## 4 Calculation
### Moment-Curvature Relation Calculation
After reaching the tensile strength, the stress-strain diagram curve (line) continues in the direction of the coordinate (\(\varepsilon_{\text{Q}}\), \(0.2\cdot\text{f}_{\text{ctm}}\)) but only up to the strain \(\varepsilon_{\text{c}}\), which is calculated as the intersection point of the negative slope line of the stress-strain relation of plain concrete, after reaching the tensile strength, and of the line of the stress-strain relation of fiber-reinforced concrete between the coordinates (\(\varepsilon_{\text{SLS}}\), \(\text{f}_{\text{Fts}}\)) and (\(\epsilon_{\text{ULS}}\), \(\text{f}_{\text{Fts}}\)) (see Figure 7b,c). The strain \(\varepsilon_{\text{Q}}\) can be calculated with:  
\[\varepsilon_{\text{Q}}=\frac{\text{G}_{\text{F}}}{\text{f}_{\text{ctm}}\cdot \text{l}_{\text{cs}}}+\left(\varepsilon_{\text{P}}-0.8\cdot\frac{\text{f}_{ \text{ctm}}}{\text{E}_{\text{cm}}}\right), \tag{7}\]  
where \(\text{G}_{\text{F}}\) is the fracture energy of plain concrete. \(\text{G}_{\text{F}}\) can be calculated with:  
\[\text{G}_{\text{F}}=0.73\cdot\text{f}_{\text{cm}}\text{{}^{0.18}}. \tag{8}\]  
Figure 7: Constitutive law of fiber-reinforced concrete according to the _fib Model Code for Concrete Structures 2010_[17] (the graphs have different scales for clarity): (a) in compression; (b) in tension for small strains of \(<1\%\); and (c) in tension for large strains of \(>1\%\). where \(G_{F}\) is calculated in N/m (f\({}_{cm}\) needs to be in MPa). The strains \(\epsilon_{SLS}\) and \(\epsilon_{ULS}\) correspond to the CMOD values of 0.5 mm and 2.5 mm. However, as explained before, the CTOD values are considered for calculating the strains (according to the relation \(\varepsilon=\frac{CTOD}{R_{B}}\)). The part of the constitutive law in Figure 7b,c with tensile strains larger than \(\epsilon_{c}\) differs from the constitutive law of plain concrete. It indicates the specifics (differences) of the mechanism of fiber-reinforced concrete's behavior compared with plain concrete. The fiber action is activated at larger strains (strain values at crack formation) and starts to dominate at larger strains (at larger crack widths). The multilayer method was chosen to calculate the moment-curvature relation (refer to [24] for a detailed explanation). It was decided to divide the cross-section into 1500 layers. The bisection method was employed to calculate the strain at the bottom edge of the cross-section for different curvature values. The corresponding loads (forces) were calculated from the resulting bending moment in the cross-section (by applying equilibrium conditions for forces and bending moments in the cross-section). The expressions of the equilibrium of forces and moments in the cross-section section were incorporated as:  
\[\sum_{i=1}^{1500}\sigma_{i}\cdot\mathrm{b}\cdot\mathrm{t}_{\mathrm{SI}}=0, \tag{9}\]  
Figure 6.0  
where \(\sigma_{i}\) is the stress in the i-th layer of the cross-section (calculated for the strain of the i-th layer \(\varepsilon_{i}\) from the stress-strain relation presented in Figure 7), and \(\mathrm{y}_{\mathrm{SI},i}\) is the distance of each layer's center of gravity to the cross-section's bottom edge. A loop calculation was carried out by varying the tensile strength of the concrete f\({}_{\mathrm{ctm}}\) and the residual strengths f\({}_{\mathrm{Fls}}\) and f\({}_{\mathrm{Ftu}}\). The calculation was repeated until the difference between the experimental and calculated maximum load-bearing capacities, the residual load capacities at a CMOD of 0.5 mm, and the residual load-bearing capacities at a CMOD of 2.5 mm were less than 1 N. The input parameters and the back-calculated tensile strength value results of the calculation are given in Figure 6.1  
The _fib Model Code for Concrete Structures 2010_[17] also gives expressions for determining the residual tensile strength values at CMOD values of 0.5 mm (f\({}_{\mathrm{Fls}}\)) and 2.5 mm (f\({}_{\mathrm{Ftu}}\)).