- Email: [email protected]
A study of the onset of flexural failure in cross-ply laminated strips
FibreScienceand Technology 13(1980)325-336
A STUDY OF THE ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
G. J. Tu~vEv
Department of Engineering, University of Lancaster, Bailrigg, Lancaster LAI 4YR, Lancs. (Great Britain)
SUMMARY
An exact solution of the equilibrium equations is combined with the Tsai-Hill failure criterion to facilitate the computation of the onset of flexural failure in antisymmetrically laminated cross-ply strips. Dimensionless data, i.e. initial failure loads and their associated centre-line deflections, are presented for uniformly loaded GFRP and CFRP simply supported and clamped strips. A significant feature is that the magnitude of both these quantities is very dependent on the loading direction, particularly for strips comprising a small number of laminae.
NOMENCLATURE
Strip width. In-plane stiffness. Coupling stiffness. Bit Reduced out-of-plane stiffness. D(=AIIDII - B~l ) Out-of-plane stiffness. DT( = ETh3(I - VLr VTL)/12) Transverse flexural rigidity. Strain components with respect to strip axes. ex, ey, exr eo, eyo, exy o Strain components of the strip mid-plane. Strain components with respect to lamina principal el,e2,el2 axes. Longitudinal and transverse lamina elastic moduli. Lamina shear modulus. GLT Strip thickness. h Curvature components with respect to strip axes. k x, ky, kxr Number of laminae in strip. N 325
a
All
Fibre Science and Technology 0015-0568/80/0013-0325/$0225 © Applied Science Publishers Ltd, England, 1980 Printed in Great Britain
326
q, qo dl( = qoa2s~, i h - 2 ) Sx, S.v, Sxy S1,$2,S12 SL, STt, c, SLT U l' W if'c( = WcDTST, lh - 2 a - 2) X, y , 2
0 YLT, YTL
()" ()'
G . J . TURVEY
Lateral pressure. Dimensionless lateral pressure. Stress components with respect to strip axes. Stress components with respect to lamina principal axes. Lamina strengths {longitudinal, transverse (tensile, compressive) and shear }. Displacement component in the x-direction. Displacement component in the y-direction. Strip deflection (z-direction). Dimensionless strip centre-line deflection. Cartesian co-ordinate directions. Inclination of fibre direction relative to the xdirection. Poisson's ratios. Total/partial derivative with respect to x. Partial derivative with respect to y.
INTRODUCTION
At the present time it is not possible to predict, by analytical means, the collapse or failure load of a laminated structural element with a satisfactory degree of certainty. The reason for this is that analytical failure criteria are not sufficiently well developed; they are unable to account for all types of failure modes and their possible interactions. In the case of laminates subjected to in-plane loading systems it has, however, been demonstrated 1- 3 that the onset of failure may be predicted analytically with an accuracy adequate for design application. Thus the design of laminated structural elements is currently based on the onset of failure in one or more of the laminae, i.e. the load that produces this state in the laminate is regarded as the failure load. Most studies concerning the initial failure of laminated structural elements, including those cited above, examine only in-plane loading configurations, since they were motivated by aeronautical structures applications for which this type of loading is dominant. The ever widening application of laminates in primary and secondary structural roles within the spectrum of engineering disciplines implies that situations must arise in which flexural rather than in-plane loading systems predominate. In these circumstances the load carrying capacity is limited by the onset offlexural failure. It appears, therefore, that a useful purpose may be served by undertaking studies of the initial flexural failure of laminated structural elements with the object of providing information suitable for direct design application; especially as such information does not appear to be readily available.
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
327
Recently, the author completed a study of this type and presented design data for the initial flexural failure of symmetrically laminated, cross-ply, rectangular plates. 4 Occasionally, however, the design constraints are such that a symmetric lay-up cannot be achieved. In these circumstances, the laminate exhibits in-plane-out-ofplane coupling and because of this the design data of the earlier study is of limited applicability. It is, therefore, of some importance to be able to assess the influence of this coupling phenomenon on the initial flexural failure of laminated plates. As a first step towards such an assessment, the somewhat simpler problem of the initial flexural failure of antisymmetrically laminated, cross-ply strips has been selected for study. The parameters of the study include the following: loading type--uniform, material type--Glass Fibre Reinforced Plastic (GFRP) and Carbon Fibre Reinforced Plastic (CFRP), support conditions--simply supported and clamped, and the number of laminae--N = 2 to 10; and numerical results for the complete range of parameters are presented as non-dimensional initial failure pressures and associated strip centre-line deflections.
STRIP GEOMETRY
The positive co-ordinate directions are shown in Fig. l(a) and a part-section through the strip is shown in Fig. l(b). In addition to showing the through-thickness position of the co-ordinate system, the latter figure also shows the lamina numbering system and the orientation of the fibres in each lamina.
FI~:~ES PARALLEL TO X-AX,~,. (a)
(b)
2 IIl[]ll[lllll]lllll[llll ~
Y,
j
-' Z,W.
Fig. 1.
llllllllgllllllllllllll
~
r
FIBRES PARALIFL TO y-AXIS
Laminated strip geometry. (a) Positive co-ordinate system, (b) part-section through strip.
MATERIAL PROPERTIES
The study is restricted to the two most common fibre reinforced plastic materials, viz G F R P and CFRP. Single laminae of both types of materials exhibit specially
328
G.J.
