The response of structures todynamic crowd loads

B R Ellis BRE Centre for Structural and Geotechnical Engineering

T Ji Manchester Centre for Civil and Construction Engineering, UMIST

Digest 426

The 1996 edition of the British StandardBS 6399-1, ‘Loading for buildings, Codeof practice for dead and imposed loads’,included guidance on dynamic loadsgenerated by synchronized crowdmovement. This Digest providesinformation that explains and supportsthe recommendations in the Standard,and provides a method for calculatingstructural response to rhythmic crowdloads.This edition of Digest 426 supersedesthe version published in 1997. Itincludes improved information on theloads generated by crowds.

dige

st

Dynamic crowd loads are generated by themovement of people. However the largest loadsare produced by synchronized rhythmicmovements which mainly arise from peopledancing or engaging in jumping type movements,usually in response to a musical beat. An exampleof a dance which involved jumping, and whichwas popular in the 1980s, was ‘pogoing’.Although usually associated with lower numbersof people, activities such as aerobics will give riseto similar dynamic effects.

It should be recognised that a crowd of peoplejumping rhythmically can generate large loads andthis may be a concern for both safety andserviceability evaluations. However this is anextreme case and the rhythmic excitation mostfrequently encountered does not involve everyonejumping in perfect synchronization but includespeople dancing and clapping, often with somepeople stationary. This latter situation willproduce smaller loads than if everyone was

jumping and usually a much smaller structuralresponse. For design it is therefore important toidentify an appropriate load scenario.

Two approaches to the design of structures toaccommodate dynamic crowd loads are given inBritish Standards and this Digest. One relies onensuring that the fundamental natural frequency of the structural system is sufficiently high thatresonance generated by the major dynamiccomponents of the load is avoided; the otherprovides a method for calculating structuralresponse to dynamic crowd loads so that structural safety and serviceability limit states can be checked.

This Digest deals primarily with the modellingof dynamic crowd loads and the calculation ofstructural response to these loads. It gives thebackground to the problem and discusses anumber of items which must be considered indesign.

2004 Edition

Background

It is possible that some designers may be unaware of thenature of the potential problems caused by dynamic crowdloads. Therefore this section provides a simple illustration ofthe problem and records related developments in UKStandards.

A simple illustrationFrom experience, everyone is aware that the peak loadproduced by jumping is significantly larger than the loadresulting from standing still. This is demonstrated by theaction of a child who, when trying to crush a cardboard box,will jump on it. Furthermore, if a person keeps jumping theresulting dynamic load will be cyclic; hence in certainsituations it can generate a resonant response of the loadedstructure. There are some human activities which involverepeated jumping, but for crowds the dynamic loads will onlybe significant when the movement is synchronized. Inpractice this usually occurs in conjunction with a strongmusical beat at events like lively pop concerts and aerobicexercise classes.

It is instructive to provide some quantitative values toillustrate the magnitude of the problem. Later in the Digest an equation is given which determines the load produced byone person jumping rhythmically; for one type of jumping,the peak load is 4.7 times the person’s weight. With structureswhich are designed for static loading, there are many safetyfactors built into the design which allow for most dynamicloads. In fact, if a group of people on a structure werejumping in an uncoordinated manner, the peak displacementinduced by the load would not be significantly larger than thatinduced by the static load; it is only when coordination isachieved that the load becomes significant. In an experimentwhich is described later, the response of a floor to 64 peoplejumping on it in time to a musical beat was monitored. Thedynamic displacements were 3.5 times the staticdisplacement for the group. It should be appreciated that thisis a significant load case.

The real problem, however, arises when a coordinatedjumping load induces a resonant response in a structure. If thefrequency of the dynamic load occurs at a natural frequencyof the structure, a resonant situation would arise whereby theresponse would be amplified greatly. For example, if astructure with a damping value of 2% critical is subjected to adynamic load coinciding exactly with the natural frequencyof the structure, then the displacement due to resonancewould be 25 times the static displacement. Hence it can beappreciated that a dynamic load occurring at a naturalfrequency of a structure can induce significant vibrations,and is therefore a situation to be considered. It is perhaps wiseto recognise that structural resonance produced by dynamiccrowd loads will be an unusual event and requires both aspecific form of loading and a structure which is vulnerableto such dynamic loads.

