E ec o soil p ope ies on he dynamic esponse o simply-suppo ed b idges unde
ailway a ic h ough coupled bounda y elemen - ini e elemen analyses
M.D. Ma ´
ınez-Rod igoa,∗, P. Gal ´
ınb, A. Dom´
enecha, A. Rome ob
aUni e si a Jaume I, Depa men o Mechanical Enginee ing and Cons uc ion, A da. Sos Bayna s/n, 12071 Cas ell´on, Spain
bUni e sidad de Se illa, Escuela T´ecnica Supe io de Ingenie ´ıa, Camino de los Descub imien os s/n, 41092 Se illa, Spain
Abs ac
Railway induced ib a ions on sho - o-medium span simply-suppo ed (SS) b idges is add essed in his con ibu ion. Such s uc-
u es may expe ience high le els o e ical accele a ion a he pla o m, leading o ad e se consequences such as a p ema u e
deg ada ion o he ballas laye and passenge discom o . In he p esen s udy, he e olu ion o he b idge dynamic esponse
when soil-s uc u e in e ac ion (SSI) is aken in o accoun is in es iga ed. To his end a coupled h ee-dimensional (3D) Bounda y
Elemen -Fini e Elemen model (BEM-FEM) o mula ed in he ime domain is implemen ed o ep oduce he soil and s uc u al
beha iou , espec i ely. Fi s , a se o soil-b idge sys ems o in e es is de ined, co e ing a wide ange o leng hs and na u al e-
quencies o he s uc u es, and an in e al o expec able elas ic p ope ies and damping le els o he soil. Then, di e en ypes o
analyses a e pe o med on he soil-b idge sys ems ex ac ing conclusions ega ding he e ec o including SSI in nume ical models
o p edic ing he b idge beha iou unde ailway a ic. In pa icula na u al equencies and modal damping le els a e iden i ied,
and he s uc u e ampli ica ion a e he passage o a mo ing load in ee ib a ion is in es iga ed. Conclusions ega ding how
esonance and cancella ion condi ions may be a ec ed by soil p ope ies a e ex ac ed. Finally, he dynamic esponse o a eal
b idge, belonging o he Spanish ailway ne wo k, is e alua ed unde he ci cula ion o ains ha induce second and hi d eso-
nances o he b idge undamen al mode. The e ec o he soil lexibili y, soil ma e ial damping and he b idge esonance o de a e
e alua ed. Conclusions ega ding he app op ia eness o he esul s p o ided by common models which do no include SSI e ec s
a e ex ac ed.
Keywo ds: Railway b idges, soil-s uc u e in e ac ion, esonance, cancella ion, mo ing loads, BEM-FEM coupled models
1. In oduc ion
The de elopmen o mode n, e icien and ope a ional ans-
po sys ems is essen ial o a sus ainable economic de elop-
men . In his con ex he cons uc ion o new High-Speed ail-
way lines and upg ading o con en ional lines o highe ope -
a ing speeds, has become a end in Asian and Eu opean coun-
ies in he las decades. Railway in as uc u es and, in pa icu-
la , ailway b idges, a e expec ed o exhibi an adequa e pe o -
mance unde hese new a ic equi emen s gua an eeing a ic
sa e y, passenge s com o , s uc u al in eg i y and accep able
en i onmen al condi ions in e ms o sound and ib a ion ans-
mi ed ampli udes.
The le el o ib a ions induced on b idges due o he ci cula-
ion o ailway con oys has become an issue o conce n among
he scien i ic and enginee ing communi y, due o he pe iodic
na u e o he ehicles axles and he ope a ing speeds app oach-
ing and exceeding 300 km/h in many lines. The pe iodic na u e
o he axle ansmi ed o ces may exci e impo an ans e se
ib a ion le els in he s uc u es, pa icula ly unde esonan
condi ions [1, 2]. Especially c i ical in his ega d a e sho -
o-medium span b idges composed by SS decks wi h usually
∗Co esponding au ho . Tel.: +34 964387473; ax: +34 964728106
Email add ess: [email p o ec ed] (M.D. Ma ´
ınez-Rod igo)
associa ed low masses. This p oblem agg a a es o low s uc-
u al damping le els, ypical in he a o emen ioned cons uc-
ions [1]. Figu e 1 shows wo examples o such s uc u es, be-
longing o he Spanish ailway ne wo k, wi h decks composed
by conc e e slabs es ing on se ies o p e-s essed conc e e gi d-
e s. E en hough his ypology is no common in High-Speed
lines o new cons uc ion, due o i s poo dynamic pe o mance
[3], hese beam- ype b idges do exis in o me con en ional
lines upg aded o High-Speed.
Resonance in ailway b idges may lead o ad e se conse-
quences such as ballas des abiliza ion, passenge discom o ,
a gene al deg ada ion o he ack and a aise in he main e-
nance cos s o he line [1, 4]. Fo his eason, acco ding o s an-
da ds, he maximum deck accele a ion mus be checked a he
Se iceabili y Limi S a e o he p e en ion o ack ins abili y,
and ega ded as a a ic sa e y equi emen [5].
Resonance akes place when he exci a ion pe iod o he
axles, i. e., he a io be ween a cha ac e is ic, o many imes
epea ed, dis ance and he ain speed is a mul iple o one na u-
al pe iod o he s uc u e. When his occu s, he ee ib a ion
oscilla ions induced by e e y load accumula e, and he ans-
e se esponse o he b idge p og essi ely inc eases, leading o
a subs an ial ampli ica ion i he numbe o axles is su icien .
