This study offers a comprehensive and advanced understanding of the torsional response of piles partially embedded in fractional-order viscoelastic unsaturated transversely isotropic soils, accurately capturing the true viscoelastic properties and particle orientation of the soil as formed during deposition. Based on Biot's threephase porous media wave equations and considering the coupling effects between the immiscible fluids (water and air) in the pores, the dynamic governing equations for fractional-order viscoelastic unsaturated transversely isotropic soil are established. The soil vibration displacement is solved using the method of separation of variables. In the frequency domain, employing the transfer matrix method and considering the continuity and boundary conditions of the pile-soil system for both the embedded and exposed portions, the analytical solution for the torsional complex impedance at the pile head of a partially embedded single pile in fractionalorder viscoelastic unsaturated transversely isotropic soil is derived. Furthermore, a semi-analytical solution for the pile head response in the time domain under half-sine pulse excitation is obtained through inverse Fourier transform and convolution theorem. Numerical examples are presented to investigate the effects of the parameters of the fractional-order viscoelastic constitutive model and the pile-soil parameters on the torsional complex impedance at the pile head.

Large-diameter pipe piles are widely applied in various civil engineering fields due to their outstanding load- bearing capability. The unsaturated characteristics, anisotropy, and heterogeneity of the soil jointly affect the dynamic response of the pipe pile. However, most previous studies were limited to single-phase or two-phase soil. This paper develops an analytical model for the torsional vibration of a pipe pile in transversely isotropic unsaturated soils considering construction disturbance. Based on transversely isotropic unsaturated soil theory, a pipe pile-soil interaction model has been developed, while the effect of construction disturbance is simulated by the radial heterogeneity of the soil. The general solution for the unsaturated soil is obtained using the separation of variables method with the boundary conditions. Then, the solution for the whole pipe pile-soil system is derived by considering the pile-soil interface conditions. The accuracy of the proposed solution is verified through comparisons with previous research results. The results show that unsaturated characteristics, construction disturbance, and transverse isotropy of the soil have significant effects on the impedance of large- diameter pipe piles. Specifically, with a low degree of saturation, there will be significant prediction errors when using previous works based on single-phase or two-phase soil theory to predict the dynamic response of large-diameter pipe piles.

Considering the soil skeleton to be statically transversely isotropic, the viscous and inertial coupling among three phases, along with the capillary pressure, the torsional dynamic response of an end-bearing pile in homogeneous unsaturated transversely isotropic soil under time-harmonic torsional load is investigated. The separation of variables method is employed to derive the torsional dynamic governing equations of unsaturated transversely isotropic soil in a cylindrical coordinate system, and pile is modeled as a 1D elastic theory. By combining the boundary conditions of the pile and soil, as well as the continuity conditions at the pile-soil interface, further employing the inverse Fourier transform and convolution theorem, fundamental analytical solutions for the torsional behavior of the pile are derived in both the frequency domain and time domain. Comparisons between the proposed solution and the two existing solutions indicate that the presented model exhibits stronger applicability, as it can simultaneously transition to both saturated transversely isotropic soil and unsaturated isotropic soil. Eventually, a parametric study is conducted to examine the influence of soil anisotropy, degree of saturation, pile-soil shear modulus ratio and pile length on the torsional dynamic response of end-bearing pile.