Vibration of the Millenium Bridge London

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VIBRATION OF THE LONDONM

MILLENNIUM FOOTBRIDGE:

PART 1 - CAUSE

David E Newland

Department of Engineering

University of Cambridge

Trumpington Street

CAMBRIDGECB2 !"# U$

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)*en t*e +ondon Mi,,ennium footbridge -as opened in .une 2///# it s-ayed a,arming,y&T*is generated *uge pub,i' interest and t*e bridge be'ame (no-n as +ondon0s 1-obb,y

bridge& !art of t*is paper e3p,ains *o- pedestrians un-itting,y 'aused t*e bridge to se,f4

e3'ite& A,t*oug* previous,y do'umented# t*is p*enomenon -as not -e,,4(no-n&��

��

INTRODUCTION

To mar( t*e Mi,,ennium# a ne- footbridge *as been bui,t a'ross t*e river T*ames in

+ondon& It is a s*a,,o- suspension bridge ,in(ing St& !au,0s Cat*edra, on t*e nort* side oft*e river -it* t*e Tate Modern art ga,,ery on t*e sout* side& T*e bridge is over 5// metres

,ong -it* t*ree spans# t*e ,ongest being t*e 'entre span of 66 metres& To meet t*e

designers0 artisti' re7uirements# t*e bridge0s suspension 'ab,es sag on,y 2&5 metres# afra'tion of t*e sag of a traditiona, suspension bridge of t*e same span& As a resu,t# t*e

'ab,es 'arry a very *ig* tension for'e for a bridge of t*is si8e# tota,,ing some 2/// tonnes&

)*en t*e bridge -as opened in .une 2///# it -as found t*at t*e it s-ayed noti'eab,y& )it*a ,arge number of pedestrians on t*e bridge# its side-ays movement -as suffi'ient to 'ause

peop,e to stop -a,(ing and *o,d on to t*e *and4rai,s& Be'ause t*ere -as danger of persona,in9ury# it -as de'ided to ',ose t*e bridge after a fe- days for remedia, -or(&

mailto:[email protected]





Figure 1. The London Millennium footbridge shortly after its completion. St Paul's Cathedral is at the top left

HISTOR%

T*e bridge opened on / .une 2///& :or t*e opening 'eremony# a 'ro-d of over ///

peop,e *ad assemb,ed on t*e sout* *a,f of t*e bridge -it* a band in front& )*en t*ey

started to -a,( a'ross -it* t*e band p,aying# t*ere -as immediate,y an une3pe'ted,y pronoun'ed ,atera, movement of t*e bridge de'(& T*is movement be'ame suffi'ient,y ,arge

for peop,e to stop -a,(ing to retain t*eir ba,an'e and sometimes to *o,d onto t*e *and rai,s

for support& ;ideo pi'tures s*o-ed ,ater t*at t*e sout* span *ad been moving t*roug* anamp,itude of about </ mm at /&= >8 and t*e 'entre span about ?< mm at >8

appro3imate,y& !robab,y *ig*er amp,itudes o''urred periodi'a,,y and severa, modes -ere

invo,ved& It -as de'ided immediate,y to ,imit t*e number of peop,e on t*e bridge# but evenso t*e de'( movement -as suffi'ient to be un'omfortab,e and to raise 'on'ern for pub,i'

safety so t*at on 2 .une t*e bridge -as ',osed unti, t*e prob,em 'ou,d be so,ved& It -as

not reopened to t*e pub,i' unti, 22 :ebruary 2//2&

T*ere -as a signifi'ant -ind b,o-ing on t*e opening days @for'e 546 and t*e bridge *ad been de'orated -it* ,arge f,ags# but it -as rapid,y 'on',uded t*at -ind buffeting *ad not

'ontributed signifi'ant,y to vibration of t*e bridge& Anot*er possib,e e3p,anation -as t*at

'oup,ing bet-een ,atera, and torsiona, de'( movements -as a,,o-ing verti'a, footfa,,e3'itation to e3'ite ,atera, modes# but t*is -as not found to be a signifi'ant fa'tor& Eviden'e

in support of t*is 'on',usion -as t*at t*e >8 mode of t*e 'entre span -as t*e span0s

se'ond ,atera, mode -it* nodes at its 'entre and at t*e t-o bridge piers# t*is mode *ad pra'ti'a,,y no torsiona, movement&



It -as rea,ised very 7ui'(,y t*at t*e prob,em -as one of ,atera, e3'itation and a,t*oug*

a,,o-an'e *ad been made for ,atera, for'es it *ad not been e3pe'ted t*at pedestrians -ou,dso easi,y fa,, into step or t*at t*e ,atera, for'e per person -ou,d be as great as -as

apparent,y proving to be t*e 'ase&

RESEARCH

An immediate resear'* programme -as ,aun'*ed by t*e bridges engineering designers ve

