Simplified calculation method for deformation of double-tower cable-stayed bridge under load action

Abstract

The invention discloses a simplified calculation method for deformation of a double-tower cable-stayed bridge under the load action. The method comprises the following steps: establishing a Cartesian coordinate system by taking the left side beam end of a side span main beam of the double-tower cable-stayed bridge as a coordinate origin; the double-tower cable-stayed bridge structure system is disassembled at the inhaul cable and main beam anchoring point to be divided into a main beam system and a bridge tower system; calculating the axial force and the vertical component of the inhaul cable of the cable-tower system while calculating the deformation of the main beam system under the action of the external load; the external load effect and the vertical force of the inhaul cable are combined, and the updated load effect of the main beam system is obtained; the updated load effect is applied to the main beam system to replace the external load of the original main bridge system, and the deformation amount of the updated main beam system at any point is obtained; the displacement value of an inhaul cable anchoring point is calculated according to the deformation of any point, and the inhaul cable axial force and the vertical component of the cable-tower system are updated according to the displacement value of the inhaul cable anchoring point; calculating the tolerance according to the result; judging whether the tolerance is less than or equal to an allowable deviation value; and outputting results such as deformation of the full-bridge main beam and cable force of each stay cable.

Description

Technical Field

[0001] This invention relates to the field of engineering, and in particular to a simplified calculation method for the deformation of a double-tower cable-stayed bridge under load. Background Technology

[0002] Cable-stayed bridges are a common type of long-span bridge structure used in highway and railway engineering to cross valleys and rivers, with double-tower cable-stayed bridges being the most widely used. Double-tower cable-stayed bridges have low structural stiffness, and the structural deformation under external loads is a key indicator controlling the structural design. The double-tower cable-stayed bridge structure is a repeatedly statically indeterminate structural system; under load, the internal forces of the bridge towers, main beams, and stay cables are coupled, making structural deformation calculations complex.

[0003] There are currently three main calculation methods, including the finite element model method, the elastic foundation beam method, and the elastically supported continuous beam method. The finite element model method generally uses finite element software to establish calculation units for each component to simulate its mechanical behavior, but the modeling process is cumbersome and the calculation efficiency is low, making it inconvenient to apply. The elastic foundation beam method treats the cables as springs supporting the main beam and the mid-span main beam as an infinitely long elastic foundation beam with equal stiffness supporting the springs to solve the structural deformation. The main shortcomings of this method are: (1) it ignores the interaction of cables at different locations, resulting in low calculation accuracy; (2) the elastic foundation beam analysis adopts the assumption of equal support stiffness, which cannot consider the difference in cable stiffness at different locations; (3) it cannot calculate the deformation of the side span under load. The elastic support continuous beam method treats the cables as springs supporting the main beam and the main beam as a multi-point elastic support continuous beam to solve the structural deformation. The main shortcomings of this method are: (1) There are many cables with different forms, the calculation formula for the equivalent support stiffness of the cables is complicated, and the interaction between cables at different positions is ignored, resulting in low calculation accuracy; (2) The simplified elastic support continuous beam system still needs to be solved by finite element method or numerical method, which is cumbersome. Summary of the Invention

[0004] This invention addresses the problems of existing technologies, such as the cumbersome modeling process and low computational efficiency of finite element methods, which are inconvenient to apply; the low computational accuracy of elastic foundation beam methods due to neglecting the interaction of cables at different locations; the failure to consider the differences in cable stiffness at different locations in analytical solutions of elastic foundation beams, which cannot calculate the deformation of the side spans under load; the complexity of calculating the equivalent support stiffness of cables in elastic supported continuous beam methods due to the large number and varied forms of cables, and the neglect of the interaction of cables at different locations, resulting in low computational accuracy; and the fact that simplified elastic supported continuous beam systems still require finite element or numerical solutions, leading to cumbersome calculation processes. This invention provides a simplified calculation method and device for the deformation of a double-tower cable-stayed bridge under load, thus solving the aforementioned problems.

[0005] This invention provides a simplified calculation method for the deformation of a double-tower cable-stayed bridge under load, comprising the following steps:

[0006] S1. Establish a Cartesian coordinate system with the left end of the main beam of the side span of the double-tower cable-stayed bridge as the origin.

[0007] S2. The double-tower cable-stayed bridge structure is separated at the anchorage point between the cable and the main beam, and divided into the main beam system, the left tower system and the right tower system.

