A 2.5D coupled FE-BE model o he p edic ion o ailway induced
ib a ions
P. Gal ´ına,b, S. F an¸coisa, M. Sche enelsa, E. Bonginic, G. Deg andea, G. Lombae a
aDepa men o Ci il Enginee ing, K.U.Leu en, Kas eelpa k A enbe g 40, B-3001 Leu en, Belgium
bEscuela T´ecnica Supe io de Ingenie os, Uni e sidad de Se illa, Camino de los Descub imien os s/n, E-41092 Se illa,
Spain
cSNCF, Di ec ion de l’Inno a ion e de la Reche che, ue de Lond es 45, 75379 Pa is Cedex 8, F ance
Abs ac
G ound ib a ions induced by ailway a ic a g ade and in unnels a e o en s udied by means o
wo-and-hal dimensional (2.5D) models ha a e based on a Fou ie ans o m o he coo dina e in he
longi udinal di ec ion o he ack. In his pape , he need o 2.5D coupled ini e elemen -bounda y ele-
men models is demons a ed in wo cases whe e he p edic ion o ailway induced ib a ions is conside ed.
A ecen ly p oposed no el 2.5D me hodology is used whe e he ini e elemen me hod is combined wi h a
bounda y elemen me hod, based on a egula ized bounda y in eg al equa ion. In he o mula ion o he
bounda y in eg al equa ion, he G een’s unc ions o a laye ed elas ic hal space a e used, so ha no dis-
c e iza ion o he ee su ace o he laye in e aces is equi ed. In he i s case, wo al e na i e models o
a ballas ed ack on an embankmen a e compa ed. In he i s model, he ballas and he embankmen a e
modelled as a con inuum using 2.5D solid elemen s, whe eas a simpli ied beam ep esen a ion is adop ed
in he second model. The ee ield ib a ions p edic ed by bo h models a e compa ed o hose measu ed
du ing a passage o he TGVA a a si e in Reugny (F ance). A e y la ge di e ence is ound o he ee ield
esponse o bo h models ha is due o he ac ha he de o ma ion o he c oss sec ion o he embankmen
is dis ega ded in he simpli ied ep esen a ion. In he second case, he ack and ee ield esponse due
o a ha monic load in a unnel embedded in a laye ed hal space a e conside ed. A simpli ied me hodology
based on he use o he ull space G een’s unc ion in he unnel-soil in e ac ion p oblem is in es iga ed. I
is shown ha he igo ous ini e elemen -bounda y elemen me hod is equi ed when he dis ance be ween
he unnel and he ee su ace and he laye in e aces o he hal space is small compa ed o he wa eleng h
in he soil.
Key wo ds: 2.5D modelling, BEM-FEM coupling, dynamic soil–s uc u e in e ac ion, embankmen ,
unde g ound a ic
Email add ess: ped oga[email p o ec ed]s (P. Gal ´ına,b)
P ep in submi ed o Soil Dynamics and Ea hquake Enginee ing Ap il 3, 2010
1. In oduc ion
The ecen deploymen o he high-speed ain (HST) ne wo k in Eu ope, he USA, and Asia has s im-
ula ed he de elopmen o se e al nume ical models o he p edic ion o ailway induced ib a ions. The
mos gene al amewo k is o e ed by he h ee-dimensional (3D) ini e elemen (FE) me hod and 3D coupled
ini e elemen -bounda y elemen (FE-BE) me hods. These me hods allow accoun ing o he ull coupling
be ween he ack and he soil and can be used o s udy singula poin s in he ack such as ansi ion zones
and swi ches. Fu he mo e, a ime domain o mula ion o e s he possibili y o accoun o non-linea con-
s i u i e beha iou in he FE pa o he model. Mohammadi and Ka abalis [1] ha e p esen ed a equency
domain FE-BE o mula ion whe e he ailway ack is modelled as a g oup o igid sleepe s es ing on a
iscoelas ic hal space. Ande sen and Nielsen [2] ha e p esen ed a FE-BE model in he equency domain
o in es iga e he ee ield esponse p oduced by e ical and ho izon al mo ing loads on an embankmen
and he educ ion o ib a ions by means o ba ie s o soil imp o emen along he ailway ack. Celebi [3]
has used a BE model o compu e he ans e unc ion be ween a poin load on he ack and he ee ield
esponse and o s udy he mi iga ion o ailway induced ib a ions by open enches. The ack is ep e-
sen ed by an equi alen homogeneous laye ha ep esen s a conc e e slab o an asphal laye o e a base
and os p o ec ion laye . O’B ien and Rizos [4] ha e p esen ed a di ec ime domain app oach o s udy
he ansien esponse o a ack-soil sys em due o he passage o ains whe e he sleepe s a e conside ed
o be igid and ails a e coupled o he sleepe s in he e ical di ec ion only. Gal ´ın and Dom´ınguez [5, 6]
and Gal ´ın e al. [7] ha e p esen ed a ime domain o mula ion o s udy he g ound mo ion due o a ain
passage. The model has been alida ed by means o ield measu emen s [5, 6] and has been used o s udy
he in luence o he ballas on a conc e e unde pass s uc u e [5] and he ib a ion isola ion by a loa ing
