Performance assessment and optimization of fluid viscous dampers through life-cycle cost criteria and comparison to alternative design approaches

Abstract

The performance assessment and optimal design of fluid viscous dampers through life-cycle cost criteria is discussed in this paper. A probabilistic, simulation-based framework is described for estimating the life-cycle cost and a stochastic search approach is developed to support an efficient optimization under different design scenarios (corresponding to different seismicity characteristics). Earthquake losses are estimated using an assembly-based vulnerability approach utilizing the nonlinear dynamic response of the structure whereas a point source stochastic ground motion model, extended here to address near-fault pulse effects, is adopted to describe the seismic hazard. Stochastic simulation is utilized for estimation of all the necessary probabilistic quantities, and for reducing the computational burden a surrogate modeling methodology is integrated within the framework. Two simplified design approaches are also examined, the first considering the optimization of the stationary response, utilizing statistical linearization to address nonlinear damper characteristics, and the second adopting an equivalent lateral force procedure that defines a targeted damping ratio for the structure. These designs are compared against the optimal life-cycle cost one, whereas a compatible comparison is facilitated by establishing an appropriate connection between the seismic input required for the simplified designs and the probabilistic earthquake hazard model. As an illustrative example, the retrofitting of a three-story reinforced concrete office building with nonlinear dampers is considered.

Appendix: State-space representation for structural system

The state space representation for a $n_{s}$-story building with output z composed of inter-story drifts and absolute accelerations is

$$\begin{aligned} \begin{aligned} \dot{\mathbf{x}}_s (t)&=\mathbf{A}_s \mathbf{x}(t)+\mathbf{B}_s \mathbf{u}(t)+\mathbf{E}_s a_g (t) \\ \mathbf{z}(t)&=\mathbf{C}_s \mathbf{x}(t)+\mathbf{D}_s \mathbf{u}(t);\quad \dot{\mathbf{x}}_D (t)=\mathbf{L}_s \mathbf{x}(t) \end{aligned} \end{aligned}$$

(24)

where $\mathbf{x}_{s}\in \mathfrak {R}^{2n_s }$ is the state vector composed or displacements and velocities of all stories (relative to the ground), $a_{g}$ is the ground acceleration and the matrices in formulation (24) are

$$\begin{aligned} \begin{array}{l} \mathbf{A}_s =\left[ {{\begin{array}{c@{\quad }c} {\mathbf{0}_{n_s xn_s } }&{} {\mathbf{I}_{n_s } } \\ {-\mathbf{M}_s ^{-1}\mathbf{K}_s }&{} {-\mathbf{M}_s ^{-1}\mathbf{C}_s } \\ \end{array} }} \right] \quad \mathbf{B}_s =\left[ {{\begin{array}{c} {\mathbf{0}_{n_s xn_s } } \\ {\mathbf{M}_s ^{-1}\mathbf{T}_s^T } \\ \end{array} }} \right] \quad \mathbf{E}_s =\left[ {{\begin{array}{c} {\mathbf{0}_{n_s xn_s } } \\ {\mathbf{R}_s } \\ \end{array} }} \right] \\ \mathbf{C}_s =\left[ {{\begin{array}{c@{\quad }c} {\mathbf{T}_s}&{} {\mathbf{0}_{n_s xn_s } } \\ {-\mathbf{M}_s ^{-1}\mathbf{K}_s }&{} {\mathbf{M}_s ^{-1}\mathbf{C}_s } \\ \end{array} }} \right] \quad \mathbf{D}_s =\left[ {{\begin{array}{c} {\mathbf{0}_{n_s xn_s } } \\ {\mathbf{M}_s ^{-1}\mathbf{T}_s^T } \\ \end{array} }} \right] \quad \mathbf{L}_S =\left[ {{\begin{array}{c@{\quad }c} {\mathbf{0}_{n_s xn_s } }&{} {\mathbf{T}_s } \\ \end{array} }} \right] \\ \end{array} \end{aligned}$$

(25)

where $\mathbf{M}_{s},\, \mathbf{K}_{s}$ and $\mathbf{C}_{s}$ are the mass, stiffness and damping matrices of the structure, $\mathbf{I}_{j}$ the identity matrix of dimension $j,\, \mathbf{0}_{jxk}$ a matrix of zeros of dimension jxk, $\mathbf{R}_{s}$ the vector of earthquake influence coefficients (corresponding to a vector of ones) and $\mathbf{T}_{s}$ a transformation matrix for defining relative responses between consecutive floors. The state-space representation for the Kanai–Tajimi filter with spectral density (14) is

$$\begin{aligned} \begin{aligned} \dot{\mathbf{x}}_f (t)&= \mathbf{A}_f \mathbf{x}_f (t)+\mathbf{E}_f \mathbf{Z}_w (t) \\ a_g (t)&=\mathbf{C}_f \mathbf{x}_f (t) \end{aligned} \end{aligned}$$

(26)

where $\mathbf{x}_{f}\in \mathfrak {R}^{2}$ is the filter state vector and the rest of the matrices are defined as

$$\begin{aligned} \begin{aligned}&\mathbf{A}_f =\left[ {{\begin{array}{c@{\quad }c} 0&{} 1 \\ {\omega _g^2 }&{} {-2\zeta _g \omega _g } \\ \end{array} }} \right] \quad \mathbf{E}_f =\left[ {{\begin{array}{c} 0 \\ 1 \\ \end{array} }} \right] \\&\mathbf{C}_f =\sigma _o \sqrt{2\pi }\left[ {{\begin{array}{c@{\quad }c} {\omega _g^2 }&{} {2\zeta _g \omega _g } \\ \end{array} }} \right] \\ \end{aligned} \end{aligned}$$

(27)

Combining (24) and (26) leads to the representation of the form (15) with the following definitions

$$\begin{aligned} \begin{aligned}&\mathbf{x}=\left[ {{\begin{array}{c} {\mathbf{x}_s } \\ {\mathbf{x}_f } \\ \end{array} }} \right] \quad \mathbf{A}=\left[ {{\begin{array}{c@{\quad }c} {\mathbf{A}_s }&{} {\mathbf{E}_s \mathbf{C}_f } \\ {\mathbf{0}_{2x2n_s } }&{} {\mathbf{A}_f } \\ \end{array} }} \right] \quad \mathbf{B}=\left[ {{\begin{array}{c} {\mathbf{B}_s } \\ {\mathbf{0}_{2xn_s } } \\ \end{array} }} \right] \quad \mathbf{E}_w =\left[ {{\begin{array}{c} {\mathbf{0}_{2n_s x1} } \\ {\mathbf{E}_f } \\ \end{array} }} \right] \\&\mathbf{C}=\left[ {{\begin{array}{c@{\quad }c} {\mathbf{C}_s }&{} {\mathbf{0}_{2n_s x2} } \\ \end{array} }} \right] \quad \mathbf{D}=\mathbf{D}_s \quad \mathbf{L}=\left[ {{\begin{array}{c@{\quad }c} {\mathbf{L}_s }&{} {\mathbf{0}_{n_s x2} } \\ \end{array} }} \right] \\ \end{aligned} \end{aligned}$$

(28)