MODELLING THE INTERACTION EFFECTS OF THE HIGH-SPEED TRAIN TRACK BRIDGE SYSTEM USING ADINA

MODELLING THE INTERACTION EFFECTS OF THE HIGH-SPEED TRAIN TRACK BRIDGE SYSTEM USING ADINA ABSTRACT Constança Rigueio Depatment of Civil Engineeing, Polytechnic Institute of Castelo Banco Potugal Calos Rebelo, Luis Simões da Silva Depatment of Civil Engineeing, Univesity of Coimba Potugal Email: cebelo@dec.uc.pt The main pupose of this pape is to pesent some esults concening the investigation of the effects of the vehicle-bidge inteaction in simply suppoted medium span viaducts, including the modelling of the ballast-ail tack system. This system is modelled using the ail stiffness in vetical and longitudinal diections, including the ail pad and the sleepes, and the ballast as a system of vetical spings, dampes and masses. A simplified vehicle model poposed by the Euopean Railway Reseach Institute (ERRI D214, 1999), taking into account the vehicle pimay suspension chaacteistics and the mass of the bogie is used with the contact algoithm implemented in the softwae ADINA to evaluate the esponse acceleation of a simply suppoted medium span concete viaduct. The esults ae compaed with those fom the moving loads model fo a wide ange of tain speeds. INTRODUCTION Due to the inceasing inteest in the high speed ailway tanspotation, the eseach on the vibation of bidges unde moving vehicles has been gowing much then eve. The investigation stated to be analytical, with appoximate solutions fo some simple but fundamental poblems (fo example, Fyba, 1999, Biggs, 1964). Nowadays, the investigation use moe ealistic and complex bidge and vehicle models to analyse the bidge vibations (Yang et al, 2004, Xia 2003). Howeve, in these woks, the effects of the ballasted tacks wee only patially accounted fo, o they have been completely neglected. The most fequently used model fo the dynamic calculations of the tain is the moving loads model. This model does not take into account the inetial effects of the tain masses and the stiffness and damping of the pimay suspension of the vehicle. As a consequence, the tain is modelled as a seies of point loads with constant values tavelling ove the bidge at a constant speed. The committee D214 of the Euopean Reseach Institute (ERRI) epots that, fo shot span bidges, the vetical acceleations of the deck pedicted by the moving loads model each unealistic values, which ae much highe than the limit of 3,5 m/s 2 imposed fo the bidges with ballasted tack by the Citeia fo Taffic Safety (EN1990-pAnnex A2, 2002). This committee also pointed out that, fo the esonance speeds, the pedicted bidge esponse when using tain/bidge inteaction is significantly lowe than the esponse acceleation obtained with the moving loads model. This ongoing eseach aims to highlight and evaluate those diffeences concening the models esults. Theefoe, the main pupose of this pape is to analyse the influence of tain/bidge Poto - Potugal, 24 26 July 2006 1

