fad analytic.pdf

Basic principles of analytical ¯aw assessment methods

U. Zerbsta,*, R.A. Ainsworthb, K.-H. Schwalbea

aInstitute of Materials Research, GKSS Research Centre, Max Planck Str., D-21502 Geesthacht, GermanybBritish Energy Generation Limited, Barnwood, Gloucester, GL4 3RS, UK

Received 10 May 2000; revised 7 October 2000; accepted 9 October 2000

Abstract

Analytical ¯aw assessment methods play an important role in the industrial realisation of fracture mechanics application. In the ®eld of

general as well as speci®c standards and guidelines there have been rapid developments in recent years. This paper gives a brief review of

some of the more important methods, which have been developed over the last decades. Descriptions are given of various design curves and

failure assessment diagrams, net section yielding and EPRI type solutions, the reference stress method and approaches derived from it, the

ETM and the recently developed European ¯aw assessment method SINTAP. The discussion of these approaches is restricted to the basic

principles of each method. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Crack driving force; Failure assessment diagram; Analytical ¯aw assessment

1. Introduction

In the common design philosophy, the applied stresses are

compared with a limit stress such as the yield strength of the

material. As long as the latter exceeds the applied stresses,

the component is regarded as safe. The implicit background

assumption is that the component is defect-free. If a real or

assumed crack or crack-like ¯aw affects the load carrying

capacity, fracture mechanics has to be applied. Then the

comparison between the applied and the material side has

to be carried out on the basis of crack tip parameters such as

the linear elastic stress intensity factor, K, the J integral or

the crack tip opening displacement (CTOD). As a result, the

fracture behaviour of the component can be predicted in

terms of a critical applied load or a critical crack size.

Standardised solutions for the crack tip parameters are

available for test specimens which are used for measuring

the material's resistance to fracture. As long as the deforma-

tion behaviour of the structural component is linear elastic,

then the relevant applied parameter in the component is K.

Comprehensive compendia of K factor solutions exist in

handbook format and as computer programs. The linear

elastic handbook solutions are usually approximations of

®nite element solutions, which have been generated for a

range of component and crack dimensions. Variations in the

loading geometry can be considered, for example, by apply-

ing the weight function method.

If the component behaves in an elastic±plastic manner,

the situation is much more complex because the crack tip

loading is additionally in¯uenced by the deformation

pattern of the material as given by its stress±strain curve.

This makes the generation of handbook solutions an expen-

sive task. To a limited extent this task has been realised for a

few component con®gurations, for example, in the Electric

Power Research Institute (EPRI) handbook (see below).

More generally, however, individual ®nite element analyses

have to be carried out. These analyses require a high level of

experience, which is not always available. Therefore,

despite the dif®culties described above, closed form analy-

tical solutions are desired in addition to ®nite element

analyses (Fig. 1). The aim of this paper is to give a brief

survey of the basic principles of the more common analy-

tical methods, some of which are discussed in more detail in

this special issue. A complete historical review is not

attempted.

Due to the inherent uncertainties in the determination of

the crack tip loading parameter analytical ¯aw assessment

methods are aimed at providing conservative results.

Consequently, an assessment leading to the result

ªunsafeº does not necessarily mean that the component

will fail. In those circumstances, a further analysis using

a numerical determination of the crack tip loading para-

meter may be carried out. However, it should be recog-

nised that uncertainties in the transferability of the

International Journal of Pressure Vessels and Piping 77 (2000) 855±867

0308-0161/00/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0308-0161(01)00008-4

www.elsevier.com/locate/ijpvp

* Corresponding author. Tel.: 149-4152-872-611; fax: 149-4152-872-

625.

E-mail address: [email protected] (U. Zerbst).

material resistance for fracture in terms of fracture tough-

ness or an R-curve are not resolved in this way. Test

specimens, which model the constraint conditions of the

component under consideration, are needed to resolve the

degree of this uncertainty.

2. Failure assessment diagram or crack driving force?

The analytical ¯aw assessment methods pursue two

different philosophies, which can be designated as failure

assessment diagram (FAD) and crack driving force (CDF)

concepts (Fig. 2). In principle, the two concepts can be made

fully compatible. However, some of the methods, which

will be discussed later, prefer one of both presentations.

What is the difference? In the FAD route, a roughly

geometry-independent failure line is constructed by norma-

lising the crack tip loading by the material's fracture resis-

tance. The assessment of the component is then based on the

relative location of a geometry-dependent assessment point

with respect to this failure line. In the simplest application,

the component is regarded as safe as long as the assessment

point lies within the shaded area below the failure line. It is

regarded as potentially unsafe if it is located on the line or

outside the shaded area. An increased load or larger crack

size will move the assessment point along the loading path

towards the failure line.

