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

Shanqing Li,Qifeng Peng,Huanliang Zhang,Xiongfei Yang,Hong Yuan

doi:10.28991/cej-2020-03091487

Abstract

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.

Highlights

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
The governing differential equations of the bending problem of supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection
According to the R-function theory [4], a normalized boundary equation of the first rank can be constructed from the following equation:

Summary

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. 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. The R-function theory and the quasi-Green’s function method (QGFM) proposed by Rvachev [4] are utilized. A quasi-Green’s function is established by using the fundamental solution and the boundary equation of the problem. The governing differential equations of the bending problem of supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The R-function theory can be used to describe any more complex domains of the plates and shells

Fundamental Equations

Integral Equations

Discrete Integral Equations

Results and Discussion

Conclusion