# Analytical Prediction of Tunnel Deformation Beneath an Inclined Plane: Complex Potential Analysis
## 3 Mathematical Analysis
### Complex Potential Analysis  
#### 3.2.1 Exact Solution  
[PERSON] [39] demonstrated that the complex potential method relies on the theory of linear elasticity in a deformation field. The elastic half-plane is not constantly stretched, which is a shortcoming of this complex and rigorous solution [20]. Using the complex potential functions \(\varphi(z)\) and \(\Psi(z)\) of [PERSON] [39], [PERSON] [40] solved a stress concentration problem for a doubly symmetric hole whose boundary consists of three crossed circles. The method enables the evaluation of complex tunnel wall stresses and displacements. The method is implemented on two systems of \"complex potential functions\" \(\varphi(z)\) and \(\Psi(z)\). The displacement is related to the analytical function established in Equation (8),  
\[2\mu\big{(}u_{x}+iu_{y}\big{)}=k_{0}\varphi(z)-z\frac{\overline{\partial \varphi}}{\partial z}-\overline{\Psi(z)} \tag{8}\]  
where \(k_{0}=3-4\upsilon\) is the elastic constant, \(u_{x}\) is the horizontal displacement, \(u_{y}\) is the vertical displacement and \(\mu\) is the shear modulus (\(z\ \overline{\partial\varphi}/\partial z\) and \(\overline{\Psi(z)}\) are shown in Appendix A). At the boundary condition, the necessary solution to the deformation is to evaluate the horizontal and vertical displacement of the ground around the tunnel. For deformation problems related to the surface traction boundary condition, the displacement can be represented in the complex \(z-\) plane (Figure 3a). The first step is based on the complex displacement, usually defined by Equation (8). The second step is described by stresses \(\sigma_{xx}-i\sigma_{yy}\) and \(\sigma_{yy}+i\tau_{xy}\). A space \(r\) in an annular \(z-\) plane is mapped into a \(\varsigma-\) plane, bounded by a circle \(|\varsigma|=\)1 and \(|\varsigma|=\alpha\) (Figure 3b). The general transcription of the complex forces is developed based on the Laurent series. The conformal transformation is defined by:  
\[z=\omega(z)=ih\frac{1-\alpha^{2}}{1+\alpha^{2}}\frac{1+\varsigma}{1-\varsigma} \tag{9}\]  
where the derivation \(\omega^{\prime}(\varsigma)=2 ih\big{(}1-\alpha^{2}\big{)}/(1-\varsigma) \big{(}1+\alpha^{2}\big{)}\), \(\varsigma\) is the mapped complex coordinate, \(h=r(1+\alpha)/2\alpha\) (\(\alpha\) is the embedment ratio parameter) and \(r^{2}=x^{2}+(y+h)^{2}\). Hence, the Goursat parameter can be expressed as:  
\[\varphi(z) =\varphi(\omega(\varsigma))=\varphi_{0}(\varsigma) \tag{10}\] \[\Psi(z) =\Psi(\omega(\varsigma))=\Psi_{0}(\varsigma) \tag{11}\]  
Figure 3: Conformal mapping of a tunnel beneath an inclined plane. The derivation of \(\varphi(z)\) can be obtained by \(\partial\varphi/\partial z=\partial\varphi/\partial\zeta.\partial\zeta/\partial z =\varphi_{0}(\zeta)/\overline{\omega^{\prime}(\zeta)}\). When \(|\zeta|=1\) and \(|\zeta|=\alpha\) are transformed into a \(\zeta-\) space, Equations (10) and (11) could be established as follows:  
\[|\zeta|=1;\ \varphi_{0}(\zeta)+\frac{\omega(\zeta)}{\overline{\omega^{\prime}( \zeta)}}\overline{\varphi^{\prime}_{0}(\zeta)}+\overline{\overline{\Psi_{0}( \zeta)}}=0; \tag{12}\]  
\[|\zeta|=\alpha;\ \varphi_{0}(\zeta)-\frac{\omega(\zeta)}{\overline{\omega^{ \prime}(\zeta)}}\overline{\varphi_{0}^{\prime}(\zeta)}-\overline{\overline{ \Psi_{0}(\zeta)}}=\text{F}(\zeta)\text{ with }F(\zeta)=ih\frac{\alpha^{2}-\zeta}{1- \zeta}\bigg{(}\frac{2}{1+\alpha^{2}}\bigg{)} \tag{13}\]  
For the stress result, \(|\omega^{\prime}(\sigma)|=\big{(}1-\alpha^{2}\big{)}\big{/}\big{(}1+\alpha^{2} \big{)}.h/(1-cos\theta)\). With \(|\zeta|=1\) and \(|\zeta|=\alpha\), boundaries can be established at the first boundary condition. The Laurent series expansion can be written as:  
\[\varphi_{0}(\zeta)=a_{0}+\sum_{k=1}^{\infty}\text{a}_{k}\zeta^{k}+\sum_{k=1}^ {\infty}\text{b}_{k}\zeta^{-k} \tag{14}\]  
\[\varphi_{0}(\zeta)=a_{0}+\sum_{k=1}^{\infty}\text{a}_{k}\zeta^{k}+\sum_{k=1}^ {\infty}\text{b}_{k}\zeta^{-k} \tag{15}\]  
where \(a_{k}\), \(b_{k}\), \(c_{k}\) and \(d_{k}\) are found by imposing the boundary condition on Equations (10) and (11).