The history of dynamic crowd loads and their inclusionin UK StandardsIt has been mentioned that BS 6399-1:1996 providesguidance on loads arising from synchronized crowdmovement. It is of interest to examine the two versions of theBritish Standard which preceded it. In the 1967 version ofBS CP3, no mention was made of dynamic loads generatedby crowds, so at this stage the problem cannot have beenconsidered to be important. In the 1984 edition of BS 6399Part 1, static design loads were detailed but with the warning‘the values for imposed loads given ... allow for smalldynamic effects ... The loads do not ... allow for dynamicloads due to crowds’. Hence at this stage the problem hadbeen recognised although no guidance was provided on howto deal with it. Unfortunately this resulted in the majority ofengineers ignoring the problem.

Guidance on dynamic crowd loads was included inBS 6399-1:1996 specifically to take account of loadsgenerated by repetitive jumping during dancing. Theguidance considers the loads generated by the activity whichare independent of the structure to which they are applied.The relevant section of the Standard states:

‘Dynamic loads will only be significant when any crowdmovement (dancing, jumping, rhythmic stamping, aerobicsetc) is synchronized. In practice this only occurs inconjunction with a strong musical beat such as occurs atlively pop concerts or aerobics. The dynamic loading is thusrelated to the dance frequency or the beat frequency of themusic and is periodical. Such crowd movement can generateboth horizontal and vertical loads. If the synchronizedmovement excites a natural frequency of the affected part ofthe structure, resonance will occur which can greatly amplifyits response.

‘Where significant dynamic loads are to be expected thestructure should be designed either:

a) to withstand the anticipated dynamic loads; orb) by avoiding significant resonance effects‘To avoid resonance effects the vertical frequency should

be greater than 8.4 Hz and the horizontal frequencies greaterthan 4.0 Hz, the frequencies being evaluated for theappropriate mode of vibration of an empty structure.’

To withstand the anticipated dynamic loads reference ismade to published work [1]. It should be noted that the abovefrequency recommendations relate to safety assessments andnot serviceability. Indeed, at that time there was noauthoritative guidance on acceptable serviceability levels forthis type of vibration.

The Standard, BS 6399-1, gave one example of a loadmodel for everyone in a crowd jumping, but did not excludethe use of other load models. Although everyone jumping ona dance floor may be possible, it was questioned whether thismodel was suitable for grandstands.

In October 2002, BS 6399-1 was amended. While it retainsthe frequency limits for synchronized excitation, it now refers to the jumping loads through reference to the 1997edition of Digest 426 where the use of the load model can beexplained. This provides an opportunity for specialistdocuments to be available for specific types of constructionand use (eg grandstands).

2

The response of grandstands to dynamic crowd loadsbecame the focus of attention towards the end of the 1990s – in particular the response of the cantilever tiers of somegrandstands. Consequently a joint working group of theInstitution of Structural Engineers, the Department forCulture Media and Sport, and the Office of the Deputy PrimeMinister was set up to consider dynamic loading of seatingdecks. As an interim measure the working group suggestedlimits for vertical frequencies of 3.5 Hz for new grandstandsfor normal, non-rhythmic loading but 6 Hz for those wherepop concerts could be held[2]. The frequency limits werebased on the experience of the committee members, but itwas recognised that these limits are not a perfect designsolution and many items affect a grandstand’s performancebesides its natural frequency. Even the distinction betweenstands subject to normal and rhythmic loading is somewhatcontentious, because there is always an opportunity for anormal crowd to start jumping rhythmically; alarmingvibrations have been observed at three major football stadiain England following rhythmic excitation[3]. Theseobservations resulted in corrective measures beingintroduced to the stadia.

The logical development, starting from the simple limitingfrequency, is to define appropriate load models for anindividual and a crowd. These models will then enableresponses to be calculated and specific structuralconfigurations to be evaluated.

General considerations

To define crowd loads, several factors relating to the crowd,crowd activity and the structure need to be considered.

Crowd densityWith many structures which are subject to crowd loads, astatic design load of 5 kN/m2 is used. This equates roughly tosix people per square metre – a relatively dense crowd.However, in many instances the number of people in a givenarea is limited, possibly due to seating or licensingrequirements. Therefore actual static loads appropriate to theactivity should be used in determining dynamic loads.

Crowd sizeCrowd size and the area a crowd occupies are also important.

First, the structure required to accommodate a large groupis likely to consist of many different structural bays and areas.The mode shapes of bays in the structure often show themotion in adjacent bays as being anti-phase; hence loading inone bay may effectively attenuate the response in other bays.The critical situation is often reached by the loading in asingle bay, and may well be found with some dance floors.

