Method for optimizing vertical rigidity of rubber elastic support of cable-stayed bridge based on influence matrix

Abstract

The invention provides a cable-stayed bridge rubber elastic support vertical stiffness optimization method based on an influence matrix, and the method mainly comprises the steps: building a cable-stayed bridge finite element model, setting the vertical stiffness of all elastic supports to be 0, and applying all dead loads except for the tensioning force of a stay cable; unit cable force and unit support vertical counter force are sequentially applied to main beam nodes at all stay cables and supports, and a tower beam bending moment influence matrix, a support vertical compression amount influence matrix and an influence matrix of other structural responses needing to set boundary conditions are obtained; based on the bending moment influence matrix, establishing a matrix equation among the tower beam bending energy, the tension force and the support reaction force vector; the tower beam bending energy is minimized under the constraint of various boundary conditions, and the optimal tensioning force and support reaction vector are obtained; and finally, determining the vertical rigidity of the elastic support according to the linear relation between the support reaction force and the vertical compression amount of the support. According to the method, the step of optimizing the vertical rigidity of the elastic support of the cable-stayed bridge is integrated into the step of optimizing the cable force by adopting the influence matrix with the purpose of minimizing the bending energy of the tower beam, and a scientific basis is provided for designing the vertical rigidity of the elastic support of the cable-stayed bridge.

Description

Technical Field

[0001] This invention relates to the field of cable-stayed bridge design, and specifically to a method for optimizing the vertical stiffness of rubber elastic bearings for cable-stayed bridges based on an influence matrix. Background Technology

[0002] Cable-stayed bridges, due to their unique structural characteristics and outstanding long-span capacity, have become an important part of the global transportation network and are widely used worldwide. Cable-stayed bridges can be classified into floating systems, semi-floating systems, and fixed systems, among others. Among these, the semi-floating system combines the advantages of both fixed and floating systems, optimizing internal force distribution while ensuring structural stability, making it the most commonly used structural form for cable-stayed bridges. A characteristic of the semi-floating system is that the bridge towers and main beams are connected by supports, without the use of a zero-cable. Commonly used support types include spherical steel supports, friction pendulum supports, and rubber elastic supports. Among these, rubber elastic supports are widely used in semi-floating cable-stayed bridges due to their economic efficiency, serving to vertically transfer loads to the substructure and optimize structural stress.

[0003] The vertical stiffness of rubber elastic bearings is a key parameter in cable-stayed bridge design. Its adjustment range is wider than that of steel bearings, and can be achieved by changing the bearing height or the properties of the rubber material. Under load, the forces and deformations between cable-stayed bridge components are highly coupled. Adjusting the bearing stiffness affects the cable forces by changing the displacement of the main girder. Changes in cable forces, in turn, redistribute the load to the bridge towers, causing a redistribution of internal forces throughout the structure. Therefore, changes in bearing stiffness directly affect the overall stiffness matrix of the cable-stayed bridge, thus altering the structure's response under various loads: insufficient stiffness leads to excessive cable forces and main girder displacement, while excessive stiffness results in uneven force distribution in the main girder and even the bridge towers. Therefore, rationally designing the vertical stiffness of the bearings can effectively improve the stress and deformation state of the main girder and bridge towers, achieving an optimal balance between structural stiffness and internal forces.

[0004] However, the stiffness of rubber elastic bearings for cable-stayed bridges is currently often determined based on the engineering experience of designers. There is an urgent need for a scientific and reasonable method for determining the stiffness of rubber elastic bearings for cable-stayed bridges to provide a basis for design. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention proposes a method for optimizing the vertical stiffness of rubber elastic supports for cable-stayed bridges based on an influence matrix. The method uses the bending energy of the main beam and the bridge tower as the objective function and integrates the cable force optimization and support stiffness optimization steps to achieve optimal balance between structural stiffness and internal forces.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] This invention provides a method for optimizing the vertical stiffness of rubber elastic supports for cable-stayed bridges based on an influence matrix, comprising the following steps:

[0008] Step 1: Use finite element software to establish a linear elastic finite element model of the cable-stayed bridge system, consisting of z elements for the main girder and bridge towers;

[0009] Step 2: Apply all necessary self-weight and external loads, including dead load, live load, and temperature load, but do not apply tension to the stay cables. Set the vertical stiffness of the elastic supports to 0. Under this condition, the bending moments at the left and right ends of the i-th element of the main beam and the bridge tower are M and M, respectively. Li and M Ri All units have their left and right ends bent into rectangles forming two column vectors m. L0 and m R0 The vertical compression of the main beam nodes at the supports forms a column vector d0, and the stresses of all the stay cables form a column vector T0.

