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

Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the seventh paper, the usual structural analysis of beams on an elastic foundation is extended to a thorough multiscale analysis for a fourth order linear differential equation for transverse deflection of the beam, where general boundary conditions and a wide spectrum of model parameters are prescribed. For this purpose, the solution function is expressed as a linear combination of the boundary function and the internal function, to ensure the series expression obtained uniformly convergent and termwise differentiable up to fourth order. Meanwhile, the internal function corresponds to the particular solution, and the boundary function corresponds to the general solution which satisfies the homogeneous form of the governing differential equation. Since the general solution has appropriately interpreted the meaning of the differential equation, the spatial characteristics of the solution of the equation are expected to be better captured. With the boundary function and the internal function selected specifically as combination of the linearly independent homogeneous solutions of the differential equation, and one-dimensional half-range Fourier sine series over the solution interval, the Fourier series multiscale solution of the bending problem of a beam 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 beam on the Pasternak foundation are demonstrated for a wide spectrum of model parameters.