Mathematical modeling of the mechanical stabilization of the sub-tie base of railroad tracks in the arctic zone

. The practice of construction and operation of railroads in the Arctic zone shows that the greatest share of track deformations occurs in the summer period during the transition of soils of the subgrade from the frozen state to the thawed state. Wetting, freezing-thawing, and vibrodynamic effects of trains have the greatest influence on the properties of clay soils composing the railroad bed. A model of a sub-tie base stabilized by geosynthetic materials is given. The influence of geosynthetic material on redistribution of stresses during thawing and change of shear deformation


Introduction
An increase of ground humidity due to water infiltration into the ground, and the rise of groundwater level due to redistribution of moisture in the ground during its freezing in winter can dramatically change such basic ground characteristics as shear resistance, and specific weight, which mainly depends on strength and stability of ground massifs. Therefore, the degree of stability of the earth bed does not remain unchanged but changes over time.
The impact of rolling stock on the subgrade causes a complex stress-strain state in the embankment body. Under the influence of vibrodynamic load from the rolling stock, the strength and deformation characteristics of soils comprising the subgrade are reduced, which in turn causes a decrease in the stability of the subgrade.
Previous studies have established that the strength of thawing soils varies over time. The lowest strength is observed at the moment of thawing [1][2][3][4][5]. During spring thawing of clayey soil, its shear characteristics are lower than in the pre-winter thawed state. The thawed soil is unstrengthen by numerous cracks formed during the thawing of ice inclusions. The underlying frozen ground has low water permeability and higher strength, so the thawed ground usually slides over the frozen ground, i.e., over the "melted ground-frozen ground" interface plane, as the ground at this interface is characterized by increased moisture and a significant change in texture. On this surface, during the mixing process, a small layer of crumpled soil is formed, which then acts as a lubricant and has reduced shear resistance.
In the process of operating on the surface of the main platform clay soils can form depressions associated with the vibrodynamic impact on it from the passing trains. One of the negative outcomes of the formation of depressions is the possible accumulation of atmospheric precipitation in this zone and, as a consequence, acceleration of the process of growth of residual deformations of the track.
Mechanical stabilization of soils using geosynthetic materials is one of the promising directions in the field of creating optimal structures of artificial foundations [6][7][8][9]. Due to the inclusion of geosynthetic materials into the soil, it is possible to purposefully change its strength characteristics, as well as to reduce the unevenness of track subsidence by changing the stiffness of the foundation.
The apparatus of linear elasticity theory is widely used in calculations when designing the sub-tie base of a railroad track. However, in the real operating conditions of structures, soils operate beyond the elasticity limits. For most soils, the property of elasticity appears under insignificant loads. At high stresses, the relationship between stresses and strains is no longer linear, with irreversible (residual) strains having a much greater proportion of the total strain than reversible strains. This property is called plasticity, and the residual deformations are called plastic deformations.
The most common nonlinear ground model is the ideal-elastic-plastic model with the limiting surface described by the Coulomb-More criterion. The main drawback of this model is that while it describes plastic shear deformations, it does not account for nonlinearities in the volumetric compression of the ground. In addition, the model is poorly adapted for the modeling unloading deformations. Taking into account that in the practice of sub-base design the development of the zones of limiting state of soils is limited, the outcomes of calculation of real problems can be close to the elastic solution.
Cam Clay-type models, which assume that the strength of the soil is a consequence of its pre-compaction, allow for the nonlinear shear and volume compression work of the medium and thus allow for the simulation of soil behavior under different loading trajectories.
In the world of geotechnical practice, quite a several mathematical models of soil have been developed. The so-called Hardening soil model (HSM) implemented in the software package "Plaxis" has recently gained great popularity. [11], which is a development of the Cam Clay model. The yield surface in this model is not fixed in the principal stress space and can expand due to plastic deformation. The model includes two types of hardening: 1) shear hardening-used to model irreversible deformations outcomeing from primary deviator loading; 2) compression hardening-used to model irreversible plastic deformations outcomeing from primary compaction under odometrical and isotropic loading.
Ideologically close to it is the elastic-viscous plastic model with independent strain hardening during compaction and forming, developed by Russian scientists A.G. Shashkin and K.G. Shashkin [9,10], which was used in the calculations presented below.

