# Study on the Vertical Stiffness of Bridge Expansion Joints Based on the Vehicle-Track-Bridge Coupled Model

Authors: Yunlu Wang; Mangmang Gao; Fei Yang
Journal: Shock and Vibration
 (2022)

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## Abstract

Bridge expansion joints (BEJs) are equipped at the girder end of long-span railway bridges to ensure the reliable transition of the track. BEJs should have suitable dynamic stiffness to ensure the running safety and stability of the vehicle. To explore the influence of dynamic vertical stiffness of BEJs on the dynamic response of vehicle, track, and bridge, a three-dimensional vehicle-track-bridge coupled model is established, and the influence of sleeper spacing and vehicle speed are taken into consideration. Taking a high-speed railway line in China as a case study, the dynamic response of the whole system is calculated. The results show that (1) the dynamic response of BEJs and the vehicle is affected by the vehicle speed and sleeper spacing. With the increase of vehicle speed and sleeper spacing, the dynamic response of the system increases; (2) the vertical stiffness of BEJs will affect the dynamic response of BEJs and the interaction between the wheel and rail: the vertical dynamic displacement of movable sleeper will exceed the limit if the vertical stiffness of pressure-bearing is insufficient, and the wheel unloading rate may exceed the limit if the vertical stiffness of the cushion plate is insufficient. Based on the results, the larger vertical stiffness of the cushion plate and pressure-bearing is recommended. These results have been applied to the design of BEJs on the studied railway line, which are in good service condition at present.


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## Body

## 1. Introduction

Nowadays, bridges are widely used in high-speed railway (HSR) to provide stable and easy maintenance line conditions [1, 2]. Vehicle, track, and bridge constitute a coupled system [3, 4], and the running safety and stability of the vehicle will be affected by the track and bridge [5, 6].Researchers have conducted a lot of work about the vehicle-track-bridge coupled system. Zhang [7] proposed an analysis method based on the pseudo-excitation method for the vehicle-bridge interaction system and obtained the probability density functions for safety factors and the probabilities of safety factors exceeding the given limits. Gao [1] proposed a vehicle-track-bridge coupled analysis method based on forced vibration, pointing out that local fastener failure will slightly affect the vibration of the track and car-body, but it will significantly intensify the interaction between wheel and rail. Jiang [8] used the new point estimate method to analyze the random dynamic responses of the 3D train-bridge coupled system with random parameters. Gong [9] studied the influence of spatially varying ground motion on the dynamic behavior of a train passing through a cable-stayed bridge. Xiang [10] investigated the influence of the track irregularity and deck deformation on the running safety of HSR trains. Chen [11] studied the influence of track stiffness on dynamic behaviors of the high-speed vehicle-track-bridge dynamic system. However, few studies have focused on the impact of the ancillary structure of the bridge on the vehicle-track-bridge coupled system.With the increasing bridge span, BEJs have become a vital device for long-span railway bridges. It is not only required to match the deformation of the girder end but also required to have enough strength and stiffness to bear the dynamic load brought by the vehicle. Since the track structure in BEJs is very different from that on the bridge, and the BEJs lack the support of lower foundation, BEJs are regarded as the weak parts of the railway and have attracted extensive interest from researchers. The performance of BEJs, including the static deformation and joint force, in service condition have been studied [12–15], and plenty of work has been conducted on the failure mechanism of BEJs [16, 17]. Since the track irregularity in BEJs can only be reduced by adjusting the height of the plate under the rail, and the adjustment range of height is very small, the deformation of BEJs should be controlled. The increase of stiffness of BEJs can reduce the deformation of the structure, but it will weaken the elasticity of the structure. Moreover, the vertical stiffness of BEJs should not differ too much from that of girders on both sides. The excessive change of track stiffness is unfavorable to running safety of the vehicle. Therefore, the stiffness of BEJs should be adopted reasonably. This study aims to provide a theoretical basis for adopting the stiffness of BEJs by analyzing the dynamic response of the vehicle and BEJs.Taking a high-speed railway line in China as a case to study, it has a total length of about 314 km, and the design speed of the vehicle is 250 km/h. In order to reduce the seismic response of the structure, hyperboloid bearings are used as seismic isolation devices for bridges, which results in a large displacement of the girder end. As the result, this railway line is equipped with BEJs at many positions to ensure the continuity of the track structure. The layout scheme of BEJs is shown in Figure1. The short vertical line represents the position of BEJs, and the passing time of the vehicle is also marked in the figure. As highlighted in the dashed box, the vehicle passes through the section with the densest arrangement of BEJs within 7.8 s.Figure 1
Layout scheme of BEJs in the high-speed railway line.In order to avoid the interference of boundary conditions, and consider the continuous impact of densely arranged BEJs on the vehicle, the section in the dashed box of Figure1 is studied rather than only analyzing one BEJ. The vehicle-track-bridge coupled model is established to investigate the influence of vertical stiffness of BEJs on the whole system by considering the influence of different displacement of the girder end and different vehicle speed. The dynamic response of the vehicle and BEJs is calculated when the vehicle passes through this section and evaluated based on the code. This study clarified the influence of BEJs' parameters on the dynamic response of the vehicle-track-bridge coupled system, and provided the suggestion of the reasonable value of vertical stiffness of BEJs based on the results.


