Semi-analytical method for ballastless track stress deformation analysis under roadbed spatial differential settlement

Abstract

The invention relates to the technical field of ballastless tracks, in particular to a semi-analytical method for ballastless track stress deformation analysis under roadbed spatial differential settlement, which comprises the following steps of: 1, importing geometric parameters, material parameters and roadbed settlement information of a double-block ballastless track structure; step 2, on the basis of simulating the contact relationship between a fastener system and a roadbed and a supporting layer, respectively establishing deformation analysis models of a steel rail and a ballast bed structure, and taking roadbed settlement as a displacement boundary condition; and step 3, determining steel rail deformation, fastener force and contact stress between the roadbed and the supporting layer by adopting a successive approximation method. According to the method, the influence of roadbed spatial differential settlement on deformation and interlayer contact of the double-block ballastless track can be better analyzed.

Description

Technical Field

[0001] The invention relates to the technical field of ballastless track, in particular to a semi-analytical method for analyzing the stress and deformation of a ballastless track under spatial differential settlement of a roadbed. Background Art

[0002] As an important type of ballastless track, bi-block ballastless track is widely used due to its high construction flexibility and structural integrity. In order to ensure the safety of high-speed train operation and the comfort of passengers, the deformation of track and subgrade must meet the extremely strict smoothness requirements of high-speed railway. However, in countries where bi-block ballastless track is applied, some high-speed railway lines pass through widely distributed soft soil areas. The high compressibility, low strength and spatial variation of soil parameters of soft soil may lead to significant post-construction settlement of high-speed railway subgrade. The post-construction settlement of high-speed railway subgrade can be divided into uniform settlement and differential settlement, and the differential settlement of subgrade can be further divided into longitudinal differential settlement (changing only in the longitudinal direction) and spatial differential settlement (changing in both the longitudinal and transverse directions), the former being a special case of the latter. The differential settlement of subgrade will significantly reduce the smoothness of the track and the contact performance between track layers, which will lead to problems such as reduced operating quality and increased maintenance costs. Therefore, it is very necessary to study the impact of subgrade differential settlement on bi-block ballastless track of high-speed railway.

[0003] Existing studies have mainly analyzed the impact of differential settlement of the roadbed on ballastless track from the aspects of track deformation, interlayer contact state, and structural material damage. Track deformation under differential settlement of the roadbed is mainly manifested in the deformation follow-up between the track and the roadbed settlement, as well as the possible arching of the track at the edge of the settlement area, which will directly affect the smoothness of the track and thus the safety and stability of high-speed trains. Differential settlement of the roadbed may also lead to excessive material strength at different positions of the track and damage to track components, destroy the main load transfer path from the track to the ground, and cause abnormal distribution of interlayer contact force. For example, when the settlement amplitude exceeds a certain threshold, debonding may occur between the roadbed and the ballastless track and between the structural layers of the ballastless track. When the train passes at high speed, the upper and lower structures in the debonding area will collide with each other, causing impact damage to the interlayer structure, further expanding the scope and degree of debonding between layers, and weakening the load transfer performance between structural layers. Therefore, high-speed railways have very strict requirements for roadbed settlement. For example, in China and Germany, the post-construction settlement of ballastless track roadbed should not exceed 15 mm, and the roadbed surface angle caused by roadbed settlement should not exceed 1 / 1000. The existing theoretical methods for analyzing the influence of roadbed differential settlement on ballastless track can be divided into numerical methods and analytical methods (or semi-analytical methods). Among the numerical methods, the finite element method (FEM) based on commercial software for modeling and analysis is the most commonly used. FEM has the disadvantages of cumbersome modeling process and high computational time consumption, especially when analyzing large-scale (such as relevant field surveys show that the roadbed settlement range may exceed 100 m and the amplitude may exceed 50 mm) or complex roadbed settlement types. In contrast, the analytical method has clear physical meaning in the analysis process, and the track and settlement information can be easily adjusted through parametric modeling, which has high computational efficiency and applicability when analyzing railway lines in areas with large-scale roadbed settlement.

