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Elastic and initial flexural failure analysis of unsymmetrically laminated cross-ply strips
PII: S 1359-8368(96)00031-5
ELSEVIER
Composites. Part B 27B (1996) 505-518 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
Elastic and initial flexural failure analysis of unsymmetrically laminated cross-ply strips
G. J. Turvey Engineering Department, Lancaster University, Bailrigg, Lancaster, LA 1 4YR, UK
and M. Y. Osman Faculty of Engineering and Technology, Nile Valley University, Atbara, Sudan (Received 5 January 1995; accepted 19 February 1996) A finite difference version of the Dynamic Relaxation (DR) method is used to generate elastic solutions of the small and the large deflection Mindlin plate equations for unsymmetrically laminated cross-ply strips subjected to uniform lateral pressure. The Maximum Stress (Independent) and Tsai-Hill failure criteria are combined with these solutions to produce initial failure analyses. It is shown that coupling between bending and stretching due to the Bit stiffness term can cause a considerable increase or decrease in the flexural stiffness of simply supported in-plane fixed strips relative to the equivalent orthotropic strips but not necessarily in the corresponding failure pressures. It is also found that the predictions of the two failure criteria are in close agreement. Copyright © 1996 Elsevier Science Limited (Keywords: composites; strips; laminates; flexure; failure)
1 INTRODUCTION In 1973 Jones I presented a linear buckling analysis of simply supported unsymmetrically laminated cross-ply plates subjected to uniform, uniaxial compression and provided numerical data for high modulus carbonepoxy, rectangular plates (b/a = 2) consisting of 3-100 layers. The fibres in all layers, except the second from the bottom of the laminate, were oriented parallel to the plate x-axis as shown in Figure 1. All of the layers were of equal thickness and thus, for a constant thickness laminate, the 90 ° layer became thinner and moved towards the bottom of the laminate as the number of layers increased. Although Jones recognised that such a laminate would probably not be used in engineering practice, he analysed it in order to demonstrate the effectiveness of his analytical model and to highlight the significant effect on the buckling load of coupling between bending and stretching. The laminate was first analysed with the lay-up shown in Figure 1, then with all of the layers parallel to the x-axis and finally as an equivalent orthotropic laminate in which coupling was ignored (i.e. Bij = 0). It was found that the buckling loads of the three laminates differed significantly when the number of layers was small, but that they tended to converge, albeit rather slowly, to roughly similar values
as the number of layers increased. Jones explained these differences as being due to the reduction in the magnitude of the coupling stiffnesses, Bij, as the number of layers increases. He also commented that approximating a general laminate by a specially orthotropic laminate can result in errors and, therefore, the use of such an approximation must be carefully proven in order to be justified for the particular case under consideration. Although Jones' buckling analysis is significant in its own right, it must be recognised that it is based on linear analysis and, therefore, the conclusions regarding the effects of coupling on the plate response may not hold to the same extent when deformations become large, i.e. in the context of geometric nonlinearity. In order to contribute to the resolution of this question, it was decided to undertake a large deflection rather than a postbuckling plate analysis. Thus, cross-ply laminated strips (i.e. large aspect ratio rectangular plates) under uniform lateral loading have been analysed with the following objectives: (i) To study the effect of bending-stretching coupling arising from unsymmetric cross-ply lay-ups typical of those used by Jones when the strip deflections are large.
505
Flexural failure analysis of cross-ply strips. G. d. Turvey and M. K Osman
O
o Ou° o: pwow I e x y-- -Oy - + ox-x + t ~ x ~ y J Ow o exz = (x -t- -Ox Ow o Cyz = Oy - ~ -Ox -
0
O* Y
0 o¸
9o
0 O
>x
b
,,OO
o
I
Z"L
0 °,
~X
Z Figure 1 A 6-layerunsymmetriccross-plylaminate [04°/90°/0°]
(ii) To determine the extent to which bending-stretching coupling affects the initial failure loads of unsymmetrically laminated cross-ply strips in the large deflection regime.
2 LAMINATED MINDLIN PLATE EQUATIONS Mindlin plate theory 2 is used to model the plate behaviour in flexure. It comprises of four sets of equations, viz. equilibrium, strain/curvature, constitutive and boundary condition equations, as set out below.
A brief summary of the large deflection plate equilibrium equations is given below. The small deflection versions are obtained by deleting the terms enclosed between curly brackets:
+
Oy
(2)
Ox
in which the superscript 0 denotes strains and curvatures of the plate mid-plane and the curly bracketed terms represent the nonlinear terms.
2.3 Constitutive equations The constitutive equations for cross-ply laminates are: 0
0
0
0
Nx = AllEx + A12Ey + BI1Xx + BI2Xy 0 0 0 0 Ny = A l : x + A22~y + B12Xx + B22Xy
Mx = B , : ° + B , : ° + D , , X ° + DI2X °
My
=- B12~°x + B22 g0 +
D12X 0 --k D22X°y
Mxy = B66e°y + D66X°y Q~ = A55e°z o Qy = A44eyz
ONx ONy Ox +--~-y = 0 ONxy FONy Ox Oy = 0 + Nx ~xz + 2Nxy -Ox - +Oy
Ny Oy2 J + q
02w Ow = Pw -~T + kw Ot OMx OMxy Ox -t Oy Qx = 0 (1)
nl
izk+1
k=l
k
C~;)(l'z'z2)dz
\Oxi J J
(4)
and Aij (i, j = 4, 5) denote the transverse shear stiffness coefficients and are calculated as follows:
k=l
The strains and curvatures of the mid-plane of the plate are:
=
where Aij, Bij and Dij (i, j = 1,2, 6) are, respectively, the membrane, coupling and flexural rigidities of the plate. The rigidities, Bij, arise due to the coupling between transverse bending and in-plane stretching. It should be noted that the constitutive equations for specially orthotropic plates are derived from equation (3) by setting the coupling stiffnesses, Bq, to zero. The rigidities are calculated as follows:
di:= Znl KiKj iz,k+I C(ijk) dz,
2.2 Strain/curvature equations
~x= ~ +
(3)
(Aij,Bij,Dq) = Z
OMxy F OMy OX -~y - Qy = 0
506
x o" _
gxy = A66~°y --FB66X °y
2.1 Equilibrium equations
Ox t-
0¢~
Xx = Ox o o~,, Xy = Oy
(i,j
=
4,5)
(5)
k
where Ci~) are the stiffnesses of a lamina referred to the plate principal axes and Ki, K: are the shear correction factors.
2.4 Boundary conditions Two types of simple support (simply supported inplane free (SS1) and fixed (SS2)) and two types of clamped
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman
Y
N., =~:,
=0
=,o = ~. = M,
N~=0 N=~y= 0 w----0 M,=0 ~ =0
N==0
N,y=0 w'-0
M==0
4'rt =0
u=0
v=0 w=0
v'-0 w=0
M~=0 ~,=0
M.=0
4'rj =0
u=v=w=¢.= My=O
(ss )
(ssO u=v=w=Ox=gp~=O
N.y = Ny = w = ~ . = Oy = 0
Nx=0 N=y = 0 w=0 ¢~, = 0
N.=0 N.y = 0 w=0 4~ = 0
4,~ =0
4,y =0 N=v = Nu = w =
,~= = 4'y =
My = 0
u=0
>2;
u., =iv, = w = ~ . =My =0
w=O==
u=v=
u=0 v=0
u=O v--0
w=0
w=0
~ =0
~ =0
4,y =0
,/,v = 0
0
W=~=-'~y
u-'v--
(ccl)
=0
(ccz)
Figure 2 Simplysupported and clamped boundary conditions
support (clamped in-plane free (CC1) and fixed (CC2)) are prescribed on the strip edges as shown in Figure 2. These conditions have been chosen because they represent limiting cases of in and out-of-plane restraint at the edges. 3 SOLUTION OF THE PLATE EQUATIONS Approximate solutions of the large deflection Mindlin plate equations, i.e. equations (I)-(3), are obtained using a finite difference implementation of the Dynamic Relaxation (DR) method (see refs 3 and 4 for further details). In order to solve equations (1)-(3) damping and inertia terms are first added to the right hand sides of equation (1) as follows:
ON,- ON v
02u
aN,:~, any
a2v
Ou kav
a~ +-aT-y=P"b7~+ vN aox aoy
r
a---x-v+ -~y + I Nx ~ x 2 + 2 N x ya- x a y + N .v ay 2 j + q aZ w aw = p,, ~-ff + kw --~
aMx aMxy ax
F Oy
~
OCx
Qx = p¢~ ~,,
+ k¢~ at
aMxv aM~, 02t~y OOy ax + ~ - Qy = p~, ~ + k<, 0---7
(6)
Then the following approximations are defined for the velocity and acceleration terms: aol a,
1 faola -
2
oqo~b] +
b-Y--h\a,
a,
] (7)
a,}
in which a --_ u, v, w, ~x, ~by.Substituting the velocity and acceleration term approximations of equation (7) into the right hand sides of equation (6) and re-arranging leads to the following expressions relating the velocity at the end of the time step, 6t, to the velocity at the end of the previous time step and the out-of-balance forces:
Oua _. 1 f Oub 6t (ON x +ONxy~] --~F;] Ot l+kT, [(1 - k~,)--~ + -p. - \~; ov a
0t
1
l + k;
(1 - k ; ) -OF +
~ \ ax
+
507
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman Table 1
M a t e r i a l p r o p e r t i e s used in the analysis
Ex/F~v
Gxy/Ey
axz/Ey
Gyz/Ey
1)xy
14.67
0.59
0.59
0.59
0.34
o"
OOO
o22
•,~~1000000 ,,~,0 0 0 ,,
lJ
M',
i
Ey = 3.9 k N / m m 2
"
s Table 2
S t r e n g t h p r o p e r t i e s o f m a t e r i a l used in th~ analysis
Xt/Yt
Xc/Yt
Yc/Yt
R/Yt
S/Yt
T/Yt
Yta2/Eyh 2
108.3
19.92
"1.0
2.83
2.83
2.83
30.77
,~0 0 0
I
o
o)
.) [¢v'~,;/o*] ~) t¢/9~/4,] e) [9O',V'o'/g&I a) [9~/o)'9~.]
Yt = 1 2 N / r a m 2
Table 3
•
Figure 3
F o u r m u l t i - l a y e r e d u n s y m m e t r i c cross-ply l a m i n a t e s
Deflection at the centre o f m u l t i - l a y e r u n s y m m e t r i c cross-ply simply s u p p o r t e d l a m i n a t e s u n d e r u n i f o r m pressure,
(b/a
= 5, h/a = 0.01,
# = 18o) SS1
SS2
Laminate
~]
~2(Bq = O)
%
wl
~2(Bq = O)
00/900/0 ° 90°/0°/90 °
2.0844 --
2,0844 --
0.0 --
0.8256 1.1546
0.8256 1.1546
0.0 0.0
0~/90°/0 ° 0°/90°/90~ 90~/0°/90 ° 90°/0°/90~
2.1669 2.1666 ---
2.1367 2.1368 ---
1.4 1.4 ---
0.8475 0.7687 1.1513 1.3769
0.8051 0.8051 1.2393 1.2393
5.0 -4.7 -7.6 10.0
0]/90°/0 °
2.2796 2.2790 ---
2.2286 2.2287 ---
2.2 2.2 ---
0.8500 0.7536 1.1800 1,5212
0.7972 0.7972 1.2989 1.2989
6.2 -5.8 - 10.1 14.6
90,]/0 °/90 ° 90°/0°/90,]
2.3455 2.3447 ---
2.2893 2.2894 ---
2.4 2.4 ---
0.8455 0.7480 1.2121 1.6165
0.7919 0.7919 1.3472 1.3472
6.3 -5.8 - 11.1 16.7
0~o/90°/0 ° 0°/90°/0~o 90~o/0°/90 °
2.3457 2.3446 --
2.3164 2.3166 --
1.2 1.2 --
90°/0°/90~o
.
0.81 O0 0.7424 1.3763 1.8372
0.7737 0.7737 1.5371 1.5371
4.5 -4.2 - 11.7 16.3
O~s/90°/O° 0°/90°/0°18
2.2488 2.2480 ---
2.2366 2.2367 ---
0.5 0.5 ---
0.7859 0.7417 1.5189 1.9147
0.7627 0.7627 1.6700 1.6699
3.0 -2.8 -9.9 12.8
2.1797 2.1792 ---
2.1740 2.1741 ---
0.3 0.2 ---
0.7720 0.7414 1.6289 1.9508
0.7561 0.7561 1,7602 1.7601
2.1 -2.0 -8.1 9.8
2.1407 2.1402 ---
2.1374 2.1374 ---
0.2 0.1 ---
0.7645 0.7412 1.6999 1.9684
0.7525 0.7525 1.8138 1.8138
1.6 - 1.5 -6.7 7.9
90~s/0°/90 ° 90°/0°/90~s
2.0990 2.0987 ---
2.0975 2.0975 ---
0.0 0.0 ---
0.7568 0.7410 1.7855 1.9860
0.7487 0.7487 1.8745 1.8745
1.1 -1.0 -5.0 5.6
0 ° (all) 90 ° (all)
2.0089 --
2.0089 --
0.0 --
0.7405 2.0241
0.7405 2.0241
0.0 0.0
0°/90°/0~ 90]/00/90 ° 90°/0°/90~ 0,~/90°/0 °
0°/90°/0°4
90~s/0°/90 ° 90°/0°/90~s 0~s/90°/0 °
O°/90°/O~s 90~s/0°/90 ° 90°/0°/90~8
O~s/90°/O° 0°/90°/0]8 90]8/0°/90 ° 90°/0°/90]8 O~s/90°/O °
O°/90°/O~s
%: 100 × (el
-
.
.
.
