A method for calculating displacement of a simply supported pier under action of a quadratic curve distributed force

Abstract

The application is a simply supported pier top displacement calculation method under the action of quadratic curve distribution force, comprising the following steps: establishing a simply supported pier calculation model; establishing a quadratic curve distribution force calculation model; converting the quadratic curve distribution force interval to a standard calculation interval; according to the obtained standard calculation interval, constructing a pending coefficient expression of the deflection calculation of the pier body under the action of the quadratic curve distribution force; substituting the constructed pending coefficient expression of the deflection calculation into the flexural curve differential equation to obtain an equation group related to the deflection of the calculation point; and solving the specific pier top displacement according to the boundary conditions. The application can be applied to any shape of equal cross-section bridge pier, and can be applied to the calculation of the pier top displacement under the action of quadratic curve distribution force with any length, distribution position and distribution shape combination, solves the problem that the calculation of the pier top displacement under the action of complex quadratic curve distribution force needs to rely on the establishment of a finite element simulation model for calculation, and has high accuracy.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bridge engineering in the transportation industry, and particularly relates to a calculation method for the displacement of the top of a simply supported pier under the action of a quadratic curve distribution force. BACKGROUND

[0002] When a pier bears water pressure, ice pressure and wave force in water, the actual load acting on the pier is relatively complex. For these complex loads, they can be simulated as a quadratic curve distribution force of a certain shape. In this case, for a simply supported pier of a high-speed railway or highway bridge, the calculation of the pier body deflection or the calculation of the pier top displacement under the action of a quadratic distribution force is also an actual engineering problem to be solved.

[0003] For a quadratic distribution force load, due to the complexity of the load, the general way to calculate the pier top displacement or pier body deflection under the action of the load is the finite element modeling analysis method. The disadvantage of this modeling analysis method is that it is relatively time-consuming, the modeling process is complex, and the modeling requirements for the design personnel are higher. For the case of a large number of piers to be designed and calculated, this disadvantage is not suitable, and for a large number of simply supported pier bridges, it will also affect the design efficiency. Therefore, in the design, it is urgent to develop a new method that can ensure the design calculation accuracy and improve the design calculation speed. SUMMARY

[0004] The present application aims to solve the problems of the prior art and provides a calculation method for the displacement of the top of a simply supported pier under the action of a quadratic curve distribution force, which is fast and accurate, easy to operate and easy to program, and can better solve the problem of calculating the displacement of the top of a simply supported pier under the action of a quadratic curve distribution force.

[0005] To achieve the above-mentioned purpose, the present application adopts the following technical solutions:

[0006] A calculation method for the displacement of the top of a simply supported pier under the action of a quadratic curve distribution force, comprising the following steps:

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

[0008] S2, establishing a calculation model of a quadratic curve distribution force;

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

[0010] S4, constructing a coefficient expression of the deflection calculation of the pier body under the action of the quadratic curve distribution force according to the standard calculation interval obtained in step S3;

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

[0012] S6, solving the specific pier top displacement according to the boundary conditions.

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

[0014] S11, an overall coordinate system is established, and the coordinate origin is set at the sectional centroid of the pier bottom;

[0015] S12, input the geometric parameters of the simply supported pier, including the pier height, the section type and the corresponding size;

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

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

[0018] The quadratic curve distribution force is characterized by the following mathematical expression, and the characterization formula is:

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

[0020] In the formula,

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

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

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

[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 [x1, x2] of the quadratic curve distribution force is converted to the standard calculation interval [-1, 1] through coordinate transformation, and the standard coordinate corresponding to the coordinate parameter x can be represented by the standard coordinate parameter ξ at this time;

[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 of the deflection calculation of the pier body under the action of the quadratic curve distribution force is as follows:

[0029]

[0030] Wherein,

[0031]

[0032] wherein,

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

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

[0035] θ1, θ2 represent the rotation angles of the pier body corresponding to the two endpoints 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) respectively represent seven different undetermined parameters for calculating the deflection of the pier body;

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

[0038] In step S5, the constructed undetermined coefficient expression for deflection calculation is substituted into the differential equation of the deflection curve to obtain an equation group related to the deflection of the calculation point, and the calculation formula is as follows:

[0039]

[0040] wherein,

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

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

[0043] F s is the shear force;

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

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

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

[0047] q ξ＝0 represents the quadratic curve distribution force intensity at the midpoint of the standard interval.

