## 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.

### Similar content being viewed by others

### Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.## References

Almazan JL, Cerda FA, De la Liera JC, Lopez-Garcia D (2007) Linear isolation of stainless steel legged thin-walled tanks. Eng Struct 29(7):1596–1611

Ang H-SA, Lee J-C (2001) Cost optimal design of R/C buildings. Reliab Eng Syst Saf 73:233–238

Balendra T, Wang CM, Rakesh G (1999) Vibration control of various types of buildings using TLCD. J Wind Eng Ind Aerodyn 83(1):197–208

Boroschek R, Contreras V (2012) Strong ground motion form the 2010 Mw 8.8 Maule Chile earthquake and attenuation relations for Chilean subduction zone interface earthquakes. In: Proceedings of the international symposium on engineering lessons learned from the 2011 Great East Japan earthquake, Tokyo, Japan.

Bozorgnia Y, Bertero V (2004) Earthquake Engineering: From Engineering Seismology to Performance-based Engineering. CRC Press, Boca Raton

Bray JD, Rodriguez-Marek A (2004) Characterization of forward-directivity ground motions in the near-fault region. Soil Dyn Earthq Eng 24:815–828

Cha EJ, Ellingwood BR (2012) Risk-averse decision-making for civil infrastructure exposed to low-probability, high-consequence events. Reliab Eng Syst Saf 104:27–35

Chakraborty S, Roy BK (2011) Reliability based optimum design of tuned mass damper in seismic vibration control of structures with bounded uncertain parameters. Probab Eng Mech 26(2):215–221

Chang CC (1999) Mass dampers and their optimal designs for building vibration control. Eng Struct 21:454–463

Coello CAC, Van Veldhuizen DA, Lamont GB (2002) Evolutionary algorithms for solving multi-objective problems, vol 242. Springer, New York

Daniel Y, Lavan O (2014) Gradient based optimal seismic retrofitting of 3D irregular buildings using multiple tuned mass dampers. Comput Struct 139:84–97

De Angelis M, Perno S, Reggio A (2012) Dynamic response and optimal design of structures with large mass ratio TMD. Earthq Eng Struct Dyn 41(1):41–60

De la Llera JC, Lüders C, Leigh P, Sady H (2004) Analysis, testing, and implementation of seismic isolation of buildings in Chile. Earthq Eng Struct Dyn 33(5):543–574

Debbarma R, Chakraborty S, Ghosh S (2010) Unconditional reliability-based design of tuned liquid column dampers under stochastic earthquake load considering system parameters uncertainties. J Earthq Eng 14(7):970–988

Den Hartog JP (1947) Mechanical vibrations. McGraw-Hill Inc, New York

EERI Special Earthquake Report (2010) Learning from earthquakes: the Mw 8.8 Chile earthquake of February 27, 2010. Earthq Eng Res Inst Newsl 44(6)

FEMA-P-58 (2012) Seismic performance assessment of buildings. American Technology Council, Redwood City

Fragiadakis M, Lagaros ND, Papadrakakis M (2006) Performance-based multiobjective optimum design of steel structures considering life-cycle cost. Struct Multidiscip Optim 32:1–11

Fujino Y, Sun L, Pacheco BM, Chaiseri P (1992) Tuned liquid damper (TLD) for suppressing horizontal motion of structures. J Eng Mech 118(10):2017–2030

Gidaris I, Taflanidis AA (2015) Performance assessment and optimization of fluid viscous dampers through life-cycle cost criteria and comparison to alternative design approaches. Bull Earthq Eng 13(4):1003–1028

Gidaris I, Taflanidis AA, Mavroeidis GP (2014) Multi-objective design of fluid viscous dampers using life-cycle cost criteria. In: 10th national conference in earthquake engineering, Anchorage.

Goulet CA, Haselton CB, Mitrani-Reiser J, Beck JL, Deierlein G, Porter KA, Stewart JP (2007) Evaluation of the seismic performance of code-conforming reinforced-concrete frame building-From seismic hazard to collapse safety and economic losses. Earthq Eng Struct Dyn 36(13):1973–1997

Hitchcock PA, Kwok KCS, Watkins RD, Samali B (1997) Characteristics of liquid column mass dampers (LCVA)-II. Eng Struct 19(2):135–144

Hoang N, Fujino Y, Warnitchai P (2008) Optimal tuned mass damper for seismic applications and practical design formulas. Eng Struct 30(3):707–715

