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AXISYMMETRIC AND BIFURCATION CREEP BUCKL ING

OF EXTERNALLY PRESSURISED SPHERICAL SHELLS

P. C. XIROUCHAKISnd NORMAN JoNEstDepartment of Ocean Engineeri ng, Massachusetts Institu te of Technol ogy, Cambridge, MA 02139, U.S.A.

(Received 21 November 1978; in revised jon 27 February 1979)

Abstract-The creep buckling behavior of a geometrically imperfect complete spherical shell subjected to a

uniform external pressure is examined using Sanders’ equilib rium and kinematic equations appropriately

modified to include the intluence of initial stress-free imperfections in the radius. The Norton-Bailey

const ituti ve equation s are used to describe the secondary creep behavior and elastic effects are retained.

The init ial imperfectio ns have the same shape as the classical axisymmetric elastic buc klin g mod e and the

initial elastic response is obtained analytically for external pressures smdler than the corresponding static

collaps e pressure. Numerical finit e-difference procedu res are used to obtai n the axisymmetrical creep

buckling behavior and to determine when a bifurcation or loss of uniqueness into a non-axisymmetric

deformation state occurs.

The numerical results for the creep buckl ing behavior of compkte spherical shells are similar for

hydros tatic and deadweight -type external pressur es, at kast for the particul ar p arameters examined h erein,

and demonstrate that initial imperfections exercise an important influence on the critical times. It turns out

from a practical viewpoint that axisymmetric creep buckling governs the behavior of the sphericaf shells

examined in this article. It was observed from the present results that the creep buckf ing times of externally

pressurised complete spherical shells are longer than those for “equivalent” axially loaded cylindrical

shells.

NOTATIONmean radius of spherical shell

defined by eqn (Mb)

defined by eqn (9i)

thickness of spherical shell

defined by eqn (9d)

d&ed by eons (9B, r)circumf erential harmoni c (eqns 38)

degree of Legendre pol ynomi al (eqn IS)

transverse pressure (Fig. I)

2Wz/{3a’(1 - Y*))‘~, classical buckfi ng pressure of an elastic perfect spherical s hell

static buctding pressure of an elastic imperfect spherical shell

defined by eqn (9h)

time

defined by qn (8a)nondimensional displacements (eqns 9a, b)

coordinate normal to mid-surface of shell (Fig. I)

Young’s modulus

membrane strains

defined by eqn (3b)

creep coefficient (eqns I, 2)

defined by qn (2b)

bending curvatures

defined by eqn (3a)

bending moments per unit length (Fig. I)

creep index (eqns l-3)

membrane forces per unit length (Fig. I)

defined by qn (8b)

transverse s hear forces per uni t kngt h (Fg I)nondimensional time (qn 91)

nondii sional creep buckling time

in-plan e and transverse disp lacements, respectively (Fig. I)ini tial transverse displacement (imperf ection)

circumf erential coord inate (Fig. 1)

radii imperfection nondimensionalised with respect to h (eqn IS)

z?f by qn %)

total strain rate (qn 4)

elastic strain (eqn 5)

Wresent address: The University of Liverp ool, Department of Mechanical Engin eering, P.O. Box 147. Liverp ool L693BX.

England.131



I32 p. C. XtHowt,.iht~ and NOHMAN Jwt !,

creep strain rate (eqn 1)

meridional coordinate (Fig. I)defined by eqn (22d)

h/aPoisson’s ratio

G&l/Estress tensor for plane stress

rotations of middle surface

rotation of surface normal

= - w, Jo

defined by eqn (14a)defined by eqns (1) and (2)

defined by eqn (14b)

d( )/de

iI ;;za( )/I%or d( )/LV

INTRODUCTION

The creep buckling behavior of cylindrical shells subjected to various external loads has been

examined by a number of authors as described in the review articles by Hoff [l] and Gerdeen

and Sazawal[2]. Some more recent studies on the creep buckling of cylindrical shells are

reported in Refs. [3-j]. By way of contrast, it is evident from Ref. [2] that relatively few

investigations have been published on the creep buckling of spherical shells. The creep buckling

behavior of spherical shells subjected to external pressures and made from materials prone to

creep (e.g. plastics, concrete, titanium alloys) is of interest for the design of various ocean

engineering structures (e.g. submersibles, habitats, underwater storage tanks).

The available theoretical studies on the creep buckling of viscoelastic spherical caps(6,

7, etc.], shallow spherical shells[8] and deep spherical shells[9, 10, etc.] were developed forshells with perfect geometries. However, it is well known[ll, 121 that the static elastic and

plastic buckling of spherical shells is sensitive to the influence of initial geometrical imper-

fections. A perturbation procedure was developed in Ref.[l3] to examine the influence of an

arbitrary initial imperfection field on the creep buckling behavior of a complete spherical shell

subjected to a uniform external pressure. However, the theoretical results are only valid for

small departures from the fundamental state and the influence of material elasticity was

disregarded. Bushnell[14] has developed a numerical procedure which may be used for the

creep buckling of shells but no results appear to have been published for spherical shells.

In this article, the creep buckling of an imperfect complete spherical shell subjected to a

uniform external pressure is examined using Sanders’ [151equilibrium and kinematic equationsappropriately modified to include the influence of initial stress-free imperfections in the radius.

The Norton-Bailey constitutive equations are used to describe the secondary creep behavior

and elastic effects are retained. The initial imperfections have the same shape as the classical

axisymmetric elastic buckling mode[ll, 16, 17, etc.] and the initial elastic response is obtained

analytically for external pressures smaller than the corresponding static collapse pressure.

Numerical finite-difference procedures are used to obtain the axisymmetrical creep buckling

behavior and to determine when a bifurcation or loss of uniqueness into a non&symmetric

deformation occurs.

The objective of this article is to reveal some aspects of the creep buckling behavior of

externally pressurised imperfect complete spherical shells rather than to develop a numerical

procedure which is suitable for a broader class of problems. The finite-difference energy

scheme of Bushnell [14], which may be used for many structural problems, could have been

employed for the present study. However, a different numerical procedure (finite-difference

formulation of equilibrium equations) is used herein which embodies a few advantageous

features for the present problem. For example, the initial elastic response is obtained analytic-

ally in the present work. Furthermore, all the equations and results are dimensionless ratherthan dimensional as in Ref. [14]. The time integration scheme utilises a variable time step,

which is more efficieni particularly for creep problems, and Gauss-Legendre quadrature, which

is more accurate than Simpson’s rule, is used to evaluate the integrals across the shell

thickness.



