Mesh Convergence and Error Analysis in Finite Element Modeling of Beam Structures

¹ Department of Uzbek Language and Teaching Languages, Fergana State Technical University, Fergana, 150100, Uzbekistan
² Department of Computer Engineering and Artificial Intelligence, Fergana State Technical University, Fergana, 150100, Uzbekistan

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Published online: 1 July 2026

Abstract

Checking for mesh convergence is important as it will give you an idea of the quality of your finite element solutions. A cantilever beam with a uniform loading is used as a test case in this study to examine the effect of mesh refinement on the numerical accuracy. Analytical results obtained from Euler–Bernoulli beam theory were used as reference values for the finite element predictions. Meshes were created and gradually refined along a series and comparisons between the displacement and the stress response were made. Relative error, root mean square error (RMSE), Richardson extrapolation and Grid convergence index (GCI) were used to investigate the discretisation error and convergence properties. The numerical results approached the analytical solution as the mesh density was refined, with the displacement approaching the solution faster than the stress. The convergence order was approximately second order for the stress quantity of interest observed at the midspan of the beam. The discretization uncertainty for the fine mesh was estimated to be about 1.3%. The study also seeks to show the impact of stress concentrations around constrained areas and to discuss practical considerations for the interpretation of convergence based on stresses. The outlined procedure here may be used as a reference procedure for verification studies using beam-type finite element models and other structural analysis techniques.