## 2. Numerical Model

### 2.1. Bridge Model

The studied section includes two bridge structures: (A) 32 m precast post-tensioned prestressed concrete simply supported box girder; (B) a 60m + 100 m + 60 m prestressed concrete double track continuous girder. MIDAS is used to establish the finite element model (FEM) of the bridge, as shown in Figure 2, while the arrowhead indicates the forward direction of the vehicle. The boxes mark the positions of BEJs.Figure 2
The MIDAS model of the bridge.The bridges are simulated by beam elements. The bottom of the pier is fixed, and the constraints between the pier and the bridge deck are simulated by master-slave nodes. And the secondary dead load is applied to the girder body by increasing the bulk density of materials. Because the concrete bridge will arch due to the creep, which will influence the running stability of the vehicle [10, 18], the creep is taken as the initial deformation of the bridge deck in the model. The uniform stiffness matrix and the uniform mass matrix are adopted in dynamic analysis with Rayleigh damping.The motion equation of the bridge is shown in equation (1), where [Mb], [Cb], and [Kb] is the mass, damping, and stiffness matrix of the bridge, respectively; Xb represents the displacement vector of the bridge; Pbt is the interaction force between the track and the bridge, which is the function of the displacement and velocity of the track and the bridge.(1)MbXb¨+CbXb˙+KbXb=Pbt.


### 2.2. BEJs Model

BEJs with sliding sleeper is applied in the railway bridge studied. The structure of this type of BEJs is simpler than other styles, which is less affected by the displacement of the girder end and can be applied to the condition with a wider range of vehicle speed. The structure of BEJs is shown in Figure3. It is mainly composed of fixed steel sleeper 2, movable steel sleeper 3, supporting frame 7, supporting beam 4, cushion plate 5 (between fixed sleepers and supporting frames), connecting rods 6, and pressure-bearing 8 (between supporting beams and sleepers). The rail and concrete sleeper are denoted in the figure 1 and 9, respectively.Figure 3
The structure of BEJs: (a) top view. (b) A-A section. (c) B-B section.
(a)(b)(c)The supporting frames are installed on the girder body and have no relative displacement with the bridge. The fixed sleepers and supporting frames are connected by bolts, and there are no longitudinal and lateral relative displacements between them. The local vertical stiffness is provided by the cushion plates between them. One end of the supporting beam is connected with fixed sleepers, and fixed sleepers at the other end and the movable sleeper can slide relative to the supporting beam in the longitudinal direction. There is no lateral relative displacement between supporting beams and sleepers, and the local vertical stiffness is provided by the pressure-bearings between them. The movable steel sleeper, which suspended on the supporting beam, is connected to fixed sleepers through connecting rods. When longitudinal displacement occurs at the girder end, fixed sleepers at the one end and the movable sleeper slide relative to the supporting beam, and sleeper spacing between fixed sleepers and movable sleepers are adjusted by connecting rods. The maximum longitudinal displacement of girder ends on both sides of BEJs is ±200 mm. The distance between the fixed sleeper and the movable sleeper is 450 mm when the longitudinal displacement of the girder end is 0 mm, and which is 350 mm or 550 mm when the longitudinal displacement of the girder end is +200 mm or -200 mm, respectively.The vertical stiffness of BEJs should not differ too much from that of girders on both sides. In order to make the stiffness under the sleeper of BEJs consistent with the stiffness under the sleeper of the ballasted track, the cushion plate is placed under the fixed steel sleeper. By adjusting the stiffness of the cushion plate, the abrupt change of track stiffness can be avoided, which is unfavorable to the running safety and stability. The pressure-bearing is placed between the sleepers and supporting beams to adjust the stiffness of BEJs too. The range of vertical stiffness of the cushion plate is 80 ∼ 120 kN/mm, and the range of vertical stiffness of the pressure-bearing is 160 ∼ 240 kN/mm.BEJs are modeled as the substructures of the bridge model through MIDAS, as shown in Figure4. Since the supporting frame are fixed with the bridge deck, it can be ignored in the model. The supporting beam and sleepers are simulated by beam elements, which are denoted in the figure. The nonlinear characteristics of track stiffness have little influence on the results and cost plenty of computational resources [19], hence, linear massless spring elements are used to simulate cushion plates and pressure-bearings. The stiffness of the spring in the BEJs model are changed to simulate different stiffness of cushion plates and pressure-bearings. And the MIDAS model of BEJs with different sleeper spacing is established to consider the influence of longitudinal displacement of the girder end on the system. Table 1 lists the details of different cases.Figure 4
The MIDAS model of BEJs.Table 1
Parameters of different cases.
ParameterCase1234Vertical stiffness of cushion plates (kN/mm)8012080120Vertical stiffness of pressure-bearings (kN/mm)160160240240Sleeper spacing (mm)350, 450, 550Figure5 shows the top view of BEJ models with different sleeper spacing, which is 350 mm, 450 mm, and 550 mm, respectively.Figure 5
The top view of the BEJs model: (a) sleeper spacing is 350 mm, (b) 450 mm, and (c) 550 mm.
(a)(b)(c)


