Method and device for determining live load cable force of bridge end suspenders in cable-stayed-suspension cable cooperative system

Abstract

This invention relates to the field of bridge engineering technology, specifically to a method and apparatus for determining the live load force of end cables in a cable-stayed-suspension cable-stayed system. The method includes the following steps: determining the live load force of the secondary end cable without considering cable elongation; determining the elongation of the secondary end cable under live load based on the live load force of the secondary end cable without considering cable elongation and the length of the secondary end cable; obtaining expressions for the first derivatives of the vertical displacement curves of the main cable at the end cable and the main cable at the end cable in the live load state, based on parabolic theory; establishing an expression for the live load force of the end cable; and obtaining the live load force of the end cable by combining the above expressions. This method solves the problem in existing technologies where the most important parameters affecting the fatigue performance of the end cable are unknown, leading to numerous calculation types, a large workload, and high time and effort, while the obtained parameter combination may not be the optimal solution.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology, specifically to a method and device for determining the live load cable force of the end suspenders in a cable-stayed-suspension cable cooperative system. Background Technology

[0002] The most prominent problem in cable-stayed-suspension bridge systems is fatigue in the end cables. Under live loads, the stress variation in the end cables is much greater than that in conventional suspension bridges. To date, the method for solving the end cable fatigue problem is to establish a finite element model, perform finite element calculations on different parameters, and select a set of parameters from all possible combinations that meets the stress requirements.

[0003] The disadvantage of this method is that since the most important parameter affecting the fatigue performance of the end sling is unknown, it requires cross-combination of many parameters such as the sag-to-span ratio, the span ratio, the bending stiffness, and the number of cross cables. This results in a wide variety of calculation types, a very large workload, and is time-consuming and labor-intensive. Moreover, the parameter combination obtained may not be the optimal solution. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method and apparatus for determining the live load cable force of end cables in a cable-stayed-suspension cable cooperative system. This method solves the problem in existing technologies where the most important parameters affecting the fatigue performance of end cables are unknown, resulting in numerous calculation types, a large workload, and significant time and effort, while the obtained parameter combinations may not necessarily be the optimal solution.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] On one hand, the present invention provides a method for determining the live load cable force of the end cables in a cable-stayed-suspension cooperative system, characterized by comprising the following steps:

[0007] Based on the horizontal force of the main cable under dead load, the horizontal force of the main cable under live load, the cable force of the secondary end suspender under dead load, and the vertical displacement curve of the main beam, the live load cable force of the secondary end suspender without considering cable elongation is determined.

[0008] Based on the fact that the live load force of the secondary sling is not considered when the sling elongates and the length of the secondary sling, the elongation of the secondary sling under live load is determined.

[0009] Based on the relationship between the elongation of the secondary end sling under live load and the vertical displacement of the upper and lower ends of the sling, as well as the parabola theory, we obtain the expression for the first derivative of the vertical displacement curve of the main cable of the empty cable section between the end sling and the main tower at the end sling under live load, and the expression for the first derivative of the vertical displacement curve of the main cable between the end sling and the secondary end sling at the end sling under live load.

[0010] Based on the equilibrium equations of the main cable branch points at the end slings under dead load and the equilibrium equations of the main cable branch points at the end slings under the combined action of dead load and live load, an expression for the live load force of the end slings is established.

[0011] By combining the expression for the live load force of the end sling, the expression for the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the main tower under live load conditions at the end sling, and the expression for the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the secondary end sling under live load conditions at the end sling, the live load force of the end sling can be obtained.

[0012] In some alternative solutions, determining the live load force of the secondary suspenders (ignoring suspender elongation) based on the horizontal force of the main cable under dead load, the horizontal force of the main cable under live load, the secondary suspender force under dead load, and the vertical displacement curve of the main beam includes:

[0013] Establish the equilibrium equations for the main cable branch points at the secondary end slings under dead load, and the equilibrium equations for the main cable branch points at the secondary end slings under the combined action of dead load and live load.

[0014] Based on the equilibrium equations of the main cable branch points at the secondary end suspenders under dead load and under the combined action of dead load and live load, an expression for the live load cable force of the secondary end suspenders is established.

[0015] Based on the expression for the live load force of the secondary suspenders and the vertical displacement curve of the main beam, the suspenders are analogized to a continuous membrane without considering elongation, thus obtaining the live load force of the secondary suspenders without considering suspender elongation.

