Paper by Budden and Ainsworth

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The shape of a strain-based failure assessment diagram

P.J. Budden*, R.A. Ainsworth 1

EDF Energy, Barnett Way, Barnwood, Gloucester GL4 3RS, UK

a r t i c l e i n f o

Article history:

Received 10 February 2011

Received in revised form

8 July 2011

Accepted 25 September 2011

Keywords:

Failure assessment diagram

Fracture

Defects

Strain-based

a b s t r a c t

There have been a number of recent developments of strain-based fracture assessment approaches,

including proposals by Budden [Engng Frac Mech 2006;73:537e52] for a strain-based failure assessment

diagram (FAD) related to the conventional stress-based FAD. However, recent comparisons with finiteelement (FE) data have shown that this proposed strain-based FAD can be non-conservative in some

cases, particularly for deeper cracks and materials with little strain-hardening capacity. Therefore, this

paper re-examines the shape of the strain-based FAD, guided by these FE analyses and some theoretical

analysis. On this basis, modified proposals for the shape of the strain-based FAD are given, including

simplified and more detailed options in line with the options available for stress-based FADs in existing

fitness-for-service procedures. The proposals are then illustrated by a worked example and by

comparison with FE data, which demonstrate that the new proposals are generally conservative.

Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Early developments of strain-based fracture assessmentmethods for structures containing crack-like flaws, for exampleBurdekin and Dawes [1], have been largely superseded by stress-

based methods, see for example [2e5], as discussed by Zerbstet al. [6]. The stress-based methods have been developedto addressa number of issues of practical importance such as constraint,strength mismatch and the treatment of combined primary and

secondary stresses [6]. This has led to comprehensive assessmentprocedures based on these methods [2e5]. However, a strain-basedapproach may be more appropriate for some practical situationswhere strain or displacement is the natural boundary conditions

and imposed plastic strains can be large; for example, pipe reelingor laying operations in the pipeline industry, or generally caseswhere applied displacements may be limited. The conservatism of

stress-based approaches based on elastic analysis of the uncrackedstructure can sometimes be significant in such cases.

Recently there has been a renewed interest in strain-basedfracture assessment methods driven by particular applications[7e9], including the development of a strain-based failure assess-ment diagram (FAD) [10] related to the FAD in the stress-based

methods [2e5]. This included proposals for both approximate

(Option 1) and detailed (Option 2) strain-based FADs similar to the

stress-based Option 1 and Option 2 FADs in R6 [2]. However, Bud-

den [11,12] has noted, from comparisons with detailed finiteelement (FE) data, that the strain-based FADs proposed in [10] canbe non-conservative in some cases, particularly for higher values of

the material strain-hardening coef ficient and for deeper cracks. Thenon-conservatism typically corresponded to a factor of 2 on theapplied strain.

In this paper, the shape of the strain-based FAD is examined tosee if this non-conservatism can be removed. The discussion is in

the context of strains due to primary loads alone as used toconstruct the FAD [2,10]; the inclusion of secondary strains is dis-cussed in a companion paper [13]. First, Section 2 briefly reviewsthe stress-based and strain-based FAD approaches. Then, revised

proposals for the shape for the strain-based FAD, relative to those in[10], are developed in Section 3. Finally, Section 4 provides

a worked example of the proposals and comparisons with FEresults in the format of the strain-based FAD.

2. Stress and strain-based FADs

2.1. Stress-based failure assessment diagram

In the stress-based FAD [2e6] approach, fracture is assessedusing two parameters Lr and K r defined as follows:

Lr ¼ P =P L

Àa; sy

Á¼ s

pref

=sy (1)

* Corresponding author. Tel.: þ44 1452 653824.

E-mail address: [email protected] (P.J. Budden).1 Now at University of Manchester, Pariser Building, Sackville Street, Manchester

M13 9PL, UK.

