A method for calculating the stiffness of ballast track in railways

Abstract

This invention relates to a method for calculating the stiffness of ballasted railway track, comprising the following steps: obtaining the ballast particle material parameters and obtaining the equivalent Young's modulus expression of the track bed from the constitutive relation of isotropic granular materials; obtaining the ballast contact stiffness expression from Hertzian contact theory; expressing the normal contact stiffness using the average normal contact force; obtaining the stress varying with depth in the vertical direction based on the load distribution pattern within the ballasted track bed as diffusing at fixed angles in all directions, and calculating the equivalent Young's modulus accordingly; calculating the track bed stiffness coefficient based on the equivalent Young's modulus; and calculating the track bed dynamic stiffness at different frequencies based on constant damping and stiffness coefficient. This invention calculates the equivalent Young's modulus and stiffness based on the internal stress of the track bed, accurately reflecting the nonlinear relationship between track bed stiffness and thickness. It can calculate the stiffness of ballasted track beds under different track bed thickness conditions and can accurately calculate the dynamic stiffness of the track bed at high frequencies, making it suitable for railway track bed dynamics analysis and rolling noise prediction.

Description

Technical Field

[0001] This invention relates to the field of rail transit technology, specifically to a method for calculating the stiffness of ballast track in railways. Background Technology

[0002] Ballast track is a crucial component of ballasted railway track structures, and its stiffness directly affects train load transfer, track system vibration and noise characteristics. In existing technologies, track stiffness is typically simplified to a constant for ease of application in vehicle-track dynamics models. However, this simplification neglects the nonlinear characteristics exhibited by the track as a granular material, particularly the impact of track thickness on the elastic properties of the underlying foundation.

[0003] Existing research has shown that under the load applied by train wheels, ballast exhibits complex interparticle contact, relative density variations, and force transmission behavior, resulting in track foundation elastic properties that vary with ballast layer thickness. Relatively thicker ballast layers exhibit higher elasticity but are more prone to track settlement and geometric irregularities. Conversely, insufficient ballast thickness leads to increased stiffness and increased contact forces between ballast particles, thereby exacerbating the risk of particle breakage. However, current methods have not yet proposed a quantifiable approach to calculating ballast stiffness based on the impact of ballast thickness, leading to discrepancies between track dynamics simulations and the actual performance of ballasted tracks. Summary of the Invention

[0004] To address the aforementioned problems in the prior art, this invention proposes a method for calculating the stiffness of ballast track using mechanical properties and vibration noise analysis, taking into account the influence of different track thicknesses, in order to improve the accuracy of calculating the mechanical parameters of the track.

[0005] This invention provides a method for calculating the stiffness of ballast track in railways, comprising the following steps:

[0006] S1: Obtain the ballast particle material parameters and derive the equivalent Young's modulus expression for the ballast bed from the constitutive relation of isotropic granular materials:

[0007] ;

[0008] In the formula: E e The equivalent Young's modulus of the track bed; c The average coordination number of ballast particles; The porosity of the track bed; S n Normal contact stiffness of ballast particles; R The average radius of the ballast particles; This represents the ratio of the tangential contact stiffness to the normal contact stiffness of the ballast particles.

[0009] S2: The parameters of the ballast granular material include Young's modulus and Poisson's ratio; the normal contact stiffness of the ballast is obtained from Hertzian contact theory. S n expression:

[0010] ;

[0011] In the formula: E Young's modulus of ballast granular material; n Poisson's ratio for ballast granular material; F 0 represents the normal contact force of the ballast particles; R The average radius of the ballast particles;

[0012] The tangential contact between ballast particles is governed by Mindlin's contact theory, therefore The calculation formula is:

[0013] ;

[0014] The normal contact stiffness is expressed by the average normal contact force. S n Based on the relationship between the average normal particle contact force and the average stress, F The formula for calculating 0 is:

[0015] ;

[0016] ;

[0017] in: p 0 represents the average effective stress of ballast particles; and The normal stress in the horizontal direction within the ballast track bed; The stress is the normal stress in the vertical direction;

[0018] S3: Obtain the equivalent Young's modulus expression for the track bed from steps S1 and S2:

[0019] ;

[0020] The above formula shows that the equivalent elastic modulus of the track bed is related to its average normal stress. Therefore, E e Depending on the stress distribution inside the ballast under the vertical load of the train, the load distribution inside the track bed is such that it diffuses in all directions at a fixed angle. Therefore, the stress in the vertical direction that varies with depth is:

[0021] ;

[0022] in: l eThe length of the sleeper bottom surface; l b This refers to the width of the sleeper's bottom surface; α The angle at which the load inside the track bed spreads outward; z It is the depth from the bottom surface of the sleeper to the top surface of the track bed; p This refers to the uniformly distributed load per unit area on the bottom surface of a single sleeper. m ( z () represents the mass of ballast under load under a single sleeper;

[0023] The m ( z The calculation formula is:

[0024] ;

[0025] In the formula: r This refers to the bulk density of the ballast.