[0016] In some alternative schemes, the equilibrium equation for the main cable branch point at the secondary end suspender under constant load is: Hq(y′) 12，2 -y′ 23，2 ) = T d，2 , where H q y′ represents the horizontal force on the main cable under constant load. 12，2 Let y′ be the first derivative of the curve of the main cable between the secondary and intermediate suspenders under constant load at the intermediate suspender. 23，2 Let T be the first derivative of the curve of the main cable between the secondary end sling and the No. 3 sling under constant load at the secondary end sling. d， 2 represents the cable force of the end suspender under constant load;

[0017] The equilibrium equation established at the secondary end suspender point of the main cable under the combined action of dead load and live load is: (H q +H p )(y′ 12，2 +η′ 12，2 -y′ 23，2 -η′ 23，2 ) = T d，2 +T p，2 , where H pη′ represents the horizontal force on the main cable under live load. 12，2 Let η′ be the first derivative of the vertical displacement curve of the main cable between the primary and secondary suspenders under live load conditions at the secondary suspender. 23，2 T represents the first derivative of the vertical displacement curve of the main cable between the secondary end sling and the No. 3 sling under live load at the secondary end sling. p，2 For the live load cable force of the secondary sling;

[0018] The established expression for the live load cable force of the secondary end sling is as follows:

[0019] The live load force of the secondary sling, neglecting sling elongation, is... Among them, T p，2v For the secondary suspenders, the live load force during suspender elongation is not considered. v′2 is the first derivative of the vertical displacement curve of the main beam at the second suspender, and v′3 is the first derivative of the vertical displacement curve of the main beam at the third suspender.

[0020] In some alternative solutions, according to the formula Determine the elongation of the secondary end sling under live load, where ΔL h，2 T represents the elongation of the secondary sling under live load. p，2 For the live load force of the secondary sling, L h，2 E represents the length of the secondary sling. h Let A be the elastic modulus of the sling. h，2 This represents the cross-sectional area of ​​the secondary suspension cable.

[0021] In some optional schemes, the expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load, based on the relationship between the elongation of the secondary end sling under live load, the vertical displacement of the upper and lower ends of the sling, and parabolic theory, and the expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load, including:

[0022] Based on the elongation of the sling, establish the relationship between the vertical displacements at the upper and lower ends of the sling;

[0023] Assuming the main cable of the empty cable section between the end suspender and the main tower is a parabola, the second derivative of the parabola curve of this section when the main cable has no vertical displacement and the second derivative of the parabola curve of this section when the vertical displacement of the main cable changes can be used to obtain the second derivative of the vertical displacement curve of the empty cable section caused by live load.

[0024] Based on the second derivative of the vertical displacement curve of the empty cable segment caused by live load, combined with the parabolic expression, the expression for the first derivative of the vertical displacement curve of the main cable of the empty cable segment between the end sling and the main tower under live load is obtained at the end sling.

[0025] Based on the relationship between the elongation of the secondary end sling under live load and the vertical displacement of the upper and lower ends of the sling, the expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load between the end sling and the secondary end sling is obtained.

[0026] In some alternative schemes, the relationship between the vertical displacements at the upper and lower ends of the sling is established as η. h，i +ΔL h，i =v h，i , where η h，i v represents the vertical displacement of the main cable at the intersection of the i-th suspender cable and the main cable under live load conditions. h，i Let ΔL be the vertical displacement of the main beam at the intersection of the i-th suspender cable and the main beam under live load conditions. h，i The elongation of the i-th sling under live load conditions;

[0027] The second derivative of the vertical displacement curve of the unloaded cable segment caused by live load is Where η″ is the second derivative of the vertical displacement curve of the empty cable segment caused by live load, q2 is the vertical load intensity of the main cable of the empty cable segment between the end suspender and the main tower, and y2″ is the second derivative of the parabolic curve corresponding to the empty cable segment between the end suspender and the main tower.

[0028] The expression for the first derivative of the vertical displacement curve of the main cable at the end suspender under live load conditions is as follows: Where, η′ 01，1 Let L be the first derivative of the vertical displacement curve of the main cable at the end suspender under live load conditions, where L is the length of the main cable in the empty cable section between the end suspender and the main tower. xl The horizontal length of the main cable's unloaded section, η 01，1 v1 represents the vertical displacement of the main cable at the end suspender, v1 represents the vertical displacement of the main beam at the end suspender, and ΔL represents the vertical displacement of the main beam at the end suspender. h，1 This refers to the elongation of the end sling under live load conditions.

[0029] The expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load conditions between the end sling and the secondary end sling is: Where, η′ 12，1 Let v'1 be the first derivative of the vertical displacement curve of the main cable at the end suspender under live load conditions, and v'1 be the first derivative of the vertical displacement curve of the main girder at the end suspender. Let D be the first derivative of the vertical displacement curve of the main girder at the end suspender. h This refers to the longitudinal distance between the secondary end sling and the end sling.

[0030] In some alternative solutions, the description of establishing the expression for the live load force of the end sling based on the equilibrium equations of the main cable branch points at the end slings under dead load and the equilibrium equations of the main cable branch points at the end slings under the combined action of dead load and live load includes:

[0031] Force analysis was performed on the end sling, and equilibrium equations were established for the main cable branch points at the end slings under dead load and under the combined action of dead load and live load.

[0032] Based on the equilibrium equations of the main cable branch points at the end slings under dead load and the equilibrium equations of the main cable branch points at the end slings under the combined action of dead load and live load, an expression for the live load cable force of the end slings is established.