Housner GW (1957) Dynamic pressures on accelerated fluid containers. Bull Seismol Soc Am 47(1):15–35

Jalayer F, Beck JL (2008) Effects of two alternative representations of ground-motion uncertainty in probabilistic seismic demand assessment of structures. Earthq Eng Struct Dyn 37(1):61–79

Kaneko S, Mizota Y (2000) Dynamical modeling of deepwater-type cylindrical tuned liquid damper with a submerged net. J Press Vessel Technol 122(1):96–104

Kareem A (1990) Reduction of wind induced motion utilizing a tuned sloshing damper. J Wind Eng Ind Aerodyn 36:725–737

Kohavi R (1995) A study of cross-validation and bootstrap for accuracy estimation and model selection. In: Proceedings of the international joint conference on artificial intelligence, pp 1137–1145

Lee CS, Goda K, Hong HP (2012) Effectiveness of using tuned-mass dampers in reducing seismic risk. Struct Infrastruct Eng 8(2):141–156

Leyton F, Ruiz S, Sepúlveda SA (2009) Preliminary re-evaluation of probabilistic seismic hazard assessment in Chile: from Arica to Taitao Peninsula. Adv Geosci 22(22):147–153

Lin C-C, Chen C-L, Wang J-F (2010) Vibration control of structures with initially accelerated passive tuned mass dampers under near-fault earthquake excitation. Comput Aided Civil Infrastruct Eng 25(1):69–75

Love JS, Tait MJ (2010) Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique. J Fluids Struct 26(7):1058–1077

Marano GC, Greco R, Trentadue F, Chiaia B (2007) Constrained reliability-based optimization of linear tuned mass dampers for seismic control. Int J Solids Struct 44(22–23):7370–7388

Marian L, Giaralis A (2014) Optimal design of a novel tuned mass-damperâ€“inerter (TMDI) passive vibration control configuration for stochastically support-excited structural systems. Probab Eng Mech 38:156–164

Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26:369–395

Matta E (2011) Performance of tuned mass dampers against near-field earthquakes. Struct Eng Mech 39(5):621–642

Matta E, De Stefano A (2009) Seismic performance of pendulum and translational roof-garden TMDs. Mech Syst Signal Process 23(3):908–921

Matta E, De Stefano A, Spencer BF (2009) A new passive rolling pendulum-vibration absorber using a non-axial-symmetrical guide to achieve bidirectional tuning. Earthq Eng Struct Dyn 38(15):1729–1750

Mavroeidis GP, Dong G, Papageorgiou AS (2004) Near-fault ground motions, and the response of elastic and inelastic single-degree-of-freedom (SDOF) systems. Earthq Eng Struct Dyn 33(9):1023–1049

Moroni MO, Sarrazin M, Herrera R (2011) Research activities in Chile on base isolation and passive energy dissipation. In: Proceedings of the 12th world conference on seismic isolation, energy dissipation and active vibration control of structures, Sochi.

Ordaz MG, Cardona O-D, Salgado-GÃlvez MA, Bernal-Granados GA, Singh SK, Zuloaga-Romero D (2014) Probabilistic seismic hazard assessment at global level. Int J Disaster Risk Reduct 10:419–427

Papadimitriou K (1990) Stochastic characterization of strong ground motion and application to structural response. Report No. EERL 90-03, California Institute of Technology, Pasadena.

Porter KA, Kiremidjian AS, LeGrue JS (2001) Assembly-based vulnerability of buildings and its use in performance evaluation. Earthq Spectra 18(2):291–312

Porter KA, Kennedy RP, Bachman RE (2007) Creating fragility functions for performance-based earthquake engineering. Earthq Spectra 23(2):471–489

Power M, Chiou B, Abrahamson N, Bozorgnia Y, Shantz T, Roblee C (2008) An overview of the NGA project. Earthq Spectra 24(1):3–21

Rezaeian S, Der Kiureghian A (2010) Simulation of synthetic ground motions for specified earthquake and site characteristics. Earthq Eng Struct Dyn 39(10):1155–1180

Ruiz RO (2015) A new type of tuned liquid damper and its effectiveness in enhancing seismic performance; numerical characterizaiton, experimental validation, parametric analysis and life-cycle based design. Pontificia Universidad Catolica de Chile, Santiago

Ruiz RO, Lopez-Garcia D, Taflanidis AA (2015a) An efficient computational procedure for the dynamic analysis of liquid storage tanks. Eng Struct 85:206–218

Ruiz RO, Lopez-Garcia D, Taflanidis AA (2015b) Modeling and experimental validation of a new type of tuned liquid damper Acta Mechan. doi:10.1007/s00707-015-1536-7

Ruiz RO, Lopez-Garcia D, Taflanidis AA (2015c) Tuned liquid damper with floating roof: A new device to control earthquake-induced vibrations in structures. In: XI Congreso Chileno de Sismologia e Ingenieria Sismica, Santiago.

