Deformation behavior of slab warping for longitudinal continuous rigid slab under temperature effect

Abstract

Longitudinally coupled prefabricated slab track is prone to slab warping under the non-uniform temperature field. Analytical expressions for the displacement field of the track slab during the warping process are developed hereby based on equilibrium differential equations, and the expressions are verified through a numerical model by finite element method in ANSYS package. Through analyzing the factors such as types of temperature distribution, slab gravity, and rail weight, regulations of warping deformation for the track slab are systematically analyzed. Research shows that the displacements in the x and z directions are linearly related to the respective components during the warping deformation. The displacement in y direction has a quadratic relationship with x and z components. Temperature gradient is the pivotal factor to lead the warping of the track slab, and the type of temperature distribution has less effect on the warping displacement. If track slab remains unwarped under the effect of slab gravity, temperature gradient should be maintained in the range of –18°C/m–13°C/m. Rail weight has less influence on slab warping.

Keywords

ballastless track equilibrium differential equations longitudinally coupled prefabricated slab track slab warping temperature gradient

Introduction

In recent years, high-speed railways have been widely developed in China (Chen et al., 2012; Feng et al., 2013; Ren et al., 2019). There are varieties of longitudinal continuous concrete structures in transport, such as rigid pavement, continuous bridge structures, and tunnel structures. As a typical structure of longitudinal continuous rigid slab, longitudinally coupled prefabricated slab track (LCPST) was introduced and reformed on the basis of the German Bögl slab. This kind of track was first put into operation in the Beijing–Tianjin intercity passenger lines in 2008. Owing to the high precision and durability of LCPST, it was then widely applied to other routes in China, including the Beijing–Shanghai, Shanghai–Hangzhou, and Shijiazhuang–Wuhan lines (Tian et al., 2016; Long et al., 2018).

The main components of LCPST consist of the rail CN60 (Chinese standard rail with a mass of 60 kg/m), elastic fastening system type WJ-8 invented in China, track slabs, cement asphalt mortar (CA mortar), and a concrete base (shown in Figure 1). Track slabs comprise prefabricated concrete slabs made of reinforced concrete (0.2 m thick, 2.55 m wide, and 6.45 m long). The slabs are connected by slab joints consisting of turnbuckles and elastic concrete mortar to lead the structure continuous in longitudinal direction (Lichtberger, 2005).

Figure 1.

A cross-section of LCPST.

As the concrete has lower thermal conductivity compared to metallic material, the track slab exists as a non-uniform temperature field under the effects of cyclical action of solar radiation, convective heat transfer, and radiant heat transfer, forming the temperature gradient, which will lead the slab warping in lateral. Though the track slab and the concrete base are bonded together by the bonding behavior of the CA mortar, the mortar layer has limited vertical restraint ability on the track slab. Under the effects of complex factors such as temperature, humidity, and vehicle load, de-bonding of the track slab and the mortar layer are inclined to occur (Cao et al., 2016; Han et al., 2015), which is shown in Figure 2. Thus, under the action of non-uniform temperature field, the track slab would further arise warping. The diseases above will reduce the stability of the longitudinal track structure, seriously threaten the quality and safety of vehicle’s high-speed operation, and have aroused widespread concerns (Song et al., 2016; Yan et al., 2015). Although there are methods such as bar anchoring and cement grouting (Ni et al., 2015; Wang et al., 2016) to reduce the warping displacement for the track slab, in order to further clarify the mechanism of slab warping, the theoretical analysis of slab warping is developed in this article.

Figure 2.

De-bonding of the track slab and the mortar layer.

In views of the warping problem for the concrete slabs, researchers have executed lots of work, especially the establishment of the theory for the concrete pavement warping in the early age have provided basis for the construction and maintenance of concrete slabs (Beckemeyer et al., 2002; Janssen and Snyder, 2000; Vandenbossche, 2006). With the development of China’s high-speed railway in recent years, problems of slab warping for the ballastless track under temperature effect have synchronously aroused concerns. The researches above have provided a theoretical basis for the study of warping deformation of the track structures.

To thoroughly study the warping characteristics of the track slab under temperature effect, it is necessary to clarify the types of temperature field inside the track structure under different environmental conditions. Liu et al. (2016) preliminarily analyzed the time domain characteristics of the temperature field for the slab track based on the principle of heat transfer. Yang et al. (2017) determined the distribution regulation of temperature field for the double-block ballastless track under different environments by a full-scale experiment in site. Wu et al. (2016) further studied the variation of the temperature gradient for the slab track through a test by arranged sensors along the track structure in depth and obtained the maximum temperature gradient. The existing researches for the temperature field of the track structure have provided data based on the determination of the temperature parameters for the studies in this article.

For the studies of slab warping for the ballastless track, Wang et al. (2010) proposed a construction method to reduce the warping displacement by analyzing the warping deformation of the precast reinforced concrete slabs for railway tracks. Liu and Zhao (2013) analyzed the regulation of displacement and stress development for the track slab during the warping process under the temperature gradient by establishing a finite element model for the track structure and proposed a suggestion to prevent early de-bonding inside slab layers. Concentrating the slab board in transitional zone, Chen et al. (2015) analyzed the effect of beam-rail interaction on the slab warping, indicating that the longitudinal beam-rail interaction would reduce the warping deformation of the track board. Yang et al. (2016) simulated the temperature field inside the track slab by combining field measurement with numerical simulation and analyzed the deformation behavior of the track slab under both amplitude and gradient effects of temperature. The current researches above legitimately pay attention to the regulation of stress development and the failure characteristics of bonding strength for the track slab after warping deformation, and are mainly based on the methods of numerical simulation and test in site, but lack the theoretical derivation for the mechanism of warping deformation under temperature effect.

To further explore the warping mechanism of the de-bonding track slab under the temperature effect, and make up for the lack of theoretical derivation, this article derives analytical expressions for the slab warping according to the coordination condition of local deformation for the concrete structure under the effects of temperature, slab gravity, and rail weight on the basis of equilibrium differential equations. The distribution of the displacement field during the warping process is clarified by solving the expressions of the stress and the strain field. On the basis of the derivation above, the pivotal factors of the warping displacement are deeply analyzed. The conclusions provide theoretical guidance for the construction and maintenance for the ballastless track, and the results can be generalized to the longitudinal continuous structures.

