# Structural Evaluation of Cable Bolts under Static Loading
## 3 Materials and Methods
### Material Properties  
Material properties were defined for the steel bolt and concrete. The parametric study included the analysis of varying steel yield and ultimate strength. Therefore, as part of the calibration, the primary material properties that were included in the ACARP C27040 paper are referenced. The ID SUMO Cable Bolt material properties are summarised below in Table 3.  
ABAQUS requires yield strength, \(f_{y}\), and ultimate strength, \(f_{u}\), as a pressure load. To convert force loads to a pressure, the yield strength was divided by the undeformed cross-sectional area. The steel bolt has been defined to fracture; hence, elastic, plastic and shear damage material properties were defined. Furthermore, the steel density, Young's modulus and Poisson's ratio were also defined. The basic steel properties have been obtained through the OneSteel 7 th edition steel catalogue [43]. Please see Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{
\begin{tabular}{c} **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} } & \multicolumn{2}{c}{**Elastic Modulus**} & \multicolumn{2}{c}{**Poisson’s**} & \multicolumn{2}{c}{**Fracture**} & \multicolumn{2}{c}{**Shear Stress**} & \multicolumn{1}{c}{**Strain**} \\  & & **(MPa)** & **Ratio** & **Strain** & **Ratio** & **Rate** \\ \hline
**Value** & 7800 & 200,000 & 0.3 & 0.13 & 0.7 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Steel bolt material properties. Figure 2: A 500 mm \(\times\) 500 mm \(\times\) 500 mm concrete box. To replicate actual stress-strain plastic deformation, the Giuffre-Menegotto-Pinto model with isotropic strain hardening was utilised to describe the transition from the elastic slope to the plastic deformation region. [PERSON] and [PERSON] (2012) [44] adopted this equation to model steel behaviour of various steels. \[\sigma^{*}=be^{*}+\frac{(1-b)\varepsilon^{*}}{\left(1+\varepsilon^{*R}\right)^{ 1/k}} \tag{1}\]  
The steel elastic and plastic behaviour was calculated in accordance with the AS4100:2020 Steel Structures codes. To accurately reflect the actual behaviour of steel undergoing plastic deformation, the true stress and strain must be utilised. The engineering stress-strain relationship fails to factor in the constantly changing cross-sectional area of the material and hence is simply applied to the initial cross-sectional area prior to any deformation. The following equations were utilised to account for the reducing cross-sectional area, thus giving the true stress and true strain equations [45,46]. \[\sigma_{T}=\sigma_{E}(1+\epsilon_{E}) \tag{2}\]  
\[\epsilon_{T}=\ln(1+\epsilon_{E})\]  
where the subscript \(E\) represents engineering stress/strain and the subscript \(T\) represents true stress/strain. As part of achieving calibration, this was necessary to ensure the accuracy of testing as it was compared with the experimental results obtained from ACARP Project C27040. A comparison of the engineering and true stress-strain curve for the calibrated model can be seen below in Figure 3.  
To induce a fracture, ABAQUS requires shear damage properties for damage for ductile metals to be defined and hence, a fracture energy is required. Fracture energy is defined as the required energy per unit area to change a fracture surface from an initial unloaded state to complete separation [47]. This is calculated as the work carried out to achieve fracture and effectively calculate the area underneath the respective stress-strain curve [48]. A Python code was written to determine the specific fracture energy for all steel grades (see Appendix A). The fracture energy for the calibrated model was calculated as 336,302 N/mm. The fracture energy will vary according to the steel grade. Figure 3: Engineering vs. true stress–strain curve. Concrete was selected as a substitute material for the rock layers as they possess relatively similar material properties. The various blocks are representative of layers of rocks which the cable bolt penetrates.