# Tunnel Failure Evolution and System Reliability Analysis Based on the \(\beta\)-unzipping Method
## 3 Analysis of the Tunnel Failure Process  
Because of the various stress distributions in the tunnel system, the different parts have unique failure probabilities. The parts with a high failure probability can be regarded as major failure parts, and those with a low failure probability are secondary failure parts. The failure process consisting of the major failure parts is the main failure mode. To reduce unnecessary calculations and omit the secondary failure mode in advance, the \(\beta\)-unzipping method can be used to specify the major failure parts of the main failure mode. The reliability of the major failure parts is defined as follows:  
\[\beta\in(\beta_{\text{min}},\beta_{\text{min}}+\Delta\beta) \tag{4}\]  
where \(\beta_{min}+\Delta\beta\) is the minimum value of reliability. The number of unnecessary calculations will increase if the threshold is too high. Conversely, certain main failure modes will be neglected. Scholars have posited that \(\Delta\beta=3\) in the initial failure process, and has a \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Part** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\
**Number** & & & & & & & & & & \\ \hline a & 0.286 & 0.29 & 0.24 & 0.521 & 0.158 & 0.407 & 0.418 & 0.39 & 0.587 \\ b\({}_{1}\) & 0.995 & 0.538 & 0.921 & 1.21 & 0.769 & 0.971 & 0.987 & 1.001 & 2.791 \\ b\({}_{2}\) & 0.271 & 0.3 & 0.214 & 0.197 & 0.224 & 0.251 & 0.243 & 0.271 & 0.176 \\ b\({}_{3}\) & \(-0.221\) & \(-0.034\) & \(-0.312\) & \(-0.111\) & \(-0.645\) & \(-0.082\) & \(-0.074\) & \(-0.054\) & \(-0.693\) \\ c\({}_{1}\) & 0.001 & 0.118 & 0.019 & \(-0.041\) & 0.06 & 0.013 & 0.005 & 0.005 & \(-0.371\) \\ c\({}_{2}\) & 0.003 & 0.003 & 0.004 & 0.005 & 0.004 & 0.004 & 0.004 & 0.003 & 0.005 \\  & 0 & 0.001 & 0 & 0.001 & 0 & 0 & 0 & 0 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The undetermined coefficients of the response surface function when \(\lambda=0.5\). Figure 4: Limit state surface of part 1 when \(\lambda=0.5\). value of 1 in the subsequent processes [35]. However, this method does not fully consider the change in failure probability during the failure process. This paper adopts the adaptive boundary method for regulating the threshold and terminates the failure process when \(\beta_{min}>3\). The benchmark failure probability is as follows [36]:  
\[P_{f}^{0}=P_{f}^{\text{min}}+(P_{f}^{\text{max}}-P_{f}^{\text{min}})\cdot d \tag{5}\]  
where \(p_{f}^{\text{max}}\) is the maximum value of the failure probability of all parts and \(p_{f}^{\text{min}}\) is the minimum value of the failure probability of all parts. Among the variables:  
\[d=\frac{\overline{P_{f}}+P_{f}^{\text{min}}}{\overline{P_{f}}+P_{f}^{\text{ max}}} \tag{6}\]  
where \(\overline{P_{f}}\) is the mean failure probability of all of the parts. The threshold can be obtained based on the failure probability. \[\beta_{0}=\Phi^{-1}(1-P_{f}^{0}) \tag{7}\]  
where \(P_{f}^{0}\) is the benchmark failure probability. The adaptive boundary condition is as follows:  
\[\beta\in(\beta_{\text{min}},\beta_{0}) \tag{8}\]  
Consider the intact state of the structure as the first stage of the failure process. When \(\lambda=0.5\), the reliability index at the first stage of the failure process is shown in Figure 5 (each reliability index is calculated using the JC method [34]). JC method, also known as design point method, is a method to convert non-normal distribution random variables into corresponding normal distribution solutions. The JC method takes the linearization point of the function as the design checking point, thereby increasing the accuracy of calculating \(\beta\) and ensuring the uniqueness of \(\beta\). Figure 5 shows that part 5 is the main failure part at the first stage of the failure process. After part 5 fails, the failure process enters the second stage. The tunnel will produce a  
Figure 5: Reliability index in at the first stage of the failure process. new shear coefficient and response surface function as a result of the stress redistribution.