A method for guaranteeing the shape of the horizontal projection of a cable net structure

Abstract

The application provides a form-finding method for ensuring the horizontal plane projection shape of a cable net structure. Firstly, the horizontal plane projection coordinates of free nodes and three-dimensional space coordinates of constrained nodes of the cable net structure are determined according to the building contour, the self-stress mode of the horizontal resultant force of the cable element is obtained by using the singular value decomposition method, and the self-stress mode combination is performed to determine the horizontal resultant force of the cable element. Then, the initial vertical coordinates of the arbitrary free nodes are assumed, and the vertical coordinates of the nodes in the balanced state are determined by using the Newton method. The application can determine the equilibrium state of the cable net structure under the premise of ensuring the horizontal plane projection shape of the cable net without repeated trial calculation and adjustment, and can select reasonable cable force size according to the needs, so that the different needs of the building and structure designers about the geometric shape and mechanics can be met. The application is based on the catenary cable element, has better accuracy and adaptability than the force density method using the bar element in theory, and has clear implementation process, high algorithm efficiency and good convergence.

Description

Technical Field

[0001] This invention relates to the fields of architectural and structural design, and particularly to a shape-finding method for ensuring the horizontal projection shape of a cable net structure. Background Technology

[0002] Cable net structures are lightweight, have high load-bearing capacity, large spanning capacity, and beautiful appearance, making them an important choice in architectural design and widely used in large public buildings such as stadiums.

[0003] Cable net structures, as a type of "flexible structure," differ from conventional "rigid structures." They require prestress to achieve structural stiffness and thus a fixed structural form. Therefore, the shape of a cable net structure has a specific relationship with the applied prestress; that is, there is a correspondence between "force" and "shape." Different prestress distributions will produce different cable net shapes. "Form finding" refers to finding the equilibrium shape of the cable net structure corresponding to a specific prestress distribution using a certain method. Previous form finding methods include the force density method, dynamic relaxation method, and finite element method. Each method has its advantages and disadvantages. For example, the force density method is based on rod element form finding, assuming the actual cable is a rod element subjected only to axial tension. This method has good accuracy when the cable is subjected to high stress, but it is no longer applicable to relaxed cable elements with lower stress. Both the dynamic relaxation method and the finite element method require determining the initial length or internal forces of the cable elements before analysis begins, and then iteratively solving for equilibrium states through dynamic or static methods. While these two methods can use catenary elements to analyze cable net structures, they have a drawback: they cannot guarantee that the equilibrium geometry of the cable net structure meets certain specific geometric requirements. For example, in architecture, it is sometimes desired that the cable net achieves "horizontal and vertical straightness" or a specific geometric shape in its planar projection. Currently, none of the three methods mentioned above can immediately satisfy the planar projection shape of the cable net; structural designers need to continuously adjust and experiment based on experience to achieve the desired effect. Summary of the Invention

[0004] The purpose of this invention is to provide a method for finding the shape of a cable net structure, which solves the problem in the prior art that the horizontal projection of the cable net structure cannot be guaranteed to meet a specific geometric shape.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A method for determining the shape of a cable net structure in the horizontal plane projection is characterized by first determining the horizontal resultant force of the cable elements of the cable net structure using singular value decomposition, and then determining the vertical coordinates of the free nodes using Newton's method.

[0007] Furthermore, the shape-finding method of the present invention includes the following steps:

[0008] Step S1: Establish a spatial rectangular coordinate system, with the horizontal plane as the Oxy plane and the vertical upward direction as the positive z-axis. The origin O should be a suitable point as needed, preferably selected within the structural boundary or area. Based on the horizontal projection shape of the cable net required for the building outline layout, determine the spatial coordinates of the boundary constraint nodes, the horizontal projection coordinates of the free nodes, and the node loads.

[0009] Step S2: Assume the node coordinates determined in step S1 are (x... i ,y i ,z i (i = 1, 2, ..., J), where J is the total number of nodes, c xy c represents the number of constrained degrees of freedom in the Oxy plane. z Let be the number of constrained degrees of freedom in the z-direction, and let the cable elements be numbered 1, 2, ..., b, where b is the number of cable elements. The nodal load is (p... ix ,p iy ,p iz (i = 1, 2, ..., J).

[0010] This step determines the resultant horizontal force of each cable element by solving the linear equation system (1).

[0011]

[0012] In the formula, Let k be the horizontal projection length of the cable element k. Equation (1) is simplified to Equation (2).

[0013] AF H =p xy (2)

[0014] In the formula, A is (2J-c xy The balancing matrix F is 1 / (1)×b. H Let p be the horizontal resultant force vector of a b×1 cable element. xy For (2J-c xy The x and y direction nodal load vectors are 1×1. To ensure the existence of a solution to equation (1), the rank R(A) of the coefficient matrix of the equation system must be equal to the rank R(A,p) of the augmented matrix during cable net design. xy ).

