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Bull Earthquake EngDOI 10.1007/s10518-013-9528-2
ORIGINAL RESEARCH PAPER
Tuned liquid dampers simulation for earthquakeresponse control of buildings
T. Novo · H. Varum · F. Teixeira-Dias · H. Rodrigues ·M. Falcao Silva · A. Campos Costa · L. Guerreiro
Received: 13 September 2012 / Accepted: 22 September 2013© Springer Science+Business Media Dordrecht 2013
Abstract This paper is focused on the study of an earthquake protection system, the tunedliquid damper (TLD), which can, if adequately designed, reduce earthquake demands onbuildings. This positive effect is accomplished taking into account the oscillation of the freesurface of a fluid inside a tank (sloshing). The behaviour of an isolated TLD, subjected to asinusoidal excitation at its base, with different displacement amplitudes, was studied by finiteelement analysis. The efficiency of the TLD in improving the seismic response of an existingbuilding, representative of modern architecture buildings in southern European countries wasalso evaluated based on linear dynamic analyses.
T. Novo · H. Varum · H. Rodrigues (B)Department of Civil Engineering, University of Aveiro, Aveiro, Portugale-mail: [email protected]
T. Novoe-mail: [email protected]
H. Varume-mail: [email protected]
F. Teixeira-DiasInstitute for Infrastructure and Environment, School of Engineering, University of Edinburgh,The King’s Buildings, Edinburgh EH9 3JL, UKe-mail: [email protected]
M. F. Silva · A. C. CostaEarthquake Engineering and Structural Dynamics Centre, Structures Department,National Laboratory for Civil Engineering, Lisbon, Portugale-mail: [email protected]
A. C. Costae-mail: [email protected]
L. GuerreiroDepartment of Civil Engineering, Architecture and Georesources, Technical Universityof Lisbon, Lisbon, Portugale-mail: [email protected]
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Keywords Tuned liquid damper · TLD · Earthquake protection systems ·Energy dissipation · Numerical simulation · Sloshing
1 Introduction
Recent earthquakes have dramatically revealed that research in earthquake engineering shouldbe directed towards the evaluation of the vulnerability of existing buildings, generally devoidof adequate structural characteristics (Vicente et al. 2012; Goda et al. 2013; Tavera et al.2007). Its strengthening should be made aiming at the reduction of its vulnerabilities, con-sequently reducing the associated risk to acceptable levels (Vicente et al. 2011). The studyand development of new strengthening techniques and/or the improvement of seismic per-formance is fundamental so as to avoid significant economic losses, as well as human lives,in future events.
Tuned liquid dampers (TLDs) have been used as passive or semi-active control devices ina wide range of applications, such as tall buildings and high-rise structures. TLDs are passiveenergy absorbing devices that have been suggested for controlling vibrations of structuresunder different dynamic loading conditions. A TLD consist on a rigid tank with a shallowfluid. This fluid, that can be water or another fluid, is inside the tank that is rigidly connected tothe structure. Tuning the fundamental sloshing frequency of the TLD to the structure’s naturalfrequency causes sloshing and breaking waves at the resonant frequencies of the combinedTLD-structure system which allows the dissipation of significant amounts of energy (FalcãoSilva 2010; Falcão Silva and Costa 2008).
TLDs have already been studied by other research groups for application as earthquakeresponse controlling device in buildings. In fact, a rectangular TLD can allow sloshing tooccur in any direction and, when properly designed, its inherent behaviour properties underdynamic loading can make it a good solution in the reduction of bi-directional earthquakedemands in buildings.
The global mechanism of controlling vibrations in structures with TLDs is based on thephenomena of fluid sloshing and wave breaking. These phenomena dissipate part of theenergy induced by earthquakes. The fundamental sloshing frequency of the fluid in the TLDshould be close to the natural frequency of the structure if the TLD is to dissipate energyefficiently.
The recent growing interest in liquid dampers for application in large structures can beassociated to their potential advantages, such as: low cost when compared to other solutions,easy installation in existing building structures, minimal maintenance costs, may be employedfor temporary use, benefit dynamic effect in uni- or bi-directional earthquake excitations, andeffectiveness even for small amplitude vibrations.
