a procedure for safety assessment of components …stress intensity factor range, ... to fulfil the...

SE9700359

A PROCEDURE FOR SAFETY ASSESSMENT OFCOMPONENTS WITH CRACKS — HANDBOOK

3rd revised edition

P. Andersson, M. Bergman, B. Brickstad, L. Dahlberg,F. Nilsson and I. Sattari-Far

SAQ/FoU-Report 96/08

Box 49306 • S-100 29 Stockholm • SwedenTelefon+46 8 617 40 00Telefax+46 8 651 70 43

A PROCEDURE FOR SAFETY ASSESSMENT OFCOMPONENTS WITH CRACKS — HANDBOOK

3rd revised edition

P. Andersson, M. Bergman, B. Brickstad, L. Dahlberg,F. Nilsson and I. Sattari-Far

SAQ/FoU-Report 96/08

ISSN 1401-5331

SAQ KONTROLL ABBox 49306 • SE-100 29 Stockholm • Sweden

Telephone +46 8 617 40 00Telefax+46 8 651 70 43

List of contents

Nomenclature 6

1. Introduction 8References 9

2. Procedure 102.1. Overview 102.2. Characterization of defect 112.3. Choice of geometry 112.4. Stress state 112.5. Material data 122.6. Calculation of slow crack growth 122.7. Calculation of K[ and K{ 132.8. Calculation of Lr 142.9. Calculation of Kr 142.10. Fracture assessment 152.11. Safety assessment 16References 17

Appendix A. Defect characterization 191. Defect geometry 192. Interaction between neighbouring defects 19References 21

Appendix R. Residual stresses 221. Butt welded joints 22

a) Thin plates 22b) Thick plates 23c) Butt welded pipes 24d)Pipe seam welds 26

2. Fillet welds 263. Stress relieved joints 274. Summary 27References 27

Appendix G. Geometries treated in this handbook 291. Cracks in a plate 29

a) Finite surface crack 29b) Infinite surface crack 29c) Embedded crack 30d) Through-thickness crack 30

2. Axial cracks in a cylinder 31a) Finite internal surface crack 31b) Infinite internal surface crack 31c) Finite external surface crack 32d) Infinite external surface crack 32e) Through-thickness crack 33

3. Circumferential cracks in a cylinder 34a) Part circumferential internal surface crack 34b) Complete circumferential internal surface crack 34c) Part circumferential external surface crack 35d) Complete circumferential external surface crack 35e) Through-thickness crack 36

4. Cracks in a sphere 37a) Through-thickness crack 37

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





References 70

Appendix L. Limit load solutions 711. Cracks in a plate 71





References 83

Appendix M. Material data for nuclear applications 841. Yield strength, ultimate tensile strength 842. Fracture toughness 84

a) Ferritic steel, pressure vessels 84b)Ferritic steel, pipes 85c) Austenitic stainless steel, pipes 85d) Cast stainless steel 86e) Stainless steel cladding 86f) Nickel base alloys 87

3. Crack growth data, fatigue 87a) Ferritic steel, pressure vessels 88b) Austenitic stainless steel 89

3. Crack growth data, stress corrosion 89References 90

Appendix S. Safety factors for nuclear applications 92References 92

Appendix B. Background 931. Assessment method 932. Secondary stress 933. Safety assessment 944. Weld residual stresses 965. Stress intensity factor and limit load solutions 976. Fit of stress distribution for stress intensity factor calculation 977. SACC 98References 98

Appendix X. Example problem 991. Solution 99

Nomenclature

a Crack depth for surface cracks2a Crack depth for embedded cracksdaldN Local crack growth rate for fatigue crack crackingdaldt Local crack growth rate for stress corrosion crackingC Constant for fatigue or stress corrosion crack growthE Elastic moduluse Eccentricity of embedded cracks/ Geometry function for stress intensity factor/R6 R6 option 1 type failure assessment curveJ J-integralJ\c Critical J-value according to ASTM E-813Kcr Critical value of stress intensity factorK\ Stress intensity factor£ I

max Maximum stress intensity factorK.™n Minimum stress intensity factorK[ Primary stress intensity factorK{ Secondary stress intensity factorK\a Fracture toughness at crack arrestK\c Fracture toughness according to ASTM E-399Kr Fracture parameter/ Crack lengthLr Limit load parametern Constant for fatigue or stress corrosion crack growthPi Limit load, local membrane stressR Stress intensity factor ratio, R = K™™ I £,max

Rei Lower yield strengthRi Inner radiusRm Ultimate tensile strengthRpO.2 0.2% elongation stressRpi o 1.0% elongation stressRTNDT Nil-ductility transition temperature5 Distance between neighbouring defectsSFj Safety factor against fracture described by J

Safety factor against fracture described by K\, SFK =Safety factor against plastic collapse

Sm Allowable design stressSr Magnitude of residual stressesT Temperature/ Plate or wall thicknessM Coordinate

Stress intensity factor range, AK, = K™3* - K™]n

Poisson's ratio

Parameter for interaction between primary and secondary stressesThrough-thickness bending stressGlobal bending stress

<jf Flow stress<jm Membrane stresscP Primary stressd Secondary stress<jY Yie ld strength, Rei or R ^ ^ay Ultimate tensile strength, Rm

X Parameter for calculation of interaction parameter p between primary andsecondary stresses

1. Introduction

In this handbook a procedure is described which can be used both for assessment ofdetected cracks or crack like defects and for defect tolerance analysis. The procedurecan be used to calculate possible crack growth due to fatigue or stress corrosion and tocalculate the reserve margin for failure due to fracture and plastic collapse. For ductilematerials, the procedure gives the reserve margin for initiation of stable crack growth.Thus, an extra reserve margin, unknown to size, exists for failure in components madeof ductile materials.

The procedure was developed for operative use with the following objectives in mind:

a) The procedure should be able to handle both linear and non-linear problems withoutany a priori division.

b) The procedure shall ensure uniqueness of the safety assessment.

c) The procedure should be well defined and easy to use.

d) The conservatism of the procedure should be well validated.

e) The handbook, that documents the procedure, should be so complete that for mostassessments, access to any other fracture mechanics literature should not benecessary.

