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German work on grp design
German work on grp design ~
GREENWOOD
A short review is given of the methods of designing grp laminates developed by Puck and his co-workers. The method is based on an analysis of the stresses and strains in the individual plies making up the laminate and is particularly applicable to filament winding. Resin cracking and fibre failure are treated separately and a fracture criterion is developed for each. The laminate design can be optimized for either failure mode. The thermal stresses which arise from the operating temperature being different from the resin curing temperature are calculated and are found to be often of the same order of magnitude as the mechanical stresses. Numerical examples are given in an appendix.
NO TA T/ON C C1, C2
E E' G
elastic stiffness auxiliary constants (Equation 4) Young's modulus augmented Young's modulus (Equation 1) shear modulus
m
=
Nx,Ny
surface forces per unit width (Fig. 6) elastic compliance thickness of laminate total thickness of plies with fibre angle cok temperature difference overexertion factor longitudinal thermal expansion coefficient shear strain shear thermal expansion coefficient position coordinates in Fig. 6 longitudinal strain correction factor (Fig. 5) Poisson's ratio longitudinal stress shear stress fibre content by volume angle
$
t tk AT
u
O~
3' I" 6,X
V O T
o±/otl
Subscripts
B C cyl f i,j k m Me o T Th x,y 1,2 3 I1 ± #
general fracture strength compressive strength values at beginning of cylindrical section (Fig. 6) fibre general suffixes for elastic constants: can take the values 1,2,3 refers to a particular fibre angle matrix caused by thermal expansion and having a mechanical effect initial value (Fig. 4, Fig. 6) tensile strength caused by thermal expansion reference axes of laminate (correspond to 1,2) reference axes of laminate (not in C1 ,C2 ) shear, referred to axes 1 and 2 parallel to fibres perpendicular to fibres shear, referred to the fibre and perpendicular directions
Double suffixes are interchangeable except in the case of Poisson's ratio, where the .first suffix denotes the direction of contraction and the second that of the applied stress.
A ccen t
^
applying to the laminate as a whole
German literature on fibre-reinforced plastics is extensive. Very little of it has been published in English or is known outside the German-speaking area. This paper is intended to describe the theory and practice of grp design developed by Puck and his co-workers at Darmstadt and Basle in order to make it more accessible to the reader who does not speak German. The most comprehensive summary is given by Puck ~ which covers all the earlier papers. 2-9 Paper 10 is in English and a recent review11:2 includes some of the experimental conclusions. The present paper does not intend to cover comparable British and N. American work.
Kent InstrumentsLtd, Luton, Bedfordshire,England.
COMPOSITES . JULY 1977
In general the same notation is used as in the original papers. The design methods used are applicable to grp plates and shells containing continuous fibres and are particularly suited to filament winding. The fracture criteria used are not bare mathematical expressions: they are founded on the observed behaviour of the material and are supported by much experimental data. Particular attention is paid to resin cracking. A N A L YSIS OF S T R E S S E S IN INDIVIDUAL PLIES
There are two mathematical theories which a designer can
175
use to calculate the stresses in a component. If he is interested solely in the ultimate fracture strength, provided that the fibre structure will hold together even after the resin has fractured, he may use the 'conventional' network theory which ignores the presence of the resin. This theory is used for designing lined pressure vessels. If the designer has to consider resin cracking, he must use the 'continuum' theory. This would be the case if he has to calculate the dimensional strains, if he has to keep the thermal expansion coefficients within certain limits, if the component is to be used at very low temperatures, or in electrical engineering where cracks in the resin can lead to dielectric breakdown of the composite. If there is a single stress state throughout the laminate he may be able to use one of the optimization techniques discussed under the heading 'optimization with uniform stress state', which can be used for either theory. If not, he must calculate the stresses and strains in the individual plies. To do this he must start by choosing a particular lay-up structure, that is a particular succession of plies k with fibre angles 6ok. His initial choice will be based on intuition and experience, influenced if necessary by the limitations of the manufacturing method. He will, however, usually choose a 'balanced' structure consisting of pairs of fibre angles -+ 6ok (German: ausgeglichene Winkelverbunde = AWV) directed symmetrically about the directions of principal stress. Most common is a structure with one set of fibres parallel to the higher principal stress and one pair at angles -+ 6ok about it. The reasons for this will be discussed below. For the sake of calculation, all plies with the same fibre angle 6ok are grouped together to give a total thickness tk, where ~ t = t, the thickness of the laminate. A strict k rotational angle convention is observed. The designer will then calculate the stresses in the individual plies using the method described below and, if necessary, the thermal stresses, which can arise when the operating temperature is different from the temperature at which the resin is cured. He will compare them with the fracture criterion and any other conditions imposed and then modify them until he reaches what he regards as the best solution. The procedure can be carried out very efficiently using a large programmable table-top calculator or a computer with direct access. To calculate the behaviour at stresses above that at which resin cracking starts, or to derive the contours of doubly curved surfaces, the designer will require a fullsize computer. To calculate the stresses in individual plies, the procedure is as follows: 1. Under the assumption that the stress state does not change across the thickness of the laminate (shell), the elastic moduli for a single ply are calculated from the fibre content using the following equationsXa :
Ell :
-
where ¢ = fibre content by volume, Ell and E± are the longitudinal and transverse Young's moduli respectively, u is Poisson's ratio (when it is subscripted, the first subscript denotes the direction of contraction and the second the direction of applied stress), and G# is the shear modulus referred to the fibre and transverse directions. The subscripts f and m refer to fibre and matrix respectively. In an earlier paper I Puck gives the following simplified formulae which assume values for the moduli of glass and a non-flexibilised resin. They are given here in SI units:
Ett = (74800¢ + 3600)MPa V±lI = 0.37 - 0.12¢ E± = (1.530¢ 2 - 1.160¢ + 0.475)EII
G# = (0.455¢ 2 - 0.360¢ + 0.170)Ell Layers of woven cloth are treated as two plies of mutually perpendicular fibres. The elastic constants of layers of glass mat are given by E = (29 5 8 0 ¢ -
G = (10780¢ + 1400)MPa Further data are given in Reference 14. Creep moduli can be substituted for the calculation of long-term strains. 2. The plane strain elastic stiffnesses for a ply are derived: Ell
Clt -
1
176
Em 1 - v2m
VJ_llVllj_
-
e± cl -
1 -
V±liVlli
(3)
C±lI = V±llC± c# :
G#
3. The elastic stiffnesses for each individual ply k are rotated to give the plane strain stiffnesses parallel to the axes of reference of the laminate, which are usually its symmetry axes; they are numbered 1 and 2. Cllk
=
Cl[
C33k
COS4(-Ok dr ¢1 sin 4 60k + 2CI sin 2 26ok
C± COS4 (.4)k -{- Cil
1 sin 4 6ok + ~-Cl sin 2 26ok
1 = c# + ~-C2 sin 2 26ok
)
(4)
(1) CI2k =
, where E m -
4740¢ 2 + 3920) MPa
v = 0.34 - 0.075¢
¢)v m
E± = E m [(1 - ¢)1.4s + CE,m/Ef ]
(2)
v~,±=v±,,(~)
C22 k =
CEf + (1 - ¢)E m
v±lI = Cvf + (1
(1) Gm(l dr 0 . 4 ~ ) G# = (1 - ¢)L4S + CGm/Gf
where C1
C±II dr
=
c±l I +
1 ~C2 sin 2 26ok 2c#,
C2 =
ctl + c± - 2C1
COMPOSITES. JULY 1977
These expressions are symmetrical for +co and -co. The subscripts refer to the directions of stress and strain and are in these calculations interchangeable. The subscript 3 represents shear with respect to the directions 1 and 2.
CX = g l l O x
6y = §126x + S22¢Sy
1 [6'2 sin 2 LOk -- (¢11 - - C I ) ]
(9)
7xy = ,~337¢xy
In the normal case, where the laminate is symmetric about the axes 1 and 2, proceed to section 4. If this is not so, it is necessary to continue as follows: C13 k =
-1- g126y
sin 2cok
(5)
7. These strains are rotated for each individual ply to give the strains parallel and perpendicular to the fibres: 1 @Ilk = @x cOS2 ('Ok + @y Sin2~-Ok -- ~-')'xy sin26°k
1
C23k = ~- [(CA -- Ca) - C2 sin 2 Wk] sin 26o k
1
@±k = @x sin= ~Ok + @y cOS2 C-Ok + ~-')'x y sin 26o k which change sign if co is changed to -co. These are then summed as in section 4 to give c13 and c2a, the corresponding stiffnesses for the laminate. The compliances s]i in section 5 are then derived by finding the inverse of the matrix cij and the strains in section 6 are deduced from the relation @x = S l l O x
+ S120y + S13Txy
When co = 90 °, c ,
= c±, c22 = ¢11
]
t,.
322 = 2~ "~¢22k
k t
= C#'~# k
(11)
Wqaenco = 90 °, Oll = cii6y + till@x, =
¢Lil@Y + ¢i6x, T# = --CAll@x,
FRACTURE CRI TER ION
cll = 2] __ tk Cll k k t
(7) C33k
312 = Z __ tk C12 k k t
Fracture in grp can take place by means of resin cracking, which includes cohesive fracture within the resin and adhesive fracture at the interface, or by fibre fracture. Experimental observation, for example the 'knee' in the stress/strain curve of a glass cloth reinforced laminate, shows that the fracture of a laminate is usually a gradual process which starts at an early stage with resin cracking in individual plies but which does not lead to catastrophic failure until the fibres themselves start to break. The fracture criterion respects this observation. Resin cracking in a ply takes place under the action of shear stresses in the plane of the ply and of tensile or compressive stresses perpendicular to the fibres. The behaviour under a combination of both is given for an epoxy resin matrix by the points in Fig. 1. An approximation for oi t> 0, which is also predicted theoretically by assuming cohesive fracture of the matrix, is the ellipse
Circumflexes (A) are used to denote laminate parameters. 5. The stiffnesses are inverted to give the compliances of the laminate: -
°Ilk ---- CIICIIk + ¢Alleik
These stresses must now be compared with the fracture criterion.
4. Continuing the calculations for a laminate symmetrical about the axes 1 and 2, the ply stiffnesses are multiplied by the proportional thicknesses and summed to give the stiffnesses of the laminate:
gH
8. The elastic stiffnesses are used to derive the stresses in each individual ply:
o L
Then continue as in section 7.
kt
When co = 90 °, ell = ey, eL = 6x, 7# = - T x y
T#k
(6)
C33 = C#, C12 ~--- Cill, C13 = C23 = 0
333 = ~ __ tk
7#k = (6x -- 6y)sin2co k + 7xy cos2co k
OAk = CillCll k + C£@Ak
ey = gl2Ox + g22Oy + g23"gxy "/xy = S130x + $230"y + S33Txy
(lo)
c22 N
g22 --
311 N 312
ffl2 --
oIT ] (8)
N
V#"}
With more data available on the effect of combined shear and compressive stresses, a4-16 a better approximation can be ma de:
1
~33
--
^ C33
w h e r e N = 3H c22 - 322 6. The applied stresses, referred to the axes of reference of the laminate fix, oy, f x y , a r e used to give the corresponding strains. If the axes of reference are identical with the axes of principal stress, f x y = 0.
