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Abstract

Review of technical literature regarding to train-induced vibrations shows that the effects of unsupported railway sleepers on this issue have been less investigated. So, the present study was devoted to numerical investigations of the mentioned issue. In this regard, first the problem of longitudinal train–track dynamic interaction was simulated in two dimensions by using the finite element method and the developed model was validated through comparison of the results with those obtained by previous researchers. In the next stage, a series of sensitivity analyses were accomplished to account for the effects of value of gap beneath the unsupported sleeper(s) and the track support stiffness on increasing the sleeper displacement and track support force. Moreover, the raised sleeper support force was introduced as applied load to a two-dimensional plane strain finite element model of track in lateral section and consequently the train-induced vibrations were assessed. As a result, a series of regression equations were established between the peak particle velocity in the surrounding environment of railway track and the sleeper support stiffness for tracks without unsupported sleepers and with one and two unsupported sleepers.

Keywords

Railway track finite element method train–track system environmental vibrations unsupported sleepers

Introduction

Among the various parameters affecting the ballasted track mechanical behaviour, track support conditions and track stiffness have major importance. One of the key problems which commonly occurs in the ballasted railway tracks is unsupporting of the sleepers due to local settlements in ballast layer. By passing the moving trains, the track components such as rail, sleepers and the fasteners are faced to early damage and deterioration when unsupported sleepers exist in the ballasted tracks system. When a sleeper hangs up from the rail, its share of vertical load in conjunction with rail is transferred to adjacent sleepers so they will undergo more deformations, forces and stresses. Some researchers have worked in the field of dynamic train–track interaction for various railway track conditions. In this regard, Sun et al.¹ examined dynamic responses of rail track and wagon by coupling two subsystems with Hertz contact theory. Nielsen and Abrahamsson² analysed the track including beam elements under the effect of moving vehicle with four degrees of freedom. Kerr³ analysed and designed the railway tracks under moving vehicles considering different track parameters including rail support modulus, track stiffness, axial stress, rail buckling, etc. Zhai et al.⁴ investigated railway ballast vibrations including pyramid model by using theoretical simulations and field experiments. Suzuki et al.⁵ measured the dynamic behaviour of railway track near to rail joints under the moving vehicle effect. Lundqvist and Dahlberg⁶ studied the effects of voided sleeper on track dynamic behaviour under a moving train wheel-set by considering computer model. Zakeri et al.^7–9 investigated the effects of unsupported sleeper, track parameters and rail irregularity by considering the train–track dynamic interaction model. Lei and Zhang¹⁰ studied the effects of track stiffness in transitions by considering train–track dynamic interaction. Koroma et al.¹¹ investigated the effects of rail pads on railway track vibrations under passing harmonic loads by utilizing the finite element model. Also in the presence of unsupported sleepers in tracks, the train-induced vibrations in surrounding environment increase and consequently produce some annoyances in surrounding structures in residential areas. Some researchers have worked in the field of ground-borne vibrations due to moving trains. For example, Cai et al.¹² investigated the influence of vehicle–track interaction with rail irregularities on ground surface vibrations by solving the equations of motions analytically. Adam and Von Estorff¹³ studied the effect of trenches on reducing the building vibrations due to passing train by utilizing a two-dimensional numerical model. Ni et al.¹⁴ and Yang and Hung¹⁵ studied the train-induced vibrations by developing a finite element model. Zakeri et al.¹⁶ investigated numerically the effects of step-shaped trench in order to reduce the train-induced vibrations on the surrounding building. Esmaeili et al.¹⁷ experimentally and numerically studied the effects of ballast layer solidification on ground-borne vibrations due to passing train. Nielsen et al.¹⁸ presented a hybrid model for evaluating the ground-borne vibrations generated by rail and wheel irregularities. Kouroussis et al.^19,20 investigated the effects of wheel and rail singular defects and irregularities on ground-borne vibrations by considering the numerical modelling. Younesian and Sadri²¹ investigated the effects of multitrenches for reducing the train-induced vibration by using the finite element method (FEM). Trollé et al.²² reviewed the effects of vibration and annoyance on building residents due to ground transportation vehicles by focusing on railway traffic.

