## 3 Experimental Results
### Flexural Strength Analysis  
Flexural strength analyses were performed for the mid-span section of the tested beams. The analysis performed in this study was different from that performed for a beam with internal tendons since the internal tendons remain in place relative to the surrounding concrete under varying deflection. However, in this study, the deflection of the external unbonded steel along the beam length is different from that of the surrounding concrete [41; 42; 43; 44]. Therefore, the analysis required the calculation of the elongation of the external rod, which was dependent on the deflection at the deviator point [41]. The elongation (\(\Delta\varepsilon_{ps}\)) of the external rod was obtained from Equations (2)-(4) and Figure 12, as reported by [PERSON] et al. [5]. \[\Delta\varepsilon_{ps}=\frac{L_{p}-L_{p}}{L_{p}} \tag{2}\]  
\[L_{p}=\sqrt{\left(L/2-d_{h}\right)^{2}+\left(d_{p}-d_{v}\right)^{2}} \tag{3}\]  
\[L_{p}=\sqrt{\left\{L/2-d_{h}-\left(\overline{y}-d_{v}\right)\theta\right\}^{2} +\left(d_{p}-d_{v}+\delta\right)^{2}} \tag{4}\]  
where \(L_{p}\) and \(L_{p}^{{}^{\prime}}\) are the rod lengths before and after deformation, respectively; \(L\) is the span length; \(d_{h}\) is the horizontal distance from the support to the center of the anchorage; \(d_{p}\) is the effective depth of the post-tensioning rod at the deviator; \(d_{v}\) is the vertical distance from the top of the beam to the center of the anchorage; \(\overline{y}\) is the neutral axis depth; \(\theta\) is the rotational angle; and \(\delta\) is the deflection at the deviator. To obtain the value of the elongation (\(L_{p}^{{}^{\prime}}\)) at a particular loading state, the deflection (\(\delta\)) and rotation (\(\theta\)) at that state in Equation (4) should be calculated. At a specific ultimate  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Specific. **} & \multicolumn{4}{c}{**Before \& after PT**} & \multicolumn{4}{c}{**At First Peak**} & \multicolumn{4}{c}{**At Second Peak**} & \multicolumn{4}{c}{**At \(\varepsilon_{sw}\) = 0.003**} & \multicolumn{4}{c}{**At \(\varepsilon_{sw}\) = 0.005**} \\ \cline{2-13}  & \(k_{\text{before}}\) & \(k_{\text{after}}\) & \(k_{\text{after}}\) & \(k_{\text{after}}\) & \(k_{\text{before}}\) &  
state, these parameters can be determined using Equations (5)-(7) and the ultimate curvature of Figure 13[5; 45]. \[\delta_{u}=(\varphi_{u}L_{0})\frac{L}{2}-(\varphi_{u}L_{0})\frac{L_{0}}{2}=\left( \frac{LL_{0}}{2}-\frac{L_{0}^{2}}{2}\right)\varphi_{u} \tag{5}\]  
\[\theta_{u}=\varphi_{u}L_{0} \tag{6}\]  
\[\varphi_{u}=\varepsilon_{cu}/c \tag{7}\]  
where \(\delta_{u}\) is the ultimate deflection, \(L\) is the span length, \(L_{0}\) is the plastic hinge length extended on either side of the beam from the beam center (note that it is considered as 0.5\(d\)), \(d\) is the effective depth of the tension reinforcing bar, \(\varphi_{u}\) is the ultimate curvature (\(=\varepsilon_{cu}/c\)), and \(c\) is the neutral axis depth at the ultimate state. To calculate the deflection and rotation in Equations (5) and (6), the neutral axis depth \(c\) in Equation (7) is required. These interrelated values (\(\delta\), \(\theta\), \(c\), and \(\Delta\varepsilon_{ps}\)) should be obtained by an iteration procedure until all the conditions, including the strain compatibility, force equilibrium, and compatibility between the rod elongation and deflection are satisfied.