Vertical seismic response of pipe piles in saturated frozen soils under Rayleigh-wave and concentrated-load excitation

The presence of the ice phase in saturated frozen soil makes the frequency dependence of dynamic impedance fundamentally different from that of saturated soil, limiting the ability of existing pile-soil dynamic models to predict seismic response in cold regions. This study aims to establish an analytical model for the vertical seismic response of pipe pile-frozen soil systems by considering the three-phase characteristics of frozen soil, and to quantify the effects of frozen-soil parameters and pile parameters under coupled Rayleigh-wave excitation and concentrated loading. The Rayleigh-wave field is derived using Helmholtz decomposition in the frozen soil layer. By coupling the resulting vertical dynamic impedance of the frozen soil with a one-dimensional longitudinal vibration equation for the pipe pile, a closed-form frequency-domain solution is obtained for the pile-soil system. The analytical solution is verified through model degradation to classical saturated soil cases and comparison with published results. The results show that Rayleigh-wave velocity in saturated frozen soil decreases markedly in the intermediate-frequency range and is strongly affected by temperature and porosity. Pile-head displacement increases with pile length and load, whereas the displacement along the pile depth attenuates through shaft resistance, frozen-soil dynamic impedance and wave attenuation. When the contact-reduction parameter increases, the pile-head displacement increases by approximately 152.1%, while the displacement at z = 30 m decreases by approximately 91.9%. The influence of the radius ratio is modest under low to moderate loads, but under high loads, a smaller inner-to-outer radius ratio significantly reduces vertical displacement while increasing the demand for interface shear transfer. These findings provide a theoretical basis for evaluating vertical seismic response and optimizing pipe-pile geometry in saturated frozen-soil foundations.