inteaction in simply suppoted medium span viaducts, including the modelling of the ballasted ail tack system on the bidge. Compaisons of the numeical simulations esults ae caied out fo a eal bidge, using the contact algoithm implemented in the softwae ADINA. To obtain the vehicle/bidge inteaction effects, the vehicle is usually modelled as a twodegee-of-feedom dynamic unit (see Figue 6), o as a igid beam suppoted on two-spingdampe systems to take into account the pitching effect of the vehicle (Yang, 2004). In this eseach only the fist simplified model is consideed to analyse the influence on the bidge esponse displacements and acceleations. The inclusion of the ballasted ail tack system in the numeical model allows of taking into account the longitudinal distibution of the loads though the sleepes and ballast laye. Seveal models (Figues 1 to 3) ae consideed fo the evaluation. In geneal, they include vetical spings, dampes and masses, which ae inteposed between the vehicle and the stuctue. Although both the dynamic esponse of the tack and the bidge esponse was computed, only the esponse of the bidge is pesented hee. DYNAMIC MODELS OF THE RAILWAY BALLASTED TRACK The ailway ballasted tack model is made of seveal elements which epesent the ails, the sleepes, the connections between ails and sleepes, and the ballast. The ails ae an impotant component in the tack stuctue, since they tansfe the wheel loads and distibute them ove the sleepes and suppots, guide the wheels in the lateal diection, povide a smooth unning suface and distibute acceleation and baking foces ove the suppots. In Euope the typical ail used in the high speeds lines is the flat-bottom ail, UIC60. The connections ail/sleepe ae mateialized by fastenings and ail pads. This system povides the tansfe of the ail foces to the sleepes, damps the vibations and impacts caused by the moving taffic and etains the tack gauge and ail inclination within cetain toleances. The sleepes ae elements positioned just below the ails usually made of timbe o concete. They povide suppot fo the ail, sustain ail foces and tansfe them as unifomly as possible to the ballast. They peseve tack gauge and ail inclination and povide adequate electical insulation between both ails. The sleepes must be esistant against mechanical and weatheing influences ove a long peiod. Finally, the ballast bed consists of a laye of a coase-sized, non cohesive, ganula mateial. Taditionally angula, cushed, had stones and ocks have been consideed good ballast mateials. The intelocking of ballast gains and thei confined condition inside the ballast bed pemit the load distibuting function and the damping. They also povide the lateal and longitudinal suppot of the tack, as well as the daining effect. The thickness of the ballast bed should allow the sub gade to be loaded as unifomly as possible. The usual depth fo the ballast is about 0.3 metes measued fom the undeside of the sleepe. In the ealy studies, the models of the ballasted tack wee developed in ode to investigate the tain/tack inteaction poblem. A eview of these studies is pesented in Fyba, 1999. In the 1900 s Timoshenko published papes on the stength of ails; late on, Inglis, was active in this issue. Knothe, 1993, and Popp, 1999, pesented an oveview of existing tacks models in the field of tain/tack inteaction. The main pupose of these studies was to evaluate the deflections of the tack and the vetical displacements of the vehicles, while the contact foce 2 Poto - Potugal, 24 26 July 2006

wheel/ail is evaluated in the calculations. Complete models of the vehicles and the effects of the wheel and ail iegulaities wee also investigated. A lage vaiety of ballasted tack models has been investigated, fom simple 2D model, whee a single ail is modeled as an infinite Benoulli-Eule o Timoshenko beam esting on suppots defined by spings, dampes and point masses, to moe complex 3D models, whee both ails ae taken into account and bending and shea defomation of the sleepes ae included. In these models, the ballast bed is included though vetical sping and dampe elements. Some of these models conside the mass of the ballast as a point mass located below each sleepe and its value is taken elative to the amount of stiffness and damping. Futhemoe, shea spings and dampes may inteconnect these masses (Zhai, 2003). The values fo the mechanical popeties of the tack components, such as mass, inetia and elasticity, ae mentioned as an essential input fo dynamic tack behavio and, of couse, fo the study of the inteaction between tain and tack. Since the focus of the pesent wok is on the influence of the ballast tack on the vetical vibations of the ailway bidges obtained by compaison of the numeical esults with the dynamic esponse obtained fom field measuements, only the 2D tacks models wee consideed, neglecting unimpotant tosion effects. Fo this pupose, thee diffeent models of ballasted tacks ae pesented in Figues 1, 2 and 3 (Man, 2002), (Yang, 2004), (ERRI, 1999). In Model I the ails ae consideed as infinite long beams with in-plane and out-of-plane flexual stiffness as well as axial stiffness. The linea spings and dampes on the vetical and longitudinal diections epesent the ballast. These thee models ae included in the finite element model of the bidge, which is acted by moving loads epesenting eal tains. The model paametes emain constant along the tack, despite some deviations due to constuction and maintenance woks. In the othe two models, Model II and Model III, the connections between ail and sleepe ae included as linea spings and viscous dampes acting in paallel. Thei elastic and damping popeties ae mainly detemined by the popeties of the mateial and the manufactuing pocesses. The sleepes ae included as igid bodies with point mass. The ballast bed is included as discete linea spings and viscous dampes. In Model III the mass of the ballast is included as point mass instead of distibuted mass, and spings and dampes ae used to simulate the connection between bidge and ballast (ERRI, 1999). The paametes used in Model II wee obtained fom (Man, 2002). The values of the mechanical popeties fo each model ae included in Table 1 to Table 3. Figue 1: Ballasted tack Model I (Yang, 2004). Poto - Potugal, 24 26 July 2006 3