[0004] However, there are few studies on the impact of differential settlement of subgrade on twin-block ballastless track, and the wavelength of differential settlement is mainly limited to within 50 m, which mainly considers the differential settlement in the longitudinal direction of the line, and it is difficult to fully reflect the impact of spatial differential settlement of subgrade on twin-block ballastless track. In addition, the current research using analytical methods to analyze the impact of foundation structure deformation on ballastless track mainly focuses on the case where the foundation structure is a bridge. In the few studies that focus on the impact of subgrade settlement on twin-block ballastless track, the dimensions of the analysis model are simplified or the nonlinearity of the contact between the subgrade and track layers is ignored. At present, there is no analytical method that can consider the nonlinear contact between layers to analyze the impact of large-scale spatial differential settlement of subgrade on twin-block ballastless track.

[0005] The analytical method has significant potential in analyzing the impact of large-scale spatial differential settlement of roadbed on twin-block ballastless track, but it needs to overcome some difficulties, mainly including: (1) the setting of spatial differential settlement of roadbed. In fact, it is usually cumbersome to implement this procedure in FEM commercial software. For example, in ABAQUS, it is necessary to use analytical field functions (with mathematical expressions of settlement) or Python scripts (without mathematical expressions of settlement); (2) reasonable simulation of the roadbed slab and supporting layer of the twin-block ballastless track, which is crucial to meet the actual structural deformation characteristics under spatial settlement; (3) the contact nonlinearity between the roadbed and the supporting layer needs to be reasonably simulated to realize potential debonding. Summary of the invention

[0006] The present invention provides a semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed, which can better analyze the influence of spatial differential settlement of roadbed on the deformation and interlayer contact of twin-block ballastless track.

[0007] According to the present invention, a semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed comprises the following steps: Step 1: Import the geometric parameters, material parameters and subgrade settlement information of the dual-block ballastless track structure; Step 2: Based on the simulation of the fastener system and the contact relationship between the roadbed and the supporting layer, the deformation analysis models of the rail and the roadbed structure are established respectively, and the roadbed settlement is used as the displacement boundary condition; Step 3: Using the successive approximation method, the rail deformation, fastener force, and contact stress between the roadbed and the supporting layer are determined.

[0008] Preferably, in step 1, the bi-block ballastless track structure includes rails, a fastener system, bi-block sleepers, a ballast plate, a supporting layer and a roadbed.

[0009] Preferably, in step 2, the ballast structure comprises a double-block sleeper, a ballast slab and a supporting layer, wherein the double-block sleeper is embedded in the ballast slab by pouring concrete.

[0010] Preferably, in step 2, establishing the model is specifically as follows: The rail is simplified into a simply supported Euler-Bernoulli beam, the track plate and the supporting layer are equivalent to a composite plate, and the composite plate is analyzed using Kirchhoff's thin plate theory. The elastic rectangular thin plate with two pairs of simply supported sides and the other two pairs of free sides can simulate the vertical deformation of any point in the composite plate and its free edges; the width of the composite plate is taken as the width of the supporting layer, its elastic modulus is determined based on the equivalent bending stiffness, and the thickness of the composite plate is determined by the equivalent moment of inertia; the spatial differential settlement of the roadbed is simulated by a functional expression and used as the displacement boundary condition of the model.

[0011] Preferably, the deformation equations of the rail and the composite plate are described by equation (6): ; in, is the number of fasteners, and the number of discrete springs between the roadbed and the supporting layer along the transverse direction of the line is , along the longitudinal direction , , and Respectively represent the left rail, the right rail and the first i The displacement of the response point; , and Respectively represent the left rail, right rail and the bottom of the composite plate j The supporting force of a spring; , , , and For example, 1, 2, and 3 represent the displacement indices of the left rail, right rail, and composite plate, respectively. The first number in the superscript represents the displacement index of the structure that undergoes deformation, while the second number represents the displacement index of another structure that interacts with it. i indicates the point number within the structure where deformation occurs, and j Indicates the serial number of the discrete support spring between it and another structure; and Using the deflection formula for a simply supported beam under concentrated load, , and The deflection formula of an elastic rectangular thin plate with two pairs of simply supported edges and two pairs of free edges under concentrated load is used for calculation. , and They represent the first unit load caused by the unit uniform load applied to the left rail, right rail and composite plate. i The vertical displacement of a point, and The calculation is done using the deflection formula for a simply supported beam under uniformly distributed load; The deflection formula of an elastic rectangular thin plate with two pairs of simply supported edges and the other two pairs of free edges under uniformly distributed load is used for calculation; and Both represent the gravity concentration of the rail. Represents the gravity concentration of the composite plate.

