Engineering Basic Mechanics I Statics 工程基础力学Ⅰ 静力学 课件 Chapter 7 Axial Tension and Compression

StaticsStaticsofdeformablebodyChapter7

AxialTensionandCompressionContents7.1Introductionofaxialtensionandcompression 7.2Internalforcesinaxialtensionorcompression7.3Stressinaxialtensionorcompression7.4Elasticdeformationinaxialtensionorcompression7.5Mechanicalpropertiesofmaterialsintensionandcompression7.6StrengthcalculationAxialtension,correspondingtoanexternalforcecalledtension,isappliedtotheaxialtensionbars.Axialcompression,correspondingtoanexternalforcecalledpressure,isappliedtotheaxialcompressionbars.7.1axialtensionandcompression7.2Internalforcesinaxialtensionorcompression

Asshowninthefigure,theinternalforcesontherodcrosssectionm-maredeterminedbythesectionmethod.Cuttherodalongthecrosssectionm-musinganimaginaryplaneAccordingtotheequilibrium,TheforceFNisshowninthefigure.mmFN{mmFN}FFTheinternalforcesinteractingbetweentheleftandrightsectionsofthebarincrosssectionm-mareadistributedforcessystem.FN}mmFmmF{FNLetcombinedforcebeFN.Accordingtotheequilibriumcondition,wecanget

(7-1)FNcoincideswiththeaxisandiscalledtheaxialforce.

Itisgenerallyspecifiedthattheaxialforceintensionispositiveandincompressionisnegative.(Thecalculationisusuallydonebyfirstsettingtheinternalforcetobepositive)7.2Internalforcesinaxialtensionorcompression

x-axis---paralleltotherodaxis,indicatingthepositionofthecrosssectionoftherod.

FN-axis---perpendiculartotherodaxis,indicatingthemagnitudeoftheaxialforce.FNFx

Plotalinerepresentingtherelationshipbetweentheaxialforceandthepositionofthesectionattheselectedscale——axialforcediagram.FFMeaningofaxialforcediagram:①Itreflectstherelationshipbetweenthevariationoftheaxialforceandthepositionofthecross-section,whichismoreintuitive;②Itreflectsthevalueofthemaximumaxialforceandthelocationofthesurfacewhereitislocated,i.e.thelocationofthedangeroussection,whichprovidesthebasisforstrengthcalculation.PositiveaxialforceTheaxialforcethatcausesthemicro-elementsegmenttohaveatendencytoelongateispositive.NegativeaxialforceExample：Therodissubjectedtotheforcesshowninfigure(a),trytodrawtheaxialforcediagram.(b)Solution：(1)CalculatetheaxialforceofeachsectionoftherodSectionAB:Theaxialforceisassumedtobethetensileforce,expressedin

.Thenweget

(Thenegativesignindicatesthattheaxialforceispressure)(a)F2FBCDABFNFAFNABSimilarly:theaxialforcesofBCandCDsectionsareobtainedasfollowsFF2FABCDFFABFF2FABC(a)(d)(c)FNBCFNCD

(2)

Theaxialforcediagramisshowninfigure(e)FNxF(e)2FFF2FABCD1.TakethepartwithlessnumberofexternalforcesastheobjectofstudyWhencalculating2.Theaxialforcediagrammustbedrawntogetherwiththedrawingoftheoriginalcomponent.Thevaluesoftheaxialforcesandthe"+"and"-"signsmustbeclearlymarked.3.Intheaxialforcediagram,startingandendingatzero,thesuddenchangeinvalue=concentratedload

Example7-1TheforceonastraightrodofequalsectionisshowninFigurea，F1=120kN，F2=90kN，F3=60kN，Thytodrawtheaxialforcediagram.3F1F2FABCD(a)1m3FFR1F2FABCDIIIIIIIIIIII(b)solution

（1）SolveforthesupportreactionforceLetthesupportreactionforcebeRasinfigure(b).Accordingtoequilibriumconditionofthewholerod,weget2m1.5m