TURVEY
orthotropic properties and the degree of orthotropy--the ratio of the longitudinal to the transverse elastic m o d u l u s - - o f C F R P is about three times greater than that of GFRP. Complete details of the stiffness and strength properties of these materials are given in Table 1. TABLE 1 STIFFNESS AND STRENGTH DATA FOR GFRP AND CFRP UNI-DIRECTIONAL LAMINAE
(a) Stiffness data Material EtE~. 1 GFRP CFRP
3.00 9-40
(b) Strength data Material szs~, ~ GFRP CFRP
37"50 17"85
GLrE r i
vLT
0.48 0"36
0'25 0"30
ST ST, 1
SLTST, 1
4"00 2"64
1 "50 1"43
ANALYSIS
An initial failure analysis, be it flexural or otherwise, requires the preservation of equilibrium at every point within the structure and the satisfaction of a failure criterion at only one of these points (structural symmetry may produce simultaneous satisfaction of the failure criterion at several points). Thus such an analysis essentially requires the scaling of an equilibrium solution to the problem under consideration to achieve single or simultaneous, multi-point satisfaction of the failure criterion. In general, this scaling procedure is not straightforward for laminated structures, even when they are one-dimensional as in the case of infinite strips. The principal features of the initial failure analysis, namely the satisfaction of both equilibrium and the failure criterion are treated separately below and then their combination, i.e. the scaling process, to provide a means of evaluating first yield loads and associated strip centre-line deflections is described.
Solution of the strip equilibrium equations It is assumed that initial failure is a small deflection phenomenon. Furthermore, as the applied loading produces a state of'cylindrical bending', the strip equilibrium may be represented by two, coupled ordinary differential equations, s AIIU'" -- Bll)¢ .... = 0 Dl114..... -- B l l u " " = q
(1)
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
329
The solution ofeqns (1) may be expressed in terms of the following polynomials. 5 (a) S i m p l y supported strip w = A l i D - lqox(X3 -- 2ax 2 + a3)/24
u = B 1xD-lqo(a3 - 6ax 2 + 4x3)/24
(2)
i.e. the boundary conditions implied by eqns (2) along x = 0, a are: v = w = w'" = 0 and u 4= 0. (b) Clamped strip w = A 11D-lqoX2(X2 - 2ax + a2)/24 u = B 11D-lqoX(2X2 - 3ax + a2)/12
(3)
i.e. the boundary conditions implied by eqns (3) along x = 0, a are: u = v = w = w" =0.
Equations (2) and (3) enable the complete stress and strain fields within the strip to be determined. Strain f i e M
The strains referred to strip co-ordinates are defined as,
[e l re'°1 zE I[ u [wl e,
=/e °/÷
=
Le°,J
e~,y
k,,r
-=
u' + v"
w"
(4)
2w"..]
As the displacement components, w and u, ofeqns (2) and (3) are independent o f y and, moreover, as v is zero everywhere, then eqn (4) reduces,
e = exr
00
J
(5)
Further substitution for the displacement derivatives in eqn (5) is left until a later stage. It is a simple matter, using the following transformation relationship, to obtain the lamina principal strains,
e 2
e12
=
S2
C2
-2cs
2cs
-
cs
( c 2 - s 2)
er
(6)
exr
in which c = cos 0 and s = sin 0. Stress field
The stresses within a lamina may be derived with the help of the following lamina constitutive relationship,
330
G . J . TURVEY
[1] S2
S12
= 2
FLT
VLTi]Eel l
0
1
e2
0
e 1,
(7)
in which 2 = ET(1 -- YLTYTL)-1, ~ ~- ELET 1 and fl = GL-rET l(1 -- VLTFTL). On substituting eqn (5) into eqn (6) and the result into eqn (7), the lamina stresses become,
S2
= /~(U' -- Z W " ) /
s12
(YLT c2 -'1"-S 2) /
k
-2tics
Es]
(8)
._]
..+ VLS2,.
NOW substituting the appropriate derivatives of eqns (2) and (3) into eqn (8) gives,
sz
= 2qoD-l(Oll - ZAll)L
s12
/ (~2LTc2 'l- S 2)
/
[_
.J
-2tics
(9)
in which]~, = ½x(x - a) for simply supported strips andf~ = (6x z - 6ax + a2)/12 for clamped strips. Since the present study is restricted to cross-ply strips, eqn (9) may be further simplified. The lamina numbering sequence of Fig. l(b) implies that c = 1 and s = 0 for odd numbered layers and vice versa for even numbered layers. Furthermore, as all the laminae are identical (differing only in their spatial locations), the strip stiffnesses may be expressed as simple functions of the degree of orthotropy, 5, and the number of laminae, N,1 Al1 = ½2h(l + ~) Bll = ¼;thZ(1 - oON- 1 Dll = 2h3(1 + cQ/24
(10)
Thus on substituting eqn (10) into eqn (9) the final form of the expression for the lamina stresses becomes, sl $2
= 12q0h Jxg~lv
l
(11)
YLT I
in which g,t¢= {(1 - ~)N -1 - 2f(1 + ~)}{(1 + ~)2 _ 3(1 - ~):N-2} -1 and the subvector to the left of the vertical dashed line applies to odd numbered laminae and that to the right to even numbered laminae, and f = z h - 1
Tsai-Hill failure criterion A number of failure criteria have been found to be more or less useful for the prediction of failure in fibre reinforced materials, e.g. the maximum stress and maximum strain failure criteria. However, for the present purpose the Tsai-Hill
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
331
failure criterion has been selected, since it has been shown to give good agreement with experimental results for G F R P materials subjected to in-plane loading systems. 6 This criterion takes the following form: 2 -2 = 1 (s~ - s l s ~ ) s ~ 2 + s~,(,tsT,) - 2 + s~:sl~T
(12)
Now as s~2 is zero in all of the laminae, eqn (12) simplifies to, 32sl(sl -- s2) + rl2s22 = s2,
(13)
in which fl = q - 1 6 = sts~,l
and 1
when s 2 is tensile
STcSTt I
when s 2 is compressive
r/=
Initial failure loads Substituting eqn (11) into eqn (13) and re-arranging the following expression for the dimensionless initial failure load results, 1
(14)
0 = ++-fx-~ g ~ ( o - ~ / 1 2
in which
L
=La
-2
= 32~(~ -- VLT) + ~:V~T
for odd numbered laminae
and (7~ = (~2yLT(YLT - -
1) + 62
for even numbered laminae
As N---, ~ , eqn (14) reduces to, !
