Bifurcation hierarchy of a rectangular plate

The mechanism of the complex recursive bifurcation behavior of a four-sides simply-supported rectangular plate is investigated. Such complex behavior is due to the "hidden" symmetry of the plate associated with the periodic nature of the solutions. The group-theoretic bifurcation theory is employed to arrive at a lattice of subgroups which expresses the rule for the behavior. This rule is shown to suffer degeneration due to the restriction by the boundaries compared with that of geometrical symmetry. The governing equation of this plate is discretized by means of Galerkin's method with the use of the double Fourier series as shape functions. The tangential stiffness matrix of the plate is shown to be block-diagonalized by appropriately permuting the order of the Fourier series following the rule presented. The bifurcation analysis of the plate is carried out to assess the validity of the rule and to demonstrate the merit of the block-diagonalization. As a result of this, the vital role of the bifurcation rule in the proper understanding and successful analysis of the complex bifurcation behavior has been demonstrated.