## 4 Numerical Results
### Statics
Figure 5: Static response of the equivalent model vs. the discrete FE model, for case study I-a and \(n=100\): (**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. The same very good accordance is also found in case I-b, that is, for different boundary conditions, as apparent in Figure 7. Here, the order of magnitude of the percentage error at \(x=\ell/2\) is about \(2\%\) for both \(v\left(x\right)\) and \(\theta_{x}\left(x\right)\). In contrast, some important quantitative differences occur for the torsional displacement in case II-a, when \(n=40\) (see Figure 8): indeed, here the error at \(x=\ell\) is about \(0.5\%\) for \(v\left(x\right)\) and \(32\%\) for \(\theta_{x}\left(x\right)\). This occurrence is attributed to the fact that, in case study II, the thinner cross-sections of the grid beam are much more prone to warping, with respect to the (ticker) ones of case I, where the fibers lying in a cross-section plane constitute a more stiff lattice. Therefore, significant macro-warping is experienced from the grid beam in case study II, which cannot be well-captured by a rigid cross-section equivalent model; on the contrary, to account for this effect, more refined beam theories are needed to be able to correctly describe the deformation modes of the cross-sections, as with, for example, the Generalized Beam Theory (see, e.g., [46; 47; 48]). Figure 6: Deformed shape of the equivalent model vs. the discrete FE model, for case study I-a and \(n=40\): (**a**) lateral view at \(z=0\); (**b,c**) top views at \(x=\ell/2\) and \(x=\ell\), respectively. Green dots: discrete FE solution. Continuous blue line: homogenized beam model. Amplification factor for displacements in the Figure is taken as equal to \(10\). As proof of what conjectured, a numerical evaluation of the cell's energy, sided on a Finite-Element analysis, has been carried out in cases study I and II, by extending to the 3D case what was done in [35] for a planar cell, namely: (i) the nodal displacements at the boundary joints, compatible with the deformation modes, have been assigned; (ii) the reactive forces at the same joints have been evaluated; and (iii) the elastic energy has been computed via the Clapeyron Theorem. In particular, the cell's energy has been here evaluated by assigning (i) the TO mode, and (ii) an axially-unrestrained version of the TO mode (au-TO), where the kinematic constraint enforcing planarity of the cross-sections \(\mathcal{D}_{i}\), is removed and warping is allowed. It was found that:  
* case I: \(c_{44}=1.90\times 10^{6}\) N \(\times\) mm\({}^{2}\) (TO mode) and \(c_{44}=1.76\times 10^{6}\) N \(\times\) mm\({}^{2}\) (au-TO mode);
* case II: \(c_{44}=4.27\times 10^{5}\) N \(\times\) mm\({}^{2}\) (TO mode) and \(c_{44}=2.36\times 10^{5}\) N \(\times\) mm\({}^{2}\) (au-TO mode). It can be seen that, in case I, the two torsional stiffnesses are almost the same, while in case II, the \(c_{44}\) given by the axially unrestrained FEM model is about half of the restrained one, thus confirming the importance of the macro-warping effect. Finally, the numerical analysis of the cell, also corroborates the assumptions here made on the analytical identification of the torsional stiffness: indeed, the percentage error made in the analytical evaluation of the (constrained model) coefficients (remember Table 2), with respect to the numerical ones, is about 2% in both the case studies. Figure 8: Static response of the equivalent model vs. discrete FE model, for case study II-a and \(n=40\): (**a**) lateral displacement; (**b**) torsional displacement. Green dots: discrete FE solution. Continuous blue line: homogenized beam model. Figure 7: Static response of the equivalent model vs. discrete FE model, for case study I-b and \(n=40\): (**a**) lateral displacement; (**b**) torsional displacement. Green dots: discrete FE solution.