mechanical analysis of carbon fiber wound high pressure vessel and prestress control

8/8/2019 Mechanical Analysis of Carbon Fiber Wound High Pressure Vessel and PreStress Control

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Section 1 Advanced Manufacturing Technology

E y = V y E l l + V x E 2 2 ~ Vy 11If the CFRC with orthogonal direction fibers bear a force F inx direction, which the area of transverse section is A, thestress in x direction is

(5)

(3)

a=-x

The CFRC layers are assumed that the loop winding fibershardly bear the longitudinal stress and the longitudinalwinding fibers hardly bear the loop stress because thetransverse elastic module is much smaller than axial 's.Therefore, CFRC layer is simplified to two submodelsbecause of its complex mechanical property and structure:Submodel 1: The loop CFRC wound liner. The thickness ofloop CFRC isSubmodel 2: The longitudinal CFRC wound liner. Thethickness of longitudinal CFRC is tThe loop stress of submodel 1 and the longitudinal stress ofsubmodel 2 are equal to those of whole model. Thetheoretical basis ofmodel simplifying is given below:The axial elastic module and transverse elastic module ofsimple direction CFRC are assumed as Ell and E22. Thefiber directions of layered configuration are and

(orthogonal). Let direction be x direction, and Letdirection be y direction. The volumetric ratios of x

direction and y direction are assumed as x and y It isobvious thatVx+ Vy=l(2)Consider the CFRC with orthogonal direction fibers, the

elastic module of and direction can be obtained:

radius and outer radius of liner and the outer radius ofCFRC are considered as a, b and c.The loop stress and radial stress r ') of linercylinder are merely relevant to the radius Thelongitudinal stress (a z ' ) is assumed as uniform stress. Theformulas of loop stress and radial stress of liner are listedbelow [3]:

ao' ar2 _a 2 2 )+ (a 2 2 2r2 ) Jx [r 2 2 _a 2 ) J-1a '= a 2 +a 2r2 )_ (a 2 2 2r2 ) Jx[r 2 2 _a 2 ) J-1

Po

Fig.1: cross section of carbon fiber wound aluminum lined vessel.

3.1 Assumption and model simplifying3 Mechanical analysis of vessel

Mechanical analysis is on the assumption that [1,2]:(1) CFRC is transversely isotropic. Macroscopically themechanical property in transverse direction is the samebecause the fibers are laid uniformly in transverse section.(2) CFRC is linear elastic. The resin is viscoelastic whilethe fiber is linear elastic and the module of resin is muchsmaller than fiber, So CFRC is assumed as linear elastic.(3) CFRC is continuous.The mechanical analysis carbon fiber wound aluminumlined vessel is based on the basic principle of elasticmechanics. The vessel is divided into two components.Fig.2 (a) is the axis section of aluminum liner with innerpressure (P) and outer pressure (Po). Fig.2 (b) is the CFRCwith inner pressure (Po). The pressure (Po) is theinternational force on the interface between liner and CFRC.The liner and the

Fig.2: the model ofmechanical analysis.CFRC are considered as axisymmetric problem. The inner

(a) (b) The strain x is

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International Technology and InnovationConference 2006

a F& (6)x Ex AVxEIf the fibers of y direction are removed and only the fibers ofx direction are left, the stress in x direction is Where

dw w&r =-'&0dr r (12)

The strain &x' is

, Fax AVx

,ax'x E ' AVEx x 11

(7)

(8)

The constitutive equations are=_I_a _ Vl2 a _ VI2 a0E 0E rE Z

11 22 22

(13)

(14)

It is obvious that the strain &x and x' are almost thesame.If the CFRC with orthogonal direction fibers bear a stressa z in z direction, the strain &x and &x' are

a ' v'= _ x__ x ax E ' E zxThey are also the same.

(9)

(10)

21 1=--a +-a --aE o ErE z11 22 22

z =-5L ao - a +_I_aE E rE z11 22 22Let a z be a constant a zo and let

=A I2 =B = V =DE ll ' E22 ' E22 ' E22Therefore

&0 =AaB-B(ar+a

(15)

(16)

3.2 Mechanical analysis ofCFRC [4-7]The loop stress and radial stress r ) of CFRCcylinder are merely relevant to the radius (r). The longitudinalstress z) is also assumed as uniform stress. Take noaccount of the gravity and the shear stress, The equilibriumequation of force about CFRC cylinder is

deB _A dao Bdar-- -- - (17)dr dr drAccording to constitutive equations (14) and (15), we can get

