Annex B Limit Load Solutions (Based on SINTAP and R6 · Limit Load Solutions (Based on SINTAP and R6) ... B.9.2 Crack in the interface between weld metal and base plate ... B.9.3

(01 May 2006) FITNET MK7

© FITNET 2006 – All rights reserved B-1

Annex B

Limit Load Solutions (Based on SINTAP and R6)

B Limit Load

B Limit Load........................................................................................................................................B-1 B.1 Nomenclature ..................................................................................................................................B-3 B.2 Introduction.....................................................................................................................................B-5 B.3 Flat plates ........................................................................................................................................B-9 B.3.1 Flat plate with through-thickness flaw ..........................................................................................B-9 B.3.2 Flat plate with surface flaw ..........................................................................................................B-10 B.3.3 Flat plate with long surface flaw ..................................................................................................B-13 B.3.4 Flat plate with embedded flaw .....................................................................................................B-14 B.3.5 Flat plate with long embedded flaw.............................................................................................B-18 B.3.6 Flat plate with edge flaw...............................................................................................................B-19 B.3.7 Flat plate with double edge flaw ..................................................................................................B-22 B.4 Curved shells ................................................................................................................................B-24 B.4.1 Spheres..........................................................................................................................................B-24 B.5 Pipes or cylinders .........................................................................................................................B-25 B.5.1 Through-thickness cracks in cylinder oriented axially ..............................................................B-25 B.5.2 Internal surface flaw in cylinder oriented axially........................................................................B-26 B.5.3 Long internal surface flaw in cylinder oriented axially ..............................................................B-28 B.5.4 External surface flaw in cylinder oriented axially.......................................................................B-29 B.5.5 Long external surface flaw in cylinder oriented axially .............................................................B-31 B.6 Pipes or cylinders with circumferential flaws.............................................................................B-32 B.6.1 Through-thickness flaw in cylinder oriented circumferentially.................................................B-32 B.6.2 Internal surface flaw in cylinder oriented circumferentially ......................................................B-34 B.6.3 Long internal surface flaw in cylinder oriented circumferentially.............................................B-37 B.6.4 External surface flaw in cylinder oriented circumferentially .....................................................B-41 B.6.5 Long external surface flaw in cylinder oriented circumferentially............................................B-44 B.7 Round bars and bolts ...................................................................................................................B-48 B.7.1 Embedded flaws in round bars ....................................................................................................B-48 B.7.2 Centrally embedded axial elliptical defects ................................................................................B-49 B.7.3 Solid round bar; centrally embedded extended defect ..............................................................B-50 B.8 Tubular joints ................................................................................................................................B-51 B.8.1 T- and Y-Joints with axial load.....................................................................................................B-51 B.8.2 T- and Y-Joints with in-plane and out-of-plane bending............................................................B-53 B.8.3 K-Joints with axial loads ..............................................................................................................B-55 B.8.4 K-Joints with in-plane and out-of-plane bending .......................................................................B-57 B.8.5 X- and DT-Joints with axial load ..................................................................................................B-59 B.8.6 X- and DT-Joints with in-plane and out-of-plane bending .........................................................B-61 B.9 Material mismatch.........................................................................................................................B-63 B.9.1 Crack in the centre line of the weld material ..............................................................................B-63 B.9.2 Crack in the interface between weld metal and base plate........................................................B-65 B.9.3 Crack in the interface of a bi-material joint.................................................................................B-67 B.9.4 Crack in the centre line of the weld material ..............................................................................B-69 B.9.5 Crack in the interface between weld metal and base plate........................................................B-72 B.9.6 Crack in the interface of a bi-material joint.................................................................................B-74 B.9.7 Crack in the centre line of the weld material ..............................................................................B-75 B.9.8 Crack in the interface between weld metal and base plate........................................................B-78 B.9.9 Crack in the interface of a bi-material joint.................................................................................B-80 B.9.10 Crack in the centre line of the weld metal...................................................................................B-81 B.9.11 Crack in the interface between weld metal and base plate........................................................B-83

FITNET FFS –MK7 – Annex B

B-2 © FITNET 2006 – All rights reserved

B.9.12 Crack in the interface of a bi-material joint .................................................................................B-85 B.9.13 Crack in the centre line of the weld metal ...................................................................................B-86 B.9.14 Crack in the interface between weld metal and base pipe.........................................................B-88 B.9.15 Crack in the interface of a bi-material joint .................................................................................B-90 B.9.16 Bi-material (clad) center through thickness cracked plate under tension................................B-91 B.9.17 Centre cracked bi-layer (clad) plate with repair weld .................................................................B-93



B.1 Nomenclature

a half flaw length for through-thickness flaw, flaw height for surface flaw or half height for embedded flaw

B specimen thickness, section thickness in plane of flaw

c half flaw length for surface or embedded flaws

D diameter

E Young’s modulus

FN applied normal load

FeN normal yield limit load

FeM yield limit load for mismatched weldments

FeB yield limit load for base material

H specimen height

L pipe or specimen length

Lrb normalised limit moment

LrN normalised limit normal load

Lrp normalised limit pressure

LrpN normalised limit combined pressure and tension load ( or nL ???)

M mismatch ratio across weldment given by ReW/Re

B

Mai, Mao applied in and out of plane moments for tubular joints

Mci, Mco fully plastic moments for cracked tubular joints calculated for in and out of plane loads

Mb applied bending moment

Meb limit bending moment

mb applied axisymmetric through wall bending moment per unit angle of cross section

meb limit axisymmetric through wall bending moment per unit angle of cross section

Pa applied axial load on tubular joint

Pc collapse load for cracked tubular joint

Pe yield limit pressure

p’ Applied pressure

Pe Limit pressure



ri inner radius

ro outer radius

rm mean radius

Re Yield stress

RFM mismatch flow stress of weldment

ReW yield or proof strength of weld metal

ReB yield or proof strength of base metal

Qf factor to allow for the presence of axial and moment loads in the chord.

Qu strength factor which varies with the joint and load type

t thickness of structural section

t’ Effective plate or cylinder thickness for defining local limit load

W Plate width or half plate width

W’ Effective half plate width or half cylinder length for defining local limit load

δ crack opening displacement

σm membrane stress

σb bending stress

σn,m membrane component of collapse stress

σn,b bending component of collapse stress

λ Load ratio for combined tension and bending

ν Poisson’s ratio

ϑ crack angle; 90° if perpendicular to the surface

α,β,η,θ,ϕ,ψ,ξ,ζ dimensionless crack size



B.2 Introduction

In classical solid mechanics the limit load is defined as the maximum load a component of elastic-ideally plastic material is able to withstand, above this limit ligament yielding becomes unlimited. In contrast to this definition, real materials strain harden with the consequence that the applied load may increase beyond the value given by the non-hardening limit load. Sometimes strain hardening is roughly taken into account by replacing the yield strength of the material by an equivalent yield strength called ‘flow strength’ (usually the mean of yield strength and ultimate tensile strength) in the limit load equation.

In a FITNET FFS analysis the plastic limit load marks the load at which the plastic zone spreads across the whole ligament ahead of the crack. Some authors prefer the term yield load instead of limit load in order to distinguish it from the higher plastic collapse load which is reached when the ligament has completely strain hardened and the component fails under stress controlled loading. The estimation of limit load for a given crack/component geometry is critical input to a fitness-for-service assessment.

This Annex B of the Volume III of the FITNET FFS compiles the K-solutions and limit load solutions along other needed information to conduct FFS analysis. Comprehensive set of limit load solutions are complied to serve as an accurate and user-friendly data. The results of the BS 7910, SINTAP and R6 sources are used to generate this Annex.



Bending stresses as functions of moments

Structure type Bending stress, σb with R6 nomenclature

Bending stress, σb with SINTAP nomenclature Location

Planar M

Wt6

2 ⎟⎠⎞

⎜⎝⎛ M

wd 6

2 ⎟⎠⎞

⎜⎝⎛

tensile stress at wall surface(W is plate width)

Pipe with internal circumferential defect (axisymmetric bend)

mAtR

6

b2

m⎟⎟⎠

⎞⎜⎜⎝

⎛

( )( )m

2m

b Rt62Rt12 = A

+−

bAm

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+4

32

312

3

32

26 1

3

1

21

wRRw

wRRwR

= A mmb

tensile stress at inner wall surface

Pipe with external circumferential defect (axisymmetric bend)

mBtR

6

b2

m⎟⎟⎠

⎞⎜⎜⎝

⎛

( )( )m

2m

b Rt62Rt12 = B

−−

bBm

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−32

3412

3

32

26 1

3

2

22

wRRw

wRRwR

= B mmb

tensile stress at outer wall surface

Pipe with internal or external circumferential defect (cantilever bend)

( )( )

M )Rt4(tR

Rt222

m2m

m⎥⎦

⎤⎢⎣

⎡

+π+

MRR

R

)(4

41

42

2⎥⎦

⎤⎢⎣

⎡−π

peak tensile stress at outer wall surface



Solid round bar with centrally embedded circular defect (axisymmetric bend)

- m

w⎟⎠⎞

⎜⎝⎛

3

192

tensile stress at centre of bar

Solid round bar with external circumferential defect (axisymmetric bend)

- m

w⎟⎠⎞

⎜⎝⎛

3

96

tensile stress at surface of bar

Solid round bar

(cantilever bend)

- M

w⎟⎠⎞

⎜⎝⎛

3

32π

peak tensile stress at surface of bar



B.3 Flat plates

B.3.1 Flat plate with through-thickness flaw

R6

Applicable clause(s):

(B.1)

Solution:

e

NeN

r WBRFL = ,

Wa2

=β , 3

2=γ

Plane stress Tresca, plane stress Mises and plane strain Tresca solutions:

10for 1 <≤−= ββNrL (B.1)

Plane strain Mises solution:

( ) 10for 1 <≤−= ββγNrL (B.2)

Validity limits:

The plate should be large in comparison to the length of the crack so that edge effects do not influence the results.

Bibliography:

[B.1] A G Miller, Review of limit loads of structures containing defects, Int J Pres Ves Piping 32, 197-327 (1988).



