# Flexural Strength of Composite Deck Slab with Macro Synthetic Fiber Reinforced Concrete
## 5 Ultimate Flexural Strength of the MFRC Steel Deck  
According to the previous test results, when MFRC is applied to a steel deck plate, the positive moment and the negative moment should be calculated. Here, the flexural strength model of the MFRC was calculated using the flexural strength of steel fiber-reinforced concrete as suggested in ACI 544.4R-88 [10]. To calculate flexural strength, the flexural tensile stress-strain relationship of concrete should be derived. This can be calculated from the load-CMOD relationship as Equations (1) and (2) [32]:  
\[f_{t}=\frac{3 PL}{2 bh^{2}} \tag{1}\]  
\[\varepsilon=\frac{f_{t}}{E_{c}}+\frac{w}{l_{c}} \tag{2}\]  
here, \(f_{t}\) is flexural tensile strength, \(P\) is load, \(L\) is length of specimen, \(b\) is width of test specimen, \(h\) is height of specimen, \(\varepsilon\) is tensile strain, \(E_{c}\) is modulus of elasticity, \(w\) is crack width, and \(l_{c}\) is \(2/3h\). As a flexural tensile strength test, the maximum tensile strain of concrete was about 0.03, which was greater than that at the bottom of the deck at the point of maximum strength. Therefore, residual stress could be applied to the region subjected to tensile force. In fib Model Code 2010 [11], the residual stress in the ultimate limit state is defined as CMOD = 2.5 mm; when applying the rigid plastic model, the ultimate residual strength \(f_{Flu}\) can be calculated as in Equation (3):  
\[f_{Flu}=\frac{f_{R,3}}{3} \tag{3}\]  
Strain and stress distribution for the calculation of flexural strength is shown in Figure 13. When calculating the flexural strength of the steel deck plate containing macro-synthetic fiber, the depth of the concrete-equivalent stress block can be calculated according to the force equilibrium, as shown in Equation (4):  
\[C_{c}=T_{c}+T_{deck} \tag{4}\]  
here, \(C_{c}\) is the compression force of the MFRC, \(T_{c}\) is the tension force of the MFRC, and \(T_{deck}\) is the tension force of the steel deck. Figure 13: Strain and stress distributions (positive moment). After calculating the depth of the concrete equivalent stress block, the flexural strength can be calculated by Equation (5):  
\[M_{n,mfrc}=T_{deck}\Big{(}d-\frac{a}{2}\Big{)}+T_{c}\bigg{(}\frac{H+c-a}{2}\bigg{)} \tag{5}\]  
here, \(H\) is the thickness of the slab, \(c\) is the depth of the neutral axis, \(d\) is the effective depth of the section, \(a\) is the depth of the equivalent rectangular stress block, and \(T_{c}=A_{ct}f_{Ftu}=(H-c)bf_{Ftu}\). When calculating the negative moment of a steel deck plate with macro-synthetic fiber, the upper strain was not large, so the flexural strength was calculated without considering the effect of the steel deck in this simple design process. The flexural strength was calculated in the same way as for the steel deck, using Equations (6) and (7):  
\[C_{c}=T_{c}+T_{rebar} \tag{6}\]  
\[M_{n,mfrc}=T_{rebar}\Big{(}d-\frac{a}{2}\Big{)}+T_{c}\bigg{(}\frac{d+c-a}{2} \bigg{)} \tag{7}\]  
However, unlike when obtaining the positive moment, the concrete did not bear the load in the part beyond the upper tensile reinforcing bar. Therefore, the area where the concrete bore the tensile force was set to \(b\) (\(d-c\)), and the strain and stress distributions for this are shown in Figure 14.  
Based on the previously proposed flexural strength evaluation model, the experimental results and the proposed model were compared, and the results are shown in \[\varepsilon=\frac{f_{t}}{E_{c}}+\frac{w}{l_{c}} \tag{2}\]0 Since the stress and strain distributions acting on the cross section were calculated by assuming the complete composite of the positive moment specimen, only P-Deck-F2.4S was considered, and the negative moment specimen was compared with the proposed model for all specimens. All the specimens predicted flexural strength more accurately when considering the flexural contribution of the macro-synthetic fiber.