D. Schaeffer and M. Golubitsky
Boundary conditions and mode jumping in the buckling of a rectangular plate
Commun. Math. Phys. 69 (1979) 209-236.
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 gourp 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.