# Analysis of the Interaction Effects of Shield Structure Oblique Passing under an Existing Tunnel
## 5 Deformation Characteristics of Existing Tunnels
### Torsional Deformation  
Figure 14 shows the local deformation of the existing tunnel after the completion of the shield double-line tunnel construction. It is clear from this that the shield tunnel passing under the existing tunnel will cause not only the longitudinal and transverse deformation of the existing tunnel but also the torsional deformation of the existing tunnel. It is evident from the figure that torsional deformation after LT excavation is greater than that after RT excavation. To explore the torsional deformation properties of the existing tunnel, four points (A\({}_{1}\), A\({}_{2}\), A\({}_{3}\), A\({}_{4}\)) located at the arch top, arch foot, and arch bottom of the existing tunnel were selected as reference points. Figure 15 shows the rotation of the different sections of the existing tunnel, as the LT and RT were excavated. After the completion of the RT  
Figure 14: Torsional deformation of the existing tunnel: (**a**) after RT excavation; (**b**) after LT excavation (f = 30,000). Figure 13: Bending moment and axial forces envelope of each section. ex excavation (as shown in Figure 15a), section 2-2 basically did not rotate, reference point A\({}_{1}\) of section 1-1 moved slightly down to the left, and reference point A\({}_{3}\) shifted slightly down to the right, which indicated that section 1-1 rotated counterclockwise. At the same time, reference point A\({}_{1}\) in section 3-3 was displaced down to the right, while reference point A\({}_{3}\) was displaced down to the left, which indicated that section 3-3 rotated clockwise. As the rotation directions of sections 1-1 and 3-3 were different, torsional deformation of the existing tunnel occurred. Following the completion of the LT excavation (as shown in Figure 15b), section 3-3 underwent counterclockwise rotation, section 5-5 underwent clockwise rotation, and section 4-4 showed basically no rotation. To further understand the torsional deformation of the existing tunnel, this paper refers to the tunnel torsion angle equation proposed by [PERSON] [8]:  
\[\omega=\tan\alpha=\frac{u_{y1}-u_{y2}}{H} \tag{1}\]  
where \(\omega\) is the rotation index; \(\alpha\) is the rotation angle of the existing tunnel; \(u_{y1}\) is the horizontal displacement at the top of the arch; \(u_{y2}\) is the horizontal displacement at the bottom of the arch (as shown in Figure 16); and \(H\) is the height of the existing tunnel, which in this paper is 11.3 m.  
Figure 16: Diagram of the existing tunnel rotation angle. Figure 15: Rotation of each section of the existing tunnel: (**a**) after RT excavation and (**b**) after LT excavation (f = 30,000). The longitudinal distribution of the rotation index \(\omega\) after RT and LT excavation is shown in Figure 17; among them, when \(\omega\) is clockwise rotation is positive, \(\omega\) is counterclockwise rotation is negative. As seen in Figure 17, the RT excavation resulted in clockwise rotation of the existing tunnel lining near the top left of the RT, counterclockwise rotation near the top right, and slight clockwise rotation above the axis. On the left side of RT, the rotation index \(\omega\) decreases as the distance from RT increases, and the minimum value is located around x = \(-\)17 m. On the right side of RT, the rotation index \(\omega\) increases as the distance from RT increases, and the maximum value is located around x = 1 m. After LT excavation, the longitudinal distribution of the rotation index \(\omega\) is similar to that of RT after excavation, but relatively large clockwise rotation occurs above the RT axis, and both the maximum value (located around x = 15 m) and the minimum value (located around x = \(-\)14 m) of the rotation index \(\omega\) tend to move toward the right and are roughly symmetrical along the center line of the two-lane tunnel. The existing tunnel rotation caused by the LT excavation is greater than that caused by the RT excavation.