## 4 Calculation
### Moment-Curvature Relation Calculation  
The moment-curvature relation of the notched cross-section was calculated to determine the mechanical properties of the fiber-reinforced concrete material. The calculation was based on the fiber-reinforced concrete constitutive model from the _fib Model Code for Concrete Structures 2010_[17]. To construct the constitutive law of fiber-reinforced concrete in tension, the crack widths (w) must be converted into strains (\(\varepsilon\)). This can be performed using the following relation:  
\[\varepsilon=\frac{\mathrm{w}}{\mathrm{l_{cs}}}, \tag{3}\]  
where \(\mathrm{l_{cs}}\) is the characteristic structural length, considered equal to the cross-section height for elements without conventional reinforcement. The crack width (w), according to the _fib Model Code for Concrete Structures 2010_[17], is considered equal to the crack mouth opening displacement (CMOD), which is measured at the bottom of the concrete prism (at the notch start), while the crack tip opening displacement (CTOD), measured at the notch end, is the more precise measure for the crack width (w). The CMOD and CTOD measurement locations are shown in Figure 6.  
This inconsistency is partially compensated by considering simplified stress distributions across the cross-section height, as discussed in [13]. Hence, in this study, the CTOD values were considered for the crack widths, and the residual strength values at different crack widths were calculated by matching the internal bending moment to the applied bending moment (calculated from the experimental results). To calculate the CTOD, the following relation from the EN14651 standard [19] was considered: CMOD = 1.2 \(\cdot\) CTOD. The constitutive model for fiber-reinforced concrete is given in Figure 7, where all the important key points (coordinates) are marked. Figure 6: CMOD and CTOD measurement locations. The parabolic constitutive model for compression in Figure 7a takes into account the strain (\(\epsilon_{\text{c1}}\)) at the maximum compressive stress (in our case, \(\text{f}_{\text{cm}}\)), calculated by the following equation from the EN1992-1-1 standard [20]:  
\[\epsilon_{\text{c1}}=-0.7\cdot\text{f}_{\text{cm}}\text{{}^{0.31}}, \tag{4}\]  
where \(\epsilon_{\text{c1}}\) is calculated in %, and \(\text{f}_{\text{cm}}\) needs to be in MPa. The experimentally determined mean compressive strength of concrete was adjusted considering different ages of the concrete at compressive (37 days) and bending testing (49 days) according to Equation (2). The modulus of elasticity at the concrete age of 28 days (\(\text{E}_{\text{cm}.28}\)) was also calculated from the mean compressive strength at the concrete age of 28 days with the following equation:  
\[\text{E}_{\text{cm}.28}\text{=22}\cdot\left(\frac{\text{f}_{\text{cm}.28}}{10 }\right)\text{{}^{0.3}}, \tag{5}\]  
where \(\text{E}_{\text{cm}.28}\) is calculated in GPa, and \(\text{f}_{\text{cm}.28}\) (the mean compressive strength of concrete at the age of 28 days needs to be in MPa). For the age of 49 days, the adjusted modulus of elasticity was calculated with:  
\[\text{E}_{\text{cm}}(\text{t})=\exp\left(0.3\cdot\text{s}\cdot\left(1-\left( \frac{28}{\text{t}}\right)^{\frac{1}{2}}\right)\right)\cdot\text{E}_{\text{ cm}.28}. \tag{6}\]  
The multilinear constitutive model for tension in Figure 7b,c considers that the tensile stress-strain relation up to 90% of the tensile strength (\(\text{f}_{\text{ctm}}\)) is linear considering the modulus of elasticity \(\text{E}_{\text{cm}}\). The strain at the tensile strength is equal to \(\varepsilon_{\text{P}}\).