Dynamic behavior of a polyurethane foam solidified ballasted track in a heavy haul railway tunnel

Abstract

Dynamic behavior of a new type of track using the polyurethane foam solidified ballast in heavy haul railway tunnels is comprehensively investigated in this study. First, a dynamic model of the vehicle–track–tunnel interaction system was developed based on the multi-body system dynamics theory and finite element method. Then, the dynamic effects of the polyurethane foam solidified ballast track on the train and the surrounding infrastructures were calculated and compared to those of the traditional ballasted track. Moreover, the effects of the elastic modulus and the solidified area size of polyurethane foam solidified ballast on the dynamic behavior were analyzed. Results show that, compared to the traditional ballast bed, polyurethane foam solidified ballast decreases the track stiffness and the vibration acceleration of the tunnel, while does not affect the vehicle safety (derailment coefficient and the rate of wheel load reduction). A larger elastic modulus of polyurethane foam solidified ballast has little effects on the wheel–rail interaction and the vibration acceleration of the tunnel, while a smaller modulus results in amplification of the displacements of rails and sleepers. Considering the vehicle–track interaction and tunnel vibration, the optimal elastic modulus of polyurethane foam solidified ballast is suggested to be 60–80 MPa. In addition, smaller solidified area of polyurethane foam solidified ballast presents lower effects on the vibration reduction and rate of wheel load reduction, while larger area leads to a higher derailment coefficient and cost. Therefore, an optimal solidified area size of polyurethane foam solidified ballast with the top width of 0.85 m is recommended.

Keywords

dynamic behavior elastic modulus heavy haul railway polyurethane foam solidified ballast solidified area vehicle–track interaction

Introduction

The ballasted track is a traditional track structure which has been widely used over the world due to its advantages including fast construction, low cost, and easy maintenance. However, this type of track structure has some drawbacks such as instability in the lateral direction, easy dirt accumulation, high settlement, and maintenance frequency (Esveld, 2001; Lim, 2004). In order to address some of the drawbacks, Woodward et al. (2007; 2012a, 2012b) proposed the Xi-TRACK technique to bond the ballast particles on the surface of the ballast bed, integrating the ballast particles into a whole body structure in order to reduce the deterioration and deformation of the ballast bed and to prolong the service life of the track system. In this technique, polyurethane foam is injected into the clean ballast bed using a pressure foaming machine (Keene et al., 2012, 2013). After hardening, a polyurethane foam solidified ballast (PFSB) is developed, which has high elasticity, as shown in Figure 1. This has drawn much attention of several researchers (Ebrahimi et al., 2015). In the design of Chinese heavy haul track, PFSB was proposed as a trial structure to reduce the maintenance in tunnels, since the ballast tamping and cleaning in tunnels are more difficult than those on the subgrade or bridges. PFSB, this new type of ballasted track, is expected to improve performance on the vehicle–track dynamic behavior.

Figure 1.

Polyurethane foam solidified ballast (PFSB).

The trial lines with a total mileage of only 7.3 km were constructed using the PFSB in China. Recently, some related research has been conducted on the PFSB at home and abroad. Wang (2015) measured the ballast settlement, lateral and longitudinal ballast resistances, and vertical load distribution of those lines on site. Zheng et al. (2015) tested the tensile, shear, compression strength, and cohesion of PFSB in the laboratory. Qie et al. (2015) investigated the effects of freeze–thawing cycles and fatigue loading on the elastic properties, accumulative deformation, and ballast resistance. Keene et al. (2012) found that the average elastic modulus of PFSB was lower than that of clean ballast bed and the cumulative plastic strain of the PFSB structure was significantly smaller than the ballast bed. Wang et al. (2014) investigated the structure size of PFSB and the stiffness ratio of PFSB to ballast particles using quasi-static analysis. However, most of the current research mainly focuses on the structural parameters and the material properties of polyurethane. Few studies have explored the dynamic effects of PFSB on the train and surrounding infrastructures which are crucial for the safety of railway operation. Hence, this study intends to conduct an in-depth investigation on the dynamic behavior of PFSB under railway operational conditions.

In the studies of traditional ballasted tracks or slab tracks, the vehicle–track coupling dynamic models are often used. There are a variety of ways to model the ballast structure. Clark et al. (1982) regarded the ballast as a spring structure, Nielsen and Igeland (1995) simulated the ballast using a spring–damper system, while Zhai et al. (2004) simulated the ballast as a mass–spring–damper system with the consideration of the weight and shear damping of the ballast structure. For slab tracks, the concrete slab has been modeled as a thin plate or beam structure (Hussein and Hunt, 2009; Xiang et al., 2008; Zhai et al., 2010). In addition, Xu et al. (2015) analyzed the environmental vibrations induced by the underground railway through modeling the steel spring floating slab with the solid element. As for PFSB, since it combines the characteristics of both ballasted track and slab track, the interaction between the solidified part and the granular part is important to their behaviors. Therefore, the particularity of PFSB should be considered during modeling.

In this study, a dynamic model of the vehicle–track–tunnel interaction system was developed based on the dynamic theory of the vehicle–track coupling system (Iwnicki, 2006; Zhai, 2007). The dynamic effects of the PFSB track on the train and surrounding infrastructures, including the wheel–rail contact forces, the rate of wheel load reduction, the derailment coefficient, the stresses and displacements of rails and sleepers, and the vibration acceleration of tunnel, were calculated respectively and then compared to those of traditional ballasted tracks. Moreover, the effects of elastic modulus and solidified area (width and depth) of PFSB on the dynamic behavior were analyzed. This research is expected to provide a theoretical foundation for the optimal design of PFSB in heavy haul railways, and to bring a novel insight into the structural design for vibration reduction in sensitive environments.

