Fast algorithm for deformation of railway simply supported beam under secondary dead load

Abstract

The present application is a rapid deformation calculation method for railway simply supported beam under the action of secondary constant load. A calculation model of railway simply supported beam and a corresponding calculation model of secondary constant load acting on the railway simply supported beam are established. Then the secondary constant load action interval is converted to the standard calculation interval. According to the obtained standard calculation interval, the parameter expression of the deformation calculation of the railway simply supported beam under the action of the secondary constant load is constructed. Then the constructed parameter expression of the deformation calculation is substituted into the corresponding differential equation of the bending member to obtain the equation group related to the deformation of the calculation point. Finally, according to the boundary conditions of the railway simply supported beam, the specific deformation of the railway simply supported beam at different positions is solved. The calculation method of the present application can quickly and directly obtain the accurate calculation results of the deformation of the calculation position point of the simply supported beam, and solves the problem that the deformation calculation of the railway simply supported beam under the action of the complex secondary constant load needs to rely on the establishment of the finite element simulation model for calculation.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bridge engineering in the transportation industry, and particularly relates to a rapid deformation calculation method for a railway simply supported beam under the action of secondary dead load. BACKGROUND

[0002] A railway simply supported beam is generally prefabricated in a prefabricated beam yard, then erected on a pier, and then uniformly paved with a bridge deck system, and then paved with a track and other processes. The load of the bridge deck system, track slab and steel rail, and other later loads acting on the railway simply supported beam are collectively referred to as secondary dead load. The specific gravity of the secondary dead load can reach one-eighth of the self-weight of the beam body. Therefore, it is a very valuable practical engineering problem to quickly and accurately calculate the deformation of the railway simply supported beam under the action of the secondary dead load.

[0003] The secondary dead load can generally be simplified as a uniformly distributed load acting on the bridge deck of the railway simply supported beam, but for the working condition of the presence of a longitudinal slope of the track, the load variation along the length direction of the bridge needs to be considered, and currently there is still a lack of such a calculation method. SUMMARY

[0004] The present application aims to solve the problems of the prior art, and provides a rapid deformation calculation method for a railway simply supported beam under the action of secondary dead load, which can quickly calculate the deformation of the simply supported beam in the transportation field such as railway, highway, municipal, etc. when subjected to the secondary dead load of bridge deck paving, ballast track, ballastless track, etc. to obtain the accurate calculation result of the deformation of the simply supported beam at the calculation position point.

[0005] In order to achieve the above-mentioned purpose, the present application adopts the following technical scheme:

[0006] The rapid deformation calculation method for a railway simply supported beam under the action of secondary dead load comprises the following steps:

[0007] S1, establishing a calculation model of a railway simply supported beam;

[0008] S2, establishing a calculation model corresponding to the secondary dead load acting on the railway simply supported beam;

[0009] S3, converting the secondary dead load action interval to a standard calculation interval;

[0010] S4, constructing a parameter expression for deformation calculation of the railway simply supported beam under the action of the secondary dead load according to the standard calculation interval obtained in step S3;

[0011] S5, substituting the parameter expression for deformation calculation constructed in step S4 into the corresponding differential equation of the flexural member to obtain an equation group related to the calculation point deformation;

[0012] S6, solving the specific deformation corresponding to different positions on the railway simply supported beam according to the boundary conditions of the railway simply supported beam.

[0013] In step S1, the specific process of establishing the railway simply supported beam calculation model is as follows:

[0014] S11, establish the overall coordinate system, and set the coordinate origin at the section centroid of the left support point position of the railway simply supported beam;

[0015] S12, input the geometric parameters of the railway simply supported beam, including the beam length of the simply supported beam, the positions of the two end support points of the simply supported beam, the section type of the beam section and the corresponding size;

[0016] S13, input the material parameters of the railway simply supported beam, including the grade and strength of the simply supported beam concrete, and the grade and strength of the steel bar or prestressed steel bar.

[0017] In step S2, the specific process of establishing the corresponding calculation model of the secondary permanent load acting on the railway simply supported beam is as follows:

[0018] The secondary permanent load on the railway simply supported beam is simplified as a uniformly distributed load and a linearly distributed load according to whether there is a longitudinal slope of the track, and the representation formula is as follows:

[0019]

[0020] In the formula,

[0021] x represents the coordinate along the length direction of the simply supported beam;

[0022] q(x) represents the load size of the secondary permanent load along the length direction of the simply supported beam;

[0023] q0 represents the value of the secondary permanent load when there is no longitudinal slope of the track;

[0024] a and b respectively represent the slope and intercept of the representation expression of the secondary permanent load when there is a longitudinal slope of the track.