（2）Calculatetheaxialforceofeachsectionoftherod

SectionAB：AhypotheticalplaneisusedtotruncatetherodinsectionABandtheleftsectionistakenastheobjectofstudy(Figurec).Assumethattheaxialforceisthetensileforce,denotedby.Fromtheequilibriumcondition,wehaveAII(c)AII(c)IIIIIIIIII3F1F2F1FABIIIIFRFN2FN33FIIIIII(e)(d)Similarly,weget：theaxialforcesinthesectionBC(Figured)andsectionCD(Figuree):

Example7-1TheforceonastraightrodofequalsectionisshowninFigurea，F1=120kN，F2=90kN，F3=60kN，Thytodrawtheaxialforcediagram.（3）DrawaxialforcediagramTheaxialforcediagramisshowninFiguref.Asseenfromtheaxialforcediagram,thevalueoftheaxialforceinsectionABisthelargest，FNmax=FN1=120kN。Axialforcesareinternalforces,whicharerelatedtoexternalforces,butdifferentfromthem.3F1F2FABCD(a)FN/kNx(f)60907.3Stressinaxialtensionorcompression1.normalstressformula：

ThespecificrelationshipbetweenσandFNcannotbedeterminedfromtheabovestaticrelationshipequationalone.Nowwestartfromstudyingthedeformationoftherodtoseekthevariationlawofσ.Asshownintherightfigure:BeforedeformationAfterdeformationThefollowingphenomenacanbeobservedafterdeformation:(1)Therodiselongated.However,eachtransverselineremainsstraight,andanytwoadjacenttransverselinesaremovedparalleltoeachotheralongtheaxisbyadistance;(2)Afterdeformation,thetransverselinesarestillperpendiculartotheaxis.Fromtheaboveobservation,Theplanesectionhypothesisforroddeformationcanbeobtained.

Thecross-sectionofthebarremainsflatandalwaysperpendiculartotheaxisduringtensionandcompression(geometricrelationship).planesectionhypothesisStressisequalatallpointsonthecrosssectionPhysicalrelationship

Thestresses(i.e.,positivestressesσ)areequalatallpointsofthecrosssectionontheisotropicrod,sowehaveThereforethenormalstressinthecrosssectionofthetension(compression)rodis

ThesignofσisspecifiedasthesameasFN.Thetensilestressispositiveandthecompressivestressisnegative.Thederivationprocessoftheabovepositivestressequationisbasedonthethreelawsofdeformationgeometry,physicsandstaticequilibrium.

AnalyticalmethodsofmaterialmechanicsSynthesisofthreetypesofanalysisGeometricanalysisMechanicalAnalysisPhysicalAnalysis1.MechanicalAnalysisStudyoftherelationshipbetweenthevariousmechanicalelements(includingexternalandinternalforces;forcesandmoments)inamember.2.PhysicalAnalysisstudyofthemechanicalpropertiesofmaterialsandtherelationshipbetweenthemechanical(andsometimesthethermal)andgeometricelementsofamember.Studyoftherelationshipbetweenthevariousgeometricelementsinmembersandstructures.Relationshipbetweenstrainanddeformationinamember.Therelationshipbetweenthedeformationofeachmemberinthestructure.3.GeometricAnalysis2.Thepremiseofusingthepositivestressformula1.Thelineofactionofthecombinedexternalforcesmustcoincidewiththeaxisoftherod.2.Therodmustbeanstraightrodwithequalsection.Ifthecross-sectionaldimensionschangealongtheaxis,forslowlychangingrods:

3.Theformulaiscorrectonlyatacertaindistancefromthepointofactionoftheexternalforce.Example7-2ThearticulatedbracketisshowninfollowingFigure.ABisarodofcircularsectionwithdiameterd=16mmandBCisarodofsquaresectionwithsidelengtha=14mm.iftheloadF=15kN,trytocalculatethestressinthecrosssectionofeachrod.ACBFSolution（1）CalculatetheaxialforceofeachrodThenodeBisinterceptedbythesectionmethod.Theaxialforceofeachrodisassumedtobeintension.Fromtheequilibriumequation,weget

BFFNABFNBCACBFSolution（2）Calculatethestressofeachrod.

