Revisiting the bi-dimensional elastic three-layer beam analytical solution: a novel solution for various loads followed by an XFEM numerical verification

Abstract

An analytical solution based on Airy’s stress function is presented for a three-layer sandwich beam in a cantilever configuration, subjected to transverse force, bending moment, and linearly distributed load. Each layer is assumed to be isotropic and homogeneous. Closed-form expressions for displacements and stresses are derived, with the displacement field exhibiting a fifth-degree polynomial dependence. Classical solutions for two-layer and homogeneous beams are recovered as particular cases. The proposed solution is implemented and made publicly available as an open-source library on GitHub (https://github.com/rodrgz/2D3LayerCantileverBeam), released under the MPL license. To assess its accuracy, the extended finite element method (XFEM)-well-suited for problems with material interfaces-is employed. Numerical results show excellent agreement with the analytical predictions, demonstrating the proposed solution’s potential as a reliable benchmark for validating numerical methods in elasticity problems involving layered media.