## 1 Introduction  
With the increasing frequency and intensity of earthquakes, interest in seismic design has grown significantly. In particular, the seismic behavior of pile-supported structures involves complex soil-structure interactions, making accurate predictions crucial. Traditional seismic design approaches often rely on simplified two-dimensional (2D) analyses, which fail to fully capture the intricate behavior of actual soil-structure systems. Consequently, the necessity of three-dimensional (3D) analysis methods has become more evident. Examining real earthquake damage cases, the 1995 Hanshin-Awaji earthquake (magnitude 6.9) in Japan resulted in severe structural failures due to foundation damage, even for pile-supported structures. Similarly, the 2011 Great East Japan earthquake (magnitude 9.0) led to widespread liquefaction, causing excessive settlement and tilting of many pile-supported structures. More recently, the 2023 Turkriye-Syria earthquake highlighted significant structural collapses due to inadequate seismic performance. These cases underscore the need for a more comprehensive analysis of pile-supported structures under seismic loads and the refinement of design criteria. The seismic behavior of pile foundations is influenced by various factors, among which the relative density of the soil and the mass of the superstructure are critical parameters. The relative density of the soil determines the magnitude and direction of kinematic forces transmitted through the piles. During seismic events, ground deformation generates soil kinematic forces, which are then transferred to the pile foundation and ultimately to the structure. In dense soils, the deformation is relatively small, and higher soil stiffness results in greater kinematic forces being transmitted to the piles. Conversely, in loose soils, significant ground deformation leads to increased seismic wave attenuation, but excessive soil movement around the piles can complicate their behavior. Low relative density increases the risk of pile buckling, whereas high relative density leads to greater internal stress, potentially elevating the likelihood of structural damage. The mass of the superstructure plays a crucial role in determining the magnitude of inertial forces generated during an earthquake. According to [PERSON]'s second law (F = ma), a larger superstructure mass results in greater inertial forces under seismic acceleration. These forces impose additional horizontal and vertical loads on the pile foundation, increasing bending moments and shear forces in the piles. However, an increased superstructure mass also lowers the structure's natural frequency, potentially reducing the likelihood of resonance with seismic waves and thereby mitigating seismic response. The interplay between inertial and kinematic forces necessitates a comprehensive understanding of soil-structure interaction. Traditional experimental and 2D numerical methods are often insufficient in capturing the full complexity of pile-soil-structure interactions. Physical model tests are constrained by scale effects, limiting their ability to replicate actual soil-structure interactions under seismic loads. Likewise, 2D numerical analyses fail to account for the three-dimensional effects of seismic loading on piles and soil. Particularly in pile groups, interactions between adjacent piles, asymmetric soil responses, and localized stress concentrations must be considered. Given these limitations, this study employs 3D numerical analysis using FLAC3D to provide a more precise evaluation of the seismic response of pile-supported structures. Several studies have investigated the seismic behavior of pile-supported structures, focusing on different influencing factors. Reference [1] reveals that kinematic interaction significantly influences pile behavior in both liquefiable and non-liquefiable sites, with inertial interaction affecting pile failure modes differently.