t~ = T-f~-x(1 - ~)zT-Xq~-~/24
(15)
For a simply supported strip initial failure occurs along the strip centre-line (x = ½a) and hence f 7 1 must be set to - 8 in eqns (14) and (15). Similarly, for the clamped strip, failure may initiate either at the supports or along the strip centreline, so that f~- 1 must be set to + 12 and - 24 respectively. Associated strip centre-line deflections Substituting eqn (14) into the first of eqns (2)and (3) the following expressions for the associated strip centre-line deflections result,
332
G.J.
TURVEY
~' = k h , u dt
(16)
in which h , u = (1 + a)-t{(1 + a) z - 3(1 - ~)2N-2}
and '5/192 k = [ 1/ 192
for simply supported edges for clamped edges
Again, as N-~ ~ , eqn (16) simplifies to, ff~ = k(l + ~x)-~
(17)
COMPUTATIONAL PROCEDURE
For a given laminated strip, eqn (14)~the expression for the initial failure load-depends directly on three variables, x, gand tp and indirectly on a fourth, ri. Thus, in general, the computation of the initial failure load requires a number of evaluations of eqn (14)--one for each of the possible values of these variables. However, for the strip boundary conditions under consideration, the value of x may only assume three distinct values, 0, ½a and a and this feature serves to reduce the computations. Once the initial failure load, ~, has been determined, the associated strip centre-line deflection may be evaluated directly from eqn (16). Using this approach, q and ~'c have been evaluated for simply supported, G F R P and CFRP strips comprising up to ten laminae. NUMERICAL RESULTS
For both strip materials the computed results are presented in dimensionless graphical form, i.e. as initial failure loads, ~, and associated strip centre-line deflections, fie, versus the number of laminae, N. The results for simply supported and clamped G F R P and CFRP strips are presented in Figs 2 and 3 respectively. Figure 2(a), which depicts ~ as a function of N, indicates that when the load is applied in the positive z-direction, the initial failure load increases slightly with increasing N and that failure always initiates in the highest numbered lamina. However, if the loading direction is reversed the strip is able to sustain a higher level of loading prior to the onset of failure, For strips consisting of very few laminae the increase in the initial failure load may be several hundred percent. It is interesting to note that when these simply supported G F R P strips are loaded in the negative z-direction failure always initiates in the second lamina. Another feature of Fig. 2(a) is that, as the number of laminae increases, the 'directional' initial failure loads approach, i.e. respectively increase or decrease
ONSET OF F L E X U R A L FAILURE IN CROSS-PLY L A M I N A T E D STRIPS
(2)
a
4
o
333
)
0
I 2
I 4
i 6 N
I 8
I 10
0
2
4
6
8
10
N
Fig. 2 Simply supported, antisymmetrically laminated, G F R P cross-ply strip subjected to a u n i f o r m lateral pressure. (a) t~ Versus N, (b) ~'c versus N. • = Load applied in positive z-direction, C) = load applied in negative z-direction, - . . . . a s y m p t o t e ( N = oo), (N) = failure initiates in lamina N.
towards, the asymptote corresponding to N = oo. However, the rate of approach is slower when the loading is applied in the negative z-direction. Figure 2(b), which shows a plot of *~'cagainst N, merits only the comment that, as might be inferred from eqn (16), the strip centre-line deflections closely follow the pattern of the initial failure loads. Corresponding results to those discussed above are shown in Figs 3(a) and (b) for CFRP strips. Comparing Figs 2(a) and 3(a), it is apparent that the results for positive z-direction loading are similar in nature, but that those for negative z-direction loading are not--the CFRP strip initial failure load is observed to increase as N increases, reaching a maximum at N = 4 and then decreasing as N is increased still further, whereas the initial failure load of G F R P strips subjected to negative loading decreases monotonically throughout the range of N. All of the results for clamped GFR P and CFRP strips are presented in Figs 4 and 5 respectively. The initial failure loads, t~, which are shown in Figs 4(a) and 5(a), indicate that their nature is similar for both material types. For the case of the load applied in the positive z-direction, the initial failure loads may be observed to rise gradually towards the asymptotic value as N increases, whereas for the reversed loading situation the initial failure loads pass through a maximum before decreasing to the same asymptotic value. Thus clamped strips exhibit greater flexural strengths, i.e. they are more efficient, when loaded in the positive z-direction--exactly the reverse of the situation for simply supported strips.
