Table Of ContentDOI: 10.5772/intechopen.81165
Provisionalchapter
Chapter 1
Introductory Chapter: An Introduction to the
Introductory Chapter: An Introduction to the
Optimization of Composite Structures
Optimization of Composite Structures
KKaarraamm MMaaaallaawwii
1.Introduction
Structural applications of composite materials are increasing in several engineering areas
wherehighstiffnessandstrength-to-weightratios,longfatiguelife,superiorthermalproper-
ties, and corrosive resistance are most beneficial [1–4]. Common types include laminated
composites[5],functionallygradedmaterial(FGM)structures,andnanocompositesaswellas
smart composite structures [6]. In fact composite structures are usually tailored, depending
uponthespecificobjectives,bychoosingtheindividualconstituentmaterialsandtheirvolume
fractions,fiberorientationangles,andlaminasthicknessandnumber,aswellasthefabrication
procedure.Toattainthebestresults,adequateoptimizationmodelshavetobeimplementedto
findpracticaloptimalsolutionssatisfyingagivensetofdesignconstraints.
Thisintroductorychapterprovidesabriefreviewontheoptimumdesignofcompositestruc-
turesandtherelevantoptimizationtechniquesthatarecapableoffindingtheneededoptimal
solutions. Several problems can be addressed, including the structural design for maximum
stability,maximumnaturalfrequencies,andminimummassormaximumstiffnesssubjectto
limits on strength, deflections, and side constraints. The relevant design variables include
geometricaldimensionsandmaterialpropertiesaswell.Anumericalexampleisgivenatthe
endofthischaptertodemonstratearealandpracticalapplicationoftheoptimumcomposite
structures.
2.Theoptimaldesignproblem
Several research papers and text books exist in the field of optimal design of composite
structures with a variety of valuable applications in civil, mechanical, ocean, and aerospace
engineering. An important stage has now been reached at which an investigation of such
2 Optimum Composite Structures
developments and their practical possibilities should be made and presented. Two distinct
review papers have been published covering the development of the optimum design of
compositesovermorethan40years.ThefirstpaperbySonmez[7]presentedacomprehensive
surveyformorethan1000journalpapers,conferencepapers,textbooks,andweblinksfrom
the year 1969 to 2009. Sonmez classified the papers according to the type of the composite
structure, loading conditions, optimization model, failure criteria, and the utilized search
algorithm.ThesecondpaperbyGanguli[8]coveredahistoricalreviewfrom1973to2013.It
providesthegrowthofthefieldbyincludingmorethan90referencesdealingwithavarietyof
optimization methods utilized for tailoring composites to achieve certain design objectives.
Applications of several optimization techniques were presented, including feasible direction
methods, sequential quadratic programming, and stochastic optimization such as particle
swarmandantcolonyalgorithms.Ganguliclassifiedthepublishedworkintofivecategories
named pioneering research for the work published in the 1970s, early research in the 1980s,
movingtowarddesigninthe1990s,thenewcenturyinthe2010s,andthecurrentresearchfor
paperspublishedafter2010.
In general, design optimization seeks the best values of design variables, X , to achieve,
nx1
withincertainconstraints,G (X)placedonthesystembehavior,allowablestresses,geome-
mx1
try, or other factors; its goal of optimality is defined by the a vector of objective functions,
F (X), for specified environmental conditions. Mathematically, design optimization may be
kx1
castinthefollowingstandardform[9]:
FindthesetofdesignvariablesX thatwill
nx1
Xk
minimize FðXÞ¼ w FðXÞ (1)
fi i
i¼1
subjectto GðXÞ≤0,j¼1,2,…I (2)
j
GðXÞ¼0,j¼Iþ1,Iþ2,…m (3)
j
wherewfiistheweightingfactorsmeasuringtherelativeimportance ofFi(x) withrespectto
theoveralldesigngoal:
0≤w ≤1
fi
Xk (4)
w ¼1
fi
i¼1
Figure1showstheoverallstructureofanoptimizationapproachtodesign.Majorobjectivesin
mechanicalandstructuralengineeringinvolveminimumfabricationcost,maximumproduct
reliability, maximum stiffness/weight ratio, minimum aerodynamic drag, maximum natural
frequencies, maximum critical shaft speeds, etc. Design variables describe configuration,
dimensionsandsizesofelements,andmaterialpropertiesaswell.Inthedesignofstructural
components,suchasthoseofanautomobilestructure,themaindesignvariablesrepresentthe
thickness of the covering skin panels and the spacing, size, and shape of the transverse and
Introductory Chapter: An Introduction to the Optimization of Composite Structures 3
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Figure1. Designoptimizationprocess.
longitudinal stiffeners. The sizes of the constituent elements of the system are measured by
such properties as the cross-sectional dimensions, section areas, area moments of inertia,
torsional constants, plate’s thicknesses, etc. If the skin and/or stiffeners are made of layered
composites,theorientationofthefibersandtheirproportioncanbecomeadditionalvariables.
