Saint Venant's torsion of homogeneous and composite bars by the finite volume method

We propose a new finite-volume based approach to the solution of Saint Venant's torsion problems of homogeneous and composite prismatic bars subjected to pure torsion. The semi-inverse approach is employed to construct an approximate displacement field characteristic of pure torsion within the subvolumes of the beam's discretized cross section such that the governing differential equation for the warping function is locally satisfied in each subvolume in an integral sense. Orthotropic subvolumes are admitted in the formulation to enable analysis of bars comprised of heterogeneous microstructures, including composite materials with orthotropic shear moduli. Continuity of both displacements and tractions across subvolumes' interfaces is satisfied in a surface-average sense, together with traction-free lateral boundary conditions. Closed-form expressions for the stiffness matrix elements that relate surface-averaged displacements to tractions are provided for ease of programming. The convergence and accuracy of the proposed method are assessed and verified upon comparison with the exact elasticity solutions for prismatic homogeneous and composite bars of rectangular cross section and the corresponding finite-difference solution for the homogeneous bar. The utility of the developed solution methodology is first illustrated by analyzing torsion of homogeneous T and box beams, focusing on the singular behavior at the re-entrant corners and the concomitant membrane analogy's applicability. Then, composite cross sections reinforced by angle braces as well as continuous fiber-reinforced wraps with graded shear moduli are investigated with the aim of mitigating the re-entrant corner shear stress singularities, not easily analyzed using the finite-element or finite-difference approaches.