None of the mentioned researchers have taken into account the ground-borne vibrations due to unsupported sleeper’s existence in the ballasted railway tracks. Consequently, the present study was allocated to numerical investigations of the effects of unsupported sleepers in increasing the ground-borne vibrations due to railway vehicle passage. In this regard, two finite element codes were used for vehicle–track interaction analysis and evaluation of train-induced wave propagation in surrounding media. In this matter, from the first code, the maximum track support force was obtained and it was introduced to the second code as input. Consequently by evaluating the effect of unsupported sleepers on increasing the sleeper support force in longitudinal direction of railway track, its effect on increasing of the peak particle velocity (PPV) of surrounding environment was numerically investigated by the second FEM code. Finally, many regression equations were derived between the parameters of track with unsupported sleeper with those are dominant in environmental vibrations.

Railway vehicle – track interaction simulation

For analysis of railway tracks behaviour under the moving vehicle, different track models can be considered. In order to calculate the displacement and support force of sleepers, a railway track including rail and sleepers is simulated by using the FEM. In this regard, using the developed models by Zakeri and Xia⁸ and Zakeri et al.,⁹ the mass, damping and stiffness matrix of rail and then sleepers are derived and assembled. During the modelling, each rail joint has two degrees of freedom including vertical displacement and rotation. Moreover, each sleeper has one degree of freedom including vertical displacement. Figure 1 illustrates the railway track model including rail and sleepers which have been connected with springs and dampers.

Figure 1.

Railway track model including two unsupported sleepers.

In Figure 1, K_F and C_F are the stiffness and damping of track fastening system, respectively. Also, K_S and C_S are the stiffness and damping of track support, respectively. In addition, M_S presents sleeper mass. The stiffness [K_Rail], mass [M_Rail] and damping [C_Rail] matrix of each rail element are determined as follows

[K_{Rail}] = \frac{EI}{L^{3}} [12 6 L - 12 6 L 4 L^{2} - 6 L 2 L^{2} Symmetric 12 - 6 L 4 L^{2}]

(1)

[M_{Rail}] = \frac{mL}{420} [156 22 L 54 - 13 L 4 L^{2} 13 L - 3 L^{2} Symmetric 156 - 22 L 4 L^{2}]

(2)

[C_{Rail}] = α_{R} [M_{Rail}] + β_{R} [K_{Rail}]

(3)

In these matrices, E, I, m and L are Young’s modulus, moment of inertia, mass per unit length and element length, respectively. Also, α_R and β_R are Rayleigh damping coefficients. After calculating the matrix of each rail element, the total matrix of rail is determined by assembling all rail elements. Then, the same matrix of each sleeper is derived based on its motion equation as follows

M_{s} {\overset{··}{Y}}_{s} + K_{F} (Y_{s} (t) - Y_{R} (x, t)) + K_{S} Y_{s} (t) + C_{F} ({\overset{\cdot}{Y}}_{s} (t) - {\overset{\cdot}{Y}}_{R} (x, t)) + C_{S} {\overset{\cdot}{Y}}_{s} (t) = 0

(4)

In this equation, Y_R and Y_S are the vertical displacement of rail and sleeper, respectively. Also, ${\overset{\cdot}{Y}}_{R}$ and ${\overset{\cdot}{Y}}_{S}$ present the vertical velocity of rail and sleeper, respectively. Moreover, the mass, stiffness and damping matrix of whole railway track are calculated by assembling rail and sleeper matrix. Moreover, the stiffness and damping of fastening system and track support are added to the total track matrix.

For railway track loading, the railway vehicle with four axle loads is passed over the track. For simulation of vehicle, first the motion equations of vehicle components are derived and then the mass, damping and stiffness matrix of railway vehicle are determined. In this regard, the carbody and each bogie have two degrees of freedom including vertical displacement and rotation and also, each wheel has one degree of freedom including vertical displacement. In general, railway vehicle has 10 degrees of freedom. Figure 2 shows the configuration of the railway vehicle model.

