A Review Limite Load Solution

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A review of limit load solutions for cylinders with axial cracks

and development of new solutions

Y. Lei*

British Energy Generation Ltd., Barnett Way, Barnwood, Gloucester GL4 3RS, UK

a r t i c l e i n f o

Article history:

Received 15 July 2008

Received in revised form 22 August 2008

Accepted 4 September 2008

Keywords:

Limit load

Axial crack

Surface crack

Through-wall crack

Thick-walled cylinder

a b s t r a c t

Limit load solutions for axially cracked cylinders are reviewed and compared with available finiteelement (FE) results. New limit solutions for thick-walled cylinders with axial cracks under internal

pressure are developed to overcome problems in the existing solutions. The newly developed limit load

solutions are a global solution for through-wall cracks, global solutions for internal/external surface

cracks and local solutions for internal/external surface cracks. The newly developed limit pressure

solutions are compared with available FE data and the results show that the predictions agree well with

the FE results and are generally conservative.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

The limit load of a component containing defects is one of the

most important inputs when a structural integrity assessment is

performed using R6 [1]. This is because the limit load gives the load

carrying capacity for plastic collapse of the defective component

and also defines the J-integral via the reference stress method for

elasticplastic fracture. The cylinder is one of the most commonly

used components in power stations, such as in pipelines and

pressure vessels. Circumferential and axial cracks are two common

types of defects found in girth and longitudinal welds in cylinders.

In this paper, limit load solutions for axial defects in cylinders will

be reviewed first and some new solutions will then be proposed.

For a part-through defect, the limit load may be defined

according to the plastic deformation behaviour of either the overall

defective structure (global limit load) or that in the crack ligament(local limit load). A global limit load is the load at which the load

point displacement becomes unbounded and is relevant to failure

of the whole structure. A local limit load corresponds to a loading

level at which gross plasticity occurs in the crack ligament and may

be relevant to ligament failure. The local limit load is always less

than or equal to the global limit load for a cracked structure and,

therefore, can yield a conservative result in an assessment. In this

paper both the global and local limit loads are considered.

Miller [2] summarised the limit load solutions for cylinders with

through-wall, surface and extended surface axial defects under

internal pressure available before 1987, such as solutions due to

Kiefner et al. [3] for through-wall and surface defects, Kitching et al.

[4,5] for surface and through-wall defects, Ewing [6] for surface

defects and Chell [7,8] for surface cracks and extended surface

cracks. Further solutions for cylinders with axial defects were

developed by Carter [9]. Staat [10,11] modified some of Carters

solutions [9] for thin-walled cylinders to extend them to thick-

walled cylinders and compared the modified solutions with

experimental data. Recently, Staat and Vu [12] further improvedthe

solutions due to Staat [10,11]. Kim et al. [13,14] carried out elastic-

perfectly plastic finite element (FE) analysis for axially cracked

cylinders and proposed some limit load solutions for extended

internal cracks and surface cracks under internal pressure based on

the FEresults. Jun et al. [15] performed 3-DFE analyses forcylinderswith axial surface defects under internal pressure and presented

results for the local limit pressure. Zarrabi et al. [16] also performed

3-D FE analyses for axial cracked cylinders, but did not give

formulae for the limit load solutions.

The layout of this paper is as follows. Section 2 defines the

geometry parameters and material properties. Recent develop-

ments in limit load solutions for cylinders containing axial cracks

under internal pressure or combined internal pressure, tension and

bending are reviewed in Section 3. New limit load solutions for

cylinders with axial through-wall and surface cracks under internal

pressure are then derived and validated in Section 4. Conclusions

are presented in Section 5. In Appendix A, the Folias factor is* Tel.: 44(0)1452652285; fax: 44(0)1452653025.

E-mail address: [email protected]

Contents lists available at ScienceDirect

International Journal of Pressure Vessels and Piping

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j p v p

0308-0161/$ see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijpvp.2008.09.001

International Journal of Pressure Vessels and Piping 85 (2008) 825850
mailto:[email protected]://www.sciencedirect.com/science/journal/03080161http://www.elsevier.com/locate/ijpvphttp://www.elsevier.com/locate/ijpvphttp://www.sciencedirect.com/science/journal/03080161mailto:[email protected]


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discussed to clarify some confusions in its use in the limit load

solution for axially cracked cylinders. The back-wall effect on the

limit pressure of a cylinder with an axial crack is discussed in

Appendix B and the calibration of the stressmagnification factorfor

cylinders with through-wall cracks is detailed in Appendix C.

2. Geometry definition and material properties

For consistency, solutions from different sources are converted

according to a uniform notation system in this paper. The dimen-

sions of a cylinder are described by its inner radius, Ri, and outer

radius, Ro. The mean radius of the cylinder, Rm, can then be simply

expressed as Rm Ro Ri=2 and the thickness of the cylinder, t,can be expressed as t Ro Ri. A cylinder can then be describedby the ratio between its outer and inner radii, k, i.e. k Ro=Ri, theratio between tand Rm, t=Rm, the ratio between tand Ri, t=Ri, or the

ratio between t and Ro, t=Ro. The relationships between these

various parameters are as follows.

t

Ri k 1; t

Rm 2k 1

k 1;t

Ro k 1

k(1)

For a thin-walled cylinder, k/1. The length and depth of an

axial surface crack in a cylinder are defined by 2c and a, respec-

tively. The crack depth, a, is measured from the inner surface of the

cylinder for internal cracks and from the outer surface for external

cracks. The surface crack becomes an extended surface crack whenc/N and a through-wall crack when a t.

The load types considered are internal pressure,p, through-wall

membrane stress, sm, and through-wall bending stress, sb. Their

limit values are denoted as pL, smL and sbL, respectively. Solu-tions corresponding to other loading types, such as axial tension, N,

and axial global bending moment, M, are also reviewed.

The material considered is an elastic-perfectly plastic type with

yield stress ofsy. Therefore, all limit load solutions for closed-end

cylinders subjected to internal pressure may be approximately

expressed as

pL gL

k;a

t;

a

c;.sy (2)

where L is a geometric function and g is the constraint factor, with

g 1 for Tresca yield condition

2ffiffiffi3

p for von Mises yield condition

8>=>>;

min

8>:

gln k;gRiR*2

1

Ma3ln

1 atk 1

1 atk 1

ln

k

1 atk 1

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 a

t

t

R*

2 1

2a

t2 t

R*

2

2

vuut 1 1

2

a

t

t

R*

2#

9>=

>;

(31)

Y. Lei / International Journal of Pressure Vessels and Piping 85 (2008) 825850 831


8/26

where

Ma3 1 1:25 c2

Ri aa0:5

1 1:25a

tk 1a

c

21 atk 1

0:5(32)

Eq. (31) may be non-conservative for short cracks for the

following two reasons. Firstly, the solution for cylinders with

through-wall cracks (Eq. (24)) is non-conservative for short cracks,

as discussed in Section 3.2 above. Secondly, a pressure magnifica-

tion factor, Ri a=Ri, is applied to the crack-free cylinder solutionassuming that the pressure applied on the outer cylinder is lower

than that applied on the inner one. However, for short cracks,

Ma3/1 and the second formula in Eq. (31) will be greater than

glnk. Although a limit gln

k

is set in Eq. (31), it can still

potentially over-estimate the limit pressure for short cracks.

Kim et al. [13] proposed a limit pressure solution, based on their

elastic-perfectly plastic FE analyses with the von Mises yield

criterion, and expressed it as

pLsy

2ffiffiffi3

p tRm

1 A1

a

tA2a

t

2!

2ffiffiffi3

p 2k 1k 1

1 A1

a

tA2a

t

2!(33)

where

&A1 0:0462 0:0589 rm 0:013 r2mA2

0:0395

0:3413 rm

0:0652 r2m

(34)

and rm is defined by Eq. (16).

Eqs. (33) and (34) are based on FE data for

t=Rm 0:2;0:1;0:05and0:025, with 50% internal pressure appliedon the crack faces, and, therefore, are valid forthin-walled cylinders

with crack face pressure. Note that Eq. (33) is inconsistent with Eq.

