Buckling analysis for design of pressurized cylindrical shell panels

Buckling analysis for design of pressurized cylindrical shell panels

Engineering Structures, Vo]. [9, No. 5, pp. 352 ~59. [997 ,~ 1997 Else~icl Science Ltd ELSEVIER Printed in Greal Britain. All right,, reserved 0141 ...

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Engineering Structures, Vo]. [9, No. 5, pp. 352 ~59. [997 ,~ 1997 Else~icl Science Ltd


Printed in Greal Britain. All right,, reserved 0141 (I296/97 $17.00 + 0.00

PII: S0141-0296(96)00093-4

Buckling analysis for design of pressurized cylindrical shell panels Seishi Y a m a d a

Department q/Architecture and Civil Engineering. Toyohashi University ¢?/ Technoh~gy. Toyohashi 441, Japan

In contrast with the pressure buckling of a complete cylinder, even a perfect cylindrical shell panel displays a complex form of nonlinear snap buckling behaviour. The prebuckling nonlinearity has been understood to be induced by the total equivalent imperfections involving the potential-load-induced imperfection and the actual geometric imperfection. Based on this concept, an alternative estimation procedure of the elastic and elastoplastic buckling load-carrying capacities is proposed and the resulting simple equations and formulae for design are represented. © 1997 Elsevier Science Ltd. All rights reserved. Keywords:buckling, imperfection, cylindrical stiffness analysis, elastoplastic collapse


panel, reduced

of the membrane stiffness components in the initial resistance to classical lineal buckling modes through detailed nonlinear numerical analysis -~. In the present paper, controlled parameter studies, using a fully validated numerical code, will establish the validity of the reduced stiffness criterion ~ s as an elastic buckling load-carrying capacity for design. Furthermore, comparisons with a series of extensive experimental programmes, will show that the present proposed formula is effective in estimating a conservative design criterion. Then using a total equivalent imperfection concept, an elastoplastic buckling load-carrying capacity, midshell failure load, for the cylindrical shell panels, will be obtained by revising all elastoplastic reduced stiffness model for complete cylinders '~'m. Finally, it will be suggested that for the cylindrical shell panels the elastoplastic criterion needs to be compared to a corner failure collapse load.


An important upper limit to the slenderness of long-span barrel vault roof constructions is provided by their buckling capacity. Also it is often necessary to design inter-stiffener panels of cylindrical shells used in marine structures to resist any tendency towards pressure induced buckling. The analysis for design is complicated because even a geometrically perfect cylindrical shell panel displays a complex form of nonlinear snap buckling behaviour in contrast to the pressure buckling of a complete cylinder. Buckling analysis for shells is further complicated by the observation that geometric imperfections have an important influence on the buckling mode as well as on the buckling load-carrying capacity ~. Buckling loads are, in general, considerably lower than the lowest critical loads predicted from the idealized classical linear modelling of shell behaviour. Consequently, there has been a tendency to abandon classical linear buckling analysis in favour of large-scale, nonlinear, finite element analyses of buckling. This has resulted in a loss of simplicity and explicitness in the design analysis of these and other shells. The author has studied the buckling of this class of shell during the last two decades and has shown through careful parameter studies of their imperfection sensitive nonlinear buckling behaviour, that the unifying framework of classical linear buckling theory is capable of explaining their, at times complex, behaviour. Nonlinear snap buckling of both the perfect and imperfect cylindrical shell panel has been shown to relate to classical linear buckling theory by Yamada and CrolF. Reduction in buckling loads has been demonstrated to result from an imperfection catalysed erosion

Notation The following symbols are used in this paper aspect ratio, = l/(rch) normalized circumferential half-wave number, =


flexural stiffnez;s, = Etch~{12( 1 - ~2)} modulus of elasticity longitudinal length /11 number of longitudinal half-waves m,,nl,.,m,~. moment resultants number of circumferential half-waves II stress resultants n ~,flr,tl ~, Q nondimensional load parameter, = qrF/(rr:D) D E /


