Mechanical performance of double-block ballastless track structures under different sleeper spacings

2.1. Vehicle–track coupled dynamic system

Based on the vehicle–track coupled dynamics theory (Zhai, 2020), a vehicle–track coupled system consisting of the car body, bogies, wheelsets, rails, fastening system, and double-block ballastless track structure was established in this study (Li et al., 2025, 2026). Specifically, the vehicle subsystem comprises one car body, two bogies, and four wheelsets. The wheel–rail contact model was solved using Hertzian nonlinear contact theory. The track subsystem consists of the rails, fastening system, and double-block ballastless track structure, as shown in Figure 1. In the vehicle subsystem, the car body, each bogie, and each wheelset each have five degrees of freedom, resulting in a total of 35 degrees of freedom for the system (Zhai, 2020). The detailed degree-of-freedom composition is listed in Table 1, and the vehicle parameters are given in Table 2.OPEN IN VIEWER

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Table 1.

Degrees of freedom for vehicle components.

Vehicle components	Degrees of freedom
	Longitudinal	Vertical	Lateral	Roll	Yaw	Pitch
Vehicle body	—	Z c	Y c	Φ c	ψ c	β c
Frame (j = 1, 2)	—	Z t j	Y t j	Φ t j	ψ t j	β t j
Wheelsets (j = 1, 2, 3, 4)	—	Z w j	Y w j	Φ w j	ψ w j	β w j

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Table 2.

EMU (electric multiple unit) parameters.

Item	Value	Unit
Vehicle body mass, frame mass, wheelset mass	40.5, 2.15, 1.65	t
Vehicle body pitch, yaw, and roll moments of inertia	116, 1850, 1925	t·m2
Frame pitch, yaw, and roll moments of inertia	1358, 3486, 2360	t·m2
Wheelset pitch, yaw, and roll moments of inertia	785, 785, 126	t·m2
Primary suspension lateral, vertical, and longitudinal stiffness	925.8, 1125, 925.8	kN/mm
Secondary suspension lateral, vertical, and longitudinal stiffness	193.1, 193.1, 178.4	kN/mm
Primary suspension lateral, vertical, and longitudinal damping	0, 45, 0	kN·s/m
Secondary suspension lateral, vertical, and longitudinal damping	30, 50, 500	kN·s/m
Half of the lateral spacing of the primary suspension	0.748	m
Half of the lateral spacing of the secondary suspension	0.978	m
Distance between bogie centers	19	m
Frame wheelbase	2.5	m
Height from the vehicle body center of gravity to the rail surface	1.75	m
Height from the wheelset center of gravity to the rail surface	0.46	m
Height from the frame center of gravity to the rail surface	0.5	m

2.2. Theory of wheel–rail contact

In this study, the trace-line method was adopted to geometrically describe wheel–rail contact (He et al., 2023). Based on the spatial relative position between the wheel flange-tread profile and the rail head profile, the contact trace line was generated along the wheel–rail profiles, and the instantaneous contact point and its local geometric characteristics were determined. The wheel–rail contact mode and the corresponding spatial displacement relationship are shown in Figure 2. The relative displacement components between the wheelset and the rail, as well as its geometric constraint equations, are expressed as follows:

{ x C = x B + l x R w tan δ R y C = y B − R w 1 − l x 2 ( l x 2 l y tan δ R + l z 1 − l x 2 ( 1 + tan 2 δ R ) ) z C = z B − R w 1 − l x 2 ( l x 2 l z tan δ R − l y 1 − l x 2 ( 1 + tan 2 δ R ) )

(1)

{ l x = − cos ϕ w sin ψ w l y = cos ϕ w cos ψ w l z = sin ϕ w

(2)

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Figure 2.

Spatial contact geometry between wheel and rail.

