Pedestrian excitation of bridges - University of Cambridgeden/papers/Pedestrian excitation of...

Pedestrian excitation of bridges

D E NewlandDepartment of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

Abstract: This paper reviews the evidence on dynamic bridge loading caused by moving pedestrians.The phenomenon of ‘synchronization’ by which people respond naturally to an oscillating bridgewhen this has a frequency close to their natural walking or running frequency is a feature of thisphenomenon. By increasing modal damping, synchronization can be prevented, but how muchdamping is needed in any particular situation?

If some simplifying assumptions about how people walk are made, it is possible to predictanalytically the minimum damping required to ensure that synchronization does not lead to highvibration levels. The main assumption is that the movement of a pedestrian’ s centre of mass has twocomponents. One is its natural movement when the person is walking on a stationary pavement. Theother is caused by movement of the pavement (or bridge) and is in proportion to pavement amplitudebut with a time delay that is arbitrary. When the time delay is a ‘worst case’ , pedestrians act as asource of negative damping.

This theory supports the adoption of a non-dimensional number which measures the susceptibilityof a bridge to pedestrian excitation. Although currently there are not many good bridge responsedata, predictions using this non-dimensional number are compared with the data that are availableand found to be in satisfactory agreement. Both lateral and vertical vibrations are considered.

Keywords: bridge, vibration, pedestrian, synchronization, self-excitation, pedestrian loading,pedestrian excitation, bridge dynamics, Scruton number, pedestrian Scruton number, LondonMillennium bridge

NOTATION

fn natural frequency (Hz)f …t† modal force (alternatively, force per unit

length of pavement, depending on thecontext)

f0…t† force per person exerted on pavementk lateral force per person/amplitude of

pavement lateral velocityK modal stiffness of bridgem modal mass of pedestrians for the relevant

bridge mode (alternatively, mass ofpedestrians per unit length of bridge)

mr non-dimensional mass ratio ˆ abm=Mm0 mass per personM modal mass of bridge (alternatively, mass

of bridge per unit length)Sc Scruton numberS cp pedestrian Scruton numbert time

x …t† natural displacement of a pedestrian’scentre of mass when walking on astationary pavement (alternatively,corresponding modal displacement for allthe pedestrians on a bridge whennormalized by the relevant bridge mode)

X …io† Fourier transform of x …t†y…t† displacement of the pavement

(alternatively, modal displacement of thebridge deck for the relevant mode)

Y …io† Fourier transform of y…t†z…t† total displacement of a pedestrian’s centre

of mass

a ratio of the amplitude of movement of aperson’ s centre of mass to the amplitudeof movement of the pavement

ay…t ¡ D† movement additional to x …t† of a person’scentre of mass caused by pavementmovement (alternatively, relevant modaldisplacement)

b correlation factor for when individualpeople’s natural movement synchronize…b ˆ g assumed)

The M S was received on 2 July 2003 and was accepted after revision forpublication on 9 February 2004.

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C12303 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part C: J. Mechanical Engineering Science

g correlation factor for when individualpeople synchronize with pavementmovement

D time lag between movement of pavementand movement of a person’ s centre ofmass

f phase angle ˆ oDfc critical phase angle for the onset of

instabilityo (angular) frequencyoc critical frequency at which instability � rst

occurson natural frequency of the relevant modeO non-dimensional frequency ratio ˆ o=on

B damping ratioBc critical damping ratio required for

stabilityBeff effective modal damping ratio

1 INTRODUCTION

Since the pedestrian-excited vibration of the LondonMillennium Bridge in June 2000, there has beenconsiderable interest in bridge vibrations caused by themovement of people. Extensive studies by the ArupPartnership concentrated on quantifying the excitationof the Millennium Bridge and devising a way ofincreasing the bridge’s damping. This they achievedwith complete success. By arti� cially adding damping tothe bridge, they were able to extinguish the self-excitation mechanism and so eliminate a vibration thatwas suf� cient to cause people to stop walking and holdon to the handrails and that was severe enough to giveconcern for the safety of less mobile walkers. Since thatexperience, there has been time to examine in moredetail the mechanics of interaction between a pedestrianand a moving pavement and, in particular, the processof synchronization. This process causes people to fallinto step with an oscillation and with each other to setup positive feedback, thereby causing an initially verysmall oscillation to build up. It will be shown below that,subject to necessary simplifying assumptions, a stabilitycriterion can be expressed in terms of a dimensionlessnumber involving the bridge’s damping and the ratio ofpedestrian mass to bridge mass. This is similar to thenon-dimensional Scruton number used to quantify thesusceptibility of a structure to wind-excited oscillations.The collection of more experimental data is needed, butthe analysis here suggests a framework by which resultscan be compared on a logical basis. F rom the datacurrently available, a variety of existing bridges areexamined using this approach, and it is found thatbridges with a low pedestrian Scruton number are those

bridges that are sensitive to pedestrian-excited vibra-tions.

2 BACKGROUND

A report in 1972 quoted by Bachmann and Ammann [1]in their International Association for Bridge andStructural Engineering book described how a new steelfootbridge had experienced strong lateral vibrationduring an opening ceremony with 300–400 people.They explained how the lateral sway of a person’scentre of gravity occurs at half the walking pace. Sincethe footbridge had a lowest lateral mode of about1.1 Hz, the frequency of excitation was very close to themean pacing rate of walking of about 2 Hz. Thus in thiscase ‘an almost resonating vibration occurred. More-over it could be supposed that in this case thepedestrians synchronised their step with the bridgevibration, thereby enhancing the vibration considerably’(reference [2], p. 636). The problem is said to have beensolved by the installation of horizontal tuned vibrationabsorbers.

