relative displacement method for track-structure interaction

Research ArticleRelative Displacement Method for Track-Structure Interaction

Frank Schanack1 Oacutescar Ramoacuten Ramos2 Juan Patricio Reyes1 and Marcos J Pantaleoacuten3

1 Institute of Civil Engineering Universidad Austral de Chile General Lagos 5111187 Valdivia Chile2 Department of Structural and Mechanical Engineering University of Cantabria Avenida Los Castros sn 39005 Santander Spain3 APIA XXI SA PCTCAN Avenida Albert Einstein 6 39011 Santander Spain

Correspondence should be addressed to Frank Schanack frankschanackuachcl

Received 5 August 2013 Accepted 27 November 2013 Published 22 January 2014

Academic Editors Z Guan and H-H Tsang

Copyright copy 2014 Frank Schanack et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The track-structure interaction effects are usually analysed with conventional FEM programs where it is difficult to implement thecomplex track-structure connection behaviour which is nonlinear elastic-plastic and depends on the vertical load The authorsdeveloped an alternative analysis method which they call the relative displacement method It is based on the calculation ofdeformation states in single DOF element models that satisfy the boundary conditions For its solution an iterative optimisationalgorithm is used This method can be implemented in any programming language or analysis software A comparison withABAQUS calculations shows a very good result correlation and compliance with the standardrsquos specifications

1 Introduction

Since the 1980s the track-structure interaction in railwaybridges has been the subject of research especially since thebeginning of the high speed railway traffic in Europe [1ndash4]These studies refer to the stresses and deformations in therail-deck system which may reach unsafe values and canaffect the serviceability of the track The rail stress may evenbe high enough to cause its rupture [5] Generally such effectsoccur in continuously welded rails which are currently beingused in high speed railway tracks because of their superiormaintainability and passenger comfort [6]

Usually the combined response of track and structureis analysed by standard finite element analysis software [57ndash10] The major challenge of this type of analysis is theimplementation of the connector element between rail andbridge deck which has a nonlinearmechanical behaviour andis elastic-plastic with irreversible deformations andmoreoverdepends on the value of the vertical load Much of thecommercial finite element software is not prepared for thesetasks especially the last one

The authors propose a differentmethod for the analysis ofthe effects of the track-structure interaction It is based on thecalculation of deformation states in single DOF finite elementmodels that satisfy the boundary conditions of the track and

structure For its solution an iterative optimisation algorithmshould be used instead of the solution of the system ofequations by means of a stiffness matrix This method canbe implemented in any programming language or analysissoftware such as FORTRAN MATLAB MathCAD or evenEXCEL Furthermore any mechanical behaviour of theconnector element can be incorporated easily The authorscall it the relative displacement method

In this work the concept of the new formulation isderived and the results of a comparison with the conven-tionalmethod for the loads creep shrinkage and temperaturevariation are presented

2 The Track-StructureInteraction Phenomenon

21 Structural Behaviour The track-structure interaction orthe combined response of the structure and track describesthe effects of the structural collaboration of the rails and thedeck in bridges by means of their connection elements Inthe beginning the analysis of the rails and bridge deck wasconducted separately However this type of analysis is notappropriate when the rails are continuously welded on top

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 397515 7 pageshttpdxdoiorg1011552014397515

2 The Scientific World Journal

Deck

Track

Nonlinearconnectors

Expansionjoint

Figure 1 Usual analysis model of the track-structure interaction

Loaded track

Unloaded track

Frozen ballast

u0

Relative displacement u

Shea

r res

istan

cek

k3

k2

k1

Figure 2 Load-displacement behaviour of ballasted tracks [12]

of the structure because then the track-structure interactionshows nonnegligible effects [6 11]

The track-structure interaction analysis is based on themodel shown in Figure 1The track and the deck aremodelledby beam elements in their respective centres of gravity Bothparts are connected by the ballast which transfers forcesbetween them It is modelled by longitudinal connectors withcertain nonlinearmechanical behaviourUsually this analysisis conducted with conventional finite element software

In the case of ballasted tracks the structural collaborationof rail and structure is not rigid It is generally acceptedthat the load-displacement behaviour of the ballast can beidealised by the bilinear law shown in Figure 2 similar tofrictional behaviour [9ndash14]

