## 4 Numerical Results  
Numerical results concern two sample grid beams, made of a thermoplastic polymer, having the elastic modulus \(E=2180\) N/mm\({}^{2}\) and the Poisson coefficient \(\
u=0\). To evaluate the effect of the number of longitudinal fibers \(m\) on the buckling response, in the first case study \(m=7\) is taken, while in the second it is \(m=2\) and in both of them \(h=10\) mm. The fibers' cross-section is squared, of dimensions \(1.5\,\mathrm{mm}\times 1.5\,\mathrm{mm}\) and its elasto-geometric characteristics are listed in Table 1 (\(A\) is the area, \(A^{*}\) the shear area, \(I\) the inertia moment). It is worth to notice that the above-mentioned case studies have been selected in order to be representative of grid beams which can be easily printed through additive manufacturing techniques, both for the chosen material and geometric characteristics. Buckling analyses are carried out by considering different boundary conditions, already defined in Section 2 and corresponding to: hinge-roller (H-R), clamp-free (C-F), clamp-roller (C-R), clamp-slider (C-S). The analytical (exact) solution of the equivalent beam model has been determined according to the procedure sketched in Section 2. Accordingly, critical loads vs. cell number and critical deformed shapes are compared with numerical results obtained by FE models. Concerning FE analyses, the planar grid beam has been modeled as a planar elastic frame, made of Timoshenko micro-beams, eventually restrained at the centroid of the terminal cross-sections, after a rigid body constraints of the same ends is enforced. In the FE discretization procedure here developed: (i) a Timoshenko beam finite-element has been adopted to model the fibers, whose stiffness matrix accounts for both elastic and geometric effects; (ii) each fiber, comprised between two nodes of the planar frame, has been meshed into five finite-elements. The elasto-geometric characteristics of the equivalent beam model, the cell's critical loads and the shear factors are listed in Table 2 for both the case studies. Throughout the next figures, the green dots and curves (labeled with A) represent the FE solution, while the continuous blue curves (label B) that of the Timoshenko beam model. Moreover, yellow (label C) and red (label D) curves are relevant to the results given by the Timoshenko and Euler-Bernoulli models, respectively, in which the geometric effect on the constitutive law is neglected, i.e., \(\mathbf{C}^{*}=\mathbf{0}\). Buckling analysis of case study I is addressed and the influence of the number of cells is first investigated. Results of this analysis are displayed in Figure 3 for the different boundary conditions described above, namely: H-R (Figure 3a), C-F (Figure 3b), C-R (Figure 3c), C-S (Figure 3d). It is seen that the critical buckling load multiplier \(\mu:=P_{c}/P_{cell}\) decreases with the number of cells \(n\), and, as it is expected, the more constrained the beam, the higher \(\mu\) for a given \(n\). Moreover, it is apparent that when \(n>\bar{n}\), \(\bar{n}\) being a 'critical' number of cell, the equivalent model gives excellent approximation of the FE results, i.e., the green curves (A) are almost superimposed to the blue (B) ones. This value \(\bar{n}\) depends on how the boundary conditions are applied in the fine model (i.e., restrained end cross-sections modeled as restrained rigid bodies) and it is critical in the sense that below it the homogenization procedure may lead to wrong results. Indeed, the fine model, when \(n<\bar{n}\) predicts \(\mu>1\), which cannot be reproduced by the equivalent beam model, due to the fact that its constitutive law becomes singular when \(P=P_{cell}\).