Functional gradient as a tool for semi-analytical optimization for structural buckling

A functional gradient for the buckling load (P) with respect to the structural shape function (S) is introduced for shape optimization. The gradient (P,S) is derived analytically by functional differentiation of the governing equation with respect to S, where both P and the deflection (W) are considered as functionals of S. It is found that P,S is a functional of W and is not dependent explicitly on S. The P,S is combined with a simple gradient projection optimization method and a finite element program. The analytical gradient expression is independent of the FE model but requires its input. At each optimization step the buckling problem is solved numerically and the gradient is calculated. Then, a new improved design is computed. In the first part of the study 1D beam problems are solved. Results are compared with exact solutions and found to be more accurate than common numerical optimizations. An example of a clamped beam with two simple supports is studied. The optimal area distribution of a general unimodal 1D beam is found to be composed of sections with the same shape as for the basic simply supported case. The proposed method is generalized to 2D plates. The optimization is combined with a commercial FE package. For the case of simple square plates the optimization produces smooth thickness distribution showing higher buckling load improvement than common numerical optimization. The method was then applied to square plates with a central circular hole, where non-uniform FE meshes were used. The optimal shapes exhibited singularities resembling commonly used stiffeners.