finite element analysis on the fatigue damage under compression of

12 Pages·2012·2.5 MB·English

VIIIInternationalConferenceonFractureMechanicsofConcreteandConcreteStructures
FraMCoS-8
J.G.M.VanMier,G.Ruiz,C.Andrade,R.C.YuandX.X.Zhang(Eds)
FINITE ELEMENT ANALYSIS ON THE FATIGUE DAMAGE UNDER
COMPRESSION OF A CONCRETE SLAB TRACK
ELISAPOVEDA∗,RENAC.YU,JUANC.LANCHA,ANDGONZALORUIZ
∗E.T.S.deIngenierosdeCaminos,C.yP.,UniversidaddeCastilla-LaMancha(UCLM)
Avda. CamiloJose´ Celas/n,13071CiudadReal,Spain
e-mail: [email protected],www.uclm.es
Keywords: Fatigueundercompression,concreteslabtrack,high-speedtrain
Abstract. With the growing use of high speed trains, non-ballasted track has become more popular,
in spite of its higher construction cost compared to ballasted track. This work studies the fatigue
behavior of a slab track using the finite element method. According to Tayabji and Bilow, our slab
track system can be classified as a two-slab layer system: a precast reinforced concrete plate and a
concrete base, separated by a cement-asphalt-mortar sandwich layer. The entire slab track structure
and the soil sub-base are modeled as 3D solid elements, the UIC60 rail represented as truss elements
is attached to the surface slab through the fastening devices. A three-slab track system is modeled to
reducetheboundaryeffects,thoughweonlyfocusontheresponseofthecentralslab. Modalanalysis
is performed to determine the natural frequencies and mode shapes of the system. Real high-speed
train pulses are applied to the rail to carry out the transient analysis. Most unfavorable nodes or
regions are identified for cycle counting, meanwhile the number of cycles causing fatigue failure at
each stress level is estimated according to the new FIB Model Code. Parametric analyses are carried
outtoevaluatetheinfluenceofdifferentgeometricalandmechanicalfactorsonaccumulateddamage.
Minimum requirement for material strength and slab thickness is proposed according to the current
study.
1 INTRODUCTION vious dynamic studies have been concentrated
on the vehicle-track coupled system [7,13]. In
Slab track, also called ballastless track, is a
this current work, we model a three-slab track
modern form of track construction which has
system,togetherwithitscement-asphaltmortar
beenusedsuccessfullythroughouttheworldfor
base, the concrete roadbed and supporting soil.
high speed lines, heavy rail, light rail and tram
Then we proceed to extract its modal response
systems, for an incomplete list see [1–12] and
and carry out the transient analysis applying a
the references within. Compared to ballasted
real time train load. Fatigue analysis is per-
track,concreteslabtrackoffersagreaterdegree
formed as the post-processing of the transient
of trackbed stability, therefore higher running
response. Since the concrete slab is heavily re-
speeds are achievable. Depending on the spec-
inforced, we consider fatigue damage at com-
trum of the train loading, the dynamic response
pressiverangeismorerelevant. Cyclecounting
of a railway slab track can be significantly
is based on the rainflow algorithm [14]. Dam-
larger than its static counterpart. Under such
age map is drawn for one passage of the train,
circumstances, a complete dynamic analysis of
slab life is calculated according to the most un-
theslabstructureisnecessaryinordertopredict
favorableregion.
the service life of the constructed track. Pre-
1
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
Therestofthepaperisstructuredasfollows: properties for this kind of profile [2]. The
the finite element method is described in Sec- cited beam element in ANSYS is well-suited
tion 2, dynamic analysis including modal anal- for linear, large rotation, it includes shear-
ysis, transient calculations and fatigue analysis deformation effects and provides options for
are all carried out in Section 3, results for real unrestrained warping and restrained warping of
high-speed trains in Section 4 and parametric cross-sections. Above all, it allows us to input
analysis in Section 5. Finally, relevant conclu- theexactcross-sectionprofileforUIC60. Arail
sionsaredrawninSection6. length of 15.53 m which covers the three-slab
track system is modeled and symmetry bound-
2 FINITEELEMENTMETHOD aryconditionsareimposedonbothends.
In this section, we set out to model the slab
tracksuperstructureusingtheANSYSParamet-
ricDesignLanguage(APDL),ascriptlanguage
to automate common tasks and build compli- Fastener
cated finite element models in terms of vari-
60 Rail
ables[15].
