Elastic Instability of Pressurized Cylindrical Shells Under Compression or Bending

7/27/2019 Elastic Instability of Pressurized Cylindrical Shells Under Compression or Bending

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at the

University of Florida

V ol . X V I I , No. 6 N 6 3 1 8 ; c 8 5 J u n e , 196

&%eTECHNICAL PAPER NO. 263

Elastic Instability of Pressurized Cylindrical Shells

Under Compression or Bending

bY

S. Y. Lu and W. A. N a s h

Advanced Engineering Mechanics Section

University of Florida

R e p r i n t e d f r o m P roc e e d ings of he Fourth U . S . N a t i o n a l C o n g r e s s of A p p l i e d

M e c h a n i c s , V o l . 1, h e l d at t h e U n i v e r s i t y of C a l i f o r n i a J u n e 1 8 - 21 , 1 9 6 2 .

C o p y r i g h t @ 1 9 6 2 by t h e A m e r i c a n S o c i e t y of M e c h a n i c a l E n g i n e e r s .

P u b l i s h e d m o n t hl y b y t h e

FLORIDA ENGINEERING AND INDUSTRIAL EXPERIMENT STATION 33‘’ 3‘’Lj

COL LEG E OF ENGINEERING UNIVERSITY OF FLORIDA GAINESVILLE

E n t e re d a s s e c o n d - c l a s s m a tt er at t h e P o s t O f f i c e a t C a i n e s v i l l e , F l o r id a

lJ &A c ; P - S o )



The elastic stability of circular cylindrical shells subject toeither axial compression or pure bending is investigated for thecase of cylinders strengthen ed by internal pressure. Nonlinearfinite deflection theory is employed an d an ap proxim ate solutionof the equilibrium and compatibility equations is obtained bym e of Gslerkin ’s metho d. Comparatively simple expressions arepresented for the various buckling stresses and these expressionsare evaluated to yield buckling stresses 89 a func tion of t he in-terna l pressure. The critical stress fo r bending is found to begreater han that in axial compression, and in approximately the

rat io indicated by recent experimental evidence.

N o m e n c l a t u r e

F l e x u r a l r i g i d i ty E t 3 / 1 2 ( l - 2 )

Young’s modu lus

4 i r y s t r e s s f u n c t i o n

R a d i u s of m i d d l e s u r f a c e o f s h e l l

Yurnber of w a v e s i n a x i a l a n d c i r c u m f e r e n t i a l

d i r e c t i o n s r e s p e c t i v e l y

I n t e r n a l p r e s s u r e

Tall t h i c k n e s s o f s h e l l

Radia l d e f l e c t i o n

C o - o r d i n a t e s o f a p o i n t i n t h e m i d d l e s u r f a c e o f

th e s h e l l , m e a s u r e d i n t h e l o n g i t u d i n a l a n d

c i r c u m f e r e n t i a l d i r e c t i o n s r e s p e c t i v e 1 y .

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P o i s s o n ’ s r a t i o , v =0.3 i n p r e s e n t s t u d y

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‘The present invrst igat ion was sponsored by the National Aero-

nau tic s and Space Administration under Res earch Grant NsG-16-59.

EPRDDUCED BY

NATIONAL TECHNICAL 1INFORMATION SERVICE !

s. Y. Lu

W. A. Nash

University o f Florida

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

The p o s t b u c k l i n g b e h a v io r of c y l i n d r i c a l shells s u b j e c t

t o a x i a l c o m p r e s s i on has b e e n s t u d i e d by s e v e r a l

i n v e s t i g a t o r s .

f i n i t e d e f l e c t i o n e q u a t i o n s ; l a t e r , i n 1 9 4 1 , t h e s e

e q u a t i o n s w e r e u s e d b y v on K C m cin a n d T s i e n [21 to

o b t a i n a n a p p r o x i m a t e s o l u t i o n to t h e p r o b le m of b u c k l i n g

of a n a x i a l l y c o m p r e s s e d c y l i n d er into a d i a m o n d s h a p e d

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

K e m p n e r E31 w h o u s ed a n a d d i t i o n a l p a r a m e t er i n the

b u c k l i n g d e f l e c t i o n f u n c t i o n p r o p o s e d i n [ 2 ] . S e v e r a l

v a r i a t i o n s of t h e s e analyses h a v e b e e n p r o p o s e d by other

i n v e s t i g a t o r s .

