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THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY WITH
INTERNATIONAL MENTION FOR THE UNIVERSITY OF SEVILLE
P oposal and Calib a ion o a Biodynamic
Model o Human-S uc u e In e ac ion by he
Resolu ion o he In e se Dynamic P oblem.
Applica ion o Pedes ian B idges.
Ja ie Fe nando Jiménez Alonso
Depa men o Con inuum Mechanics and S uc u al Analysis
School o Enginee ing
UNIVERSITY OF SEVILLE
Se ille, Spain 2015
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P oposal and calib a ion o a biodynamic model o human-s uc u e in e ac ion by
he esolu ion o he in e se dynamic p oblem. Applica ion o pedes ian b idges.
PhD’s Thesis in he Dynamic o S uc u es and Ea hquake Enginee ing p og am.
Ja ie Fe nando Jiménez Alonso
Ad iso : P o . D . And és Sáez Pé ez.
Depa men o Con inuum Mechanics and S uc u al Analysis
School o Enginee ing
Uni e si y o Se ille
Abs ac
In his hesis a biomechanical c owd-s uc u e in e ac ion model is p oposed and
u he implemen ed in o de o adequa ely es ima e he ene gy exchange be ween
pedes ians and oo b idge. The p oposed model ocuses on bo h he ib a ions in
he e ical and la e al di ec ions and i allows o ake in o accoun he change o
he modal p ope ies o he s uc u e due o he p esence o pedes ians, hus
imp o ing he nume ical es ima ion o he esponse o he s uc u e unde
pedes ian lows. I u he pe mi s o analyze in mo e de ail he la e al lock-in
phenomenon. The model in ol es wo sub-models, namely (i) a pedes ian-
s uc u e in e ac ion sub-model plus (ii) a c owd sub-model. The i s sub-model
ollows om a modal p ojec ion o a wo deg ee o eedom sys em ha simula es
he beha io o each pedes ian, on he ib a ion modes o he s uc u e. The
pa ame e s o his model a e es ima ed om he accele a ions eco ded on a eal
oo b idge by implemen ing an in e se dynamic app oach. Fo he second sub-
model, he c owd beha io is simula ed ia a mul i-agen me hod. The pe o mance
o he esul ing o e all model is assessed by co ela ing he expe imen al and
nume ical dynamic beha io o wo eal oo b idges. In pa icula wo phenomena
a e analyzed in de ailed: (i) he change in he i s e ical na u al equency o a
eal oo b idge induced by he pedes ian-s uc u e in e ac ion and (ii) he
occu ence o he la e al lock-in phenomenon due o he pedes ian ac ion. The
p oposed model leads o nume ical esul s ha exhibi good ag eemen wi h he
ob ained expe imen al alues. The e o e, i becomes a aluable ool o accoun o
he change on he modal p ope ies o a oo b idge induced by he c owd-s uc u e
in e ac ion phenomenon. The conside a ion o his ac o allows es ima ing mo e
accu a ely he dynamic esponse o he oo b idge unde he pedes ian ac ion,
analyzing in mo e de ailed he occu ence o he la e al lock-in phenomenon o
imp o ing he e iciency in he design o passi e and ac i e dampe s i hei
ins alla ion was necessa y o gua an ee an adequa e com o le el on he
oo b idge.
Keywo ds: simpli ied biomechanical model, human-s uc u e in e ac ion, c owd
dynamics, change o na u al equencies, ope a ional modal analysis, model
upda ing, pa ame e iden i ica ion, la e al lock-in phenomenon
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PREFACE
This Thesis has been ca ied ou a he Depa men o Con inuum Mechanics and
S uc u al Analysis a he Uni e si y o Se ille. The wo k has been supe ised by
Full P o esso D . And és Sáez Pe ez. I always owe my deepes g a i ude o And és
o encou aging and suppo ing me cons an ly du ing he de elopmen o his wo k.
This esea ch would no ha e been possible wi hou his in ini e pa ien and help ul
guidance o he o ganiza ion o he pape s. Thank him o his ime, e o and
iendship. I eel eally lucky o ha ing had he oppo uni y o de elop his wo k
unde his u elage.
Addi ionally, I would like o exp ess some lines o g a i ude o hose who ha e
con ibu ed o he de elopmen o he esea ch ca ied ou by he au ho o his
Thesis.
To P o . Al a o Cunha, who p o ided me a nice wo kplace du ing my esea ch s ay
a he Labo a o y o Vib a ions and S uc u al Moni o ing (ViBes ) o he Uni e si y
o Po o (Po ugal), allowing me o imp o e my knowledge o he ope a ional modal
analysis me hodology and pe o m se e al expe imen al es s o g ea impo ance
o he de elopmen o his wo k. The esul s I collec ed du ing my esea ch s ay
cons i u e a i al pa o my Thesis and would no ha e been possible wi hou he
suppo o Associa e P o esso Elsa Cae ano and Assis an P o esso Filipe
Magalhães.
To P o . Alexande Pa ic, o ecei ing me in o his g oup o he Vib a ion
Enginee ing Sec ion o he Uni e si y o Exe e (U.K.) du ing my second esea ch
s ay allowing me o in oduce mysel in he in e es ing ield o he con ol o ci il
enginee ing s uc u es.
To my pa en s, An onio and Ma ia del Ca men, who augh me he alue o he
educa ion and who made a ema kable e o o us, hei child en, so ha we could
ha e all he oppo uni ies ha hey did no ha e.
To my colleagues (pas and p esen ) a he Depa men o Building S uc u es o
he Uni e si y o Se ille and a he B idge Enginee ing Fi m, IDES, o hei
cons an suppo .
This hesis is based on scien i ic pape s which ha e al eady been accep ed o
publica ion in ele an scien i ic jou nals o p esen ed a in e na ional con e ence
wi h pee - e iew. Finally, wo addi ional pape s, cu en ly unde e iew, ha e been
included.
Las bu no leas , I am e y g a e ul o he lo ing suppo o my amily, in
pa icula my b illian and comp ehensi e wi e, Pa icia, and ou wo lo ely
daugh e s, Claudia and Lo ena. I hope ha one day I could compensa e he ime
ha I s ool hem o he de elopmen o his wo k.
To Ma ibel o he con inuous suppo , o me and my amily, du ing hese di icul
yea s.

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My g a i ude goes also o o he ela i es and iends ha ha e helped me o
o e ake success ully all he di icul ies o he achie emen o his Thesis.
Se ille, Oc obe 2015
Ja ie Fe nando Jiménez Alonso
ACKNOWLEDGEMENTS
This wo k was pa ially unded by he Spanish Minis y o Science unde esea ch
p ojec DPI2014-53947-R.
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THESIS
This Thesis consis s o an ex ended summa y and he ollowing appended pape s:
Pape A
J.F. Jiménez-Alonso and A. Sáez
A di ec -pedes ian s uc u e in e ac ion model o cha ac e ize he
human induced ib a ions on slende oo b idges
In o mes de la Cons ucción, Vol. 66 (Ex a 1). m007
Pape B
J.F. Jiménez-Alonso, A. Sáez, E. Cae ano, F. Magalhães
Ve ical c owd–s uc u e in e ac ion model o analyze he change o he
modal p ope ies o a oo b idge
Jou nal o B idge Enginee ing. ASCE (in p ess)
Pape C
J.F. Jiménez-Alonso and A. Sáez
Model upda ing o he selec ion o he e o i me hod o an ancien
b idge (Alme ia, Spain).
S uc u al Enginee ing In e na ional. IABSE (in p ess).
Pape D
J.F. Jiménez-Alonso and A. Sáez
Con olling he human-induced longi udinal ib a ions o a Nielsen- uss
oo b idge ia he modi ica ion o i s na u al equencies
Unde e iew
Pape E
J.F. Jiménez-Alonso, A. Sáez, E. Cae ano and A. Cunha
La e al c owd-s uc u e in e ac ion model o analyze he la e al lock-in
phenomenon on a eal oo b idge
Unde e iew
Pape F
J.F. Jiménez-Alonso, E. Cae ano and A. Cunha
Dynamic es ing o Ca pin ei a oo b idge a Col ihã (Po ugal)
5 h In e na ional Ope a ional Modal Analysis Con e ence (Guima ães,
Po ugal) 13-15 May 2013
Pape G
J.F. Jiménez-Alonso and A. Sáez
Assessmen o he dynamic beha io o Palmas Al as oo b idge a
Se ille (Spain)
37 h IABSE Symposium. Mad id (Spain) 3-5 Sep embe 2014
The appended pape s we e p epa ed in collabo a ion wi h co-au ho s. The au ho o
his Thesis is esponsible o he majo p og ess o wo k in hese pape s, including
he de elopmen /deduc ion o solu ions and nume ical me hods, pe o ming he
nume ical simula ions and expe imen al es s and w i ing he main pa s o he
pape s.
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Table o Con en s
I. Ex ended summa y ................................................................................... 1
1. In oduc ion. ............................................................................................ 1
1.1 Mo i a ion. .......................................................................................... 3
1.2 Objec i es. ......................................................................................... 3
2. Vib a o y p oblems due o pedes ian lows on oo b idges. ............................ 5
2.1. Load models o a single pedes ian. ...................................................... 5
2.2. Load models o c owds. ...................................................................... 8
2.3. Synch oniza ion and lock-in. ............................................................... 12
2.3.1. Models o he simula ion o he synch oniza ion and lock-in. ............. 13
2.3.2. Synch oniza ion and e ical lock-in. .............................................. 16
2.3.3. Synch oniza ion and la e al lock-in. ............................................... 17
2.4. The pe cep ion o he ib a ion. .......................................................... 18
2.5. Dynamic p ope ies o he s uc u es unde pedes ian ac ion. ................. 19
2.6. The con ol o he ib a o y esponse. .................................................. 20
2.6.1. Modi ica ion o he mass induced by pedes ian ac ion. ..................... 20
2.6.2. Modi ica ion o he s i ness induced by pedes ian ac ion. ................. 21
2.6.3. Modi ica ion o he damping induced by pedes ian ac ion. ................ 22
3. P oposal o a simpli ied biomechanical c owd-s uc u e in e ac ion model. ....... 23
3.1. Modelling he pedes ian-s uc u e in e ac ion. ...................................... 24
3.2. Modelling he c owd-beha io . ............................................................ 28
3.3. C owd-s uc u e in e ac ion. ............................................................... 33
4. In e se dynamic p oblem app oach. .......................................................... 35
4.1. In e se dynamic p oblem: pa ame e iden i ica ion in e ical di ec ion. ... 36
4.2. In e se dynamic p oblem: pa ame e iden i ica ion in la e al di ec ion. ..... 37
5. Expe imen al es ima ion o he pa ame e s o he pedes ian-s uc u e
in e ac ion model. ...................................................................................... 40
5.1. Desc ip ion and ini e elemen model o he “labo a o y” oo b idge: Viana
oo b idge. ............................................................................................. 40
5.2. Expe imen al iden i ica ion o he modal pa ame e s o he “labo a o y”
oo b idge. ............................................................................................. 42
5.3. Model upda ing o he “labo a o y” oo b idge. ...................................... 45
5.4. Expe imen al pedes ian and c owd es s. ............................................. 47
5.5. Es ablishing a sea ch domain o he pa ame e s o he pedes ian-s uc u e
model. ................................................................................................... 49
5.6. Pa ame e iden i ica ion o he pedes ian-s uc u e in e ac ion model:
e ical di ec ion. .................................................................................... 50
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2. Vib a o y p oblems due o pedes ian lows on oo b idges.
In his sec ion a summa y o he main aspec s o he ib a o y p oblems induced by
he c owd-s uc u e in e ac ion is p esen ed. The sec ion includes he di e en
models ha ha e mainly in luenced he au ho o he de elopmen o he p oposed
c owd-s uc u e in e ac ion model. On he o he hand, his sec ion cons i u es a
b ie summa y o he s a e o he a abou his subjec .
2.1. Load models o a single pedes ian.
The i s models p oposed in o de o s udy he e ec o a pedes ian c ossing a
oo b idge we e based on he assump ion ha he pedes ian’s ac ion can be
app oxima ed by a ha monic o ce. F om his app oach a ises he p oposal o he
B i ish s anda d (BSI, 2006), he single model, ha la e was adop ed by o he
coun ies, as o ins ance, Canada (On a io, 1995) and Spain (RPM-95, 1995). This
model conside ed ha he e ec o he passage o a pedes ian on he s uc u e is
equi alen o a mo ing e ical sinusoidal o ce p
F (in N), wi h a pedes ian s ep
equency p
(in Hz), a pedes ian s ep eloci y p
(m/s) a


F pp

2sin180)(

[1]
pp 9.0)(

[2]
being he ime a iable (sec.).
F om he end o he las cen u y, se e al esea ches ha e ocused hei e o s on
he cha ac e iza ion o his o ce mo e p ecisely, including addi ional e ms in he
Fou ie se ies and conside ing i s e ec in he h ee spa ial di ec ions. In he
ollowing equa ions, [3 and 4], and in Table 2 and Figu e 2 a summa y o he main
p oposals epo ed is shown (Se a, 2006, Bu z e al., 2007, Racic e al., 2009).






 

nh
i
p e is e ip e p i P F
1
,,, 2sin1)(

[3]



 nh
i
pla isla ipla p i P F
1
,,, sin)(

[4]
whe e
e p
F,is he e ical pe iodic o ce due o walking.
la p
F, is he la e al pe iodic o ce due o walking.
700
p
PN is he mean pedes ian’s weigh (Bu z, e al.,2007).

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e i,

and la i,

a e he Fou ie coe icien s o he i h ha monic o e ical and la e al
o ce, dynamic load ac o s (DLFs)
s
[Hz] is he s ep equency.
e i,

and la i,

a e phase shi o he i h ha monic.
nh is o al numbe o con ibu ing ha monics.
p

is he phase shi among pedes ians.
F om he analysis o he esul s p o ided by Table 2, i can be concluded ha a
leas wo ha monics a e necessa y o cha ac e ize adequa ely he e ical
pedes ian o ce while h ee ha monics a e necessa y o he la e al di ec ion. On
he o he hand, he la e al componen o he pedes ian o ce is cha ac e ized by
equencies ha a e he hal o he equencies ansmi ed in he e ical di ec ion.
The ela ionship be ween he pedes ian eloci y ( p
) and he pedes ian s ep
equency ( s
) has been s udied by di e en au ho s. The wo ks epo ed by Bu z
e al. (2007), Ricca delli and Pizzimen i (2007), Ricca delli e al. (2007) and
Be am and Ruina (2001) may be highligh ed. The ul ima e p oposal,
in e na ionally accep ed by he scien i ic communi y, is go e ned by he ollowing
equa ion.
32 35.059.193.2 ppps  [5]
La ely, his ela ionship will be conside ed in he c owd-s uc u e in e ac ion model
p oposed in his wo k.
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Table 2. Dynamics Load Fac o s (DLFs) acco ding o di e en au ho o walking
ac ion in e ical and la e al di ec ion (Se a, 2006; Bu z e al., 2007; Racic e al.
2009).
Au ho Fou ie Coe ./Phase Commen a ies
Ac ion-Di ec ion
Blancha d e
al. (1977) 1, e =0.257 Walking-Ve ical
Bachmann
& Ammann
(1987)
1, e =0.40-0.50;
2, e =3, e =0.10
s=2.00-4.00 Hz
Walking-Ve ical
Schulze
(1980)
1, e =0.37;2, e =0.10;
3, e =0.12;4, e =0.04;
5
,
e
=0.015;
s=2.00 Hz
Walking-Ve ical
Bachmann
e al. (1995)
1, e =0.40/0.50;
2, e =3, e =0.10;
1/2,la 1,la =3/2,la =0.10;
2=3=pi/2;
s=2.00-2.40 Hz
s=2.00 Hz
s=2.00 Hz
Walking-Ve ical
Walking-Ve ical
Walking -La e al
Walking-Ve ical-
La e al
Ke (1998)
1, e =0.40/0.50;
2, e =3, e =0.10;
1 acco ding o
he equency
Walking-Ve ical
Young
(2001)
1, e =0.37 ( p-0.95) ≤0.50
2, e =0.054+0.0088 s
3, e =0.026+0.015 s
4
,
e
=0.01+0.0204 s
Mean alues o
Fou ie Coe .
Walking-Ve ical
EC5 (2003) 1, e =0.40;2, e =0.20
1
,
la =2
,
la =0.10
Walking-Ve ical
Walking-La e al
SETRA
(2006)
1, e =0.40
2, e =3, e =0.04
2, e =3, e =pi/2;
1/2,la =3/2,la =0.05
1
,
la =2
,
la =0.01
Walking-Ve ical
Walking-Ve ical
Walking-Ve ical
Walking-La e al
Walking-La e al
SYNPEX
(2007)
1, e =0.0115 s2+0.2803 s-
0.2902
1, e =0.00 [º]
2, e =0.0669 s2+0.1067
s-0.0417
2, e = -99.76 s2+478.92 s -
387.80 [º]
3, e =0.0247 s2+0.1149
s-0.1518
I s<2.00 Hz
3, e = -150.88 s3+819.65 s2 -
1431.35 s+811.93 [º]
I s≥2.00 Hz 
3, e = 813.12 s3-5357.60 s2
+11726.00 s -8505.90 [º]
4, e =-0.0039 s2+0.0285
s-0.0082
4
,
e
= -34.19 s-65.14 [º]
Fou ie Coe .
and phases o
mean
pedes ian
loads
Walking-Ve ical
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2.2. Load models o c owds.
The main limi a ion o he abo e models is ha hey a e no able o p edic he
esponse o he oo b idge when he s uc u e is subjec ed o a pedes ian low,
being necessa y, he e o e, o de elop models ha accoun o he beha io o he
c owd. The me hodology used mo e equen ly in he li e a u e consis s in
mul iplying he esponse o a single pedes ian by a ac o ha conside s globally
he e ec o he c owd. Among he di e en p oposed models, i is p esen ed below
a summa y o he mos his o ically in luen ial, acco ding o he au ho ’s c i e ion.
The i s conside ed model was p oposed by Ma sumo o e al. (1978) and akes in o
accoun as mul iplica ion ac o he magni ude p
n, being p
n he numbe o
pedes ians on he deck a ce ain ins an . This ac o es ablishes, acco ding o a
Poisson dis ibu ion, he p opo ion o pedes ians ha due o he haza d a e
mo ing in phase, a oiding he e ec o he es o indi iduals. Howe e , his ac o
was used, wi hou success, in o de o p edic he esponse o he T- oo b idge
(Tokyo) ha unde wen ib a o y p oblems due o he la e al synch oniza ion o he
pedes ians (Zi ano ic e al., 2005).
Subsequen ly, he Swiss s anda ds, SIA 260 (2003), p esen ed as no el y he
modi ica ion o he mul iplica ion ac o in unc ion o he pedes ian densi y. In ha
sense, i p oposes: (i) up o 10 pedes ians, a linea law wi h se e al sec ions and a
maximum alue o 3; (ii) up o a pedes ian densi y o 0.30 P/m2 (Pe sons/m2) i
accep s he abo e p oposal and (iii) up o abou his poin i es ablishes a pa abolic
law wi h a maximum ac o o 20. The e o e, he pedes ian densi y de e mines a
egime change o he pedes ian beha io : while alues lowe han 0.30 P/m2
allows he ee mo emen o he pedes ians, as he pedes ian densi y inc eases
hei ee mo emen becomes di icul , so ha he pedes ians end o synch onize.
The Eu ocode (2002) es ablishes, simila ly o he Swiss s anda ds, h ee load
models acco ding o he expec ed pedes ian densi y. The i s model, DLM1, which
is he basis o he es , is used o cha ac e ize he ac ion o a single pedes ian,
being i s ac ion de ined as a mo ing sinusoidal o ce wi h wo spa ial componen s:
one e ical, o alue 280 N and one la e al, wi h magni ude 70 N. Bo h he
pedes ian s ep equency and he pedes ian eloci y ollow he c i e ion o he
B i ish s anda d (BSI, 2006). The second model, DLM2, cha ac e izes he ac ion o
a g oup o up o 15 pedes ians. The esponse o he oo b idge unde he ac ion o
he g oup is calcula ed mul iplying he esponse o he model DLM1 by a ac o ,
wi h a maximum alue o 3, which akes in o accoun he p obabili y ha a
esonance phenomenon occu s on he s uc u e due o he ac ion o he g oup o
pedes ians. The modi ica ion o he modal p ope ies o he oo b idge due o he
pedes ian e ec is quan i ied h ough he addi ion, a he poin wi h a maximum
modal de lec ion, o a poin load o 800 kg. The las model, DLM3, applicable o
scena ios unde a con inuous pedes ian low, cha ac e izes he ac ion o he c owd
h ough a ha monic load equi alen o he weigh o a pedes ian densi y 0.60 P/m2
mul iplied by wo ac o s in o de o accoun o he possible esonance be ween
he pedes ians and he s uc u e, as well as he ansi o y cha ac e o he load.
The i s ac o , adop s a a iable alue anging be ween 0.05-0.30 and he second
ac o is equal o 0.75. The equi alen load is only applied in he pa o he
s uc u e whe e he modal de o ma ion has he same sign as he load and i s e ec
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is un a o able. The pedes ian load is applied o he same equency han he
model DLM1. Finally, he model assumes ha he pedes ian low p oduces an
inc ease o he modal mass o he s uc u e o 40 kg/m2.
Figu e 2. Ve ical and La e al pedes ian walking o ce acco ding o di e en
au ho s (Se a, 2006; Bu z e al., 2007 and Racic e al., 2009).
The F ench s anda d, Se a (2006), based on bo h he esea ch conduc ed on he
Sol e ino oo b idge (Pa is, F ance) and labo a o y es s on eadmills, p esen s a
new me hodology ha has been widely adop ed by esea che s and designe s. The
me hodology has been accep ed equally by he Eu opean Resea ch P ojec , SYNPEX
(Bu z e al., 2007). The p oposed me hod simula es he e ec o he pedes ian
lows as an equi alen uni o m dis ibu ed load applied acco ding he conside ed
ib a ion mode and whose alue is equal o he e ec o he g oup o pedes ians
ha a e synch onized among hem. This magni ude is named as equi alen numbe
o pedes ians, 'n. In o de o de e mine he numbe o pedes ians on he
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
1400.00
0.00 0.20 0.40
Ve ical Load [N]
Time [sec.]
Blancha d e al. (1977)
Bachmann & Ammann (1987)
Schulze (1980)
Bachmann e al. (1995)
Ke (1998)
Young (2001)
Eu ocode 5 (2003)
Se a (2006)
Synpex (2007)
-200.00
-150.00
-100.00
-50.00
0.00
50.00
100.00
150.00
200.00
0.00 0.20 0.40 0.60 0.80 1.00
La e al Load [N]
Time [sec.]
Bachmann e al. (1995)
Eu ocode 5 (2003)
Se a (2006)
10

oo b idge, he s uc u e is classi ied acco ding o he expec ed a ic le el. In
Table 3, he ou possible a ic ypes a e speci ied (d=pedes ian
densi y=Pe sons/m2) which allow gene a ing on he s uc u e he di e en load
scena ios. A com o le el will be associa ed wi h each a ic le el.
The equi alen uni o m load applied on he deck o he s uc u e is ob ained om
he ollowing equa ion.








