CRTS III (China Railway Track System III) slab ballastless track structure is one of the highspeed railway ballastless track structures which has Chinese independent intellectual property rights. The mechanical performance of CRTS III slab ballastless track structure under temperature load has not been clear yet. Therefore, through temperature field model and temperature load values selected by statistics analysis based on longterm meteorological data, the mechanical performance of ballastless track structure is studied under two typical working conditions with different safety probability. It is found that the daily extreme values of monthly axial uniform temperature and the daily maximum temperature gradient obey certain statistical laws. In addition, the maximum tensile stress of the selfcompacting concrete layer is located in the middle and edge of the slab bottom and the side of the slab. The maximum tensile stress of the base plate is located at the edge of the surface of the layer and the inner edge of the limiting groove. The interface normal tensile stress is located at the end and corner of the interface. Furthermore, maximum stress increases with the increase of safety probability.
At present, highspeed railway is developing rapidly in China. Specifically, the ballastless track used in CRTS (China Railway Track System) III; is developed based on the main characteristics of that used in other ballastless track structure with certain improvements which has Chinese independent intellectual property rights. The CRTS III slab ballastless track structure consists of rail, fastening system, track slab, selfcompacting concrete layer, isolation layer, bed plate, support layer, and other parts, as shown in Figure
The composition of CRTS III slab ballastless track structure.
As an important environmental factor, temperature load has a significant impact on the mechanical performance and durability of slab ballastless track structure system used in highspeed railway. Therefore, the mechanical performance of ballastless track structure under temperature load has attracted the attention of many scholars. Generally, temperature load of ballastless track structure mainly includes axial uniform temperature and temperature gradient. China’s highspeed railway design specification stipulated in the temperature gradient is 45°C/m. Many scholars and research institutions took some temperature tests on ballastless track structure and found that the track plate temperature gradient was between 40∼80°C/m, maximum negative temperature gradient is half of the maximum positive temperature gradient, and the temperature change of the bed plate is not obvious [
However, these studies are generally based on the specific environment or special working conditions which the values of various parameters are not standardized and unified. Therefore, it is necessary to determine the appropriate temperature load first, and then study the mechanical performance of CRTS III platetype ballastless track structure system under the temperature load.
As a multilayer structure, the temperature change in the CRTS III slab ballastless track structure caused by the change of the external environment is mainly concentrated in the position shallower from the surface [
In addition, the temperature variation within the structure
And the temperature variation within the structure
Literature [
In order to describe the temperature load of ballastless track structure, it can be transformed into a random process, and then according to the sample function of the load or the existing load distribution characteristics, the maximum value of the load in the design reference period is determined. The specific steps are as follows:
The design reference period of ballastless track structure is determined to be
According to the periodicity and regularity of temperature change in a certain area, the design reference period can be divided into
The distribution function (
And its distribution function can be shown as follows:
Similarly, the minimum load and its distribution in the design reference period can be obtained:
Based on the time period probability distribution (
Since the CRTS III slab ballastless track structure used in this research relies on the ZhengzhouXuzhou passenger dedicated line, in order to be more representative, the meteorological data (maximum and minimum temperatures and solar radiation intensity) of every day in Zhengzhou from 2008 to 2017 were counted, and then the daily value of axial uniform temperature of ballastless track structure was calculated. However, it is difficult to obtain the distribution law of the diurnal extreme value of the axial uniform temperature in the past ten years. Considering that the temperature of the same month is basically similar in different years, the frequency histogram of the days with different extreme temperature values was obtained by taking the months of different years as the statistical period, which for some months are shown in Figures
The histogram of the daily extreme value of axial uniform temperature in February. (a) The maximum. (b) The minimum.
The histogram of the daily extreme value of axial uniform temperature in May. (a) The maximum. (b) The minimum.
The histogram of the daily extreme value of axial uniform temperature in October. (a) The maximum. (b) The minimum.
The histogram of the daily extreme value of axial uniform temperature in November. (a) The maximum. (b) The minimum.
It can be seen from Figures
The distribution probability diagram of the daily extreme value of axial uniform temperature in February. (a) The maximum. (b) The minimum.
The distribution probability diagram of the daily extreme value of axial uniform temperature in May. (a) The maximum. (b) The minimum.
The distribution probability diagram of the daily extreme value of axial uniform temperature in October. (a) The maximum. (b) The minimum.
The distribution probability diagram of the daily extreme value of axial uniform temperature in November. (a) The maximum. (b) The minimum.
The estimated values of the normal distribution parameters of the daily extreme values of the axial uniform temperature in different months.
Months  1  2  3  4  5  6  7  8  9  10  11  12  

