Table Of Content.
AIAA-2002-1579
DEFORMATION RESPONSE OF UNSYMMETRICALLYL AMINATED PLATES D - TO
INPLANE LQADING
Taoaoya T. ochiaero* and Michael W. Hyat
Engkxhg Scimcc and Mechanics, V- Polytechnic Institute and State University, Blacksburg, VA 24061
AEsxBKc If a laminate is msymme&ic, then the nepanec cbarac-
This paper disaun#s the out-of-plane definmatiim teristics SO~~WD UMH W C Ddm~ ~to~ t he COU-
behaviorofunsymmtnc ~-plycompoSiteplates pling of inplane forces with bending dcformptians and
*
compressed inplane by displscins (~letdg e of the plate bending moments with inplane defolmations, couplings
a known pmnnrt Thc p ha assumed to be initially nat present in symmetric laminates. The level of asym-
flat and Bevecol boradory 4th case^ a WAd- metry, and theref- the magnitude of the resulting elas-
- -
aedGeometrrcally lronlintsrbchaviorisassumed.The tic couplings, is determtned b y t b e ~ o n ~ ~ .
primary objedives 81t to study the Out-of-plane behav- The highezthe level ofasymmetry,t he higherthe mag-
ior as a frmction of fm&asing inplane compression and nitude of the elastically-coupledr eqxmss. The level of
to determine if bifurcaton behavior and secondary as- of a particular laminate may be evaluated by
buckling can occur. It is crbown that, dependbg on the considering the laminate stiflkss matrix, namely, the
bormdarycanditions,bothc an ocaa,thOugbthe charac- ABD matrix. specifically,t he B matrix, a submatrix of
teristics different tbaa the pa^ and post-buclliqg the ABD matrix, deterrm.n es the level of coupling
behavior of a comganim tymmtric cross-ply plate. between ule inplane and out-of-plane xlqmme& also
Furthermae, whiit a weirk cross-ply phtt h-wz Lu-AsLa,. a u-...l.&Iu & - o u ~nC.N\ .mq.-1. ;na T -innts m y
postbuckle with eitber a positive or negative out-of- be taild to exhibit certain elastic couplings to take
plane displacamnf tht unsymmcbic cross-ply plates advantageofthisrmiquecapability.unf~y,tradi-
studied deflect out-of-plane only m one dirtction tional elevated-temperatme cure -hniques make it dif-
throughouttherangeofinplane~~thedircc- ficult to produce an msymmetric laminntl? that does-
tion again dependingcQthebolmdarycmditicms. change shape as it is coolexifbnthe curetcmperatm
[3].Asadt,flatrmsymmetn 'c laminntcr Iuc difficult
I" to manu-. one approach istofmn an Imsymmet-
AoompoeiteWiSBaidtobtsymmetricif riC laminate by banding to- two eymmetric lami-
for every layer to one side of the lambate refeaeact sur- nates using a room-teanpexatm adhesive. However, due
face with a speciticthichess, specilk -proper- tothelimitationsofroom-~adhesivcsandthe
ti-, and specific fiber aaiartatia thae is aootber layer added &ofmanukbnhg, thisappsoadr is oflimited
the identical dktancc on the oppade side ofthe refer- applicability. An alternative manufactwiug method that
ence srafiice with identical thickness, material proper- may hold promise for the frdure is the elcctrm beem
ties, and fiber orientation. If the laminate is not curing process. Due to the low temperatures associated
symmetric, then it is referred to as an umymmetxic lam- with this process, the.m.all y-induced shape changes may
inate [I]. T-he teaminology a . 'candasymrmt- bexnbimhd or ehmated altogether. An even mazt
ricarealso used.ner-e-a respecialca-s-e-so f~--e tric intriging idea would be to use nanotechnology to tailor
-1 * , VUG- D- m-LU W: - O LI O-AUULUJ.--U-~L~ --e~l.y,-l2m m;- the thmiieh the thickness of a flat plate to be
nate, whereby the laminate consists of an even number UnSymmetriC.
