## 2 Algorithm of the Effective Ratio of the Transverse Bending Rigidity  
### Equivalent Equation of Transverse Bending Rigidity  
Because the bending rigidity of each section of the modified uniform rigidity ring is the same, reducing the transverse bending rigidity of a uniform rigidity ring with the same geometric dimensions of the installed segment ring can be accomplished by evenly allocating the weakness caused by longitudinal joints of the installed segment ring to all of the sections of the modified uniform rigidity ring. Thus, the equivalence of the transverse bending performance between the modified uniform rigidity ring and the installed segment ring is often a necessity. To calculate the effective ratio of the transverse bending rigidity based on the longitudinal joint bending stiffness of the installed segment ring, several assumptions are made as follows: (1) the installed segment ring and modified uniform rigidity ring are spread into straight beams by cutting them apart at the bottom position (at the position of 180\({}^{\circ}\)), as shown in Figure 1; (2) the condition that the end sections of both straight beams have the same relative rotation angle under the same pure bending moment is regarded as the equivalent condition of the transverse bending rigidity for both rings; and (3) when the equivalent condition of the transverse bending rigidity is satisfied, the ratio that the bending rigidity of the transverse section of a modified uniform rigidity ring to that of the corresponding installed segment ring is termed the effective ratio of the transverse bending rigidity \(\eta\). Spread straight beams from the installed segment ring and the modified uniform rigidity ring and their corresponding geometric dimensions are shown in Figure 1. The installed segment ring shown in Figure 1 is the tunnel lining of a shield-driven tunnel with non-staggered installation, which is widely used in Shanghai's single-line metroThe mean value of the outer and inner circumferences of the installed segment ring is taken as the length of its spread straight beam. It can be calculated using Equation (1):  
\[L=\frac{(D_{1}+D_{2})}{2}\pi \tag{1}\]  
where \(L\) is the length of the straight beam spread from the installed segment ring, \(D_{1}\) is the outer diameter of the installed segment ring, and \(D_{2}\) is the inner diameter of the installed segment ring. According to [PERSON]'s law, the elastic flexural strain of a straight beam is given by Equation (2), when subjected to the pure bending moment \(M\):  
\[\kappa=\frac{M}{EI} \tag{2}\]  
where \(\kappa\) is the flexural strain, which is defined as the rotation angle developed per unit length; \(M\) is the pure bending moment acting on the ends of the spread straight beam; \(EI\) is the bending rigidity of the transverse section of the segment; \(E\) is the elastic modulus of the segment; and \(I\) is the moment of inertia for the transverse section of the segment. Without considering the rotation angles of all joints, the relative rotation angle for the end sections of the straight beam spread from the installed segment ring may be expressed as Equation (3), when subjected to the pure bending moment \(M\):  
\[\theta_{1}=\frac{LM}{EI} \tag{3}\]  
Figure 1: Straight beams spread from the installed segment ring and the modified uniform rigidity ring: (**a**) Installed segment ring; (**b**) modified uniform rigidity ring; (**c**) straight beam spread from the installed segment ring; (**d**) straight beam spread from the modified uniform rigidity ring. where \(\theta_{1}\) is the relative rotation angle for the end sections of the straight beam spread from the installed segment ring, which is caused by the bend of the beam. However, the bending rigidity of the spread straight beam is weakened by the existence of segment longitudinal joints.