Boundary Element Analysis of a Curved Tubing with a Semi

Boundary Element Analysis of a Curved Tubing with a Semi-Elliptical Crack

by

Kirsten Irene Plante

A Thesis submitted to

the Faculty of Graduate Studies and Research

in partial fulfilment of

the requirements for the degree of

Master of Applied Science

Ottawa-Carleton Institute for

Mechanical and Aerospace Engineering

Department of Mechanical and Aerospace Engineering

Carleton University

Ottawa, Ontario, Canada

June 2008

Copyright ©

2008 - Kirsten Irene Plante

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Abstract

The three-dimensional boundary element method has been determined to be an effi

cient computational method for solving elastostatic problems, particularly those for

cracked geometries. For this reason, it is employed for the analysis of thick-walled

internally pressurized curved tubing. The stress distribution in a pipe bend is inves

tigated with respect to bend radius ratio, radius ratio of the cross-section, and angle

of curvature. Upon verifying the intrados as the most likely site for crack nucleation,

independent of the geometric parameters, a semi-elliptical crack of semi-minor to

semi-major axis ratio b/a=0.S is introduced. Stress intensity factors are determined

along the crack periphery, </>, for the internal pressure case, with pressure acting on

the crack faces.

Polynomial influence coefficients arc computed for all instances and used to verify

the stress intensity factors for the previous load case; they are shown to agree to

within 2.5% difference. Influence coefficients may be combined through superposition

to determine the stress intensity factor for any load case with the same cracked

geometry. This is demonstrated using a representative set of models for various levels

of overstrain due to partial autofrettage.

The stress factor in an uncracked internally pressurized pipe bend is observed to

increase for smaller bend radii, lower radius ratios and larger angles of curvature. The

effect of the longitudinal angle is minimal and becomes negligible at the larger bend

radius. Once a semi-elliptical crack is introduced, with pressure acting on the crack

111

faces, the normalized stress intensity factor results obtained are consistent with these

observations. Furthermore, it is found that the normalized stress intensity factor

is approximately constant up to 0=60°, after which it increases rapidly as the free

surface is approached.

IV

Acknowledgments

The author would like to acknowledge her thesis supervisor, Professor C.L. Tan, for

his patience and guidance throughout this work, as well as the kind assistance of

Fiona Warman, Michelle Thompson and Peter Klimas.

V

Table of Contents

Abstract iii

Acknowledgments v

Table of Contents vi

List of Tables ix

List of Figures xxii

Nomenclature xxvi

1 Introduction 1

1.1 Previous Work 1

1.2 Proposed Solution Method 3

1.3 Outline 3

2 Review of the Boundary Element Method 5

2.1 Formulation of the BEM in Three-Dimensions 5

2.1.1 Numerical Treatment of the BEM in Three-Dimensions . . . 8

2.1.2 Example Problem - Stress Concentrations 10

2.2 Fracture Problems 11

2.2.1 Crack-Front Boundary Elements 13

vi

2.3 Numerical Examples - Fracture Problems 16

2.3.1 Circular Embedded Crack in an Infinite Solid 16

2.3.2 Cylinder with Semi-Elliptical Crack 17

2.4 Influence Function Method for Obtaining Stress Intensity Factors . . 19

2.4.1 Numerical example 1 20

2.4.2 Numerical example 2 22

2.5 Conclusions 25

3 Stress Distributions in Pipe Bends 50

3.1 Numerical Model 51

3.2 Numerical Results 51

3.3 Conclusions 53

4 Stress Intensity Factors for a Crack in a Pipe Bend 68

4.1 Semi-Elliptical Crack In a Pipe Bend 68

4.1.1 Modeling Considerations 69

4.1.2 Mesh Refinement Study 70

4.1.3 Results 71

4.2 Stress Intensity Factors from Polynomial Influence Coefficients . . . . 73

4.2.1 Modelling Considerations 73

4.2.2 Verification 74

4.2.3 Application: Effect of Residual Stresses from Autofrcttage . . 75

4.3 Conclusions 77

5 Conclusions 100

References 102

vii

Appendix A Shape Functions 104

A.l Shape Functions for Triangular Elements 104

A.2 Shape Functions for Quadrilateral Elements 105

A.3 Shape Functions for Crack-Front Elements 106

Appendix B Stress Intensity Factors for a Crack in a Pipe Bend 110

Appendix C Influence Coefficients and Stress Intensity Factors for a

Crack in a Pipe Bend 120

C.l Influence Coefficients 120

C.2 Stress Intensity Factors from Influence Coefficients 148

Appendix D Stress Intensity Factors due to Residual Stresses from

Autofrettage 176

vm

List of Tables

2.1 Variation of stress concentration factor at side of circular cylindrical

hole in infinite plate of finite thickness with position through the plate

thickness 26

2.2 Comparison of normalized stress intensity factor, K*T = K)/'(|CT\/Wa),

for the penny-shaped crack problem 26

2.3 Normalized influence coefficients, xf]* = Kf]/[A^b/WY^], for

k=2, b/W=0A 27

2.4 Comparison of normalized stress intensity factors, Ki/(P\firt>), for

k=2, b/W=0A 27

4.1 Corrected values of 4> at nodal points on the semi-elliptical crack-front

in a curved tubing 79

4.2 Normalized stress intensity factors, ^ / / (FV^ri) , for R*=7.5, 9=90°

for simple mesh refinement convergence study 80

4.3 Normalized stress intensity factors, K[/(PV^b), at nodal points along

the semi-elliptical crack-front, as defined by <f), in a pressurized curved

tubing for R*=5, k=2.5, 9=90° 81

4.4 Normalized stress intensity factors, Kj/(Py/nb), at nodal points along

the semi-elliptical crack-front, as defined by (f>, in a pressurized curved

tubing for /T=5, k=2 82

IX

4.5 Polynomial coefficients to define hoop stress distribution in internally

pressurized uncracked curved tubing with R*=5 83


pressurized uncracked curved tubing with R*=7.b 83


pressurized uncracked curved tubing with R*=10 84

4.8 Normalized influence function coefficients, K\ —

Kj l[Ai{b/W)l\fixb], at nodal points along a semi-elliptical crack-front

as defined by 0 for R*=b, k=2, 9=90° 85

4.9 Comparison of normalized stress intensity factors, Ki/(PVTrb), ob

tained from the influence coefficient method and by direct BEM along

crack periphery, 4>, for R*=5, k=2, 0=90 ° 86

4.10 Comparison of normalized stress intensity factors, K//(Pv7r&), from

influence coefficient method and direct BEM along crack periphery, 0,

for R*=10, k=1.5, 0=90° 87

4.11 Normalized stress intensity factors, (/f/)yi/[cry\/7r6], from residual

stress due to autofrettage to different levels of overstrain with R*=5,

k=2, 0 = 90°, b/W=0.6 88

4.12 Normalized stress intensity factors, (/f/)>t/[0Yv/T&], from residual

stress due to autofrettage to different levels of overstrain with R*=7.5,

k=2, 9 = 90°, b/W=0.6 89

4.13 Normalized stress intensity factors, (/f/Wfow/vrfc], from residual

stress due to autofrettage to different levels of overstrain with i?*=10,

jfc=2, 6 = 90°, b/W=0.6 90

B.l Normalized stress intensity factors, Ki/{Pyfid>), at nodal points along

the semi-elliptical crack-front, as defined by 0, in a pressurized curved

tubing for iT=5, k=\.h I l l

B.2 Normalized stress intensity factors, Kil(P\fnb), at nodal points along

the semi-elliptical crack-front, as defined by (ft, in a pressurized curved

tubing for R*=b, k=2 112

B.3 Normalized stress intensity factors, Kj/(P\/nb), at nodal points along


tubing for R*=5, fc=2.5 113

B.4 Normalized stress intensity factors, Ki/(Pyfnb), at nodal points along


tubing for i?*=7.5, fc=1.5 114

B.5 Normalized stress intensity factors, Kj/(P\fivb), at nodal points along


tubing for R*=7.b, k=2 115

B.6 Normalized stress intensity factors, Kj/(Py/nb), at nodal points along


tubing for R*=7.b, fc=2.5 116

B.7 Normalized stress intensity factors, K[/(Py/Trb), at nodal points along


tubing for R*=10, fc=1.5 117

B.8 Normalized stress intensity factors, Ki/(Py/irb), at nodal points along


tubing for R*=10, k=2 118

B.9 Normalized stress intensity factors, Ki/{Pyfnb), at nodal points along


tubing for R*=\Q, k=2.b 119

C.l Normalized influence function coefficients. K} =

K\ /[Ai(b/Wy\fnb], at nodal points along a semi-elliptical crack-front

as defined by (ft for R*=b, k=1.5, 9=45° 121

XI

C.2 Normalized influence function coefficients. K} =

Kj /\Ai(b/W)lyfnb], at nodal points along a semi-elliptical crack-front

as defined by </> for R?=5, fc=1.5, 61-67.5 ° 122

C.3 Normalized influence function coefficients. Kj* =

Kjl /[Ai(b/W)lvnb], at nodal points along a semi-elliptical crack-front

as defined by 4> for R*=b, fc=1.5, 0=90° 123

C.4 Normalized influence function coefficients. Kj =

Kj /[Ai(b/W)l\fKb], at nodal points along a semi-elliptical crack-front

as defined by 4> for R*=5, k=2, 6>=45° 124

C.5 Normalized influence function coefficients. Kj * =

Kj I[Ai(b/Wy Vnb], at nodal points along a semi-elliptical crack-front

as defined by =67.5° 125

C.6 Normalized influence function coefficients. Kj* =

Kj /[Ai(b/Wy V7r6], at nodal points along a semi-elliptical crack-front

as defined by =90° 126


Kj /[Ai{b/W)l\fnb], at nodal points along a semi-elliptical crack-front

as defined by <f> for RT=b, fc=2.5, 0=45° 127

C.8 Normalized influence function coefficients. Kj —

Kj /[Ai(b/W)%V7r6], at nodal points along a semi-elliptical crack-front

as defined by </> for i?*=5, fc=2.5, 9=67.5° 128


Kj /[Ai(b/WyVnb], at nodal points along a semi-elliptical crack-front

as defined by tf> for R*=5, jfc=2.5, 0=90° 129


K) /[Ai(b/Wy\/Ttb\, at nodal points along a semi-elliptical crack-front

as defined by (p for R*=7.5, Jfc=1.5, 6>=45° 130

xn

C.ll Normalized influence function coefficients. Kj =

Kj /[Ai(b/W)xVTrb], at nodal points along a semi-elliptical crack-front

as defined by (j> for #*=7.5, fc=1.5, 0=67.5° 131


Kj/[Ai(b/W)ly/Trb], at nodal points along a semi-elliptical crack-front

as defined by <f> for R*=7.5, k=l.b, 9=90° 132

C.13 Normalized influence function coefficients, Kj =

Kj1'/[Ai(b/W)l\/7ib], at nodal points along a semi-elliptical crack-front

as defined by for R*=7.5, jfc=2, 0=67.5° 134


Kj /[A^b/Wyyfnb], at nodal points along a semi-elliptical crack-front

as defined by <f> for /{*=7.5, k=2, 0=90° 135


K\x)/[Aiib/wyy/vb], at nodal points along a semi-elliptical crack-front

as defined by <f> for R*=7.b, fc=2.5, 0=45° 136

C. 17 Normalized influence function coefficients, Kj* =

Kj /[Ai(b/W)'\/nb], at nodal points along a semi-elliptical crack-front

as defined by for 7?*=7.5,/c=2.5, 0=90° 138


Kj /[Ai(b/\Vyy/Ttb], at nodal points along a semi-elliptical crack-front

as defined by (p for R*=1Q, fc=1.5, 0=45° 139

xi n

C.20 Normalized influence function coefficients, K} =

Kj /[Ai(b/Wyy/nb], at nodal points along a semi-elliptical crack-front

as defined by </> for iT=10, fc=1.5, 0=67.5° 140


K} /[Ai(b/W)ly/nb], at nodal points along a semi-elliptical crack-front

as defined by ^ for R*=10, fc=1.5, 0=90° 141

C.22 Normalized influence function coefficients, Ky =

K) /\Ai{b/W)%yfnb^\, at nodal points along a semi-elliptical crack-front

as defined by <p for i?*=10, k=2, 9=45° 142

C.23 Normalized influence function coefficients, K, =

Kj /[Ai(b/W)%virb], at nodal points along a semi-elliptical crack-front

as defined by for #*=10, fc=2, 0=90° 144

C.25 Normalized influence function coefficients, Kj* =

Kj /[Ai(b/WyVnb], at nodal points along a semi-elliptical crack-front

as defined by 0 for iT=10, jfc=2.5, 0=45° 145

C.26 Normalized influence function coefficients. K}1'* —

K) /[Az(b/Wy\/nb], at nodal points along a semi-elliptical crack-front

as defined by cp for R*=10, jfc=2.5, 0=67.5° 146


K}1>/[Ai(b/Wy vnb], at nodal points along a semi-elliptical crack-front

as defined by <p for R*=1Q, fc=2.5, 0=90° 147

C.28 Comparison of normalized stress intensity factors, Kj/(Py/^b), ob


crack periphery, 0, for R*=5, k=1.5, 0=45° 149

xiv

C.29 Comparison of normalized stress intensity factors, Ki/{P\fnb), ob


crack periphery, (p, for R*=5, Ar=1.5, #=67.5° 150

C.30 Comparison of normalized stress intensity factors, Kjl{P\fnb), ob


crack periphery, <f>, for R*=5, fc=1.5, 0=90° 151

C.31 Comparison of normalized stress intensity factors, Ki/(Pvnb), ob


crack periphery, 0, for R*=5, k=2, 6>=45° 152

C.32 Comparison of normalized stress intensity factors, KI/(PVTT&), ob


crack periphery, <f>, for R*=5, k=2, 9=67.5° 153

C.33 Comparison of normalized stress intensity factors, Ki/(P\fnb). ob


crack periphery, <f>, for #*=5, k=2, 9=90° 154

C.34 Comparison of normalized stress intensity factors, Ki/(P\fnb). ob


crack periphery, (f>, for i?*=5, A;=2.5, 0=45° 155

C.35 Comparison of normalized stress intensity factors, Ki/(P\nrt>), ob


crack periphery, 0, for R*=5, jfc=2.5, 0=67.5° 156



crack periphery, 0, for R*=b, Ar=2.5, 6>=90° 157

C.37 Comparison of normalized stress intensity factors, Kj/(PV^b). ob


crack periphery, 0, for R*=7.5. fr=1.5, 6>=45° 158

xv

C.38 Comparison of normalized stress intensity factors, Kj/(PvTtb), ob


crack periphery, </>, for R*=7.5, Jfc=1.5, 0=67.5° 159



crack periphery, </>, for #*=7.5, fc=1.5, 0=90° 160

C.40 Comparison of normalized stress intensity factors, Kj/(Py/Trb), ob


crack periphery, 0, for R*=7.b, k=2, 0=45° 161

C.41 Comparison of normalized stress intensity factors, K]/(P\rnb), ob


crack periphery, 0, for fl*=7.5, k=2, 0=67.5° 162

C.42 Comparison of normalized stress intensity factors, Ki/{P\fnb), ob


crack periphery, <f>, for R*=7.b, k=2, 0=90° 163

C.43 Comparison of normalized stress intensity factors, K//(Pv7r6), ob


crack periphery, <j>, for J?'=7.5, fc=2.5, 0=45° 164

C.44 Comparison of normalized stress intensity factors, Ki/(P\rrrb), ob


crack periphery, <f>, for R*=7.5, jfc=2.5, 0=67.5° 165

C.45 Comparison of normalized stress intensity factors, Kj/{P\Txb), ob


crack periphery, <p, for R*=7.5, fc=2.5, 0=90° 166



crack periphery, <p, for i?*=10, £=1.5, 0=45° 167

xvi

C.47 Comparison of normalized stress intensity factors, Kj/(Pyirt)), ob


crack periphery, 0, for R*=10, k=1.5, 6=67.5° 168

C.48 Comparison of normalized stress intensity factors, Ki/(P\fnb), ob


crack periphery, </>, for R*=10, fc=1.5, 0=90° 169

C.49 Comparison of normalized stress intensity factors, Kj/(Py/nb), ob


crack periphery, 0, for R*=10, k=2, 6=45° 170

C.50 Comparison of normalized stress intensity factors, Ki/(P\rrrb), ob


crack periphery, <f>, for R*=10, k=2, 0=67.5° 171

C.51 Comparison of normalized stress intensity factors, Kj/(PyTrb), ob


crack periphery, <f>, for fl*=10, k=2, 0=90° 172

C.52 Comparison of normalized stress intensity factors, Kj/(PVnb), ob


crack periphery, </>, for R*=10, jfc=2.5, 0=45° 173

C.53 Comparison of normalized stress intensity factors, Kj/(P\nd>), ob


crack periphery, <£, for R*=10, ]fc=2.5, 0-67.5° 174

C.54 Comparison of normalized stress intensity factors, Ki/(PvTtb), ob


crack periphery, <j>, for fl*=10, )fc=2.5, 0=90° 175

D.l Normalized stress intensity factors, (/OWI^Yv^fr], from residual


k=2, 0 = 45°, b/W=0.2 177

xvii

D.2 Normalized stress intensity factors, (•K"/W[oYV7rb], fr°m residual


k=2, 6 = 45°, b/W=0A 178

D.3 Normalized stress intensity factors, (.K/)J4/[oY\/7r&]) from residual


k=2, 6 = 45°, b/W=0.6 179

D.4 Normalized stress intensity factors, (KÂ/loYy/^b], from residual


k=2, 0 = 45°, b/W=0.8 180

D.5 Normalized stress intensity factors, (KJ)A/[(JYV^M, from residual


fc=2, 6 = 67.5°, b/W=0.2 181

D.6 Normalized stress intensity factors, (Kj)A/[aY\fnb], from residual


fe=2, 0 = 67.5°, b/W=0A 182

D.7 Normalized stress intensity factors, (/(rj)i4/[oY-v/7r&], from residual

stress due to autofrettage to different levels of overstrain with R*=b,

fc=2, 6 = 67.5°, b/W=0.6 183

D.8 Normalized stress intensity factors, {KÂ/IOYVT^]-, from residual


k=2, 6 = 67.5°, b/W=0.8 184

D.9 Normalized stress intensity factors, (A'/)/i/[cry-\/7r6], from residual


k=2, 6 = 90°, 6/1^=0.2 185

D. 10 Normalized stress intensity factors, {KJ)A/[OY virb], from residual


k=2, 9 = 90°, b/W=0A 186

X V l l l

D.l l Normalized stress intensity factors, {Kj)A/[oY\rKb), from residual

stress due to autofrettage to different levels of overstrain with R*=b,

k=2, 9 = 90°, 6 /^=0 .6 187

D. 12 Normalized stress intensity factors, (K/)^/[ay\/7r6], from residual


k=2, 0 = 90°, b/W=0.8 188

D. 13 Normalized stress intensity factors, {Ki)A/[aYV^b], from residual

stress due to autofrettage to different levels of overstrain with li*=7.5,

k=2, 0 = 45°, b/W=0.2 189

D. 14 Normalized stress intensity factors, (ZOWI/'vV'^L from residual


k=2, 6 = 45°, b/W=0A 190

D. 15 Normalized stress intensity factors, {Kj)A/[aYV^b\, from residual


Jfc=2, 6 = 45°, b/W=0.6 191

D. 16 Normalized stress intensity factors, (/0W[oYV/7rfr], from residual

stress due to autofrettage to different levels of overstrain with i?*=7.5,

fc=2, 6 = 45°, b/W=0.8 192

D. 17 Normalized stress intensity factors, (Ki)A/[(TY^/nb], from residual

stress due to autofrettage to different levels of overstrain with i?*=7.5,

fc=2, 6 = 67.5°, b/W=0.2 193

D. 18 Normalized stress intensity factors, (K])A/[aY\/nb], from residual

stress due to autofrettage to different levels of overstrain with R*=7.h,

k=2, 9 = 67.5°, b/W=0A 194

D. 19 Normalized stress intensity factors, (A'/)4/[ayV/7r6], from residual

stress due to autofrettage to different levels of overstrain with /?*=7.5,

k=2, 6 = 67.5°, b/W=0.6 195

xix

D.20 Normalized stress intensity factors, (KI)A/\<JYy/nb], from residual


k=2, 6 = 67.5°, b/W=0.S 196

D.21 Normalized stress intensity factors, (Ki)A/[(TyV^b], from residual

stress due to autofrettage to different levels of overstrain with R*=7.b,

k=2, 0 = 90°, b/W=0.2 197

D.22 Normalized stress intensity factors, {Kj)Al[oY\fnb\, from residual


k=2, 0 = 90°, b/W=0A 198

D.23 Normalized stress intensity factors, (K{)A/[oyV^b], from residual

stress due to autofrettage to different levels of overstrain with R*=7.b,

k=2, 0 = 90°, b/W=0.6 199

D.24 Normalized stress intensity factors, (A'/)yt/[oy\/7r&], from residual


k=2, 0 = 90°, b/W=0.8 200

D.25 Normalized stress intensity factors, (K'/)/i/[o'yv/7r6], from residual


fc=2, 0 = 45°, 6 /^=0 .2 201

D.26 Normalized stress intensity factors, (Kj)A/[(Tyynrb], from residual


k=2, 0 = 45°, 6/^=0-4 202

D.27Normalized stress intensity factors, {K})Al[oy\fnb\, from residual

stress due to autofrettage to different levels of overstrain with Z?*=10,

k=2, 9 = 45°, b/W=0.6 203

D.28 Normalized stress intensity factors, (Kj)A/[cry/Kb], from residual


k=2, 0 = 45°, 6/Vv'=0.8 204

X X

D.29 Normalized stress intensity factors, {K])A/[ay V7r6], from residual


k=2, 6 = 67.5°, 6 /^=0.2 205

D.30 Normalized stress intensity factors, (K])A/[crY\/nb\, from residual


jfc=2, 6 = 67.5 °, b/W=0A 206

D.31 Normalized stress intensity factors, (KJ)A/[CTYV7r6], from residual


k=2, 9 = 67.5°, b/W=0.6 207

D.32 Normalized stress intensity factors, (Ki) A/[cry's/Kb], from residual


k=2, 6 = 67.5°, b/W=0.8 208

D.33 Normalized stress intensity factors, (A'/)J4/[ay\/7r6], from residual

stress due to autofrettage to different levels of overstrain with Z?*=10,

k=2, 9 = 90°, b/W=0.2 209

D.34 Normalized stress intensity factors, (/^^/[oyv^rfc], from residual


k=2, 6 = 90°, 6 /^=0.4 210

D.35 Normalized stress intensity factors, (/Cj),4/[ay \/7r6], from residual

stress due to autofrettage to different levels of overstrain with /?*=10,

k=2, 6 = 90°, 6 /^=0 .6 211

D.36 Normalized stress intensity factors, (K/)yi/[ay\/7r6], from residual


fc=2, 6 = 90°, b/W=0.8 212

x x i

List of Figures

2.1 Boundary value problem in elastostatics 28

2.2 Six node triangular element 29

2.3 Eight node quadrilateral element 30

2.4 Circular hole in an "infinite" plate of finite thickness 31

2.5 Discretization of the problem of a circular hole in an "infinite" plate

of finite thickness 32

2.6 Variation of stress concentration factor, <Ty/u, through plate thickness 33

2.7 Three displacement modes for crack surfaces 34

2.8 Crack-tip stresses shown with the Cartesian coordinate system . . . 35

2.9 Crack-front element 36

2.10 BEM mesh for problem of a circular embedded crack in an "infinite"

solid 37

2.11 Semi-elliptical surface crack in a thick-walled cylinder 38

2.12 BEM mesh for cylinder with an elliptical crack, fc=2.5, b/W=0A . . 39

2.13 Normalized stress intensity factors, Kj^P^fnb), for k=2, b/W=0A . 40

2.14 Normalized stress intensity factors, KrfiPy/irb), for fc=2.5, b/W=0.2 41

2.15 Normalized stress intensity factors, K]/(P\fnb), for fc=2.5, b/W=0A 42

2.16 Normalized stress intensity factors, K]/(Py/Trb), for fc=2.5, b/W=0.6 43

2.17 Principle of the weight function method 44

xxn

2.18 Normalized stress intensity factors, Kj/(Py/nb), using influence coef

ficients for k=2, b/W=0A 45

2.19 Typical residual hoop stress distribution in an autofrettaged cylinder 46

2.20 Residual hoop stress, a0, through wall thickness of cylinder autofret

taged to 50% overstrain 47

2.21 Normalized stress intensity factors, K//(ayy/nb), for 20% overstrain

using influence coefficients for k=2, b/W=0A 48

2.22 Normalized stress intensity factors, Ki/(ayy/nb), for 50% overstrain

using influence coefficients for k=2, b/W=0A 49

3.1 Curved tubing diagram 54

3.2 Quarter-model mesh for R*=5, fc=2, 0=90° 55

3.3 Normalized hoop stress, a^/a0, at the inner radius R\ at the intrados

along the circumferential direction, 9, of the pipe bend R*=b, k=1.5,

9=90° 56

3.4 Normalized hoop stress, efy/<70, at the inner radius Ri at the intrados

along the circumferential direction, 9, for i?*=7.5, k=\.5, 0=90° . . 57

3.5 Normalized hoop stress, cr^./a0, for R*=5, k—2, 0=90° in circumferen

tial direction, 9, at intrados and extrados 58

3.6 Normalized hoop stress, a$/a0, at the intrados along the circumferen

tial direction, 9, for pipe bends R*=5, 9=90° 59

3.7 Normalized hoop stress, cr^/a0, at the intrados along the circumferen

tial direction, 9, for pipe bends R*—7.b, 9=90° 60

3.8 Normalized hoop stress, a^./a0, at the intrados along the circumferen

tial direction, 9, for pipe bends i?*=10, 0=90° 61

3.9 Normalized hoop stress, a^/cr0, at the intrados along the circumferen

tial direction, 9, for pipe bends k=l.5, 0=90° 62

xxiii

3.10 Normalized hoop stress, a^/a0, at the intrados along the circumferen

tial direction, 0, for pipe bends k=2, 0=90° 63

3.11 Normalized hoop stress, a^/a0, at the intrados along the circumferen

tial direction, 9, for pipe bends k=2.5, #=90° 64

3.12 Normalized hoop stress, <r^/a0, at the intrados along the circumferen

tial direction, 0, for pipe bends R*=b, k=2 65

3.13 Normalized hoop stress, a^/ag, at the intrados along the circumferen

tial direction, 0, for pipe bends 7T=7.5, fc=2 66

3.14 Normalized hoop stress, a^/a0: at the intrados along the circumferen

tial direction, 0, for pipe bends R*=10, k=2 67

4.1 Semi-elliptical crack in curved tubing 91

4.2 Sample BEM mesh for R*=7.b, k=2.5, 0=67.5°, b/W=0A with 112

elements and 332 nodes 92

4.3 BEM discretization for mesh refinement study 93

4.4 Variation of the normalized stress intensity factor, Ki/(P\/Txb), along

the crack-front, 0, of pressurized curved tubing, i?*=5, fc=1.5,

b/W=0.6 94

4.5 Variation of the normalized stress intensity factor, Kj/(P\/TTb), along

the crack-front, (p, of pressurized curved tubing, R*=5, k=2, b/W=0.6 95

4.6 Variation of the normalized stress intensity factor, K]/{P\rnb), along

the crack-front, <j>, of pressurized curved tubing, R*=b, A:=2.5,

b/W=0.6 96

4.7 Variation of the normalized stress intensity factor, Kj/{P\fnb), along

the crack-front, <j>, of pressurized curved tubing, R*=7.5, k=2.5,

b/W=0.6 97

XXIV

4.8 Variation of the normalized stress intensity factor, Ki/(FJy/nb), along

the crack-front, 0, of pressurized curved tubing, R*=10, k=2.5,

b/W=0.6 98

4.9 Variation of the normalized stress intensity factor, KiKPyfUb), along

the crack-front, <f>, of pressurized curved tubing, R*=5, k=2, 0=90° . 99

xxv

Nomenclature

Arabic Letters

At polynomial coefficients

E Young's modulus

G Shear modulus

J Jacobian of transformation

Kj mode I stress intensity factor

K) polynomial influence coefficients

Nld, N} displacement and traction shape function respectively at ith node

P uniform hydrostatic pressure

Q surface field point

R radius of curvature

Ri inner radius

R.2 outer radius

R* R/Ri ratio

W wall thickness

semi-major elliptical axis

half-crack length

b semi-minor elliptical axis

k R2/R\ ratio

I length of crack tip element

rij unit outward normal vector

nRi depth of plastic region

p load point

q field point

distance from load point to field point r

cylindrical and toroidal coordinate radial direction rs distance through thickness from internal radius

xxvi

tractions

displacements

boundary of a body

domain of a body

Kronecker delta

strain tensor

toroidal coordinate longitudinal angle

cylindrical coordinate circumferential angle

plane stress or plane strain

Poisson's ratio

intrinsic local coordinate

applied stress

yield stress

stress tensor

angle of crack profile

toroidal coordinate circumferential angle

xxvn

Chapter 1

Introduction

With advancing technology, pipelines and piping components are carrying higher

pressure loads, and thus thick-walled tubing is increasingly being used. The most

susceptible point of any pipeline is within curved sections and therefore the stress

distribution throughout this area is investigated. Due to the vulnerability of these

components to cracking, the stress intensity factors for a semi-elliptical crack are de

termined for a range of parameters. To increase the versatility of the stress intensity

factor results, a full list of polynomial influence coefficients is also presented. These

may be applied to any load case with the same cracked geometry and this is demon

strated with an analysis of partially autofrettaged curved tubing. Autofrettage is a

technique used to extend the fatigue life of process piping and weaponry by imparting

beneficial residual compressive stresses.

1.1 Previous Work

A number of analyses have been conducted on semi-elliptical cracks in thick-walled

internally pressurized cylinders. Varying in crack geometry or other parameters in

vestigated, this has been studied by Tan & Fenner (1980), Nadiri, Tan & Fenner

(1982), and Shim (1986) by employing the boundary element method (BEM), as well

1

2

as Blackburn & Hellen (1977), Atluri & Kathiresan (1979), and Pu k Hussain (1981)

using the finite element method (FEM). This has been limited to straight circular

cylinders however.

Early experimental work on stresses in thick-walled pipe bends was performed

by Swanson & Ford (1959) under in-plane bending and internal pressure. This

study compared the experimental results from cold-rolled pipe bends with previously

published theoretical analyses by von Karman (1911), Barthelemy (1947), Thuloup

(1937), and Turner & Ford (1957), none of which applied specifically to thick-walled

pressurized pipe bends, however.

Further analysis of stress distributions in pressurized elbows and pipe bends was

presented by Lang (1984), and introduced as Toroidal Elasticity theory. The approx

imate solutions obtained were in the form of series. This was applied to a torus by

Colle, Redekop & Tan (1987) who also employed the BEM in three-dimensions to ver

ify Lang's series solution. Toroidal Elasticity theory and its use for pressurized elbows

was limited to uncracked geometries under various loading conditions, and the BEM

work of Colle et al. (1987) on a torus was also restricted to the stress concentration

problem of a cross-bore without any cracks present.

The autofrettage of pipe bends was analyzed by Rees (2004) but was similarly

limited to uncracked geometries; this work entailed bending of a pre-autofrettaged

tube and investigated the stresses created by this process. The current work assumes

a pipe bend manufactured as is, with no pre-stresses present, nor cross-sectional non-

uniformities such as localized wall thinning. It should be noted that when subjected

to internal pressure, these defects, particularly local flattening, can result in large

stresses.

3

1.2 Proposed Solution Method

Due to the complexity of the geometry, no exact theoretical solution for the physical

problem of a thick-walled pressurized curved tubing is available; this is particularly

so when a crack is present. Recourse to numerical methods is commonly adopted in

such circumstances and an established computational technique for stress analysis is

the boundary element method.

The boundary element method (BEM) was selected as the solution method for

the physical problem due to its efficiency with problems with high stress gradients

present such as those near stress concentrations and cracks. The BEM differs from the

well-known finite element method (FEM) in that for a three-dimensional component,

only the surface needs be discretized into elements as opposed to the entire volume;

if required, interior points may be solved for separately as a secondary exercise. The

treatment of only the surface instead of the volume of the problem effectively reduces

the numerical dimension of the problem by one. Thus, the number of final algebraic

equations to be solved may commonly be reduced by at least one order of magnitude,

thereby reducing data preparation and solution times, particularly for complicated

geometries.

1.3 Outline

A brief introduction to the BEM is presented in the next chapter. In addition the the

ory of linear elastic fracture mechanics and the weight function method, particularly

polynomial influence coefficients, for use with determining stress intensity factors for

cracked components will also be reviewed. These influence coefficients may be applied

for any load case with the same crack geometry. A cylinder which has been subjected

to partial autofrettage is introduced to demonstrate this concept.

4

The stress distribution in an uncracked, internally pressurized thick-walled pipe

bend is studied in Chapter 3. The location of the peak stress in such a component is

identified; it is where a crack would most likely develop in the physical problem. To

this end, stress factors are determined for a range of a number of parameters, namely

bend radius, radius ratio of the cross-section, and angle of curvature.

Once the most likely site for crack nucleation has been determined, in Chapter 4,

a semi-elliptical crack is introduced for the fracture mechanics analysis into all the

previous uncracked models mentioned above. Only one crack shape, namely, that

corresponding to a semi-minor to semi-major axis ratio, b/a, of 0.8 is chosen for this

study. The primary reason for this is that it would allow direct comparisons with

similar results which are available in the literature for a straight tubing as this crack

shape is very commonly found in thick-walled cylinders of radius ratios between 2

and 3. Stress intensity factors are determined along the crack periphery for each

case. To lend more versatility to the results, polynomial influence coefficients are

also determined for each instance. They are used to verify the stress intensity factors

computed directly from the BEM. Finally, the application of these influence coeffi

cients is demonstrated by determining the stress intensity factors due to the residual

stresses in an autofrettaged tubing which has been subjected to various percentages

of overstrain in a representative set of the numerical models.

Chapter 2

Review of the Boundary Element Method

The boundary element method (BEM) is a powerful numerical tool for solving elas-

tostatic problems, particularly those with high stress gradients. Its analytical and

numerical formulation are widely published in the literature (see, e.g. Becker (1992))

and therefore only a brief review is presented here. The BEM is particularly efficient

for obtaining accurate solutions to problems with stress concentrations; it can also

be applied to linear elastic fracture mechanics analysis for obtaining stress intensity

factors with minimal modifications.

2.1 Formulation of the BEM in Three-Dimensions

Using Einstein notation where repeated indices denote summation and a comma index

is used to represent the derivative, the stresses in a linear elastic solid domain, £1, in

static equilibrium are governed by

aijj + fi = 0 in R; i,j = 1, 2, 3 in three-dimensions (2.1)

5

6

where cr is the second order stress tensor and fa are the components of the body force

vector per unit volume. In addition, the strain-displacement relationship is given by

eu = iMij + ui,i) (2-2)

where e^ is the strain tensor and u; is the displacement component in the x r direction.

Furthermore, Hooke's law is denned as

Uij = 2G(e0- + _ 2 **fc%) (2-3)

where 6{j is the Kronecker delta, v is the Poisson's ratio and G is the shear modulus.

Substituting Equation 2.3 into 2.1 with use of Equation 2.2 yields the Cauchy-

Navier equation, which in the absence of body forces, becomes

G(uLjj + Y^ûJtii) = 0 (2.4)

Solving Equation 2.4 for a region f2 bounded by T necessitates the following boundary

conditions.

Ui{x) = gi(x) on Tu (2.5)

ti(x) = Gijiij = hi(x) on Tt (2.6)

where u,(x) and f,(x) are the displacement and traction components, respectively,

with <7J(X) and h{{x) as prescribed functions on the boundary I\ The boundary can

be represented by the summation

Vc{Tu + Tt)

7

and rij is the unit outward normal vector on T.

The fundamental solution to Equation 2.4 is given by Kelvin's unit point load

solution for an infinite body. The displacement and traction tensors for this problem

in three-dimensions are given by Cruse (1969) as

Uij(p,q) • (3 - 4v)5{j + rtirj (2.7) 1677(7(1 - u)r

T"M = 8^F{slJ«+r^rî-r''"'+r-"'} (28)

Here, r is the distance from the load point p to the field point q (or surface field point

Q) as shown in Figure 2.1 and is defined below.

r = y(xQi-xPi)(xQi-XjH) (2.9)

r,- = _ * ! : = > < ? ' - * » > (2.10) dxQi r

Substituting the fundamental solutions into Betti's reciprocal work theorem, see, e.g.

Becker (1992), and employing the divergence theorem yields Somigliana's identity:

«i(p) = J U{Q)Uij[p, Q)dT - J UjiQWjfa Q)dT (2.11)

From this relation, the displacements at an internal point may be obtained in terms

of the boundary data. Moving the interior point to the boundary through the usual

limiting process results in the boundary integral equation which relates boundary

tractions to boundary displacements. Neglecting the influence of body forces, the

boundary integral equation can be written as

GjiPfaiP) + J Tl3{P, Q)Uj(Q)dT(Q) = J U{j{P, Q)tj{Q)dT{Q) (2.12)

where

d^P) = lim / 7y(P, Q)dT (2.13) £-°yr e

If P lies on a smooth surface,

Ca(P) = l-6xj (2.14)

Stresses inside the domain, in the absence of body forces, can be determined from

the corresponding Somigliana's stress identity wich can be derived using Equation

2.11, the strain-displacement relation and Hooke's law. This stress identity is given

as follows:

°ijip) = j Dkij(p,Q)tk(Q)dT(Q)- J Sklj(p,Q)uk(Q)dT(Q) (2.15)

(2.16)

where

Dkij = g , _ , 2 3r fcr jr j + (1 - 2u)(5kirtj + 6kjrti - Si:jr, k)

G ( dr Skli = 4 (I _ ) 3 1 3 7 T ^ 1 ~ 2v)îr,k + vtfkirj + Skjr:i) - 5r fcr ,-r j]

+ 3i/(nJr fej-j + rijr^rtk) + (1 - 2i/)(3nkrtirtj + n ^ + " i4 j )

- (l-4i/)nfc<J0-} (2.17)

2.1.1 Numerical Treatment of the BEM in Three-Dimensions

The numerical solution of the boundary integral equation requires the boundary,

T, to be discretized into elements. Suitable algebraic functions are applied to the

elements with requisite values at specified nodal points. From experience with the

finite element method, it has been determined that the best trade-off of efficiency for

accuracy was with the use of quadratic isoparametric elements. The same has been

9

found to be true for BEM. Thus, the surface of a three-dimensional component is

represented with 6-node triangular and/or 8-node quadrilateral elements, as shown

in Figures 2.2 and 2.3, respectively. Also similar to the finite element method, the

interpolation functions applied to the elements to define the unknowns and geometry

are quadratic shape functions in terms of the local intrinsic coordinates £= £j, j = l , 2 ,

as follows, from Zienkiewicz (1977) and Reddy (2006)

«,-(£) = NC(£X

U(Z) = N'iffi (2.18)

where

1, 2, 3, ..., 8 for quadrilateral elements

1, 2, 3, ..., 6 for triangular elements

and Nc(£) are the shape functions which are given explicitly in Appendix A.

If the surface is represented by m surface elements and q distinct nodes, then

Equation 2.12 becomes

m k »

6=1 c = l J r

m k «

= E E f ^ p d ( M ) / UiiiP^QmN'WJMdt (2.i9) b=l c=l J r

where Pa is the ath node (a = l,q), d(b,c) is the number of the cth node of the

6th element, k=6 for a triangular element and 8 for a quadrilateral, and J(£) is the

Jacobian of transformation. Equation 2.19 represents a system of 2>q distinct linear

equations to be solved to determine the unknown boundary tractions or displacements

10

at the nodes. Once a solution for Equation 2.19 has been obtained, interior point

solutions may be determined by substituting the isoparametric representations into

Equations 2.11 and 2.15.

