## 4 Results and Discussion  
### Comparison of Numerical and Experimental Results  
Table 1 summarizes the numerical results and experimental results. The numerical and experiment results for three load stages were compared: the initial crack load where the first stiffness changes in the load-displacement curve or the initial crack is shown on the mid-span bottom of the RC strengthened beam; the yield load when the stress of the steel reinforcement reaches the yield stress; and the ultimate load at which the strain of the extreme compression fiber of the concrete reaches 0.003 according to ACI 318 [27]. Figure 7 shows the comparison of the numerical and experimental results for the load-displacement relationship of the SL-CTRL specimen, which was an experiment without \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Experiment**} & \multicolumn{3}{c}{**FEM**} & \multicolumn{3}{c}{**FEM/Experiment**} \\ \cline{2-10}
**Specimen** & **Initial Crack** & **Yield Load** & **Ultimate Load** & **Initial Crack** & **Yield Load** & **Ultimate Load** & **Initial Crack** & **Yield Load** & **Ultimate Load** \\
**ID** & **Load** & **(kN)** & **(kN)** & **(kN)** & **(kN)** & **(kN)** & **(kN)** & **Load** & **Load** & **Ulimate Load** \\ \hline SL-CTRL & 7.42 & 23.79 & 29.59 & 7.53 & 25.04 & 33.64 & 1.01 & 1.05 & 1.14 \\ SL-30S-4 & 9.42 & 28.48 & 36.78 & 10.68 & 29.38 & 39.81 & 1.13 & 1.03 & 1.08 \\ SL-60S-4 & 13.46 & 34.02 & 46.64 & 13.70 & 33.47 & 45.63 & 1.02 & 0.98 & 0.98 \\ SL-90S-4 & 14.04 & 41.42 & 55.41 & 16.68 & 40.90 & 53.20 & 1.19 & 0.99 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of experimental and numerical results. Figure 6: Constitutive model for Fe-SMA with recovery stress. the Fe-SMA strip. The numerical initial crack load of SL-CTRL was 7.53 kN, showing a slight error of about 1.48% from the experimental result. The numerical and experimental yield loads were 25.04 kN and 23.79 kN, respectively, showing a slight error of about 5.25%. On the other hand, the ultimate loads of the numerical and experimental results were 33.64 kN and 29.59 kN, respectively, showing an error of about 13.69%. In the experiment, SL-CTRL showed a typical flexural failure mode in which the concrete in the compression part was destroyed after yielding of the tensile reinforcement. At this time the vertical displacement of the mid-span of the specimen was about 80 mm. However, in the simulation of this specimen the strain of the extreme compressive concrete fiber reached 0.003 when the central vertical displacement reached 121.82 mm. This difference between the FE analysis and experiment caused a relatively large error at the ultimate load stage. Figure 8 shows the comparison of the numerical and experimental results for the load-displacement relationship of SL-30S-4, which was strengthened with one line of Fe-SMA strip. In the FE analysis, the SL-30S-4 had an initial crack load of about 10.68 kN. This value was about 1.75 kN smaller than the experimental initial crack load of 9.42 kN. In the FE analysis, the yielding load of SL-30S-4 was about 29.38 kN, which is a slight error of about 0.9 kN when compared with the experimental results. The numerical and experimental ultimate loads also showed a small difference of about 3.03 kN. SL-30S-4 reached 0.003 in strain in the numerical extreme concrete fiber when the vertical displacement was 105.73 mm. Figure 9 shows the comparison of the numerical and experimental load-displacement curves for SL-60S-4. The numerical initial crack load of SL-60S-4, a specimen strengthened with two lines of Fe-SMA strips, was 13.7 kN, which was a slightly larger value by about 0.24 kN from the experimental result. The numerical yielding load of this specimen was 33.47 kN, which is an insignificant difference of 1.62% from the experimental yielding load. The numerical ultimate load of this specimen was 45.63 kN, which is a small error of about 2.2% from the experimental ultimate load.