D. Schaeffer and M. Golubitsky
Boundary conditions and mode junping 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.