# Prediction of Ground Deformation during Pipe-Jacking Considering Multiple Factors
## 2 Review of Existing Methods
 \tag{7}\]  
\[v_{1}=\frac{Px}{16\pi G(1-\mu)}\int\limits_{0}^{2\pi}\int\limits_{0}^{R}\left( y+r\cos\theta\right)\left[\frac{1}{M_{1}^{3}}+\frac{3-4\mu}{N_{1}^{3}}-\frac{6x_{ 1}}{N_{1}^{5}}-\frac{4(1-\mu)(1-2\mu)}{N_{1}(N_{1}+z+c_{1})^{2}}\right]rdrd\theta\\ \end{array} \tag{8}\]  
\[w_{1}=\frac{Px}{16\pi G(1-\mu)}\int\limits_{0}^{2\pi}\int\limits_{0}^{R}\left[ \frac{z-c_{1}}{M_{1}^{3}}+\frac{(3-4\mu)(z-c_{1})}{N_{1}^{3}}-\frac{6x_{1}(z+ c_{1})}{N_{1}^{5}}+\frac{4(1-\mu)(1-2\mu)}{N_{1}(N_{1}+z+c_{1})}\right]rdrd\theta \tag{9}\]  
where, \(u_{1}\) is the movement of point \(B\) in \(x\) axis direction; \(v_{1}\) is the movement of point \(B\) in \(y\) axis direction; \(w_{1}\) is the movement of point \(B\) in \(z\) axis direction; \(P\) is the bulkhead additive thrust; \(G\) is the shear modulus of soil; \(\mu\) is Poisson's ratio. \(M_{1}\) and \(N_{1}\) can be expressed as:  
\[M_{1}=\sqrt{x^{2}+(y+R\cos\theta)^{2}+(z-c_{1})^{2}} \tag{10}\]  
\[N_{1}=\sqrt{x^{2}+(y+R\cos\theta)^{2}+(z+c_{1})^{2}} \tag{11}\]  
\[c_{1}=h-r\sin\theta \tag{12}\]  
The equations for ground movement induced by the friction between the jacking machine and surrounding soils are listed as follows:  
\[ \begin{array}{c}u_{2}=\frac{fR}{16\pi G(1-\mu)}\int\limits_{0}^{2\pi}\int \limits_{0}^{L}\left[\frac{3-4\mu}{M_{2}}+\frac{1}{N_{2}}+\frac{(x+l)^{2}}{M _{2}^{3}}+\frac{(3-4\mu)(x+l)^{2}}{N_{2}^{3}}+\frac{2x_{2}}{N_{2}^{3}}\right. \\ \left.\left(1-\frac{3(x+l)^{2}}{N_{2}^{3}}\right)+\frac{4(1-\mu)(1-2\mu)}{N_{ 2}+2+c_{2}}\bigg{(}1-\frac{(x+l)^{2}}{N_{2}(N_{2}+z+c_{2})}\bigg{)}\right]dld \theta\\ \end{array}  \tag{13}\]  
\[v_{2}=\frac{fR}{16\pi G(1-\mu)}\int\limits_{0}^{2\pi}\int\limits_{0}^{L}(x+l) (y+R\cos\theta)\left[\frac{1}{M_{2}^{3}}+\frac{3-4\mu}{N_{2}^{3}}-\frac{6x_{ 2}}{N_{2}^{3}}-\frac{4(1-\mu)(1-2\mu)}{N_{2}(N_{2}+z+c_{2})^{2}}\right]dld\theta \tag{14}\]  
\[w_{2}=\frac{fR}{16\pi G(1-\mu)}\int\limits_{0}^{2\pi}\int\limits_{0}^{L}(x+l) \left[\frac{z-c_{2}}{M_{2}^{3}}+\frac{(3-4\mu)(z-c_{2})}{N_{2}^{3}}-\frac{6x_ {2}(z+c_{2})}{N_{2}^{3}}+\frac{4(1-\mu)(1-2\mu)}{N_{2}(N_{2}+z+c_{2})}\right]dld\theta \tag{15}\]  
where, \(u_{2}\) (\(v_{2}\), \(w_{2}\)) is the ground movement along the \(x\) (\(y\), \(z\)) axis caused by friction on the jacking machine. \(f\) is the average friction between the jacking machine and surrounding soils. \(M_{2}\) and \(N_{2}\) can be expressed as:  
\[M_{2}=\sqrt{\left(x+l\right)^{2}+\left(y+R\cos\theta\right)^{2}+\left(z-c_{2} \right)^{2}} \tag{16}\]  
\[N_{2}=\sqrt{\left(x+l\right)^{2}+\left(y+R\cos\theta\right)^{2}+\left(z+c_{2} \right)^{2}} \tag{17}\]  
\[c_{2}=h-R\sin\theta \tag{18}\]  
where, \(L\) is the length of the jacking machine, \(R\) is the external radius of the jacking machine. The aforementioned solutions were used to calculate ground movements due to the bulkhead additive thrust and the frictions between the jacking machine and surrounding soils. [PERSON] (2016) has also proposed solutions for the evaluation of the friction between jacking segments and surrounding soils. However, because of the lubrication effect induced by the grouting process, the friction between jacking segments and surrounding soils is very small. On the other hand, [PERSON] and [PERSON] (2016) reported that the grouting process will apply additional pressure on surrounding soils, which may cause ground movement and finally affect the ground deformation. Therefore, a new method is required to improve the prediction of ground deformation induced by pipe-jacking construction.