CN110110387B - Finite element shape finding analysis method for cable structure - Google Patents
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Abstract
The invention relates to a finite element shape finding analysis method of a cable structure, which comprises the following steps: establishing a broken line-shaped cable structure finite element model, applying concentrated load on a broken point of the finite element model, and solving an initial balance state solution of the cable structure under prestress and the concentrated load; continuously superposing the dead weight and/or the external load under the initial balance state solution, and solving a working state solution of the cable structure superposed with the dead weight and/or the external load; and eliminating the concentrated load, solving a final state solution, and determining the structural form of the cable. The method adopts the broken line modeling, so that the modeling of the cable structure is more convenient, the solution of the initial balance state solution of the cable structure under the action of concentrated load can be more efficient and convenient based on the initial balance state solution obtained by the finite element model of the broken line, and the initial balance solution obtained by the solution is transited to the working state solution, so that the structural deformation under the dead weight and/or external load can be more conveniently calculated.
Description
Technical Field
The invention relates to the technical field of cable structure shape finding, in particular to a cable structure finite element shape finding analysis method.
Background
Cables or cable nets play an important role in structural engineering, such as cableways, suspended cable structures, stayed cable structures, cable dome structures, cable truss structures and the like. The cable is a large-deformation flexible structure, the stress analysis of the cable belongs to the problem of geometric nonlinearity, the cable has no rigidity when no tension is applied, the cable shape is a geometrically variable system, and the initial geometric form of the cable is related to the distribution of prestress. Therefore, the most central problem in cable and cable network analysis is shape finding analysis. The existing commonly used shape-finding analysis methods are a force density method, a dynamic relaxation method, a node balance method based on finite element analysis, a support lifting method and the like.
The methods either have the complex construction of the original model of the cable network; or the steps are complicated, the calculated amount is large, and the calculation time is long; or the problems of numerical value convergence, insufficient calculation precision and the like exist; and repeated iterative computation is needed, which is not beneficial to engineering application and popularization.
Disclosure of Invention
The invention aims to overcome at least one defect (deficiency) of the prior art and provides a finite element shape finding analysis method for a cable structure, which can enable the model establishment of the cable structure to be more convenient and the solution of the initial equilibrium state solution of the cable structure to be more efficient and simple.
The technical scheme adopted by the invention is as follows:
a cable structure finite element form finding analysis method comprises the following steps:
s1, establishing a polygonal cable structure finite element model, applying concentrated load on a break point of the finite element model, and solving an initial balance state solution of a cable structure under prestress and the concentrated load;
s2, continuously superposing the dead weight and/or the external load under the initial balance state solution, and solving a working state solution of the cable structure superposed with the dead weight and/or the external load;
and S3, eliminating the concentrated load, solving a final state solution, and determining the structural form of the cable.
The zigzag finite element model is established without a large number of cable rod units, so that the finite element model is established more simply and conveniently, and the speed and the accuracy of model calculation analysis are improved. The geometrical shape of the suspension cable under concentrated load is used as the initial balance solution of the suspension cable structure, so that the solving process of the initial balance solution can be converged more quickly, and the solution of the initial balance state solution can be completed more easily. And (3) transitioning from the initial balance solution obtained by solving to a working state solution, so that the structural deformation under the dead weight and/or external load can be calculated more conveniently, and finally, the initially applied concentrated load is eliminated, and the obtained final state solution is the required cable structure form.
Further, in step S2, the solving of the working state solution of the cable structure after the dead weight and/or the external load are superimposed specifically includes: and solving the working state solution of the cable structure after the dead weight and/or the external load are/is superposed through iterative analysis.
The initial equilibrium state is transited to the working state through iterative analysis, and the iterative convergence speed is higher when the working state solution is solved.
Further, the finite element model has a cable length equal to the cable length of the cable structure under no load.
Further, in step S2, the solving of the working state solution of the cable structure after the superposition of the dead weight and/or the external load through the iterative analysis specifically includes:
s21, forming an initial rigidity matrix [ K ] according to the initial balance state solution T ] 0 Forming a load matrix [ R ] from the dead weight and/or external load]Solving the equation [ K ] T ] 0 { δ } = { R }, yielding a shift { δ = { R }, a shift of 1 } and internal forces { F 1 Let i =1;
s22, generating displacement [ delta ] in the finite element model i In the case of (i) }, form the stiffness matrix [ K } T ] i ；
S23, calculating new node force (F (delta) i )}＝[K T ] i {δ i }；
S24, calculating unbalanced force (delta P) i }＝{R}-{F(δ i )}；
S25, solving equation [ K ] T ] i {Δδ i }＝{ΔP i Get displacement increment { delta } of i Calculate the displacement { δ } i+1 }＝{Δδ i }+{δ i }；
And S26, judging whether convergence exists, if not, enabling i = i +1 and returning to the step S22, and if yes, ending.
