FEM Stress Concepts

7/29/2019 FEM Stress Concepts

1/16

FEAConcepts:SWSimulationOverview J.E.Akin

Draft13.0.Copyright2009.Allrightsreserved. 29

3 ConceptsofStressAnalysis3.1 Introduction

Heretheconceptsofstressanalysiswillbestatedinafiniteelementcontext.Thatmeansthattheprimary

unknownwillbethe(generalized)displacements.Allotheritemsofinterestwillmainlydependonthe

gradientofthedisplacementsandthereforewillbelessaccuratethanthedisplacements.Stressanalysis

coversseveralcommonspecialcasestobementionedlater.Hereonlytwoformulationswillbeconsidered

initially.Theyarethesolidcontinuumformandtheshellform.BothareofferedinSWSimulation.Theydiffer

inthatthecontinuumformutilizesonlydisplacementvectors,whiletheshellformutilizesdisplacement

vectorsandinfinitesimalrotationvectorsattheelementnodes.

AsillustratedinFigure31,thesolidelementshavethreetranslationaldegreesoffreedom(DOF)asnodal

unknowns,foratotalof12or30DOF.Theshellelementshavethreetranslationaldegreesoffreedomaswell

asthreerotationaldegreesoffreedom,foratotalof18or36DOF.ThedifferenceinDOFtypesmeansthatmomentsorcouplescanonlybeapplieddirectlytoshellmodels.Solidelementsrequirethatcouplesbe

indirectlyappliedbyspecifyingapairofequivalentpressuredistributions,oranequivalentpairofequaland

oppositeforcesattwonodesonthebody.

Shellnode Solidnode

Figure31Nodaldegreesoffreedomforframesandshells;solidsandtrusses

Stresstransfertakesplacewithin,andon,theboundariesofasolidbody.Thedisplacementvector,u,atany

pointinthecontinuumbodyhastheunitsofmeters[m],anditscomponentsaretheprimaryunknowns.The

componentsofdisplacementareusuallycalledu,v,andwinthex,y,andzdirections,respectively.Therefore,

theyimplytheexistenceofeachother,u(u,v,w).Allthedisplacementcomponentsvaryoverspace.Asin

theheattransfercase(coveredlater),thegradientsofthosecomponentsareneededbutonlyasan

intermediatequantity.Thedisplacementgradientshavetheunitsof[m/m],orareconsidereddimensionless.

Unliketheheattransfercasewherethegradientisuseddirectly,instressanalysisthemultiplecomponentsof

thedisplacementgradientsarecombinedintoalternateformscalledstrains.ThestrainshavegeometricalinterpretationsthataresummarizedinFigure32for1Dand2Dgeometry.

In1D,thenormalstrainisjusttheratioofthechangeinlengthovertheoriginallength,x=u/x.In2Dand

3D,bothnormalstrainsandshearstrainsexist.Thenormalstrainsinvolveonlythepartofthegradientterms

paralleltothedisplacementcomponent.In2Dtheyarex=u/xandy=v/y.AsseeninFigure32(b),

theywouldcauseachangeinvolume,butnotachangeinshapeoftherectangulardifferentialelement.A

shearstraincausesachangeinshape.Thetotalanglechange(from90degrees)isusedastheengineering

definitionoftheshearstrain.Theshearstrainsinvolveacombinationofthecomponentsofthegradientthat


2/16



areperpendiculartothedisplacementcomponent.In2D,theengineeringshearstrainis=(u/y+v/x),asseeninFigure32(c).Strainhasonecomponentin1D,threecomponentsin2D,andsixcomponentsin

3D.The2Dstrainsarecommonlywrittenasacolumnvectorinfiniteelementanalysis,=(xy)T.

Figure32Geometryofnormalstrain(a)1D,(b)2D,and(c)2Dshearstrain

Stressisameasureoftheforceperunitareaactingonaplanepassingthroughthepointofinterestinabody.

Theabovegeometricaldata(thestrains)willbemultipliedbymaterialpropertiestodefineanewphysical

quantity,thestress,whichisdirectlyproportionaltothestrains.ThisisknownasHookesLaw:=E,(see

Figure33)wherethesquarematerialmatrix,E,containstheelasticmodulus,andPoissonsratioofthe

material.The2Dstressesarewrittenasacorrespondingcolumnvector,=(xy)T.Unlessstatedotherwise,theapplicationsillustratedhereareassumetobeinthelinearrangeofamaterialproperty.