[0033] In some alternative schemes, the equilibrium equation for the main cable branch point at the end suspender under dead load is H. q (y′ 01，1 -y′ 12，1 ) = T d，1 , where y′ 01，1 Let y′ be the first derivative of the curve of the main cable at the end suspender under constant load conditions, where y′ is the empty cable section between the end suspender and the main tower. 12，1 Let T be the first derivative of the curve of the main cable between the end sling and the secondary end sling under constant load at the end sling. d，1 The tension in the lower end of the suspension cable under constant load;

[0034] The equilibrium equation for the main cable branch point at the end of the suspender under the combined action of dead load and live load is (H q +H p )(y′ 01，1 +η′ 01，1 -y′ 12，1 -η′ 12，1 ) = T d，1 +T p，1 , where η′ 01，1 Let η′ be the first derivative of the vertical displacement curve of the main cable at the end suspender point in the live-load condition between the end suspender and the main tower. 12,1 T represents the first derivative of the vertical displacement curve of the main cable at the end sling under live load conditions between the end sling and the secondary end sling. p,1 For the live load cable force of the end sling;

[0035] The expression for the live load force of the end sling is:

[0036] In some alternative solutions, the live load cable force of the sling is

[0037]

[0038] On the other hand, the present invention also provides a device for calculating the live load cable force of bridge end suspenders in a cable-stayed-suspension cooperative system, comprising:

[0039] The secondary end cable force determination module is used to determine the live load cable force of the secondary end cable without considering cable elongation, based on the horizontal force of the main cable under dead load, the horizontal force of the main cable under live load, the cable force of the secondary end cable under dead load, and the vertical displacement curve of the main beam.

[0040] The secondary end elongation determination module is used to determine the elongation of the secondary end sling under live load based on the live load force of the secondary end sling without considering the sling elongation and the length of the secondary end sling.

[0041] The intermediate parameter determination module is used to obtain the expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load, based on the relationship between the elongation of the secondary end sling under live load, the vertical displacement of the upper and lower ends of the sling, and the parabolic theory.

[0042] The end cable force expression determination module is used to establish the expression of the live load cable force of the end cable based on the equilibrium equation of the main cable branch point at the end cable under dead load and the equilibrium equation of the main cable branch point at the end cable under the combined action of dead load and live load.

[0043] The end cable force determination module is used to combine the expression for the live load cable force of the end cable, the expression for the first derivative of the vertical displacement curve of the main cable of the empty cable section between the end cable and the main tower under live load conditions at the end cable, and the expression for the first derivative of the vertical displacement curve of the main cable between the end cable and the secondary end cable under live load conditions at the end cable to obtain the live load cable force of the end cable.

[0044] Compared with existing technologies, the advantages of this invention are as follows: The method for determining the live load force of the end suspenders in the cable-stayed-suspension cooperative system proposed in this solution obtains the live load force of the end suspenders through the expression of the live load force of the end suspenders, the expression of the first derivative of the vertical displacement curve of the main cable of the empty cable section between the end suspenders and the main tower under live load conditions, and the expression of the first derivative of the vertical displacement curve of the main cable between the end suspenders and the secondary end suspenders under live load conditions. The expression of the live load force of the suspenders clearly identifies the influencing factors and allows for quantitative analysis of a specific influencing factor to obtain its magnitude. Targeted optimization measures can then be taken, solving the problem that traditional methods rely on repeated parameter calculations using finite element methods, which cannot determine the fundamental influencing factors. This significantly saves time and improves efficiency. Attached Figure Description

[0045] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0046] Figure 1 A flowchart illustrating the calculation method for live load cable force of bridge end suspenders in a cable-stayed-suspension cooperative system provided in an embodiment of the present invention;

[0047] Figure 2 This is a schematic diagram of the segmented structure of a cable-stayed suspension bridge system provided in an embodiment of the present invention;

[0048] Figure 3 Illustrations of the stress and deformation of the sling provided in the embodiments of the present invention;

[0049] Figure 4 This invention provides a comparison between the calculated values ​​of the full formula for the influence line of live load cable force on the end sling and the finite element results in an embodiment of the invention.

[0050] Figure 5 This invention provides a comparison between the full formula calculation value and the simplified formula result of the influence line of live load cable force of the end sling in an embodiment of the invention.

[0051] In the diagram: 1. Main cable; 2. Suspension cable; 3. Main beam; 4. Stay cable; 5. Main tower. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0053] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0054] like Figure 2 As shown, the cable-stayed-suspension cooperative system bridge includes two main towers 5 and one main cable 1. Each main tower 5 is equipped with multiple cable stays 4 connected to the main beam 3. The cable stays 4 corresponding to the two main towers 5 are set at the closest interval on the main beam 3, and there are suspenders 2 connecting the main beam 3 and the main cable 1 at the interval. Some suspenders and cable stays 4 are staggered.

