R-function Theory for Bending Problem of Shallow Spherical Shells with 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.


Introduction
As a kind of structural forms, the shells and plates are widely used in various fields, such as, in the large-span roof, the underground foundation engineering, the hydraulic engineering, the large container manufacturing, the aviation, the shipbuilding, the missiles, the space technology, the chemical industry, and so on. Only few problems of the shells and plates with a regular geometric boundary and a simple differential equation can be solved with an analytical or a half analytical method. For most these problems with geometry of arbitrary shape and a complex boundary condition, only numerical methods can be used to solve the problems, such as the boundary element method [1], the Finite Element Method [2] and the finite difference method [3].
In the present paper, the R-function theory and the quasi-Green's function method (QGFM) proposed by Rvachev [4] are utilized. The bending problem of simply supported dodecagon shallow spherical shells on Winkler foundation with concave boundary is studied. The governing differential equation of the problem is decomposed into two simultaneous differential equations of lower order by utilizing an intermediate variable. A quasi-Green's function is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the 513 homogeneous boundary condition of the problem, but it does not satisfy the fundamental differential equation. The key point of establishing the quasi-Green's function consists in describing the boundary of the problem by a normalized equation = 0 and the domain of the problem by an inequality > 0. There are multiple choices for the normalized boundary equation. Based on a suitably chosen normalized boundary equation, a new normalized boundary equation can be established such that the singularity of the kernel of the integral equation is overcome. For any complicated domain, a normalized boundary equation can always be found according to the R-function theory. Thus, the problem can always be reduced to two simultaneous Fredholm integral equations of the second kind without the singularity. Using the R-function theory, Li and Yuan described successfully the rectangular, trapezoidal, triangular and parallelogrammic domains of plates [5][6][7] and shallow spherical shells [8,9]. For the first time, the Rfunction theory is applied to describe the dodecagon domain of the shallow spherical shells with concave boundary. The flowchart of research methodology is shown in the paper. 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 intermediate variable is introduced, and then the independent equation of radial deflection is decomposed to two Laplace operators. 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 FEM solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method. The R-function theory can be used to describe any more complex domains of the plates and shells.

Research Methodology
Flowchart of the research methodology has been presented in Figure 1.

Fundamental Equations
The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation [9,[11][12][13][14][15][16] can be expressed as follows: is the biharmonic operator, is the stress function, is the radial deflection of the shell, is the radius of curvature of the shell, is the elastic coefficient of the foundation, ) , ( ,  is the domain of the trapezoid of shallow spherical shells in Cartesian coordinates, is the radial load; and )) 1 is the flexural rigidity of the shell, in which ℎ is the thickness of the shell, and and are Young's modulus and Poisson's ratio, respectively. The simply supported boundary conditions can be written as:

Equations of stress function and radial deflection
is the Laplace operator, and     is the boundary of the domain  .Making use of Equations 1 and 3, we can easily obtain:

Substituting Equation 4 into Equation 2 yields:
To decompose Equation 5, let us introduce the following intermediate variable; Where the parameter varies within −1 < ≤. For example, if the value is equal to zero, then the whole domain can be presented using R-function. Let = 0 be the normalized boundary equation of the first-order on the boundary  , i.e. [4]: The quasi-Green's function can be established as follows: . Obviously, the quasi-Green's function ) , ( ξ x G satisfies the following condition: To reduce the boundary value problems Equations 7 and 8 into the integral equations, the following Green's formula of sets of function , is applied: 515 Where; 4 1 To make the kernel of the integral equations , A normalized boundary equation will be constructed to ensure the continuity of ) , ( ξ x K in the following. It can be easily testified that:

Discrete Integral Equations
In order to discrete the integral Equations 13 and 14 of the bending problem of shallow spherical shell on Winkler foundation, the integral domain , and in each subdomain the rectangular quadrature formula is applied. Finally, integral Equations 13 and 14 are discretized into the following homogeneous linear algebraic equations: In which x is singular item in Equation 10. In order to eliminate the singular term in the discrete Equation 18 of the integral equation, the integral formula of the subdomain is integrated, and the specific derivation process is as follows.
, the integral formula of the i  subdomain is integrated: The first item of Equation 19 is singular item, in order to eliminate the singularity of this item, divide the subdomain into four small regions and integrate the item by parts (as Figure 2). x     , then the first term on the right of Equation 19 can be written as: In which, Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and concave boundary conditions as follow.

Results and Discussion
Example 1: Figure 3 shows the shallow spherical shell with rectangular bottom on Winkler foundation, and its A-A section is shown in Figure 4. We set = 2 ⁄ = 100. The following reference parameters are used: the radius of curvature of the shell = 200, the elastic coefficient of the foundation = 200, thickness of shell ℎ = 2, Poisson's ratio = 0.3, Young's modulus = 2.1 × 10 6 , the radial load = 70. According to the R-function theory [4], a normalized boundary equation of the first rank can be constructed from the following equation:  . 1 = 0 and 2 = 0 denote various parts of the boundary of square shallow spherical shell on Winkler foundation, respectively. The radial deflection curves of line 2 = 100 by the QGFM with four kinds of different square network collocations (as shown in Figure 5) are shown in Figure 6, and the results are compared with those of ANSYS Finite Element Method (FEM). The radial deflection curves of line 2 = 0 for different and different by the QGFM with 121(11 × 11) square network and by the ANSYS Finite Element Method (FEM) are shown in Figures 7 and 8 for a comparison, respectively; a good agreement is observed between the two methods.

Conclusion
The radial deflection curves of line 2 = 100 by the QGFM with four kinds of different square network collocations (as shown in Figure 5) are shown in Figure 6, and the results are compared with those of ANSYS Finite Element Method (FEM), which shows the convergence of the present method. The radial deflection curves of line 2 = 40 and 1 = 0 by the QGFM using four kinds of different trapezoidal network collocations (as shown in Figure  8) are shown in Figures 10 and 11, and the results are compared with those of ANSYS Finite Element Method (FEM), In the present paper, the R-function theory is used to describe a shallow spherical shell on Winkler foundation with polygonal boundary, and it is applied to construct a quasi-Green's function. The numerical results of the QGFM demonstrate its feasibility, efficiency and rationality by comparing with the FEM solution. The R-function theory can also be applied to effectively solve various boundary value problems of the plates and shells by constructing a trial function that satisfies the complicated boundary shape and by combining with the other method of weighted residuals such as the variational method [22], the spline-approximation [23] and the Ritz method [24].