The2Dand3DstresscomponentsareshowninFigure34.Thenormalandshearstressesrepresentthe

normalforceperunitareaandthetangentialforcesperunitarea,respectively.Theyhavetheunitsof

[N/m^2],or[Pa],butareusuallygivenin[MPa].Thegeneralizationsoftheengineeringstraindefinitionsare

seeninFigure35.Thestrainenergy(orpotentialenergy)storedinthedifferentialmaterialelementishalf

thescalarproductofthestressesandthestrains.Errorestimatesfromstressstudiesarebasedonprimarily

onthestrainenergy(orstrainenergydensity).

Figure33Hooke'sLawforlinearstressstrain,=E


3/16



Figure34Stresscomponentsin2D(left)and3D

Figure35Graphicalrepresentationsof3Dnormalstrains(a)andshearstrains

3.2 Axialbarexample

Thesimplestavailablestressexampleisanaxialbar,showninFigure36,restrainedatoneendandsubjected

toanaxialload,P,attheotherendandtheweightisneglected.Letthelengthandareaofthebarbedenoted

byL,andA,respectively.ItsmaterialhasanelasticmodulusofE.Theaxialdisplacement,u(x),varieslinearly


4/16



fromzeroatthesupporttoamaximumofattheloadpoint.Thatis,u(x)=x/L,sotheaxialstrainisx=u

/x=/L,whichisaconstant.Likewise,theaxialstressiseverywhereconstant,=E=E/Lwhichinthe

casesimplyreducesto=P/A.Likemanyothermorecomplicatedproblems,thestressheredoesnot

dependonthematerialproperties,butthedisplacementalwaysdoes, .Youshouldalwayscarefullycheckboththedeflectionsandstresseswhenvalidatingafiniteelementsolution.

Sincetheassumeddisplacementislinearhere,anyfiniteelementmodelwouldgiveexactdeflectionandthe

constantstressresults.However,iftheloadhadbeenthedistributedbarweighttheexactdisplacement

wouldbequadraticinxandthestresswouldbelinearinx.Then,aquadraticelementmeshwouldgiveexact

stressesanddisplacementseverywhere,butalinearelementmeshwouldnot.

Theelasticbarisoftenmodeledasalinearspring.Inintroductorymechanicsofmaterialstheaxialstiffnessof

abarisdefinedask=EA/L,wherethebarhasalengthofL,anareaA,andisconstructedofamaterialelastic

modulusofE.Thentheabovebardisplacementcanbewrittenas ,likealinearspring.

=P/A,=PL/EA

Figure36Alinearlyelasticbarwithanaxialload

3.3 StructuralmechanicsModernstructuralanalysisreliesextensivelyonthefiniteelementmethod.Themostpopularintegral

formulation,basedonthevariationalcalculusofEuler,isthePrincipleofMinimumTotalPotentialEnergy.

Basically,itstatesthatthedisplacementfieldthatsatisfiestheessentialdisplacementboundaryconditionsand

minimizesthetotalpotentialenergyistheonethatcorrespondstothestateofstaticequilibrium.Thisimplies

thatdisplacementsareourprimaryunknowns.Theywillbeinterpolatedinspaceaswilltheirderivatives,and

thestrains.Thetotalpotentialenergy,,isthestrainenergy,U,ofthestructureminusthemechanicalwork,

W,donebytheexternalforces.Fromintroductorymechanics,themechanicalwork,W,donebyaforceisthe

scalardotproductoftheforcevector,F,andthedisplacementvector,u,atitspointofapplication.

Thewellknownlinearelasticspringwillbereviewedtoillustratetheconceptofobtainingequilibrium

equationsfromanenergyformulation.Consideralinearspring,ofstiffnessk,thathasanappliedforce,F,at

thefree(right)end,andisrestrainedfromdisplacementattheother(left)end.Thefreeendundergoesa

displacementof.Theworkdonebythesingleforceis

.Thespringstorespotentialenergyduetoitsdeformation(changeinlength).Herewecallthatstrainenergy.