Second, as already mentioned, synchronized crowdmovement is usually coordinated by music. For a largecrowd, those furthest from the musical source will hear themusic slightly later than those close to the source. This willinduce a phase difference in response with distance from themusical source, thereby affecting the overall coordination – aphenomenon which has been observed in a grandstand[4].

Frequency rangeThe frequency range for an individual person jumping isapproximately 1.5 to 3.5 Hz, but, for a crowd, the higherfrequency jumping cannot be sustained and an upper limit of2.8 is more realistic. This is considered in greater detail byGinty et al[5] and Littler[6]. With people dancing or exercisingwhich involves repeated jumping, the energy is not onlygenerated at the dance frequency but is produced at wholenumber multiples of the dance frequency. This is because theloading is cyclic but is not a simple sinusoid; consequently ifthe loading function is expressed in Fourier series, the cyclicnature of the activity constrains the Fourier components to beinteger multiples of the dance frequency.

Dynamic crowd effectIf a crowd of people all tried to jump at the same frequency,which may be the case for some dances, their coordinationwill not be perfect. Imperfect coordination can be due eitherto individuals not jumping at exactly the beat frequency,which may be more common at the extremes of the frequencyrange, or to individuals dancing with different styles orenthusiasm, or to phase differences between individuals. Itmeans that the loads produced by a group are not a simplesummation of the ideal loads produced by individuals. In factthere is an attenuation of the peak load with increasing groupsize which is greater for the higher Fourier components. Thisis considered later.

Human–structure interactionOne of the difficulties of modelling structures with crowds ofpeople is how to account for the mass of people since it canaffect the structural characteristics and, consequently, theresponse to any given load. In fact there is a range of differentsituations of which the two extremes are important. First, astationary crowd acts as an additional mass–spring–dampersystem on a structure and should be modelled accordingly[7].Second, for the case where people are moving (eg walkingand jumping) their body mass is not involved in the vibrationof the structure and the human involvement is simply as aload. Therefore for evaluation of structural response todance-type loads, the characteristics of the unloadedstructure should be used in the calculations. For the situationwhere many in the crowd are moving rhythmically but asignificant number are stationary, the stationary people willprovide a significant damping mechanism and a consequentreduction in any resonant response.

3

Loading

Load model for an individual jumpingThe load–time history for continuous jumping can bedescribed by a high contact force for a certain time tp (contactperiod) followed by zero force when the feet leave the floor. Ithas been proposed that the load–time function can beexpressed by a sequence of semi-sinusoidal pulses; thisaligns well with measurements for individuals jumping. The load function in one period for a single person Fs(t) isgiven by:

Fs(t) = { Kp Gs sin (π/tp) 0 ≤ t ≤ tp (Equation 1)0 tp ≤ t ≤ Tp

where:Kp = the impact factor (Fmax/Gs)Fmax = the peak dynamic load (N)Gs = the weight of the jumper (N)tp = the contact duration (s)Tp = the period of the jumping load (s)

The contact period tp can vary from 0 to Tp corresponding todifferent movements and activities. The contact ratio α isdefined as follows:

α = tp

Tp≤ 1.0 (Equation 2)

Therefore different contact ratios characterise differentrhythmic activities, which can be seen in Table 1.

It has been observed experimentally[9] that the mean valueof the time history of a vertical load corresponding tobouncing to music on toes (jouncing) or to rhythmic jumpingis always equal to the weight of the performer. By equatingthe mean value of Equation 1 over one period Tp to the weightGs the following relationship is obtained:

K = π

2α (Equation 3)

Thus the loads can be determined knowing the weight ofthe jumper, the period of the jumping and the contact ratio.Figure 1 shows a normalized load–time history for jumpingwith α = 1/3 and fp = 1/Tp = 2.0 Hz. The normalized load of 1.0corresponds to the static weight of the person. From thefigure it can be appreciated that the dynamic load can besignificantly larger than the static load. Indeed, fromEquation 3 the peak dynamic load can be calculated directlywhich for these parameters is 4.7 times the static load. It canalso be deduced that the higher the jumping, the lower thecontact ratio and the higher the peak dynamic load.