Arup# supported by a number of universities and resear'* organisations&

It -as found t*at some simi,ar e3perien'es *ad been re'orded in t*e ,iterature# a,t*oug*

t*ese -ere not -e,,4(no-n and *ad not yet been in'orporated into t*e re,evant bridge

bui,ding 'odes& A German report in ?2 7uoted by Ba'*mann and Ammann in t*eir

IABSE boo( @=?# des'ribed *o- a ne- stee, footbridge *ad e3perien'ed strong ,atera,vibration during an opening 'eremony -it* 5//46// peop,e&� T*ey e3p,ained� *o- t*e

,atera, s-ay of a person0s 'entre of gravity o''urs at *a,f t*e -a,(ing pa'e& Sin'e t*e

footbridge *ad a ,o-est ,atera, mode of about & >8# t*e fre7uen'y of e3'itation -as very',ose to t*e mean pa'ing rate of -a,(ing of about 2 >8& T*us in t*is 'ase "an almost

resonating vibration occurred. Moreover it could be supposed that in this case the pedestrians synchronised their step with the bridge vibration, thereby enhancing the

vibration considerably" @Ba'*mann# 2# p& F5F& T*e prob,em is said to *ave been so,ved

by t*e insta,,ation of *ori8onta, tuned vibration absorbers&

T*e 'on'ept of syn'*ronisation turned out to be very important# and a ,ater paper by :u9ino

et a,& @5 -as dis'overed -*i'* des'ribed observations of � pedestrian4indu'ed ,atera,

vibration of a 'ab,e4stayed stee, bo3 girder bridge of simi,ar si8e to t*e Mi,,ennium Bridge&

It -as found t*at -*en a ,arge number of peop,e -ere 'rossing t*e bridge @2#/// peop,e on

t*e bridge# ,atera, vibration of t*e bridge de'( at /& >8 'ou,d bui,d up to an amp,itude of/ mm -it* some of t*e supporting 'ab,es -*ose natura, fre7uen'ies -ere ',ose to /& >8

vibrating -it* an amp,itude of up to 5// mm& By ana,ysing video re'ordings of pedestrians0

*ead movement# :u9ino 'on',uded t*at ,atera, de'( movement en'ourages pedestrians to-a,( in step and t*at syn'*ronisation in'reases t*e *uman for'e and ma(es it resonant -it*

t*e bridge de'(& >e summarised *is findings as fo,,o-s "The growth process of the lateral

vibration of the girder under the congested pedestrians can be explained as follows. First a small lateral motion is induced by the random lateral human walking forces, and walking

of some pedestrians is synchronised to the girder motion. Then resonant force acts on the

girder, consequently the girder motion is increased. alking of more pedestrians are synchronised, increasing the lateral girder motion. !n this sense, this vibration was a self

excited nature. #f course, because of adaptive nature of human being, the girder amplitude

will not go to infinity and will reach a steady state."

A,t*oug* :u9ino re'ords t*e damping ratio of t*e /&>8 ,atera, mode as # *e foundt*at on,y 2/H of t*e pedestrians on t*e main span of t*e bridge -ere 'omp,ete,y

syn'*ronised to t*e girder vibration and t*e amp,itude of vibration -as on,y / mm

@'ompared -it* ?< mm for t*e Mi,,ennium Bridge& Impressions from video ',ips of t*eMi,,ennium bridge are t*at a good dea, more t*an 2/H of -a,(ers *ad syn'*ronised t*eir



step& A,so in :u9ino0s e3amp,e# t*e very ,arge movement of t*e suspension 'ab,es @5// mm

amp,itude may *ave made t*ese a't as dynami' vibration absorbers and so ,imit t*e e3tentand 'onse7uen'es of syn'*ronisation&

It -as ',ear t*at data spe'ifi' to t*e Mi,,ennium bridge -as urgent,y re7uired and Arup

undertoo( an e3tensive programme of testing to obtain t*is& In addition to 'ommissioningtests on *uman gait and *o- t*is is affe'ted by movement of t*e -a,(ing surfa'e# t*e maintests -ere 'arried out on t*e bridge itse,f& T*ese in',uded artifi'ia,,y s*a(ing t*e bridge to

