A method for finding force in a string structure based on catenary cable units

Abstract

The present invention discloses a method for finding the force of a string structure based on a catenary cable unit. Under the condition that the shape of the string structure is completely determined, the method can specify the internal force of some units as needed, and determine the prestress distribution of the entire string structure by Newton's method. The present invention performs force analysis based on the catenary cable unit, and has a higher precision result than the rod unit; the calculation principle is based on Newton's method, and the proposed tangent stiffness matrix is ​​derived from the equilibrium equation, which can achieve stable convergence of the calculation results. It is applicable to unidirectional, bidirectional and multi-directional beam-string structures, as well as different types of string structures such as string trusses and string lattice shells; the internal forces of redundant rods can be specified, and the force of the structure can be uniform through reasonable internal force selection, providing a variety of options for structural optimization; the construction material cutting form of the structure can be calculated.

Description

Technical Field

[0001] The invention relates to the fields of architectural design and structural design, and in particular to a method for finding the force of a string structure based on a catenary cable unit. Background Art

[0002] The cable-stretching structure is composed of the lower chord cable, the middle connecting rod, and the upper chord beam, truss or lattice shell structure. This structural form places the cable on the lower chord of the structure, making full use of the tensile strength of the high-strength cable, which can make the structure have a larger span capacity, so it is widely used in large-span space structures.

[0003] The upper chord of the cable-stayed structure is a traditional "rigid structure", and the lower chord is a "flexible structure" composed of cables. It is a typical hybrid structure. Due to the existence of cable units, the structure itself must rely on prestress to maintain a specific shape. In the past, flexible structures can generally determine the "force" and "shape" of the structure at the same time through "form-finding analysis". Among them, "force" refers to the distribution of prestress, and "shape" refers to the equilibrium shape of the structure. However, in the process of architectural design, sometimes a specific building shape is required to meet aesthetic requirements and building functions. Therefore, structural engineers need to arrange prestress reasonably to keep the structure under reasonable force. At this time, structural engineers need to solve "force-finding analysis" instead of form-finding analysis. Force-finding analysis has rarely been involved in previous studies, especially there is no clear calculation method for force-finding analysis based on catenary cable units. At present, in order to fully meet the needs of architectural form, form-finding analysis is often used to repeatedly try and adjust to approach or achieve the desired shape of the building, but the process is cumbersome and there is no clear adjustment direction. Among them, the force density method commonly used in form-finding analysis is based on the rod unit for analysis, which is an approximate replacement for the cable unit. In theory, it can only meet the calculation accuracy when the prestress of the cable unit is large. Summary of the invention

[0004] The purpose of the present invention is to provide a method for finding the force of a string structure based on a catenary cable unit, which method does not require repeated debugging, and directly solves the internal force of the structure based on the modeling needs of the architect, can meet the modeling requirements of the architect to the greatest extent, and can ensure that the internal force of the structure is as uniform and reasonable as possible with a small amount of intervention by the structural engineer. The catenary unit used in the present invention is applicable to both slack or tensioned cable units, has higher calculation accuracy and adaptability than the rod unit, and ensures the accuracy of the calculation results.

[0005] A method for finding force in a string structure based on a catenary cable unit, characterized in that the method comprises the following steps:

[0006] Step (1): Establish a spatial rectangular coordinate system, with the horizontal plane as the Oxy plane, the vertical upward as the positive direction of the z axis, and the origin O as a suitable point as needed. Decompose the string structure into two parts, one is a rigid structure composed of the upper chord beam or truss or lattice shell, and the other is a cable-strut system composed of the middle rod unit and the lower cable unit;

[0007] Step (2), determining the prestress distribution of the lower cable rod system according to the specified cable rod shape, comprises the following steps;

[0008] Step (2.1) is to separate the lower cable-strut system from the overall string structure while maintaining the cable-strut shape and the original boundary constraints, and at the same time add support constraints with three translational degrees of freedom at the connection between the lower and upper parts.

