## 3 Elastic and Inertial Constant Identifications
### Identification Algorithm for the Elastic Constants
By taking the (simplest) constant strain field, the end displacements (rigid motions removed) read:  
\[u\left(x_{i}\right) =0, u\left(x_{i+1}\right) =\varepsilon\,h,\] \[v\left(x_{i}\right) =0, v\left(x_{i+1}\right) =\gamma_{y}\,h+\frac{1}{2}\kappa_{z}\,h^{2},\] \[w\left(x_{i}\right) =0, w\left(x_{i+1}\right) =\gamma_{z}\,h-\frac{1}{2}\kappa_{y}\,h^{2},\] (11) \[\theta_{x}\left  
* An extensional mode (EX), in which \(\mathcal{D}_{i+1}\) translates axially;
* Two shear modes (SH\({}_{y}\), SH\({}_{z}\)), in which \(\mathcal{D}_{i+1}\) translates transversely along the \(y\)- and \(z\)-directions, respectively;
* two flexural modes (FL\({}_{y}\), FL\({}_{z}\)), in which \(\mathcal{D}_{i+1}\) rotates around the \(y\)- and \(z\)- axes, respectively, and translates transversely along the \(z\)- and \(y\)- axes;
* A torsional mode (TO), in which \(\mathcal{D}_{i+1}\) rotates around the \(x\)- axis.  
Moreover, it is worth noticing that the evaluation of the cell's elastic energy is the most difficult step in the identification procedure and, in general, it cannot be pursued in closed form except for few cases, where for example, the fibers are arranged in a simple and regular pattern and/or the cross-section is assumed to be rigid (see [28; 32; 34]). On the contrary, a numerical identification procedure, such as that based on a Finite Element analysis, is always viable, as done in [35].  
Analytical Identification of Elastic Constants of the Grid Beam  
The illustrated algorithm is applied to the grid beam. A cell made of \(n_{y}\times n_{z}\) longitudinal fibers, comprised between two adjacent cross-sections \(\mathcal{D}_{i}\), \(\mathcal{D}_{i+1}\), is considered. The fibers belonging to the cross-sections are taken of half of their stiffnesses, and only their nodal points are assumed to remain on the same plane, as displayed in Figure 2. However, the same fibers are free to warp around the plane--this deformation is here referred to as micro-warping.  
The previously introduced deformation modes, shown in Figure 2, are then assigned to the cell under the following assumptions:  
**Hypothesis 1**.: _In the shear modes_ SH\({}_{y}\)_, SH\({}_{z}\) _all nodes rotate of the unknown angles_ \(\varphi_{y}\) _and_ \(\varphi_{z}\)_, respectively (see [34]);_  
**Hypothesis 2**.: _In the flexural modes_ FL\({}_{y}\)_, FL\({}_{z}\) _the effect of micro-warping is neglected, since the associated elastic energy is small with respect to the extension of the longitudinal fibers (see [34]);_  
**Hypothesis 3**.: _In the torsional mode_ TO _the following energy contributions, related to the fibers belonging to the cross-section plane, are neglected:_  
* _That due to their bending in the cross-section plane;_
* _That due to their torsion._  
It is worth noticing that Hypothesis 3 has been numerically validated in what follows on case studies, as it will be shown ahead.  
The energy \(\tilde{U}_{\text{EX}}\), associated to the EX-mode reads:  
\[\tilde{U}_{\text{EX}}=\frac{1}{2}\,n_{y}\,n_{z}\,E\,A\,h\,\varepsilon^{2}, \tag{12}\]  
in which the only contribution is due to the extension of the longitudinal fibers. The energies \(\vec{U}_{\text{SH}_{y}}\), \(\vec{U}_{\text{SH}_{z}}\) associated to the correspondent shear modes can be determined via [PERSON]'s theorem, that is, as with the deformation work spent by the end-forces of each fiber in the correspondent (assigned) nodal displacements. They read (see [34] for details):  
\[ \begin{split}\vec{U}_{\text{SH}_{y}}&=n_{y}\,n_{z}\, \frac{6 EI}{h\,(1+\alpha)}\,\left(\gamma_{y}-\varphi_{y}\right)^{2}+\left(n_{y} -1\right)\,n_{z}\,\frac{6 EI}{h\,(1+\alpha)}\varphi_{y}^{2},\\ \vec{U}_{\text{SH}_{z}}&=n_{y}\,n_{z}\,\frac{6 EI}{h \,(1+\alpha)}\,(\gamma_{z}-\varphi_{z})^{2}+\left(n_{z}-1\right)\,n_{y}\,\frac {6 EI}{h\,(1+\alpha)}\varphi_{z}^{2},\end{split}  \tag{13}\]  
where the two contributions on the right-end sides of Equation (13) are related to bending of the longitudinal fibers and micro-warping of the transverse fibers, respectively.