## 4 Theoretical Solution and Numerical Validation of Mechanical Parameters
### Parameter Solving and Validation
#### 4.3.2 Longitudinal Equivalent Bending Stiffness  
The longitudinal equivalent bending stiffness equations for mode 1 (Equation (19)) and mode 2 (Equation (30)) are relatively simple, and the force states are relatively clear. In mode 3, \(\varphi\) has been derived. When \(N=0\), \(e=0\), and Equation (49) is deformed to:  
\[EI_{0}=\frac{2(1+sin\varphi)r^{3}E_{c}t}{A_{4}^{\prime}} \tag{59}\]  
Bringing Equation (45) into Equation (59) yields:  
\[EI_{3}=\frac{2(1+sin\varphi)r^{3}E_{c}t\pi(sin2\varphi-2 cos\varphi)}{2\pi sin \varphi-4 psin\varphi-4 cos\varphi} \tag{60}\]  
where the numerator can be written as \(-4\pi r^{3}E_{c}tcos^{3}\varphi\), and the denominator can be written as \(4\big{[}-cos\varphi+\big{(}\frac{\pi}{2}-\varphi\big{)}sin\varphi\big{]}\). Longitudinal equivalent stiffness:  
\[EI_{\theta eq}=E_{c}I_{s}\frac{cos^{3}\varphi}{cos\varphi+\big{(}\frac{\pi}{ 2}+\varphi\big{)}sin\varphi} \tag{61}\]  
\[I_{S}=\frac{\pi\big{(}D^{4}-d^{4}\big{)}}{64}=\frac{\pi\big{(}8r^{2}+2t^{2} \big{)}(8 rt)}{64}\approx\pi r^{3}t \tag{62}\]  
In the design of a shield tunnel, the thickness \(t\) of the segment is generally about \(10\%\) of the radius of the tunnel, and \(8r^{2}\) is about \(400\) times \(2t^{2}\), which is negligible. Therefore, \(\pi r^{3}t\) can be approximated as the moment of inertia of the segmental ring, multiplied by \(E_{c}\) which is the equivalent bending stiffness. Then Equation (61) morphs to:  
\[EI_{\theta eq}=E_{c}\pi r^{3}t\frac{cos^{3}\varphi}{cos\varphi+\big{(}\frac{ \pi}{2}+\varphi\big{)}sin\varphi} \tag{63}\]  
It can be considered that the LECM is a special case of the LNECM in terms of both the neutral axis angle and the value of the equivalent bending stiffness, i.e., a special case when the axial force is zero. From Section 3, the equations of equivalent bending stiffness in the 3 deformation modes contain the term \(\pi r^{3}E_{c}t\), which is also the bending stiffness when the shield tunnel is completely under pressure. Furthermore, the problem focused on in this paper can be regarded as the coefficient problem of \(\pi r^{3}E_{c}t\). Neglecting the error, the results of the derivation of the present model and the moment of inertia are numerically and formally unified. When the axial force is not 0, the \(e-EI\) curve is plotted in Figure 10 based on the \(\varphi-e\) curve according to Equation (49). The longitudinal bending stiffness curve shows an overall decreasing trend. For the sake of discussion, it is assumed that the bending moment is constant, and the axial force is varied, resulting in a change in \(e\). When \(e\leq-\frac{2}{r}\), which is mode 2, the segmental ring is pressurized in longitudinal direction and the longitudinal bending stiffness is constant at \(\pi r^{3}E_{c}t\). As the longitudinal axial pressure decreases, when \(-\frac{2}{r}\leq e\leq e_{\phi 0}\), the longitudinal bending stiffness progressively declines, with a corresponding gradual reduction in the curve's slope. In addition, the slope of the curve gradually decreases. As the longitudinal axial pressure decreases to 0 and then the tensile force gradually increases (\(e_{\phi 0}\leq e\leq\frac{2}{r}\)), and this interval is a concave interval. \(-\frac{2}{r}\leq e\leq\frac{2}{r}\) belongs to mode 3, and the equivalent bending stiffness is \(\frac{2(1+sin\rho)^{3}E_{c}t}{A_{4}^{\prime}-A_{5}^{\prime}\sigma^{\prime}}\). \(\varphi=0\) and \(e_{\phi 0}=\frac{-4\mu}{(2+\mu)\pi r}\) at the point of abrupt change between concave and convex intervals. As the tension continues to increase, the stiffness no longer decreases when \(e\geq\frac{2}{r}\) the bending stiffness is \(\frac{\pi r^{3}E_{c}t}{u+1}\). Traditional analyses of shield tunnel longitudinal stiffness frequently refer to the effective rate \(\eta\), which measures the ratio of the longitudinal equivalent bending stiffness from the combined segment and bolts to the uniform continuous longitudinal stiffness under identical conditions.