## 4 Theoretical Solution and Numerical Validation of Mechanical Parameters
### Parameter Solving and Validation
#### 4.3.3 Longitudinal Equivalent Tensile and Compressive Stiffness  
In mode 1, the segmental ring is in a state of tension and bending moment, satisfying \(\frac{N}{M}\geq\frac{2}{r}\), the upper and lower parts of the segmental ring are separated due to larger tension. The expression of longitudinal equivalent tensile stiffness is shown in Equation (24). In mode 2, the segmental ring is in the state of pressure and bending moment, satisfying \(\frac{N}{M}\leq-\frac{2}{r}\), the upper and lower parts of the segmental ring are in contact due to the larger pressure, and the expression of longitudinal equivalent compressive stiffness is shown in Equation (34). In mode 3, the axial force can be either tensile or pressure, or 0, but is small relative to the bending moment, satisfying \(-\frac{2}{r}\leq\frac{N}{M}\leq\frac{2}{r}\), the upper part of the segmental ring is separated, and the lower part is in contact, and the longitudinal equivalent tensile and compressive stiffness expression is shown in Equation (52). The tensile and compressive stiffness are localized as:  
\[EA=\frac{Nl_{s}}{\delta} \tag{64}\]  
When \(N=0\), \(EA=0\). The parameters \(u\) and \(r\) of the shield tunnel have been determined, and \(A_{1}^{\prime}\sim A_{4}^{\prime}\) are all functions of the neutral axis angle \(\varphi\), then \(EA_{3}\) can also be interpreted as a function of \(e\) and \(\varphi\), and the curves with \(e\) on the horizontal axis and \(EA_{3}\) on the vertical axis are plotted as in Figure 12.  
Figure 11: \(\eta-e\) curve. Figure 12: Comparison of analytic and numerical solutions for \(EA-e\) curve. Numerical simulations in Figure 12 generally align with the theoretical findings. The overall trend of the tensile and compressive stiffness curves is an inverse proportional function, and the transverse coordinate of the middle mutation is at \(\varphi=0\), i.e., \(e_{\varphi 0}=\frac{-4\mu}{(2+\mu)\pi t}\approx-0.3036\). For discussion purposes, assume constant magnitude and direction for the bending moment, with changes in axial force altering \(e\). When \(e\leq-\frac{2}{r}\), which is mode 2, the segmental ring is under pressure and the longitudinal compressive stiffness is constant and constant at \(2\pi rE_{c}t\). As the longitudinal pressure decreases, the compressive stiffness continues to increase when \(-\frac{2}{r}\leq e\leq e_{\varphi 0}\). When \(e\) is close to \(e_{\varphi 0}\), the angle of the neutral axis of the segmental ring is close to \(0^{\circ}\). At this time, there is an axial force but the displacement of the segment center very small and negative, so the value of the compressive stiffness increases dramatically due to the definition of stiffness. When \(e\) is slightly greater than \(e_{\varphi 0}\), the displacement of the segment center is very small but becomes positive. During this time, despite the segment pressure, under the combined action of axial force and bending moment, the displacement of the segment center is positive, leading to a phenomenon known as \"negative stiffness\". The negative stiffness is concentrated in the interval \(-0.3\leq e\leq-0.1\), corresponding to the numerical results of \(-11.8 TN\cdot m\leq EA\leq-0.006 TN\cdot m\). As the pressure decreases to 0 (\(e\) increases to 0), the absolute value of the stiffness decreases because of the linear decrease in pressure, and the stiffness becomes 0 when \(e=0\) (axial force is 0). After that, the longitudinal tensile force occurs, \(0\leq e\leq\frac{2}{r}\), and the tensile stiffness continues to increase as the tensile force increases. When \(-\frac{2}{r}\leq e\leq\frac{2}{r}\), the segmental ring is in mode 3, the equivalent tensile stiffness is \(\frac{4\sigma rE_{c}t}{(1+u)\left(A_{1}^{\prime}r+A_{2}^{\prime}\right)+A_{3} ^{\prime}\sigma-A_{4}^{\prime}}\). When the tension continues to increase, \(e\geq\frac{2}{r}\), the longitudinal stiffness no longer changes, and the tensile stiffness is constant at \(\frac{2\pi rE_{c}t}{u+1}\).