## 4 Strengthening Frames with External Tendons
### Load-Deflection Relation  
Figure 17 clarifies the relation of the vertical loading to the maximum vertical deflection for the seven samples from the FE tests. The figure shows that the four externally strengthened samples, in which the lower part of the frame-beam was enhanced (F\({}_{1}\), F\({}_{2}\), F\({}_{5}\), and F\({}_{6}\)), exhibited the same mechanical behavior under loading before the cracking stage until the load and the deflection reached about 47 kN and 1.8 mm, respectively. Furthermore, the frame samples, in which the lower part of the frame-beam was not enhanced (F\({}_{3}\) and F\({}_{4}\)), could not exert a highly positive effect on the load-deflection curve, especially in the case of the F\({}_{3}\) sample (tendons in the negative moment zone of the frame-beam only), as shown in Figure 17b, because the tension of the tendons, in this case, dragged the column near the beam, resulting in additional moments rather thanreversible moments. For the unstrengthened sample (F\({}_{0}\)), the deflection increased rapidly after the initial cracks and the tensile reinforcements reached the yielding. The specimen subsequently failed with high ductility. On the other hand, the rest of the strengthened samples, except for (F\({}_{3}\)), were able to resist loading better than (F\({}_{0}\)) and delay the yielding of the tensile reinforcements due to the ability of the tendons to carry part of the new tensile stress resulting from the loading test. This dragged the load-deflection curve upward. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Cracking Stage**} & \multicolumn{4}{c}{**Yielding Stage**} & \multicolumn{4}{c}{**Ultimate Stage**} \\ \cline{2-10}
**Sample** & **Load** & **Max. ** & **Load** & **Max. ** & **Maximum. Column** & **Load** & **Max. ** & **Maximum. Column** & **Load** \\
**No. ** & **kN** & **Deflection** & **mm** & **kN** & **Deflection** & **mm** & **Stress****Mpa** & **kN** & **Deflection****mm** & **Stress****Mpa** & **kN** \\ \hline F\({}_{0}\) & 23 & 0.6 & 33 & 3.41 & 6.78 & 160 & 33.7 & 33.64 & 23 \\ F\({}_{1}\) & 47 & 1.8 & 175.6 & 24.69 & 30.5 & 185.7  
Figure 17a shows the effect of the inner deviator on the load-deflection curve by comparing the results of sample (F\({}_{1}\)) with sample (F\({}_{2}\)). It can be seen that when the load was less than the yielding load, the load-deflection curves of (F\({}_{1}\)) and (F\({}_{2}\)) were almost identical, but when the load increased again, the gap between the load-deflection curves of sample (F\({}_{1}\)) and sample (F\({}_{2}\)) appeared. This was because when the load was less than the yielding load, the vertical deflection of the sample at the mid-span was very small. Therefore, the external tendons and the surrounding middle deviators in sample (F\({}_{2}\)) were separated. Next, when the load exceeded the yielding stage, the vertical deflection of the sample at the mid-span increased rapidly and led to the external tendons attaching to the surrounding middle deviators; consequently, the effects of the deviators in the sample (F\({}_{2}\)) began to clear and improve its mechanical behavior. As a result, the ultimate load of sample (F\({}_{2}\)) recorded 189.7 kN, which was increased by a slight value (2.2%) compared to sample (F\({}_{1}\)), which recorded 185.7 kN due to the short span of the model. However, it is expected for this percentage to increase for large spans. Generally, the deviator exerted a beneficial effect on the ultimate load of the strengthened samples. Figure 17c shows the effect of the tendon profile on the load-deflection curve by comparing the results of sample (F\({}_{5}\)) with sample (F\({}_{6}\)). It can be seen that the differences between the load-deflection curves of (F\({}_{5}\)) and (F\({}_{6}\)) appeared in both the yielding and the ultimate load, where the (F\({}_{6}\)) sample recorded 185.1 kN and 205 kN, respectively.