# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 3 Results and Discussions
### Structural Analysis Validation  
In this section, the structural analyses results are presented. Figure 9 shows the simulation environments of the three FEM software for the BBC lattice structure during the compression load case (\(-\)10 kN). As previously discussed, the boundary conditions consist in locking the displacements along the X- and Y-direction of the nodes of the beams connected to the big rod's end and locking the displacements and rotations of the nodes of the beams connected to the small rod's end. Figure 8: Overall results from the size optimization of the four lattice structures: slenderness ratio and relative density for the traction and the compression load cases. Figure 10 shows the maximum displacement along the Z-direction. The displacement results given by the three software are similar because, in the FE displacement-based type, the displacement is more accurate than the strain field [77]. A bending-dominated structure experiences the bending of its struts, whereas a stretching-dominated structure deforms primarily by the stretching or compressing of individual struts under loading. So, as can be seen from the results, the BC, BCC, and WG structures present a stretching-dominated behavior, whereas the DA has a bending-dominated behavior. According to the connectivity of the structure and Maxwell's law, the BC structure should be bending-dominated but appears to be the stiffest structure between the four. The BC structure shows a stretching-dominated behavior because it is mainly stressed along the main axes; different boundary conditions would lead to a different behavior [5]. The BCC and DA structures confirm [PERSON]'s rule prediction; indeed, even if they have a similar apparent density, the displacement of the DA structure is more than 10 times higher than the displacement of the BC one. Since the SC and the BCC structures come from the same unit cell, they exhibit similar mechanical behavior; the BCC is more constrained and therefore has a smaller deflection in both the load cases. Figure 11 represents the maximum direct stress of the structures. The values of the WG and DA lattice structures from the three FE software disagree, most likely due to the very small size of the elements for the WG structure and the very small slenderness ratio for the DA one. A divergence of the results due to the small ratios of slenderness  
Figure 10: Displacement results of the traction and the compression load cases, according to the FEM software. Figure 9: FEA deformation results for the BBC lattice structure during the compression load case. Deformation scale: 40 times. (**a**) MSC Patran/Nastran, (**b**) Karamba3D, and (**c**) ANSYS Mechanical. was also observed by [PERSON] and [PERSON] [44; 79]. For thicker beams, the shear and rotational inertia effects, which are taken into account in the Timoshenko theory but are neglected in the Euler-Bernoulli theory, start to play an important role; the shear and rotational inertia effects cause the beam to be less stiff, explaining a higher elastic modulus calculated using the Euler-Bernoulli rather than the solution based on the Timoshenko beam theory [43; 46; 73]. The difference between the results of Ansys and Karamba3D could be explained by the shear locking phenomenon that occurs where the application of the linear shape function leads to an over-stiffened element. To avoid this issue, the element can be subdivided into more instances, or the shape function can be changed with a quadratic or cubic one [46]. The element used in Ansys has a quadratic shape function, with an additional interpolation point in the center of the beam with respect to the Karamba3D linear element, and this could explain the difference between the direct stress result. Indeed, from the traction to the compression configuration, an increase in the diameter is observed with a consequent decrease in the minimum slenderness ratio, causing higher divergence between the FEM results.