The longitudinal shear resistance of the ballast 119896 isproportional to the displacement of the rail relative to thetop of the supporting deck 119906 until a relative displacementof 1199060is reached which corresponds to an elastic limit At

this point the ballast cannot resist any further load anda sliding phenomenon occurs while the resistance force isconstant (plastic shear resistance) When the direction ofthe displacement changes the ballast behaviour becomeselastic again but the relative displacement from sliding is

not recovered The elastic limit is different for frozen andunfrozen ballasts

Analogously to frictional behaviour the plastic shearresistance of the ballast is higher when an additional verticalload is applied which is the case when the live load is appliedto the track (Figure 2) Hence the analysis must take intoaccount for example that the connector elements that arein the sliding state before applying the live load will returnto elastic behaviour while their relative displacement andtheir connector force remain unchanged The implementa-tion of such a connector in the analysis of the interactionphenomenon with the finite element method causes certaincomplications such as the activation and deactivation ofelements in function of the presence of load and cannot berealised in many engineering FEM programs

22 Actions on the Track-Structure System It is necessaryto take into account all actions that may cause longitudinalforces or displacements both in the track and the structureThese actionsmay be of very different nature as for examplecreep and shrinkage temperature variation stress fromvertical loads or traction and braking forces Any of theseactions can cause a force transfer between the rail and deckvia the rail fasteners and the ballast [12]

The present work focuses on the actions that cause thegreatest relative displacements between the track and thestructure that is creep and shrinkage and the variation of thetemperature of the deck and rails Nevertheless the proposedmethod can be used to calculate the effects of any of theactions mentioned above

221 Creep and Shrinkage In concrete bridges part of thecreep and shrinkage phenomenon occurs after the installa-tion of the track This part has to be taken into account forthe track-structure interaction analysis It produces a deckshortening such that every point of the deck moves towardsthe fixed bearing of the bridge which usually is located atone abutment Consequently the creep and shrinkage strainshave a defined direction

The result is a permanent stress state of the rail-structureconnection which will certainly disappear in time due to thedynamic actions of the passing trains To take into accountthe most unfavourable condition it is prudent to analyse thetwo possibilities the presence and the absence of the imposedstress state due to creep and shrinkage

222 Variation of the Rail and the Deck Temperature Ingeneral the value of the constant temperature variation

The Scientific World Journal 3

depends on the bridge type and the climatic zone of itsplacement For the deck temperature variation the overallrange of the uniform temperature component according tothe Eurocode [15] is considered In the National Annexesalternative values may be specified For example in theSpanish railway bridge design code IAPF-07 the maximumdeck temperature variation is plusmn35K while the maximum railtemperature variation is plusmn50K The maximum temperaturedifference between both elements is plusmn20K [13]

23 Required Verifications The combined response of trackand structure can have unfavourable effects on the bridgestructure that have to be considered for its dimensioningAdditionally there are unfavourable effects on the track-ballast system that can affect the security and the func-tionality of the bridge According to Eurocode 1 the mainverifications to be conducted are the following [12]

(i) The additional rail stresses due to the combinedresponse of the structure and track to variable actionsshould be limited to 72Nmm2 in compression and92Nmm2 in tension In continuously welded railsthe stress increment is calculated with respect to therail stress in the rail at a sufficiently large distancefrom the bridge The given values correspond to thecommonly used UIC 60 rail with a tensile strength ofat least 900Nmm2

(ii) The absolute deck displacement at both ends of thebridge due to traction and braking shall not exceed5mm If there are rail expansion joints at both ends ofthe bridge this displacement shall not exceed 30mm

(iii) Additionally in some National Annexes a limit of4mm is specified for the relative longitudinal dis-placement of deck and rail due to traction and braking[13 14]

3 Alternative Analysis Method

31 Concept During the analysis of 15 high speed railwaybridges for Spanish AVE tracks the authors recognised thatthe implementation of the mechanical connector behaviouras described before is rather complicated even in veryadvanced FEM software such as ABAQUS In particular thestiffness change due to vertical loading requires additionalprogramming effort