Figue 2: Ballasted tack Model II (Man, 2002). Rails Railpads K p C p M s Sleepe K bs C bs M b Ballast K bb C bb Bidge Figue 3: Ballasted tack Model III (ERRI, 1999). Components of the tack model I Notation Value Units Rail UIC60 Young Modulus E 210E+09 N/m 2 Density ρ 7850 kg/m 3 Flexual moment of inetia Sectional aea Pe unit of length Vetical stiffness Pe unit of length Vetical damping Pe unit of length Hoizontal stiffness Pe unit of length Hoizontal damping Ballast Table 1: Popeties of tack Model I (Yang, 2004). I 3055E-08 m 4 A 76.9E-04 m 2 K bv 104E+06 N/m C 50E+03 N.s/m bv K 104E+06 N/m bh C 50E+03 N.s/m bh 4 Poto - Potugal, 24 26 July 2006

Components of the tack model II Notation Value Units Rail UIC60 Young Modulus E 210E+09 N/m 2 Density ρ 7850 kg/m 3 Flexual moment of inetia Sectional aea Vetical stiffness Vetical damping Mass Length between sleepes Vetical stiffness Vetical damping I 3055E-08 m 4 A 76.9E-04 m 2 Connection ail/sleepe K 300E+06 N/m Sleepe Ballast Table 2: Popeties of tack Model II (Man, 2002). p C p 80E+03 N.s/m M s 300 kg d 0.60 m s K b 120E+06 N/m C 114E+03 N.s/m b Components of the tack model III Notation Value Units Rail UIC60 Young Modulus E 210E+09 N/m 2 Density ρ 7850 kg/m 3 Flexual moment of inetia Sectional aea Vetical stiffness Vetical damping Mass Length between sleepes Vetical stiffness ballast/sleepe Vetical damping ballast/sleepe Mass Vetical stiffness bidge/ballast Vetical damping bidge/ballast I 3055E-08 m 4 A 76.9E-04 m 2 Connection ail/sleepe K 500E+06 N/m Sleepe Ballast p C p 200E+03 N.s/m M s 290 kg d 0.60 m s K bs 538E+06 N/m C 120E+03 N.s/m bs M 412 kg b K 1000E+06 N/m bb C 50E+03 N.s/m Table 3: Popeties of tack Model III (ERRI, 1999). bb Poto - Potugal, 24 26 July 2006 5

THE INTERACTION MODELLING USING ADINA In the pesent study, only plane models ae consideed. The vetical displacements of the ailway vehicles ae analyzed and the two ails ae effectively teated as a single one in the subsequent analysis. Figue 4 shows a typical vehicle element with a 2-DOF acting on the bidge/tack system. The uppe and lowe beam elements modeling the ail and the bidge deck espectively ae inteconnected by the ail tack model as mention befoe. Since the high speed tain ICE2 is consideed fo the analysis, it must be assumed that thee ae 56 moving vehicles in diect contact with the bidge o with the platfom that must be modeled befoe and afte the bidge itself. Figue 4: Vehicle/tack/bidge inteaction model. Same Consideations About the Contact Algoithm The contact algoithms in ADINA ae descibed in its manual and in othe efeences (Bathe, 1985; Bathe, 1996). This algoithm is pesented as a solution pocedue fo the analysis of plana o axisymmetic contact poblems involving sticking, fictional, sliding and sepaation whee lage defomation is pesent. The contact conditions ae imposed using the total potential of the contact foces with the geometic compatibility conditions along the contact sufaces, which leads to suface tactions fom the extenally applied foces. Same key aspects of the pocedue ae the contact matices, the use of distibuted tactions on the contact segments fo deciding whethe a node is sticking o eleasing and the evaluation of the nodal point contact foces. The numbe of equations due to the contact conditions is dynamically adjusted to solve two equations fo each node in contact, if the node is in sticking condition, and one equation, if the node is in sliding condition, which is the case fo the pesent study. Figue 5: Schematic epesentation of the contact poblem, with typical 2-D contact sufaces. 6 Poto - Potugal, 24 26 July 2006

