Sem.org IMAC XVII 17th 173003 Vibration Suppression Measures Stay Cables

7/28/2019 Sem.org IMAC XVII 17th 173003 Vibration Suppression Measures Stay Cables

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VIBRATION SUPPRESSION MEASURES FOR STAY CABLES

Habib Tabatabai and Armin B. MehrabiConstruction Technology Laboratories, Inc.5420 Old Orchard RoadSkokie, Illinois 60077

ABSTRACTIncidences of large-amplitude vibrations of stay cableshave been reported worldwide on a number of cable-stayed bridges. The phenomenon of rain-wind vibrationhas been cited as the cause of these vibrations in manycases. In this paper, the available methods forsuppression of cable vibrations are briefly reviewed. Ina research project sponsored by the Federal HighwayAdministration (FHWA), a non-dimensional finitedifference formulation for free vibration of cables hasbeen developed that includes the effects of intermediatesprings, dampers, cable bending stiffness, and cablesag. Using this formulation, the effects of mechanicalviscous dampers on enhancement of the overall cabledamping are quantified. Simplified non-dimensionalrelationships and design charts are presented that canbe used for selection of the size and location of externaldampers to achieve desired cable damping ratios. Usingthe developed design tool and based on the availablecriteria for suppression of rain-wind and gallopingvibrations, the minimum required cable damping ratioscan be determined.NOMENCLATURE

D = cable outside diameterEA = axial rigidity of cableEl = flexural rigidity of cableH = horizontal force in cableL = cable chord lengthN = number of discretized cable elementsSc = Scruton numberc = damping coefficient of mechanicalviscous damperm = mass per unit length of cableld = damper location parameter8 = damping ratio attributed to mechanicalA.z

damper (in percent)= sag-extensibility parameterp = mass density of air

= bending stiffness parameter\1' = non-dimensional damping parameter formechanical viscous damper

INTRODUCTIONIncidences of large-amplitude vibrations (on the order of1 to 2 meters) of inclined stay cables in cable-stayedbridges have been reported worldwide when certaincombinations of rain and moderate wind exist[ 11Formation and movements of water rivulets on thecables are believed to be the main contributing factorsfor this phenomenon. This issue has raised greatconcern for the bridge engineering community (due toheightened fatigue stress ranges) and has been a causeof deep anxiety for the observing public.In general, stay cables consist of a bundle of 15.2-mm-diameter, seven-wire strands with a nominal strength of1860 MPa. The strand bundle is typically encased in apolyethylene (or sometimes steel) pipe. Strands couldbe uncoated, epoxy-coated, or individually greased andcoated with polyethylene sheathing. In U.S. practice,cement grout is commonly injected into the pipeprovide additional protection for the strands. Tabataba1et al 121 developed a database of over 1400 stay cablesfrom 16 bridges. Based on this database, the averagelength of a stay cable is 128 m, the average force is4500 kN, and the average outside diameter is 0.182 m.Irwin 11 proposes the following relationship as a meansof controlling cable vibrations due to the rain-windphenomenon:

m8Sc = I00 pD2 > I 0

Where Sc = Scruton number; m = mass per unit length;8 is the damping ratio in percent; p = density of air; andD= cable diameter. Based on the information in thedatabase of stay cables, a cable damping ratio of 0.7%

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would meet the above requirement for more than 90%of stay cables in the database. The actual stay cabledamping ratios are generally in the range of 0.05% to0.5% [4J.VIBRATION CONTROL MEASURESIn general, three different types of vibration controlmeasures have been utilized in cable-stayed bridges. Inthe most common method, neoprene washers (rings)are placed in the annular space between the outsidediameter of the cable and a steel guide pipe (attached tothe bridge deck or tower) near the two cableanchorages. Although, this neoprene washer addssome level of damping to the cable, its presence hastypically not been sufficient to control the rain-windvibration phenomenon. The main purpose for thisneoprene device is to control flexural fatigue stresses atthe cable anchorages by providing partial support for thecable at a relatively short distance away from theanchorages. In lieu of the common neoprene washers,a set of proprietary viscoelastic ring dampers is alsoavailable.In the second method, cross cables (or cross ties) thattransversely connect different cables together areutilized. In such cases, special attention is required inthe design of the cable-cross tie connection, the level ofprestress in the cross cable, and fatigue considerationsfor the cable, cross cable, and the connection. Crosscables may also negatively impact the aesthetics of acable-stayed bridge. The level of damping contributedby cross cables is not currently clear. Based on a set ofsmall-scale laboratory tests, Yamaguchi r5l concludesthat there is "more or less a damping-increase function"in crossing main structural cables with secondarycables. The damping increment can be caused byadditional damping from other cables, as well as energydissipation in the cross cables r

