on the imperfection sensitivity of complete spherical shells

Computational Mechanics (1987) 2, 63-74 Computational Mechanics © Springer-Verlag 1987

On the imperfection sensitivity of complete spherical shells

M. Drmota, R. Scheidl, H. Troger and E. Weinmiiller Technical University Vienna, Karlsplatz 13, A-1040 Vienna, Austria

Abstract. The imperfection sensitivity of elastic complete spherical shells under external pressure is studied for axisymmetric deformations and qualitatively different types of imperfections by means of a numerical analysis of the Reissner shell equations. It is shown that strong reductions of the critical load are obtained for small deviations of the middle surface of the shell from the perfect spherical configuration whereas imperfections of the shell thickness do not have a substantial influence on the critical load.

1 Introduction

When dealing with buckling problems of thin-walled elastic shells it is well known (Koiter 1969; Lange and Kriegsmann 1981) that the post-bifurcation behavior is critical, i.e. an initially unstable post-buckling path is obtained in a load-displacement diagram. Hence these structures are imperfection sensitive, which means that small imperfections both in geometry or loading can result in a strong reduction of the bifurcation load obtained for the perfect system. For simple structures like beams, frames and plates where bifurcations at simple eigenvalues or multiple eigenvalues with low multiplicity (mostly double eigenvalues) occur, the imperfection sensitivity can be understood completely with the methods of classical bifurcation theory (Liapunov-Schmidt, catastrophe theory). For shell structures on the other hand the situation is considerably more complicated. The reason for this is basically due to the fact that for shell structures the methods of classical bifurcation theory do not apply because of the occurrence of closely spaced eigenvalues. Here, as it is done in Koiter (1969), Lange and Kriegsmann (1981), Gr[iff et al. (1985), singular perturbation methods have to be applied in order to be able to obtain estimates for the post-buckling path in a load-displacement diagram for the perfect structure. However, to calculate the complete post-buckling path, numerical methods have to be used in general. An approach which we shall adopt in this paper too.

While, for example, it is well known from the classical buckling theory of rods and plates that only such imperfections are dangerous in which there are contributions of the relevant buckling modes (Chow et al. 1976) the imperfection sensitivity problem of shells is not that straight forward. It is well known from experiments (Berke and Carlson 1968) that for the complete spherical shell the final buckled state is always a single dimple, which is a strongly localized inward deflection at one pole and no deflection at the other pole. This also has been verified by a numerical analysis in Gr~iff et al. (1985). Thus the question arises which imperfection is the most dangerous one in order to lead to a single dimple deformation of the shell. This question is also of great practical importance because by DAST (1980) regulations are given how for different shell structures certain reduction factors have to be selected to account for possible imperfections and to reduce the bifurcation load evaluated for the perfect structure: We shall show that such reduction factors are influenced both by qualitatively different imperfections and also by their amplitude and therefore one single reduction factor for one structure as it is given in DAST (1980) can lead to very conservative estimates. Especially, we are able to show that the form of the imperfections has an important influence on the imperfection sensitivity of the structure.

64 Computational Mechanics 2 (1987)

2 Shell equations

We restrict our investigation to the case of a linearly elastic, homogeneous, isotropic material and further on we allow only axisymmetric deformations of the shell. This latter assumption is not as academic as it might appear at the first moment, because frequently the production process of spherical shells favours axisymmetric imperfections which create a preference for axisymmetric buckling patterns. Furthermore the experiments in Berke and Carlson (1968) show that even for nonaxisymmetric buckling patterns the initially forming patterns are axisymmetric and therefore for the evaluation of the maximum load the axisymmetric theory will be valid. Often the support of the shell too creates a preference for axisymmetric deformations. Moreover it is shown in Knightly and Sather (1980) that the general case of buckling of a spherical shell without any restrictions concerning the deformation leads to a very complicated mathematical problem due to the high multiplicity of the critical eigenvalue. The multiplicity is of an order of magnitude of about 200-500. This is certainly one of the reasons why effective numerical methods capable to handle the postbuckling behavior, at least to our knowledge, are not yet available.

