計算力學Forlinearlyelasticdeformationsofalinearspring,Figure1.5

Force-deflectionrelationforalinearelasticspring.Theslopeoftheforce-deflectionlineisthespringconstantsuchthatTherefore,theworkrequiredtodeformsuchaspringis(1.40)springconstant,i.e.,stiffnessStrainenergyMechanicalwork2weobservethattheworkandresultingelasticpotentialenergyarequadraticfunctionsofdisplacementUtilizingEquation1.28,(1.41)thisresultisconvertedtoadifferentformasfollows:(1.42)whereisthetotalvolumeofdeformedmaterial[1.28]thestrainenergycanbewrittenas3thequantityisstrainenergyperunitvolume,alsoknownasstrainenergydensity.Ingeneral,foruniaxialloading,thestrainenergyperunitvolumeisdefinedby(1.43)Equation1.43representstheareaundertheelasticstress-straindiagram.Strainenergy4Castigliano’sFirstTheoremForanelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithrespecttodeflectionatapointisequaltotheappliedforceinthedirectionofthedeflectionatthatpoint.Consideranelasticbodysubjectedtoforcesforwhichthetotalstrainenergyisexpressedas(1.44)whereisthedeflectionatthepointofapplicationofforce,inthedirectionofthelineofactionoftheforce.5

Ifallpointsofloadapplicationarefixedexceptone,say,i,andthatpointismadetodeflectaninfinitesimalamountbyanincrementalinfinitesimalforce,thechangeinstrainenergyis(1.45)whereitisassumedthattheoriginalforceisconstantduringtheinfinitesimalchange.TheintegralterminEquation1.45involvestheproductofinfinitesimalquantitiesandcanbeneglectedtoobtain(1.46)ignored6whichinthelimitasapproacheszerobecomes(1.47)Review:Castigliano’sFirstTheoremForanelasticsysteminequilibrium,

thepartialderivativeoftotalstrainenergywithrespecttodeflection

atapoint

isequalto

the

appliedforce

inthedirectionofthedeflectionatthatpoint.ThefirsttheoremofCastiglianoisapowerfultoolforfiniteelementformulation,asisnowillustratedforthebarelement.7CombiningEquations1.30,1.31,and1.43,thetotalstrainenergyforthebarelementisgivenby(1.48)ApplyingCastigliano’stheoremwithrespecttoeachdisplacementyields(1.49)(1.50)whichareobservedtobeidenticaltoEquations1.33and1.34.[1.33]

[1.34][1.30][1.43][1.31]8Forrotationaldisplacements.Inthecaseofrotation,thepartialderivativeofstrainenergywithrespecttorotationaldisplacementisequaltothemoment/torqueappliedatthepointofconcerntheapplicationofthefirsttheoremofCastiglianointermsofasimpletorsionalmember,isillustratedinthefollowingexample.(1.35)

Writteninmatrixformas9EXAMPLE1.2AsolidcircularshaftofradiusRandlengthLissubjectedtoconstanttorqueT.Theshaftisfixedatoneend,asshowninFigure1.6.FormulatetheelasticstrainenergyintermsoftheangleoftwistatandshowthatCastigliano’sfirsttheoremgivesthecorrectexpressionfortheappliedtorque.■

SolutionFromstrengthofmaterialstheory,theshearstressatanycrosssectionalongthelengthofthememberisgivenbyFigure1.6Circularcylindersubjectedtotorsion.10thestrainenergyiswherewehaveusedthedefinitionofthepolarmomentofinertiawhererisradialdistancefromtheaxisofthememberandJispolarmomentofinertiaofthecrosssection.Forelasticbehavior,wehave

theangleoftwistattheendofthememberis11sothestrainenergycanbewrittenasPerCastangliano’sfirsttheorem,whichisexactlytherelationshownbystrengthofmaterialstheory

thestrainenergyforanelasticsystemisaquadraticfunctionofdisplacements;Therefore,applicationofCastigliano’sfirsttheoremresultsinlinearrelationshipbetweendisplacementstoappliedforces.Thisstatementfollowsfromthefactthataderivativeofaquadratictermislinear.12EXAMPLE1.3(a)ApplyCastigliano’sfirsttheoremtothesystemoffourspringelementsdepictedinFigure1.7toobtainthesystemstiffnessmatrix.Theverticalmembersatnodes2and3aretobeconsideredrigid.(b)Solveforthedisplacementsandthereactionforceatnode1if=4N/mm=6N/mm=3N/mm=-30N=0=50NFigure1.7Fourspringelements.■

Solution(a)

