On the Mechanics of Thin - Walled Laminated Composite Beams

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OntheMechanics
of
Thin-WalledLaminatedCompositeBeams
807
(ReceivedJune
8.
1(92)
<Revised
Deccmher
16.
1(92)
*DcparllllL'nt
Ill"
MechanICalandAerospaceEngineering.
**Ocp;,JftmentofCivilEngineering.
EVER
1.
BARBERO,*
ROBERTO
LOPEZ-ANIDO**
AND
JULIO
F.
DAVALOS**
Ui'stVirginiaUniversity
Morgantown,WV26506-6101
On
theMechanics
of
Thin-Walled
LaminatedCompositeBeams
sibletooptinlizethematerialitself
by
choosing
among
avariety
of
resins,fiber
systems,andfiberorientations.Changesinthegeometrycan
be
easilyrelatedto
changesinthebendingstiffnessthroughthemoment
ofinertia.
Changesinthe
material
do
notleadtosuchobviousresults,becausecompositeshaveproperties
thatnotonly
depend
on
theorientation
of
thefibersbutalsoexhibit
modular
ratiosthatcoulddifferconsiderablyfromusualvaluesinconventionalisotropic
materials(Barbero
[2]).
Although
beams
and
columns
are
themost
commonly
usedstructuralele-
ments,thetheory
of
laminatedbeamshasbeenlessdevelopedthanthetheory
of
laminatedplates.Laminated
beam
theorieswereinitiallyderivedasextensions
of
existingplate
or
shelltheories.BertandFrancis[3]presentedacomprehensive
review
of
theinitialbeamtheories.Berkowitz[4]pioneeredatheory
of
simple
beamsand
columns
toranisotropic1l1aterials.VinsonandSierakowski[5
J
ap-
pliedclassicallaminationtheoryalongwithaplanestrainassumptiontoobtain
theextensional,coupling
and
bendingstiffnessfor
an
Euler-Bernoullitypelami-
natedbeam
(All,B•.,Dtt
).
Atheoryfororthotropicthin-walledcomposite
beams
wasproposed
by
Bank
and
Bednarczyk[6],wherethein-planematerialproper-
tieswereobtainedusingclassicallaminationtheory
or
coupontests.AVlasov
theoryforthin-walled
open
cross
sectionsconlposed
of
planesymmetriclami-
nateswasproposed
by
BauldandTzeng[7]disregarding
shear
strains
in
the
mid-
dle
planes.Massonnet[8]addressedtheproblem
of
warpinginatransverselyiso-
tropicbeam
by
complementingamechanics
of
materialsapproachwith
correctivetermsderivedusingtheory
of
elasticity.Bauchau[9]
and
Bauchau
et
al.
[10]
providedamorecomprehensivetreatmenttotheproblem
of
warping
by
usingvariationalprinciplestomodelanisotropicthin-walled
bealTIs
withclosed
crosssections.Ageneralfinite
element
with
10
degrees
of
freedom
per
nodewas
derived
by
WuandSun
[11]
forthin-walledlaminatedcompositebeams
by
modi-
fyingtheassumptions
of
theVlasovtheory.Skudra
et
al.
112]
proposedatheory
forthin-walledsymmetricallylaminatedbeams
of
open
profile,andtheyillus-
tratedthedistribution
of
forcesinaflathomogeneousanisotropicstrip.Tsai
[13]
definedengineeringconstantsfromthelaminatecompliances,andemployed
themtoobtaindeflectionsforlaminatedbeams.Hefurtheremployedlanlinated
platetheorytodetermineplystresses.In
the
presentwork,kinematicassump-
tionsconsistentwiththeTimoshenko
beam
theoryareemployedin
order
togen-
eratebeamstiffnesscoefficients.Adistintictivefeature
of
thepresentapproach
withrespecttoexistingformulations
[7,9,10,12]
isthepossibility
of
considering
notonlymembranestressesbutalsoflexuralstressesinthewalls.
This
assunlp-
tionseemstobemoreappropriateformoderatelythicklaminated
beams
enlployedincivil
engineering-type
structures.
The
bendingextensioncouplingthatmayresultfrolnmaterial
and/or
geomet-
ricasymmetryisusuallytakenintoaccount
by
bending-extensioncoupling
stiffnesscoefficients.Inthiswork,theposition
of
theneutralaxisisdefinedin
sucha
way
thatthebehavior
of
athin-walledbeanl-columnwithasymnletric
material
and/or
cross-sectionaJ
shape
is
conlpJetely
described
by
axial,
bending,
and
shear
stiffnesscoefficients
(Az,D.v,F.v)
only.
Whiletheimportance
of
consideringaconsistent
shear
coefficient
in
the
Journal
a/COMPOSITE
MATERIALS,
Vol.
27,
No.
8/1993
0021-9983/93/080806-24$6.00/0
(0
1993TechnomicPublishingCo..Inc.
1.
INTRODUCTION
A
DVANCED
MATERIALS.
MAINLY
fiber'reinforcedplastic
(FRP)
composites,
willpartiallyreplaceconventionalmaterialsincivilengineeringtypestruc-
tures(BarberoandGangaRao
[1]).
Mostrecentapplicationsintransportation
systenls,offshorestructures,chemicalfacilities
and
communicationsystems,
showtheusefulness
of
compositestructureslikethin-walled
beams
andcolumns.
COlllparedtostandard
constructionlnaterials,
conlpositenlaterialspresentnlany
advantages,e.g..
lightweight,
corrosion
resistance,andelectromagnetictrans-
parency.Mostprominentistheproperty
of
tailoringthematerialforeachpartic-
ularapplication.Structuralpropertieslikestiffness,strength,andbucklingre-
sistancedependonthematerialsystem(composite)andtheshape
of
the
cross-section
of
themember.Likewithsteelstructuralshapes,
it
ispossibleto
optinlizethesectiontoincreasethebendingstiffnesswithoutcompromisi.ngthe
nlaximunlbendingstrength.Unlikesteelshapes,withcompositebeams
it
ispos-
806
ABSTRACT:
Af()nnalengineeringapproach
of
themechanics
of
thin-walledlaminated
beanlsbasedonkinenlaticassumptionsconsistentwithTimoshenkobeamtheoryispre-
sented.Thin-walledcompositebeamswithopen
or
closedcrosssectionsubjectedtobend-
ingandaxialloadareconsidered.Avariationalformulationisemployedtoobtainacom-
prehensivedescription
of
thestructuralresponse.Beamstiffnesscoefficients
..
which
accountforthecrosssectiongeometryandforthematerialanisotropy.areobtained.An
explicitexpressionforthestaticshearcorrectionfactor
of
thin-walledcompositebeanlsis
derivedfromenergyequivalence.AnumericalexampleinvolvingalaminatedI-beam
is
usedtodemonstratethecapability
of
thenlodelforpredictingdisplacementsandply
stresses.
