Desenvolvimento de código computacional para análise elástica linear de chapas utilizando o elemento finito CSQ

Abstract

With the advent of computers and technological advancement new techniques emerged that revolutionized the analysis of structures by engineers. One such technique was the Finite Element Method (FEM), which is a numerical method used to solve structures in which analytical solutions become unviable due to their complexity. The method consists of dividing the structures into small parts, called finite elements, which are joined by specific points called nodes. From the numerical modeling and consequent development of the equilibrium equations that govern the problem, it is possible to implement the FEM in computational form to perform the structural analysis. The greater the number of elements, the better the approximation of the results and the greater the computational effort. Within the twodimensional linear elastic analysis is the element known as Constant Strain Quadrilateral (CSQ), which is characterized by rectangular shape and 4 nodal points and 2 degrees of freedom per node, being these two coplanar displacements perpendicular to each other and towards the coordinate axes. The CSQ element is suitable for the analysis of plane stress problems, which are characterized by being flat elements and subject to loading contained in their own plane. In this context, this work aims to develop a computational code in Python language that performs linear elastic of plane stress problems using finite element CSQ. Throughout the work is presented the mathematical formulation of the method and the deduction of the stiffness matrix of the CSQ element. Finally, the developed code is used to simulate various examples, and the results obtained are compared with those provided in the literature or with those obtained in other software already renowned in the area of structural analysis. The comparison of results shows that the developed code provides adequate and satisfactory results.