## 2 Materials and Methods  
### RMR Classification System and Tunnel Support Design  
As established empirically for the tunnel support design by [PERSON] in 1973, the _RMR_ classification, also known the geomechanics classification, has been restructured since its establishment, either in its characterization or recommended support pattern or both [11; 16; 18; 19; 20; 21; 22; 23; 24]. The revised characterization criteria in different version of \(RMR\) is shown in Table 1[4]. The \(RMR_{89}\), which is the 1989 form of the \(RMR\), is still used as a design tool in rock mechanics, particularly for the classification of a rock mass during tunnel support design; furthermore, its latest version is 2014, which is known as \(RMR_{14}\). As a function of six parameters (\(R_{1}\)-\(R_{6}\)), the \(RMR_{89}\) is defined in Equation (1). These six parameters represent the rating for the intact rock uniaxial compressive strength (\(\sigma_{c}\)), rock quality designation (\(RQD\)), spacing of joints (\(\chi\)), condition of these joints, groundwater condition, and relation of tunnel orientation with respect to joint orientation, respectively. \[RMR_{89}=R_{1}+R_{2}+R_{3}+R_{4}+R_{5}+R_{6} \tag{1}\]  
\(R_{1}\)-\(R_{3}\) in Equation (1) can be calculated from Equations (2)-(4), respectively [13];  
\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]  
\[R_{1}=0.475\sigma_{c}^{0.626},\ (\sigma_{c}\geq 110\ \text{MPa}), \tag{2b}\]  
\[R_{2}=0.22 RQD-0.0002 RQD^{2}, \tag{3}\]  
\[R_{3}=2.281\times\ln(x)-3.41,\ (x=5\text{--200 mm}); \tag{4a}\]  
\[R_{3}=4.175\times\ln(x)-13.51,\ (x=200\text{--900 mm}); \tag{4b}\]  
\[R_{3}=6.250\times\ln(x)-27.55,\ (x=900\text{--2000 mm}). \tag{4c}\]  
In 2014, a revised structure was proposed for the \(RMR\) value calculation with some new parameters, as shown in Equation (5), where \(RMR_{b}\) is the basic \(RMR\); \(F_{0}\) is a parameter similar to \(R_{6}\) in Equation (1); and \(F_{s}\) and \(F_{e}\) are parameters that show the effect of the tunnel excavation method and stress-strain behavior at the tunnel face during their construction [19]. \[RMR_{14}=(RMR_{b}+F_{0})\times F_{s}\times F_{e} \tag{5}\]  
\(RMR_{b}\), \(F_{e}\), and \(F_{s}\) in Equation (5) can be calculated using Equations (6)-(8), respectively:  
\[RMR_{b}=R_{1}+R_{2-3}+R_{4}+R_{5}+R_{I} \tag{6}\]  
\[F_{e}=1+2\Big{(}\frac{RMR}{100}\Big{)}^{2},RMR<40; \tag{7a}\]  
\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]0  
\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]1  
\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]2  
\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]3  
The parameters \(R_{2-3}\) and \(R_{I}\) in Equation (6) are the ratings for the joint frequency and intact rock alterability. The joint frequency (\(\lambda\)) rating can be calculated using Equation (9). In Equation (8), \(ICE\) is the abbreviation of a Spanish word Indice de Comportamiento Elastico, which means index of elastic behavior, and this \(ICE\) is the function of the intact rock strength, in situ stresses, rock-mass quality, and tunnel shape and can be calculated using Equation (10) [25]. \[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]4  
\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]5\[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]6 \[R_{1}=0.126\sigma_{c}-0.0004\sigma_{c}^{2},\ (\sigma_{c}\leq 110\ \text{MPa}), \tag{2a}\]7  
where \(H\) = tunnel depth (m), \(K_{0}\) = virgin stress ratio, and \(F\) = shape coefficient. In terms of the support design, the major updates in the \(RMR_{89}\) application for tunnel support design include expressing the fiber-reinforced shotcrete thickness, rock bolt spacing, and length in the form of charts and equations [24] instead of tables [18].