[0008] S3. Calculate the deformation of the main beam system under external loads, and simultaneously calculate the axial force and vertical component of the cables in the cable-tower system;

[0009] S4. Based on the interaction of forces between the bridge tower system and the main beam system, the external load and the vertical force of the cables are combined to obtain the updated load on the main beam system.

[0010] S5. Apply the updated load to the main girder system to replace the original external load of the main bridge system, and obtain the deformation of the updated main girder system at any point.

[0011] S6. Calculate the displacement value of the cable anchorage point based on the deformation of the updated main beam system at any point, and update the axial force and vertical component of the cable in the cable-tower system according to the displacement value of the cable anchorage point.

[0012] S7. Calculate the tolerance based on the above results, using the following formula:

[0013]

[0014] Where Fc,new is the updated vertical component of the cable axial force, Fc,p is the vertical component of the cable axial force in the previous iteration, tol is the computational tolerance, and ||·|| represents the quadratic norm of the vector;

[0015] S8. Determine whether the tolerance is less than or equal to the allowable deviation value. If yes, proceed to step S9; otherwise, proceed to step S4.

[0016] S9 outputs the calculation results of the deformation of the main girder and the cable force of the stay cables of the entire bridge.

[0017] The simplified calculation method for deformation of a double-tower cable-stayed bridge under load described in this invention, as a preferred embodiment, includes the following specific method for calculating the deformation of the main beam system under external load in step S3:

[0018] Initially, the cable tension is set to 0.

[0019] Suppose n arbitrary external loads act on the main beam system, and the magnitude of the loads is represented as a column vector P = [p1 p2 ... p n ] TThe column vector representing the load application locations is X = [x1 x2 … x n ] T ;

[0020] Removing the constraints at the two intermediate supports in the main beam system to form a simply supported beam with a span of L = 2L1 + L2 as the basic structure, the deformation Δ of the simply supported beam at the corresponding positions of the two intermediate supports under the external load P is... 1p and Δ 2p The calculation method is shown in the following formula:

[0021]

[0022] Where L1 is the side span and L2 is the middle span;

[0023] in,

[0024] The flexibility Δ at the corresponding position of the middle support of a simply supported beam with a span of L = 2L1 + L2 11 Δ 12 Δ 21 and Δ 22 Calculate using the following formula:

[0025]

[0026] Calculate the reaction force R at the two intermediate supports. f1 and R f2 The formula is as follows:

[0027]

[0028] The deformation y(x) of the main beam system at any point can be calculated from the deflection curve u of a simply supported beam with a span of L under an external load P. p (x), the formula is:

[0029]

[0030] Middle support reaction force R f1 R f2 Deflection curve u of a simply supported beam with span L under load R (x), calculate

[0031] The formula is:

[0032]

[0033] The formula for calculating the deformation y(x) of the main beam system at any point is:

[0034] y(x)=u p (x)+u R (x).

[0035] The simplified calculation method for deformation of a double-tower cable-stayed bridge under load described in this invention, as a preferred embodiment, includes the following specific calculation method for the axial force and vertical component of the cables in the cable-tower system in step S3:

[0036] Suppose a bridge tower with a total height of H has t stay cables and a bending stiffness of E. t I t The anchorage point of the i-th cable on the bridge tower is at a height h from the bottom of the tower. i The coordinates of the anchorage point on the main beam are x ci The angle between the cable and the x-axis is denoted as α. i The axial stiffness of the cable is denoted as E. ci A ci The length of the cable is denoted as L. ci The deformation of the main beam at the cable anchorage point is y(x) ci )

[0037] Calculate the initial value Δ of the cable anchorage point displacement. ci The formula is as follows:

[0038] Δ ci =ζy(x ci );

[0039] Where ζ is the convergence relaxation factor, which is set to 0.1;

[0040] Calculate the horizontal displacement U at the top of the bridge tower under the action of the cable anchorage point displacement. t The formula is as follows:

[0041]

[0042] Calculate the axial force N in each cable. ci The formula is as follows:

[0043]

[0044] Calculate the vertical component F of the axial force of each cable. vci The formula is as follows:

[0045] F vci =N ci sinα i ;

[0046] Let F be the column vector of vertical forces in the cable-stayed bridge. c The formula is as follows:

[0047] F c =[F vc1 F vc2 …F vci …F vc2t ] T ;

[0048] Calculate the column vector X of the position of vertical force application. c The formula is as follows:

[0049] X c =[x c1 x c2 …x ci …x c2t ] T .

