## 2 Materials and Methods
### Modelling Development and Employment Methodology  
#### 2.3.1 Analytical Modelling  
The methodology of strain capture created by this research group allows for unparalleled comparison to existing analytical models. One of the older but widely used models was presented by [PERSON] and [PERSON] [16; 23; 37]. The simplified version, which can be seen in Equation (1), was compared. A modern method which aimed to present an easily applicable formula will also be compared. This model was presented by [PERSON] [11] and can be seen in Equation (3). Inherently, neither model captures alternative rib spacing; however, [PERSON]'s model does include the effects of grout annulus. These presented models will be compared to all collected strain data at 10, 50 and 100 kN to understand their performance across a wide range of loading.  
\[\varepsilon(\mathrm{x})=\varepsilon_{\mathrm{o}}\mathrm{e}^{-\alpha\mathrm{X}} \tag{1}\]  
where \(\alpha\) is a dimensionless parameter related to the deformation properties of the grout and the rock, Equation (2):  
\[\alpha^{2}=\frac{2\mathrm{G}_{\mathrm{r}}\mathrm{G}_{\mathrm{g}}}{\mathrm{E}_ {\mathrm{b}}\left[\mathrm{G}_{\mathrm{r}}\mathrm{In}\left(\frac{\mathrm{d}_{ \mathrm{g}}}{\mathrm{d}_{\mathrm{b}}}\right)+\mathrm{G}_{\mathrm{g}}\mathrm{ In}\left(\frac{\mathrm{d}_{\mathrm{o}}}{\mathrm{d}_{\mathrm{g}}}\right)\right]} \tag{2}\]  
E\({}_{\mathrm{b}}\) is the modulus of elasticity of the bolt, taken as the theoretical value of 200 GPa;  
G\({}_{\mathrm{r}}\) is the shear modulus of the confining medium, G\({}_{\mathrm{r}}\) = 77.5 GPa, from theoretical values of [PERSON]'s modulus (200 GPa) and [PERSON]'s ratio (0.29);  
G\({}_{\mathrm{g}}\) is the shear modulus of the grout, calculated for E and v, which were obtained from cement testing (GPa);  
X is the distance divided by the bolt radius (m/m);  
\(\varepsilon_{\mathrm{o}}\) is the theoretical strain at the loading point (mm/mm);  
d\({}_{\mathrm{g}}\) is the diameter of the borehole, taken as the inner diameter of the pipe (m);  
d\({}_{\mathrm{b}}\) is the diameter of the bolt (m);  
d\({}_{\mathrm{o}}\) is the diameter of the circle of influence of the rock bolt in the rock, such that outside of this diameter the influence of the bolt disappears (a representative value of ten times the dg was used, as per [16] (m))  
\begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **Property** & **Value** & **Source** \\ \hline \multirow{4}{*}{Cement grout round 2} & [PERSON]’s modulus & 12.4 GPa & Testing (1 sample unload and reload 3 times) 2 \\  & Compression strength & 39.5 MPa & Testing (3 samples) 1 \\  & Poisson’s ratio & 0.28 & Testing (1 sample unload and reload 3 times) 2 \\  & Tensile strength & 3.12 MPa & Testing (3 samples) 3 \\  & Density & 1.89 g/mL & Measured \\ \hline \multirow{4}{*}{Cement grout round 3} & [PERSON]’s modulus & 13.4 GPa & Testing (1 sample unloaded and reload 3 times) 2 \\  & Compression strength & 36.5 MPa & Testing (2 samples) 1 \\  & Poisson’s ratio & 0.29 & Testing (1 sample unloaded and reload 3 times) 2 \\  & Tensile strength & 3.28 MPa & Testing (3 samples) \\  & Density & 1.91 g/mL & Measured \\ \hline \end{tabular} \({}^{1}\) Conducted with guidance from ASTM-C39 [47], 2 from ASTM-C39 [47], 3 and from ASTM-C496 [48].  
\end{table}
Table 4: _Cont._\[\varepsilon(x)=\frac{a}{b}\frac{1}{1+e^{-\frac{x-x_{0}}{b}}} \tag{3}\]  
where,  
\[b=\frac{E\pi d_{b}^{2}}{4K_{o}} \tag{4}\]  
\[a=\frac{P_{max}}{K_{o}} \tag{5}\]  
\[x_{0}=L+b\ln(\frac{a}{b}\frac{Erd_{b}^{2}}{4P}-1) \tag{6}\]  
a and b coefficients determined from testing (mm);  
E is the modulus of elasticity of the bolt, taken as 200 GPa;  
P\({}_{Max}\) is the maximum applied load (kN), or if unavailable the expected max;  
P is the loading force (kN);  
L is the total length of the grouted bolt (mm);  
d\({}_{b}\) is the diameter of the bolt (mm);  
x is the distance along the bolt (mm);  
K\({}_{o}\) is the secant stiffness taken at 100 kN with displacement from DOS (kN/mm).