Based on the continuous medium model and the pile-soil interaction

the horizontal seismic response of a single pile subjected to vertically propagating S waves was investigated by regarding the single pile as a one-dimensional linearly elastic beam. The time-harmonic displacement of bedrock was introduced as the vertically propagating S waves

and the horizontal dynamic impedance function of the soil was derived by the governing equations of the plane strain model. Analytical solutions for the seismic response of the single pile subjected to vertically propagating S waves were obtained by subsuming soil impedance into the governing equation of the single pile and considering the boundary conditions at pile top and toe. The solution was verified by comparing it to the results of existing studies. Furthermore

as pile-soil modulus ratio increases

the minimum value of the kinematic response factor decreases. The kinematic response factor is not particularly sensitive to the large pile slenderness ratio and the soil material damping. For the horizontal amplification factor at the pile top

the increase of the pile-soil modulus ratio only suppresses the amplification at high resonance frequency

and the large pile slenderness ratio has the trivial effect on it. As the soil material damping increases

the amplification at resonance frequency gets considerably suppressed. The seismic response of the pile is obviously affected by the pile-soil modulus ratio only when the pile slenderness ratio is small

and it decreases with the increase of the pile-soil modulus ratio.