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Research ArticleResearch on Mechanical Performance of CRTS III
Plate-TypeBallastless Track Structure under Temperature Load Based
onProbability Statistics
Zhiwu Yu,1 Ying Xie ,2 and Xiuquan Tian3
1Professor, National Engineering Laboratory for High Speed
Railway Construction, Central South University, Changsha,
China2Ph.D. Candidate, Central South University, Changsha,
China3Central South University, Changsha, China
Correspondence should be addressed to Ying Xie;
[email protected]
Received 24 March 2019; Revised 28 May 2019; Accepted 4 June
2019; Published 26 June 2019
Academic Editor: Dimitris Rizos
Copyright © 2019 Zhiwu Yu et al. ,is is an open access article
distributed under the Creative Commons Attribution License,which
permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
CRTS III (China Railway Track System III) slab ballastless track
structure is one of the high-speed railway ballastless
trackstructures which has Chinese independent intellectual property
rights. ,e mechanical performance of CRTS III slab ballastlesstrack
structure under temperature load has not been clear yet. ,erefore,
through temperature field model and temperature loadvalues selected
by statistics analysis based on long-term meteorological data, the
mechanical performance of ballastless trackstructure is studied
under two typical working conditions with different safety
probability. It is found that the daily extreme valuesof monthly
axial uniform temperature and the daily maximum temperature
gradient obey certain statistical laws. In addition, themaximum
tensile stress of the self-compacting concrete layer is located in
the middle and edge of the slab bottom and the side ofthe slab. ,e
maximum tensile stress of the base plate is located at the edge of
the surface of the layer and the inner edge of thelimiting
groove.,e interface normal tensile stress is located at the end and
corner of the interface. Furthermore, maximum stressincreases with
the increase of safety probability.
1. Introduction
At present, high-speed railway is developing rapidly inChina.
Specifically, the ballastless track used in CRTS (ChinaRailway
Track System) III; is developed based on the maincharacteristics of
that used in other ballastless track structurewith certain
improvements which has Chinese independentintellectual property
rights. ,e CRTS III slab ballastlesstrack structure consists of
rail, fastening system, track slab,self-compacting concrete layer,
isolation layer, bed plate,support layer, and other parts, as shown
in Figure 1. Untilnow, in China, the total length of CRTS III slab
ballastlesstrack line under construction and operation is
nearly4000 kilometers [1]. In order to implement the strategy
ofChina’s high-speed trains going global, CRTS III slab
bal-lastless track becomes the important technology of
China’shigh-speed railway.
As an important environmental factor, temperature loadhas a
significant impact on the mechanical performance anddurability of
slab ballastless track structure system used inhigh-speed railway.
,erefore, the mechanical performanceof ballastless track structure
under temperature load hasattracted the attention of many scholars.
Generally, tem-perature load of ballastless track structure mainly
includesaxial uniform temperature and temperature gradient.China’s
high-speed railway design specification stipulated inthe
temperature gradient is 45°C/m. Many scholars andresearch
institutions took some temperature tests on bal-lastless track
structure and found that the track plate tem-perature gradient was
between 40∼80°C/m, maximumnegative temperature gradient is half of
the maximumpositive temperature gradient, and the temperature
changeof the bed plate is not obvious [2–5]. ,e long-term
ob-servation of ballastless track structure shows that the
HindawiAdvances in Civil EngineeringVolume 2019, Article ID
2975274, 16 pageshttps://doi.org/10.1155/2019/2975274
mailto:[email protected]://orcid.org/0000-0003-2378-5800https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/2975274
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temperature field inside ballastless track structure is
mainlyaffected by solar radiation, wind speed, and
atmospherictemperature and varied on a daily basis with
temperature,and the vertical temperature presents a nonlinear
distri-bution [6–11]. With the increase of depth, the amplitude
oftemperature change in ballastless track structure decreasesand
the peak value lags [12–16]. ,e study in literature [17]further
shows that both positive and negative temperaturegradients show
nonlinear distribution and their occurrencetime is basically
equal.
However, these studies are generally based on the spe-cific
environment or special working conditions which thevalues of
various parameters are not standardized andunified. ,erefore, it is
necessary to determine the appro-priate temperature load first, and
then study the mechanical
performance of CRTS III plate-type ballastless track struc-ture
system under the temperature load.
2. Temperature Field Model of BallastlessTrack Structure
As a multilayer structure, the temperature change in theCRTS III
slab ballastless track structure caused by the changeof the
external environment is mainly concentrated in theposition
shallower from the surface [10]. ,erefore, con-sidering that the
boundary condition does not change alongthe track longitudinal, the
calculation of temperature field ofballastless track can be
simplified to the solution of a one-dimensional differential
equation of heat conduction. Whenthe temperature changes
harmonically on the structuresurface, the temperature field inside
ballastless trackstructure can be regarded as the superposition of
temper-ature inside the plate which was caused by the action
ofatmospheric temperature θa and equivalent temperatureincrement
Δθ, respectively, which is given as follows:
θ(x, t) � θac(x, t) + Δθc(x, t). (1)
In addition, the temperature variation within thestructure
θac(x, t) coursed by the change of atmospherictemperature θa can be
shown as follows:
θac(x, t) �
θave + θampe−x����π/2αT
√
cosπT
t−T + Tmin
Tπ − x
����π
2αT
, Tmin ≤ t
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(i) ,e design reference period of ballastless trackstructure is
determined to be T years.
