## 2 Materials and Methods
### RMi System and Tunnel Support Design
#### 2.4.2 Tunnel Support Design by RMi System in Discontinuous Ground
where \(\beta\) is the block shape factor, which depends on the ratio of the longest and shortest dimension of the block; the common value is 36. \(J_{v}\) is the number of joints intersecting a volume of one m\({}^{3}\). Based on \(J_{v}\), the degree of jointing is classified into six classes, from a very low degree of jointing (\(J_{v}<1\)) to crushed rock (\(J_{v}>60\)). Random joints may represent a significant part of the total number of discontinuities, and neglecting them would lead to the erroneous quantification of the discontinuity nature of the rock mass [48]. A rule of thumb for the calculation of \(J_{v}\) is given in Equation (16) [49]:  
\[J_{v}=1/S_{1}+1/S_{2}+1/S_{3}+\ldots\ldots\ldots\ldots 1/S_{n}+N_{r}\Big{/}(5 \sqrt{A})\,, \tag{16}\]  
where \(S_{1}\), \(S_{2}\), and \(S_{3}\) are the average spacings for the joint sets, \(N_{r}\) is the number of random joints in the actual location, and \(A\) is the area in m\({}^{2}\). It is not possible to measure all of the joints in a unit volume. An alternate method to find \(J_{v}\) is from \(RQD\) or the joint frequency, as given in Equations (17) and (18) [22; 47]:  
\[RQD=110-2.5\times J_{v} \tag{17}\]  
\[J_{v}=K_{1}\times N_{1}, \tag{18}\]  
where \(K_{1}=2\) for the average condition, and \(N_{1}\) is the number of joints per unit length (joint frequency). Joint frequency can be calculated directly or determined indirectly. The first indirect methodology used for fracture frequency is the mathematical equation between \(RQD\) and fracture frequency [50], which is given as follows:  
\[RQD=100\times e^{-0.1\times\lambda}\times(1+0.1\times\lambda), \tag{19}\]  
where \(\lambda\) is the joint frequency. Another method to determine the joint frequency is to use \(RQD\) and the joint spacing rating, by using Equation (20) [51] as follows:  
\[N_{f}=(RQD_{rating}+Spacing_{rating}-8)\times\frac{40}{32}, \tag{20}\]  
where \(N_{f}\) is the fracture frequency rating. The joint frequency can be calculated from \(N_{f}\) using a graph of rating for the discontinuity density [52]. In this case, when the data are available both in terms of the number of joint sets and the average spacing between the joints of each joint set, the joint frequency is determined using Equation (21) as follows [46]:  
\[\lambda=\frac{\text{Number of joint sets}}{\text{Average spacing between joints (m)}}. \tag{21}\]  
The details of the ground conditions based on the \(RMi\) system for all of the sections of the four projects are summarized in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{4}{c}{**Project**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline UCS (MPa) & 37.5–75 & 40–100 & 82 and 86 & 54 and 75 \\ \(jR/jA\) & 5 & 0.375–1.5 & 5 & 5 \\ \(jL\) & 1 & 1 & 1 & 1 \\ \(J_{v}\) & 5.0769–42.769 & 6.7–49.77 & 6.8–87.0 & 7–37.05 \\ \(V_{b}\) (m\({}^{3}\)) & 0.00046–0.275 & 0.00029–0.1194 & 0.000055–0.1142 & 0.00071–0.105 \\ \(D_{b}\) & 0.0772–0.65038 & 0.066–0.492 & 0.038–0.485 & 0.0891–0.4717 \\ \(JP\) & 0.05697–0.31638 & 0.00314–0.099 & 0.032–0.25 & 0.0639–0.244 \\ CF & 17.175–159.009 & 19.659–145.9 & 19.375–247.67 & 7.84–41.52 \\ \(C_{0}\) & 1–3 & 1–3 & 1–3 & 2 and 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Ground conditions (\(G_{c}\)) and size ratios (\(S_{r}\)) for all of the tunnel sections of the four tunnels.