808
EVER
1.
BARBERO,ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
On
theMechanics
of
Thin-WalledLaminatedCompositeBeams
809
Consideringsymnlctricalbendingnormaltothe
x
axisresults
u(z)
=
O.
Assunlptioll
2.
Aplanesectionoriginallyl10rnlaltothebeamaxisremains
2.2KinematicAssumptions
FollowingTimoshenkobeamtheory.thebasicassumptionsregardingthepres-
entmechanics
of
thin-walledlalninatedbeanlsareintroduced.
ASsUlnption
1
..
The
contourdoesnotdefornl
in
itsownplane.
The
motions
u
and
v
alongthe
Si
and
ni
directionsrespectively,atapointonthemiddlesurface
of
the
ith
wall.canbeexpressedinterms
of
therigidbodynlotions
u(z)
and
v(z)
in
the
x
and
y
directionsrespectively(seeFigure2).
The
transverseloadsareappliedthroughtheshearcenterandarecontainedin
aplanenornlaltoone
of
theprincipalaxes
(x,y).
Underthisloadingconditionthe
beamissubjectedtosynlmetricalbendingdecoupledfromtorsion.
The
present
derivation
is
restrictedtosymmetricbendingforsake
of
brevitybutcouldbe
easilyextendedtonon-synlmetricbending.
The
joints
of
thecrosssectionare
nlodelledattheintersection
of
thewalls'nliddlesurfaces.Thisassumption,as
stated
by
Ng,Cheung,andBingzhang
[21],
is
amplyjustifiedduetothesmall
strainenergycontribution
of
thejoints
in
thin-walledhcanls.
V(Si,
z)
=-
u(z)
sin
cP;
+
v(z)
cos
cPi
(1)
(2)
dy
dS
i
=
sin
cPi
U(Z)
cos
<Pi
+
v(z)
sin
<Pi
U(Si,
z)
2.STRUCTURALMODEL
OF
THE
BEAMSUBJECTEDTOFLEXURE
Timoshenkobeamtheoryforanisotropicbeamswasrecognizedearly[4],acom-
prehensivetreatmentintheframework
of
aformalmechanicsapproachisnot
available.Cowper
[14]
derivedashearcoefficientforisotropicmaterialsfromthe
elasticitysolution
of
theclassicalSaintVenantflexureproblemunderthe
assumption
of
linearlyvarying
shear
force.Whileflexurefunctionsareavailable
forregularsections(Love[15]),inthecase
of
thin-walledsectionsthisapproach
requirestheevaluation
of
theshearstressdistributionfrom.mechanics
of
materialsmethods.DharmarajanandMcCutchen
[16]
extendedtheformulation
of
Cowperfororthotropicbeamswithoutaddressingthecase
of
thin-walledsec-
tions.Bank[17],applyingthesameequationsasthoseproposed
by
Dharmarajan
andMcCutchen[16],presentedaderivationbasedonthework
of
Cowper
[14]
for
thecase
of
thin-walledbeamsrestrictedtoassemblies
of
horizontalandvertical
orthotropicpanels.BankandMelehan
[18]
furtherextendedthefornlulationto
multicelledthin-walledsections.Bert
[19]
presentedaderivation
of
thestatic
shearfactorforbeams
of
nonhomogeneouscrosssection.Heconsideredrectan-
gularbeamswithlayersperpendiculartotheplane
of
bending.Tsaiet
ale
[20]
derivedashearcorrectionfactorforrectangularlaminatessubjectedtotorsion.
Inthepresentwork,theshearcorrectionfactorisobtainedfromenergyequiva-
lenceas
in
References
[19]
and
[20].
Thederivation
of
theshearfactor
is
based
oncomputingtheshearingstressdistributioninthecrosssection.
Theobjective
of
thisarticleistopresentthederivations
of
theMechanics
of
thin-walledLarninatedBeams(MLB)foropenandclosedcrosssections.A
variationaltormulation
is
employedtoobtainacomprehensivedescription
of
the
structuralresponse
of
compositebeamssubjectedtobendingandaxialload.The
examplepresented,involvingalaminatedI-beam,illustratesthecapability
of
the
modelforpredictingdisplacementsandplystresses,whileenvisioningthepoten-
tial
of
theapproachforthedesignoptimization
of
newstructuralshapes.
2.1
GeometryandLoadingDefinition
Astraightthin-walledcompositebeam-columnwithoneaxis
of
geometricand
materialsymmetrywillbeconsidered.
We
defineaCartesiancoordinatesystem
(x,y,z),
withthez-axisparalleltotheaxis
of
thebeamandone
of
theothertrans-
verseaxesorthogonaltotheplane
of
symmetry.
The
beam,made
of
assembled
flatwalls.couldhaveeitheranopen
or
closedcrosssection.Themiddlesurface
of
thebeamcrosssection
is
represented
by
apolygonallinecalledthecontour.
We
introduceforeachwallalocalcontourcoordinatesystem
(Si,fli,Z)
placedon
themiddlesurface
of
thewall,wheretheaxes
Si
and
n
i
aretangentandnornlal
tothecontourrespectively(seeFigure
1).
The
contour
is
definedparametrically
by
thestep-wiselinearfunctions
X(Si)
and
y(s;).
The
orientation
of
the
ith
wall
is
characterized
by
theangle
cPi
as
foHows
y
n
2
dx
dS
i
=
cos
cPi
x
Figure
1.
Cross-sectiongeometryandreferencesystems.
,,-
810
EVER
1.
BARBERO!ROBERTOLOPEZ-ANIDOANDJULIO
EDAVALOS
OntheMechanics
of
Thin-WalledLaminatedCompositeBeams
811
The
expressionsforthestiffnesssubmatrices
[A],[B]
and
[Dj
are
definedin
Jones
(23).