### 2.3. Vehicle Model

The vehicle model consists of 16 vehicles with 8 electric moto units (EMUs), which are composed of car-bodies, bogies, wheelsets, and the spring and damping connections. In order to simplify the calculation, the mechanics model of the vehicle adopts the following assumptions [4, 20–22]: (1) the car-bodies, bogies, and wheelsets are regarded as rigid components, the connections are represented by linear spring and linear damping. (2) The vehicles move at constant speed along the track, the influence of the longitudinal force of the system is ignored. (3) The wheelsets are always in contact with the rail.As shown in Figure6, the vehicle is modeled by multibody mechanics with 23 degrees of freedom (DOF). The car-body has 5 DOF to be concerned: lateral displacement Yc, vertical displacement Zc, roll angle θc, yaw angle Ψc, and pitch angle φc. The bogie has 5 DOF too, which is denoted by subscript t. And the lateral displacement and roll angle of wheelsets are considered, which is denoted by subscript w. Table 2 provides the main parameters of the vehicle model.Figure 6
Vehicle model: (a) right view. (b) Rear view. (c) Bottom view.
(a)(b)(c)Table 2
The main parameters of the vehicle model.
ParametersUnitMotor car/trailer carMass of car-bodykg3.96× 104/3.44 × 104Mass of bogiekg3.20× 103/2.60 × 103Mass of wheelsetkg2.40× 103/2.40 × 103Roll mass moment of the car-bodykg·m20.128× 106/0.111 × 106Pitch mass moment of the car-bodykg·m21.94× 106/1.686 × 106Yaw mass moment of the car-bodykg·m21.673× 106/1.453 × 106Roll mass moment of the bogiekg·m22.59× 103/2.106 × 103Pitch mass moment of the bogiekg·m21.752× 103/1.423 × 103Yaw mass moment of the bogiekg·m23.20× 103/2.60 × 103Roll mass moment of the wheelsetkg·m2720/756Lateral stiffness of the primary suspension systemkN/mm1× 103/1 × 103Vertical stiffness of the primary suspension systemkN/mm1× 103/1 × 103Lateral stiffness of the secondary suspension systemkN/mm0.2× 103/0.2 × 103Vertical stiffness of the secondary suspension systemkN/mm0.2× 103/0.2 × 103Lateral damping of the primary suspension systemkN·s/m50/50Vertical damping of the primary suspension systemkN·s/m20/20Lateral damping of the secondary suspension systemkN·s/m50/50Vertical damping of the secondary suspension systemkN·s/m10/10Distance of two bogiesm17.5/17.5Distance of two wheelsetsm2.5/2.5The motion equation of the vehicle is shown in equation (2), where [Mv], [Cv], and [Kv] are the mass, damping, and stiffness matrix of the vehicle, respectively; Xv represents the displacement vector of the vehicle; Pwr is the wheel-rail interaction force, which is the function of the displacement and velocity of the vehicle and the rail. The lateral wheel-rail interaction force is derived from the Kalker creep theory, and the vertical wheel-rail interaction force is obtained according to the static balance condition of the wheelsets.(2)MvXv¨+CvXv˙+KvXv=Pwr.


### 2.4. Track Model

The rail is modeled by the finite element method, and the under-rail structure is simulated by mass-spring-dashpot models. The parameters of the ballasted track are provided by Table3 [19]. The vertical stiffness of ballast is 9.50 × 107 N/m, which is the measured data of Jinan Yellow River bridge. In order to investigate the most unfavorable conditions, the track arranged near the outside of the bridge deck.Table 3
The main parameters of the ballasted track.
ParametersUnitValueElastic modulus of railN/m22.059× 1011Inertia moment of rail cross sectionm43.217× 10−5Mass of railkg/m60.64Spring stiffness of fastenerN/m1× 108Damping coefficient of fastenerN/m7.5× 104Spring stiffness of ballastN/m9.50× 107Damping coefficient of ballastN/m5.88× 104The motion equation of the track is shown in equation (3), where [Mt], [Ct], and [Kt] are the mass, damping, and stiffness matrices of the track, respectively; Xt represents the displacement vector of the track; the external load acting on the track includes two parts: the wheel-rail force and the interaction force between the track and the bridge. The force of the bridge acting on the track is calculated by forced vibration [1].(3)MtXt¨+CtXt˙+KtXt=Pwr+Pbt.


### 2.5. Track Irregularity

The track irregularity samples used in this study are generated by power spectral density (PSD) of rail irregularity of high-speed railway lines in German [23], the length of the sample is 1000 m and the range of the wavelength is 1m～80 m.The track irregularity is expressed as equation (4), where Sv, Sa, and Sc denote the PSD of height irregularity, alignment irregularity, and cross-level irregularity, respectively; Av and Aa is the constant of roughness; ωr, ωc, ωs are the truncation frequency. The parameters are shown in Table 4.(4)Svω=Avωc2ω2+ωr2ω2+ωs2,(5)Saω=Aaωc2ω2+ωr2ω2+ωs2,(6)Scω=Avωc2ω2ω2+ωr2ω2+ωc2ω2+ωs2.Table 4
The parameters of track irregularity PSD.
Parameterωc (rad/m)ωr （Rad/m)ωs (rad/m)Aa (m2⋅ rad/m)Av (m2⋅ rad/m)Ag (m2⋅ rad/m)Value0.82460.02060.43806.125× 10−710.08× 10−71.032× 10−7In this study, one specific rail irregularity sample is applied in all cases. When calculating the wheel-rail force in each step, wheel-rail relative displacement is superimposed on the track irregularity for calculation.


### 2.6. Vehicle-Track-Bridge Coupled Model

Vehicle, track, and bridge form a coupled system through the wheel-rail force and the interaction between the track and the bridge, the coupled model is shown in Figure7. Equation (1)–(3) constitute the overall motion equation of the whole system. The equation is solved by the iterative method, and the specific solving process is shown in Figure 8.Figure 7
The model of the vehicle-track-bridge coupled system.Figure 8
Solving process of the motion equation of the vehicle-track-bridge coupled system.


## 2.1. Bridge Model

The studied section includes two bridge structures: (A) 32 m precast post-tensioned prestressed concrete simply supported box girder; (B) a 60m + 100 m + 60 m prestressed concrete double track continuous girder. MIDAS is used to establish the finite element model (FEM) of the bridge, as shown in Figure 2, while the arrowhead indicates the forward direction of the vehicle. The boxes mark the positions of BEJs.Figure 2
The MIDAS model of the bridge.The bridges are simulated by beam elements. The bottom of the pier is fixed, and the constraints between the pier and the bridge deck are simulated by master-slave nodes. And the secondary dead load is applied to the girder body by increasing the bulk density of materials. Because the concrete bridge will arch due to the creep, which will influence the running stability of the vehicle [10, 18], the creep is taken as the initial deformation of the bridge deck in the model. The uniform stiffness matrix and the uniform mass matrix are adopted in dynamic analysis with Rayleigh damping.The motion equation of the bridge is shown in equation (1), where [Mb], [Cb], and [Kb] is the mass, damping, and stiffness matrix of the bridge, respectively; Xb represents the displacement vector of the bridge; Pbt is the interaction force between the track and the bridge, which is the function of the displacement and velocity of the track and the bridge.(1)MbXb¨+CbXb˙+KbXb=Pbt.


