Optimization as generator of structural instability: Koiter’s worst imperfection

This paper is focused on the interaction of local modes in pin-jointed structures, with the aim to present an algorithm for the appraisal of the lowest critical load characterizing the structure under the effect of small imperfections. This interaction phenomenon is framed within Koiter’s theory of elastic instability as a quartic system. First of all, the paper discusses the theoretical results of Koiter’s theorem about the determination of the worst imperfection with respect to the critical load. In doing this, the characterization of the fold lines in the modal space is recalled and used for the purposes of the reformulation of the theorem. Then, a FEM code aimed at the determination of the most dangerous shape of the imperfection, and at performing the related sensitivity analysis, is presented. The basis of the code is a nonlinear model of a 3D rod equipped with a planar flexural deformation, within the context of Eulerian instability. Some numerical results having a practical interest are discussed.