A general multi-scale two-level optimisation strategy for designing composite stiffened panels

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To cite this version :

Marco MONTEMURRO, Alfonso PAGANI, Giacinto Alberto FIORDILINO, Jérôme PAILHES,

Erasmo CARRERA - A general multi-scale two-level optimisation strategy for designing composite

stiffened panels - Composite Structures - Vol. 201, p.968-979 - 2018

omposite stiened panels

Mar o Montemurro a,

∗

,Alfonso Pagani b

,Gia into Alberto Fiordilino a,b ,Jér

ˆ

o

mePailhès a , Erasmo Carrera b a

ArtsetMétiersParisTe h,I2MCNRSUMR5295,F-33400Talen e,Fran e

b

MUL2 group,DIMEAS,Polite ni odiTorino,Torino,Italy

Abstra t

Thisworkdeals withthe problemoftheleast-weight design ofa ompositestiened panel

subje t to onstraints of dierent nature (me hani al, geometri al and manufa turability

requirements). To fa e this problem, a multi-s ale two-level (MS2L) design methodology

is proposed. This approa h aims at optimising simultaneously both geometri al and

me- hani al parameters for skin and stieners at ea h hara teristi s ale (mesos opi and

ma ros opi ones). Inthis ba kground,at therst level (ma ros opi s ale)the goalisto

ndtheoptimumvalueofgeometri andme hani al designvariablesofthepanel

minimis-ingits massand meetingthesetofimposed onstraints. These ond-levelproblemfo uses

on the laminate mesos opi s ale and aims at nding at least one sta king sequen e (for

ea hlaminate omposingthe panel)meetingthe geometri alandmaterialparameters

pro-vided by the rst-level problem. The MS2L optimisation approa h is based on the polar

formalismtodes ribethema ros opi behaviourofthe ompositesandonaspe ialgeneti

algorithmtoperformoptimisation al ulations. Thequalityoftheoptimum ongurations

is investigated, a posteriori, through a rened nite element model of thestiened panel

making use of elements with dierent kinemati s and a ura y in the framework of the

Carrera'sUnied Formulation (CUF).

Keywords: Composites,Finite Element Method,Bu kling, Optimisation, Lightweight

stru tures.

Thisisa pre-print of anarti le publishedinCompositeStru tures.

Thenalauthenti ated versionis available onlineat:

https://doi.org/10.1016/j. ompstru t.2018.06.119

∗

Correspondingauthor. Tel.: +33556845422, Fax.:+33 540006964.

Emailaddress: mar o.montemurroensam.eu, mar o.montemurrou-bordeaux.f r(Mar o

Anisotropi materials, su h as bres-reinfor ed omposites, are extensively used in

manyindustrialeldsthankstotheirpe uliarfeatures: highstiness-to-weightand

strength-to-weight ratiosthatleadtoasubstantialweightsaving when ompared tometalli alloys.

The problemof designing a ompositestru ture is quite umbersome and an be

on-sidered as a multi-s ale optimisation problem. The omplexity of the design pro ess is

a tually due to two intrinsi properties of omposite materials, i.e., heterogeneity and

anisotropy. Heterogeneity gets involved mainly at the mi ros opi s ale (i.e.,that of

on-stitutive phases), whilst anisotropy intervenes at both mesos opi s ale (that of the

on-stitutive lamina) andma ros opi one (thatof thelaminate).

To illustratethedi ultyofproperlydesign/optimiseatea hrelevant s alea

ompos-itestru ture thestudy presentedinthiswork fo uses ona real-world engineering problem

that an be onsidered asparadigmati : the multi-s ale design of a least-weight

ompos-ite stiened panel subje t to a given set of onstraints of dierent nature (geometri al,

me hani al, te hnologi al, et .).

Stiened panels arewidely usedinmany stru tural appli ations, mostlybe ause they

allow for a substantial weight saving. Of ourse, this point is of paramount importan e

espe ially inair raft design, where an important redu tion of thestru tural mass an be

a hievedif ompositelaminatesareusedinpla eofaluminiumalloys. Adrawba kofsu ha

hoi eisthatthedesignpro essbe omesmore umbersomethanthatofa lassi almetalli

stru ture. In fa t, though the use of laminated stru tures is not a re ent a hievement in

stru tural me hani s, up to now no general methods exist for their optimum design. In

pra ti alappli ations, engineersalways usesomesimplifyingrulesto take intoa ountfor

some relevant properties (whi hare verydi ult to be formalisedotherwise).

Several works on the optimum design of omposite stiened panels an be found in

literature. Nagendra et al. [1℄ made use of a standard geneti algorithm (GA) to nd a

solution for the problem ofminimising the massof a omposite stiened panelsubje tto

onstraints on the rst bu kling load, on maximum allowable strains and te hnologi al

onstraints on ply orientation angles. In [2℄Bisagni and Lanzi dened a single-step

post-bu klingoptimisation pro edure for the design of ompositestiened panels subje ted to

ompressionload. Thepro edurewasbasedonaglobalapproximationstrategy,wherethe

stru tureresponseisgivenbyanarti ialneuralnetwork(ANN)trainedbymeansofnite

element (FE)analyses,while theoptimisation tool onsisted ina standardGA.Lanziand

Giavotto[3℄proposedamulti-obje tiveoptimisationpro edureforthedesignof omposite

stiened panels apable to take into a ount thepost-bu klingbehaviour. Thepro edure

made use of a standard GA and three dierent methods for surrogate modelling: ANN,

Radial Basis Fun tionsand Kriging approximation. In [4℄Barkanov et al. dealt withthe

problemoftheoptimumdesignoflateralwingupper oversby onsideringdierentkindsof

stienersandloading onditions. Liuet al.[5℄utilised thesmearedstiness-basedmethod

forndingthebeststa kingsequen esof ompositewingswithblendingandmanufa turing

onstraints by onsidering a set of pre-dened bre angles, i.e.,

0

◦

_{, 90}

◦

and

±45

◦

. In [6℄

Lópezetal. proposedadeterministi andreliability-baseddesignoptimisationof omposite

stienedpanels onsideringpost-bu klingregimeandaprogressivefailureanalysis. Further

works on this topi an be found in literature. For example, and without any ambition

of exhaustiveness, the studiesof Lilli o et al. [7℄, Butler and Williams [8℄, Wiggenraad et

al. [9℄, Kaletta andWolf[10℄ an be ited too.

A ommon limitation of the previous works is the utilisation of simplifying

hypothe-ses and rules in the formulation of thestiened panel design problem. These restri tions

i.e.,toeliminatefromthetrueproblemsomeparti ularlydi ultpointsorpropertiestobe

obtained. Ontheotherhand,some ofsu h rules are onsideredto prevent thenal

stru -ture from some undesired phenomena, though this is never learly and rigorously stated

and proved. Unfortunately, the use of these simple rules has a main drawba k: the

de-signspa eisextremely shrunk,thustheirutilisation automati allydrivestheoptimisation

algorithm only towardssuboptimal solutions.