334
G.J.
--
TURVEY
a
(I) 12
I0
8
2~
........ 3
I~1(.#-) ~....~4p~_._.. --
2
0 0
~
-=- . . . .
;.
(2)
i 2
L 4
I 6
i 8
I tO
I
I
2
z,
N
N
1
I
I
6
8
10
Fig. 3. Simply supported, antisymmetrically laminated, C F R P cross-ply strip subjected to a uniform lateral pressure. (a) ~] Versus N, (b) ~c versus N. • = L o a d applied in positive z-direction, © = load applied in negative z-direction, - . . . . a s y m p t o t e ( N = oo), (N) = failure initiates in lamina N.
? -
gl
(4)
~
a
(2)
)
b
]o
I~l(xlO 3)
3-
6(2)
2-
o
5
I
I
2
4 N
I
1
I
6
8
10
3
1
1
2
4
I
I
I
6
8
10
N
Fig. 4. C l a m p e d , antisymmetrically laminated, G F R P cross-ply strip subjected to a u n i f o r m lateral pressure. (a) tj Versus N, (b) ffc versus N. • = L o a d applied in positive z-direction, © = load applied in negative z-direction, - . . . . a s y m p t o t e ( N = ~ ) , (N) = failure initiates in lamina N.
335
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
(6)
a
i6
b 14
II
lo
8 6
7
,/
6
f
4
(2) I
[
2
4
N
I
I
I
6
8
10
I
I
2
4
I
I
I
6
8
10
N
Fig. 5. Clamped, antisymmetrically laminated, C F R P cross-ply strip subjected to a uniform lateral pressure. (a) ~ Versus N, (b) ffc versus N. • = Load applied in positive z-direction, O = load applied in negative z-direction, - . . . . asymptote (N = o0), (N) = failure initiates in lamina N.
Again, as may be anticipated, the associated strip centre-line deflections for the clamped case exhibit the same trends as their corresponding initial failure loads (see Figs 4(b) and 5(b)). The final set of results is presented in Table 2. These show the ratio of the positive and negative initial failure loads and serve to illustrate how the directional TABLE 2 POSITIVE AND NEGATIVE INITIAL FAILURE LOAD RATIOS
Simply supported N
2 4 6 8 10
GFRP
4-/4+(%) 501 300 i86 155 140 100
CFRP
Clamped
Simply supported
4+/q-(%)
q-/q+ (%)
200 200 186 155 140 100
265 252 213 170 150 100
Clamped
4+/4-(%) 200 200 200 170 150 100
The + and - subscripts on ~ indicate the directions of the applied lateral pressure.
336
G . J . TURVEY
dependence of the flexural strength reduces as the number of laminae increases. An examination of Table 2 reveals that even when the number of constituent laminae is large (N-"-10) the 'directional' strengths are independent of the strip boundary condition and, moreover, may differ by 40 to 50 ~o. CONCLUSIONS
The main points emerging from this study of the onset of failure in antisymmetrically laminated, cross-ply strips may be summarised as follows: (1)
(2)
(3)
(4)
The initial failure loads are direction dependent. Changing the loading direction may cause a significant increase or decrease in the initial failure load, especially when the strip comprises a small number of laminae. This effect decreases as N increases. Simply supported strips are more efficient, i.e. they exhibit higher initial flexural failure loads, when they are loaded in the negative z-direction. The inverse situation applies in the case of clamped strips. The initial failure loads of positively loaded simply supported strips and negatively loaded clamped strips gradually increase, as the number of laminae increases, towards an asymptotic value. Failure always initiates in the highest numbered lamina in these strips. The initial failure loads of negatively loaded simply supported CFRP strips and positively loaded clamped G F R P and CFRP strips exhibit maxima at N = 4 or 6 prior to decreasing towards the asymptotic value with increasing N. ACKNOWLEDGEMENT
The author wishes to record his indebtedness to the Department of Engineering for its support during the course of this study and also to his father, Mr George Turvey, for his skill in the preparation of the figures. REFERENCES I. S. W. Ts^l, "Structural Behaviour of Composite Materials', NASA CR-71, July 1964. 2. M. B. SWELL, 'Strength and Elastic Response of Symmetric Angle-Ply Carbon Fibre Reinforced Plastics', Aeronautical Research Council, Reports and Memoranda No. 3816, July 1976. 3. M. B. SWELL, Strength and elastic response of symmetric angle-ply CFRP, Composites, 9(3) (July 1978) pp. 167-76. 4. G. J. TURVEY,An initial flexural failure analysis of symmetrically laminated, cross-ply, rectangular plates, Int. J. Solids Struct, (In press). 5. J. E. AsHro~ and J. M. WHJTNEY, Theory oJ Laminated Plate.,', Technomic Publishing Company Incorporated, Stamford, Connecticut, US~A, 1970, pp. I I I-I 6. 6. S. W. Ts^I, 'Strength theories of filamentary structures', in Fundamental Aspects o7 Fiber ReinJorced Plastic Composites, (Ed. R. T. Schwartz and H. S. Schwartz), Wiley lnterscience, New York, 1968, pp. 3-11.
