thegeneralsolutionYmoofthehomogeneousdifferentialequationandtheparticularsolutionYmoofthenohomogeneousdifferentialequation．Thus，thesolutionofEq.()canbeexpressedas..Generalsolutionofgoverningequation()TofindthesolutionofEq.(),wefirstconsiderthefollowingfourth-orderhomogeneousdifferentialequationwithvariablecoefficients:TheaboveequationcanbetransformedintoaWhittakerequation,andthesolutioncanbeexpresaterialstiffnessvaryingalongthethicknessdirection．Thestudiesontheplateswithin—planevariablestiffnessarequitefew．Shang[]studiedtherectangularplateswithbidirectionallinearstiffnesswithtwooppositeedgessimplysupportedandtheothertwoedgesarbitrarilysupportedunderthedistributedloads．Yang[]investigatedthestructuralanalysisoftheplateswithunidirectionallyvaryingflexuralrigiditybytheGalerkinlinemethod．Liueta.[]analyzedthefreevibrationofaFunctionallygradedisotropicrectangularplatewithin—planematerialinhomogeneityusingtheLevy-typesoution．Uymazeta．[]ConsideredthefunctionallygradedplateswithpropertiesVaryinginanin—planedirectionbasedonafive—degree—of-freedomsheardeformableplatetheorywithdifferentboundaryconditions．Inthispaper，theLevy-typesolution[-]ispresentedforthebendingofathinrectangularplatewithin—planevariablestiffnessunderthedistributedload．BasicequationsConsiderathinrectangularplateoflengthAandwithBwithin-planeVariablestiffness，asshowninFig..IntroduceaCartesiancoordinatesystem—XYZsuchthat≤X≤A，≤Y≤BWeassume.ThattheflexuralrigidityoftheplateD=D(X,Y)isafunctionofXandY.Thegoverningequationoftheplatewithin—planeVariablestiffnesscanbeobtainedasWhereWisthetransversedisplacement，VisPoisson′sratio，andQisthenormalpressureonTheplate．第页ItisassumedthattheflexuralrigidityoftheplatevariesonlyalongtheY-directionaccordingtothefollowingpowerform:WhereYandParetwomaterialparametersdescribingtheinhomogeneityofD,Doistheflexuralrigidityat，Y=,andDbistheflexuralrigidityatY=b.Inthis‘case,Eq.()canbereducedtoSolutionTherectangularplateisassumedtobesimplysupportedalongtwooppositeedgesparalleltotheY-direction.Tosolvethegoverningequationwiththeprescribedboundaryconditions,ageneralizedLevy-typeapproachisemployedaswhereYm(y)isanunknownfunctiontobedetermined.SubstitutingEq.()intoEq.()yieldsthefollowingdifferentialequation:第页Thesolutionoftheaboveequationconsistsoftwoparts,i.e,thegeneralsolutionYmoofthehomogeneousdifferentialequationandtheparticularsolutionYmoofthenohomogeneousdifferentialequation．Thus，thesolutionofEq.()canbeexpressedas..Generalsolutionofgoverningequation()TofindthesolutionofEq.(),wefirstconsiderthefollowingfourth-orderhomogeneousdifferentialequationwithvariablecoefficients:TheaboveequationcanbetransformedintoaWhittakerequation,andthesolutioncanbeexpresMathematicalModeling,,-()[]Jodaei,A.,Jalal,M.,andYes,M.H.Freevibrationanalysisoffunctionallygradedannularplatesbestate-spacebaseddifferentialquadraturemethodandcomparativemodelingbeANN.Composites:PartB,,-()[]Wen,P.H.andAliabad,M.H.Analysisoffunctionallygradedplates勿meshlessmethod:apurelyanalyticalformulation.Evgiv,eeringArtalysi。withBoundaryElemiarts,,-()[]Liew,K.M.,Zhao,X.,andLee,Y.Y.Postbucklingresponsesoffunctionallygradedcylindricalshellsunderaxialcompressionandthermalloads.Composites:PartB,,-()[]Malekzadeh,P.,Fiouz,A.R.,andSahraouian,M.Three-dimensionalfreevibrationoffunctionallygradedtruncatedconicalshellssubjectedtothermalenvironment.Internatiov.alJournalofPressureVesselsacedPiping,,-()[]Sadeghi,H.,Baghani,M.,andNaghdabadi,R.Straingradientelasticitysolutionforfunctionallygradedmicro-cylinders.InternationalJournalofEngineeringScience,,-()[]Zenkour,A.M.Dynamicalbendinganalysisoffunctionallygradedinfinitecylinderwithrigidcore.AppliedMathematicsandComputation,,-()[]Shang,X.C.Exactsolutiononaproblemofbendingofdouble-directionrectangularelasticplatesofvariablerigidity(inChinese).JournalofLanzhouUniversity(NaturalSciences),(),-()[]Yang,J.ThestructuralanalysisofplateswithunidirectionalvaryingrigidityonGalekinlinemethod(inChinese).JournalofWuhan,InstituteofChemicalTechslogy,(),-第页()[]Liu,D.Y.,Wang,C.Y.,andChen,W.Q.FreevibrationofFGMplateswithin-planematerialinhomogeneity.CompositeStruct,,-()[]Uymaz,B.,Aided,M.,andFile,S.VibrationanalysesofFGMplateswithin-planematerialinhomogeneitybyRitzmethod.CompositeStructures,,-()[]Badeshi,M.andSaudi,A.R.Levy-typesolutionforbucklinganalysisofthickfunctionallygradedrectangularplatesbasedonthehigher-ordersheardeformationplatetheory.AppliedMathematicalModeling,,于()[]Thai,H.T.andKim,S.E.Levy-typesolutionforbucklinganalysisoforthotropicplatesbasedontwovariablerefinedplatetheories.CrocoitesStructures,,-()aterialstiffnessvryingalongthethicknessdirection．