4

Table 1 Typical values of the contact ratio α for various activities[8]

Activity α

Low impact aerobics 2/3

Rhythmic exercises, high impact aerobics 1/2

Normal jumping 1/3

Figure 1 Load–time history for one person jumping

Nor

mal

ized

load

4

3

2

1

0

5

–1

0.0 0.2 0.4 0.6 0.8 1.0

Time (s)

Equations 5 & 6 (n = 6)Equation 1

α = 1/3

f = 2 Hz

Load model for an individual in terms of Fourier seriesFor analysis purposes it is useful to express Equation 1 interms of Fourier series:

Fs (t) = Gs { 1.0 + ∑�

n=1rn sin

2nπTp

t + φn } (Equation 4)

where:rn = Fourier coefficient (or dynamic load factor)

of the nth termn = number of Fourier termsφn = phase lag of the nth term

Equation 4 can represent the vertical load for an individualfor any form of rhythmic movement. In the Equation theterms within the summation sign are the Fourier componentsof the load and occur at multiples of the load frequency.Hence for each Fourier term the average load per cycle iszero. This aligns with the observation upon which Equation 3was derived. Also if the dancing stopped the dynamic loadswould become zero and the resulting force would simply bethe weight of the dancer. Based upon Equation 3, the valuesfor the load factors and phase lags can be determined forjumping using the following equations (Ji and Ellis[1], and Jiand Wang[10]):

π/2 if 2nα = 1rn = { 2cos (nπα)

if 2nα ≠ 1(Equation 5)

1 – (2nα)2

0 if 2nα = 1

tan–11 + cos(2nπα)

– π if sin(2nπα) < 0 sin (2nπα) 1 – (2nα)2

φn = –π

if sin(2nπα) =0 if 2nα ≠ 12

tan–11 + cos(2nπα)

if sin(2nπα) >0 sin(2nπα) 1 – (2nα)2

(Equation 6)

The normalized load–time history, calculated usingEquations 5 and 6, and including the first six Fourier terms, isshown in Figure 1 along with the equivalent load determinedusing Equation 1. Table 2 lists the first six Fouriercoefficients and phase lags for different contact ratios.

Load models for crowds jumping: experimental valuesIn the previous sections the loads produced by individualshave been considered. This can be extended to cover crowds.The load equation for a crowd jumping at a comfortablefrequency becomes:

F (x,y,t) = G (x,y) { 1.0 + ∑�

n=1rn,v sin

2nπTp

t + φn } (Equation 7)

where:F (x,y,t) = distributed force which varies with timeG (x,y) = density and distribution of human loadsrn,v = nth Fourier (load) coefficient induced by v persons

Here it is necessary to consider the spatial distribution of thepeople. However there is also a variation in the values of theFourier coefficients due to the different jumping styles andcoordination of the jumpers. This has been determinedexperimentally and reported in detail in IP 4/02. In theexperiments, tests were undertaken on two floors withdifferent sized groups jumping at selected frequencies. Thegroup sizes were selected to be powers of 2 with 64 being thelargest group that could be accommodated safely. Thejumping was coordinated using a musical beat at selectedfrequencies. For the experiments only the lowest threeFourier coefficients (FCs) were determined. The FCs wereplotted against the number of people in the group on a log/logplot together with regression lines for each FC based on apower relationship. The data are shown in Figure 2 on page 6.

The regression lines are defined by the followingequations which characterise the variation in FCs with thenumber of people v in the group:● first FC r1,v = 1.61 x v – 0.082

● second FC r2,v = 0.94 x v – 0.24

● third FC r3,v = 0.44 x v – 0.31

If only one person is considered (ie v = 1), it can be seenthat the FCs fall between the values given in Table 2 fornormal jumping (α = 1/3) and rhythmic exercises (α = 1/2).

For calculation purposes both the Fourier coefficients andthe phase angles are required. Using the FCs defined aboveand the theoretical phase angles for normal jumping (Table 2,α = 1/3), the load model predicted similar amplitudes ofacceleration and displacement to the average measuredvalues for the various group sizes up to 64 people.

5

Table 2 Fourier coefficients and phase lags for different contact ratios

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

α = 2/3rn

9/79/55

2/159/247

9/3912/63

φn–π/6

–5π/6–π/2

–π/6–5π/6

–π/2

α = 1/2rn π/2

2/3 0 2/15 0 2/35

φn 0 –π/2 0 –π/2 0 –π/2

α = 1/3rn

9/59/7

2/39/55

9/912/15

φn π/6–π/6

–π/2–5π/6

–π/6–π/2

Although these FCs reduce with increasing crowd size, itshould be recognised that these are still significant loads. Todemonstrate this point, it is worth examining the centraldisplacement of the floor measured for 64 people jumping.This is shown in Figure 3.

