## 2 Modeling Methodology
### Boundary Conditions  
Three-dimensional numerical modeling is essential for appropriate and accurate simulation of soil-pile-structure dynamic interaction; however, 3D full-scale analysis is often time consuming, particularly when applied to dynamic problems. If an infinite number of soil elements are created to simulate the semi-infinite boundary of the actual site, the numerical analysis of various conditions becomes difficult. Further, the analysis efficiency decreases significantly because the analysis time increases sharply. However, if boundary conditions are not properly set in dynamic numerical modeling, the input wave propagating from the center of the model can generate a reflection wave at the boundary, complicating the simulation of the actual behavior. Therefore, in this study, the simplified continuum modeling method shown in Figure 4 was applied to reduce the analysis time and to properly the simulate dynamic behavior at the model boundary. Figure 4 shows the schematic of the simplified continuum modeling method adopted in the present study. The entire region is divided into two parts: near-field and far-field zones. The soil motion in the near-field zone is influenced by the soil-pile interaction and modeled in finite zones. The soil motion in the far-field zone not influenced by the soil-pile interaction is replaced as the boundary motion of the near-field zone. This modeling method can reduce analysis time compared to full modeling approach of two zones. [PERSON] et al. (1992) [41], [PERSON] et al. (1998) [42], [PERSON] et al. (1999) [43], [PERSON] et al. (2013) [44], [PERSON] et al. (2014) [45], and [PERSON] et al. (2015) [46] applied similar approaches. However, their approaches replaced the far-field zone with nonlinear spring elements that simulate the interaction stress between the pile and nearby soil. In addition, these approaches were limited to 1D or 2D modeling. [PERSON] (2011) [47], [PERSON] et al. (2012) [30], and [PERSON] et al. (2013) [48] applied the simplified method for 3D modeling to model a boundary condition of the soil-pile system in dry sand and validate applicability of the boundary model which input proper level of acceleration-time histories at the horizontal boundary of the model. In the present study, the method has been extended to simulate the dynamic behavior of a pile in liquefiable sand. The soil motion in the far-field zone is obtained from the independent site response analysis, which can simulate the liquefaction behavior of sandy soil. The horizontal soil acceleration-time histories along the depth in the far-field zone are applied at the boundary nodes of the near-field zone.  
In the simplified continuum modeling method, it is necessary to determine the boundary extent of the near-field zone where the soil motion becomes identical to the far-field movement. In order to determine the optimal boundary extent of the near-field, the acceleration amplification ratio, which is the ratio of the amplitude of the maximum acceleration at the surface to the base input, was analyzed. The outline procedure of simplified continuum modeling is as follows. (1) Target system is modeled and analyzed by exerting earthquake load. (2) Obtain extent of near-field elements by plotting amplification ratio according to the distance from the pile center. (3) Only near-field is modeled and acceleration-time histories in far-field for each depth are input as boundary condition. Figure 5 shows the amplification ratio according to the distance from the center of the pile. The dynamic excitation of the pile increases the surface acceleration. Therefore, as the distance from the pile increases, the ratio decreases. The optimal distance with a constant ratio was determined to be 8 D (where D is the pile diameter). Finally, the boundary extent of the near-field was applied as 10 D from the center of the pile for the conservative modeling.  
Figure 4: Schematic of the simplified continuum modeling method.  
Figure 5: Amplification ratio with the distance from the pile center.