research article mechanical performance and parameter ...research article mechanical performance and...

Research ArticleMechanical Performance and Parameter SensitivityAnalysis of 3D Braided Composites Joints

Yue Wu1 Bo Nan1 and Liang Chen2

1 School of Civil Engineering Harbin Institute of Technology No 77 Huanghe Road Nangang District Harbin 150090 China2 CAPOL International Design Group Shenzhen 518038 China

Correspondence should be addressed to Bo Nan nb2003ccc163com

Received 16 April 2014 Revised 5 June 2014 Accepted 6 June 2014 Published 8 July 2014

Academic Editor Ashok Kumar Gupta

Copyright copy 2014 Yue Wu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

3D braided composite joints are the important components in CFRP truss which have significant influence on the reliability andlightweight of structures To investigate the mechanical performance of 3D braided composite joints a numerical method basedon the microscopic mechanics is put forward the modeling technologies including the material constants selection element typegrid size and the boundary conditions are discussed in detail Secondly a method for determination of ultimate bearing capacityis established which can consider the strength failure Finally the effect of load parameters geometric parameters and processparameters on the ultimate bearing capacity of joints is analyzed by the global sensitivity analysis method The results show thatthe main pipe diameter thickness ratio 120574 the main pipe diameter 119863 and the braided angle 120572 are sensitive to the ultimate bearingcapacity119873

1 Introduction

CFRP truss is composed by members and joints and thecomposite joint is the most important component wherethe force and deformation are very complex moreover thereexists phenomenon of stress concentration which is the weakpoint of the loading process [1 2] According to statisticsthere are 70 spacecraft structure damages that occurred atthe connecting part [3 4]Therefore in the analysis and opti-mization of composite structures the key point is the predic-tion of the joint strength and the influence on each parameter

According to forming process it can be divided intoMold-ing jointWinding joint Layer-Molding joint and 3Dbraidedjoint The first three joints belong to the laminated structureand the theoretical research and manufacturing processare relatively mature and they are commonly used in theCFRP truss joints currently [5 6] But the laminated CFRPjoint strength between layers is small delamination defectsoccurs easily While the 3D braided CFRP also has fiber inthickness direction overcoming the weakness of delamina-tion in the former three joints [7]Therefore the 3-D braided

joints are important forms to study on in all kinds of CFRPtruss joints

Recently researches on 3D braided joint are mainlyconcentrated onmaterial constants and numerical simulationand experiment Zheng et al and so forth [8] taking CFRP3-D braided spherical joint as the research object studied itsdamage mode under complex loads C-Y Yang and H-NYang [9] studied the bending stiffness on three connectedspecimens Zheng et al and so forth [10] analyzed the lug loadcapacity of 3D four-directional braided Sun et al [11] studiedthe carbonepoxy 3D multidirectional braided tubular jointsby finite element analysisThe current study provides numer-ical analysis methods but most of them do not get tested andno one do correlation analysis on the influence of the CFRPparameters in numerical simulation

The mechanical properties of K type 3D braided jointsand the influence of CFRP parameters were selected forresearch in this paper Numerical analysis of K type 3Dbraided joint is studied firstly verified through an existingtest and then various parameters were discussed includingthe effects of load parameters geometric parameters and

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 476262 9 pageshttpdxdoiorg1011552014476262

2 The Scientific World Journal

120576120576998400fm120576xm

120590998400m

Fiber

Composite

Matrix

120590

Xt

Xf

Xm

O

Figure 1 Stress strain curve of the fiber and matrix

material parameters on weighing the sensitivity impact of thejoint ultimate bearing capacity

2 Numerical Analysis MethodBased on Micromechanics

21 Material Model On the force along the fiber directionKelly and Davies put forward the hypothesis that all fibershave the same strength and more fragile than that of matrix(see Figure 1) if the composite has more than a minimumfiber volume content of119881119891 the composite is reaching its ulti-mate stress when the fiber deformation reaching its maxi-mum strength If the fiber strain along the fiber direction isequal to the matrix strain the ultimate strength of thecomposite is

120590119862max = 120590119891max119881119891 + (120590119898)120576119891max(1 minus 119881119891) (1)

In (1) 120590119862max is the ultimate stress of composite materials120590119891max is ultimate stress of fiber and (120590119898)120576119891max

is the stress ofultimate matrix strain

Because the fiber is brittle it cannot exhibit elongation asmatrix does While the fiber damaged by longitudinal ten-sion composite material would be damaged the strength ofthe composite is

119883119905 = 119883119891119888119891 + 1205901015840119898119888119898 (2)

