## 3 Numerical Model
### Governing Equations of Dynamic Simulation  
The entire structure is modelled by rod elements and the rod theory suggested by [PERSON] [39] is used. The behavior of the rod element is determined by the position of the rod centerline. The equation of motion is solved in general coordinate whose tangential direction follows the line profile; therefore, coordinate transformations, which increase computation time, are not required. In addition, geometric non-linearity is considered without specific assumptions associated with the shape or orientation of lines [34]. The equation of motion and the extensible condition are presented in Equations (1) and (2). \[-(EI\textbf{r}^{*})^{n}+(\lambda\textbf{r}^{\prime})^{\prime}+\textbf{q}=m \textbf{r} \tag{1}\]  
\[\frac{1}{2}(\textbf{r}^{{}^{\prime}}\cdot\textbf{r}^{{}^{\prime}}-1)=\frac{T} {A_{I}E}\approx\frac{\lambda}{A_{I}E} \tag{2}\]  
where \(\textbf{r}(s,t)\) is a position vector, which is a function of arc length \(s\) and time \(t\) in order to define space curve, \(E\) is Young's modulus, \(I\) is second moment of sectional area, \(\lambda\) is Lagrange multiplier, \(\textbf{q}\) and \(m\) are the distributed load and mass per unit length, \(T\) is the tension, and \(A_{I}\) is the cross sectional area filled with the material. In addition, dot and apostrophe denote time and spatial derivatives, respectively. The distributed load includes the weight of the rod and hydrostatic and hydrodynamic loads induced by the surrounding fluid. The hydrostatic load is subdivided into buoyancy and force induced by hydrostatic pressure. The hydrodynamic force is estimated by Morison equation for moving objects, which consists of linear wave inertia and nonlinear wave drag forces. Thus, Morison equation, which is given by Equation (3), enables to compute wave force per unit length at instantaneous rod-element positions at each time step. \[\textbf{F}_{\textbf{d}}=-C_{A}\rho A_{E}\textbf{r}^{-n}+C_{M}\rho A_{E} \textbf{V}^{n}+\frac{1}{2}C_{D}\rho D\Big{|}\textbf{V}^{n}-\textbf{r}^{.n} \Big{|}(\textbf{V}^{n}-\textbf{r}^{.n}) \tag{3}\]  
where \(C_{M}\), \(C_{A}\), and \(C_{D}\) are the inertia, added mass, and drag coefficients, \(\rho\) is density of water, and \(A_{E}\) is the cross-sectional area for the element, \(D\) is the outer diameter, and \(\textbf{V}^{n}\) and \(\textbf{V}^{.n}\) represent \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Wet Natural Frequency (rad/s)** & **Mode Number** \\ \hline \multirow{3}{*}{Tunnel (Horizontal direction)} & 1.92 & 1 st mode \\  & 2.70 & 2 nd mode \\  & 4.53 & 3 rd mode \\ \hline \multirow{3}{*}{Tunnel (Vertical direction)} & 3.12 & 1 st mode \\  & 3.45 & 2 nd mode \\  & 4.89 & 3 rd mode \\ \hline \multirow{2}{*}{Mooring lines \#1 and \#2 (Center)} & 5.78 & 1 st mode \\ \hline \multirow{2}{*}{Mooring lines \#3 and \#4 (Center)} & 9.04 & 1 st mode \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Wet natural frequencies of the tunnel hydro-elastic responses coupled with mooring lines. velocity and acceleration of a fluid particle normal to the rod centerline. The inertia coefficient of the tunnel and mooring lines is 2.0 considering that the added mass is the same as displaced mass [40]. The drag coefficient of the tunnel is a function of Reynolds number, KC (Keulegan-Carpenter) number, and relative surface roughness, and the representative value of 0.55 is used here based on the experimental results (e.g., [31]). The drag coefficient of mooring lines is 2.4 for stud-less chain [32]. It was shown in [PERSON] et al. [27] that the use of Morison equation for SFT dynamics is good enough compared to the case by using 3D diffraction/radiation panel program. Here, the Morison equation is further modified to include hydrodynamic force induced by vertical pressure variations during earthquake excitations i.e., the seaquake effect, as supported by [PERSON] and [PERSON] [41], [PERSON] et al. [24], [PERSON] et al. [42], and [PERSON] et al. [25].