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Buckling of unsymmetric composite laminates
Composite Structures 5 (1986) 101-123
Buckling of Unsymmetric Composite Laminates Paul A. Lagace, David W. Jensen and Douglas C. Finch* Technology Laboratory for Advanced Composites, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
A BS TRA CT An initial experimental and analvtical investigation was conducted to examine the effects of the inherent mechanical couplings exhibited in fully anisotropic (i. e. unsymmetric) graphite~epoxy laminates on the buckling loads and mode shapes. Experimental techniques were devised to test 254 mm square plates of A S1/3501-6 graphite~epoxy under uniaxial compressive load with free, simply-supported and clamped edges. The results indicate that the mechanical couplings, especially those which relate stretching and bending behavior, cause out-of-plane deflections prior to buckling and reduce the buckling load significantly, i.e. the load at which out-of-plane deflections become large. The analytical results also show that the buckled mode shapes exhibit twisting due to the mechanical couplings. Suggestions are offered for improvements in the experimental and analvtical techniques to better understand these phenomena.
l INTRODUCTION M o d e r n aircraft design has been enhanced by the introduction of advanced composite materials such as carbon fiber reinforced plastics. While offering significant strength-to-weight and aeroelastic advantages, such materials have been shown to introduce mechanical couplings which require m o r e careful analysis than is usually necessaw for isotropic or orthotropic materials, j-" The concept of employing unbalanced laminates *Present address: Aerospace Corporation. Los Angeles. California 90009, USA. 101 Composite Structures 0263-8223/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986, Printed in Great Britain
102
Paul A. Lagace, David W. Jensen, Douglas C. Finch
(which do not have a ply oriented at - 0 for every one oriented at +0, e.g. [45/.0]s) to improve aeroelastic efficiency, a procedure referred to as aeroelastic tailoring, 3 results in a coupling between the bending and twisting behavior of a laminate, as well as between the in-plane stretching and shearing. Moreover, even more unique couplings may arise in using unsymmetric laminates since the stretching and bending behavior become interrelated. In fact, the use of unsymmetric laminates may result in fully anisotropic behavior in that the bending, stretching, shearing, and twisting responses are completely coupled. Unsymmetric laminates may arise 'naturally' during the design process. This is most likely to occur in panels with tapered thickness in which individual plies are dropped off. Due to design constraints, these are generally dropped off in an unsymmetric fashion. Additionally, there may be applications where the unique couplings of unsymmetric laminates provide aeroelastic or other advantages which can be effectively utilized in a structure. An important consideration to be made during the design process involves the manner in which these laminates are loaded. The buckling behavior of unsymmetric laminates is important inasmuch as many structural components will experience compressive loads. In fact, conventional wing panels are often designed to operate predominantly in the postbuckled region. Beginning with Bryan's first analytical solution to the buckling of isotropic plates in 189l, ~ an immense amount of research has been performed regarding the buckling behavior of plates. Leissa, ~ for example, mentions the availability of approximately 2000 references in the open literature concerning isotropic plate buckling and an additional 200 publications relating specifically to composites. Classical plate buckling theory has been well defined with the solution of many cases being explicitly summarized in such texts as Timoshenko and Gere. ~ During the past few decades, this work has been extended to orthotropic plates and, recently, to a more limited extent, to anisotropic plates, e.g. Ashton, 7-1° Kicher and Mandell, ~E Starnes, ~2t3 and Chia." This early work, however, has been primarily restricted to specially orthotropic, midplane symmetric, or antisymmetric cross-ply laminates with simplysupported boundary conditions. Such laminates possess no bendingstretching coupling, resulting in simplified constitutive relations and behavior. Ashton and Love "' did test a [+45 ,,/ - 4 5 ~,]T (rotated, antisymmetric cross-ply) laminate which displays stretching-twisting
Buckling of unsymmetric composite laminates
103
coupling and found the experimental buckling load to be approximately half that of the predicted value. It is clear that a better understanding of the effects of the inherent mechanical couplings is needed. Generally, unsymmetric anisotropic plates contain fully populated elastic matrices, and, therefore, exhibit every possible type of mechanical coupling in a complex interwoven fashion. The mechanical behavior of such laminates is somewhat analogous to that of plates with an initial curvature or flat plates loaded eccentrically. In each of these cases two phenomena occur: (1) the out-of-plane deflections begin simultaneously with the application of the load and (2) the critical buckling load is reduced substantially. This paper reports the results of an initial experimental and analytical investigation into the critical buckling loads and the corresponding deflection patterns of unsymmetric laminates. Laminates which possess different types and amounts of mechanical couplings were tested in uniaxial compression with various combinations of boundary, conditions. The objective of this investigation was to isolate and study the effects of these various mechanical couplings.
2 SCOPE As previously mentioned, fully anisotropic laminates can exhibit coupling between all possible mechanical deformations caused by in-plane loads or moments. These couplings are dependent upon the existence of terms in the constitutive relations between applied loads and moments and inplane strains and curvatures. These relations are expressed with the use of contracted notation, ~ as N, = AoE~,~+ B,jKj i,j=
1.2.6
(1)
M, = Bii~.5~+ D#K i
where N, and M; are the in-plane stress and bending resultants. ~j are the in-plane strains, K/ are the curvatures, and A;j, B;,. and D;j are elements of the A, B. and D matrices, respectively. There are five possible mechanical couplings between the four deformation types: extension, shear, bending and twisting. These
104
Paul A, Lagace, David W. Jensen, Douglas C. Finch TABLE 1 Mechanical Couplings and Associated Matrix Terms
Coupling type 1. 2. 3. 4. 5. 6.
Extension-shear Extension-bending Extension-twisting Shear-bending Shear-twisting Bending--twisting
Non-zero matrix terms A 16, A 26 BH, BI2, B22 B16, B,_~
Blo, B2~ B~ D i~, D2~
couplings are identified in Table 1 along with the components of the A, B, or D matrix associated with that particular coupling. When the associated terms are zero, the coupling does not exist. Laminates were chosen so as to isolate the various couplings involved. This is done by allowing only the coupling coefficients associated with the desired mechanical coupling to be nonzero. However, such a laminate could not be constructed to isolate shear-twisting coupling. Additionally, a series of laminates was chosen with 'fully-populated' matrices. These laminates possess all of the possible mechanical couplings. The degree of coupling was varied by changing the lamination angle, 0, of a [06//06]v laminate from 15° to 90° in 15° increments. The subscript 'T' denotes that this is the total laminate, in order to distinguish this from a symmetric arrangement. The double slash '//" indicates a room temperature bondline joining previously cured symmetric sublaminates. This type of construction, described in the experimental section, is necessary in order to obtain flat unsymmetric laminates. Finally, a basic specially-orthotropic laminate. [03/903]s, was chosen to serve as a benchmark since this laminate does not exhibit any of the various mechanical couplings. In addition to this basic construction, a [03//90d/03]r laminate was included to ascertain the effect of the room temperature bondline. The laminates chosen for the study and the mechanical couplings they exhibit are summarized in Table 2. For all cases loading is uniaxial compression along clamped ends. Three side boundary conditions are considered: free, simply-supported and clamped. In addition to the experiments, a Rayleigh-Ritz energy analysis is carried out to determine the classical buckling loads and mode shapes of the various laminates under the different boundary, conditions.
"0
=
a room temperature bondline. 'T' implies total laminate.
Yes
Yes
Extension~hear
15 °, 3 0 °, 4 5 °, 6 0 o, 7 5 ° a n d 9 0 °.
h Subscript
"//indicates
[0
dl9OdlO3]v <'h [0JI90311031190~]~ [0~//452//(1~//452//02]r [0~_1145,110~11-45,_1102]-r [O~,//O~,]l'
[0d903]s
Laminate
TABLE 2
Yes
---
Yes
Extension-bending
Yes
-Yes
--
Extett~ion-twisting and shear-bending
Mechanical couplings
Test Laminates and Associated Mechanical Couplings
Yes
---
__
Shear-twisting
Yes
Yes __
Bending-twisting
2"
g
106
Paul A. Lagace, David W. Jensen, Douglas C. Finch
3 EXPERIMENTAL PROCEDURE One plate of each type listed in Table 2 was manufactured from Hercules AS1/3501-6 preimpregnated tape, resulting in a total of eleven plates. In order to avoid curved plates which would result from the cure of unsymmetric laminates due to the differing thermal expansion coefficients and mechanical couplings, the unsymmetric laminates were made in a two-part operation. The first part involves the lay-up and cure of unidirectional 305 mm by 350 mm plates of the desired angular orientation. These plates were cured in a standard cure process in an autoclave under 0,59 MPa pressure and 762 mm vacuum, including a l h hold at 116°C followed by a 2 h hold at 177°C. After postcuring and taking thickness measurements, these sublaminates were machined with a water-cooled diamond-grit cutting wheel to 292 mm square. This procedure resulted in excellent quality sublaminates with an average ply thickness of 0.135 mm. This compares favorably with the manufacturer's nominal ply thickness of 0.134 mm. This latter number is used in all associated stress calculations. The prepared sublaminates were bonded in the proper stacking order with a Shell Epon V-40 hardener and 815 resin (mixed in a 1 : l ratio bv weight) under full vacuum for 48 h. The laminates were allowed to sit an additional 5 days at room temperature after which the laminates were machined to their final 279 mm by 279 mm dimensions. This procedure resulted in flat plates, although there was some problem in controlling the bondline thickness. There was considerable variation from plate to plate, and even within a plate, from the average bondline thickness of 0.03 ram. As indicated in Table 2, in addition to the symmetric [0~/903]s laminate manufactured via a standard cure, a similar symmetric laminate was manufactured using the two-part sublaminate procedure described above. This allowed a comparison to be made to the benchmark laminate to assess the effect of the room-temperature cure bondline. The results show that the bondline can be treated as a spacer with no load-car~ing capability. A modular jig, with interchangeable boundary conditions, was designed and constructed. The jig, as illustrated in Fig. l. interfaced via steel tabs with the hydraulic grips of a 445 kN MTS 810 hydraulic testing machine upon which all tests were run. The nominal test area of a plate in the fixture is 254 mm by 254 mm (except for plates tested with free side boundary conditions, which have a nominal test width of 279 mm).
