Crack Growth Stress Intensity FunctionsNC-CG 6.02.002
Crack Growth Stress Intensity Functions
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Standard Stress Intensity Function Library.
Series 1
Standard Specimens
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1.1 Single Edge Crack in Tension (SENT)
P Reference: Murakami, page 9 (Originally from Srawley) B
W
a
PNominal Stress: S = B W
Crack ratio: = a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
P User inputs: W := 0.05
SIF: y := 1.12 0.231 + 10.55 21.72 +( ) 2 3 30.394
Net section stress:
P and = Where P = S BW a = W
B(W a)
S BW so ( ) STherefore = :=
B(W W) 1 Plot SIF and the net section stress constant: := 0.2 0.201 .. 0.8 ,
0.2 0.4 0.6 0.8 Crack ratio
610
0.2 0.4 0.6 0.8 Crack ratio
4
Net
str
ess
SIF
5
2
0 0
SIF KSN comparison
Checksums for validating main algorithms:
0.8 0.8 ( )y d = 2.149354 ( ) d = 1.386294 S
0.2 0.2
(c) nCode International 2003 Page 2 of 65
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1.2 Single Edge Crack in Pure Bending (SENB)
M Reference: Murakami, page 11 (Originally from Gross & Srawley) B
W
a
6M Nominal Stress: S = 2B W
Crack ratio: = a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
M User inputs: W := 0.05
SIF: y ( ) := 1.122 1.4 + 7.332 13.083 + 144
Net section stress is assumed to be 2/3 maximum bending stress:22 6M B W
= where M = S and a = W3 2 6B(W a)
22 6S BWTherefore = so ( ) := 2 S 3 6B (W W)2 3 (1 )2
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.8
4
0.2 0.4 0.6 0.8 Crack ratio
SIF
20
0.2 0.4 0.6 0.8
3
Net
str
ess
10
2
1 0
Crack ratio SIF KSN comparison
Checksums for validating main algorithms: 0.8 0.8
( )y ( )
d = 1.067088 d = 2.5 S 0.2 0.2
(c) nCode International 2003 Page 3 of 65
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1.3 Double Edge Crack in Tension (DENT)
P Reference: Murakami, page 6 (Originally from Nisitani)
2W
B
aa
PNominal Stress: S = 2 BW
Crack ratio: = a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
P User inputs: W := 0.05
SIF: y ( ) := 1.122 0.154 + 0.8072 1.8943 + 2.4944
Net section stress: ( ) := S As before described. 1
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.8
61.6
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
1.2
1.4
SIF
Net
str
ess 4
2
1 0
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms: 0.8 0.8
( )y d = 0.732675 ( ) d = 1.386294 S
0.2 0.2
(c) nCode International 2003 Page 4 of 65
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1.4 Centre Cracked Plate in Tension , L > 6W (CCP)
Reference: Murakami, page 3 (Originally from Fedderson & Tada)
:= As before described ( ) S
2W
B
2a
P
P
L
L > 6W
PNominal Stress: S = 2 BW
Crack ratio: = a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
User inputs: W := 0.05
(
)
2 4 y ( )SIF: 0.025 0.06 1:= + sec
2
Net section stress: 1
Plot SIF and the net section stress constant: := 0.2 0.201 .. 0.8 ,
62
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
4
Net
str
ess
SIF
1.5
2
1 0
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
0.8 0.8 ( )y d = 0.752719 ( ) d = 1.386294 S
0.2 0.2
(c) nCode International 2003 Page 5 of 65
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1.5 Centre Cracked Square Plate in Tension (CCSP)
Reference: T. F. Gray, Int. Jnl. Fracture, P vol 13, 1977, p65
PNominal Stress: S = 2 BW
Crack ratio: = a W
B
2a
2W
Limits: 0 0.8 If > 0.8 then return error.
P If < 0 then let = 0 2W
User inputs: W := 0.05
SIF: y ( ) := 1 1.1 (1 )0.9
Net section stress: ( ) := S As before described 1
Plot SIF and the net section stress constant: := 0 0.01, .. 0.8
62
1 0 0 0.5
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
0 0.5 1
4
Net
str
ess
SIF
1.5
2
0.8 0.8 ( )y d = 1.040902 ( ) d = 1.609438 S
0 0
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1.6 Three Point Bend Specimen, Span 4:1 (3SENB4)
Reference: Murakami, page 13 (originally from Srawley)
6P
a
B Nominal Stress: S =B W
W Crack ratio: = a W
Limits: 0.2 0.8 P If > 0.8 then return error.
4W If < 0.2 then let = 0.2
User inputs: W := 0.05
21.99 (1 )(2.15 3.93 + 2.7 )SIF: y( ) :=
(1 + 2)(1 )1.5
:=Net section stress: ( ) 2 S As before described 3 (1 )2
Plot SIF and the net section stress constant: := 0.2 0.201 .. 0.8 ,
10 20
Net
str
ess
10SIF
5
0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Crack ratio Crack ratio SIFKSN comparison (See note)
PNote: KSN assumes normalised stress as S = , the author disagrees with this B W
normalisation. For comparison, therefore, the KSN results should be divided by 6.
