Abstract
The excessive pumping of fines in saturated roadbed surface layer, which is induced by the fluid-solid interaction under dynamic loads from high-speed train, is a special form of high-speed railway subgrade defect reported recently. This can deteriorate the interface between nonballasted track structure bottom layer and roadbed surface layer and therefore lead to associated contact variation with the moving of trains. According to the dynamic Biot’s equations known as u-p formulation and the vehicle-track coupling dynamics theory, a vertical vehicle-slab track-subgrade coupling vibration model is developed to investigate the aforementioned contact variation-induced dynamic behavior of the whole system considering the fluid-solid interaction. Dynamic measurements from a field case study are adopted to verify the computation model proposed. Based on the numerical model validated, the effects of three contact variation statuses (continuous contact, vibrating contact, and contact loss) on dynamic responses of track subsystem and subgrade subsystem, such as dynamic pore-water pressure, vertical accelerations, and dynamic displacements both in time and frequency domains, are investigated. Also, a sensitivity analysis involving rail speeds and lengths of contact loss zone is performed, and the critical length of contact loss zone is suggested.
1. Introduction
Prefabricated slab track, which possesses the advantages of higher permanent stability and lower maintenance [1], has hitherto been considerably applied in nonballasted track systems of high-speed railway. To achieve better operating comfort and safety, high smoothness and stability of nonballasted track are strictly required. Also, the importance is increasingly attached to keep the subgrade structure in an intact condition.
Main numerical formulations modeling of the dynamic response of railway structure under moving train loads involves the finite element method and the semianalytical method [2–9]. For example, rail pad properties [10] on dynamic performance of railway track were analyzed numerically. The wheel-rail dynamic interaction was investigated based on the theory of vehicle-track coupled dynamics with a modified excitation model of rail spalling failure proposed [11]. In order to optimize slab railway tracks, a comprehensive analysis of the shakedown limit load of nonballasted railway tracks was carried out using an efficient numerical model [12].
In addition to numerical research studies, several field case studies and experimental tests have been reported. For instance, a site investigation focusing on the dynamic behavior of railway tracks at a culvert transition zone is presented [13], and the track dynamic responses of operational trains were monitored [14, 15]. Based on field measurements, the excitation of railway-induced ground vibration [16] and the vibration characteristics of underground structures [17] were comprehensively evaluated. Furthermore, laboratory tests were also made to investigate the dynamic responses of track system subjected to transient loads [18] or the vibration behavior of the slab track-subgrade with mud pumping under cyclic dynamic loading of high-speed train [19].
Local defects (e.g., rail joints, switches, and turnouts) have been considered as one of the main sources of railway-induced ground vibration [20, 21]. Due to the combined effects of dynamic train loading, temperature, and other environmental factors such as water, diseases like cracks and voids can be induced inside track structures. It has been found that the vibration responses of the track system can be adversely affected by the contact damage disease underneath track slab induced from the deteriorated cement-emulsified asphalt mortar (CA mortar).
Liu et al. [22] developed a vehicle-track vertical coupling model to study the influence of contact loss on the dynamic properties of vehicle and track subsystems at different train speeds, and the results were experimentally validated. With the same model, Shi [23] analyzed the effect of slab track defects, such as failure of fasteners, subgrade uneven settlement, and contact loss induced by deteriorated CA mortar, on dynamic performances of the slab track system. In addition, Xiang et al. [24] investigated the effect of the suspension of track slab caused by the degradation of CA mortar, such as invalidation, cracking, embrittlement, and cataclasm, on vibration responses of slab track and found that the accelerations and displacements were greatly increased compared to the normal condition of CA mortar.
To overcome the issues studied above, Ren et al. [25] proposed criteria for repairing damages of CA mortar for prefabricated framework-type slab track. The criteria were then applied to a 3D FEM of slab track verified by a field test. Also, a very recent paper of Chen et al. [26] studied the effects of high temperature-induced initial upwarp at the interface between track slab and concrete base on the stability of the CRTS II slab track by FEM and concluded that an initial upwarp with a span of 6.5 m and a rise of 15 mm should be avoided to ensure the stability of the CRTS II slab track.