A later paper by Fujino et al. [3] described observa-tions of pedestrian-induced lateral vibration of a cable-stayed steel box girder bridge of similar size to theMillennium Bridge. It was found that, when a largenumber of people were crossing the bridge (2000 peopleon the bridge), lateral vibration of the bridge deck at0.9 Hz could build up to an amplitude of 10 mm, whilesome of the supporting cables whose natural frequencieswere close to 0.9 Hz vibrated with an amplitude of up to300 mm. By analysing video recordings of pedestrians’head movement, Fujino et al. concluded that lateraldeck movement encourages pedestrians to walk in stepand that synchronization increases the human force andmakes it resonate with the bridge deck. They summar-ized their � ndings as follows: ‘The growth process of thelateral vibration of the girder under the congestedpedestrians can be explained as follows. First a smalllateral motion is induced by the random lateral humanwalking forces, and walking of some pedestrians issynchronised to the girder motion. Then resonant forceacts on the girder, consequently the girder motion isincreased. Walking of more pedestrians are synchro-nised, increasing the lateral girder motion. In this sense,this vibration was a self-excited nature. Of course,because of adaptive nature of human being, the girderamplitude will not go to in� nity and will reach a steadystate.’

Enquiries subsequent to the opening of the LondonMillennium Bridge identi� ed some other interestingexamples of pedestrian-excited bridge vibration [4],including the surprising vibration of the AucklandHarbour Bridge in New Zealand. This is an eight-lanemotorway bridge, with three separate parallel roadways.

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In 1975, one roadway, with two traf� c lanes, was closedto vehicles to allow a political march to pass over thebridge. Contemporary newsreel footage shows the largecrowd walking in step as the roadway built up a largeamplitude lateral vibration at about 0.6 Hz. Thisvibration was serious enough for stewards to go throughthe crowd calling for marchers to break step, when itsubsided naturally. It is interesting that the marchershad not intended to march in step but had naturallyfallen into step with each other, apparently after thebridge began to sway.

3 PEDESTRIAN LOADING DATA

The book by Bachmann and Ammann [1] discussedloading from human motions, distinguishing betweenwalking, running, skipping and dancing. For walkingand running, they pointed out that dynamicpavement load is dominated by the pacing frequency(Table 1).

Published data on dynamic loads are few, butBachmann and Ammann quoted an example for apedestrian walking at 2 Hz when the fundamental

component (at 2 Hz) of vertical dynamic loading is 37per cent of static weight and the fundamental compo-nent (at 1 Hz) of lateral dynamic loading is 4 per cent ofstatic weight. In the vertical case, harmonics are lessthan about 30 per cent of the fundamental in amplitude(a typical load–time history is shown in F ig. 1); in thelateral case there may be a signi� cant third harmonicand an example is quoted in which the third harmonicexceeds the lateral fundamental in amplitude.

These forces are for people walking on stationarypavements, but it was noted by Bachmann andAmmann that ‘pedestrians walking initially with indivi-dual pace on a footbridge will try to adjust their stepsubconsciously to any vibration of the pavement. Thisphenomenon of feedback and synchronisation becomesmore pronounced with larger vibration of the structure’.Also, for vertical vibration, Bachmann and Ammannnoted that displacements of the order of 10–20 mm haveto occur for the phenomenon to be noticeable, althoughthey said that it is more pronounced for lateralvibrations. ‘Presumably, the pedestrian, having noticedthe lateral sway, attempts to re-establish his balance bymoving his body in the opposite direction; the load hethereby exerts on the pavement, however, is directed soas to enhance the structural vibration.’

Table 1 Data on walking and running from Bachmann and Ammann [1]

Pacing frequency Forward speed Stride lengthVert ical fundamentalfrequency

Horizontal fundamentalfrequency

(Hz) (m/s) (m) (Hz) (Hz)

Slow walk 1.7 1.1 0.60 1.7 0.85Normal walk 2.0 1.5 0.75 2.0 1.0Fast walk 2.3 2.2 1.00 2.3 1.15Slow running (jogging) 2.5 3.3 1.30 2.5 1.25Fast running (sprinting) > 3.2 5.5 1.75 > 3.2 > 1.6

Fig. 1 Vertical load–time function from footfall overlap during walking at 1 pace/s. (After reference [1])

PEDESTRIAN EXCITATION OF BRIDGES 479


4 SYNCHRONIZATION

Fujino et al. [3] estimated from video recordings ofcrowd movement that some 20 per cent or more ofpedestrians on their bridge were walking in synchronismwith the bridge’s lateral vibration, which had afrequency of about 0.9 Hz and an amplitude of about10 mm. They computed the amplitude of steady statelateral vibration of this bridge, � rstly, using the value of23 N given by Bachmann and Ammann for theamplitude of lateral force per person and assumingthat the pedestrians walk with random phase and,secondly, using a force per person of 35 N and themeasured result that 20 per cent of them weresynchronized to bridge movement. For the randomphase case, the calculated amplitude is about 1 mmresponse; for the 20 per cent correlated case it is about15 mm (compared with the measured value of 10 mm).

These results were thoroughly investigated followingthe London Millennium Bridge’s problems, with theresults given by F itzpatrick et al. [5] (an amendedversion of this paper was subsequently published asreference [4]). Using moving platforms, data weremeasured on lateral dynamic force and on the prob-ability that a pedestrian would synchronize withpavement lateral vibration. Results obtained by Arup

using a shaking table at Imperial College are shown inF ig. 2 which has two other results added. It can be seenthat the fundamental component of lateral forceincreases with increasing platform amplitude but isinsensitive to pavement lateral frequency. However,walkers were not asked to try to ‘tune’ their stepintentionally to the platform’s motion; instead they wereasked to walk comfortably for the seven or eight pacesrequired to pass over the platform. F igure 2 has threeadded lines which show the ratio of dynamic lateralforce to static weight for a rigid mass when oscillated at0.75Hz (bottom line), 0.85Hz (middle line) and 0.95 Hz(top line) when the amplitude of oscillation increasesfrom 15 mm on the left-hand side to 35 mm on the right-hand side. This would apply if a pedestrian weremodelled as a rigid mass whose centre of mass movedthrough an amplitude of 15 mm on a stationarypavement and increased linearly with increasing pave-ment amplitude to 35 mm when the pavement amplitudebecame 30 mm.

