A comparison of the postbuckling behavior of plates and shells

0045-7949187 s3.M) + 0.00 Pcrgamon Journals Ltd. Compvlrrs 6 S~rurrvrrr Vol. 11. No. 1. pp. 157-163. 1987 Printed I” circa hilam A COMPARISON OF THE POSTBUCKLING OF PLATES AND SHELLS BEHAVIOR R. SCHEIDL and H. TRCGER TU-Wien, Karlsplatz 13, A-1040 Vienna, Austria Abstract-There is a basic difference in the mathematical treatment of postbuckling of plates and shells. Whereas for plates the classical methods of bifurcation theory (Liapunov-Schmidt method, Singularity Theory) in general are applicable, these methods fail for the analysis of thin-walled shells. For shells in general solutions of the boundary layer type are found and hence the methods of singular perturbation theory are required for an appropriate analysis. Applications are given for the postbuckling of a rectangular plate and a spherical shell. I. INTRODUCTION The analysis of the postbuckling behavior of thin walled structures constitutes a major challenge both to theoretically working engineers and to applied mathematicians. On the other hand, an understanding of the postbuckling behavior of a structure is very important to the practically working engineer because only then can be answered the question of imperfection sensitivity, both qualitatively and quantitatively. Nonlinear stability theory of elastic structures in a general setting started from an engineering point of view with the work of Koiter [I]. The ideas in [I ] are formalized and more generally presented in what is now called the Liapunov-Schmidt method [2] and Singularity Theory 124. The former furnishes a way of reduction of the infinite dimensional (differential) system of describing equations to a finite dimensional (algebraic) system of bifurcation equations which, locally, has the same solution set as the original system. The singularity theory provides the answers to two important questions: (1) up to which nonlinear terms must be included in the bifurcation equations, and (2) which solution set is the most general perturbation of the bifurcation equations. The answer to the latter question is provided by the theory of unfoldings and also solves the imperfection sensitivity problem. These methods, however, are practically only meaningful if the structure of the eigenvalues of the linearized problem at loss of stability has certain properties which are met for plate buckling problems, but unfortunately, are in general not fulfilled for shell buckling problems. Let us consider this statement in a little bit more detail. In engineering systems it is desirable to have shell structures with maximum symmetry in order to save production costs. These symmetries, however, are responsible for high multiplicities of the critical eigenvalue and, as it will be explained below, lead to a high dimensional bifurcation system. For example, for buckling of a spherical shell (Fig. 1) in the elastic range multiplicities of the critical eigenvalue of the order of ‘Y10’ occur [5]. The interaction of many buckling modes and the initially unstable postbuckling behavior also lead to a strong localization of the buckling pattern. But even if, for example, a geometrically imperfect structure is studied and the high multiplicity is strongly reduced mathematically, the eigenvalue distribution still shows a nasty property because then the eigenvalues are closely packed together. This can be best studied for a spherical shell with only axisymmetric deformations (61. Here mathematically only simple or double eigenvalues can occur. However, the double eigenvalues are closely spaced (Fig. 2). This has the 10" Fig. I. Notations for the axisymmetrical spherical shell under uniform external pressure. q' 6. 1p” 6 Fig. 2. Eigenvalue curves for the shell of Fig. 1 in a load i, and thickness/radius 6 [eqns (19). (20)] diagram showing closely spaced eigenvalues for small (realistic) values of 6. I57 158 R. SCHEIDL and unpleasant consequence that the range of variation of parameters after loss of stability is so small that ,it is practically meaningless. Furthermore it is well known that, for the shell buckling problem for small thickness (and only this case is meaningful for a completely elastic behavior), a singular perturbation problem is obtained [6-81. These singular perturbation problems are characterized by the occurrence of boundary layers, which is another explanation for the strong localization of the buckling pattern. A famous example is the diamond shaped buckling pattern for a cylindrical shell under axial compression. For rectangular plates, on the other hand, only double eigenvalues are found in the most degenerate situation for the linearized problem. Furthermore, close spacing of eigenvalues does not occur. Finally, in general, thcsc problems are not singularly perturbed. Thus the classical methods work very well. In fact only the use of these classical methods has enabled the discovery that the boundary conditions for a rectangular plate play an important role concerning its nonlinear stability behavior (Fig. 3). There exists a challenging problem, the so-called ‘mode jumping problem’, which is solved in [9]. Furthermore, a systematic study of