')2cos()( p oo n G p [6]
whe e
G is he componen o he pedes ian load ( 280

G N o he e ical walking and
35G N o he la e al walking).
oo
is he na u al equency o he s uc u e unde conside a ion.
'
p
nis he equi alen numbe o pedes ians acco ding o Table 3.

is he educ ion coe icien o ake in o accoun he p obabili y ha he oo all
equency app oaches he na u al equency unde conside a ion (Figu e 3).
Figu e 3. Reduc ion ac o  e sus he na u al equency o he s uc u e (Bu z e
al., 2007).
The equi alen numbe o pedes ian depends on he pedes ian densi y on he
s uc u e, he na u al equency unde conside a ion and he a io be ween he
s uc u al and he c i ical damping,

. In Table 3, se e al p oposals o i s
de e mina ion a e shown.
0.00
0.25
0.50
0.75
1.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
Reduc ion coe icien 
F equency [Hz]
Ve ical. 1º Ha monic
Ve ical. 2º Ha monic
La e al

11

Table 3. Equi alen numbe o pedes ians, '
p
n, acco ding o F ench code (Se a,
2006).
Range o na u al equencies [Hz]
T a ic
Class
d
[P/m2]
1.70-2.10 1.00-1.70
2.10-2.60
2.60-5.00 <1.00
>5.00
IV <0.20 --- --- --- ---
III 0.50

p
n8.10 --- --- ---
II 0.80

p
n8.10

p
n8.10

p
n8.10 ---
I 1.00
p
n85.1 p
n85.1 p
n85.1 ---
Meanwhile, he s uc u al damping a ios o di e en cons uc ion ypes can been
ob ained om Table 4 (Se a, 2006 and Bu z e al., 2007).
Table 4. Damping s uc u al a ios o Se ice and Ul ima e Limi s a es (Se a,
2006 and Bu z e al., 2007).
Cons uc ion ype
S.L.S. U.L.S.
min

[%] med

[%] max

[%]
Rein o ced conc e e 0.80 1.30 5.00
P es essed conc e e 0.50 1.00 2.00
Composi e s eel-conc e e 0.30 0.60 2.00
Welded s eel 0.20 0.40 2.00
Sc ewed s eel 0.20 0.40 4.00
Timbe 1.00 1.50 4.00
S ess- ibbon 0.70 1.00 2.00
Elas ome s ---- ---- 7.00
Thus, he mos ad anced in e na ional s anda ds (Se a, 2006 and Bu z e al.,
2007) es ima e he modi ica ion o he dynamic p ope ies o he oo b idge due o
he pedes ian lows only by conside ing he modi ica ion o he modal mass o he
s uc u e, adding di ec ly he passi e mass implied by he pedes ians. The e ec o
he second ha monic ha cha ac e izes he pedes ian s ep is only conside ed in
oo b idges unde pedes ian densi ies la ge han 0.80 P/m2.
Howe e , hese in e na ional s anda ds p esen he ollowing limi a ions: (i) a
simpli ied es ima ion o he change o he dynamic p ope ies o he oo b idge due
o he p esence o he pedes ians, (ii) he conside a ion o he synch oniza ion
phenomenon om an expe imen al ela ionship ob ained om es s on only one
eal oo b idge, (iii) DLFs ob ained om labo a o y es , (i ) a o mula ion ha does
no i well o he case whe e se e al ib a ion modes o he oo b idge a e in he
ange ha cha ac e izes he pedes ian-s uc u e in e ac ion, and ( ) hey do no
ake in o accoun he e ec o he non-synch onized pedes ians.
In he p esen Thesis a c owd-s uc u e in e ac ion will be p oposed and calib a ed
in o de o o e come he abo e limi a ions.
12

2.3. Synch oniza ion and lock-in.
In he con ex o he pedes ian-s uc u e in e ac ion, he synch oniza ion e lec s
he endency o he pedes ians o walk wi h he same spacing and phase among
hem, while he lock-in e lec s he endency o he pedes ians in coupling hei
s ep wi h he ib a o y mo emen o he s uc u e. This phenomenon can ha e a
delibe a e cha ac e ( andalism) o unin en ional. This las case, esponsible o
many o he ib a o y p oblems o oo b idges de ec ed du ing he las yea s, can
be gene a ed by wo di e en mechanisms, acco ding o he scheme shown in
Figu e 4.
Figu e 4. Flowcha o unin en ional o pedes ian-s uc u e synch oniza ion (Racic
e al., 2009).
The i s synch oniza ion mechanism may occu when he pedes ian densi y on he
oo b idge, d, is lowe han a c i ical alue, c
d (limi densi y o which he
mo emen o a pedes ian is in luenced by he es o he g oup), and he alue o
he ampli ude o he deck induced by he pedes ians, u, was uppe a limi alue,
c
u, ha ma ks he limi alue om he pedes ians end o synch onize wi h he
mo ion o he deck.
On he o he hand, a second mechanism, ha o igina es he synch oniza ion, may
occu , unde high pedes ian densi ies, i he mass p o ided by he pedes ians,
M
, is uppe a c i ical alue, c
M, equal o he mass o he pedes ians which ine ial
o ce may induce a ib a ion ampli ude, c
u.
In he ollowing sec ions, a li e a u e e iew o he main exis ing models o he
analysis o he lock-in phenomenon in oo b idges is p esen ed. The p ac ical
applica ion o hese s udies, adop ed by he cu en s anda ds, is also desc ibed.
Pedes ian dens iy d [P/m
2
]
d≤d
c
d>d
c
u≤u
c
u>u
c
M≤M
c
M>M
c
Fo ced
Vib a ion Fo ced
Vib a ion Synch oniza ion
Synch oniza ion
13

2.3.1. Models o he simula ion o he synch oniza ion and lock-in.
The s udy o he p oblem o synch oniza ion be ween pedes ians and he
oo b idge, he lock-in phenomenon, has been pe o med his o ically independen ly
o he simula ion o he beha io o he c owd, o in he bes case, an addi ional
checking c i e ion has been es ablished (Se a, 2006 and Bu z e al., 2007). The
i s epo ed phenomenon o his ype occu ed in a Ge man oo b idge (1972),
du ing i s opening, when a ib a ion mode o 1.10 Hz was exci ed by 300-400
pedes ians, as i is desc ibed by Bachmann and Ammann (1987) in hei book. The
com o le el o he oo b idge was gua an eed by he addi ion o se e al uned
mass dampe s wi hou gi ing addi ional impo ance o he phenomenon.
Nex , a li e a u e e iew o he mos ele an models o he analysis o he
phenomenon is p esen ed summa ized.
One o he i s models, p oposed by Fujino e al. (1993), was de eloped om he
esul s o he s udy o he la e al lock-in phenomenon in he T- oo b idge (Tokyo).
By he analysis o ideo images o 2000 pedes ians c ossing he s uc u e, hey
es ima ed ha 20% o he pedes ians we e synch onized wi h he oo b idge, wi h
a maximum la e al displacemen o he deck o 10 mm and a la e al na u al
equency o 0.90 Hz. I was es ablished, o he i s ime, as cause o he
ib a o y p oblem he synch oniza ion be ween he pedes ians and he s uc u e.
F om hese esul s, a gene al calcula ion ule was p esen ed, es ablishing a ixed
alue o he synch oniza ion o 20% (among pedes ian wi h each o he and wi h
he s uc u e) and a alue o he mean la e al o ce gene a ed by a pedes ian o 35
N. The model did no conside nei he he inc ease o he synch oniza ion wi h he
ampli ude o he mo emen o he deck no he e ec o he pedes ians ha only
was synch onized wi h each o he bu no wi h he deck.
This model was unsuccess ully applied o he s udy o he dynamic beha io o he
Millennium oo b idge (London) du ing i s design phase. A e he ib a o y
p oblems de ec ed in i , la ge scale and labo a o y es s we e pe o med in o de o
calib a e a new p oposal o simula ing he pedes ian beha io . The pedes ian
ac ion, acco ding o Dalla d (Dalla d e al., 2001), may be app oxima ed as a o ce
ha depends on he eloci y o he deck and a nega i e coe icien o pedes ian
damping. In his way, he inc ease o he pedes ians on he oo b idge p oduces a
educ ion o he global damping o he s uc u e, o such an ex en ha a dynamic
ins abili y s a e may be eached, which is addi ionally a o ed by he
complemen a y synch oniza ion p ocess expe ienced by he pedes ians. The
concep o he equi alen numbe o pedes ians is in oduced as he numbe o
pedes ians ha elimina e he damping o he sys em. The p ac ical applica ion o
his model is e lec ed by he A up o mula. The Millennium oo b idge expe ienced
a la e al mo emen o 50 mm wi h a na u al equency o 0.80 Hz in he la e al
span and a la e al mo emen o 75 mm wi h a na u al equency o 1.00 Hz in he
cen al span, synch onizing he mo emen o 50% o he pedes ians ha c ossed
he s uc u e. The esul ing alue o he la e al o ce ansmi ed by each
pedes ian, 30 N, is simila o he alue p oposed by Fujino e al. (1993). On he
o he hand, he model p o ided, a limi alue o he pedes ian densi y o 1.50
P/m2, om which is so ha d o walk on he deck ha he dynamic e ec s a e
negligible. Ne e heless, he model p esen s some limi a ions: (i) he model igno es
he ene gy ansmi ed by he no -synch onized pedes ians wi h he s uc u e, (ii)
14

i does no ake in o accoun he change o he modal p ope ies o he s uc u e
due o he p esence o he pedes ians, (iii) he change o he beha io o he
pedes ians wi h he ib a ion le el o he s uc u e is no conside ed and (i ) he
alue o he equi alen pedes ian damping is only es ima ed o one oo b idge and
speci ic ange o equencies.
In o de o conside in a mo e app op ia e manne he change o he beha io o
he pedes ians induced by he ib a ion le el, Nakamu a e al. (2004) modi ied
Dalla d’s p oposal (Dalla d e al., 2001) in o de o ake in o accoun ha ,
acco ding o he obse a ions pe o med on he T and M oo b idges (Tokyo), om
ce ain alue o he la e al displacemen , 10 mm, he pedes ians modi ied hei
s ep o gua an ee an adequa e com o le el, u he educing he la e al o ce
o igina ed by hei s ep. In ha way, a sa u a ion ac o is in oduced in each
pedes ian la e al o ce ha a oids ha he o ce inc eases linea ly wi h he eloci y
o he deck inde ini ely. This p oposal educes he a io o he inc ease o he
eloci y when he eloci y inc eases un il making i null. Al hough his model is an
imp o emen as compa ed o he p e ious p oposals, i has as main limi a ion he
necessi y o knowing he maximum displacemen o he s uc u e o scale he
sa u a ion a io, while sha es he o he limi a ions o he p e ious models.
Subsequen ly, he F ench s anda d (Se a, 2006) p oposed a mo e compac model
conside ing he esul s o wo ypes o es s, on eadmills a labo a o y and la ge
scale pedes ian es s on Sol e ino oo b idge (Pa is). As conclusions o hese wo
se s o es s: (i) an expe imen al ela ionship ha allows es ima ing he numbe o
synch onized pedes ians on he oo b idge was p oposed, as well as (ii) a new
c i e ion in o de o de e mine he sensi i i y o he oo b idge o he la e al lock-in
phenomenon. I was e i ied, in his sense, ha a change o egime in he
pedes ian beha io occu s, om o ced o synch onized ib a ion, wi h a maximum
synch oniza ion a io o 60% o a la e al accele a ion o he deck be ween 0.10-
0.15 m/s2. This limi accele a ion has been adop ed as a c i e ion in o de o
de e mine he sensi i i y o he oo b idge o he la e al lock-in.
In he las i e yea s, se e al mo e sophis ica ed models, no ye implemen ed in
he in e na ional s anda ds, ha e appea ed p o iding a new app oach o he
p oblem. Among hese models, he mos in luen ial ones o he de elopmen o
his Thesis a e desc ibed, in summa y, in he ollowing pa ag aphs
In he i s model, p oposed by Macdonald (Macdonald, 2008) and based on he
in e se pendulum model by Bake (Bake , 2002), he pedes ian is modelled as a
lumped mass a ached o he oo b idge by an inclined ba (Figu e 5). The
pedes ian unde he la e al ib a ions modi ies he il o his legs, sea ching his
s abili y, inc easing he la e al componen o he walking pedes ian o ce and being
able o a ain a dynamic ins abili y si ua ion. The p oposed model has been
accep ed o desc ibe he ini ia ion o he phenomenon, bu he e ec o modi ica ion
o he pedes ian s ep due o la ge la e al ib a ions is no included in he p oposal.
Despi e i s limi a ions, he e a e e olu ions o he model (Mo bia o e al., 2011)
wi h g ea e complexi y and p ecision.
21

On he basis o he spec al design model, Bu z (2006) de eloped an empi ical
exp ession o he de e mina ion o he equi ed modal mass, i
M (kg), o a gi en
pedes ian a ic o ensu e a equi ed com o le el unde he assump ion ha
some na u al equency o he s uc u e is in he ange ha cha ac e izes he
walking pedes ian ac ion.


lim
31 42 65.1
a
kkn
M
kk
p
i


 [12]
whe e
1
k o 4
k a e cons an s, as gi en in Table 8.
lim
ais he limi accele a ion acco ding o he conside ed com o le el (Table 6).
and, p
n, is he numbe o pedes ians.
Table 8. Cons an s o equi ed modal mass (Bu z, 2006).
Ve ical-To sion La e al
d [P/m2] k1 k
2 k
3 k
4 k
1 k
2 k
3 k
4
<0.50 0.7603 0.050
1.00 0.5700 0.4680 0.040 0.675 0.1205
0.4500
0.0120 0.6405
1.50 0.4000 0.035
2.6.2. Modi ica ion o he s i ness induced by pedes ian ac ion.
The alue o he na u al equencies o he oo b idge is p opo ional o he squa e
oo o he a io be ween he modal s i ness and mass o he s uc u e. In his
manne , la ge s uc u al modi ica ions a e necessa y i he na u al equencies o
he oo b idge mus be loca ed ou o he pedes ian-s uc u e in e ac ion ange.
The cu en end in he design o oo b idges, unde aes he ics, esis an and
economic equi emen s causes ha he abo e c i e ion is no always easible in
o de o gua an ee an adequa e com o le el (Slaich, 2005).
Howe e , he e a e occasions, whe e he i s na u al equency o he s uc u e is
inside he walking pedes ian ange, and he second na u al equency is ou side
ha ange, whe e i can be easonable o educe he s i ness o he s uc u e so
bo h na u al equencies a e ou side he pedes ian-s uc u e in e ac ion ange and
checking addi ionally ha he s a ic de lec ion o he oo b idge is compa ible wi h
i s use (Se a, 2006 and Bu z e al., 2007).
The mos common s a egies in o de o modi y he na u al equencies o he
oo b idge, om he iewpoin o he s i ness, a e (Se a, 2006 and Bu z e al.,
2007): (i) inc ease he deg ee o s a ically inde e mina ion, (ii) p o ide s uc u al
cha ac e is ics o p o ec ion o su ace elemen s, and (iii) use cable sys ems wi h a
s abilizing unc ion (Se a, 2006). I is ecommended in oo b idges wi h a wid h
la ge han 4.00 m and spans wi h leng hs abo e 50.00 m, o ins all a la e al load
ansmission sys em, as an e ec i e me hod o con ol he la e al lock-in
phenomenon (Low, 2008). Finally, a high seismici y a eas, he inc ease o he

22

s i ness o he s uc u e in o de o a oid ib a o y p oblems induced by
pedes ians may cause, by con as , an inc ease o he seismic ac ion (Slaich,
2005).
2.6.3. Modi ica ion o he damping induced by pedes ian ac ion.
The inc ease o he s uc u al damping has been, un il he da e, he mos used
me hod o con ol he ib a ions induced by pedes ians on oo b idges (Fujino e
al., 1993; Dalla d e al., 2001 and Cae ano e al., 2010). This inc emen can be
achie ed ei he by he ac ua ion on in e nal elemen s o he s uc u e, o by he
implemen a ion o ex e nal con ol de ices o di e en na u e acco ding o hei
pe o mance: ac i e, semi-ac i e, hyb id o passi e de ices (Mou inho e al., 2010).
The mos usual is he use o passi e dampe s as: (i) iscous dampe s (Bu z e al.,
2007 and Taylo , 2003), (ii) uned mass, liquid o liquid column dampe s (Fujino e
al., 1993; Bu z e al, 2007 and Cae ano e al., 2010) and (iii) pendulum dampe s
(Bu z e al., 2007). Howe e , he use o hese passi e dampe s mus be limi ed
since al hough hey allow achie ing a high le el o damping wi h a easonable cos ,
hey p esen se e al p oblems ha discou age hei widely use. Among hese
p oblems, he mos impo an a e: (i) he necessi y o damping all he na u al
equencies o he s uc u e inside he ange o pedes ian in e ac ion, (ii) a bad
pe o mance may be he cause o a spli ing o he na u al equency o iginally
damped, (iii) hey a e mechanical elemen s ha equi e main enance, and (i ) due
o hei weigh i can be non- iable hei placemen on exis ing oo b idges due o
s eng h easons (Meinha d , 2009).
23

3. P oposal o a simpli ied biomechanical c owd-s uc u e in e ac ion
model.
The p oposed c owd-s uc u e in e ac ion has been simula ed using wo indi idual
sub-models (Figu e 7): (i) he pedes ian-s uc u e in e ac ion sub-model and (ii)
he c owd sub-model.
In he i s sub-model, all he dynamics e ec s induced by he pedes ians on he
oo b idge a e conside ed. The la e al o e ical accele a ion, a
y
 o a
z
 ,
expe imen ed by each pedes ian is he ou pu ob ained om his model (acco ding
o he analysed di ec ion).
In he second sub-model, he c owd is simula ed as a beha iou al model, p o iding
a desc ip ion o he indi idual pedes ian posi ion, p
x, walking pedes ian eloci y,
p
, s ep pedes ian equency, s
, and phase among pedes ians, p

.
In o de o ake in o accoun he change o he pedes ian beha iou associa ed
wi h he accele a ion le el expe ienced, wo addi ional condi ions ha e been
included in his la e sub-model. In e ical di ec ion only a com o h eshold has
been included, while in la e al di ec ion a com o and la e al lock-in h esholds
ha e been conside ed. The i s condi ion modi ies he pedes ian eloci y, p
,
acco ding o he com o le el expe ienced by each pedes ian and he second
condi ion modi ies he s ep pedes ian equency, s
, and he phase among
pedes ians, p

, in o de o synch onize he mo emen o he pedes ians and he
s uc u e i ce ain limi is exceeded.
Figu e 7. Layou o he biomechanical c owd-s uc u e in e ac ion model.
Fo each in e ac ion he c owd sub-model de e mines he posi ion, eloci y s ep
equency and phase o each pedes ian. Subsequen ly, hese ou pa ame e s a e
used as inpu o de ine he walking o ce o each pedes ian-s uc u e model,
ob aining as ou pu he la e al o e ical accele a ion o each pedes ian. The
pedes ian eloci y and equency o each indi idual is modi ied acco ding o he
PEDESTRIAN-STRUCTURE MODEL
CROWD MODEL
PEDESTRIAN/STRUCTURE
PARAMETERS
CROWD-STRUCTURE MODEL
p
x
p
a
y

COMFORT THRESHOLDS
LOCK-IN THRESHOLD
s
p

a
z

24

com o le el expe imen ed by each pedes ian and addi ionally his phase shi i a
la e al lock-in h eshold is exceeded. Finally, he p ocess is epea ed by he
upda ed alues o he posi ion, he pedes ian eloci y, he s ep equency and he
phase shi (Figu e 7).
3.1. Modelling he pedes ian-s uc u e in e ac ion.
The p oposed pedes ian-s uc u e in e ac ion in each di ec ion ollows om he
applica ion o dynamic equilib ium equa ions (Clough and Penzien, 1993;
Dominguez, 2001; Xia and Zhang, 2005) o a simpli ied model o in e ac ion (Figu e
8) wi h sp ung ( a
m) and unsp ung masses ( s
m). This me hodology has been
applied sepa a ely o e ical (Pape A and Pape B) and la e al (Pape E)
di ec ions and he e i will summa ized o he case o e ical di ec ion. I s
implemen a ion in he la e al di ec ion is desc ibed in Pape E. The o mula ion o
he p oposed model may be u he gene alized o he h ee di ec ions by
acco dingly modi ying bo h he equa ion ha go e ns he conside ed pedes ian
load in each di ec ion and he alue o he modal pa ame e s ha de ine he TDOF
( wo deg ees o eedom) pedes ian model. In his way, he esul ing model would
be sui able o he mo e gene al 3-D p oblem and i could he e o ake in o
accoun he possible in e ac ion in he h ee spa ial di ec ions.
Figu e 8. Biomechanical pedes ian-s uc u e in e ac ion model in e ical di ec ion
(Pape B).
Conside ing he balance o he sys em, s uc u e and pedes ian model, he
ollowing coupled equa ions o mo ion may be w i en.
in _ )( FxzKzCzM piNUMiiiiii





 [13]




0




sapsapaa zzkzzczm  [14]




in , FFzzkzzczm e paspaspss





 [15]
m
a
c
p
k
p
m
s
z
a
z
s
F
in
L
z
xF
in
F
s
M
i
C
i
K
i
d
p
y
x
p
w(x, )
25

whe e
a
m is he sp ung mass o he pedes ian in he conside ed di ec ion [kg].
s
m is he unsp ung mass o he pedes ian in he conside ed di ec ion [kg].
as mmm  is he o al mass o he pedes ian in he conside ed di ec ion [kg].
i
zis he modal displacemen o he ib a ion mode i [m]
a
z is he absolu e e ical displacemen o he sp ung mass [m].
s
z is he absolu e e ical displacemen o he unsp ung mass [m].
p
k is he equi alen s i ness o a pedes ian [N/m].
p
c is he equi alen damping o a pedes ian [sN/m].
e p
F, is he e ical pedes ian o ce due o walking [N].
in
F is he in e ac ion o ce be ween he pedes ian and he s uc u e [N].
i
M is he modal mass o he ib a ion mode i [kg].
i
C is he modal damping o he ib a ion mode i [sN/m]
i
Kis he modal s i ness o he ib a ion mode i [N/m].
iNUM _

is he e ical componen o he nume ical ib a ion mode i.
x pxp  is he longi udinal posi ion o he pedes ian [m].
is he ime [sec.]
px
is he longi udinal componen o he pedes ian eloci y ec o [m/s].
p
dis he dis ance among pedes ians [m].
),( xw is he de lec ion o he oo b idge a he posi ion
x
[m].
Lis he leng h o he oo b idge [m].
F om Eq.(15) he ollowing exp ession is ob ained o , in
F:




aspaspss e p zzkzzczmFF






,in [16]
and subs i u ing his equa ion in o Eq.(13) yields.
26




 


aspaspss e ppiNUMiiiiii zzkzzczmFxzKzCzM 







 ,_

[17]
Applying, a he con ac poin be ween he pedes ian and he s uc u e, he
equa ions o compa ibili y o displacemen s, eloci y and accele a ion be ween he
s uc u e and he simpli ied pedes ian-model o in e ac ion:
),(),( w xwz pxps



[18]
),(),( w xwz pxps




 [19]
),(),( w xwz pxps




 [20]
These quan i ies may be exp essed in e ms o he ampli ude )( zi and he modal
shape o he n nume ical conside ed ib a ion modes )(
_x
iNUM

, neglec ing he e m
o a ia ion o he pedes ian eloci y along ime, as:


 n
i
piNUMip x z xw
1
_)()(),(

[21]
 

 n
i
piNUMpxi
n
i
piNUMip x zx z xw
1
_
1
_)()()()(),(


 [22]
 



 n
i
piNUMxpip
n
i
iNUMxpi
n
i
piNUMip x zx zx z xw
1
_
2
,
1
_,
1
_))()()()(2)()(),(


 [23]
dx
xd
xiNUM
iNUM
)(
)( _
_



 [24]
2
_
2
_
)(
)( dx
xd
xiNUM
iNUM



 [25]
whe e
)(
_x
iNUM

is he i s spa ial de i a e o he mode o ib a ion i.
)(
_x
iNUM

 is he second spa ial de i a e o he mode o ib a ion i.
The nume ical ib a ion modes, )(
_x
iNUM

, a e ob ained in a disc e e way using he
co esponding ini e elemen me hod as:


j
j
j
iiNUM xNx )()(
_

[26]
whe e )(xN ja e he shape unc ions and j
i

a e he nodal alues.
In he p e ious exp essions, he alue o he nume ical ib a ion modes is se o
ze o when he pedes ian emains ou side he s uc u e.

27

0)(
_

piNUM x

o L x
x
p
p


)(
0)( [27]
The abo e ela ions –Eqs.(18) o (23)- a e hen subs i u ed in he o e all dynamic
equilib ium equa ions -Eqs.(13) o (15)- so ha , o ganizing in o ma ion in a ma ix
o m, he ollowing model o in e ac ion is ob ained (see Pape B o u he de ails
and ma ix o mula ion).
)()()()()()()( FzKzCzM




  [28]
Fo a g oup o k pedes ians (Figu e 8), each o hem will be ep esen ed by he
abo e simpli ied in e ac ion model. In he case o a single pedes ian, he p oposed
model leads o a sys em o n+1 equa ions, co esponding o he conside ed numbe
o ib a ion modes n plus he simpli ied in e ac ion equa ion. Simila ly, when
conside ing a g oup o k pedes ians, a sys em o n+k di e en ial equa ions will
need o be sol ed.
Conside ing he na u e o he esul ing sys em, he use o a me hod o

-Newma k
in eg a ion amily is p oposed, wi h pa ame e s 41


and 21


, hus ensu ing
an uncondi ionally s able sys em.
Fu he mo e, he in eg a ion s ep,

, is es ablished acco ding o he usual
ecommenda ions (Clough and Penzien, 1993; Dominguez, 2001) o dynamics
models based on modal decomposi ion echnique, as he minimum o he ollowing
alues.
)01.0,
4
,
200
,
8
1
min( minmin
max pp n
L
L


 sec. [29]
wi h max
[Hz] being he highes conside ed ib a ion equency o he s uc u e and
min
L[m] he minimum span leng h o he pedes ian b idge.
In o de o de e mine he numbe o pedes ians ha c oss he oo b idge in phase,
a Poisson dis ibu ion has been adop ed, acco ding o he esul s by Ma sumo o e
al. (1978). Thus, when a g oup o p
n pedes ians a i e a he oo b idge, he
numbe o pedes ians andomly synch onized is p
n. This synch oniza ion c i e ion
has been adop ed o iginally in he c owd-s uc u e in e ac ion model. I s
implemen a ion is achie ed by he phase shi pa ame e , p

. Fo a gi en
gene a ion/g oup o pedes ians, he alue o he phase shi o he p
n andomly
synch onized pedes ians has been se equal o ze o. Fo he emaining pedes ians
he phase shi has been assigned andomly, using a Gaussian dis ibu ion in he
ange


2,0 . Subsequen ly, in he la e al di ec ion, his pa ame e will be modi ied
i he la e al lock-in h eshold is exceeded.
28

3.2. Modelling he c owd-beha io .
The pedes ian walking inside a c owd may be modelled using he go e ning
equa ions o pa icle dynamics (Rapapo , 2004), conside ing ha di e en social
o ces in e ac among pedes ians. This app oach has been success ully applied by
se e al au ho s (Helbing and Molná , 1995; Ca oll e al., 2012). In his way, he
di e en mo i a ion and in luences expe imen ed by he pedes ians a e desc ibed
by se e al o ce e ms. The model is based on New on dynamics and is able o
ep esen he ollowing ules in ela ion wi h he na u al pedes ian mo emen (see
(Helbing and Molná , 1995) o a mo e in ol ed desc ip ion): (i) pedes ians
no mally choose he as es ou e, (ii) each pedes ian has an indi idual speed ha
may be de ined by a Gaussian dis ibu ion (iii) and he dis ance be ween
pedes ians depends on he pedes ian densi y and he walking pedes ian speed.
Figu e 9 ske ches he social o ces ac ing on a pedes ian in a c owd, as desc ibed
in de ail in he ollowing pa ag aphs.

