## 4 Analysis of the Longitudinal Mechanical Properties of a Double-Layered Shield Tunnel
### Analysis of Longitudinal Internal Forces  
The deformation of, and internal forces on, the tunnel structure under external loads were interrelated, and they were found to be critical to the safety and stability of this shield tunnel. Therefore, apart from the deformation, it is necessary to analyze the longitudinal internal forces of double-layered shield tunnels under external load. In general, the longitudinal bending moment and shear force exert a greater influence on the tunnel structure than the longitudinal axial force [46]. Consequently, this paper primarily addressed the longitudinal bending moment and shear force of the shield tunnel, focusing on their distribution range and maximum values as the focal point of investigation. The outcomes of the calculations regarding the longitudinal internal forces within the shield tunnel under various secondary lining construction times are depicted in Figure 17.  
The longitudinal internal forces in single-layered and double-layered shield tunnels have the same distribution under load, being mainly concentrated in the mid-section of the tunnel. This is consistent with the principle that the vertical deformation of the shield tunnel is large in the middle and small at both sides. Following the completion of the secondary lining construction, a noticeable alteration occurred in the internal force of the segment. This was attributed to the shared load-bearing role assumed by the secondary lining. As depicted in Table 6, the simultaneous construction of the segment and the secondary lining led to a substantial reduction in the maximum longitudinal bending moment of the segment, decreasing by 17.2% from 425.5 KN\(\cdot\)m to 352.3 KN\(\cdot\)m. Simultaneously, the maximum longitudinal shear force of the segment experienced a 26.8% increase from 1996 KN to 2531 KN. As the secondary lining construction time was delayed, a gradual restoration of the maximum values of the two longitudinal internal forces was observed, eventually returning to the levels observed prior to the implementation of the secondary lining's construction. In addition it could be observed that the longitudinal internal forces of the segments changed gradually during the construction time of 0% to 30%, while changing rapidly during the period of 30% to 100%. At a secondary lining construction time of 80%, the maximum longitudinal bending moment and shear force of the segments had rebounded to 96.3% and 104.0%, respectively, of the corresponding maximum values seen when the secondary lining was not constructed. This alignment brought the values closer to the internal forces experienced in a single-layered lining tunnel. \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 Single-layered shield tunnel & - & 425.5 & 1996 \\ \hline \multirow{8}{*}{Double-layered shield tunnel} & 0\% & 352.3 & 2531 \\ \cline{2-4}  & 10\% & 354.2 & 2469 \\ \cline{1-1} \cline{2-4}  & 20\% & 355.4 & 2412 \\ \cline{1-1} \cline{2-4}  & 30\% &  
The distribution pattern of internal forces in the secondary lining resembled that of the segment, predominantly concentrated in the middle section of the tunnel. Additionally, the internal forces in the secondary lining were comparatively smaller than those in the segment. As depicted in Table 7, with a construction time of 0%, the maximum longitudinal bending moment and shear force in the secondary lining were 35.5% and 51.2% of the  
Figure 17: Cloud diagram of longitudinal internal force of doubled-layered lining shield tunnel. corresponding values in the segment. As the construction of the secondary lining was delayed, this proportion continued to decrease.