Dynamic analyses and seismic assessments of multi-storey buildings at the urban level require large-scale simulations and computational procedures based on simplified but accurate numerical models. For this aim, the present paper proposes an equivalent non-uniform beam-like model, suitable for the dynamic analysis of buildings with an asymmetric plan and non-uniform vertical distribution of mass and stiffness. The equations of motion of this beam-like model, which presents only shear and torsional deformability, were derived through the application of Hamilton’s principle. The linear dynamic behaviour was evaluated by discretizing the continuous non-uniform model according to a Rayleigh–Ritz approach based on a suitable number of modal shapes of the uniform shear-torsional beam. In spite of its simplicity, the model is able to reproduce the dynamic behaviour of low- and mid-rise buildings with a significant reduction of the computational burden with respect to that required by more general models. The efficacy of the proposed approach was tested, by means of comparisons with linear Finite Element Model (FEM) simulations, on three multi-storey buildings characterized by different irregularities. The satisfactory agreement, in terms of natural frequencies, modes of vibration and seismic response, proves the capability of the proposed approach to reproduce the dynamic response of complex spatial multi-storey frames.

Modern computational techniques have greatly influenced the numerical analyses of structures, not only in terms of calculation speed, but also in terms of procedural approach. In particular, great importance has been given to structural modelling, that is, the process by which a structure and the actions to which it is subjected are reduced to a simplified scheme. The use of a simplified calculation scheme is necessary since the structures are, in general, considerably complex physical systems whose behaviour is influenced by a large number of variables. The definition of a structural scheme that is at the same time simple enough to be easily computable as well as sufficiently reliable in reproducing the main characteristics of the behaviour of the analysed structure is, therefore, a crucial task. In particular, with reference to multi-storey framed buildings, the extensive use of three-dimensional finite element models (FEM) has been made in recent decades by researchers and structural engineers. However, an interesting and alternative research field concerns the possibility of studying multi-storey buildings through the use of equivalent beam-like models in which the number of degrees of freedom and the required computational effort are reduced with respect to more demanding FEM models. Several researchers have proposed single or coupled continuous beams to simulate either the static or dynamic response of multi-storey buildings assuming elastic or inelastic behaviour of the constitutive material. In this paper, a review of several scientific papers proposing elastic or inelastic beam-like models for the structural analyses of framed multi-storey buildings is presented. Considerations about limits and potentialities of these models are also included.

Dynamic analyses and seismic assessments of multi-storey buildings at urban level require large-scale simulations and computational procedures based on simplified but accurate numerical models. At this aim the present paper propos-es an equivalent non-uniform beam-like model, suitable for the dynamic analysis of buildings with asymmetric plan and non-uniform vertical distribution of mass and stiffness. The equations of motion of this beam-like model, which pre-sents only shear and torsional deformability, are derived through the application of Hamilton’s principle. The linear dy-namic behaviour is evaluated by discretizing the continuous non-uniform model according to a Rayleigh-Ritz approach based on a suitable number of modal shapes of the uniform shear-torsional beam. In spite of its simplicity, the model is able to reproduce the dynamic behaviour of low- and mid-rise buildings with a significant reduction of the computa-tional burden with respect to that required by more general models. The efficacy of the proposed approach has been tested, by means of comparisons with linear FEM simulations, on three multi-storey buildings characterized by different irregularities. The satisfactory agreement, in terms of natural frequencies, modes of vibration and seismic response, proves the capability of the proposed approach to reproduce the dynamic response of complex spatial multi storey frames.

The dynamic stiffness method is developed to analyze the natural vibration characteristics of functionally graded beams, where material properties change continuously across the beam thickness following a symmetric law distribution. The governing equations of motion and associated natural boundary conditions for free vibration analysis are derived using Hamilton’s principle and closed-form exact solutions are obtained for both Euler–Bernoulli and Timoshenko beam models. The dynamic stiffness matrix, which governs the relationship between force and displacements at the beam ends, is determined. Using the Wittrick–Williams algorithm, the dynamic stiffness matrix is employed to compute natural frequencies and mode shapes. The proposed procedure is validated by comparing the obtained frequencies with those given by approximated well-known formulas. Finally, a parametric investigation is conducted by varying the geometry of the structure and the characteristic mechanical parameters of the functionally graded material.