Figu e 9. Pedes ian-c owd in e ac ion o ces (Helbing and Molná , 1995).
All he pa ame e s o he c owd model conside ed in his s udy ha e been ob ained
om he epo ed esul s p o ided by di e en au ho s (Helbing and Molná , 1995;
Ca oll e al., 2012), as summa ized in Table 9 and b ie ly desc ibed nex .
Bounda y
Bounda y
Desi ed des ina ion
no
bou
F
an
bou
F
d i
F
Pedes ian j
Pedes ian
i
an_phy
ped
F
p
d
d
e
d
d
zx
y
p
x
p

p
Aniso opic pedes ian beha iou
Legend
Fo ce ec o
Axis o he s uc u e
Geome ic magni ude
Coo dina e sys em
Mo emen di ec ion
an_phy
ped
F
no phy
ped
soc
ped
_
FF 
p
n
b
b
n
p
p
no phy
ped
soc
ped
_
FF 
b
nb
an
bou
F
no
bou
F
0.1 5.0 1.0
p

29

D i ing o ce.
Each pedes ian has a ce ain mo i a ion o each his desi ed des ina ion, d
d, wi h
his desi ed eloci y, d
, which is ep esen ed by he d i ing o ce, d i
F, as:










p
dd
d i
m
e
F [30]
whe e d
e is he desi ed di ec ion ec o , p
is he pedes ian s ep eloci y and
is he elaxa ion ime o he pedes ian (Helbing and Molná , 1995) (Table 9). The
desi ed di ec ion o he mo emen may be ob ained om he posi ion o he
pedes ian in each ins an , p
x, and i s desi ed des ina ion acco ding o:
pd
pd
dxd
xd
e

 [31]
In e ac ions among pedes ians.
The in e ac ion among pedes ians o igina es a epulsi e o ce (Helbing and Molná ,
1995), ped
F, wi h wo componen s, a socio-psychological o ce, soc
ped
F, and a physical
in e ac ion o ce, phy
ped
F, as:
phy
ped
soc
pedped FFF  [32]
The socio-psychological o ce e lec s he ac ha he pedes ians y o main ain a
ce ain dis ance o o he pedes ians in he c owd. This socio-psychological o ce
depends on he dis ance be ween pedes ians, eaching i s maximum alue a he
lowes es ablished dis ance and ending o ze o as such dis ance inc eases. The
socio-psychological o ce is de ined as:
pp
p
pp
p
soc
ped s
B
d
A








 nF 2
exp [33]
whe e
p
A is he in e ac ion s eng h be ween pedes ians (Table 9).
p
B is he ange o he epulsi e in e ac ion be ween pedes ians (Table 9).
p
d is he dis ance be ween wo pedes ians.
p
is he so-called pedes ian adius (Table 9).
p
n is he no malized ec o poin ing be ween pedes ians.
30

p
s is a o m ac o o conside he aniso opic beha iou ( he pedes ian ac ion in
on o he pedes ian is mo e impo an han behind him) o he pedes ians,
which alue may be ob ained om:


2
cos1
)1( p
ppp
s



 [34]
whe e p

(Table 9) is a po en ial ac o ha conside s he in luence on he
pedes ian mo emen o o he pedes ians si ua ed in on o him (Figu e 9) and
p

is he angle be ween wo pedes ians (Figu e 9).
The physical in e ac ion o ce, phy
ped
F, is only conside ed in si ua ions o physical
con ac among pedes ians (i pp d


2), associa ed wi h si ua ions o high
pedes ian densi ies (≥0.80 P=Pe son/m2). The physical in e ac ion o ce is de ined
by he supe posi ion o wo componen s: (i) he body o ce, no phy
ped
_
F, ha desc ibes
he coun e ac ing body ac ion ha he pedes ians pe o m o a oid physical
damage due o hei physical con ac wi h o he indi iduals, (ii) and he sliding
o ce, an_phy
ped
F, ha ep esen s he pedes ians’ endency o a oid passing o he
indi iduals wi h a high eloci y a small dis ances (Helbing and Molná , 1995). I is
de ined as:
an__ phy
ped
no phy
ped
phy
ped FFF  [35]


pppp
no phy
ped d HC nF  2
_ [36]


p
pppp
phy
ped d HD F  2
an_ [37]
whe e
no phy
ped
_
F is he no mal componen o he physical in e ac ion o ce (body o ce).
an_phy
ped
F is he angen ial componen o he physical in e ac ion o ce (sliding o ce).
p
C is he body o ce s eng h due o he con ac be ween pedes ians (Table 9).
p
D is he sliding o ce s eng h due o he con ac be ween pedes ians (Table 9).
p
is a no malized ec o pe pendicula o p
n.
pp
p
 is he angen ial componen o he ela i e pedes ian eloci y,
wi h p
 being he di e ence o ec o eloci ies be ween wo gi en pedes ians.
and unc ion
H
is de ined as:
37
Figu e 10. Flowcha o he iden i ica ion p ocedu e in e ical di ec ion.
An i e a i e p ocess o educe he di e ences be ween he expe imen al and
nume ical e ical accele a ions was pe o med, unde he ules o gene ic
algo i hms (Koh and Pe y, 2010; Nocen al and W igh , 1999), The es ima ed
alues o pa ame e s o he pedes ian-s uc u e in e ac ion model, in e ical
di ec ion, we e co ela ed success ully wi h: (i) he walking pedes ian e ical o ce
sugges ed by di e en au ho s (summa ized in sec ion 2.1) and (ii) a p e ious
es ima ion o he modal pa ame e s (Pape A) ob ained om he analysis o he
change o he modal p ope ies o a oo b idge induced by a con olled g oup o
pedes ians (Geo gakis and Jo gesen, 2013). On he o he hand, he es ima ed
alues o he modal pa ame e s we e inside he ange, es ablished by Shahabpoo
e al. (2013).
4.2. In e se dynamic p oblem: pa ame e iden i ica ion in la e al di ec ion.
In o de o de ine he objec i e unc ion o he pa ame e iden i ica ion o he
TDOF-sys em in la e al di ec ion (Pape E), he eco ded dynamic esponse o he
Viana oo b idge du ing he expe imen al pedes ian es was analysed again by i s
ans o ma ion o he equency domain. In his case, howe e , i was checked ha
he dynamic esponse o he oo b idge was cha ac e ized by a ha monic se ies ha
con ained only he i s h ee equencies ha cha ac e izes he pedes ian s ep
wi hou a ema kable con ibu ion o he ha monics associa ed wi h he la e al
na u al equencies o he oo b idge. In his manne , he accele a ions eco ded
du ing he expe imen al pedes ian es con ained in o ma ion mainly o he walking
pedes ian la e al o ce, being necessa y o conduc a second expe imen al es , a
c owd es , in o de o cha ac e ize he modal pa ame e s o he TDOF-sys em. In
his c owd es , he dynamic esponse o Viana oo b idge unde a g oup o i y

38
pedes ians a di e en s ep equencies was eco ded in o de o s udy he change
o he i s la e al na u al equency o he oo b idge induced by he pedes ian-
s uc u e in e ac ion. Due o his ac , he iden i ica ion p ocess was di ided in wo
s eps, by sol ing wo in e se dynamic p oblems.
Fi s , as he modal pa ame e s o he pedes ian-s uc u e in e ac ion model ha e a
di ec e ec on he modal pa ame e s o he oo b idge (Pape A) as objec i e
unc ion o he i s minimiza ion p oblem, he mean squa e e o be ween he
expe imen al, exp
,1 la
, and nume ical, num
la
,1 , i s la e al na u al equency o he Viana
oo b idge, ob ained du ing he pe o mance o he expe imen al c owd es and i s
nume ical simula ion, was conside ed. Addi ionally, as design a iables o his i s
in e se p oblem, he h ee modal pa ame e s ha cha ac e izes he TDOF-sys em,
in la e al di ec ion ( he pedes ian sp ung mass, la a
m,, he pedes ian damping
a io, la p,

, and he pedes ian na u al equency, la p
,), we e conside ed.
Second, al hough he e a e e y ecen and comp ehensi e s udies o he la e al
o ce induced by pedes ians (Ingól sson and Geo gakis, 2011; Ingól sson e al.,
2011), hese esea ch do no include he e ec o he pedes ian-s uc u e
in e ac ion. The e o e i was necessa y o es ima e he walking pedes ian la e al
o ce unde his assump ion. In his manne a second in e se p oblem was sol ed.
As objec i e unc ion o he second minimiza ion p oblem, he mean squa e e o
be ween he expe imen al ( exp
,ila
psd , whe e i is he conside ed sec ion) and
nume ical ( num
ila
psd ,) powe spec al densi y ob ained om he la e al accele a ions
eco ded in he p e iously men ioned expe imen al pedes ian es s and i s
nume ical simula ion, was conside ed. As design a iables o his second in e se
p oblem, he i s h ee LDLF ( la ,1

, la,2

and la ,3

) and hei co esponding phase
shi s o he second and hi d ha monic ( la,2

and la ,3

) o he pedes ian walking
la e al o ce we e conside ed. In Figu e 11 a lowcha o he iden i ica ion
p ocedu e is shown.
39
Figu e 11. Flowcha o he iden i ica ion p ocedu e in la e al di ec ion. Ligh blue
ma ks he modal pa ame e iden i ica ion me hodology and da k blue he walking
pedes ian o ce iden i ica ion me hodology.
As in he abo e case ( e ical di ec ion), an i e a i e p ocess o educe he
di e ences be ween he expe imen al and nume ical magni udes was pe o med,
unde he ules o gene ic algo i hms (Koh and Pe y, 2010; Nocen al and W igh ,
1999), In his case, he iden i ica ion p ocedu e is pe o med in wo s eps. Fi s ,
he modal pa ame e s o he p oposed TDOF-sys em we e es ima ed by he
minimiza ion o he ela i e di e ences be ween he expe imen al and nume ical
change o he i s la e al na u al equency o he Viana oo b idge du ing an
expe imen al c owd es and i s nume ical simula ion. Second, once es ablished he
modal pa ame e s o he p oposed model, he walking pedes ian la e al o ce was
es ima ed by he minimiza ion o he ela i e di e ences be ween he expe imen al
and nume ical powe spec al densi ies ob ained in ou poin s o he Viana
oo b idge du ing an expe imen al pedes ian es and i s nume ical simula ion. The
es ima ed alues o he pa ame e s o he pedes ian-s uc u e in e ac ion model,
in la e al di ec ion, we e co ela ed success ully wi h: (i) he walking pedes ian
la e al o ce sugges ed by di e en au ho s (summa ized in sec ion 2.1) and he
ange o pedes ian modal pa ame e s sugges ed by Shahabpoo e al. (2013).
40
5. Expe imen al es ima ion o he pa ame e s o he pedes ian-s uc u e
in e ac ion model.
In his sec ion he es ima ion o he pa ame e s o he p oposed TDOF-sys em was
pe o med in e ical and la e al di ec ions. Fi s , a eal oo b idge, Viana do
Cas elo oo b idge (Ba bosa e al., 2012), was con e ed in o a “labo a o y”
oo b idge by he upda ing o i s ini e elemen model based on he expe imen al
modal pa ame e s o he s uc u e ob ained om an ope a ional modal analysis
pe o med on he measu emen s eco ded du ing an ambien es . Second, wo
expe imen al es s, a pedes ian and c owd es , we e conduc ed in o de o
es ablished a basis o he es ima ion o he pa ame e s o he p oposed TDOF-
sys em. Finally, he pa ame e s o he pedes ian-s uc u e in e ac ion model we e
es ima ed by he esolu ion o an in e se p oblem app oach.
5.1. Desc ip ion and ini e elemen model o he “labo a o y” oo b idge:
Viana oo b idge.
The Viana do Cas elo oo b idge (Ba bosa e al., 2012) is a mo eable cable-s ayed
b idge. The longi udinal s uc u al scheme o he oo b idge consis s o wo spans o
abou 36.50 m and 9.00 m espec i ely suspended by 6 amilies o wo hange s
( wo e aining ones) om an inclined mas . The deck, wi h 2.50 m o wid h, is
con igu ed by wo olled s eel beams o a iable dep h b aced by ci cula hollow
p o iles. The deck loo is co e ed wi h wood. The compensa ion o he main span
weigh is achie ed by placing 11 high densi y blocks (wi h a weigh o 800 kN)
placed in he sho e span. The mas is welded in i s base o a cylinde ha is
connec ed o a wheel gea bea ing ha allows he o a ional mo emen o he
s uc u e. The pylon is connec ed o a deep ounda ion ha balances he o ces
ansmi ed by he mas . A pe spec i e o he Viana oo b idge is shown in Figu e
12.
Figu e 12. La e al iew o he Viana oo b idge (Ba bosa e al., 2012).
A p
e
o de
o d
e
Fig
u
pa a
The
he
s
3D-
c
wi h
na u
p e
es i
m
p eli
asso
ou
num
e
limina y
n
o ha e
a
e
ine a s a
u
e 13.
F
me e s.
so wa e
p
s
uc u e
u
c
able elem
e
he hang
e
al eque
iously he
m
a ing i s
mina y F
E
cia ed nu
m
nume ical
be ).
n
ume ical
i
a
i s app
ing poin
F
ini e ele
m
p
ackage An
u
sing 3D-b
e
e
n s (LIN
K
e
s ha e
b
ncies and
s ess le
angen
s
E
model l
e
m
e ical na
u
ib a ion
m
ni e eleme
o
xima ion
equi ed o
m
en mod
e
sys (Ansy
s
e
am elem
e
K
10) we e
i
b
een consi
he ib
a
el o h
e
s
i ness
m
e
ads o
h
u
al equ
e
m
odes a e
41
e
n (Figu e
o he dyn
a
pe o m
h
e
l, ambie
n
s
, 2015) w
e
n s (BEA
M
implemen
e
de ed o
a
ion mod
e
e
hange s
m
a ix. Th
e
h
e i s o
e
ncies gi e
n
shown (
N
U
13) modal
a
mic beha
h
e ambien
n
es g
as used b
a
M
188), exc
e
e
d. The n
o
he de e
m
e
s o he
unde p
e
e
nume ic
a
u nume i
n
in
T
able
U
M_i, wi h
i
analysis w
iou o h
e
ib a ion
id and
m
a
sed on a
d
e
p o he
o
nlinea e
m
ina ion o
s uc u e,
e
manen l
a
l modal
a
cal ib a i
10. In Fig
being he
w
as conduc
e
oo b idg
e
es .
m
odel up
d
d
isc e iza i
hange s
w
ec s asso
c
he num
by calcu
l
oads and
a
nalysis o
on modes
u e 14 h
e
ib a ion
m
ed in
e
and
d
a ing
on o
w
he e
c
ia ed
e ical
l
a ing
hus
his
and
e
i s
m
ode
42
NUM_1=3.426 Hz NUM_2=4.421 Hz
NUM_3=6.907 Hz NUM_4=7.431 Hz
Figu e 14. Fi s ou nume ical ib a ion modes. Ini ial FE model.
5.2. Expe imen al iden i ica ion o he modal pa ame e s o he
“labo a o y” oo b idge.
In o de o ob ain expe imen ally he modal pa ame e s (na u al equencies,
damping a ios and modal shapes) o he oo b idge, an ambien ib a ion es was
pe o med. The measu emen s we e eco ded in ambien condi ions, wi h he
oo b idge exci ed by a ligh wind. In o de o acqui e a su icien le el o expe ise
in he applica ion o his iden i ica ion echnique, he assessmen o he dynamic
beha iou o o he s oo b idges was conduc ed by he au ho du ing he
de elopmen o his Thesis. Two ep esen a i e examples ha e been included in he
documen (Pape F and Pape G). The modal shape coo dina es we e measu ed
along wo g idlines sepa a ed ans e sally 1.68 m. A o al o 2x11 poin s equally
dis ibu ed along each longi udinal alignmen we e ins umen ed. Fou high
sensi i i y i-axial o ce balanced accele ome e s we e used (Figu e 15). Using wo
o hese de ices as e e ences, measu emen s we e successi ely made mo ing he
o he wo accele ome e s o he de ined ins umen a ion loca ions and eco ding in
each poin 1000 sec. ime se ies o accele a ion sampled a 100 Hz (Figu e 13).

43
Figu e 15. One accele ome e used du ing ambien /expe imen al es s.
The expe imen al iden i ica ion o he modal pa ame e s was done in he ime
domain using he S ochas ic Subspace Iden i ica ion me hod (Magalhães and
Cunha, 2011), implemen ed in he so wa e p og am A emis (A emis, 2015).
Figu e 16 illus a es he s abiliza ion diag am o he iden i ica ion algo i hm used.
The i s ou ib a ion modes we e iden i ied and subsequen ly used o he FE
model upda ing p ocess (see Pape B). The ob ained nume ical and expe imen al
na u al equencies and ib a ion modes shapes a e compa ed in Table 10 and he
co ela ion be ween he i s ou nume ical and expe imen al ib a ion modes is
shown in Figu e 17 (wi h he x axis co esponding o he longi udinal di ec ion o
he oo b idge). In o de o alida e he co ela ion be ween he nume ical and
expe imen al modal pa ame e s, bo h he ela i e di e ence (

) be ween he
nume ical and expe imen al equencies and he modal assu ance c i e ion (M.A.C.)
we e analysed (Zi ano ic e al., 2007). A good co ela ion be ween wo modes is
achie ed when he alue o hei M.A.C. a io is g ea e han 0.90. These wo
magni udes may be de ined acco ding o Eqs. (46 and 47) as ollows:
100
_
__ 


iEXP
iEXPiNUM
[%] [46]
whe e iNUM
_ is he nume ical na u al equency and iEXP
_ is he expe imen al
na u al equency o he ib a ion mode i.



iEXP
T
iEXPiNUM
T
iNUM
iEXP
T
iNUM
i
MAC
____
2
__




 [47]
whe e iNUM _

and iEXP _

a e he nume ical and expe imen al ib a ion modes o be
compa ed and T deno es he anspose.
44
Al hough he shapes o he iden i ied ib a ion modes a e in good ag eemen (wi h
M.A.C. a ios g ea e han 0.90 in h ee o he ib a ion modes), he ela i e
di e ences, , be ween he i s wo nume ical and expe imen al na u al
equencies a e s ill signi ican . The e o e, he ini ial es ima ion made on he
physical pa ame e s o he s uc u e is no good enough and i becomes necessa y
o pe o m a ini e elemen model upda ing (F iswell and Mo e shead, 1995;
Teughels, 2003, Pape C) o he oo b idge in o de o imp o e he co ela ion
be ween he nume ical and expe imen al modal pa ame e s.
Figu e 16. S abiliza ion diag am o he S ochas ic Subspace Iden i ica ion me hod.
Finally, in Table 10 an es ima ion o he damping a ios, i

, associa ed wi h he
iden i ied ib a ion modes is also shown (Magalhães e al., 2010). These alues will
be adop ed la e in he iden i ica ion p ocess o he pa ame e s o he TDOF-sys em
ha models he pedes ian.
Table 10. Fi s ou nume ical ( NUM) e sus expe imen al ( EXP) ib a ion modes o
he oo b idge.
Modes NUM [Hz] EXP [Hz]
i

[%]

[%]
M.A.C.
Desc ip ion
1 3.426 3.138 1.22 9.17 0.963 Fi s e ical mode
2 4.421 4.068 1.21 8.67 0.985 Fi s la e al mode
3 6.907 6.810 1.10 1.42 0.809 Second la e al mode
4 7.431 7.345 1.39 1.17 0.939 Second e ical mode
45
1s Vib a ion mode 2nd Vib a ion mode
3 d Vib a ion mode 4 h Vib a ion mode
Figu e 17. Fi s ou nume ical (Num.) e sus expe imen al (Exp.) ib a ion
modes.
5.3. Model upda ing o he “labo a o y” oo b idge.
As indica ed abo e, in o de o educe he le el o unce ain ies o he nume ical
analysis a ini e elemen model upda ing (F iswell and Mo e shead, 1995;
Teughels, 2003; Zi ano ic e al., 2007, Pape C) o he s uc u e has been
pe o med. The ou iden i ied ib a ion modes we e conside ed in he upda ing
p ocess due o he good quali y o he expe imen al da a. Bo h measu ed na u al
equencies and modal coo dina e alues we e aken in o accoun . The e o e, in
o al 48 esidual componen s we e selec ed o he model upda ing ( he ou
iden i ied na u al equencies and he ele en coo dina es o each iden i ied ib a ion
mode). A mo e de ailed desc ip ion o he me hodology used o pe o m he model
upda ing can be ound in Pape C and Pape G.
A sensi i i y analysis was pe o med in o de o adequa ely de e mine he physical
pa ame e s o he FE model wi h g ea e in luence on he iden i ied ib a ion
modes. In his manne , he modal sensi i i ies wi h espec o some possible
physical a iables ha e been ob ained nume ically (Fox and Kapoo (1968)). The
esul s o his s udy conclude ha he mos in luen ial physical pa ame e s on he
dynamic beha iou o he oo b idge a e he s i ness o he ou amilies o main
hange s and he soil-s uc u e in e ac ion, modelled by wo sp ing elemen s (in
longi udinal, 5

, and la e al, 6

di ec ions, as Figu e 13 illus a es) si ua ed a he
ex eme o he longe span. As he s i ness o he hange s is condi ioned by hei
s ess le el, he ini ial s ess s a e o each conside ed amily ( 1

, 2

, 3

and 4

)
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 10.00 20.00 30.00 40.00 50.00
X [m]
Num.
Exp.
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 10.00 20.00 30.00 40.00 50.00
X [m]
Num.
Exp.
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 10.00 20.00 30.00 40.00 50.00
X [m]
Num.
Exp.
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 10.00 20.00 30.00 40.00 50.00
X [m]
Num.
Exp.
46
has also been aken in o accoun as physical a iable. In Figu e 13 and Table 11,
he selec ed physical a iables a e shown.
The model upda ing p ocess has been conduc ed by sol ing an op imiza ion
p oblem in he so wa e p og ams Ansys (Ansys, 2015) and Ma lab (Ma lab, 2015).
As objec i e unc ion he mean squa e e o be ween he expe imen al and
nume ical modal pa ame e s (na u al equencies and ib a ion modes) o he Viana
oo b idge has been conside ed. In each i e a ion, a popula ion o 1000 ec o s has
been gene a ed ha , using he gene ic algo i hms ules o mu a ion, ep oduc ion
and c osso e , has minimized he alue o he p oposed objec i e unc ion. The
alues o he selec ed pa ame e s ha e been modi ied in o de o minimize he
conside ed objec i e unc ion. Addi ionally, a sea ch domain has been de ined o
con ol he a ia ion o each pa ame e , inc easing he e iciency o he op imiza ion
algo i hm and ye main aining he physical meaning o he ini e elemen model
upda ing. Fo he s i ness o he hange s, as a passi e beha iou is expec ed, hei
medium ension le el has been de e mined unde pe manen loads, wi h a alue o
250 kPa. Sligh a ia ions o his alue ha e been conside ed expanding he sea ch
domain be ween 0-500 kPa. Fo he s i ness o he sp ings ( 5

and 6

), gi en he
unce ain y associa ed wi h he s i ness o he soil, a wide sea ch domain was
conside ed. So conside ing, as i is es ablished by he geo echnical epo , a
a ia ion o he Young’s modulus o soil be ween 1001


m
E GPa and he
geome y o he abu men s, he a ia ion o he equi alen s i ness o he sp ings
has been de e mined 5

and


97
61010 

N/m. In Table 11 he ange o a ia ion
o each pa ame e and i s upda ed alues a e shown
Table 11. Upda ed alues o conside ed physical pa ame e s.
Pa ame e s Minimum
Value
Upda ed
Value
Maximum
Value
Tension s ess cable 1 ( 1

) 0.00 kPa 112.38 kPa 500.00 kPa
Tension s ess cable 2 ( 2

) 0.00 kPa 59.62 kPa 500.00 kPa
Tension s ess cable 3 ( 3

) 0.00 kPa 31.62 kPa 500.00 kPa
Tension s ess cable 4 ( 4

) 0.00 kPa 147.43 kPa 500.00 kPa
Longi udinal Sp ing ( 5

) 1.00E7 N/m 6.00E7 N/m 1.00E9 N/m
La e al Sp ing ( 6

) 1.00E7 N/m 1.90E8 N/m 1.00E9 N/m
The di e ences be ween he nume ical and expe imen al na u al equencies, a e
he ini e elemen model upda ing, a e e y small and he co ela ion be ween he
nume ical and expe imen al ib a ion modes a e e en highe . The ela i e
di e ences be ween he upda ed nume ical ( UPD
) and expe imen al ( EXP
) modal
pa ame e s and he M.A.C. alues achie ed a e he model upda ing p ocess a e
summa ized in Table 12, whe e he imp o emen wi h espec o he ini ial FE
model is clea (see Table 10).
53
s uc u e in e ac ion model The ollowing Gaussian dis ibu ions ha e been ob ained
(),(


N, being

he mean alue and

he s anda d de ia ion).
 La e al pedes ian sp ung mass, la a
m,,)736.2,216.73(N%.
 La e al pedes ian damping a io, la p,

,)405.5,116.49(N %.
 La e al pedes ian na u al equency, la p
,, )178.0,201.1(N Hz.
Figu e 22 illus a es he co ela ion be ween expe imen al and nume ical esul s o
he change o i s la e al na u al equency o he oo b idge induced by he c owd-
s uc u e in e ac ion phenomenon. Good ag eemen be ween bo h se s o esul s is
obse ed, wi h di e ences below 0.70 % o all he analysed pedes ian walking
equencies. The i s la e al na u al equency co esponding o he emp y
oo b idge is included in Figu e 22 o e e ence.