Dynamic model of the vehicle–track–tunnel

In this study, the commercial finite element software ABAQUS (Hibbitt, Karlsson and Sorensen, 2004) was used to establish a three-dimensional (3D) dynamic model of the vehicle–track–tunnel interaction system. The model consists of three main components: the vehicle model, the track–tunnel model, and the wheel–rail contact model.

Vehicle model

With the development of heavy haul railways, the axle loads of freight trains are gradually increasing and have exceeded 30 tons in some countries (Esveld, 2001). Therefore, a freight train with the axle load of 30 tons was considered in this study. A full vehicle model composed of a car body, two bogies, and four wheelsets (Zhang et al., 2010) was established, as shown in Figure 2(a). The bogie was simulated as a three-piece bogie including one bolster and two side frames that were connected by center suspension (Wang and Gao, 2015; Zhai, 2007). In the model, the car body, bogies, and wheelsets were regarded as rigid bodies.

Figure 2.

(a) Vehicle model, (b) model of track with PFSB, (c) tunnel model, (d) wheel–rail contact interaction, and (e) the dynamic model of vehicle–track–tunnel interaction system.

In the coordinate system of the model, z is vertically downward, x is along the rail, and y is perpendicular to the rail in the horizontal plane. The vehicle model has 39 degrees of freedom (DOFs). Specifically, the car body has 5 DOFs in the direction of y, in the direction of z, rotating x, rotating y, and rotating z (named lateral, floating, rolling, pitching, and yawing vibration), so do the four wheelsets. Each bolster had only 1 DOF in the yawing direction and all other freedoms were constrained rigidly by connecting to the car body. For the four side frames, the DOFs in the longitudinal, lateral, and yawing directions were considered, while the DOF of rolling was neglected. The main parameters of the vehicle model are shown in Table 1.

Table 1.

Main parameters of the vehicle model.

Parameter	Unit	Value
Mass of car body	kg	109,800
Mass of side frame/bolster/wheelset	kg	497/745/1277
Moment of inertia of car body about the x-, y-, and z-axis	kg·m²	142,800/1,726,700/1,746,800
Moment of inertia of bolster about the x-, y-, and z-axis	kg·m²	138/244/244
Moment of inertia of side frame about the y- and z-axis	kg·m²	190/176
Moment of inertia of wheelset about the x-, y-, and z-axis	kg·m²	740/110/740
Stiffness of first suspension about the x-, y-, and z-axis	MN/m	16/13/160
Stiffness of second suspension about the x-, y-, and z-axis	MN/m	7.0/7.0/6.0
Radius of wheel	m	0.42
Distance between two bogies	m	9.2
Distance between two wheelsets	m	1.83

Track–tunnel model

The track–tunnel model consisted of two parts: the track model and the tunnel model. The track model (Figure 2(b)) was composed of the rails, fastener systems, sleepers, and the ballast. The solid element was used to simulate the rail, the sleepers, and the ballast (Liu and Qu, 2011; Wang and Markine, 2018), while the spring–damper element was used to simulate the fastener system. The rail was modeled according to the Chinese Railway Track Dynamics Standards CHN60 (Lei, 2015). The stiffness of the type-III fastener that was specified in Lei (2015) was 120 MN/m in the vertical direction and 40 MN/m in the lateral direction. The sleeper was modeled as type-III sleeper having the material property of C50 concrete, with the length of 2.6 m, width of 0.3 m, and height of 0.23 m (Li, 2011).

To better analyze the dynamic responses of PFSB, the parameters of the ballast structure in the model should be as close as possible to the ones on site. It was noted that the tamping area of the ballast bed was more rigid than other areas (Lim, 2004). Therefore, the stiffness of the ballast in the model was varied according to its location. In general, the tamped area of ballast bed becomes denser and the whole stiffness becomes larger. Thus, a higher elastic modulus of 130 MPa is taken for the tamped areas of ballast (Nimbalkar et al., 2012; Zhai et al., 2004). On the contrary, the other areas remain loose and the whole stiffness is small. Therefore, a low elastic modulus of 65 MPa is chosen for the ballast area without tamping (Chang, 2015; Xia et al., 2016). A partly solidified PFSB track structure was designed by Wang (2015) who considered the supporting conditions and economic factors. The PFSB track is shown in Figure 1. Considering that the rail bottom width is 150 mm and the minimum width from the boundary of the solidified area to the rail bottom is 350 mm (Quante, 2001; Wang et al., 2014), the width of solidified bed top is set as 850 mm. The trapezoidal solidification with the slope of 1:0.43 is set because of the high fluidity of polyurethane during injection. The height of the solidified area is 0.35 m complying with the Code for Design of Railway Track (TB10082-2005). Since the PFSB is an elastic material, an elastic modulus of 60 MPa was chosen based on relevant literatures (Qie et al., 2015; Wang et al., 2014; Zheng et al., 2015). Rayleigh damping coefficient was calculated, and the damping coefficients (α and β) of the PFSB were 3 and 2.5 × 10⁻⁵, respectively, while those of ballast bed were 3 and 2 × 10⁻⁵, respectively.