Contents lists available at SciVerse ScienceDirect

International Journal of Pressure Vessels and Piping

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

0308-0161/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijpvp.2011.09.004

International Journal of Pressure Vessels and Piping xxx (2011) 1e8

Please cite this article in press as: Budden PJ, Ainsworth RA, The shape of a strain-based failure assessment diagram, International Journal of Pressure Vessels and Piping (2011), doi:10.1016/j.ijpvp.2011.09.004

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K r ¼ K p=K mat (2)

where P is the applied primary loading; P L (a,sy) is the corre-sponding limit load for the component with a crack of size a made

of rigid plastic material with yield stress equal to the 0.2% proof stress, sy, the ratio P /P L in turn defines a primary reference stress,sref

p ; K p is the stress intensity factor for the primary stresses andK mat is fracture toughness. In equations (1) and (2), the superscript

“p” is used to denote “primary” loading. More generally, thetreatment of secondary stresses can be included in the FAD [2e6]

but this is not considered here as it does not affect the shape of the FAD.

In the stress-based approach, Lr and K r are evaluated for theapplied loads and failure is conceded when K r ¼ f (Lr)or Lr ¼ Lr

max asdepicted in Fig.1. Here Lr

max is the ratio of a flow stress to the yield

stress and allows for strain hardening beyond yield. The failureassessment curve may be described in a number of ways and twoparticular curves are considered here for the subsequent devel-opment of the strain-based FAD. The first is that termed R6

Option 2 [2]:

f ðLrÞ ¼

264E 3

pref

spref

þ1=2

s

pref

=sy

2

E 3pref =s

pref

375

À1=2

(3)

where 3ref p is the strain on the material stressestrain curve at the

stress sref p (¼Lrsy) and E is Young’s modulus. Equation (3) corre-

sponds to an approximate J -estimate approach [14] which can beused with any description of the material stressestrain curve. Thesecond curve is termed Option 1 in R6 [2]:

f ðLrÞ ¼h

1 þ 0:5L2ri

À1=2h

0:3 þ 0:7exp

À0:6L6ri

(4)

and the shape of this is independent of material.

2.2. Strain-based failure assessment diagram

In the strain-based failure assessment diagram approach,equation (1) is replaced by a strain ratio, Dr, defined by

Dr ¼ 3pref = 3y (5)

where 3ref p is the imposed strain [10]. For small cracks which do not

affect the overall compliance of a component, 3ref p may be taken as

the uncracked-body equivalent strain at the location of the crack

[9]. Secondary strains, such as residual strains following welding forexample, are not considered here but are considered ina companion paper [13].

Thedefinition of K r inequation(2) isunchangedin the strain-basedFADbut the stress intensityfactor, K p, isdeduced froma stressdefinedin terms of the imposed strain [10]. That is, K p is defined by

K p ¼ F spref ðpaÞ

1=2 (6)

where sref p is obtained from the imposed strain 3ref

p as

spref ¼ s

3pref

(7)

where s( 3) represents the material stressestrain curve and F is the

dimensionless stress intensity factor function [10].

Nomenclature

a, c crack depth, crack half lengthDr parameter in the strain-based FADE Young’s modulus f stress-based failure assessment curve

f * strain-based failure assessment curveF stress intensity factor functionFE finite elementFAD failure assessment diagram

h1 normalised J valueh3 normalised displacement J elasticeplastic crack tip parameter J p fully plastic value of J

J el elastically calculated value of JK stress intensity factorK p stress intensity factor for primary loadsK r ordinate on the FAD

K mat fracture toughnessLr abscissa in the stress-based FADLr

max ratio of flow stress to 0.2% proof stress[1 normalised length in J estimation

[3 normalised length in displacement estimationn constant in power-law stressestrain equation

P applied loadP o normalising load proportional to so

P L limit loadRm mean radius of cylinder

t thickness of cylinder or plateU area under loadedisplacement curveU el elastic area under loadedisplacement curveW half width of platea constant in power-law stressestrain equationD displacementDel elastic displacement

3 strain

3o constant in power-law stressestrain equation 3ref

p applied strain

3y yield strain (¼sy/E )s stresssu ultimate stresssy 0.2% proof stressso constant in power-law stressestrain equation

sref p primary reference stress

Fig. 1. A schematic R6 stress-based failure assessment diagram; the function f (Lr) can

be described by Option 1, 2 or 3 curves and the plastic collapse cut-off Lr¼ Lrmax is also

shown [2].

P.J. Budden, R.A. Ainsworth / International Journal of Pressure Vessels and Piping xxx (2011) 1e82




Having evaluated K r and Dr, failure is conceded when

K r ¼ f *ðDrÞ (8)

as depicted in Fig. 2. A number of options for defining f *(Dr) areproposed in Section 3.

