Modelling the influence of sub-grade material parameters on the response of railway line under moving train loads

2.1. Geometry of the model

In order to examine the sub-grade response due to the surface vibration a model involving all components of the track structure (rail, rail-pad, sleepers, substructures (ballast, sub-ballast and sub-grade), and vehicle parameters considering the axle spacing, axle load and diameter of the wheel are required. The track was modeled in a configuration similar to that found in newly built tracks in Ethiopia railway Projects. The rail used in this simulation was UIC-54 rail supported by concrete sleepers laid at 0.60 m centers. The sleepers were then supported by 0.15 m layer of ballast, 0.15 m layer of sub-ballast, finally 16 m sub-grade was considered. All the track components are modeled as three dimensional solid elements except the rail-pads. The rail pads were modeled as spring element which have vertical stiffness ($K_{pad}$) and viscous damping ($C_{pad}$).

Due to excessive computational time to solve the 3-D dynamic problem through numerical models, only half of the track is modeled means symmetry with respect to the center line is assumed. The length of the track considered is limited to 12 sleepers based on Kumaran et al works [9] with two moving railcar wheels as indicated in [10]. Besides, the sleeper is considered as solid element. The slope of the embankment adopted in the model was $H:V=$ 3.5.

Fig. 2Model of rail (UIC-54)

Fig. 3Model of half sleeper

2.2. Material model

The rail, sleeper, wheel, sub-ballast, ballast and the sub-grade are considered as linear elastic and damped material. The Young’s modulus ($E$) of the soil is varying with depth, the sub-grade soil was divided in 3 different layers; i.e. 2.5 m of the embankment, 2 m formation layer (Ground 1) and 11.5 m of base soil layers (Ground 2). In addition, the damping ratio for the soil varies from 1 % to 20 % [11]. Note that all the sub-grade parameters in Table 1 are the base set parameters. For further investigation they will be varied to determine their influence on the surface vibration levels. Generally, the sub-grade and subsoil layers used are shown below in Table 1.

Fig. 4Wheel-rail contact interface: a) model of wheel-rail contact, b) real-wheel-rail contact

a)

b)

Fig. 5Finite element model track overview (longitudinal view)

2.3. Interface modeling

Surface-to-surface contact elements were model using master and slave surface interface elements. The basic Coulomb friction model with the penalty friction formulation was used to simulate the frictional force response at the contact interface. The model specifies the shear behavior in terms of normal and tangential [12]; i.e. ${\tau }_{crit}=\mu p$, where. ${\tau }_{crit}$ is the critical shear stress at contact surface, $\mu$ is the coefficient of friction and, $p$ is the contact pressure between the two surfaces. The maximum allowable frictional stress is related to contact pressure by coefficient of friction between contacting bodies [13].

Based on direct shear tests on various surface materials, Durgunoglu and Mitchell [14, 15], suggested using $\delta /\varphi =$ 0.5 for most penetration in practice, where $\varphi =$ Peak angle of effective internal friction and $\delta$ is the angle of interface friction. Besides, According to Susila and Hrychiw [16], coefficient of friction ($\mu$) between different components found to be a function of angle of internal friction; i.e. $\mu =\mathrm{t}\mathrm{a}\mathrm{n}\left(0.5\varphi \right)$. The coefficient of friction between the substructure components of the track is assumed to be a function of angle of internal friction. The contact between wheel and rail, sleeper and rail, sleeper and granular soils is used as indicated in [17].

2.4. Element type and mesh size

The track components were modeled using linear four-node tetrahedron and Eight-node linear brick elements with 3D deformable solid. Refined elements were used on the railhead contact with wheel sets and perimeter of the wheel tread, since the vibration is predominantly induced by the interaction between the moving wheels and rails [18]. Four mesh sizes with the element size of 0.05 m, 0.15 m and 0.75 m and 1 m were considered.

Fig. 6Mesh detail: a) wheel, b) rail, c) isometric view of the whole model

a)

b)

c)

2.5. Loading procedure

The axle load and spacing used for simulations can be different in various literatures. Nevertheless, the Ethiopia Railways Corporation used two axle loads of 250 kN and 1.83 m axle spacing for design of rail seat load for the case of freight car. Accordingly, two moving wheel loads of 125 kN are applied, which are 1.83 m apart in order to model the passage of freight car on a conventional line. The wheel loads remain for 0.03 s in its starting position during loading. After 0.03 s the wave induced by the mass loading have disappeared, do not affect the results anymore and the load variation in the initial state was removed. Afterwards the wheel starts to move along the rail for $t=$ 0.178 s. The computation must end later, in order to stabilize the vibrations in the surrounding media. Hence, the remaining load parameters $t=$ 0.292 s were used to stabilize the vibrations. The maximum necessary displacement imposed by the boundary condition manager is 4.94 m, obtained for the speed of 120 km/h. In general, the following steps were used to apply the train loads and to study the dynamic response of sub-grade of the track system.

