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

Abstract

<jats:p> 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. </jats:p>

On an Inverse Problem of Identifying an Unknown Boundary for the Biharmonic Equation from Cauchy Data

“…Which correspond to the homogeneous Dirichlet boundary condition, homogeneous Navier boundary condition also called (simply supported constraint boundary condition), and the homogeneous mixed boundary condition, respectively (see [9,13,14,10,12]). This inverse problem is usually encountered in elastic plates, for example, finding cracks in a medium from measurements of an elastic field on a surface of the medium [3,6,14].…”

“…In mechanics, physics, and many engineering applications, the biharmonic equation is used as a governing equation to describe: the deformation of thin plates, the motion of fluids, free boundary problems, non-linear elasticity, and problems related to blending surfaces. Several implementations of the 2D biharmonic problem have been consecrated in simply and multiply-connected regions (see, e.g., [3,5,1,8,9,21,13,19,22,4,10,12]). Boundary conditions are essential and extremely important constraints for solving a boundary value problem.…”

“…Boundary conditions are essential and extremely important constraints for solving a boundary value problem. In order to study the biharmonic problem, various boundary conditions are adopted, e.g., the Dirichlet problem [17,19], the Neumann problem [9], the mixed problem [6,8], the Navier boundary conditions [13], the Riquier-Neumann boundary conditions, the Robin boundary value problems [4] and the supported boundary condition, see [14,10,12]. These represent an important class of inverse problems known to be generally ill-posed, in which the existence, uniqueness, and stability of their solutions are not always guaranteed (see, e.g., [5,1,11,13,16,4]).…”

Convergence studies for static analysis of thin plates on Pasternak Foundations

“…A semi-analytical solution for the static analysis of thin skew plates on Winkler and Pasternak foundations is presented by [10]. In a recent study, a computing method for bending analysis of thin plates resting on Pasternak foundation is developed, [11]. Vibration analyses of isotropic rectangular plates resting on Pasternak foundation are also performed in many studies, [12][13][14].…”