the imperfection sensitivity of plates is only possible via Singularity Theory because in this case a universal parameter unfolding already requires eight parameters [3, p. 3171. This means that one has to embed the problem in an eight-parameter family in order to get all qualitatively different solutions. As it is not easy to find eight physically meaningful parameters we use instead the concept of Restricted Generic Bifurcation as it is developed in [IO], where an unfolding with four parameters is done, and which is perfectly suitable for engineering purposes. Whereas for plates the imperfection sensitivity problem can be solved completely, i.e. we get all solutions possible for the problem under investigation, we are far from being able to make such an assertion for the buckling problem of shells. To date there does not exist a systematic theory which would allow us to understand the imperfection sensitivity of singular perturbation problems. Thus in this case we have to make numerical simulation calculations which, however, are greatly supported by the knowledge of the buckling pattern of the perfect system. 2. ANALYTICAL METHODS TO ANALYZE BUCKLING PROBLEMS A. Derivation of the bifurcation equations We assume that the describing equations are a set of functional equations G (u, d) = 0, (1) where G is a mapping between function spaces Hand E, i.e. G: H x A + E, A being the finite dimensional H. TROGER Fig. 3. Notations for the rectangular plate with thrust p and len%h I and the imperfections r, p and p. parameter space. Examples for (1) will be given in Sec. 4. We further assume that u, is a solution of (1) for arbitrary values of 1. We call u, the fundamental solution, the stability of which is being investigated. In many applications u, is a trivial state of the system. Now we consider the pair (2) which is certainly a solution to (I) and we ask whether there exists locally a unique smooth curve u = u(A) through the point (4, &). This question is answered by the Implicit Function Theorem. If the linearization of (1) at (2), which is the linear map G,(u,,, J.,,), has an inverse then we have a unique solution u(l) and bifurcation cannot occur. Thus a necessary condition for bifurcation is that G,(h,, &) is not invertible at a certain value i, = A,, which is called the critical parameter value. Thus, in order to calculate J.,, we must solve the linear eigenvalue problem G,(u,,, 1,)~ = Au = 0, (3) where v are the eigenfunctions of the linearized problem. To solve the nonlinear problem we use the Liapunov-Schmidt method, which allows us to obtain an algebraic system of bifurcation equations, the dimension of which is equal to the multiplicity n of the critical eigenvalue. In the first step we decompose the unknown function u into two parts: u = u1 + w=i qiVi + W(q,, . . . qm), 9 (4) 1-I where u, is spanned by the eigenfunctions and includes the n critical variables qi, if we have bifurcation at an n-fold eigenvalue, and W includes functions orthogonal to the eigenfunctions. For W the following relation holds: W = 0(1qi12).Introducing (4) into (1) and projecting the resulting equation on the space spanned by its eigenfunctions and on the orthogonal complement we obtain the bifurcation equations and a second set of equations which is very important because it allows us to eliminate from the bifurcation equations the noncritical variables expressed by W in (4). This elimination process of W from the bifurcation equations distinguishes the Liapunov-Schmidt method from the Galerkin A I59 comparison of the postbuckling behavior of plates and shells method, where in (4) only the first part U, is used. We shall indicate in Sec. 4 that using the Galerkin method instead of the Liapunov-Schmidt method would lead to an incorrect bifurcation equation for the buckling problem of a spherical shell in the case where we must obtain the bifurcation equations up to third order terms. What we finally end up with is a set of n nonlinear algebraic equations then to study the algebraic bifurcation equations by path-following methods as it is done in [l I]. However, if the problem is described by ordinary differential equations then finite difference methods combined with path-following techniques work very well [6]. But there exist also numerical calculations with FEM codes [ 121 which give hope that soon localized buckling patterns will be able to be calculated numerically also for partial differential equations. in the n variables q,. Further discussion of (5) is done by means of Singularity Theory. 