## 2.2. BEJs Model

BEJs with sliding sleeper is applied in the railway bridge studied. The structure of this type of BEJs is simpler than other styles, which is less affected by the displacement of the girder end and can be applied to the condition with a wider range of vehicle speed. The structure of BEJs is shown in Figure3. It is mainly composed of fixed steel sleeper 2, movable steel sleeper 3, supporting frame 7, supporting beam 4, cushion plate 5 (between fixed sleepers and supporting frames), connecting rods 6, and pressure-bearing 8 (between supporting beams and sleepers). The rail and concrete sleeper are denoted in the figure 1 and 9, respectively.Figure 3
The structure of BEJs: (a) top view. (b) A-A section. (c) B-B section.
(a)(b)(c)The supporting frames are installed on the girder body and have no relative displacement with the bridge. The fixed sleepers and supporting frames are connected by bolts, and there are no longitudinal and lateral relative displacements between them. The local vertical stiffness is provided by the cushion plates between them. One end of the supporting beam is connected with fixed sleepers, and fixed sleepers at the other end and the movable sleeper can slide relative to the supporting beam in the longitudinal direction. There is no lateral relative displacement between supporting beams and sleepers, and the local vertical stiffness is provided by the pressure-bearings between them. The movable steel sleeper, which suspended on the supporting beam, is connected to fixed sleepers through connecting rods. When longitudinal displacement occurs at the girder end, fixed sleepers at the one end and the movable sleeper slide relative to the supporting beam, and sleeper spacing between fixed sleepers and movable sleepers are adjusted by connecting rods. The maximum longitudinal displacement of girder ends on both sides of BEJs is ±200 mm. The distance between the fixed sleeper and the movable sleeper is 450 mm when the longitudinal displacement of the girder end is 0 mm, and which is 350 mm or 550 mm when the longitudinal displacement of the girder end is +200 mm or -200 mm, respectively.The vertical stiffness of BEJs should not differ too much from that of girders on both sides. In order to make the stiffness under the sleeper of BEJs consistent with the stiffness under the sleeper of the ballasted track, the cushion plate is placed under the fixed steel sleeper. By adjusting the stiffness of the cushion plate, the abrupt change of track stiffness can be avoided, which is unfavorable to the running safety and stability. The pressure-bearing is placed between the sleepers and supporting beams to adjust the stiffness of BEJs too. The range of vertical stiffness of the cushion plate is 80 ∼ 120 kN/mm, and the range of vertical stiffness of the pressure-bearing is 160 ∼ 240 kN/mm.BEJs are modeled as the substructures of the bridge model through MIDAS, as shown in Figure4. Since the supporting frame are fixed with the bridge deck, it can be ignored in the model. The supporting beam and sleepers are simulated by beam elements, which are denoted in the figure. The nonlinear characteristics of track stiffness have little influence on the results and cost plenty of computational resources [19], hence, linear massless spring elements are used to simulate cushion plates and pressure-bearings. The stiffness of the spring in the BEJs model are changed to simulate different stiffness of cushion plates and pressure-bearings. And the MIDAS model of BEJs with different sleeper spacing is established to consider the influence of longitudinal displacement of the girder end on the system. Table 1 lists the details of different cases.Figure 4
The MIDAS model of BEJs.Table 1
Parameters of different cases.
ParameterCase1234Vertical stiffness of cushion plates (kN/mm)8012080120Vertical stiffness of pressure-bearings (kN/mm)160160240240Sleeper spacing (mm)350, 450, 550Figure5 shows the top view of BEJ models with different sleeper spacing, which is 350 mm, 450 mm, and 550 mm, respectively.Figure 5
The top view of the BEJs model: (a) sleeper spacing is 350 mm, (b) 450 mm, and (c) 550 mm.
(a)(b)(c)


## 2.3. Vehicle Model

The vehicle model consists of 16 vehicles with 8 electric moto units (EMUs), which are composed of car-bodies, bogies, wheelsets, and the spring and damping connections. In order to simplify the calculation, the mechanics model of the vehicle adopts the following assumptions [4, 20–22]: (1) the car-bodies, bogies, and wheelsets are regarded as rigid components, the connections are represented by linear spring and linear damping. (2) The vehicles move at constant speed along the track, the influence of the longitudinal force of the system is ignored. (3) The wheelsets are always in contact with the rail.As shown in Figure6, the vehicle is modeled by multibody mechanics with 23 degrees of freedom (DOF). The car-body has 5 DOF to be concerned: lateral displacement Yc, vertical displacement Zc, roll angle θc, yaw angle Ψc, and pitch angle φc. The bogie has 5 DOF too, which is denoted by subscript t. And the lateral displacement and roll angle of wheelsets are considered, which is denoted by subscript w. Table 2 provides the main parameters of the vehicle model.Figure 6
Vehicle model: (a) right view. (b) Rear view. (c) Bottom view.
(a)(b)(c)Table 2
The main parameters of the vehicle model.
ParametersUnitMotor car/trailer carMass of car-bodykg3.96× 104/3.44 × 104Mass of bogiekg3.20× 103/2.60 × 103Mass of wheelsetkg2.40× 103/2.40 × 103Roll mass moment of the car-bodykg·m20.128× 106/0.111 × 106Pitch mass moment of the car-bodykg·m21.94× 106/1.686 × 106Yaw mass moment of the car-bodykg·m21.673× 106/1.453 × 106Roll mass moment of the bogiekg·m22.59× 103/2.106 × 103Pitch mass moment of the bogiekg·m21.752× 103/1.423 × 103Yaw mass moment of the bogiekg·m23.20× 103/2.60 × 103Roll mass moment of the wheelsetkg·m2720/756Lateral stiffness of the primary suspension systemkN/mm1× 103/1 × 103Vertical stiffness of the primary suspension systemkN/mm1× 103/1 × 103Lateral stiffness of the secondary suspension systemkN/mm0.2× 103/0.2 × 103Vertical stiffness of the secondary suspension systemkN/mm0.2× 103/0.2 × 103Lateral damping of the primary suspension systemkN·s/m50/50Vertical damping of the primary suspension systemkN·s/m20/20Lateral damping of the secondary suspension systemkN·s/m50/50Vertical damping of the secondary suspension systemkN·s/m10/10Distance of two bogiesm17.5/17.5Distance of two wheelsetsm2.5/2.5The motion equation of the vehicle is shown in equation (2), where [Mv], [Cv], and [Kv] are the mass, damping, and stiffness matrix of the vehicle, respectively; Xv represents the displacement vector of the vehicle; Pwr is the wheel-rail interaction force, which is the function of the displacement and velocity of the vehicle and the rail. The lateral wheel-rail interaction force is derived from the Kalker creep theory, and the vertical wheel-rail interaction force is obtained according to the static balance condition of the wheelsets.(2)MvXv¨+CvXv˙+KvXv=Pwr.