Two examplesarethe use ofsymmetri sta king sequen es, asu ient but not

ne es-sary ondition formembrane-bending un oupling andtheuseof balan edsta kstoobtain

orthotropi laminates. When symmetri sta ks are utilised, the design is done using half

of thelayers, whi h means also half of thedesign variables. This kind of sta kimpli itly

implies a redu tion of the design spa e: it is very di ult to obtain the lightest

stru -ture under this hypothesis. Conversely, the use of balan ed sta ks, a su ient ondition

for membrane orthotropy, leads systemati ally to misleading solutions: whenever su h a

rule is used, bending orthotropy, a rather di ult propertyto be obtained [11℄, is simply

understated, assumed, but not reallyobtained, asin [1215℄.

In air raft stru tural design, some other rules areimposed to thedesign of omposite

stiened panels, although some of them are not me hani ally well justied, see for

in-stan e [12, 15℄. Among these rules, themost signi ant restri tion is represented by the

utilisationofalimitedsetofvaluesfor thelayersorientationangleswhi hareoftenlimited

to the anoni alvalues of

0

◦

,

90

◦

and

±45

◦

.

To over ome the previous restri tions, in the present study the multi-s ale two-level

(MS2L) optimisation approa h for designing anisotropi omplex stru tures [1618℄ is

utilisedintheframeworkofthemulti-s aleoptimisationof ompositestienedpanels. The

proposedMS2L approa h aimsat proposinga very general formulation of design problem

withoutintrodu ingsimplifyinghypothesesandby onsidering,asdesignvariables,thefull

set of geometri and me hani al parameters dening the behaviour of the panel at ea h

hara teristi s ale (mesos opi and ma ros opi ).

In the ontext of the MS2L methodology, the optimisation problem is split in two

distin t(butrelated)sub-problems. Attherstlevel(ma ros opi s ale)thegoalistond

theoptimumvalueofgeometri andme hani aldesignvariablesofthepanelminimisingits

massandmeetingthe setofimposed onstraints. These ond-levelproblemfo usesonthe

laminatemesos opi s ale(i.e.,theply-level)andaimsatndingatleastoneoptimumsta k

(forea hlaminate omposingthepanel)meetingthegeometri al andmaterial parameters

resulting from the rst-level problem. The MS2L approa h is based on the utilisation of

thepolar formalism[19℄aswell ason aGA previously developed by therstauthor [20℄.

Thequalityoftheoptimum ongurationsisinvestigated,aposteriori,througharened

niteelementmodelofthestienedpanelmakinguseofelementswithdierentkinemati s

anda ura y(inaglobal-lo alsense)intheframeworkoftheCarrera'sUniedFormulation

(CUF).

Thepaperisorganisedasfollows: thedesignproblemaswellastheMS2Loptimisation

strategyaredis ussedinSe tion2. Themathemati alformulationoftherst-levelproblem

is detailed in Se tion 3, while the problem of determining a suitable laminate sta king

sequen e is formulated in Se tion 4. A on ise des ription of the Finite Element (FE)

models of the stiened panel are given in Se tion 5, while the numeri al results of the

optimisation pro edure are shown in Se tion 6. Finally, Se tion 7 ends the paper with

2.1. Problem Des ription

Theoptimisationstrategypresentedinthisstudy isappliedtotherepetitiveunit(RU)

ofa ompositestiened paneltypi ally utilised inair raftwings. TheRUis omposedby

theunionofaskinandaomegashapedstringer(orstiener)asillustratedinFig.1. The

overallsize of the RU are xed:

a = 150

mm is the width of the RU, while

b = 600

mm is its length whi h represents also the distan e between two onse utive ribs. It must be

notedthatstienersareequispa edoverthepanelwithasteplengthequalto

a

. Bothskin andstieneraremadeof arbon-epoxyunidire tionalorthotropi laminaewhoseproperties

arelistedinTable1 (taken from [11,21,22℄).

The fundamental hypotheses about the ma ros opi me hani al response of the RU

fo usessentiallyon the laminate behaviour andgeometry (forboth skinandstringer).

•

Ea h laminate ismade ofidenti al plies(i.e.,same thi kness

t

ply

and material).

•

The material of the onstitutive layer has a linear elasti transverse isotropi be-haviour.

•

Ea h laminate isquasi-homogeneous and fullyorthotropi [18,2224℄.

•

Atthema ros opi s aletheelasti responseofea hlaminateisdes ribedinthe the-oreti alframeworkoftheFSDTandthe stinessmatri esoftheplateareexpressed

intermsof thelaminatepolarparameters [11,21℄.

•

Nodelamination o ursat thepliesinterfa e for both skinandstringer [25℄.

It is noteworthy that, no simplifying hypotheses are made on the geometri and

me- hani alparameters oftheRU(e.g.,onthenatureofthesta kingsequen es). Only

avoid-ingthe utilisation of a priori assumptions thatextremely shrink thesolution spa e (e.g.,

theutilisation ofsymmetri ,balan edsta kstoattain membrane/bending un ouplingand

membraneorthotropy,respe tively)one an hopeto obtainthebestoptimumsolutionfor

agiven problem: this is akey-point intheproposedapproa h.

2.2. Des riptionof the multi-s ale two-leveloptimisation strategy

The main goal of the MS2L optimisation strategy is the least-weight design of the

omposite stiened panel subje t to onstraints of dierent nature, i.e., me hani al,

geo-metri alaswell asfeasibilityand te hnologi al requirements. The optimisationpro edure

isarti ulatedinto the following two distin t (butrelated) optimisation problems.

First-level problem. The aim of this phase is the determination of the optimal value

of both me hani al and geometri parameters of the laminate omposing the RU of the

panelinorderto minimiseitsweightandto satisfy,simultaneously,thefullsetofimposed

requirements(formulatedasoptimisation onstraints). At thislevelea h laminateis

mod-elledasanequivalenthomogeneousanisotropi platewhosebehaviourisdes ribedinterms

ofthelaminatepolarparameters [11,21℄. Therefore,thedesignvariablesof thisphaseare

thegeometri parameters ofthe RU aswellasthelaminate polarparameters of both skin

and stiener.

Se ond-level problem. The se ondlevel ofthe strategy aimsat determining asuitable

lay-up for both skin and stringer laminates (i.e., the laminate mesos opi s ale) meeting

the optimum ombination of their material and geometri al parameters provided by the

optimumvalues of the polar parameters resulting from therst step. At this level of the

strategy,the design variables arethe layer orientations.

3. Mathemati al formulation of the rst-level problem

The overall features of the stru ture at the ma ros opi s ale have to be optimised

duringthis phase. The massminimisationof the stiened panelRU willbe performedby

satisfyingthe set ofoptimisation onstraintslisted below:

1. a onstraint onthe rst bu klingloadof the RU;

2. geometri andte hnologi al onstraints relatedtothegeometri al parameters ofthe

RU;

3. feasibility onstraintson thelaminate polar parameters of both skinand stringer.

Theseaspe ts aredetailedinthe following subse tions.

3.1. Geometri al design variables

Thedesignvariablesfortheproblemathandareoftwotypes: geometri al andme

han-i al. Some of the geometri al parameters of the RU of the stiened panel are illustrated

in Fig. 1. Of ourse, these parameters are not independent. The independent geometri

design variables are:

•

thelaminate thi knessfor both skinandstringer, i.e.,

t

S

and

t

B

,respe tively;

•

thewidth

a

2

ofthe stringerbottom ange;

•

thestringer height

h

;

•

thesize

a

3

.

