Force reduction factor for building structures equipped with added viscous dampers

Abstract

This research work focuses on the analysis of the hysteretic seismic behaviour of inelastic SDOF systems equipped with viscous dampers. In detail, it is aimed at obtaining a practical tool useful for the seismic design of building structures with added dampers, within the framework of the traditional seismic design based on ductility. The objective is to evaluate the appropriate force reduction factor for highly damped (i.e. damping ratio greater than 5 %) SDOF systems able to guarantee a prescribed level of structural safety.

Appendix

The design approach proposed in this paper for building structures equipped with added dampers requires the use of the reduction coefficient $\eta$ for damping ratios higher than 0.05. Figure 13 displays the principal formulations available in the scientific literature (Bommer et al. 2000; NEHRP 2003; Kawashima and Aizawa 1986; Italian SSN 1998; Tolis and Faccioli 1999; Priestley 2003) for the reduction coefficient $\eta$ as a function of the damping ratio. The plot clearly shows that these formulations lead to substantially different values of $\eta$.

Fig. 13

Reduction coefficient $\upeta$ as a function of $\upxi$ according to established formulations available in scientific literature

Fig. 14

Dynamic magnification factor for different values of damping

Furthermore, all the above-cited formulations available in the scientific literature do not depend on the natural period $T$, whilst from simple considerations of structural dynamics, it is clear that $\eta$ should theoretically depend on $T$. In order to clarify this issue, Fig. 14 provides the graphical representation of the ratios between the dynamic magnification factors of damped SDOF systems subjected to an harmonic input, $r=D_{\xi }/ D_{5}$ (one with damping ratio equal to 0.05 and the others with higher damping ratio $\xi >0.05$) as a function of $\beta$ (i.e. the ratio between the circular frequency of the harmonic load and the one of the system) (Clough and Penzien 1993). Inspection of the plot clearly shows that the influence of damping strongly increases as $\beta$ approaches to 1, while it rapidly reduces as $\beta$ travels away from 1 (i.e. for $\beta <0.5$ or $\beta >2.0, \ \eta$ is larger than 0.9).

Fig. 15

Mean reduction coefficient $\upeta$ versus T for all values of $\upxi$ considered. a displacement ratio; b absolute acceleration ratio

In the light of the above considerations, the seismic analyses performed on SDOF systems (with natural periods and damping ratios ranges summarized in Table 1) allowed also to numerically evaluate the reduction coefficient $\eta$ as a function of $T$ and $\xi$. Figure 15 displays $\eta$, in terms of displacement (Fig. 15a), and absolute acceleration (Fig. 15b), versus $T$ and for all values of $\xi$ considered in the present study. Inspection of the graphs clearly shows that natural period $T$ has a significant influence on the reduction coefficient $\eta$. Moreover, as expected from the simple observations on structural dynamics commented above, the benefit reduction due to higher damping strongly reduces as the natural period increases ($T>1.0$ s) and therefore the adoption of a unique $\eta$ for all period $T$ may lead to un-conservative results, especially for the case of structures characterized by high periods (e.g. tall buildings or isolated structures).

In summary, the preliminary results presented in this appendix reveal that the actual formulations for the reduction coefficient $\eta$ (i.e. $\eta$ as a function of $\xi$), also adopted by most of the actual seismic codes, should be revised including explicitly the dependence of $\eta$ on the natural period $T$.