SCI P076 Design Guide for Floor Vibrations

8/13/2019 SCI P076 Design Guide for Floor Vibrations

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A Steel Construction InstitutePublication 4inassociationwiththe 4Construction Industry n Association 11 1 _________1 14 ___________________________

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is The SteelConstruction Institute. Its aim is to promote he properandeffective use ofsteelinconstruction.Membership s open to all organisations and individuals hatare concernedwith the useofsteel in construction, and members includeclients, designers,contractors, suppliers,fabricators,academics and government departments. SC! is financed by subscriptionsfrom its members,revenue from researchcontracts, consultancyservices and by thesalesofpublications.SCI'sworkis initiated andguided through he involvementof its memberson advisorygroupsand technical committees. A comprehensive advisoryand consultancy service isavailable omembers on the useof teelinconstruction.SCI's research and developmentactivities covermany aspectsof steel constructionincluding multi-storey construction, industrial buildings, use of steel in housing,developmentofdesignguidance on the use ofstainlesssteel,behaviour ofsteel in fire,fire engineering, use of steel in barrage schemes, bridge engineering, offshoreengineering, development ofstructural analysissystems and theuse ofCAD/CAE.Further information is given in the SCIprospectusavailable free on requestfrom: TheMembership Secretary, The SteelConstruction Institute,Silwod Park,Ascot, BerkshireSL5 7QN.Telephone: (0990) 23345, Fax: (0990) 22944, Telex:846843.

Although care has been taken to ensure, to the best of ourknowledge, that all data and information contained herein areaccurateto the extent that they relate to either mattersoffactoracceptedpractice or mattersofopinionat the time ofpublication,the Steel Construction Institute assumes no responsibility for anyerrorsin ormisinterpretationsofsuch data and/or nformation oranylossordamage arising from or relatedto their use.

The Steel Construction Institute 1989


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Design Guideon the Vibrationof FloorsT.A. WyattBScPhD FEng FICE

ISBN: 1 870004 34 5The Steel Construction Institute 1989

SCI PUBLICATION076

The SteelConstruction InstituteSilwoodParkAscotBerkshire SL57QNConstruction Industry Researchand Information Association6Storey'sGateLondon SW1P3AU


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FOREWORDThispublication s ntended oprovideguidance fordesigners in an important area ofdesignwhereinformation is acking. Ithas been prepared byDrTAWyattof ImperialCollegewith assistance from DrA FDierof he SteelConstruction Institute.The Guide was drafted n conjunction with the support ofa steeringcommittee whichcommented onand otherwise advisedonthe draftversions. Themembers of he steeringcommittee comprised:MrBBoys British SteelStructuralAdvisory ServiceMrR Clark Skidmore Owings & MerrillMrEDibb-Fuller BuildingDesignPartnersh'ipMrEDore CIRIAMrKIrish Vibronoise LimitedMrRPovey MitchellMcFarlane &PartnersMrMWillford OveArup& Partners.The work leading tothis publicationhasbeen fundedby British SteelGeneral Steels, andtheDepartmentof heEnvironment undera CIRIA researchproject. Studies arecontinuingandfutureeditions of hepublication willbeamended asnecessary toaccountfornewresults. The SteelConstruction Institute willbepleasedtoreceiveany commentsconcerning this publication and subject area.How to Usethis GuideTheGuideis divided into sevenSections and twoAppendices as shown on the facingpage.Section 1 is intended asabroad ntroduction and has beenwritten in suchaway thatit issuitableforcopying oaClientas anaid topreliminary discussions. Thebackgroundtothe designprocedures,which are set out in Section 7, isgiven inSections 2to6 andastudyof these willbe anaid,althoughnotnormally necessary, intheapplication ofSection 7. Thedesignprocedures ofSection 7 areselfcontained as faras is practical,although nsomecases reference to Section 5.2mayberequired. Theexamples inAppendixB will be useful for following thedesignprocedures. Explanation of terms usedfordescribing dynamic behaviour, which maynot befamiliar to thenon-specialist, will befoundin Section 4.2 wheretheyarehighlightedby italicscript. Defmitions essential fortheapplicationof thedesignprocedures aregivenin Section 7.1.

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CONTENTSPage

SUMMARY ivNOTATION iv1. INTRODUCTION 12. SOURCES OFVIBRATION EXCITATION IN BUILDINGS 33. HUMAN REACTION TO VIBRATiON 63.1 Review of Factors 63.2 Specifications 74. GENERALCONSIDERATIONS 104.1 Structural and FloorConfigurations 104.2 Introductionto Dynamics 115. EVALUATION OFNATURALFREQUENCY 16

5.1 Component and System Frequencies 165.2 Practical Evaluation 176. FLOORRESPONSE 20

6.1 Low Frequency Floors 206.2 High Frequency Floors 21

7. DESIGN PROCEDURES 257.1 Definitions 257.2 General Considerations 257.3 Procedure forChecking Floor Susceptibility 267.4 Natural Frequency 267.5 Floorsof High Natural Frequency 277.6 Floorsof Low Natural Frequency 287.7 Acceptance Criteria 30

REFERENCES 31APPENDIXA: CALIBRATIONSTUDY 32APPENDIXB: DESIGN EXAMPLES 33


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SUMMARYThis publicationpresents uidanceor thedesignoffloorsin steelframedstructuresagainstunacceptable vibrations caused by pedestrian traffic.Ithasparticular elevance ocomposite loorscomprising permanent metaldecking toppedwithconcrete. Aswellasthedesignprocedures set outin Section 7, the Guidecontainsbackground ommentaryandageneral,non-technical, introduction.

Notationa acceleration mplitudeaccelerationesponse(Canadian Code)b floor beamspacingeffectivewidthbetween loorbeamsB parameter or effective width Canadian Code)GB factor fordetermining naturalfrequencyC Fouriercomponent factorC, effectivemass and lateraldistribution actorfor impulsive loadingC effectivemass and lateraldistribution actor for sustained vibrationEl flexuralrigidity(ofcomposite sectionwhereappropriate)f natural requencyf0 fundamental system requencyf1, J, f3 idealisedcomponent natural requencies of 0g acceleration due o gravityJ impulse (= forcex ime)k stiffness1, L floor beamspan lengthsL lengthofspanLeff lengthforestablishing effectivemassLm main beamspanrn distributedmassrn lumped massatmeshpoint 'i'M effectivemodalmassP static loadP forceamplitudeP1 amplitudeof fundamental Fouriercomponentofwalking forceP weightofoscillating massdistributed oadingR multiplication factorapplied ohuman reactionbase curveS widthforestablishing effective massS weighting factort timesmearedconcrete hickness5 deflection amplitudey deflection atmeshpoint 'i'y maximum valueofself-weightdeflectiony weightedaverage ofself-weightdeflectionW floorbaywidthcriticaldamping atio


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1. INTRODUCTIONThemainpurpose of hisGuide stoprovideapractical method forassessing thelikelyvibrational behaviour offloorsinsteelframedbuildings. Thesubjectoffloor vibration iscomplex and consequently the Guidecontains sectionsdealing with the current stateoftheart',the background totheproposedassessment methods and acommentary o thatthedesignermaydevelop anappreciation of hephenomenon rather thanapplythe designmethod byrote.Notwithstanding this ntention, thedesignprocedure setout inSection 7 and the workedexamples contained in Appendix Bhave beenprepared opermitaconservative designassessment tobeexecuted bythose withonlya imited knowledge ofstructural dynamics.Floorvibration s notanewphenomenon, the 'live' feel of imber floorsunderpedestrianloading iswellestablished. However,because of heincreasing rend owards lighterlonger spanfloorsinall formsofconstruction, butmostnotably insteelwork, CIRIA andSCIconsidered itanopportune timetoprovideinterimguidance on thisaspectofdesignpending urtherresearch. This Guidehasnot herefore beenprepared nresponse oanyexistingproblemsbutratheritis ntended that tsusewillpreventsuchproblemsoccurring nthe future. Vibration informsofconstruction otherthansteelwork mayalsorequireconsideration.The use ofstructural steelwork formulti-storey construction has increased dramaticallyover thepasttenyears.Suchincrease is argelydue otheresponseof hebuildingindustry to Clients'demands forbuildings thatare fasttoconstruct,have largeuninterrupted floorareas and arecapableofaccommodating highly sophisticated airconditioning and otherservices systems. Modern designandconstruction techniquesenable he industry to satisfy suchdemands andproducesteelframedstructures whicharecompetitive interms ofoverall cost. This trend owards longer span lightweight loorsystems inboth steelworkandother formsofconstruction, with their endency tolowernatural requencies and lesseffectivenaturaldamping,has createdagreaterawareness ofthedynamic natureofsome types ofsuperimposed loadings. Currently he mostpopularformof loorconstruction usedinconjunction with multi-storey steelframes is he'composite loor'. This form of loor slab comprises profiledmetaldecking spanningbetweenbeams andtopped with insitu concrete. Muchof hedesignguidance giveninthispublication isdirectly elatedtothis form ofconstruction.The vibration offloorscan arise from externalsources suchasroadand rail traffic.Wheresuchproblemsareanticipated, however, it ispreferable to isolate hebuildingas awhole.Thisaspectofvibration controlisnot akenfurtherinthisGuide,whichaddresses loorvibrations caused by internalsources.The mostusual and important internalsourceofdynamic excitation spedestrian raffic.Aperson walking ataregularpaceappliesaperiodically repeatedforce tothe floor whichmaycausea buildup ofresponse n the structural floor. Othersources of internalexcitationsuchas vigorous hythmic groupactivities are not specifically coveredinthisGuide.However, where suchactivities areenvisaged arobuststructureshouldbeprovidedwhichhas adequate ductility, andspecialattentionshouldbepaid tothebeam/column connections. Thesedesignfeatures are similar othoseconsidered whenpreparinggoodseismic-resistant designs and t istopublications dealing with this subjectthat hedesigner'sattention is directed.Human perception ofvibration is inone senseverysensitive; the criterion s ikelytobeset ata owlevel. In another senseit is very insensitive; asubstantial quantitative hangeintheamplitude ofvibration corresponds to arelatively small qualitative change inperception.If apersonisaskedtoexpress an opiniononhisperception ofvibration intwodifferent roomsonseparate occasions, hewillnotdrawadistinction unless hequantitative difference is atleasta factorof2. There are alsosubstantial differencesbetweenpersonsand heremayalsobedifferences betweennationalities. Human eactionatthese levelsissubstantially psychological, dependingpartlyon the delicacyof he


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activity beingperformed. Response tovibrations isoften affectedby other stimuli sightandsound). Although floorvibration mayinducea sense of insecurity in some people, itmustbe stressed thatperception offloorvibration doesnot mplyany lackofstructuralsafety.Once constructed, it is very difficult omodifyanexisting floor o reduce its susceptibilitytovibration, since only majorchanges othemass,stiffnessordampingof the floorsystem willproduceanyperceptible reduction invibration by peopleregularly traffickingthe floor.Itis herefore important that the levelsofacceptable vibration beestablished at heconceptual stagehaving regardto theanticipated usageof the floors. The Clientmust beinvolved n this decision, since the selecteddesign argetlevel forvibrational responsewillusuallyhaveasignificant bearing on both thecostandoverall floorconstructiondepthfortheproject.Thequestion s frequently raisedof he tolerance ofmodemcomputer equipment toambientvibration. Thesteering groupfor this studyhas beenunable to fmdanyfirmevidence ofactual problems resulting rom floorvibration. Manufacturers commonlystate that theirequipment is tolerantof the levelsofvibrations acceptable in agoodofficeenvironment. Consultation with a prominent manufacturer hasconfirmed thatvibrationswithin therangetolerable orhuman occupancy wouldcausenoproblem ocomputerequipment.In conclusion, therefore, it is intended that thepublication of this Guide will aid bothdesigners and Clients in setting sensible targetsforacceptable levelsofvibration whichcan thenbe incorporated into thedesignof he floor structure toproduceeconomic,usage-related, buildings.