Thetotalstrainenergyofthesystemoffourspringsisexpressedintermsofthenodaldisplacementsandspringconstantsas13ApplyingCastigliano’stheorem,usingeachnodaldisplacementinturn,writteninmatrixformasandthesystemstiffnessmatrixisthusobtainedviaCastigliano’stheorem.14(b)Substitutingthespecifiednumericalvalues,thesystemequationsbecomeEliminatingtheconstraintequation,theactivedisplacementsaregovernedbywesolvetheequation:toconvertthecoefficientmatrix(thestiffnessmatrix)toupper-triangularform;thatis,alltermsbelowthemaindiagonalbecomezero.Step1.Multiplythefirstequation(row)by12,multiplythesecondequation(row)by16,addthetwoandreplacethesecondequationwiththeresultingequationtoobtain15Step2.Multiplythethirdequationby32,addittothesecondequation,andreplacethethirdequationwiththeresult.Thisgivesthetriangularizedformdesired:Inthisform,theequationscannowbesolvedfromthe“bottomtothetop.”Thereactionforceatnode1isobtainedfromtheconstraintequationweobservesystemequilibriumsincetheexternalforcessumtozeroasrequired.161.5MINIMUMPOTENTIALENERGY

Theprincipleofminimumpotentialenergyisstatedasfollows:Ofalldisplacementstatesofabodyorstructure,subjectedtoexternalloading,thatsatisfythegeometricboundaryconditions(imposeddisplacements),thedisplacementstatethatalsosatisfiestheequilibriumequationsissuchthatthetotalpotentialenergyisaminimumforstableequilibrium.Thetotalpotentialenergyincludes:thestoredelasticpotentialenergy(thestrainenergy)aswellasthepotentialenergyofappliedloads

---thetotalpotentialenergy---thestrainenergy

---thepotentialenergyassociatedwithexternalforces

Thetotalpotentialenergyis(1.51)17wewilldealonlywithelasticsystemssubjectedtoconservativeforces.Aconservativeforceisdefinedasonethatdoesmechanicalworkindependentofthepathofmotionandsuchthattheworkisreversibleorrecoverable.Themostcommonexampleofanonconservativeforceistheforceofslidingfriction.Asthefrictionforcealwaysactstoopposemotion,theworkdonebyfrictionforcesisalwaysnegativeandresultsinenergyloss.Defination:18Therefore,themechanicalworkofaconservativeforceisconsideredtobealossinpotentialenergy;thatis,(1.52)whereWisthemechanicalwork,thetotalpotentialenergyisthengivenby(1.53)thestrainenergytermisaquadraticfunctionofsystemdisplacementsandtheworktermWisalinearfunctionofdisplacements.Rigorously,theminimizationoftotalpotentialenergyisaprobleminthecalculusofvariations.19Here,wesimplyimposetheminimizationprincipleofcalculusofmultiplevariablefunctions.atotalpotentialenergyexpressionthatisafunctionofNdisplacementsthatis,(1.54)thenthetotalpotentialenergywillbeminimizedif(1.55)Equation1.55willbeshowntorepresentalgebraicequations,whichformthefiniteelementapproximationtothesolutionofthedifferentialequation(s)governingtheresponseofastructuralsystem.Thiscanbeillustratedbythefollowingexample.20RepeatthesolutiontoExample1.3usingtheprincipleofminimumpotentialenergy.EXAMPLE1.4

Figure1.7Fourspringelements.=4N/mm=6N/mm=3N/mm=-30N=0=50N21Hence,thetotalpotentialenergyisexpressedas

andthepotentialenergyofappliedforcesisPerthepreviousexamplesolution,theelasticstrainenergyis■

Solution

22

theprincipleofminimumpotentialenergyrequiresthatgivinginsequence=1,4,thealgebraicequations

23whenwritteninmatrixform,areandcanbeseentobeidenticaltothepreviousresult.WenowreexaminetheenergyequationoftheExample1.4todevelopamoregeneralform,Thesystemorglobaldisplacementvectoris(1.56)________________________________________________________________24and,asderived,theglobalstiffnessmatrixis(1.57)Ifweformthematrixtripleproduct

(1.58)andcarryoutthematrixoperations,wefindthattheexpressionisidenticaltothestrainenergyofthesystem.Ifthestrainenergycanbeexpressedintheformofthistripleproduct,thestiffnessmatrixwillhavebeenobtained,sincethedisplacementsarereadilyidentifiable.25Homework:Problem1.1,1.3,1.4,1.5,1.7andForthespringassemblyofFigureP1.1,usingthesystemassemblyproceduredeterminetheglobalstiffnessmatrix.1.3ForthespringassemblyofFigureP1.3,determineforce,requiredtodisplacenode2anamount=0.75in.totheright.Alsocomputedisplacementofnode3.Given=50lb./in.and=25l