## 4 Conclusions  
In this study, an algorithm for calculating the effective ratio of the transverse bending rigidity was established, and the responses of the modified uniform rigidity ring model under loading obtained from this algorithm were compared to those obtained with the existing beam-spring model. The following conclusions are drawn:  
1. Based on the assumption of the equivalence for transverse bending rigidity between the installed segment ring and the modified uniform rigidity ring, an algorithm to determine the effective ratio of the transverse bending rigidity was obtained. Compared with the traditional methods for determining the effective ratio, the algorithm proposed in this article is simpler and more elegant conceptually. Moreover, the algorithm can express the correlation between the effective ratio and its influencing factors, such as the longitudinal joint bending stiffness, diameter of the segment ring, thickness of the segment, and number of joints. From the process of deriving the algorithm, it is noted that the essence of converting an installed segment ring into a modified uniform rigidity ring is to evenly allocate the weakness of the bending rigidity caused by longitudinal joints to all the sections of the modified uniform rigidity ring.
2. Through the comparison of the values for the effective ratio of the transverse bending rigidity, which are calculated with different methods, it is noted that the effective ratios calculated using the algorithm proposed in this article are similar to those calculated by the method proposed by [PERSON] but are visibly different from those calculated by the method proposed by [PERSON]. Because the convergence deformations of a modified uniform rigidity ring are similar to those of an installed segment ring under the same loading condition, it is verified that the algorithm for determining the effective ratio of the transverse bending rigidity proposed in this article is valid.
3. The analysis shows that the distribution modes for the bending rigidity for the two models, the beam-spring model and the modified uniform rigidity ring model, are different, which leads to a significant difference in the distribution of bending moments. Therefore, the modified uniform rigidity ring model is not applicable for the bending moment design of a tunnel structure.
4. Under the same loading condition, the axial forces, horizontal convergence deformations, and vertical convergence deformations for both models (the beam-spring model and the modified uniform rigidity ring model) are similar. Thus, the modified uniform rigidity ring model is applicable for the analysis of the structure convergence deformation and the interaction between the tunnel structure and the ground during operation.  
**Funding:** This study was supported by the National Natural Science Foundation of China, grant number 52078213 and U1934208, the Natural Science Foundation of Jiangxi Province, grant number 20192 ACBL21003.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** All data are available from the author.  
**Acknowledgments:** The authors thank all reviewers for their great help in this article.  
**Conflicts of Interest:** The authors declare no conflict of interest.