2.1 Geometryandboundaryconditions
Slab track
Taking advantage of the track line symme-
try (x-axis), half of the track geometry is rep- Cement asphalt mortar
resented. In order to reduce the boundary ef-
Concrete roadbed
fects, three slabs were modeled, but only data
from the central slab were extracted for the fa-
tigue analysis in Section 3. The slab in-plane
dimension is shown in Fig.1(down). Neighbor-
ing slabs are separated by a cylindrical bollard
to prevent lateral and longitudinal movements.
The slab, the cement asphalt mortar (CAM)
layer, the concrete roadbed and the soil sub-
basearealldiscretizedas8-nodevolumetricel-
ements(SOLID45),seeFig.1(up). Thevertical Figure 1: Components of the slab track struc-
and transversal dimensions for each constituent ture (up) and in-plane dimension in meters of
layeraregiveninTab. 1. anindividualslab(down).
It needs to be pointed out that the Voss-
loh rail-fastening device [16] has been simpli-
fied in the following way. The fastening cush- Table 1: Vertical and transversal dimension of
ion, which is a rubber square pad with an edge theFEMmodelforeachcomponentlayer
length of 0.35 m, is represented as deformable
solids, while the fastening mechanism is mod-
Layer Verticaldim. Transversaldim.
eled through connecting the rail inferior nodes
(z-axis)(m) (y-axis)(m)
totheslabsurfacebelowtherubberpad. Mean-
Slabtrack 0.22 1.25
while,no-slidingconstraintisimposedbetween CAmortar 0.10 0.85
therubberpadandtheslabsurface. Concreteroadbed 0.30 1.25
The UIC 60 rail cross-section profile, Soilsubgrade 6.30 7.55
see Fig. 2, is modeled as beam elements FasteningCushion 0.02 0.45
(BEAM188) with its proper section. Its di-
mension and section properties are the known
2
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
where
(cid:114)
k c
ω = and ζ = √ (2)
0
m 2 km
aretheundampednaturalangularfrequencyand
thedampingcoefficientrespectively. Inthecase
ofstiffness-proportionaldampingζ canbewrit-
tenas
c βk 1
Figure 2: The cross-section profile of the ζ = √ = √ = βω (3)
0
UIC60 rail-fastening device seen within the ex- 2 km 2 km 2
perimentalsetuptomeasureitsstiffness.
The input parameter β is inputed as 2ζ/ω .
0
When ζ = 1, the system is critically damped.
An example of the complete finite element Howeverforacomplexsystemwhichhasmany
model is given in Fig. 3. Note that the mesh degrees of freedom thus many modes of vibra-
size has been carefully designed to reduce the tion,wecanonlyselectcertainvibrationmodes
computationalcost. to be more damped than others. Therefore,
knowingζ isapercentagetothematerial’scrit-
icaldampingcoefficient,whoseestimatedvalue
is listed in Tab. 2 for each component material,
ω is set to be the frequency that has the maxi-
0
mum amplitude in the train load. For the pulse
showninFig.5,thecorrespondingfrequencyis
of3.3Hz.
When the concrete slab was casted, cylinder
specimens were also made for standard char-
acterization tests to measure the elastic modu-
lus, tensile strength and compressive strength,
allarelistedinTab.2.