%
e2)/~t
- - : solution u n o b t a i n a b l e due to n u m e r i c a l instability because the plate is in-plane free
Ot
l+k~
(1 - k'v) ---~- +
02w
02w
N O2W
+ Nx--~xz + 2Nxy o x O y F y Oyz + q
508
Ot
+
~] )J
1 1 + k *Ox ×
(I - kc,x)
+ -P~x I. Ox
Oy
Qx
Flexural failure analysis of cross-ply strips: G. d. Turvey and M K Osman Deflection at the centre of multi-layer unsymmetric cross-ply clamped laminates under uniform pressure, (b/a = 5, h/a = 0.01, q = 180)
Table 4
CCI
CC2
Laminate
wl
~2(Bij = O)
0o/90o/0 ° 9ff'/0°/90 °
0.4274 3.7599
0.4274 3.7599
0~/90°/0 ° 0°/90°/90~ 90~/0°/90 ° 90°/0°/90~
0.4442 0.4441 4.0689 3.9905
0~/90°/0 ° 0°/90°/0~ 90]/0°/90 ° 90°/0°/90;
%
~'~
w2(Ba = O)
%
0.0 0.0
0.4000 0.9908
0.4000 0.9908
0.0 0.0
0.4380 0.4380 3.1498 3.1498
1.4 1.4 22.6 21.1
0.4088 0.4088 1.0715 1.0711
0.4046 0.4046 1.0380 1.0380
1.0 1.0 3.1 3.1
0.4670 0.4667 3.8509 3.7554
0.4566 0.4566 2.4573 2.4573
2.2 2.2 36.2 34.6
0.4220 0.4220 1.1193 1.1185
0.4152 0.4152 1.0460 1.0460
1.6 1.6 6.5 6.5
0,]/90°/0 ° 0°/90°/0~ 90~/0°/90 ° 90°/0°/90~
0.4803 0.4802 3.5974 3.5175
0.4689 0.4689 2.1545 2.1545
2.4 2.4 40.1 38.7
0.4290 0.4290 1.1530 I. 1520
0.4218 0.4218 1.0539 1.0539
1.7 1.7 8.6 8.5
0~0/90°/0 ° 0°/90°/0~ 90%/00/90 ° 90°/0°/90%
0,4806 0.4805 3.0706 3.0126
0.4746 0.4746 2.0402 2.0402
1.2 1.2 33.6 32.3
0.4253 0.4253 1.2683 1.2665
0.4217 0.4217 1.1553 1.1553
0.1 0.8 8.9 8.8
0~8/90°/0 ° 0°/90°/0~8 90~8/0°/90 ° 90°/0°/90~8
0.4612 0.4611 3.1017 3.0523
0.4587 0.4587 2.3958 2.3957
0.1 0.5 22.8 21.5
0.4120 0.4119 1.3568 1.3549
0.4040 0.4104 1.2786 1.2789
1.9 0.4 5.8 5.6
0~8/90°/0 ° 0°/90°/0~8 90~s/0°/90 ° 90°/0°/90~8
0.4473 0.4472 3.2784 3.2323
0.4461 0.4461 2.7923 2.7923
0.0 0.2 14.8 13.6
0.4025 0.4025 1.4280 1.4262
0.4018 0.4018 1.3784 1.3784
0.2 0.2 3.5 3.4
0~8/90°/0 ° 0°/90°/0~8 90~8/0°/90 ° 90°/0°/90~8
0.4394 0.4394 3.4475 3.4040
0.4388 0.4388 3.0920 3.0920
0.0 0.1 10.3 9.2
0.3972 0.3972 1.4759 1.4743
0.3968 0.3968 1.4420 1.4420
0.1 0.1 2.3 2.2
0~8/90°/0 ° 0°/90°/0~s 90~8/0°/90 ° 90°/0°/90~8
0.4310 0.4310 3.7065 3."6679
0.4307 0.4307 3.4919 3.4919
0.0 0.0 5.8 4.8
0.3914 0.3914 1.5360 1.5347
0.3912 0.3912 1.5174 1.5174
0.0 0.0 1.2 1.1
0 ° (all) 90 ° (all)
0.4128 4.7487
0.4128 4.7487
0.0 0.0
0.3786 1.7149
0.3786 1.7149
0.0 0.0
% : 100 X (14~1 -- W 2 ) / ~ ' 1
algorithm is a simple sequential procedure consisting of the following steps:
1
at
1 + k~, × [(l_k;))0q~y
b
tft f O M x y Pe~
OMy
+ - V y - e,
}]
(8) In equations (7) and (8) the superscripts, a and b, refer, respectively, to the values of the velocities after and before the time step, 6t, and k; = (k~6tp21)/2, where the subscript, a, is defined immediately following equation
(7). The displacements at the end of each time increment, 6t, are evaluated using the following simple integration procedure: aa :
a b + 6t 0eta 0---T
(9)
where a is again defined following equation (7). Thus, equations (8), (9), (2) and (3) constitute the set of equations for solution. The DR numerical solution
(1) Set the initial conditions (all variables, i.e. displacements, stress resultants, etc are set to zero and the transverse pressure is applied). (2) Compute the velocities at the end of the time step, 6t from equation (8) using the known displacements and stress resultants/couples. (3) Compute the displacements from equation (9) using the velocities calculated in the previous step and the known displacements at the end of the preceding time step. (4) Apply the displacement boundary conditions so that any incorrectly computed boundary displacements determined in the previous step are corrected and the displacements are correctly determined throughout the strip. (5) Compute the strains and curvatures from equation (2) using the displacements calculated in the two preceding steps. (6) Compute the stress resultants/couples from equation
509
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman
2.2
5
.
1.8
WC 1.4
T/ //'
- - all 0 -,-[0n/9o/01
~-
..,,. [90n/0/90 ]
2
,90n/0/9t]
1.0
_ij=_o
0.6 0
, 10
, 20
, 30
I 40
I 50
I 60
I
70
Number of Plies
0
10
I
I
I
I
20 30 40 50 Number of Plies
I
60
70
Figure4 Deflectionsat the centre of a multi-layerunsymmetriccrossply simplysupportedin-planefixed(SS2)uniformlyloadedlaminatefor # = 180, b/a = 5 and h/a = 0.01
Figure 5 Deflectionsat the centre of a multi-layerunsymmetriccrossply clampedin-planefree(CC1)uniformlyloadedlaminatefor ~/= 180, b/a = 5 and h/a = 0.01
(3) using the strains and curvatures calculated in the preceding step. (7) Apply the stress resultant/couple boundary conditions so that incorrectly computed boundary stress resultants/couples are corrected and the stress resultants/couples are correctly determined throughout the strip. (8) Check that the velocities throughout the strip are acceptably small (say <10 -6 ) so that the strip vibrations have effectively ceased and the desired static solution has been achieved. (9) If step (8) is satisfied, print out the results, otherwise repeat steps (2)-(8) in sequence until the velocity convergence criterion is satisfied.
No details of the verification studies undertaken to demonstrate the accuracy of the DR finite difference solution algorithm are presented here. Suffice it to say that extensive checking has been undertaken. The effect of mesh size on the solution accuracy in the context of isotropic and orthotropic Mindlin plates has been reported on in refs 5 and 6, which also show that numerical results obtained with the D R program are in good agreement with the numerical solutions obtained by finite element and finite strip analyses. The same DR computer analysis has been used here to obtain the present laminated strip results by modelling the strip as a long rectangular plate, i.e. adopting an aspect ratio of 5 : 1.
It should be appreciated that the DR algorithm, as outlined above, is actually applied to the discrete equivalents of equations (8), (9), (2) and (3). The discrete system is obtained by recasting these equations into centred finite difference format. The finite difference forms of the equations are too lengthy to be included here. However, the interested reader may obtain a deeper appreciation of the general approach adopted by consulting refs 3 and 4, both of which provide detailed descriptions of various aspects of the finite difference DR procedure in the context of simple structural problems.
510
4 F A I L U R E CRITERIA In the analysis of the initial failure of unsymmetrically laminated cross-ply strips, the Tsai-Hill and Maximum Stress (Independent) criteria are used. The underlying philosophy and derivation of these criteria are described in ref. 7.
4.1 Tsai-Hill failure criterion Failure is assumed to occur when the following
Flexural failure analysis of cross-ply strips. G. J. Turvey and M. Y. Osman
1.8
2.2
1.4
1.8
m
w
We 1.0
We 1.4
all 90 all 0 -- [0.19.0/01. •-,.- tgon/o/90j
--
II'/
---
\
~-
- - all o
r-~U---u" \
1.0 ~ , _ _
0.6
0.2
I
0
10
I
I
I
I
20 30 40 50 Number of Plies
0.6
I
60
0
70
-,-[9o/oj9o.]
"~- [90n/O/90J -,- tOl9OlqnJ
. -.x.-[on/90/o]
I
I
I
I
I
I
I0
20
30
40
50
60
70
Number of Plies
Figure 6 Deflections at the centre of a multi-layer unsymmetric cross-
Figure 7 Deflections at the centre of a multi-layer unsymmetric cross-
ply clamped in-plane fixed (CC2) uniformly loaded laminate for g/= 180, b/a = 5 and h/a = 0.01
ply simply supported in-plane fixed (SS2) uniformly loaded laminate for ~/= 180, b/a = 5 and h/a = 0.01
criterion is satisfied:
index to 1, the pressure is increased to Aq(s) and, hence, the stresses are increased by the same scaling factor A > 1. As a result two equations may be written as follows:
17~
X2
1710"2
O'~
2 174
.
2 0-5.
2 0"6
j(2 + ~ - 5 + ~ - Y + ~ - ~ - ~
= 1
(10)
where X is set to Xt when 17t is tensile and to Xc when 17t is compressive. Similarly, Y is set to Yt when 42 is tensile and to Yc when 172 is compressive.
2 171(s) X-2
2 2 2 2 0-1(s)172(s) O'2(s).4- 174(s) 175(s) 176(s) ~ X q- y 2 - - R E + - ~ +- ~ = FIN(s)
(12) and
4.2 Maximum stress (independent)failure criterion
A2 [17~(s)
17t(s)°2(s) 1- °2(s)
°2(s)
°~(s)
'
172(s)] = 1
Failure is assumed to occur when one or more of the following conditions are satisfied: 171 = X,
172 ~-- Y,
0"4 = R,
45 : S,
0"6 = T (11)
where X and Y are interpreted as for the Tsai-Hill failure criterion. Two different procedures have been used to determine the initial failure pressures depending on the type of analysis used. The small deflection initial failure pressures were determined using a simple scaling procedure, whereas the large deflection initial failure pressures were determined by an iterative procedure outlined below: 4.2.1 The scaling procedure. A small pressure, say q(s), was applied to the strip and the maximum value of the failure index, FIN(s), which is the sum of the terms on the left hand side of equation (10), was computed. Because q(s) is small, FIN(s ) < 1. To increase the failure
(13) Hence, dividing equation (13) by equation (12) the scaling factor, A, can be expressed as: A=
~/~,1 i~,l(s)
(14)
In the case of the Maximum Stress (Independent) failure criterion, the scaling factor is obtained, likewise, by applying a small pressure and computing the ratios: A1 =
X o1 (s)
,
A2-
S
A4
= --,
17s(s)
Y °2(s)
,
A5
A3
T --
R --
174(s)
, (15)
0-6(s)
A is then set to the maximum value of Ai (i = 1 , . . . , 5).
511
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman Table 5
Initial failure of multi-layer unsymmetric cross-ply [0n/90°/0°] simply supported in-plane fixed (SS2) laminates under uniform pressure
(b/a = 5, h/a = 0.01) n
1
Maximum stress (independent)
Maximum stress (independent)
Maximum stress (independent)
Maximum stress (independent) Tsai-Hill
18
Maximum stress (independent) Tsai-Hill
28
Maximum stress (independent) Tsai-HiU
38
Maximum stress (independent) Tsai-Hill
58
Maximum stress (independent) Tsai-Hill
0° (all)
In
k
Mode
793.7 6500.0 788.9 6500.0
8.9143 3.1125 8.8595 3.1125
0.5 0.5 0.5 0.5
0.5 0.2 0.5 0.2
1 2 1 2
1 3 1 3
2 3 2 3
1 2 1 2
789.9 4520.8 785.1 4500.0
9.1222 2.7023 9.0663 2.6904
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 1 1 1
1 1 1 1
2 3 2 3
1 2 1 2
738.7 3774,3 734.2 3800.0
9.1601 2.4781 9.1042 2.4837
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 1 1 1
1 1 1 1
2 3 2 3
1 2 1 2
601.9 4887.0 606.9 4868.8
7.5378 2.6075 7.6007 2.6042
0.5 0.5 0.5 0.5
0.3 0.0 0.5 0.0
11 1 11 1
3 1 3 1
3 3 3 3
1 2 1 2
585.0 5842.5 589.5 6000.0
7.0667 2.7240 7.1207 2.7485
0.5 0.5 0.5 0.5
0.3 0.1 0.5 0.2
19 19 19 19
3 3 3 3
3 3 3 3
1 2 1 2
584.6 5842.5 588.5 5842.5
6.8616 2,7102 6.9077 2.7102
0.5 0.5 0.5 0.5
0.3 0.1 0.5 0.1
29 29 29 29
3 3 3 3
3 3 3 3
1 2 1 2
590.0 5842.5 590.0 5842.5
6.8069 2.6896 6.8068 2.6896
0.5 0.5 0.5 0.5
0.5 0.1 0.5 0.1
39 39 39 39
3 3 3 3
3 3 3 3
1 2 1 2
583.0 5842.5 592.6 5700.0
6.6003 2.6779 6.7094 2.6556
0.5 0.5 0.5 0.5
0.1 0.1 0.5 0.1
59 59 59 59
3 3 3 3
3 3 3 3
1 2 1 2
822.2 13 230.0 817.2 13 230.0
8.9150 3.4992 8.8602 3.4992
0.5 0.5 0.5 0.5
0,5 0.0 0.5 0.0
1 1 1 1
1 3 1 3
1 3 2 3
1 2 1 2
Tsai-Hill 10
y/b
Tsai-Hill
Tsai-Hill 4
x/a
s
Tsai-Hill 2
~f
Criterion
Maximum stress (independent)
~/r
s(1): linear results; s(2): non-linear results In: layer number starting from the top layer k = 1, 3: the upper and lower surfaces of a ply, respectively Mode (2): failure of fibre in compression Mode (3): failure of matrix in tension
4.2.2 The iterative method. The iterative procedure is used in the n o n l i n e a r analysis o f initial failure. A n arbitrary pressure is applied a n d the c o r r e s p o n d i n g failure index, F I N , is c o m p u t e d . W h e n F I N < 0.99, the load is increased by 5% a n d the analysis is c o n t i n u e d . W h e n F I N > 1.01, the pressure is reduced by a similar percentage a n d the analysis is c o n t i n u e d . This iterative procedure is c o n t i n u e d until the desired accuracy is reached, i.e. a tolerance of 1% o n the failure index is achieved. In the case o f the M a x i m u m Stress ( I n d e p e n d ent) criterion, the same tolerance is applied to the value of )~i (i --- 1 , . . , , 5) in e q u a t i o n (15).