3. Shape of the strain-based FAD

In this section, the shape of the strain-based FAD, i.e. f *(Dr) inequation (8), is examinedfirst for large-scale yielding in Section 3.1.Then small-scale yielding corrections are added in Section 3.2,

enabling a number of options for the shape of the strain-based FADto be given in Sections 3.3e3.5. Similarities and differences fromthe proposals of Budden [10] are highlighted.

3.1. Large-scale yielding

Ainsworth [14] showed that, for a given load, the plastic

component of the J -integral, J p, and the corresponding elastic value, J el, are related under large-scale yielding conditions, by

J = J ely

J p= J el ¼ E 3p

ref =s

p

ref (9)where E is Young’s modulus. Equation (9) is essentially the R6

reference stress approximation for J of equation (3) without thesmall-scale yielding correction.

For power-law materials in which strain is related to stress by

3 ¼ a 3oðs=soÞn (10)

where a, so, 3o, n are constants with a 3o/so ¼ 1/E , the fully plastic

value of J can be written

J ¼ aso 3oh1ðnÞ[1ðP =P oÞnþ1 (11)

where [1 is a convenient normalising length, P o is a normalisingload proportional to so and h1(n) is the normalised value of J which

generally depends on n, geometry and loading [15]. Equation (9) isequivalent to equation (11) when P o is chosen so that h1(n) ¼ h1(1)

with sref p ¼ P so/P o [14].

For the stressestrain curve of equation (10), the displacement,D, conjugate to the load P can be written in a similar form to

equation (11) as

D ¼ a 3oh3ðnÞ[3ðP =P oÞn (12)

where [3 is a convenient length and h3 is the normalised

displacement [15].Now, it has been observed for a number of common test spec-

imens that J can be related simply to the area, U , under theloadedeflection curve by a constant which is independent of

material behaviour and load magnitude. Thus, for these geometries

J = J el ¼ U =U el (13)

where U el ¼ P Del(P )/2 with Del(P ) the elastic displacement at load P ,and

U ¼

Z P

0

P dD (14)

Inserting equation (12) into equation (14), integrating andsubstituting equations (9) and (13) leads to

h3ðnÞ=h3ðn ¼ 1Þ ¼ ðn þ 1Þ=2n (15)

Thus the choice of P o, i.e. the choice of reference stress, which leadsto h1ðnÞ ¼ h1ðn ¼ 1Þ and hence equation (9) as a J estimate, leadsto values of h3ðnÞ which depend on n and are consistent with

a displacement estimate

D=Del ¼ðn þ 1Þ

2n

E 3pref

spref

(16)

at least for geometries for which equation (13) holds. Thus the

reference stress estimate of J in equation (9) is consistent witha modified reference stress estimate of displacement given byequation (16). Conversely, if the reference strain, 3

pref

, is estimatedfrom a displacement assuming simple scaling with elastic response

(i.e. P o is chosen so that the factor (n þ 1)/2n does not appear inequation (16)) then a modified reference stress estimate of J isobtained as

J

J el¼

2n

ðn þ 1Þ

E 3pref

spref

(17)

Thus, for large-scale yielding and low strain hardening (large n)with Dr defined by equation (5), then 2n=ðn þ 1Þy2 and equation(8) for the strain-based FAD becomes

f *ðDrÞ ¼

J el

J

1=2

¼

"2E 3

pref

spref

#À1=2

(18)

This is the equivalent of equation (9) at large Lr and is consideredmore appropriate in the strain-based route where the remotestrain/displacement boundary conditions are known.

It has to be recognised that equation (13) only holds for certaingeometries and is not general. Therefore the factor of [2 n/(n þ 1)]in eqn (17) is also not general. However, it is consistent with theobservations from [11,12] that the strain-based FAD proposed in

[10] can be non-conservative, particularly for high n, and typicallyby a factor of 2 on strain. Therefore, pragmatically equation (18) isused here to develop new proposals for strain-based FADs relative

to [10].

0

1

0 2 4 6 8 10Dr

Kr

)D(*f K rr

Fig. 2. A schematic strain-based failure assessment diagram; the function f *(Dr) can be

described by Option 1, 2 or 3 curves; a pragmatic strain limit on the Dr axis is also

imposed in [10].

