Linear variation stiffness matrix method for suspension bridge space cable bridge forming shape finding

Abstract

The invention discloses a linear variation stiffness matrix method for suspension bridge space cable forming form finding, relates to the technical field of bridge numerical simulation, and solves the defects of a stiffness matrix in the process of solving the suspension bridge forming form finding through a segmented catenary method. The method comprises the steps that a suspension bridge is simplified into a space cable system composed of a main span, a side span and an anchor span, and the main span, the side span and the anchor span are sequentially subjected to a bridge forming and shape finding process; the linear change stiffness matrix solved in the bridge forming and shape finding process of the main span is used for representing the linear relation between the displacement change of the control point and the initial force change of the main cable; according to the method, an analytical expression of a linear change stiffness matrix is deduced, and an efficient iterative algorithm is established. According to the algorithm, the initial force correction is directly calculated according to the displacement error of the control point in each iteration.

Description

Technical Field

[0001] This invention relates to the field of bridge numerical simulation technology, specifically to the linear variation stiffness matrix method for shape finding of spatial cable-stayed suspension bridges. Background Technology

[0002] The form-finding problem of suspension bridges in their completed state is a core aspect of cable structure analysis. Since cables themselves have no fixed shape, their geometry must be determined by balancing internal tensions and external loads. (Tang Maolin et al.) [1] A segmented catenary method for calculating the main cable alignment of a completed suspension bridge was proposed, pioneering the use of segmented catenary theory to calculate the planar main cable alignment. Due to its high accuracy and computational efficiency, the analytical method based on segmented catenary theory has become the main method for calculating the main cable alignment of suspension bridges.

[0003] With the development of suspension bridges towards longer spans and wider decks, the lateral spatial effects are becoming increasingly significant, making the study of spatial cable theory inevitable. (Luo Xiheng et al.) [2-3] The theory of segmented catenary in planar planes was extended to spatial cables, and a spatial analysis model considering the coupling effect between the main cable and the suspenders and the influence of the saddle was established. Numerical analytical methods were used to solve for the main cable alignment in both the bridge-completed and unsupported cable states. (Li Chuanxi et al.) [4] For the first time, a recursive formula for the stiffness matrix of the spatial main cable's linear variation was derived, significantly improving computational efficiency and accuracy compared to the traditional iterative influence matrix method. (Yang Zhiyong et al.) [5] An iterative method based on geometric correlation equations is proposed, employing a two-stage "coarse calculation-fine calculation" strategy to handle the main cable-suspender coupling problem, thereby improving the algorithm's stability. (Yang Yang et al.) [6] A calculation method based on geometric correction is proposed, which iteratively corrects the geometry of the main cable and suspenders by changing their shapes. The correction process is intuitive and easy to understand. (Zhang Wenming et al.) [7] The nonlinear GRG method is used for multivariate programming to solve for the bridge alignment. (Qi Dongchun et al.) [8] A modified influence matrix method for space cables is proposed.

[0004] Although the above methods have achieved good results in engineering applications, the computational efficiency and accuracy of iterative methods still need further optimization.

[0005] References:

[0006] [1] Tang Maolin, Qiang Shizhong, Shen Ruili. Segmented catenary method for calculating the main cable alignment of a completed suspension bridge [J]. Journal of Railway Engineering, 2003, (01): 87-91.

[0007] [2] Luo Xiheng, Xiao Rucheng, Xiang Haifan. Analysis of the main cable alignment of a spatial cable suspension bridge [J]. Journal of Tongji University (Natural Science Edition), 2004, (10): 1349-1354.

[0008] [3]Kim H ,Lee M ,Chang S . Non-linear shape-finding analysis of aself-anchored suspension bridge [J]. Engineering Structures, 2002, 24 (12):1547-1559. DOI:10.1016 / S0141-0296(02)00097-4.

[0009] [4] Li Chuanxi, Ke Hongjun, Liu Haibo, et al. Determination method of completed bridge state of spatial main cable self-anchored suspension bridge [J]. Engineering Mechanics, 2010, 27(05):137-146.

[0010] [5] Yang Yongzhi. Analysis of the main cable alignment of a single-tower suspension bridge with curved beams on a single-sided spatial cable surface [J]. Journal of Southwest Jiaotong University, 2024, 59(02):298-306.

[0011] [6] Yang Yang, Wang Feng. Research on cable alignment calculation method of suspension bridge based on geometric correction [J]. Bridge Construction, 2025, 55(05):151-158. DOI:10.20051 / j.issn.1003-4722.2025.05.020.