In sho o medium span b idges wi h nowadays maximum ain
speeds, he cha ac e is ic dis ance associa ed wi h de imen al
P ep in submi ed o Enginee ing S uc u es Augus 29, 2017
Figu e 1: Railway b idges in Spanish lines composed by simply-suppo ed
bays o sho - o-medium span
le els o ans e se accele a ions due o esonance usually co -
esponds o he leng h o he passenge s’ coaches. The e o e,
he dynamic ampli ica ion o beams o b idges a esonance de-
pends bo h on he pe iodici y o he loads and on he ampli ude
o he ee ib a ions le by e e y single load. Unde ideal SS
condi ions and in he absence o damping, he ampli ica ion o
he ee ib a ions le by e e y load depends on he a io be-
ween he s uc u al pe iods and he a elling ime o he load.
As indica ed in [6], depending on his a io he beam may ex-
pe ience subs an ial le els o ee ib a ions (maximum ee i-
b a ions) o hese may p ac ically cancel (cancella ion o ee
ib a ions).
I he limi s on he b idge deck accele a ion canno be me
in an exis ing s uc u e, s eng hening measu es may be applied
in o de o modi y i s dynamic p ope ies and, consequen ly, i s
dynamic beha iou [7]. Passi e con ol echniques could also
p o ide cos -e ec i e solu ions inc easing he o e all damping
le els o he s uc u e and educing he deck ib a ional e-
sponse a esonance [8]. In ei he case o new o exis ing s uc-
u es, i is essen ial o de elop accu a e nume ical models, able
o ealis ically p edic he ib a ion le els o he expec ed a -
ic condi ions in o de o make he bes decision in he design
s age o when a line is upg aded o highe ope a ing speeds.
Acco ding o some au ho s [9], he choice o bounda y condi-
ions o dynamic analyses appea s o cons i u e a g oup o e y
sensi i e pa ame e s which ha e a conside able in luence on he
dynamic esponse o ce ain b idge ypes.
The phenomena o esonance and cancella ion expe ienced
by beams o b idges unde he ci cula ion o mo ing loads has
been s udied by se e al esea che s [6, 10–17]. Ne e heless
in he p e ious wo ks, soil-s uc u e in e ac ion is always dis-
ega ded and classical bounda y condi ions a e assumed o he
b idge deck. Acco ding o some au ho s, in ce ain soil en-
i onmen s an inc ease in he undamen al na u al pe iods o
mode a ely lexible s uc u es due o SSI may ha e a de imen-
al e ec on he s uc u al beha iou [18]. The wo k p esen ed
he ein a ises in his con ex .
Only a ew au ho s ha e in es iga ed he dynamic esponse
o beams o b idges and, in pa icula , he condi ions o es-
onance and cancella ion phenomena aking in o accoun he
wa e p opaga ion in he soil. Lu e al. [19] p o e nume -
ically he occu ence o esonance and cancella ion in a pe-
iodic iaduc subjec o mo ing loads conside ing pile-soil-
s uc u e in e ac ion. Wu and Yang [20] apply a semi-analy ical
app oach o analyse g ound ib a ions induced by ains mo -
ing o e ele a ed b idges. The au ho s use impedance unc-
ions o ep esen he ounda ion-soil in e ac ion and an elas-
ic hal space model o he soil wa e p opaga ion p oblem.
In [21] and [22] he au ho s in es iga e g ound ib a ions in-
duced by High-Speed ains c ossing con inuous gi de b idges
and igid- ame iaduc s, espec i ely. In bo h con ibu ions he
g ound esponse is calcula ed by applying eac ion o ces on a
3D FEM wi h a i icial iscous bounda ies. Takemiya and Bian
[23] in es iga e nume ically he wa es gene a ed in he soil
nea a Japanese Shinkansen mul i-span iaduc . The au ho s
also p esen ield es s measu emen s on he ounda ions and
in he g ound a ield, showing equency con en s ela ed o
ain axle dis ances and s uc u e na u al pe iods. In [9] ¨
Ulke -
Kaus ell e al. p esen a quali a i e analysis o he dynamic
SSI phenomenon on a po al ame ailway b idge based on dy-
namic s i ness unc ions. The au ho s conclude ha he con i-
bu ion o he coupled soil-b idge sys em o he modal damping
a ios is subs an ial, especially o he lowe ange o he soil
elas ic modulus.
Mos o he p e ious wo ks ocus on he le el o ib a ions
ansmi ed h ough he soil along he ack, a he han on he
b idge beha iou i sel . In he opinion o he au ho s o his
con ibu ion, he e is a need o in es iga e how soil p ope ies,
in e ms o lexibili y and ma e ial damping, may a ec he dy-
namic esponse o sho SS b idges suscep ible o expe ience
excessi e accele a ions a he deck le el. I his kind o anal-
ysis is pe o med wi h gene ali y, i. e., conside ing expec able
anges o a ia ion o s uc u al and soil p ope ies, in e es ing
conclusions could be ex ac ed ega ding he app op ia eness o
he nume ical models usually used by enginee s when i comes
o assess he pe o mance o new s uc u es, o ha o exis ing
s uc u es subjec ed o mo e demanding ope a ing condi ions.
In his s udy he au ho s comple e he in es iga ion ini ia ed
2
in e e ence [24], ex ending he analysis o se e al soil ypes
wi h di e en le els o ma e ial damping, and pa icula izing
he conclusions ex ac ed o he case o a eal s uc u e.
