### The dimensioning of Seismic Viscous Dampers to be inserted in Moment Resisting Frame structures

*Tomaso Trombetti, Stefano Silvestri, Giada Gasparini, Michele Palermo*

Ultima modifica: 2011-06-21

#### Sommario

This paper deals with the analysis of torsional behavior of structures characterized by non coincident centre of mass and centre of stiffness under seismic excitation. In previous research works, the authors have identified a synthetic parameter, called “ALPHA”, which governs the torsional behavior of one-story asymmetric systems, and a simplified procedure, called “ALPHA method”, for the estimation of the maximum longitudinal displacement of the flexible side (the side far from the centre of stiffness) of such systems. One question still unresolved concerns the asymmetric structures presenting a torsionally flexible behavior; these structures have a longitudinal displacement of the flexible side different from the corresponding displacement at the center of mass.

With reference to one-story linear-elastic eccentric system, considering the diaphragm infinitely stiff in its own plane, the equations of motion are described by the relative eccentricity *e*, the ratio of stiffness to mass inertial radius (structures characterized by are torsionally flexible, while structures characterized by are torsionally stiff) and the longitudinal period of vibration . The study of the modal contribution factors and the periods of vibration normalized by the longitudinal period (coincident with the second period of vibration), shows that torsionally stiff structures behavior is governed by the fundamental period of vibration and the longitudinal displacement of flexible side of the eccentric system is like the non-eccentric one. In torsional flexible structures the dominant fundamental period is not the longitudinal one, as these structures are affected by the fundamental period shifting; as a consequence, the longitudinal displacement of the flexible side of the eccentric system is larger than that of the non-eccentric one. Taking into account the period shift occurring in these structures, it was identified a combination of the displacements, estimated from the displacement response spectrum next to the three periods of vibration of the structures, and of the corresponding modal contribution factors; it was shown that this combination represents a good approximation of the longitudinal displacement of centre of mass.

The ALPHA method has finally been improved on the basis of the results concerning torsionally flexible structures and the predictive capabilities of the method have been verified through 267.000 numerical simulations, performed on system subjected to uniaxial seismic excitation. The results obtained allow: (i) to determine the period shift describing the torsionally flexible asymmetric structures behavior, and (ii) to determine a synthetic formulation which allows to evaluate the maximum rotation and the maximum displacement of the flexible side as a function of the parameters of the method and of the system entities.

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