B.3.2 Flat plate with surface flaw

R6

Plates under combined tension (pin-loaded) and bending:


– global solution (B.3), (B.4), (B.5), (B.6), (B.7), (B.8)

– local solution (B.9), (B.10), (B.11), (B.12), (B.13), (B.14)

Solution:

Definition:

e

NeN

r WBRFL = ,

e

beb

r RWBML 2

4= ,

m

bN

b

BFM

σσλ

61

== , Ba

=α , W

c2=β ,

Bc

=ψ

Global solution:

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

++−−

+

≤++++

=0

22

2

0

12

1

for

1112

112

for 22

αα

ββαβλ

βαβλ

αααβλαβλ

d

dd

d

LNr (B.3)

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

++−−

+

≤++++

=0

22

2

0

12

1

for

1112

112

4

for 22

4

αα

ββαβλ

βαβλ

λ

αααβλαβλ

λ

d

dd

d

Lbr (B.4)

where

( ) ( )ββααβ −+−= 121 221d (B.5)



( ) ( )211 2 2 11

d αβαβ αβ αβ

⎡ ⎤⎛ ⎞−= − − + −⎢ ⎥⎜ ⎟−⎝ ⎠⎣ ⎦

(B.6)

β

λλλα

2112

121 2

0

−+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−= (B.7)

For pure tension (λ = 0) and pure bending (λ → ∞)

⎪⎩

⎪⎨⎧

∞→−

==

λβ

λα

for 2

10for 1

0 (B.8)

Extended surface cracks (set β = 1): equivalent to the Lr solution formerly in IV.1.8.1.

Through-wall cracks (set α = 1)

Local solution (W = B + c and B’= B):

( ) ( )( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>++−++−+

≤

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

+=

0

22

2

0

1

21

for 11212

for

12

12

ααψαψλαψλ

αα

ψαψλ

ψαψλ

d

d

d

d

LNr (B.9)

( ) ( )( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>++−++−+

≤

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

+=

0

22

2

0

1

21

for 11212

4

for

12

12

4

ααψαψλαψλ

λ

αα

ψαψλ

ψαψλ

λ

d

d

d

d

Lbr (B.10)

where

( )2

22

1 12

11

ψψα

ψαψ

++⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−=d (B.11)

( ) ( )2

2 11 1 1

1 1d

α α ψαψ ψ αψ ψ

−⎛ ⎞= − − − +⎡ ⎤⎜ ⎟ ⎣ ⎦+ +⎝ ⎠

(B.12)

( )ψλψλλα

++

+⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

212

21

21 2

0 (B.13)




⎪⎩

⎪⎨⎧

∞→++

==

λψψ

λα

for 21

0for 1

0 (B.14)

Validity limits:

For local case the solutions are limited to

ψψβ+

<1

Bibliography:

[B.2] Y Lei, J-integral and limit load analysis of semi-elliptical surface cracks in plates under bending, Int J Pres Ves Piping 81, 34-41 (2004).

[B.3] Y Lei, A global limit load solution for plates with semi-elliptical surface cracks under combined tension and bending, ASME/JSME Pressure Vessels and Piping Conference, San Diego, July 25-29 2004, PVP-Vol. 475, 125-131 (2004).



B.3.3 Flat plate with long surface flaw

Plates under combined tension (pin-loaded) and bending

R6


(B.15), (B.16)

Solution:

e

NeN

r WBRFL = ,

e

beb

r RWBML 2

4= ,

m

bN

b

BFM

σσλ

61

== , Ba

=α

Net-section collapse solution (plane stress Tresca):

( )( ) ( )

1for 122

122

2

<−++++

−= α

ααλαλ

αNrL (B.15)

( )( ) ( )

1for 122

1422

2

<−++++

−= α

ααλαλ

αλbrL (B.16)

For the case of pure tension (λ = 0) eqn. (B.15) applies and for the case of pure bending (λ = ∞) eqn.(B.16) applies.

Validity limits:

(The solution is limited to a/B ≤ 0.8. Also, the plate should be large in the transverse direction to the crack so that edge effects do not influence the results.)

Bibliography:

[B.4] A A Willoughby and T G Davey, Plastic collapse in part-wall flaws in plates, ASTM STP 1020, American Society for Testing and Materials, Philadelphia, USA, 390-409 (1989).



B.3.4 Flat plate with embedded flaw

Defects in plates under combined tension (pin-loaded) and bending

R6


IV.1.6.1

Solution:

e

NeN

r WBRFL = ,

e

beb

r RWBM

24L = ,

m

bN

b

BM

σσλ

61

F== ,

Ba

=α , W

c2=β ,

B

apB

k−−

= 2 , Bc

=ψ

For symmetrically embedded cracks, k = 0.

Global solution:

Extended embedded cracks set β = 1

Surface cracks set α = 0.5 – k

Through-wall cracks set k = 0 and α = 0.5

( ) ( )( )

( )[ ] ( )[ ]⎪⎪

⎩

⎪⎪

⎨

⎧

≤<++−++−

≤++++

=

21

22

2

21

12

1

for 1412

,minfor 42

Lααα

βλββλβ

ααααβλαβλ

ckk

cc

c

Nr (B.17)

( ) ( )( )

( )[ ] ( )[ ]⎪⎪

⎩

⎪⎪

⎨

⎧

≤<++−++−

≤++++

=

21

22

2

21

12

1

for 1412

4

,minfor 42

4

Lααα

βλββλβ

λ

ααααβλαβλ

λ

ckk

cc

c

br (B.18)

where



( )21 481 αβαβ −−= kc (B.19)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−= 2

2

2 41411 βα

βββ kc (B.20)

( )( ) ( ) ( ) ⎟⎠⎞

⎜⎝⎛ +−+−−+−−= λβλβλα kkkk 2

4111 222

1 (B.21)

( ) ( )

⎪⎪⎩

⎪⎪⎨

⎧

∞→−

=−−+−=

λβ

λβββα

bending purefor 1

0 tension purefor 2411 2

1 k

kk (B.22)

k−=21

2α (B.23)

Local solution (d = B + c and t1 = B):

( )

( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤<

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

+++

≤

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

++⎟⎟⎠

⎞⎜⎜⎝

⎛+

+

=

21

2

22

21

1

21

for

14

12

,minfor

14

12

Lααα

ψψλ

ψψλ

ααα

ψαψλ

ψαψλ

ckk

c

c

c

Nr (B.24)

( )

( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤<

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

+++

≤

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

++⎟⎟⎠

⎞⎜⎜⎝

⎛+

+

=

21

2

22

21

1

21

for

14

12

4

,minfor

14

12

4

Lααα

ψψλ

ψψλ

λ

ααα

ψαψλ

ψαψλ

λ

ckk

c

c

c

br (B.25)

where

2

1 14

181 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−+

−=ψ

αψψψα kc (B.26)

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

=ψ

ψαψψ 1

4411

1 22

2 kc (B.27)



⎟⎠⎞

⎜⎝⎛ +−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

++−

= λψλ

ψλα kkkk 2

41

112

2

1 (B.28)

( )( )

( )⎪⎩

⎪⎨

⎧

∞→+

=+

+−+

+=

λψ

λψ

ψψψα

bending purefor 1

0 tension purefor 1

241

1 2

2

1

k

kk (B.29)

k−=21

2α (B.30)

Global solution (pin-loaded):

Equations (IV.1.6.1-1) and (IV.1.6.1-3) to (IV.1.6.1-7) (set λ = 0). (IV.1.6.3-1)

For embedded extended defects, set β =1 and ψ = ∞.

Local solutions (pin-loaded):


(a) W’= B + c, B’ = B and W/2 > W’:

Equations (IV.1.6.2-1) and (IV.1.6.2-3) to (IV.1.6.2-7) (set λ = 0). (IV.1.6.3-2)

(b) ck

Bd +⎟⎠⎞

⎜⎝⎛

−−=

2121 α

, ( )kBt 211 −= and W/2 > d :

( )

ψα

ψα

+⎟⎠⎞

⎜⎝⎛

−−

+⎟⎠⎞

⎜⎝⎛

−−

=

k

kNr

2121

121

21L (B.31)

Global solution (fixed-grip tension):

Equations (IV.1.6.1-1) and (IV.1.6.1-3) to (IV.1.6.1-7) (set λ = 0 and k = 0 as limit load value does not depend on crack position in the cross section).


Local solution (fixed-grip tension):

For W’ = B + c, B’= B and W/2 > W’:


( )ψ

αψ+−+

=1

211LNr (B.32)

Global solution (bending):

Equations (IV.1.6.1-2) to (IV.1.6.1-7) (set λ = ∞) (IV.1.6.4-1)




Local solutions (bending):


(a) W’ = B + c, B’ = B and W/2 > W’:

Equations (IV.1.6.2-2) to (IV.1.6.2-7) (set λ = ∞) (IV.1.6.4-2)

(b) ck

Bd +⎟⎠⎞

⎜⎝⎛

−−=

2121 α

, ( )kBt 211 −= and W/2 > d :

( ) ( )ψα

αψα

+−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+⎟

⎠⎞

⎜⎝⎛

−−

−=

k

kkkb

r

2121

2141

2121

21L2

2

2 (B.33)

Validity limits:

Global solutions are a net-section collapse solution valid for

021

≥> k and k−≤21α .

Local solutions are valid for

021

≥> k , k−≤21α and

ψψβ+

<1

Bibliography:

[B.5] A J Carter, A library of limit loads for FRACTURE-TWO, Nuclear Electric Report TD/SID/REP/0191 (1992).