slab ack sys em [7]. The main disad an age o 3D FE o FE-BE models is hei e y high compu a ional
cos .
As an al e na i e o ull 3D models, so-called wo-and-a-hal dimensional (2.5D) models ha e been p o-
posed o he p edic ion o ailway induced ib a ions. The basic assump ion unde lying he 2.5D me hod-
ology is ha he geome y o he coupled ack-soil sys em is in a ian in he longi udinal di ec ion o he
ack. This allows o a Fou ie ans o m wi h espec o he coo dina e along he ack and leads o a
solu ion in he equency-wa enumbe domain whe e he o iginal 3D p oblem is eplaced by a 2D p oblem
o each wa enumbe . The 2.5D me hodology esul s in a conside able educ ion o he ime equi ed o se
up he model as well as he compu a ion ime. Aub y e al. [8] ha e applied a 2.5D p ocedu e o s udy he
esponse o an in ini ely long beam, coupled o an elas ic hal space, due o a mo ing load. The me hodology
has been applied by Sheng e al. [9, 10] o an in ini e laye ed beam model o he ack, coupled o a laye ed
hal space. Mo e ecen ly, his model has been elabo a ed o accoun o dynamic ain- ack in e ac ion
[11, 12]. In he model p esen ed by Sheng e al. [9, 10, 11, 12], he ac ions a he in e ace be ween he
2
[9, 10, 11, 12, 17, 19, 20]. The ans e unc ions be ween he ack and he ee ield a e compa ed and
subsequen ly used o p edic ee ield ib a ions du ing he passage o he TGVA a a si e in Reugny
(F ance). P edic ions a e compa ed o expe imen al esul s ha ha e been ob ained wi hin he ame o
a benchma k s udy o ganized by SNCF. In he second case, he ack and ee ield esponse due o a
ha monic load in a unnel embedded in a laye ed hal space a e s udied. The esul s o he FE-BE me hod
a e compa ed o hose ob ained om a simpli ied me hodology p esen ed by Hussein e al. [37] ha allows
o an app oxima e solu ion a educed compu a ional cos .
2. The 2.5D coupled FE-BE model
The dynamic in e ac ion be ween a ailway ack and he unde lying soil ( igu e 1a) o a unnel and he
su ounding soil ( igu e 1b) is a p oblem o dynamic soil–s uc u e in e ac ion. A domain decomposi ion
me hod is used o sol e he p oblem, whe e he subdomain Ωb ep esen s he s uc u e and he subdomain
Ωs he soil. The soil–s uc u e in e ac ion p oblem is sol ed by en o cing con inui y o displacemen s and
equilib ium o s esses on he in e ace Σbs be ween bo h subdomains. I is assumed ha he geome y o
he ack o he unnel is in a ian wi h espec o he coo dina e yin he longi udinal di ec ion. The soil is
modelled as a ho izon ally laye ed hal space and, he e o e, in a ian wi h espec o he di ec ion eyas well.
The dynamic soil-s uc u e in e ac ion p oblem is assumed o be linea and all equa ions a e elabo a ed in
he equency domain. The dynamic equilib ium equa ion o he s uc u e is disc e ized by means o 2.5D
ini e elemen s.
(a)
Ωb
Σbs Ωs
x
y
z
(b)
Ωb
Σbs
Ωs
x
y
z
Figu e 1: The 2.5D coupled FE-BE models: (a) a ballas ed ack a g ade and (b) a unnel in a hal space
The equilib ium equa ion o he dynamic soil-s uc u e in e ac ion p oblem is o mula ed in a a ia ional
o m. Fo any i ual displacemen ield bimposed on he s uc u e Ωb, he sum o he i ual wo k o he
in e nal and he ine ial o ces is equal o he i ual wo k o he ex e nal loads:
−ω2ZΩb
b·ρbubdΩ+ZΩb
ǫb( b) : σb(ub)dΩ = ZΩb
b·ρbbbdΩ+ZΓbσ
b· nb
bdΓ+ ZΣbs
b· nb
b(ub)dΓ (1)
4
whe e ubis he displacemen ec o in he s uc u e, ρbbbdeno es he body o ce in he domain Ωb, and
nb
b=σb·nbis he ac ion ec o on a bounda y wi h uni ou wa d no mal ec o nb. T ac ions nb
ba e
imposed on he bounda y Γbσ.
Accoun ing o he equilib ium o s esses on he in e ace Σbs and using a ini e elemen o mula ion
o he in e pola ion o he displacemen ield wi h espec o he coo dina es xand z, equa ion (1) can be
elabo a ed as ollows [22]:
−ω2Mbb +K0
bb −ikyK1
bb −k2
yK2
bb +ik3
yK3
bb +k4
yK4
bb +Ks
bb(ky, ω)˜
¯
ub(ky, ω) = ˜
¯
b(ky, ω) (2)
whe e K0
bb,K1
bb,K2
bb,K3
bb and K4
bb a e he s i ness ma ices, Mbb is he mass ma ix, ˜
¯
b(ky, ω) is he
ex e nal load ec o , and Ks
bb(ky, ω) ep esen s he dynamic soil s i ness ma ix. A ilde abo e a a iable
deno es i s ep esen a ion in he equency-wa enumbe domain. The ini e elemen ma ices Mbb and K0
bb
o K4
bb in equa ion (2) a e independen o he wa enumbe kyand he equency ωand a e only assembled
once. Equa ion (2) is now u he elabo a ed by di iding he ini e elemen deg ees o eedom ˜
¯
ub(ky, ω)
in o in e nal deg ees o eedom ˜
¯
ub1(ky, ω) and deg ees o eedom ˜
¯
ub2(ky, ω) on he soil-s uc u e in e ace:

