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.

A Quasi-Green’s Function Method for the Bending Problem of Simply Supported Trapezoidal Shallow Spherical Shells on Winkler Foundation

2012 ◽  
Vol 446-449 ◽  
pp. 3582-3586 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Method ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Green’S Function Method ◽  
Simply Supported

The quasi-Green’s function method (QGFM) is applied to solve the bending problem of simply supported trapezoidal shallow spherical shells on Winkler foundation. A quasi-Green’s 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 Green’s 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.

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.

scholarly journals R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

2020 ◽  
Vol 6 (3) ◽  
pp. 512-522
Author(s):  
Shanqing Li ◽  
Hong Yuan ◽  
Xiongfei Yang ◽  
Huanliang Zhang ◽  
Qifeng Peng
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Green's Function ◽  
Function Theory ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Laplace Operators ◽  
Independent Equation ◽  
Polygonal Boundary

The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green’s function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are 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 eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present 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.

Sign in / Sign up

Export Citation Format

Share Document