The floor area was 9 x 6 m; hence 64 people represent adensity of 1.185 people/m2 which is not densely packed.Taking the average weight of the jumpers gives a distributedload of 0.86 kN/m2. The measured displacement (static +dynamic) was actually 3.47 times the static displacement

6

Figure 2 Fourier coefficients determined for all of the tests

Figure 3 Measured displacements for 64 people on the second floor

Level for unloaded floor

3

2

6

5

4

3

2

4

2

0

–2

–4

–6

10 0.0987

6

10 –1.0987

First Fourier coefficientsSecond Fourier coefficientsThird Fourier coefficients

100 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8

0 2 4 6 8

Four

ier c

oeffi

cien

tDi

spla

cem

ent (

mm

)

Number of people

Time (s)

which, when used to factor the static load, gives an equivalentpeak dynamic load of 2.99 kN/m2. The floor was designed for3.5 kN/m2. This load scenario did not include a significantresonant component.

If the FCs for a group of 64 people (v = 64) are used inEquation 7 to define the load, then the calculated dynamicdisplacement–time history for the second floor is as shown inFigure 4.

7

Figure 5 Fourier coefficients against group size: combined model

Figure 4 Calculated displacements for 64 people on the second floor

4

2

0

–2

–4

–6

2

100 2 3 4 5 6 7 8101 2 3 4 5 6 7 8 102 2 3 4 5 6 7 8103 2 3 4 5 6 7 8104

0 2 4 6 8

6

5

4

3

2

10 0.0987

6

10 –1.0987

Four

ier c

oeffi

cien

tDi

spla

cem

ent (

mm

)

Number of people

Time (s)

First Fourier coefficientsSecond Fourier coefficientsThird Fourier coefficients

Load models for crowds jumping: numerical valuesAlthough the experiments provide some useful results theyare limited to a relatively small group of people; hence theexperiments were modelled numerically, with the objectiveof replicating the variation in loads observed in theexperiments. For an individual jumping in response to anaudible prompt at a given frequency, the following variableswere considered:● the contact ratio α and therefore the height of jumping;● the jumping frequency f which may not be perfectly

aligned with the prompt;● the phase difference between the prompt and the

individual jumping.

The load model for an individual jumping (Equation 4) wasused as the basis for the numerical modelling with theaforementioned variables introduced into this model in orderto calculate a load–time history. The distribution of variableswas derived from the experimental measurements. Groups ofpeople were represented by the combination of theappropriate number of individual load–time histories.

The results which were obtained, and which correlate wellwith the experiments described in the previous section, wereused to examine the effects of larger group sizes; the variationin loads with group size is seen in Figure 5 on page 7.

With the results for up to 64 people based upon sixcalculations for each group size, it was seen that the spread incalculated FCs became less for the larger groups. This wasprobably due to the statistical variations being smoothed by alarger number of samples. Furthermore, for the final FCsthere was no longer a reduction in FC with crowd size,suggesting that the power relationships derived from theexperimental data are only appropriate for the smallergroups. The values for the first three FCs for groups of8,192 people were 1.40, 0.47 and 0.072. Further evaluationsare given by Ellis and Ji[11].

It should be remembered that these calculations are basedupon experiments with students undertaking the jumping. Itis likely that some groups of people (eg professional dancers)will be better coordinated than the students (and thereforeproduce higher peak loads), whereas in other groups thecoordination may be far worse.

Vertical loads for synchronized movement not involvingjumpingThe previous sections have considered rhythmic jumpingwhich, it is appreciated, can generate high loads. However, inmany situations involving rhythmic excitation (eg popconcerts), the situation with everyone jumping does not arise.Usually some people will be jumping or dancing and othersstanding or sitting. This common scenario leads to lowerpeak loads than those produced by everyone jumping and asignificantly smaller structural response. One example willbe given to illustrate this.

In the 1990s, BRE monitored a number of events incantilever grandstands during concerts. Analysis of recordeddata concentrated on the largest responses which were whenapproximately 90% of the audience was standing, dancingand moving in response to music but not jumping. Similarmethods to those developed for the jumping tests were usedto determine the loads, although this required severalassumptions since the loading on the grandstands was notartificially controlled[12]. Based upon the assumption that theFCs varied with the number of people in a similar fashion tothe jumping tests, the first three FCs of the loads were 0.42,0.087 and 0.017. It can be appreciated that these are muchsmaller than those for jumping (ie 1.61, 0.94 and 0.44).