of layers with the prhipal material directions of tbe Once flat, or curved, unsymmetric laminates
layers altemathg at Oo and Wto the laminate axes. An can be manufhctud with a degree of dimensional fidel-
antisymmetric angle-ply laminate has layers oriented at ity, attention will then focus on mdentdm' P h
+8 to the laminate coordinate axes on one side oftbe responseof 'c laminates to rnecbnid, and
'
reference surhe and layers oriented at -8 to the lami- perhap& thermal loads. Tbc unique ability to mqle
nate axes on the other side of the reference surface [2]. inplane respanse with out-of-plane responst will make
unsymmetric laminates candidates for advamed struc-
tural tailoring cancepts and possible solutions to umfig-
d o n changes needed fix adaptive shctures not
achievable by othex means. Before explcsing those
*GraduateReseamhAssociate applications, however, it will be necessary to llly char-
Professor, Associate Fellow, AIAA; Fellow, ASME, actaize tbe response of llnsymmetric laminates to Bim-
Past President, ASC ple loadings, categorize their behavior, and understand
Copyright 0 ZOO2 by T.T. 0cLiMulDC.W. Hya. hWd by tLc their limits. This paper is directed at these goals and will
~r.ditrtc~rAuwuticr~,t,t,-~ie*~c.,ryLpol..IL*r
1
American Institute of Aeronautics and Astronautics
.
consider the deformation response of flat unsymmetric The standard plincipal material coordinatt system for
laminates to a simple inplane compmive loading. This the layem is located within this coordinate system, as
loading is considered because it is one of the simplest shown. The fiber, ar 1, directionmakes an ang-le 8 rela--
mechanical loads to model and one that can be consid- tive to the +x-axis. The displacemmts in the x-, p, and
eredin a Iaboratory setting. For Symmetric laminates z-coordinate directions am denostd by y v. aQd w,
such a load would be associated with the prebucklin& respectively. The rotations about the cogdinatc axes 8e
buckling (bifurcation), and postbuckling behaviar. For denoted as &, 4. and and fig. 1 showsthe sense of
+E,
unsymmetric hhtas, wbi4 due to the elastic cou- t hrot~atiOn rr as &fkd in ABAQUS. Tbe
plings, canbegin to deform out of plane as soon as they sive loading is applied by displnl?ins the right end of&
are compressed i n p ui t isnot clear ifthe coL1cept8 of plate aknownamolmt Uto sboaentbc plate. The asso-
prebucldin& budding, and postbuckliug are valid That ciated load is denoted as P. Such a hading am be pm-
very point is par& ofthis inveStigatim. Leissa [4] argued duced in the labommy, and since it makes sense to
that a flat unsymmetric laminatc will reanainflat undera begin any experimptal investigation ofthe dchmaticm
compressive inplane loading if out-of-phe rotations response of unsymmetric laminatee by amskkhg
are restricted on all four edges, i.e, the plate is clamped. small-scale laboratoq size specimens, the plates umaid-
This occm because the clamped umditions provide the ered are 0.508 m long by 0.406 m wide (20.0 in. by 16.0
moments necessBly to counter the moments induced by in.). The plates are asumed to be mnshded of eight-
the inplane compression which make the plate deform layers of graphiteepoxy and meaSure 1.016 mm (0.04
out of plane. If this flat state exists, then classic preb- in.) thick The properties of a layer m assume to be
r?ck!bg, buckling, and postbuckling responses are a pos-
sibility. Leissa [s] went to study tbe geometrklly E,=130.0 GPa (18.85 Msi)
linear responses of four types of uluymmetric laminates Ep9.70 GPa (1.407 Msi)
subjected to uniform and linearly-varing inplane loads,
G12~5.00G Pa (0.725 Msi)
including uniaxial, biaxial, and &ear loads. Twelve Bets
v,2"0300
of boundary conditions were considered. The msult of
the study was a tabulaticm of the center deflections of b-o.1270 mm (0.005 in)
the many cases, including the case of no deflection fap
the conditions discussed in ref. 4. The present pap Out-of-plane displacanent VI. endshortening results
could be considered an extension to ref. 5 m that the will be presented at the thffe locations labeledZ,H, and
compressive loading discussed here is one of the load- IZZ in fig. 1. Locations I and IZZ m qrrarta locations,
ings Leissa [SI considered, and plate response into the while l d o nI I is at the center. Plots of tbe defii
geometricaiiy nonibear range is mi&dcid % h?k geometry will be presented far specific cadsbortariog
levels. Additionally, the relationship beheen endshut-
address the issues of prebwkling, buckling, and post-
bwkling. HOW~OV~~C,Y rOS S-ply laminates IT coI1sid- ening and the associated loadP will beprcsenM This
ered here and fewer boundary conditions are relation is often used fa problems such as discussed
investigated, as the response of the laminates becomes bere to describe tbe &lane stiffness of the plate as the
complex when in the postbuckled state. response changes h m p rebuckling to podwkbg.
