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

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

B B U D I A N S K Y

Division of Applied Sciences Harvard University

W T K O I T E R

Laboratory of Applied Mechanics

University of Technology, Delft

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

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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),

1920-Elastic 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.

TA660.S5Y36 1981* 62^.1*7762 8 3-2 5 ^ 8 5

ISBN O-Wt-86857-7 (U.S.)

PRINTED IN THE NETHERLANDS

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

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

well as highly elastic cylinders sustainable fairly large de­

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

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-post-

v i i

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

of the contained liquid on the buckling and postbuckling of clamped cylindrical tanks under each of the three fundamental

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.

o f t h e book r e l a t e d t o s e c t i o n s 4 4 and 4 5 , 5 2 th r o u g h 5 7 and 6 2 t h r o u g h 6 7 , r e s p e c t i v e l y He i s e s p e c i a l l y t h a n k f u l

t o M e s s r s K Otomo and T S a to f o r p r e p a r i n g t h e d r a w i n g s , t o

M r K Asano f o r m a k in g t h e p h o t o g r a p h s , t o M r s K T s u c h i y a and M is s H H o s h i f o r t y p i n g t h e m a n u s c r i p t s and t o M e s s r s

S Kodama, K Otomo and T S a to f o r t h e i r h e l p i n e d i t i n g th e

f i n a l m a n u s c r i p t

Nob o ru YAMAKI

NO N L I N E A R T HEO RY OF CI RC U L A R CYLI ND RIC AL SHELLS

O f c o u r s e o n l y t he s t a b l e e q u i l i b r i u m s t a t e c a n b e r e a l i z e d in the p h y s i c a l world T h e r e h a v e b e e n l o n g d e b a t e s o n the c l a s s i ­

each o f t h e s e s t r u c t u r a l members, l i n e a r b e n d i n g t h e o r i e 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 t e a n a l y s e s w i t h i n th e s m a l l d e ­

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

t o g e t h e r w i t h t h e i m p e r f e c t i o n s e n s i t i v i t y o f a v a r i e t y o f

e l a s t i c s y s te m s

I n a d d i t i o 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 lo a d s [ 3 2 , 3 3 ] as w e l l as t h o s e u n d e r v a r i o u s dynamic l o a d s [ 3 4 , 3 5 ]

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

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

be c a us e o f t h e p r a c t i c a l i m p o r t a n c e o f t h e s e p r o b l e m s

The p u rp o s e o f t h i s book i s to c l a r i f y t h e w h o le a s p e c t o f

t h e b a s i c p ro b le m s c o n c e r n i n g t h e e l a s t i c s t a b i l i t y o f 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 u n d e r t y p i c a l l o a d i n g c o n d i t i o n s Num er­ ous r e s e a r c h e s have been made on t h i s s u b j e c t s i n c e t h e t h i n -

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

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

t u r e s H ow ever, owing t o i t s m a t h e m a t i c a l d i f f i c u l t y t o g e t h e r

w i t h p h y s i c a l c o m p l e x i t y , a c c u r a t e r e s u l t s , b o t h t h e o r e t i c a l and e x p e r i m e n t a l , have become a v a i l a b l e o n l y r e c e n t l y w i t h t h e

a d v e n t o f h i g h speed com p ute rs and h i g h l y e l a s t i c t e s t m a t e ­

εχ = ε χ Ο + ζ κ χ ’ ey = e y O + Z K y ’ Υχγ = YxyO + z Kxy > d · 2 ·!) displacement relations in the shell as

a r e g i v e n b y r e p l a c i n g p x , p y a n d p w i t h - p h U > t t , - p h V > t t a n d -phW t t , r e s p e c t i v e l y , w h e r e p i s t h e d e n s i t y o f t h e s h e l l a n d

J - h / 2 y

τ yx ) z d z

( 1 3 6 )

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

Ky ( W, yy + f V >y)

( 1 4 2 )

( 1 4 3 )

(Νχ> Nx v , Qx) = ;ο(σχ , τχ γ , τχζ) dz ,

(Ν , N , Q )

v yx * y * x y 7 -I!

