2 layout and modelling of damped bracing systems...2 layout and modelling of damped bracing systems...

The seismic response of damped

frames using different dissipative

braces: a parametric study

A. Vulcano/i) F.

Department of Structures, University of Calabria,

^ EM ail: [email protected]

Abstract

A comparative study is carried out with reference to damped braced framesequipped with different kinds of dissipative devices. For this purpose thenonlinear seismic analysis of a single-degree-of-freedom system whose prop-erties are suitably assumed to represent a damped braced frame is carriedout. The effects produced by the dissipative braces on the ductility demandfor the framed structure are evaluated assuming different properties of theframe members, braces and dissipative devices. Aspects concerning the be-haviour and modelling of the dissipative braces are discussed.

1 Introduction

In the conventional aseismic design it is accepted that structures can with-stand strong ground motions by mainly undergoing inelastic deformations.However, the structural damage is often difficult and expensive to repairfollowing a strong earthquake. To limit or avoid this, new techniques aimto enhance the performance of a structure by controlling its seismic responseand thus the damage. In particular, the insertion of steel braces equippedwith dissipative devices can be adopted for new or existing framed build-ings. Actually a wide variety of energy dissipating devices is available forthe passive control of vibrations [1-9].

For a widespread application of supplemental dampers practical anal-ysis and design procedures should be available. In this perspective it isconsidered very useful to evaluate the seismic performance of frames withdissipative braces which make use of different kinds of dampers.

Transactions on the Built Environment vol 38 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

2 Layout and modelling of damped bracing systems

In the last two decades different kinds of dissipative braces were proposed(for a general discussion, see e.g. Soong & Constantinou [1] and Housner& al. [2]). They differ for the particular arrangement of the braces (e.g.,single diagonal brace, cross or chevron braces, etc.) and/or for the featuresof the dissipative device (in particular, by the way of dissipating energy:friction (FR), yielding (YL), viscosity (VS), viscoelasticity (VE)). Typicalarrangements are schematically shown in Figure 1.

N, A N, A

(a) (b)

Damper Shear link

(c) (d)

Figure 1: Damped bracing systems.

More precisely, using the cross-bracing system shown in Figure la boththe devices proposed by Pall & Marsh [3] arid Ciampi [4] were adopted: thefirst device consists of a mechanism with slotted slip joints containing fric-tion brake lining pads, whereas the other one consists of an inner steel framewhose shape is such that energy can be dissipated under uniform bendingyielding. The systems shown in Figures Ib and Ic were adopted by differentauthors proposing the use of dampers dissipating energy in different ways( e.g. see [4-7]). In particular, viscoelastic solid clampers typically con-sist of constrained layers of viscoelastic polymers, with the damping effectdepending on the shear strain; viscoelastic or pure viscous fluid dampersconsist of a moving piston immersed in a more or less viscous (compressibleor incompressible) fluid. Finally, the eccentrically braced system in FigureId, proposed and tested at the University of Berkeley [5], benefits from a


Earthquake Resistant Engineering Structures 269

double defence line, because it can resort to the dissipation capacity due tothe yielding of the shear link in case the friction dampers should not slip.Apart from the system in Figure la, where the braces are assumed to beslender enough to buckle elastically (buckling load practically negligible),in all the other systems depicted in Figure 1 the braces are designed not tobuckle.

To simulate the behaviour of braced frames equipped with damping de-vices, suitable analytical models should be adopted to accurately describethe hysteretic behaviour of both the dissipative braces and the framed struc-ture. In addition, the models should be relatively simple to carry out theanalysis with a reasonable computational effort. In this paper aspects of theanalytical modelling are discussed with reference to the dissipative braces,while the framed structure is simply idealized as elastic-perfectly plastic.

The behaviour of different kinds of dampers can be simulated by adopt-ing the models and the idealized force-displacement (Nd — A^) relationsshown in Figure 2. More precisely, Figures 2a and 2b respectively refer tothe friction and the metallic-yielding dampers, which can both be classifiedas hysteretic; while Figures 2c and 2d respectively refer to viscous and vis-coelastic dampers, both considered as velocity-dependent. Other systemshave characteristics that cannot be represented by one of the basic typesshown in Figure 2: e.g. dampers made of shape memory alloys or frictional-spring assemblies with recentering capabilities [1]. In what follows, onlydampers whose idealized behaviour is depicted in Figure 2 are considered.

As shown in Figure 2a and 2b, hysteretic dampers are assumed asrate-independent with a stable hysteretic behaviour, which is idealized bya rigid-plastic law for a friction damper and by a bilinear law for a yieldingdamper. However, more refined models may be adopted.