−ω2

Mb1b1Mb1b2
Mb2b1Mb2b2

+

K0
b1b1K0
b1b2
K0
b2b1K0
b2b2

−iky

K1
b1b1K1
b1b2
K1
b2b1K1
b2b2


−k2
y

K2
b1b1K2
b1b2
K2
b2b1K2
b2b2

+ik3
y

K3
b1b1K3
b1b2
K3
b2b1K3
b2b2

+k4
y

K4
b1b1K4
b1b2
K4
b2b1K4
b2b2


+

0 0
0Ks
b2b2(ky, ω)



˜
¯
ub1(ky, ω)
˜
¯
ub2(ky, ω)
=

˜
¯
b1(ky, ω)
˜
¯
b2(ky, ω)
(3)
The dynamic soil s i ness ma ix ˜
Ks
b2b2(ky, ω) is w i en as:
˜
Ks
b2b2(ky, ω) = ZΣbs
NT
b2˜
s(Nb2)(x, ky, z, ω)dΓ (4)
and compu ed by means o a 2.5D bounda y elemen me hod. The bounda y elemen mesh is chosen o ma ch
he ini e elemen mesh on he soil-s uc u e in e ace Σbs. As a esul , he bounda y elemen in e pola ion
unc ions Ns(x, z) co espond o he ini e elemen shape unc ions Nb2(x, z) on he soil-s uc u e in e ace.
This allows in oducing he bounda y elemen ac ion disc e iza ion in equa ion (4):
˜
Ks
b2b2(ky, ω) = ZΣbs
NT
b2Nb2dΓ˜
¯
s(Nb2)(ky, ω) = ˜
Tq˜
¯
s(Nb2)(ky, ω) (5)
The ans o ma ion ma ix Tq=RΣbs NT
b2Nb2dΓ in equa ion (5) is independen o wa enumbe and e-
quency. The ac ions ˜
¯
s(Nb2)(ky, ω) a e ound by a 2.5D bounda y elemen me hod, based on he in eg al
equa ion ha ela es he displacemen s in he soil domain o he displacemen s and ac ions on he soil-
s uc u e in e ace. The 2.5D bounda y in eg al equa ion has been de i ed by Sheng e al. [29] om he
5
2.5D ecip ocal heo em. In his pape , a egula ized e sion [22] o he 2.5D bounda y in eg al equa ion is
applied, leading o he ollowing bounda y elemen sys em o equa ions:
h˜
T(ky, ω) + Ii˜
¯
us(ky, ω) = ˜
U(ky, ω)˜
¯
s(ky, ω) (6)
whe e ˜
U(ky, ω) and ˜
T(ky, ω) a e ully popula ed unsymme ic bounda y elemen sys em ma ices. The uni
ma ix Ico esponds o he in eg al- ee e m and is no p esen in he case o a bounded medium. The
e alua ion o he bounda y elemen sys em ma ices ˜
U(ky, ω) and ˜
T(ky, ω) in equa ion (6) equi es he
G een’s displacemen s ˜uG
ij(x, ky, z, ω) and ac ions ˜
G
ij (x, ky, z, ω) o he laye ed hal space. These undamen-
al solu ions a e compu ed wi h he di ec s i ness me hod [13, 30] using he MATLAB oolbox EDT 2.1
[31]. As he ac ion ee su ace o he hal space is accoun ed o in hese undamen al solu ions, only he
in e ace Σbs be ween he s uc u e and he laye ed hal space has o be disc e ized.
The ac ions ˜
¯
s(Nb2)(ky, ω) a e ound by sol ing he sys em o equa ions (6):
˜
¯
s(Nb2)(ky, ω) = ˜
U−1(ky, ω)˜
T(ky, ω) + I(7)
An exp ession o he dynamic soil s i ness ma ix ˜
Ks
b2b2(ky, ω) is ound by in oducing he solu ion (7) in
equa ion (5):
˜
Ks
b2b2(ky, ω) = ˜
Tq˜
U−1(ky, ω)˜
T(ky, ω) + I(8)
Once he equilib ium equa ion (3) o he dynamic soil-s uc u e in e ac ion p oblem has been sol ed,
he in eg al ep esen a ion heo em is applied o compu e he adia ed wa e ield om he ac ions ¯
˜
s(ky, ω)
and displacemen s ¯
˜
us(ky, ω) a he soil-s uc u e in e ace:
¯
˜
u (x, ky, z, ω) = ˜
U (x, ky, z, ω)¯
˜
s(ky, ω)−˜
T (x, ky, z, ω)¯
˜
us(ky, ω) (9)
whe e he ma ices ˜
U (x, ky, z, ω) and ˜
T (x, ky, z, ω) ollow om he in oduc ion o he bounda y elemen
disc e iza ion in he in eg al ep esen a ion heo em and he ec o ¯
˜
u (x, ky, z, ω) collec s he displacemen
componen s a n ecei e loca ions.
3. A ballas ed ack on an embankmen
In his sec ion, he p oposed 2.5D FE-BE model is used o p edic he ack and ee ield ib a ions a
a si e in Reugny (F ance) si ua ed along he high speed ailway line LGV A lan ique. The ailway ack in
Reugny is a classical ballas ed ack, si ua ed on op o an embankmen [32]. The ails ha e a UIC60 c oss
sec ion and a e con inuously welded. The ails a e suppo ed by ail pads and ixed wi h clips on win block
conc e e sleepe s [32] wi h a spacing o d= 0.60 m. The conc e e win block sleepe s ha e a o al leng h
lsl = 2.41 m and a e composed o wo ied conc e e blocks wi h a leng h lbl = 0.84 m, a wid h bbl = 0.29 m
a he base, and a heigh hbl = 0.22 m unde he ail. The o al mass o he sleepe s msl = 250 kg. The
6

