A new structural model of warping torsion. I: Formulation for prismatic elastic shafts

Abstract

Existing formulations of warping torsion rely on a direct approximation of the warping displacement of the cross sections, even when this approach leads to several limitations. For example, assuming the warping to be directly proportional to the rate of twist results in a high-order governing equation and anomalies in which the twisting components of the stress and internal torque are distorted by the restrainment of the warping. As identified recently by the author, this situation reveals the presence of a kinematic warping–twist constraint in this particular approach, whose relaxation through an independent warping field still with a fixed distribution of the section’s warping, results in non–equilibrated stresses despite its wide use in recent computational models. As an alternative, we consider in this paper a mixed treatment of the axial strain and stress appearing when warping is restrained, extending an early idea of Reissner. In fact, the paper presents a new structural mixed formulation of restrained warping, based on a non–uniform distribution of the shape of the warping on the cross sections, also controlled by an independent warping field, but crucially involving the aforementioned mixed axial kinematics. Theoretical results presented here indicate the suitability of the new formulation, resolving the aforementioned issues of existing formulations.