Life-cycle based design of mass dampers for the Chilean region and its application for the evaluation of the effectiveness of tuned liquid dampers with floating roof

Abstract

The assessment of the effectiveness of mass dampers for the Chilean region within a multi-objective decision framework utilizing life-cycle performance criteria is considered in this paper. The implementation of this framework focuses here on the evaluation of the potential as a cost-effective protection device of a recently proposed liquid damper, called tuned liquid damper with floating roof (TLD-FR). The TLD-FR maintains the advantages of traditional tuned liquid dampers (TLDs), i.e. low cost, easy tuning, alternative use of water, while establishing a linear and generally more robust/predictable damper behavior (than TLDs) through the introduction of a floating roof. At the same time it suffers (like all other liquid dampers) from the fact that only a portion of the total mass contributes directly to the vibration suppression, reducing its potential effectiveness when compared to traditional tuned mass dampers. A life-cycle design approach is investigated here for assessing the compromise between these two features, i.e. reduced initial cost but also reduced effectiveness (and therefore higher cost from seismic losses), when evaluating the potential for TLD-FRs for the Chilean region. Leveraging the linear behavior of the TLD-FR a simple parameterization of the equations of motion is established, enabling the formulation of a design framework that beyond TLDs-FR is common for other type of linear mass dampers, something that supports a seamless comparison to them. This framework relies on a probabilistic characterization of the uncertainties impacting the seismic performance. Quantification of this performance through time-history analysis is considered and the seismic hazard is described by a stochastic ground motion model that is calibrated to offer hazard-compatibility with ground motion prediction equations available for Chile. Two different criteria related to life-cycle performance are utilized in the design optimization, in an effort to support a comprehensive comparison between the examined devices. The first one, representing overall direct benefits, is the total life-cycle cost of the system, composed of the upfront device cost and the anticipated seismic losses over the lifetime of the structure. The second criterion, incorporating risk-averse concepts into the decision making, is related to consequences (repair cost) with a specific probability of exceedance over the lifetime of the structure. A multi-objective optimization is established and stochastic simulation is used to estimate all required risk measures. As an illustrative example, the performance of different mass dampers placed on a 21-story building in the Santiago area is examined.

Appendices

Appendix A: Parametric formulation for the equations of motion for the TLD-FR

The numerical model for the TLD-FR has two components (Ruiz et al. 2015b). The first component corresponds to the motion of the liquid and is described through a finite element formulation that condenses the vibratory response to the motion of the liquid-surface (Ruiz et al. 2015a), an idea first presented in (Almazan et al. 2007). The second component corresponds to the vibration of the roof and is similarly described through finite element principles, adopting a coincidental mess with the one used to describe the motion of the liquid-surface. The effect of external dampers is also incorporated in this second component. The two components are ultimately combined through the pressures created at the common interface (liquid surface) to provide the final coupled numerical model (Ruiz et al. 2015b).

$${\mathbf{M}}_{a} \,{\ddot{\varvec{\upeta}}}_{\,s} + {\mathbf{C}}_{a} \,{\dot{\varvec{\upeta}}}_{\,s} + {\mathbf{K}}_{a} \,{\varvec{\upeta}}_{\,s} = - {\mathbf{R}}_{a} \,{\ddot{u}}_{b}$$

(10)

where vector η _s contains the vertical nodal displacements at the free surface, ü _b is the acceleration at the base of the tank, matrices M _a, C _a , K _a are “equivalent” mass, damping and stiffness matrices and vector R _a is conceptually similar to an influence coefficient vector. Full derivation of these matrices is included in (Ruiz et al. 2015b). The transmitted force F a the base of the tank is:

$$F = - \rho \,d\,{\mathbf{A}}\,{{\ddot{\varvec{\upeta}}}}_{s} - \rho \,d\,B\,{\ddot{u}}_{b}$$

(11)

where ρ is the liquid density, d is the tank width, A is a row vector and B a scalar variable both obtained through the aforementioned numerical formulation.

