the circles lying in

the parallel planes and one of the generatrices is orthogonal to these planes.

Consider various boundary conditions.

H i n t . Use the results of Section 8.7. The required generatrix may be found

from the equation

drj/dip =

0.

i / u

(8.7.13)



C h a p t e r 9

T o r s i o n a n d B e n d i n g o f

C y l i n d r i c a l a n d C o n i c

She l l s

In this Chapter we will consider the buckling of shells of zero Gaussian cur

vature with the same modes stretched along the generatr ices as in Chapters 7

and 8. It is assumed that the initial stress-resultant T$ equals to zero or small

and that buckling occurs due to the shear stress-resultant 5°. Such a stress

sta te m ay ap pea r in a shell which has loads applied to i ts edge. T he inter

nal pressure that leads to the appearance of hoop,

T$

, and axial,

Tf,

tensile

stress-resultants on reinforced shells are also considered.

W hen writin g this C ha pte r we used the resu lts obtain ed in [61, 97, 98, 103,

152, 159].

9.1 T ors ion of C yl in dr ic al Shel ls

Let us consider the buckling of a cylindrical shell of moderate length under

combined loading assuming that the shear stresses are dominant.

Instead of formulae (7.1.3) we introduce new notation

(T?,T°,S°) =XE

0

hos

5

(e-H

0

1

,e t

0

2

,t°

3

),

(9.1.1)

which means that the critical value of the shear stress-resultant

S°

has order

greater than the critical value of the stress-resultant

T$ •

The orders of the

stress-resultants T°, T® and 5° are chosen in such a way that all of these stress-

resultants affect the zeroth approximation for the boundary value problem

when *?,*§,*§ — 1.

185

186 Chapter 9. Torsion and Bending of Cylindrical and Conic Shells

We assume that the values

d, g

and

t°

do not depend on

s.

The n the

boundary value problem in the zeroth approximation has the coefficients that

are constant in

s

and we can write the following equation

+

d q

4

wo

— XQN WO = 0,

2

g d

4

w

0

q

4

ds

4

n

d

Wn „ .

n

Own r,

n

Nw

0

= t1 —-± + 2iqt°

3

—^- q

2

t°

2

w

0

.

os

z

os

The general solution of equation (2) has the form

(9.1.2)

w

0

= J2

CkePk

"> p

k

= iq

1

xk,

k = \

and for x/, we get the fourth order equation

x

4

+ a2X

2

+

(I3X

+ 014

= 0,

where

a

2

=

AQ<J

0.3

2Ao^3

0 4

d

AQ^2

gk

2

' gk

2

q ' gk

2

gk

2

q

2

We represent the roots of equation (4) as

xi

t2

=

a±if,

X3,4— -a±

/?,

then

2 N

l / 2

3

a

2

= / ?

2

- f + ( 3 a

4 +

( 2 / ?

2

+ | )

:

(9.1.3)

(9.1.4)

(9.1.5)

(9.1.6)

Y = 2a

2

+ f3

2

+ a

2

,

(9.1.7)

a

3

=

la {f +

7

2

) .

(9.1.8)

We only consider the boundary condition groups of clamped and simply

supported types (see Section 8.4). If, on both edges

s =

0 and

s = I,

the same

boundary conditions apply then the equation which determines the parameter

AQ

has the form

tan/3i

coth 71

cos 2« i

sinh7i cos/?i

(<* i , / ? i ,7 i ) = 9 i (« , / ?> 7)

> qi=Q

2

l,

(9.1.9)



9.1.

Torsion of Cylindrica l She lls

187

where

2/3

7

6 a

2

+ a

2

8 a

2

/ ?7

for

w

0

= w'

0

=

0, (9.1.10)

• - ,

0

o , 0/P2 i >2 , i 2 r t 2 , \

f O T W

° =

W

0 = ° - ( 9 . 1 . 1 1 )

( 2 a

2

+ 2 p

2

+ a?,y + 4a

2

(2a

2

+ a

2

)

If there are different boundary conditions on the edges (WQ = w'

0

= 0 for

s = 0 and wo = w

0

' = 0 for s = I or vice-verse), then instead of (9) we have

the equation

4 a 7 s in 2 a i

p (2/?

2

- 2 a

2

+ a

2

)

t a n / ? 1

-

7

( 6 a

2

+ 2 / ?

2

+ a

2

) t an h 7 i +

(2/?

2

- 2 a

2

+ a

2

) cos/?i cosh 71

(9.1.12)

We will first consider the buckling of a circular cylindrical shell with con

s tan t parameter s

I, h, E

and

u

under pure torsion. In this case

t°

1

=t°

2

= a

2

= 0, d = g = k = 4 = a

4

=l.

(9.1.13)

Then equation (9) coincides with the equation obtained in [159], and equa

tion (12) coincides with the equation derived in [152]. Equations (9) and (12)

may be easily solved by taking an arbitrary value of/?, evaluating a and 7 by

formulae (7) and

fi\

from equations (9) and (12), and then evaluating

q

and

Ao by formulae (9), (5) and (8).

We note that in the neighbourhood of the critical load 71 ~ 10, and there

fore we can simplify equations (9) and (12) replacing the square brackets in the

right sides by unity. We represent the results obtained in the form proposed

in [61]

s ° , / h> \

5 / 8

/ R \

1 / 2

T

0

— — = k

s

E

h Kd-^v,

w ( 9 1 1 4 )

n

0

= — -

L

'

2^

D

6 \ l / 8

e " V h

2

L

4

where r

0

and no are the critical shear stress and the number of waves in the cir

cumferential direction. The dimensionless coefficients

k

s

and

k

n

depend on the

boundary conditions and for the clamped edge—clamped edge (CC), clamped

edge—simply supported edge (CS) and simply supported edge—simply sup

ported edge (SS) cases their values are

k

s

=

0.770, fc

n

=4.41, /? = 0.320 (C C ),

k

s

= 0.738, A r„ = 4. 15 , /? = 0.332 (C S), (9.1.15)

*, = 0.703, &

n

= 3 .8 7 , /? = 0.345 (SS),

188

Chapter 9. Torsion and Bending of Cylindrical and Conic Shells

and

A

0

= 1 2

5

/

8

* , r

1 / 2

,

q

0

= l2-

1

<*k

n

r

1

t

2

, k

n

=

12

1

/

8

g

1

1 /2

. (9.1.16)

Comparing the critical stresses for shell pure torsion in the cases of simply

supp orted and clamped b ound ary c onditions we note th at they differ by 10% .

It can be seen in Table 8.4, that the same difference for buckling under external

normal pressure is 50%.

The weak dependence on the boundary conditions noted above allows us to

use the following approximate method [61, 175] in solving the torsion problem.

In solution (3) we keep only two te rm s

(k —

3,4) and satisfy only one b ou nda ry

condition,

w

= 0, on each shell edge. Then instead of (9) and (12) we get the

equat ion

t a n / ? i = 0 , /?I =

TT.

(9.1.17)

As a result we obtain the values

fc

s

= 0 .74, * „ = 4 . 2 , (9.1 .18)

which usually are referred to as the classical values [61].

A reviews of the works on the problem under consideration are given in

[61, 149]. W e also no te th e wo rks [3, 58], in the first, th e influence of th e

boundary conditions on the critical load and in the second the influence of

non-linear terms on the critical load were studied.

We can use formulae (14) only for shells of m od era te len gth . T he inequa lity

no > 1 (for ex am ple , no ^ 5) m ay be the app lica bility criterion for formu lae

(14). W e will no t consider long shells, bu t it should b e note d t h a t for sufficiently

long shells, no = 2, and the critical load does not depend on the length but is

determined by the relation [137]

E f h

2

\

3 / 4

^

=

4 ((T w) •

(9

-

L19)

9 .2 C yl indr i ca l She ll u n de r C om bin ed Lo ad in g

We will now consider the buck ling of a circular cylindrical shell of m od era te

length under combined torsion, internal pressure and axial force. We assume

tha t the parameter s

E, v

and

h

are constant. The stress-resultants

Tf,

T%

are

assumed to be known and the stress-resultant 5° to be unknown. Instead of

(1.5) we denote

a

2

= t\, a

3

= — (4=1), a

4

= l + \

(9.2.1)

9.2. Cylindrical Shell under Com bined Lo ading

189

Re f e r r i n g t h e i n i t i a l s t r e s s - r e s u l t a n t s t o t h e c o r r e s p o n d i n g c r i t i c a l v a l u e s w e

i n t r o d u c e i n s t e a d o f ( 1 )

— 1/ 2 -*^2

a

2

= 2 i l i ,

a

3

= 7.0 R

3

q

1

, a

4

= 1 + 5 . 5 1 — ,

9i

w h e r e

R -

ai

0 '

R -

P

Po

R

3

=

TO

<T° = 2Ee\ p

0

=

4TT Ehe

e

(9.2.2)

(9 .2 .3)

H ere <Xi is th e a xi al te ns i le s tr es s , <T\ i s t h e c r i ti c a l v a l u e u n d e r a x i a l c o m p r e s

s ion ( s ee (3 .4 .3 ) ) ,

p

i s the in t e rna l p res sure , po i s the c r i t i ca l va lue o f ex te rna l

p res sure fo r a s imply suppor ted she l l ( s ee (7 .4 .9 ) fo r k

t

= 1 ) , an d TQ i s th e

cr i t i ca l va lue for the t an ge n t i a l s t r es s und er to r s io n ( s ee (1 .14) for k

s

= 0 .74) .

f l i = -0.5

O

1 2 3

R

2

-1

0

1 2

F i g u r e 9 . 1 : T h e f u n c t io n s ^ ( i ^ ) a n d fc„(i?2)-

I n F i g u r e 9 . 1 t h e d e p e n d e n c i e s o f

R3

a n d

k

n

o n

R2

for different values of

R\

a r e s h o w n . T h e s o l id c u r v e s c o r r e s p o n d t o a s h e ll w i t h c l a m p e d e d g e s , w h i l e

t h e d a s h e d c u r v e s c o r r e s p o n d t o a s h e ll w i t h s i m p l y s u p p o r t e d e d g e s . T h e

r e s u l t s w e r e o b t a i n e d b y t h e n u m e r i c a l s o l u t i o n o f e q u a t i o n ( 1 .9 ) f o ll o w e d b y

t h e m i n i m i z a t i o n b y t h e p a r a m e t e r g

x

, c o n n e c t e d w i t h t h e n u m b e r o f w a v e s ,

n, i n t h e c i r c u m f e r e n t i a l d i r e c t i o n . T h e c o m p r e s s i v e s t r e s s - r e s u l t a n t s T ° a n d

T j c o r r e s p o n d t o n e g a t i v e v a l u e s o f

Ri

a n d

R

2

.

In F igure 9 .1 one can see tha t the d i f f e rence be tween the r esu l t s fo r she l l s

w i t h c l a m p e d a n d s i m p l y s u p p o r t e d e d g e s d e c r e a s e s a s R

x

a n d R

2

i n c r e a s e .

For cases when th i s d i f f e rence i s no t t aken in to accoun t the numer ica l r esu l t s

are given in [61] .

W e c a n a l s o w r i t e a p p r o x i m a t e f o r m u l a e f o r

R3

a n d

k

n

a s s u m i n g t h a t

Ri

o r

R

2

a r e l a rge . T h e roo t s (1 .6 ) o f eq ua t io n (1 .4 ) we r epre sen t in th e fo rm of

190


part of the negative power series in the large parameters

R\

or

R?.

We find

i?3 from the eq ua tion (1.9) in the sam e form .

First, let us consider the case of a large internal pressure

(R 2

S> 1, .ff 1 = 0).

In thi s a ssu m ptio n /? <C 1. We assum e th a t /? = 0 in (1.7) and (1.8) and get

« = ( ^ )

1 / 4

, 7

2

= 2*

2

, J f e ^ I ' V / * . (9.2.4)

Minimizing R3 by q\, we find

i?

3

= 0.949

R\

12

, k

n

=

2.264

R \'

2

.

(9.2.5)

formulae (5) do not reveal the dependence of

R3

on the boundary condi

tions. To correct (5) we retain terms of order (3

2

. From eq ua tion (1.9) we find

approximately that

/? = T ( ? i - C r \ (9-2.6)

where £ = 0.47 in the case of clamped edges and £ = —0.94 in the case of

simply supported edges. Now we can write a more precise formula for

R3

R

3

= 0.9A9Rl

/2

ll + — =• I . (9.2.7)

V

^ ( l - o . s e c ^

1

)

2

;

formulae (5) and (7) show that R3 and k

n

increase as R

2

, and the relative

uence of the boundary conditions decreases as i?J .

In [61] in the case R2 <

1

i t is recomm ended to use the approx im ate formula

R

3

=

1 + 0.207

R

2

-

0.00175

R\,

(9.2.8)

obtained by the approximation of the dependence of

R3

on

R2,

found numeri

cally. We also note the similarity of formula (5) and formula

R\ — R2 =

1 (see

[61]) for large values of R2.

Now we consider buckling under torsion for large axial expanding stresses

(R i

3> 1,

R2

~ 1)- We represent the roots of equation (1.4) in the form of

(1.6).

T hen for 02 > 1 we get app roxim ately

2a

2

2a

2

From equation (1.9) we have approximately

o

L ( / g = >

/ a § - 4 a

a

o

4 >

^ ^

( g

^

g )

/ ? = — f - — , (9.2.10)

9.3.

A Shell with Non-C onstant Parameters Under Torsion

191

where the parameter rj takes into account the influence of the boundary condi

t ions and rj =

(TTRI)"

1

in the case when the shell edges are clam ped and t] ~ 0

in the case when the shell edges are simply supported.

Equating the expressions obtained for /?, we get

Minimizing Rz by

171,

we find

/ o 3 / 2 N

1

/

2

i ?

3

= 1 . 2 1 r -

1

— + 0 . 6 2 f l i i J

2

, fc„ = 6.06

itf'

2

.

(9.2.12)

It follows from the last formula that the relative difference of

R3

for shells

with clam ped and simply sup por ted edges decreases as i?J~ as

R\

increases.

9 .3 A She ll w i t h N on - C on s t an t P a r a m e t e r s U n

der Tors ion

In this Section we will assume that

t\

=

t\ —

0, and the variables

^3, k, I,

g

and

d

may depend on

ip .

In other words, we will examine the torsion of a

non-circular cylindrical shell of length, wall thickness and elastic properties all

of which vary in <p. Since these variables do not depend on

s

we may use the

results of Section 9.1.

By virtue of (1.5) and (1.8) we have

Ao = ^

g

a ( / ?

2

+ 7

2

) , (9.3.1)

where formulae (1.7) give the dependence of a and 7 on /? and tpQ ( through

04). To make the study of the dependence of

a

and 7 on

ipo

m ore convenient

we make the substi tution

(a , /? ,7 ) = 4

/ 4

(

a o

, / ?o , 7 o ) . ( 9 . 3 . 2 )

Th en Ao m ay be represented as

*o = f{q,<Po) = F(<p

0

)G[p

0

{q,<p

0

)] , (9.3.3)

where

/?o(<7, fo)

is the solutio n of the equ atio n \P

(/?Q)

= i] (<po)

q~

2

, where we

192


denote the right hand side of equation (1.9) or (1.12) by ri(/?

0

). Here

*(/3o) = ^ ( j r + a r c t a n n ^ ) ) "

1

,

G ( / ? o ) = [ V ( / ? o ) ]

_ 1 / 2

ao ( / ?

0

2

+ 7o

2

),

3«

0

2

= ^ o

2

+ ( 3 + 4 / ?o

4

)

1 /2

, To

2

= 2 ^ +

/?

0

2

,

{

™

A)

„ ,

,

1 fk

6

g

3

d

5

\

1/S

, , 1 fgk

2

\

1/4

In transit ion to appro xim ate equ ation (1.17) we should assu m e tha t \P(/?o) =

ft*

-1

.

The weakest generatrix could be evaluated after the minimization of

F (<po)

by

ipo-

Keeping the de signation

<po

for this generatrix we normalize the deter

m ining para m ete rs in such a way th at relations (1.13) are satisfied. Th en

F(fo) =

lo

1/2

, v(<Po)

= lo\ lo =

l(<Po). (9.3.5)

Minimizing now by

q,

we come to the same formulae (1.16), in which we

should take / = IQ .

To evaluate the first correction

eXi = \e(X

qq

X^-X

2

qv >

)

1/2

(9.3.6)

for the load parameter AQ we calculate the second derivatives in (6) and obtain

X

qq

= bul

Q

, X

qlfi

=

— & 1 2 ^ 0

~ j ,

d<p0

(9.3.7)

, d?F .3,2 (dr, V

A w

-

a 2 2

^ f

+ & 2 2 /

°

[dyj •

where the functions F and rj are given by formu lae (4). T he c on sta nts 022

and

bjj

depend only on the boundary conditions and are determined by the

formulae

a

2 2

=

1 2

5

/

8

^ ,

6n = 4.12

3

/

4

fc-

6

6 , 6

12

= 2.12

5

/

8

fc-

5

6,

d

2

Gfd ^

( 9

-

3

-

8 )

6

22

=

I2 k-H,

b =

dPo \dpo

where k

s

and k

n

have the values of (1.15). To finish the calculations we must

find the value of

b.

For the cases described by (1.15) and (1.18) we obtain

respectively

6 = 385.6, 6 = 265.5, 6 = 172.7, 6 = 28 3. 0. (9.3.9)

9 . 3 . A Shell with Non-C onstant Param eters Under Torsion

193

I t fo l lows f rom the form of the coef f ic ients 6 , j that the value of eAi does not

di)

d e p e n d o n t h e t e r m s c o n t a i n i n g — — .

dtpo

The f ina l r esu l t we wr i t e as

(

h2L

' \"" (F\

11

''

n l

, .

c

k*k

1/ 2

w h e r e

TQ

i s d e t e r m i n e d b y f o r m u l a ( 1 . 1 4 ) , a n d t h e m u l t i p l i e r

k*

t a k e s i n t o

a c c o u n t t h e v a r i a b i l i t y o f t h e d e t e r m i n i n g p a r a m e t e r s . Co e ff ic ie n t c d e p e n d s

o n t h e b o u n d a r y c o n d i t i o n s a n d is e q u a l t o

c = 0 .30 5 , c = 0 .310 , c = 0 .316 , c = 0 .30 8, (9 .3 .11 )

He re the va lu es o f c a r e wr i t t en in the s am e seque nce as in (9 ) . S ince the se

va lues a re c lose to each o ther , we can t ake an average va lue of c — 0.310 for

al l cases .

T h e c o r r e c t i o n t e r m O ( e

2

) i n (10) depends no t on ly on the main bu t a l so

o n t h e a d d i t i o n a l b o u n d a r y c o n d i t i o n s . A s w e n o t e d b e fo r e , fo r a c o m p l e t e

e v a l u a t i o n w e s h o u l d s t a r t w i t h t h e s y s t e m o f e q u a t i o n s w h i c h i s m o r e p r e c i s e

t h a n ( 8 . 1 . 1 ) .

By v i r tu e o f (7 .2 .3 ) th e r a t e of dec rease o f th e de p t h o f th e p i t s away f rom

t h e w e a k e s t g e n e r a t r i x ip — < p o i s d e t e r m i n e d b y t h e m u l t i p l i e r

e

\~

a

nf~f

0

) J ,

w h e r e

9 a . „ fR

4

{l-v

2

)\

1/S

(d

2

F\

U2

a i

= T =

L 7

H-b^J UiJ •

(9312)

For example , we cons ider the to r s ion o f a non-c i r cu la r cy l indr i ca l she l l o f

t h e c o n s t a n t ( m o d e r a t e ) l e n g t h w i t h c o n s t a n t p a r a m e t e r s

E, v

a n d

h.

T h e t o r s i o n o c c u r s d u e t o a x i a l m o m e n t M , a p p l i e d t o t h e b o d y t o w h i c h

t h e s h e ll i s a t t a c h e d . L e t t h e g e n e r a t r i x h a v e a n e l l i p t ic f o r m w i t h s e m i - a x e s

ao an d &o (cio > &o)- T w o ge ne ra t r ic es p as s in g th ro u g h th e po in ts of th e e l l ipse

B a n d B' ( s ee F igure 9 .2 ) a r e the weakes t .

To sa t i s fy r e l a t io ns (1 .13) we t ak e the r ad ius o f cu rv a t u r e o f the e l l ipse a t

p o i n t

B

i .e.

R = d^b^

as th e un i t l en g th . U nd er to r s ion po in t TV m ov es by an

a m o u n t p r o p o r t i o n a l t o r =

ON,

a n d t h e s t r e s s - r e s u l t a n t

S°

i s p r o p o r t i o n a l

t o t h e p r o j e c t i o n o f t h i s d i s p l a c e m e n t o n t h e t a n g e n t t o t h e e l li p se a t p o i n t

N.

I n a c c o r d a n c e w i t h t h i s f a c t w e t a k e

«g = r ^ = f

+

x

t M 1

] t V

/ a

. * = - < • (9 -3 .1 3 )

3

b

0

Vl+<*

4

tan

2

^y a

0

v ;

N JL

Figure 9.2: An elliptic cylinder in torsion.

For an ellipse we have

k = *-3

( s i n

2

e + p

c o s

2

ef

2

=

f

1 + <

?

a

j ^

3 / 2

= (4Y

V ;

\

1

+ J

4

t a n

2

V /

t a n 0 = 5

2

tan^, F (<p) = (<g)

5 /4

, ^

d

2

F

dip

2

V>=o

5 ( l - « ^

2

)

4<J

2

'

M

• /

S°r

COBp

1

* = o ~ rf

2

'

Rd6

(9.3.14)

The critical value of tangential stresses, r , on the weakest generatrix is

evaluated by means of formula (10), and the critical value of the torsional

moment is obtained after the evaluation of integral (14) and is equal to

M = 4K{S')a

0

b

0

hT, 8'

= %/ l - < *

2

,

1/8

T = k

s

E

h'X

h

K =

1 + 0.39

T T / 2

/ i

2

L

6

( l - < J

2

)

4

^

1 / 8

( l - i / ' ) a g

K{Cl) =

J

( l - a 2 s i n

2

x ) i / 2 -

+ 0(e

2

),

(9.3.15)

Here depending on the boundary conditions, constant

k

s

takes one of the

values in (1.15), and K(a) is the com plete elliptic inte gra l of the first kind (a

table for

K{a)

can be found, for exam ple , in [67]).

9.4. Bending of

a

Cylindrical Shell

195

9.4 B en d in g of a C yl in dr ic a l Shel l

Let us consider the bending by a tran sv ers e force, P i , (see Fig . 1.3) of a

console circular cylindrical shell of moderate length. The initial stress state is

considered to be membrane and

T°

=- ^ 1 cos

<p,

T ° =0,

S° =

- sin

<p,

1

TTRL

*'

L

2

*R

y

' (9.4.1)

0 ^ s ^ l = - ,

0

<C ip <C

2TT.

Parameter s E, v and h are assumed to be co nsta nt. In accordance w ith (1.1)

take

\Ehe

5

=

~ , 4=

sin (p. (9.4.2)

TTR

For a m ode rate ly long shell, buckling leads to the forma tion of pits in

the neighbourhood of the two weakest generatrices <po = if- where the shear

stresses are ma xim um . To evalu ate the critical value of force Pi we use form ulae

(1), (1.14) and (3.10). Then

P?=*RhT

0

k

t

,

k.

= 1

+0.31 {— —\ +0(e

2

), (9.4.3)

where

To

is the critical value of the tangential stresses under torsion (see (1.14)),

and k* take s into account th e variability of the shear stresses. Fo rm ula (3) does

not include the influence of the stress-resultant T

x

0

which is of order e

2

.

We note that the critical value of

P®

is asym ptotically tetra-m ultipl e, i .e .

four linearly independent buckling modes are possible and the values of P\

which are very close to P° correspond to them . As a rule, the eigenvalues

are asym ptotically double (see Rem ark 4.2) . Here the tetra-m ultiplicity

is

connected with the existence of the two weakest generatrices

<po

=± | - .

It is interesting to compare formula (3) with the results (see [61]), obtained

numerically by the Bubnov-Galerkin method which approximates bending by

a piece of

a

double trigo nom etric series. For 1

5, 100

^

<C 500 th e

difference betw een the r esu lts is 1—3 %, and fo rm ula (3) gives th e sm aller values

of stress than [61]. For

j -

=1000 the difference is 4% which probably means

that the difficulties of approximating bending by a piece of a tr igonometr ic

series increases as the shell thickne ss decreases. T he buck ling m od e given in

[61] qualitatively agrees with the form determined by expression (7.2.25).

For long shells, due to (1), the com pressive stress -resulta nts

if

increase and

shell buckling occurs in the neighbourhood of point

s

= I, <p =

IT.

Compar ing

the critical load under axial compression and torsion in the zeroth approxima

tions we get the applicability condition for formula (3)

L < 0 . 6 ( l -

I

/

2

)

1 / 4

/

l

-

1

/

2

J

R

3

/

2

, (9.4.4)



196


where, in evaluating r

0

by (1.14), we take k

s

— 7.4.

We will now consider the buckling of a cylindrical shell under bending by

transverse force, Pi, and torsion by axial moment,

M,

applied to the shell

edge,

s =

0. Instead of (1) and (2), in this case we get

0

P i . M

0

sin ip + m

TTR 2-KR

1

1

+ m

m

~ 2QR'

x

~ 1 + m **'

(9.4.5)

where TQ has the same value as in (1.14), and

Here, unlike the case when

M —

0 for

M

> 0 only one generatrix ipo

= \

(see Figure 9.3) is the weakest.

Figure 9.3: A force and a moment acting on a shell.

The numerical results obtained in solving this problem by the Bubnov-

Galerkin method can be found in [61]. They agree with formula (5) to the

same degree as in the case M = 0.

9.5 T h e T ors ion an d B en d ing of a C on ic Shell

In this Section we will consider the buckling of a straight circular conic

shell with constant values of E, v and h under bending by transverse force,

P i , and torsion by axial moment, M, (see Figure 9.3) applied to the largest

base of the shell. We denote the base radius and the shell generatrix length

9.5. The Torsion and Bending of a Conic Shell

197

by R and L respectively. Then according to (1.4.6) we have membrane initial

stress-resultants

mo _ P}_

* y c o s y

a0

Pi(s iny ? + m)

0

_

J l

~ Tripsin a (1-xy)

2

'

A

~ T r P ( l - x y )

2

'

J 2

"

U

'

[

^

A)

where designations (8.7.3) and (4.5) are used

M

,

L

s = s

2

( l -xy), l = s

2

-si, x = ls

2

l

,

m

= 2 ^ " p - '

l =

~R~' (

9

-

52

)

We also take

S° = Ehe*\t

3

, t

3

= . "

i n

^

m

(9.5.3)

(1 +

m)

(1 - x y )

2

We assume that Pi ^ 0, M ^ 0, then the cr i t ical values Pi and M are related

to A by t he expression

2RP? + M° = 2TVR

2

E he

5

X. (9.5.4)

The system of equations of the zeroth approximation (8.1.4) we can write

in the form

d

2

w

0

a

4

$o _

dy

2

(1 -

xy)

3

< 9

2

< 3 > o a

4

w

0

2 i a 6 / dw

0

(9.5.5)

(1 - xy) -x \- xw

0

] = 0,

dy

2

(1 - x y )

3

(1 -

xy)

3

V " ' "

J

" %

where

/ cos a \ ,

(m+l)b

/ c o s

3

a sin

2

a \ .„ „

„ .

q

= „ . , a,

A

= r-

1

— ~ . (9.5.6)

V

x

2

sin a / m + sin tp

\ x

z

J

For a fixed value of x, let the smallest positive eigenvalue be b = b(a). W e

then obtain

b

0

=mmb(a) =

b(a

0

). (9.5.7)

a

formulae (8.1.2) and (8.1.11) give

/ ^ • 9 \ 1/4 /

/ cos°

a

sin

a \ , I

A = -= 6

0

1

+

£

/ *»

V'

4

f baa

^

+

^

- T ^ ,

1 U

+ ° (

£

) • (

9 5 8

)

° \ s i n

2

a c o s a / \(m+l)b

0

J ''

v ;

In [98] the numerical integration of system (5) for two boundary condition

variants , i .e . clampe d— clampe d

(CC)

and s imply suppor ted— simply sup

por ted

(SS)

has been done

w

0

= —— = 0

for

y =

0,

y

= 1,

dy

w

0

= $

0

= 0 for y = 0, y = 1

(9.5.9)

d

2

b

and graphs of the functions a

0

(x), 6o(-*) and b

aa

(x)

= —-^

have been plotted

CLCLn

in Figure 9.4.

1

K O

Figure 9.4: The functions ao(x), bo(x), and b

aa

{x).

For m = 0 an d m = oo formu lae (4) and (8) give th e soluti ons of the

bending and pure torsion of a conic shell. As in the case of a cylindrical shell

(see Section 9.4), for

m =

0, the eigenvalue A is asympto tically tetra-m ultiple

and for 0 < m < oo it is asymptotically double. The number of waves in the

circumferential direction,

n,

under pure torsion (m = oo) is equal to

' 1 2 ( l - i /

2

) i i

6

c o s

2

a \

1 / 8

, ,

n=l

z ^ J

aoW

-

(9.5.10)

The area of applicability of the formulae obtained above is determined

by two inequalities, the first being n ^> 1, and the second being similar to

inequali ty (4.4) . The latter means that the init ial s tress-resultant

T°

does not

exceed its critical value under axial compression

L < 7.4 (1 - x )

2

(1 + m )

2

6 -

2

(1 - i,

2

)

1

/

4

( ^ |

r.1 \ 1/2

R^cosa\

(9.5.11)

For a = x = 0 due to xR = L sin a, formulae (4) and (8) evolve to formulae

(4.5) and (4.6) for a cylindrical shell.




199

9 .6 P ro b le m s an d Exe rc i ses

9 . 1 . De r ive ap pr o x i m at e r e l a t ion (2 .7 ) des c r ib in g th e e ffect o f the in i t i a l

p r e s s u r e o n t h e c r i t i c a l l o a d u n d e r t o r s i o n . I n t e g r a t e e q u a t i o n ( 9 . 2 ) n u m e r i c a l l y

f or s h e l ls w i t h ( i) c l a m p e d a n d ( ii ) s i m p l y s u p p o r t e d e d g e s a n d c o m p a r e t h e

n u m e r i c a l a n d a s y m p t o t i c r e s u l t s .

9 .2 . De r ive ap pr o x i m at e r e l a t ion (2 .12) desc r ib ing the e ffect o f the in i t i a l

t e n s i le s t r e s s - r e s u l t a n t o n t h e c r it i c a l l o a d u n d e r t o r s i o n . I n t e g r a t e e q u a t i o n

(9 .2 ) numer ica l ly fo r she l l s wi th ( i ) c l amped and ( i i ) s imply suppor ted edges

a n d c o m p a r e t h e n u m e r i c a l a n d a s y m p t o t i c r e s u l t s .

9 . 3 . Co n s i d e r t h e b u c k l i n g u n d e r t o r s i o n of a c i r c u l a r c y l i n d r i c a l s h e l l w i t h

a s l o p e d e d g e w i t h c o n s t a n t p a r a m e t e r s E, v a n d h.

A n s w e r

W e t a k e

/ = Zo + ( c o s p - l ) t a n a , ^ = 1,

F (cp

0

)

= / ~

1 / 2

,

l

0

=

-=•

H,

Formula (10) for the cr i t ica l value of r g ives

„ „„ / h

2

t a n

4

a \ , , , , .

wh ere a i s th e s lope ang le an d the s t r es s To i s de te rm in ed by fo rm ula (1 .14)

fo r a she l l o f cons tan t l eng th

L .

T h e c r i t i c a l v a l u e o f t h e t o r s i o n a l m o m e n t is

e q u a l t o M =

2irR

2

tiT.

T h e b u c k l i n g p i t s a r e l o c a l iz e d i n t h e n e i g h b o u r h o o d

of the weakes t genera t r ix which i s a l so the longes t .

9 . 4 . D e r i v e r e l a t i o n ( 4 . 6 ) f o r t h e c r i t i c a l l o a d u n d e r c o m b i n e d l o a d i n g b y

t o r s i o n a n d b e n d i n g b y f o r c e .

9 . 5 . S t u d y b u c k l i n g of c i r c u l a r c y l i n d r i c a l s h e ll u n d e r t o r s i o n m o m e n t M

a n d b e n d i n g m o m e n t

Mi

wh ich a re ap p l i e d on th e edge

s = I

(see Fig. 1.3) .

H i n t

Use r e l a t ions (1 .4 .6 ) and (3 .10) .

Chap te r 10

N e a r l y C y l i n d r i c a l a n d

Con ic She l l s

I n C h a p t e r s 7 t h r o u g h 9 w e c o n s i d e re d t h e b u c k l i n g m o d e s of m o d e r a t e l y l o n g

c y l i n d r i c a l a n d c o n i c s h e l l s . T h o s e m o d e s a r e c h a r a c t e r i z e d b y p i t s w h i c h a r e

s t r e t c h e d i n t h e d i r e c t i o n o f t h e g e n e r a t r i x . I t i s c l e a r t h a t a b u c k l i n g m o d e

c a n n o t v a r y g r e a t l y if t h e b e n d i n g of t h e g e n e r a t r i x i s s m a l l . I n t h i s C h a p t e r

t h e r e s u l t s o f Ch a p t e r s 7 - 9 a r e e x t e n d e d t o s h e l l s w h i c h a r e n e a r l y c y l i n d r i c a l

a n d c o n i c .

W e wi l l con s ider a she l l such th a t th e de v ia t io n , i. e . d if fe rence be tw ee n i t s

ge om et r y an d th a t o f a she l l o f ze ro cu rv a t u r e , a f fec ts t he c r i t i ca l load in th e

z e r o t h a p p r o x i m a t i o n . I t p r o v e s t o b e , t h a t u n d e r t h i s c o n d i t i o n t h e a m p l i t u d e

o f d e v i a t i o n m u s t b e o f o r d e r

(Rh)

l

l

2

.

T h e f u n c t i o n d e s c r i b i n g t h e d e v i a t i o n

s h o u l d n o t i n c r e a s e s i g n i f i c a n t l y u n d e r d i f f e r e n t i a t i o n b e c a u s e o t h e r w i s e t h e

p r o p o s e d m e t h o d o f c o n s t r u c t i o n o f t h e s o l u t i o n i s n o t v a l i d . S i m i l a r p r o b l e m s

for f ree vibrat ions are solved in [105, 106] .

10 .1 B as i c R e la t i on s

Let us f ir s t desc r ibe th e geo m et r y of th e she l l t o be cons id ered . T h e n eu

t ra l sur face of the shel l i s a near ly conic sur face and is refer red to as the

b a s i c s u r f a c e . O n t h e b a s i c c o n ic s u r f a c e w e i n t r o d u c e a s y s t e m o f o r t h o g o n a l

c u r v i l i n e a r c o o r d i n a t e s s, <p s imi la r to those used in Sec t ion 7 .1 .

L e t e ° , e\, n° b e t h e u n i t v e c t o r s o f t h i s c o o r d i n a t e s y s t e m (n° = e ' x e " ) .

We d e n o t e t h e d e v i a t i o n f r o m t h e b a s i c s u r f a c e b y Fo(s,<p) wh ich is eq ua l to

t h e d i s t a n c e a l o n g t h e n o r m a l

n°

f r o m a p o i n t

TQ

on th e bas ic su r face to a

p o i n t

r

on the ne u t r a l su r f ace o f th e she l l ( see F ig ure 10 .1 ) .

201

202

Chapter 10. Nearly Cylindrical and Conic Shells

Then we can write

r = r

0

+ F

0

n° = R(se1 + n

0

Fn°), F

0

=/i

0

RF,

(10.1.1)

where

R

is the characteristic size of the neutral surface and ^o is a small

parameter .

Figure 10.1: A nearly conic surface.

Coordina tes

s

and <p are the curvilinear coordinates in the neutral surface

(in the general case they are non-o rtho go nal) . By m ean s of formulae (1.1.4)

we get:

dr

dr

d<p

d

2

r

3

2

r

= R(e\ + u.

0

F

s

n°),

| ^ =

R[(s-n

0

kF)e

0

2

+ n

0

F

v

,n°],

=

RnoF

ss

n°,

= R (1 - nok F

s

) e\ + noF

sip

n° ,

=

R (n

0

kF-s) e\- no

(2A; F«, +

k

ip

F)e%+

+

(/iof

w

+ k(s- nok

F ) J n ° ,

(10.1.2)

dsdp

8

2

r

where function

k = k

[ip) is the sam e as in Sec tion 7.1 .

10.1. Basic Relations 203

The first quadratic form of the surface is equal to

da

2

- R

2

(A

2

ds

2

+ 2AB cos xds

dtp

+ B

2

dtp

2

) ,

A

2

= l +

n

2

0

F

2

,

B

2

= (s - p

0

k F)

2

+ fi

2

F

2

, (10.1.3)

fJ-oFsFp

C0SX

= -AB—

In the general case we obtain very complicated expressions for the radii of

curv ature which are not represen ted h ere. In order to simplify these expressions

we will take advantage of the close proximity of the shell neutral surface to the

basic surface and assume that /zo <S 1, and function

F

is of order 1 an d does

no t increase significantly un der differentiation. In othe r wo rds, we also assum e

that the derivatives F

s

, F

v

, F

ss

, F

stp

and F

w

have the orders not larger tha n

unity.