To reduce the complexity of the problem the authorsderived an alternative analysis method that is based onfinite elements with a single degree of freedom that isthe displacement in longitudinal direction Both the trackand the bridge deck are modelled with these elements Theconnection between the track and structure is taken intoaccount as forces applied to track and structure nodes Theforce value is obtained from the actual relative displacementand the relative displacement history according to Figure 2In the same way any longitudinal load and the restoringforces from piers and bearings are taken into account at therespective rail and deck nodes

i i+1

L

i i + 1

kk

120576Δt

ui ui+1

120576Δt 120576R+F

Nrailiminus1 Nraili+1

Fpieri+1

Ndeckiminus1 Ndecki+1

Flongi+1Flongi

Fpieri

Figure 3 Illustration of the alternative calculation model

Under given longitudinal loads from traction braking orseismic actions and imposed longitudinal strains from creepand shrinkage or temperature actions an infinite number ofdeformation states of such a model can be found Howeveronly one of these deformation states will satisfy the boundaryconditions of the analysis problem This special equilibriumstate can easily be determined by any iterative optimisationalgorithm without the need to solve a system of equations bymeans of a stiffnessmatrixThe authors first programmed thisanalysis method as an EXCEL worksheet and then utiliseda FORTRAN program due to the higher precision and thefaster mathematical operators

The output of this method includes all displacementsstrains and forces of the track the structure and their con-nection

Figure 3 shows a schematic representation of the alterna-tive analysis model There are two parallel elements one forthe track and one for the deck with their respective elon-gation stiffness The element length 119871 is determined in thesame manner as in usual FEM bridge models Good resultsare obtained for a length of 1m The required mathematicalprecision of this method is not altered by the element length

The ballast is represented by a connector element thatcan be defined with any mechanical behaviour in this casenonlinear and elastic-plastic as a function of the vertical loadThe connector force on the left acting between the nodes 119894 ofthe rail and of the deck depends on their relative displace-ment which is given as a result of the previous analysis of theadjacent left-hand element The relative displacement of thenodes 119894 + 1 is then obtained from the determination of thetotal element strain of the track and of the deck due to stressand imposed strain as shown in

119906119894+1= 119906119894+ (120576rail119894 minus 120576deck119894) sdot 119871

120576totalrail119894 =120590rail119894

119864rail+ 120576rail119894

120576totaldeck119894 =120590deck119894

119864deck+ 120576deck119894

(1)


rail119899 = const1199061= const119873rail0 = 0

While 1003816100381610038161003816120590rail119899 minus rail1198991003816100381610038161003816 = 0 do

For 119894 = 1 to 119899 do

120590rail119894 =119873rail119894minus1 minus 119865long119894 minus 119865ballast (119906119894)

119860 rail

120590deck119894 =119873deck119894minus1 minus 119865pier119894 + 119865ballast (119906119894)

119860deck

120576rail119894 =120590rail119894

119864rail+ 1205761015840

rail119894

120576deck119894 =120590deck119894

119864deck+ 1205761015840

deck119894


End Forchange 119906

1to minimise 1003816100381610038161003816120590rail119899 minus rail119899

1003816100381610038161003816

EndWhile

Algorithm 1

The element stresses result from the track and deck axialforces from the connection forces of the ballast and from anyadditional exterior longitudinal force 119865long as follows

120590rail119894 =119873rail119894

119860 rail=119873rail119894minus1 minus 119865long119894 minus 119865ballast (119906119894)

119860 rail (2)

The deck stress also depends on the restoring forces ofpiers and bearings 119865pier which can be determined from theirstiffness by the longitudinal displacement of the correspond-ing node Different stiffness for different vertical bearingloads can be considered

120590deck119894 =119873deck119894

119860deck=119873deck119894minus1 minus 119865pier119894 + 119865ballast (119906119894)

119860deck (3)

The imposed strains are those resulting from temperaturechange creep and shrinkage or vertical deflection of thedeck

120576rail119894 = 120572119879rail sdot Δ119879rail + 120576vert

120576deck119894 = 120572119879deck sdot Δ119879deck + 120576vert + 120576119862+119878(4)