Figue 5 shows schematically the contact poblem. Although in this figue only two bodies ae epesented, the algoithm can be applied to situations wehee moe bodies ae in contact. The figue shows two geneic bodies which come into contact along the contact sufaces, the contacto suface fom A to B, and the taget suface fom C to D, in the time t. Due to this contact, foces ae developed acting along the egion of contact, the contact segment. These foces ae equal and opposite and can be tansfomed into nomal and tangential vectos. The nomal tactions can only exet compessive action, and tangential tactions satisfy a law of fictional esistance, if it exists. Besides the elastic foces consideed in the static analysis, the dynamic analysis of the contact foces includes the inetia and the cinematic foce effects of each body, whose intefaces conditions must be satisfied at all instances of time, equiing displacement, velocity, and acceleation compatibility between the contacting segments. The softwae ADINA has explicit and implicit methods fo the computations at each time step. In the pesent case we have applied an implicit method of time integation of the equilibium equations, as we will discuss late on. Fo the high speed tain ICE2 model, a contact suface fo each vehicle is needed in addition to the contact suface fo the bidge/ail. This means a total numbe of 57 contact sufaces and 56 contact pais. The Vehicle Model A spung mass model, denoted in the bibliogaphy as a simplified vehicle model, defines the vehicle model. The spung mass model is conceived as a two-node system, with each node associated with lumped masses. The stiffness and damping of the suspension, denoted by C and k v, espectively, coespond to the pimay suspension of the tain vehicle. The mass of the wheel is M w and the lumped mass fom the ca body is M v (equal to a quate of the mass of the ca body and bogie mass). The 56 vehicles tavel at a constant speed v i. v Figue 6: Spung mass model used in ADINA. The vetical displacements of the masses M w and M v ae z 1 and z 2, espectively, and they ae defined as being vetical and measued fom the static equilibium positions. Let F ci be the contact foce between the ith vehicle and the ail. It can be expess in tems of static foce F = M + M g, with g denoting the acceleation F, the total weight of the two masses, ( ) wi of gavity, and the vaiation of contact foces dependent vaiation of the contact foce: wi v w Fci coesponding to the inteaction o time Poto - Potugal, 24 26 July 2006 7

Fci = Fwi + Fci (1) The equations of motions fo the spung mass model shown in Figue 6 can be witten as follows (Biggs, 1964): Mw 0 ɺɺ z1 Cv Cv zɺ 1 K v K v z1 Fc 0 M + v z 2 Cv C + = v z 2 K v K ɺɺ ɺ v z2 0 (2) The contact foce must be positive, Fc 0, to exclude the possibility of lost contact between the wheels and the ail. Solution of the Equilibium Equations Thee ae seveal methods available fo the time integation in softwae ADINA. The explicit time integation method implemented coesponds to the cental diffeence scheme. The main advantage of this method is the mass matix is diagonal and the solution of the displacements at time t + t does not involve a tiangula factoization of the stiffness matix in the step by step solution. Howeve, the cental diffeence method is a scheme only conditionally stable, which means that it equies the use of a time step t smalle than a citical time step t. If a time step lage than tc is used, the integation becomes unstable and the eos in the computation gow and make the calculations wothless in most cases. Consequently, the use of this scheme is limited to cetain poblems. The implicit methods of the Newmak family methods, the Wilson-θ method and the Composite method, which includes a thid paameteα fo the esolution of the equilibium equations, ae available in ADINA. The Composite method is a time integation scheme fo nonlinea analysis, whee the displacements, velocities and acceleations ae solved fo a time t α t α 0,1. Whenα = 0.5, it is the same as the Newmak method. Since these +, whee [ ] implicit methods ae unconditionally stable, the step t can be selected independent of stability consideations and thus they can esult in a substantially saving of computational effot. Howeve, these schemes equie the tiangulaization of the stiffness matix. Accoding to EN1990-pAnnex A2, 2002, fo the detemination of the maximum deck acceleation, the fequencies in the dynamic analysis should be consideed up to a maximum of: i) 30 Hz ; ii)1.5 times the fequency of the fist mode shape of the stuctual element being consideed, including at least the fist thee modes shapes. Fo that eason, in addition to being unconditionally stable, when only low mode esponse is of inteest it is desiable that the integation scheme has the ability to damp out the spuious paticipation of the highe modes though numeical dissipation. The Wilson θ method and the Newmak family of methods, esticted to paamete values of 1 2 γ > and β 0.25 ( γ 1 2) 2 +, whee the amount of dissipation, fo a fixed time step t, is inceased by inceasing γ, possess this advantage. On the othe hand, the dissipative popeties of this family of algoithms ae consideed less efficient then those of the Wilson θ method, since the lowe modes ae stongly affected. The Wilson θ method, with θ = 1.4, is highly dissipative at the highest modes, unconditionally stable and accuate when t Tn 0.01, whee T n is the lowest vibation peiod to take into account in the stuctual esponse analysis, (Bathe, 1996). c 8 Poto - Potugal, 24 26 July 2006

