## 4 Strengthening Frames with External Tendons
### Ductility Analysis  
From the tests of the externally strengthened samples and their load-deflection curves, the mechanical property of the linear elasticity was clearly seen. Although the high elasticity of these tests was notable, low ductility was shown in these tests due to the rapid failure after the yielding. The ductility of the structure was usually calculated with a displacement ductility coefficient denoted by:  
\[\mathrm{M}=\Delta\mathrm{u}/\Delta\mathrm{y} \tag{1}\]  
where \(\Delta\mathrm{y}\) was the yield deflection or the maximum vertical displacement of the structure that was recorded when the ordinary tensile reinforcement reached its yielding strength, and \(\Delta\mathrm{u}\) was the deflection or the maximum vertical displacement of the same structure that was recorded when the load reached the ultimate load of the structure (at the mid-span of the frame-beam in the single-bay frame case). The values of the displacement ductility coefficients of the studied strengthening samples and their comparisons with the unstrengthened samples are presented in Table 4.  
Table 4 shows that the externally strengthened samples were less ductile than the unstrengthened sample according to the FE results. The ductility of the five strengthened samples (F\({}_{0}\), F\({}_{1}\), F\({}_{2}\), F\({}_{4}\), F\({}_{5}\), and F\({}_{6}\)) decreased from 84.82% to 88.36% compared to the unstrengthened sample (F\({}_{0}\))but the ductility of sample (F\({}_{3}\)) decreased by 7.9% only due to the absence of the pre-stressing effect on the positive moment zone of the frame beam. It may also be noted that the inner deviators exerted a light impact on the displacement ductility coefficient. The decreased percentage of the ductility of the strengthened sample (F\({}_{2}\)) was approximately 87.55%, while the decreased percentage of ductility for sample (F\({}_{1}\)) was approximately 88.16%. Furthermore, the profile of the tendon exerted a light impact on the displacement ductility coefficient. The decreased percentage of ductility of the strengthened sample (F\({}_{5}\)) was approximately 84.82%, while the decreased percentage of ductility for the sample with the U-shaped external tendon along the frame beam (F\({}_{6}\)) was approximately 87.65%compared to the unstrengthened samples. In general, the effects of all the strengthened samples, except for (F\({}_{3}\)), on the ductility of frames were similar.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample No.** & **\(\Delta\mathrm{y}\) mm** & **\(\Delta\mathrm{u}\) mm** & **\(\mu=\Delta\mathrm{u}/\Delta\mathrm{y}\)** & **\% Decrease** \\ \hline F\({}_{0}\) & 3.41 & 33.7 & 9.9 & - \\ F\({}_{1}\) & 24.69 & 28.85 & 1.17 & 88.16 \\ F\({}_{2}\) & 24.76 & 30.43 & 1.23 & 87.55 \\ F\({}_{3}\) & 3.7 & 33.64 & 9.1 & 7.9 \\ F\({}_{4}\) & 26 & 29.97 & 1.15 & 88.36 \\ F\({}_{5}\) & 17.36 & 26.1 & 1.5 & 84.82 \\ F\({}_{6}\) & 25.73 & 31.51 & 1.22 & 87.65 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Displacement ductility coefficient results.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Cracking Stage**} & \multicolumn{2}{c}{**Yielding Stage**} & \multicolumn{2}{c}{**Ultimate Stage**} \\ \cline{2-7}
**Sample No.** & **Load kN** & **\% Increase** & **Load kN** & **\% Increase** & **Load kN** & **\% Increase** \\ \hline F\({}_{0}\) & 23 & - & 33 & - & 160 & - \\ \hline F\({}_{1}\) & 47 & 104.4 & 175.6 & 432.1 & 185.7 & 16.1 \\ \hline F\({}_{2}\) & 47 & 104.4 & 177.4 & 437.6 & 189.7 & 18.6 \\ \hline F\({}_{3}\) & 23 & 0 & 34 & 3 & 170.1 & 6.3 \\ \hline F\({}_{4}\) & 31 & 38.8 & 161 & 387.8 & 172.1 & 7.6 \\ \hline F\({}_{5}\) & 46 & 100 & 135.1 & 309.4 & 178.6 & 11.6 \\ \hline F\({}_{6}\) & 47 & 104.4 & 185.1 & 460.9 & 205 & 28.1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of the load capacity analysis.