Calculating method in mechanical for the slab warping

According to the structural characteristics of LCPST, the analysis model for the slab warping is established (shown in Figure 3), and the following assumptions are made:

The auxiliary components such as pro-cracking joints and rail platforms (shown in Figure 2) have less effect on the bending stiffness of the track slab (Ren et al., 2017), and the slab joints (shown in Figure 2) are assumed to be not damaged. Therefore, the track slab is simplified as a board with constant cross-section.

Due to the low tensile strength of the mortar layer, the bonding strength between the mortar layer and the track slab will gradually weaken during the vehicle operation process affected by multiple coupling loads such as temperature, humidity, and vehicle load (Yang et al., 2015). The mortar layer only takes effect of vertical support. Thus, the mortar layer is considered disconnected to the track slab completely, and the bonding effect of the mortar layer is ignored.

The bending stiffness of rail is small if compared to the track slab. Hence, the rail is simplified as a uniform load, and the effect of fastener system is ignored.

Figure 3.

Analysis model for the slab warping.

In Figure 3, x, y, and z are the coordinate components of the track structure in longitudinal, vertical, and lateral directions, respectively. The displacement of the track slab under the actions of temperature effect, slab gravity, and rail weight are expressed as

\begin{matrix} u = u_{T} + u_{G} + u_{P} \\ v = v_{T} + v_{G} + v_{P} \\ w = w_{T} + w_{G} + w_{P} \end{matrix}

(1)

where u, v, and w are the resultant displacements in x, y, and z directions, respectively. u_T, v_T, w_T, u_G, v_G, w_G, u_P, v_P, and w_P, respectively, represent the displacement components under temperature effect, slab gravity, and rail weight. According to the coordinate system in Figure 3, the vertical displacement exhibits an upward displacement when v < 0, and vice versa if v > 0. In order to analyze the mechanism of slab warping essentially, it is necessary to establish a warping model only affected by temperature field in the free status. In this condition, the strain components consist of the elastic strain at ambient temperature and the free thermal strain (Al Hamd et al., 2018; Lange and Jansson, 2014); thus, the slab exists internal stress during the warping process. The internal stress in the track slab should satisfy the equilibrium condition of the resultant moment, that is

\int_{- h / 2}^{h / 2} \frac{2 σ_{T} y^{2}}{h} dy - \int_{- h / 2}^{h / 2} E α yT (y) dy = 0

(2)

where E and α, respectively, represent the elastic modulus and the thermal expansion coefficient of the concrete used for the track slab. T(y) is the function of temperature field for the track slab related to the y component. σ_T represents the stress in the track slab during the warping process. On the basis of equation (2), the stress components are expressed as

\begin{matrix} σ_{Tx} = σ_{Tz} = \frac{E α}{1 - υ} \\ [- T (y) + \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ σ_{Ty} = 0 \\ τ_{Txy} = τ_{Tyz} = τ_{Txz} = 0 \end{matrix}

(3)

where σ_Tx, σ_Ty, σ_Tz, τ_Txy, τ_Tyz, and τ_Txz are the stress components during the warping process. h represents the thickness of the track slab in y-axis. According to the physical equation, the strain components are expressed as

\begin{matrix} ε_{Tx} = \frac{1}{E} [σ_{Tx} - υ (σ_{Ty} + σ_{Tz})] + α T (y) = \\ α [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ ε_{Ty} = \frac{1}{E} [σ_{Ty} - υ (σ_{Tx} + σ_{Tz})] + α T (y) = \\ - \frac{2 α υ}{1 - υ} [- T (y) + \frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ + α T (y) \\ ε_{Tz} = \frac{1}{E} [σ_{Tz} - υ (σ_{Tx} + σ_{Ty})] + α T (y) = \\ α [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ γ_{Txy} = γ_{Tyz} = γ_{Txz} = 0 \end{matrix}

(4)

where ε_Tx, ε_Ty, ε_Tz, γ_Txy, γ_Tyz, and γ_Txz are strain components affected by temperature field. υ represents Poisson’s ratio of the concrete used for the track slab. According to the basic equation of elastic deformation, the displacement components under temperature effect are expressed as

\begin{matrix} u_{T} = \int ε_{Tx} dx + f_{1} (y, z) = \\ α x [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] + f_{1} (y, z) \\ v_{T} = \int ε_{Ty} dy + f_{2} (x, z) = \\ - \frac{2 α υ y}{1 - υ} [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{6 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ + \frac{α (1 + υ)}{1 - υ} \int T (y) dy + f_{2} (x, z) \\ w_{T} = \int ε_{Tz} dz + f_{3} (x, y) = \\ α z [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] + f_{3} (x, y) \end{matrix}

(5)

where f₁(y, z), f₂(x, z), and f₃(x, y) are the free functions for the displacements under unconstrained conditions. According to the coordination equation of deformation, the relationship between the displacement components and the shear strain component is obtained as

\begin{matrix} γ_{Txy} = \frac{\partial v_{T}}{\partial x} + \frac{\partial u_{T}}{\partial y} = \frac{\partial f_{2} (x, z)}{\partial x} + \frac{\partial f_{1} (y, z)}{\partial y} \\ + \frac{12 α x}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy = 0 \\ γ_{Tyz} = \frac{\partial w_{T}}{\partial y} + \frac{\partial v_{T}}{\partial z} = \frac{\partial f_{3} (x, y)}{\partial y} + \frac{\partial f_{2} (x, z)}{\partial z} \\ + \frac{12 α z}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy = 0 \\ γ_{Txz} = \frac{\partial w_{T}}{\partial x} + \frac{\partial u_{T}}{\partial z} = \frac{\partial f_{3} (x, y)}{\partial x} + \frac{\partial f_{1} (y, z)}{\partial z} = 0 \end{matrix}

(6)

With the simultaneous solution of equations (5) and (6), the expressions for displacement components under temperature effect are determined as

\begin{matrix} u_{T} = α x [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ + ω_{1} z - ω_{2} y + C_{1} \\ v_{T} = - \frac{2 α υ y}{1 - υ} [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{6 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ + \frac{α (1 + υ)}{1 - υ} \int_{0}^{y} T (t) dt - \frac{6 α (x^{2} + z^{2})}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy \\ + ω_{2} x + ω_{3} z + C_{2} \\ w_{T} = α z [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{6 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ - ω_{1} x - ω_{3} y + C_{3} \end{matrix}