Buckling of unsymmetric composite laminates
tP I
107
f IWtTH NTERFACE GR,PS OF
ts
t
I TEST,NO .AC.,NE
) . _uJtlJL_U_u. ~u~ zu., =.J~ , , ......
-:': " " "[:
"'[
666
~L
// 4~Tm~l
•
°
°
•
•
• 1 •
FRONT
t ........
Z54~
324
,
All Oil~d~*|ion*
~"-
Z5 4
VIEW
SIDE
VIEW
DEFLECTION TRACKER
in m m
IP BUCKLING JIG Fig. 1. Experimental apparatus: buckling jig and deflection tracker.
In order to avoid problems due to friction, all sliding surfaces were lubricated with a molybdenum-lithium compound. Additionally, extraneous shear load transfer at the boundaries was partially prevented by applying two layers of 0.14 mm thick by 13 mm wide teflon tape on the borders of each test plate. The boundary, conditions were only 'loosely clamped' allowing the plate to slide in-plane along the boundaries. Prior to testing ,any of the composite specimens, the entire buckling jig was checked out using a 2.36 mm thick 5052-H32 aluminum plate. The deflection measurement system, also illustrated in Fig. 1, was designed and constructed using a -+25 mm Sanborn type 7DCDT-1000 transducer mounted on a two de~ee-of-freedom scanning system. Rigidly held within a Deirin plastic block, the D C D T assembly has vertical freedom along two parallel rods and is connected by a cord-pulley system to a rotary potentiometer. This vertical tracking unit can be positioned anywhere along the plate width bv sliding within a pair of horizontal
108
Paul A. Lagace, David W. Jensen, Douglas C. Finch
guide-channels. For convenience, nine discrete clamping positions are built into the horizontal guides at 25 mm intervals. Continuous traces of deflection profiles were produced on an MTS 43 l recorder, and simultaneously sampled discretely by a PDP-11/34 computer. This deflection tracker allowed mapping of the deflection pattern of the entire plate, A complete description of the experimental apparatus is contained in Ref. 16. The deflection profiles were used to guide the adjustment of the five ~tlignment bolts along the bottom of the test section. These bolts were adjusted until deflection readings on either side of the vertical centerline of the plate were identical at low load levels. Although this method assures proper load distribution for symmetric laminates, the inherent couplings of unsymmetric laminates may slightly bias these results. It was felt that at low loads, this technique would provide good alignment even for unsymmetric laminates. All plates were instrumented with two Micro Measurements EA-09125AD-120 strain gages mounted back-to-back, as illustrated in Fig. 2. These gages were located 13 mm off-center to allow deflection measurements to be taken at the plate center. N
l
l
l
l,
SlrQin gQ(]e$ ~(Io¢oted o n
a : 2 5 4 mm !
12 7 mm
i I i
i
t
I x
t
-
N× b=254mm
F'~. 2. Configuration of plate specimen test area.
Buckling of unsymmetric composite laminates
109
Initial tests showed that significant divergence of the strain readings of the back-to-back gages occurred at center plate deflections of less than one-half of the plate thickness. This behavior indicates that buckling has taken place. A typical load versus strain plot.is shown in Fig. 3 for a [0~//30~]T laminate. It was thus decided that the load necessary to cause the plate to deflect half of the plate thickness would be defined as the 'limit load'. After determination of the 'limit load', the plates were unloaded to 220 N (a finite load was maintained to prevent shifting). The plates were then loaded in increments of 445 N. At each increment, a complete deflection 30
[Oe//l%]T Clamped sides
25
z
20
.a=
ff m o
10
o
5 0
,
o 2o 4o 60 80 tOO 120 LongitudinGI in-plane stroin,~strain
]Fig. 3. Typical stress versus strain plot,
scan of the plate was taken consisting of vertical scans at nine horizontal positions spaced at 25 mm intervals. After the deflection mapping was completed, the load was carefully released and the strain gages were zeroed and recalibrated. Stress-strain data were taken via computer up to 120% of the 'limit load', to clearly illustrate the bifurcation and nonlinear behavior. This process was repeated for each of the three side boundary, conditions.
4 ANALYSIS General unsvmmetric laminates do not exhibit classical bifurcation buckling behavior. The bending-stretching coupling inherent in such laminates causes normal out-of-plane displacements to initiate simultaneously with the application of the load. A nonlinear analysis is thus
110
Paul A. Lagace, David W. Jensen, Douglas C. Finch
required, even for the 'elastic' (prebuckled) region. However, a 'pseudo' buckling value can be determined analytically using linear plate theory. Although this 'pseudo' elastic buckling load does not necessarily correspond to any meaningful location on the load--displacement curve, it can be related to an experimental buckling load determined from a Soutliwell plot. 4.1
Formulation
The 'pseudo' buckling loads are calculated for each of the laminates investigated using a Rayleigh-Ritz assumed modes energy method. Using contracted notation, the strain energy of a general laminated plate, the coordinates of which are defined in Fig. 2, can be written in the form U = ~
(EiA o~i + 2~iB~jKj + • D jKj) dx dv
(2)
In addition to the strain energy of the plate, consideration must be given to the work done by the applied load. For the axial case, this is given by W = -~
(-N,)
~
-dxdv
(3)
where Nx represents the magnitude of the compressive load along the x-direction. The total potential energy of the system, lip, iS the difference between the internal strain energy and the work performed; i.e. llp = U - W
(4)
Using the linear strain-displacement relations au E1
e2
~6
-
a~-w
~x
K t
c3v ay
~,
t)u c~v = --+-ay ax
-
~x 2
=
-
K6 = - 2
r~2w 3v 2
(5)
a"w axay
in the expression for the strain energy and subsequently combining this
Bucklingof unsymmetriccompositelaminates
111
with eqn (3) and placing into eqn (4) results in the expression for the total potential energy in terms of the displacements (u, v, and w):
+ 2A~ ~y \aY
Ox
+Ae6{~u+ ¢)lP)2_2nll(ald) {8'2w ~ ~ oy
~x
7x k-gr-x /
[3U 32W OV O2W~ _2B:,2~y- 32W - 28'e [ ~ -~-~ + ~ v;~-x-] .yT -- 2B,612 Ou 02w t- (Ou +--Ov~ 02w]
ox ox~y
7y ox)-~-~ j
-2B> 2~v ~x~y +
[ c]2W~2
-4B~
32W c92W
[ 32W\2
+
32W O2W
4- DI, ~7
) + 2 D I 2 T T - b D 2 2 ( 7 ) '+-4DI6-~I~ OxOy
o- w +4D> 3-~'-
Ox3-"-'~
k 3xOy ]
(0w)'Id.rdy
N~ ~
(6)
At this point in the Rayleigh-Ritz analysis, it is assumed that the displacement modes are separable functions of the form . :
Y~/7,~,lx)B,(y) i
v = Y7iT,y,(.r)8,(y)
(7)
1
"' = E ff'iXi(X)thi(y) i
where u~. v, and ~i represent the modal amplitudes and the o~i(x)through oiO') are generic representations of the assumed mode shapes. The specific shapes used herein can be found in the Appendix.
112
Paul A. Lagace, David W. Jensen, Douglas C. Finch
The stability problem is solved by invoking the Principle of Stationary Total Potential Energy: 81-1p= 0
(8)
Placing the assumed separable functions (eqn (7)) into the strain energy expression (eqn (6)) and minimizing yields a standard eigenvalue problem, in terms of the critical buckling value N~, as shown: Kq = N~Lq
(9)
where K and L represent the resulting stiffness and loading matrices and q represents the vector containing the modal amplitudes. The critical buckling load determined in this manner should correspond to that obtained experimentally via the Southwell method.
4.2 Boundary conditions In general, the following four boundary conditions must be prescribed along each boundary: (1) either the normal in-plane displacement or the corresponding applied traction, (2) either the tangential in-plane displacement or the applied shear load, (3) either the out-of-plane slope or the applied moment and (4) either the out-of-plane displacement or a combination of the applied twisting moment and transverse shear (i.e. the Kirchhoff condition). For the Rayleigh-Ritz analysis only the geometric boundary conditions need to be satisfied. Thus, along the loaded ends, the essential boundary. conditions are that the variation of the displacement, 8u, along the loading direction must be zero, as well as the out-of-plane slope. 3w/gx. and displacement, w. Along the sides, there are three possible b o u n d a ~ conditions. For the free case, there are no restrictions; for the simplysupported case, only the out-of-plane displacement, w, must be zero: and, for the clamped case, both the out-of-plane slope, ~w/~y, and deflection, w, are required to be zero.
