Railway simply supported beam deformation quantity rapid algorithm under second-phase dead load effect

Abstract

The invention relates to a rapid algorithm for deformation quantity of a railway simply supported beam under the action of a second-phase dead load, which comprises the following steps of: establishing a railway simply supported beam calculation model and a calculation model corresponding to the second-phase dead load acting on the railway simply supported beam, then converting a second-phase dead load acting interval into a standard calculation interval, and calculating deformation quantity of the railway simply supported beam according to the obtained standard calculation interval. And constructing a parameter expression for calculating the deformation quantity of the railway simply supported beam under the action of the second-stage dead load. And then substituting the constructed parameter expression for deformation quantity calculation into a differential equation corresponding to the flexural member to obtain an equation set related to the deformation quantity of the calculation point, and finally solving specific deformation quantities corresponding to different positions on the railway simply supported beam according to boundary conditions of the railway simply supported beam. According to the calculation method, the accurate calculation result of the deformation quantity of the position point, needing to be calculated, of the simply supported beam can be rapidly and directly obtained, and the problem that the deformation quantity of the railway simply supported beam needs to be calculated by building a finite element simulation model under the complex second-phase dead load effect is solved.

Description

Technical Field

[0001] The invention relates to the technical field of bridge engineering in the transportation industry, and in particular to a fast algorithm for calculating the deformation of a railway simply supported beam under a second-stage constant load. Background Art

[0002] Railway simply supported beams are generally prefabricated in the prefabricated beam yard, and then erected on the bridge piers for unified paving of the bridge deck system, followed by track laying and other processes. The loads acting on the railway simply supported beams in the later stage, such as the bridge deck system, track slabs, and rails, are collectively referred to as the second-stage dead load. The weight of the second-stage dead load can reach one-eighth of the dead weight of the beam itself. Therefore, it is a practical engineering problem with great research value to quickly and accurately calculate the deformation of railway simply supported beams under the action of the second-stage dead load.

[0003] The second-phase constant load can generally be simplified as a uniformly distributed load acting on the bridge deck of a railway simply supported beam. However, for working conditions with a longitudinal slope of the track, it is necessary to consider the load changes along the length of the bridge. Currently, there is a lack of such calculation methods. Summary of the invention

[0004] The present invention aims to address the deficiencies of the prior art and to provide a fast algorithm for the deformation of a railway simply supported beam under a secondary constant load. The algorithm can quickly calculate the deformation of simply supported beams in the transportation field such as railways, highways, and municipal engineering when subjected to secondary constant loads such as bridge deck pavement, ballasted track, and ballastless track, and obtain accurate calculation results of the deformation of the simply supported beam at the position point where calculation is required.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] The fast algorithm for the deformation of railway simply supported beams under the second-phase constant load includes the following steps:

[0007] S1. Establish a railway simply supported beam calculation model;

[0008] S2. Establish a calculation model corresponding to the second-stage dead load acting on the railway simply supported beam;

[0009] S3, convert the second-stage constant load action interval to the standard calculation interval;

[0010] S4, constructing a parameter expression for calculating the deformation of a railway simply supported beam under the second-stage constant 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 differential equation corresponding to the bending member to obtain a set of equations related to the deformation of the calculation point;

[0012] S6. According to the boundary conditions of the railway simply supported beam, solve the specific deformation corresponding to different positions on 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, establishing an overall coordinate system, with the origin of the coordinate system being set at the centroid of the cross section at the left support point of the railway simply supported beam;

[0015] S12, inputting geometric parameters of the railway simply supported beam, including the beam length, the positions of the two end supports of the simply supported beam, the cross-sectional type of the beam cross section and the corresponding dimensions;

[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 bars or prestressed steel bars.

[0017] In step S2, the specific process of establishing the calculation model corresponding to the second-phase dead load acting on the railway simply supported beam is as follows:

[0018] The second-phase dead load on the railway simply supported beam is simplified into uniformly distributed load and linearly distributed load according to whether there is a longitudinal slope of the track. The characterization 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 magnitude of the second-stage dead load along the length of the simply supported beam;

[0023] q0 represents the second-phase dead load value when the line track has no longitudinal slope;

[0024] a and b represent the slope and intercept of the expression representing the second-phase dead load respectively when the line track has a longitudinal slope.