In wha ollows a comp ehensi e ensemble o soil-b idge
sys ems is de ined co e ing ypical leng hs and s uc u al y-
pologies o sho o medium span SS ailway b idges, and a
wide ange o a ia ion o soil lexibili ies and ma e ial damp-
ing alues. A sensi i i y analysis is conduc ed on his ensemble
and he e olu ion o he b idges na u al equencies and s uc-
u al damping le els is e alua ed wi h he p ope ies o he soil.
The ampli ica ion o he b idge dynamic esponse in ee ib a-
ion unde a single mo ing load (SML) is hen p esen ed, and,
based on his analysis, conclusions ega ding he e olu ion o
he esonan and cancella ion phenomena induced by mul iple
mo ing loads (MML) wi h soil p ope ies is discussed. Finally,
he dynamic esponse o a eal b idge belonging o he Span-
ish ailway ne wo k is analysed unde ailway a ic. The e o-
lu ion o he s uc u e esponse unde di e en o de esonan
condi ions and unde no esonan condi ions wi h he lexibil-
i y and damping o he su ounding soil is e alua ed. Finally
conclusions a e ex ac ed ega ding he adequacy o nume ical
models ha dis ega d SSI e ec s.
2. Fo mula ion and app oach adop ed
2.1. App oach o he in es iga ion
The nume ical model implemen ed o he in es iga ion has
been p e iously p esen ed in [24] and i s main ea u es a e sum-
ma ized he ein. I is a ully coupled 3D BEM-FEM model in-
eg a ed in he ime domain. The SSI p oblem is analysed by
domain decomposi ion in he soil and s uc u e sub-domains,
ep esen ed wi h he BEM and he FEM, espec i ely. BEM-
FEM coupling is pe o med di ec ly. A scheme showing he
main pa s o he model is ep esen ed in Figu e 2.
The main ea u es o he BEM-FEM model a e:
•A beam FEM is used o ep esen he deck lexu al be-
ha iou unde mo ing loads, he e o e assuming ha he
maximum ans e se esponse o he s uc u e is mainly
go e ned by i s longi udinal bending de o ma ion. This
decission is jus i ied by he ac s ha : (i) acco ding o p e-
ious s udies [6, 25], sho o medium span SS ailway
decks a e expec ed o exhibi maximum e ical accele -
a ion le els a mid-span; (ii) in ein o ced conc e e slabs
o p es essed conc e e gi de decks, usual ypologies o
he ange o leng hs unde conside a ion, esonances o he
i s o sion mode a e usually no de e minan in he as-
sessmen o he Ul ima e Limi S a e o e ical accele a-
ion [25]; (iii) he obje i e o his in es iga ion is o e alu-
a e SSI e ec s on he main esonan p oblem ha ailway
decks may expe ience unde ailway a ic.
•The beam b idges a e idealised as Be noulli-Eule (BE)
beams in a ini e elemen con ex . The beam is disc e ized
using wo node beam elemen s wi h ension, comp ession,
o sion (no exci ed conside ing he 2D na u e o he ap-
plied loads), and bending capabili ies. The choice o he
Figu e 2: Schema ic ep esen a ion o he 3D BEM-FEM coupled model
Be noulli-Eule heo y is well sui ed o he analysis o
ailway b idges in his s udy due o he slende ness a ios
o ypical ailway decks [1, 26]. Mo eo e , he equency
ange o in e es in he s udy is low (unde 30 Hz) and mis-
ma ches be ween Be noulli-Eule and Timoshenko beams
a e expec ed o be ele an abo e 50 Hz [27].
•The in luence o he ack and he ballas , which can also
a ec he dynamic beha iou o he b idge [28–32], has
been aken in o accoun only by means o he associa ed
dead masses. A de ailed ehicle idealisa ion, ha would
cause a educ ion in he ib a ion le els o he b idge
[33, 34] a esonance and o he g ound [35], is also dis e-
ga ded, and a mo ing load model has been used du ing he
in es iga ion. These simpli ica ions, consis en wi h com-
mon design p ac ices, ha e also been adop ed in p io in-
es iga ions o he esonance and cancella ion phenomena
in ailway b idges [6, 11, 16, 17], and i has been consid-
e ed con enien in a i s app oach o he p oblem. Ad-
di ionally, as i will be shown in wha ollows, ehicle-
b idge in e ac ion and SSI will bo h lead o a educ ion o
he deck accele a ion a esonance. On he au ho s opinion
i is essen ial o sepa a e bo h e ec s in o de o cap u e he
e ec s caused by he soil sepa a ely and be able o ex ac
conclusions in his ega d.
•The ailway exci a ion is in oduced as a sequence o mo -
ing loads a elling a cons an speed, he e o e neglec ing
ehicle-s uc u e in e ac ion e ec s. The g adual na u e o
he wheel loads applica ion p ocess close o he abu men s
due o he dis ibu i e e ec o ails, sleepe s and ballas
mus be simula ed in o de o a oid un ealis ic high e-
quency modal con ibu ions. To his end, a load dis ibu-
3
ion unc ion based on he Zimme man-Timoshenko solu-
ion o an in ini e beam on Winkle ounda ion, is applied
o he axle load modulus in he abu men s p oximi ies. De-
ails o he o mula ion may be ound in [8].