The maximum 

12  17  25  33  40  44  48  47  40  32  23  15 

2.5  3.0  3.3  3.0  2.1  1.8  2.1  2.3  2.4  2.7  3.3  2.5  
The minimum 

3.6  6.8  12  19  25  29  33  33  28  21  14  6.7 

2.4  2.7  2.7  2.6  1.9  1.6  2.0  1.9  2.0  2.2  3.1  2.5 
According to the parameter estimation of the daily maximum distribution function of axial uniform temperature of ballastless track structure in different months, the probability distribution function of axial uniform temperature in different months in Zhengzhou can be obtained as follows:
It is considered that the daily distribution of axial uniform temperature of ballastless track structure in each month is independent from each other. Therefore, the annual probability distribution function of the daily extreme value of the axial uniform temperature is shown as follows:
The daily extreme distribution of axial uniform temperature of ballastless track structure in the design reference period is shown as follows:
The base design period of highspeed railway is 60 years in China. Therefore, the corresponding axial uniform temperature can be obtained by different safety probability percentages such as 50%, 90%, and 95%, as shown in Table
Axial uniform temperature based on different safety probability percentages.
 

Safety probability (%) 



The maximum (°C)  50.6544  52.4781  53.0506 
The minimum (°C)  0.7437  −1.3437  −1.9973 
The daily maximum temperature gradient of CRTS III slab ballastless track structure was calculated and analyzed to obtain the histograms which are shown in Figure
Histogram of daily maximum frequency of temperature gradient. (a) Positive temperature gradient. (b) Negative temperature gradient.
Therefore, the statistical parameters of the daily maximum annual distribution of temperature gradient of CRTS III platetype ballastless track structure in Zhengzhou can be fitted, as shown in Table
The statistical parameters of the daily maximum annual distribution of temperature gradient.




Positive temperature gradient  11.8  70.5 
Negative temperature gradient  7.3  35.8 
The base design period of highspeed railway is 60 years in China. Therefore, the corresponding temperature gradients can be obtained by calculating the safety probability percentages such as 50%, 90%, and 95%, as shown in Table
Temperature gradient daily maximum load based on different safety probability percentages.
 

Safety probability (%) 



Positive temperature gradient (°C/m)  88.2  92.3  93.6 
Negative temperature gradient (°C/m)  46.7  49.3  50.1 
In the finite element (FE) model, the main structure components of CRTS III slab ballastless track system consist of rail, track board, selfcompacting concrete (SCC) layer, isolation layer, bed pate, and other parts. Geometric parameters and material parameters of components are selected according to the relevant literature [
The basic specifications of main members.
Rail  Track slab  Selfcompacting concrete layer  Bed plate  Isolation layer  Support layer  

Length (mm)  5600  5600  5600  5600  
Width (mm)  2500  2500  2900  2600  
Height (mm)  200  90  200  4  
Concrete strength  C60  C40  C30  
Elasticity modulus  210 GPa  36000 MPa  34000 MPa  32000 MPa  3.32 MPa  
Poisson’s ration  0.3  0.2  0.2  0.2  0.35  
Thermal expansion coefficient (°C^{−1})  1.18 × 10^{−5}  1 ×10^{−5}  1 × 10^{−5}  1 × 10^{−5}  1 × 10^{−5}  
Stiffness  1000 MPa/m  
Density  7800 kg/m^{3}  2500 kg/m^{3}  2500 kg/m^{3}  2500 kg/m^{3}  700 g/m^{2} 
In this work, the constitutive models selection of concrete and steel bars is the same as the literature [
The constitutive models of steel bars can be shown as follows:
The FE model of the CRTS III slab ballastless track system is developed on the commercial software ANSYS, as shown in Figure
The finite element model of the CRTS III slab ballastless track system.
The finite element units of main structural layers.
Structural layers  Finite element units 

Concrete  Solid 65 
Rail  Beam 188 
Support layer  Combin 14 
Fastening system  Combin 14 
Isolation layer  Solid 45 
Interface between track slab and SCC layer  Inter205 
Interface between isolation layer and SCC layer  Target 170 
Interface between isolation layer and bed plate  Target 170 
Additionally, the slab ballastless track structure was consolidated with the support layer simulated by using the spring element in which all degrees of freedom of the lower node are constrained. The degrees of freedom in two horizontal directions of the bed plate are constrained, and all degrees of freedom of the rail are constrained. Furthermore, all degrees of freedom at both ends of the rail are constrained.
He [
Based on the FE model, calculate the deformation of CRTS III slab ballastless track under the same condition as that in literature [
The deformation under 100°C/m temperature gradient. (a) Track slab. (b) Bed plate.
The deformation under −20°C/m temperature gradient. (a) Track slab. (b) Bed plate.
Generally, in the track system, due to the high strength of the concrete material and the twodimensional prestressing structure used in the track board, the selfcompacting concrete layer and bed plate are required to be carefully examined on the performance under the temperature loading. There are two conditions that apply to the FE model: condition 1 (the daily maximum value of axial uniform temperature is combined with the daily maximum value of the positive temperature gradient) and condition 2 (the daily minimum value of axial uniform temperature is combined with the daily maximum value of the negative temperature gradient).
Under condition 1, the stress of the selfcompacting concrete layer is shown in Figures
Longitudinal stress of the selfcompacting concrete layer under condition 1. (a)
Transverse stress of the selfcompacting concrete layer under condition 1. (a)
Under condition 2, the stress of the selfcompacting concrete layer is shown in Figures
Longitudinal stress of the selfcompacting concrete layer under condition 2. (a)
Transverse stress of the selfcompacting concrete layer under condition 2. (a)
Maximum stress of the selfcompacting concrete layer.
Condition 1  Condition 2  

