A design method for the arch rib axis of an arch bridge

Abstract

The present invention discloses a method for designing the arch rib axis of an arch bridge, belonging to the technical field of bridge engineering. The method includes establishing a coordinate system based on the arch bridge and dividing the half-span arch rib into multiple nodes to obtain a multi-order transformation matrix; calculating the load concentration per meter of the arch rib and calculating the coefficient of the load concentration per meter of the arch rib expressed as a polynomial function and the horizontal component of the axial pressure of the arch rib cross section through the multi-order transformation matrix; calculating the coefficient of the arch rib axis expressed as a polynomial function to obtain the arch rib axis function; combining the arch rib axis function with the first arch rib self-weight load concentration per meter and correcting it according to the arch rib arc length concentration per meter to obtain the second arch rib self-weight load concentration per meter; calculating the maximum error between the first arch rib self-weight load concentration per meter and the second arch rib self-weight load concentration per meter, and determining whether the arch rib axis function is a reasonable arch rib axis. The method provided by the present invention is convenient for engineering application and is suitable for reasonable arch rib axis design of arch bridges.

Description

Technical Field

[0001] The present 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 arch rings or arch ribs (the arch ring cross section is called an arch rib when it is designed to be separated). Figure 1 As shown, the horizontal reaction force at the arch foot will greatly offset the bending moment caused by the load within the arch ring (or arch rib). Therefore, compared with beams of the same span, the arch has much smaller bending moments, shear forces, and deformations. Moreover, the arch ring is primarily subjected to compression, which enhances material efficiency. Therefore, arch bridges have a large span capacity. When the topography at the arch foot is suitable and the engineering geology has strong horizontal bearing capacity, they are highly economical in spans ranging from 50 to 700 meters.

[0003] The key issue in arch bridge design is determining a reasonable arch axis to ensure that the cross-section is primarily in compression under external loads, resulting in a more rational structural stress response and improved material utilization efficiency. The determination of the arch axis is related to the load distribution pattern on it. For conventional long-span arch bridges, the arch axis is primarily a catenary or quadratic parabola. Given the rise and span, the shape of the arch axis is related to the arch 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 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. This method has poor adaptability to the load distribution form on the arch and has low calculation and design efficiency. Therefore, it is necessary to propose a design method for a reasonable arch rib axis that is convenient for engineering application and suitable for any load distribution form of arch bridges. Summary of the Invention

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

[0006] The technical solutions of the present invention are 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 arch rib vertex as the origin, and the half-span arch rib from the arch crown to the arch foot is divided into multiple nodes. The multi-order transformation matrix is ​​obtained based on the coordinates of the multiple nodes.

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

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

[0011] The per-meter concentration of the load borne by the arch rib is obtained by adding the per-meter concentration of the bridge system load and the per-meter concentration of the first arch rib self-weight load.

[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 per meter according to the arch rib arc length concentration per meter to obtain the second arch rib self-weight load concentration per meter, and calculate the maximum error between the first arch rib self-weight load concentration per meter and the second arch rib self-weight load concentration per meter.

[0014] If the error is less than 0.1 to 0.01, it can be considered that convergence is achieved, and the arch rib axis function is a reasonable arch rib axis. If the error is greater than 0.1 to 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's 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 is specifically: calculating the concentration of the arch rib arc length in meters at multiple nodes according to the cross-sectional area and structural configuration 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 coordinate 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 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, the value range of j is 0~m, and the value range of i is 2~m+2.

[0023] In some embodiments of the present invention, 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:

[0024]

[0025] Where q represents the load concentration per meter of the arch rib.

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

[0027]

[0028] Where 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 the polynomial function expressed by the calculation of the rib axis are specifically:

[0030]

[0031] Among them, a0=a1=0.

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

[0033]

[0034] 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, y 、 represents the first derivative of the arch rib axis function;

[0035] The maximum error of the calculated 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 is as follows:

[0036]

[0037] Where g represents the linear concentration of the self-weight load of the first arch rib, and R represents the maximum value of the corrected errors 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 method for designing the arch rib axis for an arch bridge. By using a mathematical algorithm and deducing the mechanical equilibrium equation for the arch rib when the arch rib is axially compressed, the method determines an analytical function solution for the reasonable arch rib axis. This method can directly obtain a reasonable analytical function solution for the arch rib axis for any distributed load, filling a gap in arch bridge design methods. While ensuring structural safety, it reduces the use of arch rib materials and improves the safety and economy of bridge structures.