•The beam end sec ions a e connec ed h ough kinema ic
cons ain s o wo igid pla es ep esen ing he lowe su -
ace o shallow ounda ions a he abu men s. These pla es
a e coupled o he bounda y elemen s simula ing he in-
e ac ion wi h he soil. Wi h his simple idealiza ion, he
essence o he wa e p opaga ion p oblem is isola ed om
he ounda ions geome y, and i s in luence is e alua ed
conside ing only he b idge ib a ion esponse [36].
•Rega ding he soil ea men , a homogeneous soil wi h
cons an p ope ies is admi ed. The G een’s unc ion o
an elas ic hal -space is used as he undamen al solu ion
o displacemen s and ac ions in he BEM [37]. The e-
o e, he bounda y elemen disc e isa ion is limi ed o he
in e ace be ween he soil and he pla es. The soil is dis-
c e ised using nine node ec angula quad a ic bounda y
elemen s.
•Coupling o he BEM and FEM equa ions is ca ied ou
by imposing equilib ium and compa ibili y condi ions a
he soil-s uc u e in e ace. Bo h sys ems o equa ions a e
assembled in o a single sys em, oge he wi h he equilib-
ium and compa ibili y condi ions [38].
The desc ibed model is implemen ed in he SSIFiBo oolbox
o MATLAB p e iously de eloped by coau ho s o his con i-
bu ion Gal ´
ın and Rome o [39–41]. The FEM module o he
oolbox does no include any p e-p ocesso . Ins ead, a ga eway
o comme cial so wa e allows impo ing di ec ly he s uc u e
model. Using his model, SSI e ec s on he ans e se esponse
o beams a e sed by mo ing loads a cons an speeds a e s ud-
ied by means o he ollowing complemen a y s eps:
1. Fi s (sec ion 3.1), a p elimina y analysis is p esen ed
based on he equency esponse unc ion (FRF) o a soil-
b idge sys em unde impulse exci a ion, wi h he aim o
an icipa ing he in luence o he soil p ope ies on he
b idges esponse in he equency domain. This issue is
ela ed wi h he ela i e alues o he Rayleigh and beam
bending wa eleng hs.
2. Second (sec ion 3.2), he a ia ion o modal pa ame-
e s ( undamen al equency and modal damping) o he
b idges unde s udy conside ing SSI is analysed. I should
be ema ked ha , as explained in sec ion 2.3, he b idges
and soil p ope ies ha e been selec ed co e ing a wide
ange o ealis ic combina ions in he design o sho SS
ailway b idges.
3. Thi d (sec ion 3.3), he maximum esponse o he s uc-
u es unde he ci cula ion o a single mo ing load in e ms
o he uni o m speed is p esen ed, and he condi ions o
maximum esponse and cancella ion du ing he ee ib a-
ion phase (once he load has le he s uc u e) a e shown.
Gene al conclusions ega ding he in luence ha soil p op-
e ies may ha e on esonan speeds and associa ed ampli-
udes a e ex ac ed om hese esul s.
4. Finally (sec ion 4), he dynamic esponse o a eal SS ail-
way b idge belonging o he Spanish ailway ne wo k is
analysed unde he ci cula ion o a ain o mo ing loads
exci ing wo ele an esonan si ua ions in he ange o
speeds conside ed. The in luence o he soil lexibili y
and ma e ial damping is in es iga ed when he b idge un-
de goes esonances o di e en o de and a non- esonan
condi ions.
2.2. BEM-FEM ma hema ical o mula ion
The BEM is based on a ime ma ching p ocedu e o ob ain
he ime a ia ion o he bounda y unknowns; i. e., displace-
men s and ac ions. The k− h componen o displacemen s
and ac ions o e he bounda y is app oxima ed om he nodal
alues ja each ime s ep m,um j
kand pm j
k, using he space in-
e pola ion unc ions φj( ) and ψj( ), o ac ions and displace-
men s, espec i ely. A e in e pola ing he bounda y a iables,
he in eg al ep esen a ion o he displacemen ua a poin ion
he bounda y becomes [40]:
ci
lkui
k(xi, )=
n
X
m=1
Q
X
j=1



ZΓj
Unm
lk ψjdΓpm j
k
−ZΓj
Pnm
lk dτφjdΓum j
k




(1)
whe e Qis he o al numbe o bounda y nodes and Γj ep e-
sen s he elemen s o which node jbelongs. Time ke nels Unm
lk
and Pnm
lk a e espec i ely compu ed h ough he undamen al so-
lu ion o displacemen s and ac ions due o a poin load ac ing
a xiin he ldi ec ion. These ke nels a e analy ically in eg a ed
by pa s using cons an and linea piecewise ime in e pola ion
unc ions o ac ions and displacemen s [37], espec i ely. Eq.
(1) may be w i en in a mo e compac o m as:
ci
lkuni
k=
n
X
m=1
Q
X
j=1hGnmi j
lk pm j
k−b
Hnmi j
lk um j
ki(2)
Once he in eg al- ee e m ci
lk is included in he sys em ma ix,
he in eg al ep esen a ion o poin ia ime =n∆ becomes:
Hnnun=Gnnpn+
n−1
X
m=1Gnmpm−Hnmum(3)
whe e Hnmi j
lk collec s o ci
lk when i=jand n=m.