[0015] To solve equation (1) using singular value decomposition, matrix A must first be decomposed into the following form using singular values:

[0016] A = USV T (3)

[0017] In the formula, U is (2J-c xy )×(2J-c xyAn orthogonal matrix of size (2J-c) is called a left singular matrix. An orthogonal matrix of size (b×b) is called a right singular matrix. xy The augmented diagonal matrix S is denoted as )×b, and the number of non-zero diagonal elements in S is the rank r of the balanced matrix A. The block representation of equation (4) is as follows:

[0018]

[0019] In the formula, U r ={u1,u2,...,u r}, V r ={v1,v2,...,v r}, V s ={v r+1 ,v r+2 ,...,v b}, where V s This refers to the set of self-stress mode vectors for the cable-net structure. r =diag{s1,s2,...,s r}, s r Let F be a singular value of matrix A. The actual horizontal resultant force vector F of the cable elements in the cable net structure. H Determine according to formula (5):

[0020]

[0021] In the formula, α i Here, α is the combination coefficient. i (i=1,2,...,r) Solve the system of linear equations (6) to obtain

[0022]

[0023] In the formula, the coefficient α i (i=r+1,r+2,...,b) can be freely selected according to the structural stress requirements.

[0024] Step S3: Based on the resultant horizontal force of each cable element determined in Step S2, determine the z-coordinate of each node using Newton's method. The specific process is as follows:

[0025] Assume any initial free node z coordinate vector z 0 (It is advisable to take the value between the maximum and minimum z-coordinates of the constraint node).

[0026]

[0027] The z-coordinates of the free nodes are determined by iteratively solving the linear equation system (8).

[0028]

[0029] Equation (8) can be simplified to Equation (9).

[0030]

[0031] In the formula, k i Δd i , The stiffness matrix for the i-th iteration is [(Jc z )×(Jc z )], free node displacement vector [(Jc z )×1] and z-axis unbalanced force vector [(Jc z [1], In the i-th iteration, the stiffness matrix and unbalanced force vector are calculated based on the nodal coordinates at the end of the previous iteration. The final free nodal coordinates z are

[0032]

[0033] In the formula, n is the number of iterations. The criterion for stopping the iteration is generally that the magnitude of the unbalanced force vector at the last iteration is less than 1 / 10000 of that at the first iteration, or the magnitude of the free node displacement vector at the last iteration is less than 1 / 10000 of that at the first iteration.

[0034] unbalanced force f in the z-direction of free node i iz (i = 1, 2, ..., Jc) z Calculate according to formula (11):

[0035]

[0036] In the formula, F iz,k Let F be the nodal force at node i of cable element k. If cable element k does not have node i, then F is... iz,k =0. Establish a local coordinate system for the cable element, with one node of the cable element as the origin, the vertically upward direction as the positive z-axis, and the direction perpendicular to the z-axis and pointing to the other node as the positive x-axis. The formula for calculating the forces at both ends of the cable element is shown in equation (12-14).

[0037] F iz,k =F Hk sinh(α k (12)

[0038]

[0039]

[0040] In the formula, q k For the weight per unit length of the cable unit, l zkLet z be the z-coordinate of node j in the local coordinate system. Other symbols have been described previously.

[0041] Stiffness matrix element K ij The calculation formula is as follows: (15)

[0042]

[0043] In the formula, K ij,k (i,j=1,2,…,Jc z (k = 1, 2, ..., b) represents the stiffness of node i caused by the movement of node j by a unit length in cable element k. If cable element k does not have nodes i or j, then it is K. ij,k =0. In the local coordinate system, K... ij,k The calculation formula is

[0044] K ii,k =-K ij,k =F Hk cosh(α k )α c (16)

[0045]

[0046]

[0047] Furthermore, the horizontal resultant force of the cable element is obtained by solving the linear equation system of equation (1).

[0048] Furthermore, the stiffness matrix element values ​​are obtained through equations (15), (16), (17), and (18).

[0049] The form-finding method proposed in this invention effectively solves the technical problems in the background art. It can determine the final equilibrium shape of the cable net structure based on the required planar projection shape through a single form-finding process. During the form-finding process, designers can also select a reasonable prestress distribution based on the self-stress mode of the horizontal resultant force of the cable elements, ensuring the rationality of the structural stress. This invention's method is based on catenary element form-finding, avoiding the approximation of rod element form-finding, and has universality for general cable net structures. This invention combines the classical singular value decomposition method and Newton's method, featuring reliable method and good convergence. In summary, this invention has the following beneficial effects:

[0050] 1) Combining singular value decomposition and Newton's method for shape finding of cable net structures is a mature, reliable, and convergent method.

