SCI P076 Design Guide for Floor Vibrations

H!

4

41

I

4

l

• I 4

I

I

:,

—

1',

.,,.

4

44

4

....

1

..

'1

44 4

44

44

14

4

U

1q11,III

i.iic1e

1

1l41i 1

>4 4

;

•

I

444

I

4

4

IIJ4

_Iie

>44

1

11

h4bOfF1OOt1IS

..:

>1 44>4>

II

4!

(I

I

I

4

I

44!

44

I1

I

41

III I

II

44

1

::.

•

4 4

4

1.1 1411 4,1 44

4:

4

A Steel Construction InstitutePublication

4— 4

inassociationwiththe Construction Industry

n Association

1

1

4

1

1

11

_________

___________________________

14 4 4

1

-

.-.:..tz;1

44 4 .

•

__________________________________________________________________

___________________________________________

(7is document containS

50

pae)

—

is TheSteel Construction Institute. Its aim is to promotethe proper and effective use of steel in construction. Membershipis open to all organisations and individualsthat are concernedwith the use of steel in construction, and members include clients, designers,contractors, suppliers, fabricators,academics and government departments. SC! is financed by subscriptions from its members, revenue from research contracts, consultancy services and by the sales ofpublications.

SCI's workis initiated and guided throughthe involvementof its memberson advisory groupsand technical committees. A comprehensive advisoryand consultancy service is availableto members on the use ofsteel in construction. SCI's research and developmentactivities cover many aspects of steel construction including multi-storey construction, industrial buildings, use of steel in housing, developmentof design guidance on the use of stainless steel, behaviour of steel in fire, fire engineering, use of steel in barrage schemes, bridge engineering, offshore engineering, development ofstructural analysis systems and the use of CAD/CAE. Further information is given in the SCI prospectus available free on request from: The Membership Secretary, The Steel Construction Institute, Silwoód Park, Ascot, Berkshire SL5 7QN. Telephone: (0990) 23345, Fax: (0990) 22944, Telex: 846843.

Although care has been taken to ensure, to the best of our knowledge, that all data and information contained herein are accurate to the extent that they relate to either matters of fact or accepted practice or matters of opinion at the time of publication, the Steel Construction Institute assumes no responsibility for any errorsin or misinterpretations of such data and/orinformation or any loss or damage arising from or relatedto their use.

© The Steel Construction Institute 1989

SCI PUBLICATION076

Design Guide on the Vibration of Floors T. A. Wyatt BSc PhD FEng FICE

ISBN: 1 870004 34 5

© The Steel Construction Institute 1989

The Steel Construction Institute SilwoodPark Ascot Berkshire SL57QN Telephone 0990 23345 Fax 099022944 Telex 846843

Construction Industry Research and Information Association 6 Storey'sGate London SW1 P 3AU Telephone 01-222 8891 Fax 01-222 1708 Telex24224 Mon Ref G (prefix 2063)

FOREWORD This publicationis intendedto provideguidance for designers in an important area of design where information is lacking. It has been prepared by DrT A Wyattof Imperial Collegewith assistance from DrA F Dier ofthe Steel Construction Institute.

The Guide was draftedin conjunction with the support of a steering committee which commented on and otherwise advisedon the draft versions. The members ofthe steering committee comprised: Mr B Boys British Steel Structural Advisory Service Mr R Clark Skidmore Owings & Merrill Mr E Dibb-Fuller BuildingDesignPartnersh'ip Mr E Dore CIRIA Mr K Irish Vibronoise Limited Mr R Povey MitchellMcFarlane & Partners Mr M Willford Ove Arup & Partners.

The work leading to this publication has beenfundedby British SteelGeneral Steels, and the Departmentofthe Environment under a CIRIA researchproject. Studies are continuingandfutureeditions ofthe publication will be amended as necessary to account fornewresults. The Steel Construction Institute willbe pleasedto receiveany comments concerning this publication and subject area.

How to Use this Guide The Guide is divided into sevenSections and two Appendices as shown on the facing page. Section 1 is intended as a broadintroduction and has been written in sucha way that it is suitablefor copyingto a Clientas an aid to preliminary discussions. The background to the designprocedures,which are set out in Section 7, is given in Sections 2 to 6 and a study of thesewill be an aid, althoughnot normally necessary, in the application of Section 7. The designprocedures of Section 7 are selfcontained as far as is practical, althoughin some cases reference to Section 5.2 may be required. Theexamples in AppendixB will be useful for following the designprocedures. Explanation of terms used for describing dynamic behaviour, which maynotbe familiar to thenon-specialist, willbe found in Section 4.2 where theyare highlightedby italic script. Defmitions essential for the applicationof the designprocedures are givenin Section 7.1.

11

CONTENTS Page SUMMARY

iv

NOTATION

iv

1. INTRODUCTION

1

2. SOURCES OF VIBRATION EXCITATION IN BUILDINGS

3

3. HUMAN REACTION TO VIBRATiON 3.1 Review of Factors 3.2 Specifications

6 6 7

4. GENERALCONSIDERATIONS 4.1 Structural and FloorConfigurations 4.2 Introduction to Dynamics

10 10

5. EVALUATION OF NATURALFREQUENCY 5.1 Component and System Frequencies

16 16 17

5.2 Practical Evaluation

6. FLOORRESPONSE Low Frequency Floors 6.2 High Frequency Floors 6.1

7. DESIGN PROCEDURES 7.1

7.2 7.3 7.4 7.5 7.6 7.7

Definitions General Considerations Procedure for Checking Floor Susceptibility Natural Frequency Floorsof High Natural Frequency Floorsof Low Natural Frequency Acceptance Criteria

11

20 20 21

25 25 25 26

26 27 28 30

REFERENCES

31

APPENDIXA: CALIBRATIONSTUDY

32

APPENDIXB: DESIGN EXAMPLES

33

111

SUMMARY This publicationpresentsguidancefor the design offloorsin steelframedstructures againstunacceptable vibrations caused by pedestrian traffic.It has particularrelevanceto compositefloors comprising permanent metaldecking toppedwith concrete. As well as the design procedures set out in Section 7, the Guidecontainsbackgroundcommentary and a general,non-technical, introduction.

Notation a

accelerationamplitude accelerationresponse(Canadian Code) b floor beamspacing effectivewidthbetweenfloor beams B parameterfor effective width(Canadian Code) factor for determining naturalfrequency GB Fouriercomponent factor C effectivemass and lateral distributionfactorfor impulsive loading C, effectivemass and lateral distributionfactorfor sustained vibration C El flexuralrigidity(of composite sectionwhereappropriate) naturalfrequency fundamental systemfrequency f0 f1, J, f3 idealisedcomponent naturalfrequencies off0 acceleration dueto gravity g J impulse (= force xtime) k stiffness

f

1,

L

L Leff

Lm

rn rn

M

P P

floor beamspan lengths length of span lengthfor establishing effectivemass main beam span distributedmass

mass at mesh point 'i' effectivemodalmass lumped

static load

S

force amplitude amplitudeof fundamental Fouriercomponentof walking force weightof oscillating mass distributedloading multiplication factorappliedto human reaction base curve width forestablishing effective mass

t

weighting factor time

P1

P R

S

smearedconcretethickness 5 y y°

y W

iv

deflection amplitude deflection at mesh point 'i' maximum value of self-weight deflection weightedaverage of self-weight deflection

floor bay width critical dampingratio

1. INTRODUCTION The main purpose ofthisGuideis to provide a practical method for assessing the likely vibrational behaviour offloors in steelframedbuildings. The subjectoffloor vibration is complex and consequently the Guide contains sectionsdealing with the current'stateof the art', the background to the proposedassessment methods and a commentaryso that the designermaydevelop an appreciation ofthe phenomenon rather thanapplythe design method by rote. Notwithstanding thisintention, the designprocedure set out in Section 7 and the worked examples contained in Appendix B have been preparedto permit a conservative design assessment to be executed by those with only a limited knowledge of structural dynamics. Floorvibrationis not a new phenomenon, the 'live' feel oftimber floorsunder pedestrian loading is well established. However,because ofthe increasingtrend towards lighter longerspan floorsin all formsofconstruction, but most notably in steelwork, CIRIAand SCI considered it an opportune time to provideinterimguidance on this aspect of design pendingfurtherresearch. This Guidehas not therefore beenpreparedin responseto any existingproblemsbut rather it is intended that its use willpreventsuch problems occurringin the future. Vibration in formsofconstruction otherthan steelwork may also requireconsideration.

