Kiyohisa Takahashi Department of Polymer Engineering, Nagoya Institute of Technology, Gokiso-cho,
Showa-ku, Nagoya 466, Japan
Kikuo Ban Industrial Products Research Institute, 1-1-4 Yatabe-machi Higashi, Tsukuba-gun,
Ibaraki 305, Japan
and Tetsuya Sakai
Faculty of Engineering, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152, Japan
ABSTRACT
This paper presents an analytical approach for predicting the effective elastic moduli and breaking strength of FRP-FW pipes. The FW pipe was modelled as a laminate consisting of N laminae of unidirectional fibre composite. The elastic modulus of each lamina composed of aligned, continuous transversely isotropic fibres, and an isotropic matrix was derived based on the equivalent inclusion method of Eshelby. The effective stress-strain relation of the laminate was formulated assuming the plane stress state. The breaking strength of the laminate subjected to uniaxial tension or compression was estimated by applying the quadratic failure criterion proposed by Tsai and Wu. Numerical results were com- pared with experimental results obtained for C FRP-FW and G FRP-FW pipes. Good agreements were obtained both in Young' s modulus and in breaking strength.
1 I N T R O D U C T I O N
This paper is concerned with the mechanical properties of filament wound (FW) pipes made of carbon fibre reinforced plastics (CFRP) or glass fibre
reinforced plastics (GFRP). The authors '2 have investigated the mechanical properties of FRP-FW pipes experimentally. In this paper, the elastic moduli and breaking strengths of FRP-FW pipes are investi- gated theoretically. The objective is to establish an engineering design criterion for the effective use of FRP laminates and FW pipes as the structural members.
In the analysis, the FRP-FW pipe is modelled as a laminate consisting of N laminae of unidirectional FRP. The elastic modulus of each lamina is first derived based on the equivalent inclusion method of Eshelby. Then the effective stress-strain relation of the laminate consisting of N laminae is formulated assuming the plane stress state. The breaking strength of the laminate is estimated by applying the quadratic failure criterion proposed by Tsai and W u /
In the experiments, two types of FW pipes, (-+0) and (-+0 + 90°), are made with carbon fibre/epoxy or glass fibre/epoxy composites. Young's moduli and breaking strengths of the FW pipes subjected to uniaxial tension or compression are measured and compared with theoretical results.
2 THEORETICAL
The calculation procedure is specified in detail so that it may be easily used in the practical applications.
2.1 Fundamental relations
2.1.1 H o o k e ' s law Strain eq is defined by:
eq = l/2(uij + uj.i) (1)
where ui is the x~component of elastic displacement. Suffixes preceded by a comma denote differentiation; u~,j = Ou~/Oxj.
(a) Isotropic material (matrix resin). Stress o'q and strain ek~ are related by:
crij = Cqktekt (2)
Mechanical properties of FRP-FW pipes 93
where Cijkt is the elastic modulus, and repeated suffixes are summed over the values 1, 2, 3. For the isotropic matrix with Lame's constant h and shear modulus/x:
Cijkl = hSi]Skl + ~'~ikSjl "~ ~.£8il8jk (3)
vE E h = ( l + v ) ( l - 2 v ) ' / . t - lztt~"+v - - - ~ (4)
where E and v are Young's modulus and Poisson's ratio of the matrix, respectively. 8~ is Kronecker 's delta. The elastic compliance Sokt may be found from the following relation:
S qk,Ck~m. = 8 ~.,8/. (5)
(b) Transversely isotropic material (carbon fibre). It is known that carbon fibres have axisymmetric elasticity. When the x3 axis is taken along the fibre axis, the stress--strain relation is given by:
el l
e22
I
e33 I
623
l e31
e 12
m
1 / ~ ' 1 2 P13 0 0 0
E t E1 E3
1) 12 1 /)13 - - - 0 0 0
E1 El E3
1)31 1/31 l - - 0 0 0
El El Es
0 0 0 ----1 0 0 2bl,23
1 0 0 0 0
2~23
0 0 0 0 0
0
1
D .