Fujino et al. [3] noticed that a person’s head move-ment is typically twice that of their feet (laterally) at1 Hz and +10 mm pavement movement, and so it is notsurprising that pedestrians do not behave as rigidbodies. However, although they do not act as rigidmasses, the lateral force that a person generates must be

Fig. 2 Measured values of pedestrian lateral dynamic force/static weight as functions of pavement amplitude(after reference [4], F ig. 10) with data by Bachmann and Ammann [1] and Fujino et al. [3] added. Theplatform in these experiments was 7.3 m long and 0.6 m free width with a handrail along one side. Theamplitude of the fundamental component of lateral force is plotted after dividing by the subject’sweight. Arup’s data are for two different frequencies of pavement oscillation: 0.75 and 0.95 Hz. Itappears that subjects walked at a comfortable speed with a walking pace not intentionally ‘tuned’ tothe pavement frequency. The data point from the paper by Fujino et al. shows an estimated forceamplitude from observations of people walking on a bridge with a 1 Hz lateral mode at an amplitudeof about 10 mm. The three added lines (drawn for comparison) are for moving a rigid mass atfrequencies of 0.75 Hz (bottom line), 0.85 Hz (middle line) and 0.95 Hz (top line) for an amplitude of15 mm (on the left) to 35mm (on the right)

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reacted against the inertia of their body, so that the sumof mass6acceleration for all their component partsmust equal the lateral pavement force at all times.Therefore, if an average is calculated for each pedes-trian, the results in F ig. 2 suggest that their centre ofmass must be moving about +15 mm when walkingcomfortably on a stationary pavement, increasinglinearly to about +35 mm when the pavement’s lateralmovement is +30 mm. The effect of frequency ofpavement movement does not seem to have much effect,with measured data for 0.95 Hz suggesting a slightlylower ratio of dynamic to static force than 0.75Hz. Thisis consistent with the natural � exibility of the humanframe. Evidently, if the pavement were oscillating at ahigh frequency, the feet and legs would be expected tomove, but the upper body would not follow so muchand would move relatively less. Movement of the centreof mass of a pedestrian would then be signi� cantlydifferent from movement of the pavement.

To quantify this effect, a non-dimensional amplitudefactor a is de� ned as the ratio of the movement of aperson’s centre of mass to the movement of thepavement. Since, from Fig. 2, it is deduced that achange in pavement amplitude from 0 to 30 mm causes achange in body movement from 15 to 35 mm,a ˆ …35 ¡ 15†=30 ˆ 2=3. These data suggest that a isapproximately 2/3 at both 0.75Hz and 0.95 Hz. Becauseof the complex dynamics of the human frame, it ispossible that the effect of different frequencies in thisrange is small, as these data suggest.

Arup also studied the probability of synchronizationfor people using the walking platform at ImperialCollege and their results are shown in Fig. 3. This isthe estimated probability that people will synchronizetheir footfall to the swaying frequency of the platform.The ‘best-� t’ straight line does not pass through the

origin. It suggests that people synchronize with eachother when there is no pavement motion but that theprobability of synchronization increases as pavementamplitude increases. In calculations at the end of thispaper, it will be assumed that the probability ofsynchronization, to be given by the symbol b, is 0.4for platform amplitudes up to 10 mm.

In addition to these laboratory tests, Arup conducteda series of crowd tests on the Millennium Bridge. Theseconcluded that pedestrian movement was stronglycorrelated with lateral movement of the bridge but notwith vertical movement. This was attributed in part tothe conclusion that pedestrians are ‘less stable laterallythan vertically, which leads to them being more sensitiveto lateral vibration’ (reference [4], p. 26). However, itwas not concluded that vertical synchronization couldnot occur, and vibration control measures were added tothe bridge in the expectation that vertical synchroniza-tion was a possibility.

In the following analysis, the assumptions that will bemade about pedestrian loading are only appropriate forsmall-amplitude pavement movements (less than about10 mm amplitude). For larger amplitudes, people’snatural walking gait is modi� ed as they begin to losetheir balance and have to compensate by altering howthey walk. The staggering movement of pedestrianstrying to walk on a pavement which has large-amplitudelateral vibration (100 mm amplitude) has been studiedby McRobie and Morgenthal [6] using a swingingplatform. Pedestrian movement was followed by amotion capture system devised by Lasenby (see refer-ences [7] and [8]). The way that people walked on aplatform moving with such a large amplitude variedfrom person to person. ‘A common response was tospread the feet further apart and to walk at the samefrequency as the pre-existing oscillations such that feet

Fig. 3 Probability of synchronization estimated by Arup from moving-platform tests for the same twofrequencies of platform lateral oscillation as in F ig. 2, 0.75 Hz and 0.95 Hz. (After reference [4], F ig. 1)



and deck maintained a constant phase relation.’ How-ever, ‘Other walking patterns, some involving crossingof the feet, some involving walking in undulating lineswere also observed’ [6]. They also found that the lateralforces of the feet-apart gait are phase synchronized tothe structure and approach 300 N amplitude per person,which these researchers pointed out is four times theEurocode DLM1 value of 70 N for normal walking.