Axisymmetric and bifurcation creep buckling of externally pressurised spherical shells

BASIC EQUATIONS

133

The strain-displacement and curvature change relations for the complete spherical shell in

Fig. 1 are obtained in Ref. [18] according to Sander’s non-linear shell equations for small strains

and small rotations of the surface normal in comparison to rotations about the coordinate

lines[ 151 and appropriately modified to include the effect of initial stress-free axisymmetricimperfections (cfi). A consistent set of equilibrium equations were found using the principle of

virtual work and reduce to the corresponding equations in Ref. [9] for the perfect case. The

strain-displacement, curvature change, and equilibrium equations are given by eqns (1)-(13) in

Ref. [I81 and by eqns (10) and (11) here in dimensionless form.

Odqvist demonstrated that the Norton-Bailey law for uniaxial creep (i’ = KT~) when

generalized with the aid of the Prandtl-Reuss incremental relations takes the form

where +:a are the creep strain rates, Sac is the stress deviator given by S..,@ Tag 7,+5,,/3,

Qc = K&* “, KU = 3”+‘K/2, @a, b)

M = (N - 1)/2, and JZ = (T,,’ + ~22 - rl 1~22 37, : ) / 3 ( 3a. b)

for plane stress. The total strain rates are written as the sum of the elastic and creep

components

lb)\

Fig. I. Element of a spherical shell. fl and 6 are the circumf erential and meridi inal coord inates,

respectively. Subscrip ts I and 2 are associated with 0 and fl.

tGreck indi ces range from 1 o 2.



134 P. C. XIROUCHAKISand NORMAN ONES

where the linear elastic strains are

for an isotropic material. Equations (l), (4) and (5) give

62 = E+2/(1+ u) - E@J,2/(1+ V) (6a)

~r1=~(7j,,+~22)/(1-~~)-E~,{(2-v)~,,-(~-2~)~2~/3(1-~~), (W

and a similar expression for iz~.

Now the Love-Kirchoffassumption hat initiallyplane cross sections remain plane throughoutdeformation requires $8 = I?,,#+ zk,,,r so that defining he membrane forces (IV& and bendingmoments (k&J in the usual manner and using eqns (6) gives

N,I = hE(& + r&/(1 - v2) E Q {(2- v)7,1- (I- 2~)722}z/3( - v2), (7)

together with expressions for the remainder of &,a and A& which are presented in Ref. [18].

NON-DIMENSIONALISATION

The time to required to reach a uniaxial strain qc is lo = $/KT”’ according to eqn (1) for atime-independent uniaxial stress T. If the uniaxial stress T is identifiedwith the membrane stressin a spherical shell due to a pressure p, then r = pa/2h. Thus, the time to required for the creepstrain 11’ o reach the elastic strain given by 7“ = T/E is to = ~/(KET~-‘),or

to = (2/P)N_‘/KEN, P = pa/Eh. @a,b)

The dimensionless quantities

u. = k/h, w = W/h,cg = aq.slh, kale= aK.B,

CL= h/a, n.B = N,&?h, m,# = M ,plEh*,qa = Q,lEh,

e,B = aE,,dh, G.B qJE, 4 = z/h,and T = f/to, (9a-1)

together with eqn (8b) allow eqns (lH13) in Ref. [18]and eqns (4), (5) and (7)here to be written

41= P(~I- W,I), 42 = p(u2- w,2cosec @,&I = -p&i,

4 = p(u2 cot B+ u2,,- u,,~cosec 6)/2,

ell = w + UI,I+ (41~ 4*X$ + G&,

e22 w + u, cot 6 + u2,2 osec fI+ (#22+ 42)/2p,

et2 = @2,1+ u1,2 cosec 6 - u2 cot 6 + (4 + &M2/p}2,

kll = (blSl, k22 = 42,2 cosec e + 4I cot e,

k12= M2.,+ c51,2osec 8 - #2cot e)/2,

n,,,,+n,,cotfl+n,2,2cosec8-n22cot8+q,-(~,+&)n,,

- 42n12 - b#% + n22%2 cosec fin - WI + 6,) = 0,

n22,2 21112os 8 + n12,1in e + q2sin e - (42n22+ (4, + fjl)n12} in e

+ {f$(n,l + nd},j sin e/2 - P& sin e = 0,

q,,,-cqlcote+q2,2cosece-nll-n22-{(~~+~~)nll+~2n12),l

-I(b~+B~)n~1+~2n~2}cote-{(~~+B~)n~2+~2n22},2cosece-P =O,

(IOa-j)




m~~,~+m~~cot8+m12,2cosecB-mm22cotB-91t~=0,

m 22,2 2m12 cos e + r n l h l sin e - 92sin elp = 0,

where

and

,jc = 3’N+1)/2(2/p)N-I~t(N-l)/2/2

$2 = b:, +dt-u11~22+3u:z)/~,

(‘) = a( )DT.

13s

(lla-e)

(12ax)

(13)

(144

Wb)

(1W

Equations (lOa+, (1Oe-g)and (IOh-j) are the dimensionless rotations, membrane strains andcurvature changes, respectively, while eqns (lla-c) and (lid, e) are the respective threedimensionless force equilibrium equations and two moment equilibrium equations 15,181.

