Lecture34– ReviewforExam2
Instructor:Prof.MarcialGonzalez
Fall, 2017ME 323 – Mechanics of Materials
Readingassignment:HW5-HW10
News:Exam2onTuesday11/14.NolectureonMon.11/20.
WeeklyJoyswithexamples!!
2
Exam2- TuesdayNovember14th ,8:00-10:00p.m.,room RHPH172
(pleasearrive15minutesbeforetheexamandbringapictureID)
- Formulasheetwillbeprovidedanditisalreadyuploadedtotheblog.
- ThoseofyouwhorequireextraaccommodationpleasecometoMExxxx andbringyouraccommodationletterandpictureID.(pleasearrive15minutesbeforetheexam)
- AdditionalReviewSession:Sunday7-9p.m.,roomME1130
Announcements
3
Summaryoftopics
- Flexuralandshearstressinbeams- Deflectioninbeams(second- andfourth-ordermethods,superposition)- Castigliano’s secondtheorem
.axialloads,torsion,bending,shearforces
.dummyloadsandindeterminateproblems- Finiteelementmethod- Thinwallpressurevessels- Transformationofstress- Principalstressesandmaximumshearstress- Mohr’scircle
- Lectures16–33(absolutemaximumshearstresses isNOTincluded)
Review
Shearstressinbeams
- Jourawski Theory(orCollignon Theory)
5
Equilibriumofbeams
Whatabout?Transverseandlongitudinalshearstress!
whereQ(y) isthefirstmomentofareaA’(y) withrespecttotheneutralaxis.
Averagetransverseshearstress
Q(y) =
Z
A0(y)⌘dA = A⇤y⇤
= =
Load-deflectionequations
(constantcross-sectionandmaterialproperties)
6
Deflectionofbeams
inclinationangle (~slope)
deflection
Shear-deflectioneqn.
Load-deflectioneqn.
Moment-curvatureeqn.
(2nd order) (4nd order)
(followsignconventions)
Energymethods
Work andelasticstrainenergy
Workdonebytheforce: Workdonebythetorque: Workdonebythemoment:
AA
A
AA
A
Storedelasticstrainenergy: Storedelasticstrainenergy: Storedelasticstrainenergy:
9
v
PC
△C
Energymethods- Castigliano’s SecondTheorem
10
Consideranindeterminate linearlyelasticdeformablebodyactinguponbyforces,moments,andtorques.Amongallpossibleequilibriumconfigurations
ofthebody,theactualconfigurationistheoneforwhich:
wheregeneralizeddisplacements()correspondtoandareinthedirectionoftheload(),andtheredundantorinternalload(thatdonotdoanyexternalwork).Note:someoftheseloadscouldbedummyloads,withvaluezero,thatwillfacilitatethecalculationofageneralizeddisplacementattheirpointofapplication.
Castigliano’s SecondTheorem
(displacement– force)
(slope– bendingmoment)
(angleofrotation– torque)
(redundantorinternalload)
11
Finiteelementmethods– One-dimensionalrodelements
Energymethods- FEM
Introductiontofiniteelementmethods(FEM)
6
Finiteelementmethods – One-dimensionalrodelements- Example54(review):
Numberofnodes:4
Numberofelements:3
Boundaryconditions:
Stiffnessofeachelement:Introductiontofiniteelementmethods(FEM)
9
Finiteelementmethods – One-dimensionalrodelements- Example54,solvedin5steps+Step#5:Recoverthereactionatthesupports
Introductiontofiniteelementmethods(FEM)
8
Finiteelementmethods – One-dimensionalrodelements- Example54,solvedin5steps+Step#3:Enforceboundaryconditions
+Step#4:Solvethereducedsystemoflinearequations
Numberofnodes:4Numberofelements:3
12
Cylindricalbodywithhemisphericalendcaps
axialstressinthecylinder
Hoopstressinthecylinder
normalstressinthesphere
Thinwallpressurevessels
Stresstransformationforplanestress
13
with
(measuredcounterclockwisefromxtox’)
Transformationofstress
PrincipalstressesFindtheorientationsuchthatthenormalstressismaximum:
14
Principalstresses:
Principalstresses- Maximumshearstress
with
Principaldirections:
noshearstressontheprincipalplanes!!
maximuminplane
normalstress
minimuminplane
normalstress
shearstressonprincipal planes
Maximumin-planeshearstressesFindtheorientationsuchthatthenormalstressismaximum:
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theplanesofmaximumshearstressarenotfreeofnormalstress!!
Principalstresses- Maximumshearstress
Maximumin-planeshearstress Planesofmaximumshear:
with
normalstressonplanesofmax.shear
Mohr’scircleforplanestress
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Goal:agraphicalrepresentationoftheplanestresstransformationequations
Mohr’scircle
Alwaysformaright-handedcoordinate
systems