[B.6] Y Lei and P J Budden, Limit load solutions for plates with embedded cracks under combined tension and bending, Int J Pres Ves Piping 81, 589-597 (2004)



B.3.5 Flat plate with long embedded flaw

R6

See 2.4 when c=W/2

set β = 1 and set ψ = ∞


Solution:

Validity limits:

Bibliography:



B.3.6 Flat plate with edge flaw

R6


Compact tension specimen (CT) ; Plane Stress & Strain (Mises & Tresca) (B.34) - (B.37)

Three-point-bending specimen (TPB); Plane Strain (Mises & Tresca) (B.38), (B.39)

Single edge cracked plate under tension (SECP); Plane Stress & Strain (Mises & Tresca) (B.40) - (B.45)

Single edge cracked plate under bending (SECB); Plane Stress & Strain (Mises & Tresca) (B.46) - (B.49)

Solution:

Compact tension specimen (CT) ; Plane Stress & Strain (Mises & Tresca)

e

NeN

r WBRFL = ,

Wa

=β , 3

2=γ

Plane stress Tresca solution:

( ) 10for 122 2 <≤+−+= βββNrL (B.34)

Plane stress Mises solution:

( )( ) ( ) 10for 111 2 <≤+−++= βγβγβγNrL (B.35)

Plane strain Tresca solution:

( )⎪⎩

⎪⎨⎧

<<++

≤≤++−=

109.0for 1.7021- 599.4702.2

09.00for 25.0134.0482.1634.02

32

βββ

ββββNrL (B.36)


( )( )[ ]⎪⎩

⎪⎨⎧

<<++

≤≤++−=

109.0for 1.7021- 599.4702.2

09.00for 25.0134.0482.1634.02

32

βββγ

ββββγNrL (B.37)

Three-point-bending specimen (TPB); Plane Strain (Mises & Tresca)



e

NeN

r BRWLFL 2

2= ,

Wa

=β , 3

2=γ were L is the loaded length of the specimen

2S: Support distance


( )( )( )⎪⎩

⎪⎨⎧

<<−

≤≤−−+=

118.0for 122.1

18.00for 1194.313.112.12

22

ββ

ββββNrL (B.38)


( )( )( )⎪⎩

⎪⎨⎧

<<−

≤≤−−+=

118.0for 122.1

18.00for 1194.313.112.12

22

ββγ


Single edge cracked plate under tension (SECP); Plane Stress & Strain (Mises & Tresca)

e

NeN

r WBRFL = ,

Wa

=β , 3

2=γ ,

21−

=γη

pin-loaded:


( ) 10for 1 22 <≤−+−= ββββNrL (B.40)


( )( ) ( ) ( )( )⎪⎩

⎪⎨⎧

<<⎥⎦⎤

⎢⎣⎡ −−−−+−−

≤≤+−−=

1545.0for 1794.015876.01794.0702.1

545.00for 232.1122

32

ββββ

ββββNrL (B.41)


[ ]( )( ) ( ) ( )( )⎪⎩

⎪⎨⎧

<<⎥⎦⎤

⎢⎣⎡ −−−−+−−

≤≤+−−=

1545.0for 1794.015876.01794.0702.1

545.00for 232.1122

32

ββββγ



[ ]( )( ) ( ) ( )( )⎪⎩

⎪⎨⎧

<<⎥⎦⎤

⎢⎣⎡ −−−−+−−

≤≤+−−=

1545.0for 1794.015876.01794.0702.1

545.00for 232.1122

32

ββββγ

ββββγLn (B.43)

Fixed grip:

Plane stress Tresca and Mises, and plane strain Tresca solutions:

10for 1 <≤−= ββNrL (B.44)




( ) 10for 1 <≤−= ββγNrL (B.45)

Single edge cracked plate under bending (SECB); Plane Stress & Strain (Mises & Tresca)

e

beb

r RBWML 2

4= ,

Wa

=β , 3

2=γ


( ) 10for 1 2 <≤−= ββbrL (B.46)


( )( )( )⎪⎩

⎪⎨⎧

<<−

≤≤−−+=

1154.0for 1072.1

154.00for 1034.3934.012

22

ββ

ββββbrL (B.47)


( )( )( )⎪⎩

⎪⎨⎧

<<−

≤≤−−+=

1295.0for 12606.1

295.00for 172.2686.112

22

ββ

ββββbrL (B.48)


( )( )( )⎪⎩

⎪⎨⎧

<<−

≤≤−−+=

1295.0for 12606.1

295.00for 172.2686.112

22

ββγ

ββββγbrL (B.49)

Validity limits:

Bibliography:

[B.7] A G Miller, Review of limit loads of structures containing defects, Int J Pres Ves Piping 32, 197-327 (1988)




B.3.7 Flat plate with double edge flaw

B.3.7.1 Finite width plate

Plate under tension (DECP)

R6


(B.50) - (B.53)

Solution:

e

NeN

r WBRFL = ,

Wa2

=β , 3

2=γ


10for 1 <≤−= ββNrL (B.50)


( ) ( )( )⎩

⎨⎧

<<−≤≤+−

=1286.0for 1

286.00for 5401 1ββγ

βββ .LN

r (B.51)


( ) ( )( )⎪

⎩

⎪⎨

⎧

<<−

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+−=

1884.0for 157.2

884.00for 122ln1 1

ββ

ββββN

rL (B.52)


( ) ( )( )⎪

⎩

⎪⎨

⎧

<<−

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+−=

1884.0for 157.2

884.00for 122ln1 1

ββγ

ββββγN

rL (B.53)

Validity limits:



Reference(s):




B.4 Curved shells

B.4.1 Spheres

B.4.1.1 Through-thickness flaw in a sphere

Membrane stress

R6


(B.54)

Solution:

mrt

=η , mra

=θ

θηθ

η

2

2

cos811

4

++

=e

e

RP

(B.54)

Validity limits:

Bibliography:

[B.10] F M Burdekin and T E Taylor, Fracture in spherical vessels, J Mech Engng Science 11, 486-497 (1969)



B.5 Pipes or cylinders

B.5.1 Through-thickness cracks in cylinder oriented axially

Membrane stress

R6


(B.55)

Solution:

mrt

=η , at

=φ

205.11φη

η

+=

e

e

RP

(B.55)

Validity limits:

The cylinder should be long in comparison to the length of the crack so that edge effects do not influence the results.

Bibliography:


[B.12] J F Kiefner, W A Maxey, R J Eiber and A R Duffy, Failure stress levels of flaws in pressurised cylinders, ASTM STP 536, American Society for Testing and Materials, Philadelphia, USA, 461-481 (1973).



B.5.2 Internal surface flaw in cylinder oriented axially

Pressure-Excluding or Including Crack Faces; Global & Local Collapse

R6


(B.56) - (B.60)

Solution:

ta

=α , mrt

=η , ca

=φ

(a) Global solutions:

(i) Without defect-face pressure:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

++

⎟⎠⎞

⎜⎝⎛ −

=αηη

η

η

αη

211

211

ln

211 g

e

e

MRP

(B.56)

where

⎟⎠⎞

⎜⎝⎛ −

+=ηφ

αη

211

05.112

gM (B.57)

(ii) With defect-face pressure:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+

+−

−+

⎟⎠⎞

⎜⎝⎛ −

=αηη

η

αηη

η

η

αη

211

211

ln

211

211

211 g

e

e

MRP

(B.58)

(b) Local solutions:

(i) Without defect-face pressure (d = c + s1(1 - α) and t1 = B):



( )

( )ααηη

η

η

ηα

−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

++

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+−

=1

211

211

ln

211211

ln1

1

1

sc

cs

RP

e

e (B.59)

where

αηη

αηη

αη

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+⎟

⎠⎞

⎜⎝⎛ −

=

211

1ln211

1

gM

cs

(ii) With defect-face pressure (d = c + s2(1 - α) and t1 = B):

( )

( )ααηη

η

αηη

η

η

ηα

−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+

+−

−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+−

=1

211

211

ln

211

211

211211

ln1

2

2

sc

cs

RP

e

e (B.60)

where

αηαηη

η

αηη

η

η

ηη

αη

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+

+−

−−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+⎟⎠⎞

⎜⎝⎛ −

=

211

211

ln

211

211

211

211

ln211

2

gM

cs

Validity limits:

Bibliography:





B.5.3 Long internal surface flaw in cylinder oriented axially

Pressure-Excluding or Including Crack Faces

R6


(B.61), (B.62)

Solution:

ta

=α , mrt

=η

Without defect face pressure:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+=

αηη

η

211

211

lne

e

RP

(B.61)

With defect face pressure:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+

+−

−=

αηη

η

αηη

η

211

211

ln

211

211

e

e

RP

(B.62)

Validity limits:

Bibliography:




B.5.4 External surface flaw in cylinder oriented axially

Membrane and bending stress

R6


(B.63) - (B.68)

Solution:

e

mnNr R

L ,σ= ,

e

bnbr R

L ,

32 σ

= ,m

b

σσ

λ61

= , ta

=α , tc

=ψ , W

c2=β

Set ψ = ∞ for an extended axial external surface crack

Local solutions (W’ = t + c and t’ = t):

( ) ( )( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>++−++−+

≤

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

+=

0

22

2

0

1

21

for 11212

for

12

12

ααψαψλαψλ

αα

ψαψλ

ψαψλ

d

d

d

d

LNr (B.63)

( ) ( )( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>++−++−+

≤

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

+=

0

22

2

0

1

21

for 11212

4

for

12

12

4

ααψαψλαψλ

λ

αα

ψαψλ

ψαψλ

λ

d

d

d

d

Lbr (B.64)

where



( )2

22

1 12

11

ψψα

ψαψ

++⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−=d (B.65)

( )[ ] ( )ψ

ψαααψψ

αψ+−

+−−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=11211

112d (B.66)

( )ψλψλλα

++

+⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

212

21

21 2

0 (B.67)


⎪⎩

⎪⎨⎧

∞→λ+ψ+ψ

=λ=α

for 21

0for 1

0 (B.68)

Validity limits:

The solutions are limited to

ψ+ψ

≤β1

Bibliography:

[B.16] I W Goodall and G A Webster, Theoretical determination of reference stress for partially penetrating flaws in plates, Int J Pres Ves Piping 78, 687-695 (2001).





B.5.5 Long external surface flaw in cylinder oriented axially

Membrane and bending stress

R6


IV 1.9.3 with remark VI

„Set ψ = ∞ in the first parts of eqns. (B.63) & (B.64) and eqn. (B.65) to obtain the solution for an extended axial external surface crack in a cylinder under membrane and bending stresses.”