'onfirm mode s*apes and damping and a 'ompre*ensive series of 'ro-d tests& Detai,ed

vibration measurements and video re'ords -ere made -it* pedestrians -a,(ing at different

speeds and densities on ea'* span& T*ese a,,o-ed re,iab,e 7uantitative data on t*esyn'*ronous ,atera, e3'itation p*enomenon to be estab,is*ed and a se,f4e3'itation mode, to

be deve,oped -*i'* 'ou,d give a re,iab,e predi'tion of stru'tura, response&

ARUP&S PEDESTRIAN LATERAL LOADING MODEL

Arups ,oading mode, is des'ribed in :it8patri'( et a, @2//&� Using e3perimenta, datafrom 'ontro,,ed pedestrian ,oading tests# -it* an appro3imate,y 'onstant density of pedestrians -a,(ing at steady speed# Arup found t*at t*ere -as strong 'orre,ation bet-een

t*e amp,itude of t*e pedestrians0 @moda, e3'itation for'e and t*e amp,itude of t*e bridge

de'(0s @moda, ,atera, ve,o'ity& Measurement of de'( ve,o'ity is straig*tfor-ard# but t*e

e3'itation for'e -as 'a,'u,ated from a po-er f,o- ana,ysis based on t*e 'on'ept t*at t*e-or( done by t*e net e3'itation for'e @foot4fa,, for'e ,ess damping for'e is e7ua, to t*e

gain of (ineti' energy per 'y',e& T*is ,ed to t*e 'on',usion t*at# -*en syn'*ronisation *as

o''urred# t*e amp,itude of energy4transferring for'e per pedestrian is ,inear,y proportiona,to ve,o'ity and a'ts as a negative damping for'e& T*is a,,o-s t*e ,imiting number of peop,e

for stabi,ity �to be 'a,'u,ated and t*e effe'tive damping to be 'a,'u,ated for

NE' LINEAR FEEDBAC( MODEL

T*ese resu,ts 'an be rea'*ed by a 'omp,ete,y different approa'* using a feedba'( mode, ofsyn'*ronous ,atera, e3'itation& It is based on t*e idea t*at peop,e -i,, natura,,y fa,, into step

-it* ea'* ot*er and t*at t*ey -i,, un'ons'ious,y ad9ust t*eir group p*asing so t*at t*e

bridge vibration in'reases to a ma3imum& T*is is important& As t*e assumed p*ase of peop,e0s -a,(ing and t*e bridge0s movement '*anges# bridge amp,itude '*anges

enormous,y& :rom t*is feedba'( mode,# t*e p*ase to give pea( response 'an be 'a,'u,ated

t*eoreti'a,,y& It turns out t*at t*e ans-er agrees -it* t*e p*ase found by Arup from t*eir

e3perimenta, studies&

T*e p*ysi'a, me'*anism of syn'*ronous ,atera, e3'itation is -e,, des'ribed by Arup�

@Da,,ard et a,# 2// in t*e fo,,o-ing -ay

"Chance footfall correlation combined !ith the synchroniation that occursnaturally !ithin a cro!d may cause the bridge to start to s!ay horiontally. #f thes!ay is perceptible a further effect can start to ta$e hold. #t becomes morecomfortable for the pedestrians to !al$ in synchroniation !ith the s!aying of thebridge. The pedestrians find this ma$es their interaction !ith the bridge more



predicable and helps them maintain their lateral balance. This instincti%e beha%iour ensures that the footfall forces are applied at a resonant fre&uency of the bridgeand !ith a phase such as to increase the motion of the bridge. s the amplitude ofthe motion increases the lateral force imparted by indi%iduals increases as doesthe degree of correlation bet!een indi%iduals. The fre&uency "loc$(in" and positi%e

force feedbac$ caused the e)cessi%e motions obser%ed at the Millennium *ridge."

Figure 1. Feedbac$ system to represent synchronous lateral e)citation

T*e fre7uen'y4domain feedba'( mode, s*o-n in t*e diagram above represents t*is

be*aviour& Ea'* mode is treated separate,y& >ere � is t*e :ourier transform of t*e

moda, e3'itation for'e x$t% -it* no bridge movement# � is t*e :ourier transform of

t*e moda, disp,a'ement response y$t%& is t*e moda, fre7uen'y response fun'tion at

fre7uen'y � and t*e 'omp,e3 7uantity des'ribes t*e positive for'e feedba'( by

-*i'* t*e pedestrians0 moda, input for'e is modified by movement of t*e bridge&� T*ese

are a,, 'omp,e3 7uantities representing amp,itude and p*ase at fre7uen'y # using t*e

