Generalized stiffness matrix method for layered soils

Abstract

This article provides an effective extension to the well-known Stiffness Matrix Method (SMM) for layered soils. The modified algorithm allows considering elastic media subjected to both 2.5-D and 3-D sources that are neither plane nor axisymmetric. These sources can act on –or within– viscoelastic, horizontally layered media of either finite or infinite depth. Although a direct approach in the context of a fully 3-D formulation in Cartesian coordinates in combination with a 2-D spatial Fourier transform is possible and could be used for the solution of the problem at hand, it will be shown that a substantial saving in computational effort can be achieved by formulating the evaluation of the Green's functions with the same cost as that expended for plane-strain or for axisymmetric sources, provided the material properties are either isotropic or transversely isotropic. We demonstrate herein that the flexibility functions in the frequency-wavenumber domain for the general 3-D problem can be inferred from the exact rigidity matrices for the 2-D (i.e. plane-strain) problems of SVP and SH waves, and only these need to be inverted via Gaussian reduction to obtain the requisite flexibility functions in the frequency-wavenumber domain. This strategy relates to what in the technical literature is sometimes referred to as the “inversion of the descent of dimensions”, a principle that states that the response for point loads can be inferred from the response to line loads. It allows obtaining highly effective numerical solutions to the problem at hand.