Simple support pier top displacement calculation method under action of quadratic curve distribution force

Abstract

The invention relates to a simply-supported pier top displacement calculation method under the action of quadratic curve distribution force. The simply-supported pier top displacement calculation method comprises the following steps: establishing a simply-supported pier calculation model; establishing a quadratic curve distribution force calculation model; converting a quadratic curve distribution force interval into a standard calculation interval; according to the obtained standard calculation interval, constructing an undetermined coefficient expression of deflection calculation of the pier body of the bridge pier under the action of the quadratic curve distribution force; 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 suitable for uniform-section piers in any shapes, is suitable for calculation of pier top displacement under the action of quadratic curve distribution force of various arbitrary lengths, distribution positions and distribution shape combinations, and solves the problem that calculation of pier top displacement under the action of complex quadratic curve distribution force needs to depend on establishment of a finite element simulation model. The accuracy is high.

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 simply supported pier top under the action of a quadratic curve distributed force. Background Art

[0002] When the bridge piers are subjected to water pressure, ice pressure and wave force in water, the actual loads acting on the bridge piers are relatively complex. These complex loads can be summarized as a quadratic curve distribution force of a certain shape for simulation. In this case, for high-speed railway or highway simply supported beam bridge piers, the calculation of the pier body deflection or the calculation of the pier top displacement under the action of the quadratic distribution force will also be a practical engineering problem that needs to be solved.

[0003] For secondary distributed force loads, due to the complexity of the loads, the finite element modeling analysis is generally used to calculate the displacement of the pier top or the deflection of the pier body under its action. The disadvantage of this modeling analysis method is that it is time-consuming, the modeling process is complicated, and the modeling requirements for designers are high. This disadvantage is not suitable for situations where there are batches of bridge piers that need to be designed and calculated, and it will also affect the design efficiency for a large number of simply supported beam bridge piers. Therefore, in the design, it is urgent to study a new method that can both ensure the accuracy of design calculations and improve the speed of design calculations. 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 pier under the action of a quadratic curve distributed force. The method is fast and accurate, easy to operate and easy to be programmed, and can better solve the problem of calculating the top displacement of a simply supported pier under the action of a quadratic curve distributed force.

[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 pier top under a quadratic curve distributed force comprises the following steps:

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

[0008] S2. Establish a quadratic curve distribution force calculation model;

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

[0010] S4, constructing an expression for undetermined coefficients for calculating the deflection of the pier body under the quadratic curve distribution force 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 the simply supported pier is as follows:

[0014] S11. Establishing an overall coordinate system, with the origin of the coordinate system set at the centroid of the cross section at the bottom of the pier;

[0015] S12. Input the geometric parameters of the simply supported pier, including the pier height, cross-section type and corresponding dimensions;

[0016] S13. Input the material parameters of the simply supported pier, including the grade and strength of the pier concrete, and the grade and strength of the steel bars or prestressed steel bars.

[0017] In step S2, the specific process of establishing the quadratic curve distribution force calculation model is as follows:

[0018] The following mathematical expression is used to characterize the quadratic curve distribution force, and the characterization formula is:

[0019] q(x)=ax 2 +bx+c;

[0020] In the formula,

[0021] q(x) represents the load magnitude of the quadratic curve distributed force along the pier height direction;

[0022] x represents the coordinate along the pier height direction;

[0023] a, b, and c represent the relevant load distribution parameters of the quadratic curve distribution force.

