# Analytical Prediction of Tunnel Deformation Beneath an Inclined Plane: Complex Potential Analysis
## 3 Mathematical Analysis
### Complex Potential Analysis
#### 3.2.2 Approximate Analysis of the Tunnel Beneath the Slope  
The deformation mode of a tunnel cross-section cannot reflect its deformation behaviour when the ground surface is a slope [15]. In particular, when the boundary condition is re-specified as \(z=re^{im\theta}\) (\(\theta\) is the angular coordinate in the z-plane, and \(n\leq 1\)), the tunnel boundary undergoes a uniform radial displacement. Hence, Equation (9) is redefined as follows:  
\[2\mu\big{(}u_{x}+iu_{y}\big{)}=2\mu u_{0}e^{im\theta}\approx k\sum\limits_{k= 2}^{\infty}a_{k}\sigma^{k}+ka_{1}\sigma-\overline{a_{1}}\sigma-\sum\limits_{k =0}^{\infty}\Big{(}\overline{b_{k}}+(k+2)\overline{a}_{k+2}\Big{)}\sigma^{-k} \tag{32}\]  
where \(u_{0}=a_{1}/(\lambda+\mu)\) is the uniform convergence, \(\lambda\) is the first Lame coefficient, \(a_{1}\) is the constant (given by Equation (25)) and \(k\) is a variable. Therefore, Equations (6) and (7) can be expressed as \(\sigma_{xx}=\sigma_{yy}=2a_{1}/r\) and \(\tau_{xy}=0\). In the \(\varsigma-\) plane of the radius \(\alpha\), \(\varsigma\) can be re-described as \(\varsigma=\alpha e^{im\theta}\). At the boundary condition along the inner boundary in the \(z-\) plane, the displacement can be defined by:  
\[u_{x}|_{u_{c}}=-u_{0}\frac{x\sin\beta}{R};\;u_{y}|_{u_{c}}=-u_{0}\frac{y\cos \beta+h}{R} \tag{33}\]  
where \(R\) is the tunnel radius, \(x\) and \(y\) can be given by Appendix C and \(u_{c}\) is the uniform convergence. Following the Fourier series expansion, Equation (33) can be reformulated as:  
\[u_{x}|_{u_{c}}=-u_{0}\frac{x\sin\beta}{R}\frac{h(1-\alpha^{2})^{2}}{1+\alpha^{ 2}}\sum\limits_{k=1}^{\infty}\alpha^{k-1}\sin(k\theta)\left\{ \begin{array}{c }\alpha^{k-1},k=1,2,3,\ldots\\ 0,k=0\\ -\alpha^{-k+1},k=-1,-2,-3,...\end{array} \right. \tag{34}\]  
\[u_{y}|_{u_{c}}=-u_{0}\frac{h(y\cos\beta+h)}{R}\frac{(1-\alpha^{2})}{1+\alpha^{ 2}}\left( \begin{array}{c}\alpha-\ldots\\ -\big{(}1-\alpha^{2}\big{)}\sum\limits_{k=1}^{\infty}\alpha^{k-1}\cos(k\theta )\end{array} \right)\left\{ \begin{array}{c}\frac{1}{2}\big{(}1-\alpha^{2} \big{)}\alpha^{k-1},k=1,2,3,\ldots\\ -\alpha,k=0\\ \frac{1}{2}\big{(}1-\alpha^{2}\big{)}\alpha^{-k+1},k=-1,-2,-3,\ldots\end{array} \right. \tag{35}\]  
Hence, Equation (8) can be rewritten as follows:  
\[u_{x}+iu_{y}=-iu_{0}\left\{\frac{2h\alpha\big{(}1-\alpha^{2}\big{)}^{2}}{1+ \alpha^{2}}\right\}\sum\limits_{k=1}^{\infty}\alpha^{k-1}e^{ik\theta} \tag{36}\]\[u_{x}+iu_{y}=-iu_{0}\frac{\left(1-\alpha^{2}\right)h}{1+\alpha^{2}}\Bigg{\{}\alpha -\left(1-\alpha^{2}\right)\sum\limits_{k=1}^{\infty}\alpha^{k-1}e^{ik\theta} \Bigg{\}} \tag{37}\]  
Separating the real from the imaginary, the horizontal and vertical displacement can be rewritten as (for \(k=1\)) \(u_{x}=0\) and \(u_{y}=0.5u_{0}\). For \(y=0\), the maximum motion can be obtained as follows: \(u_{y}max=0.38(1-v)u_{0}R^{2}/h\) (where \(v\) is the Poisson ratio). Hence, the vertical translation can be written as:  
\[\Delta u_{y}\Bigg{|}u_{c}=u_{0}\left\{\frac{0.61(1-v)\Big{(}\frac{R}{h}\Big{)} +\left(0.89-0.61v\right)\Big{(}\frac{R}{h}\Big{)}^{2}-0.9\Big{(}\frac{R}{h} \Big{)}^{3}-1.089}{\Big{(}\Big{(}\frac{R}{h}\Big{)}^{2}+1.21\Big{)}^{2}} \right\} \tag{38}\]  
Then, the final shape of displacement can be expressed as:  
\[U_{f}\Bigg{|}u_{c}=u_{0}\left\{\frac{1.131-1.22v+(0.89-0.61v)\Big{(}\frac{R}{h }\Big{)}^{2}}{\Big{(}\Big{(}\frac{R}{h}\Big{)}^{2}+1.21\Big{)}^{3}}\right\} \tag{39}\]  
When \(\alpha\to 0\), the tunnel radius is almost equal to zero, which indicates that the tunnel is deep. When \(\alpha\to 1\), the tunnel is shallow. The asymmetric displacement induced by the slope can be evaluated by determining the stresses on the tunnel wall and the ground surface. Thus, we can have the following:  
\[u_{\delta}=\frac{\sigma_{r}R/(h_{1}+h_{2}+0.51(A^{\prime}0_{2}+\alpha_{2}B^{ \prime}))}{4\mu}(3-4v) \tag{40}\]  
where \(u_{\delta}\) is the ovalization parameter of the tunnel wall and A\({}^{\prime}0_{2}+\alpha_{2}\)B\({}^{\prime}\) is the distance between \(h_{1}\) and \(h_{2}\).