On the Topological Classification of Postbuckling Phenomena

Abstract

Abstract The classification of initial postbuckling behavior is known to be governed by the energy function of the critical equilibrium state, and this can be emphasized by viewing the load as a Lagrange multiplier in a constrained minimization. The energy of the critical branching point itself also uniquely determines the catastrophe theory classification of instability phenomena, and it is shown how a finer bifurcational classification arises when a load parameter, and hence an approach route in control space, is specified a priori. These ideas are illustrated for the compound, twofold, semisymmetric branching points with an examination of two pathological cases that have not been previously analyzed. The first of these is the degenerate situation typified by the rotationally-symmetric buckling of a complete spherical shell in which the load and engineering imperfections fail to fully unfold the topological singularity, and generate a trace of hill-top branching points. The second is the paraclinal point of bifurcation (parabolic umbilic catastrophe) that forms the typical transition from a homeoclinal to an anticlinal point of bifurcation, and represents an important practical problem in the interactive buckling of stiffened plates and shells. Closed form solutions for the three-dimensional engineering imperfection