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AB SVENSK ANLÄGGNINGSPROVNING

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A PROCEDURE FOR SAFETY ASSESSMENT OFCOMPONENTS WITH CRACKS - HANDBOOK

Mats Bergman, Björn Brickstad, Lars Dahlberg,Fred Nilsson, Iradj Sattari-Far

SA/FoU-REPORT 91/01

AB SVENSK ANLÄGGNINGSPROVNING

The Swedish Plant InspectorateBox 49306 • S-100 28 Stockholm • Sweden

Tel. Nat. 08-6174000Int.+46 8 6174000

December 1991

List of contents

Nomenclature 1

1. Introduction 3

2. Procedure 42.1. Overview 42.2. Characterization of defect 52.3. Choice of geometry 52.4. Stress state 52.5. Material data 52.6. Calculation of slow crack growth 62.7. Calculation of Kf and Kf 72.8. Calculation of LR 82.9. Calculation of KR 82.10. Fracture assessment 92.11. Assessment of iesults 10References 11

Appendix A. Defect characterization 121. Defect geometry 122. Interaction between neighbouring detects 12References 14

Appendix E. Residual stresses 15l.Dutt welded joint 15

a) Thin plates 15b) Thick plates 16c) Thin walled pipes 17d) Thick walled pipes 17

2. Fillet weld 183. Stress relieved joints 184. Conclusion 18References 19

Appendix G. Geometries treated in this handbook 201. Crack in a plate 20

a) Semi elliptical surface crack 20b) Long surface crack 21c) Through thickness crack 22d) Elliptical embedded crack 23

2. Crack orientated axially in a cylinder 24a) Semi elliptical surface crack on the inside of a cylinder 24b) Long surface crack on the inside of a cylinder 25c) Semi elliptical surface crack on the outside of a cylinder 26d) Long surface crack on the outside of a cylinder 27e) Through thickness crack 28

3. Crack orientated circumferentially in a cylinder 29a) Semi elliptical surface crack on the inside of a cylinder 29b) Complete circumferential surface crack on the inside of a cylinder... 30c) Through thickness crack 31

4. Crack in a sphere 32a) Through thickness crack 32

5. Crack in a nozzle 33a) Quarter circular corner crack 33

6. Crack at a hole 37a) Quarter elliptical corner crack 37

Appendix K. Stress intensity factor solutions 381. Crack in a plate 38







References 58

Appendix L. Limit load solutions 601. Crack in a plate 60







References 69

Appendix M. Material data for nuclear applications 701. Yield strength, ultimate tensile strength 702. Fracture toughness 70

a) Ferritic steel, pressure vessels 70b) Ferritic steel, pipes 70c) Austenitic stainless steel, pipes 71d) Nickel base alloys 71

3. Crack growth data - fatigue 72a) Ferritic steel, pressure vessels 72

b) Austenitic stainless steel 724. Crack growth data - stress corrosion 73

arAustenitic stainless steel 73References 73

Appendix P. Load, size and material factors for nuclear applications 75

Appendix B. Background 761. Interaction between primary and secondary stresses 762. Stress intensity factors 783. Limit load solutions 784. Safety margins 795. Linearization of stress distribution 79References 79

Appendix X. Example 811. Solution 81

Nomenclature

A area, stress polynomial coefficient

a crack depth for surface cracks

2a crack depth for embedded cracks

dc/dt local crack growth rate for stress corrosion cracking

dc/dN local crack growth rate for fatigue crack growth

Ee

f

Jlc

Kcr

Ki

Kla

Kic

Ko

KR

/

LR

n

PR

ReL

R,

Rm

Rp0.2

Young's moduluseccentricity

geometry function

critical J-value according to ASTM E-813

critical value of the stress intensity factor

stress intensity factor

fracture toughness at crack arrest

fracture toughness according to ASTM E-399

material constant for stress corrosion cracking

elastic fracture parameter

crack length

limit load parameter

material constant for fatigue crack growth

internal pressure

stress intensity factor ratio

lower yield strength

inner radius

ultimate tensile strength, mean radius

0,2% elongation stress

RTNDT nil ductility transition temperature

s

T

t

u

V

X

a

iis

distance between neighbouring defects

reference value for residual stress determination

temperature

thickness

coordinate

coordinate

coordinate

material constant for stress corrosion cracking

size factor for crack depth

size factor for crack length

load factor for primary stress

load factor for secondary stress

material factor for fracture toughness

material factor for yield strength

stress intensity factor range

material constant for fatigue crack growth

v Poisson's ratio

p interaction parameter for plastic deformation

<?b local bending stress

o?bg global bending stress

om membrane stress

o p primary stress

0 s secondary stress

Oy yield strength, ReL or Rpo.2

X parameter for calculation of interaction parameter p

\|/p load parameter for primary stress

Vs load parameter for secondary stress

1. Introduction

A procedure for assessment of components with cracks or crack like defect is describedin this handbook. The procedure can be used to decide if a certain defect can give cause tofracture of a component or otherwise be judged detrimental to the safety of thecomponent.

It is assumed in the method that the shape and the size of the considered defect aredefined. The defect can be an actually existing one or a hypothetical one in connectionwith damage tolerance assessments. The procedure can be used to estimate:

a) The load carrying capacity of the component with consideration of the assumed defect,

or

b) The maximal crack size that does not cause fracture for the assumed loadingconditions.

The assessments described under a) - b) can be performed with or without considerationof safety margins. A system for safety margins is outlined in 2.11. For application tocomponents in nuclear power facilities the numerical values of the safety factors given inappendix P can be used.

The method that is utilized in this procedure is based on the R6-method [1] developed atCEGB. The basic assumption is that fracture initiated by a crack can be described by thevariables KR and LR. KR is the ratio between the elastically calculated stress intensityfactor and the fracture toughness of the material. LR is the ratio between applied load andthe plastic limit load of the structure. The pair of calculated values of these two variablesis plotted in a diagram (figure 1). If the point is situated within the region enclosed by thecritical line fracture is assumed not to occur. If the point is situated outside this regioncrack growth and fracture can happen.

A procedure to conduct this type of assessments is described in this handbook. A briefinformation about the background and the considerations made is given in appendix B.

The method can in principal be used for all metallic materials where the materialproperties Cy and Kcr can be defined. It is, however, more extensively verified only forsteel alloys. The method is not intended for use at so high temperatures that creepdeformation is of importance.

A computer program [2] written in FORTRAN 77 has been developed which is capableof performing the calculations described in this document.

2. Procedure

2.1. Overview

A fracture assessment according to the procedure consists of the following steps:

1) Characterization of defect (appendix A).

2) Choice of geometry (appendix G).

3) Determination of stress state.

4) Determination of material data (appendix M).

5) Calculation of possible slow crack growth.

6) Calculation of Kf and Kf (appendix K).

7) Calculation of LR (appendix L).

8) Calculation of KR.

9) Fracture assessment.

KR 0.6 - -

0 0.2 0.4 0.6 0.8 1 1.2

LR

Figure 1. Diagram for fracture assessment.

The safe region is limited by

KR < (1 - 0,14LR) |(U + 0,7exp(-0,65LR)],

1.4 1.6

(1)

1, for material with yield plateau

and for acceptance assessmentsLR < LR = *

0,511 + — | , for all other casesOy

10) Assessment of results.

2.2. Characterization of defect

A fracture mechanics analysis requires that the actual defect geometry is characterized in aunique manner. For application to components in nuclear power facilities methodsaccording to appendix A should be used to define shape and size of cracks.

For assessment of an actual defect it is important to determine whether the defect remainsfrom the manufacture or has occurred because of service induced processes such asfatigue or stress corrosion cracking.

2.3. Choice of geometry

The geometries considered in this procedure are documented in appendix G. In theidealization process from the real geometry to these cases it should be carefully avoided tointroduce non-conservatism. In cases when an idealization of the real geometry to one ofthe cases considered here are not adequate, stress intensity factor and limit load solutionscan be found in the literature or be calculated by numerical methods. The use of suchsolutions should be carefully checked for accuracy.

2.4. Stress state

In this procedure it is assumed that the stresses have been obtained under the assumptionof linearly elastic material behaviour. The term nominal slicss denotes the stress state thatwould act at the plane of the crack in the corresponding crack free component.

The stresses are divided into primary op and secondary <f stresses. Primary stresses arecaused by the part of the loading that contributes to plastic collapse e.g. pressure, gravityloading etc.. Secondary stresses are caused by the part of the loading thai does notcontribute to plastic collapse e.g. stresses caused by thermal gradients, weld residualstresses etc.. If the component is cladded this should be taken into account when thestresses are determined.

All stresses acting in the component shall be considered. The stresses caused by theservice conditions should be calculated according to some reliable method. Someguidance about the weld residual stresses is given in appendix E.

2.5. Material data

To perform assessments the yield limit CTY. the ultimate tensile strength Rm and the criticalstress intensity factor Kcr of the material must be determined. If possible, data obtainedfrom testing of the actual material of the component should be used. This cannot be donein many cases and therefore minimum values for ay and Rm from codes, standards ormaterial specifications may be used. Kcr-data from testing of similar materials may alsobe used. These data should be determined at the actual temperature.

<Ty denotes the lower yield strength ReL if this can be determined and in other cases the0,27r elongation limit Rr>o.2- In "the cases when R^. can be determined the material isconsidered to have a yield plateau. This is for instance common for low alloy carbonmanganese steels.

Rm is the ultimate tensile strength of the material.

The yield and ultimate tensile strength of the base material should normally be used evenwhen the crack is situated in a welded joint. The reason for this is that the yield limit ofthe structure is not a local property but also depends on the strength properties of thematerial remote from the crack.

Kcr is the critical value of the stress intensity factor for the material at the crack front. Ifpossible, Kyr should be set to equal to the fracture toughness K k according to ASTM E-399 [3]. It is in many cases not possible to obtain such a value. Jr -values according toASTM E-813 [4] can be used instead and be convened according to equation (3).

Kcr =

Here E is the Young's modulus of the material and v its Poisson's ratio.

For application to components in nuclear power facilities fracture toughness dataaccording to appendix M can be used if the data for the considered material are lacking.

When not stated otherwise the material data for the actual temperature should be used.

2.6. Calculation of slow crack growth

The final fracture assessment as described below should be based on the estimated cracksize at the end of the service period. In cases where slow crack growth due to fatigue,stress corrosion cracking or some other mechanism can occur the possible growth mustbe accounted for in the determination of the final crack size.

The rate of crack growth due to both fatigue and stress corrosion cracking is supposed tobe governed by the stress intensity factor Ki. This quantity is calculated according tomethods described in appendix K.

For fatigue crack growth, the rate of growth c per loading cycle can be described by anexpression of the form

- £ = gu(AK,, R). (4)dN

Here

AKi = Klmax - Kimin , (5)

and

where Kimax and Kimin are the algebraic maximum and minimum, respectively, of Kjduring the load cycle. gu is a material function that can also depend on environmentalfactors such as temperature and humidity. For cases when R < 0 the influence of the R-value on the crack growth rate can be estimated by use of growth data for R = 0 and aneffective stress intensity factor range according to

AKf f f =K, r a a x , i f K t o i ^ O . (7)

For application to components in nuclear power facilities fatigue crack growth dataaccording to appendix M can be used if data for the material under consideration are notavailable.

For stress corrosion cracking, the growth rate c per time unit can be described by arelation of the form

| = &c(Ki). (8)

gsc is a material function which is strongly dependent on environmental factors such asthe temperature and the chemical properties of the environment.

For applications to components in nuclear power facilities stress corrosion cracking dataaccording to appendix M can be used if data for the material and environment underconsideration are not available.

2.7. Calculation o/Kf and Kf

The stress intensity factors Kf (caused by primary stresses op) and Kf (caused bysecondary stresses as) are calculated with the methods given in appendix K. For thecases given it is assumed that the nominal stress distribution (i.e. without considerationof the crack) is known.

Limits for the applicability of the solutions are given for the different cases. If results aredesired for a situation outside the applicability limits a recharacterization of the crackgeometry can sometimes be made. The following recharacterizations are likely to befrequent:

a) A semi elliptical surface crack with a length/depth ratio which is larger than theapplicability limit can instead be treated as a long crack.

b) A semi elliptical surface crack with a depth that exceeds the applicability limit caninstead be treated as a through thickness crack with the same length as the originalcrack.

c) A cylinder with a ratio between wall thickness and inner radius which is below theapplicability limit can instead be treated as a plate with a corresponding stress state.

Since the transition between two such cases is not continuous an assessment performedin this way may be exceedingly conservative. In the program [2] which is based on thecurrent procedure interpolation is done between the two cases.

In cases when the solutions of appendix K cannot be applied, stress intensity factors canbe obtained either by use of solutions found in the literature, see for example thehandbooks [5,6 and 7], or by numerical calculations, e.g. by the finite element method.

2.8. Calculation O

LR is defined as the ratio between the current primary l c J and the limit load Pg for thecomponent under consideration and with the presence of the crack taken into account. P»should be calculated under the assumption of a perfectly plastic material with the yieldstress ay chosen as discussed in section 2.5. Appendix L contains solutions of LR for thecases considered in this procedure.

Limits for the applicability of the solutions are given for the different cases. If results aredesired for a situation outside the applicability limits a recharacterization of the crackgeometry can sometimes be made similarly to what was discussed for the stress intensityfactor above.