(7)

Equation (7) is in the condition of free status, and only the temperature effect is considered without the effect of slab gravity and other loads. In equation (7), ω₁, ω₂, ω₃, C₁, C₂, and C₃ are the free constants associated with the boundary conditions. T′(y) represents the temperature gradient, which is the first derivative of temperature versus depth of the slab, indicating the non-uniformity extent of the temperature along the slab thickness. Equation (7) shows that the track slab will have different modes of deformations in x, y, and z directions under the temperature field. Among them, the displacements in the x and z directions are only related to the temperature degree and are linearly related to the respective components. The displacement in y direction has a quadratic relationship with x and z components affected by the temperature field. As the thickness in y direction is much smaller than the length in x direction and the width in z direction for the track slab, the non-uniformity of the temperature field would significantly affect the vertical warping deformation of the track slab. Therefore, it is necessary to concentrate the development on vertical displacement while analyzing the slab warping.

In addition, if the track slab is considered as a three-dimensional (3-D) structure, each expression for the displacement involves three components, which are complex in form and not conducive to calculation. Considering that the LCPST is continuous longitudinally, the longitudinal stress and the longitudinal deformation could be neglected. Therefore, if the slab joints are in good condition and the pro-cracking joints are uncracked, the track slab could be simplified to the plane strain model (shown in Figure 4). In Figure 4, b and h are the width and thickness of the track slab, respectively. Taking x = 0 in equation (7), the simplified displacement expressions of the track slab in y and z directions are obtained as

\begin{matrix} v_{T} = - \frac{2 α υ y}{1 - υ} [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{6 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] \\ + \frac{α (1 + υ)}{1 - υ} \int_{0}^{y} T (t) dt + z [ω - \frac{6 α z}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] + C_{1} \\ w_{T} = α z [\frac{1}{h} \int_{- h / 2}^{h / 2} T (y) dy + \frac{12 y}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy] - ω y + C_{2} \end{matrix}

(8)

Figure 4.

The plane strain model of the track slab.

Compared to equation (7), the number of displacement components in equation (8) is reduced from three to two, and the number of free constants is reduced from six to three, which improves the calculation efficiency while guaranteeing the accuracy.

Effect of temperature distribution on slab warping

According to the distribution characteristics of temperature field in vertical direction, the temperature degrees at neutral axis, upper surface, and lower surface of the cross-section are defined as T₀, T₁, and T₂, respectively. Neutral axis, upper surface, and lower surface correspond to the positions of y = 0 m, –0.1 m, and 0.1 m, respectively. Under the effects of positive and negative temperature gradients, the warping displacement will exhibit different morphological distributions and correspond to different boundary conditions (shown in Figure 5):

If the slab displacement at the middle of the lower surface is higher than the two sides, the support points of the mortar layer are mainly concentrated at both sides, and the boundary conditions are v_{(z = ±b/2, y = h/2)} = 0 and w_{(z = 0)} = 0.

If the slab displacement at both sides of the lower surface is higher than the middle portion, the support points of the mortar layer are mainly concentrated at the middle, and the boundary conditions are v_{(z = 0, y = h/2)} = 0 and w_{(z = 0)} = 0.

Figure 5.

Two types of boundary conditions: (a) condition 1 and (b) condition 2.

The expressions for vertical displacement corresponding to boundary conditions 1 and 2 are shown in equations (9) and (10), respectively

\begin{matrix} v_{T} = \frac{α υ}{1 - υ} [(1 - \frac{2 y}{h}) \int_{- h / 2}^{h / 2} T (y) dy + \frac{3}{h} (1 - \frac{4 y^{2}}{h^{2}}) \int_{- h / 2}^{h / 2} T (y) ydy] \\ + \frac{α (1 + υ)}{1 - υ} \int_{h / 2}^{y} T (t) dt + \frac{3 α}{2 h^{3}} (b^{2} - 4 z^{2}) \int_{- h / 2}^{h / 2} T (y) ydy \end{matrix}

(9)

\begin{matrix} v_{T} = \frac{α υ}{1 - υ} \\ [(1 - \frac{2 y}{h}) \int_{- h / 2}^{h / 2} T (y) dy + \frac{3}{h} (1 - \frac{4 y^{2}}{h^{2}}) \int_{- h / 2}^{h / 2} T (y) ydy] \\ + \frac{α (1 + υ)}{1 - υ} \int_{h / 2}^{y} T (t) dt - \frac{6 α z^{2}}{h^{3}} \int_{- h / 2}^{h / 2} T (y) ydy \end{matrix}

(10)

According to the structural characteristics of LCPST, b = 2.55 m, h = 0.2 m, and α = 1 × 10^–5 are taken in the following analysis. The test results for the temperature field of ballastless track from Ou et al. (2014), Wu et al. (2016), and Yang et al. (2017) show that the temperature distribution inside the track structure exhibits a complex form under the cycling effect from the solar radiation every day. The types of temperature distributions depend both on the outside temperature and the temperature of the ground (Liu et al., 2016; Yang et al., 2017). In extreme high-temperature environments, the temperature at upper surface of the track slab rises sharply, which is much higher than the temperature at neutral axis and lower surface; thus, the temperature distribution may be expressed as exponential type. Owing to the faster heat dissipation at upper surface and the lower surface than the neutral axis at night, the temperature distribution may be expressed as quadratic type. For the distribution of linear type, it is the most common and theoretical type. To discuss the effect of temperature distribution on slab warping, the temperature is approximately classified into three types of distribution, that is, linear, quadratic, and exponential distributions. If T₀, T₁, and T₂ are determined, T(y) can be expressed as equations (11) to (13), respectively

T {(y)}_{Linear} = \frac{1}{2} (T_{1} + T_{2}) - \frac{y}{h} (T_{1} + T_{2})

(11)

T {(y)}_{Quadric} = (1 - \frac{4 y^{2}}{h^{2}}) T_{0} + \frac{y}{h} [\frac{2 y}{h} (T_{1} + T_{2}) - (T_{1} - T_{2})]