A STUDY OF THE ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
G. J. Tu~vEv
Department of Engineering, University of Lancaster, Bailrigg, Lancaster LAI 4YR, Lancs. (Great Britain)
SUMMARY
An exact solution of the equilibrium equations is combined with the Tsai-Hill failure criterion to facilitate the computation of the onset of flexural failure in antisymmetrically laminated cross-ply strips. Dimensionless data, i.e. initial failure loads and their associated centre-line deflections, are presented for uniformly loaded GFRP and CFRP simply supported and clamped strips. A significant feature is that the magnitude of both these quantities is very dependent on the loading direction, particularly for strips comprising a small number of laminae.
NOMENCLATURE
Strip width. In-plane stiffness. Coupling stiffness. Bit Reduced out-of-plane stiffness. D(=AIIDII - B~l ) Out-of-plane stiffness. DT( = ETh3(I - VLr VTL)/12) Transverse flexural rigidity. Strain components with respect to strip axes. ex, ey, exr eo, eyo, exy o Strain components of the strip mid-plane. Strain components with respect to lamina principal el,e2,el2 axes. Longitudinal and transverse lamina elastic moduli. Lamina shear modulus. GLT Strip thickness. h Curvature components with respect to strip axes. k x, ky, kxr Number of laminae in strip. N 325
a
All
Fibre Science and Technology 0015-0568/80/0013-0325/$0225 © Applied Science Publishers Ltd, England, 1980 Printed in Great Britain
326
q, qo dl( = qoa2s~, i h - 2 ) Sx, S.v, Sxy S1,$2,S12 SL, STt, c, SLT U l' W if'c( = WcDTST, lh - 2 a - 2) X, y , 2
0 YLT, YTL
()" ()'
G . J . TURVEY
Lateral pressure. Dimensionless lateral pressure. Stress components with respect to strip axes. Stress components with respect to lamina principal axes. Lamina strengths {longitudinal, transverse (tensile, compressive) and shear }. Displacement component in the x-direction. Displacement component in the y-direction. Strip deflection (z-direction). Dimensionless strip centre-line deflection. Cartesian co-ordinate directions. Inclination of fibre direction relative to the xdirection. Poisson's ratios. Total/partial derivative with respect to x. Partial derivative with respect to y.
INTRODUCTION
At the present time it is not possible to predict, by analytical means, the collapse or failure load of a laminated structural element with a satisfactory degree of certainty. The reason for this is that analytical failure criteria are not sufficiently well developed; they are unable to account for all types of failure modes and their possible interactions. In the case of laminates subjected to in-plane loading systems it has, however, been demonstrated 1- 3 that the onset of failure may be predicted analytically with an accuracy adequate for design application. Thus the design of laminated structural elements is currently based on the onset of failure in one or more of the laminae, i.e. the load that produces this state in the laminate is regarded as the failure load. Most studies concerning the initial failure of laminated structural elements, including those cited above, examine only in-plane loading configurations, since they were motivated by aeronautical structures applications for which this type of loading is dominant. The ever widening application of laminates in primary and secondary structural roles within the spectrum of engineering disciplines implies that situations must arise in which flexural rather than in-plane loading systems predominate. In these circumstances the load carrying capacity is limited by the onset offlexural failure. It appears, therefore, that a useful purpose may be served by undertaking studies of the initial flexural failure of laminated structural elements with the object of providing information suitable for direct design application; especially as such information does not appear to be readily available.
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
327
Recently, the author completed a study of this type and presented design data for the initial flexural failure of symmetrically laminated, cross-ply, rectangular plates. 4 Occasionally, however, the design constraints are such that a symmetric lay-up cannot be achieved. In these circumstances, the laminate exhibits in-plane-out-ofplane coupling and because of this the design data of the earlier study is of limited applicability. It is, therefore, of some importance to be able to assess the influence of this coupling phenomenon on the initial flexural failure of laminated plates. As a first step towards such an assessment, the somewhat simpler problem of the initial flexural failure of antisymmetrically laminated, cross-ply strips has been selected for study. The parameters of the study include the following: loading type--uniform, material type--Glass Fibre Reinforced Plastic (GFRP) and Carbon Fibre Reinforced Plastic (CFRP), support conditions--simply supported and clamped, and the number of laminae--N = 2 to 10; and numerical results for the complete range of parameters are presented as non-dimensional initial failure pressures and associated strip centre-line deflections.
STRIP GEOMETRY
The positive co-ordinate directions are shown in Fig. l(a) and a part-section through the strip is shown in Fig. l(b). In addition to showing the through-thickness position of the co-ordinate system, the latter figure also shows the lamina numbering system and the orientation of the fibres in each lamina.
FI~:~ES PARALLEL TO X-AX,~,. (a)
(b)
2 IIl[]ll[lllll]lllll[llll ~
Y,
j
-' Z,W.
Fig. 1.
llllllllgllllllllllllll
~
r
FIBRES PARALIFL TO y-AXIS
Laminated strip geometry. (a) Positive co-ordinate system, (b) part-section through strip.
MATERIAL PROPERTIES
The study is restricted to the two most common fibre reinforced plastic materials, viz G F R P and CFRP. Single laminae of both types of materials exhibit specially
328
G.J.
