Method and system for determining reasonable acting force of arch bridge suspender or stand column

Abstract

The invention discloses a method and system for determining reasonable acting force of an arch bridge suspender or stand column, and relates to the field of civil (bridge) engineering. The axial compression bearing capacity is calculated, and the set multiple of the axial compression bearing capacity serves as the control horizontal force of the arch crown; calculating the self weight per linear meter in the arc length direction of the arch rib according to the sectional areas of the n points on the arch rib, and multiplying the self weight per linear meter by the self weight per linear meter to obtain a longitudinal self weight function per linear meter of the n points on the arch rib in the x coordinate direction; determining a reasonable acting force function of the suspender or the stand column according to a distribution force form; obtaining a numerical value on the reasonable acting force function, and calculating the reasonable acting force of the suspender or the stand column according to the numerical value. Based on the mechanical equilibrium equation when the axis of the arch rib is pressed, the reasonable acting force of the suspender or the stand column is determined by adopting a mathematical algorithm, the eccentric distance of the pressed section of the arch rib is reduced, the stress of the arch rib is optimized, the use of arch ring materials is reduced, and the economical efficiency of a bridge structure is improved while the reasonable stress of the structure is ensured.

Description

Technical Field

[0001] The 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 thrusts at the arch foot through an arch ring or an arch rib (the arch ring cross section is called an arch rib when it is designed in a separated form). 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 beams of the same span, the bending moment, shear force and deformation of the arch are much smaller, and the arch ring is mainly subjected to compression, which enhances the utilization efficiency of the material. Therefore, the spanning 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] In the design process of mid-span or bottom-span arch bridges with a strong beam-weak arch combination, the weaker arch rib plays an auxiliary 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 hanger or column to ensure that the cross-section of the weaker arch rib is basically compressed under the action of the hanger, so that the arch rib structure is more reasonably stressed and the material utilization efficiency is improved.

[0004] At present, the design of reasonable forces for the hangers or columns of arch bridges is still based on engineering experience, using the method of repeated trial calculation and trial design. It is impossible to obtain the reasonable forces for the hangers or columns of the 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 view of the deficiencies in the prior art, 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 that the structure is subjected to reasonable force, 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 a reasonable force of a hanger or column of an arch bridge, based on a mechanical equilibrium equation when the arch rib is axially compressed, and using a calculus mathematical algorithm to determine a reasonable force of the hanger or column, comprising:

[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 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 the 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 the arch axis at n points to obtain the value of the arch rib reasonable distribution 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 self-weight 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 deducted 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 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, equally spaced 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 discrete numerical form 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 of the axial compressive bearing capacity as the control 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 vault; 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 self-weight function g(x) of n points on the arch rib along the x-coordinate direction per linear meter is expressed as:

[0020]

[0021] Where ρ is the weight of the arch rib material; k is the weight 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 hanger or column, or p(x) is integrated with the middle of the front and rear spacing of the hanger or column as the starting and ending range to obtain the reasonable force of the hanger 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 suspender or column; x j is the x-coordinate of the jth hanger or column; ΔL is the spacing between hangers 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 the 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 as follows: multiplying the control horizontal force by the second-order derivative of the n points of 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 longitudinal self-weight function acquisition module is configured to calculate the self-weight per linear meter along the arc length direction of the arch rib according to the cross-sectional area at n points on the arch rib, and multiply the self-weight per linear meter by Obtain the longitudinal self-weight 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] The present invention 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. The reasonable force of the hanger or column can be designed for an arch axis of any shape, while ensuring the structural safety, reducing the use of arch rib materials and improving the economy of the bridge structure. At the same time, the reasonable force of the hanger or column is determined by using a calculus mathematical algorithm, which is less dependent on engineering experience, making the bridge design more scientific and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The accompanying drawings in the specification, 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 method flow chart according to one or more embodiments of the present invention;

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

[0040] Figure 3 The schematic diagram of a typical strong beam-weak arch combination bottom-through arch bridge;

[0041] Figure 4 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 A schematic diagram of the first derivative of the arch axis function according to one or more embodiments of the present invention;

[0044] Figure 7 A 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] Fig. 9 A schematic diagram of the self-weight function per meter of the longitudinal direction of the arch rib according to one or more embodiments of the present invention;

[0047] Fig.10 It is a 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 descriptions are illustrative and 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 meanings as those commonly understood by those skilled in the art to which the present invention belongs.

[0050] Embodiment 1:

[0051] Typical mid-span arch bridges include Figure 2 As shown, it includes a main beam 1, an arch rib 2, a hanger 4, a column 5, etc. The arch rib 2 is erected 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 direction of the main beam 1; a plurality of columns 5 are connected between the two ends of the main beam 1 and the ends corresponding to 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, and a calculus mathematical algorithm is used to determine the reasonable force of the hangers 4 or the columns 5, while ensuring that the structure is reasonably stressed, the eccentricity of the cross section of the arch rib 2 under compression is reduced, the stress of the arch rib 2 is optimized, the use of arch ring materials is reduced, and the economy 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 top 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, including n division points of the arch top and 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 of the axial compressive bearing capacity as the control 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 vault; f d is the axial compressive strength of the material; A is the cross-sectional area of ​​arch rib 2.

[0059] Step 5: 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; y” is the second-order derivative of the arch axis equation function.

