Method and system for determining reasonable acting force of arch bridge hangers or columns

Abstract

The present invention discloses a method and system for determining the reasonable force of an arch bridge hanger or column, which relates to the field of civil engineering (bridge engineering), including: constructing an arch axis function; calculating the axial compressive bearing capacity, and using a set multiple of the axial compressive bearing capacity as the control horizontal force of the arch crown; calculating the deadweight per meter along the arc length direction of the arch rib based on the cross-sectional area at n points on the arch rib, and multiplying the deadweight per meter by the deadweight per meter to obtain the longitudinal deadweight function per meter of the n points on the arch rib along the x-coordinate direction; determining the reasonable force function of the hanger or column in the form of distributed force; obtaining a value on the reasonable force function, and calculating the reasonable force of the hanger or column based on the value. The present invention is based on the mechanical equilibrium equation when the arch rib is axially compressed, and uses a mathematical algorithm to determine the reasonable force of the hanger or column. While ensuring that the structure is reasonably stressed, it reduces the eccentricity of the arch rib section under compression, optimizes the arch rib stress, reduces the use of arch ring materials, and improves the economy of the bridge structure.

Description

Technical Field

[0001] The present invention relates to the field of civil engineering (bridge engineering), and in particular to a method and system for determining the reasonable acting force of an arch bridge hanger or column. 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 (called arch ribs when the arch ring cross-section is designed as a separated structure). 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 bending moment, shear force, and deformation of the arch are much smaller, and the force on the arch ring is mainly compressive, which enhances material utilization efficiency. Therefore, arch bridges have a large span capacity. When the terrain conditions at the arch foot are suitable and the engineering geological conditions have strong horizontal bearing capacity, they are highly economical in spans ranging from 50 to 700 meters.

[0003] In the design of mid-through or bottom-through arch bridges with a strong beam-weak arch combination, the weaker arch ribs play a supporting role. Sometimes, for landscape purposes, the shape of the arch rib axis is relatively complex. A key issue is to determine the reasonable force of the hangers or columns to ensure that the cross-section of the weaker arch ribs is basically compressed under the action of the hangers, making the arch rib structure more reasonably stressed and improving material utilization efficiency.

[0004] At present, the design of reasonable forces for the hangers or columns of arch bridges in engineering projects is still based on engineering experience, using the method of repeated trial calculations and trial designs. However, it is impossible to obtain the reasonable forces for the hangers or columns of arch ribs with complex arch axes. The eccentricity of the arch rib section under compression is large, which brings great difficulties to the structural design. It is also unreasonable and uneconomical. Summary of the Invention

[0005] In response to the shortcomings of the existing technology, the purpose of the present invention is to provide a method and system for determining the reasonable force of the hangers or columns of an arch bridge. Based on the mechanical equilibrium equation when the arch rib is axially compressed, a mathematical algorithm is used to determine the reasonable force of the hangers or columns. While ensuring the reasonable force of the structure, the eccentricity of the compressed arch rib section is reduced, the arch rib force is optimized, the use of arch ring materials is reduced, and the economy of the bridge structure is improved.

[0006] In order to achieve the above object, the present invention is implemented through the following technical solutions:

[0007] In a first aspect, an embodiment of the present invention provides a method for determining the reasonable force of an arch bridge hanger or column. The method is based on the mechanical equilibrium equation when the arch rib is axially compressed and uses a calculus mathematical algorithm to determine the reasonable force of the hanger or column, including:

[0008] Establish an arch bridge coordinate system, divide the arch axis according to the x-coordinate, obtain the x-coordinates and y-coordinates of n points on the arch axis, and construct the arch axis function; where n is an integer;

[0009] Solve the first-order derivative and second-order derivative of the arch axis function at n points; calculate the axial compressive bearing capacity according to the cross-sectional area and material grade of the arch, and use a set multiple of the axial compressive bearing capacity as the control horizontal force of the arch;

