Numerical Methods for Stress Analysis Using Known Elasticity Solution - A. R. CHARMICHAEL

8/12/2019 Numerical Methods for Stress Analysis Using Known Elasticity Solution - A. R. CHARMICHAEL

1/234

University of Southampton Research Repository

ePrints Soton

Copyright and Moral Rights for this thesis are retained by the author and/or other

copyright owners. A copy can be downloaded for personal non-commercial

research or study, without prior permission or charge. This thesis cannot be

reproduced or uoted e!tensi"ely from without first obtaining permission in writing

from the copyright holder/s. The content must not be changed in any way or sold

commercially in any format or medium without the formal permission of the

copyright holders.

#hen referring to this wor$, full bibliographic details including the author, title,

awarding institution and date of the thesis must be gi"en e.g.

A%T&'R (year of submission) *+ull thesis title*, %ni"ersity of outhampton, name

of the %ni"ersity chool or epartment, h Thesis, pagination

http://eprints.soton.ac.uk
http://eprints.soton.ac.uk/http://eprints.soton.ac.uk/


2/234

UNIVERSITYOFSOUTHAMPTON

FACULTYOFENGINEERINGANDAPPLIED SCIENCEMECHANICAL ENGINEERING DEPARTMENT

"NUMERICAL METHODSFORSTRESS ANALYSISUSING KNOWN ELASTICITY SOLUTIONS"

by

A.R.Carmichael

Beingathesis submitted for thedegreeofDoctorofPhilosophy

April1983


3/234


4/234

CONTENTS

ABSTRACTPageiv.

ACKNOWLEDGEMENTS V .

NOTATION: (PartI)(PartIi;

VI .X .

LISTOFFIGURES XI V .

LISTOFTABLES

CHAPTER 1. GENERAL INTRODUCTION1.1 Backgroundto thework1.2 Reviewoftheoretical methodsandsolutions

1.2.1 Exact analytical methods1.2.2 Approximate analytical methods1.2.3 Numerical methods

1.3 Layoutofthesis

14457

11

PARTI Finite Element Superposition Method

CHAPTER 2. FINITE ELEMENT FORMULATION

2 .1 Introduction 132 .2 Notationfor the boundaries 142 .3 Displacementand stress fields 162, 4 Thevariational]rinciple 192, 5 Determinationof correction stress field c0 .1 222. 6 Determinationof thenodal displacements 242. Determinationof thetrial function coefficients 262. Determinationof thestressat anypoint 292. Summary 30

CHAPTER 3. TRIAL FUNCTIONSANDLOADING FUNCTION

3.1 Introduction3.2 Trial functionsforelliptical holes 3131


5/234

Page3.3 Trial functionsforcircular holes3.4 Loading functionforcircular holes

3339

CHAPTER 4. IMPLEMENTATIONOFFINITE ELEMENT METHOD

4.1 Introduction4.2 StructureofProgram

4.2.1 Data input4.2.2 Areasofelements4.2.3 Stiffness matrix4.2.4 Trialandloading functions4.2.5 Stress fieldsof4.2.6 Boundary conditions4.2.7 Strain energy term4.2.8 CorrectiontomatrixD4.2.9 Solutionof theequations4.2.10 Trial function coefficients4.2.11 Displacementsandstresses

4.3 UsingtheProgram

4242444445454546464648484848

CHAPTER 5. FINITE ELEMENT RESULTS

5.1 Introduction 505.2 Rectangular plate with central traction-free hole 51

5.2.1 Effectofmesh refinement 535.2.2 Effectofhole size 545.2.3 Effectofhole aspect ratio 565.2.4 Effectofspecial region sizeandadditional

trial functions 605.2.5 Effectoflocal mesh refinement 67

5.3 Square plate with central traction-free hole 695.3.1 Circular hole 695.3.2 Elliptical hole 72

5.4 Configurations with loaded circular holes 765.4.1 Various loadingson acircular hole 775.4.2 Pressurized hole inannulusorlarge plate 925.4.3 Symmetrical rectangularlug 935.4.4 Rectangular lugs 98

PART II Modified Boundary Element Method

CHAPTER 6. BOUNDARY ELEMENT FORMULATION

6.1 Introduction6.2 Notation

105107

11.


6/234

Page6.3 Basisof theboundary element method 1096.4 Discretizationof theequations 1116.5 Solutionof theequations 1136.6 Interior points 1136.7 Modified kernel function 1156.8 Implementation 117

CHAPTER 7. BOUNDARY ELEMENT RESULTS

7.1 Configurations analysed 1197.2 Comparison between modified andstandard kernel

functions 1207.3 Annuli with various sizesofhole 1237.4 Square plates with various sizesofhole 125

CHAPTER8. GENERAL CONCLUSIONS

8.1 Thefinite element superposition method 1288.2 Themodified boundary element method 1318.3 Comparisonof themethods 1328.4 Future work 133

REFERENCES 136APPENDIXA. Derivationof thevariational principle 154APPENDIXB. Listingsofselected program segments (FESM) 157APPENDIXC. Data preparationforfinite element program 174APPENDIXD. Running thefinite element program 181APPENDIXE. Finite element meshes 196APPENDIXF. Modified fundamental solution 202APPENDIXG. Running theboundary element program 205

111.


7/234

UNIVERSITYOFSOUTHAMPTONABSTRACT

FACULTYOFENGINEERINGANDAPPLIED SCIENCEMECHANICAL ENGINEERINGDoctorofPhilosophy

NUMERICAL METHODSFORSTRESS ANALYSISUSING KNOWN ELASTICITY SOLUTIONS

byAndrew Robert Carmichael

Twomethodsfor thedeterminationof thestress concentration near holesin twodimensional elastic componentsaredeveloped. One,which isbasedon thefinite element method extendsasuperposition technique originallydeveloped forcrack problems;theother usestheboundary element method.Both methods involve using analytical solutions which satisfy conditionson thehole boundary exactly thereby reducingor, in thecaseofboundaryelements avoiding entirely,theneed forelements modellingthehole.

Thefirst method usesamodified complementary energy functionaltodetermine thecoefficientsof thesuperimposed functionsand thefiniteelement nodal displacements, from which theestimatesofstressareobtained. Tractionson thehole boundaryarerepresented accuratelyusing Fourier series,and theformulationismodifiedby theinclusionof a"loading function" whichis thesolutionfor aninfinite regioncontainingthehole underthespecified loading. Representingthetractionson theholeinthis avoids inaccuraciesdue toapproximatemodellingof theload,forexampleaspoint forces, closeto thepointwherethestress concentration factorisrequired. Theloading functionisincorporated intotheformulation without requiring numerical integrationof thetractions overthecurved boundaryof thehole. Accuracyof themethodfor use ontraction-free circularorelliptical holesandloadedcircular holesissystematically examined. Forquite coarse finite elementmeshes (typically70degreesoffreedom with four-fold symmetry)3%accuracyorbettermay beexpected,animprovementby afactorofbetween5 and 10over conventional elements. Theeffectonaccuracyofsuchparametersas themesh refinement,thesizeandshapeof theholeandouter boundaries,theextentof theregionofsuperpositionand thetypeofloading isinvestigated. Fourier seriesarederivedfordifferentdistributionsoftractions occurringat ahole boundarydue to apin-load,andtheseareused todetermine stress concentration factorsforrectangularlugsofvarious dimensions.

Theformulationof theboundary element method forplane elasticproblems ispresented and ismodifiedbyintroducingakernel functionwhich satisfiesthetraction-free conditionson ahole boundary. It isshown that with this formulationnoelementsarerequired tomodelthehole, thus stressesat orneartheholemz^f beevaluated without beingaffectedby theproximityofelementsonthis boundary. Resultsforexternally pressurized annul andsquare plates with circular holesareobtained withthemethod, which showamarked improvementinaccuracyovertheboundary element method withanunmodified kernel function. Fromthese resultsitappears thattheconstant shape functionof theelementsis alimitationfor theapplicationof themethodtomore generalcon-figurationsandhigher order elementsarerecommnded.

IV .


8/234

ACKNOWLEDGEMENTS

Theauthor wishestoacknolwedgethereceiptofsupportbyMOD(PE) under AgreementNo.2040/0202STR. Theassistance givenbymembersof theRoyal Aircraft Establishment (RAE), Farnborough,during thecourseof thework isgratefully acknowledged. Inparticulartheauthor wishestothankDr P.Bartholomewof theStructures DepartmentRAE for hisvaluable assistanceandcooperationwith partsof thefinite element work,and Mr D.P.Rookeof theMaterials DepartmentRAE andmembersof theSub-department EngineeringMaterials and theMechanical Engineering Departmentat theUniversityofSouthampton formany useful discussionsandsuggestions. Thanksarealsodue to JanWardfor hercare intypingthemanuscriptandtoMichelle Smithfor herhelp with someof thefigures.

Aboveall mysincere gratitudeandthanks goesto Dr D.J.Cartwrightfor theguidanceandencouragementhe hasgivenas mysupervisorinthis work. It hasbeen most appreciated.

v.


9/234

NOTATION

Notation for PARTI(including AppendixA)

The following symbolsareused torepresent vectorsof thetype shown:

a stresse strainT tractionsu displacementsP nodal loadsa nodal displacements

The following superscripts specify theparticular field:

trial functions derived from exact elasticity solutionst anapproximationto thetrial function fields

linear displacements between nodeshaving

F finite element field (bar)prescribed quantitiesonboundaries

(tilde) displacement field definedonelement boundariesI stress field definedin the interiorofelementsc additional stresses, constant within elements,

from compatibility constraints ,arising

(1), (2)or (3) associated withaparticular nodea or b referringto anelement denoteda or b

The following subscriptsmayalsobeused:

0,1,2...i

N

, j associated withtheparticular trial functionequaltozero,theloading functionassociated withthe N'thelement

or, if

Componentsof theabove vectorsare notunderlinedand mayhavethefollowing additional subscripts:

X,Yx,yr,8n, s

Cartesian coordinates referred toglobal axesCartesian coordinates referred totrial function axesPolar coordinates referredtotrial function axesdirections normalandtangential toboundary

VI .