[0010] Step 3: Apply unit tension force and unit support vertical reaction force to the main beam nodes at all stay cables and support locations in sequence to obtain the influence matrix of tower-beam bending moment, the influence matrix of support vertical compression, and other influence matrices of structural response for which boundary conditions need to be set, as follows:

[0011] The bridge has a total of n stay cables and m elastic supports. A unit tension force is applied to each of the n stay cables in sequence, and a unit support reaction force is applied to each of the m elastic support nodes in sequence, for a total of n+m unit load cases.

[0012] Extract the column vectors of bending moments at both ends of all elements under all unit load conditions, and subtract m from each vector. L0 and m R0 , thus obtaining vector m L1 ,m L2 ,…m L(n+m) and m R1 ,m R2 ,…m R(n+m) This forms an influence matrix C with z rows and (n+m) columns. L and C R Extract the column vectors of vertical compression at all supports of the main beam nodes under all unit load conditions, and subtract d0 from each vector to obtain vectors d1, d2, ... d (n+m) This forms an influence matrix d with m rows and (n+m) columns. c Extract the stress column vectors of all stay cables under all unit load conditions, and subtract T0 from each vector to obtain vectors T1, T2, ... T. (n+m) This forms an influence matrix T with n rows and (n+m) columns. c ;

[0013] Step 4: Based on the bending moment influence matrix, establish the matrix equations between the tower beam bending energy and the tension force and support reaction force vectors, as follows:

[0014] Using the bending energy of the main beam and bridge tower as the objective function, the formula for the bending energy U for the truss element model is as follows.

[0015]

[0016] Among them, L i E i and I i These are the length, elastic modulus, and moment of inertia of the i-th element of the main beam and bridge tower, respectively.

[0017] The matrix form of bending energy is shown in the following equation:

[0018] U = m L T B m L +m R T B m R

[0019] Where, m L and m R Let B be the bending moment vectors at the left and right ends of the element, and let B be a diagonal matrix with each coefficient b. ii The value is:

[0020]

[0021] In a linear elastic finite element model subjected to an external load F, the structural response δ depends only on the stiffness matrix K of the structure, i.e., it satisfies the principle of linear superposition, as shown in the following equation.

[0022] δ=K -1 F

[0023] Based on the principle of linear superposition, the optimized bending moment vectors at both ends of the main beam and bridge tower elements are expressed as follows:

[0024]

[0025] Where the vector t = [v1, v2, ..., v] n u1, u2, ..., u m ] T The relationship between the unknown vector t and the bending energy U is obtained by including n applied cable tension forces v and m applied vertical reaction forces u of rubber elastic supports on the main beam.

[0026] Step 5: The response values ​​that need to be set for boundary conditions after optimization can also be obtained from the influence matrix. The optimized cable stress vector T and support compression vector d are shown in the following formula. Set the boundary conditions for each element in T and d according to the design requirements.

[0027] T = T0 + T c t

[0028] d = d0 + d c t

[0029] Minimize the bending energy U under all boundary conditions. At this point, t is the vector of cable force and support reaction force to be applied, and d is the vector of support compression.

[0030] Step 6: Based on the linear relationship between support reaction force and compression (support stiffness is the ratio of support reaction force to compression), obtain the optimal support stiffness vector V = [k1, k2, ..., k n ] T .