Figure 2.

Railway vehicle model.

In Figure 2, the vehicle model includes carbody, two bogies and four wheels. The mass and rotational inertia of carbody are M_c and J_c, respectively. Also, M_t₁ and J_t₁ are the mass and rotational inertia of bogie 1 while M_t₂ and J_t₂ belong to bogie 2. In this figure, L_t and L_c are the half distance of two wheels and two bogies, respectively. K_t₁, C_t₁ and K_t₂, C_t₂ are the stiffness and damping of bogies 1 and 2, respectively. Also, K_w₁, K_w₂, K_w₃ and K_w₄ are the stiffness of wheels 1, 2, 3 and 4, respectively. Moreover, C_w₁, C_w₂, C_w₃ and C_w₄ are the damping of wheels 1, 2, 3 and 4, respectively. The motion equations of car body, bogies and wheels are presented as follows:

Vertical motion equation of car body (Y_c)

M_{c} {\overset{··}{Y}}_{c} + K_{t 1} (Y_{c} - Y_{t 1}) + K_{t 2} (Y_{c} - Y_{t 2}) + C_{t 1} ({\overset{\cdot}{Y}}_{c} - {\overset{\cdot}{Y}}_{t 1}) + C_{t 2} ({\overset{\cdot}{Y}}_{c} - {\overset{\cdot}{Y}}_{t 2}) = 0

(5)

Rotational motion equation of car body (θ_c)

J_{c} {\overset{··}{θ}}_{c} + K_{t 1} L_{c} (θ_{c} L_{c} - Y_{t 1}) + K_{t 2} L_{c} (θ_{c} L_{c} + Y_{t 2}) + C_{t 1} L_{c} ({\overset{\cdot}{θ}}_{c} L_{c} - {\overset{\cdot}{Y}}_{t 1}) + C_{t 2} L_{c} ({\overset{\cdot}{θ}}_{c} L_{c} + {\overset{\cdot}{Y}}_{t 2}) = 0

(6)

Vertical motion equation of bogie 1 (Y_t₁)

M_{t 1} {\overset{··}{Y}}_{t 1} + K_{t 1} (Y_{t 1} - Y_{c} - θ_{c} L_{c}) + K_{w 1} (Y_{t 1} - Y_{w 1}) + K_{w 2} (Y_{t 1} - Y_{w 2}) + C_{t 1} ({\overset{\cdot}{Y}}_{t 1} - {\overset{\cdot}{Y}}_{c} - {\overset{\cdot}{θ}}_{c} L_{c}) + C_{w 1} ({\overset{\cdot}{Y}}_{t 1} - {\overset{\cdot}{Y}}_{w 1}) + C_{w 2} ({\overset{\cdot}{Y}}_{t 1} - {\overset{\cdot}{Y}}_{w 2}) = 0

(7)

Rotational motion equation of bogie 1 (θ_t₁)

J_{t 1} {\overset{··}{θ}}_{t 1} + K_{w 1} L_{t} (θ_{t 1} L_{t} - Y_{w 1}) + K_{w 2} L_{t} (θ_{t 1} L_{t} + Y_{w 2}) + C_{w 1} L_{t} ({\overset{\cdot}{θ}}_{t 1} L_{t} - {\overset{\cdot}{Y}}_{w 1}) + C_{w 2} L_{t} ({\overset{\cdot}{θ}}_{t 2} L_{t} + {\overset{\cdot}{Y}}_{w 2}) = 0

(8)

Vertical motion equation of bogie 2 (Y_t₂)

M_{t 2} {\overset{··}{Y}}_{t 2} + K_{t 2} (Y_{t 2} - Y_{c} + θ_{c} L_{c}) + K_{w 3} (Y_{t 2} - Y_{w 3}) + K_{w 4} (Y_{t 2} - Y_{w 4}) + C_{t 2} ({\overset{\cdot}{Y}}_{t 2} - {\overset{\cdot}{Y}}_{c} + {\overset{\cdot}{θ}}_{c} L_{c}) + C_{w 3} ({\overset{\cdot}{Y}}_{t 2} - {\overset{\cdot}{Y}}_{w 3}) + C_{w 4} ({\overset{\cdot}{Y}}_{t 2} - {\overset{\cdot}{Y}}_{w 4}) = 0