Figu e 22. Change o he i s la e al expe imen al (Exp.) and nume ical (Num.)
na u al equency ( 1,la ) e sus he s ep equency [Hz].
F om he p e ious esul s he ollowing conclusions may be ex ac ed: (i) he
s abili y ha he es ima ed modal pa ame e s p esen o he di e en s ep
equencies, allowing ha he p oposed c owd-in e ac ion model may be used as a
ool o he cha ac e iza ion o he e ec o he mo ing pedes ians on he dynamic
beha iou o oo b idges in la e al di ec ion and (ii) he good co ela ion be ween
he expe imen al and nume ical cu es (Figu e 22) ha show he change o he
i s la e al na u al equency o he oo b idge e i ies he abili y o he p oposed
model o cha ac e ize he pedes ian-s uc u e in e ac ion phenomenon in la e al
di ec ion.
3.750
3.800
3.850
3.900
3.950
4.000
4.050
4.100
1.30 1.50 1.70 1.90 2.10 2.30 2.50
1.la
[Hz]
s
[Hz]
Exp.
Num.
Emp y

54
Fo he es ima ion o he walking pedes ian la e al o ce o he TDOF-sys em, a
second in e se p oblem was sol ed again. In his case, as objec i e unc ion he
mean squa e e o be ween he expe imen al and nume ical powe spec al densi y
ob ained om he la e al accele a ions eco ded in he men ioned ou poin s o he
Viana oo b idge unde he c ossing o wo pedes ians a con olled s ep
equencies was conside ed. The expe imen al la e al accele a ions co espond o
he denoised measu emen s o he abo e desc ibed pedes ian es . The nume ical
la e al accele a ions ha e been ob ained om he implemen a ion o he p oposed
pedes ian-s uc u e in e ac ion model on he upda ed ini e elemen model o he
Viana oo b idge.
The expe imen al and nume ical powe spec al densi y has been ob ained om
hese men ioned accele a ions. Six pa ame e s we e adop ed as design a iables:
(i) he i s h ee LDLF ha cha ac e ize he pedes ian walking la e al o ce.
(ii) he phase shi s o he second and hi d ha monic ha cha ac e ize he
pedes ian walking la e al o ce.
(iii) a ime lag ha allows adjus ing he beginning o he c ossing o he
pedes ian be ween he expe imen al and nume ical esponse.
The es ima ion o he phase shi s has been made in a disc e e way,
selec ing in each case he op ion ha minimizes he objec i e unc ion. Figu e 23
illus a es he layou o he iden i ica ion p ocess o he pedes ian walking la e al
o ce o he p oposed TDOF-sys em.
Figu e 23. Layou o he walking pedes ian la e al o ce iden i ica ion
me hodology.
55
In Pape E, he esul s o he es ima ion p ocess a e summa ized, showing he
di e en es ima ed pa ame e s e sus he pedes ian s ep equency. Acco ding o
hese esul s, i is possible o ob ain a s a is ical es ima ion o he design a iables
ha cha ac e ize he pedes ian walking la e al o ce.
 Fi s LDLF, la ,1

, )017.0,086.0(N.
 Second LDLF, la ,2

, )009.0,094.0(N.
 Thi d LDLF, la ,3

, )019.0,040.0(N.
 Second la e al phase shi 0
,2

la

º.
 Thi d la e al phase shi 0
,3

la

º.
Figu e 24 illus a es he la e al pedes ian walking o ce ob ained om he
p oposed iden i ica ion p ocedu e. The maximum and minimum en eloped alues o
he la e al o ces shown in Figu e 2 a e also ep esen ed.
Figu e 24. La e al pedes ian walking o ce acco ding o he TDOF-sys em.
Fo he gene a ion o he pedes ian lows o he c owd-s uc u e in e ac ion model
in bo h di ec ions, he abo e Gaussian dis ibu ions we e conside ed.
-200.00
-150.00
-100.00
-50.00
0.00
50.00
100.00
150.00
200.00
0.00 0.20 0.40 0.60 0.80 1.00
La e al Load [N]
Time [sec.]
Minimum
Maximum
TDOF-sys em
56
6. Valida ion and main esul s o his Thesis.
Once ob ained he pa ame e s o he p oposed TDOF-sys em in bo h di ec ions, he
de ini ion o he p oposed model is comple e. The p oposed model is alida ed in
his sec ion by co ela ing he expe imen al and nume ical dynamic esponse o wo
eal oo b idges unde he e ec s induced by he pedes ian ac ion. In e ical
di ec ion (Pape B), he p oposed model was implemen ed o analyze he dynamic
esponse and he change o he i s e ical na u al equency o he Viana
oo b idge unde he p e iously desc ibed c owd es . In la e al di ec ion (Pape E),
he p oposed model was implemen ed o analyze he la e al lock-in phenomenon on
Ped o e Inês oo b idge (Coimb a, Po ugal), including he es ima ion o i s dynamic
esponse and he change o i s i s la e al na u al equency due o he pedes ian
ac ion.
6.1. Analysis o he change o he modal p ope ies o he Viana oo b idge.
The alidi y and applicabili y o he p oposed c owd-s uc u e in e ac ion model in
e ical di ec ion is assessed h ough i s p ac ical applica ion o he ollowing case
s udy (Pape B). F om he o ced eco ded esponse o he p e iously men ioned
c owd es , he expe imen al analysis o he change o he i s e ical na u al
equency o he s uc u e, due o he c ossing o he g oup o pedes ians a
di e en s ep equencies, was de e mined. Figu e 26 illus a es he expe imen al
analysis o he change o he i s e ical na u al equency o he Viana oo b idge.
Subsequen ly, he c owd-s uc u e in e ac ion model has been applied o he
upda ed ini e elemen model in o de o ob ain, i s , he e ical nume ical
accele a ion a he men ioned sec ions and, la e , o analyse nume ically he
change o he i s e ical na u al equencies o he oo b idge due o he p esence
o he pedes ians.
The assessmen o i s pe o mance has been done by co ela ing he abo e
expe imen al esul s wi h he nume ical es ima ions p edic ed by he model. Fo
each conside ed s ep equency en gene a ions o g oups wi h 50 pedes ians we e
simula ed. The numbe o pedes ians in phase in each new simula ion was
de e mined by he e alua ion o he pa ame e p

. The desi ed eloci y, d
, o each
pedes ian was assigned acco ding o Eq.(5). A pedes ian mass o 70 kg has been
conside ed acco ding o he F ench code (Se a, 2006). As ini ial spa ial dis ibu ion
o he pedes ians, a ec angula g id was selec ed, conside ing an ini ial dis ance
among pedes ians 50.0
p
d m wi h an equidis an dis ibu ion in he wid h o he
deck. The selec ed ime s ep is 01.0


sec. The nume ical e ical accele a ion
( o h ee o he 50 pedes ian gene a ions) a sec ion 2
So he oo b idge, o a
s ep equency o 1.60 Hz, is shown in Figu e 25.
57
Figu e 25. Expe imen al e sus nume ical accele a ion ( h ee gene a ions) a
sec ion S2 o Viana oo b idge unde a g oup o 50 pedes ians (walking equency
o 60.1
s
Hz).
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
Ve ical expe imen al accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
1s Gene a ion. Ve ical nume ical accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
2nd Gene a ion. Ve ical nume ical accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
3 h Gene a ion. Ve ical nume ical accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
58
As Figu e 25 shows, he co ela ion be ween he expe imen ally eco ded e ical
accele a ion and he nume ically es ima ed alues is adequa e, in e ms o bo h he
alue o he maximum accele a ion and i s empo al a ia ion.
Finally, he nume ically es ima ed e ical accele a ion a sec ion 2
So he
oo b idge unde a g oup o 50 pedes ians o di e en s ep equencies was used
o iden i y he i s na u al equency o he s uc u e, ollowing he p ocedu e
desc ibed in Pape B. Figu e 26 illus a es he co ela ion be ween expe imen al
and nume ical esul s o he change o he i s e ical na u al equency o he
oo b idge induced by he c owd-s uc u e in e ac ion phenomenon. The nume ical
es ima ion o he change o he i s e ical na u al equency was ob ained om
he mean alues o en simula ions o each s ep equency. Good ag eemen
be ween bo h se s o esul s is obse ed, wi h di e ences below 1.50 % o all he
analysed pedes ian walking equencies. The i s e ical na u al equency
co esponding o he emp y oo b idge is included in Figu e 26 o e e ence.
Figu e 26. Change o he i s e ical, e
,1 [Hz], expe imen al (Exp.) and
nume ical (Num.) na u al equency e sus he s ep equency s
[Hz].
6.2. Analysis o he la e al lock-in phenomenon on he Ped o e Inês
oo b idge.
The Ped o e Inês oo b idge is loca ed a Coimb a (Po ugal). The o al leng h o he
s uc u e is 274.5 m, con igu ed by one cen al a ch o 110 m, wo la e al semi-
a ches o 64 m and wo ansi ion spans o 30.5 and 6 m (Figu e 27). The main
ea u e o he oo b idge is he an i-symme ical con igu a ion o he deck and he
a ches wi h espec o he longi udinal axis o he s uc u e. The deck is a conc e e-
s eel composi e box-gi de wi h a a iable wid h be ween 4 and 8 m, wha
gene a es a pano amic squa e a mid-span o he oo b idge (Figu e 28.a). F om i s
design phase, he nume ical s udies de eloped abou he oo b idge indica ed ha
he s uc u e was p one o ib a ions induced by pedes ians in la e al di ec ion.
This ac mo i a ed he de elopmen o a p ecise and de ailed wo k o he
expe imen al assessmen o i s dynamic esponse and he implemen a ion o a
con ol sys em in o de o gua an ee an adequa e com o le el o he oo b idge.
2.940
2.960
2.980
3.000
3.020
3.040
3.060
3.080
3.100
3.120
3.140
3.160
1.30 1.50 1.70 1.90 2.10 2.30 2.50
1, e
[Hz]
s
[Hz]
Exp.
Num.
Emp y

59
This wo k was pe o med and epo ed by Cae ano e al. (2010) and i s esul s
ha e been used in his Thesis in o de o alida e he p oposed c owd-s uc u e
in e ac ion model in la e al di ec ion.
Figu e 27. Ele a ion and plan o he Ped o e Inës oo b idge (Cae ano e al.,
2010).
The oo b idge p esen ed a i s la e al ib a ion mode wi h an expe imen al na u al
equency o 0.91 Hz and an associa ed damping a io o 0.55 % ha was easily
exci ed by he pedes ian lows. In o de o de e mine expe imen ally he numbe
o pedes ians ha o igina es he la e al lock-in phenomenon an expe imen al es
was pe o med. Subsequen ly, in o de o alida e he pe o mance o he p oposed
c owd-s uc u e model, an expe imen al and nume ical analysis o he la e al lock-
in phenomenon on he Ped o e Inês oo b idge has been co ela ed. The analysis
ocused on he beginning o he ins abili y phenomenon, as i is he si ua ion whe e
he e ec o he modal pa ame e s o he pedes ians has mo e in luence in he
dynamic beha iou o he s uc u e (Dalla d e al., 2001). The nume ical la e al
lock-in simula ion is ob ained om he implemen a ion o he p oposed c owd-
s uc u e in e ac ion model on an upda ed ini e elemen model o he Ped o e Inês
oo b idge epo ed in he li e a u e (Cae ano e al., 2010).
Figu e 28. a) Pe spec i e o he oo b idge and b) expe imen al la e al lock-in
pedes ian es on his oo b idge (Cae ano e al., 2010).
a
)
b
)
60
In he expe imen al la e al lock-in es , he la e al accele a ion, la
a, a mid-span o
he oo b idge unde he c ossing o di e en g oup o pedes ians was eco ded
(Figu e 28.b). A g aphical ep esen a ion o he maximum la e al accele a ion a
his posi ion e sus he numbe o pedes ians on he oo b idge (Figu e 29) allows
iden i ying he ins abili y si ua ion associa ed wi h he la e al lock-in phenomenon.
As i is epo ed in he li e a u e (Cae ano e al., 2010) and i is illus a ed in Figu e
29 he numbe o pedes ians ha o igina es he beginning o he la e al lock-in
phenomenon is a ound 75.
Figu e 29. Expe imen al (Cae ano e al. 2010) and nume ical a ia ion o he
maximum la e al accele a ion,


max
la
a, a mid-span e sus he numbe o
pedes ians.
Subsequen ly, a nume ical la e al lock-in analysis based on he p oposed c owd-
in e ac ion model was pe o med. Each conside ed g oup o pedes ians was
simula ed conside ing as ini ial spa ial dis ibu ion a ec angula -shaped g id wi h
an ini ial dis ance among pedes ians 50.0

p
d m and a equidis an dis ibu ion in
he wid h o he deck. The coo dina es o he conside ed la e al ib a ion modes o
he s uc u e we e conside ed om he esul s p o ided by he li e a u e (Cae ano
e al. 2010). In o de o accoun o he change o he s uc u al damping o he
oo b idge acco ding o i s ib a ion le el a pa abolic unc ion has been es ablished
based on he esul s ob ained by Geo gakis and Jo gesen (Geo gakis and Jo gesen,
2014) in a labo a o y oo b idge. The ange o a ia ion o he damping a io was
comp ised be ween he expe imen al alue ob ained in he p e iously men ioned
ee ib a ion es and he limi alue unde s ong ib a ions p oposed by he mo e
ecen in e na ional s anda ds (Bu z e al., 2007; Se a, 2006). The maximum
nume ical la e al accele a ion a mid-span e sus he numbe o pedes ians on he
oo b idge is shown in Figu e 29. As Figu e 29 shows, he co ela ion be ween he
expe imen al la e al maximum accele a ions and he nume ically es ima ed
maximum alues a e adequa e. Addi ionally, he es ima ion o he nume ical
maximum accele a ion ob ained applying he me hodology p oposed by he mo e
0.00
0.10
0.20
0.30
0.40
0.50
0.60
15 25 35 45 55 65 75 85
(a
la
)
max
[m/s
2
]
Numbe o pedes ians
Lock-in c i e ion (Se a, 2006)
Exp.
Num. (Synpex)
Num. (TDOF-sys em)
61
ecen in e na ional s anda ds (Bu z e al., 2007; Se a, 2006) is shown in Figu e
29. The p oposed model allows ob aining a mo e accu a e nume ical analysis o he
la e al lock-in phenomenon han he conside ed s anda ds. The la e al lock-in
c i e ion es ablished by F ench s anda ds (Se a, 2006) is also illus a ed o
e e ence in Figu e 29.
Figu e 30. Expe imen al (Cae ano e al., 2010) and nume ical a ia ion o he i s
la e al, la
,1 , na u al equency o he oo b idge e sus he numbe o pedes ians.
Finally, he i s la e al nume ical na u al equency o he oo b idge du ing he
occu ence o he la e al lock-in phenomenon was ob ained and i is shown in
Figu e 26. The expe imen al i s la e al nume ical na u al equency (Cae ano e
al., 2010) is also shown in Figu e 30. Good ag eemen be ween bo h se s o esul s
is obse ed, wi h di e ences below 0.35 % in he s udied ange o he numbe o
pedes ians. Addi ionally, he alue o he i s la e al na u al equency
co esponding o he emp y oo b idge is illus a ed o e e ence (Figu e 30).
0.885
0.890
0.895
0.900
0.905
0.910
0.915
65 70 75 80 85
1,la
[Hz]
Numbe o pedes ians
Exp.
Num.
Emp y
62
7. Conclusions and u u e esea ch.
7.1 Conclusions.
In his wo k, a new c owd-s uc u e in e ac ion model has been p esen ed. The
p oposed model has been alida ed h ough he co ela ion be ween he
expe imen al and nume ical dynamic esponses o wo eal oo b idges unde he
pedes ian ac ion. The p oposed model is o ganized in wo sub-models: (i) a
pedes ian-s uc u e in e ac ion and (ii) a c owd sub-model. The pedes ian-
s uc u e in e ac ion sub-model is de ined in e ms o a TDOF-sys em, wi h sp ung
and unsp ung masses, whose pa ame e s ha e been es ima ed expe imen ally om
he esul s o wo expe imen al es s conduc ed on he Viana oo b idge (Viana do
Cas elo, Po ugal).
As iden i ica ion echnique he solu ion o an in e se dynamic p oblem has been
u ilized, minimizing an objec i e unc ion de ined as he mean squa e di e ences
be ween an expe imen al and nume ical magni ude. The es ima ion o he
pa ame e s o he model has been limi ed o e ical and la e al di ec ion since
he e a e ew epo ed cases o ib a o y p oblems in longi udinal di ec ion.
In e ical di ec ion, he expe imen al and nume ical accele a ions on ou poin s o
he Viana oo b idge unde he c ossing o wo pedes ians a con olled s ep
equencies ha e been conside ed as objec i e unc ion.
In la e al di ec ion, he iden i ica ion p ocess has been di ided in wo s eps. In he
i s s ep, he modal pa ame e s o he TDOF-sys em has been es ima ed
conside ing as objec i e unc ion he mean squa e e o be ween he i s
expe imen al and nume ical la e al na u al equency o he Viana oo b idge unde
he c ossing o a g oup o i y pedes ians a di e en con olled s ep equencies.
Subsequen ly, in he second s ep he walking pedes ian la e al o ce o he
p oposed model is es ima ed conside ing as objec i e unc ion he mean squa e
di e ences be ween he expe imen al and nume ical powe spec al densi y
ob ained in ou poin s o he Viana oo b idge unde he c ossing o he wo
men ioned pedes ians.
Fo he minimiza ion o he abo e objec i e unc ions, as global op imiza ion
me hod, he gene ic algo i hms ha e been used in all he cases. The es ima ed
pa ame e s a e wi hin he ange ecommended by p e ious wo ks in he li e a u e.
The c owd sub-model is de ined in e ms o a mul i-agen model based on he
ela ionships es ablished by he social o ce model. The in e ac ion be ween he wo
sub-models is achie ed by imposing wo beha iou al condi ions, a com o and
la e al lock-in h esholds. I he e ical o la e al accele a ions expe ienced by each
pedes ian a e abo e ce ain accele a ion limi s, he a ec ed pedes ian modi ies
his s ep eloci y. Addi ionally, i he la e al accele a ions exceed he limi
es ablished by he F ench s anda d in o de o cha ac e ize he la e al lock-in
phenomenon, he a ec ed pedes ian synch onizes his/he equency s ep and
phase shi wi h he mo emen o he deck.
The p oposed model is o mula ed unde he ollowing hypo hesis: (i) he
pa ame e o he pedes ian-s uc u e in e ac ion model a e assumed cons an , so
ha hey do no a y acco ding o he s ep equency o each pedes ian, (ii) he
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PART II
APPENDED PAPERS
71
II. Appended pape s.
72
Pape A: A di ec -pedes ian s uc u e in e ac ion model o cha ac e ize
he human induced ib a ions on slende oo b idges
The o iginal e sion o his pape can be ound in
doi: 10.3989/ic.2014. 66.iEx a-1
Jou nal name: In o mes de la cons ucción
ISI 2014 Classi ica ion: Q4 (53/59) Cons uc ion and Building Enginee ing. Impac
Fac o : 0.273
SCIMAGO 2014 Classi ica ion: Q3 (121/215) Ci il Enginee ing SJR: 0.345
ISSN: 0020-0883
In o mes de la Cons ucción
Vol. 66, EXTRA 1, m007
diciemb e 2014
ISSN-L: 0020-0883
doi: h p://dx.doi.o g/10.3989/ic.13.110
Recibido/Recei ed: 19/07/2013
Acep ado/Accep ed: 19/11/2013
ABSTRACT
Al hough he scien i ic communi y had knowledge o he human induced ib a ion p oblems in s uc u es since he end
o he 19 h cen u y, i was no un il he occu ence o he ib a ion phenomenon happened in he Millennium B idge
(London, 2000) ha he impo ance o he p oblem e ealed and a highe le el o a en ion de o ed. Despi e he la ge ad-
ances achie ed in he de e mina ion o he human-s uc u e in e ac ion o ce, one o he main de iciencies o he exis ing
models is he exclusion o he e ec o changes in he oo b idge dynamic p ope ies due o he p esence o pedes ians.
In his pape , he o mula ion o a human-s uc u e in e ac ion model, add esses hese limi a ions, is ca ied ou and i s
eliabili y is e i ied om p e iously published expe imen al esul s.
Keywo ds: Slende oo b idges; human induced ib a ion; pedes ian-s uc u e in e ac ion; dynamic beha iou
change.
RESUMEN
Aunque la comunidad cien í ica enía conocimien o de los p oblemas ib a o ios inducidos po pea ones en es uc u as
desde inales del siglo xix, no ue has a la ocu encia de los e en os ib a o ios acon ecidos en la pasa ela del Milenio (Lon-
d es, 2000), cuando la impo ancia del p oblema se puso de mani ies o y se le comenzó a dedica un mayo ni el de a en-
ción. A pesa de los g andes a ances alcanzados en la ca ac e ización de la ue za de in e acción pea ón-es uc u a una de
las p incipales de iciencias de los modelos exis en es es la exclusión del cambio en las p opiedades dinámicas de la pasa ela
po la p esencia de pea ones. En es e a ículo, se p esen a la o mulación de un modelo de in e acción pea ón-es uc u a
que in en a da espues a a dichas limi aciones, y su alidación a pa i de esul ados expe imen ales p e iamen e publi-
cados po o os au o es.
Palab as cla e: Pasa elas esbel as; ib aciones inducidas po se es humanos; in e acción pea ón-es uc u a; modi ica-
ción de compo amien o dinámico.
(*) Uni e si y o Se ille (España).
Pe sona de con ac o/Co esponding au ho : [email p o ec ed] (J. F. Jiménez-Alonso)
A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize
he human induced ib a ions on slende oo b idges
Un modelo di ec o de in e acción pea ón-es uc u a pa a ca ac e iza las
ib aciones inducidas po pea ones en pasa elas esbel as
J. F. Jiménez-Alonso(*), A. Sáez(*)
Cómo ci a es e a ículo/Ci a ion: Jiménez-Alonso, J. F., Sáez, A. (2014). A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize
he human induced ib a ions on slende oo b idges. In o mes de la Cons ucción, 66(ex a-1): m007, doi: h p://dx.doi.o g/10.3989/ic.13.110.
Licencia / License: Sal o indicación con a ia, odos los con enidos de la edición elec ónica de In o mes de la Cons ucción se
dis ibuyen bajo una licencia de uso y dis ibución C ea i e Commons Reconocimien o no Come cial 3.0. España (cc-by-nc).

J. F. Jiménez-Alonso, A. Sáez
In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.1102
1. INTRODUCTION
The phenomenon o in e ac ion be ween pedes ians and
b idges is known since, a he end o he 19 h cen u y (1), a
g oup o 60 soldie s exci ed, unde hei s ep, a b idge lo-
ca ed in he B i ish own o B ough on. Al hough he scien-
i ic communi y did no s op s udying his issue, i was he
occu ence o he phenomenon happened in he Millennium
B idge (London) ha s essed he impo ance o he p ob-
lem and led o a highe le el o a en ion (2). In mos cases,
he e ec ha he pedes ians induce on he oo b idge has
been idealized like a mo ing a iable o ce on he s uc u e
(3). The a iabili y o he abo e men ioned load ies o ha e
in conside a ion he a ia ion o he le el o p essu es ha
akes place be ween he pedes ian and he deck du ing he
phenomenon o he s ep. Howe e , in all hese models, ei he
he e ec ha he pedes ians ha e on he dynamic cha ac-
e is ics o he s uc u e is neglec ed, o such e ec is consid-
e ed by means o e y simpli ied inge ules. Consequen ly,
hese models do no inco po a e app op ia ely he ene ge ic
exchange ha akes place be ween bo h sys ems du ing he
s ep o he pedes ian lows on he s uc u e. Ne e heless, in
he exis ing publica ions (3) he e a e clea indica ions abou
he impo ance o he dynamic in e ac ion phenomena, wi h
e idence ha bo h he equencies and he modes o ib a-
ion o he s uc u e a e a ec ed by he s ep o pedes ian
g oups. In he case o s uc u es subjec ed o la ge pedes ian
lows, he co ec es ima ion o he change o hei dynamic
p ope ies due o he pedes ian c ossing is e y impo an
du ing he design phase, in o de o adjus as much as possi-
ble he na u al equencies o he s uc u e ou side he ange
o pedes ian s ep equencies and, in he case o an in e en-
ion on an exis ing oo b idge, in o de o imp o e i s com o
le el (4) (5).
In he p esen wo k, a me hodology o he co ec cha -
ac e iza ion o he whole dynamic beha io is p oposed,
by implemen ing a human-s uc u e in e ac ion model
wi h h ee deg ees o eedom, in o de o cha ac e ize
he mo emen o he g a i y cen e o he pedes ian in
he h ee spa ial di ec ions. The p oblem o ene ge ic ex-
change is add essed in a di ec o m, ealizing he modal
p ojec ion o he coo dina es in con ac be ween he pe-
des ian and he s uc u e, and main aining he physical
coo dina es o he g a i y cen e o he pedes ian. The
model conside s, in he same way, he local e ec o he
s ep by means o he modal p ojec ion o he co espond-
ing in e ac ion o ce.
This p ocedu e o esolu ion allows, on he one hand, o un-
couple he equa ions o he dynamic sys em ha go e ns he
beha io o he s uc u e, hus acili a ing he e ec i e ap-
plica ion o he model om he modal cha ac e is ics o he
oo b idge, as ob ained om any comme cial so wa e based
on he ini e elemen me hod; and on he o he hand, i al-
lows o es ima e in a di ec o m bo h he dynamic cha ac-
e is ics o he s uc u e du ing he pedes ian s ep, as well
as he componen s o he pedes ian cen e o g a i y ac-
cele a ion. Fu he mo e, addi ional pa ame e s, such as he
sign o pedes ian damping in oduced in o he sys em, may
be included in he model. Finally, a alida ion example o
he p oposed model is p esen ed, whe e he change o he
dynamic beha iou o a eal labo a o y oo b idge du ing a
a iable low o pedes ians is a o ably compa ed wi h he
model p edic ions.
2. ANALYSIS OF CURRENT STANDARDS
Cu en ly, he mos ad anced in e na ional codes abou he
dynamic beha iou o slende oo b idges (4) (5) de e mine
ha , in a wide way, i he na u al equencies o he s uc u e
is in he ange o pedes ian walking s ep equency (1.25-
2.30 Hz o e ical ib a ions and 0.50-1.20 Hz o ho izon-
al ib a ions) he accele a ion, in ha di ec ion, needs o be
de e mined and checked agains accele a ion limi s (Table 1)
o gua an ee an app op ia e com o le el o each design
scena io. Fu he mo e, o a oid la e al synch oniza ion he
accele a ion in his di ec ion mus be below 0.10-0.15 m/s2.
The design scena io is es ablished by he expec ed pedes ian
a ic (Table 2) and he si ua ion o impo ance o he s uc-
u e. The com o le el is de e mined by he owne o he s uc-
u e, and no mally a medium com o le el mus be gua an-
eed o all a ic classes, excep o pedes ian densi ies abo e
1.00 P (Pe son)/m2 whe e a minimum com o is accep able.
The pedes ian induced ac ion is ep esen ed as an oscilla-
o y dis ibu ed load p( ), de ined as:
[1] () cos(
2)
πψ
=⋅ ⋅⋅⋅⋅
′⋅p
G
n
p
whe e:
G, is he conside ed componen o he s ep o ce (G=280 N
e ical, 140 N longi udinal and 35 N la e al) (4) (5).
, is he na u al equency o he s uc u e unde conside a ion.
′
np, is he equi alen pedes ians numbe , de ined by
[2] 10.80
ζ
′=⋅⋅
nn
pp
o a ic classes TC1-TC3 o
[3] 1.85
′=⋅
nn
pp
o a ic classes TC4-TC5.
ψ, is he educ ion coe icien ha akes in o accoun he
p obabili y ha he oo all equency app oaches he na u al
equency unde conside a ion.
ζ, is he s uc u al damping a io.
np, is he numbe o he pedes ians on he loaded su ace S
(np = S · densi y).
S, is he loaded su ace ha depends on he shape o he no -
mal mode unde conside a ion.
Table 1. De ined com o classes wi h limi accele a ion anges (5).
Le el Deg ee Ve ical
accele a ion
Ho izon al
accele a ion
CL1 Maximum <0.50 m/s2<0.10 m/s2
CL2 Medium 0.50-1.00 m/s20.10-0.30 m/s2
CL3 Minimum 1.00-2.50 m/s20.30-0.80 m/s2
CL4 Discom o >2.50 m/s2>0.80 m/s2
Table 2. T a ic classes (5).
Classes Densi y
d [P/m2] Cha ac e is ics
TC1 < 15 P 15 single pe sons
TC2 < 0.20 P/m2Com o able and ee walking
TC3 < 0.50 P/m2Un es ic ed walking, signi ican ly
dense a ic
TC4 < 1.00 P/m2Uncom o able si ua ion, obs uc ed
walking
TC5 < 1.50 P/m2Unpleasan walking, e y dense a ic
A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges
Un modelo di ec o de in e acción pea ón-es uc u a pa a ca ac e iza las ib aciones inducidas po pea ones en pasa elas esbel as
In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.110 3
Howe e , his me hodology p esen s some limi a ions:
• he equi alen pedes ian numbe (pedes ian mo ing in
phase wi h he s uc u e) has been de e mined by he ex-
pe imen al esul s o only one oo b idge (5).
• he change in he dynamic s uc u al p ope ies ha he pedes-
ians low causes is conside ed h ough a inge ule (addi ion
o all he pedes ian mass densi y o he s uc u e mass ma ix).
• he in e ac ion be ween pedes ians and he s uc u e is only
sligh ly conside ed, so he in e na ional s anda ds do no
conside adequa ely he synch oniza ion phenomenon be-
ween pedes ians o be ween hese ones and he s uc u e.
The es ima ions ca ied ou , unde his me hodology, no -
mally o e es ima e he eal esul s (5).
3. PROPOSAL OF A HUMAN-STRUCTURE
INTERACTION MODEL
In his sec ion a me hod o he simula ion o he in e ac ion
be ween he pedes ian and he oo b idge is p oposed. I ol-
lows om he applica ion o he dynamic equilib ium equa-
ions o a simpli ied model o in e ac ion wi h sp ung and
unsp ung masses (Figu e 1).
Fo n modes o ib a ion φi (x), he o al esponse o he
s uc u e may be decomposed in e ms o he ampli ude o
he di e en modes yi ( ) as:
[4] w(x, )=yi( )⋅
ϕ
i(x)
i=1
n
∑
[5] 
w(x, )=
yi( )⋅
ϕ
i(x)
i=1
n
∑+yi( )⋅ ⋅′
ϕ
i(x)
i=1
n
∑
[6]

w(x, )=
yi( )⋅
ϕ
i(x)
i=1
n
∑+2⋅
yi( )⋅ ⋅′
ϕ
i
i=1
n
∑(x)+yi( )⋅ 2⋅′′
ϕ
i(x)
i=1
n
∑
whe e
[7] ′
ϕ
i(x)=d
dx
ϕ
i(x) is he spa ial de i a e
o he mode o ib a ion i.
[8] ′′
ϕ
i(x)=d2
dx2
ϕ
i(x) is he second spa ial de i a e
o he mode o ib a ion i.
and i is neglec ed, due o i s low magni ude, he empo al
a ia ion o he s ep speed .
Conside ing he equilib ium o he sys em, s uc u e and pe-
des ian model, he ollowing coupled equa ion sys em may
be ob ained.
[9] Mi
yi+Ci
yi+Kiyi=
ϕ
i
( )
⋅Fin
[10] 0
 
