## 4 Calculation
### Moment-Curvature Relation Calculation
Two simplified stress-crack opening constitutive laws for considering different stress distributions are defined: the rigid plastic model with constant tensile stresses in the cross-section (meant only for calculating the load-bearing capacity at a CMOD value of 2.5 mm) and the linear model (with different stress distributions at CMOD values of 0.5 mm and 2.5 mm). Based on the average experimental results for the rigid plastic model, the resulting f\({}_{\mathrm{Ftu}}\) value is 0.90 MPa, whereas for the linear model, the resulting f\({}_{\mathrm{Fts}}\) value is 0.94 MPa, and the resulting f\({}_{\mathrm{Ftu}}\) value is 0.93 MPa. Compared with the calculation results for f\({}_{\mathrm{Fls}}\) and f\({}_{\mathrm{Ftu}}\) in Table 6, there are noticeable differences. As a result, the linear model assumes almost constant tensile stresses between CMOD values of 0.5 mm and 2.5 mm. Contrary \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Unit** & **Value** \\ \hline \(\mathrm{f}_{\mathrm{cm}}\) & [MPa] & 37.10 \\ \(\mathrm{E}_{\mathrm{cm}}\) & [GPa] & 32.60 \\ \(\mathrm{\varepsilon}_{\mathrm{c1}}\) & [\(10^{-3}\)] & \(-2.15\) \\ \(\mathrm{G}_{\mathrm{F}}\) & [N/m] & 139.90 \\ \(\mathrm{\varepsilon}_{\mathrm{P}}\) & [\(10^{-3}\)] & 0.15 \\ \(\mathrm{\varepsilon}_{\mathrm{C}}\) & [\(10^{-3}\)] & 0.45 \\ \(\mathrm{\varepsilon}_{\mathrm{SLS}}\) & [\(10^{-3}\)] & 3.33 \\ \(\mathrm{\varepsilon}_{\mathrm{ULS}}\) & [\(10^{-3}\)] & 16.67 \\ \(\mathrm{f}_{\mathrm{Ctu}}\) & [MPa] & 2.82 \\ \(\mathrm{f}_{\mathrm{Fts}}\) & [MPa] & 0.75 \\ f\({}_{\mathrm{Ftu}}\) & [MPa] & 1.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Input parameters of the stress-strain diagram in Figure 7 (with tensile strength results). to the experiments, the linear model even assumes a slight stress decrease for increasing CMOD values. This is a consequence of the assumption in the _fib Model Code for Concrete Structures 2010_[17], where a value of 0.5 for the ratio of the residual flexural tensile strength values (\(\sigma_{\text{i}}\) in Table 4) at CMOD values of 2.5 mm and 0.5 mm is considered to calculate the residual tensile strength value (f\({}_{\text{Ftu}}\)) at the CMOD value of 2.5 mm (an explanation is given in [13]). The _fib Model Code for Concrete Structures 2010_[17] also relates the flexural tensile strength to the (uniaxial) tensile strength. This relation results in an estimated tensile strength (based on the average maximum load in Table 4) of 2.89 MPa (for the considered cross-section height of 125 mm, the calculated ratio of the flexural tensile strength and the tensile strength is 1.57), which is close to the tensile strength result of the inverse analysis in Table 6. Interestingly, in [13], expressions for f\({}_{\text{Ftu}}\) and f\({}_{\text{Fts}}\) are presented, which are the basis of the _fib Model Code for Concrete Structures 2010_[17] expressions; however, they are in a less-simplified form. The expressions are derived by considering a linear distribution of stresses in the cross-section (linear model) and the equilibrium conditions of forces and moments in the cross-section. The resulting residual strength values (0.77 MPa and 0.98 MPa, according to expressions from [13]) are relatively closer to the back-calculated values in Table 6.  
The resulting bending moment versus curvature relation for the notched and the gross (without notch) cross-section is given in Figure 8. The bending moment versus curvature relation for the notched cross-section, based on tensile strength values resulting from the simplified expressions of the _fib Model Code for Concrete Structures 2010_[17], is given for additional comparison. For the strains of \(\varepsilon_{\text{c}}\), \(\varepsilon_{\text{SLS}}\), and \(\varepsilon_{\text{ULS}}\), the values \(4.088\times 10^{-4}\), \(4\times 10^{-3}\), and \(2\times 10^{-2}\) were considered.