The use of structural steelwork for multi-storey construction has increased dramatically over the past ten years. Such increase is largelydueto the responseofthe building industry to Clients' demands for buildings that are fast to construct,have large uninterrupted floor areas and are capable ofaccommodating highly sophisticated air conditioning and other services systems. Modern design and construction techniques enablethe industry to satisfy suchdemands and producesteelframedstructures whichare competitive in terms of overall cost. This trend towards longer span lightweightfloor systems in both steelwork andother formsofconstruction, with theirtendency to lower naturalfrequencies and less effectivenaturaldamping,has createda greater awareness of the dynamic natureof some types ofsuperimposed loadings. Currentlythe most popular form offloor construction used in conjunction with multi-storey steel frames is the 'compositefloor'. This form offloor slab comprises profiledmetaldecking spanning betweenbeams and topped with insitu concrete. Muchofthe design guidance givenin thispublication is directlyrelatedto this form of construction. The vibration of floorscan arisefrom externalsources suchas road and rail traffic. Where suchproblemsare anticipated, however, it is preferable to isolatethe buildingas a whole. This aspectofvibration controlis not takenfurtherin this Guide, whichaddressesfloor vibrations caused by internalsources. The most usual and important internalsourceof dynamic excitationis pedestriantraffic. A person walking at a regularpace appliesa periodically repeatedforce to the floor which may causea buildup ofresponsein the structural floor. Other sources of internal excitationsuchas vigorousrhythmic groupactivities are not specifically coveredin this Guide. However, where such activities are envisaged a robuststructureshouldbe providedwhichhas adequate ductility, and specialattentionshouldbe paid to the beam/column connections. These design features are similarto those considered when preparinggood seismic-resistant designs andit is to publications dealing with this subject that the designer's attention is directed. Human perception ofvibration is in one sensevery sensitive; the criterionis likely to be set at a low level. In another sense it is very insensitive; a substantial quantitativechange in the amplitude ofvibration corresponds to a relatively small qualitative change in perception.If a personis asked to express an opinionon his perception of vibration in two different roomson separate occasions, he willnot draw adistinction unlessthe quantitative difference is at least a factorof 2. There are also substantial differences betweenpersonsandtheremay also be differences betweennationalities. Humanreaction at these levelsis substantially psychological, dependingpartlyon the delicacyofthe 1

activity being performed. Response to vibrations is often affectedby other stimuli(sight andsound). Although floor vibration may inducea sense of insecurity in some people, it must be stressed that perception of floor vibration doesnot implyany lack of structural safety. Once constructed, it is very difficultto modifyan existing floorto reduce its susceptibility to vibration, since only majorchangesto the mass, stiffness or dampingof the floor system will produceany perceptible reduction in vibration by peopleregularly trafficking the floor.

It is therefore important that the levelsofacceptable vibration be established at the

conceptual stage having regard to the anticipated usage of the floors. The Client must be involvedin this decision, since the selecteddesigntarget level for vibrational response will usuallyhave a significant bearing on both the cost andoverall floor construction depthfor the project. The questionis frequently raisedofthe tolerance ofmodemcomputer equipment to ambientvibration. The steering groupfor this studyhas been unable to fmd any firm evidence of actual problems resultingfrom floor vibration. Manufacturers commonly state that their equipment is tolerantof the levelsofvibrations acceptable in a goodoffice environment. Consultation with a prominent manufacturer has confirmed that vibrations within the range tolerablefor human occupancy would cause no problemto computer equipment. In conclusion, therefore, it is intended that the publication of this Guide willaid both designers and Clients in setting sensible targetsfor acceptable levelsofvibration which can thenbe incorporated intothe design ofthe floor structure to produceeconomic, usage-related, buildings.

2

2. SOURCES OF VIBRATION EXCITATION IN

BUILDINGS There are a numberofdistinctpossible causes of dynamic excitationof floors. The

important characteristics of these excitations vary to the extent that quite different check procedures may be appropriate depending on whichpotentialcause is most important. The obvious,almostuniversal, excitation is the effectofwalkingon the floor. The geometry ofthe human bodywalking is (to a first approximation) a straight-leg motion that necessarilycausesthe main bodymass to riseand fall with every pace (see

Figure 2.1).This rise andfall is typicallyabout50 mm,peakto peak,but is sensitiveto the angle of the leg at full stretch, and thus to the extent to whichthe walkeris forcingthe pace. One is not aware ofthis movement, becausethe brainidentifies theresulting acceleration signals as correlated with walking and disregards them; it is, however, interesting to note that these accelerations are around 3 m/s2,which is roughly 30 times the value that would be acceptable as the resonantresponseof a floor, and 100 times the valuethat wouldcommonly be set as a limitto sustained vibrations. The annoyance caused by floor vibrations is essentially psychological, and is very susceptible to expectation or familiarity; it is none the less a real problem. Directionofwalk

-——f Legs atpoint offootfall (Solid lines)

'',,

________

Rise and fall of main body mass

Legs at mid-stride (broken lines)

;''-.' "; /

Figure 2.1 Simplifiedgeometryof walking

The vertical accelerations ofthe body mass arenecessarily associated with reactions on the floor, and theywill be closely periodic, at the pace frequency. The fluctuation can be resolvedas a seriesofsinusoidal components (i.e.a Fourierseries) and it is found that the fundamental termagrees fairly well with the simple visualisation of Figure 2.1, giving a force amplitude between100 N and 300 N. Walking pace frequency can vary between1.4 Hz and 2.5 Hz, and the force amplitude tends to increase ratherseverelywith increasing frequency. However, walkingpace indoors is most commonlytowards the lower end of this range, around 1.6 Hz. The British Standard for bridges" suggests 180 N force amplitude for checking footbridgedesigns2. A typical example ofthe contactforce from a single footfallis shown as the light solid curve in Figure 2.2(a).Unless the floor structureis exceptionally sensitive to the precise locationofthe load(i.e. ifone pace-length makesamajor difference), the dynamic excitationis givenby the sum of the concurrent walker's foot forces,which takesthe form shown as the heavy solid curve in Figure2.2(a).The basicpace frequency is clearly represented but the second Fouriercomponent, representing excitationat twicethe pace frequency, is also important. The third component is smaller, and succeeding components can generally be ignored, exceptthat there is a significant impulsive effect of very short durationas the foot contacts the ground. The first three Fouriercomponents are shownin Figure 2.2(b), and the degreeofapproximation given by the summation of these three components is indicated on Figure2.2(a).This example is taken from the work of

Ohlsson3. The magnitude ofthe second Fouriercomponent varies with the walking pace in a similar wayto the basiccomponent. Unfortunately, however, thesehigherfrequency effects, especiallythe contactimpulse, vary considerably betweenpersons.The averagevalues of

3

z C.)

0

'300 E C

200

100

0

—100

-200

One pace, period 0.6 s Footfall force and (a) reaction on floor

N 300 Amplitude (N)

/,0\

200

10

o

100

0

—100

-200

(b) Fourier components of reaction on floor

Figure 2.2 Typicalwalkingexcitation

len4

the Fouriercoefficients reportedby Rainer, Pernicaand from a Canadian study directedto footbridgeloading are shown in Figure2.3. The contactimpulse is typically about3 Ns (Newton seconds). It is, ofcourse,possible for morethan one personto walk in unison, butsuch augmented excitationis notnormally regardedas sufficiently common to be taken as the designcheckcase against comfortcriteria.