Usually, railway tunnels are very long and the surrounding rocks are extremely thick. In this article, the solid element C3D8R, a three-dimensional eight-node solid element with linear reduction integration (Hibbitt, Karlsson and Sorensen, 2004) in ABAQUS is used to model the tunnel, which is very prevalent in the literature (Usman and Galler, 2013; Zhang and Huang, 2014), as shown in Figure 2(c). An inverted arch (C35 concrete), lining (C35 concrete), backfill soil (C20 concrete), and surrounding rock (1200 MPa/m) were considered in the model. The tunnel model was a 120 m double-line structure with both the depth and width of 80 m. In order to eliminate the boundary effects, the track length on both sides of the tunnel was set to be 50 m.

Wheel–rail interaction

The wheel–rail contact model is the coupling between the vehicle and the track, both in the normal and tangential direction. The normal force, which is generated by the compression of the wheel against the rail, can be calculated based on the non-linear Hertz elastic contact theory (Knothe and Grassie, 1993; Zhai, 2007)

p (t) = {[\frac{1}{G} Δ Z (t)]}^{3 / 2}

(1)

where G is the wheel–rail contact coefficient (Chen and Zhai, 2004; Zhai, 1992) and $Δ Z (t)$ represents the normal elastic compression deformation at the wheel–rail contact point.

In this model, the value of G for the wheel tread is

G = 4.57 R^{- 0.149} \times 10^{- 8} (m / N^{2 / 3})

(2)

where R is the wheel radius.

If $Δ Z (t)$ is less than 0, it indicates that the wheel and rail have been separated and wheel–rail force p(t) is equal to 0. Assuming that the displacement irregularities function of the surface wheel–rail contact is Z₀(t), then the wheel–rail force is defined as

p_{j} (t) = {\begin{matrix} {\frac{1}{G} [Z_{wj} (t) - Z_{r} (x_{rj}, t) - Z_{0} (t)]}^{3 / 2} \\ 0 \end{matrix}

(3)

where $Z_{wj} (t)$ is the displacement of wheel j at time t and $Z_{r} (x_{rj}, t)$ denotes the displacement of the rail under the wheel j at time t.

The wheel–rail contact was calculated using the penalty contact method, which is an internal contact algorithm defining the master surface and slave surface, as shown in Figure 2(d). The wheel tread and flange surfaces are defined as the master faces, while the rail head surface is considered as the slave face. The normal contact force between the master surface and the slave surface is simulated by defining a non-linear relationship.

The Coulomb friction model is applied to calculate the tangential wheel–rail interaction, and the contact surface frictional characteristic is represented by the friction coefficient (Vo et al., 2014; Wu et al., 2011; Zhao et al., 2016, 2017). A contact pair is defined using the same surface-to-surface contact algorithm (the penalty method based) as the wheel–rail contact pair. The magnitude of the friction F is equal to the product between the normal stress P(t) and the friction coefficient μ; when the friction stress reaches this limit, the related slip occurs

F = μ P (t)

(4)

The relationship between the dynamic and the static friction coefficients is simulated by the exponential decay model in ABAQUS, which is defined as

μ = μ_{k} + (μ_{s} - μ_{k}) e^{- d_{c} {\overset{\cdot}{r}}_{eq}}

(5)

where μ_k, μ_s, d_c, and ${\overset{\cdot}{r}}_{eq}$ represent the dynamic friction coefficient, the static friction coefficient, the extinction coefficient, and the relative slip rate, respectively.

Integration of the model

By combining the vehicle model and the track–tunnel model with the wheel–rail contact model, a dynamic model of the vehicle–track–tunnel interaction system was established (as shown in Figure 2(e)), which was used to analyze the dynamic effects of PFSB structure. As the excitation source in the model, the track irregularities are important. However, there is no heavy load spectrum in China. The heavy haul railway studied in the article has comparable track form, traffic volume, and maintenance mode with some American track. Moreover, considering the similar axle load around 30 tons and the operation speed around 60 km/h, an American V-level power spectral density of track irregularities (Sato, 1997) was applied.

The American V-level line irregularity power spectral is obtained based on a large amount of measured data by the US Federal Railway Administration (FRA). It is proposed to synthesize an even function with truncated frequency and roughness constant. Using the trigonometric series method, the power spectral density in the frequency domain was transformed into irregularity data along the railway line (Chen and Zhai, 1999). Because of the heavy loads and small curves, the actual operation speed is generally 55–70 km/h (Geng et al., 2008; Zhang et al., 2015), which may be further reduced to mitigate the disturbance of train vibration. Therefore, a vehicle speed of 60 km/h was selected as a basic condition to simulate the actual operation in the article.

Dynamic behavior of the system

Vehicle operational safety and the comfort for passengers could be indicated by operation safety indexes of the vehicle, dynamics indexes of the vehicle and track, and running stability indexes of the vehicle (Zhai, 2007), which are important in the design of track structure. In this article, the former two aspects were discussed to evaluate the dynamic performance of the vehicle–track coupling system. Operation safety indexes of the vehicle include the derailment coefficient and the rate of wheel load reduction, which are both calculated based on wheel–rail contact forces. The rate of wheel load reduction is defined as the ratio of the load reduction in the wheel vertical force, $Δ P$ , to the average of the left and the right wheel vertical forces, P, that is $Δ P$ /P. Based on Identification Method and Evaluation Standard for Dynamic Performance Test of Railway Locomotive (TB/T 2360-1993), safety standards of the rate of wheel load reduction are

{\begin{matrix} \frac{Δ P}{P} \leq 0.65 (Danger threshold) \\ \frac{Δ P}{P} \leq 0.60 (Allowable threshold) \end{matrix}

(6)

The derailment coefficient is defined as the ratio of wheel lateral (Q) to vertical (P) forces at the contact surface, that is, Q/P. Based on Identification Method and Evaluation Standard for Dynamic Performance Test of Railway Locomotive (TB/T 2360-1993), safety standards of the derailment coefficient are

{\begin{matrix} \frac{Q}{P} \leq 1.2 (Danger threshold) \\ \frac{Q}{P} \leq 1.0 (Allowable threshold) \end{matrix}

(7)

Dynamic indexes of vehicle and track could be evaluated based on wheel–rail contact forces, dynamic deformation of the track, vibration of track, and so on. These parameters are discussed in detail in the following sections.