Example7-2ThearticulatedbracketisshowninfollowingFigure.ABisarodofcircularsectionwithdiameterd=16mmandBCisarodofsquaresectionwithsidelengtha=14mm.iftheloadF=15kN,trytocalculatethestressinthecrosssectionofeachrod.BFFNABFNBC1.AxialdeformationHooke'slaw

AnequalstraightbarisshowninFigure.Lettheoriginallengthoftherodbelandthecross-sectionalareabeA.UndertheactionofaxialtensionF,thelengthoftherodchangesfromltol1.7.4ElasticdeformationinaxialtensionorcompressionFFTotalelongationintheaxialdirection:

(a)FFExperimentshowedthat：adoptingaProportionalityfactorE，wecanget（b）Foranequalstraightrodsubjectedtoaxialexternalforcesonlyatbothends,SinceFN=F,equation(b)canberewrittenas

ispositivewhenthebarisintensionandnegativewhenthebarisincompression.Theaboveequationistheformulaforcalculatingtheaxialdeformationofanequalstraightrodinaxialtensionandcompression,referredtoasHooke'slaw.E---relatedtothenatureofthematerial,knownasthematerial'stensileandcompressivemodulusofelasticity,itsvaluecanbedeterminedbyexperiment.EA---reflectstheabilityoftherodtoresisttensile(compression)deformation,knownasthetensile(compression)rigidity.FFSubstitutingandintotheaboveequationWegetor

ThisisanotherexpressionofHooke'slaw.Hooke'slawcanalsobeexpressedasfollows:whenthestressdoesnotexceedacertainlimitvalue,thestressisproportionaltothestrain.Becausethestrainεhasnodimension,themodulusofelasticityEhasthesamemeasureasthestress.Finally,itisnotedthattheformulacanbeappliedonlywhentheaxialforceFN,thecross-sectionalareaA,andthemodulusofelasticityEofthematerialareconstantsintherangeofrodlengthl.Forthesteppedbarsorthebarswhoseaxialforcevariesinsegments:

WhentheaxialforceFN(x)andthecross-sectionalareaA(x)varycontinuouslyalongtherodaxisx-direction,thereis

FF2.Poisson'sratio

Letthetransverselengthoftherodbeforedeformationbebandafterdeformationbeb1,thetransverselinestrainoftherodisExperimentshowedthat：Therelationshipbetweenthetransversestrainandtheaxial(longitudinal)strainεisBecausethesignsofε’

andε

areopposite，soµ---knownasPoisson'sratioortransversedeformationcoefficient,isadimensionlessquantitywhosevaluevarieswiththematerial.Eandµarebothelasticconstantsinherenttothematerialitself,andareparametersthatreflecttheelasticdeformationcapacityofthematerial.Example

1Asteppedsteelrodisshowninthefigure.Itisknownthatthecross-sectionalareaofsectionACisA1=500mm2,thecross-sectionalareaofsectionCDisA2=200mm2,andthemodulusofelasticityofthesteelrodE=200GPa.Trytofind:(1)theinternalforcesandstressesinthecross-sectionofeachsectionoftherod;(2)Thetotalelongationoftherod.BACDF1=30kNF2=10kN100100100Solution:（1）calculationinternalforces2F1FFN12FFN2

cutoffthebarsalongthesections1-1and2-2andcalculatetheaxialforce,weget:2F1F1122ABCDAxialforcediagram20+10_xFN/kNBACD（2）calculatestresses（3）totalelongationofthebarNegativeresultmeansthatthewholerodisshortened.Example2AsteelplateshowninthefigurehasamodulusofelasticityE=200GPaandaPoisson'sratioof0.25.Findthevariationofthicknessoftheplatewhenitissubjectedtoauniformloadof140kNatbothends.2501050140kN{}140kN2501050{}solution：Undertheactionoftheuniformloadatbothends,thesteelplateundergoesaxialtensiledeformation.Thenormalstressonitscrosssectioncanbecalculatedbytheformula:（a）FromHooke'slaw,weget

（b）140kN140kN2501050{}Thetransverselinestrainisso（c）140kN140kN2501050{}Substitutingequation(b)intoequation(c)andconsideringequation(a),weget

Thethicknessofthesteelplateisreducedby0.0035mm.140kN140kN3.Strainenergyinaxialtension