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2. SOURCES OF VIBRATION EXCITATION INBUILDINGSThere are anumberofdistinctpossible causesofdynamic excitationoffloors.Theimportant characteristics of these excitations varytothe extent thatquitedifferent checkprocedures maybe appropriate depending onwhichpotentialcauseismost important.Theobvious,almostuniversal, excitation is the effectofwalkingon the floor. Thegeometry of he human bodywalkingis (toafirstapproximation) astraight-leg motionthatnecessarilycauses he mainbodymassto riseandfallwithevery pace(seeFigure 2.1).This riseandfallis ypicallyabout50mm,peaktopeak,but s sensitive otheangleof the legatfullstretch, and thus to the extenttowhich he walker is forcing hepace.Oneisnotaware of his movement, because he brain dentifies theresultingacceleration signalsascorrelated with walking anddisregards them;itis,however,interesting to note that these accelerationsare around 3 m/s2,which isroughly30 timesthe value that wouldbeacceptable as the resonantresponseofafloor,and 100 timesthevalue hat wouldcommonly besetasa imitto sustained vibrations. Theannoyancecaused byfloorvibrations isessentially psychological, and is verysusceptible toexpectation or familiarity; it is none the less arealproblem.

DirectionofwalkRiseandfallof-f main body mass

________ Legsatmid-stride(broken lines)

The vertical accelerations f hebodymass arenecessarily associated withreactions onthefloor,and theywill be closely periodic, atthepacefrequency. The fluctuation can beresolvedas a seriesof inusoidal components (i.e.aFourierseries) and itisfound that thefundamental termagrees fairlywell with the simple visualisation ofFigure2.1, giving aforce amplitude between100Nand 300 N. Walking pacefrequency canvarybetween1.4Hz and 2.5Hz,and the force amplitude tends toincrease ratherseverelywithincreasingfrequency. However, walkingpaceindoors ismostcommonlyowards the lower end ofthis range, around1.6Hz. TheBritish Standard forbridges" suggests 180Nforceamplitude forchecking footbridgedesigns2.A ypical example of he contactforce froma single footfall sshown as the lightsolidcurve inFigure2.2(a).Unless the floor structure s exceptionally sensitive to thepreciselocationof he load(i.e.ifonepace-length makesamajordifference), thedynamicexcitation sgivenbythe sumof theconcurrent walker's footforces,which takes theform shown as theheavy solid curve in Figure2.2(a).The basicpacefrequency is clearlyrepresented but the second Fouriercomponent, representing excitationattwice hepacefrequency, is alsoimportant. The thirdcomponent is smaller, and succeeding componentscangenerally be ignored, except hat there is a significant impulsive effectofvery shortdurationasthe footcontacts theground. The first three Fouriercomponents are shown nFigure2.2(b), and thedegreeofapproximation given bythe summation of these threecomponents is indicated on Figure2.2(a).Thisexampleis takenfrom the work ofOhlsson3.Themagnitude of he secondFouriercomponent varies with the walkingpace ina similarwaytothe basiccomponent. Unfortunately, however, thesehigherfrequency effects,especially he contact mpulse, vary considerably betweenpersons.Theaveragevalues of

3

Legs atpointof ootfall(Solid lines)

'',, ;''-.' ";Figure 2.1 Simplified eometryofwalking


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zC.)0'300EC 200

1000

100

-200

N300200100

0100

-200(b)Fouriercomponents of eaction onfloor

Figure 2.2 TypicalwalkingexcitationtheFouriercoefficients reportedbyRainer, Pernicaandlen4 romaCanadian studydirected ofootbridge oading areshown inFigure2.3. Thecontact mpulse is typicallyabout3Ns (Newton seconds). It s,ofcourse,possible for more than onepersontowalkinunison, butsuchaugmented excitation snotnormally regardedas sufficiently commontobetaken as thedesigncheckcaseagainst comfortcriteria.Muchlargerimpulsive loadingcan ariseinthe so-called 'heeldrop'.Apersonstandingontip-toewho returnsheavily ontohisheels can deliveran impulse of ypically70Ns,within adurationofsome 0.04s.Although such actioncan occurin anoffice orresidence,forexamplewhenreaching forsomething onahigh shelf, itisprobably ofgreatersignificance asastandard design-check (orpractical measurement) input5, whichwillgiveusefulguidance onsensitivity toimpulsiveoadings from anycause, ncludingwalking.

Onepace, period 0.6 s(a) Footfallforceand reaction on floor

Amplitude (N),0/ \10 o


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N (presumingbody0.6 400 mass is67kg)

/10:. / 2Frequency (Hz)Figure2.3 Fouriercomponent amplitudes for egularwalking

Running-step frequencies can risetohighervalues,butdonotcommonlyexceed 3 Hz.The fundamentalFouriercomponent of he force exerted on the floorisof the orderof thebodyweight(i.e.perhaps three imes thecorresponding component in walking), withaperiodofzero force whileboth feet are off theground.The 'freeflight' phaseofbodymotionbecomeseven more mportant whenrhythmicalactivities, such asdancingoraerobic exercises, areconsidered. Thebody eaving heground, withnowayofaccelerating the return tokeepupwith the 'beat', imposesaclearupperbound on thecombination of impulse andfrequency that canbedeveloped6,andfor this reason he frequency willnotsignificantly exceed the valuequoted forrunning.Unfortunately, however, suchactivities clearly offer the likelihood ofa argenumber ofpersonsactinginunison, and the structural effects arepotentially severe. Usefulquantitative guidance canbefoundinthe NationalBuildingCodeofCanada.Mechanical excitation is alsopossible. Theclassic example is out-of-balance rotatingmachinery. There is little to besaid about suchexcitation; it is generally stronglypreferable totackle suchproblems atsourcerather han inthe structure, by reduction oftheout-of-balance orbyvibration-isolation mountings forthemachine.Impulsive or transient mechanicalexcitation smorecommonly external to thebuilding,possible causesbeingroad or rail traffic, or(in special cases)heavymachinery oruse ofexplosives. Where his effect s ikelyto besevere,vibrationisolation atbuilding foundation levelisgenerally preferable to usingcontrolmeasuresatspecific floors, especially because userreaction would be dependent on theinteraction ofvibration (includinghighfrequencies) andacoustic effects. The samecomment hat the solutiondoesnotreallylieinthe handsof he floordesignerappliesto theoccasional within-building impulsive mechanical loads,suchasproblems arisingfrom operationof he lifts. In thispreliminary surveyit is alsopertinent opointoutthat similarproblems can arise from vehiclemovement incar-parking areaswithin abuilding,andagain hepreferable remedy istotackle theproblematsourcebyproviding asmoothrunningsurface.


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3. HUMAN REACTION TO VIBRATION3.1 Reviewof FactorsGiven largeamplitudesofoscillation atfrequencies inthe range2Hz to 20 Hz theremaybesignificant strains within thehuman body,possibly including resonance ofspecificorgans, giving riseto acute discomfort, serious impairment ofabilitytoperformmechanical tasks,andeven njury. These problemshave been studied extensively inrelation o tasks involved in national defence, such aspilotinghigh-performance aircraft,and alsofortheestablishment ofcriteriaforworking conditions inonerous industrialsituations. Itis mmediately clear that there isavery widerange between theamplitudesofmotionassociated with suchcriteria and the threshold ofperception; thisrangeistypically one hundred times the threshold. Thecriteria appropriate toresidential orofficeenvironments areassociated with intermediate levels ofvibration at which purelyphysiological effects ake secondplace to psychological factors.The importance ofpsychological factors makes itdifficult oquantifyhuman reactionatthese levels.Anyexperiment inwhich the subjects areaware that their reaction sundertest is clearly subject to doubt.Thereare also wide variations between ndividuals, a angeofamplitude exceeding afactorof2 existsbetween he topand bottom5% of thepopulation forany given reaction.Reaction atthese levelsmaybe influencedby a numberof factors.Atthe lower endof hefrequency range,reaction is strongly linkedtoa eeling of insecurity, based on instinctiveassociation ofperceptible motion in a solid' building structurewith anexpectation ofstructural inadequacy or failure. At thehigherendof he frequency range,reaction sstrongly linkedtoassociatednoise levels. Ohlsson3hasreportedacasestudy in whichofficeworkershadmutually agreed that hard shoes wouldnot beworn,andfound thishighlybeneficial. Measurement showed hat thedifference invibration wasquiteinsufficient toaccount for thedifference inreaction,which wasattributed to theelimination ofnoise that the occupant would associate withvibration. The floor inquestion fallsseriously shortof heacceptance criteria putforward in this Guide.Because of the widerangeto becovered,it isusual toplotcontours indicating humanreaction on twin logarithmic scales of frequency and amplitude of esponse; the responsecan be expressed intermsofeitherdisplacement, velocity oracceleration. Ifamplitude ofacceleration is aken as the ordinate, aconstantvalueofdisplacement plotsasa straightline ofslope +2. A lineofslope 1 corresponds to aconstantvalueof he rate ofchangeofacceleration. Itisrational to assume hathuman reactionwouldberelated to the formeratvery high frequencies, since thebodymasswillnotfollow the floormotion and heperception willbeofstrain in the legsand spine. Atthe other extremeofvery lowfrequency, human reactionwould be related tothe rateofadjustment of the inertia forceson thebody,andthus reactioncontours shouldplot tothe slope ofi. It is thereforeapparent hat thecontours will havea troughshape.The most important rangeoffloorfrequencies covers he band where thereactioncontours arechangingfrom slope zero (acceleration criterion) to slope + (velocitycriterion). Typicalbroadqualitative contours of eaction osustained uniformvibrationare shown in Figure 3.1. Amargin ofatleastafactorof2 isrequiredbeforeanobserverwould change hisqualitative description ofreaction, naddition to thevariabilitybetweenobservers.Itiseven more difficult oextend thecriteria tonon-steady vibrations. Forcontinuousrandomoscillation (i.e. acontinuously modulated harmonic motion) it is usualtoquotecriteriaintermsof the root-mean-squarevalueof he motion. It isnotclear, however, howfar thisisauniform criterion overdifferent ratesofmodulation, oroveroscillations inburststhat are separated by intervals ofquiescence. Itiscertainlynota goodcriterion foroccasional occurrences ofoscillation, especiallywhere heoscillation is initiated sharplyanddamped outrapidly. Therapidityofdecayiswidely recognised ashaving amajoreffect;doubling the effectivedecayratemayraise the level ofagiven reaction contour(basedon thepeakoscillation amplitude)byafactorof three.6