Practically, the interface damage disease does not play an exclusive part in the track system, which also exists in the track-subgrade system. During the operation of high-speed railways, the roadbed (grain structure), particularly at expansion joints of concrete bases with cracked waterproof materials, exposed to frequent intense rainfalls and groundwater-level variations can change significantly from the condition of optimum water contents of soils to the saturated states due to nonfunctional drainage conditions. When the roadbed surface layer gets saturated, the hydromechanical coupling effect under cyclic dynamic train loads may lead to excessive pumping of fines [27], thereby deteriorating the interface between concrete base and roadbed surface (Figure 1). Consequently, the concrete base is likely not in continuous contact with roadbed surface in some local areas where a certain content of fine particles is washed out, thus affecting the dynamic smoothness of rail and vibration performances of the track system. The literature review shows that existing investigations are mainly confined to the contact loss underneath track slab, and to what extent the contact variation underneath concrete base can affect the vibration responses of track structures is seldom addressed particularly considering the fluid-solid interaction effect.
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The aim of this paper is to explore the effect of contact variation between the concrete base and roadbed surface on vibration characteristics of vehicle-slab track-subgrade system considering the fluid-solid interaction effect. Based on the vehicle-track coupling dynamics theory and the theory of fluid dynamics in porous medium, a vertical vehicle-slab track-subgrade coupling vibration model capable of describing the contact variation between concrete base and roadbed surface under high-speed train load is developed with the FEM code COMSOL Multiphysics. Also, it is validated by measured data from a filed case study published by the authors’ research group previously. The contact variation-induced dynamic responses of vehicle-track system under different contact statuses (continuous contact, vibrating contact, and contact loss) are discussed, and the critical value of contact loss zone is suggested.
2. Vertical Vehicle-Track-Subgrade Coupling Model
The vehicle-track-subgrade vertical coupling system of high-speed railway studied herein consists of three parts: (i) the top structure (train subsystem); (ii) the medium structure (nonballasted track subsystem); and (iii) the bottom structure (subgrade subsystem). Based on the vehicle-track rail system coupling dynamics theory proposed by Zhai et al. [28], the structure of train subsystem can be simplified as a car body, two bogies, and four wheelsets; the nonballasted track subsystem refers to rail, fastener structure, track slab, cement-emulsified asphalt (CA) mortar, and concrete base; the subgrade subsystem involves the roadbed (surface layer and bottom layer) and subgrade body, as shown in Figure 2. The fundamental formulation and methodology of the numerical model developed are briefly described in the following sections.
2.1. Vibration Equations for Vehicle Subsystem
The train is modeled as an assembly of aforementioned rigid bodies with 10 degrees of freedom describing for 7 vertical displacements and 3 rotations [29], interconnected by springs and dampers [30]. The mass of the car, bogies, and wheels as well as the longitudinal rotation inertia of car and bogies are included. The governing equations for the train subsystem may be expressed as follows:wherewhere and are mass and rolling moment of inertia for the rigid body of the car, respectively; and are mass and rolling moment of inertia for bogies respectively; and is the mass of ith wheel.where and are the vertical displacement and pitch angle of the car body, respectively; and are the vertical displacement and pitch angle of the front and rear bogies, respectively; and is the vertical displacement of the ith wheel.
Detailed equations of the damping matrix and the stiffness matrix can be found in [30]. The basic parameters of the vehicle simulated are listed in Table 1.
2.2. Vibration Equations for Nonballasted Subsystem
In the track subsystem, the rail is assumed as a Euler–Bernoulli beam discretely supported on the concrete slab by springs and dampers [31]. The governing equation for the rail beam can be described bywhere is the flexural stiffness of the rail beam; and are deflection magnitudes of the rail and concrete slab, respectively; is the mass of rail per unit length of the track; and are dashpot damping and spring stiffness per unit length of the track, respectively, resulting from the rail pad and fastening; is the wheel-axle contact force; and is the Dirac delta function and is the moving velocity of the train.