(ii) According to the periodicity and regularity oftemperature
change in a certain area, the designreference period can be divided
into N periods (τi),which can be shown as follows: T � Ni�1τi.
(iii) ,e distribution function (Fτi ,max(x)) of the maxi-mum
value of load (Qi,max) within the time period(τi) was obtained by
statistics. ,e distributionfunction of the maximum load in each
time periodis the same and independent. ,erefore, the max-imum
temperature load in the design referenceperiod can be shown as
follows:QT,max � max Q1,max, Q2,max, . . . , QN,max . (4)
And its distribution function can be shown asfollows:
P QN,max ≤ x � Fτi ,max(x) N
. (5)
Similarly, the minimum load and its distribution inthe design
reference period can be obtained:
QT,min � min Q1,min, Q2,min, . . . , QN,min ,
P QN,max >x � 1− 1−Fτi ,min(x) N
.
(6)
Based on the time period probability distribution(Fτi(x)) and
its statistical parameters, the distri-bution (Fτi ,max(min)(x)) of
the maximum (mini-mum) load (Qτi ,max(min)) and its
statisticalparameters in the design reference period can beobtained
according to the above formula.
3.2. Statistical Analysis of Axial Uniform Temperature.Since the
CRTS III slab ballastless track structure used in thisresearch
relies on the Zhengzhou-Xuzhou passenger dedi-cated line, in order
to be more representative, the meteo-rological data (maximum and
minimum temperatures andsolar radiation intensity) of every day in
Zhengzhou from2008 to 2017 were counted, and then the daily value
of axialuniform temperature of ballastless track structure was
cal-culated. However, it is difficult to obtain the distribution
lawof the diurnal extreme value of the axial uniform temper-ature
in the past ten years. Considering that the temperatureof the same
month is basically similar in different years, thefrequency
histogram of the days with different extremetemperature values was
obtained by taking the months ofdifferent years as the statistical
period, which for somemonths are shown in Figures 2–5.
It can be seen from Figures 2–5 that the daily extremevalues of
monthly axial uniform temperature of CRTS IIIslab ballastless track
structure in Zhengzhou show a trend ofnormal distribution, which
can be fitted by using the normaldistribution function.
Furthermore, by using the Jarque–Bera test function in Matlab, the
goodness of fit test ofnormal distribution was carried out which
the significance
level α was 5%, as shown in Figures 6–9. ,e estimatedvalues of
the normal distribution parameters of the dailyextreme values of
the axial uniform temperature in differentmonths are shown in Table
1.
According to the parameter estimation of the dailymaximum
distribution function of axial uniform temperatureof ballastless
track structure in different months, the prob-ability distribution
function of axial uniform temperature indifferent months in
Zhengzhou can be obtained as follows:
Fτ−m,i(x) �1
���2π
√σi
x
−∞e
x− μi( )2( /2σ2i dx. (7)
It is considered that the daily distribution of axialuniform
temperature of ballastless track structure in eachmonth is
independent from each other. ,erefore, the an-nual probability
distribution function of the daily extremevalue of the axial
uniform temperature is shown as follows:Fτ−y,i(x) � ω1Fτ−m,1(x) +
ω2Fτ−m,2(x) + · · · + ω12Fτ−m,12(x),
(8)
where Fτ−m,i(x) denotes the probability distribution func-tion
of the daily extreme value of axial uniform temperatureof
ballastless track structure of ith month in a year.
Fτ−y,i(x)denotes the annual probability distribution function of
thedaily extreme value of the axial uniform temperature. ωidenotes
the weight coefficient which can be 1/12.
,e daily extreme distribution of axial uniform tem-perature of
ballastless track structure in the design referenceperiod is shown
as follows:
FT,max(x) � ω1Fτ−m,1(x) + ω2Fτ−m,2(x) + · · ·
+ ω12Fτ−m,12(x)N
,
FT,min(x) � 1− 1− ω1Fτ−m,1(x) + ω2Fτ−m,2(x) + · · ·
+ ω12Fτ−m,12(x)N
.
(9)
,e base design period of high-speed railway is 60 yearsin China.
,erefore, the corresponding axial uniformtemperature can be
obtained by different safety probabilitypercentages such as 50%,
90%, and 95%, as shown in Table 2.