## 2.4. Track Model

The rail is modeled by the finite element method, and the under-rail structure is simulated by mass-spring-dashpot models. The parameters of the ballasted track are provided by Table3 [19]. The vertical stiffness of ballast is 9.50 × 107 N/m, which is the measured data of Jinan Yellow River bridge. In order to investigate the most unfavorable conditions, the track arranged near the outside of the bridge deck.Table 3
The main parameters of the ballasted track.
ParametersUnitValueElastic modulus of railN/m22.059× 1011Inertia moment of rail cross sectionm43.217× 10−5Mass of railkg/m60.64Spring stiffness of fastenerN/m1× 108Damping coefficient of fastenerN/m7.5× 104Spring stiffness of ballastN/m9.50× 107Damping coefficient of ballastN/m5.88× 104The motion equation of the track is shown in equation (3), where [Mt], [Ct], and [Kt] are the mass, damping, and stiffness matrices of the track, respectively; Xt represents the displacement vector of the track; the external load acting on the track includes two parts: the wheel-rail force and the interaction force between the track and the bridge. The force of the bridge acting on the track is calculated by forced vibration [1].(3)MtXt¨+CtXt˙+KtXt=Pwr+Pbt.


## 2.5. Track Irregularity

The track irregularity samples used in this study are generated by power spectral density (PSD) of rail irregularity of high-speed railway lines in German [23], the length of the sample is 1000 m and the range of the wavelength is 1m～80 m.The track irregularity is expressed as equation (4), where Sv, Sa, and Sc denote the PSD of height irregularity, alignment irregularity, and cross-level irregularity, respectively; Av and Aa is the constant of roughness; ωr, ωc, ωs are the truncation frequency. The parameters are shown in Table 4.(4)Svω=Avωc2ω2+ωr2ω2+ωs2,(5)Saω=Aaωc2ω2+ωr2ω2+ωs2,(6)Scω=Avωc2ω2ω2+ωr2ω2+ωc2ω2+ωs2.Table 4
The parameters of track irregularity PSD.
Parameterωc (rad/m)ωr （Rad/m)ωs (rad/m)Aa (m2⋅ rad/m)Av (m2⋅ rad/m)Ag (m2⋅ rad/m)Value0.82460.02060.43806.125× 10−710.08× 10−71.032× 10−7In this study, one specific rail irregularity sample is applied in all cases. When calculating the wheel-rail force in each step, wheel-rail relative displacement is superimposed on the track irregularity for calculation.


## 2.6. Vehicle-Track-Bridge Coupled Model

Vehicle, track, and bridge form a coupled system through the wheel-rail force and the interaction between the track and the bridge, the coupled model is shown in Figure7. Equation (1)–(3) constitute the overall motion equation of the whole system. The equation is solved by the iterative method, and the specific solving process is shown in Figure 8.Figure 7
The model of the vehicle-track-bridge coupled system.Figure 8
Solving process of the motion equation of the vehicle-track-bridge coupled system.