[0051] 2) It can ensure the horizontal projection shape of the cable net structure, and at the same time, it can select a reasonable combination of self-stress modes to ensure that the structure is in a reasonable stress state;

[0052] 3) The analysis is performed using catenary elements, which has high accuracy and good versatility, and is applicable to general cable net structures. Attached Figure Description

[0053] Figure 1 This is a schematic diagram of the local coordinate system and nodal forces of a cable element in the prior art, where 100 represents a cable element.

[0054] Figure 2 This is a schematic diagram of a two-way orthogonal cable net structure.

[0055] Figure 3 Schematic diagram of the shape after the bidirectional orthogonal cable net is used for shape finding. Detailed Implementation

[0056] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the technical solutions 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, 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.

[0057] Step S1: As Figure 2 As shown, a two-way orthogonal cable net is established within a geometric area of ​​3l×3l (l=30480mm) of the building area. This cable net consists of two horizontal cables and two vertical cables. Each cable is divided into three segments, each segment being a cable unit. The horizontal projection length of each segment is l. The establishment is as follows... Figure 2 The rectangular coordinate system shown has its origin at the center of the cable net. The positive x-axis points horizontally to the right, the positive y-axis points horizontally upward, and the positive z-axis points vertically upward. Nodes 1, 2, 3, and 4 are free nodes, while nodes 5-12 are boundary-constrained nodes. The degrees of freedom of the constrained nodes are constrained in all three directions. The z-coordinates of constrained nodes 7, 8, 9, and 10 are all 0, while the z-coordinates of constrained nodes 5, 6, 11, and 12 are all C = -9144 mm. The cable elements are numbered [1]-

[12] , see details. Figure 2 The load on the free node is 0.

[0058] Step S2: Determine the balance matrix A based on step S1.

[0059]

[0060] The singular value decomposition of the balance matrix yields...

[0061] U m =0 (3)

[0062]

[0063]

[0064] Actual horizontal resultant force vector F of cable element H Determine by the following formula:

[0065]

[0066] In the formula, α i (i = 1, 2, ..., 12) are the combination coefficients.

[0067] Wherein, coefficient α i (i=1,2,...,8) is obtained by solving the system of linear equations (5).

[0068]

[0069] This example allows us to obtain α. i =0 (i=1,2,...,8).

[0070] V s The column is simplified to

[0071]

[0072] Other coefficients α i (i = 9, 10, ..., 12) can take the value α i =100kN (i=9,10,...,12), then the resultant horizontal force vector of the cable element can be obtained.

[0073]

[0074] Step S3: Based on the horizontal resultant force F of each cable element determined in step S2 H The z-coordinate of each node is determined using Newton's method. The specific process is as follows:

[0075] Assume the initial free node z coordinate vector z 0 for

[0076] z 0 =[4572 4572 4572 4572] T (8)

[0077] The z-coordinates of the free nodes are determined by iteratively solving the linear equation system (9).

[0078]

[0079] Equation (9) can be simplified to Equation (10).

[0080]

[0081] In the formula, k i Δd i , The stiffness matrix [4×4], free nodal displacement vector [4×1], and z-axis unbalanced force vector [4×1] for the i-th iteration are given sequentially. In the i-th iteration, the stiffness matrix and unbalanced force vector are calculated based on the nodal coordinates at the end of the previous iteration. The final free nodal coordinates z are...

[0082]

[0083] In the formula, n is the number of iterations. The criterion for stopping the iteration is generally that the magnitude of the unbalanced force vector at the last iteration is less than 1 / 10000 of that at the first iteration, or the magnitude of the free node displacement vector at the last iteration is less than 1 / 10000 of that at the first iteration.

[0084] unbalanced force f in the z-direction of free node i iz (i=1,2,...,4) Calculate according to formula (12):

[0085]

[0086] In the formula, F iz,k Let F be the nodal force at node i of cable element k. If cable element k does not have node i, then F is... iz,k =0. Establish a local coordinate system for the cable element, with one node as the origin, the vertically upward direction as the positive z-axis, and the perpendicular direction to the z-axis pointing towards the other node as the positive x-axis. The formulas for calculating the forces at both ends of the cable element are shown in equation (13-15).

[0087] F iz,k =F Hk sinh(α k (13)

[0088]

[0089]

[0090] In the formula, q k For the weight per unit length of a cable element, this example q k =1.460 N / m, l zk Let z be the z-coordinate of node j in the local coordinate system. Other symbols have been described previously.

[0091] Stiffness matrix element K ij The calculation formula is as follows: (16)

[0092]

[0093] In the formula, K ij,k (i,j=1,2,…,Jc z (k = 1, 2, ..., b) represents the stiffness of node i caused by the movement of node j by a unit length in cable element k. If cable element k does not have nodes i or j, then it is K. ij,k =0. In the local coordinate system, K... ij,k The calculation formula is:

[0094] K ii,k =-K ij,k =F Hk cosh(α k )α c (17)

[0095]

[0096]

[0097] The z-coordinates of free nodes 1, 2, 3, and 4 can be obtained by three iterations, all of which are -4585.639 mm.