[0012] [7] Zhang Wenming, Chang Jiaqi, Tian Genmin, et al. Analytical calculation method for main cable alignment of three-tower spatial cable suspension bridge with unequal main span [J]. Bridge Construction, 2022, 52(01):109-115.

[0013] [8] Qi Dongchun, Shen Ruili. Calculation method of main cable shape of spatial cable in suspension bridge [J]. Railway Construction, 2013, (04): 13-16. Summary of the Invention

[0014] To address the aforementioned issues, this paper proposes an improved iterative method for the stiffness matrix of spatial cable profile variation based on the theory of spatial segmented catenary. This method involves: ① establishing a complete Jacobi matrix with the stress-free length as a variable, and rigorously eliminating this variable through partial derivative operations; ② constructing a global equation, directly extracting the stiffness matrix for three key control points, and solving for the initial force correction at once using the displacement errors of the three control points, thus simplifying the solution process and improving computational efficiency.

[0015] The linear variation stiffness matrix method for form finding of a spatial cable suspension bridge includes: simplifying the suspension bridge into a spatial cable system consisting of a main span, side spans, and anchor spans; sequentially performing the form finding process on the main span, side spans, and anchor spans; the form finding process adopts the spatial segmented catenary method; in the form finding process, the linear variation stiffness matrix is ​​directly established for the coordinates of three key control points: the Y-coordinate of the mid-span, the Y-coordinate of the right end point, and the Z-coordinate; the initial force correction is solved at once by the coordinate error of the three control points; and in the derivation of the stiffness matrix, the stress-free length is used as a variable in the differentiation. The bridge alignment process for the main span includes: dividing the main cable into N segments, initially determining the initial force of the main cable according to parabolic theory, calculating the force on each segment from cable segment 1 to cable segment N, obtaining the coordinates of the right tower top and the mid-span point to obtain the calculated coordinates of the control points, calculating the control point coordinate error in combination with the design values ​​of the control point coordinates, and determining whether the control point coordinate error meets the standard. If it does, the alignment process ends; if it does not, the alignment change stiffness matrix is ​​calculated and the initial force of the main cable is updated, and the aforementioned steps are repeated. The initial force of the main cable refers to the three-dimensional component force at the beginning of the main cable, i.e., the three-dimensional component force at the left end of cable segment 1.

[0016] Furthermore, the calculation of the linear stiffness matrix includes: based on the spatial piecewise catenary theory, a complete Jacobi matrix containing stress-free length variables is established through total differential derivation of the basic equations; stress-free length variables are strictly eliminated through partial derivative operations to obtain the main cable Jacobi matrix and the inverse Jacobi matrix of the suspension cable; a force transfer correction matrix and a cumulative transfer matrix are introduced to construct a global equation; and the linear stiffness matrix is ​​directly extracted by solving the global equation, establishing a precise linear relationship between the control point displacement change and the initial force change.

[0017] Furthermore, the Jacobi matrix of the main cable is established. Including: the three-dimensional projection lengths of the cable segment sequentially about Differentiate the force components and the stress-free length, considering the cable segment. Assuming the projected length is a design constant, the Jacobi matrix of the main cable is obtained by eliminating the stress-free length derivative through total differential. .

[0018] Furthermore, the inverse Jacobi matrix of the suspension cable is established. This includes: obtaining the Jacobi matrix of the suspension cable by differentiating the force balance equation of the cable and eliminating the differential of the stress-free length. Since the displacement of the upper endpoint is opposite to the direction of the projection change, the Jacobi matrix of the suspension cable is defined. .

[0019] Furthermore, the introduction of the force transfer correction matrix and the cumulative transfer matrix includes: differentiating the force balance equation, substituting the stress-free length differential, and defining the force transfer matrix. It fully describes the transmission relationship of force differentials at adjacent points; through piecewise recursion, it defines the cumulative transmission matrix. The product of the force transmission matrices of each segment, and the local cumulative matrix. It is the product of consecutive segments.

[0020] Furthermore, the construction of the global equation includes: differentiating the geometrically compatible equations of the branch points, where the change in the coordinates of the branch points is equal to the sum of the differentials of the projected lengths of each cable segment; substituting the Jacobi matrix of the main cable and the force recursion relation, the change in the coordinates of the branch points can be expressed as a linear combination of the differentials of the initial force and the load; and after changing the order of summation, defining a coefficient matrix with respect to the initial force. , Describes the sensitivity of the component displacements to the initial force; defines the intermediate coefficient matrix with respect to the load. , , , This describes the contribution of the load at each point to the displacement; the load differential is transformed into the displacement differential at each point using the inverse Jacobi matrix of the suspension cable, and a synergy coefficient matrix is ​​defined. , , , This describes the influence of the displacement of the preceding component point on the displacement of the subsequent component point, reflecting the spatial synergy of the main cable-suspender system; it assembles a coefficient matrix describing the relationships between all component points and displacements, and establishes a global equation: The global matrix Composed of blocks of identity matrix and synergy coefficient matrix, global coefficient matrix Depend on and composition.

