## 3 Multi-Failure Mode Tunnel System Analysis Model  
### Multi-Failure Modes  
In order to make the reliability analysis of the reinforced concrete lining structure system for high-pressure tunnels more in line with engineering practice, a structural reliability analysis under multiple failure modes was carried out. From multiple perspectives, such as surrounding rock deformation, rock bolt length, and support capacity, the three most commonly used failure modes [6; 23] were comprehensively selected. The specific forms are as follows:  
\begin{table}
\begin{tabular}{c c c c} \hline
**Random Variable** & **Average Value** & **Standard Deviation** & **COV** \\ \hline \(X_{1}\) & 0.001 & 0.0002 & 0.2 \\ \(X_{2}\) & 250 & 37.5 & 0.15 \\ \hline \end{tabular}
\end{table}
Table 2: The random variables and related information.  
Figure 4: Example 2: real response surface.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Method** & \(\mathbf{N_{call}}\) & \(\mathbf{\beta}\) & \(\boldsymbol{P_{f}(10^{-3})}\) & \(\boldsymbol{\zeta(\%)}\) \\ \hline MCS & \(10^{6}\) & 2.3471 & 9.460 & 0 \\ Improved response surface [32] & 24 & 2.3180 & 10.225 & 8.09 \\ PSO-FORM [33] & 70 & 2.3309 & 9.879 & 4.43 \\ Kriging-MCS & 200 & 2.3402 & 9.637 & 1.87 \\ ISSA-Kriging-MCS & 200 & 2.3441 & 9.537 & 0.81 \\ \hline \end{tabular}
\end{table}
Table 3: The comparison of reliability calculation results for Example 2.  
1. Failure mode 1--surrounding rock convergence failure mode: \[g_{1}(X)=\varepsilon_{max}-\frac{u_{r}^{D}}{R_{t}}\] (19) where \(X\) is a random variable vector, \(R_{t}\) is the excavation radius of the tunnel, \(u_{r}^{D}\) is the final radial displacement, and \(\varepsilon_{max}\) is the maximum allowable relative convergence value of the surrounding rock (\(\varepsilon_{max}\) = 0.2% in this study).
2. Failure mode 2--rock bolt length failure mode: \[g_{2}(X)=L-\left(R_{pl}-R_{t}+0.4L\right)\] (20) where \(L\) is the length of the rock bolt, and \(0.4L\) is selected as the rock bolt length of the undisturbed surrounding rock; \(R_{pl}\) is the radius of the plastic zone.
3. Failure mode 3--failure mode of bearing capacity of supporting structure: \[g_{3}(X)=p_{s}^{max}-p_{s}^{D}\] (21) where \(p_{s}^{max}\) is the maximum bearing capacity of the supporting structure, which can be determined by the support characteristic curve or numerical simulation results. \(p_{s}^{D}\) is the design bearing capacity of the supporting structure or the supporting force or surrounding rock pressure after the supporting structure reaches equilibrium.  
At present, the failure of structural systems can be divided into parallel, series, and series-parallel hybrid systems. For the actual hydraulic tunnel project, the above three failure modes are interrelated, and the failure of any one mode will lead to the failure of the whole system. Therefore, a series system is considered, as shown in the following equation:  
\[G(X)\begin{cases}\geq 0,\forall g_{i}(x)\geq 0\\ <0,\exists g_{i}(x)<0\end{cases}(i=1,2,3) \tag{22}\]  
where \(G(X)\) is the limit state function of the series tunnel system, \(\forall g_{i}(x)\) represents that each failure mode is greater than 0, and \(\exists g_{i}(x)\) represents that any one or more failure modes are less than 0.