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Numerical Methods in Geotechnical Engineering Schweiger (ed.) 2006 Taylor & Francis Group, London, ISBN 0-415-40822-9

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1 INTRODUCTION

Railway lines in Europe are 60 to 100 years old, and

are not designed in accordance with modern railwaytraffic. Due to faster and heavier modern trains, existingrailway bridges are experiencing problems, such asdeterioration and/or loss of ballast material, and theincrease of differential settlements within the bridgetransition zone. These issues have an adverse effect onthe safety, reliability, and economy of the railway line,and therefore, many existing bridge systems requireupgrading. Engineers are faced with the task of assess-ing the performance of existing bridges, and, if neces-sary, designing the strengthening or repair systems.

As part of the work performed for the EU project

Sustainable Bridges, the authors have evaluated thestress distribution and deflections within the transitionzone due to the passing of high speed trains. The analy-ses were performed using the finite difference com-

puter code FLAC3D; train loads were applied using thedynamic mode. A bridge geometry representative ofconcrete bridges in Europe was considered. A numer-ical parametric study was performed to study the effectsof the (1) train velocity, (2) stiffness of the ballast andsubballast material, and (3) stiffness of the backfill/embankment fill.

This paper provides a description of the methodsused in the numerical parametric analyses, and resultsof the parameter study, including a discussion of thetrends disclosed by the analyses.

2 DESCRIPTION OF NUMERICAL ANALYSES

2.1 Previous studies by others

Olofsson & Hakami (2000) performed three-dimensional analyses using FLAC3D to evaluate theinteraction between the train track, bridge abutment,and backfill at the transition zone. These analysesincluded structural beam elements to represent therail and sleepers, and structural shell elements torepresent the bridge abutments. The train load wasapplied statically by advancing a force along thestructural nodes of the beam elements. Adolfssonet al. (1999) present the results of FLAC3D analyses

performed to evaluate the critical train speed vibra-tion effects at a soft site in Ledsgrd, Sweden. These

analyses considered a railway on a simplified soft soilsection with no bridge. No structural elements wereused in the analyses, and the train load was appliedusing the dynamic mode in FLAC3D. The analysesdescribed by Olofsson & Hakami (2000) and Adolfssonet al. (1999) were made available to the authors andwere a valuable source of information.

2.2 Numerical analyses

Numerical analyses of a simplified bridge geometrywere performed using the finite different FLAC3D(Fast Langrangrian Analysis of Continua in 3Dimensions) computer program (Itasca 2002). The

Three-dimensional analyses of transition zones at railway bridges

M.E. Smith, P.-E. Bengtsson & G. HolmSwedish Geotechnical Institute, Linkping, Sweden

ABSTRACT: Numerical analyses were performed to evaluate the behaviour of a railway bridge transition zoneunder the passing of trains. A numerical parametric study was performed to study the effects of the train veloc-ity and the stiffness of the embankment materials. The parameter study was performed using the dynamic modeof FLAC3D. Results are expressed in terms of the maximum net horizontal stresses on the back of the abutment,and maximum vertical deflections behind the abutment for the case in which the train is directly above the backof the first bridge abutment. This paper provides a description of the methods used in the numerical parametric

analyses, and results of the parameter study, including a discussion of the trends disclosed by the analyses.

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base case geometry was chosen based upon a typicalbridge geometry provided by the Swedish RailroadAdministration, Banverket, and is representative ofmany concrete bridge systems. The bridge embankment

profile is shown in Figure 1, where it can be seen thatthe profile consists of an upper ballast layer 0.4m thick,which is underlain by a layer of subballast 1.2 m thick,which in turn is underlain by embankment backfill. Astiff, non-yielding, bearing layer is assumed to exist

beneath the embankment fill. As can be seen in Figure 1,the geometry is symmetrical about its centre-line.

The material property values used in the numericalanalyses are listed in Table 1. A linear-elastic perfectly

plastic model with a Mohr-Coulomb failure criterionwas used to represent the ballast, subballast, andembankment fill. No material damping was used.