In particular, to simulate the hysteretic response of steel cross-bracesequipped with the friction device (see Figure la), the authors [8] adoptedtwo different models: that simplified (SF) in Figure 3a or the refined one(RF) in Figure 3b. As shown in the work mentioned above, the SF modelcan overestimate the energy dissipated by the friction device, while the RFmodel gives a more accurate evaluation of this energy.

Also the behaviour of yielding dampers can be simulated using lawsmore sophisticated than that shown in Figure 2b (e.g. see [10]).

Viscoelastic and pure viscous dampers, rate-dependent, generally ex-hibit mechanical properties which are functions of the (circular) frequency(cu), the temperature and the amplitude of motion. In this study only thedependence on the frequence is considered by tuning the relevant param-eters to the fundamental frequency of the entire structural system. Boththe above kinds of dampers are modelled as an elastic spring and a dashpotacting in parallel (Kelvin model, KM) or in series (Maxwell model, MM).Indeed, the VS damper model can be considered as a specialization of KMor MM assuming, respectively, K^ =0 or /Q —» oc.

It is noteworthy to mention that the MM is considered more realistic


270 Earthquake Resistant Engineering Structures

especially for simulating the behaviour of a VE fluid damper. Both theidealized force-displacement relations shown in Figures 2c and 2d refer to asinusoidal motion of amplitude A<, and circular frequency LJ.

However, models and force-displacement relations more sophisticatedthan those reported above were proposed for both the VS and VE dampers(e.g. see [10]).

(l Ny

pw—

pKd

% P^dN..

(a) Friction damper (b) Metallic-yielding damper

N.

Kelvin model Maxwell model

(c) Viscous damper (d) Viscoelastic damper

Figure 2: Modelling and idealized response of dampers.



-AAA/"

N^ slippage N

slippage

slippage

(a) Simplified model (SF)

slippage

(b) Refined model (RF)

Figure 3: Simplified and refined models representing the behaviour of steelcross-braces with friction device (see Figures la and 2a).

Parameters and laws characterizing the idealized behaviour of the dampersconsidered in this study are synthetically reported below:

(a) Friction damper: A^=slip-load forceA^marr^tension-brace force at frame-yielding onsetTV* = A^ /W oz =slip- load ratio

(b) Yielding damper: JVy=yielding forceN* = 7V% /Nmaz =y ield-load ratioKd=mit,i&\ damper stiffnessp=hardening ratio

(c) Viscous damper: Gd=damping coefficientNd — Cd&d (for linear fluid damper)

(d) Viscoelastic (solid) damper:Cd= effective damping coefficientK£ = G'A//?,=storage (or effective) stiffness

stiffnessoss factor (=0.8 - 1.4)

G' and G"=shear-storage and shear-loss moduliX=shear area of polymer layersft,=total thickness of polymer layersNd = K'Aj +

In this study the single- degree-of- freedom (SDOF) system shown inFigure 4 is considered. The system intends to simulate the response of ashear-type one-storey frame with dissipative braces arranged as shown inFigures la, Ib or Ic. For the sake of clarity, referring to the scheme inFigure Ib, the properties of the system are indicated in what follows: Kb is



the horizontal elastic stiffness of the undamped brace; Kf, F,,, Ff and Cfrepresent, respectively, elastic stiffness, yielding force, elastic-plastic forceand viscous-damping constant with reference to the unbraced frame; thestrength level of the unbraced frame is characterized by the frame-strengthratio 77 = Fy/Mdmax, M being the mass of the system lumped at the toplevel and dmax the peak ground acceleration; F and u are, respectively,the inertial force and the relative displacement of the mass; Fb and Fdrepresent the horizontal components of the axial force for undamped braceand damper, respectively, while Fdb is the analogous component of the axialforce transmitted by the damped brace (e.g. Fdb = Ncos</)). According tothe above assumptions, the equation of motion can be expressed as:

where iig(t) is the ground acceleration.For all the damped braces the elastic stiffness is characterized by the

stiffness ratio K* — Kdb/Kf, Kdb being the horizontal elastic stiffness ofthe damped bracing system defined as follows:

1(hysteretic systems)

l/#6 + 1/JQ

where, in the case of a FR damper, it can be assumed thatis, Kdb = Kb);

(Kb + +(VE system)

(2)

oo (that

(3)f A\ + 7T,)2 + V^*-o ' d/ d

as shown by Fu & Kasai [9] assuming a sinusoidal motion. Eqn. (3) can bespecialized for a VS bracing system by setting K^ = 0.

h

), 4)->

Damper(horiz. comp.)

{p

Figure 4: Modelling of a SDOF damped braced system.



