Analysis of elastic flexure problems by the finite volume method

Flexure problems are an important class of elasticity problems with extensive applications to structural engineering. Solutions to these problems provide foundation for the design of prismatic bars of arbitrary cross section loaded on their end faces. Whereas elasticity solutions have been developed for flexure of prismatic bars with simple cross sections, cross sections employed in structural design typically cannot be solved analytically. Hence numerical approaches are employed, with the finite-element method the dominant approach. Herein, we present an alternative solution methodology to flexure problems based on the finite-volume method. The approach is rooted in the St. Venant’s semi-inverse formulation which provides the basis for constructing an approximate displacement field partitioned into torsion and bending modes. The displacement field is employed to construct closed-form local stiffness matrices for subvolumes of the discretized cross section that satisfy local equilibrium in a surface-average sense. Subvolume assembly into a global stiffness matrix satisfies both displacement and traction continuity in a surface-average sense, producing a linear system of equations for the unknown interfacial displacements. The methodology is validated by available elasticity solutions to homogeneous isotropic and orthotropic prismatic bars, illustrating quadratic convergence with domain discretization and excellent correlation with exact stress fields. We demonstrate the method’s capability in the analysis of practical beam cross sections, including assessment of the thin-wall assumption applicability in the shear center determination. Finally, we leverage the method’s torsional and flexural mode partition to determine local stress fields that arise when the end load is applied away from the shear center.