The following section desgibes the specific Tfnee combinations ofb-oun- d-ar-y conditions will be prc-
problem studied. The loading, boundary cmditions, D-c-.I.lL2I%L-N3 W -1-..- 6 4YL-I U-.AmY an4 &AM -*-d t- b_-- ,1_ k . will
cmrdirlate system, geometry, and aOmencl8hn.e are labeled ssss, CGSS, and CGCL to designate the par-
described. As the numerid results are &kmined using ticular bwndary umdition combinaton being COIIsid-
the commercial code ABAQUS [6], finiteelement con- ered. The termioology SS is contracted notation for
siderations are also discussed The section following simply-supparted dtiansa nd CL is a contraction for
that discusses numerical results. clamped conditions. The c4mtIa& 'on to the left of the
hyphen designates the prescribed b o w c ondition
along the ends, while the contraction to the right of tk
ltuniuu hyphen designates the prescribed boundary condition
The geometry and nomenclature of the prob- along the sides. What is impoat to rcalizc h mt k
onset of the discussions is that the sides of the plates are
lem studied are shown in fig. 1. Rectangluar plates of
dimensions L by W are considered, though me square assumed to be constrained 60 that yso there. This might
be considered a somewhat arbitrary and uIve8listic con-
plate will be discussed. The long dimensions of the plate
dition, but it is also unrealistic to consider plates that
are ref& to as the sides and the short dimensions the
might be joined with other elements to form a complek
ends. The xcoordinate originates at one end of the plate,
structure to have no restrictians on the v4ispIacements
and the y-coordinate is centered between the two sides.
2
American Institute OfAenmautics and Astronautics
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along the sides. Rather than look at a range of elastic took considerably more time with 400 elements than
restraints, adding another variable to the problem, the with 100 elements, so the 100 element mesh was used.
c8se offullrestraint is colrsidaed. This issue is impor-
tant because, through Poisson effects, inplane compres-
sion in the xdiredion can cause the plate to expand in Althoughtbe goal of this paper is to inveaigatc
the y-directim. With v=o almg the sides, Poisson theresponseoflmqmmem‘c compoGitc pla- results
expansion is restricted by an inplane compression force will first be presented f aa n initially flat cL”ss [W
in they-directionw ith the beodiqg-stretching coupling 90& symmetric cross-ply laminate. Because it is sym-
present in an unsymmetric lamb&% this inplane farce, metrically laminated and flat, this plate remains fiat
alongwiththeappliedmp~fbrce,are~. when initially loaded, and fhen ehiits classic bikca-
tim behavior. The response far this situation is well
Nine-- tw odimemid geamctrically- known, so relatively little new i n f d o nw ill be pre-
nodbear, sheardefmnable S8RS shell elements are sented regarding this particular plate. Rather, tbe results
usedtomodeltbeplates.Theplatesarempressedstat- presented serve as a baseline for comparing the behavior
ically by increasing the usingvery gmall of the initially flat [O@& unsymmetric cross-ply lam-
in~tsofu.Attachvalueofuthestaticstability
inate to be discussed. Also, by considering the symmet-
of the plate is checked by examining the eigenvalues of
ric cross-ply plate, the dynamic stability analysis
the tangent stifkss matrix. A zero or negative eigen-
implemented in ABAQUS is applied to a problem that is
value signals a statically unstable laminate d g m -
ticz A? the value of u whele tbe static solution ceases somewhat familiar, and the characten* sb’ a of the
q p - d can be wed. Tbe approach can then be
to be stable, some iteration is then done to adjust Us0
applied to the lmsymmetric laminnte with some experi-
thatthe value of Ujustatthe onset ofthe instablity can
ence and confidence.