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10 //uickly tiring/.0 stronglyperceptible tiringoverlong periods

C Clearly perceptible disacting0.1 Perceptible

0.01 .: Barelyperceptible/Frequency (Hz) (logscale)

Figure3.1 Qualitative description ofhuman eaction to sustained teadyoscillationIthas beensuggested bove thatnoise directly associatedwith the oscillation sanadverse actor. However, forhigh-quality environments (residential oroffice)where anoccupant will resent ntrusiononhismental concentration, itmaybethat the appropriatevibration limitwould actually behigherwhere here is substantialambientnoisefromothercauses.3.2 SpecificationsAs notedabove, studies ofhuman reactionhave tendedtofocus on relatively severecircumstances, and this is reflected nthe balanceofpublished specifications. Forexample, several specificationscanbeconsultedaboutsevere industrial workingconditions, but here is very littleavailable withatrack recordofsatisfactory applicationtoassessmentoffloors inoffice orresidential accommodation.TheCanadian Specification CAN3S16.1 SteelStructures orBuildings8does,however,includeaveryusefulAppendixentitled Guide for floorvibrations',although hisisnotamandatory partof he Code.Theproposed annoyance criteriafor floor vibrationsareshowninFigure3.2.Thelabellingof hese curvesneed interpretation: the curves abelled'walkingvibration' are to be usedforassessing theresponse oheeldrop impulse,and thecurve labelled continuous vibration'isto beusedfor he assessmentof hemotioncausedby apersonwalkingacross he floor. Forexample, in the lattercase,afloor ofspan14 mand frequency 6 Hzcrossed byapersonwalkingat2pacespersecond (sothatthere was significant response othe thirdharmonic in thepaceexcitation)wouldshowsustained response overabout tenpacesor30cycles.The interpretation of average peak'insuchacaseis leftopen;the averageover the worst 20cyclesmightbereasonable.The threecurves inFigure3.2labelled 'walkingvibration' are specificallylinkedintheCanadian Code8 with the 'heel drop' impact test.TheCanadian Specification suggests6%ofcriticaldamping ortypically-furnishedfloors withoutpartitions.The sensitivity tothe levelofdamping reflects thegreatly educed annoyance caused byanimpulsive ventwhen he subsequent decay is veryrapid. Unfortunately, it alsoreflects theapplication of

7


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100 : I 1 I50 /- , -," Criteria orwaiking20 Walkingvibration , vibrations: acceleration determined(12% damping) ,- byheel mpact est10

Walkingvibrationc_ (6% damping)Co - , -

2 Walkingvibration / Criterionfor1.0 (3% damping) continuous vibration

0 Continuous vibrationa0 _I_t,,_r

(10 to30 ycles)

0.1 I 11111 I1 2 4 6 10 20Frequency (Hz)Figure 3.2 Annoyance criteria for loor vibrations (residential,schoolandofficeoccupancies)

thistest to assessing the sensitivity of the floortowalkingexcitation, wheredampinghasadifferentaction. In this casehigherdampingprimarily causesa eduction of thedynamicmagnifieratresonance. The morerapid decayonce the sourceofexcitationhasmovedoffthespanisonlyofsecondary significance. Asnoted laterin Section 4.2, theeffectivedecayrate from the impulsive event is very commonly enhanced by a ateraldispersionof heenergyofoscillation. Thismay legitimatelybe included intheeffectivedampingvalueforidentifying theacceptable levelof nitialresponse o impulsiveexcitation,and ispresumably so included in theCanadian Specification. Theenergydispersioneffect is notequally effective underrepeated-pace excitation. Careis hereforerecommended inthe useof thesecurves. Impulse esponse criteria whichgivesimilarvalues have also beenpresentedby Murray; somediscussion ofhisproposals is giveninSection 6.2The Supplement to the NationalBuildingCode ofCanadapostulates limitsforhumantolerance incasesofgroupactivities, namelyanacceleration amplitude ofO.02gfordancinganddining,orO.05gforlivelyconcert orsports events.For heseactivities, thecheckisapplied o theconsiderationof he fundamental-frequencyexcitation omponentonly.The response considered is thus atfrequencies upto3Hz, and floorresonance tohigh frequencycomponents isnot aken ntoaccount. Asecond-component excitation,thusgivinganexcitation requency up to 6Hz,isgivenfor 'jumpingexercises'.The most relevantUnitedKingdom specification is BS 6472Evaluationof umanexposure tovibration nbuildings (1 Hzto 80Hz)9.This is strongly linkedtotheInternational Standard SO 2631Guide o theevaluation ofhuman exposure to wholebodyvibration>, which is inturn osome extentadescendant ofGermanspecificationsoriginally drawn upforindustrial working conditions. However, it ncorporates asubstantial recent eview in the broadercontext, including theworkofIrwin". BS6472defmesabase curveofacceleration asafunctionof frequency, withmultipliers to definethe acceptable level as afunctionofbuilding functionand the nature of heexcitation.The base curve is dentical in shape othe linesofFigure3.2 (forfrequencies exceeding4 Hz),withnumerical values one-tenth of heCanadian curve forsustained oscillation.However, hemeasureused in BS 6472 is the root-mean-square (r.m.s) valueof he


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acceleration, rather han thepeak(or average peak').Foraresponsewhich isdominatedby asinglehannonicexcitation omponent the r.m.s.valueisl// imes hepeak,andtheCanadian curveis husequivalent in this caseto 7units(or 'Curve7' inthenotationofBS6472)according totheBritish Standard.BS6472 gives (interalia)values for themultiplying factortoapplytothe base curve fortheassessment ofcontinuous vibration, as shown nTable3.1.Table3.1 Multiplying factors toapply o he basecurveEnvironment Reaction level A* Reaction level BOffices 4 8Residential day 2 to4 4 to 8Residential night 1.4 3*See textforexplanationof reaction level'The values ncolumn Aarepostulatedas 'magnitudesbelow which heprobability ofadversecomment slow',andit ispostulated that the valuesincolumnB 'mayresult inadversecomment'.Anote is addedtothe effect that tolerance nresidentialaccommodation isstrongly influenced by 'socialand cultural actors,psychologicalattitudesand theexpected degreeof intrusion'.Itwillbeseen that the levelsB andAforoffices correspond roughly othe Canadianrecommendation (Figure 3.2),and oone-half hat level, respectively. However, hereisastrong implication that the term continuous vibration' s to be interpretedrigorously inBS6472. Thesevalues are thus reasonably applicable only toveryheavilytrafficked floors withwalkers continually present.In suchcases occasional peaks duetoconcurrent xcitation bymore than one person canprobablybetradedoffagainst henumberofpeoplenotmoving egularly or at resonant-pace frequency. BS6472 offersthe suggestion that intermittent vibration canbeequated to anequivalent continuouslevelby the root-mean-quad, i.e.:

T 114aeq =(J a4(t)&)where a(t) is the valueofacceleration attimet.The root-mean-quad ofa sinusoidal vibration modulating asapersonwalksacross a floortakingsix seconds, repeatedonceperminute, isaboutone-third of thepeakamplitude. Asthisroot-mean-quad is used in substitution forthe root-mean-square valueofcontinuousoscillation, whichwouldbe 1 times hepeakamplitude, afloor subject to apersonwalkingatthe resonantfrequency onceperminute could reasonably bepermitted toshowpeak responseof twice hepeakvalueacceptable forcontinuous oscillation.BS6472 notes that theremaybe ocations where t is necessary torestrictvibrations othelevelof he base curve (factor1). 'Somehospital operating theatres'and someprecisionlaboratories' areputforwardas examples.


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4. GENERALCONSIDERATIONS4.1 Structural and Floor ConfigurationsThefollowingdiscussion ofsteelflooringconfigurations ispresented to indicate theterminology used in discussion offloorvibrationsand heapproximate parameter ranges;it is not ntended to constitute guidance on the selectionof theparameters.The essentialobjectiveof looring stoprovideaflat load-carryingsurface. The floorslabconstruction s generally eithersteel-concretecomposite, timber orconcrete, andusuallycarriessome form of fmishingorfurnishing carpeting and underlays,hardwood surfacingorsimilar,and, inthe caseofconcrete slabs,a screed).There is littleevidence thatfinisheshave much effect on vibration problems, exceptthrough theresulting ncrease ofmass.There is possibly amarginal increase in damping andamarginalcushioning of impulsiveloadsby appropriate fmishes, butafinish softenough o haveamarkedcushioning actionwillbe toosoftto have muchstructural dampingaction.However, theacousticandwalkingcomfort factors ofvariousfmishesare likelytointeractintheexpressedopinion ofusersrelating o thevibration environment as discussed nSection 3.1.Timberfloors arecertainly usceptible to vibration problems, whichhave been studied nbothCanada8andSwederi3.Itwill be shown hathighermassisgenerally favourable,and in thisrespect imber floorsare inherently moreatrisk thanconcrete loors.Nevertheless, in view of hecurrent balanceof he market n theU.K., attentionwill befocused n thisGuide on concrete loors, butwith emphasis on recentdesign rendsleading oa eduction of he massperunit area. Inparticular, thereis ncreasinguse ofpermanent steelformwork (profiled decking ofvariousconfigurations)andof ightweightconcrete, ofteninconjunction with eachother.Thedensityof ightweight oncretecommonly dopted inthe U.K. isaround1800kg/m3;lowervalues are notuncommon inNorthAmerica. Acomposite slabcomprising a 70 mmcontinuous thicknessof ightweight concreteon60 mmsteeldecking may thus haveamassofabout220 kg/m2,excluding inishes. Itmaybe notedhere thatreferences tofloorthicknessesn theU.K. generally referto the total slabdepth;a smeared' thickness equalto (massofconcreteperunitarea)/(concrete density) is oftenused in NorthAmericanliterature, including designguides. Suchaslab istypicallysupported on floorbeams(commonly called 'joists'inNorthAmerica) atabout 3 m spacing.Theshort-term modulusofelasticity shouldbeused for all dynamic calculations, andcurrentspecifications anddesign guides end topresent atherconservative (low)values,bearing inmind the influence of theageof the concreteand he areaparticipating inthecriticalcircumstances. Fornormal densityconcrete thedynamic modulus ofelasticity canbe takenas 38kN/mm2,and forlightweight oncreteataround 1800kg/rn3 the dynamicmoduluscanbetaken as 22 kN/mm.Astiffnessparameterof he form El1/L4canbeconsidered as anaid tothe appreciationofthe importance ofslab stiffness, in which El1 is he flexural rigidityperunit width.For theapplicationof hedesignguidance inSection 7, therigiditymaybecomputed rom asmeared hickness ofconcretewithdecking asappropriate seedesignexampleNo 1 inAppendixB). Theactual stiffnessunderdistributed load wouldbe obtained bymultiplying the stiffnessparameter byacoefficientdepending onsupport conditions andloaddistribution. Considering the spanbetweenadjacentfloorbeams,so thattheeffectivespanL,,issetequaltothe beamspacingb,thisparameter iscommonly inthe range30100kN/m3.On the otherhand, considering theability of the slabtosupport loadoverthe fullbay width,Le= W,thisparameter very rarelyexceeds 1 kN/m3and forwidebayscontinuous over (say) 8 floorbeams it willbe less than0.01 kN/m3.ThecorrespondingstiffnessparameterEI/bL4for the floorbeams istypically nthe range 110kN/m3.The relativestiffness ofslaband floor beams ndicated bytheseparameters has the effectthat underaglobaldistributedloading the slabdeflection betweenbeamsisrelativelysmall. The slab is also sufficient togivesignificantresistance odifferential deflectionofthe floorbeams,althoughclearly not hereby causingamajordeparture fromthebasic