The FEM equa ion a ime s ep nis de ined as [42]:
M¨
un+C˙
un+Kun= n(4)
whe e M,CyKa e he mass, damping, and s i ness ma ices,
espec i ely. un,˙
uny¨
un ep esen nodal displacemen , eloci y,
and accele a ion ec o s, espec i ely, and nis he load ec o
4
including he e ec o he cons an mo ing load a each ime-
s ep. Equa ion 4 is sol ed using an implici ime in eg a ion
GN22 Newma k me hod [42, 43]. An equi alen dynamic s i -
ness ma ix is de ined:
Dun= n+ n−1(5)
Coupling o BEM and FEM equa ions (Eqs. (3) and (5)) is ca -
ied ou by imposing equilib ium and compa ibili y condi ions
a he soil-s uc u e in e ace. Bo h sys ems o equa ions a e
assembled in o a single global sys em, oge he wi h he equi-
lib ium and compa ibili y equa ions [44].
As he pla e ounda ions ha e been de ined as igid bodies
in a i s app oach h ough kinema ic cons ain s, he BEM Eq.
(3) is exp essed in e ms o he kinema ic cons ain ma ix L
ela ing he displacemen s and ac ions o he cen al poin o
he pla e, u0and p0, espec i ely, wi h any o he poin o each
ounda ion:
HnnLun
0=GnnLTpn
0+
n−1
X
m=1hGnmLTpm
0−HnmLum
0i(6)
whe e equilib ium o o ces a he in e ace Γis ul illed in e-
g a ing nodal ac ions acco ding o he elemen shape unc ion
ma ix N:
=ZΓ
NTpN dΓ = Tp (7)
The ime s ep ∆ o he analysis is se su icien ly small o p op-
e ly in eg a e he s uc u e dynamic esponse and load exci a-
ion. This may be exp essed as:
∆ =min 2π
ω1kω
,L
Vk !(8)
whe e ω1co esponds o he undamen al equency o he
beam, Lis he beam leng h, and V he load speed. Pa ame-
e s kωand k de ine ime disc e iza ions o he s uc u e un-
damen al pe iod and he load passage ime, espec i ely.
The chosen ime s ep de e mines he spa ial bounda y ele-
men disc e iza ion acco ding o he s abili y pa ame e β=
cs∆ /∆l, whe e ∆lis he dis ance be ween wo nodes o a
bounda y elemen , and csis he shea wa e p opaga ion eloc-
i y in he soil. In his wo k, a s abili y pa ame e β=0.5 has
been conside ed.
The ini e elemen ep esen a ion is de e mined by he b idge
bending wa eleng h disc e iza ion. Minimum wa eleng h is de-
ined by he maximum equency ange and he phase bend-
ing wa e p opaga ion eloci y in he undamen al mode cb1=
4
qω2
1EIz/mb, whe e EIzis he beam c oss-sec ion bending s i -
ness and mbis he beam mass pe uni leng h. This wo k con-
side s 20 elemen s o he minimum wa eleng h.
2.3. De ini ion o an ensemble o soil-b idge sys ems
In his sec ion he ensemble o soil-b idge sys ems in es i-
ga ed in he sensi i i y analysis included in sec ion 3 is p e-
sen ed.
B idge beam models o leng hs anging om 12.5 o 25 m in
inc emen s o leng h o 2.5 m a e conside ed, co e ing he yp-
ical span leng hs suscep ible o expe ience high deck e ical
accele a ions unde esonan condi ions. The ange o unda-
men al equencies ealis ic o each span is selec ed om he
band p esc ibed by Eu ocode 1 [5] o he applica ion o simpli-
ied me hods (see Figu e 3). The e o e, he as majo i y o ex-
is ing and po en ial SS b idges undamen al equencies a e ex-
pec ed o all wi hin hese limi s. Th ee e enly-spaced sample
alues be ween 0 % and 70 % o Eu ocode 1 uppe equency
limi ha e been analysed. These equencies a e e e ed o as
1,000, 1,035 and 1,070 in wha ollows. As can be ex ac ed om
he s udies p esen ed in [33], he majo i y o ailway b idges
o con en ional and High-Speed lines all wi hin he selec ed
ange.
In a i s app oach, a single alue o mass pe uni leng h is
assigned o each beam, in pa icula mb=L(m)·1000 kg/m2.
The mass o he s uc u e will a ec he le el o e ical accel-
e a ion a esonance, bu i has no been selec ed as a pa ame e
o he sensi i i y s udy as i does no go e n he maximum ee
ib a ion and cancella ion condi ions in he absence o SSI e -
ec s [6]. S uc u al damping is no assigned o he beams in
sec ion 3 in o de o isola e he e ec s o SSI on he modal
pa ame e s o he b idges unde s udy. Rega ding he subs uc-
u e, iden ical 5m ×5m ounda ion pla es a e conside ed in all
he cases o ep esen he soil-subs uc u e in e ac ion su ace.
As pe he soil p ope ies, h ee homogeneous soil ypes a e
de ined wi h lexibili ies co e ing he AASHTO classi ica ion
[45]. In pa icula shea (s) and dila a ion (p) wa e eloci ies
o cs={150,220,365}m/s and cp=2csa e conside ed, admi -
ing a Poisson’s a io ν=1/3 o he soil. Soil densi y has been
se equal o 1800 kg/m3in all he cases. Soil ma e ial damping
le els o ζs={0,2.5,5}% a e conside ed o each shea wa e
eloci y. The e o e 180 BEM-FEM models a e e alua ed in he
ollowing sec ions (18 b idges ×9 soil ypes plus 18 b idges
wi h in ini ely igid soil condi ions).
