On the analytical approach for the bending/stretching of linearly elastic functionally graded rectangular plates with two opposite edges simply supported

2009 ◽  
Vol 223 (9) ◽  
pp. 2009-2016 ◽  
Author(s):  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rectangular Plate ◽  
Plate Theory ◽  
Functionally Graded ◽  
Mindlin Plate ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Simply Supported

In this article, a new analytical method for bending—stretching analysis of thick functionally graded (FG) rectangular plates is presented. Using this method, the governing equations of FG rectangular plates based on the first-order shear deformation or Mindlin plate theory are decoupled. Five coupled partial differential equations of the Mindlin FG plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function. It is analytically shown that by introducing an equivalent flexural rigidity, the equations of FG rectangular plate become similar to those of the homogeneous isotropic plate. Solving these equations, the solutions are obtained for the FG rectangular plate with two opposite edges simply supported. A comparison of the present results with available solutions from previous studies is made and a good agreement can be seen. Also, the numerical results for stress and deflection of the FG rectangular plate with various boundary conditions are obtained.

Effect of Shear Deformations on the Bending of Rectangular Plates

1960 ◽  
Vol 27 (1) ◽  
pp. 54-58 ◽  
Author(s):  
V. L. Salerno ◽  
M. A. Goldberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Wide Range

The three partial differential equations derived by Dr. E. Reissner2, 3 have been reduced to a fourth-order partial differential equation resembling that of the classical plate theory and to a second-order differential equation for determining a stress function. The general solution for the two partial differential equations has been applied to a simply supported plate with a constant load p and to a plate with two opposite edges simply supported and the other two edges free. Numerical calculations have been made for the simply supported plate and the results compared with those of classical theory. The calculations for a wide range of parameters indicate that the deviation is small.

An Analytical Solution for Nonlinear Vibrations Analysis of Functionally Graded Plate Using Modified Lindstedt–Poincare Method

2020 ◽  
Vol 12 (01) ◽  
pp. 2050003 ◽  
Author(s):  
S. Hashemi ◽  
A. A. Jafari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rectangular Plate ◽  
Volume Fraction ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Published Data ◽  
Power Law Distribution ◽  
Partial Differential

In this research, the nonlinear free vibrations analysis of functionally graded (FG) rectangular plate which simply supported all edges are investigated analytically using modified Lindstedt–Poincare (MLP) method for the first time. For this purpose, with the aid of von Karman nonlinearity strain-displacement relations, the partial differential equations of motion are developed based on first-order shear deformation theory (FSDT). Afterward, by applying Galerkin method, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations. The nonlinear equation of motion is then solved analytically by MLP method to determine the nonlinear frequencies of the FG rectangular plate. The material properties are assumed to be graded through the direction of plate thickness according to power law distribution. The effects of some system parameters such as vibration amplitude, volume fraction index and aspect ratio on the nonlinear to linear frequency ratio are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.

On the Thermal Buckling Response of Shear-Flexible All-Edge Clamped Rectangular Plates

2003 ◽  
Vol 9 (5) ◽  
pp. 495-506 ◽  
Author(s):  
Abdulateef M. Al-Khaleefi ◽  
Humayun R. H. Kabir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Shear Deformation Theory ◽  
Double Fourier Series ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Approximate Methods ◽  
Flexible Response

Using an analytical approach, we investigate a thermal stability response for a rectangular plate with all-edge clamped boundary conditions. We consider the first-order shear deformation theory that utilizes shear flexible response, in order to obtain three highly coupled governing partial differential equations in three unknowns: one transverse displacement, and two independent rotations of the normal. The solution functions are assumed in the form of double Fourier series that satisfy the boundary conditions, as well as the partial differential equations. The results obtained from the analytical solution are compared with available finite element solutions. These analytically obtained results can be capitalized to check the accuracy of various approximate methods.

Nonlinear forced vibration of rectangular plates by modified multiple scale method

2021 ◽  
pp. 095440622110527
Author(s):  
Farzaneh Rabiee ◽  
Ali Asghar Jafari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Plate Theory ◽  
Multiple Scales Method ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Nonlinear Forced Vibration

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply-supported opposite edges

2010 ◽  
Vol 224 (9) ◽  
pp. 1831-1841 ◽  
Author(s):  
M Mohammadi ◽  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Arbitrary Boundary ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Fg Plates

In this article, a novel analytical method for decoupling the coupled stability equations of functionally graded (FG) rectangular plates is introduced. Based on the Mindlin plate theory, the governing stability equations that are coupled in terms of displacement components are derived. Introducing four new functions, the coupled stability equations are converted into two independent equations. The obtained equations have been solved for buckling analysis of rectangular plates with simply-supported two edges and arbitrary boundary conditions along the other edges (Levy boundary conditions). The critical buckling loads are presented for different loading conditions, various thickness to side and aspect ratios, some powers of FG materials, and various boundary conditions. The presented results for buckling of moderately thick FG plates with two simply-supported edges are reported for the first time.

