计算力学.ppt
- 文档编号:5163204
- 上传时间:2023-05-08
- 格式:PPT
- 页数:30
- 大小:1.11MB
计算力学.ppt
《计算力学.ppt》由会员分享,可在线阅读,更多相关《计算力学.ppt(30页珍藏版)》请在冰点文库上搜索。
1,1.4STRAINENERGY,CASTIGLIANOSFIRSTTHEOREM,Whenexternalforcesareappliedtoabody,themechanicalworkdonebythoseforcesisconvertedintoacombinationofkineticandpotentialenergies.,Mechanicalworkkineticenergypotentialenergy,motion,elasticdeformation,Inthecaseofanelasticbodyconstrainedtopreventmotion,alltheworkisstoredinthebodyaselasticpotentialenergy,whichisalsocommonlyreferredtoasstrainenergy.,2,isadifferentialvectoralongthepathofmotion,(1.38),(1.37),where,Fromelementarystatics,themechanicalworkperformedbyaforceasitspointofapplicationmovesalongapathfromposition1toposition2isdefinedas,InCartesiancoordinates,workisgivenby,(1.39),Where,andaretheCartesiancomponentsoftheforcevector.,-strainenergy,-mechanicalwork,Define:
3,Forlinearlyelasticdeformationsofalinearspring,Figure1.5Force-deflectionrelationforalinearelasticspring.,Theslopeoftheforce-deflectionlineisthespringconstantsuchthat,Therefore,theworkrequiredtodeformsuchaspringis,(1.40),springconstant,i.e.,stiffness,Strainenergy,Mechanicalwork,4,weobservethattheworkandresultingelasticpotentialenergyarequadraticfunctionsofdisplacement,UtilizingEquation1.28,(1.41),thisresultisconvertedtoadifferentformasfollows:
(1.42),whereisthetotalvolumeofdeformedmaterial,1.28,thestrainenergycanbewrittenas,5,thequantityisstrainenergyperunitvolume,alsoknownasstrainenergydensity.,Ingeneral,foruniaxialloading,thestrainenergyperunitvolumeisdefinedby,(1.43),Equation1.43representstheareaundertheelasticstress-straindiagram.,Strainenergy,6,CastiglianosFirstTheorem,Foranelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithrespecttodeflectionatapointisequaltotheappliedforceinthedirectionofthedeflectionatthatpoint.,Consideranelasticbodysubjectedtoforcesforwhichthetotalstrainenergyisexpressedas,(1.44),whereisthedeflectionatthepointofapplicationofforce,inthedirectionofthelineofactionoftheforce.,7,Ifallpointsofloadapplicationarefixedexceptone,say,i,andthatpointismadetodeflectaninfinitesimalamountbyanincrementalinfinitesimalforce,thechangeinstrainenergyis,(1.45),whereitisassumedthattheoriginalforceisconstantduringtheinfinitesimalchange.,TheintegralterminEquation1.45involvestheproductofinfinitesimalquantitiesandcanbeneglectedtoobtain,(1.46),ignored,8,whichinthelimitasapproacheszerobecomes,(1.47),Review:
CastiglianosFirstTheorem,Foranelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithrespecttodeflectionatapointisequaltotheappliedforceinthedirectionofthedeflectionatthatpoint.,ThefirsttheoremofCastiglianoisapowerfultoolforfiniteelementformulation,asisnowillustratedforthebarelement.,9,CombiningEquations1.30,1.31,and1.43,thetotalstrainenergyforthebarelementisgivenby,(1.48),ApplyingCastiglianostheoremwithrespecttoeachdisplacementyields,(1.49),(1.50),whichareobservedtobeidenticaltoEquations1.33and1.34.,1.33,1.34,1.30,1.43,1.31,10,Forrotationaldisplacements.Inthecaseofrotation,thepartialderivativeofstrainenergywithrespecttorotationaldisplacementisequaltothemoment/torqueappliedatthepointofconcern,theapplicationofthefirsttheoremofCastiglianointermsofasimpletorsionalmember,isillustratedinthefollowingexample.,(1.35),Writteninmatrixformas,11,EXAMPLE1.2,AsolidcircularshaftofradiusRandlengthLissubjectedtoconstanttorqueT.Theshaftisfixedatoneend,asshowninFigure1.6.FormulatetheelasticstrainenergyintermsoftheangleoftwistatandshowthatCastiglianosfirsttheoremgivesthecorrectexpressionfortheappliedtorque.,SolutionFromstrengthofmaterialstheory,theshearstressatanycrosssectionalongthelengthofthememberisgivenby,Figure1.6Circularcylindersubjectedtotorsion.,12,thestrainenergyis,wherewehaveusedthedefinitionofthepolarmomentofinertia,whererisradialdistancefromtheaxisofthememberandJispolarmomentofinertiaofthecrosssection.Forelasticbehavior,wehave,theangleoftwistattheendofthememberis,13,sothestrainenergycanbewrittenas,PerCastanglianosfirsttheorem,whichisexactlytherelationshownbystrengthofmaterialstheory,thestrainenergyforanelasticsystemisaquadraticfunctionofdisplacements;Therefore,applicationofCastiglianosfirsttheoremresultsinlinearrelationshipbetweendisplacementstoappliedforces.Thisstatementfollowsfromthefactthataderivativeofaquadratictermislinear.,14,EXAMPLE1.3,(a)ApplyCastiglianosfirsttheoremtothesystemoffourspringelementsdepictedinFigure1.7toobtainthesystemstiffnessmatrix.Theverticalmembersatnodes2and3aretobeconsideredrigid.(b)Solveforthedisplacementsandthereactionforceatnode1if,=4N/mm=6N/mm=3N/mm=-30N=0=50N,Figure1.7Fourspringelements.,Solution,(a)Thetotalstrainenergyofthesystemoffourspringsisexpressedintermsofthenodaldisplacementsandspringconstantsas,15,ApplyingCastiglianostheorem,usingeachnodaldisplacementinturn,writteninmatrixformas,andthesystemstiffnessmatrixisthusobtainedviaCastiglianostheorem.,16,(b)Substitutingthespecifiednumericalvalues,thesystemequationsbecome,Eliminatingtheconstraintequation,theactivedisplacementsaregovernedby,wesolvetheequation:
toconvertthecoefficientmatrix(thestiffnessmatrix)toupper-triangularform;thatis,alltermsbelowthemaindiagonalbecomezero.,Step1.Multiplythefirstequation(row)by12,multiplythesecondequation(row)by16,addthetwoandreplacethesecondequationwiththeresultingequationtoobtain,17,Step2.Multiplythethirdequationby32,addittothesecondequation,andreplacethethirdequationwiththeresult.Thisgivesthetriangularizedformdesired:
Inthisform,theequationscannowbesolvedfromthe“bottomtothetop.”,Thereactionforceatnode1isobtainedfromtheconstraintequation,weobservesystemequilibriumsincetheexternalforcessumtozeroasrequired.,18,1.5MINIMUMPOTENTIALENERGY,Theprincipleofminimumpotentialenergyisstatedasfollows:
Ofalldisplacementstatesofabodyorstructure,subjectedtoexternalloading,thatsatisfythegeometricboundaryconditions(imposeddisplacements),thedisplacementstatethatalsosatisfiestheequilibriumequationsissuchthatthetotalpotentialenergyisaminimumforstableequilibrium.,Thetotalpotentialenergyincludes:
thestoredelasticpotentialenergy(thestrainenergy)aswellasthepotentialenergyofappliedloads,-thetotalpotentialenergy,-thestrainenergy,-thepotentialenergyassociatedwithexternalforces,Thetotalpotentialenergyis,(1.51),19,wewilldealonlywithelasticsystemssubjectedtoconservativeforces.,Aconservativeforceisdefinedasonethatdoesmechanicalworkindependentofthepathofmotionandsuchthattheworkisreversibleorrecoverable.,Themostcommonexampleofanonconservativeforceistheforceofslidingfriction.Asthefrictionforcealwaysactstoopposemotion,theworkdonebyfrictionforcesisalwaysnegativeandresultsinenergyloss.,Defination:
20,Therefore,themechanicalworkofaconservativeforceisconsideredtobealossinpotentialenergy;thatis,(1.52),whereWisthemechanicalwork,thetotalpotentialenergyisthengivenby,(1.53),thestrainenergytermisaquadraticfunctionofsystemdisplacementsandtheworktermWisalinearfunctionofdisplacements.Rigorously,theminimizationoftotalpotentialenergyisaprobleminthecalculusofvariations.,21,Here,wesimplyimposetheminimizationprincipleofcalculusofmultiplevariablefunctions.,atotalpotentialenergyexpressionthatisafunctionofNdisplacements,thatis,(1.54),thenthetotalpotentialenergywillbeminimizedif,(1.55),Equation1.55willbeshowntorepresentalgebraicequations,whichformthefiniteelementapproximationtothesolutionofthedifferentialequation(s)governingtheresponseofastructuralsystem.,Thiscanbeillustratedbythefollowingexample.,22,RepeatthesolutiontoExample1.3usingtheprincipleofminimumpotentialenergy.,EXAMPLE1.4,Figure1.7Fourspringelements.,=4N/mm=6N/mm=3N/mm=-30N=0=50N,23,Hence,thetotalpotentialenergyisexpressedas,andthepotentialenergyofappliedforcesis,Perthepreviousexamplesolution,theelasticstrainenergyis,Solution,24,theprincipleofminimumpotentialenergyrequiresthat,givinginsequence=1,4,thealgebraicequations,25,whenwritteninmatrixform,are,andcanbeseentobeidenticaltothepreviousresult.,WenowreexaminetheenergyequationoftheExample1.4todevelopamoregeneralform,Thesystemorglobaldisplacementvectoris,(1.56),_,26,and,asderived,theglobalstiffnessmatrixis,(1.57),Ifweformthematrixtripleproduct,(1.58),andcarryoutthematrixoperations,wefindthattheexpressionisidenticaltothestrainenergyofthesystem.,Ifthestrainenergycanbeexpressedintheformofthistripleproduct,thestiffnessmatrixwillhavebeenobtained,sincethedisplacementsarereadilyidentifiable.,27,Homework:
Problem1.1,1.3,1.4,1.5,1.7and1.8.,1.1ForthespringassemblyofFigureP1.1,usingthesystemassemblyproceduredeterminetheglobalstiffnessmatrix.,1.3ForthespringassemblyofFigureP1.3,determineforce,requiredtodisplacenode2anamount=0.75in.totheright.Alsocomputedisplacementofnode3.Given=50lb./in.and=25lb./in.,*,28,1.4InthespringassemblyofFigureP1.4,forcesandaretobeappliedsuchthattheresultantforceinelement2iszeroandnode4displacesanamountin.Determine(a)therequiredvaluesofforcesand,(b)dis
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 计算 力学