A method for calculating displacement of a simply supported beam bridge pier top under linear distribution of force

Abstract

The application is a kind of simply supported beam bridge pier top displacement calculation method under linear distribution force, comprising 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 to the standard calculation interval; constructing the undetermined coefficient expression of the pier body deflection calculation; substituting the constructed undetermined coefficient expression of the deflection calculation into the deflection curve differential equation to obtain the equation group related to the deflection of the calculation point; according to the boundary conditions, the specific pier top displacement is solved. The application can calculate the pier top displacement of the uniform cross-section bridge pier in the field of transportation under the linear distribution force such as soil pressure and wind pressure, can be applied to the uniform cross-section bridge pier of any shape, can be applied to the calculation of the pier top displacement under the linear distribution load of any length and distribution position combination, and solves the problem that the pier top displacement calculation under complex linear distribution force needs to rely on the approximate numerical method calculation of node division.

Description

Technical Field

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

[0002] In high-speed railway lines, bridges account for an average of over 50%, and among the total length of these bridges, standard 32m and 24m simply supported beams constitute the majority. The longitudinal and transverse displacements of the pier tops are crucial indicators used in design specifications to control traffic safety and structural safety. External forces acting on piers typically include earth pressure and wind pressure, which can be abstracted as linearly distributed forces. Previously, calculating pier top displacements in design generally required dividing the pier into nodes and elements, then converting the distributed forces into equivalent nodal forces. This numerical calculation method involves numerous steps, which can impact design efficiency for the large number of simply supported beam piers in high-speed railways. For piers with uniform cross-sections, the deflection and rotation under external loads can be solved quickly and accurately using differential equations.

[0003] In view of the practical problems existing in the design of bridge piers, it is necessary to propose a faster and more accurate algorithm to solve the problem of calculating the displacement of the pier tops of a large number of simply supported beam bridge piers under linearly distributed forces. Summary of the Invention

[0004] This invention aims to address the shortcomings of existing technologies by providing a method for calculating the displacement of the top of a simply supported beam bridge pier under linearly distributed forces. This method is applicable to the calculation of the displacement of the top of piers with uniform cross-sections in transportation sectors such as railways, highways, municipal works, and light rail when subjected to linearly distributed forces such as earth pressure and wind pressure. The calculation method of this invention can quickly and directly yield accurate calculation results of the pier top displacement.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

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

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

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

[0009] S3. Convert the linear distributed force interval to the standard calculation interval;

[0010] S4. Based on the standard calculation interval obtained in step S3, construct the expression for the undetermined coefficients for calculating the pier deflection.

[0011] S5. Substitute the expression for the undetermined coefficients of the deflection calculation constructed in step S4 into the differential equation of the deflection curve to obtain the set of equations related to the deflection at the calculation point.

[0012] S6. Solve for the specific pier top displacement based on the boundary conditions.

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

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

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

[0016] Establish a distribution diagram of the linearly distributed force and characterize the linearly distributed force using a mathematical expression. The specific formula can be expressed as follows:

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

[0018] In the formula, x represents the coordinate along the pier height direction;

[0019] q(x) represents the magnitude of the linearly 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 distributed force interval to the standard calculation interval is as follows:

[0022] The range of action of the linearly distributed force [x1, x2] is transformed 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 calculating the undetermined coefficient expression for the pier deflection based on 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 within the standard calculation range;

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

[0030] θ1, θ2, and θ3 represent the pier rotation angles corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1].

[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 bridge pier;

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

[0033] In step S5, the expression for the undetermined coefficients of the deflection calculation constructed in step S4 is substituted into the differential equation of the deflection curve to obtain the system of equations related to the deflection at the calculation point. The calculation formula is as follows:

[0034]

[0035] In the formula,

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

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

[0038] F s For shear force;

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

[0040] The first derivative of the linear distribution intensity at the midpoint of the interval with respect to the x-coordinate;

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

[0042] q| ξ＝0 The linear distribution intensity of the force at the midpoint of the interval is represented.