This paper presents a study of an earthquake protection system applied to an existing RCbuilding. In Sects. 2 and 3 is formulated the problem, with particular attention to the descrip-tion of the mathematical representation of the Liquid sloshing phenomena into a horizontaland low depth surfaces. In Sects. 4 and 5, respectively, is presented the case study and thedesign of the tuned liquid dampers system as function of the dynamic properties of the casestudy. Based on the designed solutions, in Sect. 6, the TLDs are modelled with finite elementmethod to evaluate the efficiency of the proposed model for the tuned liquid damper and tostudy the characteristics of the liquid sloshing mechanism. In Sect. 7 is presented the numer-ical model adopted to simulate the 3D response of the building, and is assessed the seismicsafety of the building, with and without the TLDs. Finally, the main conclusions are derivedconcerning the efficiency of tuned liquid dampers in the reduction of the seismic demands.
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2 Problem formulation
The inclusion of TLDs in structures can improve its behaviour, especially when subjected todynamic actions, such as wind and earthquake. When TLDs are rigidly linked to a buildingstructure, which is subjected to a dynamic action, fluid sloshing originates pressures thatchange the dynamic characteristics of the structure and its response to dynamic loads. Thehydrodynamic horizontal impulses applied by the steady state fluid on the tank side walls act asan auto-equilibrated system, neglecting higher-order terms and shear forces on the structure.When the structure is subjected to dynamic actions, fluid sloshing induces non-equilibratedimpulsive loads in both tank side walls corresponding to the development of lateral shearforces in the structure of the building, Find . These shear forces can be calculated, at anytime instant t , as a function of the hydrostatic pressure on the tank side walls perpendicularto the direction of ground motion (equivalent to hydrostatic pressure at the same instant).Therefore, these forces depend on the height of fluid close to the tank side walls (Sun 1991).
The building structure’s shear force induced by fluid sloshing can be expressed as:
Find = ρ · g · b
2
(h2
d − h2e
)(1)
where ρ is the density of the fluid, b is the tank width and, hd and he are the surface elevationsat the right and left side walls of the tank, respectively.
The lateral shear forces induced to the structure can be considered to be dependent only onthe hydrostatic pressure, because this is significantly higher than the inertial loads originatedby the horizontal acceleration of fluid sloshing in the tank and by friction forces acting onthe tank side walls and bottom. Therefore, in this study, these two loads were neglected. Itis then considered that liquid-induced horizontal loads can be calculated from the sum oftotal pressures acting on the right and left side walls of the tank. According to Fig. 1, thehydrostatic pressures are dependent on the free surface height of the fluid at each side wallof the tuned liquid damper (Sun 1991).
3 Fluid movement: mathematical model
Liquid sloshing movement into a horizontal and low depth surface involves non-linearities.Non-linear models, supported by the shallow water wave theory and based on partial dif-ferential equations under the specified initial and boundary conditions, can be solved using
f
L
z x
PE
PD
FIND=PD+PE
ηe
hf
ηe
Fig. 1 Schematic representation of the forces induced to the structure due to fluid sloshing in a TLD (Novo2008)
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numerical methods. However, these approaches frequently lead to high numerical instabili-ties. These equations are the result of both continuity relations and two-dimensional Navier–Stokes equations (Banerji et al. 2000; Fujino et al. 1992; Rodrigues et al. 2010; Uang andBertero 1998).
A linear analytical model is applied in this work, where the linear solution for liquidsloshing in the tank is simulated applying the long wave theory, based on dissipative anddispersive non-linear study-cases. This theory is used associated with integration techniques,in order to obtain the transference’s function expression in a simple analytical form.
Figure 1 shows the liquid sloshing in a tank, which is subjected to a horizontal motion x ,in OX direction. The local Cartesian coordinate system (OX Z ) has to be considered abovethe steady free fluid’s surface and its origin is placed at the left side wall of the tank. Thelength of the tank is designated by L and the average liquid depth is h f .