INITIAL ELASTIC RESPONSE

A uniformly distributed external pressure is applied “instantaneously” but quasi-statically othe entire external surface area of a complete spherical shell. Thus, the initial condition for thecreep buckling problem is the initial elastic response which is sought in this section. Onlyexternal pressures smaller than the corresponding elastic buckling pressure are of interest in

this article.The initial stress-free axisymmetric radial imperfection is assumed to have the form

G = ~P,(cos~) (15)

which is the shape of the elastic bifurcation modet [ll, 16, 17,etc], where 15 is the non-dimensional radial displacement due to the presence of the non-dimensionalradial imperfectionof magnitude l (15and e are nondimensionalised with respect to the shell thickness), and P,

(cos e) is a Legendre polynomial of degree n. The value of n is the integer which most closelysatisfies

Now,

n(n + 1)= 2ca/h, c = V[3( 1- V2)]. (16a,b)S

42 = C$ e12= kr2= n12= ml2 = 92= 712 0

for the axisymmetrical behavior of a spherical shell and all the remaining variables are

independent of /3. A solution of the governing equations is sought to first order in e using aperturbation scheme around the membrane state of deformation with

w=@++w’,andu=Eu’, (17a,b)

where I? is the radial displacement according to the membrane solution and l ’ and EU’ reperturbations of order c from the dominant membrane state. The zero order problem is the

Vrhc erturbationrocedureeveloped in Ref. [I9 demonstr ated that the critical h amtonk for the cnepbuckling of acomplete spherical shell was the same as the corresp ondi ng value for an elastic s hell reported in Ref. [11, 161 and [ l7j withY - 0.5.

$It may be show n that the governing equation s in this article also predict eqn (16a) and pr = 2Ehz/{3a’(1 - d)}l n.



136 P. C. XIROCCHAKIS and NORMAN ONES

classical membrane solution [181

G = - (1 - lJ)P/2/L, (18)

while the first order problem may be cast into the following form [ 181

(l+ Y)w’*+ v*s* - (V+ cot* e)s* + p(l+ V)W* + ${v*s* -(v +cot* fqs*

- v*w’* + (V+ cot’ e)w’*}/12- /L(l + V)CqS* WI*)

+ 2/J( 1+ V)9(S* - w* - 9*) = 0, (1%)

and

&+ 1- V)V(S - w’)/{12(1+ V)}-(2w’+v*S)/~ + EV(w’+ G - s) = 0, (19b)

where

s* = U’ , and V*( )=( )**+cot6( )*. (20a, b)

If

w’ = A,P. (cos O), and IA’ B.P, *&OS0) Wa, b)

are substituted into eqns (19) then it may be shown that the two resulting equations for A, and

B. have the solution

A, = ~~%{n(n 1) + 2u - p*A./6}/Det.,

B. = WI1 + Y+ 2(1+ u)/{n(n + I)} - ~*hn/6]/Det.,

(22a)

(22b)

where

Det. = ((1 f v)(l -@J) - p*k/12}[ 1+@ - ~*~./{12(1+ v)}]

-{(l +$/12)A, + (1 + v)~~}~~‘A./{l2(1+ v)} - 2/{n(n + 1)) -cl*], (22c)

and

An = 1 - Y n(n + 1). (224

Thus,

w = KJ+ eA,P,,(cos e), and u = eBB,PX (cos )

are the required solutions.

It may be shown[l8] by taking the appropriate limits as 0 approaches 0 or v that

(23a. )

(24a-d)

ell = KJ e{A. - n(n + l)B,,/2}, e2*= ell,

k,, = 5 +(B, - A.)n(n + 1)/2,kn = ki,,

~II= &(I - V)* cpn(n + 1)&/2(1+ v), n22 nil,

ml 1 = kq~‘(B ,, - A .)n(n + 1)/{24(1- Y)},ma = mu, (25a-h)

where the upper signs of-t or? are used at 0 = 0 when n is even or odd and at B = P for n even,

while the lower sign is used at 8 = s when n is odd.




AXISYMMETRIC CREEP BUCKLING

137

The governing equations for the creep buckling behavior of a complete spherical shellsubjected to a unif~~ external pressure are developed in Ref. [Ii31 sing eqns (10~12) here forthe case when the deformations remain axisymmetric. It turns out that these equations may be

reduced to six frrst order differential equations in six rate variables fil *, 4, mu, ic, k and d; andwritten in the form

where

and the elements of the 6 x 6 matrix A and the 6 x 1 vector P are given in Appendix I. It is

evident from Appendix 1 that the elements of matrix A depend on the vector 2, while vector iidepends on the current state of stress (rrii,a&. ObrechtfSf successfully employed a similarformtdation and choice of variables in a numerical analysis of the axial creep buckling ofcircular cylindricaf shells. It appears that the present method has not been used to investigatethe creep buckling behavior of imperfect complete spherical shells subjected to a uniformexternal pressure. A similar procedure was used previously with success to examine the plasticbehavior of complete spherical sheIls[ll] and to study the elastic behavior of shells ofrevolution in Refs.[l9-211. The advantage of this formulation is that no differentiation of the

stiffness quantities is required as noted in Refs. [S and II].

The boundary onditionsat the poles (0 = 0, n) are

BIFURCATION

Tiw conditionsunderwhich a completespherical shell which is deforming in the fundamen-tal axisymmet~c mode considered in the last section bifurcates into an ~ymmet~c shape atsome time Tb after creep commences are explored in this section. The constitutiveequations

for the incrementaldeformations rom the fundamental xisymmetric tate at the bifurcation

time 7” are assumed o be elastic since the bifurcation ccurs ns~ntaneously [~,22, etc.].

Let,

UI =~,tU;,U*=Ui,W=)2)+W’,~,=ir+~qb;,

42= #i,# = cb”, eli =&I + ehe22 = &2t 42,

ela=e;2,krr=RI,+kfl,kzz=k;2$kh,kt2=ki2,

n1t =A*l+nil,N22=~~+niz,nl2=ni*,ql=altql,

q2=qi,m= tiitl+m;t,mtz=~Titzfrnit,rnr2=rni2,

where the barred quantities now correspond o the axisymmetric tate considered in the

previous section, which inchrdesboth elastic and creep response,while the primed erms are

the linearelastic perturbationsrom the fundamental xisymmetric tate.The straindisplacement, urvaturechange and equibbrium quations for the perturbed

state are obtainedby substituti~ eqns (29) nto eqns (10)and (11)[lft].As remarked reviously,the constitutiverelationsfor the incrementalperturbationsrom the axisymmetric tate are

governedby the linearelastic relationships,The @dependencemay be eliminated rom these

equationsby usingI

u;(&8) = UT(@)in m/3,and I&& 9)= ~~~) cos rnfi Wkb)



138

together with

P. . IROUCHAKISand NORMANJONES

(w’, I$;, ei,, ei2, kih ki2, nil, ni2, qi, mix,i2)

= (W”, 4Y, 4i, 42, &I, 42, d/i, ni;, q;, rn;,, rn$J sin m/3, wd

and

(4’, 44, e iz, k i2, n it, qi, m i2) = (4”, 94, eh, 62, nh, 41, my21 03s m/3. w4

It is shown in Ref. [18] that the governing equations for the asymmetrical bifurcation of a

complete spherical Shell may be reduced to the system of four second order differential

equations

m**+m*+Gg=O ,

where the double primes are dropped for convenience,

Wd

( I* = a( Me,

jl = [UI, ~2, w, mJ, (31b)

and the elements of the 4 x 4 matrices I?, P and d are given in Appendix 2.