In cases when the solutions of appendix L cannot be applied, LR can be obtained eitherby use of solutions found in the literature, see for example [8], or by numericalcalculations, e.g. by the finite element method.

2.9. Calculation o/KR

The ordinate KR in the failure assessment diagram (figure 1) is calculated in the followingway.

Kf + KfKR = - 1 - - + p , if 0,05 < x and 0,002 < LR ,

Kcr

(9)

KR = _J i , if x < 0,05 or LR < 0,002, (10)Kcr

p is a parameter that takes into account plastic effects because of interaction betweensecondary and primary stresses, p is obtained from the diagram in figure 2 where p isgiven as a function of LR with x as a parameter.

0.25

0.15 - -

0.2 0.4 0.6 0.8

0.05

Figure 2. Diagram for calculation of p.

2.10. Fracture assessment

In order to assess the risk of fracture the point (LR, KR) calculated as described above isplotted in the diagram in figure 1. If the point is situated in the safe region no initiation ofcrack growth is assumed to happen and thus no fracture occurs. The safe region is limitedby

KR < (1 - 0,14li)[0,3 + 0,7exp(-0,65LR)J, (12)

LR £ LJT • (13)

For materials without a yield plateau the upper LR-limit LRax is defined by

= 0,5(1 + — ) . (14)

For materials with a yield plateau LJ?" shall be equal to 1.

For acceptance assessments of components in nuclear facilities LRax shall be equal to 1.

When the failure load of a component with a crack is sought, the above describedprocedure is carried out for different load levels and the crack geometry is kept constant.The critical load is then given by the load level which causes the point (LR, KR) to fall onthe border of the safe region. In cases when the component is subjected to severalindependent load systems the relation between these must be specified also for load levelsother than the current one. For cases where the loading can be described by one loadparameter y p for the primary stresses and another load parameter \|/s for the secondary

10

stresses, the program [2] belonging to the procedure can automatically perform two typesof variation. Put

(15)

0 s = v/oo , (16)

where of! and of» are the primary and secondary stress distributions caused by thecurrent loads. The load parameters \|/p and \f are thus equal to unity for the current loadconditions. The variation can now be performed according to the two specific relations:

a) \|/p and \\? are varied proportionally,

or

b) \|/p is varied while \f is kept constant.

When instead the critical crack size for given loading conditions is sought, the abovedescribed procedure is carried out for different crack sizes and the loading is keptconstant. The critical crack size is then given by the crack size which causes the point(LR, KR) to fall on the border of the safe region. In cases when the crack is defined bymore than one parameter the relation between these must be specified also for size levelsother than the current one. For cases where the crack can be described by length / anddepth a, the program [2] belonging to the procedure can automatically perform two typesof variation:

c) The depth a and the length / are varied proportionally,

or

d) The depth a is varied while the length / is kept constant.

Which of the given alternatives, a) or b) and c) or d), that should be chosen depends onthe specific conditions.

2.11. Assessment of results

An assessment of the results which in many cases may be an acceptance decision can beperformed in several ways. Simple measures of the safety margin may be the value of theload parameter \\i at the critical condition or the ratio between critical and current cracksize. A generalization of these measures may be constructed in the following way. Definedesign quantities as.

<fd = tfoB , (17)

<rsd = fi&o • (18)

Here of, and of) are the primary and secondary stress distributions caused by the currentloads. For the assessment the design stresses op d and o are used instead. The loadfactors "$ and y} are chosen so that the desired safety margin is obtained. In ananalogous way other design quantities arc defined.

ad = f g a , (19)

11

/ d = V g / . (20)

öv = oY/fm . (21)

K?r = Kcr/y* . (22)

7g and Vg are denoted size factors and Ym and Ym are denoted material factors. Thefracture assessment is performed in the way defined above using the design quantitiesinstead of the current quantities. If this assessment results in the point (LR, KR), obtainedby use of the design quantities, to fall in the safe region in figure 1, the situation isconsidered to be acceptable. By using different values of the different load, size andmaterial factors a differentiation of the margins with respect to different quantities can bemade.

In cases which are particularly difficult to assess a sensitivity analysis may be necessary.Such an analysis is simplest to perform by a systematic variation of the load, size andmaterial factors.

A set of load, size and material factors for applications to components in nuclear powerfacilities is given in appendix P.

References

[1] MILNE, I., AINSWORTH, R. A., DOWLING, A. R. and STEWART, A. T.,Assessment of the integrity of structures containing defects. Pressure Vessels andPiping, Vol. 32, pp. 3-104, 1988.

[2] BERGMAN, M., A procedure for safety assessment of components with cracks -manual for computer program, SA/FoU-Report 91/18, The Swedish PlantInspectorate, Stockholm, 1991, (In Swedish).

[3] ASTM E-399 Test method for plane-strain fracture toughness of metallicmaterials, American Society for Testing and Materials, Philadelphia, Pa.

[4] ASTM E-813 Standard test method for Jjc, a measure of fracture toughness,American Society for Testing and Materials, Philadelphia, Pa.

[5] MURAKAMI, Y. (ed.), Stress intensity factors handbook, Vol. 1-2, PergamonPress, 1987.

[6] TADA, H., PARIS, P. C. and IRW1N, G. C , The stress analysis of crackshandbook, 2nd edition, Paris Productions Inc, St. Louis, Miss., 1985.

[7] ROOKE, D. P. and CARTWRIGHT, D. J., Compendium of stress intensityfactors, Her Majesty's Stationary Office, London, 1976.

[8] MILLER, A. G., Review of limit loads of structures containing defects, PressureVessels and Piping, Vol. 32, pp. 197-327, 1988.

12

Appendix A. Defect characterization

A fracture mechanics assessment requires that the current defect geometry is characterizeduniquely. In this appendix general rules for this are given. For additional information it isreferred to ASME Boiler and Pressure Vessel Code, Sect. XI [Al] and British StandardsInst., Published Document 6493 [A2].

1. Defect geometry

Surface defects are characterized as quarter elliptical or semi elliptical cracks. Em bedde Jdefects are characterized as elliptical cracks. Through thickness defects are characterizedas rectangular cracks. The characterizing parameters of the crack are defined as follows:

a) The depth of a surface crack a corresponding to half of the minor axis of the ellipse.

b) The depth of an embedded crack 2a corresponding to the minor axis of the ellipse.

c) The length of a crack / corresponding to the major axis of the ellipse for surface andembedded cracks or the side of the rectangle for through thickness cracks.

In case the plane of the defect does not coincide with a plane normal to a principal stressdirection, the defect shall be projected on to normal planes of each principal stressdirection. The one of these projections is chosen for the assessment that gives the mostconservative result according to this procedure

2. Interaction between neighbouring defects

When a defect is situated near a free surface or is near to other defects the interaction shallbe taken into account. Some cases of practical importance are illustrated in figure Al.According to the present rules the defects shall be regarded as one compound defect if thedistance s satisfies the condition given in figure Al. In the distance s possible cladding isincluded. The compound defect size is determined by the length and depth of thegeometry described above which circumscribes the defects. The following shall be noted:

a) For surface cracks, except surface cracks al corners, and embedded cracks the ratio a//shall be less than or equal to 0,5.

b) In case of surface cracks in cladded surfaces the crack depth should be measured fromthe free surface of the cladding. If the defect is wholly contained in the cladding theneed of an assessment has to be judged on a case-by-case basis.

c) Defects in parallel planes should be regarded as situated in a common plane if thedistance between their respective planes is less than 12,7 mm.

It should be noted that for case 2, 3 and 6 in figure Al, i.e. the criteria for merging ofneighbouring defects in the length direction here is based on the defect length. This is inaccordance with the corresponding criteria in [A2]. All other cases coincide with the rules

13

Case Defect sketches Criterion

2a.;

If s < 0,4aithena = 2ai + s

. » . - h -

then/ = l\

/2)/2

+ s

then/2)/2

+ s

2 a i7

sC >C ^

If s < max(2ai,then2a = 2ai + 2a2 + s

sa2

C ^>

If s<max(2ai,a2)thena = 2aj + a2 + s

If si <max(2ai,a2)and s2 < (/j + /2)/2thena = 2aj + a2 + &\I = l\ + 1 2 + s2

Figure Al. Rules for defect characterization at interaction.

14

References

[Al] ASME Boiler and Pressure Vessel Code.Sect. XI, Rules for inservice inspectionof nuclear power plant components, The American Society of MechanicalEngineers, New York, 1989.

[A2] BS PD6493, Guidance on some methods for the derivation of acceptance levelsfor defects in fusion welded joints, British Standards Institution, 1980.

15

Appendix E. Residual stresses

Residual stresses exist in a structure when it is free from external loading. Thedistribution and magnitude of residual stresses in a component depend on the fabricationprocess and service influences. Residual stresses are normally created during themanufacturing stage but can also appear or be redistributed during service. At the hydropressure or load test of a component, peaks of residual stresses can be relaxed due tolocal plasticity if the total stress level exceeds the yield strength of the material.

Guidelines for estimation of residual stresses in steel components due to welding aregiven below for use in cases when more precise information is not available. In somecases, welding leads to formation of bainite or martensite at cooling which results in achange in volume. Such a change affects the weld residual stresses and make theguidelines given here not applicable.

The magnitude of the residual stresses are expressed in Sr which is set to the yieldstrength of the material at the actual temperature, see chapter 2.5. The same yield strengthvalues should normally be used here as in the determination of LR according to chapter2.8. For austenitic stainless steels, however, Sr should be chosen to at least 1,65 timesthe minimum value of the yield strength (Cymin) according to code or standard [El].

1. Butt welded joint

a) Thin plates

With thin plates is here meant plates with butt welds where the variation of the residualstress in the thickness direction is insignificant. This holds for butt welds with only oneor a few weld beads. In most cases the plates thickness is less than 25 mm.

The weld residual stresses acting in the length direction of the weld has a distributionacross the weld according to figure El. The width of the zone with tensile stresses / isabout four to six times the plate thickness. This distribution is obtained along the entireweld except when the weld ends at a free plate end where the stresses tend to zero.

<r/Sr 0.5 —

- 4(x//)2]exp|-2(x//)2]

-2 -1.5 -1 -0.5 0

Figure El . Distribution across the weld of residual stresses acting in thelength direction of the weld.

16

The magnitude of the weld residual stresses acting transverse to the weld direction isdependent on whether the plates have been free or fixed during welding. If for instance,the plates are fixed to each other by tack welding before the final welding, residualstresses with a magnitude up to Sr are obtained. If one of the two plates is free to moveduring cooling the residual stresses become limited to 0,2Sr. The values are valid at thecentre line of the weld. They successively diminish in sections outside the the centre line.

b) Thick plates

In thick plates the variation of the residual stresses in the thickness direction cannot beneglected. The plate thickness is usually above 25 mm in such cases. The stressdistribution is much more complicated in thick plates than in thin plates and depends verymuch on joint design and weld method. It is therefore impossible to give simple generalguidelines for estimation of residual stresses in thick plates. For more detailedinformation, see [E2, E3 and E4].

The weld residual stresses acting in the weld direction vary both in the thickness directionand across the weld. The highest stress in the symmetry plane of the weld is Sr. Thesimplest but in many cases very conservative assumption is that the weld residual stressis constant and equal to Sr throughout the entire thickness.

In a symmetric joint, for instance an X-joint or a double U-joint with many weld beads,the stress varies almost linearly from a compressive stress with magnitude Sr in the centreof the weld to a tensile stress of magnitude Sr at the surfaces of the weld, see figure E2.

1.5

a = Sr[4abs(x/t) - 1]

a/Sr o - -

-0.5

Figure E2. Thickness distribution of residual stresses acting both along andacross a weld in a symmetric joint with many weld beads. This distributionfor residual stresses acting across the weld applies only for a free plate.

The weld residual stresses acting across the weld vary in roughly the same way in thethickness direction as the stresses acting in the direction of the weld if any of the plates isfree to move during the cooling, i.e. the highest stress in the symmetry plane of the weldis Sr. In a symmetric joint with many weld beads the distribution of residual stressbecomes as shown in figure E2.

17

c) Thin walled pipes

The residual stress distribution in girth welds in both thick walled and thin walled pipesare complicated and depends on joint shape, weld method, wall thickness and radius. Therecommendations given here are very simplified and do only apply to girth welds appliedfrom the outside. More detailed information about weld residual stresses in girth welds isgiven in [E5].

With thin walled pipes is here meant pipes with a wall thickness less than 25 mm.

The local residual stresses acting along and across a girth weld in the weld metal and alsoa short distance outside the fusion line are shown in figure E3. The stresses decreasesuccessively in sections further outside the fusion line.

1 o/S r -1

Figure E3. Local distribution of residual stress acting along and across a girthweld in the weld metal and a short distance outside the fusion line.

The residual stresses acting along and across a seam weld can approximately be estimatedfrom the corresponding distribution in a thin plate if the pipe is not adjusted with respectto roundness after welding.

d) Thick walled pipes

With thick walled pipes is here meant pipes with a wall thickness larger than or equal to25 mm.

The local residual stresses acting along and across a girth weld in the weld metal and asmall distance outside the fusion line are shown in figure E4. The stresses decreasesuccessively in sections further outside the fusion line.

18

0,5 o/Sr

• x

<y/Sr

a = Sr[0,27 - 0,91(x/t) - 4,93(x/t)2

+ 8,6(x/t)3 - 2,03(x/t)4]

Figure E4. Local distribution of residual stress acting along and across a girthweld in the weld metal and a short distance outside the fusion line.