It is explained in Gr/iff et al. (1985) that we do not have to expect flutter type of instabilities under static loading and therefore the so-called Reissner shell equations (Reissner 1950) are an appropriate set of shell equations which allow large displacements, but small strains. With the notations from Fig. I we obtain two nonlinear ordinary differential equations in the two variables ~ and fl, where, as usual ~ is a stress function being proportional to the horizontal stress resultant H by (Fig. 1)

~ = r H . (1)

fi is a deformation variable, measuring the angular differences between the tangent to the deformed shell and the undeformed shell at the meridian 4. The shell equations read

fi" + fi' (ctg ~ + 3 ~ ) + ctgZ ~ c°s ( ~ - fl) (sin ( 4 - fi) - sin z

b' sin ( ¢ - f l ) - s i n 4] v cos (¢ -f l) - cos ~ t- 3 - - (2) - sin4 6 s i ~ J

1 sin(g-fl) 4 )~ cos(g-f l ) ¢ = - ~ 0 sin4 6 sin4 j" sint/cos $(t/)dt/ ,

0

3 !

1 cos (4 - c o s ¢5 sin I_c - 4 2 ~ sin tl cos ~(tl)dt 1 os (~ -fi) sin (~ -fl)

o s in2 4

v s - ~ (1 - f l ' ) s i n (4 - f l )+~- cos(4-fi)

' ( )1 -t- V-. (1 --fl ') COS (~ - - f l ) - -~ - sin (~ --fl) sin 4

(3)

+4)~ v s i n ( ~ ; - ~ + f i ) - s in4sin~ +4),(sin2~sin~;)' sin ¢

where dimensionless quantities corresponding to the following relations have been used

2=pipe with pc=4Ea~l/ /12(1-va). (4)

2 being the loading parameter and

= (t/a)/]//12(1 - v 2) (5)

being a geometry parameter. If we have a geometrically imperfect shell either due to variations of the thickness t(~) or due to variations of the radius a(~) (Fig. 1), c~ will depend weakly on 4 (6(4)=~5o +A3(4)). In (2) and (3) also a dimensionless stress function

~b =4~/(aZp,) (6)

M. Drmota et al. : Imperfection sensitivity of complete spherical shells

1

# -",\ \Beod [ood FolLower- force [ood Fig. 1. Notations for the spherical shell

65

is used. By means of the function/}(~) in (2) and (3) we are able to represent two different types of pressure loading by selecting/} appropriately

~(~)-- ~ dead loading ,

4;(~) = ~- f l (~) follower force type loading .

The boundary conditions are

(7)

(8)

(9)

(10)

(9) follows immediately from the definition of ¢ in (1), because r (0)= r(rc)= 0 (Fig. 1). (10) follows from the symmetry of the problem. (2) and (3) are more complicated than the corresponding equations in Grfiff et al. (1985) because of the dependence of t and a on 4.

3 Numerical methods

As already mentioned in Section (1) we cannot expect to obtain analytical solutions of(2), (3) with the boundary conditions (9), (10) for imperfect shell geometries or for other types of loading than constant pressure loading. Therefore we seek to solve the problem numerically.

In Grfiff et al. (1985) the buckling problem of the perfect shell has been treated by a finite difference method. However, in the course of the computat ions for the imperfect shell it turned out that this method did not always furnish the required accuracy. This, basically is doue to the relatively low convergence rate. This could be remedied by decreasing the stepsize h with the unwelcome consequence of high storage and time requirements. Hence as a second numerical method a collocation method implemented in COLSYS was applied. Below we motivate our choice of this code in detail.