Consideing all the pos and contas of the methods, the dynamic esponse of the bidge was obtained by using the Wilson θ method. A NUMERICAL EXAMPLE Bidge Model In ode to investigate the influence of the inteaction in the dynamic behavio, a simply 4 suppoted bidge was consideed, with a span length of 23.5 m, 0.4835 m of coss sectional moment of inetia and 21080 kg m of mass pe unit length along the axis. The Young modulus was taken equal to 40 GPa. This leads to a fundamental fequency nea to 2.70 Hz, a static deflection at mid span due to the Load Model 71 equal to δ = 23.04mm and a L δ 1020. The Load Model 71 is a static load patten poposed by the Euocode 1 (EN1991-2, 2003), fo the design of ailway bidges. Concening the damping, the Rayleigh matix was used, that is, C = α M + β K, with constants α = 0.235 and β = 3.640E 04, which coespond to a damping atio, fo the fist fequency of the bidge of about 1%. This is the value that is ecommended by the Euocode 1 fo pestessed concete ailway bidges with span geate than 20 m. Tain Speeds Accoding to Euocode 1, a seies of tain speeds up to the maximum design speed must be consideed. The maximum design speed is 1.2 max imum live speed at the site. Calculations wee made fo a seies fom 140 km / h up to maximum design speed of 300 km / h. The speed step used fo these calculations was 5 km / h, smalle speed step was adopted with the pupose to match with the esonant speeds. The High-speed Tain ICE2 The high-speed tain ICE2 consists of a total of 14 caiages including two powe cas located in the font and in the ea of the tain. It is a conventional tain, that is, it possesses two bogies fo each caiage. The Figue 7 epesents pat of the ICE2 tain, in plan view, in the moving foce model and in the moving vehicle model used fo the inteaction computations. Each bogie has two axles epesented by two foces; the weigh of each axle is 195 kn fo the powe cas, and 112 kn fo the intemediate caiages. Figue 7: TheICE2 tain. a) Plan; b) Moving foce model; c) Moving vehicle model. Poto - Potugal, 24 26 July 2006 9