In contrast to the FAD philosophy, in the CDF philosophy

the applied and material sides are strictly separated. The

determination of the crack tip loading, in the component

and its comparison with the fracture resistance of the

material are two separate steps. Like the failure line of the

FAD concept, the CDF curve can, by suitable normalisation

of the load, be made an approximately geometry-indepen-

dent function, which depends only on the deformation

behaviour of the material.

3. Analytical assessment methods

3.1. Introduction

Analytical ¯aw assessment methods have been developed

for more than 30 years. These are described in numerous

published papers but only a limited number can be discussed

within this brief survey. In addition, only the basic princi-

ples of the various methods will be reviewed. Industrial

realisations within guidelines and standards are mentioned

only in Table 1 at the end of this paper, but particularly in

this ®eld there have been rapid developments recently, as is

apparent from the other papers in this issue. Common to all

the methods discussed here is that they are not restricted to a

linear elastic deformation pattern of the material but cover

the whole range from linear elastic to elastic±plastic beha-

viour (Fig. 3).

It is important to note that the limit load of the cracked

component plays an important role in almost all the models

U. Zerbst et al. / International Journal of Pressure Vessels and Piping 77 (2000) 855±867856

Nomenclature

a Crack size

aeff Plastic zone corrected crack size (different

de®nitions for the EPRI and ETM

approaches)

E Young's modulus

E 0 E in plane stress; E=�1 2 n2� in plane strain

F Applied load (general for force, moment,

pressure, etc.)

FY Limit load

h Dimensionless function of geometry (EPRI

approach)

J J-integral

Je Linear elastic J-integral

Jmat Crack resistance in terms of critical J-integral

Jp Plastic component of the J-integral

Jssy Small scale yielding J-integral

JY J-integral at limit load

K Linear elastic stress intensity factor

Kmat Crack resistance in terms of critical K

Kr Ratio of the applied K to the crack resistance,

K/Kmat

L Characteristic length (EPRI approach)

Lr Ratio of applied load, F, to yield load, FY

Lmaxr Plastic collapse limit of Lr

N Strain hardening exponent (ETM de®nition)

n Strain hardening exponent (various de®ni-

tions)

P Applied load (EPRI approach)

Po Reference load (EPRI approach)

Rm Tensile strength

ReL Lower yield strength (materials with LuÈders

plateau)

Sr Ratio of applied load to plastic collapse load

a Fit parameter of the Ramberg±Osgood

formulation

De LuÈders strain

d CTOD

d e Linear elastic CTOD

dmat Crack resistance in terms of critical CTOD

dY CTOD at limit load

d 5 Speci®c de®nition of the CTOD (ETM

approach)

e Strain

eY Yield strain

e ref Reference strain (reference stress method)

n Poisson's ratio

s Stress

�s Flow strength (usually the average of yield

and tensile strength)

s ref Reference stress (reference stress method)

sY Yield strength

s o Normalising stress (EPRI approach)

discussed below. In fracture mechanics application, the

limit load based on the material yield strength marks a tran-

sition from contained to net section yielding. However,

because of the work-hardening capacity of the material

beyond yield this does not correspond to immediate failure.

The latter may not happen until the plastic collapse load is

reached. Note, in common solid mechanics the distinction

between a limit or yield and a collapse load is not usually

made because the latter is uniquely de®ned as the maximum

load a component of elastic±ideally plastic material is able

to withstand (Fig. 4).

3.2. Design curve approaches

3.2.1. General principle

The so-called design curve approaches are the earliest

analytical ¯aw assessment methods. They are based on an

idea of Wells in the sixties that was ®rst realised by Burde-

kin and Dawes [1] at TWI in 1971. The basic approach is

quite simple. It utilises the observation that the crack tip

loading in terms of the CTOD (d) or J-integral depends

on the local strain, e , in a quadratic manner in the contained

yielding regime and in a linear manner in the net section

U. Zerbst et al. / International Journal of Pressure Vessels and Piping 77 (2000) 855±867 857

Fig. 2. FAD and CDF philosophies of ¯aw assessment.

Fig. 1. Principles of analytical determination of the crack tip loading.


Table 1

Overview on some methods for analytical ¯aw assessment

Basic approach Assessment routes Special features and range of application Realisation in industrial guidelines

Design curves (various

approaches mostly for CTOD)

FAD CDF No K factor solution is required where these

are based on strain, which is an advantage

for shallow crack con®gurations.

Application is usually restricted to shallow

cracks (preferably originating from

notches) and to limited ligament plasticity.