4.3 Modal selection The most critical step in the Rayleigh-Ritz method is the selection of assumed modes. Two double trigonometric series were chosen to represent each of the displacements, u, v, and w, in order to allow the full effects of coupling to be correctly modeled. The chosen modes are listed
Buckling of unsymmetric composite laminates
113
in the Appendix. These modes allow the complex buckling patterns to be properly represented by coupling the extension, shear, bending, and twisting. The problem was formulated in such a manner as to allow additional modes to be added incrementally. Preliminary analysis indicated that single series, although incomplete, can adequately represent the predominant contributions to the in-plane behavior (i.e. the u and v displacements). However, further work is necessary in order to specifically identify which of the modes are most important and what the role of the coupling terms is. Future work will also need to employ these same general mode shapes in the postbuckling regime. This analysis approach was implemented using a F O R T R A N program. The nondimensional integrals were calculated using a second order N e w t o n - C o t e s integration technique with 100 steps. It should be noted that the solution of the eigenvalue problem (eqn (9)) can be greatly simplified by observing that careful partitioning will result in only the lower diagonal partition in the loading matrix being populated. This allows the size of the eigenvalue problem to be significantly reduced through a process involving simple matrix algebra.
5 RESULTS AND DISCUSSION 5.1 Stress-strain measurements
Because an anisotropic plate has an undefinable elastic axis due to the existence of different B matrix couplings, the definition of the longitudinal modulus of the specimen becomes arbitrary with respect to the axis system and centerline as defined. Therefore, in order to compare theoretical calculations with experimental measurements, the ratio of the applied loading. N~, to the strain at the laminate midplane, e ~. is chosen as a figure of merit. This is, in some sense, similar to the measurement of longitudinal modulus. The experimentally determined ratios are summarized in Table 3, along with the theoretical values derived using classical laminated plate theory, and assuming a uniform loading across the plate width. A representative plot is shown in Fig. 3. The elastic constants used in the analvsis are EL = 130 GPa E-r = 10.5 GPa
z,~T = 0.28 G~T = 6-0GPa
Paul A. Lagace, David W. Jensen, Douglas C. Finch
114
TABLE 3
Calculated and Experimental N ~t t ~ ~~Ratios
NsllEn(MN/m) Laminate
Side boundary condition
Calculated Experimental
[03/903]S
Free Simply-supported Clamped
113.5
86.6 101.3 91.6
[03//906//03]T"
Free Simply-supported Clamped
113.6
74.7 63"6 78-6
[03//903//031/903]T
Free Simply-supported Clamped
98.2
b 76.6 87.3
[OJI452//O2//45~//O2]T
Free Simply-supported Clamped
114.2
72.7 81-5 87-1
[0~_1145:1102//-45:1102]r
Free Simply-supported Clamped
115.3
69-0 73.6 76. l
[0d/15@r
Free Simply-supported Clamped
137.5
80.0 105.4 93.5
[0U/306]T
Free Simply-supported Clamped
81.9
84.1 7l '2 73.5
[OU/45~]T
Free Simply-supported Clamped
61.6
b 64-7 78.8
[Od/6J)~]T
Free Simply-supported Clamped
54. l
b 37.6 42.1
[0d/75.]-r
Free Simply-supported Clamped
51-6
b 98.1 119-6
[ OUIC~i<,]T
Free Simply-supported Clamped
5 I. 1
b 30"8 39-6
"//indicates a room temperature bondline. h Insufficient data.
Buckling of unsymmetric composite laminates
115
The experimental results possess a large coefficient of variation and exhibit a significant inconsistency with the analytical predictions. There are several factors that may have contributed to these discrepancies. First, the mechanical couplings inherent in the laminates studied induce internal stresses and moments (other than N~ ~) which would cause shearing~ bending and twisting were the plate not physically restrained. The presence of clamped or simply-supported boundary conditions restricts this motion by exerting additional forces and moments. Such effects were not included in the simple analytical predictions. Second, the actual strain was only on the order of a few hundred microstrain. The data are recorded by the computer in increments of 12.5 microstrain. This relatively large increment, coupled with the inherent strain gage errors at low strain levels and the difficulty of identifying an appropriate linear region, magnifies the scatter and potential for error. Third, due to the physical design of the buckling jig, the plate actually experiences a uniform end shortening during loading, as opposed to a uniform load distribution. As the center of the plate deflects outward, the in-plane stress is relieved in that region. The actual load distribution becomes sinusoidal in nature for an isotropic plate as soon as any out-ofplane deflection occurs. For an anisotropic plate, the load distribution is even further distorted. In determining the modulus, a uniform applied load was assumed. Before any further experimental work is undertaken, a m a n n e r in which to determine a laminate ~elastic constant', as a quality check of the specimen, should be established.
5.2 Buckling loads and mode shapes The "pseudo" buckling loads of each plate for each boundary, condition were back-calculated from the load-deflection data using the technique developed by Southwell. ~7The critical 'pseudo' buckling load is defined as the inverse of a least-squares fit of the deflection/load versus deflection curve. Southwell's method gives accurate results only for deflections small compared to the plate thickness. TM Plots were made of the deflection, as measured bv the tracker, versus the load at several (usually nine) locations on the plate. Such a typical load-deflection plot is shown in Fig. 4. Linear regressions of the data indicate that the useful range of this m e t h o d exists for a center deflection between 25% and 50% of the plate thickness. The experimental buckling loads presented in this paper were obtained by averaging the results from the various locations for each specific case.
116
Paul A. Lagace, David W. Jensen, Douglas C. Finch
The aluminum plate results isolate the accuracy of the buckling jig from the manufacturing quality of the plate. The tests conducted on the aluminum plate produced experimental buckling loads within 8% of the analytical results for all three side boundary conditions: free, simplysupported and clamped. This lends credibility to the overall effectiveness of the buckling jig and measurement techniques. Table 4 displays the experimental and analytical buckling loads for the special cases examined. The coefficient of variation for the experimental
Z
.~-
15
[06//756]T Simply-supported sides
10
o
0.5
Ol
O.3
O u t - o f - olane deflection, mm
F'~. 4. Typical load-deflection (Southwell) plot.
values is typically less than 4%, indicating a good uniformity across the plate. However, the composite plates show large discrepancies between the experimental measurements and the theoretical predictions, although the trends are similar. As mentioned previously, the unsymmetric laminates were manufactured by bonding symmetric sublaminates together at room temperature. The principal effect of the room-temperature bondline is to increase the buckling load by approximately one-third. This can be observed by comparing the results in Table 4 for the [03/903]s and [03//90E/03]T layups. This effect was predicted analytically by modeling the roomtemperature epoxy bondline as a uniform thickness spacer that carries no load. By introducing bending-stretching coupling into the plate through antisymmetry (i.e. [03//903//OJ/903]T), the out-of-plane deflection, w, begins with the onset of load. Therefore, the critical buckling load
Buckling of unsymmetric composite laminates
117
TABLE 4
Experimental and Analytical Buckling Loads
Buckling~ad (N/m) Laminate
Sidebounda O, condinon
Predicted Experimental
[03/903]s
Free Simply-supported Clamped
26 992 27 150 31 780
16 070 19 650 20 890
[03fI90~1103]¢
Free Simply-supported Clamped
28 741 28 870 33 630
21 670 24 110 27 340
[0311903110311903]r
Free Simply-supported Clamped
18 070 20 439 31 920
10 960 14 970 24 950
[02//452//02//452//02]T
Free Simply-supported Clamped
17 366 18 398 20 729
17 610 23 440 25 600
[02//45~_//02//--45d/O2]T
Free Simply-supported Clamped
17 339 17 770 19 720
17 920 21 480 23 780
//indicates a room temperature bondline.
becomes a "pseudo' buckling load. This 'pseudo' buckling load is over one-third less than its symmetric counterpart (for free side boundary conditions), although there is no significant change in the mode shape. The last two laminates summarized in Table 4 illustrate the effect of bending-twisting and bending-stretching couplings. Bending-stretching coupling, exhibited by the [02//45~_//02//--452//02]T laminate, can be seen to be the more severe of the two, in terms of the critical buckling lood where° it is noted, as above, that the unsymmetric laminate results in a "pseudo' buckling load due to the eccentricity effect. Finally, for the [0d/06]w family, there is also a noticeable general trend of decreasing buckling loads for increasing lamination angle, except for a slight unexplained increase in the clamped-clamped analytical case for 0 greater than 60 ° . These results are illustrated in Figs 5-7, which show the buckling loads as a function of lamination angle, 0, for the [06//06]T
[Oe//eejr Free side~
~
3O
Prediction
7
~ ao o
o o o
o
m
0
--
I 15
O
i 30
~ 45
i 60
oi
75
!
90
~9,degrees Fig. 5. Experimental and analytical buckling loads versus lamination angle for [0,tI0.]T graphite/epoxy plates with free sides f
[%//ee ]
[
Simply- supl)or ted sides
30 ~
~
-L 0
o
Prediction
0
- ~
~
o
"; 10 (.J
o
F
[ O{ ©
I 15
L $0
.L 45
I 60
i 75
i 90
8, degrees
Eig. 6. Experimental and analytical buckling loads versus lamination angle for [Od/O.]-r graphite/epoxy plates with simply-supported sides.
