## 2 Modeling Methodology  
In this study, a partial segment of a bridge structure supported by a 2 \(\times\) 2 group pile foundation was adopted as a prototype to evaluate the seismic response of superstructures supported by pile foundations. The model system was designed to represent actual bridge and subsoil conditions in Cheonan, Chungcheongnam-do, South Korea. To systematically analyze various seismic response conditions, soil conditions and input parameters of the superstructure were parametrically varied. The general specifications of the prototype structure are depicted in Figure 1. The foundation consists of a 2 \(\times\) 2 configuration of steel pipe piles, with a pile cap measuring approximately 4 m in width, 3 m in length, and 1.6 m in height. The piles are steel pipe piles with a diameter(\(\Phi\)) of 502 mm, thickness(t) of 12 mm. A three-dimensional continuum modeling method based on the finite difference method (FDM) was employed to assess the dynamic response of the bridge foundation under seismic loading. FLAC3D 7.0 [16], a widely used commercial software in geotechnical and foundation engineering, was utilized for this analysis. The soil model was implemented using the Mohr-Coulomb elasto-plastic model, and the input soil properties of the subsurface layers were derived from the site-specific geotechnical investigation report of the actual target location. The stratigraphy primarily consists of silty sand overlying  
Figure 1: Schematic drawing of prototype pile–structure system. weathered rock, and the material properties of each soil layer are detailed in Table 1 and Figure 2. In this study, the input material properties for silty sand with two different relative densities were determined according to the analysis cases described below. The estimation of material properties considered key geotechnical characteristics, including shear strength and deformation modulus, as functions of relative density. The material parameters were derived based on empirical formulas presented in previous studies and relevant literature. The total depth of the soil model was set to 11 m, consistent with the pile foundation length, to simulate the embedment condition in the weathered rock. The pile foundation and pile cap structures were modeled using an elastic constitutive model, and the key material properties for each structural component are presented in Table 2. The pile foundation was simulated as steel pipe piles (\(\Phi\)502, 12t). However, for computational convenience, they were modeled as solid circular piles. To maintain equivalent flexural rigidity to the actual steel pipe piles, the diameter was adjusted to \(\Phi\)329. The pile length was set to 11 m according to the design drawings, ensuring an embedment depth of 2 m into the weathered rock layer. The pile cap was modeled as a reinforced concrete structure with dimensions of approximately 4 m in width, 3 m in length, and 1.6 m in height. Figure 2 shows the layout of the structural model. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Pile Foundation** & **Pile Cap** \\ \hline Unit weight (kg/m\({}^{3}\)) & 7850 & 2400 \\ Shear modulus (MPa) & 45,230 & 38,460 \\ Bulk modulus (MPa) & 98,000 & 83,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input properties of the model structure. Figure 2: Layout of the structure model. Structures that are either in contact with or embedded within the soil exhibit different dynamic behaviors from the surrounding ground due to material heterogeneity under seismic excitation. When significant stiffness contrasts exist between the structure and the surrounding soil, and when substantial external loads are applied, dynamic soil-structure interaction (SSI) becomes pronounced, leading to response deviations from those predicted under fixed-base assumptions. Therefore, an interface model was incorporated to explicitly account for soil-structure interaction effects, including potential slippage and separation between the structure and the surrounding soil under seismic loading.