10/234

Boundariesaredenotedby Swith thefollowing subscripts:

none complete boundaryK kinematic boundary (displacements specified)T traction boundaryR interface boundary betweentheexternalandspecial regionsE inter-element boundaryN complete boundaryof N'thelement

Theboundariesmay befurther specifiedby thefollowing qualifiers:

' (prime) denotes that partof theboundary adjacenttothespecial region

N (subscript) denotes that partof theboundary adjacentto the N'thelement

e (superscript) denotes that partof theboundary adjacentto theexternal region only

Other symbols:

a radiusofcircular holesorsemi-major axis length ofelliptical hole

a , a' arbitrary constantsinAiry stress functionn nA strain/stress compliance matrixA coefficientsofFourier series specifyingthenormal

tractionson theholeb semi-minor axis length ofelliptical hole

b , b' arbitrary constants inAiry stress functionn n-N element strain matrixfor N'thelementB, , submatrixof B,,wheren = 1, 2 or 3-(n)c arbitrary constantc , c' arbitrary constantsinAiry stress functionC. matrix definedbyequation (2.21)1D matrixofcoefficientsof a inequation (2.45)d. . elementof thematrixDij -D' matrixofcoefficientsinequation (2.44)d . elementof thematrixD'i j D matrixofcoefficients arising from integralon S'd^. elementof thematrixDij -s


11/234

D coefficientsofFourier series specifyingthesheartractionson thehole

d , d' arbitrary constantsinAiry stress functionn ne exponential constantE Young's modulusofelasticityF vectorofright-hand sidesinequation (2.45)f. elementof thevectorF1 F' vectorofcoefficientsinequation (2.44)fl elementof thevectorF'1 G shear modulus1i integer specifying numberoftrial functionj integer specifying numberoftrial functionk total numberoftrial functionsK, stress concentration factortK stress concentration factor ininfinite regionK finite element stiffness matrixI half lengthofsymmetrical plate

distance from centreofholeto top of lug& distance from centreofhole tobottomof lugL traction/stress matrix

constants definedbyequations(3.23)and (3.26)m limitofFourier seriesin $ (equalto k/2)m limitofFourier series defining normal tractionsonholem limitofFourier series defining tangential tractionson

holelimitofFourier series equalto themaximum of m and m

n integer (orinteger subscript)N element numberN mesh size parameter0 originofglobal coordinates0' originoftrial function coordinatesp nodal loads (seeabove)P magnitudeofresultant forceon theholeP magnitudeofresultant forcedue toshear tractionson

theholep internal pressureona^nulusq nodal displacements (seeabove)Q parameter definedas K,/Kt ^

Vlll.


12/234

r radial polar coordinateR = for anelliptical holeR ratioofspecial region areatoareaofholes distance alonganelement side measured from node (1)s lengthofelement sideo5 boundaryofregion (seeabove)t thicknessoffinite elementsin thespecial regionT tractions (seeabove)u displacements (seeabove)U strain energy functionUg Q strain energy function evaluatedin thespecial regionV complete regionw half widthofplateX,Y Cartesian coordinates referredtoglobal axesx,y Cartesian coordinates referred totrial function axesz complex number= x + i ya. coefficientoftrial functionor, if i=0, of theloading

functiona vectoroftrial function coefficientsa (i=l to k)a' vectorofcoefficientsa. (i=0 to k) 16 half angle subtendedby arc ofpressureY angle between OX and O'xaxesY shear strain component (seeaboveforqualifiers)6^ typical linear dimension ofelements near holeA areaoftriangular element strain vector (seeabove)E ^ percentage difference termforcomparingtwovaluesof

maximum stressseeequation (5.1) fordefinition5 complex functionof z6 angular polar coordinate8 arbitrary angleK = (forplane stress)^ Lagrange multiplieru = r for anelliptical holea+bV Poisson's ratioC complex functionof zn pi =3.1415927n functional (complementary energy)n modified functionalco stress (seeabove)

IX .


13/234

0o nominal applied stress0max maximum tensile stress0com maximum compressive stress^ref valueofstressforcomparisonT shear stress component (seeabove forqualifiers)* angle betweenXaxisandoutward normaltoboundary.1 complex stress functionfortrial function*A Airy stress function*i complex stress functionfortrial function


14/234

E.. strainijo. . stressijC. positionof thepointatwhichthekernel function

isevaluated

Tensors referringto thekernel functionaredenotedby thesuperscript* andhaveanadditional subscript (before other subscripts) whichindicatesthedirectionof thepoint force.

Asubscript precededby acomma (e.g. u. .)means partial differentiation1 > Jwith respectto thecoordinate componentx . Arepeated suffix impliessummation.

Thefollowing superscriptsmayalsobeused:Kc

m or nP

correspondingto theKelvin solutioncomplementary partof thekernel function (added toKelvin solution yields thekernel function)pertainingto the m'th or n'thnodeorelementpertainingto aninternal point

Boundaries aredenoted by Swiththefollowing qualifiers:

none complete boundaryof theproblemn (subscript)n'thboundary element* (superscript) boundary includedin thekernel functionH (subscript) partof theboundaryof theproblem co-

inciding withS*' (prime) remainderof theboundaryof theproblem

Other symbols:

a radiusofcircular holeb (seeabove)A matrixof thecoefficientsof theunknown tractions

ordisplacementsc coefficientofdisplacementinSomigliana's identity

(6.17). Asuperscriptmaydenotethecoefficientforaparticular element.(italic) function definedbyequation (6.32)

E Young's modulusofelasticity

X I .


15/234

f arbitrary functionF vectorofright-hand sidesin thesimultaneous equations

7, coefficientof thetractions definedbyequation (6.24)k&mnG shear modulusG matrixof thetraction coefficients

h, coefficientof thedisplacements definedbyequations(6.22)and(6.23)

H matrixof thedisplacement coefficients1 / T

integers defining coordinate directionsK numberofdimensionsof theproblem (2 or 3)K stress concentration factorL complex number=& (seeabove)m element (ornode) numbern node (orelement) numberN numberofelementsandnodesr radial polar coordinateS boundary (seeabove)

(italic) function definedbyequation (6.33)T (seeabove)V regionof theproblemu (seeabove)w radiusofannulusorhalf widthofsquare plateX (seeabove)z complex number defining positionofpointinplane

(= + iCg)z complex number defining positionof thepoint force (= + iXg)

Kronecker delta. Seeequation (6.5)6(x-() Dirac delta function. Seeequations (6.13)-(6.15)

(seeabove)E percentage difference termforcomparingtwovaluesof

stress. Seeequation(5.1) fordefinition8 angular polar coordinateK = (forplane stress)1 + v

= 3-4v (forplane strain)V Poisson's ratioX Lame constant givenbyequation (6.3)X' equivalent constantforplane stress applications

seeequation (6.8)


16/234

C. (seeabove)m pi =3.1415927o.. (seeabove)ijo externally applied stress

o , Og radialandtangential componentsofstresscomplex number= 5^ +complex potentialfor thekernel function withthepointforinfinity

kj point forcein the kdirection

Other notation:

&n natural logarithmIm denotes imaginary partofcomplex numberRe denotes real partofcomplex number

(bar)denotes complex conjugate' (prime) 1 , , , ,\unless referred toabove denotes" (double prime j differentiation with respectto zI denotes summation.J (subscript) denotes partial differentiation with respectto X .J denotes "isincluded in..."{}dV(C) denotes integration overVwith respectto thevariable5

underlined symbols denote matrices

Xlll,


17/234

LISTOFFIGURES

Figure Page2.1 Twodimensional body with loaded hole 152.2 Twodimensional body showing special region 154.1 Structureof theFESM program 434.2 Areasofelementonhole boundary 444.3 Boundary conditions subroutine 475.1 Rectangular plate withacentral elliptical (or

circular) hole 525.2 Effectofmesh refinementon theaccuracyof the

stress concentration factor 555.3 Effectofhole sizeon theaccuracyof thestress

concentration factor 575.4 Effectofhole aspect ratioon theaccuracyof the

stress concentration factor 595.5 Effectofspecial region sizeonaccuracy formeshA 655.6 Effectofspecial region sizeonaccuracyformeshB 665.7 Square plate with circular hole 695.8 Stress concentration resultsforsquare plate withelliptical holesofvarying aspect ratios 745.9 Stress concentration resultsforsquare plate with

elliptical holesofvarying size 755.10 Use of asuperposition principletoderivethestress

concentration factorfor theloaded lug (ii)from theconfiguration (i) 78

5.11 Stresses around boundaryofhole ininfinite sheet.Loadinga)Pressure =[cos8+l](-n


18/234

Figure Page5.19 Stresses around boundaryofholeininfinite sheet.

ii)Loadingb) + h)cosG normal pressure&sin28 shear 885.20 Stresses around boundaryofhole ininfinite sheet.

iii)Loadingb) + j)cos8 normal pressure&sin^GcosG 88shear

5.21 Stresses around boundaryofholeininfinite sheet.iv)Loading c) + g)cos*8normal pressure&sine shear 905.22 Stresses around boundaryofholeininfinite sheet.

v)Loadingc) + h)cos*8normal pressure&sin8 shear 905.23 Stresses around boundary ofhole ininfinite sheet.

vi)Loadingc) + j)cos^Snormal pressure&sin^G cos8 91shear

5.24 Pressurized annulus 925.25 Symmetrical lug(pin-loaded) 945.26 Rectangular lug(pin-loaded) 985.27 Stress concentration factorsforsymmetrical/

asymmetrical loaded lugs 1005.28 Stress concentration factorsforloaded lugs 1015.29 Lugwith roundedend 1035.30 Comparisonofstress concentration factorsfor

rectangular lugs (FESM) with rounded lugs 1036.1 Boundaries with modified kernel function 1157.1 Externally loaded annulus 1197.2 Square plate with central circular holeinbiaxial

tension 1207.3 Comparisonofresultsfor inexternally loadedannulus 121

7.4 Comparisonofresultsfor o inexternally loadedannulus 123

7.5 Accuracyof themodifiedBEMsolutionforannuli withvarious sizesofhole 124

7.6 Accuracyof themodifiedBEMsolutionforsquare plateswith various sizesofhole 126

(Appendices)E1-E29 Finite element meshes 197G1 Boundary grid specified using themacro DRAWBEL 209

XV .


