## 3 Results and Discussion  
### Dynamic Responses on the Pile Cap  
Figure 7 presents the velocity-time history responses at the top of the pile cap under loose soil conditions (D\({}_{\rm r}\) = 30%), categorized by the peak acceleration magnitude of the input motion and the pile cap mass. It is observed that the pile cap velocity response increases as the peak acceleration of the input seismic wave increases, while the influence of pile cap mass variation is relatively minor. However, as the peak acceleration of the input motion increases, the variation in response due to the pile cap mass becomes more pronounced. This indicates that the influence of inertial force on the system intensifies Rwith increasing magnitude of input motion, though its absolute magnitude remains consRtrained due to pile cap embedment conditions. Figure 8 presents the velocity-time history responses of the pile cap under dense soil conditions (Dr = 80%), categorized by the peak acceleration magnitude of the input motion and the pile cap mass. Similar to the loose soil condition, the pile cap velocity response increases as the peak acceleration of the input motion increases, with the effect of pile cap mass variation remaining relatively minor. Additionally, the trend of increasing inertial force influence with higher peak acceleration of input motion is also observed. However, the specific response patterns exhibit some differences compared to loose soil conditions. To investigate these discrepancies and analyze the interaction between inertial and kinematic effects, the maximum responses for each case were additionally examined. Figure 9 compares the peak velocity response at the top of the pile cap for different magnitudes of input motion, categorized by cases with identical pile cap mass. When the peak acceleration of the input motion is relatively low, similar peak velocity responses at the pile cap are observed between loose and dense soil conditions, indicating that the influence of kinematic effects is relatively limited under relatively small seismic events in dry soil conditions. This phenomenon is also reported by [3], where the dynamic response of single pile foundations showed no significant differences when subjected to seismic waves with peak accelerations of 0.13 g and 0.25 g in three distinct systems with relative densities of 30%, 50%, and 80% in dry soil conditions. However, compared to the findings of [3], the present numerical analysis results exhibit a slightly greater difference in peak pile cap velocity response between loose and dense soil conditions. This discrepancy is attributed to the constraint on inertial force imposed by pile cap embedment conditions and the relative increase in kinematic effects. In [3], the system was used in which the single piles protrude beyond the ground surface to support the superstructure that is not in contact with the ground surface. Figure 6: Acceleration–time histories of input earthquake motion. Moreover, as the peak acceleration of the input motion increases, the difference in peak pile cap velocity response between loose and dense ground conditions becomes more significant. This implies that kinematic effects become relatively more influential as the seismic intensity increases. Additionally, in large seismic events, the manifestation of kinematic effects in dense soil conditions intensifies, leading to increased bending moments in the piles and greater responses in the pile cap and superstructure. Figure 10 compares the peak velocity response at the top of the pile cap for different pile cap masses, categorized by peak accelerations of input earthquake motion. The difference in peak velocity response at the pile cap between loose and dense soil conditions is more pronounced for cases with smaller upper mass. This suggests that inertial forces interact with kinematic effects in a resistive manner as the pile cap mass increases.