5 NUMERICAL RESULTS AND DISCUSSION
5.1 Large deflections in multi-layered cross-ply strips The cross-ply laminates analysed consist o f 3 - 6 0 layers. The fibres in all o f the layers, except one, are in
51 2
one direction only. The odd layer is either second from the top or second from the b o t t o m of the l a m i n a t e as s h o w n in Figure 3. I n laminates (a) a n d (b), the fibres are m a i n l y parallel to the plate x-axis, whereas in laminates (c) a n d (d) they are m a i n l y parallel to the yaxis. N u m e r i c a l results are o b t a i n e d for one q u a r t e r o f the strip using a 5 x 5 r e c t a n g u l a r finite difference mesh, with shear correction factors, K42 = K} = 5/6. The material properties used in the analysis are typical of a c a r b o n - e p o x y unidirectional l a m i n a a n d are given in Tables I a n d 2. The l a m i n a t e d strips analysed are thin h/a = 0.01 with aspect ratio 5 a n d SS1, SS2, CC1 a n d CC2 edge conditions. The centre deflections are listed in Tables 3 a n d 4 a n d are s h o w n graphically in Figures 4-7. Two other sets of results are o b t a i n e d by analysing the l a m i n a t e as specially o r t h o t r o p i c (Bij -- 0) a n d with all layers aligned in one direction. These two sets of results serve as indicators of the significance of c o u p l i n g a n d the effect o f the odd layer o n the response o f a cross-ply strip.
Flexural failure analysis of crcss-ply strips. G. J. Turvey and M. K Osman Table 6
Initial failure in multi-layer unsymmetric cross-ply (0°/90°/0~,) simply supported in-plane fixed (SS2) laminates under uniform pressure
(b/ct = 5, h/a = 0.01) n
1
M a x i m u m stress (independent)
M a x i m u m stress (independent)
M a x i m u m stress (independent)
M a x i m u m stress (independent) Tsai-Hill
18
M a x i m u m stress (independent) Tsai Hill
28
M a x i m u m stress (independent) T s a i - Hill
38
M a x i m u m stress (independent) Tsai- Hill
58
M a x i m u m stress (independent) Tsai-Hill
0 (all)
793.7 6500.0 788.9 6500.0
8.9143 3.1125 8.8595 3.1125
0.5 0.5 0.5 0.5
0.5 0.2 0.5 0.2
1 2 1 2
752.2 5221.7 747.6 5209.3
8.6866 2.7473 8.6331 2.7450
0.5 0.5 0.5 0.5
I 2 1 2
695.9 4664.5 691.6 4630.5
8.6293 2.5519 8.5761 2.5454
1 2 1 2
696.4 5731.3 692.1 5788.1
I 2 1 2
1 2 1 2
Tsai Hill 10
y/b
Tsai-Hill
Tsai-Hill 4
x/a
s
Tsai-Hill 2
wf
Criterion
M a x i m u m stress (independent)
qf
In
k
Mode
1 2 1 2
1 3 1 3
2 3 2 3
0.5 0.0 0.5 0.0
1 4 l 4
1 3 1 3
2 3 2 3
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 6 1 6
1 3 1 3
2 3 2 3
8.7214 2.6764 8.6677 2.6855
0.5 0,5 0.5 0.5
0.5 0.0 0.5 0.0
1 12 1 12
1 3 1 3
2 3 2 3
727.7 6536.7 723.2 6615.0
8.7900 2.7801 8.7359 2.7915
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
I 20 1 20
1 3 1 3
2 3 2 3
1 2 1 2
752.3 7231.5 747.6 7293.0
8.8291 2.8687 8.7748 2.8770
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 30 1 30
1 3 l 3
2 3 2 3
1 2 1 2
767.0 7683.3 762.3 7657.7
8.8497 2.9242 8.7953 2.9208
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 40 l 40
1 3 1 3
2 3 2 3
1 2 1 2
783.6 8500.0 778.8 8442.6
8.8709 3.0219 8.8163 3.0150
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 60 1 60
1 3 1 3
2 3 2 3
1 2 1 2
822.2 13 230.0 817.2 13 230.0
8.9150 3.4992 8.8602 3.4992
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 1 1 1
1 3 1 3
2 3 2 3
s(l): linear analysis; s(2): non-linear analysis In: layer number starting from the top layer k = 1,3: the upper and lower surfaces of a ply, respectively Mode (2): failure of fibre in compression Mode (3): failure of matrix in tension
It should be noted that DR solutions for the (SS1) laminates (90~,/0°/90 °) and (90°/0°/90~,) are difficult to obtain due to instability of the numerical computations because of large in-plane displacements. A study of the results reveals the following: (1) When the majority of the laminae have their fibredirections parallel to the longer side of the strip, i.e. (90~,/0°/90 °) and (90°/0°/90~,) laminates, the strip is, as anticipated, less stiff than when the fibres are aligned mainly across the strip, as in (0~,/90°/0 °) and (0°/90°/0~,) laminates. This comment holds for all boundary conditions. (2) The effect of coupling on the centre deflection of the (0~,/90°/0 °) and (0°/90°/0~,) laminates is marginal and accounts only for a maximum of 5 or 6% of the deflection for the SS2 edge condition, i.e. the odd layer does not alter the response of the strip significantly. Hence, if an odd layer is present due
to a mistake in the lay-up procedure, the analysis of the laminate as specially orthotropic does not lead to a large error. Coupling in (90~,/0°/90 °) and (90°/0°/90~) laminates is more significant, particularly when the number of layers is small, say less than 10. It should be noted, however, that laminates with such lay-ups are of little practical significance. (3) Coupling (Bij ¢ 0) in (90~/0°/90 °) laminates can make the strip stiffer or more flexible than equivalent orthotropic laminates in bending depending on the boundary conditions. Coupling makes simply supported in-plane fixed (SS2) strips stiffer as demonstrated in Figure 4 in which the deflection of simply supported cross-ply and orthotropic strips are compared. When the strip is clamped, i.e. CC1 or CC2 edge conditions apply, coupling makes the strip more flexible as Figures 5 and 6 indicate. (4) Comparing the deflections of (90°/0°/90~,) and
51 3
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman Table 7
Initial failure in multi-layer unsymmetric cross-ply (90~,/0°/90 °) simply supported in-plane fixed (SS2) laminates under uniform pressure
(b/a = 5, h/a = 0.01) n
1
Criterion
s
qr
~f
x/a
y/b
In
k
Mode
Tsai-Hill
1 2 1 2
60.5 2239.6 60.5 2239.6
6.4747 2.6819 6.4747 2.6818
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.2
3 3 3 3
3 3 3 3
3 3 3 3
1 2 1 2
72.0 685.9 72.0 684.2
6.7454 1.8540 6.7454 1.8524
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.5
4 1 4 1
3 1 3 1
3 3 3 3
1 2 1 2
107.3 412.2 106.3 406.2
7.2269 1.6509 7.1592 1.6419
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
6 1 6 1
3 1 3 1
3 3 3 3
1 2 1 2
120.4 417.4 120.4 416.5
7.3852 1.8949 7.3851 1.8934
0,5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
12 1 12 1
3 1 3 1
3 3 3 3
1 2 1 2
103.8 462.5 103.8 461.4
7.2240 2.1636 7.2239 2.1616
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
20 1 20 1
3 1 3 1
3 3 3 3
1 2 1 2
88.2 500.0 88.2 498.4
7.0596 2.3729 7.0595 2.3701
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
30 1 30 1
3 1 3 1
3 3 3 3
1 2 1 2
78.4 525.0 78.4 703.6
6.9519 2.5080 6.9518 2.7844
0.5 0.0 0.5 0.5
0.5 0.4 0.5 0.5
40 1 40 40
3 1 3 3
3 3 3 3
1 2 1 2
67.2 609.5 67.2 609.0
6,8234 2.7594 6.8233 2.7586
0.5 0.5 0.5 0.0
0.5 0,5 0.5 0.5
60 60 60 60
3 3 3 3
3 3 3 3
1 2 1 2
40.8 444.8 40.9 444.8
6.4657 2.7548 6.4657 2.7549
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
--1 1
3 3 3 3
3 3 3 3
M a x i m u m stress (independent) Tsai-Hill
2
M a x i m u m stress (independent) Tsai-Hill
4
M a x i m u m stress (independent) Tsai - Hill
10
M a x i m u m stress (independent) Tsai-Hill
18
M a x i m u m stress (independent) Tsai-Hill
28
M a x i m u m stress (independent) Tsai-Hill
38
M a x i m u m stress (independent) Tsai-Hill
58
M a x i m u m stress (independent) Tsai-Hill
90 ° (all)
M a x i m u m stress (independent)
s(1): linear analysis; s(2): non-linear analysis In: layer number starting from the top layer k = 1, 3: the upper and lower surfaces of a ply, respectively Mode (3): failure of matrix in tension
(90~/0°/90 °) laminates shows that when the odd layer is near the bottom of the strip, the laminates are stiffer than when it is near the top as is clearly shown in Figure 7. On the other hand, the bending stiffness of (00/90°/0 °) laminates is reduced by shifting the odd layer to the top.
5.2 Initial failure in unsymmetrie laminates Multi-layered unsymmetric cross-ply laminates, as shown in Figure 3, subjected to uniform pressure have been analysed for small and large deflection initial failure using the Maximum Stress (Independent) and Tsai-Hill failure criteria. The laminates were rectangular (b/a = 5) and simply supported in-plane fixed (SS2), i.e. the same edge condition which caused the large deflection effect reported in ref. 8. This work has been undertaken in order to study the effect of the odd layer as well as the number of layers on the strength of the laminates.
