## 3 Results and Discussion
### Load-Deflection Response  
The measured load-deflection relations for the UHPC beams are presented in Figure 9, while the data related to these curves are summarized in Table 7. The basic data in this table are defined in Figure 8. These data include the elasticity limit (\(P_{cr}\)), yield (\(P_{y}\)), and ultimate (\(P_{u}\)) loads, as well as their corresponding midspan displacements (i.e., \(\Delta_{cr}\), \(\Delta_{y}\), and \(\Delta_{u}\)). Notably, \(P_{y}\) was estimated based on the loads corresponding to the steel's yielding strains (0.0025) measured at the steel bar surfaces. Additionally, \(\Delta_{m}\) (Table 7) was taken as the beam's maximum deflection at total failure.  
Figure 6: The four-point loading test (dimensions are in mm).  
As shown in Figure 8, the typical load-deflection behavior of the bar-reinforced and unreinforced UHPC beams started with a linear relation and ended with the development of flexural cracks that decreased the flexural stiffness of the beams (decreasing the slope of the load-deflection curve). The end of this second linear stage of the load-deflection response was characterized by bar-reinforcement yielding. Afterward, the beam's deflection increased nonlinearly up to the peak load.  
As listed in Table 7, the plain UHPC beam A2 had the minimum \(P_{cr}\) (85 kN), while the reinforced beam C2 (with 115 mm effective depth) had the maximum \(P_{cr}\) (186 kN). The average \(P_{cr}\) (a measure of the first cracking load for all beams induced by the UHPC matrix behavior) was 136.6 kN, with a standard deviation and coefficient of variation of 34.4 kN and 18.5%, respectively. Additionally, \(P_{u}\) was dependent on the existence of bar reinforcements and their effective depths. As expected, the lowest average load-carrying capacity of beams was observed for the plain UHPC beams (172.8 kN). The maximum load of the UHPC beams was notably increased by the introduction of the bar reinforcement. Compared to the A-type beam, the use of bar reinforcements at 125, 115, 95, 85, and 75 mm enhanced \(P_{u}\) by 56, 55, 73, 96, and 94%, respectively. However, the maximum \(P_{u}\) (339.3 kN)  
Figure 7: Modes of failure for the UHPC beams.  
was recorded for the bar-reinforced UHPC beam with an effective depth of 85 mm. The ductility analysis of UHPC beams is discussed in Section 3.3.  
Figure 8: Definition of elastic, yield, and ultimate phases.  
Figure 9: Load mid-span deflection responses for (**a**) A, (**b**) B, (**c**) C, (**d**) D, (**e**) E, and (**f**) F UHPC beams.  
Based on the data given in Table 7, some robust correlations with a correlation coefficient (\(R^{2}\)) greater than 70% were observed (Figure 10). As expected, linear relations with negative slopes were obtained for \(P_{y}\), \(P_{u}\), and \(\Delta_{m}\) with \(d\). This finding indicates that the yielding and ultimate loads linearly decreased as the effective depth of bar reinforcement increased. This finding is consistent with that reported in [46] for high-performance concrete beams with overall depths of 610-900 mm. Moreover, the trendline of \(\mu_{u}\) was found to be nonlinear for \(d\) (third-order polynomial). This result indicates that the UHPC beam's ultimate ductility could be maximized effectively at various effective depths. Therefore, the typical design code recommendation that limits the zero-sectional stain depths of concrete beams to ensure satisfactory ductility is likely invalid for UHPC beams. This result contradicts the findings of [PERSON] and [PERSON] [47] for high-strength concrete beams.