In an one-dimensional approach, liquid sloshing in the tank can be expressed by twonon-dimensional partial differential equations (PDE), where a straight line is applied andthe u(x, t) variable is eliminated, and so the expression for the elevation of the free surfacebecomes (Rodrigues et al. 2010):
η(x, t) = −Re
{1
k· sin
[κ · (
x − 12
)]
cos( k
2
) ei ·σ ·t}
−2 ·∞∑
n=0
(−1)n · sin
[an ·
(x − 1
2
)]· Re
[fn · eSn ·t] (2)
With:With β as a dispersion parameter, ζf a dissipation parameter and s’ a frequency parameter,
and:
κ = σ′√
1 − 13 · β ·σ′2
·(
1 − i · ζf
2 · σ′
)(3)
an = (2 · n + 1) · π (4)
fn = 4 · Sn
S2n + S2 · 1 + 1
3 · β ·S2n
2 · Sn + ζf · (1 − 1
3 · β ·S2n
) (5)
Sn = −ζf − 2 · i · an · (1 + 1
3 · β ·a2n
)
2 · (1 + 1
3 · β ·a2n
) (6)
The previous equations are effective if β << 1 e ζf << 1 (Fujino et al. 1992) e and:
σ′ <
√3
β(7)
The solution can be described as the motion of a damped single degree-of-freedom oscillator.During the excitation phase two groups of terms contribute to the solution: (a) a linearcombination of all the free modes of oscillation of the basin representing the transients; and(b) a harmonic function with the frequency of the exciting motion corresponding to the steadystate.
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The fundamental linear sloshing frequency (w f ) of the TLD can be given by Fujino et al.(1992)
w f =√
π · g
L· tanh
(π · h f
L
)(8)
where L is the length of the tank, h f is the liquid height and g is the acceleration of gravity.
4 Case study
For the applications carried out in the scope of this paper, and having in mind a specificbuilding, the ground motion considered was defined only along one horizontal direction. Thebuilding chosen for this study is representative of the modern Portuguese architecture andwas designed and built in the 1950s, when earthquake design was not contemplated in thenational standards. This building has nine-storeys. The option for this case-study is justifiedby the moderate to high local seismic hazard of the Lisbon region and by the significantnumber of buildings with this typology designed and built in Southern European cities inthat period.
The block plan is rectangular with 11.10 m width and 47.40 m length. The building hasthe height of 8 habitation storeys plus the pilotis height at the ground floor. The “free plan”is also a reference because the house was conceived in a way of flexibility in use. But,the 12 structural plane frames define the architectural plan of the floor type, with 6 duplexapartments. The distance between frame’s axes is 3.80 m. Each frame is supported by twocolumns and has one cantilever beam on each side with 2.80 m span, resulting in 13 modules.
The building geometry and dimensions of the RC elements and infill walls were givenin the original project (1950–1956), and were confirmed in the technical. The structure ismainly composed by twelve plane frames oriented in the transversal direction (direction Y,as represented in Fig. 2).
The twelve transversal plane frames have the same geometric characteristics for all beamsand columns. A peculiar structural characteristic of the type of buildings, with direct influ-ence in the global structural behaviour, is the ground storey without infill masonry walls.Furthermore, at the ground storey the columns are 5.5 m height. All the upper storeys have aninter-storey height of 3.0 m. A detailed definition of the existing infill panels were consideredin the structural models.
For the numerical analyses, constant vertical loads distributed on beams were consideredin order to simulate the dead load of the self-weight including RC elements, and infill walls,finishing, and the correspondent quasi-permanent value of the live loads, totalising a valueof 8.0kN/m2.
Fig. 2 Case study: a general views of the building block under analyses; b structural system (plan)
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The mass of the structure was assumed concentrated at storey levels. Each storey has amass, including the self-weight of the structure, infill walls and finishings, and the quasi-permanent value of the live loads, of about 4 M tons. For the dynamic analysis, the storeymass is assumed to be uniformly distributed on the floors.
5 Design of a TLD for the case study
The procedure adopted for the design of the tuned liquid dampers for the studied building wasas follows. The procedure adopted for the design of the tuned liquid dampers for the studiedbuilding was as follows. From previous analysis (Rodrigues et al. 2010), it was found that themost flexible direction of the building corresponds to the longitudinal direction. Also, in thiscase study, the absence of infill’s in the ground story induces a dynamic behaviour mainlygoverned by the first mode. In this numerical analysis, intending to show the efficiency ofthe TLDs in the seismic protection of building structures, it was considered the design of theTLDs only relatively to the longitudinal direction and ignoring the upper modes influence.However, it is recalled that, in most building structures, the influence of various modes shouldbe taken in to account.