In (2) 119883119905 is the fiber tensile strength 1205901015840119898 is the matrixstress which is equal to the matrix stress when fiber tensilestrain reached limit 119888119891 is cross-sectional area of the fiber and119888119898 is the cross-sectional area of the matrix

For the 3D braided joints 3-cells model which was pre-sented by De-long and Shen [12] is accepted in engineering inthe prediction of elastic constants for 3D braided compositematerial The basic idea is as follows(1) Calculate the elastic modulus of unidirectional com-

posites

Reference [12] gives the semiempirical formula which hasbeen widely used in composite material area as

1198641 = 1198641198911119881119891 + 119864119898119881119898

1198642 = 1198643 =119864119898

1 minus radic119881119891 (1 minus 1198641198981198641198912)

11986612 = 11986613 =119866119898


11986623 =119866119898


]12 = ]13 = ]11989112119881119891 + ]119898119881119898

]23 =051198642

11986623

minus 1

(3)

In (3)11986411198642(1198643)11986612(11986613)11986623 12059212(12059213) 12059223 respectivelyare represented as the longitudinal elastic modulus trans-verse elasticmodulus longitudinal shearmodulus transverseshear modulus Poissonrsquos ratio transverse Poissonrsquos ratio1198641198911 1198641198912 11986611989112 11986611989123 12059211989112 respectively are represented as thelongitudinal elastic modulus of fiber transverse elastic mod-ulus longitudinal shear modulus and Transverse shear mod-ulus longitudinal Poissonrsquos ratio119864119898 119866119898 120592119898 respectively arerepresented as elastic modulus shear modulus Poissonrsquosratio 119881119891 is fiber volume content and 119881119898 = 1 minus 119881119891 is matrixvolume content(2) Calculate the axial flexibility matrix of unidirectional

compositesUnidirectional composite axial flexibility matrix is shown

in (4) Among them 1198641 1198642(1198643) 11986612(11986613) 11986623 12059212(12059213) 12059223are determined respectively by (3) Consider

[1198781015840] =

[[[[[[[[[[[[[[[[[[[[

[

1

1198641

minus]121198642

minus]121198642

0 0 0

minus]121198642

1

1198642

minus]231198642

0 0 0

minus]121198642

minus]231198642

1

1198643

0 0 0

0 0 01

11986623

0 0

0 0 0 01

11986612

0

0 0 0 0 01

11986612

]]]]]]]]]]]]]]]]]]]]

]

(4)

(3) Calculate the axial stiffness matrix of unidirectionalcomposite

[1198621015840] = [119878

1015840]minus1 (5)

The Scientific World Journal 3

=

Figure 2 Internal cell and fiber (thick solid line) direction

(4) Calculate transformation matrix of unidirectionalcomposite from the local coordinate to global coordinateTheInternal cell and fiber direction are shown in Figure 2

In order to obtain the unified material constants thestiffness matrix in the local coordinates (119883119884119885 coordinate) ofunidirectional composites needs to be transformed into theglobal coordinate system 119883101584011988410158401198851015840 Consider

[119879]119894 =

[[[[[[[[[[[[[[

[

11989721 119898

21 119899

21 211989811198991 211989711198991 211989811198971

11989722 119898

22 119899

22 211989821198992 211989721198992 211989821198972

11989723 119898

23 119899

23 211989831198993 211989731198993 211989831198973

11989721198973 11989821198983 11989921198993 11989821198993 + 11989831198992 11989921198973 + 11989931198972 11989721198983 + 11989731198982

11989731198971 11989831198981 11989931198991 11989831198991 + 11989811198993 11989931198971 + 11989911198973 11989731198981 + 11989711198983

11989711198972 11989811198982 11989911198992 11989811198992 + 11989821198991 11989911198972 + 11989921198971 11989711198982 + 11989721198981

]]]]]]]]]]]]]]

]

(6)

1198971 = cos 120579 1198972 = sin 120579 cos120573 1198973 = sin 120579 sin120573 1198982 = sin120573 1198983 =minus cos120573 1198991 = minus sin 120579 1198992 = cos 120579 cos120573 1198993 = cos 120579 sin120573120579 120573 are defined as shown in Figure 3 (for119885 coordinates of

the fiber as an example) 120579 as the fiber angle between the pro-jection of unit cell longitudinal plane and longitudinal axis120573 as the fiber angle between the projection of unit cell trans-verse plane and horizontal axis [120579 120573] of the four fibers are[120579 45∘] [120579 minus45∘] [minus120579 45∘] [minus120579 minus45∘]

(5) Calculate the unidirectional composite stiffness mat-rix of partial axis equivalent in the global coordinate system

[119862]119894 = [119879]119894 [1198621015840] [119879]119879119894 (7)