3.1 Finite differences

We rewrite (2), (3) in the following form

y"(~)=f(4, y(~),y'(~),2), 0 < ~ < r c , (11)

where y (4) = (Yl (~), Y2 (4), Y3 (~))T : = (fl (~), ~b ((), I(~)) T and 1(4) = j" sin q cos/~ (t/) dr/ . 0

(11) is a system of second order differential equations for/3, ~, and l a n d the equation for / fol lows from its above definition and reads with ~(t/) given by (8)

I" (~) = cos (2 ( - fl (4)) +/3' (~) sin ( sin (4 -/3(~)) •


Clearly, if ~ 01)= r/according to (7), 1(4)= (1 - c o s 2 4)/4 and there is no integral term in (11). The boundary conditions for (11) are

fl(0)=fl(Tt)=0 , ~(0)=~b(n)=0 , I ( 0 ) - I ' ( r 0 = 0 . (12)

For the numerical solution of (11) with (12) we divide the interval [0, rt] into N subintervals of length h = x / N . On approximating the derivatives of y by symmetric three-point finite differences we obtain the following nonlinear difference scheme

( Y , + t - 2 y , + Y i - t ) / h 2 = f ( ~ i , y , , ( y ~ + l - y i - x ) / 2 h , 2 ) , i = l , 2 , . . . , N - I , (13)

where ~i = ih are the grid points and Yi = (Yl,i , Y2.i, Y3,i) "r "~ Y(¢i), i = 1, 2 . . . . . N - 1. Since for i = N, IN = rt and (13) is not well-defined, we cannot use D ~ ( y ) = (YN + 1 - YN- t)/2h for the approximation of the first derivative of y at ¢ =Tt (Weinm~ller 1986b). In this case we would need to know

lim f (~ , y(¢), y'(~), Z)

and this may be quite difficult to obtain in practice. Thus we use/3~(y) = (3 YN - -4 Y N - t + Y s - 2 ) / 2 h . Both D~ and D~ are O (h 2) approximations for y'(rt), i fy is sufficiently smooth. Clearly, the boundary conditions for (13) are

Y0 =0 , Ya,N=Y2,N=0 , (3Ya,s --4y3,s-1 +y3 ,N-2) /2h =0 . (14)

On setting x = ( y o , Y l , . - . , YN) we can write (13), (14) as a nonlinear algebraic system for x

F ( x , 2) = 0 . (15)

It follows from (2) and (3) that the derivatives of f i n (11) with respect to y and y' are unbounded for ~ 0 and ~-~n; the problem is in this sense singular and we cannot use the classical theory to answer the questions of stability and convergence for the difference method described by (13) and (14). The study of the linear and nonlinear singular case may be found in Weinmihller (1984, 1986a). In order to solve (15), we usually apply the continuation method in ~.: if(x k, 2 k) is a solution of(15) then we choose 2 k+ t and use x k as a starting value for the iterative calculation of x k+ 1. This procedure, however, breaks down if we approach a limit (turning) point in the curve u = (x, Z), (Fig. 2), in which the Fr6chet derivative 8F/~x becomes singular. In such a case we modify the numerical algorithm by employing the method due to Schwetlick (1979). Assuming that the solution u k is known, we obtain u k+~ in a process consisting of the following two steps.

vk+l~

?ay ~kq

/ AU %~ X ~ = .1 _Xk U~ k v~ AS k : Ak4-A k

A* A

Fig. 2 and 3. 2 Notations for the numerical method to calculate a turning point. 3 Explanation of the parameter change 2 ~ cr for the numerical method

M. Drmota et al. : Imperfection sensitivity of complete spherical shells 67

First we solve the linear system (~F/~u)(uk)vk= O, where the matrix ~F/~u is assumed to have the whole rank. We choose v k in such a way that (Vk-1)Tvk>o and IIvkl[2 = 1, where 11![2 denotes the Euclidean norm of a vector.

Secondly the predictor value is defined by u k'° : = u k + ~k vk, where O-k is a real number which has to be prescribed yet. The corrector values u k'~ are now obtained from the Newton iteration

~ (uk'i)(u k'i+l --uk ' i )= - F ( u k'i) , i=0 , 1 . . . . , J - I ,

( v ~ ) ~ (u~, i +1 - u ~ ) = ~

which terminates if uk'J:=U k+~ meets the accuracy requirements. illustrated in Fig. 2.

Condition (16) means geometrically that all corrector values hyperplane

~ = {u: ( v ~ ) ~ ( u - u ~) = o~} .

