A method for shape and force finding of spatial suspension cables using a piecewise linear recursive iterative method

Abstract

This invention discloses a method for form finding and force finding of spatial suspension cables using a piecewise linear recursive iterative method, relating to the field of suspension structure technology. The piecewise linear recursive iterative method of this invention divides the cable into n segments. Based on force balance and geometric relationships, the recursive relationship between the internal forces and coordinates from the left end to the right end is obtained. The influence coefficients of the components of the tension at the left end in the x, y, and z directions on the coordinate changes at the right end are obtained. After 3-6 iterations, it converges to a stable value, reaching an equilibrium state. This invention achieves form finding and force finding of spatial suspension cables given the original length and the positions of both ends, with triaxial loads applied to the cable, considering the effects of elastic elongation and temperature. It does not require complex nonlinear calculation software; it can be quickly implemented using only Excel. The calculation operation is simple, with small errors, and is applicable and convenient.

Description

Technical Field

[0001] This invention relates to the field of suspension structure technology, and more specifically to a method for finding the shape and force of a spatial suspension cable using a piecewise linear recursive iterative method. Background Technology

[0002] Suspension structures are load-bearing structures formed by flexible tension cables and their edge members. They can achieve large spans, low self-weight, material savings, and ease of construction. China is one of the earliest countries in the world to apply suspension structures. In modern times, in addition to being used in large-span bridge projects, suspension structures are also used in large-span roof structures such as stadiums, aircraft hangars, exhibition halls, and warehouses. In recent years, flexible photovoltaic supports have adopted suspension structures and have been widely promoted.

[0003] The analysis of cable-stayed structures under static loads first requires determining the initial state and internal forces of the entire structure under the initial load, i.e., finding the form and forces. Then, the internal forces and deformations of the entire structure caused by load increments, temperature changes, and cable end displacements are calculated. Cable-stayed structures exhibit geometric nonlinearity, meaning that under load at each stage, both internal forces and deformations show a nonlinear relationship with the load. When using discrete methods to analyze cable-stayed structures, a system of nonlinear algebraic equations is derived, making calculations complex, whether using the nonlinear finite element method or an iterative method considering the nonlinearity generated by the catenary.

[0004] Therefore, how to provide a suspension cable form-finding and force-finding method that does not require complex nonlinear calculation software, is simple to operate, has small errors, and is applicable and convenient is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] In view of this, the present invention provides a method for finding the shape and force of a spatial suspension cable by means of a piecewise linear recursive iterative method, which aims to solve the above-mentioned technical problems.

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

[0007] A method for form and force finding of spatial suspension cables using a piecewise linear recursive iterative method specifically includes the following steps:

[0008] S1. Divide the suspension cable into n equal segments along its length. If the concentrated force is not located at any of the equally divided nodes, add a new node at the concentrated force location, or move the nearest node to the concentrated force location. Distribute the weight of each cable segment evenly across the nodes at both ends of the segment to obtain the external force F at node i in the x, y, and z axes. ix F iy and F iz ;

[0009] S2. Based on the components T1 of the tension force T1 at the left end point A of the suspension cable in the x, y, and z axes...1x T 1y and T 1z The component forces T of each cable segment in the x, y, and z axes are obtained recursively from left to right. ix T iy and T iz And recursively obtain the calculated coordinates (X) of the right end point B of the suspension cable after considering the effects of cable elastic elongation and temperature. BC Y BC Z BC ), and calculate the difference (△X, △Y, △Z) between the current position and the target position;

[0010] S3, the x-axis component of the tension force on the left end point A of the suspension cable, T 1x Make small changes to get T 1x Calculate the coordinates (X) of B BC Y BC Z BC The influence coefficient of the y-axis component of the tension force T at the left end A of the suspension cable. 1y Make small changes to get T 1y Calculate the coordinates (X) of B BC Y BC Z BC The influence coefficient of the tension force T at the left end point A of the suspension cable in the z-axis direction. 1z Make small changes to get T 1z Calculate the coordinates (X) of B BC Y BC Z BC The influence coefficient of )

[0011] S4. Based on the influence coefficient and the difference (△X, △Y, △Z) between the calculated location and the target location, a system of three linear equations is obtained. Solving this system yields the components T1 of the cable tension at the left end A of the suspension cable in the x, y, and z axes. 1x T 1y and T 1z The new round of revisions;

[0012] S5. After 3 to 6 iterations, the suspension cable converges to a stable position and internal force.