T h e i n c r e a s e i n s t a b i l i t y of i n t e rn a l l y p r e s s u r i z e d

c y l i n d r i c a l shells s u b j e c t t o a x is y m m e r i c l o a d i n g w a s

s t u d i e d by Lo, C r a t e , a n d S c h w a r t z [43. T h ey u s e d l a r g e-

d e f l e c t i o n t h e o r y a n d f o u n d t h a t th e c r it ic a l s t r e s s

i n c r e a s e s f ro m a v a l u e of 0.37 E t / R a t z e r o p r e s s u r e to

0.606 E t / R ( i .e . , t he va lue g ive n by c l a s s i c a l small

d e f l e c t i o n t h e o r y ) a s the p r e s s u re i n c r e a s e s t o

0 . 2 E t 2 / R 2 . V e r y r e c e n t l y , the e f fe c t o f in t e rna l

p r e s s u r i z a t i o n o n s t a b i l i t y o f a x i a l l y c o m p r e s s e d c y l i n -

d e r s w a s s t u d i ed by T h i e l e m a n n [SI. I n a d d i t i o n t o

p r e s e n t i n g a f i n i t e d e f l e c t i o n t h e o ry , he also c o n d u c t e d

t e s t s o n a lu m i nu m she l l s .

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e n e r g y c r i t e r i o n .

S e i d e [8l ha s r e c e n tl y p r e s e n t e d a l i n e a r small deflec-

t io n a n a l y s i s of t h e b u c k l i n g of c y l i n d r i c a l s h e l l s s u b j e c t

t o p u r e b e n d i n g . This s t u d y i n d i c a t e d t h a t , c o n t r ar y t o

t h e c o m m o n l y a c c e p t e d v a l u e , t h e ma ximum c r i t i c a l

Donne11 El1 f i r s t d e r i v e d t h e g o v e r n i n g

411 t h e a f o c e m e n t io n e d

197

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U.S. DEPARTMENT DF C O M M ~ R C E ISPRINGFIELD, VA. 22161



b e n di n g s t r e s s i s fo r all p r a c t i c a l p u r p o s e s e q u a l t o t he

c r i t i c a l s t r e s s fo u n d f o r a x i a l c o m p r e ss i o n . T h i s r e s u l t ,

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

e x p l a n a t i o n o f t h e e x p e r im e n t a l d i f f e r e n c e s kn o w n t o

e x i s t f o r t h e s e t w o s i t u at i o n s . For e x a m p l e , e x p e r i m e n t a l

e v i d e n c e d u e t o S u e r , H a r r is , S k e n e , a n d B e n j a m i n 191

i n d i c a t e s b u c k l in g l o a d s i n b e n d i n g t o b e f ro m 25 t o 60

p e r c e n t g r e a t e r t h a n i n c o m p r e s s io n , t h e e x a c t v a l u e

d e p e n d i n g u p o n t h e r a t i o R / t .

h a v i o r o f t h i n p r e s s u r i z e d c y l i n d e r s s u b j e c t to b e nd i n g

l o a d s . T h r o u g h o u t t h i s a n a l y s i s , t h e G a l e r k i n m e th o d i s

e m p l o y e d .

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s h e l l s s u b j e c t t o a x i s ym m e t r i c c o m p r e s si o n i s r e a c h e d

f i rs t . F o r t h i s c a s e , w h en t h e p r e s s u r e p a r a m e t e r

p R 2 / E t 2 a p p r o a c h e s u n i ty , i t i s f o un d t h a t t h e s o l u t i o n

i s t h e s a m e as t h e c l a s s i c a l s m a l l -d e f l ec t i o n s o l u t i o n .

T h e r e l a t i o n o f t h e c r i t i c a l s t r e s s t o i n t e r n al p r e s s u r e

h a s b e e n f o un d . F o r t h i s p u r p o s e i t i s c o n v e n i e n t to

i n t r o d u c e as a p a r a m e t e r t h e r a t i o b e t w e e n t h e i n c r e m e n t

of c r i t ic a l s t r e s s a n d t h e c r i ti c a l s t r e s s a t z e r o p r e s su r e .

T h i s p a r am e t e r w i l l b e e s s e n t i a l l y i n d e p e n d e n t of t h e

i m p e r f e c t i o n s in t h e s h e l l w h e n t h e i m p e r f e c t io n s d o n o t

v a r y s i g n i f i c a n t l y d u e t o c h a n g e s in p r e s s u r e . F i n a l l y ,

e x p e r im e n t a l d a t a d u e t o S u e r , H a r r i s, S k e n e , a n d

B e n j a m i n [91 a r e c o m p a re d w i th t h e r e s u l t s of t h e p r e s e n t

a n a l y s i s .

