7/21/2019 Vibration of the Millenium Bridge London http://slidepdf.com/reader/full/vibration-of-the-millenium-bridge-london 1/16 VIBRATION OF THE LONDONM MILLENNIUM FOOTBRIDGE: PART 1 - CAUSE David E Newland Department of Engineering University of Cambridge Trumpington Street CAMBRIDGE CB2 !"# U$ den%eng&'am&a'&u( A!"#a$" )*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 of t*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# a fra'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(&
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
7/21/2019 Vibration of the Millenium Bridge London
)*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&�������������������������������
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(&
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&
7/21/2019 Vibration of the Millenium Bridge London
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
7/21/2019 Vibration of the Millenium Bridge London
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$es their interaction !ith the bridge more
7/21/2019 Vibration of the Millenium Bridge London
predicable and helps them maintain their lateral balance. This instincti%e beha%iour ensures that the footfall forces are applied at a resonant fre&uency of the bridgeand !ith a phase such as to increase the motion of the bridge. s the amplitude ofthe motion increases the lateral force imparted by indi%iduals increases as doesthe degree of correlation bet!een indi%iduals. The fre&uency "loc$(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
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
7/21/2019 Vibration of the Millenium Bridge London
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
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
A" a&e 2= $% Fi"9a"!i3 e" al 72==,. A!* &ive "4ei! %$!#*la $!!e'$ndin& "$7,> a'
�
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
7/21/2019 Vibration of the Millenium Bridge London
• 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
'*)'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