CN108984841A - Strength check under the calculating of composite laminated plate concrete moduli and specified load - Google Patents
Strength check under the calculating of composite laminated plate concrete moduli and specified load Download PDFInfo
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Abstract
Strength check the present invention relates to composite Materials Design field, under specifically a kind of composite laminated plate concrete moduli calculating and specified load.Include MIM message input module, computing module and message output module.Input material performance information, laying information and load data information in MIM message input module, calculation procedure can calculate three stiffness matrix A, D, B of composite laminated plate according to the input data, and calculate laminate concrete moduli and corresponding effective stiffness matrix of the pure loads in plane effect down and under simple bending torsion effect with D matrix according to A matrix.Whole strain and the deflection rate of laminate is calculated according to laminate stiffness matrix and load information simultaneously, it may further obtain the ess-strain information of each single layer, by using three kinds of Failure Analysis of Composite Materials criterion, to check whether the laminate meets intensity requirement under ordinance load effect.The present invention greatlys improve calculating speed, and can directly and quickly obtain ess-strain exact numerical solution for the laminated plate analysis in the case of simple stress, improves design efficiency.
Description
Technical field
The present invention relates to the field of structural design of composite material, specifically a kind of calculating composite laminated plate etc.
Imitate modulus and the strength check program under specified load effect.
Background technique
It is well known that in COMPOSITE FINITE ELEMENT analytic process, due to the complexity and material of composite structure
Anisotropy, when definition material attribute, generally require to carry out cumbersome laying information input, especially for more laying
Or the composite product of more complex structure needs to take a substantial amount of time cost for assigning material category when being analyzed
Property.Due to symmetrical equilibrium laminated composite laminate do not have stretch bending, tension-torsion coupling, and draw cut the coefficient of coup again it is smaller,
Simplify material properties by equivalent modulus method and assign process, problem is turned round for plane stress problem or simple bending and had both been saved
Time cost improves design efficiency, and can guarantee higher computational accuracy.
The laminate progress intensive analysis that tension and compression shearing action in bearing plane or bending act on generally is used at present limited
Element method is realized.It can be obtained more compared with true stress in load application position when calculate analysis using finite element software
Big concentrated stress, this is often that designer is not conceivable.This concentrated stress makes Finite element analysis results can not be just
Really intuitively reflect the stress condition of laminate, and according to Saint Venant's principle, which is local effect, for whole knot
Structure stress does not influence.And it can achieve when calculating the laminate internal force under this kind of stress using numerical solution very high
Computational accuracy, and it is few to input information, and computational efficiency is high, and calculated result is able to reflect the stress of laminate entirety, therefore
For this kind of problem, numerical solution is better than finite element analysis.
Summary of the invention
It is composite laminated the object of the invention is in order to overcome the above-mentioned shortcomings of the prior art, provide a kind of calculating
Plate concrete moduli and the strength check program under specified load effect, can calculate the equivalent of symmetrical equilibrium laying laminate
Modulus and effective stiffness matrix simplify finite element analysis process, reduce complicated degree of analysis, improve analysis efficiency, and be directed to layer
Plywood bears loads in plane or the effect of bending load carries out efficient numerical solution ess-strain, quasi- using three kinds of common failures
The intensity value of laminate is then checked, composite structure design is carried out for designer and design basis and foundation is provided.
The present invention solve above-mentioned technical problem the technical solution adopted is that: a kind of composite laminated plate concrete moduli meter
Strength check under calculation and specified load includes MIM message input module, computing module and message output module.Input data is simultaneously
After operation, calculation procedure can first calculate every single layer stiffness matrix according to the data inputted in MIM message input module, further according to paving
Layer information is integrated into laminate stiffness matrix.Pure loads in plane effect and simple bending can be calculated by laminate stiffness matrix
The laminate concrete moduli and its effective stiffness matrix under load effect are turned round, and obtains bulk strain in conjunction with load information, into one
The ess-strain that step combines laying information to obtain every single layer is horizontal, according to maximum stress criterion, Tsai-Hill failure criteria and
Tsai-Wu failure criteria criterion judges whether the laminate fails under the effect of specified load.