Solution:

Validity limits:

Bibliography:

[B.19] I W Goodall and G A Webster, Theoretical determination of reference stress for partially penetrating flaws in plates, Int J Pres Ves Piping 78, 687-695 (2001).





B.6 Pipes or cylinders with circumferential flaws

B.6.1 Through-thickness flaw in cylinder oriented circumferentially

Membrane and bending stress, global and local solution

R6


Thick-walled cylinders under combined tension and bending: (B.77)

Thin-walled cylinders under combined tension and bending with internal pressure: (B.72)

“For through-wall defects, α ≡ 1”

Solution:

Thick-walled cylinders under combined tension and bending

em

NeN

r tRrFL⋅

=π2

, em

beb

r tRrML 24

= , 1=α , mrt

=η , mra

=θ , Nr

br

Nm

b

L

LFr

M

2πλ ==

Global solutions:

Whole crack inside the tensile stress zone (θ + β ≤ π):

⎟⎠⎞

⎜⎝⎛ −−= N

rLπθ

πβ 1

21

( ) ( ) θηβη sin21sinb c

br ffL −=

2

1211 η+=bf

2

611 η+=cf

Thin-walled cylinders under combined tension and bending with internal pressure:



1=α , em

NeN

r tRrFL⋅

=π2

, em

beb

r tRrML 24

= ,

2tr

c

m −=θ

e

em

em

empr tR

PrtRr

PtrL

222

2

≈⎟⎠⎞

⎜⎝⎛ −

= pr

Nr

m

N

LL

prF

=′

= 2πχ ( ) p

rNr

pr

pNr LLLL χ+=+= 1 (B.69)

( ) pNr

br

mN

m

b

L

LprFr

M

2

2 ππλ =

′+= (B.70)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−++

=2

1 1234

12

21

χχ

pNr

pNr

aLLS (B.71)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

=2

2 1234

12

21

χχ

pNr

pNr

aLLS (B.72)

Global solutions:

Whole crack inside the tensile stress zone (θ + β ≤ π)

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

121

1 1a

pNr

aa

a

SL

SSS

πθ

πβ

(B.73)

( )[ ]θβ sinsin21

121 aaabr SSSL −−= (B.74)

Validity limits:

Bibliography:

[B.22] M R Jones and J M Eshelby, Limit solutions for circumferentially cracked cylinders under internal pressure and combined tension and bending, Nuclear Electric Report TD/SID/REP/0032 (1990).

[B.23] Y Lei and P J Budden, Limit load solutions for thin-walled cylinders with circumferential cracks under combined internal pressure, axial tension and bending, J Strain Analysis 39, 673-683 (2004).



B.6.2 Internal surface flaw in cylinder oriented circumferentially

R6


Thick-walled cylinders under combined tension and bending: (B.75) - (B.84)

Thin-walled cylinders under combined tension and bending with internal pressure: (B.85) - (B.91)

Solution:

Thick-walled cylinders under combined tension and bending:

em

NeN

r tRrFL⋅

=π2

, em

beb

r tRrML 24

= , ta

=α , mrt

=η ,

2tr

c

m −=θ

Nr

br

Nm

b

L

LFr

M

2πλ == (B.75)

For through-wall defects, α ≡ 1 and for fully circumferential defects, θ ≡ π. Global solutions:


( ) ⎟⎠⎞

⎜⎝⎛ −−= N

ra Lfπθααη

πβ ,1

21

(B.76)

( ) ( ) θαηαβη sin,21sinb c

br ffL −= (B.77)

Part of the crack inside the compression zone (θ + β > π):



( )[ ]( )αη

πθαη

πβ

,2

,111

e

eNr

f

fL −−+−= (B.78)

( ) ( ) ( )( ) ⎥⎦⎤

⎢⎣⎡ −+= θαηβαηη sin,1

21sin, ddb

br fffL (B.79)

In eqns. (B.76) to (B.79)

αηη21

211 +−=af (B.80)

2

1211 η+=bf (B.81)

2222

31

21

411 ηααηαηηη +−++−=cf (B.82)

( ) ( )ηηηααηαηα bd ff ⎥⎦⎤

⎢⎣⎡ ++−+−= 2222

121

31

6111 (B.83)

ηααηα 2

21

211 −+−=ef (B.84)


ta

=α , em

beb

r tRrML 24

= , em

NeN

r tRrFLπ2

= ,

2tr

c

m −=θ

α ≡ 1 and for a fully circumferential defect θ ≡ π

e

em

em

empr tR

PrtRr

PtrL

222

2

≈⎟⎠⎞

⎜⎝⎛ −

=

pr

Nr

m

N

LL

prF

=′⋅

= 2πχ

( ) pr

Nr

pr

pNr LLLL χ+=+= 1

( ) pNr

br

mN

m

b

L

LprFr

M

2

2 ππλ =

′⋅+= (B.85)



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−++

=2

1 1234

12

21

χχ

pNr

pNr

aLLS (B.86)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

=2

2 1234

12

21

χχ

pNr

pNr

aLLS (B.87)

Global solutions:


⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

121

1 1a

pNr

aa

a

SL

SSS

πθα

πβ

(B.88)

( )[ ]θαβ sinsin21

121 aaabr SSSL −−= (B.89)

Part of the crack inside the compression zone (θ + β > π)

( )( ) ( ) ⎟⎠⎞

⎜⎝⎛ −−−−

−−= pN

raaaaaa

LSSSSSS π

θαααπ

β2211

21 11

(B.90)

( )( )[ ]θαβα sinsin121

221 aaabr SSSL −−−= (B.91)

Validity limits:

This is a net-section collapse solution. When θ + β > π, crack closure is ignored. For the cases of combined pressure and bending, this solution may be used by converting the pressure into an equivalent axial load N. However, for very shallow defects, this treatment may overestimate the limit load as the pressure induced hoop stress was ignored in the derivation of this solution.

Bibliography:







B.6.3 Long internal surface flaw in cylinder oriented circumferentially

R6



IV 1.8.1 with remark III “for a fully circumferential defect θ ≡ π” Thick Pipe under internal pressure:

IV 1.8.2


IV 1.8.4 with remark III “for a fully circumferential defect θ ≡ π” Thin-walled Cylinder under axial load: IV 1.8.5

Solution:


em

NeN

r tRrFL⋅

=π2

, em

beb

r tRrML 24

= , ta

=α , mrt

=η , πθ =

Nr

br

Nm

b

L

LFr

M

2πλ == (B.92)

Global solutions:

( )[ ]( )αη

αηπβ

,2,111

e

eNr

ffL −−+

−= (B.93)

( ) ( )[ ]βαηη sin,dbbr ffL = (B.94)

In eqns. (IV.1.8.1-2) to (IV.1.8.1-5)



αηη21

211 +−=af (B.95)

2

1211 η+=bf (B.96)

2222

31

21

411 ηααηαηηη +−++−=cf (B.97)


⎢⎣⎡ ++−+−= 2222

121

31

6111 (B.98)

ηααηα 2

21

211 −+−=ef (B.99)

Thick Pipe under internal pressure:

ta

=α , mrt

=η

With defect face pressure:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

++

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+= 1

211

211

21

211

211

ln

2

αηη

η

αηη

η

e

e

RP

(B.100)

if

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+>

+−1

211

211

21

211

2

αηη

η

αηη

αη

otherwise

αηη

η

αηη

η

+−

−−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+=

211

211

1

211

211

lne

e

RP

(B.101)

Without defect face pressure (sealed defect):

( )( )2

2

211

2121

211

211

ln

211

211

⎟⎠⎞

⎜⎝⎛ −

+−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+−=

η

αηαη

αηη

η

η

αηη

e

e

RP

(B.102)



if ( )( )

2

2

211

2121

211

211

ln

211

211

211

211

ln

⎟⎠⎞

⎜⎝⎛ −

+−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+−>

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+

η

αηαη

αηη

η

η

αηη

η

η

otherwise

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+=

η

η

211211

lne

e

RP

(B.103)

Thin-walled Cylinder:

em

NeN

r tRrFL⋅

=π2

, ta

=α

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

+>−

+≤⎟

⎠⎞⎜

⎝⎛ −−+

=

311for 1

32

311for 124

21 2

αα

αααNrL


ta

=α , em

beb

r tRrML 24

= , em

NeN

r tRrFL⋅

=π2

, πθ =

e

em

em

empr tR

PrtRr

PtrL

222

2

≈⎟⎠⎞

⎜⎝⎛ −

=

pr

Nr

m

N

LL

prF

=′⋅

= 2πχ

( ) pr

Nr

pr

pNr LLLL χ+=+= 1

( ) pNr

br

mN

m

b

L

LprFr

M

2

2 ππλ =

′⋅+= (B.104)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−++

=2

1 1234

12

21

χχ

pNr

pNr

aLLS (B.105)



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

=2

2 1234

12

21

χχ

pNr

pNr

aLLS (B.106)

Global solutions:

( )( ) ( )( )pNraaaa

aa

LSSSSSS

−−−−−−

= αααπ

β2211

21 11

(B.107)

( )( )[ ]βα sin121

21 −−= aabr SSL (B.108)

Thin-walled Cylinder under axial load:

em

NeN

r tRrFL⋅

=π2

, ta

=α

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

+>−

+≤⎟

⎠⎞⎜

⎝⎛ −−+

=

311for 1

32

311for 124

21 2

αα

αααNrL (B.109)

Validity limits:

Bibliography:





Thin-walled cylinders under combined tension and bending with internal pressure



[B.29] R A Ainsworth, Plastic collapse load of a thin-walled cylinder under axial load with a fully circumferential crack, Nuclear Electric Engineering Advice Note EPD/GEN/EAN/0085/98 (1998).