[0050] The simplified calculation method for deformation of a double-tower cable-stayed bridge under load described in this invention, as a preferred embodiment, involves the following specific method for obtaining the updated load effect of the main girder system in step S4:

[0051] The formula for calculating the load value vector is as follows:

[0052]

[0053] The formula for calculating the load application location vector is as follows:

[0054]

[0055] The simplified calculation method for deformation of a double-tower cable-stayed bridge under load described in this invention, as a preferred method, calculates the updated displacement value of the cable anchorage points using the following formula:

[0056] Δ′ ci =(1-ζ)Δ ci +ζy(x ci );

[0057] Where, Δ′ ci To update the load P new The deformation results of the cable anchorage points calculated by applying the load to the main beam system.

[0059] The beneficial effects of this invention are as follows:

[0060] 1) There is no need to establish a complex finite element analysis model and tedious stiffness calculation. The structural mechanics calculation diagram after splitting is a three-span continuous beam and a cantilever beam. The structural form is simple and the calculation is more convenient. It can realize manual calculation of the complex structure of a double-tower cable-stayed bridge.

[0061] 2) The equilibrium equations of the bridge tower system simply and effectively consider the interaction between cables, resulting in higher calculation accuracy;

[0062] 3) The calculation method of this invention can provide the calculation results of the deformation of the whole bridge and the cable force of each cable under multi-point loading at any position of the whole bridge. The results data are more complete and provide extensive and powerful data support for the selection of bridge design schemes. Attached Figure Description

[0063] Figure 1 This is a flowchart of a simplified calculation method for the deformation of a double-tower cable-stayed bridge under load. Detailed Implementation

[0064] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0065] Example 1

[0066] The cable-stayed bridge in this embodiment features a symmetrical arrangement of two towers. The side spans L1 = 145m, and the middle span L2 = 300m. The total height of the bridge towers is 100m, with the tower height above the bridge deck reaching 75m. Each tower has nine pairs of stay cables. The elastic moduli of the bridge towers, main beams, and stay cables are 3 × 10⁻⁶. 4 MPa, 2×10 5 MPa, 2×10 5 MPa, the bridge tower's moment of inertia is 636 m. 4 The moment of inertia of the main beam is 3m. 4 The cross-sectional area of ​​the cable is 14400 mm². 2 .

[0067] like Figure 1 As shown, a simplified calculation method for the deformation of a double-tower cable-stayed bridge under load is proposed.

[0068] Includes the following steps:

[0069] Make basic assumptions

[0070] 1) Treat the double-tower cable-stayed bridge as a planar structure and disregard its spatial effects;

[0071] 2) The bending stiffness of the bridge towers remains constant along the height direction, while the bending stiffness of the main beams remains constant along the bridge span direction;

[0072] 3) The effect of axial force on the deformation of the bridge tower and main beam is ignored;

[0073] Establish coordinate system

[0074] A Cartesian coordinate system is established with the left end of the main beam of the side span of the double-tower cable-stayed bridge as the origin. The coordinate axis x is established to the right along the bridge span direction, and the coordinate axis y is established vertically upward.

[0075] Structural system decomposition

[0076] The double-tower cable-stayed bridge structure is separated at the anchorage points between the cables and the main girder, into three systems: the main girder system, the left tower system, and the right tower system. The main girder system is a three-span continuous beam system with a span arrangement of L1+L2+L1, where L1 is the side span and L2 is the middle span. Both the left and right tower systems are elastically supported cantilever beam systems.

[0077] Deformation calculation of main beam system under external load

[0078] Initially, assuming the cable tension is 0, the main beam system is a three-span continuous beam structure subjected only to external loads. Suppose n arbitrary external loads act on the main beam system, with the load magnitudes represented as a column vector P = [p1 p2…p…]. n ] T The column vector representing the load application locations is X = [x1 x2 … x n ] T .

[0079] By removing the constraints of the two intermediate supports in the main beam system, a simply supported beam with a span of L = 2L1 + L2 is formed as the basic structure. Under the action of external load P, the deformation of the simply supported beam at the corresponding positions of the two intermediate supports can be calculated according to the following formulas.

[0080]

[0081] The slenderness ratio of a simply supported beam with a span of L = 2L1 + L2 at the corresponding position of the middle support is calculated using the following formula.

[0082]

[0083] Furthermore, the reaction values ​​of the two intermediate supports can be obtained by the structural mechanics force method, as shown in the following formula.