514
5.2.1 Linear initial failure analysis. The linear initial failure analysis results for laminates (0~,/90°/0°), (0°/90°/0~,), (90~,/0°/90 °) and (90°/0°/90~,) with SS2 edge conditions are presented in Tables 5-8 and shown in Figure 8. The main observations are as follows: (1) The Maximum Stress (Independent) and Tsai-HiU failure criteria predict similar failure pressures and confirm the earlier findings of Turvey and Osman 9 for predominantly symmetric cross and symmetric and antisymmetric angle-ply laminated rectangular plates. (2) Initial failure in (0°/90°/0~) laminates, where the fibres are aligned mainly parallel to the shorter side of the strip and the odd layer is near the top, is due to fibre fracture at the top centre of the strip. However, when the odd layer is near the bottom, i.e. in (0~/90°/0 °) laminates, the mode of failure depends on the number of layers. When the number of plies is small, failure is initiated by fibre fracture at the top
Flexural failure analysis of cross-ply strips. G. J. Turvey and M. Y. Osman Table 8
Initial failure of multi-layered cross-ply (90°/0°/90n) simply supported in-plane fixed (SS2) laminates under uniform pressure b/a = 5,
h/a = 0.01 n
1
Criterion
s
qf
wf
x/a
y/b
In
k
Mode
Tsai-Hill
1 2 1 2
60.5 2239.6 60.5 2239.6
6.4747 2.6819 6.4747 2.6818
0.5 0.5 0.5 0.5
0.5 0.2 0.5 0.2
3 3 3 3
3 3 3 3
3 3 3 3
1 2 1 2
61.8 651.6 61.8 651.6
5.7950 2.0456 5.7949 2.0456
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.5
4 4 4 4
3 3 3 3
3 3 3 3
1 2 1 2
78.3 386.9 77.4 386.9
5.2767 2.0185 5.2149 2.0185
0.5 0.0 0.5 0.0
0.4 0.4 0.5 0.4
6 6 6 6
3 3 3 3
3 3 3 3
1 2 1 2
85.8 396.6 85.8 396.7
5.2638 2.3255 5.2638 2.3258
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
12 12 12 12
3 3 3 3
3 3 3 3
1 2 1 2
79.6 439.4 79.6 439.4
5.5379 2.5255 5.5378 2.5254
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
20 20 20 20
3 3 3 3
3 3 3 3
1 2 1 2
72.0 480.8 72.0 486.8
5.7605 2.6674 5.7605 2.6780
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
30 30 30 30
3 3 3 3
3 3 3 3
1 2 1 2
66.5 500.0 66.5 500.0
5.9003 2.7384 5.9002 2.7384
0.5 0.4 0.5 0.5
0.5 0.5 0.5 0.5
40 40 40 40
3 3 3 3
3 3 3 3
1 2 1 2
59.7 480.2 59.7 480.2
6.0626 2.7412 6.0625 2.7412
0.5 0.4 0.5 0.5
0.5 0.5 0.5 0.5
60 60 60 60
3 3 3 3
3 3 3 3
1 2 1 2
40.9 444.8 40.9 444.8
6.4657 2.7548 6.4657 2.7549
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
1
3 3 3 3
3 3 3 3
Maximum stress (independent) Tsai- Hill
2
Maximum stress (independent) Tsai-Hill
4
Maximum stress (independent) Tsai-Hill
10
Maximum stress (independent) Tsai-Hill
18
Maximum stress (independent) Tsai-Hill
28
Maximum stress (independent) Tsai-Hill
38
Maximum stress (independent) Tsai-Hill
58
Maximum stress (independent) Tsai-Hill
90° (all)
Maximum stress (independent)
1 1
s(l): linear analysis; s(2): non-linear analysis In: layer number starting from the top layer k = 3: the lower surface of a ply Mode (3): failure of matrix in tension
c e n t r e a n d w h e n the n u m b e r o f layers is relatively large, i.e. 12 o r m o r e layers, failure is d u e to m a t r i x c r a c k i n g in the o d d layer a n d the failure l o c a t i o n c a n be off-centre. (3) T h e failure p r e s s u r e s o f (0~/90°/0 °) l a m i n a t e s are l o w e r t h a n (0°/90°/0~,) l a m i n a t e s for n > 8, i.e. shifting the 90 ° layer f r o m the b o t t o m to the t o p w e a k e n s the l a m i n a t e . (4) A l t h o u g h there is a g r e e m e n t b e t w e e n the M a x i m u m Stress ( I n d e p e n d e n t ) a n d T s a i - H i l l failure criteria p r e d i c t i o n s o f failure pressures, failed plies a n d failure m o d e s for all l a m i n a t e s , the t w o criteria p r e d i c t different failure l o c a t i o n s w h e n the n u m b e r o f layers in (0~,/90°/0 °) l a m i n a t e s is e q u a l to o r greater t h a n 12. A c c o r d i n g to the M a x i m u m Stress ( I n d e p e n d e n t ) c r i t e r i o n , failure a l w a y s o c c u r s at the c e n t r e o f the strip, w h e r e a s , a c c o r d i n g to the T s a i Hill c r i t e r i o n , failure l o c a t i o n s t e n d to shift a w a y f r o m the centre. (5) L i n e a r a n a l y s i s o f failure is o b v i o u s l y i n a c c u r a t e ,
b e c a u s e all o f the l a m i n a t e s a n a l y s e d were t h i n a n d this led to large deflections at initial failure. N e v e r theless, the a n a l y s i s d e m o n s t r a t e d t h a t the p o s i t i o n o f the o d d layer a n d the n u m b e r o f layers c a n h a v e a significant effect o n the strip s t r e n g t h . (6) A q u i c k g l a n c e at the failure p r e s s u r e s of(90°/0°/90~,) a n d (90~,/0°/90 °) l a m i n a t e s reveals the e x t e n t to w h i c h these strips are w e a k e n e d b y s u b j e c t i n g the m a t r i x r a t h e r t h a n the fibre to the large stresses t h a t act across the strips. B e c a u s e these l a m i n a t e s fail at very low pressures, the p o s i t i o n o f the o d d layer is p r a c t i c a l l y insignificant.
5.2.2 Nonlinear initial failure analysis. T h e n o n l i n e a r initial failure results for the s a m e l a m i n a t e s are given in Tables 5-8. T h e failure p r e s s u r e s are p l o t t e d a g a i n s t the n u m b e r o f plies in Figures 9 a n d 10 a n d the a s s o c i a t e d strip c e n t r e deflections are p l o t t e d a g a i n s t the n u m b e r o f plies in Figures 11 a n d 12. It c a n be seen that:
51 5
Flexural failure analysis of cross-ply strips. G. J. Turvey and M. Y. Osman
2.2
8
1.8
- - all 90
6
-,- [90/0/9.0n]
--[90n/0/90]
1.4
(xlO -2)
- - all 90
(xlO-3)
--0
4
"~" [90/0/90n]
--[9odo/ggJ ",'[O/90/OnJ
1.0
* [On/90/Oj
0.6
0
I
I
I
.I
10
20
30
40
0.2
I
I
50
60
70
0
I
I
I
I
I
I
10
20
30
40
50
60
Number of Plies
70
Number of Plies
Figure 8 Small deflection initial failure pressures of multi-layer
Figure 10 Large deflection initial failure pressures of multi-layer
unsymmetric cross-ply simply supported in-plane fixed (SS2) uniformly loadedlaminatesusingtheTsai Hill criterion (b / a = 5 and h / a = O.O1)
unsymmetric simply supported in-plane fixed (SS2) uniformly loaded laminates using the Tsai-Hill criterion (b/a = 5 and h/a = 0.01)
3.6 13
3.4 11
- - all 0
m
0 all
-,- [O/9.O/On] -,- [On/90/OJ
-
[o/9o/on]
-.-
[On/90/OJ
3.2
qf
9
Wc3.o
(xlO') 7
2.8
5-
3
2.6
I
o
10
I
I
20
30
I
40
I
50
2.4
I
60
70
Number of Plies
0
I
I
I
I
I
I
10
20
30
40
50
60
Number of Plies
Figure 9 Large deflection initial failure pressures of multi-layer
Figure 11 Non-linear strip centre deflections associated with initial
unsymmetric cross-ply simply supported in-plane fixed (SS2) uniformly loaded laminates using the Tsai-Hill criterion (b/a = 5 and h/a = 0.01)
failure versus the number of plies in multi-layer simply supported inplane fixed (SS2) laminates ([0~,/90°/0 °] and [0°/90°/0~])
516
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman (1) As in the case of linear analysis, there is close agreement between the Maximum Stress (Independent) and Tsai-Hill failure criteria in the prediction of the failure pressures. (2) The nonlinear failure pressures are 10 or more times higher than the corresponding linear failure pressures which is also supported by the failure pressure data for symmetric cross and symmetric and antisymmetric angle ply laminated rectangular plates given in ref. 9. (3) (90~,/0°/90 °) and (90°/0°/90n) laminates, i.e. the weaker lay-ups, fail at the centre or near the centre of the longer strip edges, whereas (0]/90°/0 °) and (0°/90°/0~,) laminates fail at or near the centre of the shorter strip edges. (4) Initial failure in all of the laminates is due to matrix cracking. In (0°/90°/0]) laminates, with the odd layer near the top, the matrix of the bottom layer fails in tension. When the odd layer is near the bottom, i.e. (0~,/90°/0 °) laminates, failure shifts from the bottom to the top for n < 10 and to the odd layer which fails due to matrix cracking in tension when n _> 10. Therefore, shifting the odd layer from the top to the bottom causes a decrease in strength which varies from 14 to 32% for the 4 and 60 layer laminates, respectively. (5) As the number of layers increases, the strength of a strip approaches one of two extremes. When the fibres are mainly aligned across the strip, the strength of a (0°/90°/0~,) laminate approaches that of a laminate in which all of the fibres are at 0 ° to the x-axis, whereas "the strength of a (0~/90°/0 °) laminate remains roughly unchanged. When the fibres are mainly aligned along the strip, the strength approaches that of a laminate in which the fibres are at 90 ° . The former is the upper bound on the strength, whereas the latter one is the lower bound. This comment applies equally to linear and nonlinear analyses as is evident from Figures 8-10. Hence, the position of the odd layer and the number of layers in a strip are very important factors in the strength of a strip. (6) It is demonstrated in ref. 8 that the magnitude of the centre deflection of a 2-layer simply supported in-plane fixed antisymmetric cross-ply strip may vary by more than 100% depending on the sign of Bll which changes according to the arrangement of the layers in the laminate, i.e. whether the 90 ° layer is on top of the 0 ° layer or vice versa. The initial failure pressures of similar strips, as shown in Table 9, are less dependent on the sign of the coupling stiffness. The initial failure pressure changes by only 8% when the sign of Bll is altered or the lay-up is reversed from (00/90 °) to (90°/0°). This is because initial failure occurs when the response of the strip is dominated by the stress resultants which are not significantly altered, unlike the stress couples which dominate at small loads and show considerable dependence on the sign of the coupling term.
2.9
2.7
2.5
2.3
WC 2.1
1.9
' - - 90 QII
•",-. [90/0/90n] "4- [90n/0/90]
1.7
1.5 0
I
I
10
26
I
I
30 40 Number of Plies
I
I
50
60
Figure 12 Non-linear strip centre deflectionsassociated with initial
failure versus the number of plies in multi-layersimply supported inplane fixed(SS2) laminates ([90~,/0°/90°] and [90°/0°/90~,]) Table 9 Large deflection initial failure in a 2-layer antisymmetric
cross-plysimplysupported in-planefixed(SS2)laminate under uniform pressure (h/a = 0.01, b/a = 5) Laminate
s
~]f
#f
x/a
y/b
In
k
Mode
0°/90°
1 2 1 2
617.0 617.0 671.5 671.5
1.8756 0.0 1.8756 0.0 1.4094 0.0 1.4094 0.0
0.5 0.5 0.5 0.5
2 2 1 1
3 3 1 1
3 3 3 3
90°/0°
s(l): Tsai Hillcriterion;s(2): Maximum Stress (Independent)criterion In: layer number starting from the top layer k = 3: the lower surface of a ply Mode (3): failure of matrix in tension 6 CONCLUSIONS (1) Coupling (Bll • 0) between bending and stretching is either positive or negative and generally reduces the bending stiffness of a laminate relative to an equivalent orthotropic laminate. (2) Coupling can increase or decrease the bending stiffness of a simply supported in-plane fixed unsymmetric cross-ply laminate in cylindrical bending relative to an equivalent orthotropic laminate depending on whether it is positive or negative. (3) The initial failure load of a 2-layer simply supported in-plane fixed antisymmetric cross-ply laminate in cylindrical bending is not greatly altered by the sign of the coupling term, B1 j, as is the centre deflection.
517
Flexural failure analysis of cross-ply strips: G. J. Turvey and 114.Y. Osman (4) The initial failure prediction of laminated strips in flexure according to the Maximum Stress (Independent) and Tsai-Hill failure criteria are broadly similar. ACKNOWLEDGEMENTS M. Y. Osman wishes to thank the Sudanese Ministry of Higher Education for providing a studentship to enable this research to be completed. Both authors wish to record their thanks to the Engineering Department for supporting their work.
2 3 4 5 6 7 8 9
Dij (i, j -- 1,2, 6) 0 • 0 Ex, ~y, Cxy 0
0
Exz, Eyz
ex,Ey 6xy, h~k, hzk+~ h k
REFERENCES 1
Bq (i, j = 1,2, 6) Cij
Jones, R.M. Buckling and vibration of rectangular unsymmetrically laminated cross-ply plates. AIAA Journal 1973, 11, 1626-1632 Mindlin, R.D. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics 1951, 18, 31-38 Otter, J.R.H., Cassell, A.C. and Hobbs, R.E. Dynamic relaxation, 'Proceedings of the Institution of Civil Engineers', Vol. 35, No. 4, 1966, pp. 633-656 Cassell, A.C. and Hobbs, R.E. Numerical stability of dynamic relaxation analysis of non-linear structures. International Journal for Numerical Methods in Engineering 1976, 10, 1407-1410 Turvey, G.J. and Osman, M.Y. Elastic large deflection analysis of isotropic rectangular Mindlin plates. International Journal of Mechanical Sciences 1990, 32, 1-14 Turvey, G.J. and Osman, M.Y. Large deflection analysis of orthotropic Mindlin plates with simply-supported and clamped-edge conditions. Composites Engineering 1991, 1, 235-248 Jones, R.M. 'Mechanics of Composite Materials', Scripta Book Company, Washington, DC, 1975, pp. 71-83 Sun, C.T. and Chin, H. On large deflection effects in unsymmetric cross-ply composite laminates. Journal of Composite Materials 1988, 22, 1045-1059 Turvey, G.J. and Osman, M.Y. Exact and approximate linear and nonlinear initial failure analysis of laminated Mindlin plates in flexure. Composite Structures 5 (Ed. I. H. Marshall), Elsevier Applied Science, London, 1989, pp. 133-171
o o o Xx, Xy, Xxy
k,, kv, kw, k~.~,k6y
Mx,My, Mxy n
nl
Nx,Ny, Nxy q ( = q a 4 E y l h -4)
Qx, Qy R, S, T U, I) W
(=wh-1) x, y, z 6t
Xt,c, Yt,c tri (i = 1-6)
NOMENCLATURE
a,b Aij ( i , j = 1,2,6) Aq (i, j = 4, 5)
518
plate side lengths in x and directions respectively plate extensional stiffnesses plate transverse shear stiffnesses
Uxy P,, Pv, Pw, Pox, Pxy
plate coupling stiffnesses lamina stiffnesses transformed to plate principal axes plate flexural stiffnesses extensional and shear strain components of the plate mid-plane transverse shear strain components of the plate mid-plane Young's moduli in-plane and transverse shear moduli distance of upper and lower surfaces of the kth lamina from the plate mid-plane plate thickness lamina number shear correction factors curvature and twist components of the plate mid-plane in-plane, out-of-plane and rotational damping factors stress couples number of laminae above/below the odd ply number of laminae in a strip stress resultants transverse pressure dimensionless transverse pressure transverse shear resultants in-plane and transverse shear strengths of a unidirectional lamina in-plane displacements deflection dimensionless deflection Cartesian co-ordinates time increment longitudinal and transverse tensile/ compressive strengths of a unidirectional lamina stress components with respect to material principal axes rotations of the original normal to the plate mid-plane in-plane Poisson's ratio in-plane, out-of-plane and rotational fictitious densities
ELSEVIER
Composites. Part B 27B (1996) 505-518 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
Elastic and initial flexural failure analysis of unsymmetrically laminated cross-ply strips
G. J. Turvey Engineering Department, Lancaster University, Bailrigg, Lancaster, LA 1 4YR, UK
and M. Y. Osman Faculty of Engineering and Technology, Nile Valley University, Atbara, Sudan (Received 5 January 1995; accepted 19 February 1996) A finite difference version of the Dynamic Relaxation (DR) method is used to generate elastic solutions of the small and the large deflection Mindlin plate equations for unsymmetrically laminated cross-ply strips subjected to uniform lateral pressure. The Maximum Stress (Independent) and Tsai-Hill failure criteria are combined with these solutions to produce initial failure analyses. It is shown that coupling between bending and stretching due to the Bit stiffness term can cause a considerable increase or decrease in the flexural stiffness of simply supported in-plane fixed strips relative to the equivalent orthotropic strips but not necessarily in the corresponding failure pressures. It is also found that the predictions of the two failure criteria are in close agreement. Copyright © 1996 Elsevier Science Limited (Keywords: composites; strips; laminates; flexure; failure)
1 INTRODUCTION In 1973 Jones I presented a linear buckling analysis of simply supported unsymmetrically laminated cross-ply plates subjected to uniform, uniaxial compression and provided numerical data for high modulus carbonepoxy, rectangular plates (b/a = 2) consisting of 3-100 layers. The fibres in all layers, except the second from the bottom of the laminate, were oriented parallel to the plate x-axis as shown in Figure 1. All of the layers were of equal thickness and thus, for a constant thickness laminate, the 90 ° layer became thinner and moved towards the bottom of the laminate as the number of layers increased. Although Jones recognised that such a laminate would probably not be used in engineering practice, he analysed it in order to demonstrate the effectiveness of his analytical model and to highlight the significant effect on the buckling load of coupling between bending and stretching. The laminate was first analysed with the lay-up shown in Figure 1, then with all of the layers parallel to the x-axis and finally as an equivalent orthotropic laminate in which coupling was ignored (i.e. Bij = 0). It was found that the buckling loads of the three laminates differed significantly when the number of layers was small, but that they tended to converge, albeit rather slowly, to roughly similar values
as the number of layers increased. Jones explained these differences as being due to the reduction in the magnitude of the coupling stiffnesses, Bij, as the number of layers increases. He also commented that approximating a general laminate by a specially orthotropic laminate can result in errors and, therefore, the use of such an approximation must be carefully proven in order to be justified for the particular case under consideration. Although Jones' buckling analysis is significant in its own right, it must be recognised that it is based on linear analysis and, therefore, the conclusions regarding the effects of coupling on the plate response may not hold to the same extent when deformations become large, i.e. in the context of geometric nonlinearity. In order to contribute to the resolution of this question, it was decided to undertake a large deflection rather than a postbuckling plate analysis. Thus, cross-ply laminated strips (i.e. large aspect ratio rectangular plates) under uniform lateral loading have been analysed with the following objectives: (i) To study the effect of bending-stretching coupling arising from unsymmetric cross-ply lay-ups typical of those used by Jones when the strip deflections are large.
