A 2.5D coupled FE-BE model for the prediction of railway induced vibrations

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. Re e ences [1] Mohammadi M, Ka abalis DL. Dynamic 3-D soil- ailway ack in e ac ion by BEM-FEM Ea hquake Enginee ing and S uc u al Dynamics, 24 (1995) 1177–1193. [2] Ande sen L, Nielsen SRK. Reduc ion o g ound ib a ion by means o ba ie o soil imp o emen along a ailway ack Soil Dynamics and Ea hquake Enginee ing, 25 (2005) 701–716. [3] Celebi E. 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Modelling o ailway ack and ehicle/ ack in e ac ion a high equencies Vehicle Sys ems Dynamics, 22 (1993) 209–262. [35] In e na ional O ganiza ion o S anda diza ion. ISO 2631-2:1999: Mechanical ib a ion and shock - E alua ion o human exposu e o whole-body ib a ion - Pa 2: Vib a ion in buildings (1 o 80 Hz), 1999. 22 [36] Deu sches Ins i u ¨u No mung. DIN 45672 Teil 2: Schwingungsmessungen in de Umgebung on Schienen e keh swegen: Auswe e e ah en, 1995. [37] Hussein MFM, Hun HEM, Rikse L, Gup a S, Deg ande G, Talbo JP, F ancois S, Sche enels M. Using he PiP model o as calcula ion o ib a ion om a ailway unnel in a mul i-laye ed hal -space 9 h In e na ional Wo kshop on Railway Noise, Munich, Ge many, 2007. [38] Kausel E and Roesse JM. S i ness ma ices o laye ed soils Bulle in o he Seismological Socie y o Ame ica, 71 (1981) 1743–1761. 23 View publica ion s a sView publica ion s a s