Buckling analysis for pressurized cylindrical shell panels: S. Yamada q

uniform external pressure load radius of cylindrical shell panel wall thickness displacements in x, y and normal direction, respectively bending component from quadratic term of energy membrane component from quadratic term of energy quadratic form arising from nonlinear membrane energy nondimensional normal displacement, = w/t longitudinal coordinate circumferential coordinate ,e Batdorf parameter for geometry, = \'1 - u2 12/(rt) Poisson's ratio central angle


t U,V,W

Uej, U2,,, V2,,,




related closely to many full-scale applications, or fully clamped edge conditions ~ ~J14' W =







mTrx sin /77 r v ...... cos / r&


v,,,, sin mTrx rim, • l cos r&


u=ZZu v= 2





m 7TX

w = Z Z w.,,,, sin i

Superscripts C classical linear buckling load e total equivalent imperfection ~y lirst surface yielding first lull plastic collapse lip sqh plastic squash load 0 geometric imperfection elastic reduced stiffness criterion elastic lower bound design



32w w = ;~v~


3u i)X

3v ay

= v = 0, at x = 0, 1




1)rrx sinnm , rd~





2.1. Reduced stiffhess buckling analysis The present model has a longitudinal length l, wall thickness t, radius r and central angle 05. Uniform external pressure q is taken to be positive inward as shown in Figure 1. Displacement components in the longitudinal, x, circumferential, y, and normal (positive inward), z, directions, are denoted by u, v and w, respectively. Elastic nonlinear responses have been obtained using a Ritz analysis of the Donnell approximation for geometrically imperfect cylindrical shell panels 2. The classical simply supported edge conditions 2


u ...... sin (m


(n + 1 )w3.' ,=ZZ v,,,.,, sin mTrx i sin r&


3 ~-w




Elastic buckling design criteria



sin r&

For the fully clamped edge conditions:




related to previous laboratory experiments, have been adopted as boundary conditions. Displacement functions (u. v, w) are taken as linear combinations of a large number of harmonic functions as follows. For the simply supported edge conditions:

at corner of panel at midshell




W =




m 7TX

w,..... sin /

R 71"3,~



sin r& sin / S i n r ;



Yamada and Croll 2 have shown that prebuckling nonlinearities observed for the geometrically perfect cylindrical shell panels with the simply supported edge conditions can be interpreted as being due to the effects of the translation and constraint of edges. The complex nonlinear behaviours are considered to arise from load related to the bending effects due to the circumferential translation of straight line edges (v ¢ 0 at y =0,r05). These loading imperfections in the nonlinear analysis can be interpreted by an idealized linear buckling model having a prebuckling uniform membrane state. In this case the linear buckling analysis provides upper bounds of carrying capacity and an idealized reduced stiffness model is shown to predict lower bounds. The fundamental stresses are in the idealized state given by

O, at v = O, r05 n, = 0;

n, = - qr;

n,, = 0


The buckling condition using a variational principle is given as +






3u .....


3w, .....



H = U2~, + U2m + V2,,

Notation and convention adopted for g e o m e t r y and internal stress and m o m e n t resultants Figure


where U> and U2m are, respectively, the quadratic components of the bending and membrane strain energy associated with the buckling mode. The third term V2m is the quad-

Buckling analysis for pressurized cylindrical shell panels: S. Yamada


ratic part of the interactions between the idealized prebuckling state and the nonlinear membrane stresses and strains. Using the Donnell-type formulation, these components for nonzero incremental buckling displacements (u, v, w) are W 0 2~,~, D[ f'rC 2w3 2 ( '2w32 + 2 v ~:~2 3x 2 3v 2

u2,, -- 7


,, Lt ax27 + tas 27


+ 2(1 - u) \Oxay/ J dxdy





+ ( 1 - . ) ay

g2'" = -




o tO)'/ dxdy






where E ~ the modulus of the elasticity, u =-- Poisson's ratio, D =-- Et3/[12(l - u2)] and J =- E t / ( l - u2). By substituting equation (1) or equation (2) into equation (4), we can obtain linear algebraic eigenvalue equations with respect to all the unknown coefficients (u ....... v..... w,,,,,). Solving this eigenvalue problem using a digital computer system, the classical linear buckling pressure q' spectrum and the associated buckling mode are derived, and then the strain energy components (U2h, U~_m)are computed from equation (5). The reduced stiffness critical load, associated with q<, is then approximated using U2b and U2m in the quotient form U2h q* = q<' Uzh 4" U 2 m


