## 4 Numerical Model Validation  
The proposed numerical model is validated on four-point beam bending experiments available in the literature [2; 10; 12; 36]. All experiments used for validation were performed on a beam reinforced with longitudinal reinforcement and steel fibers, loaded in thirds of the span with two vertical forces of equal intensity. The span-to-depth ratio of the beam was high enough to enable a dominant flexural behavior and all considered beam specimens failed in flexure. Beam parameters that varied among the specimens used for validation were the volumetric ratio of fibers (from 0.38% to 2%), fiber type (straight, irregular and hook-end), fiber aspect ratio (from 30 to 80), concrete strength (from 43 to 135 MPa), longitudinal reinforcement ratio (from 0.67% to 3.4%), beam span (from 1140 mm to 2100 mm) and cross-section dimensions. Details regarding specimens' properties can be found in Table 1 in the Appendix A. The flexural behavior of the beam is modeled using six finite elements of equal length with two integration points (i.e., cross-sections) per element (Figure 3). Each cross-section is discretized with 20 fibers along the height, representing SFRC and one fiber per bar of longitudinal reinforcement. Figure 4 compares the load-deflection relation obtained experimentally, using the proposed numerical model for specimen S2V1 tested by [PERSON] [10]. The first crack, yield and collapse points are shown as green, orange and red circles, respectively, showing how the assumed criteria for defining these points compare to the experimentally obtained results. In this case, the numerical model was able to simulate the initial, post-crack, and post-yield stiffness of the beam and to capture the first crack, yield, and collapse points with good accuracy. Figure 4: Numerical model results for specimen S2V1 tested by [PERSON] [10]. Figure 3: Numerical model set-up. Figure 5 compares the experimentally and numerically obtained values of first-crack force for all specimens used for validation. The correlation coefficient between these values is relatively high, at 0.95. Figure 6 presents the relation between the experimentally and numerically obtained values of the first-crack displacement. The correlation coefficient between these values is lower than for the first-crack force, at 0.67. In the case of first-crack displacements, the numerical model almost always underestimated the experimentally obtained displacements. The reason for this underestimation might be that the numerical definition of the first crack point is too conservative since it is defined as the point at which only the bottom fiber of the numerical mid-span cross-section attains residual tensile strength. In the experimental setting, such crack might not have been noticed. For a more significant crack to form, perhaps  
Furthermore, the identification of the collapse point can be difficult in ductile elements where collapse is not as clearly defined as in brittle elements. The randomness in the structural behavior of SFRC also contributed to the large scatter of experimentally obtained collapse displacements, further reinforcing the need for probabilistic analysis and identification of the most important sources of this uncertainty. Nevertheless, the proposed FEM model can be improved to increase the accuracy of collapse prediction. Potential improvements would be to consider deflection-softening behavior of beams, by, for example, considering strain-softening in compression. Furthermore, other collapse definitions can be considered (e.g., [33]). Figure 10: Comparison between numerical and experimental results of displacement-controlled experiments regarding collapse displacement. Red dots mark load-controlled experiments. Blue dots mark displacement-controlled experiments. Figure 9: Comparison between numerical and experimental results regarding collapse force. Ductility can be expressed by comparing curvatures [32], energies [37] or displacements [38] at ultimate (i.e., collapse) and yield states.