COMPOSITES . JULY 1977
--o~ + o±Olc + alT olT Oic o LxO±c
+ {T# 12 = 1 ~T#B}
(13)
where O±T and Cr±c are the transverse tensile and compressive strengths respectively (the sign convention used here is that stresses are written positively in the algebra and assume negative numerical values in compression: other conventions are found in the literature). Measurements with epoxy resins
177
g i.J
In practice, however, the most important criteria are: For resin cracking: Equation 13. For fibre fracture: Oil = 11lifT, 1111 = 1111fC
o Oh x IOOh a 500 h • IO00h
8o
%o
~ 40 .1= if)
-160
-120 -80 -40 Transverse stress, o/[MPa]
0
40
Fig. 1 Fracture strength of a grp ply subjected to combinations of transverse stress o± and shear stress r # according to Reference 16. The test specimens were cylinders, the resin system was Araldite CY232/HY951, the fibre content by volume 0.65 and the test temperature 23°C The points represent the short-term strength (0 h) and the stressescorrespondingto fracture after loading for 100, 500 and 1000 hour~ The ellipsesrepresentthe fracture criterion according to Equation 13 for fracture instantaneously and after 1000 hours
The effect of stress in the fibre direction 1111on the resin cracking behaviour, expressed geometrically by the narrowing of the ellipsoid in Fig. 2 towards its ends, is only significant when one is carrying out iterative calculations at relatively high stresses after resin cracking has taken place. Bardenheier and Schneider ~s46 have compared this criterion with the Hoffmann and Tsai and Wu criteria. They agree on the behaviour in the o±, r# plane but not in the 1111direction. The resin cracking and subsequent behaviour of glass mat layers is not predicted by this theory and has been the subject of much independent study. Recent results for laminates with high fibre content are given in Reference 17. Design safety factors for use with these criteria include long term stresses 2 thermal ageing 1.4 fibre misorientation 1.2 inhomogeneities 1.2 which must be multiplied together if more than one factor is present.
yield values close to 11±T = 40 MPa, 11±c = -150/dPa, r#B = 60 MPa giving the values REASONS FOR THE LOW TRANSVERSE STRENGTH
o1(11± + 110) + r~ _ 1 6000 3600 to Equation 13. The curve is shown in Fig. 1. F6rster and Knappe ~a show that r#B is independent of the fibre content, ~, at normal values, but that 11iT increases from 38 MPa a t ~ = 0.5 to 55 MPa at ~ = 0.7. Along the third axis, corresponding to the stress in the fibre direction 1111,one must differentiate between fibre and matrix failure. If the resin alone were to fail in this direction, it would do so at its fracture strain of about 5%. The elastic modulus of the ply in this direction, Ell, is about 40 000 MPa and the corresponding stress in the composite at resin failure, 11limB,is about 2000 MPa. Using Equation 12 and completing the ellipsoid, the criterion for resin failure be comes 9'*°
The transverse strength of 40 MPa is lower than the tensile strength of the resin; Puck ~,4'~° goes to some length to explain why. The transverse modulus of the fibres is high in comparison with that of the resin. As a result, when the laminate is subjected to an overall transverse strain, the strain in the resin at the narrowest points between neighbouring fibres is very much higher than the overall applied strain and cracking occurs at correspondingly low macroscopic stresses. This 'strain magnification' is illustrated in Fig. 3.
Modem fibres such as carbon and aromatic polyamide have an unexpected advantage over glass in this respect, as their transverse modulus is considerably lower than their longitudinal modulus and the strain magnification between
o
OIImB
1
~--111.LT
+ ~r#B/
= 1
(14)
Sheor stress,r#
which was that used by F6rster in his studies of the behaviour of a laminate at higher stresses after resin fracture has commenced. Adapting this to Equation 13, the criterion becomes
( 111_____j_1~ = 1111rnB]
0±2 1_11±(o±c__+ O±Tt+(r# O-LT111C
~ 11.I.TOIc ]
/2 = 1
~'/'#B ]
~in
c r ~
Fibre / / foilure
k
~
~
~
~
"
--
Longitudinol stress,o-j.
(15) in which case the apex of the ellipse is no longer on the main o u axis, ie the maximum strength in this direction is only reached if a small compressive stress is applied in the transverse direction. Before the stress 11limB is reached, however, the fibres break. This happens at a tensile stress, olltT ( ~ 100 MPa) or compressive stress, 1111fc,independently of the stress in the resin. Geometrically, this occurrence is represented by two planes which cut off the ends of the ellipsoid to complete the fracture surface (Fig. 2). 9
178
~w.O'1/
Fig. 2 The failure surfacefor a grp ply as given by Equation 15 for resincracking and by the criteria o u = OllfT; Oll = ourC for fibre fractur~ The vertical axis is shear stress, not tensilestressin the third dirnensiort The outer surface of the 'loaf' representsresin cracking, the cut off ends fibre fractur~ The central ellipsecorresponds to the outer ellipse in Fig. 1.
COMPOSITES. JULY 1977
O@C)
is introduced, which increases from zero at resin cracking. On this depends the correction factor ~ in Fig. 5. It is different for transverse tension and compression, because after the resin has cracked in compression it can continue to carry a certain stress, while in tension it can carry very little. The factor 77is calculated for each step in the process and the reduced values r/E±, r/l)±lI and r~G# used in the calculation. F6rster, Kraft and Knappe ~a'21'22 have carried out experiments on filament-wound tubes with a simple +-w structure and compared them with the predictions of this theory. The
Fig. 3 'Strain magnification' in grp according to Reference 10. If a given strain is imposed on the idealised composite in the direction shown, the upper strip, which passes between the fibres, will take it up without difficulty. In the lower strip, however, the rigid fibres can hardly extend and the strain has to be taken up by the narrow resin sections between the fibre~ This leads to premature resin cracking and low values of OiT.
I.o-
rt
bLcompression
neighbouring fibres correspondingly less. Unfortunately the adhesion of aromatic polyamide fibres is also less. Some practical results with carbon and glass fibres are given in Reference 18. In these theories, cohesive fracture of the resin and adhesive fracture of the interface are grouped together. Schneider 19 has confirmed that, when the fibres have the proper surface treatment, fracture takes place by cohesive failure of the resin under practically all stress conditions. Adhesive failure can only take place under direct transverse tensile stresses, and then at stresses only marginally lower than cohesive failure. BEHA VlOUR AFTER RESIN CRACKING HAS TA KEN PLACE
0
0
o'j. , o'j. CrLT °'j. c
I I0
IO
To calculate the behaviour o f the laminate once resin
cracking has started in one of the plies, an iterative procedure must be used. The simplest method is the following. As soon as the stress in a particular ply reaches the resin cracking stress, the elastic constants E±, PiLl and G# for that ply are set equal to zero, leaving only Ell unchanged. The elastic properties of the laminate are then recalculated along with the stress in each individual ply. As the external stress is increased further, the stresses in the plies increase in proportion, until a second ply reaches its resin cracking stress. The transverse elastic constants of that ply are then put equal to zero and the process is repeated. This continues until the stress in the fibre direction Oll in one of the plies reaches the fibre fracture strength OiIfT or OllfC. This is the point at which the laminate fails completely. Puck 9 and F6rster and Knappe 2°, applying this method to filament-wound cylinders, introduce two modifications. The first takes into account the non-linear elastic behaviour of the resin at high strains, as shown in Fig. 4. The second correction arises from the fact that at resin cracking the transverse elastic constants of a ply do not vanish immediately but sink gradually towards zero with increasing strain. As a measure of the 'overexertion' of the laminate the factor
\OmB]
r#B
COMPOSITES . JULY 1977
\O±a]
1.0 ~e Fig. 4 The correction terms used to take into account the nonlinearity of the resin behaviour according to Reference 20. The resin systems are (a) Araldite CY232/HY951 with fibre content 0.63 and (b) Araldite LY556/HT9721DYO32/DY062 with fibre content 0.54. As oI and T# increase towards their respective fracture strengths Oj.T, o±C and T#B, SO the relevant elastic moduli E.L and G # decrease relative to their initial or 'tangential' values Ei0 and G#o. In theory, these diminished values should be used in every calculation for resin cracking
179
1,0
The stress system is referred to the direction of its principal stresses Ox, Oy. If both are positive, the optimum stress
state is an isotensoid, that is the strain in all directions is equal. The shear strain is zero. In this case ell = e± = e and for each ply:
g-
E"
ion
oll = (ql + Cl,)e = e.E, 1 - v±llv,± ]
_g
ol = (C±lI +
(16)
( l+v±,, 1
c±)e
= eE± 1 - p±llPllt]
giving ol _ clll + c l = fill ell + c±ll
[1V l+l l ~ E 1 k 1 + PIll/Eli
= m
(17)
m Overexertion,
iJ
IO
Fig 5 Correction term ~ as a function of the overexertion 0 of a laminate after resin cracking according to Reference 20. In subsequent calculations E.L, V±lI and G# are each multiplied by ~1and the reduced values entered in the calculation
measured stress/strain curves agreed well with the calculations; the resin cracking stress was also predicted satisfactorily at low and high angles but not at -+45° and -+54° , where the measured resin cracking stress, as observed by the leakage of fluid from the centre of the tube, was much higher. At these angles the stress causing the resin to crack is predominantly shear, which in contrast to a transverse tensile stress o± does not cause the cracks formed to open. Values of resin cracking stress determined by fluid leakage are therefore artificially high; seen the other way a pipe can be used above its resin cracking stress if the fibre angle is chosen such that the cracks, once formed, do not tend to open up when the stress is further increased. The designer must judge how much resin cracking can be tolerated. A lined pressure vessel for short-term use can function at stresses well above that at which resin cracking starts. An electrical insulator for use in humid air must exhibit no resin cracking at all, while if it is to be immersed in oil a certain amount of resin cracking can be tolerated, as the cracks will fill with oil and this will have little effect on the insulating properties. At this point the continuum theory can be compared with the commonly used network theory for the design of grp laminates. The network theory assumes that the fibres alone carry the stress, ie that the transverse elastic constants El, V±ll and G# are zero. It is therefore equivalent to the limiting case when all the resin has cracked and can be used as a short cut to estimate the ultimate fibre fracture stress. F6rster has compared the two theories numerically for this limiting case and finds an average discrepancy of 15%. The network theory is simpler and can be used to give a general idea of the type of reinforcement necessary, or where a liner is used so that one is no longer interested in resin cracking; however, as mentioned above, it cannot be used for the calculation of deformations, resin cracking or thermal expansion effects.
