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International Journal of Pressure Vessels and Piping 85 (2008) 825–850

Contents lists avai

International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

A review of limit load solutions for cylinders with axial cracksand 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 2008Received in revised form 22 August 2008Accepted 4 September 2008

Keywords:Limit loadAxial crackSurface crackThrough-wall crackThick-walled cylinder

* Tel.: þ44(0)1452652285; fax: þ44(0)1452653025E-mail address: [email protected]

0308-0161/$ – see front matter � 2008 Elsevier Ltd.doi:10.1016/j.ijpvp.2008.09.001

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 internalpressure are developed to overcome problems in the existing solutions. The newly developed limit loadsolutions are a global solution for through-wall cracks, global solutions for internal/external surfacecracks and local solutions for internal/external surface cracks. The newly developed limit pressuresolutions are compared with available FE data and the results show that the predictions agree well withthe FE results and are generally conservative.

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1. Introduction

The limit load of a component containing defects is one of themost important inputs when a structural integrity assessment isperformed using R6 [1]. This is because the limit load gives the loadcarrying capacity for plastic collapse of the defective componentand also defines the J-integral via the reference stress method forelastic–plastic fracture. The cylinder is one of the most commonlyused components in power stations, such as in pipelines andpressure vessels. Circumferential and axial cracks are two commontypes of defects found in girth and longitudinal welds in cylinders.In this paper, limit load solutions for axial defects in cylinders willbe reviewed first and some new solutions will then be proposed.

For a part-through defect, the limit load may be definedaccording to the plastic deformation behaviour of either the overalldefective 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 failureof the whole structure. A local limit load corresponds to a loadinglevel at which gross plasticity occurs in the crack ligament and maybe relevant to ligament failure. The local limit load is always lessthan or equal to the global limit load for a cracked structure and,therefore, can yield a conservative result in an assessment. In thispaper both the global and local limit loads are considered.

.

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Miller [2] summarised the limit load solutions for cylinders withthrough-wall, surface and extended surface axial defects underinternal pressure available before 1987, such as solutions due toKiefner et al. [3] for through-wall and surface defects, Kitching et al.[4,5] for surface and through-wall defects, Ewing [6] for surfacedefects and Chell [7,8] for surface cracks and extended surfacecracks. Further solutions for cylinders with axial defects weredeveloped by Carter [9]. Staat [10,11] modified some of Carter’ssolutions [9] for thin-walled cylinders to extend them to thick-walled cylinders and compared the modified solutions withexperimental data. Recently, Staat and Vu [12] further improved thesolutions due to Staat [10,11]. Kim et al. [13,14] carried out elastic-perfectly plastic finite element (FE) analysis for axially crackedcylinders and proposed some limit load solutions for extendedinternal cracks and surface cracks under internal pressure based onthe FE results. Jun et al. [15] performed 3-D FE analyses for cylinderswith axial surface defects under internal pressure and presentedresults for the local limit pressure. Zarrabi et al. [16] also performed3-D FE analyses for axial cracked cylinders, but did not giveformulae for the limit load solutions.

The layout of this paper is as follows. Section 2 defines thegeometry parameters and material properties. Recent develop-ments in limit load solutions for cylinders containing axial cracksunder internal pressure or combined internal pressure, tension andbending are reviewed in Section 3. New limit load solutions forcylinders with axial through-wall and surface cracks under internalpressure are then derived and validated in Section 4. Conclusionsare presented in Section 5. In Appendix A, the Folias factor is

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Nomenclature

a Crack depthAdf Crack areac Half crack lengthc1, c2 Equivalent lengths of crack-free cylinders for defining

local limit pressureceq Equivalent through-wall half crack length for surface

cracksD Equivalent length of crack-free cylinder for defining

global limit pressurefpt, fps Crack face pressure factor for through-wall and

internal surface cracks, respectivelyFpt Stress transfer factorh1, h2 Equivalent lengths of crack-free cylinders for defining

local limit pressurek Ratio between the outer and inner radii of a cylinder,

k ¼ Ro=RiL Function of geometry for limit load definitionsM Global axial bending momentMa1wMa3 Stress magnification factors due to the curvature of

cylinders (see definitions where they first appear)Man;Maxn Stress magnification factors due to the curvature of

cylinders (see definitions where they first appear)Max1wMax2 Stress magnification factors due to the curvature of

cylinders (see definitions where they first appear)MLp Limit moment applied to the back-wall of a cylinder

due to internal pressureMt Folias factorMteq Folias factor for equivalent through-wall cracksMt1wMt4 Stress magnification factors due to the curvature of

cylinders (see definitions where they first appear)Mtn Stress magnification factor for through-wall cracks due

to the curvature of cylinders (see definition where itfirst appears)

N Axial tension force

NL Limit axial tension forcep Internal pressurep0 Limit pressure for a crack-free cylinderpI Global limit pressure for a cylinder with an internal

surface crack under internal pressurepL Limit pressurepLf Pressure corresponding to the front-wall failure of

a cylinder with a through-wall crackDpL Extra limit pressure due to the back-wall effect in

cylindersRi Inner radius of the cylinderRm Average radius of the cylinder, Rm ¼ ðRo þ RiÞ=2Ro Outer radius of the cylinderR*

1 Equivalent radius considering the crack face pressurefor surface cracks defined by Carter

R�2;R*2n Equivalent radius considering the crack face pressure

for surface cracks (see definitions where they firstappear)

R�t ;R*tn Equivalent radius considering the crack face pressure

for through-wall cracks (see definitions where theyfirst appear)

s1; s2 Equivalent lengths of crack-free cylinders for defininglocal limit pressure

t Cylinder thickness, t ¼ Ro � Rif Coefficient for defining the stress magnification factorg ¼ 2=

ffiffiffi3p

for von Mises yield criterion and¼1 for Trescayield criterion

l Load ratiori; rm; ro r factors defined by inner, mean and outer radii,

respectivelysb Through-wall bending stressðsbÞL Limit through-wall bending stresssm Through-wall membrane stressðsmÞL Limit through-wall membrane stresssy Yield stress.

Y. Lei / International Journal of Pressure Vessels and Piping 85 (2008) 825–850826

discussed to clarify some confusions in its use in the limit loadsolution for axially cracked cylinders. The back-wall effect on thelimit pressure of a cylinder with an axial crack is discussed inAppendix B and the calibration of the stress magnification factor forcylinders with through-wall cracks is detailed in Appendix C.

2. Geometry definition and material properties

For consistency, solutions from different sources are convertedaccording to a uniform notation system in this paper. The dimen-sions of a cylinder are described by its inner radius, Ri, and outerradius, Ro. The mean radius of the cylinder, Rm, can then be simplyexpressed as Rm ¼ ðRo þ RiÞ=2 and the thickness of the cylinder, t,can be expressed as t ¼ Ro � Ri. A cylinder can then be describedby the ratio between its outer and inner radii, k, i.e. k ¼ Ro=Ri, theratio between t and Rm, t=Rm, the ratio between t and Ri, t=Ri, or theratio between t and Ro, t=Ro. The relationships between thesevarious parameters are as follows.

tRi¼ k� 1;

tRm¼ 2

k� 1kþ 1

;t

Ro¼ k� 1

k(1)

For a thin-walled cylinder, k/1. The length and depth of anaxial surface crack in a cylinder are defined by 2c and a, respec-tively. The crack depth, a, is measured from the inner surface of thecylinder for internal cracks and from the outer surface for externalcracks. The surface crack becomes an extended surface crack whenc/N and a through-wall crack when a ¼ t.

The load types considered are internal pressure, p, through-wallmembrane stress, sm, and through-wall bending stress, sb. Theirlimit values are denoted as pL, ðsmÞL and ðsbÞL, 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 withyield stress of sy. Therefore, all limit load solutions for closed-endcylinders subjected to internal pressure may be approximatelyexpressed as

pL ¼ gL�

k;at;ac;.�

sy (2)

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

g ¼1 for Tresca yield condition2ffiffiffi3p for von Mises yield condition

8<: (3)

for pressurised defect-free cylinders.In this paper, the limit pressure is generally normalised by the

limit pressure of the crack-free cylinder based on the von Misesyield criterion, p0, defined by

p0 ¼2ffiffiffi3p syln k (4)

3. Review of the available solutions

In this section, limit load solutions for cylinders containing axialsurface/through-wall cracks under internal pressure, combined

Y. Lei / International Journal of Pressure Vessels and Piping 85 (2008) 825–850 827

membrane stress and through-wall bending, and combined internalpressure, tension and global bending are reviewed and discussed.

3.1. Solutions for extended axial surface cracksunder internal pressure

For an extended surface crack, the crack depth is a and the cracklength c/N (see Fig. 1). In this sub-section, only internal pressureis considered.

3.1.1. Internal defectsThe geometry and dimensions of a cylinder with an internal

extended crack, a, under internal pressure are shown in Fig. 1(a).Chell [8] proposed a limit pressure solution for an internal

extended crack in a thick-walled cylinder without crack face pres-sure, which can be expressed as

pL

sy¼ g

t � aRi þ a

¼ gðk� 1Þ

�1� a

t

�1þ a

tðk� 1Þ

(5)

where g takes the values defined in Eq. (3) for solutions based onthe von Mises and Tresca yield criteria. Note that Eq. (5) does not

Internal crack

External crack

Rm

t x

y

Ri

Ro

a

p

Rm

t x

y

Ri

Ro

a

p

a

b

Fig. 1. Geometry and dimensions of axial extended surface cracks in thick-walledcylinders under internal pressure.

reduce to the limit pressure for uncracked thick-walled cylinders(see Eq. (4) for the case g ¼ 2=

ffiffiffi3p

) when a=t/0.Carter [9] gave a limit pressure solution for thick-walled

cylinders, based on the Tresca yield criterion (g ¼ 1), in theform

pL

sy¼ Ri

R*1

ln�

Ro

Ri þ a

�¼

t

R*1

k� 1ln

k

1þ atðk� 1Þ

!(6)

where

R*1 ¼

�Ri without crack face pressureRi þ a with crack face pressure

(7)

Kim et al. [13] performed elastic-perfectly plastic FE analyses,using the von Mises yield criterion, for extended internal surfacecracks in cylinders under internal pressure and proposed a formula,based on the FE results, which can be expressed as

pL

sy¼ 2ffiffiffi

3p t

Rm

�1� 0:356

at� 1:6882

�at

�2þ1:0442

�at

�3

¼ 2ffiffiffi3p 2ðk� 1Þ

kþ 1

�1� 0:356

at� 1:6882

�at

�2þ1:0442

�at

�3 (8)

The cylinders considered were for t=Rm ¼ 0:2 and 0.05 with50% of the internal pressure applied to the crack surface. Therefore,Eq. (8) is valid for thin-walled cylinders with pressurised crackfaces.