Clomped
E
30 I'-
~_~
}
C:
O |I
:3 (]3
K
~
~ o
~
°
sides
Prediction E~p~imen~'
o
d~°
o i
O ~
I
0
t5
~
L
J
30 a5 6o 8, degrees
~
~
z5
9o
Fig. 7. Experimental and analytical buckling loads versus lamination angle for [O~//O.]-i
graphite/epoxy plates with clamped sides.
Buckling of unsymmetric composite laminates
119
specimens with flee, simply-supported, and clamped side boundary conditions, respectively. The solid line shows the theoretical prediction from the 30-mode Rayleigh-Ritz energy analysis (33 modes were used in the free case). Plots of isodeflection contours at the limit load were made for each test case using a nine by nine (internal) grid of deflection measurements for the experimental modes and the same grid for the analytical calculations. The isodeflection contour maps for a [0d/30~]T laminate for the three different side boundary conditions are shown in Fig. 8. These are typical of the contour maps of that laminate family. Several observations can be made. First, the clamped-free experimental maps often show svmmetrv about the vertical centerline, while the predicted counterparts show twisting about this centerline. The symmetry of the experimental maps is a direct consequence of the even load distribution adjustment procedure, which used deflection symmetry about the vertical centerline as the criterion for determining a proper load PREDICTED CONTOURS
MEASURED CONTOURS
FREE SIDES
SIMPLYSUPPORTED SIDES
CLAMPED SIDES
Fig. 8. Measured and predicted isodeflection contour maps for [06//306]T graphite/epoxy plates.
120
Paul A. Lagace, David W. Jensen, Douglas C. Finch
introduction. This adjustment procedure is valid only for those specimens with no twisting coupling, and produces uneven load distributions for all other laminates. Thus, the symmetric deflection map was assured bv this improper load introduction adjustment. A second observation reveals a twisting in the simply-supported and clamped side contours for the laminates with stretching-twisting and bending-twisting couplings. The angle of twist in the maps corresponds roughly to the lamination angle. 0. Possible causes for discrepancies between the experimentally determined and the analytically calculated buckling loads can be classified into two categories: (1) experimental error and (2) analytical inaccuracies due to the choice of assumed modes and/or degree of convergence. The most likely source of problems in the experiments is the load adjustment procedure described in the experimentation section. The assumption used was that a deflection pattern which was symmetric about the vertical centerline implied that the load was properly distributed across the plate. However, this is valid only for specimens without bending or twisting couplings. This is illustrated by the finding that the deflection shapes for the specimens with these couplings show a marked twisting of the plate about the centerline. Thus, a better method for assuring proper distribution of load across the plate must be determined for future experimentation. Additional improvements in agreement between the experimental and analytical results can be achieved by more carefully examining the actual mode shapes used in the Rayleigh-Ritz analysis to ensure that all of the inherent mechanical couplings are adequately represented and by checking the series for convergence. In spite of these limitations, the experimental and analytical results do give initial indications as to the importance of mechanical couplings in the buckling of fully anisotropic laminates.
6 SUMMARY An investigation has been conducted which examined unsymmetric (and thus fully anisotropic) composite laminates and the effect their inherent mechanical couplings have on buckling behavior. Although many of the effects cannot be specifically quantified this prelimina~ work has laid the
Buckling of unsymmetric composite laminates
121
groundwork for future investigations into the compressive behavior of such laminates. Experimental techniques have been developed for use with unsymmetric laminates and suggestions have been made to improve these techniques: specifically, the need to identify a method to assure proper loading of unsymmetric laminates which exhibit out-of-plane deflection simultaneously with the initial application of the load. The Rayleigh-Ritz analysis employed herein has illustrated the importance of the mechanical couplings, e.g. the twisting exhibited in the buckling modes, and is verified by the experimental results that the existence of coupling can severely reduce the resistance of composite laminates to buckling. The predictions from this technique can be improved by assuming modes which account for the inherent couplings as illustrated in the experiments. Further work should not only consider the improved experimental and analytical suggestions but should progress to the postbuckling range where the inherent mechanical couplings will probably have even greater effects.
ACKNOWLEDGEMENTS This work was supported bv the Boeing Military Airplane Company. Mr Robert Waner is the technical monitor. The authors wish to thank Mr Waner and his staff for the useful suggestions and discussions during the course of this work.
REFERENCES I. Jensen. D. W. and Crawler, E. F., Frequency determination techniques for cantilevered plates with bending-torsion coupling, AIAA ,lournal, 22 (3) (March 1984) 415-2(I. 2. ,lensen, D, W.. Crawler, E, F. and Dugundji. J,, Vibration of cantilevered graphite/epoxy plates with bending-torsion coupling, Journal of Reinforced Plastics and Composites, 1 (July 198_) .,.~4--69. 3. Sherrer. V. C., Hertz, T. J. and Shirk, M. H,, Wind tunnel demonstration of aeroelastic tailoring applied to forward swept wings, Journal of Aircraft, 18 (November 1981) 976-83. 4. Bryan. G. H.. On the stability of a plane plate under thrusts in its own plane with applications to the "bucl
122
Paul A. Lagace, David W. Jensen. Douglas C. Finch
5. Leissa, A. W., Buckling of composite plates. Composite Structures, 1(1983) 51-66. 6. Timoshenko, S. P. and Gere, J. M.. Theory of elastic stability. 2nd edn, McGraw-Hill, New York, 196l. 7. Ashton, J. E. and Waddoups. M. E., Analysis of anisotropic plates. Journal of Composite Materials, 3 (January, 1969) 148-65. 8. Ashton, J. E. and Waddoups. M. E.. Analysis of anisotropic plates II, Journal of Composite Materials, 3 (July 1969) 47[)-9. 9. Ashton, J. E., Anisotropic plate analysis--bounda~' conditions, Journal Of Composite Materials. 4 (April 1970) 163-71. 10. Ashton, J. E. and Love, T. S., Experimental studv of the stability of composite plates. Journal of Composite Materials, 3 (April 1969} 230-4~.. 1 I. Kicher, T. P. and Mandell. J. F., A study of the buckling of laminated composite plates, AIAA Journal, 9 (4) (April 197l) 61)5-13. 12. Starnes, J. H., Jr, and Rouse, M., Postbuckling and failure characteristics of selected fiat rectangular graphite-epoxy plates loaded in compression. Proceedings of the AIAA/ASME/ASCE/AHS 22nd SDM Conference. Atlanta, Georgia, April 1981. 13. Starnes, J. H., Jr, Knight, N. F., Jr. and Rouse. M.. Postbuckling behavior of selected flat stiffened graphite-epoxy panels loaded in compression, Proceedings of the A IA A/A SME/A SCE/A HS 23rd SDM Conference, Lake Tahoe, Nevada. Mav 1982. 14. Chia, C. Y., Nonlinear analysis of plates, McGraw-Hill, New York, 1980 15. Tsai, S. W. and Hahn, H. T.. Introduction to Composite ~14aterials. Technomic Publishing Company, Connecticut, 1980. 16. Finch, D. C., The buckling of symmetric and uns~:mmerric composite plates with various boundary conditions, Technology i~aboratorv for Advanced Composites, Report 84-3, Massachusetts Institute of Technology, S.M. Thesis. February 1984. 17. Southwell, R. V.. On the analysis of experimental observations in problems of elastic stability, Proceedings Oflthe Royal Society, London. Series ,4. 135 (1932) 61}1-16. 18. Donnell, L, H., On the application of Southwell's method for the analvsisof buckling tests, Stephen Timoshenko 60th Anniversary Volume. McGrawHill, New York. 1938, pp. 27-38.
APPENDIX The m o d e s used in the Rayleigh-Ritz analysis are included in this Appendix. For the case of the in-plane deformations, u and v. the same m o d e s were used for the three types of side botmdarv conditions
123
Buckling o f unsvmmetric composite laminates
U=
q.,+,sin
qt-+
mTrx
a
m7rv
cos~
b
(m - 1)rrx +
q,,,,
5
COS
(m + 1)~-x ] Cos
a
v=
-(1"5
q,,,
qm ~
÷
+q,~
11sin mTrx
.,"--,
a
mTrv/
-
a
Sill
J
"
,
h
j
-0-5
cos
retry
3
mrrx
-. + ~ q~.÷,scos h ~', a
mrrv
sin~
b
H o w e v e r . different m o d e s were assumed for the out-of-plane deflection w d e p e n d i n g upon the side b o u n d a r y conditions. These are WFR for the free b o u n d a r y conditions 14'FR
cos
= *
--
qs~,,, i ~-
a
2 [
tn77Y
E n
cos
I
qs,,,,
nrry 1
, ..... ,~;sin - --~- + q~,,,, ,,-n-2, c o s - - ~ J
[
]
J
a n d wss for the simply-supported side b o u n d a r y conditions
W,.;~, =
(]Aim
I I + n - IS
cos
CO~,
m = In = I
a
( m + I )rrx ] a
sin
(2n - k)~-y b
w h e r e k is equal to 0 for n even and is equal to 1 for n odd. Finally. the a s s u m e d s h a p e , w . , . for the cases where the sides are c l a m p e d is:
I~'( 1
=
, ,,7"" i ,~
q41m
I/~n-
{
0 1 - 1)try
I~
COS
COS a
( n + 1)~ry 1
a
Buckling of Unsymmetric Composite Laminates Paul A. Lagace, David W. Jensen and Douglas C. Finch* Technology Laboratory for Advanced Composites, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
A BS TRA CT An initial experimental and analvtical investigation was conducted to examine the effects of the inherent mechanical couplings exhibited in fully anisotropic (i. e. unsymmetric) graphite~epoxy laminates on the buckling loads and mode shapes. Experimental techniques were devised to test 254 mm square plates of A S1/3501-6 graphite~epoxy under uniaxial compressive load with free, simply-supported and clamped edges. The results indicate that the mechanical couplings, especially those which relate stretching and bending behavior, cause out-of-plane deflections prior to buckling and reduce the buckling load significantly, i.e. the load at which out-of-plane deflections become large. The analytical results also show that the buckled mode shapes exhibit twisting due to the mechanical couplings. Suggestions are offered for improvements in the experimental and analvtical techniques to better understand these phenomena.