NUMERICALPROCEDURE

(a) Axisymmetric creepbuckling

The system of eqns (26) is solved using the central finite-difference method with a meridian

of the shell (0 5 6 5 P) divided into N - 1 equally spaced intervals with N stations. Thus, eqn

(26) becomes

(j*)i-112 + (f@)i-l/2 = PIi-I/2, 02)

where

and

%I2 = (Zi - Zi-r)/Ae. (33a, b)

If the finite-difference approximations (33) are substituted into eqn (32) then

E.-r/rZi-l+ Gi-t& = Pi-m, i = 2,3,. . . NV

where

(34)

Wa, b)

and I’ is the identity matrix. The elements of the 6 X 6 matrix Ai-rn and the 6 X 1 vector Pi-\n

are given in Appendix 1.

Equations (34) may be recast in the form

and

WW



Axisymmetric and biiurcation creep buckling of externally prcssuriscd spherical shells 139

when the vector 2 is split into the two subvectors % and pi, where J?i= [ril ,4,firi]irp fi =

Ifi9*Vli’,

(37a-e)

Now, the boundary conditions at 8 = 0 and 8 = s are given by eqns (28). The boundaryconditions at 6 = 0 may be written in the form of eqn (36b)with i = 1 provided

(38a-c)

(384 e)

Similarly, the boundary conditions at 0 = n are given by eqn (36a)with i = N f 1 when

(394 d

Thus, eqns (28) may be written

The system of linear algebraic eqns (36) and (40) in the variables Ri and Pi are solved ateach time step using Potters’ method [23,24]and the initialconditions are given by the initialelasticresponse. The integration through the thickness of the shell required for the evaluation of pi-l/tgiven by eqn (37e) and in Appendix 1 is done using the ten-point formula of Gauss-Legendrequadrature[B] which is more accurate than equal-interval ormulae (e.g. Simpson’srule) for thesame number of integration points.

The time integration of 2 is performed by using Euler’s integration scheme. It may beshown that eqn (53) in Ref. [26] can be written

AT I 4(1 + v)[~V(~&)/P]‘-~/~N (4l)t

and provides an upper bound on the time step AT to ensure numerical stability of the timeintegration scheme. If at each time step j& is evaluated at every spatial point and the value of j2in eqn (41) s identified with the maximum value of f&msxJ, then the required time step is

AT = 4f(l+ u){~~\J(~~~,,,)/P}‘-~/~N, (42)t

where f is a factor less than unity. A result similar to eqn (42) was derived in Ref. [27] byrequiring the incremental equivalent creep strain at each time step to be smaller than thecorresponding equivalent elastic strain. It turns out in this case that the factor 4f(l+ v)/3Nt ineqn (42) must be smaller than unity.

(b) BifurcationThe system of eqns (31a) s solved using the central finite-differencemethod with a meridian

of the shell (0 s 0 s n) divided into N equally spaced intervals with N + 1 stations.

tN is defined by eqns (I) and (3a).



140

Thus, eqn (31a)becomes

P. C. XIRWHAKIS and NORMAN ONES

~i ~, - _1i B, j i , +~, ~, +, , =O, i =l , 2, . . . , N- l , (43)

where

The elements of the 4 X4 matrices & fi and Q are given in Appendix 2.Now, it may be shown that &r= ~2= w = w’= 0 at 8 = 0, = when m # 1 in order for the strains

and curvatures to remain finite at the poles (8 = 0, a). Furthermore, it is also required thatm,,=Oate=O,~whenmf0,2[l8].Thus,~,=u~=w=m~~=0atB=O,~withmfO,1,2which according to eqn (31b) gives rE’ 0 when 0 = 0 and B = n. These boundary conditions

may be written using eqn (43) with i = 0 for 8 = 0 and i = N for 8 = 7r as

where & and & are identity matrices and C,Jand & are null matrices.Bifurcation occurs when the determin~t of the coefficientsof the N + 1 algebraiceqns (43)

and (45)equals zero. This determinant may be expanded in the form of eqn (2) in Ref. [28]andthe required eigenvalues located using the modified residual

PI=Bi’C~,andp, = [Bi-&~iii-iJ-‘Cii,i~2,. . . ,N -2, (47% W

The modified residual .& is computed at each intention time step for every circ~erenti~harmonic (m) up to the degree n of the Legendre polynomial in the initial elastic response.

A change in sign of & indicates bifurcation.

DISCUSSION

The numerical results presented in Figs. 2-8, except for those markedain Fig. 3. wereobtained for a dead-weight-typeexternal pressure loading for which the terms related to P ineqns (1la) and (1 b) were dropped. Moreover, he initialradial mperfectionat B = 0, which equalsQaccording to eqn (15), s always directed radially inwards (i.e. D <O at 8 = 0).

Tong and Pian[29] devefoped a fi~teelement method and examined the influence ofsymmetric imperfections on the static buckling behavior of an elastic spherical shell witha/h = 82 and v = 0.3 subjected to a dead-weight-typeexternal pressure. The numerical resultsof Tong and Pian are taken from Fig. 3 of Ref. [29] and compared in Fig. 2 here with thepresent f&e-difference calculations for the elastic case, where pr = 2Eh2&a2{3(1v2)}“*]s theclassical buckling pressure of a perfect spherical shell. It is evident that the two sets of resultsare virtually indistinguishable. t turns out that the behavior of the spherical shell in Fig 2 isgoverned by axisymmetric buckling because bifurcation to a non-axisymmetric state never

occurred at lower loads.The axisymmetric static elastic buckling pressures in pig. 2 were found using the method

described in Ref. f24] and elsewhere. If the solution vectors S;? in Appendix 3t were known at

every meridional station for a pressure p, then 2 + 82 would be obtained in the usual manner

tThe governing quations or the linear elastic case have the same orm as eqn (26) but with & and .??eplacedby &?*

and SZ, respcctivety.A few elements in the matrix A and all those n the vector P are changedas shown n Appendix 3.