Staat [10,11] applied the g factor to Carter’s solution (Eq. (6)) toextend it to solutions based on both the von Mises and Tresca yieldcriteria. He also applied a pressure magnifying factor, ðRi þ aÞ=Ri, tothe right-hand side of Eq. (6) arguing that the pressure acts on theinner surface of the cylinder rather than the surface at radius Ri þ a[10,11]. Staat and Vu [12] considered the resistance of the back-wallof the cracked part of the cylinder to the internal pressure andfurther improved Staat’s solution [10,11]. The new solution [12] is asfollows

pL

sy¼ g

Ri

R*2

Ri þ aRi

ln�

Ro

Ri þ a

�

þ" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

1þ at

t

R*2

!þ 1

2

�at

�2

t

R*2

!2vuut �

�1þ 1

2at

t

R*2

!#

¼ g

t

R*2

k� 1

�1þ a

tðk� 1Þ

�ln

k

1þ atðk� 1Þ

!

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

1þ at

t

R*2

!þ 1

2

�at

�2

t

R*2

!2vuut �

�1þ 1

2at

t

R*2

!#ð9Þ

where

R*2 ¼

Ri without crack face pressure

Ri þa2

with crack face pressure

((10)

The second term in the right-hand side of Eq. (9) corresponds tothe extra pressure carried by the cylinder of inner radius Ri andthickness a with an extended penetrating crack. For a thin-walledcylinder, this term is negligible. However, it can be significant forthick-walled cylinders. It is also seen from Eq. (10) that the crackface pressure correction factor differs from Eq. (7) where theinternal pressure is assumed to be applied to a diameter 2ðRi þ aÞ.In contrast, in Eq. (10) the internal pressure is applied to 2Ri þ a,considering force equilibrium.

FE, Staat & Vu

Prediction, Carter (eqn.(6)), (

Prediction, Kim et al. (eqn. (8))

Prediction, Staat & Vu (eqn. (9)), ( 3)= 2γ

3)= 2γ

k = 1.05

k = 1.22

k = 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

FE, Kim et al.Prediction, Carter (eqn.(6)), (Prediction, Kim et al. (eqn.(8))Prediction, Staat & Vu (eqn. (9)), ( 3)= 2γ

3)2=γ

0 0.2 0.4 0.6 0.8 1

a/t

0 0.2 0.4 0.6 0.8 1

a/t

0 0.2 0.4 0.6 0.8 1

a/t

FE, Kim et al.Prediction, Carter (eqn. (6)), (Prediction, Kim et al. (eqn. (8))Prediction, Staat & Vu (eqn. (9)), ( 3)2=γ

3)= 2γ

a

b

c

Fig. 3. Comparison of normalised limit pressures between various solutions and FEresults due to Kim et al. [13] and Staat and Vu [12] for extended internal cracks underinternal pressure (with crack face pressure).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

a/t

p L/p

0

FE, Staat & Vu

Prediction, Chell (eqn. (5)),

Prediction, Carter (eqn. (6))


( 3)= 2γ

Fig. 2. Comparison of normalised limit pressures between various solutions and FEresults due to Staat and Vu [12] for extended internal cracks under internal pressure.


The solutions of Eqs. (5), (6), (8) and (9) (g ¼ 2=ffiffiffi3p

for Eqs. (5)and (9)) are compared with FE results based on the von Mises yieldcriterion due to Staat and Vu [12],a and Kim et al. [13] in Figs. 2and 3 for various k. In the figures, the limit pressure is normalisedusing p0 defined by Eq. (4) and plotted against the normalised crackdepth, a=t.

Fig. 2 is for cases without crack face pressure. There is only oneset of available FE results, for k¼ 2. From the figure, Chell’s solution(Eq. (5)) is non-conservative for shallow cracks and Carter’s solu-tion (Eq. (6)) is conservative, partly due to its use of a factor g ¼ 1.The solution due to Staat and Vu (Eq. (9)) provides the bestpredictions and is conservative compared with the FE results.

Fig. 3 is for cases with crack face pressure. Fig. 3(a) and (b) are forthin-walled cylinders with FE results from Kim et al. [13] andFig. 3(c) is for a thick-walled cylinder with FE results from Staat andVu [12].a From Fig. 3, Carter’s solution (Eq. (6)) is conservative forboth thin-walled and thick-walled cylinders. The solution due toKim et al. (Eq. (8)) is very accurate for thin-walled cylinders(Fig. 3(a) and (b)) because Eq. (8) was obtained by fitting the FE datashown in the two figures. It also gives good predictions for k¼ 2(Fig. 3(c)). The solution due to Staat and Vu (Eq. (9)) gives goodpredictions for both thin-walled and thick-walled cases. However,it may slightly over-estimate the limit pressure for thin-walledcylinders with very deep cracks.

3.1.2. External defectsThe geometry and dimensions of a cylinder with an external

extended crack of depth, a, under internal pressure are shown inFig. 1(b).

Chell [8] proposed limit pressure solutions based on both thevon Mises and Tresca yield criteria for a cylinder with an extendedexternal crack under internal pressure, which can be expressed as

pL

sy¼ g

t � aRi¼ gðk� 1Þ

�1� a

t

�(11)

This equation does not reduce to the limit pressure foruncracked thick-walled cylinders (see Eq. (4) for the caseg ¼ 2=

ffiffiffi3p

) when a=t/0.Carter’s limit pressure solution [9] for thick-walled cylinders,

based on the Tresca yield condition (g ¼ 1), can be expressed as

pL

sy¼ ln

�Ro � a

Ri

�¼ ln

�k� a

tðk� 1Þ

�(12)

a All the FE data presented in 3-D plots in Ref. [12] were provided by Prof.Manfred Staat.

The solutions due to Staat and Vu [12] based on both the vonMises and Tresca yield criteria are as follows

pL

sy¼ gln

Ro�a

Ri

!

þ

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRo

Ri

�Ro

Ri�a

ttRi

�þ1

2

�at

�2�

tRi

�2s

��

Ro

Ri�1

2at

tRi

�35

¼ gln�

k�atðk�1Þ

�þ" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k�

k�atðk�1Þ

�þ1

2

�at

�2ðk�1Þ2

r

��

k�12

atðk�1Þ

�#ð13Þ

2c

t

Ri

Ro Rm


The three solutions of Eq. (12) and Eqs. (11) and (13) withg ¼ 2=

ffiffiffi3p

are compared in Fig. 4 with FE results based on the vonMises yield criterion for k¼ 2 obtained by Staat and Vu [12]. In thefigure, the normalised limit pressure is plotted against the nor-malised crack depth, a=t. From Fig. 4, Chell’s solution (Eq. (11)) isnon-conservative for a=t < 0:6. Carter’s solution (Eq. (12)) isconservative for all crack depths. The solution due to Staat and Vugives the best predictions and is conservative for this case.

p

Fig. 5. Geometry and dimensions of an axial through-wall crack in a thick-walledcylinder under internal pressure.

3.2. Solutions for through-wall cracks under internal pressure

The geometry and dimensions of a cylinder with a through-wallcrack, a ¼ t, of length, 2c, under internal pressure are shown in Fig. 5.

For cylinders with through-wall cracks under internal pressure,the solution due to Kiefner et al. [3] must be mentioned. Based onpipe burst data, Kiefner et al. [3] found that the limit pressure ofa pipe with a through-wall crack can be approximately expressed as

pL

sy¼

tRmMt¼ 2

Mt

k� 1kþ 1

(14)

where Mt is the Folias factor [17] and can be approximatelyexpressed as (see Appendix A)

Mt ¼h1þ 1:255 r2

m � 0:0135 r4m

i0:5(15)

where ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

k� 1r

rm ¼cffiffiffiffiffiffiffiffiffiRmtp ¼ kþ 1

tc

(16)

Note that Eq. (14) is valid for thin-walled cylinders becauseit was based on thin-walled pipe burst test data and the Foliasfactor was defined for thin-walled cylinders [17]. Also, the Foliasfactor was defined for rm < 4:4 [17] and there is no definition of Mt

beyond this limit. In addition, Eq. (14) cannot be used for the caseswith crack face pressure because the pipes used in the burst testscarried out by Kiefner et al. were all sealed from the inside [3].

Another early solution was proposed by Erdogan [18], which canbe expressed as

pL

sy¼

tRmMt1

¼ 2Mt1

k� 1kþ 1

(17)

where

Mt1 ¼ 0:614þ 0:87542 rm þ 0:386 exp ð�2:275rmÞ (18)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8

a/t

p L/p

0

FE, Staat & Vu

Prediction, Chell (eqn. (11)), (

Prediction, Carter, (eqn. (12))


3)= 2γ

1

Fig. 4. Comparison of normalised limit pressures between various solutions and FE resultsdue to Staat and Vu [12] for extended external cracks under internal pressure (k¼ 2).

