# Tunnel Failure Evolution and System Reliability Analysis Based on the \(\beta\)-unzipping Method
## 2 Reliability Model of the Tunnel System  
The lower part of the tunnel is a 2 \(\times\) 4 m rectangle, and the upper part is a semicircle with a radius of 2 m. These two parts are the overall composition of the tunnel, which is modelled using ABAQUS. The volume modulus \(K\) of the surrounding rock of the tunnel is 10.52 GPa, the shear modulus \(G_{0}\) is 4.69 GPa, and the density \(\rho\) is 2846 kg/m\({}^{3}\)[12]. Because of the inhomogeneity of the surrounding rock distribution, the mean cohesive force is \(\mu_{c}\) 1.81 MPa, and its standard deviation \(\sigma_{c}\) is 0.55 MPa. The mean friction angle \(\mu_{\varphi}\) is 26.6\({}^{\circ}\), and its standard deviation is \(\sigma_{\varphi}\) 4.5\({}^{\circ}\). All random variables are assumed to follow a normal distribution. The surrounding rock of the tunnel is considered to have nine parts, as shown in Figure 1. Parts 8 and 9 are 0.5 \(\times\) 2.5 m rectangles. Parts 1, 2, 6, and 7 are 1 \(\times\) 0.5 m rectangles. Parts 3, 4, and 5 are arcs with an inner diameter of 2 m and outer diameter of 2.5 m.  
The upper boundary of the model is subject to a mean pressure \(\mu_{p}\) of 20 MPa with a standard deviation \(\sigma_{p}\) of 4.13 MPa. The constitutive rock model is the ideal elastic brittleness model [12]. At the left boundary, the lateral pressure coefficients \(\lambda\) are 0.5, 1, and 2. Lateral pressure coefficients of 0.5, 1, and 2 are obtained based on previous research and engineering experience [21; 23]. In tunnel engineering, the lateral pressure coefficients of 0.5, 1, and 2 are representative. The right and lower boundaries are fixed, as shown in Figure 2[15]. Figure 1: Different parts of the tunnel system. This paper considers the reliability of the tunnel's shear capacity. The Mohr-Coulomb criterion is used as the shear strength criterion. The performance function of the structure is as follows:  
\[g=\sigma\times\tan\varphi+c-(k\times p+G) \tag{1}\]  
where \(g\) is the performance function, \(\sigma_{p}\) is the positive pressure on the top of the rock, \(\varphi\) is the friction angle, \(c\) is the cohesive force of the rock, \(k\) is the shear coefficient of the different parts, and G is the gravity of the rock. The stress distribution state is shown in Figure 3a,b. The shear coefficient is listed in Table 1, which is obtained by the ratio of the initial vertical stress of the structure to the initial shear force, obtained using ABAQUS [26; 33]. The original limit state surface is replaced by the approximate quadratic response surface. The response surface function is expressed as follows [34]:  
\[g^{\prime}=a+b_{1}c+b_{2}\varphi+b_{3}p+c_{1}c^{2}+c_{2}\varphi^{2}+c_{3}p^{2} \tag{2}\]  
where \(a,b_{1},b_{2},b_{3},c_{1},c_{2}\) and \(c_{3}\) are the undetermined coefficients. The mean value of each random variable is adopted as the initial design point. Seven sample points are selected according to \((\mu-f\sigma,\mu+f\sigma)\), where \(f\) is 3 for the first iteration and 1 for the subsequent iterations. When the error of two iterations is less than \(10^{-4}\), the iteration is considered to be terminated. It is then possible to obtain the response surface function and to approximate the value \(X_{D}\) of the verification point. The new center point is as follows [28]: \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 \(\lambda\)**=0.5** & 0.219 & 0.01 & \(-\)0.329 & 0.088 & 0.665 & 0.064 & \(-\)0.066 & \(-\)0.065 & 0.218 \\ \(\lambda\)**=1** & 0.491 & 0.044 & \(-\)0.382 & 0.134 & 0.735 & 0.139 & \(-\)0.079 & \(-\)0.048 & 0.322 \\ \(\lambda\)**=2** & 1.024 & 0.174 & \(-\)0.565 & 0.248 & 0.984 & 0.132 & \(-\)0.237 & \(-\)0.042 & 0.531 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial shear coefficients of the tunnel under different lateral pressure coefficients. Figure 2: Tunnel boundary conditions.