Numerical Analysis of the Finite Element Method for Solving Elasticity Problems in Solid Mechanics

This study provides a systematic numerical analysis of the finite element method as applied to boundary value problems in linear elasticity, with a focus on convergence properties, error estimation, and the performance of alternative element formulations for structural problems representative of those encountered in Nigerian civil and mechanical engineering practice. The finite element method is among the most widely used computational tools in engineering analysis, yet its rigorous mathematical foundations, including variational formulations, Galerkin approximations, a priori and a posteriori error estimates, and locking phenomena in nearly incompressible materials, are frequently treated superficially in applied engineering programmes. The study derives the weak formulation of the Navier equations of linear elasticity and constructs conforming finite element approximations using triangular and quadrilateral elements with linear and quadratic shape functions. Theoretical convergence rates derived from the Céa lemma and Sobolev space interpolation theory are verified computationally on a series of benchmark problems with known analytical solutions, including a cantilever beam under distributed loading, a thick-walled pressure vessel, and a plane stress plate with a circular hole. Volumetric locking in the incompressible limit is demonstrated and addressed through reduced integration and mixed displacement-pressure formulations. The study employs the FEniCS computing environment for implementation. Results confirm theoretical convergence rates for regular meshes and document the improvement in convergence achieved by quadratic elements. Keywords: finite element method, linear elasticity, convergence analysis, error estimation, Galerkin method