Eq. (17) does not reduce to the solution for an uncracked thick-walled cylinder when the crack length becomes very small and,therefore, can only be used for thin-walled cylinders.

Carter [9] also proposed a limit pressure solution similar to thatdue to Kiefner et al. [3], but he used an older Folias factor [19], i.e.

pL

sy¼

tRi

Mt2¼ k� 1

Mt2(19)

where Mt2 is the Folias factor defined by

Mt2 ¼h1þ 1:61 r2

i

i0:5(20)

Note that Eq. (20) is slightly different from that in Ref. [19]where rm was used. However, in Eq. (20), ri is used instead, definedby

ri ¼cffiffiffiffiffiffiRit

p ¼ffiffiffiffiffiffiffiffiffiffiffiffik� 1p

tc

(21)

Carter defined the Mt2 factor using the internal radius of thecylinder, Ri, rather than the mean radius, Rm, to lead to a conser-vative prediction of the limit pressure. The use of Eq. (20) is furtherdiscussed in Appendix A. Similar to Eqs. (14) and (17), Eq. (19) alsodoes not reduce to the solution for an uncracked thick-walledcylinder when the crack length tends to zero and, therefore, isunsuitable for use with thick-walled cylinders.

Kim et al. [13] proposed a limit pressure solution as follows,based on their elastic-perfectly plastic FE analyses, using the vonMises yield criterion,

pL

sy¼ 2ffiffiffi

3p

tRmMt3

¼ 2ffiffiffi3p 2

Mt3

k� 1kþ 1

(22)

where Mt3 was obtained by fitting the FE results for t=Rm ¼ 0:2 and0.05 and can be expressed as

Mt3 ¼h1þ 0:34 rm þ 1:34 r2

m

i0:5(23)

Note that, in their FE analyses, Kim et al. applied 50% of the internalpressure to the crack face. Therefore, Eqs. (22) and (23) are appro-priate for cases with crack face pressure. Eq. (22) is also restricted to


thin-walled cylinders because of the thin-walled style of the equationitself and the thin-walled FE data used in the calibration of Mt3.

Staat and Vu [12] also proposed limit pressuresolutions based on both the von Mises and Tresca yield criteriafor thick-walled cylinders by considering the back-wall effect,that is

pL

sy¼ g

Mt4ln�

Ro

Ri

�þ

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t

Riþ 1

2

�tRi

�2s

� 12

�1þ Ro

Ri

�35¼ g

Mt4ln ðkÞ þ

" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðk� 1Þ þ 1

2ðk� 1Þ2

r� 1

2ð1þ kÞ

# (24)

The first part in the right-hand side of Eq. (24) is similar to Eq.(17), but expressed in the thick-walled form. The second partcorresponds to the back-wall correction, which has been found tobe significant for thick-walled cylinders [12]. In Eq. (24), Mt4 isdefined by

Mt4 ¼h1þ 1:25 r2

o

i0:5(25)

where

ro ¼cffiffiffiffiffiffiffiRotp ¼

ffiffiffiffiffiffiffiffiffiffiffiffik� 1

k

rtc

(26)

Eq. (25) was calibrated using FE data for k¼ 1.1, 1.25, 1.5, 1.75 and2 [12], and, therefore, can be used for both thin-walled and thick-walled cylinders. Examining Eq. (24), it can be seen that the secondterm in the right-hand side of the equation is a constant for given k.This means that the limit pressure may be greater than that for theuncracked cylinder when the crack length is very small and henceMt4/1.

The five solutions of Eqs. (14), (17), (19), (22) and (24) with g ¼2=

ffiffiffi3p

are compared in Figs. 6 and 7 with available FE results basedon the von Mises criterion due to Staat and Vu [12] (digitised fromFig. 3 in Ref. [12]) and Kim et al. [13] for various k values. In thefigures, the normalised limit pressure is plotted against r2

o, wherero is defined by Eq. (26).

The predictions for cases without crack face pressure arecompared with FE results based on the von Mises criterion due toStaat and Vu [12] in Fig. 6. From the figure, Eq. (24) gives the bestpredictions for all the five k values but is slightly non-conservativecompared with the FE results. This is because the FE data presentedin Fig. 6 were used to calibrate Eq. (25). It is also seen from thefigure that Eq. (24) over-estimates the limit pressure when ro/0,especially for large k (Fig. 6(e)). The other two solutions, that is, Eq.(14) due to Kiefner et al. and Eq. (17) due to Erdogan are conser-vative for all the five k values considered. This is probably becausethey are all based on the Tresca criterion. From Fig. 6(a) and (b),they can all predict the FE results very well if a factor of 2=

ffiffiffi3p

isapplied. But, for thick-walled cylinders (Fig. 6(e)), they maysignificantly under-estimate the limit pressure even if the factor2=

ffiffiffi3p

is applied. The solution due to Carter, Eq. (19), is conservativefor k � 1:25 (Fig. 6(a) and (b)). However, for thick cylinders(Fig. 6(c)–(e)), it over-estimates the limit pressure for small ro,although it is still conservative for large ro due to using the Trescacriterion.

Fig. 7 compares predictions with FE results for cases with crackface pressure due to Kim et al. [13] (Fig. 7(a)) and to Staat and Vu[12] (Fig. 7(b)). Among the five solutions, only Eq. (22) due to Kimet al. can predict the limit pressure when the crack face pressure isconsidered. From Fig. 7, Eq. (22) can predict the FE results fork¼ 1.22 very accurately (the FE data in Fig. 7(a) were used to

calibrate Eq. (23)), but it over-estimates the FE results for k¼ 2. Thisindicates that Eq. (22) is non-conservative for thick-walledcylinders.

3.3. Global solutions for axial surface cracks under internal pressure

The geometry and dimensions of a cylinder with an internal/external surface crack of depth, a � t, length 2c under internalpressure are shown in Fig. 8.

Currently, the limit pressure solutions have been obtained bysimplifying the surface cracked cylinder to two coaxial cylinders,Cylinder A with a through-wall crack and Cylinder B withouta crack (Fig. 9). The limit pressure is taken as the sum of thelimit pressures for the two cylinders. For the case of an internalcrack (Fig. 9(a)), for example, the inner cylinder, Cylinder A, withan internal radius Ri and thickness a contains a through-wallcrack of length 2c and the outer cylinder, Cylinder B, is crack-free with an internal radius Ri þ a and thickness t � a. Variouslimit pressure solutions have been obtained because differentlimit pressure solutions for through-wall and crack-free cylinderswere adopted.

Ewing [6] gave a limit pressure solution, based on the thin-walled limit pressure solutions for both cracked and crack-freecylinders, which can be expressed as

pL

sy¼ t

Rm

�1� a

tþ a

t1

Ma1

�¼ 2ðk� 1Þ

kþ 1

�1� a

tþ a

t1

Ma1

�(27)

where

Ma1 ¼�

1þ 1:61c2

Rma

�0:5

¼

1þ 1:612

atðk� 1Þ�

ac

�2

ðkþ 1Þ

!0:5

(28)

Eq. (27) can be used for cases of internal cracks without crackface pressure and for external cracks.

3.3.1. Internal cracksThe geometry and dimensions of a cylinder with an internal

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

Carter [9] used his limit pressure solution for a cylinder witha through-wall crack (Eq. (19)), which is for thin-walled cylinders,and the limit pressure solution for a thick-walled uncrackedcylinder and obtained the following equation

pL

sy¼ a

Ri

1Ma2þ Ri

R*1

ln�

Ro

Ri þ a

�¼ a

tðk� 1Þ

Ma2þ Ri

R*1

ln�

k

1þ atðk� 1Þ

�

(29)

where

Ma2 ¼�

1þ 1:61c2

Ria

�0:5

¼

1þ 1:61

atðk� 1Þa

c

�2

!0:5

(30)

The mis-match of thin-walled and thick-walled solutions for,respectively, the cylinder with a through-wall crack and the crack-free one may cause problems when Eq. (29) is used for thick-walledcylinders. It is also seen from Eq. (29) that the crack face pressureterm, Ri=R*

1, is applied only to the crack-free cylinder. The correctionfor the crack face pressure vanishes when a=t/1.

Staat and Vu [12] constructed limit pressure solutions based onboth the von Mises and Tresca yield criteria using their solution forcylinders with through-wall cracks (Eq. (24)) and the limit pressureexpression for crack-free thick-walled cylinders, which can beexpressed as

k = 1.1 k = 1.25

k = 1.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

FE, Staat & VuPrediction, Kiefner et al. (eqn. (14))Prediction, Erdogan (eqn. (17))Prediction, Carter (eqn. (19))Prediction, Staat & Vu (eqn. (24)), (Prediction, present work (eqn. (56)), (

2oρ

2oρ 2

oρ

0 2 4 6 8 102oρ

3)= 2γ3)= 2γ

FE, Staat & VuPrediction, Kiefner et al. (eqn. (14))Prediction, Erdogan (eqn. (17))Prediction, Carter (eqn. (19))Prediction, Staat & Vu (eqn. (24)), (Prediction, present work (eqn. (56)), ( 3)= 2γ

3)= 2γ

0 10 15


3)= 2γ

k = 1.75

k = 2


3)= 2γ

FE, Staat & VuPrediction, Kiefner et al. (eqn. (14))Prediction, Erdogan (eqn. (17))Prediction, Carter (eqn. (19))Prediction, Staat & Vu (eqn. (24)), (Prediction, present work (eqn. (56)), ( 3)2=γ

3)2=γ

5 0 10 15 205

2oρ

0 10 15 20 255

a b

c

e

d

Fig. 6. Comparison of normalised limit pressures between various solutions and FE results due to Staat and Vu [12] for cylinders with through-wall cracks under internal pressure(without crack face pressure).


pL

sy¼ min

8>><>>:

gln�Ro

Ri

�;g

Ri

R*2

�1

Ma3ln�

Ri þ aRi

�þ Ri þ a

Riln�

Ro

Ri þ a

�

þ" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ at

t

R*2

þ 12

�at

�2

t

R*2

!2vuut �

�1þ 1

2at

t

R*2

!#9>>=>>;

¼ min

8><>:

gln ðkÞ;gRi

R*2

�1

Ma3ln�

1þ atðk� 1Þ

�þ�

1þ atðk� 1Þ

�ln�

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ at

t

R*2

þ 12

�at

�2

t

R*2

!2vuut �

�1þ 1

2at

t

R*2

!#
k
1þ atðk� 1Þ

�9>=>;

(31)

k = 1.22

k = 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 8 10

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

FE, Kim et al .


