## 2 Algorithm of the Effective Ratio of the Transverse Bending Rigidity
### Reduction of Bending Rigidity by Reducing the Moment of Inertia  
There are two methods to reduce the moment of inertia; both achieve this goal by reducing the thickness based on a uniform rigidity ring with the same geometric dimensions of the installed segment ring. The first method is to increase the inner diameter and to decrease the outer diameter of the uniform rigidity ring simultaneously (the amount of increase is set to be equal to the amount of decrease). The second method is to increase only the inner diameter of the uniform rigidity ring, causing the outer diameter of the modified uniform rigidity ring and that of the installed segment ring to be the same. (1) The first method of reducing the moment of inertia. For the first method of reducing the moment of inertia, the length of the straight beams spread from the modified uniform rigidity ring and the installed segment ring is set to be equal (i.e., \(L^{\prime}=L\)). According to Equation (9), the moment of inertia for the modified uniform rigidity ring can be expressed as Equation (13), and the effective ratio of the transverse bending rigidity can be expressed as Equation (14):  
\[I^{\prime}=\frac{EL}{\frac{I}{I}+\frac{2E}{k_{1}}+\frac{2E}{k_{2}}+\frac{2E}{k _{3}}} \tag{13}\]  
\[\eta=\frac{I^{\prime}}{I}=\frac{L}{L+2 EI(\frac{1}{k_{1}}+\frac{1}{k_{2}}+\frac {1}{k_{3}})} \tag{14}\]  
It is noted that Equations (12) and (14) are the same. If the number of joints for the installed segment ring is \(n\), then the effective ratio of the transverse bending rigidity for the corresponding modified uniform rigidity ring can be expressed as Equation (15):  
\[\eta=\frac{L}{L+EI\sum_{i=1}^{n}\frac{1}{k_{i}}}=\frac{(D_{1}+D_{2})\pi}{(D_{ 1}+D_{2})\pi+2 EI\sum_{i=1}^{n}\frac{1}{k_{i}}} \tag{15}\]  
where \(n\) is the number of joints for the installed segment ring and \(\sum\limits_{i=1}^{n}\frac{1}{k_{i}}\) is the summation of all the rotation angles of the joint on the installed segment ring under a unit bending moment. If the other parameters in Equation (15) are kept constant, more joints on the installed segment ring and a larger section bending stiffness will lead to a smaller effective ratio of the transverse bending rigidity while a larger diameter of the segment ring and a larger joint bending stiffness correspond to a larger effective ratio of the transverse bending rigidity. To reduce \(I\), the thickness of the uniform rigidity ring is reduced, and the longitudinal width of the uniform rigidity ring is kept unchanged (i.e., \(b^{\prime}=b\)). According to the equation that \(E^{\prime}I^{\prime}=\eta EI\), Equation (16) can be derived, and the thickness of the modified uniform rigidity ring can be calculated by using Equation (17), because \(E^{\prime}=E\):  
\[E^{\prime}\frac{b^{\prime}h^{\prime 3}}{12}=\eta EI\frac{bh^{3}}{12} \tag{16}\]  
\[h^{\prime}=\sqrt[3]{\eta}h \tag{17}\]  
where \(b^{\prime}\) is the longitudinal width of the modified uniform rigidity ring, \(h^{\prime}\) is the thickness of the modified uniform rigidity ring, \(b\) is the longitudinal width of the installed segment ring, \(h\) is the thickness of the segment, and \(\sqrt[3]{\eta}\) is the thickness reduction coefficient for a modified uniform rigidity ring. If the installed segment ring shown in Figure 1 is converted into a corresponding modified uniform rigidity ring, the thickness reduction coefficient can be expressed as Equation (18) based on Equation (14):  
\[\alpha=\sqrt[3]{\eta}=\sqrt[3]{\frac{L}{L+2 EI(\frac{1}{k_{1}}+\frac{1}{k_{2}}+ \frac{1}{k_{3}})}}=\sqrt[3]{\frac{(D_{1}+D_{2})\pi}{(D_{1}+D_{2})\pi+2 EI\sum_{ i=1}^{n}\frac{1}{k_{i}}}} \tag{18}\]  
where \(\alpha\) is the thickness reduction coefficient for the modified uniform rigidity ring converted from the installed segment ring. (2) The second method of reducing the moment of inertia.