Research methods
The elastic behavior of both soil and geosynthetic material is described using the following parameters: • Eу-flexural modulus; • μ-Poisson's ratio; The following dependence is used to approximate the volumetric compression curve of the ground: Parameters λ and p0 can be obtained from compression tests of soils. The parameter p0 has the dimensionality of pressure and determines the curvature of the compression curve. The volumetric (or average) pressure p represents: In practical calculations, it is convenient to set the strain modulus instead of λ and the pressure interval in which it is obtained as model parameters.
Approximation of the dependence of the soil shear behavior is carried out by a power function of the form: where A= τ , τ=С+Mp, M= 3 3− n-the degree determining the curvature of the dependence τ −γ; γc-the value of plastic shear strains when reaching the limit state, is determined by the formula γc= 3 2 εmax Where εmaxis the maximum vertical deformation before reaching the limit state in the tests.
Modeling the stress-strain state of the sub-tie base is quite difficult to perform using simple formulas since the ground under loading works as a nonlinear medium, so it is possible to calculate the stress-strain state by numerical methods. One of them is the finite element method, which can be regarded as a further improvement of variational methods for solving nonlinear problems [11][12][13][14][15]. Numerical modeling in this study was performed in the FEM Models software package using two-dimensional plane deformed finite-element analysis.
The most accessible and reliable experimental study is the measurement of foundation settlement. To verify the described model for further studies, flume experiments with the loading of the base with a die were carried out, comparison of experimental and calculated plots of dependence of the die settlement on the load was carried out.
When performing tray experiments, a rectangular die with dimensions of 0.7x1.2 m was used. The dimensions of the flume in the plan are 3x4 m. Crushed stone ground 1.2 m thick was used as a base. During the verification, the effect of laying two layers of Tensar SS 30 geogrid in the base of the die on the reduction of the die settlement in comparison with the settlement of the die on the unsterilized geogrid base was studied.
Given the symmetry concerning the two mutually perpendicular vertical planes, the calculation scheme shown in Fig. 1 represents in plan a quarter of the tray filled with soil. A vertical load is transmitted through the die. Two layers of geosynthetic material are laid in the base, the first layer at a depth of 30 cm, the second at a depth of 50 cm from the bottom of the die. The load on the strain was applied in steps. Fig. 2 shows plots of the relationship between the pressure on the macadam base stabilized by two layers of geogrid and the settlement of the die.  As can be seen from the above figure, the outcomes of strain calculations obtained by numerical simulation are almost identical to the experimental data. The slope of the experimental curve is close to the calculated one. Comparison of numerical simulation outcomes with experimental data testifies to the fact that the adopted approach to modeling the operation of geosynthetic material in ground-limited conditions is acceptable for predicting the behavior of mechanically stabilized sub-tie base during its thawing.
The next step was to investigate the performance of the geogrid-stabilized sub-sleeper base.
The computational scheme of the sub-tie base model is shown in Figure 3. Considering the symmetry of the scheme relative to the vertical axis, half of the calculation scheme was considered in the calculations. Zero horizontal displacements are specified on the lateral boundaries, and zero horizontal displacements are specified along the bottom surface of the computational scheme.
The solution to the nonlinear problem is performed by the iterative method. The convergence tolerance in the calculations was taken to be 1x10 -10 , and the limit number of iterations was 1000. A detailed analysis of the sensitivity of the mesh was carried out before the adoption of discretization for different models by changing the number of elements in each component and comparing the outcomes of different combinations.
The vertical dynamic wheel load is modeled as quasi-static, representing the load on the axle of the VL80 locomotive at a speed of 90 km/h. The distribution of vertical forces on the sleeper was obtained by Stojanovich G.M. in [16] and is shown in Figure 4.

Analysis of outcomes
The calculation outcomes shown in Fig. 5 demonstrate the concentration of significant shear deformations in the main area of the earth bed in the sub-rail section. The low strength characteristics of the soil and, consequently, insufficient bearing capacity of the subgrade are the main reason for the deformation of the subgrade observed in real conditions. The figure shows that soil particles after thawing in the process of their compaction move not only downward, but also somewhat shift from the sub-rail zones towards the ends of sleepers. When the geosynthetic material is laid at a depth of 10 cm from the bottom of the sleeper, there is no significant concentration of shear deformations on the main site of the earth bed in the sub-rail section. The distribution of shear deformations during stabilization with geosynthetic material shown in Fig. 6 indicates that they are more uniformly distributed in the cross-section. The concentration of shear deformations, in this case, is localized in the ballast layer in the laying zone, which indicates the decisive role of the geosynthetic material in reducing shear deformations. The limitation of horizontal deformations leads to the formation of a composite layer with increased stiffness, and the increase in bearing capacity is one of the consequences of this cause.
The horizontal stress isolines for the above calculation cases are shown in Figures 7, and  8.  In the presence of a thawing layer, the horizontal stress and shear strain gradient in it is directed toward the embankment shoulder. In this case, the soil located between the ballast and the surface of the frozen soil is as if squeezed out towards the embankment slope. A similar pattern of horizontal stress distribution was observed during field experiments. Figures 9, and 10 show the epicures of embankment surface settlement only from the train load without taking into account the weight of the embankment. The calculated settlement of the embankment surface from its weight was not affected by the presence of the geogrid. Settlement for both cases was 3 cm. The maximum values of settlement from the external load are confined to the points where the load is applied. The calculated settlement of the embankment surface on the unsterilized geogrid base in the case under consideration was 10.1 cm, on the stabilized base-8.7 cm. The decrease in the settlement was 14%.  For comparative purposes, Figure 11 shows the distribution of horizontal forces in the geogrid laid at different depths from the bottom of the sleeper. It can be seen from the given diagrams that when stabilizing the upper part of the ballast prism, the maximum horizontal forces in the geogrid are registered in the sub-rail section, and when laying at a depth of 50 cm from the bottom of the sleeper, the maximum forces shift towards the track axis and the edge of the subgrade.

Conclusion
1. The soil model adopted in the calculations successfully describes the load transfer mechanisms, and geomaterial-soil interaction at the micro level, and the model response corresponds to the full-scale experiments. The considered model can be used when designing measures for stabilization of the deforming earth bed in the Arctic zone. 2. Numerical modeling shows that laying geogrids has a greater effect in reducing horizontal ground displacements than in reducing vertical displacements. 3. When geosynthetic material is laid in the track subgrade, tangential stresses and shear deformations are concentrated along the geosynthetic material, limiting the depth of the soil destruction process. 4. The choice of a rational type of stabilization from technological, economic, and computational points of view depends on the operating conditions of the railroad track, the physical and mechanical properties of ballast, and subgrade soil.