Spatial mapping relationship model of subgrade disease and track geometry for high-Speed railway

Abstract

A highly efficient and accurate mapping relationship model of subgrade disease and track geometry is significant for determining criterion of subgrade deformation and evaluating operation safety of High-speed Railway. In this paper, an innovative spatial mapping relationship model (SMRM) is proposed for describing track spatial geometry under subgrade disease of differential settlement and frost heave. Starting from analysing the theoretical interaction mechanism between subgrade and ballastless track, and presenting the flowchart of SMRM and its implementation process are described in detail. The good accuracy and applicability of SMRM method is verified based on the finite element model (FEM) and scale model experiment. The results indicate that it can better simulate the spatial characteristic of above mapping relationship than traditional theoretical model and avoid complicated and time-consuming FEM analysis process. It has high accuracy and reliability for the most application scenarios. The proposed method also can analyse the influences of others’ diseases on track geometry as long as the disease description function is available.

Keywords

high-speed railway subgrade spatial deformation mapping relationship track geometry track deformation

Introduction

The track geometry is an important issue for ensuring operation safety of High-speed railway (HSR) (Guo et al., 2018a; Guo et al., 2018b). The subgrade diseases is one kind of most significant effecting factor on track geometry (Guo, 2018; Ruge et al., 2009; Salcher et al., 2016; Vega et al., 2012). Some common subgrade diseases due to exploitation of underground resources (groundwater, petroleum, natural gas, et al.) and freeze-thaw cycle effect in some frigid regions may lead to local deformation for subgrade structure (Kang, 2016; Sheng et al., 2014). The deformation transferred to rail has a significant impact on track geometry (Gou et al., 2021a). Additionally, the rail deformation will increase under long time running trains, and magnify the operation risk of HSR (Zhu and Cai, 2014).

Some numerical simulation research for the effect of subgrade diseases on track geometry and safe operation of HSR has been performed (Nielsen and Li, 2018; Shan et al., 2020, 2021; Varandas et al., 2020). Jiang et al. (2019) studied the mapping relationship of subgrade differential settlement and track by building a 3D finite element model (FEM), which was validated with a full-scale physical model testing. Ramadan et al. (2021) performed a parametric study on the influence of subgrade differential settlement on rail conditions by FEM method. An integration method was introduced to simulate the dynamic response of HSR under moving trains considering subgrade diseases, and the results showed that subgrade disease is an important risk factor for comfort and safety operation of HSR (Zhang et al., 2016a). Besides numerical studies, some model experiments and field tests were also performed to study the effect of subgrade disease on HSR (Wang et al., 2015; Zhou et al., 2020a). The above literature indicates that numerical method is suitable for investigating effect of subgrade disease on HSR, and related scale model and field experiments also can provide reliable data for further theoretical research. However, generally professional engineers need to spend dozens of hours even several days on building FEM, computing and checking data to obtain final result with numerical simulation method. The model test for the above problem would consume more time and money. A higher efficient method for solving prementioned problem is necessary, and some related works have been performed (Guo et al., 2018b; Jiang et al., 2020). Guo et al. (2018a) established a mapping relationship model for the vertical deformation interaction of subgrade and track, which considered the contact condition of track slab and subgrade based on Winkler elastic foundation laminated beam theory. A theoretical model for transmitting bridge pier settlement to track deformation of HSR was presented in literature (Chen et al., 2015). Gou et al. (2019) proposed an analytical formula for mapping bridge vertical deformation to track geometry considering time-varying conditions. In addition to the track deformation, its irregularity can be obtained through a recursive Bayesian Kalman filtering method, based on vehicle dynamic responses (Xiao et al., 2021, 2022). Based on those theoretical methods, the safety operation of HSR was effectively analyzed by superposing the rail deformation due to structure diseases and the track random irregularity (Gou et al., 2021a). The presented analytical methods provide theoretical support for further studies. However, they mainly focused on the longitudinal profile of subgrade and rail, and the spatial effect of subgrade deformation was rarely considered. Actually, the track deformation often appears a spatial characteristic with elevation difference between left and right rail, when the subgrade suffers from local frost heave, or is located at slopes, mountainous areas or other spatial deformation diseases. Existing analytical methods are not enough to describe above spatial deformation feature well. Therefore, a novel analytical method is needed for the purpose of describing subgrade-track spatial deformation interaction with high accuracy and efficiency.

In this paper, an innovative spatial mapping relationship model (SMRM) of subgrade deformation and track geometry is proposed based on the subgrade and track structure interaction mechanism analysis under typical subgrade spatial diseases such as differential settlement and frozen heaven. Compared to traditional mapping relationship model, the proposed SMRM method in the present study can reveal the rail spatial deformation characteristic under some typical subgrade diseases, and describe mapping relationship more comprehensively on the spatial scale. The SMRM method is verified by FEM simulation and scale model experimental results, and can be used in wide engineering scenarios for HSR projects in higher efficiency.

Spatial mapping relationship model for HSR subgrade-track structure

A widely used typical subgrade structure in HSR projects of China is chosen. The detailed components of subgrade-track structure are shown in Figure 1, and the rail and track slab are connected by the fastener system. The track slab and concrete base slab are bonded by mortar layer. Additionally, the base slab is directly laid on the subgrade structure.

Figure 1.

Diagram of HSR subgrade-track structure and their local coordinates.

To establish the SMRM, three local coordinate systems for the subgrade-track structure are built as shown Figure 1. For the local coordinate (X_r, Y_r, Z_r) of rail module, the end section centroid of rail is defined as the origin of coordinate shown as Figure 1(a). The origin of local coordinate (X_s,i, Y_s,i, Z_s,i) for the slab module located at a corner of slab neutral plane as shown in Figure 1(b). Then, one corner of supposed rectangular deformation area on the subgrade surface is taken as the origin of subgrade local coordinates (X_g, Y_g, Z_g), which is shown in Figure 1(c).

In this study, the technical flowchart of establishing SMRM can be described as Figure 2. Firstly, the related parameters data is imported, and analytical models for subgrade, slab and rail are built respectively. Then, the subgrade deformation is calculated considering typical diseases. The spatial position and deflection of the slab model depends on the deformation of subgrade. Additionally, the deflection of slab is also affected by the interaction between rail and slab, and the spatial geometry of the rails can be obtained based on above interaction mechanism.

Figure 2.

Technology flowchart of SMRM method.

Analytical model of rail structure

For establishing the rail analytical model, some preconditions should be introduced:

(1) The steel rail is modelled by a Euler beam structure through mechanical analysis (Gou, Xie, et al., 2021b).

(2) The boundary effect of rail ends can be eliminated through taking sufficient length of rail analytical part (Zhou et al., 2020a).

(3) The fastener system is considered as a linear spring with lateral and vertical stiffness. It can provide lateral and vertical support for the rail, and transfer force between track slab and rail.

Therefore, A Euler beam with multipoint elastic support is taken as analytical model of rail, as shown in Figure 3. The spatial deformation of rails can be decomposed into vertical and transverse deformation.

