## 3 Results and Discussion
### Prediction of the Ultimate Moment Capacity of UHPC Beams
 & **(1)/(2)** \\ \hline \multirow{8}{*}{Authors} & B1 & 172.6 & 18.4 & 1.206 & 0.833 & 0.0738 &  
The prediction capability of the ultimate moment capacity of the bar-reinforced UHPC beam model (Equation (6)) is summarized in Table 10 and Figure 14. Table 10 shows that the suggested prediction model offered reasonable performance for the majority of the investigated data points, as the ratio of predictions to test results generally approached unity. Exceptions were observed for only four data points: R13-1, R14-1 [1], U-B3, and U-B5 [52]. This discrepancy was likely attributable to the size effect, casting methods, and testing configurations of these beams. Excluding the outlier data points (R13-1, R14-1, U-B3, and U-B5), the mean ratio was 0.92, with a standard deviation of 0.09; however, the overall average was 0.83, and the associated standard deviation was 0.23 (Table 10). Figure 14 indicates that the majority of the predicted \(M_{u}/\left(bd^{2}\ f_{c}^{\prime}\ \right)\) values using Equation (6) were close to the line of equality (LOE (prediction = test results)). However, few data points were underneath the LOE, while the majority of the data points were above this line. This finding indicates that the proposed model (Equation (6)) is likely to insignificantly overestimate the moment capacity of the UHPC beam. In conclusion, the proposed prediction model with a rational professional safety factor can be used to design UHPC beams in an adequate manner. Most of the predicted results occurred within an error band of \(\pm\)10%, although a few data points were close to the \(\pm\)20% model error band. This analysis demonstrated that the prediction formula (Equation (6)) can be efficiently employed to calculate the ultimate moment capacity of non-shear-strengthened and bar-reinforced UHPC beams under flexural loading conditions. Figure 15 presents the predicted contours of \(M_{u}/bd^{2}f_{c}^{\prime}\) for reinforced UHPC beams with different concrete compressive strengths (\(f_{c}^{\prime}\)), reinforcement ratios (\(\rho\)), and effective-to-overall depth (\(d/h\)) ratios using the proposed model (Equation (6)). In Figure 15a, the concrete's tensile strength (\(f_{l}^{\prime}\)) and \(\rho\) were assumed to be 17 MPa and 1.5%, respectively. However, \(f_{l}^{\prime}\) and \(d/h\) were assumed to be 30 Mpa and 0.55, respectively, as shown in Figure 15b. This figure could be used as a design aid for the UHPC beams. Notably, Figure 15a shows that at an \(f_{c}^{\prime}\) of up to about 170 Mpa, the \(d/h\) had little influence on the beam's moment capacity, as increasing the \(d/h\) at a constant \(f_{c}^{\prime}\) did not significantly change the \(M_{u}/\left(bd^{2}\ f_{c}^{\prime}\ \right)\). Figure 15b instead illustrates that the moment capacity of the UHPC beam was almost constant when \(f_{c}^{\prime}\) was in a range of 150-190 Mpa, and \(\rho\) was in a range of up to 2%. Additionally, this figure shows that increasing the \(\rho\) by more than 2% at a constant \(f_{c}^{\prime}\) will likely decrease the beam's moment capacity. This result is expected, as over-reinforcement in beams will lead to brittle failure, obviating the benefits of the increased reinforcement ratios to increase the beam's loading capacity [53]. Figure 14: Predicted vs. observed \(M_{u}/bd^{2}f_{c}^{\prime}\ \)[1; 17; 48; 49].