- h / 2h/2

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

h / 2

rh/2(My x ’ My ) = j h / 2 ( Tyx» a y ) z d z »

b ec om e

( 1 4 9 )

a n d x = L A s s u m e t h a t f o r t h e d e f l e c t i o n , t h e e d g e i s e i t h e r

c l a m p e d o r s i m p l y s u p p o r t e d , w h i c h w i l l b e d e s i g n a t e d a s c a s e s ( C ) a n d ( S ) , r e s p e c t i v e l y F o r t h e r e m a i n i n g c o n d i t i o n s c o n ­

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

( C ) Wl = W1 ; X = 0, o r ( S ) Wl = W1 ( X X = 0, 1

f> ( 2 2 1 6 ) ( P ) N x l = 0, N x y l = 0, o r ( U ) U x = V 1 = 0 J

R e f e r r i n g t o e q u a t i o n s ( 2 2 1 1 ) , t h e c o n d i t i o n s f o r t h e c a s e ( P ) b ec om e

T h e i n c r e m e n t a l i n - p l a n e d i s p l a c e m e n t s a n d a r e r e l a t e d t o W0 , Wjl a n d F l a s

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

Nx l - J f U l , x + W 0 , x Wl , x + v ^ 1 ,y ■ R -1W 1)] + D R - 1 W1 ( χχ ,

y 1 J [ V l >y - R - ^ + v ^ ^ + W o ^ W ! ^ ) ] - D R - ^ W ! + R - 2W1 )1-ν,

S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o e q u a t i o n s ( 2 3 8 ) a n d p u t t i n g

ε = 0 f o r t h e l a t e r a l p r e s s u r e l o a d i n g , we w i l l r e c o v e r t h e

s o - c a l l e d " F l ü g g e b a s i c e q u a t i o n s ” f i r s t d e r i v e d b y F l ü g g e [ 3 9 ] I t i s t o b e n o t e d t h a t t h e b a s i c e q u a t i o n s t h u s o b t a i n e d

Mxl,xx + 2Mxy 1 ,xy + M yl,yy +R_lNyl + Nx0 W l,xx + N xl W0,xx

+ Nxy0(2Wl>x y + R -1Vl>x ) + N y0(W l)y y + R ‘1Vl,y ) = 0· ^ A l )

2 5 BUCKLI NG UNDER T OR SI ON : APPROXI MATE A NA L Y SI S

m+n+ [ l + ( - l ) n ] B 2 } I ( - 1 ) P m P n a m = 0,

(m+n=even)

Equations (2.5.13) or (2.5.20) represent a set of homogeneous

tions, the determinant of the coefficients of am in these equa­ tions should vanish Noting that these coefficients depend on

tion

from which the minimum value of k s will be determined for each

similar calculations with a stepwise variation of 3, we can de­ termine the absolute minimum value of k g and the corresponding value of 3, which give the buckling load ih and the buckling

which yield the buckling waveform along with equations (2.5.6)

ously depends on the number of parameters am retained in the calculation.

It is to be added that with the same procedure as stated in

for the following two cases S3 and C2, respectively, where

spectively, in which the last terms with B ^ B2 or BJ, B^ are omitted.

Numerical Results

Assuming that v = 0.3, critical values of ks and 3 were de­ termined for various values of Z, taking ten unknown parameters

lines correspond to the present and Donnell’s results, respec­ tively Further, Batdorf's results for case S3 are shown by small circles while those for case C2 are found to be indistin­

noted that values of k s and 3 here obtained for the cases SI

long shells with Z greater than 100.

For reference, values of am here obtained for the buckling

corresponds to the infinite strip with breadth L and thickness

> (2.5.23) C2: W = W >x = U, = Nxyl = 0.

Ι Ο Ι Ο 2 I03 -7 I04 I05

h, for which the critical values of k s and 3 have been exactly determined as [16]

Excellent agreement with the present results will be noticed Further, the maximum error in the present results, for Z ranging from 0 to 105, is found to be less than 0.5 %, compared with the accurate ones to be stated in the following section.