Safety probability (%) 






Maximum longitudinal tensile stress (MPa)  1.15  1.31  1.35  1.89  2.25  2.40 
Maximum longitudinal compressive stress (MPa)  9.33  10.4  10.9  2.56  2.38  2.25 
Maximum transverse tensile stress (MPa)  1.68  1.76  1.81  2.10  2.11  2.11 
Maximum transverse compressive stress (MPa)  5.35  6.38  6.89  2.37  2.32  2.25 
Under condition 1, the stress of the bed plate is shown in Figures
Longitudinal stress of the bed plate under condition 1. (a)
Transverse stress of the bed plate under condition 1. (a)
Longitudinal stress of the bed plate under condition 2. (a)
Transverse stress of the bed plate under condition 2. (a)
Maximum stress of the bed plate.
Condition 1  Condition 2  

Safety probability (%) 






Maximum longitudinal tensile stress (MPa)  0.84  0.98  1.01  0.59  0.81  0.87 
Maximum longitudinal compressive stress (MPa)  6.53  6.81  7.21  3.21  2.90  2.89 
Maximum transverse tensile stress (MPa)  1.33  1.40  1.57  1.21  1.27  1.38 
Maximum transverse compressive stress (MPa)  5.60  5.07  5.38  1.65  1.32  1.43 
Under condition 1, the stress of interface between track slab and selfcompacting concrete layer is shown in Figures
Normal stress of interface under condition 1. (a)
Tangential stress of interface under condition 1. (a)
Normal stress of interface under condition 2. (a)
Tangential stress of interface under condition 2. (a)
Maximum stress of the interface.
Condition 1  Condition 2  

Safety probability (%) 






Maximum normal tensile stress (MPa)  0.45  0.50  0.53  0.47  0.50  0.51 
Maximum normal compressive stress (MPa)  0.44  0.49  0.51  0.16  0.16  0.16 
Maximum tangential tensile stress (MPa)  0.23  0.30  0.32  0.23  0.24  0.25 
Maximum tangential compressive stress (MPa)  0.17  0.21  0.23  0.23  0.24  0.25 
A relevant study [
Therefore, damage of the selfcompacting concrete layer and bed plate in ballastless track structure can be calculated according to the following formula:
Then, damage of the selfcompacting concrete layer and bed plate under temperature load based on 95% safety probability after different service times is calculated, as shown in Tables
Damage of the selfcompacting concrete layer under temperature load after different service times.
Damage under condition 1  Damage under condition 2  Total damage  

1 year  1.15 × 10^{−4}  9.2 × 10^{−3}  9.32 × 10^{−3} 
3 years  3.45 × 10^{−4}  2.76 × 10^{−2}  2.80 × 10^{−2} 
5 years  5.75 × 10^{−4}  4.6 × 10^{−2}  4.66 × 10^{−2} 
10 years  1.15 × 10^{−3}  9.2 × 10^{−2}  9.33 × 10^{−2} 
20 years  2.30 × 10^{−3}  0.184  0.186 
30 years  3.45 × 10^{−3}  0.276  0.280 
40 years  4.60 × 10^{−3}  0.368  0.372 
50 years  5.75 × 10^{−3}  0.460  0.466 
60 years  6.90 × 10^{−3}  0.552  0.559 
Damage of the bed plate under temperature load after different service times.
Damage under condition 1  Damage under condition 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} 
In this work, through the statistical analysis of ten years’ meteorological data, it has been found that the daily extreme values of monthly axial uniform temperature of CRTS III slab ballastless track structure obey normal distribution and the daily maximum temperature gradient obeys the extreme value Itype distribution.
In both conditions, the maximum tensile stress of the selfcompacting concrete layer is located in the middle and edge of the slab bottom and the side of the slab. The maximum tensile stress of the base plate is located at the edge of the surface of the layer and the inner edge of the limiting groove. The interface normal tensile stress is located at the end and corner of the interface. Furthermore, maximum stress increases with the increase of the fractal value.
The main damage factors for selfcompacting concrete layer and bed plate are the daily minimum value of axial uniform temperature is load combination of minimum axial temperature and maximum negative temperature gradient and load combination of maximum axial temperature and maximum positive temperature gradient.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
This study was developed within the research project SY2016G001 funded by China Railway and U1434204 funded by the National Science Foundation of China, whose assistance is gratefully acknowledged.