[0025] In step S3, the specific process of converting the secondary permanent load action interval to the standard calculation interval is as follows:

[0026] The secondary permanent load action range [x1, x2] is converted to the standard calculation interval [-1, 1] through coordinate transformation, and the standard coordinate corresponding to the coordinate parameter x is represented by the standard coordinate parameter ξ;

[0027] In the formula, ξ represents the standard coordinate parameter of the standard calculation interval [-1, 1].

[0028] In step S4, the specific process of constructing the parameter expression of the deformation variable calculation of the railway simply supported beam under the action of the secondary permanent load is as follows:

[0029] For the working condition of the track without longitudinal slope, the expression of the deformation variable calculation parameter under the action of the secondary constant load is:

[0030]

[0031] wherein,

[0032]

[0033] wherein,

[0034] z(ξ) represents the deformation variable parameter of the railway simply supported beam in the standard calculation interval;

[0035] z1, z2, z3 represent the railway simply supported beam deformation variables corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1];

[0036] θ1, θ2, θ3 represent the railway simply supported beam deformation variables corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1];

[0037] κ 10 (ξ), κ 11 (ξ), κ 20 (ξ), κ 21 (ξ), κ 30 (ξ) respectively represent five different undetermined parameters for the calculation of the railway simply supported beam deformation variable;

[0038] Δl represents the length of the secondary constant load acting on the railway simply supported beam;

[0039] For the working condition of the track with longitudinal slope, the expression of the deformation variable calculation parameter under the action of the secondary constant load is:

[0040]

[0041] wherein,

[0042]

[0043] two expressions,

[0044] z(ξ) represents the deformation variable parameter of the railway simply supported beam in the standard calculation interval;

[0045] z1, z2, z3 represent the railway simply supported beam deformation variables corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1];

[0046] θ1, θ2, θ3 represent the railway simply supported beam deformation variables corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1];

[0047] κ 10 (ξ), κ 11(ξ), κ 20 (ξ), κ 21 (ξ), κ 30 (ξ), κ 31 (ξ) respectively represent six different undetermined parameters in the deformation variable calculation of the railway simply supported beam;

[0048] Δl represents the length of the secondary dead load acting on the railway simply supported beam.

[0049] In step S5, the constructed parameter expression of the deformation variable calculation is substituted into the corresponding differential equation of the flexural member to obtain an equation group related to the deformation variable of the calculation point, and the calculation formula is as follows:

[0050] For the working condition of the track without longitudinal slope, the parameter expression of the deformation variable under the action of the secondary dead load is as follows:

[0051]

[0052] In the formula,

[0053] E is the elastic modulus of the pier;

[0054] I is the cross-sectional moment of inertia of the pier;

[0055] F s is the shear force;

[0056] F s | ξ＝0 represents the shear force at the midpoint of the section;

[0057] M| ξ＝0 represents the bending moment at the midpoint of the section;

[0058] q| ξ＝0 represents the linear distribution force intensity at the midpoint of the section;

[0059] For the working condition of the track with longitudinal slope, the parameter expression of the deformation variable under the action of the secondary dead load is as follows:

[0060]

[0061] In the formula,

[0062] represents the first-order derivative of the linear distribution force intensity at the midpoint of the section with respect to the x coordinate.

[0063] In step S6, the process of solving the specific deformation variable corresponding to different positions on the railway simply supported beam according to the boundary conditions of the railway simply supported beam is as follows:

[0064] S61, the numerical value of the bending moment and the shear force corresponding to the midpoint of each secondary dead load acting section is calculated;

[0065] S62, the calculated bending moment and shear force corresponding to the midpoint are substituted into the corresponding deformation variable calculation parameter expression in step S5 under the action of the secondary permanent load to calculate the deformation variable of each railway simply supported beam node;

[0066] S63, the deformation variable corresponding to the target node to be calculated is selected as the deformation variable calculation value of the required railway simply supported beam.

[0067] The beneficial effects of the present application are that the calculation method of the present application can quickly and directly obtain the accurate calculation result of the deformation variable of the simply supported beam at the required calculation position point, and solves the problem that the deformation variable calculation of the railway simply supported beam under the action of the complex secondary permanent load needs to rely on the establishment of a finite element simulation model for calculation. BRIEF DESCRIPTION OF DRAWINGS

[0068] Figure 1 The figure is a schematic diagram of the calculation process of the present application;

[0069] Figure 2 The figure is a schematic diagram of the secondary permanent load acting on the railway simply supported beam of the present application;

[0070] Figure 3 The figure is a schematic diagram of the conversion of the secondary permanent load action interval to the standard calculation interval of the present application;

[0071] The principles and features of the present application will be described in detail below with reference to the accompanying drawings. DETAILED DESCRIPTION

[0072] The principles and features of the present application will be described in detail below with reference to the accompanying drawings.