Thesize

a

1

an berelated tothe previous variables,

a

1

=

a

2

− a

2

− a

3

,

(1)

whilethe angle ofthe in lined wall ofthestieneris

θ = atan



h

a

3

−

a

₂

2



.

(2)

The previous design variables mustsatisfy aset of te hnologi al and geometri al

require-ments. Firstly, the overall thi kness of the laminates omposing the RU is a dis rete

variable, the dis retisation step being equal to the thi kness of the elementary layer, i.e.,

t

ply

(see Table 1):

t

α

= n

α

t

ply

, α = S, B ,

(3)

where

n

S

and

n

B

arethe number of layers of skinand stiener, respe tively. It must be highlightedthattheoptimumvalueofthelaminatethi knessdeterminesalsotheoptimum

number of layers

n

to be used during the se ond-level design problem (for both skinand stringer). Se ondly, parameters

a

i

,

(i = 1, 2, 3)

have to meetthefollowing onditions:

a

1

> 0,

a

3

≥

a

2

2

.

(4)

First inequality is ne essary to avoid onta t between two onse utive stringers, while

se ond one must be imposed in order to keep

θ

non-negative. In the framework of the mathemati alformalisationoftherst-levelproblem,itisusefultointrodu edimensionless

geometri design variables, asfollows:

c

1

= 2

a

2

a

, c

2

= 2

a

3

a

2

, c

3

=

h

a

2

.

(5)

Thedimensionlessgeometri parameters anbe olle tedintotheve torofgeometri design

variablesdened as:

ξ

_g

T

= {n

S

, n

B

, c

1

, c

2

, c

3

} .

(6)

Inthis ba kground,inequalities ofEq. (4) an be reformulated as:

g

1

(ξ

g

) = 2c

1

+ c

1

c

2

− 2 < 0,

g

2

(ξ

g

) = 1 − c

2

≤ 0.

(7)

3.2. Me hani al design variables

In the framework of the FSDT [26℄ the onstitutive law of the laminate (expressed

withinits global frame

R = {0; x, y, z}

) anbe statedas:







N

M







=





A

B

B

D











ε

0

χ

0







,

(8)

F

= Hγ

0

,

(9)

where

A

,

B

and

D

are the membrane, membrane/bending oupling and bending stiness matri es of the laminate, while

H

is the out-of-plane shear stiness matrix.

N

,

M

and

F

arethe ve tors ofmembranefor es, bending moments and shearfor es per unit length, respe tively,whilst

ε

0

,

χ

0

and

γ

0

are theve torsof in-plane strains, urvatures and out-of-planeshearstrainsofthelaminatemiddleplane, respe tively,(inthepreviousequations

Voigt'snotation hasbeen utilised [26℄).

A

∗

₌

1

t

A,

B

∗

₌

2

t

2

B,

D

∗

₌

12

t

3

D,

H

∗

₌











1

t

H

(basic),

12

5t

H

(modified).

(10)

where

t

is the totalthi knessof thelaminate.

As dis ussed in [11, 21℄, in the framework of the polar formalism it is possible to

express the Cartesian omponents ofthese matri esinterms oftheir elasti invariants. It

an be proven that, in the FSDT framework, for a fully orthotropi , quasi-homogeneous

laminate (i.e., a laminate having the same orthotropi behaviour in terms of normalised

membraneandbendingstinessmatri esandwhose membrane/bending oupling stiness

matrixisnull)theoverallnumberofindependentme hani aldesignvariablesdes ribingits

me hani al responseredu es to onlythree,i.e., the anisotropi polarparameters

R

A

∗

0K

and

R

A

∗

1

andthepolarangle

Φ

A

∗

1

(thislast representingtheorientationofthemainorthotropy axis) of matrix

A

∗

. For more details on the polar formalism and its appli ation in the

ontext ofthe FSDTthereader is addressedto [11,21,27℄.

In addition, in the formulation of the optimisation problem for the rst level of the

strategy,the feasibility onstraints on the polar parameters (whi h arise from the

ombi-nation of the layers orientations and positions within the sta k) must also be onsidered.

These onstraints ensure thatthe optimumvalues of thepolar parameters resulting from

therststep orrespondtoafeasiblelaminatethatwillbedesignedduringthese ondstep

oftheMS2L strategy,see [28℄. Sin e thelaminate isquasi-homogeneous, su h onstraints

an be written onlyfor matrix

A

∗

:























−R

0

≤ R

A

∗

0K

≤ R

0

,

0 ≤ R

A

₁

∗

≤ R

1

,

2

 R

A

∗

1

R

1



2

− 1 −

R

A

∗

0K

R

0

≤ 0 .

(11)

InEq. (11),

R

0

and

R

1

arethe anisotropi modulioftheplyredu edstinessmatrix [11℄. As in the ase of geometri design variables, it is very useful to introdu e the following

dimensionlessquantities:

ρ

0

=

R

A

∗

0K

R

0

, ρ

1

=

R

A

∗

1

R

1

.

(12)

Inthis ba kground,Eq. (11) writes:















−1 ≤ ρ

0

≤ 1 ,

0 ≤ ρ

1

≤ 1 ,

2 (ρ

1

)

2

− 1 − ρ

0

≤ 0 .

(13)

the panel RU, i.e., for both skin and stiener laminates (

ρ

0α

and

ρ

1α

with

α = S, B

). Moreover, the main orthotropy dire tion ofthegeneri laminate an be set equal to zero,

i.e.,

Φ

A

∗

1

= 0

for skin and stringer,whi h means thatthe main orthotropy axisis aligned withthedire tionoftheappliedload. Therefore,thedimensionlessme hani alparameters

dened above an be grouped into theve tor ofme hani al design variables:

ξ

_m

T

= {ρ

0S

, ρ

1S

, ρ

0B

, ρ

1B

} .

(14)

Firstandse ond onstraintsofEq.(13) anbetakenintoa ountasadmissibleintervals

for the relevant optimisation variables, i.e., on

ρ

0

and

ρ

1

. Hen e, the resulting feasibility onstraints onthe skinand stringer dimensionlesspolarparameters be ome:

g

3

(ξ

m

) = 2 (ρ

1S

)

2

− 1 − ρ

0S

≤ 0 ,

g

4

(ξ

m

) = 2 (ρ

1B

)

2

− 1 − ρ

0B

≤ 0 .

(15)

For a wide dis ussionupon thelaminate feasibilityand geometri al boundsas well as

ontheimportan e ofthequasi-homogeneity assumption thereader is addressedto [28℄.

3.3. Mathemati al statement of the problem

As previously stated,the aim of therst-level optimisation is theminimisation of the

massofthe RUofthe stienedpanelbysatisfying,simultaneously, onstraintsofdierent

nature. The design variables (both geometri al and me hani al) of the problem an be

olle ted into thefollowing ve tor:

ξ

T

=

ξ

T

g

, ξ

T

m

.

(16)

Inthis ontextthe optimisation problem anbeformulated asa lassi al onstrained

non-linearprogrammingproblem (CNLPP):

min

ξ

M (ξ)

M

ref

subje tto:











1.05 −

λ (ξ)

λ

ref

≤ 0 ,

g

i

(ξ) ≤ 0 ,

with

i = 1, · · · , 4 .