On the other hand (16) can be interpreted as a transformation ( k x k, A2k)-+(Au k, o-k), where

u k+x - -Uk=akVk+Au k and

(vk)T(u k+l --Uk)=~k , (vk)TAuk=O , or equivalently as a parameter change 2-~o', (Fig. 3).

(16)

The numerical method is

u k'i ( i = 1 , 2 , . . . , J) lie in the

3.2 Collocation

Many standard codes for solving classical boundary value problems for systems of ODE's cannot be used for a singular case, because of the possible evaluation of (2) and (3) at ~ = 0, n, in particular if one has to discretize the first derivative of y at the boundary points. We could remedy this situation by choosing a program which is written for systems of first order differential equations (most of the routines), but then we probably would have to expect storage and time difficulties, due to the larger dimension of the first order system, (at least 5 equations for follower force type loading) and possibly fine grids caused by rather rigorous tolerance requirements and strongly varying solutions, (Fig. 10 in Gr/iff et al. 1985). Finally, we wished to apply a method which remains reliable if it is used for solving singular problems.

Here we present how we have realized the "path-following" for the shell problems by a proper adaptation of an efficient collocation code. In Weinm~ller (1986b) stability and convergence rates for collocation methods have been discussed. It turns out that the stability of the method and the classical convergence order can be shown for model singular problems, which have similar mathematical properties as (2) and (3). For the numerical solution of (2), (3), (9), (10) we use COLSYS (Ascher et al. 1978; Steindl 1981), a collocation code for variable order systems of ODE's.

Let us explain the idea of collocation with the following parameterfree boundary value problem

y " ( ~ ) = f ( ~ , y ( ~ ) , y ' ( ~ ) ) , 0 < ~ < n ,

b (y (0), y'(0); y (n), y'(r0) = 0 ,

where y is a vector of dimension n, f i s a nonlinear vector-valued function of dimension n and b is a nonlinear vector-valued function of dimension 2n. We consider a nonuniform partition of the interval [0, ~]

A= {~o, ~1 , . . . , ~N;0= ~0 < ~1 < . . . < ~N=~}

and seek to approximate y by a function PAe CX[0, n] which on each subinterval [~i,~i+1], i = 0, 1 , . . . , N - 1, is a polynomial Pi of degree m + i. For this purpose we place in each subinterval m collocation points


and require collocation equations

P'i'(~ir) - - f (~ir, Pi(~ir), Pf (~ir))-----O,

continuity conditions

,

and boundary conditions

b (P0 (0), r ; (0); pN - 1 = 0

i=0 , I . . . . . N - 1 ; r = 1 , 2 . . . . ,m ,

i = 0 , 1 , . . . , N - 2 ,

to hold. This is a set of nNm + 2 n (N - 1) + 2 n = n (m + 2) N equations. Since Pi can be represented as m + l

pi(ff)= ~ c,r(~-~i) r, i=0 ,1 . . . . . N - 1 , r = 0

and each c,r is an n-vector, there are N(m + 2)n constants necessary for the determination of PA. COLSYS is based on collocation at the Gaussian points, which are zeros of Legendre poly-

nomials (Table 25.4, Handbook of mathematical functions, 1965). Since, for the Gaussian points ~i < ~it and ~i,, < ~i + 1, or equivalently, there is no evaluation of (2) and (3) at ~ = 0, re, COLSYS can be applied for singular systems.

Now we introduce

y3(~) :=I (~)=~ sin r/ cos (rt - fl(tt)) dr / , 0<~_<~ , y4(~) :=2 0

and we extend the system (2), (3) by y~ (~) = sin ~ cos (4 -fi(~)), Y~(¢) = 0, and boundary conditions Y3 (0) = 0, l [y] (a) = 0, 0 < ~ < x, where l[y] (a) is a linear functional of y (4) = (fl (~), ~P (~), I(~), 2) r which has to be specified yet and ~ is a fixed point in the interval [0, r~].