The residual stresses acting along and across a seam weld can approximately be estimatedfrom the corresponding distribution in a thick plate if the pipe is not adjusted with respectto roundness after welding.

2. Fillet weld

The residual stresses acting both along and across a fillet weld amount to Sr. The stressesdecrease outside the fillet weld.

3. Stress relieved joints

In correctly stress relieved joints the maximum residual stress decreases to 0,15 - 0,20Sr.Micro alloyed steels are sometimes stress relieved at temperatures lower than 560 °C. Therelaxation is in such cases not so effective and the remaining residual stress is larger than0,2Sr.

Local stress relief that sometimes is used gives a result that is more difficult to assess.

4. Conclusion

The above given guidelines for estimation of residual stresses are summarized in tableEl .

19

Type of joint

Butt weld in a plate

Butt weld in a pipe,girth weld

Butt weld in a pipe,seam weld

Thickness

Thin (< 25 mm)

Thick (> 25 mm)

Thin (< 25 mm)

Thick (> 25 mm)

Thin (< 25 mm)Thick (> 25 mm)

Fillet weldStress relieved weld

Residual stress,along the weldSee figure El

< S r

See also figure E2< S r

See figure E3< 0,5SrSee figure E4See thin plateSee thick plate<S r

< 0,2Sr

Residual stress,across the weld< Sr, fixed plate< 0,2Sr, free plate< S r

See also figure E2See figure E3

See figure E4

See thin plateSee thick plate

^ s r< 0,2Sr

Sr = ay, however at least 1.65oYmin f°r austenitic stainless steels

Table El. Estimation of residual stresses in welded joints.

References

[El] EPRI NP - 4690 SR, Evaluation of flaws in austenitic steel piping, SpecialReport, EPRI, 1986.

[E2] UEDA, Y., TAKAHASHI, E., SAKAMOTO, K. and NAKACHO, K.,Multipass Welding stresses in very thick plates and their reduction from stressrelief annealing, Trans. JWRI, Vol. 5, No. 2, pp. 79-88,1976.

[E3] UEDA, Y. and NAKACHO, K., Distribution of welding residual stresses invarious welded joints of thick plates, Trans. JWRI, Vol. 15, No. 1, 1986.

[E4] RUND, C. O. and DIMASCO, P. S., A prediction of residual stresses in heavyplate butt welds, J. Materials for Energy Systems, Vol. 3, pp. 62-65,1981.

[E5] SCARAMANGAS, A., Residual stresses in girth butt weld pipes, TechnicalReport No. CUED/D - Struct/TR. 109, Dept. of Engineering, CambridgeUniversity, 1984.

20

Appendix G. Geometries treated in this handbook

The term nominal stress denotes the stress state that would act in the corresponding crackfree component.

1. Crack in a plate

a) Semi elliptical surface crack

Figure Gl. Semi elliptical surface crack in a plate.

Nominal stress state perpendicular to the crack plane is given by

C = <5m +

b) Long surface crack

21

2w

2h

Figure G2. Long surface crack in a plate.


a = cm

(G2)

c) Through thickness crack

22

Figure G3. Through thickness crack in a plate.


c = cm + > (G3)

23

d) Elliptical embedded crack

Figure G4. Elliptical embedded crack in a plate.


o = cm +cb(l - Y ) - (G4)

24

2. Crack orientated axiallx in a cylinder

a) Semi elliptical surface crack on the inside of a cylinder

Figure G5. Semi elliptical surface crack orientated axially on the inside of a cylinder.


f, 2 l Mo = cm + <jb 1 - — • (G5)

b) Long sioface crack on the inside of a cylinder

25

Figure G6. Long surface crack orientated axially on the inside of a cylinder.


a = c m + Chll - — 1 . (G6)

26

c) Semi elliptical surface crack on the outside of a cylinder

Figure G7. Semi elliptical surface crack orientated axially on the outside of a cylinder.


a = am + cb\A - —). (G7)

27

d) Long surface crack on the outside of a cylinder

Figure G8. Long surface crack orientated axially on the outside of a cylinder.


2iT= om + • 0 - T ) - (G8)

28

e) Through thickness crack

Figure G9. Through thickness crack orientated axially in a cylinder..


o = (Tm +2u\ (G9)

3. Crack orientated circumferentially in a cylinder


29

1 U

Figure G10. Semi elliptical surface crack orientated circumferentially on theinside of a cylinder.


R + u= Om + Cb 1 - —

The crack is symmetrical with respect to the bending plane.

(G10)

30

b) Complete circumferential surface crack on the inside of a cylinder

Figure G11. Complete circumferential surface crack on the inside of a cylinder.


2u\ Rj + uJ (Gil)



31

B

Figure G12. Thi jgh thickness crack orientated circumferentially in a cylinder.



(G12)

32

4. Crack in a sphere

a) Through thickness crack

•

Figure G13. Through thickness crack in a sphere.


•(• • 2 T )(G13)

33

5. Crack in a nozzle

a) Quarter circular comer crack

Figure 014. Quarter circular corner crack in a nozzle in a cylindrical vessel.

Nominal stress state perpendicular to the crack plane for calculation of LR for an arbitraryload is given by

c = cm + cb 1 - (G14)

Nominal stress state perpendicular to the crack surface for calculation of Ki for anarbitrary load is given by

a = Aoo + A,0( l - u/a) + Aoi (1 - v/a) + A20(l - u/a)2 + (G15)

Ao2(l - v/a)2 + A30(l - u/a)3 + Ao3(l -

34

Figure G15. Corner crack in a nozzle in a cylindrical vessel.

Figure G16. Corner crack in a nozzle in a spherical vessel.

35

Figure G17. Corner crack in a nozzle in a cylindrical vessel, definition ofpressurized area Ap and load bearing area As for calculation of LR for aninternal pressure load only.

Nominal stress state perpendicular to the crack plane in the undisturbed region of thevessel for calculation of Ki for an internal pressure load only is given by

o = o m . (G16)

36

Figure GIS. Corner crack in a nozzle in a spherical vessel, definition ofpressurized area Ap and load bearing area As for calculation of LR for aninternal pressure load only.

Nominal stress state perpendicular to the crack plane in the undisturbed region of thevessel for calculation of Ki for an internal pressure load only is given by

o = ö m . (G17)

37

6. Crack at a hole

a) Quarter elliptical comer crack

Figure G19. Quarter elliptical corner crack at a hole.

Nominal stress state perpendicular to the crack plane for calculation of LR is given by

i , 2 v .

t(G18)

Nominal stress state perpendicular to the crack plane for calculation of Ki is given by

o = Aoo + Aio(u/R) + Aoi(v/t) + A20(u/R)2 + Aj i (u/R)(v/t) + (G19)

Ao2(v/t)2 + A30(u/R)3 + A2i(u/R)2(v/t) + Ai2(u/R)(v/t)2 +

Ao3(v/t)3.

38

Appendix K. Stress intensity factor solutions

I. Crack in a plate


K\ is given by

(Kl)

The stresses ©„, and a b are defined according to figure Gl and equation Gl. Thegeometry functions fm and fb are given as functions of a/t and a// in equations K2 - KSand table Kl. fm and fb are given at the deepest point of the crack (A) and at theintersection of the crack with the free surface (B). See figure Gl.

Region of applicability: a/t < 0,8 ,a//<0,5,//w < 0,3 ,//h < 0,3 .

Reference: [Kl and K2].

(K2)

(1

+

i r ( o 89 ~\7-rr-rrA 1,13 - 0,18a// + -0,54 + — ' (a/t)2

+ 4,595 (a//)1 •65)°'5L I 0,2 + 2a//)24)(a/t)4] ,+ 14(1 - 2a//)'

= [1 + (-1.22 - 0,24a//)a/t +

(0,55- l,05(2a//)0-75 + 0,47(23//)* >5)(a/t)2]f£ ,

r? =

= [(1,1 + 0,35(a/t)2)(2a//)05]-f£,

[1 - 0,34a/t - 0,22(a/t)(a//)]-fm .

(K3)

(K4)

(K5)

a// = 0,5a/t0

0,20,40,60,8

M'm

0,6630,6680,6810,7000,717

0,6630,4880,3130,130-0,065

fB

0,7290,7440,7880,8580,949

0,7290,6770,6460,6270,608

39

a/t0

0,20,40,60,8

fA

0,8960,9200,9851,0741,158

•all = 0,25fAIb

0,8960,6880,4950,2850,040

Mlm

0,6970,7240,8050,9311,084

fBIb

0,6970,6670,6780,7100,741

a//= 0,1a/t0

0,20,40,60,8

a/t0

0,20,40,60,8

fA

1,0591,1221,3011,5611,848

1,1031,1991,4921,9992,746

1,0590,8550,7110,5520,338

a// = 0,05

1,1030,9220,8470,7930,704

fB

0,5210,5590,6720,8561,094

fB

0,3840,4220,5460,7751,150

0,5210,5190,5750,6700,777

fBIb

0,3840,3930,4690,6120,827

a//= 0,025a/t0

0,20,40,60,8

fA

1,1201,2451,6812,6094,330

»•Ath

1,1200,9620,9761,1071,314

fB

0,2750,3100,4340,7151,282

fBIb

0,2750,2890,3740,5670,928

Table Kl. Semi elliptical surface crack in a plate, fm and ft,.

b) Umg surface crack

is given by

Kj = Jizii«ymt'm + (K6)

The stresses om and at, are defined according to figure G2 and equation G2. Thegeometry functions fm and 1'b are given as functions of a/t in table K2. fm and fb are givenat the deepest point of the crack (A). See figure G2.

Region of applicability: a/t < 0,8 ,a/h < 0,6 .

Reference: [K3|.

40

a/t0

0,20,40,60,8

•m1,1221,3672,1114,03311,96

fA

»b1,1221,0551,2611,914

4,676

Table K2. Long surface crack in a plate, fm and


is given by

Ki = /7t7/2(omfm + cbfb). (K7)

The stresses <rm and <Jb are defined according to figure G3 and equation G3. Thegeometry functions fm and fb are given in table K3. fm and fb are given at theintersections of the crack with the planes u = 0 (A) and u = t (B). See figure G3.

Region of applicability: //w < 0,3 ,//h < 0,3 .

Reference: [K4].

1

fA•b

1 1#-1

Table K3. Through thickness crack in a plate, fm and fb.


is given by

(K8)

The stresses a™ and a b are defined according to figure G4 and equation G4. Thegeometry functions fm and fb are given as functions of a/t, a// and e/t in equations K9 -K12 and tables K4 - K5. fm and l'b are given at the extreme points of the minor axis thatare closest to (A) and furthest from (B) the plane u = 0. See figure G4.

Resion of applicability: 2a/t < 0,9(1 - 2e/t),2a// < 1 ,0 < e/t < 0,5 ,//w < 0,3 .

Reference: [K5|.

,A _Im —

1,01 - 0,37-2a//

2a/t

1 - 2e/t(1 - 0 , 4 - 2 a / / - (e/t)2)

J

(K9)

41

fA _»b =

(1,01 - 0,37 • 2a//)(2e/t + 0,5 • 2a/t + 0,17 • 2a//- 2a/t)

HifB _

d - 0,4.2a//- WO*)]

1,01 - 0,37-2a//

" 4

- 0,4-2a// - 0,8(e/t)0-4)10.54 '[l - ( Y ^ -

(1,01 - 0,37 • 2a//)(2e/t - 0,5• 2a/t - 0,17 • 2a//- 2a/t)

[ l - f 2a/t V'8(l - 0,4-2a// - 0,8(e/t)°-4)l°'54

L VI - 2 e / t ; J

(K10)

(Kll)

(K12)

2a//=l,e/t = 02a/t

00,20,40,6

fA

0,6380,6490,6810,739

Im0,6380,6490,6810,739

2a// = 0,5, e/t = 02a/t

00,20,40,6

fA

0,8240,8440,9011,014

fB

0,8240,8440,9011,014

2a// = 0,25, e/t = 02a/t

00,20,40,6

fAIm

0,9170,9421,0161,166

fB

0,9170,9421,0161,166

2a// = 0, e/t = 02a/t

00,20,40,6

fA

1,0101,0411,1331,329

1,0101,0411,1331,329

2a// =1 , e/t = 0,152a/t0

0,20,40,6

fA•m

0,6380,6590,7250,870

fBlm0,6380,6460,6680,705

2a//= 0,5, e/t = 0,152a/t

00,20,40,6

fAIm

0,8240,8620,9871,332

fB

0,8240,8440,9021,016

2a//= 0,25, e/t = 0,152a/t

00,20,40,6

fAim

0,9170,9661,1291,655

fB

0,9170,9451,0291,206

2a//= 0, e/t = 0,152a/t0

0,20,40,6

fA

Im1,0101,0711,2822,093

fB•m

1,0101,0481,1621,429

2a//=1, e/t = 0,32a/t

00,20,40,6

fAim

0,6380,694

--

fBIm

0,6380,648

--

2a//= 0,5, e/t = 0,32a/t

00,20,40,6

fAIm

0,8240,932

--

fB

0,8240,866

--

2a// = 0,25, e/t = 0,32a/t

00,20,40,6

fA

0,9171,058

--

fBIm

0,9170,980

--

2a// = 0, e/t = 0,32a/t0

0,20,40,6

fAIm

1,0101,189

--

fBim

1,0101,099

--

Table K4. Elliptical embedded crack in a plate, fn

42

2a//=l,e/t = 02a/t0

0,20,40,6

fA

0,0000,0870,1820,296

'b

0,000-0,087-0,182-0,296

2a// = 0,5, e/t = 02a/t0

0,20,40,6

0,0000,0980,2100,355

0,000-0,098-0,210-0,355

2a// = 0,25, e/t = 02a/t0

0,20,40,6

MIb

0,0000,1020,2200,379

rBIb

0,000-0,102-0,220-0,379

2a// = 0,e/t = 02a/t0

0,20,40,6

0,0000,1040,2270,399

0,000-0,104-0,227-0,399

2a// =1, e/t = 0,152a/t0

0,20,40,6

r-Afb

0,1910,2860,4110,609

fBIb

0,1910,1080,022-0,071

2a// = 0,5, e/t = 0,152a/t0

0,20,40,6

fAIb

0,2470,3590,5260,866

0,2470,1550,060-0,051

2a// = 0,25, e/t = 0,152a/t0

0,20,40,6

fAIb

0,2750,3940,5841,034

fBIb

0,2750,1810,086-0,030

2a//= 0, e/t = 0,152a/t0

0,20,40,6

ft0,3030,4280,6411,256

fb"0,3030,2100,1660,000

2a//=1, e/t = 0,32a/t0

0,20,40,6

fAIb

0,3830,509

--

0,3830,303

--

2a// = 0,5, e/t = 0,32a/t0

0,20,40,6

fAIb

0,4940,668

--

fBIb

0,4940,418

--

2a// = 0,25, e/t = 0,32a/t0

0,20,40,6

fAtb

0,5500,749

--

fBIb

0,5500,482

--

2a// = 0, e/t = 0,32a/t0

0,20,40,6

0,6060,833

--

fbb

0,6060,550

--

Table K5. Elliptical embedded crack in a plate, ft,.