[0062] Step 6: Fig. 9 As shown in the figure, the deadweight per linear meter along the arc length direction of the arch rib 2 is calculated according to the cross-sectional area at n points on the arch rib 2, and the deadweight per linear meter is multiplied by The longitudinal self-weight function g(x) of n points on the arch rib 2 along the x-coordinate direction per meter is obtained:

[0063]

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

[0065] Step 7: Fig.10 As shown, the reasonable force function p(x) of the suspender 4 or column 5 in the form of distributed force is obtained by subtracting g(x) from the reasonable distributed force of the arch rib:

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

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

[0068] Step 8: According to the actual position of the suspension rod 4 or the column 5, the value of the function p(x) is obtained, and the value is multiplied by the spacing of the suspension rod 4 or the column 5, or p(x) is integrated with the middle of the front and rear spacing of the suspension rod 4 or the column 5 as the starting and ending range, so as to obtain the reasonable force of the suspension rod 4 or the column 5. The reasonable force of all suspension rods 4 or the column 5 can be obtained by calculating each suspension rod 4 or the column 5:

[0069] F j =p(x j )*ΔL, or

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

[0071] This embodiment can design the reasonable force of the hanger 4 or the column 5 for the arch axis 3 of any shape, while ensuring the structural safety, reducing the use of the material of the arch rib 2 and improving the economic efficiency of the bridge structure. In addition, the method uses a calculus mathematical algorithm to determine the reasonable force of the hanger 4 or the column 5, which is less dependent on the engineering experience of the designer, and can make the bridge design more scientific and efficient.

[0072] Embodiment 2:

[0073] This embodiment 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:

[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 the set multiple of the axial compressive bearing capacity as the control horizontal force of the arch;

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

[0077] The longitudinal self-weight function acquisition module is configured to calculate the self-weight per linear meter along the arc length direction of the arch rib according to the cross-sectional area at n points on the arch rib, and multiply the self-weight per linear meter by Obtain the longitudinal self-weight 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] Embodiment three:

[0080] This embodiment is based on the method described in Embodiment 1. Figure 3 In the arch bridge shown in the figure, the main beam 1 is a concrete box beam, the arch rib 2 is a steel box arch rib 2, and both ends of the main beam 1 are supported by the abutment and foundation 8 and the bridge pier 7; the arch rib 2 has a span of 170m and a rise of 27m. The arch rib 2 takes the landscape into consideration and adopts a spline curve approximating a 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. According to the cross-sectional area at the arch top of arch rib 2, the control horizontal force T = 12100 kn is determined. Multiply the second-order derivative of arch axis 3 by the control horizontal force to obtain Figure 8 The arch rib shown has 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, we can obtain Fig. 9 The longitudinal deadweight function g(x) of the arch rib 2 per meter is shown as follows; the longitudinal deadweight function g(x) of the arch rib 2 per meter is subtracted from the reasonable distribution force function q(x) of the arch rib to obtain the following: Fig.10 The reasonable force distribution function p(x) of the arch rib 2 and the hanger rod 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, which is brought into the structural calculation model for calculation and adjusted appropriately to obtain the reasonable tension of the suspenders 4 as shown in Table 1 below:

[0084] Table 1 Reasonable tension of the suspender

[0085]

[0086] The above description is only the preferred embodiment of the present application and is not intended to limit the present application. For those skilled in the art, the present application may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope 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 by calculus mathematical algorithm, 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 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 the 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 the arch axis at n points to obtain the value of the arch rib reasonable distribution 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 self-weight function per meter of n points on the arch rib along the x-coordinate direction; The reasonable distributed force of the arch rib is deducted 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 value.

2. A 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 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.

3. A method for determining the reasonable force of an arch bridge suspender or column according to claim 1, characterized in that: The partition is divided into equal intervals, and there are n partition points including the arch top and the 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 y=f(x) 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 of the axial compressive bearing capacity as the control horizontal force of the arch: T=(0.5~0.9)f d A; Where T is the controlling horizontal force of the vault; f d is the axial compressive strength of the material; A is the cross-sectional area of ​​the arch rib.

6. A 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: Where ρ is the density of the arch rib material; k is the density adjustment coefficient considering other structures in the arch rib structure.

7. A 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: p = q(x) - g(x); Where x is the coordinate of n points.

8. A method for determining the reasonable force of an arch bridge suspender or column according to claim 1 or 7, characterized in that: The value is multiplied by the spacing of the hanger or column, or p(x) is integrated with the middle of the front and rear spacing of the hanger or column as the starting and ending range to obtain the reasonable force of the hanger or column.

9. A method for determining the reasonable force of an arch bridge suspender or column according to claim 8, characterized in that: The reasonable force is expressed as: F j =p(x j )*ΔL; or, Among them, F j is the force acting on the jth suspender or column; x j is the x-coordinate of the jth hanger or column; ΔL is the spacing between hangers or columns.

10. 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 by calculus mathematical algorithm, 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 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 the 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 as follows: multiplying the control horizontal force by the second-order derivative of the n points of the arch axis to obtain the value of the force function of the reasonable distribution of the arch rib at the n points; The longitudinal self-weight function acquisition module is configured to calculate the self-weight per linear meter along the arc length direction of the arch rib according to the cross-sectional area at n points on the arch rib, and multiply the self-weight per linear meter by Obtain the longitudinal self-weight 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 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.

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