Method for determining the lower edge curve of the midspan and lower chord of a continuous rigid frame bridge with open web

Abstract

This invention discloses a method for determining the lower edge curves of the mid-span and lower chord beams of a hollow continuous rigid frame bridge, including determining the total height H between the upper and lower chord beams in the hollow area and the height h of the lower chord beam in the hollow area. 1 The height h of the upper chord beam in the empty area 2 Beam height at the junction of the upper and lower chords, and the range L of the mid-span beam segment. m and the height h of the mid-span beam 3 The method calculates the lower edge curve F(x) of the mid-span beam segment and the lower edge curve G(x) of the hollow lower chord beam segment; calculates the continuity of the function and first derivative of curves F(x) and G(x) at the connection point of the upper and lower chords; in this method, the lower edge curve of the mid-span beam segment conforms to the stress characteristics of the mid-span beam segment in cantilever construction; the upward arch of the lower edge curve of the hollow lower chord beam segment balances the self-weight of the lower chord beam segment and the load transmitted by the upper chord; the structural transition and stress transmission between the mid-span beam segment and the hollow lower chord beam segment are smooth; the lower edge curves of the mid-span beam segment and the hollow lower chord beam segment are optimized, making the structure of the hollow continuous rigid frame bridge more reasonable.

Description

A method for determining the lower edge curve of the mid-span and lower chord of a hollow continuous rigid frame bridge Technical Field

[0001] This invention relates to the field of concrete hollow continuous rigid frame bridge technology, specifically to a method for determining the lower edge curve of the mid-span and lower chord beams of a hollow continuous rigid frame bridge. Background Technology

[0002] Concrete hollow-web continuous rigid frame bridges, also known as arch-beam composite continuous rigid frame bridges, are a new modification of conventional continuous rigid frame structures. The main idea is to increase the height of the box girder root and hollow out the web at the box girder root to reduce self-weight, creating a beam-arch composite mechanical effect, thereby improving structural load-bearing efficiency and enhancing the bridge's spanning capacity. Hollow-web continuous rigid frame bridges share similar balanced cantilever construction characteristics with conventional continuous rigid frame bridges, and offer advantages such as lower operation and maintenance costs and lower project costs. Their economical spans range from 220m to 400m, and they are expected to fill the gap between the applicable spans of conventional continuous rigid frame bridges and cable-stayed bridges. As of 2024, approximately 10 large-span hollow-web continuous rigid frame bridges for highways, railways, urban roads, and rail transit have been built or are under construction, demonstrating significant technical and economic advantages and broad application prospects.

[0003] Concrete hollow continuous rigid frame bridges have many advantages such as relatively light weight, large span capacity, and beautiful appearance. However, in practical applications, there are problems with the uneven stress transition between the mid-span beam segment and the lower chord beam segment in the hollow area. The structural design is usually conservative and the structural size is large, which not only affects the realization of the economic advantages of this type of bridge, but also increases the difficulty of lower chord construction, thus restricting the promotion, application and development of hollow continuous rigid frame bridge structures. Summary of the Invention

[0004] To address the aforementioned technical shortcomings, the purpose of this invention is to provide a method for determining the lower edge curves of the mid-span and lower chord beams of a hollow continuous rigid frame bridge. This method optimizes the lower edge curves of the mid-span beam segment and the lower chord beam segment in the hollow area, resulting in a more rational structure, better economy, and greater spanning capacity for the hollow continuous rigid frame bridge.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] This invention provides a method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge, comprising the following steps:

[0007] S1. Determine the total height H between the upper and lower chord beams in the hollow area, the height h1 of the lower chord beam in the hollow area, the height h2 of the upper chord beam in the hollow area, the beam height at the junction of the upper and lower chords, and the range of the beam segment at mid-span. And the mid-span beam height h3;

[0008] S2. Calculate the range of the mid-span beam segment and the lower edge curve F(x) of the mid-span beam segment:

[0009] ;

[0010] ;

[0011] Where c is the exponent of the curve function at the lower edge of the mid-span beam segment, with a value of 1.8 to 2.0; The location of the connection point between the lower edge curve F(x) of the mid-span beam segment and the lower edge curve G(x) of the lower chord beam segment in the open area;

[0012] S3. Calculate the function value and its first derivative of the curve F(x) at the junction of the upper and lower chords:

[0013] ;

[0014] ;

[0015] S4. Calculate the coefficients of the 2.0~3.0 order polynomial function of the lower edge curve G(x) of the lower chord beam segment in the hollow region:

[0016] ;

[0017] Obtain the G(x) function of the lower edge curve of the lower chord beam segment in the empty zone:

[0018] ;

[0019] Where b is the function exponent of the lower edge curve of the lower chord beam segment in the hollow area, with a value of 2.0~3.0; The unit weight of the lower chord concrete is taken as 26 kN / m. 3 ; The control stress of the lower chord concrete is taken as 0.4 to 0.6 times the axial compressive design strength of the concrete.