19/234

LISTOFTABLESTable Page5.1 Thestress concentration factorforrectangular plate

intension(a/w = 0.5) fordifferent meshes 545.2 Thestress concentration factorforrectangular plateintension with various sizesofcircular hole 565.3 Thestress concentration factorforrectangular plate

intension with elliptical holeofvarying aspect ratio(a/w = 0.25) 58

5.4 Thestress concentration factorforrectangular plateintension with elliptical holeofvarying aspect ratio(a/w = 0.5) 58

5.5 Reference valuesofstressat theedgeof acircularholein aninfinite stripintension (a/w = 0.5) 61

5.6 Thestress concentration factorsfordifferent specialregions. MeshA. 62

5.7 Thestress concentration factorsfordifferent specialregions. MeshB. 63

5.8 Stress concentration factorsforplate with circular hole(a/w = 0.5)using different finite element meshes 68

5.9 Stress concentration factors givenbyreference [5.5]forsquare plateintension with circular hole. 70

5.10 Stress concentration factorsforsquare plate withcircular hole(a/w = 0.5) byFESM 71

5.11 Stress concentration factorsforsquare plate withcircular hole(a/w = 0.5) byother finite element methods71

5.12 Stress concentration factorsforsquare plateintensionwith elliptical hole 73

5.13 TractionsandFourier coefficientsforvariousdistributionsofradial load 80

5.14 TractionsandFourier coefficientsforvariousdistributionsofshear 86

5.15 Resultsforpressurized hole 935.16 Stress concentrationsforsymmetrical lugwith different

distributionsofload 955.17 Stress concentration factorforrectangular pin-loaded

lugs 99

5.18 Stress concentration factorsfor a lugwith roundedends from reference [5.9]. 102

XV I .


20/234

Table Page(Appendices)B.l Programs segments listed inthis appendix (B) 157B.2 Other program segments usedbyFESM program 158C.l Information stored in thedirect access file 179C.2 Variablesin thedirect access file 180D.l Macros used with theFESM program 181

X V I 1 .


21/234

CHAPTER 1GENERAL INTRODUCTION

1.1 Background to thework

Thepresenceofholesornotches instructural componentsincreasesthenominal stressinthese componentsby afactor,K^,knownas thestress concentration factor [l.l]. The aim ofthis workhasbeenthedevelopmentof accwide varietyofconfigurations,hasbeenthedevelopmentofaccurate methodsfordetermining in a

Theneed foraccurate estimatesofstress concentration factorshasariseninparticular from studies infracture mechanics. Fatiguedamagemay becausedby theinitiationandgrowthofcracks near stressconcentrations when thestructureissubjecttocyclic loading. Sincethese cracksmayappear earlyin therLfeof 1estructure,orindeedmayoccur inmanufacture,thefatigue life dependson therateatwhichthecrack grows,and todetermine crack growth ratesthestressfactorfor thecrack mustbeknown. Stress factors characterisethestress field near to acracktip andvaluesformost simple config-urations have been collected inreference manuals [1.2-1.4], Forshortcracks- and formostof thelifeof acomponentthecrack willbeshort-simple methodsofdeterminingthestress factor [1.5,1.6]maybeused evenforcomplex geometries, provided that dataisavailableforthestress concentration factoror, in thecaseofweight functionmethods,thestress distribution over thecrack sitein theabsenceofthecrack. Therecent compounding method fordetermining stressfactors [1.7-1.9] also requires knowledgeof thestress concentrationfactorat thesiteof thecrack ifthere issignificant interactionbetweentheboundariesof theconfiguration [1.10],(e.g.whentheholeiscloseto theedgeof thecomponent). Furthermore sincethecrackgrowth rate dependson thestress factor raised to apower(typically4) thestress concentration factor mustbeknown accurately(l%-3%)forthese methodsto be of use. Infact therearemany classesofproblem (e.g.finite plates with holes, pin-loaded lugs)forwhichstress concentration factorsof therequired accuracyareknownforrelatively few andonly thesimplest geometries. Even where theyareknownthestress distribution, which isrequiredbysome methodsforevaluatingthestress intensity factorsofcracks,may nothave been


22/234

included. Hence thereis aneedfor anaccurate, versatileandconvenient method fordetermining stress concentration factorsandthestress distribution neartoholesandnotches.

Formethodsto beapplicableto avarietyofcomplex structureswith several interacting boundariesandvaried loading conditions,anumerical methodofstress analysisisrequired. Broadly thesemay bedivided into three main types: finite difference methods, finite elementmethodsandboundary element methods. Thefinite difference method isperhapsthemost straight forwardandhistoricallywasdeveloped first[l.ll]. Bydividing theregionof theproblem with equally spaced nodesthroughout,thegoverning differential equationmay besolved intermsofvaluesofstressordisplacementat thenodal points. Howeverthemethodis notsuited toproblems where therearehigh stress gradients,suchasoccuratholesornotches, sincethenodes mustbeclosely spacedtomodeltheregionofstress concentration,andconsequentlythetotalnumber requiredfor anaccurate representationof thesolution becomesvery large. Forthis reason finite difference methods have been largelysupersededbyfinite elements [1.12-1.13]for all butspecializedapplications, and the twomethods developed in thepresent workarebasedon thefinite elementandboundary element methods.

Thefinite element method iswidely usedin allbranchesofcontinuummechanicsandsinceitsinceptionforstress analysis [1.14]it hasbeendeveloped toinclude many variants. Thebasisof themethod isthattheregionof theproblem isdivided into small elementsofsimple shape (intwodimensions usually trianglesorquadrilaterals) whichareassumedto beinterconnected onlyat adiscrete numberofnodes. A"shapefunction", forexampleapolynomial, isusedtorepresentthestressesordisplacements withintheelements intermsof thenodal valuesofeither displacements, stressesorboth, dependingon theparticularformulation. Anapproximate solutionforthese nodal unknownsisobtainedbyapplyingaweighted residual techniqueorvariationalprinciple (forexample minimizing energy)togivea set ofsymmetricbanded simultaneous equations [1.15, 1.16]. Thetechniqueisextremelypowerfuland hasbeen applied successfully tomany different problemsincluding three-dimensional, anisotropicandnon-linear cases.

- 2 -


23/234

Someof theconsiderable amountofwork done with finite elementsfor twodimensional elastic problems with high stress gradientsisreviewed, along with other methods,insection1.2. Indevelopinganewmethod forstress concentrations near holes,asdescribed inPartI ofthis work, certain drawbacksof thefinite element methodareavoided. Firstly inareasofsteep stress gradient, suchasfound atstress concentrations,thefinite element method usually requiresavery fine meshtoobtain acceptable accuracy. Thisisexpensiveinbothdata preparation timeand runtimeon thecomputer. Furthermore thesimplest finite element methods used constant strain triangular elementswith nodal displacementsasunknowns. This means thattoestimate thestressat anyboundary {theedgeof aholeforexample)thevalue mustbeextrapolated from theaverage stressin theelements nearto theboundary, introducingafurther sourceoferror. Byincorporating intothefinite element scheme known elasticity solutions, suchasthatforaninfinite sheet withahole,the newmethod proposed anddeveloped inthis thesis increasestheeffectivenessoffinite elementsforstressconcentration problems.

Inrecent years boundary element techniques have gained considerableacceptanceas apreferred alternativetofinite elements [1.17, 1.18].As thename suggests these methods require elementson theboundary oftheregion onlyandthusthedimensionalityof theelementsisreducedby one. Thesimultaneous equations, which mustbesolved togivetheunknownson theboundary,arederived from integral equations (hencethealternative name, "boundary integral equation" methods),andalthoughthematrix formedis notbandedas infinite element methods,it ismuchsmaller than would arise with finite elementsformost problems. Oncetheequationsaresolved thevaluesofstressordisplacementat anyinterior pointsmay becalculated. Howeveradisadvantageof themethodforstress concentrations, where stresses mustbeevaluatedatboundaries,isthat thisisusually more difficultandless accurate thanforinteriorpoints. InPartII ofthis workaboundary element method isformulatedincorporating modified "fundamental solutions" (elasticity solutionsforapoint forcein agiven region). These fundamental solutions becomethe"kernel functions"in theintegral equations. Theboundary conditionson thepartof theboundary wherethestress concentration factorisrequiredaresatisfied exactly inthesenewfundamental solutionsand


24/234

this enables this partof theboundaryto beincluded without usingboundary elements. Thustherequired stressesmay bedetermined asaccuratelyas atinterior pointsand thenumberofelements needed isreduced.

1.2 Reviewoftheoretical methodsandsolutionsTheexistenceofhigh stress near geometrical discontinuitieshas

been appreciated formany yearsandinvestigationsofstress concen-trations, both experimentalandtheoretical, were begun duringthelastcentury [1.19-1.21]. Since that timeanimmense volumeofworkhasbeenpublished on thesubjectand this review isaimedathighlightingsomeof themore important work. Reviewsofgeneral methodsofobtainingstress concentrations [1.22]and ofanalytical methodsinparticular,[1.23,1.24] have appearedin theliteratureandseveral collectionsofthesolutions obtained have been made [l.l,1.25-1.31] . The aim of thepresent surveyis toconsiderthevarious theoretical methods availableforobtaining stress concentration factorsand tocompare them withthefinite element superposition method (FESM)and themodified boundaryelement method (BEM)whicharedeveloped inPartsI and IIrespectively.Thereview islimited tomethods appliedto twod'trngMStOMoI configurationsofelastic, isotropicandhomogeneous materials with in-plane loadings.Anassessment ismadeof therelative meritsof themethods,theaccuracy(where known), whetherthemethodsmay beextendedtomore complexgeometriesormaterials (e.g.three-dimensional configurations, anisotropicmaterials,etc.) and, ingeneral terms, their theoretical basis.

1.2.1 Exact Analytical MethodsThesolutionbyLame [1.19] to thecaseof ahollow cylinder

subjected touniform pressureon theinnerandouter surfaces,was theprecursorofmanyof theanalytical solutionstostress concentrationproblems. It wasbasedon themathematical theoryofelasticity whichwasformulated,in asystematicway,duringthefirst partof thelastcenturybyNavier [1.32], Cauchy [1.33]andothers. TheintroductionbyAiry [1.34]of aformulation using stress functionsled toimportantsolutions, including thosefor aninfinite sheetintension containingatraction-free circular hole [1.20],atraction-free elliptical hole[1.35]and aloaded circular hole [1.36]. Theconfigurationof anelliptical holein aninfinite sheetintensionwasfirst solvedbyKolosov [1.37]whointroducedtwomost important concepts, complex

4"~


25/234

potentialsandconformal mapping. Thisled to thedevelopmentof amost powerful analytical method forelasticity problemsandmuch moreworkwasdone using this approachby theRussian school, notablyMuskhelishvili [l.38-1.40]andco-workers. Thetechnique remainedunknown outside Russiaformany yearsand waslater used independentlybyStevenson [l.41,1.42] andothers [1.43,1.44] forstress concentrationproblems. Examplesofother solutions obtained using Muskhelishvili'smethod are: infiniteorsemi-infinite platesintension containing deephyperbolicorshallow semi-elliptical notches [l.3l],andpoint forcesactingin aninfinite plate containingacircular [1.45]orelliptical[1.46] cut-out. Inspiteof thepowerful natureof themethod and theusefulnessof thesolutionssoobtained, onlya ^2wconfigurations havebeen solved inclosed form,andgeneral solutionsnot inclosed form(e.g.[1.47]) require much analysistoobtainaparticular solution,even assuming thattheseries involved converge. Forthis reasonapproximate methodsforcalculatingthestresses have been developedandapplied to amuch wider rangeofproblems than ispossible usinganexact analytical technique. Howevertheexact methodsaremostimportantin thedevelopmentofapproximate techniquesand in thepresentwork exact analytical solutions based on themethodsofAiry [1.34]orMuskhelishvili [1.38] areincorporated into numerical methodstoimprovetheir efficiency.

