## 4 Discussion of the Results
### Effect of the Circumferential Joint
#### 4.2.2 Coefficient of Bending Moment Transmission  
In this section, the coefficient of bending moment transmission for a quasi-rectangular tunnel is discussed. The concept for this coefficient is provided by I.T.A. [22] and is commonly used to determine the bending moment distribution between the joints and their neighboring segment bodies [24; 41]. As shown in Figure 15\((1-\xi)\cdot M\), is taken as the bending moment of the joint section while \((1+\xi)\cdot M\) is taken as the bending moment of the body section, namely the neighboring segment section. This coefficient is a vital parameter in evaluating the bending moment transmission performance between rings of the staggered segments. It is generally recognized that the coefficient is related to the joint's stiffness, and normally the coefficient \(\xi\) is set to 0.3 for a traditional circular tunnel in soft soils [13; 49]. According to the Formula (7), \(\xi\) can be calculated from Formula (8). \[\left\{ \begin{array}{l}M_{1}=(1-\xi)\cdot M\\ M_{2}=(1+\xi)\cdot M\\ \end{array} \right. \tag{7}\]  
\[\xi=1-2\cdot M_{1}/M_{2} \tag{8}\]  
The curves of mean bending moments match each other well in Figure 14a, which means that the total bending moments at the joint sections of the middle ring and related neighboring sections with or without circumferential shear stiffness are similar. Therefore, the coefficient of bending moment transmission can be derived to describe the level of bending moment transmission. The bending moment at the joint sections and related neighboring sections are listed in Table 6, as well as the coefficients of bending moment transmission calculated from Formula (8). Additionally, the cases with constant rotational stiffnesses from 100,000 to 400,000 kNm/rad are calculated, and their coefficients of bending moment transmission are also shown to compare the effect of the incremental method. If a range of coefficients of bending moment transmission from constant stiffnesses can cover the corresponding coefficient from the incremental method, the range is underlined for highlighting. The coefficients of bending moment transmission from the incremental method are generally smaller or than 0.3, except for JF9 in Case 1, where a small bending moment exists. This means that the adoption of the coefficient from circular shield tunnels can be proposed. JF9 and JF 10 have the \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Case**} & \multirow{3}{*}{**Joint**} & \multicolumn{6}{c}{**Incremental Method**} & \multicolumn{6}{c}{\(\xi\)} \\ \cline{3-11}  & & \multicolumn{3}{c}{**Moment (kNm)**} & \multirow{3}{*}{\(\xi\)} & \multirow{3}{*}{\(\xi\)} & \multirow{3}{*}{\(\xi\)} & \multirow{3}{*}{**100**} & \multirow{3}{*}{**150**} & \multirow{3}{*}{**200**} & \multirow{3}{*}{**250**} & \multirow{3}{*}{**300**} & \multirow{3}{*}{**350**} & \multirow{3}{*}{**400**} \\ \cline{3-11}  & & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \cline{3-11}  & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary of the coefficients of bending moment transmission. Figure 15: Illustration of bending moment transmission. largest coefficient, and the phenomenon of bending moment transmission reaches a maximum at these joints. The essence of the circumferential shear stiffness is to provide the resisting force for the uneven deformation between neighboring rings [50; 51]; the larger the relative stiffness difference between rings, the more pronounced the phenomenon of bending moment transmission.