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For c o m p a ri s on w i t h c e r t a i n e x i s t i n g r e s u l t s

B a s i c E q u a ti o n s a n d D e f l ec t i o n F u n c t i o n

For a n i n i t i a l l y p e r f e c t t h in c y l i n d r i c a l s h e l l t h e c o m -

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

r e s p e c t i v e l y , a s

(1)

aZw aZW 1 aZwV 4 F - E [ ( e r -dr~z

axas

a 2 F a Z W + a 2 F a z : ]

+ p = o (2 )

- 2 - -2 F aZwD V 4 w - - -a r 2 axas axas a x 2 as

In t h e a b o v e e q u a t i o n s , F i s t h e 4 i ry s t r e s s f un c ti o n of

t h e m e m br an e s t r e s s e s , w i s t h e r a d i a l d e fl e c t io n , t th e

s h e l l t h i c k n e s s , R t h e r a d i u s of t h e m i d d l e s u r f a c e , and

p i s i n t e r n al p r e s s u r e ( t ak e n t o b e p o s i t i v e ).

a s s u m e d :

4 n a p p r o x i m a t e f o r m of t h e d e f l e c t i o n p a t t e r n i s

m x ns 2rnr

R R Rw = b , + c o s Y - ~ C O S - C O S - + ~ ~ C O S - +

(3 )

0 fo r a s h e l l s u b j e c t t o a x i a l c o m p r e s s i o n

w h e r e k = 1 1 fo r a s h e l l s u b j e c t to e c c e n t r i c c o m p r es -

s i o n o r p u r e b e n d i n g

Y = e v e n i n t e g e r

a n d rn a n d n r e p r e s e n t the n u m b e r o f w a v e s i n t h e a x i a l

a n d c ir c u m f e r en t i a l d i r e c t i o n s r e s p e c t i v e l y , t h e n um b e r

of w a v e s i n t h e a x i a l d i r e c t i o n b e i n g w i th i n a l e n g t h

e q u a l t o t h e c i r c u m f e r e n c e of t h e c y l i n d e r . T h r o u g h o u t

t h e p r e s e n t s t u d y , y =2 . F o r m o re l o c a l i z e d b u c k l in g , a

l a r g e r v a l u e o f y c o u l d be u s e d . T h e n k =0, (3 ) i s t he

same as t h a t u s e d i n [ 2 ] . In (3) , b i s n o t a n i n d ep e n d en

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

p e r i o d i c i t y of c i r cu m f e r e n ti a l d i s p l a c e m e n t b l .

C o r r e s p o n d i n g t o (3 ) , a n e x p r e s s i o n fo r t h e s t r e s sf u n c t i o n F i s p r op o s ed

0 2 s 1 P R rnx n sF = - " s 2 + u b R c o s - +- - 2 + a l l c o s - c o s -

2 R 2 t R R

2mx 2ns 2rnx 2n sc o s- os - +az o o s- ao 2 OS-

R R R R(4

T h e s t r e s s e s a n d ob a r e d u e t o a x i a l c o m p r e s si o n a n d

b e n d i n g , r e s p e c t i v e l y , a n d a r e p o s i t i v e f o r co m p r e s s io n .

F o r s h e l l s s u b j e c t t o a x is y m m e t r i c c o m p r e s si o n o n l y ,

ab=0 a n d uc i s r e p l a c e d by u t o a v o i d a n y p o s s i b l e

c o n f u s i o n i n n o t a t i o n .

M e t h o d o f S o l u t i o n

V h e n w a n d F i n (3) a n d (4 ) a r e s u b s t i t u t e d i n (1) a n d

( 21 , t h e e q u a l i t i e s g e n e r a l l y w i l l n o t h o l d. T h e y c a n ,

h o w ev e r, b e e x p r e s s e d i n s t e a d as

aZw a z w 1 a 2wV4F-,F[(*r xa s --r 2 ----]=Q,s 2 R a x 2 (5

a n d

D V 2 w - - - - t -- 2 - - +--

R a r 2 a s 2 a x 2 a x a s drds a r 2 a s 2

+ p = Q 2 (6

4 n a p p r o x i m a t e s o l u t i o n i s o b t a i n e d b y m i n i m i zi n g Q ,

and Q2 on the r i gh t - ha n d s i d e s of t h e a b o v e e q u a t i o n s ;

t h i s i s d o n e by t h e G a l e r k in m e t h o d .

e q u a t i o n s :

Th e G a l e r k in m e t h o d e s t a b l i s h e s t h e f o l lo w i n g s e t of

rnr ns

R RQ, COS- C O S - ~ S d x =0

0

(

0

2rnx 2ns

lLZTR, COS R C OS -d s d r =0

2m1 nR Q , COS d s d x =0 .

