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Original Article

LatinAmericanJournalofSolidsandStructures,2018,15 3 ,e22

Finiteelementmodellingtoassesstheeffectofpositionandsizeofthepiezoelectriclayerofahybridbeam

AbstractAonedimensionalfiniteelementmodelispresentedtoassesstheeffectofpositionandsizeofthepiezoelectric layerofahybridbeam.Theefficientlayerwise zigzag theoryisusedformakingthefiniteelementmodel.The1Dbeamelementhaseightmechanicalandavariablenumberofelectricaldegrees of freedom. The codes are developed inMatlab based on the FEformulation.Thebeamsarealsomodelledin2DplanarmodellingspaceasadeformableshellusingFEpackageABAQUSforcomparisonofresults.An8‐nodedpiezoelectricquadrilateralelementisusedforpiezolayersandan8‐noded quadrilateral element with reduced integration is used for theelastic layers of hybrid beams for making the finite element mesh inABAQUS. The accuracy of the used elements are assessed for static re‐sponse.Cantileverhybridbeamswithapiezoelectric layerbondedontopoftheelasticsubstrateareconsideredfortheanalysis.Thebeamsaresub‐jectedtoelectromechanicalloading.Adetailedstudyisconductedtohigh‐lighttheinfluenceofpositonandsizeofpiezoelectric layeronthedeflec‐tionprofiles, tipdeflectionsandthroughthethicknessdistributionofdis‐placementsandstressesofhybridcomposite/sandwichbeams.Theshapecontrolusingvariousnumbersofpiezoelectricpatchesisalsostudied.The1D‐FEresultsarecomparedwiththe2D‐FEresults.

KeywordsZigzagTheory,FEM,HybridBeam,ABAQUS,Piezoelectricpatch,shapecon‐trol.

1INTRODUCTION

Smartmulti‐layered hybrid beamswith some sensory and actuator piezoelectric layers constitute an im‐portantelementofadaptivestructures.Thesestructuralsystemsaregenerallymadeofcompositeandsandwichlaminatesbecauseoftheirhighstiffnesstoweightratio.Considerableattentionhasbeenreceivedinliteratureonbehaviourofhybridsmartstructuresandreviewedbymanyresearchers Benjeddou,2000;SunarandRao,1999;SaravanosandHeyliger,1999;Tangetal.,1996 .SaravanosandHeyliger 1995 developedaUnifiedmechanicswith the capability tomodelboth sensoryandactive composite laminateswithembeddedpiezoelectric layers.Layerwiseformulationsenableanalysisofbothglobalandlocalelectromechanicalresponse.Theypresentedanapproximate finiteelement solutions for the staticand freevibrationanalysisofbeams.But thecomputationaleffortincreaseswiththenumberoflayers.ManyresearchersusedEquivalentsinglelayertheoryapproximationsCorreiaetal.,2000;MitchellandReddy,1995 forthestructuralanalysis.Sincethesameglobalvariationofdis‐placementisassumedacrossthethicknessindependentofmaterialpropertiesandlay‐up,suchanalysisfailstoreportthezigzagnatureofin‐planedisplacementsvariations.

Fewstudiesonstructuralanalysisoflaminatesusedhigher‐order/layer‐wisetheoriesassumingthroughthethickness variation for the electric field of piezoelectric layers Batra and Vidoli, 2002; Fukunaga et al., 2001;Yang,1999 .Kapuriaetal. 2004 developedatwonodedfiniteelementmodelforstaticelectromechanicalanal‐ysisofhybridbeambasedonefficientcoupledzigzagtheory Kapuriaetal.,2003 .Ganapathietal. 2004 devel‐opedaC1finiteelementforthebendingandtorsionalanalysisofpiezoelectriccompositebeams.Theirformula‐tion includes transverse shear,warpingdue to torsion, andelastic–electric couplingeffects.Kapuria andHage‐dorn 2007 presentedaunifiedcoupledefficient layerwisetheoryforsmartbeamsanddevelopedafiniteele‐mentwhichhastwophysicalnodesandanelectricnodefortheelectricpotentialsoftheelectrodedsurfaces.The

NajeeburRahmana*M.NaushadAlama

aDepartmentofMechanicalEngineering,ZakirHusainCollegeofEngineeringandTechnology‐Z.H.C.E.T,AligarhMuslimUniversity‐A.M.U,Aligarh,UttarPradesh,India.E‐mail:najeeba‐[email protected],naus‐[email protected]

*Correspondingauthor

http://dx.doi.org/10.1590/1679-78253959

Received:April25,2017InRevisedForm:October06,2017Accepted:October17,2017Availableonline:February05,2018

NajeeburRahmanetal.Finiteelementmodellingtoassesstheeffectofpositionandsizeofthepiezoelectriclayerofahybridbeam