[0010] Multiply the control horizontal force by the second-order derivative of n points on the arch axis to obtain the value of the arch rib's reasonably distributed force function at n points;

[0011] Calculate the deadweight per meter along the arc length of the arch rib based on the cross-sectional area at n points on the arch rib, and multiply the deadweight per meter by Obtain the longitudinal deadweight function per meter of n points on the arch rib along the x-coordinate direction;

[0012] The reasonable distributed force of the arch rib is subtracted from the longitudinal deadweight function per meter to obtain the reasonable force function of the hanger or column in the form of distributed force; the value on the reasonable force function is obtained according to the actual position of the hanger or column, and the reasonable force of the hanger or column is calculated based on the said value.

[0013] As a further implementation method, a coordinate system is established with the apex of the arch rib of the arch bridge as the origin, the longitudinal direction of the bridge as the x-direction, and the height direction of the arch rib as the y-direction.

[0014] As a further implementation method, equal-interval segmentation is adopted, wherein there are n segmentation points including the arch top and the arch foot.

[0015] As a further implementation method, the arch axis function y=f(x) expressed in a discrete numerical manner is obtained according to the x-coordinates and y-coordinates of n points on the arch axis.

[0016] As a further implementation method, considering that the arch rib is an eccentrically compressed member, it is necessary to reduce the axial compressive bearing capacity and take 0.5 to 0.9 times the axial compressive bearing capacity as the controlling horizontal force of the arch crown:

[0017] T=(0.5~0.9)f d A;

[0018] Where T is the controlling horizontal force of the arch; f d is the axial compressive strength of the material; A is the cross-sectional area of ​​the arch rib.

[0019] As a further implementation method, the longitudinal deadweight function g(x) of n points on the arch rib along the x-coordinate direction per meter is expressed as:

[0020]

[0021] Where ρ is the density of the arch rib material; k is the density adjustment coefficient considering other structures in the arch rib structure. As a further implementation method, the reasonable force function p(x) is expressed as:

[0022] p = q(x) - g(x);

[0023] Where x is the coordinate of n points.

[0024] As a further implementation method, the value is multiplied by the spacing of the boom or column, or p(x) is integrated with the middle of the front and rear spacing of the boom or column as the starting and ending range to obtain the reasonable force of the boom or column.

[0025] As a further implementation, the reasonable force is expressed as:

[0026] F j =p(x j )*ΔL;

[0027] or,

[0028] Among them, F j is the force acting on the jth boom or column; x j is the x-coordinate of the j-th boom or column; ΔL is the spacing between booms or columns.

[0029] In a second aspect, an embodiment of the present invention further provides a system for determining the reasonable force of an arch bridge suspender or column, which is based on the mechanical equilibrium equation when the arch rib is axially compressed and uses a calculus mathematical algorithm to determine the reasonable force of the suspender or column, including:

[0030] The arch axis function construction module is configured to: establish an arch bridge coordinate system, divide the arch axis according to the x-coordinate, obtain the x-coordinates and y-coordinates of n points on the arch axis, and construct the arch axis function;

[0031] The parameter solving module is configured to: solve the first-order derivative and the second-order derivative of the arch axis function at n points; calculate the axial compressive bearing capacity according to the cross-sectional area and material grade of the arch, and use a set multiple of the axial compressive bearing capacity as the control horizontal force of the arch;

[0032] The calculation module of the force function of the reasonable distribution of the arch rib is configured to: multiply the control horizontal force by the second-order derivative of n points on the arch axis to obtain the value of the force function of the reasonable distribution of the arch rib at the n points;

[0033] The module for obtaining the self-weight function per longitudinal meter is configured to calculate the self-weight per longitudinal meter along the arc length of the arch rib according to the cross-sectional area at n points on the arch rib, and multiply the self-weight per longitudinal meter by Obtain the longitudinal deadweight function per meter of n points on the arch rib along the x-coordinate direction;

[0034] The reasonable force calculation module is configured to: subtract the longitudinal self-weight function per meter from the reasonable distributed force of the arch rib to obtain the reasonable force function of the hanger or column in the form of distributed force; obtain the value on the reasonable force function according to the actual position of the hanger or column, and calculate the reasonable force of the hanger or column based on the value.

