fea plane stress analysis

8/12/2019 FEA Plane Stress Analysis

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Copyright2009. Allrightsreserved. 101

6 PlaneStressAnalysis6.1 IntroductionGenerally

you

will

be

forced

to

utilize

the

solid

elements

in

SW

Simulation

due

to

acomplicated

solid

geometry. Tolearnhowtoutilizelocalmeshcontrolfortheelementsitisusefultoreviewsometwo

dimensional(2D)problemsemployingthemembranetriangularelements. Historically,2Danalytic

applicationsweredevelopedtorepresent,orbound,someclassicsolidobjects. Thosespecialcasesinclude

planestressanalysis,planestrainanalysis,axisymmetricanalysis,flatplateanalysis,andgeneralshellanalysis.

Aftercompletingthefollowing2Dapproximationyoushouldgobackandsolvethemuchlarger3Dversionof

theproblemandverifythatyougetessentiallythesameresultsforboththestressesanddeflections.

Planestressanalysisisthe2Dstressstatethatisusuallycoveredinundergraduatecoursesonmechanicsof

materials. Itisbasedonathinflatobjectthatisloaded,andsupportedinasingleflatplane. Thestresses

normaltotheplanearezero(butnotthestrain). Therearetwonormalstressesandoneshearstress

componentateachpoint(x, y,and). Thedisplacementvectorhastwotranslationalcomponents(u_x,and

u_y).Therefore,

any

load

(point,

line,

or

area)

has

two

corresponding

components.

TheSWSimulationshellelementscanbeusedforplanestressanalysis. However,onlytheirinplane,or

membrane,behaviorisutilized. Thatmeansthatonlytheelementsinplanedisplacementsareactive. The

generalshellinplanerotationvectorsarenotusedforplanestressstudiesandshouldberestrained. To

createsuchastudyyouneedtoconstructthe2Dshapeeitherasasheetmetalpart,aplanarsurface(InsertSurfacePlanarsurface),orextrudeitasasolidpartwithaconstantthicknessthatissmallcomparedtotheothertwodimensionsofthepart. Asolidislaterconvertedtoashellmodelbycreatingamidsurfaceoran

offsetsurfacewithinthesolid.

Beforesolidelementsbecameeasytogenerateitwasnotunusualtomodelsomeshapesas2.5D. Thatis,

theywereplanestressinnaturebuthadregionsofdifferentconstantthickness. Thisconceptcanbeusefulin

validatingtheresultsofasolidstudyifyouhavenoanalyticapproximationtouse. Sincethemidsurfaceshells

extracttheir

thickness

automatically

from

the

solid

body

you

should

use

amid

plane

extrude

when

you

are

buildingsuchapart.

Oneuseofaplanestressmodelhereistoillustratethenumberofelementsthatareneededthroughthe

depthofaregion,whichismainlyinastateofbending,inordertocaptureagoodapproximationofthe

flexuralstresses. Elementarybeamtheoryand2Delasticitytheorybothshowthatthelongitudinalnormal

stress(x)varieslinearlythroughthedepth. Forpurebendingitistensionatonedepthextreme,compression

attheother,andzeroatthecenterofitsdepth(alsoknownastheneutralaxis).Whenthebendingisdue,in

part,toatransverseforcethentheshearstress()ismaximumattheneutralaxisandzeroatthetopand

bottomfibers. Forarectangularcrosssectiontheshearstressvariesparabolicallythroughthedepth. Since

theelementstressesarediscontinuousattheirinterfaces,youwillneedatleastthreeofthequadratic(6

node)membranetriangles,oraboutfiveofthelinear(3node)membranetrianglestogetareasonablespatial

approximationof

the

parabolic

shear

stress.

This

concept

should

guide

you

in

applying

mesh

control

through

thedepthofaregionyouexpect,orfind,tobeinastateofbending.