Figure 3.

Analysis diagram of rail structure.

According to current studies (Gou et al., 2019; Gou et al., 2021b), the vertical deformation ${Z_{r,}}_{i}$ at the position of ith fastener can be expressed as equation (1):

{Z_{r,}}_{i} = x_{i} \sum_{j = 1}^{s u m} \frac{l_{r}^{2} (l_{r} - x_{j}) - {(l_{r} - x_{j})}^{3}}{6 E_{r} I_{rz} l_{r}} F_{fz, j} - x_{i}^{3} \sum_{j = 1}^{s u m} \frac{(l_{r} - x_{j})}{6 E_{r} I_{rz} l_{r}} F_{fz, j} + \sum_{j = 1}^{i} \frac{{(x_{i} - x_{j})}^{3}}{6 E_{r} I_{rz}} F_{fz, j}

(1)

Where, the length of rail for analytical process is

l_{r}

. It is noted that the rail analytical part

l_{r}

should be larger than the longitudinal length of subgrade deformation area, as the subgrade deformation can affect the adjacent rails.

x_{i}

and

x_{j}

is the position of the ith and jth fastener respectively in the rail local coordinate;

F_{fz, j}

is the additional force from the jth fastener.

E_{r}

and

I_{rz}

is the elastic modulus and vertical moment of inertia of the rail. Then, the equation (1) can be rewritten as a matrices form:

Z_{r} = K_{rz} F_{fz}

(2)

Where:

F_{fz} = {[F_{fz, 1} F_{fz, 2} \dots F_{fz, j}]}^{T} K_{rz} = [\begin{array}{c} k_{1,1} & k_{1,2} & \dots & k_{1, j} \\ k_{2,1} & k_{2,2} & \dots & k_{2, j} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ k_{i, 1} & k_{i, 2} & \dots & k_{i, j} \end{array}] Z_{r} = {[Z_{r, 1} Z_{r, 2} \dots Z_{r, j}]}^{T}

Where

F_{fz}

Z_{r}

and

K_{rz}

are the fastener force, vertical deformation, and vertical deformation influence matrix of the rail respectively. The element

k_{i, j}

K_{rz}

can be expressed as:

\begin{array}{l} k_{i, j} = \frac{[l_{r}^{2} (l_{r} - x_{j}) - {(l_{r} - x_{j})}^{3} - {x_{i}}^{2} (l_{r} - x_{j})] x_{i}}{6 E_{r} I_{rz}}, i \geq j \\ k_{i, j} = \frac{[l_{r}^{2} (l_{r} - x_{j}) - {(l_{r} - x_{j})}^{3} - {x_{i}}^{2} (l_{r} - x_{j})] x_{i}}{6 E_{r} I_{rz}} + \frac{{(x_{i} - x_{j})}^{3}}{6 E_{r} I_{rz}}, i < j \end{array}

So far, the analytical model of rail vertical deformation is established.

The lateral deformation analytical model of rail considering lateral fastener force matrix $F_{fy}$ and lateral deformation influence matrix $K_{ry}$ can be obtained by replacing vertical moment of inertia $I_{rz}$ with lateral moment of inertia $I_{ry}$ in equation (1):

Y_{r} = K_{ry} F_{fy}

(3)

Analytical model of composite slab

The slab portion of ballastless track structure, including track slab, mortar layer (or self-compacting concrete filling layer). The track slab and base slab both are concrete structure. They are contact with interlayer of mortar and are bonded by the adhesion of mortar layer. Because of this structure form, the ballastless track structure is highly integrated (Guo, 2018; Li, 2010). This portion can be regarded as a composite structure as shown in Figure 4. Based on the mechanical analysis of this composite structure, some critical parameters can be calculated for establishing composite slab analytical model (Budynas, 1977; Huang et al., 2010). The neutral plane height of composited slab $z^{*}$ is determined as follows:

z^{*} = \frac{A_{1} z_{1} n_{1} + A_{2} z_{2} n_{2} + A_{3} z_{3}}{A_{1} n_{1} + A_{2} n_{2} + A_{3}}

(4)

Where A₁, A₂ and A₃ are the sectional areas of track slab, mortar layer and base slab respectively, and z₁, z₂ and z₃ are their centroid positions respectively, E₁, E₂ and E₃ are their elastic modulus. n₁ and n₂ are the proportional coefficients of elastic modulus, where

n_{1} = E_{1} / E_{3}

n_{2} = E_{2} / E_{3}

Figure 4.

Schematic diagram of composition structure.

The equivalent bending stiffness of the composited slab is:

E^{*} I^{*} = (\frac{1}{12} A_{1} {h_{1}}^{2} + n_{1} A_{1} {a_{1}}^{2} + \frac{1}{12} A_{2} {h_{2}}^{2} + n_{2} A_{2} {a_{2}}^{2} + \frac{1}{12} A_{3} {h_{3}}^{2} + A_{3} {a_{3}}^{2}) E_{3}

(5)

Where,

h_{1}

h_{2}

and

h_{3}

are the thickness of those three layers.

a_{1}

a_{2}

and

a_{3}

are the distances from the neutral planes of above three layers to the equivalent neutral plane, which can be obtained by subtracting

z^{*}

from z1, z2 and z3. According to equation (4) and equation (5) the equivalent elastic modulus

E^{*}

, the equivalent thickness

h^{*} = 2 z^{*}

and equivalent width

b = (A_{1} + A_{2} + A_{3}) / h^{*}

of the equivalent composite slab are determined.

To clarify the mapping relationship of subgrade and track deformation, the time-varying effect, such as temperature change and other hybrid effect are not considered in SMRM method. The above composite slab structure can be regarded as an elastic plate with free boundary, and its foundation can be considered as a winker elastic foundation. Therefore, the structure composed of composite plate and subgrade can be simulated by an elastic foundation plate (EFP) with free boundary as shown in Figure 5.

Figure 5.

Structural diagram of elastic foundation plate.

The governing equation of the above EFP under bending is expressed as:

D \nabla^{2} \nabla^{2} z + κ_{s} z = η (x, y)

(6)

Where

D

is the bending stiffness of the plate,

D = \frac{E^{*} {h^{*}}^{3}}{12 (1 - μ)}

;

μ

is the poisson ratio;

\nabla^{2}

is the Laplace operator,

\nabla {}^{2}= (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}})

;

κ_{s}

denotes the foundation stiffness.

η (x, y)

is the load generated by the fastener force and the subgrade deformation effect.