fun'tiona, notation -*ere & In t*is notation# is a 'omp,e3 7uantity

@amp,itude and p*ase and � is a rea, 7uantity @amp,itude on,y& T*e 'ontro,

e7uation is

�@

�

giving

�

@2

�

T*e feedba'( fun'tion �is a 'omp,e3 fun'tion -*i'* -e no- -rite as a

rea, argument and 'omp,e3 e3ponentia, p*ase fun'tion

�

@5

�



As noted above# t*e observed motion of pedestrians is t*at t*eir p*ase ad9usts itse,f so as to

increase the motion of the bridge. T*erefore it is natura, to '*oose t*e p*ase ang,e � so

t*at t*e bridge0s response is a ma3imum# t*at is t*at � is a ma3imum& T*is

'an be done by substituting @5 into @2 and t*en differentiating t*e denominator -it*

respe't to � to sea'* for a minimum to obtain t*e resu,t t*at

�

@6

�

:or a resonant mode -it* moda, stiffness# mass and damping given by k, m and c#

�

@<

�

-it* -*i'* @6 be'omes �

@F

�

)*en t*e footfa,, fre7uen'y �'oin'ides -it* t*e mode0s natura, fre7uen'y so

t*at

�@?

�

t*en @F gives

�

@=

�

or# in terms of t*e moda, ,oss fa'tor �of t*e bridge stru'ture -it*out pedestrians#

�&

@

�

t*e ma3imum non4dimensiona, response ratio is �

@/

�



In t*is formu,a# �is t*e amp,itude of t*e moda, for'e e3erted on t*e bridge by t*e

pedestrians -a,(ing on it# -*en t*eir pa'ing rate 'oin'ides -it* t-i'e # -it* t*e p*asing

of t*eir movement *aving natura,,y ad9usted itse,f to give ma3imum response&

T*e important 'on',usion from t*is ana,ysis is t*at t*e -a,(ing pedestrians a't as negativedamping and t*e effe'tive moda, ,oss fa'tor �is redu'ed -*en pedestrians -a,( over t*e

bridge& T*is 'on',usion 'onfirms t*at obtained by Arup from a pure,y e3perimenta,

approa'* @:it8patri'( et a,& 2//# s& 6&&

PHASE FOR PEA( RESPONSE

If -e -rite

�

@

�

t*en t*e va,ue of �for -*i'* t*e ma3imum response @6 is obtained is

�

@2

�

-*i'* from @< is

�

& @5

�

So# from @5# -e see t*at p*ase of t*e pedestrians0 feedba'( for'e is ,eading t*e output

disp,a'ement of t*e bridge de'( by an ang,e -*i'* be'omes e3a't,y �at t*e resonant

fre7uen'y defined by @?& T*is of 'ourse is -*at -e e3pe't for a negative damping for'e&

DETERMINING THE PROPORTIONALIT% CONSTANT

Referring to figure 2# t*e moda, feedba'( for'e �is generated by a,, t*e

pedestrians -a,(ing on t*e bridge& If t*ere are & peop,e unform,y distributed a,ong a span

of ,engt*� �-it*� mode s*ape �norma,ised so t*at

�@6

�

and if ea'* person 'ontributes an a'tua, for'e per unit de'( disp,a'ement of �and per

unit moda, disp,a'ement of � �so t*at t*e moda, for'e from pedestrians per

unit moda, disp,a'ement is



�

@<

�

>en'e t*e net moda, ,oss fa'tor from @/ is

�@F

�

-*ere

nJet moda, ,oss fa'tor

moda, ,oss fa'tor for stru'ture a,one

& number of peop,e on span

' ,engt* of span

natura, fre7uen'y @radKs

k moda, stiffness

m moda, mass

amp,itude of feedba'( for'e per person and per unit disp,a'ement of t*e bridge

de'( at fre7uen'y

By measuring t*e net moda, ,oss fa'tor -it* &(' peop,e per unit ,engt* of de'( -a,(ingsteadi,y at t*e syn'*ronous speed @footfa,, fre7uen'y t-i'e t*e natura, fre7uen'y# t*e

feedba'( for'e per person and per unit disp,a'ement# # 'an be 'a,'u,ated from @F if

t*e ,oss fa'tor of bare stru'ture *as been measured previous,y& A,ternative,y# if �

