Design method for arch rib axis of arch bridge

Abstract

The invention discloses a design method for an arch rib axis of an arch bridge, which belongs to the technical field of bridge engineering and comprises the following steps: establishing a coordinate system for the arch bridge and dividing a half-span arch rib into a plurality of nodes to obtain a multi-order conversion matrix; calculating the load linear meter set degree of the arch rib, calculating the coefficient of the load linear meter set degree of the arch rib expressed according to a polynomial function and the horizontal component force of the axial pressure of the cross section of the arch rib through the multi-order conversion matrix, and calculating the coefficient of the axis of the arch rib expressed according to the polynomial function to obtain an axis function of the arch rib; correcting according to the arch rib arc length linear meter set degree by combining an arch rib axis function with the first arch rib self-weight load linear meter set degree to obtain a second arch rib self-weight load linear meter set degree, and calculating an error maximum value of the first arch rib self-weight load linear meter set degree and the second arch rib self-weight load linear meter set degree; and judging whether the arch rib axis function is a reasonable arch rib axis or not. According to the method, engineering application is facilitated, and meanwhile the method is suitable for reasonable arch rib axis design of the arch bridge.

Description

Technical Field

[0001] The invention relates to the technical field of bridge engineering, and in particular to a design method for an arch rib axis of an arch bridge. Background Art

[0002] An arch bridge is a type of bridge that converts vertical loads into horizontal thrust at the arch foot through an arch ring or arch rib (the arch ring cross section is called an arch rib when it is designed in a separated form). Figure 1 As shown in the figure, the horizontal reaction force at the arch foot will greatly offset the bending moment caused by the load in the arch ring (or arch rib). Therefore, compared with the beam of the same span, the bending moment, shear force and deformation of the arch are much smaller, and the force on the arch ring is mainly compression, which enhances the utilization efficiency of the material. Therefore, the span capacity of the arch bridge is very large. When the terrain conditions at the arch foot are suitable and the engineering geological conditions have strong horizontal bearing capacity, it has high economic efficiency in the span range of 50 to 700 meters.

[0003] The key issue in the design of arch bridges is to determine the reasonable arch axis, so as to ensure that the cross section is basically compressed under the external load, so that the structure is more reasonably stressed and the material utilization efficiency is improved. The determination of the arch axis is related to the load distribution form on it. For conventional large-span arch bridges, the arch axis is basically based on a catenary or a quadratic parabola. When the rise and span are determined, the shape of the arch axis is related to the arch axis coefficient.

[0004] At present, the arch axis design of arch bridges in engineering still follows the arch axis design and calculation method of conventional arch bridges, that is, a specific catenary arch axis coefficient is adopted for the entire arch axis, and then the arch axis coefficient is adjusted according to the stress state of the arch foot and arch crown obtained by repeated forward trial and error calculation. It has poor adaptability to the load distribution form on the arch, and the calculation and design efficiency is low. Therefore, it is necessary to propose a design method for a reasonable arch rib axis that is easy to apply in engineering and suitable for any load distribution form of arch bridges. Summary of the invention

[0005] In view of the problems existing in the prior art, the present invention provides a design method for the arch rib axis of an arch bridge. By deriving the mechanical equilibrium equation when the arch rib is axially compressed and adopting a mathematical algorithm, an analytical function solution of a reasonable arch rib axis is determined. This method is convenient for engineering application and is suitable for reasonable arch rib axis design of arch bridges, making bridge design more scientific and efficient.