[0098] The schematic diagram of the cable net after shape finding in this example is shown below. Figure 3 .

[0099] This application can determine the equilibrium state of the cable net without repeated trial calculations and adjustments, while ensuring the horizontal projection shape of the cable net. Simultaneously, it allows for the selection of appropriate cable force magnitudes as needed, thus satisfying the diverse geometric and mechanical requirements of architects and structural designers. This invention is based on catenary cable elements, which theoretically offer better accuracy and adaptability than the force density method using rod elements.

[0100] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for finding the shape of a cable net structure projected onto a horizontal plane, characterized in that, First, the horizontal resultant force of the cable elements in the cable net structure is determined by the singular value decomposition method, and then the vertical coordinates of the free nodes are determined by Newton's method. The shape-finding method includes the following steps: Step S1: Establish a spatial rectangular coordinate system, with the horizontal plane as the Oxy plane and the vertical upward direction as the positive z-axis; determine the spatial coordinates of the boundary constraint nodes, the projection coordinates of the free nodes on the horizontal plane, and the node loads according to the horizontal projection shape of the cable net required for the building outline layout. Step S2: Assume the node coordinates determined in step S1 are as follows: Let i = 1, 2, …, J, where J is the total number of nodes, and c xy c represents the number of constrained degrees of freedom in the Oxy plane. z Let be the number of constrained degrees of freedom in the z-direction, and let the cable elements be numbered 1, 2, …, b, where b is the number of cable elements; the nodal loads are... The resultant horizontal force of each cable element is determined by solving the linear equation system (1); （1） In the formula, Let k be the horizontal projection length of the cable element k; Equation (1) is simplified to Equation (2). （2） In the formula, A is The balance matrix, for The horizontal resultant force vector of the cable element, for The nodal load vectors in the x and y directions; to ensure the existence of the solution to equation (1), the rank of the coefficient matrix of the equation system must be guaranteed during the cable net design. equal to the rank of the augmented matrix To solve equation (1) using singular value decomposition, matrix A must first be decomposed into the following form using singular values: （3） In the formula, for An orthogonal matrix is ​​called a left singular matrix. for An orthogonal matrix of is called a right singular matrix. for The augmented diagonal matrix, The number of non-zero diagonal elements is the rank r of the balance matrix A; the block representation of equation (4) is as follows: （4） In the formula, , , , ,in, This is the set of self-stress mode vectors for the cable-net structure. , The singular values ​​of matrix A; the horizontal resultant force vector of the actual cable elements in the cable net structure. Determine according to formula (5): （5） In the formula, Let be the combination coefficients, where are the coefficients. Solving the system of linear equations (6) yields the following results. （6） coefficient Then it can be freely selected according to the structural stress requirements; Step S3: Based on the horizontal resultant force of each cable element determined in Step S2, determine the z-coordinate of each node using Newton's method. The process includes: Assume any initial free node z-coordinate vector , （7） The z-coordinates of the free nodes are determined by iteratively solving the linear equation system (8). （8） Equation (8) is simplified to Equation (9). （9） In the formula, , , The stiffness matrices for the i-th iteration are as follows: Free node displacement vector and z-axis unbalanced force vector In the i-th iteration, the stiffness matrix and unbalanced force vector are calculated based on the nodal coordinates at the end of the previous iteration; the final free nodal coordinates... for （10） In the formula, n is the number of iterations; unbalanced force in the z direction at free node i Calculate according to formula (11): （11） In the formula, Let be the nodal force at node i of cable element k. If cable element k does not have node i, then is Establish a local coordinate system for the cable element, with one node of the cable element as the origin, the vertical upward direction as the positive z-axis, and the direction perpendicular to the z-axis and pointing to another node as the positive x-axis; then the formulas for calculating the forces at both ends of the cable element are shown in equations (12), (13) and (14). （12） （13） （14） In the formula, The weight per unit length of the cable unit. Let j be the z-coordinate of node j in the local coordinate system; Stiffness matrix elements The calculation formula is as follows: (15) （15） In the formula, (i,j=1,2,…,Jc z (k=1,2,…,b) represents the stiffness of node i caused by the movement of node j by a unit length in cable element k. If cable element k does not have nodes i or j, then it is... ; Local coordinate system The calculation formula is （16） （17） （18）。 2. The shape-finding method according to claim 1, characterized in that, The horizontal resultant force of the cable element is obtained by solving the linear equation system of equation (1).

3. The shape-finding method according to claim 1, characterized in that, The stiffness matrix element values ​​are obtained by equations (15), (16), (17), and (18).

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