British standard BS 7910, assessment level

1, US American and Canadian documents

for welded pipelines, API 1104 and CSA

Z662, Chinese pressure vessel code CVDA-

1984; Japanese guideline WES 2805 for

¯aws in fusion welded joints.

FAD based on Dugdale model FAD Original FAD approach with the abscissa

normalised by the plastic collapse load;

should only be used for materials with a

yield to ultimate stress ratio close to unity.

BSI document PD 6493, now revised to

BS7910; `explanation document' of WES

2805.

EPRI-handbook CDF (FAD) Fully plastic solutions for the J-integral and

the CTOD derived by ®nite element

analyses. The application is restricted to a

limited number of component

con®gurations. Ramberg±Osgood

description of the stress±strain curve may

be a problem. Modi®ed versions for

strength mismatch assessment procedures

such as the American EWI and the French

ARAMIS methods.

ASME Boiler and Pressure Vessel Code,

Section XI, Appendixes G and H for

pipelines.

R6-routine FAD Based on the reference stress method of

Ainsworth, which is a generalisation of the

EPRI approach within the frame of FAD

philosophy. The R6 document includes

numerous appendices for treating weld

residual stresses and strength mismatch,

constraint effects, probabilistic sensitivity

analysis, special applications such as leak-

before-break analysis, etc.

Today the most widely applied method,

essential part of theBritish standard

BS7910, the Draft API 579 procedure of the

US American petrochemical industry, the

Swedish SAQ method, the French RSE-M

Code (2000 Addenda) for nuclear

application, and others. A partly modi®ed

version is a constituent element of the

European SINTAP procedure.

CEA-A 16 guide CDF Also based on the reference stress method.

The procedure includes guidance for leak-

before-break assessment and high

temperature creep.

Draft document for nuclear industry in

France.

ETM CDF The model is part of the comprehensive

EFAM methodology of GKSS, which

covers toughness testing and failure

assessment. In addition to the basic

procedure special options for strength

mismatch, shallow cracks at notches, and

strain-based assessment, guidance for

treating data scatter and size effects on

toughness are available.

A modi®ed version is a constituent element

of the European SINTAP procedure.

SINTAP FAD CDF Uni®ed European procedure, based on

elements of R6 and ETM as well as on

additional elements from various sources:

BS 7910, the SAQ method and others.

SINTAP includes explicitly strength

mismatch analysis, statistical treatment of

toughness data according to the VTT master

curve, compendia for stress intensity factor

and limit load solutions as well as for

residual stress pro®les, appendices for

constraint effects, reliability analysis and

special applications such as leak-before-

break analysis.

The procedure is designated for a technical

document of the CEN

yielding regime. In general terms this may be written as

d or J � eY

C1�e=eY�2 for �e=eY� , k;

C2�e=eY�1 C3 for �e=eY� . k

(�1�

with eY being the yield strain and C1,2,3 being empirical

constants. The quantity k varies within the different design

curve approaches from 0.5 to 1. In general terms, C1,2,3

depend on a range of parameters, in particular the geometry

and the loading type, for example, bending or tension. For

simplicity, these constants were derived for the centre

cracked plate loaded under tension, this being expected to

provide conservative estimates of the crack tip loading for

other con®gurations. Modern developments in design

curves tend to overcome the restriction to tension loading

and to incorporate the detailed deformation behaviour of the

material. Although solutions for deeper cracks exist, the

preferred ®eld of application is the assessment of shallow

cracks, such as those originating from notches. An advan-

tage of the design curve approaches is that they do not

require stress intensity factor solutions for the con®gura-

tions to be assessed. A dif®culty is that the strain, e , is not

clearly de®ned. Design curves can, in principle, be applied

in the frame of the FAD or CDF philosophy (Fig. 5), but are

more commonly applied in the CDF framework.

3.2.2. The TWI design curve

The most widely applied design curve approach still is

that developed by Burdekin and Dawes [1]. In CDF terms it

can be written as

d � 2peYa�e=eY�2 for �e=eY� # 0:5;

�e=eY�2 0:25 for �e=eY� . 0:5:

(�2�

In special FAD terms, the failure assessment curve becomes

FD � dmat

2peYa� �e=eY �2 for �e=eY� # 0:5;

�e=eY �2 0:25 for �e=eY� . 0:5

(�3�

for ferritic steel applications. Eq. (2) is restricted to ligament

plasticity at about the limit load of the cracked component

and to shallow cracks. The latter is due to the fact that the

crack tip loading is correlated with the applied strain, which

is usually de®ned at the outer surface of the component. The

FAD and CDF approaches of Eqs. (3) and (2) are shown in

Fig. 5a and b, respectively.

Some years later, Dawes proposed a modi®cation of Eq.

