Buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression

Buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression

Thin-Walled Structures 12 (1991) 321-334 Buckling and Postbuckling Behaviour of Cylindrical Shells under Combined External Pressure and Axial Compres...

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Thin-Walled Structures 12 (1991) 321-334

Buckling and Postbuckling Behaviour of Cylindrical Shells under Combined External Pressure and Axial Compression

Hui-shen Shen" & Tie-yun Chen h "Department of Civil Engineering. t'Department of Naval Architects and Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China (Received 25 September 1989: revised version received August 1990: accepted 18 October 1990)

ABSTRACT Buckling and postbuckling behaviour of perfect and imperfect cylindrical shells of finite length subject to combined loading of external pressure and axial compression are considered. Based on the boundary layer theory which includes the edge effect in the buckling of shells, a theoretical analysis for the buckling and postbuckling of circular cylindrical shells under combined loading is presented using a singular perturbation technique. Some interaction curves for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, and it is shown that both agree well. The effects of initial imperfection on the interactive buckrng load and postbuckling behaviour of cylindrical shells have also been discussed

NOTATION

R,L.t Z

Radius, length a n d thickness o f cylindrical shell, respectively B a t d o r s parameter, d e f i n e d by Z = (1 - v2)mL2/Rt N o n - d i m e n s i o n a l parameter, defined by/3 = L/rrR /3 D,E,v Flexural rigidity, Young's m o d u l u s a n d Poisson's ratio o f shell P,q Axial compressive load a n d lateral pressure PcJ, qcl Classical buckling value o f P a n d q 321 Thin-Walled Structures 0263-8231/91/$03.50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

322

Hui-shen Shen, Tie-yun Chen

Number of longitudinal half waves and circumferential waves of the buckled shell Load-proportional parameter bl,b2 Perturbation small parameter 8 Imperfection parameter Xp,Xq Load parameter of axial compression and external pressure. respectively Theoretical value of A.p and ~.q for pure axial compression and pure external pressure, respectively R~, R.~, Non-dimensional form of stress, defined by R, =~.f;t~',, By = ~q/~q Imperfection sensitivity parameter, defined by 2* = )~ (imperfect)/,~p (perfect) m,/'/

1 INTRODUCTION The buckling of circular cylindrical shells subject to various types of combined loading is of current interest to engineers engaged in offshore engineering practice. It is of great technical importance to clarify the buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression. Numerous studies have been made of this problem theoretically I-4 and experimentally. 5-~ Classical approaches were obtained based on the small deflection theory. The results calculated indicate that the interaction curve is nearly a straight line. Due to initial geometric imperfections in the models, the test results did not all agree with the theoretical predictions. The above theoretical analyses did not take into account initial geometric imperfections as well as prebuckling deformations which are particularly important for short cylindrical shells. Tennyson etal. 9 studied the interactive buckling of perfect short cylindrical shells subjected to combined loading of axial compression and hydrostatic pressure, taking into consideration prebuckling deformations due to edge constraints. The effect of initial imperfections have been discussed by Hutchinson 1°in a preliminary manner and refer only to axisymmetric imperfections. Recently, more detailed theoretical and experimental studies have been made by Yamaki and his co-workers for clamped cylindrical shells under combined loading} ~Unfortunately, the analytical results are only for perfect shells. Some test results are in fair agreement with the theoretical predictions and the interaction relation given by any of the

Buckling and postbuckling hehaviour of cylindrical shells

323

Codes studied (ASME, DnV and ECCS) gave conservative predictions for practically all the test results, n Such a state means that further theoretical analysis should be done for cylindrical shells subject to combined loadings, especially for the imperfect cylindrical shells. As long ago as 1912, H. Reissner ~s pointed out the existence of a boundary layer phenomenon in the bending of thin shells. In Refs 14 and 15 the authors suggested a boundary layer theory and associated analytical method for the buckling of thin elastic shells. Based on this theory, the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and initial geometric imperfection could be considered simultaneously. This theory and the associated method will be used in the present work. The initial geometric imperfection is taken into account and the shape of the initial imperfection is assumed as the asymmetric buckling mode of cylindrical shells. The effects of initial imperfections on the interactive buckling load and postbuckling behaviour of cylindrical shells are also discussed.