2.1.2 Example Problem - Stress Concentrations

A typical stress concentration problem, that of a circular hole in an infinite plate of

finite thickness, was analyzed to demonstrate the efficiency of the BEM. This three-

dimensional problem was selected to show the accuracy of the BEM in capturing

stress concentrations with a relatively coarse mesh. The analytical solution to the

problem has been given by Sternberg & Sadowsky (1949) and was verified with the

BEM by Tan & Fenner (1978).

As shown in Figure 2.4, the plate was defined with a height, H, equal to its width,

W. In addition the thickness, 2c, was nominally set to 0.2H and the radius of the

cylindrical hole a=0.1H. Poisson's ratio of 0.3 was assumed and a tensile stress of a

was applied as indicated.

The physical problem was represented using a one-eighth model with 64 quadratic

elements and 194 nodes. The periphery of the circular hole was modelled with four

elements over 90°. It was constrained from displacement along the horizontal and

vertical planes of symmetry, the x- and y- axes respectively as shown in Figure 2.5(a).

Also, only one-half of the plate thickness was modelled and thus the lower surface

was constrained from displacement in the z- direction. The problem was also ana

lyzed with a commercial finite element method code, ABAQUS, and this model was

discretized with 760 20-node brick elements, as shown in Figure 2.5(b) for comparison.

The results of both the present BEM and FEM studies, as well as the BEM

results by Tan & Fenner (1978) and the analytical solution by Sternberg & Sadowsky

(1949), are shown in Table 2.1 at the mid-thickness plane and at the free surface at

the edge of the hole. It is evident that the present results are in good agreement

11

with those determined previously for this problem. As no explicit numerical results

are published at other points through the plate thickness by Tan &; Fenner (1978), a

direct comparison cannot be made elsewhere; Figure 2.6 shows the variation of the

stress concentration factor through the thickness of the plate for the present study as

well as the analytical solution. The percentage discrepancy between the present BEM

results and the analytical solution was less than 1.5% at the mid-thickness plane and

3.6% at the free surface, in spite of the relatively coarse mesh used.

2.2 Fracture Problems

In linear elastic fracture mechanics, the stress field near a crack-front may be char

acterized in term of the three modes of deformation. These are shown in Figure 2.7:

mode I, the opening mode; mode II, the edge-sliding mode; and mode III, the shear or

tearing mode. Although a fracture problem may consist of any combination of these

three modes, the opening mode is considered to be the most important as it is the

most common in practice. Therefore, throughout this study, only mode I deformation

was considered.

Using linear elastic theory and the Westergaard stress function as shown in Unger

(1995), the stress distribution near the crack-front under mode I conditions can be

expressed as

+ 0{r°) (2.20) 'yy

/C7cos(|)

\/2irr

1 - sin(\)sin(f)

l + sm( | ) s2n(f )

sm( | )cos( f )

and

12

&ZZ = V(0XX + (Jyy)

Cxz = &vz = 0 'yz

(2.21)

(2.22)

where K] is the mode I stress intensity factor, r and 9 are polar coordinates as

indicated in Figure 2.8, 0(r°) represents higher order terms that are not significant

to the crack tip solution and v is Poisson's ratio as follows

v ' = <

0 for plane stress

v for plane strain

(2.23)

The corresponding displacement field for the opening mode deformation is given

as

u,.

*V£(i + ") 2E

(2/ \K - l)cos(f) - cos ( f )

(2K + l ) s m ( f ) - s m ( f )

uz = -(v"z/E)(axx + ayy)

+ 0(r°) (2.24)

(2.25)

where E is the Young's modulus, and

for plane stress: v" = v. K = ( 3 - I Z )

(1 + ") for plane strain: v" = 0, K = (3 — Av) (2.26)

From Equation 2.20 and 2.24, it can be seen that the displacements vary accoding

to y/r while the stresses near the crack-front are singular and vary according to 1/y/r.

13

The stress intensity factor, Kj, defines the magnitude of the stress singularity and is

a function of geometry, crack size and the applied load.

2.2.1 Crack-Front Boundary Elements

In BEM, as in FEM modelling of crack problems, a commonly adopted means of in

troducing the yfr variation for the displacement field near the crack-front without any

modification of the quadratic shape functions is to shift the mid-side node adjacent

to the crack-front to the quarter-point position closest to the crack-front. Unlike in

FEM, however, the associated l/y/r traction singularity can be obtained by multi

plying the shape functions for the crack-front element by l/y/r (see, e.g. Aliabadi &

Rooke (1991)).

In the BEM code employed in this study, BIE3D5, the crack-front quadrilateral el

ements are geometrically akin to isoparametric quadrilateral elements but with shape

functions different from those in Section 2.1.1. As determined by Luchi & Poggialini

(1983), these shape functions for the quadrilateral elements provide the appropriate

displacement and traction variations near the crack-front. The following are the shape

functions when nodes 1-2-3 lie along the crack-front, as shown in Figure 2.3, those

14

corresponding to all sides of the element are listed in Appendix A.

x/2

^1(6, ?2) = ^(l + 6)[(2 + v^)^ / lT^- ( l + v/2)(l + £2)]

w|(6,&) = ^(i + 6)[V2(6-2-v /2)VT+^+(2 + v/2)(i + 6)]

^ ( 6 , 6 ) = ^ ( i - t f ) > / i + 6

^J(6,6) = i(i-6)[-V2(6 + 2 + ^ ) \ A T 6 + (2 + V2)(i + 6)]

w|&,6) = ^(i-6)[(2 + v /2)vTT6-(i + v/2)(i + e2)] (2.27)

where Nd denotes the shape function for the displacements of the crack-front element

for the zth node. Similarly,

^fo.fc) = Nfa&yy/T+T*

^3(6,e2) = NfcubyyfiTb

Nfaub) = Nfau&yy/i + T*

N?{£ub) = ^ / 2 W | ( 6 ^ 2 ) / v / l + ^

N?{Zi,£2) = V2N6d(^Z2)/SnT2

Nfaub) = V2N7d(^.(2)/^lTl2

W(8(£i,&) = »*&,&)/y/TTTt (2.28)

where Nlt denotes the shape function for the tractions on the crack-front elements for

the ith node.

15

By equating the variations of the displacements and tractions along the crack-

front element, as represented by the shape functions described above, to the classical

field solution in Section 2.2 simple expressions for the direct evalution of the stress

intensity factors from the computer BEM model results may be obtained. These have

been derived by Luchi & Poggialini (1983) for three-dimensions.

Mode I stress intensity factors can be obtained using the computed traction data

(the "traction formula"), as follows,

Kf = tfVrt (2.29)

where the superscript refers to the node indicated in Figure 2.9 along the crack face,

and / is the length of the crack tip element corrected to be perpendicular to the crack-

front. Similarly, when using the displacement data, the "displacement formulae" may

be used for the direct evaluation of the stress intensity factor:

K? = vkfi <"?-»{> <2M)

K'-BS^-^ (231)

Here, with reference to Figure 2.9, the superscripts indicate particular nodes.

It should be noted that Equation 2.31 relies on numerical data at twice the distance

from the crack-front as Equation 2.30. It has been found in numerical experiments

to be slightly less consistent with the results generated from the traction formula,

Equation 2.29. This is to be expected; however, with good mesh design there is

usually good agreement of the stress intensity factor results obtained using these

formulae; they thus provide a good check on the results.

It should also be noted that the above formulas are based on the 1/y/r stress

singularity which does not hold true when the crack-front intersects the free surface.

16

Thus the stress intensity factor at that location should be used with some caution.

The results from the displacement formula were presented throughout, unless stated

otherwise.

2.3 Numerical Examples - Fracture Problems

To affirm the validity of the BEM for fracture mechanics problems, a number of

test problems were analyzed. As a first example, a circular embedded crack in an

infinite solid was modelled. Once this was validated, the problem of a thick-walled

cylinder with a semi-elliptical crack as investigated by Tan &; Fenner (1979), as well

as by Shim (1986), was analyzed and the results for the stress intensity factor are

compared. Additionally, the same model was analyzed to obtain polynomial influ

ence coefficients for the stress intensity factor and its application to an autofrettaged

cylinder is demonstrated. Poisson's ratio was taken to be 0.3 throughout.

2.3.1 Circular Embedded Crack in an Infinite Solid

The well-known problem of a circular embedded crack in a theoretically infinite solid

was analyzed as a preliminary test of the application of the BEM for crack problems.

A one-eighth model was created due to the symmetry along the plane of the crack as

well as the axisymmetry of the crack itself. Thus a quarter-circle crack was modelled

on the corner of a cube face. To represent a theoretically infinite solid, the side of the

cube was taken to be ten times the radius of the crack. The model was constrained

from displacements normal to each of the three planes of symmetry. It was loaded

with a tensile stress at the bottom surface in the direction normal to the crack face.

The mesh for the model has 229 nodes and 79 elements, as shown in Figure 2.10;

four elements across 90° were placed along either side of the crack periphery. The

exact analytical solution to the circular embedded crack problem is provided in Unger

17

(1995)

Ki = -G^L (2.32) 7T

where a is the applied stress and a is the radius of the penny-shaped crack. Table

2.2 shows the comparison of the results obtained using the traction and displacement

formulas against the exact solution, Equation 2.32. It is evident that the traction

formula is more accurate in this instance with a percent error below 1%. The dis

placement formula results are between 2 and 5% lower than those from the traction

formula and yield a percent error within 5.3%. The nodes that produced a greater

percent error with the displacement formula were located at midpoints and thus were

twice the distance from the crack-front as the other nodes. The correlation between

the traction and displacement formulas was sufficient to determine that there were

no substantial errors in the data reduction and that the model was appropriately

discretized, even if the mesh appears relatively coarse.

2.3.2 Cylinder with Semi-Elliptical Crack

It is apparent that if the results for an elliptical crack in a straight tubing could

be reproduced to good accuracy, then the same modelling strategy would also be

suitable for the analysis of curved tubing. Thus, a series of test models were analyzed

to validate the method employed in solving for stress intensity factors using the BEM

code employed, BIE3D5. Figure 2.11 shows the diametrical section of the physical

problem. The radius ratio of the cylinder is denoted by k = R2/R1, and the semi-

elliptical crack is defined by the elliptical aspect ratio, b/a, of its semi-major and

semi-minor axes. The relative crack depth is defined by b/W, where IV — (R2 — R\),

and the position of points along the crack-front is defined by the angle (p as shown in

the figure.

18

Four test models were created and the results verified by comparison with pre

viously published analyses by Shim (1986) who had also used a BEM code but had

obtained the stress intensity factors by extrapolation techniques. The parameter com

binations selected to test are: fc=2.5, b/W=0.2, 0.4 and 0.6; k=2, b/W=0A; and all

for which 6/a=0.8.

Recognizing the two planes of symmetry, a quarter model was used to represent

each case. One plane of symmetry is the diametrical plane containing the crack

faces, and the other is the axial plane of the cylinder along the semi-minor axis of

the semi-elliptical crack. In addition to the symmetrical constraints, the model was

constrained using a plane strain end condition, as well as nodal constraints at each

node 90 ° to the axial plane of symmetry.

Figure 2.12 shows a typical mesh, in this instance for the crack depth of b/W=0A,

and k=2.5. All of the models were discretized with 8-node quadrilaterals and 6-

node triangles, ranging between 92 and 123 elements for each model. More elements

were applied to the case with the smallest crack depth to ensure that there was an

appropriate gradation in element size approaching the crack-front. It was determined

that it was not necessary to apply more than one element across the wall thickness

as the BEM is not particularly mesh-sensitive as long as a reasonable aspect ratio is

maintained and neighbouring elements do not vary excessively in size. In all instances,

i/=0.3 and uniform hydrostatic pressure, P, was applied to all elements on the internal

surface of the cylinder as well as the crack faces.

Due to the different mesh designs being used for this analysis from those previ

ously employed by Shim (1986) and Tan & Fenner (1980), the results are presented

graphically. The stress intensity factors have been normalized as Ki/(Py^nb)- Figure

2.13 shows the results from the k=2, b/W=0A test case. It can be seen that there is

good agreement between the test model results and the previously published results

of Shim (1986), as well as Tan & Fenner (1980). This good agreement is also seen for

19

the fc=2.5, 6/1^=0.2, 0.4 and 0.6 cases, shown in Figures 2.14 to 2.16. The percent

deviations remain below 3% in all instances, except for /c=2.5, 6 /^=0 .4 where they

are larger but are still less than 5%.

2.4 Influence Function Method for Obtaining

Stress Intensity Factors

The influence function or influence coefficient method is an extension of the well-

known weight function method in fracture mechanics analysis; it allows stress intensity

factors for a given cracked geometry to be obtained for various load cases without

requiring a completely new analysis of the problem. The method is based on the

principle of linear superposition. It allows a problem to be reduced to the summation

of the stress analysis of the uncracked component, subjected to the same external

loads, and the cracked component where the loading is only applied to the crack

faces, as shown in Figure 2.17. Therefore,

Ki = K(P + K{P (2.33)

The loads acting on the crack faces are equal and opposite to the stresses that would

exist on the crack face if the crack was not present. As K) —0, the solution for the

stress intensity factor for the complete problem is reduced to that of the load applied

directly to the crack faces in accordance with the uncracked stress distribution.

K, = I<P (2.34)

Assume the uncracked stress distribution due to external loads can be represented

by a third degree polynomial as,

20

<Trs — AQ + Airs + A2r2

s + A3r3

s (2.35)

where rs is a non-dimensionalized distance as shown in Figures 2.11, and Ai represents

the coefficients of the polynomial function with units of stress. It is possible to

obtain stress intensity factors for the cracked problem in terms of polynomial influence

coefficients, K]1 . These influence coefficients are stress intensity factors corresponding

to the problem with the crack faces subjected to the direct stress distribution as

follows: uniform (a = — lo); linear (a = —Airs); quadratic (er = — A2r^); and cubic

(a = —A^). The stress intensity factor for the problem subjected to the same

arbitrary loading would be

K, = Kf] + K{P + Kf] + Kf] (2.36)

2.4.1 Numerical example 1

The above case of k=2, b/W=0A was used with the same mesh to validate the appli

cation of polynomial influence coefficients to fracture mechanics problems. Uniform,

linear, quadratic and cubic tractions are applied to the crack face, each normalized

with rs, as defined in Figure 2.11, set to unity at r = b in each load case. The influ

ence coefficients are determined from the computed traction and displacement data

using Equations 2.29 and 2.30 and then normalizing by Ai(b/W)lVnb\ they are given

in Table 2.3.

After determining the influence coefficients, they were multiplied by the uncracked

stress distribution for an internally pressurized cylinder to re-validate the stress in

tensity factors above. To this end, the well-known Lame solution was approximated

as a cubic polynomial as follows

21

ars = P(1.6735 - 2.29rs + 2.0172r2 - 0.7312r3) (2.37)

The corresponding coefficients for the uncracked stress distribution in Equation

2.37 and the influence coefficients are multiplied together and summed to determine

the stress intensity factors as follows for 0=90°,

Kj0) = Ki0)*AoP(b/W)°Virb = 0.658(1.6735P)v'blbr" = 1.234P

K{p = K\i>A1P{b/W)1Vri = 0.119(-2.29P){0A)V0A^=-0.128P

K(2) = xj2)*yl2P(fe/H/)2\/^6 = OMG{2.0172P)(OA)2Vo^r = 0.017P

Kf] = i^j3)*yl3P(6/l¥)3v/^6 = 0.024(-0.7312P)(0.4)3 \ /a4^=-.001P

For an internally pressurized cylinder, with pressure acting on the crack faces, an

additional uniform stress of P is applied,

KT = K^PVri + J^K? i=o

= 0 .658P\ /O^ + (1.234 - 0.128 + 0.017 - 0.001)P

= 1.867P

A7/(PV^r6) = 1.665

The stress intensity factors around the crack periphery produced by this method

are compared to the results obtained by Tan & Shim (1986) in Figure 2.18. The

deviation between the two sets of results shown are less than 4%, and is 1.7% at the

free surface. In addition, the stress intensity factors calculated here using the influence

coefficient method deviate less than 2.5% from the results that were calculated by the

direct method presented above, as shown in Table 2.4. The agreement shown between

22

the influence coefficient method and the direct BEM, as well as the stress intensity

factors and influence coefficients presented here and those previously published is

sufficient to validate the use of influence coefficients for fracture mechanics problems,

particularly that of internally pressurized tubing.

2.4.2 Numerical example 2

As another example of the versatility of the influence function method, the influence

coefficients obtained from the BEM analyses are applied to the problem of an autofret-

taged cylinder with the same semi-elliptical crack. The cases of the cracked cylinder

with fc=2, b/W=0A and which has been autofrettaged to 20% and 50% overstrain

were considered.

Autofrettage is a process whereby favourable residual stresses are generated in

a component by loading it into the plastic range. The depth of the plastic zone,

terminating at the elastic-plastic interface, is shown as nR\ in Figure 2.19. For a

cylinder, the internal pressure to be applied to result in a plastic zone with a given

radial depth, nR\, can be determined from

k2 - n2

9eU + ~^^~ (2.38)

where ay is the yield stress. This depth is often quantified as the percent overstrain,

which is defined as the percentage of the wall thickness which has plastically deformed,

% Overstrain = ?—- x 100% (2.39) k — 1

The residua] hoop stress distribution in a cylinder due to partial autofrettage, as

suming plane strain end conditions, the von Mises yield criterion and that the plastic

23

deformation is strain-history independent, can be expressed as follows, Hill (1950)

ae = vf' k2-\

ffe = 7lM1 + f

2k2

1 / R2\fn2-k2 , \ n2 + k2 , n/?j 1 + - ^ I I - ^ s fo&" + — ^ 5 log,

2k2 ~- r

Ri<r< nRi (2.40)

n2 1 (n2-k2 , 2-k~2 + W^l {-2k2--l0gen

nRi < r < R2 (2.41)

A cubic polynomial can sufficiently describe the residual stress distribution in each

of the elastic and plastic regions. When nR\ is greater than the crack depth, b, the

crack faces are fully plastic and the stress distribution across this region can be ex

pressed as a single cubic polynomial as in Equation 2.35. Thus, influence coefficients

may be applied to compute the stress intensity factors. In the circumstance when

nR\ is less than b however, a cubic polynomial is not sufficient to describe the stress

distribution across the crack faces. Thus, when the elastic-plastic interface occurred

within the crack depth, the solution of the stress intensity factors was computed di

rectly from the BEM. The same BEM mesh was used, with 95 quadratic isoparametric

elements and 273 nodes.

The stress distribution to fit the polynomial Equation 2.35 was obtained using a

commercial finite element analysis software, ABAQUS. This was not really necessary

for this particular test model due to the exact solution available as given by Equations

2.40 and 2.41. However, there is no available theoretical solution for the curved tubing

analyses to follow and thus this opportunity was availed upon to check the validity

of the output of the FEM code.

A model of 20-node bricks was created and loaded until 20% of the wall thickness

had yielded. The stresses recorded at this pressure were summed with the negative of

the stresses obtained from subjecting the linear elastic cylinder to a uniform internal

24

pressure equivalent to that which was applied to cause yielding to simulate linear

unloading. The result of this summation was the residual stress in the cylinder. This

residual stress was then fit with cubic polynomials to define the elastic and plastic

regions. This process was repeated for a 50% overstrain.

Figure 2.20 shows the curves for the normalized residual hoop stress, o$l<jy, across

the cylinder wall obtained from ABAQUS and the exact solution at 50% overstrain.

It can be seen that there is excellent agreement for results in the plastic region and a

slight difference in the curves for the stresses across the wall section which remained

elastic.

The stress intensity factors for the 20% overstrain case, where nR\ is less than b,

were computed directly. The polynomials determined from ABAQUS to describe both

regions were used to apply tractions across the crack face. The output of BIE3D5 was

then post-processed and the stress intensity factors determined directly. The stress

intensity factors were normalized by K)/[ayAi(b/W)1Vnb] to preserve significant dec

imal places and compared to the published results by Shim (1986). They are shown

plotted in Figure 2.21.

The solution of the stress intensity factors at 50% overstrain was computed using

the influence coefficients provided in Table 2.3. similar to the approach used for the

pressurized cylinder. These results were then normalized and compared to those by

Shim (1986), as shown in Figure 2.22.

From Figures 2.21 and 2.22 the solutions obtained have good agreement with

those previously published. This is especially evident in Figure 2.22 which shows

50% overstrain and thus was calculated using the influence coefficients in Table 2.3.

The use of percent error here would be misleading due to the fact the data vary over

three orders of magnitude. Figure 2.21 also shows reasonable agreement with the

available solution given by Shim (1986). The slight increase in discrepancy seen in

this plot can be attributed to the stress distribution determined from ABAQUS, as

25

FEM was found to be very mesh sensitive as well as from fitting the curves into cubic

polynomials.

2.5 Conclusions

The boundary element method can be used efficiently for linear elastic fracture me

chanics problems. It has been validated for determining stress intensity factors both

directly and with the use of polynomial influence coefficients for an internally pres

surized thick-walled cylinder. These influence coefficients may be applied to any load

case for the same crack geometry and this was verified for the cylinder under internal

pressure and for a partially autofrettaged cylinder with a semi-elliptical crack. As the

cylinder test models have all been validated, a more complicated geometry, namely a

curved tubing, will be analyzed next.

26

Table 2.1: Variation of stress concentration factor at side of circular cylindrical hole in infinite plate of finite thickness with position through the plate thickness

z/c

0

1

ay/a

Sternberg k, Sadowsky (1949)

3.11

2.76

Tan & Fenner (1978)

3.00

2.60

BEM

3.07

2.66

FEM

3.10

2.70

Table 2.2: Comparison of normalized stress intensity factor, K] = Ki/^cry/na), for the penny-shaped crack problem: {K})T - Traction formula result; {KJ)D - Displacement formula result; Exact solution 7^=1.000

e

0°

11.25°

22.5°

33.75°

45°

56.25°

67.5°

78.75°

90°

(*7)T

1.0081

0.9997

1.0060

1.0046

1.0060

1.0046

1.0060

0.9997

1.0074

% Error

0.812

-0.031

0.601

0.460

0.601

0.460

0.601

-0.031

0.741

(*7)D

1.0274

1.0513

1.0270

1.0524

1.0275

1.0523

1.0268

1.0510

1.0283

% Error

2.741

5.132

2.700

5.244

2.751

5.233

2.685

5.102

2.832

27

Table 2.3: Normalized influence coefficients, K{p* = Kf /[A^b/WyVirb], for k=2, b/W=0A

A?> <t> 0°

0.693

0.449

0.345

0.283

11.25°

0.679

0.405

0.288

0.219

22.5°

0.691

0.429

0.315

0.247

33.75°

0.669

0.364

0.237

0.165

45°

0.672

0.356

0.223

0.150

56.25°

0.645

0.277

0.145

0.084

67.5°

0.649

0.235

0.107

0.055

78.75°

0.647

0.163

0.064

0.033

90°

0.658

0.119

0.046

0.024

Table 2.4: Comparison of normalized stress intensity factors, Kj/(Py/nb), for fc=2, b/W=0A: IC - Influence coefficient method; A % - % difference

Method

Direct

IC

A %

<t> 0°

1.523

1.540

1.130

11.25°

1.509

1.527

1.178

22.5°

1.516

1.546

1.997

33.75°

1.506

1.523

1.143

45°

1.518

1.536

1.184

56.25°

1.499

1.513

0.952

67.5°

1.536

1.550

0.954

78.75 °

1.605

1.600

0.303

90°

1.707

1.665

2.440

28

r t ( t • prescribed)

x2

- ^ X i

r ( u - prescribed)

r= ru+ r t

Figure 2 .1: Boundary value problem in elastostatics

29

Figure 2.2: Six node triangular element

30

2.3: Eight node quadrilateral element

31

Figure 2.4: Circular hole in an "infinite" plate of finite thickness

32

(a) BEM mesh

(b) FEM mesh

Figure 2.5: Discretization of the problem of a circular hole in an "infinite" plate of finite thickness

OB

EM

D

FE

M

—A

naly

tical

Fig

ure

2.6:

Var

iatio

n of

str

ess

conc

entr

atio

n fa

ctor

, a y

/a,

thro

ugh

plat

e th

ickn

ess

CO

34

(a) Mode I

(b) Mode 11

(c) Mode III

Figure 2.7: Three displacement modes for crack surfaces

35

crack tip

Figure 2.8: Crack-tip stresses shown with the Cartesian coordinate system

36

Crack Front

Crack Surface

Figure 2.9: Crack-front element

37

Figure 2.10: BEM mesh for problem of a circular embedded crack in an "infinite" solid

38

Figure 2.11: Semi-elliptical surface crack in a thick-walled cylinder

39

Figure 2.12: BEM mesh for cylinder with an elliptical crack, k=2.5, b/W=0A

1.8

1.7

1.6

4

1.5

<U-

*-

1.4

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2.1

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(PV

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20

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60

70

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100

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5: N

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aliz

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fact

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(Pyi

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44

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+

Figure 2.17: Principle of the weight function method

O S

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2.18

: N

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acto

rs,

Ki/

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nb),

us

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influ

ence

coe

ffic

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s fo

r k=

2, b

/W—

0.4

46

Figure 2.19: Typical residual hoop stress distribution in an autofrettaged cylinder

OS

b •

FEM

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lytic

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Dis

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roug

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0:

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roug

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all

thic

knes

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cyl

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: N

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fact

ors,

K

i/{<

j y\/

iib)

, fo

r 20

% o

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trai

n us

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infl

uenc

e co

effi

cien

ts

for

k=2,

blW

=Q

,A

00

-0.05

-0.15

to,

g -0.25

-0.35

-0.4

-0.45

10

20

30

40

50

60

70

80

90

100

O Present

• Sh

im (1986)

<p

(deg

rees

)

Fig

ure

2.

22:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

Ki/

((T

y\nr

f)),

fo

r 50

% o

vers

trai

n us

ing

infl

uenc

e co

effi

cien

ts

for

k=2,

b/

W=0

A

CO

Chapter 3

Stress Distributions in Pipe Bends

The stress distribution in a thick-walled circular pipe bend under internal pressure,

P, was analyzed to determine the location of the maximum stress; this will most

likely be the site for crack initiation. Throughout this analysis the curved tube was

assumed to have a uniform circular cross-section and be free of localized wall thinning,

or any other defect, that may be imparted during the forming process. Figure 3.1

shows the geometry of the physical problem. Treating the circular pipe bend as part

of a torus, it is convenient to introduce toroidal coordinate system (r,rp,0), as shown

in the figure. In this figure, R represents the radius of the pipe bend, Ri and R2 are

the internal and external radii of the cross-section, ip is the circumferential angle and

9 is the longitudinal angle. A total of 27 cases were analyzed from a combination of

the following geometric parameters

R* = R/Rj = 5, 7.5 and 10

k = R2/Ri = 1.5, 2.0 and 2.5

and angular extent of 9 for the pipe bend of

0 = 45°, 67.5° and 90°.

50

51

3.1 Numerical Model

Taking advantage of symmetry, only a quarter of the physical problem needs to be

modelled, as shown in Figure 3.2. An additional length of cylindrical tubing was

added at the cessation of the bend curvature to simplify the application of boundary

conditions to the numerical model. This tangential straight tube was 2/?2 in length.

It was observed that the stress distribution in the straight tube were in agreement

with the well-known Lame solution for the thick-walled cylinder within this distance

from the curvature.

The BEM mesh for each model has between 166 and 232 8-node quadrilateral

elements. In all the models, vertical displacement constraints were applied along the

horizontal plane of symmetry; also the cross-section created by the plane of symmetry

at the mid-section of the bend was constrained to displace only in the radial direction.

Plane strain end conditions were imposed at the end of the additional straight length

of tube. The model was subjected to an internal pressure load, P.

3.2 Numerical Results

The hoop stress results obtained for the BEM analysis were normalized with respect

to the corresponding Lame solution, denoted as <r0, for the cylinder of the equivalent

R2/R1 ratio. As such, the values presented may be considered as stress factors, with a

normalized value of unity corresponding to the hoop stress at the internal surface of a

cylinder with identical radii dimensions and subjected to the same internal pressure.

The results for the normalized hoop stress obtained from the BEM analysis were

also compared using a commercial code for finite element analysis, ABAQUS, for the

same parameters and boundary conditions of the physical problem. It was necessary

to use considerably more 20-node brick elements and a much more refined mesh when

52

using the FEM as opposed to the BEM. Figures 3.3 and 3.4 show there is good

agreement of the results, cr^/a0l between the FEM and BEM analyses. The R*=b,

k=l.b, 0=90° case, shown in Figure 3.3, has the greatest discrepancy between the

two numerical methods; however, it was less than 2% difference. In Figure 3.4, the

R*=7.5, A;=1.5, 0=90° case is more typical with a maximum difference of 1.1%; many

models remained well below 1% difference.

For R*=5, k=2, 9=90°, the stress factor was plotted along both the intrados and

extrados as shown in Figure 3.5. This result is consistent with the findings of Colle

et al. (1987); the stress factor is greatest at the intrados and therefore this site is

considered most of interest to be investigated further.

Figures 3.6 to 3.8 show the variation of the stress factor at the intrados along the

circumferential direction of the pipe bend for each k ratio investigated when 6=90 °

and for the different values of R*. It can be seen that the stress distribution increases

nonlinearly with decreasing wall thickness. There is less difference between the k=1.5

and k=1 trend lines than between the k=2 and &=2.5. This is consistent with the

stress state in a thick-walled cylinder which is inversely proportional to the difference

of squares of the internal and external radii.

The effects of the parameter R* on the stress factor are shown in Figures 3.9

to 3.11 for the k ratios analyzed. It is shown that the hoop stress is larger at a

given 6 position with smaller bend radius ratios, R*. From these figures it can also

be observed that increasing radius ratio, k, has little effect on R*=5 and that the

significance of the bend radius to k ratio relationship is more apparent for the larger

bend radii.

The stress increases throughout the curvature adjacent to the tangential straight

pipe but then becomes relatively constant with respect to the longitudinal angle for

the majority of the curvature. This is most evident in the 90° plots, and more

pronounced for larger values of R*. Despite this trend, the stress remains below the

53

axisymmetric solution present in a torus, as shown in Figure 3.12 which plots the

case of R*=b, k=2 over each value of 9.

The normalized hoop stress increases with smaller bend radius and similarly for

larger values of curvature, 6, as evident from Figures 3.12 to 3.14 which show the

effect of increasing 9 for a constant radius ratio, k—1 at each value of R*. Thus the

stress distribution increases as the model becomes farther removed from a straight

cylinder as one would expect. The largest stress factor occurs in the R*=5, fc=1.5,

0 = 90° model and the smallest in the /T=10, fc=2.5, 9 = 45° model.

3.3 Conclusions

The maximum circumferential stress in a curved tube under internal pressure occurs

at the intrados. For a given radius ratio k, the circumferential stress factor a^,/a0l has

been found to increase with increasing curvature of the tube bend, i.e. with decreasing

values of R*. Larger stresses have also been observed with decreasing radius ratio k,

but this was to be expected as it is consistent with the Lame relation for a simple

straight circular cylinder. Regardless of the combination of the geometric parameters,

the intrados of the pipe bend is thus the most likely site for crack nucleation.

54

Figure 3.1: Curved tubing diagram

(a) BEM mesh with 232 elements, 698 nodes

(b) FEM mesh with 2880 elements, 14109 nodes

Figure 3.2: Quarter-model mesh for R*=5, k=2, 6=90°

1.25

10

15

20

25

6 (d

egre

es)

OF

EM

D

BE

M

Fig

ure

3.3

: N

orm

aliz

ed h

oop

stre

ss,

(r^/

a 0,

at t

he i

nner

rad

ius

R\

at t

he i

ntra

dos

alon

g th

e ci

rcum

fere

ntia

l di

rect

ion,

9,

of t

he p

ipe

bend

#*

=5

, fc

=1.

5, 0

=90

° en

57

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°DJÔ 'jope; ssajjs

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3.7

: N

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a^/a

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at t

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ben

ds

R*=

7.5,

9=

90°

o

1.25

10

15

20

25

30

0 (d

egre

es)

35

40

Ok=

1.5

• k=

2 A

k=2

.5

Fig

ure

3.8

: N

orm

aliz

ed h

oop

stre

ss,

o^/<

7 0,

at t

he i

ntra

dos

alon

g th

e ci

rcum

fere

ntia

l di

rect

ion,

0,

for

pipe

ben

ds R

*—10

, 9=

90°

OS

1.25

1.2

-A-

75

a 1*

: &

A

A

A

-

*—X

—X

—X

—X

X

—X

-

10

15

20

25

0 (d

egre

es)

30

35

40

45

OR

*=5

AR

*=7.

5 X

R*=

10

50

Fig

ure

3.

9: N

orm

aliz

ed h

oop

stre

ss,

cr^/

a 0,

at t

he i

ntra

dos

alon

g th

e ci

rcum

fere

ntia

l di

rect

ion,

0,

for

pipe

ben

ds

k—1.

5,

0=

90

° to

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66

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Chapter 4

Stress Intensity Factors for a Crack in a

Pipe Bend

In the previous chapter, it was established that the maximum hoop stress in the

cross-section of a thick-walled pipe bend occurs at its intrados. This location is

likely to be the site of crack initiation. In this chapter, a semi-elliptical crack is

introduced there in the internally pressurized pipe bend, and stress intensity factors

for a range of geometric parameters are obtained. Polynomial influence coefficients

are also calculated and they are employed to obtain the stress intensity factors due

to the residual stresses for various percentages of overstrain due to autofrettage.

4.1 Semi-Elliptical Crack In a Pipe Bend

Figure 4.1 shows a thick-walled pipe bend with a semi-elliptical crack at the intrados.

In the present study, a semi-elliptical crack with an aspect ratio b/a=0.8 was inves

tigated for each of the 27 geometric cases of the pipe bend analyzed in the previous

chapter, namely

R* = R/Rx = 5, 7.5, 10

k = R2/Ri = 1.5, 2.0, 2.5

68

69

0 = 45°, 67.5°, 90°

In each of these cases, four different relative crack depths, except one situation, are

considered, namely

b/W = 0.2, 0.4, 0.6, 0.8

Stress intensity factors were obtained at nodal points lying along the crack-front for

the cracked geometries treated. The angular positions of these points on the crack-

front are defined by d>, as shown in Figure 4.1. The one geometric case which was

not modelled is that of R*=5, k=2.5, 0=45°, b/W=0.8 where the crack would extend

almost the full span in the circumferential (0) direction of the tubing, and the size of

the crack makes it unlikely to be a stable cracked geometry in practice. The choice of

one semi-elliptical crack aspect ratio of b/a=0.8 being considered was drawn largely

from the experience of straight cylindrical tubings (see, e.g. Tan & Fenner (1980));

the main aim of the present study being to investigate the influence of the various

geometric parameters for the curved tubing on the stress intensity factor for a given

crack size.

4.1.1 Modeling Considerations

For every geometry that was modelled, four elements were employed along the crack

face. The node placement at the free surface and its adjacent midpoint node do

not have constant values of </>, however. This is due to the curvature of the tube

foreshortening the crack before it reaches a full 90°, unlike in a straight circular

cylinder. The exact angle at the free surface and the midpoint were calculated and

employed in modelling as shown in Table 4.1, but will be set nominally to 90° and

78.75 ° respectively in the tabulated results in what follows. These adjusted values of

0 were found to be independent of 6.

In each case, only a quarter of the physical problem was modelled due to symmetry.

70

The number of triangular and quadrilateral quadratic elements in the BEM mesh

which were employed varied with the complexity of the problem, from 100 elements

and 288 nodes for iT=7.5, jfc=2.5, 0=45°, b/W=0.8 to 175 elements and 501 nodes

for R*=b, k=l.b, 9=90°, 6/1^=0.2. Internal pressure, P, was applied to all elements

on the internal surface of the tubing as well as the crack faces. A typical BEM mesh

employed is shown in Figure 4.2.

In BEM modelling, it is allowable to employ elements with an aspect ratio of 4 or 5

and neighbouring elements to increase in size by 2 or 3 times. Thus, what may appear

(when seen from the perspective of a FEM analysis) as a very coarse mesh is often

sufficient. For this application, small elements were used to model the crack-front

region and across the crack faces but the elements were gradually increased in size

to match the coarser and more uniformly shaped elements elsewhere. As the stress

distribution in a thick-walled component changes quite rapidly, there was concern over

whether a single element across the wall thickness, far removed from the crack, would

affect the accuracy of the stress intensity factor results. To this end, a convergence

study to ascertain this was therefore conducted.

4.1.2 Mesh Refinement Study

The boundary element method is less mesh sensitive than the finite element method.

However, it is still necessary to be mindful of the gradation between elements. The

mesh for many of the models used throughout this work employed only one element

across the wall thickness. Normally, this would be insufficient to accurately capture

the stress distribution. To ensure that a converged solution for the stress intensity

factor was being obtained even with a relatively coarse mesh discretization, though far

removed from the crack-front, a mesh refinement convergence study was conducted.

This consisted of a few models being discretized with multiple elements across the

thickness and those results compared to the stress intensity factors obtained from the

71

original mesh. For this purpose two arbitrarily chosen cracked geometries were ana

lyzed: R*=7.5, k=2, 0=90°, b/W=0A; and R*=7.5, k=2.b, 0=90°, b/W=0.6. The

BEM meshes for the former geometric case are shown in Figure 4.3. The comparison

of the results are shown in Table 4.2. It can be seen that the addition of an extra

boundary element across the wall of the tubing did not affect the computed stress

intensity factor in any significant way at the all points along the crack-front for the

two cracked geometries analyzed; the differences being less than 0.4%.

4.1.3 Results

Due to the absence of a theoretical solution for this problem, the veracity of the results

were assessed by checking the values of the stress intensity factors obtained from

the displacement formula against the corresponding values obtained by the traction

formula. Reasonable agreement of these values would suggest that the mesh design

employed was suitable and adequate, as has been demonstrated in Chapter 2. Also,

due to the voluminous amount of result data from the 107 individual cases that were

analyzed in the fracture mechanics study, only some sample sets of these results will

be presented in this Chapter. They represent the general trends observed in all the

cases treated. The complete sets of the results obtained are compiled in Appendix B.

Table 4.3 shows the correlation between the traction and displacement formulas

for R*=5, fc=2.5, 0=90°, b/W=0.2, 0.4, 0.6 and 0.8. The results shown in Table 4.3

for b/W=0.S represent the poorest agreement between formulas for any combination

of parameters investigated; the results for b/W=0.2 were more typical. It is evident

that there was less agreement between the values determined using the midpoint

displacement formula as the data used was a greater distance from the crack face

as discussed in Section 2.2.1. In general, with the exception of the free surface and

adjacent midside node, the agreement was within 5%.

The \j\fr relation of the traction formula does not hold in close proximity to the

72

free surface and this is observed in the results obtained. However, the traction formula

is conventionally considered more accurate than the displacement formula as it is less

mesh sensitive. Both the traction and displacement formula results were calculated

at every node on the crack periphery for every case and found to be in excellent

agreement, with the exception of the node at the free surface. The displacement

formula was used throughout unless otherwise indicated.

The results listed in Appendix B are in the form of Table 4.4 which gives the

normalized stress intensity factors, Kj/(Py/nb), along the crack periphery for i?*=5,

k=2. Interestingly, the normalized stress intensity factors are sufficiently consistent

to be considered constant along most of the crack-front. Approaching the free surface

however, typically beyond 0=60°, the stress intensity factor increases rapidly, as

expected.

Figures 4.4 to 4.6 show the normalized stress intensity factors observed along the

crack periphery, <f>, for values of 0 at each radius ratio, k, for R*=5, b/W=0.6; a larger

value of 9 is shown to result in a larger stress intensity factor for lower radius ratios.

