## 4 Parametric Analysis
Figure 9 shows that, for an outer diameter of 0.5 m, decreasing the inner diameter (thickening  
Figure 8: Influence of dynamic permeability coefficient on pile head displacement. Figure 7: Influence of pile core soil on pile head displacement. the wall) gradually reduced peak displacements. However, when the inner diameter fell below 0.2 m, its effect became negligible. Larger outer diameters significantly reduced displacements at both the pile head and the upper shaft. For small inner diameters, the influence of core soil on the horizontal dynamic response is minimal and can be ignored. For larger inner diameters, including core soil improves accuracy. This study employs the Timoshenko beam model, yielding results consistent with those of the Bernoulli-Euler beam model. Figure 10 illustrates the time-history curves of pile head displacement for pile lengths of 2 m and 10 m, as derived from both models. For a pile length of \(L=2\) m, the stress waves reflected at the pile base induced minor oscillations in the pile head response. In contrast, increasing the pile length to 10 m reduced the influence of the reflected waves, leading to a single-peak response at the pile head. Compared to short piles (2 m) with fixed pile bases, longer piles (10 m) exhibited lower transverse dynamic stiffness, resulting in higher and broader peak amplitudes in the horizontal response. Notably, for short piles, the Timoshenko beam model predicted slightly higher peak and trough response values than the Euler beam model. However, as the pile length increased, the results from both models converged, showing negligible differences. Figure 11 further examines the relationship between pile length and pile head displacement at \(t=10\) s, 20 s, and 40 s. The findings indicate that, when the pile length  
Figure 10: Effect of pile model on pile head displacement (\(u\)-\(t\)). Figure 9: Influence of pile diameter on pile head displacement. exceeded a critical threshold--termed the \"effective pile length\"--the pile head displacement stabilized. This is because the effective vibrating portion of the pile-soil system no longer increased with pile length. At this stage, the system can be approximated using an infinite length pile model. Moreover, the effective pile length varied with time: longer durations corresponded to greater effective pile lengths. This observation underscores the time-dependent nature of the pile-soil dynamic interaction and the necessity of considering temporal factors when assessing pile behavior. Figure 12 illustrates the displacement evolution of the pile body over time as represented by a cloud diagram. The displacement gradually propagated deeper into the soil layers as time progressed. By the end of the loading period (t = 20 ms), the pile displacement had not returned to its initial position, indicating residual deformation. Figure 13 analyzes the effect of the soil-to-pile elastic modulus ratio on pile displacement at the conclusion of loading (t = 20 m), assuming a constant pile elastic modulus (\(E_{s1}=E_{s2}=E_{s}\)). The results demonstrate that, as the soil elastic modulus increased--corresponding to stiffer soil--the pile displacement, bending moment, and shear force decreased progressively. Additionally, the depth of the pile response was reduced, eventually becoming almost unaffected by external loading. An important observation is that, when the soil-to-pile elastic modulus ratio reached \(1\times 10^{-3}\), there remained a small residual internal force at the pile base, which has not entirely diminished to zero. This residual force highlights the persistent interaction between the pile and the surrounding soil, even in highly stiff conditions. Figure 11: Effect of pile model on pile head displacement (_u-L_). Figure 2:0 Time-dependent contour plot of pile body displacement. The influence of flexible constraints at the pile head markedly differed from that of other constraint types.