Figure 3: The three-dimensional finite element Table 2: Material properties for each con-
modelforthe3-slabtrackstructure. stituentelement
f f E ν ρ ζ
c t
2.2 Materialcharacterization (MPa) (MPa) (GPa) (kg/m3) (%)
Concreteslab 35 4 35 0.2 2500 1
All the solid elements are modeled as lin-
CAmortar - - 0.1 0.3 1700 10
earelasticmaterialwithastiffness-proportional Concreteroadbed - - 35 0.2 2500 1
Rayleigh damping coefficient β. The damping Soilsubgrade - - 10-57 0.3 1800 5
UIC60Rail - - 200 0.3 7850 0.1
factor ζ is fed as the percentage of the mate-
Fasteningcushion - - 0.006 0.3 800 10
rial’s critical damping ζ . For instance, con-
c
sider an ideal mass-spring-damper system with
mass m, spring constant k and viscous damper
ofdampingcoefficientcsubjectedtoanoscilla- Thesoilsub-gradeisalsomodeledasalinear
tory force, the differential equation for the cor- elastic material but with its stiffness increases
respondinghomogeneoussystemis withthedepthasfollows
x¨+2ζω x˙ +ω2 x = 0 (1) E = E +E ×z (4)
0 0 0 h
3
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
wherez istheaveragelayerdepthinmeters,E onance to occur, both the mode shape and fre-
0
and E are of 10 MPa and 10 MPa/m respec- quency of the external excitation have to coin-
h
tively. Fixed boundary conditions are set for cidewiththatofthestructure. InFig.4,wegive
both transversal and longitudinal directions. A thefirstfourmodesofvibrationoftheslabtrack
parametricstudywascarriedouttodeterminea superstructure and the soil sub-grade. Except
reasonablesoildepthof6.3mtobeincludedin the third mode, which represents the settlement
the slab track structure. As we have mentioned ofthetrack structure,theotherthreemodesde-
before, the rail-fastening device has been sim- scribethebendingloadatdifferentwavelength.
plified,onlythefasteningcushionismodeledas Special care has to be taken if the external train
solidelements. Sincethestiffnessofthefasten- loadhassimilarshapeandspectrum.
ing device, measured through the experimental
setup as shown in Fig. 2, is around 70 kN/mm,
3.2 TransientAnalysis
an equivalent elastic modulus of 6 MPa (listed
inTab.2)isassignedtothefasteningcushionto Transient dynamic analysis is a technique
achieveacomparablestiffness. used to determine the dynamic response of a
The rest of the material properties in Tab. 2, structure under a time-varying load. In this
areallestimatedvalues. type of analysis the inertia or damping effects
of the structure are considered to be important.
3 DYNAMICANALYSIS In this section, we apply a real high-speed train
In this section, we carry out the dynamic pulsetotherailtocarryoutthetransientanaly-
analysis in three steps. First the modal analysis sis. Figure 5(a) shows the load pulse of a high-
isperformedtogettheimportantnaturalmodes speed train (type ETR-Y, see [17]), which is
and corresponding shapes, above all those with composed of two locomotives and ten coaches
a lower frequency. Second, a real high-speed and has a total length of 295.4 m. The train
train load history is applied to the rail to obtain runs with a speed of 300 km/h. By means
thetransientresponse. Third,fatiguedamageis of a Fourier transform, this load pulse is de-
evaluated as the accumulative value according composed into its constituent frequencies, see
toMiner’ssummationrule. Fig.5(b). Figure6showstheevolutionofmaxi-
mumequivalentstressanddeflectioninthecen-
3.1 ModalAnalysis tral slab, note that these maximum values may
Modal analysis is the study of the dynamic correspond to different nodes in the slab. The
properties of structures under vibrational exci- global maximum of the stress and the deflec-
tation, it uses a structure’s overall mass and tion occurs at 0.224 s and 3.732 s respectively.
stiffness to find the various periods that it will The maximum equivalent stress of 0.64 MPa is
naturally resonate at. Detailed modal analy- achieved at 0.224 s around the interior corner.
sis determines the fundamental vibration mode Thestressanddeflectiondistributionofthecen-
shapes and corresponding frequencies. For res- tralslabatthisinstantisshowninFig.7.