From (3) we find that cosx = O(nl), i.e. the coo rdina te system s, tp in

the neutral surface is almost orthogonal with an error of order (i\.

We obtain for the radii of curvature

p p

— - =

/i

0

ki +O(fil), —— = Hoku + O(fi

2

0

),

III

^ 1 2

— = - +

fi

0

k

2

+ O(nl),

ri-z s

where functions

k\, k\2,

and

k%

depend on

s, tp

and are equal to

&i =

F

ss

, k

12

= s~

2

(F^ -

sF

Sip

),

k

2

= 8-

2

(k

2

F + 8F,+F

vv

).

As the basic system we choose the same system as (7.1.2)

e

4

A(dAw) + \e

2

A

t

w-A

k

$ =

0,

e

4

A ( 5 r -

1

A $ ) + A

fc

u; = 0,

(10.1.4)

(10.1.5)

(10.1.6)

204


in which the operators A, A

t

and At unlike (7.1.3) have the form

10/1

dw 1 d / dw

AW=

TO {T

0

- )

+

B-

0

8-S{

BO

)>

5

O=«-MF,

A

1

—

(

*

2

—}

l

—

dw

\

Bo dtp \B

0

dtp

J Bo ds

\ dip J

1 d ( dw 1 d ( dw

+

To~fy{

t3

-ds~)

+

B~od~s{

Botl

-ds-)'

A

w

-

—

—

(—

— \ +

—

—

(k

— \ +

B

0

dtp

\B

0

dtp J Bo ds

\

1 2

dtp J

u

0

d /, dw 1 d („ (k , \

dw

+

kdlp-{

k

)

+

B-od-s{

B

°{-s

+

)l)s-

(10.1.7)

and the other notation is the same as in (7.1.3).

We choose

a

character ist ic deviation, FQ, so as to get th e effect of it in

the zeroth approximation of the iterative process, which has been developed

in Sections 7.2 and 8.1. We assume for that

" » =

£ l

(•• =«( -%. )• <

10L8

»

i .e. we assume that the characteristic value of

FQ

is of order

(Rho)

1

'

2

-

1 0.2 B o u n d a r y P r o b l e m in t h e Z e r o t h A p p r o x

i ma t i on

We seek a solution of system (1.6) in the same form as (8.1.2). First let us

consider the case when

t

2

> 0 (at least in par t of the n eu tra l surface), and

t\

and

tz

are not larger than

t

2

respectively. T he n in the zeroth app rox im atio n,

we obtain the system of equations

k -—£-

-Ari — $

0 T

wo + \

0

Nwo

= 0,

Os

z

s s

a

d

2

w

0

, q

2

,

q

4

, . .. q

2

t

2

w

0

fc-5-5 ki—w

0

-\ ?$o = 0,Nw

0

= ,

os

1

s gs

s

(10.2.1)

which differs from system (8.1.4) only by the second terms.

10.2.

Boundary Problem in the Zeroth Approximation

205

R e m a r k 1 0 . 1 .

If the shear stresses are do m ina nt for buckling then it

should be assumed in (1) that

where t® are introduced by formulae (9.1.1).

In solving system (1) it is necessary to define two ma in bou nd ary cond itions

at each shell edge

s = Sk{f)

(see Section 8.4). As in Section 7.2 th e sm allest

eigenvalue Ao is a function of two parameters q and ipo

A o = / ( g , ^ o ) . ( 1 0 . 2 . 3 )

We can find the zeroth and the first approximations for the load parameter

by function / under the same assumptions as in Section 7.2.

Ag + eA

1

+ 0 ( e

2

) ,

mm

qiVo

{f(q,<p

0

)} = f(qo,<P°

0

),

(10.2.4)

2 \fmfvp ~ fqtp)

Thus ,

we can obtain most of the information on the buckling type by ana

lyzing the function

f

(q,<fo)- If this function has a distinct minimum at point

(qo,<fio) (see (7.2.12)), then the buckling concavities are localized in the neigh

bourhood of line ip = <PQ, and the load para m eter , A, is determ ined by formula

(4). Th e min im um is terme d distinct if function

f (q,<fo)

has the only m in

imum in the interval

\q — qo\

~ /z

1

/

2

. At the sam e tim e the concavities are

stretched along lines <p

=

const.

If the coefficients

k, k\, d, g

and <2,

a n

d the value of

Sk

do not depend on

ip ,

then function / does not also depe nd on <po, i.e.

Ao = /(<?). (10.2 .5)

Here the buckling conc avities spread over the entire shell surface, and we should

minimize / by the values q = ne, where n is the integer wave number in

the circum ferential direction . Th is case corresponds to the buckling und er

axisymmetric loading of a shell of revolution which is nearly conic.

Let us now consider the special cases.

A

0

-

Ai

http://fmfvp/

http://fmfvp/

http://fmfvp/

206

Cha pter 10. Nearly Cylindrical and Con ic Shells

10.3 B uck l ing of a N ea r ly C yl in dr ic a l Shel l

L e t t h e s h e l l n e u t r a l s u r f a c e b e n e a r l y a c y l i n d e r , t h e p a r a m e t e r s o f w h i c h

are def ined by r e l a t ion s (7 .4 .1 ) . Sy s te m of eq ua t io ns (2 .1 ) ha s th e fo rm

k-w-^-- kiq

2

$

Q

-q

4

dw

0

+ \

0

q

2

t

2

wo = 0,

ds2

- (10.3 .1)

kiq

2

w

0

+q

i

g~

1

$o

= 0.

d

2

w

0

< 9 s

2

I f t he func t ions k\, d, g a n d t

2

d o n o t d e p e n d o n s, then sys tem (1) has an

exp l i c i t so lu t ion .

T h e f u n c t i o n k\ d o e s n o t d e p e n d o n s, if lines ip = c o n s t i n t h e n e u t r a l

s u rf a c e h a v e c o n s t a n t c u r v a t u r e

R^

1

= R~

l

e

2

ki .

In th is case

F

0

= F

00

+ s

2

F

0

2,

where Foo and F02 m a y d e p e n d o n

< p

( see (1 .1 ) an d (1 .5 ) ) . W e no te th a t for

F Q

~

[Rh)

1

'

2

th e su r f aces the c ros s s ec t ion s o f wh ich

< p

= cons t are c i rc les or

p a r a b o l a s c o u l d n o t b e d e s t i n g u i s h e d i n t h e z e r o t h a n d f i r s t a p p r o x i m a t i o n i n

e .

F u n c t i o n s

d

a n d

g

d o n o t d e p e n d o n

s,

i n p a r t i c u l a r , i f

E, v

a n d

h

d o

n o t d e p e n d o n

s

o r if t he y a re co ns ta n t ( in th e l as t case we m a y as su m e th a t

d = g = l).

L e t k\, d, g a n d t

2

n o t d e p e n d o n s. I n t h e c a s e o f s i m p l y s u p p o r t e d e d g e s

we can wr i t e

w

0

= $

0

= 0 a t s = s

k

(<p), (10.3 .2)

a n d t h e s o l u t i o n of s y s t e m ( 1) i s r e p r e s e n t e d i n t h e s i m p l e s t f o r m . W e n o w

h a v e

• a(s-si)

wo = sin , l(<p) = s

2

— si, a = 7r,

, , . dq

2

g I' , kc t

2

X o = f{q^o) = — + —

&

[k

iq + 1 5

-

W e wr i t e th e fun c t ion / i n the fo rm

7 6

2 (10.3.3)

/ = 6 F ( x , a ) , F =

X

2

+

^±^l

x

=

cq,

• - * .• ( & ) " • •

>-wr-

» ( ^ r

( 1 0

'

3 4

'

F o r a n a x i s y m m e t r i c a l l y l o a d e d s h e l l o f r e v o l u t i o n t h e f u n c t i o n s a ,

b

a n d

c d o n o t d e p e n d o n < p o - T h e c r i t i c a l l o a d is e v a l u a t e d b y m i n i m i z i n g

F

b y

q = q

n

= ne (or by x

n

). I n F i g u r e 10 .2 t h e m i n i m u m v a l u e o f F and the cor re

s p o n d i n g v a l u e o f x a r e r e p r e s e n t e d b y c u r v e SS w h i c h d e p e n d s o n p a r a m e t e r

a

c h a r a c t e r i z i n g t h e c u r v a t u r e o f t h e g e n e r a t r i x .

10.3.

Buckling of a Nearly Cylindrical Shell

207

O -5 O 5 a 0 - 5 O 5 a

Figure 10.2: Functions F(a) and x{a).

For convex shells (a > 0) both the critical load bF and the wave number

n in the circumferential direction (n = qs~

1

) increase as the curvature ki

increases. For shells of negative Gaussian curvature

(a

< 0) the load and the

wave number decrease as

\k\\

increases.

Now, let functions

a, b

and c depend on

tp

0

.

Then the parameters 90 and

<PQ

are determined from the minimum condition for function / by x, <po, i.e.

by the equations

F

s

= 0, b

ip

F + bF

a

a

ip

= 0. (10.3.5)

To evaluate the correction Ai by formula (2.4) we calculate the second

derivatives

Jq q — 0 C r xx j

f

qip

=

b

c (F

xx

q <V

+ FxaCttf.), (10.3.6)

Uv = b^F+(2a

lp

b

lp

+ a

tp



F

xx

+ ba

2

ip

F

aa

.

R e m a r k 1 0 . 2 .

For the app licab ility of form ula (2.4) it is necessary to

verify the fulfilment of two conditions:

e~

1

qo

S> 1 (for example,

s~

1

qo

^ 4

—

6)

and g

_ 4

A i / ^

1

^ 1. The first condition means that there are many concavities,

and the second means that they damp sufficiently fast in the <p direction.

E x a m p l e 1 0 . 1 . We will now consider a shell obtained as a result of slightly

bending the axis of symmetry of a circular cylindrical shell (see Figure 10.3)

under hom ogeneo us extern al pressure. We assum e th at th e shell edges are

simp ly sup po rte d. In the case und er co nside ration &i = — A;iocos^> and th e



208


other determining functions are constant. We assume that

-kiolcostp

(Ario > 0 ) , 6 = /

; - i

I

1

'

2

.

(10.3.7)

^

0

= 0

Figure 10.3: A Bent Cylindrical Shell.

It is clear (see curve

SS

for function

F(a)

in Figure 10.2) that line ipo = 0

for which the parameter a is a minimum is the weakest. Due to (4) and (6) we

get from formula (2.4) for the critical load that

A = /o + | ( / „ /

w

) ;

/ 2

+ 0 ( e

2

) ,

h = l-

x

F,

/ „ = 4 8 a

2

( a

2

+ a x

2

) x -

8

+ 8 a

2

* -

4

,

f

w

= 2Ario(a

2

+ a a ;

2

) x -

4

.

(10.3.8)

We can calculate by means of formulae (8) at

a =

TT and

a

=

— kiol-

For

the values of a we take the values of F and x from Figure 10.2.

To illustrate, we will take the following numerical values of the shell pa

rameter s :

v

= 0.3,

R/h =

500, / =

L/R =

2 and

k\o =

1- The maximum value

of deflection of the generatrix

F

m a x

is related to param eter a by the following

equation

• * m a x —

a L

8 V l 2 ( l - ^

2

) i ?

2

1 / 4

(10.3.9)

and for the shell being considered,

F

m a x

= 6.15

h.

For a = - 2 we find

F =

3.6

and

x =

1.76 from Figure 10.2. Now we can evaluate

f

qg

= 22.2,

f

w

=

0.766,

/o = 1.8 and A = 2.12 by formulae (8).

10.3. Buckling of a Nearly Cylindrical Shell 209

Let us verify th e fulfilment of the inequa lities given inRemark 10.2. We

have

e~

1

qo

=8, e~

l

\if~

1

= 0.6, from th at we find th at the second inequ ality

in Remark 10.2 is not valid, i.e. the depth of the pits decreases too slowly

in

the circumferential direction.

Compare the load parameter A

=

2.12 with the corresponding values

of

A for the other three problem s. For a simply suppo rted circular (k\ = 0)

cylindrical shell

A

=2.755. For a concave shell of revolution (h i = —1)

A

= 1.8

and lastly for a convex shell of revolution (&i = 1) A =4 . 1 . It follows from

these data that ,

at

first, th e sm all am ou nt of ben ding

of

the shell gen eratri x

causes

a

significant chan ge in the critica l load an d secondly th e critica l load of

the shell with a curved axis is close to the critical load of the correspon ding

shell with negative Gaussian curvature but slightly larger.

Now we proceed to discussion of the case of a shell with clam ped edges.

Instead of (2) w e h av e

w

0

=- p p = 0 at s

= s

k

(<p).

(10.3.10)

Calculate the function f(q,<fo) by the sam e formula (3) or (4), inwhich

a •£

7r, b u t

a

depends on the param eters of the problem and m ay be determ ined

from equation

a 8

a

tan-+/? tanh | = 0,

0

2

= a

2

+ 2ax

2

,

(10.3.11)

in which we use the same notation as in (4). To evaluate x from equ ation (5),

F

x

=

0 we do th e following. Fi rst, from eq uat ion (11) we num erica lly find

the

function a

—

a

(Q

, where £

=a x

2

and then

F = x

2

+ (a

2

+

C)

2

x~

6

(10.3.12)

and the equation

F

x

=

0 gives

z

8

=

3 ( a

2

+ C )

2

- 2 C ( o :

2

+

C) ( 2 a a

f

+ 1) . (10.3.13)

Therefore, three values, a, x and F are functions of

(,.

Changing £, we can

find the functions F (a) and x (a). These functions are marked by letters CC

in Figure 10.2.

From Figure 10.2 we find that for a > 0, i.e. for convex gene ratrices,

function F increases and becomes closer to function F for

a

simply suppo rted

shell as

a

grows. For

a <

0, function F first decreases until the value F =7.55

is reached at a = —2.3, and then it increases as \a \ increases, incontras t to

the case of simply supported shell. The wave numbers inthe circumferential

210


direction are a little larger for a clamped shell than for a simply supported

shell.

For a shell of revolution with the same parameters as in the previous ex

ample we obtain the following results. For a convex shell with

ki

= 1 we get

A

= 5. For a cylindrical shell

A

= 4.15 and for the concave shell with

ki

=

— 1

we obta in A = 3.8. Hence, for a clamp ed shell, the deform ation of its gener

atrix affects the critical load relatively less than for a simply supported shell.

10.4 To rs ion of a N ea r ly C yl in dr ic a l Shell

Now we will assume that the shear stress-resultant S° dominates buckling

and so we introd uce the load pa ram ete r A by form ula (9.1.1). Th en in the

zeroth approximation we have the system of equations

<9

2

$o

k -5-5 kiq

2

<b

0

- d q

4

w

0

+ A

0

N w

0

= 0,

u

d2w

°

u

2 ^ - i 4 *

n

t

1 0

-

4

-

1

)

k-^-p-kiq'wo+g

V $

0

= 0,

OS*

n

d

2

w

0

„ .

n

dwo

,

n

Nw

0

= <? - ^ + 2iqt°

3

^- q

2

t°

2

w

0

.

OS* OS

We assume that the functions fci,

d, g

and t% do not depend on

s.

Then

system (1) may be solved by the same method as in Section 9.1. Taking its

solution in the form of (9.1.3) and (9.1.4), we obtain the expressions (instead

of (9.1.5)) for coefficients a^

a 2 =

A o i ?

+

2fc

1 a ; j =

2 A o < ^

=

_d_

+

Xo4_

+

_k]_

gk

2

kq

2

' gk

2

q' gk

2

gk

2

q

2

k

2

q

4

To determine the critical load we may use equations (9.1.9) or (9.1.12)

according to the type of shell edge support.

We will present the results of the solution of equation (9.1.9) only for the

case of pure torsion (t° = t

2

= 0) of a shell of revolution with constant param

eters

E, v, h

and

ki.

In this case form ulae (2) have the form

a

2

= —

,

a

3

= 7.0

R

3

q-

1/2

,

a

4

=l + ^ ,

(10.4.3)

9i

1\

yhere

r

TO

q

1=

q

2

l,

R

3

= - , a = k

1

l,

(10.4.4)




211

CC

SS\ |

-10

- S O

5 10 a -10 -5

O

5 10 a

Figure 10.4: Functions

Rz(a)

and k

n

(a).

and r

0

is defined by formula (9.1.14) at

k

s

= 0.74.

Fig ure 10.4 shows pl ots of functions

Ra(a)

and

k

n

(a) , where

k

n

= 12

1

^

8

g

1

'

is the coefficient in formula (9.1.14). In this Figure the curves corresponding

to shells with simply supported and clamped edges are marked respectively by

SS

and

CC.


1 0 . 1 . Consider nearly cylindrical shell of revolution under external normal

pressu re. One of the shell edges is simp ly sup porte d, th e other is clam ped .

Plot graphics of the function

F(a)

for the critical loading and

x(a)

for the

number of waves in the circumferential direction (see 10.3.4). Compare the

graphs with those given in Figure 10.2.

H i n t

Assume a = 3.92 in (3.3) in accordance with formula (8.6.3).

1 0 . 2 . Consider a nearly conic shell of revolution w ith the ge nera trix of

con stan t sm all curv ature A;i unde r external no rm al pressure. Stu dy num erically

the effect of the problem parameters on the critical load and the number of

waves in the circumferential direction for the buckling mode.

H i n t

Solve system of equations (2.1) applying the method described in

Section 8.7.

1 0 . 3 .

Co nsider a nearly conic shell of revolu tion with th e gen era trix of

constant small curvature

ki

und er torsio n. Stu dy num erica lly th e effect of

the problem parameters on the critical load and the number of waves in the

circumferential direction for the buckling m ode . Co m pare the results w ith

those for nearly cylindrical shell under torsion given in Figure 10.4.

K

n

10

n



212 Chapter 10. Nearly Cylindrical and Conic Shells

H i n t Solve system of equations (2.1) applying the method described in

Section 9.5.



C h a p t e r 1 1

She l l s o f Revo lu t i on o f

N e g a t i v e G a u s s i a n

C u r v a t u r e

T h e s t a b i l i t y of t h e m e m b r a n e a x i s y m m e t r i c s t r e s s s t a t e o f s h e ll s of r e v o l u t i o n

wi t h n e g a t i v e Ga u s s i a n c u r v a t u r e w il l b e c o n s i d e r e d i n t h i s C h a p t e r . A s s u m i n g

t h a t t h e G a u s s i a n c u r v a t u r e is n o t s m a l l , t h e b u c k l i n g m o d e s of t h i s c la s s o f

she l l d if fe r s f rom tho se o f she l l s w h ich have pos i t i ve o r ze ro Ga us s i a n cu rv a tu re .

T h e l o c a l i z a t i o n o f t h e d e f o r m a t i o n i n t h e n e i g h b o u r h o o d o f l in e s ( C h a p t e r

4 ) o r po in t s (Chap te r 6 ) i s an impor t an t cha rac t e r i s t i c fo r she l l s o f pos i t i ve

c u rv a t u r e wh i l e t h e d e fo rm a t i o n p i t s s t r e t c h e d a l o n g t h e g e n e ra t r i c e s (C h a p t e r s

7 -10 ) a r e c h a r a c t e r i s t i c s o f s h e l l s o f z e ro c u rv a t u r e , fo r e x a m p l e , u n d e r e x t e rn a l

n o rm a l p r e s s u r e . T h i s p ro p e r t y is c a u s e d b y t h e f a c t t h a t t h e d e f o rm a t i o n s

h a v e a t e n d e n c y t o s p r e a d a l o n g t h e a s y m p t o t i c l in e s o f t h e n e u t r a l s u r f a c e .

In c o n t r a s t , s h e l l s o f r e v o l u t i o n w i t h n e g a t i v e G a u s s i a n c u rv a t u r e h a v e t w o

s y s t e m s o f a s y m p t o t i c l i n e s . T h i s f a c t l e a d s t o b u c k l i n g m o d e s , u n d e r a x i s y m

m e t r i c l o a d i n g , wh i c h s p r e a d a l o n g t h e e n t i r e n e u t r a l s u r f a c e a n d r e s u l t s i n a

s y s t e m o f b u c k l i n g p i t s wh i c h i s r e m i n i s c e n t o f a c h e s s b o a rd .

A l l o f t h e p ro b l e m s c o n s i d e r e d i n t h i s C h a p t e r ( in c o n t r a s t t o t h o s e e x a m

i n e d in C h a p t e r s 6 t h r o u g h 1 0) a l lo w s e p a r a t i o n of t h e v a r i a b l e s a n d m a y b e

r e d u c e d t o a s y s t e m o f e i g h t h o rd e r o rd i n a r y d i ff e r en t ia l e q u a t i o n s . I t is n o t

d i f f i cu l t t o so lve t h i s sy s t em by means o f a numer i ca l me thod [73 , 115 , 170 ] .

In t h i s Chap te r we wi sh to c l a r i fy t he qua l i t a t i ve s ide o f t he buck l ing o f she l l s

w i t h n e g a t i v e G a u s s i a n c u r v a t u r e b y m e a n s o f a n a s y m p t o t i c s o l u t i o n .

T h e b u c k l i n g o f a n a x i s y m m e t r i c a l l y lo a d e d s h e ll o f a l t e rn a t i n g G a u s s i a n

213

214 Chapter 11. Shells of Revolution of Negative Gaussian Curvature

curvature, the generatrix of which has a point of inflection is also considered.

In preparing this Chapter the results of [90, 91, 92, 130, 165] have been

used.

It should be noted that at the present time an asymptotic solution of the

problem of the buckling of a non-axisymmetrically loaded shell of revolution

with negative Gaussian curvature has not yet been developed.

11 .1 In i t i a l E qu a t io n s an d T he i r So lu t ions

The critical load and buckling mode of a shell of revolution with negative

Gaussian curvature depend on whether or not the tangential boundary con

ditions guarantee the absence of infinitesimally small neutral surface bending.

First , we assum e th at there is no bend ing of the neu tral surface. Th en , it

follows from (3.6.15) that the index of variation of the additional stress state

under buckling t = 1/3 and we may use the system of equations of shallow

shells (4.3.1) which we can rewrite in the form

elA{dAw) + so\A

t

w- A

fc

$ = 0,

e g A (

5

-

1

A $ ) + A

fcW

= 0,

(11.1.1)

where

£n =

hi

12(l-v$)R

2

'

(T°,T°,S°)=-\E

0

h

0

ei(t

1

,t

2

,t

3

),

(11.1.2)

and the other notation is the same as in (4.3.1) . Introducing the parameter

A

by formulae (2) we take into account the expected order of the initial stress-

resultants (see (3.6.15)).

The coefficients of system (1) do not depend on <p and that is why it is

possible to separate the variables. We find the solution of system (1) with m

waves in the circumferential direction as a formal power series in £o

w{s,<p) =

^ £ o W

n

( s ) e x p

I — (pif+

/

qds) \

, (11.1.3)

where

m

—

e^

l

p.

By virtu e of (3.6.15)

t

= 1/3, that is why

p

~ 1.

The substitution of (3) into (1) leads to a sequence of systems of equations.

We find from the system of the zeroth approximation Afc$o = A^wo = 0

1 /9

<? =

?

i,2 = ± ( -^-f j ) • < 7 i > ° <

9 2 =-qi-

( H - l - 4 )

11.2. Separation of the Boundary Conditions

215

In the first approximation we obtain the system of equations for the eval

ua tion of wo a nd <3>o

2j

,/^(v^»»)

+

£ ( | -n* »

o,

( ^*o ) -£ (£ - l )

a

" o+

(1LL5)

+A

0

(q

2

U+ 2 y (

3

+ ^ t

2

J WO

= 0.

For the evaluation of w

n

, $

n

for n ^ 1 we obta in the non-homogen eous system

of equations, the left sides of which coincide with the left sides of equations

(5), and the right sides are expressed through previously known functions

Wj

and $j where j < n.

Noting that q has two values (4), and system (5) is of the second order,

we m ay constru ct four linearly indepen dent solutions of system (1) of the typ e

of (3). We call these solution s the main solution s. Four other (additional)

solutions are edge effect solutions

(

S

, ^ ) = ^ ^ (

S

) ^

2

e x p {

£

-

3 / 2

/

(r + eor

1

) ds+^V

(11.1.6)

where

9*1

(11.1.7)

We m ay find t he eigenvalues of the load par am ete r A by su bs titu ting the

linear combination of solutions (3) and (6) into the boundary conditions which

are introdu ced on shell edges s — s\ and s = S2- However, it is more convenient,

similar to Section 8.4, to separate the boundary conditions at each of the shell

edges and to determine two

main boundary conditions

which must be satisfied

under construction of the main integrals.

11.2 S ep a r a t i on of t h e B o un da r y Co nd i t ion s

For all of the functions which determine the stress-strain state (in particular

the functions contained in the formulation of the boundary conditions) we can

repre sent the m ai n solu tions and the solu tions of the edge effect respec tively

216

Chapter 11. Shells of Revolution of Nega tive G aussian Curvature

Table 11.1: Indices of intensity

I

X

u

V

w

7 i

T i

S

Qu

Mi

II

A>

1

1

0

- 1

1

1

3

4

/ / /

—ikiq~

1

Wg

—ibk2p~

1

Wg

iqwl

IV

Pe

3

2

2

0

3

2

1

1

2

9

2

3

V

i

0

( fc i +j /Ar

2

) r -

1

i i )

e

((2 + ^) fc

2

- fe

1

)^<

0*2/>

2

e

r

2

6

2

w

o

igpfo

e

— r

- w

°

rb

-dr

3

W(j

dr

2

WQ

VI

*P

1

~ 2

- 1

0

1

2

0

1

2

3

2

1

in the form

x°(s,P,£o) - z

0

(s

>

e

0

)exp{—(p<p+qds)j,

OO

x°(s ,e

0

) = 4

o ( r )

E

e

^ S W .

n = 0

5

y

e

(s,(p,e

0

) =

j /

e

( s , e

0

) e x p | y ^

+ e^

3/2

/ (r +

e o r ^ d s j ,

J o

OO

y

e

(

S ) £

o ) = 4 '

( x )

E

e

o

/ 2

2 / n ( « ) -

(11.2.1)

(11.2.2)

Functions

x% and

J/Q

a n

^

t n e

corresponding indices

of

intensity /?o(z)

and

/?

e

(x ) are presented in Table 11.1. Table 11.1 gives the dimensionless func

tions introduced by formulae (8.2.1) and contained in the formulation of the

boundary conditions.



11.2.

Separation of the Boundary Conditions

217

Table 11.2: The boundary conditions

/

1

2

3

4

II

u = v =

0

Tj = v = 0

w = S = 0

7\ = 5 = 0

III

1111

1110

1101

1100

1011

1010

0111

0110

0101

0100

0011

0010

1001

1000

0001

0000

IV

3/2

3/2

2

5/2

1/2

1/2

1/2

1

3/2

3/2

1/2

1/2

1

3/2

1

1

We note that u°,v° ~ EQW° and all of the tangential additional stress-

resultants for the main solutions (i.e.

T°, T°

and

S°)

have the same order.

R e m a r k 1 1 . 1 . Table 11.1 differs only in notation from Table 5.3 in [57],

which contains the solutions of non-axisymmetric vibrations of shells of rev

olution with negative Gaussian curvature which have m ~ e^"

1

waves in the

circumferential direction. This is caused by the fact that both the load term

eo\A

t

w

in system (1.1) and the inertia l term s in the vibratio n problem [57],

are not the most important and are included only in the f irst approximation.

While the low-frequency vibration problem and the buckling problem for

shells of revolution with negative Gaussian curvature are very similar, differ

ential equations (1.5) for

wo

and $

0

a nd (13.10) from [57] are different. T h e

problems also differ since the main interest in the buckling problem is in the

smallest value of parameter A.

The boundary conditions can be separated using the same method as in

Section 8.4. We consider 16 boundary condition variants obtained as a result

218

Chapter 11. Shells of Revolution of Negative Gaussian Curvature

of equating to zero one of the values in each column

u v w

7i (1)

M n

«

Ti

S Q

lt

M

1

(0)

(ii.<t.a)

We mark these variants by four-digit series of ones and zeros as was done in

Section 1.2.

The results of the separation of the boundary conditions are remarkably

similar to those obta ined in Section 8.4. In the zeroth ap pro xim atio n the

boundary conditions separate into four groups: the clamped support group,

the simple support group, the weak support group and the free edge group.

Each group contains the same conditions as in Section 8.4.

The boundary conditions which form a group are presented in column

III

of Table 11.2. The numbers

a

such that the error incurred in transforming

from the complete boundary value problem to the simplified problem does not

exceed

e%

are contained in column IV . T he simplified pro ble m is solved by

substi tuting the l inear combination of the main solutions (1) into the main

boundary conditions which are shown in column II of Ta ble 11.2.

Since solutions (1) are expanded in an integer power series in eo, and so

wn

lutions (2) are expanded in a power series in

e

0

the para m eter A is also

expanded in a power series in e

0

A = A

0

+ 4

/ 2

A i + e

0

A

2

+ £o

/2

A

3

+ • • • (11-2.4)

If a ^ 1 the n Ai = 0, since the c on trib uti on of the solu tions of the edge

effect begins from terms of order EQ

.

The parameter Ai differs from zero for

five boundary condition variants for which

a =

1/2 (see Table 11.2). For these

variants the transformation to the simplified problem has a maximum error of

order e

0

.

As in Section 8.4, by eliminating the edge effect solutions we may reduce

the incurred error. W e do not consider all of the cases in deta il but limit

the discussion to one exa m ple . We consider at edge s = s

2

, the boundary

conditions 0010 or

T

1

= S = w = M

1

-0 .

(11.2.5)

As can be seen in Table 11.2, the replacement of conditions (5) by the

1/2

conditions T\ = v = 0 for the m ain solu tions has an error of order e

0

.

We can calculate th e corrected conditions by mea ns of formulae (8.4.9) and

(8.4.10).

We then obtain

-

2

iV2b\r\

elbk

2

' " '

"'

~ " '

y/£opgk

2

T? + ^-M°=0, w

0

+ ^

Z

° ' 7 5° = 0. (11.2.6)

11.3. Boundary Problem in the Zeroth Approximation

219

Taking into account the relation between the orders of

w°

and

S°

(see

Table 11.1), we note that in the second of conditions (6), the second term

has order e

0

with respec t to the first one. In the first con dition of (6) the

correction term has the relative order of

e

0

(and so it m ay be neglec ted). To

prove this we must consider that M° is propo rtional to w°, and by virtue of

•tin

the second of conditions (6), w° has the order e

0

• In the example considered,

Ai ^ 0 in expansion (4).

1 1.3 B o u n d a r y P r o b l e m in t h e Z e r o t h A p p r o x

i m a t i o n

We can reduce system of equations (1.5) to a second order equation

in which

2

g g l

6

4

* f

1

p ( * l - * 2 )

2

'

Vk

=

\fbk

2

q\ w

0

k,

In equation (1) only function fk(s) dep end s on the sign of

qk

a t t

3

^ 0, i.e.

under torsion. If t

3

= 0 then eq uation (1) coincides with the equatio n ob tain ed

in [91], if, in contrast

t\

=

t

2

= 0, then it coincides with that obtained in [92]

which considered torsional buckling. We note th at in [91, 92], in contrast to

system of equations (1.1), more precise initial system in displacements have

been taken.

We can now write the expressions for the displacements u, v and w and

stress-resultants

T\

and

S

in the zeroth approximation which are contained in

the formulation of the boundary conditions (see Table 11.2). Due to (2.1) and

, _ J _ fi}^. i

2

9fc*3 P<2

2k

2

\ p bqi

b

2

qi/

,^

3

^\

2igq

k

b

4

k% ( k

2

\

1 / 2

dy

k

* 0 f c = -

p

4

(k

2

- Ari)

2

\bqij ds

http://bqij/

http://bqij/

http://bqij/



220


Table 11.1 we find

u - u*

( j /

i e

*^

- y

2

e~

ix

>>)

e

im

f,

v = v* (

yi

e^ +

j fee- '* )

t

im

f,

w = w* (yie^ +

y

2

e-

ii

')

e

im

f

Ti = Tf

(yie '> -

y^e'^) e

im

f>

S = S* (y[e^

+ 2 /

2

e- '>)

e

im

^,

s

i> = i>(s) = qids,

(11.3.3)

Here

y\

and

y

2

are the solutions of equations (1) for

k =

1,2. The multi

pliers

u* ,v*, T*

and

S*

are cancelled after substitution into the boundary con

ditions and further we do not need to know their explicit form. The boundary

conditions which the solutions t/i and

y

2

of equ ation (1) mu st satisfy depend

on the main boundary condition variant being considered.

R e m a r k 1 1 . 2 . We indicated two tangential conditions which character ize

the group and set in column

II

of Ta ble 11.2. At th e sam e tim e, it follows

from Table 11.2 that condition v = 0 m ean s th at at least one of conditions

v

= 0 or

w

= 0 is introduced.

If at the shell edge (for example, at s = s\) u = v = 0, then by virtue of

formulae (3) we obtain

y\ = y

2

= 0.

If Ti =

S =

0, then yi = y

2

= 0.

However, if we consider one of the cases where T\ = v = 0 o r « = 5' = 0,

then the functions yi or

y

2

are not separated in formulating the boundary

conditions. In the case where

T\

=

v

= 0 at

s

=

s\ ,

the boundary conditions

have the form y\ + y

2

= 0, y[ — f

2

= 0, and in the case where u = S = 0 they

have the form

y\ — y

2

= 0,

y[ +

y'

2

= 0. In both cases we obtain the non-linear

condition

(yi fc)'

= 0 for

s = s

u

(11.3.4)

connected the functions yi or

y

2

.

t

_ -iki

p \ i i )

w* -e^

1

(bk

2

qi)~

1/2

;

1

f?(ki-k

2

)* \ b J

_ 2igb

3

kl

(k

2

qf\

^ ( * i - * 2 )

2

V

b J

11.3.

Boundary Problem in the Zeroth Approximation

221

Let us now formulate and discuss the boundary value problems in which

parameter Ao

is

the sma llest eigenvalue.

At

the sam e tim e we will consider

here the general case when the initial shear stresses differ from zero, i.e.

t$

^

0

(2). The case where

tz =

0 is considered in Section 11.4.

If the conditions of shell edges

s = s\

or

s = s

2

are clamped

u = v = 0,

then we come

to

two S turm-L iouville problems for th e functions

y\

and

y

2

,

which satisfy equations (1) and the conditions

yk=0

for s =

s i ,

s

= S2- (11.3.5)

We may also evaluate the eigenvalues for this problem, \(

k

>, by minimiza

tion of the functional

A(fc) = n

f{^|r} ' *

= 1

>

2

>

n

-

3

-

6

)

where

« 2 *2

s

2

h

=

jf{y')

2

ds, h

=

J rWda, li

k)

=

j f

k

y

2

ds. (11.3.7)

» i

»i JI

T he un kno wn value Ao is obta ined as

a

result of minim izatio n by

p

and

k,

i.e.

Ao =m i n ( A (

, £

)j. (11.3.8)

p,k

I J

Let

t

3

>

0 in

f

k

for definiteness. Then

I

{

3

1]

> if', and A'

1

) <A<

2

) . That

is why

A o = m i n ( ^ ± ^ \

( 1 L 3

.

9 )

By vir tue

of

(2) and (1.4), functions

/

and

fk

do not depend on p. Th a t

is why

in

(6) and (9),

h, I

2

and

1

do no t also explicitly depen d

on p.

Calculating in (9) the minimum by pwe obtain

A

0

=

3 - 2 -

2

/

3

m m j ^ | O ,

Po

= m

0

s

0

=

( ^ j ^ . (11.3.10)

Consequently, the minimum is attained for finite values of p

0

, different from

zero,

and buckling occurs for the integer values of m, which is the closest to

m

0

.

222

Cha pter 11. Shells of Revolution of Nega tive G aussian Curvature

I f c l a m p e d c o n d i t i o n s a r e i n t r o d u c e d at e d g e

s = si,

a n d e d g e

s =

S2

is

free (i .e .

T\

=

S —

0 ) t h en a l l o f t h e fo rm ulae w r i t t en abo ve a re va l id w i th t h e

on ly d i f fe rence be ing tha t func t ions

y

k

sa t is fy t he co nd i t i on s

V* (* i )

= 0, y'

k

{s

2

) = 0.

(11 .3 .11 )

In b o t h c a s e s c o n s i d e r e d h e r e t h e b u c k l i n g d e fo rm a t i o n int h e z e ro t h a p

p r o x i m a t i o n h a s t h e f o r m

s

w = w*yx cos/3(s,tp), (3 = m (ip - <p

0

)-i / q\ds, (11 .3 .12 )

£o J

wh e re j /i s a ti sf ie s e q u a t i o n (1 ) a n d b o u n d a ry c o n d i t i o n s ( 5)

or

(11 ) a nd

<po is

a n a r b i t r a r y c o n s t a n t .

A s s u m i n g in( 1 2 ) t h a t /? =c o n s t , we n o t e t h a t t h e f u n c t i o n w(s,<p) d e

sc r ibes t he se t of m p i t s wh ich a re i nc l ined at a v a r i a b l e a n g l e

(

k,V

/2

7 ( s ) =- a r c t a n 1- r M (11 .3 .13)

t o t h e g e n e ra t r i x .