TURVEY
orthotropic properties and the degree of orthotropy--the ratio of the longitudinal to the transverse elastic m o d u l u s - - o f C F R P is about three times greater than that of GFRP. Complete details of the stiffness and strength properties of these materials are given in Table 1. TABLE 1 STIFFNESS AND STRENGTH DATA FOR GFRP AND CFRP UNI-DIRECTIONAL LAMINAE
(a) Stiffness data Material EtE~. 1 GFRP CFRP
3.00 9-40
(b) Strength data Material szs~, ~ GFRP CFRP
37"50 17"85
GLrE r i
vLT
0.48 0"36
0'25 0"30
ST ST, 1
SLTST, 1
4"00 2"64
1 "50 1"43
ANALYSIS
An initial failure analysis, be it flexural or otherwise, requires the preservation of equilibrium at every point within the structure and the satisfaction of a failure criterion at only one of these points (structural symmetry may produce simultaneous satisfaction of the failure criterion at several points). Thus such an analysis essentially requires the scaling of an equilibrium solution to the problem under consideration to achieve single or simultaneous, multi-point satisfaction of the failure criterion. In general, this scaling procedure is not straightforward for laminated structures, even when they are one-dimensional as in the case of infinite strips. The principal features of the initial failure analysis, namely the satisfaction of both equilibrium and the failure criterion are treated separately below and then their combination, i.e. the scaling process, to provide a means of evaluating first yield loads and associated strip centre-line deflections is described.
Solution of the strip equilibrium equations It is assumed that initial failure is a small deflection phenomenon. Furthermore, as the applied loading produces a state of'cylindrical bending', the strip equilibrium may be represented by two, coupled ordinary differential equations, s AIIU'" -- Bll)¢ .... = 0 Dl114..... -- B l l u " " = q
(1)
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
329
The solution ofeqns (1) may be expressed in terms of the following polynomials. 5 (a) S i m p l y supported strip w = A l i D - lqox(X3 -- 2ax 2 + a3)/24
u = B 1xD-lqo(a3 - 6ax 2 + 4x3)/24
(2)
i.e. the boundary conditions implied by eqns (2) along x = 0, a are: v = w = w'" = 0 and u 4= 0. (b) Clamped strip w = A 11D-lqoX2(X2 - 2ax + a2)/24 u = B 11D-lqoX(2X2 - 3ax + a2)/12
(3)
i.e. the boundary conditions implied by eqns (3) along x = 0, a are: u = v = w = w" =0.
Equations (2) and (3) enable the complete stress and strain fields within the strip to be determined. Strain f i e M
The strains referred to strip co-ordinates are defined as,
[e l re'°1 zE I[ u [wl e,
=/e °/÷
=
Le°,J
e~,y
k,,r
-=
u' + v"
w"
(4)
2w"..]
As the displacement components, w and u, ofeqns (2) and (3) are independent o f y and, moreover, as v is zero everywhere, then eqn (4) reduces,
e = exr
00
J
(5)
Further substitution for the displacement derivatives in eqn (5) is left until a later stage. It is a simple matter, using the following transformation relationship, to obtain the lamina principal strains,
e 2
e12
=
S2
C2
-2cs
2cs
-
cs
( c 2 - s 2)
er
(6)
exr
in which c = cos 0 and s = sin 0. Stress field
The stresses within a lamina may be derived with the help of the following lamina constitutive relationship,
330
G . J . TURVEY
[1] S2
S12
= 2
FLT
VLTi]Eel l
0
1
e2
0
e 1,
(7)
in which 2 = ET(1 -- YLTYTL)-1, ~ ~- ELET 1 and fl = GL-rET l(1 -- VLTFTL). On substituting eqn (5) into eqn (6) and the result into eqn (7), the lamina stresses become,
S2
= /~(U' -- Z W " ) /
s12
(YLT c2 -'1"-S 2) /
k
-2tics
Es]
(8)
._]
..+ VLS2,.
NOW substituting the appropriate derivatives of eqns (2) and (3) into eqn (8) gives,
sz
= 2qoD-l(Oll - ZAll)L
s12
/ (~2LTc2 'l- S 2)
/
[_
.J
-2tics
(9)
in which]~, = ½x(x - a) for simply supported strips andf~ = (6x z - 6ax + a2)/12 for clamped strips. Since the present study is restricted to cross-ply strips, eqn (9) may be further simplified. The lamina numbering sequence of Fig. l(b) implies that c = 1 and s = 0 for odd numbered layers and vice versa for even numbered layers. Furthermore, as all the laminae are identical (differing only in their spatial locations), the strip stiffnesses may be expressed as simple functions of the degree of orthotropy, 5, and the number of laminae, N,1 Al1 = ½2h(l + ~) Bll = ¼;thZ(1 - oON- 1 Dll = 2h3(1 + cQ/24
(10)
Thus on substituting eqn (10) into eqn (9) the final form of the expression for the lamina stresses becomes, sl $2
= 12q0h Jxg~lv
l
(11)
YLT I
in which g,t¢= {(1 - ~)N -1 - 2f(1 + ~)}{(1 + ~)2 _ 3(1 - ~):N-2} -1 and the subvector to the left of the vertical dashed line applies to odd numbered laminae and that to the right to even numbered laminae, and f = z h - 1
Tsai-Hill failure criterion A number of failure criteria have been found to be more or less useful for the prediction of failure in fibre reinforced materials, e.g. the maximum stress and maximum strain failure criteria. However, for the present purpose the Tsai-Hill
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
331
failure criterion has been selected, since it has been shown to give good agreement with experimental results for G F R P materials subjected to in-plane loading systems. 6 This criterion takes the following form: 2 -2 = 1 (s~ - s l s ~ ) s ~ 2 + s~,(,tsT,) - 2 + s~:sl~T
(12)
Now as s~2 is zero in all of the laminae, eqn (12) simplifies to, 32sl(sl -- s2) + rl2s22 = s2,
(13)
in which fl = q - 1 6 = sts~,l
and 1
when s 2 is tensile
STcSTt I
when s 2 is compressive
r/=
Initial failure loads Substituting eqn (11) into eqn (13) and re-arranging the following expression for the dimensionless initial failure load results, 1
(14)
0 = ++-fx-~ g ~ ( o - ~ / 1 2
in which
L
=La
-2
= 32~(~ -- VLT) + ~:V~T
for odd numbered laminae
and (7~ = (~2yLT(YLT - -
1) + 62
for even numbered laminae
As N---, ~ , eqn (14) reduces to, !