There is one further important factor. As everyone was notmoving the damping of the occupied structure is increasedsignificantly from that of the bare structure due tohuman–structure interaction. A damping value of 16% wasdetermined from the analysis of the measurements whichwould have a significant effect upon any resonant response.

Therefore it should be recognised that it is important todefine the appropriate loading scenarios for design. Astructure which may perform adequately during concertsmay appear to be unacceptable when subjected to jumpingloads.

8

Structural response

Initial considerationsGiven the number of people involved and their spatialdistribution, and the frequency and type of the coordinatedmovement, the overall load can be determined. Thederivation of the response of a simply-supported floor todance-type loads is given by Ji and Ellis[1] and extended tocover other structures. This provides formulae for calculatinghow displacement and acceleration vary with time. Thesecalculations include all the modes of vibration of the systemand all the Fourier components of the load. In this Digest thederivation is not given and only the relevant equations arequoted.

Alternatively the loads could be used in a finite elementprogramme to determine the response of a structure.

Selection of the appropriate mode of vibrationFor many structures it will be the fundamental mode ofvibration which will be important and this term will be usedin the rest of this Digest. However, for other structures it maybe another mode of vibration which is critical; this is termedthe principal mode for that location. For example, if acantilevered tier of a grandstand is being examined, it wouldbe the vertical mode of vibration of the cantilever, or part ofthe cantilever, that should be considered and not necessarilythe fundamental mode of the whole grandstand.Consequently it is the appropriate mode or modes ofvibration for the analysis that must be determined. Forsimplicity the following sections consider the response in theprincipal mode which, for a single bay floor, will be thefundamental mode.

Considering a symmetric floor in a single bay under asymmetric load, there will be no vibration of the anti-symmetric modes. Therefore, the first higher mode involvedin the vibration is the second symmetric mode which has afrequency considerably higher than the fundamentalfrequency. Also the modal load for the second symmetricmode will be significantly lower which effectively means thatthe response will be dominated by the contributions of thefundamental mode. Consequently, only the response from thefundamental mode needs to be considered.

For a more complicated structure it may be necessary toinclude several of the lower frequency modes to determinedisplacement, although for acceleration it may be sufficientto consider just one mode excited at resonance.

Number of Fourier components to considerFor calculation purposes, it is not necessary to use all theterms in the Fourier series to describe the load. The number ofFourier terms to be taken into account depends on theircontribution to the response, including any possibleresonance. It should be noted that resonance can occur whenthe natural frequency of the structure fs is equal to the dancefrequency f or to integer multiples of the dance frequency.Basically, for serviceability assessments, the first I Fourierterms should be included in the determination of accelerationwhere I is defined as the first integer bigger than fs /f. Thisensures that any resonant excitation of the fundamental modewill be considered. As resonant excitation may produce thelargest response, this is one situation that should beevaluated. For safety assessments, where evaluation ofdisplacement and stresses is appropriate, it is recommendedthat the first three harmonic terms should be considered.

General solutionThe solutions provided by Ji and Ellis[1] can be applied tostructures with different boundary conditions. The difficultyencountered is in choosing suitable displacement functions.For a single bay floor with symmetric boundary conditions,under a uniformly distributed load, no anti-symmetric modesare involved in the vibration. Therefore, only the fundamentalmode needs to be considered, and the shape of this mode isrelatively easy to choose with sufficient accuracy for manycommon cases. The response of a structure can beapproximated by the contribution of the fundamental mode:

w(x,y,t) = A(t) Φ(x,y) (Equation 8)

where Φ(x,y) is the dimensionless fundamental mode withunit peak value, and A(t) is the amplitude of vibrationcorresponding to that mode and is a function of time.

The structural factorThe structural factor B relates to the fundamental mode anddepends on the type of structure and boundary conditions. Ifboth the structural mass and the loading are uniformlydistributed in space, the structural factor can, according to thesolution procedure, be defined as follows:

B =∫∫sΦ(x,y) dxdy

(Equation 9)∫∫sΦ2(x,y) dxdy

For a simply supported rectangular plate the mode shapewould be:

Φ(x,y) = sin (πx /Lx) sin (πy /Ly) (Equation 10)

where Lx and Ly are the length and width of the plate. Theresulting value of B would equal 1.62. This deflected shapemay be reasonable for many floors. If the correct mode shapewas used to determine B for a rectangular plate with fullyfixed boundary conditions, the value of B would be 1.72.