(2) for deeper cracks [2] and rewrote it in terms of stress in

order to ®t it into the FAD approach of the British PD 6493

[3] procedure. The latter modi®cation is still part of the

recently revised version of that document, the British

Standard Guide BS 7910 [4]. With some modi®cations,

the TWI design curve is also a constituent element of indus-

trial guidelines such as the US and Canadian documents for

welded pipelines, API 1104 [5] and CSA Z662 [6], and the

Chinese pressure vessel code CVDA-1984 [7].

3.2.3. The EnJ design curve

A J-integral based design curve was proposed by Turner

at Imperial College, London, in 1981 [8]. The basic equation

of the ªEnJ methodº was

J � JY

�ef =eY�2�1 1 0:5�ef =eY�2� for �ef =eY� # 1:2;

2:5 ��ef =eY �2 0:2� for �ef =eY � . 1:2

8<: �4�

with e f being the so-called ªcracked body structural strainº.

Although Turner gave expressions for e f for some basic

con®gurations, it is not clear how this is de®ned in general.

The quantity JY is the energy release rate, G, calculated

elastically, for a load equal to the limit load.

3.2.4. More recent developments

Based on extensive ®nite element analyses, a J-based

design curve approach was further developed by a group


Fig. 3. Application ranges of various fracture mechanics concepts. General

approaches cover all ranges within a uni®ed method.

Fig. 4. Schematics of limit and plastic collapse loads: (a) in fracture mechanics application, and (b) in common solid mechanics.

at the University of Wales, Swansea. Fitting the material's

stress±strain curve by a piece-wise power law

e=eY �s=sY for e # eY;

�s=sY�n for e . eY

(�5�

and determining the J-integral from the HRR ®eld, Lau

et al. [9] derived an expression for shallow cracks �0:05 #a=W # 0:1� and for a strain hardening exponent between

2 # n # 30 :

J � JY

�e=eY�2 for �e=eY� # 0:85;

Ce

eY

2 1:2

� ��n11�=n1JNG for �e=eY� . 1:2

8><>: �6�

with JNG and C being functions of the relative crack depth,

a/W, and the strain hardening exponent, n. J is interpolated

by a straight line between 0:85 , e=eY # 1:2: The authors

obtained a similar expression for pure bending [10], which

has recently been modi®ed for a more generalised stress±

strain curve ®t [11].

A largely independent design curve approach was devel-

oped by the Japan Welding Engineering Society (WES).

Since 1976, the WES 2805 standard and within this frame

the Japanese CTOD based design curve has been revised

several times. In the most recent version of 1997 [12] it is

given by

d �p

2eYa�e=eY�2 for �e=eY� # 1

p

8eYa�9�e=eY�2 5� for �e=eY� . 1

26643775: �7�

A special feature of the WES 2805 design curve is that the

applied strain, e , is determined from the membrane and

bending stress components across the wall and that it

depends on the strain hardening of the material if the

crack originates from a notch.

Recently Xue and Shi from the Beijing Polytechnical

University proposed a strain hardening design curve [13]

that is based on a simpli®cation of net section plastic

CTOD solutions of EPRI (see below). In a CDF format it

can be written as

d � 2peYa5n 1 3

8�n 1 1�s

sY

� �2

10:85 a ks

sY

� �n11

�8�

with s being the applied stress of the component. The

quantities a and n are the coef®cients of the Ramberg±

Osgood formulation of the stress±strain curve

e=eY � s=sY 1 a�s=sY�n �9�and k is an out-of-plane constraint factor, being

��3=2p

for

plane strain and 1 for plane stress conditions.

Finally, it has to be mentioned that the strain option of the

engineering treatment model (ETM) of the GKSS Research

Centre in Germany can also be interpreted as a design curve

approach (for more details see Ref. [14]).

3.3. Early failure assessment diagrams

3.3.1. General philosophy

The FAD for ¯aw assessment was ®rst used in the so-

called R6 approach of the former Central Electricity Gener-

ating Board (CEGB) in the UK. An overview on early

developments is given in Ref. [15]. The approach recog-

nised that, at one extreme, linear elastic fracture mechanics

was applicable and fracture occurred when the stress inten-

sity factor, K, became equal to the fracture toughness, Kmat.

At the other extreme failure occurred when the load F

reached its value, FL, at plastic collapse. The R6 approach

recognised that the use of K beyond the elastic regime

underestimated the crack tip loading and, therefore, some

plasticity correction was required, as shown in Fig. 6. In

general, this correction is a function of the material and


Fig. 5. Schematics of design curves: (a) special FAD route; (b) CDF route, e.g. according to Eqs. (2) and (3).

Fig. 6. Schematic illustration of the effect of plasticity on crack-tip loading

and normalised crack tip loading.

the component, the crack geometry and the type of loading.

