Some exact analytical solutions in structural optimization

Abstract

ABSTRACTIn 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 Szegö (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 Hölder 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.