P.J. Budden, R.A. Ainsworth / International Journal of Pressure Vessels and Piping xxx (2011) 1e8 3




3.2. Small-scale yielding

For small-scale yielding, global response is essentially elastic sothat n ¼ 1 is appropriate in equation (17). However, corrections forsmall-scale yielding are required as in the stress-based FAD of equation (3) so that a strain-based FAD [10] given by

f *ðDrÞ ¼

264E 3pref

spref

þ1=2

spref =sy

2

E 3pref

=spref

375À1=2

(19)

is again expected to hold. Budden [10] developed an Option 1

strain-based FAD by writing the R6 Option 2 stress-based FAD of equation (3) as

f À2ðLrÞ ¼ Dr=Lr þ 1=2L3r =Dr (20)

and solved this to relate Dr to Lr and f (Lr) by

Dr ¼h

Lr=2 f 2ðLrÞi2

41 þ

n1 À 2L2

r f 4ðLrÞo

1=2

3

5(21)

Budden then used the stress-based Option 1 FAD of equation (4) toderive Dr from equation (21) and hence develop a strain-basedOption 1 FAD defined by

f *ðDrÞ ¼

1 þ 1=2D2

r

À1=2; Dr < 1 (22a)

with the continuation for Dr > 1 given by equation (23), below.

However, the choice of f (Lr) clearly affects Dr through equation (21). Aconservative strain-based FADis developedby underestimating Dr foragiven Lr. As Dr ! Lr foranelasticeplastic material,in general,it isthenconservative to set Dr ¼ Lr in equations (20) or (21). This corresponds

to elastic behaviour up to yield and leads to f ðLrÞ ¼ ð1 þ 1=2L2r Þ

À1=2,that is the R6 Option 2 curve with only the small-scale yielding

correction, and hence to equation (22a). This small-scale yieldingcurve is an upper bound to the Option 2 curve for Lr < 1.

Having estimated Dr for a given Lr, it is conversely conservativeto take a lower bound f (Lr) to derive f *(Dr). Such a conservativefunction is the R6 Option 1 FAD of equation (4). This may besomewhat over conservative as an upper bound f (Lr) has been used

to derive Dr and a lower bound f (Lr) has been used to derive f *.Accepting this conservatism with Dr ¼ Lr then immediately leads to

f *ðDrÞ ¼

1 þ 1=2D2

r

À1=2h

0:3 þ 0:7exp

À0:6D6r

i; Dr <1

(22b)

3.3. Option 1 strain-based FAD

Budden [10] noted that it is conservative at large Dr to take

f *fDÀ1=2r . The factor of 2 in equation (18) does not affect this

argument. Therefore, it is proposed that equation (22b) iscontinued for Dr > 1 by

f *ðDrÞ ¼ f *ð1ÞDÀ1=2r (23)

which is identical to the proposal in [10], except that f *ð1Þ is nowevaluated from equation (22b) rather than equation (22a). Thecombination of equations (22b) and (23) is proposed here as the

Option 1 strain-based FAD.Instead of equation (22b), equation (22a) was used in [10]. At

small Dr, equations (22a) and (22b) are similar and both are

consistent with the FE data examined in [11,12]. The currentproposal however, leads to a lower value of f * at Dr ¼ 1 (0.559

instead of 0.816) and consequently a lower value for all Dr > 1 fromequation (23). In [11,12], it was noted that a factor of 2 on Dr inconjunction with the proposals in [10] led to a pragmaticallyconservative approach compared to some FE data. Applying this

factor of 2 to equations (22a)e(23) [10] leads to

f *ðDrÞ ¼ 0:816ð2DrÞÀ1=2 ¼ 0:577D

À1=2r ; Dr ! 1 (24)

whereas equations (22b) and (23) lead to

f *ðDrÞ ¼ 0:559ðDrÞÀ1=2; Dr ! 1 (25)

The ratio of equation (25) to equation (24) is 0.97 for all Dr ! 1.Thus, equations (22b) and (23) are consistent with the FE data

[11,12], at least at larger Dr. Comparisons with FE data will be dis-cussed further in Section 4.