In the finite element model adopted in this study, wheel–rail contact is described by the normal contact force and tangential contact force. Specifically, the normal contact between the wheelset and rail was calculated using Hertzian nonlinear contact theory (Hertz, 1882), while the tangential contact was calculated using Coulomb friction theory (Zhu and Cai, 2014). The wheel–rail normal force was calculated based on Hertzian nonlinear contact theory, as expressed in the following equation:

P ( t ) = [ 1 G Δ Z ( t ) ] 3 / 2

(3)

2.3. Establishment of a double-block ballastless track structure model

This paper focuses on the double-block ballastless track structure. The track structure submodel primarily consists of rails, fastening systems, double-block sleepers, track slabs, foundation slabs, and base slabs. The specific component dimensions and material parameters are shown in Table 3.

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Table 3.

Model dimensions and material parameters.

Component	Density kg/m3	Young’s modulus GPa	Poisson’s ratio	Mesh seed m	Dimensions (Length/Width/Height)/m
Rail	7850	205	0.3	0.05	12/0.15/0.176
Steel pad	7850	205	0.3	0.05	0.17/0.34/0.02
Sleeper	2440	36	0.2	0.05	0.3/0.8/0.15
Track slab	2440	32.5	0.2	0.04	12/2.8/0.26
Support layer	2370	25.5	0.2	0.1	12/3.4/0.3
Base slab	2200	0.18	0.2	0.15	12/3.4/0.3
Longitudinal steel bars	7850	200	0.3	0.1	0.018 (Diameter)
Lateral steel bars	7850	200	0.3	0.1	0.018 (Diameter)

2.4. Numerical methods in dynamics

As described in Section 2.1, the established vehicle subsystem incorporates the motion relationships of the car body, bogies, wheelsets, and other components, with a total of 35 degrees of freedom. During high-speed operation, the vehicle subsystem interacts with the track subsystem through the wheel–rail contact interface. Based on the flexible treatment of the wheelsets and rails, the wheel–rail contact force evolves dynamically with changes in relative displacement, relative velocity, and contact geometric parameters, thereby coupling the two subsystems into a unified vehicle–track overall dynamic system. Owing to variations in wheel–rail contact conditions, including contact-point migration, normal and tangential contact stiffness, and frictional characteristics, together with the nonlinear response of the track structure, the coupled system exhibits pronounced strong nonlinear characteristics. On this basis, the dynamic equations of the vehicle–track overall system can be uniformly expressed as (Zhai, 2020):

M X ¨ + C X ˙ + KX = F ( X , X ˙ , t )

(4)

The system is solved using the Zhai method in vehicle–track coupled dynamics theory. This method is a fast explicit integration method, and its specific solution form is given as follows:

{ X n + 1 = X n + X ˙ n Δ t + ( 1 / 2 + ψ ) X ¨ n Δ t 2 − ψ X ¨ n − 1 Δ t 2 X . n + 1 = X ˙ n + ( 1 + φ ) X ¨ n Δ t − φ X ¨ n − 1 Δ t

(5)

3.1. Response of rail displacement to the reaction forces of fasteners

For the extraction of rail displacements in different directions, the nodes at the rail head on one side of the track centerline were selected and extracted sequentially along the longitudinal direction according to the mesh order. For the extraction of fastener reaction forces, the fastening systems on one side of the track centerline were selected, and the reaction forces of each fastening system were extracted sequentially along the longitudinal direction. The extraction point of the fastener reaction force at the first pair of sleepers to the left of the loading point was denoted as L1, and that at the second pair of sleepers was denoted as L2, with the remaining positions labeled in the same manner; the same definition was adopted on the right side (Yun et al., 2026). Figure 4 shows the longitudinal rail displacement and the longitudinal fastener reaction force results, while Figure 5 shows the vertical rail displacement and the vertical fastener reaction force results.OPEN IN VIEWER

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Figure 5.

Vertical displacement of the rail and vertical reaction force of the fastener.

