# Numerical Simulation of a Submerged Floating Tunnel: Validation and Analysis
## 4 SFT Dynamic Frequency Response Analysis  
To investigate the frequency response characteristics of the submerged floating tunnel in water, a series of sinusoidal seismic waves, with frequencies ranging from 1 Hz to 15 Hz  
Figure 7: Comparison of numerical and experiment solutions. and an amplitude of 0.01\(\times\)\(g\), was applied to the bottom of the aforementioned numerical model. The dynamic response of the SFT model under each seismic wave frequency was analyzed, and the peak values of each response indicator were recorded and statistically analyzed. The results are presented in Figure 8.  
It is evident from the figure that, when the seismic wave frequency was 11 Hz, the horizontal hydrodynamic load, horizontal acceleration, and cable force of the tube body reached their maximum values (51.7 Pa, 0.777 m/s\({}^{2}\), and 36 N, respectively). This indicates that the horizontal self-oscillation frequency of the SFT model was approximately 11 Hz. According to the corresponding similarity constant, this implies that the horizontal self-oscillation frequency of the prototype of the tunnel model was approximately 1.5 Hz. Next, the seismic response of the SFT under the same horizontal sinusoidal seismic wave, with a frequency of 4 Hz and a vibration time of 1 s, was investigated. The acceleration amplitude was adjusted to 0.03\(\times\)\(g\), 0.05\(\times\)\(g\), 0.07\(\times\)\(g\), and 0.09\(\times\)\(g\), and the peak horizontal hydrodynamic load, horizontal acceleration, and cable force were recorded and analyzed. The results are presented in Figure 9. We provide the tension time history curves of the cable under various conditions in Supplementary Materials Figures S8 and S9, further illustrating the dynamic changes in the tension. It can be observed that the dynamic response indices of the SFT exhibited a gradual increase when the seismic acceleration amplitude ranged from 0.01\(\times\)\(g\) to 0.03\(\times\)\(g\). However, a noticeable inflection point and a sharp increase in the dynamic response indices occurred when the seismic acceleration amplitude was between 0.05\(\times\)\(g\) and 0.08\(\times\)\(g\). When the seismic acceleration amplitude reached between 0.08\(\times\)\(g\) and 0.09\(\times\)\(g\), the dynamic response indices of the SFT continued to increase, but did so more slowly than before. Based on the similarity constants between the simulation and the model test provided in Table 3, the following dynamic response indices of the prototype SFT were obtained. \[\sigma_{\text{m}}=F_{\text{m}}/A_{\text{m}} \tag{7}\]  
\[\sigma_{\text{p}}=N\sigma_{\text{m}} \tag{8}\]  
\[P_{\text{p}}=NP_{\text{m}} \tag{9}\]  
Figure 8: Peak value of each dynamic response index at different frequencies. Figure 9: Peak value of each dynamic response index under different seismic wave intensities. \[a_{\rm p}=a_{\rm m} \tag{10}\]  
where \(\sigma_{\rm m}\) is the tensile stress of the model anchor cable, \(\sigma_{\rm p}\) is the tensile stress of the prototype anchor cable, \(F_{\rm m}\) is the model cable force, \(A_{\rm m}\) is the cross-sectional area of the model cable, \(P_{\rm p}\) is the seismic horizontal water pressure load of the prototype, \(P_{\rm m}\) is the seismic horizontal water pressure load of the model, \(a_{\rm p}\) is the acceleration of the prototype tunnel tube body, \(a_{\rm m}\) is the acceleration of the model tunnel tube body, and \(N\) is the prototype/model similarity ratio. Table 4 summarizes the tensile stress of the anchor cable in the prototype SFT under the influence of seismic waves of varying intensities, the horizontal ground-vibration water pressure load on the tube body, and the horizontal acceleration of the tube body, as calculated using Equations (7)-(10). As shown in Table 4, the maximum tensile stress of the anchor cable in the prototype SFT was 535.03 MPa when the seismic wave acceleration strength was 0.03\(\times\)\(g\), which is within the ultimate tensile strength of the steel strand (1960 MPa; [PERSON] et al., 2022 [3]).