[0024] In step S3, the specific process of converting the quadratic curve distribution force interval to the standard calculation interval is as follows:

[0025] The action range of the quadratic curve distribution 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 can be expressed by the standard coordinate parameter ξ;

[0026] In the formula,

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

[0028] In step S4, the specific process of constructing the undetermined coefficient expression for calculating the deflection of the pier body under the quadratic curve distribution force is as follows:

[0029]

[0030] in,

[0031]

[0032] In the formula,

[0033] z(ξ) represents the deflection parameter of the pier body in the standard calculation range;

[0034] z1, z2, z3, z4, z5 represent the pier deflections corresponding to the two end points of the standard calculation interval [-1,1] and the quarter points of the interval;

[0035] θ1 and θ2 represent the corresponding pier body rotation angles at the two end points of the standard calculation interval [-1,1];

[0036] κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ),κ 40 (ξ),κ 50 (ξ) represent seven different undetermined parameters for calculating the deflection of the pier body;

[0037] Δl represents the length of the quadratic curve distributed force acting on the pier body.

[0038] In step S5, the constructed undetermined coefficient expression of the deflection calculation 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:

[0039]

[0040] In the formula,

[0041] E is the elastic modulus of the pier;

[0042] I is the moment of inertia of the pier cross section;

[0043] F s is the shear force;

[0044] F s | ξ＝0 represents the shear force at the midpoint of the interval;

[0045] The first derivative of the concentration of the quadratic curve distribution force at the midpoint of the standard interval with respect to the x-coordinate;

[0046] M| ξ＝0 represents the bending moment at the midpoint of the standard interval;

[0047] q ξ＝0 The quadratic curve representing the midpoint of the standard interval distributes the force concentration.

[0048] In step S6, the process of solving the specific pier top displacement according to the boundary conditions is as follows:

[0049] S61. Calculate the values ​​of bending moment and shear force corresponding to the midpoint of each quadratic curve distribution force action interval;

[0050] S62, substituting the calculated values ​​of the bending moment and shear force corresponding to these midpoints into the formula of step S5, and calculating the deflection and rotation angle of each pier node;

[0051] S63. Select the deflection value corresponding to the node at the top of the simply supported pier as the displacement at the top of the pier to be solved.

[0052] The beneficial effects of the present invention are as follows: the present invention can be applied to bridge piers of equal cross-section of any shape, and is applicable to the calculation of pier top displacement under the action of quadratic curve distribution forces of any length, distribution position and distribution shape combination, thereby solving the problem that the calculation of pier top displacement under the action of complex quadratic curve distribution forces needs to rely on the establishment of a finite element simulation model for calculation, and has high accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a schematic diagram of the calculation process flow of the present invention;

[0054] Figure 2 It is a schematic diagram of the quadratic curve distribution force acting on a simply supported pier according to the present invention;

[0055] Figure 3 It is a schematic diagram of converting the quadratic curve distribution force action interval into the standard calculation interval of the present invention;

[0056] The following is a detailed description of the embodiments of the present invention with reference to the accompanying drawings. DETAILED DESCRIPTION

[0057] 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.

[0058] 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.

[0059] 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.

[0060] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0061] A method for calculating the displacement of the top of a simply supported pier under the action of a quadratic curve distributed force, such as Figure 1 As shown, the following steps are included:

[0062] S1. Establish a calculation model for simply supported piers, such as Figure 2 As shown, the specific process is as follows:

[0063] S11. Establishing an overall coordinate system, with the origin of the coordinate system set at the centroid of the cross section at the bottom of the pier;

[0064] S12. Input the geometric parameters of the simply supported pier, including the pier height, cross-section type and corresponding dimensions;

[0065] S13. Input the material parameters of the simply supported pier, including the grade and strength of the pier concrete, and the grade and strength of the steel bars or prestressed steel bars.

[0066] S2. Establish a quadratic curve distribution force calculation model, such as Figure 2 As shown, the specific process is as follows:

[0067] The following mathematical expression is used to characterize the quadratic curve distribution force, and the characterization formula is:

[0068] q(x)=ax 2 +bx+c;

[0069] In the formula,

[0070] q(x) represents the load magnitude of the quadratic curve distributed force along the pier height direction;

[0071] x represents the coordinate along the pier height direction;

[0072] a, b, and c represent the relevant load distribution parameters of the quadratic curve distribution force.