[0009] Step (2.2), assuming that the node coordinates of the cable net system determined by the coordinate system of step (1) are (x i ,y i ,z i ), J is the total number of nodes, c is the number of constrained degrees of freedom, and the element numbers are 1, 2, …, b, where b is the number of elements. The node load is (p ix ,p iy ,p iz ) ; Establish the equilibrium equation of the cable-rod system, see equation (1) and equation (2), where i = 1, 2, ... J;

[0010]

[0011] Abbreviated as

[0012] A b t=p (2)

[0013] In the formula, l k is the length of the rod element k, L n is the length of the horizontal projection of cable element n, t k is the axial force of the rod element k, H n is the horizontal resultant force of cable element n, and other symbols are similar. t is the unit internal force vector of b×1, and p is the node load vector of (3J-c)×1. The preliminary prestress distribution mode of the cable-strut system is determined by singular value decomposition or by directly solving the linear equations of equation (1). The unique prestress distribution is obtained by specifying the internal forces of br units, where r is the equilibrium matrix A b It should be noted that for the bar element, the bar element axial force needs to be specified, and for the cable element, the cable element horizontal force needs to be specified.

[0014] Step (2.3), based on the initial prestress distribution t determined in step (2.2) 0 and specify the internal forces f of the br elements to be constanti (i=1,2,…,br), where f is the axial force t of the rod unit or the horizontal resultant force H of the cable unit. The Newton method is used to iteratively calculate equations (3)-(9) to obtain the actual prestress distribution.

[0015]

[0016]

[0017]

[0018]

[0019]

[0020]

[0021]

[0022] In the formula, q n is the weight per unit length of the cable unit, and the upper right corner of other symbols indicates the i-th iteration. The convergence criterion of the above iteration is preferably that the magnitude of the unbalanced force vector at the last iteration is less than 1 / 1000 of the first iteration, that is, |δp i | / |δp 1 |≤0.001.

[0023] Step (3): Keep the shape of the upper rigid structure separated from the overall string structure and the original boundary constraints, calculate the force of the lower cable-strut system on the upper rigid structure based on the prestress distribution of the lower cable-strut system determined in step (2), and apply it to the upper structure, and then calculate the internal force of the rigid structure under the combined action of the force transmitted from the lower part and the load on the rigid structure according to the linear elastic method.

[0024] Step (4): Based on the structural form and boundary constraints of step (3), the internal force obtained in step (3) is used as the initial stress, and the structural nonlinear analysis can be performed to obtain the zero state of the rigid structure when it is not stressed, that is, the construction cutting form. The internal force determined by step (3) of the rod unit can be used to determine the cutting length according to formula (10), and the prestress determined by step (2) of the cable unit can be used to calculate the initial original length according to formula (11), that is, the cutting length.

[0025]

[0026]

[0027]

[0028]

[0029] In the formula, l k0 is the cutting length of the bar unit, E is the elastic modulus, A is the cross-sectional area, s0 is the original length of the cable unit (cutting length), H is the horizontal force of the cable unit, q is the unit length weight of the cable unit, L is the horizontal projection length, z i and z j is the z coordinate value of the nodes at both ends of the cable.

[0030] According to a second aspect of the purpose of the present invention, the present invention provides a non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that when the computer program is executed by a processor, the steps of the method for finding force in a string structure based on a catenary thread unit are implemented.

[0031] According to a third aspect of the purpose of the present invention, the present invention is an electronic device, comprising a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the method for finding force in a string structure based on a catenary thread unit when executing the program.

[0032] Due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0033] 1) Avoid the process of repeated trial and error analysis to achieve the building form, and directly solve the structural internal forces and construction material cutting form based on the building form.

[0034] 2) The catenary unit is used to simulate the cable segment, which has higher accuracy and applicability than the rod unit.

