North holland series in applied mathematics and mechanics 27 elastic stability of circular cylindrical shells

N O R T H - H O L L A N D S E R I E S IN

A P P L IE D M A T H E M A T IC S
A N D M E C H A N IC S
EDITORS:
E . B E C K E R

Institutfür Mechanik
Technische Hochschule, Darmstadt
B . B U D IA N S K Y

Division of Applied Sciences
Harvard University
W . T. K O IT E R

Laboratory of Applied Mechanics
University of Technology, Delft
H . A .

L A U W E R IE R

Institute of Applied Mathematics
University of Amsterdam

V O L U M E

27

N O R T H -H O L L A N D
AM STERDAM

· NEW YORK · OXFORD


ELASTIC STABILITY OF
CIRCULAR CYLINDRICAL SHELLS

N. Y A M A K I
I n s t it u t e o f H ig h S p e e d M e c h a n ic s
T o h o k u U n iv e r s ity
S e n d a i, J a p a n

8
1984
N O R TH -H O LL A N D
AM STERDAM

· NEW YORK · OXFORD


® Elsevier Science Publishers Β .V ., 1984

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

ISBN: 0 444 86857 7

Published by:
ELSEVIER SCIENCE PUBLISHERS B.V.
P.O. Box 1991

1000 BZ Amsterdam
The Netherlands
Sole distributors for the U.S.A. and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC.
52 Vanderbilt Avenue
New York, N.Y. 10017
U.S.A.

Library of Congress Cataloging in Publication Data
Yamaki, N. (Noboru), 1920Elastic stability of circular cylindrical shells.
(North-Holiand series in applied mathematics and
mechanics ; v. 27)
Includes bibliographical references.
1. Shells (Engineering) 2. Cylinders. 3. Buckling
(Mechanics) I. Title. II. Series.
8 3 -2 5 ^ 8 5
TA660.S5Y36 1981*
62^.1*7762
ISBN O-Wt-86857-7 (U.S.)

PRINTED IN THE NETHERLANDS


INTRODUCTION

Buck l i n g

of

pr oblems

to e n g i n e e r i n g

the

problem

thanks
of
most

circu l a r

ma y

no w

to the e f forts

the

present

extensive

cylindrical

shells

for man y years.

be

has p o s e d ba f f l i n g

In the e l a s t i c dom ain

c o n s i d e r e d to be

of n u m e r o u s

autho rs

so lv ed complet el y,

inclu d i n g

b o o k P r o f e s s o r Y a maki wh o has
and accurate

up to the p r e s e n t

time.

for e l a s t i c stabi lit y,
is o t r o p i c circular

the o r e t i c a l

His wor k will be
buckling

cyl i n d r i c a l

the wri t e r

c o n t r i b u t e d the

and e x p e r i m e n t a l

and p o s t - b u c k l i n g b e h a v i o u r

shells

for m a n y years

of

to come.

W.T.

v

data

the stan d a r d re fer ence

Koit er

PREFACE

For the design of light-weight structures, it is of great
technical importance to clarify the elastic stability of circu­
lar cylindrical shells under various loading conditions. Hence,
numerous researches have been made on this subject since the
beginning of this century along with the development of air­
craft structures. In the early stage of the relevant research­
es, only approximate solutions were obtained under special
loading and boundary conditions, owing to the inherent mathe­
matical difficulty and physical complexity. Experimental stud­
ies had also been conducted with thin-walled metal test cylin­
ders, but the results were not precise enough to examine and to
improve the corresponding theoretical analyses, due to the de­
teriorating effect of both initial imperfections and plastic
deformations.
With the advent of high-speed digital computers in the 1960s,
it became possible to solve the buckling problem with suffi­
cient accuracy and effects of boundary conditions and further
those of prebuckling edge rotations have been pursued under
various loading conditions.
Experimental techniques have also
made a great progress , and nearly perfect test cylinders as
well as highly elastic cylinders sustainable fairly large de­
formations became available , leading to the verification of
reasonable agreement between theory and experiment , not only
for the buckling problem but also for the postbuckling behav­
iors .
This book presents a comprehensive treatise on the elastic
stability of circular cylindrical shells, which represents the

sum of the past 17 years of research conducted at the Institute
of High Speed Mechanics, Tohoku University . Only the static
conservative problems are treated concerning the unstiffened
cylinders made of homogeneous, isotropic elastic material with
constant thickness. Both theoretical and experimental studies
were performed on the buckling, postbuckling and initial-postv ii


PREFACE

v iii

b u c k l i n g pr o b l e m s

under

p a y i n g due a t t e n t i o n
pha ses

were

exten s i v e
results,
the

or

combined

give

a

of

of

to

precise

the o r e t i c a l

data

for

cylindrical

complete

c l osely r e l a t e d

both

fun d a m e n t a l

st ab i l i t y

loadings,

the ef fect of b o u n d a r y conditions.

presentations
to p r o v i d e

to

to

singl e

p l a c e d on the a c c u r a t e an al yses,

ela stic

made

typ ica l

experimental

the bas i c p r o b l e m s

shells .

bibl iog rap hy,

the sp eci fic

an d

Em­

tests an d

bu t

problems

No

atte m p t

only

the

s t udied

in

on
is

pap e r s
the b o o k

are cited at a p p r o p r i a t e places.
In the first ch apter,

cyl i n d r i c a l

typi ca l n o n l i n e a r

cal f o u n d ations of the ensu i n g analyses
C h a p t e r 2 deals
equations,

with

the

buckling

the h o m o g e n e o u s
derived

are

theories,

w h i c h are a p p l i e d

su b j e c t e d

the torsional,
sets

of

load

an d

to

on

boundary

to

the

the

and

of

of c i r c u l a r

the shell ge ometry,

for

of

compressive

considered

and

for

bas ic

the e i g e n v a l u e
nonlinear

cylindrical

three f u n d a m e n t a l

are

the

relevant

buckling

axially

condi t i o n s

the book.

First,

the

c o r r e s p o n d i n g m o d e are c l a r i f i e d

loads,

i.e.,

loads.
the

Eig h t

cr it ical

a w i d e ra nge of

ta king the ef fect of p r e b u c k l i n g edge r o t a ­

tions into consider ati on.
Donn e l l

the basis

one of

pressure

th r o u g h o u t

problem.

li ne ar e q u ations

problem,

shells

theor ies

shells are d e s c r i b e d w h i c h c o n s t i t u t e the t h e o r e t i ­

Mo s t of the anal y s e s

equations , the v a l i d i t y

of

which

is

are b a s e d on the
e x a m i n e d through

a p p l i c a t i o n of the F l ü g g e equations.
Ch a p t e r 3

isd e v o t e d to

pletely clamped
three

cylindrical

fun d a m e n t a l

loads.

are first presente d,
test cy linders,
are

gi ven ,

the

postbuckling problems

shells s u b j e c t e d

In

each

case,

to

an d then
by

the

com­
of

the

re sults

six p o l y e s t e r

c o r r e s p o n d i n g theore t i c a l res ults

applying

Do n nell n o n l i n e a r equations.

one

experimental

c a r e f u l l y c o n d u c t e d by u s i n g

obtained

of

the

Galerkin

method

to

R e a s o n a b l e agr e e m e n t s b e t w e e n

ory and e x p e r i m e n t are revealed.

Analyses

the
the­

for the initia l pos t-

b u c k l i n g b e h a v i o r s an d i m p e r f e c t i o n s e n s i t i v i t i e s c o r r e s p o n d i n g
to the same cases as in the for e g o i n g are p r e s e n t e d
4.