4. APPLICATIONS TO PLATE AND SHELL BUCKLING B. Discussion of the bifurcation equations As the functions F, in (5) are calculated by series expansions it is important to know up to which order terms in (5) have to be retained in order to have a consistent problem. This is the problem of determinacy which generally means that a system is mdeterminate if the local behavior of the solutions is not affected by the addition of terms of order m + I or higher. There exist precise criteria to check the determinacy [3]. Secondly, we do not only want to obtain the solutions of (5) i.e. of the one-parameter system, but we also want to know how this solution set changes under perturbations of the system. This goal can be achieved by a universal unfolding of the bifurcation equations which includes all possible solutions of the given class of equations. Finally, bifurcation diagrams can be calculated which are a partition of the parameter space into regions of qualitatively different behavior of the system. C. Singular perturbation analysis As the standard methods of bifurcation theory are unable to handle the behavior of strongly localized solutions a completely different approach must be used. Now a second parameter 6 appears in (I) which leads to a problem of the following form [S]: A. Plate buckling As a mathematical model we use the von Karman plate equations [ 131,which are a set of two nonlinear elliptic equations in two variables which describe the behavior of the plate up to moderate finite rotations. In nondimensional variables they have the following form (Fig. 3) (w is the displacement in z direction and/is a stress function): A*f + f[w, w] = 0 (7a) A*w - [w,/] +pw,, = 0 (7b) inn, whereR={(x,y),O<x<l,O<y<l} domain of the plate. The operators are is the On the edges of the plate dR = {(x, y), x = 0, 1, O<y<l;y=O, 1,06xgI}wehavethefollowing boundary conditions. Case a: w=O, Aw=O at y=O,l; O<x<l (simply supported) G(u,rl,d)=O. (6) If for 6 + 0 (6) is a singular perturbation problem we proceed in two steps. First we solve the so-called reduced problem which is given for 6 = 0. These solutions, however, have discontinuities which must be corrected in boundary layers. We shall explain this with an example in Sec. 4. 3. NUMERICAL ANALYSIS For complicated bifurcation problems, i.e. those with multiple eigenvalues at loss of stability, the immediate use of numerical methods such as finite element methods or finite difference methods is not always successful. Therefore, it is advisable to try to perform the reduction process as described above and 0 at x=0,/; O<y<l (clamped) af - = 0, + b (AJ) = 0 aq (here tt designates boundary). (8) at all boundaries. the direction normal (9) to the Case b: at all edges. w=O, Aw=O f=O, Af=O (10) R. 160 SCHEEDLand H. TROGER The conditions (9) are explained in [9]. From (7a) we obtain f = -fA-2[)(., w], (II) where the calculation of A-’ is explicitly given for boundary conditions of case a in [9] and for case b in [IO]. ~ntr~ucing (11) into (7b) yields G(w,p)=A% -[w, -~A-2[~~,~]]+~~~~=0, (12) with boundary conditions for w either given by case a or case b. The linear eigenvalue problem (3) is given by (wOZZ0): G,(W,,P)~ = A2c + pv,, = 0, (13) with the corresponding boundary conditions. Solutions are given in [9] and [lo] respectively. in Fig. 4 eigenvalue curves are shown for case b, where the critical load is plotted as a function of the length of the plate. It can be seen that generically simple eigenvalues occur. However, for special critical values of the length of the plate iC= JX(iCZ) in case a 1, = ,/m in case b (14) double eigenvalues occur. However, it also is obvious that the distance between these double eigenvalues converges to a constant for increasing k. Furthe~ore no small parameter 6 as in (6) appears in this problem which for d 40 wouId yield a singular perturbation problem. Performing a classical bifurcation analysis with the Liapunov-Schmidt method we can reduce (12) with the ansatz expansion w = q,o,(x, Y) + q2w7Y) + W(q,, q2, xv v), (1% where U, and u2 are the eigenfunctions of (13) at a double eigenvalue, to a system of two nonlinear algebraic bifurcation equations (I: + WS - &I + O(M5 + lulls) = 0 q: + t+?* - &2 + w?,lS + MS) = 0 (16) in the two critical variables q, and q,. a and d are constants depending on the boundary conditions and 1 is a new loading parameter which is zero for p = pc. Equation (16) in general needs eight parameters (including 1) for a universal unfolding [3]. However, we try to understand the imperfection sensitivity by a restricted generic unfolding by which we understand a physically meaningful unfolding of (16). This can be done with only four parameters, which are besides 1 the deviation r from the critical length 1, of lIc2 3 & 1 Fig. 4. Eigenvalue curves for the plate of Fig. 3 showing double eigenvalues at critical lengths 5_= &mj. the plate and two cases of transversal loads p and fi (Fig. 3). These parameters enter into (16) in the following way: q:+o7q:q~-J.qig,-rq~+ji=o. (17) The big advantage of (17) compared with (16) is that in calculating bifurcation solutions for fixed nonzero values of the parameters only limit points occur (Fig. Sb), whereas for setting some of these parameters to zero (p = 0) bifurcation points are also present (Fig. 5a). It is also very interesting to study the influence of the boundary conditions on the buckling behavior of the plate. Here, particularly the way in which the so-called secondary bifurcations occur is important in order to explain the cfassicat mode jumping phenomena explained in [9]. By this we unde~tand that for a plate being loaded with p >pc, and thus being buckled in a certain initially stable buckling pattern with n half waves into x-direction, a sudden change in the wave number to n + 1 half waves of the buckling pattern occurs