43

2. Crack orientated axially in a cylinder


is given by

Ki = /jra(amfm + abfb). (K13)

The stresses c m and <Jb are defined according to figure G5 and equation G5. Thegeometry functions fm and fb are given as functions of a/t, a// and t/Rj in table K6. fm andfb are given at the deepest point of the crack (A) and at the intersection of the crack withthe free surface (B). See figure G5.

Region of applicability: a/t £ 0,8 ,0,025 < a// < 0,5 ,0,1 <= t/Ri <, 0,25 ,//w < 0,3 .

Reference: [K1.K2 and K6].

a//= 0,5, t/Ri = 0,1a/t0

0,20,40,60,8

fA

0,6630,6470,6610,6770,694

ft0,6630,4640,2910,110-0,080

Im0,7290,7260,7600,8040,859

0,7290,6760,6490,6230,599

a//= 0,2, t/Rj = 0,1a/t0

0,20,40,60,8

0,9510,9321,0161,1091,211

ft0,9510,6980,5190,3160,090

fB

0,6620,6760,7680,8961,060

$0,6620,6320,6510,6740,700

a// = 0,1, t/Rj = 0,1a/t0

0,20,40,60,8

fA

1,0591,0621,2601,5001,783

ft1,0590,8060,6770,5150,320

0,5210,5780,695 j0,8761,123

0,5210,5480,5970,6600,737

a//= 0,05, t/Rj = 0,1a/t0

0,20,40,60,8

fA

1,1031,1721,4941,9852,737

ft1,1030,8970,8340,7650,689

0,3840,4510,5820,8201,219

#0,3840,4290,5030,6230,810

a//=0,5, t/Ri = 0,25a/t0

0,20,40,60,8

fA

0,6630,6430,6560,6770,704

fA

0,6630,4610,2880,107-0,079

fB

0,7290,7190,7450,7850,838

0,7290,6690,6380,6100,585

a//=0,2, t/Ri = 0,25a/t0

0,20,40,60,8

fA

0,9510,9190,9981,1101,255

ft0,9510,6880,5060,3110,103

fB

0,6620,6690,7590,8891,060

*0,6620,6270,6440,6660,694

a// = 0,1, t/Rj = 0,25a/t0

0,20,40,60,8

fA

1,0591,0451,2401,5141,865

fA

1,0590,7910,6630,5150,348

lm0,5210,5770,6980,8871,144

0,5210,5470,5990,6650,745

a// = 0,05, t/Rj = 0,25a/t0

0,20,40,60,8

fA

1,1031,1531,4702,0032,864

fA

1,1030,8810,8160,7650,749

0,3840,4510,5850,8301,242

0,3840,4280,5040,6270,819

44

a/t0

0,20,40,60,8

a// = 0,025, t/RfA'm

1,1201,2311,7012,6194,364

f(T1,1200,9460,9711,0801,301

0,2750,3350,4690,7651,374

fb

0,2750,3180,4060,5840,919

a/t0

0,20,40,60,8

a// = 0,025, t/RifA

1,1201,2111,6742,2853,163

1,1200,9290,9501,0791,081

= 0,25fB

0,2750,3340,4710,7741,400

0,2750,3180,4070,5870,928

Table K6. Semi elliptical surface crack orientated axially on the inside of a cylinder, fmand ft,.

b) Long surface crack on the inside of a cylinder

is given by

+abfb). (K14)

The stresses am and Ob are defined according to figure G6 and equation G6. Thegeometry functions fm and fb are given as functions of a/t and t/Rj in table K7. fm and fbare given at the deepest point of the crack (A). See figure G6.

Region of applicability: a/t < 0,8 ,0,1 < t/Ri < 0,25.

Reference: [K3, K7 and K8].

t/Ri = 0,1a/t0

0,20,40,60,8

fA'm

1,1221,3801,9302,9604,820

1,1221,0181,1431,4841,990

t/Ri = 0,25a/t0

0,20,40,60,8

fA

1,1221,3041,7842,5663,461

1,1221,0021,0331,0940,949

Table K7. Long surface crack orientated axially on the inside of acylinder, fm and fb.


is given by

Ki = /Jra(amfm + (K15)

The stresses om and Oh are defined according to figure G7 and equation G7. Thegeometry functions fm and fb are given as functions of a/t, a// and t/R, in table K8. fm andl'h are given at the deepest point of the crack (A) and at the intersection of the crack withthe free surface (B). See figure G7.

Region of applicability: a/t < 0,8 ,0,025 < a// < 0,5 ,0,1 <, t/Ri < 0,25 ,//w < 0,3 .

Reference: [K1.K2 and K6].

45

a// = 0,5, t/Ri = 0,1a/t0

0,20,40,60,8

f A

0,6630,6530,6750,6950,712

fA

0,6630,4700,3010,122-0,068

fB

0,7290,7360,7830,8460,926

0,7290,6850,6660,6490,634

a//= 0,2, t/Ri = 0,1a/t0

0,20,40,60,8

fA

0,9510,9531,0771,2131,361

0,9510,7160,5610,3770,167

fB•m

0,6620,6850,7990,9701,198

*0,6620,6410,6730,7150,769

a// = 0,l,t/R, = 0,la/t0

0,20,40,60,8

fA

1,0591,0921,3701,7352,188

fA

1,0590,8310,7500,6440,514

0,5210,5830,7060,9121,202

0,5210,5520,6060,6810,780

a//= 0,05, t/Rj = 0.1a/t0

0,20,40,60,8

fA

1,1031,2061,6242,2953,360

fA

1,1030,9260,9230,9571,108

0,3840,4550,5920,8531,305

*0,3840,4320,5100,6430,857

a// = 0,025,t/R, = 0,la/t0

0,20,40,60,8

fA'm

1,1201,2661,8492,6284,090

rAIh

1,1200,9761,0751,3491,549

0,2750,3380,4770,7961,471

0,2750,3210,4120,6020,972

a// = 0,5, t/Ri = 0,25a/t0

0,20,40,60,8

fA

0,6630,6560,6830,7100,736

fA

0,6630,4730,3070,131-0,055

0,7290,7410,7930,8640,954

0,7290,6890,6730,6590,647

a// = 0,2, t/Rj = 0,25a/t0

0,20,40,60,8

fAlm0,9510,9641,1101,2891,502

fA

0,9510,7260,5820,4170,230

fB1m0,6620,6890,8060,9821,217

0,6620,6440,6780,7210,775

a//= 0,1, t/Ri = 0,25a/t0

0,20,40,60,8

fA

1,0591,1061,4101,8382,390

fA

1,0590,8440,7760,6930,595

fB

0,5210,5830,6930,8671,105

0,5210,5520,5980,6590,736

a// = 0,05, t/Ri = 0,25a/t0

0,20,40,60,8

fA

1,1031,2221,6722,4323,670

fAM>

1,1030,9390,9551,0291,128

fB

0,3840,4550,5810,8111,199

0,3840,4320,5040,6220,809

a// = 0,025, t/Rj = 0,25a/t0

0,20,40.60,8

fA

1,1201,2821,7532,5813,839

fA

1,1200,9911,0111,1071,153

fB

0,2750,3380,4680,7571,352

0,2750,3210,4070,5830,918

Table K8. Semi elliptical surface crack orientated axially c.i the outside of a cylinder, fnand ft,.

46


is given by

obfb). (K16)

The stresses o m and <?b are defined according to figure G8 and equation G8. Thegeometry functions fm and fb are given as functions of a/t and t/Ri in table K9. fm and fbare given at the deepest point of the crack (A). See figure G8.

Region of applicability: a/t < 0,8 ,0,1 < t/Rj < 0,25 .


t/Ri = 0.1

a/t0

0,20.40,60,8

fA

1,1221,3801,9302,9604,820

ft1,1221,0181,1431,4841,990

t/Ri = 0,25a/t0

0.20,40,60,8

fA

1,1221,3041,7842,5663,461

1,1221,0021,0331,0940,949

Table K9. Long surface crack orientated axially on the outside ofa cylinder, fm and fb.


K\ is given by

Ki = /it772(crmfm + c b f b ) . (K17)

The stresses o m and <Jb are defined according to figure G9 and equation G9. Thegeometry functions fm and fb are given as functions of //t and t/Rj in table K10. fm and fbare given at the intersections of the crack with the planes u = 0 (A) and u = t (B). Seefigure G9.

Region of applicability: //t £ 25 ,0,05 < t/Ri < 0,1 ,//w < 0,3 .

Reference: [K9].

47

t/R, = 0,05ih0246810152025

fA

1,0001,1051,2951,5141,7391,9612,4692,8873,215

1,0001,0031,0010,9950,9860,9760,9480,9170,891

fB•m

1,0000,9700,9751,0351,1401,2811,7492,3282,982

-1,000-0,984-0,956-0,924-0,893-0,862-0,792-0,731-0,681

t/Ri=0,lll\0246810152025

fA

1,0001,1761,4681,7822,0882,3712,9573,3643,640

fAlb

1,0001,0030,9960,9840,9700,9540,9130,8760,840

fH

1,0000,9661,0201,1651,3791,6432,4483,4014,441

fbB

-1,000-0,973-0,931-0,887-0,845-0,805-0,721-0,653-0,597

Table K10. Through thickness crack orientated axially in a cylinder, fm and ft,.

48



K\ is given by

+ afm + Cblb • (K18)

The stresses om, ob and Cbg are defined according to figure G10 and equation G10. Thegeometry functions fm and fb are given as functions of a/t, a// and t/Ri in table Kl 1. fmand fb are given at the deepest point of the crack (A) and at the intersection of the crackwith the free surface (B). See figure G10. The solution of Ki for global bending stress ishere approximated with the solution for membrane stress using the stress at the deepestpoint of the crack. This is a good approximation for short cracks at the deepest point ofthe crack.

Region of applicability: a/t < 0,8 ,0,05 < a// < 0,5 ,0,1 < t/R, < 0,2 .

Reference: [Kl, K2, KlOand Kll].

a// = 0,5, t/R, = 0,1a/t0

0,20r40,60,8

•m0,6630,6670,6700,6860,702

0,6630,5740,3270,140-0,105

fB

0,7290,6810,7060,7330,764

0,7290,6230,5280,4310,332

a//= 0,25, t/R, = 0,1a/t0

0,20,40,60,8

fA

0,8960,9991,0311,1211,148

0,8960,7310,5040,3060,014

fB

0,6970,7310,8010,8890,993

*0,6970,6280,5630,5020,445

a/7 = 0,1, t/R, = 0,1aft0

0,20,40,60,8

fA

1,0591,1681,3751,5991,803

fA

1,0590,8700,7360,5610,269

fB

0,5210,6170,8351,0481,255

*0,5210,6230,5910,5560,519

a// = 0,5, t/Ri = 0,2a/t0

0,20,40,60,8

fA

0,6630,6670,6700,6860,702

0,6630,5820,3340,117-0,099

fBim

0,7290,6810,7060,7330,764

0,7290,6230,5280,4310,332

a// =0,25, t/R, = 0,2a/t0

0,20,40,60,8

fA

0,8961,0041,0301,1241,192

0,8960,7350,5030,3050,027

fBIjn

0,6970,7310,8010,8890,993

0,6970,6280,5630,5020,445

a//= 0,1, t/Ri = 0,2a/t0

0,20,40,60,8

fAIm

1,0591,1441,3181,5171,782

1,0590,8510,6980,5150,253

fB

0,5210,6170,8351,0481,255

fBIb

0,5210,6230,5910,5560,519

49

a// = 0,05, t/R; = 0,1a/t0

0,20,40,60,8

fA

1,1031,2191,5291,9392,411

fA»b

1,1030,9210,8290,6770,479

fB

0,3840,4820,7000,9811,363

0,3840,4870,4980,5250,570

all = 0,05, t/Ri = 0,2a/t0

0,20,40,60,8

fA

1,1031,2141,3821,6612,031

fA

1,1030,9030,7760,6240,386

fB

0,3840,4820,7000,9811,363

0,3840,4870,4980,5250,570

Table Kl 1. Semi elliptical surface crack orientated circunferentially on the inside of acylinder, fm and ft>.


is given by

fm + Gbfb • (K19)

The stresses am, ab and ab» are defined according to figure Gil and equation Gi l . Thegeometry functions fm and Fb are given as functions of a/t and t/Rj in table K12. fm and fbare given at the deepest point of the crack (A). See figure Gi l . The solution of Kj forglobal bending stress is hen approximated with the solution for membrane stress usingthe stress at the deepest point of the crack.