()()
+−+−=my cy ykyy
aa as as
[11] in
 
()()
+−+−=−my cy ykyy
FF
ss sa sa s
Thus, Fin ollows om he abo e equa ion o yield.
[12] in  
()()
=− −−
−−
FFmy cy yk
yy
ssss
as
a
And subs i u ing his equa ion in o he equilib ium equa ion
o he s uc u e.
[13]
M
i

y
i
+C
i

y
i
+K
i
y
i
=
ϕ
i
( )
⋅F
s
−m
s

y
s
−c
y
s
−
y
a
( )
−k y
s
−y
a
( )
( )
Applying he equa ions o compa ibili y o displacemen s, e-
loci y and accele a ion be ween he s uc u e and he simpli-
ied model o in e ac ion.
[14] (,)=ywx
s
[15] (,)

=ywx
s
[16] (,)
 
=ywx
s
Subs i u ing hese ela ions in he o e all dynamic equilib-
ium equa ion o he s uc u e and o ganizing in o ma ion
in a ma ix o m, he ollowing model o in e ac ion is ob-
ained.
[17] () () () () () ()
()
 
⋅+ ⋅+ ⋅=
M y C y K y F
Conside ing he na u e o he esul ing sys em, he use o a
me hod o β-Newma k in eg a ion amily is p oposed, wi h
pa ame e s β=1/4 and γ=1/2, hus ensu ing an uncondi ion-
ally s able sys em.
Figu e 1. Pedes ian-s uc u e in e ac ion model.
J. F. Jiménez-Alonso, A. Sáez
In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.1104
unde h ee con olled g oup o pedes ians, will be com-
pa ed in o de o de e mine he pa ame e s and goodness o
he p oposed model.
The alida ion will be ca ied ou , o simplici y, in he e -
ical di ec ion, al hough he ex ac ed esul s a e easily ex-
apola ed o he o he di ec ions.
4. DETERMINATION OF THE WALKING FORCES
The mo emen o he body mass and he pu -down, olling
and push-o o he ee o one pedes ian gene a e he in-
duced h ee-dimensional o ces be ween bo h elemen s, Fs,
ha acco ding o he esea ch de eloped by di e en au ho s
(4), can be de e mina ed om a Fou ie se ies decomposi ion
in he h ee-space componen s.
[20] Fp, e ( )=P1+
α
i, e sin 2
π
i s −
φ
i
( )
i=1
n
∑
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
[21] Fp,la ( )=P
α
i,la sin
π
i s −
φ
i
( )
i=1
n
∑
[22] Fp,long( )=P
α
i,long sin 2
π
i s −
φ
i
( )
i
=
1
n
∑
whe e
Fp, e e ical pe iodic o ce due o walking o unning
Fp,la la e al pe iodic o ce due o walking o unning
Fp,long longi udinal pe iodic o ce due o walking o unning
P [N] medium pedes ian weigh (in e na ionally conside ed
as P=700.00 N)
αi, e αi,la αi,long Fou ie coe icien o he i h ha monic o e -
ical, la e al and longi udinal o ces o dynamic load ac o
(DLF).
s [Hz] s ep equency
φi phase shi o he i h ha monic
n o al numbe o con ibu ing ha monics.
Among he con ibu ions o he di e en au ho s, o he
de elopmen o he p esen documen , he e ical dynamic
In he p e ious exp essions, he alue o he ib a ion modes
is ze o, when he pedes ian emains ou side he s uc u e.
[18]
ϕ
i(x)=0 o 0≥x≥L o , wi h L being he
leng h o he s uc u e
In he p oposed me hod, φi (x) is ob ained, in a disc e e way,
using he ini e elemen me hod, collec ing he modal dis-
placemen s and de i a es in each o he nodes o he s uc-
u e. To ob ain a con inuous unc ion o he modes hey a e
de e mined om he shape unc ions consis en wi h he i-
ni e elemen app oxima ion. Fo he oo b idge, he in e po-
la ion unc ions a e cubic adop ing he Be noulli hypo hesis
o he beam elemen s.
[19]
ϕ
i(x)=
ϕ
i
j⋅Nj(x)
j
∑
Whe e Nj (x) a e he shape unc ions and
ϕ
i
j a e he nodal
alues.
Fo a g oup o k pedes ians (Figu e 2), we may u he
ep esen each one by he abo e simpli ied in e ac ion
model.
When a g oup o pedes ians is conside ed in he calcula-
ions, he numbe o di e en ial equa ions o sol e inc eas-
es. In he case o a single pedes ian, he p oposed model
leads o a sys em o n+1 equa ions, co esponding o he
conside ed numbe o ib a ion modes n plus he app op i-
a e simpli ied in e ac ion mechanical elemen sys em. Simi-
la ly, when conside ing a g oup o k pedes ians, a sys em
o n+k di e en ial equa ions will need o be sol ed. I is im-
po an o no e ha he equa ions o he modes o ib a ion
o he s uc u e a y in e ms o he posi ion o pedes ians.
A e e y ins an , he numbe s o pedes ian on he de o med
shape mus be calcula ed, as well as he alue o he ampli-
ude, slope and cu a u e co esponding o hei posi ion.
As a p elimina y alida ion o he p oposed o mula ion, he
p e iously de ined pa ame e s will be es ima ed om he e-
sul s a ailable in he li e a u e o compa able s udies (6), as
summa ized in nex sec ions. Finally, he expe imen al and
nume ical dynamic cha ac e is ics o a labo a o y oo b idge,
Figu e 2. Pedes ians g oup acco ding o he pedes ian-s uc u e in e ac ion model.
A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges
Un modelo di ec o de in e acción pea ón-es uc u a pa a ca ac e iza las ib aciones inducidas po pea ones en pasa elas esbel as
In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.110 5
load ac o s p oposed by Se a (5) (see Table 3 and Figu e 3)
will be conside ed o cons uc and alida e ou model. This
c i e ion is widely accep ed by bo h he scien i ic communi y
and he designe s o his ype o s uc u es. These s anda ds
ob ained he dynamic coe icien s om expe imen al es s
pe o med on mobile pla o ms. The pedes ian load, acco d-
ing he esul s o such es s, is adequa ely cha ac e ized by
he con ibu ion o he i s h ee ha monics.
The ela ion be ween he eloci y magni ude, , and he pac-
ing equency, s, is conside ed by he empi ical ela ionship
based on he wo k o Be am and Ruina (3).
[23] s=0.35⋅ 3−1.59 ⋅ 2+2.93⋅
5. INITIAL ESTIMATION OF THE DYNAMIC
PROPERTIES OF THE HUMAN-STRUCTURE
INTERACTION MODEL
Fo he es ima ion o he dynamic cha ac e is ics o he
SDOF-sys em, as a i s app oxima ion, a wide bibliog aphic
s udy has been made. The e a e se e al s udies ha collec
he e ec o spec a o s on s adiums s ands in he dynamic
beha iou o he s uc u e by a SDOF s a ic sys em (6). In
Figu e 4 and Table 4, a scheme o he models used and he
es ima ed dynamic pa ame e s a e shown, whe e h and ζh a e
he na u al equency and he equi alen damping a io o he
human sys em.
The abo e esul s allow es ablishing a likely ange o a ia-
ion o he sys em pa ame e s. Thus, conside ing he maxi-
mum and minimum alues (Figu e 4) o he sp ung mass
(ma), he equi alen human damping a io (ζh) and he e i-
cal na u al equency ( h) o each pedes ian, i is shown in
Table 5 a possible ange o a ia ion o he pa ame e s o he
p oposed model. The sp ung mass (ma) is p esen ed as a pe -
cen age o he o al mass. A pedes ian ype wi h a mean o al
mass o 70.00 kg is conside ed, as es ablished by he Eu o-
pean s anda ds (4).
Figu e 3. Ve ical componen o walking pedes ian o ce (5).
Figu e 4. Simpli ied dynamic ep esen a ions o he s anding human
body: (1) SDOF model (2) SDOF model wi h igid suppo .
Table 4. Dynamic p ope ies o SDOF equi alen s
o s anding humans (6).
Human
Model
Modal
P ope ies
Human
Model
Modal
P ope ies
Foschi e al.
(Model 1)
h=3.30 Hz Fala i
(Model 1)
h=10.43 Hz
ζh=53.00 % ζh=50.00 %
ma=91.00 kg ma=25.00 kg
Al-Foqaha’a
(Model 1)
h=3.50 Hz Zheng and
B ownjohn
(Model 1)
h=5.24 Hz
ζh=34.00 % ζh=39.00 %
ma=83.00 kg ma=85.00 kg
Al-Foqaha’a
(Model 2)
h=3.70 Hz Ma sumo o
and G i in
(Model 1)
h=5.74 Hz
ζh=36.00 % ζh=69.00 %
ma=75.00 kg
ms=8.00 kg ma=76.10 kg
B ownjohn
(Model 1)
h=4.90 Hz
Ma sumo o
and G i in
(Model 2)
h=5.88 Hz
ζh=37.00 % ζh=61.00 %
ma=80.00 kg ma=70.60 kg
ms=7.06 kg
Table 3. Fou ie coe icien and phase shi o
e ical dynamic load ac o s (5).
α1α2α3φ1φ2φ
0.40 0.04 0.04 0.00 90.00 90.00
pedes ian-s uc u e in e ac ion sub-model plus (ii) a c owd sub-model. The i s
sub-model ollows om a modal p ojec ion o a sys em wi h wo d.o. , ha
simula es he beha io o each pedes ian, on he ib a ion modes o he s uc u e.
The pa ame e s o his model ha e been es ima ed om he accele a ions
eco ded on a eal oo b idge. Fo he second sub-model, he c owd beha iou is
simula ed ia a mul i-agen me hod. The pe o mance o he esul ing o e all
model is assessed by co ela ing he expe imen al and nume ical dynamic o a
eal oo b idge unde a g oup o pedes ians a di e en con olled s ep
equencies. In pa icula , he change in he i s na u al equency induced
by he pedes ian- oo b idge in e ac ion is discussed in de ail. The p oposed
model leads o nume ical esul s ha exhibi good ag eemen wi h he
eco ded expe imen al alues. The e o e i is a aluable ool o es ima e he
change on he modal p ope ies o a oo b idge induced by he c owd-s uc u e
in e ac ion phenomenon.
Keywo ds: simpli ied biomechanical model, human-s uc u e in e ac ion, c owd
dynamics, change o na u al equencies, oo b idge.
INTRODUCTION.
Du ing he las i een yea s, signi ican e o has been made by he scien i ic
communi y o cha ac e ize adequa ely he dynamic esponse o oo b idges unde
pedes ian lows (Racic e al., 2009; Zi ano ic a al., 2005). Al hough impo an
ad ances ha e been achie ed in he de ini ion o he pedes ian walking o ce (Bu z e
al., 2007; Se a, 2006), some aspec s o he c owd-s uc u e in e ac ion p oblem ha e
no been comple ely sol ed and s ill dese e a en ion. The s udy o he c owd-s uc u e
in e ac ion p oblem has been pe o med acco ding o h ee key aspec s in o de o: (i)
cha ac e ize he walking o ce ansmi ed by each pedes ian; (ii) cha ac e ize he
pedes ian-s uc u e in e ac ion and inally (iii) o cha ac e ize he in e ac ion among
pedes ians in he c owd. In his way, esea ch e o s ocused ini ially on he
de e mina ion o analy ical exp essions o he walking o ce induced by a pedes ian
(Zi ano ic e al., 2005; Bu z e al., 2007). Howe e , as he inc easing sophis ica ion o

his p oposed exp essions, cha ac e izing he pedes ian walking o ce, did no lead o
signi ican imp o emen s in he nume ical es ima ions o he esponse o he oo b idge
unde pedes ian lows, new ac o s we e conside ed in he models. In ha sense, as a
esul o he esea ch conduc ed a he Millennium oo b idge (Dalla d e al., 2001), i
was concluded ha he e ec o a pedes ian low on he s uc u e in ol ed no only an
equi alen pedes ian o ce bu also he modi ica ion o he dynamic p ope ies o he
s uc u e. Subsequen ly, his esul was alida ed by o he epo ed wo ks (Ingol sson e
al, 2008) whe e pedes ians we e conside ed as ac i e damping o ces ha inc eased he
o e all damping o he s uc u e. Following hese wo ks, se e al app oaches ha e been
p esen ed in he li e a u e o model de pedes ian-s uc u e in e ac ion p oblem. The i s
models main ained he idea o equa ing he pedes ian o an ac i e iscous dampe
(Geo gakis and Jo gesen, 2013). Subsequen ly, o he s modal pa ame e s we e
conside ed in he in e ac ion phenomenon leading o he appea ance o single deg ee o
eedom sys ems o cha ac e ize he beha iou o each pedes ian (Shahabpoo e al.,
2013). Acco ding o hese la e models, each pedes ian induced a modi ica ion o bo h
he damping and s i ness ma ix o he oo b idge du ing i s c ossing. On he o he
hand, he cha ac e iza ion o he dynamic esponse o he s uc u e unde pedes ian
lows mo i a ed he s udy o how he pedes ians in e ac in a c owd (Zi ano ic a al.,
2010). In ha sense, bo h s a is ical dis ibu ions o he s ep pedes ian equencies in a
c owd (Venu i e al., 2007) and ela ions be ween he pedes ian eloci y and s ep
equency (B uno and Venu i, 2009) we e es ablished. In o de o cha ac e ize he
c owd beha iou , he models ha e e ol ed om a mac oscopic o a mic oscopic
app oach. The mo emen o he c owd simula ed o iginally by he luid mechanics laws
(Venu i e al., 2007), is cu en ly modelled using pa icle dynamics (Ca oll e al.,
2012), by conside ing each pedes ian as an agen whose equilib ium is achie ed
h ough he in e ac ion o ces applied by i s en i onmen . Cu en ly, se e al p oposals
ha e eme ged (Venu i e al., 2014, Ta a es e al. 2014).in o de o o e a di ec and
join esponse o he h ee abo e men ioned key aspec s. In all hese p oposals, he
c owd-s uc u e in e ac ion has been simula ed h ough he coupling o wo sub-models:
a pedes ian-s uc u e in e ac ion model and a c owd model based on mul i-agen
heo y.
In his pape , a new c owd-s uc u e in e ac ion model in he e ical di ec ion is
p oposed. The model ep esen s an e olu ion o he exis ing p oposals and aims o
imp o e some sho comings o he p e ious epo ed wo ks. The p oposed model
in ol es wo sub-models as well. A pedes ian-s uc u e in e ac ion sub-model ollows
om he modal p ojec ion o a wo deg ee o eedom sys em, whe e he pedes ian
mass is di ided in sp ung plus unsp ung componen s, on he ib a ion modes o he
s uc u e. The c owd beha iou is simula ed ia a mul i-agen sub-model whe e he
mo emen o each pedes ian is go e ned by he expe ienced in e ac ion o ces. A
physical in e ac ion o ce has been included o assess mo e accu a ely he c owd
beha iou unde high pedes ian densi ies. The in e ac ion be ween he wo sub-models
in he e ical di ec ion is achie ed by implemen ing a s op h eshold, so ha i ce ain
accele a ion limi is exceeded he a ec ed pedes ians s op. The es ima ion o he
pa ame e s o he p oposed pedes ian-s uc u e in e ac ion model has been
expe imen ally pe o med based on he esul s o a pedes ian es conduc ed a he
Viana oo b idge (Viana do Cas elo, Po ugal). Subsequen ly, he c owd-in e ac ion
model has been assessed by co ela ing he expe imen al and nume ical esponse o he
Viana oo b idge unde a g oup o 50 pedes ians. Finally, he model has been applied
o s udy he nume ical change o he i s e ical ib a ion mode o he Viana
oo b idge due o he p esence o he g oup o pedes ians.
The p oposed model may u he be used o p edic he occu ence o he la e al lock-in
phenomenon in oo b idges o o imp o e he e iciency o he con ol de ices
in oduced in a oo b idge when ib a o y p oblems a e de ec ed.
The pape is o ganized as ollows: The p oposed c owd-s uc u e in e ac ion model is
p esen ed in sec ion 2, by desc ibing (i) he pedes ian-s uc u e in e ac ion sub-model;
(ii) he c owd sub-model as well as (iii) he in e ac ion mechanisms be ween bo h sub-
models. Sec ion 3 is de o ed o he expe imen al es ima ion o he main pa ame e s ha
cha ac e ize he pedes ian-s uc u e in e ac ion sub-model. In sec ion 4, he alidi y and
accu acy o he o e all c owd-s uc u e in e ac ion model is success ully assessed by
co ela ing bo h he expe imen al and nume ical esul s o a eal oo b idge (Viana do
Cas elo, Po ugal). Finally, some concluding ema ks a e d awn o close he pape in
sec ion 5.
PROPOSAL OF A SIMPLIFIED BIOMECHANICAL CROWD-STRUCTURE
INTERACTION MODEL IN VERTICAL DIRECTION.
The comple e c owd-s uc u e in e ac ion consis s o wo indi idual submodels (Fig. 1),
one o he pedes ian-s uc u e in e ac ion ( ha includes he pedes ian and oo b idge
dynamic beha iou ) and ano he o he c owd.
The pedes ian-s uc u e in e ac ion model is esponsible o modelling, in a simpli ied
way, all he dynamics e ec s (ine ia, damping, s i ness) induced on he oo b idge by
he c ossing o a pedes ian. The e ical accele a ion a
z
 expe imen ed by each
pedes ian is ob ained as ou pu om his model.
The c owd model is implemen ed as a beha iou al model p o iding a desc ip ion o he
indi idual pedes ian posi ion, p
x, walking pedes ian eloci y, p
, and s ep pedes ian
equency, p
, wha allows o simula ing he o e all beha iou o he c owd and i s
in luence in he dynamic beha iou o he oo b idge.
Fig.1. Layou o he biomechanical c owd-s uc u e in e ac ion model.
Fo each i e a ion he c owd model de e mines he posi ion and s ep eloci y o each
pedes ian. These wo pa ame e s a e used as inpu o de ine he s ep equency and
walking o ce o each pedes ian in o he pedes ian-s uc u e model, ob aining as ou pu
he pedes ian e ical accele a ion. The pedes ian eloci y o each indi idual is hen
modi ied acco ding o he le el o accele a ion expe imen ed by each pedes ian.
Finally, he p ocess is epea ed wi h he upda ed alues o he posi ion and he eloci y
(Fig. 1).
Modelling he pedes ian-s uc u e in e ac ion in he e ical di ec ion.
The p oposed pedes ian-s uc u e in e ac ion in he e ical di ec ion ollows om he
applica ion o dynamic equilib ium equa ions (Clough and Penzien, 1993; Dominguez,
2001) o a simpli ied model o in e ac ion (Fig. 2) wi h sp ung ( a
m) and unsp ung
masses ( s
m). This me hodology has been applied p e iously by he au ho s success ully
(Jiménez-Alonso and Sáez, 2014), and in his pape i is gene alized in o de o ake in o
INPUT OUTPUT
PEDESTRIAN-STRUCTURE MODEL
CROWD MODEL
PEDESTRIAN/STRUCTURE
PARAMETERS
CROWD-STRUCTURE MODEL
p
x
p
a
z

accoun he modi ica ion o he pedes ian eloci y due o he c owd-s uc u e
in e ac ion.
Fig.2. Biomechanical pedes ian-s uc u e in e ac ion model.
Conside ing he balance o he sys em, s uc u e and pedes ian model, he ollowing
coupled equa ions a e ob ained.
in _ )( FxzKzCzM piNUMiiiiii





 (1)



0




sapsapaa zzkzzczm  (2)



in , FFzzkzzczm e paspaspss





 (3)
whe e
a
m is he sp ung mass o he pedes ian [kg].
s
m is he unsp ung mass o he pedes ian [kg].
as mmm  is he o al mass o he pedes ian [kg].
a
z is he absolu e e ical displacemen o he sp ung mass [m].
s
z is he absolu e e ical displacemen o he unsp ung mass [m].
p
k is he equi alen s i ness o a pedes ian [N/m].
m
a
c
p
k
p
m
s
z
a
z
s
F
in
L
z
xF
in
F
s
M
i
C
i
K
i
d
p
y
x
p
w(x, )

p
c is he equi alen damping o a pedes ian [sN/m].
e p
F, is he e ical pedes ian o ce due o walking [N].
in
F is he in e ac ion o ce be ween he pedes ian and he s uc u e [N].
i
M is he modal mass o he ib a ion mode i [kg].
i
C is he modal damping o he ib a ion mode i [sN/m]
i
Kis he modal s i ness o he ib a ion mode i [N/m].
iNUM _

is he e ical componen o he nume ical ib a ion mode i.
x pxp  is he longi udinal posi ion o he pedes ian [m].
px
is he longi udinal componen o he pedes ian eloci y ec o [m/s].
F om Eq.(3) he ollowing exp ession is ob ained o , in
F,




aspaspss e p zzkzzczmFF




 
,in ……………………(4)
and subs i u ing his equa ion in o Eq.(1) yields.



 


aspaspss e ppiNUMiiiiii zzkzzczmFxzKzCzM 





  ,_

…(5)
Applying, a he con ac poin he equa ions o compa ibili y o displacemen s, eloci y
and accele a ion be ween he s uc u e and he simpli ied pedes ian-model o
in e ac ion a e ob ained.
),(),( w xwz pxps


 (6)
),(),( w xwz pxps




 (7)
),(),( w xwz pxps




 (8)
These quan i ies may be exp essed in e ms o he ampli ude )( zi and he modal shape
o he n nume ical conside ed ib a ion modes )(
_x
iNUM

, neglec ing he e m o
a ia ion o he pedes ian eloci y o e he ime, as:


 n
i
piNUMip x z xw
1
_)()(),(

(9)
 

 n
i
piNUMpxi
n
i
piNUMip x zx z xw
1
_
1
_)()()()(),(


 (10)
 



 n
i
piNUMxpip
n
i
iNUMxpi
n
i
piNUMip x zx zx z xw
1
_
2
,
1
_,
1
_))()()()(2)()(),(


 …(11)
dx
xd
xiNUM
iNUM
)(
)( _
_



 (12)
2
_
2
_
)(
)( dx
xd
xiNUM
iNUM



 (13)
whe e
)(
_x
iNUM

is he i s spa ial de i a e o he mode o ib a ion i.
)(
_x
iNUM

 is he second spa ial de i a e o he mode o ib a ion i.
The abo e ela ions -Eqs.(6) o (11)- a e hen subs i u ed in he o e all dynamic
equilib ium equa ions -Eqs.(1) o (3)- so ha , o ganizing in o ma ion in a ma ix o m,
he ollowing model o in e ac ion is ob ained (see Appendix I o ma ix o mula ion).
)()()()()()()( FzKzCzM





 (14)
In he p e ious exp essions, he alue o he nume ical ib a ion modes is ze o, when
he pedes ian emains ou side he s uc u e.
0)(
_

piNUM x

o L x
x
p
p


)(
0)( (15)
wi h L being he leng h o he s uc u e
The nume ical ib a ion modes, )(
_x
iNUM

, a e ob ained in a disc e e way using he
co esponding ini e elemen me hod as:


j
j
j
iiNUM xNx )()(
_

..(16)
whe e )(xN ja e he shape unc ions and j
i

a e he nodal alues.
Al hough he pape ocuses on ib a ions in e ical di ec ion. The o mula ion o he
p oposed model may be u he gene alized o he o he wo di ec ions, longi udinal and
la e al, by acco dingly modi ying bo h he equa ion ha go e ns he conside ed
pedes ian load in each di ec ion and he alue o he modal pa ame e s ha de ine he
TDOF ( wo deg ees o eedom) pedes ian model. In his way, he esul ing model
would be sui able o he mo e gene al 3-D p oblem and i could he e o ake in o
accoun he possible in e ac ion in he h ee spa ial di ec ions.
Fo a g oup o k pedes ians (Fig. 2), each o hem will be ep esen ed by he abo e
simpli ied in e ac ion model. In he case o a single pedes ian, he p oposed model
leads o a sys em o n+1 equa ions, co esponding o he conside ed numbe o ib a ion
modes n plus he simpli ied in e ac ion equa ion. Simila ly, when conside ing a g oup
o k pedes ians, a sys em o n+k di e en ial equa ions will need o be sol ed.
Conside ing he na u e o he esul ing sys em, he use o a me hod o

-Newma k
in eg a ion amily is p oposed, wi h pa ame e s 41


and 21


, hus ensu ing an
uncondi ionally s able sys em.
Fu he mo e, he in eg a ion s ep,

, is es ablished acco ding o he usual
ecommenda ions (Clough and Penzien, 1993; Dominguez, 2001) o dynamics models
based on modal decomposi ion echnique, as he minimum o he ollowing alues.
)01.0,
4
,
200
,
8
1
min( minmin
max pp n
L
L


 sec. (17)
wi h max
[Hz] being he highes conside ed ib a ion equency o he s uc u e (30 Hz
acco ding o Dominguez (2001)) and min
L[m] he minimum span leng h o he
pedes ian b idge.
Pedes ian e ical walking o ce.
The mo emen o he body mass and he pu -down, olling and push-o o he ee o
one pedes ian gene a e he induced e ical o ces be ween he pedes ian and he
s uc u e, e p
F,. Acco ding o di e en au ho s (Bu z e al., 2007; Se a, 2006), his
o ce can be de e mined om a Fou ie se ies decomposi ion as:






 

n
i
pis e i e p iPF
1
,, 2sin1

(18)
whe e
gmP  [N] is he medium pedes ian weigh ,
g
being he accele a ion o he g a i y.
e i,

is he Fou ie coe icien o he i h ha monic o e ical o ces o e ical
dynamic load ac o (VDLF).
s
[Hz] is he s ep equency o he pedes ian.
i

is he phase shi o he i h ha monic o he pedes ian o ce.
p

is he phase shi among pedes ians.
n is he o al numbe o con ibu ing ha monics.
In o de o de e mine he numbe o pedes ians ha c oss he oo b idge in phase, a
Poisson dis ibu ion has been adop ed, acco ding o he esul s by Ma sumo o e al.
(1978). Fu he es s pe o med on Sol e ino b idge (Se a, 2006), as well as o he
s udies, sugges ha lock-in in e ical di ec ion does no seem p obable due o he low
sensi i i y o he pedes ians o e ical ib a ions. Thus, when a g oup o p
n
pedes ians a i e a he oo b idge, he numbe o pedes ians andomly synch onized is
p
n. This synch oniza ion c i e ion has been adop ed in ou c owd-s uc u e
in e ac ion model. I s implemen a ion in he model is achie ed by he phase shi
pa ame e , p