Checksums for validating main algorithms:
0.8
0.8
( )y d = 1.881321
( ) d = 2.5 S 0.2 0.2
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1.7 Three Point Bend Specimen, Span 8:1 (3SENB8)
Reference: Murakami, page 13 (originally from Srawley)
12P
a
Nominal Stress: S = B B W
W Crack ratio: = a W
Limits: 0.2 0.8
P If > 0.8 then return error. If < 0.2 then let = 0.2
8W
User inputs: W := 0.05
SIF: y := 1.107 2.12 + 7.71 13.55 + 14.25( ) 2 3 4
Net section stress: ( ) 2 S As before described :=3 (1 )2
Plot SIF and the net section stress constant: := 0.2 0.201 .. 0.8 ,
4 20
3
Net
str
ess
SIF
2 10
1
0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Crack ratio Crack ratio SIFKSN comparison (See note)
PNote: KSN assumes normalised stress as S = , the author disagrees with this B W
normalisation. For comparison, therefore, the KSN results should be divided by 12.
Checksums for validating main algorithms:
0.8 0.8 ( )y d = 0.874356 ( ) d = 2.5 S
0.2 0.2
(c) nCode International 2003 Page 8 of 65
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1.8 Compact Tension Specimen (CTS)
Reference: Murakami, page 18 (originally from Srawley)
PNominal Stress: S = B W
Crack ratio: = a
W
B
a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
User inputs: W := 0.05
2 3 4(2 + )(0.886 + 4.64 13.32 + 14.72 5.6 )SIF: y ( ) :=( )1.5 1
Net section stress is taken as 2/3 of maximum stress:
B MNA P Where = +max Z A
W
a
N A net
max
=
=
a +W a
2
PMNA P ( + W)= = 2 a
B(W a)2 Z = A = B(W a)
6
3 P(a ) P P [3(a + W + (W a + W ) )]+ =Therefore = max
B(W a)2 B(W a) B(W a)2
2 W)So = 2 P (a + where P = W and a = W S Bmax B(W a)2
Therefore net = W 2 W So ( ) 4 S( + 2)2 2 SB (W + ) :=
3B(W W)2 3 (1 )2
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Plot SIF and the net section stress constant: 0.2 0.201 .. 0.8 := ,
100
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
SIF
20
Net
str
ess
50
10
0 0
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
0.8 0.8 ( )y d = 5.864474 ( ) d = 13.151608 S
0.2 0.2
(c) nCode International 2003 Page 10 of 65
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W
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1.9 Round Compact Tension Specimen (RCTS)
Reference: Murakami, page 13 (originally from Newman)
W
B
a
pp
PNominal Stress: S = B W
Crack ratio: = a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
Dia = 1.35W User inputs: := 0.05
( ) := 2 + ( ) 0.76 4.8+ 11.582
11.433
+ 4.084
( )
1 ( )1.5
Net section stress: ( ) := 4 S( + 2) As before described 3 (1 )2
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.8
10030
0 0 0.2 0.4 0.6 0.8
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
0.2 0.4 0.6 0.8
SIF
20
Net
str
ess
50
10
0.8 0.8 ( )y d = 6.115357 ( ) d = 13.151608 S
0.2 0.2
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1.A Wedge Opening Load Specimen (WOL)
Reference: E. F. Walker & M. J. May, BISRA Report MG/E/307/67, 1967.
W
B
a
Nominal Stress: S = P B W
Crack ratio: = a W
Limits: 0.2 0.8 If > 0.8 then return error. If < 0.2 then let = 0.2
User inputs: W := 0.05
y ( ) := 17.47 110.47 + 412.22 669.33 + 425.74
Net section stress: ( ) := 4 S( + 2) As before described 3 (1 )2
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.8
10030
0 0 0.2 0.4 0.6 0.8
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
0.2 0.4 0.6 0.8
SIF
20
Net
str
ess
50
10
0.8 0.8 ( )y d = 6.19343 ( ) d = 13.151608 S
0.2 0.2
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1.B Quarter Circular Corner Crack Tension Specimen (CCTS)
a
a
P
P
W
W
Reference: A. Pickard, Book ISBN 0 947817 22 0, 1983, p135.
PNominal Stress: S = 2W
Crack ratio: = a W
Limits: 0 0.8 If > 0.8 then return error. If < 0 then let = 0
User inputs: W := 0.05
2 3 4 5 y 0.7334 0.06746 0.9218( ) 0.2781 0.4799 2.445:= + + +
Net section stress:
P 2 = where P = and a = WS W
2 a2
W 4
:=S W2 ( ) S Therefore = so 22
2 2 1
W W 44
Plot SIF and the net section stress constant: := 0 0.001 .. 0.8 ,
2.5 2.5
2 2
Net
str
ess
SIF
1.5
1.5 1
0.5 1 0 0.5 1 0 0.5
Crack ratio Crack ratio SIFKSN comparison
Checksums for validating main algorithms:
0.8 0.8 ( )y d = 0.826304 ( ) d = 0.998766 S
0 0
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Series 2
Cracks at Holes
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2.1 Single Crack at Hole in Tension
P Reference: T. F. Gray, Int. Jnl. Fracture, vol 13, 1977, p65
PNominal Stress: S = 2W B
Crack ratio: = a W
RLimits: 0.1 1 0.001 W
If > upper then return error. If < 0.1 then let = 0.1
User inputs: W := 0.1 R := 0.01 P