Assuming yk to be given, we obtain yk+l by choosing

~ykt+X(a)-- fl if ~k + 1 ~,'~ma x l[yk+l](CC)=[yk4+l(O)--~o otherwise '

where 0 < ~ < r~, is such that max I fl~l ~- I flk(a)[, and fi, 2 are prescribed values. This means, that we l<_ i<_N- i

realize the continuation method in )~ away from the turning point and the continuation in fl(a) near to the turning point 2 ~- '~max. We have seen that for imperfect problems treated by the difference method

remains almost constant in a small neighborhood of the turning point and is located in the middle of the first hump of fl either near to ~ = 0 or ~ = re.

It must be underlined that we could successfully use COLSYS (as a black-box), to realize the "path-following" strategy in such a simple way, only because we knew quite well about the solution behavior near to the turning point. Furthermore, we have to remark that continuation methods are based on the assumption that the branch of solutions of the boundary value problem to be followed is a smooth function of the parameter

Bx= (z(,~), 2) for the continuation in 2 and

Bp =(z(fl), 2(fl)) for the continuation in fl(e) ,

where z--(fl , ~ , I ) . Clearly, the efficiency of the "path-following" procedure depends on the magnitude of dBa/d2 or dB~/dfl for the continuation in ): or/~, respectively.

4 Results

In our numerical calculations we basically take two different classes of imperfections. The first are geometric imperfections and the second are imperfections in the loading of the shell. All the results below have been obtained by the collocation method except in two cases where the difference method was used. In all figures the dotted line gives the post-buckling path of the perfect spherical shell.

M. Drmota et al. : Imperfection sensitivity of complete spherical shells 69

4.1 Geometric imperfections

Here we distinguish between imperfections of the form of the shell expressed by the radius a(~) and imperfections of the thickness of the shell t(~).

4.1.1 Imperfections of the form of the shell

We study two qualitatively different cases. First a global imperfection and secondly localized imperfections giving the deviation of the middle surface of the shell from the perfect spherical configuration.

(i) Global imperfection expressed by the relation

a(¢) =ao( l + e sin ~) . (17)

The results are given for 6o = 3.69 x 10 .3 in Fig. 4. It is clearly visible that for a shell for which at the equator the radius is larger than for the perfect sphere the maximum load is strongly reduced. For ]l fi ]lo~ we use the maximum norm given by

max [fli[ (18)

To explain the physical significance of (18) which is used in the figures we recall from Graft et al. (1985) that in the asymptotic limit 6 ~ 0 (Eq. 5) we obtain I] fl H ~ = 2 ? where ? denotes the angle giving the position of the boundary layer for the single dimple solution (Fig. 12).

(ii) Local imperfections Two different types of local imperfections are selected in accordance with Berke and Carlson (1968) in order to create a preference for the single dimple. It is assumed that these are the most critical imperfections.

a(~)=ao(1-eexp( -~ /k ) ) ; (19)

we select in this case k = 0.05 and furthermore we keep e = 0.01 fixed, but we vary 6o. The results are given in Fig. 5, from which the influence of an imperfection of constant magnitude on shells with decreasing thickness is clearly visible.

( e ) a(~)=ao 1 cosh(~/k) '

for c~0 = 3.69 x 10-3 and k = 0.05 the influence of the amplitude e is given in Fig. 6. This imperfection is strongly localized as it may happen due to an impact of the shell against a sharp corner. It is quite interesting to note that the amplitude of the imperfection does not have a very significant influence on the buckling behavior.

4.1.2 Imperfections of the shell thickness

Again we distinguish between global and local imperfections.

(i) Global changes of the shell thickness are expressed by

t (~)=to( l + e c o s 2 ~ ) . (21)

For 60 = 3.69 x 10- 3 the results are shown in Fig. 7 for positiv and negativ values of e. In both cases the influence on the critical load is very small.