The method utilized in the procedure is based on the R6-method [1] developed atNuclear Electric pic. The basic assumption is that fracture initiated by a crack can bedescribed by the variables Kr and Lr. Kr is the ratio between the stress intensity factorand the fracture toughness of the material. Lr is the ratio between applied load and theplastic limit load of the structure. The pair of calculated values of these variables isplotted in a diagram, see Fig. 1 in Chapter 2.1. If the point is situated within the non-critical region, fracture is assumed not to occur. If the point is situated outside theregion, crack growth and fracture may occur.

The method can in principal be used for all metallic materials. It is, however, moreextensively verified for steel alloys only. The method is not intended for use intemperature regions where creep deformation is of importance.

To fulfil the above given objectives, the handbook contains solutions for the stressintensity factor and the limit load for a number of crack geometries of importance forapplications. It also contains rules for defect characterization, recommendations forestimation of residual stresses, material data for nuclear applications and a safetyevaluation system. To ensure conservatism, the procedure with the given solutions ofthe stress intensity factor and the limit load has been validated [2]. Predictions of theprocedure were compared with the actual outcome of full scale experiments reported inthe literature. Some of the new solutions introduced in this third edition of the handbookare, however, not included in the validation reported in Ref. [2].

The first edition of the handbook was released in 1990 and the second in 1991. Thisthird edition has been extensively revised. A new safety evaluation system has beenintroduced. The conservatism in the method for assessment of secondary stresses hasbeen reduced. The solutions for the stress intensity factor and the limit load, therecommendations for estimation of residual stresses and the given material data fornuclear applications have been updated.

A modern Windows based PC-program SACC [3] has been developed which canperform the assessments described in this handbook including calculation of crackgrowth due to stress corrosion and fatigue. The program also has an option whichenables assessment of cracks according to the 1995 edition of the ASME Boiler andPressure Vessel Code, Section XI. Appendices A, C and H for assessment of cracks inferritic pressure vessels, austenitic piping and ferritic piping, respectively.

The revision of the handbook and the PC-program SACC was financially funded by theSwedish Nuclear Power Inspectorate (SKI), Barseback Kraft AB, Forsmarks KraftgruppAB, OKG Aktiebolag and Vattenfall AB Ringhals. The support has made this workpossible and is greatly appreciated.

References

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

[2] SATTARI-FAR, I., and F. NILSSON, "Validation of a procedure for safetyassessment of cracks." SA/FoU-Report 91/19, SAQ Kontroll AB, Stockholm,Sweden, 1991.

[3] BERGMAN, M., "User's manual SACC Version 4.0." SAQ/FoU-Report 96/09,SAQ Kontroll AB, Stockholm, Sweden, 1996.

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2. Procedure

2.1. Overview

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

1) Characterization of defect (Chapter 2.2 and Appendix A).

2) Choice of geometry (Chapter 2.3 and Appendix G).

3) Determination of stress state (Chapter 2.4).

4) Determination of material data (Chapter 2.5 and Appendix M).

5) Calculation of possible slow crack growth (Chapter 2.6 and Appendix M).

6) Calculation of Kf and ATf (Chapter 2.7 and Appendix K).

7) Calculation of Lr (Chapter 2.8 Appendix L).

8) Calculation of Kr (Chapter 2.9).

9) Fracture assessment (Chapter 2.10).

1.2

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6

Fig. 1. Diagram for fracture assessment (FAD).

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The non-critical region is limited by,

Kr < fR6 = (1 -0.14Z^)[0.3 + 0.7exp(-0.65I^)], (1)

1, for materials with a yield plateu

for all other casesaY

10) Assessment of results (Chapter 2.11).

2.2. Characterization of defect

A fracture mechanics analysis requires that the actual defect geometry is characterizedin a unique 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 defectremains from the manufacture or has occurred because of service induced processessuch as fatigue 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 care should be taken to avoidnon-conservatism. In cases when an idealization of the real geometry to one of the casesconsidered here are not adequate, stress intensity factor and limit load solutions can befound in the literature or be calculated by numerical methods. The use of such solutionsshould be carefully checked for accuracy.

2.4. Stress state

In this procedure it is assumed that the stresses have been obtained under theassumption of linearly elastic material behaviour. The term nominal stress denotes thestress state that would act at the plane of the crack in the corresponding crack freecomponent.

The stresses are divided into primary cf and secondary & stresses. Primary stresses arecaused by the part of the loading that contributes to plastic collapse e.g. pressure,gravity loading etc. Secondary stresses are caused by the part of the loading that doesnot contribute 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 weld residual stresses is given in Appendix R.

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2.5. Material data

To perform the assessments the yield strength cry, the ultimate tensile strength ou andthe critical stress intensity factor Kcr of the material must be determined. If possible,data obtained from testing of the actual material of the component should be used. Thisis not always possible and therefore minimum values for ay and ay from codes,standards or material specifications may be used. These data should be determined at theactual temperature.

ay denotes the lower yield strength Rei if this can be determined and in other cases the0.2% proof stress RpQ^- 1° the cases when Rei can be determined the material isconsidered to have a yield plateau. This is for instance common for certain low alloycarbon manganese steels at low temperatures.

G\J is the ultimate tensile strength of the material.

The yield strength and ultimate tensile strength of the base material should normally beused even when the crack is situated in a welded joint. The reason for this is that theyield limit of the structure is not a local property but also depends on the strengthproperties of the material remote from the crack.

Kcr is the critical value of the stress intensity factor for the material at the crack front. Ifpossible, Kcr should be set equal to the fracture toughness K\c according to ASTM E-399[1]. It is in many cases not possible to obtain a valid /qc-value. J\c-values according toASTM E-813 [2] can be used instead and be converted according to Eq. (3).

Here E is the elastic modulus of the material and vis Poisson's ratio.

For application on nuclear components fracture toughness data according to AppendixM can be used if actual test data for the considered material is not available.

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 K\. This quantity is calculated according tomethods described in Appendix K.