On the edge of plate, the values of bending moment and shear force are zero:

{\begin{array}{c} M_{x} = - D (\frac{\partial^{2} z}{\partial x^{2}} + v \frac{\partial^{2} z}{\partial y^{2}}) = 0 \\ V_{x} = - D [\frac{\partial^{3} z}{\partial x^{3}} + (2 - v) \frac{\partial^{3} z}{\partial x \partial y^{2}}] = 0 \end{array}

(7)

The equations (6) and (7) can be presented by polynomial approximation method (Lin et al., 2017):

L {z} + λ z = F (x, y), (x, y) \in Ω

(8)

{\begin{array}{c} B_{1} {z} = G (x, y), (x, y) \in \partial Ω \\ B_{2} {z} = H (x, y), (x, y) \in \partial Ω \end{array}

(9)

Where

L

is a linear partial differential operator,

λ = κ_{s} / D

B_{1}

and

B_{2}

are boundary operators,

Ω

is the solution domain with boundary

\partial Ω

. The boundary function can be derived by equation (7).

In this study, the solution for bending problem of plate is approximated by the sum of the basis functions (Lin et al., 2017):

z (x, y) = \sum_{i = 1}^{l} α_{i} u_{p i} (x, y)

(10)

Where the basis function

u_{p i} (x, y) = \frac{1}{λ} \sum_{t = 0}^{s} {(- 1)}^{t} {(\frac{L}{λ})}^{t} x^{m} y^{n}

; s is defined as the highest order of polynomial and

s = m + n

α_{i}

are collocation coefficients, l is the sum number of basis functions. The basis function

u_{p i} (x, y)

satisfies non-homogeneous differential equations and its inhomogeneous terms is polynomial:

L u_{p i} (x, y) + λ u_{p i} (x, y) = b_{i}^{s} (x, y)

(11)

Where

b_{i}^{s} (x, y) = {x^{m - n} y^{j} : 0 \leq n \leq m, 0 \leq m \leq s, 1 \leq i \leq l .}

The collocation coefficients $α_{i}$ , a set of $n_{k}$ as distinct points in solution domain and a set of $n_{b}$ points on the boundary are introduced into the equations (8) and (9):

{\begin{array}{c} L {\sum_{i = 1}^{l} α_{i} u_{p i} (x_{k}, y_{k})} + λ {\sum_{i = 1}^{l} α_{i} u_{p i} (x_{k}, y_{k})} = F (x_{k}, y_{k}), k = 1,2 . . ., n_{k} \\ B_{1} {\sum_{i = 1}^{l} α_{i} u_{p i} (x_{b}, y_{b})} = G (x_{b}, y_{b}), b = n_{k} + 1, n_{k} + 2 . . ., n_{k} + n_{b} \\ B_{2} {\sum_{i = 1}^{l} α_{i} u_{p i} (x_{b}, y_{b})} = H (x_{b}, y_{b}), b = n_{k} + 1, n_{k} + 2 . . ., n_{k} + n_{b} \end{array}

(12)

The equation (12) is rewritten in matrix form:

K A = C

(13)

Where:

K = [\begin{array}{c} {[b_{i}^{s}]}_{n_{i} \times l} \\ {[B_{1} u_{p i}]}_{n_{b} \times l} \\ {[B_{2} u_{p i}]}_{n_{b} \times l} \end{array}], A = [\begin{array}{c} a_{1} \\ a_{2} \\ . . . \\ a_{l} \end{array}], C = [\begin{array}{c} [F_{n_{i}}] \\ [G_{n_{b}}] \\ [H_{n_{b}}] \end{array}]

As for the linear system of equation (13), $K$ is high order polynomial coefficient matrix, and has high condition number. This high condition number may have adverse effect on the solution of linear system (Lin et al., 2017). To enhance the stability of equation (13) and make it has satisfactory solution, a robust multiple-scale technique is employed (Dangal et al., 2017). Assuming a pre-conditioning matrix $P = d i a g (p_{1}, p_{2}, . . ., p_{l})$ , and the equation (13) is changed into a new equation:

K P^{- 1} P A = C

(14)

Where:

p_{i} = \sqrt{\sum_{k = 1}^{n_{i} + 2 n_{b}} {A_{k, l}}^{2}}, i = 1, 2, . . ., l

And the

A_{k, l}

indicate the element of matrix

K

in the k-th row and l-th column. Therefore, the collocation coefficient matrix

A

can be obtained by solving the following equation:

K^{'} A^{'} = C

(15)

Where

K^{'} = K P^{- 1}

A^{'} = P A

. The collocation coefficient matrix

A

can be obtained by timing

P^{- 1}

A^{'}

The deformation function $z (x, y)$ of the EFP is obtained by substituting collocation coefficient matrix $A$ into the deflection function equation (10).

The study by Su (2014) shows a full-scale experiment of pushing track slab and analysis the mechanical under extreme lateral force. It indicated that the maximum lateral deformation of the track slab is 0.04 mm under 120kN∼151kN external force. In this case, mortar layer of slab structure is yielded and structural failure. The structural failure is not considered in SMRM. The value of lateral deformation (0.04 mm) is tiny number compared with the vertical deformation under typical subgrade diseases (e.g., 20 mm amplitude of subgrade differential settlement) and its impact on running performance of the high-speed train can be negligible. Therefore, the proposed composite slab model in the present study does not consider the lateral deformation of the slab structure.

Mathematical model of subgrade spatial deformation

The deformation of the subgrade can be caused by several reasons, such as arching up due to soil frost heave, subsidence and lateral uneven settlement due to locating in slope area. Among those factors, subgrade differential settlement and frost heave are two common diseases for HSR subgrade. We mainly consider those two diseases in this study.

Subgrade settlement

The local longitudinal settlement of subgrade often is simulated with concave full-wave cosine curve function (Guo et al., 2018b), as shown in Figure 6:

z_{sl} (x) = \frac{A}{2} (1 - \cos (\frac{2 π x}{L}))

(16)

Where

z_{sl} (x)

is the vertical settlement value at different x coordinates; x is the longitudinal coordinate of the deformation curve under the global coordinate; L is the wavelength of the curve; and A is the settlement amplitude of this differential settlement.

Figure 6.

Subgrade vertical differential settlement by cosine function.

In this study, a vertical deformation model of subgrade settlement considering spatial distribution is proposed:

ψ_{s} (x, y) = - \frac{A}{4} (1 - \cos (\frac{2 π x}{s_{sl}})) (1 - \cos (\frac{2 π (y - y_{o})}{s_{st}}))

(17)

Where, x and y are the coordinates of the longitudinal and lateral direction respectively;

s_{l}

and

s_{t}

are the longitudinal and lateral wavelength of deformation surface.

y_{o}

is the abscissa of the peak point for settlement surface, and A is the deformation amplitude.

Subgrade arching caused by frost heave

The current deformation curve model of subgrade due to frost heave for HSR mainly focuses on vertical deformation along longitudinal section (Gao et al., 2020). However, the deformation of subgrade due to frost heave usually is related closely with the distribution of temperature field (Zhang et al., 2016b), and that leads to transversal unevenness of deformation for subgrade. A new deformation curve model can present the spatial deformation feature of subgrade due to frost heave and distribution of temperature field is necessary.

