# Numerical Algorithm for Dynamic Impedance of Bridge Pile-Group Foundation and Its Validation
## 1 Introduction  
Pile-group foundation has been widely used in bridge engineering, and pile-soil interaction has a significant effect on the dynamic performance of the superstructure [1; 2; 3; 4]. [PERSON] et al. [5] investigated the influence of soil-structure interaction (SSI) on the seismic response of structures by using recorded motions, and concluded that some simplifications about SSI in design may lead to detrimental consequences. [PERSON] et al. [6] studied the effects of SSI and spatially varied ground motions on the seismic response of a cable-stayed bridge, and the results indicated that softer soils amplify the displacement response of the bridge. [PERSON] et al. [7] developed an analytical model to obtain free vibration frequencies of bridges with soil-structure interaction taken into consideration, and found that the effect of SSI is significant for soils with a shear wave velocity less than 175 m/s. [PERSON] et al. [8] discussed the contribution of SSI to the bridge dynamics, and found that SSI may significantly affect the structural dynamics, especially in the case of soft or medium soil deposits. [PERSON] et al. [9] established an analytical model of the train-bridge system considering SSI based on the substructure method, and found that it is important to consider SSI in the dynamic analysis of train-bridge systems, but such influence becomes weaker with the increase in the soil shear wave velocity. [PERSON] et al. [10] investigated the effects of soil-structure interaction and near-field seismic ground motions on cable-stayed bridges based on experiments and numerical analysis; their results indicated that the effect of foundation soil stiffness on the response of the bridge was influenced by the type of input ground motion and that the bridge response was amplified when their foundationwas in softer soils. All of the above studies indicate the importance of considering SSI in the dynamic analysis of bridges. There are two main approaches for soil-structure analysis during the dynamic analysis of bridges--namely, the direct approach, and the substructure approach [11]. Compared with the direct approach--which studies the structure, foundation, and soil simultaneously (usually by means of a finite element model)--the substructure approach, which divides the whole system into two subsystems (superstructure subsystem and soil-foundation subsystem), is less time consuming and more efficient, for the two subsystems can be treated separately with their respective optimal method. Based on the substructure method, the coupling of two interacting subsystems is realized through the concept of the soil-foundation dynamic impedance functions. Foundation impedance functions represent the dynamic stiffness of the soil medium surrounding the foundation, which characterize the compliant base restraints for the superstructure and the input motion experienced by the foundation, thus providing a simple means of accounting for pile-soil interaction [12]. Therefore, the study of the solution of dynamic impedance functions of pile-group foundation has drawn more and more attention. The generally accepted standard solutions for the dynamic response of pile groups in viscoelastic media were obtained by [PERSON] and [PERSON], by means of the boundary integral method [13]. Afterwards, [PERSON] et al. [14] proposed a method to calculate the impedance functions of pile-group foundations using a simplified beam-on-dynamic-Winkler-foundation model and [PERSON]'s function-based rigorous method. Based on the thin layer method and the flexible volume method, [PERSON] et al. [15] calculated the impedance functions of pile-group foundations from literature as well as in a practical layered site, and compared the results with those in the literature [16] and from experimental data. In recent years, more and more methods have been proposed for the impedance functions of pile-group foundations under various conditions [17; 18; 19; 20; 21; 22; 23]. For example, [PERSON] et al.