There is a much larger difference between the k=2 and 2.5 trend lines than the fc=1.5

and 2 trend lines. This is consistent with the stress factor results presented for an

uncracked curved tubing in Section 3.2.

Figures 4.6 to 4.8 each show the normalized stress intensity factor along the crack

periphery for a value of R* and all values of 9 with k=2.5, b/W=0.6. Increasing the

bend radius ratio results in a decrease of the normalized stress intensity factor which is

greater from R*=b to 7.5 than from R*=7.5 to 10. This is also consistent with earlier

findings for stress factors. These figures also indicate a decreasing relevance of the

circumferential extent of the tube bend, 9, with increasing R*. In the i?*=10, fc=2.5,

b/W=0.6 plot, Figure 4.8, the points defining each gradation of 9, 45, 67.5 and 90°,

are indiscernable. Thus, the results are almost entirely independent of the angular

span of the bend. It is evident that the effect of 9 is minimal and the normalized

73

stress intensity factor corresponding to #=90 ° would provide a reasonable estimate

for smaller angles.

Figure 4.9 shows a plot of the normalized stress intensity factors for each crack

depth when R*=5, k=2, 9=90°. It can be seen that the stress intensity factors along

the crack profile are greatest at b/W=0.2 and 0.8. Additionally, the stress intensity

factors along the crack periphery increases more steadily with increasing crack depth;

there is a notably steeper curve for b/W=0.8 as it approaches the free surface.

4.2 Stress Intensity Factors from Polynomial In

fluence Coefficients

Stress intensity factors can also be determined using polynomial influence coefficients

as described in Section 2.4. This method allows for alternate load cases to be analyzed

for a given cracked geometry using the superposition principle thus circumventing

the need to repeat the BEM analysis in its entirety. In the following, the influence

coefficients for a curved tubing with a semi-elliptical crack are determined and are

used to verify the stress intensity factors presented above. As an application of the

influence functions, the stress intensity factors due to the residual stresses resulting

from autofrettage of the tubing are also obtained.

4.2.1 Modelling Considerations

The polynomial influence coefficients for a given cracked geometry were determined

using the same mesh as for the direct boundary element method analysis. Four load

cases were applied to each model; uniform (a = — AQ), linear (a = —Airs), quadratic

(a = —A2r2s), and cubic (a = — Azrz

s) traction distributions were applied to the crack

face and each normalized to reach a value of unity at the maximum crack depth. The

74

Ai coefficients obtained are provided in Tables 4.5, 4.6 and 4.7 for cases with R*=5,

7.5 and 10, respectively.

In defining the stress distribution as a third degree polynomial, the parameter rs

was defined as indicated in Figure 4.1. The tractions applied to the crack face were

normalized to allow rs to reach unity at a depth equivalent to b. This definition of

rs would provide slightly higher stress intensity factors as opposed to if it had been

defined in a rectangular coordinate system. The stress in the pipe bend decreases

as 9 moves away from the intrados and this definition of rs would hold the stress to

be constant along the inner surface of the pipe bend for the duration of the crack

face. However, the difference between the curvilinear and rectangular definitions of

rs was determined to be minimal and the agreement between the influence coefficient

method and the previous BEM was within 2.5% for both definitions.

Once the influence coefficients were determined they were combined with the stress

distribution across the wall of the uncracked internally pressurized tubing to verify the

stress intensity factors obtained directly from the BEM analyses as presented in the

previous section above. This stress distribution was determined using the ABAQUS

finite element software and was further approximated by least squares method to

be a cubic polynomial. The use of the FEM was not absolutely necessary here and

was employed merely for expediency because of the available preprocessing for mesh

refinement required for more accurate (interior) solutions across the wall of the tubing

at different positions of 9. It was found that the resultant stress intensity factors were

very sensitive to even small changes in the stress distribution.

4.2.2 Verification

The complete set of results for the influence coefficients obtained for the BEM analysis

are listed in Appendix C; Table 4.8 shows a representative set for R*=b, k=2, 9=90 °.

All these influence coefficients were then used to produce stress intensity factors

75

for the corresponding internally pressured cracked pipe bend which were compared

with those obtained directly by the BEM to establish the veracity of these influence

coefficients. Results for the R*=5, k=2, 6>=90° case are typical, the comparison

of the results are shown in Table 4.9; the discrepancies were less than 2.5%. The

largest discrepancy between the normalized stress intensity factors obtained directly

and those from the influence coefficient method was for the case of the pipe bend

with R*=10, fc=1.5, 0=90°, and relative crack depth b/W=0.2, at just below 5% for

the point at the free surface, as shown in Table 4.10. However, for the R*=10, k=l.b,

0=90°, b/W=0.8 case shown in the same Table, there was excellent agreement, with

less than 1% difference. This suggests that the mesh used in the model in the b/W=0.2

case can perhaps be improved upon, as the same uncracked stress distribution was

applied to all of the crack depths for R*=10, fc=1.5, 0=90°. The discrepancies of the

stress intensity factors for the R*=10, k=l.5, 6=90°, b/W=0.2 case are, however,

still relatively small in general to be of concern.

4.2.3 Application: Effect of Residual Stresses from Autofret-

tage

Influence coefficients allow stress intensity factors for alternate load cases to be ob

tained using the superposition principle in a simple manner without the need to repeat

the numerical analysis in its entirety for the same cracked geometry. To demonstrate

this, the influence coefficients determined above were applied to determine the stress

intensity factors arising from the residual stresses of pipe bends, with radius ratio

k=2.0, which had undergone various degrees of overstrain from autofrettage.

76

Modelling Considerations

In this aspect of the study, each of the tube bends was analyzed at 10% increments

of overstrain from 10% to 100% in the autofrettaged process, before the onset of

crack growth. The material was assumed to be elastic-perfectly plastic with a yield

stress of ay and that von Mises yield criterion with strain-history independent plastic

deformation was applicable to this material.

ABAQUS was employed to determine the residual hoop stress in the uncracked

pipe bend. This was necessary as the BEM software used was not capable of mod

elling plasticity effects. Due to the lack of a theoretical solution for the residual

stress or required pressure to obtain a specified percent overstrain for this particular

geometry, a trial and error process was undertaken. To minimize error throughout

this procedure, each set of results was plotted to verify the trends of stresses were

consistent with those expected and the pressure adjusted accordingly.

For those cases where the elastic-plastic interface in the wall of the circular cross-

section, i.e. at r=nR\ was equal to or greater than the crack depth, the residual

hoop stress distribution in the curved tubing due to autofrettage was determined

using ABAQUS; this was then used with the influence coefficients to obtain the stress

intensity factors, as was done for the internal pressure case. When nR\ was less

than b, the influence coefficients were not employed as the stress distribution across

the crack face could not be represented with a single cubic polynomial. Therefore,

the stress distribution was obtained using ABAQUS and the appropriate tractions

were applied to the crack face so that the stress intensity factors could be computed

directly.

77

Results

The complete sets of results of stress intensity factors due to the residual hoop stresses

for different percentages of overstrain are available in Appendix D; all results shown

are from the traction formula, as opposed to the displacement formula presented

elsewhere. Normalized stress intensity factors at different percentages of overstrain

are given in Tables 4.11 to 4.13, for jfc=2, 0=90°, b/W=Q.G at J2*=5, 7.5 and 10.

It can be observed that most of the normalized stress intensity factors are nearly

constant in the elastic region, but steadily decrease towards the free surface.

Many of the normalized stress intensity factors, particularly for smaller crack

depths, have negative values. These are not physically meaningful in terms of a stress

intensity factor but indicate that the residual hoop stresses imparted due to autofret-

tage are compressive and thus any subsequent pressure load will have to overcome

this before opening the crack further. These negative values were a consistent result

across all crack depths in close proximity to the bore of the pipe bend but the stress

intensity factor was not a constant along the crack periphery. In cases where the

crack depth exceeded the elastic-plastic interface, the stress intensity factors were

positive further removed from the bore and thus the residual stresses in the pipe were

no longer beneficial in retarding crack growth.

4.3 Conclusions

Stress intensity factors were determined for a semi-elliptical crack of b/a=0.8 in an

internally pressurized curved tubing. These were investigated for: three bend radii,

R*=5, 7.5, 10; three ratios of outer to inner diameter, fc=1.5, 2, 2.5; three bend

durations, 0=45°, 67.5°, 90°; and four crack depths, b/W=0.2, 0.4, 0.6, 0.8. The

only combination of parameters omitted was R*—b, k=2.5, 0=90°, b/W=0.8. The

stress intensity factors were observed to be constant along most of the crack periphery

78

and then increased rapidly as the free surface was approached. The rate at which the

stress intensity factor increased as it approached the bore was more pronounced with

increasing crack depth.

The circumferential extent of the entire tubing, as denoted by the parameter 9,

was determined to have negligible influence on the stress intensity factors, particularly

with increasing bend radius, although larger values of 6 resulted in slightly higher Kj

values. Increasing the radius ratio or the bend radii both resulted in decreasing stress

intensity factors.

Polynomial influence coefficients were determined for the same numerical models.

These were used to validate the K\ values determined for an internally pressurized

pipe bend, to where they were shown to agree to within 2.5% difference. The use

of the polynomial influence coefficients has further been demonstrated with the de

termination of Kj values resulting from the residual stresses in a pressurized cracked

curved tubing which has been subjected to varying degrees of overstrain.

Tab

le 4

.1:

Cor

rect

ed v

alue

s of

0 a

t no

dal

poin

ts o

n th

e se

mi-

elli

ptic

al c

rack

-fro

nt i

n a

curv

ed t

ubin

g

k 1.5 2 2.5

R*

= 5

6/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0=78

.75°

78.6

16

77.8

55

77.4

08

76.9

61

77.8

55

76.9

61

76.0

67

75.1

76

77.4

08

76.0

67

74.7

30

73.3

99

^=9

0°

89.7

31

88.2

10

87.3

15

86.4

21

88.2

10

86.4

21

84.6

34

82.8

51

87.3

15

84.6

34

81.9

60

79.2

98

R*

=

7.5

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0=78

.75°

78.4

75

78.1

99

77.9

24

77.6

49

78.1

99

77.6

49

77.0

98

76.5

48

77.9

24

77.0

98

76.2

73

75.4

50

0=

90

°

89.4

49

88.8

98

88.3

47

87.7

97

88.8

98

87.7

97

86.6

96

85.5

96

88.3

47

86.6

96

85.0

46

83.3

99

R*

=

10

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0=78

.75°

78.5

51

78.3

52

78.1

53

77.9

55

78.3

52

77.9

55

77.5

57

77.1

59

78.1

53

77.5

57

76.9

61

76.3

65

0=

90

°

89.6

02

89.2

04

88.8

06

88.4

09

89.2

04

88.4

09

87.6

13

86.8

18

88.8

06

87.6

13

86.4

21

85.2

29

CO

Tab

le 4

.2:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, K

I/(P

\fnb

),

for

/?*=

7.5,

0=

90

° fo

r si

mpl

e m

esh

refi

nem

ent

conv

erge

nce

stud

y:

(KJ)

T

- T

ract

ion

For

mul

a R

esul

t; (

Kj)

D

- D

ispl

acem

ent

For

mul

a R

esul

t; A

% -

% D

iffe

renc

e; A

- 1

ele

men

t ac

ross

w

all

thic

knes

s aw

ay f

rom

cra

ck-f

ront

; B

- 2

ele

men

ts a

cros

s w

all

thic

knes

s aw

ay f

rom

cr

ack-

fron

t

k,b/

W

Jfc-

2,

b/W

-OA

fc-2

.5,

b/W

=0.

6

K\

(KI)

T

A%

A%

(*7)

r

A%

A%

<$>

0°

1.67

0

1.67

3

0.13

9

1.64

2

1.64

4

0.14

0

1.33

9

1.34

2

0.20

0

1.32

8

1.33

0

0.20

3

11.2

5°

1.68

3

1.68

5

0.13

8

1.63

3

1.63

2

-0.0

74

1.35

1

1.35

3

0.16

6

1.31

3

1.31

5

0.11

7

22.5

°

1.67

1

1.67

4

0.14

0

1.64

2

1.64

4

0.14

5

1.34

0

1.34

3

0.16

8

1.32

6

1.32

9

0.18

6

33.7

5°

1.67

7

1.68

0

0.14

1

1.62

9

1.62

7

-0.0

71

1.33

9

1.34

2

0.20

4

1.30

6

1.30

8

0.11

6

45°

1.68

0

1.68

2

0.14

3

1.64

3

1.64

5

0.15

8

1.34

1

1.34

4

0.20

7

1.32

4

1.32

7

0.21

8

56.2

5°

1.69

3

1.69

6

0.14

6

1.63

8

1.63

7

-0.0

71

1.35

9

1.36

2

0.24

4

1.31

3

1.31

5

0.13

1

67.5

°

1.70

3

1.70

6

0.17

3

1.68

3

1.68

6

0.18

2

1.37

9

1.38

3

0.28

1

1.36

8

1.37

2

0.28

4

78.7

5°

1.84

5

1.84

9

0.20

9

1.74

5

1.74

4

-0.0

64

1.53

9

1.54

4

0.32

1

1.40

4

1.40

6

0.11

8

90°

1.81

9

1.82

3

0.21

4

1.84

0

1.84

4

0.21

1

1.54

4

1.55

0

0.35

4

1.50

0

1.50

6

0.36

2

Tri

al

A

B

A

B

A

B

A

B

OO

o

Tab

le 4

.3:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, K

I/(P

\fnh

),

at n

odal

poi

nts

alon

g th

e se

mi-

elli

ptic

al c

rack

-fro

nt,

as d

efin

ed

by (

f),

in a

pre

ssur

ized

cur

ved

tubi

ng f

or R

*=5,

fc=

2.5,

0=

90

°:

(K})

T

- T

ract

ion

For

mul

a R

esul

t;

{K* t

) D

- D

ispl

acem

ent

For

mul

a R

esul

t; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

4>

0°

1.54

1

1.52

3

1.16

3

1.40

5

1.39

1

1.02

0

1.40

6

1.38

7

1.32

8

1.49

3

1.52

3

1.99

5

11.2

5°

1.55

5

1.50

7

3.10

2

1.41

7

1.37

6

2.88

3

1.41

3

1.37

2

2.92

2

1.52

5

1.50

5

1.32

5

22.5

°

1.54

1

1.52

3

1.19

0

1.40

9

1.39

3

1.13

3

1.40

7

1.38

5

1.54

4

1.51

0

1.50

5

0.32

8

33.7

5°

1.54

1

1.49

9

2.69

4

1.41

3

1.37

5

2.67

9

1.40

8

1.36

8

2.81

5

1.48

4

1.47

4

0.66

7

45°

1.53

9

1.51

9

1.31

3

1.41

9

1.40

0

1.33

5

1.41

6

1.39

0

1.81

4

1.49

2

1.47

7

1.03

7

56.2

5°

1.54

7

1.50

0

3.08

1

1.44

2

1.39

0

3.60

9

1.44

3

1.38

6

3.95

0

1.51

4

1.46

5

3.24

4

67.5

°

1.54

7

1.54

2

0.30

3

1.46

5

1.44

6

1.32

9

1.47

8

1.45

8

1.32

7

1.56

3

1.54

3

1.31

1

78.7

5°

1.67

8

1.58

4

5.62

7

1.61

2

1.48

5

7.89

7

1.63

4

1.49

7

8.33

0

1.78

0

1.58

3

11.0

92

90°

1.61

7

1.66

9

3.21

5

1.57

9

1.57

8

0.01

4

1.62

8

1.59

9

1.75

4

1.80

6

1.66

1

8.02

2

K]

A%

(*7)

r

A%

(*7)

r V

<I)

D

A%

(*7)

r

A%

Tab

le

4.4:

N

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

K//

(P\/

7rb)

, at

no

dal

poin

ts

alon

g th

e se

mi-

elli

ptic

al

crac

k-fr

ont,

as

defi

ned

by <

/>, in

a p

ress

uriz

ed c

urve

d tu

bing

for

R*=

5,

k=2

0

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<P 0

°

1.82

1

1.69

1

1.68

6

1.83

5

1.86

1

1.71

1

1.70

8

1.86

2

1.84

0

1.72

0

1.71

2

1.86

5

11.2

5°

1.80

3

1.67

5

1.67

2

1.83

1

1.84

4

1.69

6

1.69

4

1.85

4

1.82

1

1.70

5

1.69

9

1.85

8

22.5

°

1.82

0

1.69

2

1.68

6

1.82

5

1.86

1

1.71

2

1.70

8

1.85

0

1.83

8

1.72

1

1.71

4

1.85

4

33.7

5°

1.79

1

1.66

9

1.66

7

1.80

9

1.83

2

1.69

0

1.69

0

1.83

0

1.80

8

1.70

0

1.69

7

1.83

6

45°

1.81

0

1.69

3

1.69

1

1.81

3

1.85

3

1.71

5

1.71

6

1.83

3

1.82

7

1.72

5

1.72

4

1.84

1

56.2

5°

1.78

6

1.67

7

1.68

5

1.82

0

1.83

0

1.70

0

1.70

9

1.83

5

1.80

2

1.71

1

1.72

0

1.84

4

67.5

°

1.82

9

1.73

3

1.75

3

1.89

9

1.87

5

1.75

7

1.77

8

1.91

2

1.84

4

1.76

9

1.79

1

1.92

5

78.7

5°

1.88

1

1.78

2

1.81

7

1.99

3

1.93

1

1.80

8

1.84

3

1.99

9

1.89

6

1.82

1

1.85

8

2.01

3

90°

1.97

0

1.88

0

1.93

9

2.12

3

2.02

5

1.90

8

1.96

8

2.11

8

1.98

4

1.92

4

1.98

5

2.13

6

to

83

Table 4.5: Polynomial coefficients to define hoop stress distribution in internally pressurized uncracked curved tubing with R*=5

k

1.5

2

2.5

0

45°

67.5°

90°

45°

67.5°

90°

45°

67.5°

90°

A0

3.1149

3.1768

3.2087

1.8888

1.9418

1.9861

1.5376

1.5853

1.6302

A,

-1.7202

-1.8703

-1.9469

-1.9335

-2.0262

-2.1006

-2.6605

-2.7645

-2.8602

A2

0.5849

0.6352

0.6548

1.1159

1.1720

1.1969

2.6521

2.8027

2.9079

A3

-0.0465

-0.0561

-0.0591

-0.1931

-0.2129

-0.2145

-0.9936

-1.0771

-1.1292

Table 4.6: Polynomial coefficients to define hoop stress distribution in internally pressurized uncracked curved tubing with R*=7.5

k

1.5

2

2.5

e

45°

67.5°

90°

45°

67.5°

90°

45°

67.5°

90°

A0

2.9040

2.9073

2.9229

1.8498

1.8350

1.8883

1.4821

1.5133

1.5198

A,

-1.6419

-1.8602

-1.8832

-2.2673

-2.4042

-2.3433

-2.6276

-2.6881

-2.7120

A2

0.5987

0.9906

0.9926

1.8484

2.1609

1.8929

2.6308

2.6863

2.7158

A3

-0.0649

-0.2642

-0.2643

-0.6209

-0.7997

-0.6366

-0.9962

-1.0211

-1.0359

84

Table 4.7: Polynomial coefficients to define hoop stress distribution in internally pressurized uncracked curved tubing with R*=10

k

1.5

2

2.5

0

45°

67.5°

90°

45°

67.5°

90°

45°

67.5°

90°

A0

2.7984

2.8111

2.8128

1.8189

1.8066

1.7875

1.4488

1.4550

1.4566

Ai

-1.7521

-1.7724

-1.7853

-2.2506

-2.2595

-2.2603

-2.6134

-2.6281

-2.6373

A2

0.9365

0.9393

0.9509

1.8034

1.8380

1.8728

2.6204

2.6325

2.6449

^ 3

-0.2440

-0.2445

-0.2491

-0.5963

-0.6168

-0.6375

-0.9943

-0.9999

-1.0061

85

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t-~ r-o

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r o -* o

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o

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o r CO o

r-CM CO o

CO 00 CO o

*

k

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o

r CO o o

CO ^f o c

CM LO o o

! 1

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CM "<# 1—1

o

^ t^ 7—4

o

o CO CM o

b-^f CM CD

CM I—I CO CD

* CO

^

Tab

le 4

.9:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

Ki/

(Py/

irb)

, ob

tain

ed f

rom

the

inf

luen

ce c

oeff

icie

nt

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<f>

, fo

r R

*=5,

k=

2, 6

=90

°:

IC -

Inf

luen

ce C

oeff

icie

nt M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

<t> 0°

1.84

0

1.88

0

2.20

4

1.72

0

1.75

8

2.22

5

1.71

2

1.74

7

2.05

4

1.86

5

1.89

6

1.63

5

11.2

5°

1.82

1

1.85

8

2.06

9

1.70

5

1.74

4

2.26

7

1.69

9

1.73

8

2.24

5

1.85

8

1.89

1

1.75

9

22.5

°

1.83

8

1.87

8

2.17

9

1.72

1

1.76

0

2.26

9

1.71

4

1.75

1

2.15

5

1.85

4

1.88

6

1.74

4

33.7

5°

1.80

8

1.84

3

1.96

8

1.70

0

1.73

8

2.27

7

1.69

7

1.73

7

2.34

6

1.83

6

1.87

2

1.96

1

45°

1.82

7

1.86

3

1.93

5

1.72

5

1.76

4

2.30

2

1.72

4

1.76

5

2.38

4

1.84

1

1.88

0

2.12

0

56.2

5°

1.80

1

1.83

0

1.58

3

1.71

1

1.74

9

2.21

3

1.72

0

1.76

1

2.39

9

1.84

4

1.88

7

2.27

8

67.5

°

1.84

4

1.86

9

1.33

2

1.76

9

1.80

8

2.18

2

1.79

1

1.83

3

2.31

9

1.92

5

1.97

0

2.36

1

78.7

5°

1.89

6

1.91

1

0.83

2

1.82

1

1.85

5

1.84

2

1.85

8

1.89

0

1.71

8

2.01

3

2.04

9

1.79

4

90°

1.98

4

1.99

2

0.39

0

1.92

4

1.95

0

1.37

4

1.98

5

2.00

4

0.95

8

2.13

6

2.16

0

1.14

3

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

oo

05

Tab

le 4

.10:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

Ki/

(P\f

nb),

fr

om i

nflu

ence

coe

ffic

ient

met

hod

and

dire

ct

BE

M a

long

cra

ck p

erip

hery

, 0,

for

/r=

10

, £=

1.5,

0=

90

°: I

C -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A %

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

4> 0

°

2.47

5

2.55

0

3.02

7

2.38

7

2.43

5

2.04

0

2.39

7

2.42

5

1.16

8

2.61

0

2.61

9

0.35

2

11.2

5°

2.44

7

2.52

4

3.11

9

2.36

5

2.41

6

2.14

1

2.38

0

2.41

4

1.45

2

2.60

7

2.61

9

0.47

0

22.5

°

2.46

5

2.54

0

3.05

8

2.38

8

2.43

7

2.05

4

2.39

4

2.42

6

1.31

3

2.59

7

2.61

0

0.48

1

33.7

5°

2.42

1

2.49

9

3.20

3

2.35

0

2.40

2

2.20

6

2.36

7

2.40

7

1.66

9

2.58

3

2.59

7

0.53

5

45°

2.43

0

2.51

0

3.29

4

2.37

4

2.42

6

2.19

3

2.39

0

2.43

0

1.70

2

2.58

3

2.59

8

0.58

0

56.2

5°

2.38

6

2.47

0

3.51

5

2.34

1

2.39

7

2.36

9

2.37

2

2.42

3

2.12

4

2.58

9

2.60

3

0.52

9

67.5

°

2.40

7

2.50

0

3.82

8

2.39

0

2.45

0

2.47

3

2.43

5

2.49

4

2.41

9

2.65

6

2.67

2

0.60

1

78.7

5°

2.45

6

2.56

2

4.34

6

2.45

8

2.52

5

2.72

8

2.52

3

2.59

6

2.89

7

2.79

7

2.81

0

0.46

8

90°

2.57

6

2.70

2

4.90

.0

2.57

7

2.66

6

3.44

2

2.67

3

2.77

3

3.72

3

3.02

4

3.04

5

0.69

2

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

00

Tab

le 4

.11:

N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

(K

j)A

/[cr

y\/7

rb},

fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge

to

diff

eren

t le

vels

of

over

stra

in w

ith

R*=

5, k

=2,

9

= 9

0°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0

°

0.00

2

0.00

7

0.01

8

0.03

2

0.04

1

0.03

2

-0.0

07

-0.0

44

-0.0

53

-0.0

69

11.2

5°

0.00

2

0.00

7

0.01

8

0.03

2

0.04

1

0.02

6

-0.0

13

-0.0

50

-0.0

59

-0.0

75

22.5

°

0.00

2

0.00

7

0.01

8

0.03

1

0.03

7

0.00

8

-0.0

29

-0.0

66

-0.0

74

-0.0

91

33.7

5°

0.00

2

0.00

8

0.01

8

0.03

0

0.01

9

-0.0

23

-0.0

58

-0.0

95

-0.1

04

-0.1

21

45°

0.00

3

0.00

8

0.01

6

0.01

5

-0.0

23

-0.0

66

-0.1

00

-0.1

38

-0.1

47

-0.1

65

56.2

5°

0.00

3

0.00

7

-0.0

02

-0.0

41

-0.0

86

-0.1

29

-0.1

64

-0.2

03

-0.2

14

-0.2

33

67.5

°

0.00

5

-0.0

05

-0.0

56

-0.1

17

-0.1

60

-0.2

04

-0.2

41

-0.2

82

-0.2

94

-0.3

14

78.7

5°

-0.0

06

-0.0

64

-0.1

33

-0.2

02

-0i2

51

-0.2

99

-0.3

41

-0.3

88

-0.4

00

-0.4

22

90°

-0.0

43

-0.1

20

-0.1

92

-0.2

60

-0.3

11

-0.3

57

-0.4

01

-0.4

47

-0.4

58

-0.4

78

OO

oo

Tab

le

4.12

: N

orm

aliz

ed

stre

ss i

nten

sity

fac

tors

, (A

'/)^/

[cry

\/7r

6],

from

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h R

*=7.

5,

k=2,

0

= 9

0°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0°

0.00

2

0.00

7

0.01

6

0.02

6

0.03

8

0.03

0

-0.0

06

-0.0

37

-0.0

52

-0.0

68

11.2

5°

0.00

2

0.00

7

0.01

6

0.02

6

0.03

8

0.02

4

-0.0

12

-0.0

44

-0.0

58

-0.0

75

22.5

°

0.00

2

0.00

7

0.01

5

0.02

5

0.03

4

0.00

7

-0.0

27

-0.0

58

-0.0

73

-0.0

89

33.7

5°

0.00

2

0.00

6

0.01

5

0.02

5

0.01

9

-0.0

22

-0.0

55

-0.0

86

-0.1

00

-0.1

17

45°

0.00

3

0.00

5

0.01

3

0.01

5

-0.0

16

-0.0

63

-0.0

94

-0.1

26

-0.1

40

-0.1

58

56.2

5°

0.00

3

0.00

2

0.00

0

-0.0

28

-0.0

76

-0.1

24

-0.1

55

-0.1

87

-0.2

03

-0.2

22

67.5

°

0.00

4

-0.0

09

-0.0

41

-0.0

97

-0.1

47

-0.1

95

-0.2

27

-0.2

61

-0.2

77

-0.2

98

78.7

5°

-0.0

08

-0.0

69

-0.1

23

-0.1

90

-0.2

46

-0.2

99

-0.3

36

-0.3

74

-0.3

92

-0.4

14

90°

-0.0

50

-0.1

31

-0.1

90

-0.2

56

-0.3

15

-0.3

66

-0.4

05

-0.4

42

-0.4

60

-0.4

79

en

Tab

le

4.13

: N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

{K

I) A/[

a Y\f

Trb

),

from

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /J

*=10

, k=

2,

0 =

90°

, b/

W=

0.G

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<$>

0°

0.00

2

0.00

5

0.01

5

0.02

0

0.02

8

0.02

5

-0.0

12

-0.0

41

-0.0

56

-0.0

70

11.2

5°

0.00

2

0.00

5

0.01

5

0.02

0

0.02

8

0.01

9

-0.0

18

-0.0

47

-0.0

62

-0.0

76

22.5

°

0.00

2

0.00

5

0.01

4

0.02

0

0.02

4

0.00

5

-0.0

33

-0.0

61

-0.0

76

-0.0

90

33.7

5°

0.00

2

0.00

4

0.01

3

0.02

0

0.00

9

-0.0

24

-0.0

60

-0.0

88

-0.1

04

-0.1

18

45°

0.00

2

0.00

3

0.01

0

0.01

4

-0.0

24

-0.0

63

-0.0

99

-0.1

27

-0.1

42

-0.1

57

56.2

5°

0.00

2

0.00

0

-0.0

03

-0.0

19

-0.0

83

-0.1

22

-0.1

59

-0.1

87

-0.2

03

-0.2

19

67.5

°

0.00

3

-0.0

09

-0.0

44

-0.0

83

-0.1

53

-0.1

94

-0.2

30

-0.2

60

-0.2

77

-0.2

93

78.7

5°

-0.0

10

-0.0

64

-0.1

30

-0.1

77

-0.2

56

-0.3

00

-0.3

42

-0.3

75

-0.3

94

-041

1

90°

-0.0

50

-0.1

27

-0.2

01

-0.2

47

-0.3

29

-0.3

71

-0.4

15

-0.4

47

-0.4

65

-0.4

82

o

91

Figure 4 .1: Semi-elliptical crack in curved tubing

Figure 4.2: Sample BEM mesh for R*=7.b, fc=2.5, 0=67.5°, b/W=0A with 112 elements and 332 nodes

93

(a) Trial A

(b) Trial B

Figure 4.3: BEM discretization for mesh refinement study

D

0=90

O

6 =

67.5

A

0=

45

10

20

30

40

50

60

(p (

degr

ees)

100

Fig

ure

4.4:

V

aria

tion

of t

he n

orm

aliz

ed s

tres

s in

tens

ity f

acto

r,

Kj/

(PV

Trb

),

alon

g th

e cr

ack-

fron

t, </>

, of

pres

suri

zed

curv

ed t

ubin

g, f

i*=5

, k-

l.b,

b/W

=0.

6 C

O

10

D

0=

90

O 9

=67

.5

A

0=

45

40

50

<6 (

degr

ees)

Fig

ure

4.

5:

Var

iati

on o

f th

e no

rmal

ized

st

ress

int

ensi

ty

fact

or,

Ki/

(P\f

nb),

al

ong

the

crac

k-fr

ont,

<j>

, of

pr

essu

rize

d cu

rved

tub

ing,

R*=

5, k

=2,

b/W

=0.6

1.7

T

D8

=9

0 O

9 =

67.5

A

9=

45

20

40

50

<t> (

de

gre

es)

Fig

ure

4.

6:

Var

iati

on o

f th

e no

rmal

ized

st

ress

int

ensi

ty f

acto

r,

Kj/

(PV

Trb

),

alon

g th

e cr

ack-

fron

t, 0,

of

pres

suri

zed

curv

ed t

ubin

g, i

?*=5

, fc

=2.5

, 6

/^=

0.6

C

O

1.5S

10

20

30

40

50

4> (

de

gre

es)

60

P9

=9

0 O

6 =

67.5

A

9 =

45

Fig

ure

4.

7:

Var

iati

on o

f th

e no

rmal

ized

st

ress

int

ensi

ty

fact

or,

Ki/

(PV

nb),

al

ong

the

crac

k-fr

ont,

cf),

of p

ress

uriz

ed

curv

ed t

ubin

g, R

*=7.

5, £

=2.

5,

b/W

=0.

6 C

O

-a

1.55

1.45

1.35

1.25

H

a e

O 6

A e

=90

=67.

5 =4

5

20

30

40

50

60

4> (

degr

ees)

100

Fig

ure

4.8:

V

aria

tion

of t

he n

orm

aliz

ed s

tres

s in

tens

ity f

acto

r, K

i/{P

\fnb

),

alon

g th

e cr

ack-

fron

t, <

f>,

of p

ress

uriz

ed

curv

ed t

ubin

g, #

*=10

, fc

=2.

5, b

/W=

0.6

oo

2.2

r

40

50

c)> (

de

gre

es)

Ob/

W=

0.2

Db

/W=

0.4

Ab

/W=

0.6

Xb

/W=

0.8

100

Fig

ure

4.

9:

Var

iati

on o

f th

e no

rmal

ized

st

ress

int

ensi

ty

fact

or,

Ki/

(P\f

iib)

, al

ong

the

crac

k-fr

ont,

(f>

, of

pre

ssur

ized

cu

rved

tub

ing,

R*=

5, k

=2,

9=

90°

CO

C

D

Chapter 5

Conclusions

Throughout this thesis, internally pressurized thick-walled curved tubings were in

vestigated using the boundary element method (BEM). Before analyzing these com

ponents, the three-dimensional BEM was reviewed and its efficiency in solving stress

concentration and linear elastic fracture mechanics problems was demonstrated with

some examples.

In this study, a thick-walled curved tubing subjected to internal pressure has been

investigated over the parameters of bend radius ratio R*=R/R\, radius ratio of the

cross-section k=R2/R\, and angle of curvature 6. They were: R*—5, 7.5, 10; k=1.5,

2.0, 2.5; 0=45°, 67.5°, 90°. It was determined that the normalized hoop stress

factor, <7y;/cr0, was greatest at the intrados of the pipe bend, which is consistent with

previously reported toroidal analyses in the literature. It was also observed that the

normalized stress factor was greater for lower R2/R1 radius ratios, smaller bend radius

ratio R/Ri, and larger angles of curvature. Additionally, the effect of decreasing the

angle of curvature, 8, was found to be minimal, particularly for larger bend radii.

A semi-elliptical crack of b/a=0.8 has been modelled at the intrados and investi

gated over four relative crack depths, namely, b/W=0.2, 0.4, 0.6, and 0.8. Normalized

stress intensity factors have been obtained along the crack periphery, (p. It was ob

served that the normalized stress intensity factor remained fairly constant up to about

100

101

</>=60°, at which point it increased rapidly towards the free surface. The normalized

stress intensity factors obtained were also observed to be greater for relative crack

depths 0.2 and 0.8 than for 0.4 and 0.6; they were observed to increase more rapidly

with proximity to the free surface for increasing crack depth.

To ascertain the veracity of the stress intensity factor results obtained from the

direct BEM, polynomial influence coefficients have been determined for each combi

nation of geometric parameters and applied to the internally pressurized load case.

These influence coefficients may be applied to solve for the stress intensity factor of

any load case with the same cracked geometry. To demonstrate the versatility of the

influence coefficient results, stress intensity factors due to the residual stresses aris

ing from autofrettage of the curved tube have also been obtained. In this analysis,

different levels of overstain, from 10 to 100% at 10% intervals, were investigated for

a representative set of physical models.

The effects of non-uniformity of the cross-section, such as local thinning of the

wall, ovalization, and local defects which may occur during manufacture have not

been considered in this study. Clearly, they will have implications on the site of crack

initiation and propagation. These can be a focus of further study.

References

Aliabadi, M. H. k Rooke, D. P. (1991), Numerical Fracture Mechanics, Kluwer Aca

demic Publishers, Dordrecht.

Atluri, S. N. k Kathiresan, K. (1979), '3d analyses of surface flaws in thick-walled reactor pressure-vessels using displacement-hybrid finite element method', Nuclear Engineering and Design 51, 163-176.

Barthelemy, J. (1947), 'Memoire no. 867', Bull. Ass. Tech. Marit. 46, 411.

Becker, A. A. (1992), The Boundary Element Method in Engineering, McGraw-Hill.

Blackburn, W. S. k Hellen, T. K. (1977), 'Calculation of stress intensity factors in

three dimensions by finite element method', International Journal for Numerical

Methods in Engineering 11, 211-229.

Colle, A., Redekop, D. & Tan, C. L. (1987), 'Pressure loading and bending of hollow

tori', Int. J. Pres. Ves. Piping 27, 137-154.

Cruse, T. A. (1969), 'Numerical solutions in three dimensional elastostatics', Int. J.

Solids Structures 5, 1259-1274.

Hill, R. (1950), The Mathematical Theory of Plasticity, Oxford University Press.

Lang, H. A. (1984), 'Toroidal elastic stress fields for pressurized elbows and pipe bends', Int. J. Pres. Ves. Piping 15, 291-305.

Luchi, M. L. k Poggialini, A. (1983), Computation of 3-dimensional stress intensity factors using special boundary elements, in C. Brebbia, ed., 'Proc. Fifth Int. Conf. on BEM', Spring-Verlag, Berlin, pp. 461-470.

Nadiri, F., Tan, C. L. k Fenner, R. T. (1982), 'Three-dimensional analyses of surface cracks in pressurised thick-walled cylinders', Int. J. Pres. Ves. Piping 10, 159— 167.

102

103

Pu, S. L. & Hussain, M. A. (1981), 'Residual stress redistribution caused by notches and cracks in a partially autofrettaged tube', Journal of Pressure Vessel Technology AS ME 103, 302-306.

Reddy, J. N. (2006), An Introduction to the Finite Element Method, 3rd edn, McGraw-Hill, N.Y.

Rees, D. W. A. (2004), 'Autofrettage of thick-walled pipe bends', Int. J. of Mech.

Sciences 46, 1675-1696.

Shim, M. L. (1986), Boundary integral equation stress analysis of cracked thick-walled

cylinders, Master's thesis, Carleton University.

Sternberg, E. & Sadowsky, M. A. (1949), 'Three-dimensional solution for the stress concentration around a circular hole in a plate of arbitrary thickness', Journal of Applied Mechanics 16, 27-38.

Swanson, S. A. V. & Ford, H. (1959), 'Stresses in thick-walled plane pipe bends', J. Mech. Eng. Set. 1(2), 103-112.

Tan, C. L. &; Fenner, R. T. (1978), 'Three-dimensional stress analysis by the boundary

integral equation method', Journal of Strain Analysis 13(4), 213-219.

Tan, C. L. & Fenner, R. T. (1979), Elastic fracture mechanics analysis by the bound

ary integral equation method, in 'Proc. R. Soc. Lond. A369', pp. 243-260.

Tan, C. L. & Fenner, R. T. (1980), 'Stress intensity factors for semi-elliptical surface cracks in pressurized cylinders using the boundary integral equation method', International Journal of Fracture 16(3), 233-245.

Tan, C. L. & Shim, M. L. (1986), 'Stress intensity factor influence coefficients for internal surface cracks in thick-walled cylinders', Int. J. Pres. Ves. and Piping 24, 49-72.

Thuloup, A. (1937), 'Memoire no. 725', Bull. Ass. Tech. Marit. 41, 317.

Turner, C. E. k Ford, H. (1957), Examination of the theories for calculating the stresses in pipe bends subjected to in-plane bending, in '1957 Proc. Instn Mech. Engrs, Lond.', p. 513.

Unger, D. J. (1995), Analytical Fracture Mechanics, Academic Press.

von Karman, T. (1911), Z. Ver. dtsch. Ing. 55, 1889.

Zienkiewicz, O. C. (1977), The Finite Element Method, 3rd edn, McGraw-Hill, N.Y.

Appendix A

Shape Functions

A.l Shape Functions for Triangular Elements

Shape functions for a quadratic triangular element as shown in Figure 2.2 are given

below.

^ ( 6 , 6 ) = ( W i - £ 2 ) ( l - 2 6 - 2 6 )

#2(6,6) = 4 6 ( 1 - 6 - 6 )

^ ( 6 , 6 ) = 6(26-1)

N4(6,6) = 466

iv5(6,6) = 6 (26 - i )

^ ( 6 , 6 ) = 4 6 ( 1 - 6 - 6 ) (A.l)

104

105

A.2 Shape Functions for Quadrilateral Elements

Shape functions for a quadratic element as shown in Figure 2.3 are given below.