Thatstoredenergyisgivenby


5/16



Therefore,thetotalpotentialenergyfortheloadedspringis

Theequationofequilibriumisobtainedbyminimizingthistotalpotentialenergywithrespecttotheunknown

displacement,.Thatis,

Thissimplifiestothecommonsinglescalarequation

k=F,whichisthewellknownequilibriumequationforalinearspring.Thisexamplewasslightlysimplified,sincewe

startedwiththeconditionthattheleftendofthespringhadnodisplacement(anessentialorDirichlet

boundarycondition).Nextwewillconsideraspringwhereeitherendcanbefixedorfreetomove.Thiswill

requirethatyoubothminimizethetotalpotentialenergyandimposethegivendisplacementrestraint.

Figure37Theclassicandgenerallinearspringelement

Nowthespringmodelhastwoenddisplacements,1and2,andtwoassociatedaxialforces,F1andF2.The

netdeformationofthebaris=21.Denotethetotalvectorofdisplacementcomponentsas

andtheassociatedvectorofforcesas

Thenthemechanicalworkdoneonthespringis

1F1+2F2Thenthespring'sstrainenergyis

,

wherethespringstiffnessmatrixisfoundtobe

.Thetotalpotentialenergy,,becomes

.

Notethateachtermhastheunitsofenergy,i.e.forcetimeslength.Thematrixequationsofequilibriumcome

fromsatisfyingthedisplacementrestraintandtheminimizationofthetotalpotentialenergywithrespectto


6/16



eachandeverydisplacementcomponent.Theminimizationrequiresthatthepartialderivativeofallthe

displacementsvanish:

,or

.

Thatrepresentsthefirststagesystemofalgebraicequationsofequilibriumfortheelasticsystem:

.Thesetwosymmetricequationsdonotyetreflectthepresenceofanyessentialboundaryconditiononthe

displacements.Therefore,nouniquesolutionexistsforthetwodisplacementsduetoappliedforces(theaxial

RBMhasnotbeeneliminated).Mathematically,thisisclearbecausethesquarematrixhasazerodeterminate

andcannotbeinverted.Ifallofthedisplacementsareknown,youcanfindtheappliedforces.Forexample,if

youhadarigidbodytranslationof1=2=CwhereCisanarbitraryconstantyouclearlygetF1=F2=0.Ifyou

stretchthespringbytwoequalandoppositedisplacements;1=C,2=Candthefirstrowofthematrix

equationsgivesF1=2kC.ThesecondrowgivesF2=2kC,whichisequalandoppositetoF1,asexpected.

Usually,youknowsomeofthedisplacementsandsomeoftheforces.Thenyouhavetomanipulatethematrix

equilibriumsystemtoputitintheformofastandardlinearalgebraicsystemwhereaknownsquarematrix

multipliedbyavectorofunknownsisequaltoaknownvector: .3.4 Equilibriumofrestrainedsystems

Liketheoriginalspringproblem,nowassumetherightforce,F2,isknown,andtheleftdisplacement,,hasagiven(restrained)value,say.Then,theabovematrixequationrepresentstwouniqueequilibriumequationsfortwounknowns,thedisplacement2andthereactionforce.Thatmakesthislinearalgebraicsystemlookstrangebecausethereareunknownsonbothsidesoftheequals,=.Youcould(butusuallydo

not)correctthatbyrearrangingtheequationsystem(notdoneinpractice).First,multiplythefirstcolumnof

thestiffnessmatrixbytheknownvalueandmoveittotherightside:

andthenmovetheunknownreaction,,totheleftside

.

Nowyouhavetheusualformofalinearsystemofequationswheretherightsideisaknownvectorandthe

leftsideistheproductofaknownsquarematrixtimesavectorofunknowns.Sinceboththeenergy

minimizationandthedisplacementrestraintshavebeencombinedyounowhaveauniquesetofequationsfor

theunknowndisplacementsandtheunknownrestraintreactions.Invertingthe2by2matrixgivestheexact

solution:

sothatF1F2always,asexpected.If=0,asoriginallystated,thentheenddisplacementis .Thissortofrearrangementofthematrixtermsisnotdoneinpracticebecauseitdestroysthesymmetryofthe

originalequations.Algorithmsfornumericallysolvingsuchsystemsrelyonsymmetrytoreduceboththe

requiredstoragesizeandtheoperationscount.Theyareveryimportantwhensolvingthousandsofequations.