( ) R R ( ) 1 W
:= F1 :=W 1.08 )1.8 R + 1 (0.5 + G 2
( ) ( ( ( ))) ( ( )16.6 )0.9 ( )0.333 ( ( ))F2 := 1 1 + 1.45 sin sin
y ( ) :=( ) F2 ( ) ( )F1 0.5 + G
Net section stress:
P = where P = B and a = W 2S W
B(2W 2R a)
2S WBTherefore = so ( ) := S B(2W 2R a) R
1 W 2
R a
2W
G :=
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RPlot SIF and the net section stress constant: := 0.1 0.101 .. 1, 0.001 W
2.5
0 0.5 0 0.5 1
SIF
1.2 2
Net
str
ess
1 1.5
0.8 1
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
R R 0.001 0.001 1 1
W W ( )y d = 0.771211 ( ) d = 1.269757 S
0.1 0.1
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R a
2W
a
2.2 Double Crack at Hole in Tension
P Reference: T. F. Gray, Int. Jnl. Fracture, vol 13, 1977, p65
PNominal Stress: S = 2W B
Crack ratio: = a W
RLimits: 0.1 1 0.001 W
If > upper then return error. If < 0.1 then let = 0.1
User inputs: W := 0.1 R := 0.01 P
F1 :=( ) := R G := R ( ) 1 W 1.08 R + W )1.8 1 ( + G
F2 := 1 ( ) )(1 )0.78 + 1.23 sin )( ) ( sin ( ) ( )5 ( )0.19 ( ( )( ) F2( ) ( )
y :=
( ) F1 + G
Net section stress:
P = where P = B and a = W 2S W
2B (W R a)
= :=Therefore 2S WB so ( ) S 2B (W R a) R
1 W
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RPlot SIF and the net section stress constant: := 0.1 0.101 .. 1, 0.001 W
100040
0 0.5 1 0 0.5 1
Net
str
ess
500SIF
20
0 0
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
R R 0.001 0.001 1 1
W W ( )y d = 1.56331 ( ) d = 6.684609 S
0.1 0.1
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G
T R
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2.3 Surface Crack at a Hole in Tension
Reference: Newman & Raju, ASTM STP 2R
c
2a
2T P
791, 1983, p238
Nominal Stress: S = P 4B T
Crack ratio: = a T
Limits: 0.1 0.9
If > 0.9 then return error. B > T + 2R 2B If < 0.1 then let = 0.1
= parametric angle, angle of crack to longitudinal axis User inputs: := 0.01 := 0.01
B := 0.1 := 35deg Crack aspect ratio, a/c := 1.5
c ( ) := T G
( ) := 1 1
c ( ) cos 0.9( )+
R
1.65 0.5 Q := (1 + 1.464G ) if (G > 1) M1 := G if (G > 1) 1.65 1.0 otherwise (1 + 1.464G ) otherwise
40.05 0.29 ( ) ( )M2 := M3 := M4 := 1 cos 1.5 1.5 1 + 4G 0.11 + G 0.23 + G
( ) := ( ) ( )2 ( )3 ( )41 + 0.358 + 1.425 1.578 + 2.156
M5 1 + 0.08 ( )2
0.25 ( )2 + sin )2 (1 )10 )2( ) ( ( )M6 1 0.1 1 cos (
:=
( )G cos M7 :=:= + 0.5 0.5 (
( )) 2 R + c R
sec
( ) ( )M8 ( ( ))sec := ( ) 2 c 2B 4B
4
2F ( ) ( ) ( ) ( ) ( )M4 M5 M6 M7 M8 M2 M3M1 +:= +
( )y ( ) :=
QF
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Net section stress:
P T( ) where P = 4S BT a = T c = G = 4T (B R) acsin
4 S BTTherefore = 2 2T ( )4T (B R) sin G
:=so ( ) 4 S BG 2 ( )4G (B R) T sin
Plot SIF and the net section stress constant: := 0.1 0.101 .. 0.9 ,
2.5
1 0 0.5 1
SIF
1.1 0 0.5
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
2
Net
str
ess 1.14
1.5 1.12
0.9 0.9 ( )y d = 1.267756 ( ) d = 0.898035 S
0.1 0.1
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Series 4
Cracks at Corners
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4.1 Quarter Elliptical Corner Crack in Tension
Reference: Newman & Raju, ASTM STP 791, 1983, p238
PNominal Stress: S = B T
P T
c
a
Crack ratio: = a T
Limits: 0 1.0
If > 1.0 then = 1.0 B If G > 1.0 then let lo = 0.001
= parametric angle, angle of crack to longitudinal axis User inputs: T := 0.01 B := 0.1
:= 35deg Crack aspect ratio, a/c G := 1.5
1.65 (1 + 1.464G ) if (G > 1)c ( ) := T Q :=
G 1.65 (1 + 1.464G ) otherwise (0.375G )0.03
if (G > 1)
0.5 2M1 ( ) := G( )
1.08 M2 := if (G > 1)+
0.44 +1.06
0.3 + G
otherwise 1.08 0.03 otherwise +
2M3 := (0.25G ) if (G > 1)
15
0.5 + 0.25G 14.8 (1 G) otherwise +
2 )3
2 ( ( ))31 + (0.08 + 0.4 ) 1 sin otherwise ( )c
M4 ( ) ( ( ) sin 1 0.08 0.4 1 if (G > 1):= + + T
2
( ( ) cos
)3
otherwise
( )c
M5 ( )
( ( )
)3 cos 1 0.08 0.15 1 if (G > 1):= + +
T
2( + 0.15 )1 0.08 1+ 2( ( )2 ( )2)0.25 M6 := G sin + cos if (G > 1) 2( ( )2 ( )2)0.25 G cos + sin otherwise
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F ( ) := ( )M1 + 2 M2 + M34
( ) ( )M4 M5 M6
( )y :=( ) F
Q
Net section stress:
P T= where P = S BT a = T c =
ac ( ) G B T sin 4
S BTTherefore = 2 2
B T sin T ( )4G
( ) 4 S BG so :=4 GB T2 ( ) sin
Plot SIF and the net section stress constant: := 0 0.001 .. 1.0 ,
1.04 20
0 0.5 0 0.5 1
1.02
Net
str
ess
SIF
10
1
0 0.98
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
1 1 ( )y d = 0.787256 ( ) d = 1.009195 S
0.001 0.001
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4.2 Quarter Elliptical Corner Crack at a Hole in Tension NOT VALIDATED!