[0006] The technical solution of the present invention is as follows:

[0007] In a first aspect of the present invention, a method for designing an arch rib axis of an arch bridge is provided, comprising:

[0008] A coordinate system is established with the apex of the arch rib of the arch bridge as the origin, and the half-span arch rib from the arch top to the arch foot is divided into multiple nodes, and a multi-order transformation matrix is ​​obtained according to the coordinates of multiple nodes;

[0009] Calculate the bridge system weight borne by the arch rib in the completed state, and convert it into the bridge system load concentration per meter according to multiple nodes;

[0010] Calculate the concentration of the arch rib arc length in meters at multiple nodes as the concentration of the self-weight load in meters of the first arch rib, or preliminarily estimate the concentration of the self-weight load in meters of the first arch rib;

[0011] The concentration of the bridge system load per meter is added to the concentration of the first arch rib deadweight load per meter to get the concentration of the arch rib load per meter.

[0012] The coefficient of the load concentration per meter on the arch rib expressed by a polynomial function and the horizontal component of the axial pressure on the arch rib section are calculated through a multi-order conversion matrix, and the coefficient of the arch rib axis expressed by a polynomial function is calculated to obtain the arch rib axis function;

[0013] Calculate the first-order derivative of the arch rib axis function at multiple nodes, and correct the first arch rib self-weight load concentration by the arch rib arc length concentration to obtain the second arch rib self-weight load concentration, and calculate the maximum error between the first arch rib self-weight load concentration and the second arch rib self-weight load concentration;

[0014] If the error is less than 0.1-0.01, it can be considered that convergence has been achieved, and the arch rib axis function is a reasonable arch rib axis. If the error is greater than 0.1-0.01, the meter concentration of the second arch rib's own weight load is added to the meter concentration of the bridge system load to obtain a new meter concentration of the arch rib load and the calculation is repeated until convergence.

[0015] In some embodiments of the present invention, the coordinate system is established with the apex of the arch rib of the arch bridge as the origin, specifically: the apex of the arch rib of the arch bridge is the origin, the X-axis direction of the coordinate system is the longitudinal direction of the bridge, and the Y-axis direction of the coordinate system is the arch rib height.

[0016] In some embodiments of the present invention, the weight of the bridge system borne by the arch rib in the completed bridge state includes the weight of the bridge system including columns or hangers and 0-1 times the lane load.

[0017] In some embodiments of the present invention, the calculation of the concentration of the arch rib arc length in meters at multiple nodes of the arch rib is specifically: calculating the concentration of the arch rib arc length in meters at multiple nodes of the arch rib according to the cross-sectional area and the structural structure of the arch rib.

[0018] In some embodiments of the present invention, the multi-order conversion matrix is ​​specifically:

[0019]

[0020] Among them, x represents the horizontal axis coordinates of multiple nodes, and m represents the number of nodes.

[0021] In some embodiments of the present invention, the load concentration per meter of the arch rib is calculated according to a polynomial function The rib axis is expressed as a polynomial function To express;

[0022] Among them, b is the expression coefficient of the concentration function of the arch rib load per meter, a is the expression coefficient of the arch rib axis function, j ranges from 0 to m, and i ranges from 2 to m+2.

[0023] In some embodiments of the present invention, the coefficients of the polynomial function for calculating the load concentration of the arch rib in meters by the multi-order conversion matrix are specifically:

[0024]

[0025] Among them, q represents the load concentration per meter borne by the arch rib.

[0026] In some embodiments of the present invention, the horizontal component of the axial pressure of the rib section is calculated as follows:

[0027]

[0028] Among them, T represents the horizontal component of the axial pressure of the arch rib section, F represents the sagittal height of the arch rib, and l represents the half span of the arch rib.

[0029] In some embodiments of the present invention, the coefficients of calculating the rib axis expressed by a polynomial function are specifically:

[0030]

[0031] Among them, a 0 =a 1 =0.

[0032] In some embodiments of the present invention, the first-order derivative of the rib axis function at multiple nodes is calculated, and the concentration of the first rib self-weight load per meter is corrected according to the concentration of the rib arc length per meter to obtain the concentration of the second rib self-weight load per meter, specifically:

[0033]

[0034] in, represents the concentration of the deadweight load of the second arch rib in meters, d represents the concentration of the arc length of the arch rib in meters, y 、 represents the first derivative of the rib axis function;

[0035] The maximum error of the self-weight load concentration per meter of the first arch rib and the self-weight load concentration per meter of the second arch rib is calculated as follows:

[0036]

[0037] Wherein, g represents the linear concentration of the self-weight load of the first arch rib, and R represents the maximum value of the correction error of the linear concentration of the self-weight load of the first arch rib and the linear concentration of the self-weight load of the second arch rib at multiple nodes.

