ME 211 Homework 6 Solutions

ME 211 Homework 6 Solutions - Problem#18.58...

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Unformatted text preview: Problem#18.58 ME211HomeworkSet#6Solutions 10172008 DeterminetheapproximateaveragenormalstraininthewireABasafunctionofthe rotationoftherigidbarCAbyassumingissmall.Whatisthevalueif=2degrees. 1.)Drawingouttherelationshipsasitrotatesisseenbelow.Ifweuseasmallangle approximationthismeans (thetamustbeinradians).Usingthisand thestrainformula,wecanapproximateBA'asjusttheoriginalplusthesineofthe angle,dropthecomplexgeometry. 2.)Applyingtheapproximation Problem#28.62 TheguywireABofabuildingframeisoriginallynotstretched.Duetoan earthquake,thetwocolumnsoftheframetilt=2degrees.Determinethe approximatenormalstraininthewirewhentheframeisinthisposition.Assume thecolumnsarerigidandrotateabouttheirlowersupports. 1.) Drawthestructureasitisshiftedtobettervisualizewhatisgoingoninthe problem. 2.) Findtheamountthatthewireextendsinthexdirection. x a = (1m)2 = 0.03491m 180 x b = (4m)2 = 0.13963m 180 3.)Findthenewlengthofthewireinthexdirection.Addthechangeinlengthofthe wireinthexdirectiontothetotallengthofthewirebeforethestretchingoccurred. x = 4 + (x b - x a ) = 4.1047m A' B'= 32 + 4.1047 2 = 5.08416m AB = 32 + 4 2 = 5m 3.)Findtheaveragenormalstrain. AB = A' B'-AB 5.08416 - 5 = = 16.8(10-3 )m /m AB 5 Problem#38.88 Thepieceofrubberisoriginallyrectangular.Determinetheaverageshearstrainxy ifthecornersBandDaresubjectedtothedisplacementsthatcausetherubberto distortasshownbythedashedlines. 1.)Determinetheangles,1and2. 2 = 0.0066667rad 300 3 2 tan 2 = = 0.0075rad 400 1 tan 1 = 2.)Findtheshearstrainbyaddingthetwoanglestogether. xy = 1 + 2 = 0.006667 + 0.0075 xy = 0.0142rads Problem#48.68 Therectangularplateissubjectedtothedeformationshownbythedashedline. Determinetheshearstrainsxyandx'y'developedatpointA. 1.)Determinetheshearstraininthexyplane,xy. Aswecanseeinthedrawingbefore,therightanglesdonotchangeinthexyplane; therefore,thereisnoshearstraininthexyplane. xy = 0 2.)Determinetheshearstraininthex'y'plane,x'y'. Beginbyfindingthenewangleofthedeformedrectangle. 4.01 3.995 = .787272rad tan = Thenfindthechangeinangle. x' y' = x' y' x' y' - 2 2 = - 2(0.787272) 2 = -3.75(10-3 ) Problem#59.15 Abarhavingalengthof5inandcrosssectionalareaof0.7in*inissubjectedtoan axialforceof8000lbs.Ifthebarstretches0.002in,determinethemodulusof elasticityofthematerial.Thematerialhaslinearelasticbehavior. 1.)Findthenormalstress = P 8 = = 11.43ksi A 0.7 2.)Findthenormalstrain = L 0.002 = = 0.0004in /in L 5 3.)FindthemodulusofElasticity E= 11.43 = = 28.6(10 3 )ksi 0.0004 Problem#69.20 Thesupportconsistsofthreerigidplates,whichareconnectedtogetherusingtwo symmetricallyplacedrubberpads.Ifaverticalforceof50NisappliedtoplateA, determinetheapproximateverticaldisplacementofthisplateduetoshearstrains intherubber.Eachpadhascrosssectionaldimensionsof30mmand20mm.Gr= 0.20MPa. 1.)DrawtheFBDofthetopsection. 2.)Findtheshearstress 3.)ApplyHooke'sLawwiththeshearformula. Deflectionwasfoundusingsmallangleapproximation. Problem#79.21 AshortcylindricalblockofbronzeC86100,havinganoriginaldiameterof1.5inand alengthof3in,isplacedinacompressionmachineandsqueezeduntilitslength becomes2.98in.Determinethenewdiameteroftheblock. 1.)Determinetonormalstrain = L - Lo 2.98 - 3 = = -0.00667in /in Lo 3 2.)Determinethelateralstrainutilizingthepoissonratiorelationshipoflateraland longitudinalstrain. lat = -long = -0.34(-0.00667in /in) lat = 0.002267in /in 3.)Findthechangeinlengthofthediameterandaddittotheoriginaldiameterto determinethenewdiameter. d = lat d = 0.002267(1.5) = 0.00340in d = do + d = 1.5 + 0.0034 = 1.5034in Problem#810.9 Theassemblyconsistsoftworigidbarsthatareoriginallyhorizontal.Theyare supportedbypinsand0.25indiameterA36steelrods.IFtheverticalloadof5kipis appliedtothebottombarAB,determinethedisplacementatC,B,andE. 1.)DrawtheFBD. 2.)Findthetensions. 3.)FindthedeflectionatpointC. 4.)FindthedeflectionatpointE 5.)FindthedeflectionatpointB ...
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