On the derivation of Michell's theorem

Abstract

In the classical proofs (proofs along the lines of the original proof) of Michell's theorem on the limits of economy of material in frame structures [i], or in using Maxwell's lemma [2] to find the minimum weight truss for a restricted class of loadings [3], it has been assumed that the minimum weight structure will be fully stressed. The assumption is explicitly stated in the version of the proof given by Cox [3], but is only implied in the original proof [I]. The validity of this assumption is obvious for statically determinate trusses. For statically indeterminate trusses, the sizing of one member can affect the forces in other members and therefore can affect the minimum weights of other members. For this reason, if it is not known in advance that the minimum weight truss will be statically determinate, it is not obvious that the minimum weight truss will be fully stressed. However, even for statically determinate Michell or Maxwell structures, previous classical proofs are not valid. The statical determinancy of that structure, which results from a derivation that assumes the minimum weight structure will be fully stressed, cannot be used to justify the assumption which produced that result. It is felt that a proof of Michell's theorem or use of Maxwell's lemma that makes an unsubstantiated fully stressed assumption is incomplete and therefore incorrect. What follows are the modifications in the use of Maxwell's lemma and of the proof of ~ichell's theorem; following the format in Reference [3], that does not make a fully stressed assumption, but instead proves the minimum weight structure will be fully stressed. The final results in both cases are unchanged but correct proofs are provided.