0 " 1 1
0"22
0"33
0"23
O'31 I
0-t2 ]
(6)
where
V13 1231 E t /zl2 - (7)
E3 El ' 2(1 + vl2)
9 4 Kiyohisa Takahashi, Kikuo Ban, Tetsuya Sakai
Solving eqn. (6) for oiv. we obta in the elastic modulus C~7~ of the carbon fibre:
C ~ l l l = C~222 = El(-V~3El + E3)
(1 + v t2 )O
EI(v~3EI +/,'12E3) (1 + Vl2)Q
C~122 =
C~133 * - - = C 2 2 3 3 v l 3 E I E 3
e
(1 - v12)E~ C3333 Q
(s)
C2323 = /1£23, = ~['£12 = C3131 C*121:
with
Q = -2v~3 El + (1 - v12)E3
2.1.2 Eshelby's tensor In ou r calculat ion, the shape of fibres is assumed to be spheroidal with an infinite aspect ratio. Then Eshe lby ' s tensor T0kt is given by:
T I I I I = T2222 - _ _ 5 - 4 v
8(1 - v)
T1122 = T2211 - _ _ 4 v - 1
8(1 - ~,)
T I 1 3 3 = T2233 - _ _ 2(1 - v)
T3311 - - T3322 = T3333 = 0
1 T2323 = T3131 = -
4
3 - 4 v T1212 - _ _
8(1 - v )
(9)
whe re v is the Poisson 's ratio o f the matrix.
Mechanical properties of FRP-FW pipes 95
2.1.3 Equivalent inclusion method Effective elastic moduli of the unidirectional fibre composites can be calculated by applying Eshelby's equivalent inclusion method. 3 In order to replace the fibres (Cij*kt) with equivalent inclusions (Ci~k~), which have the same modulus as the matrix, an appropriate eigen strain e~is allocated to the equivalent inclusions. When the composite is subjected to a uniform stress ~A e 'should satisfy the following relation:
Cpqm.[eA. + (1 - v f)( T,..kte~t-- e*.)] , A ,
= Cpqmn[emn + (1 - vf)Tmnkte~l + Vfernn] (lo)
where vf is the volume fraction of fibres and
a A (11) era, = SmnijOr ij
Equation (10) takes into account the elastic interactions among the fibres and the presence of the free surface of the specimen. (See also Moil and Tanaka, 5 Takahashi et at. 6'7) Therefore, it can be applied to an arbitrary value of vf (0 =< v f=< 1). Solving the simultaneous equations (10) for e~, the effective compliance of the composite SqUk~ can be calculated from:
1 A A 1 A , _ 1 U A /2SijklO'ijO'kl + /2v fo'ijei j -- /2SijklO~ijOr kl (12)
Both sides of eqn. (12) represent the elastic energy per unit volume of the composite subjected to o-i~'.
2.2 Elastic moduli of ~ d i r e c t i o n a l fibre composites
The composite, in which continuous fibres are aligned unidirectionally, is effectively transversely isotropic and has five independent moduli. The calculation procedure for the five independent compliances SmU,, is shown in Fig. 1. It is assumed that the fibres are oriented along the x3 axis.
2.3 Effective modeli of FRP lamiaates
We consider a thin laminate consisting of N laminae of unidirectional fibre composites. The coordinates~j (j = 1,2, 3) are fixed in the laminate, while the coordinates xj(j = 1, 2, 3) are chosen parallel to the principal
Fig. 1. Calculation procedure for the effective compliance of unidirectional fibre com- posites. The x3 axis is taken along the fibre axis.
O N
3 111
...12 I l l " i i i
]'2"" (a)
O
X3 X3
( b ) X2= ~
o
X~
~- Xl
Fig. 2. The coordinate system xi (i = 1, 2, 3) is fixed in the FRP laminate (a), and the coordinate system xl (i = 1,2, 3) is taken along the principal axes of each lamina (b). Ti, Oi
(i = 1 - N) denote the thickness and orientation angle of the ith lamina.
Mechanical properties o f FRP-FW pipes 97
axes of each individual lamina. O,(i = 1 ~ N ) is the orientation angle of the fibres and Ti(i = 1 ~ N ) is the thickness of the ith lamina (Fig. 2).
Our calculation will be based on the following assumptions:
(1) Each lamina is perfectly bonded together. (2) Stress components vanish in the thickness direction (x2 = x2).
(~22 = 0"23 = 0"12 = 0 (plane stress state). (3) The laminae have the same strain components in the ~r-~3 plane
with each other and with the whole laminate.