[0021] Furthermore, the step of directly extracting the linear stiffness matrix by solving the global equation includes: defining the compliance matrix. Global matrix Inverse and global coefficient matrix The product of the two forces describes the sensitivity of the point displacements to the initial force; from the compliance matrix Extract the rows corresponding to key control points, including those crossing midpoints. Coordinates, right endpoint coordinates and Coordinates, forming a compliance submatrix ; for the flexibility submatrix Inverse the matrix to obtain the linearly changing stiffness matrix. Establish a linear relationship between the displacement error of key control points and the initial force correction.

[0022] The beneficial effects of this invention include:

[0023] (1) A complete analysis model of the space cable system was established. The main cable and suspenders were simplified into flexible cables and formed a spatial synergistic relationship, which can accurately describe the stress and deformation characteristics of the space cable system.

[0024] (2) An analytical expression for the linearly varying stiffness matrix was derived, and an efficient iterative algorithm was established. Based on the spatially segmented catenary theory, a complete Jacobi matrix containing stress-free length variables was established through the total differential derivation of the basic equations. The stress-free length variables were strictly eliminated through partial derivative operations to obtain the main cable Jacobi matrix and the inverse Jacobi matrix of the suspension cable. A global equation was constructed by introducing a force transfer correction matrix and a cumulative transfer matrix. The stiffness matrix was directly extracted by solving the global equation, and a precise linear relationship between the control point displacement change and the initial force change was established. In each iteration, the algorithm directly calculates the initial force correction from the control point displacement error.

[0025] (3) The accuracy and engineering applicability of the method were verified. Through the calculation example of the Xi'an Bahe Bridge project and the comparison with BNLAS software, the calculation results of the method in this paper are in good agreement with the calculation results of BNLAS, which verifies the correctness and reliability of the method in this paper. Attached Figure Description

[0026] Figure 1 The following is a simplified diagram of the spatial cable calculation involved in the embodiments of this application, wherein (a) is the main cable-suspender cable collaborative analysis model, (b) is the main cable discretization and point stress diagram, (c) is the simplified diagram of the main cable segment stress analysis, and (d) is the simplified diagram of the suspender cable stress analysis.

[0027] Figure 2 This describes the form-finding calculation steps for the main span of the bridge involved in the embodiments of this application.

[0028] Figure 3 This is the calculation process for the linear variation stiffness matrix involved in the embodiments of this application. Detailed Implementation

[0029] 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 a part of the embodiments of this application, and not all of the embodiments. Therefore, the following detailed description of the embodiments of this application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely represents selected embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0030] The following is in conjunction with the appendix Figure 1 Specific embodiments of the present invention will be described in detail;

[0031] It should be noted that: This embodiment involves several equations. By default, multiple formulas in the equations are ordered using (a), (b), (c), etc. For example, the first formula in equation (1) is formula (1-a), the second formula is formula (1-b), and so on.

[0032] The spatial cable system of the suspension bridge will be established as follows: Figure 1 The model shown in -a illustrates the main cable-suspender synergistic analysis. In this model, the main cable is simplified as a continuous flexible cable, starting from the anchorage, passing through the side span saddle and the main tower saddle, crossing the side span and the main span, and symmetrically arranged to the opposite anchorage. The suspenders are also simplified as flexible cables, hinged at the upper end to the main cable and at the lower end to the bridge deck, transferring the bridge deck load to the main cable as a concentrated force. Due to the suspenders' inclination in the transverse direction, the load they exert on the main cable includes both lateral and vertical components. Under the combined action of its own weight and the suspender load, the main cable forms a spatial catenary, while the suspenders, under their own weight and tension, also exhibit a catenary shape. The main cable alignment determines the suspender position, and the suspender tension reacts on the main cable, forming a spatial synergistic relationship between the two. Figure 1 The nodes in the text refer to the same points as the sub-points mentioned in this embodiment.

