A finite element method for the Saint-Venant torsion and bending problems for prismatic beams

• Developed a formulation for Saint Venant torsion that is valid for multiply-connected, anisotropic and compound bars.
• Developed a formulation for Saint Venant bending that is based on the theory of elasticity.
• Both formulations yields good coarse-mesh accuracy even for thin-walled structures.
• No boundary terms are involved in either formulation; hence can be used for multiply-connected domains.

In this work, we present a finite element formulation for the Saint-Venant torsion and bending problems for prismatic beams. The torsion problem formulation is based on the warping function, and can handle multiply-connected regions (including thin-walled structures), compound and anisotropic bars. Similarly, the bending formulation, which is based on linearized elasticity theory, can handle multiply-connected domains including thin-walled sections. The torsional rigidity and shear centers can be found as special cases of these formulations. Numerical results are presented to show the good coarse-mesh accuracy of both the formulations for both the displacement and stress fields. The stiffness matrices and load vectors (which are similar to those for a variable body force in a conventional structural mechanics problem) in both formulations involve only domain integrals, which makes them simple to implement and computationally efficient.