�and� �are (no-n# �'an be 'omputed& T*e app,i'ation of t*is formu,a @F is

'onsidered in !art 2&

Referen'es are at t*e end of !art 2 of t*is paper

VIBRATION OF THE LONDONM

MILLENNIUM FOOTBRIDGE:PART 2 - CURE

David E Newland

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CAMBRIDGECB2 ,P. U/

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INTRODUCTION

Pe$le-e5i"ed la"e!al )!id&e vi)!a"i$n i' li3el( "$ $*! %$! ede'"!ian )!id&e'w4i4 4ave l$w na"*!al %!e6*enie' $% 'wa(in& #$ve#en" 7le'' "4an 8 H9 and%$! w4i4 "4e la"e!al #$de' 4ave li&4" da#in&1 In "4e a'e $% L$nd$n;'Millenni*# B!id&e. )$"4 "4e'e $ndi"i$n' alied1

ne so,ution -ou,d *ave been to stiffen t*e bridge to in'rease its natura, fre7uen'ies and

ta(e t*ese outside t*e e3'itation fre7uen'y range& >o-ever t*e artisti' design of t*e bridge

-ou,d *ave been 'ompromised by stiffening and t*is -as regarded as most undesirab,e&

T*e a,ternative -as to find a -ay of in'reasing t*e bridge0s in*erent,y ,o- damping so t*at

se,f4e3'itation did not o''ur& It *as been found t*at# be,o- a t*res*o,d damping ,eve,#

motion -ou,d bui,d up# but t*at above t*e t*res*o,d damping ,eve,# se,f4e3'itation -ou,d

not o''ur& Determining -*at t*is t*res*o,d ,eve, -as and t*en providing a means of

introdu'ing t*e re7uired amount of added damping proved a '*a,,enging tas(& It *as

invo,ved adding 5? ,inear vis'ous dampers and over </ tuned mass vibration absorbers to

t*e initia, stru'ture& As a resu,t# t*is bridge is no- probab,y t*e most 'omp,e3 passive,y4

damped stru'ture in t*e -or,d&





Figure + Centre span of the Millennium *ridge

Insta,,ing dampers in a -ay t*at -as 'onsistent -it* t*e aest*eti' design of t*e bridge -as

diffi'u,t& A great dea, of effort *ad been put into '*oosing a preferred design and t*e

'on'ept of 1a b,ade of ,ig*t *ad been adopted and re'eived -idespread approva, @seeDeyan Sud9i' @ed#� )lade of 'ight* The +tory of 'ondons Millennium )ridge# 2//& So

far as possib,e dampers *ad to be mounted underneat* t*e bridge de'( so t*at t*ey -ou,d

be out4of4sig*t of everyone using t*e bridge&

RECONCILIATION <ITH ARUP;+ DAMPING CALCULATION

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�

7,?

�

A,t*oug* t*is *as been arrived at by a 'omp,ete,y different approa'*# it is identi'a, -it*

@F& T*is 'an be verified by ma(ing t*e substitutions # # #

# and & Apart from different norma,isation of t*e moda, s*ape fun'tion#

t*e main differen'e is t*at Arup defined feedba'( for'e as proportiona, to ve,o'ity -*ereas



t*e above ana,ysis begins by assuming t*at t*e feedba'( for'e is proportiona, to

disp,a'ement @at a fi3ed fre7uen'y& Arup use t*eir symbo, k not for stiffness but to re,ate

pedestrian feedba'( for'e to de'( ,atera, ve,o'ity# -*ereas � as defined above re,ates

feedba'( for'e to de'( ,atera, disp,a'ement&�

Arup0s 'omputation of t*eir proportiona,ity fa'tor k -as done by measuring t*e

a''e,eration time *istory under 'onditions of steady4state 'ro-d ,oading -it* a 'onstant

number of peop,e -a,(ing steadi,y over ea'* span @in turn at t*e 'orre't speed to resonate

-it* t*e re,evant mode& :rom t*is time4*istory# t*ey 'a,'u,ated moda, ve,o'ity @see

:it8patri'( et a,# 2//# p& 6& If F is t*e amp,itude of t*e moda, feedba'( for'e @-*i'* is

assumed proportiona, to ve,o'ity# - is t*e amp,itude of t*e moda, damping for'e @a,so

proportiona, to ve,o'ity and (no-n from previous measurements# and is t*e amp,itude of

t*e moda, ve,o'ity# t*en for 'onservation of energy#

�

7,@

�

'$ "4a"

�

7,

�

-*ere m is moda, mass# and / is t*e amp,itude of t*e moda, a''e,eration# sin'e &

By p,otting � 'a,'u,ated from @ @ & is t*e number of peop,e on t*e span against

moda, ve,o'ity # Arup arrived at an average va,ue for k. Some typi'a, va,ues are s*o-n in

figure 6 in -*i'* p*ysi'a, rat*er t*an moda, resu,ts are p,otted& T*e appro3imate,y ,inear

re,ations*ip in figure 6 appears to derive from t*e 'ombined a'tion of t-o fa'tors t*e for'e

per person in'reases @s,o-,y -it* amp,itude and more peop,e syn'*ronise -it* de'(

movement at ,arger amp,itudes @see t*e additiona, materia, in :it8patri'( et a,# 2//&

+inearity is of 'ourse a starting assumption for t*e feedba'( mode, in part &



Figure ,. Correlation of pedestrian feedbac$ force and dec$ %elocity. The straight line sho!s a%erage force e)erted per pedestrian

-e!tons/ plotted against lateral dec$ %elocity -m0s/. -rup figure reproduced from eyan Sud2ic -ed/ 3441 p. 5+/.