. Fo a gi en gene a ion/g oup o pedes ians, he alue o he phase shi
p
D is he sliding o ce s eng h due o he con ac be ween pedes ians (Table 3).
p
is a no malized ec o pe pendicula o p
n.
pp
p
 is he angen ial componen o he ela i e pedes ian eloci y, wi h p

being he di e ence o eloci ies be ween wo gi en pedes ians.
and unc ion
H
is de ined as:
00
0
)( 



 i
i
H
(27)
In e ac ions wi h bounda ies.
The in e ac ion wi h he bounda ies gi es ise o o ces, bou
F. These o ces a e
equi alen o he ones esul ing om he in e ac ion wi h o he pedes ian, so hey can
be o mula ed in a simila ashion.
an
bou
no
boubou FFF  (28)

bbpb
b
bp
b
no
bou d HC
B
d
AnF 


















 exp (29)


bbpbpbbou d HD F  ,
an (30)
whe e
no
bou
F is he no mal componen o he bounda y in e ac ion o ce.
an
bou
F is he angen ial componen o he bounda y in e ac ion o ce.
b
A is he in e ac ion s eng h be ween he pedes ian and he bounda y (Table 3)..
b
B is he ange o he epulsi e in e ac ion be ween he pedes ian and he bounda y
(Table 3)..
b
d is he dis ance be ween he pedes ian and he bounda y.

b
C is he body o ce s eng h due o he con ac wi h he bounda y (Table 3).
b
D is he sliding o ce s eng h due o he con ac wi h he bounda y (Table 3).
b
n is he no malized ec o de ined pe pendicula ly om he pedes ian o he
bounda y.
b
is he ec o pe pendicula o b
n.
deno es scala p oduc .
Resul an o ce.
Finally, he p oposed mul i-agen model ha simula es he beha iou o he c owd
consis s in he sum o all hese pa ial o ces ha ep esen he di e en in luences ha
he pedes ians su e when in e ac ing in a c owd. The e o e, he esul an o ce, pci
F,
desc ibes he mo emen and di ec ion o each pedes ian in he c owd as:
boupedd ipci FFFF



(31)
Table 3. C owd model pa ame e s conside ed (Helbing and Molná , 1995; Ca oll e al.,
2012).
Pa ame e Elemen Value
Relaxa ion ime
0.50 sec.
In e ac ion s eng h pedes ians p
A 2000 N
In e ac ion ange pedes ians p
B 0.30 m
Po en ial ac o p

0.20
Con ac s eng h pedes ians p
C 2000 N
Sliding s eng h pedes ians p
D 4800 N
In e ac ion s eng h bounda ies b
A 5100 N
In e ac ion ange bounda ies b
B 0.50 m
Con ac s eng h bounda ies b
C 2000 N
Sliding s eng h bounda ies b
D 4800 N
Radius o pedes ian p
0.20 m
Fo he gene a ion o pedes ians lows h ee pa ame e s ha e been conside ed, he
pedes ian densi y es ablished by in e na ional s anda ds (Bu z e al., 2007; Se a, 2006)
acco ding o he expec ed pedes ian a ic on he oo b idge, he alue o he desi ed
eloci y, d
, o he pedes ians and he dis ance be ween pedes ians, p
d. In his pape
a one-way a ic has been conside ed o simplici y in he gene a ion o he pedes ian
lows, al hough he model may be easily gene alized o wo-way a ic.
The alues o he desi ed eloci y o each pedes ian ha e been ob ained om he
pedes ian s ep equencies, s
. Fo he p esen c owd-s uc u e in e ac ion model he
Gaussian dis ibu ion o he pedes ian s ep equency p o ided by Zi ano ic e al
(2010) has been adop ed, )186.0,87.1(NHz (whe e ),(


N is he Gaussian
dis ibu ion,

is he mean alue and

is he s anda d de ia ion). A e assigning a
s ep equency o each pedes ian, i s desi ed eloci y is de e mined om he empi ical
ela ion gi en by Be am and Ruina (2001) (B uno and Venu i (2009)),
ppp
 93.259.135.0 23
s (32)
so ha he ini ial condi ions o each pedes ian assume ha he pedes ian eloci y, p
,
is equals o he desi ed eloci y, d
.
Finally, once he pedes ian densi y and he desi ed eloci y o each pedes ian a e
es ablished, he o iginal dis ance among pedes ians is calcula ed conside ing he wid h
o he oo b idge and assuming a ec angula -shaped mesh o pedes ians.
Solu ion p ocedu e.
The esul an c owd-s uc u e in e ac ion o ce, pci
F, ac s on each pedes ian du ing
each ime i e a ion j. The accele a ion ec o , j
p
a, ollows om
m
j
pci
j
p
F
a (33)
conside ing a pedes ian mass, m(as mm


).
The e alua ion o he emaining a iables ha go e n he c owd model, 1j
p
and 1j
p
x,
is hen pe o med using a mul i-s ep me hod based on a p edic i e-co ec i e me hod,
namely he Gea ’s algo i hm (Hee mann, 1986), due o he ac ha he social o ces
depend on he eloci y and he posi ion o he pedes ians. The algo i hm calcula es i s
an app oxima e alue, called a p edic o ha subsequen ly is co ec ed wi h a co ec o
alue. The algo i hm applied in his case is o i h o de . Fi s , he new loca ions,
eloci ies, accele a ions and highe de i a i es a e p edic ed acco ding o -Eqs.(34) o
(37)-, whe e he supe sc ip )( p indica es a p edic ed alue.
j
p
j
p
j
p
j
p
j
p
jp
p
βαa xx 







2462
432
1)( (34)
j
p
j
p
j
p
j
p
jp
p
βαa 





62
32
1)( (35)
j
p
j
p
j
p
jp
p
βαaa 



2
2
1)( (36)
j
p
j
p
jp
p βαα 
1)( (37)
whe e p
α is he i s de i a i e o p
a and p
β is he second de i a i e o p
a. F om
hese p edic ed loca ions, he di e ence be ween he accele a ion a ime s ep j+1 and
he p edic ed accele a ion 1)( jp
p
ais ob ained (Eq.(38)) om
1)(1   jp
p
j
pco aaΔ (38)
This co ec ion ac o ec o , co
Δ, allows o ob aining he co ec ed loca ions,
eloci ies and highe de i a ions acco ding o
120
19
2
2
1)(1 

 
co
jp
p
j
pΔxx (39)
4
3
2
1)(1 

 
co
jp
p
j
pΔ (40)
2
1
3
1
1)(1 

 
co
jp
p
j
pΔαα (41)
12
1
12
1
2
1)(1 

 
co
jp
p
j
pΔββ (42)
C owd-s uc u e in e ac ion.
The maximum e ical accele a ion expe ienced by each pedes ian c ossing he
s uc u e,

max
a
z
 may be compa ed agains he accele a ion h eshold alues es ablished
by in e na ional codes (Bu z e al., 2007; Se a, 2006) in o de o modi y he indi idual
pedes ian beha iou due o he esponse o he s uc u e. Due o he good ole ance o
pedes ians o e ical ib a ions (Racic e al., 2009; Zi ano ic a al., 2005) only a s op
h eshold has been es ablished. In his way, when he e ical accele a ion expe ienced
by a pedes ian exceeds he accele a ion limi e, 50.2
lim

z
 m/s2 (Se a, 2006),
pedes ians s op walking o main ain balance, and hey emain s opped un il he
accele a ion le el educes again, conside ing o bo h ac ions a eac ion ime,
00.2
ea
sec. A linea a ia ion o he pedes ian eloci y du ing he eac ion ime has
been assumed. In o de o a oid meaningless small walking eloci ies, a p ac ical lowe
limi on walking eloci y magni ude has been imposed as sugges ed by Ca ol e al.
(2012).



0
1.0 d
p
i
i


lim
max
lim
max 1.0
zz
zz
a
dpa



 (43)
EXPERIMENTAL ESTIMATION OF THE PARAMETERS OF THE
PEDESTRIAN STRUCTURE INTERACTION MODEL.
In e se dynamic p oblem me hodology implemen ed.
The es ima ion o he pa ame e s ha cha ac e ize he dynamic beha iou o he
p oposed pedes ian-s uc u e in e ac ion model in e ical di ec ion has been pe o med
om he esponse o a eal oo b idge unde he c ossing o wo pedes ians by sol ing
he co esponding in e se dynamic p oblem me hodology. In Fig. 5 he lowcha o he
iden i ica ion p ocedu e is shown.
Fig.5. Flowcha o he iden i ica ion p ocedu e.
INITIALIZATION
VARIABLES
(1, e 2, e mah h)k
NUMERICAL ANALYSIS
NUMERICAL ACCELERATIONS
EVALUATION OF OBJECTIVE FUNCTION
2
exp
,,
2
1
i
num
i aa 
MINIMIZATION STEP
UPDATED VALUES (1, e 2, e mah h)k+1
RESULT: IDENTIFIED VARIABLES
(1, e 2, e mazh h)=(1, e 2, e mah h)k+1
YES
NO k=k+1
CONVERGENCE?
TDOF MODEL PARAMETERS IDENTIFICATION
F.E.M. MODEL UPDATING
PRELIMINARY F.E.M.
AMBIENT VIBRATION TEST
O.M.A.
ESTABLISHING SEARCH DOMAIN
VERTICAL PEDESTRIAN WALKING FORCE
PEDESTRIAN MODAL PARAMETERS
m
a

h
h

1, e

2, e
UPDATED F.E.M.
EXPERIMENTAL PEDESTRIAN TEST
TWO PEDESTRIANS
p
=1.50, 2.00 AND 2.50 Hz
exp
,i
a
EXPERIMENTAL RESPONSE
num
i
a,

As iden i ica ion me hod he minimiza ion o a leas squa es p oblem has been adop ed
(Koh and Pe y, 2010). The objec i e unc ion has been de ined as he mean squa e
e o be ween he expe imen al ( exp
,i
a, whe e i is he conside ed sec ion) and nume ical (
num
i
a,) e ical accele a ions ob ained, in ou poin s o he Viana oo b idge (Ba bosa e
al., 2012), unde he c ossing o wo pedes ians a con olled s ep equencies. Gene ic
algo i hms ha e been used o ensu e a global op imiza ion and a sea ch domain o each
pa ame e has been es ablished. The cha ac e iza ion o he dynamic beha iou o he
oo b idge has been pe o med by he ini e elemen model upda ing (Teughels, 2003;
Zi ano ic e al., 2007) based on he modal pa ame e s o he s uc u e es ima ed om
he applica ion o an ope a ional modal analysis (Magalhães and Cunha, 2011). An
ambien es has been pe o med on Viana oo b idge and he measu ed signals ha e
been p ocessed by an ou pu -only iden i ica ion me hod in he ime domain, which
p o ided es ima es o he i s i s ou na u al equencies, he co esponding modal
shapes and he associa ed damping a ios (Magalhães e al., 2010). (Fig.6). La e , his
es has been ep oduced nume ically by he implemen a ion o he p oposed pedes ian-
s uc u e in e ac ion model. An i e a i e p ocess o educe he di e ences be ween he
expe imen al and nume ical e ical accele a ions has been pe o med, unde he ules
o gene ic algo i hms (Koh and Pe y, 2010; Nocen al and W igh , 1999), and
conside ing as design a iables he i s wo VDLF ( e ,1

and e ,2

) o he pedes ian
walking o ce and he h ee modal pa ame e s ha cha ac e izes he TDOF pedes ian-
s uc u e in e ac ion model; he pedes ian sp ung mass, a
m, he pedes ian damping
a io, p

, and he pedes ian na u al equency, p
.
Fig.6. Layou o he iden i ica ion me hodology.
Es ablishing a p elimina y sea ch domain o he pa ame e s o he pedes ian-
s uc u e model.
In o de o educe he unce ain y o he es ima ed alues o he i e conside ed
pa ame e s and hus p e en an ill-condi ioned in e se p oblem, a sea ch domain has
been es ablished.
Acco ding o Table 1 he minimum and maximum alues o each VDLF allow
es ablishing a sea ch domain o hei expe imen al es ima ion. In o de o ake in o
accoun ha he es ima ion o hese pa ame e s will be pe o med om measu emen s
ca ied ou on a eal oo b idge, he abo e men ioned sea ch domains ha e been
ex ended. As he alue o he hi d ha monic o he walking pedes ian o ce p oposed
by he di e en au ho s has a lowe magni ude i s con ibu ion has been neglec ed.
S1 S2 S3 S4
UPDATED F.E.M. OF THE FOOTBRIDGE
p
a
,1num
a
,2exp
a
,1exp
a
,3exp
a
,4exp
a
,2num
a
,3num
a
,4num
2
exp
,,
2
1
i
num
i aa 
EVALUATION OF OBJECTIVE FUNCTION

1, e

2, e
m
a

h
h
IDENTIFIED VARIABLES
m
a

h
h

1, e ,

2, e
X
Z
Y
Z
Y
X
TRIAXIAL
ACCELEROMETER
PEDESTRIAN
MODAL
PARAMETERS
VERTICAL
WALKING
FORCE
TDOF MODEL PARAMETERS IDENTIFICATION
EXPERIMENTAL
NUMERICAL
Simila ly, he phase shi s o he second and hi d ha monic o he e ical walking o ce
ha e no been conside ed.
Simila ly, acco ding o Table 2, he minimum and maximum alues o he pa ame e s
o he passi e pedes ian models allow de ining he sea ch domains o hei es ima ion.
The knowledge o he physical p oblem sugges s: (i) a educ ion o he damping and
s i ness o he pedes ian associa ed wi h i s mo emen and (ii) he change in he
human body mass dis ibu ion be ween he ac i e and passi e s a es. The sea ch
domains o hese pa ame e s ha e been inc eased o ake in o accoun o hese ac s.
Thus he ollowing sea ch domains ha e been adop ed:
 Fi s VDLF,

45.000.0
,1


e

.
 Second VDLF,

20.000.0
,2


e

.
 Pedes ian sp ung mass,


10080


a
m%.
 Pedes ian damping a io,


6910


p

%.
 Pedes ian na u al equency,


43.101


p
Hz.
Desc ip ion and ini e elemen model o he “labo a o y” oo b idge: Viana
oo b idge.
The Viana do Cas elo oo b idge (Ba bosa e al., 2012) is a mo eable cable-s ayed
b idge. The longi udinal s uc u al scheme o he oo b idge consis s o wo spans o
abou 36.50 m and 9.00 m espec i ely suspended by 6 amilies o wo hange s ( wo
e aining ones) om an inclined mas . The deck, wi h 2.50 m o wid h, is con igu ed by
wo olled s eel beams o a iable dep h b aced by ci cula hollow p o iles. The deck
loo is co e ed wi h wood. The compensa ion o he main span weigh is achie ed by
plac
i
mas
allo
w
oun
d
oo
b
A p
o h
a
poin
(An
s
(BE
A
impl
e
o
h
he
s
load
s
his
asso
c
i
ng 11 high
is welded
w
s he o a
d
a ion ha
b
idge is sh
o
Fig.7. Fini
elimina y
n
a
e a i s
a
equi ed
s
ys, 2014)
w
A
M188) e
x
emen e
d
.
T
h
e de e mi
n
s
uc u e,
by
s
and hus
e
p elimina
y
c
ia ed nu
m
densi y bl
o
in i s base
ional mo
balances
h
o
wn in Fig.
e elemen
m
n
ume ical
i
a
pp oxima i
o pe o
m
w
as used, b
a
x
cep o
T
he nonline
a
n
a ion o
h
y
calcula in
e
s ima ing
i
y
FE mod
e
m
e ical na
u
o
cks (wi h a
o a cylin
d
emen o
h
e o ces
a
7
.
m
odel, amb
i
ni e eleme
n
o
n o he d
y
m
he ambi
e
a
sed on a d
i
he hang
e
a
e ec s
a
h
e nume ic
a
g
p e ious
l
i
s angen
s
e
l leads
o
u
al eque
n
weigh o
8
d
e ha is
c
he s uc u
a
nsmi ed b
y
b
ien es g
i
n
(Fig.7)
m
y
namic be
h
e
n ib a i
o
i
sc e iza io
n
e
s whe e
a
ssocia ed
w
a
l na u al
l
y he s es
s
s
i ness m
a
o
he i s
n
cies gi e
n
8
00 kN) pl
a
c
onnec ed
o
e. The pyl
y
he mas .
i
d and mod
e
m
odal anal
y
h
a iou o
h
o
n es . Th
n
o he s
u
3D-cable
w
i h he ha
n
equencies
s
le el o
h
a
ix. The
n
ou num
e
n
in Table
a
ced in he
s
o
a wheel
g
o
n is con
n
A pe spec
e
l upda ing
y
sis was co
n
h
e oo b id
g
e
so wa e
u
c u e in 3
D
elemen s
(
n
ge s ha e
and he i
b
h
e hange s
u
n
ume ical
m
ical ib a
4. In Fig
.
s
ho e spa
n
g
ea bea in
g
n
ec ed o a
i e o he
V
pa ame e s
n
duc e
d
in
g
e and a s
a
package
A
D
-beam ele
m
(LINK10)
been consi
b
a ion mo
d
u
nde pe
m
m
odal anal
y
ion mode
s
.8 he i s
n
. The
g
ha
deep
V
iana
.
o de
a
ing
A
nsys
m
en s
we e
de ed
d
es o
m
anen
sis o
s
and
wo
Fig.10. Fi s wo e ical upda ed nume ical (Upd.) and expe imen al (Exp.) ib a ion
modes.
A e he de elopmen o he model upda ing, he nume ical dynamic beha iou o he
oo b idge simula es accu a ely he eal esponse o he oo b idge. This de ailed
knowledge o he dynamic beha iou o he oo b idge allows o adop ing his eal
oo b idge as a benchma k o “labo a o y” oo b idge.
Expe imen al pedes ian es .
The es ima ion o he pa ame e s ha cha ac e ize he beha iou o he p oposed
pedes ian-s uc u e in e ac ion model was made h ough he esul s o a pedes ian es .
In he es , wo pedes ians (A and B) we e selec ed. The mass o each pedes ian was
70.61
A
mkg and 50.100
B
mkg. On he deck o he oo b idge ou iaxial
accele ome e s we e placed a he in e sec ion poin be ween he hange s and he deck
(Fig.6). Th ee se ies we e eco ded o each pedes ian, c ossing he oo b idge h ee
imes a di e en s ep equencies, s
(1.50, 2.00, 2.50 Hz), con olled by a me onome.
In he ou moni o ed sec ions ( 1
S,2
S,3
S and 4
S in Fig.6) he dynamic esponse o he
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 10.00 20.00 30.00 40.00 50.00
X [m]
Second upda ed e sus expe imen al e ical ib a ion mode
Upd.
Exp.

s uc u e was eco ded. In each passage, he ini ial and end ime, ha ma ks he c ossing
o he pedes ian on he s uc u e, was eco ded as well, in o de o bo h localize he
o ced ib a ion esponse co esponding o he passage o each pedes ian and es ima e
he pedes ian eloci y, p
.
Pa ame e s iden i ica ion in ime domain.
Fo he es ima ion o he pa ame e s o he TDOF-sys em (Koh and Pe y, 2010), an
in e se dynamic p oblem was sol ed, as i has been desc ibed p e iously. As objec i e
unc ion he ela i e di e ences be ween he expe imen al and nume ical e ical
accele a ions in ou sec ions o he oo b idge ( 1
S,2
S,3
S and 4
S) has been conside ed.
As op imiza ion me hod, gene ic algo i hms ha e been used again. The expe imen al
e ical accele a ions co espond o he measu emen s o he abo e desc ibed pedes ian
es . The nume ical e ical accele a ions ha e been ob ained om he implemen a ion
o he p oposed pedes ian-s uc u e in e ac ion model on he upda ed ini e elemen
model o he Viana oo b idge. Six pa ame e s we e adop ed as design a iables:
(i) he i s wo VDLF ha cha ac e ize he pedes ian walking o ce.
(ii) he h ee modal pa ame e s (pedes ian sp ung mass, pedes ian damping
a io and pedes ian na u al equency) ha go e n he beha iou o he TDOF-sys em.
(iii) a ime lag ha allows adjus ing he beginning o he c ossing o he
pedes ian be ween he expe imen al and nume ical accele a ions.
In o de o imp o e he eliabili y o he pa ame e s es ima ion he le el o noise o he
signal has been educed (Koh and Pe y, 2010). In ha way, each measu emen eco d
has been decomposed using he Wa ele ans o m (Gopalak ishnan and Mi a, 2014),
choosing as wa ele amily, he Daubechies. A le el 7 o decomposi ion has been
applied o each signal. As h eshold selec ion ule, he p inciple o S ein's Unbiased
Risk Es ima e has been conside ed (Gopalak ishnan and Mi a, 2014). Subsequen ly, he
econs uc ion o he signal has been ca ied ou using he o iginal app oxima ion
coe icien s o le el 7 and he modi ied de ail coe icien s o le els om 1 o 7. Fo
each se ies, a gene a ion o 1000 indi iduals has been de ined. Each indi idual
modi ies, acco ding o he ules o gene ic algo i hms, he alues o i s componen s in
o de o minimize he de ined objec i e unc ion. In Table 7, he esul s o he
es ima ion p ocess a e summa ized, showing he di e en es ima ed pa ame e s e sus
he pedes ian s ep equency.
Table 7. Es ima ion o he pa ame e s o he pedes ian-s uc u al in e ac ion model.
s [Hz]
Pedes ian sp ung mass ma [%]
Minimum Medium Maximum
1.50 84.995 88.367 91.738
2.00 81.636 86.467 91.297
2.50 82.119 86.583 91.048
s [Hz] Pedes ian damping a io 
p
[%]
Minimum Medium Maximum
1.50 22.143 32.115 42.087
2.00 38.908 45.992 53.076
2.50 40.132 46.214 52.295
s [Hz]
Pedes ian equency
p
[Hz]
Minimum Medium Maximum
1.50 1.923 2.923 3.924
2.00 2.094 2.915 3.736
2.50 2.295 2.962 3.629
s [Hz] Ve ical dynamic load ac o 1
,
e
Minimum Medium Maximum
1.50 0.201 0.203 0.206
2.00 0.214 0.235 0.255
2.50 0.224 0.273 0.322
s [Hz] Ve ical dynamic load ac o 2
,
e
Minimum Medium Maximum
1.50 0.039 0.040 0.041
2.00 0.040 0.042 0.043
2.50 0.043 0.047 0.051
Acco ding o he esul s o Table 7 he pa ame e s o p oposed pedes ian-s uc u e
in e ac ion model show some dependence on he s ep pedes ian equency. Howe e
due o he numbe o pedes ians used du ing he pedes ian es , only a global mean
alue,

, and a s anda d de ia ion,

, ha e been de e mined ),(


N. Fo he
p oposed pedes ian-s uc u e model, he ollowing Gaussian dis ibu ions ha e been
conside ed.
- Fi s VDLF, e ,1

, )04.0,237.0(N.
- Second VDLF, e ,2

, )004.0,043.0(N.
- Pedes ian sp ung mass, a
m, )809.3,139.87(N%.
- Pedes ian damping a io, p

, )776.9,44.41(N %.
- Pedes ian na u al equency, p
, )728.0,933.2(N Hz.
The p oposed alues a e inside he ange, es ablished by Shahabpoo e al. (2013), ha
cha ac e izes he modal p ope ies o TDOF pedes ian-s uc u e in e ac ion model. Fo
he gene a ion o he pedes ian lows o he c owd-s uc u e in e ac ion model, he
abo e Gaussian dis ibu ions ha e been conside ed.
Once ob ained he pa ame e s, he de ini ion o he p oposed model is comple e. This
model will be alida ed in he nex sec ion by co ela ing he nume ical and
expe imen al dynamic esponse o he Viana oo b idge unde he c ossing o a g oup o
pedes ians and he expe imen al and nume ical analysis o he change o he i s
e ical na u al equency o he s uc u e induced by he pedes ian low.
MODEL VALIDATION.
The alidi y and applicabili y o he p oposed c owd-s uc u e in e ac ion model is nex
assessed h ough i s p ac ical applica ion o a case s udy. The e ical accele a ion a
h ee sec ions ( 1
S,2
S, and 3
S (Fig.6)) o he Viana oo b idge has been measu ed unde
he c ossing o a g oup o 50 pedes ians a di e en s ep equencies (1.30-2.50 Hz).
F om he o ced eco ded esponse, he expe imen al s udy o he change o he i s
e ical na u al equency o he s uc u e, due o he c ossing o he g oup o
pedes ians a di e en s ep equencies, has been de e mined. Subsequen ly, he c owd-
s uc u e in e ac ion model has been applied o he upda ed FE model in o de o ob ain
i s he e ical nume ical accele a ion a he men ioned sec ions and la e o s udy
nume ically he change o he i s e ical na u al equencies o he oo b idge due o
he p esence o he pedes ians.
Expe imen al c owd es : dynamic esponse and change o he i s na u al
equency unde pedes ian low.
In he c owd es a g oup o 50 pedes ians has c ossed he oo b idge a di e en s ep
equencies (1.30-1.40-1.60-1.75-2-00-2.50 Hz) con olled by a me onome, measu ing
he e ical dynamic esponse o he oo b idge a h ee sec ions 1
S, 2
S and 3
S wi h a
i-axial accele ome e (Fig.6). Du ing he c owd es he g oup o 50 pedes ians has
been dis ibu ed in h ee alignmen s, main aining a la e al sepa a ion among pedes ians
a ound 0.85 m and a longi udinal dis ance be ween pedes ians a ound 0.50 m. A
pedes ian wi h a me onome has led he g oup in each c ossing. A scheme o he
pedes ian dis ibu ion du ing he c owd es is shown in Fig.11.
In o
he
c
ib
a
(hig
h
and
Mi
a
coe
na u
na u
ob a
i
sign
a
me
h
de o de
e
c
ossing o
a
ion espo
n
h
e modal
d
p ocessed
a
, 2014)
b
icien s (Fi
g
al equen
c
al eque
n
i
ned acco
d
a
l used o
h
odology p
Fig.11.
E
e
mine he
c
he g oup
n
se o he s
d
e lec ion)
by he Co
b
ased on
D
g
.12), in h
e
c
y o he s
n
cy h oug
h
d
ing o he
he es ima
i
e iously d
e
E
xpe imen
a
c
hange o
h
o pedes
i
uc u e ha
s
h
a e
b
een
n inuous
W
D
aubechies
e
il e ed a
n
u
c u e. Thi
s
h
he pow
e
Pea
k
-Pick
i
i
on o he p
e
sc ibed.
a
l c owd es
h
e i s na
u
i
ans a di
e
s
been con
s
conside ed
.
W
a ele T
a
amily.
T
n
ge o eq
u
s
esul has
e
spec al
ing me ho
d
owe spec
a Viana
u
al eque
n
e
en s ep
s
ide e
d
. On
l
.
The selec
a
ns o m (
C
T
he maxi
m
u
encies, is
been alid
a
densi y o
d
(Magalh
ã
al densi y
h
o
o b idge.
n
cy o he
equencies
l
y he eco
ed signal
h
C
WT) (Go
p
m
um alue
hen co el
a
a
ed by he
e
he abo e
ã
es and Cu
n
h
as been d
e
oo b idge
u
, jus he
ds a sec i
o
h
as been i
l
p
alak ishna
n
o he w
a
a
ed wi h h
e
es ima ion
o
e
signal (F
i
n
ha, 2011)
e
noised usi
n
u
nde
o ced
o
n 2
S
l
e ed
n
and
a
ele
e
i s
o
he
i
g.12)
. The
n
g he

