relative displacement method for track-structure interaction
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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)
4 The Scientific World Journal
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
119906119894+1= 119906119894+ (120576rail119894 minus 120576deck119894) sdot 119871
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
The Scientific World Journal 5
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
6 The Scientific World Journal
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
The Scientific World Journal 7
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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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)
4 The Scientific World Journal
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
119906119894+1= 119906119894+ (120576rail119894 minus 120576deck119894) sdot 119871
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
The Scientific World Journal 5
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
6 The Scientific World Journal
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
The Scientific World Journal 7
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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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)
4 The Scientific World Journal
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
119906119894+1= 119906119894+ (120576rail119894 minus 120576deck119894) sdot 119871
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
The Scientific World Journal 5
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
6 The Scientific World Journal
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
The Scientific World Journal 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 The Scientific World Journal
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
119906119894+1= 119906119894+ (120576rail119894 minus 120576deck119894) sdot 119871
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
The Scientific World Journal 5
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
6 The Scientific World Journal
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
The Scientific World Journal 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 5
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
6 The Scientific World Journal
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
The Scientific World Journal 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 The Scientific World Journal
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
The Scientific World Journal 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
top related