## 2 Model  
The grid beam consists of a grid cylinder, that is, in a 3D frame, made of a periodic and specifically designed three-dimensional assembly of equal micro-beams. It is made of micro-beams, here referred to as fibers, forming a periodic cubic micro-structure, of side \(h\) (Figure 1a), whose pattern is repeated \(n_{x}=n\) times in \(x\)-direction. Two families of fibers are detected: (i) those parallel to the beam \(x\)-axis, referred to as longitudinal fibers, which are \(n_{y}\) in \(y\)-direction and \(n_{z}\) in \(z\)-direction; (ii) those lying in the \(n+1\) planes orthogonal to the axis-line, referred to as transverse fibers, parallel to the \((y,z)\)-plane. These latter are arranged in a square orthogonal pattern that is reinforced through bracing diagonal elements, which guarantee the indeformability of the cylinder in the \((y,z)\)-plane. It is assumed that the \((x,y)\)- and \((x,z)\)-plane are of symmetry for the grid beam, and the origin of the coordinate system is at the lower end \(A\) (see Figure 1). All the fibers are modeled as Timoshenko micro-beams, whose cross-section properties are taken as equal in the two principal inertia planes, where square, circular, and ring cross-sections, or analogous, are considered. They have axial, shear, bending, and torsional stiffnesses \(EA\), \(GA^{*}\), \(EI\), and \(GJ\), respectively. An internally unconstrained model of beam, embedded in a 3D space, see, for example, [37; 38; 39; 40; 41; 42], is formulated (Figure 1b). The beam is made of material points, equipped with orientation, which accounts for the attitude of the cross-section. It is referred to as a material abscissa  
Figure 1: Study object: (**a**) grid beam; (**b**) equivalent beam model. \(\left[0,\ell\right]\), spanning the interval \(\left(A,B\right)\) of total length \(\ell:=n\,h\). The following strain-displacement relationships hold:  
\[ \begin{split}\varepsilon&=u^{\prime},\\ \gamma_{y}&=v^{\prime}-\theta_{z},\\ \gamma_{z}&=w^{\prime}+\theta_{y},\\ \kappa_{x}&=\theta_{x}^{\prime},\\ \kappa_{y}&=\theta_{y}^{\prime},\\ \kappa_{z}&=\theta_{z}^{\prime},\end{split}  \tag{1}\]  
which link the longitudinal displacement \(u\left(x,t\right)\), the transverse displacements \(v\left(x,t\right)\), \(w\left(x,t\right)\), the in \(y\)-direction and \(z\)-direction, respectively, and the rotations of the cross-section \(\theta_{v}\left(x,t\right)\), with respect to \(\
u\)-axis (\(\
u=x,y,z\)), to the elongation \(\varepsilon\left(x,t\right)\), the shear strains \(\gamma_{y}\left(x,t\right)\), \(\gamma_{z}\left(x,t\right)\), the torsional curvature \(\kappa_{x}\), and the flexural curvatures \(\kappa_{y}\left(x,t\right)\), \(\kappa_{z}\left(x,t\right)\); here, \(t\) is the time, and a prime denotes space differentiation. The following geometric boundary conditions are considered in what follows (subscripts \(A\), \(B\) denote evaluation at the ends \(x=0\) and \(x=\ell\), respectively):  
* Clamped at \(A\) and free at \(B\): \(u_{A}=v_{A}=w_{A}=\theta_{xA}=\theta_{xA}=0\);
* Simply-supported and torsionally restrained in \(A\), \(B\): \(u_{A}=u_{B}=v_{A}=v_{B}=w_{A}=w_{B}=\theta_{xA}=\theta_{xB}=0\). The equilibrium is ruled by the following equations:  
\[ \begin{split} N^{\prime}+p_{x}-m\dot{u}&=0,\\ T_{y}^{\prime}+p_{y}-m\dot{v}&=0,\\ T_{z}^{\prime}+p_{z}-m\dot{v}&=0,\\ M_{x}^{\prime}+c_{x}-I_{x}\ddot{\theta_{x}}&=0,\\ M_{y}^{\prime}-T_{z}+c_{y}-I_{y}\ddot{\theta_{y}}& =0,\\ M_{z}^{\prime}+T_{y}+c_{z}-I_{z}\ddot{\theta_{z}}&=0, \end{split}  \tag{2}\]  
where \(N\left(x,t\right)\), \(T_{y}\left(x,t\right)\) and \(T_{z}\left(x,t\right)\) are the axial and shear internal forces, respectively, \(M_{x}\left(x,t\right)\) is the torsional moment, \(M_{y}\left(x,t\right)\) and \(M_{z}\left(x,t\right)\) are the bending moments, \(p_{v}\left(x,t\right)\), \(c_{v}\left(x,t\right)\) are the external forces and couple per unit-length (\(\
u=x,y,z\)).