(9)

Rotational motion equation of bogie 2 (θ_t₂)

J_{t 2} {\overset{··}{θ}}_{t 2} + K_{w 3} L_{t} (θ_{t 2} L_{t} - Y_{w 3}) + K_{w 4} L_{t} (θ_{t 2} L_{t} + Y_{w 4}) + C_{w 3} L_{t} ({\overset{\cdot}{θ}}_{t 2} L_{t} - {\overset{\cdot}{Y}}_{w 3}) + C_{w 4} L_{t} ({\overset{\cdot}{θ}}_{t 2} L_{t} + {\overset{\cdot}{Y}}_{w 4}) = 0

(10)

Vertical motion equation of wheel 1 (Y_w₁)

M_{w 1} {\overset{··}{Y}}_{w 1} + K_{w 1} (Y_{w 1} - Y_{t 1} - θ_{t 1} L_{t}) + C_{w 1} ({\overset{\cdot}{Y}}_{w 1} - {\overset{\cdot}{Y}}_{t 1} - {\overset{\cdot}{θ}}_{t 1} L_{t}) + F_{w 1} = 0

(11)

Vertical motion equation of wheel 2 (Y_w₂)

M_{w 2} {\overset{··}{Y}}_{w 2} + K_{w 2} (Y_{w 2} - Y_{t 1} + θ_{t 1} L_{t}) + C_{w 2} ({\overset{\cdot}{Y}}_{w 2} - {\overset{\cdot}{Y}}_{t 1} + {\overset{\cdot}{θ}}_{t 1} L_{t}) + F_{w 2} = 0

(12)

Vertical motion equation of wheel 3 (Y_w₃)

M_{w 3} {\overset{··}{Y}}_{w 3} + K_{w 3} (Y_{w 3} - Y_{t 2} - θ_{t 2} L_{t}) + C_{w 3} ({\overset{\cdot}{Y}}_{w 3} - {\overset{\cdot}{Y}}_{t 2} - {\overset{\cdot}{θ}}_{t 2} L_{t}) + F_{w 3} = 0

(13)

Vertical motion equation of wheel 4 (Y_w₄)

M_{w 4} {\overset{··}{Y}}_{w 4} + K_{w 4} (Y_{w 4} - Y_{t 2} + θ_{t 2} L_{t}) + C_{w 4} ({\overset{\cdot}{Y}}_{w 4} - {\overset{\cdot}{Y}}_{t 2} + {\overset{\cdot}{θ}}_{t 2} L_{t}) + F_{w 4} = 0

(14)

In these equations, F_wi_{(i = 1,2,3,4)} is wheel–rail force that is calculated as F_wi = K_H (Y_wi − Y_r), K_H = 3/2 C_H (Y_wi − Y_r)^1/2. Parameters of C_H, Y_wi and Y_r are Hertz spring constant, wheel and rail displacements, respectively. This linearization has been carried out in many similar research works^7–9 for simplification of numerical integration during the numerical analysis. Based on the presented equations, the mass, stiffness and damping of railway vehicle matrix are derived. Then, the mass, stiffness and damping matrix of vehicle–track system is formed by assembling the matrix of railway track and vehicle. Then, the total equation of vehicle–track system is solved by using the Newmark method. Table 1 presents the important parameters of railway track and vehicle and their values.

Table 1.