2 FUNDAMENTALS A cylindrical shell with mean radius R, length L and wall thickness t is considered. It is assumed that the shell clamped at the ends is subjected to two loads combined out of uniform pressure q and axial load P. The coordinate system will be taken as shown in Fig. 1. Denoting by W* and

T x

. . . .

.-~z

Y 2

R

~

Fig. 1. Geometry and coordinate system of a cylindrical shell.

324

Hui-shen Shen, Tie-yun Chen

W the initial a n d a d d i t i o n a l deflections, respectively, a n d by F the stress f u n c t i o n for the stress resultants, t h e n Nx = ff, y.v

N~ = ff,~x Nxv = -F,,:v

T h e n the n o n l i n e a r K a r m a n - D o n n e l l equations o f cylindrical shells are given as follows 1

--

DV4W--~F,x~

= L*(W + W*,F)+

v ~ F + K1 -W,xx =

q

1 L * ( W + -W*,W)

-

(1)

(2)

where O4 04 ~ + 2 O x 2~O y +

V4 -

02 02 Ox 2 Oy 2

L*-

2

-

O4 Oy 4

02 02 OxOy OxOy

02 0:

A s s u m i n g b o t h edges o f the shell are c l a m p e d , the b o u n d a r y conditions are X = 0,L;

W = W,,. = V = 0

f 2.n N~dy + P + arrR2q

(3a)

= ()

(3b)

w h e r e a = 0 a n d a = 1 for the lateral a n d hydrostatic pressure loadings. respectively, a n d we have the closed c o n d i t i o n 2JrR

__

V,ydy = 0

I4)

a0

T h e unit e n d - s h o r t e n i n g relationship is Ax L -

1 c2-R eL_ 2zrRL Jo Jo U ' x d x d y

(5)

In the foregoing, we have used the notations as ;7 x = ~X,

_y y

_ R'

/3

L rrR'

e -

zr:~ x/~,

(W*.W) = e~/12(1 - v 2) ( W * , W ) / t

Buckling and postbuckling behaviour of cylindrical shells

F = e2ff/D,

Xq = q/qd,

Ap = P/Pcl,

325

6p = (AxE)/(Lo~I),

6q = (~cE)/(Lo~I) where crg~and cr~lare the classical critical stresses of the cylindrical shell under pure axial compression and pure lateral pressure, respectively. They are E t ~/3(1 - v2) R

cry. t -

o~l -- 3x/-~ (1

(6a)

(6b)

v2)3/4L

-

Then eqns (1) and (2) may be written in the following non-dimensional form

~ 2 L * ( W + W*.F) + 4 (3)1/4~q E3/2

8 2 L I W - F,x.,-

L tF

+

W,x.,-

(7)

(8)

2 B 2 L * ( W + W*,W)

-

where t04

LI

04

j~4 04

O x 4 "~- 2~ 20x20y ? +

-

L* -

02 02

2

OX 2 Oy 2

02

Oy 4

02

02 02

+---OxOy OxOy Oy 2 OX2

The clamped boundary conditions become

x = O, rr:

W=

W,x = V =

l (2.

0

(9a)

2

2rr Jo B2F'xvdY + 2,~pF, + a ~ (3)1/4~q83/2 = 0

(gb)

and the closed condition becomes (F,~ a. - vj~2 F,.o,) Jr" W -

l

j~2(W,y)2

-,62Wv,W*v. dy

= 0

(lO)

The unit end-shortening relationship can be rewritten as

(11) 6q

(3):/4¢ -3/2 8/72

- W,,.W*,~]dxdy ~

.

326

Hui-shen Shen, Tie-yun Chen

Equations (7)-(11) are the governing equations for the buckling and postbuckling problem of clamped, perfect and imperfect cylindrical shells under combined external pressure and axial compression. Buckling under external pressure alone and buckling under axial compression alone are two special cases of the present interactive buckling problem. Based on the small deflection analysis, Batdorf (1947) 16 pointed out that eqn (6a) be used only for cylindrical shells with Z > 2.85. When Z > 2-85, the small parametere < 1, thus eqns (7) and (8) are the equations of boundary layer type, from which the nonlinear prebuckling deformation, the large deflection in the postbuckling range and initial geometric imperfection can be considered simultaneously.