It should be noted that assumptions (2) and (3) do not hold if the laminate is thick relative to its width or length. The thickness ratio t~ of the ith lamina is defined by:
Ti t, = N (i = I - N ) (13)
E T, i=1
For the lamina with orientation angle 0, the transformation law of the coordinates between xi and ~j is given by:
o
Xi = aOx j
X1
X2
X3
lCOS0 0sn0] L!I = 0 1 0 2
sin0 0 cos0 LX3d
(14)
The elastic compliances of the ith lamina referred to the xj and ~j co- . Oi ordinate systems are expressed by S~,~ and S k ~ , respectively. ~b, . (i = 1 -- N) is S~,,. calculated by the procedure shown in Fig. 1. And S ~,b., is given by:
o. S klm. = apkaya rmasnS~q~ (15)
Assumption (2) gives the stress-strain relation of the ith lamina:
K1 . . . . . ' ] . o, o,
° i ° i o i k e 3 l J S3133 2 S 3131 eo',,J
06)
98 Kiyohisa Takahashi, Kikuo Ban, Tetsuya Sakai
o i Solving eqn (16) for o',~, we obtain:
d-',, ] J-!,~| J-';,J
o i C I I
o i --~ (731
o i C 5 I
oi oi ] ( ' I~ C 15
° i ° i / C 33 C 35
o I o i / C 53 C 55 -1
It should be noted relation of the
- - "-] - - N o I ~ ° i
o-~ ,.., c~]ti I i = I I I u o E °1
i = 1
o 1 ['/ e~LJ
(17)
o i ° i o i ° i that ctj + C I l l l , C l 3 :~= C l 1 3 3 , etc. The stress-strain whole laminate is expressed as:
m N N
E °, E °, C t3ti C 15ti i = 1 i = 1
N N
E °, E °, C 33ti C 35ti i = 1 i = 1
N N
E ° E c~3t i °t C 55ti i = 1 i = 1 - -
1 o
_ e 3 1 J
by considering the assumption (3). Solving eqn (18) for ~,.,., we have the effective elastic compliance ~.~k? of the laminate (Fig. 3)
o][ 1 I°/ e 3, J L S},* $55" S~{4 d o-31J
13" - I Fig.1 -~ Spqrs-Spqrs
~ (15)
(19)
Eqs.(16),(17)
Eqs .(18 ),(19)
Fig. 3. Calculat ion procedure for the elastic compl iance ,~kr of the laminate .
Mechanical properties of FRP-FW pipes 99
2.4 Breaking stre~,th of FlIP laminate
In a plane stress state, the failure criterion proposed by Tsai and Wu 4 is expressed by:
1 1 1 ~,)
X X t ~ X X t YE t + YE tOrT -4- ~ - < = 1 (20)
where X, X ' , Y, Y' and S are parameters for the strength criterion of the unidirectional FRP. Subscripts L and T denote the fibre direction (x3) and the transverse direction (xl in Fig. 2), respectively.
We consider the laminate loaded uniaxially to the :~3 direction in Fig. 2, simple tension or compression. The failure criterion, eqn (20), is applied to each lamina. The breaking stress trB(i) for the ith lamina (i = 1 - N) can be calculated by the procedure shown in Fig. 4.
We set the initial value of ~3~ = o-0 at a small value, lest any lamina should fail below tr0. The increment of ~'~ = Air may be determined according to the accuracy required. We employ positive values for o-0 and Atr in the calculation of tensile strength, and negative values in the calculation of compressive strength. /3i denotes the degradation co- efficient of the ith lamina. Once any one of the laminae has failed, the degradation of the failed lamina should be taken into account in order to calculate the second lamina's failure and so forth. In our calculation, the elastic degradation is accounted for by changing the thickness ratio ti of the failed lamina to/3i ti(0 -<_/3~ <_- 1).
3 EXPERIMENTAL
3.1 Specimens
Fibres used for the FRP-FW pipes are: (1) carbon fibre TORAYCA T300 (Toray Co.) with specific gravity of 1-74, and (2) glass fibre RS-74PE-535 (Nittobo Co.) with specific gravity of 2.54.