2nsQ, CO S - d s d x =0

698



rmc n s ksJ L J 2 1 r R Q2 COS- os- os ( g ) d s d x =0

R R

2 m x +cos %)cos ' ( -$)ds d z =05

( (8 )0 0

IL[ 2 n R Q2 ( c o syR

\gain, k =0 w h e n the shell is subject to axial

compression only. In the following sections, solutions

are obtained for axial compression and bending separately,

although the approaches are the same.

A x i s y m m e t i i c C o m p r es s i on

In this section, the parameter k in (3) and ( 8 ) and a lso

ob i n (4) are zero. Also, 0, in (4) i s replaced by U . The

coefficients u 2 0 , u O 2 , , , nd u 22appearing in (4) can

be expressed in terms of 6 2 and b S through the four

integrals of (7). The relations are found to be

(9)

In the above expressions, the dimensionless parameters

are defined so that

n 2p =- - (10)

(11)

The ratio n / m evidently represents the wave-length ratio

in axial/circumferential directions.

The integration of (8) together with the relations given

in (9) eads to the following two equations:

a+=1 8

(14)

equations i s a nondimensional stress parameter defined

by the relation

U R pR 2+=-- P-

E t E t 2

For brevity, (13) and (14) may be rewritten as:

(15)

a + = A , + ( A 2 + A S q + A 4 q 2 ) a 2 +A , -b;

S

where

(1+/d2

12 (1- 2 )

(I+/d2

A , =

1A , =

1 /L 2

A , =

16

1B 2 =-

4

The parameter ( b , / t ) is eliminated between (13a) and

(14a) and the stress parameter +is expressed as

(16)

+C 2 A 2 a (17), A ,+=

a

In the above equation,

(18)

The function +on the left-hand sides of the above

699

3



4 s c a n be s e e n f ro m (15) t o (19) t h e d i m e n s i o n l e s s

v a r i a b l e u R / E c i s a f u n c t i o n o f a, 7, n d p . Theb u c k l i n g s t r e s s i s t h u s o b t a i n e d t h ro u g h m i n i m i za t io n

w i th r e s p e c t t o t h e p a r a m e t e r s a, 7, n d p . Dif fe re n t i a t ion

of +w i th r e s p e c t t o a i s c a r r i e d o u t f i r s t to o b t a i n

p = o 0.2 0 .2 5 0. 5 1 o

I = 0.59 0 . 5 3 0 .3 2 5 0.18

Sa,.,, 0 .6 0 5 0.44 0.406 0 .2 9 0 .1 9

a+ au

a a a ,- - = o

fo r p = c o n s t a n t. T h u s f ro m (17):

1.15

0.11

0 .1 6 1

(20)

(21)

a n d f i n a l l y f r o m (17) wi th a g i v e n by (21)--a =2dAI A2 dCI c 2 (22)

Th e n o t a t i o n +a t h u s d e n o t e s t h e v a l u e o f Q m i n i m i z e d

w i th r e s p e c t t o a. Th e e x p r e s s i o n s f o r c, a n d c, arefound f rom (18), (19), a nd (16 ) . F rom (16):

I .

p a r a m e t e r s

- uR

E t

0 =-

a n d

- p R Z

p=zT h e n (15) m a y be r e w r i t t e n as

(23)

(24)

(25)

w h e r e Ca r e p r e s e n t s t h e m i n i m i z e d v a l u e o f w i t h r e -,?

700

s p e c t t o a a n d JI.

s t r e s s zcr a t a g i v e n v a l u e of d i m e n s i o n l e s s p r e s s u r e ps e v e r a l d i f f e r e n t v a l u e s o f p a r e t r i e d in (25) t o g e t h e r

w i th c o r r e s p o n d i n g v a l u e s o f

r i g h t- h a n d s i d e o f (25) is m i n i m i z e d . T h e v a l u e o f p, a

w h i c h za T is m i ni m um a n d e q u a l s Zcr, i s c a l l e d per.

S o m e v a l u e s of ccr n d pcr f o r v a r i o u s v a l u e s of p a r e

g i v e n i n Table 2.

