## 4 Discussion of the Results
### Effect of the Circumferential Joint
#### 4.2.1 Effect of the Circumferential Shear Stiffness
The bending moment of the middle ring with and without the circumferential shear are illustrated (cyan and black curve in Figure 14b), and the bending moment of the neighboring ring under the condition with circumferential shear is added as well (pink curve in Figure 14b). The bending moments around the joint areas of the interior column and segments T1 and T2 are not shown. Figure 14: Comparison of bending moment distribution: (**a**) a typical bending moment distribution and mean bending moment values with and without circumferential shear; (**b**) bending moment distribution in different rings. Figure 13: Load distribution for calculation case. As shown in Figure 14a, the curves of mean values with and without circumferential shear are in good agreement with each other. This means that the circumferential shear forces have little influence on the overall bending moments in the lining structure model (including all the 3 rings). However, the bending moments on the middle ring appear to have a different distribution, depending on whether the circumferential shear stiffness is considered or not. When comparing the bending moment distribution of the staggered model to that of the model without the circumferential shear in Figure 14b, it is clear that the moment value on the middle ring with circumferential shear stiffness (cyan) is larger than that without circumferential shear stiffness (black). Moreover, the moment on the related section of the neighboring ring (pink) is smaller than that on the middle ring without circumferential shear stiffness (black) and vice versa. It is evident that a transmission of the bending moment occurs because of the existence of the circumferential shear. Comparing the bending moment distributions between the middle and its neighboring rings in the staggered model (cyan and pink), we observe that in the longitudinal joints of the middle ring (JF1 to JF10) with circumferential shear stiffness, the absolute value of the bending moments (cyan) are smaller than those of the neighboring segment bodies (pink), such as JF1, JF2, JF4, JF5, JF6, JF8, JF9, JF10. In the longitudinal joints of the neighboring rings (CJ1 to CJ10), we observe a same situation. This is because the stiffnesses of the longitudinal joints are smaller than those of their neighboring segment bodies. As a result, the bending moments transfer from the weaker area to the stiffer area. When comparing the left half (segments C2, B1, L, F) of the middle ring with circumferential shear (cyan) to its right half ring (segments C1, B3, B2, C3), it is observed that the moment in the right half ring is generally larger than that in the left half ring, especially at the sections of segment F. This is a direct proof that the right half part of the structure is stiffer than the left half. The cantilever beams of segments T1 and T2 have larger spans in the left half than those in right half (as shown in Figure 2a). The distance between JF6 and JF10 is smaller than that between JF1 to JF5. At the same time, segment F is the smallest segment. All these aspects result in a closer joint layout in the left half, which may be the reason why the right half of the structure is stiffer and a larger bending moment distribution is found in the right half. It also explains that the left convergence deformation (key deformation point of D3) is larger than the right one (key deformation point of D1) in the full-scale ring experiments. The structure is divided into two parts, left half and right half, by the interior column, and the different stiffnesses in these two parts are a unique characteristic of quasi-rectangular tunnels. It results in a non-symmetric distribution of bending moments in the same ring when the pattern of staggered assembly is adopted.