## 5 Numerical Results and Discussion  
### Identifying the Impact of Effective Velocity Duration on Maximum Relative Displacements  
Following the inelastic dynamic analysis of the structure imparted by the 16 NFGMs for the same seismic intensity level, the section at the center of the tunnel was chosen as the typical research object. At the same time, the maximum response for each key reference point during the earthquake course was extracted and studied. The effect of the EVD on the peak relative displacements was examined by using a different fixed threshold value. We note that a linear correlation analysis between the EVD and the peak relative displacement was employed herein to identify a general tendency. The linear correlation analysis was also applied to the following section of the paper. As expected, the trend lines present a positive correlation among all relative peak displacements, which indicates that the relative peak displacement during the NFGMs with a longer EVD is prone to more considerable deformation than those under a shorter EVD for the given level of seismic intensity (see Figure 6). Moreover, changing the fixed threshold value of EVD did not alter the changing law of the trend line. It was also found that the NFGMs with a longer EVD would cause a larger longitude peak displacement compared to the hydraulic arched tunnels than the shorter EVD, as shown in Figure 7. \begin{table}
\begin{tabular}{c c c c} \hline
**Constitutive Model** & **Parameter** & **Unit** & **Magnitude** \\ \hline DP model & Cohesion & MPa & 1.1 \\  & Friction angle & \(\circ\) & 41 \\  & Poisson’s ratio & - & 0.25 \\  & Mass density & kg/m\({}^{3}\) & 2625 \\  & Elastic modulus & GPa & 4.0 \\  & Compressive yield stress & MPa & 16.7 \\  & Tensile yield stress & MPa & 1.78 \\  & Poisson’s ratio & - & 0.167 \\  & Mass density & kg/m\({}^{3}\) & 2450 \\  & Elastic modulus & GPa & 28 \\ Linear elasticity & Bulk modulus & GPa & 2.07 \\  & Mass density & kg/m\({}^{3}\) & 1000 \\ \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of surrounding rock, water, and concrete lining. Figure 6: The relative peak displacement of the concrete lining under different fixed threshold values of EVD ((**a**) the right spandrel—left foot, (**b**) the left spandrel—right foot, (**c**) the top arch—bottom, (**d**) the left haunch—right haunch). Figure 7: The peak longitudinal displacement of the concrete lining under different fixed threshold values of EVD. Identifying the Impact of Effective Velocity Duration on Accumulated Damage Patterns and Cumulative Damage Dissipation Energy  
In the presented numerical analysis, the NF earthquake-induced accumulated damage crack patterns of the structure under different duration of NFGMs were generated by the CDP model, as shown in Figure 8. The color depth of the figure for the lining damage indicates the degree of damage. The darker patches were damaged first, and subsequently the cracks spread outward. Clearly, one main crack branch was triggered and propagated to the foot region through the longitudinal thickness of the concrete lining in the bottom region under the shorter NFGM excitations. The concrete lining showed slightly accumulated damage cracking for the top section, which means that shorter EVD earthquakes cannot induce severe damage to the structure. Moreover, as seen in Figure 8, the top section, the sidewall section, and the bottom section of the structure are particularly vulnerable to significant damage during prolonged EVD earthquakes. In addition, the damage energy dissipation index that can directly express the overall damage state of the concrete materials was employed to measure the effect of the EVD, as depicted in Figure 8. The connection between the cumulative damage energy dissipation of the concrete lining and the various EVDs is demonstrated in Figure 9. The damage energy dissipation of the concrete lining exhibited a strong link with EVD, despite the fact that the fixed threshold value had no influence on this effect.