(17)

Thedesignspa e ofthe rst-levelproblem, together withthetypeof ea h designvariable,

isdetailed inTable2. InEq. (17)

M

istheoverallmassof theRU,

λ

istherst bu kling loadofthestienedpanel, while

M

ref

and

λ

ref

arethe ounterpartsforareferen e solution whi hissubje ttothesameboundary onditions (BCs)asthoseapplied ontheRUofthe

panel thatwill be optimised. Theproperties of the referen e onguration of theRU are

reportedinTable3.

3.4. Numeri al strategy

Problem (17) is a non- onvex CNLPP in terms of both geometri al and me hani al

alsodue to the non-linear feasibility onstraints onthelaminate polarparameters.

The total number of design variables is nine while that of optimisation onstraints is

ve (see Eq. (17)). Furthermore, the nature of design variables is dierent (see Table 2):

integer(

n

S

and

n

B

),dis rete(

c

1

,

c

2

,

c

3

)and ontinuous(

ρ

0S

,

ρ

1S

,

ρ

0B

,

ρ

1B

) variablesare involved inthe denitionofthis CNLPP.

For the resolution of problem (17) the GA BIANCA [20, 29℄ oupled with the FE

model of the panel RU (for al ulating the rst bu kling load of the stru ture) has been

utilised as optimisation tool for the solution sear h, see Fig. 2. The GA BIANCA was

alreadysu essfullyapplied tosolvedierentkindsofreal-worldengineering problems,see

for example[3033℄.

AsshowninFig.2,forea hindividualatea hgeneration, thenumeri al toolperforms

a FE analysis for al ulating the rst bu kling load(eigenvalue problem) of thestiened

panel as well as its weight. The inputs of the FE model of the RU (implemented in

ANSYS

r

environment) are both geometri al and me hani al parameters (generated by

BIANCA).The GA elaborates the results provided by the FEmodelin order to exe ute

thegeneti operations. TheseoperationsarerepeateduntiltheGAmeetstheuser-dened

onvergen e riterion.

The generi individual (i.e., a generi point in thedesign spa e) of the GA BIANCA

representsapotentialsolutionfortheproblemat hand. Thegenotypeoftheindividualfor

problem (17) is hara terised by only one hromosome omposed of nine genes, ea h one

odinga omponent oftheve tor ofdesign variables,seeEq. (16).

4. Mathemati al formulation of the se ond-level problem

The se ond-level problem fo uses on the lay-up design of the both skin and stringer

laminates. The goalistodetermine atleastone sta king sequen esatisfyingtheoptimum

valuesofboth geometri andpolarparameters resultingfromtherstlevelofthestrategy

and having the elasti symmetries imposed to thelaminate withinthe formulation of the

rst-levelproblem, i.e.,quasi-homogeneity andorthotropy. IntheframeworkoftheFSDT

and onsidering the polar formalism for representing the laminate stiness matri es, this

problem an be stated inthe formof an un onstrained minimisationproblem[11,21℄:

min

δ

I (f

i

(δ)) ,

(18) with

I (f

i

(δ)) =

6

X

i=1

f

i

(δ) .

(19) where

δ

∈ R

n

istheve tor ofthelayerorientations,i.e., thedesign variablesofthisphase,

while

f

i

(δ)

arequadrati fun tionsinthespa eofpolarparameters,ea honerepresenting arequirementtobesatised,su hasorthotropy,un oupling,et . Fortheproblemathand

f

1

(δ) =

 |Φ

A

∗

0

(δ) − Φ

A

∗

1

(δ)|

π/4

− K

A

∗

_(opt)



2

,

f

2

(δ) =

R

A

∗

0

(δ) − R

A

∗

_(opt)

0

R

0

!

2

,

f

3

(δ) =

R

A

∗

1

(δ) − R

A

∗

_(opt)

1

R

1

!

2

,

f

4

(δ) =

|Φ

A

∗

1

(δ) − Φ

A

∗

_(opt)

1

|

π/4

!

2

,

f

5

(δ) =

 ||C(δ)||

||Q||



2

,

f

6

(δ) =

 ||B

∗

_(δ)||

||Q||



2

,

(20)

where

f

1

(δ)

represents the elasti requirement on the orthotropy of the laminate having thepres ribedshape(imposedbythevalueof

K

A

∗

whi hisrelatedtothesignof

ρ

0

atthe endoftherststepofthestrategy),

f

2

(δ)

,

f

3

(δ)

and

f

4

(δ)

aretherequirementsrelatedto thepres ribedvaluesoftheoptimalpolarparametersresultingfromtherst-levelproblem,

while

f

5

(δ)

and

f

6

(δ)

arelinked to thequasi-homogeneity ondition.

I (f

i

(δ))

is a positive semi-denite onvex fun tion in the spa e of laminate polar parameters,sin eitisdenedasasumof onvexfun tions,seeEqs.(19)-(20). Nevertheless,

su hafun tionishighlynon- onvexinthespa eofpliesorientationsbe ausethelaminate

polarparametersdependupon ir ularfun tionsofthelayersorientationangles. Moreover,

the absolute minima of

I (f

i

(δ))

are known a priori sin e they are the zeroes of this fun tion. For more detailsabout the nature of these ond-level problemsee [11,21℄. It is

noteworthythatproblem(18) mustbesolved twotimes, i.e.,for ea hlaminate omposing

theskinandthe stiener.

In order to simplify theproblem of retrieving an optimum sta k, thesear h spa e for

problem (18) has been restri ted to a parti ular lass of quasi-homogeneous laminates:

thequasi-trivial (QT)sta kingsequen es whi h onstitute exa t solutions withrespe tto

the requirements of quasi-homogeneity, i.e., fun tions

f

5

(δ)

and

f

6

(δ)

in Eq. (20) are identi ally null forQT sta ks.

QT solutions an be found for laminates with identi al plies by a ting only on the

position of the layers within the sta k. Indeed, QT sta ks are exa t solutions, in terms

of quasi-homogeneity ondition, regardless to the value of the orientation angle assigned

to ea h layer. In this way orientations represent free parameters whi h an be optimised

to full further elasti requirements, i.e., fun tions

f

1

(δ)

,

f

2

(δ)

,

f

3

(δ)

and

f

4

(δ)

. The pro edureforsear hingQTsta ksis on eptuallysimple. Let

n

bethenumberoflayersand

n

g

≤ n

the number of saturated groups. Plies belonging to a given saturated group share thesameorientation angle

θ

j

, (j = 1, ..., n

g

)

. Theideaistolookforallthepermutationsof thepositionofthepliesindexesbelonginto ea h groupwhi hmeetthequasi-homogeneity

ondition. Moredetailson this topi an befound in[34℄.

Supposenowtoxboththenumberofpliesandsaturatedgroups,namely

n

and

n

g

. As dis ussedin[34℄, theproblemofdeterminingQTsta ksforagiven oupleof

n

and

n

g

an give rise to a huge number of solutions: the number ofQT sta ks rapidly in reases along

with

n

. To thispurposeadatabase ofQT sta kshasbeen builtfor dierent ombinations of

n

and

n

g

.