(6) Calculate the unit cell overall stiffness matrixUnit cell overall stiffness is obtained by the averagematrix

stiffness of all fiber direction and this method is called stiff-ness average method Consider

[119862119911] =1

4sum [119862]119894 (8)

(7) Calculate the elastic constants of 3D braided compos-ite materials by unit cell overall flexibility matrix

3-cells model was used to predict the elastic constantsof 3D braided composite material which calculate stiffnessmatrix at each element according to the method which isshown in (3)ndash(8) then the whole stiffness matrix of themate-rial could be calculated by stiffness matrix of each elementaccording to the proportion of each unit cell by using theweighted sum method Because of [13] unit cell stiffnessmatrix which occupies the largest proportion of the com-posite materials represents the stiffness matrix of braidedmaterial Consider

[119878119911] = [119862119911]minus1 (9)

The elements of the flexibilitymatrix inwhole unit cell arecomposed of elastic constants as shown in (4)Therefore therelationship between elastic constants of 3D braided compos-ite material and the elements of unit cell overall flexibilitymatrix is as follows

119864119909 =1

11987811

119864119910 =1

11987822

119864119911 =1

11987833

119866119910119911 =1

11987844

119866119911119909 =1

11987855

119866119909119910 =1

11987866

]119909119910 = minus11987812

11987822

]119910119911 = minus11987823

11987833

]119909119911 = minus11987831

11987811

(10)

119864119909 119864119910 119864119911 are the elastic modulus of 3D braided composites119866119909119910 119866119910119911 119866119909119911 are the shear modulus of 3D braided com-posites 120592119909119910 120592119910119911 120592119909119911 are the Poissonrsquos ratio of 3D braidedcomposites 119878119894119895 is row 119894 and column 119895 in the overall flexibilitymatrix [119878119911]

After the MATLAB programming calculation materialelastic constants of this paper are 119864119909 = 7379e4Mpa 119864119910 =119864119911 = 814e3MPa 119866119909119910 = 119866119909119911 = 1256e4MPa 119866119910119911 =403e3MPa and 120592119909119910 = 120592119909119911 = 08 120592119910119911 = 033 Test119864119909 differs by415 from values in [14] (Unified listed in Table 1)

22 Geometric AnalysisModel In theK-joint size as shown inFigure 4 the diameter ofmain pipe is 120mm 4mmof thick-ness the diameter of branch pipe is 54mm 32mm ofthickness

The K-joint adopts four step 1 times 1 (making one transverseand longitudinal movement for each time) woven technology(as shown in Figure 5) which braided angle is 25∘ fibervolume content is 59 Properties of the matrix and fiber (isshown in Table 1)


Table 1 Mechanical properties of T700 and TDE85

The elastic constants T700 TDE851198641 (GPa) 221 341198642 (GPa) 138 mdash11986612 (GPa) 90 mdash11986623 (GPa) 48 mdash12059212 02 03412059223 025 mdash119883119891119905 (MPa) 3528 85119883119891119888 (MPa) 2070 165120588 (gmm3) 177 121

Table 2 The influence of unit type on bearing capacity

Unit type Ultimate load (kN) Relative errorShell93 703 147Solid92 6928

23 Unit Type and Mesh Density

231TheUnit Type SHELL93 and the SOLID92were used toanalyze the K-joint in this paper the calculation results areshown in Table 2 It showed that the calculation results of 3Dbraided joint were less influenced by the type units In orderto improve the calculation efficiency SHELL93 is used toanalyze the K-joint in this paper

232 The Mesh Dividing Woven fibers have their owndirection branch and main pipe of the fiber direction differsso as to overall performance In order to simulate orthotropiccharacter of material the unit material properties should becorresponding to the fiber orientation ANSYS provides theunit coordinate system to simulate the fiber direction thereare 2ways one is tomesh under different local coordinate theother is to mesh together then revise the unit coordinate tothe local coordinate in unit characterThe secondmethod hasbeen used in this paper branch pipe divided unit coordinatesis shown in Figure 6 in which the axial tangential andnormal represent the unit coordinate three directions119883119884 119885

The mesh precision has influence on the calculationresults The K-joints are divided along in five ways sparseslightly sparse moderate slightly dense and dense quantita-tive indicators which are shown in Table 3

It can be seen that the joints ultimate bearing capacitydiffers 2 when grid is slightly dense compared with thedensest however the error of jointsrsquo ultimate bearing capacityis larger when it divided sparsely Considering the calculationof time and accuracy the slightly dense grid is adopted in thepaper

24 Applying the Boundary Conditions and Load The com-mon used boundary conditions and loading method in anal-ysis of K-joint is shown in Figure 7 In (a) one end of themainpipe is fixed the other is connected with the sliding hinge thebranch pipe is hinged in (b) the main pipe is fixed at bothends the branch pipe is hinged in (c) one end of the main