## 3. Numerical Analysis

### 3.1. Dynamic Response of BEJs

Since the size and the mass of BEJs are much smaller than the bridge, the change of the BEJs' parameters has no effect on the dynamic response of the bridge. However, in order to show the dynamic response of BEJs more clearly, the results of the bridge are shown as a reference.Figure9 shows the maximum value of vertical displacement and acceleration of bridges and BEJs at different positions under Case 1 when sleeper spacing is 450 mm and the vehicle speed is 250 km/h. Since the limits in the code are given based on the maximum values, the values discussed in the paper are the maximum values. The horizontal ordinate represents the sequence that passed by the vehicle. It is shown that the dynamic response of bridges at different positions is basically the same. The dynamic response of BEJs at different positions is different, but it shows randomness, which may be related to the track irregularity and the state of vehicles passing by.Figure 9
The maximum value of vertical dynamic response of BEJs and bridges at different positions in Case 1.As the main part of BEJs, the supporting beam can be regarded as a continuous girder with fixed sleepers as support. Figure10 shows the vertical dynamic response of the supporting beam of the third BEJs in time history in Case 1 when sleeper spacing is 450 mm and the vehicle speed is 250 km/h, and the dynamic response of midspan of the adjacent girder is shown as well. It can be seen that although the span of the supporting beam is much smaller than that of the bridge, its vertical displacement is close to that of the bridge. Therefore, BEJs is more likely to cause the deformation of under-track foundation with a small wavelength, and has a greater impact on running safety and stability. The vertical acceleration of the supporting beam is significantly greater than that of the bridge.Figure 10
Time histories of the vertical dynamic response of BEJs and bridges adjacent to BEJs in Case 1.Figure11 shows the vertical dynamic response of the movable sleeper, fixed sleeper, and supporting beam in the same BEJs in time history in Case 1 when sleeper spacing is 450 mm and the vehicle speed is 250 km/h. Since movable sleepers are only supported by the supporting beam, the vertical dynamic displacement of the movable sleeper is significantly greater than that of the fixed sleeper and the supporting beam. Similarly, the vertical accelerations of a movable sleeper are also greater than that of the fixed sleeper and the supporting beam.Figure 11
Time histories of the vertical dynamic response of BEJs in Case 1.Figure12 shows the maximum values of the dynamic response of BEJs with different vehicle speeds in Case 1 when sleeper spacing is 450 mm. It can be seen that the dynamic response of BEJs increases with the increase of vehicle speed.Figure 12
The maximum value of dynamic response of BEJs with different vehicle speed in Case 1.Figure13 shows the vertical dynamic response of BEJs in time history in different cases when sleeper spacing is 550 mm and the vehicle speed is 250 km/h. It can be seen that the changing tendence of the dynamic response of the movable sleeper, fixed sleeper, and supporting beam are different when the vertical stiffness of BEJs changes. Figure 14 shows the maximum values of the dynamic response of BEJs under different cases when the vehicle speed is 250 km/h. The vertical displacement of the fixed sleeper is mainly affected by the vertical stiffness of the cushion plate. When the vertical stiffness of the cushion plate increases, the vertical dynamic displacement of the fixed sleeper decreases significantly. The vertical acceleration of fixed sleepers also shows the same trend. The stiffness change of pressure-bearing has no effect on the fixed sleeper. The vertical response of the movable sleeper is mainly affected by the vertical stiffness of pressure-bearing. When the vertical stiffness of pressure-bearing increases, the vertical dynamic displacement and vertical acceleration of the movable sleeper decrease significantly. With the increase of the vertical stiffness of the cushion plate, the vertical displacement of the movable sleeper decreases slightly. The vertical displacement of the supporting beam is mainly affected by the vertical stiffness of pressure-bearing. When the vertical stiffness of pressure-bearing increases, the vertical displacement of the supporting beam decreases significantly. The vertical stiffness of the cushion plate has no effect on the vertical displacement of the supporting beam, but will affect the vertical acceleration of the supporting beam. When the stiffness of the cushion plate increases, the vertical acceleration of the supporting beam decreases significantly.Figure 13
Time histories of the dynamic response of BEJs in different cases.Figure 14
The maximum value of the dynamic response of BEJs in different cases.The dynamic response of BEJs is also affected by sleeper spacing. Although the changing trend of each part is different, in general, when sleeper spacing is the largest, the dynamic response of BEJs is also the largest.According to the regulation TB 10761–2013 (technical regulations for dynamic acceptance for high-speed railways construction) [24], the vertical displacement of the sleeper should not exceed 2 mm, and the vibration acceleration of the sleeper should not exceed 500 m/s2. When the stiffness of pressure-bearing is 160 kN/mm and sleeper spacing is 550 mm, the vertical displacement of the movable sleeper will exceed the limit, which may affect the running safety. Based on the results of BEJs, the larger vertical stiffness of pressure-bearing is recommended.


### 3.2. Dynamic Response of the Vehicle

According to the specification GB5599-2019 (specification for dynamic performance assessment and testing verification of the rolling stock) [25], the dynamic response of the vehicle is evaluated by safety indexes and stability indexes: derailment factor Q/P, wheel unloading rate ΔP/P¯, the lateral wheel-rail force Q, vertical vibration acceleration of the car-body αz, lateral vibration acceleration of the car-body αy, and the Sperling index of comfort W in the vertical direction and lateral direction, while P represents the vertical wheel-rail force, ΔP represents the offload vertical wheel-rail force, P¯ represents the average static vertical wheel-rail force, and W is the frequency weighted value of acceleration of the car-body. The allowable values of indexes are listed in Table 5.Table 5
Allowable values of evaluation indexes of the vehicle.
IndexQ/PΔP/P¯Q (kN)αzαyWzWyAllowance value≤0.8Qualified:≤0.6≤19+0.3P≤0.13 g≤0.1 gExcellent:≤2.5Dangerous:≤0.65Good: 2.5 ∼ 2.75Qualified: 2.75 ∼3 .0The changes of indexes with the different speed of the vehicle in load Case 1 are shown in Figure15, while sleeper spacing is 450 mm. It can be seen that all evaluation indexes of the vehicle become worse when the speed of vehicle increases.Figure 15
The maximum value of evaluation indexes of the vehicle with different speed in Case 1.Table6 lists the maximum values of evaluation indexes under different cases at the vehicle speed of 250 km/h. On the whole, the evaluation indexes of the vehicle under different load cases meet the specification. The evaluation indexes of the vehicle are almost the same in different load cases, except the wheel unloading rate.Table 6
Evaluation indexes of the vehicle in different load cases.
Load caseSleeper spacing (mm)Q/PΔP/P¯Q (kN)αz (m/s2)αy (m/s2)WzWy13500.5480.49934.140.9211.7692.5982.8714500.5480.55034.1930.9211.7642.6032.88215500.5480.58534.1470.9211.7692.6032.87324500.5480.47634.1930.9211.7642.5682.88334500.5480.54434.1930.9211.7642.6032.88244500.5480.47634.1930.9211.7642.5682.883Figure16 shows the vertical and lateral acceleration of the last car-body in time history in Case 1 when sleeper spacing is 550 mm and the vehicle speed is 250 km/h, and the red boxes indicate the positions of the four BEJs. Due to the cushioning of the vehicle, BEJs has no obvious effect on the acceleration of the car-body, but will affect the interaction between the wheelset and the rail. Figure 17 shows the wheel unloading rate of the last wheelset in time history. It can be seen that the wheel unloading rate on BEJs will be greater than that on the bridge. The wheel unloading rate of the vehicle only changes with the stiffness of BEJs when the vehicle passing through BEJs. As shown in Figure 18, the influence of the vertical stiffness of the cushion plate on the wheel unloading rate is more obvious than that of pressure-bearing. The increase of vertical stiffness of the cushion plate will significantly reduce the wheel unloading rate. It is worth noting that the wheel unloading rate is close to the limit value 0.6 when sleeper spacing is 550 mm in load Case 1 and 3. It indicates that the wheel unloading rate may exceed the limit if the vertical stiffness of the cushion plate is insufficient. Therefore, the larger vertical stiffness of the cushion plate is recommended.Figure 16
Time histories of acceleration of the last car-body in Case1.Figure 17
Time histories of the wheel unloading rate of the last wheelset in different cases: (a) the whole time period (b) the time period when the wheelset passes the first BEJs.
(a)(b)Figure 18
The maximum value of the wheel unloading rate in different cases.