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

0 2 8 10 12 14 16

FE, Staat & Vu


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

4 6

4 6

a

b 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.


where

Ma3 ¼�

1þ 1:25c2

ðRi þ aÞa

�0:5

¼

1þ 1:25

atðk� 1Þa

c

�2ð1þ atðk� 1ÞÞ

!0:5

(32)

Eq. (31) may be non-conservative for short cracks for thefollowing two reasons. Firstly, the solution for cylinders withthrough-wall cracks (Eq. (24)) is non-conservative for short cracks,as discussed in Section 3.2 above. Secondly, a pressure magnifica-tion factor, ðRi þ aÞ=Ri, is applied to the crack-free cylinder solutionassuming that the pressure applied on the outer cylinder is lowerthan that applied on the inner one. However, for short cracks,Ma3/1 and the second formula in Eq. (31) will be greater thangln ðkÞ. Although a limit gln ðkÞ is set in Eq. (31), it can stillpotentially over-estimate the limit pressure for short cracks.

Kim et al. [13] proposed a limit pressure solution, based on theirelastic-perfectly plastic FE analyses with the von Mises yieldcriterion, and expressed it as

pL

sy¼ 2ffiffiffi

3p t

Rm

�1þ A1

atþ A2

�at

�2

¼ 2ffiffiffi3p 2ðk� 1Þ

kþ 1

�1þ A1

atþ A2

�at

�2

(33)

where�A1 ¼ 0:0462� 0:0589 rm � 0:013 r2

mA2 ¼ 0:0395� 0:3413 rm þ 0:0652 r2

m(34)

and rm is defined by Eq. (16).

Fig. 7. Comparison of normalised limit pressures between various solutions and FEresults due to Staat and Vu [12] and Kim et al. [13] for cylinders with through-wallcracks under internal pressure (with crack face pressure).

Eqs. (33) and (34) are based on FE data fort=Rm ¼ 0:2;0:1;0:05and0:025, with 50% internal pressure appliedon the crack faces, and, therefore, are valid for thin-walled cylinderswith crack face pressure. Note that Eq. (33) is inconsistent with Eq.(22) when a=t ¼ 1.

Fig. 10 compares the normalised limit pressures predictedusing Eqs. (27), (29) and (31) (g ¼ 2=

ffiffiffi3p

) with FE results basedon the von Mises yield criterion due to Staat and Vu [12] for casesof k¼ 2 without crack face pressure. From the figure, Eqs. (27)and (29) due to Ewing and Carter, respectively, are conservativefor 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-wallcracks.

Figs. 11–13 compare the normalised limit pressures predictedusing Eqs. (29), (31) (g ¼ 2=

ffiffiffi3p

) and Eq. (33) with FE results basedon the von Mises yield criterion due to Staat and Vu [12] (Fig. 11 fork¼ 2), and due to Kim et al. [13] (Fig. 12 for k¼ 1.05 and Fig. 13 fork¼ 1.22) for cases with crack face pressure. From Fig. 11 for k¼ 2,predictions using Eq. (33) due to Kim et al. are very close to the FEresults for shallow cracks but are non-conservative for deep cracks.Carter’s solution (Eq. (29)) is conservative for long and shallowcracks but over-estimates the FE results for short and deep cracks. Itis also seen from the figure that the predictions using Eq. (31) dueto Staat and Vu are reasonably close to the FE results but are slightly

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 withsurface cracks.


non-conservative for shallow cracks and some through-wall cracks.For thin-walled cylinders (Figs. 12 and 13), the solution due to Kimet al. (Eq. (33)) gives accurate predictions of the FE results up toa=t ¼ 0:8 but significantly over-estimates the limit pressure forthrough-wall cracks. This is not surprising because Eq. (33) wasfitted to the FE data presented in Figs. 12 and 13. From the figures,Carter’s solution is conservative for all crack lengths and depthsconsidered by comparison with the FE data. The solution due toStaat and Vu is reasonably close to the FE results and conservative,but it slightly over-estimates the limit pressures for through-wallcracks.

3.3.2. External cracksThe 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).

Carter’s solution [9] for a cylinder with an external crack can beexpressed as

pL

sy¼ a

Ro � a1

Max1þ ln

�Ro � a

Ri

�

¼atðk� 1Þ

k� atðk� 1Þ

1Max1

þ ln�

k� atðk� 1Þ

�(35)

where

Max1 ¼�

1þ 1:61c2 �0:5

ðRo � aÞa

¼

1þ 1:61

atðk� 1Þa

c

�2ðk� atðk� 1ÞÞ

!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-freethick-walled cylinders. This mis-match may also cause problemswhen Eq. (35) is used for thick-walled cylinders.

The limit pressure solutions based on both the von Mises andTresca yield criteria for external surface cracks due to Staat and Vu[12] can be expressed as follows

pL

sy¼ g

�1

Max2ln�

Ro

Ro � a

�þ ln

�Ro � a

Ri

�

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ro

Ri

�Ro

Ri� a

ttRi

�þ 1

2

�at

�2�

tRi

�2s

��Ro

Ri� 1

2at

tRi

�35¼ g

h 1Max2

ln�

k

k� atðk� 1Þ

�þ ln

�k� a

tðk� 1Þ

�

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kk� a

tðk� 1Þ

�þ 1

2

�at

�2ðk� 1Þ2

r�k� 1

2atðk� 1Þ

�#

(37)

where

Max2 ¼�

1þ 1:25c2

Roa

�0:5

¼

1þ 1:25ðk� 1Þa

t

kac

�2

!0:5

(38)

Eq. (37) may over-estimate the limit pressure for short cracksbecause the second term in the right-hand side of Eq. (37) does notdepend on crack length.

Figs. 14 and 15 compare normalised limit pressures predictedusing Eqs. (27), (35) and (37) (g ¼ 2=

ffiffiffi3p

) with FE resultsbased on the von Mises yield criterion due to Staat and Vu[12] for cases of k¼ 2 and due to Zarrabi et al. [16] for k¼ 1.57.From the figures, Eq. (27) due to Ewing is conservative for allcrack lengths and depths considered. Carter’s solution (Eq. (35))is also conservative, except for very short and deep cracks (seeFig. 15(e)). It is also seen from Figs. 14 and 15 that the predictionsusing Eq. (37) due to Staat and Vu are very close to the FE resultsbut slightly non-conservative for very short and through-wallcracks.

3.4. Local solutions for axial surface defects under internal pressure

The limit pressure expression for a cylinder with a surfacecrack under internal pressure given by Kiefner et al. [3] may beexpressed as

pL

sy¼ t

Rm

1� at

1� at

1Mteq

¼ 2k� 1kþ 1

1� at

1� at

1Mteq

(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 theequivalent half crack length, ceq, defined by

ceq ¼Adf

2a(40)

a/c = 0.2 a/c = 0.4

a/c = 0.6

FE, Staat & Vu

Prediction, Ewing (eqn. 27))


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

Prediction, present work (eqn.(62)), ( 3)2=γ3)2=γ

FE, Staat & VuPrediction, Ewing (eqn. (27))Prediction, Carter (eqn. (29))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)= 2γ

3)= 2γ

FE, Staat & VuPrediction, Ewing (eqn. (27))Prediction, Carter (eqn. (29))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2=γ

3)2=γ

a/c = 0.8

a/c = 1

FE, Staat & Vu

Prediction, Ewing (eqn. (27))



Prediction, present work (eqn. (62)), ( 3)= 2 γ3)= 2γ

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

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2p L

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

FE, Staat & VuPrediction, Ewing (eqn. (27))Prediction, Carter (eqn. (29))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2= γ

3)= 2γ

1

0 0.2 0.4 0.6 0.8

a/t10 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t10 0.2

a b

c d

e

0.4 0.6 0.8

a/t1

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 internalpressure (k¼ 2, without crack face pressure).


where Adf is the crack area and Adf ¼ 2ac for rectangular cracks.Eq. (39) is an empirical formula obtained from burst experimentson 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 ofinternal defects. Hence, Eq. (39) applies to cases without crack facepressure.

