scholarly journals Generalized Finite Difference Method for Plate Bending Analysis of Functionally Graded Materials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1940 ◽  
Author(s):  
Yu-Dong Li ◽  
Zhuo-Chao Tang ◽  
Zhuo-Jia Fu
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Plate Bending ◽  
Fg Plates ◽  
Bending Behavior ◽  
Resultant Matrix

In this paper, an easy-to-implement domain-type meshless method—the generalized finite difference method (GFDM)—is applied to simulate the bending behavior of functionally graded (FG) plates. Based on the first-order shear deformation theory (FSDT) and Hamilton’s principle, the governing equations and constrained boundary conditions of functionally graded plates are derived. Based on the multivariate Taylor series and the weighted moving least-squares technique, the partial derivative of the underdetermined displacement at a certain node can be represented by a linear combination of the displacements at its adjacent nodes in the GFDM implementation. A certain node of the local support domain is formed according to the rule of “the shortest distance”. The proposed GFDM provides the sparse resultant matrix, which overcomes the highly ill-conditioned resultant matrix issue encountered in most of the meshless collocation methods. In addition, the studies show that irregular distribution of structural nodes has hardly any impact on the numerical performance of the generalized finite difference method for FG plate bending behavior. The method is a truly meshless approach. The numerical accuracy and efficiency of the GFDM are firstly verified through some benchmark examples, with different shapes and constrained boundary conditions. Then, the effects of material parameters and thickness on FG plate bending behavior are numerically investigated.

Static Analysis of Moderately Thick Functionally Graded Plates With Various Boundary Conditions

2010 ◽  
Author(s):  
Nastaran Shahmansouri ◽  
Mohammad Mohammadi Aghdam ◽  
Kasra Bigdeli
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Various Boundary Conditions ◽  
Fg Plates ◽  
Ode Systems

The present study investigates static analyses of moderately thick FG plates. Using the First Order Shear Deformation Theory (FSDT), functionally graded plates subjected to transversely distributed loading with various boundary conditions are studied. Effective mechanical properties which vary from one surface of the plate to the other assumed to be defined by a power law form of distribution. Different ceramic-metal sets of materials are studied. Solution of the governing equations, including five equilibrium and eight constitutive equations, is obtained by the Extended Kantorovich Method (EKM). The system of thirteen Partial Differential Equations (PDEs) in terms of displacements, rotations, force and moment resultants are considered as multiplications of separable function of independent variables x and y. Then by successful utilization of the EKM these equations are converted to a double set of ODE systems in terms of x and y. The obtained ODE systems are then solved iteratively until final convergence is achieved. Closed form solution is presented for these ODE sets. It is shown that the method is very stable and provides fast convergence and highly accurate predictions for both thin and moderately thick plates. Comparison of the normal stresses at various points of rectangular plates and deflection of mid-point of the plate are presented and compared with available data in the literature. The effects of the volume fraction exponent n on the behavior of the normalized deflection, moment resultants and stresses of FG plates are also studied. To validate data for analysis fully clamped FG plates, another analysis was carried out using finite element code ANSYS. Close agreement is observed between predictions of the EKM and ANSYS.

scholarly journals Finite difference solution of plate bending using Wolfram Mathematica

2019 ◽  
Vol 13 (3) ◽  
pp. 241-247
Author(s):  
Katarina Pisačić ◽  
Marko Horvat ◽  
Zlatko Botak
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Rectangular Plate ◽  
Difference Method ◽  
Mapping Function ◽  
Equation System ◽  
Plate Bending ◽  
Wolfram Mathematica ◽  
Linear Equation System ◽  
Finite Difference Solution

This article describes the procedure of calculating deflection of rectangular plate using a finite difference method, programmed in Wolfram Mathematica. Homogenous rectangular plate under uniform pressure is simulated for this paper. In the introduction, basic assumptions are given and the problem is defined. Chapters that follow describe basic definitions for plate bending, deflection, slope and curvature. The following boundary condition is used in this article: rectangular plate is wedged on one side and simply supported on three sides. Using finite difference method, linear equation system is given and solved in Wolfram Mathematica. System of equations is built using the mapping function and solved with solve function. Solutions are given in the graphs. Such obtained solutions are compared to the finite element method solver NastranInCad.

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