## 4 Parametric Analysis  
To validate the accuracy of the theoretical derivation presented in this study, specific parameters were assigned: \(r_{2}=10^{-4}\) m, \(k_{41^{\prime}}=k_{42^{\prime}}=10^{-1}\) m\({}^{2}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\) and \(\rho_{f}=0\). By approximating the exclusion of pore water effects, the derived solution was simplified to represent a solid pile in a single-phase soil. This simplified solution was subsequently compared with the results of [PERSON]. (1995) [15] (Figure 2) and [PERSON]. (2016) [33] (Figure 3). In [PERSON]. (1995) [15], a sudden load was applied at the pile head and then maintained at a constant magnitude. As illustrated in Figure 2, the solution derived in this study exhibits excellent agreement with the results reported in [15], demonstrating consistency with the static load limit observed in prior analyses [34]. In reference [33], a cosine-shaped impact load was applied to the pile head. \[P(t)=\Bigg{\{} \begin{matrix}\frac{t_{0}}{t_{0}}\left(1-\cos\frac{2\pi}{t_{0}}t \right),&0\leq t\leq t_{0}\\ 0,&other\end{matrix}  \tag{48}\]  
Figure 3: Comparison with results from [PERSON]. (2016) [33]. Figure 2: Comparison with results from [PERSON]. (1995) [15]. The impulse of the excitation force, denoted as \(I_{0}\), was set to \(I_{0}\) = 1.0 N\(\cdot\)s, \(t_{0}\) = 2 ms, \(L\) = 7 m, \(r_{1}\) = 0.4 m, \(\rho_{p}\) = 2.4 g/cm\({}^{3}\), \(v_{p}\) = 0.28, \(E_{p}\) = 25.5 GPa, \(\rho_{s}\) = 2.7 g/cm\({}^{3}\), \(v_{s}\) = 0.35, and \(v_{s}\) = \(\sqrt{G/\rho}\) = 100 m/s. As illustrated in Figure 3, the displacement-time curves derived from the two methods demonstrated excellent agreement. However, during the post-loading phase, the results from reference [9] exhibited slightly larger displacements compared to the calculations presented in this study. This minor discrepancy is likely due to differences in the modeling assumptions and computational methods. Specifically, this study treats the pile as a Timoshenko beam and employs a dynamic Winkler foundation model for analysis, whereas reference [9] models the pile as a three-dimensional viscoelastic body and solves the viscoelastic wave equation using the finite difference method. The following analysis explores the effects of various parameters on the displacement and internal forces of the pipe pile shaft. Unless explicitly stated otherwise, the baseline parameters for the pile-soil system are defined as follows:  
Soil properties: \(K_{s1}\) = \(K_{s2}\) = 36 GPa, \(\rho_{s1}\) = \(\rho_{s2}\) = 2.7 g/cm\({}^{3}\). Pore water: \(K_{f}\) = 2.25 GPa, \(\rho_{f}\) = 1.0 g/cm\({}^{3}\). Soil skeleton: \(G_{1}\) = \(G_{2}\) = 10 MPa, \(v_{p}\) = 0.20, \(n\) = 0.375, \(k_{41^{\prime}}\) = \(k_{42^{\prime}}\) = 10\({}^{-10}\) m\({}^{2}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\). Pipe pile properties: \(E_{p}\) = 25.5 GPa, \(v_{p}\) = 0.20, \(\rho_{p}\) = 2.5 g/cm\({}^{3}\), \(L\) = 10 m, \(r_{1}\) = 0.50 m, \(r_{2}\) = 0.38 m. The pile head was subjected to a triangular impact load with a peak magnitude of \(P_{max}\) = 1000 kN and an impact duration of \(t_{0}\) = 20 ms. These parameters represent a typical engineering scenario in which tubular pile foundations, commonly used in soft soils in marine engineering, are subjected to dynamic impact loads. Unless otherwise stated, the pile head is considered a free boundary with no rotational constraints. Figures 4 and 5 show the effects of different impact load types on pile head and pile body displacements. All three load types had a duration of 20 ms and a peak load of 1000 kN. The figure shows that, under different impact forms, the pile head and pile body displacements followed consistent patterns with respect to time and depth. The displacement of the pile head increased over time, reaching a peak before gradually returning to zero. The displacement of the pile body decreased rapidly with increasing depth.