## 3.1. Dynamic Response of BEJs

Since the size and the mass of BEJs are much smaller than the bridge, the change of the BEJs' parameters has no effect on the dynamic response of the bridge. However, in order to show the dynamic response of BEJs more clearly, the results of the bridge are shown as a reference.Figure9 shows the maximum value of vertical displacement and acceleration of bridges and BEJs at different positions under Case 1 when sleeper spacing is 450 mm and the vehicle speed is 250 km/h. Since the limits in the code are given based on the maximum values, the values discussed in the paper are the maximum values. The horizontal ordinate represents the sequence that passed by the vehicle. It is shown that the dynamic response of bridges at different positions is basically the same. The dynamic response of BEJs at different positions is different, but it shows randomness, which may be related to the track irregularity and the state of vehicles passing by.Figure 9
The maximum value of vertical dynamic response of BEJs and bridges at different positions in Case 1.As the main part of BEJs, the supporting beam can be regarded as a continuous girder with fixed sleepers as support. Figure10 shows the vertical dynamic response of the supporting beam of the third BEJs in time history in Case 1 when sleeper spacing is 450 mm and the vehicle speed is 250 km/h, and the dynamic response of midspan of the adjacent girder is shown as well. It can be seen that although the span of the supporting beam is much smaller than that of the bridge, its vertical displacement is close to that of the bridge. Therefore, BEJs is more likely to cause the deformation of under-track foundation with a small wavelength, and has a greater impact on running safety and stability. The vertical acceleration of the supporting beam is significantly greater than that of the bridge.Figure 10
Time histories of the vertical dynamic response of BEJs and bridges adjacent to BEJs in Case 1.Figure11 shows the vertical dynamic response of the movable sleeper, fixed sleeper, and supporting beam in the same BEJs in time history in Case 1 when sleeper spacing is 450 mm and the vehicle speed is 250 km/h. Since movable sleepers are only supported by the supporting beam, the vertical dynamic displacement of the movable sleeper is significantly greater than that of the fixed sleeper and the supporting beam. Similarly, the vertical accelerations of a movable sleeper are also greater than that of the fixed sleeper and the supporting beam.Figure 11
Time histories of the vertical dynamic response of BEJs in Case 1.Figure12 shows the maximum values of the dynamic response of BEJs with different vehicle speeds in Case 1 when sleeper spacing is 450 mm. It can be seen that the dynamic response of BEJs increases with the increase of vehicle speed.Figure 12
The maximum value of dynamic response of BEJs with different vehicle speed in Case 1.Figure13 shows the vertical dynamic response of BEJs in time history in different cases when sleeper spacing is 550 mm and the vehicle speed is 250 km/h. It can be seen that the changing tendence of the dynamic response of the movable sleeper, fixed sleeper, and supporting beam are different when the vertical stiffness of BEJs changes. Figure 14 shows the maximum values of the dynamic response of BEJs under different cases when the vehicle speed is 250 km/h. The vertical displacement of the fixed sleeper is mainly affected by the vertical stiffness of the cushion plate. When the vertical stiffness of the cushion plate increases, the vertical dynamic displacement of the fixed sleeper decreases significantly. The vertical acceleration of fixed sleepers also shows the same trend. The stiffness change of pressure-bearing has no effect on the fixed sleeper. The vertical response of the movable sleeper is mainly affected by the vertical stiffness of pressure-bearing. When the vertical stiffness of pressure-bearing increases, the vertical dynamic displacement and vertical acceleration of the movable sleeper decrease significantly. With the increase of the vertical stiffness of the cushion plate, the vertical displacement of the movable sleeper decreases slightly. The vertical displacement of the supporting beam is mainly affected by the vertical stiffness of pressure-bearing. When the vertical stiffness of pressure-bearing increases, the vertical displacement of the supporting beam decreases significantly. The vertical stiffness of the cushion plate has no effect on the vertical displacement of the supporting beam, but will affect the vertical acceleration of the supporting beam. When the stiffness of the cushion plate increases, the vertical acceleration of the supporting beam decreases significantly.Figure 13
Time histories of the dynamic response of BEJs in different cases.Figure 14
The maximum value of the dynamic response of BEJs in different cases.The dynamic response of BEJs is also affected by sleeper spacing. Although the changing trend of each part is different, in general, when sleeper spacing is the largest, the dynamic response of BEJs is also the largest.According to the regulation TB 10761–2013 (technical regulations for dynamic acceptance for high-speed railways construction) [24], the vertical displacement of the sleeper should not exceed 2 mm, and the vibration acceleration of the sleeper should not exceed 500 m/s2. When the stiffness of pressure-bearing is 160 kN/mm and sleeper spacing is 550 mm, the vertical displacement of the movable sleeper will exceed the limit, which may affect the running safety. Based on the results of BEJs, the larger vertical stiffness of pressure-bearing is recommended.