Und e r

eac h

by a p p l y i n g the
line ar

equat i o n s

l o ading condit io n, the p r o b l e m is

in

Chapter

first

so lv ed

G a l e r k i n p r o c e d u r e di r e c t l y to the Do n n e l l
an d

then

asy m p t o t i c

so lut ions

are

non­

obtained


ix

PREFACE

thro u g h a p e r t u r b a t i o n pr ocedure,
effect

of

m o d e as

in itial

well

as

imperfections

the

r ange

of

thus c l a r i f y i n g the d e g r a d i n g
in

the shape

of

applicability

init ial p o s t b u c k l i n g th eory o r i g i n a t e d by

the b u c k l i n g

of

the so-ca l l e d

Koiter

and d e v e l o p e d

by Bud ian sky .

Buckling and postbuckling problems

under

combined loads are

treated in Chapter 5, in which the combined actions of hydro­
static pressure together with the torsional, axial and trans­
verse edge loads, respectively, are considered. Finally, effects
of the contained liquid on the buckling and postbuckling of
clamped cylindrical
loads are

tanks under each of

examined in Chapter

6.

the three fundamental

In each case above

stated,

the buckling problem is theoretically analysed and experimental
results are presented for typical postbuckling behaviors check­
ing the accuracy of the critical load theoretically determined.
Both theoretical and experimental results are given for the
postbuckling problems under the first two loading conditions in
Chapter 5, demonstrating fairly good agreement between theory
and experiment.
Thin-walled

circular

cylindrical

m o r e e x t e n s i v e l y u s e d in m a n y
as m o s t

efficient

sess

of

b e e n m o r e and

an d the auth o r h opes

to d eepen the b a s i c u n d e r s t a n d i n g

stab i l i t y c h a r a c t e r i s t i c s

the v a l i d i t y

have

d i f f e r e n t b r a n c h e s of e n g i n e e r i n g

struc t u r a l m e mbers,

b o o k to be b e n e f i c i a l
complex

shell s

of this

s t r ucture

o ther n u m e r i c a l p r o c e d u r e s

of

and

such

this
the

to a s ­

as those

u t i l i z i n g the f inite el e m e n t metho d.
T he

aut h o r

Professors
Seri es

Koiter

and

to

a c k n o w l e d g e hi s

tion to w r i t e

at

this

He

volume

is also

si nc ere

gratitude

to

B u d i a n s k y , E d i t o r s of the N o r t h - H o l l a n d

in A p p l i e d M a t h e m a t i c s

manusc rip t.
Editor

wishes

an d

an d

for

thankful

North-Holland,

Mecha n i c s ,
thei r

to Drs.

for their

kind

re m a r k s

Seve nst er,

for his c o u r t e o u s

and

sugges­
on

the

Mathematical

efficient c o l ­

laboration .
The auth o r
stud ent s

for

is i n d e b t e d to all
the ir contrib u t i o n s ,

d u ring the past
of Drs.

J. Tani,

two decades.

of

hi s

as so ciates,

cooperations

He a p p r e c i a t e s

S. K o d a m a and H.

Doki,

staffs and

an d a s s i s t a n c e s

the co l l a b o r a t i o n s

in w r i t i n g

the port i o n s


X

PREFACE

o f th e book r e l a t e d
and

6 .2

through

to M e s s rs .
M r.
and

H.

fo r

4 .4

and

6 .7 , r e s p e c tiv e ly .

K. Otomo and T .

K. Asano
M is s

to s e c tio n s

m aking

Hoshi f o r

S. Kodama, K. Otomo

and

S ato
th e

ty p in g
T.

He i s

fo r

S ato

th r o u g h

e s p e c ia lly
to

Mrs.

m an u scrip ts

fo r

t h e i r h e lp

and
in

5 .7

th an kfu l

p r e p a r in g th e draw ings,

photographs,
th e

4 .5 , 5 .2

to

K. T s u c h i y a
to

M e s s rs .

e d itin g

th e

f i n a l m an u s c rip t.

Noboru YAMAKI


CHAPTER 1

NONLINEAR THEORY OF CIRCULAR CYLINDRICAL SHELLS

1.1

I NTRODUCTI ON
When an e l a s t i c

body i s

s u b je c te d to a sm all d e fo rm a tio n

w h ic h d i s p l a c e m e n t s

as

are s m a ll,

d efo rm atio n w ith

s tra in s ,
-s tra in

th a t
we w i l l

n e g le c tin g

w e ll

as

d e riv a tiv e s

and

s tra in -s tre s s

o f d i s p la c e m e n t s

sm all r o t a t io n s

have l i n e a r e x p r e s s i o n s

re la tio n s

rium c o n d itio n s
tio n s

is ,

re la tio n s

and t h e e q u i l i b ­

th e e f f e c t o f d is p la c e m e n ts .
th e

d efo rm atio n

of

th e b a s ic

term s o f d i s p l a c e m e n t ,

re s u ltin g

When t h e body i s s u b j e c t e d t o

[1 ,2 ].

in

Thus,

d e fo rm a tio n in which e i t h e r

th e c l a s s i c a l
th e r o t a t io n s

n o t s m a l l enough i n c o m p a ris o n w i t h u n i t y ,
cease t o h o l d i n g e n e r a l and t h e l i n e a r
In

p a rtic u la r

but la rg e ro ta tio n s ,
v a l i d b u t th e
ered in

fo r

n o n lin e a r

e ffe c t

of ro ta tio n s

s h o u ld be exam ined

at

c o n s id e rin g th e e f f e c t o f d is p la c e m e n ts .
e q u a t i o n s w i l l be n o n l i n e a r

in

th e cases

under

the

o f s o l u t i o n as w e l l as

lin e a r

e q u ilib riu m
c o n d itio n s ,

sm all

In o th e r words,

are

in a d e ­
s tra in s

re la tio n s

r e m a in

s h o u ld be c o n s i d ­
F u rth e r,

th e

th e e q u i­

d e fo rm ed

s ta te

The r e s u l t i n g b a s i c

[3 ,4 ].

e q u ilib riu m
of

le a d in g

I n c o n t r a s t to

th e o ry o f e l a s t i c i t y ,

th e s t a b i l i t y o f

n o t be g e n e r a l l y a s s u r e d on t h e b a s i s

of e la s t ic it y .

or s tra in s

term s o f d i s p l a c e m e n t ,

to th e n o n l in e a r th e o ry o f e l a s t i c i t y

th eo ry

a la r g e or

becomes

w ith

s tre s s -s tra in

th e d is p la c e m e n t - s t r a i n r e l a t i o n s .

lib r iu m c o n d itio n s

lin e a r

in

t h e above a s s e r t i o n s

th e o ry

d efo rm atio n

th e l i n e a r

equa­

t h e body become l i n e a r

of e la s t ic ity

quate.

and s m a l l

f o r b o th d i s p l a c e m e n t

can be d e r i v e d a t t h e o r i g i n a l un deform ed s t a t e

governing

fin ite

in

u n iq u en es s
s ta te

th e n o n lin e a r

can

th e o ry

we may have s e v e r a l d i f f e r e n t

c o n f i g u r a t i o n s u n d e r t h e same l o a d i n g and bo un d ary
some o f w h i c h a r e

s t a b l e and t h e o t h e r s u n s t a b l e .


