ch轴向拉压实用

会计学1ch轴向拉压实用2.1BarsUnderSimpleTension–ElasticBehaviourCh.2AxialForcesN（kip)48x第2页/共45页第1页/共45页N=P2.2SimpleStressCalculations:squaresectionbarPPb2bPLeftsection:??=P(equilibrium)Rightsection:PxUNITS?NCh.2AxialForces第3页/共45页第2页/共45页Ch.2AxialForces2.3StressDistribution第4页/共45页第3页/共45页Ch.2AxialForces2.4DesignConsiderationUltimateStrengthofaMaterialAllowableLoadandAllowablestress;FactorofSafety强度条件第5页/共45页第4页/共45页No.8No.1030ºACB三角架：Q235；AB：两根等边角钢80807

AC：两根10号槽钢[]=120MPa,求许可载荷PCh.2AxialForces2.5SampleProblem1解：1.受力分析30PxN2N1Ay…(1)…(2)第6页/共45页第5页/共45页No.8No.1030ºACBCh.2AxialForces2.5SampleProblem130PxN2N1Ay2.计算许可轴力[N]第7页/共45页第6页/共45页2.6DeformationandStrain(变形和应变)

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页/共45页第7页/共45页RobertHookewasthefirsttoexperimentwithanddefinethestiffnessofmaterials.Hewasalso:uStiffness(RobertHooke,1648)Theforceisproportionaltotheextensionofthespring.mHesuspendedvariousmassesfromsprings,andmeasuredtheextension.

anarchitect

anastronomer

fatherofmicroscopy

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

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

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

LateralContractions:PPdLYoung’sModulus(E)andPoisson’sRatio(n)areMATERIALPROPERTIES.i.e.Canonlybedeterminedbyexperimentalmethods.第19页/共45页第18页/共45页Poisson’sRatiovMildSteel0.3Aluminium0.33Concrete0.1-0.2Wood-Nylon0.4Rubber0.45-0.5Thereexistsatheoreticallimit:0.5Constantvolume2.8DeformationandStrain(变形和应变)Ch.2AxialForces第20页/共45页第19页/共45页Sample

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