A comparison of primal- and dual-mixed finite element formulations for Timoshenko beams

Analytical and numerical comparisons of a primal-mixed, a dual-mixed and a consistent primal-dual mixed finite element model for shear-deformable beams are presented using the lowest possible order, constant and linear, polynomial approximations. The stiffness matrices and the load vectors of the mixed elements are derived and compared analytically with each other and with that of the standard displacement-based Timoshenko beam element. It is pointed out that the element stiffness matrices of the dual-mixed and the primal-dual mixed elements are the exact ones that differ from the standard Timoshenko beam element in a geometry-, material- and mesh-dependent constant factor denoted by C (s). This constant, or its reciprocal, can be considered not only as a shear locking indicator, but can, as the results for the primal-dual mixed formulation indicate, be used to transform the standard Timoshenko beam element into a shear locking-free element, independently of the slenderness ratio and the loading of the beam.