OPTIMIZATION WITH UNIFORM STRESS STATE
If the stress state in the laminate is uniform, the designer can optimize the fibre orientation without going through the calculation outlined above.
180
This ratio of transverse to parallel stress in the plies is therefore fixed by the elastic constants to be about 0.3. Since the strength ratio is in fact about 1:20 = 0.05, this shows that resin cracking takes place at a much lower stress than fibre fracture and that the fibres can only be used to the full if resin cracking is permitted. The stresses in the individual plies k are then referred to the direction of principal stress in the laminate: Oxk =
¢Yllc°s2
Wk + (71 sin z Wk
(18)
Oy k = oll sin2Wk + OLCOS~ Wk
and the total external stresses Ox, Oy are given by Ox = ~tk/,Oxk_==.. = a l I S t k / c o s 2 w k + O / s i n 2 w k "~'t -'~ k ~ t k OII
Oy =
|\ ]
°ll
Oy k ~
(19) From this one derives the optimization condition Cry_ m t k sin 2 w k = k t
Ox
(20)
(1 -- m) 1 + 17x
with m as given in Equation 17. There is a condition (l/m) i> (ay/Ox)/> m which takes into account that if the ratio of principal stresses is too anisotropic the optimum solution is a unidirectional laminate. This condition is satisfied by a range of laminates of which one consists of two angles only, ±w. This solution is, however, not recommended. Supposing that such a two-angle laminate is stressed to the point at which resin cracking occurs. The value of m decreases and the fibre angles are no longer matched to the applied stress. The fibre structure will then deform violently. A structure with two pairs of fibre angles (counting 90 ° and 0 ° as a pair each) will, however, be stable and continue to carry the applied stress. The inclusion of more than two pairs of angles does not offer any particular advantage and is more complicated to make. The result is usually a structure with fibres at three single angles: one parallel to the higher principal stress and one pair, +w, symmetrically placed about it and satisfying Equation 20. All such solutions are equivalent. The use of the optimization condition to determine winding patterns is given in Example 1 of the Appendix.
COMPOSITES. JULY 1977
The magnitude of the transverse stress o± is given by o± =
(Ox + Oy)
(21)
Filament winding theory dictates that the fibre must be laid along a geodesic line so that it does not slip. Geometrically, this yields the condition sin ~ = constant
For the classic case of a cylinder under internal pressure, oy/Ox = 2. The network theory is represented by putting m = 0. For a structure with only two fibre angles this yields the well-known fibre angle of +-54.7 ° , which is the optimum angle for a two-angle laminate taking fibre fracture as the sole criterion. If resin cracking is to be considered, the optimum angle for a two-angle laminate is about +-65°, depending on the fibre content. F6rster and Knappe obtain better results for -+54° for reasons given above. Again, however, laminates with at least three fibre angles are preferred. If the principal stresses Ox,Oy are of opposite sign the solution is more complicated? ~--2s If fibre fracture is the criterion, the best solution is to orient the fibres perpendicular to one another in the directions of the principal stresses with thickness dependent on the relevant tensile and compressive strengths. If resin cracking is the criterion, the stresses must be calculated in full and compared with the experimental fracture curve. The optimization procedure is applicable only if there is a uniform stress state. If the laminate has to support more than one stress state, for example a cylinder in bending (see Example 2 in the Appendix), then it will be necessary to calculate the stresses at the critical locations in full, compare them with the fracture criterion, and find the best compromise solution.
DOUBL Y CURVED SURFACES
F6rster and Kraft 26 have applied the theory to doubly curved surfaces, in particular the end domes of pressure vessels. For a body of rotation under internal pressure, as shown in Fig. 6 and described by the position coordinates and X, the following equation can be derived from pressure and curvature considerations alone:
},"-
l+
X'=[(Nx)x-21
(22)
where dashes ( ' ) denote differentiation with respect to the coordinate 8.
(23)
At the 'neck' of the pressure vessel there is a turn-round point for the fibre at which ~ = 90 °. If the corresponding value of k is Xo, h sin 60 = Xo
(24)
The equations so far have been entirely independent of the design theory used. For the network theory applied to a simple two-angle (+-co) laminate one can deduce from Equation 20 with tk/t = 1 and m = 0 that 5y/6 x = tan 2 60. Equation 24 then gives Ny _ f y = tan2 60 _ X~ Nx fx Xz - 3,~
(25)
which can be inserted into Equation 22 and integrated numerically to produce a suitable contour. For the continuum theory the corresponding equation is N__r = by = h2(ci + c±ll) + X~(Cll - cl) Nx fix h2(c/t + C ± l l ) - h 2 ( C l / - el)
(26)
which can be integrated in the same way. F6rster and Kraft 26 apply the further criterion that Equation 14, the fracture criterion for resin cracking, should apply over the entire dome surface, and they derive the corresponding contours. In their examples the contours calculated using the various methods differ little from one another, although the stresses and stress ratios differ widely. The contour calculated is taken as the central line of the vessel wall. The inner surface, which is that required for making the mandrel, is deduced by subtracting half the wall thickness t, which becomes greater as the radius ), decreases according to the equation t
=
A/(Xcyl/XO)2 - ' i (~k/~°) 2 _ 1
(27)
tcyl. ¥
In general, the end of a pressure vessel will be wound using a simple +co structure, while the central, cylindrical section will be strengthened with 90 ° windings in a proportion satisfying Equation 20.
T H E R M A L STRESSES
Schneider 27'28 has calculated the thermal stresses developed in a laminate at temperatures other than that at which the resin originally cured; more commonly, when it is cooled to room temperature or below. The method is similar to that used for calculating the mechanical stresses. 1. The coefficients of expansion of a ply are calculated:
pomt
oqt = a f +
C~m - c~f
1 -¢)Em
+ 1
[ 2.2¢(v~ - vm - t) otI = otm - ( o r m - °tf) li.l~(-~v2m +-~m~- ~ _ (1 + Pm)
Vm (Ef/Em)
--
Er/Em + Fig. 6 Coordinate axes and surface stresses used in calculating the end dome of a pressure vessel
COMPOSITES. JULY 1977
(1 -
1.1¢)/1.1~b
1
]
(28)
where a denotes coefficient of thermal expansion.
181
2. The coefficients are rotated through angles 60k to give the coefficients parallel to the axes of reference x, y of the laminate 0~xk = Oql COS2cok + O t l
all sin2
Otyk =
9. The strains in the individual plies which produce mechanical stresses are deduced by subtracting the normal thermal expansion: ellkMe = ellkT h -- alIAT
sin2 60k
60k + O~l cOs2 60k
(29)
Fxy k = (tXl -- all ) Sin 260k
eJ_kMe = 6J_kTh --
a±AT
(33)
3'#kMe = 3'#kTh
when 60 = 90°: ax = ix±, ay = all, Fxy = 0
10. The stresses in the individual plies are derived:
Fxy is a coefficient denoting shear strain per unit temperature in the same way that a denotes longitudinal strain; Fxy is asymmetrical for +60 and -60.
OiikTh = CllellkM e + C±II6±kMe O.LkTh = C±IlelIKMe + C±e±kMe
(34)
T#kTh = C#3'#kM e
3. The stiffnesses of the plies are derived, referred to the axes of reference of the laminate: Equations 4 and 5. 4. The 'thermally induced' stresses are calculated, referred to the axes of reference (for a totally constrained system) ~/xTh = A T 7
tk (OtxkCllk +
O~ykCl2k +
rxY k¢13k)
(0~xkCl2k + tXykC22k +
FxykC2ak)
-7,t
A T 7 tk -7,t
byTh =
~xyTh
= AT~ tk (tXxkClak +
(30)
where AT is the temperature difference with respect to the temperature of gelation of the resin and is usually negative. Oxth, Oyth are symmetrical for +co and -60, ~xyth changes sign. For 60 = 90 ° the brackets reduce to:
(0~I¢I+
Oql¢±ll)
for by : (Oq.CllI + ¢XlICll) for ~xy : 0 5, The stiffnesses of the laminate are calculated: Equation 7. If the laminate is not symmetric about the axes of reference the same remarks apply as in section 3 of the mechanical stress calculation. 6. The compliances of the laminate are calculated: Equation 8. 7. The laminate strains are derived: exTh = gn OxTh + §12 OyTh (31)
eyTh = §12bxT h + ,f220"yTh 3'xyTh = ff33TxyTh
Expressed in strain per unit temperature, these values denote the effective thermal expansion coefficients of the laminate. 8. The strains in the individual plies are derived: 1
ellkTh = 6xTh COS2 GOk + eyTh sin2 COk -- ~-3'xTh sin 26ok
1
eJ.kTh = exTh sin2 cok + eyTh cOs2 (..Ok + 5-3'xyTh sin 260k
3'#kTh = (exTh -- eyTh) Sin 260k + 3'xyTh COS2cok (32) The e's are symmetric for +¢o, 3'# changes sign. For co = 90 °, elITh = eyTh, elTh = exTh, 3'#Th = --3'xyTh.
182
The importance of the thermal stresses in a laminate should not be underestimated (see Example 3 in the Appendix). Cooling to sub-zero temperatures can produce resin cracking by thermal stresses alone which, unlike the mechanical stresses, cannot be reduced by increasing the wall thickness. The only consolation is that both the elastic modulus and the strength of the resin increase at low temperatures, while the thermal expansion coefficient decreases sharply. 29
O~ykC23k -}" FxykC33k)
k t
for bx :
11. These stresses should now be compared with the fracture criterion.
DESIGN HINTS EMERGING FROM THE WORK
The above methods should enable the designer to calculate the mechanical and thermal stresses in a laminate or shell, to compare them with the fracture criterion and to select the best laminate structure for his purpose. Some numerical examples are given in the Appendix. In conclusion a few design hints and details to be found in the various papers reviewed will be presented. There should always be at least three fibre directions present. the most commonly used structure being of the type 90 ° , -+60. The fibre structure is then capable of resisting the stresses even when resin cracking has taken place. In a cylinder to be stressed in axial tension this can be helped by placing any hoop windings on the inside to stop the diagonal windings trying to collapse towards the centre. If it is not possible to lay fibres in the direction of principal stress, two sets of diagonal fibres (+-cot, +-co=)should be used instead of the conventional (90 ° , +-co)arrangement. One such case is the cylinder with the higher principal stress in the axial direction, as winding in this direction is very difficult to achieve. Another is a cylinder with a conical section, upon which fibres slip when wound at 90 ° to the axis. In pipes and containers, the layer in which resin cracking is expected to take place last, if any, should be placed on the inside. The plies should be as thin as possible to stop resin cracking spreading from one ply to the next, or interwoven as automatically occurs with cloth laminates and filament winding. Layers of glass mat are used in this context as an internal layer to support the gelcoat or as a crack stopper around or between layers of rovings or woven cloth. 14,n Mat-cloth laminates have also been used successfully as high strength electrically insulating materials.* 7,3o The problem of buckling is treated in References 31 and 32. The principal parameter, as in the isotropic case, is the
COMPOSITES. JULY 1977
wall thickness. Reference 31 also gives details of how to achieve a void-free composite, and discusses the design of flanges.
angles and, on the other hand, filament winding is more difficult at angles below -+30°, a structure between 0.5 at -+33.1 ° and 0.8 at -+48.5 ° is recommended.
