CN112149265A - Construction method of initial tension state of cable net structure in nonlinear finite element analysis - Google Patents

Construction method of initial tension state of cable net structure in nonlinear finite element analysis Download PDF

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CN112149265A
CN112149265A CN202011084787.6A CN202011084787A CN112149265A CN 112149265 A CN112149265 A CN 112149265A CN 202011084787 A CN202011084787 A CN 202011084787A CN 112149265 A CN112149265 A CN 112149265A
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王威
吴松
唐国安
张美艳
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Abstract

The invention discloses a construction method of a cable net structure in an initial tension state in nonlinear finite element analysis, which comprises the following steps: s1, selecting parameters of the cable net and the supporting structure, and creating a model in finite element simulation software; s2, establishing an initial stress stiffness matrix of the cable unit according to the given pretension, equating the initial stress stiffness matrix with three one-way springs, and adding the initial stress stiffness matrix into a finite element model of the cable net structure according to format requirements; s3, calculating an influence matrix of temperature on tension by using a linear method; and S4, calculating the temperature of the cable net unit when the axial tension of the cable net is simulated by a cooling method under the given pretension, and finishing pretension construction by taking the temperature value as a known unit temperature condition.

Description

Construction method of initial tension state of cable net structure in nonlinear finite element analysis
Technical Field
The invention relates to a prestress structure in finite element calculation simulation of a cable net structure and the technical field of application thereof, in particular to a method for constructing an initial tension state of the cable net structure in nonlinear finite element analysis.
Background
The cable net structure is an important component of a tension structure system, has the remarkable advantages of light weight, high strength, large contraction ratio, various forms and the like, and can be applied to the fields of space deployable antennas and the like. The initial tension is a necessary condition for forming the cable net structure and an important parameter for improving the curved surface stability of the cable net structure. In the structural load working stage, both static behavior and dynamic behavior of the antenna show high nonlinear characteristics, however, as the antenna required by aerospace development tends to be large-sized, the structural full-size ground test is difficult to develop, and therefore finite element numerical analysis shows great advantages in the design stage.
In finite element analysis, the construction method of the pretension comprises an equivalent load method, a cooling method, an initial strain method and the like. For complex cable net structures, the equivalent load method is not suitable; the primary strain method is essentially the same as the cooling method, and the pretension is constructed by changing the original unit length of the cable net. The method of determining the cooling amount (or initial strain) may be classified into an iterative correction method and a direct method according to whether iteration is performed. The iterative correction method firstly takes a cooling amount (or initial strain) to calculate the pretension of the cable unit, corrects the cooling amount (or initial strain) according to the difference between the result and the design value, and then recalculates until the iteration converges. At this stage, this is a common calculation method, but requires iterative calculation, resulting in reduced efficiency. The direct method does not need iteration and has high efficiency, but the precision is not high when the direct method is applied to a complex nonlinear structure.
Therefore, there is a need for a method of constructing an initial tension state that can comprehensively consider the geometric nonlinearity of the cable net, the singularity of the stiffness matrix, the deformation of the frame, the pre-tension application accuracy, the calculation efficiency, and other factors when analyzing by the finite element numerical method.
Disclosure of Invention
The invention aims to provide a construction method of an initial tension state of a cable net structure in nonlinear finite element analysis, which integrates the advantages of an iterative correction method and a direct method, has simple and convenient calculation process and strong operability, can ensure the calculation precision and improve the efficiency, is not limited by configuration and can further carry out the calculation of nonlinear inherent frequency and response when being applied to a complex, multi-layer and elastic constrained cable net structure including a satellite deployable antenna.