[0073] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the present application belongs. The terminology used in the description of the present application herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the present application. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.

[0074] The present application will be further described below with reference to the accompanying drawings and embodiments:

[0075] The railway simply supported beam deformation variable fast algorithm under the action of the secondary permanent load, as shown in Figure 1 includes the following steps:

[0076] S1, a railway simply supported beam calculation model is established, as shown in Figure 2 The specific process is as follows:

[0077] S11, a global coordinate system is established, and the coordinate origin is set at the cross-sectional centroid of the left support point position of the railway simply supported beam;

[0078] S12, the geometric parameters of the railway simply supported beam are input, such as the beam length of the simply supported beam, the support point positions of the two ends of the simply supported beam, the cross-sectional type and corresponding size of the beam cross section, and the like;

[0079] S13, the material parameters of the railway simply supported beam are input, such as the grade and strength of the simply supported beam concrete, the grade and strength of the steel bar or prestressed steel bar.

[0080] S2, a corresponding calculation model of the secondary permanent load acting on the railway simply supported beam is established, as shown in Figure 2 The specific process is as follows:

[0081] The secondary permanent load on the railway simply supported beam is caused by structures such as ballast, bridge deck pavement, and track slab. Generally, the secondary permanent load acting on the railway simply supported beam can be represented by a uniform load, but for places where the track has a longitudinal slope, the secondary permanent load along the line direction changes and can be simplified as a linearly distributed force load.

[0082] The secondary permanent load on the railway simply supported beam is simplified as a uniform load and a linearly distributed load according to whether there is a longitudinal slope of the track, and the representation formula is as follows:

[0083]

[0084] In the formula,

[0085] x represents the coordinate along the length direction of the simply supported beam;

[0086] q(x) represents the load size of the secondary permanent load along the length direction of the simply supported beam;

[0087] q0 represents the value of the secondary permanent load when there is no longitudinal slope of the track;

[0088] a and b respectively represent the slope and intercept of the expression for representing the secondary permanent load when there is a longitudinal slope of the track.

[0089] S3, the secondary permanent load action interval is converted to a standard calculation interval, as shown in Figure 3 The specific process is as follows:

[0090] The secondary permanent load action range [x1, x2] is converted to the standard calculation interval [-1, 1] through coordinate transformation, and the standard coordinate corresponding to the coordinate parameter x is represented by the standard coordinate parameter ξ;

[0091] wherein ξ represents a standard coordinate parameter of a standard calculation interval [-1, 1].

[0092] S4, constructing a parameter expression for deformation calculation of the railway simple beam under the action of the secondary permanent load according to the standard calculation interval obtained in step S3;

[0093] For the working condition of the track without longitudinal slope, the parameter expression for deformation calculation under the action of the secondary permanent load is:

[0094]

[0095] wherein,

[0096]

[0097] wherein,

[0098] z(ξ) represents a deformation parameter (deflection) of the railway simple beam in the standard calculation interval;

[0099] z1, z2 and z3 represent the deformation (deflection) of the railway simple beam corresponding to the two end points and the midpoint of the standard calculation interval [-1, 1];

[0100] θ1, θ2 and θ3 represent the deformation (rotation angle) of the railway simple beam corresponding to the two end points and the midpoint of the standard calculation interval [-1, 1];

[0101] κ 10 (ξ), κ 11 (ξ), κ 20 (ξ), κ 21 (ξ), κ 30 (ξ) respectively represent five different undetermined parameters for deformation (deflection) calculation of the railway simple beam;

[0102] Δl represents the length of the secondary permanent load acting on the railway simple beam;

[0103] For the working condition of the track with longitudinal slope, the parameter expression for deformation calculation under the action of the secondary permanent load is:

[0104]

[0105] wherein,

[0106]

[0107] two expressions,

[0108] z(ξ) represents a deformation parameter (deflection) of the railway simple beam in the standard calculation interval;

[0109] z1, z2, z3 represent the railway simply supported beam deformation variable (deflection) corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1];

[0110] θ1, θ2, θ3 represent the railway simply supported beam deformation variable (rotation angle) corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1];

[0111] κ 10 (ξ), κ 11 (ξ), κ 20 (ξ), κ 21 (ξ), κ 30 (ξ), κ 31 (ξ) represent six different undetermined parameters for calculating the railway simply supported beam deformation variable (deflection) respectively;

[0112] Δl represents the length of the secondary constant load acting on the railway simply supported beam.