[0025] In step S3, the specific process of converting the second-phase constant load action interval to the standard calculation interval is as follows:

[0026] The second-stage dead load action range [x1, x2] is converted to the standard calculation interval [-1, 1] through coordinate transformation. At this time, the standard coordinate corresponding to the coordinate parameter x is represented by the standard coordinate parameter ξ;

[0027] Where 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 for calculating the deformation of the railway simply supported beam under the second-stage constant load is as follows:

[0029] For the working condition when the track has no longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0030]

[0031] in,

[0032]

[0033] In the formula,

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

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

[0036] θ1, θ2, θ3 represent the deformation of the railway simply supported beam corresponding to the two end points 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) represent five different undetermined parameters for calculating the deformation of railway simply supported beams;

[0038] Δl represents the length of the second-stage dead load acting on the railway simply supported beam;

[0039] For the working condition where the track has a longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0040]

[0041] in,

[0042]

[0043] Two types,

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

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

[0046] θ1, θ2, θ3 represent the deformation of the railway simply supported beam corresponding to the two end points 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) represent six different undetermined parameters for calculating the deformation of railway simply supported beams;

[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 for deformation calculation is substituted into the differential equation corresponding to the bending member to obtain a set of equations related to the deformation of the calculation point. The calculation formula is as follows:

[0050] For the working condition when the track has no longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0051]

[0052] In the formula,

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

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

[0055] F s is the shear force;

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

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

[0058] q| ξ＝0 Represents the linear distribution force concentration at the midpoint of the interval;

[0059] For the working condition where the track has a longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0060]

[0061] In the formula,

[0062] Represents the first derivative of the linear distribution force concentration at the midpoint of the interval with respect to the x-coordinate.

[0063] In step S6, the process of solving the specific deformation amounts 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. Calculate the values ​​of bending moment and shear force corresponding to the midpoint of each second-stage constant load action interval;

[0065] S62, substituting the calculated values ​​of the bending moment and shear force corresponding to these midpoints into the corresponding deformation calculation parameter expression under the second-stage constant load in step S5, and calculating the deformation of each railway simply supported beam node;

[0066] S63, selecting the deformation variable corresponding to the target node that needs to be calculated as the calculated value of the deformation variable of the railway simply supported beam to be solved.

[0067] The beneficial effect of the present invention is that the calculation method of the present invention can quickly and directly obtain the accurate calculation result of the deformation variable of the simply supported beam at the position point where the calculation is required, thereby solving the difficult problem that the calculation of the deformation variable of the railway simply supported beam under the action of complex secondary constant load needs to rely on the establishment of a finite element simulation model for calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a schematic diagram of the calculation process flow of the present invention;

[0069] Figure 2 It is a schematic diagram of the second-stage dead load acting on a railway simply supported beam of the present invention;

[0070] Figure 3 It is a schematic diagram of the conversion of the second-phase constant load action interval to the standard calculation interval of the present invention;

[0071] The following is a detailed description of the embodiments of the present invention with reference to the accompanying drawings. DETAILED DESCRIPTION

[0072] The principles and features of the present invention are described below in conjunction with the accompanying drawings. The embodiments are only used to explain the present invention and are not used to limit the scope of the present invention. The present invention is described in more detail by way of example with reference to the accompanying drawings in the following paragraphs. The advantages and features of the present invention will become clearer according to the following description. It should be noted that the accompanying drawings are all in a very simplified form and are not in precise proportions, and are only used to facilitate and clearly assist in explaining the purpose of the embodiments of the present invention.

[0073] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art of the present invention. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more related listed items.

