Analytical Solution for the Bending of a Plate on a Functionally Graded Layer of Complex Structure

Abstract

The problem of interaction between an asymmetrically loaded thin circular plate and a supporting elastic foundation is reduced to the solution of system of the dual integral equations for the unknown normal contact pressure and differential equation of plate bending. The supporting medium is an isotropic elastic functionally graded layer of complex structure of constant thickness lying, with or without friction, on a homogeneous elastic half-space. In designing raft foundations, prediction of differential settlements and of bending moments induced in the foundation is of great importance. It is common practice to assume that the raft foundation behaves like a thin elastic plate. Development of analytical methods of the solution of such problems for a foundation with elastic properties varying in-depth is based on the numerical construction of transforms for the kernels of integral equations of the corresponding contact problems [1]. For the construction of the approximate analytical solution of the corresponding integral equations the bilateral asymptotic method is used [2]. This method is found to be effective for rigid, as well as for flexible plates, in contrast to the method of orthogonal polynomials, collocation method, or asymptotic methods (of small or large parameter). Representation of plate deflections in the form of series by their own oscillations [3] is used under corresponding boundary conditions. It is possible to reduce the specified problems to the solution of the systems of linear algebraic equations. We provide the analysis of influence of various laws of variation of the elastic properties of the coating on the distribution of contact pressures under the circular plate; its deflection; surface settlement of the functionally-graded layer outside of plate; and value of the radial and tangential moments in the thin circular plate. The developed method allows one to construct analytical solutions with prescribed accuracy and gives the opportunity to conduct multiparametric and qualitative analysis of the problem with minimal computation time.