Reference: Newman & Raju, ASTM STP 2R
c
aT P
791, 1983, p238
Nominal Stress: S = P 2B T
Crack ratio: = a T
0.2 GLimits: 0 1.0 T
If > 1.0 then = 1.0 B > T + 2R 2B If G < 0 then = 0 = parametric angle, angle of crack to longitudinal axis User inputs: T := 0.01 B := 0.1
R := 0.01 := 35deg Crack aspect ratio, a/c G := 1.5
1.65 c ( ) := T ( ) :=
c
1 ( ) Q := (1 + 1.464G ) if (G > 1)G ( ) cos 0.85
1 + 1.65 (1 + 1.464G ) otherwise R
0.04 1 +
G G
if (G > 1) (0.2G ) 4M1 := M2 := if (G > 1) 0.89
0.54
otherwise +(1.13 0.09G ) otherwise 0.2 + G
4M3 := (0.11G ) if (G > 1) otherwise
10.5
if (G > 0.8 )
24
0.65 + G 1
0.5 G)14 (1 otherwise +0.65 + G
2
)2
1 + (0.1 + 0.35 ) 1 ( ) otherwise 2 ( sin )2
M4 ( ) := ( ( ) sin 1 0.1 0.35 1 if (G > 1)+ + G
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( ) := ( ) ( )2 ( )3 ( )4
M5 1 + 0.13
1 + 0.358 + 1.425 1.578 + 2.156
( )2
( )0.09
1.13 )2 0.25 ( )M6 := ( ( ) cos 0.151 0.1 1 0.85 if (G > 1)+ +G
0.25 ( ( ))2 ( ) cos 0.85 + 0.15 1 + 0.04 G)( 1 0.1 1 otherwise +
2( ( )2 ( )2)0.25 M7 := G sin + cos if (G > 1) (G cos + sin )0.25 otherwise 2 ( )2 ( )2
2 R c ( )+( ) 2 c ( )
R
( ) ( )M8 := ( ( ))sec := sec 2B 4B
M1 + M2
2 + M3
4
( )F ( ) ( ) ( ) ( )M4 M5 M6 M7 M8 :=
( )y :=( ) F
:=
Q
Net section stress: ( ) 4 S BG As before described 4 G(B R) T2 ( ) sin
Plot SIF and the net section stress constant: 0 0.001 .. 1:= ,
2
1 0 0.5 1
Crack ratio
1.1 0 0.5
Crack ratio SIF KSN comparison
(c) nCode International 2003 Page 27 of 65
1.14
Net
str
ess
SIF
1.5
1.12
1
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Series 5
Cracks in Solid Cylinders
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5.1 Circumferential crack in tension
Reference: T. F. Gray, Int. Jnl. Fracture, a
P
vol 13, 1977, p65
Nominal Stress: S = 4P
D2 Crack ratio: = 2a
D
Limits: 0.1 0.7
If > 0.7 then return error.If < 0.1 then let = 0.1D
User inputs: D := 0.05
y ( ) 1.25 :=Net section stress: ( )2.4 1.47 1 24P D D
= where P = S and a = (D 2a )2 4 2
2 Therefore = SD so ( ) := S
(D D)2 (1 )2
Plot SIF and the net section stress constant: := 0.1 0.101, .. 0.7
4 15
1 0 0 0.2 0.4 0.6 0.8
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms:
0 0.2 0.4 0.6 0.8
SIF
3 10
Net
str
ess
2 5
0.7
0.7
( )y ( )
d = 1.071026
d = 2.222222 S 0.1 0.1
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5.2 Straight crack in tension
Reference: James & Mills, Eng. Fract. Mech., vol 30, 1988, p641
Crack ratio:
Nominal Stress:
a
D =
S = 4P
D2
Limits: 0.1 0.65
D
If > 0.65 then return error. If < 0.1 then let = 0.1
a
P
User inputs: D := 0.05
y ( ) := 0.926 1.771 + 26.4212 78.4813 + 87.9114
Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0.1 0.101, .. 0.65
Checksums for validating main algorithms:
0.65
( ) y d = 1.093342 0.1
Crack ratioSIFKSN comparison
0 0.2 0.4 0.6 0.8 0
2
4
6
SIF
(c) nCode International 2003 Page 31 of 65
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5.3 Semi-circular crack in tension
Reference: James & Mills, Eng. Fract. Mech., vol 30, 1988, p641
4P Nominal Stress: S = 2
DCrack ratio: = a
D
Limits: 0.1 0.6
If > 0.6 then return error. If < 0.1 then let = 0.1 D
a
P
User inputs: D := 10
( ( ))
M1 ( ) := 1.84 tan 1( ) := ( ( )) ( ( ))2 cos
M2 ( ) ( ( ( )))3 := 0.752 + 2.02 + 0.37 1 sin
y ( ) ( ) M2 := M1 ( )
Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0.1 0.101, .. 0.6
0 0.2 0.4 0.6 0
1
2
3
SIF
Checksums for validating main algorithms:
0.6
( ) y d = 0.612199 0.1
Crack ratio SIF KSN comparison
(c) nCode International 2003 Page 32 of 65
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5.4 Crack at thread in tension
Reference: James & Mills, Eng. Fract. a Mech., vol 30, 1988, p641
4P Nominal Stress: S = 2
DCrack ratio: = a
D P Limits: 0.1 0.6
If > 0.6 then return error. If < 0.1 then let = 0.1
D User inputs: D := 0.05
y ( ) := 2.043 e 31.332 + 0.6507 + 0.5367 + 3.04692 19.5043 + 45.6474
Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0.1 0.101, .. 0.6
0
2
4
SIF
Checksums for validating main algorithms:
0.6
( ) y d = 0.718844 0.1
0 0.2 0.4 0.6 Crack ratio
