UNCORRECTED PROOF
Enginee ing S uc u es xxx (2018) xxx-xxx
Con en s lis s a ailable a ScienceDi ec
Enginee ing S uc u es
jou nal homepage: www.else ie .com
E. To oja’s b idge: Tailo ed expe imen al se up o SHM o a his o ical b idge wi h a
educed numbe o senso s
Pablo Pachón⁠a⁠, ⁠⁎, Ra ael Cas o⁠b, En ique Ga cía-Macías⁠c, Víc o Compan⁠a, Es he Pue as⁠d
aDep . o Con inuum Mechanics, Uni e sidad de Se illa, A enida Reina Me cedes, 41012 Se illa, Spain
bDep . o Mechanics, Uni e sidad de Có doba, Campus de Rabanales, 14071 Co doba, Spain
cDepa men o Con inuum Mechanics and S uc u al Analysis, Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092 Se ille, Spain
dDep . o Mechanical S uc u es and Hyd aulic Enginee ing, Uni e sidad de G anada, A enida Fuen enue a, 18001 G anada, Spain
ARTICLE INFO
Keywo ds:
Ambien ib a ion
Cul u al he i age
Gene ic algo i hm
His o ical cons uc ions
Ope a ional modal analysis
Op imal senso placemen
S uc u al Heal h Moni o ing
ABSTRACT
This pape p esen s he design o an expe imen al se up wi h a educed numbe o senso s o he s uc u al
heal h moni o ing o he his o ical b idge o Posadas (Có doba, Spain), designed by he eminen enginee Ed-
ua do To oja in 1957. The mo i a ion o his s udy s ems om he need o sa egua ding his piece o cul u al
he i age. In pa icula , he singula i y o his his o ical cons uc ion, a s eel–conc e e composi e ypology con-
sis ing o a conc e e deck slab and in e ed bows ing s eel usses, makes con inuous in-se ice condi ion as-
sessmen essen ial o i s main enance. Ne e heless, he applica ion o exis ing con inuous moni o ing sys ems
o such la ge-scale s uc u es en ails conside able in es men s as well as complex signal p ocessing algo i hms.
Whe eby he op imiza ion o he numbe o senso s and hei loca ion is o he u mos in e es . In his line, his
wo k p esen s he applica ion o an Op imal Senso Placemen (OSP) me hodology o ailo an expe imen al se up
o a cos -e icien con inuous moni o ing o he E. To oja’s b idge. Due o he ac ha mos OSP app oaches
a e model-based, i is essen ial o coun on a su icien ly accu a e nume ical model. To his aim, an ex ensi e
ib a ion-based ope a ional modal analysis is i s conduc ed wi h a la ge numbe o accele ome e s. A e wa d,
a h ee-dimensional ini e elemen model o he E. To oja’s b idge is upda ed on he basis o he expe imen-
ally iden i ied dynamic p ope ies wi h a gene ic op imiza ion algo i hm. Finally, an op imal senso placemen
me hodology is u ilized o design an expe imen al se up wi h a limi ed numbe o senso s o long- e m mon-
i o ing pu poses. The esul s demons a e ha ew senso s a e needed o accu a ely assess he main esonan
equencies and mode shapes.
1. In oduc ion
His o ical b idges cons i u e a key piece o cul u al he i age, inas-
much as hey bea wi ness o he cou se o his o y and hold an im-
po an social, cul u al, and a is ic alue. The e exis s a g ea conce n
abou hei conse a ion and, he e o e, he assessmen o hei heal h
condi ion is absolu ely c ucial. S uc u al Heal h Moni o ing (SHM) en-
compasses he applica ion o Non-Des uc i e Tes ing (NDT) and dam-
age de ec ion in o de o ex end he li e-cycle o s uc u es. In pa ic-
ula , Ope a ional Modal Analysis (OMA) is conside ed one o he mos
sui able me hods o assess he condi ion o s uc u es h ough hei i-
b a ional p ope ies [1,2]. OMA is pe o med unde condi ions o se -
ice wi hou he need o a i icial exci a ions, ea u e ha is essen ial
o moni o his o ical s uc u es whe e he use o s onge modal ex-
ci e s, such as ins umen ed hamme s o shake s, is o en inadmissible.