[0026] and Calculated from the lateral pressure coefficient of the track bed, and The calculation formula is:

[0027] ;

[0028] in: K 0 represents the track bed side pressure coefficient;

[0029] S4: The equivalent Young's modulus is directly expressed by the vertical stress that varies with depth as follows:

[0030] ;

[0031] Due to the bulk medium characteristics of the ballast bed, the number of effective contact points between ballast particles increases with increasing load, while the gaps between particles decrease. Therefore, calibration is performed by fitting experimental data. As a function of effective stress, the result is ;

[0032] S5: Deriving the track bed stiffness coefficient using the equivalent Young's modulus expression includes the following steps:

[0033] S51: Force transmitted from sleepers to ballast Q With stress q The functional relationship between them is calculated using the following formula:

[0034] ;

[0035] S52: The formula for calculating the displacement of the ballast layer is:

[0036] ;

[0037] In the formula: S This refers to the displacement of the ballast layer; e ( z () represents the depth within the ballast layer z Adapt to change; q ( z () represents the depth within the ballast layer z Stress; E e ( z The vertical stress varies with depth.

[0038] S53: Ballast layer stiffness coefficient K The expression is:

[0039] ;

[0040] S6: The stiffness of ballasted track bed is highly frequency-dependent. To obtain the stiffness amplitude at different frequencies, the track bed-sleeper system is treated as a spring-damping system. C Given a constant damping coefficient of 200 kN / m, the dynamic stiffness at different frequencies is calculated using the following formula:

[0041] ;

[0042] in: K ( oh () represents the dynamic stiffness at different frequencies; K This refers to the stiffness coefficient of the ballast layer; yes Angular frequency; C is the damping coefficient.

[0043] Preferably, the ballast material parameters obtained in S1 are: Young's modulus of granite ballast particles. E =60GPa; Poisson's ratio of granite ballast granular material n =0.25; Ballast bulk density r =1425kg / m 3 ; Track bed lateral pressure coefficient K 0 = 0.36; Depth of sleeper bottom surface from track bed top surface z =0.35m; length of sleeper bottom surface l e =2.5m, width of sleeper bottom surface l b =0.26m; the outward spread angle of the load within the track bed is 30°; the preload is the sleeper's own weight of 280kg; the average coordination number of ballast particles. c track bed porosity Average effective stress of ballast particles p The functional relationship between 0 and 0 is: ;

[0044] Based on the values ​​of the parameters of the ballast material, the stiffness coefficient of the ballast layer is calculated sequentially according to steps S1, S2, S3, S4, and S5. K =270MN / m, then according to the dynamic stiffness formula in S6. K ( oh )= K + iωC The dynamic stiffness at different frequencies was calculated. K ( oh The stiffness amplitude ranges from 270 MN / m at a low frequency of 20Hz to 1285 MN / m at a high frequency of 1000Hz.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows: On the one hand, the present invention calculates the equivalent Young's modulus and stiffness based on the internal stress of the track bed, which can accurately reflect the nonlinear relationship between the stiffness and thickness of the track bed and can calculate the stiffness of ballasted track bed under different track bed thickness conditions; on the other hand, the present invention can accurately calculate the dynamic stiffness of the track bed at high frequencies and can effectively avoid complex processes such as conducting experiments, and is applicable to the fields of railway track bed dynamics analysis and rolling noise prediction. Attached Figure Description

[0046] Figure 1 This is a schematic diagram of the calculation process of a preferred embodiment of the railway ballast track stiffness calculation method of the present invention. Detailed Implementation

[0047] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0048] This invention provides a method for calculating the stiffness of ballast track in railways, such as... Figure 1 As shown, it includes the following steps:

[0049] S1: Obtain the ballast particle material parameters and derive the equivalent Young's modulus expression for the ballast bed from the constitutive relation of isotropic granular materials:

[0050] ;

[0051] In the formula: E e The equivalent Young's modulus of the track bed; c The average coordination number of ballast particles; The porosity of the track bed; S n Normal contact stiffness of ballast particles; R The average radius of the ballast particles; This represents the ratio of the tangential contact stiffness to the normal contact stiffness of the ballast particles.