The concrete moduli and strength check program, the information that MIM message input module includes are divided into material
Performance information, laying information and load data information three parts: material property information includes that content has: the axial mode of each single layer
Measure E1, transverse modulus E2, Poisson's ratio μ12, shear modulus G12, the axial tensile strength X of each single layert, transverse tensile strength Yt, axis
To compressive strength Xc, transverse compression intensity Yc, shear strength S；Laying information includes that content has: the laying order of each single layer
Ord, thickness t, laying angle, θ；Load data information includes that content has: axial push-pull load Nd x, lateral Tensile or Compressive Loading Nd y, face
Interior shear-type load Nd xy, axial bending load Md x, deflection by lateral force Md y, torsional load Md xy(the above load is laminate
Internal force and interior moment on cross section on unit width (or length)).
The concrete moduli and strength check program, the Computing Principle of computing module are as follows:
Step 1: information E according to the input data1、E2、μ12、G12, each list is calculated using formula (2-1)~(2-3)
The stiffness matrix Q of layer.
Wherein,
Formula (2-1) is write as to the relational expression that stress is indicated with strain:
Wherein Q is two-dimentional stiffness matrix, is inverted by two-dimentional flexibility matrix S.
Step 2: the single layer stiffness matrix Q being calculated is led to by integrating laying angle, θ, sequence ord and thickness t information
Cross laminate is calculated in formula (2-4)~(2-9) Bulk stiffness matrix A, D, B.
Strain-stress relation of the single layer in whole coordinate x-y is as follows when wing flapping is θ:
Wherein, T is coordinate conversion matrix.
Consider that laminate is formed by stacking by multilayer lamina, then the strain-stress relation of kth layer single layer are as follows:
Have for laminate
In formula, Kx、KyDeflection rate, K are bent for face in platexyFor face twisting coefficient, ε in plate0 x、ε0 x、 γ0 xyFor strain in middle plane.
If Nx、Ny、Nxy、Mx、Mx、MxyFor the internal force of unit width (or length) on laminate cross section and interior moment, then
The stress of itself and laminate should meet formula (1-9):
Therefore, the transformation of the internal force of laminate, interior moment and strain is
It is abbreviated as
Then obtain laminate stiffness matrix A, D, B.
Step 3: stretch bending is not present for symmetrical equilibrium laminated composite laminate according to A, D, B matrix of acquisition
Coupling, therefore B matrix is empty matrix；Due to A16、A26、 D16、D26In there are it is positive and negative alternately item, therefore its numeric ratio other just
It is much smaller to spend coefficient, when calculating can be made to simplify.
A. under pure loads in plane operative condition, without bending load M, therefore laminate internal force is with strain stress relation
N=A ε0 (2-10)
To be computed the concrete moduli parameter E of available laminate by A matrix1、E2、 μ12、G12And equivalent stiffness
Matrix Q.
B. simple bending is turned round under load operative condition, there is no loads in plane N, therefore moment and strain stress relation in laminate
For
M=DK (2-11)
To be computed the concrete moduli parameter E of available laminate by D matrix1W、E2W、 μ12W、G12WAnd it is equivalent
Stiffness matrix QW。
Step 4: according to A, D, B matrix of acquisition, and specified external load Nd x、 Nd y、Nd xy、Md x、Md y、Md xy, lead to
It crosses formula (2-9) and counter can release internal force, the flexibility relational expression that interior moment indicates strain and curvature are as follows:
Therefore the components of strain of laminate entirety can be calculated, it then can be in the hope of different lists using formula (2-6)
The components of strain ε of layerx、εx、γxy, then by formula (2-4) and the available each laying of stress transmission formula in whole coordinate
Stress level σ under x-y and principal direction coordinate 1-2x、σy、τxy、σ1、σ2、τ12。
Step 5: according to the stress level data σ in the principal direction being calculated1、σ2、τ12, it is respectively adopted following three kinds
The strength check of intensity judgment criterion progress composite laminated plate:
A. maximum stress criterion:
In formula, Xt、YtFor tensile strength axially and transversely, Xc、YcFor compressive strength axially and transversely, S is shearing
Intensity.
B.Tsai-Hill failure criteria:
In formula, X, Y respectively correspond axially and transversely intensity (stretch with the judgement of compression depend on stress be positive value or
Negative value), S is shear strength.
