A ring-shaped tensegrity structure and optimization method

Abstract

The application provides a ring-shaped tensegrity structure, which has n tensegrity units in series connection, wherein n is an even number, and each tensegrity unit comprises three compression struts, and any two end points of the three compression struts are connected with a cable; the tensegrity unit has two bottom surfaces, i.e., an upper bottom surface and a lower bottom surface, which are formed by the same side end points of the three compression struts and are perpendicular to the central axis of the ring-shaped tensegrity structure; the structure of the tensegrity unit is determined by determining the coordinates of the six end points of the three compression struts, and the tensegrity unit satisfies the following structure requirements: the orthographic projection of the upper bottom surface and the lower bottom surface are equilateral triangles, the upper bottom surface and the lower bottom surface are not parallel and do not intersect and have an included angle β; the upper bottom surfaces or the lower bottom surfaces of adjacent tensegrity units coincide with each other, and the adjacent tensegrity units are arranged in mirror image symmetry with respect to the coincident surface. The structure of the application is reasonable in stress and has better stability.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of prestressed cable-strut structure, and more particularly to a ring-shaped tension integral structure and an optimization method. BACKGROUND

[0002] The current cable dome structure needs to be supported on the rigid ring beam of the outer ring and cannot maintain self-balance. The ring-shaped tension integral structure can transmit ring pressure and thus can replace the rigid ring beam of the outer ring of the cable dome. The new cable-strut dome structure formed by the ring-shaped tension integral structure is self-balanced and has a full tension, is aesthetically pleasing, light in weight, efficient, and has good theoretical significance and engineering application value.

[0003] The related patent CN 109255142 A of the ring-shaped tension integral structure: a ring-shaped tension integral structure topology optimization method based on a small habitat genetic algorithm, gives a ring-shaped tension integral structure topology optimization method, which needs to find a topology connection method to minimize the structure quality under the condition of given initial node coordinates, but cannot consider the size and length of the generated structure components. The patent CN 106522368B: a circular ring-shaped tension integral structure, discloses a ring-shaped tension integral structure, but does not give a further optimization scheme, and meanwhile, part of the ring-shaped tension integral structure results will also have the problem of overall torsion due to the same rotation direction of the elements. The existing ring-shaped tension integral structure for determining the topology relationship has no corresponding optimization scheme, the lengths of the components are often quite different, the prestress of the structure after forming is uneven and often cannot be controlled, which will cause trouble and unnecessary loss to the material and construction. SUMMARY

[0004] The present application discloses a ring-shaped tension integral structure and an optimization method, which solves the problems of poor stability of the ring-shaped tension integral structure in the prior art, blank optimization scheme, and uneven and difficult-to-control component length.

[0005] To achieve the above-mentioned purpose, the present application provides a ring-shaped tension integral structure, which has n tension integral units, and the n tension integral units are connected in series in a ring direction, wherein n is an even number, and each tension integral unit includes three compression struts and twelve cables, wherein any two end points of the three compression struts are connected to a cable, the tension integral unit has two bottom surfaces, i.e., an upper bottom surface and a lower bottom surface, the bottom surfaces are composed of the same side end points of the three compression struts, the ring-shaped tension integral structure is connected by the compression struts between adjacent tension integral units, and each bottom surface is perpendicular to the central axis of the ring-shaped tension integral structure.

[0006] The structure of the tension integral unit is determined by determining the coordinates of the six end points of the three compression struts, and the tension integral unit meets the following structure requirements:

[0007] The orthographic projection of the upper base and the lower base are both equilateral triangles. The upper base and the lower base are not parallel to each other, do not intersect, and have an included angle β.

[0008] The upper or lower bottom surfaces of adjacent tensioning integral units overlap, and are arranged in a mirror-symmetric manner with respect to the overlapping surfaces;

[0009] The shape of the annular tensioned integral structure is uniquely determined by five parameters that make up the tensioned integral unit: the height H of the tensioned integral unit, the radius r of the circumscribed circle of the bottom surface, the angle α0 of the bottom surface rotating horizontally counterclockwise around the center O1 as the fulcrum, the relative rotation angle α between the orthographic projection of the top surface and the bottom surface, and the included angle β.