CHAPTER

2

1

Of cou rse only the sta ble e q u i l i b r i u m state can
the phys i c a l world.
fication,

definition

[5 ,6 ,7 ]

systems

and

and alth o u g h the m a t h e m a t i c a l theory of el astic

its ex t e n s i o n

system,

i.e.,

pl i s h e d [8 ].

r e a l i z e d in

on the c l a s s i ­

c r i t e r i o n of the stabi l i t y of ela st ic

sta bility has b e e n e s t a b l i s h e d by L i a p o u n o v
system,

be

The r e hav e been lon g deba te s

and

generalization

elast ic bodies,
However,

for

does no t

a

to

a

load,

continuous

seem to have b e e n a c c o m ­

in case - w h e n an elas tic body

to a static c o n s e r v a t i v e

discrete

is s u b jected

the so - c a l l e d energ y

criterion

is g e n erally ac c e p t e d for the v e r i f i c a t i o n of stability,
require s

the total po t e n t i a l

ene rgy

of

the

bod y

which

to

assu me

a relat ive m i n i m u m at the e q u i l i b r i u m po sition.
W i t h the adve nt of air craft
nume r o u s

r esear ches

struct ure s

in w e i g h t

l i g h t-weight

and

st ruc tures

stiffness,
which

every field of industry.
tures

in the b e g i n n i n g of this century,

have been co n d u c t e d to deve lop mos t e ffective

are incr e a s i n g l y used

In general,

are c o mposed of sle nder columns

shells,

which

are

stiff

in axial

or

flex ible in b e n d i n g defor mat ion s.
bers

le ading to the p r e s e n t - d a y

can

be ea sily d e f o r m e d

tions w i t h i n the range

of

to v a rious

ins t a b i l i t y phenome na.

jec ted

axia l or

to

in- plane

at fairl y low stress
mation s.
either

levels,

and

and

thin-walled

pla tes

states

strains,

behavior

usually

are

rota­

s u s c eptible

associated
or

calle d

has

load

limit

and

be en one

the d e v e l o p m e n t of

with

b r a n c h i n g of a

respectively .

afte r b u c k l i n g

for

finite

are

they o ften lose st ab ility

e q u i l i b r i u m load

b i f u r c a t i o n buckl ing ,

impo rta nt p r oblems

with

they

r e s u l t i n g in large b e n d i n g d e f o r ­

b u c k l i n g p r o b l e m to d etermine the critical
the ensuing

defo r m a t i o n s but

In fact, w h e n they are s u b ­

forces,

n e w e q u i l i b r i u m con fig uration, w h i c h
buckling

struc­

Sin ce these st ructural m e m ­

The loss of s tability is
an ex tr emal of the

almo st

lig h t - w e i g h t

in- plane

into

small

in

the

poin t

Thus,

the

to clarif y
of

the m o s t

lig h t - w e i g h t

struc­

tures .
It is quite d ifficult

to solve

the

foregoing buckling p r o b ­

lem through a direct a p p l i c a t i o n of the gene ral n o n l i n e a r
ry of elasticity.
interes t

is

On the other hand,

g e n erally

nite deform a t i o n

of

restricted

ela sti c beams,

to

theo­

the p r o b l e m of pract i c a l
c o m p a r a t i v e l y small f i ­

plat es

and

shells,

and for


NONLI NEAR

each

of

THEORY

OF C Y L I N D R I C A L

th e s e s t r u c t u r a l members,

SHELLS

lin e a r

3

b e n d in g

th e o rie s

have b e e n e s t a b l i s h e d f o r a p p r o x i m a te a n a ly s e s w i t h i n th e s m a l l d e­
f o r m a t i o n ra n g e

[9-15] .

Hence,

as t h e b a s i c

eq u a tio n s

fo r

the

b u c k l i n g p r o b le m , t h e c o r r e s p o n d i n g n o n l i n e a r t h e o r i e s have been
d e v e lo p e d ,

ta k in g

th e e f f e c t

fo rm atio n in t o

co n s id e ra tio n .

pro b lem s

16-20]

[10,

of

th e fo r e g o in g sm all f i n i t e

Based on t h e s e ,

ary c o n d itio n s .

Thro ug h t h e s e a n a l y s e s ,

th e o ry o f th e e l a s t i c
In

1945,

lin g

K o ite r

[25]

ory,

su bje cte d

th e

e n e rg y

of

on t h e c r i t i c a l
was

e la s tic

th e

s o -c a lle d

in itia l

b u c k lin g

c o n s e rv a tiv e

a sym p to tic a n a ly s is
th r o u g h

th e

w h ic h

e ffe c t

of

of

is

d e v e lo p e d

c o n t in u o u s

and

sm all i n i t i a l

as

th is

th e ­

the t o t a l p o t e n t i a l

re fin e d

system [ 2 6 - 2 8 ]

In

e la s tic

s y s te m a tic a lly

the i n i t i a l

lo a d a r e r e a s o n a b l y p r e d i c t e d .

fu rth e r

postbuck­

of

lo ad s.

th e b i f u r c a t i o n p o in t

t h e sys tem ,

b e h a v i o r as w e l l as

l o a d i n g and bound­

however , t h e g e n e r a l

b ifu rc a tio n

s ta tic

at

th e

the

[2 1-24]

had n o t been d u l y e x p l o r e d .

o rig in a te d

to

s ta b ility

c la r if ie d w ith

ory

s ta b ility

th e o ry c oncerning

bodies

numerous b u c k l i n g

as w e l l as p o s t b u c k l i n g pro b lem s

have been f o r m u l a t e d and s o lv e d u n d e r v a r i o u s

d e­

L a te r,

in

w e ll

p o stb u c k lin g
im p e rfe c tio n s
the t h e ­

connection

w ith

as d i s c r e t e

sys­

tem w i t h g e n e r a l i z e d c o o r d i n a t e s

[ 2 9 - 3 1 ] , w h ic h have been su c­

c e s s fu lly

in itia l

a p p lie d

to g e th e r w it h th e
e la s tic

In

to

c la rify

th e

im p e rfe c tio n

p o stb u c k lin g b e h av io r

s e n s itiv ity

of

a

v a rie ty

a d d itio n

t o th e a f o r e - m e n t i o n e d t r a d i t i o n a l b u c k l i n g p r o b ­

lems, we have t h e s t a b i l i t y p ro b le m s u n d e r n o n - c o n s e r v a t i v e
[3 2 ,3 3 ]

as w e l l as th o s e u n d e r v a r i o u s

F u rth e r,

th e

p ro b le m s

t i o n have a t t r a c t e d
t u r a l researchers
c o n tra s t to
w h ic h

of

system s.

th e

th e
s ta tic

a s s o c ia te d

in c re a s in g
in

v a rio u s

dynamic lo a d s

w ith

in te re s ts
in d u s tria l

fie ld s

p ro b le m s u n d e r c o n s e r v a t i v e
e n e rg y method

is

[3 4 ,3 5 ].

s o lid -flu id
re c e n tly

in te ra c ­

among
[3 6 ,3 7 ]

s ta tic

a p p lic a b le ,

lo ad s

s tru c ­
.

lo a d in g

In
fo r

th e s e p ro b le m s

s h o u ld be s o l v e d by e x a m in in g th e dynamic re s p o n s e o f t h e system
a fte r

th e a p p l i c a t i o n

of p e rtu rb a tio n ,

much more c o m p l i c a t e d .
th e s t a b i l i t y
c u ltie s ,

of

B es id es,

it

m o tio n p r o p e r l y .

lo n g -ra n g e in t e n s i v e

because o f t h e p r a c t i c a l

w h ic h makes t h e a n a l y s i s

i s more d i f f i c u l t
In

s p ite

of

to d e f i n e

these

d iffi­

s tu d ie s a re expected to c o n tin u e ,

i m p o r t a n c e o f th e s e p r o b le m s .