Region of applicability: a/t < 0,8 ,0,1 < t/Rj < 0,2 .


t/Rj = 0,1a/t0

0,20,40,60,8

fA

1,1221,2611,5822,0912,599

1,1220,9540,9090,8100,600

a/t0

0,20,40,60,8

t/R| = 0,2fA

1,1221,2151,4461,8042,280

fA

Ib1,1220,9330,8100,6500,411

Table K12. Complete circumferential surface crack on the insideof a cylinder, fm and fb.

c) Through thickness cra^k

is given by

K, = /n//2(omfm + obfh + ob gfb g). (K20)

The stresses om , ob and cb g are defined according to figure G12 and equation G12. Thegeometry functions fm, fb and fbg are given as functions of //rcRm and t/R, in tables K13 -

50

K14. fm and f^ are approximately valid along the whole crack front, fb is given at theintersections ot the crack with the planes u = 0 (A) and u = t (B). See figure G12.

Region of applicability: //nRm < 1 ,0,05 < t/Ri < 0,2 .

Reference: [K13 and K14].

//7lRm

00,10,20,30,40,50,60,70,80,9

1

t/R,=fm

1,0001,0901,2561,4771,7512,0862,4952,9973,6194,3895,346

0,05fba

1,0001,0781,2201,4051,6251,8792,1652,4842,8383,2303,665

t/Rj =

f»1,0001,0621,1781,3301,5201,7522,0362,3842,8153,3494,012

= 0,1fbe

1,0001,0541,1521,2801,4431,6091,8072,0282,2742,5462,847

t/Ri =

fm1,0001,0531,1521,2831,4461,6441,8872,1862,5553,0123,580

= 0,15fbe

1,0001,0461,1301,2401,3711,5221,6911,8812,0912,3242,582

t/Ri

fm1,0001,0491,1381,2571,4041,5851,8052,0762,4102,8253,341

= 0,2

fb.1,0001,0421,1181,2181,3371,4731,6271,7991,9902,2012,435

Table K13. Through thickness crack orientated circumferentially in a cylinder, fmandfbg-

//7lRm

00,050,10,20,30,40,50,60,70,80,9

1

t/R, = 0,05fAth

0,8620,6880,5540,3860,2970,2490,2200,2000,1820,1660,1480,128

-0,843-0,720-0,620-0,474-0,382-0,328-0,297-0,278-0,263-0,253-0,248-0,248

t/Ri = 0,1

0,8880,7440,6230,4590,3560,2850,2520,2230,2020,1810,1630.141

«-BIh

-0,890-0,776-0,680-0,545-0,457-0,396-0,356-0,330-0,311-0,300-0,292-0,294

t/Rj = 0,15

0,9290,7850,6620,5020,3930,3220,2720,2370,2100,1880,1650,141

Ä-0,925-0,818-0,722-0,597-0,507-0,441-0,398-0,368-0,347-0,337-0,330-0,330

t/Rj = 0,2

&0,9570,8130,6880,5310,4190,3440,2870,2460,2160,1890,1650,138

*-0,964-0,851-0,749-0,628-0,548-0,478-0,432-0,400-0,381-0,369-0,363-0,363

Table K14. Through thickness crack orientated circumferentially in a cylinder, fh.

51



K\ is given by

cbfb). (K21)

The stresses o^ and (Tb are defined according to figure G13 and equation G13. Thegeometry functions fm and fb are given as functions of //t and t/R; in table K15. fm and fbare given at the intersections of the crack with the planes u = 0 (A) and u = t (B). Seefigure G13.

Region of applicability: III < 20 ,0,05 < t/Rj < 0,1.

Reference: [K9].

t/Rj = 0,05Ut02468101520

fA

1,0001,1441,4011,7002,0202,3513,1863,981

fAIb

r

1,000,020

1,0501,0801,1061,1301,1801,219

1,0000,9410,8970,8950,9321,0031,3091,799

*6-1,000-0,995-0,992-0,993-0,996-1,001-1,014-1,028

t/Ri = 0,lih

0,02,04,06,08,010,015,020,0

fA

1,0001,2401,6372,0832,5493,0164,1245,084

1,0001,0311,0741,1111,1431,1701,2261,272

1m1,0000,9190,8940,9441,0591,2311,9152,968

tt-1,000-0,993-0,993-0,997-1,003-1,011-1,031-1,050

Table K15. Through thickness crack in a sphere, fm and

52



for an arbitrary load is given by

+A3Of3O+

+ A2of2o+ (K22)

The stress state coefficients Ay are defined according to figure G14 and equation G15.The geometry functions f« are given in table K16. The geometry function f is given as afunction of a/RjS in table K17. The geometry functions are given at the deepest point ofthe crack (A), at the intersection of the crack with the free surface at the nozzle (B) and atthe intersection of the crack with the free surface at the vessel (C). See figure G14.

The Kj-solution according to equation K22 is valid only for a quarter circular crack at aright angled corner between a nozzle and a cylindrical vessel, see figure G14. Thesolution can approximately be used for round off and not perfectly right angled comers.For examples, comers not situated in the axial section and comers between a nozzle and aspherical vessel. See figures G15 and G16. With a is then meant the real crack depth.However, Ay shall be determined in the specified coordinate system given in figure G14.That is, for geometries according to figures G15 and G16, Ay is determined in the showncoordinate system. In equation G15 a shall then be replaced by a + d.

Region of applicability: a/tmjn ^ 0,8 ,a/Ris < 3 ,Rls/Rim < 0,6 .

Reference: [K15].

1J00100120023003

fA»i|

1,081,001,000,770,770,500,50

M1,400,980,980,800,650,580,55

1,400,980,980,650,800,550,58

Table K16. Quarter circular corner crack in a nozzle, fy.

53

a/RiS0

0,10,20,30,40,5

11,52

2,53

fA

0,640,600,570,550,540,530,510,490,470,460,46

0,640,640,640,640,640,640,640,640,640,640,64

0,640,590,550,540,520,520,490,470,460,450,43

Table K17. Quarter circular corner crack in a nozzle, f.

i for an internal pressure load only is given by

(K23)

The stress om is defined according to figuu G17 and equation G16 for a cylindricalvessel and according to figure G18 and equation G17 for a spherical vessel. Thegeometry function f is given as a function of a/tm in table K18. f gives an approximatemean value of Kj along the crack front, see figure G14.

The Kj-solution according to equation K23 is valid only for a quarter circular crack at aright angled corner, see figure G14. The solution can approximately be used for roundoff and not perfectly right angled corners. With a is then meant the real crack depth. Seefigures G15 and G16.

Region of applicability: a/tm < 0,8 ,R l s /R i m <0 ,4 .

Reference: [K16].

a/tm0

0,20,40,60,8

f1,881,601,381,221,13

Table K18. Quarter circular corner crack in a nozzle, f.

54

6. Crack at a hole

a) Quarter elliptical corner crack

is given by

= •v/7ta(Aoofoo + A2of2o + (K24)

The stress state coefficients A^ are defined according to figure G19 and equation G19.The geometry function fy are given as functions of a/t, a// and R/t in table K19 - K21. fyare given at the deepest point of the crack (A) and at the intersection of the crack with thefree surface (B). See figure G19.

Region of applicability: a/t< 1,1,1 < a / / < 2 ,0,5 < R/t < 1 ,(/ + R)/w < 0,3 ,(/ + R)/h < 0,3 .

Reference: [K17].

a//= 1,1, R/t = 0,5a/t0

0,20,40,60,8

1

fob0,7000,7110,7210,7510,8010,877

ffo0,0000,0910,1820,2820,3970,531

tö0,0000,0860,1750,2720,3840,518

0,0000,0170,0680,1560,2930,489

fAMl

0,0000,0100,0410,0940,1740,290

# 20,0000,0130,0510,1200,2250,376

0,0000,0040,0300,1040,2580,537

ft0,0000,0020,0140,0480,1190,345

0,0000,0010,0110,0380,0950,196

tib0,0000,0020,0160,0570,1430,298

a// =1,1, R/t = 0,5a/t0

0,20,40,60,8

1

foo0,6010,6320,6620,7040,7570,820

tfo0,0000,1440,3000,4700,6590,868

ft0,0000,0500,1030,1630,2300,308

tfo0,0000,0390,1600,3720,6901,125

rfi0,0000,0100,0420,0980,1820,296

0,0000,0050,0220,0530,0990,165

ffo0,0000,0110,0920,3190,7841,591

A0,0000,0030,0210,0720,1760,354

0,0000,0010,0080,0290,0720,145

0,0000,0010,0060,0200,0500,103

a//=1,5, R/t = 0,5a/t0

0,20,40,60,81

foo0,6280,6360,6430,6650,7030,762

tfo0,0000,0600,1200,1840,2570,345

foAi0,0000,0790,1600,2480,3470,463

tfb0,0000,0080,0330,0750,1390,232

0,0000,0070,0270,0630,1160,191

tf20,0000,0120,0480,1100,2050,341

r?o0,0000,0010,0110,0360,0900,188

ft0,0000,0010,0070,0240,0580,119

ff?0,0000,0010,0080,0260,0630,130

fft)0,0000,0020,0150,0530,1310,271

55

a//= 1,5, R/t = 0,5a/t

00,20,4

0,60,8

1

loo0,5990,6130,6280,6520,6860,728

r»1100,000

0,0970,2000,3080,4260,555

fö»01

0,0000,0470,0970,1490,2080,273

f»120

0,0000,0190,0770,1770,3240,525

,-BMl

0,0000,0070,0280,0640,1170,189

•020,0000,0050,0210,0480,0890,146

I300,0000,0040,0320,1110,2690,543

•21

0,0000,0010,0100,0340,0820,165

r»M2

0,0000,0010,0060,0190,0460,093

fB130

0,0000,0010,0050,0180,0450,091

a// = 2,0, R/t = 0,5a/t0

0,20,40,60,81

too0,5610,5650,5700,5840,6100,650

fA'10

0,0000,0400,0800,1210,1680,222

fA101

0,0000,0720,1460,2240,3100,408

120

0,0000,0040,0160,0370,0680,112

rf.0,0000,0050,0190,0420,0780,127

&0,0000,0110,0440,1010,1860,304

tfo0,0000,0010,0040,0140,0330,068

»A121

0,0000,0000,0040,0120,0290,059

fAM2

0,0000,0010,0050,0170,0430,087

0,0000,0010,0140,0490,1190,243

a// = 2,0, R/t = 0,5a/t0

0,20,4

0,6

0,81

100

0,5810,5860,5900,6040,6250,653

tfo0,0000,0670,1360,2080,2850,367

f B»01

0,0000,0450,0900,1380,1890,245

rB»20

0,0000,0100,0390,0890,1610,259

fBMl

0,0000,0050,0190,0430,0780,125

fB102

0,0000,0050,0190,0440,0810,130

«?o0,0000,0020,0120,0420,1000,200

f2H .

