## 6 Results and Discussion
### Free Rocking on a Flexible Beam Structure  
Numerical results for the elastic beam structure are presented and discussed. The goal is to investigate the effect of beam flexibility on the rocking of the block and on the numerical algorithm's prediction capability. The effect of beam flexibility on the rocking response is also analyzed by comparing it with the rocking on a rigid base; therefore, the experimental results for the rigid base rocking are also displayed in the following figures together with the numerical results.  
We will refer to \(\overline{E}\) as the Young's modulus for which the beam deflects for L/2500 under the block's weight concentrated at the beam centre. Young's modulus in the following analyses is set as \(E=\overline{E}\cdot 10^{n}\), where \(n\) is varied in order to change the beam's bending stiffness. The geometrical characteristics of the beam and all other material properties used in the model are provided at the beginning of Section 6. Figure 16 shows a set of simulations where the mesh and time-step from the rigid base model are kept while increasing the flexibility of the beam (i.e., reducing its Young's modulus). It is observed that as the flexibility is increasing, the numerical predictions start to differ, in particular when the stiffness drops from \(\overline{E}\cdot 10^{8}\) to \(\overline{E}\cdot 10^{6}\).  
Figure 15: Block B8M—experimental and numerical results for the block rotation angle in time.  
In Figure 17, the time interval containing the first two impacts from Figure 16 is extracted. It can be seen that differences start to appear after the impacts between the block and the beam base. Since the mesh discretization and time-step are unchanged, it can be observed that the impulse occurring during contact is also affected by the beam stiffness.  
In the following, mesh and time-step effects on the numerical results are analyzed. Figure 18 shows the results obtained with different meshes. It is observed that the mesh initially displays fast convergence, while for flexibility obtained with \(\overline{E}\cdot 10^{2}\), the prediction of the block's rotations becomes sensitive to the mesh.  
A similar observation is made by analyzing the time-step effect in Figure 19, but the time-step influence can be noticed for much higher bending stiffness (already for \(\overline{E}\cdot 10^{8}\)). A decrease in the time-step shows a convergent trend for all analyzed values of [PERSON]'s modulus until \(\overline{E}\cdot 10^{0}\), but it appears that rather small time-steps become necessary. However, the simulations clearly indicate that the rocking behaviour still qualitatively replicates rocking on a rigid base, i.e., smooth decay of the rotation angle until a rest state is reached and a similar time to reach it. This trend appears to happen up to the stiffness corresponding to a deflection of \(10^{-3}\cdot L/250\) under the block weight, where \(L\) is the beam span. The value \(L/250\) corresponds to the engineering limit for beam deflection in structures.  
Figure 16: Block angle time history for flexibility increase in the rigid base model.  
Figure 17: Zoomed image of the block rotation angle between the first two impacts.  
For further increases in flexibility, it was observed that convergence is lost after a few impacts, both in terms of mesh and time-step, as shown in the last graph of Figure 19. This is observed for the beam with stiffness giving a static displacement under the block weight of roughly \(L/2500\). Therefore, below this limit, the predictions do not provide convergent and reliable results for the entire simulation time.  
Figure 19: _Cont._  
Figure 18: Mesh density effect on the rocking behaviour when increasing the flexibility.