[0031] The design concept of this invention is as follows:

[0032] A finite element model of the cable-stayed bridge was established, and the vertical stiffness of all elastic supports was set to 0. All dead loads except for the tension force of the stay cables were applied. Unit cable force and unit support vertical reaction force were sequentially applied to the main beam nodes at all stay cable and support locations to obtain the influence matrix of tower-beam bending moment, the influence matrix of support vertical compression, and other influence matrices of structural responses requiring boundary conditions. Based on the bending moment influence matrix, a matrix equation was established between the tower-beam bending energy and the vectors of tension force and support reaction force. The tower-beam bending energy was minimized under various boundary conditions to obtain the optimal tension force and support reaction force vectors. Finally, the vertical stiffness of the elastic supports was determined based on the linear relationship between the support reaction force and the support vertical compression.

[0033] The beneficial effects of this invention are as follows: In the prior art, cable force and support stiffness of cable-stayed bridges are usually designed separately. Support stiffness is first determined based on the designer's experience, and then cable force optimization is performed. The determination of support stiffness lacks scientific basis, and the combination of support stiffness and cable force design values ​​may not yield optimal results. This invention uses the bending energy of the main girder and bridge towers as the objective function and integrates the cable force and support stiffness optimization steps using an influence matrix, i.e., simultaneous optimization. This avoids empiricism in the design stage and provides a scientific basis for the design of cable-stayed bridge support stiffness. Attached Figure Description

[0034] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. They are used together with the embodiments of the present invention to explain the present invention and do not constitute a limitation of the present invention.

[0035] Figure 1 This is a flowchart illustrating a method for optimizing the vertical stiffness of a cable-stayed bridge rubber elastic support based on an influence matrix, as described in an embodiment of the present invention.

[0036] Figure 2 This is a schematic diagram of the elastic finite element model of the cable-stayed bridge strut system in an embodiment of the present invention;

[0037] The following are the labels in the figure: 1-Simplified model of rubber elastic support; 2-Bridge tower element; 20-Bridge tower node; 3-Main beam element; 30-Main beam node; 31-Main beam hinge node; 4-Cable stay element; 5-Pier top node.

[0038] Figure 3 This is the reference load condition for the cable-stayed bridge model in this embodiment of the invention;

[0039] Figure 4 This is the unit load condition for the cable-stayed bridge model in this embodiment of the invention. Detailed Implementation

[0040] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0041] This invention provides a method for optimizing the vertical stiffness of rubber elastic bearings for cable-stayed bridges based on an influence matrix. The specific process is as follows: Figure 1 As shown, it includes:

[0042] Step 1: Establish a linear elastic finite element model of the semi-floating cable-stayed bridge system using finite element software. The tower and pier are fixed, and the tower and beam are connected by rubber elastic supports. There are a total of 6 pairs of stay cables, all of which are truss elements. Figure 2 As shown. The main beam has 14 elements, the bridge tower has 9 elements, and the tower and beam together have 23 beam elements. The two rubber elastic supports are simplified to springs with vertical stiffnesses of k1 and k2, connecting the main beam-pier and the main beam-tower respectively. The bending energy of the pier is not within the optimization range, so only the top nodes of the pier are shown, and the pier elements are not displayed. The bottom nodes of the bridge tower and pier are fixed.

[0043] Step 2: Apply all necessary self-weight and external loads, including dead load, live load, and temperature load, but do not apply tension to the stay cables. Set both k1 and k2 to 0. Figure 3As shown, under this working condition, the bending moments at the left and right ends of the i-th element of the main beam and the bridge tower are M and M, respectively. Li and M Ri Then the bending moments at the left and right ends of all elements can form two column vectors m. L0 and m R0 Each has 23 elements; the vertical compression of the main beam node at the support forms a column vector d0, which includes 2 elements; all cable stresses form a column vector T0, which includes 6×2=12 elements;