The building first natural frequency in the longitudinal direction measured in-situ(Rodrigues et al. 2010) was f = 1.08 Hz. The sloshing frequency of the fluid is equalled tothe first natural frequency of the building:
f = 1
T= ω
2π(9)
where T is the natural period of the structure and ω is the corresponding angular frequency.In terms of linear frequency (in Hz) the equation of sloshing frequency of the fluid (Eq. 3),
can be rewritten as follows
f f = 1
2
√g
π · Ltanh
(π · h f
L
)(10)
Imposing a ratio between the height of the fluid, h f,, and the length of the tank, L , equalto 0.15, as suggested by Fujino et al. (1992), and substituting the values for the structureunder analysis in Eq. (5), the following non-linear system of equations can be obtained as afunction of h f and L .
1.08 = 12
√9.81π ·L tanh
(π ·h f
L
),
h fL = 0.15
and (11)
Therefore, on the present case, L = 0.3000 m and hf = 0.0441 m.In the scope of this work, it was assumed that the width b of the TLD is equal to its length
L , leading to a quadrangular geometry in plan. This geometry guarantees that the TLD willbehave equally on both horizontal directions, as opposed to a rectangular tank. From theliterature review, no rule or proposal was found for the height limitation of the TLDs.
6 Numerical simulation of an isolated TLD
The finite element method (FEM) was used to evaluate the efficiency of the proposed tunedliquid damper model and to study the characteristics of the liquid sloshing mechanism in
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a TLD when subjected to a sinusoidal dynamic excitation at its base for different displace-ment amplitudes. ANSYS CFXTM was the finite element analysis (FEA) tool used to test theefficiency of the TLD with the dimensions determined in the previous section. The finite ele-ment mesh of the tank was generated and the problem boundary conditions and the particularproperties of the fluid were defined (Novo 2008).
The optimized finite element mesh (see Fig. 3) adopted in these analyses has 69131tetrahedral elements with an element average dimension of 18 mm.
The same tank dimensions and TLD parameters were used for all the analyses. Thedynamic load applied to the tank was imposed by a sinusoidal motion law of the accelerationsapplied directly on the tank base. The sinusoidal acceleration imposed along the OX direction,which simulates the direction of ground motion, is given by
ay = K · g · sin(2 · π · f · t) (12)
where K is the constant acceleration amplitude and f is the natural frequency of the build-ing. Gravity acceleration is considered along the OZ direction (az = 9.81 m/s2) and noacceleration is considered along the OY direction (ax = 0 m/s2).
Considering the acceleration law
x(t) = AC sin(2 · π · f · t) (13)
and integrating twice, leads to
x(t) =∫
x(t) = 1
4 · π2 · f 2 · AC · sin(2 · π · f · t). (14)
Thus, the displacement amplitude can be calculated with the expression
D = AC
4 · π2 · f 2 ⇔ AC = D · 4 · π2 · f 2 (15)
The values of Ac and K and listed in Table 1. According to Eqs. (7) and (8)
K · g = Ac (16)
Fig. 3 Finite element mesh of the TLD (with 69131 tetrahedral elements) (Fujino et al. 1992)
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Table 1 Amplitude values forAc and K D (m) Ac (m/s2) K
0.002 0.092 0.009
0.005 0.230 0.023
0.010 0.460 0.046
0.015 0.690 0.070
0.030 1.355 0.138
0.040 1.806 0.184
0.100 4.517 0.460
The dissipated energy and corresponding equivalent damping were calculated to evaluatethe efficiency of the tuned liquid damper (TLD). The dissipated energy for all simulations wascalculated and compared to the input energy of the system. For one specific load cycle, theenergy introduced in system is calculated with the following expression (Uang and Bertero1998):
Ei = −∫
mx(t)dt (17)
The energy dissipated per cycle can be evaluated determining the interior area of a cyclicresponse of the TLD in terms of lateral force versus horizontal displacement. For one specificcycle of the TLD’s response, the energy dissipation is determined by the following expression(Uang and Bertero 1998; Guerreiro 2003):
Ed =∫
Fx(t)dt (18)
For a chosen cycle, it is possible to estimate the equivalent damping coefficient through thefollowing expression:
ζeq = Area of the cycle
2 · π · Fmax · Dmax(19)
where Fmax . is the maximum force (due to hydrostatic pressure) and Dmax is the maximumhorizontal displacement of the tank (Guerreiro 2003; Chopra 2001).