l INTRODUCTION M o d e r n aircraft design has been enhanced by the introduction of advanced composite materials such as carbon fiber reinforced plastics. While offering significant strength-to-weight and aeroelastic advantages, such materials have been shown to introduce mechanical couplings which require m o r e careful analysis than is usually necessaw for isotropic or orthotropic materials, j-" The concept of employing unbalanced laminates *Present address: Aerospace Corporation. Los Angeles. California 90009, USA. 101 Composite Structures 0263-8223/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986, Printed in Great Britain
102
Paul A. Lagace, David W. Jensen, Douglas C. Finch
(which do not have a ply oriented at - 0 for every one oriented at +0, e.g. [45/.0]s) to improve aeroelastic efficiency, a procedure referred to as aeroelastic tailoring, 3 results in a coupling between the bending and twisting behavior of a laminate, as well as between the in-plane stretching and shearing. Moreover, even more unique couplings may arise in using unsymmetric laminates since the stretching and bending behavior become interrelated. In fact, the use of unsymmetric laminates may result in fully anisotropic behavior in that the bending, stretching, shearing, and twisting responses are completely coupled. Unsymmetric laminates may arise 'naturally' during the design process. This is most likely to occur in panels with tapered thickness in which individual plies are dropped off. Due to design constraints, these are generally dropped off in an unsymmetric fashion. Additionally, there may be applications where the unique couplings of unsymmetric laminates provide aeroelastic or other advantages which can be effectively utilized in a structure. An important consideration to be made during the design process involves the manner in which these laminates are loaded. The buckling behavior of unsymmetric laminates is important inasmuch as many structural components will experience compressive loads. In fact, conventional wing panels are often designed to operate predominantly in the postbuckled region. Beginning with Bryan's first analytical solution to the buckling of isotropic plates in 189l, ~ an immense amount of research has been performed regarding the buckling behavior of plates. Leissa, ~ for example, mentions the availability of approximately 2000 references in the open literature concerning isotropic plate buckling and an additional 200 publications relating specifically to composites. Classical plate buckling theory has been well defined with the solution of many cases being explicitly summarized in such texts as Timoshenko and Gere. ~ During the past few decades, this work has been extended to orthotropic plates and, recently, to a more limited extent, to anisotropic plates, e.g. Ashton, 7-1° Kicher and Mandell, ~E Starnes, ~2t3 and Chia." This early work, however, has been primarily restricted to specially orthotropic, midplane symmetric, or antisymmetric cross-ply laminates with simplysupported boundary conditions. Such laminates possess no bendingstretching coupling, resulting in simplified constitutive relations and behavior. Ashton and Love "' did test a [+45 ,,/ - 4 5 ~,]T (rotated, antisymmetric cross-ply) laminate which displays stretching-twisting
Buckling of unsymmetric composite laminates
103
coupling and found the experimental buckling load to be approximately half that of the predicted value. It is clear that a better understanding of the effects of the inherent mechanical couplings is needed. Generally, unsymmetric anisotropic plates contain fully populated elastic matrices, and, therefore, exhibit every possible type of mechanical coupling in a complex interwoven fashion. The mechanical behavior of such laminates is somewhat analogous to that of plates with an initial curvature or flat plates loaded eccentrically. In each of these cases two phenomena occur: (1) the out-of-plane deflections begin simultaneously with the application of the load and (2) the critical buckling load is reduced substantially. This paper reports the results of an initial experimental and analytical investigation into the critical buckling loads and the corresponding deflection patterns of unsymmetric laminates. Laminates which possess different types and amounts of mechanical couplings were tested in uniaxial compression with various combinations of boundary, conditions. The objective of this investigation was to isolate and study the effects of these various mechanical couplings.
2 SCOPE As previously mentioned, fully anisotropic laminates can exhibit coupling between all possible mechanical deformations caused by in-plane loads or moments. These couplings are dependent upon the existence of terms in the constitutive relations between applied loads and moments and inplane strains and curvatures. These relations are expressed with the use of contracted notation, ~ as N, = AoE~,~+ B,jKj i,j=
1.2.6
(1)
M, = Bii~.5~+ D#K i
where N, and M; are the in-plane stress and bending resultants. ~j are the in-plane strains, K/ are the curvatures, and A;j, B;,. and D;j are elements of the A, B. and D matrices, respectively. There are five possible mechanical couplings between the four deformation types: extension, shear, bending and twisting. These
104
Paul A, Lagace, David W. Jensen, Douglas C. Finch TABLE 1 Mechanical Couplings and Associated Matrix Terms
Coupling type 1. 2. 3. 4. 5. 6.
Extension-shear Extension-bending Extension-twisting Shear-bending Shear-twisting Bending--twisting
Non-zero matrix terms A 16, A 26 BH, BI2, B22 B16, B,_~
Blo, B2~ B~ D i~, D2~
couplings are identified in Table 1 along with the components of the A, B, or D matrix associated with that particular coupling. When the associated terms are zero, the coupling does not exist. Laminates were chosen so as to isolate the various couplings involved. This is done by allowing only the coupling coefficients associated with the desired mechanical coupling to be nonzero. However, such a laminate could not be constructed to isolate shear-twisting coupling. Additionally, a series of laminates was chosen with 'fully-populated' matrices. These laminates possess all of the possible mechanical couplings. The degree of coupling was varied by changing the lamination angle, 0, of a [06//06]v laminate from 15° to 90° in 15° increments. The subscript 'T' denotes that this is the total laminate, in order to distinguish this from a symmetric arrangement. The double slash '//" indicates a room temperature bondline joining previously cured symmetric sublaminates. This type of construction, described in the experimental section, is necessary in order to obtain flat unsymmetric laminates. Finally, a basic specially-orthotropic laminate. [03/903]s, was chosen to serve as a benchmark since this laminate does not exhibit any of the various mechanical couplings. In addition to this basic construction, a [03//90d/03]r laminate was included to ascertain the effect of the room temperature bondline. The laminates chosen for the study and the mechanical couplings they exhibit are summarized in Table 2. For all cases loading is uniaxial compression along clamped ends. Three side boundary conditions are considered: free, simply-supported and clamped. In addition to the experiments, a Rayleigh-Ritz energy analysis is carried out to determine the classical buckling loads and mode shapes of the various laminates under the different boundary, conditions.
"0
=
a room temperature bondline. 'T' implies total laminate.
Yes
Yes
Extension~hear
15 °, 3 0 °, 4 5 °, 6 0 o, 7 5 ° a n d 9 0 °.
h Subscript
"//indicates
[0
dl9OdlO3]v <'h [0JI90311031190~]~ [0~//452//(1~//452//02]r [0~_1145,110~11-45,_1102]-r [O~,//O~,]l'
[0d903]s
Laminate
TABLE 2
Yes
---
Yes
Extension-bending
Yes
-Yes
--
Extett~ion-twisting and shear-bending
Mechanical couplings
Test Laminates and Associated Mechanical Couplings
Yes
---
__
Shear-twisting
Yes
Yes __
Bending-twisting
2"
g
106
Paul A. Lagace, David W. Jensen, Douglas C. Finch
3 EXPERIMENTAL PROCEDURE One plate of each type listed in Table 2 was manufactured from Hercules AS1/3501-6 preimpregnated tape, resulting in a total of eleven plates. In order to avoid curved plates which would result from the cure of unsymmetric laminates due to the differing thermal expansion coefficients and mechanical couplings, the unsymmetric laminates were made in a two-part operation. The first part involves the lay-up and cure of unidirectional 305 mm by 350 mm plates of the desired angular orientation. These plates were cured in a standard cure process in an autoclave under 0,59 MPa pressure and 762 mm vacuum, including a l h hold at 116°C followed by a 2 h hold at 177°C. After postcuring and taking thickness measurements, these sublaminates were machined with a water-cooled diamond-grit cutting wheel to 292 mm square. This procedure resulted in excellent quality sublaminates with an average ply thickness of 0.135 mm. This compares favorably with the manufacturer's nominal ply thickness of 0.134 mm. This latter number is used in all associated stress calculations. The prepared sublaminates were bonded in the proper stacking order with a Shell Epon V-40 hardener and 815 resin (mixed in a 1 : l ratio bv weight) under full vacuum for 48 h. The laminates were allowed to sit an additional 5 days at room temperature after which the laminates were machined to their final 279 mm by 279 mm dimensions. This procedure resulted in flat plates, although there was some problem in controlling the bondline thickness. There was considerable variation from plate to plate, and even within a plate, from the average bondline thickness of 0.03 ram. As indicated in Table 2, in addition to the symmetric [0~/903]s laminate manufactured via a standard cure, a similar symmetric laminate was manufactured using the two-part sublaminate procedure described above. This allowed a comparison to be made to the benchmark laminate to assess the effect of the room-temperature cure bondline. The results show that the bondline can be treated as a spacer with no load-car~ing capability. A modular jig, with interchangeable boundary conditions, was designed and constructed. The jig, as illustrated in Fig. l. interfaced via steel tabs with the hydraulic grips of a 445 kN MTS 810 hydraulic testing machine upon which all tests were run. The nominal test area of a plate in the fixture is 254 mm by 254 mm (except for plates tested with free side boundary conditions, which have a nominal test width of 279 mm).
