Kinematic response of double-lined shafts embedded in homogeneous unsaturated soil subjected to P-wave excitation

This paper presents a novel mathematical model to investigate the kinematic response of a double-lined (DL) shaft embedded in homogeneous unsaturated soil subjected to vertically incident P-wave excitation, developed within the framework of elasto-dynamic continuum theory. The primary and secondary linings of the DL shaft are modeled as two parallel, linear elastic, undamped rods. The surrounding soil is treated as a three-phase (solid, liquid, and gas) porous viscoelastic medium, accounting for the compressibility of each constituent, the inertial and viscous coupling effects between the liquid and solid phases, and the saturation-dependent dynamic shear modulus. Closed-form series solutions for the kinematic response of the DL shaft are obtained based on the boundary conditions of the coupled DL shaft-unsaturated soil system. The accuracy of the proposed model is validated through comparison with existing solutions. The effects of key problem parameters, including soil saturation, depth-to-radius ratio, secondary lining thickness, and DL shaft-unsaturated soil modulus ratio, on the kinematic response factor and kinematic amplification factor are systematically evaluated. Additionally, equivalent static and kinematic Winkler moduli are theoretically derived for potential applications in further modeling. This study establishes a comprehensive analytical framework that advances the understanding of soil-DL shaft interaction in unsaturated ground and offers theoretical guidance for the seismic design of underground DL shafts.