3.4.1. Internal cracksCarter [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 withan extended internal surface crack of depth a (see Fig. 16(a)),that is

pL

sy¼ 1

Dþ c

�D

pL½for crack-free cylinder�sy

þ cpL½for cylinder with extended crack�

sy

(41)

where D is an equivalent length of the crack-free cylinder,which can be determined by equating Eq. (41) to Eq. (29). Thelocal 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

FE, Staat & VuPrediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)= 2γ

3)= 2γ

FE, Staat & VuPrediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)= 2γ

3)= 2γ

FE, Staat & Vu


Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. 31)), (

Prediction, present work (eqn. (62)), ( 3)= 2γ3)= 2γ

FE, Staat & Vu




Prediction,present work (eqn. (62)), ( 3)= 2γ3)= 2γ

FE, Staat & Vu





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

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

a/c = 0.6 a/c = 0.8

0 0.2 0.4 0.6 0.8

a/t1

a/c = 1

0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1

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 internalpressure (k¼ 2, with crack face pressure).


t/c = 0.894 t/c = 0.447

t/c = 0.224

FE, Kim et al.

a b

c d

Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))

Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2=γ

3)2=γ

FE, Kim et al .Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn (31)), (

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

FE, Kim et al.Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2=γ

3)2=γ

t/c = 0.149

FE, Kim et al .Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2=γ

3)2=γ

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

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(k¼ 1.22, with crack face pressure).

t/c = 0.447 t/c = 0.224

t/c = 0.112

FE, Kim et al .Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work(eqn. (62)), ( 3)= 2γ

3)= 2γ

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


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


FE, Kim et al.Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33))Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2=γ

3)2=γ

t/c = 0.075

FE, Kim et al.Prediction, Carter (eqn. (29))Prediction, Kim et al. (eqn. (33)) Prediction, Staat & Vu (eqn. (31)), (Prediction, present work (eqn. (62)), ( 3)2=γ

3)2=γ

0.0

0.2

0.4

0.6

0.8

1.0

1.2a b

c d

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1

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(k¼ 1.05, with crack face pressure).


a/c = 0.2 a/c = 0.4

a/c = 0.6

FE, Staat & VuPrediction, Ewing (eqn. (27))Prediction, Carter (eqn. (35))Prediction, Staat & Vu (eqn. (37)),Prediction, present work (eqn. (65)), ( )32=γ

( )32=γ


( )32=γ


( )32=γ


( )32=γ


( )32=γ

a/c = 0.8

a/c = 1

0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

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 internalpressure (k¼ 2).


FE, Zarrabi et al.




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

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

p L/p

0

4 51

a/t = 0.7

0 2 3 6

a/c4 51

a/t = 0.5

0 2 3

a/c41

a/t = 0.3

0 1 3

a/c2

a/t = 0.1

0 2 3

a/c1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

FE, Zarrabi et al.





FE, Zarrabi et al.





FE, Zarrabi et al.





FE, Zarrabi et al.



Prediction, present work (eqn. (65)), ( )32=γPrediction, Staat & Vu (eqn. (37)), ( )32=γ

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 internalpressure (k¼ 1.57).


Internal crack, Dt

ac1 −= 1

External crack, c2 Dt

a−= 1

D D2cc1 c1

Crack

at

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 cylinderswith surface cracks under internal pressure.


pL

sy¼ 1

c1 þ c

"c1ln

�Ro

Ri

�þ c

Ri

R*1

ln�

Ro

Ri þ a

�#

¼ 1c1

cþ 1

"c1

cln ðkÞ þ Ri

R*1

ln�

k

1þ atðk� 1Þ

�#(42)

where

c1

c¼

a1� a

t

�Ma2Ri

�ln�Ro

Ri

�� Ri

R*1

ln�

Ro

Ri þ a

�#� a

¼ðk� 1Þa

t

�1� a

t

�

Ma2

�ln ðkÞ � Ri

R*1

ln�

k

1þ atðk� 1Þ

�#� a

tðk� 1Þ

(43)

and

c1 ¼ D�

1� at

�(44)

Staat and Vu [12] defined their local limit pressures based onboth the von Mises and Tresca yield criteria, using the methodologyemployed by Carter [9] but their own limit pressure solutions fora cylinder with a through-wall crack (Eq. (24)) and a cylinder withan extended crack (Eq. (9)), as

pL

sy¼

gln�Ro

Ri

�¼ gln ðkÞ for

pI

gsy� ln ðkÞ

g

s1þ c

"s1ln

�Ro

Ri

�þ c

Riþa

R*2

ln�

Ro

Riþa

�#

¼ gs1

cþ1

"s1

cln ðkÞþ Ri

R*2

�1þa

tðk�1Þ

�

ln� k

1þatðk�1Þ

�#for

pI

gsy< ln ðkÞ

ð45Þ

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

where

s1

c¼

1� a

t

�Ri

R*2

ln�

Ri þ aRi

�

Ma3

�ln�Ro

Ri

�� Ri

R*2

�1

Ma3ln�

Ri þ aRi

�þ Ri þ a

Riln�

Ro

Ri þ a

��#

¼ð1� a

t

�Ri

R*2

ln�

1þ atðk� 1Þ

�

Ma3

�ln ðkÞ � Ri

R*2

�1

Ma3ln�

1þ atðk� 1Þ

�þ�

1þ atðk� 1Þ

�ln�

k

1þ atðk� 1Þ

��#(46)

and

pI

gsy¼ Ri

R*2

�1

Ma3ln�

Ri þ aRi

�þ Ri þ a

Riln�

Ro

Ri þ a

��

¼ Ri

R*2

�1

Ma3ln�

1þ atðk� 1Þ

�þ�

1þ atðk� 1Þ

�ln�

k

1þ atðk� 1Þ

��

(47)

The two local limit pressure solutions are nowcompared with theFE results. There is only one set of well documented FE results forlocal limit pressure available, which is the results due to Jun et al. [15]based on the von Mises yield criterion and the crack ligamentyielding. Eqs. (42) and (45) (g ¼ 2=

ffiffiffi3p

) due to Carter [9] and Staatand Vu [12], respectively, are compared with the FE results due to Junet al. [15] in Figs. 17–19 for k¼ 1.05, 1.11 and 1.22, respectively, forinternal surface cracks with crack face pressure. Eq. (39) is alsoplotted in the figures for comparison, though it is for cases withoutcrack face pressure. From Figs. 17–19, the predictions using Carter’ssolution (Eq. (42)) are reasonably close to the FE results andconservative for all the three k values except for shallow cracks ina very thin cylinder (see Fig. 17(a) and (b)). It is also seen from thefigures that the solution due to Staat and Vu [12] for g ¼ 2=

ffiffiffi3p

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

cylinder with a very thin wall (Figs. 17 and 18). The formula due toKiefner et al. (Eq. (39)) shows very good and conservative predic-tions for k¼ 1.11 (Fig. 18) and 1.22 (Fig. 19). However, it may be non-conservative for short and shallow cracks for k¼ 1.05 (see Fig. 17).

3.4.2. External cracksSimilar to the cases of internal cracks, Carter’s local limit pres-

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

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, Carter (eqn. (42))Prediction, Staat (eqn. (45)), (Prediction, present work (eqn. (67))

3)2=γ

0.0

0.2

0.4

0.6

0.8

1.0

1.2a b

c d

e

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

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

Prediction, present work (eqn. (67))






Prediction, present work (eqn. (67))Prediction, Staat & Vu (eqn. (45)), ( )32=γ Prediction, Staat & Vu (eqn. (45)), ( )32=γ

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

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 internalpressure (k¼ 1.05, with crack face pressure).


pL

sy¼ 1

c2 þ c

�c2ln

�Ro

Ri

�þ cln

�Ro � a

Ri

�

¼ 1c2

cþ 1

hc2

cln ðkÞ þ ln

�k� a

tðk� 1Þ

�i(48)

where

c2

c¼

a1� a

t

�Max1ðRo � aÞln

� Ro

Ro � a

�� a

¼ðk� 1Þa

t

�1� a

t

�Max1

k� a

tðk� 1Þ

�ln�

k

k� atðk� 1Þ

�� a

tðk� 1Þ

(49)

The local limit pressure for external crack due to Staat and Vu[12] can be expressed as

pL

sy¼ g

s2 þ c

�s2ln

�Ro

Ri

�þ cln

�Ro � a

Ri

�

¼ gs2

cþ 1

hs2

cln ðkÞ þ ln

�k� a

tðk� 1Þ

�i(50)

where

s2

c¼

1� at

Max2 � 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, Carter (eqn. (42))Prediction, Staat & Vu (eqn. (45)),Prediction, present work (eqn. 67)

( )32=γ

0.0

0.2

0.4

0.6

0.8

1.0

1.2p L

/p0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p L/p

0

0 0.2 0.4 0.6 0.8

a/t1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8

a/t

a/t

1 0 0.2 0.4 0.6 0.8

a/t1

0 0.2 0.4 0.6 0.8 1


( )32=γ


( )32=γ


( )32=γ


( )32=γ

a b

c d

e



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 tocombined membrane and through-wall bending stresses(Fig. 20) are generally obtained from solutions for cracked platesunder combined tension and bending [1,9,20,21], ignoring theeffect of curvature. In R6 [1], the limit load solution for a thin-walled cylinder with an internal axial surface crack undercombined membrane and through-wall bending stresses isa local solution based on the plate solution due to Goodall andWebster [22] and Lei [23,24]. Actually, this solution can beextended to thick-walled cylinders with internal/externalsurface cracks as long as the bending stress tends to open thecrack because the plate solution [22–24] 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 surfacecracks under combined internal pressure, axial tension and globalbending was proposed by Desquines et al. [25], followed Kitchinget al. [4]. However, the limit pressures predicted using this solutionare much lower than those predicted using the solution due toKiefner 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 of t=Rm ¼0:05 with a surface crack of a=t ¼ 0:2 and a=c ¼ 0:0224 undercombined internal pressure and global bending and concludedthat a bending load has only a slight effect on the limit pressure foraxial 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

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

p L/p

0

p L/p

0

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

(γ =2

a/t

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

( )32=γ

0.0

0.2

0.4

0.6

0.8

1.0

1.2

a/t

p L/p

0


a/c = 0.05

a/c = 0.033

a/t

p L/p

0FE, Jun et al.Prediction, Kiefner et al. (eqn. (39))Prediction, Carter (eqn. (42))Prediction, Staat & Vu (eqn. (45)),Prediction, present work (eqn. (67))

(γ = 2

a/t

p L/p

0

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

3)

3)

3)

a b

cd

e

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

(γ = 2 3)



crack-free cylinder when internal pressure only is applied. Furtherinvestigation is necessary for the limit load of axially crackedcylinders under combined loading.

4. New limit load solutions for cylinders withaxial cracks under internal pressure

The results of the review of the limit loads for axially cracked cylin-ders under internal pressure in Section 3 can be summarised as follows.