1.2.2 Approximate Analytical MethodsApproximate methods suchas the"alternating technique" have been

usedtodetermine Airy stress functions fromaseries representation,andhence toobtainthestressin, forexample,aninfinite strip withacentral circular hole [1.48]. Thebasisofthis methodwasthatseparate partsof theboundary (e.g. theholeand thestraight edgesofthestrip) were considered alternatively. Ateach iterationof themethodresidual stresses occurredon theother partof theboundary, which werethen cancelledby thenext iteration, leaving smaller residualson thefirst partof theboundary. Thiswascontinued untilanacceptableaccuracywasobtained. Approximate solutionsforplatesandstripswith circular holes [1.49-1.52 land forloaded holesinstrips [1.53-1.55]have been producedbyHowland andothersin asimilar manner.Themethodwasextendedfor astrip withanasymmetrical holebyLing [1.56]whoalso obtained thesolutionfor aninfinite sheet loaded intension,perforatedby twoequal holes [1.57, 1.58].

-5-


26/234

Isida [1.59-1.65] solved several strip configurations usinga"perturbation" method basedon thealternating technique usedbyHowland. thesolutions included thoseforstrips containinganeccentric circular hole [1.59,1.61], elliptical hole [1.62,1.63]andsymmetrical notches [l.60]. ThesolutionbyShibuyaet al [1.66]for aplate withaconical hole isbased on asimilar principlebutextended to 3dimensionsbyusingaleast-squares approximationtosatisfy theboundary conditions.

Approximate solutionsforplates with different shaped holes havealso been obtained basedonMuskhelishvili's complex variable approachwith conformal mapping. Many variantsof themethod exist,butgenerallytheholeornotch ismappedon to aunit circle,thecomplex potentialsaredetermined from theboundary conditions, usually in aseries form,andtruncationof theseries yieldsanapproximate solution. Savin[1.26,1.27] andmany other authors have obtained solutionsforplatesperforatedbycircular holes [1.67-1.69], square, rectangularortriangular holes [1.70-1.74], reinforced holes [1.75-1.77]andmultipleholes [1.78-1.80],andmany ofthese solutionsarecollectedin the twomonographs [1.26,1.27]where many anisotropic andelastic/plasticproblemsarealso treated. Thesame approachhasalso been used forsome edge notch problems [1.81-1.83].

Results from these approximate analytical methodsaregenerallyaccuratetowithin 2%buteach problem mustbeformulated individuallyandparticular mapping functions mustbefound foreach configuration.Thismay not bepossible especially iftherearediscontinuitiesin thecurvatureof thenotch. Often thereareproblemsofconvergence also,such thatanappreciable improvementofaccuracycanonlybeachievedbyincludingagreat many more termsin theseries representationofthepotentials,andthisisparticularly true whentheconfigurationsare offinite size, rather than infinite planesorstrips. Theimportanceof themethodsto thepresent work therefore,is notthatthey offeranalternative tomore general numerical techniques,butthat several accurate solutions have already been obtained thatmay beusedforcomparisonsinconfirmingtheaccuracyof any newmethod. Inparticular solutionsbyHowland [1.48], Hengst [1.52], Knight [1.54]andIsida [1.62] have been used in thepresent work forthis purpose.

- 6 -


27/234

1.2.3 Numerical MethodsTheadventofpowerful digital computers meant thattheemphasis

instress analysis moved from analytical methodstonumerical methods.Ofthese, mentionhasalready been madeof thefinite difference,finite elementandboundary element methods. The"collocation method"however isanother important technique.

Thecollocation method [1.84] consistsofusing stress functionsorcomplex potentialsinseries form, thecoefficientsof theseriesbeing unknown. Theseriesaretruncatedand thecoefficients determinedbymatching theboundary conditionsat afinite numberofpointson theboundary. Hooke [1.85,1.86]used this method fortwo-dimensionalanda^isymmetric three-dimensional notch problems under tensionandbendingloads.

Thecollocation methodhasbeen combined with conformal mappingbyBowieandothers [1.87,1.88] andfurther improvedbypartitioningtheregionof theproblem into separate sub-regions [1.89]. Solutionsforvarious shapesofedge notches insemi-infinite platesandholesininfinite plates have been obtained using this method [1.90]. Thecollocation methodmayalsobecombined with other numerical techniques,suchas thefinite element method [l.9l], which gives added flexibilityin its use. Typical accuracyfor themethod isgenerallyin theregionof 1%[1.92] butproblems with convergence, ill-conditioningorsensitivityto thenumberanddistributionof theboundary pointsmayincreasetheerror. Consequently thecollocation methodis not asversatileassome other numerical methodsand it hasreceived relativelylittle attention compared tofiniteorboundary elements. Some workiscontinuingon thecollocation method for theevaluationofstress intensityfactors,at theUniversity ofSouthampton [1.93|.

Adrawbackof theanalytical methods,and to anextentthecollocationandfinite difference methods,is alackofversatility inanalysingawide varietyofdifferent geometries. It is inthis respectparticularly thatthefinite element methodandboundary element methodare soeffectiveandthis explainsthelarge amountofwork whichhasbeendonein thelast twenty years, especiallyon thefinite element method.

-7-


28/234

General reviewsofwork infinite elements have been presented,forexample,inseveralof thestandard texts [1.12,1.13,1.94]. Here,however, having mentioned someof theproblemsofconventional finiteelements, particular attentionispaidto thedevelopmentofmethodscombining both finite elementandcontinuum concepts,ofwhichthefinite element superposition method formulatedin thepresent work isanexample.

Inapplying conventional finite element methodstoconfigurationswith steep stress gradients, several difficulties occur. Many elementsarerequired tomodel thestress field accuratelyandconsequently itisexpensivefordata preparation, computer processingandpost-processingof theresults. Inadditiontoobtainavalueofstressat theboundarysome sortofinterpolation from interior pointsmay berequired. Evenhigher order elementsare notalwaysanadvantage since althoughthenumberofelements wouldbereduced (or theaccuracy increased) morenodesareintroducedperelementandthismayleadto asimilar numberofunknownsin theproblem. Much workhasbeen doneinproposingmodificationsto thefinite element schemetoovercome these problems,particularly forcrack problems [1.95]. Isoparametric elements [1.96],different variational principles [1.97], andhybrid methods [1.98]haveallbeen used toimprovethemethod forcracked configurations.Theforerunnersof thepresent work, also using methods formulated forcrack problems, superimposed analytical trial functions, correspondingto thesingular stress field aroundacrack tip,overaregionof theconfiguration. This region varied from aspecial crack-tip element[1.99-1.103]to thewhole regionof theproblem [1.104-1.108]or, asin thepresent formulation,a"special region" including several elementsaround thenotchorcrack [1.109-1.111]. Asuperposition approachwasproposed forstress concentrationsatsmooth cut-outsby Rao [1.103]using large "primary" elementsin theregionof thenotch,andbySchnack[1.111]whocombinedthe use ofaugmenting functions with six-node hybridelements. The aim ofthese methodsis toincorporate known solutionsfor thestress field nearto acrackornotchin an region, intothefinite element scheme. Thusthefinite elements model onlythedifference betweentheinfinite region solution, scaledbyarbitrarycoefficients,and theexact solutionfor theconfiguration being analysed.


29/234

Since this difference willberelatively smallin theregionofinterestif thetrial functionsareappropriateto theparticularproblem,theerrors introducedbymodelling theregion withacoarsefinite element meshandinterpolating valuesofstresson theboundary,will alsobesmall.

Thefinite element superposition method presented inPartI isadevelopmentofthis work inthat trial functions, derived from knownelasticity solutionstoappropriate configurations,arecombined withconstant strain triangular finite elements. Loadingon ahole boundaryisincorporated into themethod using similar elasticity solutions,knownasloading functions, which remove theneed torepresent loadingsas aseriesofnodal forces-oftenafurther sourceoferrorinconventional finite element analysis. Thetrial functionsforconfig-urations with circular holesarebased on thegeneral Airy stressfunctions, rather than solutionsforinfinite regions, which means thattheeffectsof theother partsof theboundarymayalsobeincluded inthetrial functionstosome extent. Theaccuracyandsmall numberofdegreeso ffreedom that result from well chosen trialandloadingfunctions,and theversatilityof thefinite element method ingeneralcombinetomake thisapowerful method for thesolutionofstressconcentration problems.

Theboundary element methodwasproposednotlong afterthefiniteelement method,but thefirst practical applicationsof themethodbyJaswonandSymm [1.112, 1.113] appearedin 1963 andinitial developmentwasmuch less rapid than finite elements. Thismaypossiblybe due totheslightly greater mathematical complexityof theformulation,and thefact thatit isless easily understood intuitively. Howeverthemethodhasseveral advantages over finite elements,themost important beingthat since onlytheboundaryof theregion needbedivided into elementsthedimensionsof theelementsarereducedby one, e.g.fromathree-dimensional volumeto a twodimensional surface. Themethodwasfirstused forelastostatic problemsbyCruseandRizzo [1.114, 1.115] andinrecent yearsanupsurgeininterestin themethodhastaken placedueto itsclaimed superiority overthefinite element methodformanyapplications [1.116, 1.117]. Broadly themethodmay bedivided intotwomain types: direct formulations [1.118-1.121] where theunknown functions


30/234

in theboundary integral equations are thephysical variablesof theproblem (e.g.tractionsanddisplacements),andindirect formulations[1.122-1.125] inwhichtheintegral equationsareexpressed intermsof a"density function", whichinitselfhas nophysical significancebutfrom which thephysical parametersmay bederivedat anypointinthebody. Aform ofindirect method, called the"body force method"developedbyNisitani [1.126]hasbeen usedformany notchandcrackproblems [1.46, 1.127-1.129] . These includeaninfinite sheetcontainingone or tworowsofelliptical holes,asemi-infinite platecontaining variously shaped notches,a row ofelliptical holesor arow ofnotches,and aninfinite strip containingtwosymmetrical semi-elliptical notches.