2.2. Simply supported edge conditions For the simple support boundary conditions, equation (6) gives a very simple buckling load spectrum 2 as


( ,)2 B+B

upon the Donnell-type formulation is known to be given as a function of a single geometric parameter of Z. For relatively small values of Z (short or shallow shells), the results are consistent with that based on the nonshallow (Fluggetype) formulation. For relatively large values of Z (long or high-raise cylinder roofs) the classical linear buckling load is inaccurate because the associated membrane stillness component includes a non-negligible error, however, the reduced membrane stiffness buckling load would be rather accurate. In Figure 2 for Z = 1000, elastic nonlinear paths show the initial geometric imperfection sensitivity. For this result, as well as all subsequent results, the Poisson ratio ~, was taken equal to 0.3. The imperfection mode adopted is of the same form as the linear bending deflection mode. Figure 2 shows the pressure versus the total deformation in nondimensional normal displacement normalized by the thickness t, W~.5 + W~J5(which has a single half-wave in the axial direction and five half-waves circumferentially, n = 5), in the case of a = 1 or in the component W~ + W~)3 in the case of a = 2. Superimposed upon Figure 2 are the deflection modes at each buckling load in which the mode coupling with the various components (n = 1,3,5 . . . . ) is indicated. Let us define a noninteger form of buckling mode computed with the developed scale of buckling lobes. Partly as a consequence of the high levels of loading imperfection, resulting mainly from the circumferential translation of straight line edges, the shell panel under external pressure is shown to frequently buckle into modes having a contribution not just from that associated with the minimum classical linear buckling pressure, B = B<= an(, but also from the adjacent longer wave mode, (B = B ' - 1 = an< -1). Our previous extensive numerical studies have shown that the upper bounds of the fully nonlinear analytical buckling loads for various geometric imperfection amplitudes are predicted from the idealized classical linear bifurcation analysis. As lower bounds of the scattered buckling loads are the reduced stiffness critical pressure Q*(B') and Q* ( B < - 1); Q*(B<) represents the buckling resistance into the classical critical mode B< when the membrane stiffness has been eliminated; Q * ( B < - 1) is the similar prediction for mode ( B ' - 1). Figure 3 shows summaries of the two categorized lower bounds. The classical linear upper bound Q< is coincident with that of pressurized complete cylinders. Two broken lines Q* give the associated reduced stiffness critical loads for modes B< and ( B ' - 1 ) . Open dots represent the

(7) ~,



- .... a=l

Q =- qrl2/(rrZD) nondimensional pressure parameter B =-- an normalized circumferential half-wave number a =- l/(rvb) aspect ratio

In order to determine the rational value of B, we can refer to several elastic nonlinear numerical results, for instance, those by Yamada and Croll 2. Let us define the socalled Batdorf parameter F rt






4 3 2 W1,5-t-W~I,s for a = l






' I t 1 1 2 or Wl,:l+W~l,3 for a = 2

Figure2 Elastic nonlinear behaviour for imperfect cylindrical The classical buckling load of complete cylinders j2 based

shell panels, with Z= 1000 and simple supports

Buckling analysis for pressurized cylindrical shell panels: S. Yamada Non-Linear Analyses a=l a=2 arl ¢ o tx







/ o



Qc for an e

- -

/ 1~






10 t

10 °




[. :

. . . . . .

3. Elastoplastic lower b o u n d for simply s u p p o r t e d shells

..--~ "~

/~. . "o

/8 o



¢ ,o,

.... __

Q* for an ~ - 1 Q**


i Illl~J







Figure3 Present lower bounds compared with nonlinear numerical results for imperfect shell panels having simple supports

observed nonlinear snap buckling loads for incremental buckling into modes B", while closed dots represent the loads when buckling is into the subdominant mode (B' - 1 ) in nonlinear numerical experiments fully reported by Yamada and Croll 2:3. They provide safe, but not overly conservative, estimates for the elastic buckling collapse. Consequently we would be able to propose elastic design lower bound criteria for the simple support boundary condition as the solid line in Figure 3 and the following equation

logloQ** = -0.448 + 0.560 x logloZ


Q** _> 4


2.3. Fully clamped edge conditions The experimental buckling tests for fully clamped edge conditions, have been performed by Uchiyama et al. 14 and Yoked et al. ~5"~6. For all the specimens the relationship between the experimental buckling pressure and the present reduced stiffness buckling pressure by equation (6), is summarized in Figure 4 (Yamada e t al.17). Consequently the criteria by equation (6) have been formulated as IogloQ** = C~ + Ct logl(~Z + C2 (logloZ) 2 C~ = 0.895 + 0.085




