Minimum energy characterizations of Saint-Venant's solution to the relaxed Saint-Venant problem

Abstract

The singular importance, in both theory and practice, of SAINT-VENANT'S celebrated memoirs [1], [2] on what has long since become known as SAINTVENANT'S problem, requires no emphasis. Indeed, a comprehensive bibliography of the vast and varied literature to which the work contained in [1], [2] has given impetus would multiply the length of this rather limited study. With a view toward describing the aim of the present investigation we recall first that SAINT-VENANT'S problem consists in the determination of the stresses and deformations in an elastic cylinder (or prism), which - in the absence of body forces - is subjected to surface tractions arbitrarily prescribed over its ends and which is free from lateral loading. In this formulation the problem, even within the linearized equilibrium theory of homogeneous and isotropic elastic solids, has remained one of undiminished notoriety. SAINT-VENANT'S treatment of the foregoing problem rests on a relaxed formulation in which the detailed assignment of the terminal tractions is abandoned in favor of prescribing merely the appropriate stress resultants.