CN106934155B - Shape-finding method of cable truss structure - Google Patents
Shape-finding method of cable truss structure Download PDFInfo
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Abstract
The application discloses a shape finding method of a cable truss structure, and compared with the prior art, the shape finding method has the following characteristics and beneficial effects: the cable truss structure can be shaped under the premise of giving the roof or curtain wall geometry and keeping the stay bar or the sling vertical, so that the shaping result completely conforms to the expected building shape; the method has wide application range, and can be used for various cable truss arrangement modes such as different types of roof or curtain wall curved surfaces, boundary shapes, spoke type, parallel type and cross type; the concept of a force density mode and a unidirectional force density method are provided, only the z coordinate of the node is updated in the shape finding process, the x and y coordinates of the node automatically meet a balance equation, and updating is not needed, so that the brace rod or the sling automatically keeps vertical; the control parameters are few, a series of shape finding results with different sizes and corresponding internal force distribution can be quickly given according to the given geometric shape of the roof or the curtain wall, and comparison and selection of building schemes, structural optimization and the like are facilitated.
Description
Technical Field
The invention belongs to the field of building design and structural design in building engineering, and particularly relates to a shape finding method of a cable truss structure.
Background
The cable truss structure is formed by arranging and combining a series of cable trusses according to a certain rule, and common cable truss arrangement modes include a spoke type, a parallel type, a cross type and the like. The structure has the characteristics of light dead weight, large applicable span, convenient construction and the like, thereby being widely applied to various building forms such as large-span structures, curtain wall structures and the like.
The basic components of the cable truss structure can be summarized as an upper chord cable, a lower chord cable and a brace (or a sling) between two layers of inhaul cables, and the cable truss structure is a typical cable tension structure system. Such structures need to be rigid by introducing pre-stress, and thus have load-bearing capacity. The structural states before and after the introduction of the prestress are referred to as the zero state and the initial state, respectively. In the case of a cable truss structure, whether the structure can form the bearing capacity is directly related to the structure configuration, so form finding (form definition) is always one of the core problems in the research and design of the cable truss structure. The existing shape-finding method applied to the cable truss structure mainly comprises a nonlinear finite element method, a dynamic relaxation method, a force density method and the like. After the configuration of the cable truss structure is determined, the prestress distribution of the structure can be obtained by utilizing a balance matrix theory and the like, and the prestress level is determined according to the requirements of the structural rigidity and the bearing capacity, so that the initial state of the cable truss structure is obtained.
In engineering practice, building roofs or curtain walls are usually laid directly or indirectly in the plane of the cable truss structure where the upper chord or lower chord is located. To achieve architectural results, the roof or curtain wall geometry is typically determined by the architect, i.e. the upper chord (or lower chord) geometry of the lattice truss is given, while the struts (or slings) are generally held upright. Therefore, a method for finding the shape of the cable truss structure when the geometry of part of the components is given is needed in engineering, so that the initial state of the cable truss structure can be found on the premise of ensuring the building effect.
Disclosure of Invention
Aiming at the problems in the prior art, the invention aims to provide a cable truss structure form-finding method, which effectively solves the problems in the prior art.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method of shaping a cable truss structure, the method comprising the steps of:
1) establishing cable truss structure upper chord or lower chord nodes conforming to the geometric shape of a given roof or curtain wall, and connecting the nodes according to the topological relation to obtain a geometric model of the upper chord cable or the lower chord cable;
2) shifting each upper chord node or each lower chord node downwards or upwards by any distance along the vertical direction to obtain the initial position of each lower chord node or each upper chord node, and connecting the corresponding upper chord node and the corresponding lower chord node to obtain an initial geometric model of the brace rod or the sling;
3) restraining the outer ring support joint and the lower or upper end point of the stay bar or sling to obtain a force finding model named as a model A;
4) additionally establishing a form finding model B: establishing an initial geometric model of the lower chord cable or the upper chord cable according to the initial position of the lower chord node or the upper chord node in the step 2), and constraining the outer ring support node;
5) assembling a balance matrix of the model A and performing singular value decomposition to obtain a self-stress mode of the model A;