3.3. Statistical Analysis of Temperature Gradients. ,e
dailymaximum temperature gradient of CRTS III slab ballastlesstrack
structure was calculated and analyzed to obtain thehistograms which
are shown in Figure 10. As can be seenfrom Figure 10, the daily
maximum temperature gradient ofballastless track structure presents
an obvious skewed dis-tribution and tends to the extreme value
I-type distribution:
Fτ(x) � exp −exp −x− βα
,
α �σ
1.2826,
β � μ− 0.5772α.
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(10)
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,erefore, the statistical parameters of the daily maxi-mum
annual distribution of temperature gradient of CRTSIII plate-type
ballastless track structure in Zhengzhou can befitted, as shown in
Table 3.
,e base design period of high-speed railway is 60 yearsin China.
,erefore, the corresponding temperature gra-dients can be obtained
by calculating the safety probabilitypercentages such as 50%, 90%,
and 95%, as shown in Table 4.
4. Finite Element Model of BallastlessTrack Structure
4.1. 4e Selection of Parameters. In the finite element
(FE)model, the main structure components of CRTS III
slabballastless track system consist of rail, track board,
self-compacting concrete (SCC) layer, isolation layer, bed
pate,
and other parts. Geometric parameters and material pa-rameters
of components are selected according to the rel-evant literature
[19], which are listed in Table 5.
4.2. 4e Selection of Constitutive Models. In this work,
theconstitutive models selection of concrete and steel bars is
thesame as the literature [20]. ,e constitutive models ofconcrete
can be shown as follows:
σn � [I−D(n)]E0εn, (11)
where σn denotes the effective stress. εn denotes the
totalstrain, and E0 represents initial Young’s modulus. D(n)denotes
the damage variable.
,e constitutive models of steel bars can be shown asfollows:
11 12 13 14 15 16 17 18 19 20 21 22 23 24 2510Temperature
0
5
10
15
20
25
30
35
40
February
Freq
uenc
y
(a)
1 2 3 4 5 6 7 8 9 10 11 12 13 140Temperature
0
10
20
30
40
50
Freq
uenc
y
February
(b)
Figure 2: ,e histogram of the daily extreme value of axial
uniform temperature in February. (a) ,e maximum. (b) ,e
minimum.
May
0
10
20
30
40
50
60
Freq
uenc
y
36 37 38 39 40 41 42 43 44 45 46 4735Temperature
(a)
May
0
10
20
30
40
50
60
70Fr
eque
ncy
20 21 22 23 24 25 26 27 28 29 30 3119Temperature
(b)
Figure 3: ,e histogram of the daily extreme value of axial
uniform temperature in May. (a) ,e maximum. (b) ,e minimum.
4 Advances in Civil Engineering
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October
0
10
20
30
40
50
60Fr
eque
ncy
25 26 27 28 29 30 31 32 33 34 35 36 37 38 3924Temperature
(a)
October
0
10
20
30
40
50
60
Freq
uenc
y
16 17 18 19 20 21 22 23 24 25 26 2715Temperature
(b)
Figure 4: ,e histogram of the daily extreme value of axial
uniform temperature in October. (a) ,e maximum. (b) ,e minimum.
November
0
10
20
30
40
50
60
70
80
90
Freq
uenc
y
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
3214Temperature
(a)
November
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
246Temperature
0
20
40
60
80
100Fr
eque
ncy
(b)
Figure 5: ,e histogram of the daily extreme value of axial
uniform temperature in November. (a) ,e maximum. (b) ,e
minimum.
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
12 14 16 18 20 22 24 2610Data
(a)
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
2 4 6 8 10 12 140Data
(b)
Figure 6: ,e distribution probability diagram of the daily
extreme value of axial uniform temperature in February. (a) ,e
maximum.(b) ,e minimum.
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Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
36 38 40 42 44 46 4834Data
(a)
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
20 22 24 26 28 30 3218Data
(b)
Figure 7: ,e distribution probability diagram of the daily
extreme value of axial uniform temperature in May. (a) ,e
maximum.(b) ,e minimum.
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
26 28 30 32 34 36 38 4024Data
(a)
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0Pr
obab
ility
16 18 20 22 24 26 2814Data
(b)
Figure 8: ,e distribution probability diagram of the daily
extreme value of axial uniform temperature in October. (a) ,e
maximum.(b) ,e minimum.
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
14 16 18 20 22 24 26 28 30 32 3412Data
(a)
Normal probability plot
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty
6 8 10 12 14 16 18 20 22 244Data
(b)
Figure 9: ,e distribution probability diagram of the daily
extreme value of axial uniform temperature in November. (a) ,e
maximum.(b) ,e minimum.
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Table 1:,e estimated values of the normal distribution
parameters of the daily extreme values of the axial uniform
temperature in differentmonths.
