## 6 Application for Tunnel Design  
During the construction of the Zhongyi tunnel in the Yunnan province of China, over 44% of the tunnel encountered large deformation and the maximum deformation was larger than 600 mm. The main reason for this problem was the mismatch between the support stiffness and the weak surrounding rock conditions [37,40]. This section adopts the finite strain procedure to calculate the needed stiffness of the whole supporting system. The three-centered arch section is equivalent to a circular tunnel with a radius of 4.25 m. The in-situ stress and rock mass parameters are as follows: \(P_{0}=12\) MPa, \(E=0.95\) GPa, \(v=0.3\), \(\gamma^{p*}=0.035\), \(c_{\rm p}=0.48\) MPa, \(c_{\rm r}=0.25\) MPa, \(\varphi_{\rm p}=26^{\circ}\), \(\varphi_{\rm r}=20^{\circ}\), \(\psi_{\rm p}=16^{\circ}\), \(\psi_{\rm r}=12^{\circ}\)[37].  
First, the finite strain procedure is integrated into one MATLAB toolbox called SAFE (Sensitivity Analysis For Everybody) [41] to perform the sensitivity analysis of tunnel deformation. The boundary of rock mass parameters is \((\mu-3\sigma,\ \mu+3\sigma)\), where \(\mu\) and \(\sigma\) denote the mean value and the standard deviation of each parameter. The distribution type of the rock mass parameters is normal distribution [42]. It is supposed that the coefficient of variations (COVs) of the in-situ stress and of other parameters are 0.1 and 0.05, respectively. The support pressure is sampled randomly and uniformly between 0 and 0.25\(P_{0}\). According to the global sensitivity indices (Sobol indices) of each parameter in Figure 11, tunnel deformation is most sensitive to the support pressure \(p\) compared to other factors except for the inexorable geostness \(P_{0}\). Therefore, improving the support stiffness is the most direct and effective means to control the tunnel deformation.  
Furthermore, the stochastic analysis is performed by the Monte Carlo method to optimize the stiffness of the support structure for the Zhongyi tunnel. All distribution parameters are the same as the above but the range of the support pressure is \([0,\ 0.5P_{0}]\). The sampling number is 20,000 and the time for each calculation is between 0.9 s and 15 s in a computer with Intel Core I 7-6700 CPU. The tunnel displacement and support pressure are scattered in Figure 12a and the upper and lower envelope curves are depicted. As the support pressure increases, the variation range of \(u_{0}\) becomes smaller. When the deformation limit is 0.5 m, the range of the support pressure is between 0.72 MPa and 1.63 MPa.  
Figure 10: Comparison of finite strain solution, small strain solution, and [PERSON]’s self-similar solution: (**a**) \(2 Gu_{0}/R_{0}(P_{0}-p_{i}^{*})\) varying with \(p/P_{0}\); (**b**) plastic radii varying with \(p/P_{0}\).  
In the specific design process for the tunnel, calculating the needed stiffness is more practical than predicting the tunnel displacement. Hence, the combined stiffness of the primary support and the secondary lining is obtained by taking the displacement release coefficient as 0.25 before installing the support structure [43]. As shown in Figure 12b, the discrete degree of the displacement \(u_{0}\) significantly decreases with the stiffness \(k\) increasing. When the combined stiffness is larger than 4.3 MPa/m, it can be guaranteed that the tunnel displacement \(u_{0}\) will not exceed 0.5 m, smaller than the limit deformation. As for the detail design, references [44; 45; 46; 47; 48; 49; 50; 51; 52; 53] can be used to calculate the support stiffness, optimize the support parameters, analyze the tunnel stability and perform the whole stiffness design for the tunnel-support system.