The continuum theory predicts that vessels under internal pressure should, if designed for maximum stress at resin cracking, be wound with fibre angles -+65° , or, better, have a structure with three fibre directions. Against this Forster and Knappe's evidence la that an angle of -+54° yields better results because the cracks, once formed, do not open up, comes as something of a non-result. This angle is, after all, the optimum angle according to the conventional network theory. The continuum theory, however versatile and however realistic in its basic principles, is only now being fully tested and consolidated by practical application.
If the resulting design factor o±T/o± = 40/17 = 2.35 for resin cracking is regarded as too small, the wall thickness must be increased. The stress in the fibre direction Oll = o±/m = 52 MPa. Since O I I T ~ 1000 MPa, the safety factor for fibre fracture is 19.
Example 2 A cylinder with the same dimensions and fibre and matrix parameters as in Example l is subjected to an internal pressure and a bending moment producing the following stresses:
A CKNOWL EDGEMENTS This work was started while working at BBC Brown Boveri & Co. Ltd., Baden, Switzerland, in collaboration with CibaGeigy Ltd., Basle, Switzerland. I thank Dr Puck and his group for their instruction and assistance.
Pressure contribution oy (circumferential) Ox (axial): tensile surface o x (axial): compressive surface rxy (shear)
Appendix
Bending contribution
Resultant stress (MPa)
40
0
40
20
10
30
20 0
- 10 0
10 0
NUMERICAL EXAMPLES Example 1 A cylinder container of diameter 200 mm and wall thickness 2.5 mm is subjected to an internal pressure of 1 MN/m 2 and an axial tensile load of 15 kN. The fibre content by volume is 0.56. What are the ideal winding patterns? From the dimensions above, in the circumferential direction: by = 40 MPa in the axial direction: fix = 29.55 MPa hence fry/fix = 1.354 Using Equations 1 and 3 and the values Ef = 73 000 MPa, E m = 3300 MPa, ~)f = 0 . 2 5 , ~'m = 0.37, the following numerical values are derived Cll
=
=
c# =
As the stress system is not uniform, the optimization condition given in Equation 20 cannot be used alone. The stresses must therefore be derived using Equations 4-11 and compared with the fracture surface in Equation 13. The highest stresses on the cylinder occur on the 'tensile' surface where the maximum bending stress is added to the pressure contribution; the optimum solutions for this case are therefore taken as a guide. The optimum solutions for the tensile surface are approximately those in Example 1 and satisfy the condition ~_~
43 410.5 MPa
c± = 11 762.5 MPa C±lI
What is the best winding pattern for minimal resin cracking stress?
3561.7 MPa 4783.7 MPa
Turning to Equation 17, m = 0.328 and from Equation 20, the condition to be satisfied by the winding angles is
~
tksin2wk = 0.649 k t
Examples o f structures which satisfy this equation are all fibres at -+53.6 ° (the network theory, with m = 0, gives +49.3 ° ) 0.8 of fibres at -+48.5 ° , 0.2 at 90 ° 0.5 of fibres at -+33.1 °, 0.5 at 90 ° 0.351 o f fibres at 0 ° (axial), 0.649 at 90 °. The transverse stress o± is deduced from Equation 21: o± = 17MPa Since it is desirable to have more than one pair of fibre
COMPOSITES . JULY
1977
k
t_k_k sin 2 ~ k = 0.641 t
o± = 17 MPa. The expression of the left hand side of Equation 13 with O±T = 40, o±C = --150, r # = 60 MPa, takes the value 0.37. On the compressive surface the stresses in the plies at 90 ° are very small; those in the plies at +w are given by:
~(o)
tk(o~)/t
o±(MPa)
T#(MPa)
Eq. 13
32.1 39.3 44.3 47.9 50.8 53.2
0.5 0.6 0.7 0.8 0.9 1.0
13.2 11.9 10.6 9.4 8.3 7.2
- 5.6 - 7.1 - 8.2 - 9.1 - 9.8 -10.4
0.28 0.26 0.22 0.21 0.19 0.17
The total stresses in the plies at -+~oon the compressive surface are therefore always lower than the stresses on the tensile surface. Their lowest value occurs in a structure consisting of fibres lying entirely at +53.2 ° . As mentioned above, however, once resin cracking has taken place a
183
laminate is required with more than two layers. To resist any further off-axis stress an intermediate structure of the type 0.8 at +47.9 °, 0.2 at 90 °, would be preferred.
10 11
Example 3 The solutions to Example 1 being equivalent, which results in the lowest thermal stresses? W i t h $ = 0.56, off = 4.8 x 10-6°C -1 and 0tm --- 60 x 10-6°C -~, from Equation 28 the values tz± = 29.27 x 10-6°C -t andtht = 6.69 x 10-6°C -~ are derived. Putting these into Equations 29-34 gives the thermal stresses for the four possible solutions given in Example 1 for a nominal temperature difference A T of - 1 0 0 ° C . Diagonal plies, -+w
90 ° plies
O ± ( M P a ) r#(MPa)
o±(MPa)
12
13
14 15 16
t_k t
6o(°)
17 1
53.6
16
5
(13) 1
0.8 0.5
48.5 33.1
17 18
4 3
13 15
0
19
0
16
0.351
nominal value only: there is no ply in this direction. The transverse stress is tensile and increases with decreasing ~o. The variation is not large, but the m i n i m u m stresses are obtained with a laminate consisting entirely of plies at +53.6 ° .
18 19 20 21
A laminate with 0.8 of the fibres at 48.5 °, 0.2 at 90 °, is, however, preferred, and can be achieved with only a small increase in thermal stress.
22
The thermal stresses for A T = - 1 0 0 ° C , by no means an unlikely temperature difference, amount to a third of the resin cracking strength.
23
REFERENCES
24
Puck, A. 'Einfuhren in das Gestalten und Dimensionieren' in 'Konstruieren und Berechnen yon GFK-Teilen', edited by Ehrenstein, G.W. and Martin, H.-D., pp 44-66. Supplement to Kunststoff-Berater, Umschau, Frankfurt, 1969 Puck, A. 'Dimensionierung tragender Leichtbaukonstruktionen aus Glasfaser-Kunststoffen', Kunststoffe 53 (1963)pp 150157 Puck, A. 'Glasfaser/Kunststoff in hochbeanspruchten Leichtbaukonstruktionen',Kunststoffe 53 (1963) pp 722733 Puck, A. 'Zum Deformationsverhalten und Bruchmechanismus yon unidirektionalem und orthogonalem Glasfaser/Kunststoff', Kunststoffe 55 (1965) pp 913-922 Puck, A. 'Zur Beanspruchung und Verformung yon GFKMehrschichtenverbund-Bauelementen: Teil 1. Grundlagen der Spannungs- und Verformungsanalyse', Kunststoffe 57 (1967) pp 284-293 Puck, A. 'Zur Beanspruchung und Verformung yon GFKMehrschichtenverbund-Bauelementen: Teil 2. Spannungsund Verformungsanalyse an GFK-Wickelrohren unter Ueberdruck', Kunststoffe 57 (1967) pp 573-582 Puck, A. 'Zur Beanspruchung und Verformung yon GFKMehrschichtenverbund-Bauelementen: Tell 3. Versuche an Mehrschichtenverbunden', Kunststoffe 57 (1967) pp 965973 Puck, A. 'Das "Knie" im Spannungs-Dehnungs-Diagramm und Rissbildung bei Glasfaser/Kunststoffen', Kunststoffe 58 (1968) pp 886-893 Puck, A. 'Festigkeitsberechnung an Glasfaser/KunststoffLaminaten bei zusammengesetzter Beanspruchung', Kunststoffe 59 (1969) pp 780-787
184
25 26
27
28 29 30 31 32
Puck, A. and Schneider, W. 'On failure mechanisms and failure criteria of filament-wound glass-fibre/resin composites', Hast and Polymers 37 (1969) pp 33-43 Dodds,R. 'Transverse fibre debonding in reinforced thermosets - a review. Part 2: Propagation of transverse cracks and ing'. RAPRA Members Journal 4 (1976) pp 49-56 Dodds,R. 'Transverse fibre debonding in reinforced thermosets - a review. Part 2: Propagation of transverse cracks and methods of reducing transverse fibre debonding'. RAPRA Members Journal 4 (1976) pp 71-76 Forster, R. and Knappe, W. 'Experimentelle und theoretische Untersuchungen zur Rissbildungsgrenzean zweisehichtigen Wickelrohren aus Glasfaser/Kunststoff unter Innendruck', Kunststoffe 61 (1971) pp 583-588 Schneider, W. 'Berechnung und Gestaltung hochbeanspruchbarer Bauteile aus glasfaserverst~rktem Kunststoff', VDIZeitung 115 (1973) pp 375-382 Schneider, W. and Bardenheier, R. 'Versagenskriterien fttr Kunststoffe', JMater Technol 6 (1975) pp 269-280 Schneider, W. and Bardenheier, IL 'Versagenskriterien ftlr Kunststoffe. Teil II: Experimenteile Ergebnisse', JMater Technol 6 (1975) pp 339-348 Fuehs, H. and Greenwood, J.H. 'Konstruktionselemente aus bearbeiteten GFK-Presslaminaten fur den Bau elektrischer Grossapparate'. Paper 12, 13th Annual Conference of the Arbeitsgruppe Verstarke Kunststoffe, Freudenstadt, W. Germany, 1976 Micafil-Nachrichten (in press) Greenwood, J.H. 'Glas-oder kohlenstoffverst~trkte Kunststoffe fur Kleinzentrifugen', Technische Rundschau 66 (1974) No. 38 oo 41.43 Schneider, W. 'Adhesive and cohesive failure criteria of unidirectional fibre/matrix composites', International Conference on Fracture, Munich, 1973, Paper VII-234 Forster, R. and Knappe, W. 'Spannungs- und Bruchanalyse an Glasfaser/Kunststoff-WickelkOrpern',Kunststoffe 60 (1970) pp 1053-1059 F6rster, R. 'ExperimenteUe Untersuchungen zu optimal ausgelegten dreischichtigen Wickelrohren aus Glasfaser/ Kunststoff unter Innendruck', Kunststoffe 62 (1972) pp 881-886 FOrster,R. and Kraft, G. 'Spannungs- und Deformationsanalyse an geschleuderten Glasfaser/Kunststoff-Rohren und Methoden zu ihrer Dimensionierung' Kunststoff-Rundsehau 18 (1971) pp 595-600 Forster, IL 'Grundlagen der Optimierung yon Mehrschichtenverbunden aus faserverst~'rkten Werkstoffen. Teil 1. Optimierungsregeln' Kunststoffe 62 (1972) pp 57-62 FOrster,IL 'Grundlagen der Optimierung yon Mehrschichtenverbunden aus faserverstarkten Werkstoffen. Teil 2. Das Verformungsverhalten von optimal ausgelegten Mehrschichtenverbunden unter ebener Beanspruchung', Kunststoffe 62 (1972) pp 181-186 Knappe, W. and Schneider, W. 'Bruchkriterien fttr unidirektionalen Glasfaser/Kunststoff unter ebener Kurzzeit- und Langzeitbeanspruchung', Kunststoffe 62 (1972) pp 864-868 FOrster,R. and Kraft, G. 'Vergleich verschiedener Schalenformen dUnnwandiger, fasergewickelter Rotationsschalen bei Dimensionierung auf Rissbildung', Kunststoffe 62 (1972) pp 587-591 Schneider, W. 'Warmeausdehnungskoeffizienten und W~rmespannungenyon Glasfaser/Kunststoff-Verbunden aus unidirektionalen Schichten', Kunststoffe 61 (1971) pp 273277 Schni;idet, W. 'W~rmeausdehnungskoeffizienten und W~rmespannungenyon Glasfaser/Kunststoff-Verbunden', Kunststoffe 63 (1973) pp 929-933 Hartwig,G., Puck, A. and Weiss, W. 'Thermische Kontraktion yon glasfaserverst~'rktenEpoxydharzen bis zu tiefsten Temperaturen', Kunststoffe 64 (1974) pp 32-35 Fuchs, H. a~d Greenwood, J.H. 'The application of glass fibre epoxy laminates in large turbogenerators' Composites 8 (1977) pp 75-80 Lissewski,D., Moeller, J. and Puck, A. 'HOchstspannungsPrtfftransformator far 2.4 MV mit GFK/Giessharzlsoliermanter, Kunststoffe 64 (1974) pp 734-741 Puck, A. and Ruegg, Ch. 'Experimentelle Untersuchungen zur BeulstabilitAtyon gewickelten GFK-Zylindern unter axialer Druckbelastung', Kunststoffe 64 (1974) pp 718-726
COMPOSITES. JULY 1977
GREENWOOD
A short review is given of the methods of designing grp laminates developed by Puck and his co-workers. The method is based on an analysis of the stresses and strains in the individual plies making up the laminate and is particularly applicable to filament winding. Resin cracking and fibre failure are treated separately and a fracture criterion is developed for each. The laminate design can be optimized for either failure mode. The thermal stresses which arise from the operating temperature being different from the resin curing temperature are calculated and are found to be often of the same order of magnitude as the mechanical stresses. Numerical examples are given in an appendix.