7/16



3.5 Generalequilibriummatrixpartitions

Theabovesmallexamplegivesinsighttothemostgeneralformofthealgebraicsystemresultingfromonly

minimizingthetotalpotentialenergy:asingularmatrixsystemwithmoreunknownsthanequations.Thatis

becausethereisnotauniqueequilibriumsolutiontotheproblemuntilyoualsoapplytheessential(Dirichlet)

boundaryconditionsonthedisplacements.Thealgebraicsystemcanbewritteninageneralpartitionedmatrix

formthat

more

clearly

defines

what

must

be

done

toreduce

the

system

toasolvable

form

by

utilizing

essentialboundaryconditions.

Foranelasticsystemofanysize,thefull,symmetricmatrixequationsobtainedbyminimizingtheenergycan

alwaysberearrangedintothefollowingpartitionedmatrixform:

whereurepresentstheunknownnodaldisplacements,andgrepresentsthegivenessentialboundaryvalues(restraints,orfixtures)oftheotherdisplacements.ThestiffnesssubmatricesKuuandKggaresquare,whereasKugandKguarerectangular.InafiniteelementformulationallofthecoefficientsintheKmatricesareknown.TheresultantappliednodalloadsareinsubvectorFgandtheFutermsrepresenttheunknowngeneralized

reactionsforcesassociatedwithessentialboundaryconditions.ThismeansthataftertheenforcementoftheessentialboundaryconditionstheactualremainingunknownsareuandFu.Onlythendoesthenetnumberofunknownscorrespondtothenumberofequations.But,theymustberearrangedbeforealltheremaining

unknownscanbecomputed.

Here,forsimplicity,ithasbeenassumedthattheequationshavebeennumberedinamannerthatplacesrows

associatedwiththegivendisplacements(essentialboundaryconditions)attheendofthesystemequations.

Theabovematrixrelationscanberewrittenastwosetsofmatrixidentities:

.

Thefirstidentitycanbesolvedfortheunknowndisplacements,,byputtingitinthestandardlinearequationformbymovingtheknownproducttotherightside.Mostbooksonnumericalanalysisassumethatyouhavereducedthesystemtothissmaller,nonsingularform()beforetryingtosolvethesystem.Invertingthesmallernonsingularsquarematrixyieldstheuniqueequilibriumdisplacementfield:

.Theremainingreactionforcescanthenberecovered,ifdesired,fromthesecondmatrixidentity:

.Inmostapplications,thesereactiondatahavephysicalmeaningsthatareimportantintheirownright,or

usefulinvalidatingthesolution.However,thispartofthecalculationisoptional.

3.6 StructuralComponentFailure

Structuralcomponentscanbedeterminedtofailbyvariousmodesdeterminedbybuckling,deflection,natural

frequency,strain,orstress.Strainorstressfailurecriteriaaredifferentdependingonwhethertheyare

consideredasbrittleorductilematerials.Thedifferencebetweenbrittleandductilematerialbehaviorsis

determinedbytheirresponsetoauniaxialstressstraintest,asinFigure38.Youneedtoknowwhatclassof

materialisbeingused.SWSimulation,andmostfiniteelementsystems,defaulttoassumingaductilematerial


8/16



anddisplaythedistortionalenergyfailuretheorywhichisusuallycalledtheVonMisesstress,oreffective

stress,eventhoughitisactuallyascalar.Abrittlematerialrequirestheuseofahigherfactorofsafety.

Figure38Axialstressstrainexperimentalresults

3.7 FactorofSafety

Allaspectsofadesignhavesomedegreeofuncertainty,asdoeshowthedesignwillactuallybeutilized.For

allthereasonscitedabove,youmustalwaysemployaFactorofSafety(FOS).Somedesignersrefertoitasthe

factorofignorance.RememberthataFOSofunitymeansthatfailureiseminent;itdoesnotmeanthatapart

orassemblyissafe.Inpracticeyoushouldtrytojustify1


9/16



6 GeometryofMesh Defeaturingcanintroduceerrors.Elementsizesandlocation

areimportant.Lookinglikethepartisnotenough.