6.1 Numerical results and discussion
The results in Fig. 4 show the force-displacement response of the TLD for a cyclic load-ing amplitude of 15 mm. Figure 5 shows the evolution of the input and dissipated energycomponents for a cyclic loading corresponding to an imposed displacement amplitude of15 mm.
For all the analyses performed, corresponding to displacements imposed in the 2–100 mmamplitude range, the evolution of the ratio between the input energy and the dissipated energyis shown in Fig. 6. In Fig. 7 are plotted, for each displacement amplitude the maximumratio between the dissipated energy and the input energy (Fig. 7a), as well as the estimatedequivalent damping, which gives an indication of the TLD efficiency. From the analysis of theresults it is clear that, generally, the TLDs introduce an important damping effect, dissipatingpart of the total energy introduced in the system by the seismic excitation.
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Fig. 4 Force-displacement behaviour for a cyclic loading amplitude of 15 mm
Fig. 5 Input and dissipated energy evolutions for a cyclic loading corresponding to an imposed displacementamplitude of 15 mm
For small displacement amplitudes (until 30 mm), the fluid behavior is stable with a cyclic-type response. But, for higher displacement levels, sloshing is observed and an irregularresponse of the fluid starts to occur, with pronounced non-linear behavior. For these higherdemands, the fluid contacts also with the top of the tank, which introduces different boundaryconditions, if compared with the behavior for low displacement demand amplitudes. Thesefacts justifies the response observed of the TLD and, namely, the reduction of the dissipatedenergy for displacement levels upper than 30 mm.
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0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Ene
rgy
diss
ipat
ion/
Intr
oduc
ed [%
]
Time [s]
2mm5mm10mm15mm30mm40mm100mm
Fig. 6 Ratio between the input and dissipated energy
0102030405060708090
100
Displacement [mm]
0 2 5 10 15 30 40 100
0102030405060708090
100
Dam
ping
equ
ival
ent [
%]
Displacement [mm]0 2 5 10 15 30 40 100
(a) (b)
Ene
rgy
diss
ipat
ion/
ener
gy
intr
oduc
ed [J
]
Fig. 7 Efficiency of the TLD as a function of the cyclic loading amplitude, in terms of: (a) maximum ratiobetween the dissipated energy and the input energy and (b) the estimated equivalent damping
7 Structural seismic safety assessment
7.1 Design of the solution
The efficiency of the tuned liquid damper in the improvement of the seismic response ofan existing building was evaluated and the main results are presented. The studied buildingwas already presented in Sect. 4. The linear analyses were preformed with a finite elementanalysis program (SAP 2000TM).
According to the building dimensions, the structure properties and the TLD dimensionand its characteristics, the number of TLDs needed to guarantee the required performancewas estimated, in terms of energy dissipation of the structure-TLD system. The mass of thefluid in each tank is m f = ρ · h f · b · L , where ρ· is the mass density of the fluid and h f , band L are the tank dimensions. The number of tanks required, N , was determined with thefollowing relation (Banerji et al. 2000):
μ = m . ms
m f(20)
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where μ is the mass ratio between the liquid and the structure (typically varying between 1and 5 %), ms is the mass of the structure and m f is the mass of the fluid in each tank.
The fluid chosen for this analysis is water. The total mass of the building is 3863 ton. Con-sequently, according to Eq. (15), the number of tanks needed for each mass ratio consideredin this study is listed in Table 2.
In Fig. 8 is presented an example of the application of the TLDs in the top of the buildingunder study, for the case of masses relationμ = 2.5. It was considered a uniformly distributionof the TLDs in the roof slab, in order to improve the efficiency of the system and allowing abetter distribution of the vertical loads associated to the TLDs weight in the building structure.For the present case, the TLDs are assumed to be aggregated in 10 groups, each one with 20alignments of tanks in the transversal direction, 12 in the longitudinal direction, and differentnumber of levels/layers according to the mass relation intended (for μ = 1 %, 2.5 e 5 % areconsidered 4, 10 and 20 levels of TLDs, respectively—see Fig. 9).