C.Tsai-Wu failure criteria:
In formula
F12It takes
Judge whether each single layer fails by each failure criteria, to obtain the composite material under specified load effect
Whether laminate meets intensity requirement.
The packet of the concrete moduli and strength check program, message output module output contains:
A. concrete moduli E of the laminate under the effect of pure loads in plane1、E2、μ12、G12And simple bending is turned round under load effect
Concrete moduli E1W、E2W、μ12W、G12WAnd stiffness matrix Q, Q under two kinds of load conditionsW。
B. each laminated material principal stress S under specified load effect11(i.e. σ1)、S22(i.e. σ2)、 S12(i.e. τ12)；
C. the strength check based on three kinds of Failure Analysis of Composite Materials criterion is as a result, be based on maximum stress criterion, Tsai-
Hill criterion, the strength check result of Tsai-Wu criterion.
The invention has the advantages that having used three kinds of composite material strength failure criterias to judge whether laminate is full
Sufficient strength condition, and the program and composite laminated plate concrete moduli calculation procedure are systematically combined in one.The present invention
The calculation method of composite laminated plate concrete moduli and the strength check methods of laminate are incorporated, and use computer
Language has worked out the complete calculation procedure comprising input, operation and output three parts, and user only need to be in input module
Corresponding position keys in material parameter, laying information and load information, can rapidly obtain the equivalent mould of composite laminated plate
Amount data and strength check are built as a result, eliminating and carrying out modeling rendering, grid dividing, analysis step in finite element analysis software
The complex processes such as vertical, boundary condition setting, and there is higher precision by the calculated result that numerical solution obtains.
Specific embodiment
The present invention will be further described with reference to the examples below:
Embodiment 1: including MIM message input module, computing module and message output module.After input data and operation, meter
Every single layer stiffness matrix can first be calculated according to the data inputted in MIM message input module by calculating program, whole further according to laying information
Synthesize laminate stiffness matrix.Pure loads in plane effect can be calculated by laminate stiffness matrix and simple bending is turned round load and made
Laminate concrete moduli and its effective stiffness matrix under, and bulk strain is obtained in conjunction with load information, further combined with
The ess-strain that laying information obtains every single layer is horizontal, according to maximum stress criterion, Tsai-Hill failure criteria and Tsai-Wu
Failure criteria criterion judges whether the laminate fails under the effect of specified load.
The concrete moduli and strength check program, the information that MIM message input module includes are divided into material
Performance information, laying information and load data information three parts: material property information includes that content has: the axial mode of each single layer
Measure E1, transverse modulus E2, Poisson's ratio μ12, shear modulus G12, the axial tensile strength X of each single layert, transverse tensile strength Yt, axis
To compressive strength Xc, transverse compression intensity Yc, shear strength S；Laying information includes that content has: the laying order of each single layer
Ord, thickness t, laying angle, θ, ply stacking-sequence information are ranked up according to the successive of input from bottom to top, i.e. the first row laying
Information corresponds to laminate bottommost first layer, and the second row corresponds to the second layer, and so on, last line corresponds to top layer；It carries
Lotus data information includes that content has: axial push-pull load Nd x, lateral Tensile or Compressive Loading Nd y, inplane shear load Nd xy, axial curved
Qu Zaihe Md x, deflection by lateral force Md y, torsional load Md xy(the above load is that unit width on laminate cross section is (or long
Degree) on internal force and interior moment).
The concrete moduli and strength check program, input module after accomplishing the setting up, execute operation program, according to
It is secondary sequentially carry out automatically each single layer stiffness matrix, laminate Bulk stiffness matrix, concrete moduli, whole strain and stress distribution,
The calculating of strength check, by the result being calculated output in unified window.
The concrete moduli and strength check program, the Computing Principle of computing module are as follows:
Step 1: information E according to the input data1、E2、μ12、G12, each list is calculated using formula (2-1)~(2-3)
The stiffness matrix Q of layer.
Wherein,
Formula (2-1) is write as to the relational expression that stress is indicated with strain:
Wherein Q is two-dimentional stiffness matrix, is inverted by two-dimentional flexibility matrix S.
Step 2: the single layer stiffness matrix Q being calculated is led to by integrating laying angle, θ, sequence ord and thickness t information
Cross laminate is calculated in formula (2-4)~(2-9) Bulk stiffness matrix A, D, B.