[0055] like Figure 1 As shown, on one hand, the present invention provides a method for determining the live load cable force of the end cables in a cable-stayed-suspension cooperative system, comprising the following steps:

[0056] S1: Based on the horizontal force of the main cable under dead load, the horizontal force of the main cable under live load, the cable force of the secondary end suspender under dead load, and the vertical displacement curve of the main beam, determine the live load cable force of the secondary end suspender when cable elongation is not considered.

[0057] In some optional embodiments, step S1 includes the following steps:

[0058] S11: Establish the equilibrium equations for the main cable branch points at the secondary end slings under dead load, and the equilibrium equations for the main cable branch points at the secondary end slings under the combined action of dead load and live load.

[0059] In this example, the equilibrium equation established for the main cable branch point at the secondary end suspender under constant load is: H q (y′ 12，2 -y′ 23，2 ) = T d，2 , where H q y′ represents the horizontal force on the main cable under constant load. 12，2 Let y′ be the first derivative of the curve of the main cable between the secondary and intermediate suspenders under constant load at the intermediate suspender. 23，2 Let T be the first derivative of the curve of the main cable between the secondary end sling and the No. 3 sling under constant load at the secondary end sling. d，2 The tension of the end cable under constant load.

[0060] The equilibrium equation established at the secondary end suspender point of the main cable under the combined action of dead load and live load is: (H q +H p )(y′ 12，2 +η′ 12，2 -y′ 23，2 -η′ 23，2 ) = T d，2 +T p，2 , where H p η′ represents the horizontal force on the main cable under live load. 12，2 Let η′ be the first derivative of the vertical displacement curve of the main cable between the primary and secondary suspenders under live load conditions at the secondary suspender. 23，2 T represents the first derivative of the vertical displacement curve of the main cable between the secondary end sling and the No. 3 sling under live load at the secondary end sling. p，2为 Secondary end sling live load cable force.

[0061] S12: Based on the equilibrium equations of the main cable branch points at the secondary end suspenders under dead load and under the combined action of dead load and live load, establish the expression for the live load cable force of the secondary end suspenders.

[0062] In this example, the established expression for the live load cable force of the secondary end sling is:

[0063] S13: Based on the expression of the live load force of the secondary end suspender and the vertical displacement curve of the main beam, the suspender is analogous to a continuous membrane without considering elongation, and the live load force of the secondary end suspender without considering the elongation of the suspender is obtained.

[0064] In this example, assuming the slings are analogous to a continuous membrane and elongation is not considered, the live load force T of the secondary sling is calculated. p，2v The expression for the elongation of the secondary cable is obtained, and an approximate expression for ΔL is derived. h，2 .

[0065] Assuming the suspenders are analogous to a continuous membrane and elongation is not considered, the vertical displacements of the main cable and the main beam at the same longitudinal position within the suspender section are the same, and are uniformly represented by the vertical displacement v of the main beam.

[0066] The live load force of the secondary sling, neglecting sling elongation, is... Among them, T p，2v For the secondary suspenders, the live load force during suspender elongation is not considered. v′2 is the first derivative of the vertical displacement curve of the main beam at the second suspender, and v′3 is the first derivative of the vertical displacement curve of the main beam at the third suspender.

[0067] S2: Determine the elongation of the secondary sling under live load conditions based on the live load force of the secondary sling without considering the sling elongation and the length of the secondary sling.

[0068] Considering that the live load amplitude of other slings besides the end slings is generally small, whether or not the effect of sling elastic elongation is taken into account, the sling force will not change significantly. Therefore, T p，2 ≈T p，2v .

[0069] Accordingly, according to the formula Determine the elongation of the secondary end sling under live load, where ΔL h，2 T represents the elongation of the secondary sling under live load. p，2 For the live load force of the secondary sling, L h，2 E represents the length of the secondary sling. h Let A be the elastic modulus of the sling. h，2 This represents the cross-sectional area of ​​the secondary suspension cable.

[0070] S3: Based on the relationship between the elongation of the secondary end sling under live load and the vertical displacement of the upper and lower ends of the sling, as well as the parabolic theory, we obtain the expression for the first derivative of the vertical displacement curve of the main cable at the end sling in the empty cable section between the end sling and the main tower under live load, and the expression for the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the secondary end sling under live load.

[0071] like Figure 3 As shown, in some optional embodiments, step S3 includes:

[0072] S31: Based on the elongation of the sling, establish the relationship between the vertical displacements at the upper and lower ends of the sling.

[0073] In this example, the relationship between the vertical displacements at the upper and lower ends of the sling is established as η.h，i +ΔL h，i =v h，i , where η h，i v represents the vertical displacement of the main cable at the upper end of the i-th sling (at its intersection with the main cable) under live load conditions. h，i Let ΔL be the vertical displacement of the main beam at the lower end of the i-th suspender cable (at the intersection with the main beam) under live load conditions. h，i This represents the elongation of the i-th sling under live load conditions.

[0074] The formula for calculating the elongation of the sling is as follows: In the formula: T p，i Let L be the live load force of the i-th sling. h，i Let E be the length of the i-th sling. h Let A be the elastic modulus of the sling. h，i Let be the cross-sectional area of ​​the i-th sling.