[0035] 3) It is applicable to unidirectional, bidirectional and multidirectional beam-string structures, as well as different types of beam-string structures such as beam-string trusses and beam-string lattice shells, and has general applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the specific implementation methods of the present invention or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some implementation methods of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0037] Figure 1 It is a flow chart of the force finding method of the present invention.

[0038] Figure 2 This is a schematic diagram of a planar beam string structure according to Example 1 of the present invention.

[0039] Figure 3 This is a schematic diagram of the analysis of the separated lower cable rod system of Example 1 of the present invention.

[0040] Figure 4 This is a schematic diagram of the analysis of the separated upper rigid structure of Example 1 of the present invention.

[0041] Figure 5 This is the equilibrium state and construction cutting state of the planar beam string structure of Example 1 of the present invention.

[0042] Figure 6 This is a schematic diagram of a multi-directional beam string structure according to Example 2 of the present invention.

[0043] Figure 7 This is a schematic diagram of the analysis of the separated lower cable rod system of Example 2 of the present invention.

[0044] Figure 8 This is a schematic diagram of the analysis of the separated upper rigid structure of Example 2 of the present invention.

[0045] Fig. 9 This is the balanced state and construction cutting state of the multi-directional beam string structure of Example 2 of the present invention.

[0046] Among them: 1 is the upper chord beam in the equilibrium form (design form), 2 is the strut, 3 is the lower cable, 4 is the connection node, 5 is the hinged support, 6 is the vertical support support, 7 is the node load, 8 is the load of the lower structure acting on the upper structure, and 9 is the upper chord beam in the construction unloading state. DETAILED DESCRIPTION

[0047] The technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. It should be understood that the specific implementation methods described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0048] Step (1): Establish a spatial rectangular coordinate system, with the horizontal plane as the Oxy plane, the vertical upward as the positive direction of the z axis, and the origin O as a suitable point as needed. Decompose the string structure into two parts, one is a rigid structure composed of the upper chord beam or truss or lattice shell, and the other is a cable-strut system composed of the middle rod unit and the lower cable unit.

[0049] Step (2): Determine the prestress distribution of the lower cable-bar system according to the specified cable-bar shape. See steps (2.1) to (2.3) for details.

[0050] Step (2.1): Separate the lower cable-strut system from the overall string structure while maintaining the cable-strut shape and original boundary constraints, and add support constraints with three translational degrees of freedom at the connection between the lower and upper parts.

[0051] Step (2.2): Assume that the node coordinates of the cable net system determined according to the coordinate system of step (1) are (x i ,y i ,z i )(i=1,2,...,J), J is the total number of nodes, c is the number of constrained degrees of freedom, and the element numbers are 1, 2,...,b, where b is the number of elements. The node load is (p ix ,p iy ,p iz )(i=1,2,...,J). The equilibrium equations of the cable-rod system are established, see equations (1) and (2).

[0052]

[0053] Abbreviated as

[0054] A b t=p (2)

[0055] In the formula, l k is the length of the rod element k, L n is the length of the horizontal projection of cable element n, t k is the axial force of the rod element k, H n is the horizontal resultant force of cable element n, and other symbols are similar. t is the unit internal force vector of b×1, and p is the node load vector of (3J-c)×1. The preliminary prestress distribution mode of the cable-strut system is determined by singular value decomposition or by directly solving the linear equations of equation (1). The unique prestress distribution is obtained by specifying the internal forces of br units, where r is the equilibrium matrix A b It should be noted that for the bar element, the bar element axial force needs to be specified, and for the cable element, the cable element horizontal force needs to be specified.

[0056] Step (2.3): Based on the initial prestress distribution t determined in step (2.2) 0 and specify the internal forces f of the br elements to be constant i (i=1,2,…,br), where f is the axial force t of the rod unit or the horizontal resultant force H of the cable unit. The Newton method is used to iteratively calculate formula (3-9) to obtain the actual prestress distribution.