5 SMALL-AMPLITUDE PEDESTRIAN LOADINGMODEL

The previous considerations lead to the notion that theforce exerted by a walking pedestrian can be modelled(approximately) as two mass6acceleration terms. The� rst arises from the natural displacement of a person’scentre of mass while walking on a stationary pavement,and the second from the additional displacement thatoccurs as a consequence of movement of the pavement.Let x …t† be the natural movement of the centre of masson a stationary pavement and y…t† the movement of thepavement; then, if f0…t† is the force per person of mass mexerted on the pavement,

f0…t† ˆ m �xx …t† ‡ ma �yy…t ¡ D† …1†

where D is a time lag to account for the fact that aperson’ s centre of mass will generally not move in phasewith movement of the pavement and a is a proportion-ality factor relating centre-of-mass movement to pave-ment movement. For lateral forces, the data in F ig. 2suggest, for walking comfortably at about 0.85 Hz (1.7paces/s) that a suitable value for a is 2/3 and that theamplitude of x …t† would be about jx j ˆ 15 mm. Theseare the values used to plot the middle line in F ig. 2.

The same equation (1) applies for modal quantitieswhen there are many pedestrians provided that, at anypoint on the bridge, all of them are moving together.The modal force f …t† now replaces the force per personf0…t†, m becomes the modal mass of pedestrians (i.e. thedistributed pedestrian mass normalized by the square ofthe bridge’s displacement mode function), and x and ybecome modal displacements (both normalized by thebridge’s displacement mode function).

However, it has been found experimentally thatpedestrians do not always synchronize their steps (seeF ig. 3), and only forces that are synchronized (i.e.correlated in time) cause bridge vibration, the uncorre-lated forces cancelling each other out. Therefore wede� ne two correlation coef� cients: b to describe thecorrelation of the natural swaying movements thatpeople make walking on a stationary platform, and g todescribe the correlation of the pedestrian movementsthat depend on platform movement. Introducing these

factors into equation (1) gives

f …t† ˆ mb �xx …t† ‡ mag �yy…t ¡ D† …2†

For small pavement movements (up to 10 mm ampli-tude), there are currently insuf� cient experimental datato determine whether b and g are different and it will beassumed that they are the same (and constant), so that

b ˆ g ˆ constant …3†

The upshot is that it will be assumed that the followingequation applies to describe how the modal bridgeexcitation force arising from pedestrian motion dependson the modal accelerations of the people and the bridge,and the modal mass of pedestrians. It includes the twoempirical factors a, which is the ratio of movement of aperson’s centre of mass to movement of the pavement,and b, which is a correlation factor for when individualpeople’s movements synchronize. Thus, to a � rstapproximation, the bridge loading to be expected fromwalking pedestrians will be modelled by the equation

f …t† ˆ mb �xx …t† ‡ mab �yy…t ¡ D† …4†

The de� nitions of all terms are given in the notationsection and the quantities in equation (4) can beinterpreted either as modal quantities or as localquantities per unit length.

Note that the presence of pedestrians is assumed notto alter the bridge’s modal properties which aresatisfactorily described by small damping theory.Pedestrians act only as a forcing function for the bridgemodes, generating a force de� ned by equation (4). Thiswill now be used to compute bridge response.

6 ANALYSIS OF PEDESTRIAN–BRIDGEINTERACTION

The following calculation explores the interactionbetween pedestrians of effective modal mass m walkingon a bridge with a vibration mode of (modal) mass Mand stiffness K using the above (small-amplitude) forcemodel. The interaction (modal) force which is trans-mitted from the pedestrians to the bridge, and viceversa, is f. This system is shown in F ig. 4 where z…t† is

Fig. 4 Interaction between a bridge mode with modal massM and stiffness K and pedestrians with modal mass m.The (modal) force transmitted between the pavementand pedestrians is f. The large open circle recognizesthat there is a complex interaction between pedestriansand bridge recognized by the time delay D and thecorrection factors a and b in equation (7)

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the (effective) modal displacement of the pedestrians’centre of mass and y…t† measures the modal displace-ment of the bridge’s pavement or walkway.

The equations of motion are

M �yy…t† ‡ K �yy…t† ˆ ¡ f …t† …5†f …t† ˆ m �zz…t† …6†

where, using equation (4),

m �zz…t† ˆ mb �xx …t† ‡ mab �yy…t ¡ D† …7†

so that

M �yy…t† ‡ Ky…t† ‡ mab �yy…t ¡ D† ˆ ¡ mb �xx …t† …8†

where x and y are modal displacements, and a and b arede� ned above. If the modal natural frequency is on ˆ��

K=Mp

and if damping is included with a damping ratioB, then equation (8) may be written

�yy…t† ‡ mM

ab �yy…t ¡ D† ‡ 2Bon _yy…t† ‡ o2ny…t† ˆ ¡ m

Mb �xx …t†

…9†

If the natural movement of a pedestrian (on a stationarypavement), x …t†, is known, then equation (9) can besolved for the modal bridge displacement y…t†). Ofcourse, only small-amplitude movement of the pave-ment is considered so that pedestrians walk unimpededby motion of the bridge, without any pronouncedchange in gait. The time lag D allows for the possibledelay in body following feet. This is likely to be a greaterfactor in lateral vibration than vertical vibration becausethe body can sway slightly laterally and then thepedestrian may easily have a body movement which isout of phase with the motion of their feet. For brevity inthe following analysis, the mass ratio parameter mr isnow de� ned as

mr ˆ abmM

…10†

and a frequency ratio as

O ˆ oon

…11†

where m/M is the ratio of pedestrian mass to bridgemass (either modal or per unit length when theparameters are constant along the bridge), a is the ratioof movement of a person’s centre-of-mass to movementof their feet, b is the proportion of pedestrians whosemovement has synchronised with pavement movementand O is the ratio of frequency of excitation to bridgenatural frequency.