XY

Z

120573

120579

Z998400

Y998400

X998400

Figure 3 The coordinates of internal cell

380mm380mm

45∘ 280mm280mm

45∘

Figure 4 The size of K-joint

pipe is fixed the other is connected with the sliding hingethe branch pipe is free in (d) the main pipe is fixed at bothends the branch is free When the main pipe is stressed itbecomes two-way loading and is one-way loading when itdoes not stress

When loaded in one-way direction the Loading-deformation curves in different boundary conditions areshown in Figure 8 Constraint on (a) and (b) (c) and (d) theresults are similar which illustrates constraints of the jointsare less influenced the ultimate bearing capacity constraintson (a) and (c) (b) and (d) differed greatly which illustratesconstraints at end of the main pipe have less influence on thejoints ultimate bearing capacity

When the branch is free the ultimate bearing capacity ofthe joints is pretty less than the branch pipe which is hingedat the same time branch pipe of K-joint deformation is largerwhich has deviated from the K-plane as shown in Figure 9Considering the actual structure of the branch pipe endsrestrained by abdominal rod the main pipe could move tosome extent therefore the boundary conditions of constraint(a) has been chosen in this paper considering the one-wayand two-way loading modes

25 Failure Criterion and Ultimate Bearing Capacity Wanget al [13] did statistic for usage of the composite materialsfailure criterion (as shown in Figure 10)Themaximum stresscriterion application frequency is ranked in second placenext to the maximum strain criterion The maximum stresscriterion was adopted in this paper for this criterion is not


Table 3 The influence of the mesh density on bearing capacity

Mesh density Number of element Ultimate load (kN) Relative errorSparse 1039 5434 1265Slightly sparse 1520 5064 498Moderate 2645 50112 388Slightly dense 3685 4921 2Dense 4031 4824 mdash

0

3

2-3 plane

Yarns

Braided preform

Yarns carriers

Machine bed

(a) Braided indicate

1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24

(b) Initial state

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24

(c) The first step in the state

1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24

(d) The second step in the state

1 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23 24

(e) The third step in the state

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24

(f) The fourth step in the state

Figure 5 Four-step 1 times 1 3D braided process

XY

Z

Figure 6 The unit coordinates of branch pipe

only simple practical but also the test data of the maximumstress was verified in most of the reference for comparison120590 is the maximum stress of Von Mises of the joint [120590] is

the allowable material stress [120590] is solved using Zuo [15]proposed 3D braided material strength of two order Tsai-Wustrength criterion based on prediction method and the

maximum stress criterion Prediction of the joint strengthvalue is 6107MPa the test value in the reference is 629MPaand the difference between the two is minus29

When the maximum equivalent stress is higher than thematerial allowable stress joints damage and this load can beseen as the joint bearing capacity

26 Numerical Methods for Verifying the Accuracy The lugtensile test in [10] as shown in Figure 11 Carbon fibermaterialfor the test is T700-12K matrix material is TDE-85 epoxyresin the braided angle is 20∘ and fiber volume content is45 Calculation results show that the materials elastic con-stants are 119864119909 = 7198e4MPa 119864119910 = 119864119911 = 67e3MPa 119866119910119911 =292e3MPa119866119909119910 = 119866119909119911 = 781e4MPa 120592119909119910 = 120592119909119911 = 0656 and120592119910119911 = 0346 The material strength [120590] = 78755MPa

3D braided joint numerical analysis method was used tomake numerical simulation for the lug calculating the ulti-mate bearing capacity which is 749 kN testing value which is794 kN which differed 566 the maximum stress position


P P

P

(a)

P P

(b)

P P

P

(c)

P P

(d)

Figure 7 Boundary conditions and loading mode

000 05 10 15

10

20

20

30

40

50

Displacement (mm)

(a)(b)

(c)(d)

Ulti

mat

e bea

ring

capa

city

(kN

)

Figure 8 Loading-deformation curves in different boundary con-ditions

and joint breaking position are basically the same (as shownin Figure 12) that is to say the numerical simulation methodachieves certain precision which can be used to simulate the3D braided joint

3 Analysis on the Parameter Sensitivity of3D Braided Joint

After the analysis on themechanical properties of 3D braidedjoint it should make parameter sensitivity analysis on theparameters affecting the bearing capacity

Figure 9 Plane deformation of branch pipe

31 The Parameters and Value of 3D Braided Joint In thispaper the effect of load parameters geometric parametersand process parameters on the ultimate bearing capacity of119873was considered The main parameters and their values are asshown in Table 4

