# Theoretical Analysis on the Effectiveness of Pipe Roofs in Shallow Tunnels
## 4 Case Calculation and Verification  
The portal section of the Birch Ridge Tunnel [23] is dominated by silty clay. The depth of the tunnel is 2.7 m, and the degree of the surface slope is 15\({}^{\circ}\). The mechanical parameters of the surrounding rock are shown in Table 1. The design adopts a pipe shed with a length of 30 m, a circumferential spacing of 40 cm, and a pipe diameter of 108 mm (the thickness is 6 mm) for pre-support. The pre-support radius is 7 m. The actual excavation height of the upper steps is 3 m, and the excavation input per cycle is 0.6 m. According to the method of equivalent stiffness [17], the equivalent moment of inertia is 6.68 \(\times\) 10\({}^{-6}\) m\({}^{4}\), and the equivalent elastic modulus of elasticity is 7.89 \(\times\) 10\({}^{7}\) kPa. [PERSON] et al. [23] used field tests to obtain the flexural deformation of the steel pipe of the pipe roofs at the vault with the advancement of the palm face. The measured value shown in Figure 3 is the deformation of the steel pipe of the pipe roofs at the vault when the palm face is excavated to 22.5 m. Figure 3 shows the theoretical value for the deformation of the steel pipes in the pipe roofs at the vault when the palm surface is excavated to 22.5 m according to the calculational method for surrounding rock pressure proposed in this paper and the force deformation of the pipe roofs in [PERSON] et al. [24]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Base Bed Coefficient** & **Foundation Shear** & **Unit Weight (kN/m\({}^{3}\))** \\  & **(kN/m\({}^{3}\))** & **Modulus (kN/m)** & **Unit Weight (kN/m\({}^{3}\))** \\ \hline Value & 28,000 & 2800 & 17.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of the surrounding rock. Figure 3 shows that the theoretical size and distribution of the flexural deformation of the steel pipe at the vault are basically consistent with the measured data on site. Although the measured and calculated values show some differences at a distance of 22.5 to 27 m from the opening, their characteristics are basically the same. Therefore, it can be inferred that the error of the distance from the palm face to the opening of the hole is the reason for this difference and that the actual position of the palm face may be slightly farther than 22.5 m.  
Contrast analysis demonstrates that the calculational method for the surrounding rock pressure proposed in this paper and the calculational method for force and deformation proposed in [18] correspond with actual working conditions. On this basis, it is reasonable to conduct a theoretical analysis of the pre-support effects of pipe roofs. The distribution of surrounding rock pressure borne by the pipe shed on different tunnel cross-sections is obtained by calculation as shown in Figure 4.  
Figure 4: Surrounding rock pressure distribution of steel pipes on different sections. Figure 3: Comparison between the calculated value and the measured value of pipe roof deformation. Figure 4 shows that as the tunnel progresses deeper, the burial depth of the tunnel becomes larger, and the size and distribution of the surrounding rock pressure acting on the steel pipes of each pipe roof also changes. The portal section of the tunnel was buried at a depth of 2.7 m. Additionally, the surrounding rock pressure on the steel pipe at the vault was 19.2 kN/m, and the surrounding rock pressure on the steel pipe at the arch springer was 39.4 kN/m, indicating a gradual increase from the vault to the arch springer. The tunnel was buried at a depth of 10.7 m and located 30 m from the portal section. The surrounding rock pressure on the steel pipe at the vault was 76.7 kN/m, and the surrounding rock pressure on the steel pipe at the arch springer was 97.7 kN/m. With an increase in depth, the position of the maximum surrounding rock pressure has a tendency to move to the spandrel between the arch springer on both sides.