[0035] The beneficial effects of the present invention are as follows:

[0036] This method, based on the mechanical equilibrium equations for arch ribs under axial compression and employing a calculus-based mathematical algorithm to determine the appropriate forces acting on hangers or columns, can be used to design appropriate forces acting on hangers or columns for arch axes of any shape. This approach reduces arch rib material usage while ensuring structural safety, improving the economic efficiency of bridge structures. Furthermore, the use of a calculus-based mathematical algorithm to determine the appropriate forces acting on hangers or columns reduces reliance on engineering experience, making bridge design more scientific and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The accompanying drawings, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute improper limitations on the present invention.

[0038] Figure 1 is a flow chart of a method according to one or more embodiments of the present invention;

[0039] Figure 2 This is a typical schematic diagram of a mid-through arch bridge;

[0040] Figure 3 Schematic diagram of a typical through-arch bridge with a strong beam-weak arch combination;

[0041] Figure 4 is a schematic diagram of an arch axis coordinate system according to one or more embodiments of the present invention;

[0042] Figure 5 is a schematic diagram of an arch axis function according to one or more embodiments of the present invention;

[0043] Figure 6 Schematic diagram of the first derivative of the arch axis function according to one or more embodiments of the present invention;

[0044] Figure 7 Schematic diagram of the second derivative of the arch axis function according to one or more embodiments of the present invention;

[0045] Figure 8 A schematic diagram of a reasonable distribution force function of an arch rib according to one or more embodiments of the present invention;

[0046] Figure 9 A schematic diagram of the self-weight function per meter of the longitudinal arch rib according to one or more embodiments of the present invention;

[0047] Figure 10 Schematic diagram of a reasonable force function of an arch rib according to one or more embodiments of the present invention.

[0048] Among them, 1. Main beam, 2. Arch rib, 3. Arch axis, 4. Hanger, 5. Column, 6. Arch seat and foundation, 7. Pier, 8. Abutment and foundation. DETAILED DESCRIPTION

[0049] It should be noted that the following detailed description is illustrative and is intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used in the present invention have the same meaning as commonly understood by those skilled in the art to which the present invention belongs.

[0050] Example 1:

[0051] Typical mid-span arch bridges include Figure 2 As shown, the structure comprises a main beam 1, an arch rib 2, a hanger 4, and a column 5. The arch rib 2 is mounted on an arch seat and a foundation 6. A plurality of hangers 4 are connected between the arch rib 2 and the main beam 1, and the hangers 4 are arranged at a certain distance along the length of the main beam 1. A plurality of columns 5 are connected between the ends of the main beam 1 and the corresponding ends of the arch rib 2. In a typical embodiment of the present invention, a method for determining the reasonable force of the hangers or columns of an arch bridge is provided. Based on the mechanical equilibrium equation when the arch rib 2 is axially compressed, a calculus mathematical algorithm is used to determine the reasonable force of the hangers 4 or columns 5. While ensuring the reasonable force of the structure, the eccentricity of the cross-section of the arch rib 2 under compression is reduced, the force on the arch rib 2 is optimized, the use of arch ring materials is reduced, and the economic efficiency of the bridge structure is improved.

[0052] like Figure 1 As shown, the following steps are included:

[0053] Step 1, such as Figure 4 As shown, a coordinate system is established with the vertex of the arch rib 2 of the arch bridge as the origin, the longitudinal direction of the bridge as the x-direction, and the height direction of the arch rib 2 as the y-direction.

[0054] Step 2, such as Figure 5 As shown, the arch axis 3 is divided according to the x coordinate, preferably with equal intervals, and there are n dividing points including the arch top and the arch foot, where n is an integer; the x and y coordinates of n points on the arch axis 3 are obtained, and the arch axis function y = f(x) expressed in discrete numerical form is obtained.