B.6.4 External surface flaw in cylinder oriented circumferentially

R6


Thick-walled cylinders under combined tension and bending: IV 1.8.1

Thin-walled cylinders under combined tension and bending with internal pressure: IV 1.8.4

Solution:


em

NeN

r tRrFL⋅

=π2 ,

em

beb

r tRrML 24

= , ta

=α , mrt

=η ,

2tr

c

m −=θ

Nr

br

Nm

b

L

LFr

M

2πλ == (B.110)

For through-wall defects, α ≡ 1 and for fully circumferential defects, θ ≡ π. Global solutions:


( ) ⎟⎠⎞

⎜⎝⎛ −−= N

ra Lfπθααη

πβ ,1

21

(B.111)

( ) ( ) θαηαβη sin,21sinb c

br ffL −= (B.112)

Part of the crack inside the compression zone (θ + β > π):



( )[ ]( )αη

πθαη

πβ

,2

,111

e

eNr

f

fL −−+−= (B.113)

( ) ( ) ( )( ) ⎥⎦⎤

⎢⎣⎡ −+= θαηβαηη sin,1

21sin, ddb

br fffL (B.114)

In eqns. (IV.1.8.1-2) to (IV.1.8.1-5)

αηη21

211 −+=af (B.115)

2

1211 η+=bf (B.116)

2222

31

21

411 ηααηαηηη +−−++=cf (B.117)


⎢⎣⎡ ++−−−= 2222

121

31

6111 (B.118)

ηααηα 2

21

211 +−−=ef (B.119)


ta

=α , em

beb

r tRrML 24

= , em

NeN

r tRrFL⋅

=π2

,

2tr

c

m +=θ

e

em

em

empr tR

PrtRr

PtrL

222

2

≈⎟⎠⎞

⎜⎝⎛ −

=

pr

Nr

m

N

LL

prF

=′⋅

= 2πχ

( ) pr

Nr

pr

pNr LLLL χ+=+= 1

( ) pNr

br

mN

m

b

L

LprFr

M

2

2 ππλ =

′⋅+= (B.120)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−++

=2

1 1234

12

21

χχ

pNr

pNr

aLLS (B.121)



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

=2

2 1234

12

21

χχ

pNr

pNr

aLLS (B.122)

Global solutions:


⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

121

1 1a

pNr

aa

a

SL

SSS

πθα

πβ

(B.123)

( )[ ]θαβ sinsin21

121 aaabr SSSL −−= (B.124)

Part of the crack inside the compression zone (θ + β > π)

( )( ) ( ) ⎟⎠⎞

⎜⎝⎛ −−−−

−−= pN

raaaaaa

LSSSSSS π

θαααπ

β2211

21 11

(B.125)

( )( )[ ]θαβα sinsin121

221 aaabr SSSL −−−= (B.126)

Validity limits:

Bibliography:







B.6.5 Long external surface flaw in cylinder oriented circumferentially

R6



IV 1.8.1 with remark III “for a fully circumferential defect θ ≡ π” Thick Pipe under internal pressure:

IV 1.8.3


IV 1.8.4 with remark III “for a fully circumferential defect θ ≡ π” Thin-walled Cylinder under axial load: IV 1.8.5

Solution:


em

NeN

r tRrFL⋅

=π2

, em

beb

r tRrML 24

= , ta

=α , mrt

=η , πθ =

Nr

br

Nm

b

L

LFr

M

2πλ == (B.127)

Global solutions:

( )[ ]( )αη

αηπβ

,2,111

e

eNr

ffL −−+

−= (B.128)

( ) ( )[ ]βαηη sin,dbbr ffL = (B.129)

In eqns. (IV.1.8.1-2) to (IV.1.8.1-5)



αηη21

211 −+=af (B.130)

2

1211 η+=bf (B.131)

2222

31

21

411 ηααηαηηη +−−++=cf (B.132)


⎢⎣⎡ ++−−−= 2222

121

31

6111 (B.133)

ηααηα 2

21

211 +−−=ef (B.134)


ta

=α , mrt

=η

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+

−−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−+=

2

211

211

121

211

211

lnαηη

η

η

αηη

e

e

RP

(B.135)

if

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+

−−>

−+

+2

211

211

121

211

211

αηη

η

αηη

η

otherwise

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

+=

η

η

211211

lne

e

RP

(B.136)


ta

=α , em

beb

r tRrML 24

= , em

NeN

r tRrFL⋅

=π2

, πθ =

e

em

em

empr tR

PrtRr

PtrL

222

2

≈⎟⎠⎞

⎜⎝⎛ −

=



pr

Nr

m

N

LL

prF

=′⋅

= 2πχ

( ) pr

Nr

pr

pNr LLLL χ+=+= 1

( ) pNr

br

mN

m

b

L

LprFr

M

2

2 ππλ =

′⋅+= (B.137)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−++

=2

1 1234

12

21

χχ

pNr

pNr

aLLS (B.138)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

=2

2 1234

12

21

χχ

pNr

pNr

aLLS (B.139)

Global solutions:

( )( ) ( )( )pNraaaa

aa

LSSSSSS

−−−−−−

= αααπ

β2211

21 11

(B.140)

( )( )[ ]βα sin121

21 −−= aabr SSL (B.141)

Thin-walled Cylinder:

em

NeN

r tRrFL⋅

=π2

, ta

=α

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

+>−

+≤⎟

⎠⎞⎜

⎝⎛ −−+

=

311for 1

32

311for 124

21 2

αα

αααNrL (B.142)

Validity limits:



Reference(s):





Thin-walled cylinders under combined tension and bending with internal pressure



[B.35] R A Ainsworth, Plastic collapse load of a thin-walled cylinder under axial load with a fully circumferential crack, Nuclear Electric Engineering Advice Note EPD/GEN/EAN/0085/98 (1998).



B.7 Round bars and bolts

B.7.1 Embedded flaws in round bars

SINTAP


p. AII. 7-8.

Solution:

Through wall bending for infinite axisymmetric body

Embedded Defect; Through Wall Bending

2)12(8

,−

= ebn

Rσ , bebn m

t⎟⎠⎞

⎜⎝⎛= 3,192σ (B.143)

Surface Defect; Through Wall Bending

2)12(8

,−

= ebn

Rσ , bebn m

t⎟⎠⎞

⎜⎝⎛= 3,

96σ (B.144)

Validity limits:

Bibliography:

[B.36] A. J. Carter, A Library of Limit Loads for FRACTURE.TWO, Nuclear Electric Report TD/SID/REP/0191, (1992).



B.7.2 Centrally embedded axial elliptical defects

SINTAP


p. AII. 38-39

Solution:

Tension; Global & Local Collapse

Global Collapse

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=)(

21cWW

acWLRF eN

e (B.145)

Local Collapse

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=)2(

21caWW

bcWLRF eN

e (B.146)

Validity limits:

Bibliography:




B.7.3 Solid round bar; centrally embedded extended defect

SINTAP


p. AII. 33

Solution:

Radial Tension

⎟⎠⎞

⎜⎝⎛ −=

WaWLRF e

Ne

21 (B.147)

Validity limits:

Bibliography:




B.8 Tubular joints

B.8.1 T- and Y-Joints with axial load

SINTAP


p. AIII. 21-22

Description: T- and Y-Joints

Loading: Axial

Schematic:

Limit load Solution:

The characteristic strength of a welded tubular joint subjected to unidirectional loading may be derived as follows:

θsin

2ae

fucKTRQQP = (B.148)

where

CP = characteristic strength for brace axial load

eR = characteristic yield stress of the chord member at the joint (or 0.7 times the characteristic tensile strength if less). If characteristic values are not available specified minimum values may be substituted.

2sin

11 ⎟⎠⎞

⎜⎝⎛ +

= θaK (B.149)

fQ = is a factor to allow for the presence of axial and moment loads in the chord. fQ is defined as:



fQ = 1.0 - 1.638 2Uγλ for extreme conditions

fQ = 1.0 - 2.890 2Uγλ for operating conditions

where

λ = 0.030 for brace axial load

λ = 0.045 for brace in-plane moment load

λ = 0.021 for brace out-of-plane moment load

and

( )e

aoaia

TRDMMDP

U 2

222

72.023.0 ++

= (B.150)

with all forces (Pa, Mai, Ma0) in the function U relating to the calculated applied loads in the chord. Note that U defines the chord utilisation factor.

fQ = may be set to 1.0 if the following condition is satisfied:

( ) 5.022

0.23D1force tension axial chord aoai MM +≥ with all forces relating to the calculated applied loads in

the chord.

uQ = is a strength factor which varies with the joint and load type:

( ) ββ QQu 202+= (for Axial Compression) (B.151)

( )β228+=uQ (for Axial Tension)

Qβ = is the geometric modifier defined as follows

0.1=βQ for β≤0.6

( )βββ 833.013.0

−=Q for β>0.6

Bibliography:

[B.39] Offshore Installations: Guidance on Design, Construction and Certification, Fourth Edition, UK Health & Safety Executive, London (1990).



B.8.2 T- and Y-Joints with in-plane and out-of-plane bending

SINTAP


p. AIII. 23-24

Description: T- and Y-Joints

Loading: In-plane and out-of-plane bending

Schematic:



θsin

2dTRQQMM efucoci == (B.152)

where

ciM = characteristic strength for brace in-plane moment load

coM = characteristic strength for brace out-of-plane moment load


fQ = is a factor to allow for the presence of axial and moment loads in the chord.

fQ is defined as:





where




and

( )e

aoaia

TRDMMDP

U 2

222

72.023.0 ++

= (B.153)

with all forces (Pa, Mai, Mao) in the function U relating to the calculated applied loads in the chord. Note that U defines the chord utilisation factor.


( ) 5.022


the chord.


θβγ sin5 5.0=uQ (for In-Plane Bending)

( ) ββ QQu 76.1 += (for Out-of Plane Bending) (B.154)


0.1=βQ for β≤0.6

( )βββ 833.013.0

−=Q for β>0.6

Bibliography:




B.8.3 K-Joints with axial loads

SINTAP


p. AIII. 25-26

Description: K-Joints

Loading: Axial

Schematic:



θsin

2ae

fucKTRQQP = (B.155)

where

cP = characteristic strength for brace axial load


2sin

11 ⎟⎠⎞

⎜⎝⎛ +

= θaK (B.156)






where




and

( )e

aoaia

TRDMMDP

U 2

222

72.023.0 ++

= (B.157)



( ) 5.022


the chord.