Figure 7 is from the observed data of subgrade temperature field distribution in the literature (Zhang et al., 2016b), and it significantly shows that the isotherm can be fitted with cosine curve. According to the relationship between temperature field distribution and deformation of subgrade, this transversal unevenness deformation is described by a cosine function. Additionally, the soil temperature field affected by asymmetric thermal regime of embankment and groundwater migration, may lead to the change of peak point of frost heave in transversal section. Considering all the above conditions, a cosine function $z_{ht} (y)$ with a variable position of peak point is proposed to describe the transversal unevenness deformation of subgrade:

z_{ht} (y) = \frac{A_{h}}{2} (1 - \cos (\frac{2 π (y - y_{o})}{s_{ht}}))

(18)

Figure 7.

Temperature field distribution of subgrade in cold regions (unit: °C).

Then, a spatial deformation model for subgrade of HSR is developed:

ψ_{h} (x, y) = \frac{A_{h}}{4} (1 - \cos (\frac{2 π x}{s_{hl}})) (1 - \cos (\frac{2 π (y - y_{o})}{s_{ht}}))

(19)

Where,

s_{hl}

and

s_{ht}

are the longitudinal and lateral wavelength of the deformation surface.

y_{o}

is the transversal coordinate of the peak point for the settlement surface, and

A_{h}

is the amplitude value of deformation.

Since subgrade settlement and frost heave have similar functional forms, a unified function $ψ (x, y, D)$ is proposed to describe deformation of subgrade. The fixed parameter vector $D$ is used to describe deformation feature. For example, the vector $D$ of frost heave contains the longitudinal and transversal wavelengths, amplitudes and peak point position ( $s_{hl}$ , $s_{ht}$ , $A_{h}$ , $y_{o}$ ). The subgrade disease can be quantized by function $ψ (x, y, D)$ . This part can be considered as the fixed boundary with special enforced displacement applied on the bottom of entire mapping relationship model. And the displacement features can be adjusted according to specific subgrade disease.

In addition to those typical diseases, the proposed SMRM method also can be used to solve the subgrade-track interaction of other complex diseases, as long as subgrade deformation function is available.

Spatial mapping relationship model between subgrade deformation and track geometry

The SMRM between subgrade deformation and track geometry is integrated based on the above established models for each component including the rail, composited slab and subgrade, and deformation interaction among them.

Location for the spatial datum of EFP

The complex spatial deformation characteristic of subgrade affects the spatial position of slab on subgrade, and the displacement of slab follows subgrade deformation. Therefore, it is necessary to obtain the datum plane position of each composited slab in the global coordinate after the subgrade deformation is determined. Generally, the subgrade deformation of HSR mainly happens along longitudinal section, and secondary occurs along transversal section of railway. Therefore, the spatial deformation of subgrade is decomposed into two planes, and determined by superposing deformation from longitudinal section and lateral inclination angle of the datum plane. The tool named Notional Reticle is created to obtain the spatial position of datum plane, and is constructed by the horizontal and longitudinal centerlines together on the datum plane as shown in Figure 8. The details of Notional Reticle method used to calculate the position of datum plane is introduced as follows. The position can be calculated via this tool.

Figure 8.

Location diagram of datum plane in spatial scale.

Firstly, the projection reticle is defined as projection of notional reticle on the deformation area in global coordinates. The positions of four end points for projection reticle are calculated as follows:

{\begin{cases} X_{g, i}^{n} = (i - 1) (a + d_{inter}) + x_{start}, n = 1 \\ X_{g, i}^{n} = i \cdot (a + d_{inter}) + x_{start} - d_{inter}, n = 3 \\ X_{g, i}^{n} = (i - 1) (a + d_{inter}) + x_{start} + a / 2, n = 2, 4 \end{cases}

(20)

{\begin{cases} Y_{g, i}^{n} = y_{slab}, n = 1, 3 . \\ Y_{g, i}^{n} = y_{slab} + b / 2, n = 2 . \\ Y_{g, i}^{n} = y_{slab} - b / 2, n = 4 . \end{cases}

(21)

Z_{g, i}^{n} = ψ (X_{g, i}^{n}, Y_{g, i}^{n}), n = 1, 2, 3, 4 .

(22)

Where, a and b are the length and width of EFP respectively.

d_{inter}

is the distance between adjacent EFPs.

x_{start}

is the x-coordinate of the first EFP in global coordinate.

y_{slab}

is the y-coordinate of projection reticle’s longitudinal centerline.

Then, $y_{slab}$ is substitute into deformation description function $ψ (x, y)$ , which can be simplified as a unary function $ψ (x, y_{s l a b})$ of variable x. The gradient of function $ψ (x, y_{s l a b})$ is diverse within the deformation range, and it increase the difficulty of solving EFP position. Therefore, the domain of $ψ (x, y_{s l a b})$ is divided into three regions and discussed respectively:

(1) The “convex” region where the second-order derivative of function $ψ (x, y_{s l a b})$ is greater than zero;

(2) The “concave” region where the second-order derivative of function $ψ (x, y_{s l a b})$ is less than zero;

(3) The “s-shaped” region where the second-order derivative of function $ψ (x, y_{s l a b})$ exists the zero point.

The notion of three cases are shown in Figure 9. The subgrade surface touch datum plane at $(X_{gt, i}, Y_{gt, i}, Z_{gt, i})$ , and the approach of calculating touch point position depends on different cases.

Figure 9.

Three cases for different relative positions of subgrade and EFP: (a) The “convex” region; (b) The “concave” region; (c) The “s-shaped” region.

Employed the second derivative of $ψ (x, y_{s l a b})$ , region types underlayer certain EFP can be obtained as follows:

(1) If the ith EFP placed on convex region, second derivative of $ψ (x, y_{s l a b})$ satisfies equation (23):

\frac{\partial ψ^{2} (x, y_{s l a b})}{\partial x^{2}} > 0, x \in (X_{g, i}^{1}, X_{g, i}^{1} + a)

(23)

The ith EFP should be placed above subgrade surface, under the action of gravity and foundation support. And the slope of the longitudinal centerline can be calculated:

\tan (R_{sy, i}) = \frac{Z_{g, i}^{3} - Z_{g, i}^{1}}{X_{g, i}^{3} - X_{g, i}^{1}}

(24)

Draw a tangent line through a point on the profile of $ψ (x, y_{s l a b})$ , that is touch point, with same slope of $\tan (R_{sy, i})$ . The x-coordinate of this point can be obtained:

\frac{\partial ψ (X_{go, i}, y_{s l a b})}{\partial x} = \frac{Z_{g, i}^{3} - Z_{g, i}^{1}}{X_{g, i}^{3} - X_{g, i}^{1}}

(25)

And this tangent line is the longitudinal centerline of datum plane, which is expressed as:

z = \tan (R_{sy, i}) \cdot (x - X_{gt, i}) + ψ (X_{gt, i}, y_{s l a b})

(26)

(2) If the ith EFP placed on concave region, the second derivative of $ψ (x, y_{s l a b})$ satisfies equation (27):

\frac{\partial ψ^{2} (x, y_{s l a b})}{\partial x^{2}} < 0, x \in (X_{g, i}^{1}, X_{g, i}^{1} + a)

