## 4 Results and Discussion
### Comparison with Other Models
#### 4.3.2 Flexural Strength Models  
To study the reliability of the proposed model, the results from the flexural strength ANN model were compared with the values that were calculated by existing models presented in the literature [31; 32; 33]. There are rarely models for predicting the flexural strength of UHPFRC, thus, the models predicting the flexural strength of steel fiber reinforced concrete (SFRC) are summarized in Table 9.  
Where \(f_{ff}\) and \(f_{f}\) are the flexural strength of steel fiber reinforced concrete and plain concrete, respectively; \(V_{f}\) is the volume fraction of steel fibers, \(l_{f}\) is the length of steel fibers, and the \(d_{f}\) is the diameter of steel fibers.  
The analytical models that are described above were applied to calculate the flexural strength of UHPFRC based on the collected data. Table 10 illustrates the mean value, standard deviation (\(SD\)), and IAE of \(f_{predicted}/f_{experimental}\) for the analytical models of previous studies and the flexural  
Figure 5: Comparison between predicted values and experimental values from three analytical models (proposed by [PERSON], [PERSON], and [PERSON]) in previous studies and the ANN model studied in this paper. The 45 degree diagonal line (red line) denotes the predicted values equal to the experimental values and the black lines denote that the maximum ranges between predicted results and experimental results.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Analytical Model** & **Flexural Strength** \\ \hline JGJ/T 221 [31] & \(f_{ff}=f_{f}(1+1.25 Vl_{f}/d_{f})\) \\ [PERSON] [32] & \(f_{ff}=0.97f_{f}(1-V_{f})+3.41 Vl_{f}/d_{f}\) \\ [PERSON] [33] & \(f_{ff}=f_{f}+0.103 Vl_{f}/d_{f}\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Analytical models for predicting the Flexural strength of SFRC.  
strength ANN model of this study. In Table 10, the mean values for the analytical models ranged from 0.8429 to 1.1458, and the \(SD\) and the \(IAE\) were within the ranges of 0.2055-0.4431 and 3.03-4.04%, respectively. The mean values from the ANN model in this study were fairly close to one, and the \(SD\) and the \(IAE\) were quite low. Figure 6 indicates that the predicted values that were obtained from the analytical models of previous studies might underestimate the experimental results with high variation. The ranges of predicted results relative to experimental results obtained from JGJ/T 221 model, Swamy model, and Won-Kya Chai model are \(-\)35-180%, \(-\)50-30%, and \(-\)35-150%, respectively. Whereas, the range of predicted results relative to experimental results obtained from ANN model is \(-\)15-15%. In the comparison with the analytical models of flexural strength for UHPFRC, the ANN model that was proposed in this study could predict the flexural behavior of UHPFRC very well. Thus, the predictions of the flexural strength ANN model are considerably accuracy and stability for evaluating the effects of steel fibers on the flexural strength of UHPFRC.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & _Mean_ & _SD_ & _IAE_ \\ \hline ANN model & 0.9915 & 0.1509 & 1.50\% \\ JGJ/T 221 model & 1.2807 & 0.4431 & 4.04\% \\ [PERSON] model & 0.8429 & 0.2055 & 3.03\% \\ Won-Kya Chai model & 1.1458 & 0.3547 & 3.30\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: The Mean value, SD, and IAE of ANN model and analytical models.  
Figure 6: Comparison between predicted values and experimental values from three analytical models (proposed by JGJ/T 221, [PERSON], [PERSON]) in previous studies and the ANN model studied in this paper. The 45 degree diagonal line (red line) denotes the predicted values equal to the experimental values and the black lines denote that the maximum ranges between predicted results and experimental results.