7 Loading Areloadspreciseordotheycomefromwaveaction,etc.

8 Materialdata Isthematerialwellknown,orvalidatedbytests

9 Reliability Mustthereliabilityofthedesignbehigh

10 Restraints Designsaregreatlyinfluencedbyassumedsupports

11 Stresses Wasstressconcentrationconsidered,orshockloads

3.8 ElementTypeSelection

Evenwithtodaysadvancesincomputingpoweryouseemnevertohaveenoughcomputationalresourcesto

solvealltheproblemsthatpresentthemselves.Frequentlysolidelementsarenotthebestchoicefor

computationalefficiency.Theanalystsshouldlearnwhenotherelementtypescanbevalidorwhentheycan

beutilizedtovalidateastudycarriedoutwithadifferentelementtype.SWSimulationoffersasmallelement

librarythatincludesbars,trusses,beams,frames,thinplatesandshells,thickplatesandshells,andsolid

elements.Therearealsospecialconnectorelementscalledrigidlinksormultipointconstraints.

Theshellsandsolidelementsareconsideredtobecontinuumelements.Theplateelementsareaspecialcase

offlatshellswithnoinitialcurvature.Solidelementformulationsincludethestressesinalldirections.Shells

areamathematicalsimplificationofsolidsofspecialshape.Thinshells(likethinbeams)donotconsiderthe

stressinthedirectionperpendiculartotheshellsurface.Thickshells(likedeepbeams)doconsiderthe

stressesthroughthethicknessontheshell,inthedirectionnormaltothemiddlesurface,andaccountfor

transversesheardeformations.

LethdenotethetypicalthicknessofacomponentwhileitstypicallengthisdenotedbyL.Thethicknessto

lengthratio,h/L,givessomeguidanceastowhenaparticularelementtypeisvalidforananalysis.Whenh/Lis

largesheardeformationisatitsmaximumimportanceandyoushouldbeusingsolidelements.Conversely,

whenh/L

isvery

small

transverse

shear

deformation

isnot

important

and

thin

shell

elements

are

probably

the

mostcosteffectiveelementchoice.Intheintermediaterangeofh/Lthethickshellelementswillbemostcost

effective.Thethickshellsareextensionsofthinshellelementsthatcontainadditionalstrainenergyterms.

Theoverlappingh/LrangesforthethreecontinuumelementtypesaresuggestedinFigure39.Thethickness

ofthelinessuggeststhoseregionswhereaparticularelementtypeisgenerallyconsideredtobethepreferred

elementofchoice.Theoverlappingrangessuggestwhereonetypeofelementcalculationcanbeusedto

validateacalculatedresultobtainedwithadifferentelementtype.Validationcalculationsincludethe

differentapproachestoboundaryconditionsandloadsrequiredbydifferentelementformulations.Theyalso

canindirectlycheckthatauseractuallyunderstandshowtoutilizeafiniteelementcode.


10/16

FEAConcept

Draft

3.9 SWSThesymbolsu

Figure310.

T

elementsolut

representsth

areoftenrefe

displacement

enoughrestra

Al

Forsimplicit

Thatis,they

thetypeofr

frequentlye

understand

s:SWSimul

13.0.Copyri

imulationsedinSWSim

hesymbols

fo

ionsarebased

mechanical

rredtoasgen

DOFsforthe

intstopreven

odeofsolid

lthreedispla

F

manyfinite

enforceanI

straint,asw

counterthe

ymmetry

pla

Displacem

Figu

tionOvervie

ght2009.All

Figure3

Fixtureaulationtorep

rthe

correspo

onworkener

orkdoneatt

ralizeddispla

solidnodes(to

tanymodelfr

ortrusselem

cementsare

igure310Fix

elementexa

movableco

ellaswhere

commonco

ne

restraints

nt

re311Singl

rightsreser

9Overlappin

dLoadSesentasingle

ndingforces

a

gyrelations,t

epoint.Wh

cements.The

p)andshelln

omundergoin

ent:

zero.

edrestraints

plesincorr

nditionfors

thepartisre

ditionsofsy

for

solids

an

Force

components

ed.

gvalidranges

mbolstranslational

ndmoment

lo

heabovewor

namodelcan

SWSimulatio

desareseen

garigidbody

mbolsforsol

ctlyapplyco

lidsoraFix

strainedisof

mmetryora

d

shells.

