Michell cantilevers constructed within trapezoidal domains—Part II: virtual displacement fields

Abstract

The present second part of the paper deals with the virtual displacement fields associated with the optimality conditions $\varepsilon _{I} {\left( {\overline{{\mathbf{u}}} } \right)} = 1,\varepsilon _{{II}} {\left( {\overline{{\mathbf{u}}} } \right)} = - \kappa ,\kappa = {\sigma _{T} } \mathord{\left/ {\vphantom {{\sigma _{T} } {\sigma _{C} }}} \right. \kern-\nulldelimiterspace} {\sigma _{C} }$, where σ _T and σ _C represent the allowable values of the tensile and compressive stress, respectively. The displacement fields vanish along a straight segment of a line support and are constructed within an infinite domain bounded by two half-lines. The displacement fields are provided by the integral formulae involving the Lamé fields found in part I of this paper. All the results are expressed in terms of Lommel-like functions. These results make it possible to determine the volumes of the optimal cantilevers designs within the feasible domain considered. Computation of the volumes along with analyses of concrete cantilevers will be the subject of part IV of the present paper.