Wheel–rail interaction

Figure 3(a) and (b) show the history of wheel–rail contact forces of the traditional ballasted track and the PFSB track at vertical and lateral directions, respectively. It is clearly seen that the vertical wheel–rail contact forces of the traditional ballasted track and the PFSB track are 201.677 and 199.955 kN and the lateral forces are 27.397 and 26.994 kN. This demonstrates that the PFSB track has little effect on wheel–rail forces compared to the ballasted track. Moreover, it is noted that there are some fluctuations in the wheel–rail contact forces, which is believed to result from the track irregularities rather than the ballast structure.

Figure 3.

(a) Vertical wheel–rail force and (b) lateral wheel–rail force.

Based on the vertical and lateral wheel–rail contact forces, the rate of wheel load reduction and the derailment coefficient of the vehicle could be calculated to evaluate the safety of vehicle operation. Calculating results indicate that in the PFSB track, the rate of wheel load reduction decreases to 0.395 from 0.401 in ballasted track, and the derailment coefficient also decreases to 0.174 from 0.176. These results suggest that the PFSB has almost no effect on wheel–rail forces and does not affect the safety of train operation.

Stress and displacement of the track

The dynamic stresses of rail and sleeper are shown in Figure 4(a). The stress observation points of rail and sleeper are located at the rail head and in the middle of the sleeper, respectively. An increase in rail stress from 98.132 MPa in the traditional ballasted track to 99.963 MPa in the PFSB track is found, and the sleeper stress also ascends from 1.012 to 1.287 MPa. The excessive bending deformation of the concrete sleeper caused by the low stiffness of the ballast layer may be responsible for the incremental dynamic stress of the sleeper.

Figure 4.

(a) Dynamic stress of rail and sleeper and (b) dynamic displacement of rail and sleeper.

The dynamic displacements of rail and sleeper are shown in Figure 4(b). It is noted that when the PFSB is used in the heavy haul railways, the rail displacement increases to 2.632 from 2.422 mm in traditional ballasted track, and meanwhile the sleeper displacement increases to 0.781 from 0.535 mm. The rail displacement reflects the dynamic stiffness of the whole track structure, while the sleeper displacement corresponds to the compression of the ballast layer (Zakeri and Abbasi, 2012). Polyurethane foam is a low-stiffness cellular material. When it is injected into the ballast bed and fills the gaps between the ballast particles, the overall stiffness is reduced and the deformation is increased (Jmal et al., 2014; Ju et al., 2013).

Vibration of tunnel

The vibration acceleration of the tunnel is shown in Figure 5(a) and (b). In the case of the PFSB track, the vertical acceleration of the tunnel is 0.476 m/s², which is significantly lower than that in the traditional ballasted track (0.847 m/s²). The PFSB is able to significantly isolate vibration (by about 50%) and consequently reduce the transmission of vibration to substructures, which is caused by the high damping of polyurethane materials.

Figure 5.

(a) Time history curve of tunnel vibration acceleration, (b) vibration acceleration of tunnel, and (c) VLz of tunnel.

In order to further investigate the dynamic effects of the PFSB on vibration reduction, the degree of z-direction vibration level (VLz) of the tunnel was calculated and shown in Figure 5(c). The VLz was obtained by amending the vibration acceleration level (VAL) with z being the weighting factor, which could effectively evaluate the structure vibration characteristics in the frequency domain. The VAL was calculated according to equation (8) (Kato et al., 2014)

VAL = 20 \lg (\frac{a_{rms}}{a_{0}})

(8)

where a_rms is the root mean square (RMS) value of vibration acceleration corresponding to one-third of the octave center frequency, and a₀ is the reference acceleration equal to 1 × 10⁻⁶ m/s².

From Figure 5(c), it is clearly seen that the VLz of the tunnel in the PFSB is lower than that in the traditional ballasted track, especially in the frequency below 50 Hz. The reduction in VLz caused by the PFSB varies with different frequency, and it is more than 5 dB in most cases. In general, the PFSB facilitates an excellent isolation of vibration from the railway track to the tunnel, except for the slight amplification effects at the frequency of 80 Hz which is close to the natural frequency of PFSB.

According to the analysis of the tunnel vibration with and without the PFSB track, it can be found that PFSB is an effective way to reduce the vibration of substructures. Therefore, the railway lines with intensive requirement of low vibration, such as urban rail transit, tunnels in unstable rock areas, tunnels in fault zones, and the tunnels with high stress or rich water (Asakura and Kojima, 2003; Hwang and Lu, 2007; Nelson, 1996), can use the PFSB track.

Elastic modulus of PFSB

The elastic modulus of the PFSB varies highly due to the existence of polyurethane foam. Literature pointed out that the average elastic modulus of the PFSB was lower than that of the ballast bed (Keene et al., 2012). As a new type of track, the elastic modulus of the PFSB is required to match the dynamic behavior of current railway operation. Therefore, the optimal design of the PFSB is performed. During the process of optimal design of the PFSB, four elastic modulus values, which are 30, 60, 80, and 150 MPa, were selected to analyze the effects of the elastic modulus on the dynamic behavior of the PFSB. The solidified area in the PFSB is fixed, which is with the top width of 0.85 m, bottom width of 1.15 m, and depth of 0.35 m.

