Iterative algorithms for optimal design of columns subject to a stress constraint

Two algorithms based on an integral equation formulation of the buckling optimization problem are formulated and implemented. The objective of the optimization is to maximize the buckling load of an elastically restrained column by optimally designing the cross-sectional area subject to a minimum cross-section or maximum stress constraint. The first approach involves solving the resulting integral equations iteratively taking into account the boundary conditions, the optimality criterion and the imposed constraints. In the second approach an iterative finite difference approximation scheme is developed. lumn is elastically restrained at both ends which produce the simple support and clamped end conditions for the limiting cases leading to the optimal design of columns under general boundary conditions. The above problems do not have analytical solutions due to the complexity of the boundary conditions, constraints and the optimality conditions necessitating the formulation of computational schemes for their solution. Several numerical results are given and compared with available results in the literature. Moreover the accuracy of the methods is studied by comparing the iterative solutions with finite element ones and with exact results when available.