(1) For extended internal/external surface cracks, solutions due toStaat and Staat and Vu (Eqs. (9) and (13)) are for thick-walledcylinders 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, needsto be re-calibrated.

(3) For the global limit pressure of internal surface cracks, the limitpressure solution due to Staat and Vu (Eq. (31)) is for thick-walled cylinders and gives good predictions for available FEresults for both thin-walled and thick-walled cylinders.However, it over-estimates the FE results for short and shallowcracks due to the problem in the solution for through-wall cracksdescribed in (2) and the pressure magnifying factor, ðRi þ aÞ=Ri,applied to the term corresponding to the crack-free cylinder.

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

Internal crack

External crack

σm σ b

2c

at

Ri

Ro

2c

at

RiRo

a

b

σbσm

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


the FE results for short and shallow cracks due to the problemin the solution for through-wall cracks described in (2).

(5) For the local limit pressures for internal/external surfacecracks, Carter’s solutions (Eqs. (42) and (48)) are for thick-walled cylinders and give reasonably good and conservativepredictions of FE results for thin-walled cylinders. However,the expressions for the local limit pressure are based on therelevant global solutions. Therefore, they need to be re-derivedto maintain consistency with the global solutions.

New limit pressure solutions for axially cracked thick-walledcylinders under internal pressure are derived in this section. Theycan also be used for thin-walled cylinders.

4.1. Through-wall cracks under internal pressure

New limit load solutions based on both the von Mises and Trescayield criteria for a thick-walled cylinder with a through-wall crackunder internal pressure are obtained by summing the pressurecorresponding to the front-wall failure, p0=Mtn, and the back-wallcorrection, DpL (see Eq. (C1) in Appendix C). From Eq. (C1), the limitpressure without considering the crack face pressure can beexpressed as

pL

sy¼ p0

Mtnsyþ DpL

sy(52)

where Mtn is the stress magnification factor and is defined using theouter radius of the cylinder, with the coefficient being re-calibratedusing the FE data for k � 2 (see Appendix C), that is,

Mtn ¼�

1þ 1:4 r2o

�0:5¼�

1þ 1:4c2

Rot

�0:5

¼

1þ 1:4

k� 1ktc

�2

!0:5

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 furtherexpressed as

pL

sy¼ Ri

R*t

p0

Mtnsyþ DpL

sy(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) inAppendix B. Note that Eq. (54) leads to the limit pressure fora defect-free cylinder ðRi=R*

t Þp0 < p0 when c/0 because the factorRi=R*

t does not change with crack length, c, noting that the secondterm in the right-hand side of Eq. (54) tends to zero and Mtn/1. Inorder to avoid this, the R*

t in Eq. (54) may be replaced by R*tn, which

is defined as


Ri without crack face pressuret

8>8>>><

R*tn ¼ Ri þ 2

forc � t

Ri þc2

forc < twith crack face pressure

<>:>>>:

(55)

It is seen from Eq. (55) that for long cracks (c � t) R*tn ¼ R*

t andfor short cracks (c < t) it is a linear interpolation between Ri þ t=2and Ri. This allows the effect of the crack face pressure factor tovanish when the crack length tends to zero and the limit pressure ofthe crack-free cylinder to be accurately reproduced. Here, choosingc < t as ‘‘short cracks’’ is for consistency with the cases of surfacecracks with c < a and is somewhat arbitrary. Using R*

tn, the limitpressure for a thick-walled cylinder with a through-wall crackunder internal pressure can be expressed as

pL

sy¼ Ri

R*tn

g

Mtnln�

Ro

Ri

�þ

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 1

2ð1þ 1

MtnÞ t

R*tn

!2

þ14

1� 1

M2tn

! t

R*tn

!2vuut �

1þ 1

2

�1þ 1

Mtn

�t

R*tn

!35

¼ fptgMtn

ln ðkÞ þ

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ 1

2ð1þ 1

MtnÞðk� 1Þfpt

�2

þ14

1� 1

M2tn

!�ðk� 1Þfpt

�2

vuut ��

1þ 12

�1þ 1

Mtn

�ðk� 1Þfpt

�35(56)

where fpt is the crack face pressure factor and can be expressed as,from Eq. (55),

fpt ¼Ri

R*tn

¼

1 without crack face pressureRi

Riþt2

¼ 1

1þ 12ðk� 1Þ

fortc� 1

Ri

Riþc2

¼tc

tcþ 1

2ðk� 1Þ

fortc> 1


8>>>>>>><>>>>>>>:

8>>>>>>>>><>>>>>>>>>:

(57)

The new solution, Eq. (56) (g ¼ 2=ffiffiffi3p

), is compared withother 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 withall other solutions. It is slightly conservative compared with theFE 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 internalsurface crack under internal pressure are obtained by summingthe limit pressure corresponding to the cylinder of inner radiusRi and thickness a with a through-wall crack of length 2c(Cylinder A in Fig. 9(a)) and that for the crack-free cylinder ofinner radius Ri þ a and thickness t � a (Cylinder B in Fig. 9(a)),that is

pL ¼ pL½Cylinder A� þ FptRi*

pL½Cylinder B�(58)

sy sy R2nsy

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

R*2n ¼


Ri þa2

forc � a

Ri þc2

forc < awith crack face pressure

8><>:

8>>><>>>:

(59)

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

Fpt ¼ 1þ aRi

�1� 1

Man

�(60)

where Man is the stress magnification factor for Cylinder A and canbe obtained by applying Eq. (53) to Cylinder A, that is

Man ¼�

1þ1:4c2

ðRiþaÞa

�0:5

¼

1þ1:4

atðk�1Þa

c

�2ð1þatðk�1ÞÞ

!0:5

for�

1þatðk�1Þ

��2 (61)

The pressure transfer factor, Fpt, is applied to the term in Eq. (58)representing the limit pressure of the crack-free cylinder (CylinderB in Fig. 9(a)) to capture the behaviour of pressure transferring fromthe 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. anextended penetrating crack in Cylinder A in Fig. 9(a), Fpt tends toðRiþaÞ=Ri ¼ 1þða=RiÞ 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 to 1 because the fully yielded Cylinder A in Fig. 9(a) cannotbear any more pressure difference and the pressure is transferredconstantly from the inner surface of Cylinder A (Fig. 9(a)) to theinner surface of Cylinder B (Fig. 9(a)). For all other cases betweenthese two limits, the factor is estimated using linear interpolationbased on 1=Man.

Determining the limit pressure of Cylinder A in Fig. 9(a) byapplying Eq. (56) to a cylinder of inner radius Ri and outer radiusRi þ a with a through-wall crack of length 2c and the limit pressurefor the defect-free cylinder of inner radius Ri þ a and outer radiusRo (Cylinder B in Fig. 9(a)), the limit pressure of a thick-walledcylinder with an internal surface crack can be obtained from Eq.(58) and expressed as

pLsy¼

8><>:

Ri

R*2n

g

Manln�

Ri þ aRi

�

þ

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 1

2ð1þ 1

ManÞat

t

R*2n

!2

þ14

1� 1

M2an

! at

t

R*2n

!2vuut �

1þ 1

2

�1þ 1

Man

�at

t

R*2n

!35

9>=>;

þ�

1þ aRi

�1� 1

Man

��Ri

R*2n

gln�

Ro

Ri þ a

�

¼ gfps

�1

Manln�

1þ atðk� 1Þ

�þ�

1þ atðk� 1Þ

�1� 1

Man

��ln�

k1þ a

tðk� 1Þ

�

þ

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ 1

2ð1þ 1

ManÞatfpsðk� 1Þ

�2

þ14

1� 1

M2an

!�atfpsðk� 1Þ

�2

vuut ��

1þ 12

�1þ 1

Man

�atfpsðk� 1Þ

�35 for k � 2:718

(62)


where fps is the crack face pressure factor and is defined, usingEq. (59), as

fps ¼Ri

R*2n

¼

1 without crack face pressureRi

Riþa2

¼ 1

1þ12

atðk�1Þ

forac�1

Ri

Riþc2

¼ac

acþ1

2atðk�1Þ

forac>1


8>>>>>>><>>>>>>>:

8>>>>>>>>><>>>>>>>>>:

(63)

Note that Eq. (62) is valid for pL=ðgsyÞ�1 because the pressuretransfer factor for long cracks is defined based on the assumption ofan elastic Cylinder A and yielding may take place in Cylinder A evenfor the case of an extended surface crack when pL>gsy. Thiscondition is always satisfied for cylinders of k�2:718 with anycrack size.

Eq. (62) reduces to Eq. (56) for through-wall cracks, whena=t/1, and to Eq. (9) for internal extended cracks when a=c/0and a=t > 0. It also reproduces the limit pressure for crack-freethick-walled cylinders when a=t ¼ 0 or a=c/N.