Considering the relative displacement history from any pre-vious load and the actual value of the relative displacement itis possible to determine the actual connection force betweenthese nodes This force is taken as the basis for the analysis ofthe next right-hand element

In that way all connection forces and all node displace-ments of the complete bridge length can be calculated succes-sively The authors call this method the relative displacementmethod

32 Solution Algorithm The relative displacement of thefirst pair of nodes 119894 may be arbitrary Its correct value

Table 1 Parameters of Giles Viaduct Spain

Bridge length 24m + 36m + 5 times 48m + 36m +24m = 360m

Track number 2Deck cross-section 10198m2

Rail cross-section 4 times 7678mm2 = 30712mm2

Plastic shear resistance k 20 kNmRelative displacement elasticlimit 119906

0

2mm

Creep and shrinkage strain minus456119864 minus 2permilRail temperature increment Δ119879 +20KCoefficient of thermal expansion

Deck 100119864 minus 5Kminus1

Rail 120119864 minus 5Kminus1

must be determined by an iterative optimisation algorithmsuch that the boundary conditions of the bridge project arefulfilled The precision of the correct value must be veryhigh especially in long viaducts (over 500m) because smalldeviations will sum up to a large error Only one solution willfulfil the boundary conditions

Good boundary conditions are zero stress at rail or deckexpansion joints zero deck displacement at fixed bearings orany particular stress value on the embankment on a sufficientdistance from the bridge In the optimisation algorithm therelative displacement of the first pair of nodes 119894 is varied untilall of the boundary conditions are fulfilled Each iterationrequires the calculation of the complete bridge length

In Algorithm 1 the outline of the calculation algorithmis shown for the example of a bridge with two rail expansionjoints

33 Definition of the Connector Behaviour As described inSection 21 the mechanical behaviour of the rail-deck con-nection is rather complex The usual finite element programsdo not offer connector elements with such characteristics Itmust be composed of a combination of various elements andsubroutines or it might even be impossible to model

The advantage of the proposed relative displacementmethod is that the connector behaviour can be defineddirectly as a mathematical function in the chosen program-ming languageThis function can consider any parameters orresults of the analysis

For example for the analysis of creep and shrinkage andsubsequent temperature variation the six different connectorbehaviours shown in Figure 4 can be distinguished At theend of the first step the creep and shrinkage strain twodifferent states of the connector are possible elastic orplastic behaviour The subsequent temperature variation canproduce a displacement in the same direction as the beforestep or it can be contrariwise If it is in the same directionthe connector behaviour will be the same as previous and ifit is contrariwise it will be elastic but without recovering thepossible previous plastic deformation Furthermore the finalstate of the connector can be elastic or plastic This load-displacement behaviour can be described as follows


119865ballast119862+119878 =

119906119862+119878sdot119896

1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 1198961003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

119865ballastΔ119879 =

sgn (119906119862+119878) = sgn (119906

Δ119879)

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

sgn (119906119862+119878) = sgn (119906

Δ119879)

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

(sgn (119906119862+119878) sdot 1199060+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906Δ1198791003816100381610038161003816 lt 2 sdot 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906Δ1198791003816100381610038161003816 ge 2 sdot 1199060

(5)

In this manner it is possible to define any connectorbehaviour even for the more complex cases when loaded andunloaded tracks have to be considered

4 Application of the Proposed Method

To evaluate the validity of the proposed relative displacementmethod of the track-structure interaction in the following itis applied to a real bridge example The results are comparedwith those obtained from a conventional finite elementsanalysis performed in ABAQUS Standard software Figure 5shows the FEM bridge model that was used The bridgeselected for the comparison is the Giles Viaduct of the AVEhigh speed railway track from Los Gallardos to Sorbas inSpain It has a prestressed concrete box girder with a totallength of 360m divided into 8 spans This bridge has onerail joint and one deck expansion joint at each abutmentThe thermal centre is located in the centre of the bridgeThe necessary analysis parameters are taken from the Spanishrailway bridge design code [13] Table 1 shows the mostimportant of them

The loads evaluated are in the first step the deckdeformation due to creep and shrinkage at infinite time Inthe second step based on the equilibrium state of the firstload case the variation of the rail temperature is applied inthis case a temperature increase of 20K