505
Flexural failure analysis of cross-ply strips. G. d. Turvey and M. K Osman
O
o Ou° o: pwow I e x y-- -Oy - + ox-x + t ~ x ~ y J Ow o exz = (x -t- -Ox Ow o Cyz = Oy - ~ -Ox -
0
O* Y
0 o¸
9o
0 O
>x
b
,,OO
o
I
Z"L
0 °,
~X
Z Figure 1 A 6-layerunsymmetriccross-plylaminate [04°/90°/0°]
(ii) To determine the extent to which bending-stretching coupling affects the initial failure loads of unsymmetrically laminated cross-ply strips in the large deflection regime.
2 LAMINATED MINDLIN PLATE EQUATIONS Mindlin plate theory 2 is used to model the plate behaviour in flexure. It comprises of four sets of equations, viz. equilibrium, strain/curvature, constitutive and boundary condition equations, as set out below.
A brief summary of the large deflection plate equilibrium equations is given below. The small deflection versions are obtained by deleting the terms enclosed between curly brackets:
+
Oy
(2)
Ox
in which the superscript 0 denotes strains and curvatures of the plate mid-plane and the curly bracketed terms represent the nonlinear terms.
2.3 Constitutive equations The constitutive equations for cross-ply laminates are: 0
0
0
0
Nx = AllEx + A12Ey + BI1Xx + BI2Xy 0 0 0 0 Ny = A l : x + A22~y + B12Xx + B22Xy
Mx = B , : ° + B , : ° + D , , X ° + DI2X °
My
=- B12~°x + B22 g0 +
D12X 0 --k D22X°y
Mxy = B66e°y + D66X°y Q~ = A55e°z o Qy = A44eyz
ONx ONy Ox +--~-y = 0 ONxy FONy Ox Oy = 0 + Nx ~xz + 2Nxy -Ox - +Oy
Ny Oy2 J + q
02w Ow = Pw -~T + kw Ot OMx OMxy Ox -t Oy Qx = 0 (1)
nl
izk+1
k=l
k
C~;)(l'z'z2)dz
\Oxi J J
(4)
and Aij (i, j = 4, 5) denote the transverse shear stiffness coefficients and are calculated as follows:
k=l
The strains and curvatures of the mid-plane of the plate are:
=
where Aij, Bij and Dij (i, j = 1,2, 6) are, respectively, the membrane, coupling and flexural rigidities of the plate. The rigidities, Bij, arise due to the coupling between transverse bending and in-plane stretching. It should be noted that the constitutive equations for specially orthotropic plates are derived from equation (3) by setting the coupling stiffnesses, Bq, to zero. The rigidities are calculated as follows:
di:= Znl KiKj iz,k+I C(ijk) dz,
2.2 Strain/curvature equations
~x= ~ +
(3)
(Aij,Bij,Dq) = Z
OMxy F OMy OX -~y - Qy = 0
506
x o" _
gxy = A66~°y --FB66X °y
2.1 Equilibrium equations
Ox t-
0¢~
Xx = Ox o o~,, Xy = Oy
(i,j
=
4,5)
(5)
k
where Ci~) are the stiffnesses of a lamina referred to the plate principal axes and Ki, K: are the shear correction factors.
2.4 Boundary conditions Two types of simple support (simply supported inplane free (SS1) and fixed (SS2)) and two types of clamped
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman
Y
N., =~:,
=0
=,o = ~. = M,
N~=0 N=~y= 0 w----0 M,=0 ~ =0
N==0
N,y=0 w'-0
M==0
4'rt =0
u=0
v=0 w=0
v'-0 w=0
M~=0 ~,=0
M.=0
4'rj =0
u=v=w=¢.= My=O
(ss )
(ssO u=v=w=Ox=gp~=O
N.y = Ny = w = ~ . = Oy = 0
Nx=0 N=y = 0 w=0 ¢~, = 0
N.=0 N.y = 0 w=0 4~ = 0
4,~ =0
4,y =0 N=v = Nu = w =
,~= = 4'y =
My = 0
u=0
>2;
u., =iv, = w = ~ . =My =0
w=O==
u=v=
u=0 v=0
u=O v--0
w=0
w=0
~ =0
~ =0
4,y =0
,/,v = 0
0
W=~=-'~y
u-'v--
(ccl)
=0
(ccz)
Figure 2 Simplysupported and clamped boundary conditions
support (clamped in-plane free (CC1) and fixed (CC2)) are prescribed on the strip edges as shown in Figure 2. These conditions have been chosen because they represent limiting cases of in and out-of-plane restraint at the edges. 3 SOLUTION OF THE PLATE EQUATIONS Approximate solutions of the large deflection Mindlin plate equations, i.e. equations (I)-(3), are obtained using a finite difference implementation of the Dynamic Relaxation (DR) method (see refs 3 and 4 for further details). In order to solve equations (1)-(3) damping and inertia terms are first added to the right hand sides of equation (1) as follows:
ON,- ON v
02u
aN,:~, any
a2v
Ou kav
a~ +-aT-y=P"b7~+ vN aox aoy
r
a---x-v+ -~y + I Nx ~ x 2 + 2 N x ya- x a y + N .v ay 2 j + q aZ w aw = p,, ~-ff + kw --~
aMx aMxy ax
F Oy
~
OCx
Qx = p¢~ ~,,
+ k¢~ at
aMxv aM~, 02t~y OOy ax + ~ - Qy = p~, ~ + k<, 0---7
(6)
Then the following approximations are defined for the velocity and acceleration terms: aol a,
1 faola -
2
oqo~b] +
b-Y--h\a,
a,
] (7)
a,}
in which a --_ u, v, w, ~x, ~by.Substituting the velocity and acceleration term approximations of equation (7) into the right hand sides of equation (6) and re-arranging leads to the following expressions relating the velocity at the end of the time step, 6t, to the velocity at the end of the previous time step and the out-of-balance forces:
Oua _. 1 f Oub 6t (ON x +ONxy~] --~F;] Ot l+kT, [(1 - k~,)--~ + -p. - \~; ov a
0t
1
l + k;
(1 - k ; ) -OF +
~ \ ax
+
507
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman Table 1
M a t e r i a l p r o p e r t i e s used in the analysis
Ex/F~v
Gxy/Ey
axz/Ey
Gyz/Ey
1)xy
14.67
0.59
0.59
0.59
0.34
o"
OOO
o22
•,~~1000000 ,,~,0 0 0 ,,
lJ
M',
i
Ey = 3.9 k N / m m 2
"
s Table 2
S t r e n g t h p r o p e r t i e s o f m a t e r i a l used in th~ analysis
Xt/Yt
Xc/Yt
Yc/Yt
R/Yt
S/Yt
T/Yt
Yta2/Eyh 2
108.3
19.92
"1.0
2.83
2.83
2.83
30.77
,~0 0 0
I
o
o)
.) [¢v'~,;/o*] ~) t¢/9~/4,] e) [9O',V'o'/g&I a) [9~/o)'9~.]
Yt = 1 2 N / r a m 2
Table 3
•
Figure 3
F o u r m u l t i - l a y e r e d u n s y m m e t r i c cross-ply l a m i n a t e s
Deflection at the centre o f m u l t i - l a y e r u n s y m m e t r i c cross-ply simply s u p p o r t e d l a m i n a t e s u n d e r u n i f o r m pressure,
(b/a
= 5, h/a = 0.01,
# = 18o) SS1
SS2
Laminate
~]
~2(Bq = O)
%
wl
~2(Bq = O)
00/900/0 ° 90°/0°/90 °
2.0844 --
2,0844 --
0.0 --
0.8256 1.1546
0.8256 1.1546
0.0 0.0
0~/90°/0 ° 0°/90°/90~ 90~/0°/90 ° 90°/0°/90~
2.1669 2.1666 ---
2.1367 2.1368 ---
1.4 1.4 ---
0.8475 0.7687 1.1513 1.3769
0.8051 0.8051 1.2393 1.2393
5.0 -4.7 -7.6 10.0
0]/90°/0 °
2.2796 2.2790 ---
2.2286 2.2287 ---
2.2 2.2 ---
0.8500 0.7536 1.1800 1,5212
0.7972 0.7972 1.2989 1.2989
6.2 -5.8 - 10.1 14.6
90,]/0 °/90 ° 90°/0°/90,]
2.3455 2.3447 ---
2.2893 2.2894 ---
2.4 2.4 ---
0.8455 0.7480 1.2121 1.6165
0.7919 0.7919 1.3472 1.3472
6.3 -5.8 - 11.1 16.7
0~o/90°/0 ° 0°/90°/0~o 90~o/0°/90 °
2.3457 2.3446 --
2.3164 2.3166 --
1.2 1.2 --
90°/0°/90~o
.
0.81 O0 0.7424 1.3763 1.8372
0.7737 0.7737 1.5371 1.5371
4.5 -4.2 - 11.7 16.3
O~s/90°/O° 0°/90°/0°18
2.2488 2.2480 ---
2.2366 2.2367 ---
0.5 0.5 ---
0.7859 0.7417 1.5189 1.9147
0.7627 0.7627 1.6700 1.6699
3.0 -2.8 -9.9 12.8
2.1797 2.1792 ---
2.1740 2.1741 ---
0.3 0.2 ---
0.7720 0.7414 1.6289 1.9508
0.7561 0.7561 1,7602 1.7601
2.1 -2.0 -8.1 9.8
2.1407 2.1402 ---
2.1374 2.1374 ---
0.2 0.1 ---
0.7645 0.7412 1.6999 1.9684
0.7525 0.7525 1.8138 1.8138
1.6 - 1.5 -6.7 7.9
90~s/0°/90 ° 90°/0°/90~s
2.0990 2.0987 ---
2.0975 2.0975 ---
0.0 0.0 ---
0.7568 0.7410 1.7855 1.9860
0.7487 0.7487 1.8745 1.8745
1.1 -1.0 -5.0 5.6
0 ° (all) 90 ° (all)
2.0089 --
2.0089 --
0.0 --
0.7405 2.0241
0.7405 2.0241
0.0 0.0
0°/90°/0~ 90]/00/90 ° 90°/0°/90~ 0,~/90°/0 °
0°/90°/0°4
90~s/0°/90 ° 90°/0°/90~s 0~s/90°/0 °
O°/90°/O~s 90~s/0°/90 ° 90°/0°/90~8
O~s/90°/O° 0°/90°/0]8 90]8/0°/90 ° 90°/0°/90]8 O~s/90°/O °
O°/90°/O~s
%: 100 × (el
-
.
.
.