(22) when a=t 1.Fig. 10 compares the normalised limit pressures predicted

using Eqs. (27), (29) and (31) (g 2=ffiffiffi

3p

) with FE results based

on the von Mises yield criterion due to Staat and Vu [12] for cases

of k 2 without crack face pressure. From the figure, Eqs. (27)and (29) due to Ewing and Carter, respectively, are conservative

for 0 a=t 1 and 0:2 a=c 1 probably because they arebased on the Tresca yield criterion. The predictions using Eq. (31)

due to Staat and Vu are very close to the FE results but are non-

conservative for short and shallow cracks and through-wall

cracks.

Figs. 1113 compare the normalised limit pressures predicted

using Eqs. (29), (31) (g 2=ffiffiffi

3p

) and Eq. (33) with FE results based

on the von Mises yield criterion due to Staat and Vu [12] (Fig. 11 for

k 2), and due to Kim et al. [13] (Fig. 12 for k 1.05 and Fig. 13 fork 1.22) for cases with crack face pressure. From Fig. 11 for k 2,predictions using Eq. (33) due to Kim et al. are very close to the FE

results for shallow cracks but are non-conservative for deep cracks.

Carters solution (Eq. (29)) is conservative for long and shallow

cracks but over-estimates the FE results for short and deep cracks. It

is also seen from the figure that the predictions using Eq. (31) dueto Staat and Vuare reasonably close tothe FEresultsbut are slightly

k = 1.22

k = 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 8 10

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

FE, Kim et al.

Prediction, Kim et al. (eqn. (22))

Prediction, present work (eqn. (56)), ( 3)= 2

0 2 8 10 12 14 16

FE, Staat & Vu



4 6

4 6

a

b

Fig. 7. Comparison of normalised limit pressures between various solutions and FE

results due to Staat and Vu [12] and Kim et al. [13] for cylinders with through-wall

cracks under internal pressure (with crack face pressure).

Internal surface crack

External surface crack

2c

at

RiRo Rm

p

2c

at

Ri

RoRm

p

a

b

Fig. 8. Geometry and dimensions of axial surface cracks in cylinders under internalpressure.

Y. Lei / International Journal of Pressure Vessels and Piping 85 (2008) 825850832


9/26

non-conservative for shallow cracks and some through-wall cracks.

For thin-walled cylinders (Figs.12 and 13), the solution due to Kim

et al. (Eq. (33)) gives accurate predictions of the FE results up to

a=t 0:8 but significantly over-estimates the limit pressure forthrough-wall cracks. This is not surprising because Eq. (33) was

fitted to the FE data presented in Figs. 12 and 13. From the figures,

Carters solution is conservative for all crack lengths and depths

considered by comparison with the FE data. The solution due to

Staat and Vu is reasonably close to the FE results and conservative,

but it slightly over-estimates the limit pressures for through-wall

cracks.

3.3.2. External cracks

The geometry and dimensions of a cylinder with an external

surface crack, a t, under internal pressure are shown in Fig. 8(b)and the simplified model is shown in Fig. 9(b).

Carters solution [9] for a cylinder with an external crack can be

expressed as

pLsy

aRo a

1

Max1 ln

Ro a

Ri

a

tk 1

k atk 1

1

Max1 ln

k a

tk 1

(35)

where

Max1

1 1:61 c2

Ro aa0:5

1 1:61a

tk 1a

c

2k atk 1

0:5(36)

Eq. (35) was constructed using the limit pressure solution fora cylinder with a through-wall crack (Eq. (19)), which is for thin-

walled cylinders, and the limit pressure solution for crack-free

thick-walled cylinders. This mis-match may also cause problems

when Eq. (35) is used for thick-walled cylinders.

The limit pressure solutions based on both the von Mises and

Tresca yield criteria for external surface cracks due to Staat and Vu

[12] can be expressed as follows

pLsy

g

1

Max2ln

Ro

Ro a

ln

Ro aRi

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RoRi

RoRi

at

t

Ri

1

2

at

2 tRi

2sRo

Ri 1

2

a

t

t

Ri

35

gh 1Max2ln k

k atk 1

ln k atk 1!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k

k atk 1

1

2

at

2k 12

r k 1

2

a

tk 1

#

(37)

where

Max2

1 1:25 c2

Roa

0:5

1 1:25k 1a

t

ka

c

20:5

(38)

Eq. (37) may over-estimate the limit pressure for short cracks

because the second term in the right-hand side of Eq. (37) does not

depend on crack length.Figs. 14 and 15 compare normalised limit pressures predicted

using Eqs. (27), (35) and (37) (g 2=ffiffiffi

3p

) with FE results

based on the von Mises yield criterion due to Staat and Vu

[12] for cases of k 2 and due to Zarrabi et al. [16] for k 1.57.From the figures, Eq. (27) due to Ewing is conservative for all

crack lengths and depths considered. Carters solution (Eq. (35))

is also conservative, except for very short and deep cracks (see

Fig. 15(e)). It is also seen from Figs. 14 and 15 that the predictions

using Eq. (37) due to Staat and Vu are very close to the FE results

but slightly non-conservative for very short and through-wall

cracks.

3.4. Local solutions for axial surface defects under internal pressure

The limit pressure expression for a cylinder with a surface

crack under internal pressure given by Kiefner et al. [3] may be

expressed as

pLsy

tRm

1 at

1 at

1

Mteq

2k 1k 1

1 at

1 at

1

Mteq

(39)

where the factor Mteq should be evaluated using Eqs. (15) and (16).

The half crack length, c, in Eq. (16) should be replaced by the

equivalent half crack length, ceq, defined by

ceq A

df2a (40)

Internal crack

External crack

2c

at

Ri

Ro

Cylinder A

Cylinder B

p

2c

at

RiRo

Cylinder A

Cylinder B

p

a

b

Fig. 9. Mechanics models for determining the global limit pressures for cylinders with

surface cracks.



10/26

where Adf is the crack area and Adf 2ac for rectangular cracks.Eq. (39) is an empirical formula obtained from burst experiments

on thin-walled pipes with internal or external defects [3]. It is,

therefore, a solution for thin-walled cylinders with internal/

external cracks. Note that the defective pipes used in the experi-

ments were sealed from the inside of the pipes for the case of

internal defects. Hence, Eq. (39) applies to cases without crack face

pressure.

3.4.1. Internal cracks

Carter [9] defined the local limit pressure for a cylinder with

an internal surface crack under internal pressure as follows.

Firstly, the global limit pressure for a cylinder with an internalsurface crack under internal pressure (Eq. (29)) is alternatively

expressed as the average of the limit pressures of two crack-

free cylinders of length D and a cylinder of length 2c with

an extended internal surface crack of depth a (see Fig. 16(a)),

that is

pLsy

1D c

D

pLfor crack-free cylindersy

cpLfor cylinder with extended cracksy

!(41)

where D is an equivalent length of the crack-free cylinder,

which can be determined by equating Eq. (41) to Eq. (29). The

local limit pressure is then defined in a similar way to Eq. (41)

with a reduced equivalent length of the crack-free cylinder,c1 D, as

a/c = 0.2 a/c = 0.4

a/c = 0.6

FE, Staat & Vu

Prediction, Ewing (eqn. 27))Prediction, Carter (eqn. (29))

Prediction, Staat & Vu (eqn. (31)), (

Prediction, present work (eqn.(62)), ( 3)2=

3)2=

FE, Staat & VuPrediction, Ewing (eqn. (27))

Prediction, Carter (eqn. (29))



3)= 2

FE, Staat & Vu

Prediction, Ewing (eqn. (27))Prediction, Carter (eqn. (29))


Prediction, present work (eqn. (62)), ( 3)2=

3)2=

a/c = 0.8

a/c = 1

FE, Staat & VuPrediction, Ewing (eqn. (27))




3)= 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8

a/t

pL/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p

L/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

FE, Staat & Vu

Prediction, Ewing (eqn. (27))




3)= 2

1

0 0.2 0.4 0.6 0.8

a/t

10 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

10 0.2

a b

c d

e

0.4 0.6 0.8

a/t

1

Fig. 10. Comparison of normalised limit pressures between various solutions and FE results due to Staat and Vu [12] for cylinders with internal surface cracks under internal

pressure (k2, without crack face pressure).