C~ = -0.315 + 0.215 × a


C2 = 0.170 - 0.065 x a for l0 < Z < 3000 and 0.5 --< a -< 2.0. Figure5 shows the relationships between Q** by equation (10) and the experimental results for a = 1.91, 1.43, 0.95 and 0.48 picked out from Uchiyama et al. H. While the lower bound criterion for the simple support boundary is independent of a as demonstrated before, that for the clamped boundary is much affected by a.

1.5 ¸

Q Uchiyama et al. 0 Yoked et al,

0 0



-. •








0.8 10 z






Figure 4 Ratios of previous test load Oe"oto present theory O*

3.1. First surface yielding load by elastic nonlinear analysis When a nonlinear analysis is performed using a step-bystep process in which either load or displacement is used as a control parameter, the incremental analytical procedure make it convenient to compute the distributions of stress and moment resultant components at any stage along the equilibrium path. It is then possible to derive the wellknown effective stress n4r at any equilibrium state. The yon Mises yielding condition for a two-dimensional body may be written as N, tI \ ; ( N , -+- M , ) e - ( N ,

+ M,)(N,

+ M,)+(N,

: M.)'+


~ M,,) ?-


(12) where N,:,~ =- n j ( t c r v ) is nondimensional effective stress o'y uniaxial material yielding stress (N,, N,., N,,) =-- (n: n,., n~,)/(tcr r) nondimensional stress resultants (M,, M,, M,,3 =-- 6(m,, m,, m,:,)/(t2try) nondimensional moment resultants. In equation(12), the positive and negative signs are, respectively, for the inside and outside surface of the panel. The various dots in Figure 6 show the first yielding load satisfying equation (12) through elastic nonlinear stress analysis using E/err = 875 varying the shape and amplitude of geometric imperfectionS3: the path SA is for geometrically perfect but has a potential-load-induced imperfection (W*" =- w"/t ~- 0.96), SC for W~,3=-0.90 (W" ~ 0.06; nearly idealized shell panel), AG for W~).~=-0,96 and W~,2=2.0 (W" ~- 1.2), and AH for W¢,~=-0.96 and W~,2 = 2.5 (W" ~ 1.7). In this figure the geometric parameters Z and a are fixed as 200 and 1.0, respectively. The other parameters are then obtained using the ratio of radius to thickness, r/t, as l/r = ~b= 14.48/~/r~. The open dots indicate that the maximum point is at the corner on the inside surface, while the closed dots are for.the point on the outside surface at midshell. For relatively thick panels, low r/t, corner failure precedes that at midshell; this failure is largely independent of the initial geometric imperfection. As the shell panels become thinner, r/t increases, midshell failure represents the limiting condition. At midshell the maximum bending stresses are dependent upon the imperfection level. It is for this reason that first surface yield is highly affected by the imperfection levels. 3.2. First surface yielding load Jbr midshell failure One of the most difficult design aspects in shell buckling has been the prediction of an appropriate allowance for additional knockdowns arising from the interaction between elastic and plastic nonlinearity. Hutchinson ~ has discussed a general theory on the bifurcation type of the plastic buckling of plates and shells, while the simplicity of the reduced stiffness method makes it a particularly convenient basis for the prediction of more complicated problems associated with the plastic lower bound to imperfection sensitivity 'J'm. On the basis of a reduced stiffness buckling model, any

Buckling analysis for pressurized cylindrical shell panels: S. Yamada

356 100"


/ - - -•..







I t iIIJ






I l lll







Figure 5 Present design curves for fully clamped edge condition no.