Axisymmctric and bifurcation creep buckling of externally pressuriscd spherical shells

- TONG AND PIAN [29]

A PRESENT RESULTS

I 1 I I I

-0.1 -0.2 -0.3 -04 65C

Fig. 2. Comparison of present theoretical predictions for static elastic buckling of externally pressuriscdcomplete spherical shells with the finite-element results of Tong and Pian[29] (p = l/82. Y = 0.3, n = 16).

A AXISYMMETRIC CREEPBUCKLING

I

DEAD WEIGHT

x BIFURCATION ( m * 2)TYPE PRESSURE

l AXISYMMETRIC CREEP HYDROSTATICBUCKLING 1 PRESSURE

,c = -0.001

\

.\hi\\

i \’ * 0.01 ‘\\ \\ ‘A

‘A\\

\ \‘A \

i\ \

\c = 0.10 ‘A,\

\

‘r\ \ ‘A,

\

‘A.

’ ‘1

\. .\

\ ‘1\ L_, .=._, .-N---___A

‘k, ‘x:x__.

‘y,~~-‘-A__y-----d

E: = -0.50 J--_ -------__A cm -0.20---_-_A

1 I I

5 IO I5TC

Ff. 3. CrGp buck ling of externally prcssuri sed compl ete spherical shells wit h p = l/64.5 Y = l/3. N = 3

and n = 14.



142 P. C. XIRWCHAKIS and NORMAN JONES

Fi g. .Spati alnd temporal ari ati onf w ml and ni l or spherical shell with p = 1gj4.5, y = l/3, N = 3,

n = 14, t = - 0.2 and pip, = 0.06.

Fig. 5. Growth of radii displacement at pole (6 = 0) for a spherical she11with F = lj64.5, Y= 113,N = 3 andn= 14 and where@p/ p, po. 1, =

:p/pr=O.lO, E=-0.5, T’=3.0702, @p/ p, =O. 3, r=-0. 01, 7' =4. 9513S,

T' - 6.38615,@p/ pc =O. OS. =-0.5, T’=8.63307, @p/ p, =O. 2, c 1-0. 01,

T' =9. 32410, ~: pl p, ~0.1,f=-0. 1, Tr=9.91795.



*-----_-_&



144 P. C. XIKO~;CHAMS and NDKMAN JONES

1.00

t

-SPHERICAL SHELL(FROM FIGURE 31

0.75 ---CYLINDRICAL SHELL [5] T*2

P

B

0.50-

0.25-

T= I

T= 3T= 5

Fig. 8. Comparison of results from Fig. 3 for an externally pressurised spherical shell with those for an

“ equivalent” axially loaded cylindrical shell from Obrecht[S].

complete spherical shell with various magnitudes of initial imperfections. It is evident that

axisymmetric creep buckling essentially governs the response since bifurcation into a non-

axisymmetric state only occurs for a few shells with short creep buckling times and at pressures

which are quite close to the axisymmetric creep buckling results. The meridional distribution

and temporal variation of the non-dimensional radial displacement (w), dimensionless meri-

dional bending moment (ml,), and dimensionless meridional membrane force (nil) are indicated

in Fig. 4, while the typical dimensionless radial displacement-time histories in Fig. 5 reveal the

growth of the inwards displacements at the pole for several of the data points in Fig. 3. The

numerical results in Fig. 6 serve to demonstrate the influence of the creep index N on the creep

buckling pressure-critical time characteristics.

The non-dimensionalised time step given by eqn (42) ensures numerical stability of the time

integration scheme for the axisymmetric creep buckling calculations. A value of f = 0.5 was

used for all the results presented herein. Furthermore, the factor 4f(l+ v)/3N equals 0.2%.0.178 and 0.127 for N = 3, 5 and 7 respectively and v = l/3. Thus, the two different numerical

stability criteria developed in Refs. [26,271 are satisfied. Now, eqn (42) may be written in the

form

AT = 2f(l+ v)/PJMmax)],

where &(max) is defined by eqn (Ma) with $2 replaced by BArnax). The growth of the factor

&(max) is constructed in Fig. 7 for two particular cases using the largest values of $2 which

most often occurred at the integration point[25] nearest the outer surface of the spherical shell

at a pole. The associated temporal variation of the dimensionless time steps (AT) according toeqn (42) are also shown in Fig. 7. It was assumed that axisymmetric creep buckling occurred

when the criterion AT/T < 6 with 6 = 10S3was satisfied.

It should be noted that for small enough 6 then axisymmetric creep buckling could occur

prior to bifurcation for the four cases p/pC= 0.825, 0.850 and 0.875 with e = -0.001 and

plpc = 0.25 with c = -0.5 in Fii. 3. BifurcatioCnto a non-axisymmetric state occurred when

the respective nondimensional radial displacements at the poles (w(0)) of the two cases

p/pc = 0,825 and 0.875 equalled -0.07244 and -0.92249 with associated dimensionless creep

buckling times T’ of 0.87640, and 0.58761. Axisymmetric creep buckling with s = 10W3

developed for these two cases when w(O)was -3.16320 and -3.51870 with T’ equal to 0.91561

and 0.60168, respectively.



Axisymmclric and bifurcation creep buckling of cxlcrndly prcssuriacd bphcricd ~hclls I45

Some numerical creep buckling calculations for the hydrostatic pressure loading case, whichrequires the retention of the terms related to P in eqns (1 a) and (1 b) are presented in Fig. 3. Itis evident from these results that there is no practical difference between the creep bucklingbehavior of complete spherical shells subjected to hydrostatic or dead-weight-typepressures, at

least for the parameters examined.The numerical creep buckling results presented in the previous figures are replotted in Fig. 8

with the non-dimensional external prcssurc ratio (p/p,.) as the ordinate and a measure of the

imperfection (PJp,) as the abscissa. Also indicated in Fig. 8 are some numerical results for thecreep buckling of axially loaded cylindrical shells which were taken from Fig. 2 in Ref. [S].Thiscomparison is made for axially loaded cylindrical shells and externally pressurised sphericalshells with the same a/h ratios and material properties and having the same effective stresses. Itis evident that the critical times for the creep buckling of spherical shells are longer than thosefor “equivalent” cylindrical shells. A similar situation was also found in Ref. [131wherein it wasshown that the radial displacements of externally pressurised cylindrical shells grew much more

rapidly than those for externally pressurised spherical shells. In passing, it should be noted thatObrecht’s results in Fig. 2 of Ref. [5], some of which are reproduced in Fig. 8 here, predict finitecreep buckling times when p/p, = 0 which does not appear acceptable from a physicalviewpoint.