The passage of successive loadings with unifom spacing, in this case is 26.40 m, can excite the stuctue and ceate esonance. This effect can be pointed out in tems of a citical speed. In the pesent case the esonance speed can be calculated using the following fomula Dk vci. = n 0, i = 1, 2,3,..., n (3) i Fo i = 1, Dk = 26.40 m and n0 = 2.70 Hz the citical speed occu fo 257 km / h. If i > 1 is consideed, the citical speeds will be infeio to the studied speed ange. The Moving Loads Model Consideing the moving load model olling on in the ange of speeds mentioned above, fo the seveal models of bidge, with and without the tack ballast model, maximum values of the displacements and acceleations wee computed. As we can see in Figue 8 and Figue 9, the esponses of the bidge, fo the maximum values of displacements and acceleations ae vey simila. Consideing these esults, it can be concluded that the pesence of the ailway tack model does not influence the maximum esponse of the bidge fo the moving load model. Howeve, the use of the ballast tack model suppesses the contibution of fequency components of the bidge esponse acting as a low pass filte. This effect can be shown in Figue 10, fo a lowe speed, 140 km / h, whee the epesentation of the acceleations at the mid span of the bidge is made in the fequency domain. The contibution of the fequencies in the ange 10-30 Hz ae suppessed when the tack model is included. Fom this point of view all the tack models have the same behavio in filteing the highe fequencies. Displacement (m) 6.0E-02 5.0E-02 4.0E-02 3.0E-02 2.0E-02 Displacements at mid span fo moving loads Without tack model Tack model I Tack model II Tack model III 1.0E-02 0.0E+00 140 160 180 200 220 240 260 280 300 Speed (km/h) Figue 8: Maximum displacements at mid span consideing the moving foce model. The Figue 10 also allows the conclusion that the application of the Wilson θ integation method leads to a good esult in damping out the spuious paticipation of the highe modes, that is, the contibution of the fequencies highe than 30 Hz is vey low as it is ecommended by the EN1990-pAnnex A2. 10 Poto - Potugal, 24 26 July 2006

Acceleactions (m/s2) 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 Acceleactions at mid span fo moving loads Without tack model Tack model I Tack mode II Tack mode III 0.0 140 160 180 200 220 240 260 280 300 Speed (km/h) Figue 9: Maximum acceleations at mid span consideing the moving foce model. 1.60E-01 1.40E-01 1.20E-01 Without tack model Tack model I Amplitude 1.00E-01 8.00E-02 6.00E-02 4.00E-02 2.00E-02 0.00E+00 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100. Fequency (Hz) 0 Figue 10: Compaison of the acceleations of the bidge with a tack model I and without tack model in fequency domain duing the passage of the ICE2 tain at a speed of 140km/h, fo the moving load model. The Inteaction Model Consideing now the esults obtained when the vehicle/bidge o vehicle/tack/bidge inteaction is pesent, the maximum values of the displacement and acceleation esponse ae epesented in Figue 11 and Figue 12, espectively. The esonance speed is eached at about the same value as befoe, but the maximum values of displacement and acceleation obtained with these models ae much lowe than those obtained with the moving loads model. The maximum displacement is 3.60 cm, obtained fo the vehicle/tack/bidge inteaction models II and III. The maximum displacement obtained with the model without ballast tack is 3,4 cm. Consideing the esponse acceleations, all the models funish identical esults. Out of the esonance situation the vehicle/bidge model, without tack, shows modest highe values than the othe models. Poto - Potugal, 24 26 July 2006 11

Displacement (m) 4.0E-02 3.5E-02 3.0E-02 2.5E-02 2.0E-02 1.5E-02 1.0E-02 Displacements at mid span consideing inteaction Without tack model Tack model I Tack model II Tack model III 5.0E-03 0.0E+00 140 165 190 215 240 265 290 Speed (km/h) Figue 11: Maximum displacements at mid span consideing the inteaction model. Acceleactions (m/s2) 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 Acceleactions at mid span consideing inteaction Without tack model Tack model I Tack model II Tack model III 0.0 140 165 190 215 240 265 290 Speed (km/h) Figue 12: Maximum acceleations at mid span consideing the inteaction model. Analyzing the compaison (see Figue 13) of the fequency esponse acceleations at the midspan of the bidge taking into account the vehicle/tack/bidge inteaction model, with diffeent systems and the vehicle/bidge inteaction model, it can be concluded that the ballast tack model acts like a low-pas filte. The esults consideing vehicle/bidge inteaction, without any tack model, show a contibution of the highe fequencies when compaed with the equivalent system when the moving foce model is used (compae Figues 10 and 13). The diffeences among the esults of the seveal vehicle/tack/bidge inteaction models ae small. Theefoe, in Figue 13, only the esults of the effects of the tack ballast model III ae epesented, since this model seems to be the most efficient in damping the highe fequencies. Figue 14 and Figue 15 compae the esults obtained with the two diffeent loading methodologies, the moving load model methodology and inteaction model methodology. The 12 Poto - Potugal, 24 26 July 2006