However, in the early R6 approach, it was recognised that

by using the two normalising parameters

Kr � K=Kmat; �10�and

Sr � F=FL; �11�the plasticity correction could be converted to a general

purpose failure avoidance curve

Kr � f �Sr�: �12�

3.3.2. The early R6 routine of CEGB

In the early R6 routine, the problem of a conservative

plasticity correction function was solved by choosing the

Dugdale model for predicting the elastic±plastic crack tip

loading. Rewritten in terms of the J-integral this corre-

sponds to

J

Je

� 8

p2

s

sY

� �2

ln secp

2

s

sY

� �� 13�

for a defect subject to a remote stress s in an elastic

perfectly plastic material, with the linear elastic J-integral,

Je, being

Je � K2=E: �14�

In a second step, the elastic±plastic J in Eq. (13) was

replaced at failure by the fracture resistance of the material,

Jmat, which is related to Kmat by an equation similar to

Eq. (14). The applied load normalised by the plastic collapse

load, de®ned for a ¯ow stress �s ; then replaced (s /sY) in Eq.

(13). This led to a geometry independent failure line, which

formed the ®rst FAD of CEGB [16] (Fig. 7). In terms of Eq.

(12), the Dugdale plasticity correction corresponds to

f �Sr� � Sr £ 8

p2ln sec

p

2Sr

� �� 21=2

: �15�

The FAD of Eq. (15) could then be interpreted simply as an

interpolation between the two limiting failure states Ð

brittle fracture and plastic collapse Ð of a cracked

component.

In the 1980s, a number of modi®cations of the CEGB

FAD were proposed, for example Refs. [17,18]. A disad-

vantage of the original approach was that it was tailored to

elastic±ideally plastic deformation behaviour due to its

origin from the Dugdale model. One aim of the further

development was to address in more detail the strain hard-

ening capacity of real materials. An attempt based on a

power law description of the stress±strain curve yielded

separate FADs for simple bending and tension cases [19].

However, at that time the major interest was focussed on a

new development initiated by General Electric in the United

States.

3.4. Fully plastic solutions of the EPRI type

In 1976, Shih and Hutchinson [20] proposed a new

method for determining fully plastic yielding J-integral

and CTOD solutions. The method was applied to plate

and cylindrical con®gurations and summarised in the

EPRI handbook in 1981 [21]. At the beginning of the nine-

ties it was expanded to a number of additional con®gura-

tions [22]. Fully plastic solutions were not only generated at

EPRI. Alternative examples were given by other authors, for

example, solutions for semi-elliptical surface cracks were

published by Yagawa and Ueda in 1989 [23].

What is the basis of the method? First, the J-integral (or

CTOD) is split into a small scale yielding and a net-section

yielding component

J � Jssy 1 Jp: �16�The plastic part, Jp, is derived from dimensional arguments

recognising that for proportional loading, the fully plastic

component must be related to the applied load, which in the

EPRI handbook is usually designated as P, by

Jp � aeosohLP

Po

� �n11

: �17�

The load is normalised by a reference load, Po, which, in

principle, may be freely chosen provided it is proportional to

s o but is often identi®ed with the limit load of the cracked

component. The quantities e o, s o, a and n are the ®tting

parameters of the Ramberg±Osgood formulation of the

stress±strain curve (Eq. (9) is used with eY and sY replaced

by e o, s o). Like Po the characteristic dimension L can be

freely chosen. Usually it is identi®ed with the ligament

length or the crack size. The derivation of the non-dimen-

sional parameter h is discussed below. The small scale

yielding component of J, Jssy, is derived from a plastic

zone correction to K

J � K 2�aeff�=E 0 �18�for an effective crack length that depends on strain hardening

aeff � a 11

1 1 �P=Po�21

bp

n 2 1

n 1 1

K

so

� �2

: �19�


Fig. 7. FAD according to R6-Rev. 1, schematically.

The quantity b is 6 for plane strain and 2 for plane stress

conditions. The choice of so is arbitrary for the purposes of

the Ramberg±Osgood ®t to the stress±strain curve and the

subsequent use in Eq. (17) but to be of relevance to Eq. (19)

it must be of the order of the yield strength. J-integral values

were determined for different sets of stress±strain curves as

de®ned by Eq. (9) using extensive 2D ®nite element analyses.

The component and crack dimensions were varied step-wise

and the fully plastic results were ®tted by Eq. (17) to de®ne the

quantity h which of course depended on the choice of Po and L

(Fig. 8). Tables for this parameter as a function of component,

crack size and strain hardening generated for a range of plate

and cylindrical con®gurations form the main items of the EPRI

handbooks. In addition to J solutions, solutions for CTOD and

displacement are available.