A direct quantitative comparison cannot be made between the experimental results onexternally pressurised hemispherical shells reported in Ref. [30] and the predictions of thenumerical procedure presented herein because the initial imperfections of the experimentalmodels were not recorded. Moreover, the creep characteristics of lead were not measured inRef. [30] which as remarked in Ref. [31] should be obtained for the same batch of material asused in the experimental tests. In any event, the following qualitative comparisons may be of

some value.Tong and Greenstreet 1301 ound that the creep buckling of hemispherical shells was verysensitive to the presence of geometrical imperfections. The numerical results for the completespherical shells in Figs. 3 and 8 here also reveal a strong sensitivity to the magnitude of theinitial geometric imperfections. Furthermore, Tong and Greenstreet observed that the collapseof the shells was usually very violent which is in accord with the rapid growth of the radialdisplacements near the critical time indicated in Fig. 5. It appears that the buckled shells in Ref.[30]had various deformed shapes some of which were nearly axisymmetric, while others werenon-symmetric. Tong and Greenstreet uncovered no obvious correlation between the deformedshapes of the buckled shells and life and in this regard it is interesting to observe from Fig. 3that the axisymmetric creep buckling and bifurcation pressures for spherical shells with radially

inwards initial imperfections were quite close within certain ranges of the parameters.Experimental investigations into the creep buckling behavior of boss loaded hemispherical

shells and externally pressurised shallow shells are reported in Ref. [32,33], respectively, Theseparticular cases have not been examined herein but the temporal characteristics of the radialdisplacements are similar to those in Fig. 5 in that they grow very rapidly near the critical times.

The influences of primary creep, post-buckling behavior, and material plasticity have notbeen included in the numerical method developed in this article. Murakami and Tanaka[3] haveshown that primary creep can exercise an important effect on the creep buckling behavior ofcylindrical shells. The post-buckling characterics are probably not very important from apractical viewpoint for the problem examined herein because bifurcation controls the behavior

only within the restricted range of parameters indicated in Fig. 3 for which the associatedpressures are only slightly different from the corresponding axisymmetric creep bucklingvalues. The authors intend to examine the influence of material plasticity by retaining in eqn (4)the appropriate expressions for plastic strain increments 171 ince it might be quite importantfor spherical shells without large yield stresses or having relatively small a/h ratios.

CONCLUSIONS

The numerical finite-difference results for the elastic buckling of externally pressurisedimperfect spherical shells agree very closely with the numerical finite-element results of Tongand Pian[29] as shown in Fig. 2.

Ss Vd. 16. No. 2-D



146 P. C. XIROUCHAEIS and NORMAN JONES

The numerical results in the various figures indicate that initial imperfections exercise an

important influence on the critical times for the creep buckling of complete spherical shells

subjected to external pressures. It appears from the numerical calculations that the creep

buckling behavior of a complete spherical shell is very similar for hydrostatic and dead-weight-

type external pressures, at least for the particular parameters examined herein. It turns out

from a practical viewpoint that axisymmetric creep buckling governs the behavior of the

spherical shells investigated in this article with initial imperfections described by eqn (15) but

acting radially inwards at the pole.

It was observed from the present results that the creep buckling times of externally

pressurised complete spherical shells are longer than those for “equivalent” axially loaded

cylindrical shells.

Acknowkdgements-The authors are indebted to the Solid Mechanics Program of the National Science Foundation for

their suppo rt of the work r eported herein under contract number ENG 75-13642 with the Massachusetts Instit ute o f

Technology. The authors also wish to take this opportun ity to express their appreciation to Prof. T. H. H. Pian of M.I.T.

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and G. K. Mikhail ov), pp. 124-140. Spring er-Verlag, Berlin (1973).

2. J. C. Gerdeen and V. K. Sazawal, A Rcuiew of Creep Inslability in High Temperature Pip ing and Pressure Vessels,

Welding Research Council Bullclin No. 195.33-56 (1974).

3. S. Murakami and E. Tanaka, Gn the creep buckl ing o f circular cylind rical shells. Inr. I. Mech. Sci. 18, 185-194 (1976).4. N. Jones and P. F. Sullivan, Gn the creep bu ckling of a long cylindrical shell. Inl. 1. Mech. Sci . 18.209-213 (1976).

5. H. Gbrecht, Creep buckling and postbucklin g of circular cylindrical shells under axial compression. Jnr. 1. Solids

stru ctur es l&337-355 (1977).

6. N. C. Huang, Axisymmetrical creep buck ling of clamped shallo w shells. 1. Appl . Mech. 32,323-330 (1%5).

7. P. Varpasuo, Asymmetric form o f stability loss of a shallow spherical shell with limited creep of the material. Sov.

Appl. Mech. 10,~945 (1976).8. N. Miyazaki, G. Ya8awa and Y. Ando, A parametric analysis of creep buckl ing o f shallo w spheric al sh ell by the fini te

element method. contrib uted at the Joint Petroleum Mech. Eng. und Pressure Vessels and Piping Conf. Mexico (Sept.

1976).

9. N. C. Huang and G. Funk, Inelastic b uckling o f a deep spherical shell subject to external pressure. AIAA 1.12. 914-920

(1974).

IO. S. Takezono and K. Inoue, Elastic-plasti c creep deformati ons of axisymmetrical shells. Bul loli n of JSME 19,226-232

(1976).11. J. W. Hutchinson, Gn the postbucklin g behavior o f imperfection-sensitive structures in the plastic range. J. Appl.

Mech. 39, 155-162 (1972).

