## 3 Results and Discussion
### Dynamic Responses on Pile Foundations  
Figures 11 and 12 illustrate the displacement-time history response of pile foundations at various depths under different soil relative densities. In loose soil conditions, where the influence of kinematic effects is expected to be relatively significant, the displacement response of the pile foundation is generally larger than in the dense soil conditions at all depths, and the difference is approximately 1.58 to 1.69 times depending on the depth. Furthermore, in loose soil conditions and under a relatively light pile cap mass (\(\gamma\) = 2.4 t/m\({}^{3}\)), the dynamic responses at the top and bottom of the pile foundation exhibit an in-phase behavior. This suggests that the influence of kinematic effects outweighs that of inertial  
Figure 8: Velocity–time histories at the pile cap under dense soil conditions. Figure 9: Peak velocity responses at the top of the pile cap for different magnitude of input motion. effects, leading to a reduced inertial effect and a synchronized response between the top and bottom of the piles. Conversely, under dense soil conditions and a relatively heavy pile cap mass (\(\gamma\) = 12 t/m\({}^{3}\)), the dynamic responses at the top and bottom of the pile foundation tend to exhibit slightly out-of-phase behavior. This is attributed to the interaction between inertial and kinematic effects, where the dynamic behavior at the pile top, which is more influenced by inertial effects, shows a phase difference relative to the bottom, which is more affected by kinematic effects. However, due to the constraints imposed by pile cap embedment conditions, the influence of inertial effects remains limited, similar to the findings in the results of the pile cap velocity-time history response. Figure 13 presents the maximum lateral displacement response of pile foundations at different depths under varying soil relative densities. The dynamic displacement responses of the pile foundation are generally larger under loose soil conditions, with the difference gradually diminishing with increasing depth. Quantitatively examining the difference in response according to depth, it was approximately 0.22 mm to 0.43 mm under the condition of 0.1 g input motion, approximately 0.45 mm to 0.84 mm under the condition of 0.2 g input motion, and approximately 0.91 mm to 1.72 mm under the condition of 0.4 g input motion. This suggests that while kinematic effects govern the response, interaction with inertial effects at the pile top influences the variation in displacement response along depth. Figure 14 shows the maximum displacement response of pile foundations at different depths for varying pile cap weights. The dynamic displacement response is generally greater for pile foundations with heavier pile cap mass, though the relative differences are smaller compared to Figure 11. Moreover, the variation in displacement response along the depth increases with the depth, indicating that the combined effects of pile group constraints and pile cap embedment limit the manifestation of inertial effects. On the other hand, the magnitudes of overall displacement of the pile foundation remain relatively small, with the maximum pile top displacement not exceeding 5 mm even in the most extreme case. Consequently, under typical pile group bridge foundation conditions (pile groups with embedded pile caps), the limited manifestation of inertial effects suggests that the impact on superstructure serviceability is minimal. This influence is expected to be even more restricted in conditions where bridge piers are interconnected and have a greater number of piles per pier. However, separate from serviceability considerations, material stability of the pile foundation must be carefully assessed. In contrast to the dry ground conditions considered in this study, liquefiable soils may exhibit significantly different dynamic responses due to shear strength degradation, lateral spreading-induced kinematic effects, and other liquefaction-related phenomena.