Fi
To i
l
sec i
o
Fig.
1
The
oo
b
Nu
m
i s
The
expe
cons
i
g
.12. Es i
m
l
lus a e h
e
on 2
S
und
e
1
3.
expe ime
n
b
idge is su
m
m
e ical c o
w
na u al
e
assessmen
imen al
e
i
de ed s e
p
m
a ing he c
h
e
eco ded
e
e
a g oup
o
n
al s udy
o
m
ma ized i
n
w
d es : e
s
e
quenc
y
u
n
o he
m
e
sul s wi h
p
equenc
y
h
ange o h
e
e
sul s, he
m
o
50 pede
s
o
he ch
a
n
Fig.14.
s
ima ion
o
n
de pedes
m
odel pe
o
he nume i
y
en gene
a
e
i s e i
c
m
easu ed
e
s
ians a a
a
nge o i
o
he d
y
n
a
ian low.
o
mance h
cal es ima
a
ions o
g
c
al na u al
e
ical acce
s ep eque
s e ical
a
mic espo
n
as been d
o
ion p edic
g
oups wi
h
equency o
l
e a ion o
n
cy o 1.6
0
na u al
n
se and h
o
ne co el
a
e
d by he
m
h
50 pedes
he s uc
u
he oo b i
d
0
Hz is sho
w
equency
o
e chan
g
e
o
a
ing he
a
m
odel. Fo
ians ha e
u
e.
d
ge in
w
n in
o
he
o
he
a
bo e
each
been
simula ed. The numbe o pedes ians in phase in each new simula ion has been
de e mined hough he e alua ion o he pa ame e p

. The desi ed eloci y, d
, o each
pedes ian has been assigned acco ding o Eq.(32). A pedes ian mass o 70 kg has been
conside ed acco ding o he F ench code (Se a, 2006). As ini ial spa ial dis ibu ion o
he pedes ians, a ec angula g id has been selec ed, conside ing an ini ial dis ance
among pedes ians 50.0
p
d m wi h an equidis an dis ibu ion in he wid h o he
deck. The selec ed ime s ep is 01.0

 sec. The nume ical e ical accele a ion ( o
h ee o he 50 pedes ians gene a ions) a sec ion 2
So he oo b idge, o a s ep
equency o 1.60 Hz, is shown in Fig.13.
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
Ve ical expe imen al accele a ion (S2). 50 Pedes ians a 1.60 Hz
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
1s Gene a ion. Ve ical nume ical accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
Fig.13. Expe imen al e sus nume ical accele a ion ( h ee gene a ions) a sec ion S2 o
Viana oo b idge unde a g oup o 50 pedes ians (walking equency o 60.1
p
Hz).
As Fig.13 shows, he co ela ion be ween he expe imen ally eco ded e ical
accele a ion and he nume ically es ima ed alues is adequa e, in e ms o bo h he alue
o he maximum accele a ion and i s empo al a ia ion.
Finally, he nume ically es ima ed e ical accele a ion a sec ion 2
So he oo b idge
unde a g oup o 50 pedes ians o di e en s ep equencies has been used o iden i y
he i s na u al equency o he s uc u e, ollowing he p ocedu e desc ibed in he
p e ious sec ion. Fig.14 illus a es he co ela ion be ween expe imen al and nume ical
esul s o he change o i s e ical na u al equency o he oo b idge induced by he
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
2nd Gene a ion. Ve ical nume ical accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.00 10.00 20.00 30.00 40.00 50.00 60.00
m/s
2
Time [sec.]
3 h Gene a ion. Ve ical nume ical accele a ion (S
2
). 50 Pedes ians a 1.60 Hz
c owd-s uc u e in e ac ion phenomenon. The nume ical es ima ion o he change o he
i s e ical na u al equency has been ob ained om he mean alues o en
simula ions o each s ep equency. Good ag eemen be ween bo h se s o esul s is
obse ed, wi h di e ences below 1.50 % o all he analysed pedes ian walking
equencies. The i s e ical na u al equency co esponding o he emp y oo b idge
is included in Fig.14 o e e ence.
Fig.14. Change o he i s e ical, e
,1 [Hz], expe imen al (Exp.) and nume ical
(Num.) na u al equency e sus he s ep equency p
[Hz].
CONCLUSIONS.
In his pape , a new c owd-s uc u e in e ac ion model in he e ical di ec ion has been
p esen ed and u he alida ed h ough he co ela ion be ween he expe imen al and
nume ical dynamic esponse o a eal oo b idge (Viana) adop ed as benchma k. The
p oposed model has been o ganized in wo sub-models: a pedes ian-s uc u e
in e ac ion and a c owd sub-model. The pedes ian-s uc u e in e ac ion sub-model is
con igu ed by a TDOF sys em, wi h sp ung and unsp ung masses, whose pa ame e s
ha e been es ima ed expe imen ally om he solu ion o an in e se p oblem on he
2.940
2.960
2.980
3.000
3.020
3.040
3.060
3.080
3.100
3.120
3.140
3.160
1.30 1.50 1.70 1.90 2.10 2.30 2.50
1, e
[Hz]
p
[Hz]
Exp.
Num.
Emp y
Koh Ghee C., Pe y M.C.. S uc u al Iden i ica ion and Damage De ec ion using
Gene ic Algo i hms. CRC P ess, Taylo &F ancis G oup, 2010.
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Magalhães, F., Cunha, A., Cae ano, E., B incke , R. Damping es ima ion using ee
decays and ambien ib a ion es s. Mechanical Sys ems and Signal P ocessing,
Volume 24, Issue 5, pp. 1274–1290, July 2010.
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Ma sumo o, Y., Nishioka, T., Shioji i, H., Ma suzaki, K.. Dynamic design o
oo b idges. IABSE P oceedings, No. P-17/78, pp. 1-15, 1978.
Nocen al J., W igh S.J.. Nume ical Op imiza ion. Sp inge , New Yo k, USA, 1999.
Racic, V., Pa ic, A., B ownjohn, J.M.W. Expe imen al iden i ica ion and analy ical
modelling o human walking o ces: Li e a u e e iew. Jou nal o Sound and
Vib a ion, Vol. 326, pp. 1-49, Ap il 2009.
Rapapo , D.C.. The a o molecula dynamic. Camb idge Uni e si y P ess; 2004.
SETRA/AFGC. Guide mé hodologique passe elles pié onnes (Technical Guide
Foo b idges: Assessmen o ib a ion beha iou o oo b idge unde pedes ian
loading). SETRA, 2006.
Shahabpoo , E., Pa ic, A., Racic, V.. Modelling e ec o pedes ians walking on
dynamic p ope ies o s uc u es. IMAC XXXI: A Con e ence and Exposi ion on
S uc u al Dynamics, 11-14 Feb ua y, O ange Coun y, Cali o nia, USA (2013).
Ta a es, F., Leal, R., Shube M. Biodynamic single pe son and c owd pedes ian
model: a wo ked es case and simula ion. Foo b idge 2014. London, Uni ed
Kingdom (2014).
Teughels, A., In e se Modelling o Ci il Enginee ing S uc u es Based on Ope a ional
Modal Da a. Ph. D. Thesis, Ka holieke Uni e si ei Leu en, 2003.
Venu i, F., B uno, L., Bellomo, N. C owd dynamics on a mo ing pla o m:
ma hema ical modelling and applica ion o li ely oo b idges. Ma hema ical and
Compu e Modelling 45 (3-4), 252-269, Feb ua y 2007.
Venu i, F., Racic, V., Co be a, A. Pedes ian-s uc u e in e ac ion in he e ical
di ec ion: coupled oscilla o - o ce model o ib a ion se iceabili y assessmen .
P oceedings o he 9 h In e na ional Con e ence on S uc u al Dynamics,
EURODYN 2014. Po o, Po ugal, 30 June - 2 July 2014.

Zi ano ic, S., Pa ic, A., Ingol sson, E., Modelling spa ially un es ic ed pedes ian
a ic on oo b idges. Jou nal o S uc u al Enginee ing, Vol. 136 (10), pp.
1296-1308. 2010.
Zi ano ic, S., Pa ic A., Reynolds P.. Fini e elemen modelling and upda ing o a li ely
oo b idge: The comple e p ocess. Enginee ing S uc u es, Vol. 301(1-2), pp.
126-145, Ma ch 2007.
Zi ano ic, S., Pa ic A., Reynolds P.. Vib a ion se iceabili y o oo b idges unde
human-induced exci a ion: a li e a u e e iew. Jou nal o Sound and Vib a ion,
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74
Pape C: Model upda ing o he selec ion o he e o i me hod o an
ancien b idge (Alme ia, Spain).
The o iginal e sion o his pape can be ound in
doi: 10.2749/101686615X14355644771333
S uc u al Enginee ing In e na ional (in p ess)
ISI 2014 Classi ica ion: Q4 (105/124) Ci il Enginee ing. Impac Fac o : 0.414
SCIMAGO 2014 Classi ica ion: Q2 (106/215) Ci il Enginee ing SJR: 0.417
ISSN: 1683-0350
Model upda ing o he selec ion o he e o i me hod o an ancien
b idge (Alme ia, Spain).
ABSTRACT
In his pape we analyse a case s udy whe e a ini e elemen model upda ing is
conduc ed on he basis o he expe imen al modal pa ame e s eco ded o a
ein o ced conc e e uss b idge buil in Alme ia (Spain) in 1927. The inal aim o
his s udy is o help unde s anding he ac ual s a e o s uc u al conse a ion o
he b idge, in o de o selec an app op ia e e o i echnique o ein o ce i
be o e he planned widening o i s deck. A p esen , he wo mos widely used
me hods o e o i ing consis on ei he ein o cing he s uc u e wi h ex e nal
p es essing o inc easing i s lexu al s eng h by adhe ing CFRP lamina es. The
i s me hod is especially ad an ageous i he s uc u e is de e io a ed o such an
ex en ha he s eng hening needs o ocus no only on inc easing he lexu al
s eng h o he b idge, bu also on a oiding he de lec ion p oblems associa ed
wi h he educed ine ia o he s uc u e. I his we e no he case, s eng hening
wi h CFRP would be an adequa e op ion. To his end, and gi en he monumen al
cha ac e o his ancien cons uc ion, i is no possible o apply di ec ly
des uc i e o s a ic load es s o de e mine he p ope ies o i s cons i uen
ma e ials, so ha i becomes necessa y o es ima e i s de e io a ion s a e
indi ec ly wi h he use o non-des uc i e echniques. Fo his pu pose, in he
p esen pape we ca y ou a ini e elemen model upda ing o he s uc u e, based
on expe imen al modal pa ame e s, which allows o checking i s se ice
condi ion h ough he es ima ion o he alue o se e al physical pa ame e s o
he b idge. Subsequen ly, he esul ing upda ed model cons i u es a aluable ool
o help es ablishing which is he mos adequa e e o i me hod.
Keywo ds: Re o i me hods, ope a ional modal analysis, model upda ing, ancien
ein o ced conc e e b idge, non-des uc i e es ing.
1. In oduc ion.
The Molinos B idge is a ein o ced conc e e uss gi de b idge (Fig. 1) loca ed
o e he Anda ax Ri e a he ou ski s o he own o Alme ía (Spain). I was buil in
1927, being a p esen one o he bes p ese ed achie emen s o he i s ein o ced
conc e e b idges buil in Spain. The design o he s uc u e co esponds o one o he
schemes p oposed by he Spanish enginee Juan Manuel Za a in he o icial model o
oad b idges o he collec ion o ein o ced conc e e s aigh b idge o spans o 32 m,
published in 1920 [1-6].
Recen ly, he Planning Depa men o he Ci y o Alme ia called o a public
compe i ion in o de o p oceed wi h he epai , ein o cemen and widening o his
ancien conc e e b idge. The g ow h o he ci y owa ds he Eas and he a emp o
p omo ing he ou ism on he Eas Coas o he p o ince made necessa y o imp o e he
oad access o hose neighbou hoods o he ci y. Since he cons uc ion o a new
s uc u e was dismissed due o i s high cos , he p oposal ocused on adap ing he
exis ing b idge o he new se ice condi ions.
Acco ding o he echnical p o isions o he compe i ion published by he
Planning Depa men , he wo ks o he ehabili a ion and unc ional adap a ion o he
Molinos b idge should sa is y he ollowing condi ions: (i) widening he exis ing lanes
o ehicles, inc easing i s wid h un il 3.25 m pe lane; (ii) c ea ing a pa emen and a
cycle lane wi h a minimum wid h o 1.50 m; and (iii) he p ojec should main ain he
cu en ypology o he b idge and s ill ensu e i s adequa e bea ing capaci y unde he
new loading scena io.
All he p esen ed p ojec p oposals add essed h ee main aspec s, namely: (i)
design o a s uc u al sys em o inc ease he wid h o he deck; (ii) e o i o he main
uss gi de s; and (iii) ea men o he exis ing c acking on he ein o ced conc e e
elemen s. Rega ding he e o i me hod o he main gi de s, he al e na i es p esen ed
by he di e en enginee ing i ms may be g ouped in wo ca ego ies based on: (i) ei he
use ex e nal p es essing [7] o (ii) use ca bon ibe ein o ced polyme s (CFRP)
lamina es [8] o e o i he ein o ced conc e e uss gi de s. The main ac o ha
condi ions he choice o he e o i me hod is he ac ual de e io a ion le el o he
s uc u e and i s in luence in he momen o ine ia o he gi de . In his way, i he
s uc u e was se iously damaged, i would be necessa y o ocus no only on con olling
he le el o s esses bu on educing he de lec ion o he gi de as well.
The e o e, in o de o p ope ly e alua e he di e en p ojec s and selec he mos
adequa e e o i me hod, he Planning Depa men o he Ci y o Alme ia decided o
conduc a de ailed s udy o he s a e o de e io a ion o he s uc u e using non-
des uc i e echniques. The use o he b idge, be o e i s e o i , was jus limi ed o ligh
a ic, so he Planning Depa men u he es ablished as a equi emen he
impossibili y o pe o ming a s a ic load es , conduc ed wi h hea y loads, in o de o
assess he s uc u al condi ion o he b idge.
The p esen pape summa izes he wo k de eloped by he au ho s o e alua e he
damage le el o he b idge. This s udy desc ibes an inno a i e applica ion o he FE
model upda ing me hod, using i as a aluable ool in o de o ind he op imum
ein o cemen me hod o a ancien b idge. The wo k mainly ocuses on pe o ming a
FE model upda ing [9] o he b idge based on he expe imen al modal pa ame e s. These
pa ame e s a e de e mined om an ambien es , using ope a ional modal analysis [10].
This s udy ocuses on he e ical di ec ion in o de o es ima e he de e io a ion s a e o
he ancien b idge compa ing wo scena ios: (i) one de ined by he cu en s uc u e,
om a nume ical load es pe o med on he upda ed model; and (ii) a second scena io

de ined by he o iginal s uc u e, om he o iginal expe imen al load es a he ime o
comple ion o he b idge [4]. The esul s o he s udy, and in pa icula he esul ing
upda ed model, a e hen used as e e ence o he e alua ion o he di e en p ojec
p oposals and he selec ion o he mos app op ia e e o i me hod.
The pape is o ganized as ollows: In sec ion 2, a p elimina y s udy o he s uc u al
beha iou o he s uc u e is p esen ed, desc ibing he main s uc u al elemen s and he
le el o c acking obse ed on he s uc u e om isual inspec ion. Fu he mo e, a ough
es ima ion o he ange o a ia ion o he Young’s modulus o he conc e e is ob ained
om a hamme ebound es . A he end o his sec ion, a FE model o he s uc u e is
pe o med in o de o ob ain an ini ial es ima ion o he dis ibu ion o he p incipal
s esses o he s uc u e unde sel -weigh and dead load. La e , he esul s om he
hamme ebound es and he s a ic analysis we e used o es ablish bo h he physical
pa ame e s used in he upda ing p ocess and i s ange o a ia ion. To make he pape as
sel -con ained as possible, a desc ip ion and compa ison o he wo p oposed e o i
me hod is b ie ly p esen ed in sec ion 3, and u he a c i e ion o i s selec ion is
es ablished. In sec ion 4, he modal pa ame e s o a p e iously selec ed ex eme span
a e es ima ed expe imen ally h ough he applica ion o ope a ional modal analysis in
he equency domain. In sec ion 5, he FE model upda ing o he selec ed span is
pe o med. In sec ion 6, he upda ed model is employed o assess he s uc u al
condi ion o he b idge, by compa ing he esul s o he o iginal load es o he b idge
[4] wi h he nume ical simula ion o such load scena io using he upda ed model.
Finally, om he ob ained esul s, he bes e o i me hod is selec ed and se e al
conclusions a e d awn in sec ion 7.
2. P elimina y s udy o he s uc u al beha iou o he b idge.
As a p elimina y s ep o he selec ion o any e o i sys em i is o key impo ance o
unde s and he ac ual s uc u al beha iou o he di e en elemen s ha con igu e he
b idge, as well as o de e mine he mechanical cha ac e is ics o he ma e ial and o
conduc an in-dep h isual inspec ion o he cu en s a e o he b idge.
2.1. Desc ip ion o he o iginal s uc u e.
The Molinos B idge o e he Anda ax Ri e is loca ed along he oad be ween Alme ía
and Níja (Spain). I is an isos a ic s uc u e wi h i e spans (32.64 - 32.72 - 32.72 -
32.68 and 32.78 m) wi h a o al leng h o 163.54 m (Fig. 1). The wid h o he o iginal
b idge is a ound 6.20 m, composed by wo lanes o 2.20 m and wo pa emen s o 0.20
m.
Fig. 1. Ele a ion o he Molinos B idge om he Eas abu men .
Each span [2, 3 and 11] is con igu ed by wo ein o ced conc e e uss gi de s sepa a ed
2.70 m and connec ed a he op by a ein o ced conc e e slab o a iable dep h (0.18-
0.54 m) wi h a o al wid h o 6.20 m. The o al dep h o he b idge is abou 2.66 m. The
wid h o he di e en elemen s ha con igu e he uss gi de s is 0.40 m. The dep h o
he lowe cho d is 0.40 m, bu he dep h o s u s and diagonal a ies be ween 0.18 m in
he mid-span and 0.64 m o e he suppo s. Inside he lowe cho d o he uss gi de s
he e a e se e al ec angula s eel pla es 300x12 mm, in a numbe a ying om 10
pla es in each lowe cho d a he mid span o 4 pla es a he suppo s [2, 3 and 11]. The
o ce ansmission be ween he lowe cho ds and he s u s and diagonals is achie ed by
he placemen o pins o diame e 40-50 mm (Fig. 2). The usses a e join ed a he
in e sec ion be ween he lowe cho d and he s u s by se e al ein o ced conc e e
ec angula c ossba s wi h 0.40 m o dep h and a iable wid h (Fig. 3.a).
Due o he geome ic con igu a ion o he b idge he diagonal elemen s wo k in
comp ession and a e ein o ced wi h longi udinal ba s and a high densi y o s i ups.
The s u s o he uss gi de s, always in ension, a e ein o ced wi h longi udinal and
ans e sal ba s which numbe inc eases wi h he p oximi y o he suppo s.
Fig. 2. O iginal de ails o he c oss sec ion o he b idge [11].
The pie s and he abu men s ha e a ec angula c oss-sec ion wi h dimensions 1.60x5.60
m and a heigh o 6.00 m. Each o hese elemen s es on a mason y block wi h
ec angula c oss-sec ion 2.00x6.00 m and heigh a ound 5.70 m [3]. The ounda ions o
hese elemen s a e ein o ced conc e e slabs wi h 1.50 m o dep h and ho izon al
dimensions 4.00x8.00 m. The linking be ween he deck and he pie s is achie ed by
ixed (one hinge) and sliding ( wo hinges) bea ings (Fig. 3.b). To a oid es ic ing he
longi udinal mo emen s o he deck due o he heological and he mal e ec s, in each
pie hese wo ypes o bea ings ha e been ins alled [3].
Fig. 3. a) Spa ial con igu a ion o he uss gi de . b) Suppo on abu men s
2.2. Visual inspec ion o he o iginal s uc u e.
The de e io a ion s a e o he b idge was s udied, as a i s app oxima ion, h ough i s
isual inspec ion. The main damages obse ed in he s uc u e we e he ollowing:
(i) De achmen o he ein o ced conc e e along o he uss gi de s. The
in il a ions o wa e and sal s, due o he impai men o he wa e p oo ing, in he
uss gi de s ha e caused he oxida ion o he eba s and he appea ance o he
de achmen o he conc e e. De achmen s a ound 3.50 cm we e measu ed. The
oo b idge. Jou nal o B idge Enginee ing, ASCE (in p ess). doi:
10.1061/(ASCE)BE.1943-5592.0000828
Jones, C.A., Reynolds, P., Pa ic, A. (2010). Vib a ion se iceabili y o s adia s uc u es
subjec ed o dynamic c owd loads: A li e a u e e iew. Jou nal o Sound and
Vib a ion, Vol. 330, pp. 1531-1566.
Koh Ghee C., Pe y M.C. (2010). S uc u al Iden i ica ion and Damage De ec ion using
Gene ic Algo i hms. CRC P ess, Taylo &F ancis G oup.
Macdonald, J.H.C. (2008). Pedes ian-induced ib a ions o he Cli on Suspension
B idge, UK. P oceedings o he ICE-B idge Enginee ing 161 (2), pp. 69-77.
Macdonald, J.H.C. (2008). La e al exci a ion o b idges by balancing pedes ians.
P oceedings o he Royal Socie y.
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Ma sumo o, Y., G i in, M.J. (2003). Ma hema ical models o he appa en masses o
s anding subjec s exposed o e ical whole-body ib a ion. Jou nal o Sound
and Vib a ion Vol. 260 (3) pp. 431-451.
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Sound and Vib a ion, Vol. 326, pp. 1-49.
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PhD Thesis. No wegian Uni e si y o Science and Technoloy.
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77
Pape F: Dynamic es ing o Ca pin ei a oo b idge a Col ihã (Po ugal).
Con e ence name: 5 h In e na ional Ope a ional Modal Analysis Con e ence
Loca ion and da e: Guima ães (Po ugal). 13-15 May 2013.
Pape ID: 224 (pp. 1-10).
ISBN: 978-972-8692-83-4.
Scopus: h p://0-www.scopus.com. ama.us.es/inwa d/ eco d.u l?eid=2-s2.0-
84906259319&pa ne ID=40&md5=3cd5 cc0518066db 05cdacdea01d34
IOMAC'13
5 h In e na ional Ope a ional Modal Analysis Con e ence
2013 May 13-15 Guima ães - Po ugal
DYNAMIC TESTING OF CARPINTEIRA
FOOTBRIDGE AT COVILHÃ, PORTUGAL
Ja ie Fdo. Jiménez-Alonso 1, Elsa Cae ano2, Ál a o Cunha3
ABSTRACT
The oo b idge o e he Ca pin ei a s eam es ablishes a connec ion be ween wo s eep cli s, a a
heigh o 52.00 m abo e he wa e , ha ing a leng h o abou 220.00 m and being composed o h ee
s aigh sec ions wi h di e en o ien a ions in plan iew, joined by ci cula cu es and suppo ed on
ou columns. In o de o desc ibe he dynamic beha iou o he oo b idge, an ambien ib a ion es
was pe o med, complemen ed by cha ac e isa ion es s o he ib a ions induced by pedes ians in he
loading scena ios assumed as c i ical o he s uc u e. A de ailed ini e elemen model o he s uc u e
has also been de eloped o co ela ion analysis wi h he measu ed alues. This pape shows he
esul s o hese es s and he main conclusions abou he com o le el p o ided by he oo b idge
unde he se ice condi ions.
Keywo ds: Foo b idge, Ope a ional modal analysis, Human induced ib a ions, Ambien dynamic
es ing.
1. INTRODUCTION
Inse ed in he U baniza ion Plan o Ca pin ei a Valley, p omo ed by he Polis P og am, he oo b idge
o e he Ca pin ei a s eam es ablishes a link be ween he wo s eep cli s o Ca pin ei a Valley a a
heigh o 52.00 m o e he wa e . The design has been de eloped by A associados, in collabo a ion
wi h he a chi ec Ca ilho da G aça [1], and he cons uc ion, pe o med by he company CERTAR,
was inished in Sep embe 2009.
The dynamic cha ac e is ics o he oo b idge, p edic ed a he design s age, mo i a ed some conce n
owing o he possibili y o occu ence o signi ican la e al and e ical ib a ions, since se e al o he
na u al equencies o he calcula ed ib a ion modes, bo h in la e al and e ical di ec ions, would be
appa en ly loca ed a c i ical in e als om he poin o iew o he exci a ion induced by pedes ians
[2]. Acco dingly, and as i wouldn’ be possible o ac on he s i ness o mass o he oo b idge in
o de o achie e signi ican changes in e ms o modi ying he dynamic cha ac e is ics o he s uc u e,
he Designe has o eseen he need o ins all uned mass dampe s, i a e he cons uc ion o he
s uc u e, he e was e idence ha hey would be eally equi ed.
1 Assis an P o esso , Uni e si y o Se ille, Highe Technical School o Building Enginee ing, j [email protected]
2 Associa e Agg ega e P o esso , Uni e si y o Po o, Facul y o Enginee ing, [email p o ec ed]
3 Full P o esso , Uni e si y o Po o, Facul y o Enginee ing, [email p o ec ed]
Session 1, J.F. Jiménez-Alonso, Elsa Cae ano, Ál a o Cunha
2
The oo b idge was cons uc ed in Sep embe 2009 and placed a he se ice o he popula ion wi hou
in oducing any de ice o mi iga e ib a ions, which a oused he in e es in conduc ing he p esen
esea ch.
Thus, using he expe imen al esou ces a ailable a he Labo a o y o Vib a ions and Moni o ing
(ViBes , www. e.up.p ) o FEUP, an ambien ib a ion es was pe o med wi h he aim o iden i ying
he dynamic cha ac e is ics o he oo b idge. Addi ionally, measu emen s o he esponse o he
s uc u e unde se ice condi ions we e made, and also unde condi ions o use po en ially haza dous
o he oo b idge. This pape p esen s he main esul s achie ed, which show some di e ences in he
dynamic beha io o he s uc u e in ela ion o he p edic ed a he design phase and, in pa icula ,
allow he cha ac e iza ion o he com o le el o he oo b idge unde no mal use condi ions.
2. DESCRIPTION OF THE STRUCTURE: CARPINTEIRA FOOTBRIDGE
Figu e 1 Foo b idge o e he Ca pin ei a s eam, iew om he sou h side.
No a:
As co as ap e sen adas são med idas ao long o do eix o lo ngi udin al d a pon e
Escala 1/400
Plan a
42.267 48.406 49.000 49.302 31.769
Figu e 2 Ca pin ei a oo b idge, la e al and plan iews.
42.267 48.406 49.000 49.302 31.769
5 h In e na ional Ope a ional Modal Analysis Con e ence, Guima ães 13-15 May 2013
3
The oo b idge (Figu e 1) is composed o a s eel deck ha uns a a cons an le el, being o med by
h ee linea segmen s wi h di e en o ien a ion in plan , connec ed by ci cula cu es. The o al leng h
o he deck is abou 220.00 m and i is suppo ed by ou composi e s eel-conc e e columns, wi h a
a iable heigh be ween abou 18.00 and 40.00 m. The oo b idge is hus di ided in i e spans, wi h
leng hs om abou 32.00 o 50.00 m (Figu e 2).
The c oss-sec ion o he oo b idge, wi h o e all dimensions o 4.40 x 1.75 m2 and 3.50 m o e ec i e
wid h, is o med by wo longi udinal s eel welded gi de s and a wooden suppo ing loo s uc u e ha
con igu e he cha ac e is ics o he U-shaped sec ion o he oo b idge.
3. NUMERICAL MODELLING OF THE FOOTBRIDGE
Due o he complexi y o bo h he geome y and he beha iou o he s uc u e and, wi h he aim o
suppo ing he de elopmen o he dynamic es s and subsequen in e p e a ion o he ob ained esul s,
a ini e elemen model o he oo b idge was de eloped using 3-D beam elemen s [3, 4 and 5]. In
Figu e 3, he ini e elemen mesh used is p esen ed.
.
Figu e 3 Ca pin ei a oo b idge, ini e elemen mesh used in he nume ical model and de ail o disc e iza ion.
The o e all mass o he s eel deck is abou 300 on, co esponding o a linea mass o 1250 kg/m. In
he design o he oo b idge, he associa ed dynamic e ec s o a pedes ian low, wi h a densi y o 0.60
pedes ian/m2, ha e been conside ed, which co esponds o an inc ease o he mass o he deck o
abou 10 %. Though he whole pedes ian mass is no usually conside ed in he dynamic
cha ac e iza ion o he oo b idge, he calcula ion o he na u al equencies has been made assuming
ei he an emp y s uc u e o a loaded s a e wi h 0.60 Pedes ian/m2, in o de o ame hei na u al
equencies. On he o he hand, gi en he ela i ely low loading le els, i was assumed ha du ing
se ice, he elas ic suppo s a he ex emes wouldn’ be ac i a ed. Acco dingly, al e na i e condi ions