[0055] Step three, such as Figure 6 and Figure 7 As shown, find the first-order derivative y' and second-order derivative y" of the arch axis function at n points.

[0056] Step 4: Calculate the axial compressive bearing capacity according to the cross-sectional area and material grade of the arch. Considering that the arch rib is an eccentrically compressed member, it is necessary to reduce the axial compressive bearing capacity and take 0.5 to 0.9 times the axial compressive bearing capacity as the controlling horizontal force T of the arch.

[0057] T=(0.5~0.9)f d A;

[0058] Where T is the controlling horizontal force of the arch; f d is the axial compressive strength of the material; A is the cross-sectional area of ​​arch rib 2.

[0059] Step five, such as Figure 8 As shown, the control horizontal force T is multiplied by the second-order derivative of n points on the arch axis 3 to obtain the value of the arch rib reasonable distribution force function q(x) at n points:

[0060] q=Ty”;

[0061] Where T is the controlling horizontal force of the arch crown; y' is the second-order derivative of the arch axis equation function.

[0062] Step six, such as Figure 9 As shown, the self-weight per meter along the arc length of the arch rib 2 is calculated according to the cross-sectional area at n points on the arch rib 2, and the self-weight per meter is multiplied by Obtain the longitudinal self-weight function g(x) per meter of n points on arch rib 2 along the x-coordinate direction:

[0063]

[0064] Wherein, ρ is the density of the material of arch rib 2; k is the density adjustment coefficient considering other structures in the arch rib 2 structure.

[0065] Step seven, such as Figure 10 As shown, the reasonable distributed force q(x) of the arch rib is subtracted from g(x) to obtain the reasonable force function p(x) of the hanger 4 or column 5 in the form of distributed force:

[0066] p = q(x) - g(x);

[0067] Here, x is the coordinate of n points.

[0068] Step 8: Obtain the value of the function p(x) according to the actual position of the boom 4 or column 5, multiply the value by the spacing of the boom 4 or column 5, or integrate p(x) with the middle of the front and rear spacing of the boom 4 or column 5 as the starting and ending range to obtain the reasonable force of the boom 4 or column 5. Calculate each boom 4 or column 5 to obtain the reasonable force of all booms 4 or columns 5:

[0069] F j =p(x j )*ΔL, or

[0070] Among them, F j is the force acting on the jth boom 4 or column 5; x j is the x-coordinate of the j-th boom 4 or column 5; ΔL is the spacing between booms 4 or columns 5.

[0071] This embodiment can design the appropriate forces for hangers 4 or columns 5 for arch axis lines 3 of any shape, thereby ensuring structural safety while reducing the amount of material used in arch ribs 2 and improving the economic efficiency of the bridge structure. Furthermore, this method utilizes a calculus-based mathematical algorithm to determine the appropriate forces for hangers 4 or columns 5, relying less on the designer's engineering experience, resulting in more scientific and efficient bridge design.

[0072] Example 2:

[0073] This embodiment provides a system for determining the reasonable force of an arch bridge suspender or column. The system is based on the mechanical equilibrium equation when the arch rib is axially compressed and uses a calculus mathematical algorithm to determine the reasonable force of the suspender or column, including:

[0074] The arch axis function construction module is configured to: establish an arch bridge coordinate system, divide the arch axis according to the x-coordinate, obtain the x-coordinates and y-coordinates of n points on the arch axis, and construct the arch axis function;

[0075] The parameter solving module is configured to: solve the first-order derivative and the second-order derivative of the arch axis function at n points; calculate the axial compressive bearing capacity according to the cross-sectional area and material grade of the arch, and use a set multiple of the axial compressive bearing capacity as the control horizontal force of the arch;

[0076] The calculation module of the force function of the reasonable distribution of the arch rib is configured to: multiply the control horizontal force by the second-order derivative of 3n points on the arch axis to obtain the value of the force function of the reasonable distribution of the arch rib at n points;