(compression)Theenergystoredinthelinearelasticbodyasaresultofdeformationiscalleddeformationenergyorstrainenergy.ElasticdeformationEnergyStorageWorkdonebyexternalforcesExternalforcedecreasesDecreaseindeformationReleaseenergyAsshowninthefigure,thelowerendofastraightrodissubjectedtoagraduallyincreasingaxialtensionfromzerotothefinalvalueF.Thedisplacementatthepointofactionalsograduallyincreasesto∆l,andthetensionFisproportionalto∆lintherangewherethestressislessthantheproportionallimit.FF(a)(b)F

Wisequaltotheareaofthetriangleinthediagram.Clearly,dWisequaltodifferentialareacoveredbytheshadedlineinthefigure.Ignoringtheenergyloss,accordingtotheprincipleoffunction,thedeformationenergyUstoredintheelasticbodyshouldbeequaltotheworkWdonebythepullingforceF

ConsideringtheaxialforceandHooke'slaw,wecangetTheunitofstrainenergyisthejoule(J),1Joule(J)=1N∙m.Strainenergyperunitvolumeisknownasthedeformationspecificenergyorstrainenergydensity(denotedasu).

ConsideringtheHooke'slaw,wecangetTheunitofstrainenergydensityisjoulespermetre3(J/m3).

7.5Mechanicalpropertiesofmaterialsintension(compression)

Mechanicalpropertiesofmaterials——thedeformation,damageandothercharacteristicsofmaterialsduringdeformationbyforces1.Experimentalconditions:materialatroom-temperature,thetensiletestandcompressiontestinaslowandsmoothloadingmode2.Experimentalobject:astandardspecimenofcircularcross-sectionintensionasshowninthefigure:FFllFF——gagelength——diameterTherearetworatiosbetweenthelandthedinthenationalstandard.

and3.F-ΔlcurveThespecimenissubjectedtoagraduallyincreasingtensileforceFfromzero,whilethevariation∆lofthelengthlcanbemeasuredTheverticalcoordinaterepresentsthetensileforceFandthehorizontalcoordinaterepresentstheelongation∆l.TheF-∆l

curveisplottedaccordingtoaseriesofdatameasuredfromthebeginningoftheexperimentuntilthespecimenispulledoff,andiscalledatensilediagram.1.MechanicalpropertiesoflowcarbonsteelintensionLow-carbonsteelisavarietyofcarbonsteelswithacarboncontentof0.25%orless,andiscommonlyusedtorevealsomeofthemechanicalpropertiesofplasticmaterials.ThefigurebelowshowstheF-Δlcurvewhenlow-carbonsteelisstretched,andthiscurveisalsocalledatensilediagram.efgF0abcdhThe

σ-ε

curveforlow-carbonsteelthroughoutthetensiletestcanbedividedintofourstagesasfollows.

hgbd0aec1.ElasticRange2.YieldingRange3.StrainHardening

Range4.NeckingRangefInitialrange——elasticrange（sectionob）Iftheloadisslowlyremoved(unload)atthisstage,thedeformationcandisappearcompletely.Thestresscorrespondingtopointbiscalledtheelasticlimitofthematerialandisexpressedasσe,i.e.,themaximumstressthatthematerialcanwithstandintheelasticdeformationphase.Inthisrange,alargesectionofthecurveisastraightline(sectionoa),indicatingthatthestress-strainrelationshipisproportionalandsatisfiesHooke'slawThestresscorrespondingtopointaiscalledtheproportionallimit(σp)

ofthematerial.

Itisthemaximumstressthatthematerialcanwithstandwhenthestress-strainisinthepositiveproportionalrelationship.Initialrange——elasticrange（sectionob）Theexternalforcefluctuatesinasmallrange,butthedeformationincreasessignificantly.i.e.Thematerialtemporarilylosesitsabilitytoresistdeformation，Thisphenomenoniscalledyieldingorflowofthematerial.