An Option 2 strain-based FAD has been developed for large Dr inequation (18) with the corresponding result for small Dr given by

equation (19). These two equations clearly differ, with equation (18)producing a lower value at Dr ¼ 1, for example (noting that sp

ref will

generally be close to sy for Dr ¼ 1). However in [11] it was observedthat for some loading cases there is a sharp change in the strain-

based FAD in the region of Dr ¼ 1. Therefore, it is proposed herethat the Option2 curve issimplydefined by equations (18) and (19),for Dr > 1 and Dr < 1 respectively, with a material-dependentdiscontinuity at Dr ¼ 1. This discontinuity is similar to that which

occurs in Option 2 stress-based FADs for materials with a Lüdersstrain. In summary, the Option 2 strain-based FAD is described by

f *

ðDrÞ ¼264 Dr

spref

=syþ

1=2spref

=sy3

Dr

375À1=2

; Dr < 1 (26)

f *ðDrÞ ¼

"2Dr

spref

=sy

#À1=2; Dr>1 (27)

with f *(Dr) being a vertical line at Dr ¼ 1 joining the values fromequations (26) and (27). In these equations, sref

p is again the stress

on the stressestrain curve at the strain Dr 3y.At large Dr, the use of equation (27) is equivalent to the factor of

2 on Dr compared to the continued use of equation (26) which isone of the proposals in [10]. Thus the current proposals are likely to

be consistent with the FE data for low strain-hardening materials.

For higher strain-hardening materials, it might be possible toreduce conservatism by adjusting the factor of 2 in equation (27) toa value 2n/(n þ 1) if the strain-hardening exponent n can be esti-

mated or by adjusting the factor to enforce continuity in f *(Dr) atDr ¼ 1. However, the accuracy of these adjustments has not beenexamined in detail and therefore equation (27) is retained here toensure conservatism.


In [10], it was noted that an Option 3 strain-based FAD can be

deduced directly from FE data by plotting f *ðDrÞ ¼ ð J el= J Þ1=2 as

a function of Dr. This is fully consistent with the Option 3 stress-

based FAD in R6 [2] and is not discussed further here.



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4. Worked example and validation using finite element

results

Three cases are considered to illustrate the strain-based FADapproach developed in Section 3:

i. A thin-walled cylinder under tension with a semi-ellipticalexternal surface crack [16];

ii. A thin-walled cylinder under bending with a semi-elliptical,external surface crack at the position of maximum bending

stress [16];iii. A plate under tension with a semi-elliptical surface crack [17].

The first case is set out in some detail to serve as a worked

example. The other cases are set out more briefly.

4.1. Cylinder under tension

4.1.1. De fine geometry

The cylinder [16] has mean radius, Rm, and wall thickness, t ,equal to 221.1 mm and 15 mm, respectively, so that t /Rm ¼ 0.0678.

4.1.2. De fine defect

Semi-elliptical, external part-circumferential surface cracks areconsidered, each with a normalised crack depth, a, of a/t ¼ 0.2. Fourdifferent total crack lengths, 2c ¼ 25 mm, 50 mm, 75 mm and

100 mm, are analysed.

4.1.3. De fine material tensile properties

The material has a pure linear region up to the limit of pro-

portionality of 450 MPa, followed by plastic straining up to thefinalpoint on the stressestrain curve, which is taken as the ultimatetensile strength, su ¼ 588.5 MPa (see Fig. 3) corresponding toa plastic collapse limit on the R6 stress-based FAD of

Lmaxr hðsu þ syÞ=2sy ¼ 1:13. The strain-hardening coef ficient, n, is

estimated from the 0.2% proof stress, sy, and s

uusing methods in R6

[1] and is n ¼ 16.1. Hence the material has relatively low strain-

hardening capacity and there is little conservatism introduced byreplacing 2n/(n þ 1) by 2 in equation (18).

4.1.4. Calculate Dr

Dr is defined by the ratio of remote axial surface strain from theFE analysis to the yield strain. There is no redistribution of stress orstrain due to plasticity along the uncracked cylinder, so that thestress and strain at the crack surface position can be taken as the

remote values.

