# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 1 Introduction  
The advancements in computational procedures and parallel processing of the last few years enhanced the accurate dynamic analysis and seismic assessments of multi-storey buildings. The analyses can be carried out at the building or urban level and high-fidelity models or sophisticated large-scale simulations can be adopted, respectively. In particular, a dynamic analysis of the entire urban area needs optimized computation procedures and simplified although accurate numerical models. Thus, a new and renovated interest has grown in beam-like models, which were introduced in the last century. Beam-like models, which are based on the equivalence of multi-level structures to flexural-shear coupling continuum beams, aim to simulate the dynamic behaviour of multi-level buildings by drastically reducing the computational burden. Several authors demonstrated interest in beam-like models and proposed suitable simplified approaches for the dynamic analysis of multi-level structures. With regard to high-rise buildings, the discretization of multi-floor frames with coupled beams was introduced in 1982 [1]. The proposed approach was limited to the principal three modes and in the plane behaviour. so that it is capable of experiencing shear strains and torsion only. The elasto-geometric and inertial characteristics of the beam are directly identified from a discrete model of a three-dimensional frame, via a homogenization process. A more refined Timoshenko beam-like model suitable for the dynamic analysis of periodic buildings has been recently proposed [23]. As this excursus shows, the definition of an equivalent beam was not always simple and immediate, especially in case of three-dimensional, torsional coupling behaviour. The reduction of complex structural systems to equivalent beam models is still an open challenge of great interest. In this paper, a non-uniform beam-like model, which is able to take into account two different irregularities, is proposed. In particular, this model is suitable for the schematization of real buildings that do not have a uniform mass and stiffness distribution along the height and are characterized by unsymmetrical plans. The equation of motion of the proposed beam-like model is derived through the application of [PERSON]'s principle. The linear dynamic behaviour of the non-uniform beam-like element is then evaluated by discretizing the continuous model according to a Rayleigh-Ritz approach based on an appropriate number of modal shapes of a uniform beam having only shear and torsional deformability [24]. The proposed approach allows an accurate description of the dynamic behaviour of low- and mid-rise buildings with limited computational effort. Furthermore, eventual uncertainties in the parameters of the structure can be taken into account by means of an opportune calibration strategy, based on the introduction of appropriate correction stiffness coefficients. Aiming at investigating the capabilities of the proposed approach to provide reliable dynamic responses of existing multi-storey reinforced concrete (RC) frame buildings designed to resist only gravity loading, three representative buildings are taken into account. In particular, the first is characterized only by a horizontal irregularity, the second one only by a vertical irregularity while horizontal and vertical irregularities are present in the last case study. Real accelerograms of earthquakes that recently struck the Italian peninsula have been selected for comparing the seismic responses of the beam-like models to those obtained by 3D FEM models. The comparison showed a good accuracy of the non-uniform beam-like model and encourages the authors to consider the proposed procedure as a very useful and powerful tool for the seismic assessment at the urban level. It is worth noting that several residential buildings not designed to withstand earthquakes are spread throughout Italy and their seismic assessment is still argued at political and academic levels.