11/26

FE, Staat & Vu





3)= 2

FE, Staat & Vu





3)= 2

FE, Staat & Vu



Prediction, Staat & Vu (eqn. 31)), (


3)= 2

FE, Staat & Vu




Prediction,present work (eqn. (62)), ( 3)= 2

3)= 2

FE, Staat & Vu





3)= 2

a/c = 0.2 a/c = 0.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2a b

c d

e

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

a/c = 0.6 a/c = 0.8

0 0.2 0.4 0.6 0.8

a/t

1

a/c = 1

0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1

Fig. 11. Comparison of normalised limit pressures between various solutions and FE results due to Staat and Vu [12] for cylinders with internal surface cracks under internal

pressure (k2, with crack face pressure).



12/26

t/c = 0.894 t/c = 0.447

t/c = 0.224

FE, Kim et al.

a b

c d





3)2=

FE, Kim et al .



Prediction, Staat & Vu (eqn (31)), (

Prediction, present work(eqn. (62)), ( )32=

)32=

FE, Kim et al.





3)2=

t/c = 0.149

FE, Kim et al .





3)2=

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

Fig.13. Comparison of normalised limit pressures between various solutions and FE results due to Kim et al. [13] for cylinders with internal surface cracks under internal pressure(k1.22, with crack face pressure).

t/c = 0.447 t/c = 0.224

t/c = 0.112

FE, Kim et al .




Prediction, present work(eqn. (62)), ( 3)= 2

3)= 2

FE, Kim et al.



Prediction, Staat & Vu (eqn (31)), (


3)= 2

FE, Kim et al.Prediction, Carter (eqn. (29))




3)2=

t/c = 0.075

FE, Kim et al.Prediction, Carter (eqn. (29))




3)2=

0.0

0.2

0.4

0.6

0.8

1.0

1.2a b

c d

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1

Fig.12. Comparison of normalised limit pressures between various solutions and FE results due to Kim et al. [13] for cylinders with internal surface cracks under internal pressure

(k1.05, with crack face pressure).



13/26

a/c = 0.2 a/c = 0.4

a/c = 0.6

FE, Staat & Vu



Prediction, Staat & Vu (eqn. (37)),

Prediction, present work (eqn. (65)), ( )32=

( )32=

FE, Staat & Vu





( )32=

FE, Staat & Vu





( )32=

FE, Staat & Vu





( )32=

FE, Staat & Vu





( )32=

a/c = 0.8

a/c = 1

0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

a b

c d

e

Fig. 14. Comparison of normalised limit pressures between various solutions and FE results due to Staat and Vu [12] for cylinders with external surface cracks under internal

pressure (k2).



14/26

FE, Zarrabi et al.





)32=

a/t = 0.9

0.0

0.2

0.4

0.6

0.8

1.0

1.2e

dc

ba

0 2 3 6

a/c

pL

/p0

4 51

a/t = 0.7

0 2 3 6a/c

4 51

a/t = 0.5

0 2 3a/c

41

a/t = 0.3

0 1 3

a/c

2

a/t = 0.1

0 2 3

a/c

1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

FE, Zarrabi et al.





)32=

FE, Zarrabi et al.





)32=

FE, Zarrabi et al.





)32=

FE, Zarrabi et al.




Prediction, Staat & Vu (eqn. (37)), ( )32=

Fig. 15. Comparison of normalised limit pressures between various solutions and FE results due to Zarrabi et al. [16] for cylinders with external surface cracks under internal

pressure (k1.57).



15/26

pLsy

1c1 c

"c1ln

RoRi

cRi

R*1ln

Ro

Ri a#

1c1c

1

"c1c

ln k RiR*1

ln

k

1 atk 1

#(42)

where

c1c

a

1 at

Ma2Ri

lnRo

Ri

Ri

R*1ln

Ro

Ri a#

a

k 1a

t

1 a

t

Ma2

ln k Ri

R*1ln

k

1 atk 1

# a

tk 1

(43)

and

c1

D1

a

t (44)Staat and Vu [12] defined their local limit pressures based on

both the von Mises and Tresca yield criteria, using the methodology

employed by Carter [9] but their own limit pressure solutions for

a cylinder with a through-wall crack (Eq. (24)) and a cylinder with

an extended crack (Eq. (9)), as

pLsy

glnRo

Ri

gln k for pI

gsy! ln k

g

s1 c

"s1ln

RoRi

cRi a

R*2ln

Ro

Ri a#

gs1c

1

"s1c

ln k RiR*2

1 atk 1

ln k

1 atk 1

#for

pIgsy

< ln k

45

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:where

and

pIgsy

RiR*2

1

Ma3ln

Ri a

Ri

Ri a

Riln

Ro

Ri a

RiR*2

1

Ma3ln

1 atk 1

1 atk 1

ln

k

1 atk 1

(47)

The twolocal limit pressure solutionsare nowcompared withthe

FE results. There is only one set of well documented FE results for

local limitpressure available, whichis theresultsdue toJun etal. [15]

based on the von Mises yield criterion and the crack ligament

yielding. Eqs. (42) and (45) (g 2=ffiffiffi

3p

) due to Carter [9] and Staat

andVu [12], respectively, are comparedwith the FE results dueto Jun

et al. [15] in Figs. 1719 for k 1.05, 1.11 and 1.22, respectively, forinternal surface cracks with crack face pressure. Eq. (39) is also

plotted in the figures for comparison, though it is for cases without

crack face pressure. From Figs. 1719, the predictions using Carters

solution (Eq. (42)) are reasonably close to the FE results and

conservative for all the three k values except for shallow cracks in

a very thin cylinder (see Fig. 17(a) and (b)). It is also seen from the

figures that the solution due to Staat and Vu [12] for g 2=ffiffiffi

3p

is

non-conservative for short and shallow cracks, especially for the

cylinder with a very thin wall (Figs. 17 and 18). The formula due to

Kiefner et al. (Eq. (39)) shows very good and conservative predic-

tions for k 1.11 (Fig.18) and 1.22 (Fig.19). However, it may be non-conservative for short and shallow cracks for k 1.05 (see Fig. 17).

3.4.2. External cracks

Similar to the cases of internal cracks, Carters local limit pres-

sure solution [9] for an external surface crack (see Fig. 16(b)) underinternal pressure is defined as follows

s1c

1 a

t

RiR*2

ln

Ri a

Ri

Ma3ln RoRi

RiR*2

1

Ma3ln

Ri aRi

Ri aRi

ln Ro

Ri

a#

1 a

t

RiR*2

ln

1 atk 1

Ma3

ln k Ri

R*2

1

Ma3ln

1 atk 1

1 atk 1

ln

k

1 atk 1

#(46)

Internal crack, Dt

ac1 = 1

External crack, c2 Dt

a=

1

D D

2cc1 c1

Crack

a

t

Ri

Ro

p

D D

2cc2 c2

Crack

a

t

Ri

Ro

p

a

b

Fig. 16. Alternative partitions to define global and local limit pressures for cylinders

with surface cracks under internal pressure.



16/26

pLsy

1

c2 c

c2lnRo

Ri cln Ro aRi !

1c2c

1

hc2c

ln k ln

k atk 1

i(48)

where

c2c

a

1 at

Max1Ro aln

RoRo a

a

k 1a

t

1 a

t

Max1k

a

tk

1

ln k

k a

tk 1

a

tk

1

(49)

The local limit pressure for external crack due to Staat and Vu

[12] can be expressed as

pLsy

gs2 c

s2ln

RoRi

cln

Ro a

Ri

!

gs2c

1

hs2c

ln k ln

k atk 1

i(50)

where

s2c

1 a

tMax2 1

(51)

No relevant FE results have been found for local limit pressuresof cylinders with external surface cracks.

a/c = 0.33 a/c = 0.167

a/c = 0.083 a/c = 0.05

a/c = 0.033

FE, Jun et al.