Much larger impulsive loadingcan arisein the so-called 'heeldrop'. A personstanding on tip-toe whoreturns heavily onto his heelscan deliveran impulse oftypically70 Ns, within a durationofsome 0.04 s. Although such actioncan occur in an officeor residence,for examplewhenreaching for something on a high shelf, it is probably of greater significance as a standard design-check (orpractical measurement) input5, which will give useful guidance on sensitivity to impulsiveloadings from any cause,including walking.

4

N

0.6

400

/1

0:.

/

(presumingbody mass is67 kg)

2

Frequency (Hz)

Figure 2.3 Fouriercomponent amplitudes forregularwalking Running-step frequencies can rise to higher values, but do not commonlyexceed 3 Hz. The fundamentalFouriercomponent ofthe forceexerted on the floor is of the order of the bodyweight(i.e. perhaps threetimes thecorresponding component in walking), with a periodofzero forcewhileboth feet are off the ground. The 'free flight' phase of bodymotionbecomesevenmoreimportant whenrhythmical activities, such as dancingor aerobic exercises, are considered. The bodyleavingthe ground, with no way of accelerating the return to keep up with the 'beat', imposesa clear upperbound on the combination of impulse and frequency that can be developed6,and for this reasonthe frequency willnot significantly exceed the valuequoted for running. Unfortunately, however, such activities clearly offer the likelihood ofa largenumber of personsacting in unison, and the structural effects are potentially severe. Useful quantitative guidance can be found in the NationalBuildingCode ofCanada. Mechanical excitation is also possible. The classic example is out-of-balance rotating machinery. There is little to be said about suchexcitation; it is generally strongly preferable to tackle such problems at sourceratherthan in the structure, by reduction of the out-of-balance or by vibration-isolation mountings for the machine. Impulsive or transient mechanicalexcitationis more commonly external to the building,possible causesbeingroad or rail traffic, or (in special cases)heavy machinery oruse of explosives. Wherethis effectis likely to be severe, vibration isolation at building foundation level is generally preferable to usingcontrolmeasures at specific floors, especially becauseuser reaction would be dependent on the interaction ofvibration (includinghigh frequencies) and acoustic effects. The same commentthat the solutiondoes not really lie in the hands ofthe floordesignerapplies to the occasional within-building impulsive mechanical loads, suchas problems arising from operation ofthe lifts. In this preliminary surveyit is also pertinentto point out that similarproblems can arise from vehiclemovement in car-parking areas within a building,and againthe preferable remedy is to tacklethe problemat sourceby providing a smoothrunningsurface.

5

3. HUMAN REACTION TO VIBRATION 3.1 Review of Factors Given large amplitudes ofoscillation at frequencies in the range 2 Hz to 20 Hz there may be significant strains within the human body,possibly including resonance of specific organs, giving rise to acute discomfort, serious impairment ofabilityto perform mechanical tasks,andeven injury. These problemshave been studied extensively in relationto tasks involved in national defence, such as pilotinghigh-performance aircraft, and also for the establishment ofcriteriafor working conditions in onerous industrial situations. It is immediately clear that there is a very wide range between the amplitudes of motionassociated with such criteria and the threshold of perception; thisrangeis typically one hundred times the threshold. The criteria appropriate to residential or office environments are associated with intermediate levels of vibration at which purely physiological effectstake secondplace to psychological factors. The importance of psychological factors makes it difficultto quantifyhuman reactionat theselevels.Any experiment in which the subjects are aware that their reactionis under test is clearly subject to doubt.Thereare also wide variations betweenindividuals, a range of amplitude exceeding a factorof2 existsbetweenthe topand bottom5% of the population for any given reaction. Reaction at these levelsmay be influenced by a numberof factors. At the lower end ofthe frequency range, reaction is strongly linkedto a feeling of insecurity, based on instinctive association ofperceptible motion in a 'solid' building structure with an expectation of structural inadequacy or failure. At the higher end ofthe frequency range, reactionis strongly linked to associatednoise levels. Ohlsson3has reporteda case study in which officeworkershadmutually agreed that hard shoes would not be worn,and found this highly beneficial. Measurement showedthat the difference in vibration was quite insufficient to account for the difference in reaction,which was attributed to the elimination ofnoise that the occupant would associate with vibration. The floor in question falls seriously shortofthe acceptance criteria put forward in this Guide. Because of the wide range to be covered,it is usual to plot contours indicating human reactionon twin logarithmic scales of frequency and amplitude ofresponse; the response can be expressed in termsof eitherdisplacement, velocity or acceleration. If amplitude of acceleration is taken as the ordinate, a constantvalueofdisplacement plots as a straight line of slope +2. A line of slope —1 corresponds to a constantvalue ofthe rate ofchange ofacceleration. It is rationalto assumethat human reactionwouldbe related to the former at very high frequencies, sincethe body masswill notfollow the floor motion andthe perception willbe of strain in the legs and spine. At the other extremeof very low frequency, human reaction would be related to the rate ofadjustment of the inertia forces on the body, and thus reactioncontours shouldplot to the slope of—i. It is therefore apparentthat the contours will havea troughshape. The most important rangeof floor frequencies coversthe band where thereaction contours are changingfrom slope zero (acceleration criterion) to slope +1 (velocity criterion). Typicalbroad qualitative contours ofreactionto sustained uniform vibration are shown in Figure 3.1. A margin ofat least a factorof2 is requiredbeforean observer would change his qualitative description ofreaction,in addition to the variability betweenobservers. It is even more difficultto extend the criteria to non-steady vibrations. For continuous randomoscillation (i.e. acontinuously modulated harmonic motion) it is usual to quote criteriain terms of the root-mean-square value ofthe motion. It is not clear, however, how far thisis a uniform criterion over different ratesof modulation, or over oscillations in burststhat are separated by intervals of quiescence. It is certainlynot a goodcriterion for occasional occurrences ofoscillation, especiallywherethe oscillation is initiated sharply anddamped out rapidly. Therapidityofdecayis widely recognised as having a major effect; doubling the effectivedecayrate may raisethe level ofa given reaction contour (basedon the peakoscillation amplitude) by a factorof three.

6

10

Quickly tiring

// /

stronglyperceptible — tiring over

1.0

long periods C Clearly perceptible —

disacting 0.1 Perceptible

Barelyperceptible/ 0.01

.: Frequency (Hz)

(logscale)

Figure 3.1 Qualitative description ofhumanreaction to sustainedsteadyoscillation

It has been suggestedabove that noise directly associated with the oscillationis an

adversefactor. However, for high-quality environments (residential or office)where an occupant will resent intrusion on his mental concentration, it may be that the appropriate vibration limit would actually be higher wherethere is substantial ambientnoise from othercauses.

3.2 Specifications As noted above, studies ofhuman reactionhave tendedto focus on relatively severe circumstances, and this is reflectedin the balanceof published specifications. For example, several specifications can be consultedabout severe industrial working conditions, but there is very little available with a track record of satisfactory application to assessmentof floors in office or residential accommodation.

The Canadian Specification CAN3—S16.1 SteelStructuresforBuildings8does, however, includea very usefulAppendixentitled 'Guide for floor vibrations',althoughthisis not a mandatory part ofthe Code.The proposed annoyance criteriafor floor vibrationsare shown in Figure 3.2. The labellingofthesecurvesneed interpretation: the curveslabelled 'walkingvibration' are to be usedfor assessing the responseto heel drop impulse,and the curvelabelled 'continuous vibration' is to be used forthe assessment ofthe motion causedby a personwalkingacrossthe floor. For example, in the lattercase,a floor of span 14 m and frequency 6 Hz crossed by a person walkingat 2 pacesper second (sothat there was significant responseto the thirdharmonic in the pace excitation) would show sustained response over about ten paces or 30 cycles. The interpretation of'average peak' in such a case is left open;the average over the worst 20 cyclesmightbe reasonable. The threecurves in Figure 3.2 labelled 'walkingvibration' are specifically linked in the Canadian Code8 with the 'heel drop' impact test. The Canadian Specification suggests 6% of criticaldampingfor typically-furnished floors withoutpartitions. The sensitivity to the level of damping reflects the greatlyreduced annoyance caused by an impulsiveevent whenthe subsequent decay is very rapid. Unfortunately, it also reflects the application of 7

