## 2 Literature Review
### Shear Strength Prediction Equations  
The number of equations used to predict the shear strength of SFRC beams without shear reinforcement can be found in many research studies. Selected shear strength prediction equations for the SFRC beam without shear reinforcement proposed by previous researchers are shown in Equations (2)-(12). [PERSON] [10]:  
\[\text{V}_{\text{c,SF}}=\frac{2}{3}f_{t}^{\prime}(\
icefrac{{4}}{{a}})^{0.25} \text{b}_{\text{w}}\text{d} \tag{2}\]  
where \(f_{t}^{\prime}\) is the concrete tensile strength based on split-cylinder test. [PERSON] and [PERSON] [12]:  
\[\text{V}_{\text{c,SF}}=\left(\text{e}\left[0.24f_{\text{spfc}}+80\
icefrac{{ \text{d}}}{{a}}\right]+\text{v}_{\text{b}}\right)\text{b}_{\text{w}}\text{d} \tag{3}\]\[\text{With }f_{spfc}=f_{\text{\tiny{eff}}}/(\
icefrac{{20}}{{-\sqrt{\text{F}}}})+0.7+1.0 \sqrt{\text{F}};\;\text{F}=(\text{L}_{\text{f}}/\text{D}_{\text{f}})\text{V}_{ \text{f}}\text{d}_{\text{f}}\text{ and }\text{v}_{\text{b}}=0.41\tau\text{F} \tag{4}\]  
where \(f_{spfc}\) is the computed value of split-cylinder strength of fiber concrete, F is the fiber factor; \(f_{cuf}\) is the cube strength of fiber concrete, L\({}_{\text{f}}\) is the steel fiber length, \(\tau\) is the average fiber interfacial bond stress that can be taken as 4.15 MPa [17], d\({}_{\text{f}}\) is the bond factor: 0.5 for round fibers, 0.75 for crimped fibers, and 1.00 for indented fibers, v\({}_{\text{b}}\) the shear stress provided by steel fibers, e is the arch action that is equal to 1.0 for \(\frac{\text{a}}{\text{d}}\) greater than 2.80 and equal to 2.8\(\frac{\text{d}}{\text{a}}\) for \(\frac{\text{a}}{\text{d}}\) less than 2.80. [PERSON] et al. [13]:  
The first equation (Equations (5a) and (5b)):  
\[\text{for }\
icefrac{{\text{a}}}{{\text{d}}}\geq 2.5,\;\text{V}_{\text{c, SF}}=\left(2.11\sqrt[3]{f^{\prime}}_{c}+7\text{F}\right)(\
icefrac{{\text{a}}}{{ \text{d}}})^{0.333}\text{b}_{\text{w}}\text{d} \tag{5a}\] \[\text{for }\
icefrac{{\text{a}}}{{\text{d}}}<2.5,\;\text{V}_{\text{c, SF}}=\left[\left(2.11\sqrt[3]{f^{\prime}}_{c}+7\text{F}\right)(\
icefrac{{\text{a}}}{{ \text{a}}})^{0.333}\frac{2.5}{(\
icefrac{{\text{a}}}{{\text{d}}})}+\text{v}_{ \text{b}}(2.5-\
icefrac{{\text{a}}}{{\text{d}}})\right]\text{b}_{\text{w}} \text{d} \tag{5b}\]  
The second equation (Equation (6)):  
\[\text{V}_{\text{c,SF}}=\left[\left(0.7\sqrt[3]{f^{\prime}}_{c}+7\text{F} \right)\
icefrac{{\text{d}}}{{\text{a}}}+17.2\
icefrac{{\text{a}}}{{\text{d }}}/\text{a}\right]\text{b}_{\text{w}}\text{d} \tag{6}\]  
Parra-Montesinos [15]:  
[PERSON] in 2006 summarized the test data from beams subjected to shear force and reported the use of deformed steel fibers as an alternative minimum stirrups for beams subjected to factored shear forces ranging from \(0.085\sqrt[3]{f^{\prime}}_{c}\text{b}_{\text{w}}\text{d}\) to \(1.7\sqrt[3]{f^{\prime}}_{c}\text{b}_{\text{w}}\text{d}\) (in SI units). The range of those values corresponds to \(0.5\text{V}_{\text{c}}\) and \(\text{V}_{\text{c}}\). The SFRC beams with a fiber volume fraction of 0.75% or greater exhibited ultimate shear stresses that were greater than the conservative lower bound value of \(0.3\sqrt[3]{f^{\prime}}_{c}\) MPa. Therefore, the shear strength of SFRC beam can be expressed in Equation (7). \[\text{V}_{\text{c,SF}}=0.3\sqrt[3]{f^{\prime}}_{c}\text{b}_{\text{w}}\text{d }\text{ }\text{(N)} \tag{7}\]  
[PERSON] et al. [17]:  
[PERSON] et al. in 2002 conducted an experimental program and proposed equations to predict the shear strength of SFRC beams, as shown in Equation (8). The shear strength prediction equation is expressed in Equation (8a). \[\text{V}_{\text{c,SF}}=\left(3.7\;\text{e}\;f_{spfc}^{\frac{2}{3}}(\
icefrac{{ \text{p}}{{\text{d}}}}/\
icefrac{{a}}{{\text{a}}})^{1/3}+0.8\text{v}_{\text{b}} \right)\text{b}_{\text{w}}\text{d} \tag{8a}\]  
where e is shown in Equation (8b).