In this paper,a model named fictitious soil pile was introduced to solve the boundary coupled problem at the pile tip.In the model,the soil column between pile tip and bedrock was treated as a fictitious pile,which has the same properties as the local soil.The tip of the fictitious soil pile was assumed to rest on a rigid rock and no tip movement was allowed.In combination with the plane strain theory,the analytical solutions of vertical vibration response of piles in a frequency domain and the corresponding semi-analytical solutions in a time domain were obtained using the Laplace transforms and inverse Fourier transforms.A parametric study of pile response at the pile tip and head showed that the thickness and layering of the stratum between pile tip and bedrock have a significant influence on the complex impedances.Finally,two applications of the analytical model were presented.One is to identify the defects of the pile shaft,in which the proposed model was proved to be accurate to identify the location as well as the length of pile defects.Another application of the model is to identify the sediment thickness under the pile tip.The results showed that the sediment can lead to the decrease of the pile stiffness and increase of the damping,especially when the pile is under a low frequency load. In this paper, a model named fictitious soil pile was introduced to solve the boundary coupled problem at the pile tip. The model, the soil column between pile tip and bedrock was treated as a fictitious pile, which has the same properties as the local soil. The tip of the fictitious soil pile was assumed to rest on a rigid rock and no tip movement was allowed. In combination with the plane strain theory, the analytical solutions of vertical vibration response of piles in a frequency domain and the corresponding semi- analytical solutions in a time domain were obtained using the Laplace transforms and inverse Fourier transforms. A parametric study of pile response at the pile tip and head showed that the thickness and layering of the stratum between pile tip and bedrock have a significant influence on the complex impedances.Finally, two applications of the analytical model were presented. One is to identify the defects of the pile shaft, in which the proposed model was proved to be accurate to identify the location as well as the length of pile defects. Another application of the model is to identify the sediment thickness under the pile tip. The results showed that the sediment can lead to the decrease of the pile stiffness and increase of the damping, especially when the pile is under a low frequency load.