ack is suppo ed by a ballas laye wi h a hickness hbo abou 0.30 m. The densi y o he ballas laye is
si ua ed in a ange be ween 1400 kg/m3and 1700 kg/m3. The embankmen has a wid h o abou we1 = 6 m
a he op suppo ing he ailway ack, a wid h o we2 = 13 m a he soil’s su ace, and a heigh he= 2 m.
Two al e na i e ack models a e conside ed. Fo bo h models, he geome y o he coupled ack-soil
sys em is assumed o be in a ian wi h espec o he longi udinal di ec ion o he ack, so ha he 2.5D
me hodology can be applied. In he i s model, he ballas and he embankmen a e modelled as an elas ic
con inuum using 2.5D solid elemen s ( igu e 2). The second model is a simpli ied model whe e he ballas
is ep esen ed by dis ibu ed sp ings and dampe s while he embankmen is modelled as an Eule -Be noulli
beam ( igu e 3). Simila simpli ied models o he ballas and he embankmen ha e been equen ly used in
he li e a u e [9, 10, 11, 12, 17, 19, 20]. In he ollowing, he dynamic cha ac e is ics o he ack and he
soil ha ha e been p o ided by he SNCF [32] wi hin he ame o a benchma k s udy a e used o de ine
he pa ame e s o bo h models. Nex , he ack ecep ance and ee ield mobili y ob ained by bo h models
a e compa ed. Finally, a compa ison is made o he ee ield ib a ions du ing he passage o he TGV
A lan ique (TGVA).
3.1. T ack model 1
lsl
y1y2
u 1
k p c p
u 2
usl
βsl
we2
ail
ail pad
sleepe
ballas hb
he
we1
embankmen
x
y
FF02z
Figu e 2: C oss sec ion o model 1 o he ballas ed ack on he embankmen .
Figu e 2 shows he c oss sec ion o he i s model wi h he mesh o he 2.5D solid elemen s ha a e
used o model he ballas and he embankmen . The ails a e ep esen ed by Eule -Be noulli beams wi h a
bending s i ness E I and a mass ρ A pe uni leng h. The ail displacemen s a e deno ed as u 1(y, ) and
u 2(y, ). The posi ions o he ail a e de e mined by y1= 1.145 m and y2= 2.580 m, wi h y2−y1equal
o he ack gauge d. The UIC 60 ails ha e a bending s i ness E I = 6.45 ×106N/m2and a mass pe
uni leng h ρ A = 60.34 kg/m o each ail. The in e nal ene gy dissipa ion in he ail is modelled by a
loss ac o η = 0.05. The ail pads a e modelled as con inuous sp ing-dampe connec ions. The ail pad
7
s i ness k p o a single ail pad is used o calcula e an equi alen s i ness k p =k p/d = 130 ×106N/m2. A
loss ac o η p = 0.23 is used o accoun o in e nal ene gy dissipa ion in he ail pad.
The conc e e sleepe s a e assumed o be igid in he plane o he ack c oss sec ion, so ha he e ical
sleepe displacemen s along he ack a e de e mined by he e ical displacemen usl(y, ) and o a ion
βsl(y, ) a he cen e o g a i y o he sleepe . The sleepe s a e modelled as a uni o mly dis ibu ed mass
msl =msl/d o 417 kg/m. The sleepe ’s o a ional ine ia ρslIsl =ρslIsl/d has been es ima ed as 298 kgm2/m
aking in o accoun he excen ic posi ion o he wo blocks.
The ballas bed is modelled as an elas ic con inuum, using 88 2.5D solid elemen s [22]. The Young’s
modulus o he ballas bed is compu ed om he gi en e ical ballas s i ness pe sleepe kb= 180×106N/m
aking in o accoun he suppo a ea o he win block sleepe 2lblbbl = 0.48 m2. This esul s in a ballas
s i ness Kb= 370 ×106N/m3o a Young’s modulus Eb=Kbhb= 111 ×106N/m2. Addi ionally, a
Poisson’s a io νb= 0.36, a densi y ρb= 1550 kg/m3, and a loss ac o ηb= 1.00 a e assumed o he ballas
laye . The embankmen is modelled as an elas ic con inuum using 528 2.5D solid elemen s. A Young’s
modulus Ee= 170 ×106N/m2, Poisson’s a io νe= 0.36, and a densi y ρe= 1400 kg/m3a e used o he
embankmen .
The soil is modelled as a ho izon ally laye ed elas ic hal space [32], wi h a single laye wi h a hickness
