## 1 Introduction  
Research on periodic micro-structured systems and architected materials have recently received great attention in several fields of applied sciences [1, 2], also due to significant advances in 3D printing techniques [3]. Indeed, micro-structured bodies reveal some exotic characteristic which far exceed those of classical bulk materials, compared to weight, like, e.g., higher mechanical performances, resilience, impact resistance and, in general, energy absorption capability [4, 5, 6, 7, 8]. The lattice of fibers, i.e., the micro-structure, with its mechanical properties and topology, confers to the resulting (macro) structure such peculiar characteristics. However, periodic lattices are often characterized by high slenderness and can be particularly sensitive to local and global buckling phenomena. For example, in [9, 10, 11, 12, 13, 14, 15], buckling- and plasticity-driven failure mechanisms are investigated in honeycombs, under different loading conditions. In [16], buckling and post-buckling of elastic square honeycombs, under in-plane biaxial compression, is analyzed using a two-scale approach; a simple formula for the evaluation of the very-long-wave buckling stress is also provided. In [17], buckling of a square lattice is studied also to address the possibility of tuning the buckling shape by varying the cell geometry. In this framework, a very promising aspect in dealing with buckling of cellular structures, is the idea to transform the usually (historically) undesirable effects of buckling phenomenon from a negative into a positive, for their potential contribution to'smart' applications in several fields [18] as, e.g., in biology, mechanics, physics, engineering, aerospace and so on. One of the most powerful tools in modeling micro-structured lattices as equivalent continua is homogenization [19; 30; 31] and several methods, see, e.g., [20; 26], have been developed. In this field, generalized (continuous) beam models have been proposed in [32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53] to analyze statics, dynamics, buckling and aeroelasticity of different micro-structured bodies, as lattices, and civil structures as tower buildings and multi-storey frames. In a set of the mentioned papers, [43; 44; 45; 46; 47; 49; 50; 51; 52; 53], buildings and planar frames have been modeled, through a homogenization approach, as continuous beams, namely as: shear-shear-torsional beams, by neglecting macro-bending, or Timoshenko beams. In this framework, the floors and the columns of the periodic frames have been identified with the cross-sections and longitudinal fibers of the continuous beam, respectively. The reader is referred to [53], where a review of these papers has been provided together with a systematization of the homogenization procedure. In [43; 44; 45; 46; 47; 49; 50], the floors have been assumed to behave as rigid bodies. More details regarding the shear-shear-torsional beam model: in [43; 44], this model has been proposed to analyze the aeroelastic behavior of tower buildings; in [47], the main assumptions and the limits of applicability have been discussed; in [50], the beam model, under small displacement assumption, has been proved to be effective in analyzing linear free and forced dynamics of multi-storey buildings. On the other end, with reference to the Timoshenko equivalent beam model: in [45], such a model has been proposed to properly account for macro-bending of multi-storey buildings in linear statics and dynamics; in [46], the model, embedded in a 3D space, is improved to account for buckling analysis of uniformly compressed multi-storey buildings; in [49], an extension of [46] is proposed, where the same equivalent beam model is shown to be effective in describing the flexural-torsional buckling behavior of not-uniformly compressed tower-buildings, also accounting for the effect of an elastic soil.