Railway vehicle and track parameters.^7–9

Railway track			Railway vehicle
Parameter	Symbol	Value	Parameter	Symbol	Value
Rail density	ρ_R	7850 kg/m³	Carbody mass	M_c	49,500 kg
Rail length	L_R	39 m	Bogie mass	M_b	10,750 kg
Rail cross section area	A_R	76.9 cm²	Wheel mass	M_w	2200 kg
Rail Young’s modulus	E_R	2.1 × 10⁵MPa	Rotational inertia of carbody	J_c	1.7 × 10⁶kg m²
Rail moment of inertia	I_R	3.055 × 10⁻⁵m⁴	Rotational inertia of bogie	J_b	9.6 × 10³kg m²
Sleepers spacing	L_S	0.6 m	Distance of two wheels centrelines	L_t	2.5 m
Rayleigh damping coefficient	α	400	Distance of two bogies centrelines	L_c	19 m
Rayleigh damping coefficient	β	4 × 10⁻⁷	Bogie stiffness	K_t	1.7 MN/m
Stiffness of track fastening system	K_F	250 MN/m	Bogie damping	C_t	300 kN s/m
Damping of track fastening system	C_F	248 kN s/m	Wheel stiffness	K_w	4.3 MN/m
Stiffness of track sleeper support	K_S	100 MN/m	Wheel damping	C_w	220 kN s/m
Damping of track sleeper support	C_S	180 kN s/m	Vehicle speed	V	100 km/h
Sleeper mass	M_S	300 kg	Hertz spring constant	C_H	1 × 10¹¹N/m^3/2

Numerical results of vehicle–track interaction model

In this section, first the vehicle–tracks interaction model is validated and in continuation, a series of sensitivity analyses are performed.

Vehicle–track interaction model validation

For validation of presented model, the analysis results are compared with those presented by Lei and Zhang.¹⁰ Figure 3 illustrates the rail deflection under the moving vehicle effect.

Figure 3.

Rail vertical displacement due to moving vehicle.

As can be seen from Figure 3, there is a good agreement between the results of present study model and the work of Lei and Zhang.¹⁰ In continuation, a series of sensitivity analyses are performed on the railway track dynamic interaction considering the unsupported sleepers parameters.

Sensitivity analyses results

In the previous part, the modelling procedure of train–track system by using the FEM was explained and the developed model was validated with reported results of Lei and Zhang.¹⁰ In this section, numerical results of train–track system are presented for the various cases including railway track without unsupported sleepers, track with an unsupported sleeper and track with two unsupported sleepers. During dynamic analyses, a spring and damper are imposed beneath the sleeper. For modelling the unsupported (or partially supported) sleeper condition, a gap can be imposed beneath the sleeper. Consequently, the spring stiffness and damping of sleeper support can be conditionally defined as follows

K_{S} = {if Δ_{Sleeper} ≺ Gap; K_{S} = 0 if Δ_{Sleeper} \geq Gap; K_{S} = SleeperSupportStiffness} C_{S} = {if Δ_{Sleeper} ≺ Gap; C_{S} = 0 if Δ_{Sleeper} \geq Gap; C_{S} = SleeperSupportDamping}

(15)

where Δ _Sleeper is sleeper deflection, K_S and C_S are spring stiffness and damping of sleeper support, respectively.

Figures 4 and 5 show the displacements of sleeper and track sleeper support forces in adjacent sleeper, respectively, due to the analysis of railway track with an unsupported sleeper for different track support stiffness and gaps values beneath the unsupported sleeper.

Figure 4.

Displacement of sleeper with an unsupported sleeper. (a) Minimum support stiffness 50 MN/m and (b) maximum support stiffness 200 MN/m.

Figure 5.

Track support force in adjacent sleeper with an unsupported sleeper. (a) Minimum support stiffness 50 MN/m and (b) maximum support stiffness 200 MN/m.

As can be observed from Figures 4 and 5, the absolute displacement of sleeper increases when there is an unsupported sleeper in the railway track. Also, it decreases by increasing the track support stiffness. For example, the absolute displacement of track with an unsupported sleeper increases 93.2% with respect to the track without unsupported sleepers while the track stiffness is 50 MN/m. On the other hand, by increasing the support stiffness from 50 to 200 MN/m, the absolute displacement of sleeper decreases 43.3% with respect to the gap value of 1.5 mm. Moreover, the support force of adjacent sleeper increases when an unsupported sleeper exists in the track. For instance, the ratio of support force of track with an unsupported sleeper to track without unsupported sleepers is 1.39. Figure 6 depicts the maximum displacement of sleeper and support force in track with an unsupported sleeper for various support stiffness and gap values.