4
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
MX
ZZ ZZ
XX YY XX YY
MX
MN MN

(a)f=6.60Hz (b)f=6.96Hz
MX
MX
ZZ ZZ
XX YY XX YY
MN MN

(c)f=7.31Hz (d)f=8.02Hz
Figure4: Thefirstfourmodesoffreevibration.
30 15.0
25 12.5
)
20 z10.0
N) H
k N/
d ( 15 (k 7.5
a d
o a
L 10 o 5.0
L
5 2.5
0 0.0
0 1 2 3 4 1 10 100
Time (s) Frecuency (Hz)
(a) (b)
Figure 5: (a) A high-speed train (ETR-Y) load pulse measured at a rail fastening device and (b) its
frequencyspectrum.
5
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
1.0 1.0
0.8 0.5
)
m
) 0.7 m 0.0
a (
P n
M 0.5 o -0.5
(eqv 0.3 eflecti -1.0
S D
0.2 -1.5
0.0 -2.0
0 1 2 3 4 0 1 2 3 4
Time (s) Time (s)
(a) (b)
Figure 6: Evolution of the maximum equivalent stress and deflection versus time in the central slab
(loadedwiththetrainpulseETR-Y).
MN
Z Z
X Y SEQV(MPa) X Y Deflection(mm)
0.00 -1.05
0.08 MN -1.02
MX 0.16 -0.99
0.24 -0.96
0.32 -0.93
0.40 -0.90
0.48 -0.87
MX
0.56 -0.84
0.64 -0.81

Figure 7: Distribution of the equivalent stress and deflection at loading time 0.224 s (loaded with the
trainpulseETR-Y).
3.3 Fatiguedamageevaluation tor5togettheresultsinFig.8.
Cycle counting is carried out for the com-
We concentrate on evaluating the damage
pressive stress σ by means of the rain-flow al-
c
in the central slab due to compressive stresses.
gorithmdevelopedbyDowningandSocie [14].
Firstweextractthethirdprincipalstress(andits
A subroutine is developed to perform this task
principal direction) to determine the most un-
and to represent the counted cycles as the ma-
favorable node or region, the one that suffers
trixshowninFig.8(b)foreachstressrangeand
themaximumcompressivestressduringtheen-
mean. Taking the characteristic compressive
tire loading duration. Next, taking this princi-
strength f as a reference, the design fatigue
ck
pal direction as a reference, the corresponding
strengthf forcompressionisestimatedac-
cd,fat
stress at this direction σ and its time history
c cordingtothenewModelCode[18]asfollows,
is obtained and plotted in Fig. 8(a). Addition-
ally shown in Fig. 8(a) are the three principal
(cid:20) (cid:18) (cid:19)(cid:21)
stresses, the equivalent stress and the three nor- f
ck
f = 0.85β (t) f 1− /γ
mal stresses. It needs to be pointed out that, cd,fat cc ck 25f c
ck0
in order to achieve certain level of damage, the (5)
load-pulseinFig.5hasbeenamplifiedbyafac- where f is the reference characteristic
ck0
6
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
strengthof10MPa,γ isapartialsecuritycoef- Repeating this procedure for all the nodes in
c
ficientequalsto1.5,β (t)isafactordepending thecentralslab,amapforthefatiguedamageis
cc
on the age of concrete at the beginning of fa- obtained and given in Fig. 9. We observe that
tigueloadingandcanbecalculatedas the most unfavorable nodes are located around
the middle of the slab track, under the central
(cid:34) (cid:35)
(cid:18) 28(cid:19)1/2 fastening cushions (below the third, fourth and
β (t) = exp s f 1− (6)
cc ck t fifthrailfasteningdevices).
where t is the concrete age in days and s is the
coefficient which depends on the strength class
ofcementconcrete[18]. Z
X Y MX Damage[10-4]
The damage caused by those cycles counted 0.000
MN 0.013
inFig.8(b)ateachstressmeanandrangeiscal-
0.026
culated following the Model Code and listed in 0.039
0.052
the matrix shown in Fig. 8(c), the total damage
0.065
isthesumofthosevaluesshowninthedamage 0.078
0.091
matrixaccordingtotheMiner’srule[19]. 0.104
Figure 9: Fatigue damage map at the central
slab(loadedwiththetrainpulseETR-Y,ampli-
fiedbyafactorof5).
