## 1 Introduction  
It has been reported in the works by [PERSON] et al. [1] and [PERSON] et al. [2] that the vertical earthquake ground motion component of some earthquakes located near the causative fault can have an important detrimental impact in the structural behavior of different systems. For instance, [PERSON] et al. [1] attributed seismic failures to the effect of the vertical component of three studied earthquakes on structures. Besides possible compressive overstress or tension failure, the vertical response can lead to failures for the ultimate limit states of flexure moment and shear. The latter (i.e., shear) may explain observed failures perhaps associated with underlying vertical motion effects, because the compression can be reduced, or even mild tension could be reached, decreasing the contribution of concrete to shear resistance. The moment capacity and ductility of reinforced concrete (RC) columns can also be reduced. Unconsidered fluctuations in the axial force due to vertical motions may precipitate flexural failures. In another study, [PERSON] et al. [2] based on modal analysis proposed a conservative simple approach to assess vertical seismic forces for buildings. More recently, [PERSON] et al. [3] by means of experimental and analytical studies corroborated that the axial load level associated with the seismic vertical component has a detrimental effect on the shear capacity. The influence of the earthquake ground motion vertical component has been found not only important in the structural response of buildings but also in that of bridges. For instance, [PERSON] et al. [4] evaluated the effect of vertical earthquake ground motion on RC bridge piers and found that the inclusion of the vertical component importantly impacts the response of all components. They concluded that the vertical component should be taken in consideration for bridge design. Likewise, for highway overcrossings, [PERSON] et al. [5] found the vertical component of ground motions as a main cause of amplification in the axial force in the columns and flexure moment in the girders. This was reported as especially critical for midspan moments in negative bending. Ancient and historical buildings have also presented damaged attributed to the vertical component of earthquakes, as observed on 7 September 2017 in historical structures located near the epicentral zone of the great Tehuantepec Mw8.2 earthquake, Mexico, [PERSON] et al. [6]; it was suggested that the high frequency contents generated by this event and its vertical component led to damages to belfries, domes, towers and vaults of the historical structures. On the other hand, it is also reported that when a modal analysis is carried out, different bending moments, axial forces and shear forces are obtained whether the lumped or consistent matrix mass approach is followed, as indicated in [PERSON] et al. [7], being the structural response (in terms of bending moments and shear forces) significantly increased for the consistent matrix mass approach if the vertical component is included. This is possibly due to the dynamic behavior of beams, which implies carrying loads (due to the vertical component of earthquakes) perpendicular to their longitudinal direction, which in turn causes bending moment and deflections, as cited in [PERSON] et al. [8]. The two approximate methods to consider the inertial effect in the structure in general, and in the beams in particular, are the lumped mass approach (the distributed mass is concentrated in point masses) and the consistent mass approach (the point masses also include rotational effects); this latter method is consistent with the static traverse displacements of the beam, as cited in [PERSON] et al. [8]. In addition, in the studies by [PERSON] et al. [9] is reported that under certain conditions (i.e., short structural periods, especially for soft soils and near the epicentre) the vertical component becomes much more significant than the horizontal component of ground motion.