Method for calculating displacement of pier top of simply supported girder bridge under action of linear distribution force

Abstract

The invention relates to a simply supported beam bridge pier top displacement calculation method under the action of linear distribution force. The method comprises the following steps: establishing a simply supported beam bridge pier calculation model; establishing a linear distribution force calculation model; converting the linear distribution force interval into a standard calculation interval; constructing an undetermined coefficient expression for calculating the deflection of the pier body of the pier; substituting the constructed undetermined coefficient expression for deflection calculation into a deflection line differential equation to obtain an equation set related to the deflection of a calculation point; and according to the boundary condition, the specific pier top displacement is solved. The method can be used for calculating the pier top displacement of the uniform-section bridge pier in the field of traffic transportation when the uniform-section bridge pier is subjected to linear distribution force such as soil pressure and wind pressure, can be suitable for uniform-section bridge piers in any shapes, and is suitable for calculating the pier top displacement under the action of various linear distribution loads with any length and distribution position combinations; the problem that pier top displacement calculation needs to depend on an approximate numerical method of node division under the action of complex linear distribution force is solved.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge engineering in the transportation industry, and in particular to a method for calculating the displacement of a pier top of a simply supported beam bridge under the action of a linear distributed force. Background Art

[0002] In high-speed railway lines, bridges account for an average of more than 50%, and in the total length of bridges, standard 32m and 24m simply supported beams account for the majority. The longitudinal and transverse displacements of the pier tops are important indicators used by design specifications to control driving safety and structural safety. As for the external forces acting on the piers, there are usually loads such as soil pressure and wind pressure, which can be abstracted into linear distributed forces. In the past, in design, to calculate the displacement of the pier tops, it was generally necessary to divide the piers into nodes and units, and then convert the distributed forces into equivalent nodal forces. This numerical calculation method has many calculation steps, which will also affect the design efficiency for the large number of high-speed railway simply supported beam piers. For piers with equal cross-sections, the deflection and rotation problems of the piers under external loads can be quickly and accurately solved by solving differential equations.

[0003] In view of the practical problems existing in the above-mentioned bridge pier design, it is necessary to propose a faster and more accurate algorithm to solve the problem of calculating the pier top displacement caused by a large number of simply supported beam piers under the action of linear distributed forces. Summary of the invention

[0004] The present invention aims to solve the deficiencies of the prior art and provides a method for calculating the top displacement of a simply supported beam bridge pier under the action of a linear distributed force. The method can be used to calculate the top displacement of bridge piers of equal cross-section in the transportation field such as railways, highways, municipal engineering, light rail, etc. when subjected to linear distributed forces such as soil pressure and wind pressure. The accurate calculation result of the pier top displacement can be quickly and directly obtained through the calculation method of the present invention.

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

[0006] A method for calculating the displacement of a simply supported beam bridge pier top under a linear distributed force comprises the following steps:

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

[0008] S2. Establish a linear distributed force calculation model;

[0009] S3, converting the linear distribution force interval to the standard calculation interval;

[0010] S4, constructing an expression for undetermined coefficients for calculating the deflection of the pier body according to the standard calculation interval obtained in step S3;

[0011] S5, substituting the undetermined coefficient expression of the deflection calculation constructed in step S4 into the deflection curve differential equation to obtain a set of equations related to the deflection of the calculation point;

[0012] S6. Based on the boundary conditions, solve the specific pier top displacement.

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

[0014] For simply supported beam piers, a pier calculation model is established based on the pier height, cross-sectional dimensions, and material properties. The coordinate origin is set at the bottom of the pier, with the standard vertical direction being positive.

[0015] In step S2, the specific process of establishing the linear distributed force calculation model is as follows:

[0016] A distribution diagram of the linear distributed force is established, and a mathematical expression is used to characterize the linear distributed force. The specific formula can be expressed as follows:

[0017] q(x)=ax+b;

[0018] Where x represents the coordinate along the pier height direction;

[0019] q(x) represents the load magnitude of the linear distributed force along the pier height direction;

[0020] a and b represent the slope and intercept of the linear distributed load, respectively.

