(1.3.3)-ch0-3 Force and stress材料力学材料力学

Chapter0Introduction

MechanicsofMaterialsForceandStressWelcometomechanicsofmaterials.Inthelastvideo,wereviewedstatics,thisvedioisgoingtodiscussforcesandstresses.ExternalForcesForcesappliedtothestructurefromoutsideofthestructure.Asweknowit,aforceissomethingthatactsonanobject.Forexample,agravitationalforcepullsobjectstowardtheearth,andafrictionforceactsagainstthemomentumofanobject.So,whatisexternalforce?Theexternalforceisaforcethatactsfromtheoutsideofthestructure.Therearemanyexamplesofexternalforces,likegravitationalforce,buoyantforce,friction,pullandpushforces,electromagneticforce,windforceandotherforces.InternalForcesForcesproducedfromtheexternalforcesactingwithinastructure.Bycontrast,aninternalforceisaforcethatactswithinasystem.Anexternalforceoccursasaresultofinteractionbetweenasystemandthesurroundings.Ontheotherhand,aninternalforceisaninteractionthatexistswithinasystem.Externalforcescausemotioninanobject,whileaninternalforceresiststhemotion.*let’slookingatthisgif,atreeissettoswingwhenaforceisappliedbythewindonit.Sincetheforceexertedbythewindonthetreeisfromtheoutside,itisknownasanexternalforce.Onthecontrary,theforcethathelpsthetreetostayinthepositionandpreventsitfromfallingdownisknownasaninternalforce.InternalForcesFxFyFzMxMyMzmmmmmmFx-axialforce/normalforceMx-torqueFy,Fz-shearforceMy,Mz-bendingmomentFromtheStatics,weknowthattherecanbeamaximumofsixindependantinternalreactioncomponents,*threeinternalforcesand*threecouplemoments.Normally,thesesixinternalloadsareplacedonaspecialpoint,centroidofthecrosssection.Theseforcesandmomentsinfactareresultantforcesandresultantmomentsofinternalforcesaboutx,yandzaxes,respectively.*Furthermore,thesixinternalforcescanbeclassifiedintofourcategories,axialforceFX,transverseshearforcesFYandFZ,torsinalmomentMX,bendingmoment

MYandMZ,correspondingtofourtypesofdeformations,axialtensionandcompression,shear,torsionandbending,respectively.

Theclassificationoftheinternalforcesandcouplemomentsisbasedonthedeformations(physicaleffects)theycauseonabody.*Theactualinternalforcesaredistributedthroughoutthiscrosssection.Wewilllearnhowtheinternalloadsdistributeonacrosssectionlaterinthiscourse.These

forcescanbefoundatanypointalongthememberbymakinganimaginarycut,

withinternalforcesexposedonthecutend.Thismethodiscalledmethodofsections.MethodofsectionsStandardmethodtofindinternalloads.cuthereCalculationoftheinternalforcesisthefoundationtoanalyzetheproblemsofstrength,rigidity,stabilityetc.*Methodofsectionsisthestandardwaytofindinternalloads.Asaforementioned,Mechanicsofmaterialsisconcernedwiththestudyofdeformationsofdeformablebodies.

*Firstfindingtheinternalloadsatkeypointsalongthemembersofthestructure,*isthefoundationtoanalyzetheproblems

ofstrength,rigidity,stabilityandotherrelevantproblems.Methodofsectionsq(x)P1P2①Cut：Makeanimaginarysectioncutnormaltotheaxisofamemberatthepointofinterest.q(x)P1P2q(x)VNM②FBD：Dawa

free-bodydiagramforeitherpart,includingalltheexternalforces,andinternalloads

onthecutsectionexertedbyanotherpart.③Equilibrium：Applyequilibriumequationstodetermineinternalloads.VNMP1P2Tousethemethodofsections,

thefirststepiscut,thatistomakeanimaginarysectioncutnormaltotheaxisofamemberatthepointofinterest,andthememberiscutintotwoparts.Then,selecteitherparttoanalyze,drawafree-bodydiagramforoneofthetwoparts,includingalltheexternalforcesonthepart,andinternalloadsonthecutsectionexertedbytheremovedpart.Eachsegmentorpartmustbeinequilibium,soapplyequilibriumequationstothepart,andsolveforunknowninternalloads.Pleasenote,inmostcasesitispossibletowriteoneequationtosolveforoneunknowndirectly.Solookforit

