## 4 Numerical Analysis  
### Numerical Modelling  
ABAQUS nonlinear finite element software was used to simulate all the experimentally tested specimens. Figure 9 shows the testing setup of the numerical models as conducted in the experimental program. In this simulation, only a quarter of the specimens were modelled by considering the symmetry configuration of the specimen length and width. This modelling technique can reduce computation time [24] without compromising the validation of the numerical results for a full model.  
The numerical models adopted concrete damage plasticity (CDP) model for the concrete material. Table 5 summarizes the concrete damage plasticity parameters. The concrete beam was modelled in C3D8R elements (8-node linear bricks with reduced integration). The concrete compression damage behavior parameters, which are compressive stress (\(f_{c}\)), and the corresponding inelastic strain (\(\epsilon_{c}\)\({}^{in}\)), were derived from [PERSON] and [PERSON] [25]. The concrete compression damage parameter (\(d_{c}\)) and concrete tension damage parameter (\(d_{t}\)) were derived from [PERSON] et al. [26]. The tensile stress (\(\sigma_{t}\)) and corresponding inelastic strain (\(\epsilon_{t}\)\({}^{in}\)) were derived from the Modified Tension Stiffening Model [27]. The density and Poisson's ratio of concrete were adopted as 2500 kg/m\({}^{3}\) and 0.2, respectively. The cube compressive strength of concrete of 21.56 MPa adopted in the model was based on experimental testing at 28 days age, \(f_{c}\)\({}^{\prime}\). The elasticity modulus of concrete was 26,245 MPa based on the formula suggested by [PERSON] [28].  
The high yield deformed steel bar was modelled in T3D2 (2-mode linear 3-D truss) whereas the mild steel plate and loading plate at support were modelled in C3D8R elements (8-node linear bricks with reduced integration). The stress-strain relation of both the high yield deformed steel bar and the mild steel plate were modelled in an elastic-perfectly plastic model with a gentle slope, as adopted by [PERSON] et al. [29]. Table 6 summarizes the properties of the steel reinforcement used in the numerical simulation work.  
The interaction between the concrete beam and high yield steel bar was modelled as an embedded region to reflect the non-slip reinforcement condition. Meanwhile, the interaction between the concrete beam and mild steel plate was modelled as surface-based cohesive behaviour. The traction-separation behaviour was selected with stiffness coefficients in normal and two tangential directions, \(K_{nn}\), \(K_{ss}\), and \(K_{tt}\), as 65 MPa/mm, 5 MPa/mm, and 5 MPa/mm, respectively [12]. The maximum separations at damage initiation in the normal and two tangential directions, \(\delta_{n}\)\({}^{max}\), \(\delta_{s}\)\({}^{max}\), and \(\delta_{t}\)\({}^{max}\) were taken as 0.025 mm, 0.06 mm, and 0.06 mm, respectively [12]. Damage evolution was employed  
Figure 9: Numerical model assembly.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Dilation Angle** & **Eccentricity** & \(f_{bol}\)/\(\epsilon_{o}\) & K \\ \hline Value & 38.5 & 0.1 & 1.16 & 0.667 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Concrete damage plasticity parameters.  
along with a linear softening law where the total displacement at failure was adopted as 2.5 mm [12].  
The element shape of the concrete beam was assigned as tetrahedral, whereas the mild steel plate and loading plate were assigned as hexahedral and meshed with a mesh size of 25 mm. The high yield steel bars were assigned as truss elements with a mesh size of 25 mm. Two-point loads were gradually applied onto the specimens in monotonic loading conditions, as shown in Figure 9.