0,0000,0010,0050,0170,0410,081

ft0,0000,0000,0040,0130,0310,061

fB'30

0,0000,0010,0050,0170,0400,081

Table K19. Quarter elliptical corner crack at a hole, f,} l»r R/t = 0,5.

a//= 1,1, R/t = 0,75a/t0

0,20,40,6

0,81

,Aloo

0,7170,7310,7450,7790,8350,917

,AMO

0,0000,0620,1250,1930,2730,368

fAMM

0,0000.0880,1790,2800,3960,537

f A120

0,0000,0080,0310,0710,1330,222

fAMl

0,0000,0070,0280,0640,1190,198

,Ai()2

0,0000,0130,0530,1230,2310,390

fA130

0,0000,0010,0090,0310,0780,162

fA»21

0,0000,0010,0060,0220,0540,111

»•AM2

0,0000,0010,0080,0260,0640,133

fA'30

0,0000,0020,0170,0590,1460,308

a//= 1,1, R/t = 0,75a/t0

0.2

0,40,6

0,8

1

,-BMM)

0,6100,6400,6700,7160,7760,847

r"MO0,000

0,0970,2010,3160,4460,591

,-BMM

0,0000,0500,1040,1650.2360,319

1200,0000.0170,0710,1660,3100,508

fRMl

0,0000,0070,0280,0660,1230,202

,Bl()2

0,0000,0050,0230,0540,1020,171

fB'30

0,0000,0030,0270,0950,2340,477

fB121

0,0000,0010,0090,0320,0790,160

fBM2

0,0000,0010,0060,0200,0490,099

fB130

0,0000,0010,0060,0200,0510,107

56

a//=1,5, R/t = 0,75a/t0

0,20,40,60,81

loo0,6400,6530,6670,6940,7370,801

fAMO

0,0000,0410,0830,1270,1780,239

fA101

0,0000,0810,1650,2560,3600,482

0,0000,0040,0150,0340,0640,107

0,0000,0050,0190,0430,0790,131

«o\0,0000,0120,0490,1140,2120,353

&0,0000,0000,0030,0110,0270,057

fA'21

0,0000,0000,0030,0110,0260,054

ff20,0000,0010,0050,0180,0430,089

0,0000,0020,0160,0540,1350,280

a//=1,5, R/t = 0,75a/t0

0,20,40,60,81

foo0,6100,6250,6390,6650,7020,750

ffo0,0000,0660,1350,2080,2880,377

fB'010,000

0,0480,0980,1530,2130,282

%o0,0000,0080,0340,0790,1450,236

fBhi0,000

0,0050,0190,0430,0790,129

0,0000,0050,0210,0490,0910,151

# 00,0000,0010,0100,0330,0800,163

0,0000,0000,0040,0150,0370,074

f?20,0000,0000,0040,0130,0310,063

# 00,0000,0010,0050,0190,0460,094

a// = 2,0, R/t = 0,75a/t0

0,20,40,60,81

rA100

0,5690,5800,5920,6110,6430,687

fAMO

0,0000,0270,0550,0840,1170,154

fA'01

0,0000,0740,1500,2320,3230,427

tfo0,0000,0020,0080,0170,0320,052

0,0000,0030,0130,0290,0540,087

&0,0000,0110,0450,1040,1920,316

40,0000,0000,0010,0040,0100,021

fA'21

0,0000,0000,0020,0060,0130,027

&0,0000,0000,0040,0120,0300,060

tfo0,0000,0020,0140,0500,1230,252

a// = 2,0, R/t = 0,75a/t0

0,20,40,60,8

]

foo0,5910,5980,6050,6190,6420,673

ffo0,0000,0460,0920,1410,1930,250

fo H .0,0000,0460,0920,1410,1940,253

# 00,0000,0040,0170,0400,0730,117

0,0000,0030,0130,0290,0530,085

fob20,0000,0050,0200,0450,0830,135

# 00,0000,0000,0040,0120,0300,060

f2B.

0,0000,0000,0020,0080,0180,037

ft0,0000,0000,0030,0090,0210,042

# 00,0000,0010,0050,0170,0420,084

Table K20. Quarter elliptical corner crack at a hole, fy for R/t = 0,75.

a//= 1,1, R/t = 1a/t0

0.20,40,60,81

,-A'00

0,7290,7500,7700,8100,8730,965

no0,0000,0480,0970,1500,2120,288

fA'01

0,000

o,on0,1840,2890,4110,560

fA'20

0,0000,0050,0180,0410,0770,129

fA'11

0,0000,0050,0210,0490,0920,153

fA»02

0,0000,0130,0540,1260,2390,404

fA130

0,0000,0010,0040,0140,0340,070

rA121

0,0000,0000,0040,0130,0310,064

»•A' 1 2

0,0000,0010,0060,0200,0500,103

fA'30

0,0000,0020,0170,0600,1510,319

57

a// =1,1 , R/t =1a/t0

0,20,40,60,8

1

fB1000,618

0,6490,6800,7290,7950,873

fB

MO0,0000,0730,1520,2390,3400,453

fB*01

0,0000,0510,1060,1690,2420,329

fB120

0,0000,0100,0400,0940,1760,290

f-BM l

0,0000,0050,0210,0500,0940,155

,-B»02

0,0000,0060,0230,0550,1050,176

&0,0000,0010,0120,0400,1000,204

121

0,0000,0010,0050,0180,0450,091

f?20,0000,0010,0040,0150,0370,076

tto0,0000,0010,0060,0210,0530,111

a//=1,5, R/t = 1a/t0

0,20,40,60,8

1

,-A100

0,6490,6700,6900,7230,7740,846

ffo0,0000,0310,0640,0990,1390,187

fA»01

0,0000,0820,1700,2650,3740,504

»to0,0000,0020,0090,0200,0370,062

f AM l

0,0000,0040,0140,0330,0610,102

fA»02

0,0000,0120,0500,1170,2200,368

fA»30

0,0000,0000,0010,0050,0120,025

f A»21

0,0000,0000,0020,0060,0150,031

f A»12

0,0000,0000,0040,0140,0330,069

f A13O

0,0000,0020,0160,0560,1390,291

a7/=1,5, R/t =1a/t0

0,20,40,60,8

1

foo0,6190,6360,6530,6810,7210,773

tfo0,0000,0500,1020,1580,2200,289

fB

0,0000,0490,1000,1560,2190,291

tfo0,0000,0050,0200,0450,0830,135

fBMl

0,0000,0030,0140,0330,0600,098

&0,0000,0050,0220,0500,0940,156

fSfo0,0000,0010,0040,0140,0340,069

f2B.

0,0000,0000,0030,0090,0210,043

1T20,0000,0000,0030,0100,0240,048

ito0,0000,0010,0050,0190,0470,098

a// = 2,0, R/t = 1a/t0

0,20,40,60,81

too0,5750,5930,6110,6380,6760,728

MMO

0,0000,0210,0420,0660,0920,122

fA»01

0,0000,0750,1550,2400,3360,447

fA120

0,0000,0010,0040,0100,0180,030

fAM l

0,0000,0020,0100,0230,0420,068

fA»02

0,0000,0110,0460,1070,1990,330

fAI30

0,0000,0000,0010,0020,0040,009

ft0,0000,0000,0010,0030,0080,016

fAM2

0,0000,0000,0030,0090,0230,047

fA130

0,0000,0020,0150,0520,1270,262

a// = 2,0, R/t = 1a/t0

0,20,40,60,8

1

too0,5990,6090,6190,6360,6620,696

fBMO

0,0000,0350,0700,1080,1480,191

fnfoi0,000

0,0460,0940,1450,2000,262

fB120

0,0000,0020,0100,0230,0410,067

fBMl

0,0000,0020,0100,0220,0410,065

fB1<>20,000

0,0050,0200,0470,0860,140

»to0,0000,0000,0020,0050,0130,026

»210,0000,0000,0010,0040,0110,021

i?20,0000,0000,0020,0070,0160,032

fB»30

0,0000,0000,0050,0180,0430,087

Table K21. Quarter elliptical corner crack at a hole, f,j for R/t = 1.

58

References

[Kl] RAJU, I. S. and NEWMAN, J. C , Stress intensity factor equations for cracks inthree-dimensional finite bodies, NASA Technical Memorandum 83200, pp. 1-49,1981.

[K2] RAJU, I. S. and NEWMAN, J. C , An empirical stress intensity factor equationfor the surface crack, Engng. Frac. Mech., 15, pp. 185-192,1981.

[K3] JOSEPH, P. F. and ERDOGAN, F., Surface crack problem in plates, Int. J.Fract, 41, pp. 105-131, 1989.

[K4] SIH, G. C , PARIS, P. C. and ERDOGAN, F., Stress intensity factors for planeextension and plate bendings problems, J. Appl. Mech., 29, pp. 306-312, 1962.

[K5] OVCHINNIKOV, A. V. and VASILTCHENKO, G. S., The defectschematization and SIF determination for assessment of the vessel and pipingintegrity, Final report CNIITMASH project 125-01-90, pp. 1-46, 1990.

[K6] RAJU, I. S. and NEWMAN, J. C , Stress intensity factor influence coefficientsfor internal and external surface cracks in cylindrical vessels, ASME PVP, 58,pp. 37-48, 1978.

[K7] BUCHALET, C. B. and BAMFORD, W. H., Stress intensity factor solutions forcontinous surface flaws in reactor pressure vessels, Mechanics of Crack Growth,ASTM STP 590, pp. 385-402, 1976.

[K8] ANDRASIC, C. P. and PARKER, A. P., Dimensionless stress intensity factorsfor cracked thick cylinders under polynomial crack face loadings, Engng. Frac.Mech., 19, pp. 187-193, 1984.

[K9] ERDOGAN, F. and KIBLER, J. J., Cylindrical and spherical shells with cracks,Int. J. Frac. Mech., 5, pp. 229-237, 1969.

[K10] DEDHIA, D. D. and HARRIS, D. O., Improved influence functions for part-circumferential cracks in pipes, ASME PVP, 95, pp. 35-48,1984.

[Kl l ] BERGMAN, M. and BRICKSTAD, B., Stress intensity factors for circum-ferential cracks in pipes analyzed by FEM using line spring elements, Int. J. ofFract., 47, pp. R17-R19, 1991.

[K12] KUMAR, V., GERMAN, M. D. and SHIH, C. F., An engineering approach forelastic-plastic fracture analysis, NP-1931, Research Project 1237-1, pp. 4.4-4.15, 1981.

[K13] ZAHOOR, A., Closed form expressions for fracture mechanics analysis ofcracked pipes, ASME PVP, 107, pp. 203-205, 1985.

[K14] SATTARI-FAR, I., Stress intensity factors for circumferential through thicknesscracks in cylinders subjected to local bending, Int. J. of Fract., 53-1, 1992.

[K15] KOBAYASHI, A. S., POLVANICH, N., EMERY, A. F. and LOWE, W. J.,Corner crack at a nozzle, Proc. 3rd Int. Conf. on Pressure Vessel Technology,pp. 507-517, Tokyo, 1977.

[K16] MOHAMED, M. A. and SCHROEDER, J., Stress intensity factor solution forcrotch-corner cracks of tee-intersections of cylindrical shells, Int. Journal ofFracture, Vol. 14, No. 6, pp. 605-621, 1978.

59

[K 17] PEREZ, R., GRANDT. A. F.. Jr. and SAFF, C. Tabulated stress-intensityfactors for corner cracks at holes under stress gradients, Surface-Crack Growth:Models, Experiments and Structures, ASTM STP 1060, Philadelphia, pp. 49-62,1990.

60

Appendix L. Limit load solutions

1. Crack in a plate


LR is given by

U = -2—L*^ + V^+d-O2<*

where

a/(L2)

The stresses am and 0b are defined according to figure Gl and equation Gl .

Region of applicability: a/t < 0,8 ,(/ + 2t)/2w < 1 .

Reference:

LR is given

LR

where

[LI].

b) Long surface crack

by

(1 - 0 aY

t '

(L3)

(L4)

The stresses om and ab are defined according to figure G2 and equation G2.

Region of applicability: a/t < 0,8 .

Reference: [LI].


LR is given by

LR = -2 L 2 . (L5)

61

The stresses am and ab arc defined according to figure G3 and equation G3.

Region of applicability: //w « 1.


LR is given by

(L6)

where

2a/(L7)

1 a e

The stresses am and ab are defined according to figure G4 and equation G4.

Region of applicability: 2a/t < 0,8 ,e/t > 0 ,( /+2t ) /2w<l .

Reference: [LI].

62

2. Crack orientated axially in a cylinder


LR is given by

PLR = ~ a r ) 2 q • <L9>

where

(L10)' t(/+2t)

The stresses o m and Ob are defined according to figure G5 and equation G5.

Region of applicability: a/t < 0,8 ,(/ + 2t)/2w < 1.

Reference: Based on [LI].

b) Long surface crack on the inside of a cylinder

LR is given by

where

C = * • (L12)

The stresses am and Ob are defined according to figure G6 and equation G6.




LR is given by

—r-, < L 1 3 )

(1 - Q2ay

where

63

The stresses am and <Jb are defined according to figure G7 and equation G7.

Region of applicability: a/t < 0,8 ,(/ + 2t)/2w < 1 .



LR is given by

LR = (L15)

where

C = \ • (L16)

The stresses am and <?b rc defined according to figure G8 and equation G8.




LR is given by

where

LR = - = V l + 1,05X2,

X =

(L17)

(L18)

The stress am is defined according to figure G9 and equation G9. The solution is validonly for membrane stress.

Region of applicability: t/R,<O,l .

Reference: [L2].

64



LR is given by

N M(L19)LR

where

NNf

MMf

N

"Y

+

«

,n4

MMf '

a/2Rit

• (n

K

o • fa .- 2arcsin —sin

L2t

t/R:(1 + t/R:)(2 +

al \ a .

(Ml'l2R;JJ

t/Rj)

r / v

(L20)

(L21). , 7i a/ \ a . f i >

sin

The stresses am and Obg are defined according to figure G10 and equation G10. Thesolution is valid only for membrane stress and global bending stress.

Region of applicability: a/t < 0,8 ,/ / R i < 1 .

Reference: Based on [L3].


LR is given by

1 M (N\ l ( M f ,f#MvLR = + 1 — + - — , (L22)2Mf \{N 4 1 M J

where

N om (R,/t+ I)2 - (R./02

Nf aY(Ri/t + I)2 - (Ri/t + a/t)2 '

M 3nObg (Ri/t + I)4 - (R,/t)4

(L23)

(L24)

The stresses om and abg are defined according to figure Gi l and equation Gl 1. Thesolution is valid only for membrane stress and global bending stress.

Region of applicability: a/t < 0,8.

