## 5 Conclusions  
The stiffnesses of the joints in a shield tunnel are related to the level of the joints' internal forces. For joints in quasi-rectangular tunnels, the joints' rotational and shear deformations are not only affected by bending moments and shear forces in the joints but also by axial forces. Additionally, the longitudinal joints of quasi-rectangular tunnels are subjected to a wide range of both bending moments and shear forces. Considering this, a proper numerical model is required for the further study and structural improvement of this new type of tunnel. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Target** & **D3 (Max of Deformation)** & **JM2 (Max of Positive Moment)** & **JM8 (Max1 of Negative Moment)** & **JM10 (Max2 of Negative Moment)** \\ \hline Rotation & \(\downarrow\downarrow\) & \(\uparrow\) & \(\uparrow\) & \(\downarrow\downarrow\) \\ Shear & \(\downarrow\downarrow\) & \(\downarrow\) & \(\downarrow\downarrow\) & \(\uparrow\uparrow\) \\ Rigid zone & \(\downarrow\) & \(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ Column & \(\downarrow\) & \(\uparrow\uparrow\) & \(\downarrow\downarrow\) & \(\uparrow\uparrow\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Summary of the tendency and sensitive level of different parameters. (1) In this paper an iterative incremental method is presented to simulate the stiffness changes caused by the joint's moment-axial force and shear-axial force interaction behavior. The applicability of the numerical model is verified by comparing the bending moments and deformations in the lining to the results of unique indoor full-scale ring experiments;  
(2) The iterative incremental method provides acceptable results for different load levels, while a constant stiffness based analysis does not;  
(3) Investigation of the effect of the circumferential shear stiffness reveals that the half part of the tunnel lining without the smallest segment (F in Figure 2) is stiffer than the other half. Consequently, in tunnels with a staggered assembly, the bending moments in this half part increase a lot, almost doubling the bending moment in the F segment, due to the bending moment transmission caused by the uneven deformation in these two halves;  
(4) The range of the coefficients of bending moment transmission for the joints in a quasi-rectangular tunnel is similar to that of circular shield tunnels, but between each other, they deviate significantly. Consequently, they should be adopted properly, based on the location of the joints, and especially relatively large transmission coefficient values up to 0.3 are needed for the joints at the edges of the smallest segment (JF9 and JF10 in Figure 2);  
(5) A parametric study reveals that not only the joint's rotational stiffness but also the width of interior column and the joint's shear stiffness significantly affect the structural response. Optimizing the structural performance by adjusting only one parameter is difficult. A combination of the choices for the parameters is significant for practical design of quasi-rectangular tunnel linings. (6) For circular tunnels, only one geometric shape has to be considered. However, for quasi-rectangular tunnels, many geometric shapes are possible. Although the analysis presented in this paper is based on the particular geometry of a metro tunnel in Ningbo, the conclusions could be helpful for other special-section shield tunnels as well, especially where the influence of the nonlinear behavior of the joints is non-negligible. Conceptualization, methodology, writing--original draft preparation, W.Z. ; formal analysis, validation, W.Z. and W.D.C. ; software, W.Z., W.D.C. and X.L. ; writing--review and editing, supervision, W.D.C. and L.T. ; project administration, funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (grant number 51578409) and the China Scholarship Council. The financial support is gratefully acknowledged.