4

CHAPTER

The pu rp os e

of

t h i s book i s

1

to

c la rify

t h e b a s i c p ro b le m s c o n c e r n i n g th e e l a s t i c
la r c y lin d ric a l
w a lle d c i r c u l a r

c y lin d ric a l

on

th is

s h e ll

H ow ever,

owing t o i t s

w ith p h y s ic a l c o m p le x ity ,
and e x p e r i m e n t a l ,
of

of

s u b je c t sin ce

c o n s titu te s

s t r u c t u r a l e le m e n t most w i d e l y used

advent

s ta b ility

c irc u ­

s h e ll s under t y p i c a l lo a d in g c o n d itio n s .

ous r e s e a r c h e s have been made

tu res.

th e w h o le a s p e c t o f

in

a

th e

m a th e m a tic a l d i f f i c u l t y

h i g h speed com puters

and

both

o n ly

h ig h ly

th in -

fu n d a m e n t a l

the l i g h t - w e i g h t

accu rate r e s u lt s ,

have become a v a i l a b l e

Numer­

s tru c ­

to g eth er

th e o re tic a l

re c e n tly w ith
e la s tic

th e

t e s t m ate­

ria ls .
Because

of

space l i m i t a t i o n s ,

b u c k l i n g p ro b le m s
th e b u c k l i n g ,

under

s ta tic

p o s tb u c k lin g

we s h a l l d e a l w i t h

c o n s e rv a tiv e

and

in itia l

o n ly

th e

fo rc e s ,

th a t is ,

p o s tb u c k lin g

p ro b le m s

u n d e r one o f th e t h r e e f u n d a m e n t a l lo a d s as w e l l as th e b u c k l i n g
and p o s t b u c k l i n g p ro b le m s u n d e r t h e

in flu e n c e

combined lo a d s o r th e c o n t a i n e d l i q u i d .
on

th e

accu rate a n a ly s is

f o r th e b u c k l i n g p r o b le m ,
o r e t i c a l a n a ly s is
tio n

of

t h e ra n g e

f o r th e i n i t i a l
In

th is

lin e a r

fo r
of

th e o rie s

e ith e r

The emphases a r e p l a c e d

co m p reh e n s ive n u m e r i c a l

e x p e rim e n ta l v e r i f i c a t i o n

t h e p o s t b u c k l i n g p ro b le m
a p p lic a b ility

th e

re s u lts

o f th e t h e ­

and

c la rific a ­

o f t h e p e r t u r b a t i o n method

p o s t b u c k l i n g p r o b le m .

ch apter,

d e v e lo p e d by

and

of

of

we s h a l l b r i e f l y

e x p la in

c irc u la r c y lin d ric a l

D o n n e l l , F lü g g e

th e g o v e r n i n g e q u a t i o n s

fo r

and
th e

th e

s h e lls ,

t y p i c a l non­

th at is ,

S a n d e rs , w h ic h w i l l
e n s u in g

a n alyses

th o s e

p ro v id e

th r o u g h o u t

th e book.

1.2

DONNELL THEORY
D o n n e ll's n o n l i n e a r t h e o r y o f c i r c u l a r c y l i n d r i c a l

e s t a b l i s h e d by D o n n e l l i n

1933,

in conn ectio n w ith

o f to r s io n a l b u c k lin g of t h in - w a lle d
re la tiv e

tu b e s

[3 8 ].

s i m p l i c i t y and p r a c t i c a l a c c u ra c y , t h i s

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

d e s p ite

s h a ll

c ritic is m s

c o nsid er

concerning i t s

m o d erately la r g e

s h e l l s was

th e a n a ly s is
Owing t o i t s

t h e o r y has been

and

p o s tb u c k lin g

a p p lic a b ility .

d efo rm atio n

of

a c ir-


NONLI NEAR THEORY

c u la r c y lin d r ic a l
R,

le n g th L

and

s h e ll

w ith

Y o u n g 's
v.

surface

s h e ll,

system i s
and

th e

the

d is p l a c e m e n t

ly .

The

V

and
L

F ig .

1 .1 ,

X

components

w ill

0

and

D o n n e ll

E

m idd le

co o rd in a te

t a k e n as shown i n

be d en o ted by U,

e la s tic

th e

th e

is

y

The s h e l l

i . e . , h /R
(2 )

«

is

1,

s u ffic ie n tly

h /L

The s t r a i n s

«

2R

b ased on

th e f o l l o w i n g a s s u m p tio n s :
(1)

z

W, r e s p e c t i v e ­

th eo ry

5

ra d iu s

modules

A lo n g

SHELLS

w h ic h i s

is o tro p ic

P o i s s o n 's r a t i o
of

w ith

th ickness h,

made o f homogeneous,
m a te ria l

OF CY L I N D R I C A L

F i g . 1. 1 S h e l l geom etry
and c o o r d i n a t e system ,

th in ,

1.

ε are

s u ffic ie n tly

s m a ll,

ε «

1,

and H o o k e 's

la w h o l d s .
(3)

S tra ig h t

lin e s

re m a in s t r a i g h t
th e ir
(4 )

and

n o rm a l

to

n o rm a l t o

(5 )

in

in

D i s p la c e m e n t s U and V a r e

th e

surface

d i r e c t i o n n o rm a l t o t h e

i n co m p a ris o n w i t h

th e d i r e c t i o n p a r a l l e l

same o r d e r as t h e s h e l l
IWI

m idd le

l e n g t h unchanged.
The n o rm a l s t r e s s a c t i n g

m i d d l e s u r f a c e may be n e g l e c t e d
ac tin g

t h e undeform ed

th e d eform ed m i d d l e s u r f a c e w i t h

to the m id d le

th e

stresses

su rfa c e .

in f in it e s im a l, w h ile W is

th ick n e s s ,

th a t

is ,

|U|

«

h,

o f th e

|V|

«

h,

= 0 (h ) .

(6)

The d e r i v a t i v e s

of

W

are

s m a ll,

but th e ir

sq u a re s

and

p r o d u c t e s a r e o f t h e same o r d e r as t h e s t r a i n h e r e c o n s i d e r e d .
Hence,

(7)

C u r v a t u r e changes a r e

V are n e g l ig i b l e
tio n s

so t h a t

s m a l l and t h e i n f l u e n c e s

of

t h e y can be r e p r e s e n t e d by l i n e a r

U

and

fu n c­

o f W o n ly .