3 METHOD AND ASYMPTOTIC SOLUTIONS Due to the boundary layer problem, the singular perturbation technique is used to construct the asymptotic solutions. Assume the solution of eqns (7) and (8) may be written as w

~,'(x,4',y,e) + W(x,¢,y,e)

= W(x,y,e)+

Il2)

F = f i x , y , e) + F(x, ~,y, t) + F(x, ~',y, a) Where W(x, y, e),f(x, y, a~ are called outer solutions, or regular solutions of the shell, I,V(x,~, y, e), F (x, (, y, e) and I,V(x,(, y, e), F(x, ~', y, e) are the boundary layer solutions near the x = 0 and x = rr edges, respectively, and ~ and ~"are the boundary layer variables, defined as = x/x/e-,

¢ = (. - x)/ve

This shows that the width of the boundary layers is of the order vJRt. Let the regular and boundary layer solutions be asymptotic expansions and write

= X

.1 = o

j=0

= X JJ2z.j2(x,y) ./= o

i=0

(13)

F i/2 + 2(x, ¢:,y) I~'(x,(,y,c) = E eY/2+ ll,tzi/2 + ,(x, (, y), F(x, ( , y , e ) = E ein+? j=O

/~yn+ 2(x, ( , y )

j=O

Buckling and postbuckling behaviour of cylindrical shells

327

and the buckling load parameters are expanded as 2Ape = Kx = E e J k ' J '

4 ~ (3)1/4~'qe3/2 = Kv = E ejk~j

j =0

(14)

j =0

The non-dimensional initial imperfection is assumed to be in the shape of the asymmetric buckling mode of the shell, namely W*(x,y,e)

= e2A*lsinmxsinny

= eZt~AtZ)sinmxsinny

(15)

where/~ is imperfection parameter and 11 = A ~l/A ]]). By means of the singular perturbation technique the matching of the solutions on the boundary can be performed. From Refs 14 and 15 it is found that the effect of boundary layers on the solution of the compressive shell is of the ordere' and on the solution of the pressurized shell is of the order e 3/2,thus for cylindrical shell under combined loading two kinds of loading conditions should be considered. Case (1) higher values of external pressure combined with relatively low axial load Let P 7rR2q - b l

(16a)

or

2Zee _ b 4 (3)1/4~.qC3/2 2 3

(16b)

in this case, the boundary condition (9b) becomes

'f/

2~

~2F"~:vdY+ (a + b l) 2 (3)l/4Xqe3/2 = 0

(17)

By replacing (a + b~) with a~, and by using a singular perturbation procedure similar to that in Ref. 15, the asymptotic solutions satisfying clamped boundary conditions can be obtained and written as Xq = 1 (3)3/4 [~(qO) + ,~(q2)(At2 ) •)2 + . . where

"l

(18)

Hui-shenShen, Tie-yunChen

328

(

,a ~qO~=

m4

,2

(m 2 + //232)2 /'/232 + ~alm ) (1 + +

( m2 "~ H232)2

;_3j2

~el/2

1 2)(l + m) (n2fl2+~a,m 1

m4n232

f

1 (m 2 +/'/232)2 (1 + 2/~)

ACq 2)= ~(m2+n232) 2 el. 2(1 +/~)(2 +/~)+~02-7~-2~22

] .........

,, 3 [ ,, 13 + ~ a,m )

F/23 2 (m 2 + /'/232)2

(m: + n232): (n:32 + 1 aim2 ) (1 [20 + ~)2 + (2 +/1)

+/J) - 2alrn 6

(m 2 + n232) 2 (1

+ 2~) + 8m4(1 (m 2 + n232)2

+/d)

1-aim2(12+ 2/~) } +

(

1

?/23 2 + }aim

2)

e'/2

and 6q ~--~-

[(la,-v)+ ~2v(l-} l a,v) e,/2]Aq+ [(3)l/nV/2 [ 3rr 1 -~

l a,v)2e]X2+

[I-(3)3/4 -~--m 2(1+

2//)8 I/2] (At2~g)2 + . .,

(19)•

In eqns (18) and (19),A~2?eis taken as the second perturbation parameter relating to non-dimensional maximum deflection, and we have --

1

1

20,

and the non-dimensional maximum deflection is written as -W m - Wmx/12(1-V2)- ( t

1 )e'/2Z~ ~ l-~alv

(21)

329

Buckling and postbuckling behaviour of cylindrical shells

Case (2) higher values of axial compression combined with relatively low pressure

Let nR2q - b2 P

(22a)

or 4 (3)1/4,~qE 3/2

2,~,r~.