Atthisstageafterunloadingtheexternalforcetozerotheelasticwilldeformationdisappears,whilethereisstillapartofplasticdeformationisretainedpermanently.Sliplines(atanangleof45degreestothespecimenaxis)willappearonthespecimensurface.Thestresscorrespondingtothelowestpointoftheyieldingrangeiscalledtheyieldstressoryieldlimit(denotedasσs)(Usuallyregardedasanindicatorofthestrengthofthematerial)succumbThesecondrange——YieldingRange（sectionbc）SliplineAftertheyieldingrange,thematerialregainsitsabilitytoresistdeformation.Thisphenomenonisknownasstrainhardening.Thestresscorrespondingtothehighestpointeiscalledtheultimatestressandultimatestrength,expressedbyσb.TheultimatestressisthemaximumstressthatthematerialcanwithstandduringtheentiretensileprocessThethirdrange——StrainHardeningRangeWhenthestressreachestheultimatestrength,thereisasuddenandsharpreductioninthetransversedimension,formingtheneckingphenomenon.Thereafter,theaxialdeformationofthespecimenismainlyconcentratedattheneckshrinkage.Theloadontheneckalsodecreasesrapidly,andfinallythespecimenispulledoffattheneck.stresswhenthespecimenispulledoff.Thelastrange——NeckingRange(sectionef)Nec-kingPercentageelongationisthemainindexofmaterialplasticity(5%)(1)Percentageelongation：(2)percentagereductioninarea

A1

——Minimumcross-sectionalareaatthefractureofthespecimen

A

——originalcrosssectionarea

Zisalsoanindexofmaterialplasticity(1).PercentageelongationandpercentagereductioninArea(1)

Unloadinglaw:

duringtheprocessofunloading,thestressandstrainvaryinalinearrelationship.

Thetotalstraincorrespondingtoanypointdbeyondtheelasticrangecontainsbothelasticandplasticstrains.hgefOabcd(2).Unloadinglawandcoldhardening)（2）coldhardeningefhg0abcdIfunloadingisfollowedbyreloadingwithinashortperiodoftime,Itwillbeseenthatproportionallimitincreases,buttheplasticdeformationdecreases.ColdhardeningcanbeusedtoimprovethestrengthofmaterialinitselasticrangeForplasticmaterialswithnoapparentyieldrange,thenominalyieldstressσ0.2,i.e.thestresscorrespondingto0.2%oftheplasticstrain,isusedastheyieldstress.se0.2%(3).MechanicalpropertiesofotherplasticmaterialsintensionGreycastironisatypicalbrittlematerial.Thestressatfractureistheultimatestrengthandistheonlystrengthindicator.SometimeswechooseacutlinetodeterminethevalueofEandconsiderthatthematerialobeysHooke'slaw.12510075502500.150.300.452(MN/m)s(%)e2、Mechanicalpropertiesofcastironintension3、Mechanicalpropertiesofmaterialincompression(1)PlasticmaterialsYellowline-curveoflowcarbonsteelintensionGreenline-curveoflowcarbonsteelintensionFesss(2)BrittlematerialsAsshowninthefigure:thecurveofcastironincompression.Experimentsshowthat:Thecurvedoesnothavea"yieldpoint",thespecimenissuddenlydamagedunderasmalldeformation.Thedamagesurfaceisinclinedatanapproximateangleof45degreestotheaxis.FF600500(%)e2MN/m

s1100200300400423506

Proportionallimitelasticlimit

Yieldlimit

（）Ultimatelimit

Modulusofelasticity

EPercentageelongationpercentagereductioninareaThemainindicatorsofthemechanicalpropertiesofvariousmaterials:——maximumstressallowedinthematerial.

——thestressthatcausesthematerialtobreakiscalledtheultimatestressor

damagestressn——thefactorn,whichisgreaterthan1,iscalledthefactorofsafety.PlasticmaterialsBrittlematerials7.6Strengthcalculation1、Factor

of

SafetyandAllowableStress

forequal-sectionrodstheaboveformulasarethestrengthconditionsofthebarinaxialtension(compression).Thesectionwherethemaximumworkingstressisgeneratediscalledthedangeroussection.2、ThestrengthconditionsUsingstrengthconditions,threeaspectsofstrengthcalculationsinengineeringcanbesolved：1.Strengthcheck2.SectiondesignTher