[0084]

[0085] Furthermore, the deformation y(x) of the main beam system at any point can be represented by the deflection curve u of a simply supported beam with a span of L under an external load P. p (x) and the reaction force R at the middle support f1 R f2 Deflection curve u of a simply supported beam with span L under load R (x) is obtained by superposition.

[0086] y(x)=u p (x)+u R (x)

[0087] Among them, u p (x) and u R (x) is calculated using the following formulas.

[0088]

[0089]

[0090] Calculation of axial force and vertical component of cable in cable-to-tower system

[0091] Suppose a bridge tower with a total height of H has t stay cables and a bending stiffness of E. t I t The anchorage point of the i-th cable on the bridge tower is at a height h from the bottom of the tower. i The coordinates of the anchorage point on the main beam are x ci The angle between the cable and the x-axis is denoted as α. i The axial stiffness of the cable is denoted as E. ci A ci The length of the cable is denoted as L. ci The deformation of the main beam at the cable anchorage point is y(x) ci The stiffness of each cable can be accurately calculated based on actual conditions, taking into account the differences in stiffness among the cables.

[0092] Based on the deformation coordination relationship between the cable-tower system and the main beam system at the cable anchorage points, the initial displacement of the cable anchorage points is calculated.

[0093] Δ ci =ζy(x ci )

[0094] Where ζ is the convergence relaxation factor, which is set to 0.1.

[0095] Considering the interaction between the cables, calculate the horizontal displacement of the top of the bridge tower under the action of the cable anchorage point displacement, which is established by the equilibrium equation.

[0096]

[0097] Further, the axial force of each cable is calculated.

[0098]

[0099] Furthermore, the vertical component of the axial force of each cable is calculated.

[0100] F vci =N ci sinα i

[0101] The calculations for the two bridge tower systems are performed separately, and the column vectors of the vertical force on the cables and the column vectors of the positions where the vertical force acts are calculated as follows.

[0102] F c =[F vc1 F vc2 …F vci …Fvc2t ] T

[0103] X c =[x c1 x c2 …x ci …x c2t ] T

[0104] Update the load on the main beam system

[0105] Based on the force interaction between the bridge tower system and the main girder system, the bridge tower system applies a reaction force to the main girder system through the cables. The main girder has high axial stiffness, and its axial deformation is negligible; only the load effect of the cable vertical force component on the main girder is considered. Combining the external load P and the cable vertical force Fc, the updated load effect on the main girder system is obtained. The load value vector and load application location vector are shown in the following formulas.

[0106]

[0107] Update the deformation of the main beam system

[0108] The updated load P new The original external load P is applied to the main beam system to replace the main beam system. The deformation of the main beam system is calculated, and the calculation result in equation (7) is denoted as y. new (x).

[0109] Update the vertical component of the axial force of the cables in the cable-to-tower system.

[0110] Based on the above calculation results, calculate the displacement value of the cable anchorage point, where Δ ci,p This is the displacement value from the previous iteration step.

[0111] Δ′ ci =(1-ζ)Δ ci +ζy(x ci )

[0112] Using the calculated data, update the axial force N of the cable in the cable-to-tower system. ci and vertical component F c,new .

[0113] Tolerance checking and iteration

[0114] The tolerance is calculated, where Fc,new is the updated vertical component of the cable axial force, Fc,p is the vertical component of the cable axial force in the previous iteration, tol is the calculated tolerance, ||·|| represents the quadratic norm of the vector, and err is the allowable deviation value, taken as 1×10. -6 .

[0115]

[0116] If the tolerance check conditions of the above formula are met, the calculation ends, y new (x) represents the final deformation of the main beam, N ci This is the axial force of the i-th cable; if the tolerance check condition is not met, repeat the above steps until the tolerance check condition is met.

[0117] Calculation results

[0118] After the convergence condition is met, the calculation results of the deformation of the main girder of the entire bridge and the cable force of each stay cable are extracted.