## 3.2. Dynamic Response of the Vehicle

According to the specification GB5599-2019 (specification for dynamic performance assessment and testing verification of the rolling stock) [25], the dynamic response of the vehicle is evaluated by safety indexes and stability indexes: derailment factor Q/P, wheel unloading rate ΔP/P¯, the lateral wheel-rail force Q, vertical vibration acceleration of the car-body αz, lateral vibration acceleration of the car-body αy, and the Sperling index of comfort W in the vertical direction and lateral direction, while P represents the vertical wheel-rail force, ΔP represents the offload vertical wheel-rail force, P¯ represents the average static vertical wheel-rail force, and W is the frequency weighted value of acceleration of the car-body. The allowable values of indexes are listed in Table 5.Table 5
Allowable values of evaluation indexes of the vehicle.
IndexQ/PΔP/P¯Q (kN)αzαyWzWyAllowance value≤0.8Qualified:≤0.6≤19+0.3P≤0.13 g≤0.1 gExcellent:≤2.5Dangerous:≤0.65Good: 2.5 ∼ 2.75Qualified: 2.75 ∼3 .0The changes of indexes with the different speed of the vehicle in load Case 1 are shown in Figure15, while sleeper spacing is 450 mm. It can be seen that all evaluation indexes of the vehicle become worse when the speed of vehicle increases.Figure 15
The maximum value of evaluation indexes of the vehicle with different speed in Case 1.Table6 lists the maximum values of evaluation indexes under different cases at the vehicle speed of 250 km/h. On the whole, the evaluation indexes of the vehicle under different load cases meet the specification. The evaluation indexes of the vehicle are almost the same in different load cases, except the wheel unloading rate.Table 6
Evaluation indexes of the vehicle in different load cases.
Load caseSleeper spacing (mm)Q/PΔP/P¯Q (kN)αz (m/s2)αy (m/s2)WzWy13500.5480.49934.140.9211.7692.5982.8714500.5480.55034.1930.9211.7642.6032.88215500.5480.58534.1470.9211.7692.6032.87324500.5480.47634.1930.9211.7642.5682.88334500.5480.54434.1930.9211.7642.6032.88244500.5480.47634.1930.9211.7642.5682.883Figure16 shows the vertical and lateral acceleration of the last car-body in time history in Case 1 when sleeper spacing is 550 mm and the vehicle speed is 250 km/h, and the red boxes indicate the positions of the four BEJs. Due to the cushioning of the vehicle, BEJs has no obvious effect on the acceleration of the car-body, but will affect the interaction between the wheelset and the rail. Figure 17 shows the wheel unloading rate of the last wheelset in time history. It can be seen that the wheel unloading rate on BEJs will be greater than that on the bridge. The wheel unloading rate of the vehicle only changes with the stiffness of BEJs when the vehicle passing through BEJs. As shown in Figure 18, the influence of the vertical stiffness of the cushion plate on the wheel unloading rate is more obvious than that of pressure-bearing. The increase of vertical stiffness of the cushion plate will significantly reduce the wheel unloading rate. It is worth noting that the wheel unloading rate is close to the limit value 0.6 when sleeper spacing is 550 mm in load Case 1 and 3. It indicates that the wheel unloading rate may exceed the limit if the vertical stiffness of the cushion plate is insufficient. Therefore, the larger vertical stiffness of the cushion plate is recommended.Figure 16
Time histories of acceleration of the last car-body in Case1.Figure 17
Time histories of the wheel unloading rate of the last wheelset in different cases: (a) the whole time period (b) the time period when the wheelset passes the first BEJs.
(a)(b)Figure 18
The maximum value of the wheel unloading rate in different cases.


## 4. Conclusion

This study investigated the influence of the parameters of BEJs on the dynamic response of the vehicle-track-bridge coupled system. A three-dimensional vehicle-track-bridge model is established, in which the vehicle is modeled by multibody dynamics, and the track and bridges are modeled by the finite element method. The dynamic response of the system when the vehicle passes through the railway line with densely arranged BEJs is calculated by the iterative method, and the effect of vehicle speed and sleeper spacing are taken into consideration.The main conclusions of this study are:(1)
The vertical stiffness of BEJs will affect the interaction between the wheel and the rail. The wheel unloading rate may exceed the limit of specification when the vertical stiffness of the cushion plate is insufficient.(2)
The movable sleeper is the most noteworthy part of BEJs because the movable sleeper is suspended. When the vertical stiffness of pressure-bearing is insufficient, the vertical displacement of the movable sleeper will exceed the limit of regulation, which will hurt the running safety.(3)
The dynamic response of BEJs and the vehicle is affected by the vehicle speed and sleeper spacing. With the increase of vehicle speed and sleeper spacing, the dynamic response of BEJs and the vehicle increases.The influence of the vertical dynamic stiffness of BEJs on the dynamic response of the vehicle-track-bridge coupled model is clarified in this study. And this study provided a theoretical basis for adopting the stiffness of BEJs by analyzing the dynamic response of the vehicle and BEJs. Based on the research results, the larger vertical stiffness of the cushion plate and pressure-bearing is recommended. The results have been applied to the design of BEJs on the studied railway line, which are in good service condition at present.Due to the complexity and the diversity of BEJs in different bridges, the influence of BEJs on the running safety and stability will be further studied in our future work.



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*Source: 2814807-2022-07-11.xml*