Non.LineaX s







• } B~ ~ }B~_I


15 10 ¸















/m ~










Fa21ure : W =1

0 100


Figure 6






First surface yielding pressure of simply supported shell panels; Z = 200, a = 1.0, B c = 3.225, E/~rr: 875

imperfection introduction into the shell panel will, at a prescribed pressure level, provide upper bounds of the incremental deformation compared with that predicted for the exact shell panel behaviour. This represents that the incremental stress components found using the reduced stiffness model will, at this prescribed pressure level, be upper bounds of those occurring in the exact behaviour. Consequently, the load for first yield using the reduced stiffness method will be a lower bound of the exact first yield occurrence. A similar approach is possible for a cylindrical shell panel when the total equivalent imperfections w", involving the potential-load-induced imperfection and the actual geometric imperfection, are used to calculate midshell surface yielding. In the reduced stiffness model, the incremental deflection is assumed to be related to the nondimensional total equivalent imperfection amplitude W" = w"/t through Q


W=- w / t = W e Q , _ ~ s ~2 i n T s itn

nTrv r~,


The squash load q.Wh iS given by the idealized yield condition n~ = -to& -- _qW%. The associated fundamental stress state is

N, = 0;



pB2)F; M , . = ( u + B 2 ) F ;

M,,. = 0



6 W"Q F - Q,.qh ( Q , _ Q) Q,,.qh = q,,,~J, rr2D rl2 = ~;'I - v 2 12Zro-rTr2tE


N,,. = 0


Substituting equations (14) and (16) into equation (12), results in a nonlinear algebraic equation ['or Q, whose numerical solution gives a first surface yielding load, Q{2}. Figure 6 shows summaries of the two categorized lower bounds for Q{i',~). Two lines Q{),~)give those for modes B' and B ' - 1 in the case of W" = 1. The close dots are Q~;',~) from the elastic nonlinear analysis. For relatively thin panels, large r/t, the large scatter of the square and circular dots is bounded from below by Q{)I,'}for the mode B', while that of the triangular dots is bounded from below by Q!;',~ for the mode B', while that of the triangular dots is bounded from below by Q{;',~}for the mode B ' - 1.

3.3. Neglecting the periodic components of membrane stress, which would be lost in the elastic postbuckling erosion of stiffness, the maximum surface stress occurs when the fundamental membrane stress combines with the maximum flexural stresses arising from


N,. = - U.~, A,.),;

First full plastic collapse load.for midshell failure

While the first yield criterion described above would provide lower bounds to the limiting elastic behaviour, it may not have a direct relationship to the loads at which collapse is eventually precipitated. The central arch of the cylindrical panel could evidently continue to support pressure until the yielding on the crests and troughs spreads throughout the thickness. Once such a full plasticity state is attained there is no conceivable way through which the panel could continue to support the pressure without greatly increasing the deformations. With such increased deformations inevitably involving further increases in moment, the average circumferential stress and therefore pressure would need to be reduced if the stresses are to remain on the full plasticity state at a particular cross-section of the central arch. The resulting drop-off in pressure resistance for this central arch

Buckling analysis for pressurized cylindrical shell panels: S. Yamada implies that first full plasticity would be likely to provide a close indication of incipient collapse. When a simple full plastic modelling as shown in Figure 7 can be adopted, the associated nondimensional stress resultants are 3'0( 1 - '0)(¢,~ - ~,,,)




My = 3"0(1 - '0)(o-,,i= - o',,,,)




'0ff,i + ( 1 - "0)if,,,



would be possible for first corner yield to occur before that at midshell. These comer failure pressures would be affected by the load-induced imperfections, but only moderately by the geometric imperfections. Consequently, it is possible to provide estimates of the comer failure pressures through the linear bending stress analysis defined as the modified model in Yamada and Croll 2. When the nonlinear terms are omitted, the linear equations give l U = rt Q ~


nTry U ...... cosmirrx sin rd~










(17d) fly


where '0 ~ t,,/t is the elevation of the nondimensional neutral surface from the inside face. Making use of equation (17) and the von Mises criteria at both the inside and outside surface

~ ~ Win,,, s i n m 7 sin;7~ ' m


U ...... = W,,,.,, (B 2 ~Hi - °,io~ + o"~,~,i=


o'~,, - ~r,,,cr,.,. + oT.~ = oS~.