4.1.5. Calculate K r

The first value of load in the FE analyses of [16] equates to

a maximum Lr in equation (1) of close to 0.1. From the Option 1 R6stress-based failure assessment curve of equation (4), J / J e isapproximately 1.005 at that initial load point and hence the cor-responding J value at the mid-point of the surface crack is essen-

tially elastic. This, therefore defines K p in equation (2) for the firstvalue of strain. For other values of applied strain, K p is obtained byscaling this first value of K p by the ratio of the stress in Fig. 3 cor-responding to the applied strain to the stress corresponding to the

first value of strain.In these assessments, K mat is equation (2) at any applied strain is

taken as the K -equivalent of the FE value of J . Thus, each assessmentpoint (Dr, K r) should correspond to failure and the locus of assess-

ment points with increasing strain defines the failure assessmentcurve.

4.1.6. Failure assessment

The FE data of [16] are plotted as described above on the strain-based FAD in Fig. 4 and these are the Option 3 curves of Section 3.5.Fig. 4 also plots the Option 1 and Option 2 curves of equations (22b),

(23) and (26), (27), respectively, and the earlier Option 1 curve [10],

equations (22a)e(23). Note that the new Option 2 curve lies abovethe corresponding Option 1 curve, with both inside the earlierOption 1 line [10]. It can be seen that the Option 3 curves lie inside

the Option 1 curve of equations (22a)e(23) but thatthe curve for theshortest crack length, 2c ¼ 25 mm, is close to the Option 1 strain-based FAD. The amount of non-conservatism increases withincreasing crack length, 2c . The newly proposed Option 1 strain-

based FAD of equations (22b) and (23) effectively removes thenon-conservatism for all crack lengths, with only the curve for thelongest crack, 2c ¼ 100 mm, very slightly non-conservative. The newOption 2 curve of equations (26) and (27) is very close to the Option

3 curve for the shortest crack butis slightly non-conservative forthelonger cracks.

4.2. Cylinder under bending

The cylinder geometry, defect sizes and tensile properties for

the cylinder under bending are identical to those described inSection 4.1. The calculation of Dr uses the computed surface strainvalues and follows the same methodas described above for tension.The calculation of K r also follows the approach of Section 4.1 but for

the elastic J , J e, it is assumed that the stress profile is linear withdistancefrom the bending axis with the peak surface value given by

0

100

200

300

400

500

600

700

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Strain (Absolute)

S t r e s s ( M P a )

Fig. 3. Stresse

strain curve for cylinder FE analyses [16].

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

Dr

Kr

Fig. 4. Strain-based FAD for the cracked cylinder under tension.



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the stresscorresponding to the applied strain. The FE data for globalbending [16] are plotted on the strain-based FAD in Fig. 5. Similarobservations hold as for the tension case. The previous Option 1strain-based FAD [10] of equations (22a)e(23) is generally non-

conservative with respect to the FE (Option 3) curves for strainsin excess of yield, the non-conservatism increasing with cracklength. However, in each case, the Option 1 and Option 3 curvesbecome close for large strain levels. The Option 1 strain-based FAD

of equations (22b) and (23) proposed in this paper is conservativeor accurate for all crack lengths. The revised Option 2 curve of equations (26) and (27) is slightly non-conservative.

4.3. Plate under tension

Lei [17] analysed a surface-cracked semi-elliptical crack ina plate of total width W and thickness t under remote tension load.

The crack half surface length, c , was fixed in the FE analyses at30 mm, the plate half width W ¼ 4c , and the plate half length was

16c . Three ratios of crack depth to plate thickness ratio, a/t ¼ 0.2,0.5, 0.8 were considered and also three ratios of crack depth to half surface length, a/c ¼ 0.2, 0.6, 1.0. The plate material was describedby the Ramberg-Osgood expression:

3

3y¼ ssy

þ assy

n (28)

where the strain-hardening index was taken as n ¼ 5 or 10 and,without loss of generality, a¼ 1, E ¼ 500 MPa and sy ¼ 1 MPa.

Hence, the normalising strain 3y ¼ sy/E ¼ 0.002.For tension loading [17], as in the case of the cylinder, stresses in

the uncracked plate do not redistribute due to plasticity and thestrain corresponding to any particular stress level follows simply

from the stressestrain law of equation (28). The maximum valuesof J at the deepest point of the crack are used to plot Option 3curves.