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

[0044] First, calculate the bending moment and shear force corresponding to the midpoint of each linearly distributed force application range;

[0045] Then, substitute the calculated values ​​of bending moment and shear force corresponding to these midpoints into the equation set formula 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 this invention are: the calculation method of this invention can be used to calculate the displacement of the pier top of bridge piers with uniform cross-sections in the transportation field, such as railways, highways, municipal works, and light rail, when subjected to linearly distributed forces such as earth pressure and wind pressure. It is applicable to bridge piers with uniform cross-sections of any shape and to the calculation of the displacement of the pier top under linearly distributed loads of various arbitrary lengths and distribution positions. It solves the problem that the calculation of the displacement of the pier top under complex linearly distributed forces requires an approximate numerical method based on node division. Attached Figure Description

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

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

[0050] Figure 3 This is a schematic diagram illustrating the conversion of the linear distributed force interval to the standard calculation interval according to the present invention;

[0051] The following will describe in detail, with reference to the accompanying drawings, embodiments of the present invention. Detailed Implementation

[0052] The principles and features of the present invention are described below with reference to the accompanying drawings. The embodiments given are for illustrative purposes only and are not intended to limit the scope of the invention. The invention is described more specifically in the following paragraphs by way of example with reference to the accompanying drawings. The advantages and features of the invention will become clearer from the following description. It should be noted that the drawings are in a very simplified form and use non-precise proportions, and are only used to facilitate and clarify the illustration of the embodiments of the invention.

[0053] It should be noted that when a component is described as "fixed to" another component, it can be directly on the other component or may have a component in between. When a component is considered "connected to" another component, it can be directly connected to the other component or may have a component in between. When a component is considered "set on" another component, it can be directly set on the other component or may have a component in between. The terms "vertical," "horizontal," "left," "right," and similar expressions used in this document are for illustrative purposes only.

[0054] 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 this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

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

[0056] A method for calculating the displacement at the top of a simply supported beam bridge pier under linearly distributed force, such as... Figures 1 to 3 As shown, it includes the following steps:

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

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

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

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

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

[0062] Establish a distribution diagram of the linearly distributed force and characterize the linearly distributed force using a mathematical expression. The specific formula can be expressed as follows:

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

[0064] In the formula, x represents the coordinate along the pier height direction;

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

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

[0067] S3. Convert the linear distributed force interval to the standard calculation interval;

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

[0069] The range of action of the linearly distributed force [x1, x2] is transformed 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. Based on the standard calculation interval obtained in step S3, construct the expression for the undetermined coefficients for calculating the pier deflection.

[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 within the standard calculation range;

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

[0078] θ1, θ2, and θ3 represent the pier rotation angles corresponding to the two endpoints and the midpoint of the standard calculation interval [-1, 1].

[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 bridge pier;

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

[0081] S5. Substitute the expression for the undetermined coefficients of the deflection calculation constructed in step S4 into the differential equation of the deflection curve to obtain the set of equations related to the deflection at the calculation point.

[0082] The calculation formula is as follows:

[0083]

[0084] In the formula,

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

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

[0087] F s For shear force;

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

[0089] The first derivative of the linear distribution intensity at the midpoint of the interval with respect to the x-coordinate;

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

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

[0092] S6. Solve for the specific pier top displacement based on the boundary conditions;

[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 linearly distributed force application range;

[0095] Then, the calculated values ​​of bending moment, shear force and other parameters corresponding to these midpoints are substituted into the equation set formula 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 Implementation Example 1:

[0098] A high-speed railway project has a 38m high, uniformly shaped, circularly ended bridge pier with a fixed abutment at the bottom. The pier experiences earth pressure near its base, which can be simulated using linear distributed force. The displacement at the pier top caused by this earth pressure is calculated. The algorithm of this invention is incorporated into a self-developed calculation program, and the results are verified by comparing the self-developed program with a back-to-back calculation method using Midas.