NO TA T/ON C C1, C2
E E' G
elastic stiffness auxiliary constants (Equation 4) Young's modulus augmented Young's modulus (Equation 1) shear modulus
m
=
Nx,Ny
surface forces per unit width (Fig. 6) elastic compliance thickness of laminate total thickness of plies with fibre angle cok temperature difference overexertion factor longitudinal thermal expansion coefficient shear strain shear thermal expansion coefficient position coordinates in Fig. 6 longitudinal strain correction factor (Fig. 5) Poisson's ratio longitudinal stress shear stress fibre content by volume angle
$
t tk AT
u
O~
3' I" 6,X
V O T
o±/otl
Subscripts
B C cyl f i,j k m Me o T Th x,y 1,2 3 I1 ± #
general fracture strength compressive strength values at beginning of cylindrical section (Fig. 6) fibre general suffixes for elastic constants: can take the values 1,2,3 refers to a particular fibre angle matrix caused by thermal expansion and having a mechanical effect initial value (Fig. 4, Fig. 6) tensile strength caused by thermal expansion reference axes of laminate (correspond to 1,2) reference axes of laminate (not in C1 ,C2 ) shear, referred to axes 1 and 2 parallel to fibres perpendicular to fibres shear, referred to the fibre and perpendicular directions
Double suffixes are interchangeable except in the case of Poisson's ratio, where the .first suffix denotes the direction of contraction and the second that of the applied stress.
A ccen t
^
applying to the laminate as a whole
German literature on fibre-reinforced plastics is extensive. Very little of it has been published in English or is known outside the German-speaking area. This paper is intended to describe the theory and practice of grp design developed by Puck and his co-workers at Darmstadt and Basle in order to make it more accessible to the reader who does not speak German. The most comprehensive summary is given by Puck ~ which covers all the earlier papers. 2-9 Paper 10 is in English and a recent review11:2 includes some of the experimental conclusions. The present paper does not intend to cover comparable British and N. American work.
Kent InstrumentsLtd, Luton, Bedfordshire,England.
COMPOSITES . JULY 1977
In general the same notation is used as in the original papers. The design methods used are applicable to grp plates and shells containing continuous fibres and are particularly suited to filament winding. The fracture criteria used are not bare mathematical expressions: they are founded on the observed behaviour of the material and are supported by much experimental data. Particular attention is paid to resin cracking. A N A L YSIS OF S T R E S S E S IN INDIVIDUAL PLIES
There are two mathematical theories which a designer can
175
use to calculate the stresses in a component. If he is interested solely in the ultimate fracture strength, provided that the fibre structure will hold together even after the resin has fractured, he may use the 'conventional' network theory which ignores the presence of the resin. This theory is used for designing lined pressure vessels. If the designer has to consider resin cracking, he must use the 'continuum' theory. This would be the case if he has to calculate the dimensional strains, if he has to keep the thermal expansion coefficients within certain limits, if the component is to be used at very low temperatures, or in electrical engineering where cracks in the resin can lead to dielectric breakdown of the composite. If there is a single stress state throughout the laminate he may be able to use one of the optimization techniques discussed under the heading 'optimization with uniform stress state', which can be used for either theory. If not, he must calculate the stresses and strains in the individual plies. To do this he must start by choosing a particular lay-up structure, that is a particular succession of plies k with fibre angles 6ok. His initial choice will be based on intuition and experience, influenced if necessary by the limitations of the manufacturing method. He will, however, usually choose a 'balanced' structure consisting of pairs of fibre angles -+ 6ok (German: ausgeglichene Winkelverbunde = AWV) directed symmetrically about the directions of principal stress. Most common is a structure with one set of fibres parallel to the higher principal stress and one pair at angles -+ 6ok about it. The reasons for this will be discussed below. For the sake of calculation, all plies with the same fibre angle 6ok are grouped together to give a total thickness tk, where ~ t = t, the thickness of the laminate. A strict k rotational angle convention is observed. The designer will then calculate the stresses in the individual plies using the method described below and, if necessary, the thermal stresses, which can arise when the operating temperature is different from the temperature at which the resin is cured. He will compare them with the fracture criterion and any other conditions imposed and then modify them until he reaches what he regards as the best solution. The procedure can be carried out very efficiently using a large programmable table-top calculator or a computer with direct access. To calculate the behaviour at stresses above that at which resin cracking starts, or to derive the contours of doubly curved surfaces, the designer will require a fullsize computer. To calculate the stresses in individual plies, the procedure is as follows: 1. Under the assumption that the stress state does not change across the thickness of the laminate (shell), the elastic moduli for a single ply are calculated from the fibre content using the following equationsXa :
Ell :
-
where ¢ = fibre content by volume, Ell and E± are the longitudinal and transverse Young's moduli respectively, u is Poisson's ratio (when it is subscripted, the first subscript denotes the direction of contraction and the second the direction of applied stress), and G# is the shear modulus referred to the fibre and transverse directions. The subscripts f and m refer to fibre and matrix respectively. In an earlier paper I Puck gives the following simplified formulae which assume values for the moduli of glass and a non-flexibilised resin. They are given here in SI units:
Ett = (74800¢ + 3600)MPa V±lI = 0.37 - 0.12¢ E± = (1.530¢ 2 - 1.160¢ + 0.475)EII
G# = (0.455¢ 2 - 0.360¢ + 0.170)Ell Layers of woven cloth are treated as two plies of mutually perpendicular fibres. The elastic constants of layers of glass mat are given by E = (29 5 8 0 ¢ -
G = (10780¢ + 1400)MPa Further data are given in Reference 14. Creep moduli can be substituted for the calculation of long-term strains. 2. The plane strain elastic stiffnesses for a ply are derived: Ell
Clt -
1
176
Em 1 - v2m
VJ_llVllj_
-
e± cl -
1 -
V±liVlli
(3)
C±lI = V±llC± c# :
G#
3. The elastic stiffnesses for each individual ply k are rotated to give the plane strain stiffnesses parallel to the axes of reference of the laminate, which are usually its symmetry axes; they are numbered 1 and 2. Cllk
=
Cl[
C33k
COS4(-Ok dr ¢1 sin 4 60k + 2CI sin 2 26ok
C± COS4 (.4)k -{- Cil
1 sin 4 6ok + ~-Cl sin 2 26ok
1 = c# + ~-C2 sin 2 26ok
)
(4)
(1) CI2k =
, where E m -
4740¢ 2 + 3920) MPa
v = 0.34 - 0.075¢
¢)v m
E± = E m [(1 - ¢)1.4s + CE,m/Ef ]
(2)
v~,±=v±,,(~)
C22 k =
CEf + (1 - ¢)E m
v±lI = Cvf + (1
(1) Gm(l dr 0 . 4 ~ ) G# = (1 - ¢)L4S + CGm/Gf
where C1
C±II dr
=
c±l I +
1 ~C2 sin 2 26ok 2c#,
C2 =
ctl + c± - 2C1
COMPOSITES. JULY 1977
These expressions are symmetrical for +co and -co. The subscripts refer to the directions of stress and strain and are in these calculations interchangeable. The subscript 3 represents shear with respect to the directions 1 and 2.
CX = g l l O x
6y = §126x + S22¢Sy
1 [6'2 sin 2 LOk -- (¢11 - - C I ) ]
(9)
7xy = ,~337¢xy
In the normal case, where the laminate is symmetric about the axes 1 and 2, proceed to section 4. If this is not so, it is necessary to continue as follows: C13 k =
-1- g126y
sin 2cok
(5)
7. These strains are rotated for each individual ply to give the strains parallel and perpendicular to the fibres: 1 @Ilk = @x cOS2 ('Ok + @y Sin2~-Ok -- ~-')'xy sin26°k
1
C23k = ~- [(CA -- Ca) - C2 sin 2 Wk] sin 26o k
1
@±k = @x sin= ~Ok + @y cOS2 C-Ok + ~-')'x y sin 26o k which change sign if co is changed to -co. These are then summed as in section 4 to give c13 and c2a, the corresponding stiffnesses for the laminate. The compliances s]i in section 5 are then derived by finding the inverse of the matrix cij and the strains in section 6 are deduced from the relation @x = S l l O x
+ S120y + S13Txy
When co = 90 °, c ,
= c±, c22 = ¢11
]
t,.