Le t edge

s = si

bec l a m p e d ,

u

=

v =

0, a n d f u r t h e r let t h e r e beonly one

t a n g e n t i a l s u p p o r t

(u = S

= 0 or

v— T\ =

0) i n t r o d u c e d at e d g e

s = S2 -

F i r s t , c o n s i d e r t h e c a s e wh e n

v = T\ =

0. We n o w a r r i v e at t h e b o u n d

a ry v a l u e p ro b l e m inwh i c h we m u s t d e t e r m i n e t h e s m a l l e s t v a l u e ofAo such

t h a t t h e n o n - t r i v i a l s o l u t i o n s

y\

a n d

y? of

e q u a t i o n s

(1)

ex is t an d sa t i s fy th e

b o u n d a r y c o n d i t i o n s

J / I = J / 2 = 0 at

s = si,

(11.3.14)

yie

i

^+ y

2

e-

i,l

" = 0

t

y ' ^ - j /

2

e - ' ^

= 0 at

s = s

2

,

(11 .3 .15)

w h e r e

2 =4> {

s

2)

( see fo rm ula (3 ) ) .

It is

c l e a r t h a t t h e f u n c t i o n s

?/i

a n d

y

2

a r e c o m p l e x - v a l u e d . W e a s s u m e

yi=y\e- \ y

2

= y°

2

e

i

. (11 .3 .16)

T h e n fo r r e a l f u n c t i o n s

y\

a n d

y

2

,

c o n d i t i o n s ( 1 4) at

s

=

s\

a n d c o n d i t i o n

(4) at

s = s

2

m u s t b e ful fi ll ed . T h e d e fo rm a t i o n m o d e h a s t h e fo rm

w {s,

<p)

= w* (yi

c o s / ? + j/2 co s

/3

2

) ,

1

f

(11 .3 .17)

fi

k

=

m (ip - tpo) -\

/

q

k

ds

£o

J

11.4. Buckling Modes Without Torsion

223

and represents the superposition of two systems of pits which overlap and are

inclined at angles 7 and —7 to the genera trix. At the sam e time

y° + y% —

0

at

s = S2-

In the case of conditions u = S = 0a,ts = S2 for the de term ina tion of Ao

we have the same boundary value problem, however, in contrast to the case

where v = T\ = 0 one should take yj — y° = 0 at s = S2.

As in th e case when bo th shell edges are clamp ed the value Ao m ay be

determined by minimizing the functional

A

0

= m i n {

V

L \ (H.3.I8)

p ^ \

^ ( 4

1 }

+ 4

2 )

)

J

where

S

2

« 2 * 2

/<*> = J f (ylfds, 4

fc

> = J f-

1

{Vlf

ds,

/ f = Jf

k

(y°

k

)

2

ds,

«1 »1 S i

(11.3.19)

and

J/? = vl = °

a t

* = * i ,

y ? - I ^ = 0, y?' + «§' = 0 at

S

=

S 2

-

(11.3.20)

We note that the last of boundary conditions (20) is natural and may be

omitted when we choose the functions

y\

and 3/2

•

As in the case when both shell edges are clamped, it follows from (18) that

the minimum is attained for a finite value of

p

which is not equal to zero.

Boundary conditions which allow non-trivial bending of the neutral surface

are considered in Section 11.5.

11 .4 Buck l ing M od es W i t h o u t To r s ion

In th e case w here £3 = 0 by vi rtu e of (3.2) coefficient fk in (3.1) does not

depend on the choice of

q\

or

qi

and that is why the functions j/i and 7/2 satisfy

the same equation (3.1). This fact causes the buckling mode to change as will

be discussed below.

Let us consider the case when conditions

u

=

v =

0 are introduced on both

shell edges. To determ ine Ao we arrive at bou nd ary value proble m (3.1) and

(3.5), however the buckling mode is not determined uniquely and is represented

by formula

w (s, tp)

=

w*

2/1

C\

cos

(m (<p - <pi)

+

ip

(s)J +

+ C

2

c o s ( m ^ - ^ ) - ^ ( * ) ) ] , (H .4 .1 )

where C\, C2, fi and ^2 are arb itrary .

Deformation (1) represents the result of the overlapping of two systems of

pits which are stretched along the asymptotic lines of the neutral surface.

We may turn from deformation form (1) to the form

w (s,ip) — w* ycos m(ip — >po)cos(il> (s) — ipo), (11.4.2)

whe re in th e zeroth ap pro xim ati on y>o and V'o are arb itr ary . De form ation (2)

describes a chess board system of pits. The constant cpo also remains arbitrary

under the development of a higher approximation, and the constant rpo may

be found.

Let one of the four boundary condition variants for which

a

> 1 (see

Table 11.2) is introduced at each of the shell edges

s = si, s = 82-

Then

when developing the next approximation we may neglect the non-tangential

bo un da ry c on ditio ns. In exp ansio n (2.4) Ai = 0, coefficient A2 an d con sta nt ipo

have two values A

2

, V'o

a n (

i A

2

, ipg , and Aj = — X

2

• This construction

for the vibra tion proble m of a shell of nega tive cu rva ture ha s been done in [89].

Therefore, the critical value A is given by the a pp rox im ate form ula A =

Ao + £oA

2

, Aj < 0 and by virtu e of the arbitra rines s of <po, two independent

(2)

buckling mod es correspond to it. Th e value A = A

0

+ £oA

2

differs from th e

critical one by a small amount of order

EQ

and two buckling modes correspond

to it.

Now let conditions

u — v =

0 be introduced at edge

s

= «i, and condi

t ions

v

=

T\ =

0 be introduced at s = s

2

- As in Section 11.3, in th e ze roth

approximation we come to boundary value problem (3.1) with boundary con

ditions (3.14) and (3.15). However, since y\ and j/2 satisfy the sam e eq uatio n,

it follows from (3.14) t h a t j/2 =

Cy\.

Th en we ob tain from cond itions (3.15)

j/i

( e ^

2

+ Ce- ' "*

J

) = 0, j/'i ( e ^

2

- C

e-**')

= 0. (11.4.3)

As a result , the bou ndary value problem spli ts into two problem s. They

consist of equ ation (3.1) and the bou nd ary cond itions

y i ( * i ) = y i ( * 2 ) = 0 , C = e

2i

^ (11.4.4)

or conditions

yi(si) = y[(s2) = 0, C = -e

2i

^.

(11.4.5)

11.5. The Case of the Neutral Surface Bending

225

It is clear that problem (3.1), (5) gives the smallest value of A

0

.

The deformation mode in the zeroth approximation is determined uniquely

w (s, <p) = w* 1/1 cosm (cp — ipo) s in l — / qids J (11.4.6)

but ipo is still arb itrary . As in th e case of cond ition u = v = 0 on both edges,

deform ation (6) describes a system of pits resembling a chessboard . However,

in contrast to the case of condition u = v = 0 on bo th edges the b ou nda ry

value problem for fixed

m

has no eigenvalues close to the critical one.

11 .5 T h e C ase of th e N e u t r a l Su r face B en d in g

Let the boundary condit ions belong to the second or third group (see Ta

ble 11.2 and Remark 11.2) and let them be introduced at both shell edges.

Under the construction of the main solutions in the zeroth approximation we

satisfy the bo un dary condit ions

v=Ti=0

or

u = S = 0.

(11.5.1)

At some c harac teristic shell sizes

(eigensizes)

(see [20, 49, 172]) th e b en di ng

of the neutral surface is permitted, and it leads to the decrease in the order

of the critical load . T h a t is why the equatio n for the dete rm ina tion of Ao

essentially differs from the ones obtained in Sections 11.3 and 11.4.

To formulate this equation at t$ ^ 0 (i.e. unde r torsion) we intro du ce the

fundamental system of solutions

yk\

and

j/fc

2

of equation (3.1), which at

s

=

s\

satisfies the initial conditions

l/*i = 0, J/M = 1; y*

2

= l , 2/fc

2

= 0> * = l , 2 . (1 1.5 .2 )

Then, in order to determine Ao we come to the equation

( /ii fc2 + Vi2 fci)'l

= ± V T H 2 C O S 2 I / >

2

,

(11.5.3)

\s=s

2

J (S2)

in which we take the "plus" sign if the same boundary conditions (1) are

introduced at both shell edges, and the "minus" sign if the conditions are

different.

Later, we will arrive at the general case where

t

3

^ 0, bu t now let us assum e

t h a t

ts =

0. Then j/ij =

y?j

and equation (3) in the case of the same boundary

conditions as (1) has the form

yny'i2 = -Tr\sm

2

Tp

2>

(11.5.4)

/ 1*2)



226


and in the case of different boundary conditions has the form

,./ _ / ( « i ) 2 .

where

1p2

2/112/12

=

_

7 7 7 T

c o s

^

2

' (11.5.5)

32

\k&

dS

-

( 1 L 5

-

6 )

We will now consider in more detail the case when both shell edges are

simply supported

(v — T\ =

0). T he other cases m ay be considered in a

similar fashion.

Let the parameters of the neutral surface and the number

m

be such that

sin ip2 = 0. We then have again the Sturm-Liouville problem, which consists

of equation (3.1) and the boundary conditions

J/i(*i) = l/i(*2) = 0. (11.5.7)

For the given m let the minimum eigenvalue for this problem be Ao(m).

For its approximate evaluation we assume in (3.6) that

y

= 1. Then

A

0

(ra)

» 2

p

2

Li f ds

L\

- \

— ,

L

2

= fids, p = e

0

m.

(11.5.8)

To determine if the value Ao (m) is critical or not we m us t consider the other

values of m.

Let now sinV'2 ^ 0. The functions y^ i and yk i satisfy the Volterra integral

equations

s t

2/fc2=l-

[([Fkitjynttjdh

5 l Si

(11.5.9)

7 / ( 0 '

where

F

k

(s)

= X

0

p

4

f

k

(s)-p

6

f-

1

(s),

A = 1 , 2 . (11 .5 .10)

We can solve equation (9) by means of an iterative method starting with

2/fci

=

2 / f c 2

=

0- T his m eth od converges faster with sm aller values of the

11.5. The Case of the Neutral Surface Bending

Til

functions

\Fu\-

For £3 = 0 we take only the functions j/n and

y\

2

(since

y

2

j = j / i j ) . Assuming tha t \Fk\ <C 1 we lim it the discussion to ap pr ox im at e

solutions (9) in the form

s s t

* (*) = / y ^ * '

y i 2

=

l

~ j ( l

F l { i l ) d i l

) j l j ) -

(1L511)

Si Si Si

After substi tuting into (4) we obtain the approximate formula

sin

2

^2 . P

2

L I . .

^ H - i r r + T

1

(11.5.12)

p^L\L

2

L

2

which generalizes formula (8).

T he critical value of pa ram ete r Ao is Ao = m inA o(m ). T he cases when

sin i p 2 is close to zero for a small value of m are the least favourable for shell

stabil i ty . We note that s in

i\>

2

= 0 approximately corresponds to the case when

the neutral surface has bending satisfying the conditions

v

= 0 at s =

s\

and

s = s

2

.

Now we obtain a formula of the same type as (12) for £3

^

0. First let us

take the approximate solutions of equations (9) in the form

Vk\ -

Vki

--

'J fit)

Si

s t

» l s i

I U

F k { u )

I

"

dh

dn)

f

-Mdt,

f(h) V fit)

S l

. , , dt

h)dt

(11.5.13)

f i t ) '

The substitution into equation (3) after neglecting terms of order |-Ffc|

2

leads to the same equation (12) in which

L \

has the previous value and

' 2

L2

= \jih + h)ds.

(11.5.14)

» i

Recalling (3.2) we note th at the value of

L

2

does not depend on

£3,

consequently

the effect of torsion is not detected in the approximation being considered.

If the shell is loaded only under torsion (i.e. t\ = t

2

= 0, t

3

^ 0) then

L

2

=

0 and formula (12) is not valid.

To obtain a more accurate relation in the case when all tj ^ 0, we retain

terms of order

\Fk\

2

in solutions of the same type as (13). Then equation (3)

may be written in the form

A o P % L

2

+

\lp

s

L

x

L

3

=

sin

2

1/>

2

+

p

6

L \,

(11.5.15)

http://fu/-

http://fu/-

http://fu/-

where

L \

and

L ^

are given by formulae (8) and (14) and

—

/(/ 7-(/(/ ')7)

2

<»>

5 l Si Si Si

For L

2

7^ 0, L3 0 equation (15) allows us to account for the effect of

torsion on the critical load.

However, for

L 2

=0 (i.e. under pure torsion) equation (15) leads to an

unexpected result: if the neutral surface size is close to the eigensize for small

m then i t does not cause (in contrast to the case where <3 = 0 ,f i , ( 2 ^ 0 ) a

decrease in the order of the critical load. Indeed, let sin ip2 =0 for small m

(p

<C 1)- Then Ao(m) ~ p

_ 1

, i .e. decreasing m (starting from the same value)

leads to an increase of Ao

(m) .

We will discuss this question again in Sections

11.6 and 12.2.

We may make some remarks about formulae (12) and (15) . These formulae

were developed under the assumption that \Fk\ -C1, m 3> 1, which may

be not fulfilled simultaneously. For small m (for example, m = 2 or m = 3)

constructed solution (1.3) does not guarantee the required accuracy since both

system (1.1) and the solution method are not sufficiently accurate. Cases when

the neutral surface has bending for small m will be considered in Chapter 12.

It follows from (12) and (15) that the critical value of the parameter Ao

for a shell of revolution of negative curvature with support

(v

=0) at each

shell edge may change in range from values of order unity to values of order

el

depending on the size of the shell.

Let

< T

be the characteristic value of the membrane initial shell stresses

und er b uck ling. We consider the dime nsionless value <x* =

a/(Eh*),

where

h* = ho/R. T he n for strongly sup por ted shells with negative cu rva ture , er„ ~

1/3

h* (see form ula (1.2) and Sections 11.3 and 11 .4).

For shells which have one support

v

= 0 or

u

= 0 at each edge, the order

1 /3

of

<T,

varies from hjto

h*

depending on the size of the shell. It follows from

Table 11.2 that the same result is obtained if we replace condition v = 0 by

w = 0orifv = w = 0

sim ultaneou sly, however, the com position of conditions

u

=

w

— 0 gives

<T*

~

h*

.

In order to compare the results we recall that for the strongly supported

shells of positive curvature

a*

~1 (see (3.1.3)), and for the shells of zero

1

1 1

cu rva tur e un der ex tern al pre ssure <r* ~ hj(see (3.5.5)).

11.6 T h e B uck l ing of a T oru s Se c to r

To illustrate the results of Sections 11.3-11.5 we will now consider the buck

ling of a torus sector which has negative Gaussian curvature, under the axial

11.6. The Buckling of a Torus Sector

229

force

P

and the torsional moment

M,

applie d to shell edges

si

and

s

2

(

s e e

Figure 11.1). The torus is obtained by rotation of an arc of a circle of radius

R

around axis

00'.

Let

Ro

be a distance between the centre

C

of this circle

and the axis. We take R as the unit length. Then

hi = — 1, b= a — cos s, k

2

= b

1

cos

s,

£ > > •

(11.6.1)

Figure 11.1: A torus sector.

By virtue of (1.4.6) the initial stress-resultants are the following

P

T °

rp O

r

rj-,0 -»1

1

~

2nRPk

2

'

2

~ k

2

'

We take in (1.2)

lp Ip

then

P = 2irREhe%R

P

,

M = -2irR

2

Ehs

4

R

M

,

o o _

M

° 2 T T R

2

6

2

t3

=

~ W '

Rp

^

—A lp,

RM

=

A / M -

(11.6.2)

[11.6.3)

(11.6.4)

We introduce

lp

and ZM and seek the critical values of the pa ram ete r

A

> 0. We

will consider both compressive (lp > 0) and tensile (lp < 0) stress-resultants

Tj

0

. T°, however due to symmetry we can only consider the case when IM ^ 0.

The numer ical parameters in the example are :

a =

1.5,

v

= 0.3,

s

2

=

-

S i

= TT/4 and

h/R =

0.003, then e

0

= 0.0968.

230

Cha pter 11. Shells of Revo lution of Nega tive Gau ssian Curva ture

RM

^

'•te

J*^£

* r

y

0 - 3 O

3 6

R

P

F i g u r e 1 1 . 2 : T h e l o a d i n g p a r a m e t e r s a c t i n g o n t h e t o r u s a t b u c k l i n g .

Co ns id er the var ious typ es of she l l edges su pp or t s . F i r s t , le t b o th edges be

c l a m p e d . T h i s i s t h e n t h e b o u n d a r y v a l u e p r o b l e m ( 3 . 1 ) , ( 3 .5 ) f or k = 1. T h e

n u m e r i c a l s o l u t i o n is r e p r e s e n t e d b y c u r v e 1 i n F i g u r e 1 1 . 2 . Bu c k l i n g o c c u r s

f or m = 8 . W e o b t a i n f r o m a p p r o x i m a t e f o r m u l a ( 3 .1 0 ) a t

Rp

^ 3 th a t a t

y = cos 2s t he va lue o f Ao which i s l a rger t h a n the e xac t so lu t ion by no t m or e

th a n 1%. A t the s am e t im e we f ind mo = 8 .3 , an d by v i r tu e o f (3 .10) th e v a lue

m o d o e s n o t d e p e n d o n t h e p a r a m e t e r 7 =

IP/IM-

I f edge s = s\ i s c l a m p e d , a n d e d g e s = s

2

i s s i m p l y s u p p o r t e d t h e n o n

e d g e s — s

2

w e a r r i v e a t c o n d i t i o n ( 3 . 4 ) . T h e r e s u l t s o f n u m e r i c a l i n t e g r a t i o n

are show n as curv e 2 in F ig ure 11 .2 . I f

Rp

< 0 .7 , bu ck l ing oc cur s for

m

= 7,

and i f Rp > 0 .7 bu ck l ing occu r s for m = 8.

If edge s = Si is c l a m p e d , a n d e d g e s = s

2

is f re e , w e c o m e t o b o u n d a r y

va lue p rob lem (3 .1 ) , (3 .11) fo r

k

= 1 . T h e c o r r e s p o n d i n g r e s u l t s a r e s h o w n a s

cu rve 3 in Fi gu re 11.2 . I f

Rp

^ 5 , bu ck l ing occu rs for m = 7 .

As the t ens i l e ax ia l s t r es s (R p > 0) inc rea ses , curv es 1 , 2 an d 3 in Fi gu re

11 .2 approach each o ther . To exp la in th i s e f f ec t , no te tha t by v i r tue o f (3 .2 )

and (3)

fk(s) = =— c os s (a - c os s) ^-j-

1

r= . (11.6 .5)

c o s

z

s \ )

c o s s ( a

—

cos

sy

F o r lp < 0 we have f

2

(s) < 0 for all s, a n d fi(s) > 0 for all s, if 7 > - 0 . 8 4 .

For — 0.84 > 7 > — 2.8 th e fun ct io n

f\ (s)

changes i t s s ign in the in t e rva l

(s \

, s

2

)

w h i l e a t t h e s a m e t i m e n e a r t h e e n d p o i n t s o f t h i s i n t e r v a l f\(s) < 0. It follow s

t h a t t h e s o l u t i o n s o f t h e c o r r e s p o n d i n g b o u n d a r y v a l u e p r o b l e m s a r e d a m p e d

a s w e a p p r o a c h e d g e s

s\

a n d

s

2

a n d t h e s o l u t i o n i s l es s d e p e n d e n t o n t h e t y p e

o f s u p p o r t . L o c a l i z a t i o n of t h e b u c k l i n g m o d e n e a r p a r a l l e l s = 0 occurs in a

m a n n e r s i m i l a r t o t h a t d e s c r i b e d i n S e c t i o n 4 . 2 .

If w e c h a n g e t h e w a v e n u m b e r m i n t h e c i r c u m f e r e n t i a l d i r e c t i o n t h e n t h e

e i g e n v a l u e A o ( m ) o f t h e c o r r e s p o n d i n g b o u n d a r y v a l u e p r o b l e m h a s a m i n i m u m

for m = 7 or

m =

8 in a l l of th e cases cons ide red (see , for ex am pl e , curv e 4 in

11.6. The Buckling of a Torus Sector

231

Figure 11.3, which corresponds to the case when lp = IM = 1 with cla mp ed

edge

s

=

si

and free edge

s = S2).

We have another picture in the case of simply supported edges s = s\ and

s — s

2

.

It follows from a pp rox im ate formulae (5.12) and (5.15) th at the value

of A

0

(ra) depends on

smip2(m),

and the critical load depe nds on wh ether or

not sin

ip2

is close to zero for a sm all value of m . We recall t h at if sin

ip2

is

close to zero it means that the shell size is close to the eigensize.

X(m)

2

1

O

4 6 8 10

m

Figure 11.3: The values of A(m).

For simp ly sup po rted edges we come to boun dar y value proble m (3.1), (5.2),

(5.3). The solution, af ter minimization by

m

are shown as curve 4 in Figure

11.2. We have sin

Y>2

= 0.18 at

m =

3 and th at is why withou t torsion

(IM

= 0)

l

P

while under small torsion (7 = -— > 0.4) buckling occurs for m — 3. Un der

iM

pure torsion (without any axial force) shell buckling takes place for m = 7. As

was noted in Section 11.5, if the shell size is close to eigensize then the critical

load and buckling m ode are not affected. T he calculation shows th at t he sa m e

case also takes place under tensile stresses

P

and small compressive stresses

(7 < 0.4).

It follows from above that curve 4 in Figure 11.2 has two distinct parts.

The left hand portion of the curve corresponds to m = 3 and to the buckling

mode close to surface bending, and the right portion of the curve corresponds

to

m =

6

—

9.

We will now consider another shell with the same parameters and «2 =

—si

= 0.86. For this shell sin ^2 = 0 if m = 4. Let th e shell edges be sim ply

supp orted. Th e graphs of A

0

(m) for l

P

= 1 , l

M

z= Q (curve 1), for lp = 0.5,

l

M

= 1 (curve 2) an d for

lp —

0,

IM =

1 (curve 3) are plotted in Figure 11.3.

For comparison, curve 4 gives Ao(m) for a shell, one edge of which is clamped

and the other is free for

lp

=

IM — 1 •

Curves 1 and 2 have two minim um s at

m

= 4 and

m —

7, the first is cau sed

by be nd ing . On e can see th a t the effect of ben ding decreases as the torsio n

becomes relatively larger.

J I L

232 Chapter 11. Shells of Revolution of Nega tive Gau ssian Curvature

11 .7 She ll w i th G au ss ia n C u rv a t u re o f V ar iab le

Sign

There are two types of shells of revolution with alternating Gaussian cur

vature. Toroidal shells, the curvature k

2

of which changes sign at some specific

line (point A in Figure 11.4a) is the first type. The development of asymptotic

solutions of the equations of statics, dynamics and buckling of this type of

shell has been the subject of many studies (see, for example, [12, 24, 82, 119]).

W e note especially th e pap ers [20], in which the specific typ e of quas i-ben ding

of the neutral surface is introduced [53]. In the present Section we will not

discuss this problem in detail.

O

O'

(a)

Figure 11.4: The two types of shells in which the Gaussian curvature changes

sign.

We will consider shells the curvature k\ of which changes sign at some

parallel

s

= s* (point

A

in Figure 11 .4b). T he genera trix has a point of

inflection at point A. We assume tha t k[(s*) < 0, i.e. the part s» < s ^ s

2

of

the shell has negative curvature.

We begin with system of equations (1.1) and seek its solution in the form

of (1.3) or (3.3). It follows from (1.4), that

q

1

= q

2

— 0 at s = s

t

,

i.e. point

s* is the turning point for system of equations (1.1). In the neighbourhood of

s — s* by vi rtue of (1.4), qi ~ (s — s* )

1

/

2

and equation (3.1) has a singular

point

s

= s*. We consider only the case without torsion

(t^ =

0). Then, in the

neighbourhood of point s = s* equation (3.1) has two solutions of the form

y

( ) = l

+ a i

( s - S , ) +

„(2)

( * - s * )

1 / 2

[1 + 6i

(*-*. )

+ • • • ] .

(11.7.1)

11.7 . Shell with Gau ssian Curvature of Variable Sign

233

At the same time by virtue of (3.3), all solutions of system of equations (1.1)

are non-regular.

We seek the regular solutions of system (1.1) in the form

1

( * ,

< P )

w^(s,e

0

)U(r

1

) + w^(s,eo)^

(11.7.2)

where

U ( r j )

is the solution of the Airy equation

d

2

U

+

V

U = 0,

i? = e o

2 / 3

C(«) , (H-7 .3)

drf

and the functions io(

fe

) are the formal power series in

£ Q

(

f c

)

= e

7 * ( « 0 £

£

2 ;

w

( M )

( s )

. (n.7.4)

w

(=0

In the same form (2) we seek the rest of the functions, determining the stress-

strain state of the shell.

At this point we will discuss further constructions very briefly. The details

may be found in [130, 165], where the initial system of equations more accurate

than (1.1) has been considered. However, this circumstance does not effect the

zero th approximat ion .

Unknow n functions (,{s), w^

k

'

l

\s) are determined as a result of sub sti tutin g

solutions (2) into system of equations (1.1). They are regular at

s

= s„, and

n(s) =

{ i ~

0

I

q i d s

) •

(1L75)

We assume that the shell edges are sufficiently far from point

s =

s„, i.e.

e ^

1

J q i d s » l , SQ

1

J { q x l d s ^ l . (11.7.6)

S . S i

Th en

r j

(S2) 3> 1, — n (si ) 3> 1 and s ubsti tu ting solutions (2) into the bou nda ry

conditions we may use the asymptotic formulae for the Airy functions as 7 7 — >

±00 (see [15, 63]),

U\

~ 7 r

- 1

'

2

7 7

- 1

/

4

c o s 7 ,

Ui

~ 7 r

_ 1 /

'

2

j ;~

1

/

4

s i n 7 ,

r \

—>•

00;

I7 l ~ ( l /2)7T-

1

/ 2 ( _

f ?

) - l / 4

e

- «

j [ / 2

_

7 r

- l / 2 ( _

r ?

) - l / V

i

, , _ » _ o o ,

(11.7.7)

23 4 Chap ter 11. Shells of Revolution of Nega tive Gau ssian Curvature

In sea rch ing fo r

u>( •'

we a r r iv e a t second o rde r d if fe ren t ia l eq ua t io ns .

T h e re fo r e , n o t i n g t h a t e q u a t i o n (3 ) a l s o h a s two l i n e a r l y i n d e p e n d e n t s o lu

t i o n s , U\ a n d U

2

, i n t o t a l fou r so lu t ion s o f sy s t e m (1 .1 ) o f t y pe (2 ) hav e bee n

d e v e l o p e d .

I t fo l lows from (7) tha t for

s

> s„ , a l l so lu t ion s (2) os c i l la te . For s < s*

t w o s o l u t i o n s c o r r e s p o n d i n g t o

U

=

U\

, d e c r e a s e e x p o n e n t i a l l y a wa y f ro m t h e

t u r n i n g p o i n t , a n d t h e o t h e r t w o s o l u t i o n s c o r r e s p o n d i n g t o

U = U

2

,

i nc rease

e x p o n e n t i a l l y a wa y f ro m t h e t u r n i n g p o i n t .

I f a t l e as t o n e s u p p o r t u = 0 , u = 0 o r w = 0 i s i n t r o d u c e d a t e d g e s = s\,

wh i c h b o rd e r s t h e p a r t o f t h e s h e l l wh i c h h a s p o s i t i v e c u rv a t u r e , t h i s e d g e

c a n n o t c a u s e b u c k l i n g . T o s a t is fy t h e b o u n d a ry c o n d i t i o n s a t e d g e

s — s

2

we

m us t t ake two so lu t io ns (2 ) fo r

U = U\,

a s s u m i n g t h a t , s i n c e t h e y d e c r e a s e

e x p o n e n t i a l l y f o r s < s * , t h e y a p p ro x i m a t e l y s a t is fy a n y b o u n d a ry c o n d i t i o n s

a t s = s\.

We c a n n o w a l s o p r e s e n t t h e e x p l i c i t f o rm o f t h e m a i n ( r e a l ) t e rm s o f

so lu t ions (2 ) fo r

U = U\

u s i n g a s y m p t o t i c f o r m u l a e ( 7) a s

r)

—> 00 (i. e. for

s

> «*)

u

=

(-Ciy simp*+C2yW cosip*) \u*\cosm(„) \w * \ cos m (ip —

<po),

Ti = (-C

iy

M'sin

Ti>

t

+

C

2

yW

cos ^)\T?\ cos m(< p-

sin^) \S*\sinm(<p-^

0

),

ip*{s) = — qids- - ,

£o7 4

w h e r e C\, C

2

a n d <po a r e a r b i t r a r y c o n s t a n t s , t /

1

' a n d T /

2

' a re t h e so lu t io ns o f

e q u a t i o n (3 .1 ) wh i c h h a v e e x p a n s i o n s (1 ) a t s = s» and u*, v* ... a r e d e t e r

m i n e d b y fo rm u l a e (3 .3 ) .

U n d e r t h e a b o v e a s s u m p t i o n s f or d e t e r m i n i n g Ao w e s h o u l d s u b s t i t u t e s o

l u t i o n ( 8 ) i n t o t h e m a i n b o u n d a r y c o n d i t i o n s a t s = s

2

.

L e t c l a m p e d c o n d i t i o n s

u = v —

0 b e i n t r o d u c e d a t

s

=

s

2

•

T h e n t h e a b o v e

s u b s t i t u t i o n l e a d s t o t h e e q u a t i o n

2 /

( 1 ,

( s 2 , A o ) 2 /

( 2 )

( s

2

,A o) = 0 , (11 .7 .9)

f rom wh ich i t fo ll ows th a t t h e c r i t i ca l va lue Ao i s equ a l t o t he sm a l l e s t roo t s

o f e q u a t i o n s 7 /

1

) = 0, y^ = 0 , i n to wh ich eq ua t io n (9 ) m ay be sp l i t . I t can

b e s h o wn t h a t i t i s t h e e q u a t i o n

y^(s

2

,

Ao) = 0 w hic h g ives th e cr i t i ca l va lu e

of A

0

.

11.7. Shell with Gaussian Curvature of Variable Sign

235

The buckling mode in the zeroth approximation has the form

w(s,<p) = w

0

cosm{tp - ip

0

), w

0

—e

0

wi(s) U\ [rj (s , e

0

)], (11.7.10)

where w\(s) does not depen d on EQ. For the case where s > s*, by virtue of

(7)

wo = j/^

1

^|«)*|cosV'*.

(11.7.11)

For the case when

s

<

s

t

,

the function WQ exponentially decreases as \s — s*|

increases, and for s > s*, it oscillates as shown in the graph of

WQ(S)

in Figure

11.5. The portion of the shell with negative Gaussian curvature is covered by

a system of pits th at resembles a chess bo ard . In the neigh bourh ood where

s = s

t

, the pits are deepest.

s

2

5

Figure 11.5: Schematic of function WQ(S).

Now, let the conditions

v = T\ —

0 be introduced at

s

= «2- T he n th e

equ ation for the de term ina tion of Ao has the form

cos

2

tf

3

.

+ i /

( 2

V

1 )

' | sin

2

V>2*=0 ,

ih. = t/>2(s*).

(11.7.12)

) , /

2

) '

2T 'y

By virtue of the identity / ( j/

1

)* /

2

) '

— y^y^')

= const equ ation (12) is writ

ten in the form

2 /

( 1 )

V

2 )

= - T ^ C C S

2

V>2*,

cj

=

gb

3

(-k[k

2

)

1/ 2

(11.7.13)

where cj is calc ulated a t s = s

t

.

In this case as in Section 11.5, we may obtain the approximate formula for

Ao if cos

^2*

is close to zero for small

m.

Using the inequa lity

\Fi\

<g 1, we

take the approximate solutions of equation (3.1) in a form similar to (5.11)

236


Now equation (13) gives

cos

2

V2* ,

P

2

Lx*

A

° (

m

) = „

4

r

T +

0 L\

t

L

2t

IJ2*

(11.7.15)

fds

f

L\*

= / -r, L

2

* = / / i d s .

In the case of the boundary conditions

u = S

= 0 at

s = s

2

we ob tain similar

results with the only difference being that cos

2

ip

2

,

is replaced by sin

2

^2* in

(13) and (15).


11.1 .

Derive the system of equ ation s of the second ap pro xim atio n for

functions w$s) in (1.3)

11.2 . Derive equ ation s (3.1) for the pa rtic ula r case of hype rboloid of rev

olution of one sheet x

2

/ a

2

— j /

2

/ 6

2

= 1 unde r com bined axial com pression P,

external pressure

q°

and torsion

M.

H i n t To obtain the dimensionless stress-resultants use relations (1.4.6).

Geometric functions in (3.1) and (3.2) are

k

2

= a^ \l0% sin

2

9 — 6

2

co s

2

9, b = k^ sin9,

jti = - a ^ o - 2 * 2 ,

d

( )l

ds

= -

f e

i

d

( )/d9,

where

9

is the angle between the axis of revolution and the normal.

11.3 .

Fin d the critical load and the wave num ber in the circumferential

direction for hyperboloid of revolution of one sheet x

2

/ a

2

— J /

2

/6 Q

=

1 under

axial compression P. Shell edges 9\ = 2TT/3 and 9

2

= 47r/3 are clamped.

Assume

ao/h =

0.005,

&o

= «o,

v

= 0.3, and

d = g =

1.

H i n t For arbitrary

m

solve boundary value problem (3.1) with the bound

ary conditions y{9$ = y(9

2

) = 0. Then find the minumum of function A(m).

11.4 .

Use ap pro xim ate relation s (3.10) to dete rm ine Ao and mo for Prob

lem 11.3 assuming y — (9

2

—

9){9 — 9\). Co m pare the result with tha t of

Problem 11.3.

11 .5 .

Consider Prob lem 11.3 assum ing shell edges

9\

t2

= 7r/2±#o, #o < T / 4

to be simply sup po rted . Find th e eigensizes of the shell.



11.8. Problems and Exercises 237

H i n t For arbitrary

m

and

9

solve boundary value problem (3.1) with the

boundary condi t ions

y'(0\) = y' (O2)

— 0. O bta in the function A(m,

0)

and

then f ind i ts minumum.

1 1 . 6 . Use appro xim ate relation (5.12) to solve Prob lem 11.5 and com pare

the results.



C h a p t e r 1 2

S u r f a c e B e n d i n g a n d S h e l l

B u c k l i n g

In this Chapter the buckling of a membrane axisymmetric state of a weakly

su pp orte d shell of revolu tion will be considered . T he case when non -triv ial

bending of the neutral surface exists which, as a rule, leads to a decrease in

the order of the critical load, are also considered.

We will also consider cases without pure bending that satisfy the bound

ary conditions but with deformations of the neutral surface that are close to

bending. Such deformations lead to a decrease in the order of the critical load

compared to that for a well supported shell.

1 2 .1 T h e T ra n s fo rm a t io n of P o t e n t i a l E n e rg y

It is noted above that the consideration of deformations which are close to

neutral surface bending imposes a special requirement on the accuracy of the

initial system of equations. That is why the expressions which specify (2.3.10)

and (2.3.11) are introduced here.

As in Section 2.2, we represent the unknown functions in the form of sums

u° +

Ui,

w,° + Wi, T° +Ti, M° + Mi, . . . and expand potential energy (1.10.1)

as the sum of homogeneous terms with respect to the additional unknowns U{,

w, ... (hencefo rth, for brevity we will refer to the initial disp lace m ents u° and

to the additional displacements u,- and in the same manner we refer to other

unknowns) .

n

= n

£

+

n „

=

n°

+ u°

H

+

n*

+ n j , +

n

£

2

+

nl +

• •

•, (12.1.1)

where 11° and Il£. are calculated by formulae (1.10.1), (1.1.10) and (1.1.13) for

239

240

Chapter 1 2. Surface Bending and Shell Buckling

displacements u° while at the same time, 11° is non-linear in u°. Further we

can write

nj = II ( i feh +

s°e\

2

+

T

2

°4

2

) < m ,

Ui = ff (Mfxi +

2

H°T + M

2

°x

2

) dQ,

(12.1.2)

where 7^° and

M°

are the initial stress-resultants and stress-couples and Xj

and r are calculated by formulae (1.1.13) for displacements «,-, and

4 = ei + e?e* + w?w,- +

7

,

?

7 i ,

( 1 2 x 3 )

e\

2

= w + ejw

2

+ w^ei + e^wi + w Je

2

+ 7? 72 + 7

2

7 i •

Here e°, wf and 7? are calculated by formulae (1.1.7) for displacements u°,

and £,-, w,- and 7,- are calculated by formulae (1.1.7) for displacements it*.

The second order terms in u,- are equal to

n

2

= n

2 1

+ n

2 T

, (12.1.4)

n f =

\lf

[T?

(el

+

w

? +

7l

2

) +

T

2

°

(

£

2

+

W

2

+ 7

2

2

) +

n

+ 5° (eiw

2

+ £2^1 + 7172)] dfi, (12.1.5)

where II

2 1

is calculated by formula (1.10.1), in which deformations

et j

are

tak en from (3). In (2) 11^ is calc ulate d for displa cem en ts

u,.

