## 1 Introduction  
Recent advances in 3D printing techniques and associated micro-fabrication technologies have made (periodic) micro-structured systems and architected the material object of study in several fields of engineering, ranging from aerospace to biomechanics [1]. Interest in research is particularly focused on some peculiar characteristics of such objects, far exceeding those of conventional bulk materials, like a very low lightweight-stiffness ratio, accompanied with significant mechanical properties, such as resilience and impact resistance, which usually allow for important energy absorption capability, see for example, [2, 3], also through micro-scale buckling phenomena. A unit-cell of such micro-structured bodies, whose characteristic dimension is of the order of some millimeters, consists of a lattice of fibers, which interacts at the internal connections. The intrinsic extensional, shear, flexural, and torsional stiffnesses of the fibers, together with the topology of the lattice, is able to confer to the resulting structure, which is the above-mentioned properties at the macro-scale. Homogenization techniques are very useful in modeling periodic structures as equivalent continua (e.g., micropolar continua [4, 5]), such as in the case of lattice structures [6, 7], masonry walls [8], lattice beams and plates [9], reticulated structures [10], and pantographic structures [11] (the reader is also referred to the review papers [12, 13, 14]), as well in dealing with buckling problems in pantographs [15, 16] and micro-structured plates [17, 18]. Several methods have accordingly been developed in the literature,that can be roughly classified into three main categories: (i) heuristic approaches (see, e.g., [19; 20; 21]), grounded on an a priori assumption of the homogenized model, without formulating the relations between the micro and macro quantities; (ii) asymptotic homogenization approaches (see, e.g., [22; 23; 24]), in which the equivalent continuum is rigorously derived; and (iii) mixed approaches (see, e.g., [25]), in which the continuum model is heuristically predicted and, in addition, the micro-macro relationships are analytically determined on the ground of energy balances. The mixed up-scaling technique, adopted in the present work, has also been successfully applied to model tower buildings and multi-storey frames as generalized (continuous) beams in [26; 27; 28; 29; 30; 31; 32; 33; 34; 35], to describe their overall mechanical behavior in statics [28; 30; 34; 35], dynamics [28; 33; 35], buckling [29; 32], and aeroelasticity [26; 27; 31] with a lower computational effort, with respect to, for example, Finite-Element models of the correspondent fine structures. These beam-like structures have been macroscopically modeled as shear-shear-torsional beams, that is, where macro-bending has been neglected, or Timoshenko beams, by identifying the floors and the columns of the periodic frames, with the cross-sections and longitudinal fibers of the continuous beam, respectively. In a subset of them [26; 27; 28; 29; 30; 32; 33], identification has been carried out by assuming the floor could behave as a rigid body, both in its plane and out-of-plane; whereas in [25; 34; 35], this hypothesis has been partially removed and out-of-plane deformability of the floor has been taken into account on an energy average ground. In particular, in [26; 27], an equivalent shear-shear-torsional beam model embedded in a 3D-space has been developed to describe the nonlinear aeroelastic behavior of tower buildings. In [30], a rational justification of the main assumptions, heuristically adopted in [27], is provided. In this framework, the limits of applicability of the model have been discussed in terms of nonlinear static response of some buildings.