12. J. W. Hutchin son and W. T. Koiter, Postbu cklin g theory. Appl. Mech. Rev. 23, 135&1366 (1970).13. N. Jones, Creep bu ckli ng of a complete spherical shell. 1. Appl. Mech. 43,450-454 (1976).

14. D. Bushnell, Bifurcation buckling of shells of revolution includi ng large deflections. p lasticity and creep. Int. 1. So/ids

Structures 10, 1287-1305 (1974); Srms, buckling and vibration of hybrid bodies o( rev&ion. presented (II

AIAAIASMEISAE 17th slruc&res, structural dynamics and materials conference (May 1976).

IS. J. L. Sanders, Nonli near theories for thin shells. Quart. Ap pl. Math. 21.21-M (1%3).

16. W. T. Koiter. The non-li near buckl ing pro blem of a complete spherical shell under unifor m external pressure. Proc.

Ken. Ned. Akad . Werens chapp en 72B, 40-123 (1%9).

17. N. Jones and C. S. Ahn. Dvnamic elastic and elastic b uckli nE of complete spherical shells. In!. 1. Solid s SIrucrures 10,

l357-1374(1974). -18. P. C. Xirouch akis and N. Jones, Axisymmetric and Bifurcofion Creep Buckling of Externally Pnssurised Spherical

Shells, M.I.T., lkpartment of Ocean Engineering Report 78-6, (Oct. 1978).

19. A. Kalnins. Analvsis of shells of revolution subject to symmetrical and nonsymmetrical loads. 1. Appl. Mech. 33,

467-476 (l&). -20. W. B. Stephens and R. E. F&on, Axisymmetric Static and Dynamic Buckl ing o f Spherii Caps Due to Centrally

Distr ibu ted Pressur es. AlAA 1.7,212&2126 (1969).

21. G. A. Greenbaum and D. C. Conroy, Postwrinkling behavior of a conical shell of revolution subject to bending loads.

AIM 1. 8,700-707 (1970).22. E. 1. Grigo liuk and Y. V. Lipovtsev, On the creep buckli ng of shells. Inl. 1. Soli ds S~ruc ~res 5, 155-173 (1%9).

23. M. L. Potters, A Mohx Method For the Solution of a Linear Second Order Difference Equalion in Two Variables,

Reporf MR 19. Mathematisch Centrum. Amsterdam (1955).

24. W. B. Stephens, Computer progr am for static and dynamic axisymmetric nonl inear respo nse of symmetrically loaded

orthotropic shells of revolution. NASA Tech. Note TN WI58 (Dec. 1970).

25. B. Carnahan, H. A. Luth er and J. 0. Wilkes, A ppli ed Numerical Methods. Wiley, New York (1%9).

26. I. Cormeall Numerical stabili ty in quasi-static elasto/visco-pl asticity . Jnl. 1. Num. Merh. Engng 9, 109-127 (1975).

27. T. Shim& Axisymmetric creep analysis by assumed stress hybr id finite-element method , M. S. Thesis. Dept.ofAeronautic s an d Astron autics, M.I.T. (1974).

28. R. E. Blum and R. E. Fulton. A modification of Potters’ method for solving eigenvdue problems involving tridiagonalmatri ces. AIM I. 4.2231-2232 (1966).

29. P. Tong and T. H. H. Pian. Postbuckling analysis of shells of revolution by the finite+lement method, 7%~Shell

Sftudures: T!~eory, Erperimennl, and lksign (Edited by Y. C. Fung and E. E. Sechler), pp. 435-452. Prentice-Hall,

New Jersey (1974).



Axisymmetric and bifurcation cr eep uckling of externally pressurised spherical shells 147

30. K. N. Ton8 and B. L. Greenstreet, Experimental observatio ns on creep buck ling of spherical shells, collected papers

on instabi lity of shell structu res, NASA Tee/t. Note D-15 10,587-S% (1%2).31. P. T. Tarpgaard and N. Jones, The influenc e of fini te-deformatio ns upo n the creep behavior of circular plates. Jnr. J.

&c/t. Sei. 14,447&l (1972).

32. R. K. Penny and D. L. Marriott, Design for Creep. McGraw-Hil l, New York (1971).

33. J . J . J . Shi, C. D. Johnson and N. R. Bauid, Appl ication of the variational theorem for creep of shallow spherical shells,

AIAA J. 8.469-476 (1970).

APPENDIX lThe elements of the matrix A and vector F en eqn (26) are

A,,=(l-u)cotB-(di+&,A,:=l,A,,=O.A,,=-ficot*B,

A,z=- pcot&A,o= - (n, r+P), Ar, =-{l +v+(~+d)*+v(0+C)cote

t (6 + &2), AZ: =cot B t a$ + 4. Azs = - 12(1- v’)n ,,/ri,

A24 =-p cot e{l +cot e(d +d)},Az.( -/~{l +cot 8(9 +i,},

Aa=- nt , - n, , cot et vcot en, , - ( 6+B) ( n, , +P), A, , =O,

A32=- l l ~1, Azt =( l - ~) ~0te, Ay=A~~=O,

AM= - pcot zU/ 12, A, ,=-(1-v")//L, A,?= A,r*O,

Au=vcotB,Aa= l+u,A~=(~+i)/~,AI,=A~2JA~)=O,

A-W=- 1, Ar5 = O, As= lip, Ah , = A, , 2 0,

Af$. r=- 20 - v”fip,A,,, = A65 = 0, Am = Y ot @,

I

112P,=cote

_,I2 wal - 2uzz)d / 3,

P, ={ltcotB(Ot/ l j ~~~C(or, -~zr~dU3+r(al l ~~~~#~{(2-v)~~, - (1-2zh&d~/ ~,

I

I I ?

Pj = cot 8 Mu,, - 2u& d413,-111

I

112P4= _,,~C{f2-v)cr,,-(1-2ufo12}df/3~,

PI%0

I

II?P,=4 _,,z#~t(2-vhr~~-(l-2u)[1235dXICL.