In 1980, Bloom proposed that the EPRI solutions were

replotted in a FAD format [24] (Fig. 9). There were,

however, two differences to the CEGB FAD. The applied

load was normalised by the EPRI normalising load (cf.

Section 3.1) proportional to s o and not by the collapse

load proportional to �s : The failure curve became a function

of the strain hardening of the material. Both features were

direct consequences of the EPRI approach.

The main advantage of the EPRI handbook is that it is

based on ®nite element analyses. There are, however,

problems in its application. One limitation of the handbook

is that it is applicable to only a few con®gurations, those for

which extensive tables of h factors are available. A critical

point is the Ramberg±Osgood ®t to the stress±strain curve.

While this is convenient for ®nite element calculations,

most materials do not follow this ®t satisfactorily and in

particular the important region near the yield strength is

usually poorly described.

3.5. Approaches based on the CEGB reference stress

method

3.5.1. The reference stress method

Some limitations of the EPRI handbook were overcome

by the development of the reference stress method which

was ®rst applied to elastic±plastic fracture by Ainsworth in

1984 [25]. In fact, the method has been used earlier by

Ainsworth [26,27] to apply steady state creep fracture

mechanics in the absence of detailed numerical solutions

such as those in the EPRI handbook. With the production

of the handbook, it was possible to examine the reference

stress method in more detail to develop a general FAD

approach. Two modi®cations were essential: First the Jp

solution according to Eq. (17) was replaced by the expression

Jp � srefhL eref 2srefeo

so

� �: �20�

De®ning the reference stress, e ref, by

sref � so

P

Po

� ��21�

and a reference strain, e ref, such that it corresponds to s ref

via the Ramberg±Osgood formulation. Eq. (20) is identical

with Eq. (17). It was, however, possible to make the method

much more general. Introducing s ref and e ref point-wise

from the real stress±strain curve into the analysis allowed

an exact description of any material. However, the quantity

h in Eq. (20) was still a function of the strain hardening

exponent, n. This dependency was minimised by rede®ning

the reference load, Po, in Eq. (21). Having done this h could

simply be set equal to the value for n � 1; the corresponding

value for linear elastic material behaviour, which could be

related to K. In this way Ainsworth reached the expression

Jp � mK2

E

Eeref

sref

2 1

� ��22�

with m equal to 0.75 for plane strain and 1 for plane stress,

respectively. Eq. (22) allowed the determination of the plas-

tic component of the J-integral simply from the linear elastic

K-factor, which implicitly took into account the component

geometry. The deformation behaviour was re¯ected by the

stress±strain curve from which the reference stress and

strain were taken from. To determine the reference stress

as described above, it was necessary to modify the value of

EPRI reference load Po and this required the numerical h

solutions. However, it was found that the modi®ed reference

load was often close to the yield load and, therefore, the


Fig. 8. Basic principle of the EPRI approach.

Fig. 9. EPRI based FAD according to Bloom [24].

reference stress was rede®ned as

sref � LrsY � F

FY

sY: �23�

The quantity Lr, based on yield stress, rather than the ¯ow-

stress based quantity Sr of Eq. (11) forms the basis of the R6

revision 3 method, which is described in the following

sections. Minor additional corrections were introduced at

that stage to ensure conservatism. The overall accuracy of

the reference stress method and indeed the accuracy of the

EPRI method used in its development were also examined

[28]. It was concluded that the limit load based reference

stress was acceptably accurate for practical applications and

generally erred on the side of conservatism.

3.5.2. The FAD according to R6 Rev. 3

The R6 routine, rev. 3 [29], is probably the most widely

applied analytical ¯aw assessment method worldwide. It

forms Ð sometimes slightly modi®ed Ð the basis of indus-

trial codes and guidelines such as the British standard

BS7910 for assessing ¯aws in weldments [4], the draft

API 579 ®tness for service procedure of the American petro-

chemical industry [30], the Swedish SAQ method [31] and

the French RSE-M Code (2000 Addenda) for assessing

¯aws in nuclear power components [32], and others.

Introducing some additional reductions, the reference

stress method according to Eq. (22) was rewritten in terms

of the R6 parameters of Eqs. (10)±(12) but using Lr of Eq.

(23) to replace Sr to give a failure avoidance curve [25]

f �Lr� � Eeref =sref 11

2L2

r =�1 1 L2r �

� �2 12

: �24�

The second term in Eq. (24) can be interpreted as a plasticity

correction in the intermediate small scale yielding region

where the bulk behaviour of the component is still elastic

but the ligament ahead of the crack already experiences

substantial local plasticity. In order to improve the ®t to

real conditions, the denominator �1 1 L2r � was replaced

semi-empirically by the term (Ee ref/s ref) [29] to give the

equation

f �Lr� � Eeref

LrsY

1L3

rsY

2Eeref

" #; �25�

which is termed the material speci®c option 2 of R6 rev.3.

