Research On Moderately Thick Plates Using The Meshless Local Radial Point Interpolation Method

The meshless method is a new numerical method with a great prospect developed after traditional numerical methods such as Finite Element Method, Boundary Element Method et al. The meshless method possesses many advantages, among these the most outstanding advantage is independent of meshes, and thoroughly or partly eliminates meshing. By using this method, it becomes easy to solve large deformation problems, crack propagation problems and high velocity impact problems et al. A lot of the important pioneering effort has been done on the meshless methods by scholars in a recent decade. The meshless local radial point interpolation method (LRPIM) is a new numerical technique presented in recent years. It doesn't need any element or mesh for the energy integral or the purpose of interpolation. Therefore it is a truly meshless method. The shape functions have the Kronecker delta function property, and the essential boundary conditions can be easily imposed. Applications of the meshless LRPIM to bending and dynamic problems as well as elasto-plastic problems of moderately thick plates are presented in this dissertation.At the beginning of the dissertation, recent developments of the meshless method are overviewed. Several typical meshless methods are reviewed and appraised in term of their discretization schemes. Characteristics, advantages and disadvantages of all kinds of meshless methods are pointed out. Applications of the meshless methods to the plate and shell problems are introduced. The shape function of the meshless LRPIM is all constructed by using the radial basis functions with polynomial basis functions, the singularity of the system matrix is overcome. The shape functions and their derivatives are simple, consequently, lower computational cost. The efficient and accurate results can be obtained.Although a lot of achievements are obtained about meshless methods for the plate and shell problems, the solution of moderately thick plate problem is rarely reported in the use of the meshless LRPIM. In this dissertation, the meshless LRPIM is used to investigate bending and dynamic problems as well as elasto-plastic problems of moderately thick plates. Based on the equilibrium equations and dynamic equations of a moderately thick plate, the various discretized system equations for moderately thick plates are derived using locally weighted residual method. In the analysis of the bending problems for moderately thick plates with various boundary conditions and under various loads, deformations and bending (torsional) moments as well as stresses are calculated. Effects of the shape parameters of the radial basis function on the numerical results are investigated. Computing efficiency is studied when polynomials of the low and high order are used. Effects of sizes of the quadrature sub-domain and the influence domain on the numerical results are investigated. The reason of the shear locking and the measure of avoiding the shear locking are analyzed. It is found that the shear locking is easier avoided in the meshless method than in FEM. The static bending problems of a nonhomogeneous moderately thick plate are analyzed using the meshless LRPIM, too. The discretized system equation for moderately thick plates on the elastic foundation with two parameters is derived using a locally weighted residual method. Bending problems for the raft and moderately thick plates on the elastic foundation with simply supported and clamped boundary conditions are analyzed by the meshless LRPIM. The relative error and convergence rate for deflections and bending moments are studied. For the dynamic analysis of moderately thick plates, the discretized system equations of the free vibration and forced vibration for the moderately thick plate are presented. The subspace iterative method is adopted to solve the eigenvalue equation of the free vibration problem, and the Newmark method is used to discrete the time domain. The approaches of numerical implement are presented, and several numerical examples are presented for the free vibration and forced vibration of moderately thick plates with various boundary conditions. The dynamic bending problems of a nonhomogeneous moderately thick plate are analyzed using the meshless LRPIM, too. In the end, the elasto-plastic bending problems of moderately thick plates are analyzed by the meshless LRPIM. The elasto-plastic stress-strain relation of the moderately thick plate is studied, and an incremental Newton-Raphson iterative algorithm is employed to solve the nonlinear incremental discretized system equation of the moderately thick plate.Numerical results show that the present method possesses not only feasibility and validity, but also high accuracy and good performance of convergence for moderately thick plate problems including bending and dynamic problems as well as the elasto-plastic problem.