## 4 Discussion
### Balanced Prestressing Reinforcement Ratio for PSCB-IUCFRP  
The ACI 440.4R specifications introduced the balanced prestressing reinforcement ratio \(\rho_{\text{b}}\) for bonded members. With a balanced ratio, the tendon ruptures simultaneously with compression failure of the concrete. In this section, the balanced prestressing reinforcement ratio \(\rho_{\text{b}}\) is derived for PSCB prestressed with internal unbonded tendons based on the proposed prediction equations. For beams with the balanced prestressing reinforcement ratio \(\rho_{\text{b}}\), the compression block is generally within the depth of the flange at the ultimate states. Therefore, the following derivation is based on the assumption that \(c_{\text{u}}<h_{\text{f}}\). According to the definition of \(\rho_{\text{b}}\), the ultimate tendon stress reaches the tensile strength\(f_{\text{pu}}\). Therefore, Equations (11) and (12) can be rewritten as Equations (14) and (15), respectively. \[f_{\text{pu}}-\sigma_{\text{pe}}==\frac{2.1d_{\text{p}}}{L_{0}}E_{\text{p}} \varepsilon_{\text{cu}}\big{(}d_{\text{p}}/c_{\text{u}}-1\big{)} \tag{14}\]  
\[0.85f_{\text{c}}b\beta c_{\text{u}}=A_{\text{p}}f_{\text{pu}} \tag{15}\]  
Figure 16: Analysis and prediction results of PSCB-IUCFRP with different segment sizes. Figure 15: Failure modes of PSCB-IUCFRP when failure cross-section not at the mid-span. With Equation (14), we can obtain the ultimate neutral axis height \(c_{\rm u}\), as shown in Equation (16). By substituting Equation (16) into (15), the balanced prestressing reinforcement area can be denoted as Equation (17). The expression of \(\rho_{\rm b}\) is obtained as Equation (18), the proposed balanced prestressing reinforcement ratio for PSCB-IUCFRP. The value of \(\rho_{\rm b}\) is influenced by \(L_{0}/d_{\rm p}\),\(f_{c}\), and \(\sigma_{\rm pe}\) parameters. An increase in \(f_{c}\) and \(\sigma_{\rm pe}\) will increase the value of \(\rho_{\rm b}\), while an increase in \(L_{0}/d_{\rm p}\) will decrease \(\rho_{\rm b}\). In Equation (18), if we set \(\kappa=1\), we can obtain similar equations for \(\rho_{\rm b}\) to those specified in ACI 440.4R for bonded prestressed members. The parameter \(\kappa\) in Equation (18) reflects the effects of the un-bonded property, which decreases the balanced prestressing reinforcement ratio \(\rho_{\rm b}\). \[c_{\rm u}=\frac{d_{\rm p}}{\frac{(f_{\rm pu}-\sigma_{\rm pe})L_{0}}{2.1d_{\rm p }F_{\rm p}\sigma_{\rm cu}}+1} \tag{16}\]  
\[A_{\rm p}=\frac{0.85f_{\rm c}b\beta c_{\rm u}}{f_{\rm pu}}=\frac{0.85f_{\rm c} b\beta d_{\rm p}}{f_{\rm pu}\left[\frac{(f_{\rm pu}-\sigma_{\rm pe})L_{0}}{2.1d_{ \rm p}F_{\rm p}\sigma_{\rm cu}}+1\right]} \tag{17}\]  
\[\rho_{\rm b}=\frac{A_{\rm p}}{bd_{\rm p}}=\frac{0.85f_{\rm c}\beta}{f_{\rm pu} \kappa\frac{(f_{\rm pu}-\sigma_{\rm pe})}{E_{\rm p}\sigma_{\rm cu}}+1},\, \kappa=\frac{L_{0}}{2.1d_{\rm p}} \tag{18}\]  
If the PSCB-IUCFRP beam is prestressed by a lower ratio of prestressing reinforcements than \(\rho_{\rm b}\), the beam will fail by tendon rupture. For cases with a higher ratio of prestressing reinforcements than \(\rho_{\rm b}\), then the failure mode will be concrete crushing. The balanced prestressing reinforcement area \(A_{\rm p}\) calculated by Equation (17) is 116 mm\({}^{2}\) for the second parametric analysis beam (ID = 2) listed in Table 1. The numerical model is then used to verify the prediction accuracy. Eight models are analyzed, using beam (ID = 2) as the reference beam and varying the parameter \(A_{\rm p}\) from 100 mm\({}^{2}\) to 135 mm\({}^{2}\). The trend in tendon stress and mid-span deflection is shown in Figure 17. For beams prestressed by tendons with less than 120 mm\({}^{2}\) area, the structural failure is caused by tendon rupture. Concrete crushing occurred at the ultimate state for beams with over 120 mm\({}^{2}\) tendon area. The numerical results show that the balanced prestressing reinforcement area can be considered \(A_{\rm p}=120\) mm\({}^{2}\). The prediction error of Equation (17) is only \((120-116)/120=3.33\%\).