^(6 ,6 ) =

JV2(£i,6) =

^3(6,&) =

JV4(6,6) =

w5(&,6) =

JV6(6,6) =

w7(6,6) =

Af8(6,6) =

- l ( i - 6 ) ( i - 6 ) ( i + 6 + 6)

^(1-€?)(!-&)

- I ( i + e,)(i-6)(i-ei+e2)

^ ( i - 6 ) ( i - £ i )

^(i + ^)(i-Cl)

- j ( i - 6 ) ( i + e2)(i + 6 - 6 )

i(i-e?)(i+6)

-^(i + 6)(i + 6 ) ( i - 6 - 6 )

106

A.3 Shape Functions for Crack-Front Elements

The displacement and traction shape functions for the quadrilateral element shown

in Figure 2.3 are as follows for the nodes 1-2-3 along the crack-front

wi(6,6) = |(i-fi)[-ei-(i-êi)A/r+e2+â + 6)]

^ ( 6 , 6 ) = ( i -e?)[ i -Â /TT&]

WJ&.&) = ^(i + 6 ) [6 - ( i + ^ 6 ) v / ^ K 2 + ^ ( i + 6)]

^4(6,6) = ^(i + 6)[(2 + v /2)v /TT6-(i + %/2)(i + 6)]

^5(6,6) = ^(i + 6)[v /2(6-2-v /2)vT+^+(2 + ^ ) ( i + 6)]

^6(6,6) = ^ ( i - € ? ) V i + 6

wj(6,&) = ^(i-6)[-v/2(ei + 2 + y2)x/r+6 + (2 + v/2)(i + e2)]

^ (6 ,6) = ^(l-?])[(2 + ) y r T 6 - ( l + V/2)(l+e2)] (A.3)

A?(£I,6) = ^d3(6,6)/vTT6

A?(6,6) - Ard4(6,6)/v/T+6

Af(6,6) = V2Nlfo,t2)/\/i + T2

Af(6,6) = N*(h,b)/y/T+l2 (A.4)

107

If the nodes 3-4-5 are on the crack-front, the shape functions are as follows

iv](6,6) = ^(i + 6)[e2-(i + ^ 2 ) \ / r ^ + ^ ( i - 6 ) ]

^1(6,6) = ^(i + e2)[(2 + v / 2) v/ r^6- ( i + ^2)(i-6)]

Wj(fi,&) = ^(l + £ 2 ) [V2(6-2-V2) v/ r^+(2 + ^ ) ( l - 6 ) ]

#!&,&) = ^ ( i - e l ) \ / r ^ (A.5)

A?(6,£2) = Nfcukyy/T^Ti

^3(ei,e2) = Afl(6,e2)/v /iT:6"

A7(6,6) = yfiNfaubyy/T^

^8(6,£2) = v/27Vd8fe,6)/v/T:76 (A.6)

108

If the nodes 5-6-7 are on the crack-front, the shape functions are

wJ(£i,£2) = ^(i-6)hv/2(6 + 2 + v/2)x/rî-K2 + v^)(i-e2)]

ivd2(6,6) = ^ ( i - t f ) v T = 6

^1(6,6) = i ( i + 6)[V2(6-2-v/2)x/T^6 + (2 + ^ ) ( i -6 ) ]

^6(6,6) = ( i - t f ) [ i -^>/r="6]

wj&,&) = ^ ( i - 6 ) [ - 6 - ( i - ^ 6 ) v / ^ + ^ ( i - £ 2 ) ]

^Kei,e2) = ^(i-6)[(2+v/2)\/r^-(i + v )(i-e2)] (AJ)

^ ( 6 , 6 ) =

A?(6,6) =

w(3&,&) =

^ ( 6 , 6 ) =

JVt5(6,6) =

ivt6(6,6) =

iv/(6,6) =

^8(6,C2) =

V2ivi(ei,6)/\/i-e2

y/2Ni(^^)/y/l-^

V2^3(6,6)/Vl-?2

^ 4 (6 ,6 ) / \ / i -6

^5(^,6)/yi-e2

^ 6 (6 ,6 ) /V i -6

wJ(£i,6)/Vi-£2

^ (6 ,6 )7^1 -6

109

If the nodes 7-8-1 are on the crack-front, the shape functions are

tfj(6,6) = ^ ( i - 6 ) [ - 6 - ( i - ^ 6 ) V ^ + ^ ( i + 6)]

ivj(6,6) = \(i-b)[(2 + >ft)y/i + Ti- (i + v/2)(i + 6)]

w|(6,6) = J(i-6)[-V2fe + 2 + ) v/ i T 6 + (2 + V2)(i + 6)]

V2, ^ (6 ,6 ) = ^-(i-e22)x/T+6

^1(6,6) = ^(l + 6)[>/2(6-2-v^)>A :i :6 + (2 + ) ( l + ei)]

K(Zi, 6) = ^(l + 6)[(2 + ) % / T f ^ - ( l + V/2)(l + 6)]

^7(6,&) = ^(i + 6 ) [6 - ( i + ^ 2 ) y r T 6 + ( i + 6)]

^1(6,6) = ( i - C i H i - ^ V ^ + ^ l (A.9)

^3(6,e2) = V27vd3(6,e2)/\/rT^

^(4(6,£2) = v /2ivd

4(6,6)/\/rT6

^8(6,£2) - w | (6 ,6) /vT+6 (A.io)

Appendix B

Stress Intensity Factors for a Crack in a

Pipe Bend

110

Tab

le

B.l

: N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

i/{P

yfnb

),

at n

odal

poi

nts

alon

g th

e se

mi-

elli

ptic

al c

rack

-fro

nt,

as

defi

ned

by <

/>, in

a p

ress

uriz

ed c

urve

d tu

bing

for

R*=

5, k

=1.

5

0

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<t> 0

°

2.76

5

2.57

5

2.59

0

2.84

7

2.78

9

2.58

7

2.59

9

2.85

3

2.80

3

2.60

2

2.60

5

2.76

8

11.2

5°

2.73

7

2.55

3

2.57

8

2.85

2

2.76

2

2.56

8

2.59

0

2.86

0

2.77

4

2.58

2

2.59

7

2.77

7

22.5

°

2.75

4

2.57

6

2.59

2

2.84

0

2.77

9

2.59

0

2.60

3

2.84

8

2.79

2

2.60

4

2.60

9

2.76

7

33.7

5°

2.71

2

2.53

9

2.57

1

2.83

8

2.73

7

2.55

5

2.58

4

2.84

9

2.75

1

2.57

1

2.59

3

2.77

1

45°

2.72

7

2.56

5

2.59

9

2.84

7

2.75

2

2.58

2

2.61

3

2.85

9

2.76

6

2.59

8

2.62

3

2.78

5

56.2

5°

2.69

0

2.53

7

2.59

1

2.87

3

2.71

7

2.55

6

2.60

9

2.88

9

2.73

1

2.57

4

2.62

2

2.81

8

67.5

°

2.72

8

2.59

7

2.67

3

2.96

8

2.75

6

2.61

8

2.69

3

2.98

9

2.77

1

2.63

8

2.71

0

2.92

3

78.7

5°

2.80

8

2.68

0

2.78

3

3.14

3

2.83

8

2.70

3

2.80

7

3.16

7

2.85

5

2.72

6

2.82

8

3.09

8

90°

2.96

3

2.81

9

2.97

2

3.36

5

2.99

5

2.84

4

2.99

9

3.39

1

3.01

6

2.87

2

3.02

6

3.33

8

Tab

le

B.2

: N

orm

aliz

ed s

tres

s in

tens

ity f

acto

rs,

Kj/

{P\f

Kb)

, at

nod

al p

oint

s al

ong

the

sem

i-el

liptic

al c

rack

-fro

nt,

as

defin

ed b

y </>

, in

a p

ress

uriz

ed c

urve

d tu

bing

for

R*=

5, k

=2

0

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<P 0°

1.82

1

1.69

1

1.68

6

1.83

5

1.86

1

1.71

1

1.70

8

1.86

2

1.84

0

1.72

0

1.71

2

1.86

5

11.2

5°

1.80

3

1.67

5

1.67

2

1.83

1

1.84

4

1.69

6

1.69

4

1.85

4

1.82

1

1.70

5

1.69

9

1.85

8

22.5

°

1.82

0

1.69

2

1.68

6

1.82

5

1.86

1

1.71

2

1.70

8

1.85

0

1.83

8

1.72

1

1.71

4

1.85

4

33.7

5°

1.79

1

1.66

9

1.66

7

1.80

9

1.83

2

1.69

0

1.69

0

1.83

0

1.80

8

1.70

0

1.69

7

1.83

6

45°

1.81

0

1.69

3

1.69

1

1.81

3

1.85

3

1.71

5

1.71

6

1.83

3

1.82

7

1.72

5

1.72

4

1.84

1

56.2

5°

1.78

6

1.67

7

1.68

5

1.82

0

1.83

0

1.70

0

1.70

9

1.83

5

1.80

1

1.71

1

1.72

0

1.84

4

67.5

°

1.82

9

1.73

3

1.75

3

1.89

9

1.87

5

1.75

7

1.77

8

1.91

2

1.84

4

1.76

9

1.79

1

1.92

5

78.7

5°

1.88

1

1.78

2

1.81

7

1.99

3

1.93

1

1.80

8

1.84

3

1.99

9

1.89

6

1.82

1

1.85

8

2.01

3

90°

1.97

0

1.88

0

1.93

9

2.12

3

2.02

5

1.90

8

1.96

8

2.11

8

1.98

4

1.92

4

1.98

5

2.13

6

Tab

le

B.3

: N

orm

aliz

ed

stre

ss i

nten

sity

fac

tors

, K

i/(P

y/ir

b),

at

noda

l po

ints

alo

ng t

he s

emi-

elli

ptic

al

crac

k-fr

ont,

as

defi

ned

by 0

, in

a p

ress

uriz

ed c

urve

d tu

bing

for

R*=

5,

£=2.

5

0 45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

4> 0°

1.48

9

1.35

8

1.34

2

1.51

8

1.37

9

1.37

9

1.51

2

1.52

3

1.39

1

1.38

7

1.52

3

11.2

5°

1.47

5

1.34

4

1.32

6

1.50

4

1.36

5

1.36

3

1.49

4

1.50

7

1.37

6

1.37

2

1.50

5

22.5

°.

1.49

0

1.36

1

1.33

9

1.51

9

1.38

2

1.37

7

1.49

4

1.52

3

1.39

3

1.38

5

1.50

5

33.7

5°

1.46

7

1.34

1

1.32

0

1.49

7

1.38

9

1.35

9

1.46

1

1.49

9

1.37

5

1.36

8

1.47

4

45°

1.48

6

1.36

6

1.33

9

1.51

7

1.38

8

1.38

1

1.46

2

1.51

9

1.40

0

1.39

0

1.47

7

56.2

5°

1.46

8

1.35

4

1.33

2

1.50

0

1.37

7

1.37

6

1.45

1

1.50

0

1.39

0

1.38

6

1.46

5

67.5

°

1.50

9

1.40

7

1.39

7

1.54

4

1.43

1

1.44

8

1.52

6

1.54

2

1.44

6

1.45

8

1.54

3

78.7

5°

1.55

0

1.44

3

1.43

4

1.58

7

1.46

9

1.48

7

1.56

6

1.58

4

1.48

5

1.49

7

1.58

3

90°

1.63

4

1.53

1

1.53

0

1.67

6

1.56

2

1.58

7

1.64

2

1.66

9

1.57

8

1.59

9

1.66

1

Tab

le

B.4

: N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

j/(P

\fnb

),

at

noda

l po

ints

alo

ng t

he s

emi-

elli

ptic

al c

rack

-fro

nt,

as

defi

ned

by <

/>, in

a p

ress

uriz

ed c

urve

d tu

bing

for

i?*

=7.

5, A

;=1.

5

9

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<t> 0

°

2.56

6

2.46

6

2.47

5

2.67

1

2.56

2

2.48

2

2.47

2

2.66

8

2.57

2

2.45

5

2.46

9

2.66

3

11.2

5°

2.53

8

2.44

5

2.46

3

2.67

2

2.53

5

2.46

2

2.46

0

2.66

9

2.54

7

2.43

5

2.45

6

2.66

3

22.5

°

2.55

6

2.46

7

2.47

7

2.66

5

2.55

2

2.48

3

2.47

4

2.66

0

2.56

3

2.45

7

2.47

0

2.65

4

33.7

5°

2.51

3

2.43

1

2.45

5

2.65

5

2.51

0

2.44

9

2.45

3

2.65

1

2.52

1

2.42

1

2.44

7

2.64

3

45°

2.52

4

2.45

5

2.48

0

2.65

9

2.52

1

2.47

4

2.47

7

2.65

5

2.53

2

2.44

5

2.47

2

2.64

6

56.2

5°

2.48

4

2.42

7

2.47

0

2.67

0

2.48

1

2.44

8

2.46

8

2.66

7

2.49

3

2.41

6

2.46

0

2.65

8

67.5

°

2.51

2

2.48

1

2.54

2

2.74

5

2.50

8

2.50

5

2.54

2

2.74

3

2.52

1

2.47

0

2.53

0

2.73

3

78.7

5°

2.57

2

2.55

9

2.64

5

2.89

3

2.56

8

2.58

5

2.64

5

2.89

4

2.58

2

2.54

3

2.62

9

2.88

2

90°

2.71

6

2.69

6

2.81

5

3.12

5

2.70

8

2.72

4

2.81

5

3.12

8

2.72

3

2.67

3

2.79

1

3.11

2

Tab

le B

.5:

Nor

mal

ized

str

ess

inte

nsity

fac

tors

, K

jj(P

\fiT

b),

at n

odal

poi

nts

alon

g th

e se

mi-

ellip

tical

cra

ck-f

ront

, as

de

fined

by

<j>

, in

a p

ress

uriz

ed c

urve

d tu

bing

for

R*=

7.5,

k=

2

0

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<t> 0

°

1.75

8

1.64

0

1.62

8

1.74

8

1.76

5

1.64

4

1.62

6

1.76

8

1.76

6

1.64

5

1.62

6

1.76

2

11.2

5°

1.74

2

1.62

8

1.61

5

1.73

3

1.74

9

1.63

2

1.61

4

1.75

6

1.75

1

1.63

3

1.61

4

1.74

9

22.5

°

1.75

5

1.64

0

1.62

9

1.73

0

1.76

2

1.64

4

1.62

7

1.75

2

1.76

4

1.64

5

1.62

7

1.74

5

33.7

5°

1.73

0

1.62

3

1.60

9

1.70

3

1.73

7

1.62

7

1.60

9

1.72

7

1.73

9

1.62

9

1.60

9

1.72

1

45°

1.74

1

1.64

0

1.63

0

1.70

2

1.74

9

1.64

5

1.63

0

1.72

6

1.75

1

1.64

7

1.63

0

1.72

0

56.2

5°

1.72

0

1.63

1

1.62

1

1.69

2

1.72

7

1.63

6

1.62

2

1.71

9

1.72

9

1.63

8

1.62

2

1.71

2

67.5

°

1.74

9

1.67

8

1.68

3

1.74

3

1.75

7

1.68

4

1.68

5

1.77

3

1.75

9

1.68

7

1.68

5

1.76

7

78.7

5°

1.79

5

1.73

4

1.73

9

1.81

4

1.80

4

1.74

1

1.74

2

1.85

2

1.80

7

1.74

5

1.74

2

1.84

5

90°

1.89

4

1.83

2

1.85

3

1.94

0

1.90

4

1.83

9

1.85

6

1.99

1

1.90

7

1.84

5

1.85

7

1.98

2

Tab

le B

.6:

Nor

mal

ized

str

ess

inte

nsity

fac

tors

, K

i/{P

\fnb

),

at n

odal

poi

nts

alon

g th

e se

mi-

ellip

tical

cr

ack-

fron

t, de

fined

by

</>,

in a

pre

ssur

ized

cur

ved

tubi

ng f

or R

*=7.

5, k

=2.

5

9 45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<t> 0°

1.45

2

1.32

4

1.32

2

1.45

6

1.46

0

1.33

0

1.32

7

1.46

6

1.46

3

1.33

2

1.32

8

1.45

5

11.2

5°

1.44

1

1.31

0

1.30

6

1.43

7

1.44

8

1.31

7

1.31

1

1.44

6

1.45

2

1.31

8

1.31

3

1.43

6

22.5

°

1.45

3

1.32

5

1.32

0

1.43

5

1.46

0

1.33

2

1.32

5

1.44

5.

1.46

4

1.33

3

1.32

7

1.43

5

33.7

5°

1.43

4

1.30

5

1.29

8

1.40

2

1.44

2

1.31

3

1.30

4

1.41

2

1.44

6

1.31

4

1.30

6

1.40

3

45°

1.44

6

1.32

7

1.31

7

1.39

7

1.45

4

1.33

4

1.32

3

1.40

7

1.45

8

1.33

6

1.32

5

1.39

9

56.2

5°

1.43

2

1.31

3

1.30

4

1.37

8

1.44

0

1.32

1

1.31

0

1.38

9

1.44

4

1.32

3

1.31

3

1.38

2

67.5

°

1.46

3

1.36

3

1.36

0

1.42

1

1.47

1

1.37

2

1.36

6

1.43

2

1.47

5

1.37

4

1.37

0

1.42

7

78.7

5°

1.50

6

1.39

7

1.39

3

1.46

9

1.51

4

1.40

6

1.40

0

1.48

0

1.51

9

1.40

9

1.40

4

1.47

4

90°

1.60

6

1.48

5

1.49

0

1.57

7

1.61

4

1.49

6

1.49

8

1.59

2

1.62

0

1.49

9

1.50

3

1.58

7

Tab

le

B.7

: N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

j/(P

\fnb

),

at n

odal

poi

nts

alon

g th

e se

mi-

elli

ptic

al

crac

k-fr

ont,

defi

ned

by <

fr,

in a

pre

ssur

ized

cur

ved

tubi

ng f

or i

?*=

10,

fc=

1.5

e

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<t> 0°

2.49

3

2.40

8

2.41

7

2.62

2

2.49

1

2.39

4

2.41

8

2.61

7

2.47

5

2.38

7

2.39

7

2.61

0

11.2

5°

2.46

6

2.38

7

2.40

3

2.62

0

2.46

4

2.37

1

2.40

3

2.61

5

2.44

7

2.36

5

2.38

0

2.60

7

22.5

°

2.48

3

2.40

9

2.41

7

2.61

1

2.48

1

2.39

4

2.41

8

2.60

6

2.46

5

2.38

8

2.39

4

2.59

7

33.7

5°

2.44

1

2.37

3

2.39

3

2.59

8

2.44

0

2.35

6

2.39

2

2.59

2

2.42

1

2.35

0

2.36

7

2.58

3

45°

2.45

2

2.39

6

2.41

6

2.59

8

2.44

9

2.37

9

2.41

5

2.59

2

2.43

0

2.37

4

2.39

0

2.58

3

56.2

5°

2.41

1

2.36

7

2.40

3

2.60

5

2.40

9

2.34

7

2.40

0

2.59

9

2.38

6

2.34

1

2.37

2

2.58

9

67.5

°

2.43

6

2.41

9

2.47

0

2.67

3

2.43

3

2.39

6

2.46

4

2.66

6

2.40

7

2.39

0

2.43

5

2.65

6

78.7

5 °

2.49

2

2.49

2

2.56

6

2.81

6

2.48

9

2.46

2

2.55

7

2.80

6

2.45

6

2.45

8

2.52

3

2.79

7

90°

2.62

6

2.61

8

2.73

2

3.04

8

2.62

2

2.58

2

2.71

6

3.03

9

2.57

6

2.57

7

2.67

3

3.02

4

Tab

le

B.8

: N

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Kj/

(PV

Trb

),

at

noda

l po

ints

alo

ng t

he s

emi-

elli

ptic

al

crac

k-fr

ont,

as

defi

ned

by <

/>, i

n a

pres

suri

zed

curv

ed t

ubin

g fo

r R

*~10

, k=

2

0

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<P 0

°

1.72

7

1.60

9

1.59

5

1.72

5

1.72

8

1.61

1

1.58

6

1.72

2

1.72

7

1.60

7

1.59

6

1.71

4

11.2

5°

1.71

1

1.59

8

1.58

3

1.71

1

1.71

3

1.60

0

1.57

3

1.70

8

1.71

2

1.59

6

1.58

4

1.69

9

22.5

°

1.72

4

1.60

9

1.59

6

1.70

8

1.72

5

1.61

1

1.58

6

1.70

5

1.72

4

1.60

7

1.59

7

1.69

5

33.7

5°

1.69

9

1.59

3

1.57

7

1.68

2

1.70

1

1.59

5

1.56

7

1.67

9

1.70

0

1.59

0

1.57

8

1.66

7

45°

1.71

0

1.60

9

1.59

6

1.67

9

1.71

2

1.61

2

1.58

7

1.67

7

1.71

1

1.60

7

1.59

8

1.66

4

56.2

5°

1.68

8

1.60

0

1.58

6

1.66

7

1.69

0

1.60

3

1.57

6

1.66

6

1.68

9

1.59

8

1.58

8

1.65

1

67.5

°

1.71

6

1.64

5

1.64

3

1.71

3

1.71

8

1.64

8

1.63

3

1.71

2

1.71

7

1.64

3

1.64

6

1.69

5

78.7

5°

1.76

1

1.70

1

1.69

9

1.78

5

1.76

3

1.70

5

1.68

6

1.78

5

1.76

1

1.69

9

1.70

2

1.76

3

90°

1.85

8

1.80

1

1.81

4

1.92

2

1.86

1

1.80

2

1.79

6

1.92

3

1.85

7

1.79

7

1.81

8

1.88

9

Tab

le

B.9

: N

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Kj/

{P\f

nb),

at

no

dal

poin

ts a

long

the

sem

i-el

lipt

ical

cr

ack-

fron

t, as

de

nned

by

4>,

in a

pre

ssur

ized

cur

ved

tubi

ng f

or R

*=10

, k=

2.5

0

45°

67.5

°

90°

b/W

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

<\>

0°

1.43

0

1.30

4

1.30

1

1.41

5

1.43

4

1.30

6

1.29

9

1.43

1

1.43

3

1.30

6

1.29

9

1.43

4

11.2

5°

1.41

9

1.29

1

1.28

6

1.39

4

1.42

3

1.29

2

1.28

4

1.41

0

1.42

2

1.29

3

1.28

4

1.41

4

22.5

°

1.43

1

1.30

5

1.29

9

1.39

2

1.43

4

1.30

7

1.29

7

1.40

8

1.43

4

1.30

7

1.29

7

1.41

2

33.7

5°

1.41

3

1.28

6

1.27

8

1.35

8

1.41

6

1.28

8

1.27

6

1.37

5

1.41

6

1.28

8

1.27

6

1.37

9

45°

1.42

4

1.30

6

1.29

5

1.35

3

1.42

7

1.30

8

1.29

3

1.36

8

1.42

7

1.30

8

1.29

3

1.37

2

56.2

5°

1.40

9

1.29

2

1.28

1

1.33

1

1.41

3

1.29

5

1.27

9

1.34

7

1.41

2

1.29

5

1.27

9

1.35

2

67.5

°

1.43

9

1.33

9

1.33

1

1.36

5

1.44

3

1.34

2

1.33

0

1.38

2

1.44

2

1.34

3

1.33

0

1.38

7

78.7

5°

1.47

9

1.37

2

1.36

4

1.40

6

1.48

4

1.37

5

1.36

3

1.42

7

1.48

3

1.37

6

1.36

4

1.43

2

90°

1.57

8

1.46

2

1.46

5

1.51

6

1.58

3

1.46

6

1.46

4

1.54

4

1.58

2

1.46

7

1.46

5

1.55

1

Appendix C

Influence Coefficients and Stress Intensity

Factors for a Crack in a Pipe Bend

C. l Influence Coefficients

120

a3 cj

121

a o

o a "3 -d o

v

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a (!) O

effi

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( i i

fin

03 - a

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o

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m r- CO

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l O

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1—I

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o

* • * *

^

£ -o

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i n r-CO

o

CM OS CO

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as 0 0 CO

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CM

o

^F O CO

o

-*1 r-CM o

CO ^F CO

o

* CN

^

CO O

o CO o o

r-CO o o

CO F O O

CM t-~ O

o

1 — t

CO 1 — 1

o

CO F

i-H

O

1 — t

CO CM

o

1—1

o CM o

CO r CM

o

# CO * • — • - <

^

t^ o 00 o

CO 00 I--o

-F r~ t-~

o

00 CO r~-o

in a> t-o

i—i

o 00 o

as CM OO

o

00 CM OO

o

CO - 00

o

* o

^

-F CO i — i

o

en a> i — i

o

i — i

F CM

o

t--a> CM CD

i — l

O0 CO

o

00 o ^F O

CM 00 ^F O

O r-^F O

1 -i — t

in

o

* rH

^

00 CO O

o

1—1

OO o o

CM o i-H

o

CO ^F i-H

O

CO CM CM

o

CO in CM O

i-H

^F CO O

00 CM CO

o

t--00 CO

o

* cs

k

00

o

t-~ CO o o

CO ^F o o

CM

m o o

T — f

00 o o

(M ^F 1 — 1

o

-* h-r-H

o

1 — 1

CO CM

o

00 ^F CM

o

CO 1 — I

CO

o

* CO

-—-'-* ^

Tab

le C

.5:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

^*

= K

^/[A

^b/W

yVnb

],

at n

odal

poi

nts

alon

g a

sem

i-el

lipt

ical

cr

ack-

fron

t as

def

ined

by

<j>

for

R*=

5,

k—2,

6=

67.5

°

b/W

0.2

0.4

0.6

0.8

Kf*

K(2

>

<f> 0

°

0.68

6

0.43

8

0.33

2

0.26

8

0.69

2

0.43

9

0.33

2

0.26

8

0.73

3

0.45

4

0.34

0

0.27

3

0.83

8

0.51

5

0.38

6

0.31

2

11.2

5°

0.67

3

0.38

8

0.26

8

0.19

9

0.67

7

0.38

8

0.26

7

0.19

8

0.71

5

0.40

1

0.27

4

0.20

1

0.82

3

0.46

8

0.32

7

0.24

7

22.5

°

0.68

3

0.41

6

0.30

1

0.23

2

0.68

8

0.41

6

0.29

9

0.22

9

0.72

7

0.42

8

0.30

4

0.23

1

0.82

5

0.48

0

0.34

0

0.26

0

33.7

5°

0.66

3

0.34

7

0.21

8

0.14

8

0.66

6

0.34

5

0.21

5

0.14

5

0.70

2

0.35

5

0.21

9

0.14

6

0.79

7

0.40

7

0.25

5

0.17

4

45°

0.66

9

0.34

4

0.21

0

0.13

8

0.67

4

0.34

0

0.20

5

0.13

3

0.71

0

0.34

7

0.20

5

0.13

1

0.79

2

0.38

0

0.22

3

0.14

2

56.2

5°

0.64

8

0.26

4

0.13

4

0.07

5

0.65

1

0.26

0

0.13

0

0.07

2

0.68

4

0.26

6

0.13

1

0.07

2

0.76

5

0.29

7

0.14

6

0.08

1

67.5

°

0.65

6

0.22

5

0.09

9

0.05

0

0.66

1

0.21

9

0.09

4

0.04

7

0.69

5

0.22

2

0.09

4

0.04

8

0.77

2

0.24

1

0.10

2

0.05

2

78.7

5°

0.66

2

0.16

7

0.06

5

0.03

3

0.66

4

0.16

6

0.06

5

0.03

3

0.69

7

0.17

6

0.07

1

0.03

7

0.78

0

0.19

8

0.08

1

0.04

3

90°

0.68

3

0.11

9

0.04

5

0.02

3

0.68

5

0.12

4

0.04

8

0.02

5

0.72

2

0.13

9

0.05

6

0.03

0

0.80

0

0.16

2

0.06

8

0.03

7

Tab

le C

.6:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j =

Kj/

\Ai(

b/W

)l\/~T

Tb\

, at

nod

al p

oint

s al

ong

a se

mi-

elli

ptic

al

crac

k-fr

ont

as d

efin

ed b

y (/

) fo

r R

*=5,

k=

2, 0

=90

°

b/W

0.2

0.4

0.6

0.8

Kf*

K(D

*

K(2

)*

Kf)*

K(2

).

Kf]*

</>

0°

0.68

6

0.43

8

0.33

2

0.26

8

0.69

2

0.43

9

0.33

2

0.26

8

0.73

2

0.45

4

0.34

0

0.27

3

0.83

7

0.51

5

0.38

6

0.31

2

11.2

5°

0.67

3

0.38

8

0.26

8

0.19

9

0.67

7

0.38

8

0.26

7

0.19

8

0.71

5

0.40

1

0.27

4

0.20

1

0.82

2

0.46

8

0.32

7

0.24

7

22.5

°

0.68

3

0.41

6

0.30

1

0.23

2

0.68

8

0.41

6

0.29

9

0.22

9

0.72

7

0.42

8

0.30

4

0.23

1

0.82

4

0.48

0

0.34

0

0.26

0

33.7

5°

0.66

3

0.34

7

0.21

8

0.14

8

0.66

6

0.34

5

0.21

5

0.14

5

0.70

2

0.35

5

0.21

9

0.14

6

0.79

7

0.40

7

0.25

5

0.17

4

45°

0.66

9

0.34

4

0.21

0

0.13

8

0.67

4

0.34

0

0.20

5

0.13

3

0.71

0

0.34

7

0.20

5

0.13

1

0.79

1

0.38

0

0.22

3

0.14

2

56.2

5°

0.64

8

0.26

4

0.13

4

0.07

5

0.65

1

0.26

0

0.13

0

0.07

2

0.68

4

0.26

6

0.13

1

0.07

2

0.76

4

0.29

6

0.14

6

0.08

1

67.5

°

0.65

6

0.22

5

0.09

9

0.05

0

0.66

1

0.21

9

0.09

4

0.04

7

0.69

5

0.22

3

0.09

4

0.04

8

0.77

1

0.24

0

0.10

2

0.05

2

78.7

5°

0.66

2

0.16

7

0.06

5

0.03

3

0.66

4

0.16

6

0.06

5

0.03

3

0.69

7

0.17

6

0.07

1

0.03

7

0.77

8

0.19

7

0.08

1

0.04

3

90°

0.68

3

0.12

0

0.04

5

0.02

3

0.68

5

0.12

4

0.04

8

0.02

5

0.72

2

0.13

9

0.05

6

0.03

0

0.79

8

0.16

2

0.06

7

0.03

7

Tab

le C

.7:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j =

Kj

/[A

i{b/

W)%

\fii

b],

at n

odal

poi

nts

alon

g a

sem

i-el

lipt

ical

cr

ack-

fron

t as

def

ined

by

(p f

or R

*—5,

k—

2.5,

6=

45°

b/W

0.2

0.4

0.6

Kf*

Kf]*

4> 0°

0.67

3

0.43

4

0.32

9

0.26

7

0.67

7

0.43

4

0.32

9

0.26

6

0.72

0

0.45

4

0.34

4

0.27

8

11.2

5°

0.65

8

0.38

3

0.26

5

0.19

7

0.65

9

0.38

1

0.26

3

0.19

5

0.70

1

0.40

4

0.28

1

0.21

0

22.5

°

0.67

0

0.41

1

0.29

7

0.22

9

0.67

3

0.40

8

0.29

3

0.22

5

0.71

3

0.42

4

0.30

2

0.23

1

33.7

5°

0.64

8

0.34

0

0.21

4

0.14

5

0.64

7

0.33

6

0.20

9

0.14

0

0.68

6

0.35

2

0.21

8

0.14

6

45°

0.65

5

0.33

6

0.20

5

0.13

4

0.65

6

0.32

8

0.19

5

0.12

5

0.69

1

0.33

5

0.19

5

0.12

2

56.2

5°

0.63

1

0.25

6

0.12

9

0.07

2

0.62

8

0.24

7

0.12

1

0.06

7

0.66

0

0.25

3

0.12

2

0.06

6

67.5

°

0.63

9

0.21

6

0.09

3

0.04

7

0.63

7

0.20

4

0.08

5

0.04

2

0.66

9

0.20

2

0.08

2

0.04

1

78.7

5°

0.64

1

0.15

9

0.06

2

0.03

1

0.63

3

0.15

5

0.06

0

0.03

1

0.66

2

0.16

0

0.06

3

0.03

3

90°

0.66

1

0.11

4

0.04

3

0.02

2

0.65

2

0.11

6

0.04

5

0.02

3

0.68

0

0.12

7

0.05

1

0.02

7

to

Tab

le C

.8:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

^*

= K

^/{A

i{b/

W)j"/

:nb},

at

nod

al p

oint

s al

ong

a se

mi-

elli

ptic

al

crac

k-fr

ont

as d

efin

ed b

y 4>

for

R*=

5, k

—2.

5, 6

=67

.5 °

b/W

0.2

0.4

0.6

0.8

Kf*

K(2

)*

Kf] *

</>

0°

0.67

3

0.43

3

0.32

9

0.26

6

0.67

6

0.43

3

0.32

8

0.26

6

0.72

2

0.45

5

0.34

4

0.27

8

0.82

6

0.50

5

0.37

6

0.30

1

11.2

5°

0.65

8

0.38

3

0.26

5

0.19

7

0.65

8

0.38

1

0.26

3

0.19

5

0.70

2

0.40

4

0.28

1

0.21

0

0.80

4

0.45

2

0.31

1

0.23

2

22.5

°

0.67

0

0.41

1

0.29

7

0.22

9

0.67

2

0.40

8

0.29

3

0.22

5

0.71

4

0.42

5

0.30

3

0.23

1

0.80

9

0.46

5

0.32

5

0.24

5

33.7

5°

0.64

8

0.34

0

0.21

4

0.14

5

0.64

7

0.33

5

0.20

9

0.14

0

0.68

7

0.35

2

0.21

9

0.14

6

0.77

1

0.38

5

0.23

5

0.15

6

45°

0.65

4

0.33

6

0.20

5

0.13

4

0.65

5

0.32

8

0.19

5

0.12

5

0.69

2

0.33

5

0.19

5

0.12

2

0.76

4

0.35

5

0.20

1

0.12

4

56.2

5°

0.63

1

0.25

6

0.12

9

0.07

2

0.62

8

0.24

7

0.12

1

0.06

6

0.66

2

0.25

3

0.12

2

0.06

6

0.72

6

0.26

9

0.12

8

0.06

9

67.5

°

0.63

9

0.21

6

0.09

3

0.04

7

0.63

6

0.20

3

0.08

5

0.04

2

0.67

1

0.20

2

0.08

3

0.04

1

0.73

0

0.21

0

0.08

6

0.04

4

78.7

5°

0.64

0

0.15

9

0.06

2

0.03

1

0.63

2

0.15

5

0.06

0

0.03

1

0.66

3

0.16

1

0.06

4

0.03

3

0.72

2

0.17

7

0.07

3

0.03

9

90°

0.66

1

0.11

4

0.04

3

0.02

2

0.65

1

0.11

6

0.04

5

0.02

3

0.67

9

0.12

7

0.05

1

0.02

8

0.72

8

0.14

7

0.06

1

0.03

3

03

129

en d

a o

C 'o a

"3 -a o

^

^

o CJ

CD

8 m~ r j CM

.2 il 43 -« c_> S

£>

l O

OS CD o C a>

a> — .2 -a

S * O "T3

-e-

o CI

.75°

oo

o lO t-4 co

.25°

co in

°

o

m CO

co 0

in

CM

o

m CM i—1

o O

* ^ "K

^ -©

T—(

CO CD O

T—1

" CO

o

o> CO CO

o

1—1

CO CO

o

"* in CO

o

00 • < #

CO

o

a> CO CO

o

00 in CO

o

CO f-CO

o

# o "—"-* ^

r i — i

T — 1

O

a> m i—i

o

CO • — i

CM O

CO in CM

o

CO CO CO

o

o f

CO

o

I—1

1—1

Tf

o

co 00 CO

o

CO CO -«* o

* T — I

^

CO -*f o o

CM CO o o

CO CI o o

05 CM 1—1

o

m o CM o

-*r i—i

CM

o

1 -Ol CM

o

in CO CM

o

C2 CM CO

o

* ca

^

CM

o

CM CM O

o

1—1 CO CO

o

t^ ^f o o

CM t— CO

o

^ CO i — 4

o

in -* i—i

o

O CM CM

o

r-C5 1—1

o

t^ CO CM

o

* m

^

O m CO o

CM CO CO

o

CO CO CO O

f-CM CO

o

in in CO

o

CO ^ CO o

CM 1^ CO O

00 in CO

o

CO t^ CO

o

* o —"-. ^

CO i—i i—i

o

in in i — 4

o

co o CM o

f~ ^f CM

o

00 CM CO

o

m CO CO

o

00 o "d1 o

t — i

oo CO

o

CO CO Tj<

o

* ,-t

^

t

in ^f o o

o CO o o

m OO o o

r—1

CM 1—4

o

in CI i—i

o

ai o (M o

CO C7i CM

o

co CO CM

o

oo CM CO O

* <N

^

o

CO CM O

o

1—1

co o o

CM xf o o

CO CO o o

m CM 1—1

o

o r i—i

o

in CM CM

o

m Oi i—i

o

CO CO CM

o

# m

i<

o> t— CO O

(M CO CO

o

1—1

I— CO

o

1—1

CO CO

o

CM en CO

o

t--oo CO

o

"sf i—i

r-~ o

CM O f-

o

CM CM 1>

o

* o

^

r~ CM i—i

o

1—1

co i—(

o

CM o CM o

CO m CM o

in CO CO

o

CM m CO o

m CM ^ O

J< o ^t4 o

m m • ^

o

# ^ ^

I—I

in o o

M< CO o o

CO 00 o o

CM CM 1 — I

o

in en i—i

o

Oi 1—1

CM

o

CO o co o

1—4

«> CM

o

^f " CO

o

*

^

CO

o

00 CM o o

co co o o

1—1

r o o

CO CO o o

CM CM 1—1

o

CO • ^

1—1

o

1 — 1

co CM

o

o 1—1 CM

o

C5 1^ CM

o

*

^—->--. ^

CO CM t~~

o

CM CM r-o

o CO r-o

CO CM r--o

in CO r-o

1—1

1^ f-

o

CT> o 00 o

- o 00 o

CO CM OO

o

* o - *-. ^

t---# i — i

CO

00 !>• i—4

O

o 1—1

CM

o

05 CO CM

o

CO in CO

o

in 00 CO

o

m co -** o

CM in t< o

in CD in

o

* H

* •—. ^

r-t CO o o

CO r-o o

CO O0 o o

00 CM 1—1

o

1—1

o CM o

CO CO CM

o

in CM CO

o

1—1 1—1

CO

o

CO r-co o

* <M

—'*-* ^

OO

o

"* co o CO

05 CO o o

^ t< o o

02 CO o o

^f CM 1—1

o

CO in i—4

O

in xt< CM

o

CM CO CM

o

1—1

o CO o

* m —'*-* X

a> -y O

E2 •3

Tab

le

C.1

0:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j —

Kj

/[A

^b/W

yVir

b],

at

noda

l po

ints

alo

ng a

sem

i-el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

j> f

or /

?*=

7.5,

k=

1.5,

0

=4

5°

b/W

0.2

0.4

0.6

0.8

Kf*

K[3)

*

4> 0°

0.70

9

0.45

8

0.35

3

0.29

1

0.71

2

0.44

8

0.33

7

0.27

1

0.74

3

0.45

9

0.34

3

0.27

6

0.83

0

0.51

5

0.38

8

0.31

5

11.2

5°

0.69

8

0.41

8

0.30

0

0.23

1

0.70

0

0.39

7

0.27

3

0.20

2

0.73

1

0.40

9

0.27

9

0.20

6

0.82

1

0.47

1

0.33

1

0.25

2

22.5

°

0.70

4

0.43

6

0.32

1

0.25

2

0.70

9

0.42

6

0.30

6

0.23

6

0.74

0

0.43

7

0.31

2

0.23

9

0.82

3

0.48

7

0.34

9

0.27

1

33.7

5°

0.68

9

0.37

5

0.24

5

0.17

3

0.69

0

0.35

7

0.22

4

0.15

2

0.72

2

0.36

8

0.22

9

0.15

5

0.80

5

0.41

9

0.26

8

0.18

6

45°

0.69

1

0.36

2

0.22

7

0.15

2

0.69

7

0.35

5

0.21

7

0.14

3

0.72

8

0.36

4

0.22

1

0.14

5

0.80

3

0.40

1

0.24

5

0.16

2

56.2

5°

0.67

4

0.28

7

0.15

0

0.08

7

0.67

7

0.27

6

0.14

1

0.08

0

0.71

0

0.28

6

0.14

5

0.08

2

0.78

5

0.32

1

0.16

6

0.09

5

67.5

°

0.67

8

0.24

2

0.10

8

0.05

6

0.68

6

0.23

9

0.10

6

0.05

5

0.72

2

0.24

8

0.11

1

0.05

7

0.79

2

0.27

2

0.12

2

0.06

4

78.7

5°

0.68

8

0.17

7

0.06

9

0.03

5

0.69

6

0.18

0

0.07

2

0.03

7

0.73

7

0.19

2

0.07

8

0.04

1

0.81

2

0.21

5

0.08

8

0.04

6

90°

0.72

1

0.12

4

0.04

8

0.02

4

0.72

5

0.13

3

0.05

2

0.02

7

0.77

2

0.14

9

0.06

0

0.03

2

0.85

9

0.17

3

0.07

1

0.03

8

Tab

le

C.l

l:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

y*

=

Kj

/{A

i(b/

W)%

\fT

rb],

at

no

dal

poin

ts a

long

el

lipt

ical

cra

ck-f

ront

as

defi

ned

by 4

> f

or R

*=7.