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concept hat thedominant loadpathisvia the floor beamsasa one-way' span.The netresultintermsofdynamic action s hat the floor behavesbroadlyas astronglyorthotropic plate (see Section 4.2)and a strip containing oneor two floorbeams can beconsidered as thedominant structuralunit whenconsidering walkingexcitation.Precast Omniatype' planks, 5065mm n hickness withaninsitu concrete oppingandsupplementaiy continuity reinforcement, willbehave ina similarmanner to ametaldecking composite loor systemconsidered above. However, greatercaution must beexercised whenassessing the continuity and stiffening effectsofother forms ofprecastfloorconstruction. Where hollow-cored precastunits are requiredtomobilise thecompositeactionof he supporting beams, hen the endsof he units should be 'notched'andsupplementary tyingreinforcement usedinconjunction with an insitu concretetoppingshould beprovided. The implementation of hesemeasures will, inaddition, haveastiffening effectonthe floor slab such that the floorsystemwill tendtoact as anorthotropic plate.Conversely, if dry construction' precastflooring sused,withoutsuchmeasures being mplemented, then the supporting beams shouldnotbeconsidered toactcompositely with the slabnorshould the slabsbe assumed to assistinreducinganydifferential deflection betweenbeamsor indistributing any local effects. This form ofconstruction therefore, hroughIckofstiffness, contributes only by virtueof tsmass tothe vibration characteristics of the floor as awhole.Forvery long spans, or where very highstandards aresought, the floorsystemmaycomprisebeamsofcomparable stiffness inthe twoorthogonal directions,constituting neffective'two-way' span,and husanearly isotropic dynamic system. Subject to theabove limitation on deflectionof the slab betweenbeams,this mobilises he wholefloorinresisting dynamic excitation, andis husaveryfavourable configuration.The floor beams hemselves willvery oftenbe supported bymainbeams,which formpartof heprincipalstructural framing of thebuilding.Theresultingadditional deflectionunderaglobaldistributed loading maybe comparable to the floor beamdeflectionbetweenmainbeams.Itshouldbenotedthat he deflection and stress levels tolerable in dynamic responsearelow, ypical stress amplitudes beingless than 1%of the static designstress, so thatconventional designprovisions for simple supports willnotgenerally inpractice act assuchindynamic situations. Largefloor areasmay thus act as ifstructurally continuous.Thegreatereffective structural continuity, underdynamic loading, has the effectthatcolumn stiffness commonly contributes significant endrestraint, even where he beamconnections areofa form thatwouldnormally beregardedas permitting rotation. Columnstiffness sparticularly likely o be significant inhigh-risebuildings. Anadequateanalysiscancommonly be achieved by the 'substitute-frame' procedure.Cantilever formsofconstruction are relatively uncommon. Although the methodspresented inSection 5 forevaluating naturalfrequencies arebroadly applicable tocantilever onstruction, thisformgives arather ineffective mobilisation ofmassifdynamic excitation sappliednear the free end, and the evaluationofresponsepresentedinSection6 maybenon-conservative. Specialist advice should be takenifareliableestimate srequired.4.2 Introduction to DynamicsThe classicext-bookmodelofadynamic system,shown nFigure4.1, is characterisedby amass,aspringstiffness,and adamper. Formathematical convenience, thedamper susually magined odevelop aforceopposing thedirection ofmovement inproportion othevelocity. Except nveryrarecases wheresomeidentifiable damperhas been fitted totackle a specific oscillation problem, eal floors donot ncorporate suchelements, butnevertheless therewillbesome waysinwhichenergy isdissipated intheevent ofoscillation. Thisisusuallybyfrictionwhichcommonlydepends heavilyonnon-structuralcomponents suchaspartitions. It also depends onstructural behaviour differingfrom thedesigner'smodel,suchas nominally non-moment-resisting connections that actuallydevelop considerable frictional esistance. Human occupants also add damping, althoughahigh densityofoccupation would be necessary tohaveanysubstantial effectona loor


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/ /////Stiffness I iscousk

[...]damper

Mass mFigure4.1 Simpledynamicmodelwith aconcreteorcomposite slab; this effect is mostnoticeable wherehighoccupationdensityis combinedwithlowmass,as in a schoolroom with atimber loor.Damping s hus generally recognisable only as aglobalproperty, most directlymeasuredandexpressedby the logarithmicdecrement'of hedecayof hefreeoscillation afterexcitationhas ceased. For the moderateorsmall evelsofdamping nherent inengineering tructures, a logarithmic decrementof(say)0.2means that theamplitude fallsby20% ineachsuccessive cycle.An alternative measure, especially popularin NorthAmerica, is the 'fractionofcriticaldamping'or 'criticaldamping atio', whichis1/2times the logarithmic decrement. Thesequantities arenon-dimensional, and careisnecessary to avoid confusionwhenreadingdesignguidesor test reports.The frequency offreeoscillation of he system shown in Figure4.1 depends on thestiffness n comparison with the massaccording othe following equation:iFf=;;wheref s the naturalfrequency (inHz,i.e. Hertz=cycles/second).Itisusuallyconvenient to workinkN and t (tonne) units; inthis case the stiffnesskwouldbeexpressed n kN/mand themassintonnes. Damping has very littleeffecton thenaturalfrequency, orvice-versa.Itcan be seen that the static deflection caused by the weight of he mass m(presumed toact intheappropriatedirection, in line with the spring) would bey,=mg/k,and thus thefrequency equation can also be expressed in the form:-1[--

21ryThe self-weight deflection is aquantity which heengineercangenerally characterisequitecloselywithout the needfordetailed calculation, andwhichwill followaconsistentpatternasa functionofspanforany given structural form. Thisequation thus offers ausefulgeneralapproach to evaluating frequencies, and shows hatconventional staticdesignprocedures, whichincludea imitony,, actually constrain very strongly thevaluethatwill result for naturalfrequency.For 'multi-degree-of-freedom' systems with several masses elastically interconnected,andespecially hecontinuously distributed masssystem suchas the beamshown inFigure4.2,there willbeaseriesofnatural requencies, eachassociatedwithitsownmodeshape.Thevarious modesaredynamically independent (orthogonal, or 'normal'modes)so thatresponsecan be synthesised byadding modal solutions computedindependently.The lowestfrequency mode is the undamental.This mode has the simplest shape,and itsfrequencywill stillbe strongly constrained as above, and thereis aprocedure (Rayleigh'senergymethod)forestimating anappropriateweighted average of he self-weight


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3 2 1Figure4.2 Beammodeshapesdeflection.Formany beamandplateproblems, y, inthe aboveequation should be takenasabout3/4of themaximum valueof he self-weightdeflection. Forcontinuous beamsgreatercare is required nthis approach, which is discussed urtherbelow.Thehigher modes,whichmaybereferred to asharmonics (although theirfrequencies arenot ingeneralexactintegermultiples of he fundamental frequency), haveshapes ofincreasing omplexity. Forbeams, he secondmodefrequency scommonly atleastthreetimes the fundamental, depending on the support conditions, mass andstiffnessdistribution and(whereapplicable) spanratios. For the simply supported uniformbeam(Figure 4.2) thesecond frequency isfour times the fundamental.Ausefulinsight into the behaviourofsome floorsisgivenbythe behaviourofanorthotropic plate, shown nFigure4.3. Thefundamental mode shaperesembles hecorresponding beammodeshape nbothdirections. Thisprinciple applies alsoto thehighermodes,butif he stiffness is highly orthotropic, the weakdirection deformationhasrelatively littleeffectonthe frequency, andabasic amilyofmodesretaining hefundamentalshape nthestrongdirection can occuratratherclosefrequencies.