(27)

In this case, the ith EFP is embedded in deformation area. And the method for calculating the slope $\tan (R_{sy, i})$ is the same as that of convex region case, which can be obtained by equation (28). Then, the longitudinal centerline is expressed as:

z = \tan (R_{sy, i}) \cdot (x - X_{g, i}^{3}) + Z_{g, i}^{3}

(28)

(3) If the ith EFP placed on “s-shaped” region, the second derivative of $ψ (x, y_{s l a b})$ satisfies equation (29):

\max [\frac{\partial ψ^{2} (x, y_{s l a b})}{\partial x^{2}}] > 0, and \min [\frac{\partial ψ^{2} (x, y_{s l a b})}{\partial x^{2}}] < 0; x \in (X_{g, i}^{1}, X_{g, i}^{1} + a)

(29)

The longitudinal centerline of notional reticle has two endpoints, and one of it is located in the concave region of $ψ (x, y_{s l a b})$ profile with lowest position as a touchpoint $(X_{gt, i}^{low}, Y_{gt, i}^{low}, Z_{gt, i}^{low})$ , which can be calculated by the equation (20), (21), (22). The other touchpoint is tangent point of function $ψ (x, y_{s l a b})$ touched with centerline as shown in Figure 9(c). It just likes the EFP “reclines” on the subgrade surface. And the position of tangent point of $ψ (x, y_{s l a b})$ profile and centerline can be derived:

\frac{(Z_{gt, i} - Z_{gt, i}^{l o w}) \partial x}{\partial ψ (X_{gt, i}, y_{s l a b})} + X_{gt, i}^{l o w} = X_{gt, i}

(30)

The slope of this longitudinal centerline $\tan (R_{sy, i})$ can be calculated according to those two touchpoints:

\tan (R_{sy, i}) = \frac{Z_{gt, i} - Z_{gt, i}^{low}}{X_{gt, i} - X_{gt, i}^{low}}

(31)

Then, the function of longitudinal centerline is derived as equation (32):

z = \tan (R_{sy, i}) \cdot (x - X_{gt, i}) + Z_{gt, i}

(32)

Additionally, the lateral inclination angle of EFP need to be determined for its spatial position. The subgrade is a kind of row construction along its longitudinal direction, which means the lateral inclination degree is much lesser than longitudinal slope. Therefore, the lateral inclination can be approximately represented by lateral centerline slope of projection reticle. Then, the slope $\tan (R_{sx, i})$ of the datum plane lateral centerline is calculated by equation (33), as a supplementary for the datum plane spatial position.

\tan (R_{sx, i}) = \frac{Z_{g, i}^{2} - Z_{g, i}^{4}}{X_{g, i}^{2} - X_{g, i}^{4}}

(33)

Therefore, the spatial function of datum plane for ith EFP can be derived as $φ_{i} (x, y)$ :

φ_{i} (x, y) = \tan (R_{sy, i}) \cdot (x - X_{gt, i}) + \tan (R_{sx, i}) \cdot (y - Y_{gt, i}) + Z_{gt, i}

(34)

Deformation interaction between subgrade and EFP

Based on the aforementioned EFP model, the subgrade deformation will directly cause deflection of EFP, since they are bounded together with elastic support. Thus, this deformation effect can be converted to area load imposed on EFP. To obtain the area load, the conditional deformation of subgrade, which refers to the deformation relative to EFP, need to be calculated. The subgrade relative deformation under ith EFP can be obtained by subtracting $φ_{i} (x, y)$ from deformation description function $ψ (x, y)$ :

Δ_{i} (x, y) = ψ (x, y) - φ_{i} (x, y)

(35)

The effect of subgrade deformation can be considered as the enforced displacement. It is applied to the bottom of EFP structure. Thus, the area load is obtained by times the foundation stiffness coefficient:

{η_{i}}_{, s} (x, y) = κ_{s} \cdot Δ_{i} (x, y)

(36)

Where,

κ_{s}

is the foundation stiffness coefficient.

Additionally, the EFP is also subjected to fastener forces, which is represented utilizing the Dirac Delta function:

δ (x - x_{0}) = {\begin{array}{c} 0, x \neq x_{0} \\ \infty, x \neq x_{0} \end{array}, \int_{- \infty}^{\infty} δ (x - x_{0}) d x = 1

(37)

The Dirac Delta function has this attribute:

\int_{- \infty}^{\infty} f (x) δ (x - x_{0}) d x = f (x_{0})

(38)

Thus, the fastener force $F_{f, n}$ for nth fastener on track slab can be expressed as equation (39), which is at the position of $(X_{sf, n}, {Y_{sf}}_{, n})$ in composited slab local coordinates.

F_{f, n} (x, y) = F_{f, n} \cdot δ (x - X_{sf, n}) δ (y - {Y_{sf}}_{, n})

(39)

And th

η_{i, f} (x, y)

e sum force

η_{i, f} (x, y)

of N fasteners loads on ith EFP, can be expressed as:

η_{i, f} (x, y) = \sum_{n = 1}^{N} F_{f, n} (x, y) \cdot δ (x - X_{sf, n}) δ (y - {Y_{sf}}_{, n})

(40)

The total area load on ith EFP is:

η_{i} (x, y) = η_{i, s} (x, y) + η_{i, f} (x, y)

(41)

The study of literature (Guo, 2018) indicated that the flexural rigidity of rail is much smaller than that of track slab and base slab, which means the fastener force cannot be great enough to influence EFP deflection effectively. The fastener force can be omitted in normal calculation, and only consider the area load on EFP due to subgrade deformation.

Then, the deformation of ith EFP can be solved by substituting equation (41) into equation (8):

L {w_{i}} + λ w_{i} = κ_{s} \cdot [ψ (x, y, D) - φ_{i} (x, y)], (x, y) \in Ω

(42)

The EFP deformation at fastener positions can affect fastener force and influence rail geometry. For the ith EFP, deformation at all fastener locations can be obtained through EFP deflection solution and arranged as a 3×n array $U_{sf, i}$ :

U_{sf, i} = {[\begin{array}{c} X_{sf, 1} & X_{sf, 2} & . . . & X_{sf, n} \\ Y_{sf, 1} & Y_{sf, 2} & . . . & Y_{sf, n} \\ w_{i} (X_{sf, 1}, Y_{sf, 1}) & w_{i} (X_{sf, 2}, Y_{sf, 2}) & . . . & w_{i} (X_{sf, n}, Y_{sf, n}) \end{array}]}_{3 \times n}^{T}

The local deformation array $U_{sf, i}$ in slab local coordinates, need to be transformed to global coordinates. We define the fixed parameter vectors $Φ = {a / 2, b / 2, 0}$ and $Ψ_{i} = {X_{go, i}, Y_{go, i}, Z_{go, i}}$ . Where, $X_{go, i} = \sum_{k = 1}^{4} (X_{g, i}^{k} / 4)$ , $Y_{go, i} = \sum_{k = 1}^{4} (Y_{g, i}^{k} / 4)$ , $Z_{go, i} = φ_{i} (X_{go, i}, Y_{go, i})$ . As for a total number of n fasteners on ith EFP, the size of vectors can be arranged as ${[Φ_{n}]}_{n \times 3}$ and ${[Ψ_{i, n}]}_{n \times 3}$ respectively. The deformation array in global coordinates is calculated through the equation (43):

