## 3 Results and Discussion
### Flexural and First-Crack Toughness, Indicies, and Residual Strength Factor  
Table 5 contains the estimated values of flexural toughness parameters, indices (I\({}_{5}\), I\({}_{10}\), and I\({}_{20}\)), residual strength factors (R\({}_{5,10}\) and R\({}_{20,10}\)), and deflection for all concrete mixes. It can be observed that the addition of fibers into concrete had significant effects on the first crack toughness, \(FT_{fc}\). In the first set of 100% cement concrete, PVA fibers increased \(FT_{fc}\), by 13% and the CCS fibers increased it by up to 41% compared the control mix (CM). In the fly ash concrete, PVA fibers enhanced it by 41% and CCS fibers increased it by 57% compared to the control mix (FC). This is a substantial improvement in the first crack strength of concrete. The flexural toughness (FT), which was calculated at the failure conditions, showed a significant enhancement with fiber addition. In 100% cement concrete, PVA and CCS fibers enhanced the FT value 1.8 times and 3.6 times, respectively, compared to that of the mix CM. The flexural toughness ratio and the first crack toughness (\(FT\)/\(FT\)/\(FT\)) indicate the concrete's ductility. As shown in Table 5, the \(FT\)/\(FT_{fc}\) ratio of the control mixes, CM and CF, was obtained as 1.3 and 1.26, respectively. For PVA fibers, \(FT\)/\(FT_{fc}\) was obtained as 2.05 and 1.91. For CCS fibers, it was determined as 3.31 and 2.21, indicating a significant enhancement in the ductility with fiber addition. Figure 9 shows a correlation between the flexural strength, \(f_{fl}\), flexural toughness, \(FT\), and the first crack toughness, \(FT_{fc}\). The following Equations (3) and (4) were obtained:  
\[FT_{fc}=2.4515\Big{(}f_{fl}\Big{)}^{2}-22.275f_{fl}+55.38 \tag{3}\]  
\[FT=8.9728\Big{(}f_{fl}\Big{)}^{2}-80.692f_{fl}+186.42 \tag{4}\]  
Figure 10 shows the relationship between first crack strength, \(FT_{fc}\), and flexural toughness, \(FT\); a power correlation was obtained as:  
\[FT=0.0955\Big{(}FT_{fc}\Big{)}^{2.5696} \tag{5}\]  
The first index, I\({}_{5}\), is the ratio of strain energy calculated at 38 to the elastic strain energy calculated at \(\delta\) (the deformation at first crack). The region from \(\delta\) to 38 is transitionary when material behavior shifts from elastic to plastic. CCS fibers showed an I\({}_{5}\) value of 2.45 in OPC concrete, indicating high resilience of the matrix after the first crack appeared. In fly ash concrete, it was calculated as 1.84. PVA fibers showed an I\({}_{5}\) value of 1.82 and 1.59 in OPC and fly ash concrete, respectively. The indices I\({}_{10}\) and I\({}_{20}\) indicate the matrix's plastic flow behavior calculated at 5.58 and 10.58, respectively. CCS fibers showed I\({}_{20}\) values as 3.33 and 2.21 in OPC and fly ash concrete, respectively. PVA fibers showed an I\({}_{20}\) value of 2.05 and 1.91 in OPC and fly ash concrete. The high values of indices shown by CCS fibers could be the reason for the high tensile strength (2300 MPa) compared to PVA fibers. The tensile strength was 1000 MPa. In general, the addition of fibers has reasonably improved the plastic flow behavior of the composite.  
As shown in Table 5, residual strength factor, R\({}_{5,10}\), for PVA fibers in 100% cement and fly ash concrete was obtained as 4.03 and 4.84, respectively. Similarly, R\({}_{5,10}\) for CCS fibers in 100% cement concrete and the fly ash concrete was determined as 11.93 and 6.77, respectively. The residual strength factor represents the strength retained in the structural member after the first crack onset. A high value of the factor, R, indicates the top performance of concrete. In this case, a value of four and above means that the concrete mixes with these fibers can display appropriate plastic behavior.  
Figure 10: Correlation between First-Crack Toughness and Flexural Toughness.  
Figure 9: Correlation between flexural strength, \(f_{\text{H}}\), and the flexural toughness at first crack, \(FT_{\text{fc}}\), and ultimate conditions, \(FT\).