The concrete slab and concrete base are treated as elastic continuum and their governing equation can be written aswhere is the stress tensor; is the mass density; and is the displacement vector. They are connected by springs with stiffness per unit length of the track from CA mortar and dashpots with damping coefficient per unit length of the track from CA mortar. The related parameters of the nonballasted track model simulated are listed in Table 2.
The calculation of the contact force between the wheel and the rail is described by the nonlinear Hertz contact theory. A train is assumed to move at a constant speed in the positive direction. The contact force between the wheel and rail may be expressed aswhere and are the Hertz contact force and spring constant, respectively, and , and are deflections of the rail at a contact point and wheel displacement in contact with the rail, respectively.
2.3. Dynamic Equations for Subgrade Subsystem
The subgrade subsystem is a multilayer structure (see Figure 2). Owing to the drainage system of nonballasted track and the specific mechanical properties of graded gravel adopted in the roadbed surface layer, the roadbed surface can be immersed by water and get saturated with the cracked waterproof materials in expansion joint of concrete base [32]. The roadbed surface layer is accordingly modeled as a two-phase saturated porous medium, whereas the others (bottom layer of roadbed and subgrade body) are considered as single-phase elastic soil.
Based on the dynamic Biot’s equation known as formulation proposed by Zienkiewicz and Shiomi [33], the dynamic responses of saturated porous media in terms of two variables (the displacement of the solid phase, , and pore-water pressure, ) neglecting the acceleration of pore water under dynamic loading can be described through the following governing equations:where represents the effective stress tensor of the solid phase; denotes the displacement vector of the solid phase; is the pore-water pressure; is Biot’s coefficient; is the Kronecker delta; is the strain component of the solid matrix; represents the permeability; is the unit of pore water and is the pore fluid bulk modulus; and are lame constants of the solid phase; is the mixture mass density; is the fluid density; is the solid density; and is the porosity. The related mechanical and hydraulic parameters of subgrade soils are tabulated in Table 3.
2.4. Other Information
The hydraulic conductivity of well-graded gravel recommended in the design code of high-speed railway is generally between 10−3 m/s and 10−5 m/s. Based on field observations, roadbed surface where the mud pumping occurs often has a relatively high content of graded gravel fines. Therefore, the 10−5 m/s order of magnitude is selected as the value of hydraulic conductivity of roadbed surface layer in the dynamic hydromechanical coupling calculation. Generally, the surface water and rainfall flow into roadbed surface layer through nonfunctional expansion joints due to aging and cracking, and therefore it is considered as drainage boundary.
3. Model Validation
A field case study conducted by the authors’ research group [34] on Shanghai-Nanjing intercity railway line was adopted to verify the proposed computational scheme. In this field test, the piezoelectric accelerometer and displacement gauges were installed at the top of the concrete base and roadbed surface near an expansion joint between concrete bases. In this test, data acquisition and processing were performed mainly based on the running of a high-speed train at a test speed of 247 km/h, which is operated for a passenger train, with a marshalling mode of 2 trailer cars and 6 motor cars.
The calculated acceleration-time curve of the concrete base at the concerned location shows noticeable 16 group peak amplitudes corresponding to the acceleration responses of the concrete base at the moment of 8 cars with 16 bogies passing, which were induced by the effects superimposed by 2 wheelsets of the single bogie. Thus, the load characteristics of the single axle pair of either front or rear bogie are not obvious (see Figure 3(a)). The calculated acceleration of the roadbed surface at the concerned location indicates that peak amplitudes with loading and unloading processes are not as evident as those on the concrete base, particularly at the moment of bogies except for the first and last bogie passing, which is likely due to the system damping effect and the geometry effect of the vibration wave transmitted to the top surface of the roadbed (see Figure 3(b)). The peak amplitudes were mainly caused by the superimposed effect of 4 wheelset axle load of the rear bogie and the front bogie between two adjacent cars. The calculated acceleration-time curves from both the concrete base and the roadbed surface are generally consistent with measured ones.