[0033] The main cable is discretized according to the position of the suspenders. Section, such as Figure 1 -b is shown in the upper part. (Point numbering) Point 0 is the left endpoint (IP point at the top of the left tower). The right endpoint (IP point at the top of the right tower) is the dividing point. ( =1, 2, ..., -1) Anchor points on the sling. Sling segment numbers: 1, 2, ..., cable section Points from -1 to the minute The main cable segment. The longitudinal projection length of each cable segment is... , , ,..., , Sling numbers 1, 2, ..., -1, Sling Upper end and main cable branch point The connection is anchored to the bridge deck at the lower end. Force transfer occurs in the space near point i, such as... Figure 1 -b is shown in the lower half. Cable segment. -1. Cable Section By dividing points Connections, branch points Suspension cable Concentrated loads transmitted Function. The main cable is subjected to an initial force from the left end. Initially, the self-weight of the cable segment and the load at each point are applied, and the load is transferred segment by segment to the right to the point. The equilibrium condition of component forces connects the forces on adjacent cable segments and suspenders.

[0034] cable section Force analysis, such as Figure 1 As shown in -c. The coordinates of the points at both ends of the cable segment are respectively... and The left end of the cable segment is subjected to triaxial tension. Function: The right end is subjected to triaxial tension. Function. The cable segment bears the tensile force at both ends and its own weight. Its function is to appear as a catenary in space. The projected lengths of the cable segments in the three coordinate directions are... , , Cable segment material parameters include stress-free length. Elastic modulus Cross-sectional area .

[0035] slings Force analysis in the YZ plane, such as Figure 1 -d is shown. Coordinates of the upper end point of the sling. Coinciding with the main cable branch point i, the coordinates of the lower endpoint are... Determined by the location on the bridge deck. The upper end of the suspension cable is under tension. Function: The lower end is subjected to vertical loads. Function. In the completed bridge state, the suspenders are not tilted longitudinally. The projected lengths of the suspenders in the vertical and horizontal directions are respectively... , Slings Self-respect It exhibits a catenary shape within the YZ plane. The cable material parameters include stress-free length. Elastic modulus Cross-sectional area Tension at the upper end of the sling As a concentrated load acting on the branch point of the main cable .

[0036] For cable section (From the points) to the point The three-dimensional component forces at the left end are { , , Define auxiliary variables:

[0037] ,

[0038] ,

[0039] ,

[0040] .

[0041] cable section The three-dimensional projection lengths are:

[0042] (1)

[0043] Branch The spatial coordinates are determined by the coordinates of the previous sub-point and the projected length of the cable segment:

[0044] (2)

[0045] Branch The force balance relationship at the point is:

[0046] (3)

[0047] For slings (points) The coordinate difference between the upper and lower suspension points (the slings below) is the coordinate difference in equations (1-a) and (1-b). , ,and The directions are balanced by the sling forces, namely:

[0048] (4)

[0049] (5)

[0050] The form-finding problem for completed bridges can be described as follows: given the bridge's geometric parameters, material properties, and load conditions, solve for the spatial coordinates of the main cable and the stress-free length of the cable to ensure that the coordinates of the key control points meet the design requirements. The known geometric parameters include the main span, the longitudinal projected length of the cable segments, the coordinates of the lower ends of the suspenders, and the design coordinates of the control points (including the IP points at both ends and the mid-span elevation). The known material properties include the elastic modulus, cross-sectional area, and unit weight of the main cable and suspenders. The known load conditions are the vertical loads on the lower ends of each suspender.

[0051] Based on parabolic theory, the initial three-dimensional component forces at the top of the left tower were preliminarily determined. , , }, calculate segment by segment from left to right. For the th cable segment ( Given the three-dimensional component forces at the left end { , , } and longitudinal bridge length The stress-free length of the cable segment can be solved by simultaneously solving equation (1). Among them, in (1-a) Given a known quantity, consider this expression as relating to... The nonlinear equations were solved using Newton's iteration method. The results were obtained. Then, the projected length is calculated using (1-b) and (1-c). and Then, the division points are obtained from equation (2). coordinates and .

[0052] For the first root sling ( Given the coordinates of the hanging point ( , ), coordinates of the lower hanging point ( , ) and the vertical component at the lower end The calculation formula for suspension cables is the same as that for cable segments. Since suspension cables do not tilt along the bridge's longitudinal direction, therefore... Then, the same state equations (1-b) and (1-c) as the main cable segment are adopted, and the parameters are replaced with the suspender parameters. Combined with the force balance equation (5), the stress-free length of the suspender is taken as the stress-free length. and lateral force As the initial value for iteration, Newton's method is used to solve the problem. By establishing the Jacobi matrix of the suspension cable, the unknown correction is calculated based on the projection error, and the stress-free length of the suspension cable and the tension at the upper end are obtained iteratively.