o 2.0 m and a shea wa e eloci y Cs= 211 m/s on op o a hal space wi h a shea wa e eloci y o 403 m/s.
The densi y ρis equal o 1400 kg/m3 o he op laye and equal o 2650 kg/m3 o he unde lying hal space.
The Poisson’s a io νis 0.36 o he op laye and 0.16 o he hal space. The ma e ial damping a io βin
bo h de ia o ic and olume ic de o ma ion has a alue o 0.05 and 0.06 o he op laye and he hal space,
espec i ely.
3.2. T ack model 2
The c oss sec ion o model 2 is shown in igu e 3. Compa ed o model 1 ( igu e 2), he ails, ail pads,
sleepe , and unde lying soil a e modelled simila ly, whe eas a simpli ied model is used o he ballas and
he embankmen .
The ballas bed is now ep esen ed by a se o dis ibu ed linea sp ings and dampe s. The smea ed
ballas s i ness kbis compu ed om he e ical sp ing s i ness kbpe sleepe [N/m] as kb/d and equal o
300 ×106N/m2. The loss ac o ηb= 1.00. The equi alen ballas mass mbis compu ed om he ballas
mass mbsi ua ed unde each sleepe as mb/d. The ballas mass mbis es ima ed om he heigh hbo he
ballas laye and a wid h wb1 =lsl and wb2 = 3 m a he op and he bo om o he ballas laye , espec i ely,
as mb= 0.5ρbhb(wb1 +wb2)bbl. This leads o a alue o 608 kg/m o he equi alen ballas mass mb.
The embankmen is ep esen ed by an Eule -Be noulli beam wi h a bending s i ness EeIe, a o sional
igidi y GeJe, a o a ional ine ia ρeIpe, and a mass ρeAepe uni leng h. This implies ha he c oss
sec ion o he embankmen is now assumed o be igid. The embankmen has a Young’s modulus Ee=
8
l14
lsl
y1y2
u 1
k p c p
u 2
usl
βsl
kbcb
ail
ail pad
sleepe
ballas hb
he
we1
embankmen
ue
βe
Figu e 3: C oss sec ion o model 2 o he ballas ed ack on he embankmen .
170 ×106N/m2, a shea modulus Ge= 107 ×106N/m2, and a densi y ρe= 1400 kg/m3. Based on
he dimensions o he c oss sec ion o he embankmen , he ollowing sec ion cha ac e is ics ha e been
compu ed: he a ea Ae= 19 m2, he bending momen o ine ia Ie= 20.67 m4, he pola momen o ine ia
Ipe= 182.96 m4, and he o sion cons an Je= 22.03 m4.
A he in e ace be ween he embankmen and he soil, elaxed bounda y condi ions a e assumed, so
ha only con inui y o he e ical displacemen s is imposed. Due o he assump ion o a beam model,
he e ical displacemen s usz(x, y, z = 0, ) in he soil a he in e ace Σ a e de e mined by he e ical
displacemen ue(y, ) and o a ion βe(y, ) a he cen e o he in e ace.
I is expec ed ha model 1 leads o mo e accu a e esul s as i allows o a be e app oxima ion o he
s ess dis ibu ion a he in e ace be ween he embankmen and he soil. S eenbe gen and Me ikine [21]
ha e shown ha his is c ucial o an accu a e p edic ion o ailway induced ib a ions.
3.3. T ack ecep ance and ee ield mobili y
In he ollowing, he ack ecep ance and he ee ield mobili y a e compa ed o bo h models. The ack
ecep ance is compu ed om he solu ion o equa ion (3) ha go e ns he dynamic ack-soil in e ac ion,
conside ing an impulsi e poin load on bo h ails. The esponse o he ou e ail a he poin y= 0 whe e
he impulsi e poin load is applied is ob ained subsequen ly by means o an in e se wa enumbe domain
ans o m. Figu es 4a and 4b compa e he modulus and phase o he ack ecep ance o bo h models.
As an addi ional e e ence, he ack ecep ance has also been compu ed wi hou aking in o accoun he
embankmen in model 2. In his case, he in e ace be ween he ballas and he soil is assumed o be igid
o e a wid h equal o he sleepe leng h lsl = 2.41 m. This esul is also shown in igu e 4.
The mos p onounced di e ence be ween he esul s is ound a equencies below 100 Hz. In his
equency ange, he highes alue o he ack ecep ance is ound o model 1 whe e he con inuum model
