## 4 Theoretical Derivation
### Construction of Pipeline Deflection Differential Equation  
As shown in Figure 9, the left and right ends of any joint position are intercepted with dx length for force analysis. It is assumed that they act on the line axis position and are evenly distributed along the y _direction_. Figure 8: Schematic diagram of pipeline deflection caused by shield tunnel excavation. Figure 7: Z-displacement cloud image of the model after tunnel excavation. Firstly, the vertical force balance of the microelement can be obtained using  
\[(Q-\mathrm{d}Q)-q(x)b\mathrm{d}x-Q=0 \tag{1}\]  
where \(\mathrm{d}x\) is the element width (m), \(\mathrm{q}\) is the cross-section shear force (N), is the increment of cross-section shear force (N) along the length of \(\mathrm{d}x\) of the pipeline, \(\mathrm{q}(x)\) is the foundation reaction force _acting_ on the pipeline (N/m\({}^{2}\)), and \(\mathrm{b}\) is the diameter of the pipeline (m). Equation (1) can be obtained by simplification  
\[\frac{\mathrm{d}Q}{\mathrm{d}x}=-q(x)b \tag{2}\]  
Secondly, find the bending moment of the section of the left end of the element, which can be obtained by balancing the bending moment  
\[(M-\mathrm{d}M)+Q\mathrm{d}x+q(x)b\frac{(\mathrm{d}x)^{2}}{2}-M=0 \tag{3}\]  
where \(M\) is the cross-section bending moment (N\(\cdot\)m), and \(\mathrm{d}M\) is the increment of the cross-section bending moment (N\(\cdot\)m) along the \(\mathrm{d}x\) length of the pipeline. Equation (3) simplifies and omits the second-order trace, which can be obtained by  
\[\frac{\mathrm{d}M}{\mathrm{d}x}=Q \tag{4}\]  
Equation (2) is obtained in conjunction with (4):  
\[q(x)=-\frac{1}{b}\frac{\mathrm{d}^{2}M}{\mathrm{d}x^{2}} \tag{5}\]  
Suppose the force of the pipeline on the surrounding formation is \(p(x)\), and its interaction with \(q(x)\) is the interaction force, then  
\[p(x)=-q(x) \tag{6}\]  
Assuming that the bending stiffness of any cross-section of the pipeline along the direction is infinite, the pipeline will produce an overall translation along the \(y\) direction when it interacts with the soil. Combined with the assumption that the pipeline is always in contact with the surrounding strata, the vertical displacement \(\mathrm{u}(x)\) of the soil on the contact surface of any cross-section of the pipeline will not change along the direction. Therefore, when the pipeline interacts with the soil, the relationship between the pipeline force \(p(x)\) and the ground displacement \(\mathrm{u}(x)\) caused by the pipeline can be expressed as Equation (7). \[p(x)=ku(x)-G_{\mathrm{p}}\frac{\mathrm{d}^{2}u(x)}{\mathrm{d}x^{2}} \tag{7}\]  
where \(k\) is the foundation reaction coefficient (N/m\({}^{3}\)) and \(Gp\) is the foundation shear stiffness (N/m). Figure 9: Element force analysis: micro force analysis of the left end of the joint. Simultaneous Equation as (5)-(7) can be obtained  
Figure 7:0  
Similarly, for the right end of the joint and the standard pipe section, Equation (8) can be deduced. The formation displacement around the pipeline originates from two aspects, one is the formation settlements (\(x\)) caused by the formation loss in tunnel excavation, and the other is the foundation displacement u(x) caused by the interaction between the pipeline and the soil. Assuming that the axial deflection along the pipeline is \(w(x)\), the Equation (9) can be obtained based on the assumption that the pipeline is always in contact with the surrounding strata. Figure 7:1  
In the Equation, \(s(x)\) and \(u(x)\) are taken from the position of the pipeline axis. By combining Equation as (8) and (9), we can obtain  
Figure 7:2  
The deflection w(x) satisfies the Equation (11) of the extramural differential equation. Figure 7:3  
where E is the elastic modulus (N/m\({}^{2}\)) of the pipeline and I is the moment of inertia of the pipeline cross-section.