Nonlinear Free and Forced Vibrations of In-Plane Bi-Directional Functionally Graded Rectangular Plate with Temperature-Dependent Properties

2020 ◽  
Vol 20 (08) ◽  
pp. 2050097
Author(s):  
Soheil Hashemi ◽  
Ali Asghar Jafari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rectangular Plate ◽  
Forced Vibrations ◽  
Functionally Graded ◽  
Nonlinear Frequency ◽  
Temperature Dependent ◽  
Partial Differential ◽  
Free And Forced Vibrations ◽  
Temperature Dependent Properties

In this paper, the nonlinear free and forced vibrations analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with temperature-dependent properties is studied for the first time. For this purpose, with the aid of von Karman nonlinearity strain–displacement relations, the partial differential equations of motion are developed based on the first-order shear deformation theory (FSDT). Then, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations by applying the Galerkin method. The primary and super harmonic resonances are analyzed by the method of multiple scales (MMS). The material properties are assumed to be temperature-dependent and graded in the thickness direction according to the power-law distribution. The effects of some system parameters, such as vibration amplitude, volume fraction indexes, length-to-thickness ratio, temperature and aspect ratio on the nonlinear frequency and also frequency responses curve, are discussed in detail. To validate the analysis, the results of this paper are compared with the published data and good agreements are found.

Curvature Approximation Effects in the Free Vibration Analysis of Functionally Graded Shells

2016 ◽  
Vol 08 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Salvatore Brischetto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Curvilinear Coordinates ◽  
Equilibrium Equations ◽  
Partial Differential ◽  
Exact Geometry

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

scholarly journals Analysis Bending Solutions of Clamped Rectangular Thick Plate

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Yang Zhong ◽  
Qian Xu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Shear Deformation ◽  
Thick Plate ◽  
Plate Theory ◽  
Analytic Solutions ◽  
Higher Order ◽  
Numerical Comparison ◽  
Governing Equations ◽  
Partial Differential

The bending solutions of rectangular thick plate with all edges clamped and supported were investigated in this study. The basic governing equations used for analysis are based on Mindlin’s higher-order shear deformation plate theory. Using a new function, the three coupled governing equations have been modified to independent partial differential equations that can be solved separately. These equations are coded in terms of deflection of the plate and the mentioned functions. By solving these decoupled equations, the analytic solutions of rectangular thick plate with all edges clamped and supported have been derived. The proposed method eliminates the complicated derivation for calculating coefficients and addresses the solution to problems directly. Moreover, numerical comparison shows the correctness and accuracy of the results.

Vibratory Bending of Damped Laminated Plates

1969 ◽  
Vol 91 (4) ◽  
pp. 1081-1090 ◽  
Author(s):  
R. A. Ditaranto ◽  
J. R. McGraw
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Natural Frequencies ◽  
Laminated Plates ◽  
Simultaneous Equations ◽  
Laminated Plate ◽  
Partial Differential ◽  
Simply Supported ◽  
First Case ◽  
Viscoelastic Layers

The natural frequencies and associated composite loss factor have been determined for a finite-length laminated plate having alternate elastic and viscoelastic layers. Partial differential equations in terms of the variables of the plate are derived and, with the loading equation for a freely vibrating plate, a set of simultaneous partial differential equations is formed. Of two solutions considered the first is general and the second satisfies the boundary condition for a simply supported plate. In both cases, the resulting algebraic simultaneous equations are complex since the shear modulus of the viscoelastic material is a complex expression. In the first case, the expressions could not be solved directly since the value of the eigenvalues depended upon the boundary conditions, whereas the eigenvalues for the simply supported plate could be easily chosen. The simply supported case is solved and the results plotted for specific dimensionless parameters.

Nonlinear Vibration of Functionally Graded Material Cylindrical Shell Based on Reddy’s Third-Order Plates and Shells Theory

2012 ◽  
Vol 625 ◽  
pp. 18-24 ◽  
Author(s):  
Lu Dong ◽  
Yu Xin Hao ◽  
Jian Hua Wang ◽  
Li Yang
Keyword(s):  
Cylindrical Shell ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Third Order ◽  
Partial Differential ◽  
Graded Material ◽  
Plates And Shells

In this paper, an analysis on nonlinear dynamics of a simply supported functionally graded material (FGM) cylindrical shell subjected to the different excitation in thermal environment. Material properties of cylindrical shell are assumed to be temperature-dependent. Based on the Reddy’s third-order plates and shells theory[1], the nonlinear governing partial differential equations of motion for the FGM cylindrical shell are derived by using Hamilton’s principle. Galerkin’s method is utilized to transform the partial differential equations into a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitation. The effects played by different excitation and system initial conditions on the nonlinear vibration of the cylindrical shell are studied. In addition, the Runge–Kutta method is used to find out the nonlinear dynamic responses of the FGM cylindrical shell.

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