Further, the step S26 of determining whether convergence occurs specifically includes: judging whether delta is satisfied i /δ 1 Epsilon is less than or equal to the set allowable value.
Further, in step S1, the break point of the finite element model is located on a node of the suspension cable.
Compared with the prior art, the invention has the beneficial effects that:
(1) The invention can adopt broken line modeling for the cable structure, is convenient to model, only needs to satisfy that the total length of the broken line is equal to the original length of the suspension cable, does not need to use a large number of cable rod units, and improves the speed and accuracy of model calculation analysis;
(2) Compared with other methods, the initial stable solution is easily calculated in finite element software based on the initial balance state solution under the action of concentrated load obtained by the finite element model of the broken line, and corresponding cable shapes can be obtained only by continuously performing cyclic calculation in the finite element for dozens of times or more in other methods such as dead weight load shape finding, but the method only needs to perform calculation in the finite element once, does not need cyclic calculation, and has good convergence, high calculation speed and high precision;
(3) And applying the dead weight and the external working load on the basis of the initial stable solution, and easily calculating in finite element software to obtain the working state solution after the dead weight and the external working load are applied.
Drawings
FIG. 1 is a schematic diagram of a finite element modeling according to an embodiment of the present invention.
FIG. 2 is a schematic diagram of an embodiment of the present invention.
FIG. 3 is a schematic view of another embodiment of the present invention
Detailed Description
The drawings are only for purposes of illustration and are not to be construed as limiting the invention. For a better understanding of the following embodiments, certain features of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
Examples
The embodiment provides a finite element shape finding analysis method for a cable structure, which comprises the following steps:
s1, establishing a polygonal cable structure finite element model, applying a plurality of concentrated loads on a break point of the finite element model, and solving an initial balance state solution of a cable structure under prestress and the concentrated loads;
s2, under the initial balance state solution, continuously superposing the dead weight and/or the external load, and solving a working state solution of the cable structure after superposing the dead weight and/or the external load;
and S3, eliminating the concentrated load, solving a final state solution, and determining the structural form of the cable.
The zigzag finite element model is established without a large number of cable rod units, so that the finite element model is established more simply and conveniently, and the speed and the accuracy of model calculation analysis are improved. Generally, the geometrical shape of the suspension cable under the dead weight and uniformly distributed along the span length is taken as the initial balance solution of the suspension cable structure, and the geometrical shape of the suspension cable under the concentrated load is taken as the initial balance solution of the suspension cable structure in the embodiment, so that the solving process of the initial balance solution can be converged more quickly, and the solving of the initial balance state solution can be completed more easily. And (3) transitioning from the initial balance solution obtained by solving to a working state solution, so that the structural deformation under the self weight and/or external load can be more conveniently calculated, and finally eliminating the initially applied concentrated load to obtain a final state solution which is the required cable structure form.
The method provided by the embodiment has self-adaptability, and the requirement of balancing the cable force and the loading force is not required to be met.
In the implementation process, the break point of the finite element model can be located at any position on the suspension cable of the cable structure, and the number of the break points can be multiple. The position of the concentrated load to be applied can be determined, and the position of the break point can be determined according to the deformation condition of the suspension cable subjected to the concentrated load in advance.
Preferably, the number of the break points is one, and the number of the concentrated loads is one, so that constraint conditions in finite element model analysis can be reduced, and convergence efficiency is improved.
Preferably, the value of the concentrated load is smaller than the value of the dead weight and/or the value of the external load. The smaller the value of the concentrated load is relative to the value of the dead weight and/or the external load, the faster the iterative convergence speed of solving the initial balance state solution is.
In this embodiment, in step S2, the solving of the working state solution of the cable structure after the dead weight and/or the external load are superimposed specifically includes: and solving the working state solution of the cable structure after the self weight and/or the external load are/is superposed through iterative analysis.
The initial equilibrium state is transited to the working state through iterative analysis, and the iterative convergence speed is higher when the working state solution is solved.
In this embodiment, the finite element model has a cable length equal to the cable length of the cable structure under no load.
And a constraint condition is added during the establishment of the zigzag finite element model, namely the cable length of the finite element model is equal to the cable length of the cable structure under no load, so that the finite element model is more practical.
As shown in FIG. 1, the end points O and B are known positions of the support, the folding point A is any position on the suspension cable, and a folding line-shaped cable structure finite element model can be established according to the positions of the end points O and B and the position of the folding point A. The length of the broken line OAB is equal to the length of the suspension cable in an unstressed state. And after the concentrated load F is applied to the break point A, solving the solution of the initial balance state of the cable structure under the prestress and the concentrated load F. To facilitate the finite element modeling and the analytical calculations, the coordinate system can be selected as OX 'Y'.