DEPENDENCE OF NET DAMPING ON THE NUMBER OF PEDE+TRIAN+

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l$'' %a"$! %$! "4i' #$de w4en "4e!e a!e �ede'"!ian' $n "4e 'an will)e

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�

PRACTICAL DAMPING MEA+URE+

Ba'ed $n "4e'e $n'ide!a"i$n'. A!* deided "$ ai# %$! , "$ 2= $% !i"ialda#in& %$! all la"e!al and la"e!al"$!'i$nal #$de' )el$w ,1 H9 and "$ ,=$% !i"ial da#in& %$! ve!"ial and ve!"ial"$!'i$nal #$de' )el$w 8 H91 T4i' i'

a 4*&e in!ea'e in "4e $!i&inal da#in& !a"i$' $% "4e'e #$de' w4i4 we!e"(iall( , $! le''1 T$ *nde!'"and 4$w "4i' wa' a4ieved. i" i' nee''a!( "$*nde!'"and "4e $n'"!*"i$n $% "4e )!id&e1 T4i' an )e 'een %!$# "4e4$"$&!a4. &*!e 81

T*e bridge de'( is 'arried on ,atera, supports spa'ed periodi'a,,y& T*ese rea'* out to ',amp

onto t*e four para,,e, stee, 'ab,es at ea'* side of t*e de'(& To a first appro3imation# t*e

bridge vibrates ,i(e a taut string passing over supports at t*e t-o bridge piers and an'*ored



to fi3ed supports at t*e river ban(s& T*erefore ,atera, vibration invo,ves s*earing of t*e

de'( stru'ture -it* no appre'iab,e bending& A,, t*e ,o- fre7uen'y modes *ave nodes at t*e

atta'*ment of t*e 'ab,es to t*e bridge piers and to t*e river ban( an'*orages&� A,t*oug*

,inear vis'ous dampers 'an be 'onne'ted bet-een t*e bridge de'( and t*ese fi3ed

an'*orages# t*e re,ative motion *ere is sma,, and dampers fi3ed *ere 'annot be made to

-or( effi'ient,y& Ma3imum re,ative s*ear disp,a'ement o''urs bet-een t*e ,atera, supports

near anti4nodes# and t*erefore a-ay from t*e fi3ed an'*orages&

T*e so,ution adopted -as to fit A4s*aped frames to a,ternate ,atera, supports# -it* t*e

points of t-o A0s meeting at t*e intermediate supports @as s*o-n in figure <&

Figure 6. Plan %ie! of underside of dec$ sho!ing installation of dampers-from Fitpatric$ et al. 3441/

Bet-een t*e points of ea'* pair of A frames# a ,inear vis'ous damper -as mounted& It -as

possib,e to do t*is so t*at t*e moving parts -ere supported verti'a,,y on t*e upper4side of

t*e ,atera, supports& A,, t*e vis'ous dampers -ere supp,ied by t*e US firm Tay,or Devi'es#

In'& and in'orporated meta, be,,o-s sea,s so t*at t*ey are fu,,y4sea,ed to t*e environment&

:or t*e 'entre span# t*e damping introdu'ed by frame4mounted vis'ous dampers -as

supp,emented by t*e a'tion of 6 pairs of ,atera,,y4a'ting tuned4mass vibration absorbers

supp,ied by t*e German firm Gerb S'*-ingungsiso,ierungen Gmb># mounted

on t*e upper side of t*e bridge de'(0s ,atera, supports# as s*o-n in figure <&

An additiona, 2F pairs of verti'a,,y4a'ting tuned4mass vibration absorbers -ere insta,,ed in

simi,ar positions on ot*er ,atera, supports to in'rease verti'a, damping& T*is is to guard

against t*e @un,i(e,y possibi,ity t*at syn'*ronous verti'a, vibration mig*t o''ur -*en t*e