6.1 Frequency analysis

By taking Fourier transforms of both sides of equation(9), and putting

Y …io† ˆ 12p

…?

¡?y…t† exp…¡ iot† dt …12†

and similarly for Z …io†,

Y …io† ¡ o2 ‡ 2Bon io ‡ o2n ¡ mro

2 exp…¡ if†£ ¤

‡ X …io† mr

a¡ o2

¡ ¢ˆ 0 …13†

where the phase angle f is given by

f ˆ oD …14†

6.2 Singular solution for X …ix† ˆ 0

F irstly, it is assumed that x …t† ˆ 0 so that X …io† ˆ 0.This means that, on a stationary pavement, a pedestriancan walk without introducing any dynamic force. Thepedestrian’ s weight glides forwards in the direction ofwalking without any up-and-down or side-to-side move-ment of their centre of mass, and no force is exerted onthe pavement (except static weight). If there is no timedelay so that D ˆ 0 in equation (5), and therefore f ˆ 0in equation (9), the only solution is Y ˆ 0 and novibration occurs. However, if some time delay canoccur, there is the possibility of a non-zero solution forY .

This non-zero solution can be found as follows. Afterseparating the real and imaginary parts in equation (13),this becomes

Y …io†‰…o2n ¡ o2 ¡ mro2 cos f† ‡ io…2Bon ‡ mro sin f†Š

…15†

which has a non-zero solution for Y (actually anindeterminate solution) if the real and imaginary partswithin the square brackets are both zero, so that

o2n ¡ o2 ¡ mro2 cos f ˆ 0 …16†

2Bon ‡ mro sin f ˆ 0 …17†

which, on eliminating f, lead to the following expressionfor the damping ratio B:

4B2 ˆ 2 ‡ O2…m2r ¡ 1† ¡ 1

O2 …18†

For this solution, the phase angle f can be calculatedfrom either equation (16) or equation (17) to give

f ˆ sin¡ 1 ¡ 2BOmr

³ ´…19†



Results from equations (18) and (19) are plotted in F igs5 and 6.

It can be seen that, for small pedestrian-to-bridgemass ratio mr, only a small amount of damping isneeded to ensure stability and there is only a narrowband of frequencies, close to the natural frequency, atwhich self-excitation can occur. However, for highermass ratios, the required damping to maintain stabilityincreases and the range of frequencies over which self-excitation can occur increases greatly. The phase angle

by which pedestrian body movement leads bridge move-ment for this limiting motion is shown in Fig. 6. It isclose to 908 when the frequency ratio is close to unity,i.e. when the excitation frequency of pedestrian loadingis close to the natural frequency of pavement motion.

6.3 Critical damping ratio for stability

F igure 5 shows the minimum damping needed to givestability for any chosen mass ratio mr. Consider, for

Fig. 5 Damping ratio required for stability as a function of frequency ratio for different mass ratios—from0.1 to 1. The unstable region lies below the curve in each case

Fig. 6 Phase angle between pedestrian motion and pavement motion (at the stability limit) plotted againstfrequency ratio for different mass ratios mr. Since f is de� ned as positive when pedestrian motion lagspavement motion, in the graph pedestrian motion leads pavement motion by 908 when the frequencyratio is unity

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example, the curve for mr ˆ 0:5. The system will beprone to self-excite at any frequency in the range 0.8–1.4approximately. In theory, any frequency o is possible(as the only input from the pedestrian is to in� uence thephase angle f by responding to pavement movement atwhatever frequency this occurs). Therefore, unless thedamping ratio exceeds the maximum shown on thiscurve, about 0.27, self-excitation will occur at afrequency ratio close to unity.

By calculating the maxima of the curves drawn inF ig. 5, the critical damping ratio required for stability(whatever the frequency) and the frequency at which thisinstability � rst occurs if it does occur can be found.After differentiating equation (18) with respect to ousing equation (11), with mr constant, maxima are foundto occur at a critical frequency oc where

oc

on

³ ´4

ˆ 11 ¡ m2

r…20†

and the value of the critical damping ratio Bc is

B2c ˆ 1

2 1 ¡��1 ¡ m2

r

q³ ´…21†

When the mass ratio mr is small, the right-hand side ofequation (21) can be expanded by the binomial theorem,to give

2Bc&mr , mr 5 1 …22†

Similarly, from equation (20), the corresponding criticalfrequency oc is

oc

on&1 ‡ m2

r

4…23†

and, from equation (19), the critical phase angle fc is

fc& ¡ p2

‡mr

2…24†

when higher-order terms are neglected.The damping ratio [equations (21) and (22)] is the

damping for which an oscillatory motion can, in theory,occur at (any) constant amplitude. For greater damping,vibration if started, subsides. For lesser damping,vibration builds up spontaneously. These results areplotted in F igs 7 and 8.

The curve in F ig. 5 for mr ˆ 1 becomes asymptotic tothe damping ratio B ˆ 1=

��2

pˆ 0:7071, and curves for

higher mass ratios than unity are monotonicallyincreasing, with no maxima. For such high pedestrianloading ratios, according to this response model, a stablesolution is not possible, whatever the damping ratio.