The a s s u m p tio n s (3) and (4) c o n s t i t u t e
-L o v e h y p o t h e s e s w h i l e
s h a llo w s h e l l

th o s e fr o m

(5)

the s o - c a ll e d K ir c h h o f f
to

a p p ro xim atio ns a p p lic a b le

(7)

correspond to the

f o r d efo rm atio ns

domi­

n a t e d by t h e n o rm a l d is p l a c e m e n t W.
Based upon t h e f o r e g o i n g a s s u m p tio n s ,

we have

th e

s tra in -


CHAPTER

6

1

displacement relations in the shell as

εχ = ε χ Ο + ζ κ χ ’

ey = e y O + Z K y ’

Υχγ = YxyO + z Kxy > d · 2 ·!)

where
£y0 = V >y - R- 1W + j W 2y ,

εχ0 = U ,x + I W ,2x>

(1.2.2)
^xyO = U ,y + V ,x + W ,xW ,y ·
KX = ~w ,xx>

Ky

= _ W ,yy>

Kx y = - 2 w ,xy·

(1 .2 .3 )

In the foregoing, subscripts following a comma stand for partial
differentiation.
Εεχ = σχ - VV

The stress-strain relations are given by
Ε ε γ

= σγ - ν σ χ ,

Ύχγ

= Τχγ ,

from which the stresses in the shell become

0χ _ ι - v 2 (εχ + v e y } ’

ay ~ ι- v 2 (£y + ν£χ) ’

Τχ? = 2(i+v) Y x y '
(1 .2.4)

Here we define the stress resultants and stress couples per unit
length, acting along the x = const, and y = const, sections, as
rh / 2

(Νχ , NXy , Qx) -


(σχ , TXy , τχ ) dz ,
J-h/2

J

y

Qy) “ |_h/2(Tyx’ °y’ Tyz)dz ’
fh/2

(Μχ, Μ

) =
J-h/2

(σχ , τχ ) zdz ,
J

fh/2
(MyX , Μγ) = j
(xyx , σy)zdz ,

(1.2.5)

which lead to
Nx

J (exQ +

Μ χ = ϋ ( κ χ + ν κ γ ),

^(£y0
My

νεχ0^ » ^xy

= D ( K y + V K x ) ,

^yx

2

^xyO *
(1.2.6)

Mx y = M y x = D . i ^ K xy .
(1.2.7)


NONLI NEAR

In

the

fo re go in g,

we h a v e

T “ 1-v
Eh 2 ’
w hich

stand

sh e ll,

THEORY

OF CY L I N D R I C A L

in tro d u ce d

7

SHELLS

the n o t a t i o n s

D = 7^7T^-2T<
12(l-v2) ’

for

the

e x te n sio n al

(1.2.8)
and

fle x u ra l

rig id itie s

of

the

re sp e c tiv e ly .

Now we s h a l l
tio n a l

d e rive

p rin c ip le .

The

the

b a sic

e la stic

e q u ation s

stra in

through

energy,

U e (U ,

V,

a v a ria ­
W ), w i l l

be g i v e n by
I fL ^
r ^RR rrh/2
h/2

e = 2 L L

J_

( σ * ε * + ay ey + Tx y Y x y ) dx dy dz

p
fL
Γ2πR fh / 2
E
(L(2wR(h/2
2 Ü - v 2T U o
J-h/2(C" + ^
w h ile ,

under

p o te n tia l

of

the

assu m ptio n

external

of

fo rce s,

ί _λ),
1-V

- + 2V£x£y + —
the

V f (U ,

- J ^ TTR[ P j U + P * V + P*W -

where ρ χ , p y and p a r e
d istrib u te d

P J , Py
M*

c o n se rv a tiv e

V,

lo a d in g ,

the

W ) , may b e e x p r e s s e d a s

y

J0*10

the

(1 .2 .9 )

r L r2πR
= - ( p x U + p V + pW) dx dy

Vf

of

Y x y )dxdy dz>

and

P*

are

the x,

forces
the

y and z com ponents,

per u n it

com ponents

is

the

e x t e r n a l b e n d i n g moment,

a lo n g

the

edges.

by

Π = Ue + V f .

of

the

total

v irtu a l

The

p o te n tia l

d isp la ce m e n t

co n stra in t

total

When t h e

a lo n g

area
of

energy

is

of

the

the

in

assum es

c o n s is t e n t w ith

(1.2.10)

re sp e c tiv e ly ,

sh e ll.

external

each per u n i t

p o te n tia l

sh e ll

the b o u n d a r ie s .

6Π = 6Ue + <5Vf

M *W ;X] ^ Q d y ,

energy

le n g th ,

n(U,

V,

e q u ilib riu m ,

W)

a sta tio n a ry

= 0,

in

the

ge o m e trical

(1 .2.11)

ν // ο2 < σ χ δ ε * + ay 6ey + Tx y 6Yx y ) d x dy dz
■'-h
+ Ny5£y0

+ Μ χ όκχ

give n

we h a v e

γL i ^ R rh/2

"

is

v a lu e

where

6Ue = •'O-'O
η L

w h ile

a p p lie d

the v a r i a t i o n

the p r e s c r i b e d

Thus,

Further,

lo a d s,

+

+ N x y 6yxy0

6ky + M x y 0Kx y ) dx d y ,


8

CHAPTER

1

|*L f ^ R
6 Vr

(p

6U + p

J oJ 0

6V + p6W) dx dy
y

f2^R
-J

With

the

help

_T

[P *6U + Py 6V + P*6W - M j6W ) X ]^~Q dy .

of

a Gauss’s theorem, the foregoing condition

leads to an equation in the following form:
(L (2ffR

j

[ L l 6U + L 2 6V + L 3 6W] dx dy

j

r2πR
+ j

Hence,

by

se ttin g

eq u ation s

Nx ,x

[B^<5U + B2 6V + B3 6W + B4 0W( X ] ^ = q dy = 0.

Li

= 0 (i

= 1,2,3),

we o b t a i n

the

(1 .2 .1 2 )

e q u ilib riu m

as

+ N x y , y + Px = °>
(1.2.13)

^ xy,x

^ N y ,y

Py

Mx , x x + 2Mx y , x y + My , y y + R ' ^ y
+ (NXW)X + NxyW>y) (X + (NxyW)X + NyW>y) >y + p = 0 ,
where

the

la st

e q u a t io n becomes

I.y W^v
..
D V ^ W - R ^ N y - N XW >XX - 2Nx y W >xy -■N
y
y

>yy

-

ρ + Ρ χ \ν>χ + P / >y=0,
(1.2.14)

in w hich

v 2 = a 2/ 3 x 2 + 3 2/ 9 y 2 .
Further,

the n a t u r a l boundary

(i= 1 ^ 4 ),
ro ta tio n
tio n s

a lo n g
are n ot

(1.2.15)
c o n d itio n s

th e b o u n d a r y where
sp e c ifie d .

Hence,

the

w ill

be g i v e n b y B i

d isp la c e m e n ts

and/or

a p p r o p r ia te boundary

= 0
the

c o n d i­

a l o n g x = 0 a n d x = L may b e g i v e n b y

Νχ = Ρ χ

or

U = U*,

N xy = Ρ- yί

or

V = V*,
(1 .2 .1 6 )

M x ,x + 2Mx y ; y + N xW> x + N xyW>y = P *
Mx = Mx

W >x = W*.

or

W = W* ,


NONLI NEAR

THEORY

OF C Y L I N D R I C A L

9

SHELLS

y
Fig. 1.2

Forces and moments acting on the shell element.

w h e r e U * , V * , W* a n d W*, r e s p e c t i v e l y ,
of

the

ary.
in g

d isp la ce m e n t
V a rio u s

components

boundary

one c o n d i t i o n

to be added t h a t

c o n d itio n s

fo re go in g

be d e r i v e d by e x a m in in g

te sim a l

s h e ll

in te n sity

as w e ll

m o m ents a c t i n g
Then,

the

y axes

as

the

on t h e

e q u ilib riu m

y ie ld

the

after

in

the

e q u ilib riu m

of

as

and
of

se le c t­

of

the

show n i n

It

is

(1.2.14)

an

We n o t i c e

d ire c tio n

ele m ent a r e

c o n d itio n s

the bou nd­

(1.2 .1 6 ).