&r -&0 =(C+B)ar -(A+B)aB+(B-D)aAccording to equations (11), (13) and (17), we have

daa -a =r-_ro r drThe geometrical equations are

dadear +r_ r )A drdrIt is simplified as

(11)

da 1 daB _ r +B)ar +B)(ar+r-r)+(B-D)azldr r drB-Da =C eXIt C e ~ - -aI 2 A-C z

(19)

(21)

r2a n+3ra A-C a B-D ar r A r A z (20)The equation (19) is an Euler equation. The solution is

where

x xI ~ A 2 ~ (22)

and_ dar C(I ) B-Dao - a r+ r - - - } I+x} r 2 +X r +--azM A-C (23)


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Section 1 AdvancedManufacturing Technology

Let a =0, because it is assumed that loop winding fibersdon't bear the longitudinal stress. The boundary conditionson the internal and external surface are given by

a r a rand

(26)

ThereforeXl X2

= X (25)(eXl r X2 _ e X2 r X)a =--..;...0 _

r eXl bX2 _ eX2 bXl (27)

The solutions are

Here &0 ' is the hoop strain of aluminum liner and &0 isthe hoop strain of CFRC layer. Due to 0 ' 0 ' we can get

(34)

(28)

(33)0'1] [Q11 Q12 0] &10'2 = Q12 Q22 &20 0 Q66 r12The stiffness matrix of single direction offset-axis CFRClayer is

Here is the offset-axis stiffness. is the offset-axisangle. The offset-axis stiffness can be calculated as below [8]The stiffness matrix of single direction orthoaxis CFRClayer is

(29)r =a = =-P =-70MPar =b=0.104,ar '=ar =-Par =b =0.104'&0' =&0

The elastic module of liner is 70Gpa, and the mechanicalproperty of single direction CFRC layer isa b =2678.8Mpa, Ell=181Gpa, E22=E33=10.3Gpa,

G12=7.17Gpa, V21 =0.28to =0.0105m, ta =0.006m, a=O.1m, b=0.104m, c=0.1145m,

1=700mmThe value of Po can be calculated according to the boundarycondition of aluminum liner and equations of deformativeharmony. The boundary conditions and equations ofdeformative harmony are

3.3 Example

[(}"e ' - r ~ ) = 0'0 - 21 r z ) ] (30)Assuming that the longitudinal strain of aluminum liner isequal to the longitudinal strain of longitudinal fibers

The stiffness matrix of single direction offset-axis CFRClayer can be obtained by transforming the stiffness matrix ofsingle direction orthoaxis CFRC layer. The matrix oftransformation is

(31)


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offset-axis angle. When a 14 (]

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QIl m4 n4 2m2n2 4m2n2-Q22 n4 m4 2m2n2 4m2n2 QIl

m2n2 m2n2Q12 m4 +n 4 -4m 2n2 Q22 (35)Q66 m2n2 m2n2 -2m 2n2 (m2 _n 2 )2 Q12

m3n -mn3 mn3 -m 3n 2(mn3 -m 3n) Q66Q16- mn3 -m 3n m3n -mn 3 2(m 3n-mn 3 )Q26

Here the letter m cos a , n sin a . a is theE1a QIl cos 4 14n Qzz sin 4 14 c, 2QIZ cos 2 14 n X sin z14n 4Q66 cos 2 14" xsinz14n=166.42GpaThe solution can be given according to the system ofequations (30) and equation (32)

Po =57.8MPa, a z ' =180.22MPa, a z =430.64MPaWhen r=b=O.1 04mae=633.15MpaWhen r=c=O.1145mae=531.3 IMpa

The maximum hoop stress of liner can be calculated byLame formulas

( KZ 2Kae -z - -Po -z- =253.4Mpa-1The longitudinal stress of hoop winding fibers can beobtained by figuring out the longitudinal stain oflongitudinal winding fibers. The longitudinal stain oflongitudinal winding fibers isa v

z =_2 aE E430.64xl06 0.28 x(-57.8x1Q6)166.42xl0 181xl0=2.677x10.

The longitudinal stress of hoop winding fibers isEa z =E &z +V E ao 21 =18.lMPa

The hoop stress of longitudinal winding fibers can also beobtained by figuring out the hoop strain of hoop windingfibers. The hoop strain of longitudinal winding fibers is

V=-a a =3.59xIOr1 1The hoop stress of longitudinal winding fibers is

Eae =Ez6e vZ1a r VZ1 _Z a z =27.6MPaEThe ratio of the longitudinal stress of hoop winding fibers tothe longitudinal stress of longitudinal winding fibers is

18.1- - - x100% =4.2%430.64The ratio of the hoop stress of longitudinal winding fibers to