6) linearly combining the self-stress modes of the model A, extracting corresponding numerical values of the stay bars or the suspension ropes in a combined result, and directly extracting the corresponding numerical values of the stay bars or the suspension ropes in the modes when the self-stress modes are unique, wherein an obtained result is used as an external load during shape finding;
7) establishing an x-direction and y-direction balance equation set of the model B by taking the force density as a variable, assembling a coefficient matrix of the equation set, namely a force density balance matrix, and performing singular value decomposition on the matrix to obtain a force density mode of the model B meeting the x-direction and y-direction balance conditions;
8) linearly combining the force density modes of the model B, and directly multiplying the force density modes by an adjusting coefficient when the force density modes are unique, wherein the obtained result is used as the force density for shape finding;
9) adopting the force density obtained in the step 8), applying the external load obtained in the step 6) to a corresponding node of the model B, and carrying out shape finding on the model B by a force density method with load to obtain coordinates of a lower chord or an upper chord node meeting balance conditions and the vertical requirement of a brace rod or a sling;
10) establishing an integral geometric model of the cable truss structure by using coordinates of upper chord or lower chord nodes in the model A and coordinates of lower chord or upper chord nodes in the shape-finding model B, and naming the model as a model C;
11) checking whether the geometric dimension of the model C meets the requirements related to the building functions, if so, entering the subsequent step, and if not, returning to the step 8), updating the force density modal combination coefficient or the adjustment coefficient, and reshaping until a cable truss structure with the geometric dimension meeting the requirements is obtained;
12) assembling a balance matrix of the model C and performing singular value decomposition to obtain a cable truss structure self-stress mode after shape finding is completed; then the form finding result is corrected by considering the dead weight or other loads.
Further, the specific meanings and calculation methods of the force density balance matrix and the force density mode in the step 7) are as follows:
an x-direction and y-direction equilibrium equation is established for the ith node of model B:
wherein (x)i,yi) Is the x, y coordinates of the ith node, n is the number of cells connected to the ith node, Lk、fkAnd (x)k,yk) (k ═ 1,2, …, n) are the length, internal force, and x, y coordinates of the other end point, respectively, of the kth element connected to the ith node; density of induced force qk＝fk/LkEquation (1) can be transformed into:
in the formula (2), (x)i,yi)、(xk,yk) Has been given according to the horizontal position of each node, and qkIs unknown, and therefore equation (2) can be considered as relating to qkThe system of equations (1); establishing the same balance equation for all nodes of the model B, and writing the equation into a matrix form after grouping:
[Aq]{q}＝{0} (3)
matrix in formula (3)[Aq]The force density balance matrix is called as a model B, and q is the force density which meets the horizontal balance condition of each node of the model B;
will [ A ] beq]Singular value decomposition is carried out to obtain a group of vectors { sj(j ═ 1,2, …), satisfying:
[Aq]{sj}＝{0} (4)
then sjIt is a general solution of the homogeneous linear equation set of equation (3), called force density mode.
Further, the principle and implementation process of the unidirectional force density method in the step 9) are as follows:
applying an external load p in the z-direction to each node of model BiThen, the balance equation of model B at the ith node is:
wherein (x)i,yi,zi) Is the x, y, z coordinates of the ith node, n is the number of elements connected to the ith node, Lk、fkAnd (x)k,yk,zk) (k ═ 1,2, …, n) are the length, internal force, and x, y, z coordinates of the other end point, respectively, of the kth element connected to the ith node; density of induced force qk＝fk/LkEquation (5) can be written as:
as can be seen from the equations (1) to (4), when the force density obtained in the step 8) is adopted, the first two equations of the equation (6) are automatically established, and only the third equation needs to be solved;
the model B is provided with B units and m nodes, wherein the number of the nodes to be shaped and the fixed outer ring support nodes is m respectivelyfAnd mcIntroducing a topological matrix of b × m:
in [ C ]]Wherein the node to be shaped is arranged before the outer ring support node, [ C ] can be arranged]Splitting into a free node topology matrix [ C ]f]And constraint node topology matrix [ Cc]I.e. C [ [ C ]f][Cc]](ii) a The z-balance equations are listed and grouped for all nodes of model B, which can be written as:
[Cf]T[Q][Cf]{zf}+[Cf]T[Q][Cc]{zc}＝{p} (8)
wherein zfZ coordinate vector of node to be solved, { z }cZ coordinate vector of outer ring support node, { p } external load vector, [ Q ]]Is a force density diagonal matrix; solving the formula (8) to obtain the z coordinate of the free node:
{zf}＝([Cf]T[Q][Cf])-1({p}-[Cf]T[Q][Cc]{zc}) (9)
the solution means that only the z coordinate of the node needs to be updated in the shape finding process of the model B, and the x and y coordinates are not changed, so that the stay bar or the sling can be automatically kept vertical; the method realizes the shape finding only in the z direction, so the method is called a unidirectional force density method.
