Boundary Conditions and Mode Jumping in the Buckling of a Rectangular Plate

We show that mode jumping in the buckling of a rectangular plate may be explained by a secondary bifurcation as suggested by Bauer et al. [1] when "clamped" boundary conditions on the vertical displacement function are assumed. In our analysis we use the singularity theory of mappings in the presence of a symmetry group to analyse the bifurcation equation obtained by the Lyapunov-Schmidt reduction applied to the Von Karman equations. Noteworthy is the fact that this explanation fails when the assumed boundary conditions are "simply supported". Mode jumping in the presence of "clamped" boundary conditions was observed experimentally by Stein [9] "simply supported" boundary conditions are frequently studied but are difficult if not impossible to realize physically. Thus, it is important to observe that the qualitative post-buckling behavior depends on which idealization for the boundary conditions one chooses. Mode jumping [9] is perhaps the most noteworthy feature of experimental studies of the post-buckling behavior of plates. As is well known, a rectangular plate can support a number of different buckled configurations these may be distinguished by their wave number, by which we mean the number of zeroes of the (normal) deflection function along a line parallel to the leading direction. Experiments [9] have shown that the wave number need not remain constant as the load is gradually increased past the buckling load rather there are special values of the load parameter at which a sudden and violent change in buckling pattern occurs. The new mode typically has a wave number greater than the old mode by unity. A spring model proposed by Stein [8] offers an attractive explanation of this phenomenon. As observed by Bauer et al. [1] secondary bifurcation often results from splitting a double eigenvalue by perturbation; in the spring model mode * Research sponsored in part by the U.S. Army Contract DAAG29-75-C-0024 and the N.S.F. Grant MCS77-04148 ** Research partially supported by the N.S.F. Grant MCS77-03685 and the Research Foundation of C.U.N.Y. 0010-3616/79/0069/0209/$05.60 210 D. Schaeffer and M. Golubitsky jumping could occur [1] when the primary solution branch lost stability through such a secondary bifurcation. Whether or not mode jumping actually occurs depends on the value of certain parameters. In [4] we analyzed the most general form of reduced bifurcation equations for a rectangular plate at a double eigenvalue, consistent with the symmetries of the von Karman equations (or any other plate theory). It follows from this analysis that the spring model already exhibits (essentially) all possible behavior of the plate, and moreover that whether the plate exhibits mode jumping is determined by two dimensionless parameters which using the terminology of singularity theory we call modal parameters. Calculations by Matkowsky and Putnik [6], Chow et al. [2], and Magnus and Poston [5] cast doubt on this explanation of mode jumping. These authors analyzed a simply supported plate governed by the von Karman equations and (in our language) found that the modal parameter values were such that mode jumping would not occur. However simply supported boundary conditions are hard to achieve experimentally, and in fact Stein [9] suggests that clamped boundary conditions would be the most accurate approximation for the loaded ends. Therefore, in this paper we analyze a von Karman plate subject to these mixed boundary conditions clamped at the loaded edges, simply supported at the unloaded edges and we find that here mode jumping does occur. (For comparison we also consider the case of simply supported boundary conditions on all four sides.) Of course, given the many doubts surrounding the von Karman equation, the most important result of this paper is perhaps that even the qualitative behavior of a buckled plate may be changed by the choice of boundary conditions. The reader should be warned that the calculations outlined in Sect. 8 are long and tedious. However, the fact that these computations are made without the use of a computer is a simplification when compared to other work in this area. Moreover, our choice of boundary conditions made this simplification possible and this choice was prompted by physical considerations alone. 1. The Experiment of Stein and Its Mathematical Idealization Let us begin by a description of Stein's [9] apparatus. In Fig. 1 we have indicated part of a large plate divided into 11 panels, each 4.71" by 25.36". The division is performed by knife blades located on either side of the plate, as shown in the end view. The knife blades prevent any normal displacement while (in principle) not inhibiting motion in the plane. Actually one is interested in the buckling of just a single panel, but having many, more or less identical, adjacent panels provides an experimentally feasible way of achieving simply supported boundary conditions along the unloaded edges. All measurements were performed on the center panel, which for theoretical purposes is treated as being embedded in an infinite periodic array of such panels. In the experiment, when the load was first increased beyond the buckling load, the initial configuration of the plate contained 5 buckles. This pattern persisted as the load was gradually increased until a load approximately 1.7 times the buckling load was passed, when the plate jumped suddenly and violently to a new configuration with 6 buckles. Further increases in the load led to jumps to states with 7 and 8 buckles, and eventually to the complete collapse of the plate. These latter jumps occurred after the plate had begun to deform plastically and are not discussed here we consider only the jump from 5 buckles to 6. Mode Jumping in the Buckling of a Rectangular Plate