ON THE CONVEXITY AND CONCAVITY OF COMPLIANCES

Compliances, which are measures of ''inverse stiffness'', are sometimes used in the objective function, or as constraint functions, in structural optimization. It is known that if compliances are expressed as functions of thickness variables t(j), e.g. cross-sectional areas of truss elements, then these functions become convex. In this paper it is shown that if compliances are expressed as functions of reciprocal thickness variables x(j) = 1/t(j), then these functions become concave. Based on this result, it is further shown that a well-known structural optimization method is globally convergent when applied to minimum weight problems subject to constraints on compliances under multiple load cases.