## 4 Theory of Flexural Bearing Capacity Calculation  
In contrast to internally prestressed structures, which facilitate accurate estimates of structural response via sectional analysis, constructions incorporating exterior prestressing present additional complexity. The force analysis of the reinforced beam is shown in Figure 11. In beams that are reinforced with externally prestressed CFRP tendons, the strain variations in these tendons are not aligned perfectly with the deformations of the concrete. Moreover, the stress increments in these externally prestressed CFRP tendons are directly correlated with the overall deformations of the structure. Beams with lower initial prestressing levels, particularly those containing initial cracks, demonstrate greater mid-span ultimate deflections, resulting in a significant secondary effect. The aforementioned secondary effects diminish the extent to which CFRP reinforcement enhances the ultimate flexural bearing capacity of the test beams. Consequently, beams that have lower levels of initial prestress and existing initial cracks exhibit a minor decrease in their ultimate flexural capacity. \[\mathrm{M_{u}}=\mathrm{f_{y}}\cdot\mathrm{A_{s}}\cdot\left(\mathrm{h_{0}}- \frac{\mathrm{x}}{2}\right)+\mathrm{k_{2}}\cdot\mathrm{f_{ps}}\mathrm{A_{pe}} \cdot\left(\mathrm{h_{pe}}-\frac{\mathrm{x}}{2}-\Delta_{\mathrm{u}}\right), \tag{1}\]  
\[\mathrm{x}=\frac{\mathrm{f_{y}}\cdot\mathrm{A_{s}}-\mathrm{f_{y}^{\prime}} \cdot\mathrm{A_{s}^{\prime}}+\mathrm{f_{ps}}\cdot\mathrm{A_{pe}}}{\alpha_{1} \cdot\mathrm{f_{c}}\cdot\mathrm{b}}. \tag{2}\]  
The stress \(\mathrm{f_{ps}}\) of the externally prestressed CFRP tendons in the limit state can be expressed as:  
\[\mathrm{f_{ps}}=\mathrm{f_{pe}}+\Delta\mathrm{f_{ps}}, \tag{3}\]  
\[\Delta\mathrm{f_{ps}}=\frac{\Delta\mathrm{L_{cfrp}}}{\mathrm{L_{cfrp}}}. \mathrm{E_{cfrp}}. \tag{4}\]  
Figure 10: Moment–CFRP tendon stress curve. As depicted in Figure 12, the deflection-moment behavior exhibited by the test beams can be approximated as a tri-linear response. Theoretically, after cracking occurs in a section, the moment of inertia of the cracked part applies only to the cracked portion, while the gross moment of inertia pertains to the section between the cracks. Nevertheless, this methodology is deemed impractical and incapable of guaranteeing a precise evaluation of the beam's deflection, as the true rigidity of the beam falls within the range of \(\mathrm{E_{c}I_{g}}\) and \(\mathrm{E_{c}I_{cr}}\). Currently, the standard ACI 440.4R [33] uses the effective moment of inertia approach for evaluating the deflection under a specified flexural moment. The definition of the effective moment of inertia, as stated in ACI 440.4R, can be expressed mathematically by Equation (5). When exceeding the cracking moment, the beam experiences discrete cracks, leading to variations in its flexural rigidity throughout its length. The effective moment of inertia essentially reflects this rigidity variation and represents the collective flexural rigidity of the beam as a whole. As the applied moment increases, the variable \(\mathrm{I_{e}}\) represents a gradual shift from the initial moment of inertia \(\mathrm{I_{g}}\) to the cracking moment of inertia \(\mathrm{I_{cr}}\)[15].