6.4 Forced vibration solution

Now the forced vibration solution when X …io†=0 isconsidered. Suppose that the natural movement of apedestrian’ s centre of mass, when walking steadily on astationary pavement, is

x…t† ˆ X exp…iot† …25†

and that the resulting pavement response is

y…t† ˆ Y exp…iot† …26†

On substituting equations (25) and (26) into equation(9), or directly from equation (13), the result is that the

Fig. 7 Critical damping ratio required for stability (lower curve) and frequency ratio at which instability � rstoccurs (upper curve) as functions of mass ratio



amplitude of the pavement response is given by

aYX

mrO2

»‰1 ¡ O2 1 ‡ mr cos f… †Š2

‡ 2BO ‡ O2m r sin f£ ¤2

¼¡ 1=2

…27†

and the pavement response lags behind the pedestrianexcitation by angle y where

y ˆ tan¡1 2BO ‡ O2mr sin f

1 ¡ O2…1 ‡ mr cos f†

µ ¶…28†

For equations (27) and (28) to describe the total motionthat is occurring, any transient motions must havedecayed. This will only happen if the damping ratio ofthe bridge exceeds the critical value given by equations(21) and (22) above.

The amplitude of response given by equation (27)depends on the phase angle f by which pedestrian bodymovement lags movement of their feet. Figure 9 showsfour curves for the case when mr ˆ 0:1, B ˆ 0:1, for f ˆ0, ¡ p=2, ¡ p and p=2 in which jaY =X j is plottedagainst O. Since the worst-case scenario is sought, thecurve which gives the highest response is needed. Ratherthan plot all possible curves for all values of f, the upperenvelope of these curves can be plotted, as shown inF ig. 10. The equation for this envelope can be calculatedfrom equation (27) by differentiating with respect to f

to seek a maximum. This occurs when

f ˆ tan¡ 1 ¡ 2BO

1 ¡ O2

³ ´…29†

and is given by

aYX

max

ˆ mrO2

µ1 ¡ O2

¡ ¢2‡4B2O2 ‡ m2r O

4

¡ 2m rO2��1 ¡ O2

¡ ¢2‡4B2O2q ¶¡1=2

…30†

Equation (30) is used to compute the results in F igs 11and 12. These results assume that a stable solution ispossible, which is only the case if B > Bc, where Bc is thecritical damping ratio required for stability and is givenby equation (21) or, for mr 5 1 by equation (22). F igure11 has mass ratio mr ˆ 0:1 and F ig. 12 a larger ratiomr ˆ 0:3. For both graphs (F igs 11 and 12), thedamping is chosen so that each graph shows curves forB=Bc ˆ 1:1, 1:3, 1:5, 2, 3 and 5. Note that the criticaldamping cc depends on mr according to equations (21)and (22).

6.5 Effective damping ratio

From Figs 11 and 12, the forced response is clearly verysimilar in its appearance to the forced resonance of asingle-degree-of-freedom system. Provided that thebridge’s damping is only slightly higher than the critical

Fig. 8 Critical phase angle between relative movement of body and feet at the stability limit, plotted as afunction of mass ratio

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damping for stability, vibration occurs at a frequencyclose to the natural frequency of the relevant mode.

Consider a single degree-of-freedom system withdisplacement response Y exp…iot† when subjected to aharmonic force F exp…iot†. The amplitude of its steadystate response is given by

kYF

1 ¡ O2¡ ¢2‡ 2BeffO… †2

h i¡ 1=2…31†

where k is the stiffness, O the frequency ratio [equation

(11)] and Beff is the effective damping ratio. When theforce F…t† comes from the inertial loading of a mass bmmoving harmonically through distance X then

F…t† ˆ ¡ bmo2X exp…iot† ˆ F exp…iot† …32†

After substituting in equation (31) for F and sortingterms gives

aYX

mrO2 1 ¡ O2

¡ ¢2‡ 2BeffO… †2h i¡ 1=2

…33†

Fig. 9 Forced response of bridge for mr ˆ 0:1, B ˆ 0:1 and four different phase angles f. The ordinate is thenon-dimensional response jaY =X j de� ned by equation (27)

Fig. 10 The same as F ig. 9 but showing the upper envelope of all the curves



At resonance …O ˆ 1†, then

aYX

Oˆ1

ˆ mr

2Beff…34†

Compare the corresponding result obtained fromequation (30) by putting O ˆ 1:

aYX

max, Oˆ1

ˆ mr

2B ¡ mr…35†

The peak height of the response curves calculated by

equations (30) and (33) will therefore be the same if

Beff ˆ B ¡mr

2…36†

so that the effective damping ratio can be calculated bysubtracting mr=2 from the actual (structural) dampingratio of the bridge mode concerned. Using the de� nitionof Bc in equation (22), equation (36) may alteratively bewritten as

Beff ˆ B ¡ Bc …37†

Fig. 11 Forced response of bridge for mr ˆ 0:1 and six different damping ratios given by B=Bc ˆ 1:1, 1.3, 1.5,2, 3 and 5. The ordinate is the non-dimensional response jaY =X jmax de� ned by equation (30)

Fig. 12 The same as Fig. 11 except that the mass ratio mr ˆ 0:3 instead of 0.1

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provided that the damping is small and mr ˆ abm=M issmall.

7 PEDESTRIAN SCRUTON NUMBER

McRobie and Morgenthal [9] have pointed out theanalogy between wind excitation and people excitation.The tendency for vortex shedding to excite structuraloscillations is measured by the non-dimensional Scrutonnumber Sc which is a product of damping and the ratioof representative structural and � uid masses. The usualde� nition is

S c ˆ 4pBMrb2

…38†

where B is the damping ratio of the relevant mode, r isthe air density and, for a cylindrical structure ofdiameter b, M is the mass per unit length of thestructure. Large Scruton numbers are preferable.McRobie and Morgenthal suggested that the sameapproach should be taken for pedestrian-excited vibra-tion, distinguishing between vertical and lateral vibra-tion to allow for the different human responses tovertical and lateral pavement movement. The de� nitionof pedestrian Scruton number S cp is arbitrary but, forthe purpose of this paper, by comparison with equation(38), it is de� ned as

S cp ˆ 2BMm

…39†

where

S cp ˆ pedestrian Scruton number

B ˆ modal damping ratio

M ˆ modal mass or, for a uniform deck,bridge mass per unit length

m ˆ modal mass of pedestrians or, for auniform bridge deck with evenly spacedpedestrians, pedestrian mass per unit length

For this de� nition, in order to exceed the minimumdamping given by equation (22), it is necessary that

S cp > ab …40†

This analysis uses the model in equation (4). As before, ais the ratio of movement of a person’s centre of mass tomovement of the pavement, which from the measuredresults above is typically 2/3 for lateral vibration in thefrequency range 0.75–0.95 Hz, and b is the correlationfactor for individual people to synchronize with pave-ment movement, which is typically 0.4 for lateralpavement amplitudes less than 10 mm.