322 = 2~ "~¢22k
k t
= C#'~# k
(11)
Wqaenco = 90 °, Oll = cii6y + till@x, =
¢Lil@Y + ¢i6x, T# = --CAll@x,
FRACTURE CRI TER ION
cll = 2] __ tk Cll k k t
(7) C33k
312 = Z __ tk C12 k k t
Fracture in grp can take place by means of resin cracking, which includes cohesive fracture within the resin and adhesive fracture at the interface, or by fibre fracture. Experimental observation, for example the 'knee' in the stress/strain curve of a glass cloth reinforced laminate, shows that the fracture of a laminate is usually a gradual process which starts at an early stage with resin cracking in individual plies but which does not lead to catastrophic failure until the fibres themselves start to break. The fracture criterion respects this observation. Resin cracking in a ply takes place under the action of shear stresses in the plane of the ply and of tensile or compressive stresses perpendicular to the fibres. The behaviour under a combination of both is given for an epoxy resin matrix by the points in Fig. 1. An approximation for oi t> 0, which is also predicted theoretically by assuming cohesive fracture of the matrix, is the ellipse
Circumflexes (A) are used to denote laminate parameters. 5. The stiffnesses are inverted to give the compliances of the laminate: -
°Ilk ---- CIICIIk + ¢Alleik
These stresses must now be compared with the fracture criterion.
4. Continuing the calculations for a laminate symmetrical about the axes 1 and 2, the ply stiffnesses are multiplied by the proportional thicknesses and summed to give the stiffnesses of the laminate:
gH
8. The elastic stiffnesses are used to derive the stresses in each individual ply:
o L
Then continue as in section 7.
kt
When co = 90 °, ell = ey, eL = 6x, 7# = - T x y
T#k
(6)
C33 = C#, C12 ~--- Cill, C13 = C23 = 0
333 = ~ __ tk
7#k = (6x -- 6y)sin2co k + 7xy cos2co k
OAk = CillCll k + C£@Ak
ey = gl2Ox + g22Oy + g23"gxy "/xy = S130x + $230"y + S33Txy
(lo)
c22 N
g22 --
311 N 312
ffl2 --
oIT ] (8)
N
V#"}
With more data available on the effect of combined shear and compressive stresses, a4-16 a better approximation can be ma de:
1
~33
--
^ C33
w h e r e N = 3H c22 - 322 6. The applied stresses, referred to the axes of reference of the laminate fix, oy, f x y , a r e used to give the corresponding strains. If the axes of reference are identical with the axes of principal stress, f x y = 0.
COMPOSITES . JULY 1977
--o~ + o±Olc + alT olT Oic o LxO±c
+ {T# 12 = 1 ~T#B}
(13)
where O±T and Cr±c are the transverse tensile and compressive strengths respectively (the sign convention used here is that stresses are written positively in the algebra and assume negative numerical values in compression: other conventions are found in the literature). Measurements with epoxy resins
177
g i.J
In practice, however, the most important criteria are: For resin cracking: Equation 13. For fibre fracture: Oil = 11lifT, 1111 = 1111fC
o Oh x IOOh a 500 h • IO00h
8o
%o
~ 40 .1= if)
-160
-120 -80 -40 Transverse stress, o/[MPa]
0
40
Fig. 1 Fracture strength of a grp ply subjected to combinations of transverse stress o± and shear stress r # according to Reference 16. The test specimens were cylinders, the resin system was Araldite CY232/HY951, the fibre content by volume 0.65 and the test temperature 23°C The points represent the short-term strength (0 h) and the stressescorrespondingto fracture after loading for 100, 500 and 1000 hour~ The ellipsesrepresentthe fracture criterion according to Equation 13 for fracture instantaneously and after 1000 hours
The effect of stress in the fibre direction 1111on the resin cracking behaviour, expressed geometrically by the narrowing of the ellipsoid in Fig. 2 towards its ends, is only significant when one is carrying out iterative calculations at relatively high stresses after resin cracking has taken place. Bardenheier and Schneider ~s46 have compared this criterion with the Hoffmann and Tsai and Wu criteria. They agree on the behaviour in the o±, r# plane but not in the 1111direction. The resin cracking and subsequent behaviour of glass mat layers is not predicted by this theory and has been the subject of much independent study. Recent results for laminates with high fibre content are given in Reference 17. Design safety factors for use with these criteria include long term stresses 2 thermal ageing 1.4 fibre misorientation 1.2 inhomogeneities 1.2 which must be multiplied together if more than one factor is present.
yield values close to 11±T = 40 MPa, 11±c = -150/dPa, r#B = 60 MPa giving the values REASONS FOR THE LOW TRANSVERSE STRENGTH
o1(11± + 110) + r~ _ 1 6000 3600 to Equation 13. The curve is shown in Fig. 1. F6rster and Knappe ~a show that r#B is independent of the fibre content, ~, at normal values, but that 11iT increases from 38 MPa a t ~ = 0.5 to 55 MPa at ~ = 0.7. Along the third axis, corresponding to the stress in the fibre direction 1111,one must differentiate between fibre and matrix failure. If the resin alone were to fail in this direction, it would do so at its fracture strain of about 5%. The elastic modulus of the ply in this direction, Ell, is about 40 000 MPa and the corresponding stress in the composite at resin failure, 11limB,is about 2000 MPa. Using Equation 12 and completing the ellipsoid, the criterion for resin failure be comes 9'*°
The transverse strength of 40 MPa is lower than the tensile strength of the resin; Puck ~,4'~° goes to some length to explain why. The transverse modulus of the fibres is high in comparison with that of the resin. As a result, when the laminate is subjected to an overall transverse strain, the strain in the resin at the narrowest points between neighbouring fibres is very much higher than the overall applied strain and cracking occurs at correspondingly low macroscopic stresses. This 'strain magnification' is illustrated in Fig. 3.
Modem fibres such as carbon and aromatic polyamide have an unexpected advantage over glass in this respect, as their transverse modulus is considerably lower than their longitudinal modulus and the strain magnification between
o
OIImB
1
~--111.LT
+ ~r#B/
= 1
(14)
Sheor stress,r#
which was that used by F6rster in his studies of the behaviour of a laminate at higher stresses after resin fracture has commenced. Adapting this to Equation 13, the criterion becomes
( 111_____j_1~ = 1111rnB]
0±2 1_11±(o±c__+ O±Tt+(r# O-LT111C
~ 11.I.TOIc ]
/2 = 1
~'/'#B ]
~in
c r ~
Fibre / / foilure
k
~
~
~
~
"
--
Longitudinol stress,o-j.
(15) in which case the apex of the ellipse is no longer on the main o u axis, ie the maximum strength in this direction is only reached if a small compressive stress is applied in the transverse direction. Before the stress 11limB is reached, however, the fibres break. This happens at a tensile stress, olltT ( ~ 100 MPa) or compressive stress, 1111fc,independently of the stress in the resin. Geometrically, this occurrence is represented by two planes which cut off the ends of the ellipsoid to complete the fracture surface (Fig. 2). 9
178
~w.O'1/
Fig. 2 The failure surfacefor a grp ply as given by Equation 15 for resincracking and by the criteria o u = OllfT; Oll = ourC for fibre fractur~ The vertical axis is shear stress, not tensilestressin the third dirnensiort The outer surface of the 'loaf' representsresin cracking, the cut off ends fibre fractur~ The central ellipsecorresponds to the outer ellipse in Fig. 1.
COMPOSITES. JULY 1977
O@C)
is introduced, which increases from zero at resin cracking. On this depends the correction factor ~ in Fig. 5. It is different for transverse tension and compression, because after the resin has cracked in compression it can continue to carry a certain stress, while in tension it can carry very little. The factor 77is calculated for each step in the process and the reduced values r/E±, r/l)±lI and r~G# used in the calculation. F6rster, Kraft and Knappe ~a'21'22 have carried out experiments on filament-wound tubes with a simple +-w structure and compared them with the predictions of this theory. The
Fig. 3 'Strain magnification' in grp according to Reference 10. If a given strain is imposed on the idealised composite in the direction shown, the upper strip, which passes between the fibres, will take it up without difficulty. In the lower strip, however, the rigid fibres can hardly extend and the strain has to be taken up by the narrow resin sections between the fibre~ This leads to premature resin cracking and low values of OiT.
