Free-form optimization method of frame structures for elastic buckling

Abstract

In this study, we propose a shape optimization method of a frame structure for maximizing the elastic buckling load. The 1st buckling load factor is maximized under a volume constraint. The problem is formulated as a distribute parameter shape optimization problem, and the shape gradient function for this problem is theoretically derived using the Lagrange multiplier method, the adjoint variable method and the formulae of the material derivative. The derived shape gradient function is applied to the H¹ gradient method for frame structures, a gradient method in the Hilbert space, where the optimal shape variation is calculated as the displacement field of the fictitious linear elastic of the frame structure. With the proposed method, the free-form, or arbitrarily formed frame structures with large scale design variables can be optimized without any shape parametrization. The problems of repeated eigenvalue and Euler buckling of each member are also considered and solved. The results show the validity of the proposed method to determine the optimal free-form of a frame structure for the elastic buckling design.