## 4 Discussion
### Short Beams  
Figure 4 shows the load-displacement graph for the eight beams for which results were recorded. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Experimental** & **ACI** & **\(K\)** & **FIB** & **\(K\)** \\  & **(MPa)** & **(MPa)** & **(ACI)** & **(MPa)** & **(FIB)** \\ \hline HR & 68.9 & 64.8 & 0.67 & 78.4 & 0.35 \\ HM & 71.5 & 62.6 & 0.85 & 75.3 & 0.47 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison between the experimental and analytical values for reinforced test pieces. Table 6 shows the most significant results obtained from the graph in Figure 4: the maximum load reached in the test, \(P_{max}\); the displacement under this maximum load, \(\delta_{max}\); the rigidity of the beam, \(K\), obtained in the interval 0.15 \(P_{max}\)-0.35 \(P_{max}\), where the behavior is linear; and the energy absorbed in the test, \(G_{max}\), calculated as the area enclosed by the graph. The average values of the previous figures (\(\overline{x}\)) are also provided. The results in Table 6 and Figure 4 demonstrate that the behavior of the beams tested was very fragile in all cases. The reinforcement procedure is thus seen to be highly effective. The reinforced beams achieved highly significant increases in the values for breaking strain, ultimate displacement, rigidity, and energy absorbed in the tests. Furthermore, there were no differences between the behaviors of the beams reinforced using the two types of fiber. The five curves obtained for these beams are virtually overlapping; however, significant differences can be observed between the three diagrams obtained for the unreinforced beams. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam**} & \multirow{2}{*}{\(P_{max}\) (**\(\overline{x_{P}}\)**)} & \multirow{2}{*}{\(\delta_{max}\) (**\(\overline{x_{S}}\)**)} & \multirow{2}{*}{\(\overline{x_{S}}\)} & \multirow{2}{*}{\(\overline{x_{X}}\)} & \multirow{2}{*}{\(\overline{x_{X}}\)} & \multirow{2}{*}{\(\overline{x_{X}}\)} & \multirow{2}{*}{\(\overline{x_{X}}\)} \\  & & **(kN)** & & **(mm)** & **(mm)** & **(kmN/mm)** & **(kN/mm)** & **(kN-mm)** & **(kN-mm)** \\ \hline BRF1 & 46.6 & & 6.8 & 0.77 & & 160.7 & \\ BRF2 & 53.4 & 46.0 & 8.2 & 7.8 & 0.57 & 0.62 & 218.9 & 180.9 \\ BRF3 & 37.9 & & 8.5 & & 0.52 & & 163.0 & \\ BHR4 & 112.8 & & 12.8 & & 1.1 & & 723.8 & \\ BHR5 & 113.8 & & 12.0 & & 1.0 & & 685.7 & 704.8 \\ BHM7 & 109.6 & & 11.3 & & 1.0 & & 621.3 & \\ BHM8 & 113.5 & 110.8 & 10.9 & 11.3 & 1.0 & 1.0 & 620.4 & 629.8 \\ BHM9 & 109.4 & & 11.8 & & 1.0 & & 647.6 & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results obtained in the flexural strength tests. Figure 4: Load–displacement graph for the beams tested. The flexural strength tests were analyzed using ANOVA tests. This methodology treats jointly the set of eight samples that are had in this case. Two ANOVA tests were conducted: one for the breaking load and another for the maximum displacement. For the breaking load experiments, significant differences were found between the three levels of the factor. The resulting ANOVA \(p\)-value (\(p\) = 3.21 \(\times\) 10\({}^{5}\)) led us to reject the null hypothesis of equal averages. Table 7 summarizes the Tukey contrasts of pairwise mean comparisons. Therefore, BHM-BRF and BHR-BRF are significantly different, but the hypothesis that BHM and BHR averages are equal cannot be rejected. For the maximum displacement experiments, significant differences were also found between the three levels of the factor. The resulting ANOVA \(p\)-value (\(p\) = 0.00142) led us to reject the null hypothesis of equal averages (here with a lesser degree of confidence, because the \(p\)-value is larger but lies within the significant range). Table 8 summarizes the Tukey contrasts of pairwise mean comparisons. Therefore, consistent with what was found for the breaking load experiments, the BHM-BRF and BHR-BRF treatments are significantly different, while the hypothesis that BHM and BHR averages are equal cannot be rejected.