Ne e heless, OMA usually equi es a la ge numbe o senso s o p op-
e ly cha ac e ize he dynamic p ope ies, ac ha limi s he scalabili y
o long- e m OMA-based SHM o la ge-scale s uc u es. Gi en he high
cos o such sys ems, i is essen ial o coun on echniques ha allow o
ailo ing expe imen al se ups in such a way ha only a educed num-
be o senso s can accu a ely iden i y he ib a ional p ope ies o s uc-
u es.
Plen y o s udies on Fini e Elemen Modeling (FEM) and expe i-
men al in es iga ion o his o ical b idges can be ound in he li e a-
u e. A no ewo hy con ibu ion was done by Chia a Pepi e al. [3]
who s udied he s uc u al pe o mance o an ancien b idge loca ed in
Todi (Umb ia, I aly), h ough he in eg a ion o geome ic su ey p oce
⁎Co esponding au ho .
Email add ess: [email p o ec ed] (P. Pachón)
h ps://doi.o g/10.1016/j.engs uc .2018.02.035
Recei ed 26 Sep embe 2017; Recei ed in e ised o m 5 Janua y 2018; Accep ed 9 Feb ua y 2018
A ailable online xxx
0141-0296/ © 2017.
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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
du es, dynamic es ing and nume ical modeling. Conde e al. [4] con-
duc ed ambien ib a ion es s on he Vilano a b idge, a mason y s uc-
u e loca ed in Galicia (Spain), and whose o igin da es back o he
13–14 h cen u ies. Those au ho s p esen ed a mul idisciplina y ap-
p oach o he s uc u al assessmen o mason y a ch b idges by using
NDT echniques and h ee-dimensional nume ical modeling. Gen ile and
Saisi [5] conduc ed he dynamic cha ac e iza ion o wo his o ic s uc-
u es, namely he Collegia a o San Vi o e bell owe (A cisa e, I aly),
and he San Michele b idge (Milan, I aly), an a ch b idge buil in 1889.
In he la e case, hose au ho s in es iga ed he a ia ion o he dy-
namic cha ac e is ics o he b idge unde di e en a ic condi ions. Fi-
nally, i is also wo h no ing he wo k done by Al unisik e al. [6] on
a mid-nine een h cen u y b idge in Tu key, he Mik on a ch. Those au-
ho s epo ed abou he de ini ion o a h ee-dimensional FEM o he
b idge, an OMA campaign, and he upda ing o he FEM on he basis
o he expe imen ally iden i ied modal p ope ies. O e all, despi e g ea
e o s ha e been pu in o he implemen a ion o hese echniques o he
conse a ion o his o ical cons uc ions, he ele a ed cos s o hese sys-
ems s ill emain an impo an obs acle.
This pape is aimed a p esen ing a me hodology o ailo a cos -e -
icien expe imen al se -up wi h a educed numbe o senso s o he
long- e m SHM o he E. To oja’s b idge. This b idge was cons uc ed
in 1951 o e he Guadalqui i i e by he enowned ci il enginee Ed-
ua do To oja (Fig. 1). The cul u al and his o ical impo ance o his
b idge jus i ies he implemen a ion o a long- e m SHM sys em, so ha
p e en i e ac ions can be aken in o de o p e en o mi iga e s uc-
u al aging and acciden al damages. Following he p e ious discussions,
he p esen me hodology p oposes he use o a limi ed numbe o sen-
so s which a e placed a op imal loca ions de e mined by means o an
OSP algo i hm. OSP algo i hms, which a e ypically based on a nume -
ical model o he s uc u e, maximize he modal in o ma ion wi h a e-
duced numbe o deg ees o eedom and, he e o e, a limi ed numbe o
senso s. Hence, he sui abili y o he ailo ed expe imen al se -up is c i -
ically de e mined by he accu acy o he nume ical model. A ho ough
ambien - ib a ion es wi h a la ge numbe o senso s is i s conduc ed
o assess he ib a ional p ope ies o he s uc u e and o se e as a ba-
sis o he upda ing o he p elimina y nume ical model. A e wa d, he
disc epancies be ween he heo e ical and expe imen al esul s a e min-
imized by upda ing di e en a iables o he nume ical model h ough
a gene ic op imiza ion algo i hm. Once he nume ical model is p op-
e ly uned, he op imal loca ions o a educed se o senso s a e de e -
mined by an OSP algo i hm. The esul s demons a e ha ew senso s
a e needed o accu a ely assess he main esonan equencies and mode
shapes.
