Michell-like grillages and structures with locking

Abstract

The paper generalizes the Michell theory of plane pseudo-continua to the anti-plane problems in which the loading is perpendicular to the plane of the structure. The starting point is the minimum compliance problem for a two-phase Kirchhoff plate. Upon relaxation, one of the materials can be degenerated to a void (or microvoids) and by imposing the condition of the volume being small, one arrives at the Michell-like problem for a locking plate. The locking locus B is determined explicitly; it tends to a square if the Poisson ratio tends to 1. In the last case the locking locus coincides with that used in the Rozvany-Prager theory of optimal grillages. A theory of perfectly-locking and elastic-locking plates and shells, not necessarily isotropic, is formulated. Dual extremum and existence theorems are also given.