# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 3 Applications  
In order to apply the proposed procedure to real structures with different irregularities, namely planar and vertical ones, in the present paper three multi-storey RC frames representative of residential buildings designed to resist only gravity loading have been considered. The considered buildings have been chosen taking into account the results of an extensive survey of existing multi-storey RC buildings built in Catania, Italy, before the introduction of the seismic code [28; 29; 30]. The geometrical details of the typical building can be found in [30], where further information on the loading and the material properties are reported. A synthetic description of the proposed method, for a demonstrative purpose only, together with applications to simple spatial frames have been presented by the authors in previous works [31; 32]. The three models of realistic buildings here considered will be denoted as Model Type 1, Type 2 and Type 3 in the following. Type 1 refers to a building with planar asymmetry and vertical uniform distribution of mass and stiffness. Due to the peripheral staircase, which is located on one side of the building, the center of stiffness does not coincide with the center of mass. The eccentricity determinates torsional effects that are correctly identified by the proposed beam-like model. Due to the presence of planar asymmetry only (no vertical irregularity is introduced), the objective function \(O\) introduced into Equation (17) to determine the stiffness correction coefficients of the equivalent beam model for Type 1 building, accounting for the natural frequencies only (i.e., \(\beta=0\)), provided accurate results. Type 2 presents a planar symmetry and vertical non-uniform distribution of mass and stiffness due to the reduction of the cross sections of the columns along the height. Once again, due to the existence of only one of the considered irregularities (i.e., the vertical one), the minimization of the objective function \(O\) in Equation (17) has been performed by considering only the frequencies of the vibration. The structural plans of Type 1 and Type 2 are reported in Figure 4 and the inter-storey height of both models is set equal to 3.30 m at each level. In order to consider the contemporary occurrence of both the planar and vertical irregularities, Type 3 has been considered. The latter has been obtained by adding vertical asymmetries to the already-described Type 1. In this view, the height of the first floor has been incremented up to 4.3 m, and a reduction along the height of the building in the column cross-section has been introduced and a total number of six floors have been considered. In this case, due to the contemporary presence of the two types of irregularities, it will be shown that, in order to obtain a reliable dynamic response of the beam-like model, the contribution of the modal shape vector term in the objective function must also be considered; hence,\(\beta=1\). Precisely, it is worth noting that only the modal displacements of the top floor have been introduced in the objective function (Equation (17)). In all the three building types the dimensions of the cross sections of the beams on the edge were assumed to be \(30\times 50\) cm\({}^{2}\), while all the remaining beams have a \(110\times 23\) cm\({}^{2}\) size. The columns were assumed fully fixed atng, theec3 fully level. The results, in terms of seismic response, were compared to those obtained by means of a conventional FEM model developed with SAP2000 [33]. The reliability of the beam-like model was evaluated by comparing the time histories of the \(x\) and \(y\) displacements of some control points. For the sake of brevity, in the following only the results related to a control point located at the upper left corner of the highest floor are reported. Furthermore, the maximum displacements of the same corner located at each floor were calculated.