In this embodiment, in step S2, the step of superimposing a dead weight and/or an external load on the finite element model and performing an iterative analysis to solve the working state solution specifically includes:
s21, forming an initial rigidity matrix [ K ] according to the initial balance state solution T ] 0 Forming a load matrix [ R ] from the dead weight and/or external load]Solving the equation [ K ] T ] 0 { δ } = { R }, yielding a shift { δ = { R }, a shift of 1 And internal force { F } 1 Let i =1;
s22, generating displacement [ delta ] in the finite element model i In the case of { K } form a stiffness matrix T ] i ；
S23, calculating new node force (F (delta) i )}＝[K T ] i {δ i }；
S24, calculating unbalanced force (delta P) i }＝{R}-{F(δ i )}；
S25, solving equation [ K ] T ] i {Δδ i }＝{ΔP i Get displacement increment { delta } of i Calculate the displacement { δ } i+1 }＝{Δδ i }+{δ i }
And S26, judging whether convergence exists, if not, enabling i = i +1 and returning to the step S22, and if so, ending.
In this embodiment, the step S26 of determining whether to converge specifically includes: judging whether delta is satisfied i /δ 1 Epsilon is equal to or less thanAnd (6) allowing the value.
And adding the self-weight and/or external load to the cable structure at one time, calculating node displacement according to the initial rigidity matrix, and calculating the structural rigidity according to the deformed structure to obtain the cable end force. And in order to meet node balance, applying the unbalanced loads as node loads on the nodes, calculating the displacement of the nodes after deformation, and repeating the iteration process until the unbalanced loads are smaller than a set allowable value.
In this embodiment, in step S1, the break point of the finite element model may be selected at a node of the suspension cable.
As shown in figure 2, the structure of the two-end fixed cable for bearing a plurality of concentrated loads (unit: kN) is characterized in that the cross section area A =5.48 multiplied by 10 of the suspension cable -4 m 2 Elastic modulus E =1.54 × 10 of suspension cable material 5 MPa, the self weight of the suspension cable is 47.03N/m, and the original length S of the suspension cable in an unstressed state 0 =312.85m, the sag of the cable structure in the initial state is f c ＝35.25m。
In the document "static analysis method and precision research of cable structure" (highway and motor transport, 2008 (3): 148-152), a segmented catenary method is adopted to perform shape finding analysis on the cable structure shown in fig. 2, and the Y-direction coordinate values of each node are obtained through calculation, and the calculation results are counted in table 1.
The method provided by this embodiment is used to perform shape finding analysis on the cable structure shown in fig. 2, and the Y-coordinate values of the nodes are also calculated, and the calculation results are also counted in table 1.
TABLE 1 coordinate values (unit: m) of Y direction of each node of the cable structure
| Node numbering | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Catenary method of reference 2 | -9.615 | -18.538 | -26.848 | -34.612 | -30.145 | -25.211 | -19.779 | -13.805 | -7.235 |
| Examples provide methods | -9.441 | -18.427 | -26.961 | -35.045 | -30.399 | -25.275 | -19.675 | -13.598 | -7.042 |
| Error/%) | 1.8 | 0.6 | 0.4 | 1.3 | 0.8 | 0.3 | 0.5 | 1.5 | 2.6 |
As shown in table 1, the calculation results of the two methods are slightly different, which indicates that the calculation results of the method provided by the present embodiment are substantially accurate.
As shown in fig. 3, the cross-sectional area a =1.4645 × 10 of each cable segment is an orthogonal cable net structure -4 m 2 Elastic modulus E =8.2737 × 10 of suspension cable material 4 MPa, the original lengths of the rope segments with the numbers of 3, 4, 8 and 11 are 30.419m, and the original lengths of the rope segments with the numbers of 1, 2, 5, 6, 7, 9, 10 and 12 are 31.76m. Vertical load of 35.56kN acts on the nodes (4), (5), (8) and (9), and the self-weight load q =1.46 multiplied by 10 -3 kN/m。
In document 2, "Shape defining of incomplete Structures relating to the same slice and compressed elements" (Computers & Structures,2005,83 (21-22): 1767-1779), the Shape finding analysis is performed on the cable network structure shown in fig. 3 by the method of document 2, and coordinate values of nodes X, Y, and Z of node number 5 are calculated, and the calculation results are shown in table 2.
In document 3, "A current element for the analysis of cable Structures" (Computers & Structures,1981,14 (3): 325-333), the shape finding analysis is performed on the cable network structure shown in FIG. 3 by the method of document 3, and the coordinate values of nodes X, Y, and Z with the node number 5 are calculated, and the calculation results are also shown in Table 2.
