ISOPERIMETRIC-INEQUALITIES IN OPTIMAL STRUCTURAL DESIGN

Isoperimetric inequalities arising in exactly solvable structural opti mization problems are discussed in this article. Most isoperimetric in equalities of mechanical problems have been proved using one of three known methods. The first tool is the variational method, which is a po werful way to prove inequalities for systems described by ordinary dif ferential equations. For the systems described by equations with parti al derivatives, the symmetrization method can be applied. Methods base d on positivity properties of operators, for example, Hopf's maximum p rinciple, are used to prove the inequalities for local functionals. Us ing these algorithms, the new isoperimetric inequalities for engineeri ng problems were investigated. Classical examples, namely the St. Vena nt-Polya inequality rigidity, the Faber-Krahn inequality for vibrating membranes, the Banichuk-Wheeler inequality for maximum stress in a pe rforated plane and the Keller-Ting inequality for static moments of a bar are treated. The inequalities for the maximum stress in twisted is otropic and orthotropic elastic bars are proved. For the perfectly-pla stic solid, the inequalities for collapse load are considered. The oth er class of inequalities, associated with eigenvalue problems, arises in vibration, conservative and non-conservative stability problems. Th e new inequalities for the bimodal critical axial compression load for bars with clamped ends, a generalized Pfluger column and a circular r ing under hydrostatic pressure are stated.