To f in d t h e d i m e n s i o n l e s s c r i t i c al

,? f r o m T a b l e 1 u n t i l t

T A B L E 2

It c a n b e s e e n t h a t p cr d e c r e a s e s w i th i n c r e a s i n g p. T

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

C u r v e 11. The r e s u l t s f r o m [.?.I a n d 151 a n d t h e t e s t d a t a

from [61 a n d [71 a r e p l o t t e d i n t h i s f i g u r e a l s o . The

b r o k en c u r v e s h o w n t h e re r e p r e s e n t s t h e c u r v e b e s t

f i t ti n g t h e d a t a i n 151, [61, a n d (71. The p r e d i c t i o n s of

t h e p r e s e n t t h e o r y a r e s h o w n a s C u r v e I1 i n F i g . 2 s o a

t o a f fo rd a c o m p a r i s o n w i t h e x p e r i m e n t a l d a t a g i v e n i n

T h e r e l a t i o n b e t w e e n Ocr a n d i s s h o w n in F i g . 1 a

191.

In Fig . 1 , t h e b r o k e n c u r v e i n d i c a t e s that O,, is o n l

a b o u t 0.09 a t

a p p r o x i m a t e l y 0.35, t h e n l e v e l s off a t h i g h e r v a l u e s of

p.of 0.606 w h e n t h e v a l u e of T; is r e l a t i v e l y h i g h . one o

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

by t h e b r ok e n c u r v e c o r r e s p o n d i n g to t e s t d a t a l i e s in

i n i t i a l i m p e r f e c t io n s w h i c h i n g e n e r a l i n c r e a s e w i t h

i n c r e a s i n g R / t . If t h e i m p e r f e c t i o n f a c t o r d o e s n o t v a r

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

u,, a t t w o d if f e re n t p r e s s u r e s w i l l b e n e a r l y i n d e p e n d e

of t h e e f f e c t o f i m p e r f e c t i o n s . I , e t %,, r e p r e s e n t Ocr a

1- _ - _ -

= 0, a n d FCr n c r e a s e s f rom 0.09 t o

- I I o w e v e r , Ccr is e x p e c t e d t o r e a c h t h e c l a s s i c a l v a l

-

p = , a n d

flcr - ucr flu,,

T h e r a t i o , , s p l o t t ed a g a i n s t i n F i g . 3.

s h o w n i n t h i s f i g u r e , t h e p r e d i c t i o n s of t h e p r e s e n t t h e

a r e i n r e a s o n a b l y g o o d a g r e e m e n t w i th e x p e r i m e n t a l

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

F i g . 1. O n e a d v a n t a g e o f i n t r o d u ci n g t h e r a t i o

>GcrEcr,, is t h a t t h e r e l a ti o n i n F i g . 3 c a n b e u s e d

h a v i n g t e s t d a t a a t o n ly o n e p r e s s u r e t o p r e d i c t t h e

c r i t ic a l s t r e s s i n t h e s a m e i m p er f ec t s h e l l a t a n y o t h e

p r e s s u r e . F o r i n s t a n c e , a t e s t i s m ad e a t p =0.4, fo r



b RE t

0.6

0.5

0.4

0.3

0.2 P r e s e n t t h e o r y ,

axisymmetric c o m -

p r e s s i o n

Theo ry o f R ef .

P r e s e n t t h e o r y ,- c c e n t r i c compres-

s i o n a s e c c e n t r i -

c i t y a p p r oa c h eszero

C ur ve f i t t i n g t e s t

o f R e f s .

0 e s t d a t a , R ef . [b ]

A T e s t d a t a , R e f. 173

0 T e s t d a t a , R e f. [6]

-. -

0 1 I I I 1 1

1.2 1.4 1.6 1.8.4 0.6 0.8 1.0. 2

w h i c h value Ec r is found experimentally to be 0.29.

From Fig. 3, the present analysis gives AZ,, /a',,,, =

2.02. Therefore, i t can be predicted that iFm =0.096,

0.221, and 0.356 a t

while the mean test data from Fig. 1 shows that

F m =0.09, .25, a n d 0.34 at =0, 0.2, and 0.8,

respectively .

=0, 0.2, and 0.8, respectively,

FIGURE 1. POSTBUCKLING BEHAVIOR OF AXIA LLY COMPRESSEDPRESSURIZED CY L lNDR lCA L SHELLS.

where p. a, a n d 77 are given by (lo ), ( l l ) , and (12),

respectively.

From (8) and (261,

E c c e n t r i c C o m p r e s s i o n , P u r e B e n d i n g

Then a cylindrical shell is subject either to pure

bending or eccentrically applied compression k in (3) and

( 8 ) is unity. The solution in this ca se is analogous to

the solution of the previous section. The coefficients of

the 4 i r y stress function F are found to be

E t 2 2(1+d 2

(26)

6P 19 + 9 p 2 + - +-

2 m4 1+-.