(22b)

- 2b2

in this case, the boundary condition (9b) becomes 1 (2. fl2F,vvdy + (1 + ab2)2)~re = 0 2n Jo ..

(23)

By taking a2 = 2b2/(1 + ab2), the asymptotic solutions obtained in Ref. 14 may be extended to the combined loading condition. They are \ 1 / m 2 \|m 2 g~-F/2j~2/u2/ [,~(pO)__,~(p2)(~4t2)~)2_~_,~(4)(At21)E)4+ .... ] + (1 ab2) where

/l(o) = 1[

P

m2 ~-I ..~_(m2 + ?/2fl2)2 1 (m 2 + n21~2)2 (1 + p) m2(1 +/2)

m6(2 +/-/) -i _14 m2n41~4 2(m2 + n2f12)4e - m2(1 + 2p)e + (m 2 + n2B2)2 (m 2 + n2f12)2

1

A'~;2)- 8

m2

+ 2/~)] + 8m4(1 +//) (2 + O) m 2

(m2 + n2~2)2 (1 + / . / ) - 4m4 (m 2 _1_~2n2~2) X~41 =

1 m'°(1 + p ) J i m 2 + qa2n2t~2~ 128 (m 2 + ?,/2j~2)6(m2 + qn2132)2 L~ ~ Sr ~ J (1 + 3p +//2)

[m 2 + 5a2n2/32~ ] + ~ ~-2~ ~2n2~ j (4 + 2p) + (1 + p ) + (m 2 + n2#2)2 km2S,_ ~

j (6 + 8p + 2,u2) - (2,u + 3# 2 +p3)

[m 2 + qa2n2~2'~ F/21~2)2 (m2 + qn2~2)2 k m-2 ~ 2 n 2 ~ - ] - - ( m 2 + ¢1 +/-0

oe-I

330

Hui-shen Shen. Tie-yun Chen

and ( l - a 2 v ) - 2 v J 2 - v ( v n -a2)c"2 Z , +

6p = (1 + a b 2 )

.

.

.

.

.

e .2

~ - ( 1 + ab2) 2

(,4 t27 e)2 +

___ ( m 2 + ?/_2]~2)2!l...~_//) .1_ 8m4(1 .1~.//) ....... \ ] 2

m2n4]~4(1 +/1)2

(m2 + n2~2)2(1 + / / ) - - 4 m 4 ( - , Z i n U d ~ ] | \m- a_n p /3 e3

(25)

(At2?e) 4 + . . .

in eqns (24) and (25), similarly, At2)e is taken as the second perturbation parameter in this case, and we have

--

[.1

m4(1 + / / ) a - '

At2)8 = W m - [_16 n2~j 2 (m 2 -}- r/2/j2)2

m2

•~, j - - m

+ 2(v - a2)

...

~26)

and the non-dimensional m a x i m u m deflection is written as

Wm - W,, x/12(1 _ v2) + 2(v - a2) t

m 2 +a2n2/32 '~')

(27)

Eqns (18) and (24) characterize the postbuckling load-deflection curves of cylindrical shells under combined loading. By increasing b l and b2, respectively, the interaction curve of a cylindrical shell under combined loading can be constructed with these two lines. Note that since b2 = 1/b~, only one load-proportional parameter should be determined by experiment. The perfect shell buckling load can readily be obtained by setting W*/t = 0 (or/1 = 0) while taking the m a x i m u m deflection W,, = 0. It should be pointed out that, due to the effect of the boundary layer, nonlinear prebuckling deformation is taken into account; thus the result presented is different from the classical one.

4 N U M E R I C A L RESULTS A N D DISCUSSIONS A computer program was written to calculate interaction curves for perfect and imperfect cylindrical shells under combined pressure and

331

Buckling and postbuckling behaviour of cylindrical shells 1.0 " ' . .

