## 2 Model
A linear hyper-elastic law is assumed, linking incremental stresses to strains, in which the elastic constants depend on the prestress, i.e., the elastic operator is \(\mathbf{C}=\mathbf{C}\left(\dot{N}\right)\). Accordingly to what was done in [49], a linear function is chosen, namely (It is important to remark that the linear constitute law (5) can be derived by the Green law, once the existence of a suitable strain energy function, namely \(\phi=\phi\left(\varepsilon,\,\gamma,\,\kappa,\dot{N}\right)\), is postulated):  
\[\boldsymbol{\sigma}=\mathbf{C}\left(\dot{N}\right)\boldsymbol{\varepsilon}= \left(\mathbf{C}^{0}-P\,\mathbf{C}^{*}\right)\boldsymbol{\varepsilon}, \tag{5}\]  
where \(\boldsymbol{\sigma}:=\left(N,\,T,\,M\right)^{T}\), \(\boldsymbol{\varepsilon}:=\left(\varepsilon,\,\gamma,\,\kappa\right)^{T}\) and:  
\[\mathbf{C}:=\mathbf{C}^{0}-P\,\mathbf{C}^{*}=\left[ \begin{array}{cc}c_{11} &0&0\\ &c_{22}&0\\ \text{Sym}&c_{33}\end{array} \right]-P\left[ \begin{array}{cc}0&0&0\\ &c_{22}^{*}&c_{23}^{*}\\ \text{Sym}&c_{33}^{*}\end{array} \right]. \tag{6}\]  
It is worth noticing that the particular structure of \(\mathbf{C}^{0}\) and \(\mathbf{C}^{*}\) is due to the regularity of the lattice (see, also [49; 53]). Moreover, the dependency of the elastic matrix on the prestress entails the existence of two critical values of \(P\), which render it singular, namely \(\det\left(\mathbf{C}^{0}-P\mathbf{C}^{*}\right)=0\), the smallest of which is the cell critical load \(P_{cell}\). When a cell is loaded at \(P=P_{cell}\), a local buckling mode occurs; however, when the cells are assembled, as in the grid beam, they can elastically interact, this entailing that bifurcations is expected to manifests itself at a critical load, \(P_{c}<P_{cell}\). Another important aspect relies on how the geometric effects are taken into account in the equivalent model. First, as it has been previously done to enforce equilibrium, the adjacent configuration was considered, in order to describe global geometrical effects. Second, the constitutive matrix \(\mathbf{C}\), governing Equation (5), also accounts for the prestress to describe local geometrical effects, inside the cell [53]. A displacement formulation of the elastic problem is, finally, obtained by re-writing the equilibrium (4) in terms of displacements, and making use of Equations (1) and (5). The elastic problem reads:  
\[ \begin{split}&\mathbf{K}_{2}\mathbf{v}^{\prime\prime}+\mathbf{K}_{1} \mathbf{v}^{\prime}+\mathbf{K}_{0}\mathbf{v}=\mathbf{0},\\ &\mathbf{A}_{1}\mathbf{v}_{A}^{\prime}+\mathbf{A}_{0}\mathbf{v}_ {A}=\mathbf{0},\\ &\mathbf{B}_{1}\mathbf{v}_{B}^{\prime}+\mathbf{B}_{0}\mathbf{v}_ {B}=\mathbf{0},\end{split}  \tag{7}\]  
where \(\mathbf{v}=\left(v,\theta\right)^{T}\) and \(u=0\)\(\forall x\) is accounted for. Moreover, the following definitions are introduced:  
\[ \begin{split}\mathbf{K}_{2}&:=\left[\begin{array}{ cc}c_{22}-P\,c_{22}^{*}&-P\,c_{23}^{*}\\ -P\,c_{23}^{*}&c_{33}-P\,c_{33}^{*}\end{array}\right],\\ \mathbf{K}_{1}&:=\left[\begin{array}{cc}0&-c_{22}+P\,\left(c_{22}^{*}-1 \right)\\ c_{22}-P\,\left(c_{22}^{*}-1\right)&0\end{array}\right],\\ \mathbf{K}_{0}&:=\left[\begin{array}{cc}0&0\\ 0&-c_{22}+P\,\left(c_{22}^{*}-1\right)\end{array}\right],\end{split}  \tag{8}\]  
where \(\mathbf{K}_{i}\)\((i=0,1,2)\) are symmetric or skew-symmetric stiffness matrices. Equation (7) defines a boundary value problem, whose field equation is (7)-a, which is sided by geometric and mechanical boundary conditions at \(A\) and \(B\), (7)-b,c. Moreover, the following restraints at the ends are considered, for which the correspondent definitions of boundary operators in Equations (7)-b,c read:  
* hinge at \(A\), horizontal roller at \(B\) (H-R): \[\mathbf{A}_{1}=\mathbf{B}_{1}:=\left[ \begin{array}{cc}0&0\\ -P\,c_{23}^{*}&c_{33}-P\,c_{33}^{*}\end{array} \right],\;\;\;\mathbf{A}_{0}= \mathbf{B}_{0}:=\left[ \begin{array}{cc}1&0\\ 0&P\,c_{23}^{*}\end{array}