Mixed 3D beam models: Differential equation derivation and finite element solutions

In this document we illustrate the dimensional-reduction approach applied to 3D solid elastic equations in order to obtain a beam model. We start from the Hellinger-Reissner (HR) principle, in a formulation which guarantees the selection of a compatible solution in a family of equilibrated fields. Then, we introduce a semi-discretization within the cross-section, this allows to reduce the problem’s dimension from 3D to 1D and to formulate the beam model. After a manipulation of the 1D weak model (done in order to guarantee the selection of an axis-equilibrated solution in a family of axis-compatible fields), we introduce a discretization along the beam axis obtaining the related beam Finite Element (FE). On one hand, the initial HR principle formulation leads to an accurate stress analysis into the cross-section, on the other hand, the 1D model manipulation leads to an accurate displacement analysis along the beam-axis. Moreover, the manipulation allows to statically condensate stresses out at element level, improving the numerical efficiency of the FE algorithm. In order to illustrate the capability of the method, we consider a slim cross-section beam that shows non trivial behaviour in bending and for which the analytical solution is available in literature. Numerical results are accurate in description of both displacement and stress variables, the FE solution converges to the analytical solution, and the beam FE models complex phenomena like anticlastic bending and boundary effects.