Green Quasifunction Method for Bending Problem of Clamped Orthotropic Thin Plates with Trapezoidal Boundary Shape

2011 ◽  
Vol 117-119 ◽  
pp. 456-459 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Differential Equation ◽  
Mathematical Method ◽  
Integral Equation ◽  
Homogeneous Boundary Condition ◽  
Thin Plates ◽  
Biharmonic Operator ◽  
Boundary Equation ◽  
Boundary Shape ◽  
Governing Differential Equation ◽  
Suitable Form

The Green quasifunction method(GQM) is employed to solve the bending problem of clamped orthotropic thin plates with trapezoidal boundary shape. Firstly the governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Green’s formula. A Green quasifunction is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. A numerical example demonstrates the feasibility and efficiency of the proposed method, and it is a novel mathematical method.

Green Quasifunction Method for Bending Problem of Clamped Orthotropic Trapezoidal Thin Plates on Winkler Foundation

2011 ◽  
Vol 138-139 ◽  
pp. 705-708 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Integral Equation ◽  
Homogeneous Boundary Condition ◽  
Thin Plates ◽  
Winkler Foundation ◽  
Element Solution ◽  
Biharmonic Operator ◽  
Boundary Equation ◽  
Governing Differential Equation ◽  
Suitable Form ◽  
Good Agreement

The Green quasifunction method (GQM) is applied to solve the bending problem of clamped orthotropic thin plates with trapezoidal boundary shape on Winkler foundation. Firstly the governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Green’s formula. A Green quasifunction is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with ANSYS finite element solution shows good agreement. The proposed method is a novel and effective mathematical one.

A Quasi-Green Function Method for Free Vibration of Clamped Orthotropic Parallelogram Thin Plates on Winkler Foundation

2013 ◽  
Vol 397-400 ◽  
pp. 431-434
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Free Vibration ◽  
Function Method ◽  
Homogeneous Boundary Condition ◽  
Thin Plates ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Green Function Method ◽  
Suitable Form

The quasi-Green function method (QGFM) is applied to solve the free vibration of clamped orthotropic thin plates with parallelogram boundary shape on Winkler foundation. Firstly the model governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Greens formula. A quasi-Green function is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with ANSYS finite element solution shows a good agreement. The proposed method is a novel and effective mathematical one.

A Quasi-Greens Function Method for the Bending Problem of Simply Supported Polygonal Shallow Spherical Shells on Pasternak Foundation

2013 ◽  
Vol 353-356 ◽  
pp. 3215-3219
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Function Method ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Pasternak Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Simply Supported ◽  
Suitable Form ◽  
Greens Function ◽  
Good Agreement

The quasi-Greens function method (QGFM) is applied to solve the bending problem of simply supported polygonal shallow spherical shells on Pasternak foundation. A quasi-Greens function is established by using the fundamental solution and the boundary equation of the problem. And the function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Greens formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the proposed method.

Buckling and Vibration of Isosceles Triangular Plates having the Two Equal Edges Clamped and the Other Edge Simply–Supported

1955 ◽  
Vol 59 (530) ◽  
pp. 151-152 ◽  
Author(s):  
Hugh L. Cox ◽  
Bertram Klein
Keyword(s):  
Differential Equation ◽  
Axial Load ◽  
Unit Length ◽  
Approximate Solutions ◽  
The Other ◽  
Thin Plates ◽  
Critical Buckling Load ◽  
Simply Supported ◽  
Governing Differential Equation ◽  
Image Position

Approximate Solutions obtained by the method of collocation are presented for the lowest critical buckling load of an isosceles triangular plate loaded as shown in Fig. 1. Also, the fundamental frequency is given. The base of the triangle is simply supported and the other equal edges are clamped. The usual assumptions regarding the bending of thin plates are made. The governing differential equation for the plate loaded as shown in Fig. 1 is1where D is the plate stiffness, N is axial load per unit length, w is deflection, positive downward, and the quantities a and h are dimensions shown in Fig.1.

Application of the R-Function Theory for the Bending Problem of Shallow Spherical Shells with a Dodecagon Domain

2012 ◽  
Vol 468-471 ◽  
pp. 8-12 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Theory ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Suitable Form ◽  
Good Agreement

The R-function theory is applied to describe the dodecagon domain of shallow spherical shells on Winkler foundation, and it is also used to construct a quasi-Green’s function. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. A comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

On linear wave motions in magnetic-velocity shears

1977 ◽  
Vol 285 (1328) ◽  
pp. 607-636 ◽  
Keyword(s):  
Differential Equation ◽  
Critical Level ◽  
Velocity Shear ◽  
Linear Wave ◽  
Wave Amplification ◽  
Isothermal Fluid ◽  
Governing Differential Equation ◽  
Boussinesq Fluid ◽  
Propagation Properties ◽  
Background State

The propagation properties of linear wave motions in magnetic and/or velocity shears which vary in one coordinate z (say) are usually governed by a second order linear ordinary differential equation in the independent variable z. It is proved that associated with any such differential equation there always exists a quantity A which is independent of z. By employing A a measure of the intensity of the wave, this result is used to investigate the general propagation properties of hydromagnetic-gravity waves (e.g. critical level absorption, valve effects and wave amplification) in magnetic and/or velocity shears, using a full wave treatment. When variations in the basic state are included, the governing differential equation usually has more singularities than it has in the W.K.B.J. approximation, which neglects all variations in the background state. The study of a wide variety of models shows that critical level behaviour occurs only at the singularities predicted by the W.K.B.J. approximation. Although the solutions of the differential equation are necessarily singular at the irregularities whose presence is solely due to the inclusion of variations in the basic state, the intensity of the wave (as measured by A) is continuous there. Also the valve effect is found to persist whatever the relation between the wavelength of the wave and the scale of variations of the background state. In addition, it is shown that a hydromagnetic-gravity wave incident upon a finite magnetic and/or velocity shear can be amplified (or over-reflected) in the absence of any critical levels within the shear layer. In a Boussinesq fluid rotating uniformly about the vertical, wave amplification can occur if the horizontal vertically sheared flow and magnetic field are perpendicular. In a compressible isothermal fluid, on the other hand, wave amplification not only occurs in both magnetic-velocity and velocity shears but also in a magnetic shear acting alone.

scholarly journals New Bi-modular Material Approach to Buckling Problem of Reinforced Concrete Columns

2016 ◽  
Vol 6 (1) ◽  
pp. 19 ◽  
Author(s):  
Ahmad Salah Edeen Nassef ◽  
Mohammed A. Dahim
Keyword(s):  
Differential Equation ◽  
Reinforced Concrete ◽  
Differential Equations ◽  
Concrete Columns ◽  
Neutral Axis ◽  
Reinforced Concrete Columns ◽  
New Approach ◽  
Codes Of Practice ◽  
Governing Differential Equation ◽  
Transverse Deflection

<p class="1Body">This paper was investigating the buckling problem of reinforced concrete columns considering the reinforced concrete as bi – modular material. Governing differential equations was driven. The relation between the non-dimensional transverse deflection and non-dimensional distance between centroid axis and the neutral axis "eccentricity" was drawn to enable the solution of the governing differential equation. The new approach was verified with different experimental results and different codes of practice.<strong></strong></p>

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials

2014 ◽  
Vol 1030-1032 ◽  
pp. 223-227
Author(s):  
Lin Fan ◽  
Song Rong Qian ◽  
Teng Fei Ma
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Numerical Integration ◽  
Integration Method ◽  
Equation Of Motion ◽  
Numerical Integration Method ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Method Of Molecular Dynamics ◽  
Classic Theory

In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

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