# Prediction of Ground Deformation during Pipe-Jacking Considering Multiple Factors
## 2 Review of Existing Methods  
[PERSON] [67] has proposed equations for calculating the ground deformation induced by tunneling. In [PERSON]'s solution, the ground was considered as an elastic half-plane with a circular cavity. Figure 1 shows the illustration of a case with uniform stress applied at the tunnel boundary. \(\sigma_{\mathrm{p}}\) is the stress at the interface of the plastic and elastic zones, \(R_{\mathrm{p}}\) is the external radius of the plastic zone, and \(h\) is the depth of tunnel center. After dividing the deformation of tunnel boundary into a uniform radial displacement and a downward displacement, the equations can be deduced by complex variables, as listed as follows:  
\[u_{\mathrm{xA}} =\mathrm{Re}\bigg{(}\frac{1+\mu}{E}\Big{(}(3-4\upsilon)\phi(Z)-Z \overline{\phi^{\prime}(Z)}-\overline{\psi(Z)}\Big{)}\bigg{)} \tag{1}\] \[u_{\mathrm{yA}} =\mathrm{Im}\bigg{(}\frac{1+\mu}{E}\Big{(}(3-4\upsilon)\phi(Z)-Z \overline{\phi^{\prime}(Z)}-\overline{\psi(Z)}\Big{)}\bigg{)} \tag{2}\]  
where \(\mathrm{Re}\) and \(\mathrm{Im}\) means taking the real and imaginary parts respectively; \(u_{xA}\) is the displacement of point \(A\) at \(x\) direction; \(u_{yA}\) is the displacement of point \(A\) at \(y\) direction; \(u\) is [PERSON]'s ratio; \(E\) is [PERSON]'s modulus; \(Z=x+iy\); \(\varphi(Z)\) and \(\Psi(Z)\) are analytic functions, and can be determined from following equations:  
\[\phi(Z)=M_{d}\Bigg{(}-2i\Big{(}1+\xi^{2}\Big{)}+2i\frac{Z\big{(}1+\xi^{2} \big{)}+ih\big{(}1-\xi^{2}\big{)}}{Z(1+\xi^{2})-ih(1-\xi^{2})}+2i\xi^{2}\frac{ Z\big{(}1+\xi^{2}\big{)}-ih\big{(}1-\xi^{2}\big{)}}{Z(1+\xi^{2})+ih(1-\xi^{2})} \Bigg{)} \tag{3}\]  
\[ \begin{split}\psi(Z)=M_{d}\Bigg{(}-3i\big{(}1+\xi^{2}\big{)}+2i \xi^{2}\frac{Z\big{(}1+\xi^{2}\big{)}+ih\big{(}1-\xi^{2}\big{)}}{Z(1+\xi^{2})- ih\big{(}1-\xi^{2}\big{)}}+i\bigg{(}\frac{Z\big{(}1+\xi^{2}\big{)}+ih \big{(}1-\xi^{2}\big{)}}{Z(1+\xi^{2})-ih\big{(}1-\xi^{2}\big{)}}\bigg{)}^{2} \\ +2i\frac{Z\big{(}1+\xi^{2}\big{)}-ih\big{(}1-\xi^{2}\big{)}}{Z(1+ \xi^{2})+ih\big{(}1-\xi^{2}\big{)}}+i\xi^{2}_{0}\bigg{(}\frac{Z\big{(}1+\xi^{2} \big{)}-ih\big{(}1-\xi^{2}\big{)}}{Z(1+\xi^{2})+ih\big{(}1-\xi^{2}\big{)}} \bigg{)}^{2}\Bigg{)}\end{split}  \tag{4}\]  
\[M_{d}=-\frac{\xi^{2}\sigma_{\mathrm{p}}h}{(1-\xi^{2})(1-\xi^{4})} \tag{5}\]\[\xi=\frac{h-\sqrt{h^{2}-R_{\rm p}^{2}}}{R_{\rm p}} \tag{6}\]  
For three dimensional elastic scenarios, [PERSON] [69] had proposed solutions for deformation in a homogenous isotropic solid. [PERSON] [61] used [PERSON]'s solution to calculate the ground movement induced by the bulkhead additive thrust and the friction between the jacking machine and surrounding soils at arbitrary point \(B(x,y,z)\). Figure 2 shows the calculation schematic of [PERSON]'s solution. \(x\) is the horizontal distance from the excavation face of the point \(B\) along the jacking direction, \(y\) is the lateral distance from the jacking axis to the point \(B\), \(z\) is the depth of the point \(B\), \(h\) is the depth of jacking machine centerline. Figure 1: Illustration of a case with uniform stress applied at tunnel boundary (recreated based on [PERSON] et al. [11], [PERSON] et al. [68]). Figure 2: Calculation schematic of [PERSON]’s solution. The equations for calculating the ground movement due to the bulkhead additive forces are listed as follows:  
\[ \begin{array}{c}u_{1}=\frac{P}{16\pi G(1-\mu)}\int\limits_{0}^{2\pi}\int\limits _{0}^{R}\left[\frac{3-4\mu}{M_{1}}+\frac{1}{N_{1}}+\frac{x^{2}}{M_{1}^{3}}+ \frac{(3-4\mu)x^{2}}{N_{1}^{3}}+\frac{2x_{1}}{N_{1}^{3}}\bigg{(}1-\frac{3x^{2} }{N_{1}^{2}}\bigg{)}\right.\\ \left.+\frac{4(1-\mu)(1-2\mu)}{N_{1}+2+c_{1}}\Big{(}1-\frac{x^{2}}{N_{1}(N_{1} +2+c_{1})}\Big{)}\right]rdrd\theta\\ \end{array}