Byfullinversion
of
thestiffness
matrix,
Equation(4)resultsin
tionsforalaminatedwallwithrespecttothelocal
contour
coordinatesystem
depictedinFigure2
are
andthelalninatestrainsandcurvaturesare
(5)
(7)
(6)
(4)
(13)]
{I~l
1
(0)
IMI}
l.B.I]
{If}
1
[DJ
Ix)
J
{
I€")
1_
[[
ex
)
Ix)
J
-IJ3l
r
{
I~ll
=
[lAJ
IMI}
[B)
IfI
=
r;:]
Ix}
=
!;:
l
ly"
"
IN}
=
[~]
IMI
=
r
~:]
N"
lM"
wherethelaminateresultantforcesand
moments
are
x,
u(z)
.th
II
1
wa
y,
V(Z)
Figure
2.
Motionsandappliedloads.
L
plane.butnotnecessarilynormaltothebeamaxis
due
to
shear
defc)rnlation.
The
axialdisplacement
of
the
contour
canbeexpressedas
W(Sj.Z)
=
\rv(z)
-[y(s;)-
Yn]l/t.v(Z)
(3)
wherethecOlllpliancesubnlatrices
are
Tsai
[13]
employedtheelements
of
the
compliance
matricespresentedinthese
equationstodefinein-plane
and
flexuralengineeringconstants.Inthisworkthe
matrixEquation(7)forgenerallaminatesisreducedforthe
case
of
laminatesthat
are
components
of
thin-walledbeams.Consistentwith
beam
theory
and
based
on
AssumptionIwe
consider
thatforalaminatedwalltheresultantforceandmo-
mentoriginated
by
thetransverse
normal
stresses(inthe
Sj
direction)
are
negligi-
ble,then
[a)
=
[[AJ
-
[B][dHB]]-1
raj
=
(At
l
where
w(z)
representstheaxialdisplacement
of
thebeam
in
the
z
directionatthe
position
of
theneutralaxis
of
bending
-"n.
The
kinematicvariable
t/!y(z)
111easures
therotationintheplane
of
bending.Thisassumptioncouldbemodifiedtoac-
count
forresidualdisplacements
or
warping
of
thecrosssection
by
considering
additionalcorrectiveterms.
The
out-of-planewarpingcouldbeexpressedasa
seriesexpansioninterms
of
aset
of
orthogonalfunctionswhich
depend
uponthe
cross-sectionalgeonletryandthecomposite
lay~up,
andaset
of
newkinematic
variablesthataccountfortheloading
and
the
boundary
conditions.Inthissense,
Hjelmstad
[22
J
obtainedthewarpingfunctionsforisotropicmaterialsfromthe
exactsolution
of
theSaintVenanfsflexure
problem
throughtheapplication
of
the
Gram-Schnlidtorthogonalizationprocess.Bauchau[9]derivedfromenergyprin-
ciplestheeigenwarpingfunctionsforthe
case
of
curvilinear
orthotropic
Inaterials.
3.
ANALYSIS
OF
AFLATLAMINATEDWALL
AS
A
BEAMCOMPONENT
and
[oj
[13]
[[Dj
-[B][a][B]]-1
11311'
=-
laJ(B][ol
[e/)
=
[D)-I
-[d)[BJ[ex)
(8)
3.1
ConstitutiveEquations
EmployingClassicalLamination
Theory
(CLT),thegeneralconstitutiverela-
N.~=Ms=O
(9)
812
EVER
1.
BARBERO,
ROBERlO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
On
the
Mechanics
of
Thin-WaLLed
Laminated
Composite
Beams
813
Wu
and
Sun[II]showedthatforslenderthin-walledlaminated
beams
withoutribs
Equation(9)yieldsmoreaccuratenaturalfrequenciesthanthealternativeplane
strainassumption
(Es
=
)(sz
=
0).Forbendingwithouttorsionwecanfurther
statethat
Equation
(5)
weobtainthestiffnessmatrix
of
the
ith
wall
of
athin-walledlami-
natedbeamasfollows
M
sz
=
0
(10)
Ii:!
[
ii
Bi
o
Ii;
0
]1
€z
I
15i
0
)(z
o
"F..
:Yu
(16)
Incorporatingtheconditions(9)and
(10)
intoEquation(7)yieldsforthe
i
thwall
where
It
is
convenienttoderivethegoverningequationsforathin-walledlanlinated
beaOlfro
III
energyprinciples.
The
strainenergy
per
unitlengthconsideringthe
beaolasanassctllbly
of
n
wallsresults
in
[i(z)
=
~
t
r,'2
[(CXII),N~
+
2({JII),N,M,
+
2(cx,.),NR"
I-I
J-h;,2
(
all
)
15;
=
alloll
-
(3~1
;
F
i
=
(-t-),
isthebendingstiffness
is'the
shear
stiffness
isthebending-extensioncouplingstiffness
isthe
extensional
stiffness
(
-(311)
alloll
-
{3~1
;
(
011
)
altOl1-
13~1
i
Ai
Ii;
()
I)
a
16
]
[Nz
I
{316
M
z
(~66
i
N.~z
E
z
I
[all
{311
)(z
=
{311
b••
;Ysz
alt>
1316
+
(bl.)JVl~
+
2({316);M)Vsz
+
(a66)j\l~z]dsi
(12)
Foreachwalltheposition
of
themiddlesurface
is
defined
by
thefunction
Thereforethestrainenergy
per
unitlength[Equation
(12)]
expressed
in
terms
of
thewallstiffnesscoefficients[Equation
(16)]
simplifiesasfollows
Y(Si)
=
Si
sin
cP,.
+
Yi
for
hih;
-'2
:5
S,.
:5
2"
(
13)
II
Jh,l2
I
V(z)
=
~
E.
(Ai€~
+
2Bi
€,Ji,
+
15iJi~
+
Fi;Y:,)ds,
i=
I
-h;l2
(
17)
Theseconditionsaresatisfied
by
laminateswithoff-axispliesthatarebalanced
symmetric.HenceEquation
(11)
reducesto
where
hi
isthewallwidthand
Yi
istheposition
of
thewallcentroid(seeFigure
1).
We
observefrotntheexpression
(12)
thatthecoefficients
(XI6
and
{316
are
responsiblefortheshear-extensionandshear-bendingcouplingrespectively.