LatinAmericanJournalofSolidsandStructures,2018,15 3 ,e22 2/17

considerationofelectricnodesresultinsignificantreductioninthenumberofelectricdegreesoffreedom.Rah‐manandAlam 2012,2014,2015 usedlayerwise zigzag theoryfor1D‐FEmodellingofsmartlaminatedbeam.KapuriaandKulkarni 2008 proposedanefficient four‐nodequadrilateralelementusingzigzag theory for thedynamic analysis of hybrid plateswith segmented piezoelectric sensors and actuators. The theory considers athird‐orderzigzagapproximationforinplanedisplacements,alayerwisequadraticapproximationfortheelectricpotential and a layerwise variation of the deflection to account for the piezoelectric transverse normal strain.Catapanoetal. 2011 adoptedCarrera’sUnifiedFormulationforlinearstaticanalysisofcompositebeams.Theyused an N‐order polynomial approximation of the displacement unknown variables to impose kinematic fieldabovethecross‐section.Komeilietal. 2011 presentedthestaticbendinganalysisoffunctionallygradedpiezoe‐lectric beams under thermo‐electro‐mechanical load. They derived the governing equations from Hamilton’sprinciple and used the finite element method and Fourier series method as solution technique. Filippi et al.2015 presentedanewclassofrefinedbeamtheoriesforstaticanddynamicanalysisofcompositestructures.They implemented higher‐order expansions of Chebyshev polynomials for the displacement field componentsoverthebeamcross‐section.Sayyadetal. 2015b developedatrigonometricbeamtheoryforthebendinganaly‐sisoflaminatedcompositeandsandwichbeams.Theyconsideredtransversedisplacementasasumoftwopartialdisplacements intheir formulation.Trigonometric functionisusedintermsofthicknesscoordinate intheaxialdisplacementfieldtoincorporatetheeffectoftransversesheardeformation.Inanotherstudytheparabolicfunc‐tionisusedtoincludethiseffect Sayyadetal.,2015a .Giuntaetal. 2016 usedamesh‐freestrong‐formsolutiontoexaminethestaticresponseoffunctionallygradedbeams.Theyderivedalgebraicsystemviacollocationwithmultiquadricradialbasisfunctions.

Afterexhaustiveliteraturesurveyitcanbeconcludedthatlotsofresultsareavailableforcompositebeamsbasedonvarioustheories.Butthemostrecentefficientzigzagtheoryhasstillsomescope.Inthereportedlitera‐turevariousaspectsofzigzagtheoryhasnotbeenstudied.Theshapecontrolofstructureisalsoveryimportantinapplicationssuchasantennasmountedonairplaneorsatellites,whereasmalldistortioninshapeorsurfaceer‐rorsresultindegradedperformancecausingsignificantdifficultieswithrespecttoitsusability.Modernapplica‐tionsrequiresmartbehaviourofstructures.Piezoelectriclayersoftenformanintegralpartofsuchsmartstruc‐tureowingtotheirabilitytotransformelectricalenergytomechanicalenergyandviceversa.The locationandsizeof these layers is significantas it results indifferentstructural responseundervarious loadingconditions.FiniteElementmodeltoassesstheeffectofpositionandsizeofpiezoelectriclayerofahybridbeamunderstaticelectromechanicalloadingispresentedinthiswork.Theefficientlayerwise zigzag theory Kapuriaetal.,2004 isreframedformakingthefiniteelementmodelofhybridbeamwithsegmentedpiezoelectricpatches.ThecodesaredevelopedinMatlabbasedontheFEformulation.Staticresponseareobtainedforthreedifferentloadcasesfordifferentpositionsandsizesofpiezoelectriclayerbondedontopsurfaceofthebeam.Theshapecontrolisalsoobtainedusingvariousnumbersofpiezoelectricpatches.Theresultsarepresentedforthinandmoderatelythickcomposite/sandwichbeams.Toshowtheaccuracyof thepresent formulation, the1D‐FEresultsarecomparedwiththe2D‐FEresultsobtainedusingplanestresselementinABAQUS.

2GOVERNINGEQUATIONSFORHYBRIDPIEZOELECTRICBEAM

Considerahybridbeamwithpiezoelectricpatchesbondedtoitssurfaces.Thebeamisdividedintosegmentsduetothepresenceofthesepatchesofvariablesizepositionedatvariableaxiallocations Figure1 .Thehybridsegmentshavepiezoelectricpatchesbondedtothesurfaceofelasticsubstratehowevertheelasticsegmentsarewithoutthesepatchesandconstituteoftheelasticsubstrateonly.Asaresult,thethicknessofbeam,numberoflayersacrossthethicknessandtopandbottomsurfacesatanysectionmaydiffersegment‐wisealongthelength.Theplanewhichisthemid‐planeformostofthelengthofthebeamisidentifiedasthe xy plane.Atanysection,

0z isthethicknesscoordinateofthebottomsurfaceand Nz isthethicknesscoordinateofthetopsurface.Allthe

layersareconsideredtobeperfectlybonded.Thebeamistransverselyloaded.Theloaddistributionisindepend‐entofthewidthcoordinate.The z axisisthepolingdirectionforpiezoelectriclayers.