Let SA be the elementary work of external forces (1.10.4) on displacements

8ui. We represent SA as a sum

6A = 8A

1

+ SA

2

, (12.1.6)

where

8A\

and

SA

2

depe nd on w° and

Sut

while at the same time

SA\

does

not depend on u,-, and SA

2

linearly depends on M,-.

Then, in order to determine the init ial s tate,

u°,

one can use equation

(1.10.5)

6nl + SUi = SAi

(12.1.7)

and the additional state u; is determined from the equation

6n? + 6Tg

r

+ 6Tll = 6A2.

(12.1.8)



12.1. The Transformation of Potential Energy

241

In the case of a conservative external load we may introduce the work of

the external forces,

A (u° +

«,•) which depe nd on the disp lac em en ts. Let us

represent it as in (1), in the form

A = A

0

+ A

1

+ A

2

+ ---

, (12.1.9)

where

AQ

does no t d epe nd on M»,

A\

linearly dep end s on «,-, an d

A

2

has

quadratic terms in u* (i.e.

u

2

, UjUj).

If the external load is "dead", i.e. the surface and boundary loads main

tain the same value and direction when the shell deforms, then the work,

AQ,

linearly depends on w°,

A\

linearly depends on

Ui,

and the o ther ter m s in (9)

are absent (A

2

= A3 = • • • = 0).

For a conservative load we may substitute differential relation (8) in the

finite relation

n

£

21

+ n f + ul = A

2

. (12.1.10)

The discussions of this Section have had a general character until now. At

this point we will consider the buckling of a shell of revolution under a mem

brane axisymmetric stress state, with some simplifying assumptions. Namely,

we will ignore any pre-buckling shell displac em ents. We assume th at und er

buckling, the bifurcation into a non-axisymmetric form takes place. We also

assume that the external load

q

and the init ial s tress-resultants

Tf

and

S°

l inearly depend on the loading param eter A

q^XR-'Tq

0

, (Tf,S°) = -XT(t

it

t

3

), T = E

0

h

0

n

2

, (12.1.11)

where we assume that A, q° = q°(s) and U = U(s) are dimensionless.

Under these assumptions, in the case of a conservative external load we

may write the buckling problem in a variational form similar to (3.6.1)

.

l+ )

IT +

n

4

R

2

Il' . hi

,

A

= min(+>

£

*

* ,

^ = _ — 2 _ _

12.1.12

«i

ii

2

u.'

T

12(1-V

2

)RQ

where we calculate the minimum by all w,-, which satisfy the geometric bound

ary conditions and for which the value of

11^

is positive. Here n£ and Il '

H

are

the same as in (3.6.1), and

n

r

= (A

2

-II

2T

)(\T)-

1

.

(12.1.13)

Analyzing the buckling modes close to bending, we may neglect in (5) terms

containing £,• and w, and according to (11) we can write

* - * / /

«i W + w?) +

t

2

( 7 + " I ) +

2

*37i72+

r 0 ,

+ ^ - ( 7 i « + 72")

dSl,

(12.1.14)

242

Chapter 12. Surface Bending and Shell Buckling

where S

c

= 0 for dead" external loads and S

c

=

1

for a follower normal pressure

which is also a conservative load. Furthermore, we assume that S

c

= 1.

Let the additional stress-strain state under buckling have high variabili ty

(the index of variation

t

> 0, see (1.3.4)). Then with an error of order h™

we may neglect the terms with

uif

and

S

c

in (14) (see [55], an d also (2 .3.11) ,

(3.6.2)). In some of the cases considered below,

t =

0, and if we do not keep

these terms in determining the critical load we incur an error which, while

small, does not converge to zero with /i*.

12.2 P u r e B en din g Bu ckl ing M o d e of Shells of

Revo lu t ion

Let us consider the system of equations

e\

=

£2

= w = 0 and seek its

solution in the form

u = u(s)cosnvp, v = v (s) sin nvp, w = w (s) cos nvp. (12.2.1)

Then, this system has the form

^ - i * r ° '

ll

U+

B

V

--R

2

=

0

>

B

TS{B)-B

U

= ° <

12

'

2

-

2

)

and leads to the equation

= (

f l

* E ( 5 ) ) - » ' ^ - = «-

< •»>

Let non-trivial bending (i.e. the solutions of system (2) which satisfy all of

the geometrical bou nd ary con ditions) exist . Then in (1.12) II^ = 0 and

A

= //

2

m i n ( + ) j ' ( m , « j ( m ) ) , J' = ? ^ - , (12.2.4)

Ui(m) llrp

where the minimum is calculated for all possible cases of bending.

As was noted in Section 3.6, the existence of bending decreases the critical

load. In cases where there is no bending or the deformations closely resemble

bending A ~ 1 (as follows from (1 .12)). At the sam e tim e w ith bend ing A ~ /i».

The method used here is similar to the method of studying low frequency

shell vibrat ion s (see [57]). T he difference is th at in the case of vib ratio n th ere is

a quantity which is proportional to the shell kinetic energy in the denominator

of functional (4). Also, the solution of buckling problems for weakly supported

shells is less important, from a practical point of view, than the solution of

vibrat ion problems.



12.2. Pure Bending Buckling Mo de of Shells of Revolution

243

Lis t ed be low a r e t he m a in cases o f su pp or t un de r w h ich a she l l o f rev o lu t ion

h a s n o n - t r i v i a l b e n d i n g .

A shel l in the shape of a cupola has bending i f i t s edge i s f ree .

A she l l w i th tw o edges has b en d i ng i f on ly one su pp or t i s ap p l i e d a t one o f

i t s edges .

A she l l wi th two edges has bend ing i f one o f t he edges

(s = s\)

is s u p p o r t e d

by a d i a p h r ag m , wh ich i s abs o lu t e ly r i g id i n -p l an e an d abso lu t e ly flexib le ou t -

o f -p l a n e . I n t h i s c a s e t h e g e o m e t r i c a l b o u n d a ry c o n d i t i o n s h a v e t h e fo rm

ujsinfl - ucosO = 0 , v = 0 for s = s i , (12 .2 .5)

w h e r e

9

i s t he a ng le be tw een t he ax i s o f rev o lu t ion a nd th e she ll n o r m a l . Here

we shou ld ac tua l ly sa t i s fy on ly t he cond i t i on

v

= 0 . At t he same t ime the f i r s t

o f cond i t i ons (5 ) i s au tomat i ca l l y fu l f i l l ed by v i r t ue o f e

2

= 0.

A she l l wi th two f ree edges has fo r each

m

t w o - p a r a m e t r i c s e t o f b e n d i n g

an d we sho u ld m in im ize fun c t ion a l (4 ) i n t h i s se t .

A s h e l l wh i c h h a s p o s i t i v e Ga u s s i a n c u rv a t u r e w i t h t wo e d g e s , h a s n o b e n d

ing for

m

^ 0 , i f a t l ea s t one su pp or t i s i n t ro du ce d a t each of i t s edges .

The ques t ion o f t he ex i s t ence o f bend ing fo r a she l l o f revo lu t ion o f ze ro

cu rva tu re may be eas i l y so lved , s i nce one can wr i t e exp l i c i t l y t he se t o f so lu t ions

o f sy s t e m (2 ) . Fo r a cy l ind r i ca l she l l

u = C

2

~ ,

v

= C-+C

2

--, w = d + C

2

^ (12 .2 .6 )

m

A

m mR R

and for a conica l she l l

sin a

= G

1

-

1

v = Ci C

2

— , (12 .2 .7 )

ms

2

m \ ')

_ s „ si n

2

a — m

2

w = Ci + Ci = ,

s

2

cos

a m

1

cos a

where 2a i s t he ang le a t t he t op o f t he cone , C\ a n d C

2

a r e a r b i t r a r y c o n s t a n t s ,

an d s i s t h e d i s t a nc e f rom th e t o p of t h e cone .

A she ll o f neg a t iv e Ga us s i a n cu rv a tu re m ay have ben d i ng if a t each of i t s

e d g e s o n l y o n e s u p p o r t i s i n t r o d u c e d (o r t h e e d g e i s s u p p o r t e d b y a d i a p h ra g m ) .

B e n d i n g e x i s t s f o r s o m e s h e l l e i g e n s i z e s f o r wh i c h b o u n d a ry v a l u e p ro b l e m

(3 ) has e igenva lues

m

2

wh e re m is a n i n t e g e r f or c o r r e s p o n d i n g b o u n d a ry

c o n d i t i o n s wh i c h d e p e n d o n t h e s u p p o r t t y p e .

A she l l wi th t o rs iona l momen t s app l i ed t o i t s edges wi l l no t buck le by pu re

be nd ing m od es . I t fo ll ows f rom th i s fac t , t h a t by v i r t ue o f (1 .14 ) ,

U'

T

=

0 in

(4 ) .

I n o t h e r wo rd s , t h e i n i t i a l s t r e s s - r e s u l t a n t , S°, doe s no t do wo rk on t h e

ad d i t i o na l d i s p l a ce m en t s (1 ) wh ich ag rees w i th t he f i nd ing of Sec t ion 11 .5 .



244 Chapter 12. Surface Bending and Shell Buckling

Let us now consider some examples.

Let a shell be under an external follower pressure

q*(s).

W e de no te g** =

max{g*(s )} .

T he n for th e critical pressure , we find by form ulae (4) an d (1.11)

tha t

where the functional J' is the same as in (4).

E x a m p l e 1 2 . 1 . Consider a long cylindrical shell under constant external

pressure q*. Let the bou nda ry supp orts not be introduced or at least let

the ir effect be negligible. T he n, by virtu e of (5) the following are ben din g

deformations

u = 0, v=— , w-\ (12.2.9)

m

an d formu la (4) gives Ao = (m

2

—

l ) u

2

. For m = 2 we come to the well-known

Gra shof-B ress form ula (3.5.7). In this prob lem , u>2 = 0, an d th e c orrection

introduced in ITj, is not revealed.

E x a m p l e 1 2 . 2 . Ne xt, consider a mo dera tely long cylindrical shell of length

L and radius R under external pressure q*(s). Let edge s = 0 be simply

suppor ted (7 \ = v — w = Mi = 0), and edge s = L be free. T he ben ding

deformations are then

u = - L ,

v=-£-, w=l

(12.2.10)

m

2

Rm R

Then by (4) we obtain

L

J

q =

l-[

s

2

q

°(

s

)ds.

(12.2.12)

o

If, in (1.14) we omit terms with w,

2

, then we obtain instead of (12)

L L

J,

= J?f

*V (-)

*> -

L

* J * > _

X )

I ^ * > •

( 1 2 - 2 . 1 3 )

o o

In particular, for homogeneous external pressure

q°

= 1 an d for m = 2 we

find by (11) and (12) that

/ ' = 3 ( l + ^ | ^ ) J "

1

,

J

q

= L

(12.2.14)

246


We assume that w

b

~ 1, w

m

~ ^

Qfm

, w

e

~ /z

a e

. Then a

m

> 0, since

otherwise the bending deformation is not do m ina nt.

Due to (1) we can represent all the variables in (1.12) as the sum of three

te rms ,

for example,

£ l

=

e

b

+ e

m

+ £

e ( g ^ f l ^ x i ^ + x j n + X ? . ( 1 2 . 3 . 3 )

Then

Wp = n£ + nf, + n£

e

.

Here II™, 11^, IIj,, II®,

11^.

and II | , are constructed by substituting into (3.6.1),

(1.14) and (1.15) the bending and membrane deformations and deformations

due to the edge effect, II™

e

(II^

e

, IIj.

6

) are bi-linear integrals of a form which

consists of m em bran e (or bend ing) a nd edge effect deform ations.

It can be shown that the order of the last terms in (4) is always less than the

sum of the first two term . Below, we find only the m ain te rm in the expansion

A by the powers of

\i

and that is why we omitted the last terms in (4). For

the same reason, by virtue of

a

m

> 0 we om it the m em brane com pared to the

bending deformations.

As a result (1.12) has the form

A

= m i n W ^

m

t f f l

n

^

Z C

.

Z* = nt+»

4

R

2

ni,

(12.3.5)

". pi

1

(n£ + nf,)

while at the same time, the orders of the variables in (5) for m ~ 1 are the

following

n

m _

/ i

2 «

m

.

n

k , n £ ~ l ; Z

e

~n

2o

"

+1

. (12.3.6)

Es tim atin g the o rders of Ilf and 11^. we take in to account th a t the edge

effect solutions are essential only in a band whose width is of order

/i.

The

order of

Ilj,

depends on the edge conditions in the neighbourhood of the edge

s = SQ being considered, nam ely

n j ~ /i

2

«<=-* for

h

(s

0

)

±

0,

n j ~ / z

2

^ for t

l

(s

0

) = 0, | * i ( * o ) | + | M « o ) | # 0 . (12.3.7)

Hj,

~

u

20 1

^

1

in the oth er cases .

At the same time we assume that all

|

£,| < 1.

(12.3.4)

12.3. The Buckling of a Weakly Supported Shell of Revolution

247

W e can now p r e s e n t s o m e m o r e u se fu l f o r m u l a e for the edg e effect so lut io ns

tf

j

(8,ri = F

j

(8,rie"-

lr

*-°\ v)

+

R 2

^ =

0,

(12.3 .8)

w h e r e y

is any

u n k n o w n f u n c ti o n , y^

are

pow er s e r i es

in

fi,

the

h i g h e r o r d e r

t e r m s

of

w h i c h

are

c o n s t a n t , w h i le

the

o t h e r s

are

p o l y n o m i a l s

in

p o w e r s

of s.

T h e f o l l o w i n g e s t i m a t e s and r e l a t i o n s for the h i g h e r t e r m s of ser ies (8) are

t hen va l id

w

e

, 7 |

~ n

a

°; w

e

, w f , W 2 , w

e

~ / /

a e + 1

; >c

2

,f

e

~ fJ.

a

'~

1

;

r

w

e

vuf r

2

(12.3 .9)

fl

R

2

R

2

\l*

H e r e ,

the

s u b s c r i p t

j is

o m i t t e d .

W e

can

w r i te

the

s o l u t i o n

of the

edge effect

in the

z e r o t h a p p r o x i m a t i o n

w

= C\ e x p — - - + C

2

e^P—- -, (12.3 .10)

w h e r e

C\ and C

2

are

c o n s t a n t s

to be

d e t e r m i n e d l a t e r ,

so is the

edge pa ra l l e l

in

a

n e i g h b o u r h o o d

of

w h i c h

the

edge effect

(5ft r, < 0 for s

0

= si < ^2) is

c o n s t r u c t e d . T h e n

z* =»Bc

1

c

2

(j p) (I

+

OM),

fW

T

= ^ ( n C f +

Anr

2

C,C

2

( n + r,)'

1

+

(

1 2 3 1 1

)

+ r

2

C

2

2

) ( <

1

( s

0

) + O ( / i ) ) .

T h e m u l t i p l i e r 7r, w h i c h a p p e a r s in the i n t e g r a t i o n by <p in the n u m e r a t o r

a n d d e n o m i n a t o r

(5), is

o m i t t e d .

W e w r i t e Z

e

( w i t h

an

e r r o r

of

o r d e r (i)

for

d i f ferent ty pe s

of

s u p p o r t c o n

d i t i o n s

for Q]

"

B

( "

b

>

G I )

3

\ 1/2

for

Q^ = 0, 71 +7 ? =0 ;

(12.3 .12)

1/2

Z

e

=

for w

b

+ w

e

= 0, M* = 0; (12 .3 .13)

for

w

b

+ w

e

= 0, 7 + 7 = 0,

(12.3 .14)

where all of the values are calculated at s = SQ . The value of Tlj, is compara

tively small in all of the cases considered later.

Beginn ing with a shell of positive curvatu re w ith one edge, s =

SQ,

i.e.

th e shell is a cu pol a. For fixed m system (2.2) has a unique solution u

b

, v

h

,

w

b

, which is regu lar in the to p of the cup ola .

First we introduce the support 71 (s

0

) = 0. Then, by virtue of (6), (9), (12)

we find

/ / \

a

e

= l , a

m

= 2,

\=—^{l + 0(n))~n;

So

n£ =

\j [h

(«? +

7?)

+

h H +

7l) + (12.3.15)

*cq°,

+

— 5~ («7 i + V72)

K

Bds,

where Z

e

is dete rm ined by (12), and in order to calcu late life we take the

bending deformations.

R e m a r k 1 2 . 1 .

For a stron gly s up po rted convex shell A ~ 1, for a shell

which has non-triv ial bend ing A ~

fi

2

.

It follows from (15) th a t sup po rt

introduced

71

(so) = 0 leads to an increase in the order of the p ara m ete r A

( A ~ / i ) .

Now we can show t ha t the m inim um value A is obta ined for

m —

2. For

this we assume that m^> 1, and find a solution of system (2.2) in the form

x

b

(s,m)

= m^J2

m

~

kx

«^

e±mIPdS

>

P

=B\lt) '

( 1 2

'

3 1 6 )

fc=o

^

l

'

where x is any unknown function.

Let

w

b

~ 1, then

u

b

,v

b

~m-

1

;

7^,72 ~ « ; * i ,

*2, r

b

~ m

2

. ' (12.3.17)

Let us stud y the case when the sign is positive in formula (16). Th en

functions (16) increase fast with the growth of s. That is why in calculating

the integrals of the functions of type (16), the order of the integral is m times

less than the order of the integrand. Indeed, for

ip'(s)

> 0 we have [15, 37]

5 2

/

f(s)

exp(mi/ ' ( s ) )

ds

jn%l>'(s2) V

m 2

e

m</>(*2). (12.3.18)

12.3. The Buckling of

a

Weak ly Sup ported Shell of Revolution

249

It is not difficult to find the order of the variables in (14)

ll% = 0(m), \~nm,

(12.3.19)

from which it follows that under buckling, m — 2.

Now let the support w (s

0

) = 0 be introduced (or simultaneously two sup

por t s

w

(so) = 7i(«o) = 0). T he sam e developm ents show th at in this case

buckling occurs for

m

~ / /

_ 1

while at the sam e tim e A ~ 1, i.e. the existence

of the possible tangential bending (under i ts construction the non-tangential

suppor t s

w

= 0 and 71 = 0 are neglected) does not lead to a decreasing in the

order of the critical load. This means that the buckling problem differs from

the vibration problem, for which in this case a decrease in the order of the

natural frequency does take place.

The difference mentioned is related to the denominator in (1.12). In the

vibration problem, the main term in H'

T

is w

2

(see [57]), while in the case of

buckling, the dominant term is the quantity tif

2

+ 2*37172 + *272 > which is of

order m

2

w

2

.

Consider a convex shell with two edges and limit the discussion to the case

when edges

s =

si and

s = s

2

are simply supp orte d. As was note d abov e, if

we introduce one or two supports described by (2.5), the neutral surface has

non-tr ivial bending. We assume that the suppo rts do not allow non-tr ivial

bending. The free vibrations of such a shell are considered in [57] and we will

use a similar method here for the buckling problem.

Let us take the bending solution (16)

w^ = (wo(s) + O (m

*) ) exp <

m I pds >,

\ >

l

J

J (12.3.20)

w

0

= (R

1

R

2

)

1/

\

which damps far away from the free edge

s

=

s

2

•

To satisfy the boundary conditions at s = Si we add to the solution the

functions w\, w

m

and w

e

which are respectively the bending solution, mem

brane solution and the edge effect solution which damps far away from edge

s = si.

By virtue of (20),

w\

~ 1. We can calculate the values of the variables in

(5) for m 3> 1. The values of Il£. and LT^ are determined by the contribution of

Wi in the neighbourhood of edge s = s

2

, where the displacements and bending

deformations are most essential.

250


We now have

n

b

=

m

3

R

1

n

T

=

4 5

2

m R

1 +

Ri

\v

Ri

(I +

O ^ -

1

) ) ,

R\ J R\

-(to + Oim-

1

)^, t

0

= h^- + t

[12.3.21)

where Ri, B and to are calculated at s = «2- The other variables in (5) depend

on the form of the boundary conditions at

s = s±.

We will consider only the case when

u — v = 0a.ts = si.

Using relation s

developed in [57] we obtain

n?

4 B V

m R\

» 2

1 - )- — I -l - 4 —

RiJ Ri

- l

(l + Oim-

1

c =2

pds,

(12.3.22)

where Ri and B are calculated at s = s\, and th e value of II™ is de term ine d

by the contribution of the tangential deformations in the neighbourhood of

edge

s

= «i. In the case under consideration the no n-tan gen tial bou nda ry

conditions in the zeroth approximation are not essential and the values II | ,

n ^ and Ily in (5) are secondary.

We find in the zeroth app roxim ation

A (m ) = A i / / -

2

m -

2

e -

m c

+ A

2

/<

2

m

2

(12.3.23)

where the constants A\ and A^ which do not depend on /i and m are obtained

by substitution of (21) and (22) into (5).

By minimization of A (m) by m we get

\ 2 2

A

0

~ /i m

0

,

mo

~ c

x

ln (/z

4

)

(12.3.24)

For shells of positive curvature with free edges the decrease in the order of

the critical load is typical and is caused by the difference between the math

em atica l rigidity of the surfaces an d th e physical rigidity of the shells. Th is

problem has been analyzed in [54, 55].

Est imates (24) are obtained under part icular assumptions on the boundary

conditions at edge s = si and the type of load. For other boundary conditions

which do not allow pure bending, estimates (24) are valid and only the con

stants in these estimates are changed (for example, constant

M

in the estimate

m

0

~ M l n / /

- 1

) .

12.3.

The Buckling of a Weakly Supported Shell of Revolution 251

T h e ef fect o f t he i n i t i a l s t re ss - re s u l t a n t s

t{

is m o re e s s e n t i a l . F o r m u l a (2 1)

fo r I l y , an d a l so e s t im a t e (24 ) a re va l id on ly fo r

to(s

2

) ^

0 . W e n o t e t h a t t h e

v a l u e

to

is p ro p o r t i o n a l t o t h e e x t e rn a l n o rm a l p r e s s u r e .

Let £0(^2) = 0 and <o(

s

2) 7^ 0 , w hich i s va l id for an e m p ty she l l e n t i r e ly

im m er se d in a l i qu id ( see F i gu re 1 2 .1a ) . In t h i s ca se , i n s t ea d o f (21 ) an d (24 )

w e o b t a i n t h e e s t i m a t e s

n£

m

0

In//

1

,

A

0

~

H

2

ml, (12 .3 .25 )

i .e . o f o rder

AQ

i n c r e a s e s c o m p a re d t o ( 2 4 ) .

F i g u re 1 2 .1 : An e m p t y c o n v e x s h e ll i m m e r s e d i n a l i q u i d (w a t e r ) .

Now, l e t t he t op po r t i on o f t he she l l be no t l oaded , i . e .

t

1

=t

2

=t

o

= 0

t

0

>0

for s

0

<

s

^ s

2

,

for si <C s <

SQ

(12 .3 .26 )

( see F igu re 12 .1 b ) . In t h i s ca se t he un loa de d pa r t o f t he she ll ac t s a s a s u pp or t .

T h e o rd e r o f

Ao

i nc rease s a s

s

2

—

so ge t s l a rge r and unde r cond i t i on (29 ) buck l ing

m o d e s c l o s e t o b e n d i n g d o n o t o c c u r .

Indeed , i n s t ead o f (21 ) and (23 ) we now have

n§,

Co

«2

-2 pds,

(12 .3 .27)

A (m ) ~

A-m^m^e™

0

+ Ain~

ai

i

i(c-co)

where a ,- ,

j3 ,

/?, and

A,

d e p e n d o n t h e b o u n d a r y c o n d i t i o n s a n d c h a r a c t e r i s t i c s

o f t h e l o a d i n g . F o r c o n d i t i o n s

u = v = 0

a t

s = si

a n d

t

0

(so)

/ O w e h a v e

/? = 1,

a.i

=

Pi

= 2 . T h e m i n i m i z a t i o n b y m g i v e s

An

H

a

°m

n

a

0

=

0 • - fj. m

0

,

_ Q 2 ( c - C p ) - Q I C Q

m

0

~ In

fi

x

,

/?2 ( C - Co) - / ? lC

0

(12 .3 .28 )

A > =



252

Chap ter 12. Surface Bending and Shell Buckling

at th e sam e tim e estim ate s (28) are valid only for c*o > 0. B u t, if

a

0

^ 0 (12.3.29)

we then ob tain Ao ~ 1, mo ~ p

_ 1

, i .e. the loading order does not decrease.

We note that condition ao > 0 imposes restrictions only on the location of

paral lel SQ which separates the loaded and unloa ded po rtion s of the shell. In

part icular, for a\ = a? for a shell which is symmetrical with respect to plane

s

= s* = 1/2 (s2

—

«i) , condit ion a

0

> 0 has the form «o >

«* •

12 .4 W eak ly S up p or ted C y l ind r i ca l an d Con i

cal Shells

The effect of the boundary conditions on the critical value of the load

pa ram ete r A was considered in Ch ap ter 8. It follows from Tab le 8.4 th at for

the four last bo un da ry c ondition v arian ts, the order of A decreases. These

variants are considered in more detail below.

We begin with the case of a cylindrical shell. Let the parameter

A

be intro

duced by (1.11). Then by virtue of (7.1.3) for a "well" supported cylindrical

shell

\~ft- \= i3.r, + 0(ji

2

),

m o - / "

1

'

2

, (12.4.1)

where the parameter n is given in the last c olu m n of Ta ble 8.4. In thi s case

the shell edge condition is considered to be "well" supported if

77

^ 0.

The other variants of (weak) support we divide into two groups: supports

which do and do not, allow pure bending. For the first group we get

A ~ / J

2

,

mo = 2,

\ = n

2

J' + 0{ft

3

),

(12.4.2)

where J' is introd uced by (2.4). If the shell is und er an extern al no rm al pressure

(see examples 12.1 and 12.2) then

J'

= 3

J~

l

or

J'

is repr esen ted by (2.11)

depending on whether bending case (2.9) or (2.10) satisfies the given supports.

It occurs t h at all of the varia nts in lines 8— 10 of Table 8.4, and the varia nts

in line 7, without restriction on the angle of rotation 71 neither for

s —

0 nor

for s = L allow the b en din g. In th e first case we consider b en din g (2.9), in

the second case ben ding (2.10). Read ing Table 8.4 one should use formulae

(8.4.13-16) in which we list the corresponding complete boundary conditions.

Next let us consider the type of support for which rj = 0 and which, on

the other hand, do not allow pure bending. Let one of the boundary condition

varian ts of s imple sup port be introduced on edge

s

— 0, and o ne of th e varia nts

12.4. Wea kly Supported Cylindrical and Conical Shells

253

of boundary conditions from the free edge group be introduced on edge

s

=

L

(see line 7 of Table 8.4 and formulae (8.4.14) and (8.4.16)).

We take bending (2.10) which satisfies all of the supports except support

by the angle of rot atio n (71 = 0), if it is introdu ced . T h at is why we add to

bending (2.10) (if necessary) the edge effect solutions in the neighbourhood of

edges

s

= 0 and

s = L

w- w

b

+ w

e0

+ w

eL

,

1,

fim.

(12.4.3)

We are considering a moderately long shell and that is why we can neglect

the interaction of the edge effect solutions. Unlike (3.1) in (3) the membrane

deformation

w

m

is not introdu ced since it does not con tribu te the zeroth ap

proximation, the only one which we are considering here.

Substituting (3) into (1.12) or into (3.5) and keeping the main terms we

obtain

Z

e0

+ Z

eL

+ p

4

i ?

2

n

b

n

2

n

b

(12.4.4)

That by virtue of (3.12) and (3.14) gives us

A(m) =

+ l / 6 ( m

2

- l )

2

/ V

1

+

6 ( 1 - " )

m

2

/

2

(W)~\

(12.4.5)

where I = LjR,

<*o

= <

OIL

for 0110, 0100,0010,

for 0101, 0011,

for 0111,

for 0000,

for 0001

(12.4.6)

(see the notation for the boundary conditions introduced in Section 8.4 and

Table 1.1).

In (5) the first term is proportional to the energy of deformation caused by

the edge effect solutions in the neighbourhood of edges s = 0 and s = L, and

the second term is proportional to the bending deformation energy.

Let

a

0

+ OCL >

0. By virtue of (3.15) for

t

2

>

0 we have

L

n £ = m

2

( n % + 0 ( m -

2

) ) ,

U

b

T0

= js

2

t

2

ds.

(12.4.7)



254


Now, minimization (5) by m yields

A o = ^

/ 3 ( Q

3 ^

b

^

3 / 2

( l

+ 0

(^))

m

0

n

T

°

x

i/4 (12-4.8)

Thus ,

we obtain the estimates

A o ~ / x

3 / 2

, m

0

~ / i -

1 / 4

, (12.4.9)

which falls between estimates (1) and (2).

T he accurac y of (8) for Ao is not hig h. Fo rm ula (5) for integer m which is

the closest to mo, is more accurate and has an error of order /i.

We can now write the expression for Ily in the case when the shell is loaded

by external pressure q* (s) a nd axia l force P

0

where

t\

is introduced by (1.11) for

P = 2nRT®.

In the case of compression,

t\

> 0. If we add itionally load the shell by a torsional m om ent th en it does not

effect the parameter A within the accepted limits of accuracy since the integral

of the term 2*37172 (1-14) is cancelled (7172 ~ sin

rrvp

cos

mcp

and the integral

by

ip

is equal to zero).

Now let us consider conical shells. As in the case of cylindrical shell esti

mates (1) are also valid for "well" supported conical shell. If the supports allow

pure bending then estimates (2) are valid. We will consider the intermediate

cases. The total displacement we will represent in form (3) again.

For different boundary conditions for

s = s\

and

s

=

s

2

we can tak e

different linear combinations of bending (2.7)

(12.4.11)

sin a

9

s -

s

k

, v =

msk

( 0 1 1 1 , 0 1 1 0 ,

sin a

m

2

—

• 2 ' " -

sin a

s — Sk

s in

2

a

, w= +

Sk

cos a m

z

cos

a

0 1 0 1 , 0 1 0 0 f o r s = s

k

);

s m s

—

Sk

msk m

2

- s in

2

a ' s

k

cos a

(0011,0010 for

s = s

k

);

(12.4.12)

u =

0,

v

= ,

w

m,S2 «2

cos

a

(100 1,10 00 for * = * i, s = s

2

); (12.4.13)

12.4. Weak ly Supported Cylindrical and Conical Shells

255

We will use formulae (11) and (12) in the case when the conditions of the

simple support group are introduced on edge

s

=

Sk

and the conditions of the

free edge group apply on the other edge, i.e. conditions 0000 or 0001. We use

be nd ing (11) when sup po rt v(sfc) = 0 is intro du ced , and be ndi ng (12) when

it is not, but support w(sfc) = 0 does exist. We use bending (13) in the case

when weak support conditions are introduced on both edges

s

=

si

and s = s

2

-

Let us now write the expressions for the potential energy due to bending

(11)-(13).

R

2

ni

R

2

lt

(ra

2

1)

2 sin a cos

2

a

+ x -

2

<

2

- * W -

1

+

l - *

2

2(1 - x )

2

^ 2 <>

H

3-k ^

( m

2

- l )

2

2 sin a c o s

2

a

ln;>

( l -

J

/ ) s i n

2

a ( l - x

2

) C

2

2m'

2

x

2

« 2 '

; (12.4.14)

(12.4.15)

Formula (14) is obtained for bending (11) and (12) where in case (11) £ = 1,

and in case (12) £ = m

2

(m

2

— sin

2

a) . If simple support conditions apply

on the wide edge (s = S2), we assume in (14) that k = 2, and if they apply on

the narrow edge we assume that k = 1. Formula (15) corresponds to bending

(13).

We can now begin to calculate the potential energy due to the edge effect

solutions by assuming that

R = B

(s

2

) =

«2

sin

a.

(12.4.16)

Using formulae (3.12-14) we find the expressions for Z\

2

for different supports



256


(they are marked in the parentheses)

I

Z\ ~ —

r=/jx

_1

/

2

u)

2

(cosa)

3

/

2

+ u.

2

Rww'cos a+

v 2

+ - L x

1

/ y

5

f i V

2

( c o s a )

1 / 2

>

V 2

Z\

=

—-= [iw

2

(cos

a)

z

I

2

+ fi

2

Rww'cos a+

v 2

+ ^=u

3

R

2

w'

2

( c o s a)

1

1

2

(0111);

A/2

Z\ =

- L x - ' / >

2

(cos a )

3

/

2

, (12.4.17)

2 v 2

Z | =

^ f ^

2

(

c o s a

)

3 / 2

(

0 1 1

° ) ;

Zf =

-^

r

x

1

/

2

u.

3

R

2

w'\cosa)

1

/

2

,

2\/2

Z§ = ~u.

z

R

2

w'

2

( c o s

a)

1

1

2

( 0 1 0 1 , 0 0 1 1 , 1 0 0 1 , 0 0 0 1 ) ;

2 v 2

Zf = Z | = 0 (0100, 0010, 00 00 ,, 1000).

We calculate the value of Z

e

for each edge separately and that is why the

contribution of the edge effect in the neighbourhood of shell edges s — s\ and

s

=

S2

are denoted respectively by

Z\

and

Z % .

Here

w

and

w'

are taken by

formulae (11)-(13).

For the p ara m ete r A by virtu e of (4) we have

A ( m ) = ( ^ + Z

2

e

+ / f i

2

n ^ ) ( / /

2

n ^ )

_ 1

, (12.4.18)

where Ily. is calculated by formula (3.15), and also IlJj. ~ m

2

.

If

Z\ = Z\

= 0 then es tim ates (2) are valid.

If w = 0 in (17) then Zf + Z\ ~ y? and we come to estimates (9).

In cases 0111 and 0110 we have

w(sk) ^

0 and by virtu e (11)

w(sk) =

m

_ 2

t an a . M in imiz in g A by m we come to the estimates

A

0

~ / /

5 / 4

, m

0

~ ^ -

3 / 8

, (12.4.19)

which fall between estimates (1) and (9).

Estimates (19) are not valid for cylindrical shells since, by virtue of (2.6)

for the bending of a cylindrical shell, w(0) = 0 for v(0) = 0.



12.4. Weakly Supported Cylindrical and Conical Shells

257

E x a m p l e 1 2 . 3 . Cons ider the buck l ing o f a cy l indr i ca l she l l and two con ica l

she l l s , t he top edge o f which a re f r ee and on the lower edge one cond i t ion f rom

t h e s i m p l e s u p p o r t g r o u p is i n t r o d u c e d ( se e F i g u r e 1 2 . 2 ) .

T h e r a d i u s

R

o f th e cy l ind r i ca l she l l i s eq ua l to th e r ad iu s o f th e l a rger ba se

of the conical shel l . The height of a l l of the shel l s i s equal to

L .

The she l l s a r e

l o a d e d b y a n e x t e r n a l h y d r o s t a t i c p r e s s u r e

q*(s)

w h i c h l i n e a r l y c h a n g e s w i t h

t h e h e i g h t . W e a s s u m e t h a t q*(s) = q**q°(s), w h e r e q** is t h e m a x i m u m v a l u e

of the p res sure on the lower she l l edge .

L --

L-E

- -

a o c

F igure 12 .2 : Cy l indr i ca l and con ica l she l l s in a l iqu id .

For the cy l indr i ca l she l l , case "a"

q° = l - - , h= 0, t

2

= q°,

fo r the con ica l she l l , case "b"

n _ * 2 - *

q°s

S2 - * 1 '

n

S -

S i

S2 cos a

S 2 - « 1

The fo l lowing r e l a t ions a re va l id

Eh

2

X

_ (

S2

-s)

2

(s

2

+2s)

H —

—-X

/ r — — , *2

0

SS2

(«2

— s

i ) cos a

_ (s-s

1

)*(2s + s

1

)

n — -z -. r , t

2

=

o

SS2

(«2

—

* i ) cos a « 2C os a

q°s

(12 .4 .20)

(12.4 .21)

(12 .4 .22)

L =

R

2

^/l2(l-u

2

)'

i i ( l - x )

R = S2 s in a,

« i

(12 .4 .23)

t a n a s

2

W e s u p p o s e t h a t

R/h =

4 0 0 ,

v

= 0 .3 ,

x =

0 .5 , t a n a = 0 . 5 . T h e n

L = R.



2 58 Chapter 12. Surface Bending and Shell Buckling

Ta ble 12 .1 : T he c r i t i ca l va lues of AQ

No.

1

2

3

4

5

6

Boundary Conditions

0111

0110

0101

0011

0100

0010

v = w

= 7i = 0

v = w

— 0

v = 71 = 0

w = 71 = 0

v =

0

w = 0

Cylinder (a)

m

4

2

3

3

2

2

Ao

0.0888

0.0186

0.0650

0.0650

0.0186

0.0186

Cone (b)

m

3

3

2

3

2

2

Ao

0.0798

0.0350

0.0472

0.0531

0.0168

0.0181

Cone (c)

m

4

3

2

2

2

2

Ao

0.372

0.229

0.132

0.105

0.073

0.063

T h e c r i t i ca l va lues o f Ao an d th e cor r esp on din g va lues of m for d i ff e ren t

b o u n d a r y c o n d i t i o n s a r e r e p r e s e n t e d i n T a b l e 1 2 . 1 .

