Considering the viscous damping of the soil and soil-pile vertical coupled vibration

the differential equations and boundary conditions about vertical vibration of cast-in-situ concrete thin-wall pipe piles in axisymmetric homogeneous viscoelastic soil are established. The analytical solution is obtained by Laplace transformation method. A case study is given; and the results of analytical solution can well match FEM and measured waves. Moreover

by analyzing the effects of viscous damping coefficient on velocity admittance and complex dynamic stiffness

and the effect of damping coefficient of the soil below pile bottom on velocity admittance

the regularity of variations of vertical vibration characteristic with parameters is obtained.