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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external structure is beneficial for the frame structure. In dark grey regions, α 1 or α 2 are higher than unity and the coupling is detrimental with respect to the stand-alone frame structure. Figure. 5 shows the α 1 and α 2 maps obtained for the Christchurch and Kobe records. As it can be observed from the values of the contour levels, the coupling is e ff ective in wide ranges of ρ e and ρ c . The performance maps also show the boundaries that separate the parameter plane in the two regions characterized by Mode 1a and Mode 2a (dotted thick line). The best performances are obtained in regions where the Mode 1a exists (on the left of the dotted thick lines). Christchurch Kobe 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r e r c a 1 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.78 0.8 0.8 0.82 0.82 0.84 0.84 0.84 0.86 0.88 0.88 0.88 0.9 0.9 0.92 0.92 0.94 0.96 0.96 0.98 0.98 1 1 0.62 0.64 0.65 0.66 0.66 0.66 0.67 0.69 0.71 0.72 0.73 0.74 0.75 0.76 0.78 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.95 0.96 0.97 0.98 1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r e r c a 2 0.68 0.7 0.72 0.74 0.74 0.76 0.76 0.78 0.78 0.8 0.8 0.82 0.84 0.86 0.86 0.88 0.88 0.9 0.9 0.92 0.92 0.92 0.94 0.94 0.96 0.96 0.98 1 1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r e r c 0.57 0.58 0.6 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.72 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.88 0.89 0.9 0.91 0.91 0.92 0.92 0.93 0.94 0.95 0.96 0.97 D 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r e r c B A Fig. 5. Performance α 1 and α 2 seismic maps for Christchurch and Kobe earthquakes (4-storey frame, Kelvin-Voight connection: ψ = 10, µ = 0 . 1, Maxwell connection: ψ = 10, µ = 5 . 0). To understand the e ff ectiveness of the coupling between the frame and external structures, the time-histories of the points labelled with A in Fig. 5 are shown in Fig. 6. The e ff ectiveness of the coupling can be explained by comparing the time-history of u e to those of u 1 and ∆ u . Point A is located in a relative minimum point in the α 1 map. Therefore, it is expected a significant reduction in u 1 . The time-histories in Fig. 6 show that, in the initial phase of the motion, u e is out of phase with both u 1 and ∆ u . These occurrences explain the reduction of both displacement and drift. The external structure works similarly to a tuned mass damper.

6. Concluding remarks

In this paper, a multi-storey frame structure, which is modelled as a two degree of freedom system, was coupled with a short external structure to improve its seismic behaviour. The frame and the external structures were connected through a visco-eastic device at the first storey of the frame structure. An inerter device was applied to the external structure to modify its inertial force. The coupled structure was described by a three-degree-of-freedom linear elastic mechanical system, whose equations of motion were obtained by a direct approach.