Analytical solution for torsional vibration of an end-bearing pile in nonhomogeneous unsaturated soil

Incorporation of nonhomogeneous and the three-phased nature of soils (composed of particles, water and gas) can more rationally reflects the dynamic response of pile. Therefore, this work proposes an analytical solution for investigating the torsional vibration response of an end -bearing pile in nonhomogeneous unsaturated soil. Among them, the nonlinear variation of shear modulus of the soil with depth is considered, as well as the inertial coupling effect between pore water and gas components. Utilizing the method of separation of variables, boundary and continuous conditions of the pile-soil system, the analytical solution is derived in the frequency domain. Correspondingly, the quasi-analytical solution for pile head subjected to a semi-sine excitation torque is derived by employing of the inverse Fourier transform and convolution theorem in the time domain. After verifying the derived solution against with the two published analytical solutions for pile in two-phase medium, arithmetical examples are presented to demonstrate the influence of relevant parameters on the torsional vibration of the pile.