In order to achieve the purpose, the invention is realized by the following technical scheme:
a method for constructing a cable net structure in an initial tension state in nonlinear finite element analysis is characterized by comprising the following steps:
s1, selecting parameters of the cable net and the support structure, and creating a model in finite element simulation software;
s2, establishing an initial stress stiffness matrix of the cable unit according to the given pretension, equating the initial stress stiffness matrix with three one-way springs, and adding the initial stress stiffness matrix into a finite element model of the cable net structure according to format requirements;
s3, calculating an influence matrix of temperature on tension by using a linear method;
and S4, calculating the temperature of the cable net unit when the axial tension of the cable net is simulated by a cooling method under the given pretension, and finishing pretension construction by taking the temperature value as a known unit temperature condition.
The step S2 includes:
establishing an initial stress stiffness matrix of each cable unit according to a given pretension N
Figure BDA0002719978100000022
Will be
Figure BDA0002719978100000021
Three one-way springs are equivalent and are superimposed on the finite element model of the cable mesh structure according to the format requirement.
The step S3 includes:
assume that the number of cable elements is 1 to ncApplying a unit negative temperature to the string element j, and calculating the value of 1, …, n for each string element icTaking the opposite number of the tension ofIs marked as cij(i＝1,…,nc) When the number j is from 1 to ncAfter traversing all the cable elements, the cable element is obtained as cijIs the influence matrix C of element, temperature and tension.
Compared with the prior art, the invention has the following advantages:
1. in calculating the tension c of the wire unitijIn the process, a large deformation rigidity matrix is omitted, and an initial stress rigidity matrix and a linear rigidity matrix which are dominant in a tangential rigidity matrix are reserved, so that the calculation accuracy is high.
2. Under the condition that the tension is known, the initial stress stiffness matrix is a constant matrix, each column of the calculated C matrix can be used as the same model and different working conditions for solving, and the tangential stiffness matrix does not need to be regenerated and decomposed, so that the method has high efficiency.
Drawings
FIG. 1 is a flow chart of a method of constructing a cable mesh structure in an initial tension state in a nonlinear finite element analysis in accordance with the present invention;
FIG. 2 is a schematic view of a one-way spring unit;
FIG. 3 is a schematic diagram of a cable net and structure combination structure illustrating an algorithm.
Detailed Description
The present invention will now be further described by way of the following detailed description of a preferred embodiment thereof, taken in conjunction with the accompanying drawings.
As shown in fig. 1, a method for constructing a cable net structure in an initial tension state in a nonlinear finite element analysis includes the following steps:
s1, selecting parameters of the cable net and the supporting structure, and creating a model in finite element simulation software;
s2, establishing an initial stress stiffness matrix of the cable unit according to the given pretension, equating the initial stress stiffness matrix with three one-way springs, and adding the initial stress stiffness matrix into a finite element model of the cable net structure according to format requirements;
s3, calculating an influence matrix of temperature on tension by using a linear method;
and S4, calculating the temperature of the cable net unit when the axial tension of the cable net is simulated by a cooling method under the given pretension, and finishing pretension construction by taking the temperature value as a known unit temperature condition.
The step S1 includes:
selecting material parameters, geometric parameters, a cable net topological structure, initial positions of cable net nodes and constraint conditions of the cable net and the supporting structure, and creating a model in finite element simulation software.
The step S2 includes:
establishing an initial stress stiffness matrix of each cable unit according to a given pretension N
Figure BDA0002719978100000031
Will be provided with
Figure BDA0002719978100000032
Three one-way springs are equivalent and are superimposed on the finite element model of the cable mesh structure according to the format requirement.
Specifically, when the geometric nonlinearity is considered, the equilibrium equation of the unit under the action of the equilibrium force system is
Figure BDA0002719978100000033
Wherein B isLIs a linear strain matrix independent of deformation; b isNIs related to the displacement q(e)The associated nonlinear strain matrix, D is the elastic matrix of the material,
Figure BDA0002719978100000034
is the initial stress vector of the unit cell,(e)is the strain vector of the cell, F(e)Is the equivalent nodal force.