70

\ \

0.8 \ \ e : u.w \

\

! ?X i O.B \ / /

c:~ c:n c:5 c::~ c 5 ~ ' c:5 c:5

i-{~>

Computational Mechanics 2 (1987)

7.@1~_, \ \ \

0.8 \ \ \ \ \\ 5 = 3.69.10 .3

0 .6 X / / o \ /

/"'~}.~'" / ~So = 2.39.10 -s <0.4- / ~ \ ~ \ /

\ x " hO =1'08'10-3

@ ~'~r- I ~ r ~ ~

c:5 c:5 c:5 c 5 ~ ~"

ili~il oo ~ -

5

112:~

QI \ \\

0.8 xx

\

e = 0.005 \

/ . . . . :o.oz

O- i l i r i I i C >

IllSlloo 6

1.0 / e = 0.1

0.8. ~ e = - 0 . 1

\

O.g- \\ \ \

0.4 2 ~\ \ \ N

O.Z-

@

\ \

j j j I 3 > L o

Ii,alco

1.0 -

0 .8 -

0 .5 -

0.4-

0.2 -

- - , e = 0.1

O[ ~ ~ ~ L ~ , ~[>" ¢s:) ~ - - a - ~ c-4 L o o

c D o c::; c::5 cs; c:z; , : ~

115il~ ,,,-

Figs. 4-8. 4 Influence of the amplitude e on the load carrying capacity of the globally shape- imperfect shell according to (17). 5 Influence of the geometric parameter 6o (5) on the maximum load for a locally shape-imperfect shell according to (19). 6 Influence of the amplitude e on the maximum load for a locally shape imperfect shell according to (20). 7 Influence of the globally imperfect shell thickness according to (21) on the load carrying capacity. 8 Influence of the locally imperfect shell thickness according to (22) on the maximum load


Pfl 9

Figs. 9 and 10. 9 Notations for the supported sphere. 10 Imperfection sensitivity of the spherical shell carrying its own weight

1.0-

0.8-

I 0.5

0.4

0.2-

10

\ \ \

\ \

71

\

] ] i i ~ I I I ~ >

I1}1[~ . . . .

(ii) A local imperfection of the thickness of the shell which creates a preference for the single dimple is assumed to be

t(~) = too - e exp ( -50~z) ) . (22)

For c~ o =3.69 x 10 -3 the results are given in Fig. 8 for two values of e=0.1 and e=0.2. Again only a minor imperfection sensitivity is noticeable.

4.2 Imperfections in the load&9 of the shell

We treat here one, especially from the practical standpoint, important case, where we include the pressure distribution resulting from supporting the weight of the shell (Fig. 9) by a ring shaped support.

For Pa we obtain for ~ ~ 1

pa = 7t/(E sin cz cos c0. (23)

We see from Fig. 9 that

Pn = - P sin ~ (4) , Pv =P cos ~ -7t +p j ( 4 ) , (24)

wheref(~) = 1 for c~ -E _< ~ < c~ + e andf (~) = 0 elsewhere. Introducing Q = 7t/pc and O=pa/Pc we have to replace the term 2cos ~; by 2cos ~ - ~ + O f ( t / ) , everywhere it shows up in (2) and (3).

For c~ = n/6, ~ = 5 x 10- s, 0 = 2 x 10- 3 we obtain the result given in Fig. 10. It is quite interesting to see how the deformation pattern on the shell changes during the loading process (Fig. 11). For increasing values of 2 in Fig. 10 and 2 sufficiently far away from 2ma x the influence of the support of the shell is clearly visible in the deformation/? in Fig. 11 (2 = 0.4). However, if 2 takes values in the neighborhood of 2max (2max = 0.9893) a buckling pattern is created on the shell which is similar to the spherical harmonics, which are the solutions of the linearized eigenvalue problem for a perfect sphere. After moving through the limit point we reach the unstable part in the load-displacement diagram of Fig. 10 and the corresponding deformations are given in Fig. l i b for several values of 2. For example for 2=0.5 or 2=0:0708 it is clearly visible that in the scale of these figures no influence of the support of the shell is noticeable and furthermore for 2 = 0.0708 a very localized deformation is shown. It corresponds to the single dimple in Fig. 12.

