# Structural Evaluation of Cable Bolts under Static Loading
## 3 Materials and Methods
### Mesh Generation  
Detailed meshing is crucial in developing an accurate model that will produce accurate results. Effectively, by reducing the size of the mesh, ABAQUS can process the behaviour of individual elements more precisely as interpolation is utilised across a lesser distance to calculate stresses and deformations, thus eliminating imprecise estimations. However, by increasing the fineness of the mesh, more elements are created, and thus the computational time is inevitably increased as the FEA needs to run more equations. Therefore, is it important to differentiate between parts of the model which require a fine mesh and parts that require a coarse mesh.  
The inbuilt ABAQUS auto-meshing function generates a mesh based on the seeding size that is specified. This creates a mesh size of specified seed size that does not consider the irregularity of the mesh. Therefore, partitions were manually added around areas of greater concern such as contact points, areas of significant deformation and high stress concentrations to create an ideal mesh. Figure 4 displays the difference in mesh regularity between the auto-meshing function with and without any partitions.  
Figure 4: Without partitions (**left**); with partitions (**right**).  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Density (kg/m3)**} & \multicolumn{2}{c}{**Elastic**} \\ \cline{2-3}  & **Elastic Modulus (MPa)** & **Poisson’s Ratio** \\ \hline
**Value** & 2400 & 33,346 & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Concrete material properties.  
As the mid-section of the bolt is the primary region of concern as it deflects, the mesh around this area was made twice as fine as the first and last third of the bolt. Furthermore, this was extended to the 50 mm gap between the concrete blocks. The mesh can be seen in Figure 5 below.  
When modelling steel and concrete members with ABAQUS, one of the main issues is addressing convergence due to the extensive number of contacts. This is particularly prevalent in the double shear test model at the contacts between the cable bolt and the concrete boxes. Therefore, the 8-node linear brick element, C3D8R, with reduced integration and built-in hourglass control will be used. The linear brick element contains a single integration point at the centre of the brick with 8 nodes forming the \(1\times 1\times 1\) cube. The reduced integration model is ideal when subject to plastic behaviour [52]. However, in conjunction with this, it is necessary to apply hourglass control because of the reduced stiffness of the reduced integration model. This will aid the meshing process and the overall run time of the model. Figure 6 shows the schematic diagram of the C3D8R model whilst highlighting the effects of hourglass control.  
Mesh size was determined by comparing the trends of numerical and experimental data. The coarsest mesh that would minimise computational time and produce accurate results was selected. Seeding sizes of 10%, 15%, 20% and 25% of the bolt's diameter were selected as the fine mesh portion, whilst the outer boundary was assigned double this value for static testing. For dynamic testing, an optimal mesh was selected by considering seeding sizes of the dynamic load. The results of various seeding were compared against the experimental results and can be seen in Figure 7.  
Figure 5: Overall mesh of the bolt (**left**); cross-section of the bolt (**right**).  
Figure 6: C3D8R linear brick element (**a**) and effects of hourglass control (**b**).  
The computational times for various seeding sizes were recorded in Table 6 below.  
Final mesh sizes were chosen based on the effects of a finer mesh and computational times. Excess deflections were observed when meshes of 5% and 10% were applied to the sphere. Despite reduced computational times, the coarse mesh presented too many inaccuracies. A 2% mesh size was selected due to the excessive computational time of 1% with little difference in deflection. A summary of seed sizing of the calibrated model is given below in Table 7.