Ro

ymbolsforre

ofelementt

androtational

adingsare

sh

correspondi

involveeithe

nodalsymbo

inFigure311.

ranslationor

Nodeof

Allthreedi

ro

ids(top)and

mpleterestr

dcondition

tenthemost

tisymmetry

tation

straints(fixtu

pes

DOFatanod

wnpink

in

th

ngmeansth

translations

lsfortheunk

.Youalmosta

rigidbodyrota

frameorshel

splacements

tationsarez

hellnodes

aintsatafac

orshells.Ac

difficultpart

restraints.Y

Coup

es)andloads

J

3

areshowngr

tfigure.

Sinc

ttheirdotpr

rrotationsas

owngenerali

lwaysmustsu

tion.

lelement:

andallthree

ro.

,edgeorno

tuallydeter

ofananalys

oushouldu

le

.E.Akin

8

eenin

finite

duct

DOFthey

ed

pply

de.

ining

is.You

der


11/16



3.10SymmetryDOFonaPlane

Aplaneofsymmetryisflatandhasmirrorimagegeometry,materialproperties,loading,andrestraints.

Symmetryrestraints\iareverycommonforsolidsandforshells.Figure312showsthatforbothsolidsand

shells,thedisplacementperpendiculartothesymmetryplaneiszero.Shellshavetheadditionalconditionthat

theinplanecomponentofitsrotationvectoriszero.Ofcourse,theflatsymmetryplaneconditionscanbe

statedinadifferent

way.

For

asolid

element

translational

displacements

parallel

tothe

symmetry

plane

are

allowed.Forashellelementrotationisallowedaboutanaxisperpendiculartothesymmetryplaneandits

translationaldisplacementsparalleltothesymmetryplanearealsoallowed.

Nodeofasolidortrusselement:

Displacementnormaltothesymmetryplaneiszero.

Nodeofaframeorshellelement:

Displacementnormaltothesymmetryplaneandtwo

rotationsparalleltoitarezero.

Figure312Symmetryrequireszeronormaldisplacement,andzeroinplanerotation

3.11AvailableLoading(Source)Options

Mostfiniteelementsystemshaveawiderangeofmechanicalloads(orsources)thatcanbeappliedtopoints,

curves,surfaces,andvolumes.ThemechanicalloadingterminologyusedinSWSimulationisinTable32.

Mostofthoseloadingoptionsareutilizedinlaterexampleapplications.

Table32Mechanicalloads(sources)thatapplytotheactivestructuralstudy

LoadType Description

BearingLoad Nonuniformbearingloadonacylindricalface

CentrifugalForce Radialcentrifugalbodyforcesfortheangularvelocityand/ortangential

bodyforcesfromtheangularaccelerationaboutanaxis

Force Resultantforce,ormoment,atavertex,curve,orsurface

Gravity Gravity,orlinearaccelerationvector,bodyforceloading

Pressure Apressurehavingnormaland/ortangentialcomponentsactingona

selectedsurface

RemoteLoad/

Mass

Allowsloadsormassesremotefromtheparttobeappliedtothepart

bytreatingtheomittedmaterialasrigid

Temperature Temperaturechangeatselectedcurves,surfaces,orbodies(see

thermalstudiesformorerealistictemperaturetransfers)

3.12AvailableMaterialInputsforStressStudies

Mostapplicationsinvolvetheuseofisotropic(directionindependent)materials.Theavailablemechanical

propertiesfortheminSWSimulationarelistedinTable33.Itisbecomingmorecommontohavedesigns

utilizinganisotropic(directiondependent)materials.Themostcommonspecialcaseofanisotropicmaterialsis

theorthotropicmaterial.Anyanisotropicmaterialhasitspropertiesinputrelativetotheprincipaldirectionsof

thematerial.Thatmeansyoumustconstructtheprincipalmaterialdirectionsreferenceplaneorcoordinate

axesbeforeenteringorthotropicdata.Mechanicalorthotropicpropertiesaresubjecttosometheoretical


12/16



relationshipsthatphysicallypossiblematerialsmustsatisfy(suchaspositivestrainenergy).Thus,

experimentalmaterialpropertiesdatamayrequireadjustmentbeforebeingacceptedbySWSimulation.