[0052] S2: The parameters of the ballast granular material include Young's modulus and Poisson's ratio; the normal contact stiffness of the ballast is obtained from Hertzian contact theory. S n expression:

[0053] ;

[0054] In the formula: E Young's modulus of ballast granular material; n Poisson's ratio for ballast granular material; F 0 represents the normal contact force of the ballast particles; R The average radius of the ballast particles;

[0055] The tangential contact between ballast particles is governed by Mindlin's contact theory, therefore The calculation formula is:

[0056] ;

[0057] The normal contact stiffness is expressed by the average normal contact force. S n Based on the relationship between the average normal particle contact force and the average stress, F The formula for calculating 0 is:

[0058] ;

[0059] ;

[0060] in: p 0 represents the average effective stress of ballast particles; and The normal stress in the horizontal direction within the ballast track bed; The stress is the normal stress in the vertical direction;

[0061] S3: Obtain the equivalent Young's modulus expression for the track bed from steps S1 and S2:

[0062] ;

[0063] The above formula shows that the equivalent elastic modulus of the track bed is related to its average normal stress. Therefore, E e Depending on the stress distribution inside the ballast under the vertical load of the train, the load distribution inside the track bed is such that it diffuses in all directions at a fixed angle. Therefore, the stress in the vertical direction that varies with depth is:

[0064] ;

[0065] in: l eThe length of the sleeper bottom surface; l b This refers to the width of the sleeper's bottom surface; α The angle at which the load inside the track bed spreads outward; z It is the depth from the bottom surface of the sleeper to the top surface of the track bed; p This refers to the uniformly distributed load per unit area on the bottom surface of a single sleeper. m ( z () represents the mass of ballast under load under a single sleeper;

[0066] The m ( z The calculation formula is:

[0067] ;

[0068] In the formula: r This refers to the bulk density of the ballast.

[0069] and Calculated from the lateral pressure coefficient of the track bed, and The calculation formula is:

[0070] ;

[0071] in: K 0 represents the track bed side pressure coefficient;

[0072] S4: The equivalent Young's modulus is directly expressed by the vertical stress that varies with depth as follows:

[0073] ;

[0074] Due to the bulk medium characteristics of the ballast bed, the number of effective contact points between ballast particles increases with increasing load, while the gaps between particles decrease. Therefore, calibration is performed by fitting experimental data. As a function of effective stress, the result is ;

[0075] S5: Deriving the track bed stiffness coefficient using the equivalent Young's modulus expression includes the following steps:

[0076] S51: Force transmitted from sleepers to ballast Q With stress q The functional relationship between them is calculated using the following formula:

[0077] ;

[0078] S52: The formula for calculating the displacement of the ballast layer is:

[0079] ;

[0080] In the formula: S This refers to the displacement of the ballast layer; e ( z () represents the depth within the ballast layer z Adapt to change; q ( z () represents the depth within the ballast layer z Stress; E e ( z The vertical stress varies with depth.

[0081] S53: Ballast layer stiffness coefficient K The expression is:

[0082] ;

[0083] S6: The stiffness of ballasted track bed is highly frequency-dependent. To obtain the stiffness amplitude at different frequencies, the track bed-sleeper system is treated as a spring-damping system. C Given a constant damping coefficient of 200 kN / m, the dynamic stiffness at different frequencies is calculated using the following formula:

[0084] ;

[0085] in: K ( oh () represents the dynamic stiffness at different frequencies; K This refers to the stiffness coefficient of the ballast layer; yes Angular frequency; C is the damping coefficient.

[0086] Optionally, the material parameters of the ballast in S1 are obtained as follows: Young's modulus of granite ballast particles. E =60GPa; Poisson's ratio of granite ballast granular material n =0.25; Ballast bulk density r =1425kg / m 3 ; Track bed lateral pressure coefficient K 0 = 0.36; Depth of sleeper bottom surface from track bed top surface z =0.35m; length of sleeper bottom surface l e =2.5m, width of sleeper bottom surface l b =0.26m; the outward spread angle of the load within the track bed is 30°; the preload is the sleeper's own weight of 280kg; the average coordination number of ballast particles. c track bed porosity Average effective stress of ballast particles p The functional relationship between 0 and 0 is: ;

[0087] Based on the values ​​of the parameters of the ballast material, the stiffness coefficient of the ballast layer is calculated sequentially according to steps S1, S2, S3, S4, and S5. K =270MN / m, then according to the dynamic stiffness formula in S6. K ( oh )= K + iωC The dynamic stiffness at different frequencies was calculated. K ( oh The stiffness amplitude ranges from 270 MN / m at a low frequency of 20Hz to 1285 MN / m at a high frequency of 1000Hz.

[0088] As can be seen from the above preferred embodiments and calculation examples, the present invention calculates the equivalent Young's modulus and stiffness based on the internal stress of the track bed, which can accurately reflect the nonlinear relationship between the stiffness and thickness of the track bed and can calculate the stiffness of ballasted track bed under different track bed thickness conditions. It can also accurately calculate the dynamic stiffness of the track bed at high frequencies and effectively avoid complex processes such as conducting experiments. It is applicable to the fields of railway track bed dynamics analysis and rolling noise prediction.