Ifoneoptimizesforconfiguration,thedesignvariableswillincludespatialcoordinates.Also,
in dynamic problems, the location of nonstructural masses and their magnitudes can be
additionaldesignvariables.
3.Optimizationtechniques
TheclassofoptimizationproblemsdescribedbyEqs.(1)–(3)maybethoughtofasasearchin
ann-dimensionalspaceforapointcorrespondingtotheminimumvalueoftheoverallobjec-
tivefunctionandsuchthatitlieswithintheregionboundedbythesubspacesrepresentingthe
constraintfunctions.Iterativetechniquesareusuallyusedforsolvingsuchoptimizationprob-
lemsinwhichaseriesofdirecteddesignchanges(moves)aremadebetweensuccessivepoints
in the design space. Several optimization techniques are classified according to the way of
4 Optimum Composite Structures
selectingthesearchdirection[9].Themostcommonlyusedapproachesaretherandomsearch,
conjugate directions, and conjugate gradients methods. Other algorithms for solving global
optimizationproblemsmaybeclassifiedintoheuristicmethodsthatfindtheglobaloptimum
onlywithhighprobabilityandmethodsthatguaranteetofindaglobaloptimumwithsome
accuracy.Thesimulatedannealingtechniqueandthegeneticalgorithms(GAs)belongtothe
former type, where analogies to physics and biology to approach the global optimum are
utilized.Thesimulatedannealingtechniqueisaniterativesearchmethodbasedonthesimu-
lationofthermalannealingofcriticallyheatedsolids.Hasancebietal.[10]appliedittofindthe
optimumdesignoffibercompositestructuresasanefficientmethodtosolvemulti-objective
optimizationmodels.Ontheotherhand,theGAs[11,12]arebasedontheprinciplesofnatural
genetics and natural selection. GAs do not utilize any gradient information during the
searchingprocess.NarayanaNaiketal.[12]usedGAandvariousfailuremechanismsbased
ondifferentfailurecriteriatoreachanoptimalcompositestructure.Anotherrobustalgorithm
insolvingcomplexproblemsofoptimalstructuraldesignisnamedparticleswarmoptimiza-
tion algorithm (PSOA). This algorithm is based on the behavior of a colonyof living things,
suchasaswarmofinsectslikeants,bees,andwasps,afolkofbirds,oraschooloffish.Omkar
etal.[13]appliedPSOAtoachieveaspecifiedstrengthwithminimizingweightandtotalcost
of a composite structure under different failure criteria. To the author’s knowledge, GA has
beenthemostefficientstochasticmethodforobtainingtheglobaloptimumdesignofcompos-
itestructures.
4.Application:bucklingoptimizationofanisotropiccylindricalshells
Structuralbucklingfailureduetohighexternalhydrostaticpressureisamajorconsiderationin
designingcylindricalshell-typestructures.Thissectionpresentsadirectapproachforenhanc-
ingbucklingstabilitylimitsofthin-walledlongcylindersthatarefabricatedfrommulti-angle
fibrous laminated composite lay-ups. The mathematical formulation employs the classical
lamination theory for calculating the critical buckling pressure, where an analytical solution
thataccountsfortheeffectiveaxialandflexuralstiffnessseparatelyaswellastheinclusionof
the coupling stiffness terms is presented. The associated design optimization problem of
maximizingthecriticalbucklingpressurehasbeenformulatedinastandardnonlinearmath-
ematicalprogrammingproblemwiththedesignvariablesencompassingthefiberorientation
anglesandtheplythicknessesaswell.Thephysicalandmechanicalpropertiesofthecompos-
itematerialaretakenaspreassignedparameters.Theproposedmodeldealswithdimension-
less quantities in order to be valid for thin shells having different thickness-to-radius ratios.