Buckling of unsymmetric composite laminates
tP I
107
f IWtTH NTERFACE GR,PS OF
ts
t
I TEST,NO .AC.,NE
) . _uJtlJL_U_u. ~u~ zu., =.J~ , , ......
-:': " " "[:
"'[
666
~L
// 4~Tm~l
•
°
°
•
•
• 1 •
FRONT
t ........
Z54~
324
,
All Oil~d~*|ion*
~"-
Z5 4
VIEW
SIDE
VIEW
DEFLECTION TRACKER
in m m
IP BUCKLING JIG Fig. 1. Experimental apparatus: buckling jig and deflection tracker.
In order to avoid problems due to friction, all sliding surfaces were lubricated with a molybdenum-lithium compound. Additionally, extraneous shear load transfer at the boundaries was partially prevented by applying two layers of 0.14 mm thick by 13 mm wide teflon tape on the borders of each test plate. The boundary, conditions were only 'loosely clamped' allowing the plate to slide in-plane along the boundaries. Prior to testing ,any of the composite specimens, the entire buckling jig was checked out using a 2.36 mm thick 5052-H32 aluminum plate. The deflection measurement system, also illustrated in Fig. 1, was designed and constructed using a -+25 mm Sanborn type 7DCDT-1000 transducer mounted on a two de~ee-of-freedom scanning system. Rigidly held within a Deirin plastic block, the D C D T assembly has vertical freedom along two parallel rods and is connected by a cord-pulley system to a rotary potentiometer. This vertical tracking unit can be positioned anywhere along the plate width bv sliding within a pair of horizontal
108
Paul A. Lagace, David W. Jensen, Douglas C. Finch
guide-channels. For convenience, nine discrete clamping positions are built into the horizontal guides at 25 mm intervals. Continuous traces of deflection profiles were produced on an MTS 43 l recorder, and simultaneously sampled discretely by a PDP-11/34 computer. This deflection tracker allowed mapping of the deflection pattern of the entire plate, A complete description of the experimental apparatus is contained in Ref. 16. The deflection profiles were used to guide the adjustment of the five ~tlignment bolts along the bottom of the test section. These bolts were adjusted until deflection readings on either side of the vertical centerline of the plate were identical at low load levels. Although this method assures proper load distribution for symmetric laminates, the inherent couplings of unsymmetric laminates may slightly bias these results. It was felt that at low loads, this technique would provide good alignment even for unsymmetric laminates. All plates were instrumented with two Micro Measurements EA-09125AD-120 strain gages mounted back-to-back, as illustrated in Fig. 2. These gages were located 13 mm off-center to allow deflection measurements to be taken at the plate center. N
l
l
l
l,
SlrQin gQ(]e$ ~(Io¢oted o n
a : 2 5 4 mm !
12 7 mm
i I i
i
t
I x
t
-
N× b=254mm
F'~. 2. Configuration of plate specimen test area.
Buckling of unsymmetric composite laminates
109
Initial tests showed that significant divergence of the strain readings of the back-to-back gages occurred at center plate deflections of less than one-half of the plate thickness. This behavior indicates that buckling has taken place. A typical load versus strain plot.is shown in Fig. 3 for a [0~//30~]T laminate. It was thus decided that the load necessary to cause the plate to deflect half of the plate thickness would be defined as the 'limit load'. After determination of the 'limit load', the plates were unloaded to 220 N (a finite load was maintained to prevent shifting). The plates were then loaded in increments of 445 N. At each increment, a complete deflection 30
[Oe//l%]T Clamped sides
25
z
20
.a=
ff m o
10
o
5 0
,
o 2o 4o 60 80 tOO 120 LongitudinGI in-plane stroin,~strain
]Fig. 3. Typical stress versus strain plot,
scan of the plate was taken consisting of vertical scans at nine horizontal positions spaced at 25 mm intervals. After the deflection mapping was completed, the load was carefully released and the strain gages were zeroed and recalibrated. Stress-strain data were taken via computer up to 120% of the 'limit load', to clearly illustrate the bifurcation and nonlinear behavior. This process was repeated for each of the three side boundary, conditions.
4 ANALYSIS General unsvmmetric laminates do not exhibit classical bifurcation buckling behavior. The bending-stretching coupling inherent in such laminates causes normal out-of-plane displacements to initiate simultaneously with the application of the load. A nonlinear analysis is thus
110
Paul A. Lagace, David W. Jensen, Douglas C. Finch
required, even for the 'elastic' (prebuckled) region. However, a 'pseudo' buckling value can be determined analytically using linear plate theory. Although this 'pseudo' elastic buckling load does not necessarily correspond to any meaningful location on the load--displacement curve, it can be related to an experimental buckling load determined from a Soutliwell plot. 4.1
Formulation
The 'pseudo' buckling loads are calculated for each of the laminates investigated using a Rayleigh-Ritz assumed modes energy method. Using contracted notation, the strain energy of a general laminated plate, the coordinates of which are defined in Fig. 2, can be written in the form U = ~
(EiA o~i + 2~iB~jKj + • D jKj) dx dv
(2)
In addition to the strain energy of the plate, consideration must be given to the work done by the applied load. For the axial case, this is given by W = -~
(-N,)
~
-dxdv
(3)
where Nx represents the magnitude of the compressive load along the x-direction. The total potential energy of the system, lip, iS the difference between the internal strain energy and the work performed; i.e. llp = U - W
(4)
Using the linear strain-displacement relations au E1
e2
~6
-
a~-w
~x
K t
c3v ay
~,
t)u c~v = --+-ay ax
-
~x 2
=
-
K6 = - 2
r~2w 3v 2
(5)
a"w axay
in the expression for the strain energy and subsequently combining this
Bucklingof unsymmetriccompositelaminates
111
with eqn (3) and placing into eqn (4) results in the expression for the total potential energy in terms of the displacements (u, v, and w):
+ 2A~ ~y \aY
Ox
+Ae6{~u+ ¢)lP)2_2nll(ald) {8'2w ~ ~ oy
~x
7x k-gr-x /
[3U 32W OV O2W~ _2B:,2~y- 32W - 28'e [ ~ -~-~ + ~ v;~-x-] .yT -- 2B,612 Ou 02w t- (Ou +--Ov~ 02w]
ox ox~y
7y ox)-~-~ j
-2B> 2~v ~x~y +
[ c]2W~2
-4B~
32W c92W
[ 32W\2
+
32W O2W
4- DI, ~7
) + 2 D I 2 T T - b D 2 2 ( 7 ) '+-4DI6-~I~ OxOy
o- w +4D> 3-~'-
Ox3-"-'~
k 3xOy ]
(0w)'Id.rdy
N~ ~
(6)
At this point in the Rayleigh-Ritz analysis, it is assumed that the displacement modes are separable functions of the form . :
Y~/7,~,lx)B,(y) i
v = Y7iT,y,(.r)8,(y)
(7)
1
"' = E ff'iXi(X)thi(y) i
where u~. v, and ~i represent the modal amplitudes and the o~i(x)through oiO') are generic representations of the assumed mode shapes. The specific shapes used herein can be found in the Appendix.
112
Paul A. Lagace, David W. Jensen, Douglas C. Finch
The stability problem is solved by invoking the Principle of Stationary Total Potential Energy: 81-1p= 0
(8)
Placing the assumed separable functions (eqn (7)) into the strain energy expression (eqn (6)) and minimizing yields a standard eigenvalue problem, in terms of the critical buckling value N~, as shown: Kq = N~Lq
(9)
where K and L represent the resulting stiffness and loading matrices and q represents the vector containing the modal amplitudes. The critical buckling load determined in this manner should correspond to that obtained experimentally via the Southwell method.
4.2 Boundary conditions In general, the following four boundary conditions must be prescribed along each boundary: (1) either the normal in-plane displacement or the corresponding applied traction, (2) either the tangential in-plane displacement or the applied shear load, (3) either the out-of-plane slope or the applied moment and (4) either the out-of-plane displacement or a combination of the applied twisting moment and transverse shear (i.e. the Kirchhoff condition). For the Rayleigh-Ritz analysis only the geometric boundary conditions need to be satisfied. Thus, along the loaded ends, the essential boundary. conditions are that the variation of the displacement, 8u, along the loading direction must be zero, as well as the out-of-plane slope. 3w/gx. and displacement, w. Along the sides, there are three possible b o u n d a ~ conditions. For the free case, there are no restrictions; for the simplysupported case, only the out-of-plane displacement, w, must be zero: and, for the clamped case, both the out-of-plane slope, ~w/~y, and deflection, w, are required to be zero.