,atera, prob,em *ad been removed& T*e tuned4mass vibration absorbers *ave masses



bet-een and 5 tonnes and t*ey are ,o'ated as ',ose as possib,e to t*e antinodes of t*e

modes t*at t*ey are damping&

EPERIMENTAL VALIDATION

In addi"i$n "$ a !an&e $% la)$!a"$!( "e'"' "$ '"*d( 4*#an &ai" and "4ein"e!a"i$n $% ede'"!ian' and #$vin& la"%$!#'. "4e #ain e5e!i#en"al "e'"'we!e a!!ied $*" $n "4e )!id&e1 T4e'e $n'i'"ed $% "w$ e''en"iall( die!en""(e' $% "e'"'1 Te'"' wi"4 n$ e$le. *'in& a #e4anial '4a3e! "$ !$videe5i"a"i$n. we!e a!!ied $*" "$ #ea'*!e #$dal %!e6*enie' and da#in&1 T4i'wa' d$ne ini"iall( %$! "4e )a!e )!id&e. and "4en %$! "4e )!id&e wi"4 'ei#envi'$*' and "*ned-#a'' da#e!' in'"alled. "$ ve!i%( "4ei! a"i$n1� Te'"' wi"4wal3in& e$le $n'i'"ed #$'"l( $% !ei!*la"in& "e'"' w4e!e a #e"e!ed n*#)e!$% ede'"!ian' wal3ed in $ne di!e"i$n a!$'' a 'in&le 'an. and "4eni##edia"el( "*!ned !$*nd and wal3ed )a3 "$ "4ei! '"a!"in& $in"1 Re'*l"' %!$#"4e'e "e'"' we!e *'ed "$ &ene!a"e da"a li3e "4a" in &*!e and "$ $n!# "4e

e''en"iall( *n'"a)le %ea"*!e $% la"e!al '(n4!$n$*' e5i"a"i$n1 A "(ial !e'*l"%$! "4e n$!"4 'an. wi"4$*" an( added da#in&. i' '4$wn in &*!e >1 A #e"e!edn*#)e! $% e$le we!e in'"!*"ed "$ wal3 '"eadil( a" "4e 'eed needed "$'(n4!$ni'e wi"4 "4e !'" la"e!al #$de $% "4e n$!"4 'an1� P!$&!e''ivel( "4en*#)e! $% e$le wal3in& wa' in!ea'ed a' '4$wn )( "4e '"ai!a'e &!a41 T4e)!id&e de3 aele!a"i$n 7l$""ed )el$w "4e '"ai!a'e &!a4 in!ea'ed 'li&4"l(*n"il ,>> e$le we!e wal3in&. w4en "4e!e wa' a '*dden in!ea'e in de3la"e!al !e'$n'e w4i4 wa' '*ien"l( vi$len" "$ '"$ "4e "e'"1 +ine. w4en%*ll(-laden. "4e n$!"4 'an an a$##$da"e e!4a' ?== e$le. "4e !ea'$n%$! "4e !$)le#' $n $enin& da( i' aa!en"1

T*e performan'e targets for t*e modified bridge -ere e3pressed as rms a''e,eration ,eve,s

measured at t*e 7uarter and *a,f4span points -it* a minute averaging time& T*e ,atera,

target# after fi,tering -it* a passband of /&2 to 2&6 >8 -as t*at t*e rms s*ou,d not e3'eed

2<3/45g ,atera,,y t*e verti'a, target in a passband of /&2 to 6&= >8 -as t*at t*e rms s*ou,d

not e3'eed </3/45g verti'a,,y& T*ese targets -ere to be met in t*e presen'e of a test in

-*i'* 2#/// peop,e -a,(ed over t*e bridge t*ree times at /&F mKs# /& mKs and &2 mKs

appro3imate,y -it* t*e bridge 'omfortab,y fu,, of peop,e& A great dea, of p,anning -ent

into t*e organisation and 'arrying out of t*is test -*i'* -as su''essfu,,y 'omp,eted on 5/

.anuary 2//2& Measured a''e,eration ,eve,s -ere substantia,,y be,o- t*e target ,imits for

a,, t*e tests# typi'a,,y ,ess t*an one si3t* of t*e agreed ,imits&



Figure 7. 8nset of instability in cro!d test on undamped north span fundamental mode. s the number of people !al$ing on the

span -upper graph/ increased to 177 progressi%ely the bridge lateral acceleration -lo!er graph/increased only slo!ly until instability

!as reached. The scale for acceleration is not gi%en but the pea$ acceleration reached !as about 94)14 (+ g at the right(hand side