Results have been obtained for cases of filament wound cylinders fabricated from different
typesofcompositematerials.
Thebasicanalysisandanalyticalformulationpresentedinthischapterarebasedonthework
givenbyMaalawi[14],whichprovidesgoodsensitivitytolaminationparametersandallows
the search for the needed optimal stacking sequences in a reasonable computational time.
Referring to the structural model depicted in Figure 2, the significant strain components are
Introductory Chapter: An Introduction to the Optimization of Composite Structures 5
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Figure2. Laminatedcompositecylindricalshellunderexternalpressure(udisplacementintheaxialdirectionx,vinthe
tangentialdirections,wintheradialdirectionz).
the hoop strain (ε0) and the circumferential curvature (Kss) of the mid-surface. The reduced
ss
formofthestress-strainrelationshipsinmatrixformis
(cid:2) (cid:3) (cid:4) (cid:5)(cid:2) (cid:3)
N A B εo
ss ¼ 22 22 ss (5)
M B D κ
ss 22 22 ss
where N and M are the resultant distributed force and moment and (A , B , D ) are the
ss ss ij ij ij
extensional,coupling,andbendingstiffnesscoefficients,respectively[1].
4.1.Analyticalbucklingmodel
Thegoverningdifferential equationsofanisotropiclongcylinderssubjectedtoexternalpres-
surearecastinthefollowing:
(cid:6) (cid:7)
M0 þR N0 (cid:2)βN ¼βpR2 (6.1)
ss ss ss
h (cid:6) (cid:7) (cid:6) (cid:7)i
M00 (cid:2)R N þ βN 0þp w þv0 ¼pR2 (6.2)
ss ss ss o o
whereu,v,andw arethedisplacementsofa genericpoint(x,s) ontheshellmiddlesurface
o o o
(z=0)inx,s,andzdirection(cid:6)s,respec(cid:7)tively.Theprimedenotesdifferentiationwithrespecttothe
angularpositionφandβ¼ v (cid:2)w0 =R:Forthecaseofthincylinderswiththickness-to-radius
o o
ratio(h/R)≤0.1,thecriticalbucklingpressurecanbedeterminedusingthemathematicalexpres-
sion[14]:
6 Optimum Composite Structures
(cid:4) (cid:5)" (cid:6) (cid:7)#
D 1(cid:2) ψ2=α
p ¼3 22 (7.1)
cr R3 1þαþ2ψ
(cid:8) (cid:9)(cid:8) (cid:9)
1 B
ψ¼ 22 (7.2)
R A
22
(cid:8) (cid:9)(cid:8) (cid:9)
1 D
α¼ 22 (7.3)
R2 A22
4.2.Definitionofthebaselinedesign
Itisconvenientfirsttonormalizeallvariablesandparameterswithrespecttoabaselinedesign,
which has been selected to be a unidirectional orthotropic laminated cylinder with the fibers
parallel to the shell axis x. Optimized designs shall have the same material properties, mean
radiusR,andtotalshellthicknesshofthebaselinedesign.Expressionsforcalculatingthecritical
bucklingpressure(P )ofthebaseline designaredefinedinTable1,whichdependuponthe
cro
typeofcompositematerialutilizedandtheshellthickness-to-radiusratio(h/R)aswell.
4.3.Optimizationmodel
Thesearchfortheoptimizedlaminationcanbeperformedbycouplingtheanalyticalbuckling
shell model to a standard nonlinear mathematical programming procedure. The resulting
optimizationproblemmaybecastinthefollowingstandardformto
minimize —^p (8.1)
cr
subject to h ≤^h ≤h , (8.2)
L k U
θ ≤θ ≤θ k¼1,2,…:n (8.3)
L k U
Xn
h^ ¼1 (8.4)
k
k¼1
where ^p ¼p =p isthedimensionlesscriticalbucklingpressure,and(h ,h )arethelower
cr cr cro L U
andupperboundsimposedontheindividualdimensionlessplythicknesses ^h ¼h =h.
k k
Materialtype Orthotropicmechanicalproperties*(GPa) Pcro(cid:3)(h/R)3(GPa)
E E G ν
11 22 12 12
E-Glass/vinylester 41.06 6.73 2.5 0.299 1.708
Graphite/epoxy 130.0 7.0 6.0 0.28 1.757
S-Glass/epoxy 57.0 14.0 5.7 0.277 3.567
*E11=longitudinalmodulus,E22=hoopmodulus,ν12=Poisson’sratioforaxialload,ν21=ν12E22/E11.