[0077] The module for obtaining the self-weight function per longitudinal meter is configured to calculate the self-weight per longitudinal meter along the arc length of the arch rib according to the cross-sectional area at n points on the arch rib, and multiply the self-weight per longitudinal meter by Obtain the longitudinal deadweight function per meter of n points on the arch rib along the x-coordinate direction;

[0078] The reasonable force calculation module is configured to: subtract the longitudinal self-weight function per meter from the reasonable distributed force of the arch rib to obtain the reasonable force function of the hanger or column in the form of distributed force; obtain the value on the reasonable force function according to the actual position of the hanger or column, and calculate the reasonable force of the hanger or column based on the value.

[0079] Example 3:

[0080] This embodiment is based on the method described in Example 1. Figure 3 The arch bridge shown in the figure has a main beam 1 as a concrete box beam and an arch rib 2 as a steel box arch rib. Both ends of the main beam 1 are supported by abutments and foundations 8 and piers 7. The span of the arch rib 2 is 170m, and the rise of the arch rib 2 is 27m. The spline curve of the arch rib 2, which takes the landscape into consideration, adopts an approximate sine function. Figure 4 The arch axis coordinate system shown in Figure 1 gives Figure 5 The arch axis function is shown.

[0081] like Figure 6 As shown, the first-order derivative of the arch axis function is obtained; Figure 7 As shown, the second-order derivative of the arch axis function is calculated. Based on the cross-sectional area at the arch top of arch rib 2, the control horizontal force T = 12100 kN is determined. Multiplying the second-order derivative of arch axis 3 by the control horizontal force gives Figure 8 The arch ribs shown have a reasonable distribution force function q(x).

[0082] According to the cross-sectional area, structural characteristics and first-order derivative of the arch axis 3 of the arch rib 2, the following is obtained: Figure 9 The longitudinal deadweight function g(x) of the arch rib 2 per meter is shown as follows; the arch rib reasonable distribution force function q(x) is subtracted from the longitudinal deadweight function g(x) of the arch rib 2 per meter, and the following is obtained: Figure 10 The reasonable force distribution function p(x) of the arch rib 2 and the hanger 4 is shown.

[0083] According to the actual position and spacing of the suspenders 4, the reasonable force distribution function of the suspenders 4 is converted into the concentrated force of each suspender 4, and then brought into the structural calculation model for calculation. After appropriate adjustments are made, the reasonable tension of the suspenders 4 can be obtained as shown in Table 1 below:

[0084] Table 1 Reasonable tension of the boom

[0085]

[0086] The above description is merely a preferred embodiment of the present application and is not intended to limit the present application. Various modifications and variations are possible for those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present application shall be included within the scope of protection of the present application.

Claims

1. A method for determining the reasonable force of an arch bridge suspender or column, characterized in that: Based on the mechanical equilibrium equation when the arch rib is axially compressed, the reasonable force of the hanger or column is determined using calculus mathematical algorithms, including: Establish an arch bridge coordinate system, divide the arch axis according to the x-coordinate, obtain the x-coordinates and y-coordinates of n points on the arch axis, and construct the arch axis function; where n is an integer; Solve the first-order derivative y' and second-order derivative y'' of the arch axis function at n points; calculate the axial compressive bearing capacity according to the cross-sectional area and material grade of the arch, and use a set multiple of the axial compressive bearing capacity as the control horizontal force of the arch; Multiply the control horizontal force by the second-order derivative of n points on the arch axis to obtain the value of the arch rib's reasonably distributed force function at n points; Calculate the deadweight per meter along the arc length of the arch rib based on the cross-sectional area at n points on the arch rib, and multiply the deadweight per meter by Obtain the longitudinal deadweight function per meter of n points on the arch rib along the x-coordinate direction; The reasonable force function p(x) of the suspender or column in the form of distributed force is obtained by subtracting the longitudinal deadweight function per meter from the reasonable distributed force of the arch rib; the value on the reasonable force function is obtained according to the actual position of the suspender or column, and the reasonable force of the suspender or column is calculated based on the value; The reasonable force of the boom or column can be obtained by multiplying the value by the spacing of the boom or column, or integrating p(x) with the middle of the spacing between the boom or column as the starting and ending range. The reasonable force can be expressed as: ； or, ; in, is the force acting on the jth boom or column; is the x-coordinate of the j-th boom or column; The spacing between booms or columns.

