## 3 Numerical Analysis
### Numerical Analysis Results  
Figures 19-22 show the simulation results for the vertical and horizontal displacement contours generated in the lower tunnel when the allowable loads were applied to the pile. As the offset between the tunnel and piles increased by 1.0 D, the effect on the tunnel by the pile installation decreased. As Figure 22 shows, the tunnel was farther outside of the influence area due to the pile installation compared to other cases where the distance was close. Figures 23 and 24 show the simulation results for the axial force generated in the lower tunnel when the allowable and ultimate loads were applied to the pile. When the allowable load was applied, the axial force acting on the tunnel according to the horizontal pile-tunnel offset decreased as the vertical offset decreased. When the ultimate load was applied, the magnitude of the axial force acting on the side wall of the tunnel changed significantly as the horizontal pile-tunnel offset increased.  
Figure 23: Cont.  
Figure 21: Displacement contours for allowable load (case 2-2).  
Figure 22: Displacement contours for allowable load (case 3-3).  
As shown in Figures 25 and 26, when the allowable load was applied to the pile, the axial force acting on the crown of the tunnel gradually decreased. Compared to case 1-0,  
Figure 23: Axial force on tunnel by allowable load in 3D numerical analysis.  
Figure 24: Axial force on tunnel by ultimate load in 3D numerical analysis.  
cases 1-1, 1-2, and 1-3 were lower by 6.9, 37.7, and 78.7%, respectively. In the case of the side wall, the axial force was larger than that acting on the crown. Compared to case 1-0, where the pile was immediately above the tunnel, the decrease rates of cases 1-1, 1-2, and 1-3 were reduced to be 21.6%, 49.7%, and 69.3%, respectively. When the vertical offsets were 2.0 D and 3.0 D, as the pile moved from the position immediately above the tunnel (0.0 D) to 1.0 D, 2.0 D, and 3.0 D in the horizontal direction, the axial force acting on the crown of the tunnel showed a tendency to sharply decrease by 8.3, 43.9, and 88.1% and by 8.1, 26.7, and 50.1%, respectively. The axial force acting on the side wall also decreased by 42.0, 59.9, and 61.9% and by 8.2, 31.6, and 47.9%, respectively, indicating that the influence of the pile on the tunnel decreased when the pile was farther away. This tendency was also observed when the ultimate load was applied to the pile. The axial force acting on the crown decreased by 3.31, 25.4, and 84.3% (vertical offset: 1.0 D); by 7.4, 65.5, and 71.1% (vertical offset: 2.0 D); and by 6.1, 29.5, and 49.8% (vertical offset: 3.0 D) compared to the case with the pile immediately above the tunnel. The axial force acting on the side wall decreased by 30.3, 49.7, and 70.0% (vertical offset: 1.0 D); by 29.9, 55.6, and 64.6% (vertical offset: 2.0 D); and by 7.7, 31.8, and 50.9% (vertical offset: 3.0 D). Therefore, securing a sufficient horizontal offset from the existing tunnel is necessary, and the appropriate separation distance found by analysis is consistent with the standards of Seoul.  
Figure 25: Normalized axial force by allowable load according to horizontal offset (3D numerical analysis).