Displacement and Stress Function-based Linear and Quadratic Triangular Elements for Saint-Venant Torsional Problems

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections.  On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.