I.o-
rt
bLcompression
neighbouring fibres correspondingly less. Unfortunately the adhesion of aromatic polyamide fibres is also less. Some practical results with carbon and glass fibres are given in Reference 18. In these theories, cohesive fracture of the resin and adhesive fracture of the interface are grouped together. Schneider 19 has confirmed that, when the fibres have the proper surface treatment, fracture takes place by cohesive failure of the resin under practically all stress conditions. Adhesive failure can only take place under direct transverse tensile stresses, and then at stresses only marginally lower than cohesive failure. BEHA VlOUR AFTER RESIN CRACKING HAS TA KEN PLACE
0
0
o'j. , o'j. CrLT °'j. c
I I0
IO
To calculate the behaviour o f the laminate once resin
cracking has started in one of the plies, an iterative procedure must be used. The simplest method is the following. As soon as the stress in a particular ply reaches the resin cracking stress, the elastic constants E±, PiLl and G# for that ply are set equal to zero, leaving only Ell unchanged. The elastic properties of the laminate are then recalculated along with the stress in each individual ply. As the external stress is increased further, the stresses in the plies increase in proportion, until a second ply reaches its resin cracking stress. The transverse elastic constants of that ply are then put equal to zero and the process is repeated. This continues until the stress in the fibre direction Oll in one of the plies reaches the fibre fracture strength OiIfT or OllfC. This is the point at which the laminate fails completely. Puck 9 and F6rster and Knappe 2°, applying this method to filament-wound cylinders, introduce two modifications. The first takes into account the non-linear elastic behaviour of the resin at high strains, as shown in Fig. 4. The second correction arises from the fact that at resin cracking the transverse elastic constants of a ply do not vanish immediately but sink gradually towards zero with increasing strain. As a measure of the 'overexertion' of the laminate the factor
\OmB]
r#B
COMPOSITES . JULY 1977
\O±a]
1.0 ~e Fig. 4 The correction terms used to take into account the nonlinearity of the resin behaviour according to Reference 20. The resin systems are (a) Araldite CY232/HY951 with fibre content 0.63 and (b) Araldite LY556/HT9721DYO32/DY062 with fibre content 0.54. As oI and T# increase towards their respective fracture strengths Oj.T, o±C and T#B, SO the relevant elastic moduli E.L and G # decrease relative to their initial or 'tangential' values Ei0 and G#o. In theory, these diminished values should be used in every calculation for resin cracking
179
1,0
The stress system is referred to the direction of its principal stresses Ox, Oy. If both are positive, the optimum stress
state is an isotensoid, that is the strain in all directions is equal. The shear strain is zero. In this case ell = e± = e and for each ply:
g-
E"
ion
oll = (ql + Cl,)e = e.E, 1 - v±llv,± ]
_g
ol = (C±lI +
(16)
( l+v±,, 1
c±)e
= eE± 1 - p±llPllt]
giving ol _ clll + c l = fill ell + c±ll
[1V l+l l ~ E 1 k 1 + PIll/Eli
= m
(17)
m Overexertion,
iJ
IO
Fig 5 Correction term ~ as a function of the overexertion 0 of a laminate after resin cracking according to Reference 20. In subsequent calculations E.L, V±lI and G# are each multiplied by ~1and the reduced values entered in the calculation
measured stress/strain curves agreed well with the calculations; the resin cracking stress was also predicted satisfactorily at low and high angles but not at -+45° and -+54° , where the measured resin cracking stress, as observed by the leakage of fluid from the centre of the tube, was much higher. At these angles the stress causing the resin to crack is predominantly shear, which in contrast to a transverse tensile stress o± does not cause the cracks formed to open. Values of resin cracking stress determined by fluid leakage are therefore artificially high; seen the other way a pipe can be used above its resin cracking stress if the fibre angle is chosen such that the cracks, once formed, do not tend to open up when the stress is further increased. The designer must judge how much resin cracking can be tolerated. A lined pressure vessel for short-term use can function at stresses well above that at which resin cracking starts. An electrical insulator for use in humid air must exhibit no resin cracking at all, while if it is to be immersed in oil a certain amount of resin cracking can be tolerated, as the cracks will fill with oil and this will have little effect on the insulating properties. At this point the continuum theory can be compared with the commonly used network theory for the design of grp laminates. The network theory assumes that the fibres alone carry the stress, ie that the transverse elastic constants El, V±ll and G# are zero. It is therefore equivalent to the limiting case when all the resin has cracked and can be used as a short cut to estimate the ultimate fibre fracture stress. F6rster has compared the two theories numerically for this limiting case and finds an average discrepancy of 15%. The network theory is simpler and can be used to give a general idea of the type of reinforcement necessary, or where a liner is used so that one is no longer interested in resin cracking; however, as mentioned above, it cannot be used for the calculation of deformations, resin cracking or thermal expansion effects.
OPTIMIZATION WITH UNIFORM STRESS STATE
If the stress state in the laminate is uniform, the designer can optimize the fibre orientation without going through the calculation outlined above.
180
This ratio of transverse to parallel stress in the plies is therefore fixed by the elastic constants to be about 0.3. Since the strength ratio is in fact about 1:20 = 0.05, this shows that resin cracking takes place at a much lower stress than fibre fracture and that the fibres can only be used to the full if resin cracking is permitted. The stresses in the individual plies k are then referred to the direction of principal stress in the laminate: Oxk =
¢Yllc°s2
Wk + (71 sin z Wk
(18)
Oy k = oll sin2Wk + OLCOS~ Wk
and the total external stresses Ox, Oy are given by Ox = ~tk/,Oxk_==.. = a l I S t k / c o s 2 w k + O / s i n 2 w k "~'t -'~ k ~ t k OII
Oy =
|\ ]
°ll
Oy k ~
(19) From this one derives the optimization condition Cry_ m t k sin 2 w k = k t
Ox
(20)
(1 -- m) 1 + 17x
with m as given in Equation 17. There is a condition (l/m) i> (ay/Ox)/> m which takes into account that if the ratio of principal stresses is too anisotropic the optimum solution is a unidirectional laminate. This condition is satisfied by a range of laminates of which one consists of two angles only, ±w. This solution is, however, not recommended. Supposing that such a two-angle laminate is stressed to the point at which resin cracking occurs. The value of m decreases and the fibre angles are no longer matched to the applied stress. The fibre structure will then deform violently. A structure with two pairs of fibre angles (counting 90 ° and 0 ° as a pair each) will, however, be stable and continue to carry the applied stress. The inclusion of more than two pairs of angles does not offer any particular advantage and is more complicated to make. The result is usually a structure with fibres at three single angles: one parallel to the higher principal stress and one pair, +w, symmetrically placed about it and satisfying Equation 20. All such solutions are equivalent. The use of the optimization condition to determine winding patterns is given in Example 1 of the Appendix.
COMPOSITES. JULY 1977
The magnitude of the transverse stress o± is given by o± =
(Ox + Oy)
(21)
Filament winding theory dictates that the fibre must be laid along a geodesic line so that it does not slip. Geometrically, this yields the condition sin ~ = constant
For the classic case of a cylinder under internal pressure, oy/Ox = 2. The network theory is represented by putting m = 0. For a structure with only two fibre angles this yields the well-known fibre angle of +-54.7 ° , which is the optimum angle for a two-angle laminate taking fibre fracture as the sole criterion. If resin cracking is to be considered, the optimum angle for a two-angle laminate is about +-65°, depending on the fibre content. F6rster and Knappe obtain better results for -+54° for reasons given above. Again, however, laminates with at least three fibre angles are preferred. If the principal stresses Ox,Oy are of opposite sign the solution is more complicated? ~--2s If fibre fracture is the criterion, the best solution is to orient the fibres perpendicular to one another in the directions of the principal stresses with thickness dependent on the relevant tensile and compressive strengths. If resin cracking is the criterion, the stresses must be calculated in full and compared with the experimental fracture curve. The optimization procedure is applicable only if there is a uniform stress state. If the laminate has to support more than one stress state, for example a cylinder in bending (see Example 2 in the Appendix), then it will be necessary to calculate the stresses at the critical locations in full, compare them with the fracture criterion, and find the best compromise solution.
DOUBL Y CURVED SURFACES
F6rster and Kraft 26 have applied the theory to doubly curved surfaces, in particular the end domes of pressure vessels. For a body of rotation under internal pressure, as shown in Fig. 6 and described by the position coordinates and X, the following equation can be derived from pressure and curvature considerations alone:
},"-
l+
X'=[(Nx)x-21
(22)
where dashes ( ' ) denote differentiation with respect to the coordinate 8.
(23)
At the 'neck' of the pressure vessel there is a turn-round point for the fibre at which ~ = 90 °. If the corresponding value of k is Xo, h sin 60 = Xo
(24)
The equations so far have been entirely independent of the design theory used. For the network theory applied to a simple two-angle (+-co) laminate one can deduce from Equation 20 with tk/t = 1 and m = 0 that 5y/6 x = tan 2 60. Equation 24 then gives Ny _ f y = tan2 60 _ X~ Nx fx Xz - 3,~
(25)
which can be inserted into Equation 22 and integrated numerically to produce a suitable contour. For the continuum theory the corresponding equation is N__r = by = h2(ci + c±ll) + X~(Cll - cl) Nx fix h2(c/t + C ± l l ) - h 2 ( C l / - el)
(26)
which can be integrated in the same way. F6rster and Kraft 26 apply the further criterion that Equation 14, the fracture criterion for resin cracking, should apply over the entire dome surface, and they derive the corresponding contours. In their examples the contours calculated using the various methods differ little from one another, although the stresses and stress ratios differ widely. The contour calculated is taken as the central line of the vessel wall. The inner surface, which is that required for making the mandrel, is deduced by subtracting half the wall thickness t, which becomes greater as the radius ), decreases according to the equation t
=
A/(Xcyl/XO)2 - ' i (~k/~°) 2 _ 1
(27)
tcyl. ¥
In general, the end of a pressure vessel will be wound using a simple +co structure, while the central, cylindrical section will be strengthened with 90 ° windings in a proportion satisfying Equation 20.
T H E R M A L STRESSES
Schneider 27'28 has calculated the thermal stresses developed in a laminate at temperatures other than that at which the resin originally cured; more commonly, when it is cooled to room temperature or below. The method is similar to that used for calculating the mechanical stresses. 1. The coefficients of expansion of a ply are calculated:
pomt
oqt = a f +
C~m - c~f
1 -¢)Em
+ 1
[ 2.2¢(v~ - vm - t) otI = otm - ( o r m - °tf) li.l~(-~v2m +-~m~- ~ _ (1 + Pm)
Vm (Ef/Em)
--
Er/Em + Fig. 6 Coordinate axes and surface stresses used in calculating the end dome of a pressure vessel
COMPOSITES. JULY 1977
(1 -
1.1¢)/1.1~b
1
]
(28)
where a denotes coefficient of thermal expansion.
181
2. The coefficients are rotated through angles 60k to give the coefficients parallel to the axes of reference x, y of the laminate 0~xk = Oql COS2cok + O t l
all sin2
Otyk =
9. The strains in the individual plies which produce mechanical stresses are deduced by subtracting the normal thermal expansion: ellkMe = ellkT h -- alIAT
sin2 60k
60k + O~l cOs2 60k
(29)
Fxy k = (tXl -- all ) Sin 260k
eJ_kMe = 6J_kTh --
a±AT
(33)
3'#kMe = 3'#kTh
when 60 = 90°: ax = ix±, ay = all, Fxy = 0
10. The stresses in the individual plies are derived:
Fxy is a coefficient denoting shear strain per unit temperature in the same way that a denotes longitudinal strain; Fxy is asymmetrical for +60 and -60.
OiikTh = CllellkM e + C±II6±kMe O.LkTh = C±IlelIKMe + C±e±kMe
(34)
T#kTh = C#3'#kM e
3. The stiffnesses of the plies are derived, referred to the axes of reference of the laminate: Equations 4 and 5. 4. The 'thermally induced' stresses are calculated, referred to the axes of reference (for a totally constrained system) ~/xTh = A T 7
tk (OtxkCllk +
O~ykCl2k +
rxY k¢13k)
(0~xkCl2k + tXykC22k +
FxykC2ak)
-7,t
A T 7 tk -7,t
byTh =
~xyTh
= AT~ tk (tXxkClak +
(30)
where AT is the temperature difference with respect to the temperature of gelation of the resin and is usually negative. Oxth, Oyth are symmetrical for +co and -60, ~xyth changes sign. For 60 = 90 ° the brackets reduce to:
(0~I¢I+
Oql¢±ll)
for by : (Oq.CllI + ¢XlICll) for ~xy : 0 5, The stiffnesses of the laminate are calculated: Equation 7. If the laminate is not symmetric about the axes of reference the same remarks apply as in section 3 of the mechanical stress calculation. 6. The compliances of the laminate are calculated: Equation 8. 7. The laminate strains are derived: exTh = gn OxTh + §12 OyTh (31)
eyTh = §12bxT h + ,f220"yTh 3'xyTh = ff33TxyTh
Expressed in strain per unit temperature, these values denote the effective thermal expansion coefficients of the laminate. 8. The strains in the individual plies are derived: 1
ellkTh = 6xTh COS2 GOk + eyTh sin2 COk -- ~-3'xTh sin 26ok
1
eJ.kTh = exTh sin2 cok + eyTh cOs2 (..Ok + 5-3'xyTh sin 260k
3'#kTh = (exTh -- eyTh) Sin 260k + 3'xyTh COS2cok (32) The e's are symmetric for +¢o, 3'# changes sign. For co = 90 °, elITh = eyTh, elTh = exTh, 3'#Th = --3'xyTh.
