## 2 A Hypothetical Example Application of the V-GSI Chart in a Tunnel
### Vertical and Horizontal Axis Quantification on the V-GSI Chart  
The overall V-GSI rating for the tunnel rock mass shown in Figure 2 can be calculated using Equation (10) [1]:  
\[\text{V-GSI rating}=1.5\times\text{J}\text{Cond89}\ +(50-8.5\ln(\text{VFC})) \tag{10}\]  
The latter half of Equation (10), \((50-8.5\ln(VFC))\), provides the vertical axis rating on the V-GSI chart using the _VFC_ data in Table 3. The former half of the equation provides the horizontal axis rating, when the [PERSON] [22] joint condition assessment table is used to calculate _JCond89_, and this is multiplied by the correlation factor of 1.5 on the V-GSI chart (_JCond89_). A correlated Jr/Ja scale is also provided in Figure 1 for practitioners that prefer the Q classification system [1,23].  
In practice, it is easier to (a) plot the mean rock mass _VFC_ and the confidence limits on the scaled vertical column, as shown in Figure 1, and (b) read off the rating in the adjacent vertical rating column included on the V-GSI chart.  
This quantification is overlaid on Figure 1. The mean _VFC_ of 1.09 fractures/m\({}^{3}\) with 95% confidence on the mean of 0.72 fractures/m\({}^{3}\) and 2.56 fractures/m\({}^{3}\) results in a mean vertical axis rating of 49, and lower and upper 95% confidence ratings of 42 and 53, respectively, as shown on the vertical axis on Figure 1.  
In this example, \(1.5\times JCond89\) has been assumed to have an average rating of 30 and a range of between 25 and 35 as shown in Figure 1, recalling that _JCond89_ includes assessments of joint persistence, aperture, roughness, infill and weathering.  
The overall mean V-GSI rating of 79 is obtained at the intersection of the vertical and horizontal axes ratings (30 + 49), as shown on Figure 1. A box of associated V-GSI limits for the tunnel rock mass example is also shown on Figure 1 with a lower V-GSI limit of 67 and an upper limit of 88, respectively. These results are also summarized in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & _VFC_ & _VFC_ & Rating & _JCond89 Rating_ & **V-GSI Rating** \\ \hline Lower Limit & 0.70 & 53 & 35 & 88 \\ Mean & 1.09 & 49 & 30 & 79 \\ Upper Limit & 2.56 & 42 & 25 & 67 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Summary of _VFC_ rating for the example tunnel heading shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Discontinuity Set**} & _Lower Limit VFC_ & _Mean VFC_ & _Upper Limit VFC_ \\  & _Contribution_ & _Contribution_ & _Contribution_ \\ \hline J1 & 0.40 & 0.65 & 1.72 \\ B & 0.30 & 0.44 & 0.83 \\
**0.702 (LL VFC)** & **1.09 (Mean VFC)** & **2.56 (ILL VFC)** \\ \hline \hline \end{tabular} Note: If 1.96 \(\times\) standard error is greater than or equal to the mean St for a set, then the smallest true spacing value can be substituted in place of the standard deviation (\(\partial S_{ti}\)) to avoid a negative or zero upper limit value is not meaningful. Alternatively, when there are too few data measurements, \(\partial S_{ti}\) may be approximated as the highest conceivable spacing value minus the lowest conceivable spacing value divided by Nsigma (where Nsigma is less than 6) to avoid a negative number [1]. LL—Lower Limit; UL—Upper Limit.  
\end{table}
Table 3: _VFC_ data for the example shown in Figure 3.