3. Sensi i i y analysis
3.1. SSI e ec on he b idges beha iou in he equency do-
main. P elimina y analysis
In o de o ge some insigh ega ding how SSI may a ec he
b idges dynamic esponse depending on he equency ange, a
p elimina y analysis is included in his subsec ion. Fi s , beams
and soil wa eleng hs a e compu ed and ep esen ed in o de o
es ima e he equency ange in which he in e ac ion be ween
he s uc u e and he soil could be app eciable. Second, he im-
pulse esponse o a pa icula beam is p esen ed and he e ec
o he soil lexibili y and damping a e shown in he equency
domain.
Figu e 4 ep esen s he Rayleigh wa eleng h o he soil λR=
cR/ and he beam bending wa eleng h λb=cb/ in e ms o
he equency .cRs ands o he Rayleigh wa e p opaga ion
eloci y in he soil, app oxima ed as in [46], and cb o he beam
bending wa e p opaga ion eloci y:
5

ζs=5.00 %
ζs=2.50 %
5
10
15
1,000
1,035
1,070
1,100 ζs=0.00 %
cs=in
1(Hz)
ζs=5.00 %
ζs=2.50 %
1,000
1,035
1,070
1,100 ζs=0.00 %
cs=150 m/s
ζs=5.00 %
ζs=2.50 %
510 15 20 25 30
5
10
15
1,000
1,035
1,070
1,100 ζs=0.00 %
cs=220 m/s
L(m)
1(Hz)
ζs=5.00 %
ζs=2.50 %
510 15 20 25 30
1,000
1,035
1,070
1,100 ζs=0.00 %
cs=365 m/s
L(m)
Figu e 3: Ensemble o soil-b idge sys ems unde s udy. ◦F equency and span leng h o analyzed b idges o pa icula soil p ope ies (cs, ζs)
cR=0.87 +1.12ν
1+νcscb=4
qω2EIz/mb(9)
Bo h wa eleng hs ha e been no malised wi h espec o he
beam wa eleng h in i s undamen al mode o SS condi ions
(λ1,ss =2L).
In Figu e 4 each g aph co esponds o a se o b idges
wi h na u al equencies in he SS case co esponding o le els
1,000, 1,035, 1,070 and 1,100 in he Eu ocode equency band.
1,100 is conside ed only in his subsec ion o compa ison pu -
poses, as i does no ep esen common s uc u es o he ypolo-
gies o in e es . In each plo , di e en cu es associa ed o he
same soil ype (same g ay colou aces) co espond o di e -
en span leng hs L=12.5 m o L=25 m. Mo eo e , in he
ho izon al axis he equency has been no malised by he un-
damen al equency o he SS beam ( / 1,ss). This no malisa-
ion allows o ep esen all he beams wi h a single cu e gi en
ha λb/λ1,ss =( 2/ 2
1,ss)0.25. Two equency egions may be
dis inguished o each soil-beam sys em: (i) a egion whe e he
Rayleigh wa eleng h o he soil is highe han he beam bending
wa eleng h (low equency ange); and (ii) a egion whe e he
Rayleigh wa eleng h o he soil is lowe han he wa eleng h o
he beam. The cu -o equency be ween bo h egions may be
easily ob ained equa ing bo h wa eleng hs (λR=λb), en ailing
ha cR=cb.
No ice ha he beam bending wa eleng h s a s o exceed he
soil Rayleigh wa eleng h a a equency ha inc eases wi h he
soil s i ness and he beam undamen al pe iod. I should be
he e o e expec ed SSI o be mo e pe cep ible a low equen-
cies on he b idge esponse in he case o mo e lexible soils
(lowe alues o cs) and o beams wi h highe undamen al e-
quencies. Mo eo e , highe modal con ibu ions o he beams
should be mo e a ec ed by SSI e ec s han he esponse asso-
cia ed o he undamen al mode.
As an example, Figu e 5 shows he equency esponse unc-
ion a L/4 o a b idge span L=15 m, conside ing ou b idge
equencies co e ing he comple e Eu ocode ange o ha pa -
icula leng h, and di e en soil condi ions (wa e p opaga ion
eloci ies and damping a ios). The FRF is compu ed loading
he s uc u e wi h an impulsi e o ce ac ing on he same sec-
ion. The FRF shows peaks a he equencies co esponding
o he i s h ee bending modes o he b idge. The b idge e-
quencies and he peaks ampli udes mo e owa d lowe alues
as he soil becomes so e , and SSI e ec s become mo e im-
po an . Also, b idges wi h highe na u al equencies a e mos
a ec ed by SSI. Mo eo e , i can be concluded om his anal-
ysis ha he in luence o he soil ma e ial damping is almos
impe cep ible a he undamen al equency o he s uc u e and
i is much mo e no iceable in he equency ange abo e he
a o emen ioned cu -o equency.
In he ollowing sec ions modal p ope ies o he b idge ca -
alogue unde s udy a e iden i ied, and SSI e ec s on he con-
di ions o maximum ee ib a ion and cancella ion o he
b idges unde a SML a e e alua ed.