Figure 6.

Maximum displacement of sleeper and support force in track with an unsupported sleeper. (a) Maximum displacement of sleeper and (b) maximum track support force in adjacent sleeper.

Based on Figure 6, the absolute maximum sleeper displacement and support force in adjacent sleeper increase when there is an unsupported sleeper in the track with respect to track without unsupported sleepers. In continuation, the effects of two unsupported sleepers are investigated. Figures 7 and 8 indicate the displacements of sleeper and track sleeper support forces in adjacent sleeper, respectively, due to the analysis of railway track with two unsupported sleepers for different track support stiffness and gap values beneath unsupported sleepers.

Figure 7.

Displacement of sleeper with two unsupported sleepers. (a) Minimum support stiffness 50 MN/m and (b) maximum support stiffness 200 MN/m.

Figure 8.

Track support force in adjacent sleeper with two unsupported sleepers. (a) Minimum support stiffness 50 MN/m and (b) maximum support stiffness 200 MN/m.

As can be seen from Figures 7 and 8, the sleeper displacement increases when there are two unsupported sleepers in the railway track compared to track with an unsupported sleeper while it decreases by increasing the sleeper support stiffness. For example, the absolute displacement of sleeper with two unsupported sleepers increases 53.88% with respect to the track with an unsupported sleeper in the case of track stiffness 50 MN/m. Also by increasing the support stiffness from 50 to 200 MN/m, the sleeper deflection decreases 36.8% for a gap of 2 mm. Furthermore, the support force of adjacent sleeper increases 1.28 times when two unsupported sleepers exist in the track with respect to the track with an unsupported sleeper. Figure 9 illustrates the maximum sleeper displacements and support forces in adjacent sleeper of track with two unsupported sleepers for various support stiffness and gap values.

Figure 9.

Maximum displacement of sleeper and support force in track with two unsupported sleepers. (a) Maximum displacement of sleeper and (b) maximum track support force in adjacent sleeper.

According to Figure 9, the maximum displacement of sleeper and support force in adjacent sleeper increase when there are two unsupported sleepers in the track with respect to the track with an unsupported sleeper and track without unsupported sleepers. Figure 10 shows the effects of vehicle speed on track support force under the various conditions of railway track for support stiffness 50 MN/m.

Figure 10.

Maximum track support force in adjacent sleeper.

As can be observed from Figure 10, track support force increases 4.6, 13.2 and 25.6% due to increase in vehicle speed from 100 to 200 km/h for track without unsupported sleepers, track with one and two unsupported sleepers, respectively. In continuation, the environmental vibrations are investigated for various conditions of railway track including unsupported sleeper(s).

Numerical model for assessing the train-induced vibration model

In this section, the numerical model for evaluation of ground-borne vibrations is presented. This model which is developed by plane strain elements of ABAQUS as well-known finite element code has two parts including the near and far fields. The near field contains railway and surrounding media and the far field includes infinite space. The near field is modelled by using plane strain finite elements and the far field is simulated by using the dashpot elements as absorbing boundaries.²³ According to theory of elasticity, the elastic settlement of subgrade (Se) under a rectangular foundation is calculated based on the following equation

S_{e} = \frac{qB (1 - μ_{S}^{2})}{E_{S}} (I_{1} + \frac{1 - 2 μ_{s}}{1 - μ_{s}} I_{2}) I_{F}

(16)

In this equation, E_S, μ_S, q and B are subgrade elastic modulus, subgrade Poisson’s ratio, mean stress value beneath the foundation and foundation width, respectively. Also, I₁, I₂ and I_F are correction factors.²⁴ So, the subgrade elastic modulus can be using the following equation

E_{S} \approx \frac{B (1 - μ_{S}^{2})}{A} K_{S}

(17)

In this equation, K_S and A are the subgrade stiffness and base area of sleeper, respectively. In this model, the load is applied to track centreline and consequently the environmental vibrations are calculated in the ground surface for various bench points.^15–17 Figure 11 shows the model of environmental vibrations.

