A numerical method for bending problem of slip clamped shallow spherical shell

Abstract

Abstract. In this paper, the R-function theory(RFT) is applied to solve the f bending problem of slip clamped shallow spherical shell. Firstly the fundamental solution of the biharmonic operator, the boundary equation and the R-function are used to construct the quasi-Green's function. Then the model governing differential equation of the problem is reduced to the Fredholm integral equation of the second kind by Green's formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation by the R-function. A numerical example shows that this method is an effective numerical method. IntroductionAs a kind of structural forms, a shell is 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.In the analysis and calculation of various physical and mechanical problems in engineering, the governing differential equation describing their physical state and process needs firstly to be built. Only few problems with a regular geometric boundary and a simple differential equation can be solved with an analytical or a half analytical method. For most complex engineering problems, it is difficult to find an analytical solution so that an approximate method is used to analyze and calculate the problems. In many calculation problems of engineering, although geometry of arbitrary shapes, complex boundary conditions, various properties and inhomogeneous of materials, and so on, but a numerical solution can be obtained directly by using a numerical method from a mathematical model. The main numerical methods are the boundary element method, the finite element method, the finite difference method and the coupling method.In the paper, the R-function theory proposed by Rvachev[1] are utilized. The bending problem of slip clamped shallow spherical shell is studied. 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, but it does not satisfy the fundamental differential equation. The key point of establishing the quasi-Green function consists in describing the boundary of the problem by normalized equation 0 = ω and the domain of the problem by inequality 0 > ω . There are multiple choices for the normalized boundary equation. Based on a suitably chosen form of the 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 area, a normalized boundary equation can always be found according to the R-function theory. Thus, the problem can always be reduced to the Fredholm integral equation of the second kind without singularity. Using the present method, Li and Yuan solved successfully the free vibration o...