The dimensional reduction approach for 2D non-prismatic beam modelling : a solution based on Hellinger-Reissner principle

The present paper considers a non-prismatic beam i.e., a beam with a cross-section varying along the beam axis. In particular, we derive and discuss a model of a 2D linear-elastic non-prismatic beam and the corresponding finite element. To derive the beam model, we use the so-called dimensional reduction approach: from a suitable weak formulation of the 2D linear elastic problem, we introduce a variable cross-section approximation and perform a cross-section integration. The satisfaction of the boundary equilibrium on lateral surfaces is crucial in determining the model accuracy since it leads to consider correct stress-distribution and coupling terms (i.e., equation terms that allow to model the interaction between axial-stretch and bending). Therefore, we assume as a starting point the Hellinger-Reissner functional in a formulation that privileges the satisfaction of equilibrium equations and we use a cross-section approximation that exactly enforces the boundary equilibrium. The obtained beam-model is governed by linear Ordinary Differential Equations (ODEs) with non-constant coefficients for which an analytical solution cannot be found, in general. As a consequence, starting from the beam model, we develop the corresponding beam finite element approximation. Numerical results show that the proposed beam model and the corresponding finite element are capable to correctly predict displacement and stress distributions in non-trivial cases like tapered and arch-shaped beams.