[0013] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method for finding the shape and force of a spatial suspension cable using a piecewise linear recursive iterative method. The piecewise linear recursive iterative method of the present invention divides the cable into n segments. Based on the force balance and geometric relationship, the recursive relationship between the internal force and coordinates from the left end to the right end is obtained, and the influence coefficients of the x, y, z components of the tension at the left end on the coordinate change at the right end are obtained. After 3-6 iterations, it can converge to a stable value and reach an equilibrium state. It does not require complex nonlinear calculation software, but can be quickly implemented with just Excel. The calculation operation is simple, the error is small, and it is applicable and convenient. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0015] Figure 1 The attached figure is a schematic diagram of the coordinate system and the endpoint coordinates of the cable provided by the present invention;

[0016] Figure 2 The attached figure is a schematic diagram of the decomposition and direction of cable force provided by the present invention;

[0017] Figure 3 The attached figure is a simplified schematic diagram of the suspension cable curve segment and the straight line provided by the present invention;

[0018] Figure 4 The attached figure is a schematic diagram of the simplified curve coordinates and straight lines (unit: m) provided by the present invention;

[0019] Figure 5 The attached figure is a flowchart of the method for finding the shape and force of a spatial suspension cable using a piecewise linear recursive iterative method provided by the present invention. Detailed Implementation

[0020] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0021] This invention discloses a method for form-finding and force-finding of spatial suspension cables using a piecewise linear recursive iterative method:

[0022] 1. Prerequisites and objectives for shape and force identification:

[0023] Given a suspension cable with an initial length L under stress-free conditions, a cross-sectional area A, an elastic modulus E, and a coefficient of thermal expansion α, and two ends fixed at points A and B respectively, the cable bears external forces in arbitrary directions at multiple points in addition to its own weight distributed uniformly along the cable. Considering the temperature change ΔT, determine the shape and internal forces at the equilibrium position of the cable.

[0024] 2. Coordinate system and piecewise linearity:

[0025] Choose xy as the horizontal plane, and the direction perpendicular to the ground upwards as the z-direction. The x-direction should ideally point from the left endpoint A of the cable to the right endpoint B. In principle, the origin of the coordinate system can be anywhere; for convenience, the origin can be placed at the left endpoint A of the cable. See [reference needed]. Figure 1 Thus, the coordinates A(X) of point A are determined before shape finding. A Y A Z A The shape-finding target location B(X) and point B. BT Y BT Z BT ).

[0026] Divide the suspension cable into n equal straight segments; the shape-finding accuracy is related to n. 2 Inversely proportional, sufficient accuracy is achieved when n=100, with a calculation error of only 0.005% relative to the sag f. For example, in a 500-meter suspension bridge, the sag f is approximately 50 meters, resulting in a calculation difference of about 2.5 mm. The self-weight of the suspension cable is applied to the nodes. When the concentrated force on the suspension cable is not at a segment node, a new node can be added at the concentrated force location, or the nearest equally spaced node can be moved to the concentrated force location. The self-weight of the left and right halves of the cable segment at node i is applied to node i.

[0027] 3. Linear recursion:

[0028] Based on the balance at node i, we obtain three equilibrium equations in component form:

[0029]

[0030] After considering the elongation under tension and temperature Δt, the stress-free length of the i-th cable segment is L. i The coordinate increments in the x, y, and z directions are:

[0031]

[0032] The coordinates of the left endpoint A of the cable are (X... A ,Y A Z A This allows us to obtain the calculated coordinates of each node of the suspension cable and the calculated coordinates B(X) of the right endpoint B. BC ,Y BC Z BC ):

[0033]

[0034] The target coordinates of point B are B(X). BT ,Y BT Z BT The design is defined; the difference between the calculated coordinates and the target coordinates is then determined.

[0035]

[0036] 4. Iterative solution:

[0037] The iterative solution will be performed using the piecewise linear recursive formula mentioned earlier.

[0038] For the initial calculation, first estimate and give the initial value of the tension in the first segment of the cable at the left end, denoted as T. 0x T 0y T 0z Initial value estimation method:

[0039]

[0040] T 0x =max(|T 0y |,|T 0z |) (15)

[0041] Step a: Take T 1x =T 0x T 1y =T 0y T 1z =T 0z The tension of n cable segments is obtained by recursion through equations (1)(2)(3), and the coordinates B(X) of point B are obtained by equations (7)(8)(9). BC ,Y BC Z BC ), denoted as X respectively. B0 =X BC Y B0 =Y BC and Z B0 =Z BC ;

[0042] ΔX, ΔY and ΔZ are calculated using equations (10), (11) and (12). When ΔX, ΔY and ΔZ meet the accuracy requirements, the cable force and the coordinates of each node are the target values ​​for form finding. Otherwise, the following calculations are performed.