[0089] The above-described technical solutions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for calculating the stiffness of ballast track bed in railways, characterized in that, Includes the following steps: S1: Obtain the ballast particle material parameters and derive the equivalent Young's modulus expression for the ballast bed from the constitutive relation of isotropic granular materials: ; In the formula: E e The equivalent Young's modulus of the track bed; c The average coordination number of ballast particles; Φ The porosity of the track bed; S n Normal contact stiffness of ballast particles; R The average radius of the ballast particles; ξ This represents the ratio of the tangential contact stiffness to the normal contact stiffness of the ballast particles. S2: The parameters of the ballast granular material include Young's modulus and Poisson's ratio; the normal contact stiffness of the ballast is obtained from Hertzian contact theory. S n expression: ; In the formula: E Young's modulus of ballast granular material; ν Poisson's ratio for ballast granular material; F 0 represents the average normal contact force; The tangential contact between ballast particles is governed by Mindlin's contact theory, therefore ξ The calculation formula is: ; Using average normal contact force F 0 represents the normal contact stiffness. S n Based on the relationship between the average normal particle contact force and the average stress, F The formula for calculating 0 is: ; ; in: p 0 represents the average effective stress of ballast particles; σ x and σ y The normal stress in the horizontal direction within the ballast track bed; σ z The stress is the normal stress in the vertical direction; S3: Obtain the equivalent Young's modulus expression for the track bed from steps S1 and S2: ; The above formula shows that the equivalent elastic modulus of the track bed is related to its average normal stress. Therefore, E e Depending on the stress distribution inside the ballast under the vertical load of the train, the load distribution inside the track bed is such that it diffuses in all directions at a fixed angle. Therefore, the stress formula in the vertical direction that varies with depth is: ; in: l e The length of the sleeper bottom surface; l b This refers to the width of the sleeper's bottom surface; α The angle at which the load inside the track bed spreads outward; z It is the depth from the bottom surface of the sleeper to the top surface of the track bed; p This refers to the uniformly distributed load per unit area on the bottom surface of a single sleeper. m ( z () represents the mass of ballast under load under a single sleeper; The m ( z The calculation formula is: ; In the formula: ρ This refers to the bulk density of the ballast. σ x and σ y Calculated from the lateral pressure coefficient of the track bed, σ x and σ y The calculation formula is: ; in: K 0 represents the track bed side pressure coefficient; S4: The equivalent Young's modulus is directly expressed by the vertical stress that varies with depth as follows: ; Due to the bulk medium characteristics of the ballast bed, the number of effective contact points between ballast particles increases with increasing load, while the gaps between particles decrease. Therefore, calibration is performed by fitting experimental data. c (1- Φ ) is a function of effective stress, and the result is ; S5: Deriving the track bed stiffness coefficient using the equivalent Young's modulus expression includes the following steps: S51: Force transmitted from sleepers to ballast Q With stress q The functional relationship between them is calculated using the following formula: ; S52: The formula for calculating the displacement of the ballast layer is: ; In the formula: S This refers to the displacement of the ballast layer; ε ( z () represents the depth within the ballast layer z Adapt to change; q ( z () represents the depth within the ballast layer z Stress; E e ( z The vertical stress varies with depth. S53: Ballast layer stiffness coefficient K The expression is: ; S6: The stiffness of ballasted track bed is highly frequency-dependent. To obtain the stiffness amplitude at different frequencies, the track bed-sleeper system is treated as a spring-damping system. C Given a constant damping coefficient of 200 kN / m, the dynamic stiffness at different frequencies is calculated using the following formula: ; in: K ( ω () represents the dynamic stiffness at different frequencies; K This refers to the stiffness coefficient of the ballast layer; iω Angular frequency; C is the damping coefficient.

2. The method for calculating the stiffness of ballast track bed in railways according to claim 1, characterized in that, The ballast material parameters obtained in S1 are as follows: Young's modulus of granite ballast particles. E =60GPa; Poisson's ratio of granite ballast granular material ν =0.25; Ballast bulk density ρ =1425kg / m 3 ; Track bed lateral pressure coefficient K 0 = 0.36; Depth of sleeper bottom surface from track bed top surface z =0.35m; length of sleeper bottom surface l e =2.5m, width of sleeper bottom surface l b =0.26m; The load spreads outward at a 30° angle within the ballast bed; the preload is the sleeper's own weight of 280 kg; the average particle size distribution of the ballast is... c track bed porosity Φ Average effective stress of ballast particles p The functional relationship between 0 and 0 is: ; Based on the values ​​of the parameters of the ballast material, the stiffness coefficient of the ballast layer is calculated sequentially according to steps S1, S2, S3, S4, and S5. K =270MN / m, then according to the dynamic stiffness formula in S6. The dynamic stiffness at different frequencies was calculated. K ( ω The stiffness amplitude ranges from 270 MN / m at a low frequency of 20Hz to 1285 MN / m at a high frequency of 1000Hz.

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