[0010] The inner diameter span d of the annular tensioned monolithic structure, wherein the tensioned monolithic unit satisfies the following constraints:

[0011] Simultaneously satisfying the requirement that the three compression bars do not intersect;

[0012] The included angle β is a factor of 360° and satisfies n = 360° / β;

[0013] The included angle β and the radius r of the circumscribed circle of the bottom surface satisfy the inner diameter span d = H / tanβ-2r;

[0014] All the tensioning units can meet the conditions of compression of the compression bar and tension of the cable.

[0015] This invention also discloses an optimization method for a ring-shaped tensioned integral structure, the optimization method comprising the following steps:

[0016] S1, set the lower base as an equilateral triangle and located in the plane formed by the X-axis and Y-axis, with center O1 as the origin of the coordinate axis, Y-axis pointing to the center of the ring tensioning whole, X-axis perpendicular to the plane where the central axis of the ring is located, and Z-axis pointing to the center O2 of the upper base;

[0017] The six endpoints of the three compression rods are numbered as follows: the three endpoints of the lower bottom surface are numbered A, B, and C counterclockwise, and the three endpoints of the upper bottom surface are numbered D, E, and F counterclockwise.

[0018] Within the range of variation of parameters H, r, α0, α, β, R H R r , R α R β Generate initial parameter values ​​internally, R H R r , R α R β Set according to actual needs;

[0019] S2, according to the initial parameter value, coordinates of 6 end points are calculated in the following way:

[0020] A=[r*cos a0 r*sin a0 0];

[0021]

[0022]

[0023] D=[r*cos(a+a0) r*sin(a+a0) H-tan b*(y D +r)];

[0024]

[0025]

[0026] S3, according to the coordinates, a balanced matrix of the tension integral unit is calculated, and a self-stress mode is obtained by singular value decomposition of the balanced matrix;

[0027] According to the coordinates of 6 end points and the self-stress mode, whether the tension integral unit meets the structural requirements and constraint conditions is judged according to the expression of the constraint condition in the optimization model, and a target function value is calculated;

[0028] The target function is:

[0029]

[0030] The optimization model is:

[0031]

[0032] In the formula, i is a compression rod or a cable, c1 and c2 are combination coefficients of the target function, S cable is a cable component set, S strut is a rod component set, t i is a component prestress, is an average internal force of the cable component, is an average internal force of the rod component, LN cable is the length type number of the cable component, LN strut is the length type number of the rod component, P is a penalty function item, and its value is much larger than A is a balanced matrix of the structure, t is a component internal force vector, D ij is the distance between two rods, d is the required span of the ring tension integral structure, and n is the required number of tension integral units of the ring tension integral structure;

[0033] S4, using Matlab software to import the optimization model in S3 and solve the above 5 parameters H, r, a0, a, b, and through the optimization search algorithm to adjust the values of the 5 parameters, repeating steps S2 and S3, the target function value of the structure is constantly reduced until the optimal solution of the tensegrity unit meeting the structural requirements and constraint conditions is searched out.

[0034] As a further improvement of the application, if the tensegrity unit meets the structural requirements and constraint conditions, the target function value thereof is calculated;

[0035] If not, the target function value is calculated again after adding the corresponding penalty function term.

[0036] As a further improvement of the application, c1 and c2 are set according to the optimization target:

[0037] If the optimization target is to make the internal force of the component as uniform as possible, c1=1 and c2=0.

[0038] If the optimization target is to make the length of the component as consistent as possible, c1=0 and c2=1.

[0039] As a further improvement of the application, the optimization search algorithm includes a genetic algorithm.

