# Dynamic Response of Rectangular Tunnels Embedded at Various Depths in Spatially Variable Soils
## 4 Results and Discussion
### Effect of Tunnel Embedment Depth on Liquefied Zone  
According to the research of [PERSON] et al. (2016), the index of the liquefied zone is defined as:  
\[A_{80}(t)=\frac{A(\frac{u(t)}{\sigma_{v0}^{\prime}}>0.8)}{A} \tag{4}\]  
where \(u(t)\) is the EPWP at a time instant \(t\) after the seismic loading is applied, \(\sigma_{v0}^{\prime}\) is the initial effective stress of a spatial position, \(A(\frac{u(t)}{\sigma_{v0}^{\prime}}>0.8)\) and \(A\) represents the area in which the EPWP ratio is greater than 0.8 and the total analysis domain, respectively. The comparison of the liquefied zone of both deterministic and stochastic analyses for various tunnel embedment depths is shown in Figure 7. It can be seen that the deterministic maximum liquefied zone is larger than the peak value of the mean liquefied zone of the stochastic analysis for different tunnel embedment depths, indicating that without consideration of soil spatial variability the maximum liquefied zone can be overestimated. The maximum liquefaction range of the soil increases with the increase of tunnel embedment depth for both the deterministic analysis and the condition considering the spatial variability of soil shear modulus. This was mainly due to the fact that the spatial variability of soil shear modulus decreased around the tunnel. In general, soil closer to the surface is easier to liquefy. The soil modulus in shallow depth was less affected by the local averaging effect of tunnel with the increase of tunnel buried depth. Figure 6: Influence of the tunnel embedment depths on the EPWP ratio at Point A. Figure 8 presents the comparison of the time history curve of the mean liquefaction range of randomness analysis under different embedment depths. It can be seen that the soil does not basically liquefy within the first 4 s of the earthquake action; with the increase of earthquake time, local liquefaction occurs in the soil, and the liquefaction range of soil rises rapidly to the peak; the liquefaction range of soil gradually decreases after 20 s of the earthquake. With the increase of tunnel embedment depth, the liquefaction range in the ascending period rises at an increasing rate; the liquefaction range in the descending period shrinks at an increasing rate; and the mean of liquefaction range in randomness analysis under different conditions is maintained at a similar level until the end moment of dynamic calculation. To analyze the effects of tunnel depth on the distribution of peak values of the liquefaction range, the peak probability density curves and cumulative distribution curves of the liquefaction range under different tunnel depths are shown in Figures 9 and 10, respectively. As shown in Figure 9, the distribution range of peak value of soil liquefaction range in random analysis remains unchanged as the tunnel embedment depth increases from 6 to 12 m. However, the peak value of the liquefaction range corresponding to the peak point of the PDF curve gradually increases with the tunnel embedment depth, and the kurtosis gradually decreases. It is shown that the mean values given in Figure 8 can reflect the characteristics of stochastic simulations to a certain extent. As shown in the cumulative distribution of peak value of liquefaction range, the probability of peak value of liquefaction range randomly calculated smaller than deterministic results is 74.5% when the tunnel  
Figure 7: Variation of \(A_{80}\) with time history under different embedment depths of tunnel: (**a**) \(H\) = 6 m; (**b**) \(H\) = 8 m; (**c**) \(H\) = 10 m; (**d**) \(H\) = 12 m.  
-embedment depth is 6 m. Then, this probability reduces to 67.3% as the tunnel embedment depth increases to 8 m. However, when the tunnel embedment depth increases to 10 m and 20 m, 18.3% and 19.9% of the peak values of the liquefaction range in the random calculations are higher than deterministic results under the corresponding embedment depth. The maximum value of the liquefaction range exceeds most of the random analyses results when considering the soil as homogeneous.