Low bendingstiffness

Forcontinuous beams the fundamental frequency isclearly associatedwithashapeof heform shown nFigure4.4. The inertial loads actinthe sense shownandenhance thedeflections, whereas nthe static designprocess heself-weight effectsonadjacent panscombine toreduce he corresponding stressesand deflections. Thus,ifdesigned tothesamestatic criteria, continuous construction withfairlyclosely uniformspansmayhaveasignificantly lowerfundamental frequency thanasimple structure. For he self-weightdeflectionapproach,effective elf-weight loads should be applied inan upwards directioninalternate spans.For eachmode, it ispossible oestablishan effectivemassand stiffness, whichcan beusedinbroadly he same wayas the massand stiffness of thesimple systemofFigure4.1.These are referred to as the modalgeneralisedvalues.The modalgeneralisedmassforeach modeofa simple beam is one-halfof theactual mass. Forplatesand forcontinuousbeams the fraction is smaller, but he effectivemassofacontinuous beamsystem maystillbelarger han thecorresponding simple structure, because the factorisapplicable o

I 1A

..-/ -.-..'"/ / / Modeshapes

High bendingstiffness

Figure4.3 Orthotropic platemodeshapesOrthotropic platesimply supported on edges


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Inertial oad

Fundamental modeshapeFigure 4.4 Continuous beamfundamentalmodeshapethetotalmass in motion, i.e. all spans. Awordofcaution snecessary: thevalues referredtoherepresume hat the modeshape unctions are eachdefmed to giveamaximum valueofunity, but some authors and some computer programs adoptother criteriaby which toscale theshapefunctions.Theresponse ovarioussources ofvibration is discussed in detail later (Section 6).In thisintroduction to dynamics itissufficient tonote theanalytical solutions to two classes ofexcitation: impulsive oads,andharmonic (sinusoidal variation withtime)loads. Inbothcases the motionisapproximately a sinusoidal functionof ime,atthe free vibrationfrequency and the loading frequency respectively. Theamplitude is hen thepeakvalue inthe currentcycle,and the responseenvelopeis he smoothcurve indicated bythepeaks,as shown inFigure4.5. Note that the amplitude s 'mean-to-peak', .e.closelyequal oone-halfof hepeak-to-peak value.Amplitudes maybe quoted fordisplacement, velocityoracceleration; displacement maybe implied fnoother ndication isgiven. Changewithtimeof heresponseenvelope ordinate isreferred to as modulation. Progressive reductionof theenvelope ordinate (usually by damping) isreferred to asattenuation.a)C00.Caa)

Time

impulse isdefmedas achangeofmomentum; itisusually mplied that thisisproducedbya argeforceofshortduration. Forasingle-degree-of-freedom system (Figure 4.1)thesolution is very simplefor an impulse (valueJ, say, with unitsNsconsistentwithexpressingmassinkg, orkNsforuse with tonne) ofshortdurationby comparison withthe naturalperiodofvibration. Within heduration of the impulse the massacquiresvelocity im. Subsequently it is infreevibration and, ignoring theattenuation due todamping, hedisplacement ycan be written in terms of he initial amplitude9:y =ysin(2irft)By differentiating theabovedisplacement equation themaximum velocity is obtained as2rf9 and equating this toJimyields:J= 2irfm

. Response envelopeResponse

Figure 4.5 Definition of esponse envelope


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Precisecalculations are neither ustifiednor equiredfor floorvibrations, and this impulsesolution sadequate or forcepulses ofdurationuptoatleast one-thirdof thenaturalperiodofvibration, and thusforthe impulsive components ofwalking orcesandforheel-dropexcitation or natural requencies upto about 10 Hz.Unfortunately theresponseofapractical distributed-mass floor system is not so simple.Potentially, all modesare excitedand, as theeffective masses (modal generalisedmasses)of he variousmodes areofsimilarorder, he solutionby summation of themodalresponses willconverge ratherslowly. Response inhighermodeswill makeamajorcontribution totheaccelerations of the floor but theadequacy of he impulsive modelofthe excitation becomes questionable for suchfrequencies. Modal analysis snotoftenrecommended for impulsive actions, forwhich a stress-wave solution would inprinciplebepreferable, but there is no simple answerfor this structural form.Asnotedearlier,thesimple floor whichapproximates dynamically to anorthotropic platewill haveafamilyofmodes as illustrated inFigure4.3. Theresponse n each modemaybepredicted rom thesimple mpulse responsesolutiongivenabove,and the totalobtainedby summation. Several membersof hisfamily maystart withsimilaramplitudes,and theywill start in phase.However, theywill rapidly getout ofphaseaccording to the frequency differences, and he resultmaybearapidattenuation of theresponseenvelope, perceptually equivalent to highdamping.The solutions forharmonic excitationareperhaps morefamiliar. The mostimportant caseis resonance,when he frequency of the load(orofaperiodic component in the load)coincides withanaturalfrequency of thestructure. Inthiscase the responsebuildsupoversuccessive cycles. Thesteady-statedisplacement amplitude for long-continuedexcitation s givenby:

P5 = x (magnificationfactor)where .P = amplitude of resonant Fourier component of forcek = stiffness.Themagnificationfactor s:

amplitude _______________ 0.5static deflection damping logarithmic criticaldampingby same force decrement ratioThis magnification may typicallybefifteen-fold. Amagnification of en-fold(ortwo-thirdsof he steady-state value,ifsmaller) wouldbereached within fivecyclesofexcitation.Itwouldnormally be quiteunacceptable for afloortohaveafundamental frequencywithin the range ofwalking- or running-pace frequency. However,asnotedin Section 2,walking contains significant secondand third Fouriercomponents whichmaycoincidewith the natural requency and cause resonantresponse. Response as the sum ofseveralmodes creates essofaproblemwith continued xcitation, becauseexactresonance willclearly onlyoccur in onemode. Repeated impulsive effectsatthe naturalfrequency or atintegersub-multiplesof hatfrequency (one-half, one-third, etc.) can also causeresonantbuild-upof esponse.


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5. EVALUATION OF NATURALFREQUENCY5.1 Component and System FrequenciesIthas been notedabove thatafloorusuallycomprises three identifiable elasticcomponents: aconcrete orcomposite slab, floorbeams,and mainbeams.Thesecomponents arebasically connected inseries,and for theevaluation of tatic deflections(forexample) it is appropriate toconsidereachcomponent separately and estimate hetotaldeflection as the sum of he component deflections. A similardivision maywellbeusefulfordynamic analyses, but with greatercaution because the interactions betweencomponent deformations arecommonly moresubtle.Once the components areconnected to form afloorstructure twillgenerally no longer bepossible to identify specific component frequencies. Floor frequencies areapropertyofthe assembled structure, andinprincipleeach mode nvolves motion ofall parts of thesystem. However, idealised component natural frequencies can be defined: the frequency ofoscillation of heslab,presuming nodeflection of the floorbeams; the frequencyof the floorbeams,presuming no deflection of the mainbeamsandthat amassassociated withastripof labofwidthequaltothe floor beamspacingmoveswith eachbeam; the frequency of he mainbeams,presuming that the motion of he floorcorrespondstothe deflectionof the mainbeamsonly.These component frequencies canbeusedforan approximate evaluation of thefundamental frequencyof the total floorsystemby Dunkerly'smethod. Denoting thecomponent requencies byf, f2, andf3 (Hz)respectively, the fundamental systemfrequencyf0 isobtained from:

1 1 1 1= + +f f12 f22 f2With theexercise ofsome engineering judgementconcerning support conditions, thisprocedure cangive goodestimates. The support conditions assumed foreachcomponentmustbecompatible with theconceptof inertia loading'whichacts inthedirection of hetotal systemdeflection. Forexample, fora simple floorcomprising aslabcontinuous overanumberoffloorbeams supported by stiff mainbeams, thereareperhaps wopossibilitiesthatmaysensibly be considered for he fundamental modeshape.The interaction of hefloorbeams and the slab wouldnormally givea undamental systemmode as shown nFigure 5.1(a)and the slope of the slabisonly small where t is supported on the floorbeams. The slab component frequency hould thus be basedonfixed-end conditions.L(a)

(b) k. Deck aloneFigure 5.1 Possible deckand loor beam nteractions

Floorbeams

Main /eamsIDeck and floorbeaminteraction Mode shapes onSectionAA: stiffmain beams


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The lowestpossible requency fortheslab alone wouldarise with the shapeshown inFigure5.1(b), corresponding tono rotational restraint. This wouldnotcombine with thefloor beamdeflection in thisexample; itmightnevertheless correspond tothe lowestnaturalfrequencyof the floor, but this isveryrare inpractice. Similarconsiderationsapplytothe interaction between loor beamsandmainbeams,andin thiscaseitmaybenecessary o sketch differentpossibilities. The valid solution s hecombination giving thelowestvalueoff0,givencompatible assumptions forsupport conditions foreachcombination.Itcanbeseen from the formofDunkerly'sequation that the result isnotespeciallysensitive othe interactions. Anycomponent frequency that s more than twice he lowestcomponent has littleeffect. Bearing in mind thecomments on relativestiffness nSection 4.1, togetherwith the relationbetweenstiffnessandfrequency, itcanbe seen thatthecomposite labcomponent requency generally haslittle influence on the floorfundamental frequency.The component frequency for the floor beamisclosely constrained (asafunctionofspan)inconventional designs bytheapplication ofconventional limitsondeflection oronspan/depth ratiosincombination with normaldesignstresses. However, these limits arebasedprincipallyon live loading, whereas thegoverning factorfornaturalfrequency isthe stiffness nrelationtomassorself-weight. It s generally appropriate toassessfloordynamics on the assumption thatonly a smallfraction,say 10%, of the specified(characteristic) live load(including10% forpartitions whereallowed) willbeoperative inaddition to the massof theslab,ceiling,servicesand anyraisedfloor. Thereisalsoaconsistent trend for ahigh specified live loading toresult inrelatively highnaturalfrequency becausebeamstiffness increases bya argeramount than thecorrespondingincrease inoscillating mass.Since the main beam ayoutandstructural form variesgreatlyfrombuilding tobuilding,the importance of he main beamcomponent frequency isvariable, ranging rombeingnegligible tobeingsimilar o thatof he floorbeams.5.2 Practical EvaluationFour levelsofapproach orevaluating natural frequencies canusefullybe discussed. Inincreasingorderof refmement:a)fromaglobalestimateof heself-weight deflection;b)fromacombination ofcomponent requencies estimated from self-weightdeflectionor tabulatedfrequency formulae;c)by iterative application of tatic analysis, usingcommon static analysis software atthedesk-top;d)by useofdynamic analysis software packages, possibly including finiteelementmodelling of he structure.The first three levels aregenerally limitedto anevaluation of he fundamental requencybut anextended sequence ofmodes willbe output by the fourthapproach. Whereresponseamplitudes are tobestudied, mostpractical floorsshow the modesequenceeffect discussed nSection4.2 and llustrated byFigure4.3.Toa elatively poorapproximation thisproblemcanbecircumvented by usinganempirical estimateofaneffectivestrip widthoffloor (see Section 6). ifabetter estimateof esponse srequired,dynamic analysismust be moredetailed,generallycalling foran appropriate establishedsoftware package.Detailsof the fourapproaches aregivenbelow:

a)Theself-weight deflection approach was introduced nSection 4.2.Taking hesuggestedweightedaveragevalueof he deflectiony= -y0,wherey0 is themaximum value,theequation canberewritten forconvenience:1 flj 18