%
e2)/~t
- - : solution u n o b t a i n a b l e due to n u m e r i c a l instability because the plate is in-plane free
Ot
l+k~
(1 - k'v) ---~- +
02w
02w
N O2W
+ Nx--~xz + 2Nxy o x O y F y Oyz + q
508
Ot
+
~] )J
1 1 + k *Ox ×
(I - kc,x)
+ -P~x I. Ox
Oy
Qx
Flexural failure analysis of cross-ply strips: G. d. Turvey and M K Osman Deflection at the centre of multi-layer unsymmetric cross-ply clamped laminates under uniform pressure, (b/a = 5, h/a = 0.01, q = 180)
Table 4
CCI
CC2
Laminate
wl
~2(Bij = O)
0o/90o/0 ° 9ff'/0°/90 °
0.4274 3.7599
0.4274 3.7599
0~/90°/0 ° 0°/90°/90~ 90~/0°/90 ° 90°/0°/90~
0.4442 0.4441 4.0689 3.9905
0~/90°/0 ° 0°/90°/0~ 90]/0°/90 ° 90°/0°/90;
%
~'~
w2(Ba = O)
%
0.0 0.0
0.4000 0.9908
0.4000 0.9908
0.0 0.0
0.4380 0.4380 3.1498 3.1498
1.4 1.4 22.6 21.1
0.4088 0.4088 1.0715 1.0711
0.4046 0.4046 1.0380 1.0380
1.0 1.0 3.1 3.1
0.4670 0.4667 3.8509 3.7554
0.4566 0.4566 2.4573 2.4573
2.2 2.2 36.2 34.6
0.4220 0.4220 1.1193 1.1185
0.4152 0.4152 1.0460 1.0460
1.6 1.6 6.5 6.5
0,]/90°/0 ° 0°/90°/0~ 90~/0°/90 ° 90°/0°/90~
0.4803 0.4802 3.5974 3.5175
0.4689 0.4689 2.1545 2.1545
2.4 2.4 40.1 38.7
0.4290 0.4290 1.1530 I. 1520
0.4218 0.4218 1.0539 1.0539
1.7 1.7 8.6 8.5
0~0/90°/0 ° 0°/90°/0~ 90%/00/90 ° 90°/0°/90%
0,4806 0.4805 3.0706 3.0126
0.4746 0.4746 2.0402 2.0402
1.2 1.2 33.6 32.3
0.4253 0.4253 1.2683 1.2665
0.4217 0.4217 1.1553 1.1553
0.1 0.8 8.9 8.8
0~8/90°/0 ° 0°/90°/0~8 90~8/0°/90 ° 90°/0°/90~8
0.4612 0.4611 3.1017 3.0523
0.4587 0.4587 2.3958 2.3957
0.1 0.5 22.8 21.5
0.4120 0.4119 1.3568 1.3549
0.4040 0.4104 1.2786 1.2789
1.9 0.4 5.8 5.6
0~8/90°/0 ° 0°/90°/0~8 90~s/0°/90 ° 90°/0°/90~8
0.4473 0.4472 3.2784 3.2323
0.4461 0.4461 2.7923 2.7923
0.0 0.2 14.8 13.6
0.4025 0.4025 1.4280 1.4262
0.4018 0.4018 1.3784 1.3784
0.2 0.2 3.5 3.4
0~8/90°/0 ° 0°/90°/0~8 90~8/0°/90 ° 90°/0°/90~8
0.4394 0.4394 3.4475 3.4040
0.4388 0.4388 3.0920 3.0920
0.0 0.1 10.3 9.2
0.3972 0.3972 1.4759 1.4743
0.3968 0.3968 1.4420 1.4420
0.1 0.1 2.3 2.2
0~8/90°/0 ° 0°/90°/0~s 90~8/0°/90 ° 90°/0°/90~8
0.4310 0.4310 3.7065 3."6679
0.4307 0.4307 3.4919 3.4919
0.0 0.0 5.8 4.8
0.3914 0.3914 1.5360 1.5347
0.3912 0.3912 1.5174 1.5174
0.0 0.0 1.2 1.1
0 ° (all) 90 ° (all)
0.4128 4.7487
0.4128 4.7487
0.0 0.0
0.3786 1.7149
0.3786 1.7149
0.0 0.0
% : 100 X (14~1 -- W 2 ) / ~ ' 1
algorithm is a simple sequential procedure consisting of the following steps:
1
at
1 + k~, × [(l_k;))0q~y
b
tft f O M x y Pe~
OMy
+ - V y - e,
}]
(8) In equations (7) and (8) the superscripts, a and b, refer, respectively, to the values of the velocities after and before the time step, 6t, and k; = (k~6tp21)/2, where the subscript, a, is defined immediately following equation
(7). The displacements at the end of each time increment, 6t, are evaluated using the following simple integration procedure: aa :
a b + 6t 0eta 0---T
(9)
where a is again defined following equation (7). Thus, equations (8), (9), (2) and (3) constitute the set of equations for solution. The DR numerical solution
(1) Set the initial conditions (all variables, i.e. displacements, stress resultants, etc are set to zero and the transverse pressure is applied). (2) Compute the velocities at the end of the time step, 6t from equation (8) using the known displacements and stress resultants/couples. (3) Compute the displacements from equation (9) using the velocities calculated in the previous step and the known displacements at the end of the preceding time step. (4) Apply the displacement boundary conditions so that any incorrectly computed boundary displacements determined in the previous step are corrected and the displacements are correctly determined throughout the strip. (5) Compute the strains and curvatures from equation (2) using the displacements calculated in the two preceding steps. (6) Compute the stress resultants/couples from equation
509
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman
2.2
5
.
1.8
WC 1.4
T/ //'
- - all 0 -,-[0n/9o/01
~-
..,,. [90n/0/90 ]
2
,90n/0/9t]
1.0
_ij=_o
0.6 0
, 10
, 20
, 30
I 40
I 50
I 60
I
70
Number of Plies
0
10
I
I
I
I
20 30 40 50 Number of Plies
I
60
70
Figure4 Deflectionsat the centre of a multi-layerunsymmetriccrossply simplysupportedin-planefixed(SS2)uniformlyloadedlaminatefor # = 180, b/a = 5 and h/a = 0.01
Figure 5 Deflectionsat the centre of a multi-layerunsymmetriccrossply clampedin-planefree(CC1)uniformlyloadedlaminatefor ~/= 180, b/a = 5 and h/a = 0.01
(3) using the strains and curvatures calculated in the preceding step. (7) Apply the stress resultant/couple boundary conditions so that incorrectly computed boundary stress resultants/couples are corrected and the stress resultants/couples are correctly determined throughout the strip. (8) Check that the velocities throughout the strip are acceptably small (say <10 -6 ) so that the strip vibrations have effectively ceased and the desired static solution has been achieved. (9) If step (8) is satisfied, print out the results, otherwise repeat steps (2)-(8) in sequence until the velocity convergence criterion is satisfied.
No details of the verification studies undertaken to demonstrate the accuracy of the DR finite difference solution algorithm are presented here. Suffice it to say that extensive checking has been undertaken. The effect of mesh size on the solution accuracy in the context of isotropic and orthotropic Mindlin plates has been reported on in refs 5 and 6, which also show that numerical results obtained with the D R program are in good agreement with the numerical solutions obtained by finite element and finite strip analyses. The same DR computer analysis has been used here to obtain the present laminated strip results by modelling the strip as a long rectangular plate, i.e. adopting an aspect ratio of 5 : 1.
It should be appreciated that the DR algorithm, as outlined above, is actually applied to the discrete equivalents of equations (8), (9), (2) and (3). The discrete system is obtained by recasting these equations into centred finite difference format. The finite difference forms of the equations are too lengthy to be included here. However, the interested reader may obtain a deeper appreciation of the general approach adopted by consulting refs 3 and 4, both of which provide detailed descriptions of various aspects of the finite difference DR procedure in the context of simple structural problems.
510
4 F A I L U R E CRITERIA In the analysis of the initial failure of unsymmetrically laminated cross-ply strips, the Tsai-Hill and Maximum Stress (Independent) criteria are used. The underlying philosophy and derivation of these criteria are described in ref. 7.
4.1 Tsai-Hill failure criterion Failure is assumed to occur when the following
Flexural failure analysis of cross-ply strips. G. J. Turvey and M. Y. Osman
1.8
2.2
1.4
1.8
m
w
We 1.0
We 1.4
all 90 all 0 -- [0.19.0/01. •-,.- tgon/o/90j
--
II'/
---
\
~-
- - all o
r-~U---u" \
1.0 ~ , _ _
0.6
0.2
I
0
10
I
I
I
I
20 30 40 50 Number of Plies
0.6
I
60
0
70
-,-[9o/oj9o.]
"~- [90n/O/90J -,- tOl9OlqnJ
. -.x.-[on/90/o]
I
I
I
I
I
I
I0
20
30
40
50
60
70
Number of Plies
Figure 6 Deflections at the centre of a multi-layer unsymmetric cross-
Figure 7 Deflections at the centre of a multi-layer unsymmetric cross-
ply clamped in-plane fixed (CC2) uniformly loaded laminate for g/= 180, b/a = 5 and h/a = 0.01
ply simply supported in-plane fixed (SS2) uniformly loaded laminate for ~/= 180, b/a = 5 and h/a = 0.01
criterion is satisfied:
index to 1, the pressure is increased to Aq(s) and, hence, the stresses are increased by the same scaling factor A > 1. As a result two equations may be written as follows:
17~
X2
1710"2
O'~
2 174
.
2 0-5.
2 0"6
j(2 + ~ - 5 + ~ - Y + ~ - ~ - ~
= 1
(10)
where X is set to Xt when 17t is tensile and to Xc when 17t is compressive. Similarly, Y is set to Yt when 42 is tensile and to Yc when 172 is compressive.
2 171(s) X-2
2 2 2 2 0-1(s)172(s) O'2(s).4- 174(s) 175(s) 176(s) ~ X q- y 2 - - R E + - ~ +- ~ = FIN(s)
(12) and
4.2 Maximum stress (independent)failure criterion
A2 [17~(s)
17t(s)°2(s) 1- °2(s)
°2(s)
°~(s)
'
172(s)] = 1
Failure is assumed to occur when one or more of the following conditions are satisfied: 171 = X,
172 ~-- Y,
0"4 = R,
45 : S,
0"6 = T (11)
where X and Y are interpreted as for the Tsai-Hill failure criterion. Two different procedures have been used to determine the initial failure pressures depending on the type of analysis used. The small deflection initial failure pressures were determined using a simple scaling procedure, whereas the large deflection initial failure pressures were determined by an iterative procedure outlined below: 4.2.1 The scaling procedure. A small pressure, say q(s), was applied to the strip and the maximum value of the failure index, FIN(s), which is the sum of the terms on the left hand side of equation (10), was computed. Because q(s) is small, FIN(s ) < 1. To increase the failure
(13) Hence, dividing equation (13) by equation (12) the scaling factor, A, can be expressed as: A=
~/~,1 i~,l(s)
(14)
In the case of the Maximum Stress (Independent) failure criterion, the scaling factor is obtained, likewise, by applying a small pressure and computing the ratios: A1 =
X o1 (s)
,
A2-
S
A4
= --,
17s(s)
Y °2(s)
,
A5
A3
T --
R --
174(s)
, (15)
0-6(s)
A is then set to the maximum value of Ai (i = 1 , . . . , 5).
511
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman Table 5
Initial failure of multi-layer unsymmetric cross-ply [0n/90°/0°] simply supported in-plane fixed (SS2) laminates under uniform pressure
(b/a = 5, h/a = 0.01) n
1
Maximum stress (independent)
Maximum stress (independent)
Maximum stress (independent)
Maximum stress (independent) Tsai-Hill
18
Maximum stress (independent) Tsai-Hill
28
Maximum stress (independent) Tsai-HiU
38
Maximum stress (independent) Tsai-Hill
58
Maximum stress (independent) Tsai-Hill
0° (all)
In
k
Mode
793.7 6500.0 788.9 6500.0
8.9143 3.1125 8.8595 3.1125
0.5 0.5 0.5 0.5
0.5 0.2 0.5 0.2
1 2 1 2
1 3 1 3
2 3 2 3
1 2 1 2
789.9 4520.8 785.1 4500.0
9.1222 2.7023 9.0663 2.6904
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 1 1 1
1 1 1 1
2 3 2 3
1 2 1 2
738.7 3774,3 734.2 3800.0
9.1601 2.4781 9.1042 2.4837
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 1 1 1
1 1 1 1
2 3 2 3
1 2 1 2
601.9 4887.0 606.9 4868.8
7.5378 2.6075 7.6007 2.6042
0.5 0.5 0.5 0.5
0.3 0.0 0.5 0.0
11 1 11 1
3 1 3 1
3 3 3 3
1 2 1 2
585.0 5842.5 589.5 6000.0
7.0667 2.7240 7.1207 2.7485
0.5 0.5 0.5 0.5
0.3 0.1 0.5 0.2
19 19 19 19
3 3 3 3
3 3 3 3
1 2 1 2
584.6 5842.5 588.5 5842.5
6.8616 2,7102 6.9077 2.7102
0.5 0.5 0.5 0.5
0.3 0.1 0.5 0.1
29 29 29 29
3 3 3 3
3 3 3 3
1 2 1 2
590.0 5842.5 590.0 5842.5
6.8069 2.6896 6.8068 2.6896
0.5 0.5 0.5 0.5
0.5 0.1 0.5 0.1
39 39 39 39
3 3 3 3
3 3 3 3
1 2 1 2
583.0 5842.5 592.6 5700.0
6.6003 2.6779 6.7094 2.6556
0.5 0.5 0.5 0.5
0.1 0.1 0.5 0.1
59 59 59 59
3 3 3 3
3 3 3 3
1 2 1 2
822.2 13 230.0 817.2 13 230.0
8.9150 3.4992 8.8602 3.4992
0.5 0.5 0.5 0.5
0,5 0.0 0.5 0.0
1 1 1 1
1 3 1 3
1 3 2 3
1 2 1 2
Tsai-Hill 10
y/b
Tsai-Hill
Tsai-Hill 4
x/a
s
Tsai-Hill 2
~f
Criterion
Maximum stress (independent)
~/r
s(1): linear results; s(2): non-linear results In: layer number starting from the top layer k = 1, 3: the upper and lower surfaces of a ply, respectively Mode (2): failure of fibre in compression Mode (3): failure of matrix in tension
4.2.2 The iterative method. The iterative procedure is used in the n o n l i n e a r analysis o f initial failure. A n arbitrary pressure is applied a n d the c o r r e s p o n d i n g failure index, F I N , is c o m p u t e d . W h e n F I N < 0.99, the load is increased by 5% a n d the analysis is c o n t i n u e d . W h e n F I N > 1.01, the pressure is reduced by a similar percentage a n d the analysis is c o n t i n u e d . This iterative procedure is c o n t i n u e d until the desired accuracy is reached, i.e. a tolerance of 1% o n the failure index is achieved. In the case o f the M a x i m u m Stress ( I n d e p e n d ent) criterion, the same tolerance is applied to the value of )~i (i --- 1 , . . , , 5) in e q u a t i o n (15).