In document 4, "partitioned initial-value analysis of cable nets" (International Journal of Solids & Structures,1973,9 (11): 1403-1420), the cable net structure shown in FIG. 3 was subjected to shape finding analysis by the method of document 4, and coordinate values of nodes X, Y, and Z in the node number 5 were calculated, and the calculation results were also shown in Table 2.
TABLE 2 coordinate values (unit: m) of the nodes 5 of the cable network structure in X, Y and Z directions
| Direction | X | Y | Z |
| Method of |
15.2805 | 15.2805 | -9.5945 |
| Method of |
15.2802 | 15.2802 | -9.5922 |
| Method of |
15.2804 | 15.2804 | -9.5920 |
| The method provided by the embodiment | 15.28039 | 15.28039 | -9.59287 |
As shown in table 2, the calculation results of the four methods are relatively small, further indicating that the calculation results of the method provided by the present embodiment are substantially accurate.
On the basis of meeting the shape finding accuracy, the calculation method of the initial balance state solution is optimized, the calculation efficiency of the initial balance state solution is improved, the finite element model of the cable structure is more convenient to establish, units adopted in the finite element model are simpler, and the calculation method is favorable for further calculating the structural deformation under the dead weight and/or external load.
It should be understood that the above-mentioned embodiments of the present invention are only examples for clearly illustrating the technical solutions of the present invention, and are not intended to limit the specific embodiments of the present invention. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention claims should be included in the protection scope of the present invention claims.
Claims (6)
1. A cable structure finite element form finding analysis method is characterized by comprising the following steps:
s1, establishing a polygonal cable structure finite element model, applying a concentrated load on a break point of the finite element model, and solving an initial balance state solution of a cable structure under prestress and the concentrated load;
s2, continuously superposing the dead weight and/or the external load under the initial balance state solution, and solving a working state solution of the cable structure superposed with the dead weight and/or the external load;
and S3, eliminating the concentrated load, solving a final state solution, and determining the structural form of the cable.
2. A finite element form finding analysis method of a cable structure according to claim 1, wherein in the step S2, the solving of the working state solution of the cable structure after the dead weight and/or the external load are/is superimposed specifically includes: and solving the working state solution of the cable structure after the dead weight and/or the external load are/is superposed through iterative analysis.
3. A finite element form finding analysis method for cable structure as claimed in claim 2, wherein the cable length of the finite element model is equal to the cable length of the cable structure under no load.
4. A finite element form finding analysis method of a cable structure according to claim 2 or 3, wherein in the step S2, the solving of the working state solution of the cable structure after the cable structure is superposed with the dead weight and/or the external load through the iterative analysis specifically comprises:
s21, forming an initial rigidity matrix [ K ] according to the initial balance state solution T ] 0 Forming a load matrix [ R ] from the dead weight and/or external load]Solving the equation [ K ] T ] 0 { δ } = { R }, yielding a shift { δ = { R }, a shift of 1 And internal force { F } 1 Let i =1;
s22, generating displacement [ delta ] in the finite element model i In the case of { K } form a stiffness matrix T ] i ；
S23, calculating new node force (F (delta) i )}＝[K T ] i {δ i }；
S24, calculating unbalanced force [ delta P ] i }＝{R}-{F(δ i )}；
S25, solving equation [ K ] T ] i {Δδ i }＝{ΔP i Get the displacement increment [ Delta ] delta i Calculate the displacement { δ } i+1 }＝{Δδ i }+{δ i }
And S26, judging whether convergence exists, if not, enabling i = i +1 and returning to the step S22, and if so, ending.
5. A finite element form finding analysis method of a cable structure as claimed in claim 4, wherein the step S26 of determining whether to converge specifically comprises: judging whether the requirements are metΔδ i /δ 1 ε is a set tolerance value.
6. A finite element form finding analysis method for cable structure as claimed in claim 1, wherein in step S1, the break point of the finite element model is located at the node of the suspension cable.
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| WO2013149592A1 (en) * | 2012-04-04 | 2013-10-10 | 中国航空规划建设发展有限公司 | Cable dome prestressing force determination method based on overall analysis of tension and bearing |
| CN106096257A (en) * | 2016-06-06 | 2016-11-09 | 武汉理工大学 | A kind of non-linear cable elements analyzes method and system |
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| CN101582095A (en) * | 2009-06-19 | 2009-11-18 | 东南大学 | Non-linear dynamic finite element method for determining cable-strut system static balancing state |
| WO2013149592A1 (en) * | 2012-04-04 | 2013-10-10 | 中国航空规划建设发展有限公司 | Cable dome prestressing force determination method based on overall analysis of tension and bearing |
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