100

:

I

1

I

50

,,

• —

20

—

Walking vibration —— (12% damping)

, ,"

/ -

Criteriaforwaiking

—

vibrations: acceleration determined

byheel impacttest

,-

10 Walking vibration c_

• -

Co

—

2

(6% damping)

,

-

/

Walking vibration

—

—

Criterionfor continuous vibration

(3% damping)

1.0

a0

Continuous vibration

0 •

I

0.1 1

_I_t,,_r

(10 to30cycles)

2

4

11111 6

10

I

20

Frequency (Hz)

Figure 3.2 Annoyance criteria forfloor vibrations (residential, schoolandofficeoccupancies)

this test to assessing the sensitivity of the floor to walkingexcitation, wheredampinghas a differentaction. In this case higherdampingprimarily causesa reduction of the dynamicmagnifierat resonance. Themorerapid decayoncethe sourceofexcitationhas movedoff the span is only of secondary significance. As notedlater in Section 4.2, the effectivedecayrate from the impulsive event is very commonly enhanced by a lateral dispersionofthe energyof oscillation. This may legitimately be included in the effective dampingvaluefor identifying the acceptable level ofinitialresponseto impulsive excitation,and is presumably so included in the Canadian Specification. Theenergy dispersioneffect is not equally effective underrepeated-pace excitation. Careis therefore recommended in the use of these curves. Impulseresponse criteria which give similar values have also been presentedby Murray; some discussion of his proposals is given in Section 6.2 The Supplement to the NationalBuildingCode ofCanada°postulates limitsfor human tolerance in cases ofgroupactivities, namelyan acceleration amplitude of O.02gfor dancinganddining,or O.05g for lively concert or sports events.Forthese activities, the check is appliedto the consideration ofthe fundamental-frequency excitationcomponent only. The response considered is thus at frequencies up to 3 Hz, and floor resonance to high frequencycomponents is not takeninto account. A second-component excitation, thus givingan excitationfrequency up to 6 Hz, is givenfor 'jumpingexercises'. The most relevantUnitedKingdom specification is BS6472Evaluationofhuman exposure to vibrationin buildings (1 Hz to 80 Hz)9. This is strongly linkedto the International StandardISO 2631 Guideto the evaluation ofhuman exposure to whole bodyvibration°>, which is in turnto some extenta descendant ofGermanspecifications originally drawn up for industrial working conditions. However, it incorporates a substantial recentreview in the broadercontext, including the work of Irwin". BS6472 defmesa base curveofacceleration as a functionof frequency, with multipliers to define the acceptable level as a functionofbuilding functionand the nature ofthe excitation. The base curve is identical in shapeto the linesof Figure 3.2 (for frequencies exceeding 4 Hz), with numerical values one-tenth ofthe Canadian curve for sustained oscillation. However,the measureused in BS6472 is the root-mean-square (r.m.s) value ofthe 8

acceleration, ratherthan the peak (or 'average peak'). For a responsewhich is dominated timesthe peak, and by a singlehannonicexcitationcomponent the r.m.s.value is the Canadian curveis thus equivalent in this caseto 7 units (or 'Curve 7' in the notation of BS6472)according to the British Standard.

l//

BS6472 gives (interalia) values for the multiplying factorto applyto the base curvefor the assessment ofcontinuous vibration, as shownin Table 3.1. Table 3.1 Multiplying factors to applyto the basecurve Reaction level A*

Environment Offices Residential —day Residential — night

Reaction level B

4 2 to 4

8

1.4

3

4 to 8

*See text forexplanation of'reaction level'

The valuesin column A are postulatedas 'magnitudesbelow whichthe probability of adversecommentis low',andit is postulated that the valuesin columnB 'may result in adversecomment'.A note is addedto the effect that tolerancein residential accommodation is strongly influenced by 'social and culturalfactors, psychological attitudesand the expected degreeof intrusion'. It will be seen that the levelsB and A for offices correspond roughlyto the Canadian recommendation (Figure 3.2),andto one-halfthat level, respectively. However,there is a strong implication that the term 'continuous vibration'is to be interpreted rigorously inBS 6472. These values are thus reasonably applicable only to very heavily trafficked floors with walkers continually present. In suchcases occasional peaks due to concurrentexcitation by more thanone person can probablybe tradedoff againstthe numberofpeoplenot movingregularly or at resonant-pace frequency. BS6472 offers the suggestion that intermittent vibration can be equated to an equivalent continuous level by the root-mean-quad, i.e.: aeq =

T

114

(J a4(t)&)

where a(t) is the value ofacceleration at time t.

The root-mean-quad of a sinusoidal vibration modulating as a personwalks across a floor taking six seconds, repeatedonceper minute, is aboutone-third of thepeak amplitude. As thisroot-mean-quad is used in substitution for the root-mean-square value of continuous timesthe peak amplitude, a floor subject to a person oscillation, whichwouldbe 1 at the resonant walking frequency once per minute could reasonably be permitted to show of twice the peak response peak valueacceptable for continuous oscillation. BS6472 notes that there maybe locations whereit is necessary to restrictvibrationsto the level ofthe base curve (factor 1). 'Some hospital operating theatres' and'some precision laboratories'are put forwardas examples.

9

4. GENERAL CONSIDERATIONS 4.1

Structural and Floor Configurations

The followingdiscussion of steelflooringconfigurations is presented to indicate the terminology used in discussion of floor vibrations andthe approximate parameter ranges; it is notintended to constitute guidance on the selection of the parameters. The essentialobjectiveofflooringis to providea flat load-carrying surface. The floor slab constructionis generally either steel-concrete composite, timber or concrete, and usually carries some form of fmishing or furnishing(carpeting and underlays, hardwood surfacing or similar, and, in the case of concrete slabs,a screed). There is little evidence that finishes have mucheffect on vibration problems, except through the resultingincrease ofmass. There is possibly a marginal increase in damping and amarginalcushioning of impulsive loads by appropriate fmishes, but a finish softenoughto haveamarkedcushioning action will be too soft to havemuchstructural dampingaction. However, the acoustic andwalking comfort factors ofvariousfmishesare likely to interactin the expressedopinion ofusers relatingto the vibration environment as discussedin Section 3.1.

Timberfloors are certainlysusceptible to vibration problems, whichhavebeen studiedin both Canada8 andSwederi3. It will be shownthat highermass is generally favourable, andin thisrespecttimber floorsare inherently more at risk thanconcretefloors. Nevertheless, in view ofthe current balance ofthe marketin the U.K., attentionwill be focusedin thisGuide on concretefloors, but with emphasis on recent designtrends leadingto a reduction ofthe mass perunit area. In particular, thereis increasinguse of permanent steel formwork (profiled decking ofvarious configurations) and oflightweight concrete, often in conjunction with each other. The densityoflightweightconcrete commonlyadopted in the U.K. is around 1800kg/m3; lower values are not uncommon in North America. A composite slab comprising a 70 mm continuous thickness oflightweight concrete on 60 mm steeldecking may thus have a massofabout 220 kg/m2,excludingfinishes. It may be notedherethat references to floor thicknessesin the U.K. generally referto the total slab depth;a 'smeared' thickness equal to (massof concreteper unitarea)/(concrete density) is often used in NorthAmerican literature, including designguides. Such a slab is typicallysupported on floor beams (commonly called 'joists' in NorthAmerica) at about 3 m spacing. The short-term modulusof elasticity shouldbe used for all dynamic calculations, and currentspecifications and design guidestend to presentratherconservative (low) values, bearing in mindthe influence of the age of the concreteandthe area participating in the criticalcircumstances. For normal densityconcrete the dynamic modulus ofelasticity can be taken as 38 kN/mm2,and for lightweightconcrete at around 1800kg/rn3 the dynamic moduluscan be taken as 22 kN/mm. A stiffnessparameter ofthe form El1/L4 canbe considered as an aid to the appreciation of the importance of slab stiffness, in which El1 is the flexural rigidity per unit width.For the applicationofthe design guidance in Section 7, the rigiditymay be computedfrom a smearedthickness ofconcretewith decking as appropriate(see designexampleNo 1 in AppendixB). The actual stiffnessunder distributed loadwouldbe obtained by multiplying the stiffnessparameter by a coefficientdepending on support conditions and load distribution. Considering the spanbetweenadjacentfloor beams,so that the effective spanL,, is set equal to the beam spacing b, thisparameter is commonly in the range 30—100 kN/m3. On the other hand, considering the ability of the slab to support load over the full bay width,Le = W, thisparameter very rarelyexceeds 1 kN/m3and for wide bays continuous over (say) 8 floor beams it willbe less than 0.01 kN/m3. The corresponding stiffnessparameterEI/bL4 for the floor beams is typicallyin the range 1—10 kN/m3. The relative stiffness of slab and floor beamsindicated by theseparameters has the effect that under aglobaldistributedloading the slab deflection betweenbeamsis relatively small. The slab is also sufficient to give significant resistanceto differential deflection of the floor beams,althoughclearly not thereby causinga majordeparture from the basic 10