Muchof therecent interestinboundary elements,aswith finiteelements, centredonimprovingthemethod forconfigurations withcracks. Cruse [1.130] proposed includingthe crack explicitlyin thefundamental solution from whichtheintegral equationsarederived sothatthecrack neednot bemodelledbyboundary elements,andthisproved most successful. In thecaseof thebody force methodasimilarapproachwasadopted byMurakamiandNisitani forelliptical holes[1.131, 1.132]and,usingadirect boundary element method, TellesandBrebbia [1.133-1.134] usedtheapproachforconfigurations containingalong straight boundary. Thesuccessofthese methods suggested thatasimilar approach couldbeusedfor adirect boundary elementformulation withafundamental solution which satisfied theboundaryconditionsof acircular hole. This ideais thebasisof theworkpresented inpartII ofthis thesis. Notonly does this approach reducethenumberofelements requiredbutalsothestressesat theholemaybecalculated directlyandwith greater accuracy than wouldbepossiblewith standard boundary elements.

- 1 0 -


31/234

1. 3 Layoutof thethesisThemain bodyof thethesisisdivided intotwoparts: PartI

comprising Chapters2 to 5 isconcerned withtheworkonfinite elementsandPartIIcomprising Chapters6 and 7concernstheboundary elementwork. Chapters1 and 8 aregeneraltoboth aspectsof thework.

CHAPTER2presentstheformulationof thefinite element super-position method. Theconceptoftrial functions derived from knownelasticity solutions isintroducedforconfigurations with loadedortraction-free holes. Inadditionto thetrial functionsthe newloadingfunctionisincorporated intothemethod whichis the(known) solutionfor aninfinite sheet with thespecified loadingon thehole. Avariational principle isused todeterminethearbitrary coefficientsof thetrial functions,thefinite element unknowns (nodal displace-ments)andcertain correction stresses which ariseinelements nearboundaries.

CHAPTER 3deals withtheanalytical elasticity solutions whicharerequiredby thefinite element superposition method,i.e. thetrialfunctionsandloading function. Twotrial functionsaregivenforelliptical holes basedon ananalytical solution using complex stressfunctionsand aconformal mapping function. Forcircular holesthegeneralised solutionfor theAiry stress functionin twodimensionalpolar coordinatesisused tospecifyageneralset oftrial functions.Thegeneralised solution isalso used togive theloading function,withadistributionoftractions roundthehole boundary specified usingaFourier expansion.

InCHAPTER4 the way inwhich themethod isimplementedon thecomputerisexplained. Thestructureand themain processes occurringin theprogramarediscussedand abrief resumeisgivenof how theprogram and itsperipheral facilitiesmay beused inpractice.

Theresults obtained using thefinite element superposition methodarepresented inCHAPTER 5. Confirmationof theaccuracyof themethodfortraction-free holes iscarriedout bycomparison with other estimatesforstress concentration factorsinrectangular plates with holes. The

- 1 1 -


32/234

effectonaccuracyofsuch parametersas thefinite element mesh size,numberoftrial functionsandsizeof thehole isdeterminedand newresultsfor thestress concentration factorsofelliptical traction-free holesinsquare platesaregiven. Various distributionsfor thetractionsonloaded holesaresuggestedandcompared. Estimatesforstress concentration factorsforloaded holes determinedby thefiniteelement superposition method arecompared with some known valuesandfinallynewresultsareobtainedforrectangular lugs wilWh loaded holes.

InCHAPTER 6 theformulationof theboundary element method ispresented and themodificationto themethod,byusing fundamentalsolutions which include theboundary near thestress concentration,isexplained. Theimplementationofthese modificationsin thecomputerprogram isalso discussed.

Theresults given inCHAPTER 7were obtained usingthemodifiedboundary element method. Theadvantagesandlimitationsof themodifiedmethodareshownbycomparingtheresults fromthe twomethodsfor anexternally pressurized annulus. Theaccuracyof themodified methodforannuli andsquare plates with various sizesofcircular hole isshownbycomparingtheestimatesforstress concentration factors fromtheboundary element program with known values.

CHAPTER8, thefinal chapter, containsasummaryof theconclusionsfrom boththefinite elementandboundary element work. Comparisonbetween themethodsandtheir relative meritsismadeandsome possibledirectionsforfuture workaresuggested.

- 1 2 -


33/234

PART I

THE FINITE ELEMENT SUPERPOSITION METHOD


34/234

CHAPTER 2FINITE ELEMENT FORMULATION

2.1 Introduction

Thefinite element superposition method (FESM) used inthis workisbasedon amethod originatedbyMorley[2.l] andextendedbyBartholomew [2.2, 2.3]. basisof themethod isthatWiepiece-wiselinear displacement fieldofconstant strain finite elementsmay beaugmentedby thesuperpositionof one ormore known elasticity solutions,referredto as the"trial functions", whichareweightedbyarbitrarycoefficients.* Thetrial functionsareelasticity solutions whichsatisfy exactly conditionsofequilibrium andcompatibilitybut not alltheboundary conditionsof theproblem. Theyarechosen such that theygive risetostressesanddisplacements closely matching thosein theregionof thestress concentration. Forexampletheknown solutionforauniformly stressed infinite sheet containingacircular holemay beused as atrial functionfor afinite plate loaded insome manner withasimilar hole.

Bartholomewhasformulated this method fortraction-free cracks.In thepresent workthemethod isextended toapplytoconfigurationswith circularorelliptical holes whichmay betraction-freeorsubjectedtospecified tractions. Todeal with loaded holes another knownelasticity solution referredto as the"loading function"hasbeenintroduced. The use ofthis function removestheneedtorepresentthetractionsat theholein thepiece-wise constant form usually employedby thefinite element method, thus avoiding theintroductionofinaccuraciesat thevery point wherethestress concentration factoristo bedetermined.

Theloading function correspondsto theelasticity solutioninwhichtheholeissubjectedto thetractionsforwhichasolutionis

" Theterm "trial function"may beusedof anyfunction used toapprox-imatetheexact solution. Inthis sensethepiece-wise linear displace-ment field isalsoatrial function, however heretheterm isusedoftheknown elasticity solutions with whichtheconstant strain finiteelement field isaugmented.

-13-


35/234

requiredbut theextentof thesheetisassumedto beinfinite. Thetrial functions,on theother hand,areelasticity solutionsfor aplate withastress-yreg hole under various different remote boundaryconditions which neednotnecessarily correspondto aninfinite region.Thesuperpositionof theloading functionandthese trial functionstherefore, resultsin thetraction boundary conditionson theholebeing satisfied exactlyandleaves residuals remote from thel 3le whicharecorrectedby theconstant strain finite elements.

It hasbeen shown [2.3]thatin thecaseofcracked configurationsit isadvantageoustolimitthesuperpositionto a"special region"whichislarger thanasingle special elementbutsmaller thanthecomplete regionof theproblem. Forthis reasonthepresent formulationcontinuesthe use of aspecial region over whichthetrial functionsandloading functionaresuperimposed, constant strain elements alone beingusedin theexterior region.

Thetrial functionsandloading functionforspecific classesofproblem aredetermined inChapter3. Thedetailsof how themethod isformulated areoutlined in theremainderofthis chapter, with additionalmaterialinAppendixA.

2.2 Notationfor theboundariesTheconfigurationto beanalysedbyFESMmay berepresented

diagramatically as infigure2.1. The twodimensional body containingahole isdenotedV and theboundary, includingthehole boundary,isdenotedby S. TheboundaryS ismadeup of S ,where traction boundaryconditionsareapplied,and S ,where kinematic boundary conditionsareapplied. Sinceasuperposition principleis to beusedin therepresent-ationofloadon thehole boundary,it isassumed thatthehole formspartof S . Thegeometryof theconfigurationisdefined relativetoCartesian coordinates (X,Y)withanoriginat 0.

Thevariational principle, from whichthefinite element solutionisderived, willberepresented intermsofvolumeandboundary integrals

various partsof theconfiguration. Sinceaspecial regionisintroduced, over whichthetrial functionsare to besuperimposed,further subdivisionof theboundaryS isrequired (seefigure2.2).

-14-


36/234

Y

0

Loaded holeApplied Tractions (T)

Prescribeddisplacement

(u)X

Figure 2.1 Two dimensional body with loaded hole

Exterior region

O Special Region( S p . R )Figure 2.2 Two dimensional body showing special region

-] 5-


37/234

Boundaries adjacentto thespecial regionaredenoted withaprime ('),hence forexample. Thetraction boundary adjacentto thespecialregionisdivided intotwoparts, thehole boundaryand theremaining traction boundaries. This distinctionisnecessary in thepresent workdue to theintroductionof theloading functiontorepresenttheloadingon thehole. The"interface boundary" betweenthespecialregionandexterior region isdenoted S^. Thenotationfor theboundariesmay besummarized thereforeasfollows:

and

S = 5% + S,

S'

^T =

( 2 . 1 )

where S denotes that partof thetraction boundaryin theexteriorregion. Thecomplete regionV isdivided into triangular elements.

2.3 Displacementandstress fieldsThebasisofFESM isthatthedisplacements, u , andstresses,

Ia , of theapproximate solutionare asuperpositionof aconstant strainfinite element fieldand a set ofknown elasticity solutions withdis-

* *

placementsandstresses denoted u and j respectively. Theintegeritakesthevalues1 to k for thetrial functions,kbeingthenumberoftrial functions,and i=0 for theloading function. (Underlinedsymbolsareused throughoutthetexttodefine both vector fieldsandmatrices).

Theassumed formof thedisplacement field u on theboundariesof thefinite elementsmaythereforebeexpressedasfollows:

kIi=0

u = u" + J a. (u. - u^; 1 1

withinthespecial region,on and

in theexternal region,on and

> ( 2 . 2 )

on SK

- 1 6 -


38/234

where a. is thecoefficientof the i'thtrial function,1uand t

are theprescribed displacementson S ,u areconstant strain finite element fields which takethe

*same valuesas u. at thefinite element nodesand arelinear between-1 tnodes. Thereasonforintroducing these fields,u., isthat(u. - u.)1 1 1iszeroat thenodes thus nodal displacementsaregiven simplybypu inboththeexteriorandspecial regionsand thedisplacement field,asdefinedbyequation(2.2) iscompatible acrossallelement boundaries.