[0043] Step b: Give T 1x A tiny increment δT x Take T 1x =T 0x +δT x T 1y =T 0y T 1z =T 0z The tension of n cable segments is obtained by recursion through equations (1)(2)(3), and the coordinates B(X) of point B are obtained by equations (7)(8)(9). BC ,Y BC Z BC ), denoted as X respectively. Bx=X BC Y Bx =Y BC Z Bx =Z BC , get T 1x Influence coefficient on the coordinates of point B:

[0044]

[0045] Step c: Give T 1y A tiny increment δT 1y Take T 1y =T 0y +δT 1y ;T 1x =T 0x T 1z =T 0z The tension of n cable segments is obtained by recursion through equations (1)(2)(3), and the coordinates of point B (X) are obtained by equations (7)(8)(9). BC ,Y BC Z BC ), denoted as X respectively. By =X BC Y By =Y BC Z By =Z BC , get T 1y Influence coefficient on the coordinates of point B:

[0046]

[0047] Step d: Give T 1z A tiny increment δT 1z Take T 1z =T 0z +δT 1z ;T 1x =T 0x T 1y =T 0y The tension of n cable segments is obtained by recursion through equations (1)(2)(3), and the coordinates of point B (X) are obtained by equations (7)(8)(9). BC ,Y BC Z BC ), denoted as X respectively. Bz =X BC Y Bz =Y BC Z Bz =Z BC , get T 1z Influence coefficient on the coordinates of point B:

[0048]

[0049] Step e: T obtained from steps b, c, and d 1x T 1y T 1z The influence coefficients on the coordinates of point B and the differences ΔX, ΔY, and ΔZ between the calculated coordinates of point B and the target coordinates are used to obtain a system of three linear equations:

[0050]

[0051] Solving for T 1x T 1y T 1z Iterative correction amount:

[0052]

[0053] With T 0x +ΔT 1x For the new T 0x T 0y +ΔT 1y For the new T 0y T 0z +ΔT 1z For the new T 0z A new round of iterative calculations will begin from step a.

[0054] When the calculated coordinates of the right endpoint B of the cable in step b differ from the target coordinates to meet the accuracy requirements (e.g., not exceeding 0.01 mm), the iteration is complete, and the cable internal forces and coordinates of each node are obtained. This piecewise linear recursive iteration method transforms the nonlinear problem of the suspension structure height into a linear problem with only 3 parameters per iteration, significantly reducing the computational difficulty. The iteration convergence process can be observed through equation (26). The example shows that by determining the initial values ​​of the iteration according to equations (13), (14), and (15), it generally converges to a form-finding coordinate difference of less than 0.01 mm after 3 to 5 iterations.

[0055] 5. Error Analysis

[0056] Assuming the cable lies in the xz plane and is divided into n equal segments, select any segment and see... Figure 3 For a theoretical curve, the coordinates of the left endpoint are (x, z), and the coordinates of the curve within this segment can be calculated using Taylor expansion.

[0057] When the load is uniformly distributed along the curve, the curve is a catenary, which can be approximated as a quadratic curve. Let the mid-span sag be f, and considering deformation compatibility within a span, the maximum mid-span form-finding error is approximately:

[0058]

[0059] The larger n is, the smaller the error between the calculated result and the curve. When n = 100, the mid-span error is only 0.005% relative to the sag f.

[0060] After the shape finding is completed, the suspension cable remains in equilibrium under its own weight and the initial external load. The suspension cable segment n does not affect the magnitude of the deformation value under the external load, which will be verified in Example 1.

[0061] Calculation example 1:

[0062] The cable length is L = 120m. Ignoring elastic elongation, the left endpoint A is (-50, 0, 0), the right endpoint B is (50, 0, 0), and the cable's weight is 11 N / m. (See...) Figure 4 This example uses a standard catenary to verify the feasibility and accuracy of the method.

[0063] The equation of the catenary is:

[0064]

[0065] The parameter α is a solution to the length constraint equation sinh(50α)=60α, and we obtain α=0.02129737, f=29.23433m=29234.33mm.

[0066] Taking n=100, each segment length 1200mm, and nodal force 13.2N, the initial estimated values ​​of the tension at the left end of the suspension cable are obtained according to formulas (21)(22)(23). The row with 0 iterations is shown in Table 1. Since the initial values ​​in the y and z directions are correctly determined, point B remains at the target position in both the y and z directions. The initial distance from point B in the X direction is 5554mm, which reaches 0.35mm after 3 iterations. After the 4th iteration, the distance from the target position is only 0.000002mm.

