Mechanical Properties of Materials | MechaniCalc

Remember that the plastic strain at failure can be calculated from the percent elongation, eL, by εf = eL/100%. Luckily all of these properties are commonly ... 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RelatedPages EngineeringMaterials StrengthofMaterials MaterialsDatabase PDHClassroom AffordablePDHcreditsforyourPElicense ViewCourses MailingList Subscribeforoccasionalupdates: Go RelevantTextbooks RelatedPages: • EngineeringMaterials • StrengthofMaterials • MaterialsDatabase StressandStrain Therelationshipbetweenstressandstraininamaterialisdeterminedbysubjectingamaterialspecimentoatensionorcompressiontest.Inthistest,asteadilyincreasingaxialforceisappliedtoatestspecimen,andthedeflectionismeasuredastheloadisincreased.Thesevaluescanbeplottedasaload-deflectioncurve.Thedeflectioninthetestspecimenisdependentonboththematerial'selasticmodulusaswellasthegeometryofthespecimen(areaandlength).Sinceweareinterestedmaterialbehaviorwithoutregardtogeometry,itisusefultogeneralizethedatatoremovetheeffectofgeometry.Thisisdonebyconvertingtheloadvaluestostressvaluesandconvertingthedeflectionvaluestostrainvalues: Stress: Strain: Intheequationforstress,PistheloadandA0istheoriginalcross-sectionalareaofthetestspecimen.Intheequationforstrain,ListhecurrentlengthofthespecimenandL0istheoriginallength. Stress-StrainCurve Thevaluesofstressandstraindeterminedfromthetensiletestcanbeplottedasastress-straincurve,asshownbelow: Stress-StrainCurveCalculator CheckoutourStress-StrainCurvecalculatorbasedonthemethodologydescribedhere. Thereareseveralpointsofinterestinthediagramabove: P:Thisistheproportionalitylimit,whichrepresentsthemaximumvalueofstressatwhichthestress-straincurveislinear. E:Thisistheelasticlimit,whichrepresentsthemaximumvalueofstressatwhichthereisnopermanentset.Eventhoughthecurveisnotlinearbetweentheproportionalitylimitandtheelasticlimit,thematerialisstillelasticinthisregionandiftheloadisremovedatorbelowthispointthespecimenwillreturntoitsoriginallength. Y:Thisistheyieldpoint,whichrepresentsthevalueofstressabovewhichthestrainwillbegintoincreaserapidly.Thestressattheyieldpointiscalledtheyieldstrength,Sty.Formaterialswithoutawell-definedyieldpoint,itistypicallydefinedusingthe0.2%offsetmethodinwhichalineparalleltothelinearportionofthecurveisdrawnthatintersectsthex-axisatastrainvalueof0.002.Thepointatwhichthelineintersectsthestress-straincurveisdesignatedastheyieldpoint. U:Thispointcorrespondstotheultimatestrength,Stu,whichisthemaximumvalueofstressonthestress-straindiagram.Theultimatestrengthisalsoreferredtoasthetensilestrength.Afterreachingtheultimatestress,specimensofductilematerialswillexhibitnecking,inwhichthecross-sectionalareainalocalizedregionofthespecimenreducessignificantly. F:Thisisthefracturepointorthebreakpoint,whichisthepointatwhichthematerialfailsandseparatesintotwopieces. Stress-straincurvesarecommonlyneededwhenanalyzinganengineeredcomponent.However,stress-straindatamaynotalwaysbereadilyavailable.Inthiscase,itisfairlystraightforwardtoapproximateamaterial'sstress-straincurveusingtheRamberg-Osgoodequation. TrueStressandStrain