Session 1, J.F. Jiménez-Alonso, Elsa Cae ano, Ál a o Cunha
4
o connec ion we e simula ed: elas ic beha iou associa ed o neop ene bea ings, o cons ain
mo emen in longi udinal di ec ion.
Table 1 p esen s he alues o se e al na u al equencies co esponding o ou nume ical simula ions
de eloped in o de o ame he b idge na u al equencies: M1 - emp y oo b idge, elas ic suppo s;
M2 - ull oo b idge wi h 0.60 pedes ian/m2, elas ic suppo s; M3 - emp y oo b idge, ixed suppo s;
M4 - ull oo b idge wi h 0.60 pedes ian/m2, elas ic suppo s. I is desc ibed in Table 1 he main
cha ac e is ics o he mos impo an ib a ion modes (L: longi udinal; T: ans e se; V: e ical; To:
o sion). On he o he hand, Figu e 4 shows he ou mos ele an ib a ion modes, based on he M3
model.
Table 1 Nume ical na u al equencies ob ained unde di e en assump ions in ela ion o mass (emp y/ ull
deck wi h a pedes ian densi y o 0.60 pedes ian/m2) and he s i ness o he ex eme suppo s (elas ic/ ixed).
Modes M1
(elas ic/ emp y)
M2
(elas ic/ ull)
M3
( ixed/emp y)
M4
( ixed/ ull)
1 1.15 (T) 1.11 (T) 1.21 (T) 1.16 (T)
2 1.34 (L) 1.29 (L) 1.6 (T local) 1.52 (T local)
3 1.38 (T local) 1.3 (T local) 1.97 (To) 1.88 (To)
4 1.79 (T) 1.71 (T) 2.02 (To+ T) 1.93 (To+T)
5/ 6(M3+M4) 2.01 (V+T) 1.9 (V+T) 2.33 (To+T) 2.20 (T+To)
7 (M3+ M4) 2.44 (To+T) 2.32 (T+To)
… … … … …
14/ 13 (M3+ M4) 3.78 (V) 3.54 (V) 3.83 (V) 3.58 (V)
F eq. 1= 1.21 Hz
F eq. 3= 1.97 Hz
F eq. 7= 2.44 Hz
F eq. 13= 3.83 Hz
Figu e 4 Se e al modal shapes ob ained om ini e elemen model M3 (emp y oo b idge wi h ixed
longi udinal suppo a abu men s).
4. IDENTIFICATION OF THE DYNAMIC PROPERTIES
The iden i ica ion o he na u al equencies, ib a ion modes and modal damping a ios was made
pe o ming an ambien ib a ion es , conduc ed in July 2011. In his es , 5 seismog aphs p o ided
wi h iaxial accele ome e s, we e used. These elemen s we e successi ely placed along he posi ions
indica ed in Figu e 5, keeping wo o he de ices in sec ions 6 and 15, in he wes side o he
oo b idge. In each posi ion, eco ds o ambien accele a ion ha e been collec ed in 15 channels wi h
13- minu e du a ion, sampled a 100 Hz.
5 h In e na ional Ope a ional Modal Analysis Con e ence, Guima ães 13-15 May 2013
5
Figu e 6 shows wo images o he es s pe o med, held du ing a no mal day and in ol ing he
occasional passage o pedes ians, unde no mal condi ions o use.
Figu e 5 Ins umen ed poin s in he ambien ib a ion es .
Figu e 6 Images o he ambien ib a ion es s.
Acco ding o he image o Figu e 6, and gi en he peculia cha ac e is ics o he oo b idge, i was
decided o dispose sys ema ically he senso s, so ha he ans e se axis would coincide wi h he
No h di ec ion. This pe mi ed he use o a common e e ence and acili a ed he signal p ocessing.
The iden i ica ion o he modal pa ame e s was made using he so wa e ARTEMIS [6]. Table 2 shows
a lis o he mos ele an iden i ied na u al equencies and modal damping a ios, whe eas Figu e 7
cha ac e izes se e al o he co esponding ib a ion modes. I is se led in Table 2 a co espondence
be ween some iden i ied and calcula ed ib a ion modes, conside ing as basis he p e iously p esen ed
M3 model.
The analysis o Table 2 and Figu es 4 and 7 shows ha , al hough he main aspec s o he dynamic
beha io o he oo b idge a e cha ac e ized by he nume ical ini e elemen model, in e ms o he
na u al equencies alues o he ib a ion modes ( ans e se, e ical and o sion), signi ican
di e ences exis be ween he modes calcula ed and iden i ied.
Indeed, we can no ice ha he i s iden i ied ans e se ib a ion mode has a na u al equency
sligh ly highe han he calcula ed one, which is indica i e o he beha io o he ex eme suppo s,
almos ixed o he abu men s. Al hough he nume ical modeling has jus in oduced longi udinal
cons ic ion, he na u al equency o he second mode o ib a ion, o local cha ac e , is also highe
han he calcula ed, which ce ainly also in ol es he cons ic ion o he ans e se mo ion in he
ex eme suppo s a he abu men s.
Mo eo e , i is also obse ed ha he eal ans e se lexibili y o he columns was in e io o he
calcula ed one, since he e ical ib a ion modes in ol e gene ally he o sion o he deck, bu no he
ans e se bending o he columns, con a y o he esul s o he calcula ion. Finally, i is no iced some
p oximi y be ween he na u al equencies o he modes wi h e ical and o sional componen s, which
is ce ainly de e mined by he mechanical cha ac e is ics o he deck.
Session 1, J.F. Jiménez-Alonso, Elsa Cae ano, Ál a o Cunha
6
Table 2 Nume ical na u al equencies
Modes Iden i ied na u al equency
(Hz)
Nume ical na u al equency
(Hz)
Damping a io (%)
1 1.37 (T) 1.21 (T) 0.28
2 2.20 (T local) 1.6 (T local) 0.21
3 2.47* (V+ To) 0.17
4 2.76 (V+To) 0.13
5 2.95 (V+To) 0.33
6 3.59 (V+T) 3.83 (V) 0.07
* Se e al nume ical modes wi h close equencies.
F1= 1.37 Hz
F2= 2.20 Hz
F3= 2.47 Hz F4= 2.76 Hz
F5= 2.95 Hz
F6= 3.59 Hz
Figu e 7 Iden i ied ib a ion modes using ARTEMIS so wa e.
5. CHARACTERIZATION OF THE DYNAMIC BEHAVIOUR
F om inspec ion o he iden i ied na u al equencies o he oo b idge, i can be concluded, i s ly, ha
he oo b idge is no ulne able o he la e al synch oniza ion phenomenon. Indeed, his phenomenon
ypically occu s wi h na u al equencies close o 1 Hz. The in e na ional codes and ecommenda ions
[7, 8] de ine his synch oniza ion as possible in he ange o equencies om 0.5 o 1.2 Hz, wi h a
c i ical scena io when he undamen al equencies a e si ua ed be ween 0.7 and 1.0 Hz. The ini ial
nume ical s udies [2] sugges ed a undamen al equency o 1.14 Hz, so he isk o occu ence o his
phenomenon was eal. No e ha , despi e he possible sophis ica ion o he cu en nume ical models,
he eal bounda y condi ions o he oo b idge and, some imes, he addi ion o elemen s assumed as
non-s uc u al, can g ea ly in luence i s dynamic beha iou , and he e a e o en di e ences in he
undamen al equencies o he buil s uc u e agains he nume ical esul s o he design. In his case,
he na u al equency o 1.37 Hz measu ed in he ans e se di ec ion educes he isk o la e al
5 h In e na ional Ope a ional Modal Analysis Con e ence, Guima ães 13-15 May 2013
7
synch oniza ion, no a oiding howe e ques ions abou he com o le el p o ided by he s uc u e,
bo h in e ms o he e ical and ho izon al ib a ions. This p oblem is pa icula ly ele an aking in
mind he loca ion o he b idge, a a e y high le el, and also aking in o accoun he cha ac e is ics o
he pa emen , wi h a sla ed wooden ela i ely spa se.
I is e e ed, on he o he hand, he e y low damping iden i ied by he ambien ib a ion es . I is
impo an o no e ha he quali y o he damping es ima es ob ained in his way is ques ionable,
pa icula ly since he e was no oppo uni y o alida e hem using o he me hod. I is no ed, howe e ,
ha he modal damping a ios iden i ied a e lowe han expec ed.
In ela ion o he e ical ib a ions, i was also ound a he design s age [2] ha he e we e se e al
ib a ion modes wi h na u al equencies close o 2 Hz, which could lead o esonan phenomena. In
he buil oo b idge, i was obse ed ha he equencies o he e ical modes a e gene ally a li le
highe han calcula ed, being se led ypically abo e 2.5 Hz.
In con as , he lexibili y o he columns is lowe han he modelled one, which educes hei
pa icipa ion in e ms o ans e se bending in he main ib a ion modes. This ac leads o a mo e
signi ican o sional beha iou o he deck.
Also hese cha ac e is ics become bene icial o he s uc u e, since he esonan phenomena a
equencies a ound 2 Hz don’ occu . In con as , he dynamic e ec s induced by pedes ians jogging
become mo e ele an . Mo eo e , and gi en ha he o sional beha iou o he deck u ns ou o be
e iden , also he la e al modes o ib a ion a e associa ed gene ally o he componen s o e ical
ib a ion modes, which may esul in mo e se e e ib a ion le els.
Taking in o accoun he obse ed dynamic cha ac e is ics o he oo b idge, p e iously discussed, i is
assumed ha he dynamic esponse may be c i ical in he ollowing si ua ions:
1) Slow walking o la ge lows o pedes ians, wi h equencies om 1.4 o 1.5 Hz (exci a ion o
ib a ion mode 1 and modes 3 o 6 (Figu e 7) h ough he second ha monic);
2) Jogging by a pedes ian o a g oup o pedes ians, wi h na u al equencies o 2.47 Hz, 2.76 Hz o
2.95 Hz (exci a ion o ib a ion modes 3, 4 o 5 (Figu e 7)).
Al hough i has been obse ed ha he use o he oo b idge is no in ense, i was possible o es a
no mal ope a ing si ua ion ela i ely close o he desc ibed condi ion (1) o he abo e pa ag aph,
al hough mobilizing a educed densi y o pedes ians. The dynamic es s we e pe o med in wo days
o he summe o 2011. I was ound ha he oo b idge was used me ely occasionally du ing he day.
Con e sely, and gi en he wa m empe a u e obse ed in he la e e ening, om 19:00 h, a con inuous
use by pedes ians was de ec ed, cha ac e ized by slow walking.
Al hough an accu a e quan i ica ion o he densi y o pedes ians has no been made, i is possible
howe e o say ha a densi y o a leas o 0.1 pedes ian/m2 has been eached, since mo e han 80
pedes ians we e o e he s uc u e. Unde hese condi ions, he la e al and e ical ib a ions we e
clea ly pe cep ible, also mobilizing high and low equencies. Taking as basis he ange o 0-8 Hz,
bo h le els o ans e se and e ical accele a ions we e collec ed, wi h alues o 0.08 m/s2 and 0.17
m/s2, espec i ely, cha ac e is ics o he maximum com o le el o he oo b idge, in line wi h he
ecommenda ions men ioned abo e [7, 8].
Figu e 8 shows examples o he eco ds o he ans e se and e ical accele a ion collec ed in he
ma k numbe 16 (see Figu e 5) unde hese condi ions, oge he wi h hei co esponding spec al
con en .
Due o he slende ness o he s uc u e, he join be ween he deck and he pie s has been specially
s i ened.
Fig. 2: P elimina y Fini e Elemen Model.
3. P elimina y nume ical modal analysis.
Fi s ly, a nume ical modal analysis was de eloped (Figu e 2), whe e he e ec o he s eel oo was
conside ed as a passi e mass uni o mly dis ibu ed on he deck (app oxima ely 500 kg/m). The
model o he s uc u e was ca ied ou by he ini e elemen so wa e Au odesk Robo S uc u al
Analysis P o essional [4]. Unde his hypo hesis, he nume ical ib a ion modes (Figu e 3) o he
oo b idge has been de e mined in wo scena ios, absence o pedes ians ( emp) and he si ua ion
whe e a pedes ian low o 1.00 P/m2 (Pedes ians/m2) c oss he s uc u e ( ul) [2].
Nume ical e ical mode 1 ul/emp=2.34/2.43 Hz
Nume ical la e al mode 1 ul/em
p
=2.13/2.20 Hz
Nume ical e ical mode 2 ul/em
p
=2.56/2.66 Hz
Nume ical la e al mode 2 ul/em
p
=5.08/5.25 Hz
Fig. 3: Fi s ou ib a ion modes. Emp y (emp) and ull( ul) oo b idge.
Gi en he si ua ion o he oo b idge and assuming ha on he s uc u e a e no expec ed pedes ians
densi ies abo e 0.80 P/m2, he nume ical es ima ed na u al equencies a e ou side o he no mal
anges ha cha ac e ize he pedes ian walking s ep. On he o he hand, he na u al equencies o
he s uc u e a e in he ange ha cha ac e izes he ac ion o jogging o unning. The nume ical
esponse ha p oduces he c ossing o he p e iously p ede ined ha monic load on he s uc u e (1)
eaches i s maximum alue unde a c ossing jogge a 2.56 Hz. In Figu e 4 he e ical dynamic
esponse (accele a ion) o he s uc u e unde he passage o his pedes ian is shown. The
maximum alue is less han he limi es ablished by he com o le el (1.00 m/s2).

Nume ical e ical acele a ion. 1 Jogge =2.56 Hz
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0 5 10 15 20 25 30 35 40 45 50
ime [sec]
[m/s2]
Fig. 4: Nume ical e ical accele a ion a mid-span due o a jogge s ep equency =2.56 Hz.
4. Ambien and pedes ian es s.
4.1 Ambien es .
The dynamic pa ame e s o he s uc u e ha e been de e mined by he measu es ob ained om an
ambien es . The deck o he s uc u e has been di ided in o a 2x15 g id, being he poin s sepa a ed
longi udinally 5.65 m and ans e sally 3.15 m (see Figu e 5, blue a ows a e e e ence
accele ome e s). Two se ies o measu emen s we e ca ied ou , each one consis ing on 14 se -up,
he i s one co esponds o he de e mina ion o e ical dynamic pa ame e s and he second o
es ima e he la e al dynamic pa ame e s. The measu es we e made wi h 4 uniaxial accele ome e s,
sensi i i y 10 V/g, ype Episenso and p oduced by he company Kineme ics. The du a ion o each
se -up was 900 seconds and he sampling equency was 100.00 Hz [5].
Ve ical Ambien Tes Layou
La e al Ambien Tes Layou
Fig. 5: Measu emen g id o he ambien es .
F om he abo e se ies, he dynamic pa ame e s o he s uc u e ha e been de e mined, p ocessing
he signals by wo di e en me hods, one in he equency domain, Enhanced F equency Domain
Decomposi ion (E.F.D.D.), and ano he in he ime domain, S ochas ic Subspace Iden i ica ion
(S.S.I.). Fo he alida ion o he esul s [5], he M.A.C. a io (Modal Assu ance C i e ion) o
ce ain ib a ion modes, has been calcula ed (Table 1) p esen ing all he iden i ied ib a ion modes
a M.A.C. g ea e han 0.90.
In Figu e 6 he g aphical ep esen a ion o he i s ou de e mined ib a ion modes is shown. The
p ac ical applica ion o he abo e algo i hms was pe o med using he ARTeMIS Ex ac o P o 2012
so wa e de eloped by SVS A/S [6].
Table 1: Expe imen al na u al equencies
Mode EFDD [Hz] SSI [Hz] Desc ip ion M.A.C.
1 2.372 2.370 La e al 0.999
2 3.026 3.024 Ve ical 1.000
3 3.830 3.844 Ve ical 0.890
4 5.508 5.601 La e al 0.901
Expe imen al e ical mode 1 =3.026 Hz
Expe imen al la e al mode 1 =2.372 Hz
Expe imen al e ical mode 2 =3.844 Hz Expe imen al la e al mode 2 =5.508 Hz
Fig. 6: Expe imen al i s ou modes o ib a ion (E.F.D.D.).
4.2 Pedes ian es .
Finally, i is pe o med a es wi h i e di e en pedes ians, measu ing he dynamic esponse o he
oo b idge unde di e en s ep equencies ( = 1.50, 2.00, 2.50, 3.00, 3.50 and 4.00 Hz). In Figu e
7, he measu ed maximum e ical accele a ion in he cen al mid-span o a pedes ian wi h a
weigh o 114.00 kg and a s ep equency o 3.00 Hz is shown. The maximum measu ed
accele a ion is less han he limi es ablished by he medium com o le el and he nume ical one
es ima ed p e iously.
Expe imen al e ical acele a ion. 1 Jogge
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
0 5 10 15 20 25 30 35 40 45 50
ime [sec]
[m/s
2
]
Fig. 7: Expe imen al e ical accele a ion a mid-span due o a jogge s ep equency =3.00 Hz
The s i ening o he s uc u e caused by he p esence o he s eel oo o igina es an imp o emen o
he com o le el o he s uc u e.
5. Model upda ing.
5.1 De ailed ini e elemen model o he whole s uc u e.
In o de o ha e a mo e accu a e unde s anding o he dynamic beha iou o he s uc u e a de ailed
ini e elemen model o he whole s uc u e has been de eloped. Nume ical modal analysis has been
de eloped h ough he applica ion o he ini e elemen me hod [7]. In he ini e elemen model o
he s uc u e has been necessa y o model all he elemen o he oo b idge and he co e o
cha ac e ize as p ecisely as possible he mass and s i ness ma ices. The model has been ca ied ou
using 3D-beam (BEAM188) elemen s excep in he case o he s eel co e we e 2D-shell
(SHELL63) elemen s has been conside ed.
5.2 Model upda ing o he de ailed ini e elemen model.
A model upda ing o he abo e de ailed ini e elemen model has been de eloped [8] om he
esul s o he abo e ope a ional modal analysis in o de o cha ac e ize mo e adequa ely he
dynamic beha iou o he oo b idge. In his sense, 7 physical pa ame e s o he s uc u e
(acco ding o Table 2) ha e been modi ied in o de o minimize he mean squa e e o be ween he
expe imen al and nume ical pa ame e s, conside ing he iden i ied na u al equencies and hei
co esponding modal coo dina es. A e a sensi i i y s udy o he main physical pa ame e s o he
ini e elemen model, i was ound ha he physical pa ame e s wi h g ea e in luence on he
dynamic beha iou o he oo b idge a e he s i ness o bea ings. The s i ness o hese elemen s
has been simula ed h ough h ee sp ings, one in each di ec ion (longi udinal, la e al and e ical).
The objec i e unc ion, in his case, was de ined as he sum o he ela i e di e ences be ween he
na u al equencies and he modal coo dina es ob ained expe imen ally and nume ically. As
op imiza ion me hod he gene ic algo i hms ha e been chosen. In Figu e 8 he esul s o he
adjus men made on he i s ou ib a ion modes a e shown.
A e he adjus men o he selec ed physical pa ame e s, high co ela ions be ween expe imen al
and nume ical modal shapes (M.A.C. abo e 95 %) ha e been eached in he ou modes iden i ied.
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
x [m]
Exp
Num
Expe imen al&nume ical 1s e ical mode
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
x [m]
Exp
Num
c
Expe imen al&nume ical 1s la e al mode
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
x [m]
Exp
Num
Expe imen al&nume ical 2nd e ical mode
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
x [m]
Exp
Num
Expe imen al&nume ical 2nd la e al mode
Fig. 8: Compa ison be ween expe imen al (Exp.) and nume ical (Num.) ib a ion modes
Table 2: Upda ed alues o conside ed physical pa ame e s
Pa ame e s Ini ial Value Upda ed Value
E ec i e co e hickness 0.003 mm 0.0015 mm
E ec i e slab conc e e hickness 0.15 m 0.10 m
E ec i e abu men s i ness 30000 MPa 33240 MPa
Soil s i ness 5.00E8 kN/m 4.64E8 kN/m
Longi udinal bea ing s i ness 1.00E9 kN/m 2.00E8 kN/m
La e al bea ing s i ness 1.00E10 kN/m 2.64E9 kN/m
Ve ical bea ing s i ness 1.00E11 kN/m 1.28E11 kN/m
Finally, in Figu e 9 he upda ed ou i s ib a ion modes om he de ailed ini e elemen me hod
a e shown.
Upda ed e ical mode 1 =3.026 Hz
Upda ed la e al mode 1 =2.372 Hz
Upda ed e ical mode 2 =3.844 Hz
Upda ed la e al mode 2 =5.508 Hz
Fig. 9: Upda ed ou i s ib a ion modes.
6. Nume ical es ima ion o he e ec o he s eel co e cons uc ion.
In Table 3 he a ia ion in he na u al equencies o he oo b idge due o he cons uc ion o he
oo b idge has been es ima ed. F om he upda ed ini e elemen model, i has been possible o
simula e he beha iou he dynamic beha iou o he oo b idge wi hou he s eel oo ( NUM_INI) and
he cu en si ua ion ( NUM_COV). Bo h alues ha e been ob ained nume ically. The pe cen age
alues a e ep esen a i es o he s i ening e ec ha he co e p esen s in each di ec ion.
The s eel co e inc ease he s i ness o he oo b idge in he e ical di ec ion, howe e , in he
la e al di ec ion he cons uc ion o he s eel co e educes he alue o he na u al equencies in
ha di ec ion.
F om he poin o iew o he maximum accele a ion alues achie ed, he e is a sligh imp o emen
in he com o le el due o he s i ening o he s uc u e. I p esen s ce ain sa e y ma gin, ensu ing
ha he oo b idge eaches a medium com o le el, h ough e en, unde a e y a e load case as he
Table 3: Es ima ion o he change in he na u al equencies o he oo b idge
Mode NUM_INI [Hz] NUM_COV [Hz] Desc ip ion  [%].
1 2.569 2.372 La e al -7.69
2 2.839 3.026 Ve ical 6.55
3 3.863 3.844 Ve ical -0.51
4 5.722 5.508 La e al -3.74
ci cula ion on he oo b idge o se e al jogge s in pa allel (Figu e 10).
Fig. 10: Change in he i s wo e ical ib a ion modes. – wi hou co e -- wi h co e
7. Conclusions.
In his pape , i has been es ima ed expe imen ally and nume ically, he change o he dynamic
beha iou o a slende oo b idge due o he cons uc ion o a s eel oo o e he o iginal s uc u e.
The s eel oo inc eases he s i ness o he s uc u e in e ical di ec ion bu educes he alue o
he na u al equencies o he s uc u e in he la e al di ec ion. This e ec is especially ele an in
he i s wo na u al equencies o he s uc u e. Howe e , he alues o he cu en na u al
equencies ensu e ha he s uc u e will no su e om com o p oblems due o pedes ians low
walking. In ela ion o jogging o unning, i has been shown ha he s uc u al s i ening imp o es
i s beha iou unde hese ypes o human ac ion.
8. Re e ences.
[1] SETRA, Guide mé hodologique passe elles pié onnes (Technical guide oo b idges:
Assemen o ib a ional beha iou o oo b idges unde pedes ian loading), Se a, 2006.
[2] SYNPEX Guidelines, Eu opean P ojec on Ad anced Load Models o synch onous
Pedes ian Exci a ion and Op imized Design Guidelines o S eel Foo b idges, 2007.
[3] CLOUGH, R and PENZIEN, J. Dynamics o S uc u es, 2nd. Edi ion, Mc G aw-Hill, 1993.
[4] AUTODESK ROBOT STRUCTURAL ANALYSIS PROFESSIONAL 2011.
[5] MAGALHÃES, F., CUNHA, A. "Explaining Ope a ional Modal Analysis wi h da a om an
a ch b idge", Mechanical Sys ems and Signal P ocessing, In i ed Tu o ial Pape , Volume 25,
Issue 5, pp. 1431-1450 , 2011.
[6] ARTeMIS Ex ac o P o 2012.
[7] ANSYS Mechanical Release 11.0.
[8] ZIVANOVIC, S., PAVIC, A. REYNOLD, P., “Fini e elemen modelling and upda ing o a
li ely oo b idge: The comple e p ocess”, Jou nal o Sound and Vib a ion, Vol. 301,. nº 1-2,
pp. 126-145,2007.