Mf 16 aY (R/t + I)4 - (R,/t + l)(R,/t + a/t)3

65


LR is given by

N MLR = — + — , (L25)

Nf Mf

where

N _ om n

* ~ 2R~

1 - " *M. = a ^ n (l ^ t/Rj)(2H , . , ,

sin| - - — I - -sin|

The stresses am and Obg are defined according to figure G12 and equation G12. Thesolution is valid only for membrane stress and global bending stress.

Region of applicability: UnRm £ 1.


66



LR is given by

1 + 1/1 += , (L/S)

where

X = — l — . (L29)

The stress am is defined according to figure G13 and equation G13. The solution is validonly for membrane stress.

Region of applicability: t/Rj < 0 ,1 .

Reference: [L4].

67



LR for an arbitrary load is given by

—=-. , (L30)(1 - Q2Oy

where

(L31)•min

The stresses om and Ob are defined according to figure G14 and equation G14.

Region of applicability: a/tmjn < 0,8 .


LR for an internal pressure load only is given by

LR = - 2 - — ^ . . (L32)<Jy . « a 2

Pressurized area Ap and load bearing area As are defined according to figure G17 for acylindrical vessel and according to figure G18 for a spherical vessel where

/, = 0 , 4 / R ~ £ , (L33)

|0,4VR~t^/m = minoRis . for a cylindrical vessel. (L34)

i/m = m i n j 0 ' 4 ^ 1 " 1 " 1 , for a spherical vessel. (L35)

[R.S

Other definitions of the extension along the nozzle /s and the vessel /m are given byASME III [L5J.

Region of applicability: sJ'\j4As/n < 0,8 .


68

6. Crack at a hole

a) Quarter elliptical corner crack

LR is given by

LR = J \r-^i . (L36)

where

a/(L37)

The stresses am and <Jb are defined according to figure G19 and equation G18.

Region of applicability: a/t < 0,8 ,(/ + t)/(w - R) <, 1.


69

References

[LI] WILLOUGHBY, A. A. and DAVEY, T. G., Plastic collapse in part-wall flaws inplates, ASTM STP 1020, ASTM, Philadelphia, pp. 390-409, 1989.

[L2] KIEFNER, J. F., MAXEY, W. A., EIBER, R. J. and DUFFY, A. R., Failurestress levels of flaws in pressurized cylinders, ASTM, STP 536, AmericanSociety of Testing and Materials, Philadelphia, Pa., pp. 461-481, 1973.

[L3] KANNINEN, M. F., BROEK, D., MARSCHALL, C. W., RYBICKI, E. F.,SAMPATH, S. G., SIMONEN, F. A. and WILKOWSKI, G. M., Mechanicalfracture predictions for sensitized stainless steel piping with circumferentialcracks, Final Report, EPRI NP-192, 1976.

[L4] BURDEKIN, F. M. and TAYLOR, T. E., Fracture in spherical pressure vessels,J. Mech. Eng. Sci., 11, pp. 486-497, 1969.

[L5] ASME Boiler and Pressure Vessel Code, Sect. III, Div. 1 - NB-3000, TheAmerican Society of Mechanical Engineers, New York, 1989.

[L6] LIND, N. C , Approximate stress-concentration analysis for pressurized branchpipe connections, ASME Paper 67-WA/PVP-7, The American Society ofMechanical Engineers, New York, pp. 951-958, 1967.

70

Appendix M. Material data for nuclear applications

In order to perform fracture mechanics assessments according to this handbookknowledge about yield strength <7y, ultimate tensile strength Rm and critical stressintensity factor Kcr is needed. Crack growth calculations require in addition knowledgeabout the growth rate per load cycle or per time unit for fatigue cracking and stresscorrosion cracking, respectively. All material data should preferably be determined bytesting of the material of the considered component in the environment and at thetemperature for which the fracture assessment is to be performed. Below, somerecommendations are given for steels common in nuclear applications. The data isintended for use in cases when test data for the actual steel are lacking. In most cases thedata given below are conservative estimates.

1. Yield strength, ultimate tensile strength

For most steels used in nuclear applications information about minimum levels of ay andRm as functions of the temperature can be found in ASME III, Appendices [Ml].

2. Fracture toughness

In cases when the fracture toughness could not be directly determined, Jic-data convertedaccording to equation 3, chapter 2 have been used

a) Ferritic steel, pressure vessels

For the materials SA-553 Grade B Class 1, SA-508 Class 2 and SA-508 Class 3 thefracture toughness Kia (conservative fracture toughness value at crack arrest) is given as afunction of the difference between actual temperature T and the nil ductility transitiontemperature RTNDT »n ASME XI, Appendix A, figure A-4200-1 [M2]. For temperaturesabove the transition region higher values than 220 MPaVm are usually not assumed. It isto be noted that this level can be decreased as well as that RTNDT

c a n ^ increased due toneutron irradiation. Figure A-4200-1 corresponds to the analytic expression (T in °C)

KIa = 29,43 + 1,355exp[0,0261 (T - RTNDT) + 2,32] MPaVm . (Ml)

b) Ferritic steel, pipes

The following data are taken from Norris, EPRI NP-6045 [M3] and are intended for thefollowing material classes:

Class 1: Seamless or welded carbon steel piping with a minimum yield strength lowerthan or equal to 276 MPa (base material) and welds with electrodes of typeE7O15, E7016 or E7O18 (basic electrodes with a yield strength of the order of500 MPa, Charpy-V toughness 27 J at -29 °C and weldable in all weldingpositions).

Class 2: All other welded ferritic piping with shielded metal arc welds (SMAW) orsubmerged arc welds (SAW) with a minimum ultimate tensile strength lowerthan or equal to 552 MPa.

The table below differentiates between temperatures above, in or under the transitionregion. In addition, in |M3) it is distinguished between circumferential and axial cracks.The fracture properties are often worse for cracks orientated along the texture direction

71

(axial cracks) than for cracks orientated across the texture direction (circumferentialcracks).

Material class

1122

1,21,2

Temperatureinterval

T > trans, reg.T < trans, reg.T > trans, reg.T < trans, reg.T > trans, reg.T < trans, reg.

Crackorientation

CircumferentialCircumferentialCircumferentialCircumferential

AxialAxial

KcJMPaVm](min value)

150421154210642

Table Ml. Fracture toughness data for pipe of ferritic steel.

c) Austenitic stainless steel, pipes

The following data are taken from Ländes and McCabe, EPRI NP-4768 [M4] and areapplicable at the temperature 288 °C.

The materials are base material of type 304 or 316 and shielded metal arc welds (SMAW)or submerged arc welds (SAW). The heat affected zone (HAZ) often shows betterproperties than the weld material according to [M4].

Material typeBase material

SMAWSAW

Kcr [MPaVm]350182117

Table M2. Fracture toughness data for austenitic stainless steel.

According to [M4] the fracture toughness at room temperature are not lower than at 288°C. The fracture toughness for TIG-welds are only slightly below those of the basematerial.

d) Nickel base alloys

Fracture toughness data for nickel base alloys are rare in the literature. For the weldmaterial Alloy 182 data are published by Yoshida et al [M5j. [M5] gives only data atroom temperature. The fracture toughness value corresponding to the Jic-value is

Kcr=154MPaVm. (M2)

According to |M4| the Jit-value decreases with approximately 30% in average forstainless weld material (SMAW and SAW) at 288 °C in comparison with roomtemperature. This corresponds to an approximately 20% decrease of the fracturetoughness value. If the same decrease is supposed to be valid for Alloy 182, thefollowing fracture toughness value can be estimated at 288 °C

Kc r= 123MPa\(m . (M3)

72

3. Crack growth data -fatigue

The amount of crack growth c per load cycle is assumed to be expressed by equation(M4).

de

dNr. ' m/cykel. (M4)

Here n and AKQ are material constants.

a) Ferritic steel, pressure vessels

Fatigue crack growth data for the materials SA-533 Grade B Class 1, SA-508 Class 2and SA-508 Class 3 can be found in ASME XI, Appendix A, figure A-4300-1 [M2] forcracks both in air and reactor water environment. In the latter cases the growth ratedepends strongly on the R-value. Values of the constants n and AKo are given in tableM3 for reactor water environment and in table M4 for air environment.

AKi [MPaVm]< 23,4 - 15.8R> 23,4 - 15.8R

n

5,951,95

AK0 [MPaVm]R <, 0,25

20,723,42

0,25 < R < 0,65l l , 9 ( R - 0 , 2 1 3 ) 0 1 6 8

ll,9(R + 0,016)0-513

0,65 < R13,5614,65

Table M3. Growth data for fatigue of ferritic steel in reactor water environment.

n3,726

AK0 [MPaVm]49,72

Table M4. Growth data for fatigue of ferritic steel in air environment.

b) Austenitic stainless steel

Fatigue crack growth data for austenitic stainless steel including welds and castings in airenvironment are given in ASME XI, Appendix C, figure C-3210-1 [M2]. In analyticalform holds

n = 3 ,3 ,

AK0 = O,05[Qexp(-23 + 3,08- 10"3T - 7,7- 10'6T2

l ,37-10- 8T 3 ) r° ' 3 0 3MPa/m,

where T is the temperature in °C and Q is given by table M5.

(M5)

(M6)

73

R0 < R

<0<0,79

0,79 < R $ 1 -43

Q1

1 + 1,8R,35 + 57.97R

Table M5. Influence of R-value on growth constant for austeniticstainless steel in air environment.

Growth data for fatigue of austenitic stainless steels in reactor water environment are stillnot available in ASME XI. Here the results from an experimental investigation byÖstensson and Gott [M6] are suggested. The report presents fatigue data for SS 2333-steel in reactor water environment. A water environment with 8 ppm oxygen at 288 °Cgives for 0 < R < 0,6

n = 3,45 ,

AK0= 31,2(1 -R)°-sMPaVm.

(M7)

(M8)

4. Crack growth data - stress corrosion

The amount of crack growth c per time unit is assumed to be expressed by equation(M9).

dt(M9)

a) Austenitic stainless steel

For weld sensitized austenitic stainless steels in reactor water environment (0,2 ppmoxygen) growth data are recommended by U.S. Nuclear Regulatory Commission (NRC)in a NUREG-report of Hazelton [M7]. Reference [M7] gives

a = 2,161 ,

Ko = 6,02 MPaVm .

(M10)

(Mil)

Note that the growth rate may be strongly dependent on the degree of sensitization andthe current water chemistry.

References

[Mil ASME Boiler and Pressure Vessel Code, Sect. Ill, Div. 1, The American Societyof Mechanical Engineers, New York, 1989.

[M2| ASME Boiler and Pressure Vessel Code, Sect. XI, Rules for inservice inspectionof nuclear power plant components, The American Society of MechanicalEngineers, New York, 1989.

|M3j NORRIS, D. M., Evaluation of flaws in ferritic piping. Final Report EPRI NP-6045, 1988.

74

[M4] LÄNDES, J. D. and McCABE, D. E., Toughness of austenitic stainless steel pi/ewelds, Topical Report EPR1 NP-4768, 1986.

[M5] YOSHIDA, K., KOJIMA, M., IIDA, M. and TAKAHASHI, I., Fracturetoughness of weld metals in steel piping for nuclear power plants, Int. J. Pres.Ves. & Piping, Vol. 43, pp. 273 - 284, 1990.

[M6] ÖSTENSSON, B. and GOTT, K., Fatigue crack growth in austenitic stainlesssteel in simulated BWR environment, Studsvik Report El-79/116,1979.

[M7] HAZELTON, W. S., Technical report on material selection and processing guide-lines for BWR coolant pressure boundary piping, NUREG-0313, Rev. 2,USNRC, 1986.

75

Appendix P. Load, size and material factors for nuclear applications

For applications to components in nuclear facilities partial safety factors according to tablePI are applied.

Type of eventNormal and upsetEmergencyFaulted,/<?raric steelsFaulted, austeniticstainless steelsOrNormal and upsetEmergency and faulted

Yf1.01.01,01,0

1.01,0

1,01.01,01,0

1,01,0

t1,01.01.01,0

10,02.0

i1,01,01,0

1,0

10.02,0

iLUSe1

min(l,25se, 1,0)l

l,43Rpo.2(T)/Rm(T)l

max(0,63se,l^Rpo^TVIWT))1

1,01,0

3,161,411,411,41

1,01,0

1) This value may be divided by 1,5 if the only primary stress is a local membrane stress.

Table PI. Load, size and material factors for nuclear applications.

The coefficient se is given by

_ 2RpO,2(T)Se = -

3 Sm

T is the design temperature. Sm is given by

Sm =

for ferritic steels and

Sm = minQ

(PI)

(P2)

, 0,9Rpo.2(T)j, (P3)

for austenitic stainless steels.

Appendix B. Background

The methodology of this handbook i based on the CEGB R6-method [Bl] which hasbeen widely used. This method has been verified against numerical calculations as well asexperiments for many cases, see for example [B2]. The most recent revision of the R6-method, rev. 3 contains three different options for determination of the safe region in thefracture assessment diagram. In addition, the R6-method allows for three differentalternatives, categories, for how stable crack growth shall be taken into account.

The goal of the work with this handbook and its accompanying computer program [B3Jhas been to simplify the usage as far as possible and at the same time assure safe but notnecessarily accurate failure assessments. One requirement is thus that the method shouldbe as uniquely defined as possible. For this reason only one option has been chosen todefine the safe region. This coincides mainly with the option 1 of the R6-method.Furthermore, the assessment is done with respect to initiation of crack growth and anystable crack growth may not be credited.