[0021] In step S3, the specific process of converting the linear distribution force interval to the standard calculation interval is as follows:

[0022] The range of action of the linear distributed force [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 ξ.

[0023] In step S4, the specific process of constructing the undetermined coefficient expression for calculating the pier body deflection according to the standard calculation interval obtained in step S3 is as follows:

[0024]

[0025] in,

[0026]

[0027] In the above two formulas,

[0028] z(ξ) represents the deflection parameter of the pier body in the standard calculation range;

[0029] z1, z2, z3 represent the pier deflection corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval;

[0030] θ1, θ2, θ3 represent the pier body rotation angles corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval;

[0031] κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ),κ 31 (ξ) represent six different undetermined coefficients for calculating the deflection of the pier body;

[0032] Δl represents the length of the distributed load acting on the pier.

[0033] In step S5, the undetermined coefficient expression of the deflection calculation constructed in step S4 is substituted into the deflection curve differential equation to obtain the equation group related to the deflection of the calculation point, and the calculation formula is as follows:

[0034]

[0035] In the formula,

[0036] E is the elastic modulus of the pier;

[0037] I is the moment of inertia of the pier cross section;

[0038] F s is the shear force;

[0039] F s | ξ＝0 represents the shear force at the midpoint of the interval;

[0040] It represents the first derivative of the linear distribution force concentration at the midpoint of the interval with respect to the x-coordinate;

[0041] M| ξ＝0 represents the bending moment at the midpoint of the interval;

[0042] q| ξ＝0 Represents the linearly distributed force concentration at the midpoint of the interval.

[0043] In step S6, the process of solving the specific pier top displacement according to the boundary conditions is as follows:

[0044] First, calculate the bending moment and shear force corresponding to the midpoint of each linear distribution force action interval;

[0045] Then, the calculated values ​​of the bending moment and shear force corresponding to these midpoints are substituted into the equations in step S5 to calculate the deflection and rotation angle of each pier node;

[0046] Finally, the deflection value corresponding to the pier top node is selected as the pier top displacement to be solved.

[0047] The beneficial effects of the present invention are as follows: the calculation method of the present invention can be used to calculate the displacement of pier tops of equal-section bridge piers in the field of transportation, such as railways, highways, municipal engineering, light rail, etc., when subjected to linear distributed forces such as soil pressure and wind pressure. It can be applied to equal-section bridge piers of any shape, and to the calculation of the displacement of pier tops under the action of linear distributed loads of any length and distribution position combination, thus solving the problem that the calculation of the displacement of pier tops under the action of complex linear distributed forces needs to rely on the approximate numerical method of node division. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the steps of the present invention;

[0049] Figure 2 It is a schematic diagram of the linear distributed force acting on the simply supported beam bridge pier of the present invention;

[0050] Figure 3 It is a schematic diagram of converting the linear distribution force interval to the standard calculation interval of the present invention;

[0051] The following is a detailed description of the embodiments of the present invention with reference to the accompanying drawings. DETAILED DESCRIPTION

[0052] 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.

[0053] It should be noted that when a component is referred to as being "fixed to" another component, it may be directly on the other component or there may also be a component centered. When a component is considered to be "connected to" another component, it may be directly connected to the other component or there may also be a component centered. When a component is considered to be "set on" another component, it may be directly set on the other component or there may also be a component centered. The terms "vertical", "horizontal", "left", "right" and similar expressions used herein are for illustrative purposes only.

[0054] 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.

[0055] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0056] A method for calculating the displacement of the top of a simply supported beam bridge pier under linear distributed forces, such as Figures 1 to 3 As shown, the following steps are included:

[0057] S1. Establish a calculation model for simply supported beam piers;

[0058] like Figure 2 As shown, the specific process is as follows:

[0059] For simply supported beam bridge piers, a pier calculation model is established based on the pier height, cross-sectional dimensions, material properties and other characteristics of the pier. The coordinate origin is set at the bottom of the pier, and the standard vertical upward is positive.

