ch軸向拉壓實(shí)用

會(huì)計(jì)學(xué)1ch軸向拉壓實(shí)用2.1BarsUnderSimpleTension–ElasticBehaviourCh.2AxialForcesN（kip)48x第2頁/共45頁第1頁/共45頁N=P2.2SimpleStressCalculations:squaresectionbarPPb2bPLeftsection:??=P(equilibrium)Rightsection:PxUNITS?NCh.2AxialForces第3頁/共45頁第2頁/共45頁Ch.2AxialForces2.3StressDistribution第4頁/共45頁第3頁/共45頁Ch.2AxialForces2.4DesignConsiderationUltimateStrengthofaMaterialAllowableLoadandAllowablestress;FactorofSafety強(qiáng)度條件第5頁/共45頁第4頁/共45頁No.8No.1030oACB三角架：Q235；AB：兩根等邊角鋼80807

AC：兩根10號(hào)槽鋼[]=120MPa,求許可載荷PCh.2AxialForces2.5SampleProblem1解：1.受力分析30PxN2N1Ay…(1)…(2)第6頁/共45頁第5頁/共45頁No.8No.1030oACBCh.2AxialForces2.5SampleProblem130PxN2N1Ay2.計(jì)算許可軸力[N]第7頁/共45頁第6頁/共45頁2.6DeformationandStrain(變形和應(yīng)變)

ElasticStiffness(RobertHooke,1648)Elasticdeformationofmaterialswillbethefocusofthiscourse.Plentyofchallengingproblemsinthisarea…

Hooke’sLaw(Young&Poisson)

MaterialProperties(ThomasYoung,1810)

TemperatureStrains(WilliamRankine,1870)

Mat.Props(Cont.)(SimonPoisson,1825)

StrainEnergy(CarloCastigliano,1881)Asinallareasofscience,avarietyofpeoplehavemadesignificantcontributionstotheMechanicsofMaterials.TheA-list:Ch.2AxialForces第8頁/共45頁第7頁/共45頁RobertHookewasthefirsttoexperimentwithanddefinethestiffnessofmaterials.Hewasalso:uStiffness(RobertHooke,1648)Theforceisproportionaltotheextensionofthespring.mHesuspendedvariousmassesfromsprings,andmeasuredtheextension.

anarchitect

anastronomer

fatherofmicroscopy

aphysicist第9頁/共45頁第8頁/共45頁WAuLoad(W)Extension(u)BarBBarAElasticBehaviourLimitofProportionalityKHooke’sFindings:LOADEXTENSIONi.e.AxialStiffness(inUnitsofN/m)Kdependson:i)MaterialPropertiesii)GeometryofBar(i.e.LandA)第10頁/共45頁第9頁/共45頁ThomasYounghelpedtodevelopthetheoryofhowmaterialsdeformelastically.Inparticular,hedefinedanimportantmaterialconstant,“Young’sModulus”.MaterialProperties(ThomasYoung,1810)LetAxialStress,LetAxialStrain,PPL(inUnitsofN/m2orPascals(Pa))(Dimensionless)第11頁/共45頁第10頁/共45頁Considerasmallelementofthisbar:PPLPPN(NormalForce=P)NNuN(NormalForce=P)第12頁/共45頁第11頁/共45頁P(yáng)PPoissonmadeimportantobservationsandtheoriesaboutlateraldeflectionsofmaterials.Arealmathnut.MaterialProperties(Cont.)(SimonPoisson,1825)“Hisonlypassionhasbeenscience:helivedandisdeadforit.”Whenabarisplacedintension,lateralcontractionsaccompanytheextension.InitialShapeFinalShape第13頁/共45頁第12頁/共45頁SimpleStrainCalculation:Consideracableofconstantdiameter,withanappliedtensionP:PPIfthecablehasaninitiallengthof1m,andstretches10mm,calculatethestrain:(NOTE:nounits)dxxLo第14頁/共45頁第13頁/共45頁DefinitionofStrainxyzdXdYdZ(Undeformedgeometry)Previousconceptofstrain:FnFnLoLLet’sconsiderstrainsintheXYplane.第15頁/共45頁第14頁/共45頁Considerabarunderuniaxialtension:xyzuvwuwvPoissonfoundthatunderuniaxialtension:StrainsinYandZ-directionsStraininX-directionE0nistermed“Poisson’sRatio”i.e.&第16頁/共45頁第15頁/共45頁Example:Considertherectangularbarbelowundertension.xyzuvw