[0020] S5. Check the second derivative of the lower edge curve G(x) of the lower chord beam segment in the hollow area, adjust the parameters H, h1, and h2, and repeat the aforementioned steps S1-S4 to ensure that the value of the second derivative of the lower edge curve G(x) of the lower chord beam segment in the hollow area is not less than / And within a certain range.

[0021] Preferably, in step S1, the total height H between the upper and lower chord beams in the hollow area is:

[0022] ;

[0023] Where L is the net span of the main span of the hollow continuous rigid frame bridge.

[0024] Preferably, in step S1, the height h1 of the lower chord beam in the hollow area is:

[0025] .

[0026] Preferably, in step S1, the height h2 of the upper chord beam in the hollow area is:

[0027] ;

[0028] The beam height at the junction of the upper and lower chords is .

[0029] Preferably, in step S1, the mid-span beam segment range :

[0030] .

[0031] Preferably, in step S1, the mid-span beam height h3 is:

[0032] .

[0033] Preferably, the second derivative value of the lower edge curve G(x) of the lower chord beam segment in the hollow region is no greater than 3 to 5 times. / This ensures that the curvature of the lower chord beam segment is not too large, and the maximum eccentricity of the lower chord beam segment under eccentric compression is controlled.

[0034] Thus, the lower edge curve F(x) of the mid-span beam segment and the lower edge curve G(x) of the hollow lower chord beam segment are determined. The functions and their first derivatives of curves F(x) and G(x) at the connection point of the upper and lower chords are continuous. The second derivative of the lower edge curve G(x) of the hollow lower chord beam segment is within a certain range, which makes the structural transition between the mid-span beam segment and the hollow lower chord beam segment smooth, the stress transfer smooth, and the eccentricity of the hollow lower chord beam segment under eccentric compression small and the stress reasonable.

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

[0036] 1. The lower edge curve F(x) of the mid-span beam segment adopts a parabola of degree 1.8 to 2.0. The length of the mid-span beam segment is 12 to 15 times the beam height at the junction of the upper and lower chords, which is consistent with the stress characteristics of the mid-span beam segment in cantilever construction and the stress is safe and reasonable.

[0037] 2. The lower edge curve G(x) of the lower chord beam segment in the hollow area adopts a polynomial function of order 2.0 to 3.0, which meets the requirement of small eccentricity under eccentric compression when subjected to the internal forces transmitted by the meeting point of the upper and lower chords, the self-weight of the lower chord beam segment, and the vertical forces that may be transmitted by the upper chord. The stress on the lower chord beam segment is safer and more reasonable.

[0038] 3. The continuity of the functions and their first derivatives of curves F(x) and G(x) at the connection point of the upper and lower chords makes the structural transition between the mid-span beam segment and the lower chord beam segment in the open area smooth, and the stress transfer smooth, which solves the problem of easy cracking of concrete at the connection point and improves the safety and durability of the connection point.

[0039] 4. The second derivative value of the curve G(x) at the lower edge of the lower chord section in the control zone shall not be less than [value missing]. / Within a certain range, the upper arch of the lower chord beam segment can balance its own weight and the load transmitted by the upper chord, and the curvature of the upper arch is not too large, so that the maximum eccentricity of the lower chord beam segment under eccentric compression can be controlled.

[0040] 5. The lower edge curves of the mid-span beam segment and the lower chord beam segment in the open section have been optimized, making the structure of the open continuous rigid frame bridge more reasonable, more economical, and with greater spanning capacity. Attached Figure Description

[0041] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0042] Figure 1 is a structural schematic diagram of a concrete hollow continuous rigid frame bridge provided in an embodiment of the present invention.

[0043] Figure 2 is a schematic diagram of the beam height and curve function coordinate system of the mid-span beam segment and the lower chord beam segment in the open area in Figure 1.

[0044] In the diagram: 1. Upper chord beam segment, 2. Lower chord beam segment, 3. Meeting point of upper and lower chords, 4. Mid-span beam segment, 5. Connection point, 6. Lower edge curve of mid-span beam segment, 7. Lower edge curve of lower chord beam segment. Detailed Implementation

[0045] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0046] Taking the hollow concrete continuous rigid frame bridge in Figure 1 as an example, this embodiment provides a method for determining the lower edge curves of the mid-span and lower chord beams of a hollow continuous rigid frame bridge, including the following steps:

[0047] S1. Referring to Figure 2, determine the total height H between the upper chord beam 1 and the lower chord beam 2 in the hollow area, the height h1 of the lower chord beam in the hollow area, the height h2 of the upper chord beam in the hollow area, the beam height at the junction of the upper and lower chords 3, and the range of the beam segment 4 at mid-span. And the mid-span beam height h3;

[0048] The total height H between the upper and lower chord beams in the hollow section:

[0049] ;

[0050] Where L is the net span of the main span of the hollow continuous rigid frame bridge.

[0051] The height h1 of the lower chord beam in the empty zone:

[0052] .

[0053] The height h2 of the upper chord beam in the empty zone:

[0054] ;

[0055] The beam height at the junction of the upper and lower chords is .

[0056] Range of the middle beam segment :

[0057] .

[0058] Mid-span beam height h3:

[0059] .