U_{gf, i} = [U_{sf, i} - Φ_{n}] T_{R, i} + Ψ_{i, n}

(43)

Where,

T_{R, i}

is rotation matrix, which can be expressed as:

T_{R, i} = [\begin{array}{c} \cos R_{sz, i} & \sin R_{sz, i} & 0 \\ - \sin R_{sz, i} & \cos R_{sz, i} & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & \cos R_{sx, i} & \sin R_{sx, i} \\ 0 & - \sin R_{sx, i} & \cos R_{sx, i} \end{array}] [\begin{array}{c} \cos R_{sy, i} & 0 & - \sin R_{sy, i} \\ 0 & 1 & 0 \\ \sin R_{sy, i} & 0 & \cos R_{sy, i} \end{array}]

Therefore, the deformation array for fastener positions of each EFP can be obtained, and these arrays can form a total deformation array $U_{gf}$ , including deformations of every fastener positions within subgrade disease area:

U_{gf} = {[\begin{array}{c} {U_{gf, 1}}^{T} & {U_{gf, 2}}^{T} & . . . & {U_{gf, i}}^{T} \end{array}]}_{3 \times l}^{T}

Where,

l = n \times i

, since there are i EFPs in subgrade disease area, and each EFP has n fasteners on them.

Deformation interaction between rail and EFP

Firstly, we analyse vertical deformation interaction between rail and EFP. Vertical deformation elements in the third column of array $U_{gf}$ are extracted as a new vector ${[U_{gf, 3 l}]}_{1 \times l}$ , and the interaction can be described as equation (44):

F_{f} = κ_{fz} \cdot (Z_{r} - U_{gf, 3 l})

(44)

Where,

κ_{fz}

is vertical stiffness of fastener. Thus, rail vertical deformation

Z_{r}

can be obtained by substituting equation (44) into equation (2), and it is expressed as:

Z_{rz} = K_{r} κ_{fz} {([I] + κ_{fz})}^{- 1} U_{gf, 3 l}

(45)

Where, I is a unit diagonal matrix.

As for rail lateral deformation, the second column elements of array $U_{gf}$ are extracted as the vector ${[U_{gf, 2 l}]}_{1 \times l}$ , which includes lateral deformation information. And the interaction between the rail and EFP is obtained by same method as above. Thus, the lateral deformation of the rail is obtained:

Y_{r} = K_{ry} κ_{fy} {([I] + κ_{fy})}^{- 1} U_{gf, 2 l}

(46)

Hitherto, the SMRM has been established. As for a certain subgrade disease, subgrade spatial deformation model generates a specific deformation with certain disease parameters. Then, the composite slab model determines its spatial position according to subgrade deformation, and the effect of subgrade disease transfers to composite slab model through their deformation interaction. Simultaneously, the composite slab interacts with rail by fastener, and finally the rail spatial geometry can be calculated based on fastener force.

SMRM verification by numerical simulation and experiment

Numerical simulation verification

The CRTS I ballastless track slab and typical subgrade structure as shown in Figure 10 widely used in HSR in China is chosen as research object to validate the proposed SMRM. The track slab is 4.97 m × 2.5 m × 0.21 m with 0.07 m interval between adjacent slabs. The width and thickness of base slab are 2.9 m and 0.2 m respectively, and mortar layer with 0.07 m thickness locates between track slab and base slab. As for the rail, CN60 rail is taken as study object, and the section moment of inertia is 3.217 × 10⁷ mm⁴. The fastener matching to CN60 rail are used with stiffness of 3 × 10⁷ N/m and fastener interval of 0.63 m. The material parameters of track and subgrade structure are shown in Table 1 (Tian, 2021).

Figure 10.

Typical cross section of HSR in China.

Table 1.

Parameters of track and subgrade structure.

Element	Elasticity modulus E (MPa)	Poisson ν	Density ρ(kg/m³)
Track slab	3.5 × 10⁴	0.16	3000
Mortar layer	92	0.3	2000
Base slab	3.0 × 10⁴	0.16	270
Subgrade surface layer	400	0.3	2400
Subgrade deep layer	180	0.3	1950
Embankment	40	0.35	1800
Rail	2.1 × 10⁴	0.3	7800

The FEM of subgrade-track structure with length of 60m is built in ANSYS as shown in Figure 11 with total 15,791 elements. The rail and fasteners are simulated by beam element and spring element. The subgrade and track including track slab, mortar layer and base slab are modelled by solid element, and fixed boundary is applied on the bottom of the FEM. The subgrade diseases are simulated by applying enforced deformations on the bottom of subgrade surface layer, as shown in Figure 11(c). The proposed SMRM is verified by FEM simulation considering typical diseases such as differential settlement and frost heave.

Figure 11.

FEM of subgrade structure: (a) Overall structure; (b) Local details of track; (c) Structure deformation simulation.

Verification of SMRM considering subgrade settlement along the longitudinal section

The total of 4 subgrade settlement cases with wavelengths of 30m and 40m, and the amplitudes of 20 mm and 30 mm are set up respectively as listed in Table 2. The vertical deformations of track are analysed by FEM numerical simulation and SMRM under above settlement conditions. The results are demonstrated in Figure 12 and Table 2.

Table 2.

Deviation analysis of FEM and SMRM for rail deformation amplitude under different settlement cases.

Cases (wavelength/amplitude)	30 m/20 mm	30 m/30 mm	40 m/20 mm	40 m/30 mm
SMRM solution (mm)	−19.98	−29.97	−19.99	−29.98
FEM solution(mm)	−19.99	−29.98	−19.99	−29.99
Deviation value (mm)	0.01	0.01	0.01	0.01
Relative error (%)	0.04	0.04	0.02	0.02

Figure 12.

Comparison of rail deformations results based on FEM and SMRM under different amplitude and wavelength: (a) Amplitude 20 mm and wavelength 30m; (b) Amplitude 30 mm and wavelength 30m; (c) Amplitude 20 mm and wavelength 40m; (d) Amplitude 30 mm and wavelength 40m.

The results of Figure 12 indicate that the rail deformation induced by subgrade settlement descripted by the SMRM method agree well with that of FEM numerical solution for all the cases. The relative error is less than 0.04% and decreases with the increase of the wavelength, and the amplitude deviation is less than 0.01 mm as listed in Table 2. All these results indicate that the proposed method can effectively calculate the change of track geometry under subgrade disease with high accuracy.

