## 3 Results and Discussion
### Prediction of the Ultimate Moment Capacity of UHPC Beams  
In this research, the ultimate moment capacity of UHPC beams was predicted using the Minitab programming package [49; 50]. Here, the prediction approach relied upon the design of an experimental model that correlates the biased impacts of individual independent variables to the nonlinear observed response surfaces. This numerical modeling approach can yield continuous formulas for measured responses, taking into consideration its input limitations. Additionally, the iso-responsive contours (surfaces) can be assessed in terms of the designed investigational properties by employing this method. Figure 12: Load crack width responses for UHPC beams: (**a**) A, (**b**) B, (**c**) C, (**d**) D, (**e**) E, and (**f**) F.  
Figure 13: The moment–curvature: (**a**) typical curve and (**b**) comparison between different UHPC beams. In this investigation, four input parameters were used to predict the normalized ultimate moment capacities of beams (i.e., \(M_{u}/\left(bd^{2}\ f_{c}^{\prime}\ \right)\)), where \(b\) and \(d\) are the width and effective depth of the beam, respectively, and \(f_{c}^{\prime}\) is the compressive strength of the concrete. These parameters were \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\), which represent \(f_{c}^{\prime}\) (MPa), \(f_{t}^{\prime}\) (the concrete's tensile strength in MPa), \(\rho\) (the percentage of reinforcement, %), and \(d/h\) (the effective-to-overall depth ratio), respectively. The normalized ultimate-moment-model prediction equation and coefficients are given by Equation (6) and presented in Tables 8 and 9, respectively. A wide-ranging database (Table 10) based on the present experiments and independent data was prepared for the development of this model. The collected data points encompass 25 bar-reinforced UHPC concrete beams. Notably, the previous data were carefully selected to match the current data. The primary matching criteria included concrete's comparable compressive strength and tensile-bar arrangement, a lack of transverse reinforcement, and a similar test setup under flexural loading. Accordingly, some of the test results from the four datasets [1, 20, 51, 52] are included in Table 10.  
\[ \begin{array}{c}\frac{M_{u}}{bd^{2}f_{c}^{\prime}}=\alpha+\beta_{1}x_{1}+ \beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{4}x_{4}+\beta_{11}x_{1}^{2}+\beta_{22}x_{ 2}^{2}+\beta_{33}x_{3}^{2}+\beta_{44}x_{4}^{2}+\beta_{12}x_{1}x_{2}+\beta_{13 }x_{1}x_{3}+\beta_{14}x_{1}x_{4}\\ \hskip 14.226378 pt+\beta_{23}x_{2}x_{3}+\beta_{24}x_{2}x_{4}+\beta_{34}x_{3}x_{4 }\left(R^{2}=98.5\%\right)\end{array}  \tag{6}\]  
A wide-ranging database (Table 10) based on the present experiments and independent data was prepared for the development of this model. The collected data points encompass 25 bar-reinforced UHPC concrete beams. Notably, the previous data were carefully selected to match the current data. The primary matching criteria included concrete's comparable compressive strength and tensile-bar arrangement, a lack of transverse reinforcement, and a similar test setup under flexural loading. Accordingly, some of the test results from the four datasets [1, 20, 51, 52] are included in Table 10.  
\[ \begin{array}{c}\frac{M_{u}}{bd^{2}f_{c}^{\prime}}=\alpha+\beta_{1}x_{1}+ \beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{4}x_{4}+\beta_{11}x_{1}^{2}+\beta_{22}x_{ 2}^{2}+\beta_{33}x_{3}^{2}+\beta_{44}x_{4}^{2}+\beta_{12}x_{1}x_{2}+\beta_{13 }x_{1}x_{3}+\beta_{14}x_{1}x_{4}\\ \hskip 14.226378 pt+\beta_{23}x_{2}x_{3}+\beta_{24}x_{2}x_{4}+\beta_{34}x_{3}x_{4 }\left(R^{2}=98.5\%\right)\end{array}  \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Beam** & \(f_{c}^{\prime}\) (MPa) & \(f_{t}^{\prime}\) (MPa) & \(\rho\) & \(\frac{d}{h}\) & \ \begin{tabular}{c} **Test** \\ \(M_{u}/(bd^{2}f)\) \\ \end{tabular}  & \begin{tabular}{c} **Predicted** \\ \(M_{u}/(bd^{2}f)\) \\ \end{tabular}