Figure 6 shows the results for the rail stress of the firstload case for both the ABAQUS and the relative displacementanalysis Both graphs are plotted in the same diagram butcannot be distinguished because they are virtually the sameThe minimum rail stress value of minus8578Nmm2 is identicalfor both analysis methods

The rail stress for the second load case a rail temperatureincrement of +20K is obtained by applying a subsequentrail deformation to the analysis model equilibrium state aftercreep and shrinkage Figure 7 shows the resulting rail stressboth for the ABAQUS model and for the relative displace-mentmethod As before the corresponding graphs cannot bedistinguished in the diagram because they are virtually the

same The minimum rail stress values 13533Nmm2 fromABAQUS and 13566Nmm2 from the proposed method areidentical in practical terms (03 deviation)

In this example and as experienced in 14 other railwayviaducts with lengths from 123m to 25255m the resultsof the conventional FEM analysis and of the relative dis-placement method are of equal quality The CPU time wasinstantaneous for bothmethods while themodel preparationtime before analysis for an experienced user was about halfa day for the ABAQUS model and less than half an hour forthe relative displacementmethodThis comparison takes intoaccount that a general model of the bridge is already availablein ABAQUS from the bridge design process

5 Summary and Conclusions

The track-structure interaction in railway bridges is com-monly calculated with finite element analysis software In thecase of ballasted tracks the connection between track andstructure has a nonlinear plastic and irreversible mechanicalbehaviour that dependsmoreover on the vertical load appliedto the viaduct Most of the commercial software is notprepared for the implementation of such elements

To find a less complex method the problem was reducedto single DOF finite elements and an iterative optimisationalgorithm was proposed in place of the solution of theequilibrium equation system bymeans of the stiffnessmatrixThis method can be programmed in any language or evenin spreadsheet applicationsThe definition of any mechanicalbehaviour of the track-structure connector is easily possible

In the proposedmethod an initial relative track-structuredisplacement is assumed at one node and subsequently allnode forces and displacements of the deck and the trackare calculated Exterior forces acting on the track or onthe structure such as traction and braking force or bearingrestoring force can be taken into account Furthermoreall imposed deck or track deformations such as creepshrinkage or thermal expansion are implemented

The correct value of the initial relative track-structuredisplacement is determined by an iterative optimisation


k

k1

uu0

(a)

k

u

k1

k

k1

u0uu0

minusk1minusk1

minusu0

2u0

(b)

Figure 4 Rail-deck connection behaviour (a) for creep and shrinkage and (b) for subsequent temperature variation

Figure 5 FEM bridge model used in ABAQUS

050 100 150 200 250 300 350

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus100

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Proposed method (min = minus8619Nmm2)

Bridge length (m)

ABAQUS model (min = minus8578Nmm2)

Figure 6 Rail stress due to creep and shrinkage deformation

algorithm It is obtained when the calculated deformationstate of the model fulfils all the boundary conditions of theviaduct for example zero stress at expansion joints

The comparison of this proposed relative displacementmethod with an ABAQUS analysis model shows that bothresults are of the same quality and that their rail stress valuesare virtually identical In terms of time consumption the

050 100 150 200 250 300 350

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Bridge length (m)

ABAQUS model (min = minus13533Nmm2)Proposed method (min = minus13566Nmm2)

Figure 7 Rail stress due to creep shrinkage and temperaturedeformation

relative displacement method is very advantageous becausethe preparation time before analysis is less than half an hourwhile it is half a day for the ABAQUS analysis model

The proposed method has certain limitation because thedeformation of the whole bridge is calculated starting fromone node A very high precision of the deformation values isnecessary otherwise small deviations will sum up to a largeerror The precision of EXCEL spreadsheets is sufficient forup to 500m long viaducts with FORTRAN a 25255m longbridge was calculated successfully

Notations

119899 Total number of nodes119896 Plastic shear resistance of the track119906 Relative track-structure displacement1199060 Elastic limit of the relativetrack-structure displacement119860 Cross-section area


119864 Youngrsquos modulus119865 Longitudinal force119871 Element length119873 Axial force120572119879 Coefficient of thermal expansion119909 Strain120590 Stress Boundary condition stressΔ119879 Temperature variation