Months 1 2 3 4 5 6 7 8 9 10 11 12
,e maximum μ 12 17 25 33 40 44 48 47 40 32 23 15σ 2.5 3.0 3.3
3.0 2.1 1.8 2.1 2.3 2.4 2.7 3.3 2.5
,e minimum μ 3.6 6.8 12 19 25 29 33 33 28 21 14 6.7σ 2.4 2.7 2.7
2.6 1.9 1.6 2.0 1.9 2.0 2.2 3.1 2.5
Table 2: Axial uniform temperature based on different safety
probability percentages.
T� 60 yearsSafety probability (%) P � 50 P � 90 P � 95,e maximum
(°C) 50.6544 52.4781 53.0506,e minimum (°C) 0.7437 −1.3437
−1.9973
0
100
200
300
400
500
600
Freq
uenc
y
40 50 60 70 80 90 10030Temperature gradient
(a)
0
200
400
600
800
1000
Freq
uenc
y
–40 –30 –20 –10 0–50Temperature gradient
(b)
Figure 10: Histogram of daily maximum frequency of temperature
gradient. (a) Positive temperature gradient. (b) Negative
temperaturegradient.
Table 3: ,e statistical parameters of the daily maximum annual
distribution of temperature gradient.
A βPositive temperature gradient 11.8 70.5Negative temperature
gradient 7.3 35.8
Table 4: Temperature gradient daily maximum load based on
different safety probability percentages.
T� 60 yearsSafety probability (%) P � 50 P � 90 P � 95Positive
temperature gradient (°C/m) 88.2 92.3 93.6Negative temperature
gradient (°C/m) 46.7 49.3 50.1
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fy(N) � σmin + Δσ( 1−lgNlgNf
1−σmin + Δσ
fy ,
σ(N) �Esε(N), Δεr(N− 1)< ε(N)≤ εy(N),
fy(N), ε(N)> εy(N),
⎧⎪⎨
⎪⎩
εr(N) � Δεr(N− 1) +fy
Es,
Δεr(N− 1) �fy(N)−fy(N− 1)
Es,
(12)
where Es denotes the initial Young’s modulus of rebar. fy
andfy(N) denote the initial yield strength and yield strength
afterN times fatigue load. εr(N) and Δεr(N) denote the yield
strainand residual strain after N times fatigue load.
4.3. Development of FE Model of CRTS III Slab BallastlessTrack
Structure. ,e FE model of the CRTS III slab bal-lastless track
system is developed on the commercialsoftware ANSYS, as shown in
Figure 11. ,e FE units ofmain structural layers are determined as
shown in Table 6.In addition, the interfaces between slabs should
be con-sidered. ,e interfaces between track slab and
self-com-pacting concrete layer can be modelled by
cohesiveelements. And the interfaces between
self-compactingconcrete layer and isolation layer and the
interfaces be-tween isolation layer and bed plate can be modelled
bycontact interface element with detailed parameters de-termined
based on the internal frictional force of these twointerfaces
[21].
Additionally, the slab ballastless track structure
wasconsolidated with the support layer simulated by using thespring
element in which all degrees of freedom of the lowernode are
constrained. ,e degrees of freedom in two hori-zontal directions of
the bed plate are constrained, and alldegrees of freedom of the
rail are constrained. Furthermore,all degrees of freedom at both
ends of the rail are constrained.
4.4. Model Verification. He [22] studied the deformationbehavior
of the CRTS III slab ballastless track system underthe temperature
effect on the rigid support. Research shows
that under the action of 100°C/m temperature gradient,
themaximum displacement of track plate relative to the baseplate is
0.951mm which located at the angle of plate. Fur-thermore, under
the action of −20°C/m temperature gradient,the maximum displacement
of the track plate relative to thebase plate is 0.180mmwhich also
located at the angle of plate.
Based on the FE model, calculate the deformation ofCRTS III slab
ballastless track under the same condition asthat in literature
[22], as shown in Figures 12 and 13. ,emaximum displacement of the
track plate relative to the baseplate is 0.861mm under the action
of 100°C/m temperaturegradient, which is 9.5% smaller than the
measured value inreference literature [22]. In addition, the
maximum dis-placement of the track plate relative to the base plate
is0.192mm under the action of −20°C/m temperature gra-dient, which
is 3.2% smaller than the measured value inreference literature
[22]. ,erefore, the FE model proved tobe reasonable and
reliable.
5. Mechanical Performance of Key StructuralLayers of Ballastless
Track Structure
Generally, in the track system, due to the high strength of
theconcrete material and the two-dimensional prestressingstructure
used in the track board, the self-compacting con-crete layer and
bed plate are required to be carefully examinedon the performance
under the temperature loading.,ere aretwo conditions that apply to
the FE model: condition 1 (thedaily maximum value of axial uniform
temperature iscombined with the daily maximum value of the
positivetemperature gradient) and condition 2 (the daily
minimumvalue of axial uniform temperature is combined with the
dailymaximum value of the negative temperature gradient).