As isusualin thedevelopmentoffinite element methodsthepiece-F twise linear displacement fieldsu and u. ;&reexpressed intermsof

F t - -1vectors,q and q. respectively, containingthecomponentsofdisplace-mentat thenodes. Thestrain withinanelementisconstant from thesefieldsandthusmay beexpressed intermsof thenodal displacementsas:

and

-N %

1 -N.1

(2.3)

Fwhereq' and q_, contain displacement componentsfor the N'thelementN only, B is the"element strain matrix"and and are thecons ant strain vectors (three components inplane stress}for therespective fields. Thematrix isobtained from thegeometryoftheelement and isgiven; as inreference [2.4],by:

-^il)'^^2)' -(3) (2.4)

whereatypical sub-matrix, say, is:

:i) 2 A ' V s '

(X3-X2)(X3-X^) (2.5)

and A is theareaof theelement,X , Y are thecoordinatesof then nn'thnode and thenodesarenumbered anti-clockwise round theelement.

-17-


39/234

Thestress field o may now bedefinedin theintertorofeach elementasfollows:

I F r + T 4 - k - 1 -0 = a + ^ a. (o. - a. + o. J In thespecial regioni=0 1 ^ 1 ^I Fa = a In theexternal region.

( 2 . 6 )

F * tThestress fields a , 0. and a. correspond toexact solutionsfor^ F * ^ t cthedisplacement fields u , u. and u. respectively. Thetermsa. 1 1 1constant within elements, mustbeincludedin thespecial regiondue tolinear displacements being definedon theboundaries and toensure compatibility. Theterms thereforearenon-zero only inspecial region elements adjacenttokinematicorinterface boundaries.Inspiteof themany termsinequation (2.6a)it may beseen thatit isF tsimply thesuperpositionofconstant finite element fields, a , 0.c * "and a. , withthetrial function fields 0. .1 1

Again following standard finite element methods,theconstantstress fieldsfor anelementmay beexpressed intermsofstrain. Thus:

and

A 0Nt-N.1

A 0N.1

-N.1

-N.1(2.7)

, F t c F twhere 0^ , , containthestressandstrain components respectively forthe^N'th element,and

1E v 0

1 00 2(l+v) ( 2 . 8 )

forplane stress. E isYoung's modulusand v isPoisson's ratio.

- 1 8 -


40/234

Inordertodetermine displacementsandstresses from theFequations(2.2) and(2.6),thenodal displacements, q , thetrial

function coefficients, a., and thecorrection stress fields, o? ,1 1mustbeknown. Thesearedetermined usingavariational principle.

2.4 Thevariational principleAvariational principle usesascalar quantity (a"functional")

whichmay bedefined intermsofintegralsof theunknown parametersin acontinuum problem- inthis case displacementandstress. Thefunctionsof theparameters which makethefunctional stationary isthesolutionto theproblem. Bylimitingthepossible functionsofdisplacementandstressto a set oftrial functions (asabove) withfinite degreesoffreedom theproblemmay bereducedto a set ofsimultaneous equations.

Thespecificationofboth displacementsandstressesbyequations(2.2) and (2.6)means thatthevariational principle must allowfortrial functionsto bespecified forboth parameters. Suchavariationalprinciple derived fromamodified principleofminimum complementaryenergy (seeAppendixA) wasgivenbyPianandTong [2.5]: .

n = I { (o^) + I c (u)^ dS - I (u)T f dS } (2.9)N N - J - - Js^ - -

where U (j^) is thestrain energyof thespecified fieldfor the N'thT I

element, a and T arerespectively theinterior stressandcorrespondingtractionsfor theelement,u is thedisplacementon thecomplete elementboundary, which isdenotedby S , and T are theprescribed tractions' Ton theelement traction boundary S . Thesuperscript ( ) denotestheNthetransposeof avectorofmatrix. Thesummationiscarriedoutoverall theelements.

Ingeneralthefunction U (o ,02) isdefinedas thestrain energy,in theregionVboundedby thesurfaceS, due to twostress fieldsG_ 1and On . Thus:

1 [ , - T-1^ ^2

V^ 2 ' Og) = 2 J dV (2.10)

where is thestraindue to thestress field

-19-


41/234

Ifonlyoneparameter isspecified to thefunctionthe twofieldsinequation (2.10)areunderstoodto be thesame. Similarly thefunctionmay bewritten with displacement rather than stress fieldsas thevariables. Provided thatfor onefield, o say, thestressesareinequilibrium overV and for theother thestrain field iscompatibleoverV, thevolume integralmay bereducedto asurface integral usingthedivergence theorem:

1 f sT" 2

SU(o^,a^) = - j (T^) dS (2.11)

whereT_, and u are thetractionsanddisplacements respectively of1 rithe twofieldsandthereare nobody forces. Using these propertiesofthestrain energy functionandsubstituting equations (2.2) and (2.6)into equation (2.9) thefollowing formof thefunctionalisobtained;

F F T -n = U( u ) - I (u ) T dSC - Jo ~T

kIi=0 ^ ^ '"T "HI - Js>s. 'h*- I

1 ^ ^ f * t T *- I I a. a. I (u. - 2uT) T. dS" lio jio ' J's'.s'

^ F T * '' * + T FI (u) T. dS + I (u ) T. dS - I (u.-LL^ T dS

^ ^ I f * T * r * + T ++ f f a.o. [ - % I (u.) T. dS + I (u.-u.l T? dS1 = 0 j ' o ^ J ' 2 - 1 - J - 1 - 1 - J

( u * - T " : d S I ( 2 . 1 2 )

where withtheappropriate subscriptsandsuperscripts, denotesthetractionson aboundarydue to thecorresponding stress fieldand Ugp ^is thestrain energy function evaluatedin thespecial region only.

- 2 0


42/234

Theprescribed tractionson thehole boundary areappliedexplicitly usingtheloading function. This functionisdefined suchthat:

= f (on thehole boundaryS') (2.13)0 0 H

Since a , thecoefficientof theloading function,isconstantit isnot atrial function. It isincludedin thefunctional withthetrialfunctionsbut thecoefficientisdeterminedby themagnitudeof theloadingon theholeandthusitdoesnotappearas anunknownin thefinal system ofequations.

Thetrial functionsarechosentosatisfy zero traction conditionson thehole boundary,i.e.:

T. = 0 (on S', for i=l to k) (2.14)1 HBysubstituting equation (2.13)in thefunctional, equation (2.12),theterms which relateto thetraction boundariesmay berewrittenas:

r p T f F T * f f T T - S -- I (u ) T dS - On I (u ) T_ dS + a_ I (u ) T dS- - 0 Jc, - 0 0 Jc, - -0

% a. [ (u ) ^T* dS + % a. [I (u^)^ T* - f I dS]

^ f k k ^I Go*i I (Hi-Hi) lo dS + I I OiO. j (Hi-Hi) I. dSi=0 ^ i=0 j=0 1 J J

^ ^ f * + T * -I r * T *f % a.a. [ I (u. - ul) T. dS - - I (u.) T. dS ] (2.15)i io j i o ' J Js' ' -J 2 -J

Further substituting equation (2.14)in theabove expression eliminatesall but thefive underlined termsandthusthefunctionalmay bewritten:

- 2 1 -


43/234

"c ' " i % . H * J o j o "sp.R'aI-^]'-%.H'2l'Hr

f F T - ^ f F T * I" * t T -- I (u ) T dS + I a. [I (u ) T. - | (u.-u.) T dSi=o 1 1 1 1

^ ^ r * f T * 1 f * T *+ % X a.a. [I (u.-u.) T. dS - - I (u.) T. dS]i'o jlo ' : 's- -J : 's' -J

+^ r _ ' r * r f% a. [| (u) T. dS + (u ) T. dS - (u.-u.) T dS].=0 ' i s - - J s . " - -

y ^ r * + T r * + T r+ 2 X a.a. [ I (u.-u.) T. dS - I (u.-u.) T. dSi=0 j=0 1 J ^ ^ ^ ^ ^ J(2.16)

2.5 Determinationof thecorrection stress fieldsa?1

Thecorrection stress fieldsaredeterminedbyvariationof thefunctional equation (2.16), with respectto Theterms which dependcon a. are:1

I I (,^o5)- I (h'-HI)'1= dS I (2.17)- j-u

Sincethestresses2^ areconstant over each element,theexpression(2.17)may beexpressed:

(Sp.R) ^ ^1 , , C\T . c , c/ra a [ -N i=0 j=0I { I I a.Oj [- tA(o^^l'A 2^^ -(o^])^L C^] } (2.18)

wherethesummation iscarriedoutoverallelementsin thespecialregion, t is thethicknessof theelementsin thespecial regionand

is thevectorofcomponentsof thecorrection stress fieldin theNiX and Ycoordinate directionsfor the N'thelement.

- 2 2 -


44/234


45/234

F2.6 Determinationof thenodal displacements qTheconstant strain finite element partof thedisplacement field,

F Fu , isdefined by thevectorq of itsnodal components. Thetermsin" Fequation (2.16) which dependon u are asfollows:

F C F T - r F T *U(u ) - I (u ) T dS + I a. { I (u ) T. dS

r * t T F F t(u. - u.) T dS - 2 U_ a (u ,U.) } (2.24)c,,q, -1 -1 - Sp.R - -1

FIntermsof q thismay bewritten;

i (gF)T K qF _ (gF)T & + f o (qf)? p (2.25)i=0whereK is thestiffness matrix for theconstant strain finite elementschemeand p and p. arevectors whichmay beconsidered asequivalent 1nodal loads. K isassembled from thestiffness matricesofindividualelementsas forconventional finite element methods. Thusthefirst termofequation (2.24)may bewritten:

I tA(c;)?A-l c? (2.26)N

I t6(c|F)T [ ] gF (2.27)

where thesummationisoverallelements. Theelement stiffness matrixthereforeisgivenby tA [B^ A ^ B^] and bysumming thecontributionsfrom each elementthestiffness matrix K isobtained.

Thevectorp isdetermined simply from theprescribed tractionsT(excludingthetractionson thehole boundary Auniform loadonanelement traction boundary isdistributed equally betweenthe 2nodeson theelement side.

Contributionsto thevectorp.however arise from thelast three1termsinexpression (2.24). From thefirstofthese termsthefollowingequivalent nodal loads arisein anelement withapartof theboundary

or betweentwo of itsnodes, node (1) andnode (2) say:24


46/234

:2)

f (2 )t I (1

^ ( 1 )--) T. dSs 1o

-i T- dS:i) ^

> on or

2 .28 )

wheres is thedistance along theelement side measured from node (1),is thelengthof theelement side and thesuffix to pj ^ denotes

thenode atwhich theequivalent load isconsidered to act. Theintegralis of aknown function and may becarried outnumerically. Theremainingtwotermsofequation (2.24)may bewritten:

Sp. RC.1

tA -Ni * (2.29)

where thesummation iscarried out forelements in thespecial regionand"t" tthevector containsthecomponentsof o. for the N'thelement.-Ni -1

Theexpression (2.29)may berearranged, using equation (2.22)and thedefinitionof thematrix (equations (2.3) to (2.5)),as:

SP'R c t T FI t - tA (o^i + EN i) SN } (2.30)

Thusthecontributionsto thenodal loads p. in the N'thelement from1these termsaregivenby:

and (2.31)

which may beevaluated whenthestresses and arecalculatedforeach element in thespecial region.