maximum values obtained with the inteaction model ae about 33% lowe than the equivalent esults obtained with the moving foces model. 1.8E-01 1.6E-01 1.4E-01 1.2E-01 Without tack model Tack model III Amplitude 1.0E-01 8.0E-02 6.0E-02 4.0E-02 2.0E-02 0.0E+00 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100. 0 Fequency (Hz) Figue 13: Compaison of the acceleations of the bidge with a tack model III and without tack model in fequency domain duing the passage of the ICE2 tain at a speed of 140km/h, fo the inteaction model. Displacement (m) 6.0E-02 5.0E-02 4.0E-02 3.0E-02 2.0E-02 Displacements at mid span consideing inteaction Without tack model Tack model I Tack model II Tack model III Moving loads 1.0E-02 0.0E+00 140 165 190 215 240 265 290 Speed (km/h) Figue 14: Compaison of the maximum displacements of the bidge fo the inteaction and moving loads models Acceleactions (m/s2) 16.0 14.0 12.0 10.0 8.0 6.0 4.0 Acceleactions at mid span consideing inteaction Without tack model Tack model I Tack model II Tack model III Moving loads 2.0 0.0 140 165 190 215 240 265 290 Speed (km/h) Figue 15: Compaison of the maximum displacements of the bidge fo the inteaction and moving loads models Poto - Potugal, 24 26 July 2006 13

CONCLUSIONS The main pupose of this investigation was the study of the behavio of simply suppoted ailway bidges with medium span and low stiffness, subject to the high speed tain ICE2, using two diffeent methodologies fo the loading models: the vehicle/tack/bidge inteaction methodology and the moving loads methodology. Additionally, thee types of tack models wee consideed Accoding to the esults obtained fo the acceleation in the fequency domain, it can be concluded that the use of the Wilson-θ method in ailway poblems shows to be suitable in filteing the high fequency components. The esults eveal a good numeical dissipation of the spuious paticipation of the highe modes. The esponse of the system tack/bidge when subject to the moving loads model shows that the diffeent tack models do not influence the maximum displacements and acceleations. The esults obtained fo the esponse acceleations in the fequency domain show that those models act as a filte in the high fequency components. Compaing the esults obtained fo the maximum displacements and acceleations at the mid span fo the two diffeent methodologies, inteaction model and moving load model, it can be concluded that the use of the inteaction model esults in 33% lowe displacements and acceleations. Theefoe, the inclusion of the inetia effects of the moving vehicles contibutes decisively to the eduction of the peak esponse. REFERENCES Biggs J.M. Intoduction to stuctual dynamics. McGaw-Hill. London, 1964. ERRI D214, Rail Bidges fo speeds >200km/h. Final Repot. Euopean Rail Reseach Institute, 1999. Fyba L. Vibations of solids and stuctues unde moving loads. Thomas Telfod, 1999. Yang Y.B., Yau J.D., Wu Y.S. Bidge inteactions dynamics with applications to high-speed ailways. Wold Scientific, 2004. Xia H., Zhang N., Roeck G. Dynamic analysis of high speed ailway bidge unde aticulated tains. Computes and Stuctues 81, 2003, p. 2467-2478. EN1991-2, Actions on stuctues Pat 2: Geneal actions Taffic loads on Bidges. Euopean Committee fo Standadization, CEN, 2003. EN1990-pAnnex A2, Basis of Stuctual Design Annex A2: Application fo bidges (nomative), Final Daft. Euopean Committee fo Standadization, CEN, 2002. Bathe K J. Finite Element Pocedues, pentice-hall, 1996. Bathe K.J., Chaudhay A., A Solution method fo plana and axisymmetic contact poblems, Intenational Jounal fo Numeical Methods in Engineeing, vol. 21, 61-88, 1985. Zhai M, Wang Y, Lin H. Modelling and expeiment of ailway ballast vibations, Jounal of Sound and Vibation, 270, pp. 673-683, 2003. Man A. A suvey of dynamic ailway tack popeties and thei quality, PhD. Thesis, TU Delft, DUP Science, Delft, 2002. 14 Poto - Potugal, 24 26 July 2006