Figure1:Geometryofhybridbeamdividedintoelasticandhybridsegments

Usingcoupledzigzagtheoryapproximations Kapuriaetal.,2004 forapiezoelectricmedium,theconstitu‐tiveequationsrelatingthestresses x , zx andelectricdisplacements xD , zD withthestrains x , zx and

electricfields xE , zE aregivenby:

11 31

1555

15 11

31 33

ˆ ˆ

0

ˆ ˆ

0 0

00

0 ˆ

ˆ 0

ˆ 0

ˆ

x x x

zx zx z

xx x

zxz z

Q e E

e EQ

eD E

eD E

1

where 11Q and 55Q are the coefficients of reduced stiffnesses; 31e and 15e are stress constants of piezoelectric

materialand 11 and 33 aretheelectricpermittivities.Thepotentialfieldisapproximated Kapuriaetal.,2004 as

, l lx z z x 2

where ,l lx x z , l z arelinearinterpolationfunctionsfor .

Considering only electric contribution in transverse strain, z Kapuria et al., 2004 and integrating it

throughthethickness,thedeflection zu isestimatedas

0

, zl l

zu x z u x z x 3

where 0zu x is the deflection of mid‐plane z 0 defined as

0 0,0z zu ux x and l z is a piecewise linear

functiongivenby 33

0

z ll d z

z d dzdz

.

Theaxialdisplacement xu isapproximated Kapuriaetal.,2004 as

0

0 0,l

z k klx x

uu x z u x z x S z x S z x

x x

4

where , k klS z S z are the cubic functions of z;0xu and 0 are the translation and the rotation variables

respectively for the 0thk layer. The functions , k klS z S z are so chosen that the assumed displacement field

ensures the continuity of xu and zx at the interfaces of layers across the thickness of the laminate and alsosatisfiesthezeroshearstressconditionatthebottomandtopsurfacesofthebeam.

Thedisplacementsfieldmaybeexpressedintermsofgeneralizeddisplacements 1u and 2 u as:



1 1x2 1

1x4 2 2

x

z

g z Ou u

O g zu u

5

with

1 2 1 , 1 k kl lg z z S z S z g z z 6

0

0 01 0 2,

Tl Tz lx z

uu u u u

x x

7

where 1x2O is null matrix of order 1x2 and 1x4O is a null matrix of order 1x4, the index 1,2, , l n .

Substituting the assumed displacement field and potential field in partially nonlinear strain‐displacementrelationsandelectricfield‐potentialrelationsrespectively,thestrainsandtheelectricfieldsmaybeobtainedas:

0 1 2 3

0 1 2 3

4 5 6

4 5 6

x x x x xk kl

zx zx zx zx zx

k klx x xl

zx zx zx

z S z S z

S z S zz

z z

8

and

1 2

1 2

lx x xl

z z z

zE E Ez

E zE E

9

wherethevariousstraincomponentsare:

2 20 1 2 00 0 0

20 1 2

23 4 5

23 4 5

0

12 , , ,

000

0, ,

0

x z zx x x

zx zx zx

l

x x x

zx zx zx

u u u

xx x x

x

6

6

0 0

, xl l

zxx x

10

andtheelectricfieldcomponentsare:

1 2

1 2

0,

0

l

x xl

z z

E Ex

E E

11

Thestrainsmayalsobeshownintermsofthegeneralisedbeammechanicalstrains 1 and 5 as:

20

11 1x2

51x4 5

1 / 2

0

zx

zx

ug z Ox

O g z

12

where 1 and 5 aregivenby:

2 20 0 0

1 5 02 2,

T Tl lx zu u

x x xx x

13



andthevector 5g z isdefinedas

5

k kllS z S z

g z zz z

14

Thevariationalequationforthebeamisobtainedas:

0 0

20

1 1 5 5

2 4

**

* * * * * * * * * * *0 000 0

0

l TT T l l l

T zT l

a

zx

F F H Gx

dxu u

u q q Nx x

au lz l l l lN u V u M P H V Rx x x z x x xx x

15

wherethesuperscript*meansvaluesattheends. 1 5 ,, xF F V and lV arethebeamstressresultantsgivenby:

1 1

1 51 1

1

, k k

k k

k

z zN Nk k

k x zxkllk kz z

kl

S zz z

F bdz F bdzS z S z

zzS z

16

1 11 1

, k k

k k

z zN Nk kl l

x zx zxk kz z

V bdz V z bdz

17

and lH and lG arethebeamelectricdisplacementresultantsgivenby:

1 11 1

, k k

k k

z z lN Nk kl l l

x zk kz z

zH D z bdz G D bdz

z

18

2 q and 4lq arethemechanicalloadandelectricalloadrespectively.