+ 256

1 (1+3 p +'d 2 =

2 (1- 6 m 2 4m 2

(28)

1

512 (1 +p)'

The stress parameters $ 1 and # 2 are defined by the re-

lations701

5



U b R 3 U R 1Q ,=-+-.c--

E t 2 E t 2(29)

Q 2 = - + - . C - - p P + - ) - - @ -bR 3 a R 1E t 2 E t 2 2 m 2

1 p R 2

T h e e x p e r im e n t s of S u e r, H a r r i s, S k e n e , a n d B e n j a m i n

[91 i n d i c a t e t h a t s h e l l s s u b j e c t t o c o m p r e s si o n or b e n d i n g

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

d i r e c t io n . N u m e r i c a l r e s u l t s f ro m t h e p r e s e n t a n a l y s i s

a l s o i n d i ca t e t h a t m h a s a m a g n i t u d e g r e a t e r t h a n 10

w h e n R / t i s g r e a t e r th a n 500. T h e s e c a l c u l a t io n s a r e t o o

l e n g t h y t o p r e s e n t , b u t f o r e x a m p l e : 4 t T;=0.48 a n d

R / t = 1,000, i t w a s f o u n d t h a t l / m 2 =0 .00204 , w h i c h i s

m u ch l e s s t h a n u n i ty . T h e v a l u e of a i s u s u a l l y i n t h e

n e i g h b o r h o o d o f u n i t y ; h e n c e , f r o m (11) m 2 v a r i e s

a p p r o x i m a t e l y as R / t . I n t h e p r e s e n t a n a l y s i s w e a r e

c o n c e r n e d o n ly w i t h e x tr e m e ly t h in s h e l l s ; h e n c e t h i s

r a t i o i s l a rg e . T h e r e f o r e , f or p r a c t i c a l p u r p o s e s l / m 2

a n d l / m 4 a r e n e g l i g i b l e c o m p a r e d t o u n i t y . T h u s ,

(30)

+-P '1'1 a' +-' (sy ( 3 1 )(1+ d 2

FIGURE 3. PREDICTIONS OF PRESENT THEORY AND TEST DAFO R A X A L L Y COM PR ESSE D PR E SSU R IZ ED CY L ND R l CA LSHELLS

a n d

For b r e v i t y , ( 3 1 ) a n d ( 3 2 ) m ay b e r e w r i t t e n a s :

- - - -( 3 1

FIGURE 2. PO ST BU C KL IN G BEH AVIO R O F PR ESSU R IZ EDC YL IN D R IC AL SH EL L S SU BJ EC T T O BEN D IN G OR AX IALCOMPRESSION.

E t

702



a n d

c1 = o-9

($b)a,71 =0.74

+(E2+B,92 )a2+-sB +-"x

0.25 0.5 1.0 1 .15

0.66 0.46 0 .23 0 .15

0 .5 2 0.34 0 .2 1 0 .1 9 5

w h e r e

6

- 9p2

- 9 ( 1 + p " 2 )

- 1+p2

A, =

(1 + p ) z

A, =

256

B , =2 (1-2)

- 1B, =

F u r t h e r ,- -A 5 - -- B , - Bs --

9

-

A1

c,=

- B6-A , - BS --

9

- O bO b =

E t

a n d

- 0u =-

E t

T h u s , f r o m ( 3 0 )

( 3 8 )

(39)b + gGC ) = ( + b ) a , q '3 -pa, 71

To fin d t h e d i m e n s i o n l e s s c r i t i c a l st ress

a t a g i v e n v a l u e o f d i m e n s i o n l e s s p r e s s u r e jT, s e v e r a l dif-

f e re n t v a l u e s of p a r e t r i e d i n (39), t o g e t h e r w i th

c o r r e s p o n d i n g v a l u e s o f (4

r i g h t- h a n d s i d e o f (39) i s m i n i m i z e d .

f ro m T a b l e 3 u n t i l t h e4s9 '7

Th e v a l u e of p a t w h i ch (b +: E c ) a , ~s minimum

- B 6A 5 - , --

77 a n d e q u a l s ( c b +;Zc,) cr i s c a ll e d per. T a b l e 4 i n d i c a t e

s o m e n u m e r i c al r e l a t i o n s i n t e rm s of t h e p r e s s u r e .he s i m u l t a n e o u s s o l u t i o n o f ( 3 1 a ) a n d ( 3 2 a ) leads t o

- -Ci A i - -

4 b =7 C2 A2 a (35) T A B L E 4

F r o m t h e r e l a t i o n d q j b / d a =0 n d ( 3 5 ),

and f ina l ly f r om (35) w i t h a g i v e n b y ( 3 6 ) I t c a n b e s e e n t h a t pcr d e c r e a s e s w i th i n c r e a s i n g p. In

t h e a b o v e t a b l e , ('',& sta nd s f o r t h e v a l u e o f (Ec)crw h e n

'b +0. T h e r e l a t i o n s b e t w e en ( C b + ..>r a n d as37)

703



FIGURE 4. IN C R EMEN T O F CRIT ICA L STRESS AS A FUNCTION OFPRESSURE FOR CYLIND RICA L SHELLS SUBJECT TO ECCEN TRlCCOMPRESSION OR PURE BENDING.

w e l l a s ( F c ) *

data from [9f:s shown a l s o .