- -

?n b'gF/2) 7/'8 B

V,,.,, = -W,,,.,, {B e + (2 + u)m 2} wS

may be shown to result in the nondimensional forms,

16Z Win.it

ao + 3a,'0 + 90¢2"02 = 0


ao - 3a,( I - "0) + 9o¢2( 1 - "0)2 = 0


\:1 - u 2 r# m n T S = (B 2 + rn 2)2 12Z2m4




ceo = M~, - M,M, + The wavenumbers corresponding with the function representations of equation(21) are m=1,3,5,...,31 and n = 1,3,5,...,31. At the comer (x = y = 0), only shear stress components exist for the present boundary condition. In this case, equation (12) predicts a first surface corner yielding load

a, = 2N,M, + 2N,M,.- N , M , . - N,M~ a e = N~, - N,.N,. + N ~ - 1 Equation (19) gives the relation,

"0 = 2



In this case, to use equations(14) and (16) results in a nonlinear algebraic equation for Q, whose numerical solution gives a first full plastic collapse load, Q~,,it,'~. 3.4. First surface corner yielding load In the case of panel structures, it would be necessary to consider a corner failure mechanism. Also for cylindrical panels having the simple support boundary conditions, it

Q/~ v



\ 3 ( M , , + N,,.) where M,, = 6(1 - v) ~'~ ~ m

N,=. 12\,I-~Z~

W,,,.,, mB





Win,, w2S

3.5. First full plastic collapse load at the corner A first full plastic collapse load at the corner, Q~!];, can be derived in a similar way to that at midshell. Using the nondimensional stresses at the corner, M,~, and N,,, of equation (23), Q~'d'would be given as \/ 3 "0Qwh Q(~[" = /14,,+ 3"0N, I Figure 7 Stress block on circumference attainin 9 fully plastic state




Buckling analysis for pressurized cylindrical shell panels: S. Yamada "0=~

1 + ~:+ V'I + ~


= 3N,,. 3.6.


equation (19) or determine it using Figure 8 through computing the ratio, q./qWh or q**/qWJ, (6) Ensure that the midshell criterion is smaller than the corner failure criterion q(~]; from equation(24) or using Figure 8.

Results" and discussion

The first surface yielding criteria are superimposed on Figure 6. The present estimations are in good agreement with the lower bound of nonlinear numerical experiments. The first full plastic collapse loads obtained for W '~= 1 are also shown in Figure 8 using the same plotting as in CrolP. In this figure, MD1 means the midshell failure for a = 1, MD2 the midshell failure for a = 2, CN1 the corner failure for a = 1 and CN2 the corner failure for a = 2. In the cylindrical shell panels having simple support boundary conditions, the critical load-calTying capacity is much smaller than the idealized squash yielding load, q"~J'. This kind of plotting would be available for an estimate of design. The steps involved are: (1) Obtain the classical linear buckling load q' and mode n', and determine the reduced stiffness buckling mode parameter B = an ~ - 1 (2) Calculate the reduced stiffness buckling load as the elastic lower bound to initial imperfection sensitivity, q* (from equation (6)) or q** (from equation (9) or equation (10)) (3) Calculate the squash yielding load q,qh from equation (15) (4) Determine the total equivalent imperfection amplitude W~ (5) Calculate the elastoplastic midshell criterion q({![tfrom


It had been widely recognized that in contrast to the pressure buckling of a complete cylinder, even a perfect cylindrical shell panel displays a complex form of nonlinear snap buckling behaviour. This prebuckling nonlinearity has been understood to be induced by the total equivalent imperfections involving the potential-load-induced imperfection and the actual-geometric imperfection. Based upon this mechanical concept, an alternative estimation procedure of the elastoplastic buckling load-carrying capacities for the pressurized cylindrical shell panels has been proposed. At the first stage of the present procedure, the elastic buckling load-carrying capacity has been obtained to be of the reduced stiffness critical load. In the present paper the proposed elastic criterion has been compared with nonlinear Ritz analytical results or previous experimental results. At the second stage, determining the total equivalent imperfection amplitude, an elastoplastic buckling load-carrying capacity (midshell failure load), for the cylindrical shell panels, has been obtained from revising an elastoplastic reduced stiffness model for complete cylinders. Finally it has been suggested that the elastoplastic criterion needs to be compared to a corner failure collapse load. References I


