Kinematic response of end-bearing piles in nonhomogeneous unsaturated soils subjected to seismic P-waves

This study proposes an analytical model to investigate the kinematic response of end-bearing piles in nonhomogeneous unsaturated soils subjected to vertically propagating seismic P-waves. The soil profile exhibits continuous heterogeneity characterized by power-law variations in Lamé constants, density, porosity, and bulk modulus of three-phase along the pile shaft, while the pile is modeled as a standard beam with fixed-base boundary conditions. Governing equations for the coupled soil-pile system are derived employing the three-phase poroelastic theory within a continuum mechanics framework. The mathematical formulation utilizes Laplace transforms combined with operator decomposition techniques to decouple wave propagation equations, with potential functions employed to address compressional wave interactions. Frequency-domain solutions are obtained through rigorous enforcement of displacement-stress continuity conditions at the pile-soil interface and predefined boundary constraints under P-wave excitation. Validation through degenerate case comparisons demonstrates consistency with classical solutions for homogeneous saturated and single phase soils profiles. Parametric analyses systematically explore how gradient variation patterns, soil saturation, air-entry value, pile slenderness ratios, and pile-soil modulus ratios on seismic response characteristics.