Prediction, Kiefner et al. (eqn. (39))


Prediction, Staat (eqn. (45)), (

Prediction, present work (eqn. (67))

3)2=

0.0

0.2

0.4

0.6

0.8

1.0

1.2a b

c d

e

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

10.0

0.2

0.4

0.6

0.8

1.0

1.2

pL/p0

FE, Jun et al.Prediction, Kiefner et al. (eqn. (39))



FE, Jun et al.Prediction, Kiefner et al. (eqn. (39))



FE, Jun et al.

Prediction, Kiefner et al. (eqn. (39))Prediction, Carter (eqn. (42))


FE, Jun et al.




Prediction, Staat & Vu (eqn. (45)), ( )32= Prediction, Staat & Vu (eqn. (45)), ( )32=

Prediction, Staat & Vu (eqn. (45)), ( )32=Prediction, Staat & Vu (eqn. (45)), ( )32=

Fig. 17. Comparison of normalised local limit pressures between various solutions and FE results due to Jun et al. [15] for cylinders with internal surface cracks under internal

pressure (k1.05, with crack face pressure).



17/26

3.5. Limit load solutions for axial cracks in cylinders subjected tocombined membrane and through-wall bending stresses

Limit load solutions for axially cracked cylinders subjected to

combined membrane and through-wall bending stresses

(Fig. 20) are generally obtained from solutions for cracked plates

under combined tension and bending [1,9,20,21], ignoring the

effect of curvature. In R6 [1], the limit load solution for a thin-

walled cylinder with an internal axial surface crack under

combined membrane and through-wall bending stresses is

a local solution based on the plate solution due to Goodall and

Webster [22] and Lei [23,24]. Actually, this solution can be

extended to thick-walled cylinders with internal/external

surface cracks as long as the bending stress tends to open the

crack because the plate solution [2224] was derived for anythickness of the plate.

3.6. Limit load solutions for axially cracked cylindersunder combined loading

A limit load solution for thin-walled cylinders with axial surface

cracks under combined internal pressure, axial tension and global

bending was proposed by Desquines et al. [25], followed Kitching

et al. [4]. However, the limit pressures predicted using this solution

are much lower than those predicted using the solution due to

Kiefner et al. [3] for the limiting case of a cylinder with a through-

wall crack under internal pressure alone.

Kim et al. [14] performed an FE analysis for a cylinder oft=Rm 0:05 with a surface crack of a=t 0:2 and a=c 0:0224 undercombined internal pressure and global bending and concluded

that a bending load has only a slight effect on the limit pressure for

axial cracks. This might not be true for short cracks where thelimit pressure of the cylinder approaches the limit pressure of the

a/c = 0.33 a/c = 0.167

a/c = 0.083 a/c = 0.05

a/c = 0.033

FE, Jun et al.




Prediction, present work (eqn. 67)

( )32=

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL/

p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

pL

/p0

0 0.2 0.4 0.6 0.8

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8

a/t

a/t

1 0 0.2 0.4 0.6 0.8

a/t

1

0 0.2 0.4 0.6 0.8 1

FE, Jun et al.





( )32=

FE, Jun et al.





( )32=

FE, Jun et al.




( )32=

FE, Jun et al.





( )32=

a b

c d

e





18/26

crack-free cylinder when internal pressure only is applied. Furtherinvestigation is necessary for the limit load of axially cracked

cylinders under combined loading.

4. New limit load solutions for cylinders with

axial cracks under internal pressure

The results of the review of the limit loads for axially cracked cylin-

ders under internal pressurein Section 3 can be summarised as follows.

(1) For extended internal/external surface cracks, solutions due to

Staat and Staat and Vu (Eqs. (9) and (13)) are for thick-walled

cylinders and give good predictions of the available FE results.

(2) For through-wall cracks, the solution due to Staat and Vu (Eq.

(24)) is for thick-walled cylinders and gives good predictions ofavailable FE results for both thin-walled and thick-walled

cylinders. However, Eq. (24) is non-conservative for short andshallow cracks because the back-wall correction in the equa-

tion is incorrect and the stress magnification factor, Mt4, needs

to be re-calibrated.

(3) For the global limit pressure of internal surface cracks, the limit

pressure solution due to Staat and Vu (Eq. (31)) is for thick-

walled cylinders and gives good predictions for available FE

results for both thin-walled and thick-walled cylinders.

However, it over-estimates the FE results for short and shallow

cracksdue to the problem in the solutionfor through-wall cracks

described in (2) and the pressure magnifying factor, Ri a=Ri,applied to the term corresponding to the crack-free cylinder.

(4) For the global limit pressureof external surface cracks, the limit

pressure solution due to Staat and Vu (Eq. (37)) is for thick-

walled cylinders and gives good predictions for available FEresults for thick-walled cylinders. However, it over-estimates

a/c = 0.33 a/c = 0.167

a/c = 0.083

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

a/t

pL/p0

pL/p0

FE, Jun et al.




( =2

a/t

FE, Jun et al.




( )32=

0.0

0.2

0.4

0.6

0.8

1.0

1.2

a/t

pL/p0

FE, Jun et al.Prediction, Kiefner et al. (eqn. (39))Prediction, Carter (eqn. (42))

a/c = 0.05

a/c = 0.033

a/t

pL/p0

FE, Jun et al.Prediction, Kiefner et al. (eqn. (39))Prediction, Carter (eqn. (42))Prediction, Staat & Vu (eqn. (45)),Prediction, present work (eqn. (67))

( = 2

a/t

pL/p

0

FE, Jun et al.Prediction, Kiefner et al. (eqn. (39))Prediction, Carter (eqn. (42))Prediction, Staat & Vu (eqn. (45)), ( =2Prediction, present work (eqn. (67))

3)

3)

3)

a b

c d

e

Prediction, Staat & Vu (eqn. (45)),Prediction, present work (eqn. (67))

( = 2 3)





19/26

the FE results for short and shallow cracks due to the problem

in the solution for through-wall cracks described in (2).

(5) For the local limit pressures for internal/external surface

cracks, Carters solutions (Eqs. (42) and (48)) are for thick-

walled cylinders and give reasonably good and conservative

predictions of FE results for thin-walled cylinders. However,

the expressions for the local limit pressure are based on the

relevant global solutions. Therefore, they need to be re-derived

to maintain consistency with the global solutions.

New limit pressure solutions for axially cracked thick-walled

cylinders under internal pressure are derived in this section. They

can also be used for thin-walled cylinders.

4.1. Through-wall cracks under internal pressure

New limit load solutionsbased on both the von Mises and Tresca

yield criteria for a thick-walled cylinder with a through-wall crack

under internal pressure are obtained by summing the pressure

corresponding to the front-wall failure, p0=Mtn, and the back-wall

correction,DpL (see Eq. (C1) in Appendix C). From Eq. (C1), the limit

pressure without considering the crack face pressure can be

expressed as

pLsy

p0Mtnsy

DpLsy (52)

where Mtn is the stressmagnification factorand is defined using the

outer radius of the cylinder, with the coefficient being re-calibrated

using the FE data for k 2 (see Appendix C), that is,

Mtn

1 1:4 r2o0:5 1 1:4 c2

Rot

0:5

1 1:4k 1

ktc

20:5

fork 2 (53)

The crack face pressure can be considered, following Staatand Vu [12], by applying a factor Ri=R

*t for the pressure corre-

sponding to the front-wall failure and Eq. (52) can be further

expressed as

pLsy

RiR*t

p0Mtnsy

DpLsy

(54)

where R*t is defined in Eq. (B6) (see Appendix B) and the second

term in the right-hand side of Eq. (54) is given by Eq. (B7) in

Appendix B. Note that Eq. (54) leads to the limit pressure for

a defect-free cylinder Ri=R*t p0 < p0 when c/0 because the factorRi=R

*t does not change with crack length, c, noting that the second

term in the right-hand side of Eq. (54) tends to zero and Mtn/1. In

order to avoid this, the R

*

t in Eq. (54) may be replaced by R

*

tn, whichis defined as

Internal crack

External crack

m b

2c

at

Ri

Ro

2c

at

RiRo

a

b

bm

Fig. 20. Geometry and dimensions of axial surface cracks in thick-walled cylinders subjected to membrane stress and through-wall bending.



