Lecture34– ReviewforExam2

Instructor:Prof.MarcialGonzalez

Fall, 2017ME 323 – Mechanics of Materials

Readingassignment:HW5-HW10

News:Exam2onTuesday11/14.NolectureonMon.11/20.

WeeklyJoyswithexamples!!

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Exam2- TuesdayNovember14th ,8:00-10:00p.m.,room RHPH172

(pleasearrive15minutesbeforetheexamandbringapictureID)

- Formulasheetwillbeprovidedanditisalreadyuploadedtotheblog.

- ThoseofyouwhorequireextraaccommodationpleasecometoMExxxx andbringyouraccommodationletterandpictureID.(pleasearrive15minutesbeforetheexam)

- AdditionalReviewSession:Sunday7-9p.m.,roomME1130

Announcements

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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

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EquationsheetforExam2(draft)

Shearstressinbeams

- Jourawski Theory(orCollignon Theory)

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Equilibriumofbeams

Whatabout?Transverseandlongitudinalshearstress!

whereQ(y) isthefirstmomentofareaA’(y) withrespecttotheneutralaxis.

Averagetransverseshearstress

Q(y) =

Z

A0(y)⌘dA = A⇤y⇤

= =

Load-deflectionequations

(constantcross-sectionandmaterialproperties)

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Deflectionofbeams

inclinationangle (~slope)

deflection

Shear-deflectioneqn.

Load-deflectioneqn.

Moment-curvatureeqn.

(2nd order) (4nd order)

(followsignconventions)

Boundaryconditions

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Deflectionofbeams

(followsignconventions)

= |

= |

Continuityconditions

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Deflectionofbeams

>0

>0

= |

= |

Energymethods

Work andelasticstrainenergy

Workdonebytheforce: Workdonebythetorque: Workdonebythemoment:

AA

A

AA

A

Storedelasticstrainenergy: Storedelasticstrainenergy: Storedelasticstrainenergy:

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v

PC

△C

Energymethods- Castigliano’s SecondTheorem

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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)

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Finiteelementmethods– One-dimensionalrodelements

Energymethods- FEM

Introductiontofiniteelementmethods(FEM)

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Finiteelementmethods – One-dimensionalrodelements- Example54(review):

Numberofnodes:4

Numberofelements:3

Boundaryconditions:

Stiffnessofeachelement:Introductiontofiniteelementmethods(FEM)

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Finiteelementmethods – One-dimensionalrodelements- Example54,solvedin5steps+Step#5:Recoverthereactionatthesupports

Introductiontofiniteelementmethods(FEM)

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Finiteelementmethods – One-dimensionalrodelements- Example54,solvedin5steps+Step#3:Enforceboundaryconditions

+Step#4:Solvethereducedsystemoflinearequations

Numberofnodes:4Numberofelements:3

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Cylindricalbodywithhemisphericalendcaps

axialstressinthecylinder

Hoopstressinthecylinder

normalstressinthesphere

Thinwallpressurevessels

Stresstransformationforplanestress

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with

(measuredcounterclockwisefromxtox’)

Transformationofstress

PrincipalstressesFindtheorientationsuchthatthenormalstressismaximum:

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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

Studyhard!

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Reviewsession– Exam2