SIFKSN comparison
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5.5 Straight Crack in Bending
a
Diameter DDM
M
M iameter D
M
Reference: James & Mills, Eng. Fract. Mech., vol 30, 1988, p641
32MNominal Stress: S = 3
DCrack ratio: = a
D
Limits: 0.1 0.6
If > 0.6 then return error. If < 0.1 then let = 0.1
User inputs: D := 0.05
y ( ) := 1.04 3.64 + 16.862 32.593 + 28.44
Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0.1 0.10, .. 0.6
0 0.2 0.4 0.6 0.5
1
1.5
2
SIF
Checksums for validating main algorithms:
0.6
( ) y d = 0.477819 0.1
Crack ratio SIF KSN comparison
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5.6 Semi-circular Crack in Bending
Reference: James & Mills, Eng. Fract. Mech., vol 30, 1988, p641
32MNominal Stress: S = 3MM D
Crack ratio: = a D
Limits: 0.1 0.6
If > 0.6 then return error. If < 0.1 then let = 0.1MM Diameter DDiameter D
User inputs: D := 0.05
( ) 1.84
M1 ( ))
( )(tan M1 ( ) := M1 ( ) := ( ) := ( ( ))2 cos
M2 ( ) ( ( ( )))4
y ( ) := M1 ( ):= 0.923 + 0.199 1 sin
( )M2
Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0.1 0.10, .. 0.6
Checksums for validating main algorithms:
0.6
( ) y d = 0.366788 0.1
Crack ratioSIFKSN comparison
a
0 0.2 0.4 0.6 0.6
0.8
1
1.2
SIF
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Series 6
Cracks in Hollow Cyclinders
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6.1 Internal Surface Crack Under Hoop Stress
Reference: Murakami page 751 (originally = Parametric angle, angle of crack to longitudinal axis from Newman & Raju)
RInternal pressure Nominal Stress: S = T
2c
a
R
Crack ratio: = a T
Limits: 0 1.0
If > 1.0 then return error. If < 0 then let = 0
Wall thickness T User inputs: R := 0.02 T := 0.01
:= 90deg Outer radius: Ro := R + T Crack aspect ratio, a/c
0.2 G 2.0 G := 1
1.65 1 + 1.464 GQ := 0.89
M1 := 1.13 0.09G M2 := 0.54 + 0.2 + G
1
M3 := 0.5
if [(G > 0.8 )(G < 1.2 )]
24
0.65 + G 1
G)0.5 14 (1 otherwise +0.65 + G
2M4 ( ) ( ) ( ( ))2 := 1 + 0.1 + 0.35 1 sin
2M5 ( ) ( ( )2 ( )2)0.25 := sin + cos
:=
+ 1 0.5
T R
2 2Ro + R2 2Ro R
M6 ( )
M1 + M2
2 + M3
4 F ( ) := ( ) ( ) ( )M4 M5 M6 0.97
( )y ( ) :=
QF
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Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0 0.001, .. 1.0
0 0.5 1 1.05
1.1
1.15
SIF
Checksums for validating main algorithms:
1
( ) y d = 1.094941 0
Crack ratio SIF KSN comparison
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6.2 Circumferential Crack in Thin Walled Tube in tension
Reference: Rooke and Cartwright
R
a
P
Wall thickness T
PNominal Stress: S = 2
(T )2 RT Crack ratio: = a
T
Limits: 0 1.0
If > 1.0 then return error. If < 0 then let = 0
User inputs: R := 0.02 T := 0.002
y ( ) := 1.2114 1.6578 + 11.7432 16.67293 + 9.77084
2 2 RTNet section stress: ( ) := S(T + )(R + T T)2 R2
Plot SIF and the net section stress constant: := 0 0.001, .. 1.0
6 1500
0 0.5 1
SIF
0 0.5 1
4
Net
str
ess 1000
5002
0 0
Crack ratio Crack ratio SIF KSN comparison
Checksums for validating main algorithms: 1
0.999
( )y ( )
d = 2.082768
d = 7.201966 S 0 0
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Series 8
Cracks in Welded Tubular Joints
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8.1 Cracks in Welded Tubular Joints (All dimensions for this in mm!)
D T
d
t a
Reference: J. C. Kam and W. D. Dover, in proc. 6th OMAE Conference, vol 3, 1987.
PNominal Stress: S = 2
(d t t )Crack ratio: = a
T
Limits: 0.1 1.0
If > 1.0 then return error. If < 0.1 then let = 0.1
User inputs: 16 T 45mm T := 20
= d/D ratio 0.48 0.76 := 0.5 Maximum Stress concentration factor: 2.66 SCF 9.4 SCF := 3
Average Stress Concentration Factor: 1.1 aveSCF 2.22 aveSCF := 2
0.11
T 16
M2 := (0.669 0.1625 aveSCF)
0.099
T 16
M3 := (0.353 + 0.057 aveSCF)
1.71 M4 := 0.231
T 16
0.31 0.18 SCF
M1 ( ) := 1.0 if ( > 0.25 ) M4
0.25
otherwise
y ( ) := ( )M2 M3
M1
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Net Section Stress:
Net Section Stresses are not currently available for this section as the formulae would be excessivly complicated for cycle-by-cycle calculation.
