The dimensional reduction modelling approach for 3D beams: Differential equations and finite-element solutions based on Hellinger-Reissner principle

This paper illustrates an application of the so-called dimensional reduction modelling approach to obtain a mixed, 3D, linear, elastic beam-model. We start from the 3D linear elastic problem, formulated through the Hellinger-Reissner functional, then we introduce a cross-section piecewise-polynomial approximation, and finally we integrate within the cross section, obtaining a beam model that satisfies the cross-section equilibrium and could be applied to inhomogeneous bodies with also a non trivial geometries (such as L-shape cross section). Moreover the beam model can predict the local effects of both boundary displacement constraints and non homogeneous or concentrated boundary load distributions, usually not accurately captured by most of the popular beam models. We modify the beam-model formulation in order to satisfy the axial compatibility (and without violating equilibrium within the cross section), then we introduce axis piecewise-polynomial approximation, and finally we integrate along the beam axis, obtaining a beam finite element. Also the beam finite elements have the capability to describe local effects of constraints and loads. Moreover, the proposed beam finite element describes the stress distribution inside the cross section with high accuracy. In addition to the simplicity of the derivation procedure and the very satisfying numerical performances, both the beam model and the beam finite element can be refined arbitrarily, allowing to adapt the model accuracy to specific needs of practitioners.