Verification of SMRM considering subgrade frost heave

The frost heave of subgrade with inhomogeneity in transversal section may lead to the elevation difference on left and right rail of one HSR line. A serious of spatial frost heave conditions with longitudinal deformation wavelength 30m, transversal wavelength 15m, and amplitude 20 mm coupled with −2.5 m, 0m and 2.5 m offsetting centerline of peak point position is designed to validate the accuracy of SMRM method in spatial scale as shown in Figure 13.

Figure 13.

Schematic diagram of a serious frost heave coupled with three cases of offsetting the peak point position from centerline: (a)The offset distance y_o is −2.5 m; (b)The offset distance y_o is 0m; (c)The offset distance y_o is 2.5 m.

Both left and right rail geometries are calculated by FEM and SMRM method under above three cases and shown in Figure 14 and Table 3.

Figure 14.

Comparison of rail (both left and right rail) deformation results based on FEM and SMRM under serious frost heave coupled with three offset cases: (a)The offset distance y_o is −2.5 m; (b)The offset distance y_o is 0m; (c)The offset distance y_o is 2.5 m.

Table 3.

Deviation analysis of FEM and SMRM for rail (both left and right rail) deformation amplitude under serious frost heave coupled with three offset cases.

Offset cases	y_o = −2.5 m		y_o = 0m		y_o = 2.5 m
Offset cases	Left rail	Right rail	Left rail	Right rail	Left rail	Right rail
SMRM solution (mm)	19.61	19.35	16.99	11.81	7.40	2.46
FEM solution (mm)	19.57	19.51	17.46	12.08	7.92	2.55
Deviation value (mm)	0.04	0.16	0.47	0.27	0.52	0.09
Relative error (%)	0.20	0.82	2.69	2.24	6.57	3.53

The results shown as Figure 14 and Table 3 illustrate that the rail geometries obtained by SMRM match well with that by FEM with relative errors 0.2%∼6.57% and deviation value 0.04–0.52 mm, and SMRM method can be used to describe the influence of subgrade spatial deformation on the track. Additionally, the deformation wavelength and amplitude of left and right rails decrease with deviating from the centerline of peak point position under the subgrade frost heave conditions with the same wavelength and amplitude. The decrease is different for left and right rails. The results indicate that SMRM method can provide a more comprehensive and detailed description for spatial scale mapping relationship between subgrade and track deformations than traditional method just considering vertical deformation along subgrade longitudinal section.

On the other hand, the professional engineers generally need to spend dozens of hours or even longer time on modelling, computing and checking data to obtain final track spatial geometry by FEM method. When more analysis cases need to be included, the work process will consume longer time. Compared with FEM analysis, the proposed SMRM method can obtain satisfactory high-precision calculation results within tens of seconds, and just need to modify some parameters for different disease conditions with significant advantages on efficiency and simplicity.

Verification of SMRM based on the measured data from scale model experiment

In this part, the measured data from a scale model experiment is used to verify the proposed SMRM method. The data was observed from a model experiment of the subgrade-slab ballastless track structure with a quarter scale of actual structure (Zhao et al., 2020b). This experiment was performed to study the influence of subgrade settlement on the track geometry. The layout of test device and observing sensors is shown in Figure 15 (Zhao et al., 2020a). The subgrade deformation (settlement) was simulated by controlling the adjuster to change the height of segment deformation plate. Displacement sensors and other facilities are used to monitor the deformation of subgrade and track as shown in Figure 16 (Zhao et al., 2020c).

Figure 15.

Diagram of scaled model experiment: (a) Transverse section; (b) Longitudinal section.

Figure 16.

Facilities and sensors of scaled model experiment: (a) Adjustment claw; (b) Dial indicator; (c) Displacement Sensors.

The sizes of the track slab and base slab model are 1.4 m×0.65 m × 0.075 m and 4.235 m×0.775 m × 0.075 m respectively, and the thickness of subgrade is 0.5 m. The 0.075 m thickness of self-compacting concrete is used to bond the track slab and base slab. The material of track structure is bentonite cement mortar with the elastic modulus of 7.5∼8 GPa. The deep and shallow layer subgrade is modelled by fine sand and medium sand with elastic modulus 25 MPa and 15 MPa, respectively. In addition, this study designed model materials, which has a quarter of elastic modulus compared with the actual structure material, and the density is same as it. In this experiment the subgrade deformation profiles were simulated by adjusting the position of segmental deformation plate, and the deformation cases with amplitude of 15 mm under settlement wavelengths of 2.8 m and 5.6 m are designed. The measured deformation data of experimental model was compared with numerical results of SMRM method, as shown in Figure 17. The deformation amplitudes of track under each case also were analysed and listed in Table 4.

Figure 17.

Comparison of rail deformation based on scale model experiment and SMRM under different settlement cases: (a) The deformation wavelength is 2.8 m; (b) The deformation wavelength is 5.6 m.

Table 4.

Deviation analysis of tested data and SMRM results for rail deformation amplitude under different settlement cases.

Cases (wavelength)	Observed data (mm)	Numerical solution (mm)	Deviation value (mm)	Relative Error (%)
2.8	−10.86	−12.45	−1.59	14.94
5.6	−14.90	−14.65	0.25	1.70

The root means square error (RMSD) of the observed data is employed to quantify the fitting degree of continuous data of track deformation profile from SMRM and discrete tested data from experiment:

σ = \sum_{m = 1}^{N} \sqrt{{(z_{o, m} - z_{c, m})}^{2}} / N

(47)

Where,

z_{o, m}

is tested data from experiment and

z_{c, m}

is the calculated data attained by the SMRM. N is the total number of tested data.

The RMSD are 0.637 mm and 0.627 mm corresponding to the two cases of settlement wavelength of 2.8 m and 5.6 m, respectively. These two wavelength conditions (2.8 m and 5.6 m) of scale model corresponds to about 10m and 20m wavelength of actual structure. It shows that the deformation calculated by SMRM method agree well with the tested data, and match better with the increase of deformation wavelength. The experimental data proves that the proposed SMRM can well fit the actual deformation.

The results shown in Figure 17 and Table 4 indicates that the relative error of experiment and SMRM method decreases from 14.94% to 1.70% with the increasing of wavelength from 2.8 m to 5.6 m. This trend has also been witnessed in FEM verification of 3.1 Numerical simulation verification. The error of 14.94% can be explained as follows. The differential settlements of scale model test are simulated by adjusting deformation plate with fixed space and forming the polyline deformation profile which has some deviation from the assumed cosine curve. On the other hand, the possible separation between slab and subgrade may occur when the wavelength of the subgrade disease is extremely small or the amplitude is large, and lead to significant error. According to the literature (Jiang et al., 2019), when the settlement wavelength is longer than 20m, the concrete base of track structure keeps good contact with the subgrade and this error can be reduced. The result also confirms this conclusion with only 1.70% relative error under wavelength of 5.6 m (corresponding to 20m of actual structure). Combined with the verification results that the error decreases with wavelength, we concluded that the SMRM method has higher precision for the condition of wavelength more than 20m. and the application scope can satisfy most conditions in real engineering scenarios. For example the differential settlement control standard is 12.5 mm for a 20m length of track (i.e. a chord length of 20 m) in Japan (Zhang et al., 2016b).