Fortheproblemathand,andforea h onsidered ase(i.e.,skinandstringerlaminates),

theoptimumnumberofplies

n

α

, (α = S, B)

onstitutes aresultof therst-levelproblem, while the numberof saturated groups

n

g

hasbeen xed a priori. Let be

n

sol

the number ofQTsta ksfor aparti ular ombination of

n

α

and

n

g

. Ea hsolution olle tedwithinthe database is uniquely dened bymeans of an identier

ID

sol

(i.e.,an integer) whi h varies

inthe range

[1, n

sol

]

. Therefore,

ID

sol

represents a further design variable alongwith the

n

g

orientation angles ofthe dierent saturatedgroups, i.e.,

θ

∈ R

n

g

. The design variables

an be thus olle tedinto the following ve tor,

η

T

=

ID

sol

, θ

1

, ..., θ

n

g

,

(21)

and problem(18) an bereformulated as

min

η

4

X

i=1

f

i

(η) ,

(22)

f

5

(η)

and

f

6

(η)

beingidenti ally null.

In thisba kground,thesolution sear hfor problem (22) isperformedbymeansof the

GABIANCA. In the aseof QT sta ksthestru ture of theindividualgenotype issimple

be ause it is omposed ofa single hromosome with

n

g

+ 1

genes: the rstone odesthe variable

ID

sol

whilst the remaining genes ode the orientation angles of every saturated groupwhi h aredis retevariables intherange[-89

◦

,90

◦

℄withastep lengthequal to 1

◦

.

5. Finite element modelsof the stiened panel

In this se tiontwo FE models ofthe stiened panelRU aredis ussed: therst one is

used in the framework of the rst-level problem of the MS2L approa h while the se ond

one isutilised for veri ation purposes.

5.1. The nite element modelfor the optimisation pro edure

TheFEmodelofthepanelRUusedattherst-leveloftheMS2Lstrategyisbuiltusing

theFE ommer ial ode ANSYS

r

. A linear eigenvalue bu klinganalysis is ondu ted to

determinethe valueoftherstbu klingloadforea h individual,i.e., forea hpointinthe

design spa e, at the urrent generation.

The need to analyse, within the same generation, dierent geometri al ongurations

(RUs with dierent geometri al and me hani al properties), ea h one orresponding to

an individual, requires the reation of an ad-ho input le for the FE ode that has to

be interfa ed with BIANCA.The FEmodelmust be on eived to take into a ount for a

variablegeometry,materialandmesh. Indeed,forea hindividualatthe urrentgeneration,

theFE odehastobeabletovaryinthe orre twaythepreviousquantities,thusaproper

parametrisationof themodelhasto bea hieved.

The FE model of the RU is illustrated in Fig. 3. The model has been built by using

a ombinationofeight-nodesshell elements(ANSYSSHELL281 elements) andnon-linear

multi-point onstraints elements (ANSYS MPC184 elements) both with six Degrees Of

Freedom (DOFs)pernode.

As far as on erns SHELL281 elements, their me hani al behaviour is des ribed by

deningdire tlythe homogenisedstiness matri es

A

∗

,

B

∗

,

D

∗

and

H

∗

.

The ompatibility of the displa ement eld between skin and stringer is a hieved

throughANSYSMPC184elementswhoseformulationisbasedupona lassi almulti-point

onstraintelements heme[35℄. MPC184elementsaredenedbetweenea h oupleofnodes

belonging to ontiguous shell elements as depi ted in Fig. 3. In parti ular, MPC184

ele-mentsaredenedbetween nodesof themiddleplaneof theskin(masternodes)andthose

tions of the RU, in order to simulate the presen e of ribs (these last having an in-plane

stinessone/two orderof magnitudehigher than the exuralstinessof theRU).In

par-ti ular, two pilot nodes A

= {0, 0, ˆ

z}

and B

= {b, 0, ˆ

z}

have been dened a ordingto the RUglobalframedepi ted inFig. 3(

ˆ

z

isthe

z

omponent ofthebary entreoflines belong-ing to a given transverse se tion). Then, nodes A and B have been onne ted (through

MPC184 elements) to those lo ated on lines of the orresponding transverse se tion, i.e.,

lines belonging to the planes

x = 0

and

x = b

, respe tively (see Fig. 3). The BCS for nodesAand Bare

node A:

u

i

= 0, β

i

= 0;

node B:

F

x

= −1

N

, u

y

= u

z

= 0, β

i

= 0,

(i = x, y, z).

(23)

InEq. (23)

u

i

and

β

i

arenodaldispla ements androtations,respe tively,whilst

F

x

isthe

x

omponent ofthe nodalfor e.

It is noteworthy that in problem (17) the rst-bu kling load of the stiened panel

is al ulated by onsidering pertinent BCs on its RU. This fa t impli itly implies the

hypothesis of a panel having an innite length along

y

-axis, a ording to the frame depi ted in Fig. 3. To take into a ount for this aspe t, periodi boundary onditions

(PBCs) mustbe onsidered:

u

i



x, −

a

2

, 0



− u

i



x,

a

2

, 0



= 0, ∀x ∈ ]0, b[ ,

β

i



x, −

a

2

, 0



− β

i



x,

a

2

, 0



= 0, ∀x ∈ ]0, b[ ,

(i = x, y, z).

(24)

PBCs of Eq. (24) must be dened for ea h ouple of nodes belonging to the skinlateral

edges(i.e.,lineslo atedat

y = ±a/2

)ex eptthosepla edon thelinesat

x = 0

and

x = b

, these last being already onne ted to the pilot nodes A and B, respe tively. PBCs are

denedthrough ANSYS onstraintequations (CEs)[35℄between homologousnodesofthe

skinlateraledges

Finally,beforestartingtheoptimisationpro ess,asensitivitystudy (notreportedhere

for the sake of brevity) on the proposed FE model with respe t to the mesh size has

been ondu ted: itwasobserved thata mesh having

56959

DOFs issu ient to properly evaluate the rstbu klingloadof the stiened panel.

5.2. The enhan ed nite element model for the veri ation phase

Thevalidityanda ura yoftheANSYSmodelutilisedwithin theoptimisation

pro e-dureis veried a-posteriori in this work, by using an advan ed higher-order formulation.

This rened solutions make use of the Carrera Unied Formulation (CUF), a ording to

whi h the three-dimensional displa ement eld

u(x, y, z)

an be expressed as a general expansionof the primary unknowns. In the aseof one-dimensionaltheories,one has:

where F

τ

are arbitrary fun tions of the oordinates

y

and

z

on the ross-se tion of thebeamstru ture,

u

τ

is theve tor of the generalized displa ements whi h layalong the beam axis

x

and

M

stands for thenumberof termsused inthehigh-order expansion. To be remarked that in Eq. (25) (as well as in the rest of the equations of this subse tion)

Einsteinsummation onvention onrepeatedindi esis ta itlyassumed.

The hoi eofF

τ

determinesthe lassofthe1DCUFmodelthatisrequiredand subse-quently to be adopted. For example,ifLagrange polynomials are usedasF

τ

,Layer-Wise (LW)theoriesfor ompositestru tures anbeeasilyimplemented,see[36℄. Unlike lassi al

modelsforlaminateswhi hareavailablein ommer ialsoftwaretools,theunknownsofthe

problem(and, thus, the numberof DOFs) arelayer-dependent inthe ase of LW models.

In this manner, it is possible to satisfy the ontinuity of the transverse stresses and the

zig-zag behaviour of the displa ements along thethi kness of the omposite stru ture, in

a ordan e withthe equilibriumand ompatibilityequations of elasti ity.