7.1.1 Lumped mass method
Simplified methods can be found in the literature to simulate the behaviour of tuned liquiddampers. The lumped mass method and the linear wave theory are such methods. The lumpedmass method used in this work was originally suggested by Housner (1963). With this method,
Table 2 Number of tanksestimated for each mass ratioanalysed
μ (%) Number of tanks
1.0 9,600
2.5 24,000
5.0 48,000
Fig. 8 TLDs distribution for μ = 2.5 %: a plan, b frontal view and c left view
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Fig. 9 Representation of one group of TLDs for a mass relation of μ = 2.5 % (10 levels)
h
L
H
H
1
0
M0
M1
k2
k2
f
Fig. 10 Lumped mass model for a rectangular TLD (Housner 1963; Newmark and Rosenblueth 1971)
the wall of the TLD is assumed to be rigid. Hydrodynamic pressure caused by liquid sloshingin the tank due to the dynamic loading is considered separately as impulsive pressure andsloshing pressure. The impulsive pressure is proportional to the tank acceleration, but withopposite direction. The sloshing pressure is related to the height of the wave and to thesloshing frequency of the liquid. Therefore, both hydraulic pressures can be simulated bytwo equivalent masses linked to the tank. Figure 10 shows a schematic representation of thelumped mass model, in which Mo is the mass rigidly connected to the tank at an elevationH0 above its base and M1 represents the impulsive mass attached to springs with stiffness kat elevation H1.
For rectangular tanks, these parameters can be estimated by Newmark and Rosenblueth(1971), Jin et al. (2007):
Mo = tanh (1.7∗(L/2)/hf )
1.7∗(L/2)/hf· mf and M1 = 0.83 · tanh (1.6∗hf/(L/2))
1.6∗hf/(L/2)· M (21)
H0 = 0.38 · hf ·[
1 + α
(mf
M0− 1
)](22)
H1 = h f ·⎡
⎣1 − 0.33 · m f
M1·(
L/2
H f
)2
+ 0.63 · β · (L/2)
h f·√
0.28 ·(
m f · L/2
M1 · h f
)2
− 1
⎤
⎦
(23)
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k = 3 · g · M21 · hf
mf · L2 (24)
where M is the total mass of the contained fluid, β = 2.0 and α = 1.33 are parameters dueto the hydrodynamic moment on the tank base and hf is the liquid height.
7.1.2 Damping of surface wave in a rectangular tank
The damping of surface waves in a rectangular tank was studied by Miles (1967), whosuggested that the dissipation coefficient can be corrected by a factor 1 + 2h/b + S, where bis the tank width. With this correction factor, the energy dissipation due to the fluid frictionon the lateral side walls and the liquid surface contamination is taken into account. Frictiondue to the side wall boundary layer is assumed to be the same as that of the bottom boundarylayer. S is a surface contamination factor that can vary between 0 and 2. S = 1 will be usedin this study, which corresponds to a fully contaminated surface, as suggested by Sun (1991).
ζf =√
νf ·ωf
2·[
1 +(
2 · hf
b
)+ S
]· L
hf · √g · hf
. (25)
7.1.3 Proposed model
The steps adopted for the simplified simulation of the passive energy dissipation systembased on TLDs in the Structural Analysis Program (SAP2000TM) are briefly described inthe next paragraphs (see also Fig. 11). Each TLD group is simulated by a macro-model, asfollows:
(i) A damper type link was defined in order to simulate the equivalent damping (ζ) prop-erties of the TLD;
(ii) Each link is connected to the structure with bar elements with stiffness k (according toexpression 19) and
(iii) Concentrated masses were introduced at the ends of the macro-model to simulate thestatic mass and at the centre to simulate the dynamic mass.
To simplify the analysis, and due to the large number of TLDs needed for the passiveenergy dissipation system designed in this study, each macro-model corresponds to a set ofTLDs. The schematic representation of the macro-model developed in SAP to simulate agroup of TLDs is shown in Fig. 11.
The model was implemented in SAP. The 3D structure representation with each TLD groupis simulated by a macro-model/element, as described before in the longitudinal direction (seeFig. 12).
