Fourier series (based) multiscale method for computational analysis in science and engineering: V. Fourier series multiscale solution for elastic bending of Reissner plates on Pasternak foundations

TL;DR

This paper develops a Fourier series multiscale analytical method for solving the elastic bending problem of Reissner plates on Pasternak foundations, extending classical analysis to a comprehensive multiscale framework with convergence analysis.

Contribution

It introduces a novel Fourier series multiscale solution for Reissner plates on foundations, incorporating boundary conditions and parameter variations with proven convergence.

Findings

Derived a Fourier series multiscale solution for plate bending

Demonstrated convergence and multiscale characteristics numerically

Extended analysis to a wide range of model parameters

Abstract

Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the fifth paper, the usual structural analysis of plates on an elastic foundation is extended to a thorough multiscale analysis for a system of a fourth order linear differential equation (for transverse displacement of the plate) and a second order linear differential equation (for the stress function), where general boundary conditions and a wide spectrum of model parameters are prescribed. For this purpose, the solution function each is expressed as a linear combination of the corner function, the two boundary functions and the internal function, to ensure the series expression obtained uniformly convergent and termwise differentiable up to fourth (or second) order. Meanwhile, the sum of the corner function and the internal function corresponds to the particular solution, and the two boundary functions correspond to the general solutions which satisfy the homogeneous form of the equation. With the corner function, the two boundary functions and the internal function selected specifically as polynomials, one-dimensional half-range Fourier series along the y (or x)-direction, and two-dimensional half-range Fourier series, the Fourier series multiscale solution of the bending problem of a Reissner plate on the Pasternak foundation is derived. And then the convergence characteristics of the Fourier series multiscale solution are investigated with numerical examples, and the multiscale characteristics of the bending problem of a Reissner plate on the Pasternak foundation are demonstrated for a wide spectrum of model parameters.