To function Ψ(e)(q(e)) Find its dependent variable q(e)Gradient of (a) to obtain a tangential stiffness matrix of the cells of
Figure BDA0002719978100000035
Wherein
Figure BDA0002719978100000041
Is a cellular linear stiffness matrix;
Figure BDA0002719978100000042
is a unit nonlinear stiffness matrix;
Figure BDA0002719978100000043
is a unit initial stress rigidity matrix.
For a length of l(e)The 2-node cable unit of (2) is set to have node coordinates of sum (x)2 y2,z2) The node displacements after deformation are respectively (u)1,v1,w1) And (u)2,v2,w2). Then the nonlinear strain matrix of the cable elements is
Figure BDA0002719978100000044
For equation (6)
Figure BDA0002719978100000045
Calculating the variation of the node displacement component, and obtaining
Figure BDA0002719978100000046
Wherein the gradient of the nonlinear strain matrix is
Figure BDA0002719978100000047
Here I3Representing a 3 x 3 unit array.
Given the pretension N of the cable element(e)According to the formulas (5) and (8), the initial stress rigidity matrix of the pre-tension cable unit can be calculated
Figure BDA0002719978100000048
Consider three one-way springs as shown in figure 2. Connected at the spring ends are a pair of identical nodes A and B, with the left spring (a) providing horizontal x-direction stiffness with a coefficient of stiffness of
Figure BDA0002719978100000049
The right middle spring (b) provides stiffness in the vertical y direction with a stiffness coefficient of
Figure BDA00027199781000000410
The right spring (c) provides a stiffness in the vertical z-direction with a stiffness factor of
Figure BDA00027199781000000411
In the finite element method, the matrix of the element stiffness of the three springs is
Figure BDA0002719978100000051
Wherein B isx＝[1 0 0 -1 0 0],By＝[0 1 0 0 -1 0],Bz＝[0 0 1 0 0 -1]。
As can be seen from the equations (9) and (10), when taken
Figure BDA0002719978100000052
Then there is an equation
Figure BDA0002719978100000053
It shows that the initial stress stiffness matrix generated by the tension of the cable units can be equivalently simulated by using the springs only by superposing three units which have the same geometric parameters and material properties and different displacement directions on each original cable unit, thereby simplifyingAnd (5) modeling.
The step S3 includes:
assume that the number of the wire units is 1 to ncApplying a unit negative temperature to the string element j, and calculating the value of 1, …, n for each string element icThe tension of (2) is expressed as cij(i＝1,…,nc) When the number j is from 1 to ncAfter traversing all the flexible cable units, the total number c is obtainedijIs the influence matrix C of element, temperature and tension.
Specifically, the initial stress stiffness matrix KσAll the components are equivalent by using a one-way spring, and are superposed into a finite element model of the cable-net structure according to format requirements, so that the linear stiffness matrix of the cable-net structure after treatment has no singularity any more, is a constant matrix irrelevant to deformation or displacement, and can be calculated by using a linear method. When general finite element program is adopted for calculation, the serial numbers of the flexible cable units are assumed to be 1 to ncSequentially applying unit negative temperature to the flexible cable unit j, and calculating each flexible cable unit i to be 1, …, n through a linear algorithmcThe negative value of the tension of (c) is taken asij(i＝1,…,nc). When the number j is from 1 to ncAfter traversing all the flexible cable units, the total number c is obtainedijIs an element, temperature influence matrix C on tension, where C is a size nc×ncNon-singular order matrix of (a).
The step S4 includes:
using the formula T ═ C-1N, calculating the unit temperature of the cable net when the axial tensile stress of the cable net is simulated by a cooling method under the given pretension, and taking the temperature value as the load structure of the initial tension state when the calculation is finished under the known unit temperature load condition.
In particular, if the design value of the cable element tension is N(e)(e＝1,…,nc) Then the negative temperature T to be applied to each cable unit(e)(e＝1,…,nc) Should satisfy the relationship
T＝C-1N (11)
Wherein N and T are each independently of the otheriAnd TiThe aligned column vectors.