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

^ R). (4)

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Here

AiCj =£ Im a x-A: I

m i n , (5)

and

where ArImax and A^11111 are the algebraic maximum and minimum, respectively, of K\

during the load cycle, gf 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

For application on nuclear components fatigue crack growth data according to AppendixM can be used if actual test data for the considered material is not available.

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

^ (8)

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

For application on nuclear components stress corrosion crack growth data according toAppendix M can be used if actual test data for the material and environment underconsideration is not available.

2.7. Calculation of K[ and Kf

The stress intensity factors K[ (caused by primary stresses aP) and Kf (caused bysecondary stresses o*) 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 resultsare desired for a situation outside the applicability limits a recharacterization of thecrack geometry can sometimes be made. The following recharacterizations arerecommended:

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

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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.

In cases when the solutions of Appendix K cannot be applied, stress intensity factorscan be obtained either by use of solutions found in the literature, see for example thehandbooks [3], [4] and [5], or by numerical calculations, e.g. by the finite elementmethod.

2.8. Calculation o/Lr

Lr is defined as the ratio between the current primary load and the limit load Pi for thecomponent under consideration and with the presence of the crack taken into account.Pi should be calculated under the assumption of a perfectly-plastic material with theyield strength ay chosen as discussed in Chapter 2.5. Appendix L contains solutions ofLr for the cases considered in this procedure.

Limits for the applicability of the solutions are given for the different cases. If resultsare desired for a situation outside the applicability limits a recharacterization of thecrack geometry can sometimes be made similarly to what was discussed for the stressintensity factor 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 [6], or by numericalcalculations, e.g. by the finite element method.

2.9. Calculation ofKr

The ordinate Kr in the failure assessment diagram (Fig. 1) is calculated in the followingway.

Kr= l l + p, (9)

where p is a parameter that takes into account plastic effects because of interactionbetween secondary and primary stresses, p is obtained from the diagram in Fig. 2 wherep is given as a function of Lr and the parameter x defined as

do)

X is set to zero if x fells below zero. Also, p is restricted to non-negative values asdefined in Fig. 2.

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0.25

0.2-4

0.15 —

0 . 1 -

0.05 -

Lr

Fig. 2. Diagram for calculation of p.

2.10. Fracture assessment

In order to assess the risk of fracture the assessment point (Lr, Kr) calculated asdescribed above is plotted in the diagram in Fig. 1. If the point is situated within thenon-critical region no initiation of crack growth is assumed to occur and thus nofracture. The non-critical region is limited by the R6 option 1 type failure assessmentcurve according to

Kr <fR6 =(l-0. (H)

rmax (12)

For materials which shows a continuos stress-strain curve without any yield plateau, theupper limit of Lr is defined by

j max _L,r -

<Ty(13)

where c^ is the uniaxial flow stress. Depending on application and type of material, ay isgiven by

(oy

for ferritic nuclear components

for austenitic nuclear components .

for all other cases

(14)

16

Here Sm is the allowable design stress defined by

Sm = min(-^ \ -^—, %& , \-J,

for ferritic materials and by

c minf 2 g y ( 2 ° C) OQ^ (T\ ^ ' ( 2 ° C ) ^ < ^ 1 n «Sm=min ,0.9crY(T), , — - — I , (16)

for austenitic materials. 2" is the temperature of the material.

For materials that exhibit a discontinuous yield point L™™ is restricted to 1.0. This isconservative and should be regarded as a compromise when applying the option 1 typeR6 failure assessment curve to a problem that is actually better described by an option 2type failure assessment curve, cf. [7].

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 me point (Ln Kr) to fall onthe border to the critical region. Similarly, the limiting crack size is obtained by keepingthe loads fixed and calculating the point (Lr, Kr) for different crack sizes until it falls onthe border to the critical region.

2.11. Safety assessment

The following conditions should be fulfilled to determine if a detected crack of a certainsize is acceptable, cf. [8]:

^ , (17)

Eqs. (17) and (18) account for the failure mechanisms fracture and plastic collapse andSFj and SFi are title respective safety factors against these failure mechanisms. Plasticcollapse is assumed to occur when the primary load P is equal to the limit load Pi. Thisoccurs when the remaining ligament of the cracked section becomes fully plastic andhas reached the flow stress a/. J is the path-independent ./-integral which is meaningfulfor situations where J completely characterizes the crack-tip conditions. J should beevaluated with all stresses present (including residual stresses) and for the actualmaterial data. J\c is the value of the ./-integral at which initiation of crack growth occurs.

In this procedure J i s estimated using the option 1 type R6 failure assessment curve. TheR6-estimation of J i s given by

V £ \]2

17

where f^ is defined by Eq. (11). The second fraction on the right hand side of Eq. (19)can be interpreted as a plasticity correction function, based on the limit load, for thelinear elastic value of J determined by the stress intensity factor K\.

Combining Eqs. (3), (17) and (19) gives the following relation for the acceptance of acrack:

(20)

The left-hand side of Eq. (20) represents the parameter Kr used for safety assessment.Eq. (20) implies that the assessment point (Lr, Kr) should be located below the R6failure assessment curve divided by the safety factor V&F/. The maximum acceptablecondition is obtained in the limit when the assessment point is located on the reducedfailure assessment curve, expressed by Eq. (20) with a sign of equality. In addition asafety factor against plastic collapse, corresponding to Eq. (17), is introduced as a safetymargin against the cut-off of Lr as

i-max

Eqs. (20) and (21) represent the safety assessment procedure used in this handbook andin the PC-program SACC [9]. In Appendix S, a set of safety factors are defined to beused for nuclear applications, cf. [8]. The safety factor SFR is introduced which is thesafety factor on Kcr corresponding to the safety factor SFj on J\c. They are relatedthrough SF]i = "JSFj. Critical conditions are obtained when all safety factors are set tounity and when the assessment point is located on the failure assessment curve.

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, crack sizeand material properties.

References

[1] ASTM E-399 Test method for plane-strain fracture toughness of metallicmaterials. American Society for Testing and Materials, Philadelphia, U.S.A..