5 Conclusions and summary

By means of numerical methods explained in Section 3 we have calculated the load displacement diagrams for various imperfections accounting for changes both in geometry and loading. There is a significant difference in the load carrying capacity of the shell whether the geometric imperfection

?2 Computational Mechanics 2 (1987)

Z = 0.4000

1.51 =4

!0 _1.51-4

0 0,785/+ 1.571 2,350 3,142

4.01 4

2, = 0.8300

_4.81-4 0

0

0,7854 1,571 2,356 3,142

0.4004

t O

-0.4004

1.31-3

0.7854 1.571 2.356 ~ - - - - -

Z = 0 . 9 7 5 0

3,142

I0 AA

_1.31-3 i 0 0.7854

A •

k / v

I I 1.571

~ - - - - - . . 2.356 3.142

0.8303

i0

-0.%03 0.785/+ 1.571 2.356 3.142

~ __.__.._

Z = 0.9893 7.02 4

I _7,02 -3

d v - ~ --- v -.-,, V

0.7854

A A A , ~ , A / I

I _ _ I I i I ~ l 1.571 2.356 3.142

0.g755

t0 "I -0.9755

O 0.7854 1,571 2.355 3,142

09902 I

_0.99021 i ~ i i O 0.7854 1.571 2.356 3.142

Fig. 1 la and b. Deformation pattern on the shell of Fig. 10 for rising and falling values of Z. ~. = 0.9893 gives the maximum load. For )~ = 0.0708 the single dimple solution is clearly visible (there are different scales for different values of Z used)


=0.9777 0.0109 0.0236

Z = 0.9200

73

! 0 ~ v " ~ -I -0.0109 I ! P t I -0.0236

0 0.7854 1.571 2.356 3.142

0.9782

i0 -0.9782

0 0.7854 1.571 2.356 3.142

~, = 0.5000 0.144

T

I / 0 0.7854

I [ 1.571

...A

P I I

2.356 3.142

O . 9 2 O 3 [ ~ ,

-0.9203f r ~ ! ~ r i I 0 0.7854 1.571 2.356 3.142

Z--- 0.0708 2.001

i -0.144 I

0 I F I

0.7854 1.571 ~L

I I " - - ~ 2.356 3.14.2

-2.001 I

0.7854 1.571 2.356 3.142

0'50021 i

I01

-0.5002 0 0.7854 1.571 2.356 3.142 ~ -____,._

5

0.0511

C-

~.0511 0

I I F r I [

0.7854 1.571 2.356 3.142

concerns the form of the shell or the thickness. In case of an imperfection of the form of the middle surface (17, 19, 20) a much stronger imperfection sensitivity is found than in the case of an imperfection of the thickness (21, 22) where only a minor influence on the maximum sustainable load could be noted. The example of an imperfection of loading, which we considered showed also only an influence of minor significance.


Fig. 12. Single dimple solution with a boundary layer at y corresponding to 2 = 0.0708 of Fig. 11

The main conclusion which we are able to draw is that one has to avoid deviations from the perfect form of the middle surface both globally and locally because in both cases a strong imperfection sensitivity has been found (Figs. 4-6) whereas deviations in the shell thickness do not have a big influence.

Finally also a comment on the worst imperfection concerning the buckling problem should be made. Whereas in the buckling of rods and plates only imperfections which contribute to the buckling modes obtained from the linearized problem are relevant; the situation is quite different for shell buckling problems. For the present problem the buckling modes of the linearized problem are spherical harmonics (Fig. 11, 2 = 0.9893) of high order which do not have any significant influence on the imperfection sensitivity. This is due to the fact that we have closely spaced eigenvalues and the classical buckling theory must be replaced by singular perturbation theory. The resulting deformation is furthermore characterized by the occurrence of boundary layers (e.g. Fig. 11, )~=0.0708 and Fig. 12) with strong variations of the variables in small domains.

Thus in conclusion one could say that not the eigenfunctions of the linearized buckling problem determine the shape of the most dangereous imperfections but rather the final shape of the deformed shell which is the single dimple as it is shown in Gr~iff et al. (1985).

Acknowledgements

We would like to thank the referees for their valuable comments. This research project has been partly supported by the Austrian Fonds zur F~rderung der wissenschaftlichen Forschung under project P5519,

References

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Berke, L. ; Carlson, R.L. (1968) : Experimental studies of the postbuckling behavior of complete spherical shells. Experimental Mech. 8, 548-553

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Communicated by H.A. Mang, May 19, 1986

on the imperfection sensitivity of complete spherical shells

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