The mesh geometry used in the numerical analysesis shown in Figure 2. The total length of the model is56 m. The length of the bridge is 10 m. Zone sizes areon average 0.36 0.31 0.28m, and a total of 63,376zones were used. In FLAC3D, the spatial element sizeshould be smaller than approximately one-tenth to one-eighth of the wavelength associated with the highestfrequency component of the input wave (Itasca 2002).For the analyses presented herein, the wavelength ofthe input wave is equal to 7.25 m, which correspondsto the length of the simplified load associated withone bogie.

No structural elements were used to model thebridge abutments, or the rail and sleepers. It wasdesired to model the train dynamically, and since the

timestep in the dynamic mode is determined by thelargest material stiffness, using structural elementswould require unrealistically long model run times,and may cause numerical problems. Therefore, the

bridge structure was represented by fixed gridpointsat the bridge-soil interface.

For these analyses, a hypothetical train load wasused. The embankment system is relatively stiff and

in order to obtain observable behavourial patterns, arelatively high train load was used. The applied trainload was calculated based upon an axle load of 250kN.For the base cases analyses, the maximum applied trainload, qmax, is approximately 72kPa.

The train load was applied dynamically along thetop of the model over a width of 1.25m from thecentre-line. To gain a clear understanding of the behav-iour of the system under a dynamic load, only one bogieof the train was considered. The approximated loadrepresenting one bogie is schematically shown inFigure 3. The shape of the load in Figure 3 is a simpli-fied approximation of the load beneath two wheels ofone bogie. This load was input as a cosine-wave, witha maximum magnitude of 72 kPa. The cosine-wave

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Figure 1. Cross-section of numerical model.

Table 1. Base case material property values used in thenumerical analyses.

Ballast Subballast Fill

Dry density (kg/m3) 1900 1900 1700Elastic modulus (MPa) 193 160 47.9Poissons ratio 0.30 0.30 0.31Bulk modulus (MPa) 161 133 42Shear modulus (MPa) 74 42 18.3Friction angle (deg) 40 37 31Dilation angle (deg) 4 3 2

Figure 2. Mesh geometry of numerical model.

Figure 3. Approximation of the load from one bogie.

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was applied successively over each zone. One loadvariable was created for each successive zone. Each

load variable is a function of time, and has a durationthat is equal to the length of one bogie (7.25m) divided

by the velocity of the train.The first step in the numerical analyses was to ini-

tialize the vertical and horizontal stresses so that theysatisfy both equilibrium and the gravitational gradient.To avoid shear stresses from developing in the soil atthe bridge abutments during the initialisation process:(1) the model profile was built up in incremental layersfor a total of four layers, with each layer being equili-

brated prior to adding the subsequent layer, and(2) the gridpoints at the bridge abutments were fixed in

the horizontal directions only, and not in the verticaldirection. Once the model was brought to initial equi-librium, all gridpoints at the bridge abutments werefixed all directions prior to applying the train load.

3 PARAMETER STUDY

A parametric study was performed to study the effectsof (1) the train velocity, (2) the stiffness of the ballastand subballast material, and (3) the stiffness of theembankment fill. The approach adopted for the numer-

ical parameter studies relies on a base case analysis,with systematic variation of parameter values from thebase case.

The train velocity was varied over a range of 50 to350 km/hr. The stiffness of the ballast, subballast, andembankment fill were varied systematically accordingto the values listed in Table 2.

Results and conclusions presented in this sectionpertain to the behaviour of the system at the first bridgeabutment when the train is directly above the back ofthe abutment. Results are expressed in terms of nethorizontal stress, which represents the additional hori-zontal stress applied to the back of the structure dueto the train load, and maximum vertical deflections

behind the abutment.