6.2 SimplysupportedbeamAsimplerectangularbeamplanestressanalysiswillbeillustratedhere.Considerabeamofrectangularcross

sectionwithathicknessoft=2cm,adepthofh=10cm,andalengthofL=100cm. Letauniformly

distributeddownwardverticalloadofw=100N/cmbeappliedatitstopsurfaceandletbothendsbesimply

supported(i.e.,haveu_y=0attheneutralaxis)byarollersupport. Inaddition,bothendsaresubjectedto


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PlaneStressAnalysis J.E.Akin

Draft10.2Copyright2009. Allrightsreserved. 102

equalmomentsthateachdisplacesthebeamcenterdownwards. TheendmomenthasavalueofM=1.25e3

Nm.Thematerialisaluminum1060. Thisisaproblemwherethestressesdependonlyonthegeometry.

However,thedeflectionsalwaysdependonthematerialtype.

Figure61 Asimplysupportedbeamwithlineloadandendmoments

Itshouldbeclearthatthisproblemissymmetricalabouttheverticalcenterline(whythatistruewillbe

explainedshortlyshoulditnotbeclear). Therefore,nomorethanhalfthebeamneedstobeconsidered(and

halftheload). Selecttherighthalf. Thebeamtheoryresultsshouldsuggestthatanevenmoresimplified

modelwouldbevalidduetoantisymmetry(ifweassumehalfthelineloadactsonboththetopandbottom

faces). The3Dflatfacesymmetryrestraintwasdescribedearlier. The2Dnatureofthisexampleprovides

insightintohowtoidentifylines(orplanesin3D)ofsymmetryandantisymmetry,asshowninFigure62.

Figure62 Onequarterofthebeam

6.2.1 SymmetryandantisymmetryrestraintsAprocessforidentifyingdisplacementrestraintsonplanesofsymmetryandantisymmetrywillbeoutlined

here. Assumethatthehorizontalcenterlineofthebeamcorrespondstothe

Antisymmetric,ub= ua Symmetric,vb=va

Figure63 Antisymmetric(u=0,v=?),andsymmetric(u=?,v=0)displacementstates

dashedcenterlineoftheantisymmetricimageattheleftinFigure63. Thequestionis,what,ifany,restraint

shouldbeappliedtotheuorvdisplacementcomponentonthatline. Toresolvethatquestionimaginetwo

mirrorimagepoints,aandb,eachadistance,,aboveandbelowthedashedline. Notethatboththeupper

andlowerhalfportionsareloadeddownwardinanidenticalfashion,andtheyhavethesamehorizontalend

supports. Therefore,youexpectvaandvbtobeequal,buthaveanunknownvalue(sayva=vb=?). Likewise,


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thehorizontalloadapplicationisequalinmagnitude,butofoppositesignintheupperandlowerregions.

Therefore,youexpectub= ua. Nowletthedistancebetweenthepointsgotozero(0). Thelimitgivesv=

va=vb=?,sovisunknownandnorestraintisappliedtoit. Thelimitonthehorizontaldisplacementgivesu=

ub= ua0,sothehorizontaldisplacementcanberestrainedtozeroifyouwithtouseahalfdepthanti

symmetricmodel. Anotherwaytosaythatis:onalineorplaneofantisymmetrythetangential

displacementcomponent(s)isrestrainedtozero. Forashellorbeamtherotationalcomponentnormaltoan

antisymmetry

plane

is

also

zero.

Theverticalcenterlinesymmetrycanbejustifiedinasimilarway. ImaginethattherightimageinFigure63is

rotated90degreesclockwisesothedashedlineisparalleltothebeamverticalsymmetryline. Nowu

representsthedisplacementcomponenttangenttothebeamcenterline(i.e.,vertical). Theverticalloadingon

bothsidesisthesame,asaretheverticalendsupports,sotheverticalmotionataandbwillbethesame(say

ua=ub=?). Inthelimit,asthetwopointsapproacheachotheru=ua=ub=?,sothebeamverticalcenterline

hasanunknowntangentialdisplacementandisnotsubjecttoarestraint. Nowconsiderthedisplacement

normaltothebeamverticalcenterline(herev). Atanyspecifieddepth,theloadingsanddeflectionsinthat

directionareequalandopposite. Therefore,inthelimitasthetwopointsapproacheachotheru=ub= ua

0,sothedisplacementcomponentnormaltothebeamverticalcenterlinemustvanish. Anotherwaytostate

thatis:onalineorplaneofsymmetrythenormaldisplacementcomponentisrestrainedtozero.Forashell

orbeam

the

rotational

components

parallel

to

asymmetry

plane

are

also

zero.