Structural responseFor this section it is assumed that the structural mass and

9

loading are uniformly distributed (ie m(x,y) = m and G(x,y) = G)

The structural displacement and acceleration can beexpressed as:

A = BG

(1.0 + D) (Equation 11)m (2πf)2

Ä = BG

Da (Equation 12)m

D and Da are defined as dynamic magnification factors forthe displacement and acceleration;

D = ∑3

n=1Dn Dn =

rn,v sin (2πnft – θn + φn)(Equation 13)

√(1– n2 β2)2 + (2nς β)2

Da = ∑I

n=1D a

n Dan = n2 β2 Dn (Equation 14)

tan–1

2nςβ if 1 – n2β2>0 1 – n2β2

θn = πif 1 – n2β2=0 (Equation 15)

2

tan–1

2nςβ + π if 1 – n2β2<0 1 – n2β2

β = fp

fs

Equations 11 and 12 give the steady state response whichexcludes transient terms; this is because the transientresponse decays quickly in a damped system and generally isof little interest[13]. At resonance, the acceleration inEquation 14 will be dominated by the term that occurs at thenatural frequency of the structure.

Using dynamic measurements in the analysisFor checking the safety and serviceability of existingstructures or improving the accuracy of theoreticalpredictions, it is desirable to perform dynamic tests. This willprovide feedback from the actual structure including accuratevalues of the fundamental frequency, damping, modalstiffness and mode shape. The fundamental frequency ofmost structures is relatively easy and inexpensive tomeasure[14]. On the other hand, it may prove difficult orinconvenient to measure the actual response when a crowd ofpeople is involved.

Equations 11 and 12 can be expressed in the followingform to accommodate the measurements.

A = G ∫∫

sΦ(x,y) dxdy

(1.0 + D) (Equation 16)k*

Ä = G(2πfs)

2 ∫∫sΦ(x,y) dxdy

Da (Equation 17)k*

To use the above equations the measured modal stiffnessk*, natural frequency f, damping ratio ζ and mode shapeF(x,y) are required. Alternatively, Equations 9 and 10 can beused where the floor mass density needs to be estimated. It isalso preferable to use the measured damping value.

If one person with a weight of Gs jumps at the centre of afloor the response is simply:

A = Gs (1.0 + D) (Equation 18)k*

Ä = Gs (2πfs)2

Da (Equation 19)k*

10

Simple measures to avoid safety problems

As dynamic crowd loading is confined to a narrow frequencyrange, and as it is only necessary to consider a limited number of Fourier terms to evaluate displacements, it ispossible to define minimum frequencies for structures whichshould avoid safety problems from dynamic loads. This mayprovide simple criteria which are convenient for the design of many structures; however, it does not mean that allstructures with lower frequencies will have problems, and,for some structures, calculation of the actual response will beappropriate. If the calculations show an unacceptableresponse, then the likely design solution will be to increasethe frequencies towards the given limits, or to limit the crowd size or crowd activities. Solutions which seek toincrease the damping may have some attraction; however,they recognise that resonance will occur and attempt to limitthe resonant response.

The structural frequency above which vertical vibrationshould not pose a safety problem for jumping on floors is8.4 Hz (ie 3 x 2.8 Hz, avoiding resonance from the first threeFourier components of the load). For grandstands whereconcerts may be held, a frequency of 6 Hz[2] has been given asan interim measure (primarily avoiding resonance from thefirst two Fourier components of the load). As mentionedpreviously, these frequencies are for the appropriate mode ofvibration of an empty structure (ie without a crowd)[7].

Serviceability limits

For serviceability assessments related to human perception ofvibrations, it is the acceleration levels which should bedetermined. However, the question of what are acceptablevibration levels must then be considered. A lot of work hasbeen conducted on what vibration levels can be perceived,and it is known that the frequency of the vibration isimportant; therefore the UK Standards, BS 6472 andBS 6841, related to perception of vibrations providefrequency weighting recommendations for motion in variousdirections. The frequency weighting effectively defines afilter that can be applied to the acceleration to compensate forthe finding that human perception of vibration varies withfrequency. In addition, the number of cycles of the vibrationis important when assessing acceptability of such vibrations.Therefore, a vibration level with a duration of one cycle (ieimpulsive), will have a different effect to a vibration levelwhich is continuous. These British Standards thereforerecommend using vibration dose values (VDVs) which takeaccount of these factors.