Extension(inX-direction):bdLPPFindexpressionsforaxialstiffness,andtheaxialandlateraldeflections.Whatarethenormalstressandstrain?第17頁/共45頁第16頁/共45頁xyzuvwbForhighK:PdL

AxialStiffness(inX-direction):PX-sectionalareaYoung’sModulusLengthExample:Considertherectangularbarbelowundertension.第18頁/共45頁第17頁/共45頁xyzuvwb

LateralContractions:PPdLYoung’sModulus(E)andPoisson’sRatio(n)areMATERIALPROPERTIES.i.e.Canonlybedeterminedbyexperimentalmethods.第19頁/共45頁第18頁/共45頁P(yáng)oisson’sRatiovMildSteel0.3Aluminium0.33Concrete0.1-0.2Wood-Nylon0.4Rubber0.45-0.5Thereexistsatheoreticallimit:0.5Constantvolume2.8DeformationandStrain(變形和應(yīng)變)Ch.2AxialForces第20頁/共45頁第19頁/共45頁Sample

ProblemExtensionofabarduetoSelfWeightL,A,r,E,nvConsideralongbarhangingvertically,beingdeformedbygravity.IfthemaximumstressinthebarissMax,whatisthemaximumlengthofbar,LMax,andthetotalextensionofthisbar?xygABRAyFyy(y)yFBD:第21頁/共45頁第20頁/共45頁gvxyABRAyFyy(y)yFyy0RAyFyyismaximumaty=L(Fyy=RAy)第22頁/共45頁第21頁/共45頁ABRay=(rAg)LExtension?vydyFBD:FyyFyydvBydefinition:dy第23頁/共45頁第22頁/共45頁ForaMildSteelWire,r=7850kg/m3,smax=280MPa,E=200GPaWhataboutlateralcontraction?Ifthebariscircularincross-section,andhasadiameterD:SampleProblemExtensionofabarduetoSelfWeight第24頁/共45頁第23頁/共45頁TherigidbarBDEissupportedbytwolinksABandCD.LinkABismadeofaluminum(E=70GPa)andhasacross-sectionalareaof500mm2.LinkCDismadeofsteel(E=200GPa)andhasacross-sectionalareaof(600mm2).Forthe30-kNforceshown,determinethedeflectiona)ofB,b)ofD,andc)ofE.SampleProblem第25頁/共45頁第24頁/共45頁SOLUTION:Applyafree-bodyanalysistothebarBDEtofindtheforcesexertedbylinksABandDC.EvaluatethedeformationoflinksABandDCorthedisplacementsofBandD.WorkoutthegeometrytofindthedeflectionatEgiventhedeflectionsatBandD.SampleProblem第26頁/共45頁第25頁/共45頁Freebody:BarBDESOLUTION:SampleProblemDisplacementofB:第27頁/共45頁第26頁/共45頁DisplacementofD:SOLUTION:SampleProblem第28頁/共45頁第27頁/共45頁DisplacementofE:SampleProblem第29頁/共45頁第28頁/共45頁FBarsABandACeachhaveacrosssectionalareaA，amodulusofelasticityEandthedimensionsh.IfadownwardforceFisappliedatAwhatistheresultinghorizontalandverticaldisplacementsofpointA?伸長量？位移？SampleProblem第30頁/共45頁第29頁/共45頁SampleProblemSOLUTION:第31頁/共45頁第30頁/共45頁BSampleProblemSOLUTION:第32頁/共45頁第31頁/共45頁CSampleProblemSOLUTION:第33頁/共45頁第32頁/共45頁u=?