(12)

\begin{matrix} T {(y)}_{Exponential} = \frac{{T_{0}}^{2} - T_{1} T_{2}}{2 T_{0} - (T_{1} + T_{2})} \\ + \frac{(T_{0} - T_{1}) (T_{0} - T_{2})}{2 T_{0} - (T_{1} + T_{2})} {(\frac{T_{0} - T_{2}}{T_{1} - T_{0}})}^{\frac{2 y}{h}} \end{matrix}

(13)

Under the effect of solar radiation during the daytime, the temperature in the track slab shows a distribution of positive temperature gradient due to the exposure of the surface to a high-temperature environment. In extreme high-temperature environments, the temperature gradient would reach 100°C/m (Yang et al., 2017). According to the experiment results (Liu et al., 2016; Yang et al., 2017), T₁ = 55°C and T₂ = 35°C are taken, and the T₀ are 45°C, 33°C, and 37°C corresponding to the three types of temperature distributions (shown in Figure 6(a)). For the heat dissipation at the upper surface is faster than the lower surface at night, the track slab exhibits a negative temperature gradient distribution, with a gradient of almost –30°C/m (Wu et al., 2016). Thus, T₁ = 30°C and T₂ = 36°C are taken, and T₀ are 33°C, 39°C, and 35°C corresponding to the three types of curves (shown in Figure 6(b)).

Figure 6.

Three types of vertical temperature distributions: (a) positive temperature gradient and (b) negative temperature gradient.

With the simultaneous solution of equation (8) and equations (11) to (13), warping displacement of the track slab in vertical direction is determined, and the boundary conditions are appropriately selected according to the forms of displacement at the lower surface. The warping displacements for the three types of temperature distributions under the positive and negative temperature gradients are shown in Figures 7 and 8, respectively.

Figure 7.

Warping displacements of the track slab in vertical direction under positive temperature gradient: (a) linear distribution, (b) quadratic distribution, and (c) exponential distribution.

Figure 8.

Warping displacements of the track slab in vertical direction under negative temperature gradient: (a) linear distribution, (b) quadratic distribution, and (c) exponential distribution.

Results show that warping displacements of the track slab under three types of temperature distribution are similar. Track slabs under positive and negative temperature gradients correspond to boundary conditions 1 and 2, respectively. If the track slab is under the temperature of linear distribution, the stress in the track slab is 0 because T′(y) is a constant, and the track slab has the largest warping displacement in vertical direction. Correspondingly, owing to the large temperature stress during the warping deformation process, the track slab under the temperature of exponential distribution has a smaller warping displacement in vertical direction compared to other two types of temperature distribution. The maximum and minimum values of the vertical displacement for the three types of temperature distributions appear at the middle and the side of the slab. If the difference between the maximum displacement and the minimum displacement of the track plate is defined as Δv_T, then Δv_T reaches the maximum value Δv_T_max at the upper surface under the three types of temperature distributions. According to Figure 8, the value of Δv_T_max for the three types of temperature distributions under positive temperature gradient are 0.81 mm, 0.81 mm, and 0.63 mm, respectively, and that under negative temperature gradient are 0.24 mm, 0.24 mm, and 0.21 mm, respectively. Therefore, temperature gradient is the pivotal factor to lead the slab warping, but the type of temperature distribution has less effect on the warping displacement.

Effect of slab gravity on slab warping

Slab gravity is also a pivotal factor that restricts warping deformation; thus, the warping results based on non-gravity conditions would not precisely reflect the actual situation. Thus, the displacements of the track slab affected by gravity under the two boundary conditions above are analyzed separately in the following. According to the cross-section characteristics of the track slab in Figure 5, the stress function ϕ_G (Timoshenko and Goodier, 1970) under the action of gravity is expresses as

ϕ_{G} = A y^{3} + B y^{5} + Cy z^{2} + D z^{2}

(14)

where A, B, C, and D are the coefficients needed to be determined. Equation (14) is supposed to satisfy the compatibility equation ∇⁴ϕ_G = 0, which is expanded as

\frac{\partial^{4} ϕ_{G}}{\partial y^{4}} + 2 \frac{\partial^{4} ϕ_{G}}{\partial y^{2} \partial z^{2}} + \frac{\partial^{4} ϕ_{G}}{\partial z^{4}} = 0

(15)

According to the inhomogeneous characteristics of the equilibrium differential equation, the stress components under the action of slab gravity are expressed as

σ_{Gy} = \frac{\partial^{2} ϕ_{G}}{\partial z^{2}} - γ gy, σ_{Gz} = \frac{\partial^{2} ϕ_{G}}{\partial y^{2}}, τ_{Gyz} = - \frac{\partial^{2} ϕ_{G}}{\partial y \partial z}

(16)

where σ_Gy, σ_Gz, and τ_Gyz are the stress components under the action of gravity. γ and g represent the density and gravity parameters of the track slab, respectively.

Boundary condition 1

Figure 5(a) shows that with the constraint of boundary condition 1, the equilibrium condition of the stress at the boundary of the cross-section is

\begin{matrix} {σ_{Gy}}_{(y = - h / 2)} = {τ_{Gyz}}_{(y = - h / 2)} = 0 \\ \int_{- h / 2}^{h / 2} {σ_{Gz}}_{(z = \pm b / 2)} dy = \int_{- h / 2}^{h / 2} y {σ_{Gz}}_{(z = \pm b / 2)} dy = 0 \\ \int_{- h / 2}^{h / 2} {τ_{Gyz}}_{(z = \pm b / 2)} dy = - \frac{1}{2} γ g \end{matrix}

(17)

With the simultaneous solution of equations (14) to (17), the stress components subjected to gravity are determined as

\begin{matrix} σ_{Gy} = \frac{γ gy}{2} (1 - \frac{4 y^{2}}{h^{2}}) \\ σ_{Gz} = \frac{3 γ gy}{2 h^{2}} (b^{2} - 4 z^{2}) + γ gy (\frac{4 y^{2}}{h^{2}} - \frac{3}{5}) \\ τ_{Gyz} = - \frac{3 γ gz}{2} (1 - \frac{4 y^{2}}{h^{2}}) \end{matrix}

(18)