[0075] S32: Assuming the main cable of the empty cable section between the end suspender and the main tower is a parabola, the second derivative of the parabola curve of this section when the main cable has no vertical displacement and the second derivative of the parabola curve of this section when the main cable has a change in vertical displacement are obtained, thus obtaining the second derivative of the vertical displacement curve of the empty cable section caused by live load.

[0076] Assuming the main cable of the empty cable section between the end suspender and the main tower is a parabola, and the vertical load intensity of this section of the main cable is q2, then the second derivative of the parabolic curve of this section is... When the vertical displacement of the main cable changes, since the vertical load intensity of the empty cable section remains constant, the second derivative of the parabolic curve of this section after deformation will be... Subtracting the two equations, we obtain the second derivative of the vertical displacement curve of the unloaded cable segment caused by live load: Where η″ is the second derivative of the vertical displacement curve of the empty cable segment caused by live load, q2 is the vertical load intensity of the main cable of the empty cable segment between the end suspender and the main tower, and y2″ is the second derivative of the parabolic curve corresponding to the empty cable segment between the end suspender and the main tower.

[0077] S33: Based on the second derivative of the vertical displacement curve of the empty cable segment caused by live load, combined with the parabolic expression, the expression for the first derivative of the vertical displacement curve of the main cable of the empty cable segment between the end sling and the main tower under live load is obtained.

[0078] From the parabolic expression, the expression for the first derivative of the vertical displacement curve of the main cable at the end suspender under live load conditions is as follows: Where, η′ 01，1 Let L be the first derivative of the vertical displacement curve of the main cable at the end suspender under live load conditions, where L is the length of the main cable in the empty cable section between the end suspender and the main tower. xl The horizontal length of the main cable's unloaded section, η 01，1v1 represents the vertical displacement of the main cable at the end suspender, v1 represents the vertical displacement of the main beam at the end suspender, and ΔL represents the vertical displacement of the main beam at the end suspender. h，1 This refers to the elongation of the end sling under live load.

[0079] S34: Based on the relationship between the elongation of the secondary end sling under live load and the vertical displacement of the upper and lower ends of the sling, the expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load between the end sling and the secondary end sling is obtained.

[0080] The expression for the vertical displacement η at the upper and lower ends of the sling h，i +ΔL h，i =v h，i The expression for the first derivative of the vertical displacement curve of the main cable at the end sling under live load conditions between the end sling and the secondary end sling can be obtained as follows: Where, η′ 12，1 Let v'1 be the first derivative of the vertical displacement curve of the main cable at the end suspender under live load conditions, and v'1 be the first derivative of the vertical displacement curve of the main girder at the end suspender. Let D be the first derivative of the vertical displacement curve of the main girder at the end suspender. h This refers to the longitudinal distance between the secondary end sling and the end sling.

[0081] S4: Based on the equilibrium equations of the main cable branch points at the end slings under dead load and the equilibrium equations of the main cable branch points at the end slings under the combined action of dead load and live load, establish the expression for the live load force of the end slings.

[0082] In some optional embodiments, step S4 includes:

[0083] S41: Perform stress analysis on the end sling, establish the equilibrium equations of the main cable branch points at the end slings under dead load, and the equilibrium equations of the main cable branch points at the end slings under the combined action of dead load and live load.

[0084] In this example, the equilibrium equation for the main cable branch point at the end sling under constant load is H. q (y′ 01，1 -y′ 12，1 ) = T d，1 , where y′ 01，1 Let y′ be the first derivative of the curve of the main cable at the end suspender under constant load conditions, where y′ is the empty cable section between the end suspender and the main tower. 12，1 Let T be the first derivative of the curve of the main cable between the end sling and the secondary end sling under constant load at the end sling. d，1 This represents the cable force at the lower end of the suspension cable under constant load.

[0085] The equilibrium equation for the main cable branch point at the end of the suspender under the combined action of dead load and live load is (H q +H p )(y′ 01，1 +η′ 01，1 -y′ 12，1 -η′ 12，1) = T d，1 +T p，1 , where η′ 01，1 Let η′ be the first derivative of the vertical displacement curve of the main cable at the end suspender point in the live-load condition between the end suspender and the main tower. 12，1 T represents the first derivative of the vertical displacement curve of the main cable at the end sling under live load conditions between the end sling and the secondary end sling. p，1 For the live load cable force of the end sling;

[0086] S42: Based on the equilibrium equations of the main cable branch points at the end slings under dead load and the equilibrium equations of the main cable branch points at the end slings under the combined action of dead load and live load, establish the expression for the live load force of the end slings.

[0087] The expression for the live load force of the end sling is:

[0088] S5: By combining the expression for the live load force of the suspender, the expression for the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end suspender and the main tower under live load conditions at the end suspender, and the expression for the first derivative of the vertical displacement curve of the main cable between the end suspender and the secondary end suspender under live load conditions at the end suspender, the live load force of the end suspender is obtained.