6.2.2 Beamloadcase1,momentloadingsFromtheabovearguments,the2Dapproximationcanbereducedtoonequarteroftheoriginaldomain. The

othermaterialisremovedandreplacedbytherestraintsthattheyimposeontheportionthatremains. Now

yourattentioncanfocusontheappliedloadstates. Thetop(andbottom)lineloadcanbereplacedeither

withatotalforceonthetopsurface,oranequivalentpressureonthetopsurface,sinceSWSimulationdoes

notofferaloadperunitlengthoption. Unfortunately,eitherrequiresahandcalculationthatmightintroduce

anerror. Thelessobviousquestionishowtoapplytheendmoment(s).

Sincethegeneralshellelementhasbeenforcedtolieinaflatplane,andhavenoloadsnormaltotheplane,its

twoinplanerotationalDOFwillbeidenticallyzero. However,thenodalrotationsnormaltotheplanearestill

active(intheliteraturetheyarecalldrillingfreedomsin2Dstudies). Thatmaymakeyouthinkthatyoucould

applyamoment,Mz,atanodeontheneutralaxisofeachendofthebeam. Intheory,thatshouldbepossible,

butinpracticeitworkspoorly(tryit)andtheendmomentshouldbeappliedinadifferentfashion. Oneeasy

waytoapplyamomentistoformacouplebyapplyingequalpositiveandnegativetriangularpressuresacross

thedepthoftheendsofthebeam. Thatapproachworksequallywellfor3Dsolidsthatdonothaverotational

degreesoffreedom.

Themaximumrequiredpressureisrelatedtothedesiredmomentbysimplestaticequilibrium. Theresultant

horizontalforceforalinearpressurevariationfromzerotopmaxisF=Apmax/2,whereAisthecorresponding

rectangulararea,A=t(h/2),soF=thpmax/4. Thatresultantforceoccursatthecentroidofthepressure

loading,soitsleverarmwithrespecttotheneutralaxisisd=2(h/2)/3=h/3(forthetopandbottomportions).

The

pair

of

equal

and

opposite

forces

form

a

combined

couple

of

Mz

=

F

(2d)

=

t

h

2

pmax

/

6.

Finally,

the

requiredmaximumpressureis pmax=6Mz/th2.

ToapplythispressuredistributioninSWSimulationyoumustdefinealocalcoordinatesystemlocatedatthe

neutralaxisofthebeamanduseittodefineavariablepressure. However,theSWSimulationnonuniform

pressuredatarequiresapressurescale,pscale,timesanondimensionalfunctionofaselectedlocalcoordinate

system. Hereyouwillassumeapressureloadlinearlyvaryingwithlocalyplacedattheneutralaxis:p(y)=

pscale*y(withynondimensional). Thismustmatchpmaxaty=h/2,so

pscale =2pmax/h=12Mz/th3.


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Sinceitisoftennecessarytoapplymomentstosolidsinthisfashionthismomentloadingwillbechecked

independentlyagainstbeamtheoryestimatesbeforeapplyingthelineload.Here,pmax=3.75e7N/m2,pscale =

7.5e8.

6.2.3 SWSimulationplanestressmodelThebeamtheorysolutionforasimplysupportedbeamwithauniformloadiswellknown,asisthesolutionfor

theloading

by

two

end

moments

(called

pure

bending).

In

both

cases

the

maximum

deflection

occurs

at

the

beammidspan. Thetwovaluesarevmax=5wL4/384EI,andvmax=ML

2/8EI,respectively. Herethe

centerlinedeflectiondueonlytotheendmomentisvmax=1.36e3m. Foralinearanalysisandthesumof

thesetwovaluescanbeusedtovalidatethecenterlinedeflection. Next,theonequartermodel,shownin

Figure62,willbebuilt,restrained,andloaded:

1. BuildtherectanglesketchandconvertittoaplanarsurfaceviaInsertSurfacePlanarSurfaceandselecttheCurrentSketchasthePlanarSurface.

