Análisis de la viga de Timoshenko mediante el MEF a partir de la variante de Galerkin

Abstract

The beam is one of the fundamental structural elements in Civil Engineering, which due to the loads it must support presents a vertical displacement or deflection as its main characteristic. Its study is based on theories such as the Euler-Bernoulli theory and the Timoshenko theory, which has been of interest for the present work. This theory is based on the solution of a differential equation for which the use of numerical methods is required, among which the Finite Element Method (FEM) stands out, not being the only one, but the most used at present. One of the variants is that of Galerkin, which stands out for its efficiency and is well known in other fields of Physics-Mathematics, but not in Structural Analysis, where the solution to the discrete problem can be found explicitly as a linear combination of the basis functions or shape functions with unknown coefficients, i.e., a system of n linear equations with n unknowns is obtained, which allows determining the coefficients. Therefore, the implementation of the Galerkin variant for the finite element formulation of the Timoshenko theory in MATLAB is performed for the solution of different cases of beams subjected to different load (point, distributed) and support (embedded, simply supported) states. The results show the difference between the value of deflections obtained by the classical beam deformation method and the Timoshenko theory, mainly due to the consideration of the shear action by the latter, which is valid for beams where the slenderness values are low.