4.3 Modal selection The most critical step in the Rayleigh-Ritz method is the selection of assumed modes. Two double trigonometric series were chosen to represent each of the displacements, u, v, and w, in order to allow the full effects of coupling to be correctly modeled. The chosen modes are listed
Buckling of unsymmetric composite laminates
113
in the Appendix. These modes allow the complex buckling patterns to be properly represented by coupling the extension, shear, bending, and twisting. The problem was formulated in such a manner as to allow additional modes to be added incrementally. Preliminary analysis indicated that single series, although incomplete, can adequately represent the predominant contributions to the in-plane behavior (i.e. the u and v displacements). However, further work is necessary in order to specifically identify which of the modes are most important and what the role of the coupling terms is. Future work will also need to employ these same general mode shapes in the postbuckling regime. This analysis approach was implemented using a F O R T R A N program. The nondimensional integrals were calculated using a second order N e w t o n - C o t e s integration technique with 100 steps. It should be noted that the solution of the eigenvalue problem (eqn (9)) can be greatly simplified by observing that careful partitioning will result in only the lower diagonal partition in the loading matrix being populated. This allows the size of the eigenvalue problem to be significantly reduced through a process involving simple matrix algebra.
5 RESULTS AND DISCUSSION 5.1 Stress-strain measurements
Because an anisotropic plate has an undefinable elastic axis due to the existence of different B matrix couplings, the definition of the longitudinal modulus of the specimen becomes arbitrary with respect to the axis system and centerline as defined. Therefore, in order to compare theoretical calculations with experimental measurements, the ratio of the applied loading. N~, to the strain at the laminate midplane, e ~. is chosen as a figure of merit. This is, in some sense, similar to the measurement of longitudinal modulus. The experimentally determined ratios are summarized in Table 3, along with the theoretical values derived using classical laminated plate theory, and assuming a uniform loading across the plate width. A representative plot is shown in Fig. 3. The elastic constants used in the analvsis are EL = 130 GPa E-r = 10.5 GPa
z,~T = 0.28 G~T = 6-0GPa
Paul A. Lagace, David W. Jensen, Douglas C. Finch
114
TABLE 3
Calculated and Experimental N ~t t ~ ~~Ratios
NsllEn(MN/m) Laminate
Side boundary condition
Calculated Experimental
[03/903]S
Free Simply-supported Clamped
113.5
86.6 101.3 91.6
[03//906//03]T"
Free Simply-supported Clamped
113.6
74.7 63"6 78-6
[03//903//031/903]T
Free Simply-supported Clamped
98.2
b 76.6 87.3
[OJI452//O2//45~//O2]T
Free Simply-supported Clamped
114.2
72.7 81-5 87-1
[0~_1145:1102//-45:1102]r
Free Simply-supported Clamped
115.3
69-0 73.6 76. l
[0d/15@r
Free Simply-supported Clamped
137.5
80.0 105.4 93.5
[0U/306]T
Free Simply-supported Clamped
81.9
84.1 7l '2 73.5
[OU/45~]T
Free Simply-supported Clamped
61.6
b 64-7 78.8
[Od/6J)~]T
Free Simply-supported Clamped
54. l
b 37.6 42.1
[0d/75.]-r
Free Simply-supported Clamped
51-6
b 98.1 119-6
[ OUIC~i<,]T
Free Simply-supported Clamped
5 I. 1
b 30"8 39-6
"//indicates a room temperature bondline. h Insufficient data.
Buckling of unsymmetric composite laminates
115
The experimental results possess a large coefficient of variation and exhibit a significant inconsistency with the analytical predictions. There are several factors that may have contributed to these discrepancies. First, the mechanical couplings inherent in the laminates studied induce internal stresses and moments (other than N~ ~) which would cause shearing~ bending and twisting were the plate not physically restrained. The presence of clamped or simply-supported boundary conditions restricts this motion by exerting additional forces and moments. Such effects were not included in the simple analytical predictions. Second, the actual strain was only on the order of a few hundred microstrain. The data are recorded by the computer in increments of 12.5 microstrain. This relatively large increment, coupled with the inherent strain gage errors at low strain levels and the difficulty of identifying an appropriate linear region, magnifies the scatter and potential for error. Third, due to the physical design of the buckling jig, the plate actually experiences a uniform end shortening during loading, as opposed to a uniform load distribution. As the center of the plate deflects outward, the in-plane stress is relieved in that region. The actual load distribution becomes sinusoidal in nature for an isotropic plate as soon as any out-ofplane deflection occurs. For an anisotropic plate, the load distribution is even further distorted. In determining the modulus, a uniform applied load was assumed. Before any further experimental work is undertaken, a m a n n e r in which to determine a laminate ~elastic constant', as a quality check of the specimen, should be established.
5.2 Buckling loads and mode shapes The "pseudo" buckling loads of each plate for each boundary, condition were back-calculated from the load-deflection data using the technique developed by Southwell. ~7The critical 'pseudo' buckling load is defined as the inverse of a least-squares fit of the deflection/load versus deflection curve. Southwell's method gives accurate results only for deflections small compared to the plate thickness. TM Plots were made of the deflection, as measured bv the tracker, versus the load at several (usually nine) locations on the plate. Such a typical load-deflection plot is shown in Fig. 4. Linear regressions of the data indicate that the useful range of this m e t h o d exists for a center deflection between 25% and 50% of the plate thickness. The experimental buckling loads presented in this paper were obtained by averaging the results from the various locations for each specific case.
116
Paul A. Lagace, David W. Jensen, Douglas C. Finch
The aluminum plate results isolate the accuracy of the buckling jig from the manufacturing quality of the plate. The tests conducted on the aluminum plate produced experimental buckling loads within 8% of the analytical results for all three side boundary conditions: free, simplysupported and clamped. This lends credibility to the overall effectiveness of the buckling jig and measurement techniques. Table 4 displays the experimental and analytical buckling loads for the special cases examined. The coefficient of variation for the experimental
Z
.~-
15
[06//756]T Simply-supported sides
10
o
0.5
Ol
O.3
O u t - o f - olane deflection, mm
F'~. 4. Typical load-deflection (Southwell) plot.
values is typically less than 4%, indicating a good uniformity across the plate. However, the composite plates show large discrepancies between the experimental measurements and the theoretical predictions, although the trends are similar. As mentioned previously, the unsymmetric laminates were manufactured by bonding symmetric sublaminates together at room temperature. The principal effect of the room-temperature bondline is to increase the buckling load by approximately one-third. This can be observed by comparing the results in Table 4 for the [03/903]s and [03//90E/03]T layups. This effect was predicted analytically by modeling the roomtemperature epoxy bondline as a uniform thickness spacer that carries no load. By introducing bending-stretching coupling into the plate through antisymmetry (i.e. [03//903//OJ/903]T), the out-of-plane deflection, w, begins with the onset of load. Therefore, the critical buckling load
Buckling of unsymmetric composite laminates
117
TABLE 4
Experimental and Analytical Buckling Loads
Buckling~ad (N/m) Laminate
Sidebounda O, condinon
Predicted Experimental
[03/903]s
Free Simply-supported Clamped
26 992 27 150 31 780
16 070 19 650 20 890
[03fI90~1103]¢
Free Simply-supported Clamped
28 741 28 870 33 630
21 670 24 110 27 340
[0311903110311903]r
Free Simply-supported Clamped
18 070 20 439 31 920
10 960 14 970 24 950
[02//452//02//452//02]T
Free Simply-supported Clamped
17 366 18 398 20 729
17 610 23 440 25 600
[02//45~_//02//--45d/O2]T
Free Simply-supported Clamped
17 339 17 770 19 720
17 920 21 480 23 780
//indicates a room temperature bondline.
becomes a "pseudo' buckling load. This 'pseudo' buckling load is over one-third less than its symmetric counterpart (for free side boundary conditions), although there is no significant change in the mode shape. The last two laminates summarized in Table 4 illustrate the effect of bending-twisting and bending-stretching couplings. Bending-stretching coupling, exhibited by the [02//45~_//02//--452//02]T laminate, can be seen to be the more severe of the two, in terms of the critical buckling lood where° it is noted, as above, that the unsymmetric laminate results in a "pseudo' buckling load due to the eccentricity effect. Finally, for the [0d/06]w family, there is also a noticeable general trend of decreasing buckling loads for increasing lamination angle, except for a slight unexplained increase in the clamped-clamped analytical case for 0 greater than 60 ° . These results are illustrated in Figs 5-7, which show the buckling loads as a function of lamination angle, 0, for the [06//06]T
[Oe//eejr Free side~
~
3O
Prediction
7
~ ao o
o o o
o
m
0
--
I 15
O
i 30
~ 45
i 60
oi
75
!
90
~9,degrees Fig. 5. Experimental and analytical buckling loads versus lamination angle for [0,tI0.]T graphite/epoxy plates with free sides f
[%//ee ]
[
Simply- supl)or ted sides
30 ~
~
-L 0
o
Prediction
0
- ~
~
o
"; 10 (.J
o
F
[ O{ ©
I 15
L $0
.L 45
I 60
i 75
i 90
8, degrees
Eig. 6. Experimental and analytical buckling loads versus lamination angle for [Od/O.]-r graphite/epoxy plates with simply-supported sides.