[0060] S2. Calculate the range of the mid-span beam segment and the lower edge curve F(x) of the mid-span beam segment:

[0061] ;

[0062] ;

[0063] Where c is the function exponent of the lower edge curve of the mid-span beam segment, with a value of 1.8 to 2.0; The location of point 5 is the connection point between the lower edge curve F(x) of the mid-span beam segment and the lower edge curve G(x) of the lower chord beam segment in the open area;

[0064] S3. Calculate the function value and its first derivative of the curve F(x) at the junction of the upper and lower chords:

[0065] ;

[0066] ;

[0067] S4. Calculate the coefficients of the 2.0~3.0 order polynomial function of the lower edge curve G(x) of the lower chord beam segment in the hollow region:

[0068] ;

[0069] Obtain the G(x) function of the lower edge curve of the lower chord beam segment in the empty zone:

[0070] ;

[0071] Where b is the function exponent of the lower edge curve of the lower chord beam segment in the hollow area, with a value of 2.0~3.0; The unit weight of the lower chord concrete is taken as 26 kN / m. 3 ; The control stress of the lower chord concrete is taken as 0.4 to 0.6 times the axial compressive design strength of the concrete.

[0072] S5. Check the second derivative of the lower edge curve G(x) of the lower chord beam segment in the hollow area, adjust the parameters H, h1, and h2, and repeat the aforementioned steps S1-S4 to ensure that the value of the second derivative of the lower edge curve G(x) of the lower chord beam segment in the hollow area is not less than / And within a certain range.

[0073] The second derivative of the curve G(x) at the lower edge of the lower chord segment in the empty zone is no greater than 3 to 5 times. / This ensures that the curvature of the lower chord beam segment is not too large, and the maximum eccentricity of the lower chord beam segment under eccentric compression is controlled.

[0074] Thus, the lower edge curve 5 (F(x)) of the mid-span beam segment and the lower edge curve 6 (G(x)) of the hollow lower chord beam segment are determined. The functions and their first derivatives of curves F(x) and G(x) at the connection point of the upper and lower chords are continuous. The second derivative of the lower edge curve G(x) of the hollow lower chord beam segment is within a certain range, which makes the structural transition between the mid-span beam segment and the hollow lower chord beam segment smooth, the stress transfer smooth, and the eccentricity of the hollow lower chord beam segment under eccentric compression small and the stress reasonable.

[0075] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge, characterized in that, The steps include: S1, determining the total height H between the upper and lower chord beams in the hollow area, the height h1 of the lower chord beam in the hollow area, the height h2 of the upper chord beam in the hollow area, the beam height at the junction of the upper and lower chords, and the range of the beam segment at mid-span. And the mid-span beam height h3; S2, calculate the range of the mid-span beam segment and the lower edge curve F(x) of the mid-span beam segment: ； Where c is the exponent of the curve function at the lower edge of the mid-span beam segment, with a value of 1.8 to 2.0; S3. Determine the connection point between the lower edge curve F(x) of the mid-span beam segment and the lower edge curve G(x) of the lower chord beam segment in the open section; Calculate and determine the function value and first derivative of curve F(x) at the connection point of the upper and lower chords: ； S4. Calculate the coefficients of the 2.0~3.0 order polynomial function of the lower edge curve G(x) of the lower chord beam segment in the hollow region: ; Obtain the G(x) function, which represents the lower edge curve of the lower chord beam segment in the hollow region: Where b is the function exponent of the lower edge curve of the lower chord beam segment in the hollow area, with a value of 2.0~3.0; The unit weight of the lower chord concrete is taken as 26 kN / m. 3 ; The control stress of the lower chord concrete is taken as 0.4 to 0.6 times the axial compressive design strength of the concrete; L is the net span of the main span of the hollow continuous rigid frame bridge; S5, check the second derivative of the lower edge curve G(x) of the lower chord beam segment in the hollow area, adjust the parameters H, h1, and h2, and repeat the aforementioned steps S1-S4 to ensure that the value of the second derivative of the lower edge curve G(x) of the lower chord beam segment in the hollow area is not less than / 。 2. The method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge as described in claim 1, characterized in that, In step S1, the total height H between the upper and lower chord beams in the hollow area is: 。 3. The method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge as described in claim 2, characterized in that... In step S1, the height h1 of the lower chord beam in the hollow area is: 。 4. The method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge as described in claim 3, characterized in that... In step S1, the height h2 of the upper chord beam in the hollow area is: ； The beam height at the junction of the upper and lower chords is 。 5. The method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge as described in claim 3, characterized in that... In step S1, the range of the mid-span beam segment ： 。 6. The method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge as described in claim 5, characterized in that... In step S1, the mid-span beam height h3 is: 。 7. The method for determining the lower edge curve of the mid-span and lower chord beam of a hollow continuous rigid frame bridge as described in claim 6, characterized in that, The second derivative of the curve G(x) at the lower edge of the lower chord segment in the empty zone is no greater than 3 to 5 times. / This ensures that the curvature of the lower chord beam segment is not too large, and the maximum eccentricity of the lower chord beam segment under eccentric compression is controlled.

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