be closely determmed. ForavalueofUslightlygreater
*
tban this anset value, and far this value of Uheld fixed,
a dynamic analysis, usinga small amount ofdampiug, is
then initiated. Far a statically &le dtiian,s imply
initiating a dynamic analysis should be emugh to allow Figure 2 presQltst he ncnmd‘m d load VS. ad-
S h d g r elation, the normalized out-of-plane dis-
the plate to begin to move toward a stabk equilibrium
placement vs. endshortening relations, and & f d
configuration. However, for some mstable conditions,
plate canfigurations at selected values of endsbortenisg
this requires integrating fiJnvard in time far long periods
for the CL-SS [02/90& symmetric croskply laminate as
of time, as the plate may move slowly drom its unstable
configuration. f%x43rfjiugiy,a s an d a v ctG s@!y the endshcdming increases k r n zero to a nomulid
value of 10. “be noRnalizing factor Uw is ihe dticd,
initiating the dynamic analysis, the plate is given a very
small pressure pulse perpendicular to the plate at time or buckling, value of endshoxtening computed by
zero of the dynamic analysis. The dynamic analysis is assuming a linear, non-sheardeformable eigenvalue
d e do ut for small pressure pulses in both the +z and - solution to the classic SS-SS uniaxial buckling problem
z directions (see fig. 1). In general, if press= pulses iae [2]. The term ‘classic SS-SS’ implies that the bod-
used, the plate moves to the same canfiguration inde- aries are fixe to move inplane (&) and are only con-
pendent of the sign of the pressure pulse. As the plate *&$ !%= nitt-nf-plane deformation (-0). Tbc
motion is decaying a r dth is unique CanfiguratioIb the value of Ucr for the specific plate here is 0.00516 mm
decaying being due to the light damping, a new static (0.000302 in.) and this n ** g factor was chosen
analysis is inititated using a decayed dynamic canfigu- simply because it is easily calculated The CGSS
boundary conditions inrreases the &tical gldsboaening
ration as a starting point. The static analysis then con-
verges within a reasonable number of iterations to what with respect to the classic value, and therefon, a nor-
is to be the statically stable configuration for malized endshortening value of 10 does not imply that
that value of enshortening. lhis nunerid scenario cor- the plate being discussed is compressed uniaxially to 10
responds to a displacementantrolledc ompression test times its own critical value. Strain levels in the plate
in the laboratory. Ten elements in each direction, for a were checked at UNct=lO and they were not particu-
total of 100 dements, are used to model the plates. To larly high. The nannalizing factor Pcr used to normalize
P is the load associated with Ucr.
test the robustness of this mesh, for several specific test
The three figures for the out-of-plane displace
cases the mesh was doubled in each direction, far a total
ments at points I, 11, and III show that as the endshoxten-
of 400 elements. All aspects of plate response were
identical for the two meshes, including the dynamic sta- ing is increased from zero, point A, the plate remains
flat. At endshortening values past point B, the plate
bility analysis. Of course, the dynamic stability analysis
3
American Institute of Aeronautics and Astronautics
remains flat andis in equilibriumbut it is unstable, as is tions at selected values of endshortening shown in fig. 3
well known. At a point just past point B in the flat con- result The format offig. 3 is identical to fig. 2 to aidina
figuration, the dynamic analysis is initiated. Since thc direct c o e nof plate responses. A quick glance at
plate is very close to being in a stable condition, simply the out-of-plane displacements at locaticm~I, II, and III
integrating the equations of motion with M) small pres- shows that unlike the [o$!loi]s symmetric cross-ply
sure pulse results in a large number of time intepticm case witb multiple stable solution patbs (BW,B 'C'D'.