conceptthat the dominant load pathis via the floor beamsas a 'one-way' span. The net resultin terms ofdynamic actionis that the floor behavesbroadlyas a strongly orthotropic plate (see Section 4.2) and a strip containing one or two floor beams can be considered as the dominant structural unit whenconsidering walkingexcitation. Precast'Omnia type' planks, 50—65 mm in thickness with an insituconcretetoppingand supplementaiy continuity reinforcement, will behave in a similarmanner to a metal decking compositefloor system considered above. However, greater caution must be exercised when assessing the continuity and stiffening effectsof other forms ofprecast floor construction. Where hollow-cored precastunits are requiredto mobilise the compositeactionofthe supporting beams,then the ends ofthe units should be 'notched' andsupplementary tyingreinforcement used in conjunction with an insitu concrete toppingshould be provided. The implementation ofthese measures will, in addition, have a stiffening effect on the floor slab such that the floor systemwill tendto act as an orthotropic plate. Conversely, if 'dry construction' precastflooringis used, withoutsuch measures being implemented, then the supporting beams shouldnotbe considered to act compositely with the slab nor should the slabsbe assumed to assistin reducingany differential deflection betweenbeams or in distributing any local effects. This form of construction therefore,throughIck of stiffness, contributes only by virtueofits mass to the vibration characteristics of the floor as a whole. For very long spans, or where very high standards are sought, the floor systemmay comprisebeamsof comparable stiffness in the two orthogonal directions,constitutingan effective'two-way' span,andthus anearly isotropic dynamic system. Subject to the above limitation on deflectionof the slab betweenbeams, this mobilisesthe whole floor in resisting dynamic excitation, and is thus a very favourable configuration. The floor beamsthemselves willvery often be supported by main beams,which form part ofthe principalstructural framing of thebuilding.The resultingadditional deflection undera globaldistributed loading may be comparable to the floor beamdeflection betweenmain beams. It shouldbe notedthatthe deflection and stress levels tolerable in dynamic responseare low,typical stress amplitudes beingless than 1% of the static designstress, so that conventional design provisions for simple supports willnot generally in practice act as such in dynamic situations. Largefloor areasmay thus act as if structurally continuous. The greatereffective structural continuity, underdynamic loading, has the effect that column stiffness commonly contributes significant endrestraint, even wherethe beam connections are ofa form that would normally be regardedas permitting rotation. Column stiffnessis particularly likelyto be significant in high-risebuildings. An adequateanalysis can commonly be achieved by the 'substitute-frame' procedure. Cantilever forms ofconstruction are relatively uncommon. Although the methods presented in Section 5 for evaluating naturalfrequencies are broadly applicable to cantileverconstruction, thisform gives a rather ineffective mobilisation ofmass if dynamic excitationis appliednear the free end, and the evaluationofresponsepresented in Section6 may be non-conservative. Specialist advice should be taken if a reliable estimateis required.

4.2 Introduction to Dynamics The classictext-bookmodelofa dynamic system,shownin Figure 4.1, is characterised

by a mass, a springstiffness,and a damper. For mathematical convenience, the damperis usuallyimaginedto develop a forceopposing the direction ofmovement in proportionto the velocity. Exceptin very rare cases wheresome identifiable damperhas been fitted to tackle a specific oscillation problem,real floors do not incorporate suchelements, but nevertheless therewill be some ways in which energy is dissipated in the event of oscillation. This is usuallyby frictionwhichcommonlydepends heavily on non-structural components suchas partitions. It also depends on structural behaviour differingfrom the designer'smodel, suchas nominally non-moment-resisting connections that actually develop considerable frictionalresistance. Human occupants also add damping, although a high densityof occupation would be necessary to have any substantial effect on a floor 11

/ Stiffness

///// I

Viscous damper

k

[...] Mass m

Figure 4.1 Simpledynamicmodel

with a concreteor composite slab; this effect is most noticeable wherehigh occupation densityis combinedwith low mass, as in a schoolroom with a timberfloor. Dampingis thus generally recognisable only as a globalproperty, most directlymeasured andexpressedby the 'logarithmicdecrement'ofthe decay ofthe free oscillation after excitationhas ceased. For the moderateor smalllevelsof dampinginherent in engineeringstructures, a logarithmic decrementof (say)0.2 means that the amplitude falls by 20% in each successive cycle.An alternative measure, especially popularin North America, is the 'fractionof criticaldamping' or 'critical dampingratio', which is 1/2 times the logarithmic decrement. Thesequantities are non-dimensional, and care is necessary to avoid confusionwhenreadingdesignguidesor test reports. The frequency of freeoscillation ofthe system shown in Figure4.1 depends on the stiffnessin comparison with the massaccordingto the following equation:

iF f=;;

wherefis the natural frequency (in Hz, i.e. Hertz= cycles/second). It is usuallyconvenient to workin kN and t (tonne) units; in this case the stiffnessk would be expressedin kN/m and the mass in tonnes. Damping has very littleeffect on the natural frequency, or vice-versa. It can be seen that the static deflection caused by the weight ofthe mass m (presumed to act in the appropriatedirection, in line with the spring) would be y, = mg/k,and thus the frequency equation can also be expressed in the form:

-1[-21ry

The self-weight deflection is a quantity whichthe engineercan generally characterise quitecloselywithoutthe needfor detailed calculation, andwhichwill followa consistent pattern as a functionof span for any given structural form. This equation thus offers a usefulgeneral approach to evaluating frequencies, and showsthat conventional static design procedures, which include a limitony,, actually constrain very strongly the value that will resultfor natural frequency. For 'multi-degree-of-freedom' systems with several masses elastically interconnected, andespeciallythe continuously distributed mass system suchas the beam shown in Figure 4.2, there will be a seriesof naturalfrequencies, eachassociatedwith its own modeshape. The various modesare dynamically independent (orthogonal, or 'normal' modes) so that responsecan be synthesised by adding modal solutions computed independently.

The lowest frequency mode is thefundamental.This modehas the simplest shape,and its frequencywillstill be strongly constrained as above, and there is a procedure (Rayleigh's energymethod)for estimating an appropriateweighted average ofthe self-weight 12

I

1

A

/

—

/ / /

3

2

1

— ..—-—

-.-..'