In
order
todecouplethevariationalproblem.andwithinthescope
of
theengineer-
ingapplications,werestrict
our
formulationtolatl1inatesthatsatisfy
3.2Strain-DisplacementRelations
The
wallstrains
are
derivedfromthekinematicrelations(2)and(3)
d~'
dz-
(y(s,)
-
YH)
d1/;,
dz
(18)
1/;,
)
sin
<Pi
(
dV
dz
ow
oz
~(Si'Z)
=
ail
ow
-(z)--
+-
'Ysz
S;,.
-
GZ
as,.
(14)
al.6
=
(316
-
0
The
motions
of
apointawayfromthemiddlesurface
of
thewallfollowsfrom
Assumption2.Consequentlythewallcurvatureresultsin
!
~z
I
[a
11
1311
)(z
=
1311
bl.
;Yn
00
cx~.H~:!
(15)
_
dt/;.v
d.x
dl/;.v
)(z(s;,z)
=-
-d
-d'
=-
-d'"
cos
cP;
z
.\,.
~.
(
19)
wherethecompliancecoefficient
1311
accountsforbending-extensioncoupling
duetounsymnletricorthotropiclayers.Byinvertingthecompliancenlatrixin
Equation
(19)
impliesthatflexuralstrains,whichvarylinearlyinthedirection
of
814
EVER
1.
BARBERO,
ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
On
the
Mechanics
of
Thin-Walled
Laminated
Composite
Beams
815
the
wall
thickness,willbegenerated
in
addition
to
thetypicailypredominant
membranestrains[Equation
(18)].
Thus,thiskinematicmodelcanbeapplied
eithertothin
or
thicklaminatedwalls.Usingthestrain-displacementrelation-
ships
(18)
and
(19),
andtheparametricdefinition
of
thecontour[Equation
(13)],
thestrainenergyperunitlength[Equation
(17)]
becomes
4.DERIVATION
OF
THE
BEAM
GOVERNING
EQUATIONS
4.1
BeamStiffnessCoefficients
ThetotalpotentialenergytobeminimizedfollowsfromEquation
(20),
with
theintroduction
of
beanlstiffnesscoefficients,yielding
U(Z)
=
~
thi
{A;
(tlw)
2
1=1
tlz
-
dl-t'd1/;y
2[A.(v.-
VII)
+
B;
COS
c/>;l
-d
7
-:-,
..
'
."
.
~
"
...,
!
L
I
dw
2
dM-'
d1/;y
d1/;y
2
n
="2
JA'(dZ)
-
2B
y
dzdz
+
DY(dZ)
[
-(
bf)
-
-]
+
Ai
(:Vi
-
y,,)2
+
12
sin2
c/>;
+
2B;(
Yi
-
Yn)
cos
c/>i
+
Di
cos2
c/>i
3.3
Ply
StrainsandStresses
Theaxialandshearstrainsevaluatedthroughthethickness
of
the
i
th
wall
result
in
where
L
is
thelength
of
thebeam,and
qy
and
qz
aretheappliedtransverseand
axialloadsrespectively(depicted
in
Figure
2).
Thebeamstiffnesscoefficientsare
defined
by
+
KyFy
(~~
-
1/;y)
2]dZ
-
!~
(qyv
+
q.w)dz
(23)
EA;b;
A
z
(20)
(
dl/;y)
2
_.
(til,'
)2}
x
dz.
+
[F;
~ln2
c/>,.J
dz
-
l/;y
Ez(S;~~~Z)
=
Ez(S;~Z)
+
~j(z(s;,z)
i=1
(21
)
1'...z(S;~Z)
=
;.ysz(s;~z)
1/
By
='
E
[A;(
y;
-
Yn)
+
Ii,.
cos
c/>;]b;
i=l
Theshearcorrectionfactor
K
y
is
introduced
in
order
to
accountfortheactual
shearstressdistribution
in
thecrosssection.Anexpressionfor
Ky
basedonen-
ergyequivalence
is
derived
in
thisarticle.Theset
of
equationsobtainedforthe
beamstiffnesscoefficients[Equation
(24)]~
reducesforthecase
c/>;
=
0
to
the
parallelaxistheorempresented
by
Tsai
andHahn
[24].
Areduction
to
apure
membrane
case~
wheretheflexuralstrains
in
the
wall
arenegligible,
is
obtained
by
setting
Ii,.
=
15;
=
O.
where
~
is
thethicknesscoordinate(inthe
11,.
direction).Althoughthelaminae
in
alanlinated
wall
areconstrainedandinteractwithone
another~
in
order
to
obtain
anapproximation,fortheplystressesandfollowingEquation
(9)~
we
further
assumethatthetransversenormalstressesareinsignificant
as
~
O.
Thiscondi-
tionyieldsthefollowingexpressionsfortheaxialandshearstresses
in
the
kth
layer.
fa.}
[QII
An}
QI6
Ez
(22)
asz
=
QI6
Q66
'Y.~z
where
A-
Q;2Qlj
for
i.)
=
1.6
Q;}
=
Q;J
-
--0-
22
and
Q;}
arethetransformedreducedstiffnesscoefficientsemployed
in
CLT
(Jones
[23
D.
Fortheparticularcaseofalayerwithfibersoriented
in
thedirection
of
thebeamaxis.thenlodifiedstiffness
co~fficients
of
Equation
(22)
reduce
to
the
correspondinglanlinaelasticconstants:
Ql1
=
£1.
066
=
G12~
and
016
=
O.
~
[_
(
bf)
Dy
=
~
Ai
(Yi
-
Yo)2
+
12
sin
2
t/Ji
+
2Bi
(Yi
-
Yo)
COS
t/Ji
+
15i
COS
2</>i]
hi
"
Fy
=
EF;b;
sin
2
c/>;
i=l
(24)
816
EVER
1.
BARBERO,
ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
OntheMechanics
of
Thin-WalledlAminatedCompositeBeams
817
4.2EquilibriumEquations
Wedefinetheposition
of
theneutralaxis
of
bending
of
the
cross
section
by
set-
ting
By
=
0,whichyields
4.3ConstitutiveEquations
ExpressingthetotalpotentialenergyEquation(26)in
terms
of
thewallstress
resultantsleadstothefollowingdefinitionsforthe
beam
resultantforces
and
mo-
ments
II
E
(y;A;
+
cos
</>J{)b;
v
=
~
n
;=1
Az
(25)
Nz(z)
"f
h;l2
~
j
-b;12
N,ds
i
Introducingthecoordinate
y'
=
y-
Yn,
we
are
ableto
decouple
theextensional
and
bendingresponsesinEquation
(23),
asfollows.