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4. Three Different Contact Statuses
The progressive mud pumping on the roadbed surface layer induced from the coupling action of water and cyclic loading deteriorates the interface between concrete base and roadbed surface, which may lead to voids in some areas particularly close to expansion joints. According to field investigations, the contact variation between the concrete base and the roadbed surface may be presented as three stages, i.e., the continuous contact, vibrating contact, and contact loss, respectively, corresponding to no contact damage, incipient contact damage, and full contact damage. Based on site observations and simulation results, it is found that the elliptical-shaped form can be selected as the most suitable form to describe the contact loss zone realistically, which is governed by the displacement function and spring-damping element variation. As is shown in Figure 4, H and L are, respectively, the depth and length of contact damaged area. Following discussion is all based on the numerical model just validated in the previous section.
Figure 5 shows the dynamic stress distribution of the roadbed surface under such three kinds of contact statuses at the moment of the rear bogie passing with a rail speed of 250 km/h. The length of the contact damaged zone of both vibrating contact and contact loss considered was 10 spans of fastener spacing (0.625 m), and the depth of the contact damaged area was 0.16 mm and 1 mm, respectively. The longitudinal distribution of the dynamic stress is remarkably different when the rear bogie travels through the damaged area. The dynamic stress under the state of continuous contact smoothly distributes along the longitudinal direction, while under the other contact statuses, the dynamic stress has great changes between seriously large stress concentration and zero in a certain range in the contact damaged area highlighted with green lines. Further, there are four contact positions (two corners and two contact points with vibration within the contact damaged zone) for the case in vibrating contact but only two contact positions (two corners of contact damaged zone) for the case with contact loss. The maximum dynamic stress of each contact status (continuous contact, vibrating contact, and contact loss) is 13 kPa, 21.7 kPa, and 65.6 kPa, respectively. It is clear that the dynamic stress at the corner of contact loss zone exceeds five times of that in the normal contact condition, which is the most detrimental to the train operation safety.
Figure 6 illustrates the dynamic pore-water pressure distribution in different contact statuses for the same case as in dynamic stress distribution. Compared with these three cases, the distribution of the dynamic pore-water pressure outside the contact damaged zone is basically the same while the three cases show significant difference in the contact damaged zone. For the case of continuous contact, the distribution of dynamic pore-water pressure is pretty smooth longitudinally as does the dynamic stress, except for a sudden decrease at the drainage boundary (the expansion joint). For the case of contact loss, when the rear bogie passes through the contact damaged zone, the dynamic pore-water pressure increases sharply at the two corners of the contact damaged zone and reduces to almost zero inside the contact loss zone, which is consistent with the finding of dynamic stress. For the case of vibrating contact, the dynamic pore-water pressure shows a steep increase at two vibrating contact positions and then decreases to almost zero at the expansion joint. This indicates that the transient impact and stress concentration occur at the corner of the contact damaged zone and vibrating contact positions, and the dynamic behavior in these three cases is worth being further discussed.
5. Vibration Characteristic for the State of Vibrating Contact
5.1. Time-Frequency Analysis of Vehicle-Track System
It can be observed from Figure 7(a) that both cases clearly show four interaction points corresponding to 4 wheelset axle loading at the moments of the front and rear bogies passing the expansion joint. Also, the dynamic rail displacement in the middle of contact damaged zone in the state of vibrating contact is slightly larger than that at the same location in the case of continuous contact. Vibrating contact positions between the concrete base and the roadbed surface, which can be considered as extremely short narrow strips, dynamically change with the vehicle movement, leading to the short interval deformation uncoordinated. Before reaching the critical status of contact loss, the dynamic shortwave irregularity excitation induced from time-varying contact stiffness between the concrete base and the roadbed surface has been largely dissipated by the spring-damping system made up of the fasteners and CA mortar, with little effect to the rail vibration, which may be explained for this very slight difference of rail displacements between two cases. As depicted in Figure 7(b), the dominant excitation frequency of both conditions mainly ranges below 40 Hz with a main excitation frequency being approximately 4 Hz, which is consistent to the excitation frequency of the center-center spacing (17.375 m) between the front and the rear bogie of a car. According to the comparison, the rail vibration caused by the transient impact in the state of vibrating contact marginally enlarges the vibration response at the fundamental excitation frequency.