[0113] S5, the parameter expression of the deformation variable calculation constructed in step S4 is substituted into the corresponding differential equation of the bending member to obtain the equation group related to the calculation point deformation variable;

[0114] The calculation formula is as follows:

[0115] For the working condition of the track without longitudinal slope, the deformation variable calculation parameter expression under the action of the secondary constant load is:

[0116]

[0117] In the formula,

[0118] E is the elastic modulus of the pier;

[0119] I is the cross-sectional moment of inertia of the pier;

[0120] F s is the shear force;

[0121] F s | ξ＝0 represents the shear force at the midpoint of the interval;

[0122] M| ξ＝0 represents the bending moment at the midpoint of the interval;

[0123] q| ξ＝0 represents the linear distribution force intensity at the midpoint of the interval;

[0124] For the working condition of the track with longitudinal slope, the deformation variable calculation parameter expression under the action of the secondary constant load is:

[0125]

[0126] In the formula,

[0127] The first derivative of the linear distribution force intensity of the interval midpoint with respect to the x coordinate.

[0128] S6, according to the boundary conditions of the railway simply supported beam, the specific deformation variable corresponding to different positions on the railway simply supported beam is solved;

[0129] S61, the numerical value of the bending moment, shear force and other parameters corresponding to the midpoint of each secondary permanent load acting interval is calculated;

[0130] S62, the numerical value of the bending moment, shear force and other parameters corresponding to the midpoint calculated is substituted into the deformation variable calculation parameter expression in step S5 under the corresponding secondary permanent load, and the deformation variable of each railway simply supported beam node is calculated;

[0131] S63, the deformation variable corresponding to the target node to be calculated is selected, which is the deformation variable calculation value of the railway simply supported beam to be solved. Specific embodiment 1:

[0133] A test simply supported beam, beam length 4m, no longitudinal slope in the first quarter section (1m long) of the track, assuming that it bears a secondary permanent load of uniform load, and the load intensity is 1KN / m. There is a longitudinal slope in the last quarter section (1m long) of the track, and it is assumed that the secondary permanent load it bears is a linear distribution load, and the load intensity of the two endpoints is 1KN / m and 2KN / m respectively. In the middle position of the beam, there is a concentrated load for simulating the load, and the size of the load is 1KN. The deformation variable size of the beam span center under this calculation condition is calculated.

[0134] The algorithm of the present application is compiled into an executable program to calculate this engineering case, and the back-to-back comparison calculation method with the commercial finite element software midas is used for verification. The comparison of the calculation results of the present application and the calculation results of midas is shown in Table 1:

[0135] Table 1 Comparison of deformation variables of test simply supported beam under secondary permanent load

[0136] Test content Invention algorithm Midas calculation result Invention / Midas (%) Beam midpoint deformation (m) 0.1023 0.1025 99.8％

[0137] As shown in Table 1, the calculation results of the present application are very close to the calculation results of the commercial software midas, and the error is within 1%, which indicates that the calculation results of the method in the present application are very accurate for the deformation variable calculation of the test simply supported beam under the secondary permanent load of the beam midpoint, and meet the engineering calculation requirements.

[0138] The present application can quickly calculate the deformation of a simple supported beam in the field of transportation such as railway, highway, municipal, etc. under the action of the second stage constant load such as bridge pavement, ballast track, ballastless track, etc.

[0139] The method of the present application can be applied to the working conditions of the presence or absence of longitudinal slope of the track, and the distribution range length of the uniform load or linear distribution load is not limited, and the starting point or the end point of the action on the railway simple supported beam is not limited.

[0140] The calculation method of the present application can quickly and directly obtain the accurate calculation result of the deformation of the simple supported beam at the required calculation position, and solves the problem that the calculation of the deformation of the railway simple supported beam under the action of the complex second stage constant load needs to rely on the establishment of a finite element simulation model for calculation.

[0141] The present application has been described above in conjunction with the drawings, and it is obvious that the specific implementation of the present application is not limited by the above manner, and various improvements using the method concept and technical solution of the present application or direct application to other occasions without improvement are within the protection scope of the present application.

