Optimal Shape of Column with Own Weight: Bi and Single Modal Optimization

The optimality conditions, via Pontryagin’s maximum principle, in the case of bimodal optimization of columns are derived. When these conditions are applied to the stability of a compressed column with own weight, the problem of determining the optimal cross–sectional area function is reduced to the solution of a nonlinear boundary value problem. Two specific problems are analyzed in detail. In Problem 1, that is new, the shape of a heavy compressed column with clamped ends stable against buckling and having minimal volume is determined. In Problem 2, formulated by Keller and Niordson, optimal shape of a vertical column with one end clamped the other end free is determined.