The emainde o his pape is o ganized as ollows. Sec ion 2 con-
cisely desc ibes he his o ical e olu ion o he b idge and i s geome -
ic con igu a ion. Sec ion 3 in oduces he p elimina y FE model o he
b idge. Sec ion 4 p esen s he dynamic cha ac e iza ion o he b idge
by means o an expe imen al OMA campaign. Sec ion 5 de ails he up
da ing p ocess o he p elimina y FE model. Sec ion 6 o e iews he he-
o e ical o mula ion o he u ilized OSP echnique and epo s he p o-
posed op imal se -up o a long- e m SHM o he E. To oja’s b idge. Fi-
nally, Sec ion 7 d aws he main conclusions o his s udy.
2. E. To oja’s b idge: Cons uc ion and e olu ion
Edua do To oja Mi e (1899–1961) is conside ed one o he majo
igu es o Spanish ci il enginee ing, wi h a undamen al con ibu ion o
he design o hin-shell conc e e s uc u es. In his book “Razón y Se de
ipos es uc u ales”[7], his oeu e is concei ed as a ques o s uc u al
u h, a concep h ough which E. To oja ad oca es ha beau y lies
in a ionali y and no in a i icial o namen a ion. Wi h his in mind, E.
To oja de eloped new ways o designing s uc u es whe eby aes he ics
a ises om he a ionali y o he di e en s uc u al membe s. Fu he -
mo e, E. To oja showed in e es in undamen al a o ms and a e p e-
sen in mos o his s uc u es. Nowadays, nume ous b idges designed by
his eminen enginee a e conside ed as his o ical cons uc ions, such as
he Muga b idge (Fig. 2a) o he Ped ido b idge (Fig. 2b).
A e he Spanish ci il wa , E. To oja was commissioned o subs i-
u e he b idge o e he Guadalqui i i e in Posadas (Có doba, Spain)
which had been se iously damaged in he con lic . The o iginal s uc-
u e, which consis ed o ein o ced conc e e a ches, was eplaced by i e
isos a ic s eel-conc e e composi e spans o 43m (Fig. 3). The new so-
lu ion was de ined wi h a 7m wid h ein o ced conc e e deck and wo
in e ed bows ing s eel usses, whose bo om cho ds we e de ined as
pa abolic a ches wi h a maximum ise o 6m. Fig. 4 shows wo pho-
og aphs o he o iginal b idge unde cons uc ion.
In 1983, some new deck epai s we e conduc ed, including he con-
s uc ion o new dila a ion join s, injec ions, as well as epai s o he
piles’and abu men s’walls. In addi ion, some damaged s eel compo-
nen s we e eplaced and, e en ually, he me allic s uc u e and he
hand ails we e epain ed. None heless, he s uc u al ypology was kep
unal e ed (see Fig. 5).
In 1991, E. To oja’s g andson, An onio To oja, was en us ed o
ca y ou an ex ension o he deck wid h om 6.5m (Fig. 6(a)) o
11m (Fig. 6(b)). To his end, wo new a ches we e added and connec ed
o he o iginal ones by a ubula uss s uc u e. The o iginal deck had o
be comple ely emo ed because new uppe ein o cemen s we e needed
o bea he ans e se nega i e momen s (Fig. 7).
