Effect of the damper property variability on the seismic reliability of linear systems equipped with viscous dampers

Abstract

Viscous dampers are dissipation devices widely employed for seismic structural control. To date, the performance of systems equipped with viscous dampers has been extensively analysed only by employing deterministic approaches. However, these approaches neglect the response dispersion due to the uncertainties in the input as well as the variability of the system properties. Some recent works have highlighted the important role of these seismic input uncertainties in the seismic performance of linear and nonlinear viscous dampers. This study analyses the effect of the variability of damper properties on the probabilistic system response and risk. In particular, the paper aims at evaluating the impact of the tolerance allowed in devices’ quality control and production tests in terms of variation of the exceedance probabilities of the Engineering Demand Parameters (EDPs) which are most relevant for the seismic performance. A preliminary study is carried out to relate the variability of the constitutive damper characteristics to the tolerance limit allowed in tests and to evaluate the consequences on the device’s dissipation properties. In the subsequent part of the study, the sensitivity of the dynamic response is analysed by harmonic analysis. Finally, the seismic response sensitivity is studied by evaluating the influence of the allowed variability of the constitutive damper characteristics on the response hazard curves, providing the exceedance probability per year of EDPs. A set of linear elastic systems with different dynamic properties, equipped with linear and nonlinear dampers, are considered in the analyses, and subset simulation is employed together with the Markov Chain Monte Carlo method to achieve a confident estimate of small exceedance probabilities.

Appendix: Details of the Atkinson–Silva ground motion model

The Atkinson–Silva ground motion model (Atkinson and Silva 2000) used in this work is characterized by the radiation spectrum S(f) and the time modulating function e(t). The radiation spectrum gives a spectral representation of the ground motion at the construction site, accounting for several physical contributions influencing the wave propagation. Its analytical expression is

$$ S(f) = \varepsilon_{\bmod } S_{0} (f) \cdot S_{n} (f) \cdot S_{f} (f) \cdot V(f). $$

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The (two corner frequencies) point-source spectrum is represented by $S_{0} \left( f \right)$

$$ S_{0} \left( f \right) = C \cdot M_{0} \cdot (2\pi f)^{2} \cdot \left( {(1 - \varepsilon ) \cdot \frac{1}{{1 + \left( {f/f_{a} } \right)^{2} }} + \varepsilon \cdot \frac{1}{{1 + \left( {f/f_{b} } \right)^{2} }}} \right) $$

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where, $M_{0}$ is the seismic moment (expressed in dyne·cm), related to the moment magnitude $M_{m}$ by

$$ M_{0} = 10^{{\frac{3}{2}\left( {M_{m} + 10.70} \right)}} $$

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and C is a constant given by

$$ C = 10^{ - 20} \cdot \frac{{\hat{R} \cdot V \cdot F_{s} }}{{4\pi \rho \beta^{3} }} $$

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where $\hat{R}$, $V$, $F_{s}$ are respectively the radiation pattern ($\hat{R} = 0.55$), a factor partitioning the total shear-wave energy into 2 horizontal components ($V = 0.71$) and the free-surface amplification factor ($F_{s} = 2.0$); $\rho$ and $\beta$ represent the soil density ($\rho = 2.8\,{\text{t/m}}^{3}$) and wave velocity ($\beta = 3.5\,{\text{km/s}}$) near the source; the multiplicative factor 10⁻²⁰ is in order to obtain cm as unit dimension for the ground motion (cm/s² for accelerations). The two corner frequencies $f_{a}$ and $f_{b}$, and the $\varepsilon$ parameter are related to the magnitude by

$$ \log \left( {f_{a} } \right) = 2.181 - 0.496 \cdot M_{m} $$

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$$ \log \left( {f_{b} } \right) = 1.380 - 0.227 \cdot M_{m} $$

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$$ \log \left( \varepsilon \right) = 3.223 - 0.670 \cdot M_{m}. $$

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The $S_{n} (f)$ function, characterizing the path effects of seismic waves, is given by

$$ S_{n} (f) = \frac{1}{R}e^{{\frac{ - \pi fR}{Q(f)\beta }}} $$

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where the 1/R term represents the geometrical spreading effect. The effect of the waves-transmission is accounted by the quality factor $Q(f)$,, defined as

$$ Q(f) = Q_{0} f^{n} $$

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whit $Q_{0} = 180$ and $n = 0.45$ regional parameters. The $S_{f} (f)$ function accounts for the path-independent loss of high-frequency in the ground motion and it is defined by

$$ S_{f} (f) = e^{{\left( { - \pi kf} \right)}} \left[ {1 + \left( {\frac{f}{{f_{\hbox{max} } }}} \right)^{8} } \right]^{ - 0.5} $$

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whit k = 0.03 and f _max  = 100 Hz. The soil amplification factor V(f) is taken according to (Boore and Joyner 1997) for generic soil (V_S,30 = 310 m/s). The model-error parameter $\varepsilon_{\bmod }$ is the adding lognormal random variable (μ_lnε = 0, σ_lnε = 0.5), according to Jalayer and Beck (2008), used for increasing the record-to-record variability. For which concerns the envelope function $e\left( t \right)$, it is given by

$$ e(t) = a \cdot \left( {\frac{t}{{T_{n} }}} \right)^{b} \exp \left( { - c \cdot \left( {\frac{t}{{T_{n} }}} \right)} \right) $$

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with parameters $b = - \varepsilon \cdot \frac{\ln (\eta )}{{\left[ {1 + \varepsilon \left( {\ln (\varepsilon ) - 1} \right)} \right]}}$, $c = \frac{b}{\varepsilon }$, $a = \left( {\frac{\exp \left( 1 \right)}{\varepsilon }} \right)^{b}$ and η = 0.05, ε = 0.2 [0].

The ground-motion total duration is equal to $T_{n} = 2T_{w}$ with $T_{w}$ defined as

$$ T_{w} = 0.05 \cdot R + \frac{1}{{2f_{a} (M_{m} )}}. $$

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The hypo-central distance $R$ from the earthquake source to the site can be defined as follow, in function of the epicentral distance r _e and the moment dependent nominal pseudo-depth h ($\log \left( h \right) = 0.15M_{m} - 0.05$)

$$ R = \sqrt {r_{e}^{2} + h^{2} }. $$

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