# Analytical Prediction of Tunnel Deformation Beneath an Inclined Plane: Complex Potential Analysis
## 6 Conclusions  
The deformation of a tunnel under a slope is usually caused by the asymmetric stress distribution over the tunnel cross-section. Fourier series expansion is used to determine the contour line surrounding the tunnel. This study proposes an approximate solution beneath the slope for predicting the ground deformation around the tunnel. The analytical method of ground displacement is proposed by the complex potential method, obtained by using the elastic problem in a circular domain. A definition of the exact and approximate solutions beneath the tunnel is given. The mathematical derivations and integrations for modelling the ground displacements due to the tunnel excavation under an inclined plane are based on continuum mechanics. The soil model resulting from the deformation is studied, and the factors influencing the stress distribution in the soil are identified and redefined in a simple soil compression form. A definition of the exact and approximate solutions under the slope for the maximum soil displacement is established. Conformal convergence, ovalization and vertical translation establish control parameters for the displacement distribution. It is found that both solutions give similar results. This implies that the control factors for the ground displacement parameters are established based on the presence of the internal friction angle and the slope-induced friction force. The effect of tunnel geometry was also proposed to calculate the weight of the load on the ground surface. The results show that the effect of the geometry has more influence on the direction of the downslope because the contact between the ground surface and the tunnel wall is small. To verify the analytical solutions numerically, a series of numerical simulations for different deformation modes are proposed. The analysis of the parameters shows that the model used for the cross-section has considerable effects on the behavior of the displacement field below the slope limit. Furthermore, the ground behavior at the cross-section can determine the ground movement around the tunnel and the surface settlement by equilibrating the stresses at the boundary conditions. The deformation type is also influenced by the tunnel geometry and the angle of the inclined plane. Data from the Qijiazhuang tunnel in China were used for numerical modelling of the tunnel cross-section. Keeping in mind that the continuum theory would work perfectly with the asymmetric stress distribution, the complex potential and displacement theory is proposed, which is a second-order justification in the classical ground deformation theory, to take into account the stress variation and the downslope proximity effect. It is argued that these prediction-based theories exhibit a conjunction with the basic criterion of maximum displacement, even when the angle of the inclined plane is variable. Thus, with this approach, the results show that the proposed analytical method is compatible with the quantitative data of the Qijiazhuang tunnel. This indicates that the analytical solutions could be an excellent alternative model for tunnel deformation risk assessment. The proposed method can also predict inelastic ground deformation in preliminary tunnel studies. [PERSON] developed the ideas, reformulated the research objectives, designed the research methodology, analyzed the results and revised the original version. [PERSON] supervised the work, acquired the funds, planned the research and revised the manuscript. [PERSON]-[PERSON], [PERSON] and [PERSON] revised the manuscript by checking the proposed formulas. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by National Key R&D Program of China, grant number 2017 YC0805300. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.