I should be no ed ha all he modi ica ions expe ienced by he
s uc u e en ail a high le el o unce ain y, a ec ing bo h he ma e ial
p ope ies and he s uc u al beha io . Mo eo e , due o he singula
geome y ha cha ac e izes he b idge, he me allic pa o he s uc-
u e is no physically accessible (Fig. 8) and, he e o e, he expe imen al
modal cha ac e iza ion o he b idge mus be ca ied ou on he oad. In
his sense, he nume ical model ep esen s a undamen al ool o e al-
ua e he cu en s a e o conse a ion o he b idge in gene al and he
s eel s uc u e in pa icula .
Fig. 1. E. To oja’s b idge in Posadas, Có doba (Spain).
2
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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
Fig. 2. Views o (a) Muga b idge, 1939 (Gi ona, Spain), and (b) To de a b idge, 1939 (Ba celona, Spain).
Fig. 3. F on iew o E. To oja’s b idge (dimensions in m).
Fig. 4. Pho og aphs o he o iginal b idge unde cons uc ion (1940–1951).
Fig. 5. Pho og aphs o he o iginal b idge in se ice (1966).
3. Fini e elemen modal analysis
This sec ion de ails he p elimina y FEM o he E. To oja’s b idge.
Fi s ly, o p o ide some insigh in o he s uc u al e olu ion o he
b idge, he esul s epo ed in a p e iously published wo k by he au-
ho s [8] on he s udy o he a ia ion o he dynamic p ope ies o
he b idge along i s his o y a e b ie ly p esen ed. Then, he p elimina y
h ee-dimensional FEM o he b idge, which se es as a basis o he ol-
lowing upda ing p ocess, is p esen ed in de ail.
3.1. S uc u al e olu ion o he b idge
As p e iously indica ed, he b idge has expe ienced impo an modi-
ica ions since i s cons uc ion. Some o he o iginal elemen s we e p e-
se ed, whils some o he s we e la e inco po a ed. In o de o shed
3
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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
Fig. 6. C oss sec ions o he b idge (dimensions in m).
Fig. 7. Views o (a) ini ial and (b) inal ope a ions o deck’s wid h inc emen (1992).
Fig. 8. Views o he cu en b idge in se ice.
some ligh on he e olu ion o he s uc u al beha io o he b idge,
he au ho s ca ied ou a p elimina y esea ch ha was p esen ed a
he In e na ional Modal Analysis Con e ence (IMAC 2015) [8]. In ha
s udy, bo h he o iginal and he cu en designs o he b idge (see Fig.
6) we e simula ed by p elimina y FEMs. Then, a compa ison o bo h
models in e ms o hei dynamic p ope ies was made in o de o aid
in he unde s anding o he s uc u al e olu ion. Beam elemen s we e
used o all he componen s excep o he deck o which hick shell
elemen s we e u ilized. Finally, he esonan equencies o bo h s uc-
u es we e compu ed by modal analysis, and i was obse ed ha only
small di e ences a e ound in he i s i e na u al modes. This ac in-
dica es ha , despi e he conside able modi ica ions expe ienced by he
s uc u e, he dynamic beha io o he b idge has no been signi ican ly
al e ed. The e o e, i was concluded ha he inc ease in he s i ness
o he b idge has been coun e ac ed by a simila inc ease in i s mass,
whe e he change o weigh o s eel pe a ea o he deck om 66.22 o
79.33kg/m⁠2 is indica i e o his conclusion. I was also concluded ha ,
since he dynamic cha ac e is ics ha e no been appa en ly al e ed, he
di e en ope a ions conduc ed in he b idge may ha e in oduced un-
ce ain ies de i ed om di e en ial aging p ocesses.