3FINITEELEMENTMODEL

Aonedimensionalfiniteelementmodel 1D‐FE basedonefficientlayerwise zigzag theoryisusedfortheanalysis.Twonodedelements Kapuriaetal.,2004 areconsideredfortheelectromechanicalvariables Figure2 .The1Dbeamelementhaseightmechanicalandavariablenumberofelectricaldegreesoffreedom.

Figure2:Twonoded1Dbeamelementwithdegreesoffreedom



Thehighestorderofthederivativesofthevariables0z

u and l ,inthevariationalEq. 15 whenexpandedin

termsoftheprimaryvariables0 0 0, ,x zu u and l ,is2andthatoftheothertwovariables

0xu and 0 is1.There‐

fore,inordertosatisfytheconvergencerequirementsofthefiniteelementprocedure 0

0 0,,z

x

uu

x

and

l

x

are

requiredtobecontinuousattheboundariesoftheelementconsidered.HencecubicHermiteinterpolation Kapu‐

riaetal.,2004 isusedforexpanding0z

u and l whereas0xu and 0 areexpandedusinglinearLagrangeinterpo‐

lation.

00

0 0

0 0

0 0 0

ˆ0 0 0

0 0 0

0 0 0 ˆ e

exx

ez z

e

l l

uu N

u uN

N

N

19

where N and ˆ N are vectors of linear interpolation functions and cubic Hermite interpolation functions

respectively;0 0 0, ,e e e

x zu u andel arethenodalvaluevectors.

TheintegrandinthevariationalEq. 15 maybeshownas

0 02

z zT Tu x

u uF u g N

x x

20

where F is thegeneralised stressvector, is thegeneralised strainsvector and ug is theelectromechanical

loadvectorofthebeamdefinedas

1 5 1 5 2 4, ,

TlT TT T l l T T l luF F F H G g q q

x

21

Substitutingfor F and xN ,theparticipationin eT ofoneelementtotheintegralfrom0to l inEq. 15 isat‐tainedas

0

0 0 0

2

2

20

1 1 1 11

1

2

1

2

zT Tn u

le

z z zT l l

uD g u g

xT dx

u u uA A

x x x

22

where D isthegeneralisedstiffnessmatrix, ng isthematrixthatrelates F withnonlinearstrainand 1A isthefirstcolumnofbeamstiffnessmatrix A .

Assumingsmalldisplacementsandstrainsandconsidering linearelasticbehavior, thenonlineartermsareignored.ThusEq. 22 maybesimplifiedforlinearanalysisas

2

0

le T T

uT D u g dx 23

Thegeneraliseddisplacementvector, ed fortheelementisdefinedas

0 0 0

TT T T T ee e e e lx zd u u 24



Using Eqs. 7 , 13 , 21 and 24 , the generalised displacements 2u and strains can be expressed in

termsof ed as

22 ,e emu B d Bd 25

where2mB and B aredisplacementinterpolationandstraindisplacementmatrices.UsingEq. 25 for 2u and

inEq. 23 , eT maybeshownas

2

0

le eT T e T

m u

eT e e e

T d B DBd B g dx

d K d P

26

with

2

0 0

, l l

e T e Tm uK B DBdx P B g dx 27

Thecontributionsfromalltheelementsaresummeduptoobtainthesystemequationas

Kd P 28

in which K is the assembled stiffnessmatrix and d and P are the assembled displacement and load vectorsrespectively.Theboundaryconditionsforclampedendandfreeendare

0

0 0 0Clamped end: 0, 0, 0, 0,

Free end: 0, 0, 0, 0.

zx z

x x x x

uu u

xN V M P

29

Thebeamismodelledwithseparatesegmentsforelasticpartsandhybridparts.ThehybridpartshavePZT‐5Apatchbondedon the topsurfaceof theelastic substratewhereas theelasticpartsarewithoutpiezoelectricpatch.Theelasticpartshavefournumbersofnodalelectricdegreesoffreedomhoweverhybridpartshave12,asfoursub‐layersareconsideredthroughthethicknessofthepiezoelectriclayer.

Toshow theaccuracyandefficiencyof thepresent formulation, the resultsof1D‐FEmodelare comparedwith the converged2D‐FE ABAQUS results Khanet al., 2016 .Thehybridbeamsaremodelled in2DplanarmodellingspaceasadeformableshellusingFEpackageABAQUS 2010 whereinthelayupwithdifferentmateri‐alpropertiesaredefinedacrossthebeamthicknessandsuitableelementtypesforelasticsubstrateandpiezoe‐lectriclayersareassigned.Thethicknessdirectionisalsodiscretisedfordisplacements.An8‐nodedbiquadraticplanestresspiezoelectricquadrilateralelementisusedforpiezolayersandan8‐nodedbiquadraticplanestressquadrilateralelementwithreducedintegrationisusedfortheelastic layersofhybridbeamsforgeneratingthefiniteelementmeshinABAQUS.Thegeometricorderisconsideredlinearforbothtypeofelements.