5.1. Stress Condition of Self-Compacting Concrete Layer.Under
condition 1, the stress of the self-compacting concretelayer is
shown in Figures 14 and 15. ,e maximum longi-tudinal tensile stress
of the self-compacting concrete layer islocated in the middle of
the bottom surface and the side ofthis layer.,emaximum transverse
tensile stress is located atthe bottom edge of the layer and the
side of the layer.Furthermore, the maximum stress increases with
the in-crease of the fractal value.
Under condition 2, the stress of the self-compactingconcrete
layer is shown in Figures 16 and 17. ,e maximum
Table 5: ,e basic specifications of main members.
Rail Track slab Self-compactingconcrete layer Bed plate
Isolation layer Support layer
Length (mm) 5600 5600 5600 5600Width (mm) 2500 2500 2900
2600Height (mm) 200 90 200 4Concrete strength C60 C40 C30Elasticity
modulus 210GPa 36000MPa 34000MPa 32000MPa 3.32MPaPoisson’s ration
0.3 0.2 0.2 0.2 0.35,ermal expansion coefficient (°C−1) 1.18×10−5
1×10−5 1× 10−5 1× 10−5 1× 10−5
Stiffness 1000MPa/mDensity 7800 kg/m3 2500 kg/m3 2500 kg/m3 2500
kg/m3 700 g/m2
8 Advances in Civil Engineering
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longitudinal tensile stress of the self-compacting concretelayer
is located at the side of the layer and the maximumtransverse
tensile stress is located at the bottom edge of thelayer. In
addition, the maximum longitudinal tensile stressincreases with the
increase of the fractal value, but themaximum transverse tensile
stress remained basically stable.Furthermore, both longitudinal and
transverse maximumcompressive stress decreases slightly with the
increase of thefractal value, as shown in Table 7.
5.2. Stress Condition of Bed Plate. Under condition 1, thestress
of the bed plate is shown in Figures 18 and 19. emaximum
longitudinal tensile stress of the bed plate islocated at both ends
of the bottom surface of the layer and
the inner edge of the limiting groove. e maximumtransverse
tensile stress is located at the side of the layerand the inner
edge of the limiting groove. Under con-dition 2, the stress of the
bed plate is shown in Figures 20and 21. Both longitudinal and
transverse maximumtensile stresses are located at the edge of the
upper surfaceof the base plate. Furthermore, the maximum stress
in-creases with the increase of the fractal value, as shown inTable
8.
5.3. Stress Condition of Interface between Track Slab and
Self-CompactingConcrete Layer. Under condition 1, the stress
ofinterface between track slab and self-compacting concretelayer is
shown in Figures 22 and 23. e maximum normal
Table 6: e nite element units of main structural layers.
Structural layers Finite element unitsConcrete Solid 65Rail Beam
188Support layer Combin 14Fastening system Combin 14Isolation layer
Solid 45Interface between track slab and SCC layer
Inter205Interface between isolation layer and SCC layer Target
170Interface between isolation layer and bed plate Target 170
Fastening systemPrestressed steel bars
Interface
Isolation layer
Support layer
Rail
Track slab
Self-compactingconcrete layer
Bed plate
Figure 11: e nite element model of the CRTS III slab ballastless
track system.
–0.9
29E
– 03
–0.5
80E
– 03
–0.2
32E
– 03
0.11
7E –
03
0.46
5E –
03
0.81
4E –
03
0.00
1163
0.00
1511
0.00
186
0.00
2208
(a)
–0.1
16E
– 03
–0.1
02E
– 03
–0.8
87E
– 04
–0.7
52E
– 04
–0.6
17E
– 04
–0.4
82E
– 04
–0.3
47E
– 04
–0.2
11E
– 04
–0.5
88E
– 05
–0.7
64E
– 05
(b)
Figure 12: e deformation under 100°C/m temperature gradient. (a)
Track slab. (b) Bed plate.
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–0.7
98E
– 04
–0.4
97E
– 04
–0.1
95E
– 04
0.10
6E –
04
0.40
8E –
04
0.70
9E –
04
0.10
1E –
03
0.13
1E –
03
0.16
1E –
03
0.19
2E –
03
(a)
–0.5
05E
– 04
–0.4
43E
– 04
–0.3
81E
– 04
–0.3
18E
– 04
–0.2
56E
– 04
–0.1
94E
– 04
–0.1
31E
– 04
–0.6
91E
– 05
–0.6
82E
– 06
0.55
5E –
05
(b)
Figure 13: e deformation under −20°C/m temperature gradient. (a)
Track slab. (b) Bed plate.