182
The importance of the thermal stresses in a laminate should not be underestimated (see Example 3 in the Appendix). Cooling to sub-zero temperatures can produce resin cracking by thermal stresses alone which, unlike the mechanical stresses, cannot be reduced by increasing the wall thickness. The only consolation is that both the elastic modulus and the strength of the resin increase at low temperatures, while the thermal expansion coefficient decreases sharply. 29
O~ykC23k -}" FxykC33k)
k t
for bx :
11. These stresses should now be compared with the fracture criterion.
DESIGN HINTS EMERGING FROM THE WORK
The above methods should enable the designer to calculate the mechanical and thermal stresses in a laminate or shell, to compare them with the fracture criterion and to select the best laminate structure for his purpose. Some numerical examples are given in the Appendix. In conclusion a few design hints and details to be found in the various papers reviewed will be presented. There should always be at least three fibre directions present. the most commonly used structure being of the type 90 ° , -+60. The fibre structure is then capable of resisting the stresses even when resin cracking has taken place. In a cylinder to be stressed in axial tension this can be helped by placing any hoop windings on the inside to stop the diagonal windings trying to collapse towards the centre. If it is not possible to lay fibres in the direction of principal stress, two sets of diagonal fibres (+-cot, +-co=)should be used instead of the conventional (90 ° , +-co)arrangement. One such case is the cylinder with the higher principal stress in the axial direction, as winding in this direction is very difficult to achieve. Another is a cylinder with a conical section, upon which fibres slip when wound at 90 ° to the axis. In pipes and containers, the layer in which resin cracking is expected to take place last, if any, should be placed on the inside. The plies should be as thin as possible to stop resin cracking spreading from one ply to the next, or interwoven as automatically occurs with cloth laminates and filament winding. Layers of glass mat are used in this context as an internal layer to support the gelcoat or as a crack stopper around or between layers of rovings or woven cloth. 14,n Mat-cloth laminates have also been used successfully as high strength electrically insulating materials.* 7,3o The problem of buckling is treated in References 31 and 32. The principal parameter, as in the isotropic case, is the
COMPOSITES. JULY 1977
wall thickness. Reference 31 also gives details of how to achieve a void-free composite, and discusses the design of flanges.
angles and, on the other hand, filament winding is more difficult at angles below -+30°, a structure between 0.5 at -+33.1 ° and 0.8 at -+48.5 ° is recommended.
The continuum theory predicts that vessels under internal pressure should, if designed for maximum stress at resin cracking, be wound with fibre angles -+65° , or, better, have a structure with three fibre directions. Against this Forster and Knappe's evidence la that an angle of -+54° yields better results because the cracks, once formed, do not open up, comes as something of a non-result. This angle is, after all, the optimum angle according to the conventional network theory. The continuum theory, however versatile and however realistic in its basic principles, is only now being fully tested and consolidated by practical application.
If the resulting design factor o±T/o± = 40/17 = 2.35 for resin cracking is regarded as too small, the wall thickness must be increased. The stress in the fibre direction Oll = o±/m = 52 MPa. Since O I I T ~ 1000 MPa, the safety factor for fibre fracture is 19.
Example 2 A cylinder with the same dimensions and fibre and matrix parameters as in Example l is subjected to an internal pressure and a bending moment producing the following stresses:
A CKNOWL EDGEMENTS This work was started while working at BBC Brown Boveri & Co. Ltd., Baden, Switzerland, in collaboration with CibaGeigy Ltd., Basle, Switzerland. I thank Dr Puck and his group for their instruction and assistance.
Pressure contribution oy (circumferential) Ox (axial): tensile surface o x (axial): compressive surface rxy (shear)
Appendix
Bending contribution
Resultant stress (MPa)
40
0
40
20
10
30
20 0
- 10 0
10 0
NUMERICAL EXAMPLES Example 1 A cylinder container of diameter 200 mm and wall thickness 2.5 mm is subjected to an internal pressure of 1 MN/m 2 and an axial tensile load of 15 kN. The fibre content by volume is 0.56. What are the ideal winding patterns? From the dimensions above, in the circumferential direction: by = 40 MPa in the axial direction: fix = 29.55 MPa hence fry/fix = 1.354 Using Equations 1 and 3 and the values Ef = 73 000 MPa, E m = 3300 MPa, ~)f = 0 . 2 5 , ~'m = 0.37, the following numerical values are derived Cll
=
=
c# =
As the stress system is not uniform, the optimization condition given in Equation 20 cannot be used alone. The stresses must therefore be derived using Equations 4-11 and compared with the fracture surface in Equation 13. The highest stresses on the cylinder occur on the 'tensile' surface where the maximum bending stress is added to the pressure contribution; the optimum solutions for this case are therefore taken as a guide. The optimum solutions for the tensile surface are approximately those in Example 1 and satisfy the condition ~_~
43 410.5 MPa
c± = 11 762.5 MPa C±lI
What is the best winding pattern for minimal resin cracking stress?
3561.7 MPa 4783.7 MPa
Turning to Equation 17, m = 0.328 and from Equation 20, the condition to be satisfied by the winding angles is
~
tksin2wk = 0.649 k t
Examples o f structures which satisfy this equation are all fibres at -+53.6 ° (the network theory, with m = 0, gives +49.3 ° ) 0.8 of fibres at -+48.5 ° , 0.2 at 90 ° 0.5 of fibres at -+33.1 °, 0.5 at 90 ° 0.351 o f fibres at 0 ° (axial), 0.649 at 90 °. The transverse stress o± is deduced from Equation 21: o± = 17MPa Since it is desirable to have more than one pair of fibre
COMPOSITES . JULY
1977
k
t_k_k sin 2 ~ k = 0.641 t
o± = 17 MPa. The expression of the left hand side of Equation 13 with O±T = 40, o±C = --150, r # = 60 MPa, takes the value 0.37. On the compressive surface the stresses in the plies at 90 ° are very small; those in the plies at +w are given by:
~(o)
tk(o~)/t
o±(MPa)
T#(MPa)
Eq. 13
32.1 39.3 44.3 47.9 50.8 53.2
0.5 0.6 0.7 0.8 0.9 1.0
13.2 11.9 10.6 9.4 8.3 7.2
- 5.6 - 7.1 - 8.2 - 9.1 - 9.8 -10.4
0.28 0.26 0.22 0.21 0.19 0.17
The total stresses in the plies at -+~oon the compressive surface are therefore always lower than the stresses on the tensile surface. Their lowest value occurs in a structure consisting of fibres lying entirely at +53.2 ° . As mentioned above, however, once resin cracking has taken place a
183
laminate is required with more than two layers. To resist any further off-axis stress an intermediate structure of the type 0.8 at +47.9 °, 0.2 at 90 °, would be preferred.
10 11
Example 3 The solutions to Example 1 being equivalent, which results in the lowest thermal stresses? W i t h $ = 0.56, off = 4.8 x 10-6°C -1 and 0tm --- 60 x 10-6°C -~, from Equation 28 the values tz± = 29.27 x 10-6°C -t andtht = 6.69 x 10-6°C -~ are derived. Putting these into Equations 29-34 gives the thermal stresses for the four possible solutions given in Example 1 for a nominal temperature difference A T of - 1 0 0 ° C . Diagonal plies, -+w
90 ° plies
O ± ( M P a ) r#(MPa)
o±(MPa)
12
13
14 15 16
t_k t
6o(°)
17 1
53.6
16
5
(13) 1
0.8 0.5
48.5 33.1
17 18
4 3
13 15
0
19
0
16
0.351
nominal value only: there is no ply in this direction. The transverse stress is tensile and increases with decreasing ~o. The variation is not large, but the m i n i m u m stresses are obtained with a laminate consisting entirely of plies at +53.6 ° .
18 19 20 21
A laminate with 0.8 of the fibres at 48.5 °, 0.2 at 90 °, is, however, preferred, and can be achieved with only a small increase in thermal stress.
22
The thermal stresses for A T = - 1 0 0 ° C , by no means an unlikely temperature difference, amount to a third of the resin cracking strength.
23
REFERENCES
24
Puck, A. 'Einfuhren in das Gestalten und Dimensionieren' in 'Konstruieren und Berechnen yon GFK-Teilen', edited by Ehrenstein, G.W. and Martin, H.-D., pp 44-66. Supplement to Kunststoff-Berater, Umschau, Frankfurt, 1969 Puck, A. 'Dimensionierung tragender Leichtbaukonstruktionen aus Glasfaser-Kunststoffen', Kunststoffe 53 (1963)pp 150157 Puck, A. 'Glasfaser/Kunststoff in hochbeanspruchten Leichtbaukonstruktionen',Kunststoffe 53 (1963) pp 722733 Puck, A. 'Zum Deformationsverhalten und Bruchmechanismus yon unidirektionalem und orthogonalem Glasfaser/Kunststoff', Kunststoffe 55 (1965) pp 913-922 Puck, A. 'Zur Beanspruchung und Verformung yon GFKMehrschichtenverbund-Bauelementen: Teil 1. Grundlagen der Spannungs- und Verformungsanalyse', Kunststoffe 57 (1967) pp 284-293 Puck, A. 'Zur Beanspruchung und Verformung yon GFKMehrschichtenverbund-Bauelementen: Teil 2. Spannungsund Verformungsanalyse an GFK-Wickelrohren unter Ueberdruck', Kunststoffe 57 (1967) pp 573-582 Puck, A. 'Zur Beanspruchung und Verformung yon GFKMehrschichtenverbund-Bauelementen: Tell 3. Versuche an Mehrschichtenverbunden', Kunststoffe 57 (1967) pp 965973 Puck, A. 'Das "Knie" im Spannungs-Dehnungs-Diagramm und Rissbildung bei Glasfaser/Kunststoffen', Kunststoffe 58 (1968) pp 886-893 Puck, A. 'Festigkeitsberechnung an Glasfaser/KunststoffLaminaten bei zusammengesetzter Beanspruchung', Kunststoffe 59 (1969) pp 780-787
184
25 26
27
28 29 30 31 32
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COMPOSITES. JULY 1977