Typical data have been assembled from the sourcesavailable (which are somewhat meagre and generally

incomplete) and is reproduced in F ig. 13 (for lateralvibration) and Fig. 14 (for vertical vibration). Each� gure has two horizontal lines showing S cp calculatedfrom equation (40) for the case when a ˆ 2=3 and b ˆ0:4 (lower limit) and when a ˆ 1 and b ˆ 1 (upper limit).Depending on the values of these empirical factors, thehorizontal lines show the minimum pedestrian Scrutonnumber required for stability.

It can be seen that for typical modes of the LondonMillennium Bridge the pedestrian Scruton numberswere initially very low, less than the lower limit. Aftermodi� cation to increase arti� cially the bridge’s damp-ing, the corresponding S cp are much higher, well abovethe upper limit drawn and also well above an alternativelimit (42) suggested by Arup (see below). Of course atpresent there is, as already mentioned, only limitedexperimental data and the ‘best’ values to use for a andb remain to be established. Although b (the correlationfactor for when people synchronize with pavementmovement) cannot exceed unity, it is possible that a(the ratio of body movement to pavement movement)may be greater than unity and then the upper limitdrawn for a ˆ b ˆ 1 would be higher. Interestingly, arecent study of lateral vibration by Roberts [10],published while this paper was in press, suggests thatit is plausible that the limit of stability will occur whena ˆ 1. This (unproven) assumption leads to an expres-sion for the maximum number of pedestrians permis-sible on a bridge of given length and dynamic propertiesif instability is to be prevented. Although apparentlydifferent, when expressed as a non-dimensional Scrutonnumber, Roberts’ stability criterion can be shown toreduce to S cp ˆ 1, the upper limit in F ig. 13. At the timeof completion of this paper, various new studies of thedynamics of pedestrian bridges are taking place whichmay provide much needed further data but, until theydo, uncertainty remains.

For lateral vibration (Fig. 13), the estimated pedes-trian Scruton numbers for both the bridge in Japanstudied by Fujino et al. and the Auckland HarbourBridge lie below the lower limit from equation (40). Inthe case of vertical vibration (F ig. 14), additional datawere given by McRobie and Morgenthal [9] for someother bridges that have caused concern, all of which fallbelow the limit. The data for Auckland Harbour Bridgeare interesting because they fall between the upper andlower limits from equation (40). This relates to datameasured during the course of a marathon race in 1992when a large number of runners crossed one of the two-lane roadways. It is recorded that vertical amplitudes ofup to 3 mm were experienced in a frequency range 2.6–3 Hz, which is noticeable by runners. F rom this it maybe concluded that vertical bridge oscillation of seriousamplitudes could be excited by the natural synchroniza-tion of a large enough crowd of runners (as distinct froma marching army in the traditional sense). That is whythe decision was taken to increase arti� cially the



damping of vertical as well as lateral modes for theLondon Millennium Bridge.

8 ARUP’S ANALYSIS

As a result of a series of crowd tests, and an energyanalysis of vibrational power � ow, Arup concluded thatthe correlated lateral force per person is related to thelocal velocity by an approximately linear relationshipwhich was found to hold for lateral frequencies in therange 0.5–1 Hz (pacing frequency 1–2 Hz). It is interest-ing that, within this frequency range, the results again

appear to be insensitive to frequency. If k is de� ned asthe slope of a graph of the amplitude of average lateralforce per person plotted against the amplitude ofpavement lateral velocity, Arup found thatk&300 N s=m for a bridge mode in the frequency range0.5–1 Hz (see reference [4], p. 27).

By assuming that each person generates a velocity-dependent force which acts as negative damping, andmaking a modal calculation, they also concluded thatvibrational energy in the mode would not increase if

B >Nk

8pfnM…41†

where B is the modal damping ratio, fn is the natural

Fig. 13 Some collected data on the pedestrian Scruton number for lateral modes. The critical value fromequation (42) is plotted for comparison

Fig. 14 Some collected data on the pedestrian Scruton number for vertical modes

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frequency and M is the modal mass [4, equation (9)].For this condition, the positive modal damping exceedsthe negative damping generated by pedestrian move-ment. On substituting equation (41) into equation (39)to calculate the required pedestrian Scruton number, itis found that this limiting condition can be expressed as

S cp >k

2pfnm0…42†

where m0 is the mass per person for whom k ˆ300N s=m in the frequency range 0.5–1.0 Hz. This resulthas been added to Fig. 13 for the case whenm0 ˆ 63:5 kg. Equation (42) crosses the upper limitfrom equation (40) at a frequency of 0.75 Hz approxi-mately.

If S cp is less than the limit in equation (42), vibrationself-excites as people begin by walking normally andthen progressively fall into synchronism with thepavement motion until this builds up to a level at whichsteady walking becomes impossible and a staggeringmovement takes over.

9 CONCLUSIONS

The analysis in this paper turns on assuming that, in theprocess of synchronization, the time lag that people taketo respond to bridge movement (represented by D)naturally adjusts itself to have the greatest effect, subjectto a correction factor to allow for the fact that only aproportion b of all pedestrians make this synchroniza-tion. By selecting the value of D to give the greatestresponse, it is concluded that bridge vibration willbecome unstable when the live load, represented bypeople of mass m per unit length, is too great aproportion of the bridge mass M per unit length.

