## 2 Model Setup
### Joint Parameters: Establishment of Joint Parameter Database
#### 2.3.2 Rotational Stiffness  
As mentioned previously, full-scale bending moment resistance experiments on longitudinal joints were conducted to investigate the \(M-\theta\) curve under different loading conditions. Furthermore, a 3D finite element simulation approach was developed, taking into account the cast iron parts, and the reinforcement in the precast concrete segments. The details of the bending moment resistance experiments and the 3D simulation model are provided in Appendix A. The simulation model provides a verified approach for obtaining the joint's rotation under a given axial force and bending  
Figure 7: Joint experiments: (**a**) bending moment resistance experiment; (**b**) shear resistance experiment. moment. Numerically predicted and experimentally verified moment-rotation curves, based on the joint configuration of the quasi-rectangular shield tunnel, are presented in Figure 8. The results provide insight into the rotational stiffness for axial forces up to 1200 kN and absolute bending moments up to 400 kNm. There are two categories of bending moments in the segments. One category relates to positive bending moments. These moments create compression at the outer side of the segment and tension at the inner side. The other category relates to negative bending moments, which create the opposite stress distribution. Based on the variation of the joint's opening at the inner and outer side, joint rotations can be calculated by dividing this variation by the total joint height. Clearly, the rotational stiffness decreases with increasing bending moment, and the stiffness changes quickly in the range of bending moments from 100 to 300 kNm. Since the bending moments of many joints occur in this range, as shown in Section 4.2 for a case with a 17 m buried depth, the relevance of the nonlinear approach is evident. In addition, the influence of the axial forces, which increase the rotational stiffness over the full range of bending moments, is also apparent. Considering the fact that different axial forces occur in different joints, one \(M-\theta\) curve cannot represent the rotational behavior of all the joints in a quasi-rectangular tunnel lining. Hence a surface function of the joint's rotational behavior is needed. This function should include the axial force and bending moment as variables and cover different axial force and bending moment combinations. From the verified numerical model and for the joint geometry of the quasi-rectangular shield tunnel, two surface functions for positive and negative bending moments are derived. These functions are used as an input in the full-ring structural calculations. A cubic polynomial function of bending moment \(M\) (unit: kNm) and axial force \(N\) (unit: kN) according to Formula (4), can fit the rotations well. The polynomial coefficients corresponding to the best-fitting results are listed in Table 1.  
\[\theta(N,M)=aM^{3}+bM^{2}N+cM\cdot N^{2}+dM^{2}+eM\cdot N+f\cdot M. \tag{4}\]  
The authors are aware that these values are related to the specific geometry of the quasi-rectangular shield tunnel under consideration but are confident that these can be integrated in a lining joint database \begin{table}
\begin{tabular}{c c c} \hline
**Coefficient** & **Positive** & **Negative** \\ \hline a & \(9.33\times 10^{-11}\) & \(6.58\times 10^{-10}\) \\ b & \(-1.18\times 10^{-11}\) & \(-2.03\times 10^{-10}\) \\ c & \(8.34\times 10^{-12}\) & \(2.40\times 10^{-11}\) \\ d & \(1.30\times 10^{-7}\) & \(9.17\times 10^{-8}\) \\ e & \(-4.80\times 10^{-8}\) & \(-3.29\times 10^{-8}\) \\ f & \(3.79\times 10^{-5}\) & \(2.36\times 10^{-5}\) \\ SSE & \(5.116\times 10^{-6}\) & \(1.092\times 10^{-4}\) \\ Adjusted R-square & 0.999 & 0.998 \\ \hline \end{tabular}
\end{table} 
Table 1: List of coefficients and goodness of fitting. Figure 8: Calculated results of bending moment resistance experiments: (a) positive moment; (b) negative moment. in the future.