For the cy l indr i ca l she l l we t ake bend ing (2 .10) which exac t ly s a t i s f i es sup

p o r t s n u m b e r s 2 , 5 a n d 6 a n d t h e c o r r e s p o n d i n g v a l u e of Ao m a y b e f o u n d b y

( 2 . 4 ) .

T h e i n t r o d u c t i o n o f t h e a d d i t i o n a l s u p p o r t 7 1 = 0 i n c r e a s e s A o, a n d

buck l ing occur s fo r m ^ 3 ( see for m ula e (5) an d (8 ) ) .

For th e con ic a l she l l we t ak e be nd ing (11) o r (12) de pe nd ing on the bo un d

ary co nd i t io ns a n d Sfc = s\ c o r r e s p o n d s t o c a s e " b " ( s u p p o r t a t t h e n a r r o w

e d g e ) .

W e f in d t h e p a r a m e t e r Ao b y m e a n s o f f o r m u l a ( 1 8 ) . H e r e , p u r e b e n d i n g

tak es p lac e on ly in cases num be r 5 an d 6 .

I n t h e o t h e r c a s e s , t h e a p p e a r a n c e o f t h e d e f o r m a t i o n e n e r g y (Z \ o r Z\ in

(18)) as so cia ted w i th th e edge effect le ad s to an increa se in th e value of Ao.

C o n d i t i o n n u m b e r s 1 a n d 2 f or t h e c o n i c a l s h e l l l e a d t o e s t i m a t e s ( 1 9 ) ,

a n d c o n d i t i o n n u m b e r s 3 a n d 4 l e a d t o e s t i m a t e s ( 9 ) . I n t h e e x a m p l e u n d e r

c o n s i d e r a t i o n / i

1

/

4

= 0 .4 and the d i f f e rence be tween these es t imates i s no t

r e v e a l e d s i n ce t h e o t h e r p a r a m e t e r s o f t h e p r o b l e m a r e d o m i n a n t .

The o rder o f the c r i t i ca l load fo r the cy l indr i ca l she l l and fo r the con ica l

s h e l l s u p p o r t e d a t t h e n a r r o w e d g e ( c a s e " b " ) , a r e a p p r o x i m a t e l y t h e s a m e .

The con ica l she l l suppor ted a t the wide edge i s more r ig id .

I t can be s een f rom a compar i son o f numbers 3 and 4 , and a l so 5 and 6 in

T a b l e 1 2 .1 t h a t i n s o m e ca s e s , s u p p o r t v = 0 i s m or e r ig id , wh i le in the oth er

c a s e s s u p p o r t

w —

0 is m or e r ig id .



12.5. Weakly Supported Shells of Negative Gaussian Curvature

259

1 2 .5 W e a k l y S u p p o r t e d S h e l l s of N e g a t i v e G a u s

s i a n C u r v a t u r e

I t i s a ch a ra c t e r i s t i c o f she l ls o f rev o lu t io n wi th n ega t ive G au ss i an cu r

va tu r e t h a t if on ly one su pp or t a t each edge i s app l i ed t he she l l m ay h ave

be nd in g w h ich sa t is f i es t he se su pp or t s . I f t h i s i s t h e case t h e si ze of t h e she l l

is t e r m e d t h e

eigensize.

T h i s p ro p e r t y of s h e ll s w i t h n e g a t i v e c u rv a t u r e h a s

bee n ex ten sive ly an aly ze d (see [4 , 31 , 51 , 52 , 56 , 57 , 9 1 , 119]) . B elow w e wi l l

cons ide r t h e l e ss o f t en s tud i ed p ro b le m of t h e e ffec t o f no n - t a ng en t i a l su p

po r t s and sm a l l va r i a t i on s f rom the e igens i ze on the c r i t i ca l l oad ( see a lso

[50, 90, 118, 128, 129]).

S i n c e t h e r e e x i s t m a n y di ff e re n t b o u n d a r y c o n d i t i o n v a r i a n t s we w i ll c o n

s i d e r o n l y t h e c a s e wh e n c o n d i t i o n s s i m i l a r t o t h e s i m p l e s u p p o r t g ro u p a r e

app l i ed a t bo th she l l edges ,

s — s\

a n d

s — s

2

.

F o r t h i s g ro u p o f s u p p o r t s

t h e c o n d i t i o n wh e re T\ = 0 ( see (8 .4 .14 ) ) i s ch a ra c t e r i s t i c . As w as no t ed abo ve ,

we c a n r e p r e s e n t t h e a d d i t i o n a l s t r e s s - s t r a i n s t a t e a s a s u m o f t h r e e t e rm s (3 .1 )

a n d u s e v a r i a t i o n a l f o rm u l a (3 .5 ) t o d e t e rm i n e t h e c r i ti c a l l o a d .

Le t u s d i scuss t he e s t im a t e of t he v a lue II™, in fo r m ula (3 .5 ) by con s ide r ing

t h e a u x i l i a r y s y s t e m o f e q u a t i o n s wh i c h i s e q u i v a l e n t t o e q u a t i o n (2 .3 )

dx R

2

,

„

dy x ,

„

-r-

m

id-y

=

0

' ; r -

m

^ 7 T = °-

1 2

-

5 1

as BRi as BR

2

W e c a n r e p r e s e n t t h e d i s p l a c e m e n t s

u

b

,

v

b

a n d

w

b

of th e she l l und er

b e n d i n g a n d s t r e s s - r e s u l t a n t s 7 \ ,

T

2

a n d

S

( i n a c c o rd a n c e w i t h t h e s t a t i c -

g e o m e t r i c a l a n a l o g y ) , wh i c h a r e t h e s o l u t i o n s o f s y s t e m (3 .2 ) , i n t e r m s of s o

l u t i o n s x a n d y of sy s te m (1)

w

b

= - ^ , v

b

= By, w

b

= B'x + mR

2

y;

J \ _

=

_R2i

j2___R|y_

s _ x

E

0

h

0

B

2

'

E

0

h

0

~ RiB

2

'' E

0

h

0

~

S

2

T h e d i s p l a c e m e n t s

u

m

, v

m

a n d

w

m

,

wh i c h a r e g e n e ra t e d b y s t r e s s - r e s u l t a n t s

(3) ,

m a y b e fo u n d f ro m t h e s a m e fo rm u l a e (2 )

Bx

m

u

m

= - — , v

m

= By

m

, w

m

= B'x

m

+ mR

2

y

m

,

(12 .5 .4)

B2

where , by v i r t ue o f (3 .2 )

x

m

a n d

y

m

s a ti s fy n o n -h o m o g e n e o u s s y s t e m (1) a n d

(12 .5 .2)

(12 .5 .3 )

260


are equal to

x

m

= d(s)

Xl

+ C

2

(s) x

2

, y™ = Ci(«)

2/1

+ C

2

(s) y

2

,

S

- I d

,'2{l + v)xx

2

, f, 1vR

2

, R\\ R\yy

2

\ ,

ci = -i ( —

W 3

— ^ - - R T ^ - B ^ ;

(12

-

5

-

5)

« 1

C 2 =

-2(1 +

, ) ^ / ^ ^ ^

^

• / (

2

5

3

V Ri R\) B

3

S i

Here

x

an d j / are the solutions of system (1) dete rm ining stress -resu ltants

(3). Th e fundam enta l set of the solutions of system (1) is formed by Xk and

j/fe,

which satisfies (for definiteness) at

s = s\

the conditions

x\-

3/2 = 1,

x

2

= yi=0.

(12.5.6)

The multiplier g is determined by (4.3.2), and for constants E, v and h we

have g = 1.

After integration by parts the expression

II? = -±— f

(Tiei + T

2

s

2

+ Sw) B ds

(12.5.7)

Z-C/Ofto J

» 1

we obtain by virtue of (3.2)

B

™ =

3

— (Tx t . - + S ,

m

) |

j = s 2

= I (</

m

z - x™y) \

s=s2

, (12.5.8)

2£,

since u

m

= v

m

= 0 for s = «i.

Let the support i> = 0 be introduced at s — s\ and s = s

2

and let for some

m ^ 2 be

yi(s

2

) = S

and

\5 \

-C 1.

Then bending (2) for x — m~

l

x\, y = m~

l

y\ exa ctly satisfies the con dition s

v

b

(si)

= 0 and appro xim ately satisfies the condition

v

h

(s

2

)

= 0 (the m ultiplier

m

_ 1

is introduced to get w

b

~ 1).

To compensate for the error in condition «(s2) = 0we introduce the mem

brane deformation determining by (3) and (5)

x = D

l

x

1

+ D

2

x

2

, y = D

lV l

+D

2

y

2

, (12.5.9)

where

D\

and

D

2

are unknown cons tants. We take

D

2

= 0, then by virtue of

(3) and (6), condition

T\

= 0 is fulfilled (exa ctly at

s

— «i and approxim ately

12.5. Weakly Supported Shells of Negative G aussian Curvature 261

at s = s

2

)- We can determine the constant D\ from the condition 2/

b

(s

2

) +

y

m

( s

2

) = 0. With an error of order

S

we then obtain [128].

A =

- J * M - n? = JL rf(.

a)> c =

£ i M (12.5.10)

a3/

2

(s2) m

J

la

where a = Ci{s2) for ar = x i an d y = y

1

.

Let edges s = si and s = s

2

D e

free in the tangential directions (Xi = S

—

0),

however, support w = 0 is introduced, i.e. due to (2)

(

k = y

+ f3

k

x = 0, & = - - ? - for s = s

k

, k=l,2. (12.5.11)

mR

2

We take the bending given by (2) for

x

= m -

1

(x i

- Pix

3

),

y

= m '

1

(

y i

- / ?

l 2 / 2

) , (12.5.12)

then w (s i ) = 0.

Let £2 = tf/m and |tf| <g 1, i.e. the condition

w (s

2

)

= 0 is fulfilled ap prox

imately. As above, to compensate for the error in satisfying this condition we

introduce the membrane deformation determined by (3), (5) and (9). In order

to determine constants D\ and D

2

in (9) we have equation w° + w

m

= 0 or

y

m

+ (l

2

x

m

+ G = 0 for s = s

2

. (12.5.13)

The second equation for

D\

and

D

2

we obtain, accounts for the requirement

that the energy, II™, be minimal. This condition has the form D

2

+ 0\D

X

— 0.

Finally, we obtain

2 (2/2 + /?2*

2

)

2

( «

2

i + 2 A « n - / ? i V

2

)

^ = J / i - / ? i 2 /

2

+ / ? 2 ( * i - / ? i ^ 2 ) , ( 1 2 . 5 . 1 4 )

aik = Ci(s

2

) for x = x

k

, y = yk, i,k= 1,2,

at the same t ime, in calculat ing

c\

and

8

we take the functions

x

k

and y

2

a t

s = s

2

- For /?i = /?

2

= 0 form ula (14) turn s into (10).

Consider formula (3.5)

A (m ) = (LT

£

m

+ Z^ + Z

2

e

+ fi

4

R

2

Ul) ( / i

2

n ^ )

_ 1

. (12.5.15)

For all of the boundary condition variants considered

iTf ~

^ , Ul

~ m

4

, n§ , ~

m

2

,

(12.5.16)

262

Cha pter 12. Surface Bending and Shell Buckling

however, the order Zf depends on the boundary condition variant and by virtue

of (3.12-14) is equal to

Zf = 0

Zf ~ n

3

m

2

Zf

~

urn'

2

Zf ~ \im~

2

+ n

3

m

2

We use here the estimates

w

h

~

(0100,0010);

(0101,0011);

(OHO);

(0111),

x = 1,2.

(12.5.17)

1 and 7^ ~

m .

For these boundary condition variants, the orders of the terms in the ex

pression for A(m) (see (15)) are presented in Table 12.2.

Table 12.2: The orders of the terms

1

0100,0010

0101,0011

0110,0111

A(m)

2

5 2

2 2

62

2 2

- j — j -

+ n + H

2

m

2

S2

1 2 2

\ \ < f

Ao

3

H

2

m

4

2

2

1

V/

1

H

2

-C

(5

< 1

Ao

5

( ^ ) l

// + (M)»

p t ^ - f / / )

1

m

6

(0

(0

For fixed values of m, the value of

S

is a m eas ure of the de viat ion of the shell

size from the eigensize. It is clear th at

S

depends on

m

and that is why the

minimization of A(ra) by

m

may be carried out only in sorting by

m =

2, 3 , . . . .

We recall that A ~ ju

2

/

3

and m ~ n~

2

l

z

, if for all m < / i

- 2

/

3

the shell sizes are

not close to the eigensizes (S ~ 1).

We formally assume that S does not depend on m, and minimize the ex

pressions in the second column of Table 12.2 by m. For S = 0 or \S \ < fi

2

the

results of the m inimiz ation are presented in columns 3 and 4 of Tab le 12.2, and

for larger values of

S

the results are given in columns 5 and 6 of that table.

It follows from T able 12.2 th a t if only one sup po rt,

v

= 0 or

w

= 0, is applied

at each of the edges, and the shell size is close to the eigensize (\S\ < p

2

) for

m = 2, then buckling occurs for m = 2, and also A ~ / /

2

.

The introduction of the additional support 71 = 0 keeps the wave number

(m = 2) und er th e buckling value. How ever, it leads to an increase in the



12.5. Weakly Supported Shells of Nega tive Gaussian Curvature

263

order of th e c ritic al load (A ~ /i). If, simultaneously the order of the quantity

S increases (\6\ ^$> fi

2

) and (or) supports are introduced of type v = w = 0,

then buckling will occur for m > 2.

R e m a r k 1 2 . 2 .

The term 11™/

(^U^)

for

m

» 1 becom es the first ter m

in (11.5.12). A value close to zero indicates that the size of the shell is close

to the eigensize.

If, in stea d of the c ondition of simple supp ort, we impo se clamp ed conditions

(see (8.4.13)), then in this case the existence of the possible tangential bending

does not lead to a decrease in the order of the critical load compared to A ~

For example, if for some shell eigensizes there exists bending satisfying the

suppor ts , u = 0, at both edges, then the introduction (even at one edge) of

an additional non-tangential support w = 0 immediately increases the order

of the critical load from A ~ \j? to A ~ /i

2

/

3

.

E x a m p l e 1 2 . 4 . Consider the buckling of a hyperboloid of revolution under

a homogeneous external pressure (see Figure 12.3). Let it be formed by the

revolution of hyperbola x

2

a~

2

— z

2

b~

2

= 1 around the z axis. We have [119]

Rl

= -a-

A

c

2

R\, R

2

= a

2

(a

2

-(a

2

+c

2

)co

S

2

e)~

1/2

. (12.5.18)

After change of the independent variables due to formula

9

c f

R

2

(9)d9

$ . 1 +

TCOS0

)) = / — — = - arcsin —

;

—

'

a

2

J

s ine 2 V r ( l + cos(?)

T T / 2

. 1 - 7 - C O S 0 \ C

2

,.„,,„,

- a r c s i n — , r = 1 + - j (12.5.19)

T

(1 - cos9) J a

z

the coefficients of system of equations (1) become constant.

Let the supports

v (Oi) = v (9

2

) = 0 (12.5.20)

be introduced. Then the shell eigensize (for which system (1) has a non-zero

solution satisfying the conditions y{9$ = y (9

2

) = 0) are determined by the

equation

smm(

m

-

m

) = 0, ri

k

=T,(9

k

),

(12.5.21)

from which it follows in particular that if the shell sizes are the eigensizes for

some

m ,

then they are also the eigensizes for 2m, 3m, ....




z

\

J

A

Figure 12.3: The hyperboloid of revolution.

Let us consider a numerical example

[128].

As the

eigensize

of the surface

we will take

R - c.

Let

u -

0.3,

c/a

= 2, cos6» = 1/3 an d

q°

= 1 (in (1.11)).

Then, due to (19) and (20) rji =

TT/4

and the shell sizes are the eigensezes for

m =

2, if 0

2

=

7T

-

#1

=

01

. V2

= - T /

4

-

Let only supports (20) be introduced, however, the shell differs from the

eigensize (02 = 0*, + A). In Figure 12.4 the dependence of A on A is shown for

m = 2 and

m =

4 and for

h/R =

0.003 (solid lines) and

h/R =

0.01 (dotted

lines).

160-

80

O

-0 .04

O

0.04

A

Figure 12.4: The load parameter for hyperboloid buckling.

These graphs agree with the results represented in the first line of Table

12.2.

Nam ely, for sm all devia tions in shell geom etry from its eigensize, b uckling

occurs for m = 2, while for large deviations, buckling occurs for m = 4.

The value of A which separates these cases, increases with the dimensionless

thickness.

J I I I L




265

1 2 . 6 P r o b l e m s and Exerc ises

1 2 . 1 . Cons t ruc t for m = 2 ,3 ,4 ,5 bend ings of the neutral surface of the

shell

of

revolution that

has the

form

of the

elliptic cupola.

H i n t . Co nstruct num erically solutions of equation (2.3) regular at the apex

of the cupola . Use geometric functions from (4.4.8).

1 2 . 2 .

Cons t ruc t the solution that has a form of bend ing for the spherical

cupola.

H i n t

Use relat ions (2.2) and (2.3) for

R

X

= R

2

= R,B

=

Rsm6.

A n s w e r

9

8 0

u=

( l - c o s 0 ) t a n

m + 1

- , v = s i n 0 t a n

m

- , « = (m + cos0) tan

TO

- .

1 2 . 3 . Consider buckling

of the

weightless semi-spherical shell

of the

radius

R

and thickness

h

entirely immersed in a container of liquid of the density 7.

The shell edge is free. The force P = (2/3)7r7?

3

7 balancing the Archimedean

force is applied at the apex of the shell.

H i n t . Use relation (2.4) and the bending obtained in Problem 12.2 for

m

= 2. Under calculation stresses

t\

and

t

2

in (1.14) use relations (1.4.6) for

q°

=

7_R(1

—

s in#)

and

P

=

(2/3)irR

3

*f.

The

rest loads

are

equal

to

zero.

1 2 . 4 .

Find the exact eigensizes of the hyperboloid of revolution of one

sheet described

in

Problem

11.3 and

Problem 11.5

for the

bou ndary condi t ions

v(6i) — v(02)

= 0. Com pare wi th the results with the approximate values

obtained from equation s in^2 = 0, where ^2 is given by (11.5.6).



C h a p t e r 1 3


a t an Edge

In th i s Chap te r we wi l l d i s cus s the buck l ing modes o f she l l s o f r evo lu t ion under

a m e m b r a n e s tr e s s s t a t e l o c a l iz e d in t h e n e i g h b o u r h o o d o f t h e s h el l e d g e . T h e

i n fl u e nc e o f m o m e n t s o n t h e i n i t i a l s t r e s s s t a t e a n d p r e - b u c k l i n g d e f o r m a t i o n s

w i ll b e c o n s i d e r e d i n C h a p t e r 1 4 . W e a k e d g e s u p p o r t a n d v a r i a b i l i t y o f t h e

d e t e r m i n i n g p a r a m e t e r s c au s e t h e a p p e a r a n c e of t h e s e b u c k l i n g m o d e s .

Modes of th is type are poss ible for convex shel ls and a lso for shel l s of zero

G a u s s i a n c u r v a t u r e u n d e r a x i a l p r e s s u r e . L o c a l i z a t i o n o f b u c k l i n g m o d e s i n

t h e n e i g h b o u r h o o d o f a n e d g e f or sh e l ls o f z e r o G a u s s i a n c u r v a t u r e u n d e r o t h e r

t y p e s o f l o a d i n g ( e x t e r n a l p r e s s u r e , t o r s i o n ) a n d fo r s h e l ls o f n e g a t i v e G a u s s i a n

c u r v a t u r e d o e s n o t t a k e p l a c e (s ee C h a p t e r s 7 - 1 2 ) . A s w i ll b e s h o w n b e l o w , a

weak edge suppor t may s ign i f i can t ly lower the c r i t i ca l load whi l e a t t he s ame

t i m e ,

t h e v a r i a b i l i t y of d e t e r m i n i n g p a r a m e t e r s c h a n g e s i t o n l y s l i g h t l y .

1 3.1 R e c t a n g u l a r P l a t e s U n d e r C o m p r e s s io n

As an ex am pl e i l lu s t r a t in g the e ffect o f loca l i za t io n o f th e buc k l ing m o d e

d u e t o a w e a k e d g e s u p p o r t , c o n s i d e r t h e b u c k l i n g of a r e c t a n g u l a r p l a t e u n d e r

p r e s s u r e . T h e b u c k l i n g e q u a t i o n h a s t h e f o r m [ 1 6 ]

E d g e s y ~ 0 a n d y — b ( se e F i g u r e 1 3 .1 ) a r e a s s u m e d t o b e s i m p l y s u p p o r t e d

w=—^-

= 0

a t

y = Q, y = b.

(13.1 .2)

ay

1

267

268

Chapter 13. Buckling M odes Localized at an Edge

y

b

W w w w w ¥

o

I I I I I

W a x

Figure 13.1: A rectangular plate under comp ression.

Under cylindrical bending of the plate, the deflection does not depend on

x

and has the form

w =

WQ sin — . Th e critical value of the load is equal to

6

T,=T:

n

2

D

b

2

'

(13.1.3)

If edges

x

= 0,

x = a

are also sim ply sup po rte d, t he n th e deflection (for

b

<

a)

has the form

w —

WQ

s in

^~

sin ^ , and the critical value of T2 is equal

to

r

2

= Ar,

A =

1

+

(13.1.4)

where T is defined in (3). As the plate length a increases (b/a —> 0) A —> 1, i.e.

the critical load approaches (from above) value (3) under cylindrical bending.

The same result takes place for all other type of supports of edges x = 0 and

x

= a except in the case when at least one of the edges is free.

For example, let edge x = 0 be free, and x = a be clamped, i.e.

d

2

w d

2

w _

dx

2

dy

2

d

3

w ,

n

. 8

z

w

^ - 3 + ( 2 - ^ ) ^ ^

= 0

da;

3

w =

0

dxdy

2

dw

dx

We seek the buckling mode in the form

for

x

— 0,

for

x

=

a.

> (x, y) = ( C i e

-

T * + C

2

e~ " " + C

3

e ' " + C

4

e '" ) si

i n •

ny

(13.1.5)

(13.1.6)

13.1. Rectangular Plates Under Com pression

269

where

C{

are arbitrary constants and

8=jl-V\,

r = \fl + y/\ = y/2 - s

2

(A < 1). (13.1.7)

The substitution of these expressions into boundary conditions (5) leads to

an equation for evaluation of A

s

( r

2

-

v)

2

- r (v - s

2

)

2

-\

(s

( r

2

-

u)

2

+

+ r(v-s

2

)

2

y-^ = 0 . (13 .1 .8)

We s tudy equat ion (8) for a > i . T ha t is why we neglected term s of order

exp{— 2a/6} in deriving this equa tion. If

we

express r through

s

by form ula (7),

we find that for sufficiently large a/b, equation (8) has the single root s > 0.

Pa ram et er A can be expressed over this root due to (7) by the expression

A = (1 — s

2

)

2

. The values of s and A for different ra tio s a/b for i = 0.3 a re

given in Table 13.1.

Ta ble 13.1: Th e values of s and A for different ra tio s

a/b

na/b

s

A

25

0.0186

0.9993

30

0.0324

0.9979

35

0.0375

0.9972

40

0.0400

0.09968

0 0

0.0436

0.9962

For v = 0.3, equation (8) has a real root if iva/b > 22.7. If ira/b < 22.7', the

solution should be sought in a form which differs from (6). This case will not

be considered here. It may be notes that the buckling of the rectangular plates

unde r different bo un da ry c onditions and values of ratio a/b is con side red, for

example, in [16, 175].

The buckling of a plate with two free edges

x =

0,

x = a

is considered in

[65] and the effect of decreasing the load compared to (3) is revealed. In spite

of the difference between the boundary conditions at

x

=

a

(clamped edge in

our case and free edge in [65]), for

a/b =

oo the resu lts (A = 0.9962, see T ab le

13.1) are identical. This fact is related to the localization of the deflection near

the free edge x — 0. As a result, the bo un dar y con ditions at x = a do not

effect the critical load.

Indeed, for sufficient large values of a/b we may assume that C3 = C4 = 0

in (6) and satisfy only the first two conditions (5) at

x =

0. As a result we

come to the equation

s(r

2

-v )

2

-r(v-s

2

)

2

= 0,

(13.1.9)

270

Chapter 13. Buckling Modes L ocalized at an Edge

which fol lows f rom (8) as a/b —• oo .

I t f o ll ow s f r o m T a b l e 1 3 .1 t h a t t h e l o c a l i z a t i o n m e n t i o n e d a b o v e t a k e s p l a c e

only for suf f ic ient ly long pla tes , s ince in th is case

s

is s m a l l c o m p a r e d t o u n i t y .

I n d e e d , f r o m ( 9 ) w e c a n a p p r o x i m a t e t h a t

- * *

- * = < ' (13 .1 -10)

- {2-i>y - (2

f rom which i t fo l lows tha t the decrease in the load i s connec ted wi th the in

e q u a l i t y v ^ 0.

N ote th a t m or e s ign i f ican t d ecre as ing o f th e c r i t i ca l loa d ( f rom 4 3 % to 75%)

takes p lace fo r the p la t e , when the buck l ing mode i s loca l i zed a t the f r ee edge

y = 0 (see [43]).

I n t h e w o r k b e l o w , c o n s i d e r i n g s h e l l b u c k l i n g w e r e v e a l t h e s a m e l o c a l i z a t i o n

of th e def l ec t ion w i th a s im ul t a ne ou s dec rease o f loa d . Ho we ver , fo r she l l s , t h i s

e ffect i s m or e im p o r t a n t th a n i t i s fo r p la t es .

13 .2 C yl indr i ca l She lls an d P an e l s U n d e r A xia l

C o m p r e s s i o n

Let us con s ider the in f luence o f th e b o un da ry co nd i t io ns on th e buck l in g o f

a c i r cu la r cy l indr i ca l she l l o f r ad ius

R

u n d e r a h o m o g e n e o u s m e m b r a n e s t r e s s

s t a t e c a u s e d b y a x i a l p r e s s u r e . A s a n i n i t i a l s y s t e m o f e q u a t i o n s f or m o d e r a t e l y

long she l l s we t ake sys tem (3 .1 .1 )

w h e r e

2

. . „ . d

2

w

u.

2

AAw+2\^—-

ox-

1

d

2

d

2

-

dx

2

dy

2

'

1

-

dx 2

=0,

^ A 4

+ f e J

= 0 ,

h

2

- ° \ F / , / /

2

;;

4

-

. \ h h i i , p.

-

1 2 ( 1

_

J

,

2 ) i J 2

.

0 ^

x

^ / =

L/R.

(13.2 .1)

(13.2 .2)

H e r e

x

=

x'/R

is t h e d i m e n s i o n l e s s l e n g t h o f t h e g e n e r a t r i x a n d

ip

i s th e ang le

i n t h e c i r c u m f e r e n t i a l d i r e c t i o n .

I t was no ted ( s ee Sec t ion 3 .4 ) , t ha t the c l as s i ca l c r i t i ca l va lue o f the load

p a r a m e t e r A

c

i = 1, o b t a i n e d b y L o r e n z a n d T i m o s h e n k o , is v a l i d f or m a n y

b u c k l i n g m o d e s , i n c l u d i n g a x i s y m m e t r i c m o d e s . T h e s e m o d e s s a t i s f y t h e c o n

d i t i o n s of s i m p l e s u p p o r t a t x = 0 a n d x = I.

For more r ig id suppor t o f the she l l edges , the c r i t i ca l load inc reases , bu t

on ly s l igh t ly , due to the loca l charac te r o f the buck l ing p i t s . The c r i t i ca l load

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13.2. Cylindrical Shells and Panels Under Axial Com pression 271

increases only for ra th er sho rt shells (see [61, 66]) and we will no t consider th e

short shell case here.

Let us consider cases when weaker support leads to a decrease in load. We

will begin with the case of axisymmetric buckling modes of a shell with free

edge

x =

0. We seek the non-trivial solutions of the equation

/ ^ + 2 A / i

2

^ +

W

= 0, (13.2.3)

ox* ox

z

satisfying at

x —

0 the cond itions

Q i - 7 ? 7 i = 0, M i = 0,

that may be writ ten as

,

d

3

w

„ ,

dw d

2

w

/ , o o ^ \

" ft?

+

2 A

* r

=

0

' a ^

= 0

-

(13

-

2

-

4)

For that we assume that the external pressure at edge x = 0 does not

change direction under shell deformation, and that the term — T®^\ in the

first of equations (4) gives the projection of this loading on the (internal) shell

nor m al after deform ation.

We seek the solution satisfying the condition

w - > 0 for a : -> oo , (13 .2 .5 )

which replaces the boundary conditions at the other shell edge.

The direct verification shows that for A = 1/2 the function

w{

x) = e-f

sm

N - (13.2.6)

satisfies bo th (3) and also (4) and (5). Th is solution is co nstru cted in [74].

Due to the presence of // in the denominator, deflection (6) damps fast away

from edge

x =

0.

Under at least one of the supports

w

= 0 or for 71 = 0 there are no

axisymmetric buckling modes localized in the neighbourhood of this edge and

A = 1.

Let us try to find the non-axisymmetric buckling modes localized in the

neighbo urhoo d of edge

x

= 0. We will consider 16 bo un dar y con dition varia nts

in which generalized displacements (1.2.9) or corresponding to them general

ized forces (1.2.10) are equal to zero except that the deflection,

w,

corresponds

to the generalized shear stress resultant

1 a IT

272

Chapter 13, Buckling Mo des Localized at an Edge

We seek the so lu t ion

w (x,

<p) of sy s te m (1) in th e fo rm

4

'(

X

,P) = ^C

k

w

k

e x p j -

(p

k

x + q<p)j, (13.2.8)

[

w h e r e

C

k

a r e a r b i t r a r y c o n s t a n t s ,

w

k

a r e f ix e d c o n s t a n t s ( b e l o w w e a s s u m e

t h a t

w

k

=

1) and

p

k

a r e u n k n o w n c o n s t a n t s s a ti s fy i n g t h e r e l a t i o n s

(Pl + q

2

)

4

-2\pl(pl + q

2

)

2

+pi=0, 9 p

f c

> 0 . ( 1 3 .2 . 9)

T h e l a s t o f t h e a b o v e r e l a t i o n s p r o v i d e s t h e d a m p i n g of s o l u t i o n ( 8 ) a w a y f r o m

e d g e x = 0 . For A < 1 eq ua t io n (9 ) ha s four so lu t io ns , s a t i s fy ing ine qu a l i ty

(9) .

For a she l l which i s c losed in the c i r cumferen t i a l d i r ec t ion

q = /im.

For

a c y l i n d r i c a l p a n e l w i t h s i m p l y s u p p o r t e d r e c t i l i n e a r e d g e s <p = 0 , <p = <po,

q — ^^-

(m i s an in te ge r ) . T h e va lue o f

q

m a y b e f o u n d f ro m t h e m i n i m u m

condi t ions fo r A.

O t h e r u n k n o w n f u n c ti o n s ( $ ,

u, v,

7 1 ,

T\, S, Q\*

an d M i ) we seek in th e

s a m e f o r m a s ( 8 ) w i t h c o r r e s p o n d i n g s u b s t i t u t i o n s f o r w. W e w r i t e t h e v a l u e s

$fc, u

k

, . . . in (8 ) ( sub scr ip t k i s o m i t t e d )

p

2

w u-

p (u p

2

—

q

2

) w

$ = — — o , u =

(p

2

+ q

2

) i(p

2

+ q

2

)

fiq(q

2

+ {2 + v) p

2

) w

v = i j — — >

i (p

2

+ q

2

)

rr, Eh

0

^ ^ Eh ^ „ Eh

T

1

=

-— q

2

^> ,

T

2

= - — p

2

$ , S= — pq$,

P

w

9

W

», p . 2 / 2 , 2 \ ( 1 3. 2. 1 0)

7 l

= -f—-, 72 = ^ — , M i = -Ehp,

2

(p

2

+ vq

2

)w , < '

1

\ i K

1

fi R

M

2

= - Eh/i

2

(q

2

+ up

2

)w,

Q

U

= EJL?i(jP

+

(2-v)pq*-2\p),

Q

2

. = ^ti(

q

3 + (2-v)p

2

q-2\qt

2

),

w h i c h a r e i n c l u d e d i n t h e f o r m u l a t i o n o f t h e b o u n d a r y c o n d i t i o n s .

If w e s u b s t i t u t e s o l u t i o n s (8 ) i n t o t h e b o u n d a r y c o n d i t i o n s a t x = 0 then

we co m e to a sy s te m of four l inear eq ua t io ns in C

k

• If t h e d e t e r m i n a n t of t h e

13.2.

Cylindrical Shells and Panels Under Axial Co mpression 273

system is equal to zero then

A(A,g) = 0

(13.2.11)

we get the equation for the evaluation of A.

The case, when

q

2

<C 1 is the s imp lest. For the seven bou nd ary cond ition

variants

(0000),

(1000),

(0100),

(0010),

(0001),

(1100),

(1010),

A

= 1/2

A = 1/2

A

= 1/2

A = 1/2

A = 1/2

A = 1/2

A = 1/2

Ti = S = Qi , = Mi = 0

u =S = Q

u

=M

1

=0

Ti = u = Qi, = Mi = 0

Ti = S = w = M

1 =

0 (0010), A = 1/2; (13.2.12)

Tj = S = Q

u

= 7 i = 0

u — v

—

Q

it

= M i = 0

u = S = w = M i = 0

eq ua tion s (11) ha s a roo t A < 1, mo reover in all cases this root is equa l to 1/2

(the designations for the boundary conditions introduced in Section 8.4 and in

Table 1.1 are in the parentheses). For supports more rigid than (12), equation

(11) for

q

2

<C 1 ha s no roots (except for bo und ary cond itions (15), see be low).

It follows from the above that for boundary conditions (12), a decrease in

the critical load takes place. However, it remains unclear whether this decrease

would be larger than a factor of two at q ~ 1.

We turn to the case q ~ 1. Let A = c o s ^ and 0 < ip < ^ . Th en roo ts of

(9) in which we are interested are the following

P i , 3 = 2 {

rcoS

2

±COS

2j

+

2{

rSm

2

±Sm

2

i

2

\ 1/2

r

= ( (cos ip

- 4q

2

)

+ s i n

2

^ J , tan

0 =

0 <

9 <

7T.

sin V'

c o s % l >

— Aq

2

'

(13.2.13)

The roots p2,4 differs from P13 by the sign of the real part.

If q

2

<C 1, then roots pk are equal approximately to

pi,

2

= ± c o s

—

+ i s in—,

P3,4 9

2

P i , 2 -

(13.2.14)

T he use of expression (14) give us dou ble th e decrease in A for bo un da ry

conditions (12).



274

Chapter 13. Buckling Modes Localized at an Edge

A

1

0.5

O 0.5 1 q

F i g u r e 1 3 . 2 : T h e l o a d i n g p a r a m e t e r f o r a c y l i n d r i c a l s h e l l w i t h a w e a k l y s u p

po r t e d cur v i l ine ar edge un de r ax ia l com pres s ion (1 - 000 0 , 2 - 0100 , 3 - 1000 ,

4 - 1100, 5 - 0010, 1010, 0001, 6 - 0101) .

T h e r o o t s A = A (q) of equat ion (11) for q ^ 0 are g iven in Fig ur e 1 3.2 . In

a d d i t i o n w e a l s o h a v e f o u n d a n e w t y p e o f b o u n d a r y c o n d i t i o n

7 \ =v = Q

u

= 7 i = 0 (010 1) , (13 .2 .15)

fo r whic h a decrea se in the c r i t i ca l load is a l so poss ib le (b u t by no t mo re th an

3 % ) . I t s h o u l d b e n o t e d , t h a t A —> 1 a s q

—>

0 , ther efor e th i s case do es no t f it

(12 ) .

Cu r v e s 1 , 2 a n d 6 i n F i g u r e 1 3 .2 h a v e m i n i m u m s a t

q = 0 . 3 1 ,

0.20 and 0.36

r e s p e c t i v e l y . O t h e r c u r v e s h a v e m i n i m u m s a t

q

= 0 . Th e g rea tes t decrease in

th e cr i t ica l loa d tak es plac e in the f ree edge case (c urve 1) an d is eq ua l to 38 %

of the c l as s i ca l va lue . Th i s r esu l t was ob ta ined in [117]. W e n o t e t h a t c u r v e 5

c o r r e s p o n d s t o t h r e e d if fe re n t b o u n d a r y c o n d i t i o n v a r i a n t s .

R e m a r k 1 3 . 1 . The app l i cab i l i ty domain fo r the r esu l t s o f Sec t ion 13 .2 i s

r e s t r i c t e d t o t w o c i r c u m s t a n c e s .

F i r s t , t h e she l l i s a s s um ed to be s em i - in f in i t e . T h a t i s w hy i t i s neces sa ry

to de te r m in e wh e th er th e she l l l en g th is su ff ic ien t fo r th e nece s sa ry da m pi n g

of the def lect ion away f rom edge x = 0 . At the s ame t ime , the she l l canno t

be too long , s ince fo r ve ry long she l l s under p res sure , t he beam buck l ing mode

can occur .