Thestudiesontheplateswithin—planevariablestiffnessarequitefew．Shang[]studiedtherectangularplateswithbidirectionallinearstiffnesswithtwooppositeedgessimplysupportedandtheothertwoedgesarbitrarilysupportedunderthedistributedloads．Yang[]investigatedthestructuralanalysisoftheplateswithunidirectionallyvaryingflexuralrigiditybytheGalerkinlinemethod．Liueta第页Analyticalsolutionnonrectangularplatewithin-planeVariablestiffnessTian-chongYU，Guo-junNIE，ZhengZHONG，Fu-yunCHU(SchoolofAerospaceEngineeringandAppliedMechanics,TongjiUniversity,Shanghai,P.R.China)Abstract:Thebendingproblemofathinrectangularplatewithin-planevariablestiffnessisstudied.Thebasicequationisformulatedforthetwo-opposite-edgesimplysupportedrectangularplateunderthedistributedloads.Theformulationisbasedontheassumptionthattheflexuralrigidityoftheplatevariesintheplanefollowingapowerform，andPoisson’sratioisconstant．Afourth-orderpartialdifferentialequationwithvariablecoefficientsisderivedbyassumingaLevy-typeformforthetransversedisplacement.ThegoverningequationcanbetransformedintoaWhittakerequation，andananalyticalsolutionisobtainedforathinrectangularplatesubjectedtothedistributedloads．Thevalidityofthepresentsolutionisshownbycomparingthepresentresultswiththoseoftheclassicalsolution．Theinfluenceofin-planevariablestiffnessonthedeflectionandbendingmomentisstudiedbynumericalexamples．Theanalyticalsolutionpresentedhereisusefulinthedesignofrectangularplateswithin-planevariablestiffness.Keywords:in-planevariablestiffness,powerform，Levy-typesolution，rectangularplateChineseLibraryClassificationMathematicsSubjectClassificationB第页IntroductionTheterm”variablestiffness”impliesthatthestiffnessparametersvaryspatiallythroughout.Thestructure[]．Functionallygradedmaterials(FGMs)areinhomogeneouscomposites，inwhich.themechanicalpropertiesvarysmoothlywiththepositiontomeetthepredeterminedfunctional.performance．ThestructurescomposedoftheFGMsareofvariablestiffness．Thereareextensiveliteraturesonthebending，vibration，andfractureoftheFGMstructures[-]．Thedeformationofafunctionallygradedbeamwasstudiedbythedirectapproach[]．AnefficientandsimplyrefinedtheorywaspresentedforthebucklinganalysisoffunctionallygradedplatesbyThaiandChoi[]．Jodaeieta[]dealtwiththethree—dimensionalanalysisoffunctionallygradedannularplatesusingthestate—spacebaseddifferentialquadraturemethod(SSDQM)．WenandAiabadi[]investigatedfunctionallygradedplatesunderstaticanddynamicloadsbythelocalintegralequationmethod(LIEM)．TherearealsosomeworksontheFGMshellsandcylinders[-]．However，mostofthestudiesontheFGMsdealwithmaterialstiffnessvaryingalongthethicknessdirection．Thestudiesontheplateswithin—planevariablestiffnessarequitefew．Shang[]studiedtherectangularplateswithbidirectionallinearstiffnesswithtwooppositeedgessimplysupportedandtheothertwoedgesarbitrarilysupportedunderthedistributedloads．Yang[]investigatedthestructuralanalysisoftheplateswithunidirectionallyvaryingflexuralrigiditybytheGalerkinlinemethod．Liueta.[]analyzedthefreevibrationofaFunctionallygradedisotropicrectangularplatewithin—planematerialinhomogeneityusingtheLevy-typesoution．Uymazeta．[]ConsideredthefunctionallygradedplateswithpropertiesVaryinginanin—planedirectionbasedonafive—degree—of-freedomsheardeformableplatetheorywithdifferentboundaryconditions．Inthispaper，theLevy-typesolution[-]ispresentedforthebendingofathinrectangularplatewithin—planevariablestiffnessunderthedistributedload．BasicequationsConsiderathinrectangularplateoflengthAandwithBwithin-planeVariablestiffness，asshowninFig..IntroduceaCartesiancoordinatesystem—XYZsuchthat≤X≤A，≤Y≤BWeassume.ThattheflexuralrigidityoftheplateD=D(X,Y)isafunctionofXandY.Thegoverningequationoftheplatewithin—planeVariablestiffnesscanbeobtainedasWhereWisthetransversedisplacement，VisPoisson′sratio，andQisthenormalpressureonTheplate．第页ItisassumedthattheflexuralrigidityoftheplatevariesonlyalongtheY-directionaccordingtothefollowingpowerform:WhereYandParetwomaterialparametersdescribingtheinhomogeneityofD,Doistheflexuralrigidityat，Y=,andDbistheflexuralrigidityatY=b.Inthis‘case,Eq.()canbereducedtoSolutionTherectangularplateisassumedtobesimplysupportedalongtwooppositeedgesparalleltothe        第1页Analyticalsolutionnonrectangularplatewithin-planeVariablestiffnessTian-chongYU，Guo-junNIE，ZhengZHONG，Fu-yunCHU(SchoolofAerospaceEngineeringandAppliedMechanics,TongjiUniversity,Shanghai200092,P.R.China)Abstract:Thebendingproblemofathinrectangularplatewithin-planevariablestiffnessisstudied.Thebasicequationisformulatedforthetwo-opposite-edgesimplysup

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