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

[0100] Table 1 Comparison of pier top displacements under earth pressure on bridge piers with uniform cross-sections

[0101] Test content Algorithm of this invention Midas calculation results This invention / midas(%) Pier top displacement (m) 3.56E-06 3.56E-06 100.00％

[0102] As shown in Table 1, the calculation results of the present invention are completely consistent with those of the commercial software Midas, with zero error between the two. This indicates that the calculation results of the method of the present invention are very accurate for the case of a bridge pier with uniform cross-section bearing linear distributed force, and meet the engineering calculation requirements.

[0103] Using the calculation method of this invention, a calculation model for a simply supported beam bridge pier and a linear distributed force calculation model are established. Then, the linear distributed force interval is transformed into a standard calculation interval. Based on the obtained standard calculation interval, an expression for the undetermined coefficients of the pier body deflection is constructed. Then, the constructed expression for the undetermined coefficients of the deflection calculation is substituted into the differential equation of the deflection curve to obtain a set of equations related to the deflection at the calculation point. Finally, the specific pier top displacement is solved according to the boundary conditions.

[0104] The calculation method of this invention can be used to calculate the displacement of the pier top of bridge piers with uniform cross-sections in the transportation field, such as railways, highways, municipal works, and light rail, when subjected to linearly distributed forces such as earth pressure and wind pressure. It is applicable to bridge piers with uniform cross-sections of any shape and to the calculation of the displacement of the pier top under linearly distributed loads of various arbitrary lengths and distribution positions. It solves the problem that the calculation of the displacement of the pier top under complex linearly distributed forces requires an approximate numerical method based on node division.

[0105] The present invention has been described above by way of example with reference to the accompanying drawings. Obviously, the specific implementation of the present invention is not limited to the above-described manner. Any improvements made using the inventive concept and technical solution of the present invention, or direct application to other occasions without modification, are all within the protection scope of the present invention.

Claims

1. A method for calculating the displacement at the top of a simply supported beam bridge pier under linearly distributed force, characterized in that, Includes the following steps: S1. Establish a calculation model for simply supported beam bridge piers; S2. Establish a linear distributed force calculation model; The specific process is as follows: Establish a distribution diagram of the linearly distributed force and characterize the linearly distributed force using a mathematical expression. The specific formula can be expressed as follows: ； In the formula, Represents the coordinates along the pier height; This indicates the magnitude of the linearly distributed force along the pier height direction; , These represent the slope and intercept of the linear distributed load, respectively. S3. Convert the linear distributed force interval to the standard calculation interval; The specific process is as follows: The range of action of linearly distributed force Transform the coordinates to the standard calculation range. This corresponds to the coordinate parameters. Standard coordinates express; S4. Based on the standard calculation interval obtained in step S3, construct the expression for the undetermined coefficients for calculating the pier deflection. The specific process is as follows: ； in, ； In the above two formulas, This represents the deflection parameters of the pier body within the standard calculation range; , , Indicates the standard calculation interval The pier deflection corresponding to the two endpoints and the midpoint of the interval; , , Indicates the standard calculation interval The rotation angles of the pier body corresponding to the two endpoints and the midpoint of the interval; , , , , , These represent six different undetermined coefficients for calculating the deflection of the bridge pier. Indicates the length of the distributed load acting on the pier; S5. Substitute the expression for the undetermined coefficients of the deflection calculation constructed in step S4 into the differential equation of the deflection curve to obtain the set of equations related to the deflection at the calculation point. The calculation formula is as follows: ； In the formula, The elastic modulus of the bridge pier; Let be the moment of inertia of the pier cross section; For shear force; Indicates the shear force at the midpoint of the interval; The linear distribution of force intensity pairs at the midpoint of the interval The first derivative of the coordinates; Indicates the bending moment at the midpoint of the interval; The linear distribution intensity of the force at the midpoint of the interval; S6. Solve for the specific pier top displacement based on the boundary conditions.

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

3. The method for calculating the displacement at the top of a simply supported beam bridge pier under linearly distributed force as described in claim 2, characterized in that, In step S6, the process of solving for the specific pier top displacement based on the boundary conditions is as follows: First, calculate the bending moment and shear force corresponding to the midpoint of each linearly distributed force application range; Then, substitute the calculated values ​​of bending moment and shear force corresponding to these midpoints into the equation set formula 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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