steps foa the plate to converge to a stable equilibrium EF and E%'), there is only one stable solutim path. As
configuration. A anall pressure pulse, e.g., 689 Pa (0.1 the ends- is increased fiom ZQD, tbe plate
psi), applied in a ramp-up/&wn MOhDast,en s deforms in a stable %&ion out of plane in the negative
the motion to a stable configuration &r&emed' bya z-direction a very small am& Standard expeaimd
mallo utsf-plane displacement, as indicated by point C techniques probably could not detect any displacemmt,
in the three figures and as shown by the leftmost but the displacements are nonzero and the canfiguration
deformed configuration at the bottom of fig. 2. This at B is one with two halfivaves in eacb dk&on, as
deformed configuration bas one halfhave in the loading shown at the lower left in fig. 3. For endshaaau* ng val-
direction and one halfivave pexperdicular to the loading ues past point B, the plate is unstable and the dynamic
direction. This, of course, is very much like the first lin- analysis results in the plate moving to the canrfigrwtion
ear buckling mode far this plate, but at point C the plate given by point C. Point C represents a large increasC in
is in a postbuckled state. If the pressure pulse is positive, the out-of-plane displacement relative to point B, but the
the plate moves to wnfiguration C, and if the pulse is wnfiguration at Point C, shown at the bottom ofthe fig-
negative, the plate moves to configuration C'. Inmas- ure, has a cuufiguration like point 3. Many attempts
ing the endshortening toward point D, the smgle haif- were made to determine if& gap *between phtt I 4
wave configuration deepens, as shown at the bottom of cis actually closed. This does not seem to be the case.
the figure. Past point D equilibrium wnfigurations exist With the finite-elementa nalysis it is not possible to have
but they are again unstable. Initiating a dynamic analy- the md&orkhg values in- cantinuously. HOW-
sis just past point D results in the plate moving to can- ever, even using iine increments inencfshortcning
figuration E. The plate could also move to configuration beyondpointB. there appears to be a jump h nBt o C.
E'. If after point B the plate response is given by path This suggests that this particular mspmetn'c laminatt,
CP', then initiating a dynamic analysis at D' would witb these particular bormdary conditiw does not
result in the plate also moving to configuration E or E.' exhibit bibtion. hxeased e n d s h d gt o point D
Ascanbeseen,theconfigurationsassociatedwithEand further deepens the halfwave configuration. Far an end-
E' are represented by.. ..t - w-o halfivaves in the loading shortening value just psst point D, a dynamic analysis is
uAI2I--u-d\I-I l O--AL U a..-v uh-L *lfw;y.vep q~&ic~~ltao tth e 10- again initiated and stable configutatim E results, inde-
directioa This change of configuration is o hre felled pendent of the sign of the small pressrae pulse. Tbe con-
to as secondary buckling and is discussed for composite figuation at E, shown at the bottom of the figure, is not
plates by Tiwari and Hya [A.I ncreasing the end-- symmetric about the FLD line, as can be seen by com-
ening to point F deepars the configuration of point E. paring the outsf-plane displacement at location I with
After point B, the slope of the normalized load vs. end- th~atlocationIf~Increased~~gbeyondco~
ashndo rttheen irnegm reolanti boenc, oinm tehse s iuipgpheury l enfotn oiifn ftizgr.r ,2 m, d?hmhgs G, fpiig-muraatbio n E, &w haich *ueLxu--hw- ib iIlt--s--- "Gn-,-e -g- ative +-ohnua-twdw--hp l +an&,ee d Eis-
configuration D, or D, is approached. The change to relatively small at location IZI, results in a gradual lran-
configuration E orG is accompanied by a slight drop in sition to the configuration at point F. This mnfi&on,
load. Beyond point E, and E', the slope of the relation as shown at the lower right, has a large negative outsf-
decreases again. plane displacement at location I, a smaller negative out-
of-plane displacement at location II, and positive out-of-
plane displacement at location I... The configurationh as
two balfivaves in the loading direction, but one that
If the eight layers of the [Ofl& symmetric skewed to one end of the plate rather than being anti-
cross-ply lamin& are reawnged to form a [Od90& symmetric with respect the the ~ L fla2ca tion.