According to the physical equation, the strain components ε_Gy, ε_Gz, and γ_Gyz under slab gravity are expressed as

\begin{matrix} ε_{Gy} = \frac{γ gy}{E'} [\frac{1}{2} (1 - \frac{4 y^{2}}{h^{2}}) - \frac{3 υ}{2 h^{2}} (b^{2} - 4 z^{2}) - υ' (\frac{4 y^{2}}{h^{2}} - \frac{3}{5})] \\ ε_{Gz} = \frac{γ gy}{E'} [\frac{3}{2 h^{2}} (b^{2} - 4 z^{2}) + (\frac{4 y^{2}}{h^{2}} - \frac{3}{5}) - \frac{υ'}{2} (1 - \frac{4 y^{2}}{h^{2}})] \\ γ_{Gyz} = - \frac{3 γ gz (1 + υ')}{E'} (1 - \frac{4 y^{2}}{h^{2}}) \end{matrix}

(19)

where E′ and υ′, respectively, represent the elastic modulus and Poisson’s ratio under the plane strain model, which are expressed as

E' = \frac{E}{1 - υ^{2}}, υ' = \frac{υ}{1 - υ}

(20)

According to the geometric equation and the coordination condition of deformation, the vertical displacement v_G with boundary condition 1 under slab gravity is expressed as

\begin{matrix} v_{G} = \frac{γ g y^{2}}{E'} \\ {\frac{1}{4} (1 - \frac{2 y^{2}}{h^{2}}) - υ' [\frac{3}{4 h^{2}} (b^{2} - 4 z^{2}) + (\frac{y^{2}}{h^{2}} - \frac{3}{10})]} \\ - \frac{γ g z^{2}}{E'} [\frac{3}{2} (1 + υ') + \frac{1}{4 h^{2}} (3 b^{2} - 2 z^{2})] \\ - \frac{γ g h^{2}}{4 E'} (\frac{1}{8} + \frac{υ'}{20}) + \frac{γ g b^{2}}{4 E'} [\frac{3}{2} (1 + υ') + \frac{5 b^{2}}{8 h^{2}}] \end{matrix}

(21)

Boundary condition 2

Figure 5(b) shows that with the constraint of boundary condition 2, the equilibrium condition of the stress at the boundary of the cross-section is

\begin{matrix} {σ_{Gy}}_{(y = - h / 2)} = {τ_{Gyz}}_{(y = - h / 2)} = 0 \\ \int_{- h / 2}^{h / 2} {σ_{Gz}}_{(z = \pm b / 2)} dy = \int_{- h / 2}^{h / 2} y {σ_{Gz}}_{(z = \pm b / 2)} dy = 0 \\ \int_{- h / 2}^{h / 2} y {(σ_{Gz})}_{z = 0} dy = - \frac{1}{8} γ gh b^{2} \end{matrix}

(22)

With the simultaneous solution of equations (14) to (16) and equation (22), the stress components subjected to gravity are determined as

\begin{matrix} σ_{Gy} = \frac{γ gy}{2} (1 - \frac{4 y^{2}}{h^{2}}) \\ σ_{Gz} = \frac{2 γ gy}{h^{2}} (2 y^{2} - 3 z^{2}) - γ gy (\frac{3 b^{2}}{2 h^{2}} + \frac{3}{5}) \\ τ_{Gyz} = - \frac{3 γ gz}{2} (1 - \frac{4 y^{2}}{h^{2}}) \end{matrix}

(23)

The strain components under slab gravity are expressed as

\begin{matrix} ε_{Gy} = \frac{γ gy}{E'} {\frac{1}{2} (1 - \frac{4 y^{2}}{h^{2}}) - υ' [\frac{2}{h^{2}} (2 y^{2} - 3 z^{2}) - (\frac{3 b^{2}}{2 h^{2}} + \frac{3}{5})]} \\ ε_{Gz} = \frac{γ gy}{E'} [\frac{2}{h^{2}} (2 y^{2} - 3 z^{2}) - (\frac{3 b^{2}}{2 h^{2}} + \frac{3}{5}) - \frac{υ'}{2} (1 - \frac{4 y^{2}}{h^{2}})] \\ γ_{Gyz} = - \frac{3 γ gz (1 + υ')}{E'} (1 - \frac{4 y^{2}}{h^{2}}) \end{matrix}

(24)

The vertical displacement v_G with boundary condition 2 under slab gravity is expressed as

\begin{matrix} v_{G} = \frac{γ g y^{2}}{E'} {\frac{1}{2} (\frac{1}{2} - \frac{y^{2}}{h^{2}}) - υ' [\frac{1}{h^{2}} (y^{2} - 3 z^{2}) - \frac{1}{2} (\frac{3 b^{2}}{2 h^{2}} + \frac{3}{5})]} \\ + \frac{γ g z^{2}}{E'} [\frac{z^{2}}{2 h^{2}} + \frac{3 b^{2}}{4 h^{2}} - (\frac{6}{5} + \frac{5}{4} υ')] \\ - \frac{γ g h^{2}}{4 E'} [\frac{1}{8} + υ' (\frac{3 b^{2}}{4 h^{2}} + \frac{1}{20})] \end{matrix}

(25)

Effect analysis of gravity action

Equations (21) and (25) show that v_G has a four-time relationship with z component under both boundary conditions. Take γ = 2400 kg/m³, g = 9.81 N/kg, E = 3.55×10¹⁰ Pa, and υ = 0.2 to analyze the two equations above, and results show that the vertical displacement under the action of slab gravity is almost independent of the depth since the thickness of the track slab is much smaller than the width. The maximum displacement in vertical direction appears at the middle of the cross-section if the track slab is under the boundary condition 1, and the maximum displacement difference Δv_G_max = 0.11 mm. If the track slab is under the boundary condition 2, the maximum displacement in vertical direction appears at both sides and with the value of Δv_G_max = 0.15 mm. Combined with the results above, displacement at upper surface is selectively analyzed in the following. Thus, comparisons of the vertical displacements after being affected by slab gravity for three types of temperature distributions are shown in Figure 9.

Figure 9.

Vertical displacement at the upper surface for three types of temperature distribution before and after gravity action: (a) positive temperature gradient and (b) negative temperature gradient.

