crack growth

Crack Growth Stress Intensity FunctionsNC-CG 6.02.002

Crack Growth Stress Intensity Functions

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nCode Confidential

Standard Stress Intensity Function Library.

Series 1

Standard Specimens

(c) nCode International 2003 Page 1 of 65

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Dr. Andrew Halfpenny 17/11/2004

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


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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


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


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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


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


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


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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


Crack ratio: = a W


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



0.8 0.8 ( )y d = 0.752719 ( ) d = 1.386294 S

0.2 0.2


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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


Crack ratio: = a W

B

2a

2W

Limits: 0 0.8 If > 0.8 then return error.

P If < 0 then let = 0 2W


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



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


1

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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


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


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.


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)


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


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


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.


0.8 0.8 ( )y d = 0.874356 ( ) d = 2.5 S

0.2 0.2


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1.8 Compact Tension Specimen (CTS)



Crack ratio: = a

W

B

a W



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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30


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



0.8 0.8 ( )y d = 5.864474 ( ) d = 13.151608 S

0.2 0.2


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1.9 Round Compact Tension Specimen (RCTS)

Reference: Murakami, page 13 (originally from Newman)

W

B

a

pp


Crack ratio: = a W


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


10030

0 0 0.2 0.4 0.6 0.8



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



y ( ) := 17.47 110.47 + 412.22 669.33 + 425.74

Net section stress: ( ) := 4 S( + 2) As before described 3 (1 )2


10030

0 0 0.2 0.4 0.6 0.8



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


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


0.8 0.8 ( )y d = 0.826304 ( ) d = 0.998766 S

0 0


1

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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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1.4


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



R R 0.001 0.001 1 1

W W ( )y d = 0.771211 ( ) d = 1.269757 S

0.1 0.1


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



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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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


2.5

1 0 0.5 1

SIF

1.1 0 0.5



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


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



1 1 ( )y d = 0.787256 ( ) d = 1.009195 S

0.001 0.001


1

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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



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


4 15

1 0 0 0.2 0.4 0.6 0.8



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


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.



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


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5.3 Semi-circular crack in tension


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:



0 0.2 0.4 0.6 0

1

2

3

SIF


0.6

( ) y d = 0.612199 0.1



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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


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:



0

2

4

SIF


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


32MNominal Stress: S = 3

DCrack ratio: = a

D

Limits: 0.1 0.6



y ( ) := 1.04 3.64 + 16.862 32.593 + 28.44

Net Section Stress:



0 0.2 0.4 0.6 0.5

1

1.5

2

SIF


0.6

( ) y d = 0.477819 0.1



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5.6 Semi-circular Crack in Bending


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


( ) 1.84

M1 ( ))

( )(tan M1 ( ) := M1 ( ) := ( ) := ( ( ))2 cos

M2 ( ) ( ( ( )))4

y ( ) := M1 ( ):= 0.923 + 0.199 1 sin

( )M2

Net Section Stress:




0.6

( ) y d = 0.366788 0.1


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:



0 0.5 1 1.05

1.1

1.15

SIF


1

( ) y d = 1.094941 0



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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


6 1500

0 0.5 1

SIF

0 0.5 1

4

Net

str

ess 1000

5002

0 0


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


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:



0.2

0.4

0.6

0.8

SIF


1

( ) y d = 0.455746 0.1

0 0.5 1 Crack ratio

SIFKSN comparison


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Blank


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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.



1

y d( ) = 10.687032 0.1


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


3.1 Surface Cracks in Tension



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



0.2 0.4 0.6 0.8

0.7

0.2

0.65


SIF 0.9

( )y d = 0.489581


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3.2 Surface Cracks in Bending


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


0.6 Checksums for validating main algorithms:

0.4

0.2 0.4 0.6 0.8

SIF

0.2 0.2

0


0.9 ( )y d = 0.124839


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3.5 Surface Cracks in Combined Tension and Bending


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:



4

3 Checksums for validating main algorithms:

0 0.2 0.4 0.6 0.8



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



0.75

0.7

0.2 0.4 0.6 0.8

SIF

0.2

0.65


0.9 ( )y d = 0.498878


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7.2 Weld Toe Surface Cracks in Bending


Nominal Stress: S = 6M

B T2 Crack ratio: = a

T


B

L

a

2c

M

Limits: 0.2 0.9


User inputs: T = 0.01 B = 0.1 = 90 deg L = 0.02


y ( ) ( )y' if ( )c B

2

:=

( ) MKB H ( )otherwise


0.6

Checksums for validating main algorithms: 0.4

0.2 0.4 0.6 0.8

0.2

SIF

0.2

0


0.9 ( )y d = 0.124839


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7.5 Weld Toe Surface Cracks in Combined Tension and Bending


Nominal Stress: St = P

B TSb =

6M

B T2 Crack ratio: = a

T


B

L

a

2c

M

Limits: 0.2 0.9


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


4

3 Checksums for validating main algorithms:

0 0.2 0.4 0.6 0.8


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


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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 ,


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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.


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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



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


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


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3.4 & 7.4 Embedded Cracks in Bending NOT VALIDATED!

2c

2a

B

P


6M Nominal Stress: S = 2B T

Crack ratio: = a T

T

Limits: 0.2 0.9 M


= 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


y ( ) ( )y' if ( )c B

2

:=

MB , ' ( ( ))otherwise


10


0.2 0.4 0.6 0.8

0.2

0

Crack ratioSIFKSN comparisonWelded KSN comparison

SIF 5 0.9

( )y d = 0


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3.6 Embedded Cracks in Combined Tension and Bending NOT VALIDATED!


0.2 0.9 Limits:

a

T =Crack ratio:

Sb 6M

B T2 =St

P B T

=Nominal Stress:


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:



40

0.2 0.4 0.6 0.8


SIF

20 0.9

( )y d = 0.446623 0.2

0



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crack growth

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