5 NUMERICAL RESULTS AND DISCUSSION
5.1 Large deflections in multi-layered cross-ply strips The cross-ply laminates analysed consist o f 3 - 6 0 layers. The fibres in all o f the layers, except one, are in
51 2
one direction only. The odd layer is either second from the top or second from the b o t t o m of the l a m i n a t e as s h o w n in Figure 3. I n laminates (a) a n d (b), the fibres are m a i n l y parallel to the plate x-axis, whereas in laminates (c) a n d (d) they are m a i n l y parallel to the yaxis. N u m e r i c a l results are o b t a i n e d for one q u a r t e r o f the strip using a 5 x 5 r e c t a n g u l a r finite difference mesh, with shear correction factors, K42 = K} = 5/6. The material properties used in the analysis are typical of a c a r b o n - e p o x y unidirectional l a m i n a a n d are given in Tables I a n d 2. The l a m i n a t e d strips analysed are thin h/a = 0.01 with aspect ratio 5 a n d SS1, SS2, CC1 a n d CC2 edge conditions. The centre deflections are listed in Tables 3 a n d 4 a n d are s h o w n graphically in Figures 4-7. Two other sets of results are o b t a i n e d by analysing the l a m i n a t e as specially o r t h o t r o p i c (Bij -- 0) a n d with all layers aligned in one direction. These two sets of results serve as indicators of the significance of c o u p l i n g a n d the effect o f the odd layer o n the response o f a cross-ply strip.
Flexural failure analysis of crcss-ply strips. G. J. Turvey and M. K Osman Table 6
Initial failure in multi-layer unsymmetric cross-ply (0°/90°/0~,) simply supported in-plane fixed (SS2) laminates under uniform pressure
(b/ct = 5, h/a = 0.01) n
1
M a x i m u m stress (independent)
M a x i m u m stress (independent)
M a x i m u m stress (independent)
M a x i m u m stress (independent) Tsai-Hill
18
M a x i m u m stress (independent) Tsai Hill
28
M a x i m u m stress (independent) T s a i - Hill
38
M a x i m u m stress (independent) Tsai- Hill
58
M a x i m u m stress (independent) Tsai-Hill
0 (all)
793.7 6500.0 788.9 6500.0
8.9143 3.1125 8.8595 3.1125
0.5 0.5 0.5 0.5
0.5 0.2 0.5 0.2
1 2 1 2
752.2 5221.7 747.6 5209.3
8.6866 2.7473 8.6331 2.7450
0.5 0.5 0.5 0.5
I 2 1 2
695.9 4664.5 691.6 4630.5
8.6293 2.5519 8.5761 2.5454
1 2 1 2
696.4 5731.3 692.1 5788.1
I 2 1 2
1 2 1 2
Tsai Hill 10
y/b
Tsai-Hill
Tsai-Hill 4
x/a
s
Tsai-Hill 2
wf
Criterion
M a x i m u m stress (independent)
qf
In
k
Mode
1 2 1 2
1 3 1 3
2 3 2 3
0.5 0.0 0.5 0.0
1 4 l 4
1 3 1 3
2 3 2 3
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 6 1 6
1 3 1 3
2 3 2 3
8.7214 2.6764 8.6677 2.6855
0.5 0,5 0.5 0.5
0.5 0.0 0.5 0.0
1 12 1 12
1 3 1 3
2 3 2 3
727.7 6536.7 723.2 6615.0
8.7900 2.7801 8.7359 2.7915
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
I 20 1 20
1 3 1 3
2 3 2 3
1 2 1 2
752.3 7231.5 747.6 7293.0
8.8291 2.8687 8.7748 2.8770
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 30 1 30
1 3 l 3
2 3 2 3
1 2 1 2
767.0 7683.3 762.3 7657.7
8.8497 2.9242 8.7953 2.9208
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 40 l 40
1 3 1 3
2 3 2 3
1 2 1 2
783.6 8500.0 778.8 8442.6
8.8709 3.0219 8.8163 3.0150
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 60 1 60
1 3 1 3
2 3 2 3
1 2 1 2
822.2 13 230.0 817.2 13 230.0
8.9150 3.4992 8.8602 3.4992
0.5 0.5 0.5 0.5
0.5 0.0 0.5 0.0
1 1 1 1
1 3 1 3
2 3 2 3
s(l): linear analysis; s(2): non-linear analysis In: layer number starting from the top layer k = 1,3: the upper and lower surfaces of a ply, respectively Mode (2): failure of fibre in compression Mode (3): failure of matrix in tension
It should be noted that DR solutions for the (SS1) laminates (90~,/0°/90 °) and (90°/0°/90~,) are difficult to obtain due to instability of the numerical computations because of large in-plane displacements. A study of the results reveals the following: (1) When the majority of the laminae have their fibredirections parallel to the longer side of the strip, i.e. (90~,/0°/90 °) and (90°/0°/90~,) laminates, the strip is, as anticipated, less stiff than when the fibres are aligned mainly across the strip, as in (0~,/90°/0 °) and (0°/90°/0~,) laminates. This comment holds for all boundary conditions. (2) The effect of coupling on the centre deflection of the (0~,/90°/0 °) and (0°/90°/0~,) laminates is marginal and accounts only for a maximum of 5 or 6% of the deflection for the SS2 edge condition, i.e. the odd layer does not alter the response of the strip significantly. Hence, if an odd layer is present due
to a mistake in the lay-up procedure, the analysis of the laminate as specially orthotropic does not lead to a large error. Coupling in (90~,/0°/90 °) and (90°/0°/90~) laminates is more significant, particularly when the number of layers is small, say less than 10. It should be noted, however, that laminates with such lay-ups are of little practical significance. (3) Coupling (Bij ¢ 0) in (90~/0°/90 °) laminates can make the strip stiffer or more flexible than equivalent orthotropic laminates in bending depending on the boundary conditions. Coupling makes simply supported in-plane fixed (SS2) strips stiffer as demonstrated in Figure 4 in which the deflection of simply supported cross-ply and orthotropic strips are compared. When the strip is clamped, i.e. CC1 or CC2 edge conditions apply, coupling makes the strip more flexible as Figures 5 and 6 indicate. (4) Comparing the deflections of (90°/0°/90~,) and
51 3
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman Table 7
Initial failure in multi-layer unsymmetric cross-ply (90~,/0°/90 °) simply supported in-plane fixed (SS2) laminates under uniform pressure
(b/a = 5, h/a = 0.01) n
1
Criterion
s
qr
~f
x/a
y/b
In
k
Mode
Tsai-Hill
1 2 1 2
60.5 2239.6 60.5 2239.6
6.4747 2.6819 6.4747 2.6818
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.2
3 3 3 3
3 3 3 3
3 3 3 3
1 2 1 2
72.0 685.9 72.0 684.2
6.7454 1.8540 6.7454 1.8524
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.5
4 1 4 1
3 1 3 1
3 3 3 3
1 2 1 2
107.3 412.2 106.3 406.2
7.2269 1.6509 7.1592 1.6419
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
6 1 6 1
3 1 3 1
3 3 3 3
1 2 1 2
120.4 417.4 120.4 416.5
7.3852 1.8949 7.3851 1.8934
0,5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
12 1 12 1
3 1 3 1
3 3 3 3
1 2 1 2
103.8 462.5 103.8 461.4
7.2240 2.1636 7.2239 2.1616
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
20 1 20 1
3 1 3 1
3 3 3 3
1 2 1 2
88.2 500.0 88.2 498.4
7.0596 2.3729 7.0595 2.3701
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
30 1 30 1
3 1 3 1
3 3 3 3
1 2 1 2
78.4 525.0 78.4 703.6
6.9519 2.5080 6.9518 2.7844
0.5 0.0 0.5 0.5
0.5 0.4 0.5 0.5
40 1 40 40
3 1 3 3
3 3 3 3
1 2 1 2
67.2 609.5 67.2 609.0
6,8234 2.7594 6.8233 2.7586
0.5 0.5 0.5 0.0
0.5 0,5 0.5 0.5
60 60 60 60
3 3 3 3
3 3 3 3
1 2 1 2
40.8 444.8 40.9 444.8
6.4657 2.7548 6.4657 2.7549
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
--1 1
3 3 3 3
3 3 3 3
M a x i m u m stress (independent) Tsai-Hill
2
M a x i m u m stress (independent) Tsai-Hill
4
M a x i m u m stress (independent) Tsai - Hill
10
M a x i m u m stress (independent) Tsai-Hill
18
M a x i m u m stress (independent) Tsai-Hill
28
M a x i m u m stress (independent) Tsai-Hill
38
M a x i m u m stress (independent) Tsai-Hill
58
M a x i m u m stress (independent) Tsai-Hill
90 ° (all)
M a x i m u m stress (independent)
s(1): linear analysis; s(2): non-linear analysis In: layer number starting from the top layer k = 1, 3: the upper and lower surfaces of a ply, respectively Mode (3): failure of matrix in tension
(90~/0°/90 °) laminates shows that when the odd layer is near the bottom of the strip, the laminates are stiffer than when it is near the top as is clearly shown in Figure 7. On the other hand, the bending stiffness of (00/90°/0 °) laminates is reduced by shifting the odd layer to the top.
5.2 Initial failure in unsymmetrie laminates Multi-layered unsymmetric cross-ply laminates, as shown in Figure 3, subjected to uniform pressure have been analysed for small and large deflection initial failure using the Maximum Stress (Independent) and Tsai-Hill failure criteria. The laminates were rectangular (b/a = 5) and simply supported in-plane fixed (SS2), i.e. the same edge condition which caused the large deflection effect reported in ref. 8. This work has been undertaken in order to study the effect of the odd layer as well as the number of layers on the strength of the laminates.