o he embankmen allows accoun ing o he de o ma ion o he c oss sec ion. In model 2, he de o ma ion
9
o he c oss sec ion o he embankmen is dis ega ded and a subs an ially lowe ack ecep ance is ound.
A be e ag eemen wi h model 1 is ound when he embankmen is dis ega ded in model 2.
(a)
0 50 100 150 200
0
0.5
1
x 10−8
F equency [Hz]
Modulus [m/N]
(b)
0 50 100 150 200
−4
−3
−2
−1
0
F equency [Hz]
Phase [ ad]
Figu e 4: (a) Modulus and (b) phase o he ack ecep ance compu ed by model 1 (da k g ey line), model 2 (g ey line), and
model 2 wi hou embankmen (ligh g ey line).
(a)
0 50 100 150
0
1
2
3
4x 10−7
F equency [Hz]
Mobili y [m/sN]
(b)
0 50 100 150
0
2
4
6
x 10−8
F equency [Hz]
Mobili y [m/sN]
(c)
0 50 100 150
0
0.5
1
x 10−8
F equency [Hz]
Mobili y [m/sN]
(d)
0 50 100 150
0
1
2
x 10−9
F equency [Hz]
Mobili y [m/sN]
Figu e 5: F ee ield mobili y a (a) 2 m, (b) 12 m, (c) 32 m, and (d) 72 m om he ou e ail compu ed om model 1 (da k
g ey line), model 2 (g ey line), and model 2 wi hou embankmen (ligh g ey line).
Based on he solu ion o equa ion (3), he adia ed wa e ield due o an impulsi e poin load on bo h ails
is compu ed. The ee ield mobili y is ob ained as he in e se wa enumbe domain ans o m o he ee
ield eloci y in he wa enumbe - equency domain. Figu e 5 compa es he ee ield mobili y a a dis ance
o 2 m, 12 m, 32 m, and 72 m om ail 2 as compu ed wi h model 1 and model 2. The poin loca ed a 2 m
om he ou e ail is si ua ed on he embankmen ( igu e 2). The ee ield mobili y o he case whe e he
embankmen is omi ed in model 2 is shown as well.
A e y la ge di e ence is ound be ween he esul s o model 1 wi h he solid embankmen model and
model 2 whe e he embankmen is ep esen ed by an Eule -Be noulli beam. The assump ion o an Eule -
Be noulli model o he embankmen leads o high ac ions nea he edges o he embankmen , whe eas in
he case o he solid embankmen model, he ac ions a e dis ibu ed mo e smoo hly along he in e ace.
When he wa eleng h in he soil is e y la ge compa ed o he wid h o he in e ace be ween he embankmen
and he soil, he in luence o he ac ion dis ibu ion on he ee ield mobili y is small. In he p esen case
10
x
z
2 m
dep h
R
Figu e 10: C oss sec ion o he unnel.
4.2. The ack ecep ance and ack-soil ans e unc ions
In he ollowing, he ack ecep ance and ans e unc ions be ween he ack and he soil a e compu ed
by sol ing equa ion (3) ha go e ns he dynamic in e ac ion be ween he ack, he unnel and he soil.
The load consis s o an impulsi e load on bo h ails.
Figu es 11a and 11b show he modulus and phase o he ack ecep ance and he co esponding esponse
o he slab in he equency ange be ween 1 Hz and 100 Hz . A equencies below he esonance equency
o he slab mass on he s i ness o he esilien suppo , he ail and slab mo e in phase. A he esonance
equency, a peak is obse ed in he modulus o he esponse o he slab and he ail ( igu e 11a), while he
phase equals π/2. A highe equencies, he ail is dynamically uncoupled om he slab and he esponse
o he slab becomes much smalle .
(a)
20 40 60 80 100
−200
−190
−180
−170
−160
−150
F equency [Hz]
Displacemen [dB e m/N]
(b)
20 40 60 80 100
−4
−3
−2
−1
0
F equency [Hz]
Phase [ ad]
Figu e 11: (a) Modulus and (b) phase o he ack ecep ance (black line) and he co esponding esponse o he slab (g ey
line).
Figu e 12a shows he displacemen a a poin {0,0,−10 m}Tloca ed in he hal space a a heigh o
10 m abo e he unnel axis and a poin {0,0,0}Ta he ee su ace, which is a a heigh o 20 m abo e
he unnel axis. The e ical displacemen s show clea oscilla ions as a unc ion o he equency due o
in e e ence be ween di e en ypes o wa es ha a e adia ed in o he soil by he unnel. Gup a e al.
17