The application of Eq. (25) requires knowledge of the

complete stress±strain curve. In particular the region around

the yield point has to be available in a detailed manner.

However, there are many cases where this information is

not available to the user. Therefore, Eq. (25) was applied to

a number of materials to generate a material independent

lower bound curve, which is the more conservative option 1

curve of R6 rev. 3

f �Lr� � �1 2 0:14L2r ��0:3 1 0:7 exp�20:65 L6

r ��: �26�In Fig. 10 both R6 failure curves are plotted schemati-

cally. Note, that in contrast to the original CEGB FAD the

ligament plasticity is referred to the limit load and not to the

collapse load of the cracked component. As a consequence,

a limit line against plastic collapse had to be introduced

separately. In the ®gure this is marked as Lmaxr : Usually

this is determined as

Lmaxr � �s

sY

with �s � 1

2�sY 1 Rm� �27�

although different de®nitions for special classes of materials

also exist in R6.

Currently a fourth revision of the R6 routine is in prepara-

tion, which includes an update with respect to the recently

developed European ¯aw assessment method SINTAP (see

below).

3.5.3. The Js method of CEA

Another method based on the reference stress approach

is the so-called Js method, which was ®rst proposed by

Moulin et al. [33] in 1993. It became a constituent part of

the French CEA-A16 guide [34] in 1997. With the elastic

part of J estimated as

Je � mK2=E �28�

Eq. (22) becomes

J � Je 1 Jp � Je

Eeref

sref

� ��29�

for plane stress conditions �m � 1�: J is the complete

J-integral, in A16 designated as Js.

3.6. The engineering treatment model

The ETM has been developed at the GKSS Research

Centre in Germany since the mid eighties [35]. In 1997/

1998, the method was provided both for homogeneous

components and for heterogeneous con®gurations such as

strength-mismatched welded joints in handbook format

[36,37]. As with the development of the EPRI and reference

stress approaches, the motivation was to derive a fracture

mechanics based ¯aw assessment method including the

effects of material hardening on crack tip loading.


Fig. 10. FAD according to R6-Rev. 3, schematically.

The basic concept realised in the ETM is shown in Fig. 11.

The deformation behaviour of the ligament ahead of the

crack is assumed to be satisfactorily described by the uni-

axial tensile test ®tted by the piece-wise power law

s �Ee for s , sY

sY�e=eY�N for s $ sY

(: �30�

In Eq. (30) the strain hardening exponent is designated by a

capital N in order to distinguish it from the n (� 1/N) used

in Eq. (5). More importantly, it is de®ned from the engineer-

ing stress±strain curve, whereas all the methods reviewed

above have used the true stress±strain curve for describing

the deformation behaviour of the material. Another feature

of the ETM is that it prefers a special de®nition for the

CTOD Ð the CTOD-d5 Ð which is locally measured at

the crack tip (see Ref. [14]). The J-integral is applicable

within the ETM approach too.

The ETM curve which, like the R6 FAD curve, is an

approximately geometry independent master curve is

given by

d5 � b1

EK 1

1

mEsY

K2eff

F

FY

� �for F , FY �31�

and

d5 � d5Y

F

FY

� �1=N

for F $ FY; �32�

or Ð in terms of the J-integral Ð by

J � 1

EK2

eff for F , FY �33�

and

J � JY

F

FY

� ��11N�=Nfor F $ FY; �34�

respectively. The quantities d 5Y and JY designate the crack

tip loading at the limit load, FY, the coef®cients b 1 and m

have constant values b1 � 2:09 and m � 2 for plane strain

and b1 � 2:41 and m � 1 for plane stress conditions. Keff is

the plastic zone corrected stress intensity factor. The method

follows exclusively the CDF philosophy, i.e. the ETM curve

represents the crack tip loading. Failure conditions are given

by the intersection of the ETM curve with the fracture

resistance of the material, represented by Jmat or d 5Ymat,

see Fig. 12.

In addition to the basic ETM, special options have been

derived for shallow cracks originating from notches and for

strain-based assessment, which are discussed in more detail

in Ref. [14]. An update with respect to the European ¯aw

assessment method SINTAP (see below) is planned.

3.7. The European ¯aw assessment method SINTAP

The SINTAP procedure [38] is the outcome of a recently

®nished Brite±Euram project in which seventeen institu-

tions from nine European countries were participants.