[0038] One or more technical solutions of the present invention have the following beneficial effects:

[0039] The present invention provides a design method for the arch rib axis of an arch bridge. The method uses a mathematical algorithm to determine the analytical function solution of the reasonable axis of the arch rib based on the derivation of the mechanical equilibrium equation when the arch rib axis is under pressure. The method can directly obtain the reasonable function analytical solution of the arch rib axis for any distributed load, fills the gap in the arch bridge design method, reduces the use of arch rib materials while ensuring the structural safety, and improves the safety and economy of the bridge structure.

[0040] In addition, this design method uses calculus mathematical algorithms to determine the reasonable functional analytical solution of the arch rib axis, which is less dependent on the engineering experience of designers. By writing calculation programs, bridge design can be made more scientific and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 A flow chart of a design method for an arch rib axis of an arch bridge proposed in Embodiment 1 of the present invention;

[0042] Figure 2 This is a schematic diagram of the arch bridge structure proposed in Example 1 of the present invention;

[0043] Figure 3 This is a schematic diagram of the coordinate system direction proposed in Example 1 of the present invention.

[0044] In the figure: 1. Main beam; 2. Column; 3. Arch seat and foundation; 4. Arch rib; 5. Arch rib axis. DETAILED DESCRIPTION

[0045] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0046] Example 1

[0047] like Figure 2-3 As shown, the arch bridge provided in this embodiment includes a main beam 1, columns 2, arch seats and foundations 3, arch ribs 4 and arch rib axes 5.

[0048] In a typical embodiment of the present invention, a design method for an arch rib axis of an arch bridge is provided, comprising:

[0049] Step 1: Establish a coordinate system with the apex of the arch rib of the arch bridge as the origin, x as the longitudinal direction of the bridge, and y as the height direction of the arch rib. The sagittal height of the arch rib is F, and the span of the arch rib is L.

[0050] Step 2: For the half-span arch rib, the points from the arch crown to the arch foot are divided into m points, preferably m is 3 to 5, and the spacing is close. The m+1 order transformation matrix C is calculated according to the x coordinates of the m points:

[0051]

[0052] where x j , j = 0...m is the x coordinate of the point where the sequence is divided from the dome;

[0053] Step 3: Calculate the weight of the bridge system including columns or hangers borne by the arch ribs in the completed state, taking into account part of the vehicle load (preferably 0.5 times the lane load), and convert it into load concentration per meter p according to m points. j ;

[0054] Step 4: Calculate the arc length of the arch rib at m points based on the cross-sectional area of ​​the arch rib and the structural structure. j , initially estimate the arch rib deadweight load concentration g j , or directly take g j =d j As initial value;

[0055] Step 5: Add the arch rib self-weight load concentration g and the bridge system load concentration p to get the load concentration q:

[0056] q j =g j +p j

[0057] Step 6: Calculate the load on the arch rib by the conversion matrix C according to the polynomial function The coefficient of expression and the horizontal component of the axial pressure T of the arch rib section are:

[0058]

[0059] where j = 0...m, q j is the arch rib load concentration per meter calculated according to step 5; c j is the coefficient of the arch rib load expressed by a polynomial function; l = 0.5L is the half span of the arch rib;

[0060] Step 7: According to the load polynomial coefficient c j , calculate the arch rib axis according to the polynomial function The coefficients of expression are:

[0061]

[0062] a 0 =a 1 =0

[0063]

[0064] Step 8: Find the arch axis function The first-order derivative y' at m points, and the concentration g of the arch rib self-weight load per meter is corrected according to the concentration d of the arch rib arc length per meter, and the error R is calculated:

[0065]

[0066] Where j = 0...m, R is the maximum value of the correction error of the arch rib deadweight load per meter at m points;

[0067] Step 9: If R≤0.1~0.01, it can be considered that convergence has been achieved, and the arch axis function is obtained is the reasonable arch rib axis; otherwise, Repeat steps 5 to 8 until convergence.