5,

k=l.5

, 9=

67.5

°

b/W

0.2

0.4

0.6

0.8

KP*

K(°>

0 0

°

0.70

8

0.45

8

0.35

3

0.29

1

0.71

2

0.44

8

0.33

7

0.27

1

0.74

2

0.45

9

0.34

3

0.27

5

0.82

9

0.51

5

0.38

8

0.31

5

11.2

5°

0.69

8

0.41

8

0.30

0

0.23

1

0.70

0

0.39

7

0.27

3

0.20

2

0.73

1

0.40

9

0.27

9

0.20

6

0.82

0

0.47

1

0.33

1

0.25

2

22.5

°

0.70

4

0.43

5

0.32

1

0.25

2

0.70

9

0.42

6

0.30

6

0.23

6

0.73

9

0.43

7

0.31

2

0.23

9

0.82

1

0.48

6

0.34

9

0.27

1

33.7

5°

0.68

8

0.37

4

0.24

5

0.17

2

0.69

0

0.35

7

0.22

4

0.15

2

0.72

1

0.36

7

0.22

9

0.15

5

0.80

4

0.41

9

0.26

8

0.18

6

45°

0.69

1

0.36

2

0.22

7

0.15

2

0.69

6

0.35

5

0.21

7

0.14

3

0.72

7

0.36

4

0.22

1

0.14

5

0.80

1

0.40

0

0.24

4

0.16

2

56.2

5°

0.67

4

0.28

7

0.15

0

0.08

6

0.67

7

0.27

6

0.14

1

0.08

0

0.71

0

0.28

6

0.14

5

0.08

2

0.78

4

0.32

1

0.16

5

0.09

5

67.5

°

0.67

8

0.24

2

0.10

8

0.05

5

0.68

6

0.23

9

0.10

6

0.05

5

0.72

2

0.24

8

0.11

0

0.05

7

0.79

1

0.27

2

0.12

1

0.06

3

78.7

5°

0.68

7

0.17

7

0.06

9

0.03

5

0.69

6

0.18

0

0.07

2

0.03

7

0.73

6

0.19

2

0.07

8

0.04

0

0.81

1

0.21

5

0.08

8

0.04

6

90°

0.72

1

0.12

4

0.04

8

0.02

4

0.72

5

0.13

3

0.05

2

0.02

7

0.77

2

0.14

9

0.06

0

0.03

2

0.85

8

0.17

2

0.07

1

0.03

8

Tab

le

C.1

2:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j *

=

K^/

[Ai{

b/W

)i\fnb

] :

at

noda

l po

ints

alo

ng a

el

lipt

ical

cra

ck-f

ront

as

defi

ned

by 

K(i

>

K(o

>

K(i

>

K(2

>

4> 0

°

0.70

8

0.45

8

0.35

3

0.29

1

0.71

1

0.44

7

0.33

7

0.27

1

0.74

3

0.45

9

0.34

3

0.27

5

0.82

7

0.51

5

0.38

8

0.31

4

11.2

5°

0.69

9

0.41

8

0.30

0

0.23

1

0.69

9

0.39

7

0.27

3

0.20

2

0.73

1

0.40

9

0.27

9

0.20

6

0.81

8

0.47

0

0.33

1

0.25

1

22.5

°

0.70

4

0.43

5

0.32

0

0.25

2

0.70

9

0.42

6

0.30

6

0.23

6

0.73

9

0.43

7

0.31

2

0.23

9

0.81

9

0.48

6

0.34

9

0.27

0

33.7

5°

0.68

8

0.37

4

0.24

5

0.17

2

0.69

0

0.35

7

0.22

4

0.15

2

0.72

2

0.36

7

0.22

9

0.15

5

0.80

1

0.41

8

0.26

8

0.18

6

45°

0.69

1

0.36

2

0.22

7

0.15

2

0.69

6

0.35

4

0.21

7

0.14

3

0.72

8

0.36

4

0.22

1

0.14

5

0.79

8

0.39

9

0.24

4

0.16

1

56.2

5°

0.67

4

0.28

7

0.15

0

0.08

7

0.67

7

0.27

6

0.14

1

0.08

0

0.71

1

0.28

6

0.14

5

0.08

2

0.78

1

0.32

0

0.16

5

0.09

5

67.5

°

0.67

8

0.24

2

0.10

8

0.05

6

0.68

5

0.23

9

0.10

6

0.05

5

0.72

2

0.24

8

0.11

1

0.05

7

0.78

7

0.27

1

0.12

1

0.06

3

78.7

5°

0.68

8

0.17

7

0.06

9

0.03

5

0.69

5

0.17

9

0.07

2

0.03

7

0.73

7

0.19

2

0.07

8

0.04

1

0.80

7

0.21

3

0.08

7

0.04

6

90°

0.72

1

0.12

4

0.04

8

0.02

4

0.72

5

0.13

2

0.05

2

0.02

7

0.77

3

0.14

9

0.06

0

0.03

2

0.85

4

0.17

1

0.07

0

0.03

8

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Tab

le

C.1

4:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

^*

=

K\l)

/[A

i{b/

W)i\fT

rb},

at

no

dal

poin

ts a

long

a

elli

ptic

al c

rack

-fro

nt a

s de

fine

d by

<\>

for

R*=

7.5,

k=

2,

6=67

.5°

b/W

0.2

0.4

0.6

0.8

Kf*

Kf]*

K(2

>

K?]*

4> 0°

0.69

2

0.45

3

0.35

0

0.28

9

0.69

7

0.45

4

0.35

0

0.28

9

0.72

9

0.45

5

0.34

1

0.27

4

0.83

2

0.51

6

0.38

8

0.31

4

11.2

5°

0.68

1

0.41

1

0.29

6

0.22

9

0.68

4

0.41

2

0.29

6

0.22

8

0.71

2

0.40

2

0.27

5

0.20

3

0.81

5

0.46

9

0.32

9

0.25

0

22.5

°

0.68

8

0.42

9

0.31

7

0.24

9

0.69

3

0.43

0

0.31

6

0.24

8

0.72

4

0.43

0

0.30

7

0.23

5

0.81

8

0.48

3

0.34

6

0.26

6

33.7

5°

0.67

1

0.36

7

0.24

1

0.16

9

0.67

3

0.36

7

0.23

9

0.16

8

0.69

9

0.35

8

0.22

3

0.15

0

0.78

8

0.41

1

0.26

2

0.18

0

45°

0.67

4

0.35

5

0.22

2

0.14

9

0.67

7

0.35

3

0.21

9

0.14

5

0.70

6

0.35

3

0.21

3

0.13

8

0.78

2

0.38

8

0.23

4

0.15

2

56.2

5°

0.65

4

0.27

9

0.14

5

0.08

3

0.65

6

0.27

6

0.14

2

0.08

1

0.67

9

0.27

2

0.13

6

0.07

6

0.75

2

0.30

4

0.15

4

0.08

7

67.5

°

0.65

8

0.23

3

0.10

3

0.05

3

0.66

2

0.22

9

0.10

0

0.05

1

0.68

9

0.23

1

0.10

1

0.05

1

0.75

3

0.25

0

0.10

9

0.05

6

78.7

5°

0.66

4

0.16

9

0.06

5

0.03

3

0.66

4

0.16

8

0.06

5

0.03

3

0.68

7

0.17

7

0.07

1

0.03

7

0.75

6

0.19

7

0.08

0

0.04

2

90°

0.69

1

0.11

7

0.04

5

0.02

3

0.68

6

0.12

1

0.04

7

0.02

4

0.70

9

0.13

6

0.05

5

0.02

9

0.78

4

0.15

7

0.06

5

0.03

5

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o LO 1—1

o

LO CO CM

o

CO

o CM CD

T * f~ CM

o

*

*<

CM

oo 1—

o

co LO t—

o

o LO t—

CD

OS

^ c-o

as I— !> o

LO 00 t^

o

LO 1—1 00

o

CM 1—1 00

CD

o CO 00

o

*

^

CO LO 1—1

o

CO o> 1—1

o

as - CM

o

CO o CO

o

t^ 00 CO

o

o 1—1 -tf

o

CM 00 -xt<

o

oo CO TJ<

<=:

CO I—l LO

o

#

^

LO CO

o o

o 00

o o

OS

o 1—1 CO

- LO 1—1

o

CO co CM

o

1—1 CO CM

o

CO

^ CO o

as CM CO

CD

00 00 CO

o

*

^

00

o

LO

co o o

CM

^ o o

CO LO

o o

r-~ 00

o o

CM LO I—l

c

o 00 1—I

o

CO CO CM

o

CO LO CM

CD

^f I—l CO

o

*

X

Tab

le

C.1

6:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

^*

=

K^/

[Ai(

b/W

yVnb

},

at

noda

l po

ints

alo

ng a

sem

i-el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

j> f

or R

*=7.

5, f

c=2.

5, 0

=4

5°

b/W

0.2

0.4

0.6

0.8

K(*

)*

Kf]"

K?]*

K(2

)*

Kf)*

K(2

)*

<t> 0°

0.67

8

0.44

8

0.34

8

0.28

7

0.67

7

0.43

5

0.33

0

0.26

7

0.72

0

0.45

2

0.33

9

0.27

3

0.83

4

0.51

6

0.38

8

0.31

4

11.2

5°

0.66

6

0.40

6

0.29

3

0.22

7

0.65

9

0.38

2

0.26

5

0.19

7

0.70

0

0.39

8

0.27

2

0.20

1

0.81

2

0.46

7

0.32

8

0.24

8

22.5

°

0.67

5

0.42

4

0.31

3

0.24

7

0.67

3

0.41

1

0.29

7

0.22

8

0.71

4

0.42

5

0.30

3

0.23

1

0.81

5

0.48

0

0.34

2

0.26

2

33.7

5°

0.65

6

0.36

1

0.23

7

0.16

7

0.64

6

0.33

9

0.21

3

0.14

4

0.68

3

0.35

1

0.21

8

0.14

6

0.77

8

0.40

4

0.25

5

0.17

4

45°

0.65

9

0.34

8

0.21

7

0.14

5

0.65

5

0.33

5

0.20

3

0.13

2

0.69

0

0.34

3

0.20

5

0.13

1

0.76

9

0.37

7

0.22

3

0.14

3

56.2

5°

0.63

8

0.27

1

0.14

1

0.08

0

0.62

5

0.25

3

0.12

7

0.07

1

0.65

6

0.26

0

0.12

9

0.07

2

0.73

0

0.29

0

0.14

4

0.08

1

67.5

°

0.64

2

0.22

5

0.09

9

0.05

0

0.63

4

0.21

3

0.09

2

0.04

6

0.66

4

0.21

7

0.09

2

0.04

7

0.72

5

0.23

1

0.09

9

0.05

0

78.7

5°

0.64

2

0.16

1

0.06

2

0.03

1

0.62

5

0.15

6

0.06

1

0.03

1

0.65

0

0.16

5

0.06

6

0.03

5

0.71

4

0.18

3

0.07

5

0.04

0

90°

0.66

8

0.11

1

0.04

2

0.02

1

0.64

1

0.11

4

0.04

4

0.02

3

0.66

6

0.12

7

0.05

1

0.02

8

0.72

9

0.14

6

0.06

1

0.03

3

Tab

le

C.1

7:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

y*

=

Kj

/[A

i(b/W

)i\fii

b\,

at

noda

l po

ints

alo

ng a

el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

j> f

or R

*=7.

5, f

c=2.

5, 0

=67

.5°

b/W

0.2

0.4

0.6

0.8

Kf*

K(2

>

0 0°

0.67

8

0.44

8

0.34

8

0.28

7

0.67

7

0.43

5

0.33

0

0.26

7

0.72

0

0.45

2

0.33

9

0.27

3

0.83

5

0.51

7

0.38

8

0.31

4

11.2

5°

0.66

6

0.40

6

0.29

3

0.22

7

0.65

9

0.38

2

0.26

5

0.19

6

0.70

0

0.39

8

0.27

3

0.20

1

0.81

3

0.46

8

0.32

8

0.24

8

22.5

°

0.67

5

0.42

4

0.31

3

0.24

7

0.67

2

0.41

1

0.29

7

0.22

8

0.71

4

0.42

6

0.30

3

0.23

1

0.81

6

0.48

1

0.34

2

0.26

2

33.7

5°

0.65

6

0.36

1

0.23

7

0.16

6

0.64

6

0.33

9

0.21

3

0.14

4

0.68

3

0.35

1

0.21

8

0.14

6

0.78

0

0.40

4

0.25

6

0.17

5

45°

0.65

9

0.34

8

0.21

7

0.14

5

0.65

4

0.33

5

0.20

3

0.13

2

0.69

1

0.34

3

0.20

5

0.13

1

0.77

0

0.37

7

0.22

4

0.14

3

56.2

5°

0.63

8

0.27

1

0.14

1

0.08

0

0.62

5

0.25

3

0.12

7

0.07

1

0.65

6

0.26

0

0.12

9

0.07

2

0.73

1

0.29

0

0.14

5

0.08

1

67.5

°

0.64

1

0.22

5

0.09

9

0.05

0

0.63

3

0.21

3

0.09

2

0.04

6

0.66

4

0.21

7

0.09

2

0.04

7

0.72

5

0.23

2

0.09

9

0.05

1

78.7

5°

0.64

2

0.16

1

0.06

2

0.03

1

0.62

4

0.15

6

0.06

1

0.03

1

0.65

0

0.16

5

0.06

6

0.03

5

0.71

5

0.18

3

0.07

5

0.04

0

90°

0.66

8

0.11

1

0.04

2

0.02

1

0.64

1

0.11

4

0.04

4

0.02

3

0.66

6

0.12

7

0.05

1

0.02

8

0.72

9

0.14

6

0.06

1

0.03

4

Tab

le

C.1

8:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j —

K,

/[A

i(b/

WY

\/nb

],

at

noda

l po

ints

alo

ng a

sem

i-el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

j> f

or /

2*=

7.5,

&=2

.5,

0=90

°

b/W

0.2

0.4

0.6

0.8

K{p*

<t> 0°

0.67

8

0.44

8

0.34

7

0.28

7

0.67

7

0.43

5

0.33

0

0.26

7

0.72

0

0.45

2

0.33

9

0.27

3

0.83

0

0.51

5

0.38

7

0.31

3

11.2

5°

0.66

6

0.40

6

0.29

3

0.22

7

0.65

9

0.38

2

0.26

5

0.19

7

0.70

0

0.39

8

0.27

3

0.20

1

0.80

7

0.46

6

0.32

7

0.24

8

22.5

°

0.67

5

0.42

4

0.31

3

0.24

7

0.67

2

0.41

1

0.29

7

0.22

8

0.71

4

0.42

6

0.30

3

0.23

2

0.81

1

0.47

8

0.34

1

0.26

1

33.7

5°

0.65

6

0.36

1

0.23

7

0.16

7

0.64

6

0.33

9

0.21

3

0.14

4

0.68

3

0.35

1

0.21

8

0.14

6

0.77

4

0.40

2

0.25

5

0.17

4

45°

0.65

9

0.34

8

0.21

7

0.14

5

0.65

5

0.33

5

0.20

3

0.13

2

0.69

1

0.34

3

0.20

5

0.13

1

0.76

6

0.37

6

0.22

3

0.14

3

56.2

5°

0.63

8

0.27

1

0.14

1

0.08

0

0.62

5

0.25

3

0.12

7

0.07

1

0.65

6

0.26

0

0.12

9

0.07

2

0.72

6

0.28

8

0.14

4

0.08

0

67.5

°

0.64

1

0.22

5

0.09

9

0.05

0

0.63

3

0.21

3

0.09

2

0.04

6

0.66

4

0.21

7

0.09

2

0.04

7

0.72

1

0.23

0

0.09

8

0.05

0

78.7

5°

0.64

2

0.16

1

0.06

2

0.03

1

0.62

4

0.15

6

0.06

1

0.03

1

0.65

0

0.16

5

0.06

6

0.03

5

0.71

0

0.18

1

0.07

4

0.03

9

90°

0.66

8

0.11

1

0.04

2

0.02

1

0.64

1

0.11

4

0.04

4

0.02

3

0.66

6

0.12

7

0.05

1

0.02

8

0.72

4

0.14

4

0.06

0

0.03

3

Tab

le

C.1

9:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j =

K

/'/[A

i(b/

W)lV

nb],

at

no

dal

poin

ts a

long

el

lipt

ical

cra

ck-f

ront

as

defi

ned

by 0

for

R*=

10,

&=

1.5,

#=

45

°

b/W

0.2

0.4

0.6

0.8

K?'

K(2

)*

K( 0

)*

K(D

*

R(2

>

K<f

i).

4> 0°

0.70

8

0.45

8

0.35

3

0.29

1

0.71

1

0.44

8

0.33

7

0.27

2

0.74

2

0.45

9

0.34

3

0.27

6

0.83

2

0.51

7

0.38

9

0.31

6

11.2

5°

0.69

8

0.41

8

0.30

0

0.23

1

0.69

9

0.39

7

0.27

3

0.20

2

0.73

0

0.40

9

0.27

9

0.20

6

0.82

2

0.47

2

0.33

2

0.25

3

22.5

°

0.70

4

0.43

6

0.32

1

0.25

2

0.70

9

0.42

6

0.30

7

0.23

6

0.73

8

0.43

7

0.31

2

0.24

0

0.82

3

0.48

8

0.35

1

0.27

2

33.7

5°

0.68

8

0.37

5

0.24

5

0.17

3

0.69

0

0.35

7

0.22

5

0.15

3

0.72

0

0.36

8

0.23

0

0.15

6

0.80

5

0.42

1

0.27

0

0.18

8

45°

0.69

1

0.36

3

0.22

7

0.15

3

0.69

6

0.35

5

0.21

8

0.14

4

0.72

6

0.36

5

0.22

3

0.14

6

0.80

1

0.40

3

0.24

7

0.16

4

56.2

5°

0.67

4

0.28

7

0.15

1

0.08

7

0.67

6

0.27

7

0.14

2

0.08

0

0.70

8

0.28

6

0.14

6

0.08

3

0.78

3

0.32

3

0.16

8

0.09

7

67.5

°

0.67

7

0.24

2

0.10

9

0.05

6

0.68

5

0.24

0

0.10

7

0.05

5

0.71

9

0.24

9

0.11

2

0.05

8

0.78

9

0.27

5

0.12

4

0.06

5

78.7

5°

0.68

7

1.77

1

0.06

9

0.03

5

0.69

5

0.18

0

0.07

2

0.03

7

0.73

2

0.19

1

0.07

8

0.04

0

0.80

9

0.21

5

0.08

8

0.04

6

90°

0.72

0

0.12

4

0.04

8

0.02

4

0.72

2

0.13

2

0.05

2

0.02

7

0.76

8

0.14

8

0.06

0

0.03

1

0.85

6

0.17

2

0.07

1

0.03

8

Tab

le

C.2

0:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

, =

K

, /[

A^b

/Wyy

fnb]

, at

no

dal

poin

ts a

long

a s

emi-

elli

ptic

al c

rack

-fro

nt a

s de

fine

d by

(f>

for

/?*

=10

, &

=1.5

, 0=

67.5

°

b/W

0.2

0.4

0.6

0.8

KW

*

Kf)*

Kf)*

K(2

)*

<P 0

°

0.70

8

0.45

8

0.35

3

0.29

1

0.71

0

0.44

7

0.33

7

0.27

1

0.74

2

0.45

9

0.34

4

0.27

6

0.83

1

0.51

6

0.38

9

0.31

6

11.2

5°

0.69

8

0.41

8

0.30

0

0.23

1

0.69

7

0.39

7

0.27

3

0.20

2

0.73

0

0.40

9

0.27

9

0.20

6

0.82

1

0.47

2

0.33

2

0.25

3

22.5

°

0.70

4

0.43

5

0.32

1

0.25

2

0.70

7

0.42

5

0.30

6

0.23

6

0.73

9

0.43

7

0.31

2

0.24

0

0.82

2

0.48

8

0.35

1

0.27

2

33.7

5°

0.68

8

0.37

5

0.24

5

0.17

3

0.68

8

0.35

7

0.22

4

0.15

3

0.72

1

0.36

8

0.23

0

0.15

6

0.80

4

0.42

0

0.27

0

0.18

8

45°

0.69

0

0.36

3

0.22

7

0.15

3

0.69

4

0.35

5

0.21

8

0.14

4

0.72

7

0.36

5

0.22

3

0.14

6

0.80

0

0.40

2

0.24

7

0.16

4

56.2

5°

0.67

4

0.28

7

0.15

1

0.08

7

0.67

5

0.27

6

0.14

1

0.08

0

0.70

9

0.28

7

0.14

6

0.08

3

0.78

2

0.32

3

0.16

7

0.09

7

67.5

°

0.67

7

0.24

2

0.10

9

0.05

6

0.68

3

0.23

9

0.10

7

0.05

5

0.72

0

0.25

0

0.11

2

0.05

8

0.78

8

0.27

4

0.12

4

0.06

5

78.7

5°

0.68

7

0.17

7

0.06

9

0.03

5

0.69

2

0.17

9

0.07

1

0.03

6

0.73

4

0.19

2

0.07

8

0.04

0

0.80

6

0.21

5

0.08

8

0.04

6

90°

0.72

0

0.12

4

0.04

8

0.02

4

0.72

1

0.13

1

0.05

2

0.02

7

0.77

0

0.14

9

0.06

0

0.03

2

0.85

5

0.17

1

0.07

1

0.03

8

Tab

le

C.2

1:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j =

K

j /[

Ai(

b/W

)ly/nb

],

at

noda

l po

ints

alo

ng a

sem

i-el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

f> f

or i

?*=

10,

A;=

1.5,

#=

90

°

b/W

0.2

0.4

0.6

0.8

Kf*

Kf)*

K(o

>

Kf]*

K(2

>

0 0°

0.70

8

0.45

8

0.35

3

0.29

0

0.71

0

0.44

7

0.33

7

0.27

1

0.73

8

0.45

8

0.34

3

0.27

5

0.82

9

0.51

6

0.38

9

0.31

5

11.2

5°

0.69

8

0.41

7

0.30

0

0.23

1

0.69

8

0.39

7

0.27

3

0.20

2

0.72

6

0.40

7

0.27

8

0.20

6

0.81

9

0.47

1

0.33

1

0.25

2

22.5

°

0.70

4

0.43

5

0.32

1

0.25

2

0.70

8

0.42

5

0.30

6

0.23

6

0.73

4

0.43

5

0.31

1

0.23

9

0.82

0

0.48

7

0.35

0

0.27

2

33.7

5°

0.68

8

0.37

5

0.24

5

0.17

3

0.68

9

0.35

7

0.22

4

0.15

3

0.71

6

0.36

6

0.22

9

0.15

5

0.80

1

0.41

9

0.26

9

0.18

8

45°

0.69

0

0.36

3

0.22

7

0.15

3

0.69

5

0.35

5

0.21

8

0.14

4

0.72

2

0.36

4

0.22

2

0.14

6

0.79

8

0.40

2

0.24

6

0.16

4

56.2

5°

0.67

3

0.28

7

0.15

0

0.08

7

0.67

5

0.27

6

0.14

1

0.08

0

0.70

4

0.28

5

0.14

5

0.08

3

0.78

0

0.32

2

0.16

7

0.09

7

67.5

°

0.67

7

0.24

2

0.10

9

0.05

6

0.68

3

0.24

0

0.10

7

0.05

5

0.71

5

0.24

8

0.11

1

0.05

7

0.78

6

0.27

3

0.12

3

0.06

5

78.7

5°

0.68

8

0.17

7

0.06

9

0.03

5

0.69

3

0.17

9

0.07

1

0.03

7

0.72

8

0.19

0

0.07

7

0.04

0

0.80

5

0.21

4

0.08

8

0.04

6

90°

0.72

0

0.12

4

0.04

8

0.02

4

0.72

2

0.13

2

0.05

2

0.02

7

0.76

3

0.14

6

0.05

9

0.03

1

0.85

2

0.17

0

0.07

0

0.03

8

Tab

le

C.2

2:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

^*

=

K^/

[A^b

/WyV

irb}

, at

no

dal

poin

ts a

long

a

elli

ptic

al c

rack

-fro

nt a

s de

fine

d by

(j)

for

i?*

=10

, k=

2,

6=45

°

b/W

0.2

0.4

0.6

0.8

Kf*

K^

Kf)*

K(2

)*

Kf).

K(2

h

4> 0°

0.69

2

0.45

2

0.35

0

0.28

9

0.69

7

0.45

4

0.35

1

0.28

9

0.73

1

0.45

6

0.34

2

0.27

5

0.82

9

0.51

6

0.38

9

0.31

5

11.2

5°

0.68

1

0.41

1

0.29

7

0.22

9

0.68

4

0.41

2

0.29

7

0.22

9

0.71

4

0.40

3

0.27

6

0.20

4

0.81

1

0.46

9

0.33

0

0.25

1

22.5

°

0.68

8

0.42

9

0.31

7

0.25

0

0.69

2

0.43

0

0.31

7

0.24

9

0.72

6

0.43

2

0.30

9

0.23

7

0.81

4

0.48

4

0.34

8

0.26

9

33.7

5°

0.67

0

0.36

8

0.24

1

0.17

0

0.67

3

0.36

8

0.24

1

0.16

9

0.70

0

0.36

0

0.22

5

0.15

2

0.78

4

0.41

2

0.26

4

0.18

3

45°

0.67

3

0.35

6

0.22

3

0.15

0

0.67

7

0.35

5

0.22

1

0.14

8

0.70

7

0.35

6

0.21

6

0.14

1

0.77

8

0.39

1

0.23

8

0.15

7

56.2

5°

0.65

4

0.27

9

0.14

6

0.08

4

0.65

5

0.27

8

0.14

4

0.08

3

0.68

0

0.27

5

0.13

9

0.07

9

0.74

7

0.30

7

0.15

8

0.09

0

67.5

°

0.65

8

0.23

4

0.10

4

0.05

3

0.66

0

0.23

1

0.10

2

0.05

2

0.68

9

0.23

6

0.10

4

0.05

3

0.74

6

0.25

5

0.11

3

0.05

8

78.7

5°

0.66

3

0.16

9

0.06

5

0.03

3

0.66

2

0.16

8

0.06

6

0.03

3

0.68

7

0.17

8

0.07

2

0.03

8

0.74

7

0.19

6

0.08

0

0.04

2

90°

0.69

0

0.11

7

0.04

5

0.02

3

0.68

4

0.12

0

0.04

7

0.02

4

0.71

0

0.13

6

0.05

5

0.02

9

0.77

7

0.15

4

0.06

4

0.03

4

Tab

le

C.2

3:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

, =

K

, /[

Ai(

b/W

y\fn

b},

at

noda

l po

ints

alo

ng a

sem

i-el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

f> f

or R

*=10

, k=

2,

8=67

.5°

b/W

0.2

0.4

0.6

0.8

K?]*

Kf)*

Kf)*

4> 0°

0.69

2

0.45

2

0.35

0

0.28

9

0.69

7

0.45

4

0.35

1

0.28

9

0.72

7

0.45

4

0.34

1

0.27

4

0.82

8

0.51

6

0.38

8

0.31

5

11.2

5°

0.68

1

0.41

1

0.29

6

0.22

9

0.68

4

0.41

2

0.29

7

0.22

9

0.70

9

0.40

2

0.27

6

0.20

4

0.81

0

0.46

9

0.33

0

0.25

0

22.5

°

0.68

8

0.42

9

0.31

7

0.25

0

0.69

2

0.43

0

0.31

7

0.24

9

0.72

1

0.43

1

0.30

8

0.23

6

0.81

3

0.48

3

0.34

7

0.26

8

33.7

5°

0.67

0

0.36

8

0.24

1

0.17

0

0.67

3

0.36

8

0.24

1

0.16

9

0.69

6

0.35

9

0.22

4

0.15

2

0.78

3

0.41

2

0.26

4

0.18

3

45°

0.67

3

0.35

5

0.22

3

0.15

0

0.67

7

0.35

5

0.22

1

0.14

8

0.70

3

0.35

5

0.21

5

0.14

1

0.77

7

0.39

1

0.23

8

0.15

7

56.2

5°

0.65

4

0.27

9

0.14

6

0.08

4

0.65

5

0.27

8

0.14

4

0.08

3

0.67

5

0.27

3

0.13

8

0.07

8

0.74

6

0.30

7

0.15

8

0.09

0

67.5

°

0.65

7

0.23

4

0.10

4

0.05

3

0.66

1

0.23

1

0.10

2

0.05

2

0.68

4

0.23

4

0.10

3

0.05

3

0.74

5

0.25

4

0.11

3

0.05

8

78.7

5°

0.66

3

0.16

9

0.06

5

0.03

3

0.66

2

0.16

8

0.06

6

0.03

3

0.68

1

0.17

6

0.07

1

0.03

7

0.74

6

0.19

6

0.08

0

0.04

2

90°

0.69

0

0.11

7

0.04

5

0.02

3

0.68

4

0.12

0

0.04

7

0.02

4

0.70

4

0.13

4

0.05

4

0.02

9

0.77

7

0.15

4

0.06

4

0.03

4

Tab

le

C.2

4:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

, =

K

, /[

Ai{

b/W

)i\fK

b],

at

noda

l po

ints

alo

ng a

sem

i-el

lipt

ical

cra

ck-f

ront

as

defi

ned

by (

p fo

r i?

*=10

, k=

2,

8=90

°

b/W

0.2

0.4

0.6

0.8

Kf*

K(o

>

<$>

0°

0.69

2

0.45

3

0.35

0

0.28

9

0.69

6

0.45

4

0.35

1

0.28

9

0.73

1

0.45

6

0.34

2

0.27

5

0.82

4

0.51

5

0.38

8

0.31

5

11.2

5°

0.68

1

0.41

1

0.29

7

0.22

9

0.68

4

0.41

2

0.29

7

0.22

9

0.71

4

0.40

4

0.27

6

0.20

4

0.80

6

0.46

8

0.32

9

0.25

0

22.5

°

0.68

8

0.42

9

0.31

7

0.25

0

0.69

2

0.43

0

0.31

7

0.24

9

0.72

6

0.43

2

0.30

9

0.23

7

0.80

9

0.48

2

0.34

7

0.26

8

33.7

5°

0.67

0

0.36

8

0.24

1

0.17

0

0.67

3

0.36

8

0.24

0

0.16

9

0.70

1

0.36

0

0.22

5

0.15

2

0.77

8

0.41

0

0.26

3

0.18

2

45°

0.67

3

0.35

5

0.22

3

0.15

0

0.67

7

0.35

5

0.22

1

0.14

8

0.70

7

0.35

7

0.21

6

0.14

1

0.77

1

0.38

9

0.23

7

0.15

6

56.2

5°

0.65

4

0.27

9

0.14

6

0.08

4

0.65

5

0.27

7

0.14

4

0.08

2

0.68

0

0.27

5

0.13

9

0.07

9

0.73

9

0.30

5

0.15

7

0.09

0

67.5

°

0.65

7

0.23

4

0.10

4

0.05

3

0.66

0

0.23

1

0.10

2

0.05

2

0.68

9

0.23

6

0.10

4

0.05

3

0.73

8

0.25

4

0.11

2

0.05

8

78.7

5°

0.66

2

0.16

9

0.06

5

0.03

3

0.66

1

0.16

8

0.06

6

0.03

3

0.68

7

0.17

8

0.07

2

0.03

8

0.73

8

0.19

3

0.07

9

0.04

1

90°

0.69

0

0.11

7

0.04

5

0.02

3

0.68

3

0.12

0

0.04

7

0.02

4

0.71

0

0.13

6

0.05

5

0.02

9

0.77

0

0.15

1

0.06

2

0.03

3

Tab

le

C.2

5:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

y =

K

, /{

Ai(

b/W

)i\fnb

},

at

noda

l po

ints

alo

ng a

el

lipt

ical

cra

ck-f

ront

as

defi

ned

by <

f> f

or R

*=10

, k—

2.5,

9=

45°

b/W

0.2

0.4

0.6

0.8

K?*

K(D

*

Kf]*

Kf]*

4> 0

°

0.67

8

0.44

8

0.34

8

0.28

7

0.67

8

0.43

6

0.33

1

0.26

8

0.72

1

0.45

3

0.34

0

0.27

4

0.82

6

0.51

6

0.38

8

0.31

4

11.2

5°

0.66

6

0.40

6

0.29

4

0.22

7

0.65

9

0.38

3

0.26

5

0.19

7

0.70

0

0.39

9

0.27

4

0.20

2

0.80

2

0.46

6

0.32

8

0.24

9

22.5

°

0.67

4

0.42

4

0.31

4

0.24

8

0.67

3

0.41

2

0.29

8

0.23

0

0.71

4

0.42

8

0.30

6

0.23

4

0.80

5

0.48

0

0.34

4

0.26

5

33.7

5°

0.65

6

0.36

2

0.23

8

0.16

8

0.64

7

0.34

1

0.21

5

0.14

6

0.68

4

0.35

3

0.22

1

0.14

9

0.76

8

0.40

5

0.25

8

0.17

8

45°

0.65

9

0.35

0

0.21

9

0.14

7

0.65

5

0.33

8

0.20

6

0.13

5

0.69

0

0.34

8

0.21

0

0.13

6

0.76

0

0.38

1

0.23

0

0.14

9

56.2

5°

0.63

7

0.27

3

0.14

2

0.08

1

0.62

5

0.25

6

0.13

0

0.07

3

0.65

6

0.26

5

0.13

3

0.07

4

0.71

9

0.29

4

0.14

9

0.08

4

67.5

°

0.64

1

0.22

7

0.10

0

0.05

1

0.63

3

0.21

7

0.09

5

0.04

8

0.66

2

0.22

3

0.09

7

0.04

9

0.71

3

0.23

7

0.10

3

0.05

3

78.7

5°

0.64

0

0.16

2

0.06

2

0.03

1

0.62

3

0.15

7

0.06

2

0.03

1

0.64

7

0.16

6

0.06

7

0.03

5

0.70

0

0.18

1

0.07

4

0.03

9

90°

0.66

7

0.11

1

0.04

2

0.02

1

0.64

0

0.11

3

0.04

4

0.02

3

0.66

4

0.12

6

0.05

1

0.02

7

0.72

4

0.14

2

0.05

9

0.03

2

Tab

le

C.2

6:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

y*

=

K,

'/[A

i(b/

WyV

7rb]

, at

no

dal

poin

ts a

long

el

lipt

ical

cra

ck-f

ront

as

defi

ned

by c

p fo

r R

*=10

, k=

2.5,

6=

67.5

°

b/W

0.2

0.4

0.6

0.8

Kf*

Kf)*

K(o

>

K?]*

K(2

)*

4> 0°

0.67

8

0.44

8

0.34

8

0.28

7

0.67

7

0.43

6

0.33

1

0.26

8

0.71

9

0.45

2

0.34

0

0.27

3

0.83

3

0.51

8

0.39

0

0.31

5

11.2

5°

0.66

6

0.40

6

0.29

4

0.22

7

0.65

9

0.38

3

0.26

5

0.19

7

0.69

8

0.39

8

0.27

4

0.20

2

0.81

0

0.46

9

0.33

0

0.25

0

22.5

°

0.67

5

0.42

4

0.31

4

0.24

8

0.67

3

0.41

2

0.29

8

0.23

0

0.71

2

0.42

7

0.30

6

0.23

4

0.81

3

0.48

3

0.34

6

0.26

6

33.7

5°

0.65

6

0.36

2

0.23

8

0.16

8

0.64

6

0.34

1

0.21

5

0.14

6

0.68

2

0.35

3

0.22

1

0.14

9

0.77

6

0.40

8

0.26

0

0.17

9

45°

0.65

9

0.35

0

0.21

9

0.14

7

0.65

5

0.33

8

0.20

6

0.13

5

0.68

9

0.34

7

0.21

0

0.13

6

0.76

6

0.38

3

0.23

1

0.15

0

56.2

5°

0.63

7

0.27

3

0.14

2

0.08

1

0.62

5

0.25

6

0.13

0

0.07

3

0.65

5

0.26

4

0.13

3

0.07

4

0.72

6

0.29

6

0.15

1

0.08

5

67.5

°

0.64

1

0.22

7

0.10

0

0.05

1

0.63

3

0.21

7

0.09

5

0.04

8

0.66

1

0.22

3

0.09

7

0.04

9

0.71

9

0.24

0

0.10

4

0.05

4

78.7

5°

0.64

0

0.16

2

0.06

2

0.03

1

0.62

3

0.15

7

0.06

2

0.03

1

0.64

6

0.16

6

0.06

7

0.03

5

0.70

5

0.18

4

0.07

5

0.04

0

90°

0.66

7

0.11

1

0.04

2

0.02

1

0.64

0

0.11

3

0.04

4

0.02

3

0.66

3

0.12

6

0.05

1

0.02

7

0.72

3

0.14

4

0.06

0

0.03

3

Tab

le

C.2

7:

Nor

mal

ized

inf

luen

ce f

unct

ion

coef

fici

ents

, K

j =

K

jl'/[A

i(b/

W)l\/

nb],

at

no

dal

poin

ts a

long

a s

emi-

elli

ptic

al c

rack

-fro

nt a

s de

fine

d by

</>

for

/?*

=10

, k=

2.5,

0

=9

0°

b/W

0.2

0.4

0.6

0.8

Kf*

K(2

>

<t> 0°

0.67

8

0.44

8

0.34

8

0.28

7

0.67

7

0.43

5

0.33

1

0.26

8

0.71

9

0.45

2

0.34

0

0.27

3

0.83

5

0.51

9

0.39

0

0.31

6

11.2

5°

0.66

6

0.40

6

0.29

4

0.22

7

0.65

9

0.38

3

0.26

5

0.19

7

0.69

8

0.39

8

0.27

4

0.20

2

0.81

2

0.47

0

0.33

0

0.25

0

22.5

°

0.67

4

0.42

4

0.31

4

0.24

8

0.67

3

0.41

2

0.29

8

0.23

0

0.71

2

0.42

7

0.30

6

0.23

4

0.81

5

0.48

4

0.34

6

0.26

6

33.7

5°

0.65

6

0.36

2

0.23

8

0.16

8

0.64

6

0.34

1

0.21

5

0.14

6

0.68

2

0.35

3

0.22

0

0.14

9

0.77

8

0.40

9

0.26

1

0.18

0

45°

0.65

9

0.35

0

0.21

9

0.14

7

0.65

5

0.33

8

0.20

6

0.13

5

0.68

9

0.34

7

0.21

0

0.13

6

0.76

9

0.38

4

0.23

2

0.15

1

56.2

5°

0.63

7

0.27

3

0.14

2

0.08

1

0.62

5

0.25

6

0.13

0

0.07

3

0.65

4

0.26

4

0.13

3

0.07

4

0.72

8

0.29

7

0.15

1

0.08

6

67.5

°

0.64

1

0.22

7

0.10

0

0.05

1

0.63

3

0.21

7

0.09

5

0.04

8

0.66

1

0.22

3

0.09

7

0.04

9

0.72

1

0.24

1

0.10

5

0.05

4

78.7

5°

0.64

0

0.16

2

0.06

2

0.03

1

0.62

3

0.15

7

0.06

2

0.03

1

0.64

6

0.16

6

0.06

7

0.03

5

0.70

9

0.18

5

0.07

6

0.04

0

90°

0.66

7

0.11

1

0.04

2

0.02

1

0.64

0

0.11

3

0.04

4

0.02

3

0.66

3

0.12

6

0.05

1

0.02

7

0.73

1

0.14

6

0.06

1

0.03

3

148

C.2 Stress Intensity Factors from Influence Coef

ficients

Tab

le

C.2

8:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

j/(P

vTtb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, cf

), fo

r R

*=5,

fc=

1.5,

(9=

45°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

4>

0°

2.76

5

2.73

9

-0.9

24

2.57

5

2.62

6

2.00

3

2.59

0

2.63

2

1.62

0

2.84

7

2.81

9

-0.9

67

11.2

5°

2.73

7

2.71

2

-0.9

12

2.55

3

2.60

6

2.05

7

2.57

8

2.62

3

1.74

9

2.85

2

2.82

2

-1.0

73

22.5

°

2.75

4

2.73

0

-0.8

80

2.57

6

2.62

9

2.05

3

2.59

2

2.63

6

1.68

9

2.84

0

2.81

1

-1.0

30

33.7

5°

2.71

2

2.68

4

-1.0

25

2.53

9

2.59

2

2.07

6

2.57

1

2.61

9

1.86

0

2.83

8

2.80

7

-1.1

01

45°

2.72

7

2.69

7

-1.0

80

2.56

5

2.61

9

2.08

3

2.59

9

2.64

8

1.90

2

2.84

7

2.81

9

-0.9

80

56.2

5°

2.69

0

2.65

4

-1.3

44

2.53

7

2.58

9

2.05

3

2.59

1

2.64

4

2.03

3

2.87

3

2.84

2

-1.0

81

67.5

°

2.72

8

2.68

6

-1.5

32

2.59

7

2.65

0

2.04

4

2.67

3

2.73

0

2.15

5

2.96

8

2.94

1

-0.9

16

78.7

5°

2.80

8

2.75

4

-1.9

05

2.68

0

2.73

2

1.93

9

2.78

3

2.84

3

2.14

8

3.14

3

3.10

6

-1.1

54

90°

2.96

3

2.89

3

-2.3

70

2.81

9

2.87

6

2.03

1

2.97

2

3.03

9

2.24

0

3.36

5

3.33

6

-0.8

62

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.2

9:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

j/{P

\fnb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, (f

>,

for

/?*=

5, f

c=1.

