## 4 Numerical Results
It is also apparent in Figure 3 that, when the geometric effect of the prestress is not taken into account in the elastic law (5), both the Timoshenko and Euler-Bernoulli models, yellow (C) and red (D) curves, are ineffective in describing as the critical load varies with \(n\), even if the former accounts for the flexibility of the transverse fibers via the corrective factor \(\chi\), Equation (18). This sort of failure becomes much more evident the more constrained is the beam for a given \(n\). As a final remark, the importance of the shear factors, defined in Equation (18), can be better stressed by looking at their numerical values in Table 2: indeed, it is seen that, not accounting for the transverse fibers flexibility would lead to (elastic and geometric) shear factors equal to 1, which are very different from those here found. Therefore, both the elastic and geometric corrective factors of the constitutive constants (17) play a crucial role in the constitutive law definition, the former producing the much more significant correction. \begin{table}
\begin{tabular}{c c c} \hline \hline \(EA\) & \(GA^{*}\) & \(EI\) \\ \([\mathbf{N}]\) & \([\mathbf{N}]\) & \([\mathbf{N}\times\mathbf{mm}^{2}]\) \\ \hline
4905 & 2043.75 & 919.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elasto-geometric properties of the fibers for the case studies I and II. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Case Study I**} & **Case Study II** \\ \hline \(c_{11}\) & \([\mathbf{N}]\) & \(3.43\times 10^{4}\) & \(9.81\times 10^{3}\) \\ \hline \(c_{22}\) & \([\mathbf{N}]\) & \(3.38\times 10^{2}\) & 69.81 \\ \hline \(c_{33}\) & \([\mathbf{N}\times\  
In order to compare the critical mode of the equivalent model with that furnished by FE discretization on the fine model, \(n=50\) is taken. Comparison is shown in Figure 4, where the critical modes of the two models are superimposed, for different boundary conditions, namely: H-R (Figure 4a), C-F (Figure 4b), C-R (Figure 4c), C-S (Figure 4d). It is seen that the green curves, representing axis-line and cross-sections sampled at different abscissas of the [PERSON] equivalent model (remember Equation (13)), are almost coincident with the dots, relevant to the correspondent FE solutions. It is worth mentioning that the critical mode of the C-S grid beam (Figure 4d) resembles the one provided in [16] for the long-wavelength buckling of a square honeycomb with square cells, under uniaxial compression. Figure 3: Non-dimensional critical load \(\mu\) vs. the number of cells \(n\), for the case study I and for different boundary conditions: (**a**) Hinge–roller (H-R); (**b**) clamp–free (C-F); (**c**) clamp–roller (C-R); (**d**) clamp–slider (C-S). Green curve (A): Timoshenko model; blue curve (B): Finite-element (FE) solution; yellow (C) and red (D) curves: Timoshenko and Euler–Bernoulli models without constitutive geometric effect. Same qualitative considerations are apparent for the buckling analysis of case study II, whose results, in terms of \(\mu\) vs. \(n\), are reported in Figure 5 for all the above-mentioned beam models and for the following boundary conditions: H-R (Figure 5a), C-F (Figure 5b), C-R (Figure 5c) and C-S (Figure 5d). Moreover, Figure 6 shows the superimposition of the analytical and numerical critical modes, namely those relevant to the equivalent beam and FE models, respectively, when \(n=20\) and for the same boundary conditions. In addition, in this case, the accordance is very good both in terms of critical loads and modes, providing \(n>\bar{n}\). Therefore, the effectiveness of the [PERSON] equivalent beam model, corrected by elastic and geometric shear factors, is confirmed also for a low number of longitudinal fibers. Finally, in order to better investigate how the critical loads depend on \(m\), a buckling analysis on a sample grid beam is carried out. The grid beam's fibers have the same elasto-geometric characteristics described above and, moreover, \(n=50\) and H-R boundary conditions are considered.