## 3 Results
### Flexural Strength or Modulus of Rupture  
Figure 10 illustrates the relationship between the PPFRC modulus of rupture and fiber volume. The lines shown represent the modulus of rupture (\(f_{r}\)) trend obtained from a linear regression at the fiber lengths of 40, 50, and 60 mm. Pearson's correlation coefficients, \(r\), were 0.62, 0.74, and 0.87, respectively, showing, considerable, and strong positive correlations. The determination ratios, \(r^{2}\), were 0.62, 0.74 and 0.87, respectively. Therefore, good concordance existed between the lines and behavior of the strength. The three trend lines showed that the modulus of rupture increased as the fiber length increased. The strength increase ratios were 0.36, 0.44, and 0.46 MPa, for each 0.40% increase in fiber volume, respectively. These were considerable values considering that the standard deviation of the beam flexural test was 0.23 MPa. Tables 5 and 12 show the hypothesis \"t\" and Pearson tests. The \"t\" hypothesis test showed that the null hypothesis was rejected in most cases, which stated that the results were not statistically different from the standard mixture without fibers. Concerning the Pearson test, in mixtures with 60 mm fibers, the null hypothesis was rejected, stating that there was no correlation between the results. On the other hand, this hypothesis could not be rejected in mixtures with 40 and 50 mm fiber length fibers. A strength increase was verified when the dosage increased to at least 60 mm fiber lengths. The strength increase for the 40 and 50 mm fiber length was negligible for the evaluated cases in this study. Figure 11 denotes the results of the present study compared with other studies [53; 54; 55]. The observed trend was consistent with the results from previous studies. [PERSON] et al. [51] reported that synthetic macrofibers have no apparent effects on flexural strength, which is dominated by the properties of the concrete matrix. Furthermore, low PP fiber volumes have no effects on flexural strength [61]. Other authors argue that the improvement in splitting-test tensile strength is similar to the improvement in the modulus of rupture [60]. The main advantage of using macrofibers is an improvement in the behavior of concrete post-cracking, i.e., on ductility and toughness. This is a paramount property regarding improving concrete behavior in the presence of cracking, as concrete is characterized as a quasi-fragile material. Cracks occur when the exerted stress overcomes the concrete flexural strength. When the concrete fails, loads are transmitted to fibers, which prevent cracks from spreading and delay collapse. This mechanism is called the bridging effect [64; 65]. Figure 10: PPFRC modulus of rupture based on fiber dosage. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**MV** & **IV** & **DV** & \(r^{2}\) & \(r\) & \(p\) & **Null Hypothesis** \\ \hline \(l=40\) mm & \(V_{f}\) & \(f_{r}\) & 0.622 & 0.789 & 0.211 & no rejection \\ \(l=50\) mm & \(V_{f}\) & \(f_{r}\) & 0.740 & 0.860 & 0.140 & no rejection \\ \(l=60\) mm & \(V_{f}\) & \(f_{r}\) & 0.869 & 0.932 & 0.067 & rejected \\ \hline \hline \end{tabular}
\end{table} 
Table 12: [PERSON]’s hypothesis test of the correlation between \(f_{r}\) and \(V_{f}\). Figure 12 denotes the relationship between the modulus of rupture of PPFRC and fiber length. The lines shown represent the modulus of rupture (\(f_{r}\)) trend, which arose from a linear regression at fiber dosages of 0.40%, 0.80%, and 1.20%; [PERSON]'s correlation coefficients, \(r\), were 0.59, 0.96, and 0.95, respectively, showing medium, very strong, and strong positive correlations. The determination ratios, \(r^{2}\), were 0.35, 0.92 and 0.90, respectively. Therefore, a good fit existed between the lines and strength behavior. Therefore, a good match existed between these lines and the strength behavior. The three trend lines indicated different behaviors against the flexural tensile strength.