5, 0

=67

.5°:

IC

- I

nflu

ence

Coe

ffic

ient

Met

hod;

A %

-

% D

iffe

renc

e b/W

0.2

0.4

0.6

0.8

4>

0°

2.78

9

2.76

9

-0.7

10

2.58

7

2.64

5

2.24

2

2.59

9

2.64

1

1.59

5

2.85

3

2.81

4

-1.3

84

11.2

5°

2.76

2

2.74

3

-0.6

75

2.56

8

2.62

7

2.29

9

2.59

0

2.63

5

1.75

7

2.86

0

2.81

9

-1.4

55

22.5

°

2.77

9

2.76

1

-0.6

50

2.59

0

2.64

9

2.27

7

2.60

3

2.64

7

1.67

6

2.84

8

2.80

8

-1.4

35

33.7

5°

2.73

7

2.71

6

-0.7

83

2.55

5

2.61

4

2.32

3

2.58

4

2.63

3

1.88

0

2.84

9

2.80

8

-1.4

47

45°

2.75

2

2.72

9

-0.8

42

2.58

2

2.64

2

2.31

7

2.61

3

2.66

3

1.91

1

2.85

9

2.82

1

-1.3

37

56.2

5°

2.71

7

2.68

7

-1.0

92

2.55

6

2.61

5

2.30

9

2.60

9

2.66

3

2.08

8

2.88

9

2.85

0

-1.3

68

67.5

°

2.75

6

2.72

1

-1.2

79

2.61

8

2.67

8

2.30

8

2.69

3

2.75

3

2.22

3

2.98

9

2.95

4

-1.1

78

78.7

5°

2.83

8

2.79

2

-1.6

36

2.70

3

2.76

4

2.24

3

2.80

7

2.87

1

2.26

6

3.16

7

3.12

5

-1.3

40

90°

2.99

5

2.93

3

-2.0

76

2.84

4

2.91

2

2.40

5

2.99

9

3.07

2

2.44

0

3.39

1

3.35

9

-0.9

45

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

0:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

i/(P

\fnb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, cf

), fo

r R

*=5,

k—

1.5,

0=

90

°: I

C -

Inf

luen

ce C

oeff

icie

nt M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

<t> 0

°

2.80

3

2.79

3

-0.3

40

2.60

2

2.66

2

2.34

1

2.60

5

2.65

2

1.83

1

2.76

8

2.82

2

1.94

5

11.2

5°

2.77

4

2.76

6

-0.3

11

2.58

2

2.64

4

2.39

6

2.59

7

2.64

8

1.96

8

2.77

7

2.83

1

1.94

8

22.5

°

2.79

2

2.78

4

-0.2

83

2.60

4

2.66

6

2.38

6

2.60

9

2.65

9

1.89

8

2.76

7

2.82

1

1.94

5

33.7

5°

2.75

1

2.73

9

-0.4

19

2.57

1

2.63

3

2.41

7

2.59

3

2.64

7

2.07

4

2.77

1

2.82

4

1.90

6

45°

2.76

6

2.75

3

-0.4

73

2.59

8

2.66

1

2.41

6

2.62

3

2.67

8

2.10

3

2.78

5

2.83

9

1.90

9

56.2

5°

2.73

1

2.71

1

-0.7

31

2.57

4

2.63

5

2.37

7

2.62

2

2.68

0

2.23

0

2.81

8

2.86

7

1.73

6

67.5

°

2.77

1

2.74

5

-0.9

24

2.63

8

2.70

0

2.36

0

2.71

0

2.77

3

2.32

8

2.92

3

2.97

0

1.62

1

78.7

5°

2.85

5

2.81

9

-1.2

92

2.72

6

2.78

7

2.25

3

2.82

8

2.89

3

2.30

3

3.09

8

3.13

4

1.17

4

90°

3.01

6

2.96

3

-1.7

32

2.87

2

2.93

9

2.34

5

3.02

6

3.09

8

2.38

2

3.33

8

3.36

7

0.85

6

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

1:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Kj/

(P\n

rb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

4>,

for

R*=

5, k

=2,

9=

45°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

%

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

<t> 0

°

1.82

1

1.82

7

0.34

6

1.69

1

1.71

7

1.52

5

1.68

6

1.71

3

1.63

3

1.83

5

1.88

2

2.51

8

11.2

5°

1.80

3

1.80

5

0.11

7

1.67

5

1.70

1

1.53

8

1.67

2

1.70

1

1.78

1

1.83

1

1.87

4

2.32

6

22.5

°

1.82

0

1.82

5

0.27

0

1.69

2

1.71

8

1.56

3

1.68

6

1.71

5

1.74

8

1.82

5

1.86

8

2.35

5

33.7

5°

1.79

1

1.79

0

-0.0

51

1.66

9

1.69

5

1.54

8

1.66

7

1.69

9

1.90

1

1.80

9

1.84

8

2.12

7

45°

1.81

0

1.80

9

-0.0

91

1.69

3

1.72

0

1.59

6

1.69

1

1.72

5

2.01

0

1.81

3

1.85

2

2.16

7

56.2

5°

1.78

6

1.77

6

-0.5

76

1.67

7

1.70

2

1.48

2

1.68

5

1.71

8

1.94

0

1.82

0

1.85

4

1.86

3

67.5

°

1.82

9

1.81

2

-0.9

01

1.73

3

1.75

8

1.47

7

1.75

3

1.78

5

1.87

3

1.89

9

1.93

1

1.72

6

78.7

5°

1.88

1

1.85

3

-1.5

12

1.78

2

1.80

2

1.12

1

1.81

7

1.83

9

1.20

9

1.99

3

2.00

9

0.82

9

90°

1.97

0

1.92

9

-2.0

37

1.88

0

1.89

3

0.69

3

1.93

9

1.94

8

0.44

7

2.12

3

2.12

4

0.03

8

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

2:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

KI/

(PV

TT

B),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, (j

), fo

r i?

.*=5

, k=

2,

0=67

.5°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

<t> 0°

1.86

1

1.85

6

-0.2

62

1.71

1

1.73

9

1.64

6

1.70

8

1.73

4

1.50

3

1.86

2

1.88

5

1.26

6

11.2

5°

1.84

4

1.83

4

-0.5

28

1.69

6

1.72

4

1.67

8

1.69

4

1.72

2

1.68

1

1.85

4

1.87

9

1.38

7

22.5

°

1.86

1

1.85

4

-0.3

64

1.71

2

1.74

1

1.68

5

1.70

8

1.73

6

1.59

0

1.85

0

1.87

5

1.38

9

33.7

5°

1.83

2

1.81

9

-0.7

34

1.69

0

1.71

9

1.68

7

1.69

0

1.72

0

1.77

7

1.83

0

1.85

9

1.59

9

45°

1.85

3

1.83

8

-0.7

90

1.71

5

1.74

4

1.72

1

1.71

6

1.74

7

1.82

8

1.83

3

1.86

6

1.79

7

56.2

5°

1.83

0

1.80

5

-1.3

50

1.70

0

1.72

7

1.62

4

1.70

9

1.74

1

1.85

2

1.83

5

1.87

0

1.94

7

67.5

°

1.87

5

1.84

3

-1.7

35

1.75

7

1.78

5

1.60

7

1.77

8

1.81

0

1.81

0

1.91

2

1.95

1

2.06

4

78.7

5°

1.93

1

1.88

4

-2.4

29

1.80

8

1.83

1

1.26

5

1.84

3

1.86

6

1.23

1

1.99

9

2.02

9

1.50

2

90°

2.02

5

1.96

3

-3.0

50

1.90

8

1.92

4

0.82

9

1.96

8

1.97

8

0.51

8

2.11

8

2.13

8

0.91

6

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

3:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Ki/

(P\f

irb)

, ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

(ft,

for

R*=

5, k

=2,

8=

90°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

%

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

<P 0°

1.84

0

1.88

0

2.20

4

1.72

0

1.75

8

2.22

5

1.71

2

1.74

7

2.05

4

1.86

5

1.89

6

1.63

5

11.2

5°

1.82

1

1.85

8

2.06

9

1.70

5

1.74

4

2.26

7

1.69

9

1.73

8

2.24

5

1.85

8

1.89

1

1.75

9

22.5

°

1.83

8

1.87

8

2.17

9

1.72

1

1.76

0

2.26

9

1.71

4

1.75

1

2.15

5

1.85

4

1.88

6

1.74

4

33.7

5°

1.80

8

1.84

3

1.96

8

1.70

0

1.73

8

2.27

7

1.69

7

1.73

7

2.34

6

1.83

6

1.87

2

1.96

1

45°

1.82

7

1.86

3

1.93

5

1.72

5

1.76

4

2.30

2

1.72

4

1.76

5

2.38

4

1.84

1

1.88

0

2.12

0

56.2

5°

1.80

1

1.83

0

1.58

3

1.71

1

1.74

9

2.21

3

1.72

0

1.76

1

2.39

9

1.84

4

1.88

7

2.27

8

67.5

°

1.84

4

1.86

9

1.33

2

1.76

9

1.80

8

2.18

2

1.79

1

1.83

3

2.31

9

1.92

5

1.97

0

2.36

1

78.7

5 °

1.89

6

1.91

1

0.83

2

1.82

1

1.85

5

1.84

2

1.85

8

1.89

0

1.71

8

2.01

3

2.04

9

1.79

4

90°

1.98

4

1.99

2

0.39

0

1.92

4

1.95

0

1.37

4

1.98

5

2.00

4

0.95

8

2.13

6

2.16

0

1.14

3

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

4:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Ki/

{P\f

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<f)

, fo

r /?

*=5,

k=

2.5,

9=

45°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

4> 0

°

1.48

9

1.51

1

1.43

2

1.35

8

1.38

0

1.59

2

1.34

2

1.37

1

2.19

2

11.2

5°

1.47

5

1.49

3

1.27

7

1.34

4

1.36

7

1.70

6

1.32

6

1.35

8

2.36

8

22.5

°

1.49

0

1.51

0

1.36

8

1.36

1

1.38

3

1.68

2

1.33

9

1.37

1

2.38

4

33.7

5°

1.46

7

1.48

4

1.14

2

1.34

1

1.36

5

1.78

0

1.32

0

1.35

5

2.68

0

45°

1.48

6

1.50

3

1.13

0

1.36

6

1.39

1

1.86

4

1.33

9

1.37

9

2.98

0

56.2

5°

1.46

8

1.47

8

0.70

3

1.35

4

1.37

9

1.83

6

1.33

2

1.37

4

3.19

8

67.5

°

1.50

9

1.51

6

0.42

7

1.40

7

1.43

3

1.89

2

1.39

7

1.44

6

3.53

5

78.7

5°

1.55

0

1.54

7

-0.1

97

1.44

3

1.46

5

1.50

5

1.43

4

1.47

6

2.99

1

90°

1.63

4

1.62

1

-0.7

89

1.53

1

1.54

8

1.10

4

1.53

0

1.56

7

2.39

7

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

5:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity f

acto

rs,

Kij

{P\r

nb),

ob

tain

ed f

rom

the

inf

luen

ce

coef

ficie

nt

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

cf>

, fo

r B*=

5, k

=2.

5, 0

=67

.5°:

IC

- I

nflu

ence

Coe

ffic

ient

Met

hod;

A %

-

% D

iffer

ence

b/W

0.2

0.4

0.6

0.8

<£

0°

1.51

8

1.53

5

1.12

2

1.37

9

1.39

9

1.40

2

1.37

9

1.39

3

1.08

6

1.51

2

1.52

7

0.98

3

11.2

5°

1.50

4

1.51

8

0.93

3

1.36

5

1.38

6

1.52

6

1.36

3

1.38

0

1.20

7

1.49

4

1.50

9

1.00

1

22.5

°

1.51

9

1.53

5

1.04

4

1.38

2

1.40

2

1.49

0

1.37

7

1.39

3

1.19

6

1.49

4

1.50

9

1.05

8

33.7

5°

1.49

7

1.50

9

0.76

6

1.38

9

1.38

5

-0.2

99

1.35

9

1.37

8

1.37

4

1.46

1

1.47

8

1.15

4

45°

1.51

7

1.52

8

0.74

2

1.38

8

1.41

1

1.64

5

1.38

1

1.40

2

1.53

2

1.46

2

1.48

3

1.38

0

56.2

5°

1.50

0

1.50

3

0.24

4

1.37

7

1.39

9

1.62

3

1.37

6

1.39

8

1.59

8

1.45

1

1.47

2

1.49

1

67.5

°

1.54

4

1.54

2

-0.0

97

1.43

1

1.45

5

1.63

6

1.44

8

1.47

3

1.69

8

1.52

6

1.55

3

1.76

5

78.7

5°

1.58

7

1.57

4

-0.7

98

1.46

9

1.48

7

1.24

4

1.48

7

1.50

3

1.09

6

1.56

6

1.58

2

1.01

7

90°

1.67

6

1.65

1

-1.4

78

1.56

2

1.57

3

0.71

3

1.58

7

1.59

1

0.19

6

1.64

2

1.64

9

0.39

5

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

6:

Com

pari

son

of

norm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Kj/

(P\n

rb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<f>

, fo

r /?

,*=5

, fc

=2.

5, 0

=9

0°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

4> 0

°

1.52

3

1.55

9

2.35

2

1.39

1

1.41

7

1.87

6

1.38

7

1.41

0

1.64

6

1.52

3

1.54

2

1.29

2

11.2

5°

1.50

7

1.54

1

2.27

1

1.37

6

1.40

4

2.01

6

1.37

2

1.39

6

1.79

2

1.50

5

1.52

5

1.32

8

22.5

°

1.52

3

1.55

8

2.34

4

1.39

3

1.42

1

1.97

0

1.38

5

1.41

0

1.75

6

1.50

5

1.52

6

1.34

3

33.7

5°

1.49

9

1.53

2

2.18

3

1.37

5

1.40

3

2.07

6

1.36

8

1.39

5

1.96

0

1.47

4

1.49

5

1.43

8

45°

1.51

9

1.55

2

2.17

7

1.40

0

1.43

0

2.12

1

1.39

0

1.42

0

2.10

1

1.47

7

1.50

0

1.60

1

56.2

5°

1.50

0

1.52

7

1.84

7

1.39

0

1.41

9

2.10

9

1.38

6

1.41

7

2.25

9

1.46

5

1.49

1

1.76

8

67.5

°

1.54

2

1.56

7

1.61

3

1.44

6

1.47

7

2.12

1

1.45

8

1.49

3

2.38

7

1.54

3

1.57

4

2.03

1

78.7

5°

1.58

4

1.60

1

1.06

6

1.48

5

1.51

0

1.71

9

1.49

7

1.52

5

1.82

9

1.58

3

1.60

4

1.34

4

90°

1.66

9

1.67

8

0.51

3

1.57

8

1.59

7

1.15

2

1.59

9

1.61

5

0.97

0

1.66

1

1.67

3

0.70

4

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

7:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

i/(P

\fnb

),

obta

ined

fr

om t

he

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, cf

>,

for

i?*=

7.5,

/c=

1.5,

#—

45°:

IC

- I

nflu

ence

Coe

ffic

ient

Met

hod;

A %

-

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

<t> 0°

2.56

6

2.62

4

2.28

6

2.46

6

2.51

6

2.03

4

2.47

5

2.51

9

1.75

4

2.67

1

2.70

2

1.13

7

11.2

5°

2.53

8

2.59

6

2.31

0

2.44

5

2.49

6

2.07

8

2.46

3

2.51

0

1.88

0

2.67

2

2.70

6

1.24

2

22.5

°

2.55

6

2.61

4

2.30

4

2.46

7

2.51

8

2.07

4

2.47

7

2.52

2

1.81

0

2.66

5

2.69

8

1.24

4

33.7

5°

2.51

3

2.57

1

2.29

0

2.43

1

2.48

2

2.08

8

2.45

5

2.50

3

1.96

7

2.65

5

2.68

9

1.29

6

45°

2.52

4

2.58

3

2.31

9

2.45

5

2.50

7

2.08

6

2.48

0

2.52

9

1.98

6

2.65

9

2.69

5

1.33

8

56.2

5°

2.48

4

2.54

0

2.24

8

2.42

7

2.47

6

2.03

4

2.47

0

2.52

2

2.10

3

2.67

0

2.70

4

1.26

6

67.5

°

2.51

2

2.56

9

2.26

9

2.48

1

2.53

1

2.00

3

2.54

2

2.59

7

2.18

3

2.74

5

2.78

0

1.28

9

78.7

5°

2.57

2

2.62

8

2.15

0

2.55

9

2.60

7

1.89

5

2.64

5

2.70

3

2.19

5

2.89

3

2.92

1

0.99

8

90°

2.71

6

2.77

5

2.16

1

2.69

6

2.74

9

1.97

5

2.81

5

2.87

9

2.29

1

3.12

5

3.15

4

0.91

6

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

8:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

i/(P

vftb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, </>

, for

i?.

*=7.

5, f

c=1.

5, 9

=67

.5°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce b/

W

0.2

0.4

0.6

0.8

4> 0°

2.56

2

2.61

1

1.91

0

2.48

2

2.49

7

0.58

9

2.47

2

2.49

5

0.91

8

2.66

8

2.67

6

0.30

7

11.2

5°

2.53

5

2.58

5

1.96

5

2.46

2

2.47

8

0.62

2

2.46

0

2.48

6

1.05

2

2.66

9

2.67

9

0.38

2

22.5

°

2.55

2

2.60

2

1.93

2

2.48

3

2.49

9

0.64

9

2.47

4

2.49

8

0.96

9

2.66

0

2.67

1

0.40

0

33.7

5°

2.51

0

2.55

9

1.98

2

2.44

9

2.46

4

0.62

4

2.45

3

2.48

1

1.14

8

2.65

1

2.66

2

0.40

2

45

°

2.52

1

2.57

2

2.02

6

2.47

4

2.48

9

0.62

2

2.47

7

2.50

6

1.16

6

2.65

5

2.66

7

0.44

2

56.2

5°

2.48

1

2.53

2

2.04

8

2.44

8

2.46

2

0.54

8

2.46

8

2.50

2

1.34

7

2.66

7

2.67

6

0.34

0

67.5

°

2.50

8

2.56

2

2.13

9

2.50

5

2.51

8

0.53

0

2.54

2

2.57

9

1.48

3

2.74

3

2.75

4

0.38

0

78.7

5°

2.56

8

2.62

3

2.14

9

2.58

5

2.59

7

0.45

7

2.64

5

2.68

8

1.64

6

2.89

4

2.90

0

0.18

0

90°

2.70

8

2.77

1

2.32

6

2.72

4

2.74

3

0.68

3

2.81

5

2.87

0

1.93

0

3.12

8

3.13

7

0.30

0

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.3

9:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

j/(P

\/nb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, (j

>,

for

i?.*

=7.5

, A

r=1.

5, 0

=9

0°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

<t> 0

°

2.57

2

2.62

0

1.83

6

2.45

5

2.50

1

1.86

9

2.46

9

2.50

1

1.28

8

2.66

3

2.67

4

0.41

0

11.2

5°

2.54

7

2.59

5

1.88

1

2.43

5

2.48

3

1.96

0

2.45

6

2.49

4

1.52

3

2.66

3

2.67

7

0.50

0

22.5

°

2.56

3

2.61

0

1.85

3

2.45

7

2.50

4

1.88

9

2.47

0

2.50

5

1.38

5

2.65

4

2.66

8

0.51

5

33.7

5°

2.52

1

2.56

9

1.88

8

2.42

1

2.47

0

2.01

7

2.44

7

2.48

9

1.70

0

2.64

3

2.65

8

0.55

4

45°

2.53

2

2.58

1

1.93

3

2.44

5

2.49

5

2.01

0

2.47

2

2.51

4

1.72

8

2.64

6

2.66

2

0.59

5

56.2

5°

2.49

3

2.54

1

1.93

7

2.41

6

2.46

7

2.13

4

2.46

0

2.51

1

2.10

2

2.65

8

2.67

2

0.53

0

67.5

°

2.52

1

2.57

2

2.01

4

2.47

0

2.52

4

2.21

4

2.53

0

2.59

0

2.35

8

2.73

3

2.74

9

0.59

7

78.7

5°

2.58

2

2.63

3

2.00

3

2.54

3

2.60

4

2.37

1

2.62

9

2.70

1

2.71

9

2.88

2

2.89

5

0.44

1

90°

2.72

3

2.78

3

2.18

2

2.67

3

2.75

1

2.90

6

2.79

1

2.88

4

3.32

7

3.11

2

3.13

1

0.60

5

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

0:

Com

pari

son

of n

orm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Kj/

{P\f

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

</>, f

or R

*=7.

5,

k=2,

9=

45°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

4>

0°

1.75

8

1.79

2

1.91

7

1.64

0

1.66

7

1.64

8

1.62

8

1.65

5

1.70

8

1.74

8

1.77

5

1.56

7

11.2

5°

1.74

2

1.77

5

1.89

8

1.62

8

1.65

5

1.64

8

1.61

5

1.64

5

1.80

6

1.73

3

1.76

1

1.63

1

22.5

°

1.75

5

1.78

9

1.91

9

1.64

0

1.66

7

1.68

1

1.62

9

1.65

7

1.76

1

1.73

0

1.75

7

1.59

2

33.7

5°

1.73

0

1.76

2

1.84

6

1.62

3

1.65

1

1.70

9

1.60

9

1.63

9

1.86

9

1.70

3

1.73

2

1.71

1

45°

1.74

1

1.77

4

1.87

6

1.64

0

1.66

9

1.76

1

1.63

0

1.66

1

1.89

1

1.70

2

1.73

3

1.80

0

56.2

5°

1.72

0

1.74

9

1.69

5

1.63

1

1.65

8

1.67

2

1.62

1

1.65

2

1.93

3

1.69

2

1.72

5

1.97

5

67.5

°

1.74

9

1.77

8

1.67

1

1.67

8

1.70

6

1.70

7

1.68

3

1.71

6

1.96

4

1.74

3

1.78

2

2.26

6

78.7

5°

1.79

5

1.82

0

1.38

5

1.73

4

1.75

9

1.44

0

1.73

9

1.76

8

1.65

7

1.81

4

1.85

7

2.33

5

90°

1.89

4

1.92

0

1.34

0

1.83

2

1.85

7

1.37

4

1.85

3

1.87

7

1.30

5

1.94

0

1.99

8

2.98

5

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

1:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Ki/

(Py/

irb)

, ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<p,

for

R*=

7.5,

k=

2,

(9=6

7.5

°: I

C -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A %

-

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

</>

0°

1.76

5

1.77

3

0.46

8

1.64

4

1.64

6

0.13

4

1.62

6

1.62

9

0.18

0

1.76

8

1.77

4

0.32

5

11.2

5°

1.74

9

1.75

7

0.47

3

1.63

2

1.63

4

0.13

1

1.61

4

1.61

8

0.22

1

1.75

6

1.76

1

0.31

7

22.5

°

1.76

2

1.77

0

0.48

8

1.64

4

1.64

7

0.16

1

1.62

7

1.63

0

0.19

4

1.75

2

1.75

8

0.31

9

33.7

5°

1.73

7

1.74

4

0.44

2

1.62

7

1.63

1

0.19

8

1.60

9

1.61

2

0.23

3

1.72

7

1.73

3

0.29

9

45°

1.74

9

1.75

7

0.47

6

1.64

5

1.64

9

0.24

1

1.63

0

1.63

4

0.23

2

1.72

6

1.73

2

0.31

4

56.2

5°

1.72

7

1.73

3

0.35

4

1.63

6

1.63

9

0.20

2

1.62

2

1.62

6

0.24

6

1.71

9

1.72

4

0.32

2

67.5

°

1.75

7

1.76

3

0.35

8

1.68

4

1.68

8

0.28

3

1.68

5

1.68

9

0.27

5

1.77

3

1.78

1

0.45

9

78.7

5°

1.80

4

1.80

6

0.14

0

1.74

1

1.74

3

0.12

2

1.74

2

1.74

3

0.05

6

1.85

2

1.85

9

0.38

2

90°

1.90

4

1.90

7

0.17

0

1.83

9

1.84

2

0.19

5

1.85

6

1.85

3

-0.1

63

1.99

1

1.99

8

0.35

0

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

2:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

i/{P

\nrb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, cf

>,

for

R*-

7.5,

A

;=2,

0=

90

°: I

C -

Inf

luen

ce C

oeff

icie

nt M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

<t> 0°

1.76

6

1.81

2

2.61

4

1.64

5

1.68

3

2.28

2

1.62

6

1.66

2

2.21

4

1.76

2

1.79

7

1.98

5

11.2

5°

1.75

1

1.79

6

2.57

4

1.63

3

1.67

1

2.28

2

1.61

4

1.65

1

2.33

2

1.74

9

1.78

5

2.06

5

22.5

°

1.76

4

1.80

9

2.57

8

1.64

5

1.68

3

2.32

0

1.62

7

1.66

4

2.26

4

1.74

5

1.78

1

2.03

4

33.7

5°

1.73

9

1.78

2

2.49

8

1.62

9

1.66

7

2.34

5

1.60

9

1.64

7

2.39

6

1.72

1

1.75

8

2.14

6

45°

1.75

1

1.79

5

2.54

5

1.64

7

1.68

6

2.39

3

1.63

0

1.66

9

2.40

6

1.72

0

1.75

8

2.22

2

56.2

5°

1.72

9

1.77

0

2.35

8

1.63

8

1.67

6

2.29

8

1.62

2

1.66

2

2.47

6

1.71

2

1.75

3

2.36

9

67.5

°

1.75

9

1.80

0

2.33

1

1.68

7

1.72

6

2.31

4

1.68

5

1.72

8

2.50

5

1.76

7

1.81

2

2.58

0

78.7

5°

1.80

7

1.84

4

2.04

3

1.74

5

1.78

0

2.02

4

1.74

2

1.78

1

2.21

8

1.84

5

1.89

2

2.50

9

90°

1.90

7

1.94

5

1.98

0

1.84

5

1.88

0

1.93

4

1.85

7

1.89

3

1.91

1

1.98

2

2.03

3

2.57

7

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

3:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Ki/

(P\f

irb)

, ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<p,

for

R*=

7.5,

k=

2.5,

0=

45°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

4> 0

°

1.45

2

1.48

2

2.06

1

1.32

4

1.34

6

1.65

2

1.32

2

1.33

8

1.26

4

1.45

6

1.47

6

1.38

5

11.2

5°

1.44

1

1.47

0

1.98

8

1.31

0

1.33

2

1.73

1

1.30

6

1.32

4

1.39

6

1.43

7

1.45

7

1.42

0

22.5

°

1.45

3

1.48

3

2.08

4

1.32

5

1.34

8

1.70

4

1.32

0

1.33

8

1.34

7

1.43

5

1.45

5

1.42

8

33.7

5°

1.43

4

1.46

2

1.91

1

1.30

5

1.32

8

1.75

7

1.29

8

1.31

8

1.51

2

1.40

2

1.42

4

1.53

4 .

45°

1.44

6

1.47

5

1.98

9

1.32

7

1.35

0

1.78

1

1.31

7

1.33

8

1.59

1

1.39

7

1.42

0

1.67

6

56.2

5°

1.43

2

1.45

5

1.57

7

1.31

3

1.33

5

1.66

5

1.30

4

1.32

6

1.68

0

1.37

8

1.40

4

1.87

4

67.5

°

1.46

3

1.48

4

1.44

8

1.36

3

1.38

4

1.54

5

1.36

0

1.38

4

1.78

1

1.42

1

1.45

2

2.18

7

78.7

5°

1.50

6

1.51

5

0.60

4

1.39

7

1.41

0

0.94

4

1.39

3

1.41

0

1.19

7

1.46

9

1.49

4

1.75

1

90°

1.60

6

1.60

4

-0.0

77

1.48

5

1.48

9

0.20

7

1.49

0

1.49

6

0.43

0

1.57

7

1.58

9

0.72

1

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

4:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Kj/

(P\/

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

cj),

for

R*=

7.5,

k=

2.5,

8=

67.5

°:

IC -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A

%

- %

Dif

fere

nce b/

W

0.2

0.4

0.6

0.8

<t> 0°

1.46

0

1.49

8

2.66

1

1.33

0

1.35

8

2.10

8

1.32

7

1.35

0

1.74

2

1.46

6

1.49

1

1.69

6

11.2

5°

1.44

8

1.48

6

2.59

0

1.31

7

1.34

5

2.18

4

1.31

1

1.33

6

1.89

8

1.44

6

1.47

1

1.73

8

22.5

°

1.46

0

1.50

0

2.68

7

1.33

2

1.36

0

2.14

7

1.32

5

1.34

9

1.82

3

1.44

5

1.46

9

1.71

8

33.7

5°

1.44

2

1.47

8

2.51

5

1.31

3

1.34

1

2.20

1

1.30

4

1.33

0

2.01

9

1.41

2

1.43

7

1.82

8

45°

1.45

4

1.49

2

2.59

8

1.33

4

1.36

4

2.21

8

1.32

3

1.35

1

2.07

6

1.40

7

1.43

4

1.92

6

56.2

5°

1.44

0

1.47

1

2.19

1

1.32

1

1.34

9

2.09

6

1.31

0

1.33

9

2.22

4

1.38

9

1.41

8

2.15

3

67.5

°

1.47

1

1.50

2

2.06

6

1.37

2

1.39

9

1.96

7

1.36

6

1.39

8

2.32

8

1.43

2

1.46

8

2.46

0

78.7

5°

1.51

4

1.53

2

1.22

8

1.40

6

1.42

5

1.36

4

1.40

0

1.42

5

1.78

7

1.48

0

1.51

1

2.07

5

90°

1.61

4

1.62

4

0.58

6

1.49

6

1.50

6

0.67

5

1.49

8

1.51

3

1.05

3

1.59

2

1.60

6

0.88

4

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

5:

Com

pari

son

of n

orm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Kj/

(Py/

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

</>, f

or R

*=7.

5,

&=

2.5,

0=

90 °

: IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

<t> 0

°

1.46

3

1.50

1

2.59

9

1.33

2

1.36

0

2.13

1

1.32

8

1.35

1

1.67

0

1.45

5

1.48

0

1.72

5

11.2

5°

1.45

2

1.48

9

2.53

5

1.31

8

1.34

7

2.22

2

1.31

3

1.33

7

1.82

1

1.43

6

1.46

1

1.74

6

22.5

°

1.46

4

1.50

2

2.63

4

1.33

3

1.36

2

2.18

1

1.32

7

1.35

0

1.74

9

1.43

5

1.45

9

1.71

7

33.7

5°

1.44

6

1.48

1

2.45

6

1.31

4

1.34

4

2.24

6

1.30

6

1.33

1

1.93

3

1.40

3

1.42

8

1.79

8

45°

1.45

8

1.49

5

2.54

2

1.33

6

1.36

6

2.25

5

1.32

5

1.35

2

1.98

7

1.39

9

1.42

5

1.87

8

56.2

5°

1.44

4

1.47

5

2.13

0

1.32

3

1.35

1

2.14

3

1.31

3

1.34

1

2.11

7

1.38

2

1.41

0

2.09

1

67.5

°

1.47

5

1.50

5

1.99

5

1.37

4

1.40

2

2.01

1

1.37

0

1.40

0

2.20

8

1.42

7

1.46

0

2.37

3

78.7

5°

1.51

9

1.53

6

1.15

2

1.40

9

1.42

8

1.40

4

1.40

4

1.42

7

1.64

8

1.47

4

1.50

4

2.00

5

90°

1.62

0

1.62

8

0.45

4

1.49

9

1.50

9

0.64

2

1.50

3

1.51

6

0.88

3

1.58

7

1.59

9

0.72

1

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

6:

Com

pari

son

of n

orm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Kj/

(P\Z

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

</>, f

or R

*=1Q

, fc

=1.5

, 0

=4

5°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

<t> 0

°

2.49

3

2.54

3

2.01

1

2.40

8

2.43

4

1.05

2

2.41

7

2.43

5

0.75

5

2.62

2

2.62

8

0.25

4

11.2

5°

2.46

6

2.51

7

2.06

9

2.38

7

2.41

4

1.12

5

2.40

3

2.42

5

0.89

0

2.62

0

2.62

9

0.31

2

22.5

°

2.48

3

2.53

3

2.02

7

2.40

9

2.43

5

1.09

3

2.41

7

2.43

7

0.80

8

2.61

1

2.62

0

0.33

6

33.7

5°

2.44

1

2.49

2

2.08

6

2.37

3

2.40

0

1.16

5

2.39

3

2.41

7

0.98

2

2.59

8

2.60

6

0.32

6

45°

2.45

2

2.50

4

2.13

0

2.39

6

2.42

4

1.15

8

2.41

6

2.44

0

0.99

7

2.59

8

2.60

7

0.34

7

56.2

5°

2.41

1

2.46

3

2.17

1

2.36

7

2.39

5

1.20

3

2.40

3

2.43

2

1.17

7

2.60

5

2.61

1

0.22

8

67.5

°

2.43

6

2.49

2

2.27

9

2.41

9

2.44

9

1.23

3

2.47

0

2.50

2

1.31

2

2.67

3

2.67

9

0.24

4

78.7

5°

2.49

2

2.55

0

2.32

7

2.49

2

2.52

3

1.27

4

2.56

6

2.60

4

1.48

8

2.81

6

2.81

7

0.02

1

90°

2.62

6

2.69

5

2.62

4

2.61

8

2.65

9

1.56

6

2.73

2

2.78

1

1.78

4

3.04

8

3.05

0

0.09

3

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

-J

Tab

le

C.4

7:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

j/{P

\fiT

b),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, (f

>,

for

i?*=

10,

fc=1

.5,

8=67

.5°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce b/

W

0.2

0.4

0.6

0.8

4>

0°

2.49

1

2.55

0

2.35

4

2.39

4

2.43

4

1.69

9

2.41

8

2.44

2

0.98

0

2.61

7

2.62

7

0.39

4

11.2

5°

2.46

4

2.52

4

2.42

1

2.37

1

2.41

4

1.81

3

2.40

3

2.43

2

1.20

8

2.61

5

2.62

8

0.48

0

22.5

°

2.48

1

2.54

0

2.37

7

2.39

4

2.43

5

1.72

8

2.41

8

2.44

4

1.07

8

2.60

6

2.61

8

0.48

7

33.7

5°

2.44

0

2.49

9

2.45

5

2.35

6

2.40

0

1.89

0

2.39

2

2.42

5

1.38

4

2.59

2

2.60

5

0.52

0

45°

2.44

9

2.51

0

2.51

3

2.37

9

2.42

4

1.87

7

2.41

5

2.44

9

1.40

8

2.59

2

2.60

6

0.55

0

56.2

5°

2.40

9

2.47

1

2.57

8

2.34

7

2.39

5

2.05

3

2.40

0

2.44

2

1.75

6

2.59

9

2.61

2

0.47

9

67.5

°

2.43

3

2.49

9

2.70

6

2.39

6

2.44

7

2.15

7

2.46

4

2.51

4

2.00

0

2.66

6

2.68

0

0.52

8

78.7

5°

2.48

9

2.55

8

2.77

1

2.46

2

2.52

2

2.41

0

2.55

7

2.61

8

2.36

4

2.80

6

2.81

6

0.36

1

90°

2.62

2

2.70

2

3.06

4

2.58

2

2.66

3

3.10

6

2.71

6

2.79

6

2.93

7

3.03

9

3.05

3

0.47

9

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

8:

Com

pari

son

of n

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

K

i/{P

\fnb

),

obta

ined

fr

om

the

infl

uenc

e co

effi

cien

t m

etho

d an

d by

dir

ect

BE

M a

long

cra

ck p

erip

hery

, cf

>,

for

/?,*

=10,

£=

1.5,

#=

90

°: I

C -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A %

-

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

<$>

0°

2.47

5

2.55

0

3.02

7

2.38

7

2.43

5

2.04

0

2.39

7

2.42

5

1.16

8

2.61

0

2.61

9

0.35

2

11.2

5°

2.44

7

2.52

4

3.11

9

2.36

5

2.41

6

2.14

1

2.38

0

2.41

4

1.45

2

2.60

7

2.61

9

0.47

0

22.5

°

2.46

5

2.54

0

3.05

8

2.38

8

2.43

7

2.05

4

2.39

4

2.42

6

1.31

3

2.59

7

2.61

0

0.48

1

33.7

5°

2.42

1

2.49

9

3.20

3

2.35

0

2.40

2

2.20

6

2.36

7

2.40

7

1.66

9

2.58

3

2.59

7

0.53

5

45°

2.43

0

2.51

0

3.29

4

2.37

4

2.42

6

2.19

3

2.39

0

2.43

0

1.70

2

2.58

3

2.59

8

0.58

0

56.2

5°

2.38

6

2.47

0

3.51

5

2.34

1

2.39

7

2.36

9

2.37

2

2.42

3

2.12

4

2.58

9

2.60

3

0.52

9

67.5

°

2.40

7

2.50

0

3.82

8

2.39

0

2.45

0

2.47

3

2.43

5

2.49

4

2.41

9

2.65

6

2.67

2

0.60

1

78.7

5°

2.45

6

2.56

2

4.34

6

2.45

8

2.52

5

2.72

8

2.52

3

2.59

6

2.89

7

2.79

7

2.81

0

0.46

8

90°

2.57

6

2.70

2

4.90

0

2.57

7

2.66

6

3.44

2

2.67

3

2.77

3

3.72

3

3.02

4

3.04

5

0.69

2

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.4

9:

Com

pari

son

of n

orm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Ki/

(P\f

irb)

, ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

</», f

or /

?* =

10,

k—

2, 6

=4

5°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

% -

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

4>

0°

1.72

7

1.77

0

2.52

7

1.60

9

1.64

6

2.25

5

1.59

5

1.63

0

2.19

3

1.72

5

1.75

9

1.97

2

11.2

5°

1.71

1

1.75

4

2.51

3

1.59

8

1.63

4

2.24

8

1.58

3

1.62

0

2.31

8

1.71

1

1.74

6

2.04

8

22.5

°

1.72

4

1.76

7

2.53

0

1.60

9

1.64

6

2.29

3

1.59

6

1.63

2

2.25

4

1.70

8

1.74

2

1.99

9

33.7

5°

1.69

9

1.74

1

2.46

4

1.59

3

1.62

9

2.29

7

1.57

7

1.61

4

2.37

6

1.68

2

1.71

7

2.10

7

45°

1.71

0

1.75

3

2.49

0

1.60

9

1.64

7

2.33

6

1.59

6

1.63

4

2.38

7

1.67

9

1.71

5

2.17

8

56.2

5°

1.68

8

1.72

7

2.32

6

1.60

0

1.63

5

2.20

3

1.58

6

1.62

5

2.45

1

1.66

7

1.70

6

2.33

3

67.5

°

1.71

6

1.75

6

2.31

1

1.64

5

1.68

1

2.19

7

1.64

3

1.68

4

2.48

8

1.71

3

1.75

7

2.56

6

78.7

5°

1.76

1

1.79

6

2.04

0

1.70

1

1.73

2

1.85

1

1.69

9

1.73

6

2.23

1

1.78

5

1.83

1

2.55

2

90°

1.85

8

1.89

6

2.05

7

1.80

1

1.83

2

1.74

1

1.81

4

1.84

9

1.93

0

1.92

2

1.97

7

2.87

1

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.5

0:

Com

pari

son

of n

orm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Kj/

(Pvi

rb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<p,

for

B*=

W,

k=2,

6=

67.5

°:

IC -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A %

-

% D

iffe

renc

e

b/W

0.2

0.4

0.6

0.8

4> 0°

1.72

8

1.76

1

1.91

6

1.61

1

1.63

7

1.64

1

1.58

6

1.61

2

1.68

1

1.72

2

1.74

8

1.49

5

11.2

5°

1.71

3

1.74

6

1.90

2

1.60

0

1.62

5

1.58

9

1.57

3

1.60

1

1.76

6

1.70

8

1.73

5

1.54

6

22.5

°

1.72

5

1.75

8

1.91

8

1.61

1

1.63

7

1.62

3

1.58

6

1.61

3

1.69

3

1.70

5

1.73

1

1.50

5

33.7

5°

1.70

1

1.73

2

1.85

0

1.59

5

1.62

1

1.60

9

1.56

7

1.59

5

1.79

4

1.67

9

1.70

5

1.57

6

45°

1.71

2

1.74

4

1.87

1

1.61

2

1.63

8

1.64

5

1.58

7

1.61

5

1.79

5

1.67

7

1.70

4

1.62

8

56.2

5°

1.69

0

1.71

9

1.70

5

1.60

3

1.62

7

1.49

6

1.57

6

1.60

5

1.86

1

1.66

6

1.69

5

1.73

2

67.5

°

1.71

8

1.74

7

1.67

8

1.64

8

1.67

3

1.48

4

1.63

3

1.66

4

1.90

2

1.71

2

1.74

5

1.92

9

78.7

5°

1.76

3

1.78

8

1.42

2

1.70

5

1.72

4

1.16

7

1.68

6

1.71

5

1.72

1

1.78

5

1.81

9

1.90

0

90°

1.86

1

1.88

7

1.45

0

1.80

2

1.82

4

1.17

3

1.79

6

1.82

7

1.72

5

1.92

3

1.96

7

2.29

7

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.5

1:

Com

pari

son

of n

orm

aliz

ed

stre

ss i

nten

sity

fa

ctor

s,

Ki/

(P\f

Ob)

, ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<f>

, fo

r i?

