Primal- and Dual-Mixed Finite Element Models for Geometrically Nonlinear Shear-Deformable Beams – A Comparative Study
The relationships between the system matrices of the displacement-based, a primal-mixed, a dual-mixed and a consistent primal-dual mixed finite element model for geometrically nonlinear shear-deformable beams are investigated. Employing Galerkin-type weak formulations with the lowest possible order, constant and linear, polynomial approximations, the tangent stiffness matrices and the load vectors of the elements are derived and compared to each other in their explicit forms. The main difference between the standard and the dual-mixed element can be characterized by a geometry-, material- and meshdependent constant that can serve not only as a locking indicator but also to transform the displacement-based element into a shear-locking-free dual-mixed beam element. The numerical performances of the four different elements are compared to each other through two simple model problems. The superior performance of the mixed, and especially the dual-mixed, beam elements in the nonlinear case is demonstrated, not only for the deflection, but also for the force and moment computations.