One ofthe most important advantages of CUFisthatitallows to writethegoverning

equations and therelatednite element arrays of low-order to high-delityLWmodels in

anuniedmanner. Generallyspeaking,CUF anbeusedtogenerateniteelementswhose

formalmathemati alexpressionsareindependentofthetheorykinemati s. Forexample,in

this work,the riti albu kling loads are al ulated bylinearising thegeometri nonlinear

governing equations and evaluating the loads that make the linearised tangent stiness

matrix singular; i.e.

|K

T

| ≈ |K + K

σ

| = 0

, where

K

isthe linearstinessmatrix and

K

σ

isthe geometri stiness matrix.

The linear stiness matrix an be evaluated fromthe virtual variation of the internal

work,whi h holds

δL

int

=

Z

l

Z

Ω

δǫ

T

σdV ,

(26)

where

ǫ

and

σ

arethestrainandstress ve tors(Voigt'snotation),

Ω

isthe ross-se tionof thebeamstru ture and

l

isthe beam length. By substituting the onstitutive and linear geometri alrelationsaswellasCUF(Eq.(25))anda lassi alniteelementapproximation

along the beam axis

x

, su h that

u

τ

(x) = N

i

(x)u

τ i

, the virtual variation of the strain energy reads:

δL

int

= δ

u

T

τ i

K

ijτ s

u

sj

,

(27)

where u

τ i

is the ve tor of the nite element unknowns and

i

represents summation on the nodes of the beam element. K

ijτ s

represents the

3 × 3

fundamental nu leus of the stiness matrix, whi h an be expanded a ording to

(i, j)

and

(τ, s)

to obtain thenite element array of the generi beam theory [37℄. Similarly, the gometri stiness matrix

K

σ

an be expressed in terms of fundamental nu leus by evaluating the linearisation of the virtual variation of the strain energy and, subsequently, by linearising the nonlinear

geometri relations [38℄. Thismatrix,infa t, represents the ontributionof thepre-stress

on thestinessof thesystem. It isimportant to underlinethat, inthis work, asa urate

LWmodelsofthe reinfor ed ompositepanelsareimplemented,thefullthree-dimensional

stress eldis taken into a ount for evaluating thegeometri stiness matrix

K

σ

. This is not true inthe ase of the ANSYSmodel employed intheoptimisation pro edure, whi h

Before starting themulti-s ale optimisation pro ess a referen e stru ture must be

de-nedinordertoestablishreferen evaluesfortheRU massaswellasfortherstbu kling

loadofthe stienedpanel: both material andgeometri al propertiesofthereferen e

solu-tion arereported in Tables 1 and 3, respe tively. Thereferen e solution is subje t to the

samesetofBCs,i.e., Eqs. (23)and(24),asthoseapplied ontheRUofthepanelthatwill

be optimised. One an noti e thatthe referen estru ture hasalaminated skin omposed

of 28 plies and disposed a ording to a symmetri , balan ed sta k (therefore the

result-ing laminate is un oupled and orthotropi in membrane, but not inbending), whilst the

stringer laminateis madeof 32plieswitha symmetri quasi-isotropi sta k(thelaminate

isun oupledandthemembranestinessmatrixisisotropi ,butthebendingoneistotally

anisotropi ). This referen e solution orresponds to a lassi al onguration utilised in

theaeronauti al eld: itsmassand its stinessproperties(in termsof bu kling load)still

represent a good ompromisebetween weight andstinessrequirements.

Regarding the setting of the geneti parameters for the GA BIANCA utilised to

per-formthesolutionsear hfor both rstandse ond-levelproblemstheyarelistedinTable4.

Moreover, on erningthe onstraint-handling te hniquefor therst-levelproblemthe

Au-tomati Dynami Penalisation (ADP) method has been onsidered, see [29℄. For more

detailsonthe numeri al te hniquesdeveloped withinthenewversionofBIANCAandthe

meaning of the values of the dierent parameters tuning the GA the reader is addressed

to [20℄.

6.1. Optimum ongurations of the panel

Theoptimumvaluesofbothgeometri andme hani al designvariables(dimensionless

variables) resulting from the rst-level of the optimisation strategy are listed inTable 5.

When omparingtheoptimumsolutionofthe rst-levelproblemwiththereferen e

ong-uration,one annoti ethenumberofpliesredu esfrom28to20fortheskinlaminateand

from32 to28 for the stringer. Moreover,both laminatesarequasi-homogeneousand fully

orthtropi (both membrane and bending stiness matri es) with an ordinary orthotropy

shape(parameter

K

A

∗

= 0

be ausethe anisotropi polarmodulus

R

A∗

0K

ispositivefor both ases, see [11℄). However, skin laminate gets a lower value of polar parameter

R

A∗

1

(an order of magnitude lower than the orresponding value of

R

A∗

0K

) whi h means that this solutions tendstoexhibit asquare symmetri behaviour(forboth membrane andbending

stinessmatri es), asillustrated in the polar diagrams ofFig. 4. For a deeperinsight on

theseaspe tsthe interested reader isaddressedto [11,21℄.

Table 6reportsthersttwo beststa kingsequen es, forbothskinandstringer,whi h

represents just as many solution for problem (22). As stated in Se tion 4 the

se ond-level problem is solved in the spa e of QT sta ks. In this ba kground, after xing the

number of plies

n

and the number of saturated groups

n

g

the design variables are the identier of the QT solution as well as theorientation angle of ea h saturatedgroup, see

Eq. (21). Be auseproblem(22) ishighlynon- onvexinthespa e oftheorientation angles

ofsaturatedgroups, itispossibletondseveralsolutions(theoreti allyaninnitenumber)

meeting the optimum value of the laminate polar parameters provided by the rst-level

problem.

For the problem at hand, the number of plies for both skin and stringer laminates,

(

n

S

and

n

B

,respe tively)is adire t resultof therstlevel problem,while thenumber of saturatedgrouphasbeen setequal to

•

four for sta kS2,

•

ve forsta kB2.

Asit an be easily inferredfrom theresults listed inTable 7,by ombining the

previ-oussta ksitispossibleto getfour dierent optimum ongurations ofthestienedpanel.

Indeed, these optimumpanels really represent equivalent solutions. Sin e they share the

same ma ros opi geometri al parameters they have the same mass, i.e.,

M = 0.814

Kg whi hrepresents asigni ant redu tion(

−11.5%

) when ompared tothereferen e ong-uration. Furthermore, these optimal ongurations dier only in terms of the optimum

sta k omposing skinand stiener laminatesbut they show almost thesame bu kling

re-sponse: the per entage in rement of therst bu kling load(with respe tto the referen e

value

λ

ref

)ranges from

9%

to

9.5%

,seeTable7.

Therefore, ea h optimum onguration is simultaneously lighter and stier than the

referen e one and this result has been a hieved only by abandoning the usual

engineer-ing rules and hypotheses related to the nature of the sta king sequen e of the laminates

omposing the panel.