2.6. Boundary conditions

To minimize unrealistic reflections of the waves from the boundary, Artificial Non-Reflecting Boundary was defined at the far end of the model. For the implementation of Artificial Non-Reflecting boundary, a region in 8 m width is added to the end of the model in each direction. These regions were divided into four sections with varying damping ratios as shown in Fig. 8. The Rayleigh damping coefficients are specified as gradually increasing from the first section to the last one. These boundary consisted of soil elements with high damping ratio ($\zeta =$ 0.7-0.8). To calibrate the model, the contour plots of velocities are plotted at different times with and without an Artificial Non-Reflecting boundary. Surface velocity field move with the train along model at specific time is used for comparison that corresponds to the end of the movement of the train loads. It is observed that the waves reflect back in the regular model. However, in the model with artificial boundary only very small portion of waves reflect back. See Fig. 9 below.

Fig. 7Sequence of application of the wheel loadings in the model

a) Applying gravity and wheel load (the wave induced by the mass loading)

b) The wave induced by the mass loading disappeared after $t=$ 0.03 s

c) Rolling and translating wheels along the rail for $t=$ 0.178 s

d) End of the movement of the wheels

e) The vibrations were stabilized at total time $t=$ 0.5 s

The boundary conditions considered here was translation degree of freedom with restricting both vertical and horizontal movements on the bottom part of the sub-grade and horizontal movements on the far end of the model and along the axis of symmetry were used. The finite length boundary condition was used at the far end of the rail and the sleeper is constrained such that the displacements were restrained orthogonal to the lateral boundaries of the model. The profiles of the cross-section and the boundary conditions for the model were demonstrated in Fig. 8.

Fig. 8Schematics of the finite element model

Fig. 9The contour plot of total velocity in: a) the original model, b) the model with artificial non-reflecting boundary at t= 0.2 s

a)

b)

5.1. The effect of varying the unit weight of the soil

The unit weight was varied between two reasonable and expected values. For example, from 1330 kg/m³ to 1500 kg/m³ for soft clay 1470 kg/m³ to 1570 kg/m³ for loose sand 1730 kg/m³ to 1900 kg/m³ for the stiff clay and 1800 kg/m³ to 2000 kg/m³ for the dense sand i.e. in step of 50-100 kg/m³. As the results shown in Fig. 12-14 and 15, it can be concluded that unit weight has a minor influence on the response of the model with the PVV only varying to 1.62 %, 4.16 %, 1.2 % and 4.25 % for the soft clay, stiff clay, loose sand and dense sand, respectively.

Fig. 12Vibration levels in the evaluation point A1 for different density of the sub-grade (soft clay)

Fig. 13Vibration levels in the control point A1 for different density of the sub-grade (loose sand)

5.2. The effect of varying the Young’s modulus of the soil

The Young’s modulus of the soil may have different values and was varied between 10 MPa to 30 MPa for loose sand and 50 MPa to 80 MPa for dense uniform sand. In addition, the Young’s modulus of the soft clay varied between 2.5 MPa to 8 MPa and 15 MPa to 40 MPa for the stiff clay [4].

Fig. 14Vibration levels in the evaluation point A1 for different density of the sub-grade (stiff clay)

Fig. 15Vibration levels in the evaluation point A1 for different density of the sub-grade (dense sand)

Fig. 16Vibration levels in the evaluation point A1 for different Young’s modulus of the sub-grade (clay)

Fig. 17Vibration levels in the evaluation point A1 for different Young’s modulus of the sub-grade (sand)

5.3. The effect of varying the Poisson’s ratio of the soil

The Poisson’s ratio $\left(\nu \right)$ of the soil was varied between 0.1 to 0.3 for clay soil and 0.3 to 0.4 for commonly used sand soil [21]. Results are shown in Fig. 18 and Fig. 19. Similar to the effect of the density, the Poisson’s ratio of the sub-grade shows only small influence on the PVV. An approximate variation of 6.93 % for the soft clay and 12 % for the stiff clay was observed. Similarly, the PVV variation for the loose sand and dense is calculated as 5.56 % and 8.16 %, respectively.

Fig. 18Vibration levels in the evaluation point A1 for different Poisson’s ratio of the sub-grade (soft and stiff clay)

Fig. 19Vibration levels in the evaluation point A1 for different Poisson’s ratio of the sub-grade (loose and dense sand)

5.4. The effect of varying the damping ratio of the soil

The analyses were performed for the damping ratio of sub-grade as 1 %; 5 %; 10 %; 15 %; 20 % and comparison for similar soil type are made.

When compared with the other parameters, soil damping does have very high influence on the vibration levels of all soil types. Fig. 20 and 21, show that the variation of PVV for $\zeta =$ 1 % and $\zeta =$ 20 % is rounded to 42.5 %, 58.6 %, 60.66 % and 71.11 % for the soft clay, loose sand, stiff clay, dense sand, respectively.

Fig. 20Vibration levels in the evaluation point A1 for different damping ratio of the sub-grade (soft and stiff clay)

Fig. 21Vibration levels in the Evaluation point A1 for different damping ratio of the sub-grade (loose and dense sand)