ZZZ YYY
4 RESULTS FOR REAL HIGH-SPEED
XXX
TRAINPULSES
In this section, we show the results obtained
(a) for each kind of train that we have analyzed:
AVE Class 103 and RENFE Class 120 -Alvia-.
First the high-speed train load pulse is applied
to the model in the transient analysis. Second,
thecorrespondingstressanditstimehistoryare
obtained for a node at the central slab and fi-
nally the fatigue damage map due to compres-
sionisplotted.
(b)
4.1 AVEClass103Train
The AVE Class 103 is a spanish high-speed
train (trajectory Madrid-Barcelona), made and
named as Velaro E by Siemens. It is of
200.84 m in total length and is composed of
eight cars. Although the train could run with
a speed of 350 km/h, the track is limited to
(c) 300 km/h. Figure 10 shows a typical load pulse
ZZZ YYY Figure 8: (a) Stress history; (b) counted cycles of this high-speed train. Amplifying this pulse
XXX forσ and(c)thecorrespondingdamagecaused with a factor of 4, we obtain the results shown
ZZZ YYY c in Fig. 11. The most damaged region is located
(loaded with the train pulse ETR-Y, multiplied
XXX under the fourth fastening device. The stress
by5).
history for this damaged node is represented in

7
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
Fig. 11(a). The damage map due to fatigue is 4.2 AlviaClass120Train
showninFig.11(b). Asimilardamagedistribu- The Alvia Class 120 is another type of train
tion to that of ETR-Y, see Fig. 9, is observed, used for long distance service in Spain. It is
though the maximum damage amounts to ten composedoffourcoachesandhasatotallength
timesmorethanthatofETR-Y. of 106.23 m. It can run up to 250 km/h. A typ-
ical load pulse for this type of train is shown in
Fig.12.
50
40
50
N) 30 40
k
(
ad 20 ) 30
o N
L 10 (k
d 20
a
0 Lo 10
-10
0 1 2 3 0
Time (s)
-10
0 1 2 3
Figure 10: A typical load pulse for the train
Time (s)
AVES103.
Figure12: Alvia’strainloadpulse.
ZZZ YYY ZZZ YYY
XXX
XXX

(a) (a)
Z Z
X Y MX Damage[10-3] X Y MX Damage[10-5]
0.00 0.00
MN 0.02 MN 0.08
0.04 0.17
0.06 0.25
0.08 0.34
0.10 0.42
0.12 0.50
0.14 0.59
0.16 0.67
(b) (b)
Figure 11: Results for AVE S103, load pulse Figure 13: Results for Alvia, load pulse ampli-
amplifiedbyafactor4: (a)stresshistoryforthe fiedbyafactor4: (a)stresshistoryforthemost
mostunfavorablenodeand(b)damagemap. unfavorablenodeand(b)damagemap.
8
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
Afteramplifyingwithafactor4,theresulted while keeping the rest unchanged. The influ-
stresshistoryforthemostunfavorablenodeand encesofthefasteningstiffness,thelengthofthe
the fatigue map are respectively represented in fastening device, the modulus of the CA mor-
Fig.13(a)andFig.13(b). Asimilardamagedis- tar and the slab thickness are respectively illus-
tribution to that of ETR-Y and AVE S103, see trated in the subfigures a, b, c and d of Fig. 15.
