## 3 Elastic and Inertial Constant Identifications
### Identification Algorithm for the Elastic Constants
Moreover, \(\alpha:=12 EI/\left(GA^{*}h^{2}\right)\) is a nondimensional stiffness ratio. By requiring \(\frac{\partial\Omega_{\text{SH}_{y}}}{\partial\varphi_{y}}=0\), \(\frac{\partial\Omega_{\text{SH}_{z}}}{\partial\varphi_{z}}=0\) the unknowns \(\varphi_{y}\) and \(\varphi_{z}\) are eliminated, and Equation (13) reads:  
Figure 2: Single cell of the grid beam, and its deformation modes (bracings not shown): (**a**) 3D view of the EX mode; (**b**) top view of the TO mode; (**c**) lateral view of the SH\({}_{\
u}\) mode, \(\
u=y,z\); (**d,e**) lateral view of the FL\({}_{\
u}\) mode, \(\
u=y,z\). \[\tilde{U}_{\text{SH}_{y}} =\frac{1}{2}\chi_{y}\,n_{y}\,n_{z}\frac{12 EI}{(1+\alpha)\,h}\gamma_ {y}^{2}, \tag{14}\] \[\tilde{U}_{\text{SH}_{z}} =\frac{1}{2}\chi_{z}\,n_{y}\,n_{z}\frac{12 EI}{(1+\alpha)\,h}\gamma_ {z}^{2},\]  
where:  
\[\chi_{y}:=\frac{n_{y}-1}{2n_{y}-1},\quad\chi_{z}:=\frac{n_{z}-1}{2n_{z}-1}, \tag{15}\]  
are nondimensional quantities, here referred to as shear factors, which, as shown in [34], account for the micro-warping of the fibers (i.e., a warping which leaves the joints aligned) of the fine model in the same way they account for warping of the solid cross-section in the Timoshenko beam model. The energies \(\tilde{U}_{\text{FL}_{y}}\), \(\tilde{U}_{\text{FL}_{z}}\) associated to the correspondent flexural modes are:  
\[\tilde{U}_{\text{FL}_{y}} =\frac{1}{2}\,n_{y}\left(EI\,h\,n_{z}+EA\,h\,\sum_{j=1}^{n_{z}}z_ {j}^{2}\right)x_{y}^{2}, \tag{16}\] \[\tilde{U}_{\text{FL}_{z}} =\frac{1}{2}\,n_{z}\left(EI\,h\,n_{y}+EA\,h\,\sum_{k=1}^{n_{y}}y_ {k}^{2}\right)\kappa_{z}^{2},\]  
where the two contributions on the right-end sides of Equation (16) accounts for bending and extension of the longitudinal fibers. Here, according to that discussed in [34], micro-warping of \(y\)- and \(z\)-fibers is neglected. The energy \(\tilde{U}_{\text{TO}}\), associated with the torsional mode TO, is the sum of the torsional energies: (i) of its longitudinal fibers, which, according to the Hypothesis 3, experience the same twist angle; and (ii) of the two orthogonal orders of planar frames parallel to \(y\) and \(z\). Concerning this latter contribution, it is found by evaluating the torsional energy of a single planar frame, undergoing a shear mode, namely \(\gamma_{yj}=-\kappa_{x}z_{j}\) (\(y\) -parallel) or \(\gamma_{zk}=\kappa_{x}y_{k}\) (\(z\)-parallel), analogously to what was done to determine Equation (14). Accordingly, \(\tilde{U}_{\text{TO}}\) reads:  
\[\tilde{U}_{\text{TO}}=\frac{1}{2}\left[n_{y}\,n_{z}\,G\,J\,h+\chi_{y}\,n_{y}\, \frac{12 EI}{h\,(1+\alpha)}\sum_{j=1}^{n_{z}}z_{j}^{2}+\chi_{z}n_{z}\,\frac{12 EI}{h\,(1+\alpha)}\sum_{k=1}^{n_{y}}y_{k}^{2}\right]\,\kappa_{x}^{2}, \tag{17}\]  
where the terms in the square brackets are the torsional stiffnesses of the longitudinal fibers, and of the frames parallel to the \((x,z)\)- and \((y,z)\)-planes, respectively. Moreover, since \(\tilde{U}=\tilde{U}_{\text{EX}}+\tilde{U}_{\text{SH}_{y}}+\tilde{U}_{\text{SH} _{z}}+\tilde{U}_{\text{FL}_{y}}+\tilde{U}_{\text{FL}_{z}}+\tilde{U}_{\text{TO}}\), Equation (10) supplies \(\tilde{\phi}\); then, the equality \(\phi=\tilde{\phi}\quad\forall\,\left(\epsilon,\gamma_{y},\gamma_{z},\kappa_{x },\kappa_{y},\kappa_{z}\right)\) is enforced and the Green law supplies the following elastic constants, Equation (4), namely:  
\[c_{11} =n_{y}\,n_{z}\,EA, \tag{18}\] \[c_{22} =\chi_{y}\,n_{y}\,n_{z}\frac{12 EI}{h^{2}(1+\alpha)},\] \[c_{33} =\chi_{z}\,n_{y}\,n_{z}\frac{12 EI}{h^{2}(1+\alpha)},\] \[\tilde{U}_{\text{SH}_{y}} =\frac{1}{2}\chi_{y}\,n_{y}\,n_{z}\frac{12 EI}{(1+\alpha)\,h}\gamma_ {y}^{2}, \tag{14}\]0 \[\tilde{U}_{\text{SH}_{y}} =\frac{1}{2}\chi_{y}\,n_{y}\,n_{z}\frac{12 EI}{(1+\alpha)\,h}\gamma_ {y}^{2}, \tag{14}\]1 \[\tilde{U}_{\text{SH}_{y}} =\frac{1}{2}\chi_{y}\,n_{y}\,n_{z}\frac{12 EI}{(1+\alpha)\,h}\gamma_ {y}^{2}, \tag{14}\]2As a final remark, it is important to observe that, since bracing elements are assumed to be hinged, they do not energetically contribute to \(\tilde{U}\).