[0119] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A simplified calculation method for deformation of a twin-tower cable-stayed bridge under load, characterized in that: The following steps are involved: S1. Establish a Cartesian coordinate system with the left end of the main beam of the side span of the twin-tower cable-stayed bridge as the coordinate origin; S2, disassemble the double-tower cable-stayed bridge structure system at the anchorage points of the cables and the main beams, and divide it into the main beam system, the left bridge tower system and the right bridge tower system; S3. Calculate the deformation of the main beam system under external loads and calculate the axial force and vertical component of the cable in the cable-tower system; S4. According to the interaction between the bridge tower system and the main beam system, the external load and the vertical force of the cable are combined to obtain the updated load of the main beam system; S5, applying the updated load to the main beam system to replace the original external load of the main bridge system, and obtaining the deformation of the updated main beam system at any point; S6. Calculate the displacement value of the cable anchor point based on the updated deformation of the main beam system at any point, and update the axial force and vertical component of the cable of the cable-tower system according to the displacement value of the cable anchor point; S7. Calculate the tolerance based on the above results. The formula is: Where, Fc,new is the updated vertical component of the cable axial force, Fc,p is the vertical component of the cable axial force in the previous iteration step, tol is the calculation tolerance, and ||·|| represents the quadratic norm of the vector; S8, determine whether the tolerance is less than or equal to the allowable deviation value, if yes, proceed to step S9; otherwise, proceed to step S4; S9. Output the calculation results of the deformation of the main beam of the whole bridge and the force of the inclined cables.

2. The simplified calculation method for deformation of a twin-tower cable-stayed bridge under load according to claim 1 is characterized in that: The specific method for calculating the deformation of the main beam system under the external load in step S3 is: In the initial state, the cable force is taken as 0; Assume that n arbitrary external loads act on the main beam system, and the load magnitude is expressed as a column vector in the form of P = [p1 p2…p n ] T , the column vector of the load position is expressed as X = [x1 x2…x n ] T ; Remove the constraints of the two middle supports in the main beam system, and form a simply supported beam with a span of L = 2L1 + L2 as the basic structure. Under the action of external load P, the deformation of the simply supported beam at the corresponding positions of the two middle supports is Δ 1p and Δ 2p , the calculation method is shown in the following formula: Among them, L1 is the side span, and L2 is the middle span; The flexibility Δ of a simply supported beam with a span of L = 2L1 + L2 at the corresponding position of the middle support 11 , Δ 12 , Δ 21 and Δ 22 Calculated according to the following formula: Calculate the reaction force R of the two middle supports f1 and R f2 , the formula is as follows: The deformation y(x) of the main beam system at any point can be calculated by the deflection curve u of the simply supported beam with a span of L under the action of the external load P. p (x), the formula is: Support reaction force R in calculation f1 , R f2 Deflection curve u of a simply supported beam with a span of L under action R (x), the formula is: The deformation y(x) of the main beam system at any point is given by: calculate y(x)=u p (x)+u R (x)。 3. The simplified calculation method for deformation of a twin-tower cable-stayed bridge under load according to claim 1 is characterized in that: The specific calculation method of the axial force and vertical component of the cable of the cable-tower system described in step S3 is: Assume that a bridge tower with a total height of H is equipped with t inclined cables and the bending stiffness of the bridge tower is E. t I t The height of the anchor point of the i-th cable on the bridge tower from the bottom of the tower is h i , the coordinate of the anchor point on the main beam is x ci , the angle between the cable and the x-axis is denoted as α i , the axial stiffness of the cable is recorded as E ci A ci , the length of the cable is recorded as L ci ; The deformation of the main beam at the cable anchor point is y(x ci ) Calculate the initial displacement of the cable anchor point Δ ci , the formula is as follows: D ci =ζy(x ci ); Among them, ζ is the convergence relaxation factor, which is taken as 0.1; Calculate the horizontal displacement U of the top of the bridge tower under the displacement of the cable anchor point t , the formula is as follows: Calculate the axial force N of each cable ci , the formula is as follows: Calculate the vertical component F of the axial force of each cable vci , the formula is as follows: F vci =N ci sinα i ; The vertical force vector F of the central and decentralized cables is c , the formula is as follows: F c =[F vc1 F vc2 …F vci …F vc2t ] T ; Calculate the vertical force position column vector X c , the formula is as follows: X c =[x c1 x c2 …x ci …x c2t ] T 。 4. The simplified calculation method for deformation of a twin-tower cable-stayed bridge under load according to claim 1 is characterized in that: Said The specific method for obtaining the updated load action of the main beam system in step S4 is: Calculate the load value vector using the following formula: Calculate the load position vector using the following formula:

5. The simplified calculation method for deformation of a twin-tower cable-stayed bridge under load according to claim 1 is characterized by: Calculate the updated cable anchor point displacement value using the formula: D′ ci =(1-ζ)D ci +ζy(x ci ); Among them, Δ′ ci To update the load effect P new Calculated deformation results of the cable anchorage points applied to the main beam system.

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