Kinematics of single end-bearing piles in partially treated soil subjected to vertically incident P-waves

An analytical model is proposed to investigate the kinematic response of a single endbearing pile in partially treated soil under vertical P-waves. Region-matching method is adopted to divide the physical domain of the soil-pile system into three parts. The total vertical displacement of the system is assumed to be the sum of two parts as eigenfield and free-vibration field. The boundary value problem is formulated as an infinite system of linear algebraic equations with unknown coefficients. The displacement of each divided region is obtained numerically by matrix form after series truncation. Parametric study shows that the geometric and material properties of the treated soil have notably effects on the kinematic response of the soil-pile system. The amplified factors and friction force along the shaft increase with the increase of the range of the treated soil for low incident frequency. The curves of friction along the shaft depth exhibit peaks near the interface between the treated and untreated soils. The concentration of the friction force above the bottom of the soil-socket interface decreases with the increase of the depth of the socket. The effect of the modulus of the treated soil is negligible on the response of the pile when the modulus ratio of the treated soil to the natural soil exceeds to 10. The friction force increases with the increase of the value of the soil modulus ratio.