This
is
done
tosimplifytheformulationalongthelines
of
classicalstructural
analysis
as
usedbythemajority
of
structuralengineers.
The
Timoshenko
beam
solutionisobtained
by
minimizingthetotalpotentialenergy[Equation(26)]with
respecttothefunctions
w,
v,
l/;yo
Integratingby
parts
and
applyingthefundamen-
tal
lemma
of
calculus
of
variationsweobtainthe
equilibrium
equations
(31
)
(30)
(29)
dl/;y
dz
dw
€~(z)
=
dz
}(y(Z)
=
Nz(z)
=
Az€~
dv
l'yz(z)
=
dz-
l/;y
II
f
",/2
~
j
-bJ2
[N,y'
(s,)
+
M,
COS
t/>,]ds,
"r
h;l2
V,(z)
=
K,t
j
-b;/2
N.,
sin
t/>,ds,
My(z)
My(z)
=
Dyx
y
Vy(z)
=
Ky
Fy
l'yz
For
the
ith
wall,thestrains[Equation
(18)]
and
the
curvature
[Equation
(19)]
in
terms
of
the
beam
resultantforces
and
moments
become
Thereforethe
beam
constitutive
equations
can
be
expressed
as
and
forthegeneralized
beam
strains
(26)
(27)
~
[K,F,
(~;
-
1/;,)]
+
q,
=
0
d
(dl/;y)
(dV)
dzDy
dZ
+
KyFy
dz-
l/;y
=
0
d(dW)
dz
Az
dz
+
qz
=
0
-
JL
(q,v
+
q,w)dz
o
1
fL
[
(dW)
2
(dl/;y)
2
(dV
)2]
n
=
"2
j
0
A,
dz
+
D,
dZ
+
K,F,
dz
-
1/;,
dz
and
the
boundary
conditions
of
thesystem
dw
Az-owlk
=
0
dz
_Nz
,
My
€z(s;,z)
=
-A
+
Y
(05;)
D
zy
D
dl/;y
~.I,
IL-
0
Y
dz
U'l/y
0-
(28)
_
My
x
z(s;,z)
=D
cos
cP;
y
(32)
K,F,
(~;
-
1/;,
)
ov
I
~
=
0
v
;Y.n;(Si,Z)
=
K
~
sin
<Pi
yy
818
EVER
1.
BARBERO,ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
OntheMechanics
of
Thin-WalledLaminatedCompositeBeams
819
Giventhebendingandaxialstressresultantsforacertainaxialposition
z
of
the
beanl,theneutralaxis
of
combinedbendingandaxialforce,i.e.,theaxisfor
which
€:
is
zero,followsfromEquation
(32)
Foracrosssectionhavingafreeedge,
or
havingsomeotherpointsuchthat
N.~(
-b./2,z)
=
0,wecanintegrateoverthecontouruptothe
rth
wall,yielding
Y~(Z)
=
Yn
Dy
Nz(z)
A
z
My(z)
(33)
-*
..,.
__
V
y
[y
!-"(
2_
b~)
N.~l.(.5n4.)
-
D
y
S
r
+
2
A
r
SIn
$r
Sr
4
--(br
)]
+
(Ary:
+
Br
cos
$r)
Sr
+
2
(38)
,.-1
S~
=
E
[A;y:
+
Ii;
cos
$;]b;
;=1
5.
EVALUATION
OF
SHEARSTRAINEFFECTS
5.1
ShearingStressDistribution
The
shearingstressdistribution(shearflow)inthecrosssection
of
thethin-
walledbeamisobtainedhereinfromequilibriumineachwall,interms
of
theax-
ialstressresultant
Nl.
.
Thustheshear
flow
evaluatedineachwall
(f£~)
consti-
tutesarefinementoverthelaminateshearstressresultant
(N.~l.)
calculatedfrom
constitutiveEquations
(16).
From
Equations
(16)
and
(32)
follows
where
for
hr
br
-2
~
Sr
~
"2
The
in-planeequilibriumequationforthe
ith
laminatedwall,intheabsence
of
bodyforces,inthe
z
directionis
-N
l.
-
My
-,-
Nz(Si,Z)
=
-A
Ai
+
D
[A,)'
(s;)
+
Bi
cos
$;]
zy
N
l.
.l.
+
Ns~.s;
=
0
(34)
(35)
is
theweightedstaticmoment
of
theportion
of
thecrosssectioncorresponding
tothefirst
r-
I
walls,and
y:
=
Yi
-
Yn.
However,forageneralclosedcross
sectiontheshearingstressdistributioncannotbeobtainedemployingEquation
(38)
alone,sincewe
do
notknow
apriori
where
N'!z
vanishes.
The
procedurefor
aone-cellcrosssectionistogeneratefreeedges
by
introducingaslitinthecom-
partment,andthencloseitagainbyobtainingtheshearflowinthecompartn1ent
thatproduceszerounitangle
of
twist.ApplyingBredt'sformulaforthin-walled
hollowbeams(CookandYoung
[25]),
wecanwriteforacompartmentcomposed
of
n'
walls
Furthermore,inbeamtheorythefollowingresultantequilibriumequationsare
employed
II'
I
J";I"1-
EFi
N"f,(Si.Z)dsi
=
0
1=
1
-b;/2
(39)
dM
y
=
Vv
dz
.
(36)
dN
z
=
0
dz
The
netshear
flow
is
N.~(Si'Z)
=
[N~(si,z)l'JX'n
+
N~l.
where
[N.~l'JX'n
is
thevari-
ableopen-cellshearflowobtainedfromEquation
(38),
and
N~z
istheuniform
closed-cellshearflowreleased
by
theslit.Formulticellcrosssectionstheabove
procedurehastoberepeated
by
satisfyingEquation
(39)
foreachcompartment.