[0074] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0075] A fast algorithm for the deformation of a railway simply supported beam under the action of a second-stage constant load, such as Figure 1 As shown, the following steps are included:

[0076] S1. Establish a railway simply supported beam calculation model, such as Figure 2 As shown, the specific process is as follows:

[0077] S11, establishing an overall coordinate system, with the origin of the coordinate system being set at the centroid of the cross section at the left support point of the railway simply supported beam;

[0078] S12, inputting geometric parameters of the railway simply supported beam, such as the beam length of the simply supported beam, the positions of the two end supports of the simply supported beam, the cross-sectional type of the beam cross section and the corresponding dimensions, etc.;

[0079] S13. Input the material parameters of railway simply supported beams, such as the grade and strength of simply supported beam concrete, and the grade and strength of steel bars or prestressed steel bars.

[0080] S2. Establish a calculation model corresponding to the second-stage dead load acting on the railway simply supported beam, such as Figure 2 As shown, the specific process is as follows:

[0081] The secondary constant load on a railway simply supported beam is caused by structures such as ballast, bridge deck pavement, and track slabs. Under normal circumstances, the secondary constant load acting on a railway simply supported beam can be expressed as a uniformly distributed load. However, for places where the track has a longitudinal slope, the secondary constant load varies along the direction of the line and can be simplified as a linear distributed force load.

[0082] The second-phase dead load on the railway simply supported beam is simplified into uniformly distributed load and linearly distributed load according to whether there is a longitudinal slope of the track. The characterization 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 magnitude of the second-stage dead load along the length of the simply supported beam;

[0087] q0 represents the second-phase dead load value when the line track has no longitudinal slope;

[0088] a and b represent the slope and intercept of the expression representing the second-phase dead load respectively when the line track has a longitudinal slope.

[0089] S3. Convert the second-stage constant load action interval to the standard calculation interval, such as Figure 3 As shown, the specific process is as follows:

[0090] The second-stage dead load action range [x1, x2] is converted to the standard calculation interval [-1, 1] through coordinate transformation. At this time, the standard coordinate corresponding to the coordinate parameter x is represented by the standard coordinate parameter ξ;

[0091] Where ξ represents the standard coordinate parameter of the standard calculation interval [-1,1].

[0092] S4, constructing a parameter expression for calculating the deformation of a railway simply supported beam under the second-stage constant load according to the standard calculation interval obtained in step S3;

[0093] For the working condition when the track has no longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0094]

[0095] in,

[0096]

[0097] In the formula,

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

[0099] z1, z2, and z3 represent the deformation (deflection) of the railway simply supported 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 simply supported 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) represent five different undetermined parameters for calculating the deformation (deflection) of railway simply supported beams;

[0102] Δl represents the length of the second-stage dead load acting on the railway simply supported beam;

[0103] For the working condition where the track has a longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0104]

[0105] in,

[0106]

[0107] Two types,

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

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

[0110] θ1, θ2, and θ3 represent the deformation (rotation angle) of the railway simply supported beam corresponding to the two end points 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 deformation (deflection) of railway simply supported beams;

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

[0113] S5, substituting the parameter expression for deformation calculation constructed in step S4 into the differential equation corresponding to the bending member to obtain a set of equations related to the deformation of the calculation point;

[0114] The calculation formula is as follows:

[0115] For the working condition when the track has no longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0116]

[0117] In the formula,

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

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

[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 concentration at the midpoint of the interval;

[0124] For the working condition where the track has a longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is:

[0125]

[0126] In the formula,

[0127] Represents the first derivative of the linear distribution force concentration at the midpoint of the interval with respect to the x-coordinate.