Fig. 4shows thedeformed shape relatedto therst bu klingmode aswell astherst

omponent of the normalised stiness matri es of the laminate, i.e.,

A

∗

,

B

∗

and

D

∗

for

both skinand stringer for the onguration S1-B1: the solidline refersto the membrane

stiness matrix, the dashed one to the bending stiness matrix, while the dash-dotted

one is linked to themembrane/bending oupling stiness matrix. It an be noti ed that

thelaminate is un oupled asthe dash-dotted urve disappears,homogeneous as thesolid

and dashed urves are oin ident and orthotropi be ause there are two orthogonal axes

of symmetry in the plane. In addition, for both laminates the main orthotropy axis is

oriented at

Φ

A

∗

1

= 0

◦

a ording to the hypothesis of the rst-level problem. The same onsiderations an be repeatedalso for therestof theoptimumsolutions.

6.2. Veri ation of the optimum ongurations

Aone-dimensional, high-ordermodelbasedonCUFisusedfor validatingthereferen e

andoptimisedRU analyses. Thepresent CUFmodelemploysa LWrenedkinemati sfor

the a urate des ription of the pre-stress state of the RU subje ted to ompression and,

thus, for enhan ed evaluation of bu kling loads. The CUF-LW models of the referen e

and optimised RU panels have

372588

and

333792

DOFs, respe tively. As inthe ase of the ANSYS model, PBCs are imposed by using the dire t penalty approa h. However,

it is important to underline that, be ause the employed LW CUFmodels have only pure

translational displa ements asunknowns,only therstline of Eq. (24)is enfor ed.

The rst bu kling mode ofthe optimum onguration S1-B1is shownin Fig.5. That

of the referen e onguration as well as those asso iated to the other optimum solutions

areequivalent,thustheyarenotdepi tedforthesakeofbrevity. For ompletenessreasons,

however, the through-the-thi kness stressdistributions (seeFig.6)a ordingto CUFand

ANSYSaregiveninFigs.7and8. Theseguresshowthedistributionsofaxial,

σ

xx

, trans-verse shear,

σ

xz

, and transverse normal,

σ

zz

,stress omponents. It should be underlined thattheadoptedANSYSmodelprovidesagooddistributionof axialstresses. In ontrast,

and a ording to CUF referen esolutions, the ANSYSFE model is not able to take into

a ount shear and transverse normal stresses and this would dire tly ae t the a ura y

ofthebu kling al ulation.

Table 8 summarises the rst riti al bu kling loads given by CUF high-order beam

models and they are ompared to those resulting from ANSYS model. The dieren es

panels range from

7.4 %

to

7.9 %

, while for the referen e onguration the per entage dieren e is signi ant (up to 14

%

). This higher dis repan y is probably related to the anisotropi bending behaviourof the referen e solution. These dieren es arereasonable

andarerelatedtothe3Dstressdistributionswithinea h onstitutivelayerandthedierent

order of a ura y hara terising the CUF LW beam model. Of ourse, this stress eld

strongly ae ts the geometri stiness matrix and annot be a quired by ANSYS shell

elementswhi h arebasedon the FSDThypotheses.

Itisnoteworthythat,a ordingto CUFnumeri alresults,thegainintermsofstiness

is even higher than that foreseen by ANSYS, ranging from

15.2%

for solution B1-S1 to

15.8%

for solutionB2-S2, assummarised inTable 9.

7. Con lusions

The design strategy presented in this paper is a numeri al optimisation pro edure

hara terisedby several featuresthat make itan innovative,ee tive and general method

for the multi-s ale design of omposite stru tures. In the present work this strategy has

been applied to themulti-s ale optimisation ofthe repetitive unitof a ompositestiened

panel.

On the one hand, the design pro ess is not submitted to restri tions: any parameter

hara terisingthestru ture(atea hrelevants ale)isanoptimisationvariable. Thisallows

sear hingfor atrue global minimum withoutmakingsimplifyinghypothesesonthenature

ofthelaminate sta king sequen e. Ontheotherhand,themulti-s ale design problemhas

been split into two optimisation sub-problems whi h are solved subsequently within the

same numeri al pro edure.

The rst-level problem fo uses on the ma ros opi s ale of the panel: ea h laminate

omposing thestru tureis onsidered asan equivalenthomogeneous anisotropi plate (for

both skinand stringer) and its ma ros opi me hani al responseis des ribed in terms of

polar parameters. Furthermore, alsogeometri design variablesdes ribing thetopology of

both skin andstiener areinvolved at this level. At this stage, theme hani al properties

of the multilayer plates are represented by means of the polar formalism, a

mathemati- alrepresentation basedontensorinvariantswhi his hara terisedbyseveraladvantages.

The main features of thepolar method arethe possibility to represent inan expli it and

straightforward waytheelasti symmetriesof thelaminate stinessmatri esandto

elimi-natefromthe optimisation pro edureredundant me hani al properties.

The se ond level of the pro edure is devoted to the laminate mesos opi s ale: the

goalisto nd at leastoneoptimum sta k(forboth stringerand skin)meeting ontheone

handtheelasti requirementsimposedtothelaminate(quasi-homogeneityandorthotropy)

during the rst-level problem and on the other hand the optimum value of the laminate

polar parameters resulting fromtherst step.

The utilisation of an evolutionarystrategy, together with thefa tthat theproblem is

statedinthemostgeneral sense,allowsndingsomenon- onventional ongurationsmore

e ient than the standard ones. In fa t, the onsidered numeri al example proves that,

when standard rules for tailoring laminate sta ks are abandoned and all the parameters

hara terising the stru ture are in luded within the design pro ess, a signi ant weight

saving anbeobtained: upto11.5

%

withrespe ttothereferen estru turewithenhan ed me hani alpropertiesintermsofrstbu klingload(theper entagein rementrangesfrom

9

%

to 9.5

%

dependingon the onsideredoptimumsolution).

Inase ondtime,bothreferen eandoptimum ongurationsofthestienedpanelhave

(whi h is built by using shell elements based on FSDT) is overestimated and that the

per entagedieren erangesfrom

7.4÷7.9%

foroptimumsolutionsto14

%

forthereferen e onguration. Thisdis repan y isrelated to the al ulation of the3D stresseld inea h

layerwhi hstronglyae tsthegeometri stinessmatrixusedtoevaluatetherstbu kling

loadof the panel.

Nevertheless,despitethesedis repan ies, lassi alshellelementsbasedonFSDT anbe

reliablyemployedintheframework oftheMS2Loptimisation strategy be ause theyallow

nding true optimumsolutions without using expensive models,in terms of both

num-ber of DOFs and omputational ost. Moreover, a ording to CUF results, the optimum

ongurations arereally e ient when ompared to the referen e one: the weight saving

isalways the same, but thegainin termsof stinessis even higher than thatforeseenby

ANSYS,ranging from

15.2%

to

15.8%

dependingon theoptimumsolution.

These results unquestionably prove the ee tiveness and the robustness of the

opti-misation approa h proposed in this work and provide onden e for further resear h in

this dire tion. Asan example,future worksmay fo uson oupling thepresent MS2L

op-timisation strategy with high-order models based on CUF. These onsiderations remain

still valid if further requirements (e.g., strength, fatigue, delamination, et .) have to be

in ludedinto thedesign problemformulation. Alloftheseaspe ts an beeasily integrated

within the MS2L optimisation strategy without altering its overall ar hite ture and they

do not represent a limitation to the proposed strategy,on the ontrary they ould be an

interesting hallengefor futureresear hes onreal-world engineering appli ations.

