## 2 Governing Equations of Motion
### Equations of Motion  
The equation of motion for the SFT moored by the mooring cables and subjected to gravity, hydrodynamic and seismic loads can be written in matrix form as follows:  
\[[M+M_{a}]\{\ddot{q}\}+[C]\{\dot{q}\}+[K_{e}+K_{m}]\{q\}=\{f\}+\{f(q,t)\}-[M+M _{a}]\{I\}\Big{\{}\ddot{q}_{g}\} \tag{8}\]  
where \([M]\) is the structural mass matrix of SFT; \([M_{a}]\) is the added mass matrix, which is the mass contribution by water and is obtained from the third term on the right hand side of Equation (6). \([C]\) is the system damping matrix. \([K_{e}]\) and \([K_{m}]\) are the elastic stiffness of the SFT and the mooring stiffness matrices, respectively. \(\{f(q,t)\}\) is the vector of time-dependent hydrodynamic forces. \(\{f\}\) is the vector showing hydrostatic and live loads acting on the SFT. \(\{\ddot{q}_{g}\}\) is the vector of ground motions acceleration. \(\{I\}\) is the influence coefficient vector, having one for elements corresponding to the degrees of freedom in the direction of the applied ground motion and zero for the other degrees of freedom. \(\{\ddot{q}\}\), \(\{\dot{q}\}\) and \(\{q\}\) are the vectors representing structural acceleration, velocity and displacement, respectively. The vector \(\{q\}\) for an element is given as follows:  
\[\{q\}=\Big{[}\begin{array}{cccccccc}q_{X1}&q_{Y1}&q_{Z1}&\theta_{Y1}&\theta_ {Z1}&q_{X2}&q_{Y2}&q_{Z2}&\theta_{Y2}&\theta_{Z2}\end{array}\Big{]}^{T} \tag{9}\]  
where \(q_{X}\), \(q_{Y}\) and \(q_{Z}\) are the displacements along the \(X\), \(Y\) and \(Z\) directions, respectively; while \(\theta_{Y}\) and \(\theta_{Z}\) are the rotations about the \(Y\)-axis and \(Z\)-axis, respectively. The subscripts \(1\) and \(2\) represent the first and second nodes of an element, respectively.  
The structural mass \([M]\) and the added mass \([M_{a}]\) are lumped at the nodes. The added mass is applied in the traverse and vertical directions only. The elastic stiffness \([K_{e}]\) is calculated using the 3D beam element, and the nonlinear mooring stiffness \([K_{m}]\) is calculated using the truss element. We used both cable elements and truss elements for modeling the mooring cables, and there was almost no difference in the response of the SFT due the fact that mooring cables always remain in tension under the input conditions of the present study. Therefore, we adopted the truss element. However, cable elements should be used in general when severe environmental conditions exist. In modeling the mooring cables, the residual buoyancy, which is the net hydrostatic force, was used as pretensions in the cable elements. The damping matrix \([C]\) is calculated using the Rayleigh damping model as follows:  
\[[C]=\beta_{1}[M+M_{a}]+\beta_{2}[K_{e}+K_{m}] \tag{10}\]  
where:  
\[\beta_{1}=\zeta\frac{2\omega_{i}\omega_{j}}{\omega_{i}+\omega_{j}} \tag{11}\]  
\[\beta_{2}=\zeta\frac{2}{\omega_{i}+\omega_{j}} \tag{12}\]  
where \(\beta_{1}\) and \(\beta_{2}\) are the Rayleigh damping coefficients; \(\zeta\) is the modal damping ratio; \(\omega_{i}\) and \(\omega_{j}\) are the \(i\)th and \(j\)th modal natural frequencies of the structure, respectively. The mass and stiffness proportional Rayleigh damping coefficients are chosen based on the first two modes of SFT, so that the modal damping ratio is 2.5%. The same damping ratios have been used for the structure and for the cables due to the fact that SFT is a tubular structure and resembles a pipe [28].  
The numerical procedure for the static and dynamic analysis is briefly described here. In the static analysis of SFT, the maximum wave and current forces were applied as uniformly-distributed static loads. In the dynamic analysis, the equations of motion were solved by [PERSON]'s direct time integration method. A computer program was developed to perform the dynamic analysis of SFT subjected to net residual buoyancy and time-dependent hydrodynamic and seismic forces. The tunnel was modeled using FEM with 30 3D-beam elements, and the torsional effect was neglected because it was very small.