"

Modeshapes

Figure 4.2 Beammodeshapes

deflection.For many beamand plate problems, y, in the above equation should be taken as about 3/4 of the maximum value ofthe self-weight deflection. For continuous beams greater care is requiredin this approach, which is discussedfurtherbelow. The highermodes,whichmay be referred to as harmonics (although their frequencies are notin generalexact integermultiples ofthe fundamental frequency), have shapes of increasingcomplexity. For beams,the second modefrequencyis commonly at least three times the fundamental, depending on the support conditions, mass and stiffness distribution and (where applicable) spanratios. For the simply supported uniformbeam (Figure 4.2) the second frequency is four times the fundamental. A useful insight intothe behaviourof some floorsis givenby the behaviourof an orthotropic plate, shownin Figure4.3. The fundamental mode shaperesemblesthe corresponding beammode shapein both directions. This principle applies also to the highermodes,but ifthe stiffness is highly orthotropic, the weak direction deformationhas relatively littleeffect on the frequency, and a basicfamily ofmodesretainingthe fundamentalshapein the strongdirection can occurat ratherclosefrequencies. Low bending stiffness

High bending

stiffness

Orthotropic plate simply supported on edges

Figure 4.3 Orthotropic plate modeshapes

Forcontinuous beams the fundamental frequency is clearly associatedwith a shapeofthe form shownin Figure 4.4. The inertial loads act in the sense shownandenhance the deflections, whereasin the static designprocessthe self-weight effectson adjacentspans combine to reducethe corresponding stressesand deflections. Thus,if designed to the

samestatic criteria, continuous construction with fairly closely uniformspans may have a significantly lower fundamental frequency than a simple structure. Forthe self-weight deflectionapproach,effectiveself-weight loads should be applied in an upwards direction

in alternate spans. For each mode, it is possibleto establishan effective mass and stiffness, whichcan be used in broadlythe same way as the massand stiffness of the simple systemofFigure4.1. Theseare referred to as the modalgeneralisedvalues.The modalgeneralisedmass for each modeof a simple beam is one-halfof the actual mass. For platesandforcontinuous beams the fraction is smaller, but the effectivemass ofa continuous beam system may still be largerthan the corresponding simple structure, because the factoris applicableto 13

Inertialload

— —

Fundamental modeshape

Figure 4.4 Continuous beam fundamental modeshape

the total mass in motion, i.e. all spans. A word ofcautionis necessary: the values referred to here presumethat the mode shapefunctions are each defmed to givea maximum value of unity, but some authors and some computer programs adoptothercriteriaby which to scale the shape functions.

The responseto various sources of vibration is discussed in detail later (Section 6). In this introduction to dynamics it is sufficient to note the analytical solutions to two classes of excitation: impulsiveloads, and harmonic (sinusoidal variation with time) loads. In both cases the motionis approximately a sinusoidal functionoftime, at the free vibration frequency and the loading frequency respectively. The amplitude is then the peak value in the currentcycle, and the responseenvelope is the smoothcurve indicated by the peaks, as shown in Figure 4.5. Note that the amplitudeis 'mean-to-peak',i.e. closelyequalto one-halfofthe peak-to-peak value.Amplitudes may be quoted for displacement, velocity or acceleration; displacement may be impliedifno otherindication is given. Changewith time ofthe responseenvelope ordinate is referred to as modulation. Progressive reduction of the envelope ordinate (usually by damping) is referred to as attenuation. a) C

0 0. Ca

a) .—

Response envelope

Response

Time

Figure 4.5 Definition ofresponse envelope

impulse is defmedas a changeofmomentum; it is usually implied that this is producedby a largeforce of short duration. For a single-degree-of-freedom system (Figure 4.1) the solutionis very simplefor an impulse (valueJ, say, with units Ns consistentwith expressingmass in kg, or kNs foruse with tonne) of shortdurationby comparison with the natural periodof vibration. Withinthe duration of the impulse the mass acquires velocityJim. Subsequently it is in free vibration and, ignoring the attenuation due to damping,the displacement y can be written in terms ofthe initial amplitude9:

y = ysin(2irft) By differentiating the abovedisplacement equation the maximum velocity is obtained as

2rf9 and equating this toJim yields: =

J 2irfm

14

Precisecalculations areneitherjustifiednorrequiredfor floor vibrations, and this impulse solutionis adequatefor force pulses ofdurationup to at least one-thirdof the natural periodof vibration, and thus for the impulsive components ofwalkingforces and for heel-dropexcitationfor naturalfrequencies up to about 10 Hz. Unfortunately the responseofa practical distributed-mass floor system is not so simple. Potentially, all modesare excited and, as the effective masses (modal generalisedmasses) ofthe variousmodes are of similarorder,the solutionby summation of the modal responses will converge rather slowly. Response in higher modeswill make a major contribution to the accelerations of the floor but the adequacy ofthe impulsive modelof the excitation becomes questionable for such frequencies. Modal analysisis not often recommended for impulsive actions, for which a stress-wave solution would in principle be preferable, but there is no simple answerfor this structural form.

As notedearlier,the simple floor whichapproximates dynamically to an orthotropic plate willhave a family of modes as illustrated in Figure4.3. Theresponsein each modemay be predictedfrom the simpleimpulse responsesolution givenabove,and the total

obtainedby summation. Several membersofthis family maystart with similar amplitudes,and theywill start in phase.However, theywill rapidly get out ofphase according to the frequency differences, andthe result may be a rapid attenuation of the responseenvelope, perceptually equivalent to high damping. The solutions for harmonic excitationare perhaps morefamiliar. The most important case is resonance,whenthe frequency of the load(or of a periodic component in the load) coincides with anatural frequency of the structure. In thiscase the responsebuilds up over successive cycles. Thesteady-statedisplacement amplitude for long-continued excitationis givenby:

5=

P

x (magnificationfactor)

where .P = amplitude of resonant Fourier component of force k = stiffness. The magnification factoris: 0.5 amplitude — — _______________ — — static deflection damping logarithmic criticaldamping decrement ratio by same force

This magnification maytypicallybe fifteen-fold. A magnification often-fold(or twothirds ofthe steady-state value,if smaller) wouldbe reached within five cyclesof excitation.

It wouldnormally be quiteunacceptable for a floor to have a fundamental frequency

within the range of walking- or running-pace frequency. However,as noted in Section 2, walking contains significant secondand third Fouriercomponents whichmay coincide with the naturalfrequency and cause resonantresponse. Response as the sum of several modes createsless ofa problemwith continuedexcitation, becauseexact resonance will clearlyonly occur in one mode. Repeated impulsive effectsat the naturalfrequency or at integersub-multiples ofthat frequency (one-half, one-third, etc.) can also causeresonant build-upofresponse.

15

5. EVALUATION OF NATURAL FREQUENCY 5.1 Component and System Frequencies It has beennoted above that a floor usuallycomprises three identifiable elastic

components: a concrete or composite slab, floor beams,and mainbeams.These components are basically connected in series,and for the evaluation ofstatic deflections (for example) it is appropriate to considereach component separately and estimatethe total deflection as the sum ofthe component deflections. A similardivision maywell be usefulfor dynamic analyses, but with greatercaution because the interactions between component deformations are commonly more subtle. Once the components are connected to form a floor structureit will generally no longer be possible to identify specific component frequencies. Floor frequencies are a propertyof the assembled structure, and in principleeach modeinvolves motion of all parts of the system. However, idealised component natural frequencies can be defined: • the frequency of oscillation ofthe slab, presuming no deflection of the floor beams; • the frequencyof the floor beams,presuming no deflection of the mainbeamsand that a mass associated with a strip ofslab ofwidth equalto the floor beam spacing moves with each beam; the frequency ofthe mainbeams,presuming that the motion ofthe floor corresponds to the deflectionof the mainbeamsonly. These component frequencies can be used for an approximate evaluation of the fundamental frequencyof the total floor systemby Dunkerly'smethod. Denoting the componentfrequencies byf, f2, andf3 (Hz)respectively, the fundamental system frequencyf0 is obtained from:

•

f 1

1 1 1 =— +— +—

f12

f22

f2

With the exercise of some engineering judgementconcerning support conditions, this procedure can give goodestimates. The support conditions assumed for each component must be compatible with the conceptof 'inertia loading' whichacts in the direction ofthe total system deflection. For example, for a simple floor comprising a slab continuous over a numberof floor beams supported by stiff main beams, thereare perhapstwo possibilities that may sensibly be considered forthe fundamental modeshape.The interaction ofthe floor beams and the slab wouldnormally give a fundamental systemmode as shownin Figure 5.1(a) and the slope of the slab is only small whereit is supported on the floor beams. The slab component frequencyshould thus be basedon fixed-end conditions.

L Main beams

I (a)

/

Floor beams

Deck and floor beam interaction

Mode shapes on Section AA: stiff main beams (b)

k.

