## 3 Finite Element Simulation
### Finite Element Simulation Results
#### 3.3.3 Deformation Trend of Existing Tunnel  
Table 6 shows the cross-sectional deformation trend of the existing tunnel for the different horizontal spacings between the two tunnels, vertical spacings, and tunnel diameter ratios. The table shows that for numerical simulation scenarios 2-4, when the existing tunnel is located in front of (\(L\)/\(D\) = 1) or directly above (\(L\)/\(D\) = 0) the excavation surface, the cross-section of the existing tunnel is deformed elliptically. This is mainly due to the formation of the stress-unloading region in front of the destabilized excavation surface; the side of the existing tunnel near the excavated surface tends to move toward the stress-unloading region, which gradually evolves from a circle to an ellipse. Different horizontal spacings of the two tunnels cause the tunnel cross-section to deform along different angles. The angles between the long axis of the deformation ellipse and the horizontal centerline of the tunnel are about 109.3\({}^{\circ}\) and 57.9\({}^{\circ}\) when \(L\)/\(D\) = 1 and \(L\)/\(D\) = 0, respectively. Due to the different locations of the relative stress unloading zones, the deformation directions of the two conditions differ significantly. The long axis of the elliptic section is longer when \(L\)/\(D\) = 1, and the existing tunnel is most affected by the destabilization of the excavation surface. Although the existing tunnel is closer to the unloading zone of the excavation surface when \(L\)/\(D\) = 0, its inclination to form an 'ellipse-shaped' deformed cross-section is limited by the newly built tunnel underneath. When \(L\)/\(D\) = \(-\)1, the existing tunnel is farther away from the excavation surface and thus is least affected.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Working Condition** & \\ \hline \hline
**Schemes 2–4** & \\ \(h\)/\(D\) = 1 & \\ \(S\) = 3 mm & \\ \hline \hline
**Schemes 5–7** & \\ \(h\)/\(D\) = 0.75 & \\ \(d\)/\(D\) = 1 & \\ \(S\) = 3 mm & \\ \hline \hline
**Schemes 8–10** & \\ \(h\)/\(D\) = 1 & \\ \(d\)/\(D\) = 0.5 & \\ \(S\) = 3 mm & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Radial deformation of the existing tunnel cross-section.  
When \(h/D=0.75\), the 'ellipse' trend of the existing tunnel becomes more noticeable. This indicates that the existing tunnel is now closer to the destabilization zone of the excavation surface and is greatly affected by the stress unloading zone. The angle between the deformed ellipse's long axis and the tunnel's horizontal centerline increases for \(L/D=1\) and \(L/D=0\), and the maximum deformation of the cross-section varies with the spacing between the two tunnels.  
When \(d/D=0.5\), the existing tunnel for \(L/D=1\) is affected by the stress unloading zone and moves downward. The lower half of the cross-section essentially shows expansion deformation towards the excavation surface, while the upper half of the tunnel undergoes inward depression deformation due to the extrusion of the soil body on both sides and the loading of the soil body from above. When \(L/D=0\), the deformation trend of the cross-section of the existing tunnel mainly shows 'elliptical' deformation, inclining towards the stress unloading zone, wherein the radial deformation increment of the lower half of the cross-section is larger.