HOMOGENIZATION OF PERIODIC BUILDINGS: REFINED BEAM THEORIES AND EFFECTS OF THE RESONANCE OF THE FLOORS

Abstract eng:
Very frequently buildings are made up of identical stories and their structure is periodic in height. Moreover, the experimental mode shapes of regular buildings suggest using continuous beam models to describe their first modes of vibration. The interest of such a modelling lies in the reduction of the calculation and in the efficiency of the analytical formulation to identify the key parameters which govern the dynamic behaviour of the structure. The problem is addressed by applying the homogenization method of periodic discrete media (HPDM) to idealized buildings obtained by repeating unbraced frames. The advantage of this approach is that the equivalent beam model is derived rigorously from the characteristics of the basic frame without any assumption on the macroscopic behaviour. Contrary to "massive media", the studied structures can present a very low stiffness in shear, which opens the possibility of enriched local kinematics with inner deformation or resonance. As a result, several beam models can be generated by varying the properties of the frame elements. The transverse vibration can be described either by classical beam theories like the Euler-Bernoulli beam theory and the Timoshenko beam theory or by atypical beam theories with inner bending. A generic beam model governed by a sixth-order differential equation includes all the possible mechanisms: shear, global bending and inner bending. The stiffness of these mechanisms depend only on the elastic properties of the basic frame and provide simple criteria to identify the relevant model for a real structure. The longitudinal vibration is governed by the tension-compression of the walls at the scale of the structure and is described by the classical equation. However, the resonance in bending of the floors can also occur in the same frequency range. In this case, the real mass must be replaced in the macroscopic description by an effective mass which depends on the frequency. The structure behaves as a metamaterial and exhibits unusual properties. In particular, the same macroscopic mode shape is associated with several resonant frequencies (but the deformation of the elements at the local scale is different). These atypical behaviours are first established theoretically and then, they are confirmed by numerical simulations.

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