32 Analysis on the Global Parameter Sensitivity of 3D BraidedJoint According to whether to consider the interactionbetween parameters parameter sensitivity analysis methodscan be divided into local sensitivity analysis and global sensi-tivity analysis Local sensitivity analysis refers to one param-eter changed and the other parameters remain unchangedwhich can test the degree of influence on certain parametersvariation to target results Global sensitivity analysis refers to


Table 4 Joint parameters and the parameters values

Parameter name Parameter valueLoad parameter119899 minus09 minus08 0 08 09

Geometric parameters120591 06 07 08 09 1120574 20 30 40 50 mdash120573 035 04 045 05 055120579 30 35 40 45 50119889 40 60 80 100 120

Process parameters119881119891 35 45 55 65 mdash120572 5 15 25 35 45 55119899 The ratio of pipe force and material strength120574 The ratio of main pipe diameter and thickness120573 The ratio of main pipe diameter and branch pipe diameter120579 Branch pipe axis and main pipe axis angle (∘)120591 The ratio of main pipe wall thickness and branch pipe wall thickness120572 Fiber braiding angle (∘)119881119891 Fiber volume content ()119889 The main pipe diameter (mm)

1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion

Figure 10 Frequency of application of composite material failurecriteria 1 Principal strain criterion 2 strain energy density criterion3 the maximum strain criterion 4 the maximum stress criterion 5Tsai-Hill criterion 6 Tsai-Wu criterion 7 strain-strain rate criterion8 other criterion

the influence on target results by changing one certain param-eter when all the parameters changed Because the globalsensitivity analysis method considers the interaction betweenparameters it is generally believed to be more accurate andscientific than the local sensitivity analysis method

Since the parameter units of each node are not consistentit needs to normalize the various parameters then makethe parameters normalized to the range [0 1] by using theEquation (119896 minus 119896min)(119896max minus 119896min) [16] to get the curve ofnormalized parameters and target results The Equation is asfollows

119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)

119878119873(119896)mdashsensitivity coefficient Δ119873mdashvariation target takingthe ultimate bearing capacity of 119873 as a target 1198730mdashtargetreference value Δ119896mdashvariations of parameter 119870 and 119896max minus119896minmdashdesigned domain of parameter 119870

Then it takes nonparametric statistical methods to makesensitivity analysis which is proposed byMarivoet and Saltelliin 1990 The method carried out regressing to analyze theparameters and the results calculating the parameters and thestandard variance and relative coefficient between the twousing the following equation to obtain sensitivity coefficients119878119894 under different parameters

119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)

119878119894mdashcoefficient parameter of parameter 119909119894 cov(119910 119909119894)mdashcoefficient parameter of parameter 119909119894120590119909119894mdashstandard vari-ance of parameter119909119894 and120590119910mdashstandard variance of parameter119909119894

This paper selected 120574 = 30 120573 = 045 120579 = 45 120591 = 08 119899 =0119881119891 = 59 120572 = 25∘ and 119889 = 120mm as the joints referencestate

Figure 13 shows an example of ultimate bearing capacitychanged under different parametersThe slope of the curve isgreater the change of parameters has more influence on theultimate bearing capacity and the sensitivity is stronger

Linear fitting for each parameter then take the averagevalue calculating sensitivity coefficient 119878119873(119896) in Table 5

Equation (13) defined the parameter 119870 (the relative con-tribution rate) as target sensitivity that is the ratio of eithersensitivity coefficient or the sum of all the parameter sensitiv-ity

119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash

Figure 11 Single ear three-dimensional woven joint specimen and size

Figure 12 Lug joint failure pattern and the stress distribution

00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572

Normalized parameters

Ulti

mat

e bea

ring

capa

city

(kN

)

Figure 13 Relationship between ultimate bearing capacity and thenormalized parameters

Table 5 The sensitivity coefficients of parameters on the119873

119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08

According to the definition above the relative contribu-tion of various parameters on the sensitivity of119873 is the ratioof the sensitivity coefficient and the sum of all parameterssensitivity coefficient which is shown in Figure 14

4 The Conclusion

(1) By studying the ultimate bearing capacity of 3D brai-ded composite joint with numerical analysis methodfrom the material model element type mesh sizeboundary conditions loads and failure criteria sixaspects and compared the 3D braided compositeswith the lug the numerical analysis method is provedfeasible in this essay

(2) Based on these results the main results about thesensitivity analysis of parameters on the 3D braided