If the boundary conditions on the ends of the
unsymmetric croas-ply laminate and the numerical ad-
plate are changed to be simply supported, then the
ysis ofthe previous section far CL-SS boundary condi-
responses shown in fig. 4 result. As can be seen, chang-
tiom is repeated,t he normalized load vs. endshortening
ing of the boundary conditions has a profound influence
relaticq the normalized outsf-plane displacement VS.
on the response of the plate. Specifically, as the end-
endshortening relations, and deformed plate configm-
shortening is increased fiom zero the plate deforms out
4
American Institute of A e r ~ nadnd ~As~ttan autics
.
of plane a significant amount. Tbis is in contrast to the CGSS and SS-SS boundary conditims offigs. 3 and 4.
CL-ss case previously discussed in fig. 3, where the As was the case earlier, changing the bolmdsry cod-
plate remained relatively flat fm low end shortening lev- tions relative to the CL-SS case of fig. 3, in this case
els. For the SS-SS case, as seen in the out-of-plane dis- making the sides clamped, bas an important effect an
placement vs. relations at locations I, Lt, plate response. In particular, it would appear that the
*
and III in fig. 4, up to point B the plate deforms out of plate exhibits bifurcation behavim at B, much like a flat
plane in a single halfwave d w t i o n .A t point B, the symmetrically laminated plate of fig. 2. However,
load w. endshmtgll* ng relation deaeases slope slightly. instead of bifurcating to positive and negative out-of-
Astheeodsbartenmg is the plate configm- plane displacement branches, only one out-of-plane dis-
*
tionbeginstocbangt.wbentheendshortening~h placement branch OCCULF.
the level of point C,t he plate appears to have two half- 'Ibe psuedo-halfhave response thughout the
waves in the loading direction. A close look at tbe out- -ne grange for the CLCL boundary d t i m
of-plane ctisplacement s at location fI, however, reveals shown in fig. 5 is a result of the rectangular geometry of
thatthedisplacanent is not zero in the center of the the plate. If the sidelengtb of the plate in fig. 5 is
plate. At point C the plate becomes unstable, and a decreased so the plate is square, theresponsc shown in
dynamic analysis is used to affect the transition to point fig. 6 results. As the endshorte-nbg is increase firnn zem
D. At point D the deformed configuration looks like that to point B, the plate remains flat, to within the numerical
at point C, but the out-of-plane dispacement at location accuracy of the analysis. At point B the flat dgura-
I1 is positive, while the magnit& of the out-of-plane tion becomes unstable. The dynamic analysis is initiated
&s!~cemmts at locations I and I17 has decreased rela- for a flat configuration just beyond point B and the con-
tive to point C. A frather incaeasc in endshortening to figuration of point C results. configuration hap a
point E does not produce any configuration changes. It single balfivave in both directions. This configuration
is clear there is nothing that resembles bifurcation remains as the endshortening is increased to point D. At
behavior ne8r point B in fig. 4, ratha the plate configu- point D the single halfhave configuration becames
ration makes a Bmootb continuous shape change as tbe unstable and the dynamic analysis is agaiu initiated. The
endshorknhg inaeaseS h n Ato Bto C. Furtherma plate assumes a configuration with two halfwaves in the
tbe plate deflects out of plane in a direction opposite to loading direction and one halfwave perpendicular to that
the CLSS case m fig. 3. direction. Noting the nonzero displacement at location
If iIl&ead of enforcing simply-suppoxt bound- 11, it is seen that the configuration is not truly a two half-
ary conditions an all four sides, clamped conditions are wave one. However, as the endshortening k inmased
e n f uth e responses illustrated in fig. 5 OCCUT. As the beyond point E, the out-of-plane displacement of tbe
endsbortezllng is h m m ,*the pkic qpii cxzter cf !!t p!zte zppaches zero and tbe displace
to remain flat According to Leissa [4], this should be ments at locations f and I11 approach being equal and
the case, though the finiteelementr esults indicate there opposite. For this specific configllratian, bifurcation
are very ~nalolu tof-plane displacements. As they are occurs at point B, and secondary buckling occurs at
many orders ofmagnitude less than the out-of-plane dis- point D. The bifurcation results in just one out-of-plane
placements along path AB in fig. 3, these are felt to be a solution branch. ---- - .