The parametric formulation is then established through modal reduction, keeping only the first mode of the TLD-FR (the mode that is tuned to the vibration of the structure). Let Φ denote the eigenvector for the eigenvalue problem corresponding to mass and stiffness matrices M _a and K _a . Then (10) is transformed into:

$$M_{m} \,{\ddot{y}} + C_{m} \,\dot{y} + K_{m} \,y = - R_{m} \,{\ddot{u}}_{b}$$

(12)

where

$$M_{m} = {\varvec{\Phi}}^{T} {\mathbf{M}}_{a} {\varvec{\Phi}};\quad C_{m} = {\varvec{\Phi}}^{T} {\mathbf{C}}_{a} {\varvec{\Phi}}\,;\quad K_{m} = {\varvec{\Phi}}^{T} {\mathbf{K}}_{a} {\varvec{\Phi}};\quad R_{m} = {\varvec{\Phi}}^{T} {\mathbf{R}}_{a} ;\quad {\varvec{\upeta}}_{s} = {\varvec{\Phi}}\,y$$

(13)

The modal coordinate y can be further normalized as $y_{n} = y\,M_{m} /R_{m}$, and by defining the natural frequency in the fundamental sloshing mode as $\omega_{m} = \sqrt {K_{m} /M_{m} }$ and the damping ratio as $\xi_{m} = C_{m} /2\,M_{m\,} \omega_{m}$, (12) yields (1). It is evident through this formulation that $2\xi_{m} \omega_{m} \dot{y}_{n}$ and $\omega_{m}^{2} y_{n}$ can be treated as damping and spring forces, respectively, for the mass damper.

The expression for the transmitted force (11) also simplifies to

$$F = - \rho \,d\left[ {{\mathbf{A\Phi }}\frac{{R_{m} }}{{M_{m} }}\,} \right]\,{\ddot{y}}_{n} - \rho \,d\,B{\ddot{u}}_{b}$$

(14)

and setting as $m = \rho \,dB$ the liquid mass and as $\gamma = {\mathbf{A\Phi }}\,\left( {R_{m} /B\,M_{m} } \right)$ the efficiency index yields ultimately (2). Note that term γm can be equivalently considered as the convective mass, i.e. the portion of the mass that has a dynamic contribution to the liquid vibration.

Appendix B: Details on stochastic ground motion model

According to the adopted stochastic ground motion model, the discretized time series of the ground motion, ${\ddot{a}}_{g} (t)$, is expressed as

$${\ddot{a}}_{g} (t) = e(t,{\varvec{\uptheta}}_{g} )\left\{ {\sum\limits_{i = 1}^{k} {\frac{{h[t - t_{i} ,{\varvec{\uptheta}}_{g} (t_{i} ))]}}{{\sqrt {\sum\limits_{j = 1}^{k} {h[t - t_{j} ,{\varvec{\uptheta}}_{g} (t_{j} ))]^{2} } } }}w_{w} (i\Delta t)} } \right\} \, k\Delta t < t < (k + 1)\Delta t$$

(15)

where [w _w (iΔt): i = 1,2,…, N _T ] is a white noise sequence, Δt = 0.005 s is the chosen discretization interval, e(t,θ _g ) is the time-modulating function, and h[t − τ,θ _g (τ)] is an impulse response function corresponding to the pseudo-acceleration response of a single-degree-of-freedom (SDOF) linear oscillator with time varying frequency ω _f (τ) and damping ratio ζ _f (τ), in which τ denotes the time of the pulse

$$\begin{aligned} h[t - \tau ,{\varvec{\uptheta}}_{g} (\tau ))] = \frac{{\omega_{f} (\tau )}}{{\sqrt {1 - \zeta_{f}^{2} (\tau )} }}\exp \left[ { - \omega_{f} (\tau )\zeta_{f} (\tau )(t - \tau )} \right]\sin \left[ {\omega_{f} (\tau )\sqrt {1 - \zeta_{f}^{2} (\tau )} (t - \tau )} \right];\quad \tau \le t \hfill \\ \, = 0;{\text{ otherwise}} \hfill \\ \end{aligned}$$

(16)

The time varying characteristics are

$$\begin{aligned} \omega_{f} (\tau ) = \omega_{r} + (\omega_{p} - \omega_{r} )\left( {\frac{{\omega_{s} - \omega_{r} }}{{\omega_{p} - \omega_{r} }}} \right)^{{\tau /t_{\hbox{max} } }} \;\;\; \hfill \\ \zeta_{f} (\tau ) = \alpha_{f} (\tau )/\omega_{f} (\tau )\quad {\text{where }}\alpha_{f} (\tau ) = \omega_{p} \zeta_{p} + (\omega_{r} \zeta_{r} - \omega_{p} \zeta_{p} )\tau /t_{r} \hfill \\ \end{aligned}$$

(17)

with ω _p (primary wave frequency), ω _s (secondary wave frequency), ω _r (surface wave frequency), ζ _p (primary wave damping), and ζ _r (surface wave damping) ultimately corresponding to the primary model parameters for the filter, t _max corresponding to the time at which maximum intensity of the ground motion is achieved and $t_{r} = \alpha_{dur} t_{95}$ corresponding to a sufficiently large time, chosen to be proportional to the time that 95 % of the Arias intensity is reached, denoted $t_{95}$.

