计算力学.ppt
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计算力学.ppt
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1.4STRAINENERGY,CASTIGLIANOSFIRSTTHEOREMWhenexternalforcesareappliedtoabody,themechanicalworkdonebythoseforcesisconvertedintoacombinationofkineticandpotentialenergies.MechanicalworkkineticenergypotentialenergymotionelasticdeformationInthecaseofanelasticbodyconstrainedtopreventmotion,alltheworkisstoredinthebodyaselasticpotentialenergy,whichisalsocommonlyreferredtoasstrainenergy.1isadifferentialvectoralongthepathofmotion(1.38)(1.37)whereFromelementarystatics,themechanicalworkperformedbyaforceasitspointofapplicationmovesalongapathfromposition1toposition2isdefinedasInCartesiancoordinates,workisgivenby(1.39)Where,andaretheCartesiancomponentsoftheforcevector.-strainenergy-mechanicalworkDefine:
2Forlinearlyelasticdeformationsofalinearspring,Figure1.5Force-deflectionrelationforalinearelasticspring.Theslopeoftheforce-deflectionlineisthespringconstantsuchthatTherefore,theworkrequiredtodeformsuchaspringis(1.40)springconstant,i.e.,stiffnessStrainenergyMechanicalwork3weobservethattheworkandresultingelasticpotentialenergyarequadraticfunctionsofdisplacementUtilizingEquation1.28,(1.41)thisresultisconvertedtoadifferentformasfollows:
(1.42)whereisthetotalvolumeofdeformedmaterial1.28thestrainenergycanbewrittenas4thequantityisstrainenergyperunitvolume,alsoknownasstrainenergydensity.Ingeneral,foruniaxialloading,thestrainenergyperunitvolumeisdefinedby(1.43)Equation1.43representstheareaundertheelasticstress-straindiagram.Strainenergy5CastiglianosFirstTheoremForanelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithrespecttodeflectionatapointisequaltotheappliedforceinthedirectionofthedeflectionatthatpoint.Consideranelasticbodysubjectedtoforcesforwhichthetotalstrainenergyisexpressedas(1.44)whereisthedeflectionatthepointofapplicationofforce,inthedirectionofthelineofactionoftheforce.6Ifallpointsofloadapplicationarefixedexceptone,say,i,andthatpointismadetodeflectaninfinitesimalamountbyanincrementalinfinitesimalforce,thechangeinstrainenergyis(1.45)whereitisassumedthattheoriginalforceisconstantduringtheinfinitesimalchange.TheintegralterminEquation1.45involvestheproductofinfinitesimalquantitiesandcanbeneglectedtoobtain(1.46)ignored7whichinthelimitasapproacheszerobecomes(1.47)Review:
CastiglianosFirstTheoremForanelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithrespecttodeflectionatapointisequaltotheappliedforceinthedirectionofthedeflectionatthatpoint.ThefirsttheoremofCastiglianoisapowerfultoolforfiniteelementformulation,asisnowillustratedforthebarelement.8CombiningEquations1.30,1.31,and1.43,thetotalstrainenergyforthebarelementisgivenby(1.48)ApplyingCastiglianostheoremwithrespecttoeachdisplacementyields(1.49)(1.50)whichareobservedtobeidenticaltoEquations1.33and1.34.1.331.341.301.431.319Forrotationaldisplacements.Inthecaseofrotation,thepartialderivativeofstrainenergywithrespecttorotationaldisplacementisequaltothemoment/torqueappliedatthepointofconcerntheapplicationofthefirsttheoremofCastiglianointermsofasimpletorsionalmember,isillustratedinthefollowingexample.(1.35)Writteninmatrixformas10EXAMPLE1.2AsolidcircularshaftofradiusRandlengthLissubjectedtoconstanttorqueT.Theshaftisfixedatoneend,asshowninFigure1.6.FormulatetheelasticstrainenergyintermsoftheangleoftwistatandshowthatCastiglianosfirsttheoremgivesthecorrectexpressionfortheappliedtorque.SolutionFromstrengthofmaterialstheory,theshearstressatanycrosssectionalongthelengthofthememberisgivenbyFigure1.6Circularcylindersubjectedtotorsion.11thestrainenergyiswherewehaveusedthedefinitionofthepolarmomentofinertiawhererisradialdistancefromtheaxisofthememberandJispolarmomentofinertiaofthecrosssection.Forelasticbehavior,wehavetheangleoftwistattheendofthememberis12sothestrainenergycanbewrittenasPerCastanglianosfirsttheorem,whichisexactlytherelationshownbystrengthofmaterialstheorythestrainenergyforanelasticsystemisaquadraticfunctionofdisplacements;Therefore,applicationofCastiglianosfirsttheoremresultsinlinearrelationshipbetweendisplacementstoappliedforces.Thisstatementfollowsfromthefactthataderivativeofaquadratictermislinear.13EXAMPLE1.3(a)ApplyCastiglianosfirsttheoremtothesystemoffourspringelementsdepictedinFigure1.7toobtainthesystemstiffnessmatrix.Theverticalmembersatnodes2and3aretobeconsideredrigid.(b)Solveforthedisplacementsandthereactionforceatnode1if=4N/mm=6N/mm=3N/mm=-30N=0=50NFigure1.7Fourspringelements.Solution(a)Thetotalstrainenergyofthesystemoffourspringsisexpressedintermsofthenodaldisplacementsandspringconstantsas14ApplyingCastiglianostheorem,usingeachnodaldisplacementinturn,writteninmatrixformasandthesystemstiffnessmatrixisthusobtainedviaCastiglianostheorem.15(b)Substitutingthespecifiednumericalvalues,thesystemequationsbecomeEliminatingtheconstraintequation,theactivedisplacementsaregovernedbywesolvetheequation:
toconvertthecoefficientmatrix(thestiffnessmatrix)toupper-triangularform;thatis,alltermsbelowthemaindiagonalbeco
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