## 2 Mechanical Model Establishment and Equations Derivation
### Longitudinal Deformation of Shield Tunnel Adjacent to Foundation Pit  
#### 2.3.1 Collaborative Deformation Mode of Shield Tunnel with Rotation and Dislocation  
This paper cites the cooperative deformation model that can comprehensively consider the two deformation effects of segment ring rotation and dislocation of the shield tunnel [13]. As shown in Figure 5, relative rotation and relative dislocation will occur between adjacent segments. The longitudinal deformation of the shield tunnel is considered to be formed by combining shear dislocation and rigid body rotation between adjacent segments. As shown in Figure 6, the total relative displacement between adjacent ring segments numbered \(m\) and \(m+1\) is \(\delta_{\chi}\), the relative horizontal displacement induced by the rigid rotation is \(\delta_{\chi 1}\), and the relative horizontal displacement caused by the dislocation is \(\delta_{\chi 2}\). Then, it satisfies the equation \(\delta_{\chi}=\delta_{\chi 1}+\delta_{\chi 2}\). Set \(\delta_{\chi 1}=j_{\chi}\delta_{\chi}\), where \(j_{\chi}\) is the proportionality coefficient for the rotation effects of the segment rings. When \(j_{\chi}=0\), there is no relative rotation between adjacent segments. It is consistent with the deformation model of dislocation proposed by [PERSON] et al. [21]. When \(j_{\chi}=1\), the horizontal displacement is completely caused by the rotation of adjacent segments, which is the deformation mode of complete rotation. Figure 5: The schematic diagram for the collaborative deformation mode of a shield tunnel [13]. Assuming that the adjacent shield tunnel meets the condition of deformation coordination with the surrounding soil, the deformation of the shield tunnel is the same as the soil deformation at the corresponding position:  
\[u(l)=u_{\rm t}(l) \tag{11}\]  
where \(l\) is the calculated position of the tunnel axis, take the center of foundation pit excavation as the zero point, and it is positive along the \(Y\)-axis; \(u(l)\) is the horizontal deformation of adjacent shield tunnel, and \(u_{\rm t}(l)\) is the horizontal displacement of the soil at the tunnel axis. The difference between the tunnel horizontal displacement values of adjacent segments is the relative horizontal displacement, which is:  
\[\delta_{\rm x}=u((m+1)D_{\rm t})-u(mD_{\rm t})=u_{\rm t}((m+1)D_{\rm t})-u_{\rm t }(mD_{\rm t}) \tag{12}\]  
where the serial numbers of the adjacent segment rings are \(m\) and \(m+1\), respectively, and \(D_{\rm t}\) is the width of the segment ring. It can be assumed that \(\sin\left(\theta_{\rm x}\right)=\theta\) and \(\cos\left(\theta_{\rm x}\right)=1\) when the rotation angle \(\theta_{\rm x}\) between the segment ring is small. Therefore, the calculation equation for the horizontal displacement caused by the rotation of the rigid body between the lining rings is:  
\[\delta_{\rm x1}=D_{\rm t}\theta_{\rm x} \tag{13}\]  
Substituting \(\delta_{\rm x1}=j_{\rm x}\delta_{\rm x}\) into Equation (13):  
\[\theta_{\rm x}=\frac{j_{\rm x}\delta_{\rm x}}{D_{\rm t}} \tag{14}\]  
The shear force between the segment rings can be obtained as follows:  
\[Q_{\rm x}=k_{\rm s1}\delta_{\rm x2}=(1-j_{\rm x})k_{\rm s1}\delta_{\rm x} \tag{15}\]  
The maximum tension between rings is:  
\[F_{\rm tx}=k_{\rm t}\theta_{\rm x}D \tag{16}\]  
The strata resistance is:  
\[F_{\rm tx}=kDa_{\rm t}(l)=kDa(l) \tag{17}\]  
Figure 6: Calculation model for the collaborative deformation of rotation and dislocation between shield tunnel segment rings [13]. In Equations (15) and (16): \(k_{\rm sl}\) and \(k_{\rm t}\) are the inter-ring shear stiffness and inter-ring tensile stiffness of the tunnel, which can be calculated by [PERSON] et al. [22]. In Equation (17), [PERSON]'s formula [23] is used to calculate \(k\), which is the foundation reaction coefficient.