[0053] Calculate the cable segment using the mechanical equilibrium equation (3) Right end (dividing point) The three-dimensional component of the force { , , }, as a cable segment The left-hand force. Repeat the above calculation process until the first... Cable section, obtain the top of the right tower (branch point) )coordinate( , ) and coordinates of the midpoint (point M) .

[0054] Forward calculations can be performed using the initial force { , , The calculated coordinates of control points often have a certain error compared to the design values. The control point coordinate error is defined as follows:

[0055] (6)

[0056] in,{ , , } represents the calculated coordinates of the control points. , , } represents the design values ​​for the control point coordinates.

[0057] The shape-finding problem is transformed into: adjusting the initial force Reduce control point coordinate error The force approaches zero. The key lies in how to quickly calculate the initial force correction based on the control point displacement error. .

[0058] The linear variation stiffness matrix method proposed in this embodiment establishes a linear relationship between the control point displacement change and the initial force change. In the k-th iteration, the current control point error is used to determine the relationship between the control point displacement change and the initial force change. The initial force correction can be calculated directly using the stiffness matrix:

[0059] (7)

[0060] in The vector representing the change in displacement of the control point. Let the initial force change vector be... This is the linearly varying stiffness matrix; the negative sign indicates that the correction direction is opposite to the error direction. Then, the initial force is updated:

[0061] (8)

[0062] Perform the next forward calculation until the error meets the convergence criterion. The main span bridge form-finding steps are as follows: Figure 2 As shown.

[0063] The projected length of the cable segment is calculated using equation (1). , , Consider it as tension at the left end of the cable segment and stress-free length Functions:

[0064] (9)

[0065] Linearizing nonlinear functional relationships through total differentials, utilizing... Elimination of constraints for constants The Jacobi matrix is ​​established, and finally the linear change stiffness matrix is ​​derived.

[0066] (1) Calculate the Jacobi matrix of the main cable

[0067] For equation (1-a) with respect to the force components { , ,} and stress-free length Find the partial derivative, define partial derivative:

[0068]

[0069]

[0070] Towards and Z direction The partial derivatives follow the same logic, and the process of solving for the partial derivatives is omitted.

[0071] because For design constants, Taking the total differential of (1-a), we get:

[0072] (10)

[0073] in , , .

[0074] Substituting equation (10) into the total differential forms of equations (1-b) and (1-c), we can eliminate... Jacobi matrix of the main cable (2×3):

[0075] (11)

[0076] The matrix elements are:

[0077] ,

[0078] ,

[0079] ,

[0080] ,

[0081] ,

[0082] .

[0083] For equation (1-a) with respect to the force components { } and stress-free length Find the partial derivative. Because As a design constant, it is eliminated by total differential. The Jacobi matrix of the main cable is obtained. (2×3) Establish the relationship between the differentials of the projected lengths of the cable segment in the Y and Z directions and the differentials of the three-dimensional forces at the left end.

[0084] (2) Calculate the inverse Jacobi matrix of the sling.

[0085] The state equations for the suspenders are the same as those for the main cable. For equations (1-b) and (1-c) with respect to the force components... , and stress-free length Find the partial derivative, define partial derivative:

[0086]

[0087] partial derivatives And so on, the process of solving for partial derivatives is omitted.

[0088] Differentiating the equilibrium equation (5) for the sling force, we get:

[0089] (12)

[0090] Similar to the main cable processing method, substitute equation (12) into the total differential of equation (1-b) to eliminate... ,have to:

[0091] (13)

[0092] Similarly, processing equation (1-c) yields:

[0093] (14)

[0094] Define the Jacobi matrix of the suspension cable (2×2), its elements are:

[0095] ,

[0096] ,

[0097] ,

[0098] .

[0099] Because the relationship between the displacement of the upper endpoint and the change in projection is as follows: , Define the inverse Jacobi matrix of the suspension cable. :

[0100] (15)

[0101] The equation of state for the suspender is the same as that for the main cable. By differentiating the suspender force balance equation and eliminating the stress-free length differential, the corrected suspender Jacobi matrix [J*h,i] (2×2) is obtained. Since the displacement at the upper end point changes in the opposite direction to the projection, the suspender Jacobi matrix [Gh,i] = -[J*h,i]^(-1) is defined, establishing the relationship between the differential force and the differential displacement at the upper end point of the suspender.