[0067] Table 1 Iteration process data

[0068]

[0069] Calculations were performed with n = 40, 60, 80, and 100 respectively. All four cases converged to within 0.01 mm after the 5th iteration. The mid-span deflection calculated by the four segmentation methods was compared with the theoretical mid-span deflection of the catenary curve, 29234.35 mm, by the difference Δz, as shown in Table 2. It can be seen that the form-finding error estimated by formula (27) has sufficient accuracy.

[0070] Table 2 Comparison of calculation errors for different segments

[0071]

[0072] The total weight of the suspension cable is 1320N. A concentrated force of 13.2N is applied at mid-span, which is 1% of the total vertical force on the cable. Calculations and analyses are performed for divisions of n=40, 60, 80, and 100, yielding increases in vertical deformation at mid-span of 149.6mm, 149.7mm, 149.8mm, and 149.8mm respectively, differing by only 0.1%. This verifies that the cable's form-fitting process generates deformation related to 1 / n. 2 The error is proportional, but the deformation caused by the added load after the shape finding is almost unrelated to the number of segments n.

[0073] Calculation example 2:

[0074] A 12000mm long cable is divided into 10 segments. The triaxial forces borne by the 9 intermediate nodes are shown in Table 3, with coordinates A(0, 0, 0) and B(10000, 1000, 1500). EA = 6.28 × 10⁶ N / mm². 2 Calculate the internal forces of each segment and the position of each node, αΔt = 0.0001.

[0075] This example uses n=10, and the calculation results are shown in Table 3. The example only uses n=10 to manually verify the correctness of the calculation results.

[0076] Table 3. Results of cable segment length, nodal force, and form-finding calculations.

[0077] node <![CDATA[L i ]]> <![CDATA[F ix ]]> <![CDATA[F iy ]]> <![CDATA[F iz ]]> <![CDATA[T ix ]]> <![CDATA[T iy ]]> <![CDATA[T iz ]]> <![CDATA[x i ]]> <![CDATA[y i ]]> <![CDATA[z i ]]> A(0) / / / / 459.64 105.83 -383.17 0.00 0.00 0.00 1 1200 10 10 -100 459.64 105.83 -383.17 907.82 209.02 -756.79 2 1200 10 20 -100 449.64 95.83 -283.17 1907.31 422.04 -1386.23 3 1200 10 10 -100 439.64 75.83 -183.17 3001.43 610.75 -1842.08 4 1200 10 20 -100 429.64 65.83 -83.17 4166.65 789.29 -2067.64 5 1200 10 10 -100 419.64 45.83 16.83 5358.80 919.49 -2019.82 6 1200 10 20 -100 409.64 35.83 116.83 6508.92 1020.09 -1691.80 7 1200 10 10 -100 399.64 15.83 216.83 7563.22 1061.85 -1119.78 8 1200 10 20 -100 389.64 5.83 316.83 8494.36 1075.78 -362.62 9 1200 10 10 -100 379.64 -14.17 416.83 9302.29 1045.62 524.45 B(10) 1200 / / 369.64 -24.17 516.83 10000.00 1000.00 1500.00

[0078] This invention enables spatial suspension cable shape finding and force finding under given original length and end positions, triaxial load applied to the cable, and considering the effects of elastic elongation and temperature. This invention also provides a method for shape finding using Excel software, which is simple to operate, has a fast convergence speed, and small error.