120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf

Figure 14 The relative contribution of various parameters on thesensitivity of 119873 119899 the ratio of pipe force and material strength 120574the ratio of main pipe diameter and thickness 120573 the ratio of mainpipe diameter and branch pipe diameter 120579 branch pipe axis andmain pipe axis angle (∘) 120591 the ratio of main pipe wall thicknessand branch pipe wall thickness 120572 fiber braiding angle (∘) 119881119891 fibervolume content () and 119889 the main pipe diameter (mm)

composite joint is as follows the main truss diameterthickness ratio 120574 gt fiber braiding angle 120572 gt load para-meters 119899 gt competent diameter 119889 gtmain and branchdiameter ratio 120573 gt fiber volume content 119881119891 gt branchand main thickness ratio 120591 The ratio of thicknessbetween the branch andmain pipe has small influenceon the ultimate bearing capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work described in this paper was jointly funded by theNational Natural Science Foundation of China under Grantnos 51378150 and 91215302

References

[1] J W H Yap M L Scott R S Thomson and D HachenbergldquoThe analysis of skin-to-stiffener debonding in composite aero-space structuresrdquo Composite Structures vol 57 no 1-4 pp 425ndash435 2002

[2] A Mills ldquoAutomation of carbon fibre preform manufacturefor affordable aerospace applicationsrdquo Composites A AppliedScience and Manufacturing vol 32 no 7 pp 955ndash962 2001

[3] S Ju J Zeng and D Jiang ldquoStudy progress in composite truss-jointrdquo Materials Review vol 20 no 12 pp 28ndash32 2006(Chinese)

[4] Q Guo G Zhang and J Li ldquoProcess parameters design of athree-dimensional and five-directional braided composite jointbased on finite element analysisrdquo Materials amp Design vol 46pp 291ndash300 2013

[5] J M Yang C L Ma and T W Chou ldquoFiber inclination modelof three-dimensional textile structural compositesrdquo Journal ofComposite Materials vol 20 no 5 pp 472ndash484 1986

[6] L D Zhou and Z Zhuang ldquoStrength analysis of three-dimen-sional braided T-shaped composite structuresrdquo CompositeStructures vol 104 pp 162ndash168 2013

[7] D-L Wu and Z-P Hao ldquo5-D braided structural compositesrdquoJournal of Astronautics vol 3 no 7 pp 40ndash50 1993 (Chinese)

[8] B Zheng S Zhang P He J Ju and Y Sun ldquoExperiments andanalyses for carbon fiber reinforced composite multi-joints ofthe satellite antenna trussrdquo Acta Materiae Compositae Sinicavol 22 no 6 pp 172ndash177 2005 (Chinese)

[9] C-Y Yang and H-N Yang ldquoBending rigidity of a satelliteantenna truss joint made of 3D woven compositesrdquo MaterialsScience amp Technology vol 16 no 6 pp 810ndash813 2008

[10] X Zheng Y Guo Q Sun Y Chai and X Li ldquoStrength pred-ication for load-bearing joints of three-dimensional braidedcompositesrdquo Journal of Mechanical Strength vol 28 no 6 pp923ndash926 2006 (Chinese)

[11] Y Sun L Chen and J-L Li ldquoFinite element analysis on loadingproperties of intersecting tubular joint with carbonepoxy 3Dmultidirectional braided compositerdquo Journal of Solid RocketTechnology vol 31 no 3 pp 266ndash274 2008 (Chinese)

[12] W U De-long and H R Shen ldquoOn the study of mechanicalbehavior of textile structural compositesrdquo Advances In Mechan-ics vol 31 no 4 pp 583ndash591 2011 (Chinese)

[13] BWang SWu and J Liang ldquoStrength theory and failure analy-sis of composite materialsrdquo Failure Analysis and Prevention vol1 no 2 pp 13ndash19 2006

[14] BWang Study on theMechanical Behavior ofThreeDimensionalBraided Composites Northwestern Polytechnical UniversityXian China 2003 (Chinese)

[15] W-W Zuo Research on Mechanical Properties and EngineeringApplication of 3-D Braided Composites Huazhong University ofScience amp Technology Wuhan China 2006 (Chinese)

[16] M-HWei andD-P Yu ldquoLandslide susceptibility factor analysisand stability evaluationrdquoChinaWater Transport vol 5 no 8 pp97ndash98 2007 (Chinese)

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120576120576998400fm120576xm

120590998400m

Fiber

Composite

Matrix

120590

Xt

Xf

Xm

O

Figure 1 Stress strain curve of the fiber and matrix

material parameters on weighing the sensitivity impact of thejoint ultimate bearing capacity

2 Numerical Analysis MethodBased on Micromechanics

21 Material Model On the force along the fiber directionKelly and Davies put forward the hypothesis that all fibershave the same strength and more fragile than that of matrix(see Figure 1) if the composite has more than a minimumfiber volume content of119881119891 the composite is reaching its ulti-mate stress when the fiber deformation reaching its maxi-mum strength If the fiber strain along the fiber direction isequal to the matrix strain the ultimate strength of thecomposite is