A knowledgements

G.A.FiordilinoisgratefultotheNouvelle-Aquitaine regionfor its ontributiontothis

paperthrough the SMARTCOMPOSITE proje t. A. Pagani also a knowledges nan ial

support from the Compagnia di San Paolo and Polite ni o di Torino trough the proje t

ADAMUS.

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Te hni al onstants Polarparameters of

Q

a

Polarparameters of

ˆ

Q

b

E

1

[MPa℄

161000.0

T

0

[MPa℄

23793.3868

T

[MPa℄

5095.4545

E

2

[MPa℄

9000.0

T

1

[MPa℄

21917.8249

R

[MPa℄

1004.5454

G

12

[MPa℄

6100.0

R

0

[MPa℄

17693.3868

Φ

[deg℄

90.0

ν

12

0.26

R

1

[MPa℄

19072.0711

ν

23

0.10

Φ

0

[deg℄

0.0

Φ

1

[deg℄

0.0

Densityand thi kness

ρ

[Kg/mm

3

℄

1.58 × 10

−6

t

ply

[mm℄

0.125

a

In-planeredu ed stinessmatrix of theply.

b

Out-of-plane shearstiness matrixof the ply.

Table1: Materialpropertiesofthe arbon-epoxyplytakenfrom[11,21,22℄.

Designvariable Type Lowerbound Upperbound Dis retisationstep

ρ

0

S

ontinuous

−1.0

1.0

-ρ

1S

ontinuous

0

1.0

-ρ

0

B

ontinuous

−1.0

1.0

-ρ

1

B

ontinuous

0

1.0

-c

1

dis rete

0.1

0.45

0.001

c

2

dis rete

1.00

3.00

0.01

c

3

dis rete

1.00

3.00

0.01

n

S

integer

20

32

1

n

B

integer

20

32

1

Table2: Designspa eoftherst-levelproblem.

a

[mm℄

150.00

b

[mm℄

600.00

a

2

[mm℄

15.00

a

3

[mm℄

21.50

h

[mm℄

30.00

M

ref

[Kg℄

0.92

λ

ref

[N℄

445074

Sta king sequen e Part N.of plies

[(45/ − 45/90

2

)

2

/(45/ − 45)

3

]

s

skin(S) 28

[45

2

/0

2

/ − 45

2

/90

4

/ − 45

2

/0

2

/45

2

]

s

stringer (B) 32 Table3: Referen esolutionforthestienedpaneldesignproblem.

1

st

levelproblem 2

nd

levelproblem N. ofpopulations

1

1

N. ofindividuals

200

500

N. ofgenerations

150

500

Crossoverprobability

0.85

0.85

Mutation probability

0.005

0.002

Sele tion operator roulette-wheel roulette-wheel

Elitism operator a tive a tive

Table4: Geneti parametersoftheGABIANCAforrstandse ond-levelproblems.

Geometri parameters

a

2

[mm℄

a

3

[mm℄

h

[mm℄

n

S

n

B

21.300 29.607 31.950 20 28 Polarparameters

R

A∗

0K

[MPa℄

R

A∗

1

[MPa℄ Skin(S) 3511.00 242.36 Stringer(B) 9391.51 12080.84

Table5: Numeri alresultsoftherst-leveloptimisationproblem.

ID Beststa kingsequen e N.ofplies

Skin(S) S1

[−63/0/63/0/63/ − 63/0/0/63/ − 63/63/ − 63/0/0/63/ − 63/0/ − 63/0/63]

20 S2

[43/90/0/0/ − 43/90/ − 43/90/0/ − 43/43/90/0/43/0/43/90/90/0/ − 43]

20 Stringer(B) B1

[1/61/1/1/1/ − 51/1/1/ − 51/1/1/1/61/1/1/ − 51/1/1/1/61/1/1/61/1/1/1/ − 51/1]

28 B2

[0/59/ − 1/ − 54/2/0/2/2/2/0/ − 54/ − 1/59/2/0/0/ − 54/ − 1/0/59/0/2/59/2/ − 1/ − 54/2/0]

28

Table6:Numeri alresultsofthese ond-levelproblem(rsttwooptimumsta ksforbothskinandstringer).

Panel ongurations

REF S1-B1 S1-B2 S2-B1 S2-B2

M

[Kg℄

0.920

0.814

(

−11.5%

)

λ

[N℄

445074

483951

(

9%

)

483838

(

9%

)

487493

(

9.5%

)

487386

(

9.5%

) Table7: Propertiesoftheoptimumsolution(intermsofmassandbu klingload)fordierentskin-stringer

ongurations; for ea h property the per entage dieren e between the optimum onguration and the

λ

[N℄ REF S1-B1 S1-B2 S2-B1 S2-B2

CUF

390870

450323

450430

451843

452615

ANSYS

445074

(

14%

)

483951

(

7.5%

)

483838

(

7.4%

)

487493

(

7.9%

)

487386

(

7.7%

) Table8: Comparisonofthebu klingload

λ

al ulationbetweenANSYSFEmodelandhigh-orderbeam CUFmodelforbothreferen eandoptimumsolutions;theper entagedieren ebetweenANSYSandCUF

modelsisindi atedinparentheses.

Panel ongurations

REF S1-B1 S1-B2 S2-B1 S2-B2

λ

[N℄

390870

450323

(

15.2%

)

450430

(

15.2%

)

451843

(

15.6%

)

452615

(

15.8%

) Table9: Comparisonofthebu klingloadprovidedbythehigh-orderbeamCUFmodelforbothreferen e

andoptimumsolutions; theper entagedieren e betweenea hoptimum onguration andthereferen e

Figure1: (a)Geometryandoverallsizeofthestienedpanel(onlytworepetitiveunitsarehererepresented

mode(normalizeddispla ement)and polardiagramof therst omponent ofthehomogenizedlaminate

in-planestinessmatri es[MPa℄for(b)skinand( )stringer.

A'

A

B'

B

Figure6: Cross-se tionofthepanelRU.

−1000

−800

−600

−400

−200

0

0

0.5

1

1.5

2

2.5

σ

xx

[MPa]

t

s

[mm]

(a)Skin(A-A')-

σ

xx

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0

0.5

1

1.5

2

2.5

σ

xz

[MPa]

t

s

[mm]

(b)Skin(A-A').

σ

xz

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0

0.5

1

1.5

2

2.5

σ

zz

[MPa]

t

s

[mm]

( )Skin(A-A').

σ

zz

Figure7:Mid-spandistributionsofstresses omponentsthroughtheskinthi kness(A-A')oftheoptimum

panelS1-B1;solidline isCUFsolution, ir les

◦

representANSYSsolution.

−1000

−800

−600

−400

−200

0

0

0.5

1

1.5

2

2.5

3

3.5

σ

xx

[MPa]

t

b

[mm]

(a)Stringer(B-B').

σ

xx

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

3

3.5

σ

xz

[MPa]

t

b

[mm]

(b)Stringer(B-B').

σ

xz

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0

0.5

1

1.5

2

2.5

3

3.5

σ

zz

[MPa]

t

b

[mm]

( )Stringer(B-B').

σ

zz

Figure 8: Mid-span distributions of stresses omponents through the stringer thi kness (B-B') of the