ANALYSIS MODELS OF SUBGRADE FROST HEAVING FOR CRTSⅠSLAB TRACK
Wenbo ZHAO1，2，Xiubo LIU2，Yutao LIU3
1China Academy of Railway Sciences，Beijing 100081
2Infrastructure Inspection Research Institute，China Academy of Railway Sciences，Beijing 100081
3Southwest Jiaotong University，MOE Key Laboratory of High-speed Railway Engineering，Chengdu 610031
Abstract：The subgrade frost heaving is one of the main defects of high-speed railway in seasonal frozen region，where CRTSⅠslab track is often used，so it is necessary to establish reasonable analysis models to research the subgrade frost heaving for CRTSⅠ slab track.This paper has established three models including composite beam model，beam-plate model and beam-solid finite element model.The deformation results of track structure layers under the same subgrade frost heaving condition are compared by using the three models.The contrast results show that some certain degree of subgrade frost heaving will lead to voids of the track slab and concrete roadbed，which will vary with the location and amplitude of the subgrade frost heaving.From the engineering point of view，the results from the three models are very close，so the three models all can be used to research the subgrade frost heaving in CRTSⅠslab track.The beam-solid model is complicated in calculation and low in efficiency；the beam-plate model can be better to simulate the subgrade frost heaving in CRTSⅠslab track，but is lower in efficiency than the composite beam model.
Keywords：slab track，subgrade frost heaving，composite beam model，beam-plate model，beam-solid model
Email：baishenyingxiong@126.com
1 Introduction
In China，the area of frozen earth is about 7.2million square kilometers with about 5.28million square kilometers are seasonal frozen[1].The data from frost heaving test of Haerbin-Dalian Line indicate that the proportion of frost heaving on subgrade is more than 98％[2].The subgrade frost heaving can lead to track vertical irregularity，and there may be voids between base plate and subgrade also between slab and base plate in severe cases.The voids will have an effect on the security，comfort and durability of high speed railway operation，so it is important to study the effect of subgrade frost heaving on CRTSⅠslab track for the construction and maintenance during operation of high speed railway.
At present，there are mainly three models for the slab track：composite beam model，beam plate model and beam solid model.The article[3-4]compares the models and the results show that the structural stress and the deformation of ballastless track can be better simulated by beam plate model.A three-dimensional static model of subgrade frost heaving was established in article[5]with the finite element software ABAQUS to study the transitive relationship between subgrade frost heaving and track irregularity of high speed railway.
All the researches above showed that the three-dimensional solid model is accurate yet complicated in modeling and time consuming.To get the optimum model，the paper establishes composite beam model，beam plate model and beam solid model based on CRTSⅠ slab track of high speed railway in seasonal frozen region.By comparing the deformation of the track structure layers under the same subgrade frost heaving condition，the paper gets the most optimum model.
2 Models of CRTSⅠ Slab Track
As shown in Figure 1，CRTSⅠslab track is mainly composited of rail，fastener，track slab，CA mortar，and concrete roadbed.The paper using the finite element method establishes composite beam model，beam plate model and beam solid model of subgrade frost heaving for CRTSⅠslab track.
Figure 1 CRTSⅠslab track
2.1 Composite beam model
The composite beam model is proposed based on traditional theory of elastic foundation beam combining the characteristics of ballastless track.Considering the structural symmetry，only half of the track cut along the longitudinal center section is researched.As shown in Figure 2，two-dimensional beam element is adopted to simulate the rail，track slab and concrete roadbed whose element length is equal to the interval between fasteners which are simultaneously simplified to linear stiffness.Due to the faint adhesion stress，the tensile properties of CA mortar are neglected，so it is simulated by a nonlinear spring which can only bear pressure.Subgrade is simplified to two-dimensional solid element and separable contact is introduced between the subgrade and concrete roadbed for the purpose of discovering whether voids appear or not.Overall，based on the simplification above，the track along the longitudinal direction can be seemed as a composite beam model.
The bollard set in CRTSⅠslab track is to restrict the longitudinal and lateral displacement of the track slab.As shown in Figure 2，during the modeling，a rigid arm is applied to simulate the bollard with a spring set between the rigid arm and track slab to simulate the resin layer.
