## 4 Analysis of the Longitudinal Mechanical Properties of a Double-Layered Shield Tunnel
### Determination of the Optimal Construction Time of the Secondary lining  
In this paper, the reasonable construction time for the secondary lining of the shield tunnel was determined by amalgamating tunnel deformation, the internal force of both the segment lining and secondary lining, and the reasonable degree of internal force. As depicted in Figure 13, tunnel deformation reached a minimum at 0% construction time and subsequently increased with the postpement of the secondary lining's construction. Hence, the earlier the secondary lining was constructed, the more advantageous it had  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Type of Shield Tunnel**} & \multirow{2}{*}{\ \begin{tabular}{c} **Construction Time of the** \\ **Secondary Lining** \\ \end{tabular}  } & \multicolumn{2}{c}{**Longitudinal Bending**} & \multirow{2}{*}{ \begin{tabular}{c} **Longitudinal Shear Force** \\ **/(kN\(\cdot\)m)** \\ \end{tabular}  } \\ \hline \multirow{6}{*}{Double-layered shield tunnel} & 0\% & 180.5 & 898.8 \\ \cline{2-4}  & 10\% & 169.6 & 883.2 \\ \cline{2-4}  & 20\% & 150.0 & 701.1 \\ \cline{2-4}  & 30\% & 140.4 & 676.4 \\ \cline{2-4}  & 40\% & 117.5 &  
in mitigating tunnel deformation. To attain an optimal deformation control effect, the reasonable construction time of the secondary lining was defined as the period when segment displacement reduced by at least 50% post-construction. Drawing from the curve illustrating the vertical deformation of the segment concerning the secondary lining's construction time in Figure 14, it was deduced that the reasonable construction time for the secondary lining, while considering the segment's deformation control effect, fell within the range of 0% to 40%. Figure 18: Curve of internal force of double-layered lining structure with the construction time of secondary lining. As depicted in Figure 18a, after the secondary lining was constructed, the bending moment and shear force of the segment exhibited a distinct change pattern. Specifically, the bending moment increased while the shear force decreased. Consequently, it was essential to account for the disparity in the segment's load-bearing capacity concerning both bending moment and shear force. This study adopted the total safety factor method and introduced the reasonable degree \(W\) of internal force as the evaluation criterion. In contrast to the longitudinal axial force within the tunnel, the longitudinal bending moment and shear force wielded a more significant influence on the tunnel's structure, leading to more prevalent instances of damage [46]. Hence, this paper employed the longitudinal bending moment and shear force as the key indicators for calculations. \[W=(1-\frac{n^{\prime}}{n})^{\frac{1}{3}}+(1-\frac{m^{\prime}}{m})^{\frac{1}{3}} \tag{9}\]  
\(n\) and \(m\) represented the maximum longitudinal bending moment and shear force achievable by segments during various construction times, while \(n^{\prime}\) and \(m^{\prime}\) denoted the corresponding maximum values at a specific construction time. Subsequently, the alteration trend of the segment's internal force reasonable degree \(W\) with varying construction times of the secondary lining was depicted in a graph (Figure 19). A heightened reasonable degree of internal force indicated a more judicious distribution of internal forces within the segment. Figure 19 illustrated that the internal force reasonable degree peaked within the 20-100% construction time range. During this interval, the segment displayed a well-balanced internal force distribution, devoid of substantial shear forces or bending moments. This scenario showcased the segment's optimal utilization of its bending and shear resistance capacities. The curve depicting the change in longitudinal internal force of the secondary lining with respect to its construction time is illustrated in Figure 18b. It can be observed from the figure that as the construction time was extended, the internal force of the secondary lining experienced a continuous reduction.