t~ = T-f~-x(1 - ~)zT-Xq~-~/24
(15)
For a simply supported strip initial failure occurs along the strip centre-line (x = ½a) and hence f 7 1 must be set to - 8 in eqns (14) and (15). Similarly, for the clamped strip, failure may initiate either at the supports or along the strip centreline, so that f~- 1 must be set to + 12 and - 24 respectively. Associated strip centre-line deflections Substituting eqn (14) into the first of eqns (2)and (3) the following expressions for the associated strip centre-line deflections result,
332
G.J.
TURVEY
~' = k h , u dt
(16)
in which h , u = (1 + a)-t{(1 + a) z - 3(1 - ~)2N-2}
and '5/192 k = [ 1/ 192
for simply supported edges for clamped edges
Again, as N-~ ~ , eqn (16) simplifies to, ff~ = k(l + ~x)-~
(17)
COMPUTATIONAL PROCEDURE
For a given laminated strip, eqn (14)~the expression for the initial failure load-depends directly on three variables, x, gand tp and indirectly on a fourth, ri. Thus, in general, the computation of the initial failure load requires a number of evaluations of eqn (14)--one for each of the possible values of these variables. However, for the strip boundary conditions under consideration, the value of x may only assume three distinct values, 0, ½a and a and this feature serves to reduce the computations. Once the initial failure load, ~, has been determined, the associated strip centre-line deflection may be evaluated directly from eqn (16). Using this approach, q and ~'c have been evaluated for simply supported, G F R P and CFRP strips comprising up to ten laminae. NUMERICAL RESULTS
For both strip materials the computed results are presented in dimensionless graphical form, i.e. as initial failure loads, ~, and associated strip centre-line deflections, fie, versus the number of laminae, N. The results for simply supported and clamped G F R P and CFRP strips are presented in Figs 2 and 3 respectively. Figure 2(a), which depicts ~ as a function of N, indicates that when the load is applied in the positive z-direction, the initial failure load increases slightly with increasing N and that failure always initiates in the highest numbered lamina. However, if the loading direction is reversed the strip is able to sustain a higher level of loading prior to the onset of failure, For strips consisting of very few laminae the increase in the initial failure load may be several hundred percent. It is interesting to note that when these simply supported G F R P strips are loaded in the negative z-direction failure always initiates in the second lamina. Another feature of Fig. 2(a) is that, as the number of laminae increases, the 'directional' initial failure loads approach, i.e. respectively increase or decrease
ONSET OF F L E X U R A L FAILURE IN CROSS-PLY L A M I N A T E D STRIPS
(2)
a
4
o
333
)
0
I 2
I 4
i 6 N
I 8
I 10
0
2
4
6
8
10
N
Fig. 2 Simply supported, antisymmetrically laminated, G F R P cross-ply strip subjected to a u n i f o r m lateral pressure. (a) t~ Versus N, (b) ~'c versus N. • = Load applied in positive z-direction, C) = load applied in negative z-direction, - . . . . a s y m p t o t e ( N = oo), (N) = failure initiates in lamina N.
towards, the asymptote corresponding to N = oo. However, the rate of approach is slower when the loading is applied in the negative z-direction. Figure 2(b), which shows a plot of *~'cagainst N, merits only the comment that, as might be inferred from eqn (16), the strip centre-line deflections closely follow the pattern of the initial failure loads. Corresponding results to those discussed above are shown in Figs 3(a) and (b) for CFRP strips. Comparing Figs 2(a) and 3(a), it is apparent that the results for positive z-direction loading are similar in nature, but that those for negative z-direction loading are not--the CFRP strip initial failure load is observed to increase as N increases, reaching a maximum at N = 4 and then decreasing as N is increased still further, whereas the initial failure load of G F R P strips subjected to negative loading decreases monotonically throughout the range of N. All of the results for clamped GFR P and CFRP strips are presented in Figs 4 and 5 respectively. The initial failure loads, t~, which are shown in Figs 4(a) and 5(a), indicate that their nature is similar for both material types. For the case of the load applied in the positive z-direction, the initial failure loads may be observed to rise gradually towards the asymptotic value as N increases, whereas for the reversed loading situation the initial failure loads pass through a maximum before decreasing to the same asymptotic value. Thus clamped strips exhibit greater flexural strengths, i.e. they are more efficient, when loaded in the positive z-direction--exactly the reverse of the situation for simply supported strips.