[23] ha e shown ha o a unnel embedded in a ull space, he equency spacing be ween he oscilla ions
can be compu ed om he shea wa e eloci y Cs, he longi udinal wa e eloci y Cpand he dis ance
be ween he obse a ion poin and he sou ce as CsCp/( (Cp−Cs)). In he p esen case o a laye ed
hal space, he in e e ence pa e n is mo e complica ed, howe e , as i esul s om he in e ac ion be ween
Rayleigh wa es and shea and longi udinal wa es a eling in he di e en soil laye s. Figu e 12b shows
he e ical displacemen a wo poin s {10 m,0,0}Tand {20 m,0,0}Tloca ed a he ee su ace a a
ho izon al dis ance o 10 m and 20 m om he unnel axis, espec i ely. The esul s in igu es 12a and 12b
show ha he la ges esponse is no necessa ily ound a he smalle dis ance om he unnel axis due o
he in e e ence and he di ec i i y o he wa es emi ed by he unnel.
(a)
20 40 60 80 100
−290
−270
−250
−230
−210
F equency [Hz]
Displacemen [dB e m/N]
(b)
20 40 60 80 100
−290
−270
−250
−230
−210
F equency [Hz]
Displacemen [dB e m/N]
Figu e 12: Ve ical displacemen in he soil a (a) wo poin s {0,0,−10 m}T(black line) and {0,0,0}T(g ey line) loca ed
abo e he unnel axis and (b) wo poin s {10 m,0,0}T(black line) and {20 m,0,0}T(g ey line) a he ee su ace.
4.3. Simpli ied me hodology
The p oposed 2.5D me hodology allows o igo ously accoun o he laye ed s uc u e o ho izon ally
s a i ied soils. This equi es a compu a ional cos which may be highe han o o he models whe e
app oxima e me hods a e used o accoun o he laye ed s uc u e o he soil. Such an app oach has
ecen ly been p oposed by Hussein e al. [37] o ex end he capabili ies o he PiP model ha o iginally
p edic s ib a ions om unde g ound a ic o a unnel embedded in a ull space [26, 27, 28]. Fi s , he
unnel-soil in e ac ion p oblem in equa ion (3) is sol ed conside ing he unnel embedded in a homogeneous
ull space wi h he same p ope ies as he laye in which he unnel is loca ed. This implies ha he dynamic
s i ness ma ix o he soil in equa ion (5) is compu ed by sol ing he bounda y elemen sys em o equa ions
(6) in e ms o he ull space G een’s unc ions. Second, he displacemen s and ac ions a he unnel-soil
in e ace a e used o compu e he ee ield esponse acco ding o equa ion (9) using he G een’s unc ion o
a laye ed hal space [31, 38]. In his subsec ion, i is in es iga ed unde which condi ions a simila app oach
can be ollowed o educe he compu a ional cos in he 2.5D amewo k.
Fi s , he esul s p esen ed in igu es 12a and 12b a e ecompu ed wi h he app oxima e me hod, based
on he solu ion o he dynamic unnel-soil in e ac ion p oblem in a homogeneous hal space. Figu e 13
compa es he e ical displacemen in he soil a 10 m and 20 m ( he ee su ace) abo e he cen e o he
unnel p e iously shown in igu e 12a wi h he app oxima ion based on he ull space solu ion. I can be
18
obse ed ha a ela i ely good app oxima ion o he solu ion has been ob ained. A simila good ag eemen
(a)
20 40 60 80 100
−290
−270
−250
−230
−210
F equency [Hz]
Displacemen [dB e m/N]
(b)
20 40 60 80 100
−290
−270
−250
−230
−210
F equency [Hz]
Displacemen [dB e m/N]
Figu e 13: Ve ical displacemen in he soil a wo poin s (a) {0,0,−10 m}Tand (b) {0,0,0}Tloca ed abo e he unnel axis
compu ed wi h he o iginal 2.5D me hodology (black line) and he simpli ied 2.5D me hodology (g ey line).
is obse ed o he wo poin s {10 m,0,0}Tand {20,0,0}Ta he ee su ace ( igu e 14) o which he
esponse has been p e iously shown in igu e 12b.
In his case, he p oposed simpli ied me hodology allows o a good app oxima ion o he esponse in
he soil a a conside ably educed compu a ional cos . I is clea , howe e , ha he ee su ace and he
laye ed s uc u e o he soil can only be dis ega ded in he solu ion o he unnel-soil in e ac ion p oblem
when he ee su ace and he laye in e aces a e su icien ly a om he unnel. In o de o demons a e
he limi a ions o he p oposed simpli ied me hodology, a second case is conside ed whe e he cen e o he
unnel is si ua ed a a dep h o 5.5 m below he ee su ace. In his case, he dis ance om he unnel apex
o he ee su ace is 2.5 m and he dis ance om he unnel apex o laye in e ace is 0.5 m.
(a)
20 40 60 80 100
−290
−270
−250
−230
−210
