## 7 Proposed Prediction Equations
\[\mathrm{V_{cSF}}=\left(0.16\sqrt{f^{\prime}}_{c}+17.2\rho_{\mathrm{w}}\frac{ \mathrm{V_{u}d}}{\mathrm{M_{u}}}\right)\mathrm{F_{eff}}b_{\mathrm{w}}\mathrm{d} \tag{15}\]  
\[,\text{ where }\mathrm{F_{eff}}=1+\frac{\tau_{\mathrm{eq}}\mathrm{V_{f}\frac{ \mathrm{Lf}}{\mathrm{d}_{f}}}}{0.75\sqrt{f^{\prime}}_{c}} \tag{16}\]  
Figure 9 shows the relationship between the normalized shear strength and multiplication between the tensile longitudinal reinforcement ratios and the ratios of shear-span-to-effective depth divided  
Figure 8: Proposed fiber effectiveness factor equation and comparison between SFRC shear strength-to-normal concrete shear strength ratios and proposed fiber effectiveness factor. by the square-root of concrete strength \(\left(\rho.\frac{\text{V.d.}}{\text{M}\sqrt{f^{\prime}{{}_{c}}}}\right)\). The upper bound for SFRC beam is proposed as multiplication between \(0.29\sqrt{f^{\prime}{{}_{c}}}\) (MPa) and the fiber effectiveness factor instead of setting the value to be nearer the average value given by test data. This consideration is based on the limitation of the SFRC beam database, where no test data with a higher value of \(\left(\rho.\frac{\text{V.d. }}{\text{M}\sqrt{f^{\prime}{{}_{c}}}}\right)\) appeared. Therefore, in the absence of steel fibers, the limit of concrete stress remains equal to \(0.29\sqrt{f^{\prime}{{}_{c}}}\) (MPa), as set in ACI 318-14 and previous ACI 318 codes. Using the shear stress of RC \(0.29\sqrt{f^{\prime}{{}_{c}}}\) (MPa) also reconfirms test data constructed by [PERSON] [15], that set the lower bound shear stress as \(0.30\sqrt{f^{\prime}{{}_{c}}}\) (MPa) for SFRC beam design. The proposed shear strength equation is given in Equation (17). Two lines corresponding to the fiber volume fraction of 0.75% and 1.50% are shown as well. The upper limit of 0.3 in. dashed line, that corresponds to fiber effectiveness factor for fiber volume fraction of 0.75%, is only passed by few data points, which correspond with specimens with fiber volume fraction in the range of 0.75% to 0.80%. Meanwhile, only two data points with a fiber volume fraction of 1.50% pass the upper limit of 0.1 in. dashed line, that corresponds to the fiber effectiveness factor for the fiber volume fraction of 1.50%. For practical application, it is highly recommended to use a fiber volume fraction of not greater than 1.50%. According to [PERSON] et al. in 2017 and [PERSON] in 2019 [44], the SFRC material needs fiber volume fraction to be less than 2% to meet tensile strain hardening criteria. With tensile strain hardening behavior, a loaded SFRC member experiencing bending moment and shear has more resistance to crack opening, can avoid strain localization, and improve energy absorbing capability through showing multiple cracks on its surface. Furthermore, the fiber effectiveness factor shall not exceed 2.6. The maximum number of the fiber effectiveness factor is taken based on the nearest number to the largest value, shown in Figure 8a. \[\text{V}_{\text{c,SF}}=\left(0.16\sqrt{f^{\prime}{{}_{c}}}+17.2\rho_{\text{w}} \frac{\text{V}_{\text{w}}\text{d}}{\text{M}_{\text{w}}}\right)\text{F}_{\text {eff}}\text{b}_{\text{w}}\text{d}\leq 0.29\sqrt{f^{\prime}{{}_{c}}}\text{F}_{ \text{eff}}\text{b}_{\text{w}}\text{d}\ (\text{N}) \tag{17}\]  
Figure 9: The relationship normalized beam shear strengths and beam parameters. The evaluation results show that the proposed shear strength prediction equation can be applied to predict the shear strength of RC or SFRC beams with a concrete compressive strength, fiber volume fraction, yield strength of longitudinal reinforcing bar, and shear-span-to-effective depth ratio ranging from 20 MPa to 194 MPa, 0% to 1.5%, 420 MPa to 785 MPa, and 2.5 to 6.0. The shear strength of a beam with shear-span-to-effective depth ratio less than 2.5, particularly a beam without steel fibers, shall be calculated or predicted by using other methods, since that beam is classified as deep beam and has different behavior with beams tested in this study.