3.2. P elimina y FEM o he E. To oja’s b idge
In o de o inco po a e all he geome ical de ails o he s uc u e,
a sophis ica ed h ee-dimensional FEM o he E. To oja’s b idge is de-
eloped. Due o he la ge size o he b idge, including i e simply sup-
po ed spans, a FEM o he comple e s uc u e would esul in an ex-
cessi e compu a ional cos . Mo eo e , since he e is no s uc u al con-
nec ion be ween adjacen spans, each span o he b idge beha es inde-
penden ly o he o he s. The e o e, only one single span is modeled as
4
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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
common p ac ice in he s udy o his ype o b idges. Only shell ele-
men s a e de ined in such a way ha he connec ions be ween he s eel
a ches and he conc e e desk can be accu a ely simula ed (Fig. 9). The
ho izon al b aces a e de ined as buil -up sec ions wi h UPN-120 p o-
iles and ba en pla es o dimensions 400×50×8mm e e y 50cm. The
a ches a e also buil -up sec ions wi h IPN-200 p o iles and ba en pla es
o dimensions 210×50×8mm e e y 50cm. Wi h ega d o he nume -
ical de ini ion o he s uc u al membe s, 4-node iangula shell ele-
men s a e used o he conc e e slab and all he s eel membe s. O e -
all, he comple e model has 374.928 elemen s, 388.679 nodes and
2.332.074 deg ees o eedom. The bounda y condi ions a e de ined as
cons ained displacemen s and ee o a ions, as can be obse ed in he
de ails o Fig. 9. Table 1 summa izes he ma e ial p ope ies used in he
modeling. No e ha all he selec ed p ope ies a e common alues o
he design o his ype o s uc u es, wi h he excep ion o he mass den-
si y o s eel, whose alue is inc eased by 1.9% wi h espec o he s an-
da d alue o 7850kg/m⁠3. This decision is aken o accoun o all hose
ac o s ha ha e no been explici ly included in he nume ical model,
such as welds, bol s, and o he ancilla y elemen s. The weigh o he
ba ens and he pa emen is also included in e ms o added mass.
A e wa d, he modal p ope ies o he s uc u e a e compu ed by
a modal analysis in Abaqus CAE [9]. The i s eigh mode shapes a e
shown in Fig. 10. I is obse ed ha he b idge exhibi s a na ow band
o low esonan equencies wi h na u al modes ha a e highly coupled,
wha makes he dynamic ea u es o he s uc u e qui e complex.
4. Ambien ib a ion es and ope a ional modal analysis
The dynamic es ing o he s uc u e can p o ide an accu a e p edic-
ion o i s global modal pa ame e s. In o de o ensu e an e icien iden-
i ica ion o he modal p ope ies o he b idge, an ex ensi e ambien
ib a ion campaign wi h a la ge numbe o senso s was i s pe o med
on Ma ch 14, 2017, on he E. To oja’s b idge. In his way, he expe i-
men ally iden i ied dynamic p ope ies se e as a basis o he upda ing
o he p elimina y FEM o he b idge.
4.1. Ambien ib a ion es
The expe imen al layou o he p elimina y ambien ib a ion es
is schema ically ep esen ed in Fig. 11. Accele a ions we e egis e ed
in h ee o hogonal di ec ions wi h he aim o iden i ying he ib a-
ion modes in he la e al, longi udinal and e ical di ec ions o he
b idge. Twel e se -ups we e de ined wi h ou mobile senso s, whils
one was kep ixed as a e e ence (placed a poin 6, Fig. 11), wha
amoun ed o a o al o 36 measu ing poin s. The posi ion o he e e -
ence accele a ion senso was chosen acco ding he esul s p o ided by
he nume ical model [10]. In each one o hese se -ups, a sampling ime
o 10min and a sampling a e o 100Hz we e selec ed, aking in o ac
Fig. 9. FEM o he E. To oja’s b idge.
5

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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
Table 1
Ma e ial p ope ies used in he FEM.