[0079] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0080] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for form finding and force finding of spatial cable structures by piecewise linear recursive iteration, characterized in that, Specifically comprising the following steps: S1, divide the suspension cable into n sections along the length, and average the self-weight in each cable section to the two end nodes of the cable section to obtain the external force F of node i in the x, y, and z axis directions ix , F iy , and F iz ; S2, the first segment cable tension T1 at the left end point A of the suspension cable in the x, y, z axis direction 1x , T 1y and T 1z , recursively from left to right to obtain the force T ix , T iy and T iz of each cable segment in the x, y, z axis direction, and recursively to obtain the calculated coordinates (X BC , Y BC , Z BC ) of the right end point B of the suspension cable considering the cable elastic elongation and temperature effect, and the difference (△X, △Y, △Z) between the calculated position and the target position; According to the balance of node i, three balance equations in component form are obtained: After considering the elongation under tension and temperature Δt, the unstressed length of the i-th segment is L i The coordinate increments in the x, y, z directions are: Wherein: a is the temperature expansion coefficient The left end point A of the cable has coordinates (X A ,Y A ,Z A ), and the calculated coordinates of each node of the cable and the right end point B of the cable are B(X BC ,Y BC ,Z BC ): The target coordinates B(X BT ,Y BT ,Z BT ) of point B are determined by design, and the difference between the calculated coordinates and the target coordinates is calculated. S3, the x-axis direction component of the tension of the left end point A of the suspension cable 1x Make a small change to get T 1x The influence coefficient of B on the coordinates (X BC , Y BC , Z BC ); the y-axis direction component of the tension of the left end point A of the suspension cable 1y Make a small change to get T 1y The influence coefficient of B on the coordinates (X BC , Y BC , Z BC ); the z-axis direction component of the tension of the left end point A of the suspension cable 1z Make a small change to get T 1z The influence coefficient of B on the coordinates (X BC , Y BC , Z BC ); First, the initial value of the tension of the first segment of the left end is estimated and given, which is marked as T 0x , T 0y , T 0z The initial value estimation method: T 0x = max(|T 0y |, |T 0z |) (15) Step a: take T 1x = T 0x , T 1y = T 0y , T 1z = T 0z , the tension of n segment cable is obtained by recursion of (1) (2) (3) formula, and the calculation coordinates B (X BC , Y BC , Z BC ) of B point are obtained by (7) (8) (9) formula, which are respectively recorded as X B0 = X BC , Y B0 = Y BC and Z B0 = Z BC ; Calculate ΔX, ΔY and ΔZ through (10), (11) and (12), when ΔX, ΔY and ΔZ meet the accuracy requirement, the cable force and the coordinates of each node are the target values of the shape finding, otherwise the following calculation is performed. Step b: give T 1x a small increment δT x , take T 1x = T 0x + δT x , T 1y = T 0y , T 1z = T 0z , the tension of n-section cable is obtained by recursion of (1) (2) (3), and the calculation coordinates B (X BC , Y BC , Z BC ) of B point are obtained by (7) (8) (9), which are respectively recorded as X Bx = X BC , Y Bx = Y BC , Z Bx = Z BC , and the influence coefficient of T 1x on the coordinates of B point is obtained. Step c: give T 1y a small increment δT 1y , take T 1y = T 0y + δT 1y ; T 1x = T 0x , T 1z = T 0z ; through (1) (2) (3) formula recursion to get n section cable tension, through (7) (8) (9) formula get B point calculation coordinates B (X BC , Y BC , Z BC ), respectively, X By = X BC , Y By = Y BC , Z By = Z BC , get T 1y on the influence coefficient of B point coordinates: Step d: give T 1z a small increment δT 1z , take T 1z = T 0z + δT 1z ; T 1x = T 0x , T 1y = T 0y ; the tension of n-section cable is obtained by recursion of (1) (2) (3) formula, and the calculation coordinates B (X BC , Y BC , Z BC ) of B point are obtained by (7) (8) (9) formula, which are respectively recorded as X Bz = X BC , Y Bz = Y BC , Z Bz = Z BC , and the influence coefficient of T 1z on the coordinates of B point is obtained. S4, according to the influence coefficient and the difference (△X, △Y, △Z) between the calculated position and the target position, a ternary linear equation group is obtained, and the component forces T of the cable tension T1 at the left end point A in the x, y and z axis directions are solved 1x , T 1y and T 1z new round of correction values; T obtained according to steps b, c, d 1x , T 1y , T 1z The influence coefficient of the coordinates of point B and the difference ΔX, ΔY and ΔZ between the calculated coordinates of point B and the target coordinates are obtained, and a three-element linear equation group is obtained. Solving for T 1x , the iterative correction for T 1y , the iterative correction for T 1z , the iterative correction for T T 0x + ΔT 1x = new T 0x , T 0y + ΔT 1y = new T 0y , T 0z + ΔT 1z = new T 0z , and a new iteration is started from step a. S5, after 3-6 iterations, the suspension cable converges to a stable position and internal force.

2. The method for form-finding and force-finding of spatial catenary according to claim 1, wherein, In step S1, when the concentrated force position is not at the equal division node, a new node is added at the concentrated force position, or the nearest node is moved to the concentrated force position.

3. The method for form-finding and force-finding of spatial catenary according to claim 1, wherein, When the calculated coordinates of the right end point B of the cable in step b and the target coordinates differ by an accuracy requirement, such as not more than 0.01 mm, the iteration is completed, and the internal force of the cable and the coordinates of each node are obtained.

4. The method for form-finding and force-finding of spatial catenary according to any one of claims 1-3, characterized in that, Excel software is used for shape finding.

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