[0012] As a preferred embodiment, , and Determined by formula (7): ; In formula (7), , represent the vertical stiffness of the discrete springs between the rail and the composite slab and between the composite slab and the roadbed surface, respectively; , and Respectively , and Displacement of the projection point on the corresponding substructure; Based on the Heaviside function, it represents the composite plate j The contact state at each discrete spring position.

[0013] Preferably, when the composite plate is displaced beyond the lower roadbed surface, The value of is 1, otherwise it is 0, as shown in formula (8): .

[0014] Preferably, by substituting equation (7) into equation (6), equation (9) is obtained, and equation (9) is organized into a matrix form, as shown in equation (10): ; In formula (10), , , , , ,in i and j denote the row and column indices of these matrices respectively; , for matrix, , , for A matrix with 4 non-zero elements in each row; , , , ; , , and They represent the displacements of the left rail, right rail, composite slab and roadbed surface respectively; Represents the identity matrix.

[0015] As a preferred option, formula (10) is written as , we get formula (11) as follows: ; in is called the settlement influence coefficient matrix, is the displacement matrix of the twin-block ballastless track. is the generalized loading matrix.

[0016] Preferably, in step 3, specifically: The displacement of the twin-block ballastless track under differential subgrade settlement is obtained by solving equation (11) and the successive approximation method is used to solve it. The solution process is as follows: (1) Assume that the roadbed and the composite slab are in full contact, that is, All elements in are 1; (2) Calculate the displacement of the twin-block ballastless track using equation (11): ; (3) According to the calculation , use equation (8) to update the contact state between the roadbed and the composite plate, denoted as ; (4) Check whether If satisfied, stop the iteration and complete the calculation, otherwise , and return to step (2) for further calculation and repeat until the contact state matrix no longer changes.

[0017] The beneficial effects of the present invention are as follows: The present invention provides a semi-analytical method for analyzing the influence of spatial differential settlement of roadbed on twin-block ballastless track, and fully considers interlayer contact nonlinearity. The present invention has high accuracy in evaluating the static response of twin-block ballastless track under spatial differential settlement of roadbed. In addition, it also has high computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It is a flow chart of a semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed in an embodiment; Figure 2 is a schematic diagram of an equivalent cross section of a double-block ballastless track in an embodiment; FIG3 (a) is a schematic diagram of longitudinal differential settlement of the roadbed in the embodiment; FIG3( b ) is a schematic diagram of spatial differential settlement of the roadbed in the embodiment; Figure 4 Schematic diagram of the analysis model of double-block ballastless track under spatial differential settlement of roadbed in the embodiment; FIG5( a ) is a force diagram of the rail in the embodiment; FIG5( b ) is a force diagram of the composite plate in the embodiment; FIG5( c ) is a force diagram of the roadbed in the embodiment; Figure 6 It is a finite element model diagram of the roadbed and the double-block ballastless track in the embodiment; Figure 7 Schematic diagram of spatial differential settlement data of roadbed in the embodiment; Figure 8 It is a schematic diagram comparing the rail deformation results based on the finite element method and the analytical method in the embodiment. DETAILED DESCRIPTION

[0019] In order to further understand the content of the present invention, the present invention is described in detail in conjunction with the accompanying drawings and embodiments. It should be understood that the embodiments are only for explaining the present invention and are not intended to limit it.

[0020] Example

[0021] like Figure 1 As shown, this embodiment provides a semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed, which includes the following steps: Step 1: Import the geometric parameters, material parameters and subgrade settlement information of the dual-block ballastless track structure; Step 2: Based on the simulation of the fastener system and the contact relationship between the roadbed and the supporting layer, deformation analysis models of the rail and the roadbed structure (including the double-block sleepers, the roadbed slab and the supporting layer, where the double-block sleepers are embedded in the roadbed slab by concrete pouring) are established respectively, and the roadbed settlement is used as the displacement boundary condition; Step 3: Using the successive approximation method, the rail deformation, fastener force, and contact stress between the roadbed and the supporting layer are determined.

[0022] In step 1, the bi-block ballastless track structure includes rails, fastening system, bi-block sleepers, ballast plate, supporting layer and roadbed.

