PSI - Issue 44

A.Di Egidio et al. / Procedia Structural Integrity 44 (2023) 2136–2143 A. Di Egidio, S. Pagliaro, A. Contento / Structural Integrity Procedia 00 (2022) 000–000

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isolation as a non-conventional TMD and introduced a method for the seismic design of such a system, whereas in Pagliaro and Di Egidio (2022), a 3-DOF model was used to investigate the dynamic and seismic benefits of an intermediate discontinuity for a general frame structure. Finally, low-dimensional models were used also to describe the behaviour of rigid block-like structures. For instance, in Di Egidio et al. (2016) a dynamic mass absorber was used to improve the dynamic behaviour of rigid blocks. The present paper assesses the improvement of the behavior of a frame structure due to the coupling with an external structure shorter than the frame structure and connected only to the first storey, so it has a limited visual impact. The purpose of this study is to understand if the connection with the external structure can reduce both the displacement of the connected storey (i.e., the first storey of the structure) and the drift of the structure above the connection, without changing the geometrical and the mechanical properties of the frame structure to protect. The connection between the frame and the external structures is performed with visco-elastic device. A general multi-storey frame is described with a 2-DOF model, whereas a 1-DOF model is used to describe the external structure. Moreover, an inerter device is applied to the external structure to modify its inertial force. The coupled structure is described by a 3-DOF linear elastic mechanical model. The coupling with the external structure is considered beneficial for the frame when there is a reduction in the displacements and drifts of the coupled structure with respect to the stand-alone frame structure. The e ff ects of the coupling with the external structure and with the inerter device are shown in terms of modification of the frequencies and modal shapes. It is observed that, if the mechanical characteristics of the coupling system are suitably chosen, the first mode of the coupled system maximizes the modal displacement of the external structure and minimizes the modal displacements of the frame structure. Moreover, this mode acquires a modal mass participating factor that overcomes the 90%, thus making the dynamics of the system very regular. Further parametric analysis is performed exciting the coupled structure with several ground motion records. The results are arranged in performance maps that compare the displacements and drifts of the coupled structure with those of the stand-alone frame structure. Each map is built by varying the sti ff ness of the external structure and the connection device, for fixed values of the other characteristics of the coupled structure. In wide regions of the parameter plane the coupling appears to be beneficial for the frame structure. The frame structure, whatever number of storeys, is modelled by an equivalent 2-DOF system as in Fabrizio et al. (2019). The first DOF, u 1 , is the displacement of the substructure in correspondence of the connection with the external structure, whereas the second one, u 2 , is the displacement of the superstructure, above the connection point. The external structure is described by a 1-DOF system, whose displacement is named u e (Fig. 1a,b). An inerter device with virtual mass (or inertance) m Re increases the inertial force of the external structure to which is connected through a sti ff chevron structure, as in Makris and Kampas (2016). The damping coe ffi cient of the external structure, c e , is obtained by assuming that the damping ratio of the 1-DOF system that models the external structure is ξ e = 0 . 02, whereas the damping coe ffi cients of the equivalent 2-DOF structure are derived from the Rayleigh formulation by assuming a damping ratio ξ = 0 . 05 for both the oscillation modes. The equations of motion are obtained with a direct approach by imposing the equilibrium of the forces acting on m 1 , m 2 , and m e as m 1 ¨ u 1 ( t ) + c 1 ˙ u 1 ( t ) − c 12 ˙ u 2 ( t ) + ( k 1 + k 2 ) u 1 ( t ) − k 2 u 2 ( t ) − c c [ ˙ u e ( t ) − ˙ u 1 ( t )] − k c [ u e ( t ) − u 1 ( t )] = − m 1 ¨ x g ( t ) m 2 ¨ u 2 ( t ) − c 12 ˙ u 1 ( t ) + c 2 ˙ u 2 ( t ) − k 2 u 1 ( t ) + k 2 u 2 ( t ) = − m 2 ¨ x g ( t ) ( m e + m Re ) ¨ u e ( t ) + c e ˙ u e ( t ) + k e u e ( t ) + k 2 u 2 ( t ) + c c [ ˙ u e ( t ) − ˙ u 1 ( t )] + k c [ u e ( t ) − u 1 ( t )] = − m e ¨ x g ( t ) (1) where ¨ x g is the external base excitation. It is worth observing that when the Rayleigh method is applied, a full sym metric damping matrix C = α M + β K is generated, where c 12 = c 21 in Eq. 1 are the o ff -diagonal terms of C . The inerter is a linear mechanical device that develops a resisting force F ID proportional to the relative acceleration between its terminals. The inerter device considered in this paper is of the rack-pinion-flywheel type, as shown in Fig. 1c. The system consists of two (or more) flywheels of radius R i and mass m ω i that are free to rotate and connected 2. Mechanical model