# Dynamic Response of Rectangular Tunnels Embedded at Various Depths in Spatially Variable Soils
## 4 Results and Discussion
### Effect of Tunnel Embedment Depth on Ground Displacement

Table 3: Comparison of maximum vertical displacement between surface and tunnel. Figure 17: Variation of the maximum uplift displacement with the embedment depth of the tunnel for the stochastic and deterministic analysis: (**a**) Mean of random cases; (**b**) deterministic case. Figure 20 shows the comparison of vertical displacements at each position of the ground under different embedment depths of tunnels by deterministic analysis and stochastic analysis. The soil above the tunnel shows uplift displacement, while the settlement deformation occurred in the surface far away from the tunnel. This finding were accordant to the research of [PERSON] and [PERSON] [20]. It can be found that the difference between the displacement mean value of stochastic analysis and the deterministic calculation result is small in the area of surface uplift, while the difference is large in the area of surface settlement far from the tunnel structure. With the increase of tunnel embedment depth, the maximum uplift displacement of surface soil decreases continuously, and the maximum settlement also decreases. When the depth of the tunnel is \(H\) = 12 m, the difference between the stochastic calculation and the deterministic analysis is the largest, with a difference of 28.8%. The largest difference in surface subsidence occurs when \(H\) = 6 m, which is 67.9%. It can be explained that the spatial variability of soil far from the tunnel was less reduced by the averaging effect of tunnel. Figure 21 shows a set of random analysis of seismic vertical displacement contours for tunnel embedment depth \(H\) = 6 m. As can be found from the figure, after the earthquake effect, the foundation is in different degrees of liquefaction, leading to the surface of the tunnel structure, squeezing soil above the tunnel and, at the same time, making the soil around the tunnel supplement the bottom of the region. As illustrated by [PERSON] et al. [18], it is an imbalance of vertical pressures arising from the development of excess pore water pressure beneath the underground structures that leads to the underground structures uplift, which is accompanied by the surrounding soil settlement as a result of the soil  
Figure 18: PDF of the peak value of vertical surface displacement at different embedment depths: (a) \(H\) = 6 m; (b) \(H\) = 8 m; (c) \(H\) = 10 m; (d) \(H\) = 12 m.  
movement, which is consistent with the results of this study. From the displacement vector diagram of soil around the tunnel shown in Figure 22, it can be more intuitively seen that the soil above the tunnel is squeezed and moves away from the tunnel area, while the soil on both sides forms a circular displacement trajectory to complement the lower part of the tunnel. A similar phenomenon was also revealed by [PERSON] et al. [51], that the displacement vectors were directed toward the bottom of the structure. In addition, [PERSON] et al. [52] illustrated that soil deformation around the circular tunnel formed wide circular loops on both sides of the structure, displacing soil from the crown to the invert of the structure. However, [PERSON] et al. [21] presented that the soil deformation around the structure did not form wide circular loops at the two sides of the structure, which may be attributed to the rectangular shape of the metro tunnel. The greater the depth of the tunnel, the thicker the upper floating soil layer; the greater the resistance to its upward movement, the smaller the corresponding upward displacement and surface uplift of the tunnel; and the less soil needed to replenish beneath the tunnel, the smaller the settlement of the soil in the area away from the tunnel.