## 3 Verification of Developed 3D FDEM for Modeling of RC Structure Collapse
### Simulation Results of the Three-Point Bending Test  
Figure 8 shows the numerical simulation results of the three-point bending test for the RC beam in which the RC model's stress distribution and crack propagation are presented. At first, the point load is applied to the RC beam through contact with the top-loading rod with the constant velocity, while the bottom-loading rod remains fixed. After further displacement of the loading rod, the vertical loading to the RC specimen linearly increases due to the specimen's elastic deformation, as shown in Figure 8a. The tensile stress is concentrated at the bottom part of the beam, while the microscopic crack is not presented at this stage. With the additional displacement of the loading rod, the axial loading to the reinforced concrete specimen continues to increase linearly, and tensile fracturing finally occurs at the bottom of the beam, as presented in Figure 8b. With the further loading, the tensile crack propagates upward (Figure 8c). After that, additional fracture initiation and propagation occurs along with the rebar due to the further deflection of the rebar around the center (Figure 8d). Then, the reinforced concrete beam collapses completely, resulting in significant concrete fracturing and separation. Here, the crack is propagated along with the rebar model due to severe plastic deformation of the rebar (Figure 8e). The resultant crack patterns for four different element sizes (\(h_{ave}\) = 100 mm, 50 mm, 25 mm and 12.5 mm) are shown in Figure 9. The case of \(h_{ave}\) = 100 mm shows that the RC beam is split into two halves (Figure 9a), and fragments move almost parallel to the applied loading, which is not reasonable compared with the experimental observation [34]. On the other hand, the remaining cases with \(h_{ave}\leq 50\) mm exhibit a similar crack pattern, with numerous cracks distributed throughout the lower part in the model (Figure 9b,c), which shows better agreement with the experiment [34]. Hence, it is indicated that the value of \(h_{ave}\leq 50\) mm is at least required to obtain an acceptable fracture pattern in the case of this three-point bending simulation. Figure 10 illustrates the applied load (\(P\))--displacement (\(d\)) curves for the RC beam model with various \(h_{ave}\)s during three-point bending testing simulation. As shown in Figure 10, when \(h_{ave}\) = 100 mm, the RC beam loses its bearing capacity at \(d\) = 1 mm, while the remaining cases exhibit nearly identical behavior. The sudden drop in the \(P\) in the case of \(h_{ave}\) = 100 mm is owing to the crack propagation mode identified as splitting mode (Figure 9a). On the other hand, for the cases with \(h_{ave}\) = 50 mm, 25 mm and 12.5 mm, the shapes of the \(P\)-\(d\) curves are almost similar both for the elastic regime and plastic regime for \(d\) > 1 mm. Thus, at least from this problem, it may be concluded that the FDEM simulations  
Figure 7: Numerical simulation model for the three-point bending test of the RC beam: (**a**) the specimen geometry and (**b**) the rebar inserted in the specimen. incorporating the rebar element are less sensitive to \(h_{ave}\) as long as \(h_{ave}\) is taken small enough in the way that the fracture process and pattern are accurate. Figure 8: Spatial distribution of maximum principal stress (**left column**) and macroscopic crack (**right column**) in the RC beam subjected to the three-point loading. Figure 9: The resultant crack pattern of RC beam as a result of numerical simulations with various finite concrete element sizes: (**a**) \(h_{ave}=100\) mm, (**b**) \(h_{ave}=50\) mm, (**c**) \(h_{ave}=25\) mm and (**d**) \(h_{ave}=12.5\) mm. The fracture behavior of the plain concrete without rebar is also analyzed to investigate the reinforcement effect of the applied rebar model (i.e., RE and RJE). Figure 11 presents the mid-span loading-displacement curves for plain and reinforced concrete structures obtained from FDEM simulations.