Therefore,thecorrespondingshearingstraindistributioninthe
i
thwallis
SubstitutingEquation
(34)
inthewallequilibriumEquation
(35),
andaccounting
forthebeamequilibriumconditions
(36),
weobtaintheshearflowvariation
in
the
ith
wall
I-
:V*(l'
~)
--
N*
,n
Ji,~
-
F
i
n
(40)
N'fz,S,(Si,Z)
.
~
Vv
--
-D
[Aiy'(Si)
+
Bi
cos
$i]
y
(37)
5.2Location
of
theShearCenter
Thelocation
of
theshearcenter
S(x.~,y'~)
isdefinedin
order
todecouplebending
andtorsion.Foracrosssectionhavingoneaxis
of
symmetry,one
of
thecoor-
820
EVER
1.
BARBERO,ROBERTO
LOPEZ-ANIDO
ANDJULIO
F.
DAVALOS
OntheMechanics
of
Thin-WalledLaminatedCompositeBeams
821
(41)
dinates
of
theshearcenterisknown.Hencetheothercoordinateisobtainedfrom
thefollowingmomentequilibriumequation
n
!b;/2
1:
N;t:(Si,Z)[y(Si)
-
y,]
cos
q,ids
i
==
0
1=
I
-b;/2
"
!b
i/2
1:
N;':(Si,Z)[X(Si)
-
x,]
sin
q,ids
i
1=
I
-hi
/2
(47)
(46)
[t
hi
sin
ep,(S':
+
cr)]2
F
~
hi
.
2
y
I..J
Fi
sIn
cPi
[(Sn2
+
2crSr
+
drj
1=1
K
y
wherethestiffnessparameters
of
the
ith
wall,
cr
and
dr,
aredefinedasfollows
cr
==
~
hi[
;r.(:w
-
i
hi
sin
q,i)
+
B
i
cos
q,i]
l[
-
(bt
b
i
)
dr
=
"3
b~
(A
i
)2
40
sin2
cPa
-
4
Y:
Sfn
<P;
+
<y:)2
--(b
i
)-]
+
2A;B;
cos
<P;
y:
-
8
sin
<Pi
+
(B
i)2
COS2
<Pi
Anexplicitexpressionfor
Ky
isobtained
by
replacingtheexpressionfortheshear
flowEquation(38)
in
Equation(45)andperformingtheintegrals,yielding
(42)
n
Jb;/2
1_1
"2
1:
N;t:(Si,zfy1;(s"z)
sin2
q,ids
i
==
"2
Vy(z)-yy,(z)
1=
I
-b;l2
5.3Derivation
of
theShearCorrectionFactor
Astaticshearcorrectionfactorisintroduced
by
equatingtheshearstrainen-
ergypredicted
by
thepresentTimoshenkobeamtheory,andtheshearstrainen-
ergyobtainedfromtheshearingstressdistributioninthecrosssection
Aftersubstitution
of
theexpressionsfor
'¥yz
fromconstitutiveEquations(31),and
for
;y;,:
obtainedfromequilibriuminEquation
(40),
weobtain
The
variation
of
K.y
withrespecttothegeometricdimensionsforthecase
of
an
I-beamcomposed
of
homogeneouswalls
is'
shown
in
Figure
3.
bJ/b~
=
bl/b~
=
1/4
/
~~~.~~~~.~
-
-
~
~
-:
~:
-
~
~
:-
:
-
~
-:.

'1
""
:,I
",
:
,
",
l-
/..
..:.;;
"..
.-
L
•••1.-
••
j
f
,,'
/~!:h
:
.,
,"~
1
:'/I1
.,
1I
r/I'
i
~.-
...
··:·1~2
.....
?~(~~
...~
..
_~(~
..
_.
:'I
II
.,
II
:
'
"
,
••
II
,/
I
----b~'
"II
:,
:1:
hl/ba
=
0
[.~.~.~.~?~~il:f~~g~;.~~-~~~~~~._'
0.fJ5
0.85
~}
0.9
(44)
(43)
"f
b;/2
Vy(z)
==
~
J
-h;l2
N;t:
sin
q"ds
i
I"
fb;/2
1_11
"2
~
J-h;12
Fi
(N;t:(Si'Z)
sin
q,Ydsi
==
"2
K"Fy
(Vy(Z»2
IntroducingthisexpressioninEquation(43),wearriveat
wherethebeamshearforceresultant
(V
y
)
canbeexpressedas
43
2
h./h
2
0.80
I
J
o
Figure
3.
Variation
of
theshearcorrectionfactorwithrespecttothegeometricdimensions
foranhomogeneousJsectionwith
a
webheighttothicknessratio
b
2
/h
2
=
16.
(45)
[
"(h;/2
)]2
1:
sin
q,i
r.
N1;ds
i
1=1
J
-h;/2
"J
b
;/2
Fy
1:
~i
sin2
q,i
(N;t:)2ds
i
1=
I
-h;/2
K
y
822
EVER
1.
BARBERO,ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
0.15
y,v
j
B
=
H/2
~
:..
..:
0.03
c
,
0.12
~...
...........,......j.....pi"<da
I"
.;.
•••
..Iit'.
.......
0.09
t
~
i:tt.'.~
~
.
C
I"I
o
I,I
;:;

tt'
:...
MLB
C,)
~
,
I
~-
0.06
••
Cl
••
~.a/~.
.
..
;
I:J.~NS~~.~~
~
I
(1)
"'0
~
E=
'3
'>
I
H
=1
in
x,u
..
T
w
=
H/16
y,v
L
:4
~~
J.
T
,
,
,
,
-~
,
._._._._._._._._._._._._._._._._._._._._._._._._._._.-._.
,
"
Py
,
~
,
Figure
4.
CantileverI-beamsubjectedtotip-shearforce.
o
o
30
60
90
Ply
angle
e
[deg]
6.
NUMERICALEXAMPLE
Aclamped-freeI-beam
of
length
L
subjectedtoatip-shearforce
p.l'
=-
100
lbsisanalyzed(seeFigure
4).
The
dimensions
of
thebeamaredefinedrelative
tothebeanlheight
H
=
1
in.
The
materialemployedisaralnid(Kevlar49)-
epoxywiththefollowingelasticconstants,
£1
=
11.02
X
106
psi(76.0GPa),
£2
=
0.80
X
106
psi(5.50GPa),
G
12
=
0.33
X
106
psi
(2.30
GPa),and
\'11
=
0.34.