Figure 9 shows that gravity action will partly restrict the warping deformation of the track slab. Among them, the degree of reduction in warping displacement under the positive temperature gradient is lower than the negative temperature gradient. k_G = Δv_G_max/Δv_T_max is defined as the rate of the warping displacement reduction affected by slab gravity. The reduction in warping displacement is higher if k_G is larger. Among the three types of temperature distributions, the maximum k_G corresponds to the exponential distribution, and the values are 0.174 and 0.714 under positive and negative temperature gradients, respectively. In addition, it is determined that the track slab remains unwarped if k_G≥ 1. Because of the representative form of the linear temperature distribution, it is applied to analyze the limitation of temperature gradient under gravity action. Simultaneous solution of equations (9) to (11) shows that if the track slab remains unwarped under the action of gravity, the critical temperature differences between the upper and lower surfaces with positive and negative temperature gradients are 2.6°C and –3.6°C, respectively, and the corresponding temperature gradients are 13°C/m and –18°C/m. Therefore, if the track slab affected by slab gravity remains unwarped, the temperature gradient should be maintained within the range of –18°C/m–13°C/m.

Effect of rail weight on slab warping

Similar to the slab gravity, the rail weight is also a factor that restricts warping deformation. Thus, it is necessary to analyze the warping displacement of the track slab under rail weight. An analytical model with rail weight is established as Figure 10 shows, where b₁ indicates the rail spacing. Figure 10 shows that the action of the rail on the track slab is a positive symmetrical load. Thus, the displacement field could be only analyzed in zone 0 ≤z≤b/2. Due to the rail weight being simplified to a concentrated force, it is necessary to perform the segmentation analysis of two parts, that is, 0 < z≤b₁/2 and b₁/2 < z≤b/2.

Figure 10.

Analytical model under the rail weight.

Boundary condition 1

With the constraint of boundary condition 1, the stress components σ_Py, σ_Pz, and τ_Pyz under rail weight can be approximated as

\begin{matrix} σ_{P y} = 0, σ_{P z} = {\begin{matrix} \frac{P y}{2 I} (b - b_{1}), (0 \leq z \leq b_{1} / 2) \\ \frac{P y}{2 I} (b - 2 z), (b_{1} / 2 < z \leq b / 2) \end{matrix}, \\ τ_{P y z} = {\begin{matrix} 0, (0 \leq z \leq b_{1} / 2) \\ \frac{P}{2 I} (\frac{h^{2}}{4} - y^{2}), (b_{1} / 2 < z \leq b / 2) \end{matrix} \end{matrix}

(26)

where I = h³/12 is the moment of inertia for the cross-section of the track slab. According to the physical equation, the strain components ε_Py, ε_Pz, and γ_Pyz under the rail weight are expressed as

\begin{matrix} ε_{P y} = {\begin{matrix} - \frac{P υ' y}{2 E' I} (b - b_{1}), (0 \leq z \leq b_{1} / 2) \\ - \frac{P υ' y}{2 E' I} (b - 2 z), (b_{1} / 2 < z \leq b / 2) \end{matrix}, \\ ε_{P z} = {\begin{matrix} \frac{P y}{2 E' I} (b - b_{1}), (0 \leq z \leq b_{1} / 2) \\ \frac{P y}{2 E' I} (b - 2 z), (b_{1} / 2 < z \leq b / 2) \end{matrix} \\ γ_{P y z} = {\begin{matrix} 0, (0 \leq z \leq b_{1} / 2) \\ \frac{P (1 + υ')}{E' I} (\frac{h^{2}}{4} - y^{2}), (b_{1} / 2 < z \leq b / 2) \end{matrix} \end{matrix}

(27)

Based on the continuity behavior, v_P and w_P should be unified when z = b₁/2. Thus, the expression of vertical displacement subjected to rail weight under the constraint of boundary condition 1 could be obtained as

v_{P} = {\begin{matrix} \frac{P}{4 E' I} [- υ' y^{2} (b - b_{1}) - z^{2} (b - b_{1}) + \frac{1}{12} ({b_{1}}^{3} + 2 b^{3} - 3 {b_{1}}^{2} b)], (0 \leq z \leq b_{1} / 2) \\ \frac{P}{4 E' I} [- υ' y^{2} (b - 2 x) - \frac{1}{3} z^{2} (3 b - 2 z) + \frac{1}{12} (6 {b_{1}}^{2} z + 2 b^{3} - 3 {b_{1}}^{2} b)], (b_{1} / 2 < z \leq b / 2) \end{matrix}

(28)

Boundary condition 2

With the constraint of boundary condition 2, the stress components under the rail weight can be approximated as

\begin{matrix} σ_{P y} = 0, σ_{P z} = {\begin{matrix} - \frac{P y}{I} (\frac{b_{1}}{2} - z), (0 \leq z \leq b_{1} / 2) \\ 0, (b_{1} / 2 < z \leq b / 2) \end{matrix}, \\ τ_{P y z} = {\begin{matrix} \frac{P}{2 I} (\frac{h^{2}}{4} - y^{2}), (0 \leq z \leq b_{1} / 2) \\ 0, (b_{1} / 2 < z \leq b / 2) \end{matrix} \end{matrix}

(29)

The strain components under the rail weight are expressed as

\begin{matrix} ε_{P y} = {\begin{matrix} \frac{P υ'}{E' I} (\frac{b_{1}}{2} - z), (0 \leq z \leq b_{1} / 2) \\ 0, (b_{1} / 2 < z \leq b / 2) \end{matrix}, \\ ε_{P z} = {\begin{matrix} - \frac{P y}{E' I} (\frac{b_{1}}{2} - z), (0 \leq z \leq b_{1} / 2) \\ 0, (b_{1} / 2 < z \leq b / 2) \end{matrix} \\ τ_{P y z} = {\begin{matrix} \frac{P (1 + υ')}{E' I} (\frac{h^{2}}{4} - y^{2}), (0 \leq z \leq b_{1} / 2) \\ 0, (b_{1} / 2 < z \leq b / 2) \end{matrix} \end{matrix}

(30)

Since the stress and strain components are both 0 at the part b₁/2 < z≤b/2, the vertical displacement of this part could be expressed as

v_{P} = v_{P (z = b_{1} / 2)} + \int_{b_{1} / 2}^{z} {\frac{\partial v_{P}}{\partial z} |}_{z = b_{1} / 2} dt

(31)