334
G.J.
--
TURVEY
a
(I) 12
I0
8
2~
........ 3
I~1(.#-) ~....~4p~_._.. --
2
0 0
~
-=- . . . .
;.
(2)
i 2
L 4
I 6
i 8
I tO
I
I
2
z,
N
N
1
I
I
6
8
10
Fig. 3. Simply supported, antisymmetrically laminated, C F R P cross-ply strip subjected to a uniform lateral pressure. (a) ~] Versus N, (b) ~c versus N. • = L o a d applied in positive z-direction, © = load applied in negative z-direction, - . . . . a s y m p t o t e ( N = oo), (N) = failure initiates in lamina N.
? -
gl
(4)
~
a
(2)
)
b
]o
I~l(xlO 3)
3-
6(2)
2-
o
5
I
I
2
4 N
I
1
I
6
8
10
3
1
1
2
4
I
I
I
6
8
10
N
Fig. 4. C l a m p e d , antisymmetrically laminated, G F R P cross-ply strip subjected to a u n i f o r m lateral pressure. (a) tj Versus N, (b) ffc versus N. • = L o a d applied in positive z-direction, © = load applied in negative z-direction, - . . . . a s y m p t o t e ( N = ~ ) , (N) = failure initiates in lamina N.
335
ONSET OF FLEXURAL FAILURE IN CROSS-PLY LAMINATED STRIPS
(6)
a
i6
b 14
II
lo
8 6
7
,/
6
f
4
(2) I
[
2
4
N
I
I
I
6
8
10
I
I
2
4
I
I
I
6
8
10
N
Fig. 5. Clamped, antisymmetrically laminated, C F R P cross-ply strip subjected to a uniform lateral pressure. (a) ~ Versus N, (b) ffc versus N. • = Load applied in positive z-direction, O = load applied in negative z-direction, - . . . . asymptote (N = o0), (N) = failure initiates in lamina N.
Again, as may be anticipated, the associated strip centre-line deflections for the clamped case exhibit the same trends as their corresponding initial failure loads (see Figs 4(b) and 5(b)). The final set of results is presented in Table 2. These show the ratio of the positive and negative initial failure loads and serve to illustrate how the directional TABLE 2 POSITIVE AND NEGATIVE INITIAL FAILURE LOAD RATIOS
Simply supported N
2 4 6 8 10
GFRP
4-/4+(%) 501 300 i86 155 140 100
CFRP
Clamped
Simply supported
4+/q-(%)
q-/q+ (%)
200 200 186 155 140 100
265 252 213 170 150 100
Clamped
4+/4-(%) 200 200 200 170 150 100
The + and - subscripts on ~ indicate the directions of the applied lateral pressure.
336
G . J . TURVEY
dependence of the flexural strength reduces as the number of laminae increases. An examination of Table 2 reveals that even when the number of constituent laminae is large (N-"-10) the 'directional' strengths are independent of the strip boundary condition and, moreover, may differ by 40 to 50 ~o. CONCLUSIONS
The main points emerging from this study of the onset of failure in antisymmetrically laminated, cross-ply strips may be summarised as follows: (1)
(2)
(3)
(4)
The initial failure loads are direction dependent. Changing the loading direction may cause a significant increase or decrease in the initial failure load, especially when the strip comprises a small number of laminae. This effect decreases as N increases. Simply supported strips are more efficient, i.e. they exhibit higher initial flexural failure loads, when they are loaded in the negative z-direction. The inverse situation applies in the case of clamped strips. The initial failure loads of positively loaded simply supported strips and negatively loaded clamped strips gradually increase, as the number of laminae increases, towards an asymptotic value. Failure always initiates in the highest numbered lamina in these strips. The initial failure loads of negatively loaded simply supported CFRP strips and positively loaded clamped G F R P and CFRP strips exhibit maxima at N = 4 or 6 prior to decreasing towards the asymptotic value with increasing N. ACKNOWLEDGEMENT
The author wishes to record his indebtedness to the Department of Engineering for its support during the course of this study and also to his father, Mr George Turvey, for his skill in the preparation of the figures. REFERENCES I. S. W. Ts^l, "Structural Behaviour of Composite Materials', NASA CR-71, July 1964. 2. M. B. SWELL, 'Strength and Elastic Response of Symmetric Angle-Ply Carbon Fibre Reinforced Plastics', Aeronautical Research Council, Reports and Memoranda No. 3816, July 1976. 3. M. B. SWELL, Strength and elastic response of symmetric angle-ply CFRP, Composites, 9(3) (July 1978) pp. 167-76. 4. G. J. TURVEY,An initial flexural failure analysis of symmetrically laminated, cross-ply, rectangular plates, Int. J. Solids Struct, (In press). 5. J. E. AsHro~ and J. M. WHJTNEY, Theory oJ Laminated Plate.,', Technomic Publishing Company Incorporated, Stamford, Connecticut, US~A, 1970, pp. I I I-I 6. 6. S. W. Ts^I, 'Strength theories of filamentary structures', in Fundamental Aspects o7 Fiber ReinJorced Plastic Composites, (Ed. R. T. Schwartz and H. S. Schwartz), Wiley lnterscience, New York, 1968, pp. 3-11.