An algorithm for the nonlinear analysis of compound bifurcation

Abstract

The paper presents a procedure for the localized analysis of compound bifurcations. A full range of phenomena are embraced, including loci of equilibria, secondary bifurcation, and imperfection sensitivity, the scheme showing how to generate the appropriate lowest-order non-trivial equations of interest. A number of aids to solution are presented, including the concepts of generalized imperfection and generalized loading parameter. The scheme is developed by using a general non-diagonalized format suitable for numerical analysis, but the special diagonalized form can also be used to good effect. This is illustrated in the application to the interactive buckling of stiffened plates and shells, when local and overall buckling occur simultaneously or nearly so. The modelling relies heavily on the elimination-of-passive-coordinates routine of the general scheme. The study shows that the parabolic umbilic catastrophe is the key phenomenon for most such problems. Finally, the branching analysis is fully illustrated for semi-symmetric branching, where one of the contributing bifurcations is symmetric and the other is asymmetric. In all, ten different loci are treated, including the full imperfection sensitivity at complete and near coincidence plotted in three-dimensional form; these relate to an earlier stiffened-plate formulation. The general scheme is thus made directly accessible for any problem that exhibits a bifurcational manifestation of either the elliptic or hyperbolic umbilic catastrophe.