P ope y Uni Value
Mass o secu i y ba ie s kg/m 14.66
Mass o asphal kg/m⁠2 110
Young’s modulus o conc e e slab MPa 30,000
Poisson’s a io o conc e e slab –0.2
Densi y o conc e e slab kg/m⁠3 2500
Young’s modulus o s eel MPa 210,000
Poisson’s a io o s eel –0.3
Densi y o s eel kg/m⁠3 8000
coun he empi ical ule p oposed by J. Rod igues in his Phd Thesis
[11]. These assump ions ensu e ha equencies om 1 o 50Hz a e
p ope ly eco ded.
The equipmen used in he es s included i e sel -con ained eco de
ins umen s manu ac u ed by he company GeoSIG Measu ing Sys ems.
These ins umen s ha e h ee in e nal senso s, an i-aliasing il e s, a
bandwid h anging om 0.01 o 250Hz, a dynamic ange o 146dB, a
sensi i i y o 10V/g, and 4.70kg o weigh (model GMSplus) (Fig. 12).
The same condi ions o empe a u e and humidi y we e aken in o
accoun du ing he whole campaign in o de o a oid a ia ions in he
modal pa ame e s [12]. In addi ion, he modal exci a ion o he b idge
was always caused by en i onmen al loads, such as wind o a ic (Fig.
13).
Fig. 10. Fi s eigh nume ical eigenmodes compu ed by he FEM o he E. To oja’s b idge.
Fig. 11. Plan iew o he accele ome e loca ions ( e e ence accele ome e in ed). (Fo in e p e a ion o he e e ences o colo in his igu e legend, he eade is e e ed o he web
e sion o his a icle.)
6
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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
Fig. 12. Sel -con ained eco de ins umen (GMSplus).
Fig. 13. Time his o y o accele a ion egis e ed by accele ome e numbe wo, 4 h se -up.
4.2. Ope a ional modal analysis
The A emis So wa e [13] is used o analyze he da a ob ained
du ing he expe imen al campaign. The Enhanced F equency Domain
Decomposi ion (EFDD) echnique [14,15] and he S ochas ic Subspace
Iden i ica ion (SSI) me hod [16,17] a e he wo di e en iden i ica ion
me hods used o ob ain he modal pa ame e s o he b idge (Fig. 14).
Wi h ega d o he signal p ocessing o he eco dings, a decima ion
ac o o 5 is applied in o de o ake in o conside a ion ha he ex-
pec ed na u al equencies a e below 10Hz, acco ding o he esul s o
he nume ical modal analysis p e iously shown in Fig. 10. In addi ion,
Fig. 14. Expe imen al iden i ica ion o he esonan equencies by he EFDD and SSI echniques.
7
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P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
he esolu ion o he spec al densi y es ima ion is de ined as 1024, wha
esul s in a equency line spacing o 0.005Hz. Ha monic de ec ion al-
go i hms a e also applied in o de o check all he equencies in he
spec um.
In o de o alida e he iden i ied modal pa ame e s, he Modal As-
su ance C i e ion (MAC) [18] is applied. Gi en wo mode shapes, φjand
φk, hei MAC alue w i es:
(1)
whe e supe index T designa es ansposi ion. When he MAC alue is
highe han 0.80, a good co ela ion be ween he wo modes is consid-
e ed [18]. Finally, Table 2 p esen s he ob ained esul s, including he
s anda d de ia ion o modal equencies and he damping a ios.
As can be obse ed in Table 2, he i s eigh ib a ion modes in a
equency ange up o 10Hz ha e been iden i ied. The di e ences be-
ween he equencies iden i ied by he SSI and EFDD me hods a e al
ways lowe han 1.5%, aking he esul s o he SSI me hod as e e -
ence alue. The esul s o he damping a ios show an highe a iabil-
i y, wi h a e age modal damping a ios o 0.61% and 1.64% o SSI
and EFDD echniques, espec i ely. Such la ge di e ences a e ypically
ound in p ac ice, ac ha indica es he need o highe le els o exci a-
ion o accu a ely cap u e he damping a ios [19]. The damping alues
ob ained by he EFDD me hod a e e en less eliable han hose ob ained
by he SSI me hod, as can be ex ac ed om hei highe s anda d de-
ia ion alues. Fu he , he MAC indica es a good co ela ion be ween
he modes iden i ied by bo h me hods wi h alues highe han 0.80. The
i s and he ou h modes a e o sional modes, while he o he modes
co espond o bending modes o he b idge (see Fig. 15).