A cross-section of the net horizontal stresses at thefirst bridge abutment is shown in Figure 4 for the casein which base case material properties were used andthe train velocity was equal to 350 km/hr. A few generalobservations can be made based on the results shownin Figure 4. First, below a depth of 1.2m, the net hori-zontal stresses in the fill are relatively small. Thisstress distribution pattern is not the same, however,for the case in which a soft embankment fill is used,as discussed below. Second, the maximum net horizon-

tal stress for this case is on the order of 80 kPa, andoccurs over a depth of 0.2 to 0.45 m. For all the analysesin this study, the net maximum horizontal stresses wereobserved in the ballast layer or at the ballast-subballastlayer interface.

The variation in the maximum net horizontalstresses, net h,max, with train velocity and material

property values is given inTable 3. In all cases, themaximum net horizontal stresses were observed

beneath the centre of the train load. The values pre-sented in Table 3 are for the case in which the train isdirectly above the back of the first abutment. The

variation of net horizontal stresses with train velocityand material property values is discussed in the fol-lowing sections.

A cross-section of the vertical deflections 1m behindthe first bridge abutment are shown inFigure 5for thecase in which base case material properties were usedand the train velocity was equal to 350 km/hr. Thedeflections in Figure 5 are the vertical deflectionscalculated when the train is directly above the back ofthe first abutment. When the train is at this location,the maximum vertical deflections occur approximately1 m behind the train (unless noted otherwise). Forthe base case analysis, the maximum vertical deflec-tion behind the first bridge abutment is 2.87mm(Figure 5).

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Table 2. Variable material property values.

E (MPa) Poissons ratio

Ballast 160 0.30193 0.30

300 0.30Subballast 47.9 0.31

160 0.30

300 0.30

Embankment fill 20 0.3147.9 0.31

70 0.30

Bold values are base case values.

Figure 4. Distribution of net horizontal stresses behind firstbridge abutment (v 350 km/hr). (Contours given in kPa.)

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A few general observations can be made based onthe results shown in Figure 5. First, most of the verticaldeflections occur in the ballast and subballast layers.Second, as would be expected, the maximum verticaldeflections are concentrated beneath the width of thetrain.

The variation in the maximum vertical deflections,

v,max

, with train velocity and material stiffness valuesare given in Tables 4 and 5, respectively, and are dis-cussed in the following paragraphs.

3.1 Train velocity

The train velocity was varied over a range of 50 to350km/hr. The base case material property values,listed inTable 1, were not varied. In general, as thetrain velocity increases, the net horizontal stresses inthe ballast and subballast layer increase, as can be seenin Table 3. However, the relationship between train

velocity and maximum net horizontal stress on the backof the abutment is not linear. The largest incrementalincrease in the maximum net horizontal stresses at thefirst bridge abutment occurs when the train velocityincreases from 250 to 350km/hr.

As can be seen in Table 4, as the train velocityincreases, the maximum vertical deflection behind thetrain increases. The largest increase in maximum ver-tical deflections occurs for the case in which the trainvelocity increases from 250 to 350km/hr.

The goal of the Sustainable Bridges project is toevaluate the behaviour of bridge systems under highspeed trains, which may have velocities up to 350 km/hrin the future. The remainder of the parametric analyseswere performed using a train velocity of 350km/hr.

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Table 3. Maximum net horizontal stresses at the first abutment.

Net h,max % ChangeCase (kPa) Depth (m) from base case

v 50 km/hr 36.6 0.2

v

150 km/hr 38.5 0.2v 250 km/hr 52.1 0.2v 350 km/hr 79.4 0.20.4

Ballast (E 160 MPa) 87.3 0.4 10Ballast (E 300 MPa) 79.2 0.2 neg.

Subballast (E 47.9 MPa) 107.4 0.2 35Subballast (E 300 MPa) 42.6 0.4 46

Fill (E 20 MPa) 78.5 0.2 neg.Fill (E 70 MPa) 49.2 0.2 38

Bold value is base case value.

Figure 5. Distribution of vertical deflections 1 m behindfirst bridge abutment (v 350km/hr).

