RBF-based Meshless Method for Large Deflection of Plates on Nonlinear Foundations

Abstract

Modeling the interaction between the soil and the structural foundation is a major concern in many engineering applications. The soil foundations very often represent a complex medium and therefore it is extremely difficult to obtain analytical-based solutions for this interaction. The problem becomes more difficult if the structure is undergoing large deformation. Numerical methods are powerful alternative tools to solve such complicated problems. There are two main categories of these methods: element-based methods such as the finite element (FEM) and the boundary element (BEM) and element free-based or mesh-less methods like radial basis functions (RBF). The FEM and BEM have been studied extensively so far, whereas the RBF still needs to be investigated for the use in various engineering application areas. Free-element or free-mesh methods are advantageous for problems that require frequent re-meshing such as those arising in non-linear analysis. In this work, the use of multi-quadratic radial basis function (MQ-RBF) is proposed to solve the problem of large deflection of thin elastic plates resting on nonlinear foundations. The study covers several variables, including the plate shape, the loading type, the boundary conditions, the foundation type and the foundation parameter. Two types of foundation models (discrete and continuous) are adopted. The discrete foundation model consists of Winkler, non-linear Winkler, Pasternak and non-linear Winkler-Pasternak model whereas the Boussinesq model is used to represent interaction of the plate with the continuous foundation. WALFRAM MATHEMATICA software was utilized for the development of the computer codes. The accuracy and efficiency of the method is demonstrated by comparing the obtained solution with readily available and corresponding numerical solutions. From the comparison, the obtained results from MQ-RBF method are close to the ones obtained from FEM. As a main conclusion, the proposed meshless-based method (MQ-RBF) can be utilized as a good alternative numerical method for the element based numerical methods such as FEM and BEM in the solution of the large deflection of thin elastic plates on nonlinear foundations. The developed computer codes have been utilized to generate correlations between discrete and continuous foundation parameters. The obtained correlations of the foundation parameters have been verified through several examples which revealed that the Pasternak and non-linear Winkler-Pasternak discrete models are the most in agreement with the continuous foundation model.