Engineerstypicallyworkwithengineeringstress,whichistheforcedividedbytheoriginalareaofthespecimenbeforeloading:σ=P/A0.However,asamaterialisloaded,theareadecreases.Thetruestress,,isthevalueofstressinthematerialconsideringtheactualareaofthespecimen.Becausetheareadecreasesasamaterialisloaded,truestressishigherthanengineeringstress. Thefigurebelowshowsanengineeringstress-straincurveascomparedtoatruestress-straincurve.Becausetheengineeringstressiscalculatedasforcedividedbyoriginalarea(whichisaconstant),theengineeringstress-straincurvehasthesameshapeastheload-deflectioncurve.Theengineeringstress-straincurvedropsaftertheultimatestrengthisreachedbecausetheforcethatcanbesupportedbythematerialdropsasitbeginstoneckdown.However,thestressvalueinthetruestress-straincurvealwaysincreasesasthestrainincreases.Thisisbecausetheinstantaneousvalueofareaisusedwhencalculatingtruestress.Evenwhentheforcesupportedbythematerialdrops,thereductioninthespecimenareaoutweighsthereductioninforce,andthestresscontinuestoincrease. Itshouldbenotedthattheengineeringstressandthetruestressareessentiallythesameinthelinear-elasticregionofthestress-straincurve.Becauseengineerstypicallyoperatewithinthislinear-elasticregion(itisuncommontodesignastructurethatisintendedtooperatebeyondtheelasticlimit),itisvalidtoworkwithengineeringstressasopposedtotruestress. Engineeringstrainisthechangeinlengthdividedbytheoriginallength:ε=ΔL/L0.InsteadofjustcalculatingasinglevalueofΔL,considerthatthechangeinlengthisdividedamongmanysmallincrements,ΔLj.Thestrainisalsocalculatedinsmallincrements:εj=ΔLj/Lj,whereΔLjisthechangeinlengthforanincrement,andLjisthelengthatthestartoftheincrement.Astheseincrementsbecomeinfinitesimallysmall,thesummationofthestrainsapproachesthetruestrain,: Ifitisassumedthatthevolumeisconstantthroughoutthedeflection,thentruestressandstraincanbecalculatedas: TrueStress: TrueStrain: whereandarethetruestressandstrain,andσandεaretheengineeringstressandstrain. Hooke'sLaw Belowtheproportionalitylimitofthestress-straincurve,therelationshipbetweenstressandstrainislinear.Theslopeofthislinearportionofthestress-straincurveistheelasticmodulus,E,alsoreferredtoastheYoung'smodulusandthemodulusofelasticity.Hooke'slawexpressestherelationshipbetweentheelasticmodulus,thestress,andthestraininamaterialwithinthelinearregion: σ=Eε whereσisthevalueofstressandεisthevalueofstrain. Hooke'sLawinShear Hooke'slawalsohasaformrelatingshearstressesandstrains: τ=Gγ whereτisthevalueofshearstress,γisthevalueofshearstrain,andGistheshearmodulusofelasticity.Theelasticmodulusandtheshearmodulusarerelatedby: whereνisPoisson'sratio. MoreinformationonHooke'slawcanbefoundhere. Poisson'sRatio Asloadisappliedtoamaterial,thematerialelongatesandthecross-sectionalareaisreduced.Thisreductionincross-sectionalareaiscalledlateralstrain,anditisrelatedtotheaxialstrainbyPoisson'sratio,ν.Foracircularspecimenthisreductioninareaisrealizedasareductionindiameter,andthePoisson'sratioiscalculatedas: Poisson'sratioonlyapplieswithintheelasticregionofthestress-straincurve,anditistypicallyabout0.3formostmetals.ThetheoreticalmaximumlimitofPoisson'sratiois0.5. NeedStructuralCalculators? Wehaveanumberofstructuralcalculatorstochoosefrom.Herearejustafew: BeamCalculator BoltedJointCalculator BoltPatternForceDistribution LugCalculator ColumnBucklingCalculator FatigueCrackGrowthCalculator StrainHardening Afteramaterialyields,itbeginstoexperienceahighrateofplasticdeformation.Oncethematerialyields,itbeginstostrainhardenwhichincreasesthestrengthofthematerial.Inthestress-straincurvesbelow,thestrengthofthematerialcanbeseentoincreasebetweentheyieldpointYandtheultimatestrengthatpointU.Thisincreaseinstrengthistheresultofstrainhardening. Theductilematerialinthefigurebelowisstillabletosupportloadevenaftertheultimatestrengthisreached.However,aftertheultimatestrengthatpointU,theincreaseinstrengthduetostrainhardeningisoutpacedbythereductioninload-carryingabilityduetothedecreaseincrosssectionalarea.BetweentheultimatestrengthatpointUandthefracturepointF,theengineeringstrengthofthematerialdecreasesandneckingoccurs. Inthestress-straincurveforthebrittlematerialbelow,averysmallregionofstrainhardeningisshownbetweentheyieldpointYandtheultimatestrengthU.Notehoweverthatabrittlematerialmaynotactuallyexhibitanyyieldingbehaviororstrainhardeningatall--inthiscase,thematerialwouldfailonthelinearportionofthecurve.Thisismorecommoninmaterialssuchasceramicsorconcrete. Becausethestrainhardeningregionoccursbetweentheyieldpointandtheultimatepoint,theratiooftheultimatestrengthtotheyieldstrengthissometimesusedasameasureofthedegreeofstrainhardeninginamaterial.Thisratioisthestrainhardeningratio: strainhardeningratio=Stu/Sty AccordingtoDowling,typicalvaluesofstrainhardeningratioinmetalsrangefromapproximately1.2to1.4. Ifamaterialisloadedbeyondtheelasticlimit,itwillundergopermanentdeformation.Afterunloadingthematerial,theelasticstrainwillberecovered(returntozero)buttheplasticstrainwillremain. Thefigurebelowshowsthestress-straincurveofamaterialthatwasloadedbeyondtheyieldpoint,Y.Thefirsttimethematerialwasloaded,thestressandstrainfollowedthecurveO-Y-Y',andthentheloadwasremovedoncethestressreachedthepointY'.Sincethematerialwasloadedbeyondtheelasticlimit,onlytheelasticportionofthestrainisrecovered--thereissomepermanentstrainnowinthematerial.Ifthematerialweretobeloadedagain,itwouldfollowlineO'-Y'-F,whereO'-Y'isthepreviousunloadingline.ThepointY'isthenewyieldpoint.NotethatthelineO'-Y'islinearwithaslopeequaltotheelasticmodulus,andthepointY'hasahigherstressvaluethanpointY.Therefore,thematerialnowhasahigheryieldpointthanithadpreviously,whichisaresultofstrainhardeningthatoccurredbyloadingthematerialbeyondtheelasticlimit. Bystrainhardeningthematerial,itnowhasalargerelasticregionandahigheryieldstress,butitsductilityhasbeenreduced(thestrainbetweenpointsY'-FislessthanthestrainbetweenpointsY-F). ElasticandPlasticStrain Uptotheelasticlimit,thestraininthematerialisalsoelasticandwillberecoveredwhentheloadisremovedsothatthematerialreturnstoitsoriginallength.However,ifthematerialisloadedbeyondtheelasticlimit,thentherewillbepermanentdeformationinthematerial,whichisalsoreferredtoasplasticstrain. Inthefigureabove,bothelasticandplasticstrainsexistinthematerial.Iftheloadisremovedattheindicatedpoint(σ,ε),thestressandstraininthematerialwillfollowtheunloadinglineasshown.Theelasticstrainandplasticstrainareindicatedinthefigure,andarecalculatedas: ElasticStrain: εe=σ/E PlasticStrain: εp=ε−εe whereσisthestressattheindicatedpoint,εisthestrainattheindicatedpoint,andEistheelasticmodulus. Ductility Ductilityisanindicationofhowmuchplasticstrainamaterialcanwithstandbeforeitbreaks.Aductilematerialcanwithstandlargestrainsevenafterithasbeguntoyield.Commonmeasuresofductilityincludepercentelongationandreductioninarea,asdiscussedinthissection. Afteraspecimenbreaksduringatensiletest,thefinallengthofthespecimenismeasuredandtheplasticstrainatfailure,alsoknownasthestrainatbreak,iscalculated: whereLfisthefinallengthofthespecimenafterbreakandLoistheinitiallengthofthespecimen.Itisimportanttonotethatafterthespecimenbreaks,theelasticstrainthatexistedwhilethespecimenwasunderloadisrecovered,sothemeasureddifferencebetweenthefinalandinitiallengthsgivestheplasticstrainatfailure.Thisisillustratedinthefigurebelow: Inthefigure,itcanbeseenthattheplasticstrainatfailure,εf,isthestrainremaininginthematerialaftertheelasticstrainhasbeenrecovered.Theultimatestrain,εu,isthetotalstrainatfailure(theplasticstrainplustheelasticstrain). Thepercentelongationiscalculatedfromtheplasticstrainatfailureby: Thepercentelongationisacommonlyprovidedmaterialproperty,sotheplasticstrainatfailureistypicallycalculatedfrompercentelongation: εf=eL/100% Theultimatestrainaccountsforbothplasticandelasticstrainatfailure: εu=εf+Stu/E Anotherimportantmaterialpropertythatcanbemeasuredduringatensiletestisthereductioninarea,whichiscalculatedby: Rememberthatpercentelongationandreductioninareaaccountfortheplasticcomponentsoftheaxialstrainandthelateralstrain,respectively. NeedStructuralCalculators? Wehaveanumberofstructuralcalculatorstochoosefrom.Herearejustafew: BeamCalculator BoltedJointCalculator BoltPatternForceDistribution LugCalculator ColumnBucklingCalculator FatigueCrackGrowthCalculator DuctileandBrittleMaterials Aductilematerialcanwithstandlargestrainsevenafterithasbeguntoyield,whereasabrittlematerialcanwithstandlittleornoplasticstrain.Thefigurebelowshowsrepresentativestress-straincurvesforaductilematerialandabrittlematerial. Inthefigureabove,theductilematerialcanbeseentostrainsignificantlybeforethefracturepoint,F.ThereisalongregionbetweentheyieldatpointYandtheultimatestrengthatpointUwherethematerialisstrainhardening.ThereisalsoalongregionbetweentheultimatestrengthatpointUandthefracturepointFinwhichthecrosssectionalareaofthematerialisdecreasingrapidlyandneckingisoccurring. Thebrittlematerialinthefigureabovecanbeseentobreakshortlyaftertheyieldpoint.Additionally,theultimatestrengthiscoincidentwiththefracturepoint.Inthiscase,noneckingoccurs. Becausetheareaunderthestress-straincurvefortheductilematerialaboveislargerthantheareaunderthestress-straincurveforthebrittlematerial,theductilematerialhasahighermodulusoftoughness--itcanabsorbmuchmorestrainenergybeforeitbreaks.Additionally,becausetheductilematerialstrainssosignificantlybeforeitbreaks,itsdeflectionswillbeveryhighbeforefailure.Therefore,itwillbevisuallyapparentthatfailureisimminent,andactionscanbetakentoresolvethesituationbeforedisasteroccurs. Arepresentativestress-straincurveforabrittlematerialisshownbelow.Thiscurveshowsthestressandstrainforbothtensileandcompressiveloading.Notehowthematerialismuchmoreresistanttocompressionthantotension,bothintermsofthestressthatitcanwithstandaswellasthestrainbeforefailure.Thisistypicalforabrittlematerial. StrainEnergy Whenforceisappliedtoamaterial,thematerialdeformsandstorespotentialenergy,justlikeaspring.Thestrainenergy(i.e.theamountofpotentialenergystoredduetothedeformation)isequaltotheworkexpendedindeformingthematerial.Thetotalstrainenergycorrespondstotheareaundertheloaddeflectioncurve,andhasunitsofin-lbfinUSCustomaryunitsandN-minSIunits.Theelasticstrainenergycanberecovered,soifthedeformationremainswithintheelasticlimit,thenallofthestrainenergycanberecovered. Strainenergyiscalculatedas: GeneralForm: U=Work=∫FdL (areaunderload-deflectioncurve) WithinElasticLimit: (areaunderload-deflectioncurve) (springpotentialenergy) Notethattherearetwoequationsforstrainenergywithintheelasticlimit.Thefirstequationisbasedontheareaundertheloaddeflectioncurve.Thesecondequationisbasedontheequationforthepotentialenergystoredinaspring.Bothequationsgivethesameresult,theyarejustderivedsomewhatdifferently. StrainEnergyDensity Itissometimesmoreconvenienttoworkwithstrainenergydensity,whichisthestrainenergyperunitvolume.Thisisequaltotheareaunderthestress-straindiagram: wherethelimitsintheintegralabovearefromastrainof0toεapp,whichisthestrainexistingintheloadedmaterial. NotethattheunitsofstrainenergydensityarepsiinUSCustomaryunitsandPainSIunits. ModulusofResilience Themodulusofresilienceistheamountofstrainenergyperunitvolume(i.e.strainenergydensity)thatamaterialcanabsorbwithoutpermanentdeformationresulting.Themodulusofresilienceiscalculatedastheareaunderthestress-straincurveuptotheelasticlimit.However,sincetheelasticlimitandtheyieldpointaretypicallyveryclose,theresiliencecanbeapproximatedastheareaunderthestress-straincurveuptotheyieldpoint.Sincethestress-straincurveisverynearlylinearuptotheelasticlimit,thisareaistriangular. Themodulusofresilienceiscalculatedas: generalform triangularform whereσelandεelarethestressandstrainattheelasticlimit,Styisthetensileyieldstrength,andEistheelasticmodulus. Notethattheunitsofthemodulusofresiliencearethesameastheunitsofstrainenergydensity,whicharepsiinUSCustomaryunitsandPainSIunits. ModulusofToughness Themodulusoftoughnessistheamountofstrainenergyperunitvolume(i.e.strainenergydensity)thatamaterialcanabsorbjustbeforeitfractures.Themodulusoftoughnessiscalculatedastheareaunderthestress-straincurveuptothefracturepoint. Anaccuratecalculationofthetotalareaunderthestress-straincurvetodeterminethemodulusoftoughnessissomewhatinvolved.However,aroughapproximationcanbemadebydividingthestress-straincurveintoatriangularsectionandarectangularsection,asseeninthefigurebelow.Theheightofthesectionsisequaltotheaverageoftheyieldstrengthandtheultimatestrength. Themodulusoftoughnesscanbeapproximatedas: whereStyisthetensileyieldstrength,Stuisthetensileultimatestrength,εyisthestrainatyield,εuistheultimatestrain(totalstrainatfailure),andEistheelasticmodulus. AbettercalculationofthemodulusoftoughnesscouldbemadebyusingtheRamberg-Osgoodequationtoapproximatethestress-straincurve,andthenintegratingtheareaunderthecurve. Itshouldbenotedhowgreatlytheareaundertheplasticregionofthestress-straincurve(i.e.therectangularportion)contributestothetoughnessofthematerial.Sinceaductilematerialcanwithstandmuchmoreplasticstrainthanabrittlematerial,aductilematerialwillthereforehaveahighermodulusoftoughnessthanabrittlematerialwiththesameyieldstrength.Eventhoughstructuresaretypicallydesignedtokeepstresseswithintheelasticregion,aductilematerialwithahighermodulusoftoughnessisbettersuitedtoapplicationsinwhichanaccidentaloverloadmayoccur. Notethattheunitsofthemodulusoftoughnessarethesameastheunitsofstrainenergydensity,whicharepsiinUSCustomaryunitsandPainSIunits. NeedStructuralCalculators? Wehaveanumberofstructuralcalculatorstochoosefrom.Herearejustafew: BeamCalculator BoltedJointCalculator BoltPatternForceDistribution LugCalculator ColumnBucklingCalculator FatigueCrackGrowthCalculator Stress-StrainCurveApproximation Stress-straincurvesformaterialsarecommonlyneeded;however,withoutrepresentativetestdataitisnecessarytocomeupwithanapproximationofthecurve.TheRamberg-Osgoodequationcanbeusedtoapproximatethestress-straincurveforamaterialknowingonlytheyieldstrength,ultimatestrength,elasticmodulus,andpercentelongationofthematerial(allofwhicharecommonandreadilyavailableproperties). Stress-StrainCurveCalculator CheckoutourStress-StrainCurvecalculatorbasedonthemethodologydescribedhere. TheRamberg-Osgoodequationfortotalstrain(elasticandplastic)asafunctionofstressis: whereσisthevalueofstress,Eistheelasticmodulusofthematerial,Styisthetensileyieldstrengthofthematerial,andnisthestrainhardeningexponentofthematerial,whichcanbecalculatedfromknownmaterialpropertiesasshownlaterinthissection.(Note1) AnexplanationofthederivationoftheRamberg-Osgoodequationisprovidedinthefollowingsections. Ramberg-OsgoodEquation ArelationshipwasproposedbyRambergandOsgoodthatisfrequentlyusedtoapproximatethestress-straincurveforamaterial.Thisrelationshipisexponentialandisusedtodescribetheplasticstraininamaterial.Thestress-straincurveintheplasticregioncanbeapproximatedby: σ=Hεpn ⟹ where,intheequationabove,εpistheplasticstrain,Histhestrengthcoefficientwiththesameunitsasstress,andnisthestrainhardeningexponentandisunitless. Theelasticstraininamaterialislinearlyproportionaltostress: σ=Eεe ⟹ εe=σ/E where,intheequationabove,εeistheelasticstrainandEistheelasticmodulus. Thetotalstraininamaterialisthesummationoftheelasticstrainandtheplasticstrain: ε=εe+εp ⟹ DeterminingConstantsforRamberg-Osgood FortheRamberg-Osgoodequationtobeuseful,valuesfortheconstantsnandHmustbeknown.Adiscussionofhowtodeterminetheconstantsforanexponentialequationisgivenhere. Theconstantsarefoundby: where,intheequationsabove,(σ1,ε1)and(σ2,ε2)correspondtotwopointswithintheplasticregionofthestress-straincurve.Thetaskathandthenistofindthosetwopointssothattheconstantsmaybecalculated. Iftheyieldstrength(Sty),ultimatestrength(Stu),elasticmodulus(E),andplasticstrainatfailure(εf)foramaterialareknown,thentwopointswithintheplasticregioncanbedetermined(theyieldandultimatepoints),andfromthosepointstheplasticregioncurvecanbecalculated.Rememberthattheplasticstrainatfailurecanbecalculatedfromthepercentelongation,eL,byεf=eL/100%.Luckilyallofthesepropertiesarecommonlyknownforamaterial. Itisimportanttonotethattheequationfortheplasticregioncurve,σ=Hεpn,isdependentonplasticstrain,andsowewillneedtodeterminethevaluesofplasticstrainforthetwopointsofinterest.Plasticstraincanbecalculatedfromtotalstrainusing: εp=ε−εe=ε−σ/E where,intheequationabove,εisthetotalstrainandεeistheelasticstrain. Thetablebelowisusedtodeterminetheyieldpointandtheultimatepoint: Stress,σ TotalStrain,ε ElasticStrain,εe PlasticStrain,εp YieldPoint: Sty Sty/E+0.002 Sty/E 0.002 UltimatePoint: Stu Stu/E+εf Stu/E εf Notethatwhendeterminingthestrainattheyieldpoint,aplasticstrainof0.002wasassumed.Thisisconsistentwiththe0.2%offsetmethod,asdescribedpreviously.Thisassumptionisnecessaryinordertoplacetheyieldpointwithintheplasticregionofthecurve.Fromthetableabove,itcanbeseenthattheyieldpointandultimatepointwithintheplasticregionaregivenby: • YieldPoint: (Sty,0.002) • UltimatePoint: (Stu,εf) Fromthetwopointsintheplasticregionofthecurve,theconstantsnandHfortheRamberg-Osgoodequationcanbecalculated.Thestrainhardeningexponent,n,iscalculatedas:(Note1) ThevalueforHiscalculatedusingtheyieldpoint,(Sty,0.002),asthepointofreference,althougheitherpointwoulddo: NowthattheconstantsnandHhavebeendetermined,theequationforthetotalstrainasafunctionofstressisknown: TheequationabovecanbesimplifiedbysubstitutingtheexpressionforH.Thefinalequationfortotalstrainasafunctionofstressis: EarnContinuingEducationCreditforReadingThisPage PDHClassroomoffersacontinuingeducationcoursebasedonthismechanicalpropertiesofmaterialsreferencepage.ThiscoursecanbeusedtofulfillPDHcreditrequirementsformaintainingyourPElicense. Nowthatyou'vereadthisreferencepage,earncreditforit! Viewthecoursenow: ViewCourse Notes Note1:StrainHardeningExponentinRamberg-OsgoodEquation Thestrainhardeningexponent,denotedbyn,shouldnotbeconfusedwiththeRamberg-Osgoodparameter,whichisalsodenotedbyn.Thetwoparametersarereciprocalsofoneanother,whichonlyaddstotheconfusion.WeusethestrainhardeningexponentintheRamberg-OsgoodequationratherthantheRamberg-Osgoodparameter.ThereasonthatweusethestrainhardeningexponentisthatitisageneralmaterialpropertythatisusefuloutsidethecontextoftheRamberg-Osgoodequation. References GeneralReferences: Budynas-Nisbett,"Shigley'sMechanicalEngineeringDesign,"8thEd. Dowling,NormanE.,"MechanicalBehaviorofMaterials:EngineeringMethodsforDeformation,Fracture,andFatigue,"3rdEd. Gere,JamesM.,"MechanicsofMaterials,"6thEd. Hibbeler,RussellC.,"MechanicsofMaterials,"10thEd. Lindeburg,MichaelR.,"MechanicalEngineeringReferenceManualforthePEExam,"13thEd. SpecificationsandStandards: ASTME8,"StandardTestMethodsforTensionTestingofMetallicMaterials,"AmericanSocietyforTestingandMaterials,2011.