## 6 Results and Discussion  
In this section, we investigate if a correlation between rocking on a rigid and flexible base exists. First, the case of rocking on a rigid base is presented, for which convergence analyses and experimental validations are performed. Then, the base flexibility is gradually increased and its influence on rocking and on the algorithm predictions is tested. In that process, the influence of mesh density and time-step size are checked. Moreover, comparisons between the numerical predictions of free rocking on a flexible beam and rocking on a rigid base are provided.  
The structural base on which the block is rocking is implemented as a simply supported beam. For this type of boundary conditions, the simplification that contact is happening only at the block vertices adequately represents the real case since the beam will bend downward in a convex shape. Other boundary conditions and complex loading excitations could require the generalization of the contact algorithm in order to model more generic contact conditions, such as contact occurring between the block edge and beam. In all examples, it is assumed that the beam has a span of 30 cm, cross-section width and height of 40 mm and 1.71 mm, respectively, Poisson's ratio of 0.334, and density of 2795 \(\frac{\text{kg}}{\text{m}^{3}}\). In order to introduce damping effects, as they are present in real structures, Rayleigh damping was used, with the values of damping ratios \(\xi_{1}=0.05\) (for the first natural frequency) and \(\xi_{1}=0.2\) (for the third natural frequency) [42; 45]. The friction coefficient was set to 0.364.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Block** & \(m\) **[g]** & \(b\) **[m]** & \(h\) **[m]** & \(\frac{h}{b}\) & \(\alpha\) **[rad]** & \(s\) **[m]** \\ \hline B4M & 727.2 & 0.045 & 0.135 & 3 & 0.3218 & 0.0712 \\ B6M & 1089.6 & 0.045 & 0.2025 & 4.5 & 0.2187 & 0.1037 \\ B8M & 1634.0 & 0.045 & 0.30375 & 6.75 & 0.1471 & 0.1535 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric characteristics and masses of the tested blocks.  
Figure 9: Experimental setup according to [14]: (**a**) system of tapes to assure pure rocking, (**b**) edge contact between the base and the block.  
This value was obtained as an average from several simple tests involving a block laying on a beam. The beam was initially in a horizontal position, and then its inclination was gradually increased until the sliding of the block occurred. The inclination angle at which the block starts to slide was measured, and the coefficient of friction is obtained as the tangent of this angle. The weight of the beam is taken into account throughout the simulation, and the beam is given an initial displacement under its own weight and the block's weight, which corresponds to the actual initial conditions. For time integration, the \(\theta\)-parameter was set to 1, leading to the backward Euler method. The block material and geometry characteristics are reported in Table 1. The Young's modulus value is provided in the specific examples since the value is varied in order to change the bending stiffness of the beam structure. All analyzed stiffnesses are such that the beam displacement caused by the block weight is less than \(1/250\) of the beam span. This value is chosen in accordance with most building codes that limit the maximum value of the deflection as \(1/360\) to \(1/240\) of the span length [46]. Moreover, this limitation justifies the application of linear beam theory that has been implemented in the numerical code. Nonetheless, we point out that the NSCD approach can be applied also to problems in which large displacements and deformations occur [32].