One hundred parts by weight (pbw) of epoxy resin EPIKOTE 828 (Shell Chemical Co.) was mixed with 80 pbw of curing agent HHPA (hexahydrophthalic anhydride) and one pbw of promoting catalyser
I00 Kiyohisa Takahashi, Kikuo Ban, Tetsuya Sakai
t i= t i "~ i (0~_ai~_l)
IoA C~33 = C~O J
"-{ Eq.(19)" ~rnrl
@ °' Eq.( 17 ) -,- Okt J
½ oi Yes I 0"In= amkant0kI }
= 0
o A OA 033=(~#3 +~0 I J 0a(i)= d33 ]
Fig. 4. Calculation procedure for the breaking strength trB(i) of the ith lamina (i = 1 - N). After the failure of the ith lamina, the thickness ratio ti is multiplied by the degradation coefficient/3i. Atr is set as positive or negative in accordance with tr0 in the calculation of
tensile or compressive strength.
K61B. The fibres were penetrated with the epoxy resin compound and were formed into FRP pipes by the filament winding method. The FRP- FW pipes were cured at 130°C for 3 h. The inside diameter and thickness of the FW pipes were about 19 mm and 1"5 mm, respectively.
3.1.1 (4-0) Specimen FW pipes of helical winding ( - 0 ) without hoop winding; 0 = 20 °, 30 °, 45 °, and 60 °. The volume fraction of carbon fibres in CFRP-FW pipes is
Mechanical properties of FRP-FW pipes 101
0-455 on average, and that of glass fibres in GFRP-FW pipes is nearly equal to 0-5.
3.1.2 (+-0 + 90 °) Specimen Both of the inner and outer laminae are helical winding ( - 0) and the middle lamina is a hoop winding (90°). Thickness ratio of the hoop lamina is 0.16. The volume fraction of the fibres is about 0.48. 0---20 °, 30 °, 45 °, and 60 °. In this report only the results of GFRP specimens are shown for (___0+9o°).
3.2 Measurements
The length of the FRP-FW pipes was 300 ram. Tensile and compressive tests were carried out using an Instron 1332 testing machine. The gauge length of the specimens was 200 mm and the rates of extension and compression were both 2 mm/min. The strain of the specimen was measured by the strain gauge KFR-5-C1-65 (Kyowa Co.). In these tests, steel rods of 19 mm diameter and 70 mm length were inserted in both ends of the FW pipes to protect against the chuck failure.
4 RESULTS
Experimental results of the FRP-FW pipes were compared with numerical calculations. Elastic constants and parameters for the failure criterion employed in our calculations are shown in Tables 1 and 2, respectively. These values were determined by the preliminary experiments.
TABLE 1 Elastic Moduli of Constituents
Young' s modulus Poisson' s ratio Shear modulus (kglmm 2) (kg/mm 2)
Young's moduli are shown in Figs. 5-7. Theoretical values ET* were = US** Breaking strengths o-B are derived from $3"_* in eqn (19); ET* / ~3 .
shown in Figs. 8 and 9. The degradation coefficient/3~ for the hoop (90 °) lamina was assumed to be 0.8. In all those figures the angle 0 is in degrees, and Young's modulus and strength are in kg/mm2; 1 kg/mm 2 = 9.8 × 106
N/m 2. Theoretical values agree satisfactorily with experimental results both in Young's modulus and in breaking strength.
The theoretical value in Fig. 9 represents the failure stress 0-B of the ---0 lamina in the (-+ 0 + 90 °) specimen. Preceding the failure of the ---0 lamina,
O A the hoop (90 °) lamina fails at a lower value of 0"33. For example, the calculation of the tensile strength of the (---20 ° + 90 °) specimen, assuming the degradation coefficients/3i = 0 for all laminae, predicts that the hoop
o A lamina fails at 033--12.5 kg/mm 2 and the ---20 ° lamina fails at
~-EIO000 E
aooc
6000 I o E
4000
2000
0 90
i i i i i i i i
CFRP('_8)
= ,4 5
I 1 3 1 0 l 1 6 0 1 I I
Winding angle 0 (deg. )
400C
t'4 E E
300C
"~ 200C E P O~ c moc > -
f I i o 3'0
Winding
r I i i 1 i i r
GFRP (z O) ~ vf=o.5
~ o
60 90 angle O (deg.)