Conflict of Interests

Theauthors of the paper do not have a direct financial relationwith the French corporation Dassault Systemes distributerand developer of ABAQUS that might lead to a conflicts ofinterest for any of the authors

References

[1] M Muller D Jovanovic and P Haas ldquoTracks-gravel-bridgeinteractionrdquoComputers and Structures vol 13 pp 607ndash611 1981

[2] L Fryba ldquoThermal interaction of long welded rails withrailways bridgesrdquoRail International vol 16 no 3 pp 5ndash24 1985

[3] A M Cutillas ldquoTrack-bridge interaction problems in bridgedesignrdquo in Track-Bridge Interaction on High-Speed Railways RCalcada R Delgado A Campos e Matos J M Goicolea and FGabaldon Eds pp 19ndash28 Taylor amp Francis London UK 2009

[4] J M Goicolea-Ruigomez ldquoService limit states for railwaybridges in new design codes IAPF and Eurocodesrdquo in Proceed-ings of the Track-Bridge Interaction on High-Speed RailwaysFEUP Porto Portugal October 2007

[5] M Cuadrado and P Gonzalez ldquoTrack-structure interaction inrailway bridges Step-by-step calculation algorithmsrdquoRevista deObras Publicas vol 156 pp 38ndash48 2009

[6] L Fryba Dynamics of Railway Bridges Thomas Telford Lon-don UK 2nd edition 1996

[7] H Freystein ldquoTrackbridge-interactionmdashstate of the art andexamplesrdquo Stahlbau vol 79 no 3 pp 220ndash231 2010

[8] A Reguero ldquoTypes of viaduct on theMadrid-Barcelona-Frenchborder high speed railway linerdquo Revista de Obras Publicas vol151 no 3445 pp 109ndash114 2004

[9] P Ruge and C Birk ldquoLongitudinal forces in continuouslywelded rails on bridgedecks due to nonlinear track-bridgeinteractionrdquo Computers and Structures vol 85 no 7-8 pp 458ndash475 2007

[10] P Ruge D R Widarda G Schmalzlin and L BagayokoldquoLongitudinal track-bridge interaction due to sudden change ofcoupling interfacerdquo Computers and Structures vol 87 no 1-2pp 47ndash58 2009

[11] Union International des Chemins de Fer (UIC) Code 774-3-R Trackbridge interaction Recommendations for calculations2nd edition Paris France 2001

[12] European Committee for Standardization (CEN) EN1991-2Eurocode 1 Actions on structures Part 2 General actionsTraffic Loads on Bridges Brussels Belgium 2003

[13] Ministerio de Fomento (MF) Instruccion Sobre las Acciones aConsiderar en el Proyecto de Puentes de Ferrocarril (IAPF-07)Direccion General de Ferrocarriles Madrid Spain 2007

[14] Deutsches Institut fur Normung (DIN) DIN-Fachbericht 101Einwirkungen auf Brucken Beuth Berlin Germany 2nd edi-tion 2003

[15] European Committee for Standardization (CEN) EN1991-1-5Eurocode 1 Actions on structures Part 1ndash5 General actionsThermal actions Brussels Belgium 2003

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



Deck

Track

Nonlinearconnectors

Expansionjoint

Figure 1 Usual analysis model of the track-structure interaction

Loaded track

Unloaded track

Frozen ballast

u0

Relative displacement u

Shea

r res

istan

cek

k3

k2

k1

Figure 2 Load-displacement behaviour of ballasted tracks [12]

of the structure because then the track-structure interactionshows nonnegligible effects [6 11]

The track-structure interaction analysis is based on themodel shown in Figure 1The track and the deck aremodelledby beam elements in their respective centres of gravity Bothparts are connected by the ballast which transfers forcesbetween them It is modelled by longitudinal connectors withcertain nonlinearmechanical behaviourUsually this analysisis conducted with conventional finite element software

In the case of ballasted tracks the structural collaborationof rail and structure is not rigid It is generally acceptedthat the load-displacement behaviour of the ballast can beidealised by the bilinear law shown in Figure 2 similar tofrictional behaviour [9ndash14]

