Mechanics of Materials: Stress - Boston University

There are two types of stress that a structure can experience: 1. Normal Stress and 2. Shear Stress. When a force acts perpendicular (or "normal") to the ... ThisSiteBUMedicalAllBUBUMapsBUDirectoryGoogle MechanicsofMaterials:Stress research people courses blog WelcometotheMechanicsofMaterials.ThiscoursebuildsdirectlyonthefundamentalswelearnedinStatics–calculatingthestaticequilibriumofvariousstructuresundervariousloads.Instatics,weconsidertheexternalforces actingonrigidbodies.Inreality,allbodiesaredeformableandthoseexternalforcesgenerateinternalstresses.Wellthen,what'sastress?  Stressisthemeasureofanexternalforce actingoverthecrosssectionalareaofanobject.Stresshasunitsofforceperarea:N/m2 (SI)orlb/in2(US).TheSIunitsarecommonlyreferredtoasPascals,abbreviatedPa.Sincethe1Paisinconvenientlysmallcomparedtothestressesmoststructuresexperience,we'lloftenencounter103Pa=1kPa(kiloPascal),106Pa=aMPa(megaPascal),or109Pa=GPa(gigaPascal). Therearetwotypesofstressthatastructurecanexperience:1.NormalStressand2.ShearStress.Whenaforceactsperpendicular(or"normal")tothesurfaceofanobject,itexertsanormalstress.Whenaforceactsparalleltothesurfaceofanobject,itexertsashearstress. Let'sconsideralightfixturehangingfromtheceilingbyarope.Thecrosssectionoftheropeiscircular,andtheweightofthelightispullingdownward,perpendiculartotherope.Thisforceexertsanormalstresswithintherope. Okay,howdidwearriveatthisequation.Therearealotofassumptionsbehindthescenes.Throughoutthiscourse,wewillassumethatallmaterialsarehomogenous,isotropic,andelastic.Wewillalsoassumethattheobjectis"prismatic"–meaningthecrosssectionsarethesameallalongitslength(e.g.acucumberisprismatic,whileabutternutsquashisnot). Alltheseassumptionsallowustostatethattheobjectwilldeformuniformlyateverypointofitscrosssection.Thenormalstressatapointonacrosssectionisdefinedas(withsimilarequationsinthe x and y directions).: Eachsmallareaofthecrosssectionissubjectedtothesameforce,andthesumofalltheseforcesmustequaltheinternalresultantforce P.IfweletΔAgotodA,and ΔFgotodF,thenwecansimplyintegratebothsidesoftheequation,andwearriveatourrelationshipfornormalstress. Thisrelationshipforthenormalstressismoreaccuratelyan averagenormalstress,sincewe'veaveragedtheinternalforcesovertheentirecrosssection. Stressisoftenadifficultconcepttograspbecauseyoucan’teasilyobserveit.Asitturnsout,placingatransparentobjectthroughcrosspolarizedlightallowsyoutodirectlyobservestresswithinamaterial,basedonaconceptcalledphotoelasticity: Stresscanactuallyexistinamaterialintheabsenceofanappliedload.Thisisknownasresidualstress,anditcanbeusedasawaytotoughenmaterials,suchasinthefabricationoftheJapaneseKatanasword.Conversely,undesiredresidualstressescanencouragecrackgrowthandleadtofracture,suchaswiththecollapseoftheSilverBridgeofWestVirginiain1967.Perhapsthemoststrikingexampleofresidualstressisrelatedtotherapidcoolingofmoltenglass,knownas“PrinceRupert’sDrop”: Let'slookatanotherexample.Consideraboltthatconnectstworectangularplates,andatensileforceperpendiculartothebolt.Fromafreebodydiagram,weseethattheexternallyappliedforceexertsaforceparalleltothecircularcrosssectionofthebolt.Thisexternalforceresultsinashearstresswithinthebolt. Now,theformaldefinitionsofshearstresstakeonasimilarformasthosedescribedabove.Let'sconsidertheshearstressactingonthe z-faceofanelement: Theshearstressisthestressactingtangenttothecrosssection,andittakesonanaveragevalueof: It'simportanttonotethatthestresseswehavejustdescribedareaveragestresses.Wehaveassumedthatalloftheexternalforcehasbeenevenlydistributedoverthecrosssectionalareaofthestructure–thisisnotalwaysthecase,andwewillrevisitthisassumptionthroughoutthecourse. Whenyoulookatanelementundershear,thingsseemtobeabitmorecomplicated.Considerasmallcubicelementwithinastructureundershear,asshownbelow. Now,equilibriumrequiresashearstressactingonτzy tobeaccompaniedbyshearstressesontheotherplanes.But,let'sconsidertheforceequilibriuminthe y-direction.Knowingthatforcecanbewrittenasstress(tau)timesarea(ΔxΔy),wecanwritethisforceequilibriumas: Sincetheareasofthecubearebydefinitionthesame,thatmeansτzy =τ'zy. Asimilarforceequilibriuminthe z-directionleadstoτyz =τ'yz.Let'sconsideramomentequilibriumaboutthe x-axis.Knowingthatwecanwritetheforceasbefore,andthemomentarmwillbeΔz,thismomentbalancebecomes: Thissimplerelationshiptellsusthatτzy =τyz, andthereforeallfourshearstresseshaveequalmagnitudes,andmustpointtowardorawayfromeachotheratoppositeedgesoftheelement.Thisrelationshipisknownas"pureshear". 1.2FactorofSafety Engineersusestresstoaidinthedesignofstructures.Theexternalloadandthegeometryofthestructuretellsuswhatstressisbeingexertedwithinthematerial,butittellsusnothingaboutthematerialitself.Eachmaterialhasanultimatestress –ameasureofhowmuchstressthematerialcanwithstandbeforefailing.Toproperlydesignasafestructure,weneedtoensurethattheappliedstressfromtheexternalloadingneverexceedstheultimatestressofthematerial. Partofthedifficultywiththistaskisthatwedon'talwaysknowexactlywhattheexternalloadis–itmayvaryunpredictably,andthestructuremayhavetowithstandunexpectedlyhighloads.Toaccountforthisuncertainty,weincorporateaFactorofSafetyintoourdesign.Thefactorofsafetyisjustaratioofthefailureloadorstresstotheallowableloadorstress.Thefailureorultimatevalueisamaterialproperty whiletheallowablevalueisdeterminedbytheexternalforceandthegeometryofthestructure.    Summary We'veintroducedtheconceptofstressinthislecture.Stressistheameasureofwhatthematerialfeelsfromexternallyappliedforces.Itissimplyaratiooftheexternalforcestothecrosssectionalareaofthematerial.Forcesthatareappliedperpendiculartothecrosssectionare normalstresses,whileforcesappliedparalleltothecrosssectionare shearstresses.Whiletheconceptsintroducedherearenottooforeign,muchofthedifficultywiththismaterialcomesfromthechallengeofcalculatingthe staticequilibrium correctly.Calculatingstaticequilibriumwilltellusthemagnitudeanddirectionoftheappliedforces,whichwecanthenusetocalculatethestresses.Ifthefollowingexamplevideosandthehomeworkcauseyoudifficulty,nowwouldbeagoodtimetogobackandreviewsomeoftheconceptsfromyourStaticscourse. ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunderGrantNo.1454153.Anyopinions,findings,andconclusionsorrecommendationsexpressedinthismaterialarethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNationalScienceFoundation. BostonUniversitySearchDirectoryContactBUToday