The procedure offers both a FAD and a CDF route. They

are complementary and yield identical results. The basic

failure line of the FAD route is

Kr � f �Lr� �35�with Kr being K/Kmat as in Eq. (10) for primary loading and

Lr being F/FY as in Eq. (23). The CDF expressions for

primary loading are

J � Je�f �Lr��22 �36�with

Je � K2=E 0 �37�

for the J-integral and

d � de�f �Lr��22 �38�with

de � K2=E 0sY �39�

for CTOD. The effective Young's modulus, E 0, is E/

(1 2 n 2) for plane strain and E 0 � E for plane stress


Fig. 11. Basic principle of the ETM.

Fig. 12. ETM curve, J-integral option, schematically.

conditions, respectively. It is of importance that the function

f(Lr) is identical for the FAD and CDF routes.

In Fig. 13, the overall structure of the procedure is

presented. SINTAP consists of various analysis levels

which are constituted by the quality and completeness of

the input information required. Higher levels are more

complex than the lower ones and need improved input infor-

mation; however, the user is `rewarded' by less conservative

results. An unacceptable result does not necessarily mean

failure of the component analysed. Instead, it rather

provides a motivation for repeating the analysis at the

next higher level. On the other hand, if a lower level analysis

indicates safety the user does not need to provide improved

input information and to perform a more complex analysis.

The SINTAP standard analysis levels are:

Level 1. This forms the basic level. Only the toughness

and the yield strength and the ultimate tensile strength

of the material need to be known. The function f (Lr)

includes elements from R6, rev. 3 and the ETM, both

being slightly modi®ed in order to provide compatible

FAD and CDF solutions. A SINTAP speci®c feature is

that it offers different sets of equations for materials with

and without LuÈders plateau. This is an important modi®-

cation of both original methods.

Level 2. The second level is a modi®cation of level 1

for inhomogeneous con®gurations such as strength-

mismatched weldments which is also based on R6, rev. 3

and the ETM. Combinations of materials where both, none

or only one showing a LuÈders plateau can be analysed.

Level 3. This level is identical to R6 option 2 which requires

toughness data and the complete stress±strain curve of the

material. Both homogeneous and strength mismatch

components can be handled, the treatment of strength

mismatch being based on the ETM mismatch handbook

[36] and on Appendix 16 of R6 [39].

There are additional levels:

Level 0. This is a default option which requires the yield

strength of the material only and which allows the fracture

resistance of the material to be estimated from Charpy data.

A level 0 analysis will usually be highly conservative but

it enables a fracture mechanics analysis even with a mini-

mum of input information Ð a case that is not unusual in

industrial practice.

The advanced levels are:

Level 4. Within this level, loss of constraint in thin

sections or predominately tensile loading is considered.

Level 5. This is based on complete numerical analysis,

e.g. ®nite element analysis.

Level 6. This is a special option for evaluating leak before

break in pressurised components.

The SINTAP procedure is described in more detail

elsewhere in this issue [40]. Here, only the level 1 equations

for materials showing a LuÈders plateau will be given for


Fig. 13. Overall structure of the SINTAP procedure.

illustration. These are

f �Lr� � 1 11

2L2

r

� �21=2

; for 0 # Lr , 1; �40�

f �1� � l 11

2l

� �21=2

for Lr � 1 �41�

with

l � 1 1EDe

ReL

�42�

and

f �Lr� � f �Lr � 1�L�N21�=2Nr for 1 # Lr , Lmax

r : �43�Eq. (41) leads to a step change in f(Lr) in the LuÈders

plateau. If the LuÈders strain is unknown it can be estimated

conservatively from an empirical correlation

De � 0; 0375 1 2ReL

1000

� �: �44�

The strain-hardening exponent can also be determined on

a conservative, empirical basis:

N � 0; 3 1 2ReL

Rm

� ��45�

with ReL being the lower yield strength and Rm the tensile

strength of the material. The plastic collapse limit is de®ned

as

Lrmax � 1

2

ReL 1 Rm

ReL

� �: �46�

The SINTAP procedure has been developed as a contri-

bution towards the development of a European Committee

for Standardisation (CEN) ®tness for service standard.

Presently it is planned as an input to a technical document

of the CEN.

4. Conclusions

The basic principles of various analytical ¯aw assessment

methods such as the design curve approaches of TWI and

WES, the EPRI handbook, the R6 routine, the CEA-A16

guide, the ETM method of GKSS and the recently devel-

oped European SINTAP procedure have been reviewed.

The aim has been to give an impression of the historical

development of all these methods and to point to some

common features, which may be important in the context

of any future development towards an international uni®ed

¯aw assessment method.

Acknowledgements

This paper is published with permission of GKSS

Research Centre and British Energy Generation Ltd.

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