[0040] Compared with the prior art, the application has the following beneficial effects:

[0041] (1) The ring-shaped tensegrity structure is composed of multiple tensegrity units, is light and beautiful, and has reasonable structural stress. The existing cable dome structure needs to be supported on the rigid ring beam of the outer ring and cannot maintain self-balance, while the ring-shaped tensegrity structure can balance the ring pressure, serves as the outer pressure ring beam of the cable dome, and together forms a self-balanced large-span roof structure, which has good application prospects.

[0042] (2) The ring-shaped tensegrity structure is arranged in a mirror image symmetrical manner through the upper bottom surface or the lower bottom surface of adjacent tensegrity units, so that the members in the ring-shaped tensegrity structure are continuous and the pressure can be easily transmitted, and the adjacent tensegrity units are in a mirror image relationship and have opposite spiral directions, so that the torsion of the ring-shaped tensegrity structure can be avoided and the structural stability is enhanced.

[0043] (3) The ring-shaped tensegrity structure can uniquely determine the specific shape of the ring-shaped tensegrity structure and the relative positions of the compression bars and the cables in the structure by setting 5 parameters of the tensegrity unit, and a new form of the ring-shaped tensegrity structure is provided; meanwhile, the number of parameters is small and the parameters are simple to define, so that the problems of difficult description of the connection relationship between the cables and the bars and indirect calculation of the node coordinates of the existing ring-shaped tensegrity structure are solved.

[0044] (4) The annular tensegrity structure is based on the unique correspondence between parameters and structures, and further provides an optimization scheme of the annular tensegrity structure, and establishes a complete optimization model, so that the annular tensegrity structure with the uniform internal force of components or the consistent length of components can be found according to the actual engineering needs, such as span, unit number, etc., and the annular tensegrity structure is convenient for practical engineering application. BRIEF DESCRIPTION OF DRAWINGS

[0045] Figure 1 It is a top view of the annular tensegrity structure of the application;

[0046] Figure 2 It is an axonometric view of the tensegrity unit in the application;

[0047] Figure 3 It is a projection view of the tensegrity unit in the application in the xy plane;

[0048] Figure 4 It is a projection view of the tensegrity unit in the application in the yz plane;

[0049] Figure 5 It is an axonometric view of the adjacent annular tensegrity structure obtained by mirroring the tensegrity unit above the bottom surface;

[0050] Figure 6 It is a yz plane projection view of the adjacent annular tensegrity structure obtained by mirroring the tensegrity unit above the bottom surface;

[0051] Figure 7 It is an axonometric view of the adjacent annular tensegrity structure obtained by mirroring the tensegrity unit below the bottom surface;

[0052] Figure 8 It is a yz plane projection view of the adjacent annular tensegrity structure obtained by mirroring the tensegrity unit below the bottom surface;

[0053] Figure 9 It is the annular tensegrity structure obtained by continuously mirroring the tensegrity unit;

[0054] Figure 10 It is the convergence curve of the objective function with the iteration number in the optimization process in Example 1;

[0055] Figure 11 It is the annular tensegrity structure optimized in Example 1;

[0056] Figure 12 It is the convergence curve of the objective function with the iteration number in the optimization process in Example 2;

[0057] Figure 13 It is the annular tensegrity structure optimized in Example 2;

[0058] In the diagram: 1. Top surface; 2. Bottom surface; 3. Tensioned integral unit; 4. Positive direction; 5. Mirror surface; 6. Central axis of the ring. Detailed Implementation

[0059] The present invention will now be described in detail with reference to the embodiments shown in the accompanying drawings. However, it should be noted that these embodiments are not intended to limit the present invention. Equivalent changes or substitutions in function, method, or structure made by those skilled in the art based on these embodiments are all within the scope of protection of the present invention.

[0060] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0061] The following is combined Figures 1-13 The annular tensioning system and its optimization method, as shown, further describe the specific embodiments of the present invention.