3.2. Iden i ica ion o modal pa ame e s
In iew o he esul s o he p e ious sec ion and ollowing
he app oach in [24], a pa ame e κ=EIzπ3/(K L3) is de ined
as he a io o he lexu al igidi y o he b idges o he e ical
6
12345678910
/ 1,ss
0
0.5
1
1.5
2
2.5
3
λ/λ1,ss
(a) 1,000
12345678910
/ 1,ss
0
0.5
1
1.5
2
2.5
3
λ/λ1,ss
(b) 1,035
12345678910
/ 1,ss
0
0.5
1
1.5
2
2.5
3
λ/λ1,ss
(c) 1,070
12345678910
/ 1,ss
0
0.5
1
1.5
2
2.5
3
λ/λ1,ss
(d) 1,100
Figu e 4: Rayleigh wa eleng h (λR/λ1,ss) o di e en b idge spans (12.5 o 25 m) and soil p ope ies:
cs=150 m/s, cs=220 m/s and cs=365 m/s. Beam bending wa eleng h (λb/λ1,ss)
s i ness o he soil- ounda ion suppo s unde s a ic loading,
K . In Figu e 6 he alues o κ o he soil-b idge sys ems unde
s udy a e ep esen ed (no ice ha soil ma e ial damping does
no a ec his pa ame e ). κ=0 co esponds he e o e o an
in ini ely igid soil.
The na u al equency and modal damping associa ed o he
undamen al mode a e ob ained om he beam esponse sub-
jec ed o an impulse load o he comple e ensemble o b idges
unde s udy (108 BEM-FEM models). The a ia ion o hese
wo modal pa ame e s in e ms o κ o all he soil-b idge sys-
ems a e included in Figu es 7 and 8, espec i ely.
3.2.1. E ec o soil p ope ies on iden i ied na u al equencies
In Figu e 7 he undamen al equency a ia ions, wi h e-
spec o in ini ely igid soil condi ions, expe ienced by he
b idges a e ep esen ed wi h ci cles. These esul s a e calcu-
la ed using he BEM-FEM model desc ibed in sec ion 2.2. Fig-
u es in he same ow co espond o he same soil shea -wa e
eloci y, while igu es in he same column co espond o he
same alue o soil ma e ial damping. In all he plo s, he an-
aly ical solu ion o he undamen al equency a ia ion o an
elas ically suppo ed (ES) Be noulli-Eule beam wi h iden ical
elas ic suppo s o K e ical s i ness has been ep esen ed in
hick black ace [6]. Finally, in each g aph di e en equency
bands a e dis inguished in shaded a eas, and ci cle sizes a e
p opo ional o he leng hs o he b idges.
The e ical lexibili y o he soil- ounda ions leads o a e-
duc ion in he undamen al equency o he b idges unde
s udy. This educ ion is mo e e iden in he case o b idges
wi h highe na u al equencies (and he e o e, highe alues o
κ). This is consis en wi h he esul s p esen ed in sec ion 3.1.
Fo each equency g oup, b idges wi h longe spans a e mos
a ec ed by soil condi ions. This is due o he ac ha longe
b idges p esen highe κ alues [24].
F om he analysis o he esul s p esen ed i may be con-
cluded ha : (i) he equency a ia ion expe ienced by he
s uc u es when SSI is included ollows he gene al end shown
by he ES BE beam in e ms o he s a ic ela i e s i ness pa-
ame e κ. The equency dependence o he soil- ounda ion
s i ness is no ele an , especially o low κ alues and long
b idges; (ii) soil-b idge sys ems wi h simila κ alues show
simila equency a ia ions, independen ly o he soil p ope -
ies and he beam na u al equency in he absence o SSI; (iii)
o each soil ype and equency band, sho e b idges show a
sligh ly highe de ia ion wi h espec o he analy ical solu ion
o he ES beam; (i ) hese endencies ake place o di e en
soil ma e ial damping le els, and he in luence o his pa am-
e e is almos negligible ega ding he a ia ion o he b idges
undamen al equency. This issue was an icipa ed in sec ion
3.1 o low equency anges.
7
12345678910
/ 1,ss
10−12
10−10
10−8
10−6
FRF (m/N)
(a) 1,000
12345678910
/ 1,ss
10−12
10−10
10−8
10−6
FRF (m/N)
(b) 1,035
12345678910
/ 1,ss
10−12
10−10
10−8
10−6
FRF (m/N)
(c) 1,070
12345678910
/ 1,ss
10−12
10−10
10−8
10−6
FRF (m/N)
(d) 1,100
Figu e 5: FRF o a b idge o L=15 m and soil p ope ies:
cs=150 m/s, cs=220 m/s and cs=365 m/s. B idge esponse in SS case.
Conside ing he ollowing damping a ios: ζs=0.000 (solid lines), ζs=0.025 (dashed lines) and ζs=0.050 (do ed lines)
1,000 1,035 1,070
0
0.2
0.4
0.6
25.0 m
22.5 m
20.0 m
17.5 m
15.0 m
12.5 m
cs=150 m/s
cs=220 m/s
cs=365 m/s
1,band
κ
Figu e 6: Dimensionless a io κ o he soil-b idge sys ems unde s udy
3.2.2. E ec o soil p ope ies on iden i ied modal dampings
In Figu e 8 he alues o s uc u al damping in he undamen-
al mode iden i ied om he b idges esponse o he anges o
soil p ope ies unde conside a ion a e ep esen ed wi h ci cles.