Influence on the wheel–rail interaction

The variations of the rate of wheel load reduction and the derailment coefficient of the vehicle with the elastic modulus are presented in Figure 6(a). The rate of wheel load reduction varies from 0.391 to 0.401, and the derailment coefficient changes in the range of 0.171–0.178. Their slight changes indicate that the elastic modulus of the PFSB has little influence on the vehicle safety. Although the impacts of the elastic modulus of PFSB on the train operation is not significant, lower elastic modulus values are beneficial to the safety of train operation.

Figure 6.

(a) Vehicle safety indexes with different elastic modulus, (b) stress with different elastic modulus, (c) track displacement with different elastic modulus, and (d) tunnel vibration acceleration with different modulus.

Influence on the stress and displacement of the track

The effects of elastic modulus on the stress of rail and sleeper are shown in Figure 6(b). With the increase in elastic modulus from 30 to 150 MPa, the rail stress varies from 101.146, 99.963, 99.270 to 98.113 MPa, and the sleeper stress varies from 1.535, 1.287, 1.263 to 0.962 MPa. It is clearly seen that the elastic modulus has no obvious effect on the stress of rail. By contrast, the stress of sleeper sharply drops as the elastic modulus increases from 30 to 60 MPa.

However, the variations of the dynamic displacements of the rail and sleeper with the elastic modulus are shown in Figure 6(c). In the four cases, the rail displacement varies from 2.852, 2.632, 2.560 to 2.410 mm, respectively, and the sleeper displacement decreases from 1.036, 0.781, 0.690 to 0.518 mm, respectively.

Clearly, with the increasing elastic modulus, a slight decrease in the rail displacement and a larger decrement in sleeper displacement at the elastic modulus of 30–60 MPa are noted. Under train loads, a lower elastic modulus definitely leads to larger compressive deformation of the PFSB structure, and hence brings larger dynamic displacements of the rail and sleeper. In the range of 30–60 MPa, the PFSB is softer than the untreated area (65 MPa) and the tamping area (130 MPa), which means that in the middle of the concrete sleeper, the displacement is larger than other areas under the wheel–rail load. To prevent the large dynamic deformation of tracks and the damage to sleepers, the elastic modulus of PFSB is suggested to be more than 60 MPa.

Influence on the vibration of the tunnel

The effects of the elastic modulus of PFSB on the vertical acceleration of the tunnel are shown in Figure 6(d). It is noted that with the modulus of the PFSB changing from 30 to 150 MPa, the tunnel acceleration grows rapidly from 0.249 to 0.925 m/s². A high elastic modulus of the PFSB leads to an increase in the track stiffness, which is detrimental to the vibration reduction in particularly at the elastic modulus of more than 80 MPa. Under the wheel–rail interaction, a low elastic modulus of the PFSB could effectively reduce the tunnel vibration, especially when the elastic modulus is less than 80 MPa. Taken the vibration reduction into account, the elastic modulus of the PFSB should be less than 80 MPa.

When the elastic modulus of the PFSB is lower than 60 MPa, the stresses and displacements of the rail and sleeper are very high, which may cause large track deformation. Thus, to avoid the deformation, the elastic modulus of the PFSB should be larger than 60 MPa. However, the PFSB contributes little to the vibration reduction when its modulus exceeds 80 MPa. Hence, the elastic modulus of the PFSB is suggested to be in the range of 60–80 MPa. During railway design, reasonable parameters of the PFSB should be selected according to the various types of rock and environment requirements to reduce the tunnel vibration. For the high requirements of vibration reducing in the tunnel and the vulnerable rock around the tunnel, the PFSB with an elastic modulus around 60 MPa is recommended. There are no special requirements for other areas. The elastic modulus of the PFSB can be adjusted by changing the apparent density of the polyurethane foam.

Solidified cross-sectional area of the PFSB

Solidified cross-sectional area case

The sizes of the solidified area of the PFSB determine the supporting conditions for the sleepers, which consequently affects the dynamic behaviors of the vehicle–track–tunnel interaction system. Moreover, considering the high cost of the PFSB material and the engineering difficulty during the construction, it is necessary to study the effects of the solidified area in the cross-section of the PFSB structure. On site, the slope of the solidified area of the PFSB is usually 1:0.43, and the bottom of each solidified area is wider than the top.

Based on the dynamic model, four cases were analyzed with the aim of optimizing the solidified cross-sectional area of the PFSB. Case 1 is the minimum dimension with the top width of 0.45 m and the bottom width of 0.75 m, as seen in Figure 2(b). Case 2 is the medium size with the top width of 0.85 m and the bottom width of 1.15 m. Case 3 is the critical size, where the top width is 1.20 m and two solidified areas are intersected at the bottom. Case 4 is the maximum size, where all ballast is solidified as the PFSB structure. In the four cases, the thickness of the ballast bed is set as 0.35 m according to the Code for Design of Railway Track (TB10082-2005). In the following dynamic analysis, the elastic modulus of the PFSB was set to be 60 MPa.

Wheel–rail interaction

Figure 7(a) shows the rate of wheel load reduction in the four cases. It is noted that the rate of wheel load reduction almost keeps constant at the beginning and then slightly increases with the increase in the solidified area. The variation trend is not obvious and all the values fall in the range of 0.395–0.401. The calculated derailment coefficients in the four cases are 0.167, 0.174, 0.175, and 0.189, respectively, which presents an evident growth trend with the increase in the solidified area. Combining the train operation safety and cost, case 2 is proposed to be a reasonable structure size.

