## 3 Parameter Optimization of PRPS and CGPR
### Parameter Optimization of PRPS  
#### 3.2.1 Pipe Diameter Optimization  
The pipe roof consisted of the outer steel pipe and inner grouted concrete, the outer diameter of the steel pipe was D, and the steel pipe thickness was T (Figure 10). The density and the elastic modulus of the steel pipe were 7850 kg/m\({}^{3}\) and 210 GPa, respectively, with those of the inner grouting material being 2300 kg/m\({}^{3}\) and 25 GPa, respectively. Note that the surrounding rock was also reinforced by grouting, and the elastic modulus of the reinforced zone was 12 GPa [26]. The diameter of the steel pipes to be selected ranged from 89 to 186 mm.  
The mechanical parameters of the pipe roof could be calculated according to the equivalence principle of bending stiffness and mass [27]:  
\[E_{0}I_{0}=E_{1}I_{1}+k_{w}E_{2}I_{2}, \tag{1}\]  
\[\rho_{0}S_{0}=\rho_{1}S_{1}+\rho_{2}S_{2}, \tag{2}\]  
where \(E_{0}\), \(I_{0}\), \(\rho_{0}\), \(S_{0}\) are the equivalent elastic modulus, the moment of inertia, the equivalent density, and the area of the pipe roof; \(E_{1}\), \(I_{1}\), \(\rho_{1}\), \(S_{1}\) are the elastic modulus, the moment of inertia, the density, and the area of the grouting concrete; \(E_{2}\), \(I_{2}\), \(\rho_{2}\), \(S_{2}\) are the elastic modulus, the moment of inertia, the density, and the area of the steel pipe. \(k_{w}\) is the reduction factor considering the cracking of the grouting concrete, \(k_{w}=0.6\).  
The pipe roof parameters of six schemes are shown in Table 3. During the pipe diameter optimization, the arrangement angle of the pipe roof was preset to 180\({}^{\circ}\) (\"180\({}^{\circ}\)\" was not the final optimized arrangement angle). The MCS and the PPR with different pipe diameters are shown in Figure 11.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(p\) (kg/m\({}^{3}\)) & \(E\) (GPa) & \(\
u\) & \(\epsilon\) (kPa) & \(\varphi\) (\(\uparrow\)) & **H (m)** \\ \hline Miscellaneous soil filling & 1750 & 0.1 & 0.42 & 20 & 20 & 5.0 \\ Completely wauthered & 1860 & 1.0 & 0.40 & 100 & 24 & 25.0 \\ monozonite granite & 2000 & 1.5 & 0.37 & 160 & 25 & 170.0 \\ Tunnel lining & 2400 & 26.4 & 0.20 & - & - & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties in the numerical simulation.  
Figure 10: Arrangement sketch of pipe roof pre-support (PRPS).  
Based on numerical results, we observed that the pipe roof played a more critical role in reducing MCS when compared with that in minimizing PPR. When the pipe diameter increased from 0 to 186 mm, the MCS decreased from the 56.2 to 26.5 mm, while the PPR slightly decreased from 100% to 89.5%. It could be expressed that the role of the pipe roof was mainly to form a strong arch structure above the tunnel and to bear loose rock pressure. In comparison, it had less ability in reinforcing the rock pillar due to the limited arrangement area. Therefore, the MCS should be the major evaluation parameter in the optimization of PRPS.  
With the increase of pipe diameter, the pipe roof possessed larger stiffness and showed better performance in controlling crown settlement. When the pipe diameter was larger than 108 mm, the downward trend of the MCS became no longer prominent. Therefore, 108 mm could be selected as the optimized pipe diameter.