Singular perturbations for sensitivity analysis in symmetric bifurcation buckling

A direct procedure for the evaluation of imperfection-sensitivity in bifurc ation problems is presented. The problems arise in the context of the gener al theory of elastic stability (Koiter's theory) for discrete structural sy stems, in which the total potential energy is employed together with a stab ility criterion based on energy derivatives. The imperfection sensitivity o f critical states, such as bifurcations and trifurcations, is usually repre sented as a plot of the critical load versus the amplitude epsilon of the i mperfection considered. However, such plots have a singularity at the point with. epsilon = 0, so that a regular perturbation expansion of the solutio n is not possible. in this work, we describe a direct procedure to obtain t he sensitivity of the critical load (eigenvalue of the bifurcation problem) and the sensitivity of the critical direction (eigenvector of the bifurcat ion problem) using singular perturbation analysis. The perturbation expansi ons are constructed as a power series in terms of the imperfection amplitud e, in which the exponents and the coefficients are the unknowns of the prob lem. The solution of the exponents is obtained by means of trial and error using a least degenerate criterion, or by geometrical considerations. To co mpute the coefficients a detailed formulation is presented, which employs t he conditions of equilibrium and stability at the critical state and their contracted forms. The formulation is applied to symmetric bifurcations, and the coefficients are solved up to third-order terms in the expansion. The algorithm is illustrated by means of a simple example (a beam on an elastic foundation under axial load) for which the coefficients are computed and t he imperfection-sensitivity is plotted. Copyright (C) 2001 John Wiley & Son s, Ltd.