Conclusions

In this study, a SMRM method of solving the track spatial geometry under complex subgrade diseases is proposed and verified by numerical simulation and a scale model experimental data. Some conclusions are drawn as follows:

(1) A new spatial mapping relationship model (SMRM) is proposed. Compared to previous analytical model that mainly consider mapping structure vertical deformation to rail geometry, it can well describe the influence of subgrade diseases on track geometry in spatial scale. Even the elevation diffidence of left and right rail caused by subgrade diseases also can be quantified with SMRM. This method has been verified by the results of FEM numerical simulation and scale model experiment.

(2) The SMRM method has great advantage of high efficiency, simple process and sufficient accuracy compared to numerical simulation methods such as FEM. It can also comprehensively analyse spatial deformation of track structure without spending a lot of time on building FEM, computing and checking data to obtain result.

(3) The SMRM method can be used to analyse other subgrade diseases’ effect on track as long as the description function of diseases is available based on the present study and verification for differential settlement and frost heave cases.

(4) The proposed SMRM method is mainly applicable to subgrade deformation with the wavelength more than 20m, which can satisfy most practical applications.

(5) The analytical mapping relationship of hybrid effect on track geometry will be considered in the future study, such as the effect of temperature change and action of train coupling with subgrade diseases.

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: The work was support provided by National Key Research and Development Plan of China (2018YFE0207100).

ORCID iD

Yan Li

References

Budynas

(1977) Advanced Strength and Applied Stress Analysis. New York, NY: McGraw-Hill Science, Engineering & Mathematics.

Chen

Zhai

Cai

, et al. (2015) Safety threshold of high-speed railway pier settlement based on train-track-bridge dynamic interaction. Science China Technological Sciences 58(2): 202–210. DOI: 10.1007/s11431-014-5692-0.

Dangal

Chen

Lin

(2017) Polynomial particular solutions for solving elliptic partial differential equations. Computers & Mathematics with Applications 73(1): 60–70. DOI: 10.1016/j.camwa.2016.10.024.

Gao

Zhao

Hou

, et al. (2020) Analysis of influencing mechanism of subgrade frost heave on vehicle-track dynamic system. Applied Sciences 10(22): 8097. DOI: 10.3390/app10228097.

Gou

Ran

Yang

, et al. (2019) Mapping vertical bridge deformations to track geometry for high-speed railway. Steel and Composite Structures 32(4): 467–478. DOI: 10.12989/SCS.2019.32.4.467.

Gou

Xie

Liu

, et al. (2021b) Analytical study on high-speed railway track deformation under long-term bridge deformations and interlayer degradation. Structures 29: 1005–1015. DOI: 10.1016/j.istruc.2020.10.079.

Gou

Liu

Hua

, et al. (2021a) Mapping relationship between dynamic responses of high-speed trains and additional bridge deformations. Journal of Vibration and Control 27(9–10): 1051–1062. DOI: 10.1177/1077546320936899.

Guo

(2018) Effects of Differential Subgrade Settlement and Its Evolution in High-Speed Railways on Mechanical Performance of Vehicle-Track Coupled System. China: Southwest Jiaotong University.

Guo

Zhai

Sun

(2018b) A mechanical model of vehicle-slab track coupled system with differential subgrade settlement. Structural Engineering & Mechanics 66(1): 15–25. DOI: 10.12989/SEM.2018.66.1.015.

10.

Guo

Zhai

Gao

(2018a) Effect of differential subgrade settlement on Dynamic Performance of High-Speed Vehicle and Double-Block Ballastless track coupled system ICRT 2017, China: American Society of Civil Engineers, pp. 553–562. DOI: 10.1061/9780784481257.055.

11.

Huang

Luo

, et al. (2010) Analysis on distribution of dynamic stresses of ballastless track subgrade surface under axle loading of vehicle. Journal of the China Railway Society 32(2): 6.

12.

Jiang

Xin

, et al. (2019) Geometry mapping and additional stresses of ballastless track structure caused by subgrade differential settlement under self-weight loads in high-speed railways. Transportation Geotechnics 18: 103–110. DOI: 10.1016/j.trgeo.2018.10.007.

13.

Jiang

Zheng

Feng

, et al. (2020) Mapping the relationship between the structural deformation of a simply supported beam bridge and rail deformation in high-speed railways. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 234(10): 1081–1092. DOI: 10.1177/0954409719880668.

14.

Kang

(2016) Influence and control strategy for local settlement for high-speed railway infrastructure. Engineering 2(3): 374–379. DOI: 10.1016/J.ENG.2016.03.014.

15.

(2010) An Analysis on structural characteristics of CRTSⅠ and CRTSⅡ Slab Ballastless Track. Journal of East China Jiaotong University.

16.

Lin

Chen

Wang

, et al. (2017) Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems. Applied Mathematical Modelling 49: 452–469. DOI: 10.1016/j.apm.2017.05.012.

17.

Nielsen

JCO

(2018) Railway track geometry degradation due to differential settlement of ballast/subgrade – numerical prediction by an iterative procedure. Journal of Sound and Vibration 412: 441–456. DOI: 10.1016/j.jsv.2017.10.005.

18.

Ramadan

Jing

Zhang

, et al. (2021) Numerical analysis of additional stresses in railway track elements due to subgrade settlement using FEM simulation. Applied Sciences 11(18): 8501. DOI: 10.3390/app11188501.

19.

Ruge

Widarda

Schmälzlin

, et al. (2009) Longitudinal track–bridge interaction due to sudden change of coupling interface. Computers & Structures 87(1–2): 47–58. DOI: 10.1016/j.compstruc.2008.08.012.

20.

Salcher

Pradlwarter

Adam

(2016) Reliability assessment of railway bridges subjected to high-speed trains considering the effects of seasonal temperature changes. Engineering Structures 126: 712–724. DOI: 10.1016/j.engstruct.2016.08.017.

21.

Shan

Zhou

Wang

, et al. (2020) Differential settlement prediction of ballasted tracks in bridge-embankment transition zones. Journal of Geotechnical and Geoenvironmental Engineering 146(9): 04020075.

22.

Shan

Wang

Zhang

, et al. (2021) The influence of dynamic loading and thermal conditions on tram track slab damage resulting from subgrade differential settlement. Engineering Failure Analysis 128: 105610. DOI: 10.1016/j.engfailanal.2021.105610.

, et al. (2014) A potential new frost heave mechanism in high-speed railway embankments. Géotechnique 64(2): 144–154. DOI: 10.1680/geot.13.P.042.

24.

(2014) Preliminary Study on Interface Performance for Longitudinal Ballastless Track on High-Speed Railway. Central South University.

25.

Tian

(2021) Mechanical Behavior of Coarse-Grained Materials and Dynamic Response of High-Speed Railway’s Embankment in Freezing-Thawing Regions of High Latitude. China: Harbin Institute of Technology.

26.

Varandas