The
lay-upsequence
is
[
±
O/O]s,
wheretheplyangle
0
isselectedas
thedesignvariable.
The
tipdeflectionisobtainedfromthesolution
of
thegovern-
ingEquation
(27)
forthespecifiedboundaryconditions.
The
tipdeflectionisevaluatedapplying
MLB
fortwodifferentaspectratios:
L/H
=
6(Figure5).and
L/H
=
12
(Figure6).
The
resultsarecOlnparedwith
thevaluesobtainedfromarefinedFiniteElement
(FE)
analysiswithANSYS[26]
enlploying8-nodeisoparametriclaminatedshellelements.
The
minimunltip
deflectiontor
L/H
=
6isobtainedfor
0
=
16
0

andfor
L/H
=
12
isexhibited
t()r
0
=
9
0

Theratiobetweenthetipsheardeflection
(\'s(L»
andthetipbending
deflection
(vb(L»
isdepictedinFigure
7.
The
resultsobtainedwith
MLB
(Figure
7)provideinsightintothedeflectioncomponents(bendingandshear).thatisnot
availablefromthe
FE
solution.Plystressesarecomputedbasedontheplystrains
obtainedfromEquation
(21)
followingtwodifferentapproaches.
The
firstap-
proachconsidersthestiffnesscoefficients
Qi).
and
thesecondapproachemploys
themodifiedstiffnesscoefficients
Qi}
introducedinEquation(22).Plyaxial
stresses
in
thetopflangecalculatedatadistance
z
=
L/48
of
thefixedendare
presented
in
Figures8and
9.
Averageply
shear
stressesinthewebevaluatedat
Figure
6.
Variation
of
thetipdeflectionwithrespecttothe
ply
anglefor
a
cantileverI-beam
withaspantoheightratio
L/H
=
12.
90
60
30
o',
o
Ply
angle
a
[deg]
0.2
/-
------.-,-.
..-
--
,-
-
~
0.8
~
....-
....
__
.J
.....:.:.:;~
~.~~
~~:;
0;;;.....~~
"U
"I
0.6
~..
J./.~
!.....
0.4
~.....
,"
;
•••.
MLB
•.
.....
14'
..L
....
:
D
'.'0'"
..
......,
~NSYS
FE
.
.
...
..
..
.
..
~
c
o
c
...,..;
C,)
(l)
~
(l)
"0
0..
E=
-J
'>
I
Figure
5.
Variation
of
thetipdeflectionwithrespecttothe
ply
anglefor
a
cantileverI-beam
withaspantoheightratio
L/H
=
6.
(48)
P
L
J
PyL
v(L)
=
vb(L)
+
v.~(L)
=
3~.v
+
K.l'F.v
823
Ply
angle
e
[deg]
Figure
7.
Variation
of
theratiobetweenthetipsheardeflectionandthetipbendingdeflec-
tionwithrespecttothe
ply
anglefor
a
cantileverI-beam.
90
r'
----
__r---------.-------.
90
(:I
60
30
-10000
'
I
o
'[,30000
N
b
20000
CJ)
~
fJ)
CJ)
~
10000
~
...,>
if)
(lj
~
0
<t:
o
I
~~----~--;,
:
.....
~.~.~
.,
~
-
~\.
_~.
_
-
.
,
,..
,
-----
',l
\:
Qij
,LJ
..
t
'I
\I
••••
......
--
\\~.-
~\
~
..
QiJ
.
"
..
t
"\I
0
I
""
0
..
:
AN
SYS
FE
"
\.1
,,
..'
_.
__",
J.
_
..
"',~
~.
"<
"'d
~-_.'::~...--
40000
dJ
c
0
tP
Stresses
at:
+e
layer
fac~
farthest
from
the
f:lange
middle:
surface
Ply
angle
e
[deg]
Figure
9.
Variation
of
plyaxialstresses
at
a
+
0layerinthetopflangewithrespecttothe
ply
angle,measured
at
z
=
U48,
for
a
cantileverI-beamwith
L/H
=
12.
90
1
••
:
L/H
=
12
:
L/H
=
6
60
30
,
,I
..•.
\:.•.
""1
,
....•,
.•
,,'
....••....
-
.,.
..
~,
"
"-""...
":
I'
,,"~
I,'
o
I
'
..·_·r:::.:::::.:j:::~~_·
..
..
..
-
o
J
30
60
.........
.....J
-'
en
;>
'3
~
;> 100000
2000
,
i
90
60
30
500
~-
.....
Ply
angle
e
[deg]
t
1.0
\I:
I
II,
1500
.\
'"
.
J
.•...
J..
.
..
/._.
\
:
-----
:
,I
1\
I
I
\I
\:tij
I
I
~
:
......:
/
1000
~"
Qij
0
1
/-
\ANSYSFE
i
"
/
)g
t
I
r:f
····,····~····-····-·-····-·-···-·~---··r-···-···-·-··
,,~
~
,
---0---8'-""I
II
Stresses
at:
a
layer
face;
farthest
o
I
from
the
'feb
middle
syrface
I
o
en
~
N
VI
b
CJ)
~
if)
CJ)
~
~
.....,J
if)
~
co
~
~
en
90
60
30
Ply
angle
e
[deg]
o
a
N
b
.~
80000
~···~·/;t!f"'~t~
...
..r:I...:....
1!J
~
:
1/
I
60000
-.
-.--
---.-.--
..
~.;..
.
--~
..
--
..
-.-
..
"-"-'
CJ)
/J
~
~:
I_
CJ)
,
I,
Q
..
~
40000
-.-
-.....-..
.
;af
...
--
~
...-
..
-.
--
-.....
~.
'.'~~.'.'.-.'
....-...
.....,J
n./'
,
A
CJ)
..
----:
:
\:tij
I
0
.~
::
ANSYS
FE
><
20000
--1----
..
-
.....
~
..
-.-
.....-.....
-
....
--.
<
II
Stresses
at,
a
layer
face,
farthest
from
the
f;lange
middle:
surface
Figure
8.
Variation
of
ply
axialstresses
at
a
0
layer
in
thetopflangewithrespecttotheply
angle,measured
at
z
=
U48.
for
a
cantileverI-beamwith
L/H
=
12.
Figure10.Variation
of
plyaverageshearstresses
at
a
0
layer
in
thewebwithrespecttothe
ply
angle,measured
at
z
=
U2,for
a
cantileverI-beamwith
L/H
=
12.