The invention realizes the rapid construction of the initial tension state in the nonlinear finite element analysis of the cable net structure, and the result shows that for the cable units with remarkable geometric nonlinear effect and dominant tension in the transverse rigidity, the initial displacement rigidity matrix caused by the change of configuration is omitted in the tangent rigidity matrix of the units, and the initial stress rigidity matrix generated by the tension is reserved, so that the initial tension of the flexible cable units can be effectively constructed, and the calculation of the initial tension belongs to the linear problem because the reserved initial stress rigidity matrix is a constant, thereby having high calculation efficiency. The initial tension obtained is compared by a temperature reduction method, and the tension of the flexible cable calculated by using the thermoelastic finite element can be highly consistent with a design value.
The specific implementation process comprises the following steps:
1. model definition
FIG. 3 is a schematic diagram of a cable net and structure combination structure for illustrating the algorithm. The numbers with #, which represent the node numbers of the finite element model of the composite, are the element numbers in the parentheses. The units (1) - (5) form a cable net by five flexible cables, the flexible cables are made of nylon ropes, the elastic modulus of the flexible cables is measured to be 1.5GPa, and the flexible cables are modeled by a rod unit, wherein the lengths L of the flexible cables (1) - (4) in the horizontal direction10.2m, tensile stiffness EA in cross section13000N, length L of the flexible cable (5) in the vertical direction20.1m, tensile stiffness in section EA2150N, the supporting springs (6) and (7) have a stiffness coefficient k of 105N/m of grounding spring. Considering the deformation in the plane, the degree of freedom of the set of cord-mesh structures is 6, and the corresponding displacement components are labeled in fig. 3.
The calculation example was calculated using the finite element analysis program NASTRAN, the model being defined as in table 1 under the file name demomodel. The data defining the model includes node coordinates, cell composition information, material constants, cell properties, and constraints. For ease of analysis, the coefficient of thermal expansion of the wire material was set to α -1. With this arrangement, the temperature load T applied to the wire unit(e)Is the elongation per unit length of the cell
Figure BDA0002719978100000061
As can be seen from fig. 3, the nodes 2 and 3 in the model have no stiffness in the longitudinal direction, and the overall stiffness matrix of the cable net and structure combination is singular.
TABLE 1 finite element model definition of Cable-Net structural example
Figure BDA0002719978100000071
2. Equivalence of initial stress stiffness matrix
Based on the node coordinates, constraints, and tension configuration requirements for the units (1) to (5) in Table 1, the initial stress stiffness matrix of the units (1) to (4) and the unit (4) can be calculated as
Figure BDA0002719978100000072
Figure BDA0002719978100000073
Wherein I2Is a second order unit matrix. The initial stress stiffness matrix is converted into equivalent spring units, and the file name is CELAS1.bdf, and the content is shown in Table 2.
TABLE 2 spring unit model equivalent to initial stress stiffness matrix
Figure BDA0002719978100000074
3. Temperature to tension influence matrix generation
To obtain the matrix C of the effect of temperature on tension, an input file for NASTRAN was created, the contents of which are shown in table 3. Table 3 is supplemented with row 17 include 'cell as1. bdf', the contents of which are spring elements equivalent to the initial stress stiffness matrix to be superimposed. The input file contains three calculation conditions, which respectively correspond to applying unit negative temperature on the flexible cable units (1), (2) and (5) and solving the tension of the 5 flexible cable units. Because the influence matrix and the structure have symmetry, the working condition of applying unit negative temperature on the units (3) and (4) can be replaced by the working condition 1 and the working condition 2. After NASTRAN operation, the product is obtained by finishing
Figure BDA0002719978100000081
Only variable names, elements without specific numbers in the matrix may be substituted with data for structure or matrix symmetry.
TABLE 3 NASTRAN input document for calculating temperature versus tension influence matrix
Figure BDA0002719978100000082
4. Calculation of cooling value and application of temperature
Substituting the matrix C given by the formula (12) into the equation (11) according to the design values 100N, 100N and 1N of the tension of the 5 flexible cables can calculate the temperature of each unit as
T(3)＝T(1)＝-3.58272×10-2℃
T(4)＝T(2)＝-3.58270×10-2℃
T(5)＝-2.62745×10-2℃
This set of temperature values was used as a known unit temperature load condition to build a NASTRAN input file as in table 4.
