Minimum weight design of non-linear elastic structures with multimodal buckling constraints

It well known that multimodal instability is an event particularly relevant in structural optimization. Here, in the context of non-linear stability theory, an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints. Given an initial design, the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight. Multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem, and the difficulties related to the non-differentiability in the common sense of state variables in multimodal critical states, are overcome by means of the Lagrange multiplier method. Further constraints impose that only the first critical equilibrium states (local maxima or bifurcation points) on the initial equilibrium path of the actual designs are taken into account.