Bifurcations and catastrophes of a two-degrees-of-freedom nonlinear model simulating the buckling and postbuckling of rectangular plates

The nonlinear buckling and postbuckling behavior of rectangular plates in symmetric and antisymmetric modes is re-examined, in the context of Bifurcation and Catastrophe Theories, using a two-degrees-of-freedom model, which has been adopted for the same purpose in the pioneer literature. At first the perfect system is dealt with in detail, symbolically utilizing the exact as well as the approximate equilibrium equations, the latter being products of a universal unfolding of the original total potential energy function. Conditions for the existence of remote secondary bifurcations are fully assessed and the stability of critical states is determined, revealing sudden qualitative changes in the postbuckling response of the perfect system, which have been also reported for the actual continuous structural system—the rectangular plate—using the von Kárman equations. Thereafter, the imperfection sensitivity is dealt with, introducing symmetric as well as asymmetric imperfections, considered as individual or consecutive perturbations of the perfect system. It is found that symmetry breaking bifurcations give birth to complicated cusp singularities, which may lead to unexpected jumps from one to two-mode remote postbuckling behavior. Finally, considering the general case of random imperfections, higher order two-mode singularities are revealed, mainly of the double-cusp catastrophe type, which have been also discovered in the postbuckling response of rectangular plates, a fact validating the choice of the foregoing nonlinear simulation.