[2] ASTM E-813 Standard test method for Jio a measure of fracture toughness.American Society for Testing and Materials, Philadelphia, U.S.A..

[3] MURAKAMI, Y. (ed.), Stress intensity factors handbook. Vol. 1-3, PergamonPress, Oxford, U.K., 1987-1991.

[4] TADA, H., PARIS, P. C. and G. C. IRWIN, The stress analysis of crackshandbook. 2nd edition, Paris Productions Inc., St. Louis, U.S.A., 1985.

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

[6] MILLER, A. G., "Review of limit loads of structures containing defects." TheInternational Journal of Pressure Vessels and Piping. Vol. 32, pp. 197-327,1988.

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

[8] BRICKSTAD, B. and M. BERGMAN, "Development of safety factors to beused for evaluation of cracked nuclear components." SAQ/FoU-Report 96/07,SAQ Kontroll AB, Stockholm, Sweden, 1996.

[9] BERGMAN, M., "User's manual SACC Version 4.0." SAQ/FoU-Report 96/09,SAQ Kontroll AB, Stockholm, Sweden, 1996.

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Appendix A. Defect characterization

A fracture mechanics assessment requires that the current defect geometry ischaracterized uniquely. In this appendix general rules for this are given. For additionalinformation it is referred to ASME Boiler and Pressure Vessel Code, Sect. XI [Al].

1. Defect geometry

Surface defects are characterized as semi-elliptical cracks. Embedded defects arecharacterized as elliptical cracks. Through thickness defects are characterized asrectangular 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 close to other defects the interactionshall be taken into account. Some cases of practical importance are illustrated in Fig.Al. According to the present rules the defects shall be regarded as one compound defectif the distance 5 satisfies the condition given in the figure. The compound defect size isdetermined by the length and depth of the geometry described above whichcircumscribes the defects. The following shall be noted:

a) The ratio I/a shall be greater than or equal to 2.

b) In case of surface cracks in cladded surfaces the crack depth should be measuredfrom the free surface of the cladding. If the defect is wholly contained in the claddingthe need 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.

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Case Defect sketches Criterion

If s < 0.4a!thena = 2a\ + s

If s < min(/i, h)then/ = /] + /2 + s

If s < min(/i,then

2a,;

Si

C O\fs< max(2fli,then2a = 2a\ + 2a2 + s

2a,

Si

a2;

CO

If s < max(2a], athena = 2a\ + a2 + s

lfs\ < max(2ai, a2)and 52 <min(/i, /2)thena = 2ai + a2 + 5i/ = l\ + h + S2

Fig Al. Rules for defect characterization at interaction.

21

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, U.S.A., 1995.

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Appendix R. Residual stresses

Residual stresses is defined as stresses existing in a structure when it is free fromexternal loading. The distribution and magnitude of residual stresses in a componentdepend on the fabrication process and service influences. Residual stresses are normallycreated during the manufacturing stage but can also appear or be redistributed duringservice. At pressure or load test of a component, peaks of residual stresses can berelaxed due to local plasticity if the total stress level exceeds the yield strength of thematerial.

It is anticipated that with increasing amount of external load (increasing limit loadparameter Lr) the contribution to the risk of fracture from the weld residual stresses isdiminished. There is experimental evidence [Rl], which demonstrates that for ductilematerials, the influence of weld residual stresses on the load carrying capacity is quitelow. In [Rl] the influence of the weld residual stresses beyond Lr of 1.0 was almostzero. The ASME Code, Section XI, has dealt with this problem by simply ignoring weldresidual stresses for certain materials, for instance for austenitic stainless steels.However, this treatment is questionable for situations that is dominated by secondaryloads (e.g. for thermal shock loads) and when the material is not sufficiently ductile thatthe failure mechanism is controlled only by plastic collapse. At the present time there isnot sufficient information to give a quantitative recommendation on how to treat weldresidual stresses for high Zr-values in a R6 fracture assessment.

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 volume change affects the weld residual stresses and make itdifficult to perform accurate predictions. A more comprehensive compendium ofresidual stress profiles are given in [R2]. The residual stresses are given in thetransverse and longitudinal directions, corresponding to stresses normal and parallel tothe weld run. Weld residual stresses acting in the through thickness direction areassumed to be negligible.

The magnitude of the residual stresses are expressed in Sr which is set to the 0.2% yieldstrength of the material at the actual temperature, see Chapter 2.5. For austeniticstainless steels, however, Sr should be chosen to the 1% proof stress at the consideredtemperature. This is mainly due to the large strain hardening that occurs for stainlesssteels. If data for the 1% proof stress is missing, it may be estimated as 1.3 times the0.2% proof stress. The actual yield strength values rather than minimum values shouldbe used for a realistic estimation.

For an overmatched weld joint, the yield strength is referred to the weld material for theresidual stress in the longitudinal direction in the weld centerline. For the transverseresidual stress, the yield strength is in general referred to the base material.

1. Butt welded joints

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.

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The weld residual stresses acting in the longitudinal direction of the weld has adistribution across the weld according to Fig. Rl. The width of the zone with tensilestresses / is about four to six times the plate thickness. This distribution is obtainedalong the entire weld except when the weld ends at a free surface where the stresses tendto zero.

a/Sr

-0.5

1.5

Fig. Rl. Distribution across the weld of residual stresses actingin the longitudinal direction of the weld.

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, see e.g. Ref. [R8]. Thevalues are valid at the centerline of the weld. They successively diminish in sectionsoutside the centerline.

b) Thick plates

In thick plates welded with many weld beads the variation of the residual stresses in thethickness direction cannot be neglected. The stress distribution is much morecomplicated in thick plates than in thin plates and depends very much on joint designand weld method. It is therefore difficult to give simple general guidelines forestimation of residual stresses in thick plates. It is not possible to give a specific value ofthe plate thickness that represents the transition from a thin to a thick plate. For moredetailed information, see Refs. [R3], [R4] and [R5].