4RESULTSANDDISCUSSIONS

4.1Validation

The1D‐FEformulationisfirstvalidatedbyconsideringtheflexuralanalysisproblem Problem2 ofGanapa‐thietal. 2004 forapiezoelectric sandwichbeam.Mechanicalandpotential loadcasesareconsidered for theproblembeamwithcantileverendconditions.Apointloadof1Nisapplied,atthefreeend,fortheformerwhere‐asforlatteranelectricpotential 100V isappliedatthetop/bottomsurfaces.ThegeometricalconfigurationofbeamandthepropertiesofthematerialaretakenfromGanapathietal. 2004 .Theresultsarecompared Table1 forspantothicknessratio / 10S l h .The%ageofdifferenceofthepresent1D‐FEresultswithrespectto

theresultspresentedinGanapathietal. 2004 isinarangeof2.5%‐2.8%forthecaseofmechanicalloadand3.3%‐8.6%forthecaseofpotentialload.



Table1:Comparisonof1D‐FEresultsforcantileverpiezoelectricsandwichbeamwithGanapathietal. 2004 .

Mechanicalloadcase Potentialloadcase

S.No.

Entity Units

1D‐FE

Ganapathietal.2004 *

1D‐FE

Ganapathietal.2004 *

1 710 0.2, 0.05zu m 1.50 1.54 ‐3.17 ‐2.92

2 710 0.2, 0.05xu m 0.72 0.70 ‐1.45 ‐1.50

3 0.2,-0.05x N/m2 23164 23750 ‐5410 ‐6000

*Resultsreadfromgraphs

4.2NumericalExample

Toassesstheeffectofsizeandpositionofpiezoelectricpatchonstaticresponseofasmartbeam,twobeamconfigurations, b and c areconsidered.Thesubstrate b isafourlayerssymmetricgraphiteepoxycompositelaminatewithlayup 0ᴼ,90ᴼ,90ᴼ,0ᴼ .Thethicknessofeachlayerofthecompositelaminateistakenas 0.25h ,where h is the total thicknessofelasticsubstrateatanysection.Thesubstrate c isa three layerssymmetricsandwichwithtopandbottomfacesheetsofthicknesses 0.08h andanin‐betweensoftcoreofthickness 0.84h .Thetotalthicknessofelasticsubstrateatanysectionremainsthesame.Piezoelectricpatches PZT‐5A ofvaria‐blesizearebondedonthetopsurfacesoftheelasticsubstrateatvariedaxial locations.Thethicknessofpiezo‐patches is takenas 0.1h .ThePZT‐5Apatcheshavepolling in zdirection.The topandbottomsurfacesof theelastic substrate are grounded. The properties of thematerials used are listed in Table 2 Rahman andAlam,2015 .

Thefollowingloadcasesareconsidered(1) A uniform pressure 0tq q on the top surface over entire length with open circuit condition.

(2) A uniform pressure 0tq q on the top surface over entire length with closed circuit condition.

(3) A uniform applied potential 0n on the top surface of the piezoelectric patch layer.

Table2:Materialpropertiesforsubstrates b and c .

Material Y1 Y2 Y3 G12 G23 G13 ν12 ν13 ν23 ρGPa kgm‐3

Graphiteepoxy 181 10.3 10.3 7.17 2.87 7.17 0.28 0.28 0.33 1578Face 131.1 6.9 6.9 3.588 2.3322 3.588 0.32 0.32 0.49 1000Core 22.08x10‐5 20.01x10‐5 2.76 0.01656 0.4554 0.5451 0.99 3x10‐5 3x10‐5 70PZT‐5A 61 61 53.2 22.6 21.1 21.1 0.35 0.38 0.38 7600

PZT‐5Ad31 d32 d33 d15 d24 η11 η22 η33

x10‐12mV‐1 x10‐8Fm‐1 ‐171 ‐171 374 584 584 1.53 1.53 1.5

Thenon‐dimensionalisedresultsareobtainedas:

1) 4 3 2 4 2

0 0 0 0 0 0 0 0100 / , 100 / , / , 10 /z z x x x xu u Y hS q u u Y hS q S q Y d hS q

2) 2

0 0 0 0 0 0 0ˆ 10 / , 10 / , 0ˆ 1ˆ /z z x x x xu u S d u u Sd h Y d

where /S l h isthespantothicknessratio, 0Y forbeam b istakenas10.3 GPa andforbeam c 6.9 GPa . 0 d istakenas374x10‐12CN‐1.

4.2.1Staticresponsewithvariationofpiezo‐patchlayerposition

Cantileverbeams b and c ,withapiezoelectricpatchoflength 0.4l ,bondedontopoftheelasticsubstrateatadistance px fromthefixedend Figure3 areconsideredfortheanalysis.Thebeamsaremodelledandana‐

lyzedwithdifferentvaluesof px .