Strain-stress relation of the single layer in whole coordinate x-y is as follows when wing flapping is θ:
Wherein, T is coordinate conversion matrix.
Consider that laminate is formed by stacking by multilayer lamina, then the strain-stress relation of kth layer single layer are as follows:
Have for laminate
In formula, Kx、KyDeflection rate, K are bent for face in platexyFor face twisting coefficient, ε in plate0 x、ε0 x、 γ0 xyFor strain in middle plane.
If Nx、Ny、Nxy、Mx、Mx、MxyFor the internal force of unit width (or length) on laminate cross section and interior moment, then
The stress of itself and laminate should meet formula (1-9):
Therefore, the transformation of the internal force of laminate, interior moment and strain is
It is abbreviated as
Then obtain laminate stiffness matrix A, D, B.
Step 3: stretch bending is not present for symmetrical equilibrium laminated composite laminate according to A, D, B matrix of acquisition
Coupling, therefore B matrix is empty matrix；Due to A16、A26、D16、 D26In there are it is positive and negative alternately item, therefore its numeric ratio other just
It is much smaller to spend coefficient, when calculating can be made to simplify.
A. under pure loads in plane operative condition, without bending load M, therefore laminate internal force is with strain stress relation
N=A ε0 (2-26)
To be computed the concrete moduli parameter E of available laminate by A matrix1、E2、 μ12、G12And equivalent stiffness
Matrix Q.
B. simple bending is turned round under load operative condition, there is no loads in plane N, therefore moment and strain stress relation in laminate
For
M=DK (2-27)
To be computed the concrete moduli parameter E of available laminate by D matrix1W、E2W、 μ12W、G12WAnd it is equivalent
Stiffness matrix QW。
Step 4: according to A, D, B matrix of acquisition, and specified external load Nd x、 Nd y、Nd xy、Md x、Md y、Md xy, lead to
It crosses formula (2-9) and counter can release internal force, the flexibility relational expression that interior moment indicates strain and curvature are as follows:
Therefore the components of strain of laminate entirety can be calculated, it then can be in the hope of different lists using formula (2-6)
The components of strain ε of layerx、εx、γxy, then by formula (2-4) and the available each laying of stress transmission formula in whole coordinate
Stress level σ under x-y and principal direction coordinate 1-2x、σy、τxy、σ1、σ2、τ12。
Step 5: according to the stress level data σ in the principal direction being calculated1、σ2、τ12, it is respectively adopted following three kinds
The strength check of intensity judgment criterion progress composite laminated plate:
A. maximum stress criterion:
In formula, Xt、YtFor tensile strength axially and transversely, Xc、YcFor compressive strength axially and transversely, S is shearing
Intensity.
B.Tsai-Hill failure criteria:
In formula, X, Y respectively correspond axially and transversely intensity (stretch with the judgement of compression depend on stress be positive value or
Negative value), S is shear strength.
C.Tsai-Wu failure criteria:
In formula
F12It takes
Judge whether each single layer fails by each failure criteria, to obtain the composite material under specified load effect
Whether laminate meets intensity requirement.
The packet of the concrete moduli and strength check program, message output module output contains:
A. concrete moduli E of the laminate under the effect of pure loads in plane1、E2、μ12、G12Or simple bending is turned round under load effect
Concrete moduli E1W、E2W、μ12W、G12WAnd stiffness matrix Q or Q under two kinds of load conditionsW。
B. each laminated material principal stress S under specified load effect11(i.e. σ1)、S22(i.e. σ2)、 S12(i.e. τ12)；
C. the strength check based on three kinds of Failure Analysis of Composite Materials criterion is as a result, be based on maximum stress criterion, Tsai-
Hill criterion, the strength check result of Tsai-Wu criterion.
Embodiment 2: on the basis of described in embodiment 1, in order to meet the case where part requires the rigidity of structure, increase
Add export structure rigidity data, including:
A. tensible rigidity EA:
Wherein, wherein EA is tensible rigidity, and E is elasticity modulus, and A is area of section；F is pulling force, and l is rod length, Δ
The elongation of L rod piece.
B. bending stiffness EI:
Wherein EI is bending stiffness, and E is elasticity modulus, and I is the moment of inertia of the section about rotation axis；F is cantilever rod end collection
Middle power, l are rod length, and ω is rod end amount of deflection.