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Cu

3.0

2.0

1.0

Itshould be noted that thisisadimensional form, inwhichy0must be expressed inmm. The foregoing comments concerning theappropriate loadpatternforcontinuousspans andconcerning theassumptions to bemadeonsupport conditions, Young'smodulusofconcreteand the contribution ofsuperimposed ('live') loads should beborneinmind. Long-term effects suchas shrinkage andcreepdeflections areexcluded. Thismethod is ikely o bequitesufficient for he estimation of thefundamental frequency ofaslab and floor beamsystemon stiffmainbeams.b)Thecomponent frequency approach is ikelyto behelpful where here is asignificant interaction with mainbeamdeflections, especially where this results nafundamental modeshapewithsignificant deflections infurther bays.In suchcasesacarefulsketch of he mode shape srecommended. if hecomponent requencies areestimated bytheself-weight deflection method, this becomeseffectively the same astheglobalself-weight deflection approachbutaidsaclearer udgementof thecriticalmode shape.In some cases the analytical solution forthe natural requency ofuniformbeams canbe used; thiswillgenerally bepreferable forregularcontinuous beams.Theanalytical solutionmaybe written as:El 1/2f=CB()wheremis he massperunitlength(units nt/mifEl isexpressed nkNm2, orkg/mifEl sexpressed nNm2)L is he spanin m (forcontinuous beams take the longestspan).ValuesofGB forasingle spanwith variousendconditions are:pinned/pinned ('simplysupported') 1.57fixed/pinned 2.45fixed both ends 3.56

fixed/free (cantilever) 0.56Valuesforcontinuous beamsaregiveninFigure5.2. Thecomponent requencies arecombinedby the formulagiveninSection 5.1, namely:Jo= 11

1)1/2

0.6 0.8 1.0Span ratio, ilLFigure5.2 Frequency factorCB forcontinuous beams

c)Where he layoutis insufficiently regulartopermit dealisation asuniformbeamcomponents acting n series,and/oraconvincing pictureof the fundamental modeshapecannot be obtained bysimple udgement, he fundamental modeshapecanbefoundby successive approximation usingdesk-topstatic analysis procedures. Thefundamental frequency can thenbeobtained with excellent accuracy by a subsequent

(1I ++1,2 f2 2i J2 J3

0 0.2 0.4


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summation ornumerical integration stage that is amenable to either spreadsheet'computation orhandcalculation.Theobjective sto discover adistribution of loading, q (say), whichproducesdeflections, y(say),such hat theproductym isinthe sameproportion tothe loadqat allpoints.The loadsq and themasses m canberegardedascontinuous variables orfunctions ofco-ordinates defmingpositionon the floor, ortheycanbediscretisedasasetofpoint loads andcorresponding 'lumped' masses. Ifattention s focused ononebayof the floor, comprising anumberof loorbeams,alumpedmodelwiththreepoints(i.e.atthe quarter andmid-spanpositions)on eachfloor beaminthisbayand anequalnumber on deckpointsmidwaybetween he floor beamswouldusuallybeappropriate. In the adjoiningbaysacoarsermesh shouldbeacceptable.Onepointatmid-span oneach floor beamisoftensufficient,butmorepointsaredesirable ifitis anticipated that the modeshapewill show nearly equaldeflections inthe variousbays.Themode shapeis sketched by udgementandvalues,y1(say),assigned ateachpoint. Loadsinproportion tom,y, (where m,is helumpedmassattributed oload-point )can then be estimated. The constantofproportionality isarbitraryatthis stage, so the loads canbewritten asPq1 wherePis aconvenient value(say1 kN)andq1arenon-dimensional coefficients expressing the variationaccording othevariation of the targetn1y1.The staticdeflections resulting from this loading arenowcomputed; these will be abetterapproximation tothe modeshape han theinitialset. Inprinciple, this process canbecontinued oconvergence atthe trueshape.Inpractice,itwillbesufficient whenvaluesofy11y0at allpoints(wherey0isthebiggestvalue) arechanging by less than0.1 in onecycleof thisprocess oproceed toestimate he frequency from:

1/21 (Pq1y,27rlEmyThis will oftenonlyrequire two or threecyclesof iteration. IfPisexpressed nkN,then m1mustbe expressed in tonne andyinmetre; alternatively, N,kg, mcanbeusedrespectively.d)All he established ommercial structural analysispackages (e.g. ASAS,NASTRAN,PAFEC,STRUDL, etc.)includeappropriate dynamic capability, generallywithprovision forfmiteelement modelling, and hese are readilyavailable throughcomputerbureaux. Thedegreeof refinement inmodelling should generallybesomewhat superior to that indicated for the iterative approach, above.Themathematical solution canbeobtained to anydesiredaccuracy. It should beborne in mind, however, that this willoutstrip hequalityof he inputdata,includingjoint andsupport continuity, stiffness prediction for concreteelements and modellingof he excitation rocesses.


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6. FLOOR RESPONSE6.1 Low Frequency FloorsWhen any personwalks ontoanyfloor,afulldescription of theresponseof he structureinevitably involves dynamic (thatis to say, inertial) effects, because hebasic process ofwalking inescapably produces forces whichvary througheachpace,asdescribed inSection 2. The likelihood that the floor is strongly orthotropic, and has abasic family ofmodessharing hepropertyofasimilarmode shapealongthedirection parallel tothefloorbeams, dentifiesa potentially criticalevent when heperson walks atasteady paceparallelto afloor beam.This is especially so whenasmall integermultiple (i.e. less than4,say)of hepace frequencycomes within the closebandof requencies of thebasicfamilyofmodes. Thecorresponding Fouriercomponent of hepaceforce will thengiverise toa esonant, or nearly resonant, response. Theeffective modal input varies with thelocation of thewalker in proportion tothe modeshapefunction atthat location, andthustypically ncreases as the walker movesonto thespan,reaches amaximum whenheisnearmid-span and will then fall off. Withpractical structural dampingvalues and thewalker advancing (say) 0.8 mperpace,themaximum responsewilloccurafewpacesafterpassingmid-span, andof the orderof 10paceswill have beenapplied. If he floorhasa easonant frequency betweenabout 4.8 Hz and 7 Hz, i.e.it issusceptible tothe thirdFouriercomponent of hepace,it willbe subjected to some 30cycles of that component.This wouldbe sufficient to givea esponse very nearly equaltothe steady-state responsetoa sinusoidal forceof hegivenamplitude applied continuously atmid-span.Itwasnotedin Section 4.2that thedisplacement amplitude, 5(say),was thengivenby:P1where is the damping, expressedas the criticaldamping atio.Nowthe response will be nearlysinusoidal, so theacceleration amplitude, a(say),is4ir2f025;but he stiffness kis 4ir2f02M whereM is heeffective modal mass(compare thiswith the firstequation inSection 4.2).Thus:

- PCsa= mWL 2where P = amplitudeofnear-resonant Fouriercomponent of forcem = massperunit areaof loorW= bay widthL = floor beamspanand C, isafactorwhich takes accountof theratioof he effectivemodalmass,M, tothe valuemWLandalsoof the interaction of the modesmaking upthebasicfamily.Thebay width,W, is difficult odefinebothconcisely and rigorously, but theobjective isvery simple: todefine he masswhichmust be significantly set in motion. Thebaythusdefined is very commonly argerthan the rectangle marked offbythegrid ofadjacentcolumns. Thecouplingof themotion of the floorbeams clearly dependson the relativestiffnessof the slab. For floorsofconventional proportions, couplingwill be effectivelyinterrupted by afloor beamgivinga stiffness exceeding 2.5 times hat of itsneighbour; itshouldbenoted that effectivestiffnessescommonly varybetweenadjacentbeams due toelastic supporton mainbeams by comparison with direct raming intocolumns.Fora basic floor baywhichapproximatesasimply supported orthotropic plate,thegeneralisedmassisapproximately mWL. If he bay is roughly square, andgivenconventional slab and beamstiffnesses, the interaction betweenmodes sweak.ThusC. 4. Forwiderbays, he frequency differential between the modesshown inFigure4.3becomesvery narrow and significant dynamic magnification mayoccur in more thanonemode.C,is hus increased. A restricted parametric study of he steady-state response of


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floorsofconventional proportions hassuggested hat theeffective massofwidesimplysupported orthotropic platesis approximately --mSL,in whichS canbecomputedasafunctionof he relative orthogonal flexural rigidities. This has beengeneralised forcaseswherethe actual floor beamstiffnessesneed to bemodified according otheirsupportconditions by substituting theirstiffnessas4ir2f02M (seeabove), thusmaking use of theallowance madeforsupport conditions incomputinghenatural frequency. The result issetout in Section 7.6.The floor beamsupport conditions andcontinuity mayalso directly affect heeffectivemassby bringing intoplaymore than one floor beamspan. This mayoccur in twoways.Firstly, he floor beammaybe continuous such hat the fundamental mode takes the formshown inFigure4.4. With twoequalspans the deflectionsof he twospansareequalinmagnitude and the effectivemass is doubled.The response is thus halved.Thiseffect,however, fallsoffrapidly if thespansareofdissimilarength; the design procedure givenin Section 7.6 postulates areduction of0.6(= 1/1.7) provided theadjoining spanis notless than 0.8L Alternatively, themain beammaybe sufficiently flexible hat thefundamental mode has similar deflections inboth floor beamspans. Asimilar ncrease intheeffectivemass is applicable.This condition generally onlyapplies where hecolumnlayout has been selected to givelongclear spans inboth directions. In eithercase, floorbeamcontinuity is accountedfor inSection 7.6 through an effective engthparameter, LCff.Itwillberecalled from earliersections: that he Fouriercomponentsofpaceforcesare identifiable uptothe third(frequencyupto about 7 Hz)butwith diminishing amplitude, roughly in inverse proportion totheirrespective frequencies, and that the reaction orcriterion) curve foracceleration isflatupto about 8 Hz.Itfollows that thecriterion ofreaction to sustained oscillation setup by regularwalkingbecomesincreasingly onerous in the frequency bands4.87 Hz(third-componentfrequency), and 34.8Hz (second-component frequency).Anaturalfrequency intherange 44.8 Hz isparticularly likelytoresultinperceptible response o walking.Itcanbeseenthatforagivennaturalfrequency hiscriterion ineffect eadsto aminimumacceptable participating massoffloor. However, itshouldbenoted thatan ncrease ofmassmust be accompanied by apro-rata increase ofstiffness if he same naturalfrequency isto be maintained.Simplified rules basedonthis analysis aregiven in Section 7.6. ThedimensionsS andLth,dependent on the relativeorthogonal stiffness, aregivenexplicitly byincorporating C, =4into the overallnumerical factor.Thebasevalues of he exciting force and theaccelerationperception criterionhavealso beentaken into the numerical factor.Itmaybe noted thata ow frequency floor will alsorespond to the transient forces due toheelstrikes. Indeed, his maybemore oftenperceived oractual floors han the resonanteffectdiscussedabove. However, fromadesignpointofview, resonance defmesamoreonerous event.6.2 High Frequency FloorsThe behaviourdescribed bove snot seenas anappropriate model for floorswhere thenatural requency exceeds that of the third Fourierharmonic of thewalking pace.Forhigher frequencies, an impulsiveexcitationcan be considered.The simple expression or heresponse oan impulsive excitationgivenin Section 4.2 canlikewise readilybeexpressed ntermsofacceleration. The effectof ateral ontinuity onthisresponse srelatively weak, because the criterion sbasedonthefirst esponsepeak,before lateral dispersion takessubstantial effect.It s therefore sensible to base theequation on the massof onepanel(i.e. widthb, the floorbeamspacing) togetherwithacoefficient, C (say), which isto be determinedempirically:Ja= 2irfC,mbL

