Some exact analytical solutions in structural optimization

In this article, the isoperimetric inequalities arising in exactly solvable structural optimization problems of stability are discussed. The purpose of this article is to review some types of inequalities that may be regarded as isoperimetric. This type of inequalities is long known in geometry and physics; see, e.g., Polya and Szego (1951), Banichuk (1977), Bandle (1980), and Chavel (2001). The variational method is a powerful way to prove inequalities for systems described by ordinary differential equations. The proof of isoperimetric inequalities exploits the variational method and the Holder inequality. The applications of this method for stability problems are illustrated in this article. The inequalities for Euler's column with boundary conditions of mixed type, for a twisted rod with periodic simple supports, and for a ring acted upon by a uniformly distributed, compressive hydrostatic load are rigorously verified.