Since thematrices 2 ^^d p in theexpression (2.25)can bedetermined, thestationary pointmay befoundbysetting thevariation

Fof theexpression with respect to q aszero, thus:

K qi =0 Ei

(2.32)

-25-


47/234

Thevariables q and q. are nowdefined such that: 1

q = p (2.33)

and q. = K ^ p. (2.34)

These simultaneous equations, subjectto thekinematic constraints

on 5%,and q. = 0 J (2.35)1

aresolvedon thecomputer using Choleski factorisationof thematrixinbanded form,asimplementedbyMorley [2.6] andothers [2.7, 2.8].Thedisplacementsg, asdefinedbyequations (2.33)arethose whichwould arise from constant strain finite elementsif noaugmenting trialfunctions were used. This "basic" solutionmay beusedforcomparisoninassessing theimprovementin thefinal augmented solution.

FThenodal displacements, q ,writtenintermsof q and q follow

from equations (2.32)to(2.34)and aregivenby:q^ = q - % a q. (2.36)

i=0 ^

2.7 Determinationof thetrial function coefficientsEquation (2.36)may besubstituted back into equation (2.25)and

hence intothefunctional, equation (2.16). Furthermorethefollowingterms from equation (2.16)mayalsobecombined:

J o J o "i-j'"sp.n'Hl.Mjl -"sp.R'Hi,M])

f * + T t f * t T r+ I (u.-u.) T. dS - I (u.-u.) T. dS } (2.37)

1" cSincethestresses o. and o' areconstant over each elementthefinal1 1twotermsof theexpression (2.37)may bewritten:

- 2 6 -


48/234

t T * t f T(T ) I (u -u ) dS = % (o ) L CJ S^+S^ N J

rp r * c Tand -(T.) I (u.-u.)dS = - X (& ) L C. (2.38)

Since, from equation (2.22)

L C. = - tA A (2.39) 1 NIand U(G ,c_^^f) = ^ tA (a^C)T^

theexpression (2.37)may bewritten:

^ - 2"sp.RlZi'zj)+Usp,R(2i,2j)' (2-41)i=0.^0

or intermsof thestress fields:

2 f *ij "sp.R(^I-H;. sj-f;) (2.42)i=0:k0Thusthefinal formof thefunctionalto bevaried with respect

to theremaining unknownsa (i=l to k) is asfollows:

i ^ a - & + I % q +(g.)^2+ (q)^ p.]i=0

k kI Ii=0 j=06 6 " 1" L (Qi)^ K - (q.)T Pj ]

kI [i=o 's; ^ ^

f * + T _ f T *I (u.-u^) T dS + I (u) T. ]J - 1 - 1 - J c , ~1

k k+ f * + T * i f * T *Z I GiOj [ I (Hi-Hi) I. dS - - I (u ) T dSi=0 j=0 J J ^S' J

"sp.R ] (2.43)

-27-


49/234

Thisis aquadratic expressionin a whichmay bewritteninmatrixform:

^ (a'fo'a' - F' a' + c (2.44)

whereD' is asquare matrix,Fy avectorand thevectorofcoefficients,allhaving (k+1)rows. Thefinal term cdoesnotvary with a'.Variationwith respectto (i = 1 to k) todeterminethestationary point yieldsa set ofsimultaneous equations:

D a = F (2.45)

whereD, F and a inthis case have onlykrows (omittingtheloadingfunction coefficient a froma' ). D is asymmetric matrix whereanelemento _ _dL , of thematrixmay begivenintermsof theelements, of D', as:

d . + d;.ij.. = (i, j = 1 to k) (2.46)

Equation (2.46)may beverifiedbyexpandingtheexpression (2.44)anddifferentiating. Thecoefficientsof a in D' , d' . and d ( i = 1 to k) ,^ o Ol lOcontributeto theright hand sideofequation (2.45) sincea is aconstant,andthusanelementof F, f., intermsof thecorrespondingelementof F', fj, isgivenby:

d + d'.f^ = f^ - 2 ( i = 1 to k) (2.47)

Theline integralsinequation (2.43)arecarried outusing Gaussquadrature overtheboundariesS^, and inordertoform therequiredmatricesD and Fy Aparticular advantageof thepresent formulationofthis method, however,isthatnoapproximate integration needbecarriedoutoverthehole boundary S^,which sinceit iscurved might have intro-duced either greater complicationorinaccuracy. Theboundary appearsinequation (2.43)aspartof S', butexplicit integrationmay beavoidedasfollows. Let therelevant terms, denoted d^ , bestoredin thematrixD ,wheres

d- = - i I (u'l^ T* dS (l,j = 0 to k) (2.48)

Since from equation (2.14)- 2 8 -


50/234

(u.)^ T. dS = 0 for fi = 0 to ks; -1 -J j = 1 to k (2.49)then:

= - ^ I ( u j " T. dS for f i = 0 to kj = 1 to k (2.50)

1 f , *\Tij = ^ 5 Ij

Thus only thetermsd. do notgivetheintegral requiredbyequation* - X -(2.43). Since u. and T. areexact elasticity solutionsand S' is a-Jclosed contour

I (u.) T. dS = I (u.) T. dS (i,j = 0 to k) (2.51)Jc, -1 -J Jc, "J "1S

Therefore thetermsd^. may besubstituted for thetermsdf togiveoi lOtherequired integralin thematrixD^. Theonly term whichmay notbecorrected inthisway is d^ which is acoefficientof in theoo ofunctional. Sincea doesnotvary this termis notrequiredandthusthepotentially complicated integrationof thetrial functionson thecurved hole boundary isavoided.

Thesimultaneous equations (2.4f)may now besolved togivethecoefficients a..1

2.8 Determinationof thestressat anypointF FThenodal displacementsq , andhencethedisplacement fieldu

aredetermined from equations (2.36). Equation (2.2)givesthe dis-placements,u,should theyberequiredatintermediate points,and thestressesmaythenbecalculated from equation (2.6). Thestress fieldF F

o isdetermined foreach element from thenodal displacementsq , the*trial function stresseso. may beevaluatedat anypoint fromtheknownanalytical expressionsfor thetrial functions,and theterm (o.-o.),1 1whichisconstantforeach elementin thespecial region would havebeen calculated previously foreach elementinorder toevaluatethefinal termof thefunctional (2.43). All thetermsinequation (2.6)thereforeareknownand areused inevaluatingthestressesat therequired points.

-29-


51/234

2.9 SummaryInconclusion,theformulationofthis method isbasedon a

variational principle which requiresthespecificationofdisplacements(onelement boundaries)andstresses (within elements)intermsoftrial functions whichareknown elasticity solutions. Tractionsonpartof theboundary (inthis casethehole boundary)may bespecifiedusing another elasticity solution,theloading function. Variationofthefunctional with respectto theunknownsof theproblem yieldsasystemofsimultaneous equations. Some line integralsof thetrialfunctionsandloading functions mustbeevaluatedfor thecoefficientsofthese equationsbut nointegrationson thehole boundaryarenecessaryas aresultof theloading function. Thestressat anypointin anelementmay bedetermined fromthesolutionto thesimultaneous equations,thetrialandloading functionsand thecorrection termsfor theelements which will have been evaluated inconstructingtheequation.

-30-


52/234

CHAPTER 3

TRIAL FUNCTIONSANDLOADING FUNCTION

3.1 IntroductionThetrial functionsfor usewith thefinite element superposition

method (FESM)arederived from two-dimensional elasticity solutionsforconfigurations which identically satisfy zero-traction conditionsattheholeornotch wherethestress concentrationis to becalculated.Boundary conditions remote from theholeare notspecified for thetrialfunctions. If thehole isloaded insome mannerthetractionson thehole mustberepresentedby aloading function whichis theexactelasticity solutionfor aninfinitesheet withthesame hole and thesame loadingon theholeas theproblemto besolvedandzero stressremote from thehole. Thesuperpositionoftrial functionsandloadingfunction therefore satisfiestheboundary conditionson theholeexactly. For themethod towork welltheconstant strain finite elementfield, which ensures thattheremote boundary conditionsareapproximatelysatisfied, should introduce only small correctionstothat partof thesolution resulting fromtheloadingandtrial functions. Thusanimportant aspectof themethodis theselectionofsuitable trialfunctions.

Insection3.2 twotrial functionsaregivenforconfigurationswith elliptical (orcircular) holes. Amore generalset oftrial functionsforconfigurations with circular holesisgiven insection3.3 and theloading functionforcircular holesisgiveninsection3.4. Thetrialfunctionsandloading functionaredefined eitherincartesian coordinates(x,y) orpolar coordinates (r,8)relativetoaxes withtheorigin,0',at thecentreof thehole. Thefinite element geometry isdefined incartesian coordinates (X,Y)relativeto the"global" axes wiUhadifferentorigin0 anddifferent orientation. Thepositionof theholeand theangle,y,betweenthe OX and O'xaxes must thereforebespecified ineach case.

3.2 Trial functions forelliptical holesTwosatisfactory trial functionsmay beobtained forconfigurations

with ellipticalorcircular holes from theknown solutionfor auniformlystressed infinite sheet containingatraction-free hole [3.1].

-31-


53/234

Theinfinite region boundedby theellipse:

= 1 (3.1)

wherea and b are thesemi-majorandsemi-minor axesof theellipserespectively,may betransformedto theunit circle, |cl


54/234

Fortrial function i = 2:a R 1Ogfc) = [ Y - G(2 + M) ]

"zf 1 cVgfc) = - - - [ - ^ + p + pz + ;=) ] (3.7)

wherethesubscripti (i = 1 or 2)denotesthenumberof thetrial* *function. Thecomponentsof thetrial function stresses (a a . * * ri 01

and T ) anddisplacements (u , and u .), inpolar coordinates (r,e),01 1 01areobtained from thefollowing general formulae:

"ri * ^ "ei " - zOjfz)- 4^(z) ]o . + o . = 4 Re r o (z) 1ri 8 1 L 1 ^

*0^- - o . + 2i T n- = 2 e^^^ [z oV(z) +^^(z)] (3.8)ei ri rei L ? J

wherethe bar ( )here denotesthecomplex conjugate, prime (')denotesdifferentiation with respectto z and Redenotesthereal part. Also

K = (forplane stress)1 + vand G = (3.9)

wherev isPoisson's ratio,E isYoung's modulusand G is theshearmodulus.