Figure3:Cantileverbeamwithpiezoelectricpatchlayeratvariableposition.

ThedeflectionprofileofcentrelinesarecomparedinFigure4underloadcase3.Herethemid‐surfacedeflec‐tionismoreforpiezo‐patchpositionsnearertothefixedend.Therefore,thepiezoelectricpatchactuatorsshouldbeplacednearertothefixedendforcontroloftipdeflectionincantileverbeams.However,thetipdeflectionismoreasthepiezo‐patchpositionismovedfartherfromthefixedendofcantileverbeamforpressureloadcases1and2 Figure5 .Thisisduetoincreasedbendingmomentresultingfromlargerdistanceofpiezo‐patchfromthefixedend.Theaxialdisplacementandnormalstressdistributionacrossthethicknessatmidpositionofpiezoelec‐tricpatcharecomparedinFigure6forbeams b and c .

Figure7showscomparisonofthroughthethicknessdistributionofaxialdisplacementandnormalstressatthe centre of the piezoelectric patch for pressure load case 2 and potential load case 3.Themaximumnormalstressoccurinloadcase3.

Thethroughthethicknessdistributionprofilesofaxialdisplacementandnormalstressesatmidpositionofpiezoelectricpatch forpressure load cases show increasing trend for xu anddecreasing trend for x with in‐creaseindistanceofpiezo‐patchfromthefixedend.Howeverforloadcase3nosignificantvariationisobservedfortheaxialdisplacementatthetopsurfaceofpiezoelectricpatchandnormalstressatthebottomsurfaceofpie‐zoelectricpatchwithvariationinpiezo‐patchpositionsfromthefixedend.TheseobservationsareclearerinFig‐ure8,showingvariationoftheseparameterswithpositionofpiezo‐patchlayer.

Figure4:Deflectionprofileofcentrelinesofcantileverbeamsforvariouspiezopatchpositions.

Figure5:Variationoftipdeflectionofcantileverbeamwithpiezopatchposition.



Figure6:Throughthethicknessdistributionofaxialdisplacementandnormalstressatthecentreofpiezoelectricpatch

forvariouspatchpositions.

Figure7:Comparisonofaxialdisplacementandnormalstressacrossthethicknessinbeams b and c .



Figure8:Variationofaxialdisplacementandnormalstresswithpositionofpiezo‐patchlayer.

Tables3and4showthemid‐surfacedeflectionatthefreeend,theaxialdisplacementandnormalstressatthecentreofpiezoelectricpatchlayer ( 0.2 )px x l forbeams b and c respectively.Theresultsarepresented

innon‐dimensionalformbutthebarandhatareomitted.Thetworesultsareingoodagreementwitheachother.



Table3:Comparisonof1D‐FEand2D‐FEresultsforbeam b withvariationofpiezo‐patchposition.

Loadcase1 Loadcase2 Loadcase3

S px

Entity 1D‐FE

2D‐FE

1D‐FE

2D‐FE 1D‐FE

2D‐FE

10 0 ,0zu l ‐10.191 ‐10.191 ‐10.274 ‐10.268 ‐3.031 ‐3.006

0.2 ,0.6xu l h 3.450 3.520 3.504 3.568 1.635 1.626

0.2 , 0.5x l h ‐1.990 ‐2.010 ‐2.006 ‐2.015 ‐0.416 ‐0.434

0.2 ,0.5ex l h 1.710 1.750 1.771 1.779 1.269 1.301

0.2 ,0.6x l h 0.770 0.760 0.714 0.715 ‐2.221 ‐2.205

0.2l ,0zu l ‐10.730 ‐10.901 ‐10.770 ‐10.963 ‐2.067 ‐2.029

0.4 ,0.6xu l h 5.850 6.110 5.876 6.140 1.399 1.443

0.4 , 0.5x l h ‐1.080 ‐1.090 ‐1.105 ‐1.099 ‐0.418 ‐0.443

0.4 ,0.5ex l h 0.980 0.960 0.984 0.971 1.269 1.289

0.4 ,0.6x l h 0.420 0.410 0.389 0.380 ‐2.221 ‐2.215

0.4l ,0zu l ‐11.071 ‐11.230 ‐11.088 ‐11.242 ‐1.378 ‐1.352

0.6 ,0.6xu l h 7.151 7.340 7.167 7.358 1.399 1.443

0.6 , 0.5x l h ‐0.450 ‐0.460 ‐0.462 ‐0.461 ‐0.417 ‐0.438

0.6 ,0.5ex l h 0.420 0.410 0.423 0.413 1.269 1.287

0.6 ,0.6x l h 0.170 0.160 0.158 0.150 ‐2.221 ‐2.215

Table4:Comparisonof1D‐FEand2D‐FEresultsforbeam c withvariationofpiezo‐patchposition.