Plot SIF and the net section stress constant: := 0.1 0.11, .. 1.0
0.2
0.4
0.6
0.8
SIF
Checksums for validating main algorithms:
1
( ) y d = 0.455746 0.1
0 0.5 1 Crack ratio
SIFKSN comparison
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Series 9
Cracks at Spot Welds in Tension
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9.1 Cracks at Spot Welds in Tension (All dimensions for this in mm!)
Reference: R. F. Smith, Unpublished Research, Sheffield University, 1993.
PNominal Stress: S = T W
Crack ratio: = a T
Limits: 0.1 1.0
P T If > 1.0 then return error. If < 0.1 then let = 0.1
W
User inputs: 0.5 T 4 T := 2 3 D 10 D := 5
( ) := T base := T a 1.5 D
2 3 M3 := 7.407 base M4 := 11.2 baseM2 := 1.877 base
y' := M1 M2 + M3 M4 y ( ) := ( )
y' WT
a
Net Section Stress:
Net Section Stresses are not currently available for this section.
Plot SIF and the net section stress constant: := 0.1 0.101, .. 1.0
Checksums for validating main algorithms:
1
y d( ) = 10.687032 0.1
Crack ratioSIFKSN comparison
D
M1 0.2608 :=
W 100 :=
0 0.5 1 0
10
20
30
SIF
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Series 3 & 7
Elliptical & Semi-elliptical Surface Cracks in
Welded and Unwelded Plates
The following calculations are based on the work by Newman & Raju, ASTM STP 791, 1983, p238. These provide a standard set of formulae pertaining to surface cracks in plates subjected to tension and bending stresses. The work was subsequantly extended by BSI Published Document PD6493, 1991 (originally from TWI and based on Newman & Raju) to include the effects of welds.
The following section derives a basic set of equations and then determines the SIFs for the common set of geometries contained in the KRAKEN database.
Note:
Net Section Stress values are not currently available for these geometries!
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Basic Cracks in Tension
2c
a
B
T P
= Parametric angle, angle of crack to longitudinal axis B > T
Surface Cracks
T
B
L
a
2c
P
P 6M Nominal Tension Stress: St = Nominal Bending Stress: Sb = B T 2 B TCrack ratio: = a Limits: 0.2 0.9
T If > 0.9 then return error. If < 0.2 then let = 0.2
User inputs: T := 0.01 B := 0.1 := 90deg L := 0.02
Crack aspect ratio, a/c G := 1.0 Ratio of bending to tension: U := 5
Derived properties: Crack depth: a ( ) := T Crack half width: c ( ) := T
G
Coefficient common to all plates: := 1 + 1.464G1.65 if (G 1.0 ) 1.65 1 + 1.464G otherwise
LWeld length ratio, L/T :=T
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SIF calculations:
B1 Special case for Full Width Cracks where C 2
This special case is common for all plates whether welded or unwelded.
y ( ) ( ( ) )' := 0.89127 if 0.999B < a < 1.001B ( )
2 T
cos a
otherwise
1 y
2. Partial Width Surface Cracks in Unwelded Plates in Tension:
2M1 := (1.13 9.000001 10 G) if ( G 1.0 ) 0.5 G
0.04
1 otherwise +G
M2 0.54 0.2
G4
:=0.89
+0.2 + G
1.0 )
24
if (G
otherwise
10.5 G) if (G 1.0 )M3 := 14 (1+0.65 + G 0.11
otherwise 4G
0.25 ( )2sin )2(
( )G cos 1.0 )M4 := if (G+
0.25
( )2 cos 2( )sin
otherwise +G
2( + 0.35 )
)2
( sin ( ) )2
M5 ( ) ( ( ) sin 1.0 )1 0.1 1 if (G:= +
2
1 + 0.1 0.35 + 1 otherwise
G
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1( ) :=M6 cos
( ) c
B
M1 + M2
2 + M3
4 M4
( )MM := ( ) ( )M5 M6
3. Partial Width Surface Cracks in Unwelded Plates in Bending:
M7 ( ) := 1 0.34 0.11G if (G 1.0 ) 1
0.04 +
0.41 +
0.55 1.93 1.38 2 otherwise +
0.75 1.5 G G
G
M9 := 1.22 0.12G if (G 1.0 ) 0.77
2.11 + otherwise G
M10 := 0.55 1.05G0.75
+ 0.47G1.5 if (G 1.0 ) 0.72 0.14
0.55 + otherwise 0.75 1.5 G G
M8 ( ) := 1 + M9 + M10 2
( )P1 := 0.2 + G + 0.6 if (G 1.0 ) 1
0.2 + 0.6 + otherwise G
H ( ) := 0 if ( < 1deg ) + ( > 179deg ) sin otherwise ( )H ( ) + ( ) ( ) P1( ) M7 (M8 M7 )max H 0( , )
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4. Modification Factors for Surface Cracks in the Weld Toe:
Uniaxial Tension:
0.55 1 := 0.05
( )MKM := if ( 2.0 )MKM 0.51 0.27 0.31 if ( 1)
Copyright 2010 HBM
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3.1 Surface Cracks in Tension
Reference: Newman & Raju, ASTM STP 791, 1983, p238
PNominal Stress: S = B T
a
Crack ratio: = a T
2c
B
P
T Limits: 0.2 0.9
If > 0.9 then return error. If < 0.2 then let = 0.2 = Parametric angle, angle of
crack to longitudinal axis User inputs: T 0.01 = B 0.1 =
90 deg =Crack aspect ratio, a/c G 1=
B > T
if ( )c B
y ( ) ( )y' :=
2
MM ( )otherwise
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
Checksums for validating main algorithms:
0.2 0.4 0.6 0.8
0.7
0.2
0.65
Crack ratioSIFKSN comparison
SIF 0.9
( )y d = 0.489581
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3.2 Surface Cracks in Bending
Reference: Newman & Raju, ASTM STP 791, 1983, p238
2c
B
P
6M Nominal Stress: S = 2
a B TCrack ratio: = a
T T
Limits: 0.2 0.9
If > 0.9 then return error. If < 0.2 then let = 0.2 = Parametric angle, angle of
crack to longitudinal axis User inputs as above: T 0.01 = B 0.1 =
90 deg =Crack aspect ratio, a/c G 1=
B > T
if ( )c B
2
y ( ) ( )y' :=H ( )
otherwise
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
0.6 Checksums for validating main algorithms:
0.4
0.2 0.4 0.6 0.8
SIF
0.2 0.2
0
Crack ratioSIFKSN comparison
0.9 ( )y d = 0.124839
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3.5 Surface Cracks in Combined Tension and Bending
Reference: Newman & Raju, ASTM STP 791, 1983, p238
2c
B
T M
P
P 6M Nominal Stress: St = =SbB T 2a B T
Crack ratio: = aT
Limits: 0.2 0.9
If > 0.9 then return error. If < 0.2 then let = 0.2 = Parametric angle, angle of crack to
longitudinal axis U = Bending / Tension ratio User inputs as above: T = 0.01 B = 0.1 B > T
= 90 degCrack aspect ratio, a/c G = 1
SbRatio of Bending stress to tension stress, 0 10: U = 5 St
SIF is calculated by the weighted sum of the above formulae following the expression:
y ( ) ( )y' (1 + U) if ( )c B
2
:=
( )( ) ( )MM + U H MM
otherwise
Net Section Stress:
Net Section Stresses are not currently available for this section.
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
4
3 Checksums for validating main algorithms:
0 0.2 0.4 0.6 0.8
Crack ratioSIFKSN comparison
(c) nCode International 2003 Page 54 of 65
SIF
2 0.9 ( )y d = 1.157583
1 0.2
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7.1 Weld Toe Surface Cracks in Tension
Reference: BSI Published Document PD6493, 1991 (originally from TWI and based on Newman & Raju) to include the effects of welds.
Nominal Stress: S = P B T
Crack ratio: = a T
T If < 0.2 then let = 0.2
B
L
a
2c
P
Limits: 0.2 0.9
If > 0.9 then return error.
User inputs: T = 0.01 B = 0.1 = 90 deg L = 0.02
Crack aspect ratio, a/c G = 1
y ( ) ( )y' if ( )c B
2
:=
( ) MKM MM ( )otherwise
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
Checksums for validating main algorithms:
0.75
0.7
0.2 0.4 0.6 0.8
SIF
0.2
0.65
Crack ratioSIFKSN comparison
0.9 ( )y d = 0.498878
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7.2 Weld Toe Surface Cracks in Bending
Reference: BSI Published Document PD6493, 1991 (originally from TWI and based on Newman & Raju) to include the effects of welds.
Nominal Stress: S = 6M
B T2 Crack ratio: = a
T
T If < 0.2 then let = 0.2
B
L
a
2c
M
Limits: 0.2 0.9
If > 0.9 then return error.
User inputs: T = 0.01 B = 0.1 = 90 deg L = 0.02
Crack aspect ratio, a/c G = 1
y ( ) ( )y' if ( )c B
2
:=
( ) MKB H ( )otherwise
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
0.6
Checksums for validating main algorithms: 0.4
0.2 0.4 0.6 0.8
0.2
SIF
0.2
0
Crack ratioSIFKSN comparison
0.9 ( )y d = 0.124839
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7.5 Weld Toe Surface Cracks in Combined Tension and Bending
Reference: BSI Published Document PD6493, 1991 (originally from TWI and based on Newman & Raju) to include the effects of welds.
Nominal Stress: St = P
B TSb =
6M
B T2 Crack ratio: = a
T
T If < 0.2 then let = 0.2
B
L
a
2c
M
Limits: 0.2 0.9
If > 0.9 then return error.
P User inputs: T = 0.01 B = 0.1 = 90 deg L = 0.02
Crack aspect ratio, a/c G = 1 Bending ratio: U = 5
y ( ) ( )y' (1 + U) if ( )c B
2
:=
( ) ( ) ( ) ( )otherwise
MM + MM ( ) MKM U H MKB
Plot SIF and the net section stress constant: := 0.2 0.21, .. 0.9
4
3 Checksums for validating main algorithms:
0 0.2 0.4 0.6 0.8
Crack ratio SIF KSN comparison
SIF
2
0.9 ( )y d = 1.166879
0.2 1
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Series 3 & 7
Elliptical & Semi-elliptical Embedded
Cracks in Welded and Unwelded Plates
The following calculations are based on the work by Newman & Raju, ASTM STP 791, 1983, p238. These provide a standard set of formulae pertaining to embedded cracks in plates subjected to tension and bending stresses. The results apply similarly for welded details following an assumption that the weld has negligable effect on the SIF.
The following section derives a basic set of equations and then determines the SIFs for the common set of geometries contained in the KRAKEN database.
Note:
Net Section Stress values are not currently available for these geometries!