2. Startanewstudyusingashellmesh:Simulation

New

StudyStatic

.NameitAntisymmbeam.

3. UsePartEditDefinitiontosettheelementTypetoThin,andtheShellThicknesstobe0.02m. AlsousePart Apply/EditMaterialtoselectthelibrarymaterialof1060aluminum.

Sincethestressesthroughthedepthofthebeamaregoingtobeexaminedhere,youshouldplanaheadand

insertsomesplitlinesonthefrontsurfacetobeusedtolistand/orgraphselectedstressanddeflection

components:

1. Rightclickonthefrontface,InsertSketch.2. Insertalinesegmentthatcrossesthefaceattheinteriorquarterpoints.Includingtheendlines,fivegraphingsectionswillbeavailable. Alsoaddarightcornerarcfortheverticalsupportedge.


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3. InsertCurveSplitLineselectthebodyfaces,clickOK.

6.2.4 EdgerestraintsandloadsRememberthatshellsdefinedbyplanarsurfacesmusthavetheirrestraintsandloadsapplieddirectlytothe

edgesoftheselectedsurface. Firstthesymmetryandantisymmetryrestraintswillbeapplied. Sincetheshell

meshwillbeflatitiseasytouseitsedgestodefinedirectionsforloads,orrestraints:

1. RightclickingonFixturesopenstheFixturepanel.2. Thezerohorizontal(x)deflectionisappliedasaverticalsymmetryconditionontheedgecorresponding

to

the

beam

centerline;

AdvancedUse

Reference

Geometry,

select

vertical

Edge1

to

restrainandhorizontalEdge2forthedirection.

3. Applytheantisymmetryconditionalongtheedgeoftheneutralaxis.Usereferencegeometry,selectthefivebottomedgesformedbythesplitlinesandEdge7forthedirection.


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Atthesimplysupportedenditisnecessarytoassumehowthatsupportwillbeaccomplished. Beamtheory

treatsitasapointsupport,butin2Dor3Dthatcausesafalseinfinitestressatthepoint. Anothersplitlinearc

wasintroducedsoaboutonethirdofthatendcouldbepickedtoprovidetheverticalrestraintrequired. This

servesasareminderthatwhere,andhow,partsarerestrainedisanassumption. Soitiswisetoinvestigate

morethanonesuchassumption. Softwaretutorialsareintendedtoillustratespecificfeaturesofthesoftware,

andusuallydonothavethespacefor,orintentionof,presentingthebestengineeringjudgment. Immovable

restraintsare

often

used

in

tutorials,

but

they

are

unusual

in

real

applications.

Applytherightverticalendsupportrestraint:

1. SelectFixturestoopentheFixturepanel.2. PickAdvancedUseReferenceGeometry;selectthelowerrightfrontverticaledgelinetorestrain,andtheupperverticaledgeasthedirection.


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6.2.5 RigidbodymotionrestraintSinceageneralshellelementisbeingusedinaplanestress(membraneshell)applicationitstillhastheability

totranslatenormaltoitsplaneandtorotateabouttheinplaneaxes(xandy). Thosethreerigidbodymotions

shouldalsobeeliminatedinanyplanestressanalysisonmostFEAsystems. Ifnothingelse,sucharestraint

avoidscalculatingthreezerovaluesateachnodeandmakesyouranalysismoreefficient. Atworst,duringthe

solvephaseyoumaygetafatalerrormessage(duetoroundofferrors):

Forgoodmodelingpracticeapplythoserestraintsvia:

1. SelectFixturestoopentheFixturepanelandpickAdvancedFixtures.2. PickSymmetry;selectthefivefrontfacesegmentsofthebeamtorestrain.ChangetheSymbolColor.Thatcompletesthesymmetry,antisymmetryandrigidbodymotionrestraintsforthismodel.