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For the simple panel asabove,roughlysquare nplan,C, 1.7.Longitudinal floorbeam)continuity should be beneficialbut his hasnotbeenexplored o thestage atwhichpositive recommendationscanbemade. It is therefore suggested thatC1= 1.7isanappropriatedefaultvalueforall cases. For the design ules set outin Section 7.5, ithas,however, been feltprudent o setanupperlimit on theeffectivewidthtoensurecoverageofslabs whichare moreslender han currentnormalpractice.Impulsive excitationhas directsignificance forrelatively highfrequency floors. A regularpaceimpulse would eadtoapace-by-pace response whicheachtime would besubstantially damped (including any lateraldispersion ffect) during heensuing paceinterval, only to berenewed oasimilar evel. This effectisnotdependenton theregularity ofpacing or he exact synchronism, but the repeateddecay andrenewal wouldmoderate the subjectivereactioncomparedwithsustained oscillation atthepredicted peakacceleration amplitude. Applying the root-mean-quadprocedure Section3.2) wouldgivetheeffective acceleration amplitude as between0.6 and 0.75times hemaximum.Withapaceimpulse of34Ns andC = 1.7, the effectiveacceleration amplitude wouldthus be:

4 200f 2a= 2irf= (25%; m/s givenmblinkg);mbL mbL 8This effectisprincipally significantat frequencies above 8 Hz where heacceptable levelofacceleration increases in proportion to frequency. This implies that the massmbLshouldbe not ess thansome limitingvaluewhich sin turnproportional tothemultiplying factor,R(say),(orthe selected curveR')asusedinBS6472.Thepermittedacceleration is 0.005R rn/s2r.m.s.or0.007R rn/s2 amplitude atf= 8 Hz, and thus therequirement is:

200 30000mbL< = (kg)0.007R RWhereR=7 is acceptable:

mbL


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Figure3.1 (which ncluded, interalia,the resultspublished byReiherand Meister orcontinuous oscillation) but setatlevels5 to 10 timeshigher toreflect thedifferencebetween apidly-damped impulsive vents andconstant-amplitude motion. Theabsence ofthechange toaspecific acceleration criterion forfrequency below8Hzhasthe effectthattheMurray/Reiher-Meister criteriabecome relatively moreonerous forfrequencies below8 Hz.Murraydoes not suggest thatthis isviewedas anallowance for thepossiblecumulative effectsof uccessive paces ncluding thepotential esonance with aFouriercomponent of hepaceforce as discussedabove, butitwouldseemtoprovideadegreeofcoverforthiseffect.Murray'srecommendations areparticularly influential intheU.S.A. The mostwidelyusedacceptance level (forthepeakpredicted velocity response o the 70Nsimpulse) s14mm/s.in the middleof thequalitatively 'distinctlyperceptible' ange. For frequenciesexceeding 8 Hzthis ismarginally above the level setfor6% damping ntheCanadianSpecification butasnoted aboveitbecomes moreonerous atlowerfrequencies (Figure6.1).Murray has alsodiscussed he importance ofdamping'31.

C05)C.)C.,(3Co1)0.

Velocity1mm/s/ (R=7)/1 2 5 10 20 50

Frequency (Hz)Figure6.1 Impulse reactioncriteriaAdifferent approach oassessmenton the basisof esponse oasingle forcepulsehasbeenpresentedbyUngarand White4. In this case theforceresulting fromasingle foot ncontactwith the floor, asifinregularwalkingmotion, isconsidered as theexcitation. Thepulseduration s husrelatively long, beingmore than thepace period,and he durationofthe initialpartof hepulseinwhich the foot force isrisingrapidly provestohaveadominant influence on thepredicted esponse. Ungarand Whiteuse an dealisation of thisphasewhich s still of elatively longduration about 0.2 s) bycomparison with thecontact-impulse effects considered earlier(0.030.05 s),anddonotconsider heconcurrent liftingof the otherfootwhich sclearly inherent inwalking. The effectof hisdifference inthe force model is thatUngarand Whiteconclude that stiffness atherthanmass isthe governing parameter forsatisfactory behaviourofhighfrequency floors. Thecitedreferenc&t4includes valuable discussion ofvibration acceptance criteriafor heoperation ofprecision scientific equipment.

(m/s')5Suggested2 criterion/

(%g)50

20

10 -5.2-10.5

/ // // Specification(6% damping)///0.1 //Acceleration 50mm/s2(R=7)


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It should be noted that thevarious procedures allinvolve assumptions andgeneralisationsbothonprediction and assessment ofreaction, andare intended by theirauthors obeused as individual packages, validated by experience.Partsof thevarious proceduresshouldnot betaken in isolation, orrecombined.

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7. DESIGN PROCEDURES7.1 DefinitionsBay: An areaof floorsubstantially participating inthe motion. Thewidthof thebay shouldbedelimited by stiff inesof upport orbya ree

edgewhereappropriate. Externalcladdingand internalshear wallsshouldnormallyberegardedasprovidingstiff upport. Anyfloorbeamof tiffnesssuch hatitsdeflection undera uniformlydistributed loadapplied tothe wholeflooris ess than 0.4 times thedeflection of headjacentparallel floor beamshould ikewiseberegardedas delimiting thebay.Baywidth: Thebaydimension perpendicular tothe floorbeams.Floorbeam: The beamdirectlysupporting the floor slab andwhich,forcomposite slabs,isorthogonal tothe deckspan. Thismaybe thesecondary beam whenpartofagrillageofbeams.Floorslab: The two-dimensional element of he floorstructure: forexample, aconcrete slabcomposite with steeldecking,precastpiankswithtopping,precast prestressed floorunits.Floorstructure: Anassembly generally comprising afloorslab,floorbeams andmainbeams.Main beam Anybeamperpendicular toandprincipally loadedbythe floorbeams.This maybetheprimary beamwhenpartofagrillageofbeams.Relativeflexibility: The relative contribution bydeflection ofagivenstructural elementtothe maximumdeflection nthegoverning naturalmodeofoscillation. Thismaybeexpressed interms of the respectivedeflections intheself-weight approach or he respectivecontributions (1/f ) inDunkerly'smethod (discussed inSection 5)for heevaluation of the naturalfrequency. Forexample, if heself-weight deflections ofslab, floor beamand main beam are 1, 6and 3 mm respectively, the relative flexibilityof the floor beam is6/(l 6+3)=0.6.7.2 General ConsiderationsFloorsshouldnotnormally bedesigned ohaveanatural frequency less than 3 Hz.Floorsthatmaybe subject torhythmic groupactivities (dancing, 'pop' concerts, exercises)shouldnotnormally bedesigned ohaveanatural requency less than5 Hz.All suchfloors must be designed to reduce to anegligible level thepossibility ofanyrisk ofpersonal njury resulting romexcessive vibration, including thepossibilityofwilfulresonantexcitation. Thiscondition canbemetby ensuring robustconnection of the floormembers o theirsupports and attention tothegeneral requirements forductile behaviourathighdeformations. Attention should also be paidtorobustattachmentofany items thatcould falltothestoreybelowand o theprotection ofservices from the effects ofsustained vibration. Someguidance canbe foundintheNational BuildingCodeofCanada.Wherepossible, floorlayoutsforrhythmic groupactivities shouldbearranged ominimise he transmission ofvibration topartsof the flooroccupied bypeoplenotdirectlyparticipating intheactivity; forexample, diners na egionadjoining adancefloor. Whereaquantitative assessment ofvibration isrequired n suchcases,adetailedprocedure srequired hat will take accountofmodeshapeordinatesatthepointofreceptionas well asatthepointofexcitation.

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Thepossible perception ofvibration by theoccupants innormal officeandresidentialbuildings should becheckedbyevaluating the response that maybe setupbyasinglepersonwalkingon the floor, as setout below.Thegivenprocedure for theevaluation ofresponsemayalso be used forbuildings forspecialised occupancy (laboratories,hospitals, etc.)but heacceptance criteriamaybe considerably moreonerous and detailedliaison with theClientmay benecessary toestablishsuchcriteria.The response predictedby theequations in Sections 7.5and7.6maybe exceeded ocallynearunsupported edges,whether thesebe associated withcantilevered floorbeams orwitha floor beamatthe edgeofafloor area thatisnotdynamically restrained, bycladding orotherwise. The floor edgebeside an atrium maybeparticularly atrisk inthisrespect. Reservations as tothe applicability of hisGuide o certain formsofprecast-unitfloor slabs are discussed n Section 4.1.7.3 Procedure for Checking Floor SusceptibilityThefirststepistoidentify hecritical ocations to bechecked. For his purpose the flooris divided nto 'bays' (seedefinition above), eachrepresenting an areaofwhich themasswillbe substantially mobilisedby resonant dynamic excitation within that area.Where abay isdynamically continuous (see lastparagraph ofSection 7.4 below) withabayhaving longer spanfloorbeams (spanratioexceeding 1.2), thecheckmaybebased on thebay with the longerspans.The nextstepistoestimate he fundamental natural frequency ofvibration, as setout nSection 7.4. Where this naturalfrequency is greater han7 Hz, theresponseparameterRshouldbeevaluatedaccording toSection 7.5. Where his naturalfrequency is less than7 Hz,R should be evaluated according toSection 7.6.Theacceptable values of heparameterRare indicated nSection 7.7,according othefunctionof he floor. A flexible approach isdesirable to ensure that the correctbalance smaintained betweenstructure costand the refinement ofenvironmental quality inthebuilding. Aclear appreciation of he needsandwishes of he Clientmaybeassistedbyspecific discussion on thispoint. If hepredicted valueofRexceeds theacceptable value,it willcommonlybe preferable toincrease the massof he floor.7.4 Natural FrequencyBeforeproceedingo anumerical evaluation, asketch should bemadeof the likelyrelevantmodeshapeorshapes. The deflectedshapeproduced by auniformly distributedload on the single floor beamhaving thelongest spanin the floorunderconsideration willgenerally giveausefulguideto thisshape.Inmostcases the floor naturalfrequency, f0, canbe evaluated fromthemodifiedsell-weightdeflection using he formula:

18f0= (Hz)where y0 is the maximum short-term deflection in mmresulting romloadsSPunilormlydistributed over eachareabounded byadjacentcolumn centrelines(note that thecentrelines may run parallel withornormal to the floorbeams,depending on the assumed mode shapeunderconsideration see Example2 inAppendixB) inwhich:P isself-weight inthegivenarea,comprising thestructure deadload,thesuperimposed deadload,and 10% of hecharacteristic imposed oadS. isaweighting factor,equalto unity inthe area thatincludes hepointof

maximum deflection andselected nother areasaccording othe magnitudeof themaximum deflection in those areas.Theoptimum solution isgiveniffactors S aredirectly proportional tothe totalrespectivedeflections inthe areas (including negative values asappropriate); this condition can beapproached bysuccessive modification. The fulldevelopment of thisprinciple as an