S e c o n d , c u r v e s 3 , 4 a n d 5 i n c r e a s e m o n o t o n i c a l l y w i t h

q

an d for shel l s whic h

a r e c l o s e d i n t h e c i r c u m f e r e n t i a l d i r e c t i o n , t h e m i n i m u m i s a t t a i n e d a t

m —

2 , where m i s the number o f the ha l f -waves in the c i r cumferen t i a l d i r ec t ion .

However , fo r smal l va lues o f m, th e ac cu rac y of sy s t em (1) i s insuf f ic ient .

13.3. Cylindrical Panel with

a

Weakly Supported Edge 275

Let

us

es t imate

the

shell length

for

which

the

deflection decrea ses

by

N

t imes. We have

l l

0

= n\aN(Qp

3

)~

1

.

(13.2.16)

For that , due to (14) 9 p

3

= q

2

sin

f

~ 1/2 q

2

, which corresponds to A ~ 1/2.

For example, for

q

= 0.31, v = 0.3 and

TV

= 10 we have

. = £ = » (A )

1

". ,13,-17,

If condition (16) is not fulfilled then we cannot neglect the mutual influence

of shell edges x — 0 and x = I and we m ust satisfy the bo un da ry c onditions at

both edges. These cases will not be considered here.

Many works have been devoted to the problem of the influence of the bound

ary conditions and shell length on the value of the critical load for cylindrical

shells un de r a xial com pression . T he references m ay be found in [21, 22, 61 , 149].

Let us consider a cylindrical panel which is simply suppo rted at the recti

linear edges. For this panel

q =

fin/tpo. If angle <po is sufficiently sm all (for

example ,

<po

<; 7r/6), then we can use (after verifying con dit ion (16)) equa

tions

(1) and the

resul ts following from these eq ua tion s, w hich

are

shown

in

Figure 13.2.

The results for a shell which is closed in the circum ferential direction qual

itatively corresponding to curves 3, 4, 5 (the curve num ber depends on the

introduced boundary condition) with minimums at q = 0 are shown in Fig

ure 13.2. Quantitatively, the results in Figure 13.2 m ust be m ad e more precise.

However, the fact that

the

critical load decreases rem ains tru e.

13 .3 C yl indr i ca l P an e l w i th

a

W e a kly S u p p o r t e d

E d g e

Here the problem is solved under the same assumptions as in Section 13.2.

It is assumed tha t the curvilin ear edges

x

= 0 and

x = I

are simply supported

and various boundary conditions at <p = 0 are considered. A tten tion is con

centrated on the boundary conditions for which the buckling is localized in the

neighbourhood of thi s edge a nd also for which the critical load decrea ses.

We seek

the

solution

of

system (2.1)

in a

form similar

to

(2.8)

W

{

X

,<P) = ^CfcWfcexp J - (px

+ q

k

y) I

(13.3.1)

fc=i ^ '

and formulae (2.10) are still valid.

276


at an

Edge

Th e roles of par am eters p and q are different th a n th ey w ere in Section 13.2.

Here, p = ^j^, n = 1,2,..., and qk are the roots of equation (2.9), for which

9<7fc > 0. We then have

gi,2 = r i ( ± c o s Y + is in Y J ,

q

3> A

= r

3

( ± c o s

-^ +

i s in

~

j , (13.3.2 )

r i ,3 = ( p

2 =

F p

3

c o s - + p

4

j , tan ^ i

| 3

=

sin

± cos p

2

F

where the upper sign correspond s to the first subscrip t and A = cos^i.

For p <C 1 in formulae (2) we m ay assum e ap prox im ately th at

ri = r

3

=

y

/fi, 0

1

= - , 6»3 = 7 r - - .

(13.3.3)

As in Section 13.2 we search for the values of

A

by equating the fourth order

determinant to zero

A i( A ,p ) = 0 (13.3.4)

Let firs t p < l . We then take approx imate formulae (3) and find tha t the

decrease in the critical load takes place for six boundary condition variants at

<p = 0

T

2

= 5 = Q

2

* = M

2

= 0

T

2

= u = Q

2

* =

M

2

= 0

T

2

=

S = Q

2t

=

7 2

= 0

T

2

= u - Q

2

* = 72 = 0

T

2

= S = w = M

2

= 0

v = S = Q

2

* = M

2

= 0

(0000),

(0100),

(0001),

(0101),

(0010),

(1000),

A

0

= 0 . 1 1 3 ;

A

0

= 0 . 2 2 3 ;

A

0

= 0.223;

A

0

= 0 . 4 1 9 ;

A

0

= 0.809;

A

0

= 0 . 8 0 9 .

(13.3.5)

Let root A (p) of equation (4) converge to Ao as p —> 0. Then for the other

10 boundary condition variants, equation (4) has no roots, for which A

0

< 1.

It follows from (5), that the free edge tp — 0 gene rates a decrease in th e

critical load by a factor of nine. The least rigid are supports u = 0 and 72 = 0,

and the most r igid are supports

v =

0 and

w =

0. If we simu ltaneou sly

introduce two or more supports (except the case when

u = j

2

= 0 and one

more case, which is shown below) it prevents the appearance of the buckling

mode which is localized in the neighbourhood of edge <p

=

0.

13.3. Cylindrical Panel with a Weakly Supported Edge

111

Figure 13.3: The load parameter for a cylindrical panel with a weakly sup

ported rectilinear edge under axial compression (1 - 0000, 2 - 0001, 3 - 0100,

4 - 0101, 5 - 0010, 6 - 1000, 7 - 1100).

The numerical solution of equation (4) for

p

> 0 is shown in Figure 13.3.

As in Section 13.2, we find here one more variant of the boundary condition

(curve 7)

Q

2

, = M

2

= 0 (1100),

(13.3.6)

for which a buckling mode localized near

ip =

0 is possible and A(0) = 1. The

maximum decrease in the critical load is 3.7 % and is obtained for p ~ 0.5. For

the other six boundary condition variants which are listed in (5), the functions

X(p) (curves 1-6 in Figure 13.3) increase monotonically with p. Hence, the

buckling mode occurs for the least possible value of

p,

namely for

7T/Z

T'

(13.3.7)

For shells of moderate length, p -C 1, and we take from the left side of the

graph the value of A(p). The value p ~ 1 correspond s to a sho rt shell.

In the case p < 1 we represent the buckling mode in the form

TTX

w(x,y) =

y

(77)

s i n — ,

i) =

VVP-

For six variants of the weak support of the rectilinear edge y = 0 listed in (5)

the functions

Y(rj)

are shown in Figure 13.4. One can see that for the larger

values of

Ao

th e deflection decrease slower away from the shell edge.

Let us consider the problem of buckling mode localization for p <C 1 • By

278


a t an

Edge

0

Y

0

Y

0000 X

0

=0.113

l

_ 0001

0100

10

1

X

0

=0.223

10

*1

0010 X

0

=0.809 j I

0101

X

o

=0.223

10

n

X

0

=0.419

F i g u r e 1 3 . 4 : F u n c t i o n s

Y(rj).

vir tue of (2) and (3)

m i n { 9 ?

f c

}

fc

VP-

4>

and the def l ec t ion decreases by N t imes for

f

filnN

V ^ s i n -

o.4i

( i - ^ ) -

1 7 8

^

Vi

1 / 4

I n

TV

"7T'

sin —

4

(13.3 .8)

(13.3 .9)

w he re cos '0 = Ao an d th e valu e of Ao is def ined by (5) de pe nd in g on th e

b o u n d a r y c o n d i t i o n s a t <p = 0 . T h e b u c k l i n g m o d e s w h i c h d e c r e a s e m o r e

s l o w l y w i t h t h e g r o w t h o f (p cor resp on d to th e l a rg er v a lues o f Ao-

R e m a r k 1 3 . 2 . I t i s e a s y t o s a ti s fy t h e b o u n d a r y c o n d i t i o n s a t b o t h e d g e s

> = 0 a n d tp = tp

0

t ak ing in so lu t ion (1 ) a l l e igh t roo t s q^ o f equa t ion (2 .9 ) .

Ho we ver , th e res ul t s g iven here for a shel l , whic h i s in fac t , se m i- inf in i te in

t h e c i r c u m f e r e n t i a l d i r e c t i o n , a l l o w u s t o f i n d t h e m a i n p a r t o f t h e s u p p o r t

in f luence on the c r i t i ca l load .

13.4. Shallow Shell with

a WeaA-

Edge Support

279

In [35] the depe nden ce of the c ritical value A on the p ane l wid th in the

circumferential direction, <po, for various boundary conditions on the rectilinear

shell edges is studied. In this work the case of the clamped curvilinear edges is

also studied. It is interesting to note that in the last case the rectilinear edge

is considered to be weakly supported only for four first boundary conditions

(5) and the critical load is ~ 1.5 times larger than in (5).

13 .4 Sha llow She ll w i th a W eak Ed ge S u p p o r t

Let us consider the buckling of a convex shallow shell under a membrane

stress state using the same assumptions and designations as in Section 3.1. As

an initial system we will take (3.1.1)

p

2

AAw

+ \A

t

w- A

k

<f> = 0, p

2

AA$ + A

k

w = 0,

(13.4.1)

where

d

2

w d

2

w d

2

w d

2

w

d

2

w

„

d

2

w d

2

w

A

t

w = * - — + 2

1

3

-z-T- + h-—.

ox

1

ox oy oy

z

(13.4.2)

We will consider edge

x

= 0 and seek the boundary condition variants for

which buckling modes localized in the neighbourhood of this edge are possible.

As before, we seek the solution of system (1) in the form

w

l-(px + qy)l.

0

exp< -(px + qy) >.

(13.4.3)

In the same form we seek other unknown functions, which are included in

the boundary conditions. We have

(k

2P

2

+ k

iq

2

)w Eh

2

Eh ,

$ = i— , Ti = —q

2

$ , 5 = — - p q $ ,

{p

2

+ q

2

f R R

Qu = ~ ^ (p

3

+ {2-v)pq

2

-2\ (t

lP

+ t

3

q))w,

H

v

'

(13.4.4)

Ehp,

i

R

M i =

— Ehp?

(p

2

+

vq

2

) w, u= £-(kiw—

(g

2

-

up

2

)

$ J,

v ~ — (k

2

w- (p

2

-vq

2

)$), 7x = w,

iq \

) p,

280


a t an

Edge

w h e r e Q i * is t h e g e n e r a l i z e d s h e a r s t r e s s r e s u l t a n t

1

f)H

Q

lt

= Q

1

- -

T °

7 l

- < ? V ( 1 3. 4. 5)

C o n d i t i o n Qi* = 0 i s a n a t u ra l co nd i t io n for fun c t io na l (3 .6 .1 ) in th e case wh en

the def l ec t ion

w

i s not g iven.

P a r a m e t e r q > 0 in (3) i s fixed, an d for pa ra m e te r p we ge t the e igh th o rder

e q u a t i o n

( p

2

+

9

2

)

4

- A

(*

lP

2

+ 2t

3

pq +

t

2

q*) ( p

2

+ g

2

)

2

+ (fc

2

p

2

+

k

iq

>)

2

= 0.

(13.4 .6)

W e wi ll cons id er th e va lu es of A, fo r which e qu a t i on (6 ) ha s no r ea l roo t s .

We s e e k u n k n o w n f u n c t i o n s i n t h e f o r m o f ( 2 . 8 ) , w h e r e Qpk > 0 . A s in Sect ion

13 .2 we come to the equa t ion fo r A

A

0

(A ,g ) = 0 , (13.4 .7)

b y e q u a t i n g t h e f o u r t h o r d e r d e t e r m i n a n t t o z e r o .

T h e r o o t s of e q u a t i o n ( 7 ) , d e p e n d o n p a r a m e t e r q a n d a l so o n t h e p a r a m

e t e r s ki a n d ti w h i c h c h a r a c t e r i z e t h e s h e l l c u r v a t u r e a n d m e m b r a n e i n i t i a l

s t r e s s e s , a n d a l s o o n t h e b o u n d a r y c o n d i t i o n v a r i a n t i n t r o d u c e d a t e d g e

x

= 0.

A n u m e r i c a l s t u d y o f t h e r o o t s of e q u a t i o n ( 7) s h o w s t h a t f or e i g h t b o u n d a r y

c o n d i t i o n v a r i a n t s t h e r e a r e d o m a i n s o f p a r a m e t e r s

ki

a n d

ti

fo r which buck l ing

m o d e lo c a l i z a t i o n o c c u r s i n t h e n e i g h b o u r h o o d o f e d g e x = 0. F or t h a t A < Ao,

w here Ao i s th e va lu e o f the load pa ra m et e r fou nd in Sec t ion 3 .1 as su m in g th a t

p a n d q i n ( 3) a r e r e a l a n d t h e b o u n d a r y c o n d i t i o n s a r e n e g l e c t e d .

I t fo l lows f rom (2) th a t we can ch oos e A or one of th e va lue s of ti ( t e r m s

of the fo rm Xti a r e u n k n o w n ) . I f w e a s s u m e , i n p a r t i c u l a r , t h a t Ao = 1 t h e n

th e va lu e A < 1 i s a m ea su re o f th e decrease o f th e c r i t i ca l load ca used by th e

w e a k s u p p o r t o f e d g e

x

= 0.

T h e e i g h t b o u n d a r y c o n d i t i o n v a r i a n t s m e n t i o n e d a b o v e a r e t h e f o l l o w i n g :

f r e e e d g e ( 0 0 0 0 ) , c o n d i t i o n s w i t h o n e s u p p o r t ( 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 ) , a n d

t h r e e b o u n d a r y c o n d i t i o n v a r i a n t s e a c h of w h i c h c o n t a i n s t w o s u p p o r t s ( 1 1 0 0,

1010 , 0101) .

T h e p r o b l e m u n d e r c o n s i d e r a t i o n c o n t a i n s f i v e p a r a m e t e r s

t\, t

2

, t

3

,

&i and

k

2

. W e ca n f ix one of th e value s of ti by choos ing A, and we can f ix one of the

ki

b y c h o o s i n g a c h a r a c t e r i s t i c li n e a r s i ze . T h r e e e s s e n t i a l p a r a m e t e r s r e m a i n

a s a r e s u l t . A f ul l i t e m - b y - i t e m e x a m i n a t i o n o f t h e s e p a r a m e t e r s i s b e y o n d

t h e s c o p e o f t h i s b o o k , a l t h o u g h t h e n u m e r i c a l s o l u t i o n o f e q u a t i o n ( 7 ) i s n o t

d i ff i cu l t . Th ere fo re , we wi l l l im i t th e d i s cus s ion to the con s id era t ion o f th r ee

load cases .




a Weak

Edge Support

281

E x a m p l e 1 3 . 1 . Let us consider th e buck ling of a convex shell of revo lution

under a xial pressure. Takin g into account the expected local cha racter of

buckling, we assume that the initial stress-resultants

t\

, and

ti

and curvatures

k\

and

k^

are approximately constant. We take

t\ = fa

= 1,

t

3

= 0,

t

2

= -ki,

where k\ J> 0 is the only par am et er of the pro ble m .

(13.4.8)

Figure 13.5: The load parameter for a convex shell of revolution under axial

compression (the values oi k\ are given in the Figure).

We can see in Figure 13.5 a number of curves of A (q ) which depend on the

values of

hi

and the boundary conditions (shown in the Figure).

For the free edge (0000) and ki > 0, all of the curves have a minimum

A = 0 at

q —

0, which tells us abo ut th e decrease in th e critical load (see

Section 12.3, where we construc ted pseudo -bending for a convex shell. T he

existence of pseudo-bending leads to a decrease in the critical load).

282

Chapter 13. Buckling Modes L ocalized at an Edge

For boundary cond i t ions 1000 , 0100 , and 1100 ( as in Sec t ion 13 .2 ) the

c u r v e s X(q) h a v e a m i n i m u m A = 1 /2 a t q = 0.

In the case o f b ou nd ar y co nd i t ion s 0001 an d 1010 , the curve s of

\(q)

m a y

h a v e a m i n i m u m a t q — qo > 0 . In the se case s eq ua t io n (7) ha s a ro ot A < 1

only for k\ < k° ( see Figure 13.5) .

For condi t ion 0101 equat ion (7) has a root a t &i = 0 , which i s c lose to uni ty

(see curve 6 in Fig ur e 1 3.2) . For k\ J> 0 .1 eq ua t io n (7 ) ha s no roo t s such th a t

A ^ 1.

Le t us r e tu rn to the d i s cus s ion o f case 0001 . As was shown in Sec t ion 12 .3 ,

th e in t r od uc t io n o f su pp or t 71 = 0 does no t e ffect the inc re ase o f th e c r i t i ca l

load ( i . e . i t mus t be

A

(0) = 0 as in case 00 00 ).

Ho we ver , i t fo l lows f rom F ig ur e 13.5 th a t A ^ 1/2 . T h e di f f icul ty i s th a t

f or p a r a m e t e r s ( 8 ) , £0 = 0 i n f o r m u l a ( 1 2 . 3 . 2 1 ) a n d t h e e s t i m a t e s o b t a i n e d

in Se ct io n 12.3 are no t val id . In case 000 1 th e curve s A

(q )

for

t

2

= 0

a n d

t

2

= k\ a r e co ns t ru c te d . In b o th cases A (0) = 0 wh ich agre es w i th th e r esu l t s

of Sect ion 13.2 s ince to ^ 0.

E x a m p l e 1 3 . 2 . U s i n g th e s a m e a s s u m p t i o n s a s i n E x a m p l e 1 3 . 1 , c o n s id e r

th e buc k l ing o f a conv ex she l l o f r ev o lu t ion un de r t ens i l e loa d in g . Le t us t ake

t

2

= k

x

= l,

<

3

= 0 , h=-k

2

, (13.4 .9)

w h e r e

k

2

> 0 i s t h e p a r a m e t e r o f t h e p r o b l e m . I t s h o u l d b e n o t e d t h a t t h e

n o r m a l i z a t i o n o f t h e v a l u e s o f t, a n d &,- in (9) is n o t th e sa m e as in (8 ) , ho we ver

as ear l ier , A

0

= 1 (for q < 1).

T h e c u r v e s A

(q )

for the different values of

k

2

a n d d if fe re n t b o u n d a r y c o n

d i t ions a re shown in F igure 13 .6 .

F o r b o u n d a r y c o n d i t i o n s 0 0 0 0 a n d 0 0 0 1 , t h e c u r v e s of A (q ) i n c r e a s e m o n o -

t o n i c a l l y a n d h a v e a n o n - z e r o m i n i m u m a t

q =

0 . H ere we ag ain h av e <o = 0

( se e t h e e x p l a n a t i o n in E x a m p l e 1 3 .1 ) . F o r b o u n d a r y c o n d i t i o n s 0 1 0 0 a n d

0 1 0 1 , t h e m i n i m u m i s r e a c h e d a t q = qo > 0 . In th e o the r cases eq ua t io n (7)

ha s no roo t s A < 1 a t

k

2

^ 0 . 1 , a l t h o u g h i t d o e s h a v e su c h r o o t s a t

k

2

—

0 in

cases 1000, 00 10 , an d 1100.

E x a m p l e 1 3 . 3 .

Le t us cons ider the buck l ing o f a convex she l l o f r evo lu t ion

u n d e r t o r s i o n . We t a k e

t

1 =

t i = 0, t

3

= \ A i , *2 = l , ( 13 .4 .1 0)

w h e r e k\ > 0 i s t h e p a r a m e t e r of t h e p r o b l e m . A g a i n , A

0

= 1, i.e . A < 1 is a

m e a s u r e o f t h e d e c r e a s e i n t h e c r i t i c a l l o a d c o n n e c t e d w i t h t h e w e a k s u p p o r t

of the edge.

T h e d e c r e a s e o f t h e c r i t i c a l l o a d o c c u r s o n l y f o r t w o b o u n d a r y c o n d i t i o n

v a r i a n t s (0 0 0 0 a n d 0 0 0 1 ) ( se e F i g u r e 1 3 . 7 ) . T h e m i n i m u m is r e a c h e d a t q = 0

in bo th cases .




a

Weak Edge Support

283

Figure 13.6: The loading parameter for a convex shell of revolution under axial

tension (the values of

&2

are given in th e Fig ure).

0001

0.1

0.5

2.0

1

o

1

Figure 13.7: The load parameter for a convex shell of revolution under torsion

(the values of k\ are given in the Figure).

Let us mak e a rem ark s im ilar to R em ark 13.1 in connection w ith the results

of this Section.

R e m a r k 1 3 . 3 . The minimum value of A (q) is reached at q = 0 in most of

the cases considered. Th is mean s th at we should take the m inim um possible

value of

q

to evaluate the critical load. However, equ ation (6) has the ro ot

p =z iq^P^- for sm all values of q and this root determines the s low damping

of the buckling mode, which may lead to necessity to consider the boundary

conditions at the other edge

x = x^.

Therefore it is not a reliable practice to take the value

A

(0) as the critical

284


a t an

Edge

l oad . However , t he f ac t tha t the c r i t i ca l load decreases fo r th i s suppor t a t edge

x

= 0 r e m a i n s a n d i n d e e d , w e c a n i n c r e a s e t h e r a t e o f d a m p i n g o f t h e b u c k l i n g

m o d e b y t a k i n g l a r g e r v a l u e s o f q a n d a t t h e s a m e t i m e w e c a n g e t A (q ) < 1.

I f t h e m i n i m u m v a l u e o f A (q ) i s r eac hed a t q = go > 0 , th en we ca n tak e A (q

0

)

a s t h e a p p r o x i m a t e c r i t i c a l v a l u e .

13.5 M o d es of Shell s of R ev olu t io n Lo cal ized

nea r an E dge

L e t u s c o n s i d e r t h e b u c k l i n g of t h e m e m b r a n e a x i s y m m e t r i c st r e s s s t a t e o f

a convex she l l o f r evo lu t ion which i s l imi t ed by two para l l e l s s = s\ a n d s = S2-

W e w i l l u s e t h e e q u a t i o n s a n d d e s i g n a t i o n s f r o m S e c t i o n 4 . 3

w h e

H

2

A(dAw) + XA

t

w- A

k

$ = 0,

fi

2

A(g-

1

A$)+A

k

w = 0,

1 d / , dw\ 1 d

2

Aw = - —

[b — ) +

(13.5.1)

b ds \ ds ) b

2

d<p

2

'

1 d A dw\ hd

2

w

AtW

= bds{

bh

^)

+

¥W

+

1 d ( dw\ t

3

d

2

w

b ds \ dip J b dipds

1 5 / , , dw\ ki d

2

w

(13.5 .2)

b ds \ ds J b

2

dip

2

A s i n S e c t i o n 4 . 3 , w e a s s u m e t h a t t h e d e t e r m i n i n g f u n c t i o n s 6 , d, g, ki a n d ti

d o n o t d e p e n d o n <p, b u t m a y d e p e n d o n s .

I n C h a p t e r 4 w e d e v e l o p e d t h e b u c k l i n g m o d e s f or w h i c h t h e w e a k e s t p a r a l

l el i s s i t ua te d c om ple te ly ins id e th e she l l . Here w e wi l l cons ider cases in w hich

t h e b u c k l i n g m o d e is l o c a l iz e d i n t h e n e i g h b o u r h o o d o f e d g e s = s i . T h i s

loc a l i za t ion m ay occu r due to th e weak su pp or t o f an edge o r if t h e weak es t

para l l e l co inc ides wi th the edge .

Le t us show f i r s t t ha t the r esu l t s o f Sec t ion 13 .4 a re app l i cab le to a d i s cus

s ion o f the weak suppor t o f an edge . Indeed , the so lu t ion o f equa t ion (1 ) we

seek in the form of a formal ser ies in / /

w(s,ip) = Y^CfcWJ*exp I — / pkds + imip J , (13.5 .3)

13.5. Modes of Shells of Revolution L ocalized near an Edge

285

where

oo

^ = £ > " t 4

n )

(

s

) ,

Q

Pk

(

Sl

)>0,

(13.5.4)

n = 0

and m is the number of waves in the circumferential direction.

Other unknowns we seek in the form (3), and for which have the lowest

order in p te rm s in (4) we have the sam e relation ship (4.4). For th a t case

Pk

= Pk

(s)

are the roots of the equation

+ g(k

2P

2

+ k

1

^) = 0, p = fim. (13.5.5)

If we normalize the parameters

E

and

h

in such a way that

d = g

= 1

at s = si and assume that q = p (6 (s i) ) , then equ ation (5) coincides with

equ atio n (4.6). As before, if we su bs titu te solution (3) into the bo un da ry

condition at s = «i, then the fourth order determinant is equal to zero

A (A,

q, p) =

A

0

(A,

q) + O (p) =

0, (13.5.6)

The elements of this determinant differ by values order of p from those

of determinant (4.7). If we neglect this difference we get the same results as

in Section 13.4, namely that the same boundary conditions allow the desired

buckling modes. Concerning the critical value of A, we have

\ = \o + p\i(p),

(13.5.7)

where Ao is the roo t of equ ation (4.7), and the correction te rm p Ai d epe nds

on the variability of the determining functions and may have any sign.

Now let the edge support at s = Si not be weak, i.e. the boundary condi

tions at

s = si

are such that for 0 ^

q <

oo equ atio n (6) ha s no roots A for

A < Ao, where Ao = m i n / at s = s\ (see below). Let the weakest parallel be

when s •= s\ . T hi s will be discussed below (see also Section 4.3 ).

If we resolve equation (5) in A, we get

X =

f ( p , p , s ) .

(13.5.8)

We seek the s m allest value of this function Ao > 0 in the do m ain — oo <

p < oo, 0 sC p < oo, si ^ s ^ s

2

- Let the value of A = A

0

be reached for

real

p = p

0

, p

=

p

0

and

s — s

0

.

Th e weakest parallel is

si

if so = s i- W e

additionally assume that the parallel s

0

is unique and

f°- — >Q

for

p = p

0

, p = po, s =

s j . (13.5.9)

286 Chapter 13. Buckling M odes Localized at an Edge

For

X = X

0

, p

—

p

0

equation (5) has real root

p = po

a t

s

= si and has no

real roots for

s

^

s\ .

Th en we take

A

= A

0

+ A A , A A > 0 . (1 3.5 .1 0)

and equation (5) has real roots for

s\

<C

s

^ s* and has no real roots for

s

> s*. Indeed, if we fix

p

= po and expand (8) into a Taylor series we obtain

approximately

AX

=

f°(s-

Sl

)

+

lf

0

pp

(p-po)

2

, /

P

°

P

= 0 , (13-5.11)

where

f°

p

is calc ulate d for th e sam e variab les as in (9). Fro m (10) and (11)

we get

*. =*i + 4 £ . (13.5.12)

Point

s

= s , is called a tu rni ng po in t. For si <C

s

< s* by virtue of (3)

and (11), system (1) has oscillating integrals, and for s > s* all of its solutions

increase or decrease expon entially. Th e existence of the oscillations in direction

s means that buckling is possible.

Selecting AA in (10) one must select the phase for the oscillating solution

such that together with the other solutions of system (1) for which

Qpk >

0

satisfies the boundary conditions introduced at

s = s\.

For th at we take the

oscillating solution which transforms to a damped solution for

s

> s* and as a

result we get the buckling mode localized in the neighbourhood of edge

s

= s i .

The critical value of

A

does not depend on the boundary conditions at

s

=

S2

,

if this edge is not weakly supported.

The technical realization of the outlined plan proves to be rather complex

and is discussed in Section 13.6. Here we only note that the critical value of A

can be expanded in powers of

p

1

/

3

A = A

0

+ AA , AA = / j

2 / 3

A

2

+ / i A

3

+ --- , (13.5.13)

wh ere Ao is defined by form ulae (4.3.7) an d (4.3.9), and A2, A3 are o bta ine d in

Section 13.6.

In direction

s

the size of the buckling domain is of order

p?l

z

'.

The last cir

cum stan ce m ay be som e justification for not tak ing into accoun t pre-buckling

displacements and moment stresses, since the boundary effect solutions pene

trate only to a depth of order

p

(smaller than

p

2

^

3

)

• Their influence on the

critical load is discussed in Section 14.4.

13.6. Buckling Modes with Turning Points

287

13.6 Buck l ing M od es w i t h T ur n i n g Po i n t s

The difficulty of developing analytical solutions of system (1) with a turning

point, s», is caused by the fact that in the neighbourhood of

s =

s* exponential

expression (5.3) becomes invalid. This is related to the fact that at

s

= s»

equation (5.5) has multiple roots and the coefficients w£ in (5.4) ap pro ach

infinity at s = s ,.

We note th at using the orthogona l sweep me thod of num erical integration of

the system of equations obtained from (1) after separation of the angle coordi

na te

ip ,

the presence of the t urn ing p oint does not bring ad ditio nal difficulties.

Therefore, the aim of the derivations given below is to clarify qualitatively the

phenomenon and to obtain formula (5.13) .

Let us limit the discussion to two cases, when (3 ^ 0 at s = si if at least

one of the inequalities

t\

> 0,

ti

> 0,

t\

>

t\ti

(case 1) is valid or £3 = 0 an d

simultaneously t 0, £2^2 > *i&i (case 2).

In case 1 th e pits are inclined a t an a cu te ang le to the edge (see (3.1.12 )),

and in case 2 the pits are orthogonal to the edge. The case defined by formula

(4.3.10) (£3 = 0, t\ > 0, < & > £2^2), in which the pits are stretch ed along t he

edge is not considered here.

Under the above assumptions, equation (5.5) has two roots which coincide

at s = s„ : pi and p2- By virtue of (5.11) we have approximately

P i .

3

= P o ± (

2 y ? (

p " '

)

)

1 / a

. (13.6.1)

The solution of system (5.1) is damped for s > s* and has the form similar

to (11.7.2) (see also [157])

*

f* P1+P 2, . .— I

—

ds + imtp

w(s,ip,n)= \w^Ai{ri) + w^Ai'(Tj) I e'* •'*•

2

, (13.6.2)

where Ai(r;) is the Airy function, Ai (r)) = U {—rj), U (rj) is the solution of

equation (11.7.3)

(13.6.3)

u /

1

' , w(

2

> are th e form al po wer series in /<,

0 0

u

, (

f c

) =

A J

T " = ( « ' ) ^ ^

w

(

f c

. " ) (

s

) , Jf e= l,2 . (13.6.4)

n = 0

288

Chapter 13. Buckling Modes Localized at

an

Edge

The other variables included in the boundary condit ions have the same form

as (2).

The substitution of the linear combination of solution (2) and the other

three solutions of type (5.3) into the boundary conditions leads to the approx

imate equation for the determination of

A

Ai(

m

) = ^

3

F(X)Ai'(

m

),

i»i(A) = ,(* ) < 0 , (13.6.5)

where the rather awkward function

F

(A) depends on the boundary condit ions

a t s = s\. For a clam ped edge, the function F is given in [157].

The solution of equation (5) leads to formula (5.13) where

AA=

UHf,y

n,

V"

Co-n

1,3

F(\o)], Co = 2.33 8. (13.6.6)

Here Co is the first ro ot of th e equ atio n Ai (—Co) = 0- It s hou ld be n ote d th at

only the second term in the brackets of (6) depends on the specific form of the

boundary condit ions.

E x a m p l e 13 .4. Let us consider th e torsion of a convex shell of revolution

by moments

M

applied to shell edges

s = s\

and

s = S2-

The pa ramete rs

E,

v

and

h

are assumed to be constant.

Then, due to (4.3.3) and Section 4.5

T?=T§=0, S°

= — ^ — =

-\Tt

3

, T = Ehf,

2*R

2

b

2

(13.6.7)

*3 = - T j, A

0

= m i n

s

{ 7 ( s ) } , j (s) =2b

2

^k

1

k

2

.

Let us assume that the function

j(s)

at ta ins i ts m inim um value at

s = si.

Then, by virtue of (3.1.14) and (4.5.5)

(13.6.8)

f2 = 2L , ^ = 2 6 V

/

V

/ a

( 4 * ? + ( * i- t

a

)

2

)- (13.6.9)

<W2Ar

2

« i + k

2

The derivatives in (6) are equal to

6 i W 2 Ari

A*(* i + ^ 2 )

ds\

The critical value for the torsional moment



13.6. Buckling Modes with Turning Points

289

Let us consider, for example, the torsion of an oblate ellipsoid of revolution

with semi-axes a =

R

and 6 =

R/V2

(see Fig ure 4.4).

Let

R/h

= 500,

v =

0.3, 0i ^ 0 ^ 0

2

, #i = V

4

and 0

2

=

T / 2 ,

where 0 is an

angle between the shell normal and the axis of symmetry. Taking into account

formulae (4.4.8) 7 (s) = 2 \/ 2 si n

2

0, hence the weakest parallel is 0 =

0\

and

con ditio n (5.9) is fulfilled. T he critica l value for th e torsiona l mo m en t is equal

to

M

•

2nEh

2

R

v / 6 ( l - ^

2

)

l + 6 . 1 / i

2 / 3

- 1.2//

1.48

2irEh

2

R

^ 6 ( 1 - ^ ) '

(13.6.11)

Here the second term in the brackets does not depend on the boundary con

dit ions at

s = s\,

and the third term corresponds to the case of a clamped

edge.

Figure 13.8: An ellipsoid of revolution under torsion.

On th e shell surface 21 pit s (see Fig ure 13.8) are form ed. Th ese p its a re

inclined at an angle ipo to the generatrices such, that

M i

tan ^0 =

\ T~ —

1-2.

V

h

The shape of the pits is approximately defined by the expression

(13.6.12)

(s,v)=[Ai(T,) + 0(u.

1

/

3

)

cos mo

(13.6.13)

which is ob tai ne d after simplifying th e real pa rt of function (2). T he values

k\, &2 and b are ca lculated at 0 = 9\.

E x a m p l e 1 3 . 5 .

Let us consider the buckling of a convex shell of revolution

under tensile loading by an axial force

P.

The parameter s

E, v

and

h

are

290

Chap ter 13. Buckling M odes Localized at an Edge

assumed to be constant. As in 4.4 we have

p

Ac

Po

= 2TT

REh/j,

2

X,

= m i n , { Y ( s ) } ,

= 0,

q

0

=

-^/kl,

h--

j(s)-.

rn

0

=

1

k

2

b

2

= 2 b

2

k\ =

_ bqo_

t

2

2 sin

2

Ai

k

2

b

2

'

e,

(13.6.14)

The min imum 7

(s)

for a convex shell is reached at one of the edge parallels.

Let it be parallel

s — s\ .

We then obtain

f°=

1

'

s

=4k

1

smeco

S

9 = 4k

1

k

2

bb',

/

p

°

p

= ^ A . (13.6.15)

Now we obtain the critical value by formulae (5.13) and (6)

/

l

[ \ l /8

J

I

1 3

6

-

1 6

)

c= * W ) Co.

The values of 6, k\ and k

2

in formulae (15) and (16) are calculated at s = si.

We get the approximate expression for the buckling mode

w{s,<p)= [Ai(r/ ) + o ( / i

1 / 3

) ] c o s m

0

(vP-Vo) , (13 .6 .17)

and the pits are s tretched along the meridians.

The initial stress state in the considered example unlike Example 13.4, is

not membrane. The influence of the initial bending stresses and pre-buckling

deformations will be considered in Chapter 14.

Here, as in (13), the O -term includes th e sum of the edg e effect integ rals

damping away from edge

s = Si

faster than the first term in the brackets

of (17).

13 .7 M od es Loca li zed ne a r th e W eak es t Po in t

on an Edge

As in Section 6.1, we will consider the buckling of a convex shell under a

m em br an e stress sta te . Let us fix the coefficients of the syste m of eq uatio ns

(6.1.1) and seek a solution in the form of (6.1.4)

w

= w

0

exp[ifi

x

( p i o i +p

2

a

2

)] . (13.7.1)

13.7. Mod es L ocalized near the Weakest Point on an Edge

291

T h e n , t h e p a r a m e t e r s

pi, pi

an d A a re r e l a t e d by exp res s ion (6 .1 .5 )

, . , ,

d {q\

+

q

2

2

)

A

+

g {k

2

q\

+

fri<?

2

2

)

A = /

(p i

:

ai)

= 1

( « l + « l ) (tiq

2

i+2t

3q i

q2 + t

2

q%)

(13.7 .2)

where g,- =

Pi/Ai.

T h e a p p r o x i m a t e v a l u e of t h e l o a d p a r a m e t e r , A

0

, t he wave

n u m b e r s

p°, p\

a n d t h e w e a k e s t p o i n t

a°, u\

a r e e v a l u a t e d b y m i n i m i z i n g

(6.1 .6)

A

0

= m i n { / (p,-, a,-)} = / (p °, a ° )

(13.7 .3)

In Sec t ions 6 .1 -6 .3 i t was as sumed tha t the weakes t po in t i s p l aced su f f i

c i en t ly f a r f rom t h e she ll edge . H ere we con s ider th e case whe n th i s po in t l ie s

at one of the shel l edges .