6.2.6 MomentapplicationasanonuniformpressureUnlikegeneralshells,membraneshellsdonothaveactiverotationaldegreesoffreedomthatallowforthe

directapplicationofacouple. Alinearvariationofequalandoppositepressures,relativetotheneutralaxis,

canbeusedtoapplyastaticallyequivalentmomenttoacontinuumbodythatdoesnothaverotational

degreesoffreedom. Suchaloadingalsohasthesidebenefitofmatchingthetheoreticalnormalstress

distributionin

abeam

subjected

to

astate

of

pure

bending.

A

varying

pressure

loading

usually

requires

the

usertodefinealocalcoordinatesystemattheaxisaboutwhichthemomentacts. Inthiscase,itmustbe

locatedattheneutralaxisofthebeam:

1. SelectInsertReferenceGeometryCoordinateSystemtoopentheCoordinateSystempanel.RightclickontherightendoftheneutralaxistosettheoriginofCoordinateSystem1.


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2. Acceptthedefaultdirectionsfortheaxesasmatchingtheglobalaxes.ToapplytheexternalmomentuseExternalLoadsPressure. TheapplicationofthenonuniformpressureisappliedatthefrontverticaledgeatthesimplesupportinthePressurepanelofFigure64. Aunitpressure

valueisusedtosettheunitsandthemagnitudeisdefinedbymultiplyingthatvaluebyanondimensional

polynomialofthespatialcoordinatesofapoint,relativetolocalCoordinateSystem1definedabove.

Figure64 Applyingtherequiredlinearpressure

6.2.7 MeshandrunthemomentstudyHavingcompletedtherestraintsandmomentloading,thedefaultnamesinthemanagermenuhavebeen

changed(byslowdoubleclicks)toreflectwhattheyareintendedtoaccomplish(leftofFigure65). Nowyou

canRunthemomentloadcasestudy.


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compressionwithastressvaluex=pmax=3.75e7N/m2. Thatseemstoagreewiththecontourrangein

Figure68andindeed,astressprobetheregivesavalueofx=3.77e7N/m2. Beamtheorygivesalinear

variation,throughthedepth,fromthatmaximumtozeroattheneutralaxis. Tocomparewiththat,agraphof

alongthequarterpointsplitlineisgiveninFigure68. Itshowsthatthesevennodesalongtheedgesofthe

threequadraticelementshavepickedupthepredictedlineargraphquitewell. Forthenextloadcaseofafull

spanlineloadtheshearstress(thatiszerohere)willbeparabolicandthecorrespondinggraphwillbeless

accuratefor

such

acrude

mesh.

Figure67HorizontalstressalongtherightL/8spansegmentanditsprobevalue

Figure68 GraphofhorizontalstressatverticallineL/8fromthesupport

6.2.9 ReactionrecoveryFor

this

first

load

case,

the

only

external

applied

load

is

the

horizontal

pressure

distribution.

It

caused

aresultantexternalhorizontalforcethatwasshownabovetobeF=18,750N. Youshouldexpectthefinite

elementreactiontobeequalandoppositeofthatexternalresultantload. CheckthatintheManagermenu:

1. RightclickResults ListResultForceReactionForcetoopenthepanelwiththereactionforces.


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2. Examinethehorizontal(x)reactionforceaboveandverifythatitssumis18,750N.(Thesumofthemomentsisoftenconfusingbecausetheyarecomputedwithrespecttotheoriginoftheglobalcoordinatesystem,andmostprogramsnevermentionthatfact.)

TotestyourexperiencewithSWSimulation,youshouldnowrunthisspecialcasestudyasafull3Dsolid

subjecttothesameendpressures. Youwillfindthismodelwasquiteaccurate.Whileplanning3Dmeshes

youcangetusefulinsightsbyrunninga2Dstudylikethis. Also,a2Dapproximationcanbeausefulvalidation

toolifnoanalyticresultsorexperimentalvaluesareavailable. Theycanalsobeeasiertovisualize. Ofcourse,

manyproblemsrequireafull3Dstudybut1Dor2Dstudiesalongthewayareeducational.

6.2.10Beamloadcase2,thetransverselineloadHavingvalidatedthemomentloadcase,thelineloadwillbevalidatedandthenbothloadcaseswillbe

activatedto

obtain

the

results

of

the

original

problem

statement.

