## 2 Materials and Methods
### Modelling Development and Employment Methodology
#### 2.3.2 Numerical Modelling  
To support this research investigation, a 3D finite element analysis (FEA) of a rock bolt was conducted. The aim of the model was to create a better representation of observed laboratory behaviour. Our research group previously utilized a 2D model in RS2 [50], but we struggled to capture the response of full-length members [18]. Although the short embedment model (100 mm) performed well, when the length increased to 500 mm, the model did not accurately match the collected results. This is because there is a shift between linear and exponential decay, as seen in the laboratory results [9], which was not captured by the model. For the RS2 model, the joint's strength and stiffness are explicitly defined; therefore, to improve the model a frictional contact model was utilized, which defines the interaction of the two bodies by interpreting the interaction of the two surfaces where they can slide and separate. This preliminary investigation to utilizing this joint definition was conducted on ANSYS. For these purposes, a 3D linear elastic solution was developed and compared to laboratory collected results. The methodology, results and recommendations are provided. Although not within the scope of this investigation, when an understanding of the behaviour is established, laboratory-collected parameters can be integrated into excavation modelling software, which ultimately leads to better representation [51].  
This modelling was split into three stages. Models of 100 mm, 250 mm, 500 mm and 750 mm (featuring two different rib spacings) were analyzed. Each stage features the creation of a small, ribbed section to establish and verify meshing, loading, boundary conditions and contact analysis. All stages of the model featured three materials: the bolt, the grout and the confining pipe. The selected material properties can be seen in Table 5. Load was applied to the bolt and load was transferred through material contact, as shown in Figure 3. The bearing rig utilized in the laboratory testing was represented by a boundary condition. Due to the elastic loading, no failure of the material should occur and no cracking can occur as this model utilizes a constitutive relationship that is linear elastic. Due to this, it was expected that the movement of the rebar will be associated with dilation of the grout and pipe, where high stress will be seen at the grout-rebar interface. Models were created over various lengths to compare their behaviour against previous models and existing laboratory results [18], and to results collected during this investigation. The coefficient of friction between the rebar and the grout was set at 0.57 [52] and the contact between the pipe and the grout was modelled as a bonded contact as typically minimal movement occurs at this interface both in the laboratory and in the field [13; 18]. The models were generated in SolidWorks [53] using geometric simplifications, which were investigated. Solid modelling was conducted to capture the gradient stress response through the thickness of the materials. Taking advantage of the symmetry, a half model was built along the longitudinal axis. The materials were meshed with higher-order tetrahedral and brick elements (Solid186, Solid 187, load applied using Surf 154 and contact surfaces were Targe170 and Conta174). Mesh sizing was selected by convergence studies and optimized by splitting materials into sections in which meshing size matched the complexity. Contact was conducted using an unsymmetrical interface and an augmented LaGrange method, with a stiffness factor of 3 for the frictional (grout-rebar contact) and 10 for bonded (and 15 for later models). The models were run as static non-linear analyses. Load steps were utilized both to collect results at specific loads and to aid in convergence. However, pre-tensioning was not required to aid in convergence due to the model constraints and construction method. Weak springs were turned off and large deflections were left on. An iterative solver was used.