Constrained torsion of prismatic bars

A formulation is developed for the determination of three dimensional displacements and stresses within finite length beams of arbitrary cross-section experiencing restrained torsion. This restraint arises from the twisting of a prismatic bar where one of its cross-sections is subject to displacement and slope constraints. The deformations within this beam are represented by an exponentioy decaying residual solution superimposed on the classical Saint-Venant solution. Galerkin's Method is used to solve the boundary value problem resulting from a simultaneous system of three quadratic eigenproblems subject to the stress free boundary condition arising from the traction-free lateral surface of the bar. The eigenvalues represent decay rates of the residual solutions and their associated eigenvectors are expanded to form an approximate solution for the three dimensional displacements and corresponding stresses. A three dimensional finite element model is used for validation. Relationships between stress and stress resultants for this constrained bar are developed.