C. torsion stiffness GIp:
Wherein GIpFor torsion stiffness, G is modulus of shearing, IpFor polar moment of inertia；T is torque, and l is rod length,For phase
To torsion angle.
It should be noted that above-mentioned three kinds of rigidities of structure represent be simple unit area laminate Deformation Resistance,
The cross sectional shape of actual products is not considered.In actual use, especially torsion stiffness generally requires to consider really to be product
Cross section structure, therefore the rigidity of structure can be converted as needed when in use.
Embodiment 3: on the basis of as described in example 2, professional computing staff checks calculating details for convenience, increases
A, D, B matrix information are exported, the ess-strain information σ under each laying x-y coordinate and under 1-2 coordinatex、σy、τxy、σ1、σ2、τ12。
But it should be noted that due to data type in calculating process, 4 only retained after decimal point for short type,
Therefore data deviation can be generated for calculated result accordingly, data 0 are often calculated as 10-10Magnitude number below.
Embodiment 4: on the basis of described in embodiment 3, in order to more intuitively show laminate stress in thickness direction
Distribution situation increases output stress distribution information, including axial stress, lateral stress and shear stress.
Embodiment 5: on the basis of as described in example 4, to ignored event A in stiffness matrix A and D16、A26、D16、
D26With A11、D11Between carry out size comparison, i.e. evaluationAnd percentage numeral system is converted to, pass through knot
Fruit examines or check A16、A26、D16、 D26Ignored reasonability.
Claims (8)
1. the strength check under a kind of composite laminated plate concrete moduli calculating and specified load, it is characterised in that: include letter
Cease input module, computing module and message output module；After input data and operation, calculation procedure can be according to MIM message input module
The data of middle input first calculate every single layer stiffness matrix, are integrated into laminate stiffness matrix further according to laying information；By laminated
Pure loads in plane effect can be calculated in panel stiffness matrix and simple bending is turned round the laminate concrete moduli under load effect and its waited
Stiffness matrix is imitated, and obtains bulk strain in conjunction with load information, obtains the ess-strain of every single layer further combined with laying information
Level judges that the laminate is referring to according to maximum stress criterion, Tsai-Hill failure criteria and Tsai-Wu failure criteria criterion
Determine whether to fail under load effect.
2. according to claim 1 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
The information that sign is that the MIM message input module includes is divided into material property information, laying information and load data information three
Part；Wherein material property information includes that content has: the axial modulus E of each single layer1, transverse modulus E2, Poisson's ratio μ12, shearing mould
Measure G12, the axial tensile strength X of each single layert, transverse tensile strength Yt, axial compression strength Xc, transverse compression intensity Yc, shearing
Intensity S, laying information include that content has: laying order ord, thickness t, the laying angle, θ of each single layer, load data information include
Inside have: axial push-pull load Nd x, lateral Tensile or Compressive Loading Nd y, inplane shear load Nd xy, axial bending load Md x, transverse curvature
Load Md y, torsional load Md xy；The above load is unit width or internal force and interior moment in length on laminate cross section.
3. according to claim 1 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
Sign is that the stiffness matrix Q of each single layer is calculated using formula (1-1)~(1-3) according to the data information of the input；
Wherein,
Formula (1-1) is write as to the relational expression that stress is indicated with strain:
Wherein Q is two-dimentional stiffness matrix, is inverted by two-dimentional flexibility matrix S.
4. according to claim 3 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
Sign is to be calculated the single layer stiffness matrix Q by integrating laying angle, θ, sequence ord and thickness t information, passes through public affairs
Bulk stiffness matrix A, D, B of laminate is calculated in formula (1-4)~(1-9)；
Strain-stress relation of the single layer in whole coordinate x-y is as follows when wing flapping is θ:
Wherein, T is coordinate conversion matrix；
Consider that laminate is formed by stacking by multilayer lamina, then the strain-stress relation of kth layer single layer are as follows:
Have for laminate
In formula, Kx、KyDeflection rate, K are bent for face in platexyFor face twisting coefficient, ε in plate0 x、ε0 x、γ0 xyFor strain in middle plane；
If Nx、Ny、Nxy、Mx、Mx、MxyFor the internal force of unit width (or length) on laminate cross section and interior moment, then itself and layer
The stress of plywood should meet formula (1-7):
Therefore, the relationship of the internal force of laminate, interior moment and strain can be converted into
It is abbreviated as
Then obtain laminate stiffness matrix A, D, B.