Table 4. Maximum vertical deflections,v,max, behind the first abutment as afunction of train velocity.

Velocity (km/hr) v,max (mm)

50 1.45

150 1.49250 1.93350 2.87

Table 5. Maximum vertical deflections,v,max, behind the firstabutment as a function of material stiffness (v 350km/hr).

E (MPa) v,max (mm) % Change

Ballast 160 3.03 5.6193 2.87

300 2.71 5.6

Subballast 47.9 5.46 90.2160 2.87

300 1.83 36.2

Embankment fill 20 9.10 217*47.9 2.87

70 1.43 50.1

*Observed 3m behind train.

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3.2 Ballast stiffness

The elastic modulus of the ballast was varied over arange of 160 to 300MPa. Changing the stiffness of

ballast material has only a small effect on the net hori-zontal stresses and the calculated vertical deflections

behind the first bridge abutment (seeTables 3 and5,respectively). This is likely due to the fact that thethickness of the ballast is relatively thin compared to

the thicknesses of the subballast and embankment fillmaterials.

3.3 Subballast stiffness

The elastic modulus of the subballast was varied overa range of 47.9 to 300MPa. When the stiffness of thesubballast layer is equal to 47.9 MPa, it is equal to thestiffness of the embankment fill, and for this case, it isas if there is no subballast layer and the embankmentfill extends to the bottom of the ballast layer. When thestiffness of the subballast layer is equal to 300MPa,

the subballast is 1.6 times stiffer than the ballast layer.The stiffness of the subballast layer greatly affectsthe calculated maximum net horizontal stresses, whichoccur at the ballast-subballast interface. Of all the

parameters evaluated, the greatest decrease in max-imum net horizontal stress occurs for the case in whichthe stiffness of the subballast is greater than that ofthe ballast layer. This decrease in maximum net hori-zontal stresses is greater than the decrease that corres-

ponds to the decrease in train velocity from 350 to250km/hr.

A cross-section of the net horizontal stresses at thefirst bridge abutment are shown in Figure 6 for thecase in which the stiffest (E 300MPa) subballastlayer was used and the train velocity was equal to

350 km/hr. As can be seen in Figure 6, the maximumnet horizontal stresses are on the order of 40kPa andoccur at a depth of about 0.4 m. The net horizontalstresses below a depth of 1m are relatively small.

In general, as the stiffness of the subballast layerincreases, vertical deflections behind the first abut-ment decrease.

3.4 Embankment fill

The elastic modulus of the embankment f ill was var-ied over a range of 20 to 70 MPa. An increase in thestiffness of the fill corresponds to an decrease in thenet horizontal stresses in the ballast and subballastlayers.

A cross-section of the net horizontal stresses at thefirst bridge abutment are shown in Figure 7 for the casein which the softest fill layer was used and the trainvelocity was equal to 350 km/hr. A decrease in stiff-ness of the embankment fill has little affect on the

calculated magnitude of maximum net horizontalstress. As can be seen in Figure 7, the maximum nethorizontal stress occurs at about a depth of 0.2m andis approximately 79 kPa. However, net horizontalstresses on the order of 30kPa occur at the subballast-fill interface, and net horizontal stresses on the orderof 20 kPa are observed to a depth of approximately2.7 m. This behaviour is different than that indicatedinFigure 4, in which the net horizontal stresses belowa depth of 1.2m are relatively small.

The greatest vertical deflection behind the firstbridge abutment was calculated for the case in whichthe soft embankment fill was used. This high value ofcalculated deflection occurs approximately 3 m behindthe train.

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Figure 6. Net horizontal stresses behind first bridge abut-ment (v 350 km/hr) for stiff subballast case. (Contours

given in kPa.)

Figure 7. Net horizontal stresses behind first bridge abut-ment (v 350 km/hr) for soft fill case. (Contours given

in kPa.)

