## 2 Calculation Method for Surface Settlement
### Surface Settlement Induced by Pipe Roof Construction (\(u_{2}\))  
First, the surface settlement caused by the jacking of a single pipe is calculated, followed by the determination of surface settlement induced by the superimposed effect of adjacent pipe jacking. Surface settlement induced by single-pipe construction  
The mechanical calculation model for a single pipe constructed using a micro shield machine is illustrated in Figure 3.  
The surface settlement resulting from single-pipe construction is determined using the Peck formula:  
\[s(x)=\frac{AV}{i\sqrt{2\pi}}e^{-\frac{\alpha^{2}}{2\pi^{2}}} \tag{11}\]  
where \(s(x)\) is the surface settlement calculated by the Peck formula; V is the ground loss ratio; \(x\) is the distance from the tunnel centerline; i is the distance from the symmetrical center of the settlement trough to the inflection point; A is the excavation area, A = \(\pi r^{2}\). \[V=\frac{4 wr-w^{2}}{4r^{2}} \tag{12}\]  
where w is the maximum settlement at the pipe crown caused by single-pipe construction; r is the radius of the pipe roof. Since the vault settlement w is much smaller than r, V can be simplified as follows:  
\[V\approx\frac{w}{r} \tag{13}\]  
\[i=\frac{H}{\sqrt{2\pi\tan\left(45^{o}-\frac{\varphi}{2}\right)}} \tag{14}\]  
where H is the depth of the pipe roof; c is the cohesion; \(\varphi\) is the internal friction angle. The Levy-Mises theory describes the plastic flow laws of ideal elastic-plastic materials:  
\[de_{ij}^{p}=d\lambda\cdot\frac{\partial f}{\partial\sigma_{ij}} \tag{15}\]  
where \(de_{ij}^{p}\) is the increment of plastic strain, \(d\lambda\) is the plastic multiplier, \(f\) is the yield function (here, the Hoek-Brown criterion is used). The Hoek-Brown criterion describes rock mass strength:  
\[\sigma_{1}=\sigma_{3}+\sigma_{c}\left(m\frac{\sigma_{3}}{\sigma_{c}}+s\right) ^{a} \tag{16}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the principal stresses, \(\sigma_{c}\) is the uniaxial compressive strength of the rock, \(m\), \(s\), and \(a\) are parameters related to the Geological Strength Index (GSI). Assuming that pipe jacking is an axisymmetric problem, the axial stress and radial stress satisfy the equilibrium equations:  
\[\frac{d\sigma_{r}}{dr}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \tag{17}\]  
Figure 3: Mechanical calculation model for a single pipe. By substituting the Hoek-Brown criterion into the equilibrium equations and introducing the [PERSON] flow rule, we obtain:  
\[\frac{d\sigma_{r}}{dr}=\frac{\beta}{r}\bigg{(}\frac{\sigma_{r}-p_{0}}{\alpha} \bigg{)}^{1/\alpha} \tag{18}\]  
where \(\alpha\) and \(\beta\) are coefficients related to the Hoek-Brown parameters and material properties. By integrating using the separation of variables method, the expression for radial displacement ww is obtained:  
\[s(x)=\frac{AV}{i\sqrt{2\pi}}e^{-\frac{\alpha^{2}}{2\pi^{2}}} \tag{11}\]0  
where p is the grouting pressure; G is the shear modulus of the soil; p\({}_{0}\) is the earth pressure, p\({}_{0}\) = \(\gamma H\); and since the burial depth is small. Substituting Equation (19) into Equation (13) and further into Equation (11) yields the maximum surface settlement value at x = 0:  
\[s(x)=\frac{AV}{i\sqrt{2\pi}}e^{-\frac{\alpha^{2}}{2\pi^{2}}} \tag{11}\]1  
(2) Surface settlement caused by the construction of multiple pipes  
The surface settlement caused by the superposition of multiple pipes is expressed as:  
\[s(x)=\frac{AV}{i\sqrt{2\pi}}e^{-\frac{\alpha^{2}}{2\pi^{2}}} \tag{11}\]2  
where \(f(x)\) is the surface settlement correction coefficient, and its calculation formula is as follows:  
\[s(x)=\frac{AV}{i\sqrt{2\pi}}e^{-\frac{\alpha^{2}}{2\pi^{2}}} \tag{11}\]3  
where x is the horizontal distance between the centerlines of two jacked pipes; M is the maximum surface settlement correction coefficient; and \(\lambda\) is generally assigned a value of either 2.5 or 3.