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For the case of continuous contact, there are only 2 peak amplitudes of dynamic displacements corresponding to 2 bogies of a car, which clearly indicates the superposition of two wheelsets of the same bogie (see Figure 8(a)). This is consistent with the measurement in the field shown before. A similar finding applies to the case in vibration contact although this superposition is not as strong as that in continuous contact but much noticeable than that for the rail displacements in the same case (see Figure 7(a)). Also, due to the effects of dashpot damping from fasteners and CA mortar, the vibration effect on the concrete base is effectively attenuated, with the maximum dynamic displacement being smaller than that of rail directly sustaining the wheelset axle loading. As the front wheel approaches the contact damaged area, the depth of contact damaged zone is filled by the relative distance of the concrete base and the roadbed surface, resulting in a sudden change of the contact status from being loss to being contact. Under the cyclic action of four wheels in the calculation time, the stress concentration caused by the transient impact excites the excess acceleration in vibrating contact positions within the contact damaged area as shown in Figure 8(b).
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5.2. Time-Frequency Analysis of Roadbed Surface
The acceleration curves on the roadbed surface at the expansion joint (in the middle of contact damaged area) in the states of continuous contact and vibrating contact are plotted in Figure 9. For the case of vibrating contact, it can be clearly seen from Figure 9(a) that the number of the acceleration amplitudes is equivalent to that of the wheelsets as well as the vibrating contact positions. For the case of continuous contact, the acceleration almost varies evenly with time when the wheelsets pass through the same zone, with the amplitude being 0.22 m/s2, while for the case of vibrating contact, the acceleration increases sharply in an extremely short time when the transient impact occurs at the vibrating contact position on the roadbed surface, with an amplitude of acceleration being approximately 7 times as that in continuous contact, which demonstrates that the vibration response on the roadbed surface is significantly magnified by an abrupt change of interface interaction at vibrating contact locations. As shown in Figure 9(b), the dominant frequency of acceleration distributed in the case of continuous contact mainly ranges below 40 Hz with a main excitation frequency of 20 Hz. In comparison, the main excitation frequency of acceleration for the case of vibrating contact is 23 Hz but with a much wider dominant frequency range exceeding 150 Hz. It may be concluded that the high frequency induced from the aforementioned transient impact enlarges the vibration response of the roadbed surface layer in spite of being the similar main excitation frequency.
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6. Vibration Characteristic for the State of Contact Loss
6.1. Vibration Responses of Vehicle-Track System
Figure 10 shows the acceleration of the car body for the cases of full contact loss with a depth of contact loss zone (H = 1 mm) and different typical lengths of contact loss zone (L = 3 spans, 7 spans, and 10 spans) with a comparison case of continuous contact (H = 0 mm; L = 0 span) at a rail speed of 250 km/h, where the middle of the contact damaged area is selected as the observation point. The vehicle acceleration varies with the length of the contact loss zone to some extent when the wheelsets pass through the contact loss area (see Figure 10(a)). The main reason for this is that the uneven deformation generated in the contact loss zone due to its irregular shape triggers some irregularity excitation to the upper track structure. As the length of the contact loss zone increases, greater influences are exerted to the vehicle acceleration. Despite the influence of the contact loss length, the total difference in the acceleration between the contact loss with 10 spans length and the continuous contact is less than 0.05 m/s2, and the dominant frequencies of the acceleration are almost coincided as shown in Figure 10(b), which are likely owing to the attenuation effects of the damping from the CA mortar and the fastener as well as the primary and secondary suspension of the vehicle. Therefore, it may be concluded that the vehicle acceleration variations are not governed by the length of contact loss.
