## 4 Theoretical Approach and Discussion
### Constitutive Law and Bending Moment Prediction  
The _fib_ Model Code [34] proposes the following relationship for the ultimate residual flexural strength \(f_{ult}\) based on SFRC for a rigid plastic model (Equation (3)):  
\[f_{ult}=f_{R3}/3 \tag{3}\]  
While this equation is suitable for FRC with softening behavior, it requires the notch beam test results and is developed based on the performance of SFRC. It would be of great interest to predict the ultimate strength or moment capacity of a FRC beam by knowing the split tensile strength of FRC obtained from a simple indirect tensile strength testing only. Therefore, this study proposes a technique to relate the ultimate flexural strength of synthetic fiber reinforced SCC using the split tensile strength.  
To relate the ultimate residual flexural strength (\(f_{ult}\)) of the SCC with split tensile strength \(f_{ctsp}\), it can be logically assumed that the fiber content contributes to the residual flexural strength. This is due to the fact that even though the split tensile strength of plain concrete and FRC is very similar, the flexural failure of plain concrete occurs at a CMOD of less than 0.5 mm, as depicted in Figure 7a. In contrast, significant residual stress is obtained for FRC from CMOD 0.5 to 2.5 mm (Figure 9). Since the ultimate residual flexural strength (\(f_{ult}\)) is associated with 2.5 mm (CMOD3), it can be related to the split tensile strength using Equation (4).  
\[f_{ult}=k_{f}f_{ctsp} \tag{4}\]  
where, \(f_{ctsp}\) = split tensile strength of FRC, \(f_{ult}\) = ultimate residual flexural strength corresponding to ultimate crack mouth opening (2.5 mm), and \(k_{f}\) = factor related to fiber dosage.  
As discussed above, the fiber contributes to the gaining of residual flexural strength. As a result, \(f_{ult}\) should be related to fiber dosage, which is taken into account by the factor \(k_{f}\). This factor is calibrated by Equation (5).  
\[k_{f}=f_{R1}/f_{L}\]  
Figure 9: Simplified stress vs. CMOD model behavior for twisted-bundle macro-synthetic fiber reinforced self-compacting concrete (SCC). A trilinear constitutive model was adopted.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Mixture** & \multicolumn{4}{c}{**FF4**} & \multicolumn{4}{c}{**FF6**} & \multicolumn{4}{c}{**FF8**} \\ \hline
**Samples** & **S1** & **S2** & **S3** & **S4** & **S1** & **S2** & **S3** & **S4** & **S1** & **S2** & **S3** & **S4** \\ \hline \(f_{ult}/f_{L}\) & 0.31 & 0.34 & 0.38 & 0.36 & 0.43 & 0.31 & 0.21 & 0.27 & 0.6 & 0.58 & 0.63 & 0.66 \\ Av [Std] & & 0.35 [0.03] & & & 0.31 [0.09] & & & & 0.62 [0.03] & \\ Av + Std & & 0.38 & & & 0.40 & & & & 0.65 & \\ \hline \hline \end{tabular}
\end{table}
Table 8: \(f_{R1}/f_{L}\) ratio of all tested samples for three different mixtures.  
The determination of the ratio \(f_{R1}/f_{L}\) is discussed in Table 8. It can be noted here that the authors are currently working on developing a fiber dosage vs. _kf_ curve that will eliminate the use of the notch beam testing; thus the ultimate residual flexural strength can be determined by simply using Equation (4).  
At failure, as seen in Figure 5, the crack reached the top surface of the concrete beam where the crack width is considered to reach the serviceability limit (0.5 mm), while the crack width at the bottom is considered to reach the ultimate limit (2.5 mm). Figure 9 shows that the stress is constant between the CMOD of 0.5 to 2.5 mm. As plastic deformation is observed between 0.5-2.5 mm CMOD (Figures 9 and 10a), the whole critical cross-section of the beam is subjected to the tensile post-cracking residual stress (Figure 10b). Again, as the crack reached the top surface of the beam, it can be assumed that the compressive stress is concentrated at the top fiber. Hence, the ultimate moment capacity can be determined using Equation (6).  
\[M_{u.tho}=T_{f}\frac{h_{sp}}{2}=\big{(}f_{ult}bh_{sp}\big{)}\frac{h_{sp}}{2}= \frac{f_{ult}bh_{sp}^{2}}{2} \tag{6}\]