[0089] In this example, the expressions for the live load force of the end sling, the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the main tower under live load conditions at the end sling, and the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the secondary end sling under live load conditions are substituted into the expression for the live load force of the end sling.

[0090] η ′ 01,1 and η ′ 12,1 Substituting the expression into T p,1 have to, Continue and After substituting and rearranging, we get the following expression:

[0091]

[0092] Among them: intermediate parameters

[0093] In some embodiments, the above equation is appropriately simplified. Under live load, the horizontal component of the main cable is typically much smaller than under dead load, and H is approximated as... q ≈H q +H p The simplified formula for calculating the live load cable force of the end sling, after appropriately considering the magnitude of various influences, is as follows:

[0094]

[0095] Calculations show that this formula has very high computational accuracy.

[0096] In one specific embodiment, L xl =353.5m, q2=71kN / m, H q =520678kN, elastic modulus E of the sling h =2×10 8 kPa, the area of ​​the sling is A h，1 =A h，2 =0.0707m 2 The sling spacing is D. h =7m, the lengths of the end sling and the No. 2 sling are L respectively. h，1 =75.18m, L h，2 =72.53m. The dead load forces of the end sling and the No. 2 sling are respectively T d，1 =2854kN, T d，2 =2893kN.

[0097] When a unit vertical concentrated force is applied at different positions x on the main beam, H p v1, v′1, v′2, and v′3 are all different, therefore the live load force T of the end sling is different. p，1 They are also different. x~T p，1 The curve is the influence line of the live load cable force of the end suspender cable.

[0098] Figure 4 The comparison between the calculated values ​​of the full formula (unsimplified formula) and the finite element method (FEM) calculations of the influence line of the live load force of the end sling in this embodiment of the invention shows that the linear shape of the calculated formula and the FEM calculation results are in good agreement. The maximum values ​​of the calculated formula and the FEM results are 0.0778 kN and 0.0928 kN, respectively, with a difference of 12.6%; the minimum values ​​are -0.0736 kN and -0.0807 kN, respectively, with a difference of 8.8%; and the maximum values ​​in the positive mid-span interval are 0.0365 kN and 0.0395 kN, respectively, with a difference of 7.6%. This indicates that the proposed calculation formula has high calculation accuracy.

[0099] Figure 5 The comparison between the full formula (unsimplified formula) and the simplified formula of the influence line of live load force of the end sling in this embodiment of the invention shows that the two curves basically overlap, which means that the simplified calculation formula has sufficient calculation accuracy.

[0100] On the other hand, the present invention also provides a device for calculating the live load cable force of bridge end cables in a cable-stayed-suspension cooperative system, comprising: a secondary end cable force determination module, a secondary end elongation determination module, an intermediate parameter determination module, an end cable force expression determination module, and an end cable force determination module.

[0101] The secondary end cable force determination module is used to determine the live load cable force of the secondary end suspender (ignoring cable elongation) based on the horizontal force of the main cable under dead load, the horizontal force of the main cable under live load, the cable force of the secondary end suspender under dead load, and the vertical displacement curve of the main beam. The secondary end elongation determination module is used to determine the elongation of the secondary end suspender under live load based on the live load cable force of the secondary end suspender (ignoring cable elongation) and the length of the secondary end suspender. The intermediate parameter determination module is used to obtain the expression for the first derivative of the vertical displacement curve of the main cable at the secondary end suspender under live load based on the relationship between the elongation of the secondary end suspender under live load, the vertical displacement at both ends of the suspender, and parabolic theory. The module provides an expression for the first derivative of the vertical displacement curve of the main cable between the end sling and the secondary end sling under live load conditions at the end sling. The end cable force determination module is used to establish the expression for the live load force of the end sling based on the equilibrium equations of the main cable branch points at the end sling under dead load and the equilibrium equations of the main cable branch points at the end sling under the combined action of dead load and live load. The end cable force determination module is used to obtain the live load force of the end sling by combining the expression for the live load force of the end sling, the expression for the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the main tower under live load conditions, and the expression for the first derivative of the vertical displacement curve of the main cable between the end sling and the secondary end sling under live load conditions.

[0102] In summary, the proposed method for determining the live load force of end cables in a cable-stayed-suspension cable-stayed system obtains the live load force of the end cables through expressions for the live load force of the end cables, expressions for the first derivative of the vertical displacement curve of the main cable at the end cable in the empty cable section between the end cable and the main tower under live load conditions, and expressions for the first derivative of the vertical displacement curve of the main cable at the end cable between the end cable and the secondary end cables under live load conditions. The expressions for the live load force of the end cables clearly identify the influencing factors and allow for quantitative analysis of specific influencing factors to determine their magnitude. Targeted optimization measures can then be taken. This method solves the problem of traditional methods that rely on repeated parameter calculations using finite element methods, which fail to determine the fundamental influencing factors, significantly saving time and improving efficiency.