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

[0049] S61, the bending moment and shear force values corresponding to the midpoint of each quadratic curve distribution force action interval are calculated;

[0050] S62, the calculated bending moment and shear force values corresponding to the midpoint are substituted into the formula of step S5 to calculate the deflection and rotation angle of each pier node;

[0051] S63, the deflection value corresponding to the simply supported pier top node is selected as the pier top displacement to be solved.

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

[0053] Figure 1 The flow chart of the calculation process of the present application is shown in the figure;

[0054] Figure 2 The schematic diagram of the quadratic curve distribution force acting on the simply supported pier of the present application is shown in the figure;

[0055] Figure 3 The schematic diagram of the conversion of the quadratic curve distribution force action interval to the standard calculation interval of the present application is shown in the figure;

[0056] The principles and features of the present application will be described in detail below with reference to the accompanying drawings. DETAILED DESCRIPTION

[0057] The principles and features of the present application will be described in detail below with reference to the accompanying drawings.

[0058] It should be understood that when a component is referred to as being "on" another component, it can be directly on the other component or intervening components can also be present. When a component is referred to as being "connected" to another component, it can be directly connected to the other component, or intervening components can be present. When a component is referred to as being "disposed" on another component, it can be directly disposed on the other component, or intervening components can be present. The terms "vertical", "horizontal", "left", "right", and similar terms as used herein are for purposes of description only.

[0059] 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 application belongs. The terminology used in the description herein is for describing particular embodiments only and is not intended to be limiting of the application. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.

[0060] The application will be further described with reference to the drawings and embodiments:

[0061] A simply supported pier top displacement calculation method under the action of a quadratic curve distribution force, as shown in Figure 1 , includes the following steps:

[0062] S1, a simply supported pier calculation model is established, as shown in Figure 2 , the specific process is as follows:

[0063] S11, an overall coordinate system is established, and the coordinate origin is set at the sectional centroid of the pier bottom;

[0064] S12, the geometric parameters of the simply supported pier are input, including the pier height, the section type, and the corresponding size;

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

[0066] S2, a quadratic curve distribution force calculation model is established, as shown in Figure 2 , the specific process is as follows:

[0067] The quadratic curve distribution force is characterized by the following mathematical expression, and the characterization formula is:

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

[0069] In the formula,

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

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

[0072] a, b, c respectively represent the related 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 [x1, x2] of the quadratic curve distribution force 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, the undetermined coefficient expression of the deflection calculation of the pier shaft under the action of the quadratic curve distribution force is constructed, as follows:

[0078]

[0079] Wherein,

[0080]

[0081] In the formula,

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

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

[0084] θ1, θ2 represent the pier shaft rotation angle corresponding to 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) respectively represent seven different undetermined parameters of the pier shaft deflection calculation;

[0086] Δl represents the length of the quadratic curve distribution force acting on the pier shaft.

[0087] S5, the pending coefficient expression of the deflection calculation of step S4 is substituted into the deflection curve differential equation to obtain an equation related to the deflection of the calculation point, and the calculation formula is as follows:

[0088]

[0089] In the formula,

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

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

[0092] F s is the shear force;

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

[0094] denotes the first derivative of the bending moment of the standard interval midpoint with respect to the x coordinate;

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

[0096] q ξ＝0 denotes the quadratic curve distribution force intensity of the standard interval midpoint.

[0097] S6, according to the boundary conditions, the specific pier top displacement is solved, and the process is as follows:

[0098] S61, the numerical value of the bending moment, shear force and other parameters corresponding to the midpoint of each quadratic curve distribution force action interval is calculated;

[0099] S62, the numerical value of the bending moment, shear force and other parameters corresponding to the midpoint calculated is substituted into the formula of step S5, and the deflection and rotation angle of each pier node are calculated;

[0100] S63, the deflection value corresponding to the simply supported pier top node is selected as the pier top displacement to be solved. Specific embodiment 1

[0102] A highway equal cross section rectangular pier with a pier height of 26m, a pier bottom consolidation, located in a deep water area, subjected to a large water flow impact force, a limit ice pressure and the like, these complex forces can be simplified as a quadratic curve distribution force with specific coefficients acting on the middle part of the pier, and the pier top displacement caused by the quadratic curve distribution force is calculated.