[0068] The method of the present invention is verified by taking the Wumengshan Bridge of Naqing Expressway as an example, as follows:

[0069] Step 1: Establish a coordinate system with the arch rib vertex as the origin, x as the longitudinal direction of the bridge, and y as the height direction of the arch rib. The arch rib sagitta is F = 54m, and the span of the arch rib is L = 270m.

[0070] Step 2: For the half-span arch rib, from the arch crown to the arch foot, there are 4 points, and the x-coordinates are used to calculate the 5th-order transformation matrix C:

[0071]

[0072] Step 3: Calculate the weight of the bridge system including the columns borne by the arch ribs in the completed state, consider 0.5 times the lane load, and convert it into load concentration per meter according to the four points. j =481.6kN / m;

[0073] Step 4: Calculate the arc length concentration d of the arch rib at 4 points based on the cross-sectional area of ​​the arch rib and the structural structure. j =340.9kN / m, initial estimate of the arch rib deadweight load concentration per meter g j =d j =340.9kN / m as the initial value;

[0074] Step 5: Add the initial arch rib self-weight load concentration g and the bridge system load concentration p to obtain the initial load concentration q = 822.5 kN / m:

[0075] Step 6: Calculate the load on the arch rib by the conversion matrix C according to the polynomial function The coefficients expressed are as follows:

[0076] Table 1 Polynomial function of arch rib load

[0077]

[0078] according to

[0079]

[0080] Calculate the horizontal component of the axial pressure of the arch rib section T = 138796.875 kN;

[0081] Step 7: According to the load polynomial coefficient c j , calculate the arch rib axis according to the polynomial function The coefficients of expression are:

[0082] Table 2 Polynomial function of arch rib axis

[0083]

[0084] Step 8: Find the arch axis function The first-order derivative y' at the four points, and the concentration g of the arch rib self-weight load per meter is corrected according to the concentration d of the arch rib arc length per meter:

[0085]

[0086] Table 3 Arch rib deadweight load linear meter correction concentration g

[0087]

[0088] The error R is calculated according to the following formula:

[0089]

[0090] R = 0.219, which does not meet the convergence criteria in step 9. Continue to repeat the process from step 5 to step 8, and finally obtain the polynomial function as shown in Table 4:

[0091] Table 4 Polynomial functions of the arch rib axis that meet the convergence criteria

[0092]

[0093] Although the above describes the specific implementation mode of the present invention in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art on the basis of the technical solution of the present invention without creative work are still within the scope of protection of the present invention.

Claims

1. A method for designing the rib axis of an arch bridge, characterized in that: include: A coordinate system is established with the apex of the arch rib of the arch bridge as the origin, and the half-span arch rib from the arch top to the arch foot is divided into multiple nodes, and a multi-order transformation matrix is ​​obtained according to the coordinates of multiple nodes; Calculate the bridge system weight borne by the arch rib in the completed state, and convert it into the bridge system load concentration per meter according to multiple nodes; The concentration of the length of the arch rib at multiple nodes is initially estimated as the concentration of the self-weight load of the first arch rib. The concentration of the bridge system load per meter is added to the concentration of the first arch rib deadweight load per meter to get the concentration of the arch rib load per meter. The coefficient of the load concentration per meter on the arch rib expressed by a polynomial function and the horizontal component of the axial pressure on the arch rib section are calculated through a multi-order conversion matrix, and the coefficient of the arch rib axis expressed by a polynomial function is calculated to obtain the arch rib axis function; Calculate the first-order derivative of the arch rib axis function at multiple nodes, and correct the first arch rib self-weight load concentration by the arch rib arc length concentration to obtain the second arch rib self-weight load concentration, and calculate the maximum error between the first arch rib self-weight load concentration and the second arch rib self-weight load concentration; If the error is less than 0.1-0.01, it can be considered that convergence has been achieved, and the arch rib axis function is a reasonable arch rib axis. If the error is greater than 0.1-0.01, the meter concentration of the second arch rib's own weight load is added to the meter concentration of the bridge system load to obtain a new meter concentration of the arch rib load and the calculation is repeated until convergence.

