## 3 Results and Discussions  
### Effect of the Tunnel Footage  
Under the condition of a certain thickness of the rock and soil layer, the new tunnel penetrates through existing tunnel a and crosses existing tunnel b. The displacement changes in the rock and soil around the two existing tunnels are selected for analysis when the tunnel excavation device travels to the front, middle, and back of the artificial observation spot. As can be seen from Figure 8, the shield jacking length \(L\) is selected to be analyzed, such as \(L=1.9D\), \(2.2D\), and \(2.5D\). Among them, the \(L\) of \(2.2D\) represents the position of the artificial spot observation surface. The displacement changes in the rock and soil bodies around the existing tunnels can be obtained during the advance of the shield tunnel by importing the original figures taken at each stage of the test into the PIV software (PIVview2 CDemo_v359) for analysis and processing.  
Figure 8 shows the displacement of the rock around the two existing tunnels when the new tunnel crosses above and beneath the existing tunnels with the same thickness as the rock and soil interlayer (\(h=0.5D\)) at the same time. When the tunneling device moved ahead of the artificial spot observation surface (\(L=1.9D\)), there was a small settlement below existing tunnel a and a small uplift above existing tunnel b. At this moment, the maximum vertical displacement-to-diameter ratio of existing tunnel a was about 0.0022, and that of existing tunnel b was approximately 0.0026. When the tunneling device moved to the artificial spotting observation surface (\(L=2.2D\)), the settlement below existing tunnel a and the uplift above existing tunnel b was larger than before the spotting surface. Also, the maximum vertical displacement-to-diameter ratio for existing tunnel a and tunnel b equaled about 0.0038 and 0.0045, respectively. Additionally, the maximum vertical displacement-to-diameter ratio increased by 0.0016 and 0.0019 compared with that at \(1.9D\). When the tunneling device reached the back of the artificial spot observation surface (\(L=2.5D\)), maximum deformation occurred under existing tunnel a and above existing tunnel b. The maximum vertical displacement-to-diameter ratio was about 0.0048 and 0.0059, respectively. These two values were 0.001 and 0.0014 mm larger than that at \(L=2.2D\). With the advance of the newly excavated tunnel, the rock deformation around the existing tunnels gradually decreased from the center to the two sides along the direction of the existing tunnel length. The deformation pattern conformed to the Gaussian distribution curve, and some rock bodies bulged at both ends of the existing tunnel bulge. This was because the simulated existing tunnels were not completely horizontal and there were small warps at both ends of the tunnels, which tended to deform upwards. Figures 9 and 10 show the displacement of the surrounding rock as the new tunnel passes through existing tunnel a and existing tunnel b. In Figure 9, the thickness of the rock and soil interlayer above existing tunnel a is \(0.3D\) and \(0.7D\) below existing tunnel b. In contrast, these two values in Figure 10  
are equal to \(0.7D\) and \(0.3D\), respectively. Since the deformations are consistent with the thickness (i.e., \(0.3D\)) of the rock and soil interlayer, we will not repeat this.  
Generally, the settlement deformation and have deformation of the rock and soil around the existing tunnels increase gradually with the increasing excavation footage of the new tunnel when the new tunnel crosses the upper and lower two existing tunnels at the same time. And the maximum settlement deformation and have deformation occur when crossing the existing tunnel above or below, and the increasing deformation decreases after crossing and tends to become gradually stable.  
Figure 8: Vertical displacement of the rock and soil around the existing tunnels. (\(h_{1}=h_{2}=0.3D\)).  
Figure 9: Vertical displacement of the rock and soil around the existing tunnels (\(h_{1}=0.3D\), \(h_{2}=0.7D\)).