TABLE 4 NASTRAN input document for coupled deformation and tension calculations
Figure BDA0002719978100000083
Figure BDA0002719978100000091
5. Applying precision verification
The calculated flexible cable sheetThe element tension was completely consistent with the design value. The displacement component with respect to the deformation can be analytically estimated under the assumption of a small displacement. Referring to fig. 3, the lateral displacement of the node 3 is in the tension N of the wires (1) and (2)1By the amount of elongation of the supporting spring (6), i.e.
Figure BDA0002719978100000092
The contraction amounts of the flexible cables (1) and (2) are the same, and the sum of the contraction amounts is equal to u3And thus. The vertical displacement of the node 2 causes the tension N of the wires (1) and (2)1Change direction, its vertical component and the tension N of the flexible cable (5)2Is balanced, thereby obtaining
Figure BDA0002719978100000093
The comparison of the finite element calculation of the displacement component and the estimation result of the analytic method is shown in the table 5, and the difference between the two results is almost the same, which shows that the method has high calculation precision.
When the equivalent temperature of the initial tension is determined, the coupled deformation of the cable net-structure can also be calculated by nonlinear analysis as the known load problem. At this time, the horizontal direction flexible cable and the vertical direction flexible cable are respectively equivalent to the side length
Figure BDA0002719978100000094
And
Figure BDA0002719978100000095
the square section beam unit avoids the singularity of the cable net stiffness matrix under the zero tension state by using the weaker transverse stiffness of the slender beam, and realizes the initial iteration of nonlinear calculation. The NASTRAN solution sequence 106 is used for calculation, the obtained cable net tension in the horizontal direction is 100.0N and is completely consistent with the design value, the flexible cable tension in the vertical direction is 0.9744N, the error is about-2.56%, and the accuracy requirement of engineering calculation can be met. The results of the nonlinear calculation of the displacement are also shown in Table 5. Analytical methods are comparable to linear and nonlinear methods in terms of error in results.
TABLE 5 calculation results of example coupling deformation of cable-network structure
Figure BDA0002719978100000096
The analysis result of the example is integrated, and the invention has no configuration limitation on the elastic constrained cable net structure, and can effectively construct the initial tension of the flexible cable unit when the rigidity matrix is odd. And the configured equivalent temperature drop of the flexible cable and the subsequent coupling deformation and tension calculation are linear processes without iteration, and the calculation efficiency is high.
While the present invention has been described in detail with reference to the preferred embodiments, it should be understood that the above description should not be taken as limiting the invention. Various modifications and alterations to this invention will become apparent to those skilled in the art upon reading the foregoing description. Accordingly, the scope of the invention should be determined from the following claims.

Claims (2)

1. A method of constructing a cable net structure in an initial tension state in a nonlinear finite element analysis, comprising the steps of:
s1, selecting parameters of the cable net and the support structure, and creating a model in finite element simulation software;
s2, establishing an initial stress stiffness matrix of the cable unit according to the given pretension, equating the initial stress stiffness matrix with three one-way springs, and adding the initial stress stiffness matrix into a finite element model of the cable net structure according to format requirements;
s3, calculating an influence matrix of temperature on tension by using a linear method;
and S4, calculating the temperature of the cable net unit when the axial tension of the cable net is simulated by a cooling method under the given pretension, and finishing pretension construction by taking the temperature value as a known unit temperature condition.
2. The method of claim 1, wherein the step S2 includes:
establishing an initial stress stiffness matrix of each cable unit according to a given pretension N
Figure FDA0002719978090000011
Will be provided with
Figure FDA0002719978090000012
Three one-way springs are equivalent and are superimposed on the finite element model of the cable mesh structure according to the format requirement.
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