[0073] S3. Convert the quadratic curve distribution force interval to the standard calculation interval, such as Figure 3 As shown, the specific process is as follows:

[0074] The action range of the quadratic curve distribution 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 can be expressed by the standard coordinate parameter ξ;

[0075] In the formula,

[0076] ξ represents the standard coordinate parameter of the standard calculation interval [-1,1].

[0077] S4. According to the standard calculation interval obtained in step S3, an expression for the undetermined coefficient for calculating the deflection of the pier body under the quadratic curve distribution force is constructed as follows:

[0078]

[0079] in,

[0080]

[0081] In the formula,

[0082] z(ξ) represents the deflection parameter of the pier body in the standard calculation range;

[0083] z1, z2, z3, z4, z5 represent the pier deflections corresponding to the two end points of the standard calculation interval [-1,1] and the quarter points of the interval;

[0084] θ1 and θ2 represent the corresponding pier body rotation angles at the two end points of the standard calculation interval [-1,1];

[0085] κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ),κ 40 (ξ),κ 50 (ξ) represent seven different undetermined parameters for calculating the deflection of the pier body;

[0086] Δl represents the length of the quadratic curve distributed force acting on the pier body.

[0087] S5. Substitute the undetermined coefficient expression of the deflection calculation constructed in step S4 into the deflection curve differential equation to obtain the equation group related to the deflection of the calculation point. The calculation formula is as follows:

[0088]

[0089] In the formula,

[0090] E is the elastic modulus of the pier;

[0091] I is the moment of inertia of the pier cross section;

[0092] F s is the shear force;

[0093] F s | ξ＝0 represents the shear force at the midpoint of the interval;

[0094] The first derivative of the concentration of the quadratic curve distribution force at the midpoint of the standard interval with respect to the x-coordinate;

[0095] M| ξ＝0 represents the bending moment at the midpoint of the standard interval;

[0096] q ξ＝0 The quadratic curve representing the midpoint of the standard interval distributes the force concentration.

[0097] S6. According to the boundary conditions, solve the specific pier top displacement. The process is as follows:

[0098] S61, calculating the values ​​of parameters such as bending moment and shear force corresponding to the midpoint of each quadratic curve distribution force action interval;

[0099] S62, substituting the calculated values ​​of the bending moment, shear force and other parameters corresponding to the midpoints into the formula of step S5, and calculating the deflection and rotation angle of each pier node;

[0100] S63. Select the deflection value corresponding to the node at the top of the simply supported pier as the displacement at the top of the pier to be solved. Specific embodiment 1

[0102] A highway pier with equal cross-section and rectangular shape is 26m high and has a consolidated bottom. It is located in a deep water area and is subject to large water flow impact force, extreme ice pressure, etc. These complex forces can be simplified into quadratic curve distribution forces with specific coefficients acting on the middle part of the pier. The displacement of the pier top caused by this quadratic curve distribution force is calculated.

[0103] The algorithm of the present invention is compiled into an executable program, and the calculation of this engineering case is carried out, and the calculation result of the present invention and the calculation result of the midas are compared and verified by back-to-back comparison calculation method. The comparison of the calculation result of the present invention and the calculation result of the midas is shown in Table 1 below.

[0104] Table 1 Comparison of pier top displacements of equal-section bridge piers subjected to quadratic distribution forces

[0105] Test content Algorithm of the present invention Midas's calculation results The present invention / midas(%) Pier top displacement (m) 2.68E-05 2.60E-05 103.08％

[0106] 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 4%, indicating that the calculation results of the present invention are very accurate for the working condition of equal-section bridge piers subjected to quadratic curve distributed forces and meet the engineering calculation requirements.

[0107] 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 administration, light rail, etc., when they are subjected to quadratic curve distribution forces such as water pressure, ice pressure, and wave force.

[0108] The present invention can be applied to bridge piers of equal cross-section of any shape, and is applicable to the calculation of pier top displacement under the action of quadratic curve distribution forces of any length, distribution position and distribution shape combination, thus solving the problem that the calculation of pier top displacement under complex quadratic curve distribution forces needs to rely on the establishment of a finite element simulation model for calculation.