5J. Yamaguchi r5lsuggests that the damping contribution of the crosscables would be increased if "more flexible and moreenergy-dissipative ties were used". Failure of crosscables has been noted on at least one prominent cable-stayed bridge r6l.The third method for vibration control of stay cablesinvolves the use of mechanical viscous dampersattached to the cables and supported by the bridgedeck. Such devices are generally attached to the cableat a distance of 2 to 6% (of cable length) from the decklevel anchorage. This paper addresses the designprocedures for these types of cable dampers in detail.DESIGN OF MECHANICAL VISCOUSDAMPERSExternal viscous dampers have been used to suppresstransverse cable vibrations, mostly induced by windexcitation. However, simple and accurate damperdesign provisions that concurrently consider allimportant cable parameters such as cable sag andbending stiffness are lacking. In most previoustreatments of this problem, stay cables have beenidealized as taut strings. In this way, the combined

effects of some of the parameters affecting dynamicbehavior of cables were ignored. The most important ofthese parameters are sag-extensibility and bendingstiffness parameters which have been shown toinfluence dynamic characteristics of stay cables [7].In a research project sponsored by the Federal HighwayAdministration (FHWA) r2l, the governing differentialequation for cable vibration was first converted to acomplex eigenvalue problem containing non-dimensional parameters including those related to cablebending stiffness, sag-extensibility, and viscousdampers. Figure 1 shows the cable model with aviscous damper attached at a distance Ld from one end.The discretized non-dimensional form of the differentialequation greatly facilitated parametric studies for a vastrange of non-dimensional parameters for stay cables.The database of stay cables r2J was utilized to identifythe range of relevant parameters in this study. Onlyviscous dampers with constant damping factors wereconsidered. The non-dimensional complex eigenvalueproblem was repeatedly solved within the selectedrange of parameters. Based on the results of theseparametric studies, the effects of dampers on the firstmode vibration frequencies and the additional first modedamping ratios associated with external viscousdampers were presented in non-dimensional format.Furthermore, simplified relationships were proposed torelate first mode damping ratios to non-dimensionalcable and damper parameters. Based on the proposedsimple relationships, design equations were presentedfor use in the selection of dampers in stay cables. Thefollowing section summarizes the proposed designprocess based on the results of this research.The non-dimensional parameters used to develop andpresent the results of this study are as follows:

ljl = C/(Hm)05= L (H/Eil5

where L = cable chord length; g = acceleration ofgravity; 8 = cable inclination angle; H = cable forcealong the chord; EA =axial rigidity of cable; El = flexuralrigidity; C = damping factor for the damper; and,(mgLcose IH) 2Le:: L [ I+ S ]

For any damper location, there is an optimum damperparameter (ljle). This means that as the damping factorof the damper (C) is increased, the damping ratio of thecable increases up to a maximum value beyond whichthe damping contribution decreases. Therefore, over-design of dampers may be counter-productive. For adamper located at 2, 4, or 6% of cable length from the

