# Experimental Research on Bond Behavior between GFR Bars and Stirrups-Confined Concrete
## 5 Bond Stress-Slip Relationship Model Between GFRP Bar and Stirrup-Confined Concrete  
The cracking behavior of RC structures under tension is mainly governed by the relative slip relationship between the bar and the concrete. Moreover, most of structural problems are dealt with at the serviceability limit state. Thus, a refined modeling of the \(\tau\)-\(s\) curve is only needed for the ascending branch (i.e., for slip less than \(s_{\text{m}}\)) [21]. In previous research, primarily the Malvar model [18], BPE, and CMR models [19; 20] are used for the analysis of the bond behavior of FRP bars. When compared with the Malvar model, the BPE model and CMR models are simple and convenient for application [44], thus these two models will be the primary focus in this paper. Equation (2) describes the ascending branch of of the BPE model and the same branch of the CMR model by Equation (3). \[\frac{\tau}{\tau_{\text{max}}}=\left(\frac{s}{s_{m}}\right)^{\alpha} \tag{2}\]  
\[\frac{\tau}{\tau_{\text{max}}}=\left(1-\exp\!\left(-\frac{s}{s_{r}}\right) \right)^{\beta} \tag{3}\]  
where \(\alpha\) is the parameter for the BPE model and \(\beta\)\(s_{r}\) are the parameters for CMR model, which can be determined from curve fitting of test results. Many studies indicate that the parameters that were outlined in the BPE and CMR models, as obtained from curve fitting based on experimental data, are affected by the surface characteristics of GFRP bars. Table 4 shows the values of BPE and CMR models parameters that are based on surface characteristics. The values reported in Table 4 also confirm that the trend of the \(\tau\)-\(s\) constitutive law is strongly dependent on the surface characteristics of GFRP bars. At present, no scholar in this field is yet to study the \(\tau\)-\(s\) model between the GFRP bar and the stirrup-confined concrete. With the results of the pullout tests that were obtained from specimens with C1 strength, curve fitting has been performed utilizing the BPE and CMR models. Figure 14 shows the distribution of fitting parameters \(\alpha\)\(\beta\)\(s_{r}\) based on different surface characteristics. The high variation in Figure 14 revealed that the values of these parameters in the BPE and CMR models are highly affected by both failure mode and stirrups. The parameters \(\alpha\)\(\beta\)\(s_{r}\) which were obtained from specimens with stirrups (R1-C1-20-Y, R2-C1-16-Y, R2-C1-20-Y, R3-C1-16-Y, R3-C1-20-Y, R4-C1-16-Y, and R4-C1-20-Y) are ostensibly different to those that were obtained from specimens without stirrups (R1-C1-20-N, R2-C1-16-N, R2-C1-20-N, R3-C1-16-N, R3-C1-20-N, R4-C1-16-N, and R4-C1-20-N) and  
Figure 13: The ratio of average bond strength and corresponding average slip of specimens with and without stirrups (**a**) \(t^{*}\) (Y) / \(t^{*}\) (N); and, (**b**) \(s^{*}\) (Y) / \(s^{*}\) (N). the specimens that underwent pullout failure (R1-C1-8-N, R1-C1-12-N, R1-C1-16-N, R2-C1-8-N, R2-C1-12-N, R3-C1-8-N, R3-C1-8-N, R3-C1-12-N, and R4-C1-12-N). Figure 15 shows the box plots for parameters \(\alpha\)\(\beta\)\(s_{\mathrm{r}}\) fitted by stirrup-containing specimens test results with regard to specific surface characteristics. Medians of \(\alpha\)\(\beta\)\(s_{\mathrm{r}}\) were obtained as 0.2205, 0.3468, and 0.775 for R1 bars, 0.2501, 0.445, 0.715 for R2 bars, 0.18, 0.635, 0.72 for R3 bars. Comparisons between the experimental and analytical \(\tau\)-s curves are shown in Figure 16. The analytical curves that were obtained by using the parameters fitted to its experimental counterpart are represented by BPE and CMR models, and the analytical curves obtained by using the medians of parameter are represented by BPE* and CMR* Models.