2. The method for determining the reasonable force of an arch bridge suspender or column according to claim 1, characterized in that: A coordinate system is established with the arch rib vertex of the arch bridge as the origin, the longitudinal direction of the bridge as the x-direction, and the arch rib height direction as the y-direction.

3. The method for determining the reasonable force of an arch bridge suspender or column according to claim 1, characterized in that: The equal-interval segmentation is adopted, where there are n segmentation points including the arch top and arch foot.

4. A method for determining the reasonable force of an arch bridge suspender or column according to claim 1 or 3, characterized in that: According to the x-coordinates and y-coordinates of n points on the arch axis, the arch axis function expressed in discrete numerical form is obtained. .

5. The method for determining the reasonable force of an arch bridge suspender or column according to claim 1, characterized in that: Take 0.5 to 0.9 times the axial compressive bearing capacity as the control horizontal force of the arch: ； in, is the controlling horizontal force of the vault; is the axial compressive strength of the material; is the cross-sectional area of ​​the arch rib.

6. The method for determining the reasonable force of an arch bridge suspender or column according to claim 1, characterized in that: The longitudinal self-weight function g(x) of n points on the arch rib along the x-coordinate direction per meter is expressed as: ； in is the weight of the arch rib material; It is the weight adjustment factor to consider other structures in the arch rib structure.

7. The method for determining the reasonable force of an arch bridge suspender or column according to claim 1, characterized in that: The reasonable force function p(x) is expressed as: ； Where x is the coordinate of n points.

8. A system for determining the reasonable force of an arch bridge suspender or column, characterized in that: Based on the mechanical equilibrium equation when the arch rib is axially compressed, the reasonable force of the hanger or column is determined using calculus mathematical algorithms, including: The arch axis function construction module is configured to: establish an arch bridge coordinate system, divide the arch axis according to the x-coordinate, obtain the x-coordinates and y-coordinates of n points on the arch axis, and construct the arch axis function; The parameter solving module is configured to: solve the first-order derivative y' and the second-order derivative y'' of the arch axis function at n points; calculate the axial compressive bearing capacity according to the cross-sectional area and material grade of the arch, and use a set multiple of the axial compressive bearing capacity as the control horizontal force of the arch; The calculation module of the force function of the reasonable distribution of the arch rib is configured to: multiply the control horizontal force by the second-order derivative of n points on the arch axis to obtain the value of the force function of the reasonable distribution of the arch rib at the n points; The module for obtaining the self-weight function per longitudinal meter is configured to calculate the self-weight per longitudinal meter along the arc length of the arch rib according to the cross-sectional area at n points on the arch rib, and multiply the self-weight per longitudinal meter by Obtain the longitudinal deadweight function per meter of n points on the arch rib along the x-coordinate direction; The reasonable force calculation module is configured to: subtract the longitudinal deadweight function per meter from the reasonable distributed force of the arch rib to obtain a reasonable force function p(x) of the suspender or column in the form of distributed force; obtain a value on the reasonable force function according to the actual position of the suspender or column, and calculate the reasonable force of the suspender or column based on the value; The reasonable force of the boom or column can be obtained by multiplying the value by the spacing of the boom or column, or integrating p(x) with the middle of the spacing between the boom or column as the starting and ending range. The reasonable force can be expressed as: ； or, ; in, is the force acting on the jth boom or column; is the x-coordinate of the j-th boom or column; The spacing between booms or columns.

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