Table33Isotropicmechanicalproperties

Symbol Label Item

E

EX

Elastic

modulus

(Youngs

modulus)

NUXY PoissonsratioG GXY Shearmodulus

DENS Massdensity SIGXT Tensilestrength(Ultimatestress) SIGXC Compressionstresslimit SIGYLD Yieldstress(yieldstrength) ALPX Coefficientofthermalexpansion

Table34Orthotropicmechanicalpropertiesinprincipalmaterialdirection

Symbol Label Item

Ex EX ElasticmodulusinmaterialXdirection

Ey EY ElasticmodulusinmaterialYdirection

Ez EZ ElasticmodulusinmaterialZdirection

xy NUXY PoissonsratioinmaterialXYdirectionsyz NUYZ PoissonsratioinmaterialYZdirectionsxz NUXZ PoissonsratioinmaterialXZdirectionsGxy GXY ShearmodulusinmaterialXYdirections

Gyz GYZ ShearmodulusinmaterialYZdirections

Gxz GXZ ShearmodulusinmaterialXZdirections

DENS Massdensity SIGXT Tensilestrength(Ultimatestress) SIGXC Compressionstresslimit SIGYLD Yieldstress(Yieldstrength) ALPX ThermalexpansioncoefficientinmaterialX ALPY ThermalexpansioncoefficientinmaterialY ALPZ ThermalexpansioncoefficientinmaterialZNote:NUXY,NUYZ,andNUXZarenotindependent

Partscanalsobemadefromorthotropicmaterials(asshownlater).However,theirutilizationismost

commoninlaminatedmaterials(laminates)wheretheyeachplylayerhasacontrollableprincipalmaterial

direction.TheconceptforconstructinglaminatesfromorthotropicmaterialplysisshowninFigure.

Understandingthefailuremodesoflaminatesusuallyrequiresspecialstudy.


13/16



Figure313Exampleofafourplylaminatematerial

3.13StressStudyOutputs

Asuccessfulrunofastudywillcreatealargeamountofadditionaloutputresultsthatcanbedisplayedand/or

listedinthepostprocessingphase.DisplacementsaretheprimaryunknowninaSWSimulationstressstudy.

Theavailabledisplacementvectorcomponentsarecited inTable35andTable36,alongwiththereactions

theycreateifthedisplacementisusedasarestraint.Thedisplacementscanbeplottedasvectordisplays,or

contourvalues.Theycanalsobetransformedtocylindricalorsphericalcomponents.

Table35Outputresultsforsolids,shells,andtrusses

Symbol Label Item Symbol Label Item

Ux UX Displacement(Xdirection) Rx RFX Reactionforce(Xdirection)

Uy UY Displacement(Ydirection) Ry RFY: Reactionforce(Ydirection)

Uz UZ Displacement(Zdirection) Rz RFZ Reactionforce(Zdirection)

Ur URES: Resultantdisplacement

magnitude

Rr RFRES Resultantreactionforce

magnitude

Table36Additionalprimaryresultsforbeams,plates,andshellsSymbol Label Item Symbol Label Item

x RX Rotation(Xdirection) Mx RMX: Reactionmoment(Xdirection)y RY Rotation(Ydirection) My RMY Reactionmoment(Ydirection)z RZ Rotation(Zdirection) Mz RMZ: Reactionmoment(Zdirection) Mr MRESR Resultantreactionmoment

magnitude

The strains and stresses are computed from the displacements. The stress components available at an

elementcentroidoraveragedatanodearegiven inTable37. Thesixcomponents listedonthe left inthat

tablegivethegeneralstressatapoint(i.e.,anodeoranelementcentroid).Thosesixvaluesareillustratedon

theleftofFigure314.TheycanbeusedtocomputethescalarvonMisesfailurecriterion.Theycanalsobe

usedtosolveaneigenvalueproblemfortheprincipalnormalstressesandtheirdirections,whichareshownontherightofFigure314. Themaximumshearstressoccursonaplanewhosenormal is45degreesfromthe

direction of P1. The principal normal stresses can also be used to compute the scalar vonMises failure

criterion.

