# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 4 Conclusions  
The study proposes a beam-like model suitable for the schematization of real buildings that do not have a uniform mass and stiffness distribution along the height and are characterized by asymmetric plans.  
The linear dynamic behaviour of the beam-like element was evaluated by discretizing the continuous model according to a Rayleigh-Ritz approach based on a suitable number of modal shapes of the uniform shear torsional beam clamped at the base. An optimization procedure was adopted for estimating the appropriate stiffness correction coefficients in order to take into account possible uncertainties in the structural parameters and the out-of-plane floor deformability. These correction coefficients were identified by minimizing the objective function that measures the difference between a fixed number of eigen-properties of the beam-like model and the correspondent target values. The latter represent the natural frequencies and the modes of vibration of the real structure and could be directly measured by means of dynamic tests. In the present paper, however, without limiting the general applicability of the proposed procedure, these target values were calculated by means of three-dimensional linear FEM models.  
The proposed procedure was applied in the evaluation of the dynamic response of three multi-storey RC frames representative of residential buildings designed for vertical loads. Only the first three modes have been considered for calibrating the correction factors. The first two case studies report only one irregularity, either in the planar or in the vertical distribution of mass and stiffness. In the last analyzed building, both the planar and vertical irregularities are present. It is highlighted that an increase in the number of irregularities require the knowledge of more data in order to seek an appropriate set of correction coefficients for the stiffness of the building. The results reported in the applicative section show that, in spite of the simplicity of the proposed beam-like model and the small computational burden, satisfactory results were obtained. In particular, the frequencies and modes of vibration, time histories of the displacements of some control points and maxima displacement at each floor level of the 3D FEM models were compared to those calculated by means of the proposed beam-like model, showing, for all the considered buildings, good agreements.  
Although the study is limited to the elastic range of the dynamic behaviour of beam-like structures, the obtained results show good accuracy, indicating the proposed approach has potential in modelling real buildings. The elastic model could either be useful in the definition of an operativity limit state index, based on ambient vibration, or represent the first step needed for the implementation of a powerful tool for the seismic assessment of urban areas with an extremely low computational burden.  
Conceptualization, A.G. and I.C.; methodology, A.G. and S.C.; software, G.O. and I.F.; formal analysis, G.O. and I.F.; investigation, D.S.; supervision, I.C. and S.C.; funding acquisition, I.C. and S.C. All authors have read and agreed to the published version of the manuscript.  
The authors gratefully acknowledge the financial support of the Ministero dell'Istruzione, dell'Universita e della Ricerca (National Research Project PRIN 2015 JW9 NJT \"Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges\").  
The authors declare no conflict of interest.