20/26

R*tn Ri without crack face pressure

Ri t

2forc! t

Ri c

2forc< t

with crack face pressure

8>:

8>>>>>:

(55)

It is seen from Eq. (55) that for long cracks (c! t) R*tn R*t andfor short cracks (c< t) it is a linear interpolation between Ri t=2and Ri. This allows the effect of the crack face pressure factor to

vanish when the crack length tends to zero and the limit pressure of

the crack-free cylinder to be accurately reproduced. Here, choosing

c< t as short cracks is for consistency with the cases of surface

cracks with c< a and is somewhat arbitrary. Using R*tn, the limit

pressure for a thick-walled cylinder with a through-wall crack

under internal pressure can be expressed as

where fpt is the crack face pressure factor and can be expressed as,

from Eq. (55),

fpt RiR*tn

1 without crack face pressure

Ri

Rit

2

11 1

2k 1

fortc

1

Ri

Ric

2

t

ct

c 1

2k 1

fort

c> 1


8>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>>>:

(57)

The new solution, Eq. (56) (g 2=ffiffiffi

3p

), is compared with

other existing solutions and the FE data due to Staat and Vu [12]

and Kim et al. [13] in Figs. 6 and 7. From Figs. 6 and 7, Eq. (56)

provides the best predictions of the FE results compared with

all other solutions. It is slightly conservative compared with the

FE data for cases without crack face pressure (Fig. 6) and accu-

rate or slightly non-conservative for cases with crack pressure(Fig. 7).

4.2. Surface cracks under internal pressure

4.2.1. Internal cracks (global)

New limit load solutions based on both the von Mises and

Tresca yield criteria for a thick-walled cylinder with an internal

surface crack under internal pressure are obtained by summing

the limit pressure corresponding to the cylinder of inner radius

Ri and thickness a with a through-wall crack of length 2c

(Cylinder A in Fig. 9(a)) and that for the crack-free cylinder of

inner radius Ri a and thickness t a (Cylinder B in Fig. 9(a)),that is

pLsy

pLCylinder Asy

Fpt RiR*2n

pLCylinder Bsy

(58)

where Ri=R*

2n is the crack face pressure factor defined for Cylinder

A. The equivalent radius R*2n is the R*tn for Cylinder A and can be

obtained by applying Eq. (55) to Cylinder A, that is

R*2n Ri without crack face pressure

Ri a

2forc! a

Ri c

2forc< a


8>:

8>>>>>:

(59)

In Eq. (58), Fpt is the pressure transfer factor and is defined as

Fpt 1 aRi

1 1

Man

(60)

where Man is the stress magnification factor for Cylinder A and can

be obtained by applying Eq. (53) to Cylinder A, that is

Man

11:4 c2

Ri aa0:5

11:4a

tk1a

c

21atk1

0:5for

1atk1

2 (61)

The pressure transfer factor, Fpt, is applied totheterm inEq. (58)

representing the limit pressure of the crack-free cylinder (Cylinder

B in Fig. 9(a)) to capture the behaviour of pressuretransferring from

the inner surface of Cylinder A to the inner surface of Cylinder B

(Fig. 9(a)). For an extreme case c/N and hence Man/N, i.e. an

extended penetrating crack in Cylinder A in Fig. 9(a), Fpt tends to

Ri a=Ri 1a=Ri because Cylinder A in Fig. 9(a) is almostelastic and the pressure transfer is based on radial force equilib-

rium. Another extreme case is c/0 and hence Man/1. In this case,

Fpt tends to1 because the fullyyielded Cylinder A in Fig. 9(a) cannot

bear any more pressure difference and the pressure is transferred

constantly from the inner surface of Cylinder A (Fig. 9(a)) to the

inner surface of Cylinder B (Fig. 9(a)). For all other cases between

these two limits, the factor is estimated using linear interpolation

based on 1=Man.

Determining the limit pressure of Cylinder A in Fig. 9(a) by

applying Eq. (56) to a cylinder of inner radius Ri and outer radius

Ri a with a through-wall crack of length 2cand the limit pressurefor the defect-free cylinder of inner radius Ri a and outer radiusRo (Cylinder B in Fig. 9(a)), the limit pressure of a thick-walled

cylinder with an internal surface crack can be obtained from Eq.(58) and expressed as

pL

sy

Ri

R*

tn

g

Mtn

ln Ro

Ri 24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1

21

1

Mtn

t

R*

tn

2

1

41

1

M2tn

t

R*

tn

2vuut

1

1

21

1

Mtn

t

R*

tn35

fptgMtnln k 24ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 121 1

Mtnk 1fpt

21

4

1 1

M2tn

k 1fpt

2vuut 1 12

1 1

Mtn

k 1fpt

35(56)



21/26

where fps is the crack face pressure factor and is defined, using

Eq. (59), as

fps

Ri

R*

2n

1 withoutcrackfacepressureRi

Ri a

2

111

2

a

tk1

fora

c1

Ri

Ri c

2

a

ca

c1

2

a

tk1

fora

c>1

with crack facepressure

8>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>>>:

(63)

Note that Eq. (62) is valid for pL=gsy 1 because the pressuretransfer factor for long cracks is defined based on the assumption of

an elastic Cylinder A and yielding may take place in Cylinder A even

for the case of an extended surface crack when pL>gsy. This

condition is always satisfied for cylinders of k 2:718 with anycrack size.

Eq. (62) reduces to Eq. (56) for through-wall cracks, when

a=t/1, and to Eq. (9) for internal extended cracks when a=c/0

and a=t> 0. It also reproduces the limit pressure for crack-free

thick-walled cylinders when a=t 0 or a=c/N.The new solution, Eq. (62) with g 2=

ffiffiffi3

p, is compared with

other existing solutions and the FE data due to Staat and Vu [12] in

Figs.10 and 11 and those due to Kim et al. [13] in Figs.12 and 13. For

cases without crack face pressure (Fig. 10), Eq. (62) has largely

removed the non-conservatism of the solution due to Staat and Vu

[12] for short cracks. From the figure, the predictions using Eq. (62)

are close to the FE results and conservative. For cases with crack

face pressure (Figs. 1113), Eq. (62) has also improved the non--

conservatism of the solution of Staat and Vu [12] for short

and shallow cracks for thick-walled cylinders (Fig. 11) and gives

reasonably good and conservative predictions for both thick-walled

(Fig. 11) and thin-walled (Figs. 12 and 13) cylinders.

4.2.2. External cracks (global)

New limit load solutions based on both the von Mises and

Tresca yield criteria for a thick-walled cylinder with an external

surface crack under internal pressure are obtained by directly

summing the limit pressure corresponding to the cylinder of inner

radius Ro a and outer radius Ro with a through-wall crack oflength 2c (Cylinder A in Fig. 9(b)) and that for the crack-free

cylinder of inner radius Ri and outer radius Ro a (Cylinder B inFig. 9(b)), that is

pLsy

pLCylinder Asy

pLCylinder Bsy

(64)

In Eq. (64), a simple addition for the limit pressures for the two

cylinders is used because Cylinder B in Fig. 9(b) is defect-free and

the pressure transfer factor from the inner surface of Cylinder B

(Fig. 9(b)) at Ri tothe inner surface of Cylinder A (Fig. 9(b)) at Ro ais unity (see Section 4.2.1 above).

Determining the limit pressure of Cylinder A of Fig. 9(b) by

applying Eq. (56) to a cylinder of inner radius Ro a and outerradius Ro with a through-wall crack of length 2c and the limit

pressure for the defect-free cylinder of inner radius Ri and outer

radius Ro a (Cylinder B in Fig. 9(b)), the limit pressure of a thick-walled cylinder with an external surface crack can be obtained from

Eq. (64) and expressed as

pLsy

8>:

gMaxn

ln

RoRoa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RoRi

121 1Maxnat tRi214

1 1

M2axn

at

tRi

2s RoRi

12

1 1

Maxn

a

t

t

Ri

359>=>;

gln

RoaRi

gh

1Maxn

ln

kkatk1

ln k atk 1i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k 121 1Maxnatk 1214

1 1M2axn

atk 1

2

s

k 12

1 1

Maxn

a

tk 1

3

5

(65)

pLsy

8>:

RiR*2n

g

Manln

Ri a

Ri

24ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 121 1

Mana

t

t

R*2n

21

4

1 1

M2an

a

t

t

R*2n

2vuut

1 12

1 1

Man

a

t

t

R*2n

35

9>=>;

1

a

Ri1

1

Man

Ri

R*

2n

gln Ro

Ri a gfps

1

Manln

1 atk 1

1 atk 1

1 1

Man

ln

k

1 atk 1!

24ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 121 1

Mana

tfpsk 1

21

4

1 1

M2an

atfpsk 1

2vuut 1 12

1 1

Man

a

tfpsk 1

35 for k 2:718

(62)



22/26

where the stress magnification factor, Maxn, can be obtained by

applying Eq. (53) to Cylinder A in Fig. 9(b) and expressed as

Maxn

1 1:4 c2

Roa

0:5

1 1:4k 1a

t

ka

c

2

0:5

fork

k a

tk 1 2 (66)

Eq. (65) reduces to Eq. (56) for through-wall cracks when a=t/1

and to Eq. (13) for external extended cracks when a=c/0 and

a=t> 0. It also reproduces the limit pressure for crack-free thick-

walled cylinders when a=t 0 or a=c/N.The new solution, Eq. (65) with g 2=

ffiffiffi3

p, is compared with

other existing solutions and the FE data due to Staat and Vu [12] in

Fig. 14 and those due to Zarrabi [16] in Fig. 15. From the figures, Eq.

(65) has largely removed the non-conservatism of the solution due

to Staat and Vu [12] for deep and short cracks. It is also seen from

the figures that the predictions using Eq. (65) are very close to the

FE data and conservative for all cases shown in Figs. 14 and 15

except for the cases with very shallow cracks, where the FE results

are slightly over-estimated by Eq. (65).

4.2.3. Internal cracks (local)

A new local limit pressure solution for a thick-walled cylinder

with an internal surface crack under internal pressure is obtained

from the methodology used by Carter [9] (see Section 3.4.1

above) based on the new limit load solutions for thick-walled

cylinders with internal surface cracks (Eq. (62)) and the limit

load solution for thick-walled cylinders with internal extended

cracks under internal pressure due to Staat and Vu [12] (Eq. (9)).

Following Carter [9], the local limit pressure for a thick-walled

cylinder with an internal surface crack of depth a and length 2c

can be expressed as the weighted sum of the limit pressures of

a cylinder of length 2c with an internal extended crack of depth

a and two crack-free cylinders of length h1

(refer to Fig. 16(a)

with c1 replaced by h1), that is

pLsy

1h1 c

"h1ln

RoRi

cRi

R*2

Ri aRi

ln

Ro

Ri a#

z

1

h1c

1

h1c

ln k fps

1 atk 1

ln

k

1 atk 1

! (67)

where Ri=R*

2 fps for c! a (see Eq. (63)) and Ri=R*2zfps for c< ahave been adopted. The normalised equivalent length of the crack-

free cylinder, h1=c, can be obtained by following Eqs. (41)(44) but

using Eq. (62) as the global limit pressure for a thick-walled

cylinder with an internal surface crack and Eq. (9) as the limit

pressure for a thick-walled cylinder with an internal extended

crack. The result can be expressed as

Note that the back-wall correction terms in Eqs. (9) and (62)have been omitted as only local ligament yielding is considered.

The g factor is also set to unity because the comparison with

the FE data below shows the solution based on the von Mises

yield criterion may be non-conservative for short and shallow

cracks.

The new solution, Eq. (67), is compared with other existing

solutions and the FE data due to Jun et al. [15] in Figs. 1719 for

k 1.05, 1.11 and 1.22, respectively. From the figures, the limitpressure obtained using Eq. (67) is very close to, but slightly

higher than that predicted using Carters solution. It is also

seen from the figures that the predictions using Eq. (67) are

reasonably close to and conservative compared with the FE

results for all cases shown in Figs. 1719. The conservatism of

Eq. (67) may increase with increase of k, noting the trends

shown in Figs. 1719.

4.2.4. External cracks (local)

A new local limit pressure solution for a thick-walled cylinder

with an external surface crack under internal pressure is obtained

from the methodology used by Carter [9] (see Section 3.4.2 above)

based on the new limit load solutions for thick-walled cylinders

with external surface cracks (Eq. (65)) and the limit load solution

for thick-walled cylinders with external extended cracks underinternal pressure due to Staat and Vu [12] (Eq. (13)). Following

Carter [9], the local limit pressure for a thick-walled cylinder with

an external surface crack of depth a and length 2ccan be expressed

as the weighted sum of the limit pressures of a cylinder of length 2c

with an external extended crack of depth a and two crack-free

cylinders of length h2 (refer to Fig. 16(b) with c2 replaced by h2),

that is

pLsy

1h2 c

h2ln

RoRi

cln

Ro a

Ri

!

1h2c

1

h2c

ln k ln

k atk 1

!(69)

The normalised equivalent length of the crack-free cylinder,h2=c, can be obtained by following Eqs. (48)(50) but using Eq. (65)

as the global limit pressure for a thick-walled cylinder with an

external surface crack and Eq. (13) as the limit pressure for a thick-

walled cylinder with an external extended crack. The result can be

expressed as

h2c

1 a

tMaxn 1 (70)

Note that, again, the back-wall correction terms in Eqs. (13) and

(65) have been omitted as only local ligament yielding is consid-

ered. The g factor is also set to unity because of the same reason

given in Section 4.2.3 for internal cracks.

No relevant FE results have been found for local limit pressuresof cylinders with external surface cracks.

h1c

1 atfps

ln

RiaRi

a

t

t

Riln

Ro

Ri a

Manh

ln

RoRi

fps

1Man

ln

RiaRi

1 aRi

1 1Man

ln

RoRiai

1 at

fps

ln

1 atk 1 atk 1ln k1a

tk1

Man

hln k fps

1

Manln

1 atk 1 1 atk 11 1Man

ln

k1a

tk1i

(68)



23/26

5. Conclusions

1. The limit load solutions for axially cracked cylinders have been

reviewed and compared with available FE results. The findings

are as follows.

(1) For extended internal/external cracks under internal pres-

sure, solutions due to Staat and Vu (Eqs. (9) and (13)) are for

thick-walled cylinders and give the best predictions of the

available FE results.

(2) For through-wall cracks under internal pressure, the solu-

tion due to Staat and Vu (Eq. (24)) is for thick-walled

cylinders and gives the best predictions of available FE

results for both thin-walled and thick-walled cylinders.

However, it is non-conservative for short cracks because

the back-wall correction in the equation is incorrect and

the stress magnification factor needs to be re-calibrated.

(3) For the global limit pressure of internal surface cracks, the

solution due to Staat and Vu (Eq. (31)) is for thick-walled

cylinders and gives the best prediction of available FE results

for both thin-walled and thick-walled cylinders. However, it

over-estimates the FE results for short and shallow cracks due

to the problems in the solution for through-wall cracks

addressed in (2)and thepressure amplifyingfactor, Ri a=Ri,applied to the term corresponding to the crack-free cylinder.

(4) For the global limit pressure of external surface cracks, the

solution due to Staat and Vu (Eq. (37)) is for thick-walled

cylinders and gives the best prediction of available FE results

for thick-walled cylinders. However, it over-estimates the FE

results for short and through-wall cracks due to the problem

in the solution for through-wall cracks addressed in (2).

(5) For the local limit pressures of internal/external surface

cracks, Carters solutions (Eqs. (42) and (48)) are for thick-

walled cylinders and give reasonably good and conserva-

tive predictions of available FE results for thin-walled

cylinders. However, the expressions for the local limit

pressure are based on the corresponding global solutions.

Therefore, they need to be re-derived to maintain consis-tency with the global solutions. The solutions due to Staat

and Vu (Eqs. (45) and (50)) are for thick-walled cylinders.

However, the solution for internal cracks (Eq. (45)) with

g 2=ffiffiffi

3p

is non-conservative for short and shallow

cracks, especially for the cylinder with a very thin wall

compared with the available FE results.