(a) ""

× 0.5

xx~

.

w,ro "X-x

T 0 1.0

r~

:'~-'x,, \"

(b)

m=3

m=l

I

", '~ Y~"Y "~\%.

~..~..

0-5 Ry

.,,

^ 1.0 0

m:l

' \

(d)

(f)

z =1000 R,t=405

, _

i

~.\~

x

× 0"5

w, l o

\

×\

to-o5

tx

z =5 0 R/t=405

~'.-.

t

~\

I

x\

wt o

",,,\

~

R,t_-,o5

^~

m=2

(e) z=2034 R/t=533

z=200

%\

m:1

',,,,

(c)

"X~,'\

z=20 R/t=405

~X~". ~ ' \ \ ¢r

"-.

__ I

0.5 Ry

\x

m:3

z=3434 Ritz400 ~/rn=l

"z3-...,~ \m:.

×\ X

1.0 0

0.5

10

Ry

Fig. 2. Interaction buckling curves for perfect and imperfect cylindrical shells under combined hydrostatic pressure and axial compression. axial load. Numerical calculations were carried out taking Poisson's ratio v as 0.3. All shells buckled in the asymmetric mode. Numerical results are s h o w n in Fig. 2 which will n o w be discussed in detail. (1) For perfect cylindrical shells, the interaction curve is constructed with two lines, both m = 1 (solid and dotted lines); the transition from one to another is smooth, and they appear as one line. Since the circumferential wave number n increases as the loadproportional parameter b2 increases, the buckling mode will change as usual. The interaction curve approaches a straight line w h e n the shell geometric parameter Z is large enough. (2) For short cylinders with Z < 50, the shell with and without an initial imperfection usually buckles in the m = 1 mode. By computing, it is not found that the interaction curves change from concave to convex as the Z parameter decreases. (3) In general, w h e n Z > 100 the shell buckles in the asymmetric mode m = 2 , 3 , . . . under loading condition case (2), and buckles in the m = 1 mode under loading condition case (1). For an imperfect shell, the shape of the interaction curve differs from that of the perfect one. The interaction curve is constructed with two lines (solid and dashed lines) o f m = 1 and m = 2 (or 3), and the intersection of these two lines exists.

Hui-shen Shen, Tie-yun Chen

332

(4) The effect of an initial imperfection in reducing the buckling loads of cylindrical shell under axial compression is a major factor. In Fig. 2 the experimental results are from Yamaki !Land Weingarten and SeidC. It is found that reasonable agreement between theory and experiment is achieved when the initial imperfection is taken into account. The curves of imperfection sensitivity of cylindrical shells under loading condition case (2) are shown in Fig. 3. It may be seen that, for most practical values of Z, the imperfection sensitivity decreases as Z increases and the load proportional parameter b2 increases. In this case the cylindrical shell is imperfection-sensitive, and an initial imperfection causes a large scatter in test results and reduction in the buckling load. Meanwhile, we should point out that for very low values of Z, the cylinder is imperfection-insensitive. In addition, it is noted that the effect of initial imperfection on the cylindrical shell is insensitive under loading condition case (1), except for very short cylinders. Thus in this case the experimental data follow the theoretical interaction curve reasonably well. Figure 4 presents typical postbuckling load-shortening curves and load-deflection curves for perfect and imperfect cylindrical shells under combined loading. It shows that postbuckling behaviour of a shell depends not only upon the geometric parameter Z, but also on the load1"2 1.0 0"8

Z=1000 (m=2) " ~ . ~ z =500 (m=2)

0"6 0"4 0.2

0

z=lOO(m=l)

~

~

o}

0"05 = b2

----

[ o~z

I 0.4

I 0.6

I o~

I 1.0

w*/t

Fig. 3. Comparisonsof imperfection sensitivitiesof cylindricalshells under combined loading.

