## 4 Numerical Results
### Statics  
The static analysis here developed leads to a comparison of the fine and coarse models in terms of: (i) lateral displacement \(v\left(x\right)\) and torsional angle \(\theta_{x}\left(x\right)\), and (ii) vertical displacements of the grid's cross-sections at selected abscissa \(x=\bar{x}\) and \(z=0\), namely, \(\bar{u}\left(\bar{x},y,0\right)\). The aim of this analysis is to check the effectiveness of the Timoshenko beam model in reproducing the static behavior of the grid beam. A very good accordance is found in cases I-a, irrespectively from the number of cells, that is, \(n=40\) (Figure 4) and \(n=100\) (Figure 5). In the first case, the deformed shape of the equivalent model resembles that of a shear-deformable beam, while in the second case, it resembles that of a slender (Euler) beam. In the former case, the very good agreement between the coarse and fine models is also shown in Figure 6 by comparing their deformed shapes (displacements are amplified by a factor equal to 10 in the Figure). Here, only the joints of the fine model (green dots) are represented, while the blue lines are related to the current configuration of the equivalent model, whose displacements are evaluated by using Equation (9). In Figure 6a, the lateral view at \(z=0\) is displayed and some of the cross-sections of the Timoshenko beam, at sampled abscissas \(x_{i}\), are shown to compare the two models. Figure 6b,c show two top views of the same deformed shape, at \(x=\ell/2\) and \(x=\ell\), respectively, and only the edges of the cross-section of the coarse model are represented for comparison. The order of magnitude of the percentage error, evaluated at \(x=\ell\), is about: \(4\%\) for \(v\left(x\right)\) and \(5\%\) for \(\theta_{x}\left(x\right)\), when \(n=40\); \(0.5\%\) for \(v\left(x\right)\) and \(7\%\) for \(\theta_{x}\left(x\right)\), when \(n=100\). In all the cases, the vertical displacements, given by the FE analysis, at different levels, are very well-described by the Timoshenko beam, by remembering the fact that the coarse model is a rigid cross-section model and, therefore, it furnishes the mean-value of the local rotation of the deformable sections of the grid beam (see Figures 4b and 5b), which, indeed, also undergo macro-warping (i.e., misalignment of the joints). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Case Study I** & **Case Study II** \\ \hline \(c_{11}\) & \(\left[\mathrm{N}\right]\) & \(2.40\times 10^{5}\) & \(1.03\times 10^{5}\) \\ \(c_{22}\) & \(\left[\mathrm{N}\right]\) & \(2.37\times 10^{3}\) & \(8.80\times 10^{2}\) \\ \(c_{33}\) & \(\left[\mathrm{N}\right]\) & \(2.37\times 10^{3}\) & \(1.01\times 10^{3}\) \\ \(c_{44}\) & \(\left[\mathrm{N}\times\mathrm{mm}^{2}\right]\) & \(1.93\times 10^{6}\) & \(4.36\times 10^{5}\) \\ \(c_{55}\) & \(\left[\mathrm{N}\times\mathrm{mm}^{2}\right]\) & \(9.62\times 10^{7}\) & \(4.12\times 10^{7}\) \\ \(c_{66}\) & \(\left[\mathrm{N}\times\mathrm{mm}^{2}\right]\) & \(9.62\times 10^{7}\) & \(6.89\times 10^{6}\) \\ \(m\) & \(\left[\mathrm{kg/mm}\right]\) & \(5.44\times 10^{-4}\) & \(2.01\times 10^{-4}\) \\ \(I_{x}\) & \(\left[\mathrm{kg}\times\mathrm{mm}\right]\) & \(3.78\times 10^{-1}\) & \(8.13\times 10^{-2}\) \\ \(I_{y}\) & \(\left[\mathrm{kg}\times\mathrm{mm}\right]\) & \(1.89\times 10^{-1}\) & \(6.99\times 10^{-2}\) \\ \(I_{z}\) & \(\left[\mathrm{kg}\times\mathrm{mm}\right]\) & \(1.89\times 10^{-1}\) & \(1.15\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Elasto-geometric and mass coefficients of the equivalent beam model. Figure 4: Static response of the equivalent model vs. the discrete FE model, for case study I-a and \(n=40\): (**a**) lateral and torsional displacements; (**b**) vertical displacements of the cross-section joints at \(x=h\), \(\ell/2\), \(\ell\), respectively. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.