Static and Dynamic Analysis of Framed Buildings by Means of an Equivalent Multi-stepped Beam

In order to evaluate the static and dynamic response of multi-storey buildings many researchers have focused, in the last thirty years, their attention on simplified models which allow to drastically reduce the computational effort without losing the required accuracy. In this context, with the aim of analyzing buildings characterised by a decreasing size of the column cross sections along the height of the structure, the present study refers to a simplified model represented by a multi-stepped beam. Therefore, in the proposed approach, each inter-storey of the building is modelled by means of a beam segment with uniform stiffness and mass properties. The model is adopted for the evaluation of the elastic response of multi-storey buildings either to horizontal static forces or to seismic accelerations at the base. The equations of motion of the considered multi-stepped beam, which exhibits only shear and torsional deformability, have been derived by means of the Hamilton’s principle and its deformed shapes have been evaluated through a Rayleigh–Ritz approach based on a certain number of mode shapes of the uniform shear beam. Numerical applications to a reinforced concrete multi-storey building have been performed by comparing the static and dynamic responses, evaluated by means of the proposed approach, to the results obtained according to 3D FEM models. Furthermore, the static response of the considered multi-stepped beam has been also validated by means of an original closed form solution for step-wise shear-torsional elastic beams.