Fall 2017 - Spring 2022

On mechanical solution of composite plates and shells using smooth basis functions and mesh free methods

In the present thesis, a number of meshfree methods have been developed in the context of Trefftz approaches to analyze the mechanical behavior of composite plates and shallow shells including bending, free vibration and buckling analysis. Due to the sophistication of applying 3D elasticity analysis for plate and shell problems, alternative deformation theories have been introduced in order to simulate plate and shell behavior in the form of simpler equations. The well-known Equivalent Single Layer (ESL) theories such as Classical Plate Theory (CLPT) based on Kirchhoff assumptions, First order Shear Deformation Theory (FSDT) proposed by Mindlin-Ressiner and the Third order Shear Deformation Theory (TSDT) proposed by Reddy are used to obtain acceptable results in the thickness range of thin to moderately thick plates and shells, respectively. The governing Partial Differential Equations (PDE) and boundary conditions corresponding to these theories may be derived by using the energy principle (virtual work). The main idea of the proposed meshfree methods is based on splitting the PDE solution into homogenous and particular parts, to avoid any integration process throughout the solution procedure. The methods have been developed in boundary (global) and local forms. In the boundary form, the homogenous and particular solutions are interpolated over the total domain by using smooth shape functions, and the unknowns can be found by satisfaction of the boundary conditions. Although the boundary form has advantages such as high accuracy and low computational expenses, it also has limitations regarding the solution complexity (e.g. concave domains, internal holes, singularities, highly oscillating loads, etc.). To overcome such deficiencies and include a wider range of problems in the realm of plates and shells, the research develops a domain decomposition technique as well as a meshless local approach. In the local form, the solution domain is discretized by a set of nodes at which the degrees of freedom are defined. A sub-domain (cloud) centered at each node and containing several adjacent nodes is considered, over which the corresponding PDE is split into homogenous and particular parts. The meshless methods presented in this thesis are based on Exponential Basis Functions (EBFs) and Equilibrated Basis functions (EqBFs), where the formulations in each form (boundary and local) may be developed in the same manner. These methods have significant advantages such as using smooth interpolation functions, avoiding integration throughout the solution procedure, application of the boundary conditions in a point-wise manner without the need for boundary meshes, and also needlessness of complicated mesh generation and assembling procedures. In addition, high convergence rate, sufficient overlap between the clouds to ensure complete continuity of deformation as well as the stress components, compatibility for high orders of continuity and the use of more accurate shape functions compared to the ordinary Finite Element Method (FEM) are some other advantages of the present local method. To demonstrate the accuracy and the efficiency of the methods, solution of various problems such as composite plates and shells with different shapes and boundary conditions has been presented. The numerical results of these analyses have been compared with that of other researchers (if available) or those obtained by FEM using commercial codes.