[0023] In step 2, the model is established as follows: Simplifying the rail into a simply supported Euler-Bernoulli beam can meet the deformation analysis requirements. Since the prefabricated double-block sleepers are embedded in the ballast slab by pouring concrete on site, the ballast slab is tightly combined with the supporting layer. Therefore, the track slab and the supporting layer are equivalent to a (homogeneous) composite slab (such as Figure 2 As shown). In addition, the ballast plate of the double-block ballastless track in the roadbed section is longitudinally integrated, and the length and width of the ballast plate and the supporting layer are much greater than the thickness. The composite plate is analyzed using Kirchhoff's thin plate theory. The elastic rectangular thin plate with two pairs of simply supported edges and the other two pairs of free edges can simulate the vertical deformation of any point in the composite plate and its free edges. The width of the composite plate is taken as the width of the supporting layer, and its elastic modulus is determined based on the equivalent bending stiffness, and the thickness of the composite plate is determined by the equivalent moment of inertia; the spatial differential settlement of the roadbed can be simulated by a functional expression and used as the displacement boundary condition of the model.

[0024] Figure 2 middle, represents the centroid coordinates of the composite plate, calculated according to formula (1): ; in, and Represent the cross-sectional areas of the ballast slab and the supporting layer respectively; and Represent the centroid coordinates of the ballast slab and the supporting layer respectively; ; and represent the elastic modulus of the ballast slab and the supporting layer respectively; and Respectively represent the thickness of the ballast slab and the supporting layer; and Respectively represent the width of the ballast slab and the supporting layer; and Represent the distance from the centroid of the ballast slab and supporting layer to the centroid of the equivalent cross section, respectively.

[0025] Considering the direct contact between the supporting layer and the roadbed, the width of the composite plate is taken as the width of the supporting layer, and its elastic modulus is Based on the equivalent bending stiffness, as shown in formula (2); then, the thickness of the composite plate Determined by the equivalent moment of inertia, as shown in formula (3): ; The spatial differential settlement of the roadbed is simulated by equations (4) and (5), where equation (4) is the longitudinal differential settlement (a special case of spatial differential settlement) and equation (5) is the spatial differential settlement; as shown in Figure 3 (a) and Figure 3 (b), respectively.

[0026] ; In formula (4) and formula (5), and They represent the amplitude and wavelength of roadbed settlement respectively; Indicates the longitudinal coordinate of the starting point of the settlement; Indicates the width of the roadbed surface; It represents the longitudinal direction of the response point on the roadbed surface. Indicates the lateral coordinate of the response point on the roadbed surface. The settlement amplitude of the left shoulder is zero, and the settlement amplitude of the right shoulder is ; Represents the Heaviside function.

[0027] The interlayer contact relationship between the subgrade and the bi-block ballastless track system involves the rails and track slabs as well as the subgrade and the supporting layer. The fasteners connect the rails and track slabs and can be simulated as linear elastic springs under subgrade settlement conditions. The surface of the high-speed railway subgrade is composed of graded crushed stone to meet the compaction criteria, which indicates that its bond strength is extremely low. Therefore, it is assumed that the subgrade and the supporting layer are only in compression, and debonding will occur when in tension. In this case, discrete compression-only springs are used to simulate the interaction between the subgrade and the supporting layer, as well as the potential debonding under subgrade settlement. In addition, in order to eliminate the influence of the boundary conditions at both ends of the model, the bi-block ballastless track analysis model needs to have sufficient length.

[0028] Figure 4 The schematic diagram of the analysis model of the twin-block ballastless track under the spatial differential settlement of the roadbed is shown. , , represent the discrete spring forces under the left rail, right rail, and composite plate, respectively; , represent the vertical stiffness of the discrete springs between the rail and the composite slab and between the composite slab and the roadbed surface, respectively; , , and Represent the displacements of the left rail, right rail, composite slab and roadbed surface respectively.

[0029] In the analytical model, the longitudinal direction of the track is defined as X direction, the length of the ballastless track is l , the number of fasteners is , the sleeper spacing is , the number of discrete springs between the roadbed and the supporting layer along the transverse direction of the line is , along the longitudinal direction The force diagrams of the subgrade and the double-block ballastless track are shown in Figure 5 (a), Figure 5 (b), and Figure 5 (c).