The weld residual stresses acting in the weld vary both in the thickness direction andacross the weld. The highest stress in the symmetry plane of the weld is Sr. The simplestbut in many cases very conservative assumption is that the weld residual stress isconstant and equal to Sr throughout the entire thickness.

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In a symmetric joint, for instance an X-joint or a double U-joint with many weld beads,the residual stress through the thickness t varies as shown in Fig. R2.

-1.5-0.5

Fig. R2. Thickness distribution of residual stresses acting in awelded thick plate in a symmetric joint with many weld beads.This distribution for residual stresses acting transverse the weldapplies for a free plate.

The weld residual stresses acting transverse the weld vary as shown in Fig. R2 if any ofthe plates is free to move during the cooling.

c) Butt welded pipes

The residual stress distribution in girth welds in both thick walled and thin walled pipesare complicated and depends on joint shape, weld method, heat input, wall thicknessand pipe radius. The recommendations given here are based on a numericalinvestigation on austenitic stainless steel pipes [R6] and apply to girth welds appliedfrom the outside (single sided V- or U-preparation). They are valid for a radius tothickness ratio RJt of approximately 8 but can conservatively be used for higher ratiosof Rj/t. The heat input relative to the pipe thickness was between 75 and 101 MJ/m2

which is considered to be high. More detailed information is given in [R6]. Aparametric study of a large number of pipes can also be found in [R7].

The local residual stresses acting longitudinal and transverse to a girth weld are shownin Fig. R3 for a pipe thickness up to 30 mm. The stresses decrease successively insections further outside the fusion line.

25

-1

Fig. R3. Local distribution of residual stress acting longitudinaland transverse to a girth weld for an austenitic pipe up to athickness of 30 mm.

In Table Rl the specific values of the membrane stress am and local bending stress cri,are given. They are taken from [R6] and determined at 288 °C which is a commonoperating temperature for nuclear LWR plants. The residual stresses at roomtemperature, if the weld earlier has been subjected to a temperature cycle of 288 °C, canbe obtained by multiplying the values in Table Rl with a factor of approximately 1.2.This does not just correspond to the increase in yield strength with decreasingtemperature. When the temperature is raised to 288 °C after cool down to roomtemperature after the last weld pass, also some plastic deformation occurs, see Ref.[R6].

The local residual stresses acting longitudinal and transverse to a girth weld are shownin Fig. R4 for a pipe thickness greater than 30 mm. The stresses decrease successively insections further outside the fusion line.

Cm —

+ 3 . 8 1 1 6 ( X / O - 9 9 . 8 2 0 ( X / / ) +

+339.97(x//)3 -404.59(x/O4+158.16(x/O5

Fig. R4. Local distribution of residual stress acting longitudinaland transverse to a girth weld for an austenitic pipe with athickness greater than 30 mm.

26

In Table Rl the specific values of the membrane stress am and stress amplitude oo aregiven. They are determined at 288 °C. The residual stresses at room temperature can beobtained by multiplying the values in Table Rl with a factor of approximately 1.2.

In [R6] the stainless steel pipes considered were overmatched with a weld material yieldstrength at 1% strain of 348 MPa and base material yield strength at 1% strain of 187MPa at 288 °C. Sr in Table Rl is referred to the 1% proof stress at 288 °C for the basematerial except for the longitudinal stress at the weld centerline where Sr is referred tothe 1% proof stress of the weld material. For other combinations of weld and basematerial yield properties, appropriate adjustments of these recommendations can bemade according to Table Rl.

Pipes with a thickness up to 40 mm were studied in [R6]. For a pipe wall thicknessgreater than this, the recommendations in Table Rl should be conservative.

Transverse stress (MPa)/(mm)

t<l

7 < t < 2525<f<30

t>30

Weld centerline and HAZ

1.219Sr[l-2(jc//)](1.5884 - 0.05284/)Sr[l - 2(x//)]

0.2674Sr[l - 2(x//)]

0.42465r[l + 3.81 \6(x/t) - 99.%2{xlt)2 + 339.97(x/r)3 -404.59(x/04+ 158.16(x/Y)5]

Longitudinal stress (MPa)/(mm)f<30/>30

Weld centerline0.9255r

0.925SV

HAZ0.8615,0.6465r

Table Rl. Recommended longitudinal and transverse residualstress distributions for austenitic stainless steel pipe welds at 288°C. x = 0 at the inside of the pipe wall. See Figs. R3 and R4.

For ferritic steel piping, predictions based on numerical methods are more difficult dueto volume changes during certain phase transformations. Based on the investigation in[R7], the principal feature for the residual stresses for girth welds should be similar asfor austenitic piping. Thus if no other data for a specific case exists, it is proposed to useTable Rl also for ferritic piping. Proper adjustment has to be made for the actual yieldproperties of the ferritic pipe weld. Also, the bending stress <Jb for thinner pipes in TableRl, should be limited to the yield strength for a ferritic pipe.

d) Pipe seam welds

The residual stresses acting longitudinal and transverse to a seam weld canapproximately be estimated from the corresponding distribution in a plate if the pipe isnot adjusted with respect to roundness after welding.

2. Fillet welds

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

27

3. Stress relieved joints

In correctly stress relieved joints the maximum residual stress decreases to 0.15-0.205r.Micro alloyed steels are sometimes stress relieved at temperatures lower than 560 °C.The relaxation is in such cases not so effective and the remaining residual stress is largerthan 0.25r.

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

4. Summary

The above given guidelines for estimation of residual stresses are summarized in TableR2.

Type of joint

Butt weld in a plate

Butt weld in a pipe,

girth weld

Butt weld in a pipe,

seam weldFillet weld

Stress relieved weld

Thickness

Thin plate with onlya few weld beads

Thick plate withmany weld beads,symmetric X-joint

Thin (< 30 mm)

Thick (> 30 mm)

ThinThick

Residual stress,longitudinal

Fig. Rl

Fig.R2

Fig. R3 and TableRlFig. R4 and TableRlSee thin plateSee thick plate

<Sr

< 0.2Sr

Residual stress,transverse

< Sr, fixed plate< 0.2Sr, free plate

Fig.R2

Fig. R3 and TableRlFig. R4 and TableRlSee thin plateSee thick plate

<Sr

< 0.2Sr

Sr = 0.2% proof stress for ferritic steels and 1% proof stress for austenitic stainless steels at the temperature ofassessment.