Sakai F, Takaeda S, Tamaki T (1989) Tuned liquid column damper-new type device for suppression of building vibrations. In: Proceedings of the international conference of high-rise buildings, Nanjing. pp 926–931

Salvi J, Rizzi E, Gavazzeni M (2014) Analysis on the optimum performance of tuned mass damper devices in the context of earthquake engineering. In: EURODYN 2014-IX international conference on structural dynamics, Porto, Portugal, 30 June–2 July. Universidade do Porto, Faculdade de Engenharia, FEUP, pp 1729–1736

Scherbaum F, Cotton F, Staedtke H (2006) The estimation of minimum-misfit stochastic models from empirical ground-motion prediction equations. Bull Seismol Soc Am 96(2):427–445

Shin H, Singh MP (2014) Minimum failure cost-based energy dissipation system designs for buildings in three seismic regions Part II: Application to viscous dampers. Eng Struct 74:275–282

Soto MG, Adeli H (2013) Tuned mass dampers. Arch Comput Methods Eng 20(4):419–431

Spall JC (2003) Introduction to stochastic search and optimization. Wiley-Interscience, New York

Taflanidis AA, Beck JL (2009) Life-cycle cost optimal design of passive dissipative devices. Struct Saf 31(6):508–522

Taflanidis AA, Beck JL, Angelides DA (2007) Robust reliability-based design of liquid column mass dampers under earthquake excitation using an analytical reliability approximation. Eng Struct 29(12):3525–3537

Tait MJ, Isyumov N, El Damatty AA (2007) Effectiveness of a 2D TLD and its numerical modeling. J Struct Eng 133(2):251–263

Tributsch A, Adam C (2012) Evaluation and analytical approximation of tuned mass damper performance in an earthquake environment. Smart Struct Syst 10(2):155–179

Tse KT, Kwok KCS, Tamura Y (2012) Performance and cost evaluation of a smart tuned mass damper for suppressing wind-induced lateral-torsional motion of tall structures. J Struct Eng 138(4):514–525

Vamvatsikos D, Cornell CA (2005) Direct estimation of seismic demand and capacity of multidegree-of-freedom systems through incremental dynamic analysis of single degree of freedom approximation 1. J Struct Eng 131(4):589–599

Vetter CR, Taflanidis AA, Mavroeidis GP (2015) Tuning of stochastic ground motion models for compatibility with ground motion prediction equations. Earthq Eng Struct Dyn. doi:10.1002/eqe.2690

Wang D, Tse TKT, Zhou Y, Li Q (2015) Structural performance and cost analysis of wind-induced vibration control schemes for a real super-tall building. Struct Infrastruct Eng 11(8):990–1011

Wong KK (2008) Seismic energy dissipation of inelastic structures with tuned mass dampers. J Eng Mech

Zemp R, de la Llera JC, Roschke P (2011) Tall building vibration control using a TM-MR damper assembly: Experimental results and implementation. Earthq Eng Struct Dyn 40(3):257–271

## Acknowledgments

Financial support was provided by the Pontificia Universidad Catolica de Chile, by the University of Notre Dame and by the National Research Center for Integrated Natural Disaster Management CONICYT/FONDAP/15110017 (Chile). This support is gratefully acknowledged. The dynamic properties of the structure considered in the case study described in Sect. 6 were provided by VMB Ingenieria Estructural (Santiago, Chile).

## Author information

### Authors and Affiliations

### Corresponding author

## 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).

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:

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:

where

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

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

where [*w*
_{
w
}(*i*Δ*t*): *i* = 1,2,…, *N*
_{
T
}] is a white noise sequence, Δ*t* = 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

The time varying characteristics are

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

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

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.

## Rights and permissions

## About this article

### Cite this article

Ruiz, R., Taflanidis, A.A., Lopez-Garcia, D. *et al.* 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.
*Bull Earthquake Eng* **14**, 943–970 (2016). https://doi.org/10.1007/s10518-015-9860-9

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10518-015-9860-9