In this document we illustrate the dimensionalreduction approach applied to 3D solid elastic equations in order to obtain a beam model. We start from the HellingerReissner (HR) principle, in a formulation which guarantees the selection of a compatible solution in a family of equilibrated fields. Then, we introduce a semidiscretization within the crosssection, this allows to reduce the problems dimension from 3D to 1D and to formulate the properly called beam model. After a manipulation of the 1D weak model (done in order to guarantee the selection of an axisequilibrated solution in a family of axiscompatible fields), we introduce a discretization also 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 crosssection, on the other hand, the 1D model manipulation leads to an accurate displacement analysis along the beamaxis. Moreover, the manipulation allows to statically condensate stresses at element level, improving the numerical efficiency of the FE algorithm. In order to illustrate the capability of the method, we consider a slim crosssection beam that shows interesting 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.