–0.9
33E
+ 07
–0.8
17E
+ 07
–0.7
00E
+ 07
–0.5
84E
+ 07
–0.4
67E
+ 07
–0.3
51E
+ 07
–0.2
34E
+ 07
–0.1
18E
+ 07
0.11
5E +
07
–145
46.9
(a)
–0.1
04E
+ 08
–0.9
11E
+ 07
–0.7
81E
+ 07
–0.6
51E
+ 07
–0.5
20E
+ 07
–0.3
90E
+ 07
–0.2
60E
+ 07
–0.1
29E
+ 07
0.13
1E +
07
1003
4.3
(b)
–0.1
09E
+ 08
–0.9
56E
+ 07
–0.8
19E
+ 07
–0.6
83E
+ 07
–0.5
47E
+ 07
–0.4
10E
+ 07
–0.2
74E
+ 07
–0.1
37E
+ 07
0.13
5E +
07
–111
74.8
(c)
Figure 14: Longitudinal stress of the self-compacting concrete
layer under condition 1. (a) P� 50%. (b) P� 90%. (c) P� 95%.
–0.5
35E
+ 07
–0.4
57E
+ 07
–0.3
79E
+ 07
–0.3
00E
+ 07
–0.2
22E
+ 07
–0.1
44E
+ 07
–661
887
1191
03
0.16
8E +
07
9000
93
(a)
–0.6
38E
+ 07
–0.5
47E
+ 07
–0.4
57E
+ 07
–0.3
66E
+ 07
–0.2
76E
+ 07
–0.1
86E
+ 07
–952
088
–480
09.3
0.17
6E +
07
8560
70
(b)
–0.6
89E
+ 07
–0.5
92E
+ 07
–0.4
96E
+ 07
–0.3
99E
+ 07
–0.3
02E
+ 07
–0.2
06E
+ 07
–0.1
09E
+ 07
–122
362
0.18
1E +
07
8447
37
(c)
Figure 15: Transverse stress of the self-compacting concrete
layer under condition 1. (a) P� 50%. (b) P� 90%. (c) P� 95%.
10 Advances in Civil Engineering
-
tensile stress is located at both ends of the interface, and
themaximum tangential stress is located at the edge of
theinterface. In addition, under condition 2, the stress of
in-terface between track slab and self-compacting concretelayer is
shown in Figures 24 and 25. e maximum normaltensile stress is
located at the corner of the interface and themaximum tangential
stress is located at the edge of theinterface Figure 25.
Furthermore, both normal and tan-gential maximum stress increases
with the increase of thefractal value, as shown in Table 9.
6. Fatigue Performance of Key StructuralLayers of Ballastless
Track Structure
A relevant study [17] shows that the positive
temperaturegradient and the negative temperature gradient occur
al-ternately, and the time of these two gradients in a day
isbasically the same.erefore, it can be assumed that
workingconditions 1 and 2 all work 365 times in a year. e
fatiguelife under temperature can be calculated by using the
fol-lowing formula [23]:
Table 7: Maximum stress of the self-compacting concrete
layer.
Condition 1 Condition 2Safety probability (%) P� 50 P� 50 P� 90
P� 50 P� 90 P� 95Maximum longitudinal tensile stress (MPa) 1.15
1.31 1.35 1.89 2.25 2.40Maximum longitudinal compressive stress
(MPa) 9.33 10.4 10.9 2.56 2.38 2.25Maximum transverse tensile
stress (MPa) 1.68 1.76 1.81 2.10 2.11 2.11Maximum transverse
compressive stress (MPa) 5.35 6.38 6.89 2.37 2.32 2.25
–0.2
56E
+ 07
–0.2
07E
+ 07
–0.1
57E
+ 07
–0.1
08E
+ 07
–581
773
–868
17.7
4081
37
9030
92
0.18
9E +
07
0.14
0E +
07
(a)
–0.2
28E
+ 07
–0.1
77E
+ 07
–0.1
27E
+ 07
–769
262
–266
447
2363
68
7391
83
0.12
4E +
07
0.22
5E +
07
0.17
4E +
07
(b)
–0.2
25E
+ 07
–0.1
73E
+ 07
–0.1
22E
+ 07
–699
491
–183
671
3321
48
8479
68
0.13
6E +
07
0.24
0E +
07
0.18
8E +
07
(c)
Figure 16: Longitudinal stress of the self-compacting concrete
layer under condition 2. (a) P� 50%. (b) P� 90%. (c) P� 95%.
–0.2
37E
+ 07
–0.1
87E
+ 07
–0.1
38E
+ 07
–878
277
–381
542
1151
92
6119
26
0.11
1E +
07
0.21
0E +
07
0.16
1E +
07
(a)
–0.2
32E
+ 07
–0.1
82E
+ 07
–0.1
33E
+ 07
–840
386
–348
131
1441
25
6363
80
0.11
3E +
07
0.21
1E +
07
0.16
2E +
07
(b)
–0.2
25E
+ 07
–0.1
76E
+ 07
–0.1
28E
+ 07
–794
912
–310
036
1748
39
6597
15
0.11
4E +
07
0.21
1E +
07
0.16
3E +
07
(c)
Figure 17: Transverse stress of the self-compacting concrete
layer under condition 2. (a) P� 50%. (b) P� 90%. (c) P� 95%.