Fig. 9 and Fig. 11(b), is observed, though the It can be observed that, the damage value in-
maximumdamageisthesmallestofthethree. creases with the increase of the fastening stiff-
ness,whiledecreaseswiththelengthofthefas-
5 PARAMETRICANALYSIS tening device. It needs to be remarked that, the
In this section we analyze the influence of damage value decreases to an asymptotic value
the concrete compressive strength, the fasten- withtheincreaseofthemodulusoftheCAmor-
ing stiffness, the length of the fastening device, tar layer; meanwhile, the damage does not vary
the elastic modulus of the cement-asphalt mor- much when the slab thickness increased from
tar and the slab thickness on the damage of the 0.17 m to 0.36 m. This implies that, if the
mostunfavorablenode. rest were kept the same, the CA mortar layer
First we look into the influence of the com- can have a designed modulus as low as 60 to
pressive strength of concrete by keeping con- 80 MPa whereas the the slab track can be as
stant the rest of the material parameters in thinas0.17m.
Tab. 2. The load pulse is a more critical vari-
ant of the pulse shown in Fig. 10 multiplied by 1x10-4 100
a factor 4 or 5. In Fig. 14 we show the result of Life 4
Life 5
this study by plotting the damage (solid lines) 8x10-5 Damage 4 80
Damage 5
and fatigue life (dashed lines) of the most un-
L
favorable node with respect to the compressive ge6x10-5 60 ife
a (
strength f . The average traffic is assumed to m y
ck e
be60trainsperday. Noticethatthedamagede- Da4x10-5 40 ar
s
)
creases rapidly with the increase of f . If a se-
ck 2x10-5 20
curity coefficient of 4 is employed, f should
ck
be no less than 50 MPa to guarantee a fatigue
0 0
lifeof100years. 35 40 45 50 55 60 65 70
Next we study the sensitivity of the rest of f c k (MPa)
the parameters on the damage of the most un-
Figure 14: Damage and fatigue life of the most
favorable node. In particular, the variation of
unfavorable node versus concrete compressive
the fastening stiffness is carried out by vary-
strength.
ingtheelasticmodulusofthefasteningcushion
9
ElisaPoveda,RenaC.Yu,JuanC.LanchaandGonzaloRuiz
4x10-4 4x10-4
3x10-4 3x10-4
e e
g g
ma2x10-4 ma2x10-4
a a
D D
1x10-4 1x10-4
0 0
25 35 45 55 65 75 85 95 0.3 0.4 0.5
Fastening stiffness (kN/mm) Fastening length (m)
a) b)
2x10-3 1x10-3
8x10-4
ge e6x10-4
a g
m1x10-3 a
m
Da a4x10-4
D
2x10-4
0 0
40 80 120 160 200 0.1 0.2 0.3 0.4
CA Mortar modulus (MPa) Slab track thickness (m)
c) d)
Figure 15: Damage of the most unfavorable node versus the fastening stiffness and the length of the
fasteningdevice,theelasticmodulusofthecementasphaltmortarandthethicknessoftheslab.
6 CONCLUSION predicting the slab fatigue life and is partic-
ularly useful in optimizing the slab geometry
We have modeled a three-slab track super-
and the superstructure. The current parametric
structure and the soil sub-grade in ANSYS to
studyhassuggestedthat
evaluatethefatiguebehaviorinthecentralslab.
Numerical procedures are automated using the • for a security coefficient of four, the im-
APDL script language. Modal analysis is per- pliedconcreteshouldhaveacompressive
formed to determine the fundamental modes strength no less than 50 MPa to give a
andshapes. Transientanalysisiscarriedoutfor 100-yearfatiguelifeguarantee;
a high-speed train load applied to the rail. Fa-
tigue damage due to compression is evaluated • the slab track thickness can be as thin as
followingtheModelCode. Sinceonetrainpas- 0.17mforalowleveloffatiguedamage;
sage causes no damage, real train pulses from
• fatigue damage increases with the in-
ETR-Y, AVE Class 103 and Alvia Class 120
creaseofthestiffnessofthefasteningde-
multiplied by a factor 4 or 5 are applied to the
vice;
rail for parametric studies. The damage map
obtainedshowsthattheunfavorableregionsare • for a low level of fatigue damage, the
locatedunderthethirdtofifthfasteningdevice. elastic modulus of the cement-asphalt-
The developed methodology is promising in mortarlayercanbeaslowas60-80MPa.
10