Analytical derivation of a general 2D non-prismatic beam model based on the Hellinger-Reissner principle

Abstract

This paper presents an analytical model for the study of 2D linear-elastic non-prismatic beams. Its principal aim is to accurately predict both displacements and stresses using a simple procedure and few unknown variables. The approach adopted for the model derivation is the so-called dimensional reduction starting from the Hellinger–Reissner functional, which has both displacements and stresses as independent variables. Furthermore, the Timoshenko beam kinematic and appropriate hypotheses on the stress field are considered in order to enforce the boundary equilibrium. The use of dimensional reduction allows the reduction of the integral over a 2D domain, associated with the Hellinger–Reissner functional, into an integral over a 1D domain (i.e., the so-called beam-axis). Finally, through some mathematical manipulations, the six ordinary differential equations governing the beam structural behaviour are derived. In order to prove the capabilities of the proposed model, the solution of the six equations is obtained for several non-prismatic beams with different geometries, constraints, and load distributions. Then, this solution is compared with the results provided by an already existing, more expensive, and refined 2D finite element analysis, showing the efficiency of the proposed model to accurately predict both displacements and stresses, at least in cases of practical interest.