[0030] The deformation equations of the rail and composite plate are described by equation (6): ; in, , and Respectively represent the left rail, the right rail and the first i The displacement of the response point; , and Respectively represent the left rail, right rail and the bottom of the composite plate j The supporting force of a spring; , , , and For example, 1, 2, and 3 represent the displacement indices of the left rail, right rail, and composite plate, respectively. The first number in the superscript represents the displacement index of the structure that undergoes deformation, while the second number represents the displacement index of another structure that interacts with it. i indicates the point number within the structure where deformation occurs, and j Indicates the serial number of the discrete support spring between it and another structure; and Using the deflection formula for a simply supported beam under concentrated load, , and The deflection formula of an elastic rectangular thin plate with two pairs of simply supported edges and two pairs of free edges under concentrated load is used for calculation. , and They represent the first unit load caused by the unit uniform load applied to the left rail, right rail and composite plate. i The vertical displacement of a point, and The calculation is done using the deflection formula for a simply supported beam under uniformly distributed load; The deflection formula of an elastic rectangular thin plate with two pairs of simply supported edges and the other two pairs of free edges under uniformly distributed load is used for calculation; and Both represent the gravity concentration of the rail. Represents the gravity concentration of the composite plate.

[0031] In formula (6) , and Determined by formula (7): ; In formula (7), , and Respectively , and displacement of the projection point on the substructure; Based on the Heaviside function, it represents the composite plate j The contact state at the discrete spring position is 1 when the composite plate displacement exceeds the lower roadbed surface, otherwise it is 0, as shown in formula (8): .

[0032] By substituting formula (7) into formula (6), we get formula (9). For the convenience of calculation, formula (9) is organized into a matrix form, as shown in formula (10): ; In formula (10), , , , , ,in i and j denote the row and column indices of these matrices respectively; , for matrix, , , for A matrix with 4 non-zero elements in each row; , , , ; Represents the identity matrix.

[0033] Formula (10) can be written as , we get formula (11) as follows: ; in is called the settlement influence coefficient matrix, is the displacement matrix of the twin-block ballastless track. is the generalized loading matrix.

[0034] In step 3, specifically: The displacement of the twin-block ballastless track under differential settlement of the roadbed is obtained by solving equation (11). Since there is nonlinear contact between the roadbed and the supporting layer, the actual contact area is unknown in advance, which makes it impossible to solve equation (11) directly. Therefore, the successive approximation method is used to solve it. The solution process is as follows: (1) Assume that the roadbed and the composite slab are in full contact, that is, All elements in are 1; (2) Calculate the displacement of the twin-block ballastless track using equation (11): ; (3) According to the calculation , use equation (8) to update the contact state between the roadbed and the composite plate, denoted as ; (4) Check whether If satisfied, stop the iteration and complete the calculation, otherwise , and return to step (2) for further calculation and repeat until the contact state matrix no longer changes.

[0035] Model Validation In order to verify the accuracy and efficiency of the semi-analytical model established in this embodiment, a subgrade-double-block ballastless track finite element model was established in Abaqus (see Figure 6 ). Figure 7 The deformation of the roadbed with a length of 299.65 m is shown, of which 50-250 m of data comes from the measured high-speed railway roadbed elevation. Figure 7 The subgrade settlement in the example was used as input, and the rail deformation was calculated using the finite element method (based on Abaqus and Python scripts) and the semi-analytical model established in this embodiment (based on Matlab) on a workstation equipped with an AMD3900X CPU and 32 GB RAM. Figure 8 As shown in the figure, the rail deformation results obtained by the two methods are in good agreement, which shows that the proposed semi-analytical method has high accuracy in evaluating the static response of twin-block ballastless track under spatial differential settlement of roadbed. In addition, the calculation time of the finite element method and the semi-analytical method is about 582 s and 23 s, respectively, which shows that the proposed semi-analytical method has high computational efficiency.

[0036] The present invention and its embodiments are described schematically above, and the description is not restrictive. The drawings show only one embodiment of the present invention, and the actual structure is not limited thereto. Therefore, if a person skilled in the art is inspired by it and designs a structural method and an embodiment similar to the technical solution without creativity without departing from the purpose of the invention, they shall all fall within the protection scope of the present invention.

Claims

1. A semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed, characterized by: The following steps are involved: Step 1: Import the geometric parameters, material parameters and subgrade settlement information of the dual-block ballastless track structure; Step 2: Based on the simulation of the fastener system and the contact relationship between the roadbed and the supporting layer, the deformation analysis models of the rail and the roadbed structure are established respectively, and the roadbed settlement is used as the displacement boundary condition; Step 3: Using the successive approximation method, the rail deformation, fastener force, and contact stress between the roadbed and the supporting layer are determined.

2. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 1 is characterized by: In step 1, the bi-block ballastless track structure includes rails, fastening system, bi-block sleepers, ballast plate, supporting layer and roadbed.

3. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 2 is characterized by: In step 2, the ballast structure includes a double-block sleeper, a ballast slab and a supporting layer, wherein the double-block sleeper is embedded in the ballast slab by pouring concrete.

4. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 3 is characterized by: In step 2, the model is established as follows: The rail is simplified into a simply supported Euler-Bernoulli beam, the track plate and the supporting layer are equivalent to a composite plate, and the composite plate is analyzed using Kirchhoff's thin plate theory. The elastic rectangular thin plate with two pairs of simply supported sides and the other two pairs of free sides can simulate the vertical deformation of any point in the composite plate and its free edges; the width of the composite plate is taken as the width of the supporting layer, its elastic modulus is determined based on the equivalent bending stiffness, and the thickness of the composite plate is determined by the equivalent moment of inertia; the spatial differential settlement of the roadbed is simulated by a functional expression and used as the displacement boundary condition of the model.

5. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 4 is characterized by: The deformation equations of the rail and composite plate are described by equation (6): ； in, is the number of fasteners, and the number of discrete springs between the roadbed and the supporting layer along the transverse direction of the line is , along the longitudinal direction , , and Respectively represent the left rail, the right rail and the first i The displacement of the response point; , and Respectively represent the left rail, right rail and the bottom of the composite plate j The supporting force of a spring; , , , and For example, 1, 2, and 3 represent the displacement indices of the left rail, right rail, and composite plate, respectively. The first number in the superscript represents the displacement index of the structure that undergoes deformation, while the second number represents the displacement index of another structure that interacts with it. i indicates the point number within the structure where deformation occurs, and j Indicates the serial number of the discrete support spring between it and another structure; and Using the deflection formula for a simply supported beam under concentrated load, , and The deflection formula of an elastic rectangular thin plate with two pairs of simply supported edges and two pairs of free edges under concentrated load is used for calculation. , and They represent the first unit load caused by the unit uniform load applied to the left rail, right rail and composite plate. i The vertical displacement of a point, and The calculation is done using the deflection formula for a simply supported beam under uniformly distributed load; The deflection formula of an elastic rectangular thin plate with two pairs of simply supported edges and the other two pairs of free edges under uniformly distributed load is used for calculation; and Both represent the gravity concentration of the rail. Represents the gravity concentration of the composite plate.

6. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 5 is characterized by: In formula (6) , and Determined by formula (7): ； In formula (7), , represent the vertical stiffness of the discrete springs between the rail and the composite slab and between the composite slab and the roadbed surface, respectively; , and Respectively , and Displacement of the projection point on the corresponding substructure; Based on the Heaviside function, it represents the composite plate j The contact state at each discrete spring position.

7. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 6 is characterized by: When the composite slab displacement exceeds the lower roadbed surface, The value of is 1, otherwise it is 0, as shown in formula (8): 。 8. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 7 is characterized by: By substituting equation (7) into equation (6), we get equation (9), which is then organized into a matrix form, as shown in equation (10): ； In formula (10), , , , , ,in i and j denote the row and column indices of these matrices respectively; , for matrix, , , for A matrix with 4 non-zero elements in each row; , , , ; , , and They represent the displacements of the left rail, right rail, composite slab and roadbed surface respectively; Represents the identity matrix.

9. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 8 is characterized by: Formula (10) can be written as , we get formula (11) as follows: ； in is called the settlement influence coefficient matrix, is the displacement matrix of the twin-block ballastless track. is the generalized loading matrix.

10. The semi-analytical method for analyzing the stress and deformation of ballastless track under spatial differential settlement of roadbed according to claim 9 is characterized in that: In step 3, specifically: The displacement of the twin-block ballastless track under differential subgrade settlement is obtained by solving equation (11) and the successive approximation method is used to solve it. The solution process is as follows: (1) Assume that the roadbed and the composite slab are in full contact, that is, All elements in are 1; (2) Calculate the displacement of the twin-block ballastless track using equation (11): ; (3) According to the calculation , use equation (8) to update the contact state between the roadbed and the composite plate, denoted as ; (4) Check whether If satisfied, stop the iteration and complete the calculation, otherwise , and return to step (2) for further calculation and repeat until the contact state matrix no longer changes.

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