On Saint-Venant Torsion and the Plane Problem of Elastostatics for Multiply Connected Domains

Abstract

Michell [1] (1899), in a classical paper, showed that the determination of the plane-strain and generalized plane-stress solutions associated with the three-dimensional plane traction-problem of the linearized equilibrium theory of homogeneous and isotropic elastic solids—in the absence of body forces—is reducible to a Dirichlet problem for the two-dimensional biharmonic equation on the cross-sectional domain at hand.¹ This reduction is achieved with the aid of a general solution of the two-dimensional stress equations of equilibrium in terms of Airy’s stress function. That the generating Airy function must be biharmonic follows from the requirement that the corresponding strains satisfy the appropriate compatibility condition, while the accompanying boundary data for the Airy function and its normal derivative result from an integration of the original boundary conditions for the given surface tractions.