[0102] (3) Force transmission and accumulation matrix

[0103] Differentiating the mechanical equilibrium equation (3), and substituting equation (10) into the differential form of equation (3-b), we get:

[0104] (16)

[0105] Combining the differential forms of equations (3-a) and (3-c), the force transfer matrix is ​​defined. (3×3):

[0106] (17)

[0107] The differential form of equation (3) can be simplified to:

[0108] (18)

[0109] in .

[0110] Starting from equation (18), proceed step by step from the first segment to the i-th segment, and define:

[0111] (19)

[0112] The cumulative transfer matrix is ​​defined as follows:

[0113] (20)

[0114] Define the local cumulative matrix (j < i):

[0115] (twenty one)

[0116] By differentiating the force equilibrium equation and substituting the stress-free length differential relationship, a force transfer matrix [Ti] (3×3) is defined to fully describe the transfer relationship of force differentials at adjacent points. Through segment-by-segment recursion, the cumulative transfer matrix [Φi] is defined as the product of the force transfer matrices of each segment, and the local cumulative matrix [Ψij] is the product of the partial segments, thus establishing the cumulative relationship between the force differentials at each point and the initial force differentials and load differentials.

[0117] (4) Global equation

[0118] Differentiate equation (2) and divide the points. The coordinate changes are as follows:

[0119] (twenty two)

[0120] Substituting the Jacobi matrix relation (11) of the main cable into equations (22-a) and (22-b):

[0121] (twenty three)

[0122] The following is based on Taking direction as an example, we derive the expression for the change of coordinates of the component points with respect to the changes of initial force and concentrated load. Substituting the force recursive relation (19) into equation (23), we get:

[0123] (twenty four)

[0124] In the formula, the subscripts X, Y, and Z represent the corresponding components of the vector. Taking X as an example, the subscript X represents the X-axis component of the vector, that is, the element in the first row of the vector; and so on, the subscripts Y and Z represent the elements in the second and third rows, respectively.

[0125] After changing the order of the summation and simplifying, we get:

[0126] (25)

[0127] In the formula, This represents the element in the second row (corresponding to the Y direction) of the main cable Jacobi matrix.

[0128] Define the coefficient matrix with respect to the initial force. (N×3), No. Behavior:

[0129] (26)

[0130] Define the intermediate coefficient matrix for concentrated loads. , (Scalar), the element in the i-th row and j-th column (j < i-1) is:

[0131] (27)

[0132] (28)

[0133] because Containing only the Y and Z components (with no longitudinal inclination of the suspenders in the completed bridge state), the concentrated load term in equation (25) can be expanded as follows:

[0134] (29)

[0135] Substituting into equation (29) and using equations (26)-(28), equation (23-a) can be written as:

[0136] (30)

[0137] Similarly, define (N×3) , ,get Coordinate transformation expression.

[0138] From the inverse Jacobi matrix equation (15), the concentrated load changes as follows:

[0139] (31)

[0140] (32)

[0141] Substituting equations (31) and (32) into equation (30) and rearranging the terms, we get:

[0142] (33)

[0143] Define the coefficient matrix , (N×N lower triangular matrix), the first Line number The column elements (j < i-1) are:

[0144] (34)

[0145] (35)

[0146] Among them, the diagonal elements and the upper triangular elements are 0.

[0147] Similarly, define , .

[0148] By rearranging and simplifying the displacement relationships of all points, we obtain the global equation:

[0149] (36)

[0150] in, For all The vertical coordinate change vector of each division point For all The lateral coordinate change vector of each point.

[0151] Equation (36) can be written in block matrix form:

[0152] (37)

[0153] Define a global matrix ( ) and global coefficient matrix ( ), simplified to:

[0154] (38)

[0155] For the differential of the geometric compatibility equation, the change in the coordinates of the component points is equal to the sum of the differentials of the projected lengths of each cable segment. Substituting the Jacobi matrix of the main cable and the force recursion, the change in the coordinates of the component points can be expressed as a linear combination of the differentials of the initial force and the load. After changing the order of summation, we define coefficient matrices [AY] and [AZ] (N×3) for the initial force to describe the sensitivity of the component point displacement to the initial force; and intermediate coefficient matrices [BY,ij], [CY,ij], [BZ,ij], and [CZ,ij] for the load to describe the contribution of the load to the displacement at the component points. By using the inverse Jacobi matrix of the suspenders, we transform the differential of the load into the differential of the component point displacement, and define the coordination coefficient matrices [KYY], [KYZ], [KZY], and [KZZ] (N×N lower triangular matrices) to describe the influence of the displacement of the previous component point on the displacement of the subsequent component points, reflecting the spatial coordination of the main cable-suspender system. Assemble all the displacement relationships at each point and establish the global equation: [Ω]×[{Y}; {Z}] = [A]×{dF0}, where the global matrix [Ω] (2N×2N) is composed of the identity matrix and the synergistic coefficient matrix, and the global coefficient matrix [A] (2N×3) is composed of [AY] and [AZ]. This establishes a linear relationship between the displacement at each point and the differential of the initial force.

