## 2 Model  
A planar grid beam is considered, consisting of a planar frame, made of two order of micro-beams (namely the fibers), which are arranged in a periodic square pattern of side \(h\), as displayed in Figure 1a. The families of fibers are: (i) that of the longitudinal \(x\)-fibers, which are \(m\) in \(y\)-direction; (ii) that of the transverse \(y\)-fibers, which are \(n+1\) in \(x\)-direction. The nodal points of the \(y\)-fibers constitute the cross-section \(\mathcal{D}_{i}\). Throughout the paper it is assumed that the \(x\)-axis is a symmetry axis for the grid beam and the origin of the coordinate system is placed at the left end \(A\) (see Figure 1a). All the fibers are modeled as Timoshenko micro-beams, having axial, shear and bending stiffnesses \(EA\), \(GA^{*}\) and \(EI\), respectively. The \(x\)-fibers of the grid beam are uniformly compressed by axial forces, whose resultant magnitude is \(P\). The coarse model of the grid beam is (heuristically) assumed to be the Timoshenko beam model, embedded in a 2D space (Figure 1b), whose constitutive law needs to be suitably defined, as described in what follows. Accordingly to this choice, the internally unconstrained model is referred to the material abscissa \(x\in[0,\ell]\), belonging to the interval \((A,B)\), of length \(\ell:=n\,h\), and the following strain-displacement relationships hold [55; 56]:  
\[ \begin{split}\varepsilon&=u^{\prime},\\ \gamma&=v^{\prime}-\theta,\\ \kappa&=\theta^{\prime},\end{split}  \tag{1}\]  
where a prime denotes space-differentiation and, moreover, \(u\left(x\right)\), \(v\left(x\right)\) and \(\theta\left(x\right)\) are the longitudinal and transverse displacements, and the rotations of the cross-section; these displacements are linked by Equation (1) to the elongation \(\varepsilon\left(x\right)\), the shear strain \(\gamma\left(x\right)\) and the flexural curvature \(\kappa\left(x\right)\). Different restraints at the boundary \(H=A,\ B\), namely at \(x=0\) and \(x=\ell\), are considered, for which the relevant boundary conditions read as follows:  
* clamp (C) at \(H\): \(u_{H}=v_{H}=\theta_{H}=0\);
* hinge (H) at \(H\): \(u_{H}=v_{H}=0\);
* horizontal roller (R) at \(H\): \(v_{H}=0\);
* horizontal slider (S) at \(H\): \(v_{H}=\theta_{H}=0\). Figure 1: Study object: (**a**) grid beam; (**b**) equivalent beam model in the reference and current configuration. To analyze the buckling problem, equilibrium of the Timoshenko beam is enforced in the adjacent (slightly varied) configuration. This is done via the virtual work principle (VWP), which, as usual, also accounts for the virtual work spent by the prestress \(\dot{N}=-P\) on the second-order part of the unit-extension \(\varepsilon^{(2)}\). Once the exact unit-extension \(\varepsilon\) is expanded in series, it reads [55; 57; 58; 59; 60; 61]:  
\[\varepsilon:=\varepsilon^{(1)}+\varepsilon^{(2)}+\ldots=u^{\prime}-\frac{1}{2 }\theta^{2}+v^{\prime}\theta+\ldots. \tag{2}\]  
In the absence of incremental loads, the VWP reads:  
\[\int\limits_{0}^{\ell}\left(N\delta u^{\prime}+T\delta\left(v^{\prime}-\theta \right)+M\delta\theta^{\prime}\right)dx-\int\limits_{0}^{\ell}P\delta\left(v^ {\prime}\theta-\frac{1}{2}\theta^{2}\right)dx=0,\quad\forall\left(\delta u, \delta v,\delta\theta\right), \tag{3}\]  
which supplies the following equilibrium equations:  
\[ \begin{split}& N^{\prime}=0,\\ & T^{\prime}-P\theta^{\prime}=0,\\ & M^{\prime}+T+P\left(v^{\prime}-\theta\right)=0,\end{split}  \tag{4}\]  
\(N\left(x\right)\), \(T\left(x\right)\) and \(M\left(x\right)\) being the (incremental) axial and shear internal forces, and bending moment, respectively. The mechanical boundary conditions, considered in what follows, are determined from the principle (3); they are:  
* free (F) at \(H\): \(N_{H}=T_{H}-P\theta_{H}=M_{H}=0\);
* hinge (H) at \(H\): \(M_{H}=0\);
* horizontal roller (R) at \(H\): \(N_{H}=M_{H}=0\);
* horizontal slider (S) at \(H\): \(N_{H}=0\).