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iterative procedure is presentedasMethod (c) inSection 5.2. For initialassessmentScanbe takenasunity (positiveornegative according to thedirectionofdeflection) inareaswhere thecentral deflection exceeds one-third of themaximum deflectiony0,andzero inallotherareas.Theassumed relative values ofS houldbere-examined if the frequency thusestimatedis less than3.6 Hz (6.0 Hz for floorswhichmaybe subject torhythmic groupactivities).Theyshould alsobere-examined if he frequency thusestimated is either nthe range4.55.5Hz orin the range 7.08.0Hz andadditionally the valueofR huspredictedexceeds one-halfof the permitted value. If heevaluation ofy0issensitive ovariationsofthe relative valuesofS, theevaluation shouldberepeated withvalues morecloselyapproaching the optimum.Where he fundamental mode isgoverned by floor beamflexibility, and thebay widthextends overanumberof floor beams having a epeated sequence of tiffnesses (forexample, beams alternatelyframingdirectly ocolumns andframing tomainbeams), heaverageof hecomputed loor beamdeflections maybeused,subject to theaddition ofthe slabdeflection asappropriate. For this purpose thedeflection of he mainbeams canbe computedas iffixedended.Frequencies mayalternativelybeestimated bysummation ofanalytical solutions forthecomponent lements according to Dunkerly'smethod(this procedure isdescribed, withpertinentanalytical results,as Method(b) in Section 5.2) orbyanyof the recognisedcomputerpackagesofferingdynamic eigenvalue analysis forwhichqualityassurance canbe obtained.Calculation of the naturalfrequency should be basedonthedynamic Young'smodulus,typically38 kN/mm2fornormal weightconcreteor22 kN/mmfor lightweight concrete(densityaround 1800kg/rn3). Support conditions should beassessedbearing inmind thesmallmagnitude ofdynamic deflections.Composite floor slabs with steeldecking cannormally be regarded as dynamicallycontinuous over the floorbeams irrespectiveof heprovision of opreinforcement. Conventionalend-plate or cleatedbeamconnections atpointswhere he slab is continuouscannormally beregardedas giving continuity. Edgesofbaysco-linearwith walls orcladding cannormally beregardedasstifflysupported.7.5 Floors of High Natural FrequencyIf he fundamental natural frequency exceeds 7 Hz the floorshouldbe assessed on theresponse otheheel-contactmpulse component of thewalkingpace.The response factor R is thengivenby:30000R= mbLwhere m = floor masskg/rn2) includingallowance foroccupancy loading as above

(Section7.4)b= the lesserofeither the floor beamspacingb(m) or 40times the averageslabthickness (m)L = floor beam panm). Forcontinuous construction Lcan be takenas the lengthof he spanadjoining thespanunderconsideration, if his is larger.This formulation mayberatherconservative forconcreteslab floors wherebL


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7.6 Floorsof Low Natural FrequencyIf he fundamental natural requency is less than 7 Hz,the floorshould be assessed on thebasisof response totheappropriate near-resonant component ofregularwalking orces.Writing hecomponent forceamplitude,P,intermsof he typical basic fundamentalFouriercomponent amplitude P1=240 N:P=C1P1=240C1

The valueof he Fouriercomponent factorCfshouldbetaken asafunctionof the floorfrequencyf0 (Hz), .e: ifjisbetween3Hzand4Hz Cf=0.4 ifj s between4Hz and 4.8Hz C1= 1.40.25f0 ifj s over 4.8Hz C1= 0.2Fornormal office orresidential environments theoccurrence ofcriticalexcitationonfloorsof requencyf0 between4.0 Hzand 4.8 Hz willbesufficiently uncommon thatassessment canbebased on the reduced valueofCf shown; this reduction (fromCf=0.4)shouldnot be allowed if hecircumstances ofuse are such that veryvigorous walking(pacefrequency exceeding 2.0Hz)is ikelyto beasignificantly frequentoccurrence, andit should alsobenoted thatrunning can leadtoexcitation significantly inexcessofCf=0.4 n this frequency range.The naturaldamping of he floor, expressedas thecritical damping ratio, should ingeneral be takenas = 0.03 fornormal, open-plan, well-furnished floors. Alowervaluemaybeappropriate or exceptionallybarefloors,noting that thedampingofanunfumished floorofcomposite-deck construction maybe as low as =0.015. Ahighervalue, suchas =0.045 forafloor withpartitions, hould only beassumed if he designerisconfident thatpartitions will be appropriately located o interrupt all relevant modes ofvibration.Theresponse actorR is then given by:

68000CfR= mSLCffcwhere m = floor masskg/m2) including allowance for occupancy loading as above(Section7.4)S = flooreffective width(m)LCff= floor beam ffective panm)= structuraldamping critical dampingratio).

Thedimensions S andLCffaccount fortheeffective distribution of he effectof theexcitation. Theirvaluesshould be takenfrom Table7.1, in which:RF = relative lexibility see Section 7.1)(El1 \h/4S*=4.5__j (m)mf0)( EIbL* = 3.81 I (m)m f02jwhere El1 = dynamic lexuralrigidityofslab (Nm2 perm width)Elb = dynamic lexuralrigidityofcomposite floor beam (Nm2)b = floor beampacing (m)W = baywidth(m)Lm = lengthofmain beam(m)= totallengthofcontinuous floor beam(m).

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Table71 Values fordimensions L ndSIndicative floor ayout Qualifying conditions L (m) S (m)Case(1)

w_______S.

RFnt,amO.2 L GreaterofS*orLbutWCase(2)

t4

L

l=L 2LAsforCase(1)aboveO.8L


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Forallcases inTable7.1 it is assumed that the relative flexibility of he slab is less than0.2, suchaswouldbeapplicable for metal deckcomposite labs. Where thiscondition isnotmet,it is suggested thatSshould be takenas the beamspacing,b, forrelativeflexibilities equalto orgreater han0.3 and alinearly nterpolated valueforrelativeflexibilities betwen0.2and0.3. For 'dryconstruction' precast flooring systems (asdiscussed inSection 4.1)Sshould also be takenas thebeamspacing, b.Floors inwhich he flexibility of he slab islargerthan the flexibility of he floorbeams(asexpressedby the respective self-weight deflection inthe frequency evaluation), andwhichhaveafrequency less than 7 Hz, should be assessed byadetailed analytical methodthat takesproperaccount of hevariations ofdeflection of the slab frompointtopoint inbothdirections. This willgenerally require afmite element computer model.7.7 Acceptance CriteriaIn officeaccommodation the response factorR should notexceed hevalues given inTable7.2.Table7.2 R actors forofficesTypeofoffice Response factor, RGeneral office 8Specialoffice 4Busyoffice 12The 'general' officeclassification providesasuitable environment fornormalofficeactivities, including the use ofcomputers andnormal textoperations on VDUs.The'special' office is suitable for technical tasksrequiring prolonged specialconcentration,including precisionoperations on VDUs. The 'busy' office is oneaccessible toa argenumberofpersons, withvisual and audibledistractions concurrent with anyvibration.Occasional adversecommentmayariseatlevelsattributed tothe 'general'and 'busy'office above.Forareas subject tomuch walking rafficwithpeoplewalkingbrisklyandpurposefully,suchasmaybeassociatedwith largepubliccirculation reas (i.e.pedestrianmalls,extensive lobbiesas at floor level in atria, banking halls, etc.)the response actor(computed for single-person excitation as setout in thisSection 7) should not exceedR=4.Thedaytime esponse actorforresidential accommodation should notexceed hevalueindicated or the special' office. Ifwalkingexcitation can occuratnightona section offloor supporting bedrooms, itmaybe necessary torestrict hevalueofR tohalfthisvalue.Where hehighestenvironmental quality isrequired(precision manual operations) lowervalues maybe specified. Much lowervalues ofRmaybe appropriate forcertaincriticalprocesses, notably inthe semi-conductor manufacturing industry. In thesecasesguidanceonacceptable criteriamustbesoughtfrom theClient.Itshouldbenoted thatalthough he factorRhasbeenpresented inconjunction withBS6472,the latter isperhapsmore focusedon reaction to vibration fromexternal sources.Inallcasesanopen-minded approach shouldbemade to selecting the valueofR,andarealistic balancestruckbetween the risk andpotentialconsequences of reaction atthe'some adversecomment' evel, and the costof he floor. Inmanycases adversecommentwouldnot beassociatedwithtangible loss. Changing R by afactorof2 is equivalent onlytothe mostmarginal changeofhuman reaction.


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REFERENCES1. BRITISH STANDARDS INSTITUTIONBS 5400: 1978SteelConcrete andComposite Bridges. Part2Specification for

LoadsBSI,19782. BLANCHARD, J., DAVIES, B.L. and SMITH, J.W.Design CriteriaandAnalysisBehaviour ofBridgesTransportand Road ResearchLaboratory, Supplementary Report SR275, 19773. OHLSSON, S.V.FloorVibrations and Human Discomfort

Department ofStructural Engineering, Chalmers University ofTechnology, Goteborg,Sweden, 19824. RAINER, J.H.,PERNICA, G. andALLEN, D.E.Dynamic LoadingandResponseofFootbridgesCanadianJournalofCivilEngineering, Vol. 15,No. 1, February 1988,pp66715. ALLEN,D.E. andRA1NER,J.H.Vibration CriteriaforLong-span FloorsCanadianJournalofCivilEngineering, Vol.3,No.2,June 1976,pp 1651736. WYATF, TA.FloorExcitationby RhythmicVerticalJumpingEngineering Structures, Vol.7,July 1985,pp2082107. TheSupplement to theNational BuildingCode ofCanada

National ResearchCouncil ofCanada, Ottawa,19858. CANADIAN STANDARDS ASSOCIATIONCAN3S16.1 M84:SteelStructures forBuildings LimitStates DesignCanadian Standards Association,December19849. BRITISH STANDARDS INSTITUTIONBS 6472: 1984 Evaluation ofHumanExposure to Vibration in Buildings (1 Hz to80Hz)BSI,198410. INTERNATIONAL STANDARDS ORGANISATIONISO 2631: Guide otheEvaluation ofHumanExposure toWholeBodyVibrationISO, 198511. IRWIN, A.W.Human Response toDynamicMotionofStructuresThe StructuralEngineer, Vol56A,September1978,pp 23724412. MURRAY, T.M.DesigntoPrevent FloorVibrationsEngineering Journal,A.I.S.C., Vol.12, No.3, 197513. MURRAY, T.M.Acceptabil

SCI P076 Design Guide for Floor Vibrations

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