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anchorage (!d of 0.02, 0.04, and 0.06), the optimumdamper parameters are 20, 8, and 6, respectively.First mode damping ratios (in percent) were calculatedand plotted for the ranges of "'A2 and parametersdiscussed earlier. The complete sets of plots are givenby Tabatabai et al. [21. Figure 2 shows one of these plotsobtained for !d = 0.04 and \Jf = 8. The intrinsic dampingof cable has been ignored in these analyses. Therefore,the entire calculated damping ratio is directly attributableto the damper.These analyses showed that the effect of mechanicaldampers is not very sensitive to sag-extensibilityparameter () .2) for the range of A.2 up to 1. This rangecovers more than 95% of cables in the stay cabledatabase. For A.2 values greater than 1, the dampingratio, 8 (in percent) decreases slightly as ),2 increases.This reduction is most noticeable for bending stiffnessparameters larger than 100. Therefore, for A.2smallerthan 1, the damping ratio attributable to the damper canbe assumed to be independent of the sag-extensibilityparameter.Figures 3, 4, and 5 show plots for damper locations of 2,4, and 6 percent of cable length from one end,respectively. For clarity, only a portion of calculateddata has been reported in these figures. For a damperlocated at 2 percent (1d = 0.02), the optimum dampingparameter (\Jf8 ) is 20. It can be noted in Fig. 3 that,when bending stiffness parameters are less thanabout 100, higher \Jf values result in higher dampingratios than that of the optimum damping parameter.However, since the vast majority of cables (more than82 percent) in the database have parameters greaterthan 100, the optimum \Jf value in this case is stillconsidered to be about 20. A more accurate selectionof optimum \Jf can be made for l o w e r ~ values for eachspecific case by referring directly to relationship curvespresented. For !d of 0.04 and 0.06, the optimumdamping parameters are 8 and 6, respectively.For damping parameters less than or equal to \Jfe, therelationship between the first mode cable damping ratioand the bending stiffness parameter can beapproximated by the following equation:

where, a, b, c, and d are defined for !d of 0.02, 0.04,and 0.06 in Table 1. The above equation was derivedbased on a series of regression analyses with aminimum coefficient of determination, R2 , of 0.96. For 0< \Jf < 2, a linear interpolation can be used between 8calculated for \Jf =2 using the above equation and 8 = 0for \Jf = 0.

Kovacs [SJ concluded that the maximum possibledamping ratio attainable in any mode is approximately0.5 !d. This indicates maximum damping ratios of 1, 2,and 3 percent for 1d = 0.02, 0.04, and 0.06,respectively. This relationship is a simple yet accuraterule-of-thumb.In this paper, the intrinsic damping of the cable isconservatively ignored. The developed equations canbe used to design a damper (for specific attachmentlocations near the deck anchorage).Design EquationThe following design relationships are based on theabove equation:

when 2S \jf S \Jfed (" b + )= 2 89 \jf c; c


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With 8 = 0.57 and a trial and error process a lj.l of 8.43is found using the above equation.2.0 < lj.l = 8.43 < lj./e = 20

Similar results can be obtained directly from Fig. 3 using8 = 0.57 a n d ~ = 119.For a cable length of 93 m, a damper with a C value of202 kN s/m (C = lj.l .jHiTi , lj.l = 8.43) should be attached1.86 meters from the cable end.SUMMARY AND CONCLUSIONSTo prevent excessive vibration in stay cables (rain-windinduced or galloping vibrations), a minimum dampingratio is required for cables. Stay cables possess some(relatively small) level of intrinsic damping. Thisdamping may not be adequate for suppressingexcessive vibrations in some cables. In this paper,simple design equations were proposed for determiningthe location and size of viscous dampers that can beused for suppression of stay cable vibrations includingrain-wind induced vibrations. In the derivation ofrecommended damping parameters for external viscousdampers, the intrinsic damping of cables have beenconservatively ignored. When a damper is located inthe general vicinity of the specified locations in theproposed equations, a linear interpolation may be usedto specify the required damper properties.ACKNOWLEDGMENTSThe study presented in this paper was supported by theFederal Highway Administration under Contract No.DTFH-61-96-C-00029. Opinions expressed in thispaper are those of the writers and do not necessarilyrepresent those of the Federal Highway Administration.