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As shown in Figure 11(a), when the length of contact loss zone is less than 7 spans, the number of peak amplitudes of the rail displacement is clearly consistent with the number of four wheelsets although the peak value increases gradually. As the length increases to 7 spans, the vibration effects caused by the front and rear wheelsets of the same bogie are gradually superimposed and the number of displacement amplitudes is almost merged to two, corresponding to the two bogies (front and rear bogies) with the length up to 10 spans. This indicates that the support stiffness under the rail has been greatly reduced. In Figure 11(b), the dominant frequencies of the acceleration in the states of contact loss are basically the same with the state of continuous contact focusing near 4 Hz, 8 Hz, 12 Hz, 16 Hz, 20 Hz, 23 Hz, 28 Hz, and 32 Hz, while the amplitudes are clearly enlarged with the increase of the contact loss length. Comparing these conditions, it can be concluded that the contact loss between the concrete base and roadbed surface enlarges the vibration response of rail and the rail deflection increases significantly with the length of contact loss zone.
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It can be seen in Figure 12(a) that the vibration effect caused by the front and rear wheelsets of the same bogie on the concrete base is clearly superimposed with only 2 amplitudes corresponding to 2 bogies of a car and is amplified with the increase of the contact loss length. As the length of the contact loss zone exceeds 3 spans, the increment of the vertical dynamic displacement of the concrete base is great in the middle position of the contact loss zone. When the length reaches 10 spans, the peak value of the vertical displacement is 1.52 mm, approximately 8 times of that in the normal condition. This shows that the support stiffness and constraint from the underlying roadbed is largely decreased. Meanwhile, the acceleration of the concrete base is increased simultaneously with the increase of contact loss length as depicted in Figure 12(b). When the length of contact loss zone is smaller than 7 spans, four amplitude values of the acceleration can be observed as four wheelsets pass through the contact loss area. When the length reaches 10 spans, the high vertical accelerations caused by the front and rear wheelsets of the same bogie are almost superimposed in the contact loss zone where only two peak amplitudes of the acceleration appear.
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6.2. Vibration Responses of Roadbed Surface
Figure 13 shows the comparison of the accelerations between in the middle and at the corner of contact loss zone on roadbed surface for the case of 10-span-long and 1 mm-deep contact loss area. For the middle of contact loss zone, the acceleration remains to be stably fluctuated because no stress interaction can be found due to the detachment of the concrete base and the roadbed surface. In contrast, a much higher vertical acceleration is mobilized at the corner of contact loss zone due to the stress concentration generated at this position as depicted in Figure 13(a). Similarly, in the middle of contact loss zone, the dominant frequency of vibration acceleration is approximately 7 Hz and 20 Hz while the dominant frequencies at the corner of contact loss zone are near 7 Hz, 48 Hz, and 150 Hz with a much wider range (see Figure 13(b)). The transient impact resulting from an abrupt change from being contact to being loss at the corner of the contact likely induces high frequencies with the vibration amplitudes increased largely.
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6.3. Index Analysis
Figure 14 presents the vibration responses of the rail and the concrete base with different lengths of the contact loss zones (3 spans, 7 spans, 10 spans, and 15 spans) and the rail speeds (150 km/h, 250 km/h, and 350 km/h) in the middle of contact loss zone. As can be seen from Figure 14(a), when the length of the contact loss zone is smaller than 10 spans, the vertical displacement peak amplitude of the rail has a relatively mild increase, while the length is greater than 10 spans, a sharp increase can be found, which even exceeded the limit (2 mm, required by Chinese technical regulations for dynamic acceptance for high-speed railways construction [35]) of the rail vertical displacement at the speed of 350 km/h. Also, the vertical displacement peak amplitude is not sensitive about the rail speed despite a general increase with the increase of rail speed. It can be seen in Figure 14(b) that the vertical acceleration of the rail tends to be stable after a slight increase within 10 spans of the contact loss length. With the same length, the rail speed has a significant effect on the rail acceleration. As shown in Figure 14(c), the vibration displacement peak amplitude of the concrete base shares the same trend as that of rail. Figure 14(d) shows that the vertical acceleration amplitude of the concrete base increases almost linearly with the length of contact loss at the rail speeds of 150 km/h and 250 km/h, while at the rail speed of 350 km/h, the vertical acceleration increases steeply as the length of contact loss exceeds 7 spans. From the analysis above, it may be concluded that the vibration responses of the track system are greatly enlarged with the length of contact loss zone more than 10 spans, and this critical value may be considered as a threshold for treatment of the contact damage disease.