(1.2.13)

d efo rm atio n .

p o sitiv e

sh e ll

a lo n g

e q u ation s

e q u ation s

can a l s o

ele m ent

the p r e s c r i b e d v a lu e s

can be c o n s t r u c t e d by

from each p a i r s
the

are

and the r o t a t i o n

in fin i­
that

the

forces

and

F ig .

t h e m o m ents a b o u t

1.2.

the x and

e x p re ssio n s

Qx (1 .2 .1 7 )
Qy = M x y . x
w h ile
w ith

those
these

of

+ My , y
the

The a f o r e s t a t e d
d itio n s

are

m o d e rate ly
e q u ation s

force s

e x p re ssio n s,

for

t h e x, y a n d
to

e q u ilib riu m

the r e q u ir e d
large

in

le a d

the

eq u ation s

D o n n e ll b a s ic

d eform ation s

of

z d ire c tio n s,

same r e s u l t s

and the b o u n d ary

eq u ation s

c y lin d r ic a l

a n a ly sin g n o n lin e a r

free

together

as before.

for

sh e lls.

v ib ra tio n s

of

con­

a n a ly sin g
The b a s i c
the

sh e ll

CHAPTER

10

are

give n

by

re p la c in g

p x , p y and p w it h

-phW t t , r e s p e c t i v e l y ,
t

is

tim e.

represent
tio n s
In
are

in

It
a

set

term s

is

where

p

to be n o t e d

of

is

the

that

- p h U > t t , - p h V >tt

d e n sity

the

three n o n lin e a r

o f U,

of

the

e q u ilib riu m

p a rtia l

and

sh e ll

and

e q u ation s

d iffe re n tia l

equa­

V a n d W.

c a s e when p x = p y = 0,

id e n tic a lly

1

sa tisfie d

both

w ith

e q u ilib riu m

the use

of

eq u ation s

the

stress

(1.2.13)

fu n ction

F

d e f in e d by
N
LNx = F
x ,yy ’
w h ile

the

N
λ y = F ,x x ’

fo llo w in g

re latio n s

N"xy = - F ,xy ’
w ill

(1 2

be o b t a i n e d

from

18)

(1.2 .6 ).

E h [ U )X + ( 1 / 2 ) W^x ] = Nx - v Ny = F >yy - v F > x x ,
E h [ V >y - R _1W + ( 1 / 2 ) W

]y ]

= Ny - v N x = F >xx - v F >yy ,

(1.2.19)

E h ( U >y + V jX + W ;X W >y ) = 2 ( l + v ) N xy = - 2 ( l + v ) F > x y .

E lim in a tin g
c o n d itio n

U and V

from th e s e ,

we o b t a i n

thec o m p a t i b i l i t y

as

V"*F + E h ( R " 1W >xx
xx - W2
χχ W , yy ) = 0 ,
jx y + W »xx
w h ile

the r e m a i n i n g

e q u ilib riu m

DV 4W - R
ix- 1F , x x - F ,y y W, x x

eq u ation

+ 2 F ,x y

W, x y

(1.2.20)

is

re w ritte n

- rF , x x vv,
W yy

as

- ^d = υ0.
·

(1.2.21)
Eq u atio ns
D o n n e ll's
w hich

(1 .2.20)
b a sic

and

seem t o b e m o re

the p r e c e d in g

ones.

when R b e c o m e s
the w e ll-k n o w n

(1.2.21)

eq u ation s

w ith

co n ve n ien t
F in a lly ,

in d e fin ite ly
Kärmän

co n stitu te

another

set

of

the

F

and

W,

two u nkn o w n f u n c t i o n s

it

la rg e ,

e q u ation s

in

p ra c tic a l

is

these
fo r

a p p lic a tio n s

to be n o t e d
e q u ation s

la rg e

that

in

reduce

d e fle c tio n s

of

than
case
to
th in

p la te s.

1.3

MODI FIED
D o n n e ll's

FLÜGGE THEORY
theory has

a d e fic ie n c y

s h e l l ap p ro x im a tio n th a t i s

in h e re n t

to

the

n o t a p p l i c a b l e to the a n a l y s i s

sh allo w
of

the


NONLI NEAR

THEORY

OF C Y L I N D R I C A L

d e f o r m a tio n s o f a c y l i n d e r in w hich
d isp la ce m e n t
ex am ple,

is

of

ben ding

the

d e rive d b a s ic

lin d ric a l

sh e lls

resort

the

to

a p p lic a b le
c lu d in g

sh a llo w

to

the

the

state,
m ain

E u le r

the

In

of

of

to

ce d ing

the

(1)

not

prob le m ,

adopted

to

re ta in

d e riv in g

term s

w ith

The r o t a t i o n s

product

and

is

in

squares

t o be

the

those

for

to

are
in ­

th in n e ss

orders

for

up

to

are m o d e rate ly
on t h e

com pres­

the

go ve rn in g
the

sense

of

F lü gge

the

stress

the

eq u ation s
b a sic

assu m ptio n s
e q u ation s.

stated

of

stress
W ith

n o n lin e a r

h y p o t h e s e s , we a s s u m e

ex p re ssio n s

in

a membrane

(4)

cy­

w ith o u t

e q u ation s

on t h e b a s i s

(1)

o th e r hand,

d efo rm atio n .

fore -stated

c ir­

c irc u la r

under a x ia l

d e rive

sh e ll,

the

c o n fig u ra tio n ,

accurate

accurate

we s h a l l

in -p la n e

[39],

These

sh e lls

assum ed

m ore

of

c o n d itio n s

o f b en ding

the a ssu m p tio n s
i.e .,

On t h e

any b u c k l i n g
lon g

e ffe ct

and K i r c h h o f f - L o v e
In

of

state

the

four.

lo a d in g

the

the d e f l e c t i o n , f o r

the b u c k l i n g

su ffic ie n tly

c y lin d ric a l

those

se c tio n ,

stra in s

than

of

of

c y lin d e r w ith

ap p ro x im a tio n .

o b tain in g

b u c k lin g

a d d itio n

(2)

are

p re b u c k lin g

eq u ation s
s im ila r

sh e ll

b u c k lin g

n e g le c tin g
object

for

ty p ica l

that

a lo n g

for

the p ro b le m w it h

s i o n . Howe ve r , t h e y
that

e q u ation s

under

as

of

c u m f e r e n t i a l w av e n u m b e r N l e s s
F lü gge

the m a gn itu d e

same o r d e r

deform ation s

11

SHELLS

in

the p r e ­

sh e ll,

the

sm all

fo llo w in g

re su lta n ts,

:

we

( h / R ) 2 from u n i t y .
sm all but

m id -su rface

the

stra in s

e ffe c t of th e ir
w ill

be c o n s i d ­

ered.
(3)

The c u r v a t u r e

a riz e d

e x p re ssio n s

The f o r e g o i n g
d e fo rm atio n s
We t a k e
F ig .

