## 4 Discussion of the Results
### Effect of the Circumferential Joint  
#### 4.2.1 Effect of the Circumferential Shear Stiffness  
A staggered assembly is commonly adopted in shield tunnel construction [45; 46]. Due to the limits of the testing facility, an experiment with a staggered full-scale prototype could not be carried out. In the following section, the effect of staggering is discussed using the incremental model. Herein, the shear stiffnesses in radial and tangential direction are taken into account in the circumferential joints of the model. The total jack thrust force during the construction of a quasi-rectangular shield tunnel is about 100,000 kN according to the construction data. The longitudinal force in shield tunnels in soft soils is approximately 10-20% of the total thrust of the tunnel boring machine [30; 47]. Hereafter, the mean value, i.e., 15%, is adopted. Formula (5) is used to give an estimated value of \(288\times 10^{3}\) kN/m\({}^{2}\) for both the radial and tangential shear stiffnesses. In order to better simulate the real soil and water pressure on the linings, a uniform load distribution is used to replace the point loads in the full-scale ring experiments in Figure 13. Herein, q\({}_{1}\) is the vertical soil and water pressure, q\({}_{2}\) is the ground overload, which is 20 kPa according to Code for Design of Metro [GB50157] [37], q\({}_{3}\) is the bottom counter force, q\({}_{4}\) is the arc-shaped pressure caused by the curve of the linings' outline, g\({}_{1}\) is the structural dead weight, e\({}_{1}\) and e\({}_{2}\) are the lateral pressures, e\({}_{3}\) is the resisting pressure from lateral soils caused by the structural horizontal deformation. The value of q\({}_{1}\) is determined by multiplying the unit weight of the soil by the height of the overburden, and q\({}_{3}\) is balanced by the total load of q\({}_{1}\), q\({}_{2}\) and g\({}_{1}\). The lateral pressure is calculated by multiplying the vertical load with the coefficient of lateral pressure. The lateral soil resisting pressure is calculated by multiplying the structural horizontal deformation with the horizontal coefficient of the soil resistance, which is 6 MPa/m according to tests on the construction site. In previous studies, similar load distributions have been used by [PERSON] et al. [5], [PERSON] et al. [14], [PERSON] et al. [15], [PERSON] et al. [17], [PERSON] and [PERSON] [48]. Two typical design cases with an 8 and 17 m overburden are used for the analysis in the current study. All relevant parameters are listed in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Calculation** & **Soil** & **Coefficient** & **Vertical Soil** & **Ground** & **Bottom** & **Top Value** & **Bottom Value** \\
**Case** & **Overburden** & **of Lateral** & **Pressure** & **q\({}_{1}\)** & **Overload** & **Source q\({}_{3}\)** & **Pressure** & **Pressure** \\
**(m)** & **Pressure** & **(kPa)** & **q\({}_{2}\)** & **(kPa)** & **(kPa)** & **(kPa)** & **(kPa)** \\ \hline
1 & 8 & 0.7 & 144.0 & 20 & 208.9 & 114.8 & 202.7 \\
2 & 17 & 0.6 & 306.0 & 20 & 370.9 & 195.6 & 271.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters for calculation cases. Calculations are made with and without shear stiffness in the circumferential joints. Taking Calculation Case 2 as example, the bending moments on all sections are unfolded clockwise (in the direction of \(\alpha\) in Figure 14a) along the circumference and compared in Figure 14. A typical bending moment distribution example is also shown. The joint locations are based on the middle ring (Pattern A) and longitudinal joints JF1 to JF10 and segments in the middle ring are marked, as well as the longitudinal joints of the neighboring rings CJ1 to CJ10. The mean values of the bending moments (green and red curves in Figure 14a) are the average bending moment of a segment section and its related sections in the neighboring rings, namely the average bending moment in the middle ring and its front half ring and back half ring.