Optimal design of elastically clamped columns subject to distributed axial load

The optimum designs are given for columns which are elastically clamped at both ends and under concentrated and distributed axial loads. The objective is to maximize the buckling load subject to volume and maximum stress constraints. The results for a minimum area constraint are also obtained for comparison. Under a distributed axial load, the stress constraint leads to different minimum cross-sections at different locations as the stress changes along the column. An iterative solution method is developed based on finite elements and the results are obtained for n=1, 2, 3, defined as I=αnA with I being the moment of inertia, and A the cross-sectional area. Numerical results show that the optimal areas become larger in the direction of the distributed load and the effect of the concentrated load is to make the optimal shape more uniform. Results are given for uniformly and triangular distributed loads which have distinct effects on the optimal shape. The unsymmetry in the optimal shapes is noted to be due to the distribution of the axial load and the unsymmetry in the boundary conditions.