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MechanicsofLaminated

CompositeStructures

NachiketaTiwari

Indian Institute of Technolo Kan ur

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Lecture35

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BucklingofaPlateSSonTwoSides

Inthislecture,weshallexploretheconceptofbuckling,andhowitrelatestoconceptsofminimumpotentialenergytheoremandgeometricnonlinearity.

ConsideraplateasshowninFig.35.1ofdimensions2a X2b,withtheor g n ocate atcentero p ate.

SS SS

x

y

Fig.35.1

Thisplatehasthefollowingboundaryconditions.Noconstraintsonedgesy=+b.

wo =0onedges,x=+a

= =

Externalconstantforceperunitlength(N)appliedatedge x=+a.Itshouldbenotedherethatloadingdirectioniscompressive,andhenceloadisnegative.

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BucklingofaPlateSSonTwoSides

Further,weassumethattheplateis:

Symmetric:Thus[B]=[0]

Speciallyorthotropic:Thus,A16

=A26

=D16

=D26

=0.

ThisplatewillremainflatatverylowvaluesofN.AtsmallvaluesofN,the

platewilldemonstrateonlyinplanedisplacements,andnooutofplane

displacement.Wecallsuchaconfigurationoftheplateitsprimary

configuration.

However,beyondacriticalvalueofNcr theplatewillbuckleandshiftmove

toanalternativeconfiguration,i.e.itssecondaryconfiguration.

Inthisexample,wewanttofindthestabilityofitsprimaryconfiguration.

Weexpectthatinitiallythisconfigurationwillbestable,butoncewecross

athresholdNcr,instabilitywillsetin,therebycausingplatetobuckle.

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BucklingofaPlateSSonTwoSides

Toconductastabilityanalysisofplatesprimaryconfiguration,weplanto:

1. Solvedisplacementfieldforprimaryconfiguration.Asthisisaneasyproblem,wewill

directlyuseequilibriumequationstosolveit,andgettheexactsolution.

2. Evaluatetotalpotentialenergy(initial)inthesystemassociatedwithprimary

.

3. Next,wewillperturbthisconfigurationwithsomeinfinitesimallysmalldisplacements,

whichareconsistentwithessentialBCsoftheproblem.

4. Oncethesystemisperturbed,wewillcalculatethenewPEofthesystem,i.e. new.

Whilecalculating new,nonlineareffectsduetomoderaterotationswillbeincorporated

in the definition of strains.

5. Fromthesetwovaluesofpotentialenergy,wewillcalculate .

6. Finally,wewilldevelopconditionsunderwhich willnolongerremainpositive.This

willbethepointatwhichtheplatewillbuckle.

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BucklingofaPlateSSonTwoSides

1. Solvefordisplacementfieldforprimaryconfiguration,i.e.whenplate

remainsflatundercompression. Thisisdonebelow:

(Eq.35.1)

. .

Giventhatthisisahomogenousplateatamacroscopiclevel,andin

uniformlyloadsymmetricplatethereisnobending,strainsintheplate

wou econstantover tsent rearea. ence,a oveexpress ons orstra n

arevalidnotonlyattheedgeoftheplate,butalsooveritsentirearea.

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BucklingofaPlateSSonTwoSides

Thus,solvingfordisplacementfieldforprimaryconfiguration,weget:

(Eq.35.3)

Here,itshouldbenotedthat:(i)wehaveplacedtheoriginofcoordinate

systematthe

geometric

center

ofplate,

(ii)

plates

dimensions

are

2a and

2b and iii u dis lacement at x=a is zero.

2. NextwefindthetotalP.E.associatedwiththisconfiguration.

(Eq.35.4)

Here,weuseEq.35.3inEq.35.4tocalculate initial.

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BucklingofaPlateSSonTwoSides

3. Nextweapplyperturbationstoexistingdisplacementfield.The

perturbationfieldisassumedtobe:

u(x,y)= v(x,y)=0,and wo(x,y)= W11(x2a2)(y2b2)

Suchaperturbationfieldischosensuchthatitsatisfiesessential

boundaryconditions,andalsodifferentiabilityrequirements.Sincewe

areusingweakformulation,weneedC2 differentiabilityoutofplane

displacementfield.

Addin thisfieldtoori inaldis lacementfield E .35.3 we et:

(Eq.35.5)

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BucklingofaPlateSSonTwoSides

4. Now,usingEq.35.4wefindthenewPEofthesystem,i.e. new inthe

vicinityofitsoriginalconfiguration.Whilefinding new,nonlineareffects

ue omo era ero a onsw e ncorpora e n e e n ono

strains.Theserevisedstraindefinitionsare:

(Eq.35.6)

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BucklingofaPlateSSonTwoSides

Equations35.6provideuswithdefinitionsofnewmidplanestrainsandcurvatures.Fromthesewenowcalculateexpressionsfornewforceandmomentresultants.Theseare:

Finally,wenowfindrevisedP.E.(new

)ofthesystemusingthefollowingexpression.

E .35.7

duetoapplicationofvariationinthedisplacementfieldsincevariations

inuo

andvo

fieldsarezero.

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BucklingofaPlateSSonTwoSides

5. Next,wefindthevalueof.Itiscomputedtobe:

(Eq.35.8)

6. Fromaboveexpressions,wemakethefollowingobservations.

Theplateisunderequilibrium,since1st variationof iszero.

2 . , cr,

ofP.E.isnolongerpositive.ThisvalueofNcr canbecalculatedbysettingthesecond

variationofP.E.tobezero.Thus,

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BucklingofaPlateSSonTwoSides

Thus,theplatesinitialconfigurationasdefinedinEq.35.3is:

StableaslongasN Ncr.

Also,whenN =Ncr,weseefromEq.35.8that:

1=0 2=0

3=0 4>0

Thisimplies,thatwhenN equalsNcr, isapositiveentity.

Hence, whenNequalsNcr theplatesinitialconfigurationisstable.Itshouldbenotedherethatourobservationsarevalidforachosensetof

trialfunctions.