## 3 Numerical Model
### Simplified Calculation of Ship Impact Loads  
Ship impact loads are typically simplified as half-wave sine forms. In professional sectors such as aviation, automotive, or mechanical engineering, the commonly utilized time-history curve for impact loads exhibits a high amplitude and short duration [11]. However, during normal ship berthing, the impact load exhibits a relatively low velocity, usually below 0.45 m/s, with an extended duration, often around 1 s. This study focuses on a high-piled wharf designed to accommodate vessels of up to 20,000 t. Considering one specific working condition, a 20,000 t design ship is employed. As per the \"Port Engineering Technical Code-SL260-2023\" [30], the typical berthing speed is considered as 0.2 m/s, and the impact load duration is approximated at 1 s. Given the structural configuration of the high-piled wharf in this study, a partition coefficient of 0.31, representing the most adverse single-bent condition, is utilized.  
When a ship docks, the magnitude of its impact force on the wharf structure depends on the effective ship impact energy, the performance of rubber fenders in absorbing ship impact energy, and the overall stiffness of the structure. According to the \"Load Code for Harbour Engineering-JTS144-1-2010\" [31], the effective ship impact energy is calculated using Equation (1).  
\[E_{0}=\frac{\rho}{2}mv_{n}^{2} \tag{1}\]  
where \(E_{0}\) is the effective ship impact energy in kilojoules (kJ); \(\rho\) is the effective energy coefficient, which considers factors such as the influence of the water body-added mass, the eccentricity of berthing point, the interaction with the dock structure, and the deformation of the ship side plates. In this paper, a value of 0.8 is adopted; \(m\) is the ship's laden displacement (t); \(v_{n}\) represents the ship's normal berthing velocity (m/s).  
After calculating the effective ship impact energy using Equation (1), the corresponding performance curve of the rubber fender selected for the dock in this paper is consulted to determine the reactive force of the fender, which represents the theoretical value of the berthing ship's impact force. To underscore general applicability, this article considers two working conditions: the berthing of oversized ships (40,000 t) and an atypical, accidental berthing (lasting 1.0 s). Ship impact loads for these three illustrative working conditions,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & **Normal Stiffness** & **Tangential** & **Normal** & **Tangential** & **Normal Friction** & **Tangential** \\  & **(kPa/m)** & **Stiffness (kPa/m)** & **Cohesion (kPa)** & **Cohesion (kPa)** & **Angle (\({}^{\circ}\))** & **Friction Angle (\({}^{\circ}\))** \\ \hline \⃝ & 5900 & 1000 & 120 & 28.7 & 70 & 6.65 \\ ⃝ & 5900 & 1000 & 120 & 28.7 & 70 & 6.65 \\ ⃝ & 28,000 & 10,000 & 548 & 10.5 & 70 & 9.45 \\ ⃝ & 53,500 & 10,000 & 197 & 13.3 & 81 & 14.70 \\ ⃝ & 67,100 & 10,000 & 315 & 28.7 & 81 & 13.65 \\ ⃝ & 72,300 & 10,000 & 216 & 5.6 & 81 & 22.05 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coupled spring parameters at the pile-soil interface.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Components**} & \multirow{2}{*}{**Concrete Grade**} & **Elastic Modulus** & **Density** & **Poisson Ratio** \\  & & **(GPa)** & **(kg/m\$)** & **Poisson Ratio** \\ \hline Beam & C40 & 32.5 & 2500 & 0.167 \\ Pile & C50 & 34.5 & 2500 & 0.167 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Model parameters of the transverse bracing structure for high-piled wharves.  
computed using the aforementioned methods, are presented in Table 4. This computation yields a calculated amplitude of 409.2 kN for the berthing ship's impact force borne by a single bent. The time-history curve illustrating the impact force is illustrated in Figure 4.