Figure 11.

Environmental vibration model.

Validation of ground-borne vibration numerical model

For FEM model validation, the results of present study are compared with the works of Yang and Hung¹⁵ and Ni et al.¹⁴ due to harmonic load excitations. Table 2 shows the railway and soil specifications for validation purposes.

Table 2.

Specifications of vibration model parts.¹⁵

Materials	Density (kg/m³)	Young’s modulus (MPa)	Poisson’s ratio	Damping ratio
Railway	2400	13,500	0.25	0.02
Soil	1800	46	0.25	0.05

The index of train environmental vibrations, i.e. amplitude reduction ratio (A_r) is calculated as follows

A_{r} = \frac{Ground Surface Displacement with Trench}{Ground Surface Displacement without Trench}

(18)

Figure 12 illustrates the obtained amplitude reduction ratio (A_r) in the present study and other researchers’ studies.

Figure 12.

Validation of the results in present study.

As shown in Figure 12, the presented results have a good agreement with the mentioned works. Consequently, it can be claimed that the finite element model of wave propagation has enough validity. In continuation, a series of sensitivity analyses are accomplished on the environmental vibration model for track without unsupported sleepers, track with an unsupported sleeper and track with two unsupported sleepers.

Effects of unsupported sleeper on ground-borne vibrations

In the previous section, the FEM modelling procedure for assessing the train-induced environmental vibrations and its validation were described. In this part, the environmental vibration results with respect to various track support conditions including track without unsupported sleepers, track with an unsupported sleeper and track with two unsupported sleepers are presented. In this regard, the maximum track support forces in adjacent sleeper which were calculated for different support conditions and stiffness in previous section are applied to the vibration model as input force. Consequently, the environmental vibrations are determined for various bench points. Figures 13 and 14 indicate the time histories of vibration velocities for different points in the environmental vibration model.

Figure 13.

Vibration velocity for minimum support stiffness 50 MN/m at various distances from applied load. (a) Vibration velocity at distance of 3 m from applied load, (b) vibration velocity at distance of 6 m from applied load and (c) vibration velocity at distance of 10 m from applied load.

Figure 14.

Vibration velocity for maximum support stiffness 200 MN/m at various distances from applied load. (a) Vibration velocity at distance of 3 m from applied load, (b) vibration velocity at distance of 6 m from applied load and (c) vibration velocity at distance of 10 m from applied load.

As can be observed from Figures 13 and 14, at a distance of 3 m from applied load, the maximum vibration velocity increases 39.9 and 47.4%, respectively, for track support stiffness 50 and 200 MN/m in the case of presence of an unsupported sleeper with respect to track without unsupported sleepers. In addition, these values increase to 85 and 101.15% for track with two unsupported sleepers. It should be emphasized that the oscillation of PPV considerably depends on soil media damping. In the present study, the soil damping has been considered based on Table 2. So, considering the damping ratio as 0.05, semi-non-oscillating behaviour is observed in the results. But, by setting the damping ratio equal to zero, the oscillating behaviour can be observed for vibration velocity for various distances from railway track. Figure 15 shows the effects of vehicle speed on environmental vibrations for the various conditions of railway track.

Figure 15.

Peak particle velocity.

As can be observed from Figure 15, PPV increases 80.3, 96.5 and 116.8% when two unsupported sleepers exist in railway track in comparison to track without unsupported sleepers for vehicle speed 100, 150 and 200 km/h, respectively. In continuation, regression equations are derived between PPV and track parameters.

Derivation of regression equations between PPV and sleeper support parameters

Figure 16 illustrates the PPV versus track support stiffness for various track conditions.

Figure 16.

Peak particle velocity for various distances. (a) Peak particle velocity at distance of 3 m from applied load, (b) peak particle velocity at distance of 6 m from applied load and (c) peak particle velocity at distance of 10 m from applied load.