The longitudinal shear resistance of the ballast 119896 isproportional to the displacement of the rail relative to thetop of the supporting deck 119906 until a relative displacementof 1199060is reached which corresponds to an elastic limit At

this point the ballast cannot resist any further load anda sliding phenomenon occurs while the resistance force isconstant (plastic shear resistance) When the direction ofthe displacement changes the ballast behaviour becomeselastic again but the relative displacement from sliding is

not recovered The elastic limit is different for frozen andunfrozen ballasts

Analogously to frictional behaviour the plastic shearresistance of the ballast is higher when an additional verticalload is applied which is the case when the live load is appliedto the track (Figure 2) Hence the analysis must take intoaccount for example that the connector elements that arein the sliding state before applying the live load will returnto elastic behaviour while their relative displacement andtheir connector force remain unchanged The implementa-tion of such a connector in the analysis of the interactionphenomenon with the finite element method causes certaincomplications such as the activation and deactivation ofelements in function of the presence of load and cannot berealised in many engineering FEM programs

22 Actions on the Track-Structure System It is necessaryto take into account all actions that may cause longitudinalforces or displacements both in the track and the structureThese actionsmay be of very different nature as for examplecreep and shrinkage temperature variation stress fromvertical loads or traction and braking forces Any of theseactions can cause a force transfer between the rail and deckvia the rail fasteners and the ballast [12]

The present work focuses on the actions that cause thegreatest relative displacements between the track and thestructure that is creep and shrinkage and the variation of thetemperature of the deck and rails Nevertheless the proposedmethod can be used to calculate the effects of any of theactions mentioned above

221 Creep and Shrinkage In concrete bridges part of thecreep and shrinkage phenomenon occurs after the installa-tion of the track This part has to be taken into account forthe track-structure interaction analysis It produces a deckshortening such that every point of the deck moves towardsthe fixed bearing of the bridge which usually is located atone abutment Consequently the creep and shrinkage strainshave a defined direction

The result is a permanent stress state of the rail-structureconnection which will certainly disappear in time due to thedynamic actions of the passing trains To take into accountthe most unfavourable condition it is prudent to analyse thetwo possibilities the presence and the absence of the imposedstress state due to creep and shrinkage

222 Variation of the Rail and the Deck Temperature Ingeneral the value of the constant temperature variation










i i+1

L

i i + 1

kk

120576Δt

ui ui+1

120576Δt 120576R+F


Fpieri+1


Flongi+1Flongi

Fpieri







120576totalrail119894 =120590rail119894

119864rail+ 120576rail119894

120576totaldeck119894 =120590deck119894

119864deck+ 120576deck119894

(1)




For 119894 = 1 to 119899 do


119860 rail


119860deck

120576rail119894 =120590rail119894

119864rail+ 1205761015840

rail119894

120576deck119894 =120590deck119894

119864deck+ 1205761015840

deck119894




1003816100381610038161003816

EndWhile

Algorithm 1


120590rail119894 =119873rail119894


119860 rail (2)


120590deck119894 =119873deck119894


119860deck (3)












0

2mm











119865ballast119862+119878 =

119906119862+119878sdot119896

1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 1198961003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

119865ballastΔ119879 =

sgn (119906119862+119878) = sgn (119906

Δ119879)

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

sgn (119906119862+119878) = sgn (119906

Δ119879)

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

(sgn (119906119862+119878) sdot 1199060+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906Δ1198791003816100381610038161003816 lt 2 sdot 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906Δ1198791003816100381610038161003816 ge 2 sdot 1199060

(5)















k

k1

uu0

(a)

k

u

k1

k

k1

u0uu0

minusk1minusk1

minusu0

2u0

(b)



050 100 150 200 250 300 350

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus100

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom


Bridge length (m)





050 100 150 200 250 300 350

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Bridge length (m)





Notations






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















i i+1

L

i i + 1

kk

120576Δt

ui ui+1

120576Δt 120576R+F


Fpieri+1


Flongi+1Flongi

Fpieri







120576totalrail119894 =120590rail119894

119864rail+ 120576rail119894

120576totaldeck119894 =120590deck119894

119864deck+ 120576deck119894

(1)