3 Numerical results

To study the effects produced by the insertion of different dissipative bracesinto the framed structure, a numerical investigation has been carried outconsidering the nonlinear behaviour of the system in Figure 4 under strongground motions. For this purpose different values of the parameters char-acterizing the behaviour of the dissipative braces were assumed; moreover,different models were adopted with reference to FR and VE bracing sys-tems. The step-by-step procedure already adopted by the authors [8] wasused to integrate the nonlinear motion eqn. (1).

All the results presented below refer to the displacement ductility de-mand for the framed structure, which can be considered representative of theframe damage. The results have been obtained as an average of those corre-sponding to three artificial accelerograms whose average response spectrummatches the design spectrum adopted by Eurocode 8 [11] with reference tosubsoil class B and peak ground acceleration dmax = 0.35#.

In Figure 5 results for braced systems with hysteretic dampers areshown assuming different values of the hardening ratio p (p=Q correspondsto the FR damper) and two values of the frame strength ratio 77. It is inter-esting to note that the framed structure performed better when adoptinghigher p values; this effect, evident for the lowest rj value (i.e. r/=0.4), ispractically negligible for 77=0.6. However, in all the cases the control of theframe damage has been evident, provided that a suitable value of the slip- oryield-load ratio TV* (TV*=0 corresponds to the unbraced frame) is assumedin the range 0.5-=-l, which is suggested for practical applications [8].

As shown in Figure 6 considering the FR damped systems, the selectionof a suitable TV* value is more important in the cases corresponding tounbraced frames with a low period of vibration Tf (e.g. less than 0.6 sec).

12.00

8.00-

4.00-

0.00 n • r— 1 ' r0.00 0.20 0.40 0.60 0.80 LOO 1.20 1.40 1.60

N*

Figure 5: Results for braced systems with hysteretic damper.



12.00

8.00-

4.00-

0.00

.5; K*=0.5

UFN*=0.4N*=1.0

1 , 1 , 1 • , • 1 1 1 1 1 r-0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20

Figure 6: Results for unbraced frame (UF) and friction-damped bracedsystems.

12.00

8.00-

4.00-

0.000.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Figure 7: Results for the unbraced frame (UF) and braced systems withviscous damper.

In Figure 7 curves obtained for braced systems with a VS damper arecompared with that for the unbraced frame (UF). More precisely, the curvesfor the damped braced systems have been obtained assuming K%/Kf=2.2and the values 1 or 2 for the undamped-brace stiffness ratio K£(= Kb/Kf)]these assumptions correspond, respectively, to the values 0.10 or 0.23 ofthe added damping ratio £db due to the VS damped brace and calculatedas indicated in Reference [9], where further detail can be found. As can beobserved, for the given frame strength ratios (i.e. 77—0.4 or 0.6) the effective-ness of the VS damped brace is more evident for structures correspondingto relatively stiff braces and/or weak framed structures.



12.00

I8.00-

4.00-

0.00

TfO.Ssec

tang§=0.8tang5=1.0tang6=1.2tang5=1.4

} Ti=0.3

)n=0.6

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Figure 8: Results for systems with the viscoelastic damper idealized bythe Maxwell model.

12.00

8.00-

4.00-

0.00

rj=0.4;tang§=lK*=0(UF)K*=0.25K*=0.50K*=1.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Figure 9: Results for the unbraced frame (UF) and braced systems withthe viscoelastic damper idealized by the Maxwell model.

Results obtained for systems with the VE damper, idealized by theMaxwell model, are shown in Figures 8 and 9. As shown in Figure 8, thevariation of the loss factor in the range indicated above (i.e. 0.8-4-1.4) canbe important for the effectiveness of the damped bracing system only inthe case of a relatively weak frame (e.g. for 77=0.8); moreover, in the caseof a relatively strong frame (e.g. for 77=0.6) the performance of the framedstructure is not appreciably enhanced, especially when the damper is so stiff(e.g. Kd/Kf greater than 1.5) that the frame behaves practically as elastic.

To show the effectiveness of the VE damped bracing, in Figure 9 curvesobtained for damped braced systems are compared with that for unbraced



frames: it is interesting to note the better performance of the framed struc-ture obtained by stiffening the damped bracing, besides some analogy withthe results for FR damped systems previously shown in Figure 6.

25.00

O.OO-i ' 1 • 1 ' 1 i 1 1 • r0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80

K*

Figure 10: Results obtained adopting the simplified (SF) or refined (RF)model of cross-braces with friction device (see Figure 3).

10.00

8.00-

§ 6.00-j5»"5 4.00 H

2.00-

0.000.00 0.50 1.00 1.50

K*

2.00 2.50

Figure 11: Comparison between results for systems with the viscoelasticdamper idealized by the Maxwell (MM) or the Kelvin (KM)model.