Table R2. Estimation of residual stresses in welded joints.

References

[Rl] SHARPLES, J. K., SANDERSON, D. J., BOWDLER, B. R., WIGHTMAN, A.P. and R. A. AINSWORTH, "Experimental programme to assess the effect ofresidual stresses on fracture behaviour.", ASME PVP. Vol. 304, pp. 539-551,1995.

[R2] SANDERSON, D. J., "Recommendations for revised compendium of residualstress profiles for R6." AEA/TSD/0554, AEA Technology, Risley, U.K., 1996.

[R3] UEDA, Y., TAKAHASHI, E., SAKAMOTO, K. and K. NAKACHO,"Multipass welding stresses in very thick plates and their reduction from stressrelief annealing." Trans. JWRI. Vol. 5, No. 2, pp. 79-88,1976.

28

[R4] UEDA, Y. and K. NAKACHO, "Distribution of welding residual stresses invarious welded joints of thick plates." Trans. JWR1, Vol. 15, No. 1, 1986.

[R5] RUND, C. O. and P. S DIMASCO, "A prediction of residual stresses in heavyplate butt welds." J. Materials for Energy Systems. Vol. 3, pp. 62-65, 1981.

[R6] BRICKSTAD, B. and L. JOSEFSON, "A parametric study of residual stresses inmultipass butt-welded stainless steel pipes." SAQ/FoU-Report 96/01, SAQKontroll AB, Stockholm, Sweden, 1996.

[R7] SCARAMANGAS, A., "Residual stresses in girth butt weld pipes.", TechnicalReport No. CUED/D-Struct./TR. 109, Dept. of Engineering, CambridgeUniversity, U.K., 1984.

[R8] ANDERSSON, B. and L. KARLSSON, "Thermal stresses in large butt-weldedplates." J. Thermal Stresses. Vol. 4, pp. 491-500, 1981.

29

Appendix G. Geometries treated in this handbook

1. Cracks in a plate

a) Finite surface crack

Fig. Gl. Finite surface crack in a plate.

b) Infinite surface crack

u

Fig. G2. Infinite surface crack in a plate.

30

c) Embedded crack

2a

Y\B

Fig. G3. Embedded crack in a plate.

d) Through-thickness crack

B

u

Fig. G4. Through-thickness crack in a plate.

2. Axial cracks in a cylinder

a) Finite internal surface crack

31

u

B

Ri

Fig. G5. Finite axial internal surface crack in a cylinder.

b) Infinite internal surface crack

u

Ri

Fig. G6. Infinite axial internal surface crack in a cylinder.

32

c) Finite external surface crack

B

Ri

1 •

Fig. G7. Finite axial external surface crack in a cylinder.

d) Infinite external surface crack

a

u

Ri

Fig. G8. Infinite axial external surface crack in a cylinder.

33

e) Through-thickness crack

uB

Ri

Fig. G9. Axial through-thickness crack in a cylinder.

34

3. Circumferential cracks in a cylinder

a) Part circumferential internal surface crack

Fig. G10. Part circumferential internal surface crack in acylinder.

b) Complete circumferential internal surface crack

Fig. G i l . Complete circumferential internal surface crack in acylinder.

35

c) Part circumferential external surface crack

Fig. G12. Part circumferential external surface crack in acylinder.

d) Complete circumferential external surface crack

Fig. G13. Complete circumferential external surface crack in acylinder.

36


B

Fig. G14. Circumferential through-thickness crack in a cylinder.

37

4. Cracks in a sphere

a) Through-thickness crack

B

Fig. G15. Circumferential through-thickness crack in a sphere.

38

Appendix K. Stress intensity factor solutions



K\ is given by

(Kl)(=0

• (/ = 0 to 5) are stress components which define the stress state a according to

5

i=0

for 0<u<a. (K2)

a is to be taken normal to the prospective crack plane in an uncracked plate, o; isdetermined by fitting <rto Eq. (K2). The coordinate u is defined in Fig. Gl.

fi(i = 0 to 5) are geometry functions which are given in Tables Kl and K2 for thedeepest point of the crack (/^), and at the intersection of the crack with the free surface( 8 ) , respectively. See Fig. Gl.

Remarks: The plate should be large in comparison to the length of the crack so that edgeeffects do not influence.

Ref.: [Kl].

I/a = 2

alt

00.20.4

0.60.8

/oA

0.6590.6630.6780.6920.697

/ . A

0.4710.4730.4790.4860.497

/ 2 A

0.3870.3880.3900.3960.405

/ 3 A

0.3370.3370.3390.342

0.349

ft0.2990.2990.3000.304

0.309

/ 5 A

0.2660.2690.2710.274

0.278

11 a = 5/2

alt

00.20.4

0.60.8

/oA

0.741

0.7460.771

0.800

0.820

/ i A

0.5100.512

0.519

0.5310.548

/2A

0.4110.413

0.4160.422

0.436

/ 3 A

0.3460.3520.356

0.362

0.375

ft0.300

0.3060.309

0.317

0.326

/ 5A

0.2660.2700.2780.284

0.295

39

I/a = 10/3alt

00.20.4

0.6

0.8

ft0.8330.8410.885

0.930

0.960

/ , A

0.5490.5540.568

0.587

0.605

/2A

0.4250.4300.4420.454

0.476

/ 3 A

0.351

0.3590.371

0.3810.399

ft0.301

0.3090.320

0.331

0.346

/5A

0.2670.2710.285

0.295

0.310

11 a = 5

alt

00.20.4

0.6

0.8

ft0.9390.9571.057

1.146

1.190

/ , A

0.5800.595

0.631

0.6680.698

/2A

0.434

0.4460.475

0.4950.521

/ 3 A

0.353

0.3630.389

0.4070.428

ft0.302

0.3100.3320.350

0.367

/ 5 A

0.2680.2730.292

0.3090.324

I/a = 10alt

00.20.4

0.60.8

ft1.053

1.1061.3061.5721.701

/ , A

0.6060.6400.724

0.8150.880

/2A

0.4430.467

0.5250.5710.614

/3A

0.3570.374

0.4200.448

0.481

ft0.3020.314

0.3480.3770.399

ft0.2690.2770.304

0.3270.343

I)'a -»oo

a//

00.2

0.40.60.8

/oA

1.123

1.3802.1064.02511.92

ft0.682

0.784

1.0591.7504.437

/ 2 A

0.524

0.5820.7351.1052.484

/ 3 A

0.4400.478

0.5780.8141.655

ft0.3860.414

0.4850.6511.235

ft0.344

0.3690.4230.5480.977

Table Kl . Geometry functions for a finite surface crack in aplate — deepest point of the crack.

