## 4 Theoretical Solution and Numerical Validation of Mechanical Parameters
### Parameter Solving and Validation
#### 4.3.1 Neutral Axis Angle  
In the longitudinal nonlinear equivalent continuous model, mode 3 gives an expression for the transcendental equation satisfied by the neutral axis angle \(\varphi\), as in Equation (46). When the segment is subjected only to bending moments, the axial force is 0. The \(n_{2}\) in Equation is 0. Equation (46) can be transformed into Equation (53). \[(4 cos\varphi+2\pi sin\varphi+4\varphi sin\varphi)m_{2}u+4\pi sin\varphi m_{2}=0 \tag{53}\]  
Simplify to obtain:  
\[cot\varphi+\varphi=\pi\bigg{(}\frac{1}{u}+\frac{1}{2}\bigg{)} \tag{54}\]  
At this point, \(1/u=-(4\text{cos}\varphi+2\pi\text{sin}\varphi+4\varphi\text{sin}\varphi)/4 \pi\text{sin}\varphi=-(\text{cot}\varphi/\pi+1/2+\varphi/\pi)\). Comparing with Equation (3) for calculating the neutral axis angle in the longitudinal equivalent continuous model, it can be seen that the neutral axis angle \(\varphi\) is constant, and \(1/u\) can be solved as \(k_{r}l_{s}/E_{c}t\). This implies that the longitudinal equivalent continuous model represents a specific case within the longitudinal nonlinear equivalent continuous model, specifically when the axial force is zero. At this moment, the neutral axis angle remains constant and does not vary with changes in the bending moment. When the axial force is nonzero, combining Equations (16) and (17) helps determine the longitudinal axial force, bending moment, and other tunnel parameters. The corresponding neutral axis position angle \(\varphi\) is also determined. However, this is a complex equation containing several trigonometric functions, and it is difficult to obtain the solutions of \(\varphi\) and \(N\), \(M\). Moreover, the expressions of the mechanical property-related parameters such as equivalent bending stiffness, equivalent tensile stiffness, and opening are also related to \(\varphi\) in mode 3. During the iterative calculation, it was found that the results of Equation (46) were often difficult to converge, and the calculation was large enough to analyze the results of multiple \(M\) and \(N\) combinations. Considering that \(\varphi\) is both the main focus of the study and an important intermediate variable in the calculation of other parameters related to mechanical properties, and the iterative method is difficult to meet the demand, it is crucial to find a simple and convenient solution for \(\varphi\). Bringing Equations (16) and (17) into (46) gives:  
\[e=\frac{N}{M}=-\frac{(4 cos\varphi+2\pi sin\varphi+4 psin\varphi)u+4\pi sin\varphi}{ [(sin2\varphi+2\varphi+\pi)u+2\pi]r} \tag{55}\]  
When the axial force is not 0, there exists a relationship between \(e=\frac{N}{M}\), \(u\), \(\varphi\) and, \(r\) as in Equation (55). At the critical point of the change from mode 2 to mode 3, the segmental ring is in overall contact with a tendency to separate in the upper part, i.e., the neutral axis is about to appear, at which point \(\varphi=-\frac{\pi}{2}\), which carries over into Equation (55) and has:  
\[e_{2}=-\frac{(0-2\pi+2\pi)u-4\pi}{[(0-\pi+\pi)u+2\pi]r}=\frac{2}{r} \tag{56}\]  
Similarly, at the critical point of the change from mode 3 to mode 1, where most of the segmental ring is pulled apart and the last bit of contact area in the lower part is about to disappear, there will be no neutral axis, and by definition, at this point \(\varphi=\frac{\pi}{2}\), which carries over into Equation (56) has:  
\[e_{1}=-\frac{(0+2\pi+2\pi)u+4\pi}{[(0+\pi+\pi)u+2\pi]r}=\frac{4\pi u+4\pi}{[2 \pi u+2\pi]r}=-\frac{2}{r} \tag{57}\]  
In Mode 3, in addition to the \(e_{0}\) point, there is a special point: during the change, when the neutral axis moves to be just at the center of the segmental ring, at which time \(\varphi=0\). \[e_{\varphi 0}=-\frac{(4+0+0)u+0}{[(0+0+\pi)u+2\pi]r}=-\frac{4u}{(2+u)\pi r} \tag{58}\]  
Equation (55) is the transcendental equation. Combined with the delineation of each mode and the computational analysis in this section, it is initially believed that \(\varphi\) and axial bending moment ratio \(e\) have a regular correspondence.