514
5.2.1 Linear initial failure analysis. The linear initial failure analysis results for laminates (0~,/90°/0°), (0°/90°/0~,), (90~,/0°/90 °) and (90°/0°/90~,) with SS2 edge conditions are presented in Tables 5-8 and shown in Figure 8. The main observations are as follows: (1) The Maximum Stress (Independent) and Tsai-HiU failure criteria predict similar failure pressures and confirm the earlier findings of Turvey and Osman 9 for predominantly symmetric cross and symmetric and antisymmetric angle-ply laminated rectangular plates. (2) Initial failure in (0°/90°/0~) laminates, where the fibres are aligned mainly parallel to the shorter side of the strip and the odd layer is near the top, is due to fibre fracture at the top centre of the strip. However, when the odd layer is near the bottom, i.e. in (0~/90°/0 °) laminates, the mode of failure depends on the number of layers. When the number of plies is small, failure is initiated by fibre fracture at the top
Flexural failure analysis of cross-ply strips. G. J. Turvey and M. Y. Osman Table 8
Initial failure of multi-layered cross-ply (90°/0°/90n) simply supported in-plane fixed (SS2) laminates under uniform pressure b/a = 5,
h/a = 0.01 n
1
Criterion
s
qf
wf
x/a
y/b
In
k
Mode
Tsai-Hill
1 2 1 2
60.5 2239.6 60.5 2239.6
6.4747 2.6819 6.4747 2.6818
0.5 0.5 0.5 0.5
0.5 0.2 0.5 0.2
3 3 3 3
3 3 3 3
3 3 3 3
1 2 1 2
61.8 651.6 61.8 651.6
5.7950 2.0456 5.7949 2.0456
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.5
4 4 4 4
3 3 3 3
3 3 3 3
1 2 1 2
78.3 386.9 77.4 386.9
5.2767 2.0185 5.2149 2.0185
0.5 0.0 0.5 0.0
0.4 0.4 0.5 0.4
6 6 6 6
3 3 3 3
3 3 3 3
1 2 1 2
85.8 396.6 85.8 396.7
5.2638 2.3255 5.2638 2.3258
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
12 12 12 12
3 3 3 3
3 3 3 3
1 2 1 2
79.6 439.4 79.6 439.4
5.5379 2.5255 5.5378 2.5254
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
20 20 20 20
3 3 3 3
3 3 3 3
1 2 1 2
72.0 480.8 72.0 486.8
5.7605 2.6674 5.7605 2.6780
0.5 0.0 0.5 0.0
0.5 0.4 0.5 0.4
30 30 30 30
3 3 3 3
3 3 3 3
1 2 1 2
66.5 500.0 66.5 500.0
5.9003 2.7384 5.9002 2.7384
0.5 0.4 0.5 0.5
0.5 0.5 0.5 0.5
40 40 40 40
3 3 3 3
3 3 3 3
1 2 1 2
59.7 480.2 59.7 480.2
6.0626 2.7412 6.0625 2.7412
0.5 0.4 0.5 0.5
0.5 0.5 0.5 0.5
60 60 60 60
3 3 3 3
3 3 3 3
1 2 1 2
40.9 444.8 40.9 444.8
6.4657 2.7548 6.4657 2.7549
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
1
3 3 3 3
3 3 3 3
Maximum stress (independent) Tsai- Hill
2
Maximum stress (independent) Tsai-Hill
4
Maximum stress (independent) Tsai-Hill
10
Maximum stress (independent) Tsai-Hill
18
Maximum stress (independent) Tsai-Hill
28
Maximum stress (independent) Tsai-Hill
38
Maximum stress (independent) Tsai-Hill
58
Maximum stress (independent) Tsai-Hill
90° (all)
Maximum stress (independent)
1 1
s(l): linear analysis; s(2): non-linear analysis In: layer number starting from the top layer k = 3: the lower surface of a ply Mode (3): failure of matrix in tension
c e n t r e a n d w h e n the n u m b e r o f layers is relatively large, i.e. 12 o r m o r e layers, failure is d u e to m a t r i x c r a c k i n g in the o d d layer a n d the failure l o c a t i o n c a n be off-centre. (3) T h e failure p r e s s u r e s o f (0~/90°/0 °) l a m i n a t e s are l o w e r t h a n (0°/90°/0~,) l a m i n a t e s for n > 8, i.e. shifting the 90 ° layer f r o m the b o t t o m to the t o p w e a k e n s the l a m i n a t e . (4) A l t h o u g h there is a g r e e m e n t b e t w e e n the M a x i m u m Stress ( I n d e p e n d e n t ) a n d T s a i - H i l l failure criteria p r e d i c t i o n s o f failure pressures, failed plies a n d failure m o d e s for all l a m i n a t e s , the t w o criteria p r e d i c t different failure l o c a t i o n s w h e n the n u m b e r o f layers in (0~,/90°/0 °) l a m i n a t e s is e q u a l to o r greater t h a n 12. A c c o r d i n g to the M a x i m u m Stress ( I n d e p e n d e n t ) c r i t e r i o n , failure a l w a y s o c c u r s at the c e n t r e o f the strip, w h e r e a s , a c c o r d i n g to the T s a i Hill c r i t e r i o n , failure l o c a t i o n s t e n d to shift a w a y f r o m the centre. (5) L i n e a r a n a l y s i s o f failure is o b v i o u s l y i n a c c u r a t e ,
b e c a u s e all o f the l a m i n a t e s a n a l y s e d were t h i n a n d this led to large deflections at initial failure. N e v e r theless, the a n a l y s i s d e m o n s t r a t e d t h a t the p o s i t i o n o f the o d d layer a n d the n u m b e r o f layers c a n h a v e a significant effect o n the strip s t r e n g t h . (6) A q u i c k g l a n c e at the failure p r e s s u r e s of(90°/0°/90~,) a n d (90~,/0°/90 °) l a m i n a t e s reveals the e x t e n t to w h i c h these strips are w e a k e n e d b y s u b j e c t i n g the m a t r i x r a t h e r t h a n the fibre to the large stresses t h a t act across the strips. B e c a u s e these l a m i n a t e s fail at very low pressures, the p o s i t i o n o f the o d d layer is p r a c t i c a l l y insignificant.
5.2.2 Nonlinear initial failure analysis. T h e n o n l i n e a r initial failure results for the s a m e l a m i n a t e s are given in Tables 5-8. T h e failure p r e s s u r e s are p l o t t e d a g a i n s t the n u m b e r o f plies in Figures 9 a n d 10 a n d the a s s o c i a t e d strip c e n t r e deflections are p l o t t e d a g a i n s t the n u m b e r o f plies in Figures 11 a n d 12. It c a n be seen that:
51 5
Flexural failure analysis of cross-ply strips. G. J. Turvey and M. Y. Osman
2.2
8
1.8
- - all 90
6
-,- [90/0/9.0n]
--[90n/0/90]
1.4
(xlO -2)
- - all 90
(xlO-3)
--0
4
"~" [90/0/90n]
--[9odo/ggJ ",'[O/90/OnJ
1.0
* [On/90/Oj
0.6
0
I
I
I
.I
10
20
30
40
0.2
I
I
50
60
70
0
I
I
I
I
I
I
10
20
30
40
50
60
Number of Plies
70
Number of Plies
Figure 8 Small deflection initial failure pressures of multi-layer
Figure 10 Large deflection initial failure pressures of multi-layer
unsymmetric cross-ply simply supported in-plane fixed (SS2) uniformly loadedlaminatesusingtheTsai Hill criterion (b / a = 5 and h / a = O.O1)
unsymmetric simply supported in-plane fixed (SS2) uniformly loaded laminates using the Tsai-Hill criterion (b/a = 5 and h/a = 0.01)
3.6 13
3.4 11
- - all 0
m
0 all
-,- [O/9.O/On] -,- [On/90/OJ
-
[o/9o/on]
-.-
[On/90/OJ
3.2
qf
9
Wc3.o
(xlO') 7
2.8
5-
3
2.6
I
o
10
I
I
20
30
I
40
I
50
2.4
I
60
70
Number of Plies
0
I
I
I
I
I
I
10
20
30
40
50
60
Number of Plies
Figure 9 Large deflection initial failure pressures of multi-layer
Figure 11 Non-linear strip centre deflections associated with initial
unsymmetric cross-ply simply supported in-plane fixed (SS2) uniformly loaded laminates using the Tsai-Hill criterion (b/a = 5 and h/a = 0.01)
failure versus the number of plies in multi-layer simply supported inplane fixed (SS2) laminates ([0~,/90°/0 °] and [0°/90°/0~])
516
Flexural failure analysis of cross-ply strips: G. J. Turvey and M. Y. Osman (1) As in the case of linear analysis, there is close agreement between the Maximum Stress (Independent) and Tsai-Hill failure criteria in the prediction of the failure pressures. (2) The nonlinear failure pressures are 10 or more times higher than the corresponding linear failure pressures which is also supported by the failure pressure data for symmetric cross and symmetric and antisymmetric angle ply laminated rectangular plates given in ref. 9. (3) (90~,/0°/90 °) and (90°/0°/90n) laminates, i.e. the weaker lay-ups, fail at the centre or near the centre of the longer strip edges, whereas (0]/90°/0 °) and (0°/90°/0~,) laminates fail at or near the centre of the shorter strip edges. (4) Initial failure in all of the laminates is due to matrix cracking. In (0°/90°/0]) laminates, with the odd layer near the top, the matrix of the bottom layer fails in tension. When the odd layer is near the bottom, i.e. (0~,/90°/0 °) laminates, failure shifts from the bottom to the top for n < 10 and to the odd layer which fails due to matrix cracking in tension when n _> 10. Therefore, shifting the odd layer from the top to the bottom causes a decrease in strength which varies from 14 to 32% for the 4 and 60 layer laminates, respectively. (5) As the number of layers increases, the strength of a strip approaches one of two extremes. When the fibres are mainly aligned across the strip, the strength of a (0°/90°/0~,) laminate approaches that of a laminate in which all of the fibres are at 0 ° to the x-axis, whereas "the strength of a (0~/90°/0 °) laminate remains roughly unchanged. When the fibres are mainly aligned along the strip, the strength approaches that of a laminate in which the fibres are at 90 ° . The former is the upper bound on the strength, whereas the latter one is the lower bound. This comment applies equally to linear and nonlinear analyses as is evident from Figures 8-10. Hence, the position of the odd layer and the number of layers in a strip are very important factors in the strength of a strip. (6) It is demonstrated in ref. 8 that the magnitude of the centre deflection of a 2-layer simply supported in-plane fixed antisymmetric cross-ply strip may vary by more than 100% depending on the sign of Bll which changes according to the arrangement of the layers in the laminate, i.e. whether the 90 ° layer is on top of the 0 ° layer or vice versa. The initial failure pressures of similar strips, as shown in Table 9, are less dependent on the sign of the coupling stiffness. The initial failure pressure changes by only 8% when the sign of Bll is altered or the lay-up is reversed from (00/90 °) to (90°/0°). This is because initial failure occurs when the response of the strip is dominated by the stress resultants which are not significantly altered, unlike the stress couples which dominate at small loads and show considerable dependence on the sign of the coupling term.
2.9
2.7
2.5
2.3
WC 2.1
1.9
' - - 90 QII
•",-. [90/0/90n] "4- [90n/0/90]
1.7
1.5 0
I
I
10
26
I
I
30 40 Number of Plies
I
I
50
60
Figure 12 Non-linear strip centre deflectionsassociated with initial
failure versus the number of plies in multi-layersimply supported inplane fixed(SS2) laminates ([90~,/0°/90°] and [90°/0°/90~,]) Table 9 Large deflection initial failure in a 2-layer antisymmetric
cross-plysimplysupported in-planefixed(SS2)laminate under uniform pressure (h/a = 0.01, b/a = 5) Laminate
s
~]f
#f
x/a
y/b
In
k
Mode
0°/90°
1 2 1 2
617.0 617.0 671.5 671.5
1.8756 0.0 1.8756 0.0 1.4094 0.0 1.4094 0.0
0.5 0.5 0.5 0.5
2 2 1 1
3 3 1 1
3 3 3 3
90°/0°
s(l): Tsai Hillcriterion;s(2): Maximum Stress (Independent)criterion In: layer number starting from the top layer k = 3: the lower surface of a ply Mode (3): failure of matrix in tension 6 CONCLUSIONS (1) Coupling (Bll • 0) between bending and stretching is either positive or negative and generally reduces the bending stiffness of a laminate relative to an equivalent orthotropic laminate. (2) Coupling can increase or decrease the bending stiffness of a simply supported in-plane fixed unsymmetric cross-ply laminate in cylindrical bending relative to an equivalent orthotropic laminate depending on whether it is positive or negative. (3) The initial failure load of a 2-layer simply supported in-plane fixed antisymmetric cross-ply laminate in cylindrical bending is not greatly altered by the sign of the coupling term, B1 j, as is the centre deflection.
517
Flexural failure analysis of cross-ply strips: G. J. Turvey and 114.Y. Osman (4) The initial failure prediction of laminated strips in flexure according to the Maximum Stress (Independent) and Tsai-Hill failure criteria are broadly similar. ACKNOWLEDGEMENTS M. Y. Osman wishes to thank the Sudanese Ministry of Higher Education for providing a studentship to enable this research to be completed. Both authors wish to record their thanks to the Engineering Department for supporting their work.
2 3 4 5 6 7 8 9
Dij (i, j -- 1,2, 6) 0 • 0 Ex, ~y, Cxy 0
0
Exz, Eyz
ex,Ey 6xy, h~k, hzk+~ h k
REFERENCES 1
Bq (i, j = 1,2, 6) Cij
Jones, R.M. Buckling and vibration of rectangular unsymmetrically laminated cross-ply plates. AIAA Journal 1973, 11, 1626-1632 Mindlin, R.D. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics 1951, 18, 31-38 Otter, J.R.H., Cassell, A.C. and Hobbs, R.E. Dynamic relaxation, 'Proceedings of the Institution of Civil Engineers', Vol. 35, No. 4, 1966, pp. 633-656 Cassell, A.C. and Hobbs, R.E. Numerical stability of dynamic relaxation analysis of non-linear structures. International Journal for Numerical Methods in Engineering 1976, 10, 1407-1410 Turvey, G.J. and Osman, M.Y. Elastic large deflection analysis of isotropic rectangular Mindlin plates. International Journal of Mechanical Sciences 1990, 32, 1-14 Turvey, G.J. and Osman, M.Y. Large deflection analysis of orthotropic Mindlin plates with simply-supported and clamped-edge conditions. Composites Engineering 1991, 1, 235-248 Jones, R.M. 'Mechanics of Composite Materials', Scripta Book Company, Washington, DC, 1975, pp. 71-83 Sun, C.T. and Chin, H. On large deflection effects in unsymmetric cross-ply composite laminates. Journal of Composite Materials 1988, 22, 1045-1059 Turvey, G.J. and Osman, M.Y. Exact and approximate linear and nonlinear initial failure analysis of laminated Mindlin plates in flexure. Composite Structures 5 (Ed. I. H. Marshall), Elsevier Applied Science, London, 1989, pp. 133-171
o o o Xx, Xy, Xxy
k,, kv, kw, k~.~,k6y
Mx,My, Mxy n
nl
Nx,Ny, Nxy q ( = q a 4 E y l h -4)
Qx, Qy R, S, T U, I) W
(=wh-1) x, y, z 6t
Xt,c, Yt,c tri (i = 1-6)
NOMENCLATURE
a,b Aij ( i , j = 1,2,6) Aq (i, j = 4, 5)
518
plate side lengths in x and directions respectively plate extensional stiffnesses plate transverse shear stiffnesses
Uxy P,, Pv, Pw, Pox, Pxy
plate coupling stiffnesses lamina stiffnesses transformed to plate principal axes plate flexural stiffnesses extensional and shear strain components of the plate mid-plane transverse shear strain components of the plate mid-plane Young's moduli in-plane and transverse shear moduli distance of upper and lower surfaces of the kth lamina from the plate mid-plane plate thickness lamina number shear correction factors curvature and twist components of the plate mid-plane in-plane, out-of-plane and rotational damping factors stress couples number of laminae above/below the odd ply number of laminae in a strip stress resultants transverse pressure dimensionless transverse pressure transverse shear resultants in-plane and transverse shear strengths of a unidirectional lamina in-plane displacements deflection dimensionless deflection Cartesian co-ordinates time increment longitudinal and transverse tensile/ compressive strengths of a unidirectional lamina stress components with respect to material principal axes rotations of the original normal to the plate mid-plane in-plane Poisson's ratio in-plane, out-of-plane and rotational fictitious densities