[0057]

[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064] In the formula, q n is the weight per unit length of the cable unit, and the upper right corner of other symbols indicates the i-th iteration. The convergence criterion of the above iteration is preferably that the magnitude of the unbalanced force vector at the last iteration is less than 1 / 1000 of the first iteration, that is, |δp i | / |δp 1 |≤0.001.

[0065] Step (3): Keep the shape of the upper rigid connection structure separated from the overall string structure and the original boundary constraints, calculate the force of the lower cable-strut system on the upper rigid structure based on the prestress distribution of the lower cable-strut system determined in step (2), and apply it to the upper structure, and then calculate the internal force of the rigid structure under the combined action of the force transmitted from the lower part and the load on the rigid structure according to the linear elastic method.

[0066] Step (4): Based on the structural form and boundary constraints of step (3), the internal force obtained in step (3) is used as the initial stress, and the structural nonlinear analysis can be performed to obtain the zero state of the rigid structure when it is not stressed, that is, the construction cutting form. The rod unit is determined by the internal force determined in step (3) according to formula (10) to determine the cutting length, and the cable unit is determined by the prestress determined in step (2) according to formula (11) to calculate the initial original length, that is, the cutting length.

[0067]

[0068]

[0069]

[0070]

[0071] In the formula, l k0 is the cutting length of the bar unit, E is the elastic modulus, A is the cross-sectional area, s0 is the original length of the cable unit (cutting length), H is the horizontal force of the cable unit, q is the unit length weight of the cable unit, L is the horizontal projection length, z i and z j is the z coordinate value of the nodes at both ends of the cable.

[0072] Example 1

[0073] Figure 2-5 The present invention is described by taking a planar beam string structure as an embodiment.

[0074] Figure 2The figure shows an obliquely placed plane beam string structure, where both the lower and upper chords are parabolic in shape, and are composed of an upper chord beam 1, an intermediate strut 2, and a lower chord 3. The connection node 4 is a hinged node, the boundary node 5 is a hinged node, the boundary node 6 is a vertical support, and the node load 7 is the load on the beam element.

[0075] Establish a plane rectangular coordinate system, with the horizontal right as the positive x-axis, the vertical upward as the positive z-axis, and the origin O as the support node 5. Decompose the string structure into two parts, one of which is a cable-strut system consisting of a middle rod unit and a lower cable unit (see Figure 3 ), part of which is a rigid structure consisting of a beam at the top chord (see Figure 4 ).

[0076] The node coordinates of the cable net system determined according to the established plane coordinate system are (x i ,0,z i )(i=1,2,...,J), J is the total number of nodes, c is the number of constrained degrees of freedom, and the element numbers are 1, 2,...,b, where b is the number of elements. The node load is (p ix ,0,p iz )(i=1,2,...,J). The equilibrium equations of the cable-strut system are established, see equations (14) and (15).

[0077]

[0078] Abbreviated as

[0079] A b t=p (15)

[0080] Directly solve the linear equations of (14) to determine the preliminary prestress distribution mode of the cable-strut system. By specifying the internal forces f of br units i (i=1,2,…,br) determines the unique prestress distribution t 0 .

[0081] Keeping the internal forces of the specified br units unchanged, iteratively solve equations (16)-(22) until the magnitude of the unbalanced force vector at the last iteration is less than 1 / 1000 of that at the first iteration.

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088]

[0089] based on Figure 4 The structural form and boundary constraints are taken into consideration, and the calculated internal force of the rod unit is applied to the superstructure as a reaction force. Then, the internal force of the rigid structure under the combined action of the force transmitted from the lower part and the load on the rigid structure is obtained according to the linear elastic method.

[0090] Still based on Figure 4 The structural form and boundary constraints, the load is only the structural internal force obtained in the previous step as the initial stress, and the structural nonlinear analysis can be performed to obtain the zero state of the rigid structure when it is not stressed, that is, the construction material cutting form. Figure 5 Shown are the design form and construction cutting form of the plane tensioned beam.