As can be observed from this figure, the ratio of PPV for track with two unsupported sleepers to track with an unsupported sleeper is 1.32 and 1.36 for track support stiffness 50 and 200 MN/m, respectively, at distance of 3 m from applied load. Moreover, the mathematical relation between the PPV and track support stiffness is non-linear in bench points close to the applied load but it tends to linear by increasing the distance of bench points. Tables 3 to 5 present the regression equations between PPV and track support stiffness for different track conditions and various distances of bench points. In these tables, parameters of PPV and SSS are the peak particle velocity (mm/s) and sleeper support stiffness (MN/m), respectively.

Table 3.

Derived regression equations at distance of 3 m from applied load.

Track conditions	Power equation	R-squared value
Without unsupported sleepers	PPV = 13.81(SSS)^−0.18	R²= 0.91
With unsupported sleeper	PPV = 18.77(SSS)^−0.17	R²= 0.94
With two unsupported sleepers	PPV = 21.46(SSS)^−0.14	R²= 0.94

PPV: peak particle velocity.

Table 4.

Derived regression equations at distance of 6 m from applied load.

Track conditions	Power equation	R-squared value
Without unsupported sleepers	PPV = 7.38(SSS)^−0.08	R²= 0.7
With unsupported sleeper	PPV = 13.06(SSS)^−0.13	R²= 0.9
With two unsupported sleepers	PPV = 13.78(SSS)^−0.07	R²= 0.67

PPV: peak particle velocity.

Table 5.

Derived regression equations at distance of 10 m from applied load.

Track conditions	Linear equation
Without unsupported sleepers	PPV ≈ 4.63
With unsupported sleeper	PPV ≈ 6.66
With two unsupported sleepers	PPV ≈ 8.97

PPV: peak particle velocity.

As can be seen from these tables, the mathematical relations between PPV and support stiffness are non-linear and linear at distances of 3 and 10 m from applied load, respectively. Also, the track with unsupported sleepers has more vibration velocities than track without unsupported sleepers for all distances.

Conclusions

In this paper, the effects of unsupported railway sleepers on the environmental train-induced vibrations were investigated. In this regard, first the model of train–track dynamic interaction was described and validated. In continuation, the effects of track sleeper support on maximum support force were studied. Subsequently, a validated model of train-induced vibrations was presented based on the continuum FEM. The output of the first FEM model as the total support force including spring and dashpot forces was introduced to the second one as an input force. Subsequently, the effects of the mentioned track support conditions on the ground-borne vibrations were investigated. The results showed that the displacement of railway sleeper under the moving vehicle increased 93.2 and 232.9% when there was an unsupported sleeper in the railway track with respect to track without unsupported sleepers for track support stiffness 50 and 200 MN/m, respectively. Also, these values rose to 197.3 and 471% for track with two unsupported sleepers. The ratio of track support force in adjacent sleeper with two and one unsupported sleepers to it without unsupported sleepers under the moving vehicle was 1.4 and 1.78, respectively, when track support stiffness was 50 MN/m. Also, the maximum vibration velocity of ground surface increased 39.9 and 47.4% when an unsupported sleeper existed in the track comparing to track without unsupported sleepers for track support stiffness 50 and 200 MN/m, respectively, at distance of 3 m from applied load. As well as, these values rose to 85 and 101.15% for track with two unsupported sleepers. Moreover, the regression equations between the PPV and track support stiffness in bench points close to applied load were non-linear whereas by increasing the distance of bench points from the applied load, they tended to linear form.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Numerical investigation of the effects of unsupported railway sleepers on train-induced environmental vibrations

Abstract

Keywords

Introduction

Railway vehicle – track interaction simulation

Numerical results of vehicle–track interaction model

Vehicle–track interaction model validation

Sensitivity analyses results

Numerical model for assessing the train-induced vibration model

Validation of ground-borne vibration numerical model

Effects of unsupported sleeper on ground-borne vibrations

Derivation of regression equations between PPV and sleeper support parameters

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References