## 2 Experimental Program
### Methods
#### 2.2.4 Prediction of the Ultimate Moment Capacity  
In the current research, an ACI 544-based approach [41] was used to predict the ultimate moment capacity of hybrid fiber-reinforced UHPC beams. This method is, however, initially recommended for conventional fiber-reinforced concrete. This method employs uniform tensile and compressive stress blocks. In this prediction model, the tensile properties of the UHPC were evaluated in terms of the fiber's characteristics and fiber-matrix bonding strength. The fundamentals of this approach have been employed by different investigators [10, 31, 35, 36, 37, 38] for calculating the capacity of singly reinforced beams prepared by fiber-reinforced concrete. The ultimate load of the beam was predicted based on the assumption that it is governed by the yielding of reinforcement bars. Figure 3: Compressive test (**a**) machine and (**b**) control unit. Flexural test (**c**) schematic diagram and (**d**) setup (dimensions are in mm). Moreover, the idealization of the sectional compression and tensile stress distributions by the uniform stress blocks was adopted in this approach (Figure 5). In ACI 544 [41], the uniform tensile stress is evaluated by Equation (1). This equation assumes that the composite's tensile strength (\(f^{\prime}_{t_{1}}\)) is a function of the fiber-matrix bonding strength. It is noteworthy that this strength (\(\tau_{f}\)) has been assumed by the ACI 544 [41] as 2.3 MPa for conventional fiber-reinforced concrete. Generally, \(\tau_{f}\) is in the range of 1 to 9 MPa for normal and high-strength concrete and various fibrous configurations [42]. [PERSON] et al. [38] have employed a fiber-matrix interfacial shear strength of 4.15 MPa for high-performance concrete, as suggested by [PERSON] [43]. The factor of \(7.72\times 10^{-3}\) in Equation (2) was thus increased to \(20\times 10^{-3}\) in Equation (3). For UHPC beams, [PERSON] and [PERSON] [37] adjusted Equations (2)-(4), where \(\tau_{f}\) was adopted as 7.7 MPa for UHPC with a mean compressive strength of 136 MPa. It is worth noting that this adjustment (Equation (4)) was recently employed by [PERSON] et al. [10], when they assumed \(\tau_{f}\) as 8.15 MPa for hybrid fiber-reinforced UHPC beams. In the current study, \(\tau_{f}\) was taken as 3.7 MPa, due to the use of hybrid smooth fibers. \[f^{\prime}_{t_{1}}=7.72\times 10^{-3}\Big{[}a_{f}\rho_{f}F_{be}\Big{]} \tag{2}\]  
\[f^{\prime}_{t_{2}}=20\times 10^{-3}\Big{[}a_{f}\rho_{f}F_{be}\Big{]}. \tag{3}\]  
\[f^{\prime}_{t_{3}}=0.85\Big{[}a_{f}\rho_{f}\tau_{f}\Big{]} \tag{4}\]  
where,  
\[a_{f} = \begin{array}{lcl}a_{f}&=&\text{the fiber's aspect ratio ($=l_{f}/d_{f}$), and $l_{f}$ and $d_{f}$ are the fiber's length and diameter,}\\ \rho_{f}&=&\text{the content of fiber (vol.); and}\\ F_{be}&=&\text{the fiber-matrix bonding efficiency factor.}\end{array}  \tag{5}\]  
Figure 4: Testing of beams under four-point loading: (**a**) schematic diagram and (**b**) setup and instrumentations (dimensions are in mm). In the current investigation, the tensile strength of the UHPC was predicted by Equation (5), originally developed by [PERSON] and [PERSON] [44] and then adopted by [PERSON] and [PERSON] [37] and [PERSON] et al. [10]. This formula (Equation (5)) accounts for the different fiber properties (e.g., dispersion, cohesion, dosage, etc.). In this equation, the efficiency coefficients \(\zeta_{o}\), \(\zeta_{b}\), and \(\zeta_{l}\) stand for the fiber' dispersion, bond, and length, considered as 0.86, 1.0, and 0.41, respectively [10; 41; 45; 46]. It is noteworthy that these coefficients do not depend on the concrete strength class, but rely on the properties of the fibrous reinforcement. According to ACI 544 [41], the tensile stress distribution starts at a distance \(e\) from the sectional extreme compression fiber (Figure 5) to the ultimate fibrous strain position.