[0044] Step 3: The entire bridge has 12 stay cables and 2 elastic supports. A unit tension force of 1kN is applied sequentially to each of the 12 stay cables, and a unit support reaction force of 1kN is applied sequentially to the main beam nodes at the 2 elastic supports, resulting in a total of 14 unit load cases. Figure 4 As shown. Extract the left and right moment column vectors of all elements under all unit load cases and subtract m from each. L0 and m R0 , to obtain m L1 ,m L2 ,…m L14 and m R1 ,m R2 ,…m R14 This results in an influence matrix C consisting of 23 rows and 14 columns. L and C R Extract the column vectors of vertical compression at all supports of the main beam nodes under all unit load conditions, and subtract d0 from each vector to obtain vectors d1, d2, ... d0. 14 This forms a 2x14 influence matrix d. c Extract the stress column vectors of all stay cables under all unit load conditions, and subtract T0 from each vector to obtain vectors T1, T2, ... T. 14 This forms a 12-row, 14-column influence matrix T. c ;

[0045] Step 4: Use the bending energy of the main beam and bridge tower as the objective function. For the truss element model, the formula for the bending energy U is as follows, where L... i E i and I i These are the length, elastic modulus, and moment of inertia of the i-th element of the main beam and bridge tower, respectively.

[0046]

[0047] The matrix form of bending energy is shown in the following equation:

[0048] U = m L T B m L +m R T B mR

[0049] Where, m L and m R Let B be the bending moment vectors at the left and right ends of the element, and let B be a diagonal matrix, with each element b. ii The value is:

[0050]

[0051] When a linear elastic finite element model is subjected to an external load F, the structural response δ is only related to the stiffness matrix K of the structure, that is, it satisfies the principle of linear superposition, as shown in the following formula, where δ can be displacement, internal force, stress, etc.

[0052] δ=K -1 F

[0053] Based on the principle of linear superposition, the bending moment vectors at the left and right ends of the optimized main beam and bridge tower elements can be expressed as follows.

[0054]

[0055] Where the vector t = [v1, v2, ..., v 12 ,u1,u2] T It includes 12 applied cable tension forces v and 2 applied rubber elastic support vertical reaction forces u on the main beam, thus obtaining the relationship U(t) between the unknown vector t and the bending energy U;

[0056] Step 5: The response values ​​for setting boundary conditions after optimization are also obtained from the influence matrix. The optimized cable stress vector T and support compression vector d are shown in the following formula. According to the design requirements, the range of each element in T is limited to [MinT,MaxT] = [0,1000] MPa; the range of each element in d is limited to [Mind,Maxd] = [0,300] mm.

[0057] T = T0 + T c t

[0058] d = d0 + d c t

[0059] Minimize the bending energy U under all boundary conditions. At this point, t is the vector of the cable force and support reaction force to be applied, and d is the vector of support compression.

[0060] Step 6: Based on the linear relationship between support reaction force and compression (support stiffness is the ratio of support reaction force to compression), obtain the optimal support stiffness vector V = [k1, k2]. T .

[0061] Steps 4-6 can be implemented using MATLAB. This embodiment provides a MATLAB program example:

[0062]

[0063]

[0064] In this program, the fmincon function outputs the local minimum of the function. The structural responses such as the target cable force and the target support compression obtained may be unreasonable. It is necessary to continuously adjust the initial search point T00 vector based on the output bending energy and structural response data to determine the most reasonable optimization target.

[0065] This invention, based on an influence matrix, overcomes the limitations of empirical design by integrating cable force and support stiffness optimization. Traditional methods often design cable force and support stiffness independently, lacking a synergistic analysis of the overall structural response; the combination of support stiffness and cable force design values ​​may not yield optimal results. This invention, however, uses minimizing tower-beam bending energy as the objective function, achieving coordinated optimization of cable force and support stiffness. While ensuring the safe range of cable stress and controllable support compression, it improves the balance of internal force distribution in the main beam and bridge tower, providing a scientific basis and practical solution for the design of cable-stayed bridge support stiffness.

[0066] The above description is merely a specific embodiment of the present invention and is described in detail. However, the scope of the present invention is not limited thereto. Any changes or modifications made by those skilled in the art within the scope of the present invention are covered by the patent scope of the present invention.