Fig. 5. Young's modulus versus winding angle 0 of CRFP-FW pipes (-+0). The volume fraction of the carbon fibres is 0.455. - theoretical . O =
experimental.
Fig. 6. Young's modulus versus winding angle O of GFRP-FW pipes (-+0). The volume fraction of the glass fibres is 0.5.
- theoretical, 0 = experimental.
Mechanical properties of FR P-FW pipes 103
4000
~ 300C
-5 X:) o 200t E
E~ ~oo > . .
I I I I I 1 I I
GFRP(-*e + 90*) Vf :0 .4B
I I I I I I I I
30 60 90 Winding angle O (deg.)
Fig. 7. Young's modulus versus winding angle 0 of GFRP-FW pipes ( - 0 + 90°). The volume fraction of the glass fibres is 0.48. The thickness ratio of the hoop (90 ° ) lamina is
0-16. = theoretical, O = experimental.
- - I i I I I I 1 I
Exper. Theor. '~ CFRP(ZO)
120- '~ Vf=0.455 : • . . . . . .
~. GFRP(ZO) . E lO0 \Vf=0.5 . o
B0 \
60 tn
20
30 60 90 Winding &ngte 0 (deg.)
Fig. 8. Tensile strength versus winding angle O of CFRP (*O) and GFRP (---O) FW pipes. The volume fractions of the carbon fibres and the glass fibres are 0-455 and
0-5, respectively.
I I I I I I I T ~
Exper. Theor. Tension : o
BO Compression: • . . . . . "
E ~ GFRP (*-e.90*) 6o ,,\o v,:0.48
"c 40 ~ ~ ~'\~0 ..
u3 20 -.~.
0 30 60 90 Winding angle 0 (deg.)
Fig. 9. Tensile and compressive strengths versus winding angle 0 of GFRP-FW pipes ( - 0 + 9 0 ° ) . The volume fraction of the glass fibres is 0.48. The thickness ratio of the hoop (90 °) lamina is 0-16. The degradation coefficient /3i of the hoop
lamina is assumed to be 0.8.
46"5 kg/mm 2. On the other hand, if/3i for the hoop lamina is assumed to be 0.8, the +-20 ° lamina fails at t ~ = 49-3 kg/mm 2. Generally, the calcu- lationswithfli = 0- 8 for the hoop lamina agree weU with the experimental results (i.e. the decrease of Young's modulus after the failure of the hoop lamina and the breaking strength of the +--0 lamina). It may be interpreted
104 Kiyohisa Takahashi, Kikuo Ban, Tetsuya Sakai
that the failure of the hoop lamina represents only the local crack and does not affect so significantly the deformation of FW pipes (+0 + 90°).
In tensile tests, specimens containing the +45 ° lamina exhibit a peculiar large elongation. It may be due to the nonlinearity of the shear stress- strain relation of the FRP lamina, s Although the large elongation cannot be predicted, the analytical procedure developed here would provide a useful measure for the engineering design of FRP laminates and FW pipes.
R E F E R E N C E S
1. Ban, K., Otsuka, H. and Takahashi~ K., Development of light wheelchair with carbon/glass hybrid FRP-FW pipes~ Trans. JSCM, 6 (1980) 9-13.
2. Ban, K., Otsuka, H. and Takahashi, K., The effect of hoop winding on mechanical properties of GFRP-FW pipes (in Japanese), J. Society of Fibre Science and Technology, Japan (Sen-i GakkaishiL 37 (1981) 420-5.
3. Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. London, Set. A, 241 (1957) 376-96.
4. Tsai, S. W. and Wu, E. M., A general theory of strength for anisotropic materials, J. Composite Materials, 5 (1971) 58-80.
5. Mori, T. and Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., 21 (1973) 571-4.
6. Takahashi, K., Harakawa, K., Tanaka, K. and Sakai, T., Analysis on the effect of filler orientation distribution in elastic reinforcement theory (in Japanese), J. Society of Materials Science, Japan (Zairyo ), 26 (1977) 1232-43.
7. Takahashi, K., Harakawa, K. and Sakai, T., Analysis of the thermal expansion coefficients of particle-filled polymers, J. Composite Materials Supplement, 14 (1980) 144-59.
8. Rotem, A. and Hashin, Z., Failure modes of angle ply laminates~ J. Composite Materials, 9 (1975) 191-206.
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