andtakeadvantageofsuchashortcut!Internalforcesonthecutsectionq(x)VNMVNMP1P2-Equalinmagnitude-OppositeindirectionNote:Internalloadsappearasexternalloadsonthefree-bodydiagramofthesegments.LeftsideRightsideSectioningthememberresultsintwosegments,*leftsideand*rightside.Theforcesandmomentappliedbythe*left-handsideontotheright-handsideareequalinmagnitudebutoppositeindirectiontotheforcesappliedbythe*right-handsideontheleft-handside.InternalloadingsatapointappearasexternalloadsontheFBDoftheassociatedsegments.Signconvention-InternalloadsOnleftsideofcut：+N:out+V:down+M:anti-clockwiseVNMVNMVNMVNMOnrightsideofcut：+N:out+V:up+M:anti-clockwise①DeformationSignconventionAfterthememberiscut,afree-bodydiagramcanbedrawnforeithersection,acorrectandconsistentsignconventionfortheinternalloadsmustbeusedwhenwedrawthefree-bodydiagram.Asignconventionaugmentedtoaforceistoexpressthephysicalnatureortypeoftheforce.Insummary,therearetwosignconversionsforaforce:deformationsignconvensionandpointsignconvention.*Forthedeformationsignconvenstion,itissettoassignsignstothe(internal)forcesofamemberaccordingtotheirphysicaleffects.*Thesignconventionshowndescribeswhatismeantbyapositiveforceormoment.Thisconventionisthemostcommonsignconventionforforcesandmoments.**Forcesandmomentsactinginthedirectionshownareconsideredtobepositive.Signconvention-InternalloadsYXM+②Staticsignconvention(Pointsignconvention)-Basedonthecoordinatesystemutilized.-Setforwritingtheequilibriumequations.

-Assigns‘+

’and‘

-’signstothecomponentsoftheforces.

*Forthepointsignconvenstion,*itissetforwritingtheequilibriumequations.*Thissignconventionisbasedonthecoordinatesystemutilizedand*assignspositiveandnegativesignstothecomponentsoftheforcesinvolvedintheequationsofequilibrium*Forcesandmomentsactinginthedirectionshownareconsideredtobepositive.Axialforce:Internalforceoftherodinaxialtension/compression,designatedbyN.APPPANAPPCut:Free-bodydiagram(Left)：Equilibrium：Example1:DeterminetheaxialforceNatsectionAusingmethodofsections.

Nowlet’slookatthissimpleexample,askingyoutodetermineinternalaxialforceNatlocationAusingthemethodofsection.Followthethreestepsofthismethod.Firststep,makinganimaginarycutatpointA;secondly,selecttheleftside,drawthefreebodydiagramfortheleftsegment.Laststep,writeequilibriumequationfortheleftsegment,solveittofindaxialforceNwhichequalstoP.StressNormalstressσ:producesachanginthelength

oftheelement.Shearstressτ:

producesachangeintheshapeoftheelement.Stressesproducedinsideabodycanbeclasssifiedintonormalstressandshearstress.Normalstressesσproduceachanginthelength,Whiletheshearstressesτ

produceachangeintheshapeoftheelement.

F

AM

Fx

Fz

FyStressNormalstressShearstressUnits:AveragenormalstressAverageshearstressAsaforementioned,theactualinternalforcesaredistributedthroughoutacrosssection.Let’stakeasmallareaonthecrosssection,withareaΔA,andaforceΔFonit.ΔFcanbedecomposedintothreecomponentsalongx,yandzaxis,ΔFx,ΔFyandΔFz,whereΔFxisnormalfroce,ΔFyandΔFzareshearforces.AnaveragenormalstressandshearstesscanbedefinedasΔFx/ΔA,ΔFy/ΔAorΔFz/ΔA,thatisforcedividedbyarea.Whentheareaapprocheszero,Mbecomesapoint,thenormalstressσandshearstressesτatthispointaredefinedbytheseexpresions.Inmostcases,stressesvarieswithlocationononecrosssection,thatmeansbothmagnitudeanddirectionofnormalstressσandshearstressesτmayvaryatdifferentlocationsonthecrosssection.Sincestressesaredefinedbyforceoverarea,Consequently,stresshasunitsofnewtonspersquaremeter(N/m2),whichisequaltoapascal(Pa)intheSIunitssystem.However,thepascalissuchasmallunitofstressthatitisnecessarytoworkwithlargemultiples,usuallythemegapascal(MPa).

F

AM

Fx

Fz

Fy

AσxτxzτxyStateofStresszxyσzτzxτyzσyτxyσx*Therefore,foraninfinitesimalareaΔAonthecrosssection,*itcouldhaveamaximumofthreeindependentstressesactingonit,thenormalstressσandtwoshearstressesτ,corresppondingtothreeinternalforces.Butdoesthisparticlehaveonlyonesurface?TheanswerisNO.Therearecountlessplanesthroughthepointinspace,crosssectionplaneisoneofthem.Similartothecrosssectionplane,thereareamaximumofthreeindependentstressesonanyotherplane,onenormalandtwoshearst