-rup figure from eyan Sud2ic -ed/ 3441 p. 5+/

CONCLU+ION+

T4e in"!$d*"i$n $% da#in& )( a $#)ina"i$n $% %!a#e-#$*n"ed vi'$*'da#e!' and "*ned-#a'' vi)!a"i$n a)'$!)e!' 4a' *!ed "4e L$nd$n Millenni*#B!id&e;' %a#$*' w$))le1 I" wa' a*'ed )( '(n4!$n$*' la"e!al e5i"a"i$n %!$#ede'"!ian'. a 4en$#en$n "4a" wa' n$" well-3n$wn a" "4e "i#e )*" %$! w4i4

"4e!e i' n$w a &$$d *nde!'"andin& and &$$d da"a1

AC/NO<LEDGEMENT+

T4e )!id&e;' en&inee!in& de'i&ne!' we!e "4e Ove A!* Pa!"ne!'4i and "4e(de'i&ned. "e'"ed and '*e!vi'ed "4e $n'"!*"i$n $% "4e vi)!a"i$n $n"!$l'('"e# de'!i)ed in "4i' ae!1 T4e a*"4$!;' !$le wa' a' indeenden" "e4nialadvi'$! "$ "4e L$nd$n Millenni*# B!id&e T!*'" 7"4e )!id&e;' !inial %*ndin&)$d( %$! "4e d*!a"i$n $% "4e !e#edial w$!3 de'!i)ed a)$ve1 D*!in& "4i'e!i$d. J*l( 2=== "$ Jan*a!( 2==2. 4e wa' lea'ed "$ w$!3 wi"4 "4e Ove A!*Pa!"ne!'4i. and wi"4 "4e <1 +1 A"3in' G!$* w4$ we!e advi'in& "4e L$nd$n

B$!$*&4 $% +$*"4wa!3 and "4e C$!$!a"i$n $% L$nd$n1� Tw$ de"ailed ae!' )(Ove A!* a)$*" "4e )!id&e a!e li'"ed )el$w1 T4e in"e!!e"a"i$n $% e5e!i#en"al!e'*l"' in "e!#' $% "4e %eed)a3 #$del de'!i)ed in a!" , $% "4i' ae! i'$!i&inal "$ "4e a*"4$!1

REFERENCE+



• H Ba4#ann. KCa'e +"*die' $% +"!*"*!e' wi"4 Man-Ind*ed Vi)!a"i$n'. Journal of Structural Engineering. T!an' A+CE. 118. >8,->? 7,21

• H*&$ Ba4#ann and <al"e! A##ann. KVi)!a"i$n' in +"!*"*!e' Ind*ed)( Man and Ma4ine'. Structural Engineering Document 3e.In"e!na"i$nal A''$ia"i$n %$! B!id&e and +"!*"*!al En&inee!in& 7IAB+E.

41 2: Man-Ind*ed Vi)!a"i$n'. and Aendi5 A: Ca'e Re$!"' 7,@?1

• Pa" Dalla!d. T$n( Fi"9a"!i3. An"4$n( Flin". An&*' L$w. R$&e! Rid'dill+#i"4. Mi4ael <ill%$!d and Ma!3 R$4e. KL$nd$n Millenni*# B!id&e:Pede'"!ian-Ind*ed La"e!al Vi)!a"i$n. J. of Bridge Engineering. T!an'1A+CE. 6. ,2-,? 72==,1

• T$n( Fi"9a"!i3. Pa" Dalla!d. +$4ie Le B$*!va. An&*' L$w. R$&e! Rid'dill+#i"4 and Mi4ael <ill%$!d. KLin3in& L$nd$n: T4e Millenni*# B!id&e.R$(al Aade#( $% En&inee!in&. L$nd$n. Pae! I+BN , @?,>8 ?72==,1 An a#ended ve!'i$n $% "4i' ae! wi"4 addi"i$nal #a"e!ial wa'

'*)'e6*en"l( *)li'4ed a' P1 Dalla!d e" al1 KT4e L$nd$n Millenni*#F$$")!id&e. The Structural Engineer . 79. N$1 22. ,?-88 72==,1

• $9$ F*in$. Beni"$ M Pa4e$. +4*n-i4i Na3a#*!a and Penn*n&<a!ni"4ai. K+(n4!$ni9a"i$n $% H*#an <al3in& O)'e!ved d*!in& La"e!alVi)!a"i$n $% a C$n&e'"ed Pede'"!ian B!id&e. Earthquake Engineeringand Structural Dynamics. 22. ?,-?@ 7,81

• De(an +*di 7ed. Blade of Light: The Story of London’s illenniumBridge. Pen&*in B$$3' L"d. L$nd$n 72==,. 4a"e! )( David E1 Newland.KVi)!a"i$n: !$)le# and '$l*"i$n. @@-81

Vibration of the Millenium Bridge London

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