(6) Little information for the effect of other load types, such as

axial tension and global bending moment, on the limit

pressure of a cylinder with an axial crack can be found.

Limit load solutions for axially cracked cylinders under

combined internal pressure, tension and global bending are

currently lacking.

2. New limit pressure solutions for thick-walled cylinders with

axial cracks under internal pressure have been developed toovercome the problems addressed in Conclusion 1, above. The

new solutions are

(1) global solution for through-wall cracks,

(2) global solutions for internal/external surface cracks,

(3) local solutions for internal/external surface cracks.

3. The newly developed limit pressure solutions have been

compared with available FE data and the results show that the

predictions using the new solutions are conservative and agree

well with the FE results.

Acknowledgements

The author wishes to acknowledge Dr. P.J. Budden of BritishEnergy Generation Ltd. for his comments on this paper and Prof.

Manfred Staat of Aachen University of Applied Sciences (Germany)

for providing FE data. This paper is published by permission of

British Energy Generation Ltd.

Appendix A. Folias factor

The Folias factor is a stress magnification factor due to the

curvature of shells and was first reported by Folias [26] to addressthe stress increase in the near crack tip area in a thin-walled

spherical vessel with a fully penetrating crack under internal

pressure. Folias [19] then derived the factor for a thin-walled

cylindrical vessel with a penetrating axial or circumferential crack

under internal pressure, based on elastic thin-shell theory. At that

time, Folias [19] obtained a theoretical solution for the stress

magnification factor for axial cracks only for rm 0:55 andexpressed it as

Mt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 fr2mq

(A1)

with

f

1:61 forrm

0:55 (A2)

where

rm cffiffiffiffiffiffiffiffiffi

Rmtp (A3)

Later, Erdogan and Kibler [27] solved the problem numerically

and obtained the solution for axial cracks for rm 4:4. Theresults are tabulated in Table A1. Folias [28] found that the

numerical results could still be expressed in the form of Eq. (A1),

but the coefficient f 1:05 provided a good fit for the data,that is,

f 1:05 forrm 4:4 (A4)Kiefner et al. [3] found that the limit pressure data from burst

tests of pipes with through-wall defects could be well correlatedusing a Folias factor. In their paper [3], Kiefner et al. fitted the Folias

Table A1

Numerical solution of Folias factor [27,17].

rm Mt

0.110011 1.0096

0.220022 1.0371

0.330033 1.0795

0.440044 1.1344

0.550055 1.1993

0.660066 1.2723

0.770077 1.3519

0.880088 1.4367

0.990099 1.5256

1.10011 1.61771.210121 1.7122

1.320132 1.8085

1.430143 1.906

1.540154 2.0045

1.650165 2.1035

1.787679 2.2276

1.925193 2.3519

2.062706 2.4761

2.20022 2.5999

2.337734 2.7232

2.475248 2.8459

2.750275 3.0895

3.025303 3.3303

3.30033 3.5681

3.575358 3.8029

3.850385 4.0347

4.125413 4.2637

4.40044 4.4895



24/26

factor data shown in Table A1 [27,17] and found the data could be

well represented by the following equation

Mt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 1:255 r2m 0:0135 r4mq

(A5)

Fig. A1 compares the three equations with the numerical data in

Table A1. From Fig. A1, Eqs. (A5) and (A1) with f 1:61 or 1.05 canpredict the numerical data very well in the region r

m 0:55. It is

also seen that Eq. (A1) with f 1:05 is a good representation andEq. (A5) isthe bestfit of the datain the regionrm 4:4. However, Eq.(A1) with f 1:61 is very conservative in the region 1 < rm 4:4.

Several factors should be clarified when using the Folias factor.

Firstly, the Folias factor was derived for elastic material properties.

It was used in the limit load solutions because Kiefner et al. found

that it could correlate their experimental data very well. The author

has not found any theoretical proof for elastic plastic materials.

Secondly, the Folias factor was obtained for thin-walled shells.

There is no solution for thick-walled shells. Finally, the theoretical

solution for the Folias factor is available only for rm 4:4. Specialcare should be made for problems beyond this limitation.

Appendix B. Back-wall effect on the limit pressure of

a cylinder with an axial crack

For a cylinder with an axial defect under internal pressure, the

global limit load of the defective cylinder is the pressure corre-

sponding tothe plastic collapse of both thefront-wall of thecylinder

containing the defect and the defect-freeback-wall. The front-wall is

weaker than the back-wall due to the defect. Denoting the pressure

corresponding to the collapse of the front-wall, pLf, the total global

limit pressure can be expressed as pLf DpL, where DpL is the extrapressure the back-wall can bear after the onset of the front-wall

collapse. For thin-walled cylinders, DpL is negligible. However, it

may become significant for cylinders with very thick walls. In this

Appendix, DpL for through-wall and surfacecracks will be estimated.

The back-wall of a cracked cylinder can be treated as a plate

of thickness tsubjected to combined tension force, NL, and bendingmoment, MLp, due to the internal pressure,pLf DpL. The limit loadof an uncracked plate with a thickness t and unit width under

combined tension and bending can be expressed as [29]

NLsyt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4l2 1

q 2l (B1)

l MLptNL

(B2)

where l is the load ratio.

Cylinder with through-wall cracks

For a cylinder with an axial through-wall crack of length 2 c

subjected to internal pressure, the tensile force, NL, andthe moment,

MLp, in the back-wall due to the internal pressure, pLf DpL, are asfollows (see Fig. B1). The resultant force and moment in the back-

wall can be obtained by taking the force equilibrium along the

direction normal to the crack face and moment equilibrium in the

back-wall, assuming that the back-wall only bears half of the force

due to pLf but the full force due to DpL, and expressed as

NL 2R*tDpL R*tpLf R*t

2DpL pLf

(B3)

MLp

NL R*tpLf

R*t t

2

2R*tDpL

R*t

t

2

(B4)

The load ratio, l, following Eq. (B2), for this geometry is

l

MLp

NLt

2DpL

1 1

2

t

R*t

2DpL pLf

tR*t

(B5)

In Eqs. (B3)(B5), R*t is the equivalent radiusto includethe effect of

the crack face pressure and is defined, for long cracks (c! t), as

R*t Ri without crack face pressure

Ri t

2with crack face pressure

((B6)

The normalised limit pressure increase due to the back-

wall effect, DpL=sy, can be obtained by inserting Eqs. (B3) and

(B5) into Eq. (B1) and solving for DpL=sy. The result can be

expressed as

DpLsy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 121 1

Mtn tR*t

21

4

1 1

M2tn

t

R*t

2vuut

1 12

1 1

Mtn

t

R*t

B7

In Eq. (B7), the following assumption has been adopted

pLfsyz

t

R*t

1

Mtn(B8)

using Eq. (19), replacing Mt2 by Mtn defined by Eq. (53).

0

1

2

3

4

5

6

0 1 2 3 4 5

m

Mt

Folias Factor, data

Kiefner equation (eqn. (A5))

Eqn. (A1) with = 1.61

Eqn. (A1) with = 1.05

Fig. A1. Comparison of Folias factor between numerical data [27,17] and threeequations.

Ri

t

pLpLf +

Front wall with a

through-wall crackBack wall

NL

MLp2R*t

Fig. B1. Back-wall loads for a cylinder with a through-wall crack (R*

t shown for the caseof crack face pressure).



25/26

Cylinder with internal surface crack

For a cylinder with an axial internal surface crack of length 2 c

and depth a subjected to internal pressure, the back-wall effect is

only from the cylinder of inner radius Ri and thickness a with

a through-wall crack of length 2c (Fig. B2). The normalised

pressure increase due to the back-wall effect, DpL=sy, for this case

can be obtained directly from Eq. (B7) by replacing t, R*

tand Mtn

in Eq. (B7) by a, R*2 and Man, respectively. The result can be

expressed as

DpLsy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1

21 1

Mana

t

t

R*2

21

4

1 1

M2an

a

t

t

R*2

2vuut

1 12

1 1

Man

a

t

t

R*2

B9

where R*2 is defined in Eq. (10) and Man is defined in Eq. (61).