Figure 7.

(a) Vehicle safety indexes in different cases, (b) stress of track in different cases, (c) track dynamic displacement in different cases, and (d) tunnel vibration acceleration in different cases.

Stress and displacement of track in different cases

The effects of solidified area size on the stress of rail and sleeper are presented in Figure 7(b). In the four cases, the dynamic stress of both rail and sleeper increases slightly. The former varies from 99 to 101 MPa and the latter changes from 1.219 to 1.287 MPa. This is reasonable since the area beneath the rail bears the dynamic load through the sleepers, which is not larger than the solidified area in the above four cases. Thus, the solidified area of the PFSB has no obvious influence on the stress of the track structure.

However, the variations of dynamic displacements of the rail and sleeper with the solidified area are shown in Figure 7(c). The rail displacement in the four cases is 2.544, 2.632, 2.643, and 2.648 mm, respectively, and the sleeper displacement gradually increases from 0.675, 0.781, 0.793 to 0.799 mm. It is clearly seen that the displacement increases rapidly in the first two cases and then becomes to be stable when the solidified area exceeds the medium size of 0.85 m (case 2). That is, the PFSB only slightly affects the stiffness of the integral track when the top width of the solidified area is larger than 0.85 m.

Vibration of the tunnel in different cases

The vibration acceleration of the tunnel in the four cases is presented in Figure 7(d). It is clear that an increase in the solidified area size results in an evident decrease in the vibration acceleration. At the top width of 0.45 m (case 1), the vibration acceleration of the tunnel is maximum which reaches to be 0.539 m/s². When the top width is larger than 0.85 m, the tunnel vibration decreases from 0.476 to 0.462 m/s² at a slow rate. This demonstrates that the effects in case 2, case 3, and case 4 on the vibration reduction of the tunnel are quite similar, and more significant than that in case 1.

Based on the above discussions, it can be concluded that case 1 presents the smallest effects on vibration reduction and the rate of wheel load reduction, while case 4 shows a high derailment coefficient and cost. The dynamic behaviors of case 2 and case 3 are almost similar, but case 2 is more economical due to its smaller size. Therefore, considering both the dynamic behavior and the cost, case 2 with a top width of 0.85 m is recommended as the optimal solidified size of the PFSB.

Verification and discussion

According to the above analysis, the optimal elastic modulus of the PFSB is suggested to be 60–80 MPa and the solidified area with a top width of 0.85 m is recommended as the optimal solidified size of the PFSB. The results have been applied to the Nanling Mountain Tunnel of Watang Town–Rizhao Harbor heavy haul railways in China. The length of the test line with the PFSB is 1100 m and the train running speed is 60 km/h, as shown in Figure 8(a).

Figure 8.

(a) Field test chart and (b) safety indexes.

Based on the Standard of the Testing Method of Horizontal and Vertical Force for Wheel and Rail (TB/T2489-94) and other standards, the dynamic properties of the PFSB and traditional ballasted track were tested. Train safety index was calculated based on wheel–rail forces. The derailment coefficient and the rate of wheel load reduction test results are shown in Figure 8(b). The rate of wheel load reduction and the derailment coefficient are far less than the threshold of safety index (0.6 and 1.0), indicating the safety of the train operation. In addition, it can be seen that the test value is smaller than the theoretical calculation value in this work. This is because the track is a newly built line featured with better regularity. Meanwhile, it also proves that the structure proposed in this study is reasonable.

In addition, the supporting stiffness of the sleepers was tested. The test results show that the supporting stiffness of the sleepers on the ballast bed and PFSB is 27.4–58.7 kN/mm and 16.4–40.8 kN/mm, respectively, indicating that the elasticity of the PFSB is better than the ballast bed.

The dynamic influences of different train speeds were analyzed based on the dynamic model of vehicle–track–tunnel interaction system. When the train speed increases from 60 to 120 km/h, the rate of wheel load reduction increases from 0.395 to 0.486 and the derailment coefficient increases from 0.174 to 0.219. Meanwhile, it can be clearly seen from Figure 9 that the acceleration of the tunnel wall grows obviously. However, the acceleration of the PFSB track at 120 km/h is still lower than that of traditional ballasted track at 60 km/h. Thus, it can be concluded that the PFSB track performs a good damping effect under different train speeds. It can also be found from the calculations and the related literature (Zhao et al., 2015) that the wheel–rail forces, and the vertical displacements of the sleeper and the ballast increase slightly, while the vibration acceleration of the sleeper and ballast increases significantly with the increase in train speed.

Figure 9.

Acceleration of the tunnel wall under different speeds.

Besides, the PFSB has been applied to the Beipan River Special Bridge of Shanghai–Kunming high-speed railway, the Wuhan Tianxingzhou Bridge in Beijing–Guangzhou high-speed railway, and the Urumqi subways in Xinjiang province. However, because the cost of the PFSB is about three times higher than that of the traditional ballasted track, the PFSB is only used in special sections, such as tunnels in heavy haul railways, large span bridges in high-speed railway, and tracks in earthquake fault zones.

