## 3 Simulation Analysis
### Interaction between Steel and Concrete  
This study modelled the interaction between the steel bar and concrete block using the contact modelling option with cohesive behavior provided in ABAQUS. In this modelling technique, a bond-slip law calibrated from the experimental results was used to represent the interaction between the steel bar and concrete block. In reality, the interaction between the steel bar and concrete block was induced by the locking phenomenon between the ribs of steel bar with surrounding concrete and bond strength. In this research, the proposed modelling approach allowed us to carry out the simulations in a simple manner wherein these two effects were accounted for in a cohesive model. Hence, there was no need to model the ribs of steel bars, which could complicate the simulation model. To fully describe the bond strength-slippage response, the interaction condition was modelled by adopting the normal surface-based cohesive behavior model and the normal contact behavior and damage option. The concrete and steel surfaces were assigned to master and slave surfaces, respectively, as illustrated in Figure 5. The hard contact condition was selected to ensure that there was no penetration between the two surfaces. At the same time, the damage parameters were defined to represent the softening stage in the stress-slip diagram, wherein a reduction in bond strength was observed. Figure 5: Bond stress observed in the contact surfaces between the concrete and steel bar. In ABAQUS, the bond-slip relation could be modelled using a two-stage traction-separation law as described in Figure 6. In the first stage, the traction stresses evolved linearly with the increase of separations and reached the ultimate values \(\left(t_{n^{\prime}}^{0},t_{s^{\prime}}^{0},t_{t}^{0}\right)\). After that, bond strength damage commenced and was characterized by damage parameters to represent descending trends until the tractions became null. The first stage could be considered an elastic response and was described by an elastic constitutive matrix relating tractions stresses and separations. The uncoupled elastic constitutive matrix was chosen and represented as follows  
\[\mathbf{t}=\left\{ \begin{aligned} t_{n}\\ t_{s}\\ t_{t}\end{aligned} \right\}=\left[ \begin{aligned} K_{nn}& 0& 0\\ 0& K_{ss}& 0\\ 0& 0& K_{tt}\end{aligned} \right]\left\{ \begin{aligned} \delta_{n}\\ \delta_{s}\\ \delta_{t}\end{aligned} \right\}=\mathbf{K}\mathbf{\delta} \tag{8}\]  
where \(\mathbf{t}\) is the nominal traction stress vector, its components \(t_{n}\), \(t_{s}\), and \(t_{t}\) represent the normal and the two shear tractions corresponding to the displacements related to the normal (\(\delta_{n}\)) and transversal directions (\(\delta_{s}\), \(\delta_{t}\)), respectively, in the separation vector \(\mathbf{\delta}\). \(K_{nn}\), \(K_{ss}\) and \(K_{tt}\) are the stiffness coefficients in stiffness matrix \(\mathbf{K}\). The values of these coefficients could be obtained by the following approximation formulations [42; 43]. \[K_{ss}=K_{tt}=\frac{\tau_{\max}}{s_{1}} \tag{9}\]  
\[K_{nn}=100\frac{\tau_{\max}}{s_{1}} \tag{10}\]  
where \(\tau_{\max}\) is the maximum shear stress and \(s_{1}\) is the displacement when \(\tau_{\max}\) is reached. These values are calibrated with respect to reinforcing bar diameter \(d\) and concrete compressive strength \(f_{c}\) based on regression analysis of experimental results. The calibrated formulas used to predict the maximum shear stress and corresponding slip are given as follows:  
\[\tau_{\max}=1.1132\bigg{(}\frac{c}{d_{b}}\bigg{)}^{1/5}\sqrt{f_{c}^{\prime}} +1.8868\ \ (\text{MPa}) \tag{11}\]  
\[s_{1}=-0.0074f_{c}^{\prime}\sqrt{\frac{d_{b}}{c}}+1.2825\ \ (\text{mm}) \tag{12}\]  
where \(f_{c}^{\prime}\) is in MPa and \(d_{b}\) is in mm. The damage initiation, which marked the beginning of the cohesive response's degradation, was defined using a maximum nominal stress  
Figure 6: Traction-separation response. criterion.