As shown in Figure 4(a), under static wheel load, the longitudinal rail displacement exhibits a local fluctuating distribution along the track direction, with the maximum value occurring near the loading point and gradually decreasing toward both sides, and the overall distribution is basically symmetric. As the sleeper spacing increases from 400 mm to 900 mm, both the peak and trough values of the longitudinal rail displacement increase significantly, and the curve becomes steeper, indicating that the longitudinal deformation and relative sliding tendency of the rail in the loading region are enhanced after the discrete support becomes weaker. Specifically, the longitudinal rail displacement is 0.0739 mm under the 400 mm case and increases to 0.0930 mm under the 900 mm case, corresponding to an increase of 25.8%. The longitudinal fastener reaction force also shows a symmetric distribution, but its distribution range shrinks markedly with increasing sleeper spacing, and the reaction force gradually concentrates near the loading point. Under the 400, 500, and 600 mm cases, the peak reaction forces at Z1 adjacent to the loading point are 0.78, 0.95, and 1.02 kN, respectively, while the reaction force at Z2 is larger, exceeding that at Z1 by 0.67, 0.59, and 0.43 kN, respectively. As the sleeper spacing further increases to 700 mm and 800 mm, the difference between Z1 and Z2 gradually decreases. When the sleeper spacing reaches 900 mm, the longitudinal fastener reaction force is mainly concentrated in a few fasteners near Z2, Z1, and their symmetric positions, with similar peak values reaching about 2.29 kN, whereas the reaction forces at the fourth pair of sleepers and beyond decay rapidly and approach zero.

As shown in Figure 5(a), under static wheel load, the vertical rail displacement presents a symmetric distribution along the track longitudinal direction centered near the loading point, and is overall characterized by the coexistence of a main downward deflection region and local upward arching on both sides. As the sleeper spacing increases, both the downward deflection and the upward arching increase simultaneously, and the vertical rail displacement under the 900 mm sleeper-spacing case is 44.4% greater than that under the 400 mm case. As shown in Figure 5(b), the vertical fastener reaction force is symmetrically distributed along the support points, and because the loading point is located between Z1 and R1, the reaction forces at these two fasteners are equal and serve as the main load-bearing points. With increasing sleeper spacing, the vertical fastener reaction force at Z1 rises from 21.297 kN under the 400 mm case to 35.057 kN under the 900 mm case, representing an increase of 64.6%. In contrast, the reaction force at the adjacent support point Z2 decreases from 13.668 kN to 10.308 kN. The reaction forces at support points farther from the loading point decay more rapidly and gradually approach zero, indicating that the number of support points effectively sharing the load decreases and that the load transfer range shrinks significantly.

Figures 4 and 5 collectively show that increasing sleeper spacing weakens the multi-point load-sharing capacity of the track structure in both the longitudinal and vertical directions, causing load transfer and structural deformation to become more concentrated in a few support points near the loading point. The underlying reason is that a larger sleeper spacing reduces the support density and equivalent stiffness of the track structure, making it more difficult for rail deformation to be effectively distributed through adjacent support points. As a result, the distribution range of the fastener reaction force contracts and the local span effect is intensified. These results indicate that changes in sleeper spacing significantly affect the force distribution characteristics of the rail and fasteners near the loading region.

3.2. Response of rails and track slabs to shear forces and bending moments

When extracting the shear force and bending moment, 100 sections were defined for the rails on both sides by taking the loading point as the center and slicing longitudinally toward both directions. The extracted results were then averaged to reduce the influence of local geometric errors during loading. For the track slab, 100 sections were directly defined by taking the loading point as the center and slicing toward both sides. Figure 6 shows the shear forces of the rail and track slab under single-point loading by the wheelset, and Figure 7 shows the corresponding bending moments.OPEN IN VIEWER

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Figure 7.

Bending moment in the rail and track slab.