Conclusion

In this study, a dynamic model of the vehicle–track–tunnel interaction system was developed using FEM software ABAQUS. In the model, the PFSB was simulated and track irregularities were used as the excitation source. The dynamic effects of the PFSB track on the wheel, rail, sleeper, and tunnel were calculated and compared with those of the traditional ballasted track. Moreover, the effects of the elastic modulus and solidified cross-sectional area on the dynamic behavior of the PFSB were discussed. Based on the obtained numerical results, the following conclusions can be drawn:

Compared to the traditional ballasted track, the PFSB track obviously decreases the track stiffness, resulting in an increase in the vertical displacements of rail and sleepers. By applying the PFSB in heavy haul railways, the safety index of vehicle changes slightly; however, the vibration acceleration of tunnel is significantly decreased (more than 5 dB in most frequency bands). That is, the vehicle–track system with the PFSB has better dynamic behavior and can be used in the sensitive environments with intensive requirements of vibration reduction.

The rate of wheel load reduction and the derailment coefficient of the vehicle almost keep constant when the elastic modulus of the PFSB is higher than 60 MPa, indicating the elastic modulus of the PFSB has little influence on the vehicle safety. With the increase in elastic modulus from 30 to 60 MPa, the stress and displacement of rail slightly decrease while those of sleeper drop significantly. A high elastic modulus of the PFSB leads to an increase in track stiffness, which is detrimental to the vibration reduction. From the viewpoints of both improving vehicle–track interaction and reducing tunnel vibration, the optimal elastic modulus of the PFSB is suggested to be 60–80 MPa.

The solidified area of the PFSB track has little effects on the rate of wheel load reduction and the stress of rail and sleeper. The derailment coefficient presents an evident growth trend with the increase in the solidified area. The displacements of rail and sleeper increase rapidly with the increasing solidified area size, and then becomes to be stable as the top width of solidified area is larger than 0.85 m. Lower solidified area of the PFSB presents smaller effects on vibration reduction and the rate of wheel load reduction, while larger area increases the derailment coefficient and cost. Therefore, the solidified area with a top width of 0.85 m is recommended as the optimal solidified size for the PFSB.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Natural Science Foundation of China (NSFC; Grant Nos 51578053 and 51778050), Joint Funds of Beijing Natural Science Foundation Committee and Beijing Academy of Science and Technology (Grant No. L150003), and the Fundamental Research Funds for the Central Universities (Grant No. 2018JBM041).

ORCID iDs

Xiaopei Cai

Yanglong Zhong

References

Asakura

Kojima

(2003) Tunnel maintenance in Japan. Tunnelling and Underground Space Technology 18(2–3): 161–169.

Chang

(2015) Study on key parameters of heavy haul railway track under 35.7t axle load. Railway Standard Design 59(8): 47–50+54.

Chen

Zhai

(1999) Numerical simulation of the stochastic process of railway track irregularities. Journal of Southwest Jiaotong University 34(2): 138–142.

Chen

Zhai

(2004) A new wheel/rail spatially dynamic coupling model and its verification. Vehicle System Dynamics 41(4): 301–322.

Clark

Dean

Elkins

et al . (1982) An investigation into the dynamic effects of railway vehicles running on corrugated rails. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 24(2): 65–76.

Ebrahimi

Tinjum

Edil

(2015) Deformational behavior of fouled railway ballast. Canadian Geotechnical Journal 52(3): 344–355.

Esveld

(2001) Modern Railway Track. 2nd ed. Zaltbommel: MRT-Productions.

Geng

Zhang

(2008) Simulation study of heavy haul train operation on Datong-Qinhuangdao railway. China Railway Science 29(2): 88–93.

Hibbitt, Karlsson and Sorensen (2004) ABAQUS/Standard User’s Manual. Pawtucket, RI: Hibbitt, Karlsson and Sorensen.

10.

Hussein

MFM

Hunt

HEM

(2009) A numerical model for calculating vibration due to a harmonic moving load on a floating-slab track with discontinuous slabs in an underground railway tunnel. Journal of Sound and Vibration 321(1): 363–374.

11.

Hwang

(2007) A semi-analytical method for analyzing the tunnel water inflow. Tunnelling and Underground Space Technology 22(1): 39–46.

12.

Iwnicki

(2006) Handbook of Railway Vehicle Dynamics. Leeds: CRC Press.

13.

Jmal

Dupuis

et al . (2014) Generalization of the memory integer model for the analysis of the quasi-static behaviour of polyurethane foams. Journal of Mechanical Science and Technology 28(11): 4651–4662.

14.

Mezghani

Jmal

et al . (2013) Parameter estimation of a hyperelastic constitutive model for the description of polyurethane foam in large deformation. Cellular Polymers 32(1): 21–40.

15.

Kato

Kamohara

Yokoyama

et al . (2014) Influence of variation in track support rigidity around overbridges on ground vibration. Quarterly Report of RTRI 55(4): 241–248.

16.

Keene

Edil

Tinjum

(2012) Mitigating ballast fouling and enhancing rail freight capacity. Report no. CFIRE 04–07, National Center for Freight and Infrastructure Research and Education and University of Wisconsin–Madison, Madison, WI, November.

17.

Keene

Edil

Fratta

et al . (2013) Modeling the effect of polyurethane stabilization on rail track response. In: Geo-Congress, San Diego, CA, 3–7 March, pp. 1410–1419.

18.

Knothe

Grassie

(1993) Modelling of railway track and vehicle/track interaction at high frequencies. Vehicle System Dynamics 22(3–4): 209–262.

19.

Lei

(2015) High Speed Railway Track Dynamics: Model, Algorithm and Application. Leeds: Beijing Science Press.

20.

(2011) Random vibration of track structure and reliability research of type III sleeper. PhD Thesis, Southwest Jiaotong University, Chengdu, China.