The b r o k e n c u r v e s h o w n t h e r e b o u n d s t e s t data

o b t a i n e d b y S u e r , H a r r i s , S k e n e , a n d B e n j a m i n [9I fo r

a x i a l c o m p r es s io n . R e n d in g t e s t d a t a d ue t o t h e s e same

a u t h o r s i s s ho w n b y i n d i v i d u a l p o i n t s i n F i g . 2 .

a n d p a r e s ho w n in F i g . 2, i n wh ic h the

L e t u s i n t r o d u c e th e n o ta ti o n

w h e r e ( Z +? c) r e p r e s e n t s t h e v a l u e ofb 2 cr,o

3 - 13 (Z b .

a t =0. The r a t i o 7

t y ~ c )ri s p l o tt e d a g a i n s t p.

i b + - u c- c r , o

i n F i g . 4 .

s t r e s s i n a n i m p e r fe c t s h e l l f ro m t e s t d a t a a t o n ly o n e

p r e s s u r e .

I t h a s b e e n o b s e r v e d t h a t g e n e r a l l y (Zc ) tr is n o t

e q u a l t o t h e v a l u e o f Ccr f o u n d f o r a x i s y m m e t r i c

c o m p r e s s i o n . The d i f f e r e n c e is d u e t o t h e d e f l e c t i o n

p a t t e r n s e m p l o y e d. C u r v e s I1 a n d 111 of F i g . 2 i n d i c a t e

t h e e f f e c t o n a x i a l c o m p r e ss i o n of t h o s e d i f f e r e n t

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

a p p l i c a t i o n of l o a d w i l l g r e a t l y r e d u c e t h e b u c k l in g

s t r e s s . (Cc)zf v e r s u s T; is s h o w n i n F i g . 1 a s C u r v e I V .

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

D i s c u s s i o n a n d C o n c l u s i o n s

I t c a n b e o b s e r v e d t h a t p = 0 i n t h e c a s e o f r i n g

b u c k l i n g a n d f u r t h e r , b 3 = q = 0 i n t h e c a s e o f s m a l l d e-f l e c t ions . If e i t h e r p or b 3 i s z e ro , t h e a bo v e a n a l y s i s

r e d u c e s t o a s m a l l d e f l e c t i o n s o l u ti o n .

3T h e r a t i o b e t w e en s t r e s s e s , ( b +? Zc) r/Ccf, is

a p p r o x i m a t e l y 1.25 a n d v a n e s o n l y s l i g h t l y w it h p r e s s u r e .

T h e r e f o r e , t h e r a t i o (o,)*,r i s 0.833. T h e p r o c e d ur e

i n d i c a t e d c a n b e e m p l o y ed w h e n t e s t d a t a a t o n e

p r e s s u r e a r e a v a i l a b l e t o p r e d i c t th e c r i t ic a l s t r e s s in

* cr

704

t h e s a m e i m p e rf e c t s h e l l a t a n y o th e r pr e s su r e . S i n c e

m o re c o n s i s t e n t t e s t r e s u l t s c a n b e e x p e c t e d f ro m s h e l l

u n d er h i g h er p r e s s u r e s . F i g s . 3 a n d 4 a r e a v a i l a b le t o

p r e d ic t t h e b u c kl in g s t r e s s e s w h e n a t least o n e t e s t ha s

b e e n m a d e o n s o m e m o d e r a t e ly p r e s s u r i z e d c y l i n d e r s.

F o r i n s t a n c e , i f t h e c r i t i c a l p ur e b e n d i n g s t r e s s ,

(Ob R /E t )= , h a s b e e n f o u n d e x p e r i m e n t a l l y a s 0.53 a t

p =0.4, t h e n f r o m F i g . 4 o n e c a n f i n d ( U , R / E ~ ) ~ ,0.163

a n d 0.603 a t =0, a n d 0.8, r e s p e c t i v e l y . T h i s e v a l u a t

i s a p p l i e d o nl y t o s h e l l s h a v i n g t h e same r a t i o R / t . The f f e c t s d ue t o a c h a n g e o f R / t w i ll b e d i s c u s s e d i n a

l a t e r p a p e r .