First,

go

to

the

manager

menu,

right

click

on

themomentpressureloadandsuppressit. Nextyouopenanewforcecasetoaccountforthelineload.

Recallthatthelineloadtotaled10,000N. Sincetheparthasbeenreducedtoonefourth,throughtheuseof

symmetryandantisymmetry,youonlyneedtodistribute2,500Noverthismodel. Therearetwowaystodo

thatforselectedsurfaceshellformulationofanyplanestressproblem. Theyaretoapplythattotalaseithera

lineload,ortodistributeitoverthemeshfaceasatangentialsheartraction(whichisthebetterway). Figure

69(left)showstheApplyForceapproach. Thatapproachhasbeenmadelessclearbythewaythesplitlines

wereconstructed. Thetopofthebeamhasbeensplitintofoursegmentsandthismethodappliesaforceper

entity. Therefore,aresultantforceof625Nperedgesegmentisspecified. Hadthesplitlinesnothadequal

spacingyouwouldhavetomeasureeachoftheirlengthsandgothroughthisprocedurefourtimes(the

pressureapproachavoidsthatpotentialcomplication)..


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Figure69 Secondbeamloadcaseofalineload

Withthissecondloadcaseinplacethestudyissimplyrunagainwiththesamerestraintsandmesh. Aseries

ofquickspotchecksoftheresultsarecarriedoutbeforemovingontothetrueproblemwherebothloadcases

areactivated.

Thebeamtheoryvalidationresult,forthislineload,predictedamaximumverticalcenterlinedeflectionofvmax

=1.13e3m. Theplanestressmaximumdeflectionwasextracted:

1. DoubleclickonPlot1underdisplacements. Thecontouredmagnitudeshowsarotationalmotionaboutthesimplesupportend,andverticaltranslationatthebeamcenterline,asexpected.

1. RightclickinthemanagermenuResultsListStress,Displacement,StraintoopentheListResultspanel. SelectDisplacementsandunderAdvancedOptionsselectAbsoluteMax,andSortbyvalue,

clickOK.


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2. Whenthelistappearsnotethatthemaximumdeflectionis1.16e3mattheverticalcenterlineposition. Thatisveryclosetotheinitialvalidationestimate.

Thenumericalvalueofthemaximumhorizontalfiberstresswaslistedinasimilarmanner. Themaximum

compressionvalue,inFigure610,ofx=4.04e7N/m2compareswellwiththesimplebeamtheoryvalueof

3.75e7N/m2,beingabouta7%difference. Sincethemeshissocrudethebeamstressisprobablythemost

accurateandtheplanestressvaluewillmatchitasareasonablyfinemeshisintroduced. Thepurposeofthe

crudemeshistoillustratetheneedformeshcontrolissolidsundergoingmainlyflexuralstresses. Toillustrate

thatpoint,

Figure611presentsthenormalstressandshearstress,fromtheneutralaxistothetop,attheL/4andL/8

positions.

Beamtheorysaysthenormalstressislinearwhiletheshearstressisparabolic. Thebeamtheoryshearstress

shouldbezeroatthetopfiberand,forarectangularcrosssection,hasamaximumvalueattheneutralaxisof

1.88e6N/m2. Thegraphvaluesin

Figure611showsaplanestressmaximumshearstressof1.84e6N/m2andaminimumof0.08e6N/m2atthe

quarterspansection. Thatisquitegoodagreementwithavalidationestimatefrombeamtheory.


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Figure610 Horizontalstresses(SX)forthelineloadcase

Figure611 Normal(SX)andshearstresses(SXY)atL/4(left)andL/8forthelineloadcase

6.3 CombinedloadcasesHavingvalidatedeachofthetwoloadcasestheyarecombinedbyunsuppressingtheendmomentcondition

(Figure612left)andrunningthestudyagainwiththesamemesh. Here,thetwosetsofpeakdeflectionsand

stressessimplyaddbecauseitisalinearanalysis. Aquickspotcheckverifiestheexpectedresults. The

reactionforcecomponentswereverified(Figure612right)beforelistingthemaximumdeflectionandfiber

stress(Figure613).