1.1

the

sh e ll

surface,

L e ttin g

the

U,

enough

to

a llo w

lin e ­

at

for

fin ite

le a st

b u c k lin g .

system

of

the

c y lin d e r

d isp la c e m e n t com ponents
V

and

surface

as
by

show n
U,

in

V a n d W,

W be the d i s p l a c e m e n t com pone nts o f

w hich

is

d ista n t

z

from

the m id d le

we h a v e

fin ite

e ra lly

the

sm all

seem t o b e v a l i d
after

c o o rd in a te

a lo n g

are

t h e b e n d i n g moment.

im m e d ia te ly

the

Ü = U - zW
For

fo r

a ssu m p tio n s

and denote

as before.

changes

yx

,

V =

d e fo rm atio n s,

expressed as

κ

V
the

- zW

*y

V) = W.

c o rre sp o n d in g

stra in s

(1 .3 .1 )
may b e g e n ­


CHAPTER

12

1

εχ = ö fX+ i (ο:χ + ^ χ + » ϊ χ).
'y -

Ä

? .y - K e + l (R ^ )Il0:y + (,. y - | ö), + (ä.yt 5',)· 1'

γ...
•xy. = v »χ„ + r ~- z ϋ

„ + ^ - [ 0 ν ΰ „ + ν „ ( t f „ - j i J ) + t . xT ( 8 »yv + ^
) l ·
R
»y
r-z
>x »y
>x
»y

R
(1 .3 .2 )

R e ta in in g
a lo n g

the

n o n lin e a r

the m id d le

surface

term s
of

εχ

= U ,x - zW,x x + 4 θ >

p

= v

y

-

»y

— zW
r-z

- ¥

o nly

sh e ll,

the

stra in

components

we h a v e

(1.3.3)

1 « + ε<20>.

»yy

, ,i + K

fo r

the

r-z

u .y - (1 + s h > z H ,*y + 4y0.

w here
ε<« = 4 (U 2
"x 0

2

+ V2

v u » x ^

-(2) == iI [ U 2 +
ε.(3?
'yO
2 1' >y
4y0

v , x

+W2 ) ,

^

(V

- R" 1W) 2 +

(W

= U , x U ,y + V . x ^ . y - R " ^ )

T he c o r r e s p o n d i n g s t r e s s e s
= ■=----- 5- ( ε γ + ν ε ν ) ,
1-ν
χ

σ

(1 .3 .4 )

+ R - 1V ) 2 ],

+ W, χ ( W;y + R ~ 1V ) .

are

y

=

----- 5"(ε__ + νε__),
1-ν2 y
χ

τ

Ιχγ

Ε

za+vf^y ’
(1.3.5)

w h ile
(Νχ

the

>

stre ss re su lta n ts
fh/2
Nx y ) =
<σχ> V
J-h/2
fh/2

( N y , N yx>

= l - h / 2
and

stress

(1 -

c o u p le s

are

d e f in e d by

f ) dz

Tyx)dz ,
(1 .3 .6 )

(Μχ, Mxy> = ( h / !
J-h/2

x

TxyMl- f ) z d z

fh/2

(My, My x } V
( σ ν> τ yx ) z d z
J-h/2
y
Performing integration, we finally obtain


NONLI NEAR

THEORY

OF CY L I N D R I C A L

13

SHELLS

N x = J [ U >x + v ( V >y - R " ^ ) + ε ^ ο ) + ν ε ^ ο )] + R - 1 DW ) X X ,
N y = J [ V >y - R - 1W +

vU

(X + £ ^20) +

N xy = i ^ f J d J . y + V ^ + Y ^ )

Nyx

Mx

=

=

^

[

v

£ ^20)] - R _1D ( W j y y + R - 2W) ,

+ R - 1D ( R - 1V > x + W ( X y ) ] (

J ( U >y + V ) x + Y x 2y O ) + R ' l D ( R ‘ 1 U , y - W ) x y ) i >

-D IW .xx + ^ . y y + R - ^ U ^

My = -D(W(yy + R ' 2W + VW >XX)
MX y = - ( l - v ) D ( W

+ W . y ) ] ,

,

X y + R - 1V X ) ,

Myx = - ( l - v ) D [ W > x y + ( 1 / 2 R ) ( V >X - U f y ) ] ,

where J and D h a ve been
Th e e q u i l i b r i u m
d itio n s

w ill

ple

of

the

the

e la stic

d e fin e d by

eq u ation s

be o b t a i n e d w i t h

total

p o te n tia l

stra in

(1 .2 .8 ).

and the

2 J0j 0

of

the

boundary

sta tio n a ry

as before.

con­

p rin c i­

The v a r i a t i o n

of

e n e r g y U0 i s

rL
|zttR
^ R ffhh/
/2
1 f
Lf
Z
6Ue

a p p ro p ria te

the u se

energy

(1 .3 .7 )

J _ h / 2 ( a x <5£x + c' y
2
Txy6Yxy)(1 ' r } dx dy dz ’
8)

w h ile

that

of

the

p o te n tia l

of

the

external

fo rce s

Vf

may b e

expressed as

6Vf

= " 0 ^ 7ΓΚί ρ χ δυ + ρ γ ,5ν + ρ [ ' 1ί, χ δυ '
+

(1 + U (X + V

( W > y + R " 1V ) S V

- R - 1 W)6W] } d x dy

- | 27TR[P *6U + P*<5V + P*SW - M^6W( X ] * ” q d y ,
where
the

p x and p y , r e s p e c t i v e l y ,

d istrib u te d

m id d le

su rfa c e w h ile

a c t in g norm al
area.
ponents

force

to

Further,
of

the

the

a p p lie d
p is

the

are

in te n sity

deform ed m id d le

P * , P* , P* and Μ * ,
external

the

per u n it

(1 .3 .9 )

x and y
area

of

of
the

surface

com ponents o f
the

undeform ed

la te ra l

per u n it

re sp e c tiv e ly ,

are

pressure
deform ed
t h e com ­

l o a d a n d b e n d i n g moment a p p l i e d p e r u n i t


CHAPTER

14

o rigin a l

le n gth

of

the v a r i a t i o n a l

sh e ll

edges x = 0 and x = L.

Then,

from

p rin c ip le

6Ue +
we f i n a l l y

the

1

6 Vf

o b tain

= 0,

the

(1.3.10)

e q u ilib riu m

eq u ation s

as

fo llo w s:

[Nx ( l + U >x) ] > x + [ N y x ( l + U >x) ] >y + <NyU >y) >y + (N xy U >y) >x
+ P x - PW )X = 0,

( 1. 3 . 11a)

[Nx y ( l + V >y - R- 1 W) ] >x + [Ny ( l + V >y - R " l W)] >y
- R " 1 ( Μ y>y + Μ xy ,x ) + v( Ν x V , x 7
) , x + v( Ν y x V , x ') , y - R- 1 N yxW ,x
+ Py -

(p + R_1Ny ) ( W >y + R_1V)

Μ χ , χ χ + v( Μ x y + My x ' ), xy

+ M
y>yy

= 0,

(1 .3 .1 1 b )

+ R- 1 Ny ( 1 + V >y - R _1W)

+ [NXW)X + N x y (W>y + R ” 1V) ] >x + [NyxW>x + N y (W)Y + R“ 1V) ] >y
+ R- 1 NyxV ;X + p ( l + U > X + V >y - R_1W) = 0.

(1 .3 .1 1 c )

The a p p r o p r i a t e b o un d ary c o n d i t i o n s a t x = 0 and x = L a r e a l s o
o b t a i n e d as
N x < 1 + U ,x>

+

K

Nx y u , y =

°r

U = U* ’

Nx y ( l + V (Y - R_1W) + NXV >X - R_1Mxy = P * o r
Mx , x + ( Μ χ γ + Μ γ χ } ^

+ NXW;X + Nxy (W (y + R " 1V)

Μ χ

= M*

In th e fo re g o in g ,

U * , V*,

scrib e d

the

a lo n g

va lu e s

of

the b o u ndary.

are

the m o d i f i e d

the

c y lin d r ic a l

n o n lin e a r

co rre sp o n d in g
tin g

(1.3 .7 ).