120590119862max = 120590119891max119881119891 + (120590119898)120576119891max(1 minus 119881119891) (1)

In (1) 120590119862max is the ultimate stress of composite materials120590119891max is ultimate stress of fiber and (120590119898)120576119891max

is the stress ofultimate matrix strain

Because the fiber is brittle it cannot exhibit elongation asmatrix does While the fiber damaged by longitudinal ten-sion composite material would be damaged the strength ofthe composite is

119883119905 = 119883119891119888119891 + 1205901015840119898119888119898 (2)

In (2) 119883119905 is the fiber tensile strength 1205901015840119898 is the matrixstress which is equal to the matrix stress when fiber tensilestrain reached limit 119888119891 is cross-sectional area of the fiber and119888119898 is the cross-sectional area of the matrix

For the 3D braided joints 3-cells model which was pre-sented by De-long and Shen [12] is accepted in engineering inthe prediction of elastic constants for 3D braided compositematerial The basic idea is as follows(1) Calculate the elastic modulus of unidirectional com-

posites

Reference [12] gives the semiempirical formula which hasbeen widely used in composite material area as

1198641 = 1198641198911119881119891 + 119864119898119881119898

1198642 = 1198643 =119864119898


11986612 = 11986613 =119866119898


11986623 =119866119898


]12 = ]13 = ]11989112119881119891 + ]119898119881119898

]23 =051198642

11986623

minus 1

(3)

In (3)11986411198642(1198643)11986612(11986613)11986623 12059212(12059213) 12059223 respectivelyare represented as the longitudinal elastic modulus trans-verse elasticmodulus longitudinal shearmodulus transverseshear modulus Poissonrsquos ratio transverse Poissonrsquos ratio1198641198911 1198641198912 11986611989112 11986611989123 12059211989112 respectively are represented as thelongitudinal elastic modulus of fiber transverse elastic mod-ulus longitudinal shear modulus and Transverse shear mod-ulus longitudinal Poissonrsquos ratio119864119898 119866119898 120592119898 respectively arerepresented as elastic modulus shear modulus Poissonrsquosratio 119881119891 is fiber volume content and 119881119898 = 1 minus 119881119891 is matrixvolume content(2) Calculate the axial flexibility matrix of unidirectional

compositesUnidirectional composite axial flexibility matrix is shown

in (4) Among them 1198641 1198642(1198643) 11986612(11986613) 11986623 12059212(12059213) 12059223are determined respectively by (3) Consider

[1198781015840] =

[[[[[[[[[[[[[[[[[[[[

[

1

1198641

minus]121198642

minus]121198642

0 0 0

minus]121198642

1

1198642

minus]231198642

0 0 0

minus]121198642

minus]231198642

1

1198643

0 0 0

0 0 01

11986623

0 0

0 0 0 01

11986612

0

0 0 0 0 01

11986612

]]]]]]]]]]]]]]]]]]]]

]

(4)

(3) Calculate the axial stiffness matrix of unidirectionalcomposite

[1198621015840] = [119878

1015840]minus1 (5)


=




[119879]119894 =

[[[[[[[[[[[[[[

[

11989721 119898

21 119899

21 211989811198991 211989711198991 211989811198971

11989722 119898

22 119899

22 211989821198992 211989721198992 211989821198972

11989723 119898

23 119899

23 211989831198993 211989731198993 211989831198973

11989721198973 11989821198983 11989921198993 11989821198993 + 11989831198992 11989921198973 + 11989931198972 11989721198983 + 11989731198982

11989731198971 11989831198981 11989931198991 11989831198991 + 11989811198993 11989931198971 + 11989911198973 11989731198981 + 11989711198983

11989711198972 11989811198982 11989911198992 11989811198992 + 11989821198991 11989911198972 + 11989921198971 11989711198982 + 11989721198981

]]]]]]]]]]]]]]

]

(6)




[119862]119894 = [119879]119894 [1198621015840] [119879]119879119894 (7)



[119862119911] =1

4sum [119862]119894 (8)



[119878119911] = [119862119911]minus1 (9)


119864119909 =1

11987811

119864119910 =1

11987822

119864119911 =1

11987833

119866119910119911 =1

11987844

119866119911119909 =1

11987855

119866119909119910 =1

11987866

]119909119910 = minus11987812

11987822

]119910119911 = minus11987823

11987833

]119909119911 = minus11987831

11987811

(10)
















XY

Z

120573

120579

Z998400

Y998400

X998400


380mm380mm

45∘ 280mm280mm

45∘









0

3

2-3 plane

Yarns

Braided preform

Yarns carriers

Machine bed


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24

(b) Initial state

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24


1 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23 24


1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24



XY

Z









P P

P

(a)