The new solution, Eq. (62) with g ¼ 2=ffiffiffi3p

, is compared withother existing solutions and the FE data due to Staat and Vu [12] inFigs. 10 and 11 and those due to Kim et al. [13] in Figs. 12 and 13. Forcases without crack face pressure (Fig. 10), Eq. (62) has largelyremoved 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

pLsy¼

8><>:

gMaxn

ln�

RoRo�a

�

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

RoRi� 1

2ð1�1

MaxnÞat t

Ri

�2þ1

4

�1� 1

M2axn

��at

tRi

�2s

��Ro

Ri� 1

2

�1

þgln�

Ro�aRi

�¼ g

h1

Maxnln�

kk�a

tðk�1Þ

�þ ln

k� a

tðk� 1Þ�i

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

k� 12ð1�

1MaxnÞatðk� 1Þ

�2þ1

4

�1� 1

M2axn

�atðk� 1Þ

�2

s��

k� 12

�1�

face pressure (Figs. 11–13), Eq. (62) has also improved the non--conservatism of the solution of Staat and Vu [12] for shortand shallow cracks for thick-walled cylinders (Fig. 11) and givesreasonably 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 externalsurface crack under internal pressure are obtained by directlysumming the limit pressure corresponding to the cylinder of innerradius Ro � a and outer radius Ro with a through-wall crack oflength 2c (Cylinder A in Fig. 9(b)) and that for the crack-freecylinder of inner radius Ri and outer radius Ro � a (Cylinder B inFig. 9(b)), that is

pL

sy¼ pL½Cylinder A�

syþ pL½Cylinder B�

sy(64)

In Eq. (64), a simple addition for the limit pressures for the twocylinders is used because Cylinder B in Fig. 9(b) is defect-free andthe pressure transfer factor from the inner surface of Cylinder B(Fig. 9(b)) at Ri to the 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) byapplying Eq. (56) to a cylinder of inner radius Ro � a and outerradius Ro with a through-wall crack of length 2c and the limitpressure for the defect-free cylinder of inner radius Ri and outerradius Ro � a (Cylinder B in Fig. 9(b)), the limit pressure of a thick-walled cylinder with an external surface crack can be obtained fromEq. (64) and expressed as

� 1Maxn

�at

tRi

�359>=>;

1Maxn

�atðk� 1Þ

�35

(65)


where the stress magnification factor, Maxn, can be obtained byapplying Eq. (53) to Cylinder A in Fig. 9(b) and expressed as

Maxn ¼�

1þ 1:4c2

Roa

�0:5

¼

1þ 1:4ðk� 1Þa

t

kac

�2

!0:5

fork

k� atðk� 1Þ

� 2 (66)

Eq. (65) reduces to Eq. (56) for through-wall cracks when a=t/1and to Eq. (13) for external extended cracks when a=c/0 anda=t > 0. It also reproduces the limit pressure for crack-free thick-walled cylinders when a=t ¼ 0 or a=c/N.

The new solution, Eq. (65) with g ¼ 2=ffiffiffi3p

, is compared withother existing solutions and the FE data due to Staat and Vu [12] inFig. 14 and those due to Zarrabi [16] in Fig. 15. From the figures, Eq.(65) has largely removed the non-conservatism of the solution dueto Staat and Vu [12] for deep and short cracks. It is also seen fromthe figures that the predictions using Eq. (65) are very close to theFE data and conservative for all cases shown in Figs. 14 and 15except for the cases with very shallow cracks, where the FE resultsare 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 obtainedfrom the methodology used by Carter [9] (see Section 3.4.1above) based on the new limit load solutions for thick-walledcylinders with internal surface cracks (Eq. (62)) and the limitload solution for thick-walled cylinders with internal extendedcracks under internal pressure due to Staat and Vu [12] (Eq. (9)).Following Carter [9], the local limit pressure for a thick-walledcylinder with an internal surface crack of depth a and length 2ccan be expressed as the weighted sum of the limit pressures ofa cylinder of length 2c with an internal extended crack of deptha and two crack-free cylinders of length h1 (refer to Fig. 16(a)with c1 replaced by h1), that is

pL

sy¼ 1

h1 þ c

"h1ln

�Ro

Ri

�þ c

Ri

R*2

Ri þ aRi

ln�

Ro

Ri þ a

�#

z1

h1

cþ 1

�h1

cln ðkÞ þ fps

�1þ a

tðk� 1Þ

�ln�

k

1þ atðk� 1Þ

� (67)

where Ri=R*2 ¼ 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) butusing Eq. (62) as the global limit pressure for a thick-walledcylinder with an internal surface crack and Eq. (9) as the limitpressure for a thick-walled cylinder with an internal extendedcrack. The result can be expressed as

h1

c¼

1� a

t

�fps

�ln�

RiþaRi

�� a

ttRi

ln�

Ro

Ri þ a

��

Man

hln�

RoRi

�� fps

�1

Manln�

RiþaRi

�þ�

1þ aRi

�1� 1

Man

��ln�

RoRiþa

��i

¼ð1� a

t

�fps

�ln1þ a

tðk� 1Þ�� a

tðk� 1Þln�

k1þa

tðk�1Þ

��Man

hln ðkÞ � fps

�1

Manln1þ a

tðk� 1Þ�þ�

1þ atðk� 1Þ

�1� 1

Man

��ln�

k1þa

tðk�1Þ

��i(68)

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 withthe FE data below shows the solution based on the von Misesyield criterion may be non-conservative for short and shallowcracks.

The new solution, Eq. (67), is compared with other existingsolutions and the FE data due to Jun et al. [15] in Figs. 17–19 fork¼ 1.05, 1.11 and 1.22, respectively. From the figures, the limitpressure obtained using Eq. (67) is very close to, but slightlyhigher than that predicted using Carter’s solution. It is alsoseen from the figures that the predictions using Eq. (67) arereasonably close to and conservative compared with the FEresults for all cases shown in Figs. 17–19. The conservatism ofEq. (67) may increase with increase of k, noting the trendsshown in Figs. 17–19.

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 obtainedfrom the methodology used by Carter [9] (see Section 3.4.2 above)based on the new limit load solutions for thick-walled cylinderswith external surface cracks (Eq. (65)) and the limit load solutionfor thick-walled cylinders with external extended cracks underinternal pressure due to Staat and Vu [12] (Eq. (13)). FollowingCarter [9], the local limit pressure for a thick-walled cylinder withan external surface crack of depth a and length 2c can be expressedas the weighted sum of the limit pressures of a cylinder of length 2cwith an external extended crack of depth a and two crack-freecylinders of length h2 (refer to Fig. 16(b) with c2 replaced by h2),that is

pL

sy¼ 1

h2 þ c

�h2ln

�Ro

Ri

�þ cln

�Ro � a

Ri

�

¼ 1h2

cþ 1

�h2

cln ðkÞ þ ln

�k� a

tðk� 1Þ

�(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 anexternal surface crack and Eq. (13) as the limit pressure for a thick-walled cylinder with an external extended crack. The result can beexpressed as

h2

c¼

1� at

Maxn � 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 reasongiven in Section 4.2.3 for internal cracks.

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

Table A1Numerical solution of Folias factor [27,17].

rm Mt

0.110011 1.00960.220022 1.03710.330033 1.07950.440044 1.13440.550055 1.19930.660066 1.27230.770077 1.35190.880088 1.43670.990099 1.52561.10011 1.61771.210121 1.71221.320132 1.80851.430143 1.9061.540154 2.00451.650165 2.10351.787679 2.22761.925193 2.35192.062706 2.47612.20022 2.59992.337734 2.72322.475248 2.8459


5. Conclusions

1. The limit load solutions for axially cracked cylinders have beenreviewed and compared with available FE results. The findingsare as follows.(1) For extended internal/external cracks under internal pres-

sure, solutions due to Staat and Vu (Eqs. (9) and (13)) are forthick-walled cylinders and give the best predictions of theavailable FE results.

(2) For through-wall cracks under internal pressure, the solu-tion due to Staat and Vu (Eq. (24)) is for thick-walledcylinders and gives the best predictions of available FEresults for both thin-walled and thick-walled cylinders.However, it is non-conservative for short cracks becausethe back-wall correction in the equation is incorrect andthe stress magnification factor needs to be re-calibrated.

(3) For the global limit pressure of internal surface cracks, thesolution due to Staat and Vu (Eq. (31)) is for thick-walledcylinders and gives the best prediction of available FE resultsfor both thin-walled and thick-walled cylinders. However, itover-estimates the FE results for short and shallow cracks dueto the problems in the solution for through-wall cracksaddressed in (2) and the pressure amplifying factor, ðRiþaÞ=Ri,applied to the term corresponding to the crack-free cylinder.

(4) For the global limit pressure of external surface cracks, thesolution due to Staat and Vu (Eq. (37)) is for thick-walledcylinders and gives the best prediction of available FE resultsfor thick-walled cylinders. However, it over-estimates the FEresults for short and through-wall cracks due to the problemin the solution for through-wall cracks addressed in (2).

(5) For the local limit pressures of internal/external surfacecracks, Carter’s solutions (Eqs. (42) and (48)) are for thick-walled cylinders and give reasonably good and conserva-tive predictions of available FE results for thin-walledcylinders. However, the expressions for the local limitpressure 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 Staatand Vu (Eqs. (45) and (50)) are for thick-walled cylinders.However, the solution for internal cracks (Eq. (45)) withg ¼ 2=

ffiffiffi3p

is non-conservative for short and shallowcracks, especially for the cylinder with a very thin wallcompared with the available FE results.

(6) Little information for the effect of other load types, such asaxial tension and global bending moment, on the limitpressure of a cylinder with an axial crack can be found.Limit load solutions for axially cracked cylinders undercombined internal pressure, tension and global bending arecurrently lacking.

2. New limit pressure solutions for thick-walled cylinders withaxial cracks under internal pressure have been developed toovercome the problems addressed in Conclusion 1, above. Thenew 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 beencompared with available FE data and the results show that thepredictions using the new solutions are conservative and agreewell with the FE results.

2.750275 3.08953.025303 3.33033.30033 3.56813.575358 3.80293.850385 4.03474.125413 4.26374.40044 4.4895

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 ofBritish Energy Generation Ltd.

