## 3 Flexural Capacity Prediction Methods and Applicability Analysis
### Proposals for the Flexural Capacity Prediction Equation  
The proposed flexural capacity prediction equation is based on an analysis of the failure mechanism. The failure mode of PSCB-IUCFRP, as determined by numerical analysis and experimental observation, is presented in Figure 12(a). Under ultimate states, the failure joint cross-section will open significantly, and concrete crushing will occur at the compression flange. Structural deformation is mainly caused by plastic region deformation at the failure joint cross-section. The other region is still elastic, and the joints are almost closed. Following [PERSON] et al. [43], tendon elongation mainly occurs at the failure joint cross-section. It is approximately equal to the width of the joint opening at the location of equal height. Therefore, the tendon elongation \(\Delta\) can be calculated by Equation (7), and the tendon strain increment can be expressed as Equation (8). \[\Delta=\theta\big{(}d_{\mathrm{P}}-c_{\mathrm{u}}\big{)} \tag{7}\]  
\[\Delta\epsilon_{\mathrm{P}}=\theta\big{(}d_{\mathrm{P}}-c_{\mathrm{u}}\big{)} /L_{0} \tag{8}\]  
According to the ultimate curvature distribution obtained from the numerical model, the actual distribution mode shown in Figure 5(a) can be simplified into Figure 12(b). The simplified curvature curve is uniformly distributed along the plastic region \(l_{\mathrm{P}}\) and set to zero for the other region. The structural rotation \(\theta\) can therefore be expressed as,  
\[\theta=\phi_{\mathrm{u}}l_{\mathrm{P}}=l_{\mathrm{P}}\epsilon_{\mathrm{cu}}/c _{\mathrm{u}} \tag{9}\]  
Substituting Equation (9) into Equation (8) and following [PERSON]'s law, we can obtain the tendon stress increment, as shown in Equation (4). This means that the derivation by [PERSON] et al. [43] is applicable to PSCB-IUCFRP. As mentioned above, their unsatisfactory predictions are mainly caused by the assumption of plastic hinge length \(l_{\mathrm{P}}=10.5\)\(c_{\mathrm{u}}\). This value was put forward by [PERSON] and [PERSON] [49] through the back-calculating method based on only eight monolithic specimens. [PERSON] and [PERSON] [52] collected the test results of 148 specimens and concluded that the ratio of \(l_{\mathrm{P}}/c_{\mathrm{u}}\) ranged from 4 to 45 and showed a sizeable scattering distribution. The verification in Section 3.2 shows that the plastic hinge length value is unsuitable for PSCB-IUCFRP. We assumed the plastic hinge length as a multiple of effective depth \(d_{\mathrm{P}}\), \(l_{\mathrm{P}}=\alpha d_{\mathrm{P}}\). The tendon stress increment can be denoted as Equation (10), where \(\epsilon_{\mathrm{cu}}\) can be taken as  
Figure 13: Failure mechanism analysis of PSCB-IUCFRP. 0.003 and \(c_{\rm u}\) can be calculated by combining Equation (10) and the axial force equilibrium equation [44]. \[\Delta\sigma_{\rm p}=E_{\rm p}\alpha d_{\rm p}\frac{\varepsilon_{\rm cu}}{c_{\rm u }}\big{(}\big{(}d_{\rm p}-c_{\rm u}\big{)}/L_{0}\big{)}=\frac{\alpha d_{\rm p}}{ L_{0}}E_{\rm p}\varepsilon_{\rm cu}\big{(}d_{\rm p}/c_{\rm u}-1\big{)} \tag{10}\]  
Interestingly, we find that Equation (10), from the failure mechanism analysis, shows a similarity with the bond-reduction method Equation (3). [PERSON] and [PERSON] [39] obtained \(\alpha=2.6\) for one-point loading and \(\alpha=5.4\) for two-point/uniform loading based on the statistical analysis of collected specimen test results. For safety reasons, they conservatively modified the values to \(\alpha=1.5\) for one-point loading and \(\alpha=3.0\) for two-point/uniform loading. The modified values were adopted for code-type format in ACI 440.4R [48]. The predictions in Figure 12b show that the \(\alpha\) value is conservative for one-point loading and overestimated for two-point loading. Moreover, the parametric analysis study in Section 2.3 shows that the loading types have a minor influence on the tendon stress increments and flexural capacity. The optimal \(\alpha\) value is back-calculated based on the derived Equation (10) and the numerical analysis results.