[0062] like Figure 1 The diagram shows a top view of the annular tensioned integral structure. The annular tensioned integral structure consists of n tensioned integral units 3 arranged in a ring direction, where n is an even number. Each tensioned integral unit 3 has two bottom surfaces: an upper bottom surface 1 and a lower bottom surface 2. The upper bottom surface 1 and the lower bottom surface 2 are not parallel to each other and have an included angle β, that is, the central angle corresponding to each tensioned integral unit 3 is β, where β = 360° / n.

[0063] The central axis 6 of the annular tensioned integral structure is perpendicular to each bottom surface. The origin of the coordinate system is the center O1 of the lower bottom surface 2. The y-axis points to the center of the central axis 6 of the annular structure, the z-axis points to the center of the upper bottom surface 1, and the x-axis is perpendicular to the plane containing the central axis. The span (inner diameter) of the annular structure is d = H / tanβ-2r.

[0064] The upper bottom surface 1 or lower bottom surface 2 of adjacent tensioning integral units 3 overlap, and the overlapping surfaces are mirror images of each other. Therefore, by determining a tensioning integral unit 3, a ring-shaped tensioning integral structure can be obtained by continuously mirroring.

[0065] like Figures 2-4 As shown, a schematic diagram of the tensioning unit 3 is presented. In the diagram, the thick lines represent the compression members, and the thin lines represent the tension cables. The lower base 2 is an equilateral triangle located in the plane formed by the X-axis and Y-axis, with the center O1 being the origin of the coordinate axes. The Y-axis points to the center of the annular tensioning unit, the X-axis is perpendicular to the plane containing the central axis 6 of the annulus, and the Z-axis points to the center O2 of the upper base 1.

[0066] The tensioning unit 3 is uniquely determined by 5 parameters: height H, radius r of the circumscribed circle of the bottom surface, the initial rotation angle α0 of the lower bottom surface 2 rotating counterclockwise relative to the X-axis with the center O1 as the rotation fulcrum, the relative rotation angle α between the orthographic projection of the upper bottom surface 1 and the lower bottom surface 2, and the included angle β between the upper and lower bottom surfaces 2.

[0067] The three nodes on the lower base 2 are numbered A, B, and C counterclockwise. The angle between O1A and the x-axis is α0, and its positive direction 4 is shown in the figure. The three nodes on the upper base 1 of the tensioning integral unit 3 are numbered D, E, and F counterclockwise. The projections of points D, E, and F onto the xy plane form an equilateral triangle D'E'F'. The equilateral triangles ABC and D'E'F' are rotated by an angle α relative to the center O1, and their positive direction 4 is shown in the figure.

[0068] Because the upper bottom surface 1 and the lower bottom surface 2 of the tensioned integral unit 3 form an angle β, the z coordinates of points D, E, and F are reduced by tanβ·(y D +r), tanβ·(y E +r), tanβ·(y F +r). It is necessary to ensure z D ,z E ,z F It should be greater than 0; otherwise, the two bottom surfaces will intersect and cannot form a tensioned whole.

[0069] like Figures 5-8 As shown, a schematic diagram illustrates how adjacent tensioning units 3 are obtained by using either their upper bottom surface 1 or lower bottom surface 2 as mirror surfaces 5. Adjacent tensioning units 3 share the same bottom surface. Once the tensioning unit 3 is determined, a ring-shaped tensioning assembly can be obtained by continuously mirroring it.

[0070] like Figure 9 As shown, the formed annular tensioned integral structure is displayed.

[0071] The specific form of the equilibrium matrix A of the tensioned integral unit 3 is as follows:

[0072]

[0073] In the formula, A km A 3×1 block matrix:

[0074]

[0075] In the formula, []T denotes the transpose of the matrix, k and m are the node numbers, and l km Let k be the length of the component between points k and m, and let x, y, and z represent the x, y, and z coordinates of the points.

[0076] According to the singular value decomposition theory of matrices, the balanced matrix A can be decomposed into the following form:

[0077]

[0078] where V S = [v1 v2...v s ] T , v i is the self-stress mode;

[0079] The internal force of the member is a linear combination of the self-stress modes v i :

[0080]

[0081] where a i is the combination coefficient of the self-stress modes, t km is the internal force of the member between k and m, and t i > 0 (i∈S cable ) and t i < 0 (i∈S strut ).