F equency [Hz]
Displacemen [dB e m/N]
(b)
20 40 60 80 100
−290
−270
−250
−230
−210
F equency [Hz]
Displacemen [dB e m/N]
Figu e 14: Ve ical displacemen in he soil a wo poin s (a) {10 m,0,0}Tand (b) {20 m,0,0}Tloca ed a he ee su ace
compu ed wi h he o iginal 2.5D me hodology (black line) and he simpli ied 2.5D me hodology (g ey line).
Fig. 15 compa es he e ical displacemen in he soil a a heigh o 3.5 m abo e he cen e o he unnel
(a he in e ace be ween he laye and he hal space) and a a heigh o 5.5 m ( he ee su ace) as compu ed
wi h bo h me hods. In his case, he ag eemen be ween bo h esul s is less good and di e ences highe han
10 dB a e obse ed. An accu a e solu ion o he unnel-soil in e ac ion p oblem is he e o e only ob ained
by he o iginal 2.5D me hodology.
Figu e 16 compa es he e ical displacemen a wo poin s {10 m,0,0}Tand {20,0,0}Ton he ee
su ace. Fo hese poin s, he simpli ied me hodology esul s in a be e app oxima ion han o he poin s
on op o he unnel in igu e 13. Based on a mo e ex ensi e pa ame ic s udy, i has been concluded ha he
19
(a)
20 40 60 80 100
−270
−250
−230
−210
−190
F equency [Hz]
Displacemen [dB e m/N]
(b)
20 40 60 80 100
−270
−250
−230
−210
−190
F equency [Hz]
Displacemen [dB e m/N]
Figu e 15: Ve ical displacemen in he soil a wo poin s (a) {0,0,−2}Tand (b) {0,0,0}Tloca ed abo e he unnel axis
compu ed wi h he o iginal 2.5D me hodology (black line) and he simpli ied 2.5D me hodology (g ey line) o a unnel dep h
o 5.5m.
(a)
20 40 60 80 100
−280
−260
−240
−220
−200
F equency [Hz]
Displacemen [dB e m/N]
(b)
20 40 60 80 100
−280
−260
−240
−220
−200
F equency [Hz]
Displacemen [dB e m/N]
Figu e 16: Ve ical displacemen in he soil a wo poin s (a) {10 m,0,0}Tand (b) {20 m,0,0}Tloca ed a he ee su ace
compu ed wi h he o iginal 2.5D me hodology (black line) and he simpli ied 2.5D me hodology (g ey line) o a unnel dep h
o 5.5 m.
simpli ied me hodology is alid when he dis ance be ween he unnel and he su ace o he hal space spans
a numbe o wa eleng hs so ha he e ac ed wa e does no signi ican ly modi y he unnel-soil in e ac ion
p oblem. The same applies i he unnel is loca ed nea an in e ace be ween wo soil laye s.
5. Conclusions
In his pape , he need o 2.5D coupled ini e elemen -bounda y elemen models is demons a ed in wo
cases whe e he p edic ion o ailway induced ib a ions is conside ed.
In he i s case, wo al e na i e models o a ballas ed ack on an embankmen a e conside ed. In
he i s model, 2.5D solid elemen s a e used o model he ballas and he embankmen as a con inuum,
whe eas in he second model a simpli ied ep esen a ion is used. The ack ecep ance and ee ield mobili y
a e ound o di e conside ably a highe equencies whe e di e ences in he ac ion dis ibu ion a he
in e ace be ween he embankmen and he soil a e esol ed by he smalle wa eleng hs in he soil. A
compa ison o he p edic ed and measu ed ee ield eloci y shows ha he con inuum model o he ballas
and he embankmen leads o a ela i ely good app oxima ion a low equencies whe e he quasi-s a ic
con ibu ion o he esponse domina es. The less good ag eemen a high equencies may be due o an
o e es ima ion o he dynamic axle loads.
In he second case, he esponse o a unnel embedded in a laye ed hal space is conside ed. The p oposed
20
2.5D me hodology allows igo ously accoun ing o he laye ed s uc u e, bu may lead o ela i ely high
compu a ional cos . I has he e o e been in es iga ed unde which condi ions he p oposed 2.5D me hod-
ology can be eplaced by a simpli ied me hodology based on he use o he ull space G een’s unc ions in
he solu ion o he unnel-soil in e ac ion p oblem.
6. Acknowledgemen s
The i s au ho would like o hank he Minis e io de Educaci´on y Ciencia o Spain (JC2008-00136) and
he Jun a de Andaluc´ıa (IAC08-II-3343) o hei inancial suppo o his esea ch s ay a he Depa men
o Ci il Enginee ing o he K.U.Leu en.
The second and hi d au ho a e holde o a Pos doc o al Fellowship o he Resea ch Founda ion -
Flande s (FWO-Vlaande en). The inancial suppo o FWO-Vlaande en is kindly acknowledged.
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1743–1761.
23
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