P P

(b)

P P

P

(c)

P P

(d)


000 05 10 15

10

20

20

30

40

50

Displacement (mm)

(a)(b)

(c)(d)

Ulti

mat

e bea

ring

capa

city

(kN

)













1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion




119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)



119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)






119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










=




[119879]119894 =

[[[[[[[[[[[[[[

[

11989721 119898

21 119899

21 211989811198991 211989711198991 211989811198971

11989722 119898

22 119899

22 211989821198992 211989721198992 211989821198972

11989723 119898

23 119899

23 211989831198993 211989731198993 211989831198973

11989721198973 11989821198983 11989921198993 11989821198993 + 11989831198992 11989921198973 + 11989931198972 11989721198983 + 11989731198982

11989731198971 11989831198981 11989931198991 11989831198991 + 11989811198993 11989931198971 + 11989911198973 11989731198981 + 11989711198983

11989711198972 11989811198982 11989911198992 11989811198992 + 11989821198991 11989911198972 + 11989921198971 11989711198982 + 11989721198981

]]]]]]]]]]]]]]

]

(6)




[119862]119894 = [119879]119894 [1198621015840] [119879]119879119894 (7)



[119862119911] =1

4sum [119862]119894 (8)



[119878119911] = [119862119911]minus1 (9)


119864119909 =1

11987811

119864119910 =1

11987822

119864119911 =1

11987833

119866119910119911 =1

11987844

119866119911119909 =1

11987855

119866119909119910 =1

11987866

]119909119910 = minus11987812

11987822

]119910119911 = minus11987823

11987833

]119909119911 = minus11987831

11987811

(10)
















XY

Z

120573

120579

Z998400

Y998400

X998400


380mm380mm

45∘ 280mm280mm

45∘









0

3

2-3 plane

Yarns

Braided preform

Yarns carriers

Machine bed


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24

(b) Initial state

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24


1 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23 24


1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24



XY

Z









P P

P

(a)

P P

(b)

P P

P

(c)

P P

(d)


000 05 10 15

10

20

20

30

40

50

Displacement (mm)

(a)(b)

(c)(d)

Ulti

mat

e bea

ring

capa

city

(kN

)













1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion




119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)



119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)






119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















XY

Z

120573

120579

Z998400

Y998400

X998400


380mm380mm

45∘ 280mm280mm

45∘









0

3

2-3 plane

Yarns

Braided preform

Yarns carriers

Machine bed


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24

(b) Initial state

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24


1 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23 24


1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24



XY

Z









P P

P

(a)

P P

(b)

P P

P

(c)

P P

(d)


000 05 10 15

10

20

20

30

40

50

Displacement (mm)

(a)(b)

(c)(d)

Ulti

mat

e bea

ring

capa

city

(kN

)













1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion




119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)



119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)






119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0

3

2-3 plane

Yarns

Braided preform

Yarns carriers

Machine bed


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24

(b) Initial state

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24


1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

23 24


1 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23 24


1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223 24



XY

Z









P P

P

(a)

P P

(b)

P P

P

(c)

P P

(d)


000 05 10 15

10

20

20

30

40

50

Displacement (mm)

(a)(b)

(c)(d)

Ulti

mat

e bea

ring

capa

city

(kN

)













1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion




119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)



119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)






119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










P P

P

(a)

P P

(b)

P P

P

(c)

P P

(d)


000 05 10 15

10

20

20

30

40

50

Displacement (mm)

(a)(b)

(c)(d)

Ulti

mat

e bea

ring

capa

city

(kN

)













1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion




119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)



119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)






119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














1 2 3 4 5 6 7 80

5

10

15

20

25

30

()

Criterion




119878119873 (119896) =

1003816100381610038161003816Δ11987311987301003816100381610038161003816

|Δ119896| (119896max minus 119896min) (11)



119878119894 =cov (119910 119909119894)120590119909119894120590119910

(12)






119896 =119878119873 (119896)

sum 119878119873 (119896119894) (13)


AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










AA

AndashA

7

22W

H

L

L = 150mm H = 22mmW = 44mm

Oslash



00 020408060

10

20

40

60

80

100

n 120574

120573 120579120591 dVf 120572


Ulti

mat

e bea

ring

capa

city

(kN

)



119899 120591 120574 120573 120579 120572 119881119891 119889

119878119873(119896) 09 01 14 06 05 1 02 08


4 The Conclusion




120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120574

26

120573

10

n16

12057218

4

d

16

120591

1120579

9 Vf





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article mechanical performance and parameter ...research article mechanical performance and...

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