In

th is

b a sic

term s
case,

fo r

in
the

a

eq uation s

eq u ation s w i l l
the

fin ite
set

of

in

U,

th e p r e -

the r o t a t i o n
(1.3.12)

defo rm atio n
three

(1 .3 .3 )

eq uation s

T he
om it­

as w e ll

bec om e

of

co u p le d

V a n d W.

be o b t a i n e d b y

ex p re ssio n s

e q u ilib riu m

are

and

to g e th e r w ith

the

w hich r e p r e s e n t

d iffe re n tia l

lin e a r

the n o n l i n e a r

components

(1 .3.11)

e q u ation s

W = W* ,
(1 .3 .1 2 )

W* and W * , r e s p e c t i v e l y ,

Eq u atio n s

F lü g ge

= P* or

W)X = W*.

d isp lace m e n t

sh e ll,

p a rtia l

or

V = V *,

as


NONLI NEAR

THEORY

OF CY L I N D R I C A L

SHELLS

15

Nx , x + Ny x , y + Px = 0,

xy,x + N y,y “ R

^ x y ^ ^yx^ ,xy ^ ^y,yy + & 1^y + P = 0 ,

x,xx
w h ile

the b o u n d a ry

c o n d itio n s

or
N

(1.3.13)

0»

+ Py

^ x y , x + M y,y^

at

x = 0 and x = L a re

give n

by

U = U *,

V = V*,

xy - R _1M xy = P*y

(1.3.14)
Μ

+

M

1 .4

(Μ

)

= M*

or

W = W*,

W ,x = Wx ‘

SANDERS THEORY
In

th is

se ctio n ,

for fin it e
to

+ M

the

and m o d e r a t e ly
on

W ith

rather

the

re la tio n s

of

sm all

d e riv in g

re aso n in gs,

εχ

c y lin d ric a l

d e fo rm a tio n s

p laced

present

deform ations o f th in s h e l l s

c irc u la r

fin ite

we s h a l l

ro ta tio n s
sim p lifie d

than

the

are

in

w hich

is

sh e lls

w ith

e q u ation s

theory

sp e c ia liz e d

the p r e c e d in g

co n sid e re d but

b a sic

exact

S a n d e rs-K o ite r

[40],

As

n o n -sh allo w

same n o t a t i o n

are

sh e ll.

the

sm all

se ctio n ,
stra in s

emphases

through

are

ra tio n a l

ones.

as before,

the

stra in -d isp la c e m e n t

assum ed as

= εχ0 + ζκ>

£y = £y O + z V

^xy = Ύχγ0 + Z K xy.

where

'xO = U , x + i w : x + τ


£ y0 = V ,y

,y

YxyO = U ,;

κ, = -W

+ h ν)2 + I
r

(u

“ V,x)

(1.4.2)

+ W ,x(W ,y + S V >>
Ky

( W, yy + f V >y)
(1 .4 .3 )

cxy

2 [ W>xy +

U >y) ] .


CHAPTER

16

The c o r r e s p o n d i n g

σχ

stress

= Τ ^ Γ ( ε χ + ν ε γ ),

1

com ponents a re

ay = I ^ T ( £y + v e x ) (

τχ γ =

^

+ ν ) γ*
Ι χ γ ,
(1.4.4)

w h ile

the

stress

re su lta n ts

and

stress

co u p le s

are

d e f in e d by

ff h
h// 2

(Νχ>
Qx) “=
''x » "Nxv,
x y » ^x^

;ο(σχ , τχγ, τχζ) dz ,

-h/2

h/2

-I!

(Ν
)
v yx *, N y *, Q
xy7

( T y X , a y , Ty z ) dz ,
h/2
(1 .4.5)

h/2

(Μχ, Mxy) = 1

(σχ , xxy) zdz ,
y
J-h/2
rh/2
(My x ’ My )
w hich

le a d

= j h / 2 ( Tyx»

ay) zdz »

to

N x “ J ^£ x O + V E :yO^»

Ny _ J ( £y 0 + V e x0)»
(1 .4.6a)

XT

XT

T

xy - ^ y x

~ ^* 2

Μχ = D(kx + νκγ) ,

^xyO »

My = D(Ky +

VKx ) ,

( 1 . 4 . 6 b)

Μ

= M

xy

yx

= D *·ί~- ir
2

’

where

J = Eh/(1 -v 2) ,
W ith
stra in

from

fo re go in g

energy

g ive n
that

the

by
is,

the

the

x »x

Ue

same e x p r e s s i o n s
(1.2 .9 )

sta tio n a ry

+ N

Τ

the

x y ,y

4

[(V

,χ

and

p rin c ip le

e q u ilib riu m

2R
- U

) (N

x

as

+

that

of external

those in

the e l a s t i c

forces

the

of

the t o t a l

e q u ation s w i l l

N ) ]
+
y
,y

—

p

Vf

are

D o n n e lltheory,

(1 .2 .1 0 ), r e s p e c t iv e ly .

+

xy >y

>y

(1 .4 .6 c)

e x p r e s s i o n s , we a s s u m e

and the p o t e n t i a l

e q u ation s

Π = U0 + V f ,
Ν

D = Eh3/ 1 2 ( l - v 2) .

Then,

p o t e n t ia l energy,
be

ob tain e d as

M

=

0

,

(1.4.7a)


NONLI NEAR

N

+ N

-

y,y

—

2R

M

THEORY

OF CY L I N D R I C A L

SHELLS

17

xy > x

(1.4.7b)

(1.4.7c)

Further,

the bou nd ary

c o n d itio n s

or

It

Kx

= M X*

is

to be added

o b tain e d
the

x = 0 a n d x = L be co m e a s

U = U*

(1 .4 .8 )

or

from

sh e ll

at

that

the

the

e x p re ssio n s

e q u ilib riu m

for

c o n d itio n s

Qx a n d Qy

of

w ill

be

th e moments a b o u t

ele m ent a s
(1 .4 .9 )

Eq u atio n s

(1 .4.7)

tio n s

fin ite

for

w hich w i l l
D o n n e ll

be

to

theory

is

structural
The

to be

but

to

a n a ly sis

lin e a riz e d

fin d

u sin g

sim p le r

the n o n l i n e a r
(1 .4.6a).

In

than

m ak e s

in

than
th at

it

fin ite

case,

Sanders

that

of

equa­
sh e ll,

of

d ire c tly

a p p lica b le

c o n fig u ra tio n ,

future,

the

the m o d ifie d

e sp e c ia lly

the
in

e le m e n t method.

Sanders

term s o f the
th is

the

of

c y lin d ric a l

any g e o m e tric

v e r s i o n o f the

(1.4.2)
bec om e

a c irc u la r

favo u r

the

a set

more c o m p le x

w ith

by o m i t t i n g
and

of

ge n e ra lity

sh e lls

lik e ly

represent

much

so m e w h a t

S in ce i t s

n o n -sh a llo w

(1 .4.8)

d eform ation s

seen

theory

F lü gge the ory.

and

theory w i l l

d isp lace m e n t
the

be o b t a i n e d
in

eq uation s

e q u ilib riu m

eq u ation s