Centroids and Internal Forces - Assignment PDF

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Centroids and Internal Forces - Assignment...

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OREGONSTATEUNIVERSITY CollegeofEngineering

ENGR211 Instructor:K.Martin

ENGR211:Statics CentroidsandInternalForces Completeeachproblemonengineeringpaper.YoudonotneedtousetheGiven/Find/Solutionformat,butyour solutionsmustbefullydevelopedandpresentedprofessionally.Showallyourwork,includingequationsused,and pertinentsketches.Finalanswersshallbeboxedorcircled,mustincludecorrectunits,andshowconcernforsignificant digits(usually3or4).Yourworkistobecompletedandsubmittedindividually.

Problem1:The90‐degreebarABhasamassof50kg,actingthroughitscentroid.Whatarethe reactionsatA&B?ThesupportatBisaroller.

 a. LocatethecenterofgravityofbarABrelativetopointA.Duetosymmetry,wedon’tneedtosolve forbothx&ycoordinates.Usethetabularmethodtofind x .Useunitsof“mm”inyourtableand toexpressyourfinalanswer.Usetheshapesshownbelow.

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OREGONSTATEUNIVERSITY CollegeofEngineering

ENGR211 Instructor:K.Martin

b. Applyequilibriumequationsandsolvefortheunknownreactions(showdirection,too).Assume thetotalweightofthebaractsthroughthecentroidyoujustcomputed. c. IsbarABatwo‐forcememberinthisproblem?Explainyouranswer. d. ConsiderbarABtobecomprisedoftwopiecesasshownbelow.Assumeeachpieceweighshalf thetotalweight(actingthrougheachrespectivecentroid).Re‐computethereactionswiththis systemandcomparetoyourpreviousanswers.

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OREGONSTATEUNIVERSITY CollegeofEngineering

ENGR211 Instructor:K.Martin

Problem2:Theshaftissubjectedtotheaxialloadsshown.Determinetheinternalforceswithinthe shaft(seequestionsbelow).

 a. Firstdetermineiftheshaftisinfactinequilibrium.Checkthesumoftheforcesinthex‐direction. Assumeitisfineinthey‐direction.Wedon’tknowanythingaboutthesupportconditions,andwe canassumetheweightoftheshaftisnegligibleincomparisontotheappliedloads.Therefore, neglecttheself‐weight.

b. Iftheentireshaftisinequilibrium,anypiecewebreakoff(orcutthrough)mustbeinequilibrium. MakeacutsomewherebetweenAandBasshown,andsolvefortheinternalnormal(axial)force, NAB,betweenAandB.Seespecificinstructionsbelow...

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i.

FirstusetheleftsidetosolveforNAB=?

ii.

Thencheckyouranswerusingtherightside,NAB=?

iii.

IssectionABoftheshaftbeingstretched(tension)orsquashed(compressed)?

iv.

Ifweneglecttheweightoftheshaft,whatdoyougetfortheinternalshear,V,and moment,M? 

OREGONSTATEUNIVERSITY CollegeofEngineering

ENGR211 Instructor:K.Martin

c. NowrepeattheprocessforthepartoftheshaftbetweenBandC.Takeasectioncutsomewhere betweenpointsBandC,usethesamesignconventionasshownpreviously,andanswerthe followingquestions.

 i.

UsetheleftsidetosolveforNBC =?

ii.

Thencheckyouranswerusingtherightside,NBC =?

iii.

IssectionBCofthisshaftbeingstretched(tension)orsquashed(compressed)?

d. RepeattheprocesswithinthepartoftheshaftbetweenCandD.Takeasectioncutsomewhere betweenpointsCandD,usethesamesignconvention,andanswerthefollowingquestions.

 i.

UsetheleftsidetosolveforNCD=?

ii.

Thencheckyouranswerusingtherightside,NCD=?

iii.

IssectionCDofthisshaftbeingstretched(tension)orsquashed(compressed)?

e. Youcancheckyouranswersbytakingthesectioncutshownbelow.Anyportionoftheshaftyou cutshouldbeinequilibrium.

      Isthesectionshowninequilibrium,usingthevaluesyoucomputedforNBCandN CD?

OREGONSTATEUNIVERSITY CollegeofEngineering

ENGR211 Instructor:K.Martin

f. Ifweneglecttheweightoftheshaft,doyouexpecttheshafttobendatall(eitherupordown)? Ignorebuckling.Assumetheshaftwillnotbuckleifsubjectedtoaxialcompressiveforces. 

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NegativeBending 

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

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PositiveBending

   g. WhatdoesthissayaboutthevalueforVandMalongthelengthoftheshaft?   

Problem3:Computetheinternalforces(normalforce,shearforce,andbendingmoment)inthe cantileverbeamatpointC.Youwilldothisproblemtwodifferentwaysandcompare youranswers(bothmethodsarerequired).

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Method1:Usetherightsideofthebeam 1. ComputetheintensityofthetriangularloadatpointC.Youcanusesimilartrianglesorany othermethodyoulike. 2. CutthebeamatpointCandkeeponlytherightside(i.e.drawaFBDofthesectionofthebeam fromCtoB).YoumustshowthetriangularloadinyourFBD,andyoumustshowtheinternal forceswiththecorrectsignconvention.Hint:Seeexample7.2onpage335foraverysimilar problem. 3. Sumforcesinthex‐directionandy‐directiontocomputeN candVcrespectively. 4. Summomentsaboutthecut(pointC)tocomputeM c.Itisokaytoboxanegativevaluesince we’redealingwithinternalforceshere.

 Continuedonnextpage

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OREGONSTATEUNIVERSITY CollegeofEngineering

ENGR211 Instructor:K.Martin

Method2:Usetheleftsideofthebeam 1. First,computetheexternalreactionsatthefixedendusingtheoverallFBD.YouneedAy,Ax, andMA . 2. NowcutthebeamatpointCandkeeponlytheleftside(i.e.drawaFBDofthesectionofthe beamfromAtoC).YoumustshowthedistributedloadinyourFBD,andyoumustshowthe internalforceswiththecorrectsignconvention.Hint:Theinternalforceswillnothavethe sameorientationaswhatyoudrewforMethod1.Youmustalsoshowtheexternalreactions actingatpointA.Seefigurebelow.

 3. Breakthetrapezoidalloadshapeintoarectangleandtriangle,andcomputetheareaofeach. 4. Nowyoucansumforcesinthex‐directionandy‐directiontocomputeN candVcrespectively. 5. Summomentsaboutthecut(pointC)tocomputeM c.Again,itisokaytoboxanegativevalue sincewe’redealingwithinternalforceshere. 6. Compareyouranswerstowhatyoucalculatedusingmethod1.Iftheyarethesame,doyou expectthemtobethesame?Iftheyaredifferent,doyouexpectthemtobedifferent?...