The time envelope $e(t,{\varvec{\uptheta}}_{g} )$is parameterized by

$$e(t,{\varvec{\uptheta}}_{g} ) = e(t,I_{a} ,\alpha_{ 2} ,\alpha_{ 3} ) = \sqrt {I_{a} } \left[ {\sqrt {\frac{2}{\pi }\frac{{(2\alpha_{3} )^{{2\alpha_{2} - 1}} }}{{\Gamma (2\alpha_{2} - 1)}}} }\, \right]t^{{\alpha_{2} - 1}} \exp ( - \alpha_{3} t)$$

(18)

where Γ(.) is the gamma function, I _a is the Arias intensity expressed in terms of g, and {α ₂, α ₃} are additional parameters controlling the shape and total duration of the envelope that can be related to the strong motion duration, D _5-95 (defined as the duration for the Arias intensity to increase from 5 to 95 % of its final value), and the peak of the envelope function, λ _p . The latter is defined as the ratio of time corresponding to the peak of the envelope to the time corresponding to 95 % of its peak value. The pair {α ₂, α ₃} can be easily determined based on the values of {D _5-95, λ _p } (Vetter et al. 2015).

Ultimately, the ground motion model has as parameters ${\varvec{\uptheta}}_{g} = \{ I_{a} ,D_{5 - 95} ,\lambda_{p} ,\alpha_{dur} ,\omega_{p} ,\omega_{s} ,\omega_{r} ,\zeta_{p} ,\zeta_{r} \}$ and the functional form for their predictive relationships are chosen as

$$\begin{aligned} \ln \left( {I_{a} } \right) = c_{1,1} + c_{1,2} M + c_{1,3} r_{rup} + c_{1,4} r_{rup} M + c_{1,5} M_{{}}^{2} + c_{1,6} r_{rup}^{2} + c_{1,7} \ln \left( M \right) + c_{1,8} \ln \left( {r_{rup} } \right); \hfill \\ \ln \left( {D_{5 - 95} } \right) = c_{2,1} + c_{2,2} M + c_{2,3} \ln \left( {\sqrt {r_{rup}^{2} + c_{2,4}^{2} } } \right) ; { }\ln \left( {\lambda_{p} } \right) = c_{3,1} + c_{3,2} M + c_{3,3} r_{rup} ; \hfill \\ \ln \left( {\alpha_{dur} } \right) = c_{4,1} + c_{4,2} M + c_{4,3} r_{rup} ; \hfill \\ \ln \left( {\omega_{p} /2\pi } \right) = c_{5,1} + c_{5,2} M + c_{5,3} r_{rup} ; { }\ln \left( {\omega_{s} /2\pi } \right) = c_{6,1} + c_{6,2} M + c_{6,3} r_{rup} ;\hfill \\ \ln \left( {\omega_{r} /2\pi } \right) = c_{7,1} + c_{7,2} M + c_{7,3} r_{rup} ; { } \hfill \\ \ln \left( {\zeta_{p} } \right) = c_{8,1} + c_{8,2} M + c_{8,3} r_{rup} ; \, \ln \left( {\zeta_{r} } \right) = c_{9,1} + c_{9,2} M + c_{9,3} r_{rup} \hfill \\ \end{aligned}$$

(19)

with the coefficients $c_{i,l}$ i = 1,…,9, l = 1,…,8 formulating the (regression) coefficient vector ${\mathbf{c}}$, representing ultimately the vector optimized to establish the desired hazard compatibility. For the model tuning discussed in Sect. 5 (with results also presented in Fig. 4) the optimized coefficients are shown in Table 2.

Table 2 Optimized coefficients (Coef.) for the predictive relationships of the stochastic ground motion model parameters to achieve GMPE compatibility