Equivalent dynamic stiffnesses and 3D wave propagations of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation

Abstract

An advanced theoretical formulation along with a robust numerical evaluation method are employed to study the 3 D wave propagations and equivalent dynamic stiffnesses of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation. Using appropriate dynamic Green’s functions presented in the literature, the rocking and torsional harmonic oscillations of the foundation are formulated analytically in terms of several Fredholm integral equations of the second kind. Then the dynamic stiffnesses and flexibilities are presented analytically for rocking and torsional oscillations. To validate the analytical results, they are reduced to two simpler cases reported in the literature: the static problem and the case of isotropic materials. The semi-infinite integrals appeared in the solutions are evaluated numerically by utilizing the contour integration and residue theorem along with several commands of the Mathematica software. A complete discussion on the elastic wave propagations due to oscillations of the foundation are presented and a detailed comparison is performed between different waves propagate within the half-space and full-space elastic grounds including the P, SV, SH and the Rayleigh waves. Our results showed that due to rocking oscillation of the foundation, the Rayleigh wave does not propagate in the full-space medium but it propagates in the half-space medium. However due to torsional oscillation, the Rayleigh wave does not propagate neither in the full-space nor in the half-space elastic medium. In addition, due to rocking oscillation, the body waves including P, SV and SH propagate within both half-space and full-space elastic mediums; but due to torsional oscillation, only SH-wave propagates within both full-space and half-space elastic mediums. In some tables, numerical values and mathematical formulas for the dimensionless wave numbers and wave velocities are provided for the P, SV, SH and the Rayleigh waves. The Gaussian quadrature method is implemented for definite integrals appeared in the dynamic stiffnesses. Some graphical figures are provided to represent the dimensionless rocking and torsional dynamic stiffnesses and flexibilities versus the dimensionless frequency for different transversely isotropic materials. The effects of anisotropy of materials and the frequency of excitations on the responses are discussed. The results presented in this paper including the dynamic stiffnesses and wave propagations have direct applications in dynamic soil-structure interaction analyses in civil engineering.