Further, the specific process of correcting the shape finding result by considering the self weight or other loads in the step 12) is as follows: after the shape finding is finished, the specification and the prestress of each component of the cable truss structure are determined through primary design, and a structural design model called model D is obtained; extracting the prestress of each brace rod or sling of the model D; all the z-direction degrees of freedom of the nodes obtained by shape finding of the constraint model D are subjected to static calculation under the action of the self weight of the structure or other loads, and each constraint reaction value is superposed to the corresponding prestress of the stay bar or the sling to serve as an external load adopted for correction; applying an external load to a corresponding node of the model B; linearly combining the force density modes of the model B, and directly multiplying the force density modes by an adjustment coefficient when the force density modes are unique, wherein the obtained result is used as the force density for correction; carrying out shape finding on the model B by a unidirectional force density method with load; correcting the shape finding node coordinates of the model D according to the shape finding result, if the geometric dimension of the corrected model D does not meet the requirement, updating the force density modal combination coefficient or the adjustment coefficient, and correcting again until the geometric dimension of the model D meets the requirement; repeating the process until the single correction amplitude is smaller than a preset limit value; and finally, calculating the corrected prestress of the shape-finding part according to the cable length of the shape-finding part of the model D and the corresponding force density value.
The invention has the following beneficial technical effects:
1. the cable truss structure can be shaped under the premise of giving the roof or curtain wall geometry and keeping the stay bar or the sling vertical, so that the shaping result completely conforms to the expected building shape;
2. the method has wide application range, and can be used for various cable truss arrangement modes such as different types of roof or curtain wall curved surfaces, boundary shapes, spoke type, parallel type and cross type;
3. the concept of a force density mode and a unidirectional force density method are provided, only the z coordinate of the node is updated in the shape finding process, the x and y coordinates of the node automatically meet a balance equation, and updating is not needed, so that the brace rod or the sling automatically keeps vertical;
4. the control parameters are few, a series of shape finding results with different sizes and corresponding internal force distribution can be quickly given according to the given geometric shape of the roof or the curtain wall, and the comparison and selection of building schemes and the structural optimization are facilitated;
5. the shape finding process is relatively independent from the correction process, and static calculation is not needed in the shape finding process, so that the shape finding process is independent of the size and the load of the section of the component, and the shape finding is quicker and more convenient;
6. after the form-finding is finished, the geometrical configuration of the cable truss structure in the initial state and the zero state can be basically consistent by taking the self weight or other loads into consideration for correction.
Drawings
FIG. 1 is a flow chart of a shape-finding method of the present invention;
FIG. 2 is a flow chart of a correction method of the present invention;
FIG. 3 is a force-finding model, namely a model A, formed by a geometric model of an upper chord cable of a cable truss structure with given geometry and an initial geometric model of a strut;
FIG. 4 is a form-finding model formed based on an initial geometric model of a lower chord cable of a cable truss structure, namely a model B;
FIG. 5 is a view showing a model B after the shape finding is completed;
FIG. 6 shows the completed cable truss structure in form, model C;
FIG. 7 is a schematic view of a partial structure of model A;
FIG. 8 is a partial structural diagram of model B;
FIG. 9 is a partial structural view of the model B after the shape-finding process;
FIG. 10 is a partial structural view of model C;
FIG. 11 is a schematic view of the geometric relationship between the horizontal projection of the inner ring and the horizontal projection of the cable truss;
wherein: 1 is an upper chord node, 2 is an upper chord cable, 3 is a lower chord node, namely a lower end point of a strut, 4 is a strut, 5 is an outer ring support node, 6 is a lower chord cable, and 7 is an external load adopted by model B in shape finding; 8 is inner ring horizontal projection; and 9 is a horizontal projection of the cable truss.
Detailed Description
The present invention now will be described more fully hereinafter with reference to the accompanying drawings, in which exemplary embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the exemplary embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.