( ) ββ QQQ gu 202+= (for Axial Compression)

( ) gu QQ β228+= (for Axial Tension) (B.158)


0.1=βQ for β≤0.6

( )βββ 833.013.0

−=Q for β>0.6

5.09.07.1 ζ−=gQ but should not be taken as less than 1.0.

Bibliography:




B.8.4 K-Joints with in-plane and out-of-plane bending

SINTAP


p. AIII. 27-28

Description: K-Joints


Schematic:



θsin


where





fQ is defined as:





where




and

( )e

aoaia

TRDMMDP

U 2

222

72.023.0 ++

= (B.160)



( ) 5.022


the chord.





0.1=βQ for β≤0.6

( )βββ 833.013.0

−=Q for β>0.6

Bibliography:




B.8.5 X- and DT-Joints with axial load

SINTAP


p. AIII. 29-30

Description: X- and DT-Joints

Loading: Axial

Schematic:



θsin

2ae

fucKTRQQP = (B.162)

where

cP = characteristic strength for brace axial load


2sin

11 ⎟⎠⎞

⎜⎝⎛ +

= θaK (B.163)






where




and

( )e

aoaia

TRDMMDP

U 2

222

72.023.0 ++

= (B.164)



( ) 5.022


the chord.


( ) ββ QQu 145.2 += (for Axial Compression)

( ) ββ QQu 177 += (for Axial Tension) (B.165)


0.1=βQ for β≤0.6

( )βββ 833.013.0

−=Q for β>0.6

Bibliography:




B.8.6 X- and DT-Joints with in-plane and out-of-plane bending

SINTAP


p. AIII. 31-32

Description: X- and DT-Joints


Schematic:



θsin


where







fQ is defined as:



where




and

( )e

aoaia

TRDMMDP

U 2

222

72.023.0 ++

= (B.167)



( ) 5.022


the chord.





0.1=βQ for β≤0.6

( )βββ 833.013.0

−=Q for β>0.6

Bibliography:

[B.44] AIII.6. Offshore Installations: Guidance on Design, Construction and Certification,Fourth Edition, UK Health & Safety Executive, London (1990).



B.9 Material mismatch

B.9.1 Crack in the centre line of the weld material

RBe

RWe

SINTAP


p. AIV. 4-6

Solution:

(i) Plane Stress

The limit load for the plate made wholly of material b is

( )aWBRF Be

Be −= 2 (B.169)

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

43.1,min

43.1021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.170)

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

ψ43.1

332

321

MF

FB

e

Me (B.171)

( )( )

ψ43.111

2

MF

FB

e

Me −−= (B.172)

Overmatching (M>1)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

3

(B.173)



( )( )( ) ( )

( ) ( )( ) ( )⎪⎩

⎪⎨⎧

⋅+=≥+

+⋅−

⋅+=≤= −−−−

−−−−

5/1151

1

5/11513

43.0125

2425

12443.01

MM

MM

Be

Me

eeforMMeeforM

FF

ψψψψ

ψψ (B.174)

(ii) Plane Strain


( )aWBRF Be

Be −=

34

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

1,min

1021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.175)

( )( )

ψ111

1

MF

FB

e

Me −−= (B.176)

( )

( ) ( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤⎥⎦

⎤⎢⎣

⎡++

≤≤⎥⎦

⎤⎢⎣

⎡−

≤≤⎥⎦

⎤⎢⎣

⎡ −−

−+

=

ψψ

ψ

ψψ

ψψ

ψψ

ψ

5019.0291.1125.0

0.56.3254.3571.2

6.31104.01462.00.132

2

forM

forM

forM

FF

Be

Me (B.177)

Overmatching (M>1)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

3

(B.178)

( )( )

( ) ( )⎪⎩

⎪⎨⎧

=≥+

+⋅−

=≤= −−

−−

5/11

1

5/113

2524

25124 M

M

Be

Me

eforMMeforM

FF

ψψψψ

ψψ (B.179)

Bibliography:

[B.45] H. Schwalbe, Y.-J. Kim, S. Hao, and A. Cornec, ETM-MM – The Engineering Treatment Model for Mis-Matched Welded Joints, Mis-Matching of Welds, ESIS 17, Edited by K.-H. Schwalbe and M. Koçak, Mechanical Engineering Publications, London, 539-560 (1994).



B.9.2 Crack in the interface between weld metal and base plate

RBe

RWe

SINTAP


p. AIV. 7-8

Solution:

(i) Plane Stress


( )aWBRF Be

Be −= 2 (B.180)

Undermatching (M<1)

( )[ ][ ]MMMFF

Be

Me 108.01exp095.0095.1 −−−= (B.181)

Overmatching (M>1)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

1

(B.182)

( )( )[ ]108.01exp095.0095.1

1

MF

FB

e

Me −−−= (B.183)

(ii) Plane Strain


( )aWBRF Be

Be −=

34

(B.184)



Undermatching (M<1)

( ) ( )

⎭⎬⎫

⎩⎨⎧

= Be

Me

Be

Me

Be

Me

FF

FF

FF 21

,min (B.185)

( ) ( )( )[ ]( ) ( )( )[ ]⎪⎩

⎪⎨⎧

−−−=≥−−

−−−=≤≤= Mforf

Mforf

FF

Be

Me

1221211

122120

11

11

ψψψψ

ψψ (B.186)

⎪⎩

⎪⎨

⎧

≤

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−⎟⎠⎞

⎜⎝⎛ −

+=

5.030.1

15.0122.0152.012

MforM

MforM

MM

MMf (B.187)

( )

( ) ( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤⎥⎦

⎤⎢⎣

⎡++

≤≤⎥⎦

⎤⎢⎣

⎡−

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−

−+

≤≤

=

ψψ

ψ

ψψ

ψψ

ψψ

ψ

ψ

2.6909.0294.1125.0

2.62.4123.4881.2

2.422027.02394.03.1

2030.132

2

forM

forM

forM

forM

FF

Be

Me (B.188)

Overmatching (M>1)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

3

(B.189)

( )

⎪⎩

⎪⎨

⎧

≥+

+⎥⎦

⎤⎢⎣

⎡−

−−⎟

⎠⎞

⎜⎝⎛ +

−

≤≤=

225

24142exp

2524

203

ψψψ

forMM

Mf

forf

FF

Be

Me (B.190)

( ) ( )⎩⎨⎧

≥≤≤−−−+

=230.1

21122.0152.01 2

MforMforMMf (B.191)

Bibliography:




B.9.3 Crack in the interface of a bi-material joint

RBe

RWe

RBeRW

e

SINTAP


p. AIV. 9

Solution:

(i) Plane Stress


( )aWBRF Be

Be −= 2 (B.192)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

1

(B.193)

( )( )[ ]108.01exp095.0095.1

1

MF

FB

e

Me −−−= (B.194)

(ii) Plane Strain


( )aWBRF Be

Be −=

34

(B.195)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

1

(B.196)

( ) ( ) ( )⎩⎨⎧

≥≤≤−−−+

=230.1

21122.0152.01 21

MforMforMM

FF

Be

Me (B.197)



Bibliography:





RBe

RWe

SINTAP


p. AIV. 10-12

Solution:

(i) Plane Stress


( )aWBRF Be

Be −= 2β ;

⎪⎪⎩

⎪⎪⎨

⎧

≤≤

≤≤⎟⎠⎞

⎜⎝⎛+

=1286.0

32

286.0054.01

wafor

wafor

wa

β (B.198)

Undermatching (M<1)

ψallforMFF

Be

Me = (B.199)

Overmatching (M>1)

( )

( )⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

1

β (B.200)

( )( )

( ) ( ) ( ) ( )⎪⎩

⎪⎨

⎧

=≥⎟⎟⎠

⎞⎜⎜⎝

⎛−−⋅⎥⎦

⎤⎢⎣⎡ −+

−+

+

=≤≤= −−

−−

5/121

11

5/1211

11.011.025

12425

240

MM

M

Be

Me

eforMMMMeforM

FF

ψψψψ

ψψ

ψψ (B.201)



(ii) Plane Strain


( )aWBRF Be

Be −=

34β ; ( )

⎪⎪⎩

⎪⎪⎨

⎧

≤≤+

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

1884.02

1

884.0022ln1

wafor

wafor

awaw

πβ (B.202)

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

5.0,min

5.0021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.203)

( )( )

ψ5.011

1

MF

FB

e

Me −−= (B.204)

( ) ( ) ( )[ ]( )⎩

⎨⎧

≥+≤≤−+−+

=0

022

2172.225.05.05.05.0

ψψβψψψβψψβ

forMforBAM

FF

Be

Me (B.205)

( )⎪⎪⎩

⎪⎪⎨

⎧

<−

−−

<<−

−−

=

wafor

wafor

A35.0

5.03422.2225.0

35.005.0

3422.225.0

0

0

ψβ

ψβ

(B.206)

( )⎪⎪⎩

⎪⎪⎨

⎧

<−

−

<<=

wafor

wafor

B35.0

5.03422.2

35.000

20ψ

β (B.207)

( ) ( )20 9.192.353.16 wawa +−=ψ

Overmatching (M>1)

( )

( )⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

3

β (B.208)

( )( )

( ) ( )⎪⎩

⎪⎨⎧

+=≥+

+−

+=≤= −−

−−

2.03.050

4950

1492.03.0

5/11

1

5/113

M

M

Be

Me

eforMMeforM

FF

ψψψψ

ψψ (B.209)



Bibliography:





RBe

RWe

SINTAP


p. AIV. 13-14

Solution:

(i) Plane Stress


( )aWBRF Be

Be −= 2β ;

⎪⎪⎩

⎪⎪⎨

⎧

<<

<<⎟⎠⎞

⎜⎝⎛+

=1286.0

32

286.0054.01

wafor

wafor

wa

β (B.210)

Undermatching (M<1)

ψallforMFF

Be

Me = (B.211)

Overmatching (M>1)

ψallforFF

Be

Me 1= (B.212)

