# Bonding Performance of Steel Rebar Coated with Ultra-High-Performance Concrete
## 3 Results and Discussion
### Analytical Modelling of UHPC-Coated Steel Rebars
#### 3.4.2 Predicting Bond Stress-Slip Behavior  
Bond behaviors between the reinforcing bar and concrete matrix generally describes three kinds of properties [43]. First, the chemical bond created in the interface causes adhesion resistance. Second, the frictional resistance against the slip growth in the interface is created. Third, the irregularity of the interface from the mechanical interlocking attributes to the difference in behavior of the bond-slip curve. In this study, an additional interface from the UHPC coating was created and its effect on the debonding mechanism against the bond-slip behavior was addressed by analyzing previously used bond behavior models. The BPE model [44] for steel bars was developed to explain the bond-slip behavior, as shown in Figure 13:  
\[\mathbf{\tau}=\mathbf{\tau}_{max}\Big{(}\frac{S}{S_{1}}\Big{)}^{\alpha}\ \ \text{for}\ S\leq\mathbf{S_{1}} \tag{3}\]  
\[\mathbf{\tau}=\mathbf{\tau}_{max}\ \text{for}\ \mathbf{S_{1}}\leq\mathbf{S}\leq\mathbf{S_{2}} \tag{4}\]  
\[\mathbf{\tau}=\mathbf{\tau}_{max}-\mathbf{K_{d}}\ (S-\mathbf{S_{2}})\ \text{for}\ \mathbf{S_{2}}<S\leq\mathbf{S_{r}} \tag{5}\]  
\[\mathbf{\tau}=\mathbf{\tau}_{r}\ \text{for}\ S\geq\mathbf{S_{r}} \tag{6}\]  
where \(\tau\) and \(S\) are defined as the bond stress and the corresponding slippage, respectively; \(\alpha\) and \(K_{d}\) are the parameters that can be determined from the curve fitting of the experimental results; \(\tau_{max}\) and \(\tau_{r}\) are the maximum bond stress and the residual bond stress, defined as the bond stress and the corresponding slippage, respectively; \(S_{1}\), \(S_{2}\), and \(S_{r}\) are the corresponding slippage values for the maximum bond stress, the border of the maximum bond stress plateau, and the residual bond stress, respectively. The other model for representing the bond-slip behavior is the CMR model [45], defined as:  
\[\mathbf{\tau}=\mathbf{\tau}_{max}\Big{(}\mathbf{1}-\mathbf{e}^{\frac{-\alpha}{\alpha_{r}}} \Big{)}^{\beta}\ \text{for}\ \mathbf{S}\leq\mathbf{S_{1}} \tag{7}\]  
where \(\mathbf{s_{r}}\) and \(\beta\) are parameters that can be determined from the curve fitting coefficients. Figure 14 compares the results of the predicted BPE and CMR models with the experimental bond-slip behavior. The increase in the structural steel fibers reduced the slippage at the maximum load and increased the maximum bond stress. The predicted CMR models proposed by [PERSON] et al. [45] matched well with the bond stress-slip response for the UHPC-coated steel rebars. The results using UHPC as a concrete matrix agreed well with the results reported by [PERSON] et al. [15]. The difference from [PERSON] et al. 's results was that the UHPC was thinly coated on the steel rebar or composed the entire matrix. It was judged  
Figure 12: Normalized bond strengths of UHPC-coated steel rebars. that the experimental value from the CMR model was predicted well, regardless of the composition of the concrete matrix in the section where the maximum load was reached. The parameters used for predicting BPE and CMR models are arranged in Table 6.  
The slip values (\(S_{1}\), \(S_{2}\)) corresponding to 95% of the maximum bond stress were determined to define the second stage of the BPE model using Equation (8) by considering the effects of the UHPC coating and steel fiber volume in the concrete matrix. These points represent the borders between stage 1 and stage 2 as well as stage 2 and stage 3. The parameters of the proposed stages 2-4 (Equations (9) and (10)) were calculated and are reported in Table 7. Figure 15 presents a proposed model from the ascending stage to final the stage for the pullout behavior of the UHPC-coated steel rebar, while Figure 16 shows the predicted pullout curve compared to the experimental results for all specimens. Figure 14: Comparison of BPE (a) and CMR (b) models for steel rebar coated and reinforced with UHPC.