Dynamic interactions, impedance functions and elastic wave propagation due to horizontal and vertical vibrations of a rigid foundation in a transversely isotropic elastic ground

Abstract

The analytical and numerical treatments of horizontal and vertical forced time-harmonic vibrations of a rigid circular foundation in a transversely isotropic elastic half-space ground are presented. A complete discussion on the boundary conditions and elastic wave propagation due to horizontal and vertical vibrations of the foundation on the half-space is presented and a comparison is made with the full-space problem. With the aid of appropriate dynamic Green’s functions reported in the literature, the horizontal and vertical vibrations of the foundation are treated analytically, and the results are expressed in terms of the solutions of several Fredholm integral equations. For these two vibration modes, the relations for the contact normal and shear stresses, the resultant forces acting on the foundation, and the dimensionless impedance and compliance functions are given. To validate the analytical formulations, the results obtained in this paper are reduced to static responses for transversely isotropic half-space, and to dynamic responses for isotropic half-space; and compared with the results reported in the literature. An efficient numerical procedure is presented to evaluate the semi-infinite integrals appeared in the analytical solutions utilizing some features of the Mathematica software. The Fredholm integral equations and impedance and compliance functions are solved and evaluated numerically using the Gaussian quadrature method. Some graphical results are provided to present the impedance and compliance functions for different transversely isotropic materials. The effects of material anisotropy, as well as the frequency of excitation on the responses are discussed. The differences between vibrations of rigid foundations in the half-space and full-space mediums are emphasized in view of the Rayleigh wave propagation. Moreover, the difficulties related to numerical procedures needed for handling the strong singularities due to propagation of Rayleigh waves in the half-space media are presented.