[0091] The rod element determines the cutting length according to formula (23), and the cable element calculates the initial original length, i.e. the cutting length, according to formula (24).

[0092]

[0093]

[0094]

[0095]

[0096] Example 2

[0097] Figure 6-9 The present invention is described by taking a multi-directional beam string structure as an embodiment.

[0098] Figure 6 The multi-directional beam string structure is shown, which is composed of an upper string beam 1, an intermediate strut 2, and a lower string 3. The connection node 4 is a hinge node, the boundary node 5 is a hinge node, and the node load 7 is the load on the beam element.

[0099] Establish a spatial rectangular coordinate system, with the horizontal plane as the Oxy plane, the vertical upward as the positive direction of the z axis, and the origin O as a suitable point as needed. Decompose the multi-directional string structure into two parts, one of which is a cable-strut system consisting of a middle rod unit and a lower cable unit (see Figure 7 ), part of which is a rigid structure consisting of a beam at the top chord (see Figure 8 ).

[0100] The node coordinates of the cable net system determined according to the established spatial coordinate system are (x i ,y i ,z i)(i=1,2,...,J), J is the total number of nodes, c is the number of constrained degrees of freedom, and the element numbers are 1, 2,...,b, where b is the number of elements. The node load is (p ix ,p iy ,p iz )(i=1,2,...,J). The equilibrium equations of the cable-strut system are established, see equations (27) and (28).

[0101]

[0102] Abbreviated as

[0103] A b t=p (28)

[0104] Directly solve the linear equations of (27) to determine the preliminary prestress distribution mode of the cable-strut system. i (i=1,2,…,br) determines the unique prestress distribution t 0 .

[0105] Keeping the internal forces of the specified br units unchanged, iteratively solve equations (29)-(35) until the magnitude of the unbalanced force vector at the last iteration is less than 1 / 1000 of that at the first iteration.

[0106]

[0107]

[0108]

[0109]

[0110]

[0111]

[0112]

[0113] based on Figure 8 The structural form and boundary constraints are taken into consideration, and the calculated internal force of the rod unit is applied to the superstructure as a reaction force. Then, the internal force of the rigid structure under the combined action of the force transmitted from the lower part and the load on the rigid structure is obtained according to the linear elastic method.

[0114] Still based on Figure 8 The structural form and boundary constraints, the load is only the structural internal force obtained in the previous step as the initial stress, and the structural nonlinear analysis can be performed to obtain the zero state of the rigid structure when it is not stressed, that is, the construction material cutting form. Fig. 9Shown are the design form and construction cutting form of the plane tensioned beam.

[0115] The rod element determines the cutting length according to formula (35), and the cable element calculates the initial original length, i.e. the cutting length, according to formula (37).

[0116]

[0117]

[0118]

[0119]

[0120] Through the description of the above embodiments, it can be clearly understood by those skilled in the art that the facilities of the present invention can be implemented by means of software plus the necessary general hardware platform. The embodiments of the present invention can be implemented using an existing processor, or by a dedicated processor used for this purpose or other purposes for an appropriate system, or by a hard-wired system. The embodiments of the present invention also include a non-transitory computer-readable storage medium, which includes a machine-readable medium for carrying or having a machine-executable instruction or data structure stored thereon; such a machine-readable medium can be any available medium that can be accessed by a general or special-purpose computer or other machine with a processor. For example, such a machine-readable medium can include RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, disk storage or other magnetic storage device, or any other medium that can be used to carry or store the required program code in the form of machine-executable instructions or data structures, and can be accessed by a general or special-purpose computer or other machine with a processor. When information is transmitted or provided to a machine via a network or other communication connection (hard-wired, wireless, or a combination of hard-wired or wireless), the connection is also considered a machine-readable medium.

[0121] So far, the technical solutions of the present invention have been described in conjunction with the preferred embodiments shown in the accompanying drawings. However, it is easy for those skilled in the art to understand that the protection scope of the present invention is obviously not limited to these specific embodiments. Without departing from the principle of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will fall within the protection scope of the present invention.