For 119894 = 1 to 119899 do


119860 rail


119860deck

120576rail119894 =120590rail119894

119864rail+ 1205761015840

rail119894

120576deck119894 =120590deck119894

119864deck+ 1205761015840

deck119894




1003816100381610038161003816

EndWhile

Algorithm 1


120590rail119894 =119873rail119894


119860 rail (2)


120590deck119894 =119873deck119894


119860deck (3)












0

2mm











119865ballast119862+119878 =

119906119862+119878sdot119896

1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 1198961003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

119865ballastΔ119879 =

sgn (119906119862+119878) = sgn (119906

Δ119879)

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

sgn (119906119862+119878) = sgn (119906

Δ119879)

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

(sgn (119906119862+119878) sdot 1199060+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906Δ1198791003816100381610038161003816 lt 2 sdot 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906Δ1198791003816100381610038161003816 ge 2 sdot 1199060

(5)















k

k1

uu0

(a)

k

u

k1

k

k1

u0uu0

minusk1minusk1

minusu0

2u0

(b)



050 100 150 200 250 300 350

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus100

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom


Bridge length (m)





050 100 150 200 250 300 350

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Bridge length (m)





Notations






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












For 119894 = 1 to 119899 do


119860 rail


119860deck

120576rail119894 =120590rail119894

119864rail+ 1205761015840

rail119894

120576deck119894 =120590deck119894

119864deck+ 1205761015840

deck119894




1003816100381610038161003816

EndWhile

Algorithm 1


120590rail119894 =119873rail119894


119860 rail (2)


120590deck119894 =119873deck119894


119860deck (3)












0

2mm











119865ballast119862+119878 =

119906119862+119878sdot119896

1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 1198961003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

119865ballastΔ119879 =

sgn (119906119862+119878) = sgn (119906

Δ119879)

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

sgn (119906119862+119878) = sgn (119906

Δ119879)

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

(sgn (119906119862+119878) sdot 1199060+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906Δ1198791003816100381610038161003816 lt 2 sdot 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906Δ1198791003816100381610038161003816 ge 2 sdot 1199060

(5)















k

k1

uu0

(a)

k

u

k1

k

k1

u0uu0

minusk1minusk1

minusu0

2u0

(b)



050 100 150 200 250 300 350

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus100

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom


Bridge length (m)





050 100 150 200 250 300 350

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Bridge length (m)





Notations






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119865ballast119862+119878 =

119906119862+119878sdot119896

1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 1198961003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

119865ballastΔ119879 =

sgn (119906119862+119878) = sgn (119906

Δ119879)

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906119862+119878) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

sgn (119906119862+119878) = sgn (119906

Δ119879)

1003816100381610038161003816119906119862+1198781003816100381610038161003816 lt 1199060

(119906119862+119878+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906119862+119878 + 119906Δ119879

1003816100381610038161003816 lt 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906119862+119878 + 119906Δ1198791003816100381610038161003816 ge 1199060

1003816100381610038161003816119906119862+1198781003816100381610038161003816 ge 1199060

(sgn (119906119862+119878) sdot 1199060+ 119906Δ119879) sdot119896

1199060

1003816100381610038161003816119906Δ1198791003816100381610038161003816 lt 2 sdot 1199060

sgn (119906Δ119879) sdot 119896

1003816100381610038161003816119906Δ1198791003816100381610038161003816 ge 2 sdot 1199060

(5)















k

k1

uu0

(a)

k

u

k1

k

k1

u0uu0

minusk1minusk1

minusu0

2u0

(b)



050 100 150 200 250 300 350

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus100

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom


Bridge length (m)





050 100 150 200 250 300 350

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Bridge length (m)





Notations






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










k

k1

uu0

(a)

k

u

k1

k

k1

u0uu0

minusk1minusk1

minusu0

2u0

(b)



050 100 150 200 250 300 350

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus100

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom


Bridge length (m)





050 100 150 200 250 300 350

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

Long

itudi

nal r

ail s

tress

(Nm

m2)

Zoom

Bridge length (m)





Notations






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









relative displacement method for track-structure interaction

Documents