As shown in fig. 1, the invention provides a method for finding a cable truss structure, which comprises the following steps:
step 1: establishing a cable truss structure upper chord node 1 which accords with the geometric shape of a given roof, and connecting each node according to a topological relation to obtain a geometric model of an upper chord cable 2;
step 2: downwards offsetting each upper chord node 1 by any distance along the vertical direction to obtain the initial position of each lower chord node 3, and connecting the corresponding upper chord node and the corresponding lower chord node to obtain an initial geometric model of the brace 4;
and step 3: restraining the outer ring support node 5 and the lower end point 3 of the support rod 4 to obtain a force finding model shown in the figures 3 and 7, and naming the force finding model as a model A;
and 4, step 4: additionally establishing a form-finding model B shown in the figures 4 and 8, establishing an initial geometric model of the lower chord cable 6 according to the initial position of the lower chord node 3 in the step 2, and constraining the outer ring support node 5;
and 5: assembling a balance matrix of the model A and performing singular value decomposition to obtain a unique self-stress mode of the model A;
step 6: extracting the corresponding numerical value of the strut in the self-stress mode of the model A, and taking the obtained result as the external load 7 during shape finding;
and 7: establishing an x-direction and y-direction balance equation set of the model B by taking the force density as a variable, assembling a coefficient matrix of the equation set, namely a force density balance matrix, and performing singular value decomposition on the matrix to obtain a unique force density mode of the model B meeting the x-direction and y-direction balance conditions;
and 8: multiplying the force density mode of the model B by an adjusting coefficient, and taking the obtained result as the force density for shape finding;
and step 9: the force density obtained in the step 8 is adopted, the external load 7 obtained in the step 6 is applied to the corresponding node 3 of the model B, the shape finding is carried out on the model B by a force density method with load, the coordinates of the lower chord node 3 which meets the balance condition and the vertical requirement of the stay bar 4 and is shown in the figures 5 and 9 are obtained, and the x and y coordinates of each node 3 automatically meet the balance in the shape finding process and only the z coordinate can be updated, so the method is called as a unidirectional force density method;
step 10: establishing a cable truss structure integral geometric model shown in figures 6 and 10 by using the coordinates of an upper chord node 1 in the model A and the coordinates of a lower chord node 3 in the shape-finding model B, and naming the model C;
step 11: checking whether the geometric dimension of the model C meets the requirements related to the building functions, if so, entering the subsequent step, if not, returning to the step 8, updating the force density modal adjustment coefficient, and reshaping until a cable truss structure with the geometric dimension meeting the requirements is obtained;
step 12: and (4) assembling the balance matrix of the model C and performing singular value decomposition to obtain the cable truss structure self-stress mode after shape finding is completed.
The above steps do not take into account the effect of the dead weight of the structure or other loads. After the shape finding is finished, the specification and the prestress of each component of the cable truss structure are determined through primary design, and a structural design model called model D is obtained. And then correcting the shape finding result according to the steps shown in figure 2, wherein the correction process comprises the following steps: extracting the prestress of each stay bar 4 of the model D; the z-direction freedom degrees of all the lower chord nodes 3 of the constraint model D are subjected to static calculation under the action of the self weight of the structure or other loads, and each constraint reaction force value is superposed to the corresponding prestress of the stay bar 4 to serve as an external load adopted for correction; applying an external load to the corresponding node 3 of model B; multiplying the force density mode of the model B by an adjusting coefficient, and taking the obtained result as the force density adopted by correction; carrying out shape finding on the model B by a unidirectional force density method with load; correcting the coordinates of the lower chord node 3 of the model D according to the shape finding result, if the geometric dimension of the corrected model D does not meet the requirement, updating the force density modal adjustment coefficient, and correcting again until the geometric dimension of the model D meets the requirement; repeating the process until the single correction amplitude is smaller than a preset limit value; and finally, calculating the corrected prestress of the lower chord 6 according to the length of each lower chord 6 of the model D and the corresponding force density value. The correction process only finely adjusts the shape and prestress of the lower chord 6 of the cable truss structure, does not change the geometry of the upper chord 2, and the stay 4 remains vertical. After correction, the geometrical configuration of the initial state and the zero state of the cable truss structure are basically consistent.