In this section, we shall present accurate solutions of the title problem obtained by integrating the basic equations di­

boundary conditions of practical interest will be treated and

in each case the critical load as well as the corresponding wave number will be clarified for a wide range of shell geometries.

2 6 1 A n a l y s i s B a s e d on t h e F l ü g g e E q u a t i o n s

Basic Equations

We shall solve the buckling problem of circular cylindrical

this case, the basic equations (2.3.8) become

Method of Solution

Noting that equations (2.6.1) are a set of homogeneous linear differential equations with constant coefficients and assuming

solution in the form

introduction of this solution into equations (2.6.1), we have

(2.6.4)

Equations (2.6.6) represent a set of homogeneous linear equa­

solutions, the determinant of the coefficients should vanish, which leads to the following eighth-degree equation for r:

equation (2.6.7) and putting

( 2 6 6 )

( sin Γ^φ - i cos Γ^φ) exp (tN0) ,

j

With the general solution thus obtained, we have the follow­

I n t h e f o r e g o i n g , w e h a v e i n t r o d u c e d

When the boundary conditions are given, for example in case Cl,

we obtain the homogeneous linear equations in C^ as

determinant of the coefficients, Δ, should vanish, that is,

When the Poisson's ratio v as well as the shell dimensions, R/h and L/R, are given, the value of Δ depends only on N and Xg

represent the required critical load as well as the number of buckled waves, respectively With these values of Xg and N, the

then the incremental displacements during buckling will be ob­

As the roots r^ are generally complex, elements of the deter­ minant Δ will be also complex However, since the coefficients

of equation (2.6.7) are all real, the complex roots have always their conjugate counterparts and the corresponding columns in Δ will become also complex conjugate Hence, it is not difficult

is found from actual calculations that equation (2.6 .7) has always two real roots and three pairs of complex conjugate roots in the vicinity of the ciritical load.

pressed in terms of the displacements as

(2.6.17)

(2.6.18)

with which equation (2.6.17) can be rewritten as

ρ8 -I- 4γ2ρ6 + -^TT2yksp5 + (öy1 * + -ξ- Z 2) ρ4 + TT2y 3k sp3

the general solution may be expressed as

U — I C.a [ s m p L ν η ί r <— ) - t cos Pj (— ) ] exp — t 2x x , 2X v , i v y

When the boundary conditions at x = ± L/2 are prescribed, we will have a set of eight homogeneous linear equations in C^, and the requirement for existence of non-trivial solutions leads to

fixed values of the Poisson's ratio v and shell geometric pa­ rameter Z, the value of the determinant Δ depends on the load

With this stability equation , we can determine the critical

foregoing It is to be noted that in this case, the shell geo­ metric property is specified by only one parameter Z instead of R/h and L/R in the preceding case, which is of great practical advantage in representing the buckling characteristics for a wide range of shell geometries.

In the limiting case of Z = 0 where the shell reduces to an infinite strip of breadth L and thickness h with the indefinite increase in the radius R, equation (2.6.20) becomes

(ρ2 + γ2 ) 2 [ (ρ2 + γ2 ) 2 + TT2k syp] = 0 .

equation, the general solution in this case becomes

(2.6.25)

buckling problem will be solved as before by considering only the boundary conditions for the deflection.

2 6 3 N u m e r i c a l R e s u l t s f o r t he B u c k l i n g Un d e r T o r s i o n

Solutions Based on the Donnell Equations

and 3 for various values of Z were determined for each bounda­

and 2.2b [48] The results for the cases Cl, C4, SI and S4 are also illustrated in Fig 2.3 It will be seen that for extreme­

straint has the predominant effect on the buckling while for long

divided into two groups, the first for Cl, C2, SI, S2 and the second for C3, C4, S3, S4, indicating that the dominant role of the edge condition is whether the axial displacement is con­ strained or not during buckling.