~,~ 0.3-

0.2 0.1


M D 2 "--'-"- " ~




q, qh/q*



q. / qsqh q






, MD2/X

0 0

~ 0.5

qSqh/ q,

~ 1.0

I 0.5


q* / qsqh _q

Figure8 EDastoplastic critical loads for shell panels having simple supports in case of B= an~ 1 and VV~= 1: (a) (h-~r/6; (b) & = Ir/3


Yamada, S., Uchiyama, K. and Yamada, M. "Experimental invcsti gation of the buckling of shallow spherical shells', Int. J. Non-Linear Mech. 1983, 18, 37-54 2 Yamada, S. and Croll, J. G. A. "Buckling behavior of pressure loaded cylindrical panels', ,I. Engng Mech., ASCE 1989, 115, 327 344 3 Yamada, S. and Croll. J. G. A. 'Buckling and postbuckling characteristics of pressure loaded cylinders'. ,1. Appl. Mech., ASME 1993, 60, 290-299 4 Croll, J. G. A. 'Lower bound methods in shell buckling', m Derelol)merits ill Thin-Walled Structures-3, Elsevier. 1987, pp 117 163 5 Batista, R. C. and Croll, J. G. A. 'A design approach lbr axially compressed unstiffened cylinders', Prec. of Slabilio' Problems in Engineering, London, 1978, Applied Science, pp 377-399 6 Batista, R. C. 'Lower bounds estimates [br cylindrical shell buckling', PhD thesis, University College London, 1979 7 Batista, R. C. and Croll, J. G. A. 'Simple buckling theory lbr pressmized cylinders', J. Engng Mech., ASCE 1982, 108. 927-944 8 Batista, R. C. and Goncalves, P. B. 'Non-linear lower bound for shell buckling design', J. Constr. Steel Res. 1994, 28, 101-120 9 Croll, J. G. A. 'Lower bound elasto-plastic buckling of cylinders', Prec. lnsm Cir. Engrs 1981, 71, 235 261 10 Croll, J. G. A, 'Elasto-plastic buckling of pressure and axial loaded cylinders', Prec. h~sm Civ. Engrs 1982, 73, 633 652 11 Uchiyama, K. and Yamada, S. 'Theoretical studies of the buckling and nonlinear deflection behavior of clamped partial cylindrical shells under external pressure', J. Struct. Constr. Engng, Arch. In~l. .]pn 1986, 364, 80-90 12 Brush, D. O. and Almroth, B. O. Buckling of bars, plate,~, amt ,shell,s, McGraw-Hill, 1975, p 164 13 Yamada, S. and Croll, J. G. A. "Elasto-plastic buckling design for shallow cylindrical shells under external pressure', Prec. IASS-ASCF, Syrup. on Spatial, Lattice attd Tension Structures. 1994, pp 936 945 14 Ucbiyama, K., Uematsu, Y. and Minami, H. 'Experimental study of the asymmetric buckling of cylindrical shell roofs', Prec. 1ASS Syrup, on Shells, Membranes and Space Frames. Osaka. 1986, Vol 1, pp 249-256 15 Yoked, Y., Nakamura, T. and Matsuda, A. "Experimental investi gallon of the stability of clamped partial cylindrical shells subjected

Buckling analysis for pressurized cylindrical shell panels: S. Yamada to external pressure ~, Proc. Conf Recent Research on Structural Mechanics. Uno Shoten, Tokyo, 1968, pp 333-343 16 Yokoo, Y., Nakamura, T., Miyazaki, J. and Kano, A. "Effect of boundary conditions on buckling of circular cylindrical roof shells', Proc. RILEM Int. Syrup., Buenos Aires, 1971, pp 501-523



Yamada, S., Uchiyama, K. and Croll, J. G. A. ~Theoretical and experimental correlations of the buckling of partial cylindrical shells'. Proc, SEIKEN-IASS Symp., Tokyo, 1993, pp 151-158 18 Hutchinson, J. W. 'Plastic buckling'. Adv. Appl. Mech.. 1974, 14. 67-144