Therefore, the expression of the vertical displacement subjected to rail weight under the constraint of boundary condition 2 is obtained as

v_{P} = {\begin{matrix} \frac{P}{2 E' I} [\frac{2}{3} z^{3} - (z^{2} + υ' y^{2}) (z - \frac{b_{1}}{2})], (0 \leq z \leq b_{1} / 2) \\ \frac{P}{2 E' I} [\frac{b_{1}^{3}}{12} + (\frac{b_{1}^{2}}{4} - υ' y^{2}) (z - \frac{b_{1}}{2})], (b_{1} / 2 < z \leq b / 2) \end{matrix}

(32)

Effect analysis of rail weight

Take P = 588.6 N/m and b₁ = 1.5 m to analyze equations (28) and (32), and results show that the vertical displacement of the track slab under the rail weight is also independent of the slab depth. The distribution of the vertical displacement is similar to that under the effect of slab gravity. Thus, effect of rail weight on the warping deformation can be studied by the mode same as that in analyzing the effect of slab gravity. Analysis shows that vertical displacement of the track slab under rail weight is minimal if compared with that under slab gravity. The maximum displacement differences Δv_P_max corresponding to the boundary conditions 1 and 2 are 0.0096 mm and 0.0069 mm, respectively. Table 1 shows the maximum displacement differences of the track slab before and after imposing the rail weight, that is, Δv_T_max and Δv_T_max+Δv_P_max, respectively, where Δv_P_max represents the maximum displacement differences under the effect of rail weight.

Table 1.

Warping displacements before and after imposing the rail weight.

Types oftemperaturedistributions	Boundary condition 1			Boundary condition 2
	Δv_T_max (mm)	Δv_T_max+Δv_P_max (mm)	Displacementreduction (%)	Δv_T_max (mm)	Δv_T_max+Δv_P_max (mm)	Displacementreduction (%)
Linear	0.81	0.80	1.23	0.24	0.23	4.17
Quadratic	0.81	0.80	1.23	0.24	0.23	4.17
Exponential	0.63	0.62	1.59	0.21	0.20	4.76

Table 1 shows that the displacement reductions after imposing the rail weight are not more than 5% under the three types of temperature distributions. k_P = Δv_P_max/Δv_T_max is defined as the rate of the warping displacement reduction under rail weight. Results show that k_P is the largest when the type of temperature distribution is exponential, and the values are 0.015 and 0.033 under the positive and negative temperature gradients, respectively. Since the maximum of k_P is close to 0, it is determined that under the effect of non-uniform temperature, rail weight has less effect on the warping displacement of the track slab. Thus, if the track slab arise warping, it is not necessary to consider the effect of the rail weight.

Model verification

The reasonableness and usability of the analytical expressions are verified by an analysis model based on finite element method (FEM) in ANSYS package. The model for LCTPS is shown in Figure 11. The displacement of the track slab after warping is shown in Figure 12.

Figure 11.

The FEM model for LCTPS.

Figure 12.

The displacement after slab warping.

As Figure 11 shows, the model is mainly made up of rail, track slabs, CA mortar, and concrete base layer. First, the rail is established into two-node Timoshenko beam element with gravity by BEAM188. Track slabs and concrete base layer are established as 3-D solid elements in the FEM model to describe the shape of pre-cracking joints. The units are established as eight-node hexahedron elements by SOLID45 at high accuracy with a size of 0.1 m. To eliminate the boundary effect, track slabs are established into three pieces and the middle piece is taken for data extraction (shown in Figure 12). Then, the fastener system and the foundation structure are simulated as linear springs by COMBIN14. Due to the limited tensile strength of the mortar layer, the adhesion of the mortar layer is neglected. Therefore, to simulate the stiffness of CA mortar, constraints between the track slabs and the hydraulic material layer are simplified as nonlinear springs with generalized force–deflection (F–δ) capability by COMBIN39 in the FEM model. The F–δ curve is obtained by equivalent conversion of the elastic modulus for CA mortar (Zhou and Dai, 2015), as shown in Figure 13 (F is positive when the spring is subjected to drag force, and vice versa). Finally, the temperature field is applied to simulate the warming effect of the track slab.

Figure 13.

The F–δ curve of mortar layer.

The displacement curves at the upper surface for the three types of temperature with positive and negative temperature gradient under the effect of both slab gravity and rail weight by the analytical expressions and FEM model are shown in Figure 14.

Figure 14.

Vertical displacement at the upper surface for three types of temperature by the analytical expressions and FEM model: (a) linear distribution, (b) quadratic distribution, and (c) exponential distribution.

Results show that the shapes of warping displacement are almost the same for both methods, but there was a better agreement if temperature distribution is linear type. The deformation of slab warping under the FEM model is a little higher because the presence of pre-cracking joints in FEM model releases a certain amount of temperature stress. The maximum value of displacement under analytical expressions and FEM model and the error analysis are shown in Table 2.

Table 2.

Maximum value of displacement.

Types oftemperaturedistributions	Maximum displacement under positive temperature gradient (mm)			Maximum displacement under negative temperature gradient (mm)
	Analyticalexpression	FEM model	Displacementerror	Analyticalexpression	FEM model	Displacementerror
Linear	0.78	0.79	0.01	0.15	0.19	0.04
Quadratic	0.77	0.88	0.11	0.16	0.25	0.09
Exponential	0.59	0.70	0.11	0.13	0.18	0.05

FEM: finite element method.

The displacement error between analytical expression and FEM model are almost within the range of 0.1 mm for the three types of temperature distribution. Therefore, it was verified that the analytical expressions are satisfied with further analysis of the slab warping under the temperature effect.

Conclusion

This article presents the analytical expression of slab warping for LCPST under non-uniform temperature field, which could be generalized to longitudinal continuous structures. Through the theoretical researches, some effecting factors such as temperature distribution, slab gravity, and rail weight are discussed. The following conclusions can be drawn:

If only affected by temperature field, the displacements in the x and z directions are linearly related to the respective components during the warping deformation. The displacement in y direction has a quadratic relationship with x and z components.

Temperature gradient is the pivotal factor to lead the warping of the track slab. Track slabs under positive and negative temperature gradients correspond to boundary conditions 1 and 2, respectively. The type of temperature distribution has less effect on the warping displacement.

Slab gravity will partly restrict the warping deformation of the track slab. If the track slab remains unwarped by slab gravity, the temperature gradient should be maintained within the range of –18°C/m–13°C/m.

