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

Shanqing Li (MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Institute of Applied Mechanics, Jinan University, Guangzhou 510632,)
Hong Yuan (MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Institute of Applied Mechanics, Jinan University, Guangzhou 510632,)
Xiongfei Yang (MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Institute of Applied Mechanics, Jinan University, Guangzhou 510632,)
Huanliang Zhang (MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Institute of Applied Mechanics, Jinan University, Guangzhou 510632,)
Qifeng Peng (MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Institute of Applied Mechanics, Jinan University, Guangzhou 510632,)

Civil Engineering Journal
Vol 6, No 3 (2020): March

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.