Le t us first ma ke som e s impl i fy ing as su m pt io ns . Le t th e she l l edge in

w h i c h w e a r e i n t e r e s t e d c o i n c i d e w i t h t h e l i n e o f c u r v a t u r e . I n s t e a d o f a i , a.i

w e i n t r o d u c e a s y s t e m o f t h e c u r v i l i n e a r c o o r d i n a t e s

s,

<p c o i n c i d i n g w i t h t h e

l in e s o f c u r v a t u r e s u c h t h a t l in e s = 0 co inc ide s w i th th e she l l edge , an d th e

w e a k e s t p o i n t i s

s =

<p = 0 . W e d e n o t e

p\

=

p, p

2

=

q

a n d i n t r o d u c e t h e

m a t r i x

Jpp Jpq Jptp

A

= |

Jqp Jqq Jqip

Jtpp Jtpq Jifiip

(13.7 .4)

w h e r e t h e p a r t i a l d e r i v a t i v e s c a l c u l a t e d a t

p = p

,

q — q

 0 for the s a m e va lue s o f

t h e v a r i a b l e s , a n d t h a t m a t r i x A i s p o s i t i v e d e f i n i te . U n d e r t h e s e a s s u m p t i o n s

f u n c t i o n s / h a v e a s t r i c t m i n i m u m a t t h e c o n s i d e r e d p o i n t .

T h e n w e a s s u m e t h a t e d g e

s

= 0 , and a l so the o ther she l l edges a re no t

wea k ly su pp or te d ( see Sec t ion 13 .4 ) , i . e . t h a t the edge i t s e lf can no t c ause

b u c k l i n g .

Le t us cons ider the p rob lem wi th the in i t i a l s t r es ses o f one o f the types

wh ich were def ined a t the be g in n in g of Sec t ion 13 .6 .

U n d e r t h e s e a s s u m p t i o n s t h e b u c k l i n g m o d e a n d t h e l o a d p a r a m e t e r a r e

found in [96] and fo r the buck l ing mode the fo l lowing fo rmula i s ob ta ined

w(s, ip) = ex p

-(P°s + q\)

_

H 2/i

(ki{rj)w

a

+ / i

1 / 3

A i ' (77)u>i) +

1

r .

+ (i

1/3

^2

u>j

exp - (p^s + q°<pj

j = 2

(13.7 .5)

292

Chapter 13. Buckling Modes Localized at an Edge

where

oo

™j =

5> '

/ 6

w f ,

j = 0 , l , . . . , 4 ;

r,

= Co

+ ksfi-

2

'

3

.

(13.7.6)

/=o

Here WQ , u^ are poly nom ials in the powers of s/i

2

/

3

, ^>/z

1

/

2

, and w^ ,

w

(') ...(')

, W4 are po lyno m ials in powers

S/J,

1

, ipfi

1

'

2

; Ai (77) is the A iry function,

damping as

77

—>• 00 (see (6.2), (11.7.7)),

Co

= 2.338 is the roo t of the e qu ation

Ai(-Co) = 0 .

T h e second term in (5) is a sum of the edge effect solu tions ad justed with

the first term at

s =

0.

The approximate representation for the buckling mode we get by taking

1 (s, <p) =

exp

(A

+

, V ) - ^

Aifo) .

(13.7.7)

Function (7) describes the system of elongated pits with width of order /i,

inclined to the edge by angle

S

where (tan<J =

—q°/p°).

The domain occupied by the pits is localized in the neighbourhood of the

weakest point

s = <p =

0 and has s ize of order ^ ^ x ^

1

/

2

(see Figure 13.9 below,

where 8 = 7r/2). In direction s, function (7) is similar to functions (6.2) and

(6.13) describing the localization of the deflected shape in the neighbourhood

of the weakest line coinciding with the shell edge. In direction <p function (7)

coincides with function (4.2.6) describing the localization near the line lying

completely inside the shell.

We can give the expression for

k

and a in (6) and (7)

2 73 \ _

1 (fpqfptp — Jppjqip) —

y

Jpp

d e t

A

\T~ '

a=

f f - f*

' (13-7.8)

V Jpp / JppJqq Jpq

Th e para m eter A can be expanded in powers of fi

1

'

6

A = A

0

+ ^

2 / 3

A i + AI A

2

+ / /

/ 6

A

3

+ . . . , (13.7.9)

where AQ is defined in (3),

X

1=(:O

[IIAL)

,

(13.7.10)

and /x

2/,3

Ai coincides with the first term in AA (see (6.6)) and does not depend

on the boundary conditions at

s =

0. The next terms /iA2,

/J .

7

'

6

^3

and .. . in

(9) do however depend on the boundary conditions at s = 0.

13.7. Modes Localized near the Weakest Point on an Edge

293

_ 0.75

0.5

6

0.25

A

U

i

o

0.3

<P

Figure 13.9: The buckling m ode of a convex shell in the case w hen the weakest

point lies on the edge of the shell.

As earlier (see Section 4.2, the case B; Section 6.1), the eigenvalues are

asymptotically double.

E x a m p l e 1 3 . 6 . Let us consider the buckling of a convex shell of revolution

with constant parameters E, v

and

h being stretched

by an

axial force, P,

and

a bend ing moment ,

Mi

(see Figure 1.2).

For only one force P, one of the edge parallels is the weakest parallel (see

Example 13.5) . If we add the mo men t

M\

then we have the si tuation when

the points of this parallel are loaded non-homogeneously and buckling occurs

in

the

neighbourhood

of

the w eakest po int.

The

init ial s tress-resultants

Tf, 5°

are determined by formulae (1.4.6). We then assume

1

m cos

 ^3 = 0,

b

2

k

2

b

3

k

2

'

b

2

H

bH

2

2

Mi =

_PRm P = 27rREh

t

i

2

X.

(13.7.11)

If we calculate the m in imu m of (3) in

p

and

q,

we f ind that the weakest

poin t

is the

point where

the

function

(see

(4.4.2))

7 («, <p) = -j

1

= 2s in

2

9(1+

B

Y

)

a t ta ins a min im um va lue . For t h a t

p°

= 0 and

q° = b\fk\.

(13.7.12)

294

Chapter 13. Buckling M odes Localized at an Edge

Let m > 0, then ip — 0 at t he w eakest poin t, and this point is situ ate d at

an edge paralle l where 7 (s, 0) is a min im um by s.

Keeping in (9) the first two terms we get the critical value

A = 2 s i n

2

0 i ( l + y ) '

+ /i

2/3

Co

f f,

p

1/ 3

+ 0(fi),

(13.7.13)

where

fs

fpp

d_

ds _

8

k

2

sin

2

0

(>

s i n ^ ( l + y ) '

(13.7.14)

in 0i /„ m\-

1

For

m

= 0 formu la (13) transform s to formula (6 .16).

Parameter a determines the rate of decrease of the depth of the pits in the

circumferential direction and is equal to

fffi

fqq

1/ 2

mki

4 (m + 6)

1/ 2

(13.7.15)

In [96] for an ellipsoid of revolution the term ^1X2 in (9) taking into account

the edge support at s = 0 is obta ined . For a shell with para m ete rs 60/ao = Vo ,

7r/3 ^

0

<; 7r/2,

R/h —

500,

v

= 0.3 for different m, it occurs that this effect

does not exceed 1 %.

The contour lines for the buckling deflection, w = ±0.75 , ±0 .50 and ±0.25

for

m =

4 are shown in Figure 13.9. The point with the largest deflection,

w = 1, is m ark ed by a sm all cross. T he deflection da m ps very fast away

from generatr ix <p = 0 and therefore there is no contour line for the third pit

w =

0.75, and, likewise, there is no contour line for the fourth pit

w

= —0.75

and w = —0.50.


1 3 . 1 . Consider a long tube with square cross-sectional area, each four faces

of which are rectangular plates of width a and length 6 (a <C

b).

Both plate

edges

y

= 0 and

y = b

are subjected to the uniformly d istrib ute d load

her,

where

h

is the pla te thickness, and cr is a pressure. Using asy m pto tic m eth od

find critical axial pressure.

A n s w e r

""cr

TT

2

Eh

2

Z + v

12a

2

1 + j /

(13.8.1)




295

1 3 . 2 . C o n s i d e r b u c k l i n g o f c y l i n d r i c a l p a n e l w i t h we a k l y s u p p o r t e d r e c

t i l i n e a r e d g e y = 0 a n d c l a m p e d e d g e y = y

0

u n d e r a x ia l c o m p r e s s i o n . T h e

c u rv i l i n e a r e d g e s a r e s i m p l y s u p p o r t e d . F o r d if fe r en t b o u n d a r y c o n d i t i o n s o n

t h e e d g e

x

= 0 an a ly ze t he poss ib i l i t y of ex i s t ence o f t h e l oca l i zed b uck l ing

m o d e s n e a r t h e e d g e

x

= 0 a n d f in d t h e c r i ti c a l b u c k l i n g r e s u l t a n t s

T°.

C o m

pare re su l t s fo r

y

0

= R

w i th g iven in fo rm ula e (3 .5 ) fo r t h e sem i - in f in i t e i n

c i r c u l a r d i r e c t i o n p a n e l .

1 3 . 3 .

P l o t g r a p h s A(y o) f or 6 v a r i a n t s o f t h e we a k b o u n d a ry c o n d i t i o n s

(3 .5 ) f o u n d i n p ro b l e m 1 3 .2 .

1 3 . 4 .

F i n d t h e c r i t ic a l l o a d for t h e p a r a b o l o i d of r e v o l u t i o n u n d e r e x t e rn a l

n o r m a l p r e s s u r e . T h e i n i t ia l a x i s y m m e t r i c s tr e s s - s t r a i n s t a t e is d e t e r m i n e d b y

t h e n o n d i m e n t i o n a l s t r e s s e s

1

l - t i h

w h e r e k

2

= /•,\

h2

a n

d fci = fcf-

1 3 . 5 .

D e t e rm i n e t h e c a se s o f t h e we a k s u p p o r t of t h e e d g e x = 0 of th e

c o n v e x sh e l l o f r e v o l u t i o n u n d e r c o m b i n e d a x i a l c o m p re s s i o n a n d t o r s i o n for

t h e p a r t i c u l a r c a s e , w h e n T° = S°. T h e r e s u l t s r e p r e s e n t is t h e f o rm s i m i l a r

to F igu re 13 .4 .

H i n t .

U n d e r s o l u t i o n o f ( 4 .6 ) a n d (4.7 ) a s s u m e

t\ = t

3

= fc

2

= 1,

t

2

= -h.

C h a p t e r 1 4

She l l s o f Revo lu t ion under

G e n e ra l S t r e s s S t a t e

In contrast to the previous Chapters, here we will assume that the initial shell

stress -strain st at e is a sum of a m em bran e stat e an d an edge effect. We will

refer to this as a general stress-strain state.

We will also assu m e tha t th e shell is sufficiently long , i.e. th a t we ma y

neglect th e m u tu a l influence of th e edge effects. In cases when th e influence of

the initial edge effect and pre-buckling deformations is small, the order of this

influence on the critic al load will be found . If the influence of these factors

is significant then for the critical load parameter in the zeroth approximation

the standard boundary value problem which does not contain the relative shell

thickness is developed.

14 .1 T h e Ba s i c E q u a t io n s an d E dg e Effect So

lu t ions

Assume that the initial general shell stress-strain state is a sum of a mem

brane s ta te to

0

, 7}° and

S°

and an edge effect

w

e

, T?

and

S

e

, the n we can write

system (2.4.2) in the form

H

2

A{dAw) + X(A

t

w + SA

e

t

w)- A

k

<f> - \ 6 A

e

k

 = 0,

^Aig^A^ + AkW + XSAlw

= 0,

where the multiplier

S

= 1 is intro du ce d for the convenience of discussion.

Below, we will study only the case of an axisymmetrically loaded shell of

revolution. Th en , for the operato rs A, A

t

and A^ we may use expressions

297

298

Chapter 14. Shells of Revolution under General Stress State

(4.3.2), (4.3.3)

1 d/, dw\ 1 d

2

w

b ds \ds J b

2

dip

2

1 d / , dwki d

2

w

(14.1.2)

ds \ ds J b

2

dip

2

1

d

/ 0«A1

d ( dw

AtW

bd-

S

{

bt

^)

+

bd-s{

ta

^

] +

t

3

d

2

w t

2

d

2

w

b

ds

d<p

b

2

dtp

2

(T°,T°,S) = -X E

0

h

0t

i

2

(t

u

t

2

,t

3

), / =

n{ l

_

l/

2

)R

2- (

14

-

L3

)

Here the generatr ix length s and the deflection w are referred to R.

The terms with

the

m ultiplier S take into accou nt

the

influence

of

the initial

edge effect stress-resultants

and the

corresponding pre-buckling deform ations

A

e „ ,

_

d

(

hi

e

9W

\

,t

e

2

d

2

W

6 <9s V 5«7 &

2

d<p

2

(14.1.4)

1 5

/ ' j . / o , e\

dw

\ < xl + *l d

2

w

* - = ** ('« + »«£) +

6

2

<V

where

the

t\

are the

dimensionless initial edge effect stress-resultants.

As before,

the

loading

is

assumed

to

have only

one

param eter , A

> 0. We

represent

the

pre-buckling deflection W°

as a sum

W° — \w° + Xw

e

where

Xw° and

Xw

e

are the

membrane

and the

edge effect deflection co m po ne nts .

The

pre-buckling change

of

curvatures

are

also

in the

form

of

sums ,

A (x° +

xf),

i — 1,2. For

high intensity (large) loads

(A ~ 1), the

functions

u>

e

, xf, and tf

each depend non-linearly

on

A

(see

equation

(6)

below).

We find

the

membrane init ial s tress-resultants T°,

and 5° by

formulae

(4.3.8). Let

m a x { | t , | } ~

1, and the

intensity

of the

loading

be

defined

by

the parameter

A.

Taking into account that

for the

membrane s ta te , unknown

functions

do not

increase

in

differentiation

we

obtain estim ates

w°, x? ~

fi

2

.

(14.1.5)

For the edge effect solutions we use equation (1.5.8) which we write in the

form

d + Xh+gklwO (14.1.6)

14.1.

The Basic Equa tions and Edge Effect Solutions

299

w h e r e d, g, t\, a nd &2 are fun ct io ns in s. T h e a p p l i c a b i l i t y d o m a i n o f e q u a t i o n

(6) i s d iscussed below.

We s e e k s o l u t i o n s w

e

i n t h e n e i g h b o u r h o o d o f e d g e s =

SQ

(S \ ^ s ^ S2 =

so) in the form

w

e

(s,n) = J^Cj

f > "

W

j

n

( * ) e x p U

fqj (s) ds \ ,

(14.1 .7)

w h e r e C\ a n d C2 a r e a r b i t r a r y c o n s t a n t s w h i c h a r e d e f in e d b e l o w f r o m t h e

b o u n d a r y c o n d i t i o n s . T h e f u n c t i o n s

q\{s)

a n d <72(

s

) s a t is fy eq ua t io ns

dq

4

+ Xt

iq

2

+gkj = 0, (14 .1.8 )

a n d

sftqjX), j = 1,2. (14 .1.9 )

Co n d i t i o n ( 9 ) w h i c h g u a r a n t e e s t h e d a m p i n g of t h e s o l u t i o n s (6 ) a w a y f r o m

the edge is fulfilled if for t\ > 0 and

A < m i n {

7

( s ) } = A ° ,

7

( s ) = ^ M )

1 / 2

- (14.1 .10)

F or

ti

^ 0 co nd i t i on (9) i s a lw ay s fulf i lled .

Wi th an e r ro r o f o rder / i so lu t ions (7 ) may be r ep laced by

2

wC

{

s

^)

= ^ C

i

e x p { / i -

1

?

i ( s o ) ( s - s o ) } . (1 4 .1 .1 1 )

i = i

T h e t a n g e n t i a l b o u n d a r y c o n d i t i o n s a r e f ulf ill ed w h e n w e d e v e l o p t h e m e m

b r a n e i n i t i a l s t r e s s s t a t e . T o fin d t h e c o n s t a n t s

C\

and C2 in (11) we use

n o n - t a n g e n t i a l b o u n d a r y c o n d i t i o n s

«,• = «, , or g

=

g

1 0 > ( i 4 i i 2 )

H

= 71* o r

M{ =

M10 at

s = s

0

,

i n which AQ10 and AM10 a re the shear s t r es s r esu l t an t and the bend ing s t r es s -

c o u p l e i n t r o d u c e d a t e d g e s =

SQ

a n d Xw

t

an d A71,, are th e di f ferences be tw ee n

t h e g i v e n a n d t h e m e m b r a n e d e f l e c t i o n a n d t h e a n g l e o f r o t a t i o n .

We can g ive the exp l i c i t expres s ions fo r

w

e

fo r v a r i o u s b o u n d a r y c o n d i t i o n s

(12)

w

e

= aw,{C~S)-fS

(14 .1 .13)

300 Chapter

14. Shells of Revolution under General Stress State

for

w

e

w*, it = 7 1* =A«

1

T,

w

e

= w

t

[a C JT- -S +^-5

2a

I

2a

(14.1.14)

for

w

e

=

w

t

, M{

= Mio =

E

0

h

0

Rd^

2

m;

1/2

= i ( £ ) K ^

2

-

3 a 2

)

c

+ (

Q 2

-

3

^

2

)

s

) + i

c

- s ) }

(14.1.15)

for 7f = 7

l t

, Q | = Q

1 0

=

Eohondq;

gkl

> ( ( 3 a

3

- a/?

2

) C - ( a

3

- 3a/?

2

) s ) +

for Qf = Q

10

, M{ = M

1 0

;

w

e

=

d

d

x ti

y

1

0*2 V ^ V P /

+ g ( 2 a

2

C - ( a

2

- / ?

2

) 5 ) ] (14.1.16)

q(2a

2

C-(a

2

-p

2

)s) +

A t r

+ m ( ( 3 a

3

- a / ?

2

+

a

^ ) C - ( a

3

- 3 a / ?

2

+

a^-)s

f o r Q ; - 7 ? 7 f = g

1 0

, M

1

e

= M

1 0

,

wh ere g (so) = a + i p\ a > 0,

C = a -

1

e

a i

c o s / ?

1

,

a ( s -

s

0

)

" l = ,

5 = ^ -

1

e

a i

s i n / ?

1

,

Pi — •

(14.1.17)

(14.1.18)

For large tensile stress-resultants

(ti

< 0) the value /? is im agin ary bu t the

formulae given above are still valid in this case and give real values of

w

e

(s)

(see Example 14.5).

In formulae (13)-(18) the values

w„,

7 ,

q,

and

m

have the dimension of a

length while the rest of the quantities are dimensionless.

By choosing the values

R, E

0

, ho,

an d A it is possible (w itho ut loss of

generality at t\ > 0) to get

d = g = k2=l, ti=2

at

s

=

s

0

,

(14.1.19)

302


1/2) and it is not acceptable for the development of the membrane part of

the deflection. In particular, for b' ^ 0, form ula (22) gives th e w rong value of

deflection

w°.

From the system of equations

du

n

kiw = e\ = ut

2

-ti,

ds

(14.1.23)

6'

— U — k

2

W = £ o

=

^ 1 ~~ ^ 2

0

we find the expressions for u° and w°

u

0

= bk2

l l'

k

^\~^

£

2

ds + C

W° =

k

^-

k

^

ds +

c

(14.1.24)

bk\

where

C

is co nsta nt. For

b'

= 0, according to the third e quilibrium equa tion

Tffci +

T$k

2

+ Rq* =

0 expression (24) for

w°

coincides with (22) and for

b' ^ 0, we ought to use only formulae (24).

By excluding w° we find from (21) the equation for w

e

/x

4

d f + A^

2

(ij + t\) ^f+9 k\w

e

+ n

2

t\ (* + v k

2

) = 0,

(14.1.25)

in which the first approximation as /z —>• 0, all values except

w

e

and

t\

are

considered to be given and con stan t. W e seek the solution

w

e

that is decreasing

for

s

< so. The value

t\

is related to

vf

by equ ation (20).

We will now try to determine the conditions at which equation (25) trans

forms into linear equation (6). Let the values of

d, g, t\,

and

k

2

be of order 1

and \ki\ < 1 in (25) .

The last term in (25) may be neglected with an error of order

fi

for

b'

^ 0

and with an error of order /z

2

for 6' = 0.

The order of the non-linear term in (25)

z = \ft\ - (14.1.26)

coincides for b' ^ 0 with the order of the value of \b'fi~

1

(w

e

)

2

. T ha t is why

term (26) is of the same order as the rest of the terms in equation (25), if

w

e

~

n

for

b' ^

0 and if

w

e

~ 1 for 6' = 0. Ac cording to form ulae (13)-(17)

14.2.

Buckling with Pseudo-bending Mod es

303

the order of

w

e

depends on the boundary conditions and on the values of w*,

7, in , and q and it is equal to

W

e

~ m a x { K | , |

7

| , H > M } > (14.1.27)

where the maximum is calculated by two of the four values listed above, which

appear in boundary conditions at

s = so.

Using equation (6) and formulae (13)-(17) incurs an error of order fi if the

order of the value of

fiz

does not exceed the orders of the remaining terms in

equation (25) (namely if w

e

< fi

2

for 6' ^ 0 and if w

e

< n for b' = 0) . We will

assume later that these conditions are fulfilled.

But now let us consider an example in which these conditions are violated.

We will study the buckling of a truncated circular conical shell with angle 2a

at the vertex under axial compression in the case when its edge is free in the

radial direction. In this case

6' = s in a , Q io = jP t a n a ,

q

~ /i,

w

e

~ /* (14.1 .28)

and term (26) has the same order as the remaining terms in (25) and so it

may not be neglected. Here, we expect buckling of the limiting point type (see

Section 2.1).

At present we will not consider such cases since jf ~ 1 for w

e

~ \i and

equation (6) and equations (25) and (21) and some of the relations of Chapter

1 are no t app lica ble since in thei r deri vat ion we assu m ed t h a t 7? <C 1. T he

problems in which geometric non-linearity is modelled exactly are studied for

example in [95, 170].

1 4.2 B u c k l in g w i t h P s e u d o - b e n d i n g M o d e s

In this Section we will consider problems in which the influence of the edge

effect initial stress resultants and pre-buckling deformations on the critical

load is small and may be found (or estimated) by the perturbation method.

These are buckling problems of a shell of revolution with negative Gaussian

curvature (A ~

ft

2

/

3

)

and for shells with zero Ga ussian curv atu re under an

external normal pressure (A ~

fi)

and under torsion (A ~ (i

1

?

2

). For all of

these problem s a relatively low level of the mem bran e stress-resultants ( A < 1 )

is typical.

Let us temporar i ly assume that the parameter

S

is small and we consider

the self-adjoint boundary value problem L g consisting of two equations (1.1)

and of four boundary conditions at each of the shell edges s = si and s = «2 •

We will refer to the problem

L $

as the general problem.

304

Chap ter 14. Shells of Revolution under General Stress State

F o r

S

= 0 w e g e t t h e p r o b l e m

L Q

( d i s c u s se d a b o v e , s e e C h a p t e r s 7 - 1 1 )

o n t h e m e m b r a n e s t r e s s s t a t e b u c k l i n g w i t h o u t t a k i n g i n t o a c c o u n t a n y p r e -

b u c k l i n g d e f o r m a t i o n s o r e d g e e f f e c t s t r e s s - r e s u l t a n t s ( s h o r t l y t o b e a m e m

b r a n e p r o b l e m ) . We s e e k a s o l u t i o n o f t h e p r o b l e m

L $

in the fo rm of a fo rmal

ser ies

ws = w

0

+ S wi + ...

A«5 = A

0

+ S X i + . . .

$6 - $0 + S $ + .

(14.2.1)

where Ao , wo, a nd 4>o a re r espec t iv e ly th e e igenva lue an d th e e igenfun c t ion o f

p r o b l e m LQ.

To f ind the f i r s t approx imat ion fo r

w\

a n d $ i w e g e t t h e n o n - h o m o g e n e o u s

p r o b l e m " o n s p e c t r u m " . F r o m i t s c o m p a t i b i l i t y c o n d i t i o n w e find t h e v a lu e

of Ai which is th e m ai n pa r t o f th e cor rec t ion to A an d we r ep res en t i t i n th e

f o r m

- l

where

h =

h =

I I

I I

Ai = J/A

0

, T)

=

(I

2

-h)I

0

fdwo\

,yh_ ®

w

°

d ^ o ,

h_

d

2

w p

\ ds J b ds dip b

2

dip

2

(14.2 .2)

b ds dip,

dw

0

~ds~

+

dw

0

b

2

\~~frp~

bdsdip,

(14.2 .3)

2

/ / [ (xS

+

x 3 ) ^ ^

+

^

+ X f a W 0 a

*

0

'

b

2

dip dip

b

ds dip.

H e r e t h e d o m a i n o f i n t e g r a t i o n i s t h e e n t i r e s h e ll n e u t r a l s u r fa c e . I n e v al

u a t i n g 7 i a n d

I?

w e t a k e i n t o a c c o u n t t h a t t h e p r e - b u c k l i n g d e f o r m a t i o n i s

a x i s y m m e t r i c .

T h e e i g e n f u n c t i o n f o r p r o b l e m

L Q

we r ep rese n t in th e fo rm of a s um

wo

w°

0

+w

e

0

, * o = * g +

*S,

(14.2.4)

w h e r e WQ a n d $ Q d e s c r i b e t h e m a i n s t r e s s s t a t e a n d

WQ

a n d

<£Q

c o r r e s p o n d t o

th e edge ef fec t. Le t us ha ve the fo l lowing es t im at es o f th e o rde r s of th e t e rm s

in (4) and of thei r indices of var ia t ion

7/1° * °

U

0 I

^ 0

ds

*

w

dip

(14.2 .5)

14.2. Buckling with Pseudo-bending Mod es

305

F o r s h e l l s w i t h z e r o G a u s s i a n c u r v a t u r e u n d e r e x t e r n a l p r e s s u r e o r t o r s i o n

w e o b t a i n a

s

= 0 a n d a

v

= 1 / 2 a n d f o r s h e l l s w i t h n e g a t i v e G a u s s i a n c u r v a t u r e

w e h a v e a

s

= a

v

= 2 / 3 . For the edge e ffect so lu t io ns , w^ a n d w

e

, we ge t

8WQ/8S ~ / /

_ 1

u > o , dw

e

/d s ~ fi~

1

w

e

.

The var i ab les in (3 ) due to (1 .20) a r e o f o rder

2 P i P 2 v r /» (14 .2.6 )

x ? , x

2

° ~ ^ » , x f ~ / i

a

° « -

2

, x ^ / i "

0

^

1

( & ' ^ 0 ) .

If 6'(

So

) = 0 then ff ~ ^

2

< | a n d

x%

~ p

a

° « .

W e rep rese n t t he in t eg ra l 7j in th e fo rm

/ , • = / ? + / ? ,

j

= 0 ,1 ,2 , ( 1 4 . 2 . 7 )

where J ? does no t con ta in the edge e f f ec t so lu t ions and H c o n t a i n s t h e e d g e

effect so lu t ion s de sc r ib ing the in i t i a l s t r es s s t a t e an d /o r e igen func t ion (4 ) o f

p r o b l e m L Q. I n p a r t i c u l a r , IQ i s ca lcu la t ed by fo rm ula (3 ) fo r wo —

WQ

a n d 1°

is c al cu la te d for ioo = i t)§, $ o = $o>

a n

d ^f

=

0 , an d 1° = 0 .

No w we wi l l con s ider bu ck l in g spec i fi ca lly . F i r s t , we wi l l s tu dy t he bu ck l ing

o f a c y l i n d r i c a l o r a c o n i c a l s h e ll of r e v o l u t i o n o f m o d e r a t e l e n g t h u n d e r e x t e r n a l

n o r m a l p r e s s u r e . W e a s s u m e t h a t t h e s h el l i s " w e l l " s u p p o r t e d a n d t h a t t h e

l o a d p a r a m e t e r Ao s a ti sf ie s t h e e s t i m a t e

A

0

~A*, (14 .2 .8 )

an d t he po ss ib le va r i a n t s o f " we l l " sup po r te d edges a re g iven in the first s ix

l ines of Table 8 .4 .

U n d e r t h e s e a s s u m p t i o n s t h e f o ll ow i n g e s t i m a t e s

aoo = 2, / ^ - / i "

1

,

I°

2

=0{»), a

e

>\.

(14 .2 .9 )

are val id .

T h e i n t e n s i t y , a

e

, of th e edge effect so lu t io ns , n ig , coin cide s w i th th e er ro r

o r d e r t h a t o c c u r s i n t h e t r a n s i t i o n f r o m t h e f u l l b o u n d a r y v a l u e p r o b l e m , La,

t o t h e s i m p l i f i e d p r o b l e m ( s e e Ch a p t e r 8 ) . I n p a r t i c u l a r , a

e

= 1 if w — 0 is

o n e o f t h e b o u n d a r y c o n d i t i o n s .

T h e o r d e r o f

1%

d e p e n d s o n w h e t h e r o r n o t t h e e q u a l i t y x° = 0 is va l id . I f

x° = 0 t h e n

1%

~ /J .

2

, o t h e r w i s e 1$ ~ ft-

T h e in i t i a l edg e ef fect in te ns i ty , c*oe, d ep en ds on th e ty p e of edg e s — so

s u p p o r t . F o r e x a m p l e , b y i n t r o d u c i n g a p r e - s tr e s s e d s u p p o r t o r w i t h e d g e

s t r e s s - r e s u l t a n t s a n d / o r s tr e s s -c o u p l e s w e m a y o b t a i n a r b i t r a r y v a l u e s o f t h e

r i g h t s i d e s o f b o u n d a r y c o n d i t i o n s i n ( 1 . 1 3 ) - ( 1 . 1 7 ) . Be l o w w e a s s u m e t h a t

«0e = 2,

(14 .2 .10)

306

Cha pter 14. Shells of Revolution under Genera l Stress State

i.e. the deflection corresponding to the edge effect is of the same order as the

m em bra ne initial deflection. In partic ular , estim ate (10) is valid if the initial

edge effect is only caused by the fact that the initial membrane deflection did

not satisfy the condition

w

= 0 (and m ay be the condition 71 = 0 ) .

We will neglect the mutual influence of the edge effects, that is why the

integrals, 7J, are the sums of terms corresponding to shell edges s = si and

s = s

2

• We will study one of these edges s = s

2

= «o •

The integrals IJ are sums of terms of the form

«o

ds,

« 2

F{n) = Jf{s,n)g

f = H<*(f

0

(s)+nf

1

(*) + • • • ) ,

(14.2.11)

where the function

f(s,fi)

m ay be represented as the as ym pto tic series (11)

and g (£) —> 0 as ( —»• —

00.

By expanding functions /,• in a Taylor series at

s = so we get

F (ft)

= / /*

+ 1

[/„ (s

0

) co + H (/1 (so) co + /o (so) ci) +

+ V

2

(/2 (so) co + /{ (so) ci + - f'

0

' (so) c

2

) -

where

u

c

k

= J

C

k

9

(0

dC-

(14.2.12)

(14.2.13)

Studying the boundary condition variants from the clamped and simple

support groups (see Section 8.4), we find that at s = so the following estimates

are valid

w°

(so)

Wo = 0(fi),

——9- ~

/J,

(clamped), ——2- ~ 1 (simple sup po rt) .

<9s

0

o s

0

(14 .2 .14 )

It is important to recall that the ~ symbol indicates the exact order of the

variable and the symbol O gives the upper order estimate. In particular, for

some boundary condition variants we get

WQ

~ u? or

w%

~ p

5

'

2

(see Section

8.4).

In estimating the integrals 7 | we assume that £

2

~ 1; ^i> ^3, an d b' — 0 ( 1 ) .

Then according to (12), (6), (10) and (14) we find

7

0

e

= O ( / i ) ,

/ f ~ / i

2

,

I

e

2

=0 {u),

(14.2.15)

14.2.

Buckling with Pseudo-bending Modes

307

and then due to (2 ) we ge t

V =

0{n

2

),

(14.2 .16)

i .e . t h e r e l a t i v e o r d e r of t h e i n i t i a l m o m e n t s t r e s s - r e s u l t a n t s a n d t h e i n fl u e n c e

o f p r e - b u c k l i n g d e f o r m a t i o n s o n t h e c r i ti c a l l o a d u n d e r t h e a b o v e a s s u m p t i o n s

does no t exceed / z

2

. I n a d d i t i o n , t h e t e r m

z —

z

^ MM

( 1 4 . J . 1 7 )

in l\ is t h e m o s t s ig n i f ic a n t . F o r b o u n d a r y c o n d i t i o n s o f t h e c l a m p e d s u p p o r t

g r o u p w e h a v e ufy (so)

~

A*, $ 0 (

s

o )

~

1 and

z

e

~ / i ,

? ?

= ( z +O ( ^ ) ) ( /

0

0

)

_ 1

~

/ i

2

. (14.2 .18)

F o r s i m p l e s u p p o r t g r o u p b o u n d a r y c o n d i t i o n s we o b t a i n

w%

(SQ), $ 0 (

s

o)

~

A*

and fo r

x\ = 0

we have

z~ n

2

, V~V

3

- (14.2 .19)

F o r m o d e r a t e l y l o n g s h e l l s t h e c a l c u l a t i o n o f rj by form ula e (2) i s scarce ly

prac t i ca l s ince the cor rec t ion i s ve ry smal l . However , fo r shor t she l l s the cor

r e c t i o n , t], becomes su f f i c i en t .

In

[22, 61, 149] one may f ind

a

su rve y o f t he

n u m e r i c a l r e s u l t s f o r a c y l i n d r i c a l s h e l l w i t h v a r i o u s b o u n d a r y c o n d i t i o n s w h i c h

a g r e e w i t h t h e e s t i m a t e s o b t a i n e d h e r e .

L e t u s n o w s t u d y t h e b u c k l i n g o f a n a x i s y m m e t r i c a l l y l o a d e d s h e l l o f r e v

o l u t i o n w i t h n e g a t i v e G a u s s i a n c u r v a t u r e . L e t t h e r e b e n o t o r s i o n (<3 = 0)

a n d l e t t h e b o u n d a r y c o n d i t i o n s b e o f t h e c l a m p e d o r s i m p l e s u p p o r t g r o u p .

T h e c a s e w h e n b o t h e d g e s a r e s i m p l y s u p p o r t e d i s n o t c o n s i d e r e d . T h e n t h e

e s t i m a t e s

2 2

A

0

~^

2 / 3

,

a

s

=a

tp

= - , a

00

= a

0e

= 2, a

e

= - a, (14.2 .20)

o

o

are va l id ( s ee (5 ) ) . Here the va lue ad e p e n d s o n t h e b o u n d a r y c o n d i t i o n s

at

s

= s

0

a n d it i s g ive n in T ab le 11.2 .

We can f ind the value of rj b y f o r m u l a ( 2 ) . K e e p i n g t h e m a i n t e r m s ( w i t h

respec t to fi) we get

n = I>

2

(l°

0

y

1

, I>

2

= 2Jjx{

d

-

d

- \dsdv. (14 .2 .21)

F o r a l l b o u n d a r y c o n d i t i o n v a r i a n t s c o n s i d e r e d w e h a v e I®

~

/ i

_ 4 /

'

3

.

As

before , we ca lcu la t e the va lue o f I'

2

s e p a r a t e l y f o r e a c h e d g e , s = s\ a n d s

=

s ,

308 Chapter 14. Shells of Revolution under General Stress State

and denote the edge under consideration by s = so. By taking into account

buckling modes (11.4.2) and (11.4.6) and by using an expansion of type (12)

we obtain the estimates

j]

= O

( V '

3

+

ti

1+ 2

-r)

(clamped),

) «±2a\ (14.2.22 )

T] = O l/i

2

+ H

3 J (simple su pp ort ) .

where the values of a are given in Table 11.2.

A com parison w ith estim ate s (18) and (19) reveals tha t for shells of negative

Gaussian curvature, the use of a membrane formulation is accompanied by a

greater error than for cylindrical and conical shells under external pressure.

14.3 T h e C as es of Signif icant Effect of P re -b u c k -

l ing s t ra ins

The buckling of convex shells of revolution under arbitrary axisymmetric

loading and the buckling of shells of revolution of zero Gaussian curvature

und er axial compression is studied in this Section for the case where

A

~ 1. The

corresponding membrane problems have already been discussed in Chapters 3

and 4 and it was shown that in the zeroth approximation, the critical value

of A is a function in 7 (s) (see formulae (4.3.7), (4.3.9), and (4.3.10)). We will

discuss several related cases here.

C a s e 1. Let 7 (s) = A

0

= const.

Then, in the membrane statement, al l points on the neutral surface are

equally predisposed to buckling and buckling pits cover the entire surface.

This case is important particularly for the buckling of cylindrical and conical

shells under axial pressure and for the buckling of a spherical shell subject to

a homogeneous external pressure. Th e init ial m om ent stress-resultants and

the pre-buckling deformations infringe on this homogeneity near the edges and

buckling may occur for A < A

0

and the buckling mode is localized near one of

the shell edges.

C a s e 2. Let there be no m em bra ne stress (7 = 0) or let it not co ntain

compressive membrane stress-resultants in any direction (see (3.1.11)).

Th en , only the general sta tem en t of the buckling proble m is valid. Th e

initial edge effect stress-resultants and the pre-buckling deformations which

are generated by the local loads at s = s

0

, are the only cause for buckling and

the buckling mode is localized near the line

s

= so-

C a s e 3. Let the function 7 (s) be variable and a tta in i ts m inim um value

a t s = so — «2-

14.3. The C ases of Significant Effect of P're-buck ling strains

309

The edge parallel s — s

0

is the weakest line and the buckling mode is again

localized in i ts neighbo urhood. In the mem brane state m en t the solution of this

problem for 7'(«o) ^ 0 is given in Section 13.6 and in the general statement it

is given in Section 14.4.