2. A method for designing the arch rib axis of an arch bridge according to claim 1, characterized in that: The coordinate system established with the apex of the arch rib of the arch bridge as the origin is specifically established by taking the apex of the arch rib of the arch bridge as the origin, the X-axis direction of the coordinate system is the longitudinal direction of the bridge, and the Y-axis direction of the coordinate system is the height of the arch rib.

3. A method for designing the arch rib axis of an arch bridge according to claim 1, characterized in that: The bridge system weight borne by the arch rib in the completed bridge state includes the bridge system weight including columns or hangers and 0-1 times the lane load.

4. A method for designing the arch rib axis of an arch bridge according to claim 1, characterized in that: The method of calculating the concentration of the arch rib arc length in meters at multiple nodes of the arch rib is specifically as follows: calculating the concentration of the arch rib arc length in meters at multiple nodes of the arch rib according to the cross-sectional area and the structural structure of the arch rib.

5. A method for designing the arch rib axis of an arch bridge according to claim 1, characterized in that: The multi-order conversion matrix is ​​specifically: Among them, x represents the horizontal axis coordinates of multiple nodes, and m represents the number of nodes.

6. A method for designing the arch rib axis of an arch bridge as claimed in claim 1, characterized in that: The load concentration per meter of the arch rib is a polynomial function. The rib axis is expressed as a polynomial function To express; Among them, c is the expression coefficient of the concentration function of the arch rib load per meter, a is the expression coefficient of the arch rib axis function, j ranges from 0 to m, and i ranges from 2 to m+2.

7. A method for designing the arch rib axis of an arch bridge as claimed in claim 1, characterized in that: The coefficients of the polynomial function for calculating the load concentration per meter of the arch rib by the multi-order conversion matrix are specifically: Among them, q represents the load concentration per meter borne by the arch rib.

8. A method for designing the arch rib axis of an arch bridge as claimed in claim 1, characterized in that: The horizontal component of the axial pressure of the arch rib section is calculated as follows: Among them, T represents the horizontal component of the axial pressure of the arch rib section, F represents the sagittal height of the arch rib, and l represents the half span of the arch rib.

9. A method for designing the arch rib axis of an arch bridge as claimed in claim 1, characterized in that: The coefficients of the calculation of the arch rib axis expressed by the polynomial function are specifically: Among them, a0=a1=0.

10. A method for designing an arch rib axis for an arch bridge as claimed in claim 1, characterized in that: The first-order derivative of the arch rib axis function at multiple nodes is calculated, and the concentration of the first arch rib self-weight load per meter is corrected according to the concentration of the arch rib arc length per meter to obtain the concentration of the second arch rib self-weight load per meter, which is specifically: in, represents the concentration of the second arch rib self-weight load in meters, d represents the concentration of the arch rib arc length in meters, and y` represents the first-order derivative of the arch rib axis function; The maximum error of the self-weight load concentration per meter of the first arch rib and the self-weight load concentration per meter of the second arch rib is calculated as follows: Wherein, g represents the linear concentration of the self-weight load of the first arch rib, and R represents the maximum value of the correction error of the linear concentration of the self-weight load of the first arch rib and the linear concentration of the self-weight load of the second arch rib at multiple nodes.

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