Figure 2 Composite beam finite element models
2.2 Beam plate model
The thickness of track slab and concrete roadbed of CRTSⅠslab track is far less than the size of other two directions，which conforms to the plate element theory，so the beam plate model can be used to calculate the deformation of slab track.As shown in Figure 3，the rail is meshed into beam element，the length of which is as long as the interval of fasteners，while the track slab and concrete roadbed are meshed into plate element which is as long as the interval of fasteners along the longitudinal direction.In the lateral direction，there are nine and eleven elements symmetrically arranged on track slab and concrete roadbed respectively.The subgrade is simulated by three-dimensional solid element.The simplification of fastener，CA mortar and bollard is just as same as the composite beam model.
Figure 3 Beam plate finite element model
2.3 Beam solid model
Not only does the beam solid model can reflect the spatial mechanical characteristics of the slab track but also cansimulate various loads better such as gravity load and the subgrade surface uneven frost heaving load，etc[6].As shown in Figure 4，the rail is meshed into beam element the length of which is as long as the interval of fasteners.The track slab and concrete roadbed are meshed into solid element whose length is as long as the interval of fasteners along the longitudinal direction.In the lateral direction there will be sixteen and twenty elements symmetrically arranged on track slab and concrete roadbed respectively to make the element close to regular hexahedron.The simplification of fastener，CA mortar and the subgrade is just as same as beam plate model.
Figure 4 Beam solid finite element model
3 Model Parameters and Frost Heaving Condition
3.1 Model parameter
Gravity of the track is applied to the static model in the form of body load to explore the influence made on the track structure.In order to eliminate the boundary effect，the model is as long as 200 meters，the sizes of track structure and material parameters used in the process of calculation are shown in Table 1.
Table 1 Sizes and parameters of track structure
3.2 Frost heaving condition
The subgrade frost heaving mechanism can be ignored when simulating the influence made on track structure by subgrade frost heaving and the frost heaving load will be applied on the subgrade surface as a boundary condition directly.Similar to subgrade uneven settlement which is often simulated by a cosine curve line[7]，the frost heaving also introduce a convex cosine curve line，the expression is
In this expression，A stands for frost heaving magnitude，X stands for the longitudinal coordinates in the frost heaving section，L stands for the frost heaving wavelength.
As shown in Figure 5 and Figure 6，the frost heaving is located in the middle of the base plate（Location A）and in the base plate joint（Location B）.The frost heaving wavelength and amplitude are 10 m and 5 mm respectively.
Figure 5 Location A
Figure 6 Location B
4 Results and Analysis
The deformation clouds of track structure layers shown in Figure 7 and Figure 8 are the results calculated by ABAQUS when a 5 mm/10m scale frost heaving located at Location A or Location B occurs on subgrade.Figure 7 and Figure 8 show that when the subgrade frost heaving occurs，vertical deformation will appear on track structure layers，which will vary with the frost heaving location.
Figure 7 Track deformation cloud at Location A
Figure 8 Track deformation cloud at Location B
The vertical deformation of slab track structure layers along the route longitudinal direction are shown in Figure 9 and Figure 10when the frost heaving occurs at Location A and B respectively.The Figure 9 and Figure 10 show that the deformation curves of track structure layers are basically coincide in the scope of subgrade frost heaving which is in the form of cosine.When the frost heaving is Located at Location A，the track deformation peak is close to the frost heaving peak，while at Location B，the track deformation peak is slightly larger，but overall，track deformation and subgrade frost heaving curve tend to be consistent.
Figure 9 Deformation of track structure layers at Location A
Figure 10 Deformation of track structure layers at Location B
The composite beam model，beam plate model and beam solid model under subgrade frost heaving Located at A and B are established to calculate the deformation of slab track structure layers which are shown in Figure 11-Figure 16.The figures show that when the frost heaving is located at Location A，the track deformation peak got from beam solid model is slightly smaller than that from composite beam model and beam plate model，while at Location B，the track deformation curves got from the three models are basically the same.
