## 4 Calculation
### Moment-Curvature Relation Calculation
Many studies state different values for the appropriate plastic hinge lengths, as shown in [25], where different studies' plastic-hing  
Figure 9: Stress distributions for different load stages (CRACK–load of crack occurrence (\(\epsilon_{\text{bot}}=\epsilon_{\text{P}}\)), \(\text{M}_{\text{max}}\)—maximum bending moment (\(\epsilon_{\text{bot}}=0.262\cdot 10^{-3}\)), SLS—load of CMOD = 0.5 mm (\(\epsilon_{\text{bot}}=\epsilon_{\text{SLS}}\)), and ULS—load of CMOD = 2.5 mm (\(\epsilon_{\text{bot}}=\epsilon_{\text{ULS}}\))). lengths range from half of the cross-section height to twice the cross-section height for steel fiber-reinforced concrete. The plastic hinge length (L\({}_{\text{p}}\)) applied for the deflection calculation in this study was equal to the height of the notched cross-section. After all, this value was already used (as the characteristic or structural characteristic length) for transforming the stress versus crack width relation to the stress versus strain relation in Figure 7.  
For load steps corresponding to smaller curvature values of the notched cross-section (\(\kappa<\kappa(\text{M}_{\text{max}})\)), the notch effect was considered by linearly varying the curvature over the plastic hinge between the curvature of the gross cross-section (cross-section without the notch) and the curvature of the notched cross-section (Figure 10a). For load steps corresponding to larger curvatures of the notched cross-section (\(\kappa>\kappa(\text{M}_{\text{max}})\)), it was considered that the curvature over the plastic hinge is constant and equal to the notched cross-section curvature (Figure 10b). On the interval \(0\leq\kappa\leq(\text{L}-\text{L}_{\text{p}})/2\), the curvature distribution (\(\kappa(\text{x})\)) for each load step was determined from the gross cross-section bending moment versus the curvature relation. It must be noted that if the flexural deflection for larger curvatures (\(\kappa>\kappa(\text{M}_{\text{max}})\)) was also calculated by integrating the curvatures and the virtual bending moments, a discontinuity in the load-displacement diagram would occur. Therefore, an approximation was used, where the contribution of the plastic hinge rotation to the displacement of a load step was accounted for by considering a plastic hinge rotation (\(\varphi_{\text{pl}}\)) equal to:  
\[\varphi_{\text{pl}}=(\kappa-\kappa_{1})\cdot\frac{\text{L}_{\text{p}}}{2}, \tag{11}\]  
Table 6.0 Considered curvature distributions over the prism length prior to reaching the maximum load (**a**) and after reaching the maximum load (**b**), and definition of the angle of rotation at the notch after reaching the maximum load (**c**). where \(\kappa\) is the curvature of the notched cross-section, and \(\kappa_{1}\) is the curvature of the notched cross-section read from the positive slope part (with \(\kappa_{1}<\kappa(M_{\rm max})\)) of the moment-curvature diagram in Table 6.1 As shown in Figure 11, \(\kappa_{1}\) corresponds to a bending moment equal to the bending moment of the current load step (M(\(\kappa_{1}\)) = M(\(\kappa\))). As the (quasi)elastic rotation (\(\varphi_{\rm el}\)) of the prism segment of the plastic hinge is already counted in by the integration of the curvature over the prism length (\(\kappa(x)\)) with virtual bending moments, the curvature \(\kappa_{1}\) is subtracted from the total curvature of the notched cross-section \(\kappa\) in Equation (11). Finally, the flexural deflection (\(\Delta_{\rm fl}\)) of a load step is calculated with:  
Table 6.2  
Table 6.3  
where \(\overline{\rm M(x)}\) are the virtual bending moments, L is the span length, \(\kappa(x)\) is the curvature distribution over the prism length (with maximum value at the notch equal to \(\kappa(M_{\rm max})\)), and \(\kappa\) is the curvature of the notched cross-section. Note that the curvature distributions (\(\kappa(x)\)) considered in Equations (12) and (13) both correspond to the curvature distribution in Figure 10a.