A computing method for bending problem of thin plate on Pasternak foundation

Jiarong Gan,Huanliang Zhang,Hong Yuan,Qifeng Peng,Shanqing Li

doi:10.1177/1687814020939333

Abstract

The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

Highlights

In the bending problem of Pasternak foundation plate, its governing differential equation is a higher order equation and contains biharmonic operator.[1]
Vibration analysis of a thin functionally graded plate having an out-of-plane material inhomogeneity resting on Winkler–Pasternak foundation under different combinations of boundary conditions was researched by Piyush et al.[9]
A wavelet collocation method was proposed by Chen et al.[26] for solving the linear boundary integral equations reformulated from the modified Helmholtz equation with Robin boundary conditions

Summary

Introduction

In the bending problem of Pasternak foundation plate, its governing differential equation is a higher order equation and contains biharmonic operator.[1] it is difficult to study the plate with complex boundary conditions in depth. R-function theory and Green’s function method can always get the corresponding boundary normalization equation for the thin plate with complex boundary conditions and transform the high-order control differential equation into the low-order equations which are a Helmholtz equation[2] and a Laplace equation[3] and get the effective corresponding solution. R-function theory and quasiGreen function methods of Helmholtz equation as shown in Appendix 1 and Laplace equation are used to study the bending of thin plates on Pasternak foundation.

Method

Conclusion