APPLICATION OF THE FINITE ELEMENT METHOD FOR SOLVING PRACTICAL ENGINEERING PROBLEMS

Abstract

 This paper presents a journal-ready study on the application of the finite element method (FEM) to practical engineering boundary-value problems. A one-dimensional axial bar with a spatially varying distributed load is adopted as a benchmark model to demonstrate the complete solution chain: variational formulation, domain discretization, element matrix derivation, global assembly, imposition of boundary conditions, and numerical post-processing. The resulting finite element model is validated against a closed-form analytical solution. In addition to displacement and stress fields, convergence is quantified through domain L2 and energy-norm errors. The results show that mesh refinement steadily improves the approximation and that the FEM solution accurately reproduces the analytical response while retaining the modularity required for extension to more complex structural systems. The manuscript is formatted to align with the current submission expectations of Results in Engineering, including a concise abstract, keywords, in-text citations, editable tables, separate highlights, and an optional graphical abstract file.