Advances in Civil Engineering 11
-
–0.653E+07
–0.571E+07
–0.489E+07
–0.407E+07
–0.325E+07
–0.243E+07
–0.161E+07
–795765
23406.8
842579
(a)
–0.681E+07
–0.594E+07
–0.508E+07
–0.421E+07
–0.335E+07
–0.248E+07
–0.162E+07
–751020
113846
978713
(b)
–0.721E+07
–0.630E+07
–0.539E+07
–0.447E+07
–0.356E+07
–0.265E+07
–0.173E+07
–818498
94851.6
0.101E
+07
(c)
Figure 18: Longitudinal stress of the bed plate under condition
1. (a) P� 50%. (b) P� 90%. (c) P� 95%.
–0.560E+07
–0.483E+07
–0.406E+07
–0.329E+07
–0.252E+07
–0.175E+07
–980341
–210121
560098
0.133E
+07
(a)
–0.507E+07
–0.435E+07
–0.363E+07
–0.291E+07
–0.220E+07
–0.148E+07
–758985
–40357.2
678070
0.140E
+07
(b)
–0.538E+07
–0.460E+07
–0.383E+07
–0.306E+07
–0.229E+07
–0.152E+07
–743982
27925.2
799833
0.157E
+07
(c)
Figure 19: Transverse stress of the bed plate under condition 1.
(a) P� 50%. (b) P� 90%. (c) P� 95%.
–0.321E+07
–0.279E+07
–0.237E+07
–0.194E+07
–0.152E+07
–0.110E+07
–678665
–256820
165025
586871
(a)
–0.290E+07
–0.249E+07
–0.207E+07
–0.166E+07
–0.125E+07
–836233
–423783
–11332.4
401118
813569
(b)
–0.289E+07
–0.247E+07
–0.205E+07
–0.163E+07
–0.121E+07
–797262
–379689
37883.7
455456
873029
(c)
Figure 20: Longitudinal stress of the bed plate under condition
2. (a) P� 50%. (b) P� 90%. (c) P� 95%.
12 Advances in Civil Engineering
-
–0.165E+07
–0.134E+07
–0.102E+07
–699350
–380823
–62296.4
256231
574757
893284
0.121E
+07
(a)
–0.132E+07
–0.103E+07
–742453
–454627
–166800
121026
408856
696679
984506
0.127E
+07
(b)
–0.143E+07
–0.112E+07
–803657
–491657
–179657
132344
444344
756344
0.107E
+07
0.138E
+07
(c)
Figure 21: Transverse stress of the bed plate under condition 2.
(a) P� 50%. (b) P� 90%. (c) P� 95%.
Table 8: Maximum stress of the bed plate.
Condition 1 Condition 2Safety probability (%) P� 50 P� 90 P� 95
P� 50 P� 90 P� 95Maximum longitudinal tensile stress (MPa) 0.84
0.98 1.01 0.59 0.81 0.87Maximum longitudinal compressive stress
(MPa) 6.53 6.81 7.21 3.21 2.90 2.89Maximum transverse tensile
stress (MPa) 1.33 1.40 1.57 1.21 1.27 1.38Maximum transverse
compressive stress (MPa) 5.60 5.07 5.38 1.65 1.32 1.43
–440207
–340695
–241183
–141671
–42158.6
57353.5
156866
256378
355890
455402
(a)
–494000
–383939
–273878
–163817
–53756.1
56304.9
166366
276427
386488
496549
(b)
–512629
–396382
–280134
–163887
–47639.3
68608.1
184856
301103
417350
533598
(c)
Figure 22: Normal stress of interface under condition 1. (a) P�
50%. (b) P� 90%. (c) P� 95%.
–166
043
–122
582
–791
20.4
–356
58.9
7802
.68
5126
4.2
9472
5.7
1381
87
1816
49
2251
10
(a)
–166
043
–122
582
–791
20.4
–356
58.9
7802
.68
5126
4.2
9472
5.7
1381
87
1816
49
2251
10
(b)
–231
726
–169
961
–108
195
–464
29.1
1533
6.7
7710
2.4
1388
68
2006
34
2624
00
3241
65
(c)
Figure 23: Tangential stress of interface under condition 1. (a)
P� 50%. (b) P� 90%. (c) P� 95%.
Advances in Civil Engineering 13
-
logN± � 14.7− 13.5σ±max − σ±minf±cu − σ±min
, (13)
where f±cu denotes the tensile (+) and compressive (−)strength
of concrete.N denotes the fatigue life. σmax and σmindenote the
maximum and minimum stress under temper-ature load.
erefore, damage of the self-compacting concrete layerand bed
plate in ballastless track structure can be calculatedaccording to
the following formula:
D �n
N, (14)
where D denotes the damage variable. n denotes the fatigueload
cyclic number.
en, damage of the self-compacting concrete layer andbed plate
under temperature load based on 95% safetyprobability after dierent
service times is calculated, asshown in Tables 10 and 11. It can be
seen that, for self-
compacting concrete layer, damage under condition 2accounts for
98.7% of the total damage which meanscondition 2 is the main damage
factor. In addition, for bedplate, damage under condition 1
accounts for 88.5% of the
Table 9: Maximum stress of the interface.