Figure 11 Vertical deformation of rail in three models at Location A
Figure 12 Vertical deformation of track slab in three models at Location A
Figure 13 Vertical deformation of concrete roadbed in three models at Location A
Figure 14 Vertical deformation of rail in three models at Location B
Figure 15 Vertical deformation of track slab in three models at Location B
Figure 16 Vertical deformation of concrete roadbed in three models at Location B
The deformation of track structure layers and void height for the three models under subgrade frost heaving of 5 mm/10 m are compared in Table 2.The table shows that the vertical deformation of rail got from composite beam model and beam plate model are similar while the beam solid model is 6％ smaller at Location A and 5％ smaller at Location B.For track slab and concrete roadbed，vertical deformations of the three models are nearly same.
Voids of track slab and concrete roadbed will both appear in the three models.At Location A，the void heights of concrete roadbed got from composite beam model and beam solid model are both 0.58 mm，while the beam plate model is 0.66 mm，about 12％ larger.Besides，two void peaks will appear on track slab，the one is at the maximum amplitude of frost heaving，that is the center of the concrete roadbed，in this case，the concrete roadbed arecurves towards the subgrade，track slab and concrete roadbedare tangent，so void appears.The other is at the center of track slab in the scope of frost heaving，in this case，due to the deflection of concrete roadbed；the track slab is supported on concrete roadbed like a simply supported form，which will lead to a void at the center of track slab.The comparison of the three models can be found that the track slab voids at the center of the track slab are near，at the center of concrete roadbed，the void height of beam plate model and beam solid model are closer，respectively 0.38 mm and 0.40 mm，but the composite beam model is 0.46 mm，a little larger than others.At Location B，since the subgrade frost heaving appears at the joint of concrete roadbed，the deformation of concrete roadbed is tangent along subgrade arc surface.Thus，the deformation of track slab and concrete roadbed are coincide，and the track slab voids only appear at the center of track slab，the heights of which are close for the three models.
Table 2 Comparison of three models for CRTSⅠslab track
5 Conclusions
As described above，some conclusions can be reached from the results of the three models.
（1）The vertical deformation of track bearing layers varies with the subgrade frost heaving location，but overall，track deformation and subgrade frost heaving curve tend to be consistent.
（2）From the engineering point of view，the results from the three models are very close，so the three models can all beused to research the subgrade frost heaving in CRTSⅠ slab track.The beam-solid model is complicated in calculation and low in efficiency；the beam-plate model can be better to simulate the subgrade frost heaving in CRTSⅠ slab track，but is lower in efficiency than the composite beam model.
Acknowledgement
The authors wish to acknowledge the support and motivation provided by National 973Project of China（No.2013CB036205）and the project of Chinese railway corporation（No.Z2013-038-4-1）.
References
[1]Qu Z.X.，2009.The Analysis and Measurement of Subgrade Frost Heaving Characteristics for High Speed Railway in Seasonal Frozen Region[J].Railway Standard Design，12：8-11.
[2]Qiao L.J.，2014.Maintenance and Repair Technologies of Frost-heaving Subgrade on Haerbin-Dalian Passenger-dedicated Railway[J].Railway Engineering，1：72-74.
[3]Zhao P.R.，Zhang Y.A.，Liu X.Y.，et al.，2009.Beam-plate Model on the Elastic Foundation of Ballastless Track[J].China Railway Science，30（3）：1-4.
[4]Sun L.，Duan Y.F.，Gao P.P.，2013.Comparison of Structural Analysis Models for Slab Ballastless Track in High-speed Railway[J].Journal of Southeast University：Science and Technology，43（5）：938-943.
[5]Wu Z.L.，2015.Analysis of Track Irregularity and Dynamic Response for the Influence of High Speed Railway Subgrade Frost Heaving[D].Beijing：Beijing Jiaotong University.
[6]Liang F.，2013.Mechanical Characteristic Research of CRTSⅠ Slab Ballastless Track[D].Changsha：Central South University.
[7]Zhao L.，2015.Spatial Refinement Analysis Method of High Speed Railway Ballastless Track and its Application Research[D].Beijing：Beijing Jiaotong University.
ICRE2016-International Conference on Railway Engineering