REFERENCES[1] Matsumoto, M., Shiraishi, N., and Shirato, H. (1992)."Rain-wind induced vibration of cables of cable-stayed bridges." J. Wind Engrg. and /ndust.Aerodynamics, 41-44, 2011-2022.[2] Tabatabai, H., Mehrabi, A.B., Morgan, B.J., and Lotti,

H.R. (1998). "Non-destructive bridge evaluationtechnology: bridge stay cable conditionassessment." Report submitted to the FederalHighway Administration, Construction TechnologyLaboratories, Inc., Skokie, IL.[3] Irwin, P.A. (1997). "Wind vibrations of cables oncable-stayed bridges." Proc., ASCE StructuresCongress, Vol. 1, 383-387.[4] PTI (1998), "Recommendations for Stay CableDesign, Testing and Installation," 4th Draft, Post-Tensioning Institute Committee on Cable-StayedBridges, March 1998.[5] Yamaguchi, H., "Passive Damping Control in CableSystems." Building for the 21st Century, Y.C. Loo,Editor, EASEC-5 Secretariat, Griffith University,Australia.[6] Poston, R.W. (1998). "Cable-Stay Conundrum."Civil Engineering, ASCE, August 1998, pp. 58-61.[7] Mehrabi, A.B., and Tabatabai, H. (1998). "A unifiedfinite difference formulation for free vibration ofcables." J. Struc. Engrg., ASCE, November 1998.[8] Kovacs, I. (1982). "Zur frage der seilschwingungenund der seildampfung." Die Bautechnik, 10, 325-332(in German).

Table I: Parameters used in design equation.fct a b c d

0.02 0.259 2.080 5,613 -0.1290.04 1.276 1.795 841 0.0810.06 3.278 1.700 259 0.305

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H ... I ( ~ - I

a

L

Figure 1. Cable with viscous damper

2.22.01.81.61.41.21.00.80.60.40.20.0

Figure 2. Damping ratio (%) for damper at r ct=0.04, \j/=81241

~ H

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---\j/=2 - - - -\j/=4 - - - - - - - \j/=6 ---- -\j/=8 ----- \j/=10---\j/=20 - - \ j /=30 - - - -\j/=40 - - -\j/=50 - - -\j/=60

1.00.90.80.70.6

r.() 0.5

---------- ~ - - ~ ~ .;..:..-.:.:.-..:.:.--=.:.-=-=~~ - - - - --------- --- ... - -- -- -- -- - ----- - -- --- ,..,;; - -- --, -' -- -..::,., ~ ~ ...-G'"' ; . . _ -- -- - - - - - - - - - - - - - -~ --, . /...::;;.;.- - - - --- ------- ---- -- - -- -- - - - -- -- -- ~ ________ ,.._._.._----------,--- -- ...........______ - - - - - - - - -0.4 , - - - - - - - - - - - - - - ---------------.30.20.10.0

0 50 100 150 200 250 300 350 400 450 500

Figure 3. Damping ratio for different bending stiffness parameters, damper at r c ~ = 0 . 0 2 , 1.?=0.01

---\j/=2 - - - - \ j / =4 - - - - - - - \ j/=6 \j/=8 ---- -\j/=10- - - - - \j/=20 - - \ j /=30 - - - - \j/=40 - - - \j/=50 -- -\j/=60

2.5 . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

---------------------------------------------------2.01.5 . ------------------- - - - - - - - - - - - .. ----------------1.0 ------- - - - - - - - - - - - - - --- -- -- -- -- -- -- ------------- -- - - - - - - - - - - - - - - - - - -0.5 - - - - - - - - - - - - - - - - - -0.0 F - - - - - - - - + - - - - - - - - - - - - ~ - - - - ~ - - - - - - ~ - - - - ~ - - - - - + - - - - - - ~ - - - - ~

0 50 100 150 200 250 300 350 400 450

Figure 4. Damping ratio for different bending stiffness parameters, damper at rct=0.04, 1}=0.011242

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---\j/=2 - - - - \ j / =4 ---\j/=6 ---- -\j/=8 ----- \j/=10- - - - - - - \j/=20 - - \ j /=30 - - - \j/=40 - - -\j/=50 - - -\j/=60

3.5

3.0

------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -2.52.0

tA:)1.5 --------- --------------------------------------------1.0 ----------------- -- -- -- - - - - - - ----- - - -0.5

0.00 50 100 150 200 250 300 350 400 450 500

Figure 5. Damping ratio for different bending stiffness parameters, damper at r ct=0.06, A?=O.O 1

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Sem.org IMAC XVII 17th 173003 Vibration Suppression Measures Stay Cables