The lightest plane structures of a bounded stress level transmitting a point load to a circular support

The paper refers to the problem of Michell (1904) of finding the lightest fully stressed structures, composed of possibly infinite number of members, trasmitting a given load to a support forming a circle. The point load can be located within or outside the circle. The known analysis by Hemp (1973) is enhanced here by disclosing the explicit formulae for the weights of the ribs and the interior (fibrous domain). The optimal weight can be found by two manners: by applying the primal integral formula involving the density of the reinforcement or by computing the work of the load on the adjoint displacement. One of the aims of the paper is to show that both these formulae are equivalent. This identity is essential since in the case of point loads the equivalence of the primal and dual formulations has not been proved till now. Tlie analytically found layouts are confirmed by analysis of trusses (of finite number of joints) approximating the exact Michell-like solutions.