For guidance on acceptable vibration levels in structuressubject to jumping loads, it seems appropriate to examinedata obtained at such events. A number of experiments havebeen undertaken in Germany[15] on cantilevered grandstands;recommended peak accelerations for low frequency vibrationare given in Table 3. Ellis and Littler[16] explain how theacceleration levels given in the table can be linked to theVDVs suggested in the British Standards, and show that theyprovide a reasonable correlation with the observed audiencereaction and measured response on a number of grandstandsat a range of events. From the Table it can be seen that, forrepetitive jumping, 35% g is a significant limit as above thislevel panic may occur. However, below this level thevibration levels will certainly be felt by anyone remainingstationary on the structure. The authors experiencedvibrations of 15% g at 4 Hz on one structure and this certainlyfelt uncomfortable, even with knowledge of the source of thevibrations.

It is also likely that visible motion of a structure and itemsfixed to it would concern those immediately below thestructure.

11

Table 3 Reaction to various peak acceleration levels on grandstands[15] (for a frequency range <10 Hz)

Vibration level Reaction

<5% g Reasonable limit for passive persons<18% g Disturbing<35% g Unacceptable>35% g Probably causing panic

12

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References

[1] Ji T and Ellis B R. Floor vibration induced by dance type loads – theory. The Structural Engineer (1 February1994), 72 (3) 37–44.

[2] The Institution of Structural Engineers. Dynamic performance requirements for permanent grandstandssubject to crowd action – interim guidance for assessment and design. December 2001.

[3] Rogers D. Two more ‘wobbly’ stands. Construction News, 17 August 2000.

[4] Littler J D. Measured phase shifts in the dynamic response of a large permanent cantilever grandstand tohuman loading. 5th European Conference on Structural Dynamics, Munich, September 2002.

[5] Ginty D, Derwent J M and Ji T. The frequency range and distribution of dance type loads. The StructuralEngineer (20 March 2001), 79 (6) 27–31.

[6] Littler J D. Frequencies of synchronised human loading for jumping and stamping. The Structural Engineer(18 November 2003), 82 (6) 27–35.

[7] Ellis B R and Ji T. Human–structure interaction in vertical vibrations. Procs of the Institution of CivilEngineers. Structures & Buildings (February 1997), 122 (1) 1–9.

[8] Ellis B R and Ji T. Floor vibration induced by dance type loads – verification. The Structural Engineer(1 February 1994), 72 (3) 45–50.

[9] Tuan C Y and Saul W E. Loads due to spectator movements. ASCE Journal of Structural Engineering,(February 1985), 111 (2) 418–434.

[10] Ji T and Wang D. A supplementary condition for calculating periodical vibrations. Journal of Sound andVibration (2001), 241 (5) 920–924.

[11] Ellis B R and Ji T. Loads generated by jumping crowds: numerical modelling. (Accepted for publication inThe Structural Engineer.)

[12] Ellis B R and Littler J D. The response of cantilever grandstands to crowd loads. Part 2: Load estimation.Accepted for publication in the Procs of the Institution of Civil Engineers, Structures & Buildings.

[13] Clough R W and Penzien J. Dynamics of structures. New York, McGraw-Hill, 1975.

[14] Ellis B R. Dynamic testing. Chapter 5, Monitoring building structures (Ed J F A Moore). Glasgow, Blackie &Son Ltd, 1991.

[15] Kasperski M. Actual problems with stand structures due to spectator-induced vibrations. EURODYN’96,Florence, 1996.

[16] Ellis B R and Littler J D. The response of cantilever grandstands to crowd loads. Part 1: Serviceabilityevaluation. Accepted for publication in the Procs of the Institution of Civil Engineers, Structures & Buildings.

BREIP 4/02 Loads generated by jumping crowds: experimental assessment

British Standards InstitutionBS 6399:Part 1:1984. Loading for buildings. Code of practice for dead and imposed loads.BS 6399-1:1996. Loading for buildings. Code of practice for dead and imposed loadsBS 6472:1992 Guide to evaluation of human exposure to vibration in buildings (1 Hz to 80 Hz)BS 6841:1987 Guide to measurement and evaluation of human exposure to whole-body mechanical vibration

and repeated shockCP3:Chapter V:Part 1:1967. Code of basic data for the design of buildings. Loading. Dead and imposed loads

Acknowledgements

The authors would like to acknowledge the support of the Office of the Deputy Prime Minister(for the work undertaken by BRE) and the Engineering and Physical Sciences Research Council(for the work undertaken by University of Manchester Institute of Science and Technology)