by further increasing the load p. The way mode jumping occurs depends on the bifurcation solutions. For the one-parameter problem (16) we obtain at the bifurcation point nine solutions [9]. If, however, we use (17) with p = fi = 0 and i # 0, i.e. we have a plate which is a little bit shorter or longer than I, where we have the double eigenvalue. Then at the first bifurcation we only get three solutions but increasing d secondary bifurcations occur. The way in which these secondary bifurcations occur determines whether mode jumping can occur or not. This effect, which was first found experimentally, could not be explained with the boundary conditions (10) which usually have been used because in this case the secondary bifurcations occur supercritically and at the wrong solution branch. Only if the boundary conditions (8) and (9) are used, i.e. clamped boundary conditions at the short edges where the load is applied, then mode jumping can be predicted theore~~1~y (91 because now the secondary bifurcation is subcritical and occurring at the right solution branch. A comparison of the postbuckling behavior of plti~cs and shells q1 -*_ --__ -. The big impact of this result is given by the fact that a definite answer can be given and no un~rta~nty remains as to whether higher order terms or imperfections might have been neglected in an unjust way. 1 In order to be able to give as complete an analytic analysis as possible we study a geometricaily simple problem, namely the buckling of a complete spherical shell under uniform external pressure (Fig. l), with the restriction that only axisymmetric buckling is allowed. As mathematical model we use Reissner’s shell equations [6], which are two nonlinear ordinary differential equations in two unknowns @ and & where JI is a stress function and B a defo~ation variable, measuring the angular difference between the tangent to the buckled shell and the perfect shell at the independent variable 5. For constant shell thickness the equations read [6] sin(C - /I) - sin ( sin{ -v cos 5 ctgrr $@ctg< --$I* ( [ =i i ‘,i --__ ---__ /I ~ Fig. 5a. Mode jumping diagram showing the amplitudes q, and q2 for a plate with I = 1.4, p = 0, p > pC and for variation of p. cos(l; - 8) - cos ( X -v(l-8’) ‘., ‘. B. Shell buckling 6 +‘“-I- 161 sir% - P) sin e 1 cos2tr - 8) co8c )I Fig. Sb. Restricted generic diagram for the same data as Fig. Sa but for p # 0. Only limit points occur. cos({ - p) - cosg sin c +6[ X -4lctgC jO’cos(i(n))sinsds sin 2(e - 8) + v(l sin 2c + [ Furthermore 6 = C: for dead loading and r$ = iF - /l(e) for follower force loading. The boundary conditions are [6] B,) cos(C - 8) cos t 1 j(O) = j!?(n)= 0. (21) 41~ sin(f$ - t: + fl) + 4A (sin2< sin IQ sin c 1’ (18) where dimensionless quantities for the loading parameter 1 and a geometry parameter 6 have been used corresponding to the following relations: 09) The fundamental state, which is a solution for all values of A, is the geometrically perfect concentrically contracted sphere, which is given by MC) = 0 tio+(i’)= -rl sin 2C. (22) As /Ia is equal to zero we keep /I as one variable. Instead of $* we use and (20) C.&S. Il,l-_L * =JI*-$:. (23) 162 R. SCHEIDL and For the variables (b, JI) the trivial solution is now (0,O). The question now is for which values of I. do we have a unique solution path through (22). Thus, again we have a problem in the form of (1); however, now a parameter d (20) as it is introduced in (6) is also present. It is obvious from (18) that for b + 0 we have a singular perturbation problem, because the small parameter 6 is a coefficient of the highest derivative. Starting with a classical bifurcation analysis we have to solve first the linear eigenvalue problem (3). This is done in detail in [I41 and the result is given in Fig. 2, where the close spacing of the eigenvalues is obvious. Mathematically, however, we have generically simple and in nongeneric situations double eigenvalues, which in a classical bifurcation analysis would result in a two dimensional system of bifurcation equations. It is shown in [14] that second, respectively third, order terms in the bifurcation equations govern completely the local bifurcation behavior. It is interesting to note that in the case of a simple eigenvalue when one has to retain third order terms in the bifurcation equations the Liapunov-Schmidt reduction gives qualitatively different results to the Galerkin method which are incorrect and cannot be used as an approximation to the Liapunov-Schmidt method. The bifurcation equations can be analyzed in the spirit of Catastrophe Theory [14]. However, as already mentioned, the domain of parameter variation for which such an analysis is valid is so small that it does not have any practical meaning. Furthermore, it is well known from experiments [ 151that, due to the unstable post-bifurcation behavior (solutions exist only for subcritical parameter values), a strong localization of the buckling pattern with a single dimple occurs. Thus, by performing a singular perturbation analysis we concentrate on this type of solution. In the first step we solve the reduced problem S B 0 which leads to a solution given in Fig. 6, where at the angle