The permissible m/M ratio depends on the amount ofdamping present in the appropriate vibrational modesthat will be excited by the pacing rate of pedestriansbecause problems only arise when this excitationfrequency is close to a natural frequency of a lightlydamped vibrational mode. Speci� cally, according to theanalysis, it is necessary for stability that, by combiningequations (22) and (10),

2B >abmM

…43†

where B is the damping ratio in the mode and a and b areexperimentally determined factors. From the data so faravailable, it appears satisfactory to assume that a ˆ 2=3relates movement of a person’s centre of mass tomovement of their feet (from the slope in F ig. 2) andb ˆ 0:4 for bridge amplitudes up to about 10 mm (fromFig. 3). However, these numbers derive from a limitednumber of experiments on lateral vibration and can onlybe regarded as provisional for lateral vibration and a

� rst indication of possible numbers for vertical vibrationfor which measurements have not yet been made.

The dependence of a damping stability criterion on amass ratio is consistent with the experience of vortex-excited oscillations in aeroelasticity. Application of thepedestrian Scruton number de� ned by equation (39)allows the criterion (43) to be expressed alternatively byequation (40). The results plotted in F igs 13 and 14 showthat troublesome bridges all have pedestrian Scrutonnumbers that fall below the limit given by equation (40)with a ˆ 2=3 and b ˆ 0:4. Similarly, the LondonMillennium Bridge after modi� cation [11–13] lies wellabove the upper limit of the pedestrian Scruton numberdrawn by assuming that the factors a and b are bothunity. This is a worst-case assumption for the correla-tion factor b and may be a pessimistic assumption forthe value of the amplitude ratio a.

The collection of more experimental data is needed toverify these conclusions but, so far as the author knows,no unstable bridges whose pedestrian Scruton numberslie above the upper limit shown in F igs 13 and 14 haveyet been found.

ACKNOWLEDGEMENTS

The author is grateful to many people who have beeninvolved, directly or indirectly, in this study includingparticularly Pat Dallard, Michael Willford, RogerRidsdill Smith and the late Tony F itzpatrick of Arup,David Malam of Atkins, Shayne Gooch at theUniversity of Canterbury, New Zealand, and colleaguesin Cambridge, Allan McRobie and Joan Lasenby. Allthese and many more with whom the author has notworked personally have contributed greatly to learningabout how pedestrians and bridges interact. Research onthis problem has built on the seminal work of HugoBachmann in Switzerland and Yozo Fujino in Japanwhom, it is believed, were the � rst to identify and writeabout the issues discussed in this paper.

REFERENCES

1 Bachmann, H. and Ammann, W. Vibrations in structuresinduced by man and machines. IABSE Structural Engi-neering Document 3e, International Association for Bridgeand Structural Engineering, Zurich, 1987, Ch. 2 andAppendix A.

2 Bachmann, H. Case studies of structures with man-inducedvibrations. Trans. ASCE, J. Struct. Engng, 1992, 118, 631–647.

3 Fujino, Y., Pacheco, B. M., Nakamura, S.-I. andWarnitchai, P. Synchronization of human walking observedduring lateral vibration of a congested pedestrian bridge.Earthquake Engng Struct. Dynamics, 1993, 22, 741–758.



4 Dallard, P., Fitzpatrick, A. J., Flint, A., Le Bourva, S., Low,A., Ridsdill-Smith, R. M. and Willford, M. The LondonMillennium Footbridge. S truct. Engr, 2001, 79(22), 17–33.

5 Fitzpatrick, T., Dallard, P., Le Bourva, S., Low, A., Smith,R. R. and Willford, M. Linking London: the MillenniumBridge. Paper, Royal Academy of Engineering, London,2001.

6 McRobie, A. and Morgenthal, G. Full-scale section modeltests on human-structure lock-in. In Proceedings of theInternational Conference on Footbridges 2002, Paris,F rance, 2002.

7 Gamage, S. S. H. U. and Lasenby, J. Inferring dynamicalinformation from 3D position data using geometricalgebra. In Applications to Geometric Algebra in ComputerScience and Engineering (Eds L. Dorst, C. Doran and J.Lasenby), 2002 (Birkhauser, Basel).

8 Ringer, M. and Lasenby, J. Modelling and trackingarticulated motion from multiple camera views. In Analysisof Biomedical Signals and Images: 15th Biennial Interna-

tional EURASIP Conference Biosignal 2000 (Eds J. Jan, J.Kozumplik, I. Provazbik and Z. Szabo), 2000 (VutiumPress, Brno, Czech Republic).

9 McRobie, A. and Morgenthal, G. Risk Management forpedestrian-induced dynamics of footbridges. In Proceed-ings of the International Conference on Footbridges 2002,Paris, F rance, 2002.

10 Roberts, T. M. Synchronized pedestrian excitation offootbridges. Proc. ICE, Bridge Engng, 2003, 156, 155–160.

11 Dallard, P., Fitzpatrick, T., Flint, A., Low, A., Smith, R. R.,Willford, M. and Roche, M. London Millennium Bridge:pedestrian-induced lateral vibration. Trans. ASCE, J.Bridge Engng, 2001, 6, 412–417.

12 Newland, D. E. Vibration of the London Millennium Bridge:cause and cure. Int. J. Acoust. V ibr., 2003, 8(1), 9–14.

13 Newland, D. E. Vibration: Problem and solution. In Bladeof L ight: The S tory of London’s M illennium Bridge (Ed. D.Sudjic), 2001, pp. 88–93 (Penguin, London).

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