The specific meanings and the calculation method of the force density balance matrix and the force density mode in the step 7 are as follows:
establishing an x-direction and y-direction balance equation for the ith lower chord node of the model B:
wherein (x)i,yi) Is the x, y coordinate of the ith node 3, n is the number of the lower chord 6 connected to the ith node 3, Lk、fkAnd (x)k,yk) ( k 1,2, …, n) are the length, internal force and x, y coordinates of the other end point of the kth lower chord 6 connected to the ith node 3, respectively. Density of induced force qk＝fk/LkEquation (1) can be transformed into:
in the formula (2), (x)i,yi)、(xk,yk) Has been given according to the horizontal position of each node 3, and qkIs unknown, and therefore equation (2) can be considered as relating to qkThe system of equations of (1). The same equilibrium equation is established for all nodes 3 of the model B, and the equation is written into a matrix form after being assembled:
[Aq]{q}＝{0} (3)
matrix [ A ] in formula (3)q]The force density balance matrix called model B, { q } is the force density that satisfies the horizontal balance condition of each node 3 of model B;
will [ A ] beq]Singular value decomposition is carried out to obtain a group of vectors { sj(j ═ 1,2, …), satisfying:
[Aq]{sj}＝{0} (4)
then sjIt is a general solution of the homogeneous linear equation set of equation (3), called force density mode. From the properties of the common solution of the homogeneous system of linear equations, the vector sjThe linear combination of (c) } is still a solution of equation (3), so the force density modes can be linearly combined to obtain a force density that satisfies the equilibrium condition in the x-direction and the y-direction. Through reasonable structural arrangement, the model B can have a unique force density mode, and the force density can be obtained by multiplying the force density mode by an adjusting coefficient.
For the spoke type cable truss structure of the embodiment, the structure is projected to a horizontal plane, and the projection of each lower chord node 3 on the inner ring is connected with the inner ring projection 8 and a cable truss projection 9 on two sides, as shown in fig. 11. When each cable truss projection 9 is arranged along the direction of the angular bisector of the projection included angle of the corresponding inner ring 8, the model B has a unique force density mode, and the mode B is a sufficient condition for the model A to have a unique self-stress mode in the step 5.
The principle and implementation process of the unidirectional force density method in step 9 are as follows:
matched mouldEach lower chord node 3 of type B applies an external load 7 (with p) in the z-direction (i.e. strut direction)iExpressed), the balance equation for model B at the ith node 3 is:
wherein (x)i,yi,zi) Is the x, y, z coordinates of the ith node 3, n is the number of lower chords 6 connected to the ith node 3, Lk、fkAnd (x)k,yk,zk) ( k 1,2, …, n) are the length, internal force, and x, y, z coordinates of the other end point of the kth lower chord connected to the ith node 3, respectively. Density of induced force qk＝fk/LkEquation (5) can be written as:
as can be seen from equations (1) to (4), when the force density obtained in step 8 is used, the first two equations of equation (6) automatically hold, and only the third equation (i.e., the z-direction equilibrium equation) needs to be solved.
The model B is provided with B lower chord cables 6 and m nodes, wherein the number of the lower chord nodes 3 to be shaped and the number of the fixed outer ring support nodes 5 are m respectivelyfAnd mcIntroducing a topological matrix of b × m:
in [ C ]]Before arranging the lower chord node 3 at the outer ring support node 5, [ C ] can be arranged]Splitting into a free node topology matrix [ C ]f]And constraint node topology matrix [ Cc]I.e., [ C ]]＝[[Cf][Cc]](ii) a The z-balance equations are listed and grouped for all nodes 3 of model B, which can be written as:
[Cf]T[Q][Cf]{zf}+[Cf]T[Q][Cc]{zc}＝{p} (8)
wherein zfZ coordinate vector of node 3 to be solved, { z }cZ coordinate vector of outer ring support node 5, { p } vector of node outer load 7, [ Q ]]Is a force density diagonal matrix; solving the formula (8) to obtain the z coordinate of the free node:
{zf}＝([Cf]T[Q][Cf])-1({p}-[Cf]T[Q][Cc]{zc}) (9)
the solution means that only the z coordinate of the node 3 needs to be updated in the shape finding process of the model B, and the x and y coordinates are not changed, so that the support rod can be automatically kept vertical; the method realizes the shape finding only in the z direction, so the method is called a unidirectional force density method.