[0128] S6. According to the boundary conditions of the railway simply supported beam, solve the specific deformation corresponding to different positions on the railway simply supported beam;

[0129] S61. Calculate the values ​​of the bending moment, shear force and other parameters corresponding to the midpoint of each second-stage constant load action interval;

[0130] S62, substituting the calculated values ​​of the bending moment, shear force and other parameters corresponding to these midpoints into the corresponding deformation calculation parameter expression under the second-stage constant load in step S5, and calculating the deformation of each railway simply supported beam node;

[0131] S63, selecting the deformation variable corresponding to the target node that needs to be calculated as the calculated value of the deformation variable of the railway simply supported beam to be solved. Specific embodiment 1:

[0133] A test simply supported beam is 4m long. There is no longitudinal slope in the first quarter (1m long) of the track. It is assumed that the second-stage constant load it bears is a uniformly distributed load with a load concentration of 1KN / m. There is a longitudinal slope in the last quarter (1m long) of the track. It is assumed that the second-stage constant load it bears is a linearly distributed load. The load concentrations at the two end points are 1KN / m and 2KN / m respectively. In the middle of the beam, there is a concentrated load to simulate the heap load. The size of the load is 1KN. Calculate the deformation size of the beam span center under this calculation condition.

[0134] The algorithm of the present invention is compiled into an executable program to perform calculations for this engineering case, and is verified by back-to-back comparison calculations with the commercial finite element software MIDAS. The comparison of the calculation results of the present invention and the calculation results of MIDAS is shown in Table 1:

[0135] Table 1 Comparison of deformation of the midpoint of the simply supported beam under the second-stage constant load

[0136] Test content Algorithm of the present invention Midas's calculation results The present invention / midas(%) Beam midpoint deformation (m) 0.1023 0.1025 99.8％

[0137] As can be seen from Table 1, the calculation results of the present invention are very close to those of the commercial software midas, with an error within 1%, indicating that the calculation results of the method in the present invention are very accurate for calculating the deformation of the midpoint of the simply supported beam under the second-stage constant load, and meet the engineering calculation requirements.

[0138] The present invention can quickly calculate the deformation of simply supported beams in the field of transportation, such as railways, highways, municipal engineering, etc., when they are subjected to secondary constant loads such as bridge deck pavement, ballasted track, ballastless track, etc.

[0139] The method of the present invention is applicable to two working conditions: a line track with a longitudinal slope or a line track without a longitudinal slope; the distribution range length of the uniformly distributed load or the linearly distributed load is not limited; and the starting point or the ending point of action on the railway simply supported beam is not limited.

[0140] The calculation method of the present invention can quickly and directly obtain accurate calculation results of the deformation variables of the simply supported beam at the position points where calculation is required, thus solving the problem that the calculation of the deformation variables of the railway simply supported beam under the action of complex secondary constant load needs to rely on the establishment of a finite element simulation model for calculation.

[0141] The present invention is described above by way of example in conjunction with the accompanying drawings. It is obvious that the specific implementation of the present invention is not limited to the above-mentioned method. As long as various improvements are made using the method concept and technical solution of the present invention, or are directly applied to other occasions without improvement, they are all within the protection scope of the present invention.

Claims

1. A fast algorithm for the deformation of railway simply supported beams under the action of the second-stage constant load, characterized in that: The steps include: S1. Establish a railway simply supported beam calculation model; S2. Establish a calculation model corresponding to the second-stage dead load acting on the railway simply supported beam; S3, convert the second-stage constant load action interval to the standard calculation interval; S4, constructing a parameter expression for calculating the deformation of a railway simply supported beam under the second-stage constant load according to the standard calculation interval obtained in step S3; S5, substituting the parameter expression for deformation calculation constructed in step S4 into the differential equation corresponding to the bending member to obtain a set of equations related to the deformation of the calculation point; S6. According to the boundary conditions of the railway simply supported beam, solve the specific deformation corresponding to different positions on the railway simply supported beam.

2. The fast algorithm for the deformation of railway simply supported beams under the second-stage constant load according to claim 1 is characterized in that: In step S1, the specific process of establishing the railway simply supported beam calculation model is as follows: S11, establishing an overall coordinate system, with the origin of the coordinate system being set at the centroid of the cross section at the left support point of the railway simply supported beam; S12, inputting geometric parameters of the railway simply supported beam, including the beam length, the positions of the two end supports of the simply supported beam, the cross-sectional type of the beam cross section and the corresponding dimensions; 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 bars or prestressed steel bars.