5. FEM upda ing
On he basis o he expe imen ally iden i ied dynamic p ope ies,
he p elimina y FEM has been upda ed o ep oduce he ac ual beha -
io o he b idge and assis he OSP me hodology. S anda d FEMs o
his kind o buildings usually include unce ain ies de i ed om un
Table 2
OMA esul s: na u al equencies ( ), damping a ios (ξ) and s anda d de ia ion (s d).
Mode N⁠o SSI EFDD MAC
(Hz) s d( )ξ(%) s d(ξ) (Hz) s d( )ξ(%) s d(ξ)
1 3.28 0.02 0.85 0.40 3.31 0.01 1.46 0.66 0.92
2 3.68 0.07 0.79 0.46 3.72 0.03 2.14 1.17 0.94
3 3.80 0.04 0.78 0.31 3.84 0.07 1.60 1.01 0.82
4 5.14 0.01 0.60 0.17 5.16 0.01 0.63 0.22 0.98
5 5.61 0.10 0.38 0.27 5.68 0.09 2.78 1.16 0.81
6 5.64 0.05 0.41 0.26 5.69 0.07 2.01 1.22 0.80
7 7.79 0.05 0.58 0.31 7.68 0.14 1.70 1.06 0.86
8 8.02 0.02 0.49 0.27 8.07 0.05 0.84 0.39 0.84
Fig. 15. Expe imen ally iden i ied na u al ib a ion modes.
8
UNCORRECTED PROOF
P. Pachón e al. Enginee ing S uc u es xxx (2018) xxx-xxx
known ma e ial p ope ies, exis ing damage, complex in e nal compo-
si ion o s uc u al elemen s, and modeling app oxima ions. The e o e,
he calib a ion o he model wi h he aid o expe imen al in o ma ion
becomes essen ial o app op ia ely model he ac ual beha io o he
s uc u e.
Following he same p ocedu e as in p e ious wo ks by he au ho s
[20], he FEM upda ing is pe o med by means o i e a i e me hods
[21], in which he use in oduces changes di ec ly on some o he
physical pa ame e s ha de ine he s uc u e. To his aim, a sensi i i y
s udy is i s conduc ed o de ec hose s uc u al pa ame e s ha ha e
a g ea e in luence on i s dynamic beha io . In his wo k, he Young’s
modulus (Ec) and he densi y o he conc e e desk (ρc) a e selec ed as
design a iables.
