Least-weight design of perforated elastic plates—I

Abstract

The paper investigates implications of some recent mathematical developments in the fields of shape optimization and relaxation of variational problems. Considering the least-weight design of perforated elastic plates in either flexure or plane stress for a prescribed compliance, it is shown that at low rib densities microstructures consisting of a combination of first- and second-order infinitesimal ribs is superior to those consisting of purely first-order infinitesimal ribs. Moreover, it is indicated that thin ribs of infinite length/thickness ratio do not contribute significantly to the stiffness in a direction normal to their plane. On the basis of this conclusion, a simple specific cost function is derived and then it is used in the design of circular, uniformly loaded perforated plates with zero value of Poisson's ratio. As a basis for comparison several intuitively selected topographies are optimized for the case of simply supported plates, and in Part II of this study a variational analysis is used to obtain the optimal solutions for plates with simply supported, clamped or loaded edges.