# Numerical Modeling for Tunnel Lining Optimization
## 6 Optimization of Tunnel Lining Performance  
In order to study the sensitivity of the primary and secondary lining thickness on the tunnel performance, a parametric analysis was conducted. Four different cases were considered, as summarized in Table 4. The lining performance was established based on interaction diagrams [29; 50; 51], comparing the computed bending moment and the normal force (M-N) combination with the limit defined by the boundary in the interaction diagram. These diagrams are a simplified graphical representation of the critical fault surface, separating acceptable load combinations and those that exceed the allowed limits. The \(\sigma\)-\(\varepsilon\) method was used to derive the M-N interaction diagrams (i.e., bending moment-normal force) [51; 52]. Figures 34a and 34b show the constitutive model considered for the secondary lining (i.e., concrete and steel bars, respectively), and Figure 34c, the constitutive model for the primary lining (i.e., shotcrete reinforced with steel fibers), where the contribution of the steel fibers in the tension portion of the curve can be seen.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Case**} & **Primary Lining Thickness** & **Secondary Lining Thickness** \\  & **(m)** & **(m)** \\ \hline
1 & 0.2 & 0.4 \\ \hline
2 & 0.3 & 0.3 \\ \hline
3 & 0.4 & 0.2 \\ \hline
4 & 0.6 & — \\ \hline \hline \end{tabular}
\end{table}
Table 4: Cases of primary lining thickness analyzed.  
Figure 33: Displacement contours around the tunnel calculated with (**a**) the idealized conditions of the analytical methods, and (**b**) real conditions of the calibrated numerical model.  
Figure 35a,b shows the interaction diagrams and acting forces on the first three cases, on the primary and secondary linings, respectively. In all cases, the acting forces on the secondary lining are minimal and are concentrated in the center of the diagram. On the other hand, the acting forces on the primary lining are more important; nevertheless, they still have a good safety factor gap, which is more evident in the case with the primary lining thickness of 60 cm (Figure 36). In addition, Figure 37 shows the surface vertical displacement profiles in a cross-section perpendicular to the tunnel axis, from all the studied cases. As expected, with the primary lining thickness of 60 cm, the magnitude of settlements is minor even without the secondary lining, which in this case has a negligible contribution. These results suggest that the best alternative for tunneling in these types of materials is the use of only one robust lining, because it can support the forces from the ground stress redistribution during tunneling while decreasing the impact of the excavation on the surface. Such an alternative has been used in past tunnel projects in nearby zones with similar geotechnical conditions [47], and around the, highlighting its rapid construction as well as its practical installation compared to a composite lining. Although several problems have been identified in the usage of single linings, like high temperatures during strength gaining, high deformation, and shrinkage, it has been observed that these limitations can be overcome if the lining is sprayed in two to three layers.  
Figure 34: Constitutive laws used in the definition of interaction diagrams: (**a**) concrete, (**b**) steel bars, and (**c**) steel fiber-reinforced shotcrete (SFRS).  
Figure 35: _Cont._  
Figure 35: Interaction diagrams and acting loads at **(a)** the primary lining and **(b)** secondary lining.  
Figure 36: Interaction diagram and acting loads at primary lining.