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7. Conclusions
Based on the vehicle-track coupling dynamics theory and the theory of fluid dynamics in porous medium, a vertical vehicle-slab track-subgrade coupling vibration model capable of describing concrete base-roadbed surface contact variation under high-speed train load was developed. Dynamic data measured from a field case study were adopted for validating the computation model proposed. The influence of contact variation on the detailed dynamic responses of vehicle-track-subgrade system under different contact statuses is discussed. Following conclusions can be drawn based on the cases studied.
In the state of vibrating contact, the vibration responses of the concrete base and the roadbed surface were strongly enlarged by the transient impact and stress concentration induced from the concrete base-roadbed surface contact status changing from being detached to being interacted in comparison with the state of continuous contact. However, the vibrating contact-induced vibration responses of the vehicle and the rail were very limited due to the attenuation of damping from the CA mortar, the fasteners, and the primary and secondary suspension of a car body.
In the state of contact loss, the vibration responses of the key components (rail and concrete base) in the track system were remarkably increased with the increase of the contact loss length. For the roadbed surface, the stress concentration at the endpoint of the contact loss zone exerted greater influence on vibration responses than in the middle position. With the length of the contact loss zone more than 10 spans, the vertical displacement amplitude of the rail exceeded the standard value specified in the code of practice. In this case, it is inadvisable for the length of contact loss zone to exceed 10 spans for better structural durability and operation safety of the vehicle-track slab-subgrade system.
Notations
| : | Car body mass |
| : | Bogie mass |
| : | Axle mass |
| : | Mass of ith wheel |
| : | Mass of rail beam |
| : | Car rolling moment |
| : | Bogie rolling moment |
| : | Vertical displacement of car body |
| : | Vertical displacement of front bogie |
| : | Vertical displacement of rear bogie |
| : | Vertical displacement of ith wheel |
| : | Pitch angle of car body |
| : | Pitch angle of front bogie |
| : | Pitch angle of rear bogie |
| : | Secondary suspension stiffness of front bogie |
| : | Secondary suspension stiffness of rear bogie |
| : | Primary suspension stiffness |
| : | Stiffness of wheel-rail contact |
| : | Rail pad stiffness |
| : | CA mortar stiffness |
| : | Secondary suspension damping of front bogie |
| : | Secondary suspension damping of rear bogie |
| : | Primary suspension damping |
| : | Wheelbase |
| : | Distance between bogie centers |
| : | Axle load |
| : | Flexural stiffness of rail beam |
| : | Deflection of rail |
| : | Deflection of concrete base |
| : | Wheel displacement |
| : | Elastic modulus of track slab |
| : | Rail pad damping |
| : | Hertz contact force |
| : | Train’s moving velocity |
| : | Stress tensor |
| : | Effective stress tensor |
| : | Displacement vector |
| : | CA mortar damping |
| : | Elastic modulus |
| : | Elastic modulus of concrete base |
| : | Kronecker delta |
| : | Strain component |
| : | Unit of pore water |
| : | Pore fluid bulk modulus |
| : | Constants |
| : | Poisson’s ratio |
| : | Density |
| : | Solid density |
| : | Density of concrete base |
| : | Liquid density |
| : | Fluid density |
| : | Solid density |
| : | Pore-water pressure |
| : | Porosity |
| : | Compressibility parameter |
| : | Fluid viscosity |
| : | Permeability. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the financial support of this work sponsored by the National Natural Science Foundation of China (grant nos. 51578467 and 51608461), the Technology Research and Development Program of Chinese Railway Corporation (grant no. 2016G002–E), and China University of Mining & Technology (grant no. SKLGDUEK1726).