Claims

1. A method for optimizing the vertical stiffness of rubber elastic supports for cable-stayed bridges based on an influence matrix, characterized in that, Includes the following steps: Step 1: Establish a linear elastic finite element model of the cable-stayed bridge system, which consists of z elements for the main girder and the bridge tower; Step 2: Apply all self-weight and external loads to be considered except for the tension force of the stay cables, including dead load, live load, and temperature load, and set the vertical stiffness of the elastic support to 0. Under this condition, the bending moments at the left and right ends of the i-th element of the main beam and the bridge tower are M and M, respectively. Li and M Ri All units have their left and right ends bent into rectangles forming two column vectors m. L0 and m R0 The vertical compression of the main beam node at the support forms a column vector d0, and the stress of all the stay cables forms a column vector T0. Step 3: Apply unit tension force and unit support vertical reaction force to the main beam nodes at all stay cables and support locations in sequence to obtain the tower-beam bending moment influence matrix, support vertical compression influence matrix, and other structural response influence matrices that require setting boundary conditions; Step 4: Establish the matrix equations between the tower beam bending energy and the tension force and support reaction force vectors based on the bending moment influence matrix; Step 5: Minimize the tower beam bending energy under various boundary conditions to obtain the optimal tension force and support reaction force vectors; Step 6: Based on the linear relationship between support reaction force and compression, obtain the optimal support stiffness vector V = [k1, k2, ..., k m ] T .

2. The method for optimizing the vertical stiffness of rubber elastic supports for cable-stayed bridges based on an influence matrix as described in claim 1, characterized in that, Step 3 is described in detail below: The bridge has a total of n stay cables and m elastic supports. A unit tension force is applied to each of the n stay cables in sequence, and a unit support reaction force is applied to each of the m elastic support nodes in sequence, for a total of n+m unit load cases. Extract the column vectors of bending moments at the left and right ends of all elements under all unit load conditions, and subtract m from each vector. L0 and m R0 , thus obtaining vector m L1 ,m L2 ,…m L(n+m) and m R1 ,m R2 ,…m R(n+m) This forms an influence matrix C with z rows and (n+m) columns. L and C R Extract the column vectors of vertical compression at all supports of the main beam nodes under all unit load conditions, and subtract d0 from each vector to obtain vectors d1, d2, ... d (n+m) This forms an influence matrix d with m rows and (n+m) columns. c Extract the stress column vectors of all stay cables under all unit load conditions, and subtract T0 from each vector to obtain vectors T1, T2, ... T. (n+m) This forms an influence matrix T with n rows and (n+m) columns. c .

3. The method for optimizing the vertical stiffness of rubber elastic supports for cable-stayed bridges based on an influence matrix according to claim 1, characterized in that, Step 4 is described in detail below: Using the bending energy of the main beam and bridge tower as the objective function, the formula for the bending energy U for the truss element model is as follows. Among them, L i E i and I i These are the length, elastic modulus, and moment of inertia of the i-th element of the main beam and bridge tower, respectively. The matrix form of bending energy is shown in the following equation: U=m L T B m L +m R T B m R Where, m L and m R Let B be the bending moment vectors at the left and right ends of the element, and let B be a diagonal matrix with each coefficient b. ii The value is: In a linear elastic finite element model subjected to an external load F, the structural response δ depends only on the stiffness matrix K of the structure, i.e., it satisfies the principle of linear superposition, as shown in the following equation. δ=K -1 F Based on the principle of linear superposition, the optimized bending moment vectors at both ends of the main beam and bridge tower elements are expressed as follows: Where the vector t = [v1, v2, ..., v] n u1, u2, ..., u m ] T It includes n applied cable tension forces v and m applied vertical reaction forces u from rubber elastic supports on the main beam, thus obtaining the relationship U(t) between the unknown vector t and the bending energy U.

4. The method for optimizing the vertical stiffness of rubber elastic supports for cable-stayed bridges based on an influence matrix according to claim 1, characterized in that, Step 5 is described in detail below: The optimized cable stress vector T and support compression vector d are shown in the following formula. Boundary conditions for each element in T and d are set according to design requirements. T=T0+T c t d=d0+d c t Minimize the bending energy U under all boundary conditions. At this point, t is the vector of cable force and support reaction force to be applied, and d is the vector of support compression.

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