## Case Study
### Parametric Study  
In the stability analysis of the tunnel using the NGNRM, thirteen parameters of non-stationary viscoelastic-plastic model are shown in Table 2.  
In this paper, the orthogonal experimental design method (OEDM) is used to study the sensitivity of nine parameters of NGNRM (\(E_{0}\), \(\
u\), \(c_{0}\), \(\varphi\), \(E_{1}\), \(\eta_{1}\), \(\sigma_{\ell}\), \(\lambda\), \(E_{1}^{\prime}\)) on the vault settlement and horizontal displacement of the side wall of the tunnel. The nine parameters are selected as test factors. The values of these nine parameters in Table 2 and the values of their upper and lower 20% are taken as the level of the test factors. In addition, four empty columns are added to test the significance of the test factors. Therefore, \(L_{27}(3^{13})\) is selected as the orthogonal design test table.  
For vault settlement and horizontal displacement of the side wall, the variances of the nine parameters are calculated, respectively, and the statistics \(F\) is constructed.  
Variance of column \(j\) in the orthogonal table can be expressed by Equation (23).  
\[MS_{j}=\frac{SS_{j}}{df_{j}} \tag{23}\]  
where \(SS_{j}\) is the sum of squares of deviation; \(df_{j}\) is the degrees of freedom (DOF) of column \(j\) in the orthogonal table.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\rho\)/kg/m\({}^{3}\) & \(E_{0}\)/GPa & \(\
u\) & \(E_{1}\)/GPa & \(\eta_{1}\)/GPa.h & \(c_{0}\)/MPa & \(\varphi\)\({}^{\circ}\) \\ \hline
2370 & 2.3 & 0.33 & 8.01 & 12,050 & 0.21 & 33 \\ \hline \(\sigma_{\ell}\)/MPa & \(A\) & \(T\) & \(\lambda\) & \(\sigma_{\bullet}\)/MPa & \(E_{1}^{\prime}\)/GPa & \\ \hline
8.0 & 4.2 & 1650 & 1.188 & 35.62 & 26.311 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numerical simulation parameters of the model.  
Figure 7: Comparison of measured and calculated values.  
Variance of the error term in the orthogonal table may be expressed by Equation (24).  
\[MS_{\text{error}}=\frac{SS_{\text{error}}}{df_{\text{error}}} \tag{24}\]  
where \(SS_{\text{error}}\) is the sum of squares of deviation of error term; \(df_{\text{error}}\) is the sum of DOF of error term in the orthogonal table.  
As thus, \(F\) of factor \(j\) can be calculated by Equation (25).  
\[F_{j}=\frac{MS_{j}}{MS_{\text{error}}} \tag{25}\]  
In order to improve the sensitivity of the \(F\)-test, all factors of \(MS_{j}\leq 2 MS_{\text{error}}\) together with four empty columns are included in the error term, expressed by \(e^{\Delta}\); \(F_{\alpha}\) is the critical value of \(F\); \(e\) is the error term considering only two empty columns; SOV is source of variation; Sig. denotes the significance of the influencing factor.  
Finally, the significance test of the effect of these test factors is quantitatively assessed by their variances.  
Based on the significance (Sig.) of the influencing factors on the vault settlement of the tunnel, we obtained the sensitivity ranking of the influencing factors when \(F_{j}>F_{0.01(8,12)}\) as follows: \(\
u>\varphi>E_{0}>\lambda\). Based on the significance of the influencing factors on the sidewall horizontal displacement of the tunnel, we obtained the sensitivity ranking of the influencing factors when \(F_{j}>F_{0.01(8,14)}\) as follows: \(\
u>\varphi>\lambda>E_{0}\).  
Therefore, \(\
u\), \(\varphi\), \(E_{0}\), \(\lambda\) may be determined as the main parameters of the NGNRM by the above significance analysis of the influencing factors.