# Dynamic Response of Rectangular Tunnels Embedded at Various Depths in Spatially Variable Soils
## 4 Results and Discussion
### Effect of Tunnel Embedment Depth on Tunnel Displacement  
In this section, the displacement responses of the tunnel under different embedment depths under both deterministic and stochastic conditions are calculated. Figure 11 shows the comparison of the time history curve of the uplift displacement of the tunnel under different embedment depths. As can be seen from Figure 11a-c, when the embedment depth of the rectangular tunnel is not more than 10 m, the mean value of uplift displacement of the tunnel obtained by 200 random calculations is smaller than that obtained by deterministic calculations. When the tunnel embedment depth is 6 m, the maximum uplift displacement of the tunnel obtained by the deterministic calculation is 37.42 cm, and the maximum uplift displacement of the tunnel obtained by random analysis is 34.46 cm, which is 7.9% lower than the deterministic calculation. When the tunnel embedment depths are 8 m and 12 m, the gap is reduced to 4.2% and 4.1%, respectively. As the embedment depth of the tunnel continues to increase, when \(H=12\) m, the mean value of the maximum uplift displacement of the tunnel by random analysis exceeds the deterministic calculated displacement value under this embedment depth. This may be owing to the liquefaction degree of the soil above the tunnel considering spatial variability was less affected when the tunnel was located deep in the foundation, which was beneficial to the uplift of tunnel compared with the deterministic case. The above phenomena indicate that, with the increase of tunnel embedment depth, the safety and reliability of underground structures, such as tunnels, will be lower when the foundation soil is regarded as a uniform material to analyze the uplift displacement response of tunnels and other underground structures due to liquefaction. The comparison between the stochastic and deterministic analyses of the time history of tunnel uplift displacement under different embedment depths is illustrated in Figure 12. The maximum uplifts of tunnel buried in various embedment depths in the deterministic case given in Figure 12b is consistent with the study of [PERSON] and [PERSON] [19], which can be described as the increase of buried depth reduces the uplift of underground structures in liquefiable soil. Accordingly, the mean values of stochastic cases illustrated in Figure 12a have the same tendency. The reason is that the thickness of the overlying soil above the tunnel increases with the embedment depth of the tunnel. The soil gravity induced vertical force applied to the tunnel increases, since the parameters of the whole model soil are the same. The resistance force against the tunnel uplift displacement increases, resulting in the decrease of the tunnel uplift displacement. Figure 10: Cumulative distribution function (CDF) of \(A_{80}\) under different embedment depths of tunnel. In order to further analyze the influence of tunnel embedment depth on the uplift displacement, the maximum uplift displacement of the tunnel is calculated, as shown in Figure 13. As can be seen from the figure, the maximum uplift displacement value of the  
Figure 11: Comparison of time history curve of the uplift displacement of the tunnel under different embedment depths: (**a**) \(H\) = 6 m; (**b**) \(H\) = 8 m; (**c**) \(H\) = 10 m; (**d**) \(H\) = 12 m.  
Figure 12: Comparison between the stochastic and deterministic analyses of the time history of tunnel uplift displacement: (**a**) random cases; (**b**) deterministic cases. tunnel is approximately in line with the tunnel embedment depth in the range of 8 m to 12 m. However, under the shear modulus random field model with variation coefficient COV = 0.3 and horizontal and vertical fluctuation ranges of 50 m and 10 m, respectively, the decreasing rate of the mean displacement of the tunnel with the increase of the embedment depth of the random analysis is greater than that calculated in the deterministic condition of uniform soil field.