Cylinder with external surface cracks

For a cylinder with an axial external surface crack of length 2cand depth a subjected to internal pressure, the back-wall effect is

only from the cylinder of inner radius Ro a and thickness a witha through-wall crack of length 2c(Fig. B3). The normalised pressure

increase due to the back-wall effect, DpL=sy, for this case can be

obtained directly from Eq. (B7) by replacing t, R*t and Mtn in Eq. (B7)

by a, Ro a and Maxn, respectively, and then applying a factorRo a=Ri to the right-hand side of Eq. (B7). The result can beexpressed as

DpLsy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k 121 1

Maxna

t

t

Ri

21

4

1 1

M2axn

a

t

t

Ri

2vuut

k 1

21

1

Maxna

t

t

Ri B10

where Maxn is defined in Eq. (66).

Appendix C. Calibration of the stress magnification factor

for cylinders with through-wall cracks

The limit pressure, pL, for a cylinder with a through-wall crack

subjected to internal pressure may generally be expressed as

pL DpLp0

1Mtn

(C1)

where p0 is the limit pressure for crack-free cylinders, DpL is thepressure increase due to the back-wall effect (see Appendix B) and

Mtn is the stress magnification factor. Staat and Vu have shown that

Mtn can be expressed by the following equation

Mtn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 fr2oq

(C2)

The factor ro is a function of crack length, c, cylinder outer radius,

Ro, and cylinder wall thickness, t, and is expressed by Eq. (26). The

coefficient f may be calibrated from FE or experimental data.

Combining Eqs. (C1) and (C2), the relationship between

p0=pL DpL and ro is as follows

p0

pL

DpL

2

1 fr2o (C3)

This equation may be used to calibrate f. Fig. C1 shows the FE

limit pressure data for cylinders with through-wall cracks under

internal pressure (without crack face pressure) due to Staat and Vu

[12], plotted as p0=pL2 against r2o. From Fig. C1, the data for variousk are widely scattered with increasingro and the coefficient, f, may

depend on k. Moreover, the relationship between p0=pL2 and r2o isnon-linear for big k values. The FE data are then re-plotted in Fig. C2

considering the back-wall effect, DpL. From the figure, the FE data

for all k values considered tend to collapse to one line and can be

represented by a straight line with a slopef 1:4. Note that DpL isa function ofMtn (see Eq. (B7) in Appendix B) and, therefore, f. The

result f 1:4 was obtained by increasingf gradually and checking

Ri

t

pLpLf +

NL

MLp

2R*2

Front wallBack wall

a

Fig. B2. Back-wall loads for a cylinder with an internal surface crack ( R*

2 shown for thecase of crack face pressure).

R0

NL Front wallBack wall

a

t

MLp

pLpLf +

Fig. B3. Back-wall loads for a cylinder with an external surface crack.

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

(p0/pL)2

k = 1.1

k = 1.25

k = 1.5

k = 1.75

k = 2

Fig. C1. FE data [12] plotted in the form of Eq. (C3) for DpL 0.



26/26

the agreement with the FE data to obtain an upper-bound esti-

mation of Mtn for all the FE data.

References

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[2] Miller AG. Review of limit loads of structures containing defects. InternationalJournal of Pressure Vessels and Piping 1988;32:197327.

[3] Kiefner JF, Maxey WA, Eiber RJ, Duffy AR. Failure stress levels of flaws inpressurised cylinders. ASTM STP 536. Philadelphia, USA: American Society forTesting and Materials; 1973. p. 461481.

[4] Kitching R, Davis JK, Gill SS. Limit pressures for cylindrical shells with unre-inforced openings of various shapes. Journal of Mechanical EngineeringScience 1970;12:31330.

[5] Kitching R, Zarrabi K. Limit and burst pressures for cylindrical shells withpart-through slots. International Journal of Pressure Vessels and Piping 1982;10:23570.

[6] Ewing DJF. On the plastic collapse of a thin-walled pressurized pipe with anaxial crack. CEGB Report TPRD/L/2566/N83; 1984.

[7] Chell GG. ADISC: a computer program for assessing defects in spheres andcylinders. CEGB Report TPRD/L/MT0237/M84; 1984.

[8] Chell GG. Elasticplastic fracture mechanics. CEGB Report RD/L/R2007; 1979.[9] Carter AJ. A library of limit loads for FRACTURE.TWO. Nuclear Electric Report

TD/SID/REP/0191. Berkeley; 1991.[10] Staat M. Plastic collapse analysis of longitudinally flawed pipes and vessels.

Nuclear Engineering and Design 2004;234:2543.

[11] Staat M. Local and global collapse pressure of longitudinally flawed pipes andcylindrical vessels. International Journal of Pressure Vessels and Piping2005;82:21725.

[12] Staat M, Vu Duc Khoi. Limit analysis of flaws in pressurized pipes andcylindrical vessels. Part I: axial defects. Engineering Fracture Mechanics2007;74:43150.

[13] Kim Yun-Jae, Shim Do-Jun, Huh Nam-Su, Kim Young-Jin. Plastic limit pressuresfor cracked pipes using finite element limit analyses. International Journal ofPressure Vessels and Piping 2002;79:32130.

[14] Kim Yun-Jae, Shim Do-Jun, Nikbin Kamran, Kim Young-Jin, Hwang Seong-Sik,

Kim Joung-Soo. Finite element based plastic limit loads for cylinders withpart-through surface cracks under combined loading. International Journal ofPressure Vessels and Piping 2003;80:52740.

[15] Jun HK, Choi JB, Kim YJ, Park YW. The plastic collapse solutions based on finiteelement analyses for axial surface cracks in pipelines under internal pressure.ASME PVP 1998;373:5238.

[16] Zarrabi K, Zhang H, Nhim K. Plastic collapse pressure of cylindrical vesselscontaining longitudinal surface cracks. Nuclear Engineering and Design1997;168:3137.

[17] Folias ES. On the effect of initial curvature on cracked flat sheets. InternationalJournal of Fracture Mechanics 1969;5:32746.

[18] Erdogan F. Ductile failure theories for pressurized pipes and containers.International Journal of Pressure Vessels and Piping 1976;4:25383.

[19] Folias ES. An axial crack in a pressurized cylindrical shell. International Journalof Fracture Mechanics 1965;1:10413.

[20] API Recommended Practice 579 Fitness-for-service. 1st ed. American Petro-leum Institute; 2000.

[21] Dillstrom P, Bergman M, Brickstad B, Zang W, Sattari-Far I, Sund G, et al. Acombined deterministic and probabilistic procedure for safety assessment ofcomponents with cracks handbook. DET Norske Veritas; 2004. RSE R&DReport 2004/01, Revision 41.

[22] Goodall IW, Webster GA. Theoretical determination of reference stress forpartially penetrating flaws in plates. International Journal of Pressure Vesselsand Piping 2001;78:68795.

[23] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks inplates under bending. International Journal of Pressure Vessels and Piping2004;81:3141.

[24] Lei Y. A global limit load solution for plates with semi-elliptical surface cracksunder combined tension and bending. ASME PVP 2004;475:12531.

[25] Desquines J, Poette C, Michel B, Wielgosz C, Martelet B. Limit load of an axiallycracked pipe under combined pressure bending and tension. In: Petit J, deFouquet J, Henaff G, Villechaise P, Dragon A, editors. Mechanisms andmechanics of damage and failure, ECF 11 Proceedings; 1996. p. 21692174.

[26] Folias ES. A finite line crack in a pressurized spherical shell. InternationalJournal of Fracture Mechanics 1965;1:2046.

[27] Erdogan F, Kibler JJ. Cylindrical and spherical shells with cracks. InternationalJournal of Fracture Mechanics 1969;5:22937.

[28] Folias ES. On the fracture of nuclear reactor tubes. Paper C4/5, SMiRT III.London; 1975.[29] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks in

plates under combined tension and bending. International Journal of PressureVessels and Piping 2004;81:4356.

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16

k = 1.1

k = 1.25

k = 1.5

k = 1.75

k = 2

= 1.4

1

1.4

(p

0/(pL-

pL)

)2

Fig. C2. FE data shown in Fig. C1 re-plotted in the form of Eq. (C3) with considering the

back-wall correction.


A Review Limite Load Solution

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