The s uc u al damping a io including SSI e ec s (ζ1) is
iden i ied om he ee damped esponse o he b idges h ough
Loga i hmic dec emen . As in Figu e 7, di e en equency
bands a e dis inguished in shaded a eas, and ci cle sizes a e p o-
po ional o he b idges leng hs. The modal damping measu ed
om he b idge esponse s ongly depends on he alue aken by
he ela i e lexibili y κ. As he lexibili y o he soil inc eases
(highe κle els o he same s uc u e), so does he iden i ied
damping due o he wa e adia ion h ough he soil. Again,
o he same soil p ope ies, s uc u es wi h highe undamen al
equencies in he absence o soil exhibi highe inc emen s o
s uc u al damping when SSI is conside ed. This is again con-
sis en wi h he analysis p esen ed in sec ion 3.1. On he o he
hand, he in luence o he soil ma e ial damping on he iden-
i ied s uc u al damping is minimal in he undamen al mode.
As i was exposed in he p e ious sec ion, soil damping only
modi ies he s uc u al esponse du ing a sho ansien , due o
he ela i e wa eleng hs o he soil and he b idges. The e o e,
he in luence o soil damping is expec ed o be signi ican only
a highe equencies han he b idge undamen al one.
8
0.7
0.8
0.9
1.0
1,070
1,035
1,000
ζEC
0=1.00 %
1/ 1,SS
ζs=0.00 %
1,070
1,035
1,000
ζs=2.50 %
1,070
1,035
1,000
ζs=5.00 %
0.7
0.8
0.9
1.0
1,070
1,035
1,000
1/ 1,SS
1,070
1,035
1,000
1,070
1,035
1,000
0.1 0.30.5
0.7
0.8
0.9
1.0
1,070
1,035
1,000
κ
1/ 1,SS
0.1 0.30.5
1,070
1,035
1,000
κ
0.1 0.30.5
1,070
1,035
1,000
κ
0.1 0.30.5
cs=365 m/s
cs=220 m/s
cs=150 m/s
Figu e 7: B idge iden i ied undamen al equency s. κ. Analy ical ES beam case
3.3. F ee ib a ion esponse unde a SML
Acco ding o [6] he dynamic ampli ica ion o a SS o ES
beam a esonance caused by he ci cula ion o MML is closely
ela ed o he ee ib a ions ha he same beam expe iences
a e he passage o each single load a elling a he same
speed. The load a elling a ce ain speeds, induces on he
beam a ema kably high esponse (maximum ee ib a ions)
and, a some o he speeds, he oscilla ions when he load lea es
he beam a e almos negligible (cancella ion o ee ib a ion).
These wo phenomena a e independen o he pe iodici y o he
loads, and ake place o a single mo ing load. The aim o
his sec ion is o e alua e how SSI a ec s hese wo condi ions.
In [24] p elimina y esul s we e p esen ed in his ega d. Now
he esponse o he comple e ensemble o soil-b idge sys ems
de ined in sec ion 2.3 is ob ained in ee ib a ion a e he ci -
cula ion o a SML in a wide ange o speeds.
Le us de ine a dimensionless speed KS S
1 e e ed o he
b idges undamen al equency in he absence o soil,
KSS
1=ΩSS
1
ωSS
1
=πV
ωSS
1L(10)
In Eq. (10), ωSS
1is he undamen al ci cula equency o he
b idge wi h SS bounda y condi ions, while ΩSS
1=πV/Lis used
o ep esen he o cing equency o he SML.
The b idges unde analysis a e hose indica ed in Fig-
u e 3, conside ing soil shea wa e eloci ies cs=
{150,220,365,∞}m/s. S uc u e and soil ma e ial damping is
neglec ed in his s udy. Fo each soil-b idge sys em, 70 e enly
spaced alues o KS S
1ha e been selec ed be ween 0.1 and 0.5.
As de ailed in [6] his ange su ices o co e he ci cula ion
speeds expec ed in nowadays ailway sys ems. In o de o accu-
a ely cap u e he a ia ion o he cancella ion condi ions when
SSI is included, wen y addi ional speeds a e compu ed wi hin
he anges [0.85,1.15]KS S
1,ci, whe e KS S
1,ci ep esen s he i h non-
dimensional cancella ion speed o he i s mode in he SS case.
Fo each ci cula ion speed, he maximum e ical displacemen
a he b idge mid-span sec ion is compu ed, once he load has
le he s uc u e. This esul , di ided by he s a ic displace-
men , leads o he dimensionless quan i y R ep esen ed in Fig-
u e 9. As he esponse is ob ained a mid-span and due o he
ime-s ep used in he nume ical in eg a ion, Rbasically co e-
sponds o he con ibu ion o he undamen al mode o he beam
o he o al esponse.
In Figu e 9 all he cu es ob ained o he 72 soil-b idge sys-
ems unde s udy (18 b idges ×4 soil ypes) a e plo ed simul a-
neously. The cu es a e dis inguished using a colou code based
on he alue o pa ame e κ. F om he analysis pe o med, he
ollowing can be concluded: (i) maximum ee ib a ion and
cancella ion condi ions al e na e wi h he inc ease o he load
speed, in he same way ha happens in he absence o soil; (ii)
a om cancella ion condi ions, models wi hou SSI always
p edic a highe esponse han hose including SSI; (iii) as κ
inc eases o so e soils and b idges wi h highe na u al e-
quencies, he ampli ica ion educes be ween wo cancella ion
condi ions; (i ) he cancella ion speeds sligh ly dec ease as he
ela i e s i ness κinc eases. This a ia ion is associa ed wi h
he descen o he s uc u e undamen al equency wi h he soil
lexibili y.
An accu a e p edic ion o he cancella ion speeds is c ucial,
o ins ance, when planning an expe imen al es wi h he aim
o measu ing s uc u al pa ame e s i. e., damping. In Fig-
9