On the optimal shape of a column with partial elastic foundation

By using Pontryagin's principle we study the optimal shape of an elastic column with clamped ends and positioned on elastic foundation of Winkler type. Two problems were treated. In the first one, which is a generalization of our previous work, we consider the case of a partially supported column. In the second problem we determine the optimal shape of a column on elastic foundation subjected to restrictions on minimal cross-sectional area. It is shown that in this case the optimization can be both bimodal and unimodal. We determine the transition value between unimodal and bimodal optimization for specified values of parameters.