Finally, the curves depicted in Figure 10 for the case of the FR systemand in both the Figures 11 and 12 for that of the VE system show how thechoice of different models can affect the numerical results.

More precisely, Figure 10 emphasizes the importance of using the RF



model instead of the SF one in case of a rather weak framed structure(e.g. for 77=0.3). Moreover, Figure 11 shows that the difference betweencurves obtained using KM and MM models is more evident in the case of arelatively weak frame (e.g. for r/=0.3); while Figure 12 shows that the framedamage can be underestimated (e.g. especially for Kb = K^) when, as inpractice is often done, the brace deformability is neglected (i.e. assuming

12.00

8.00-

4.00-

0.000.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80

K*

Figure 12: Results for systems with the viscoelastic damper idealized bythe Kelvin model assuming different values of the brace stiffness.

•n=0.4; T,=0.5sec; tang5=1.0

4 Conclusions

The following conclusions can be drawn from the results:- when using braces with a hysteretic damper, a higher value of the harden-ing ratio produces a better performance of the framed structure; this effectis more evident in the case of weaker framed structures, provided that asuitable value of the slip- or yield-load ratio TV* is assumed in the range0.54-1, which is suggested for practical applications;- in the case of FR damped systems, the selection of a suitable value of theslip-load ratio is more important in cases corresponding to unbraced frameswith a relatively low period of vibration T/;- the effectiveness of the VS damped bracing is more evident for systemscorresponding to relatively stiff braces and/or weak framed structures;- when using VE dampers the variation of the loss factor in the practicalrange (i.e. 0.8—1.4) can be important for the effectiveness of the dampedbracing system only in the case of a relatively weak frame; moreover, in thecase of a relatively strong frame the performance of the framed structure isnot appreciably enhanced, especially using a very stiff damper;



- the use of the refined model (instead of that simplified) for the FR dampedbracing is recommended in the case of a rather weak framed structure;- when modelling VE damped bracing, the difference between results ob-tained by the Kelvin and Maxwell models is more evident in the case of arelatively weak frame; moreover, the frame damage can be underestimatedwhen, as in practice is often done, the brace deformability is neglected.

References

1. Soong, T.T. & Constantinou, M.C., Passive and active structural vi-bration control in civil engineering, Springer Verlag, Wien, 1994.

2. Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G.,Glaus, R.O., Masri, S.F., Skelton, R.E., Soong, T.T., Spencer, B.F.& Yao, T.P., Structural control: past, present and future, J. of Eng.Mechanics, ASCE, 123(9), pp. 899-905, 1997.

3. Pall, A.S. & Marsh, C., Response of friction damped braced frames, J.of the Struct. Div., ASCE 108(ST6), pp. 1313-1323, 1982.

4. Ciampi, V., Development of passive energy dissipation techniques forbuildings, Procs. of the Int. Post-SMIRT Conf. Seminar on Isolation,Energy Dissipation and Control of Vibrations of Structures, Capri(Italy), pp. 495-510, 1993.

5. Aiken, I.D. & Whittaker, A.S., Development and application of passiveenergy dissipation techniques in the U.S.A., Procs. of the Int. Post-SMIRT Conf. Seminar on Isolation, Energy Dissipation and Controlof Vibrations of Structures, Capri (Italy), pp. 511-525, 1993.

6. Constantinou, M.C. & Symans, M.D., Experimental study of seismicresponse of buildings with supplemental fluid dampers, J. Struct. De-sign of Tall Buildings, 2, pp. 93-132, 1993.

7. Chang, K.C., Soong, T.T., Oh, S.-T. & Lai, M.L., geisrmc Wmmorof steel frame with added viscoelastic dampers, J. of the Struct. Div.,ASCE 121(10), pp. 1418-1426, 1995.

8. Vulcano, A. & Mazza, F. Seismic analysis and design of RC frameswith dissipative braces, Procs. of the 11^ European Conf. on Earth.Eng., Paris, Balkema, Rotterdam, 1998.

9. Fu, Y. & Kasai, K., Comparative study of frames using viscoelasticand viscous dampers, J. of Struct. Eng., ASCE 124(5), pp. 513-522,1998.

10. Valles, R.E., Reinhorn, A.M., Kunnath, S.K., Li, C. & Mad an, A.,IDARC2D Version 4-0: A computer program for the inelastic damageanalysis of buildings, Technical Report NCEER-96-0010, State Univ.of New York at Buffalo (U.S.A.), 1996.

11. Eurocode 8, European Prestandard, Design provisions for earthquakeresistance of structures, 1994.


2 layout and modelling of damped bracing systems...2 layout and modelling of damped bracing systems...

Documents