Formulação computacional e análise comparativa elástico-linear de vigas utilizando o método dos elementos finitos

Abstract

This work is based on the elastic-linear structural analysis of beams using numerical methods to compare with the solutions obtained with the classical theories. In this sense, it aimed to develop a computational routine for calculating stress and displacement in continuous beams by Euler-Bernoulli, based on the Finite Element Method (FEM). Given the numerical formulation developed according to the theory of elasticity and through the MEF, the study of the elastic line was performed according to the approximation in degree three polynomial function, considering the energy analysis through the concept of the minimum total potential energy and the principle of functional minimizations of variational calculus. Thus, the matrix relations between the structure stiffness, its loading and the displacements caused by them were established. From the results of nodal displacement, the values of internal stresses and reactions in the beams can be obtained. From this numerical and mathematical formulation, the computational implementation was performed in high procedural language, assembling the global and local stiffness matrices, as well as the global and local equivalent load vectors. The main system whose unknowns are the nodal displacements had been solved by the Gauss-Jordan method and the internal efforts and support reactions obtained with stiffness-based subroutines. The program was applied in 3 cases totaling 10 examples of beams in different situations of bonds, external loading, flexural stiffness and materials. Based on the results, it showed accuracy and validation between the routine application to the analyzed examples, based on comparative analytical results and / or values from other authors and structural analysis software. The nodal results of efforts, support reaction and dislocations were consistent in terms of isostatic and hyperstatic beams. Finally, it was verified that, although the values of efforts and displacements were obtained discreetly (nodal), it is possible to obtain the approximation of the internal stress diagrams and the elastic line of a beam according to a greater refinement of the stress matrix. Elements.