## 4 Parametric Analysis
Among the three types of loading forms, the rectangular impact produced the largest displacement for both the pile head and pile body, the triangular impact resulted in the smallest, while the half-sine impact lied in between. The duration of the impact load applied at the pile head plays a critical role in shaping the structural response. Figure 6 illustrates the effect of varying impact durations on pile head displacement. The results clearly indicate that, for a given impact profile, a longer impact duration leads to a larger peak displacement at the pile head. This behavior  
Figure 4: Pile head displacement. reflects the direct relationship between the transverse excitation impulse--represented by the area under the load curve--and the amplitude of horizontal vibrations in the pile. Furthermore, the time at which the pile head displacement reached its peak consistently lagged behind the moment of peak load application. Notably, this lag became increasingly pronounced as the impact duration shortened. The underlying reason for this phenomenon lies in the energy dynamics of the pile-soil system. When the impact duration is short, the elastic potential energy generated by the deformation remains largely undissipated. This recoverable energy temporarily accumulates within the system, leading to a delayed response and amplifying the observed lag effect. The presence of core soil within the pipe pile significantly reduced the amplitude of the pile head displacement. Figure 7 compares the effects of core soil under triangular impact and sudden step loads. The pile head displacement closely followed the load profile, peaking and then decaying to zero under triangular impact, while stabilizing at a constant value under sudden step loading. The excitation impulse from the sudden step load was twice that of the triangular impact load, resulting in a peak displacement ratio of approximately 2. This highlights the direct correlation between excitation impulse and displacement magnitude. Figure 5: File shaft displacement. Figure 6: Influence of impact duration on pile head displacement. Figure 8 shows the effect of soil dynamic permeability coefficients (\(k_{d1^{\prime}}\), i = 1,2) on pile head displacement. The lower axis represents \(k_{d1^{\prime}}=k_{d2^{\prime}}\) from \(10^{-10}\)-\(10^{-3}\) m\({}^{2}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\), corresponding to Darcy permeability coefficients of \(10^{-6}\)-\(10\) m/s. The upper horizontal axis assumes \(k_{d1^{\prime}}=10^{-10}\) m\({}^{2}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\) and \(k_{d2^{\prime}}=(10^{0}\)-\(10^{\prime})\times k_{d1^{\prime}}\), with \(t\) = 20 ms. For discussion purposes, \(k_{dif}=10^{-3}\) m\({}^{2}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\) is considered, although real-world values are typically much smaller. The results reveal two stable regions at high and low \(k_{d1^{\prime}}\), where the pile head displacement remained constant and was unaffected by \(k_{d1^{\prime}}\). However, at intermediate \(k_{d1^{\prime}}\) values (\(10^{-7}\)-\(10^{-5}\) m\({}^{2}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\) ), displacement increased as \(k_{d1^{\prime}}\) rose. This occurred because higher permeability facilitates pore water flow, reducing its resistance to deformation and amplifying the horizontal dynamic response of the pile. During installation, lateral soil compression decreased \(k_{d1^{\prime}}\) while increasing \(k_{d2^{\prime}}\), as loose soil formed around the pile core. The figure also shows that variations in \(k_{d2^{\prime}}\) had a minimal influence on pile displacement when \(k_{d1^{\prime}}\
eq k_{d2^{\prime}}\). Additionally, lower soil porosity (\(n\)) reduced displacement and diminished the effect of permeability, as denser soils with less pore water provide greater confinement, limiting pile movement. The pile diameter plays a crucial role in pile head and shaft displacements.