Arnold tongue predictions of secondary buckling in thin elastic plates

The stability of post-buckled states for simply-supported flat elastic plates under compression is investigated for a range of in-plane boundary conditions. The von Karman plate equations are reduced to a series of ODEs which are solved numerically under parametric variation of both load and length. Results are checked against full numerical solutions of the PDEs, and comparison with a modal analysis highlights the dominant passive contaminations. The nondimensional amplitude at secondary bifurcation, for any combination of modes and all plate lengths, is presented in a concise form using the parameter space of Arnold tongues. This demonstrates that compound bifurcation represents a worst case for post-buckling reserve, and that long plates have inherently more such reserve than short plates. It is also shown that stiffening the boundaries against in-plane movement is destabilizing, in that it induces mode jumping at secondary bifurcation to occur at an earlier stage in the post-buckling regime.