Condition 1 Condition 2Safety probability (%) P� 50 P� 90 P� 95
P� 50 P� 90 P� 95Maximum normal tensile stress (MPa) 0.45 0.50 0.53
0.47 0.50 0.51Maximum normal compressive stress (MPa) 0.44 0.49
0.51 0.16 0.16 0.16Maximum tangential tensile stress (MPa) 0.23
0.30 0.32 0.23 0.24 0.25Maximum tangential compressive stress (MPa)
0.17 0.21 0.23 0.23 0.24 0.25
–163
778
–938
25.2
–238
71.9
4608
1.3
1160
35
1859
88
2559
41
3258
94
3958
48
4658
01
(a)
–163
362
–892
02.6
–150
42.8
5911
7
1332
77
2074
37
2815
96
3557
56
4299
16
5040
76
(b)
–162
663
–877
03.6
–127
43.7
6221
6.2
1371
76
2121
36
2870
96
3620
56
4370
16
5119
75
(c)
Figure 24: Normal stress of interface under condition 2. (a) P�
50%. (b) P� 90%. (c) P� 95%.
–235
020
–182
793
–130
566
–783
39.9
–261
13.3
2611
3.3
7833
9.9
1305
66
1827
93
2350
20
(a)
–241
873
–188
123
–134
374
–806
24.3
–268
74.8
2687
4.8
8062
4.3
1343
74
1881
23
2418
73(b)
–248
402
–193
201
–138
001
–828
00.6
–276
00.2
2760
0.2
8280
0.6
1380
01
1932
01
2484
02
(c)
Figure 25: Tangential stress of interface under condition 2. (a)
P� 50%. (b) P� 90%. (c) P� 95%.
Table 10: Damage of the self-compacting concrete layer
undertemperature load after dierent service times.
Damage undercondition 1
Damage undercondition 2 Total damage
1 year 1.15×10−4 9.2×10−3 9.32×10−33 years 3.45×10−4 2.76×10−2
2.80×10−25 years 5.75×10−4 4.6×10−2 4.66×10−210 years 1.15×10−3
9.2×10−2 9.33×10−220 years 2.30×10−3 0.184 0.18630 years 3.45×10−3
0.276 0.28040 years 4.60×10−3 0.368 0.37250 years 5.75×10−3 0.460
0.46660 years 6.90×10−3 0.552 0.559
14 Advances in Civil Engineering
-
total damage which means condition 1 is the main
damagefactor.
7. Conclusion
In this work, through the statistical analysis of ten
years’meteorological data, it has been found that the daily
extremevalues of monthly axial uniform temperature of CRTS IIIslab
ballastless track structure obey normal distribution andthe daily
maximum temperature gradient obeys the extremevalue I-type
distribution.
In both conditions, the maximum tensile stress of
theself-compacting concrete layer is located in the middle andedge
of the slab bottom and the side of the slab. ,emaximum tensile
stress of the base plate is located at theedge of the surface of
the layer and the inner edge of thelimiting groove.,e interface
normal tensile stress is locatedat the end and corner of the
interface. Furthermore, max-imum stress increases with the increase
of the fractal value.
,e main damage factors for self-compacting concretelayer and bed
plate are the daily minimum value of axialuniform temperature is
load combination of minimum axialtemperature and maximum negative
temperature gradientand load combination of maximum axial
temperature andmaximum positive temperature gradient.
Data Availability
,e data used to support the findings of this study are in-cluded
within the article.
Conflicts of Interest
,e authors declare that they have no conflicts of interest.
Acknowledgments
,is study was developed within the research projectSY2016G001
funded by China Railway and U1434204funded by the National Science
Foundation of China, whoseassistance is gratefully
acknowledged.
References
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Table 11: Damage of the bed plate under temperature load
afterdifferent service times.
Damage undercondition 1
Damage undercondition 2 Total damage
1 year 5.16×10−5 5.78×10−6 5.83×10−5
3 years 1.55×10−4 1.73×10−5 1.72×10−4
5 years 2.58×10−4 2.89×10−5 2.87×10−4
10 years 5.16×10−4 5.78×10−5 5.74×10−4
20 years 1.03×10−3 1.15×10−4 1.15×10−3
30 years 1.55×10−3 1.73×10−4 1.72×10−3
40 years 2.06×10−3 2.31× 10−4 2.29×10−3
50 years 2.58×10−3 2.89×10−4 2.87×10−3
60 years 3.10×10−3 3.47×10−4 3.45×10−3
Advances in Civil Engineering 15
-
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