*=10

, jfc

=2,

0=

90

°:

IC -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A %

-%

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

4> 0°

1.72

7

1.74

8

1.24

2

1.60

7

1.62

4

1.10

5

1.59

6

1.61

1

0.98

2

1.71

4

1.73

0

0.89

4

11.2

5°

1.71

2

1.73

3

1.20

5

1.59

6

1.61

3

1.07

2

1.58

4

1.60

1

1.04

8

1.69

9

1.71

4

0.92

0

22.5

°

1.72

4

1.74

5

1.21

3

1.60

7

1.62

5

1.11

2

1.59

7

1.61

3

1.00

4

1.69

5

1.71

0

0.87

8

33.7

5°

1.70

0

1.71

9

1.14

9

1.59

0

1.60

8

1.10

3

1.57

8

1.59

5

1.06

4

1.66

7

1.68

2

0.92

5

45°

1.71

1

1.73

1

1.17

2

1.60

7

1.62

5

1.14

1

1.59

8

1.61

5

1.06

8

1.66

4

1.68

0

0.94

5

56.2

5°

1.68

9

1.70

6

1.01

8

1.59

8

1.61

4

0.99

9

1.58

8

1.60

6

1.08

2

1.65

1

1.66

8

1.04

3

67.5

°

1.71

7

1.73

4

1.00

7

1.64

3

1.65

9

0.98

9

1.64

6

1.66

4

1.08

4

1.69

5

1.71

3

1.04

4

78.7

5°

1.76

1

1.77

5

0.78

6

1.69

9

1.71

0

0.66

0

1.70

2

1.71

6

0.81

7

1.76

3

1.78

8

1.41

1

90°

1.85

7

1.87

3

0.91

1

1.79

7

1.80

8

0.60

9

1.81

8

1.82

8

0.54

4

1.88

9

1.93

8

2.60

2

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.5

2:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Ki/

{P\f

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

</>, f

or /

?*=

10,

fc=

2.5,

6>=

45°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

4>

0°

1.43

0

1.46

0

2.08

0

1.30

4

1.32

6

1.63

1

1.30

1

1.31

7

1.20

4

1.41

5

1.43

6

1.43

2

11.2

5°

1.41

9

1.44

8

2.00

2

1.29

1

1.31

3

1.70

7

1.28

6

1.30

3

1.34

0

1.39

4

1.41

4

1.42

8

22.5

°

1.43

1

1.46

1

2.08

6

1.30

5

1.32

7

1.67

6

1.29

9

1.31

6

1.27

1

1.39

2

1.41

1

1.38

4

33.7

5°

1.41

3

1.44

0

1.92

2

1.28

6

1.30

8

1.72

5

1.27

8

1.29

6

1.42

4

1.35

8

1.37

8

1.44

6

45°

1.42

4

1.45

3

2.00

1

1.30

6

1.32

8

1.73

7

1.29

5

1.31

4

1.46

5

1.35

3

1.37

3

1.51

3

56.2

5°

1.40

9

1.43

2

1.61

4

1.29

2

1.31

3

1.62

7

1.28

1

1.30

1

1.57

5

1.33

1

1.35

4

1.74

0

67.5

°

1.43

9

1.46

0

1.50

5

1.33

9

1.35

9

1.51

2

1.33

1

1.35

3

1.66

8

1.36

5

1.39

5

2.16

8

78.7

5°

1.47

9

1.49

0

0.69

4

1.37

2

1.38

4

0.93

0

1.36

4

1.38

0

1.15

5

1.40

6

1.43

8

2.24

5

90°

1.57

8

1.57

9

0.08

5

1.46

2

1.46

6

0.23

5

1.46

5

1.47

0

0.36

9

1.51

6

1.55

9

2.85

5

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.5

3:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Kj/

(PyT

rb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

0, f

or R

*=10

, k=

2.5,

9=

67.5

°:

IC -

Inf

luen

ce C

oeff

icie

nt

Met

hod;

A

%

- %

Dif

fere

nce b/

W

0.2

0.4

0.6

0.8

<t> 0°

1.43

4

1.46

3

2.04

8

1.30

6

1.32

8

1.66

0

1.29

9

1.31

5

1.27

9

1.43

1

1.45

0

1.34

8

11.2

5°

1.42

3

1.45

1

1.98

5

1.29

2

1.31

5

1.71

2

1.28

4

1.30

2

1.39

5

1.41

0

1.43

0

1.35

5

22.5

°

1.43

4

1.46

4

2.07

0

1.30

7

1.32

9

1.67

7

1.29

7

1.31

4

1.32

4

1.40

8

1.42

7

1.32

2

33.7

5°

1.41

6

1.44

3

1.90

9

1.28

8

1.31

0

1.71

2

1.27

6

1.29

5

1.47

8

1.37

5

1.39

4

1.38

2

45°

1.42

7

1.45

6

1.98

8

1.30

8

1.33

1

1.72

2

1.29

3

1.31

3

1.51

4

1.36

8

1.38

8

1.44

2

56.2

5°

1.41

3

1.43

6

1.59

5

1.29

5

1.31

5

1.59

3

1.27

9

1.30

0

1.63

0

1.34

7

1.36

9

1.62

6

67.5

°

1.44

3

1.46

4

1.48

3

1.34

2

1.36

2

1.46

9

1.33

0

1.35

3

1.71

6

1.38

2

1.40

9

1.94

0

78.7

5°

1.48

4

1.49

3

0.65

9

1.37

5

1.38

8

0.88

5

1.36

3

1.38

0

1.22

7

1.42

7

1.45

2

1.72

7

90°

1.58

3

1.58

4

0.05

4

1.46

6

1.47

0

0.28

3

1.46

4

1.47

1

0.50

5

1.54

4

1.55

7

0.79

6

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Tab

le

C.5

4:

Com

pari

son

of n

orm

aliz

ed

stre

ss

inte

nsit

y fa

ctor

s,

Ki/

{P\f

nb),

ob

tain

ed

from

th

e in

flue

nce

coef

fici

ent

met

hod

and

by d

irec

t B

EM

alo

ng c

rack

per

iphe

ry,

<f>

, fo

r i?

,*=1

0, f

c=2.

5, 0

=9

0°:

IC

- I

nflu

ence

Coe

ffic

ient

M

etho

d; A

%

- %

Dif

fere

nce

b/W

0.2

0.4

0.6

0.8

$

0°

1.43

3

1.46

3

2.10

8

1.30

6

1.32

7

1.59

6

1.29

9

1.31

5

1.22

9

1.43

4

1.45

4

1.35

6

11.2

5°

1.42

2

1.45

1

2.02

6

1.29

3

1.31

4

1.65

7

1.28

4

1.30

1

1.35

4

1.41

4

1.43

3

1.36

1

22.5

°

1.43

4

1.46

4

2.10

7

1.30

7

1.32

9

1.62

3

1.29

7

1.31

3

1.28

0

1.41

2

1.43

1

1.33

2

33.7

5°

1.41

6

1.44

3

1.94

1

1.28

8

1.31

0

1.66

1

1.27

6

1.29

4

1.43

4

1.37

9

1.39

8

1.37

7

45°

1.42

7

1.45

6

2.02

6

1.30

8

1.33

0

1.66

9

1.29

3

1.31

2

1.46

5

1.37

2

1.39

2

1.44

4

56.2

5°

1.41

2

1.43

6

1.63

8

1.29

5

1.31

5

1.54

3

1.27

9

1.30

0

1.58

3

1.35

2

1.37

4

1.62

1

67.5

°

1.44

2

1.46

4

1.53

1

1.34

3

1.36

2

1.42

2

1.33

0

1.35

2

1.66

1

1.38

7

1.41

3

1.93

7

78.7

5°

1.48

3

1.49

3

0.72

3

1.37

6

1.38

8

0.84

4

1.36

4

1.38

0

1.17

9

1.43

2

1.45

8

1.82

1

90°

1.58

2

1.58

4

0.12

1

1.46

7

1.47

0

0.22

8

1.46

5

1.47

1

0.43

9

1.55

1

1.57

4

1.48

7

Met

hod

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

Dir

ect

IC

A%

-J

en

Appendix D

Stress Intensity Factors due to Residual

Stresses from Autofrettage

176

Tab

le D

.l:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

KÂ

/icr

yV^b

],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

diff

eren

t le

vels

of

ove

rstr

ain

wit

h R

*=5,

k=

2,

8 =

45°

, b/

W=

0.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<f>

0°

-0.0

02

-0.0

27

-0.0

65

-0.1

80

-0.2

35

-0.2

87

-0.3

29

-0.3

62

-0.3

89

-0.4

18

11.2

5°

-0.0

02

-0.0

30

-0.0

69

-0.1

85

-0.2

40

-0.2

93

-0.3

35

-0.3

69

-0.3

96

-0.4

26

22.5

°

-0.0

02

-0.0

35

-0.0

73

-0.1

88

-0.2

43

-0.2

95

-0.3

37

-0.3

70

-0.3

97

-0.4

26

33.7

5°

-0.0

03

-0.0

45

-0.0

84

-0.1

99

-0.2

53

-0.3

06

-0.3

48

-0.3

81

-0.4

08

-0.4

37

45°

-0.0

04

-0.0

60

-0.0

99

-0.2

14

-0.2

68

-0.3

20

-0.3

62

-0.3

95

-0.4

22

-0.4

51

56.2

5°

-0.0

08

-0.0

82

-0.1

21

-0.2

36

-0.2

91

-0.3

43

-0.3

85

-0.4

19

-0.4

45

-0.4

74

67.5

°

-0.0

20

-0.1

08

-0.1

46

-0.2

60

-0.3

16

-0.3

67

-0.4

10

-0.4

43

-0.4

70

-0.4

97

78.7

5°

-0.0

49

-0.1

47

-0.1

87

-0.3

10

-0.3

72

-0.4

26

-0.4

74

-0.5

09

-0.5

37

-0.5

66

90°

-0.0

71

-0.1

65

-0.2

04

-0.3

21

-0.3

82

-0.4

33

-0.4

79

-0.5

13

-0.5

39

-0.5

66

Tab

le D

.2:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(KI)

A/[

CT

Y\M

\, fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge t

o di

ffer

ent

leve

ls

of o

vers

trai

n w

ith

#*

=5

, fc

=2,

6 =

45°

, 6

/W=

0.4

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0

°

0.00

1

0.00

3

0.04

3

-0.0

18

-0.0

69

-0.1

18

-0.1

55

-0.1

84

-0.2

07

-0.2

32

11.2

5°

0.00

2

0.00

3

0.04

3

-0.0

22

-0.0

74

-0.1

23

-0.1

60

-0.1

89

-0.2

12

-0.2

38

22.5

°

0.00

1

0.00

3

0.04

0

-0.0

33

-0.0

84

-0.1

34

-0.1

70

-0.2

00

-0.2

23

-0.2

48

33.7

5°

0.00

1

0.00

2

0.03

3

-0.0

53

-0.1

04

-0.1

54

-0.1

91

-0.2

20

-0.2

44

-0.2

70

45°

0.00

1

0.00

0

0.02

0

-0.0

82

-0.1

33

-0.1

83

-0.2

20

-0.2

50

-0.2

74

-0.3

01

56.2

5°

0.00

1

-0.0

09

-0.0

20

-0.1

26

-0.1

77

-0.2

27

-0.2

65

-0.2

96

-0.3

21

-0.3

49

67.5

°

-0.0

01

-0.0

38

-0.0

67

-0.1

76

-0.2

27

-0.2

78

-0.3

17

-0.3

49

-0.3

74

-0.4

02

78.7

5°

-0.0

19

-0.0

92

-0.1

26

-0.2

46

-0.3

04

-0.3

58

-0.4

04

-0.4

38

-0.4

66

-0.4

96

90°

-0.0

52

-0.1

33

-0.1

67

-0.2

93

-0.3

54

-0.4

08

-0.4

54

-0.4

89

-0.5

16

-0.5

45

Tab

le D

.3:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

A'/)

^/[c

ryv/7r

5],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

diff

eren

t le

vels

of

ove

rstr

ain

wit

h R

*=5,

k=

2,

6 =

45

°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<fi

0°

0.00

1

0.00

4

0.05

3

0.02

9

0.02

5

0.01

2

-0.0

27

-0.0

56

-0.0

77

-0.0

98

11.2

5°

0.00

1

0.00

4

0.05

3

0.02

9

0.02

5

0.00

6

-0.0

34

-0.0

62

-0.0

83

-0.1

04

22.5

°

0.00

1

0.00

3

0.05

2

0.02

7

0.02

1

-0.0

10

-0.0

49

-0.0

77

-0.0

98

-0.1

20

33.7

5°

0.00

1

0.00

3

0.05

2

0.02

3

0.00

5

-0.0

40

-0.0

77

-0.1

06

-0.1

27

-0.1

50

45°

0.00

1

0.00

3

0.05

0

0.00

8

-0.0

33

-0.0

81

-0.1

18

-0.1

47

-0.1

69

-0.1

93

56.2

5°

0.00

2

0.00

2

0.03

5

-0.0

43

-0.0

93

-0.1

43

-0.1

81

-0.2

11

-0.2

35

-0.2

61

67.5

°

0.00

3

-0.0

08

-0.0

14

-0.1

14

-0.1

66

-0.2

17

-0.2

56

-0.2

88

-0.3

13

-0.3

41

78.7

5°

-0.0

07

-0.0

61

-0.0

82

-0.1

97

-0.2

55

-0.3

10

-0.3

56

-0.3

91

-0.4

19

-0.4

50

90°

-0.0

42

-0.1

13

-0.1

38

-0.2

52

-0.3

12

-0.3

66

-0.4

13

-0.4

48

-0.4

75

-0.5

04

-J

CO

Tab

le D

.4:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

Ki)

A/[

cry/

Kb]

, fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge t

o di

ffer

ent

leve

ls

of o

vers

trai

n w

ith

R*=

5, f

c=2,

6 =

45°

, 6

/^=

0.8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0

°

0.00

1

0.00

5

0.05

7

0.04

3

0.03

6

0.05

3

0.06

6

0.06

0

0.03

4

0.01

6

11.2

5°

0.00

1

0.00

4

0.05

7

0.04

2

0.03

4

0.05

1

0.06

2

0.05

0

0.02

4

0.00

5

22.5

°

0.00

1

0.00

4

0.05

6

0.04

0

0.03

2

0.04

8

0.05

3

0.02

7

0.00

3

-0.0

16

33.7

5°

0.00

1

0.00

3

0.05

6

0.03

7

0.02

9

0.03

6

0.02

1

-0.0

12

-0.0

35

-0.0

56

45°

0.00

1

0.00

2

0.05

5

0.03

1

0.01

9

0.00

2

-0.0

35

-0.0

67

-0.0

90

-0.1

12

56.2

5°

0.00

2

0.00

2

0.04

8

0.00

6

-0.0

34

-0.0

77

-0.1

15

-0.1

46

-0.1

70

-0.1

96

67.5

°

0.00

3

-0.0

04

0.01

7

-0.0

73

-0.1

23

-0.1

72

-0.2

12

-0.2

45

-0.2

71

-0.3

00

78.7

5°

-0.0

06

-0.0

46

-0.0

54

-0.1

64

-0.2

22

-0.2

77

-0.3

24

-0.3

62

-0.3

93

-0.4

25

90°

-0.0

40

-0.1

04

-0.1

21

-0.2

36

-0.2

98

-0.3

54

-0.4

04

-0.4

43

-0.4

73

-0.5

05

Tab

le D

.5:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

Ki)

A/[

oy\f

nb},

fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge t

o di

ffer

ent

leve

ls

of o

vers

trai

n w

ith

fl!*

=5,

A=

2, 9

= 6

7.5°

, b/

W=

0.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

 0

°

-0.0

01

-0.0

26

-0.1

07

-0.1

99

-0.2

34

-0.2

88

-0.3

17

-0.3

54

-0.3

81

-0.4

03

11.2

5°

-0.0

01

-0.0

29

-0.1

10

-0.2

04

-0.2

39

-0.2

94

-0.3

23

-0.3

61

-0.3

88

-0.4

11

22.5

°

-0.0

01

-0.0

34

-0.1

15

-0.2

07

-0.2

41

-0.2

96

-0.3

25

-0.3

62

-0.3

89

-0.4

11

33.7

5°

-0.0

02

-0.0

45

-0.1

26

-0.2

18

-0.2

52

-0.3

07

-0.3

36

-0.3

73

-0.4

00

-0.4

22

45°

-0.0

03

-0.0

60

-0.1

41

-0.2

32

-0.2

67

-0.3

21

-0.3

51

-0.3

88

-0.4

15

-0.4

36

56.2

5°

-0.0

09

-0.0

83

-0.1

64

-0.2

55

-0.2

91

-0.3

44

-0.3

75

-0.4

12

-0.4

39

-0.4

60

67.5

°

-0.0

23

-0.1

09

-0.1

89

-0.2

79

-0.3

16

-0.3

69

-0.4

00

-0.4

37

-0.4

63

-0.4

84

78.7

5°

-0.0

54

-0.1

48

-0.2

36

-0.3

31

-0.3

73

-0.4

29

-0.4

63

-0.5

03

-0.5

31

-0.5

53

90°

-0.0

77

-0.1

67

-0.2

52

-0.3

42

-0.3

84

-0.4

37

-0.4

70

-0.5

08

-0.5

35

-0.5

55

Tab

le D

.6:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

Ki)

A/[

oY\f

nb} )

fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge t

o di

ffer

ent

leve

ls

of o

vers

trai

n w

ith

R*=

5, k

=2,

B

= 6

7.5°

, b/

W=

0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4>

0°

0.00

2

0.00

6

0.00

6

-0.0

56

-0.1

50

-0.2

65

-0.3

16

-0.3

90

-0.4

41

-0.4

84

11.2

5°

0.00

3

0.00

6

0.00

6

-0.0

66

-0.1

59

-0.2

75

-0.3

26

-0.4

01

-0.4

52

-0.4

96

22.5

°

0.00

2

0.00

6

0.00

3

-0.0

95

-0.1

82

-0.2

96

-0.3

47

-0.4

20

-0.4

71

-0.5

15

33.7

5°

0.00

2

0.00

5

-0.0

04

-0.1

45

-0.2

25

-0.3

38

-0.3

88

-0.4

62

-0.5

14

-0.5

57

45°

0.00

2

0.00

4

-0.0

17

-0.2

10

-0.2

83

-0.3

94

-0.4

46

-0.5

19

-0.5

71

-0.6

15

56.2

5°

0.00

2

-0.0

07

-0.0

58

-0.3

01

-0.3

70

-0.4

80

-0.5

33

-0.6

07

-0.6

59

-0.7

03

67.5

°

-0.0

01

-0.0

37

-0.1

07

-0.3

97

-0.4

66

-0.5

73

-0.6

29

-0.7

03

-0.7

55

-0.7

98

78.7

5°

-0.0

22

-0.0

92

-0.1

72

-0-5

36

-0.6

15

-0.7

29

-0.7

92

-0.8

72

-0.9

28

-0.9

73

90°

-0.0

56

-0.1

34

-0.2

13

-0.6

08

-0.6

89

-0.7

97

-0.8

61

-0.9

37

-0.9

90

-1.0

31

Tab

le D

.7:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

K7W

[oY

v/7r

fc],

fro

m r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge t

o di

ffer

ent

leve

ls

of o

vers

trai

n w

ith

/?*=

5, f

c=2,

0 =

67.

5°,

6/W

=0

.6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0

°

0.00

2

0.00

5

0.01

4

0.02

8

0.03

5

0.01

9

-0.0

13

-0.0

45

-0.0

66

-0.0

82

11.2

5°

0.00

2

0.00

5

0.01

4

0.02

8

0.03

5

0.01

2

-0.0

19

-0.0

51

-0.0

72

-0.0

89

22.5

°

0.00

2

0.00

5

0.01

4

0.02

6

0.03

1

-0.0

05

-0.0

34

-0.0

66

-0.0

87

-0.1

04

33.7

5°

0.00

2

0.00

6

0.01

4

0.02

2

0.01

4

-0.0

37

-0.0

63

-0.0

95

-0.1

17

-0.1

35

45°

0.00

2

0.00

1

0.01

2

0.00

5

-0.0

28

-0.0

80

-0.1

04

-0.1

37

-0.1

59

-0.1

78

56.2

5°

0.00

3

0.00

5

-0.0

03

-0.0

53

-0.0

90

-0.1

43

-0.1

67

-0.2

01

-0.2

25

-0.2

45

67.5

°

0.00

4

-0.0

05

-0.0

53

-0.1

30

-0.1

64

-0.2

17

-0.2

43

-0.2

79

-0.3

04

-0.3

25

78.7

5°

-0.0

07

-0.0

60

-0.1

27

-0.2

15

-0.2

53

-0.3

12

-0^3

43

-0.3

83

-0.4

11

-0.4

34

90°

-0.0

45

-0.1

14

-0.1

85

-0.2

72

-0.3

13

-0.3

69

-0.4

01

-0.4

41

-0.4

68

-0.4

90

Tab

le D

.8:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

Kj)

A/[

o-y\

/7rb

},

from

res

idua

l st

ress

due

to

auto

fret

tage

to

diff

eren

t le

vels

of

ove

rstr

ain

wit

h /J

*=5,

k=

2,

6 =

67.

5°,

6/V

K=0

.8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0°

0.00

2

0.00

5

0.01

5

0.03

4

0.04

3

0.06

6

0.07

8

0.07

3

0.04

7

0.03

2

11.2

5°

0.00

2

0.00

5

0.01

4

0.03

3

0.04

2

0.06

4

0.07

6

0.06

3

0.03

8

0.02

2

22.5

°

0.00

2

0.00

5

0.01

4

0.03

3

0.04

1

0.06

2

0.06

8

0.04

1

0.01

6

0.00

1

33.7

5°

0.00

2

0.00

5

0.01

3

0.03

3

0.04

0

0.04

8

0.03

7

0.00

0

-0.0

23

-0.0

40

45°

0.00

2

0.00

4

0.01

3

0.02

9

0.03

0

0.00

9

-0.0

19

-0.0

55

-0.0

78

-0.0

96

56.2

5°

0.00

3

0.00

5

0.00

7

0.00

0

-0.0

28

-0.0

73

-0.0

99

-0.1

34

-0.1

59

-0.1

79

67.5

°

0.00

5

-0.0

01

-0.0

25

-0.0

86

-0.1

19

-0.1

70

-0.1

96

-0.2

34

-0.2

60

-0.2

82

78.7

5°

-0.0

06

-0.0

45

-0.1

03

-0.1

81

-0.2

19

-0.2

77

-0.3

09

-0.3

51

-0.3

81

-0.4

06

90°

-0.0

42

-0.1

04

-0.1

73

-0.2

56

-0.2

98

-0.3

56

-0.3

90

-0.4

34

-0.4

64

-0.4

88

Tab

le D

.9:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s, (

Ki)

A/[

ay\/

nb},

fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge t

o di

ffer

ent

leve

ls

of o

vers

trai

n w

ith

R*=

5,

Jfc=

2, 9

= 9

0°,

b/W

=Q

.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4>

0°

0.00

0

-0.0

30

-0.1

13

-0.1

87

-0.2

32

-0.2

76

-0.3

15

-0.3

59

-0.3

71

-0.3

92

11.2

5°

0.00

0

-0.0

32

-0.1

17

-0.1

91

-0.2

37

-0.2

82

-0.3

22

-0.3

66

-0.3

78

-0.3

99

22.5

°

0.00

0

-0.0

38

-0.1

22

-0.1

95

-0.2

40

-0.2

84

-0.3

24

-0.3

68

-0.3

79

-0.4

00

33.7

5°

-0.0

01

-0.0

50

-0.1

33

-0.2

05

-0.2

51

-0.2

95

-0.3

34

-0.3

78

-0.3

90

-0.4

11

45°

-0.0

02

-0.0

66

-0.1

48

-0.2

20

-0.2

66

-0.3

10

-0.3

50

-0.3

94

-0.4

05

-0.4

26

56.2

5°

-0.0

07

-0.0

90

-0.1

71

-0.2

43

-0.2

90

-0.3

34

-0.3

74

-0.4

18

-0.4

30

-0.4

50

67.5

°

-0.0

21

-0.1

16

-0.1

97

-0.2

68

-0.3

15

-0.3

59

-0.4

00

-0.4

43

-0.4

55

-0.4

74

78.7

5°

-0.0

52

-0.1

56

-0.2

44

-0.3

19

-0.3

73

-0.4

19

-0.4

64

-0.5

11

-0.5

22

-0.5

42

90°

-0.0

75

-0.1

75

-0.2

60

-0.3

32

-0.3

85

-0.4

28

-0.4

71

-0.5

16

-0.5

26

-0.5

45

00

Tab

le

D.1

0:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Ki)

A/[

<T

yV^b

],

from

re

sidu

al s

tres

s du

e to

au

tofr

etta

ge

leve

ls o

f ov

erst

rain

wit

h R

*=5,

k-

2,

0 =

90°

, b/

W=

0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0

°

0.00

3

0.00

8

0.00

9

-0.0

13

-0.0

63

-0.1

05

-0.1

37

-0.1

76

-0.1

86

-0.2

04

11.2

5°

0.00

3

0.00

8

0.00

8

-0.0

18

-0.0

67

-0.1

10

-0.1

43

-0.1

82

-0.1

92

-0.2

11

22.5

°

0.00

3

0.00

7

0.00

5

-0.0

30

-0.0

78

-0.1

20

-0.1

53

-0.1

91

-0.2

02

-0.2

20

33.7

5°

0.00

3

0.00

7

-0.0

03

-0.0

53

-0.0

99

-0.1

41

-0.1

74

-0.2

13

-0.2

24

-0.2

43

45°

0.00

3

0.00

5

-0.0

18

-0.0

85

-0.1

28

-0.1

70

-0.2

04

-0.2

44

-0.2

54

-0.2

74

56.2

5°

0.00

2

-0.0

07

-0.0

62

-0.1

30

-0.1

73

-0.2

15

-0.2

51

-0.2

92

-0.3

03

-0.3

23

67.5

°

0.00

0

-0.0

41

-0.1

12

-0.1

81

-0.2

24

-0.2

67

-0.3

04

-0.3

46

-0.3

57

-0.3

77

78.7

5°

-0.0

20

-0.0

98

-0.1

78

-0.2

53

-0.3

03

-0.3

49

-0.3

91

-0.4

38

-0.4

49

-0.4

70

90°

-0.0

54

-0.1

41

-0.2

21

-0.2

92

-0.3

44

-0.3

87

-0.4

30

-0.4

74

-0.4

85

-0.5

04

Tab

le

D.l

l:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(KÂ

/lcr

yV^b

],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h fl

*=5,

k=

2,

6 =

90°

, 6

/^=

0.6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<P 0°

0.00

2

0.00

7

0.01

8

0.03

2

0.04

1

0.03

2

-0.0

07

-0.0

44

-0.0

53

-0.0

69

11.2

5°

0.00

2

0.00

7

0.01

8

0.03

2

0.04

1

0.02

6

-0.0

13

-0.0

50

-0.0

59

-0.0

75

22.5

°

0.00

2

0.00

7

0.01

8

0.03

1

0.03

7

0.00

8

-0.0

29

-0.0

66

-0.0

74

-0.0

91

33.7

5°

0.00

2

0.00

8

0.01

8

0.03

0

0.01

9

-0.0

23

-0.0

58

-0.0

95

-0.1

04

-0.1

21

45°

0.00

3

0.00

8

0.01

6

0.01

5

-0.0

23

-0.0

66

-0.1

00

-0.1

38

-0.1

47

-0.1

65

56.2

5°

0.00

3

0.00

7

-0.0

02

-0.0

41

-0.0

86

-0.1

29

-0.1

64

-0.2

03

-0.2

14

-0.2

33

67.5

°

0.00

5

-0.0

05

-0.0

56

-0.1

17

-0.1

60

-0.2

04

-0.2

41

-0.2

82

-0.2

94

-0.3

14

78.7

5°

-0.0

06

-0.0

64

-0.1

33

-0.2

02

-0.2

51

-0.2

99

-0.3

41

-0.3

88

-0.4

00

-0.4

22

90°

-0.0

43

-0.1

20

-0.1

92

-0.2

60

-0.3

11

-0.3

57

-0.4

01

-0.4

47

-0.4

58

-0.4

78

Tab

le

D.1

2:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(/f/

)x/[

<7yV

7r&

], f

rom

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h R

*=5,

k=

2, 0

= 9

0°,

b/W

=0.

8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0°

0.00

2

0.00

6

0.01

7

0.03

4

0.04

7

0.06

8

0.08

8

0.07

6

0.06

3

0.04

7

11.2

5°

0.00

2

0.00

6

0.01

7

0.03

3

0.04

7

0.06

8

0.08

6

0.06

6

0.05

3

0.03

7

22.5

°

0.00

2

0.00

6

0.01

7

0.03

4

0.04

7

0.06

7

0.07

6

0.04

3

0.03

1

0.01

5

33.7

5°

0.00

2

0.00

6

0.01

7

0.03

5

0.04

7

0.05

7

0.04

4

0.00

2

-0.0

08

-0.0

24

45°

0.00

2

0.00

6

0.01

7

0.03

4

0.03

7

0.02

3

-0.0

13

-0.0

53

-0.0

63

-0.0

80

56.2

5°

0.00

3

0.00

7

0.01

0

0.01

0

-0.0

22

-0.0

59

-0.0

94

-0.1

35

-0.1

45

-0.1

64

67.5

°

0.00

5

0.00

0

-0.0

26

-0.0

74

-0.1

15

-0.1

56

-0.1

93

-0.2

36

-0.2

48

-0.2

69

78.7

5°

-0.0

04

-0.0

49

-0.1

07

-0.1

68

-0.2

15

-0.2

62

-0.3

06

-0.3

55

-0.3

68

-0.3

92

90°

-0.0

41

-0.1

10

-0.1

79

-0.2

43

-0.2

95

-0.3

42

-0.3

88

-0.4

39

-0.4

52

-0.4

75

Tab

le

D.1

3:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(KJ)

AJ{

CT

YV

^>Y

from

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /?

*=7.

5, /

c=2,

6 =

45°

, b/

W=

0.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<P 0°

-0.0

05

-0.0

37

-0.1

25

-0.2

06

-0.2

49

-0.2

94

-0.3

33

-0.3

66

-0.3

90

-0.4

33

11.2

5°

-0.0

06

-0.0

39

-0.1

28

-0.2

10

-0.2

54

-0.2

98

-0.3

38

-0.3

71

-0.3

95

-0.4

38

22.5

°

-0.0

06

-0.0

45

-0.1

33

-0.2

14

-0.2

58

-0.3

02

-0.3

42

-0.3

74

-0.3

98

-0.4

41

33.7

5°

-0.0

07

-0.0

56

-0.1

44

-0.2

25

-0.2

68

-0.3

13

-0.3

52

-0.3

85

-0.4

09

-0.4

52

45°

-0.0

08

-0.0

72

-0.1

60

-0.2

40

-0.2

84

-0.3

27

-0.3

67

-0.4

00

-0.4

24

-0.4

66

56.2

5°

-0.0

14

-0.0

94

-0.1

82

-0.2

61

-0.3

06

-0.3

49

-0.3

90

-0.4

22

-0.4

46

-0.4

88

67.5

°

-0.0

28

-0.1

21

-0.2

08

-0.2

86

-0.3

31

-0.3

74

-0.4

15

-0.4

47

-0.4

70

-0.5

12

78.7

5°

-0.0

60

-0.1

59

-0.2

52

-0.3

33

-0.3

82

-0.4

26

-0.4

70

-0.5

04

-0.5

28

-0.5

72

90°

-0.0

84

-0.1

81

-0.2

70

-0.3

47

-0.3

96

-0.4

37

-0.4

81

-0.5

13

-0.5

36

-0.5

78

Tab

le

D.1

4:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(/G

W[c

ry\/

7r6]

, fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge

to

diff

eren

t le

vels

of

over

stra

in w

ith

/?*=

7.5,

k=

2,

0 =

45°

, b/

W=

0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0

°

0.00

2

0.00

1

-0.0

01

-0.0

29

-0.0

75

-0.1

15

-0.1

51

-0.1

80

-0.2

01

-0.2

37

11.2

5°

0.00

2

0.10

1

-0.0

02

-0.0

33

-0.0

79

-0.1

20

-0.1

56

-0.1

86

-0.2

06

-0.2

43

22.5

°

0.00

2

0.00

1

-0.0

04

-0.0

44

-0.0

89

-0.1

30

-0.1

66

-0.1

95

-0.2

16

-0.2

52

33.7

5°

0.00

1

0.00

0

-0.0

10

-0.0

66

-0.1

10

-0.1

51

-0.1

87

-0.2

16

-0.2

37

-0.2

74

45°

0.00

1

-0.0

03

-0.0

25

-0.0

96

-0.1

38

-0.1

79

-0.2

15

-0.2

45

-0.2

66

-0.3

04

56.2

5°

-0.0

01

-0.0

13

-0.0

66

-0.1

40

-0.1

81

-0.2

23

-0.2

59

-0.2

89

-0.3

11

-0.3

51

67.5

°

-0.0

03

-0.0

41

-0.1

16

-0.1

91

-0.2

32

-0.2

74

-0.3

11

-0.3

42

-0.3

64

-0.4

05

78.7

5°

-0.0

24

-0.1

00

-0.1

84

-0.2

64

-0.3

11

-0.3

55

-0.3

97

-0.4

30

-0.4

54

-0.4

98

90°

-0.0

62

-0.1

48

-0.2

32

-0.3

09

-0.3

57

-0.3

99

-0.4

42

-0.4

75

-0.4

98

-0.5

40

Tab

le

D.1

5:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(/G

W[c

ryv/7r

6],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h R

*=7.

