A finite element solution for Saint-Venant torsion

Abstract

A solution of the Saint-Venant Torsion Problem is developed using the finite element method. Difficulties with multiply-connected regions are completely avoided by the introduction of a force variable that explicitly defines any traction-free boundary. A direct matrix relation is established between the forces and displacements such that the solution for any section is obtained in terms of the displacements at discrete points. In addition to discrete or distributed inhomogeneity the method is developed to treat any anisotropic effect that can be characterized by arbitrarily oriented orthotropy of individual elements. Matrix relations are derived and discussed for triangular and square elements. Applications to a solid and a hollow isotropic square section are presented and the results are shown to converge to the exact elasticity solution. A computer program using arbitrary quadrilateral elements is briefly described in an application to the analysis of cooled gas turbine blades.