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Figure612 Verifyingthemodelreactionsforthecombinedloadings

Figure613 ProbevaluesforcombinedloadingmaximumdisplacementandvonMisesstress

Whatremainstobedoneistoexaminethelikelyfailurecriteriathatcouldbeappliedtothismaterial. They

includethe

Von

Mises

effective

stress,

the

maximum

principle

shear

stress,

and

the

maximum

principle

normal

stress. TheVonMisescontourvaluesareshowninFigure614. Twicethemaximumshearstress(thestress

intensity)isgiveninthetopofFigure615,whilethebottomportiondisplaysthemaximumprinciplestress

(P3). Actually,P3iscompressiveherebutitcorrespondstothemirrorimagetensiononthebottomfiberof

theactualbeam. Allthreestressvaluesneedtobecomparedtotheyieldpointstressof2.8e7N/m2. The

arrowinthefigurehighlightswherethatfallsonthecolorbar. Allofthecriteriaexceedthatvalue,sothepart

willhavetoberevised. AtthispointfailureisdeterminedevenbeforeaFactorofSafety(FOS)hasbeen

assigned. Forductilematerials,thecommonvaluesfortheFOSrangefrom1.3to5,ormore[9,12]. Assumea

FOS=3. Thecurrentdesignisafactorofabout3.3overtheyieldstress. CombiningthatwiththeFOSmeans

thatthestressesneedtobereducedbyaboutafactorof10.

Thecrosssectionalmomentofinertia,I=th3/12,isproportionaltothethickness,t,sodoublingthethickness

cutsthe

deflections

and

stresses

in

half.

Changing

the

depth,

h,

is

more

effective

for

bending

loads.

It

reduces

thedeflectionby1/h3andthestressesbyafactorof1/(2h2). Thedesiredreductionofstressescouldbe

obtainedbyincreasingthedepthbyafactorof2.25. Theabovediscussionassumedthatbucklinghasbeen

eliminatedbyabucklinganalysis. Sincebucklingisusuallysuddenandcatastrophicitwouldrequireamuch

higherFOS.


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Figure614 VonMisesstressinthebeamwithalineload.

Figure615 Beamprincipalstressandmaximumshearcontours

6.3.1 AdvancedoutputoptionsTherearetimeswhenthesoftwarewillnotprovidethegraphicaloutputyoudesire. Forexample,youmay

wishtographtheplanestressdeflectionagainstexperimentallymeasureddeflections. TheSWSimulationList

Selectedfeatureforanycontouredvalueallowsthedataonselectededges,splitlines,orsurfacestobesaved

toafileinacommaseparatedvalueformat(*.csv). SuchafilecanbeopenedinanExcelspreadsheet,or

Matlab,to

be

plotted

and/or

combined

with

other

data.

To

illustrate

the

point,

when

the

beam

deflection

valueswerecontouredthebottomedgewasselectedtoplaceitsdeflectionsinatable:

1. WithadisplacementplotshowingrightclickonthePlotnameListSelected.2. Selectthefourbottomlinesofthebeam.Update.3. ThebottomofthelistingwindowhasaSummaryofthedata.


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Draft 10.2 Copyright 2009. All rights reserved. 117

4. ThelowerReportOptionregiondoesnotincludetheGraphIcon,butdoesshowaSaveicon. Thatisbecausethepathhasmultiplelines. PickSavetohavethelisteddata(nodenumber,deflectionvalue,

andx,y,zcoordinates)tobeoutputasacommaseparatedvalues(csv)file.

5. Nameandsavethedataforuseelsewhere.

SWSimulation

did

not

offer

aplot

option

along

all

the

selected

lines

since

could

not

identify

which

item

to

sort. Youknowthatthemultiplelinesegmentsshouldbesortedbythexcoordinatevalue. Therefore,the

datawereopenedinExcel,sortedbyxcoordinate,andgraphedasdeflectionversusposition(Figure616).

Youcouldaddexperimentaldeflectionvaluestothesamefileandaddasecondcurvetothedisplayfor

comparisonpurposes.

Figure616 ExceldisplacementgraphfromsavedCSVfileforcombinedloads

fea plane stress analysis

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