Table1. Materialpropertiesandcriticalbucklingpressureofthebaselinedesign(Pcro).
Introductory Chapter: An Introduction to the Optimization of Composite Structures 7
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According to the filament-winding manufacturing process, each ply is characterized by its
angleθ withrespecttothecylinderaxisx.Thestackingsequenceisdenotedby[θ /θ /…/θ ],
k 1 2 n
wheretheanglesaregivenindegrees,startingfromtheoutersurfaceoftheshell.Inaddition,
inareal-worldmanufacturingprocess,thefilament-windinganglesθ mustbechosenfroma
k
limited range of allowable lower (θ ) and upper (θ ) values according to technology refer-
L U
ences.Itisimportanttomentionherethatthevolumefractionsoftheconstituentmaterialsof
thecompositestructureisassumedtonotsignificantlychangeduringoptimization,sothatthe
totalstructuralmassremainsconstantatitsreferencevalueofthebaselinedesign.
4.4.Optimalsolutions
Thefunctionalbehaviorofthecandidateobjectivefunction,asrepresentedbymaximizationof
thedimensionless bucklingpressure ^p , isthoroughly investigated in order tosee howit is
cr
changed with the optimization variables in the selected design space. The final optimum
designs recommended by the model will directly depend on the mathematical form and
behavioroftheobjectivefunction.
4.4.1.Two-layeranisotropiclongcylinder
The first case study to be considered herein is a long thin-walled cylindrical shell fabricated
fromE-glass/vinylestercompositeswiththelay-upcomposedofonlytwoplies(n=2)having
equalthicknesses(^h ¼ ^h ¼0:5)anddifferentfiberorientationangles.Consideringthecase
1 2
of (cid:4)63(cid:5) angle ply, the present model gives ^p = 4.23, i.e., P = 4.23 (cid:3) 1.708 (cid:3) (h/R)3 GPa,
cr cr
dependingontheshellthickness-to-radiusratio.Theactualdimensionalvaluesofthecritical
bucklingpressureforthedifferentthicknessratiosaregiveninTable2forthecasesofbaseline
design [0(cid:5)], helically wound [(cid:4)63(cid:5)], and [(cid:4)90(cid:5)] hoop layers. The unconstrained maximum
valueof ^p =6.1occursatthedesignpoints[θ /θ ]=[(cid:4)90,(cid:4)90].
cr 1 2
For a two-ply long cylinder fabricated from graphite/epoxy composites, Figure 3 shows the
developed level curves of the dimensionless buckling pressure, ^p (also named isomerits or
cr
isobars)inthe(θ (cid:2)θ )designspace.Asseeninthefigure,themaximumvalueof ^p reachesa
1 2 cr
valueof18.57forahoopwoundconstruction.Table3presentsthesolutionsforthe[(cid:4)45(cid:5)]angle-
(cid:5)
plyandthe[90 ]cross-plyconstructionsfordifferentthickness-to-radiusratios.Thesesolutions
Baseline[0(cid:5)] Helicallywound[(cid:4)63(cid:5)] Hoopplies[(cid:4)90(cid:5)]
^p =1.00 4.23 6.10
cr
(h/R)
(1/15) 506.07 2140.69 3087.05
(1/20) 213.50 903.11 1302.35
(1/25) 109.31 462.39 666.80
(1/50) 13.66 57.80 83.35
[Pcr=^pcr(cid:3)1.708(cid:3)106(h/R)3KPa].
Table2. CriticalbucklingpressureforE-glass/vinylestercylinderwithdifferentlay-ups.
8 Optimum Composite Structures
Figure3. ^pcr-isomeritsforagraphite/epoxy,two-layercylinderin[θ1/θ2]designspace(h^1¼h^2¼0:5).
are also valid for lay-ups [0 (cid:5)], [90 (cid:5)], [45 (cid:5)/(cid:2)45 (cid:5)], and [45(cid:5)/(cid:2)45(cid:5)/45(cid:5)/(cid:2)45(cid:5)]. The case of a
3 s 3 s 2 2 s s
helicallywoundlay-upconstruction[+θ/(cid:2)θ]withunequalplaythicknesses ^h and^h ,suchthat
1 2
theirsumisheldfixedatavalueofunity,hasalsobeeninvestigated.Computersolutionshave
shownthatnosignificantchangeintheresultingvaluesofthecriticalbucklingpressurecanbe
remarked in spite of the wide change in the ply thicknesses. This is a natural expected result
sincethestiffnesscoefficientsA ,B ,andD remainunchangedforsuchlay-upconstruction.