5,

k=2,

8

= 4

5°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

(p 0°

0.00

2

0.00

5

0.01

4

0.02

1

0.02

8

0.02

0

-0.0

17

-0.0

45

-0.0

64

-0.0

93

11.2

5°

0.00

2

0.00

5

0.01

4

0.02

1

0.02

8

0.01

4

-0.0

23

-0.0

52

-0.0

71

-0.1

01

22.5

°

0.00

2

0.00

4

0.01

3

0.02

0

0.02

4

-0.0

01

-0.0

38

-0.0

66

-0.0

85

-0.1

15

33.7

5°

0.00

2

0.00

4

0.01

1

0.01

7

0.00

8

-0.0

28

-0.0

64

-0.0

93

-0.1

12

-0.1

43

45°

0.00

2

0.00

2

0.00

8

0.00

3

-0.0

28

-0.0

67

-0.1

03

-0.1

32

-0.1

51

-0.1

85

56.2

5°

0.00

2

0.00

0

-0.0

07

-0.0

45

-0.0

87

-0.1

27

-0.1

63

-0.1

93

-0.2

13

-0.2

50

67.5

°

0.00

3

-0.0

10

-0.0

52

-0.1

16

-0.1

57

-0.1

99

-0.2

35

-0.2

66

-0.2

87

-0.3

26

78.7

5 °

-0.0

10

-0.0

66

-0.1

35

-0.2

10

-0.2

56

-0.3

02

-0.3

44

-0.3

78

-0.4

02

-0.4

46

90°

-0.0

51

-0.1

26

-0.2

02

-0.2

76

-0.3

24

-0.3

67

-0.4

11

-0.4

45

-0.4

68

-0.5

12

Tab

le

D.1

6:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Kj)

^/'[o

y\fn

b],

from

re

sidu

al s

tres

s du

e to

au

tofr

etta

ge

leve

ls o

f ov

erst

rain

wit

h /?

,*-7

.5,

k=2,

0

= 4

5°,

b/W

=0.

8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0°

0.00

0

0.00

8

0.02

0

0.03

6

0.04

8

0.06

4

0.08

2

0.08

1

0.05

8

0.03

2

11.2

5°

0.00

0

0.00

8

0.01

9

0.03

5

0.04

7

0.06

3

0.08

0

0.07

2

0.04

9

0.02

3

22.5

°

0.00

0

0.00

7

0.01

8

0.03

4

0.04

4

0.06

0

0.07

2

0.05

1

0.02

9

0.00

2

33.7

5°

0.00

0

0.00

6

0.01

6

0.03

2

0.03

9

0.05

0

0.04

6

0.01

4

-0.0

07

-0.0

35

45°

0.00

1

0.00

5

0.01

3

0.02

8

0.02

9

0.02

6

-0.0

07

-0.0

37

-0.0

57

-0.0

87

56.2

5°

0.00

1

0.00

4

0.00

7

0.00

7

-0.0

15

-0.0

46

-0.0

82

-0.1

12

-0.1

32

-0.1

66

67.5

°

0.00

3

-0.0

01

-0.0

13

-0.0

61

-0.0

98

-0.1

37

-0.1

73

-0.2

03

-0.2

25

-0.2

63

78.7

5°

-0.0

09

-0.0

42

-0.1

00

-0.1

65

-0.2

08

-0.2

51

-0.2

92

-0.3

27

-0.3

52

-0.3

96

90°

-0.0

50

-0.1

11

-0.1

85

-0.2

52

-0.2

99

-0.3

42

-0.3

87

-0.4

23

-0.4

48

-0.4

93

Tab

le

D.1

7:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(/G

W[0

VV

7r6]

, ^

rom

re

sidu

al s

tres

s du

e to

au

tofr

etta

ge

leve

ls o

f ov

erst

rain

wit

h R

*=7.

5, k

=2,

9

= 6

7.5°

, b/

W=

0.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4>

0°

-0.0

04

-0.0

39

-0.1

20

-0.2

02

-0.2

45

-0.2

90

-0.3

30

-0.3

62

-0.3

84

-0.4

00

11.2

5°

-0.0

04

-0.0

42

-0.1

24

-0.2

06

-0.2

49

-0.2

94

-0.3

34

-0.3

68

-0.3

89

-0.4

05

22.5

°

-0.0

04

-0.0

48

-0.1

29

-0.2

10

-0.2

53

-0.2

98

-0.3

38

-0.3

71

-0.3

92

-0.4

08

33.7

5°

-0.0

05

-0.0

60

-0.1

40

-0.2

21

-0.2

65

-0.3

09

-0.3

49

-0.3

82

-0.4

03

-0.4

19

45°

-0.0

07

-0.0

76

-0.1

55

-0.2

36

-0.2

79

-0.3

23

-0.3

63

-0.3

96

-0.4

17

-0.4

33

56.2

5°

-0.0

12

-0.0

99

-0.1

78

-0.2

58

-0.3

02

-0.3

46

-0.3

86

-0.4

19

-0.4

40

-0.4

55

67.5

°

-0.0

25

-0.1

25

-0.2

04

-0.2

82

-0.3

28

-0.3

70

-0.4

11

-0.4

44

-0.4

65

-0.4

79

78.7

5°

-0.0

57

-0.1

64

-0.2

48

-0.3

29

-0.3

79

-0.4

23

-0.4

67

-0.5

01

-0.5

23

-0.5

37

90°

-0.0

82

-0.1

86

-0.2

67

-0.3

44

-0.3

93

-0.4

35

-0.4

79

-0.5

11

-0.5

31

-0.5

44

Tab

le

D.1

8:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(A'jr

) /i/[<

7yv/7r

6],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /?

,*=7

.5, J

fe=2

, 9

= 6

7.5°

, b/

W=

0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0°

0.00

3

0.00

3

0.00

3

-0.0

23

-0.0

69

-0.1

11

-0.1

46

-0.1

75

-0.1

94

-0.2

09

11.2

5°

0.00

3

0.00

3

0.00

2

-0.0

28

-0.0

74

-0.1

16

-0.1

51

-0.1

81

-0.1

99

-0.2

14

22.5

°

0.00

3

0.00

2

0.00

0

-0.0

39

-0.0

84

-0.1

25

-0.1

61

-0.1

90

-0.2

09

-0.2

24

33.7

5°

0.00

2

0.00

1

-0.0

06

-0.0

61

-0.1

05

-0.1

46

-0.1

82

-0.2

11

-0.2

30

-0.2

45

45°

0.00

2

-0.0

01

-0.0

20

-0.0

91

-0.1

33

-0.1

75

-0.2

10

-0.2

40

-0.2

59

-0.2

75

56.2

5°

0.00

0

-0.0

12

-0.0

61

-0.1

36

-0.1

77

-0.2

19

-0.2

55

-0.2

86

-0.3

05

-0.3

21

67.5

°

-0.0

02

-0.0

43

-0.1

11

-0.1

87

-0.2

28

-0.2

70

-0.3

07

-0.3

38

-0.3

58

-0.3

74

78.7

5°

-0.0

21

-0.1

04

-0.1

80

-0.2

61

-0!3

07

-0.3

52

-0.3

94

-0.4

27

-0.4

48

-0.4

64

90°

-0.0

60

-0.1

52

-0.2

28

-0.3

06

-0.3

54

-0.3

96

-0.4

40

-0.4

72

-0.4

93

-0.5

06

Tab

le

D.1

9:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Ki)

A/[

&Y

\/nb

},

from

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /?

.*=7

.5, k

=2,

9 =

67.

5°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0°

0.00

2

0.00

6

0.01

6

0.02

6

0.03

5

0.02

8

-0.0

10

-0.0

38

-0.0

55

-0.0

68

11.2

5°

0.00

2

0.00

6

0.01

6

0.02

5

0.03

5

0.02

2

-0.0

15

-0.0

44

-0.0

61

-0.0

74

22.5

°

0.00

2

0.00

6

0.01

5

0.02

4

0.03

1

0.00

6

-0.0

30

-0.0

59

-0.0

76

-0.0

89

33.7

5°

0.00

2

0.00

5

0.01

4

0.02

2

0.01

5

-0.0

22

-0.0

58

-0.0

86

-0.1

03

-0.1

17

45°

0.00

2

0.00

4

0.01

2

0.00

9

-0.0

21

-0.0

61

-0.0

97

-0.1

25

-0.1

43

-0.1

57

56.2

5°

0.00

3

0.00

1

-0.0

03

-0.0

39

-0.0

81

-0.1

22

-0.1

57

-0.1

87

-0.2

05

-0.2

20

67.5

°

0.00

4

-0.0

10

-0.0

47

-0.1

11

-0.1

51

-0.1

93

-0.2

30

-0.2

60

-0.2

79

-0.2

95

78.7

5°

-0.0

09

-0.0

68

-0.1

30

-0.2

05

-0.2

50

-0.2

96

-0.3

38

-0.3

72

-0.3

94

-0.4

10

90°

-0.0

49

-0.1

29

-0.1

97

-0.2

71

-0.3

19

-0.3

63

-0.4

07

-0.4

40

-0.4

61

-0.4

76

Tab

le

D.2

0:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(AT

/)^/

[cry

v/7r6]

, fr

om

resi

dual

str

ess

due

to a

utof

rett

age

to

diff

eren

t le

vels

of

over

stra

in w

ith

/?.*

=7.5

, k=

2, 0

= 6

7.5°

, b/

W=

0.8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<P 0°

0.00

3

0.00

9

0.02

0

0.03

6

0.04

9

0.06

8

0.08

9

0.08

7

0.06

5

0.05

2

11.2

5°

0.00

3

0.00

8

0.01

9

0.03

6

0.04

8

0.66

8

0.08

7

0.07

9

0.05

8

0.04

4

22.5

°

0.00

2

0.00

8

0.01

8

0.03

5

0.04

6

0.06

4

0.08

0

0.05

7

0.03

7

0.02

3

33.7

5°

0.00

2

0.00

7

0.01

7

0.03

4

0.04

3

0.05

6

0.05

2

0.01

8

-0.0

01

-0.0

14

45°

0.00

2

0.00

7

0.01

5

0.03

1

0.03

4

0.03

2

-0.0

01

-0.0

34

-0.0

52

-0.0

65

56.2

5°

0.00

3

0.00

5

0.01

0

0.01

1

-0.0

10

-0.0

42

-0.0

77

-0.1

08

-0.1

26

-0.1

41

67.5

°

0.00

4

0.00

0

-0.0

09

-0.0

57

-0.0

94

-0.1

33

-0.1

69

-0.2

01

-0.2

21

-0.2

37

78.7

5°

-0.0

07

-0.0

44

-0.0

96

-0.1

61

-0.2

04

-0.2

47

-0.2

89

-0.3

25

-0.3

47

-0.3

64

90°

-0.0

47

-0.1

14

-0.1

81

-0.2

49

-0.2

96

-0.3

39

-0.3

85

-0.4

21

-0.4

43

-0.4

59

Tab

le

D.2

1:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Kj)

A/[

<Jy

\/T

rb},

fro

m r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge

to

diff

eren

t le

vels

of

over

stra

in w

ith

/?.*

=7.5

, k=

2,

9 =

90°

, b/

W=

0.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0°

-0.0

04

-0.0

40

-0.1

12

-0.1

86

-0.2

41

-0.2

92

-0.3

27

-0.3

64

-0.3

82

-0.4

03

11.2

5°

-0.0

04

-0.0

43

-0.1

16

-0.1

90

-0.2

45

-0.2

96

-0.3

32

-0.3

69

-0.3

87

-0.4

08

22.5

°

-0.0

04

-0.0

49

-0.1

21

-0.1

94

-0.2

50

-0.3

00

-0.3

36

-0.3

73

-0.3

90

-0.4

12

33.7

5°

-0.0

05

-0.0

61

-0.1

32

-0.2

05

-0.2

60

-0.3

11

-0.3

46

-0.3

83

-0.4

01

-0.4

22

45°

-0.0

07

-0.0

77

-0.1

48

-0.2

20

-0.2

76

-0.3

25

-0.3

61

-0.3

98

-0.4

15

-0.4

36

56.2

5°

-0.0

12

-0.1

00

-0.1

71

-0.2

43

-0.2

98

-0.3

48

-0.3

84

-0.4

21

-0.4

38

-0.4

59

67.5

°

-0.0

26

-0.1

27

-0.1

97

-0.2

68

-0.3

24

-0.3

73

-0.4

10

-0.4

46

-0.4

63

-0.4

83

78.7

5°

-0.0

58

-0.1

66

-0.2

40

-0.3

14

-0.3

75

-0.4

26

-0.4

66

-0.5

04

-0.5

21

-0.5

41

90°

-0.0

83

-0.1

88

-0.2

60

-0.3

29

-0.3

90

-0.4

38

-0.4

78

-0.5

14

-0.5

30

-0.5

48

Tab

le

D.2

2:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Ki)

A/[

a Y\f

Jib]

, fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge

to

diff

eren

t le

vels

of

over

stra

in w

ith

R*=

7.5,

k=

2, 0

= 9

0°,

b/W

=0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0

°

0.00

3

0.00

3

0.00

5

-0.0

12

-0.0

65

-0.1

13

-0.1

44

-0.1

76

-0.1

91

-0.2

11

11.2

5°

0.00

3

0.00

3

0.00

5

-0.0

16

-0.0

69

-0.1

17

-0.1

48

-0.1

81

-0.1

96

-0.2

16

22.5

°

0.00

3

0.00

3

0.00

3

-0.0

27

-0.0

79

-0.1

27

-0.1

58

-0.1

90

-0.2

06

-0.2

25

33.7

5°

0.00

2

0.00

2

-0.0

02

-0.0

48

-0.1

00

-0.1

48

-0.1

79

-0.2

12

-0.2

27

-0.2

47

45°

0.00

2

-0.0

01

-0.0

15

-0.0

77

-0.1

29

-0.1

77

-0.2

08

-0.2

42

-0.2

57

-0.2

78

56.2

5°

0.00

1

-0.0

12

-0.0

54

-0.1

21

-0.1

73

-0.2

21

-0.2

53

-0.2

87

-0.3

03

-0.3

23

67.5

°

-0.0

02

-0.0

44

-0.1

04

-0.1

72

-0.2

24

-0.2

72

-0.3

05

-0.3

40

-0.3

56

-0.3

77

78.7

5°

-0.0

22

-0.1

05

-0.1

72

-0.2

45

-0.3

03

-0.3

54

-0.3

92

-0.4

29

-0.4

47

-0.4

67

90°

-0.0

61

-0.1

53

-0.2

21

-0.2

91

-0.3

51

-0.3

99

-0.4

38

-0.4

74

-0.4

91

-0.5

10

Tab

le

D.2

3:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(A/W

foys

/Trf

t],

from

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h K

*=7.

5, k

=2,

0 -

90

°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<P 0°

0.00

2

0.00

7

0.01

6

0.02

6

0.03

8

0.03

0

-0.0

06

-0.0

37

-0.0

52

-0.0

68

11.2

5°

0.00

2

0.00

7

0.01

6

0.02

6

0.03

8

0.02

4

-0.0

12

-0.0

44

-0.0

58

-0.0

75

22.5

°

0.00

2

0.00

7

0.01

5

0.02

5

0.03

4

0.00

7

-0.0

27

-0.0

58

-0.0

73

-0.0

89

33.7

5°

0.00

2

0.00

6

0.01

5

0.02

5

0.01

9

-0.0

22

-0.0

55

-0.0

86

-0.1

00

-0.1

17

45°

0.00

3

0.00

5

0.01

3

0.01

5

-0.0

16

-0.0

63

-0.0

94

-0.1

26

-0.1

40

-0.1

58

56.2

5°

0.00

3

0.00

2

0.00

0

-0.0

28

-0.0

76

-0.1

24

-0.1

55

-0.1

87

-0.2

03

-0.2

22

67.5

°

0.00

4

-0.0

09

-0.0

41

-0.0

97

-0.1

47

-0.1

95

-0.2

27

-0.2

61

-0.2

77

-0.2

98

78.7

5°

-0.0

08

-0.0

69

-0.1

23

-0.1

90

-0.2

46

-0.2

99

-0.3

36

-0.3

74

-0.3

92

-0.4

14

90°

-0.0

50

-0.1

31

-0.1

90

-0.2

56

-0.3

15

-0.3

66

-0.4

05

-0.4

42

-0.4

60

-0.4

79

CD

Tab

le

D.2

4:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(A'/)

/t/[c

'Y"v

/7rfr

], fr

om

resi

dual

str

ess

due

to a

utof

rett

age

to

diff

eren

t le

vels

of

over

stra

in w

ith

/?.*

=7.5

, k=

2,

0 =

90°

, 6

/^=

0-8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<t> 0

°

0.00

2

0.00

9

0.01

9

0.03

3

0.04

9

0.07

1

0.09

2

0.09

0

0.07

1

0.05

5

11.2

5°

0.00

2

0.00

9

0.01

8

0.03

3

0.04

9

0.07

1

0.09

2

0.08

1

0.06

3

0.04

7

22.5

°

0.00

2

0.00

8

0.01

7

0.03

2

0.04

7

0.06

9

0.08

5

0.05

9

0.04

2

0.02

6

33.7

5°

0.00

2

0.00

8

0.01

6

0.03

2

0.04

4

0.06

0

0.05

8

0.02

1

0.00

5

-0.0

11

45°

0.00

2

0.00

7

0.01

5

0.03

1

0.03

7

0.03

4

0.00

3

-0.0

31

-0.0

46

-0.0

63

56.2

5°

0.00

3

0.00

6

0.01

1

0.01

6

-0.0

05

-0.0

42

-0.0

73

-0.1

07

-0.1

23

-0.1

41

67.5

°

0.00

4

0.00

0

-0.0

05

-0.0

45

-0.0

89

-0.1

34

-0.1

65

-0.2

00

-0.2

17

-0.2

37

78.7

5°

-0.0

07

-0.0

44

-0.0

90

-0.1

47

-0.1

99

-0.2

49

-0.2

86

-0.3

25

-0.3

44

-0.3

66

90°

-0.0

48

-0.1

15

-0.1

74

-0.2

34

-0.2

91

-0.3

41

-0.3

82

-0.4

22

-0.4

40

-0.4

62

to

o

o

Tab

le D

.25:

N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

(A

'/)^/

[cry

V/vr

6],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h R

*=10

, k=

2,

0 =

45°

, b/

W=

0,2

% O

vers

trai

n

in

20

30

40

50

60

70

80

90

100

4> 0°

-0.0

05

-0.0

39

-0.1

22

-0.1

86

-0.2

54

-0.3

01

-0.3

39

-0.3

73

-0.3

91

-0.4

08

11.2

5°

-0.0

05

-0.0

42

-0.1

25

-0.1

90

-0.2

58

-0.3

05

-0.3

44

-0.3

78

-0.3

96

-0.4

13

22.5

°

-0.0

05

-0.0

47

-0.1

31

-0.1

95

-0.2

63

-0.3

09

-0.3

47

-0.3

81

-0.3

99

-0.4

16

33.7

5°

-0.0

06

-0.0

58

-0.1

42

-0.2

06

-0.2

74

-0.3

20

-0.3

58

-0.3

92

-0.4

10

-0.4

27

45°

-0.0

07

-0.0

74

-0.1

57

-0.2

21

-0.2

88

-0.3

34

-0.3

73

-0.4

06

-0.4

24

-0.4

40

56.2

5°

-0.0

12

-0.0

97

-0.1

80

-0.2

43

-0.3

11

-0.3

56

-0.3

95

-0.4

29

-0.4

46

-0.4

62

67.5

°

-0.0

23

-0.1

23

-0.2

06

-0.2

69

-0.3

37

-0.3

81

-0.4

20

-0.4

54

-0.4

71

-0.4

86

78.7

5°

-0.0

54

-0.1

63

-0.2

50

-0.3

14

-0.3

88

-0.4

33

-0.4

76

-0.5

11

-0.5

29

-0.5

43

90°

-0.0

79

-0.1

85

-0.2

69

-0.3

29

-0.4

02

-0.4

44

-0.4

87

-0.5

21

-0.5

38

-0.5

50

to

o

Tab

le D

.26

: N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

(K

Â/I

^YV

T^]

, fr

om

resi

dual

str

ess

due

to a

utof

rett

age

to

diff

eren

t le

vels

of

over

stra

in w

ith

/i*=

10,

Jfc=

2, 0

= 4

5°,

b/W

=0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

</>

0°

0.00

2

0.00

0

-0.0

01

-0.0

16

-0.0

75

-0.1

18

-0.1

53

-0.1

84

-0.1

99

-0.2

15

11.2

5°

0.00

2

0.00

0

-0.0

02

-0.0

20

-0.0

80

-0.1

23

-0.1

58

-0.1

89

-0.2

04

-0.2

21

22.5

°

0.00

2

0.00

0

-0.0

04

-0.0

29

-0.0

90

-0.1

33

-0.1

68

-0.1

98

-0.2

14

-0.2

30

33.7

5°

0.00

1

-0.0

02

-0.0

09

-0.0

48

-0.1

10

-0.1

53

-0.1

88

-0.2

19

-0.2

35

-0.2

51

45°

0.00

1

-0.0

04

-0.0

22

-0.0

76

-0.1

39

-0.1

82

-0.2

17

-0.2

48

-0.2

64

-0.2

81

56.2

5°

0.00

0

-0.0

14

-0.0

60

-0.1

19

-0.1

82

-0.2

26

-0.2

61

-0.2

92

-0.3

09

-0.3

25

67.5

°

-0.0

02

-0.0

41

-0.1

10

-0.1

70

-0.2

33

-0.2

77

-0.3

13

-0.3

44

-0.3

61

-0.3

77

78.7

5°

-0.0

19

-0.1

02

-0.1

81

-0.2

44

-0.3

14

-0.3

60

-0.4

00

-0.4

34

-0.4

52

-0.4

68

90°

-0.0

58

-0.1

51

-0.2

30

-0.2

90

-0.3

62

-0.4

05

-0.4

47

-0.4

81

-0.4

98

-0.5

11

Tab

le

D.2

7:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Kj)

A/[

aY\f

nb\,

from

res

idua

l st

ress

due

to

auto

fret

tage

to

le

vels

of

over

stra

in w

ith

/2*=

10,

k=2,

0

= 4

5°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<f> 0°

0.00

2

0.00

5

0.01

3

0.01

8

0.02

8

0.02

0

-0.0

15

-0.0

45

-0.0

59

-0.0

74

11.2

5°

0.00

2

0.00

5

0.01

3

0.01

8

0.02

7

0.01

4

-0.0

21

-0.0

51

-0.0

66

-0.0

80

22.5

°

0.00

2

0.00

4

0.01

2

0.01

8

0.02

3

0.00

0

-0.0

34

-0.0

64

-0.0

79

-0.0

93

33.7

5°

0.00

2

0.00

4

0.01

1

0.01

7

0.00

8

-0.0

27

-0.0

62

-0.0

92

-0.1

06

-0.1

21

45°

0.00

1

0.00

2

0.00

8

0.00

9

-0.0

24

-0.0

65

-0.1

00

-0.1

30

-0.1

45

-0.1

60

56.2

5°

0.00

1

-0.0

01

-0.0

05

-0.0

27

-0.0

82

-0.1

25

-0.1

60

-0.1

90

-0.2

06

-0.2

22

67.5

°

0.00

2

-0.0

10

-0.0

43

-0.0

92

-0.1

52

-0.1

96

-0.2

31

-0.2

62

-0.2

79

-0.2

95

78.7

5°

-0.0

10

-0.0

65

-0.1

29

-0.1

87

-0.2

55

-0.3

02

-0.3

42

-0.3

77

-0.3

95

-0.4

12

90°

-0.0

49

-0.1

28

-0.1

99

-0.2

56

-0.3

27

-0.3

71

-0.4

14

-0.4

48

-0.4

66

-0.4

81

Tab

le

D.2

8:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(K})

A/[

a YV

^b},

fr

om

resi

dual

str

ess

due

to a

utof

rett

age

to

diff

eren

t le

vels

of

over

stra

in w

ith

R*=

10,

fc=

2, 9

= 4

5°,

b/W

=0.

8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4>

0°

0.00

3

0.00

9

0.02

0

0.03

1

0.04

9

0.06

5

0.09

1

0.08

4

0.06

5

0.05

1

11.2

5°

0.00

2

0.00

9

0.02

0

0.03

0

0.04

8

0.06

3

0.09

1

0.07

6

0.05

7

0.04

3

22.5

°

0.00

2

0.00

8

0.01

8

0.02

8

0.04

4

0.06

0

0.08

5

0.05

4

0.03

6

0.02

2

33.7

5°

0.00

2

0.00

7

0.01

6

0.02

6

0.03

9

0.05

1

0.06

2

0.01

7

0.00

0

-0.0

13

45°

0.00

2

0.00

5

0.01

3

0.02

3

0.03

0

0.03

0

0.01

0

-0.0

32

-0.0

48

-0.0

63

56.2

5°

0.00

2

0.00

4

0.00

9

0.01

0

-0.0

08

-0.0

40

-0.0

65

-0.1

05

-0.1

21

-0.1

37

67.5

°

0.00

3

0.00

0

-0.0

07

-0.0

39

-0.0

89

-0.1

29

-0.1

56

-0.1

96

-0.2

13

-0.2

30

78.7

5°

-0.0

08

-0.0

39

-0.0

91

-0.1

42

-0.2

04

-0.2

49

-0.2

82

-0.3

24

-0.3

43

-0.3

61

90°

-0.0

48

-0.1

13

-0.1

81

-0.2

34

-0.3

02

-0.3

46

-0.3

86

-0.4

27

-0.4

46

-0.4

62

Tab

le

D.2

9:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(K\)

AI\

oy\

fnb

],

from

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /?

*=10

, k~

2,

0 =

67.

5 °,

b/

W=

0.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<P 0

°

-0.0

04

-0.0

36

-0.1

23

-0.1

79

-0.2

54

-0.3

02

-0.3

39

-0.3

72

-0.3

90

-0.4

10

11.2

5°

-0.0

05

-0.0

39

-0.1

26

-0.1

82

-0.2

59

-0.3

06

-0.3

44

-0.3

77

-0.3

96

-0.4

15

22.5

°

-0.0

05

-0.0

45

-0.1

31

-0.1

87

-0.2

63

-0.3

10

-0.3

48

-0.3

80

-0.3

99

-0.4

18

33.7

5°

-0.0

06

-0.0

56

-0.1

42

-0.1

98

-0.2

74

-0.3

20

-0.3

58

-0.3

91

-0.4

10

-0.4

28

45°

-0.0

07

-0.0

71

-0.1

58

-0.2

13

-0.2

89

-0.3

35

-0.3

73

-0.4

05

-0.4

24

-0.4

42

56.2

5°

-0.0

12

-0.0

94

-0.1

81

-0.2

36

-0.3

12

-0.3

57

-0.3

96

-0.4

28

-0.4

46

-0.4

64

67.5

°

-0.0

24

-0.1

21

-0.2

07

-0.2

62

-0.3

37

-0.3

82

-0.4

21

-0.4

53

-0.4

71

-0.4

88

78.7

5°

-0.0

55

-0.1

60

-0.2

52

-0.3

07

-0.3

89

-0.4

35

-0.4

77

-0.5

11

-0.5

29

-0.5

46

90

°

-0.0

81

-0.1

83

-0.2

71

-0.3

22

-0.4

04

-0.4

46

-0.4

89

-0.5

20

-0.5

38

-0.5

53

Tab

le D

.30:

N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

(•

K/W

[0Y

V/7r

6],

fr°m

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h #

"=1

0,

k=2,

9 =

67.

5°,

b/W

=0A

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<P 0

°

0.00

2

0.00

1

0.00

0

-0.0

10

-0.0

75

-0.1

19

-0.1

53

-0.1

82

-0.1

98

-0.2

16

11.2

5°

0.00

2

0.00

1

-0.0

01

-0.0

14

-0.0

80

-0.1

24

-0.1

58

-0.1

87

-0.2

03

-0.2

22

22.5

°

0.00

2

0.00

1

-0.0

03

-0.0

23

-0.0

90

-0.1

34

-0.1

67

-0.1

96

-0.2

13

-0.2

31

33.7

5°

0.00

2

-0.0

01

-0.0

08

-0.0

41

-0.1

10

-0.1

54

-0.1

88

-0.2

17

-0.2

34

-0.2

52

45°

0.00

1

-0.0

03

-0.0

21

-0.0

68

-0.1

39

-0.1

83

-0.2

17

-0.2

46

-0.2

63

-0.2

82

56.2

5°

0.00

0

-0.0

12

-0.0

60

-0.1

11

-0.1

82

-0.2

27

-0.2

61

-0.2

91

-0.3

08

-0.3

27

67.5

°

-0.0

02

-0.0

39

-0.1

10

-0.1

62

-0.2

33

-0.2

78

-0.3

13

-0.3

43

-0.3

61

-0.3

79

78.7

5°

-0.0

19

-0.0

99

-0.1

81

-0.2

37

-0.3

15

-0.3

61

-0.4

01

-0.4

34

-0.4

52

-0.4

70

90°

-0.0

59

-0.1

49

-0.2

31

-0.2

83

-0.3

63

-0.4

07

-0.4

48

-0.4

80

-0.4

98

-0.5

14

Tab

le

D.3

1:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(Kj)

^/[c

rYV

^b],

fr

om

resi

dual

str

ess

due

to a

utof

rett

age

to

diff

eren

t le

vels

of

over

stra

in w

ith

R*=

10,

k=2,

8

= 6

7.5°

, b/

W=

0.6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0°

0.00

2

0.00

5

0.01

5

0.02

0

0.03

2

0.02

4

-0.0

12

-0.0

41

-0.0

56

-0.0

71

11.2

5°

0.00

2

0.00

5

0.01

5

0.02

0

0.03

2

0.01

8

-0.0

18

-0.0

47

-0.0

62

-0.0

77

22.5

°

0.00

2

0.00

5

0.01

4

0.01

9

0.02

8

0.00

3

-0.0

32

-0.0

60

-0.0

76

-0.0

91

33.7

5°

0.00

2

0.00

4

0.01

3

0.01

9

0.01

2

-0.0

25

-0.0

60

-0.0

88

-0.1

03

-0.1

19

45°

0.00

2

0.00

3

0.01

0

0.01

3

-0.0

22

-0.0

64

-0.0

98

-0.1

26

-0.1

42

-0.1

59

56.2

5°

0.00

2

0.00

0

-0.0

03

-0.0

21

-0.0

81

-0.1

24

-0.1

58

-0.1

87

-0.2

03

-0.2

21

67.5

°

0.00

3

-0.0

09

-0.0

43

-0.0

84

-0.1

51

-0.1

95

-0.2

29

-0.2

59

-0.2

76

-0.2

95

78.7

5 °

-0.0

09

-0.0

63

-0.1

28

-0.1

79

-0.2

54

-0.3

01

-0.3

41

-0.3

74

-0.3

93

-0.4

12

90°

-0.0

50

-0.1

25

-0.1

99

-0.2

48

-0.3

26

-0.3

71

-0.4

13

-0.4

45

-0.4

63

-0.4

80

Tab

le

D.3

2:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(KÂ

/l^y

V^b

],

from

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h R

*=W

, k=

2,

0 =

67.

5°,

b/W

=0.

8

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0°

0.00

3

0.00

9

0.02

1

0.03

0

0.05

1

0.06

9

0.09

1

0.08

7

0.06

8

0.05

4

11.2

5°

0.00

3

0.00

8

0.02

0

0.02

9

0.05

0

0.06

8

0.08

9

0.07

9

0.06

0

0.04

6

22.5

°

0.00

2

0.00

8

0.01

9

0.02

7

0.04

7

0.06

5

0.08

1

0.05

7

0.03

9

0.02

5

33.7

5°

0.00

2

0.00

7

0.01

7

0.02

5

0.04

3

0.05

6

0.05

4

0.02

1

0.00

3

-0.0

11

45°

0.00

2

0.00

6

0.01

4

0.02

3

0.03

4

0.03

3

0.00

1

-0.0

30

-0.0

46

-0.0

61

56.2

5°

0.00

2

0.00

4

0.01

1

0.01

3

-0.0

06

-0.0

39

-0.0

73

-0.1

03

-0.1

19

-0.1

36

67.5

°

0.00

3

0.00

1

-0.0

05

-0.0

33

-0.0

88

-0.1

29

-0.1

63

-0.1

94

-0.2

11

-0.2

30

78.7

5°

-0.0

08

-0.0

36

-0.0

91

-0.1

35

-0.2

03

-0.2

49

-0.2

88

-0.3

22

-0.3

42

-0.3

62

90°

-0.0

48

-0.1

11

-0.1

82

-0.2

27

-0.3

03

-0.3

48

-0.3

91

-0.4

26

-0.4

45

-0.4

64

Tab

le D

.33

: N

orm

aliz

ed s

tres

s in

tens

ity

fact

ors,

(/

GW

[°Y

V7r

6],

from

res

idua

l st

ress

due

to

auto

fret

tage

to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /?

.*=1

0, f

c=2,

6 =

90

°,

b/W

=Q

.2

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<f>

0°

-0.0

04

-0.0

37

-0.1

24

-0.1

76

-0.2

55

-0.2

98

-0.3

38

-0.3

71

-0.3

89

-0.4

07

11.2

5°

-0.0

05

-0.0

40

-0.1

27

-0.1

80

-0.2

60

-0.3

03

-0.3

43

-0.3

76

-0.3

95

-0.4

12

22.5

°

-0.0

05

-0.0

45

-0.1

32

-0.1

84

-0.2

64

-0.3

07

-0.3

47

-0.3

79

-0.3

97

-0.4

15

33.7

5°

-0.0

06

-0.0

56

-0.1

43

-0.1

96

-0.2

75

-0.3

18

-0.3

58

-0.3

90

-0.4

08

-0.4

26

45°

-0.0

07

-0.0

72

-0.1

59

-0.2

11

-0.2

90

-0.3

32

-0.3

72

-0.4

05

-0.4

23

-0.4

40

56.2

5°

-0.0

12

-0.0

95

-0.1

82

-0.2

34

-0.3

13

-0.3

55

-0.3

95

-0.4

27

-0.4

45

-0.4

62

67.5

°

-0.0

24

-0.1

22

-0.2

08

-0.2

60

-0.3

38

-0.3

79

-0.4

20

-0.4

52

-0.4

70

-0.4

86

78.7

5°

-0.0

56

-0.1

61

-0.2

52

-0.3

05

-0.3

90

-0.4

32

-0.4

77

-0.5

10

-0.5

28

-0.5

44

90°

-0.0

81

-0.1

83

-0.2

71

-0.3

20

-0.4

04

-0.4

44

-0.4

88

-0.5

19

-0.5

36

-0.5

51

Tab

le

D.3

4:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(X/W

bv

V^

]'

fr°m

re

sidu

al s

tres

s du

e to

aut

ofre

ttag

e to

di

ffer

ent

leve

ls o

f ov

erst

rain

wit

h /?

,*=1

0, k

=2,

0

= 9

0°,

6/^

=0

.4

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0

°

0.00

2

0.00

1

0.00

0

-0.0

08

-0.0

76

-0.1

16

-0.1

52

-0.1

81

-0.1

97

-0.2

13

11.2

5°

0.00

2

0.00

1

0.00

0

-0.0

11

-0.0

80

-0.1

20

-0.1

57

-0.1

86

-0.2

02

-0.2

18

22.5

°

0.00

2

0.00

1

-0.0

02

-0.0

20

-0.0

90

-0.1

30

-0.1

66

-0.1

95

-0.2

11

-0.2

27

33.7

5°

0.00

2

0.00

0

-0.0

08

-0.0

39

-0.1

11

-0.1

51

-0.1

87

-0.2

16

-0.2

32

-0.2

49

45°

0.00

1

-0.0

03

-0.0

21

-0.0

66

-0.1

39

-0.1

80

-0.2

16

-0.2

45

-0.2

62

-0.2

78

56.2

5°

0.00

0

-0.0

12

-0.0

61

-0.1

09

-0.1

83

-0.2

24

-0.2

60

-0.2

90

-0.3

07

-0.3

24

67.5

°

-0.0

02

-0.0

39

-0.1

11

-0.1

60

-0.2

34

-0.2

75

-0.3

12

-0.3

42

-0.3

60

-0.3

76

78.7

5°

-0.0

19

-0.1

00

-0.1

82

-0.2

34

-0.3

15

-0.3

58

-0.4

00

-0.4

33

-0.4

51

-0.4

68

90°

-0.0

59

-0.1

49

-0.2

32

-0.2

81

-0.3

64

-0.4

04

-0.4

47

-0.4

79

-0.4

96

-0.5

12

Tab

le

D.3

5:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(/O

Wfa

vVT

rb],

fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge

to

diff

eren

t le

vels

of

over

stra

in w

ith

R*=

10,

k=2,

9

= 9

0°,

b/W

=0.

6

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

<fi

0°

0.00

2

0.00

5

0.01

5

0.02

0

0.02

8

0.02

5

-0.0

12

-0.0

41

-0.0

56

-0.0

70

11.2

5°

0.00

2

0.00

5

0.01

5

0.02

0

0.02

8

0.01

9

-0.0

18

-0.0

47

-0.0

62

-0.0

76

22.5

°

0.00

2

0.00

5

0.01

4

0.02

0

0.02

4

0.00

5

-0.0

33

-0.0

61

-0.0

76

-0.0

90

33.7

5°

0.00

2

0.00

4

0.01

3

0.02

0

0.00

9

-0.0

24

-0.0

60

-0.0

88

-0.1

04

-0.1

18

45°

0.00

2

0.00

3

0.01

0

0.01

4

-0.0

24

-0.0

63

-0.0

99

-0.1

27

-0.1

42

-0.1

57

56.2

5°

0.00

2

0.00

0

-0.0

03

-0.0

19

-0.0

83

-0.1

22

-0.1

59

-0.1

87

-0.2

03

-0.2

19

67.5

°

0.00

3

-0.0

09

-0.0

44

-0.0

83

-0.1

53

-0.1

94

-0.2

30

-0.2

60

-0.2

77

-0.2

93

78.7

5 °

-0.0

10

-0.0

64

-0.1

30

-0.1

77

-0.2

56

-0.3

00

-0.3

42

-0.3

75

-0.3

94

-0.4

11

90°

-0.0

50

-0.1

27

-0.2

01

-0.2

47

-0.3

29

-0.3

71

-0.4

15

-0.4

47

-0.4

65

-0.4

82

Tab

le

D.3

6:

Nor

mal

ized

str

ess

inte

nsit

y fa

ctor

s,

(/^^

/[oy

-v/T

rft]

, fr

om r

esid

ual

stre

ss d

ue t

o au

tofr

etta

ge

to

diff

eren

t le

vels

of

over

stra

in w

ith

/?.*

=10,

k=

2,

0 =

90°

, b/

W=

0.S

% O

vers

trai

n

10

20

30

40

50

60

70

80

90

100

4> 0

°

0.00

3

0.00

9

0.02

1

0.03

0

0.04

0

0.07

1

0.09

3

0.09

1

0.07

1

0.05

7

11.2

5°

0.00

3

0.00

9

0.02

0

0.03

0

0.03

9

0.07

0

0.09

2

0.08

3

0.06

3

0.05

0

22.5

°

0.00

2

0.00

8

0.01

9

0.02

9

0.03

8

0.06

7

0.08

4

0.06

1

0.04

3

0.02

9

33.7

5°

0.00

2

0.00

7

0.01

7

0.02

7

0.03

7

0.05

9

0.05

8

0.02

4

0.00

7

-0.0

06

45°

0.00

2

0.00

6

0.01

5

0.02

5

0.03

1

0.03

7

0.00

5

-0.0

26

-0.0

42

-0.0

56

56.2

5°

0.00

2

0.00

5

0.01

1

0.01

5

-0.0

06

-0.0

35

-0.0

70

-0.1

00

-0.1

16

-0.1

31

67.5

°

0.00

4

0.00

1

-0.0

04

-0.0

30

-0.0

89

-0.1

25

-0.1

60

-0.1

91

-0.2

08

-0.2

24

78.7

5°

-0.0

08

-0.0

37

-0.0

91

-0.1

32

-0.2

04

-0.2

44

-0.2

85

-0.3

19

-0.3

38

-0.3

56

90°

-0.0

49

-0.1

11

-0.1

82

-0.2

25

-0.3

03

-0.3

44

-0.3

88

-0.4

23

-0.4

42

-0.4

59

Documents

Boundary Element Analysis of a Curved Tubing with a Semi