Deck alone

Figure 5.1 Possible deckand floor beaminteractions 16

The lowestpossiblefrequency for the slab alone wouldarisewith the shapeshown in Figure 5.1(b), corresponding to no rotational restraint. This wouldnot combine with the floor beam deflection in thisexample; it mightnevertheless correspond to the lowest naturalfrequencyof the floor, but this is very rare in practice. Similarconsiderations applyto the interaction betweenfloor beamsandmainbeams,and in thiscase it maybe necessaryto sketch differentpossibilities. The valid solutionis the combination giving the lowestvalue off0,givencompatible assumptions for support conditions for each combination.

It can be seen from the form of Dunkerly'sequation that the result is notespecially sensitiveto the interactions. Any component frequency that is morethan twicethe lowest

component has little effect. Bearing in mindthe comments on relativestiffnessin Section 4.1, togetherwith the relationbetweenstiffness andfrequency, it can be seen that the compositeslab componentfrequency generally has little influence on the floor fundamental frequency. The component frequency for the floor beam is closely constrained (as a functionof span) in conventional designs by the application ofconventional limitson deflection or on span/depth ratios incombination with normaldesignstresses. However, these limits are basedprincipallyon live loading, whereas the governing factorfornatural frequency is the stiffnessin relationto mass or self-weight. It is generally appropriate to assess floor dynamics on the assumption that only a small fraction,say 10%, of the specified (characteristic) live load (including10% for partitions where allowed) will be operative in addition to the mass of the slab, ceiling,servicesand any raised floor. Thereis alsoa consistent trend for a high specified live loading to result in relatively high natural frequency becausebeam stiffness increases by a largeramount than the corresponding increase in oscillating mass.

Sincethe main beamlayout and structural form variesgreatlyfrom building to building, the importance ofthe main beam component frequency is variable, rangingfrom being negligible to being similarto that ofthe floor beams.

5.2 Practical Evaluation Four levelsof approachfor evaluating natural frequencies can usefullybe discussed. In increasingorder of refmement:

a) from a globalestimateofthe self-weight deflection; b) from a combination ofcomponentfrequencies estimated from self-weight deflection or tabulatedfrequency formulae; c) by iterative application ofstatic analysis, using common static analysis software at

the desk-top; d) by use ofdynamic analysis software packages, possibly including finite element modelling ofthe structure. The first three levels are generally limitedto an evaluation ofthe fundamentalfrequency butan extended sequence ofmodes willbe output by the fourthapproach. Where responseamplitudes are to be studied, most practical floors show the mode sequence effect discussedin Section4.2 andillustrated by Figure4.3. To a relatively poor approximation this problemcan be circumvented by usingan empirical estimateof an effectivestrip width offloor (see Section 6). if a better estimateofresponseis required, dynamic analysismust be more detailed,generallycalling for an appropriate established software package.

Detailsof the four approaches are givenbelow: a) The self-weight deflection approach wasintroducedin Section 4.2. Takingthe suggestedweightedaveragevalueofthe deflection = -y0, wherey0 is the maximum value, the equation can be rewritten for convenience:

y

1

flj

18

17

It should be notedthat this is a dimensional form, in whichy0 must be expressed in

mm. The foregoing comments concerning the appropriate loadpatternfor continuous spans andconcerning the assumptions to be madeon support conditions, Young's modulusofconcreteand the contribution of superimposed ('live') loads should be borne in mind. Long-term effects suchas shrinkage and creepdeflections are excluded. This method is likelyto be quite sufficient forthe estimation of the fundamental frequency of a slab and floor beam system on stiffmain beams. b) The component frequency approach is likely to be helpful wherethere is a significant interaction with mainbeamdeflections, especially where this resultsin a fundamental mode shapewith significant deflections in further bays. In suchcases a carefulsketch ofthe mode shapeis recommended. ifthe componentfrequencies are estimated by the self-weight deflection method, this becomeseffectively the same as the globalself-weight deflection approachbut aids aclearerjudgementof the critical mode shape. In some cases the analytical solution for the naturalfrequency ofuniformbeams can be used; thiswill generally be preferable for regularcontinuous beams.The analytical solutionmay be written as:

El

f= CB()

1/2

where m is the mass per unit length (unitsin t/m ifEl is expressedin kNm2, or kg/m ifElis expressedin Nm2)

L is the spanin m (for continuous beams take the longestspan). ValuesofGB for a singlespan with variousend conditions are: pinned/pinned ('simply supported') 1.57 fixed/pinned

2.45

fixedboth ends

3.56

0.56 fixed/free (cantilever) Valuesfor continuous beamsare givenin Figure 5.2. The componentfrequencies are combinedby theformulagivenin Section 5.1, namely: Jo =

1

(1 —+— +— 1

I

2

1,2 f2 J31)1/2 Vi J2 Cu

3.0

2.0

1.0 0.2

0

0.4

0.6

0.8

1.0

Span ratio, ilL

Figure 5.2

Frequency factorCB for continuous beams

c) Wherethe layout is insufficiently regularto permitidealisation as uniformbeam components actingin series,and/or a convincing pictureof the fundamental mode shape cannot be obtained by simplejudgement,the fundamental modeshape can be

foundby successive approximation usingdesk-topstatic analysis procedures. The fundamental frequency can then be obtained with excellent accuracy by a subsequent

18

summation or numerical integration stage that is amenable to either 'spread sheet' computation or hand calculation. Theobjectiveis to discover a distribution of loading, q (say), whichproduces deflections, y (say), suchthat the productym is in the same proportion to the load q at all points.The loadsq and the masses m can be regardedas continuous variables or functions ofco-ordinates defmingpositionon the floor, or they can be discretisedas a set ofpoint loads andcorresponding 'lumped' masses. Ifattentionis focused on one bayof the floor, comprising a numberoffloor beams,a lumpedmodelwith threepoints (i.e. at the quarter andmid-spanpositions)on eachfloor beam in this bay andan equalnumberon deck pointsmidwaybetweenthe floor beams would usuallybe appropriate. In the adjoining bays a coarsermesh shouldbe acceptable. One pointat mid-span on each floor beamis oftensufficient,but morepoints are desirable if it is anticipated that the mode shapewill show nearly equaldeflections in the variousbays. The mode shape is sketched byjudgementand values,y1 (say), assigned at each point. Loads inproportion to m,y, (where m, is the lumpedmassattributedto load-pointi) can then be estimated. The constantof proportionality is arbitraryat this stage, so the loads can be written as Pq1 whereP is a convenient value (say 1 kN) and q1 are non-dimensional coefficients expressing the variationaccordingto the variation of the targetn1y1. The static deflections resulting from this loading are now computed; these will be a better approximation to the modeshapethan the initial set. In principle, this process can be continuedto convergence at the true shape.In practice,it will be sufficient whenvaluesofy11y0 at all points (wherey0 is the biggestvalue) are changing by less than 0.1 in one cycle of this processto proceed to estimatethe frequency from: —

1

(Pq1y, 27rlEmy

1/2

This willoften only require two or threecyclesof iteration. IfP is expressedin kN, then m1 must be expressed in tonne and y in metre; alternatively, N, kg, m can be used respectively.

d)Allthe establishedcommercial structural analysispackages (e.g. ASAS,NASTRAN,

PAFEC,STRUDL, etc.) includeappropriate dynamic capability, generallywith provision for fmite element modelling, andthese are readilyavailable through computerbureaux. The degreeof refinement in modelling should generallybe somewhat superior to that indicated for the iterative approach, above. The mathematical solution can be obtained to any desiredaccuracy. It should be borne in mind, however, that this willoutstripthe qualityofthe inputdata, including joint and support continuity, stiffness prediction for concreteelements and modelling ofthe excitationprocesses.