5. according to claim 4 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
Sign is A, D, B matrix according to obtaining, and for symmetrical equilibrium laminated composite laminate, coupling between bending and extension is not present,
Therefore B matrix is empty matrix；Due to A16、A26、D16、D26There are positive and negative alternately items, therefore its numeric ratio other stiffness coefficients in
Much smaller, when calculating, can be made to simplify；
A is under pure loads in plane operative condition, without bending load M, therefore laminate internal force is with strain stress relation
N=A ε0 (1-10)
To be computed the concrete moduli parameter E of available laminate by A matrix1、E2、μ12、G12And effective stiffness matrix
Q；
B turns round under load operative condition simple bending, does not have loads in plane N, therefore moment with strain stress relation is in laminate
M=DK (1-11)
To be computed the concrete moduli parameter E of available laminate by D matrix1W、E2W、μ12W、G12WAnd equivalent stiffness square
Battle array QW。
6. according to claim 4 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
Sign is A, D, B matrix and defined external load N according to obtainingd x、Nd y、Nd xy、Md x、Md y、Md xy, pass through formula
(1-9) counter can release internal force, interior moment indicates the flexibility relational expression of strain and curvature are as follows:
Therefore the components of strain of laminate entirety can be calculated, it then can be in the hope of different single layers using formula (1-6)
Components of strain εx、εx、γxy, then by formula (1-4) and the available each laying of stress transmission formula in whole coordinate x-y and
Stress level σ under principal direction coordinate 1-2x、σy、τxy、σ1、σ2、τ12。
7. according to claim 6 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
Sign is according to the stress level data σ being calculated in the principal direction1、σ2、τ12, following three kinds of intensity is respectively adopted and sentences
Disconnected criterion carries out the strength check of composite laminated plate:
A maximum stress criterion:
In formula, Xt、YtFor tensile strength axially and transversely, Xc、YcFor compressive strength axially and transversely, S is shear strength；
B Tsai-Hill failure criteria:
In formula, X, Y respectively correspond axially and transversely intensity (stretching and depending on stress with the judgement of compression is stretching or compression),
S is shear strength；
C Tsai-Wu failure criteria:
In formula
Judge whether each single layer fails by each failure criteria, to obtain the composite laminated plate under specified load effect
Whether intensity requirement is met.
8. according to claim 1 composite laminated plate concrete moduli calculate and specified load under strength check, it is special
Sign is that the packet of the message output module output contains:
Concrete moduli E of a laminate under the effect of pure loads in plane1、E2、μ12、G12And simple bending turns round the equivalent mould under load effect
Measure E1W、E2W、μ12W、G12WAnd effective stiffness matrix Q, Q under two kinds of load conditionsW；
Stiffness matrix, B coupling between bending and extension stiffness matrix and D bending stiffness matrix are cut in the A drawing of b laminate；
Each laminated material principal stress S of the c under specified load effect11(i.e. σ1)、S22(i.e. σ2)、S12(i.e. τ12)；
Strength check of the d based on three kinds of Failure Analysis of Composite Materials criterion as a result, i.e. based on maximum stress criterion, Tsai-Hill criterion,
The strength check result of Tsai-Wu criterion.
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| CN111125615A (en) * | 2020-01-09 | 2020-05-08 | 暨南大学 | Method for calculating residual bearing strength of laminated structure of composite material containing delamination |
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| CN111539107B (en) * | 2020-04-23 | 2023-05-09 | 北京玻钢院复合材料有限公司 | Rapid prediction method for critical axial pressure load of grid reinforced composite cylindrical shell |
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| CN112800538A (en) * | 2021-01-15 | 2021-05-14 | 中国商用飞机有限责任公司北京民用飞机技术研究中心 | Laminated analysis method for any thin-wall beam structure |
| CN112800538B (en) * | 2021-01-15 | 2023-12-22 | 中国商用飞机有限责任公司北京民用飞机技术研究中心 | Lamination analysis method for arbitrary thin-wall beam structure |
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