As shown in Figure 6(a), the rail shear force exhibits a symmetric distribution along the track longitudinal direction with the loading point as the center. A rapid sign change occurs near the loading point, and the shear force decays rapidly to nearly zero away from the loading point, indicating that the internal force equilibrium under vertical wheel load is mainly completed within a limited range near the loading region. As the sleeper spacing increases, the peak values of rail shear force near the loading point continue to rise, the peaks become sharper, and the shear transition zone becomes more concentrated. The peak rail shear force increases from 27.64 kN under the 400 mm case to 40.77 kN under the 900 mm case, corresponding to an increase of about 47.5%. This indicates that increasing sleeper spacing strengthens the local span effect of the rail so that the wheel load must be transferred through fewer support points, thereby significantly amplifying the sectional shear force of the rail. As shown in Figure 6(b), the track slab shear force also shows a symmetric distribution centered at the loading point, but more pronounced local fluctuations and multiple peaks appear near the loading point. This reflects that, after the wheel load is transferred to the track slab in a discrete manner through the double-block sleepers and fastening system, a relatively large shear gradient is generated in the local load-transfer region. The peak shear force of the track slab increases from 12.43 kN under the 400 mm case to 20.78 kN under the 900 mm case, corresponding to an increase of about 67.2%.

As shown in Figure 7(a), the rail bending moment exhibits a symmetric bending pattern along the track longitudinal direction that is typical of a beam under a single-point vertical load. A pronounced negative bending moment trough appears at the loading point, while positive bending moment peaks develop at a certain distance on both sides, after which the bending moment gradually decays to nearly zero. As the sleeper spacing increases from 400 mm to 900 mm, the most unfavorable negative bending moment at the loading point increases from −12.28 kN·m to −16.08 kN·m, an increase of about 30.9%. Meanwhile, the positive bending moment peak on both sides increases from 2.96 kN·m to 3.98 kN·m, an increase of about 34.4%, indicating that increasing sleeper spacing enhances the bending-moment concentration effect in the rail. As shown in Figure 7(b), the bending moment of the track slab also exhibits a symmetric distribution centered at the loading point, but more pronounced short-range fluctuations occur near the loading point, reflecting the non-uniform local bending behavior under discrete nodal load transfer. Under different sleeper spacing cases, the peak negative bending moment of the track slab generally shows a trend of first decreasing and then slightly increasing, decreasing from −2.95 kN·m under the 400 mm case to −1.87 kN·m under the 700 mm case, and then increasing to −2.17 kN·m under the 900 mm case. The peak positive bending moment of the track slab also decreases overall, from 0.92 kN·m under the 400 mm case to 0.75 kN·m under the 900 mm case. This indicates that increasing sleeper spacing weakens the overall longitudinal bending level of the track slab, although the local nodal control effect becomes somewhat stronger under very large spacing conditions.

As indicated by Figures 6 and 7, increasing sleeper spacing enhances the local force concentration effect near the loading region. Specifically, the peak values of the shear force in both the rail and the track slab increase while their distribution range contracts, the bending-moment concentration in the rail becomes more pronounced, and the bending moment of the track slab shows a trend of first decreasing and then locally increasing. This phenomenon is mainly attributed to the discrete load transfer path and the continuous support condition of the track slab. The wheel load is transmitted to the track slab through the rail–fastener–sleeper system, causing local downward bending near the loading point; meanwhile, the adjacent slab regions are restrained by the foundation support and slab continuity, resulting in curvature reversal and the formation of positive bending moments on both sides. These results indicate that changes in sleeper spacing significantly affect the internal force distribution characteristics of the rail and track slab under discrete support conditions.

4.1. Dynamic response of rails

Based on the vehicle–track coupled dynamic model and the finite element model of the double-block ballastless track structure established above, six sleeper-spacing cases ranging from 400 mm to 900 mm were considered in this study, with an interval of 100 mm between adjacent cases. The vertical rail vibration acceleration responses were then calculated at train speeds of 100 km/h, 200 km/h, and 300 km/h. Figure 11 presents the comparison of peak rail vibration acceleration and the typical time-history responses, while Figure 12 shows the corresponding rail displacement responses.OPEN IN VIEWER

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Figure 12.