[0082] The optimization steps of the ring tension integral structure are shown below with specific examples.

[0083] Example 1: Optimize the structure parameters: span d = 50 m, equal number n = 12, and the optimization goal is to make the internal force of the member as uniform as possible.

[0084] First, according to the optimization structure parameters, determine the variation range R H ∈(0,100m、R r ∈(0,50m)、 R α ∈(0°,120°)、R β = 360° / n = 30°.

[0085] According to the span d = 50 m, the constraint condition d = H / tanβ - 2r can be converted to: 50 = H / tanβ - 3r.

[0086] According to the optimization goal of making the internal force of the member as uniform as possible, c1 = 1 and c2 = 0 are obtained, and the objective function is

[0087] Use Matlab software to write code to solve the optimization model, and the convergence curve of the objective function with the iteration number in the optimization process is as follows: Figure 10As shown, the program stops at generation 250, reaching the minimum objective function value of 0.2295, with the corresponding five parameter values: H = 69.57, r = 35.26, α0 = 129.38, α = 61.88, and β = 30°. The internal forces of each component in the structure are [0.1240 0.2799 0.1481 0.2892 0.0975 0.1955 0.2735 0.1902 0.0130 0.2720 0.2551 0.1920 -0.5220 -0.4380 -0.6687].

[0088] Based on the five parameters corresponding to the final minimum objective function value, the overall annular tensioned structure is obtained as follows: Figure 11 As shown.

[0089] Example 2, optimizing structural parameters: span d = 100m, number of equal parts n = 18, optimization goal is to make the component lengths as consistent as possible.

[0090] First, based on the optimized structural parameters, determine the variation range R of parameters H, r, α0, α, and β. H ∈(0,50m), R r ∈(0,20m), R α ∈(0°,120°), R β =360° / n=20°;

[0091] Based on the span d = 50m, the constraint d = H / tanβ - 2r can be transformed into: 50 = H / tanβ - 2r;

[0092] Based on the optimization objective of making the internal forces of the components as uniform as possible, we obtain c1 = 0, c2 = 1, and the objective function is f = (LN cable +LN strut )+P

[0093] The optimization model was solved using code written in Matlab. The convergence curve of the objective function with the number of iterations during the optimization process is shown below. Figure 12 As shown, the program stops at generation 250, reaching the minimum objective function value of 8, with the corresponding five parameter values: H = 7.59, r = 5.43, α0 = 303.75, α = 86.25, β = 20°. The lengths of each component in the structure are [9.4030 9.4030 9.4030 9.4030 9.8591 9.8591 8.7467 5.5990 8.7467 5.5990 10.6170 8.21981 1.5423 13.0170 11.5423].

[0094] According to the 5 parameters corresponding to the final minimum objective function value, the ring-shaped tensioned whole structure is obtained as shown in Figure 13

[0095] In addition, it should be understood that, although the present specification is described in terms of embodiments, not every embodiment contains only one independent technical solution, and the description of the specification is only for the sake of clarity, and those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can be appropriately combined to form other embodiments that those skilled in the art can understand.​