40

lla = 2

alt

00.20.4

0.6

0.8

fBJO

0.7160.729

0.777

0.839

0.917

f?0.118

0.1230.133

0.1480.167

f?0.041

0.045

L 0.050

0.0580.066

/ 3 B

0.022

0.0230.026

0.029

0.035

fB

J40.0140.014

0.015

0.0180.022

/ 5 B

0.0100.010

0.011

0.0120.015

11 a = S/2

alt

00.20.4

0.6

0.8

ft0.7300.749

0.7950.9010.995

/ i B

0.124

0.1260.1440.167

0.193

f?0.041

0.0460.054

0.0660.076

/ 3

0.021

0.023

0.0280.0330.042

0.0130.014

0.0170.021

0.026

/ 5 B

0.010

0.0100.012

0.0150.017

lla = 10/3

alt

00.20.4

0.60.8

ft0.7230.747

0.8030.934

1.070

/ i B

0.1180.1250.1450.1800.218

/ 2B

0.0390.044

0.0560.072

0.087

/ 3 B

0.0190.022

0.0290.0370.047

ft0.0110.014

0.0180.0230.029

/ 5 B

0.0080.0100.0120.0160.020

lla = 5

a/f

00.20.4

0.60.8

ft0.6730.7040.7920.9211.147

/ i B

0.104

0.1140.139

0.1830.244

/ 2B

0.032

0.0380.0530.0740.097

/ 3 B

0.0150.0180.0270.0380.052

ft0.0090.0110.0160.0240.032

/ 5 B

0.0060.0070.0110.0170.021

lla = 10a/f

00.20.4

0.60.8

/oB

0.5160.554

0.6550.840

1.143

/ , B

0.0690.076

0.0990.1570.243

ft0.017

0.022

0.0390.0630.099

/ 3 B

0.009

0.0110.0190.032

0.055

ft0.0050.0070.0120.0200.034

/ 5 B

0.004

0.005

0.0080.0130.023

Table K2. Geometry functions for a finite surface crack in aplate — intersection of crack with free surface.


K\ is given by

41

Ha -><x>

alt

0

0.20.4

0.6

0.8

ft0.000

0.0000.0000.000

0.000

/ l B

0.000

0.0000.0000.000

0.000

f?0.0000.0000.0000.000

0.000

ff0.0000.000

0.0000.000

0.000

ft0.0000.0000.000

0.000

0.000

/ 5B

0.000

0.0000.0000.000

0.000

(K3)

The stress state a = o(u) is to be taken normal to the prospective crack plane in anuncracked plate. The coordinate u is defined in Fig. G2.

ft (/ = 1 to 5) are geometry functions which are given in Table K3 for the deepest pointof the crack (/*). See Fig. G2.

Remarks: The plate should be large in the transverse direction to the crack so that edgeeffects do not influence.

Ref.: [K2].

alt

00.10.20.30.4

0.5

0.60.7

0.8

0.9

ft2.0002.0002.0002.0002.0002.000

2.0002.000

2.0002.000

ft0.9771.4192.5374.2386.636

10.0215.04

29.52

38.8182.70

/•A

1.1421.1381.2381.6802.8055.500

11.88

45.5178.75351.0

ft-0.350-0.355-0.347-0.410

-0.611-1.340

-3.607

-18.93-36.60-207.1

ft-0.091-0.076-0.056-0.019

0.0390.218

0.7864.883

9.87160.86

Table K3. Geometry functions for an infinite surface crack in aplate.

42

c) Embedded crack

K\ is given by

AT, ='J^(amfm(2a/t,l/2a,e/t) a/t, Ilia, e11)). (K4)

<jm and Ob are the membrane and bending stress components respectively, which definethe stress state cr according to

2Ma = <r(u) = am + C7b[l - — for 0 < u < t. (K5)

cr is to be taken normal to the prospective crack plane in an uncracked plate. am and abare determined by fitting a to Eq. (K5). The coordinate u is defined in Fig. G3.

fm andyj, are geometry functions which are given in Tables K4 and K5 for the points ofthe crack closest to (/*), and furthest (Z8) from u = 0, respectively. See Fig. G3.


Ref.: [K3].

Ilia = 1

lalt

00.20.40.6

elt = 0

fAJ m

0.6380.649

0.6810.739

ft0.0000.0870.182

0.296

e/f = 0.15fA

J m0.6380.6590.7250.870

ft0.191

^ 2 8 60.4110.609

eft = 0.3

fA

J m0.6380.694

-

-

ft0.3830.509

-

-

//2a = 2

lalt

00.20.4

0.6

elt = 0

fAJ m

0.8240.844

0.9011.014

ft0.000

0.0980.210

0.355

e// = 0.15fA

J m0.824

0.8620.9871.332

ft0.241

0.3590.526

0.866

eft = 0.3

fA

Jm0.8240.932

-

-

ft0.494

0.668-

-

Ilia = 4

lalt

00.20.4

0.6

elt = 0

fA

J m0.917

0.942

1.0161.166

ft0.0000.102

0.2200.379

elt = 0.15fA

J m0.917

0.966

1.1291.655

ft0.2750.394

0.5841.034

e// = 0.3

fA

J m0.9171.058

-

-

fA

Jb0.550

0.749-

-

43

Ilia -> oo

lalt

00.20.4

0.6

elt = 0

J m

1.0101.041

1.133

1.329

0.0000.104

0.227

0.399

elt =

J m

1.0101.0711.282

2.093

0.15fA

Jb0.3030.4280.641

1.256

elt =

fA

J m1.010

1.189-

-

= 0.3

fbA

0.6060.833

-

-

Table K4. Geometry functions for an embedded crack in a plate— point closest to u = 0.