The results are shown in Figs. 6 and 7. Fig. 6(aed) first shows the

results for the 3 different values of a/t for fixed ratios of a/c ¼ 0.2and 1.0 (the results for a/c ¼ 0.6 are similar), for the cases n ¼ 5 andn ¼ 10. Fig. 7(a,b) then plots the data for the shallowest crack, a/t ¼ 0.2, for the different values of a/c and n. In each case, calculation

of the abscissa, Dr, uses the strain from equation (28) at the appliedstress level. It can be seen that the FE data are generally accuratelyor conservatively represented by the Option 1 original strain-basedFAD [10] for the shallow crack, a/t ¼ 0.2. However, for the deeper

cracks, a/t ¼ 0.5 and a/t ¼ 0.8, the results are non-conservative. Thenon-conservatism increases with increasing a/t or strain-hardeningexponent, n. Using the new Option 1 curve of equations (22b) and(23), it can be seen that the Option 1 curve is consistently conser-

vative when n ¼ 5 (Fig. 6a, c). However, for n ¼ 10, equations (22b)

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16 18 20

Dr

Kr

Fig. 5. Strain-based FAD for the cracked cylinder under global bending.

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

Dr

Kr

FE (a/t=0.2)

Option 1 (Eqns 22a, 23)

FE (a/t=0.5)

FE (a/t=0.8)

Option 1 (Eqns 22b, 23)

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Dr

Kr

FE (a/t=0.2)Option 1 (Eqns 22a, 23)

FE (a/t=0.5)

FE (a/t=0.8)


0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

Dr

Kr

FE (a/t=0.2)

Option 1 (Eqns (22a, 23)

FE (a/t=0.5)

FE (a/t=0.8)


0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60Dr

Kr

FE (a/t=0.2)Option 1 (Eqns 22a, 23)

FE (a/t=0.5)

FE (a/t=0.8)


a b

c d

Fig. 6. Strain-based FAD for the cracked plate under tension (a) a/c ¼

0.2, n¼

5; (b) a/c ¼

0.2, n¼

10; (c) a/c ¼

1, n¼

5; (d) a/c ¼

1, n¼

10.



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and (23) are conservative when a/t ¼ 0.2, 0.5 but are slightly non-conservative when a/t ¼ 0.8 (Fig. 6b, d). The conservatism of equations (22b) and (23) for the shallowest crack, a/t ¼ 0.2, can be

seen from Fig. 7 to be greater than that from the original formu-lation of equations (22a) and (23) [10].

5. Discussion

The revised proposals for the shape of a strain-based FADpresented in this paper represent increased conservatism relativeto earlier proposals by Budden [10]. This increased conservatismhas been shown in Section 4 to lead to generally conservative

assessments compared to FE data for surface cracks in cylindersunder tension or bending and plates under tension. However,even with the increased conservatism, the new proposals do notbound the finite element data for deep cracks. This is not

surprising as the input to the strain-based FAD has been based onthe remote strain in the uncracked structure. The new Option 1strain-based FAD is generally more conservative than Option 2and bounds more of the FE data; appropriate sensitivity studies

should be considered when performing assessments usingOption 2.

For practical assessments using the strain-based FAD, the FEdata suggestthat thecrack depth should be limited to about 20%of

the wall thickness, that is a/t < 0.2, when basing the definition of reference strain on the response of the uncracked body. For deeper

cracks, the effect of the defect on the compliance of the structure

will be significant and methods for treating this case on the strain-based FAD have not been considered here. The new strain-based

approach tends to be in closer agreement with the FE data forthe cylinder compared with that for the plate, where the newapproach is generally more conservative. It is considered that thisis due to the lower strain-hardening capacity of the cylinder

material.Another limitation of the approach presented here is that

secondary strains, such as those due to the welding process, havenot been considered. This is addressed in a companion paper [13]

where it is shown that the current methods used in stress-basedFADs [2] can be extended to the strain-based FADs developed inthis paper for the combined loading case. It is expected that otherissues such as loss of constraint and weld mismatch could also be

addressed to formulate comprehensive strain-based failureassessment diagram methods but such developments are notconsidered here.

6. Conclusions

Guided by some theoretical analysis and results from FE anal-

ysis, this paper has developed proposals for the shape of a strain-based failure assessment diagram. The proposals represent

increased conservatism compared to earlier proposals of Budden[10]. The new proposals have been compared to FE data for surfacecracks in plates and cylinders and shown to be generally conser-vative. As the inputs to the strain-based FAD are based on the

strains in the uncracked body, it is suggested that practical appli-cations of the methods proposed here should be limited to flawsizes no greater than 20% of the wall thickness.