3.3 Trial functionsforcircular holesInsection3.2 a set of twotrial functionsaregivenfor

configurations with traction-free ellipticalorcircular holes. Amoregeneralset offunctionsmay beobtainedforcircular holesbyusingthegeneralised solutionfor theAiry stress function, for the two-dimensional problem inpolar coordinates [3.2]. Thisis inseries form:

-33-


55/234

D = a log r + b + c r* log r + d r*8 + a'8A o o o o o^ 1 - 1+ r 8 sin 8+ (b + a^ r + b^r log r) cos

- 1- - r 8 cos 8+ (d + c^ r + d^r log r) sin

+ L - n , n+2 , -n -n+2,(a r + b r + a'r + b'r ) cos n_ n n n nn=2r / n n+2 n n+2, . , n\) (c r + d r + c r + d' r ) sin n 8 (3.10)n n n nn=2

wherea , b , c , d , a', b', c', d' (n = 0 to ) areconstantsn n n n n n n ndeterminedby theboundary conditionsof theproblem. Bylimitingtheproblemsto besolved tothose withatleastoneaxisofsymmetry(about9 = 0 say), thefollowing constantsmay beeliminated:

d = a' = c = d = c' = d' = 0 (n = 1 to ) (3.11)o o n n n n

since these coefficients would give riseto o&ymmetric stressesanddisplacements. Furthermore for thedisplacementsto besingle-valuedit isfound that:

c = 0oa (1-v)

and b^ = - (3.12)

Thesummationmay betruncated to afinite numberofterms,m, andthusthestress functionisreduced to:

aj a (1-v)= a^ log r + b r^ + r e sin e + (b^r' + - r log r) cos

r / n n+2 n , ,n+2, ,^ 1^\+ ) (a r + b r + a'r + b'r ) cos n 8 (3.13)n n n nn=2* * *Thestresses (o , o and T ) aregivenby:r 8 r@ ^ ^

-34-


56/234

* _ 1 2 ^ 1 ^r ' r ar * r* 98=

* ^8 = 3r=

V e = - & ' ; i T ) '3-^4'

which yield:

-K- _2 1 r -1 -3o = a r + 2b + -[a.(3+v)r + 8b_r - 8a'r J cosr o o 4 1 1 1

- /{n(n-l)ar +(n+l)(n-2)br+n(n+l)a'r +(n-l)(n+2)br } cos nEn n n nn=2

Og = -a r ^ + 2b + ^ [-a(l-v)r 24b^r+ 8a^r cos

m _ _+ y {n(n-l)ar" +(n+l)(n+2)b r"+n(n+l)a'r" +(n-l)(n-2)b'r"} cos nEn n n nn=:2

T = ^ [-a.(l-v)r^ + 8b.r - Ba'r ^ ] sinr0 4 1 1 1

m+ y {n(n-l)ar"^+n(n+l)b r"-n(n+l)a'r ^-n(n-l)b'r sin n0 (3.15)n n n nn=2

Bysubstituting intotheequations (3.15) Uhezero traction boundaryconditionson thehole,but not theremote boundary conditions, halfof theunknown coefficientsmay bedetermined. Theremaining coefficients,scaled to aconvenient level,maythenbeusedas thetrial functioncoefficients, denoted a ,whicharedetermined fromthevariationalprinciple (seesection2^^ . Thestress function givenbyequation (3.13)therefore yieldsa set of 2mtrial functions.

Theexpressionsfordisplacementsareobtained from thestresses* * *usingtheformulafor thestrain components E^, '

-35-


57/234

* 1 * *,E = ( O - V O ]

* 1 * *= 8 = E )

= & "re (3-16)

and thestrain-displacement relations:*

* 9urE'r 9r* *

E = +8 r r38* * *au^ 3Ug Ug

^re rae * ar r (3.17)

For anunloaded circular hole,ofradiusa, theboundary conditionson theholeare:

a (r = a) = 0r

(r = a) = 0 (3.18)

Thesemay besubstituted into (3.15)togivethefollowing expressions* * * * *for o o^-' t u . and u . whereidenotesthevariable associatedri 01 r8i ri eiwiththe i'thtrial function.

Fortrial functioni = 1:

"l az.rl = 2 (1 -

81 = 2 (1 + F')

rei

"rl = "1% [ (1-"^ a + ;

Ugi = 0 (3.19)

-36-


58/234

Fortrial functioni = 2:

* r

* 2 .3r a^.62 ' 1 * ?'

* .r a^. .'res = 3 'S - r-'* GpS 2 2"r2 = 8E" [(l-3v) + (1+v) +2(3+v)]cos 8

u82 " 8E a(5+v) + (1+v) -2(3+v)] sin 8 (3.20)

Fortrial function i =[2n-l],n)2:

ri2n_l]= ^^-2 (""1) " ^n+2 " ^n ("-2)] cos n8a r a

* ^\2n~l 1 . r " ^ r " - ,6[2-l = - 4 l-n-2 '"-1' - -n*2 " "n cos nea r a

Orn n-2 n+2 n[2n-lJ rr , .. a nr .'re[2n-l]= - 4 [-n -2 '""1' + "n^a " " n 1 " " ""a r a

3[2n-ll r"-l a"+l r"+l'rl2-l]= 4E n+1) ' 1 < ("n-l ' " n t l 1 -|n-2+v(n+2)] cos n0a r a

- 4 L cos e }n

aor _ n-1 n+1 n+1,[2n_l]=- 4E n+I) ("+ ) " ^n+l^ "["+4+"^]^n+l]a r a

- 4 L sin 9 }n(3.21)

-37-


59/234

Fortrial functioni = [2n], n)2:

au r 1 n-2 n+2 n

- 7 ^ [ - o + - o (n+1) - - (n+2)] cos nfr[2nj 4 n-2 n+2 na r r

Br 1 n-2 n+2 n9[2]' -n_ 2 + - -n "a r r

ar_ 1 n-2 n+2 n[2nj r r a , na i're[2l = - T - [ -n-2 " -t2 ^a r r^ aor 1 n-1 n+1 n-1

"r[2n]= - ""n+lf"- )]+[n+2+v(n-2)]:^_^]cos nEa r r+ 4 L cos 6 }n

* BOrp T n-l n+1 n-1"8[2n]= +^n+l("-l)]- [n-4+nv] sin n8a r r

+ 4 L sin 6 }n(3.22)

where L = 0 whenn isevennil 13-23)

L = (-1) 2 whenn is odd

In theabove expressionsthetrial function coefficients havebeen scaled such that whena. isunity themaximum hoop stressat thehole Og^ isalso unity. This isuseful inassessingtherelativeimportanceof thevarious trial functionsinparticular solutions.Thedisplacements givenby thetrial functionsaredefined such that* *u^. iszero when8 = 0 or n and u . iszeroon thehole boundaryGi 81when 8 = n/2.

Physically these trial functionsmay beunderstoodto besolutionsfor asheetorannular region containingatraction-free hole withvarious boundary conditions remote from thehole. Forexamplethesolutionfor aninfinite sheet withahole, stressedatinfinityintwodirections,as wasused insection3.2, may beobtainedfor a

-38-


60/234

circular hole from alinear combinationof thetrial functions1 and4 ofthis section. It may beseen from equations (3.19)to(3.22)that theseare theonly trial functions thatdo notgive large stressesas rtendstoinfinity. Forthis reasonit isfound that these functionshavethegreater contributionto thesolutionofconfigurations withspecial regions whicharelarge relativeto thesizeof thehole.

3.4 Loading functionforcircular holesForconfigurations with loaded holes,aloading function, mustbe

specified which satisfiesthetraction boundary conditionson theholeexactly (seeequation (2.13), section2.4). Theconstant coefficientoftheloading function, a , ishere assumedto be UBity and thetractionson theholearerepresented asFourier series. Thustheboundaryconditionsof theloading functionare:

/ A cos n8 at r = an=0o n n

T " 2V D sin n8 at r = a (3.24)r8o 2 nn=lwherem. and are thefinite limitsof theFourier seriesand A and1 2 nD are theknown coefficients, determined from thespecified tractionson thehole. Afurther requirementof theloading function isthatthestresses approach zero remote from thehole, i.e.:

* *G = T n = 0 as r + (3.25)ro r8o

With theboundary conditions (3.24)and (3.25)all thearbitraryconstantsofequation (3.15)may bedeterminedand theloadingfunctionisgivenby:

-39-


61/234


62/234

and m , thelimitofFourier series inthis case,isequalto thegreaterof m and (A^ = 0 for n > = 0 for n > m^).

TheFourier coefficientsA and D mustbespecified togivethen nmagnitudeanddistributionof theloadingon theholeandthisisdiscussed insection 5.4.1. Forseveral standard cases (e.g.cosinepressure distribution, constant pressure overan arc,combined cosinepressureandsine shear distributions) calculation of therequiredFourier coefficientshasbeen programmedon thecomputersothat onlythemagnitudeofresultant force needbespecified.

-41-


63/234

CHAPTER4

IMPLEMENTATIONOFFINITE ELEMENT METHOD

4.1 IntroductionThecomputer program developed from theformulationof the

method described inchapter2, isbasedon aprogram developedattheRoyal Aircraft Establishment, Farnborough (R.A.E.)[4.1, 4.2],for thedeterminationofstress intensity factorsatcracks, whichwasmade availablefor thepresent work. Theprogramwasimplementedon the ICL2970 computeratSouthampton Universityand hasbeensub-stantially extended, incorporatingthe newtrial functionsandloadingfunctionofchapter3, and therevisions necessary fortheiruse indetermining stress concentration factorsattraction-freeorloadedholes. Themodifications include chantgesincalculatingtheareasofelements,toallowfor thecurvatureof thehole,andmodificationsto thecontour integrals,asdiscussed inchapter2, due to theinclusionof theloading function. Up to 8trial functionsmay beused with thisversionof theprogramand up to atotalof 54Fourier coefficientsmay bespecifiedfor theloading function. Further extensionsto theprogram enable various additional meansofdata inputandoutputto beused, including graphical techniques.

Section4.2givesanoutlineof how themethod iscarriedout intheprogram. Specific reference ismadetoseven segmentsof theprogramwhicharelisted inAppendixB.Theseare:NOTCH, AREAS, TRLFNSl, TRLFNS2,LOADFN, B

Numerical Methods for Stress Analysis Using Known Elasticity Solution - A. R. CHARMICHAEL

Documents

Transcript of Numerical Methods for Stress Analysis Using Known Elasticity Solution - A. R. CHARMICHAEL