This study develops a comprehensive analytical solution for predicting three-dimensional ground motions induced by the vibration of large-diameter end-bearing piles subjected to vertically distributed uniform loads. Both the pile and the surrounding soil are treated as elastic continuum media to capture the coupled effects of P-SV and Rayleigh waves accurately. General solutions for wave potentials, displacements, and stresses are derived using small-strain theory and continuum elasticity. Modal wave numbers are determined through a root-searching approach employing the argument principle and subdivision method. Bi-orthogonality relationships are reorganized using Betti's theorem. Soil–pile interactions are rigorously modeled through continuity conditions at the soil–pile interface. Mode-matching method is used to solve the unknown coefficients. The boundary-value problem is reduced to a system of linear algebraic equations with series truncation to ensure convergence and computational efficiency. Parametric studies reveal that excitation frequencies significantly influence the distribution of soil and pile responses. Shear waves corresponding to pile frictions dominate near-field responses and Rayleigh waves resulting from surface load propagate at larger distances. Transient displacement responses show the significant influence of complex Rayleigh wave propagation and the secondary subsurface scattering on the ground motions. The particle motions reveal the Rayleigh waves are generating and propagating through the ground surface induced by pile vibrating. This study contributes to the accurate prediction of ground motions and supports the design of vibrating grounds which ensures the safety of pile-supported structures in urban and displacement-sensitive environments.

An analytical solution for the scattering of harmonic P1 and SV waves in a poroelastic half-plane with a shallow lined tunnel is obtained using the plane complex theory in elastodynamics. In light of the wave function expansions, the wave fields of the poroelastic medium and the liner with unknown coefficients are obtained based on Biot's theory and Helmholtz decomposition. Complex-valued expressions of the effective stresses, the fluid stress, and the displacements of the poroelastic medium and the liner are expressed by the complex variable function method and the conformal transformation technique. With the boundary conditions and the continuity of the medium-liner interface, the boundary value problem results in a series of algebraic equations. The unknown coefficients in the infinite set of algebraic equations can be solved numerically by truncating the series number. A parametric study for the incident SV waves is performed to investigate dynamic stress concentrations and fluid stress of the medium and the liner. Numerical results show that the embedment depth of the tunnel, the incident angle of the excitations, and the porosity of the medium have considerable influence on the dynamic responses of the medium and the liner. The shielding effect of the tunnel on the incident SV waves is obvious. For the big embedment depth of the tunnel, the scattered waves contribute little to the displacements and dynamic stress concentration of the medium and the liner. For a high porosity close to the critical value, the response of the medium-liner system to the incident waves is great.

A wave function expansion method for out-of-plane scattering of SH waves by two symmetrical circular cavities in two bonded exponentially graded half spaces is presented by using the plane elastic complex variable theory. The medium is composed of a semi-infinite homogeneous space with a circular cavity and a bonded exponentially graded half-space with a symmetrical circular cavity along the boundary interface. In terms of Helmholtz decomposition to the wave propagation equation, the stresses and displacements of the medium are expressed by the displacement potentials. The scattered waves by the boundary interface are assumed to transmit from the images of the origins of the respective two cavities. A conformal mapping function is introduced to convert the physical plane of two bonded semi-infinite spaces into two jointed concentric annular regions. The boundary value problem is formulated as a series of infinite algebraic equations. The convergence of the present solution is examined by investigating the variations of the solution results with the truncation of the series number. Parametric study shows that the interface displacements are almost independent to the inhomogeneous coefficient βR of the right medium. The amplification of the interface displacement becomes great for high incident frequency and with the decreasing of the distance-to-radius ratio h/R. The distribution of stress concentration of the right cavity depends significantly on the inhomogeneous coefficient βR, the distance-radius ratio h/R, the incident angle and frequency of the excitations.