Clomped
E
30 I'-
~_~
}
C:
O |I
:3 (]3
K
~
~ o
~
°
sides
Prediction E~p~imen~'
o
d~°
o i
O ~
I
0
t5
~
L
J
30 a5 6o 8, degrees
~
~
z5
9o
Fig. 7. Experimental and analytical buckling loads versus lamination angle for [O~//O.]-i
graphite/epoxy plates with clamped sides.
Buckling of unsymmetric composite laminates
119
specimens with flee, simply-supported, and clamped side boundary conditions, respectively. The solid line shows the theoretical prediction from the 30-mode Rayleigh-Ritz energy analysis (33 modes were used in the free case). Plots of isodeflection contours at the limit load were made for each test case using a nine by nine (internal) grid of deflection measurements for the experimental modes and the same grid for the analytical calculations. The isodeflection contour maps for a [0d/30~]T laminate for the three different side boundary conditions are shown in Fig. 8. These are typical of the contour maps of that laminate family. Several observations can be made. First, the clamped-free experimental maps often show svmmetrv about the vertical centerline, while the predicted counterparts show twisting about this centerline. The symmetry of the experimental maps is a direct consequence of the even load distribution adjustment procedure, which used deflection symmetry about the vertical centerline as the criterion for determining a proper load PREDICTED CONTOURS
MEASURED CONTOURS
FREE SIDES
SIMPLYSUPPORTED SIDES
CLAMPED SIDES
Fig. 8. Measured and predicted isodeflection contour maps for [06//306]T graphite/epoxy plates.
120
Paul A. Lagace, David W. Jensen, Douglas C. Finch
introduction. This adjustment procedure is valid only for those specimens with no twisting coupling, and produces uneven load distributions for all other laminates. Thus, the symmetric deflection map was assured bv this improper load introduction adjustment. A second observation reveals a twisting in the simply-supported and clamped side contours for the laminates with stretching-twisting and bending-twisting couplings. The angle of twist in the maps corresponds roughly to the lamination angle. 0. Possible causes for discrepancies between the experimentally determined and the analytically calculated buckling loads can be classified into two categories: (1) experimental error and (2) analytical inaccuracies due to the choice of assumed modes and/or degree of convergence. The most likely source of problems in the experiments is the load adjustment procedure described in the experimentation section. The assumption used was that a deflection pattern which was symmetric about the vertical centerline implied that the load was properly distributed across the plate. However, this is valid only for specimens without bending or twisting couplings. This is illustrated by the finding that the deflection shapes for the specimens with these couplings show a marked twisting of the plate about the centerline. Thus, a better method for assuring proper distribution of load across the plate must be determined for future experimentation. Additional improvements in agreement between the experimental and analytical results can be achieved by more carefully examining the actual mode shapes used in the Rayleigh-Ritz analysis to ensure that all of the inherent mechanical couplings are adequately represented and by checking the series for convergence. In spite of these limitations, the experimental and analytical results do give initial indications as to the importance of mechanical couplings in the buckling of fully anisotropic laminates.
6 SUMMARY An investigation has been conducted which examined unsymmetric (and thus fully anisotropic) composite laminates and the effect their inherent mechanical couplings have on buckling behavior. Although many of the effects cannot be specifically quantified this prelimina~ work has laid the
Buckling of unsymmetric composite laminates
121
groundwork for future investigations into the compressive behavior of such laminates. Experimental techniques have been developed for use with unsymmetric laminates and suggestions have been made to improve these techniques: specifically, the need to identify a method to assure proper loading of unsymmetric laminates which exhibit out-of-plane deflection simultaneously with the initial application of the load. The Rayleigh-Ritz analysis employed herein has illustrated the importance of the mechanical couplings, e.g. the twisting exhibited in the buckling modes, and is verified by the experimental results that the existence of coupling can severely reduce the resistance of composite laminates to buckling. The predictions from this technique can be improved by assuming modes which account for the inherent couplings as illustrated in the experiments. Further work should not only consider the improved experimental and analytical suggestions but should progress to the postbuckling range where the inherent mechanical couplings will probably have even greater effects.
ACKNOWLEDGEMENTS This work was supported bv the Boeing Military Airplane Company. Mr Robert Waner is the technical monitor. The authors wish to thank Mr Waner and his staff for the useful suggestions and discussions during the course of this work.
REFERENCES I. Jensen. D. W. and Crawler, E. F., Frequency determination techniques for cantilevered plates with bending-torsion coupling, AIAA ,lournal, 22 (3) (March 1984) 415-2(I. 2. ,lensen, D, W.. Crawler, E, F. and Dugundji. J,, Vibration of cantilevered graphite/epoxy plates with bending-torsion coupling, Journal of Reinforced Plastics and Composites, 1 (July 198_) .,.~4--69. 3. Sherrer. V. C., Hertz, T. J. and Shirk, M. H,, Wind tunnel demonstration of aeroelastic tailoring applied to forward swept wings, Journal of Aircraft, 18 (November 1981) 976-83. 4. Bryan. G. H.. On the stability of a plane plate under thrusts in its own plane with applications to the "bucl
122
Paul A. Lagace, David W. Jensen. Douglas C. Finch
5. Leissa, A. W., Buckling of composite plates. Composite Structures, 1(1983) 51-66. 6. Timoshenko, S. P. and Gere, J. M.. Theory of elastic stability. 2nd edn, McGraw-Hill, New York, 196l. 7. Ashton, J. E. and Waddoups. M. E., Analysis of anisotropic plates. Journal of Composite Materials, 3 (January, 1969) 148-65. 8. Ashton, J. E. and Waddoups. M. E.. Analysis of anisotropic plates II, Journal of Composite Materials, 3 (July 1969) 47[)-9. 9. Ashton, J. E., Anisotropic plate analysis--bounda~' conditions, Journal Of Composite Materials. 4 (April 1970) 163-71. 10. Ashton, J. E. and Love, T. S., Experimental studv of the stability of composite plates. Journal of Composite Materials, 3 (April 1969} 230-4~.. 1 I. Kicher, T. P. and Mandell. J. F., A study of the buckling of laminated composite plates, AIAA Journal, 9 (4) (April 197l) 61)5-13. 12. Starnes, J. H., Jr, and Rouse, M., Postbuckling and failure characteristics of selected fiat rectangular graphite-epoxy plates loaded in compression. Proceedings of the AIAA/ASME/ASCE/AHS 22nd SDM Conference. Atlanta, Georgia, April 1981. 13. Starnes, J. H., Jr, Knight, N. F., Jr. and Rouse. M.. Postbuckling behavior of selected flat stiffened graphite-epoxy panels loaded in compression, Proceedings of the A IA A/A SME/A SCE/A HS 23rd SDM Conference, Lake Tahoe, Nevada. Mav 1982. 14. Chia, C. Y., Nonlinear analysis of plates, McGraw-Hill, New York, 1980 15. Tsai, S. W. and Hahn, H. T.. Introduction to Composite ~14aterials. Technomic Publishing Company, Connecticut, 1980. 16. Finch, D. C., The buckling of symmetric and uns~:mmerric composite plates with various boundary conditions, Technology i~aboratorv for Advanced Composites, Report 84-3, Massachusetts Institute of Technology, S.M. Thesis. February 1984. 17. Southwell, R. V.. On the analysis of experimental observations in problems of elastic stability, Proceedings Oflthe Royal Society, London. Series ,4. 135 (1932) 61}1-16. 18. Donnell, L, H., On the application of Southwell's method for the analvsisof buckling tests, Stephen Timoshenko 60th Anniversary Volume. McGrawHill, New York. 1938, pp. 27-38.
APPENDIX The m o d e s used in the Rayleigh-Ritz analysis are included in this Appendix. For the case of the in-plane deformations, u and v. the same m o d e s were used for the three types of side botmdarv conditions
123
Buckling o f unsvmmetric composite laminates
U=
q.,+,sin
qt-+
mTrx
a
m7rv
cos~
b
(m - 1)rrx +
q,,,,
5
COS
(m + 1)~-x ] Cos
a
v=
-(1"5
q,,,
qm ~
÷
+q,~
11sin mTrx
.,"--,
a
mTrv/
-
a
Sill
J
"
,
h
j
-0-5
cos
retry
3
mrrx
-. + ~ q~.÷,scos h ~', a
mrrv
sin~
b
H o w e v e r . different m o d e s were assumed for the out-of-plane deflection w d e p e n d i n g upon the side b o u n d a r y conditions. These are WFR for the free b o u n d a r y conditions 14'FR
cos
= *
--
qs~,,, i ~-
a
2 [
tn77Y
E n
cos
I
qs,,,,
nrry 1
, ..... ,~;sin - --~- + q~,,,, ,,-n-2, c o s - - ~ J
[
]
J
a n d wss for the simply-supported side b o u n d a r y conditions
W,.;~, =
(]Aim
I I + n - IS
cos
CO~,
m = In = I
a
( m + I )rrx ] a
sin
(2n - k)~-y b
w h e r e k is equal to 0 for n even and is equal to 1 for n odd. Finally. the a s s u m e d s h a p e , w . , . for the cases where the sides are c l a m p e d is:
I~'( 1
=
, ,,7"" i ,~
q41m
I/~n-
{
0 1 - 1)try
I~
COS
COS a
( n + 1)~ry 1
a