21.

Lim

(2004) Mechanics of railway ballast behavior. PhD Thesis, University of Nottingham, Nottingham.

22.

Liu

(2011) Study on selection of sleepers for ballasted track on bridges of high-speed railway. High Speed Railway Technology 2(3): 38–42.

23.

Nelson

(1996) Recent developments in ground-borne noise and vibration control. Journal of Sound and Vibration 193(1): 367–376.

24.

Nielsen

JCO

Igeland

(1995) Vertical dynamic interaction between train and track influence of wheel and track imperfections. Journal of Sound and Vibration 187(5): 825–839.

25.

Nimbalkar

Indraratna

Dash

et al . (2012) Improved performance of railway ballast under impact loads using shock mats. Journal of Geotechnical and Geoenvironmental Engineering 138(3): 281–294.

26.

Qie

Wang

et al . (2015) Research on dynamic test of the polyurethane foam solidified ballast. Railway Architecture 55(1): 107–112.

27.

Quante

(2001) Innovative track systems technical construction. European Commission Competitive and Sustainable Growth Programme Reports 6:17–18.

28.

Sato

(1997) Study on high-frequency vibrations in track operated high-speed trains. Railway Technical Research Institute Quarterly Reports 18(3): 109–114.

29.

Usman

Galler

(2013) Long-term deterioration of lining in tunnels. International Journal of Rock Mechanics and Mining Sciences 64(6): 84–89.

30.

Tieu

Zhu

et al . (2014) A 3D dynamic model to investigate wheel-rail contact under high and low adhesion. International Journal of Mechanical Sciences 85(8): 63–75.

31.

Wang

(2015) Development and application of the railway polyurethane foam solidified ballast technology. Railway Architecture 55(4): 135–140.

32.

Wang

Markine

(2018) Methodology for the comprehensive analysis of railway transition zones. Computers and Geotechnics 99: 64–79.

33.

Wang

et al . (2014) Structure and parameters of a polyurethane foam solidified ballasted track in a heavy haul railway. Railway Engineering 54(7): 129–132.

34.

Wang

Gao

(2015) Numerical simulation of wheel wear evolution for heavy haul railway. Journal of Central South University 22(1): 196–207.

35.

Woodward

Kacimi

Laghrouche

et al . (2012a) Application of polyurethane geocomposites to help maintain track geometry for high-speed ballasted railway tracks. Journal of Zhejiang University SCIENCE A 13(11): 836–849.

36.

Woodward

Kennedy

Medero

et al . (2012b) Maintaining absolute clearances in ballasted railway tracks using in situ three-dimensional polyurethane geo-composites. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 226(3): 257–271.

37.

Woodward

Thompson

Banimahd

(2007) Geocomposite technology: reducing railway maintenance. Proceedings of the ICE-Transport 160(3): 109–115.

38.

Wen

et al . (2011) Thermo-elastic-plastic finite element analysis of wheel/rail sliding contact. Wear 271(1): 437–443.

39.

Xia

Dou

et al . (2016) Numerical analysis of deformation performance of geogrid-reinforced railway ballast under train load. Journal of Railway Engineering Society 33(09): 24–28.

40.

Xiang

Zeng

(2008) Analysis theory of spatial vibration of high-speed train and slab track system. Journal of Central South University of Technology 15(1): 121–126.

41.

Xiao

Liu

et al . (2015) Comparison of 2D and 3D prediction models for environmental vibration induced by underground railway with two types of tracks. Computers and Geotechnics 68: 169–183.

42.

Zakeri

Abbasi

(2012) Field investigation on variation of rail support modulus in ballasted railway tracks. Latin American Journal of Solids and Structures 9(6): 643–656.

43.

Zhai

(1992) The vertical model of vehicle-track system and ITS coupling dynamic. Journal of the China Railway Society 14(3): 10–21.

44.

Zhai

(2007) Vehicle-Track Coupling Dynamics. 2nd ed. Leeds: Beijing Science Press.

45.

Zhai

Song

(2010) Prediction of high-speed train induced ground vibration based on train-track-ground system model. Earthquake Engineering and Engineering Vibration 9(4): 545–554.

46.

Zhai

Wang

Lin

(2004) Modelling and experiment of railway ballast vibrations. Journal of Sound and Vibration 270(4–5): 673–683.

47.

Zhang

Wang

Zhai

(2015) Effect of wheel flats on wheel/rail dynamic interaction in 30-ton heavy-haul railway. Journal of Southwest University of Science and Technology 30(4): 15–24.

48.

Zhang

Xia

Guo

et al . (2010) Vehicle-bridge interaction analysis of heavy load railway. Procedia Engineering 4: 347–354.

49.

Zhang

Huang

(2014) Geotechnical influence on existing subway tunnels induced by multiline tunneling in Shanghai soft soil. Computers and Geotechnics 56(3): 121–132.

50.

Zhao

Zhai

(2015) A discrete element simulation of high-speed railway ballast vibration. Chinese Journal of Computational Mechanics 32(5): 674–680.

51.

Zhao

et al . (2017) Local rolling contact fatigue and indentations on high-speed railway wheels: observations and numerical simulations. International Journal of Fatigue 103: 5–16.

52.

Zhao

Liu

Zhao

et al . (2016) A study on dynamic stress intensity factors of rail cracks at high speeds by a 3D explicit finite element model of rolling contact. Wear 366–367: 60–70.

53.

Zheng

Zeng

et al . (2015) The effect of the apparent density to the dynamic character of polyurethane foam solidified material. China Railway Science 36(3): 12–16.