3. The fast algorithm for the deformation of railway simply supported beams under the second-stage constant load according to claim 2 is characterized in that: In step S2, the specific process of establishing the calculation model corresponding to the second-phase dead load acting on the railway simply supported beam is as follows: The second-phase dead load on the railway simply supported beam is simplified into uniformly distributed load and linearly distributed load according to whether there is a longitudinal slope of the track. The characterization formula is as follows: In the formula, x represents the coordinate along the length direction of the simply supported beam; q(x) represents the magnitude of the second-stage dead load along the length of the simply supported beam; q0 represents the second-phase dead load value when the line track has no longitudinal slope; a and b represent the slope and intercept of the expression representing the second-phase dead load respectively when the line track has a longitudinal slope.

4. The fast algorithm for the deformation of railway simply supported beams under the second-stage constant load according to claim 3 is characterized in that: In step S3, the specific process of converting the second-phase constant load action interval to the standard calculation interval is as follows: The second-stage dead load action range [x1, x2] is converted to the standard calculation interval [-1, 1] through coordinate transformation. At this time, the standard coordinate corresponding to the coordinate parameter x is represented by the standard coordinate parameter ξ; Where ξ represents the standard coordinate parameter of the standard calculation interval [-1,1].

5. The fast algorithm for the deformation of railway simply supported beams under the second-stage constant load according to claim 4 is characterized in that: In step S4, the specific process of constructing the parameter expression for calculating the deformation of the railway simply supported beam under the second-stage constant load is as follows: For the working condition when the track has no longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is: in, In the formula, z(ξ) represents the deformation parameter of railway simply supported beam in the standard calculation range; z1, z2, z3 represent the deformation of the railway simply supported beam corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval; θ1, θ2, θ3 represent the deformation of the railway simply supported beam corresponding to the two end points and the midpoint of the standard calculation interval [-1,1]; κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ) represent five different undetermined parameters for calculating the deformation of railway simply supported beams; Δl represents the length of the second-stage dead load acting on the railway simply supported beam; For the working condition where the track has a longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is: in, Two types, z(ξ) represents the deformation parameter of railway simply supported beam in the standard calculation range; z1, z2, z3 represent the deformation of the railway simply supported beam corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval; θ1, θ2, θ3 represent the deformation of the railway simply supported beam corresponding to the two end points and the midpoint of the standard calculation interval [-1,1]; 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 deformation of railway simply supported beams; Δl represents the length of the secondary dead load acting on the railway simply supported beam.

6. The fast algorithm for the deformation of railway simply supported beams under the second-stage constant load according to claim 5 is characterized in that: In step S5, the constructed parameter expression for deformation calculation is substituted into the differential equation corresponding to the bending member to obtain a set of equations related to the deformation of the calculation point. The calculation formula is as follows: For the working condition when the track has no longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is: In the formula, E is the elastic modulus of the pier; I is the moment of inertia of the pier cross section; F s is the shear force; F s | ξ＝0 represents the shear force at the midpoint of the interval; M| ξ＝0 represents the bending moment at the midpoint of the interval; q| ξ＝0 Represents the linear distribution force concentration at the midpoint of the interval; For the working condition where the track has a longitudinal slope, the deformation calculation parameter expression under the second-stage constant load is: In the formula, Represents the first derivative of the linear distribution force concentration at the midpoint of the interval with respect to the x-coordinate.

7. The fast algorithm for the deformation of railway simply supported beams under the second-stage constant load according to claim 6 is characterized in that: In step S6, the process of solving the specific deformation amounts corresponding to different positions on the railway simply supported beam according to the boundary conditions of the railway simply supported beam is as follows: S61. Calculate the values ​​of bending moment and shear force corresponding to the midpoint of each second-stage constant load action interval; S62, substituting the calculated values ​​of the bending moment and shear force corresponding to these midpoints into the corresponding deformation calculation parameter expression under the second-stage constant load in step S5, and calculating the deformation of each railway simply supported beam node; S63, selecting the deformation variable corresponding to the target node that needs to be calculated as the calculated value of the deformation variable of the railway simply supported beam to be solved.

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