Loadcase1 Loadcase2 Loadcase3

S px

Entity 1D‐FE

2D‐FE 1D‐FE

2D‐FE 1D‐FE

2D‐FE

10 0 ,0zu l ‐20.754 ‐20.66 ‐20.754 ‐20.874 ‐6.265 ‐6.338

0.2 ,0.6xu l h 5.283 5.311 5.434 5.473 3.983 3.475

0.2 , 0.5x l h ‐4.899 ‐4.365 ‐4.410 ‐4.377 ‐0.437 ‐0.443

0.2 ,0.5ex l h 2.219 2.586 2.661 2.681 3.400 3.401

0.2 ,0.6x l h 1.421 1.548 1.481 1.484 ‐2.276 ‐2.254

0.2l ,0zu l ‐23.748 ‐24.093 ‐23.799 ‐24.157 ‐4.211 ‐4.221

0.4 ,0.6xu l h 10.183 11.630 10.565 11.693 3.394 3.474

0.4 , 0.5x l h ‐2.486 ‐2.418 ‐2.454 ‐2.425 ‐0.438 ‐0.445

0.4 ,0.5ex l h 1.408 1.454 1.493 1.506 3.400 3.401

0.4 ,0.6x l h 0.812 0.819 0.785 0.785 ‐2.274 ‐2.269

0.4l ,0zu l ‐24.876 ‐25.142 ‐24.856 ‐25.161 ‐2.828 ‐2.814

0.6 ,0.6xu l h 13.320 14.454 13.686 14.485 3.394 3.463

0.6 , 0.5x l h ‐1.077 ‐1.034 ‐1.055 ‐1.037 ‐0.438 ‐0.442

0.6 ,0.5ex l h 0.617 0.646 0.659 0.668 3.400 3.402

0.6 ,0.6x l h 0.301 0.302 0.291 0.288 ‐2.275 ‐2.261



4.2.2Staticresponsewithvariationofsizeofpiezo‐patchlayer

Cantileverbeamswithapiezoelectricpatchoflength pl ,bondedontopoftheelasticsubstrateatthefixed

end Figure9 areconsideredfortheanalysis.Thebeamsaremodelledandanalysedforloadcase3,withdiffer‐entvaluesof pl forgraphiteepoxycompositesubstrate b andsandwichsubstrate c .Thewidthofthebeam

andpiezopatcharemodelledasunity.

Figure9:Cantileverbeamwithapiezoelectricpatchlayerofvariablesize.

Figure10:Deflectionprofileofcentrelinesofcantileverbeamsforvariouspiezopatchlengths.

ThedeflectionprofilesofcentrelinearecomparedinFigure10.Thecentrelinedeflectionismoreasthepie‐zo‐patchsizeisincreased.Figure11showsvariationofmidsurfacedeflectionatthefreeendandaxialdisplace‐mentat the topof theelastic substrateat the freeend,with sizeofpiezoelectric layer.Theabsolutenumericalvalueofboththeseparametersreflectanincreasingtrend.

Figure11:Variationofaxialandtransversedisplacementswithsizeofpiezo‐patchlayer.

Themid‐surfacedeflectionatthefreeend,theaxialdisplacementandthenormalstressatthecenterofpie‐zoelectricpatch ( / 2)px l arepresentedinTable5forbeams b and c .Theresultsareingoodagreementwith

the2D‐FEresultsobtainedusingABAQUSforboththebeams.



Table5:Comparisonof1D‐FEand2D‐FEresultswithvariationofpiezo‐patchsize.