We assume that the characteristic values of the functions describing the

initial state satisfy the following estimates (see (1.20))

w°, w

e

~ fi

2

; x{,tl~l; x

e

2

,tl =

0(fi). (14.3.1)

Now we will give the solution algorithm for this problem in Cases 1 and 2.

The critical value

A

we seek from the bifurcation cond ition into the adjacent

non-symmetric mode with m waves in a circumferential direction in the form

O O

$

(s,

<p)

= Y^^^J

(0

el

'

m v

\ (14.3.2)

j=0

C = / i "

1

( s - « o ) , A = A

0

+ fi\i + •••

,

where in the neighbourhood of the parallel s = s

0

the scaled variable £ is

introduced.

To find Ao we come to the system

d

° \dO~

p2 l

J

Wo+X

°

(

 i

<

3 0 - ^ - / > ? ( i 2 0 + ^ ( C ) ) ^ o j -

- f e

2

o ^ + p ? ( f c i o + Aox

1

e

(C))$o = 0, />i = ^ >

so

-1

f ^ -

PIJ

*O + * 2 o ^ r - (*m + Ao*f(C))p?™

0

= o,

(14.3.3)

where by do, <7o, &o,

ho,

and <

l0

we denote the values of the corresponding

functions at s = SQ.

Functions wo and <J>o satisfy the four homogeneous boundary conditions at

s — SQ

(see below) and also the damping conditions

w

0

—

>

0, $

0

— > 0 at C —

>

- c o - (14.3.4)

310

Chap ter 14. Shells of Revolution under General Stress State

We can express by

w

a n d $ t h e v a r i a b l e s w h ic h a r e c o n t a in e d i n t h e b o u n d

a r y c o n d i t i o n s

6w b

2

( , b' , , /w\'\

u = — + —2 \e

2

-

T

£ l + b k

i

T J >

sm m

z

\ b \ b I J

v = -r— I £2 + ^ w - T

u

) > M i = / i

4

d (x i + J / X

2

) ,

i r a \ 6 /

1

62

+

6 '

.*HL11L>-*-*

QI*- - -T- (<#> (* i - ^ 2 ) ) -

' 2 ( 1 - v)pMmd

(14 .3 .5)

(x r

2

+ w x i ) - A<

2

(*i + <i) Aiu' + A {w

e

y Ti - n

2

\t

3

i

w h e r e

e i = - ( 7 i - i / T

2

) - A ( u ;

e

) V ,

a

2

(

1

+

J /

) C W e x '

i m

u = — -b— $w

e

) —rw,

9 b

m

2

b' ,

xr = w", x

2

= - - p - w+ —w ,

£2 = - {T

2

- i / T i ) ,

9

T

2

= fiH",

(14 .3 .6)

r = —— w —w

b \ b

Here wi t h the (/) sym bo l we de no te the de r iva t iv e w i th re sp ec t to s. T h e

error in formulae (5) is of order / i

2

. In the r ight s ides of (5) in the formula t ion

of the boundary cond i t ions fo r the sys tem we pass to the va r iab le £ and keep

th e m a in t e r m s w i th r e s p e c t t o fi.

Then we ge t

u = fi

v = fi

1 d

3

<&

0

2 + 1 / r f$o &20 dw

0

f dw

Pi9o dC

3

9o dC p\ dC, ° \ d( J

Q

W

°

1

f

d

^ • 2 * A , *20

" -77 2- +

VPl®0 )+—W

0

tPi 9o V

d

C J »Pi

d $ o

T

1

=-p{$o,

S=

~

l

P

l

~j7'' M

x

= fi

2

d

0

l - ^ - - vp{w

0

d

2

i

Qu= - p.

d

0

d

3w

° <* \ 2

dw

o\ ,

-de-

{2

-

u)p

-dZ^

+

dwo

~dZ

A

0

( (<io +

t$

^ +

p\

( ^ ) $o +

ipihowo

dw

e

(14 .3 .7)



14.3. The Cases of Significant Effect of F're-buck ling strains

311

where we take i^f) for C = 0.

The value of A

0

may be found after minimization by the number of waves,

m, in the circumferential direction

(p\ = umb^

) .

The next approximat ion

w\,

$i satisfies the non-homogeneous system of

equations with the same left sides as in system (3) and the non-homogeneous

boundary conditions with the left sides of form (7). The decreasing conditions

(4) are still valid. From the existence condition for the solution

w\,

$ i we can

find Ai.

The construction method for Ai proposed here is interesting only to vali

da te the expan sion (2) for the load pa ram ete r A = A (/i) . If we want to find

the numerical value of

A

(/j.) then it is more convenient to solve system (1.1)

numerically with boundary conditions of type (5).

From system (3) it follows that the initial stress-resultant, t%, and the pre-

buckling change of curvature,

xf,

affect equally Ao (at least from th e as ym p

totic point of view) and the functions,

t\

and

x\,

ought to be found from the

equ atio n of th e non -linea r edge effect (1.6).

If we neglect one of the values, t\ or xf, or evaluate the m from the sim ple

edge effect eq ua tion (equ ation (1.6) for A = 0) we incur an error which do es

not go to zero with the shell thickness. In [114] for the buckling problem of a

hemisphere under external pressure (see Example 14.3) with various boundary

conditions, the numerical results for three statements of the problem are given:

(i) for

x\

and

t\

which are found from the non-linear edge effect equation,

(ii) for x\ = 0,

(iii) for

x\

and

t\

which are found from the sim ple edge effect eq ua tio n.

For all boundary conditions considered, the minimum value of

A

is obtained

under the exact formulation of problem (i) . In the approximate formulations,

(ii) and (iii), the values of

A

are greater by 10 % and 2 0% respectively.

R e m a r k 1 4 .1 . Th e term /iAi in formula (2) takes into account the effect

of some factors which are neglected in the zeroth approximation (3) (namely

the variability of coefficients

b, d, g, ki,

and

tj

in system (1.1) and the effect of

relatively small values of

t\

and

x^).

If, at s = so the derivatives of the listed coefficients are equal to zero

then Ai = 0 in formula (2) and the error in system (3) and of the boundary

conditions which are introduced by formulae (7) is of order fi

2

.

In order to solve system (1.1) and approximate system (3), which have

variable coefficients, various numerical methods (such as numerical integra

tion, finite element m eth od s or varia tional m etho ds) m ay be used. We note

also the method of invariant immersion [179] which is a type of numerical inte

gration technique in which the decreasing conditions (4) may be satisfied very

conveniently.

Exa m ples 1 4.1-14.3 considered below illustr ate C ase 1 and E xam ple 14.4

312


illustrates Case 2. In these Examples <3 = 0 and E, v, and h are constant and

therefore

d

=

g

= 1.

E x a m p l e 1 4 .1 . C onsider the buckling of a cylindrical shell under axial

com pression. In this case ki = ti — 0 and we assume th at b = A; 2 = 1 and

h

= 2 , then due to (1.2)

T? = -2Eh/i

2

\, P = 2irRT?. (14.3.8)

Th e value A = 1 gives the classical Lo renz -Tim osh enk o c ritical load (see

(3.4.3)).

The value of

A

< 1 is a m easu re of the decrease of the c ritical load.

The edge effect stresses and deformations near shell edge

s = so

depend on the

type of edge support. Let

u = v = w = M

1

-0

(th at is 1110) at

s = s

0

.

(14.3.9)

Then by formulae (1.14), (1.20) we find

w

e

= Ivp? (aC-

a

~P S

\

2 a

t\ =

fi-

2

k2W

e

,

X'

where we use the designations from (1.18).

Now, if we let

v

= 0.3 , we ob ta in Ao = 0.867 (see [2]).

T he pro blem unde r conside ration h as been extensively stu died [2, 21 , 22,

44, 61, 112, 149] for the various relationships among the parameters and for

various boundary conditions.

In all cases except 0010 and 1010 the buckling mode occurs with a large

number of waves, m, in the circumferential direction (m ~ p

_ 1

) . Since Ao is

close to 1, the parameter a, which causes the decrease of functions t\, x\ is

small (see formulae (10) and (1.18)) and for relatively short shells the mutual

influence of the edge effects at s = si and at s = S2 may be im por tan t . This

question has been studied in [2, 22, 44, 61]. In cases 0010 and 1010 where the

support is weak, buckling occurs for m — 2. The same value of A = 0.5 is also

found without taking into account initial moment stresses (see (13.2.12)).

T he decrease in the critical value of Ao noted is related to th e fact th at

Poisson 's ratio u ^ 0 (see (10) ). In Tab le 14.1 (the d a ta for which has been

obtained in [71]) for six boundary condition variants, the relationships of Ao(f)

and p {v) are given. The last of these, due to the expression p = fim, allows

us to estimate the expected number of waves, m, under buckling.

) •

? =

/ ( 1 + A

0

) fc

2

\

1 / 2

^

(14.3.10)

1 —

I '

a



14.3. The Cases of Significant Effect of Pie-buckling strains

313

Table 14.1:

X

0

(u) and p(v)

1111

1011

0111

0011

1110

0110

V

Ao

P

Ao

P

Ao

P

Ao

P

Ao

P

Ao

P

0.1

0.968

0.48

0.968

0.48

0.957

0.45

0.956

0.45

0.928

0.48

0.907

0.45

0.2

0.946

0.48

0.946

0.48

0.930

0.45

0.929

0.45

0.894

0.48

0.870

0.44

0.3

0.927

0.47

0.927

0.48

0.910

0.44

0.909

0.44

0.868

0.47

0.844

0.44

0.4

0.911

0.47

0.911

0.47

0.893

0.44

0.892

0.44

0.848

0.47

0.824

0.44

0.5

0.897

0.47

0.897

0.47

0.879

0.44

0.877

0.44

0.831

0.47

0.808

0.44

E x a m p l e 1 4 . 2 .

Let us now stu dy the buck ling of a conical shell un de r

axial compression. For a conical shell we have

k\

= 0, &2 =

b

1

c o s a o , 6 = s s i n a o ,

P = 27r

J

Rtcosa

0

T

1

0

)

T° = S° = 0,

(14.3.11)

where 2ao is the cone vertex angle and

Rs

is the distance between a point

on the shell and the vertex. In relation (1.2)

T° = —\Ehp?t\

we take

t\ =

2 6

_ 1

cos «o .

Then

7 ( « ) = - r — = 1-

(14.3.12)

T he m inim izatio n of function 7 (s) gives the weakest parallel (see formu la

(4.3.10)). It follows from (12) that in the membrane statement this Example

(as in Ex am ple 14.1) corresponds to Case 1 and the buckling mod e occupies

the whole shell surface. The critical value of the force is equal to

_ 2ir h

2

cos

2

a

0

(14.3.13)

314


For A = 1, formula (13) is similar to the Lo renz -Tim oshe nko formula (3.4.3)

and has been obtained in [140].

Now we will examine the buckling mode which is localized near shell edge

s = so-

Let this edge be simply supported by a rigid body which may progres

sively move in the axial direction. Th e bound ary c ondition s und er b uckling

we take in form (9). The initial stress-strain state is determined by relations

u° = -2n

2

cot a

Q

(Ins + c), w° = -2 f? [v + In s + c), (14.3.14)

and w

e

, <|>

a n

d x\ m ay be calc ulate d by the sam e form ulae (10) as for a

cylindrical shell.

System of equations (3) for a conical shell differs from system (3) for a

cylind rical shell by only two par am et er s: &o an d Ar

2

o- We can therefore make

the substi tutions

C = C&2(>

2

>

Pi= Pi^o

/2

=

f

irnb

o

lk

2o

/2

- (14.3.15)

Then, both system (3) , and boundary conditions (9) , which are writ ten

using formulae (7), coincide for cylindrical and conical shells. Hence, for

v =

0.3, Ao = 0.874. Here, in co ntra st to a cylind rical shell,

X\ ^ 0

i.e. the error

of the zeroth approximation is greater.

If edges

s = si

and

s

= s

2

are supported identically then (in the zeroth

approximation) buckling in the neighbourhood of each of them corresponds to

equal values of P . Bu t th e num ber of waves, m , in the circum ferential direction

are different. By virt ue of (15) we can w rite

m

= p

1

( l 2 ( l - 0 )

1 / 4

( ^ )

1 / a

>

(14.3.16)

where the B is the radius of circle s — so. For larger radius th e nu m ber m is

larger.

Taking into account the noted agreement of equations (3), the values of

Ao obta ined for a cylindrical shell with various bou nd ary condition s m ay be

used in formula (13) also for a conical shell. Boundary condition variants 0010

and 0110 for w hich A = 0.5 and the bu ckling m od e is axisy m m etric [44] are

exceptions.

E x a m p l e 1 4 . 3 .

Consider the buckling of a hemisphere under a homoge

neous external pressure. For a spherical shell

k

1 =

k

2

= l, T» = T°= -

q

-£= -2\Ehfi*,

2Ehk (

1 4

-

3 1 ?

)

316 Chapter 14. Shells of Revolution under General Stress State

14 .4 T h e W eakes t Pa r a l l e l Co i nc id i ng W i t h an

E d g e

T he weakest parallel , s = so, is determ ined from the m inim um condition for

the function 7 (s) which was introduced in Section 4.3. Let parallel So coincide

with the edge of the shell (SQ = S2). Here we will study problems with the

same init ial membrane stress-resultants as in Section 13.6 but in addition, the

m om ent stress-resultants and any pre-buckling deform ations will also be taken

into acco unt. We will exa m ine, in pa rticu lar, buc kling problem s for convex

shells under torsion and under tensile load by an axial force.

Depending on the loading and type of edge support, one of two possibilities

for buc kling ma y occur: A < A

0

= 7 (so) or A > A

0

. For the case when A < A

0

we m ay use expan sion (3.2) to find A. In thi s expan sion Ao is eva luate d using

the same system (3.3) .

Now let A > A

0

and examine the characteristic equation (13.5.5) of system

(1.1)

F(p,8,p,\) = 0 (14.4.1)

assuming that

t?

= x f = 0.

The roots of equation (1) for which

3 p < 0 . (14.4.2)

correspond to a solution of system (1.1) which exponentially decreases with

s.

As was shown in Section 13.6, system (1.1) for t\ = xf = 0 has a turning

poin t , s = s„, such that for s < s« equation (1) has four roots satisfying

condition (2).

Therefore system (1.1) for

t\

, xf ^ 0 ha s four linearly inde pen den t so lutions

which exponentially decrease with

s

for

s

< s» (since the functions <?, and x?

decrease away from edge s = S2). Four such solutions allow satisfaction of four

boundary conditions introduced at s = «2-

For

t\ =

xf = 0 the cha racteristic equ ation of system (3.3) is

F(p,S2,p,X) = 0. T his equatio n for A > A

0

has only three roots satisfying

condition (2) and therefore system (3.3) generally has no solutions satisfying

the four conditions for £ = 0 and decreasing conditions (4) and so it cannot be

used as the zeroth approximation.

In Sections 13.5 and 13.6 it was shown that in the membrane formulation

for 7 '(SQ) < 0, para m eter A has the expansion

X = \°+H

2

'

3

\

1

+0(fi),

(14.4.3)

14.4. The Weakest Parallel Coinciding With an Edge

317

where

A

0

= 7

(so), Xi =

Q ( / ° )

2

/

P

°

P

) Co, Co = 2.338 , (14 .4.4)

and f° and f° are described in Section 13.5. In form ula (3) only ter m s of

order /z,

(0(H)),

depend on the bo und ary condit ions.

Let us solve the general formulation of the problem. As in Section 14.2 it

can be shown that j] = O(fi), where rj is a m easu re of the relativ e influence

of the ini t ia l moment s tress-resultants and the pre-buckling deformations on

the critical load (see (2.2)). In other words, in formula (3) the first two terms

remain the same in the general formulation.

To prove this fact we return to formulae (2.2) and (2.3) taking in them the

deflection

WQ

in the form of (13.6.13)

w

Q

= (Ai(afi~

2/3

(s-s

t

)

>

j +

3

JPk(s- s

2

)

+ H

1/3

J2C

k

e P ) cos (m(/3s-<p + <p

0

)), (14.4.5)

fc=i '

where a ,

Ck,

and

ft

do not depend on

\i

and

s,

for th a t /? = 0 if the re is no

torsion. Formula (5) is approximate but its accuracy is sufficient to obtain the

following estimates. By substituting (5) into (2.3) we find

/ o ~ ^ -

4 / 3

, / i , /

2

~ / i -

1 / 3

, (14.4.6)

and due to (2.2) it follows that T] ~

ft.

E x a m p l e 1 4 . 5 .

Let us consider the buckling of a spherical belt, (a portion

of a hemisphere with edges parallel but not on the equator), si ^

s

^ «2

with constant values of E, u, and h under an axial tensile force P. By using

formulae (13.6.14) we obtain

(14.4.7)

P

=2irREhfi

2

X,

t

2

= -t

1

=b~

2

, b-coss,

y(s)=2sin

2

s, m = 6

0

/i

_1

, hi = fc

2

= 1,

where s is the angle between the point and the equator.

Let s

2

> j* i | , then the paral lel

s

=

s

2

is the weakest.

Let «2 = ""/4. Then, by formulae (3) and (13.6.16) we find

P= ll

k2X

2V

X=l+2(oH

2/3

+ 0(n).

(14.4.8)

\ / 3 (1 - v

2

)

318

Cha pter 14. Shells of Revolution under Genera l Stress State

Table 14.2: Values of A (m) ob tained by num erical integration

m

28

29

30

31

32

33

membrane

simple

support

1

1.476

1.464

1.458

1.460

1.467

1.482

clamped

2

1.500

1.486

1.479

1.479

1.483

1.495

general

simple

support

3

1.549

1.539

1.535

1.538

1.549

1.566

clamped

4

1.590

1.577

1.571

1.573

1.581

1.595

formula (8)

1.426

In order to estimate the influence of the boundary conditions and initial

moment stress state, we take

R/h

= 400 and

v

= 0.3. Then

fi —

0.0275 and

the first two terms in formula (8) give

A

=

1.426.

In Table 14.2 we have presented the values of A (m) ob tained by num erical

integ ration (with orthog onaliz ation) of system (1.1). Colu m ns 1 and 3 cor

respond to a simply supported edge

s= S2 {T\

=

v =w = M\ =

0), and

columns 2 and 4 correspond to a clamped edge (u = v = w = 71 = 0).

In colum ns 1 and 2 we place the resu lts, obtaine d for th e m em bran e for

mulation (£ | = xf = 0), and evaluating Columns 3 and 4 we take into account

the stress-resultants

t\

and the change of the curvature changes

x\.

For the

simply supported edge we can write

tl = -2(l +

V

)(c-f-s), xt=

1J

S (14.4.9)

\ 2 o r / or

and for the clamped edge

< = - 2 ( l + i / ) ( C - S ) . x

1

e

= 2 ( l + zv)(C + S ), (14.4.10)

14.5. Problems

and

Exercises

319

where

C*

= a -

1

e

a i

c h / ? i , 5 =

p~

l

e

ai

sh/?i ,

1 + A \

1 / 2

„ / A - 0

1 / 2

It follows from the Table that buckling occurs for m = 30 which is slightly

larger tha n the value m = 26 predicted by formula (7). Th is is because the

value of

6

in the buckling pit c entre is larger th an

&o

at th e shell ed ge.

From the results presented in the Table it follows that the comparatively

low critical load depends on both the boundary conditions and whether or not

we take into account the initial moment stress state. It agrees with formula (8).

If we include the moment stress-resultants in the calculation then we obtain

a greater value of

A

since near th e edge

t\

< 0 th e effect of th e co mpres sive

m em bra ne stres s-resu ltants <2 > 0 decreases.


14 . 1

Consider an axisym m etrically loaded shell of revolution with n egative

Gaussian curvature. Find relative the order of the first term in expression for

the critical value (14.2.2) for different boundary conditions of the free support

and c lamped group.

A n s w e r

t] = 0{n

A

l

z

) (1011,1010) ,

T)

= O (/r

5

/

3

) (1111, 1110, 110 1, 1100, 0111, 0011, 0010 ),

rt = 0(n

2

) (0110 ,0101 ,0100) ,

where, in parenthesis, we list the boundary condition variants for which these

estimates are valid.

1 4 . 2 .

Consider the buckling of a cylindrical shell with th e Poisson ra tio

v —

0.3 under axial compression for th e following bo und ary cond itions at the

edge s = s

0

([2]):

a) 1111, 1011

b) 0111

c) 0011



320


d) 0110

e) 0010, 1010

A n s w e r

Boundary conditions

1111, 1011

0111

0011

0110

0010, 1010

Ao

0.926

0.910

0.908

0.844

0.5

14.3 Consider buckling of the semi-infinite circular cylindrical shell at the

free edge of which the following load is applied

(i) radial compressive load,

(ii) radial tensile load.

14.4 Consider buckling of the infinite circular cylindrical shell under radial

load distributed homogeneously along the parallel.



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Lis t o f F igu re s

1.1 T he stres s-resu ltants and stress-couples acting on a shell ele

ment. 11

1.2 T he slan ted edge. 15

1.3 T he large ben ding of the ne utra l surface 20

1.4 Shell of R ev olu tion 24

2.1 Re lat ion shi p betw een load and deflection for an idea l shell. . . 39

2.2 T he rela tion shi p betwee n the load and deflection for an ideal

shell an d for a shell wi th imp erfection s 41

3.1 W ave nu m be rs for the specia l case of a convex shell 55

3.2 W ave num be rs for axial comp ression of a cylindrical shell. . . . 58

3.3 Dim ensionless stress-re sultan ts for a cylindrical shell und er com

bined loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1 T h e eigenfunction wh en po = 0 76

4.2 T he even and odd eigenfunctions when p

0

^ 0 77

4.3 T he bu ckling m od e of a convex shell of revo lution at £2^2 > ^i&i- 81

4.4 An oblate

(S

< 1) and a prola te

(S

> 1) ellipsoid of rev olu tion . 82

4.5 Cr itic al value do m ain s for the buckling of an ellipsoid nea r an

equator 85

4.6 T he buckling m ode of a convex shell of revolution unde r torsion. 87

4.7 T h e dom ain s of the critica l values in the buckling of an ellipsoid

nea r an equ ator 89

5.1 Fun ction s A

0

(A;) and A\(k) 97

5.2 Th e non-hom ogeneo us axial comp ression of a cylindrical shell. . 98

5.3 Load para m ete rs for even and odd buckling m ode s of a cylin

drical shell unde r non-hom ogene ous axial com pression 101

5.4 T he load pa ram ete r for a cylindrical shell und er non-hom ogeneo us

axial com pression and an inte rnal pressure 103

335

336

L ist of Figures

5.5 Th e genera trix of a cylindrical shell with the curva ture of vari

able sign . 105

5.6 Th e function 6 - ^ 0 ( 6 ) . 106

6.1 T he buckling m od e of a convex shell localized ne ar the weak est

point 115

6.2 T he buck ling m od e of a elliptic shell localized nea r the weakest

point 122

6.3 T he localized buckling m od e of a cylind rical shell ne ar the weak

est po in t 129

6.4 T he localized buck ling m od e of a cylin drica l shell at a/?

2

>

( 1 - a )

2

129

6.5 T he localized buck ling m od e of a cylin drical shell at a/?

2

<

( 1 - a )

2

.

130

8.1 Th e parallel coo rdinate s near a slanted edge 157

8.2 T he functions

b°(x), ao(x), b

aa

{x),

and

b

>ci(

(x )

181

8.3 T he slan ted conic shell 182

8.4 T he functions

b*(x)

and

d(x)

182

9.1 Th e functions

R

3

{R

2

)

and

k

n

(R

2

)

189

9.2 An elliptic cylinde r in torsio n. 194

9.3 A force an d a m om en t act ing on a shell 196

9.4 T he functions

ao(x), bo(x),

and

b

aa

(x )

198

10.1 A nea rly conic surface 202

10.2 Functions

F(a)

an d a;(a) 207

10.3 A Ben t Cy lindric al Shell 208

10.4 Functions i?3(a) and

k

n

(a )

211

11.1 A tor us sector 229

11.2 Th e loading pa ram ete rs acting on the torus at buckling 230

11.3 T he values of A(m) 231

11.4 The two types of shells in which the Gaussian curvature changes

sign. 232

11.5 Schematic of function w

0

(s) 235

12.1 An em pty convex shell im m ersed in a liquid (wa ter) 251

12.2 Cy lindric al and conical shells in a liquid 257

12.3 T he hyp erbo loid of revolu tion 264

12.4 Th e load pa ra m ete r for hyperbo loid buckling . 264

13.1 A recta ngu lar pla te unde r com pression 268



List

of Figures

337

13.2 The loading parameter for a cylindrical shell with a weakly sup

ported curvilinear edge under axial compression (1 - 0000, 2 -

0100, 3 - 1000, 4 - 1100, 5 - 0010, 1010, 00 01 , 6 - 0101) 274

13.3 The load parameter for a cylindrical panel with a weakly sup

ported rectilinear edge under axial compression (1 - 0000, 2 -

0001, 3 - 0100 , 4 - 01 01 , 5 - 0010 , 6 - 1000, 7 - 1100) 277

13.4 Functions

Y(r))

278

13.5 The load parameter for a convex shell of revolution under axial

compression (the values of k\ are given in th e Fig ure) 281

13.6 The loading parameter for a convex shell of revolution under

axial tension (the values of kz are given in the F i g u r e ) . . . . . . 283

13.7 T he load p ara m ete r for a convex shell of revolution und er torsion

(the values of k\ are given in the Figu re) 283

13.8 An ellipsoid of revo lution und er torsion 289

13.9 The buckling mode of a convex shell in the case when the weak

est po int lies on the edge of the shell 293



L is t o f T a b l e s

1.1 T he bo un da ry cond itions 17

5.1 Critic al buckling loads and wave num bers for a cylindrical shell

und er com bined loading 99

5.2 Criti cal buckling loads and wave nu m be rs for a cylindrical shell

und er comb ined loading 101

8.1 T he inte nsity indices 162

8.2 Parameter 77* co rrespond ing to the boundary condi tions . . . . 177

8.3 T he sim ply sup po rted edge 177

8.4 M ain bou nda ry conditions 178

11.1 Indice s of inte nsit y 216

11.2 Th e bou nda ry conditions 217

12.1 T he critic al values of Ao 258

12.2 T he orders of the term s 262

13.1 The values of

s

and

A

for different ra tio s a /6 269

14.1

\

0

{v )

and

p(u)

313

14.2 Values of

A

(m) obtained by num erical integration 318

339



Index

bending

infinitesimal

surface, 63

boundary condition

addi t ional ,

161

clamped edge, 14, 17, 18

clamped group, 165, 168

free edge group, 165, 168

geometric, 63, 64

homogeneous, 42

main ,

161, 163, 207, 217

Navier,

18, 53

non- tangent ia l ,

168, 226, 251,

252,

26 1, 265, 301

self-adjoint, 71

separa t ion,

151, 163, 168

simple support,

17, 18, 48, 53,

122, 154, 212

simple support

group, 165, 168

special group, 168, 171

tangential, 166

weak

edge, 54

weak support,

23

weak support

group, 165

boundary value problem

"on

spec t rum",

138, 153

coefficient

Lame, 5

effect

Dubjaga-Karman-Brazier, 91

eigensize, 227, 230, 233, 239, 245,

261,

264-266

energy

potential, 34, 35, 38

addi t ional , 45

extension,

34

of

tangential deformation,

52

under bending, 52

equation

Volterra , 228

Weber,

107

equilibrium state, 37, 39-41

adjacent,

39, 42, 45

axisymmetric, 40

basic, 39

bifurcation,

39

branching, 39

buckling, 39

non-axisymmetr ic ,

40

seconary bifurcation, 40

stability

analysis, 37

stable, 37, 39, 40

unstable, 39, 40

force

Archimedean, 267

generalized,

15-17

principal vector, 26

formula

Euler,

58, 108

Gra shof-B ress, 61 , 246, 247

Lorenz-Timoshenko,

57, 123,

272,

314, 316

Southwel l -Papkovich,

60, 144,

145

341



Index

normal, 44, 85

normal

homogeneous, 81

static, 37

conservative edge, 38

conservative surface, 38

surface, 14, 26, 45

dead, 44, 45

external, 11, 20, 35, 79

tangential , 29

torsion,

213

wind, 26

method

Bubnov-Ga le rk in ,

9 1, 146, 147,

198

energy,

49, 62

Goldenveizer,

2

Lagrange, 37

Maslov,

3, 70

mitred joint

pipe, 147

moment

bending, 23

principal, 26

parallel

weakest,

69, 79, 81-84, 87-89,

161,

295

pit, 121, 128, 130

amplitude, 127

center

of, 127

depth , 114, 121, 127

domain ,

129

incl inat ion, 127

inclination

angle, 114

phase, 127

size, 114

point

bifurcation, 39, 40

turnin g, 71 , 72

weak,

111, 122

weakest, 70-72, 103, 112, 115,

119-123,128,129,131,163

343

polynomial

Hermite, 74, 125, 140

problem

Sturm-Liouvi l le ,

223, 228

relations

Codazzi-Gauss, 8 , 65, 88

constitutive, 7, 12, 17, 19, 20,

22, 25, 27, 33

Novozhi lov-Balabukh,

13

shell

closed,

133

conic, 26, 70, 133, 135, 143,

158, 176, 180, 183, 200,

203

circular,

183, 185

straight circular, 199

truncated,

185

convex,

111, 115, 133, 209

shallow,

49

cupola ,

245, 250

cylindrical,

21, 26, 32, 33, 111,

122, 128-130,133 , 135,143,

144

cantilever,

146

circular,

33, 122, 129, 143-

146

elliptic, 104, 148, 196

long, 33, 34

moderate length ,

20, 147

non-circular,

32, 33, 70, 122,

133, 145, 147

elliptic, 82- 86 , 88, 89, 119,122,

148

oblate, 82, 90

prolate, 82, 119

elliptical, 291 , 296

oblate, 291

in a liquid, 253

nearly

co nic, 20 3, 204, 207, 213

nearly cylindrical, 203, 213



344

non-closed,

140

of rev olu tion , 20, 2 1, 23, 25, 30

reinforced, 187

shallow, 31

spherical, 56, 57, 80

toroidal, 234

with impe rfections, 32, 41 , 42,

47,48

with slanted edges, 70

with variable thickness, 133

stability criterion

bifurcation, 37

dynamic, 37

Euler,

37, 48

limit point, 37

stress-couple, 11, 35

vector,

11, 35

stress-resultant, 11, 15, 16, 35

hoop, 80, 85

membrane, 29

projection,

16

shear, 22

vector, 11, 35

stress-strain

state, 20, 46

addi t ional ,

133

chacteristic, 20

character, 22

classification, 19, 22

combined,

23

edge effect,

23, 29, 134

non-linear,

24, 30

simple, 20, 24, 30, 31

fast varying, 34, 47

general, 17, 19, 20, 47, 299

infinitesimal pure bendings, 23

initial, 42, 48, 122, 133

main, 134, 135

membrane, 22, 27, 42, 45, 47,

48, 111, 133

moment, 24, 47

momentless, 22

partial, 19

post-buckling,

41

pre-buckling, 41

pseudo-bendings, 23, 66

pure bending, 23

pure moment,

23

semi-momentless, 23, 33

134

theorem

Kirchhoff,

38

theory

elasticity, 17

Liapunov, 37

shell, 7, 17

errors, 19

Kirchhoff-Love, 17

linear, 19, 20

technical, 29, 43 , 47

two-dimensional, 12

surface, 7

transform

Fourier,

96, 106, 123



A b o u t t h e A u t h o r s a n d

E d i t o r s

P e t r E . T o v s t i k

was born in Leningrad, Russia, in 1935. He presently is

professor and Head of the Department of Theoretical and Applied Mechanics of

the Faculty of Mathematics and Mechanics at St . Petersburg State University .

He graduated in Applied Mathematics from Leningrad State University (now

St. Petersburg State University - St. PSU) in 1958, obtained his Ph.D. in

Mechanics of Solids from the same university in 1963 and the degree of Doctor

of Science in 1968 for the thesis: Free Vibrations and the Stability of Thin

Elastic Shells.

Prof.

Petr Tovstik is a leading Russian specialist in the application of

asym ptotic m etho ds in thin shell and plate theory. He continues the tra

dition of Russian (Soviet) study in mechanics of thin shell structures which

is associated with the names Goldenveizer, Mushtari, Novozhilov, Vlasov, and

Vorovich. He is the author of a monograph entitled Stability of Thin Shells, the

co-author, with A.L. Goldenveiser and V.B. Lidsky, of a monograph entitled

Free Vibrations of Thin Elastic Shells

and with his pupils of books

Asymp

totic Methods in Mecha nics, A symptotic Methods in Problems and Examples,

Asymptotic Methods in Thin Shell Structures

and also num erou s pap ers on

asymptotic methods in mechanics of thin structures published in

Doklady of

the Academ y of Science of the U SSR, M echanics of Solids, Transactions of

the Canadian Society of Mech anical Engineers, Differential Equations, Soviet

Applied Ma thematics, Technische Mecha nik

and many other journals. He re

ceived First Prize for Research at Leningrad State University in 1970. In 1999

he got the highest scientific award in Russia, the State Prize. He is member of

the National Committee on Theoretical and Applied Mechanics and the editor

of the journal Vestnik Sankt-Peterburgskogo Universiteta, serie Mathematics,

Mechanics, Astronomy. He is a m em ber of AM S.

345



346

About the Authors and Editors

A n d r e i L . S m i r n o v

was born in Leningrad, USSR, in 1956. He is As

sociate Professor at the Department of Theoretical and Applied Mechanics of

the Faculty of Mathematics and Mechanics at St . Petersburg State University

(formerly Leningrad Sta te University) . He grad uate d in Applied M athem atics

from Leningrad State University in 1978 and obtained his Ph.D. in Mechanics

of Solids from the sa m e univ ersity in 1981 for th e the sis:

Vibrations o f the

Rotating Shells of Revolution .

He is the editor a nd /o r the auth or of the books

Asymptotic Methods in Mechanics, Asymp totic Methods in Problems and Ex

amples, Asymptotic Methods in Thin Shell Structures, Problems in Stability

Theory, In troduction to the Theory of Stability

and papers on asymptotic and

numerical methods in mechanics of thin structures published in

Transactions

of the ASM E, Transactions of the CSM E, T echnische Mechanik, M echanics Re

search Com munications, Vestnik L eningradskogo U niversiteta

and other jour

nals .

He is a member of AMS.

P e t e r R . F r i se was born in Peterborough, Canada in 1958. Since 1997 he

has been a Professor of Mechanical and Automotive Engineering and has held

the NSERC/DaimlerChrysler Canada Industr ia l Research Chair in Mechani

cal Design Chair at the Faculty of Engineering of the University of Windsor,

Canada. He graduated in Mechanical Engineering from Queen's University in

Kingston in 1981 and, after working in the oil industry in West Africa, ob

tained his M.Sc. also in Mechanical Engineering from the same university in

1984.

He spent a year and a half at Husky Injection Molding Systems Ltd. in

Bolton, Canada working as an R&D engineer and engineering group leader and

then left Husky to attend Carleton University where he obtained his Ph.D. in

1991 for the thesis:

F atigue C rack Growth and Coalescence in a Notch Region.

He was presented with the Ralph R. Teetor Engineering Education Award by

the Society of Automotive Engineers (SAE) in 1993. Dr. Frise is the author of

papers on modelling fatigue crack growth and non-destructive testing of struc

tural steels and zirconium-niobium alloys in

The Journal of Pressure Vessels

and Piping, The International Journal of Fatigue

and

The British Journal of

Non-Destructive Testing

and has spoken at a number of international confer

ences on Design Engineering including ICED and the ASME-DETC confer

ences in 1999. In 2000 he was elected a Fellow of the Canadian Academy of

Engineering.

A r d e s h i r G u r a n

was bor n in Te hra n, Iran , in 1957. He is the direc

tor of the Ins titu te of Struc tronic s in C an ad a. After enjoying his childhood

in the Savadkooh Mountains and the Caspian seashores, he studied Medicine at

Te hran U niversity and Structura l E ngineering at Ary am ehr University (Teh ran) .

In 1979 he emigrated to Canada, received his M.Eng.(in Applied Mechanics)

from M cGill U niversity(M ontrea l) in 1982. Bo th his M .Sc.(in M athe m atics )



347

and PhD (for the thesis C ontributions to Study of Instabilities in a Class of

Conservative Systems) were obta ined from the University of To ron to. Professor

Guran served as the associate editor of the International Journal of Mo deling

and Simulations

durin g 1993-95 . His books on

Theory of Elasticity for Scien

tists and Engineers

(2000) and

Adap tive Con trol of Non linear Systems

(2001)

have been published by Birkhauser .

Dr. G ur an chaired several inte rna tion al scientific conferences includin g:

The First International Symposium on Impact and Friction of Solids, Struc

tures and Intelligent Machines(Ottawa Congress Center, 1998), the First In

ternational Congress on Dynamics and Control of Systems (Chateau Laurier ,

1999) and the First International Conference on Acoustics, Noise and Vibra

tions (McGill University, 2000). He is the editor-in-chief of series on Stability,

Vibration and Control of Systems.

asymptotic methods in the buckling theory of elastic shells

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