[0103] In the description of this application, it should be noted that the terms "upper," "lower," etc., indicating the orientation or positional relationship are based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this application. Unless otherwise expressly specified and limited, the terms "installed," "connected," and "linked" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication between two elements. For those skilled in the art, the specific meaning of the above terms in this application can be understood according to the specific circumstances.

[0104] It should be noted that in this application, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0105] The above description is merely a specific embodiment of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. A method for determining the live cable force of an end suspension cable of a cable-suspender hybrid bridge, characterized in that, The method comprises the following steps: According to the horizontal force of the main cable under the dead load, the horizontal force of the main cable under the live load, the cable force of the secondary end sling under the dead load, and the vertical displacement curve of the main beam, the live cable force of the secondary end sling is determined without considering the elongation of the sling; Based on the live cable force of the secondary end sling without considering the elongation of the sling and the length of the secondary end sling, the elongation of the secondary end sling under the live load is determined; Based on the elongation of the secondary end sling under the live load, the relationship between the vertical displacements of the upper and lower ends of the sling, and the parabola theory, the expression of the first derivative of the vertical displacement curve of the main cable in the empty cable segment between the end sling and the main tower at the end sling under the live load is obtained, and the expression of the first derivative of the vertical displacement curve of the main cable between the end sling and the secondary end sling at the end sling under the live load is obtained, including: Based on the elongation of the sling, the relationship between the vertical displacements of the upper and lower ends of the sling is established; Assuming that the main cable in the empty cable segment between the end sling and the main tower is a parabola, according to the second derivative of the parabola curve of the main cable without vertical displacement and the second derivative of the parabola curve of the main cable with vertical displacement, the second derivative of the vertical displacement curve of the empty cable segment caused by the live load is obtained; Based on the second derivative of the vertical displacement curve of the empty cable segment caused by the live load, the expression of the first derivative of the vertical displacement curve of the main cable in the empty cable segment between the end sling and the main tower at the end sling under the live load is obtained in combination with the parabola expression; Based on the elongation of the secondary end sling under the live load and the relationship between the vertical displacements of the upper and lower ends of the sling, the expression of the first derivative of the vertical displacement curve of the main cable between the end sling and the secondary end sling at the end sling under the live load is obtained; According to the balance equation of the main cable at the end sling under the dead load and the balance equation of the main cable at the end sling under the dead load and the live load, the expression of the live cable force of the end sling is established; In combination with the expression of the live cable force of the end sling, the expression of the first derivative of the vertical displacement curve of the main cable in the empty cable segment between the end sling and the main tower at the end sling under the live load, and the expression of the first derivative of the vertical displacement curve of the main cable between the end sling and the secondary end sling at the end sling under the live load, the live cable force of the end sling is obtained.

2. The method of claim 1, wherein the method is characterized by: The method comprises the following steps: The balance equation of the main cable at the secondary end sling under the dead load and the balance equation of the main cable at the secondary end sling under the dead load and the live load are established; Based on the balance equation of the main cable at the secondary end sling under the dead load and the dead load and the live load, the expression of the live cable force of the secondary end sling is established; Based on the expression of the live cable force of the secondary end sling and the vertical displacement curve of the main beam, the sling is approximated as a continuous membrane without considering the elongation, and the live cable force of the secondary end sling without considering the elongation of the sling is obtained.

3. The method according to claim 2, wherein The established balance equation of the main cable at the sub-hanger cable under the action of the constant load is: wherein, is the horizontal force of the main cable under the action of the constant load, is the first-order derivative of the curve of the main cable between the end hanger cable and the sub-hanger cable at the sub-hanger cable under the constant load state, is the first-order derivative of the curve of the main cable between the sub-hanger cable and the 3# hanger cable at the sub-hanger cable under the constant load state, is the cable force of the sub-hanger cable under the action of the constant load. The established balance equation of the main cable at the sub-hanger under the combined action of dead load and live load is: wherein, is the horizontal force of the main cable under the action of live load, is the first derivative of the vertical displacement curve of the main cable between the end hanger and the sub-hanger at the sub-hanger under the live load state, is the first derivative of the vertical displacement curve of the main cable between the sub-hanger and the 3# hanger at the sub-hanger under the live load state, is the live load cable force of the sub-hanger. The expression of the secondary end sling live load cable force is ; The live cable force of the secondary end sling not considering the elongation of the sling is wherein, the live cable force of the secondary end sling not considering the elongation of the sling is the first-order derivative of the vertical displacement curve of the main beam at the second sling is the first-order derivative of the vertical displacement curve of the main beam at the third sling is 4. The method of claim 3, wherein the method is characterized by: The elongation of the secondary end sling under live load is determined according to the formula , wherein, is the elongation of the secondary end sling under live load, is the live cable force of the secondary end sling, is the length of the secondary end sling, is the elastic modulus of the sling, is the cross-sectional area of the secondary end sling.

