Michell structures made from two materials carrying their own weight

Paper in proceeding, 2026

Let us imagine that we have a circular support of known radius in 2 dimensions and that we want to construct a minimum weight cantilever of a given span to support a mouse at the tip and the weight of the cantilever itself. The cantilever is made from 2 materials, one for tension and one for compression, with known strengths. We will make the same assumptions that Anthony Michell made in his seminal paper of 1904, 'The limits of economy of material in frame-structures', except that we will include the own weight of the structure, which Michell did not. However we follow Michell in ignoring buckling and thus the structure will consist of a fine network of pin ended struts and ties in which compression members can be very slender. This assumption means that if we replace our mouse by a rat that weights 10 times as much, then the whole structure will have to weight 10 times a much. We offer an alternative derivation of Michell's result, again using virtual power (also known as virtual work) as he did. Including the own weight of the structure in the formulation means that the equations almost certainly require a numerical solution, and we present numerical results for the cantilever, which we compare to Michell's for the special case of a weightless cantilever.