Claims

1. A fast algorithm for the deformation of a simply supported railway beam under two-stage dead load, characterized in that, Includes the following steps: S1. Establish a calculation model for a simply supported railway beam; S2. Establish a calculation model for the secondary dead load acting on a simply supported railway beam; The specific process is as follows: The secondary dead load on a simply supported railway beam is simplified into uniformly distributed load and linearly distributed load, depending on whether there is a longitudinal gradient of the track. The characterization formula is as follows: ； In the formula, Represents the coordinates along the length of the simply supported beam; This indicates the magnitude of the second-phase dead load along the length of the simply supported beam; This indicates the second-phase dead load value when the track has no longitudinal gradient; , These represent the slope and intercept of the expression for the second-phase dead load characterization when the track has a longitudinal gradient; S3. Convert the second-phase dead load range to the standard calculation range; The specific process is as follows: The range of the second phase of constant load Transform the coordinates to the standard calculation range. This corresponds to the coordinate parameters. Standard coordinates using standard coordinate parameters express; In the formula, Indicates the standard calculation interval Standard coordinate parameters; S4. Based on the standard calculation interval obtained in step S3, construct the parametric expression for calculating the deformation of a simply supported railway beam under the second-stage dead load. The specific process is as follows: For the condition where the track has no longitudinal gradient, the expression for the deformation calculation parameters under the second-stage dead load is as follows: ； in, ； In the formula, The deformation parameters of a simply supported railway beam within the standard calculation range; , , Indicates the standard calculation interval The deformation of the simply supported railway beam corresponding to the two endpoints and the midpoint of the interval; , , Indicates the standard calculation interval The deformation of the simply supported railway beam corresponding to the two endpoints and the midpoint of the interval; , , , , These represent five different undetermined parameters for calculating the deformation of a simply supported railway beam; This indicates the length of the secondary dead load acting on a simply supported railway beam. For the working condition where the track has a longitudinal gradient, the expression for the deformation calculation parameters under the second-stage dead load is as follows: ； in, ； Two styles, The deformation parameters of a simply supported railway beam within the standard calculation range; , , Indicates the standard calculation interval The deformation of the simply supported railway beam corresponding to the two endpoints and the midpoint of the interval; , , Indicates the standard calculation interval The deformation of the simply supported railway beam corresponding to the two endpoints and the midpoint of the interval; , , , , , These represent six different undetermined parameters for calculating the deformation of a simply supported railway beam; This indicates the length of the secondary dead load acting on a simply supported railway beam. S5. Substitute the parametric expression for the deformation calculation constructed in step S4 into the differential equation corresponding to the bending member to obtain the equation set related to the deformation at the calculation point. The calculation formula is as follows: For the condition where the track has no longitudinal gradient, the expression for the deformation calculation parameters under the second-stage dead load is as follows: ； In the formula, For shear force; Indicates the shear force at the midpoint of the interval; Indicates the bending moment at the midpoint of the interval; The linear distribution intensity of the force at the midpoint of the interval; For the working condition where the track has a longitudinal gradient, the expression for the deformation calculation parameters under the second-stage dead load is as follows: ； In the formula, The linear distribution of force intensity pairs at the midpoint of the interval The first derivative of the coordinates; S6. Based on the boundary conditions of the simply supported railway beam, solve for the specific deformations at different locations on the simply supported railway beam.

2. The fast algorithm for deformation of a simply supported railway beam under two-stage dead load as described in claim 1, characterized in that, In step S1, the specific process of establishing the calculation model of the simply supported railway beam is as follows: S11. Establish an overall coordinate system, with the origin set at the centroid of the cross section at the left support point of the railway simply supported beam. S12. Input the geometric parameters of the simply supported railway beam, including the beam length, the positions of the two end supports, the cross-sectional type and corresponding dimensions of the beam. S13. Input the material parameters of the railway simply supported beam, including the grade and strength of the concrete of the simply supported beam, and the grade and strength of the steel bars or prestressed steel bars.

3. The fast algorithm for deformation of a simply supported railway beam under two-stage dead load as described in claim 2, characterized in that, In step S6, the process of solving for the specific deformations at different locations on the simply supported railway beam based on the boundary conditions is as follows: S61. Calculate the values ​​of bending moment and shear force corresponding to the midpoint of each secondary dead load action interval; S62. Substitute the calculated values ​​of bending moment and shear force corresponding to these midpoints into the corresponding deformation calculation parameter expression under the second-stage dead load in step S5 to calculate the deformation of each railway simply supported beam node. S63. Select the deformation values ​​corresponding to the target nodes that need to be calculated, which are the deformation values ​​of the simply supported railway beam to be solved.

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