## Appendix E Effect of Considering Shear Deformation on the Forces/Moments of a Single Strut  
To show the shear deformation effect on the final results of analytical relationships of a single strut, we defined different \(\alpha_{i}\) ratios each being defined as the ratio of the resultant forces and moments (required to create a displacement without rotation or rotation without displacement) in the free end of a cantilever beam based on Euler-Bernoulli to the resultant forces and moments required based on Timoshenko beam theory. The ratios \(\alpha_{1}\), \(\alpha_{2}\) and \(\alpha_{3}\) respectively represent the ratio of the following parameters calculated based on the [PERSON] and Timoshenko beam theories: the force required for lateral displacement without rotation, \(T\), the moment required for lateral displacement without rotation (and the force required to create rotation without displacement), \(V\), and the moment required to create rotation without displacement, \(U\). The relationships for \(\alpha_{i}\) ratios are derived below:  
\[\alpha_{1}=\frac{T_{Euler-Bernoulli}}{T_{Timoshenko}}=\frac{\left(\frac{12E_{ s}I}{P}\right)}{\left(\frac{1}{\frac{1}{P2E_{s}I}+\frac{1}{\kappa GA_{s}}} \right)}=\frac{\frac{12E_{s}I}{\beta}}{\frac{\frac{1}{P2E_{s}I}}{\frac{1}{2E_ {s}I}+\frac{1}{\kappa GA_{s}}}}=\frac{12E_{s}I\left(\frac{\beta}{12E_{s}I}+ \frac{I}{\kappa AG_{s}}\right)}{\frac{1}{P2E_{s}I}+\frac{12E_{s}I}{\kappa AG_{ s}}}=\frac{12E_{s}I}{\frac{1}{\kappa AG_{s}}} \tag{39}\]\[\alpha_{2}=\frac{V_{Euler-Bernoulli}}{V_{[PERSON]}}=\frac{\left(\frac{6E_{E}I}{I^{2}} \right)}{\left(\frac{1}{\frac{2}{6E_{E}I}+\frac{2}{2E\alpha_{5}}}\right)}= \frac{\frac{6E_{E}I}{I^{2}}}{\frac{2}{6E_{E}I}+\frac{2}{2E\alpha_{5}}}=\frac{6E_ {S}I\left(\frac{I^{2}}{6E_{S}I}+\frac{2}{\kappa AG_{s}}\right)}{l^{2}}=\frac{l ^{2}+\frac{12E_{S}I}{\kappa AG_{s}}}{l^{2}}=1+\frac{12E_{S}I}{\kappa AG_{s}l^{2}}\] (A40)  
\[\alpha_{3}=\frac{U_{[PERSON]}}{U_{[PERSON]}}=\frac{\left(\frac{4E_{E} I}{I}\right)}{\left(\frac{\frac{2E_{E}I}{I^{2}}+\frac{2}{2E\alpha_{5}}}{\frac{2}{6E _{S}I}+\frac{2}{2E\alpha_{5}}}\right)}=\frac{\frac{4E_{S}I}{I^{2}}}{\frac{2E_ {S}I}{\frac{2}{6E_{S}I}+\frac{2}{2E\alpha_{5}}}}{\frac{2}{6E_{S}I}+\frac{2}{ \frac{2}{2E\alpha_{5}}}}=\frac{4E_{S}I\left(\frac{I^{2}}{6E_{S}I}+\frac{2}{ \kappa AG_{s}}\right)}{l\left(\frac{2E_{S}I}{\kappa AG_{s}l^{2}}+\frac{2}{ \kappa AG_{s}}\right)}\] (A41)  
The variation of ratios \(\alpha_{1}\), \(\alpha_{2}\) and \(\alpha_{3}\) versus the parameter \(r/l\) is presented in the Figure 10. The first important result revealed from the figure is that \(\alpha_{1}\) and \(\alpha_{2}\) are equal for all values of \(r/l\). Therefore, considering the shear deformation effect has the same effect on the force required to create lateral displacement without rotation, \(T\), on the one hand and moment required to create lateral displacement without rotation (and the force required to create rotation without displacement), \(V\), on the other hand. On the other hand, the shear deformation has smaller effect on \(\alpha_{3}\) in comparison with \(\alpha_{1}\) and \(\alpha_{2}\). Although the \(r/l\) is not the ultimate parameter for evaluating the effect of shear deformation in lattice structures for different relative densities, but it is a good measure to predict the difference between the resultant force and moments obtained based on Timoshenko and Euler-Bernoulli beam theories. According to Figure 10, without considering the shear deformation effect in the beam theory, the forces and moments required to create a particular deformation could be predicted by 15-20% higher for \(r/l\) as large as 0.15.