Abstract
Abstract The three-dimensional Theory of Elasticity equations lead to a complex solution for most problems in engineering. Therefore, the solutions are typically developed for reduced systems, usually symmetrical or two-dimensional. In this context, computational resources allow the reduction of these simplifications. The most recognized methods of algebraic approximation of the differential equations are the Finite Differences Method and the Finite Element Method (FEM). However, they have limitations in mesh generation and/or adaptation. As follows, Meshless Methods appear as an alternative to these options. The present work uses the Radial Point Interpolation Method (RPIM) to evaluate the stress in two-dimensional beams in regions close to loading (Saint Venant's Principle). Formulations based on the Fourier Series Theory and the RPIM are presented. Multiquadrics Radial Basis Functions were used to obtain the stiffness matrix. Two numerical examples demonstrate the validity of the RPIM for the proposed theme. The results were obtained from the formulations cited and the Finite Element Method for comparison.
Highlights
The analytical solution to most problems of the Theory of Elasticity is difficult due to the complexity of the equations
The Radial Point Interpolation Method (RPIM) code was written in FORTRAN language and divided into modules to make the management of the main program easier
This study presented the Radial Point Interpolation Method (RPIM) to evaluate the stress in two-dimensional beams
Summary
The analytical solution to most problems of the Theory of Elasticity is difficult due to the complexity of the equations. The resolutions are typically designed for reduced systems, usually symmetrical or two-dimensional (Saad, 2005) In this way, the computational analysis has considerable relevance in the solution of these problems. The most known methods of numerical analysis are the Finite Differences Method and the Finite Element Method (FEM), the latter being the most used It includes limitations, mainly in mesh generation and adaptation. An important fact to be observed in the structure is the effect of loading in regions close to the point of application This effect is called the SaintVenant's Principle. Petrolo and Casciaro (2004) investigated the use of the Saint-Venant's general rod theory for deriving the stiffness matrix in three-dimensional beam elements with a general cross-section. The proposed research aims to demonstrate the Saint-Venant Principle for two-dimensional beams using the Radial Point Interpolation Method (RPIM). The results are compared with the analytical solution and numerical solution of the Finite Element Method utilizing the SAP2000© software
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