(ii) Plane Strain




( )aWBRF Be

Be −=

34β ; ( )

⎪⎪⎩

⎪⎪⎨

⎧

<<+

≤<⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

1884.02

1

884.0022ln1

wafor

wafor

awaw

πβ (B.213)

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

1,min

1021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.214)

( )( )

ψ111

1

MF

FB

e

Me −−= (B.215)

( ) ( ) ( )[ ]( )⎩

⎨⎧

≥+≤≤−+−+

=0

022

2172.2125.0111

ψψβψψψβψψβ

forMforBAM

FF

Be

Me (B.216)

( )⎪⎪⎩

⎪⎪⎨

⎧

<−

−−

<<−

−−

=

wafor

wafor

A35.0

13422.22125.0

35.001

3422.2125.0

0

0

ψβψ

β

(B.217)

( )⎪⎪⎩

⎪⎪⎨

⎧

<−

−

<<=

wafor

wafor

B35.0

13422.2

35.000

20ψ

β (B.218)

( ) ( )20 8.394.706.32 wawa +−=ψ

Overmatching (M>1)

ψallforFF

Be

Me 1= (B.219)

Bibliography:





RBe

RWe

RBeRW

e

SINTAP


p. AIV. 15

Solution:

(i) Plane Stress

( )aWBRF Be

Be −= 2β ;

⎪⎪⎩

⎪⎪⎨

⎧

<<

<<⎟⎠⎞

⎜⎝⎛+

=1286.0

32

286.0054.01

wafor

wafor

wa

β (B.220)

(ii) Plane Strain

( )aWBRF Be

Me −=

34β ; ( )

⎪⎪⎩

⎪⎪⎨

⎧

<<+

<<⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

1884.02

1

884.0022ln1

wafor

wafor

awaw

πβ (B.221)

Bibliography:





RWe

RBe

SINTAP


p. AIV. 16-18

Solution:

(i) Plane Stress


( )23

4641.0 aWBRFBeB

e −= (B.222)

Undermatching (M<1)

ψallforMFF

Be

Me = (B.223)

Overmatching (M>1)

( )

( ) ⎭⎬⎫

⎩⎨⎧

−= 2

1

11,min

waFF

FF

Be

Me

Be

Me (B.224)

( )( ) ( )⎪

⎩

⎪⎨

⎧

≥⎟⎟⎠

⎞⎜⎜⎝

⎛−++⎟

⎠⎞

⎜⎝⎛ −−+

−+

+

≤≤=

111

11

111150

14950

490

ψψψψ

ψψ

ψψ

forMMMMforM

FF M

Be

Me

( )( ) ( ) 8/111 7.00.2 −−−−+= MM eeψ (B.225)



(ii) Plane Strain


( )23

aWBRFBeB

e −= β ;

⎪⎪⎩

⎪⎪⎨

⎧

<<

<<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=13.0631.0

3.00245.1808.05.02

wafor

wafor

wa

wa

β (B.226)

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

0.2,min

0.2021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.227)

( ) ( )( ) ( )

1092

1201exp

10191 +

+⎥⎦

⎤⎢⎣

⎡−

−−⎥⎦

⎤⎢⎣⎡ −

=M

MM

FF

Be

Me ψ (B.228)

For 0<a/w≤0.3

( )

⎪⎪⎩

⎪⎪⎨

⎧

≥⎟⎠⎞

⎜⎝⎛ +

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

+−⎟

⎠⎞

⎜⎝⎛ −

+−+

=

1510

623.01345.1

0.150.22.0102.2

33.322.01069.1

4.53132

2

ψβψ

ψψβ

βψβ

β

forM

forM

FF

Be

Me (B.229)

For 0.3<a/w

( )

⎪⎪⎩

⎪⎪⎨

⎧

≥⎟⎠⎞

⎜⎝⎛ +

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−

=

0.710

494.0900.0

0.70.210

952.110

129.310

017.1094.132

2

ψψ

ψψψψ

forM

forM

FF

Be

Me (B.230)

Overmatching (M>1)

( )

( ) ⎭⎬⎫

⎩⎨⎧

−= 2

3

121,min

waFF

FF

Be

Me

Be

Me

β (B.231)

( )

⎪⎩

⎪⎨

⎧

≥⎟⎟⎠

⎞⎜⎜⎝

⎛++

≤≤=

111

130

ψψψψ

ψψ

ψψ

forCBA

forM

FF M

Be

Me (B.232)

( ) ( )

( )⎩⎨⎧

<≤<

=−−

−−

waforewafore

M

waM

4.024.002

8/1

10/1

1ψ (B.233)

5049+

=MA ;

( ) CMB −−

=50

149; ( ) 113.0 −−= MMC



Bibliography:





RWe

RBe

SINTAP


p. AIV. 19-20

Solution:

(i) Plane Stress


( )23

4641.0 aWBRFBeB

e −= (B.234)

Undermatching (M<1)

( )[ ] ψallforeMFF MM

Be

Me 13.0104.004.1 −−−= (B.235)

Overmatching (M>1)

( ) ψallforeFF MM

Be

Me 13.0104.004.1 −−−= (B.236)

(ii) Plane Strain


( )23

aWBRFBeB

e −= β ;

⎪⎪⎩

⎪⎪⎨

⎧

<≤

<<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=13.0631.0

3.00245.1808.05.02

wafor

wafor

wa

wa

β (B.237)



Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

4,min

4021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.238)

( )( ) CAe

FF B

Be

Me += −− 4

1ψ

[ ]⎩⎨⎧

≤−⋅≤<

= −− waforeMwaforM

f MM /3.006.006.13.0/0

3.0/)1( (B.239)

( ) ( )[ ]41 −+−= ψBCfA ;M

B−

=15.81

;10

9+=

MC (B.240)

For 0<a/w≤0.3

( )

⎪⎪⎩

⎪⎪⎨

⎧

≥⎟⎠⎞

⎜⎝⎛ +

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+−

+⎟⎠⎞

⎜⎝⎛−

+=

1410

623.0377.1

0.140.410

377.5310

377.32132

2

ψβψ

ψψβ

βψβ

β

forM

forM

FF

Be

Me (B.241)

For 0.3≤ a/w

( )

⎪⎪⎩

⎪⎪⎨

⎧

≥⎟⎠⎞

⎜⎝⎛ +

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=

0.1410

494.01

0.140.410

133.010

522.006.132

2

ψψ

ψψψ

forM

forM

FF

Be

Me (B.242)

Overmatching (M>1)

( )⎪⎩

⎪⎨

⎧

≤+−

<<≈

−−

wafore

wafor

FF

MB

e

Me

3.006.106.0

3.001

3.0/1 (B.243)

Bibliography:





RWe RB

e

RBeRW

e

SINTAP


p. AIV. 21

Solution:

(i) Plane Stress

( )23

4641.0 aWBRFBeM

e −= β ; ( ) 13.0104.004.1 −−−= Meβ (B.244)

(ii) Plane Strain

( )23

aWBRFBeB

e −= β ;( ) ( ) ( )

( ) ( )⎪⎩

⎪⎨

⎧

≤<+−

≤<+−=

∞−−

∞

∞−−

∞

13.0

3.00

3.011

11

wafore

wafore

M

waM

βββ

ββββ (B.245)

⎪⎪⎩

⎪⎪⎨

⎧

≤<

≤<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=13.0631.0

3.00245.1808.0500.02

1

wafor

wafor

wa

wa

β (B.246)

⎪⎪⎩

⎪⎪⎨

⎧

≤<

≤<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=∞

14.0670.0

4.00165.1890.0500.02

wafor

wafor

wa

wa

β (B.247)

Bibliography:




B.9.10 Crack in the centre line of the weld metal

L

L/2 L/2

RBe

RWe

SINTAP


p. AIV. 22-24

Solution:

(i) Plane Stress


( )23

960.02

LaWBRF

BeB

e−

= (B.248)

Undermatching (M<1)

ψallforMFF

Be

Me = (B.249)

Overmatching (M>1)

( ) ( )

⎭⎬⎫

⎩⎨⎧

= Be

Me

Be

Me

Be

Me

FF

FF

FF 21

,min (B.250)

( )( ) ( )⎪

⎩

⎪⎨

⎧

≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎠⎞

⎜⎝⎛ −−

−+

+

≤≤=

111

11

12.012.050

14950

490

ψψψψ

ψψ

ψψ

forMMMMforM

FF M

Be

Me

( )( ) ( ) 4/111 5.05.2 −−−−+= MM eeψ (B.251)

( )

( )22

11

960.0 waFF b

Be

Me

−=

β;

2

254.0260.200.4 ⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

WH

WH

bβ (B.252)



(ii) Plane Strain


( )23

2

LaWBRF

BeB

e−

= β ;

⎪⎪⎩

⎪⎪⎨

⎧

<≤⎟⎠⎞

⎜⎝⎛+

<<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=1172.0096.0199.1

172.00238.2892.0125.12

wafor

wa

wafor

wa

wa

β (B.253)

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

<<= ψ

ψ

0.2,min

0.2021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.254)

( )

⎪⎪⎩

⎪⎪⎨

⎧

≥⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−+⎟

⎠⎞

⎜⎝⎛ −

+−+

=

122.010

616.0384.1

0.120.22.010

384.322.010

384.53132

1

ψβψ

ψψβ

βψβ

β

forM

forM

FF

Be

Me (B.255)

( ) ( )( ) ( )

1092

1201exp

10192 +

+⎥⎦

⎤⎢⎣

⎡−

−−

−=

MM

MF

FB

e

Me ψ (B.256)

Overmatching (M>1)

( ) ( )

⎭⎬⎫

⎩⎨⎧

= Be

Me

Be

Me

Be

Me

FF

FF

FF 43

,min (B.257)

( ) ( ) ( ) ( )M

Be

Me MMMMMMF

F⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟

⎠⎞

⎜⎝⎛ −−−

−+

+=

ψψ

ψψ 11

3

113.0113.050

14950

49 (B.258)

( ) ( )

( )⎩⎨⎧

<≤<<

=−−

−−

1172.02172.002

8/1

4/1

1 waforewafore

M

waM

ψ (B.259)