824
825
826
EVER
1.
BARBERO,ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
On
theMechanics
of
n,ill-Walled'LaminatedCompositeBeams
8Tl
~3000
7.CONCLUSION
Ply
angle
e
[deg]
Figure
11.
Variation
of
ply
averageshearstresses
at
a
+
()
layerinthewebwithrespectto
the
ply
angle,measured
atz
=
U2,for
a
cantileverI-beamwith
L/H
=
12.
adistance
Z
=
L/2
areshowninFigures
10
and
11.
The
axialand
shear
stresses
atthe

layer(Figures8and
10)
obtained
by
bothapproachescoincidewiththe
FE
results.Atthe
+0
layer(Figures9and
11),
thestress.computationwiththe
OiJ
coefficientsfollowsapproximatelythetrend
of
the
FE
solution.
The
differenceobservedisduetothelimitation
of
abeamtheorytonl0delthefully
anisotropicresponseatthelaminalevel.Nevertheless,wenoticethatthepro-
posedapproachemployingthe
Qi}
coefficientsyieldsabetterapproximationto
the
FE
solutionthantheclassicalapproachwiththe
Q;J
coefficients.
.V/
'
[A),[B).[D]
Ai

iii
,
D,.

F;
warpingeffectsinthedesign
of
innovativecompositestructuralshapes.Warping
wasnotincluded
in
thisworktolimittheconlplexityoriginated
by
additional
kinematicvariables.
8.
NOTATION
laminatestiffnesssubmatrices
stiffnesscoefficients
of
the
ith
wall
of
athin-walled
laminatedbeam
Az

B.v

D.v
,
F..,
=
stiffnesscoefficients
of
athin-walledlaminatedbeam
[aJ,[dJ
=
symmetriclaminateconlpliancesubmatrices
h;
=
width
of
theithwall
E..E2
.GJ2
,
~'12
=
laminaelasticconstants
K.v
=
shear
correctionfactor
L
=
length
of
thebeam
tNt
=
tNz.f£,N.nl
=
laminateresultantforces
IMt
=
(M%'M.nlVl.~zJ
=
laminateresultantmoments
Nz(z).
M.v(z),
~v<z)
=
beamresultantforcesandmoment
N.~
=
shearingstressdistribution(shearflow)
N~z
=
uniformclosed-cell
shear
flow
n
=
nunlber
of
wallsinthecrosssection
11'
=
number
of
wallsinaclosedcell
Qij
=
transformedreducedstiffnesscoefficients
o.il
=
modifiedstiffnesscoefficientsforlaminatedbeams
qy,
qz
=
transverseandaxialappliedloads
S(x.~
•yJ
=
location
of
the
shear
center
Sr
=
weightedstaticmoment
(s;
.11;.
z)
=
local
contour
coordinatesystemforthe
ith
wall
U(z)
=
strainenergy
per
unitlength
u(z).v(z)
=
rigidbodymotions
in
the
x,y
directions
w(z)
=
motion
of
theneutralaxisinthe
z
direction
u(s;.
z),
v(s;.
z).
W(Si,
z)
=
motions
of
the
contour
in
the
(si,n;,Z)
directions
(x.y

.::)
=
Cartesiancoordinatesystem
x(s;)
•.\'(s,-}
=
parametricfunctionsthatdefinethecontour
y'
=
coordinatewithrespecttotheneutralaxis
y"
=
coordinate
of
theneutralaxis
of
bending
y:r
=
coordinate
of
theneutralaxis
of
combinedbending
and
axialforce
5\
=
centroidcoordinate
of
the
i
thwall
centroidcoordinate
of
the
ith
wallwithrespecttothe
neutralaxis
(aJ.[IJJ.(oJ
=
laminateconlpliancesubmatrices
IfI
=
If
z•fso
;Y.~zl
=
lanlinatestrains
(xl
=
p(z.x.nx~zl
=
laminatecurvatures
€~(.::),
}(
.•.(z).l'yz(z)
=
beamstrainsandcurvature
;Y.~
=
shearingstraindistributionfromequilibrium
90
o
if)
~
II
_----;-------7
.......
-
/I
-.-••
I'...
,
-
,
(.,
:Q
ij
:
..
b';,
2000
r--/'\1-Q~~~~~~-'~,
...
c!r"
",
lJ
I

,I,
',0
I
\
ANSYS
FE
::
"I
I'
I
I'I
I
I
"M
'
I
1000
~""..
.".Q
..
"'~
...
,....
,..
"
..
~
..
-.
...
/.
"""'1
o
I
',0
I
~'
0
',_
I"
-
..
,.
...
In
Stresses
at
+9
layer
face
farthest
fron1
the
wet>
middle
surf,ace
A'
c
o
30
60
if)
<l,)
if)
if)
<l,)
~
.->
if)
~
ro
(l)
..c
if)
The
mechanics
of
laminated
beams
presentedhereinintendstobridgeagapbe-
tweensophisticatedexistingmodelsandtherequirement
of
asimplebutconsis-
tenttooltorengineeringdesign.
For
theexamplepresented.theperformance
of
theproposedbeammodel
compared
satisfactorilywithshellfiniteelements.In
particular.theprediction
of
deflections.whichtypicallycontrolthedesignin
manycivilengineeringapplications,isremarkablyaccurate.Thisapproach
allow'sthedesignertooptimizeboththecross-sectiongeometryandthematerial
systemforagivenobjectivefunction.
The
presentformulationcouldbeim-
plementedasaspecializedbeamfiniteelement,providingapreprocessortocom-
putethebeatnstiffnesscoefficients,andapostprocessortocalculateplystresses.
Furtherresearchisadvisablein
ordet
toevaluatetheimportance
of
including
828
EVER
1.
BARBERO,ROBERTO
LOPEZ-ANIDO
AND
JULIO
F.
DAVALOS
OntheMechanics
of
Thin-WalledLaminated.CompositeBeams
829
~
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f
z
,
'Ysz
az,asz
fI
~
4>;
ifiY{z)
plystrains
plystresses
totalpotentialenergy
thicknesscoordinate
anglebetweenthe
x
axisandthe
Sj
axis
rotation
of
thecrosssection
ACKNOWLEDGEMENTS
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