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4 LIMITATIONS

The results discussed in this paper are only for theconditions investigated. One limitation of the currentanalyses is that structural elements were not used torepresent the rail and sleepers. As a result, the truedistribution of the train load through the rail andsleepers was not taken into account. Future analyseswill include structural elements to represent the railand sleepers and/or the bridge, with a statically appliedtrain load. Another limitation of the current analysesis that a stiff bearing layer was assumed to exist beneaththe embankment fill. It is expected that a soft bearinglayer beneath the embankment fill would have an effecton the distribution of stresses and deflections in thesystem.

The analyses presented in this paper constitute onlya small portion of the analyses being performed for

the Sustainable Bridges project. Analyses are currentlyunderway to evaluate the effects of combinations ofvarious realistic train loads and velocities, and multipletrain passes. Future analyses will also be performedusing a typical masonry arch bridge geometry.

5 SUMMARY AND CONCLUSIONS

Numerical analyses were performed to evaluatethe behaviour of a bridge system under the passing oftrains. A numerical parametric study was performed

to study the effects of (1) the train velocity, (2) thestiffness of the ballast and subballast material, and(3) the stiffness of the backfill/embankment fill.The parameter study was performed using three-dimensional, dynamic analyses using the computer

program FLAC3D.Results were expressed in terms of the maximum

net horizontal stresses on the back of the abutment,and maximum vertical deflections behind the abutmentfor the case in which the train is directly above the

back of the f irst bridge abutment. The train velocitywas varied initially, and then, for the remainder of the

parameter study analyses, a train velocity of 350 km/hrwas used.For the analyses performed, it was observed that

the maximum net horizontal stresses occur in the bal-last or at the ballast-subballast interface. The magnitudeof the maximum net horizontal stresses is affected themost by the stiffness of the subballast layer. However,the vertical deflections measured behind the train atthe first abutment are affected the most by the stiffnessof the embankment fill. Other key findings from theanalyses include:

As the train velocity increases, the net horizontal

stresses in the ballast and subballast increase.

There is a substantial increase in the maximum nethorizontal stresses when the train velocity increasesfrom 250 to 350km/hr.

Net horizontal stresses in the embankment f ill arerelatively small, except for the case in which a soft

fill is used. When the train is located directly above the back ofthe first bridge abutment, the maximum verticaldeflections are observed to occur approximately 1m

behind the train, except for the case in which a softfill is used.

Changing the stiffness of ballast material has littleeffect on the net horizontal stresses and the calcu-lated vertical deflections behind the first bridgeabutment.

As the stiffness of the subballast layer increases,vertical deflections behind the first abutmentdecrease.

An increase in the stiffness of the fill correspondsto a decrease in the net horizontal stresses in the

ballast and subballast layers.

When a soft embankment fill is used, the magni-tude of the maximum net horizontal stress is notaffected; however, greater net horizontal stressesare observed at greater depths.

When a soft embankment fill is used, verticaldeflections increase substantially. These deflec-tions are observed to occur 3m behind the train.

ACKNOWLEDGEMENTS

The authors acknowledge the important contributionsof Alexander Smekal, Bo Andrasson and OlofOlofsson. The analyses described in this paper were

performed within the EU Sixth Framework Programme;the project title is Sustainable Bridges Assessmentfor Future Traffic Demands and Longer Lives, TIP3-CT-2003-001653.

REFERENCES

Adolfsson, K., Andrasson, B., Bengtsson, P.-E., Bodare, A.,Madshus, C., Massarsch, R., Wallmark, G., & Zackrisson,P. 1999.Evaluation and Analyses of Measurements fromthe West Coast Line. Report.

Itasca Consulting Group 2002.FLAC3D Fast LangrangianAnalysis of Continua is 3 Dimensions, Itasca ConsultingGroup, Minnesota, Minn., USA.

Olofsson, S.s-O. & Hakami, E. 2000. Interaction Spr-Bro-Grund-Jord, Sttningar i vergng mellan bro ochtill fartsbank fr lanserade broar. Itasca Geomekanik,Borlnge.

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