19

6. FLOOR RESPONSE 6.1

Low Frequency Floors

a full description of theresponseofthe structure involves is to inevitably dynamic (that say, inertial) effects, becausethe basic process of forces which walking inescapably produces vary througheach pace,as described in Section 2. The likelihood that the floor is strongly orthotropic, and has a basic family of modes sharingthe propertyof a similarmode shapealongthe direction parallel to the floor beams,identifiesa potentially criticalevent whenthe person walks at a steady pace parallelto a floor beam.This is especially so whena small integermultiple (i.e. less than 4, say) ofthe pace frequencycomes within the close band offrequencies of the basic family of modes. The corresponding Fouriercomponent ofthe pace forcewill thengive rise to a resonant, or nearly resonant, response. The effective modal input varies with the location of the walker in proportion to the mode shape function at that location, and thus typicallyincreases as the walker movesonto the span, reaches a maximum whenhe is near mid-span and willthen fall off. With practical structural dampingvalues and the walker advancing (say) 0.8 m per pace, the maximum responsewilloccur a few paces afterpassingmid-span, and of the order of 10 paceswill havebeen applied. Ifthe floor has a reasonant frequency betweenabout4.8 Hz and 7 Hz, i.e. it is susceptible to the third Fouriercomponent ofthe pace, it willbe subjected to some 30 cycles of that component. This wouldbe sufficient to givea response very nearly equalto the steady-state response to a sinusoidal forceofthe given amplitude applied continuously at mid-span. It was notedin Section 4.2 that thedisplacement amplitude, 5 (say),was thengivenby: When any person walks onto any floor,

P1 is the damping, expressedas the criticaldampingratio. Now the response will be nearlysinusoidal, so the acceleration amplitude, a (say), is 4ir2f025;but the stiffness k is 4ir2f02M whereM is the effective modal mass (compare this with the first equation in Section 4.2). Thus: where

a=

PCs —

mWL

2

where P = amplitudeof near-resonant Fouriercomponent of force m = mass per unit area offloor

W= bay width L = floor beam span

and

C, is a factorwhich takesaccountof the ratio ofthe effectivemodalmass,M, to the value mWLand alsoof the interaction of the modesmaking up the basic family.

The bay width, W, is difficultto defineboth concisely and rigorously, but the objective is very simple: to definethe masswhich must be significantly set in motion. The bay thus defined is very commonlylarger than the rectangle marked off by the grid of adjacent columns. The couplingof the motion of the floor beams clearly dependson the relative stiffnessof the slab. For floorsofconventional proportions, couplingwill be effectively interrupted by a floor beamgivinga stiffness exceeding 2.5 timesthat of its neighbour; it shouldbe noted that effectivestiffnesses commonly vary betweenadjacentbeams due to elastic supporton main beams by comparison with directframing intocolumns. For a basicfloor bay which approximates a simply supported orthotropic plate, the generalisedmass is approximately mWL. Ifthe bay is roughly square, andgiven conventional slab and beam stiffnesses, the interaction betweenmodesis weak. Thus C. 4. For wider bays,the frequency differential between the modesshown in Figure 4.3 becomesvery narrow and significant dynamic magnification mayoccur in more thanone mode.C, is thus increased. A restricted parametric study ofthe steady-state response of 20

floors ofconventional proportions has suggestedthat the effective massofwide simply supported orthotropic plates is approximately --mSL, in which S can be computedas a functionofthe relative orthogonal flexural rigidities. This has been generalised for cases where the actualfloor beam stiffnesses need to be modified accordingto their support conditions by substituting their stiffnessas 4ir2f02M (see above), thus making use of the allowance made for support conditions in computingthe natural frequency. The result is set out in Section 7.6. The floor beam support conditions andcontinuity may also directly affectthe effective mass by bringing intoplay morethan one floor beam span. This mayoccur in two ways. Firstly,the floor beammay be continuous suchthat the fundamental mode takes the form shown in Figure4.4. With two equal spans the deflections ofthe two spans are equal in magnitude and the effectivemass is doubled. The response is thus halved.This effect, however, falls off rapidly if the spans are of dissimilarlength; the design procedure given in Section 7.6 postulates a reduction of 0.6 (= 1/1.7) provided the adjoining span is not less than 0.8L Alternatively, the main beammay be sufficiently flexiblethat the fundamental mode has similar deflections in both floor beam spans. A similarincrease in the effectivemass is applicable. This condition generally only applies wherethe column layout has been selected to give long clear spans in both directions. In eithercase, floor beamcontinuity is accounted for in Section 7.6 through an effectivelengthparameter, LCff. It will be recalled from earliersections: that the Fouriercomponents ofpace forcesare identifiable up to the third(frequency up to about 7 Hz) but with diminishing amplitude, roughly in inverse proportion to theirrespective frequencies, and that the reaction(or criterion) curve for acceleration is flat up to about 8 Hz. It follows that the criterion ofreactionto sustained oscillation set up by regularwalking becomesincreasingly onerous in the frequency bands4.8—7 Hz (third-component frequency), and 3—4.8 Hz (second-component frequency). Anatural frequency in the range 4—4.8 Hz is particularly likely to result in perceptible responseto walking. It can be seen thatfora given naturalfrequencythiscriterion in effectleadsto a minimum acceptable participating massof floor. However, it shouldbe noted that an increase of mass must be accompanied by a pro-rata increase of stiffness ifthe same natural frequency is to be maintained. Simplified rules basedon this analysis are given in Section 7.6. The dimensions S and Lth, dependent on the relativeorthogonal stiffness, are given explicitly by incorporating C, =4 into the overallnumerical factor.The base values ofthe exciting force and the acceleration perception criterionhave also been taken intothe numerical factor. It may be notedthat a low frequency floor will also respond to the transient forces due to heel strikes. Indeed,this maybe moreoftenperceivedfor actual floorsthanthe resonant effectdiscussedabove. However, from a designpointof view, resonance defmesa more

• •

onerous event.

6.2 High Frequency Floors The behaviourdescribedaboveis not seen as an appropriate model for floorswhere the naturalfrequency exceeds that of the thirdFourierharmonic of the walking pace.For higher frequencies, an impulsive excitationcan be considered. The simple expressionforthe responseto an impulsive excitation givenin Section 4.2 can likewise readilybe expressedin terms ofacceleration. The effectoflateralcontinuity on this responseis relatively weak, because the criterionis basedon the firstresponsepeak, before lateral dispersion takes substantial effect.It is therefore sensible to base the equation on the mass ofonepanel (i.e. width b, the floorbeam spacing) togetherwith a coefficient, C (say), which is to be determined empirically:

J a= —2irfC, mbL

21

For the simple panel as above,roughlysquarein plan, C, 1.7. Longitudinal(floorbeam) continuity should be beneficialbut this has not been exploredto the stage at which positive recommendations can be made. It is therefore suggested that C1= 1.7 is an appropriatedefaultvaluefor all cases. For the designrules set out in Section 7.5, it has, however, been felt prudentto set an upper limiton the effective width to ensurecoverage of slabs whichare more slenderthan currentnormalpractice. Impulsive excitationhas direct significance for relatively high frequency floors. A regular pace impulse wouldlead to a pace-by-pace response whicheach time would be substantially damped (including any lateral dispersioneffect) duringthe ensuing pace interval, only to be renewedto a similarlevel. This effect is notdependenton the regularity of pacing orthe exact synchronism, but the repeateddecay and renewal would moderate the subjective reactioncomparedwith sustained oscillation at the predicted peak acceleration amplitude. Applying the root-mean-quad procedure(Section3.2) wouldgive the effective acceleration amplitude as between0.6 and 0.75 timesthe maximum. Witha pace impulse of 3—4Ns andC = 1.7, the effectiveacceleration amplitude would thus be: 4 a= —2irf= 200f— (±25%; m/s2 given mbl inkg); mbL mbL 8 This effect is principally significant at frequencies above 8 Hz wherethe acceptable level of acceleration increases in proportion to frequency. This implies that themass mbL shouldbe not less than some limiting value whichis in turnproportional to the multiplying factor, R (say), (or the selected'curve R') as used in BS6472. The permitted acceleration is 0.005R rn/s2 r.m.s.or 0.007R rn/s2 amplitude atf= 8 Hz, and thus the requirement is:

mbL<

200

0.007R

= 30000 R

(kg)

WhereR=7 is acceptable: mbL