The thin-layer method in a cross-anisotropic 3-D space

Abstract

This article presents a generalization of the thin-layer method to three dimensions, a tool that allows assessing layered media subjected to loads eliciting nonsymmetrical wave fields. It is based on a formulation that fully couples the three components of motion, and allows finding effective solutions to either stationary or moving loads of arbitrary shape that act on (or within) horizontally layered media. In particular, it is an ideal tool for finding analytical solutions to the so-called 2.5D problem, which entails loads with arbitrary distribution in one horizontal coordinate direction together with a harmonic (sinusoidal) variation in the other. Inasmuch as the Green's functions for the latter case are found explicitly in the spatial domain without recourse to numerical integration, it allows using such functions — most likely in the context of the boundary element method — as fundamental solutions for problems of soil–structure interaction where the structure is invariant in one horizontal direction, such as a railroad track resting on an embankment. The method entails solving at each frequency of interest two uncoupled eigenvalue problems for generalized SH and SVP waves (i.e. horizontally and vertically polarized shear and pressure waves), after which the fundamental solutions are obtained in closed-form at any desired point in space. Inasmuch as the proposed technique dispenses with at least one of the two inverse Fourier transforms into the spatial domain, in due time the methodology presented is likely to become the preferred tool for a wide class of problems. The technique is first benchmarked against the known solution for a point load and then applied to a rectangular and triangular load distribution. Copyright © 2011 John Wiley & Sons, Ltd.