Appendix A. Folias factor

The Folias factor is a stress magnification factor due to thecurvature of shells and was first reported by Folias [26] to addressthe stress increase in the near crack tip area in a thin-walledspherical vessel with a fully penetrating crack under internalpressure. Folias [19] then derived the factor for a thin-walledcylindrical vessel with a penetrating axial or circumferential crackunder internal pressure, based on elastic thin-shell theory. At thattime, Folias [19] obtained a theoretical solution for the stressmagnification factor for axial cracks only for rm � 0:55 andexpressed it as

Mt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ fr2

m

q(A1)

with

f ¼ 1:61 forrm � 0:55 (A2)

where

rm ¼cffiffiffiffiffiffiffiffiffiRmtp (A3)

Later, Erdogan and Kibler [27] solved the problem numericallyand obtained the solution for axial cracks for rm � 4:4. Theresults are tabulated in Table A1. Folias [28] found that thenumerical results could still be expressed in the form of Eq. (A1),but the coefficient f ¼ 1:05 provided a good fit for the data,that is,

f ¼ 1:05 forrm � 4:4 (A4)

Kiefner et al. [3] found that the limit pressure data from bursttests of pipes with through-wall defects could be well correlatedusing a Folias factor. In their paper [3], Kiefner et al. fitted the Folias

Front wall with aBack wall N


factor data shown in Table A1 [27,17] and found the data could bewell represented by the following equation

Mt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1:255 r2

m � 0:0135 r4m

q(A5)

Fig. A1 compares the three equations with the numerical data inTable A1. From Fig. A1, Eqs. (A5) and (A1) with f ¼ 1:61 or 1.05 canpredict the numerical data very well in the region rm � 0:55. It isalso seen that Eq. (A1) with f ¼ 1:05 is a good representation andEq. (A5) is the best fit of the data in the region rm � 4:4. However, Eq.(A1) with f ¼ 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. foundthat it could correlate their experimental data very well. The authorhas 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 theoreticalsolution 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 ofa cylinder with an axial crack

For a cylinder with an axial defect under internal pressure, theglobal limit load of the defective cylinder is the pressure corre-sponding to the plastic collapse of both the front-wall of the cylindercontaining the defect and the defect-free back-wall. The front-wall isweaker than the back-wall due to the defect. Denoting the pressurecorresponding to the collapse of the front-wall, pLf , the total globallimit pressure can be expressed as pLf þ DpL, where DpL is the extrapressure the back-wall can bear after the onset of the front-wallcollapse. For thin-walled cylinders, DpL is negligible. However, itmay become significant for cylinders with very thick walls. In thisAppendix, DpL for through-wall and surface cracks will be estimated.

The back-wall of a cracked cylinder can be treated as a plateof thickness t subjected to combined tension force, NL, and bendingmoment, MLp, due to the internal pressure, pLf þ DpL. The limit loadof an uncracked plate with a thickness t and unit width undercombined tension and bending can be expressed as [29]

NL

syt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4l2 þ 1

q� 2l (B1)

l ¼MLp

tNL(B2)

where l is the load ratio.

0

1

2

3

4

5

6

0 1 2 3 4 5ρm

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.

Cylinder with through-wall cracks

For a cylinder with an axial through-wall crack of length 2csubjected to internal pressure, the tensile force, NL, and the moment,MLp, in the back-wall due to the internal pressure, pLf þ DpL, are asfollows (see Fig. B1). The resultant force and moment in the back-wall can be obtained by taking the force equilibrium along thedirection normal to the crack face and moment equilibrium in theback-wall, assuming that the back-wall only bears half of the forcedue to pLf but the full force due to DpL, and expressed as

NL ¼ 2R*t DpL þ R*

t pLf ¼ R*t

�2DpL þ pLf

�(B3)

MLp ¼�

NL � R*t pLf

��R*

t þt2

�¼ 2R*

t DpL

�R*

t þt2

�(B4)

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

l ¼MLp

NLt¼

2DpL

�1þ 1

2t

R*t

!�

2DpL þ pLf

� t

R*t

(B5)

In Eqs. (B3)–(B5), R*t is the equivalent radius to include the effect of

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

R*t ¼


Ri þt2


((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 beexpressed as

DpL

sy¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 1

2ð1þ 1

MtnÞ t

R*t

!2

þ14

1� 1

M2tn

! t

R*t

!2vuut

�

1þ 12

�1þ 1

Mtn

�t

R*t

!ðB7Þ

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

pLfsy

zt

R*t

1Mtn

(B8)

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

Ri

t

pLpLf Δ+

through-wall crackL

MLp

2R*t

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

of crack face pressure).

R0

NL Front wallBack wall

a

t

MLp

pLpLf Δ+

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


Cylinder with internal surface crack

For a cylinder with an axial internal surface crack of length 2cand depth a subjected to internal pressure, the back-wall effect isonly from the cylinder of inner radius Ri and thickness a witha through-wall crack of length 2c (Fig. B2). The normalisedpressure increase due to the back-wall effect, DpL=sy, for this casecan be obtained directly from Eq. (B7) by replacing t, R*

t and Mtn

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

expressed as

DpL

sy¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 1

2ð1þ 1

ManÞat

t

R*2

!2

þ14

1� 1

M2an

! at

t

R*2

!2vuut

�

1þ 12

�1þ 1

Man

�at

t

R*2

!ðB9Þ

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 isonly from the cylinder of inner radius Ro � a and thickness a witha through-wall crack of length 2c (Fig. B3). The normalised pressureincrease due to the back-wall effect, DpL=sy, for this case can beobtained 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 factorðRo � aÞ=Ri to the right-hand side of Eq. (B7). The result can beexpressed as

DpL

sy¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k� 1

2ð1� 1

MaxnÞat

tRi

�2

þ14

1� 1

M2axn

!�at

tRi

�2vuut��

k� 12

�1� 1

Maxn

�at

tRi

�ðB10Þ

where Maxn is defined in Eq. (66).

Appendix C. Calibration of the stress magnification factorfor cylinders with through-wall cracks

The limit pressure, pL, for a cylinder with a through-wall cracksubjected to internal pressure may generally be expressed as

Ri

t

pLpLf Δ+

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 the

case of crack face pressure).

pL � DpL

p0¼ 1

Mtn(C1)

where p0 is the limit pressure for crack-free cylinders, DpL is thepressure increase due to the back-wall effect (see Appendix B) andMtn is the stress magnification factor. Staat and Vu have shown thatMtn can be expressed by the following equation

Mtn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ fr2

o

q(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). Thecoefficient f may be calibrated from FE or experimental data.

Combining Eqs. (C1) and (C2), the relationship betweenp0=ðpL � DpLÞ and ro is as follows

�p0

pL � DpL

�2

¼ 1þ fr2o (C3)

This equation may be used to calibrate f. Fig. C1 shows the FElimit pressure data for cylinders with through-wall cracks underinternal pressure (without crack face pressure) due to Staat and Vu[12], plotted as ðp0=pLÞ2 against r2

o. From Fig. C1, the data for variousk are widely scattered with increasing ro and the coefficient, f, maydepend on k. Moreover, the relationship between ðp0=pLÞ2 and r2

o isnon-linear for big k values. The FE data are then re-plotted in Fig. C2considering the back-wall effect, DpL. From the figure, the FE datafor all k values considered tend to collapse to one line and can berepresented by a straight line with a slope f ¼ 1:4. Note that DpL isa function of Mtn (see Eq. (B7) in Appendix B) and, therefore, f. Theresult f ¼ 1:4 was obtained by increasing f gradually and checking

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 ¼ 0.

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 = 2φ = 1.4

1

1.4

(p0

/(p L

- p

L ))

2Δ

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


the agreement with the FE data to obtain an upper-bound esti-mation of Mtn for all the FE data.

References

[1] R6, Assessment of the Integrity of Structures Containing Defects, Revision 4,British Energy Generation Ltd., 2001, with amendments to May 2007.

[2] Miller AG. Review of limit loads of structures containing defects. InternationalJournal of Pressure Vessels and Piping 1988;32:197–327.

[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. 461–481.

[4] Kitching R, Davis JK, Gill SS. Limit pressures for cylindrical shells with unre-inforced openings of various shapes. Journal of Mechanical EngineeringScience 1970;12:313–30.

[5] Kitching R, Zarrabi K. Limit and burst pressures for cylindrical shells withpart-through slots. International Journal of Pressure Vessels and Piping 1982;10:235–70.

[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. Elastic–plastic 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;234:25–43.

[11] Staat M. Local and global collapse pressure of longitudinally flawed pipes andcylindrical vessels. International Journal of Pressure Vessels and Piping2005;82:217–25.

[12] Staat M, Vu Duc Khoi. Limit analysis of flaws in pressurized pipes andcylindrical vessels. Part I: axial defects. Engineering Fracture Mechanics2007;74:431–50.

[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;79:321–30.

[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;80:527–40.

[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;373:523–8.

[16] Zarrabi K, Zhang H, Nhim K. Plastic collapse pressure of cylindrical vesselscontaining longitudinal surface cracks. Nuclear Engineering and Design1997;168:313–7.

[17] Folias ES. On the effect of initial curvature on cracked flat sheets. InternationalJournal of Fracture Mechanics 1969;5:327–46.

[18] Erdogan F. Ductile failure theories for pressurized pipes and containers.International Journal of Pressure Vessels and Piping 1976;4:253–83.

[19] Folias ES. An axial crack in a pressurized cylindrical shell. International Journalof Fracture Mechanics 1965;1:104–13.

[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 4–1.

[22] Goodall IW, Webster GA. Theoretical determination of reference stress forpartially penetrating flaws in plates. International Journal of Pressure Vesselsand Piping 2001;78:687–95.

[23] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks inplates under bending. International Journal of Pressure Vessels and Piping2004;81:31–41.

[24] Lei Y. A global limit load solution for plates with semi-elliptical surface cracksunder combined tension and bending. ASME PVP 2004;475:125–31.

[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. 2169–2174.

[26] Folias ES. A finite line crack in a pressurized spherical shell. InternationalJournal of Fracture Mechanics 1965;1:20–46.

[27] Erdogan F, Kibler JJ. Cylindrical and spherical shells with cracks. InternationalJournal of Fracture Mechanics 1969;5:229–37.

[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 inplates under combined tension and bending. International Journal of PressureVessels and Piping 2004;81:43–56.

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