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

[0041] Figure 1 This is a flow chart of a design method for an arch rib axis of an arch bridge proposed in Example 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 with reference to 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 method for designing an arch rib axis of an arch bridge is provided, comprising:

[0049] 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, and the span of the arch rib is L.

[0050] Step 2: Divide the half-span arch rib from the arch crown to the arch foot into m points, preferably with m being 3 to 5 and with close spacing. Calculate the m+1 order transformation matrix C based on the x coordinates of the m points:

[0051]

[0052] where x j , j = 0...m is the x-coordinate of the point where the numbering starts 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 arch rib arc length in meters at m points based on the arch rib cross-sectional area and structural structure. j , preliminarily estimate the arch rib self-weight load concentration per meter 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 obtain the load concentration q:

[0056] q j =g j +p j

[0057] Step 6: Calculate the load on the arch rib using 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 expressed are:

[0061]

[0062] a0=a1=0

[0063]

[0064] Step 8: Find the arch axis function The first-order derivative y' at m points is calculated, 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 self-weight load concentration 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, let 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, divide it into 4 points from the arch crown to the arch foot, and calculate the 5th-order transformation matrix C with x coordinates:

[0071]

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

[0073] Step 4: Calculate the arch rib arc length in meters at the four points based on the arch rib cross-sectional area and structural structure. j =340.9kN / m, preliminary 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 per meter g and the bridge system load concentration per meter p to obtain the initial load concentration per meter q = 822.5 kN / m:

[0075] Step 6: Calculate the load on the arch rib using 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 = 138796.875kN;

[0081] Step 7: According to the load polynomial coefficient c j , calculate the arch rib axis according to the polynomial function The coefficients expressed 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 arch rib self-weight load concentration g per meter is corrected according to the arch rib arc length concentration d per meter:

[0085]

[0086] Table 3 Corrected concentration of arch rib deadweight load per meter (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 embodiments 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 any creative work are still within the scope of protection of the present invention.

Claims

1. A method for designing the arch rib axis of an arch bridge, characterized in that: include: A coordinate system is established with the arch rib vertex as the origin, and the half-span arch rib from the arch crown to the arch foot is divided into multiple nodes. The multi-order transformation matrix is ​​obtained based on the coordinates of the multiple nodes. Calculate the weight of the bridge system borne by the arch ribs in the completed state, and convert it into the per-meter concentration of the bridge system load according to multiple nodes; The concentration of the length of the arch rib at multiple nodes is initially estimated to be the concentration of the self-weight load of the first arch rib. The per-meter concentration of the load borne by the arch rib is obtained by adding the per-meter concentration of the bridge system load and the per-meter concentration of the first arch rib self-weight load. 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 per meter according to the arch rib arc length concentration per meter to obtain the second arch rib self-weight load concentration per meter, and calculate the maximum error between the first arch rib self-weight load concentration per meter and the second arch rib self-weight load concentration per meter. If the error is less than 0.1 to 0.01, it can be considered that convergence is achieved and the arch rib axis function is a reasonable arch rib axis. If the error is greater than 0.1 to 0.01, the second arch rib self-weight load concentration per meter is added to the bridge system load concentration per meter to obtain a new arch rib load concentration per meter and the calculation is repeated until convergence is achieved. 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: Where q represents the load concentration per meter of the arch rib; The horizontal component of the axial pressure on the arch rib section is calculated as follows: Where T represents the horizontal component of the axial pressure on the arch rib section, F represents the sagittal height of the arch rib, and l represents the half span of the arch rib; The coefficients of the polynomial function used to calculate the arch rib axis are specifically: Among them, a0=a1=0.

2. A method for designing the arch rib axis of an arch bridge according to claim 1, characterized in that: The coordinate system is established with the apex of the arch rib of the arch bridge as the origin. Specifically, the coordinate system is established with 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. The method for designing the arch rib axis of an arch bridge according to claim 1, wherein: The bridge system weight borne by the arch rib in the completed bridge state includes the bridge system weight including the columns or hangers and 0-1 times the lane load.

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

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

6. The method for designing the arch rib axis of an arch bridge according to claim 1, wherein: The load concentration per meter of the arch rib is determined by 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, the value range of j is 0~m, and the value range of i is 0~m+2.

7. The method for designing the arch rib axis of an arch bridge according to claim 1, wherein: 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 derivative of the arch rib axis function; The maximum error of the calculated 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 is as follows: Where g represents the linear concentration of the self-weight load of the first arch rib, and R represents the maximum value of the corrected errors 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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