[0156] (5) Stiffness matrix extraction

[0157] Solving equation (38), we get:

[0158] (39)

[0159] Define the compliance matrix (2N×3):

[0160] (40)

[0161] The relationship between the global displacement and the initial force is:

[0162] (41)

[0163] Flexibility matrix The sensitivity of all component displacements to the initial force is described.

[0164] For the shape-finding problem, focus on the displacement of three control points: the top of the right tower. coordinate transformation Changes in the Z-coordinate of the right tower top Crossing the midpoint coordinate transformation ( (Numbering the midpoints). Extract the corresponding rows from equation (42):

[0165] (42)

[0166] Equation (42) can be written in matrix form, and the compliance submatrix can be defined. (3×3):

[0167] (43)

[0168] The relationship between the control point displacement and the initial force is as follows:

[0169] (44)

[0170] Inverting equation (43) yields the linearly varying stiffness matrix. (3×3):

[0171] (45)

[0172] Solving the global equation yields the relationship between the displacements at all points and the differentials of the initial force. The compliance matrix [S] (2N×3) is defined as the product of the inverse of the global matrix and the global coefficient matrix, describing the sensitivity of the displacements at each point to the initial force. Rows corresponding to the key control points are extracted from the compliance matrix, including the Y-coordinate of the midpoint (row M), the Y-coordinate of the right endpoint (row N), and the Z-coordinate (row N+N), forming a compliance submatrix [F] (3×3). Inverting the compliance submatrix yields the linear change stiffness matrix [K] (3×3), establishing a linear relationship between the displacement error of the key control points and the initial force correction.

[0173] The steps for calculating the linear variation stiffness matrix are as follows: Figure 3 As shown.

[0174] The main span of the Xi'an Bahe Bridge is 300 m, with a rise of 74.63 m, a rise-to-span ratio of 1 / 4.02, and a bridge deck width of 56 m. The left and right tower tops... The coordinates of the points are (1442, 467.95, 0.25) m and (1742, 467.95, 0.25) m respectively, and the design elevation of the mid-span is 393.32 m. Main cable = 205 GPa, =0.1626 m², = 12.99 kN / m. Sling = 205GPa, a total of 29 pairs, divided into three models: ϕ7-283 ( =0.010891 m², =0.9118 kN / m) is used for #1 and #29 slings, ϕ7-187 ( =0.007197 m², =0.6028 kN / m) is used for slings #2~#14 and #16~#28, ϕ7-139 ( =0.005349 m², =0.4483 kN / m) is used for sling #15. The main cable is divided into 32 sections with a total of 33 sub-points (numbered 0~32). Sub-points 1 and 31 only have cable clamps and no slings. Sub-points 2~30 correspond to slings #1~#29 respectively. The elevation distribution of the lower suspension points is as follows: the elevation of the lower suspension point of sling #1 is 385.92 m. The elevations of the lower suspension points of slings #2~#9 increase by 0.03 m each (straight section). The elevations of the lower suspension points of slings #10~#15 are 386.189 m, 386.214 m, 386.234 m, 386.247 m, 386.256 m, and 386.259 m respectively (circular section). The elevations of the lower suspension points of slings #16~#29 are symmetrically distributed with those of slings #14~#1. The weight of the cable clamps at each point and the vertical force at the lower suspension point are shown in Table 1.

[0175] The iterative convergence criterion is that the 2-norm of the control point displacement error vector is less than 1×10-9 m (0.001μm).

[0176] The algorithm was implemented in C++ and its results were compared with those from BNLAS software simulation. The computing hardware configuration was an AMD Ryzen 9 8940HX processor (16 cores) and 32GB of memory. Table 2 shows that the spatial coordinates of the main cable are in high agreement with those of the BNLAS software, verifying the correctness of the method. Convergence accuracy was achieved after 23 iterations, taking approximately 1-2 seconds.

