# Analytical Prediction of Tunnel Deformation Beneath an Inclined Plane: Complex Potential Analysis
## 5 Application to Engineering Cases  
The evaluation of the potential of the approximate method under a slope based on the complex potentials of the above Qijiazhuang tunnel predicts the ground displacement field. The field data of the section of National Highway 109 of the Sixth Western Bypass Section (SPO-SL) of the city is used for this analysis. The start and end of the straight line of the tunnel are from AK70+076286 to AK70+375, with a total length of 298.714 m (Figure 20). The tunnel is built beneath a slope with an angle of inclination of \(\beta=27^{\circ}\). The slope surface varies from 0.5 m to 2.0 m (Figure 20), and the thickness separating the central valley is about 4 m. The lithology is andesitic with three weathering phases (Table 1). The same process adopted for the mining excavation (New Austrian tunnel method: NATM) is used for the tunnel excavation, effectively controlling the ground deformation around the tunnel. Furthermore, grout injections are used to stabilize the rocks surrounding the tunnel. Although the stability measurement data on the ground surface is monitored, the complete ground displacement can only be analyzed with monitoring devices. However, with the approximate method under a slope proposed in this study, the parameters inducing the deformation of the tunnel wall (uniform convergence, ovalization, vertical and horizontal translation and shape of the final displacement) are determined. Figure 20: 2–d longitudinal section of Qijiazhuang Tunnel (S4a). The cross-section is used to predict the ground deformation around the tunnel (Figure 21). The resulting analytical data of the tunnel geometry is represented by \(\ A^{\prime}\ o_{1}=o_{1}\ B^{\prime}=47.44\ m\), \(h_{1}\)= 55.03 m, \(h_{2}\)= 6.69 m and \(R/h=0.31\). The ground deformation is analyzed taking into account the slope angle \(\beta\) = 27\({}^{\circ}\), E = 21 \(\times\) 10\({}^{-3}\) MPa, \(v\) between 0.00 and 0.50, \(\gamma_{d}=18\) kN/m\({}^{3}\), the slope of the tunnel longitudinal axis 2.5% and \(u_{0}\)= -0.04 m (\(v_{i}=0.6\%\)). These geometric and in situ data were used to digitize Figure 21. The same data were also used to build the model shown in Figure 22, which presents a comparative pattern between the analytical solution and the numerical simulation. The difference in vertical displacement is due to the angle of the inclined plane, which gives it an asymmetric displacement on the negative image pole (\(y=h\)). Thus, due to the asymmetric stress distribution, the load applied in the numerical simulation is influenced by the nature of the weathered material, hence the presence of an asymmetry in the negative part of Figure 22.  
When Equation (3) is used as the exact boundary condition in the tunnel cross-section, the approximate method beneath a slope obtained in Equations (39), (48) and (58) is used to model and justify the results given by the exact solution considering the asymmetric compression. A degree of rationality is always necessary in defining a design analysis, and it is essential that the design engineer is able to assess the overall accuracy of a numerical analysis as far as possible [41]. These values must be defined by determining the thickness of each soil layer level on the tunnel roof. At this stage, [PERSON] still needed to present the complete solution for the elastic stresses and displacements around a doubly symmetric tunnel. Thus, the effect of internal pressure alone cannot change the tunnel equilibrium. It would be necessary for the two pressures \(p_{0}\) and \(p_{1}\) to interact on the tunnel, equilibrating the pressure of the excavation trough under the effect of the stress gradient. Moreover, the equilibrium equations and the geometric application of the tunnel are known (in an elastic plane). In this way, [PERSON] et al. [15] estimate that the boundary  
Figure 21: Numerical representation of the meshes around a tunnel beneath an inclined plane. Figure 22: Comparative analysis between the analytical solution and the numerical simulation.