[0060] S2. Establish a linear distributed force calculation model;

[0061] like Figure 2 As shown, the specific process is as follows:

[0062] A distribution diagram of the linear distributed force is established, and a mathematical expression is used to characterize the linear distributed force. The specific formula can be expressed as follows:

[0063] q(x)=ax+b;

[0064] Where x represents the coordinate along the pier height direction;

[0065] q(x) represents the load magnitude of the linear distributed force along the pier height direction;

[0066] a and b represent the slope and intercept of the linear distributed load, respectively;

[0067] S3, converting the linear distribution force interval to the standard calculation interval;

[0068] like Figure 3 As shown, the specific process is as follows:

[0069] The action range of the linear distributed force [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 ξ;

[0070] S4, constructing an expression for undetermined coefficients for calculating the deflection of the pier body according to the standard calculation interval obtained in step S3;

[0071] The specific process is as follows:

[0072]

[0073] in,

[0074]

[0075] In the above two formulas,

[0076] z(ξ) represents the deflection parameter of the pier body in the standard calculation range;

[0077] z1, z2, z3 represent the pier deflection corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval;

[0078] θ1, θ2, θ3 represent the pier body rotation angles corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval;

[0079] κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ),κ 31 (ξ) represent six different undetermined coefficients for calculating the deflection of the pier body;

[0080] Δl represents the length of the distributed load acting on the pier;

[0081] S5, substituting the undetermined coefficient expression of the deflection calculation constructed in step S4 into the deflection curve differential equation to obtain a set of equations related to the deflection of the calculation point;

[0082] The calculation formula is as follows:

[0083]

[0084] In the formula,

[0085] E is the elastic modulus of the pier;

[0086] I is the moment of inertia of the pier cross section;

[0087] F s is the shear force;

[0088] F s | ξ＝0 represents the shear force at the midpoint of the interval;

[0089] It represents the first derivative of the linear distribution force concentration at the midpoint of the interval with respect to the x-coordinate;

[0090] M| ξ＝0 represents the bending moment at the midpoint of the interval;

[0091] q ξ＝0 Represents the linear distribution force concentration at the midpoint of the interval;

[0092] S6. According to the boundary conditions, solve the specific pier top displacement;

[0093] The process is as follows:

[0094] First, calculate the values ​​of parameters such as bending moment and shear force corresponding to the midpoint of each linear distribution force action interval;

[0095] Then, the calculated values ​​of the bending moment, shear force and other parameters corresponding to these midpoints are substituted into the equations in step S5 to calculate the deflection and rotation angle of each pier node;

[0096] Finally, the deflection value corresponding to the pier top node is selected as the pier top displacement to be solved. Specific embodiment 1:

[0098] A high-speed railway project has a uniform cross-section round-end bridge pier with a height of 38m. The bottom of the cap is consolidated and the pier is subjected to earth pressure near the bottom of the pier. The earth pressure can be simulated by linear distributed force to calculate the displacement of the pier top caused by the earth pressure. The algorithm of the present invention is incorporated into a self-compiled calculation program, and the test is verified by a back-to-back comparison calculation method between the self-compiled program and midas.

[0099] The comparison between the calculation results of the present invention and those of midas is shown in Table 1:

[0100] Table 1 Comparison of pier top displacement under soil pressure on bridge piers with equal cross-section

[0101] Test content Algorithm of the present invention Midas's calculation results The present invention / midas(%) Pier top displacement (m) 3.56E-06 3.56E-06 100.00％

[0102] As can be seen from Table 1, the calculation results of the present invention are completely consistent with the calculation results of the commercial software MIDAS, and the error between the two is zero, indicating that the calculation results of the method of the present invention are very accurate for the working condition of equal-section bridge piers subjected to linear distributed forces, and meet the engineering calculation requirements.

[0103] By adopting the calculation method of the present invention, a calculation model of a simply supported beam pier and a linear distributed force calculation model are established, and then the linear distributed force interval is converted into a standard calculation interval. According to the obtained standard calculation interval, an expression of undetermined coefficients for calculating the deflection of the pier body is constructed, and then the constructed expression of undetermined coefficients for calculating the deflection is substituted into the deflection curve differential equation to obtain a group of equations related to the deflection of the calculation point, and finally, according to the boundary conditions, the specific pier top displacement is solved.