22 22 22
Baseline[0(cid:5)] Helicallywound[(cid:4)45(cid:5)] Hoopplies[(cid:4)90(cid:5)]
^p =1.00 5.9 18.57
cr
(h/R)
(1/15) 520.59 3071.50 9667.40
(1/50) 14.06 82.93 261.02
(1/120) 1.02 5.99 18.88
[Pcr=^pcr(cid:3)1.757(cid:3)106(h/R)3KPa].
Table3. Criticalbucklingpressure,Pcrforgraphite/epoxycylinderwithdifferentlay-ups.
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Baseline[0(cid:5)3] [0(cid:5)/90(cid:5)/0(cid:5)] [90(cid:5)/0(cid:5)/90(cid:5)]
^p =1.00 1.651 17.92
cr
(h/R)
(1/15) 520.59 859.57 9331.19
(1/50) 14.06 23.21 251.94
(1/120) 1.02 1.68 18.23
[Pcr=^p (cid:3)1.757(cid:3)106(h/R)3KPa].
cr
Table4. Criticalbucklingpressure,Pcrforgraphite/epoxycylinder[θ1/θ2/θ1].
4.4.2.Three-layeranisotropiclongcylinder
Results for a cylinder constructed from three, equal-thickness layers with stacking sequence
denotedby[θ /θ /θ ]aregiveninTable4.Thesamebehaviorcanbeobservedasbeforebut
1 2 1
withslightchangeintheattainedvalues.Itwasfoundthatfortherange(cid:2)30(cid:5) >θ >30(cid:5) the
1
criticalbucklingpressureisnotmuchaffectedbyvariationintheplyangleθ .Asubstantial
2
increaseinthecriticalbucklingpressurebychangingtheplyanglescanbeobserved.Similar
(cid:5) (cid:5) (cid:5) (cid:5)
solutionswereobtainedforthestackingsequences[0 /90 ] and[90 /0 ].
2 s 2 s
4.4.3.Four-layersandwichedanisotropiccylinder
Thesamegraphite/epoxycylinderisreconsideredherewithchangingthestackingsequenceto
become(cid:4)20(cid:5) equal-thickness layerssandwichedinbetweenouterandinner90(cid:5) hooplayers
with unequal thicknesses, i.e., (^h ¼ ^h ) and (^h 6¼ ^h ), such that the thickness equality
P 2 3 1 4
constraint 4k¼1 ^hk¼1 isalwayssatisfied.Figure4showsthedevelopedp^cr -isomeritsinthe
^ ^
(h , h )designspace.Thecontoursinsidethefeasibledomain,whichisboundedbythethree
1 2
lines^h ¼0 and ^h ¼0 and ^h þ2^h ¼1 (i.e., ^h ¼0), are obliged to turn sharply to be
1 2 1 2 4
asymptotestotheline ^h ¼0,inordernottoviolatethethicknessequalityconstraint.Thisis
4
^ ^
whytheyappearinthefigureaszigzaggedlines.Atthedesignpoint(h , h )=(0.25,0.25),the
1 2
dimensionlessbucklingpressure ^p =16.43(seeFigure4andTable5).Asageneralobserva-
cr
tion,asthethicknessofthehooplayersincreases,asubstantialincreaseinthecriticalbuckling
pressurewillbeachieved,e.g.,at(^h ,^h )=(0.33,0.17), ^p =17.92representingapercentage
1 2 cr
increaseof(17.92(cid:2)16.43)/16.43=9.1%.
Finally,theobtainedresultshaveindicatedthattheoptimizedlaminationsinducesignificant
increases,alwaysexceedingseveraltensofpercent,ofthebucklingpressureswithrespectto
the reference or baseline design. It is assumed that the volume fractions of the composite
materialconstituentsdonotsignificantlychangeduringoptimization,sothatthetotalstruc-
turalmassremainsconstant.Ithasbeenshownthattheoverallstabilitylevelofthelaminated
compositeshellstructuresunderconsiderationscanbesubstantiallyimprovedbyfindingthe
optimalstackingsequencewithoutviolatinganyimposedsideconstraints.Thestabilitylimits