result of numerid romd-off in the fmiteelement anal-
-
ysis. Increasing the endhrhmg pastpointBltxultsm -x,
a statically unstable condition. The dynamic analysis This study has shown that the behavior of unsymmctri-
leads to a statically stable equiliium umfiguration cally laminated cross-ply plates subjected to inplane
whereby the plate k &farmed out of plane in what compression can exhibit a range of out-of-plane defor-
appears to be a two halfwave configuration, as shown by mation behavior, depending on the level of compres-
the inset of the deformed geometry of point C. However, sion, boundary conditions, and geometry. For this
the configuration is not exactly a two halfivave one, particular study the codtion 4 was enforced along
rather there is a d out-of-plane displacement at the the sides, further contributing to the response. Except
center of the plate, locationIf, as seenin the upperpor- for the CL-CL boundaly conditiq the initially flat
tion of fig. 5. With incmasing endshortening past point plates did not remain flat of low levels of compressive
B, the displacement at location If is at first negative, loading, though for the CGSS case the out-of-plane dis-
tben positive, though it remains small. At locations f and placements were very small and could be inkrpreted as
IfI the displacments are much larger and have opposite zero in an experiment. For the CLSS case the very
signs. This psueddmlfwave umfiguration remains snsll out-of-plane displacement configuration did
throughout the range of endshortening studied. There is become unstable and an alternative stable configuration
no secondary buckling behavior, as was the case for the was found which did have large out-of-plane displace-
5
American Institute of Aeronautics and Astronautics
.
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mats. Secondary buckling could occur, resulting in 4 Leissa, A.W., "Conditions fa Laminated plates to
another stable equilibpium configdon. Remain Flat under Inplane Loading," &mpsh? strut-
W~S6, ,p p. 261-70,1986
ACI("TS
-
The work discussed berein was SUP@ by grant 5 @tu, M.S. and A.W. Leissa, "Buckling a Trans-
NAG-1-2298 fi.om the Mechanics and Durability verse Deflections of Unsymmetrically Lamhted Plate13
Branch Of tk NASA-h&Y Research Center. h. Subjected to Inplane L-oads."AUA J., 31.1, n.1 89-94,
Damodar Ambur was the grant monitor. 1993
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1- Hyer, UW. Stress Analysis ofFiber-Reinforced Corn-
piteM aterials, wBc/Mdjraw-Hill, New Y& 1998,
p. 2%
-
7 TiweN . and M.W. Hyer, 'secapldary Buclding of
Laminated Composite Plates," AIAA Paper 2000-1457,
presented at 41st A I A A / A S W A S C C Struc-
tures, Stnrctural Dynamics, and Mateds Conference,
Atlanta GA, April 2000, also AUA J, to be published
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Fig. 1 Problem description, geometry, and nomenclature
6
American Institute of Aeronautics and Astronautics
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Fig. 2 -ResponseC haracteristics of CLSS [@#)O~]S Plate
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American InstitUte of Aeronautis and Astronautics
Location II
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Fig. 3 Response Characteristicr of CLSS [0@4]T Plate
8
American Institute of Aeronautics and Astronautics
Location ZZ
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Fig. 4 Response Characteristicr of SSSS [O@~]]T Plate
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hencanI nstitute of Aeronautics and Astronautics
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Fig. 5 Response Characteristics of CGCL [OJ90& Plate
10
American Institute of Aeronautics and Astronautics