Vibration displacement response of the rail.

As shown in Figure 11, the vertical rail vibration acceleration is jointly influenced by train speed and sleeper spacing, and its sensitivity to sleeper spacing becomes more pronounced under high-speed conditions. Figure 11(a) and (b) show that, at a train speed of 100 km/h, the peak rail acceleration increases gradually with increasing sleeper spacing, and the overall variation remains relatively small. Specifically, when the sleeper spacing increases from 500 mm to 700 mm, the peak value rises from 5.4 g to 5.8 g, corresponding to an increase of 7.41%. When the sleeper spacing is further increased to 900 mm, the peak value reaches 6.2 g, which is 14.81% higher than that at 500 mm. This indicates that, under low-speed conditions, the effect of sleeper spacing on rail acceleration remains relatively limited. When the train speed increases to 200 km/h and 300 km/h, however, the peak rail acceleration increases significantly, and the increasing trend with sleeper spacing becomes more evident. At 200 km/h, the peak values corresponding to sleeper spacings of 400, 500, 600, 700, 800, and 900 mm are 20.8, 21.8, 22.8, 23.7, 24.2, and 24.6 g, respectively, which are approximately 3.97 to 4.09 times those at 100 km/h for the same sleeper spacing; among them, the values at 400 mm and 700 mm are 4.00 and 4.09 times higher, respectively. At 300 km/h, the peak values further increase to 24.5, 26.2, 27.9, 29.1, 30.5, and 31.2 g, respectively, corresponding to approximately 4.71 to 5.08 times those at 100 km/h, with the 800 mm case reaching 5.08 times. These results indicate that the increase in train speed is the dominant factor driving the growth of vertical rail vibration acceleration, while a larger sleeper spacing further amplifies the vibration response under high-speed operating conditions.

As shown in Figure 12(a) and (b), the peak rail vertical displacement increases steadily with increasing sleeper spacing, and this trend remains essentially consistent at different train speeds, indicating that the displacement response is more sensitive to sleeper spacing. At a train speed of 100 km/h, when the sleeper spacing increases from 400 mm to 900 mm, the peak displacement rises from 0.7881 mm to 1.5144 mm, corresponding to an increase of 92.16%. At 200 km/h and 300 km/h, the peak displacement increases from 0.8358 mm to 1.5915 mm and from 0.8719 mm to 1.6276 mm, with increases of 90.42% and 86.67%, respectively. By contrast, under the same sleeper spacing, increasing the train speed from 100 km/h to 300 km/h leads only to a slight increase in the peak displacement, with an increase ranging from about 7.47% to 10.63%. This indicates that, when track irregularity is not considered, the rail vertical displacement is still dominated by the quasi-static deflection induced by wheel load, whereas the contribution of inertial amplification caused by high-speed operation is relatively limited. From the time-history curves, as the operating conditions change from L = 500&v = 100 to L = 700&v = 200 and then to L = 900&v = 300, the rail displacement waveform becomes progressively steeper. The main reason is that increasing sleeper spacing enlarges the effective rail span and reduces the number of support points per unit length, thereby decreasing the equivalent vertical support stiffness and causing the peak rail vertical displacement and fluctuation amplitude to increase simultaneously.

4.2. Wheel–rail force response

Wheel–rail force is an important indicator for characterizing the dynamic interaction of the vehicle–track coupled system. Changes in sleeper spacing alter the discrete support conditions of the rail, thereby affecting the response characteristics of the wheel–rail lateral force and vertical force. Therefore, this study further analyzes the response patterns of wheel–rail force under different sleeper-spacing conditions. Figures 13 and 14 show the wheel–rail lateral force and vertical force under different sleeper-spacing conditions.OPEN IN VIEWER

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Figure 14.

Moving wheelset–rail vertical force responses.