[0104] The calculation method of the present invention can be used to calculate the displacement of pier tops of equal-section bridge piers in the field of transportation, such as railways, highways, municipal engineering, light rail, etc., when subjected to linear distributed forces such as soil pressure and wind pressure. It can be applied to equal-section bridge piers of any shape and to the calculation of the displacement of pier tops under the action of linear distributed loads of any length and distribution position combination, thus solving the problem that the calculation of pier top displacement under the action of complex linear distributed forces needs to rely on the approximate numerical method of node division.

[0105] 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 method for calculating the displacement of the top of a simply supported beam bridge pier under linear distributed force, characterized in that: The steps include: S1. Establish a calculation model for simply supported beam piers; S2. Establish a linear distributed force calculation model; S3, converting the linear distribution force interval to the standard calculation interval; S4, constructing an expression for undetermined coefficients for calculating the deflection of the pier body according to the standard calculation interval obtained in step S3; S5, substituting the undetermined coefficient expression of the deflection calculation constructed in step S4 into the deflection curve differential equation to obtain a set of equations related to the deflection of the calculation point; S6. Based on the boundary conditions, solve the specific pier top displacement.

2. The method for calculating the top displacement of a simply supported beam bridge pier under the action of a linear distributed force according to claim 1, characterized in that: In step S1, the specific process of establishing the calculation model of simply supported beam piers is as follows: For simply supported beam piers, a pier calculation model is established based on the pier height, cross-sectional dimensions, and material properties. The coordinate origin is set at the bottom of the pier, with the standard vertical direction being positive.

3. The method for calculating the displacement of the top of a simply supported beam bridge pier under the action of a linear distributed force according to claim 2, characterized in that: In step S2, the specific process of establishing the linear distributed force calculation model is as follows: A distribution diagram of the linear distributed force is established, and a mathematical expression is used to characterize the linear distributed force. The specific formula can be expressed as follows: q(x)=ax+b; Where x represents the coordinate along the pier height direction; q(x) represents the load magnitude of the linear distributed force along the pier height direction; a and b represent the slope and intercept of the linear distributed load, respectively.

4. The method for calculating the top displacement of a simply supported beam bridge pier under the action of a linear distributed force according to claim 3 is characterized in that: In step S3, the specific process of converting the linear distribution force interval to the standard calculation interval is as follows: The range of action of the linear distributed force [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 ξ.

5. The method for calculating the top displacement of a simply supported beam bridge pier under the action of a linear distributed force according to claim 4, characterized in that: In step S4, the specific process of constructing the undetermined coefficient expression for calculating the pier body deflection according to the standard calculation interval obtained in step S3 is as follows: in, In the above two formulas, z(ξ) represents the deflection parameter of the pier body in the standard calculation range; z1, z2, z3 represent the pier deflection corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval; θ1, θ2, θ3 represent the pier body rotation angles corresponding to the two end points of the standard calculation interval [-1,1] and the midpoint of the interval; κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ),κ 31 (ξ) represent six different undetermined coefficients for calculating the deflection of the pier body; Δl represents the length of the distributed load acting on the pier.

6. The method for calculating the displacement of the top of a simply supported beam bridge pier under the action of a linear distributed force according to claim 5, characterized in that: In step S5, the undetermined coefficient expression of the deflection calculation constructed in step S4 is substituted into the deflection curve differential equation to obtain the equation group related to the deflection of the calculation point, and the calculation formula is as follows: 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; It represents the first derivative of the linear distribution force concentration at the midpoint of the interval with respect to the x-coordinate; M| ξ＝0 represents the bending moment at the midpoint of the interval; q| ξ＝0 Represents the linearly distributed force concentration at the midpoint of the interval.

7. The method for calculating the top displacement of a simply supported beam bridge pier under the action of a linear distributed force according to claim 6, characterized in that: In step S6, the process of solving the specific pier top displacement according to the boundary conditions is as follows: First, calculate the bending moment and shear force corresponding to the midpoint of each linear distribution force action interval; Then, the calculated values ​​of the bending moment and shear force corresponding to these midpoints are substituted into the equations in step S5 to calculate the deflection and rotation angle of each pier node; Finally, the deflection value corresponding to the pier top node is selected as the pier top displacement to be solved.

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