Rail weight has less effect on the slab warping. Thus, if the track slab undergoes warping deformation, it is not necessary to consider the effect of the rail weight.

The displacement error between analytical expression and FEM model are almost within the range of 0.1 mm. Thus, the analytical expression could quantitatively analyze the warping characteristics of the track slab.

It is worthwhile to point out that this work has established an analytical expression for the slab warping of LCPST under the effect of non-uniform temperature field and provided some insights into the quantitative analysis of the factors such as temperature distribution, slab gravity, and rail weight. In future work, influence of temperature distribution on the boundary conditions during the warping process needs to be further analyzed to clarify the regulation of stress variation in the track slab, and effect of the slab warping on the vehicle dynamic response should be further considered to clarify whether the slab warping will affect the safety during the operation process.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially sponsored by the National Natural Science Foundation of China (Projects U1434208, 51778543, U1534203, and 51678506).

ORCID iD

Zui Chen

References

Al Hamd

RKS

Gillie

Warren

, et al. (2018) The effect of load-induced thermal strain on flat slab behaviour at elevated temperatures. Fire Safety Journal 97: 12–18.

Beckemeyer

Khazanovich

(2002) Determining amount of built-in curling in jointed plain concrete pavement: case study of Pennsylvania 1–80. Transportation Research Record: Journal of Transportation Research Board 1809(1): 85–92.

Cao

Yang

, et al. (2016) Damage mechanism of slab track under the coupling effects of train load and water. Engineering Fracture Mechanics 163: 160–175.

Chen

Qin

(2012) Influence of pile cap effect in piled embankment supporting high-speed railway. Advances in Structural Engineering 16(8): 1447–1456.

Chen

Xing

, et al. (2015) Effects of the interaction between beam and rail on slab warping on subgrade-bridge transition section. Journal of Railway Engineering Society 32(11): 37–42. (in Chinese)

Feng

Fang

, et al. (2013) Study on the mechanical behavior of lining structure for underwater shield tunnel of high-speed railway. Advances in Structural Engineering 16(8): 1381–1400.

Han

Zhao

Xiao

, et al. (2015) Effect of softening of cement asphalt mortar on vehicle operation safety and track dynamics. Journal of Zhejiang University-science A 16(12): 976–986.

Janssen

Snyder

(2000) Temperature-moment concept for evaluating pavement temperature data. Journal of Infrastructure Systems 6(2): 81–83.

Lange

Jansson

(2014) A comparison of an explicit and an implicit transient strain formulation for concrete in fire. Fire Safety Science 11: 572–583.

10.

Lichtberger

(2005) Track Compendium: Formation, Permanent Way, Maintenance, Economics. Hamburg: Eurail Press.

11.

Liu

Zeng

, et al. (2016) Study on temperature field of continuous ballastless track for high-speed railway. Journal of the China Railway Society 38(12): 86–93. (in Chinese)

12.

Liu

Zhao

(2013) Analysis of early gap between layers of CRTS II slab ballastless track structure. China Railway Science 34(4): 1–7. (in Chinese)

13.

Long

Liu

, et al. (2018) Development of high performance self-compacting concrete applied for filling layer of high-speed railway. Journal of Materials in Civil Engineering 30(2): 04017268.

14.

Ren

Zhao

(2015) Study on anchoring scheme of lifted track slab in repairing of CRTS II slab-type ballastless track. Railway Engineering 2: 132–135. (in Chinese)

15.

Sun

Cheng

(2014) Analysis on temperature field of ballastless track structure based on meteorological data. Journal of the China Railway Society 36(11): 106–112. (in Chinese)

16.

Ren

Deng

Jin

, et al. (2017) Energy method solution for the vertical deformation of longitudinally coupled prefabricated slab track. Mathematical Problems in Engineering 2017(1): 8513240.

17.

Ren

Deng

Wei

, et al. (2019) Mechanical property deterioration of the prefabricated concrete slab in mixed passenger and freight railway tracks. Construction and Building Materials 208: 622–637.

18.

Song

Zhao

Zhu

(2016) Temperature-induced deformation of CRTS II slab track and its effect on track dynamical properties. Science China Technological Sciences 57(10): 1917–1924.

19.

Tian

Zhang

Xia

(2016) Temperature effect on service performance of high-speed railway concrete bridges. Advances in Structural Engineering 20(6): 865–883.

20.

Timoshenko

Goodier

(1970) Theory of Elasticity. 3rd ed. New York City: McGraw-Hill.

21.

Vandenbossche

(2006) Quantifying built-in construction gradients and early-age slab shape to environmental loads for jointed plain concrete pavements. International Journal for Pavement Engineers 7(4): 275–289.

22.

Wang

You

Wang

, et al. (2010) Research on the slab temperature warping of the unit slab track system. China Railway Science 31(3): 9–14. (in Chinese)

23.

Wang

Shao

, et al. (2016) Plate-lift filling double-layer mortar technology for rapidly restoring subgrade settlement of CRTS I slab track. China Railway Science 37(4): 28–33. (in Chinese)

24.

Liu

Zeng

, et al. (2016) Research on the temperature field characteristic of CRTSII slab ballastless track. Journal of Railway Engineering Society 33(3): 29–33. (in Chinese)

25.

Yan

Dai

Guo

, et al. (2015) Longitudinal force in continuously welded rail on long-span tied arch continuous bridge carrying multiple tracks. Journal of Central South University 22(5): 2001–2006.

26.

Yang

Zeng

Liu

, et al. (2015) Stability of CRTS-II slab track based on energy norm. Journal of Central South University (Science and Technology) 2015(12): 4707–4712. (in Chinese)

27.

Yang

Zhang

Gao

, et al. (2016) Temperature warping and it’s impact on train-track dynamic response of CRTSII ballastless track. Engineering Mechanics 33(4): 210–217. (in Chinese)

28.

Yang

Kang

, et al. (2017) Temperature characteristics analysis of the ballastless track under continuous hot weather. Journal of Transportation Engineering, Part A: Systems 143(9): 04017048.

29.

Zhou

Dai

(2015) Stability of longitudinally connected ballastless slab track on simply-supported beam bridges of high-speed railway. Journal of the China Railway Society 37(8): 60–65. (in Chinese)