Kinematic response of single piles with arbitrary cross-sections to vertically incident P-waves

The kinematic response of piles to seismic waves differs from the free-field motion due to soil–pile interaction. The modified Vlasov model is a powerful tool for evaluating the seismic response of pile foundations. However, difficulties in determining the attenuation function hinder its application to piles with arbitrary cross-sections. Herein, an analytical solution is proposed to examine the kinematic response of single piles with arbitrary cross-sections under vertically incident P-waves. A novel displacement model is formulated by utilizing conformal mapping to characterize the motion of the soil–pile system under vertical seismic excitation. Then, the governing equations for the displacement function and the attenuation function are derived by using Hamilton’s principle. The governing equation for the displacement function is solved explicitly. Instead, the governing equation for the attenuation function is addressed by designing a one-dimensional central difference scheme. The unknowns in these functions are determined by satisfying the relevant boundary conditions and following an iterative procedure. The model is validated by comparing the pile responses with those obtained by the available results and with those by the Finite Element Method (FEM). Focusing on rectangular single piles, the study reveals that an increase in the aspect ratio enlarges the amplitude of the kinematic response factor for a low stiffness ratio, the amplitude of the axial force, and the influence radius.