[0109] 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 a simply supported pier top under a quadratic curve distributed force, characterized in that: The steps include: S1. Establish a calculation model for simply supported piers; S2. Establish a quadratic curve distribution force calculation model; S3, converting the quadratic curve distribution force interval to the standard calculation interval; S4, constructing an expression for undetermined coefficients for calculating the deflection of the pier body under the quadratic curve distribution force 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 displacement of the top of a simply supported pier under the action of a quadratic curve distributed force according to claim 1, characterized in that: In step S1, the specific process of establishing the calculation model of the simply supported pier is as follows: S11. Establishing an overall coordinate system, with the origin of the coordinate system set at the centroid of the cross section at the bottom of the pier; S12. Input the geometric parameters of the simply supported pier, including the pier height, cross-section type and corresponding dimensions; S13. Input the material parameters of the simply supported pier, including the grade and strength of the pier concrete, and the grade and strength of the steel bars or prestressed steel bars.

3. The method for calculating the displacement of the top of a simply supported pier under the action of a quadratic curve distributed force according to claim 2, characterized in that: In step S2, the specific process of establishing the quadratic curve distribution force calculation model is as follows: The following mathematical expression is used to characterize the quadratic curve distribution force, and the characterization formula is: q(x)=ax 2 +bx+c; In the formula, q(x) represents the load magnitude of the quadratic curve distributed force along the pier height direction; x represents the coordinate along the pier height direction; a, b, and c represent the relevant load distribution parameters of the quadratic curve distribution force.

4. The method for calculating the displacement of the top of a simply supported pier under the action of a quadratic curve distributed force according to claim 3, characterized in that: In step S3, the specific process of converting the quadratic curve distribution force interval to the standard calculation interval is as follows: The action range of the quadratic curve distribution 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 can be expressed by the standard coordinate parameter ξ; In the formula, ξ represents the standard coordinate parameter of the standard calculation interval [-1,1].

5. The method for calculating the displacement of the top of a simply supported pier under the action of a quadratic curve distributed force according to claim 4, characterized in that: In step S4, the specific process of constructing the undetermined coefficient expression for calculating the deflection of the pier body under the quadratic curve distribution force is as follows: in, In the formula, z(ξ) represents the deflection parameter of the pier body in the standard calculation range; z1, z2, z3, z4, z5 represent the pier deflections corresponding to the two end points of the standard calculation interval [-1,1] and the quarter points of the interval; θ1 and θ2 represent the corresponding pier body rotation angles at the two end points of the standard calculation interval [-1,1]; κ 10 (ξ),κ 11 (ξ),κ 20 (ξ),κ 21 (ξ),κ 30 (ξ),κ 40 (ξ),κ 50 (ξ) represent seven different undetermined parameters for calculating the deflection of the pier body; Δl represents the length of the quadratic curve distributed force acting on the pier body.

6. The method for calculating the displacement of the top of a simply supported pier under the action of a quadratic curve distributed force according to claim 5, characterized in that: In step S5, the constructed undetermined coefficient expression of the deflection calculation 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; The first derivative of the concentration of the quadratic curve distribution force at the midpoint of the standard interval with respect to the x-coordinate; M| ξ＝0 represents the bending moment at the midpoint of the standard interval; q| ξ＝0 The quadratic curve representing the midpoint of the standard interval distributes the force concentration.

7. The method for calculating the displacement of the top of a simply supported pier under the action of a quadratic curve 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: S61. Calculate the values ​​of bending moment and shear force corresponding to the midpoint of each quadratic curve distribution force action interval; S62, substituting the calculated values ​​of the bending moment and shear force corresponding to these midpoints into the formula of step S5, and calculating the deflection and rotation angle of each pier node; S63. Select the deflection value corresponding to the node at the top of the simply supported pier as the displacement at the top of the pier to be solved.

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