APPEXDIX2

The elements of the matrices J?, F and G in eqn (3la) are

~ , =~( l - ~*~ ~ ~=0, E, , =- ~( j , +~ ) / ( 1- ~*) , E, , =0,

&=O, E??=p sin e(l+p*/ 12) ~{2( i +~) }+p(n' , , +~t z) ~ ~e/ 4,

Ep=- m~~/ (12(l t~)},E~. =0,El l =E,~, E~2P: E23COSeCe,

E~~=1~~{~ +~~~0t ~e+2m~cose~~e) i {l 2~ t ~) }t ~( ~ +~ ~*/ ~~- ~~~

+ds 8. M= p, E,t = E.2 = 0, E,> = - v, E,, = 0.

F,,=Bcot~/(l-v’),F,2=- w cosecW/tl 4 + ~Wl20 f ~Mi2 +*(,I,, t I ~~~WSCCe/ 4,

~, J ~~~-~~, +(i , ~*tt~, +i , x~QI +6,~-~l -~~~te~tl t~~i ~l -~2~

t~,~t~~{(l~~)cot2e+m2cosecze~/{12(l-~)~t~P,

s4 = p, F2I = - F12 ine, Fz2 = IIs2 cot e +&i t , t fi r21in e/4,FV - mp#& +&)/{2(1- v)}+K’ cot e/12], F2, = 0,

FZ, =-~‘{cot2etm2cosec*e/(l+~)}/12-p{lt~+(j,tdj,)*

+(~,+~,~&;,+4;,+(l+v)cotB}~/(l-v~-~t,,F~~~-FrtcosceB,

FM= - ~~{t +~+( ~+~+2m2) cos~*e) c0t ei {~2( ~+~) ~

+pti,+rdi,x4Q,+i,)*+tB,+8,)c0te~i(l-~2)+p(ii~,+iillc0te),

F~~(2-v)~cotB,F ,,=v,F~=0,F ,,=-~2c~~,F~=0.

G,,=pI(~,+d;,)*+(i,+i,)~(l-tv)cotB-(~,+6,~~-v

-cot’ e- (1- v)m2 cosec2 ei 2 j i c i' ) ?( ( 14 cot’ e

+m*c0sec” Uf2)/(12(1+~)]-~~,-~~(A,,+~~2~~0~~~~e/4-~,



P. C. XIROUCHAMS and NORMAW ONES

GI? = W cosec e[& - Ml + i,, +(3 - v)cot epci v)

+ ?3 + 2v) cOt W]/{ZCI + v)) + pm@,, + &) cot il cosec 1914,

G11=~( ~~+d; ~) [ m cosec' 8/ ( 2~1tv) ) - I ~( l - ~) ]

- ~~~ f ?~~) c0t f f co~e~~8/ ( 12f l +1)) , G~g=~~( j - ~) ~0f 8,G: I =GC sin 0 -pm(At~ + ti5114, Gz2 = PI- m’co sec 8 + (I- ~){(2cos 8

- t& + ii! sin @)& +(a* -cot 8) t C& + &)+ sin 6 t cosec@}/2]/(1 - v’)

t~~sin~[-~cot~~+(I/ ?-m~(lt~)}cosec~8]/12(1t~)}

-Pi22 sin ~~~{-(~,~+~zz)c~ec @+(stt t d&)cos 9)/s-# sin 8,

G:~=m~ll/(t-v)-~t6,+C,)(Zcote-ci;,-~,)t(~,+~,)*

-(& + 4,) cot et/tzc1+ v)fl+ m/I”[ cot~ B + {m$l + v)

- II cosec2 tW2(1+ ~1) + WE&:+ m@,Gzc = pm,

G~,~~~~l+(i-m’)cosec~e}cote~12-p[(l+v){~~~~

t~~It~l~~t~ti,t~,~*t~~,+~,~cOteHd;,t~,t”cole)

+(~,+d;tH(~,+d;O*-vcosef8}

-(I - v)m7& +didcosec~8/211(1 -v9-pbit t +til r cot e),

Gc=- p’m cosec e{2 t Y+ (It v)(cot? e -d cosecz e)}/{lz(t f v)}

tpmcosec8(ltut~(~Itd;,)L

+(I - Y#& •~,~~~+ i, -cot ff)~2}~(1 v’)+pi cosec ef&>,

Gn = dm’ cosecZ 8(2 cosec’ 0 +(I + v)(l+2cotS B -m’cosec? e))/{12(1+ v))

-CLm’cosec’122t~1-(2+(6;,tcjri)*

+(&I + &cot @Ml -VI- m*(& +&)‘cosec’6/{2(1 t u)]],

C&4=-9

&( I - u + vm’ cosec- 6). Gdt= Y? ot 8.

G,z=- v’rn cosec 6, GM = u2m2 cosec’ 0, G+, = - IZu( I - u’)/F?.

APPENDIX 3

me rate formu lation whic h was used to derive eqn (26) is not suitable for the static axisymmetrk elastic bucklin g case.

Thus, it is assumed that n,~~‘~‘=n,~“+~n~,, where ntt(” is the value at the pressure p, whik &II, is the incremental

change in n,, when the pressures increased by an incremental amount t0 p + 6p. simib ulY+ q” **‘= q” ‘+ & m~%~~+@‘~

ml1 P’+&,, u’p+W’= u’P’+ &u, ,,JP+b) = @“+ &wzad ,““p = 4(pk+ &$. In this circumst ance, eqn (26) for dead-weight

type pressure loadi ng is replaced by S.$!* + &.Z? = P, where 82 = I& 1, 8q, ckn,,, 6~. 8w, &#]‘, i is given in Appendi x 1

except

~,~=-~~,,Aa=-(ltv+~*t~COt~t(~td~~~~~~~-~’~mc,~~c).

&= - {&$ + J)n,, - q t gu cot’ B+ pv cat Q).

and

PIr-n~,-{(1-“)COfe-~~tdHnr,-qt~uCo~2e+P~~~fe~

p~~~-q*t{1tY+b;*tvd;cotet~#t~~*~~,~

-(co~~t~t~)q~12(1-~zfnr,m~~/~.t~{l+~~+~)CO~~8)~~~~~+~l~~~~~~~~o~~~w~

P2= -m~,tq~p-(l-v)m,,cote+~co1*e/12,

PA=--U’S (l-v’)“,,lp-“ucote-~1~V~W-(rb12~~~~/~~

P5 =-w*+U-#&

P~=-~*+l2~1-Y~~m,l~p-~cot~~

where the superscripts p have been omitted for ConvenienCe.

xirouchakis1980

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