Application of CQC Method to Seismic Response Control with Viscoelastic Dampers

Abstract

The complete quadratic combination (CQC) modal combination rule for seismic analysis of structures is a great achievement of Professor Armen Der Kiureghian. Among the other various uses of the CQC method, this chapter introduces the application of the CQC method to seismic response control of buildings with dampers to highlight the practical value of the CQC method. The expanded CQC method can accurately estimate the maximum response of buildings installed with response control damping devices and enables performance-based placement design of dampers. In this chapter, the performance-based design procedure of a viscoelastic damper (VED) is introduced for finding the storywise distribution of VEDs in a building such that each peak interstory drift coincides with the corresponding prescribed value. The mechanical properties of the employed VED’s dependence on amplitude and frequency of the excitation as well as material temperature are taken into account and a mechanical nonlinear four-element model that comprises two dashpot elements and two spring elements is proposed for the VED. The developed performance-based design procedure utilizes equivalent linearization of the VED and the expanded CQC method, which involves modal analysis with complex eigenvalue analysis. Seismic response analyses are carried out for high-rise building models with optimally placed VEDs, with the results demonstrating the effectiveness of the expanded CQC method and the validity of the proposed performance-based placement-design procedure.

Appendices

Appendix I

1.1 Earthquake Interstory Drifts of Hypothetical 15- and 24-story Buildings

The dynamic loading tests of the VE material and the VED make use of simulated earthquake interstory drifts of hypothetical 15- and 24-story buildings subjected to Hachinohe 1968 EW earthquake or El Centro 1940 NS earthquake. The fundamental natural periods of the 15- and 24-story buildings are assumed to be 2 s and 3 s, respectively. Fig. 22 shows the waveforms of the earthquake interstory drifts in Figs. 4b, 11 and 12.

Fig. 22

Earthquake interstory drifts of hypothetical 15- and 24-story buildings

Appendix II

2.1 Voigt Model of VED Considering the Stiffness of the Mounting Component

For consideration of the mounting component’s stiffness, K _M the equivalent Voigt model including the mounting component can be constructed. As shown in Fig. 23, the sum of the maximum shear deformation of the VED, $\delta_{D} = \gamma_{\hbox{max} } \cdot d$, and the maximum deformation of the mounting component, δ _M , is equal to the specified peak interstory drift, $\bar{\delta } = L_{S} \cdot \bar{\beta }$.

Fig. 23

Equivalent linearization of VED considering the mounting component’s stiffness

The following equation holds for $\gamma_{\hbox{max} }$, $\bar{\delta }$, K _M , and $K_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)$ and $C_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)$ of the VED given by Eqs. (3a) and (3b) for a specified set $\left\{ {A_{s} ,\;d,\;T_{e} } \right\}$:

$$\gamma_{\hbox{max} } \cdot d = \bar{\delta } \cdot \left\{ {\left( {1 + \frac{{K_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right)^{2} + \omega^{2} \left( {\frac{{C_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right)^{2} } \right\}^{{ - \frac{1}{2}}}$$

(12)

While Eq. (12) cannot be analytically solved in terms of $\gamma_{\hbox{max} }$, the value of $\gamma_{\hbox{max} }$ can be estimated through iterative computations by substituting $L_{S} \cdot \bar{\beta }/d$ as an initial value of $\gamma_{\hbox{max} }$ on the right-hand side of Eq. (12). Once a converged value of $\gamma_{\hbox{max} }$ is obtained, $K_{D}$ and $C_{D}$ of the equivalent Voigt model considering K _M are obtained according to

$$K_{D} = \frac{{K_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)\left( {1 + \frac{{K_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right) + \omega^{2} \frac{{C_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)^{2} }}{{K_{M} }}}}{{\left( {1 + \frac{{K_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right)^{2} + \omega^{2} \left( {\frac{{C_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right)^{2} }}$$

(13)

$$C_{D} = \frac{{C_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{\left( {1 + \frac{{K_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right)^{2} + \omega^{2} \left( {\frac{{C_{eq} \left( {\omega ,\gamma_{\hbox{max} } } \right)}}{{K_{M} }}} \right)^{2} }}$$

(14)

where the fundamental natural circular frequency of a building, $\omega = 2\pi /T_{1}$, is assigned to ω in Eqs. (13) and (14) in the same way as in Sect. 3.1.