[0177] Table 1 Weight of the cable clamp at each point and vertical force at the lower suspension point

[0178]

[0179] Table 2 Comparison of coordinates of key points in the main cable bridge alignment

[0180]

[0181] The embodiments described above merely illustrate specific implementation methods of this application, and while the descriptions are detailed, they should not be construed as limiting the scope of protection of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the technical solution of this application, and these modifications and improvements all fall within the scope of protection of this application.

Claims

1. A linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges, characterized in that, include: The suspension bridge is simplified into a spatial cable system consisting of a main span, side spans and anchor spans, and the bridge form-finding process is performed sequentially on the main span, side spans and anchor spans. The bridge forming process adopts the spatial segmented catenary method. In the bridge forming process, the linearly changing stiffness matrix is ​​directly established for the coordinates of three key control points: the mid-span Y-coordinate, the right end point Y-coordinate, and the Z-coordinate. The initial force correction is solved at once by the coordinate error of the three control points. In the derivation of the stiffness matrix, the stress-free length is used as a variable in the differentiation.

2. The linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges according to claim 1, characterized in that, For the main span, the calculation process of the linear variation stiffness matrix includes: based on the spatial piecewise catenary theory, a complete Jacobi matrix containing stress-free length variables is established through total differential derivation of the basic equations; stress-free length variables are strictly eliminated through partial derivative operations to obtain the main cable Jacobi matrix and the inverse Jacobi matrix of the suspension cable; a force transfer correction matrix and a cumulative transfer matrix are introduced to construct a global equation; the linear stiffness matrix is ​​directly extracted by solving the global equation, establishing a precise linear relationship between the control point displacement change and the initial force change.

3. The linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges according to claim 2, characterized in that, Establish the main cable Jacobi matrix Including: the three-dimensional projection lengths of the cable segment sequentially about Differentiate the force components and the stress-free length, considering the cable segment. Assuming the projected length is a design constant, the Jacobi matrix of the main cable is obtained by eliminating the stress-free length derivative through total differential. .

4. The linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges according to claim 2, characterized in that, Establish the inverse Jacobi matrix of the suspension cable. This includes: obtaining the Jacobi matrix of the suspension cable by differentiating the force balance equation of the cable and eliminating the differential of the stress-free length. Since the displacement of the upper endpoint is opposite to the direction of the projection change, the inverse Jacobi matrix of the suspension cable is defined. .

5. The linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges according to claim 2, characterized in that, The introduction of the force transfer correction matrix and cumulative transfer matrix includes: differentiating the force balance equation, substituting the stress-free length differential, and defining the force transfer matrix. It fully describes the transmission relationship of the force differentials at adjacent component points; through segment-by-segment recursion, it defines the cumulative transmission matrix. The product of the force transmission matrices of each segment, and the local cumulative matrix. It is the product of consecutive segments.

6. The linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges according to claim 2, characterized in that, The construction of the global equation includes: differentiating the geometric compatibility equation of the branch points, where the change in the coordinates of the branch points is equal to the sum of the differentials of the projected lengths of each cable segment; substituting the Jacobi matrix of the main cable and the force recursion relationship, the change in the coordinates of the branch points can be expressed as a linear combination of the differentials of the initial force and the differentials of the load; after changing the order of the linear combination summation, a coefficient matrix with respect to the initial force is defined. , Describes the sensitivity of the component displacements to the initial force; defines the intermediate coefficient matrix with respect to the load. , , , This describes the contribution of the load at each point to the displacement; the load differential is transformed into the displacement differential at each point using the inverse Jacobi matrix of the suspension cable, and a synergy coefficient matrix is ​​defined. , , , This describes the influence of the displacement of the preceding component point on the displacement of the subsequent component point, reflecting the spatial synergy of the main cable-suspender system; it assembles a coefficient matrix describing the relationships between all component points and displacements, and establishes a global equation: The global matrix Composed of blocks of identity matrix and synergy coefficient matrix, global coefficient matrix Depend on and composition, This is the differential of the initial force.

7. The linear variation stiffness matrix method for form finding of spatial cable-stayed suspension bridges according to claim 6, characterized in that, The method of directly extracting the linear stiffness matrix by solving the global equation includes: defining the compliance matrix. Global matrix Inverse and global coefficient matrix The product of the two forces describes the sensitivity of the point displacements to the initial force; from the compliance matrix Extract the rows corresponding to key control points, including those crossing midpoints. Coordinates, right endpoint coordinates and Coordinates, forming a compliance submatrix ; for the flexibility submatrix Inverse the matrix to obtain the linearly changing stiffness matrix. Establish a linear relationship between the displacement error of key control points and the initial force correction.

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