Claims

1. A method for finding force in a string structure based on a catenary cable unit, characterized in that: The method comprises the following steps: Step (1), establishing a spatial rectangular coordinate system, with the horizontal plane as the Oxy plane and the vertical upward as the positive direction of the z axis; decomposing the string structure into two parts, one part is a rigid structure composed of an upper chord beam or truss or lattice shell, and the other part is a cable-rod system composed of a middle rod unit and a lower cable unit; Step (2), determining the prestress distribution of the lower cable rod system according to the specified cable rod shape; comprising the following steps: Step (2.1), keep the shape of the lower cable-strut system separated from the overall string structure and the original boundary constraints, and add support constraints with three translational degrees of freedom at the connection between the lower and upper parts; Step (2.2), assuming that the node coordinates of the cable net system determined according to the coordinate system of step (1) are (x i ,y i ,z i ), J is the total number of nodes, c is the number of constrained degrees of freedom, the element numbers are 1, 2, …, b, where b is the number of elements; the node load is (p ix ,p iy ,p iz ), establish the equilibrium equation of the cable-rod system, see equation (1) and equation (2), where i = 1, 2, ... J; Abbreviated as A b t=p (2) In the formula, l k is the length of the rod element k, L n is the length of the horizontal projection of cable element n, t k is the axial force of the rod element k, H n is the horizontal resultant force of cable unit n; t is the unit internal force vector of b×1, and p is the node load vector of (3J-c)×1; the preliminary prestress distribution mode of the cable-strut system is determined by singular value decomposition or by directly solving the linear equations of formula (1), and the unique prestress distribution is obtained by specifying the internal forces of br units, where r is the equilibrium matrix A b rank; Step (2.3), based on the initial prestress distribution t determined in step (2.2) 0 and specify the internal forces f of the br elements to be constant i , where f is the axial force t of the rod element or the horizontal resultant force H of the cable element, i = 1, 2, …, br, and the actual prestress distribution is obtained by iteratively calculating equations (3)-(9) using Newton's method; In the formula, q n is the weight per unit length of the cable unit, and the upper right corner of other symbols indicates the i-th iteration; Step (3), maintaining the shape of the upper rigid structure separated from the overall string structure and the original boundary constraints, calculating the force of the lower cable-rod system on the upper rigid structure based on the prestress distribution of the lower cable-rod system determined in step (2), and applying the force to the upper structure, and then calculating the internal force of the rigid structure under the combined action of the force transmitted from the lower part and the load on the rigid structure according to the linear elastic method; Step (4), based on the structural form and boundary constraints of step (3), the internal force obtained in step (3) is used as the initial stress, and a structural nonlinear analysis is performed to obtain the zero state of the rigid structure when it is not subjected to stress, that is, the construction cutting form; the cutting length of the rod unit is determined by the internal force determined in step (3) according to formula (10), and the initial original length of the cable unit is calculated according to formula (11) based on the prestress determined in step (2), that is, the cutting length; In the formula, l k0 is the blanking length of the bar unit, E is the elastic modulus, A is the cross-sectional area, s0 is the original length of the cable unit, H is the horizontal force of the cable unit, q is the unit length weight of the cable unit, L is the horizontal projection length, z i and z j is the z coordinate value of the nodes at both ends of the cable.

2. A method for finding force in a string structure based on a catenary cable unit as claimed in claim 1, characterized in that: The convergence criterion of the iteration in step (2.3) is that the magnitude of the unbalanced force vector at the last iteration is less than 1 / 1000 of that at the first iteration, that is, |δp i | / |δp 1 |≤0.

001.

3. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method for finding force of a string structure based on a catenary cable unit as described in any one of claims 1 to 2 are implemented.

4. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the program, the steps of the method for finding force of a string structure based on a catenary thread unit as described in any one of claims 1 to 2 are implemented.

Images