Once he upda ing pa ame e s o he nume ical model a e selec ed,
and conside ing he good quali y o he expe imen al measu emen s, he
i s h ee iden i ied modes a e selec ed as a ge modes in he upda -
ing p ocess. Taking in o accoun he alues o he na u al equencies
and he modal coo dina es, a o al o 111 esidual componen s a e ad-
jus ed and minimized h oughou he model upda ing. In o de o con-
side he g ea e c edibili y o he iden i ied equencies wi h espec o
he modal displacemen s, a weigh ac o w = 1.00 is de ined o he
esonan equencies, whils a smalle ac o ws= 0.05 is assigned o
he modal coo dina es. Finally, he FEM upda ing is pe o med using a
gene ic algo i hm in Ma lab en i onmen [22], acco ding o an objec-
i e unc ion de ined as he ela i e di e ences be ween he expe imen-
al and he nume ical modal pa ame e s. This unc ion is usually o mu-
la ed as a leas -squa es p oblem as ollows:
(2)
whe e a e he alues ela ed o he physical pa ame e s o he
nume ical model, θ(Ecand ρc), while he a iables zEXP,ja e he same
magni udes ob ained om he expe imen al campaign. The di e ences
be ween hese a iables a e se as esidues, . I mus be no ed ha
he numbe o esidues, m=m +ms(wi h m being he numbe o con-
side ed na u al equencies, and ms he numbe o he coo dina es o
he conside ed ib a ion modes), is g ea e han he numbe o adjus ed
a iables, θ. The weigh ac o s, wj, a e he alues es ablished o each
esidue. In he case o na u al equencies, he esidues ead:
(3)
whe e and EXP,ja e he alues o he ob ained equencies
om he nume ical and expe imen al model. In a simila way, he
esidues in e ms o mode shapes co espondingly w i e:
(4)
whe e and a e he selec ed and e e ence compo-
nen o he nume ical mode j, while and a e he same mag-
ni udes ob ained om expe imen al mode j. Then, in o de o minimize
his objec i e unc ion, a gene ic algo i hm is applied.
Fig. 16 shows he con e gence o he i ness unc ion using a ge-
ne ic algo i hm. A ange o a ia ion o e e y upda ing pa ame e
(Table 3) is selec ed o p e en he algo i hm om de ining physically
un ealis ic es ima es. In each i e a ion, a popula ion o 1000 ec o s
is conside ed by using he p inciples o gene ic algo i hms (as imple-
men ed in Ma lab so wa e), and he objec i e unc ion in Eq. (2) is
minimized. Such calib a ion p ocess inishes when he di e ences be-
ween he mean alues (blue do s in Fig. 16) and he bes alues (g een
Fig. 16. Con e gence o he i ness unc ion using a gene ic algo i hm. Fi ness alue
e sus he numbe o gene a ions. Blue poin s: Mean alues o he objec i e unc ion o
all he popula ion o he co esponding gene a ion. G een poin : Bes alues esul o an
indi idual o he popula ion.(Fo in e p e a ion o he e e ences o colou in his igu e
legend, he eade is e e ed o he web e sion o his a icle.)
Table 3
Summa y o upda ed pa ame e s o he FEM by a gene ic algo i hm.
Pa ame e Ini ial alue Range o a ia ion Upda ed alue
Lowe Uppe
Ec(MPa) 25,000 20,000 30,000 27790.51
ρc(kg/
m⁠3)
2500 2000 4000 3774.71
do s in Fig. 16) o he i ness unc ion o wo i e a ions a e less han
1×10⁠−3.
Table 3 p esen s he summa y o he upda ed pa ame e s, whe e
la ge di e ences can be obse ed be ween he upda ed and ini ial pa a-
me e s.
Table 4 shows he compa ison be ween he expe imen al esul s
and hose p o ided by bo h he p elimina y and he upda ed FEMs.
Table 4
Compa ison o he expe imen ally iden i ied na u al equencies, SSI, he nume ical esul s
by he p elimina y FEM, , and hose p o ided by he upda ed FEM, , as
well he MAC alues o he mode shapes de e mined by he expe imen s and he upda ed
FEM, .
Mode
n⁠o
SSI
(Hz) (Hz) (Hz)
1 3.28 3.52
(7.31%)
3.26
(0.60%)
0.96
2 3.68 3.82
(3.81%)
3.67
(0.27%)
0.93
3 3.80 4.55
(19.73%)
3.90
(2.63%)
0.83
4 5.14 5.91
(14.98%)
5.26
(2.33%)
0.94
5 5.61 5.92
(5.52%)
5.77
(2.85%)
0.88
6 5.64 6.15
(9.04%)
5.82
(3.19%)
0.82
7 7.79 9.14
(16.81%)
7.83
(0.51%)
0.92
8 8.02 9.40
(17.20%)
8.08
(0.74%)
0.84
The pe cen ages in pa en hesis co espond o he ela i e di e ences be ween equencies.
9