The Dynamical Ground Compliance of a Rectangular Foundation on a Viscoelastic Stratum

Abstract

As a basic study to find the effect of ground-structure interaction on the earthquake responses and the anti-seismic safety of structural systems, it is important to describe the dynamic properties of the sub-soil ground under the foundations of such structures. Based on the wave propagation theory, this paper deals with the "Dynamical Ground Compliance (D.G.C.)" of a model of foundation-ground systems more realistic than those used in any previous studies on foundation vibrations, namely, a rectangular foundation resting on a viscoelastic stratum over a rigid half-space for the cases where the vertical, horizontal, or rotational harmonic excitations act on the foundation. Defined in this paper as the transfer function describing the ratio between the dynamic complex displacement of a massless foundation and a harmonic exciting force, D.G.C. is a collective representation of the dynamic characteristics of the sub-soil ground under a foundation. Its limiting expression in zero frequency results in the "Statical Ground Compliance", which means the inverse of the statical ground stiffness. It is assumed that the stratum is composed of a three-dimensional, homogeneous, isotropic, Voigt solid, and lies in welded contact with the supporting rigid half-space. The analytical expressions of D.G.C. as well as the displacements at the ground surface are obtained through the multiple Fourier transformation technique in the double integral representation including an improper infinite integral. The poles of the integrand are the roots of the frequency equations connected with Rayleigh and Love waves. The detailed discussions on the properties of these free waves are also developed in relation to the attenuation and the resonance phenomena of the viscoelastic stratum. Some numerical calculations of D.G.C. as well as the equivalent coefficients of a Voigt model evaluated according to D.G.C. are given to clarify the effects of various parameters related to the frequency of excitation, the constants of the viscoelastic stratum, and the shape and size of the foundation. Compared with the results of similar problems obtained in the authors' preceding works for a perfectly elastic half-space and stratum, the present study has led to several principal conclusions on: the effects of energy attenuation owing to mechanisms of both wave radiation and internal dissipation, the comparison between the cases of a stratum and a half-space especially as to the resonance and energy attenuation phenomena, and the condition under which a stratum can be practically treated as a half-space.