Beamb Beamc

S pl

Entity 1D‐FE

2D‐FE

1D‐FE

2D‐FE

10 0.4l ,0zu l ‐3.031 ‐3.006 ‐6.265 ‐6.338

0.2 ,0.6xu l h 1.635 1.626 3.983 3.956

0.2 , 0.5x l h ‐0.417 ‐0.430 ‐0.437 ‐0.441

0.2 ,0.5ex l h 1.268 1.286 3.400 3.402

0.2 ,0.6x l h ‐2.221 ‐2.214 ‐2.276 ‐2.273

0.6l ,0zu l ‐4.028 ‐4.010 ‐8.430 ‐8.462

0.3 ,0.6xu l h 2.453 2.438 5.974 5.947

0.3 , 0.5x l h ‐0.416 ‐0.418 ‐0.437 ‐0.437

0.3 ,0.5ex l h 1.268 1.270 3.400 3.400

0.3 ,0.6x l h ‐2.222 ‐2.220 ‐2.276 ‐2.274

0.8l ,0zu l ‐4.642 ‐4.630 ‐9.788 ‐9.778

0.4 ,0.6xu l h 3.270 3.254 7.966 7.938

0.4 , 0.5x l h ‐0.416 ‐0.416 ‐0.437 ‐0.437

0.4 ,0.5ex l h 1.268 1.268 3.400 3.400

0.4 ,0.6x l h ‐2.222 ‐2.221 ‐2.276 ‐2.274

l ,0zu l ‐5.875 ‐5.864 ‐12.442 ‐12.389

0.5 ,0.6xu l h 4.497 4.479 10.953 10.924

0.5 , 0.5x l h ‐0.457 ‐0.457 ‐0.480 ‐0.480

0.5 ,0.5ex l h 1.395 1.395 3.740 3.740

0.5 ,0.6x l h ‐2.440 ‐2.441 ‐2.503 ‐2.501

4.2.3Shapecontrolusingpiezoelectricpatches

Theproblembeams are further analyzed for electromechanical load case load case4 ,where a pressure

0 tq q isappliedonthetopsurfaceoverentirelengthwithuniformappliedpotential 0n onthetopsurface

of thepiezoelectricpatch layer.Keeping thepressureas same, theappliedpotential is increased to reduce thecentrelinedeflection.Theresultsareobtainedwithlengthsofpiezoelectricpatch, 0.4 ,0.6p l ll .Asthesizeofthe

patchincreases,thevalueoftheappliedpotential,requiredtominimisethetipdeflection,decreases.Butatthesametimethecurvatureofthedeflectedbeamalsoincreasesfortheportionwherepiezoelectricpatchisbondedonthetopsurface Figure12 .Thisshowsthatasinglepiezopatchisnotsuitableforcontrollingthedeflectedshapeofbeam.Theproblemmaybeovercomebyusingmultiplepiezoelectricpatcheswithdifferentappliedvolt‐agesoneach.Figure13showstheshapecontrolforbeam b havingfournumbersofpiezoelectricpatchesbond‐ed on top surface of the beam. Their positions 1 2 3 4( , , , )x x x x , sizes 1 2 3 4( , , , )l l l l and the applied voltages

1 2 3 4, , , arementionedinthefigure.



Figure12:Deflectionprofileofcentrelinewithvariationofsizeofpiezo‐patchlayerforloadcase4.

Figure13:Shapecontrolforbeam b withfournumbersofpiezopatchesbondedontopsurfaceofthebeam.

Theexcellentagreementwiththe2D‐FE ABAQUS resultsprovestheaccuracyofthepresent1D‐FEformu‐lationbasedonefficientzigzag theory.Furthermore, theadvantageofusingpresent formulation is the smallerproblemsize in comparison to the2D‐FE ABAQUS model for the same computational accuracy.As thebeamthicknessisalsodiscretisedinthe2D‐FE ABAQUS model,itleadstoquitelargenumbersofdegreesoffreedomincomparisontothepresent1D‐FEmodel.Forexamplebeamb,with 0.2pl l andS 10,hasbeendiscretised

into704elements for2D‐FEanalysisas againstonly40elements in thepresent1D‐FE formulation.Themeshseedsandhencethetotaldegreesoffreedomincreasefurtherwithbeamlengthandnumberoflayersacrossthethicknesstoaccommodatethehigheraspectratioandmaterialheterogeneity.Thecomputationaltimetakenfor2D‐FEstaticanalysisoftheproblembeamisabout10timesthatof1D‐FEanalysis.Thisshowstheefficiencyandaccuracyofthepresent1D‐FEmodel.

5CONCLUSIONS

Insmartbeamstructures,piezoelectriclayersoftenformanintegralpartofthestructureowingtotheirabil‐ity to transformelectrical energy tomechanical energy andvice versa.The location and sizeof these layers issignificantas it results indifferent structural responseundervarious loadingconditions.Theeffectofpositionandsizeofpiezoelectriclayeronstaticresponseofahybridbeamisassessedusinga1DFEmodelbasedoneffi‐cientlayerwise zigzag theory.Theresultsareobtainedforcantilever clamped‐free endconditions.Theaccu‐racy andefficiencyof thepresent1DFEmodelhasbeenestablishedby comparing the resultswith the2DFEmodeldevelopedinABAQUS.Thecentrelinedeflectionismoreasthepiezo‐patchpositionismovedfartherfromthefixedendofcantileverbeamforpressureloadcases.Theinverseistrueforpotentialloadcase.Therefore,thepiezoelectricpatchactuatorsshouldbeplacednearer to the fixedend forcontrolof tipdeflection incantileverbeam.Thecentre linedeflectionismoreasthepiezo‐patchsize is increasedinpotential loadcase.Singlepiezopatchisnotsuitableforcontrollingthedeflectedshapeofbeam.Usingmultiplepiezo‐patches,withdifferentap‐pliedvoltagesoneach,ismorefeasibleforshapecontrolofbeaminanelectromechanicalloadcase.Theseobser‐vationsmaybeusedasabenchmarkforexperimentalinvestigationanddesignforcompositeandsandwichhy‐bridstructures.



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