y, which still has to be determined, a boundary layer occurs, correcting the discontinuities of the reduced solutions. Representing the variables /3, $ and the loading parameter I in the boundary layer by asymptotic series in fi we obtain from the zero order equations the result that 1= @ O(fi) H. TROCER Fig. 7. Postbuckling path for the single dimple solutions calculated by singular perturbation theory. for the single dimple solution and from the first order equations the relationship between loading A and the angle y of the position of the boundary layer (Fig. 7). In order to obtain this final result a numerical shooting procedure (161 had to be applied. With these results describing the postbuckling behavior for strong displacements two questions still remain open; firstly, how does the buckling pattern change during the deformation, which is more of academic interest; and secondly, which is of greater practical interest, which are the worst (i.e. most dangerous) imperfections leading to the single dimple solution. We can only answer both questions by a numerical analysis. We use a finite difference method and path-following method, both of which are explained in [6] and [I 71. The result is given in Fig. 8 for the perfect shell structure. The question of imperfection sensitivity is answered in the next section. 5. IMPERFECI-ION SENSITIVITY As in industrial production, and in engineering in general, small imperfections of arbitrary nature will always be present, one of the most important points in buckling investigations of thin-walled structures is the question of which imperfections are dangerous and which are not. A -.2819 -2 boundarylayer Fig. 6. Reduced solutions for the singular shell buckling problem with boundary layers. Fig. 8. Buckled states of the shell (calculated by a finite difference method). A 163 comparison of the postbuckling behavior of plates and shells Here the general information is supplied by the analysis of the buckling behavior of the perfect structure. Whereas for the plate we obtain a supercritical bifurcating path [I I], for the shell a subcritical bifurcating path is found. The first basic consequence is that plates are not imperfection sensitive whereas shells are imperfection sensitive. The second practically important question is which imperfections are dangerous and which are not. For the plate this question can be definitively answered by the fact that only those imperfections which include a contribution of the buckling modes are of relevance. This is also manifested in the coefficients p, j in (17), which stand for these influences and give a restricted generic unfolding. For the shell the question of the most dangerous imperfections is much more complicated and cannot be solved in such an elegant way as it is given in (17) for the plate. The problem is that the initially forming buckling pattern, i.e. spherical harmonics of high order and the finally obtained postbuckling pattern, i.e. the single dimple, do not relate to each other very much [6, Fig. IO]. Thus numerical calculations for different types of imperfections must be made in order to find out which imperfections are dangerous. In [I 61 such calculations are performed and the result is that those imperfections which resemble the final state of the shell are more dangerous than those which resemble the initially forming buckling pattern. This is quite obvious because for practical imper- fections the maximum loads are far helow the classical buckling loads and therefore the corresponding deformations are those of the final state of the shell (61. 6. CONCLUSIONS The difference in the buckling behavior of plates and shells is determined by several facts. First, the structure of the solutions of the linear eigenvalue problem is quite different, especially for shells with high multiplicities of the critical eigenvalue, or if close spacing of the eigenvalues is found. Secondly, shell buckling problems are in the elastic range singular perturbation problems with the consequence of the occurrence of boundary layers. Consequently completely different approaches must be used for the analysis. Whereas for plates classical methods of bifurcation theory (Liapunov-Schmidt, Singularity Theory) work very well, for shells methods of singular perturbation theory must be used. Finally the different qualitative type in the postbuckling behavior plays an important role. For plates the postbuckling behavior is noncritical whereas it is critical for the shell. The consequence is the completely different imperfection these two structures. sensitivity behavior of AcknoH,ledgemenr-This work is partly supported by the ‘Fonds zur Fiirderung der wissenschaftlichen Forschung’ in Austria under project P 5519. REFERENCES I. W. T. Koiter, On the stability of elastic equilibrium. Thesis. Delft (1945): NASA IT-F-10833 fl967). 2. M. Gelubitsky and’D. G. Schaeffer. Singularities and Groups in BiJiircarion Theory. Vol. I, Appl. Math. Sciences 51, pp. 455459. Springer, New York (1985). 3. T. Poston and I. Stewart, Catastrophe Theory and its Applicafions. Pitman, London (1978). 4. M. Golubitsky and D. 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