5. The method according to claim 1, wherein The relationship between the vertical displacements of the upper and lower ends of the established sling is wherein, is the vertical displacement of the main cable at the intersection of the first sling and the main cable under the live load state, is the vertical displacement of the main cable at the intersection of the first sling and the main cable under the live load state, is the vertical displacement of the main beam at the intersection of the first sling and the main beam under the live load state, is the vertical displacement of the main beam at the intersection of the first sling and the main beam under the live load state, is the elongation of the first sling under the live load state, and is the elongation of the first sling under the live load state. The second derivative of the vertical displacement curve of the empty cable section caused by the live load is wherein, is the second derivative of the vertical displacement curve of the empty cable section caused by the live load, is the vertical load intensity of the main cable of the empty cable section between the end suspender and the main tower, is the second derivative of the corresponding parabolic curve of the empty cable section between the end suspender and the main tower. The expression of the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the main tower at the end sling under the live load state is wherein, is the first derivative of the vertical displacement curve of the main cable in the empty cable section between the end sling and the main tower at the end sling under the live load state, is the horizontal length of the empty cable section of the main cable, is the vertical displacement of the main cable at the end sling, is the vertical displacement of the main beam at the end sling, is the elongation of the end sling under the live load state; The expression of the first derivative of the vertical displacement curve of the main cable at the end suspender between the end suspender and the secondary end suspender under the live load state is wherein, is the first derivative of the vertical displacement curve of the main cable at the end suspender between the end suspender and the secondary end suspender under the live load state, is the first derivative of the vertical displacement curve of the main beam at the end suspender, is the longitudinal distance between the secondary end suspender and the end suspender.

6. The method of claim 1, wherein The balance equation of the main cable branch point at the end sling under the action of the dead load, the balance equation of the main cable branch point at the end sling under the action of the dead load and the live load, the expression of the live load cable force of the end sling are established, including: The force analysis of the end sling is performed, the balance equation of the main cable branch point at the end sling under the action of the dead load is established, and the balance equation of the main cable branch point at the end sling under the action of the dead load and the live load is established; The expression of the live load cable force of the end sling is established based on the balance equation of the main cable branch point at the end sling under the action of the dead load and the balance equation of the main cable branch point at the end sling under the action of the dead load and the live load.

7. The method for determining the live load cable force of the end sling of the cable-suspender cooperation system bridge according to claim 6, characterized in that: The balance equation of the main cable at the end sling under the action of the dead load is wherein, is the first-order derivative of the curve of the main cable in the empty cable section between the end sling and the main tower at the end sling under the dead load state, is the first-order derivative of the curve of the main cable at the end sling between the end sling and the secondary end sling under the dead load state, is the end sling force under the action of the dead load; The balance equation of the main cable at the end sling under the combined action of the dead load and the live load is wherein, is the first-order derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the main tower in the empty cable section under the live load state, is the first-order derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the secondary end sling under the live load state, is the live load cable force of the end sling. The expression of the end sling live load cable force is .

8. The method of claim 6, wherein the method is characterized by: The end sling live load cable force is wherein, is the longitudinal distance of the secondary end sling from the end sling, is the end sling cross-sectional area.

9. A device for calculating the live cable force of an end suspension cable of a cable-suspender hybrid bridge, characterized in that, including: The secondary end cable force determination module is configured to determine the live load cable force of the secondary end sling without considering the elongation of the sling according to the horizontal force of the main cable under the action of the dead load, the horizontal force of the main cable under the action of the live load, the secondary end sling cable force under the action of the dead load, and the vertical displacement curve of the main beam. The secondary end elongation determination module is configured to determine the elongation of the secondary end sling under the live load state based on the live load cable force of the secondary end sling without considering the elongation of the sling and the length of the secondary end sling. The intermediate parameter determination module is configured to obtain the expression of the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the main tower under the live load state and the expression of the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the secondary end sling under the live load state based on the elongation of the secondary end sling under the live load state, the relationship between the vertical displacements of the upper and lower ends of the sling, and the parabola theory, including: The relationship between the vertical displacements of the upper and lower ends of the sling is established based on the elongation of the sling. The main cable between the end sling and the main tower is assumed to be a parabola, the second derivative of the parabola curve of the main cable without vertical displacement is obtained, and the second derivative of the parabola curve of the main cable with vertical displacement is obtained, thereby obtaining the second derivative of the vertical displacement curve of the empty cable segment caused by the live load. The expression of the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the main tower under the live load state is obtained based on the second derivative of the vertical displacement curve of the empty cable segment caused by the live load and the parabola expression. The expression of the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the secondary end sling under the live load state is obtained based on the elongation of the secondary end sling under the live load state and the relationship between the vertical displacements of the upper and lower ends of the sling. The end cable force expression determination module is configured to establish the expression of the live load cable force of the end sling based on the balance equation of the main cable branch point at the end sling under the action of the dead load and the balance equation of the main cable branch point at the end sling under the action of the dead load and the live load. The end cable force determination module is configured to obtain the live load cable force of the end sling by combining the expression of the live load cable force of the end sling, the expression of the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the main tower under the live load state, and the expression of the first derivative of the vertical displacement curve of the main cable at the end sling between the end sling and the secondary end sling under the live load state.

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