Another goal has been to make this document as complete and self contained as possible.The handbook should provide all necessary information needed to perform defectassessments for the cases considered with the exception of stress analysis and specificmaterial data. This handbook therefore contains, in contrast to the R6-method, Ki andLR-solutions for a number of cases that are of practical importance. For all geometrycases defined Kj and LR-solutions are provided. For a number of cases accurate solutionsare difficult to obtain, especially as regards LR-solutions. The strategy has been to includesolutions that are safely conservative even if they are not particularly accurate. As bettersolutions become available they are to be included in the handbook and accompanyingcomputer program. Both are structured so that new solutions can be added or existingones exchanged easily.

A number of particular questions that have been subject to special discussions in thecourse of the work are discussed below. For some cases this has led to deviations fromthe R6-method.

1. Interaction between primary and secondary stresses

Earlier applications of the R6-method have revealed certain uncertainties in the treatmentof secondary stresses and their interaction with primary stresses. One particular difficulthas been how handle cases when the primary stresses are very low so that the parameterX given by equation 11, chapter 2, becomes undefined. The problem is caused by the aimof keeping the R6-method as user friendly as possible.

The method on which the R6-method is based has been developed by Ainsworth [B4|.This method utilizes the concept of reference stresses. Some of the basic assumptions inAinsworth's work are firstly that purely secondary loads can be treated with elastic theoryand secondly that for combined primary and secondary loads the result should beconsistent with the R6-method for purely primary loads. From these assumptions thefactor p defined above is determined as a function of the secondary reference stress 0^1and LR. Cref in turn is determined from the parameter x by the following equation.

where

1R(LR) = (I - 0, 14LR)|0,3 + (),7exp(-(),65LR)| . (B2)

77

In [B4] p is calculated as a function of LR with o^f/OY as a parameter. In figure Bl thisfunction is shown for <fref/Oy = 0,8. In the R6-method the curve in figure Bl is thenapproximated by the maximum value of p in the interval 0 < LR< 0,8 and for largervalues of the straight line which decreases from the maximum value to zero at LR = 1,05.

0.1

0.05 - -

0

-0.05 - -

-0.1

0.3 0.6 0.9 1.2 1.5

Figure B1. p as a function of LR.

One problem with this construction has been the discontinuous behaviour at LR = 0. Inorder to eliminate this difficulty another approximation has been introduced in thishandbook. The beginning of Ainsworth's curve is approximated by the tangent in theorigin and the p value is calculated according to this up to the maximum value. As can beseen from figure B1 rte conservatism of the R6-method is somewhat decreased. Themost important improvement however, is a continuous behaviour as the primary stressestend to zero.

The slope of the tangent is given by

_dp_dLR L R = 0

= -r-N - h«(/cY)] -

dfR_dLR (B3>

where the equation of the derivative of fR is given by

— = -0.0X4L + |exp(-0,65LR))(0,382LR - 2,73LR - 0,196LR). (B4)dLR

The maximum p-value is given by

pmax = 0, lx° ' 7 1 4 - 0,()07x2 + 0,00()03xS. (B5)

78

2. Stress intensity/actors

For most of the cases discussed here, well investigated and commonly accepted resultsfor Ki exist, e.g. case la-c, 2a-d, 3b. For some cases, however, special considerationshave been made and sometimes complementing calculations have been performed.

For the case 3a, i.e. surface crack orientated circumferentially in a cylinder, which isimportant for nuclear applications, few results are found in the literature. The mostly usedones are those by Dedhia and Harris [B5]. Certain doubts exist about the accuracy ofthese solutions especially for deep cracks. In order to investigate this and to widen theapplicability region, Recalculations havr been performed using the line-spring facility inthe program ABAQUS [B6]. These calculations are presented in [B7].

In cases with through thickness cracks where the solution is based on shell theory thequestion >i which shell theory that should be used arises. In the classic Kirchhoff theorythe influence of the shear deformation is neglected. The shear deformation can be takeninto account in different ways each leading to a different shell theory. One problem is thattheories where the shear deformation is accounted for are usually not consistentapproximations. With the concept consistent is meant that all terms of a certain order in apower series expansion in characteristic shell thickness should be considered. Theclassical theory is such a consistent approximation when the shell thickness tends to zero.It turns out that Recalculations based on shear deformation theory give different and insome cases unrealistic results. For the cases lc, 2e and 4a results from classic theoryhave therefore been used. This is in most cases conservative as compared with publishedresults based on shear deformation theory. For the case 3c, i.e. through thickness crackorientated circumferentially in a cylinder, numerical calculations by the finite elementmethod have been performed in order to complete with the load case through thicknessbending. This solution, however, is based on shear deformation theory since a solutionbased on classical theory was not obtained with this calculation method. The calculationsare presented in [B8].

3. Limit load solutions

The limit load solutions are considerably less developed than the corresponding solutionsfor Ki and approximations have to be relied upon to a greater extent. Since a guarantiedconservatism is sought a certain over conservatism has to be accepted in some cases.

For surface cracks the problem of choosing between so called local or global solutions,respectively, emerges. With a local solutions is meant that the ligament in front of thecrack is subjected to plastic deformation but the limit load of the component, i.e. theglobal limit load, has not yet been reached. Results from several investigations, e.g. [B9]suggest that use of the global solution is not necessarily conservative as regards initiationof crack growth. In this procedure local solutions have therefore been incorporated withthe exception for case 3a, surface crack orientated circumferentially in a cylindersubjected to tension and global bending. For this case, extensive investigations haveshown that global solutions are sufficient.

For the cas^s la-b, surface crack in a plate, and 2a-d, surface crack orientated axially in acylinder, a special validation have been performed. The yield surfaces presented in [LI]were compared to numerical calculations by the finite element method. The calculationsare presented in (B10].

The yield surface given in [L3] for case 3, crack orientated circumferentially in a cylinder,has been approximated by a straight line in order to simplify the expression for LR.

79

4. Safety margins

In appendix P a set of numerical values of load, size and material factors for applicationsto components in nuclear facilities is given. The load factors and the material factor 7^follows the margins applied in ASME XI, IWB 3600 <P>\ 1]. For the material factor 7^,which gives the design yield strength, the criteria given in ASME HI, NB 3000 [B12] fordesign of nuclear components subjected to different loading conditions are followed.

For acceptance assessments of components in nuclear facilities L R " shall be set equal to1. This is due to the criteria used to ensure safety against plastic deformation and collapsein nuclear facilities which are based on an elastic perfectly plastic material behaviour.

5. Linearization of stress distribution

The computer program [B3] that has been developed in order to perform the calculationsdescribed in this document can handle an arbitrary stress distribution. From such adistribution an equivalent linear stress distribution is calculated with the condition ofpreserving equilibrium. In the Ki-calculation the stress distribution is linearized over thecrack surface while for the LR-calculation the linearization is done with respect to theentire section. The particular procedure is described in [B3].

For cases described by a two dimensional stress distribution, case 5a and 6a, are a leastsquare fit to the desired polynomial used.

References

[Blj MILNE, I., AINSWORTH, R. A., DOWLING, A. R. and STEWART, A. T.fAssessment of the integrity of structures containing defects, Pressure Vessels andPiping, Vol. 32, pp. 3-104, 1988.

[B2] SATTARI-FAR, I. and NILSSON, F., Validation of a procedure for safetyassessment of cracks, SA/FoU-Report 91/19, The Swedish Plant Inspectorate,Stockholm, 1991.

[B3] BERGMAN, M., A procedure for safety assessment of components with cracks -manual for computer program, SA/FoU-Report 91/18, The Swedish PlantInspectorate, Stockholm, 1991, (In Swedish).

[B4] AINSWORTH, R. A., The treatment of thermal and residual stresses in fractureassessments, Engng. Frac. Mech., 24, pp. 65-76, 1986.

[B5] DEDHIA, D. D. and HARRIS, D. O., Improved influence functions for part-circumferential cracks in pipes, ASME PVP, 95, pp. 35-48, 1984.

|B6] ABAQUS, Users Manual 4.7, Hibbit, Karlsson and Sorensen Inc, Providence,Rhode Island, 1988.

[B7] BERGMAN, M. and BRICKSTAD, B., Stress intensity factors for circum-ferential cracks in pipes analyzed by FEM using line spring elements, Int. J. ofFract., 47, pp. R17-R19, 1991.

[B8] SATTARI-FAR, I., Stress intensity factors for circumferential through thicknesscracks in cylinders subjected to local bending, Int. J. of Fract., 53-1, 1992.

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[B9] NILSSON, F., FALESKOG, J., ZAREMBA, K. and ÖBERG, H., Elastic-plastic fracture mechanics for pressure vessel design, SKI TR 89:20, TheSwedish Nuclear Power Inspectorate, Stockholm, 1989.

[BIO] SATTARI-FAR, I., Analysis of limit loads for surface cracks using FEM,SA/FoU-Report 91/16, The Swedish Plant Inspectorate, Stockholm, 1991.

[B11 ] ASME Boiler and Pressure Vessel Code, Sect. XI, Rules for inservice inspectionof nuclear power plant components, The American Society of MechanicalEngineers, New York, 1989.

[B12] ASME Boiler and Pressure Vessel Code, Sect. III, Div. 1, The American Societyof Mechanical Engineers, New York, 1989.

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Appendix X. Example

In a plate a defect has been detected, see figure XI. The plate constitute a component in anuclear power plant. The plate is made of a ferritic steel. At room temperature (20 °C) andat design temperature (150 °C) the material has the following data according to test results

K Ic(150°C)=160MPaVm,

RpO,2(2O °C) = 300 MPa ,

Rm(20 °C) = 490 MPa ,

RpO,2(150°C) =

R,n(150°C) = 490MPa.

The plate is subjected to a tension stress and a thermal transient. A stress analysis givesthat the nominal tension stress is 100 MPa and that the thermal transient causes a bendingstress with a size of 180 MPa at the cracked surface. The stresses act perpendicularly tothe crack plane.

Assess whether or not the defect can be accepted at normal and upset conditions.

Figure XL Plate with defect.

Solution

a) Characterization of defect

The defect is characterized as a semi elliptical surface crack with depth a = 0,009 m andlength / = 0,036 m. At normal and upset conditions the size factors ^ and "/g are equalto unity according to table PI. This gives the design quantities

ad = 0,009 m ,

/° = 0,036 m .

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b) Choice of geometry

Surface crack in a plate. Plate thickness t = 0,04 m.

c) Determination of stress state

The primary stress state consists of a membrane stress og, = 100 MPa. The secondarystress state consists of a bending stress Ob = 180 MPa. At normal and upset conditionsthe load factors "yf and f{ are equal to unity according to table PI. This gives the designquantities

< d = 100 MPa,

o£d = 180 MPa.

d) Determination of material data

At design temperature are Rpo,2 = 280 MPa and Kk = 160 MPaVin. At normal and upsetconditions table PI gives the material factors

Ym = l » J S e ,

7^ =3,16 .

The se-value is calculated according to equations PI and P2.

Sm = mint- • 490, - • 490, - • 300, - • 28o] MPa = 163 MPa,

2 280 . , .

This gives the design quantities

<4 = 280/(1,5-1,15) MPa = 163 MPa ,

Kdr = 160/3,16 MPaVm = 50 MPavm .

e) Calculation of Kf and Kf

Ki is calculated according to equation Kl. The region of applicability is fulfilled.Equations K2 - K5 give with a// = 0,009/0,036 = 0,25 and a/( = 0,009/0,04 = 0,225 thegeometry functions

fm = 0,926 ,

tf = 0,663 ,

f£ = 0,732 ,

f£ = 0,667 .

Kf and K\ at the deepest point of the crack (A) become

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KfA = /rc- 0,009(100- 0,926 + 0-0,663) MPa/m = 15,6 M P a / m ,

KfA = v/jt-0,009 (0-0,926 + 180- 0,663) MPa/iti = 20,1 MPa/m .

Kf and Kf at the intersection of the crack with the free surface (B) become

KfB = V JC- 0,009 (100 -0,732 + 0 • 0,667) MPa/iti = 12,3 M P a / m ,

KfB = /TC- 0,009 (0-0,732 + 180-0,667) MPa/m = 20 ,2MPa /m.

f) Calculation of LR

LR is calculated according to equation LI. The region of applicability is fulfilled.Equation L2 gives

0.009-0,036

0,04(0,036 + 2-0,04)

LR becomes

02 + ^ T ( i .0,070)2.1002 100

LR = ! — = 0,66 .(1-0,070)?163 (1-0,070)163

g) Calculation of KR

KR is calculated according to chapter 2.9. KR at the deepest point of the crack becomes

_A _ 20,1-0,661 15,6

pA = 0,084 ,

K A = 1 5 ' 6 + 2 Q ^ + 0,084 = 0,80.

KR at the intersection of the crack with the free surface becomes

pH -- 0,098 ,

Kg= 1 2 '3 + 2 ° ' 2 +0,098 = 0,75.

J\ f

Maximum value of KR is obtained at the deepest point of the crack and becomes

KR = 0,80 .

h) Fracture assessment

The design point for fracture assessment becomes

(LR, K R ) = (0,76, 0,80).

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KR 0.6 - -

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

LR

Figure X2. Fracture assessment.

Since the assessment point is situated in the safe region the defect is acceptable under thegiven circumstances.

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