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In this article, the problem of the optimal design of simply supported columns under loadings controlled by displacements is investigated. It seeks a cross-sectional area, varying along the axis of the column, which leads to the maximal axial displacement caused by compression before the structure buckles. The uni-modal and bi-modal problems are considered. The optimal solutions for different constraints of minimal and maximal cross-sectional area and various elastic foundations are presented in this article. The results are obtained by solving an analytical formulation of the optimization problem and by using numerical optimization methods, namely the method of moving asymptotes and the simulated annealing method. [ABSTRACT FROM AUTHOR]