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.

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.

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.

scholarly journals Invariant measures for the flow of a first order partial differential equation

1985 ◽  
Vol 5 (3) ◽  
pp. 437-443 ◽  
Author(s):  
R. Rudnicki
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Invariant Measures ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order

AbstractWe prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.

scholarly journals Buckling and Vibration of Non-Homogeneous Rectangular Plates Subjected to Linearly Varying In-Plane Force

2013 ◽  
Vol 20 (5) ◽  
pp. 879-894 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Material ◽  
Classical Plate Theory ◽  
Simply Supported

The present work analyses the buckling and vibration behaviour of non-homogeneous rectangular plates of uniform thickness on the basis of classical plate theory when the two opposite edges are simply supported and are subjected to linearly varying in-plane force. For non-homogeneity of the plate material it is assumed that young's modulus and density of the plate material vary exponentially along axial direction. The governing partial differential equation of motion of such plates has been reduced to an ordinary differential equation using the sine function for mode shapes between the simply supported edges. This resulting equation has been solved numerically employing differential quadrature method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. The effect of various parameters has been studied on the natural frequencies for the first three modes of vibration. Critical buckling loads have been computed. Three dimensional mode shapes have been presented. Comparison has been made with the known results.

Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Roshan Lal ◽  
Dhanpati
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
The Other ◽  
Thickness Variation ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Varying Thickness ◽  
Aspect Ratios ◽  
Displacement Mode

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges (y=0 and y=b) and resting on two-parameter foundation (Pasternak-type) have been studied on the basis of classical plate theory. The other two edges (x=0 and x=a) may be any combination of clamped and simply supported edge conditions. The nonhomogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli and density along one direction. By expressing the displacement mode as a sine function of the variable between simply supported edges, the fourth order partial differential equation governing the motion of such plates of exponentially varying thickness in another direction gets reduced to an ordinary differential equation with variable coefficients. The resulting equation is then solved numerically by using the Chebyshev collocation technique for two different combinations of clamped and simply supported conditions at the other two edges. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters such as nonhomogeneity, density, and thickness variation on the frequencies of the plate with different aspect ratios. Normalized displacements are presented for a specified plate. A comparison of results with those obtained by other methods shows the computational efficiency of the present approach.

scholarly journals Behaviour under Moving Distributed Masses of Simply Supported Orthotropic Rectangular Plate Resting on a Constant Elastic Bi-Parametric Foundation

2020 ◽  
pp. 64-92
Author(s):  
T. O. Awodola ◽  
S. Adeoye
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourth Order ◽  
Closed Form Solution ◽  
Form Solution ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Simply Supported ◽  
Orthotropic Rectangular Plate

This work investigates the behavior under Moving distributed masses of orthotropic rectangular plates resting on bi-parametric elastic foundation. The governing equation is a fourth order partial differential equation with variable and singular co-efficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al[1]. This is then simplified using modified asymptotic method of Struble. The closed form solution is analyzed, resonance conditions are obtained and the results are presented in plotted curves for both cases of moving distributed mass and moving distributed force.

scholarly journals Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

2010 ◽  
Vol 7 (1) ◽  
pp. 829-832
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Uniqueness Theorems ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Partially Order ◽  
Global Uniqueness

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

scholarly journals A Stable Difference Scheme for a Third-Order Partial Differential Equation

2018 ◽  
Vol 64 (1) ◽  
pp. 1-19 ◽  
Author(s):  
A Ashyralyev ◽  
Kh Belakroum
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

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