Claims

1. An optimization method for a ring-shaped tensioned integral structure, wherein the ring-shaped tensioned integral structure has n tensioned integral units, and the n tensioned integral units are connected in a ring direction, wherein, n is an even number. A single tensioning unit includes 3 pressure bars and 12 cables. Any two ends of the 3 pressure bars are connected to a cable. The tensioning unit has two bottom surfaces, an upper bottom surface and a lower bottom surface. The bottom surface is formed by the same-side ends of the three pressure bars. The annular tensioning structure is formed by connecting the pressure bars of adjacent tensioning units one by one, and each bottom surface is perpendicular to the central axis of the annular tensioning structure. The structure of the tensioning unit is determined by defining the coordinates of the six endpoints of the three pressure bars, and the tensioning unit meets the following structural requirements: The orthographic projection of the upper base and the lower base are both equilateral triangles. The upper base and the lower base are not parallel to each other, do not intersect, and have an included angle β. The upper or lower bottom surfaces of adjacent tensioning integral units overlap, and are arranged in a mirror-symmetric manner with respect to the overlapping surfaces; The shape of the annular tensioned integral structure is uniquely determined by five parameters that make up the tensioned integral unit: the height H of the tensioned integral unit, the radius r of the circumscribed circle of the bottom surface, the angle α0 of the bottom surface rotating horizontally counterclockwise around the center O1 as the fulcrum, the relative rotation angle α between the orthographic projection of the top surface and the bottom surface, and the included angle β. The inner diameter span d of the annular tensioned monolithic structure, wherein the tensioned monolithic unit satisfies the following constraints: Simultaneously satisfying the requirement that the three compression bars do not intersect; The included angle β is a factor of 360° and satisfies n = 360° / β; The included angle β and the radius r of the circumcircle of the base satisfy the inner diameter span. ; All the tensioning units can meet the conditions of compression bars being compressed and cables being tensile; The optimization method is characterized by comprising the following steps: S1, set the lower base as an equilateral triangle and located in the plane formed by the X-axis and Y-axis, with center O1 as the origin of the coordinate axis, Y-axis pointing to the center of the ring tensioning whole, X-axis perpendicular to the plane where the central axis of the ring is located, and Z-axis pointing to the center O2 of the upper base; The six endpoints of the three compression rods are numbered as follows: the three endpoints of the lower bottom surface are numbered A, B, and C counterclockwise, and the three endpoints of the upper bottom surface are numbered D, E, and F counterclockwise. Within the range of variation of parameters H, r, α0, α, and β , , , , Generate initial parameter values ​​internally. , , , , Set according to actual needs; S2, Based on the initial parameter values, calculate the coordinates of the six endpoints as follows: ； ； ； ； ； ； S3, calculate the equilibrium matrix of the tensioned integral unit based on the coordinates, and perform singular value decomposition on the equilibrium matrix to obtain the self-stress modes; Based on the coordinates of the six endpoints and the self-stress mode, determine whether the tensioned integral element meets the structural requirements and constraints according to the expression of the constraint conditions in the optimization model, and calculate the objective function value; The objective function is: ， The optimization model is as follows: ， In the formula, i is the compression bar or tension cable. and The combination coefficients of the objective function For a set of cable components, A collection of rod components. For prestressing of components, For the average internal forces of the cable-stayed member, The average internal force of the rod member, The number of length types of cable members. Let P be the number of different length types of rod components, and let P be the penalty function term, whose value is much greater than... A is the equilibrium matrix of the structure, and t is the internal force vector of the component. Let ij be the distance between the two members, d be the span required for the ring-tensioned integral structure, and n be the number of tensioned integral units required for the ring-tensioned integral structure. S4. Import the optimization model from S3 into Matlab software and solve for the five parameters H, r, α0, α, and β. Adjust the values ​​of the five parameters through an optimization search algorithm. Repeat steps S2 and S3. The objective function value of the structure will decrease continuously until the optimal solution of the tensioned integral unit that meets the structural requirements and constraints is found.

2. The optimization method for a ring-shaped tensioned integral structure according to claim 1, characterized in that, If the tensioned integral unit satisfies the structural requirements and constraints, calculate its objective function value; If the objective function is not satisfied, then after calculating the objective function value, a corresponding penalty function term is added.

3. The optimization method for a ring-shaped tensioned integral structure according to claim 1, characterized in that, The and Based on the optimization objectives: If the optimization objective is to make the internal forces of the components as uniform as possible, then ; If the optimization objective is to make the component lengths as consistent as possible, then .

4. The optimization method for a ring-shaped tensioned integral structure according to claim 1, characterized in that, The optimization search algorithm includes a genetic algorithm.

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