Fig. 11 Schematic representation of the macro-model used to represent a set of TLDs with the proposedmodel
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Fig. 12 3D model of the RC building with the TLDs (for the mass relation μ = 2.5 %)
0
1
2
3
4
5
6
7
8
9
Sto
rey
Displacement [m]
0
1
2
3
4
5
6
7
8
9
0 0.005 0.01 0.015 0 0.1 0.2
Sto
rey
Maximum inter-storey Drift [%]
Originalstructure
μ=1%
μ=2,5%
μ=5%
Fig. 13 Envelope lateral displacement and maximum inter-storey drift profiles (73 yrp)
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0
1
2
3
4
5
6
7
8
9
Sto
rey
Displacement [m]
0
1
2
3
4
5
6
7
8
9
0 0.02 0.04 0.06 0 0.5 1
Sto
rey
Maximum inter-storey Drift [%]
Originalstructure
μ=1%
μ=2,5%
μ=5%
Fig. 14 Envelope lateral displacement and maximum inter-storey drift profiles (475 yrp)
7.2 Ground motion
This study was performed for a set of synthetic ground motions artificially generated, fora moderate risk scenario, according to a non-stationary stochastic finite fault seismologicalsimulation model based on random vibration (Carvalho et al. 2005). The considered groundmotions reflect both the close and distant earthquake scenarios and were defined for severalreturn periods. Distant earthquake scenarios were considered for this study with four differentreturn periods (yrp = 73, 475, 975 and 2000 years) and three artificial earthquakes weregenerated for each return period (Carvalho et al. 2008). A total of twelve earthquakes wereadopted in this study.
7.3 Numerical results
The most relevant numerical results, namely the envelope lateral displacement profilesand the maximum inter-storey drift profile, are shown in Figs. 13, 14, 15 and 16. Eachprofile represents the average result for each return period considered (73, 475, 975 and2000 years).
The present results were obtained with a numerical model based on linear elastic analysis.In fact, a more reliable analysis would be obtained considering the non-linear behaviour of theRC concrete elements. However, it is highlighted that the building under analysis concentratesthe deformation demands mainly in the elements at the first storey, so the damage expected
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0
1
2
3
4
5
6
7
8
9
Sto
rey
Displacement [m]
0
1
2
3
4
5
6
7
8
9
0 0.02 0.04 0.06 0 0.5 1
Sto
rey
Maximum inter-storey Drift [%]
Originalstructure
μ=1%
μ=2,5%
μ=5%
Fig. 15 Envelope lateral displacement and maximum inter-storey drift profiles (975 yrp)
would be concentrated in these elements. In Figs. 13, 14, 15 and 16, it can be observe that forthe lower seismic demands (for signals with return period up to 975 years), the assumptionof linear elastic behaviour gives a good approximation since the maximum drift observed isinferior to 1 %. For larger earthquake demands, a more precise analysis should include thenon-linear modelling of the RC elements.
8 Concluding remarks
The main concern of this paper is to analyse the efficiency of quadrangular tuned liquiddampers (TLDs) in controlling the response of building structures when subjected to earth-quake ground motions.
To characterise the TLDs, as well as the liquid sloshing mechanism, an isolated TLDwas studied using a refined finite element (FE) model. The TLD studied was designed fora frequency of 1.08 Hz and subjected to sinusoidal excitations at its base with displacementamplitudes between 2 and 100 mm. From the analysis of the obtained results it can be con-cluded that, for a sinusoidal cyclic loading, the behaviour of the studied TLD is globallyefficient, significantly increasing the energy dissipation and the corresponding equivalentdamping of the global system. The TLD is more effective in reducing the structural demandsfor structures with lower natural periods, as the building structure here studied with a natural
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0
1
2
3
4
5
6
7
8
9
Sto
rey
Displacement [m]
0
1
2
3
4
5
6
7
8
9
0 0.05 0.1 0 1 2
Sto
rey
Maximum inter-storey Drift [%]
Originalstructure
μ=1%
μ=2,5%
μ=5%
Fig. 16 Envelope lateral displacement and maximum inter-storey drift profiles (2000 yrp)
period close to 1 s, representative of a large number of existing buildings in many countriesaround the world.
From the current analysis, it can be concluded that TLDs can be adopted as effectivemeasures for reducing building structural demands to earthquake input motions. It is worthemphasising that many questions are still open in the field of the TLD’s application, partic-ularly in what concerns the behaviour of TLDs. Therefore, additional numerical and exper-imental research, namely by means of shaking table tests and in-situ evaluation of theirperformance on real structures should be developed in this domain to validate the designprocedure. Nevertheless, the research work briefly reported herein is expected to contributeto a better understanding of these innovative structural protection systems. Additionally, thearchitectonic impact associated to this rehabilitation measure should be considered case bycase.
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