The above description is only for the purpose of illustrating the present invention, and it should be understood that the present invention is not limited to the above embodiments, and various modifications conforming to the spirit of the present invention are within the scope of the present invention.
Claims (1)
1. A method for forming a cable truss structure, the method comprising the steps of:
1) establishing cable truss structure upper chord or lower chord nodes conforming to the geometric shape of a given roof or curtain wall, and connecting the nodes according to the topological relation to obtain a geometric model of the upper chord cable or the lower chord cable;
2) shifting each upper chord node or each lower chord node downwards or upwards by any distance along the vertical direction to obtain the initial position of each lower chord node or each upper chord node, and connecting the corresponding upper chord node and the corresponding lower chord node to obtain an initial geometric model of the brace rod or the sling;
3) restraining the outer ring support joint and the lower or upper end point of the stay bar or sling to obtain a force finding model named as a model A;
4) additionally establishing a form finding model B: establishing an initial geometric model of the lower chord cable or the upper chord cable according to the initial position of the lower chord node or the upper chord node in the step 2), and constraining the outer ring support node;
5) assembling a balance matrix of the model A and performing singular value decomposition to obtain a self-stress mode of the model A;
6) linearly combining the self-stress modes of the model A, extracting corresponding numerical values of the stay bars or the suspension ropes in a combined result, and directly extracting the corresponding numerical values of the stay bars or the suspension ropes in the modes when the self-stress modes are unique, wherein an obtained result is used as an external load during shape finding;
7) establishing an x-direction and y-direction balance equation set of the model B by taking the force density as a variable, assembling a coefficient matrix of the equation set, namely a force density balance matrix, and performing singular value decomposition on the matrix to obtain a force density mode of the model B meeting the x-direction and y-direction balance conditions;
8) linearly combining the force density modes of the model B, and directly multiplying the force density modes by an adjusting coefficient when the force density modes are unique, wherein the obtained result is used as the force density for shape finding;
9) adopting the force density obtained in the step 8), applying the external load obtained in the step 6) to a corresponding node of the model B, and carrying out shape finding on the model B by a force density method with load to obtain coordinates of a lower chord or an upper chord node meeting balance conditions and the vertical requirement of a brace rod or a sling;
10) establishing an integral geometric model of the cable truss structure by using coordinates of upper chord or lower chord nodes in the model A and coordinates of lower chord or upper chord nodes in the shape-finding model B, and naming the model as a model C;
11) checking whether the geometric dimension of the model C meets the requirements related to the building functions, if so, entering the subsequent step, and if not, returning to the step 8), updating the force density modal combination coefficient or the adjustment coefficient, and reshaping until a cable truss structure with the geometric dimension meeting the requirements is obtained;
12) assembling a balance matrix of the model C and performing singular value decomposition to obtain a cable truss structure self-stress mode after shape finding is completed; then, correcting the shape finding result by considering the dead weight or other loads;
the specific meanings and the calculation method of the force density balance matrix and the force density mode in the step 7) are as follows:
an x-direction and y-direction equilibrium equation is established for the ith node of model B:
wherein (x)i,yi) Is the x, y coordinates of the ith node, n is the number of cells connected to the ith node, Lk、fkAnd (x)k,yk) (k ═ 1,2, …, n) are the length, internal force, and x, y coordinates of the other end point, respectively, of the kth element connected to the ith node; density of induced force qk＝fk/LkEquation (1) can be transformed into:
in the formula (2), (x)i,yi)、(xk,yk) Has been given according to the horizontal position of each node, and qkIs unknown, and therefore equation (2) can be considered as relating to qkThe system of equations (1); establishing the same balance equation for all nodes of the model B, and writing the equation into a matrix form after grouping:
[Aq]{q}＝{0} (3)
matrix [ A ] in formula (3)q]The force density balance matrix is called as a model B, and q is the force density which meets the horizontal balance condition of each node of the model B;
will [ A ] beq]Singular value decomposition is carried out to obtain a group of vectors { sj(j ═ 1,2, …), satisfying:
[Aq]{sj}＝{0} (4)
then sjThe solution is the general solution of the homogeneous linear equation set formula (3), called force density mode;
the principle and the implementation process of the unidirectional force density method in the step 9) are as follows:
applying an external load p in the z-direction to each node of model BiThen, the balance equation of model B at the ith node is:
wherein (x)i,yi,zi) Is the x, y, z coordinates of the ith node, n is the number of elements connected to the ith node, Lk、fkAnd (x)k,yk,zk) (k ═ 1,2, …, n) are the length, internal force, and x, y, z coordinates of the other end point, respectively, of the kth element connected to the ith node; density of induced force qk＝fk/LkEquation (5) can be written as:
as can be seen from the equations (1) to (4), when the force density obtained in the step 8) is adopted, the first two equations of the equation (6) are automatically established, and only the third equation needs to be solved;
the model B is provided with B units and m nodes, wherein the number of the nodes to be shaped and the fixed outer ring support nodes is m respectivelyfAnd mcIntroducing a topological matrix of b × m:
in [ C ]]Wherein the node to be shaped is arranged before the outer ring support node, [ C ] can be arranged]Splitting into a free node topology matrix [ C ]f]And constraint node topology matrix [ Cc]I.e. C [ [ C ]f][Cc]](ii) a The z-balance equations are listed and grouped for all nodes of model B, which can be written as:
[Cf]T[Q][Cf]{zf}+[Cf]T[Q][Cc]{zc}＝{p} (8)
wherein zfZ coordinate vector of node to be solved, { z }cZ coordinate vector of outer ring support node, { p } external load vector, [ Q ]]Is a force density diagonal matrix; solving the formula (8) to obtain the z coordinate of the free node:
{zf}＝([Cf]T[Q][Cf])-1({p}-[Cf]T[Q][Cc]{zc}) (9)
the solution means that only the z coordinate of the node needs to be updated in the shape finding process of the model B, and the x and y coordinates are not changed, so that the stay bar or the sling can be automatically kept vertical; the method realizes the shape finding only in the z direction, so the method is called as a unidirectional force density method;
the specific process of correcting the shape finding result by considering the dead weight or other loads in the step 12) is as follows: after the shape finding is finished, the specification and the prestress of each component of the cable truss structure are determined through primary design, and a structural design model called model D is obtained; extracting the prestress of each brace rod or sling of the model D; all the z-direction degrees of freedom of the nodes obtained by shape finding of the constraint model D are subjected to static calculation under the action of the self weight of the structure or other loads, and each constraint reaction value is superposed to the corresponding prestress of the stay bar or the sling to serve as an external load adopted for correction; applying an external load to a corresponding node of the model B; linearly combining the force density modes of the model B, and directly multiplying the force density modes by an adjustment coefficient when the force density modes are unique, wherein the obtained result is used as the force density for correction; carrying out shape finding on the model B by a unidirectional force density method with load; correcting the shape finding node coordinates of the model D according to the shape finding result, if the geometric dimension of the corrected model D does not meet the requirement, updating the force density modal combination coefficient or the adjustment coefficient, and correcting again until the geometric dimension of the model D meets the requirement; repeating the process until the single correction amplitude is smaller than a preset limit value; and finally, calculating the corrected prestress of the shape-finding part according to the cable length of the shape-finding part of the model D and the corresponding force density value.
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| CN108388698A (en) * | 2018-01-26 | 2018-08-10 | 东南大学 | A kind of improved force density equilibrium figure analysis method |
| CN108319800B (en) * | 2018-03-20 | 2021-10-26 | 中国建筑西南设计研究院有限公司 | Tension structure general body type design method based on functional target |
| CN112800521A (en) * | 2021-01-28 | 2021-05-14 | 北京市建筑设计研究院有限公司 | Low cable clamp sliding force shape finding method suitable for shape finding of cable net structure |
| CN112800520A (en) * | 2021-01-28 | 2021-05-14 | 北京市建筑设计研究院有限公司 | Shape finding method of cable net structure |
| CN114781142B (en) * | 2022-04-11 | 2023-03-10 | 北京市建筑设计研究院有限公司 | Morphological analysis method for single-ring cable wheel spoke type cable truss structure |
| CN114996793B (en) * | 2022-04-11 | 2022-11-18 | 北京市建筑设计研究院有限公司 | Suspension cable morphology analysis method |
| CN115270233B (en) * | 2022-05-16 | 2023-07-11 | 北京市建筑设计研究院有限公司 | Expanded generalized balance matrix singular value decomposition method for calculating integral prestress mode of cable structure |
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