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Embedded Cracks in Tension
T
2c
2a
B
P
P
T L
2a
2c
P = Parametric angle, angle of crack tolongitudinal axisP = Minimum clearance to surface
BB > T
P 6M Nominal Tension Stress: St = Nominal Bending Stress: Sb = B T 2 B TCrack ratio: = a Limits: 0.2 0.9
T If > 0.9 then return error. If < 0.2 then let = 0.2
User inputs: T := 0.01 B := 0.1 := 90deg P := 0.05
Crack aspect ratio, a/c G := 1 Ratio of bending to tension: U := 5 (Crack length L not required)
Derived properties: Crack depth: a ( ) := T Crack half width: c ( ) := T
G
( ) T T + P
Effective crack ratio required for embedded cracks: ' :=
Coefficient common to all plates: := 1 + 1.464G1.65 if (G 1.0 ) (as surface cracks)
1.65 1 + 1.464G otherwise
(c) nCode International 2003 Page 60 of 65
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SIF calculations:
B1 Special case for Full Width Cracks where C 2
This special case is common for all plates whether welded or unwelded. (As surfacce cracks)
y ( ) ( ( ) )' := 0.89127 if 0.999B < a < 1.001B ( )
2 T cos
a
otherwise
1 y
2. Partial Width Embedded Cracks in Unwelded Plates in Tension:
The modified M1-M6 are given as:
M1 := 1.0 if (G 1.0 ) 0.5 G otherwise
0.05 0.29 M2 := M3 :=1.5 1.5 0.11 + G 0.23 + G
0.25 ( )2sin )2(
( )G cos if (G 1.0 )M4 := +
0.25
( )2 cos 2( )sin
otherwise +G
4 2.6 2' M5 '( ) ( )cos ':= 1 1 + 4G
1
' cos
M6( ') :=,
( ) c
B
:= M1 + M2'2 + M3'
4MM ( ') ( ) ( , )M5 ' M6 'M4 ,
(c) nCode International 2003 Page 61 of 65
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3. Partial Width Embedded Cracks in Unwelded Plates in Bending:
MB ( , ') := MB 0 (MB 1.044
) ( )( )P P a
if > 89deg < 91deg
if
P T
< 0.1841 2.44 3.166 T 2T
otherwise
MB 0.94 + 1.875 P 0.1146' 1.844 P if (' < 0.125 )T 2T
P PMB 1.06 2.2 0.6666 ' 0.6666 otherwise
T 2T
MB max MB 0( , ) MB 1.0 otherwise MB
4. Modification Factors for Embedded Cracks in the Weld Toe:
No modification required. Welds do not affect the SIF.
(c) nCode International 2003 Page 62 of 65
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3.3 & 7.3 Embedded Cracks in Tension
2c
2a
P
P
Limits: 0.2 0.9
B If > 0.9 then return error. If < 0.2 then let = 0.2
Reference: Newman & Raju, ASTM STP 791, 1983, p238
PNominal Stress: S = B T
Crack ratio: = a TT
= Parametric angle, angle of crack to longitudinal axis P = Minimum clearance to surface User inputs as above: T = 0.01 B = 0.1 B > T
90 deg = P 0.05 =Crack aspect ratio, a/c G 1=
y ( ) ( )y' if ( )c B
2
:=
MM , ' ( ( ))otherwise
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
0.64 Checksums for validating main algorithms:
0.639
0.2 0.4 0.6 0.8
SIF
0.2 0.638
0.637
Crack ratio SIF KSN comparison Welded KSN comparison
0.9 ( )y d = 0.446623
(c) nCode International 2003 Page 63 of 65
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3.4 & 7.4 Embedded Cracks in Bending NOT VALIDATED!
2c
2a
B
P
Reference: Newman & Raju, ASTM STP 791, 1983, p238
6M Nominal Stress: S = 2B T
Crack ratio: = a T
T
Limits: 0.2 0.9 M
If > 0.9 then return error. If < 0.2 then let = 0.2
= Parametric angle, angle of crack to longitudinal axis User inputs as above: T = 0.01 B = 0.1 P = Minimum clearance to surface B > T = 90 deg P = 0.05
Crack aspect ratio, a/c G = 1
y ( ) ( )y' if ( )c B
2
:=
MB , ' ( ( ))otherwise
Plot SIF and the net section stress constant: := 0.2 0.201, .. 0.9
10
Checksums for validating main algorithms:
0.2 0.4 0.6 0.8
0.2
0
Crack ratioSIFKSN comparisonWelded KSN comparison
SIF 5 0.9
( )y d = 0
(c) nCode International 2003 Page 64 of 65
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3.6 Embedded Cracks in Combined Tension and Bending NOT VALIDATED!
If > 0.9 then return error.
0.2 0.9 Limits:
a
T =Crack ratio:
Sb 6M
B T2 =St
P B T
=Nominal Stress:
Reference: Newman & Raju, ASTM STP 791, 1983, p238
2c
2a
B
P
M
P
T
If < 0.2 then let = 0.2 = Parametric angle, angle of crack to longitudinal axis P = Minimum clearance to surface User inputs as above: T = 0.01 B = 0.1 U = Bending / Tension ratio
= 90 deg P = 0.05 Crack aspect ratio, a/c G = 1
SbRatio of Bending stress to tension stress, 0 10: U = 5 St
SIF is calculated by the weighted sum of the above formulae following the expression:
y ( ) ( )y' (1 + U) if ( )c B
2
:=
( , ( )) ( , '( )) MM ( '( ))MM ' + U MB , otherwise
Net Section Stress:
Net Section Stresses are not currently available for this section.
Plot SIF and the net section stress constant: := 0.2 0.201 .. 0.9 ,
40
0.2 0.4 0.6 0.8
Checksums for validating main algorithms:
SIF
20 0.9
( )y d = 0.446623 0.2
0
Crack ratioSIFKSN comparison
(c) nCode International 2003 Page 65 of 65
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