## 2 Materials and Methods
### From Euler-Bernoulli to Timoshenko  
According to the relationships obtained for Euler-Bernoulli and Timoshenko beam theories in Sections 2.1 and 2.2, the analytical relationships for elastic modulus and Poisson's ratio of any structure based on Euler-Bernoulli theory can be converted into relationships based on [PERSON] beam theory by making the replacements suggested by Table 1 in the stiffness matrix or in the derivation formulas. To evaluate the effectiveness of the conversion approach presented in Table 1, six well known strut-based lattice structures including BCC, hexagonal packing, rhombicuboctahedron, diamond, truncated cube, and truncated octahedron (Figures 4 and 5) were considered. In the following, the procedure of converting Euler-Bernoulli formulas [26; 30; 31; 32; 35; 39] into equivalent Timoshenko ones is described for each of the six unit cell types. Figure 4: Unit cells of (**a**) body-centred cubic (BCC), (**b**) diamond, (**c**) hexagonal packing, (**d**) rhombicaboctahedron, (**e**) truncated cube, and (**f**) truncated octahedron. Figure 5: Cont. The commonly known geometries for the six noted unit cell types are presented in Figure 4, and the actual geometries used for both the analytical and numerical analyses in this study are illustrated in Figure 5. The reason of altering the unit cell shape for the BCC case is obvious as the molecular structure of BCC (Figure 4a) is composed of spheres rather than struts (Figure 5a). In the case of diamond, the unit cell shown in Figure 4b is rotated, and the unit cell in a specific direction (for which all the struts have similar angles with respect to the horizontal plane) has been considered for analysis (Figure 5b). In the other cases, the unit cell position has been shifted in order to avoid neighbouring cells having adjacent struts. If the neighbour cells have adjacent side-by-side edges, the analytical relationship obtained for the unit cell does not represent that of a lattice structure. More explanations regarding this can be found in Section 2.1.2 of [32]. Although in this paper, all the analytical relationships for the lattice structures and unit cells are presented in a normalized manner and, hence, the dimensions of the unit cells do not affect the results, an equal volume of \(5\times 5\times 5\) mm\({}^{3}\) was considered for all the unit cell types. Moreover, in each unit cell, the strut radius was increased from very small values (\(r\cong 0\)) to high values up to the point where relative density of the unit cell reached \(\mu=0.5\). The conversion procedure from Euler-Bernoulli beam theory into Timoshenko beam theory for each unit cell is described in the following:  
1. BCC  
The relationships for elastic modulus and Poisson's ratio of BCC unit cell have been presented in Equations (A18) and (A19) of Appendix B (extracted from [31]). In these equations, the axial extension (\(\frac{AE_{i}}{I}\)) and lateral bending (\(\frac{12E_{i}I}{8}\)) terms of the Euler-Bernoulli  
Figure 5: Unit cells used for analytical and numerical analysis: (**a**) BCC, (**b**) diamond, (**c**) hexagonal packing, (**d**) rhombicuboctahedron, (**e**) truncated cube, and (**f**) truncated octahedron. theory can be identified easily. By substituting the term \(\frac{12E_{s}I}{l^{3}}\) with \(\left(\frac{1}{\frac{\beta}{lE_{s}I}+\frac{l}{eAG_{s}}}\right)\), the relationships for elastic modulus and Poisson's ratio can be converted. (b). Diamond  
For transforming relationships of elastic modulus and Poisson's ratio from Euler-Bernoulli into Timoshenko beam theory, the basic Equations (A20) and (A21) of Appendix B (extracted from [26]) for elastic modulus and (A22) and (A23) of Appendix B (extracted from [26]) for Poisson's ratio were considered. Axial extension (\(\frac{AE_{s}}{l}\)) and lateral bending (\(\frac{12E_{s}I}{l^{3}}\)) terms of Euler-Bernoulli theory can be identified easily.