Formulação do método dos elementos finitos utilizando o método de Rayleigh-Ritz para vigas

Abstract

The Finite Element Method (FEM) is one of the most widely used numerical tools in contemporary structural analysis, forming the basis of several commercial software packages employed in civil engineering projects. Despite its relevance, an introductory study of the FEM is not systematically addressed in the undergraduate curriculum of Civil Engineering at several Brazilian institutions, which motivated this work as a supporting material for students beginning their studies in the field. This Final Course Work aims to present, in an introductory manner, the mathematical formulation of the FEM based on the Rayleigh-Ritz Method, covering the concepts of variational calculus necessary for its understanding. To this end, the concepts of functional, variational operator δ, extrema of a functional, and the derivation of the Euler-Lagrange equation were systematically developed. Subsequently, the Rayleigh-Ritz Method was formulated and applied to a simply supported beam subjected to a uniformly distributed load, recovering the classical analytical expression of the deflection. Finally, the method was applied to the formulation of the one-dimensional beam finite element with four nodal displacements, obtaining the 4 × 4 stiffness matrix K(e) and the equivalent nodal force vector f (e) characteristic of the classical FEM beam element. The results obtained demonstrate the theoretical consistency of the Rayleigh-Ritz Method as the variational foundation of the FEM and reinforce its role as a didactic bridge between the classical methods of structural analysis and modern numerical methods.