The p r e d i c t i o n s of t h e p r e s e n t t h e o ry f o r p r e s s u r i z e d

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

a g r ee m e n t w i th t e s t d a t a f o r a r a t h e r w i d e r a n g e of v a l u

o f i n t e r n a l p r e s s u r e . P r e d i c t i o n s o f t h e t h e or y f o r

p r e s s u r i z e d c y l i n d r i c a l s h e l l s i n p u r e b e n d in g a r e i n

r e a s o n a b l e a g r e e m e n t w it h e x p e r i m en t a l e v i d e n c e f or

d i m e n s i o n l e s s in t e rn a l p r e s s u r e s i n e x c e s s of 0.1 b u t a

c o n s e r v a t i v e fo r s m a l l e r v a l u e s of i n t e r n a l p r e s s u r e .

R e f e r e n c e s

1. D o n n e l l , L . H., ' A N e w T h e o r y f o r t h e B u c k l i n g o

T h i n C y l i n d e r s U n d e r A x i a l C o m p r e s s io n a n d B e n d i n g , "

Transactions of th e ,4SME, V o l . 56 , No. 11, p p . 7 9 5 - 8 0 6

N o v e m b e r , 1 9 3 4 .

T h i n C y l i n d r i c a l 5 h e l l s U n d e r A x i al C o m p r e s s i o n , "

Jo~unal f the Aeronautical Sciences, Vol. 8 , No. 8, pp .

3 0 3 - 3 1 2 , J u n e , 1 9 4 1 .

3. K e m p n e r, J . , ' P o s t b u c k l i n g B e h a v io r of Axia l ly

C o m p r e s s e d C y l i n d r i c a l S h e l l s ," j o w n a l ofthe ,Aero-

nautical Scienc es, Vol . 21 , No . 5 , p p . 3 2 9 - 3 3 5 , M a y ,

1 9 5 4 .

o f T h i n - W a l l e d C y l i n d e r I J n d e r A x i a l C o m p r e s s i o n a n dI n t e r n a l P r e s s u r e , " N.4CA TN 2021, J a n u a r y , 1950.

l i n e a r T h e o r i e s of t h e B u c k l i n g of T h i n C y l i n d r l c a l

S h e l l s , " i n Aeronautics

P r e s s , p p . 7 6 - 1 2 1, Y e w Y o r k , 1960.

6 . F u n g , Y . C . , a n d S e c h l e r , E. E., 'B uc k l in g o f Th i

W a ll ed C i r c u l a r C y l i n d e r s I J n d er A x i a l C o m p r e s s i o n a n d

I n t e r n a l P r e s s u r e , " Journal o f the Aeronautical Science

V o l . 24, NO . 5, p p . 351-356, \ l a y , 1957.

7. I , o f b l a d , R. P., J r . , ' E l a s t i c S t a b i l i t y o f T h i n -

W a ll ed C y l i n d e r s a n d C o n e s w i th I n t e r n a l P r e s s u r e U n d

, A xi al C o m p r e s s i o n , " MlT Technical Report 25-29, \ l a y

1959.

of C i r c u l a r C y l in d r ic a l S h e l l s I Jn d e r P u r e h n d i n g , "

Journal of Applied Mechanics, V o l . 2 8, Y o. 1, pp . 112-

116, !larch, 1961.

9. S u e r , H. S., H a r r i s , L . A . , S k e n e , W. T., a n d

B e n j a m i n , R. J . , ' T h e B e n d i n g S t a b i l i t y of T h i n - W a l l e d

U n s t if f e ne d C i r c u l a r C y l i n d e r s I n c l u d i n g t h e E f f e c t o f

I n t e r n a l P r e s s u r e , " lournu1 of the .Aeronautical Scienc e

V ol . 2.5,

2. von YBrmBn, T., a n d T s i e n , I I . S . , ' T h e R u c k l i n g

4. I,o, H., C r a t e , H . , a n d C c h w a r t z, E. R . , ' B u c k l i n g

5. T h i e l e m a n n , W . F., ' N ew I l e v e l o p m e n t in t h e U o n

and Astronautics, P e r g a m o n

8. S e i d e , P. , a n d W e i n g a r t e n , v. I . , 'On th e R uc k l in

No. 5 , p p . 2 8 1 - 2 8 7 , M a y , 1958.



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Documents

Elastic Instability of Pressurized Cylindrical Shells Under Compression or Bending