The hoop stress of hoop winding fibers is27.6 xl00%=4.4%633.15

is obvious that loop winding fibers hardly bear thelongitudinal stress and the longitudinal winding fibers hardlybear the loop stress. The correctness of model simplifyingand theoretic basis is proved.4 Prestress control in manufacturing processThe aluminum liner is the key component of the vessel. Notonly does it bear high stress, it also contacts with hydrogengas. The bad environment will probably lead to stresscorrosion. In order to decrease the stress of the liner underthe operating pressure, the liner is prestressed when it is inmanufacturing process. After the manufacturing process isdone, the liner has the compressing prestress. The actualstress of liner can be obtained when the compressingprestress plus the stress calculated under the operatingpressure without prestress. The method is as followingl9lThe buckling pressure is

2.59EtZPer = rnJ : =-2.76MPaLDo-VDo ItTake 3 as safety factor, the Po is

Po=-2.76/3=-O.92MPaBecausePo(l 2 )(c rx, cx, r'" )a = - - - - ~ - - - - ~() eXl bX2 _ eX2 bXl

Put the Po into the equation and the stress distribution offibers at any radius will be obtained. The tensile forcedistribution can also be calculated when the stress multiplyby the single layer thickness of fibers. If the CFRC layer iswound according to the tensile force distribution inmanufacturing process, the pressure between the liner andthe CFRC layer is Po, and the loop prestress of liner is

2PrK za _0__ =-24.4MPa2The single layer thickness of fibers is 0.42mm, and thequantity ofwinding layers can be obtained as 10.5/0.42=25.The tensile force is changed every 2 layers in manufacturingprocess. The winding tensile forces are given in Table I.


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Section 1 AdvancedManufacturing Technology

No. r (mm) Stress Tensile force(MPa) (N/mm)1 104.42 9.97 4.192 105.26 9.78 4.113 106.1 9.60 4.034 106.94 9.43 3.965 107.78 9.27 3.906 108.62 9.13 3.847 109.46 9.00 3.788 110.3 8.89 3.739 111.14 8.78 3.69

10 111.98 8.68 3.6511 112.82 8.60 3.6112 113.66 8.52 3.5813 114.5 8.46 3.55

Table 1: hoop winding tensile force distribution of fibers.5 ConclusionsIt is important to improve the fatigue life of carbon fiberwound aluminum lined pressure vessel because it needs tobe charged or discharged many times in its life. Based onthe mechanical analysis of carbon fiber wound aluminumlined high pressure hydrogen storage vessel, this papergives the stress formula of carbon fiber wound layer on thecylinder. The winding tensile force of fibers can becalculated easily by the stress formula. When the pressurevessel is in manufacturing, the liner gets the compressingprestress through controlling the winding tensile force. Theaverage stress of the liner decreases up to 10% and it canimprove the fatigue life of liner, while the average stress offibers increase 1.6% and it never lost effect on the fatiguelife of fibers. This method can improve the fatigue life ofcarbon fiber wound aluminum lined pressure vessel[lO].References[1] H.H. Chen, Deng, M. Li, X.S. Lin, "ModemComposite Materials", Beijing: China LogisticsPublishingHouse, 1997(In Chinese).[2] L. Zhou, F.Q. Fan, "Mechanics of Composite Materials",Beijing: Higher Education Press, 1991(1n Chinese).[3] 1.Y. Zheng, Q.W. Dong, Z.F. Sang, "Design of ProcessEquipmenf', Beijing: Chemical Industry Press, 2001 (InChinese).

[4] Y.C. Wu , 1. Hu, P. Li, "Stress analysis of metal linedfiber reinforced composite material pressure vessel",Chemical Equipment Technology, 24 (5): 46-49, 2003.

[5] Viktor E. Verijenko, Sarp Adali, Pavel Y, "Tabakov.Stress distribution in continuously heterogeneous thicklaminated pressure vessels", Composite Structure, 54,

371-377, 2001.[6] M. Xia, H. Takayanagi, K. Kemmochi, "Analysis ofmulti-layered filament-wound composite pipes underinternal pressure", Composite Structure, 53, 483-491,2001.

[7] P.M. Wild, G.W.Vickers, "Analysis of filament-woundcylindrical shells under combined centrifugal, pressureand axial loading'" Composites: Part A, 28A,47-55,1997.

[8] E.L. Lv, "Mechanics Of Composite Materials",Chongqing: Chongqinq University Press, 1992 (InChinese).

[9] C.X. Zheng, "Composite Material Pressure Vessel",Beijing: Chemical Industry Press, 2006(In Chinese).

[10] C.X. Zheng, Cao Kun, "Lightweight FilamentAluminum Liner High Pressure Vessel", ChinesePatent: 03150968.1.

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mechanical analysis of carbon fiber wound high pressure vessel and prestress control

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