ThevonMiseseffectivestressiscomparedtothematerialyieldstressforductilematerials.Failureispredicted

tooccur(basedonthedistortionalenergystoredinthematerial)whenthevonMisesvaluereachestheyield

stress.Themaximumshearstressispredictedtocausefailurewhenitreacheshalftheyieldstress.SW

Simulationusestheshearstressintensitywhichisalsocomparedtotheyieldstresstodeterminefailure


14/16



(becauseitistwicethemaximumshearstress).ThefirstfourvaluesontherightsideofTable37areoften

representedgraphicallyinmechanicsasa3DMohrscircle(seeninFigure315).

Table37:Nodalandelementstressresults

Symbol Label Item Symbol Label Item

x SX Normalstressparalleltoxaxis 1 P1 1stprincipalnormalstressy SY Normalstressparalleltoyaxis 2 P2 2ndprincipalnormalstressz SZ Normalstressparalleltozaxis 3 P3 3rdprincipalnormalstressxy TXY ShearinYdirectiononplanenormaltoxaxis

INT Stressintensity(P1P3),twicethemaximumshearstress

xz TXZ ShearinZdirectiononplanenormaltoxaxis

yz TYZ ShearinZdirectiononplanenormaltozaxis

vm VON vonMisesstress(distortionalenergyfailurecriterion)

Figure314Thestresstensor(left)anditsprincipalnormalvalues

Figure315ThethreedimensionalMohr'scircleofstressyieldtheprincipalstresses


15/16



Ifdesired,youcanplotallthreeprincipalcomponentsatonce.Thethreeprincipalnormalstressesatanode

orelementcentercanberepresentedbyanellipsoid.Thethreeradiioftheellipsoidrepresentthemagnitudes

ofthethreeprincipalnormalstresscomponents,P1,P2,andP3.Thesignofthestresses(tensionor

compression)arerepresentedbyarrows.ThecolorcodeofthesurfaceisbasedonthevonMisesvalueatthe

point,ascalarquantity.Ifoneoftheprincipalstressesiszero,theellipsoidbecomesaplanarellipse.Ifthe

threeprincipalstresseshavethesamemagnitude,theellipsoidbecomesasphere.Inthecaseofsimple

uniaxialtensilestress,theellipsoidbecomesaline.

Figure316AprincipalstressellipsoidcoloredbyvonMisesvalue

TheavailablenodaloutputresultsinTable37areobtainedbyaveragingtheelementvaluesthatsurroundthe

node.Youcanalsoviewthemasconstantvaluesattheelementcentroids.Thatcangiveyouinsighttothe

smoothnessoftheapproximation.Forbrittlematerialsyoucanalsobeinterestedintheelementstrain

results.TheyarelistedinTable38.Table39showsthatyoucanalsoviewtheelementerrorestimate,ERR

whichisused

todirect

adaptive

solutions,

and

the

contact

pressure

from

an

iterative

contact

analysis.

Additionaloutputsareavailableifyouconductanautomatedadaptiveanalysistoreducethe(mathematical)

errortoaspecificvalue,ortorecoverresultsfromthedevelopedpressurebetweencontactingsurfaces.They

arelistedinTable39.

Table38Elementcentroidalstraincomponentresults

Sym Label Item Sym Label Item

x EPSX Normalstrainparalleltoxaxis

1 E1 Normalprincipalstrain(1stprincipaldirection)

y EPSY Normalstrainparalleltoyaxis

2 E2 Normalprincipalstrain(2ndprincipaldirection)

z EPSZ Normalstrainparalleltozaxis

3 E3 Normalprincipalstrain(3rdprincipaldirection)

xy GMXY ShearstraininYdirectiononplanenormaltoxaxis

r ESTRN Equivalentstrainxz GMXZ ShearstraininZdirectionon

planenormaltoxaxis

SED SEDENS Strainenergydensity(per

unitvolume)

yz GMYZ ShearstraininZdirectiononplanenormaltoyaxis

SE ENERGY Totalstrainenergy


16/16



Table39Additionalelementcentroidstressrelatedresults

Label Item

ERR Elementerrormeasuredinthestrainenergynorm

CP Contractpressuredevelopedonacontactsurface

FEM Stress Concepts

Documents

Transcript of FEM Stress Concepts