As shown in Figure 13(a), the peak wheel–rail lateral force remains within the range of 7.15 kN to 7.24 kN, and its variation with sleeper spacing and train speed is small. This indicates that, under the conditions where track irregularity is not considered and the wheel–rail contact parameters remain unchanged, the peak wheel–rail lateral force is mainly governed by the wheel–rail geometric relationship and contact state, whereas the influence of sleeper spacing is limited. At 100 km/h, the wheel–rail lateral force increases from 7.152 kN for the 400 mm case to 7.180 kN for the 900 mm case, corresponding to an increase of only 0.39%. At 200 km/h, it increases from 7.178 kN to 7.232 kN, with an increase of 0.75%; at 300 km/h, it increases from 7.217 kN to 7.242 kN, with an increase of 0.34%. Under the same sleeper spacing, the peak increase at 300 km/h relative to that at 100 km/h generally does not exceed approximately 1.3%, indicating that the amplifying effect of train speed on the peak wheel–rail lateral force is also insignificant. As shown in Figure 13(b), with increasing sleeper spacing, the fluctuation amplitude and envelope variation of the time-history curve of wheel–rail lateral force gradually become more pronounced, and the oscillation characteristics in local segments are more evident.

As shown in Figure 14(a), the peak wheel–rail vertical force remains generally stable within the range of 65.10 kN to 66.49 kN, and its variation with sleeper spacing and train speed is likewise small. This indicates that, under the conditions where track irregularity is not considered and the wheel–rail contact parameters remain unchanged, the wheel–rail vertical force is still dominated by the static load component, and the influence of sleeper spacing on its peak level is limited. At 100 km/h, the wheel–rail vertical force increases from 65.0964 kN for the 400 mm case to 65.945 kN for the 900 mm case, corresponding to an increase of 1.30%. At 200 km/h, it increases from 65.62 kN to 66.2562 kN, with an increase of 0.97%; at 300 km/h, it increases from 65.748 kN to 66.489 kN, with an increase of 1.13%. Under the same sleeper spacing, the peak increase caused by raising the train speed from 100 km/h to 300 km/h generally does not exceed approximately 1.0% to 1.5%, indicating that the amplifying effect of train speed on the peak wheel–rail vertical force is also limited. As shown in Figure 14(b), with increasing sleeper spacing, the oscillation amplitude and envelope variation of the time-history curve of wheel–rail vertical force become significantly stronger. Meanwhile, as the operating conditions change from L = 500&v = 100 to L = 700&v = 200 and then to L = 900&v = 300, the increments of wheel–rail force relative to the static vertical force are 0.172 kN, 0.571 kN, and 1.0302 kN, respectively, indicating that the increase in wheel–rail force is manifested not in a substantial rise in peak value, but in the continuous enhancement of its dynamic fluctuation component.

Considering the influence of sleeper spacing on wheel–rail force as a whole, the effect of sleeper spacing on the peak levels of wheel–rail lateral force and vertical force is relatively small. Within the sleeper-spacing range from 400 mm to 900 mm, the peak increases of both forces generally do not exceed approximately 1.5%, and the variation trends under different train speeds remain basically consistent. However, sleeper spacing has a much more significant influence on the time-domain fluctuation characteristics of wheel–rail force, as reflected by the increasing oscillation amplitude and envelope variation during operation with increasing sleeper spacing. In other words, the mean level of wheel–rail force changes little, whereas its dynamic fluctuation becomes markedly stronger. The reason for this pattern is that, under the conditions where track irregularity is not introduced and the wheel–rail contact parameters remain unchanged, the peak wheel–rail force is mainly determined by the static load component of the vehicle as well as the wheel–rail contact geometry and contact state, so changes in sleeper spacing cannot significantly increase the peak value. However, increasing sleeper spacing reduces the continuity of the equivalent support along the track and enlarges the discrete span, making the deformation coordination of the rail and the redistribution of support reactions between discrete supports more intense. As a result, the dynamic component of wheel–rail contact force is more easily excited, and its time-history fluctuation is further amplified.