[0103] The algorithm of the application is compiled into an executable program, the engineering case is calculated, and the back-to-back comparison calculation method with the commercial finite element software midas is used for verification. The comparison between the calculation results of the application and the calculation results of midas is shown in Table 1.

[0104] Table 1 Equal cross-section pier top displacement comparison under quadratic curve distribution force

[0105] Test content Invention algorithm Midas calculation result Invention / Midas (%) Pier top displacement (m) 2.68E-05 2.60E-05 103.08％

[0106] From Table 1, it can be seen that the calculation result of the application is very close to the calculation result of the commercial software midas, and the error is within 4%, indicating that the calculation result of the application is very accurate for the working condition of the equal cross-section pier under quadratic curve distribution force, and meets the engineering calculation requirement.

[0107] The application can calculate the pier top displacement of the equal cross-section pier in the field of transportation such as railway, highway, municipal, light rail, etc. when subjected to water pressure, ice pressure, wave force and other quadratic curve distribution forces.

[0108] The application can be applied to any shape of equal cross-section pier, and can be applied to the calculation of the pier top displacement under the action of quadratic curve distribution force with various lengths, distribution positions and distribution shape combinations, and solves the problem that the calculation of the pier top displacement under the action of complex quadratic curve distribution force needs to rely on the establishment of a finite element simulation model for calculation.

[0109] The application has been described above in conjunction with the drawings, and obviously the specific implementation of the application is not limited by the above manner, and various improvements using the method concept and technical solution of the application, or direct application in other occasions without improvement, are all within the protection scope of the application.

Claims

1. A method for calculating the displacement at the top of a simply supported pier under a quadratic curve distributed force, characterized in that, Includes the following steps: S1. Establish a calculation model for a simply supported pier; S2. Establish a calculation model for the distributed force of a quadratic curve; The specific process is as follows: The distributed force of a quadratic curve is characterized by the following mathematical expression: ； In the formula, This indicates the magnitude of the load along the pier height direction due to the quadratic curve distribution force; Represents the coordinates along the pier height; , , These represent the relevant load distribution parameters of the quadratic curve distributed force; S3. Convert the quadratic curve force distribution interval to the standard calculation interval; The specific process is as follows: The range of action of the force distributed in the quadratic curve Transform the coordinates to the standard calculation range. This corresponds to the coordinate parameters. Standard coordinates can be obtained using standard coordinate parameters. express; In the formula, Indicates the standard calculation interval Standard coordinate parameters; S4. Based on the standard calculation interval obtained in step S3, construct the expression for the undetermined coefficients for calculating the deflection of the bridge pier body under the action of quadratic curve distributed force. The specific process is as follows: ； in, ； In the formula, 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 quarter points of the interval; , Indicates the standard calculation interval The rotation angle of the pier body corresponding to the two endpoints; , , , , , , These represent seven different undetermined parameters for calculating the deflection of the bridge pier. It represents the length of the quadratic force distributed 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 quadratic curve distribution of the force intensity pair at the midpoint of the standard interval The first derivative of the coordinates; Indicates the bending moment at the midpoint of the standard interval; The intensity of the quadratic curve distribution at the midpoint of the standard interval; S6. Solve for the specific pier top displacement based on the boundary conditions.

2. The method for calculating the displacement of the top of a simply supported pier under a quadratic curve 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 pier is as follows: S11. Establish an overall coordinate system, with the origin set at the centroid of the cross section at the bottom of the bridge 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 reinforcing steel or prestressed steel.

3. The method for calculating the displacement of the top of a simply supported pier under a quadratic curve distributed force according to 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: S61. Calculate the values ​​of bending moment and shear force corresponding to the midpoint of the force distribution range of each quadratic curve. S62. Substitute the calculated values ​​of bending moment and shear force corresponding to these midpoints into the formula in step S5 to calculate the deflection and rotation angle of each pier node. S63. Select the deflection value corresponding to the top node of the simply supported pier as the pier top displacement to be solved.

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