Acknowledgements

This paper is published by permission of EDF Energy. Theauthors gratefully acknowledge the provision of detailed finite

element data by W. Xu, W. He (TWI) and Y. Lei (EDF Energy). Thework developed here was the subject of a review by a TAGSI (The

UK Technical Advisory Group on the Structural Integrity of HighIntegrity Plant) sub-group chaired by Prof J.W. Hancock; the inputfrom that group is gratefully acknowledged.

References

[1] Burdekin FM, Dawes MG. Practical use of linear elastic and yielding fracturemechanics with particular reference to pressure vessels. In: Conference on thepractical application of fracture mechanics to pressure vessel technology.London: I Mech E; 1971. p. 28e37.

[2] R6Revision4, Assessmentof theintegrityof structurescontainingdefects, BritishEnergy, 2001, with Amendments to March 2010, British Energy, Gloucester.

[3] BS7910:2005. Guide to methods for assessing the acceptability of flaws inmetallic structures, incorporating amendment 1. London: British Standards

Institution; 2007.[4] Koçak M, Webster S, Janosch JJ, Ainsworth RA, Koers R, editors. FITNET fitness

for service procedure. Geesthacht, Germany: GKSS; 2008.[5] API 579-1/ASME FFS-1. Fitness-for-service. Washington, DC: American

Petroleum Institute/ASME; 2007.[6] Zerbst U, Schödel M, Webster S, Ainsworth RA. Fitness-for-service fracture

assessment of structures containing cracks, a workbook based on the Euro-pean SINTAP/FITNET procedure. Amsterdam: Elsevier; 2007.

[7] Østby E, Jayadevan KR, Thaulow C. Fracture response of pipelines subjectto large plastic deformation under bending. Int J Pres Ves Piping 2005;82:201e15.

[8] Wang Y-Y, Rudland D, Denys R, Horsley D. A preliminary strain-based designcriterion for pipeline girth welds. In: Proc 4th int. pipeline conference IPC’02,Calgary. New York: ASME; 2002.

[9] Linkens D, Formby CL, Ainsworth RA. A strain-based approach to fractureassessment e example applications. In: Proc conf engng structural assess-ment. Cambridge: EMAS; 2000.

[10] Budden PJ. Failure assessment diagram methods for strain-based fracture.

Engng Fract Mech 2006;73:537e

52.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10Dr

Kr

a/c=0.2

Option 1 (Eqns (22a,23)

a/c=0.6

a/c=1

Option 1 (Eqns (22b,23)

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Dr

Kr

a/c=0.2

Option 1 (Eqns 22a,23)

a/c=0.6

a/c=1

Option 1 (Eqns 22b,23)

a

b

Fig. 7. Strain-based FAD for the cracked plate under tension (a) a/t ¼0.2, n¼5; (b) a/

t ¼0.2, n¼10.



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[11] Budden PJ. Validation of the new R6 strain-based failure assessment diagramapproach using finite element data. British Energy Report E/REP/BBGB/0044/GEN/08 Revision 001. Gloucester: British Energy; 2009.

[12] Budden PJ, Smith MC. Numerical validation of a strain-based failure assess-ment diagram approach to fracture. In: Proc ASME PVP2009, PVP2009-77377,Prague, July 26e30, 2009.

[13] Ainsworth RA, Budden PJ, Oh C-Y, Kim Y-J. The treatment of secondary strainswithin a strain-based failure assessment diagram, submitted for publication.

[14] Ainsworth RA. The assessment of defects in structures of strain hardeningmaterial. Engng Fract Mech 1984;19:633e42.

[15] Kumar V, German MD, Shih CF. An engineering approach for elasticeplasticfracture analysis. EPRI Report NP-1931. Palo Alto: EPRI; 1981.

[16] Xu W, He W. Strain-based engineering critical assessment of pipeline girthwelds. TWI Report 16772.01/2007/1297.2. Cambridge: TWI; 2007.

[17] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks inplates under tension. Int J Pres Vessels Piping 2004;81:21e30.


Please cite this article in press as: Budden PJ, Ainsworth RA, The shape of a strain-based failure assessment diagram, International Journal of Pressure Vessels and Piping (2011) doi:10 1016/j ijpvp 2011 09 004

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