( )

( )24

11

waFF b

Be

Me

−=ββ

; (B.260)

32

21818.023095.126072.35557.4 ⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

WH

WH

WH

bβ (B.261)

Bibliography:





L

L/2 L/2

RBe

RWe

SINTAP


p. AIV. 25-26

Solution:

(i) Plane Stress


( )23

960.02

LaWBRF

BeB

e−

= (B.262)

Undermatching (M<1)

ψallforMFF

Be

Me = (B.263)

Overmatching (M>1)

ψallforFF

Be

Me 1= (B.264)

(ii) Plane Strain


( )23

2

LaWBRF

BeB

e−

= β ;

⎪⎪⎩

⎪⎪⎨

⎧

<≤⎟⎠⎞

⎜⎝⎛+

<<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=1172.0096.0199.1

172.00238.2892.0125.12

wafor

wa

wafor

wa

wa

β (B.265)



( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

<<= ψ

ψ

0.4,min

0.4021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.266)

( )

⎪⎪⎩

⎪⎪⎨

⎧

≥⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+

≤≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−+⎟

⎠⎞

⎜⎝⎛ −

+−+

=

124.010

616.00.2

0.120.44.01016

616.24.0108

08.93132

1

ψβψ

ψψβ

βψβ

β

forM

forM

FF

Be

Me (B.267)

( ) ( )( ) ( )

1094

1201exp

10192 +

+⎥⎦

⎤⎢⎣

⎡−

−−

−=

MM

MF

FB

e

Me ψ (B.268)

Overmatching (M>1)

ψallforFF

Be

Me 1= (B.269)

Bibliography:





LL/2 L/2

RBeRW

e

RBeRW

e

SINTAP


p. AIV. 27

Solution:

(i) Plane Stress

( )23

960.02

LaWBRF

BeM

e−

= (B.270)

(ii) Plane Strain

( )23

2

LaWBRF

BeM

e−

= β ; ( ) ( )∞

−−∞ +−= ββββ 23.01

1Me (B.271)

⎪⎪⎩

⎪⎪⎨

⎧

≤<⎟⎠⎞

⎜⎝⎛+

≤<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=1172.0096.0199.1

172.00238.2892.0125.12

1

wafor

wa

wafor

wa

wa

β (B.272)

⎪⎪⎩

⎪⎪⎨

⎧

≤<⎟⎠⎞

⎜⎝⎛+

≤<⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

=∞

1172.0107.1238.1

172.00072.2108.1125.12

wafor

wa

wafor

wa

wa

β (B.273)

Bibliography:




B.9.13 Crack in the centre line of the weld metal

RBe

RWe

ri

ro

SINTAP


p. AIV. 28-29

Solution:

The limit load for the pipe made wholly of material b is

( )[ ]22

32 arrRF io

BeB

e +−= π (B.274)

Undermatching (M<1)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

1,min

1021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.275)

( )

⎥⎦

⎤⎢⎣

⎡ −+=

3311

1 ψMF

FB

e

Me (B.276)

( )( )

ψ111

2

MF

FB

e

Me −−= (B.277)

Overmatching (M>1)

( )

⎭⎬⎫

⎩⎨⎧

−=

waFF

FF

Be

Me

Be

Me

11,min

3

(B.278)



( )( )

( ) ( )⎪⎩

⎪⎨⎧

=≥+

+⋅−

=≤= −−

−−

5/121

1

5/1213

2524

25124 M

M

Be

Me

eforMMeforM

FF

ψψψψ

ψψ (B.279)

Bibliography:




B.9.14 Crack in the interface between weld metal and base pipe

RBe

RWe

ri

ro

SINTAP


p. AIV. 30

Solution:

The limit load for the pipe made wholly of material b is

( )[ ]22

32 arrRF io

BeB

e +−= π (B.280)

( ) ( )

⎪⎩

⎪⎨⎧

≤⎭⎬⎫

⎩⎨⎧

≤≤= ψ

ψ

2,min

2021

forF

FF

FforM

FF

Be

Me

Be

MeB

e

Me (B.281)

( )

⎥⎦

⎤⎢⎣

⎡ −+=

3621

1 ψMF

FB

e

Me (B.282)

( )( )

ψ211

2

MF

FB

e

Me −−= (B.283)

Overmatching (M>1)

ψallforFF

Be

Me 1= (B.284)



Bibliography:





SINTAP


p. AIV. 31

Solution:

( )[ ]22

32 arrRF io

BeM

e +−= π (B.285)

Remarks: Solutions are valid for thin-walled pipes with deep cracks, a/t≥0.3

Bibliography:




B.9.16 Bi-material (clad) center through thickness cracked plate under tension

+=

2

F F

FF F

F

Hom.1 2

B=b1+b2

b 1

b 2

2 a

2 w

F

F


For a centre cracked clad plate, Alexandrov and Kocak proposed a closed-form expression for the tensile limit load under plane stress condition.

Solution:

( )( )221

2)( 2141 kRMbBbFF e

he

layerbie ++= −− (B.286)

where

layerbieF − is the tensile limit load of the whole bi-layer system

( )aWBRF eh

e −= 2)( 2 is the tensile limit load of the homogeneous centre cracked plate made of the softer

material (here material 2) with yield strength 2eR , thickness of B=b1+b2, half width of W (Fig. (11.9.1)) and

shear yield strength of k2

21 bbb = ,

21ee RRM = is strength mis-match ratio with 1

eR the yield strength of the higher yield strength material (here Material 1).

Alternatively, limit load of the bi-layer plate can be derived using rule of mixtures as AK. Motarjemi, M. Kocak proposed.

By substituting the corresponding values of b, M, k2, )(h

eF and B into the equation above, and after some

simplifications, based on the Tresca yield criteria ( 22 2k=σ ), a simple tensile yield load solution, can be found for the bi-layer plate as:



⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

−

2

1

21

2)( 1

e

eh

e

layerbie

FF

bbb

FF

(B.287)

where

( )( )aWbbRF eh

e −+= 212)( 2 (B.288)

and after simplification:

∑=

− =n

i

ie

layerbie FF

1

(B.289)

in which:

layerbieF − is as defined earlier and i

eF is the tensile limit load for the centre-cracked configuration of each

constituent of the bi-layer system, i.e., 1eF and 2

eF .

References

[B.60] S. Alexandrov, M. Kocak, “Limit load solutions for bilayer plates with a through crack subject to tension”, Engineering Fracture Mechanics 64 (1999) 507-511.

[B.61] AK. Motarjemi, M. Kocak, “Tensile yield load solutions for centre cracked bilayer (clad) plates with and without repair welds”, Science and Technology of Welding and Joining, 2002, Vol.7, No 5, 299-305

[B.62] AK. Motarjemi and M. Kocak, “Fracture assessment of a clad steel using various SINTAP defect assessment procedure levels”, 2002 Fatigue Fract Engng Mater Struct 25, 929-939



B.9.17 Centre cracked bi-layer (clad) plate with repair weld

Repairweld

Hom.

+=

III II

F

FF

F

F

F

2W

2h

2W 2W

2a2a

2a

2h

B=bI+bII

+

IVHom.

FF

FF

2W2W

2a2a2h

V

F

F

2W

2a2h

=VIII

Repairweld

IV

F

F

2W

2a2h

Repairweld

III IV

F

F

2W

2h

2a +

B=bIII+bIV

Case1 Case2


By using this approach, the limit load of the repair welded bi-layer plates has been derived by Moterjemi and Kocak and SINTAP fracture assessment is verified.

Solution:

For all these cases, it is assumed that the centre cracked bi-layer plate is made of two parts; one without a repair weld (homogeneous), e.g., parts I and III, and the other one with a repair weld (mis-matched), e.g., parts II, IV and V. Based on this assumption, the tensile yield load solution for the whole system, layerbi

eF − , can be written as:

Me

he

layerbie FFF +=− )( (B.290)

Where )(heF and M

eF are the tensile yield load values respectively for the homogeneous and mis-matched parts of the bi-layer system.

For the yield load solution for the homogeneous part it can be written:

( )( )aWbbRF IIIIe

he −+= 2)( , (for the case 1) (B.291)

and



( )( )aWbbRF IVIIIIIIe

he −+= 2)( , (for the case 2) (B.292)

where all the parameters have been defined in figure above.

For the mis-matched parts (II, IV and V), tensile yield load solutions for these configurations, the plane stress tensile yield load solution for an over-matching condition, M

eF , is as follows:

{ }βα ,MinFF

Be

Me =

where

( )( ) ( )

( ) ( )( ) ( )⎪⎩

⎪⎨⎧

+=>+

+⋅−

+=≤= −−−−

−−−−

51151

1

51151

43.0125

2425

12443.01

MM

MM

eeforMMeeforM

ψψψψ

ψψα (B.293)

( ) haW −=ψ .with 2h equal to the total width of the weld.

Be

We RRM = is the strength mis-match ratio, with B

eR as the yield strength of the base and WeR as the yield

strength of the weld materials in the mis-matched (repair welded) parts

Wa−=

11β (B.294)

( )aWBRF Be

Be −= 2 (B.295)

References

[B.63] S. Alexandrov, M. Kocak, “Limit load solutions for bilayer plates with a through crack subject to tension”, Engineering Fracture Mechanics 64 (1999) 507-511.

[B.64] AK. Motarjemi, M. Kocak, “Tensile yield load solutions for centre cracked bilayer (clad) plates with and without repair welds”, Science and Technology of Welding and Joining, 2002, Vol.7, No 5, 299-305

[B.65] AK. Motarjemi and M. Kocak, “Fracture assessment of a clad steel using various SINTAP defect assessment procedure levels”, 2002 Fatigue Fract Engng Mater Struct 25, 929-939



Bibliography

[B.66] Ref

[B.67] Ref

[B.68] Ref

[B.69] Ref

Documents

Annex B Limit Load Solutions (Based on SINTAP and R6 · Limit Load Solutions (Based on SINTAP and R6) ... B.9.2 Crack in the interface between weld metal and base plate ... B.9.3