Buckling and postbuckling behaviour of cylindrical shells (a)

z=20 a=O

R/t=200

R/t=200

333

z=20 a=O

(b)

1.0 b2=0(1,12)

~ ~ ~ ~ j / / b 2

= 0 (1,21 b2= 0.03 (1,13)

0.5

_%fo

/ /

t

I

--

- ~ - LO.1

I

0"5 R/t=300

W ~'_ [0

L04

I

0

1.0

I

1.0

1

1'5 (d)

(c)

z=200 a=l

0"5 Ritz300

z=200 a=l

1.o

- w, t

/ LO.1- - ~ z ~

b2:0 (2,13) b2= 0 (2,13)

0.5 k

/

~

0.5

=0 0 2 ( 2 ' 1 4 )

1.0

0

6p

I

1.0

I

2.0 Wit

1

3.0

Fig. 4. Postbucklingequilibriumpathsfor perfectand imperfectcylindricalshells under combined loading. proportional parameter b2. It is apparent that the buckling loads as well as the lowest postbuckling loads decrease as b2 increases, and the postbuckling equilibrium paths tend to be gently smoothed out as well. Next, it is evident that the buckling loads and the lowest postbuckling loads are reduced when initial imperfections are taken into account, and the effect of initial imperfection on the buckling loads is stronger than that on the lowest postbuckling loads.

5 CONCLUSIONS A boundary layer theory and associated method have been extended to the interactive buckling problem of cylindrical shells subject to combined loading of external pressure and axial compression. With the developed procedure, the interaction curves and postbuckling equilibrium paths for the perfect and imperfect cylindrical shells under combined loading have been obtained. It is concluded that for most practical shells,

334

Hui-shen Shen. Tie-yun Chen

as the load-proportional parameter b2 increases, the circumferential wave n u m b e r n increases, imperfection sensitivity decreases and postbuckling behaviour changes from unstable to stable. As a result, the buckling a n d postbuckling behaviour of cylindrical shells u n d e r c o m b i n e d loading primarily depends on three factors: the geometric parameter, the load proportional parameter and the imperfection parameter.

REFERENCES 1. Sharman, P. W., A theoretical interaction equation for the buckling of circular shells under axial compression and external pressure. J. Aero. Sci., 29 (1962) 878-9. 2. Lackshmikantham, C. & Gerard, G., Elastic stability of cylindrical shells under axial and lateral loads. J. Roy. Aero. Soc., 68 (1964) 773-5. 3. Zyczkowski, M. & Bucko, S., A method of stability analysis of cylindrical shells under biaxial compression. AZ4A J., 9 (1971) 2259-63. 4. Kirshnamoorthy, G., Buckling of thin cylinders under combined external pressure and axial compression. J. Aircraft, 11 (1974) 65-8. 5. Weingarten, V. I. & Seide, P., Elastic stability of thin-walled cylindrical and conical shells under combined external pressure and axial compression. A/AA J., 3 (1965) 913-20. 6. Mungan, I., Buckling stress states of cylindrical shells. Proc. ASCE, lt~(ST11) (1974) 2289-306. 7. Foster, C. G., Interaction of buckling modes in thin-walled cylinders. Exp. Mech., 21 (1981) 124-129. 8. Walker, A. C., Segal, Y. & McCall, S., The buckling of thin-walled ringstiffened steel shells. In Buckling of Shells, ed. E. Ramm. 1982, pp. 275-304. 9. Tennyson, R. C., Booton, M. & Chan, K. H., Buckling of short cylinders under combined loading. J. Appl. Mech., 45 (1978) 574-8. 10. Hutchinson, J., Buckling of imperfect cylindrical shells under axial compression and external pressure. A/AA J., 3 (1965) 1968-70. 11. Yamaki, N., Elastic Stability of Circular Cylindrical Shells. Elsevier Science Publishers B. V., 1984. 12. Galletly, G. D., James, S., Kruzelecki, J. & Pemsing, K,, Interactive buckling tests on cylinders subjected to external pressure and axial compression. Proc. 15th Int. Conf. on OMAE, 2 (1986) 114-22. 13. Reissner, H., Spannungen in Kugetschalen (Kuppeln). Festschrift MueUerBreslau, 1912, p. 181. 14. Shen, H. S. & Chen, T. Y., A boundary layer theory for the buckling of thin cylindrical shells under axial compression. Adv. Appl. Math. & Mechl in China, 2 1990, 155-172. 15. Shen, H. S. & Chen, T. Y., A boundary layer theory for the buckling of thin cylindrical shells under external pressure. Appl. Math. & Mech. 9(6) (1988) 557-71. 16. Batdorf, S. B., A simplified method of elastic stability analysis for thin cylindrical shells. NACA TR-874, 1947.