//2a = 1

2a//

00.20.4

0.6

e/t = 0fBJm

0.638

0.6490.6810.739

fbB

0.000-0.087-0.182-0.296

e// =fBJ m

0.6380.6460.6680.705

0.15

fb0.1910.1080.022-0.071

e// = 0.3fBJ m

0.6380.648

--

fbB

0.3830.303

--

//2a = 2

2a//

00.20.40.6

e/t = 0

Jm

0.8240.844

0.9011.014

ft0.000-0.098-0.210-0.355

e//=0.15

J m

0.8240.8440.902

1.016

fB

0.2470.1550.060-0.051

e//=0.3

J m

0.824

0.866-

-

ft0.494

0.418-

-

//2a = 4

2a//

00.20.4

0.6

e/t = Q

fB

J m0.9170.9421.016

1.166

fbB

0.000-0.102-0.220

-0.379

e/f=0.15fBJ m

0.9170.945

1.0291.206

fB

0.2750.1810.086

-0.030

elt = 0.3

f*Jm

0.9170.980

-

-

fbB

0.5500.482

-

-

44

112a -> oo

2alt

00.2

0.4

0.6

eltfBJ m

1.0101.041

1.133

1.329

= 0

ft0.000-0.104

-0.227

-0.399

elt =fBJ m

1.010

1.0481.162

1.429

0.15

ft0.303

0.210

0.1660.000

elt =fBJ m

1.0101.099

-

-

= 0.3fB

Jb0.6060.550

-

-

Table K5. Geometry functions for an embedded crack in a plate— point furthest from u = 0.


K\ is given by

(K6)

<jm and &b are the membrane and bending stress components respectively, which definethe stress state a according to

2" for 0 < u < t. (K7)

<ris to be taken normal to the prospective crack plane in an uncracked plate. <rm and at,are determined by fitting crto Eq. (K7). The coordinate u is defined in Fig. G4.

fm and/t are geometry functions which are given in Table K6 for the intersections of thecrack with the free surface at u - 0 (/A), and at u = t (^). See Fig. G4.


Ref.: [K4].

J m

1.000

/•AJb

1.000

fBJ m

1.000ft

-1.000

Table K6. Geometry functions for a through-thickness crack in aplate.

45



K\ is given by

Malt, I la, Rt/t). (K8)i=0

jj (i = 0 to 3) are stress components which define the stress state a according to

3

i=0

forO< u<a. (K9)

a is to be taken normal to the prospective crack plane in an uncracked cylinder, o; isdetermined by fitting crto Eq. (K9). The coordinate u is defined in Fig. G5.

/• (/ = 0 to 3) are geometry functions which are given in Tables K.7 and K8 for thedeepest point of the crack (/*), and at the intersection of the crack with the free surface

8 , respectively. See Fig. G5.

Remarks: The cylinder should be long in comparison to the length of the crack so thatedge effects do not influence.

Refs.: [Kl] and [K5].

l/a = 2,Ri/t = 4alt

00.20.5

0.8

/oA

0.6590.6430.6630.704

/ i A

0.4710.454

0.4630.489

/ 2 A

0.3870.3750.3780.397

/ 3 A

0.3370.326

0.3280.342

I/a = 2, /?///= 10

alt

00.20.5

0.8

/oA

0.6590.6470.6690.694

/ , A

0.4710.4560.4640.484

/ 2 A

0.3870.3750.3800.394

/ 3 A

0.337

0.3260.328

0.339

lla = 5,Ri/t = 4alt

00.20.5

0.8

/oA

0.939

0.9191.037

1.255

/ , A

0.580

0.5790.622

0.720

/2A

0.4340.4520.4740.534

/ 3 A

0.3530.382

0.3950.443

46

lla = 5,RJt=\0

alt

0

0.2

0.5

0.8

/oA

0.9390.932

1.058

1.211

/ i A

0.5800.584

0.6290.701

/2A

0.434

0.4550.477

0.523

/ 3 A

0.3530.383

0.397

0.429

I/a = 10, Ri/t= 4

alt

0

0.2

0.5

0.8

/•AJO

1.0531.045

1.3381.865

/ i A

0.6060.634

0.739

0.948

/2A

0.4430.487

0.5400.659

/ 3 A

0.357

0.4060.4380.516

lla = 10, R-,lt= 10

«//

0

0.20.5

0.8

/oA

1.0531.0621.359

1.783

/ i A

0.6060.641

0.7460.914

/2A

0.4430.4890.544

0.639

/ 3 A

0.3570.417

0.4400.504

Table K7. Geometry functions for a finite axial internal surfacecrack in a cylinder—deepest point of crack.

lla = 2, Ri/t = 4

alt

0

0.2

0.50.8

ft0.716

0.7190.7590.867

/ i B

0.1180.124

0.1360.158

/ 2 B

0.041

0.0460.0520.062

/ 3 B

0.0220.0240.0270.032

lla = 2, Rj/t= 10

alt

0

0.2

0.50.8

ft0.7160.7260.777

0.859

/ i B

0.118

0.1260.141

0.163

/ 2 B

0.041

0.0470.054

0.063

/ 3 B

0.0220.024

0.0280.033

l/a = 5,Rjlt = 4

alt

00.2

0.50.8

/oB

0.6730.670

0.8031.060

/ i B

0.104

0.107

0.1510.229

/ 2 B

0.032

0.037

0.0590.095

/ 3 B

0.0160.018

0.0310.051

47

I/a = 5, R,/t = 10

alt

0

0.2

0.50.8

/oB

0.6730.6760.814

1.060

/ , B

0.104

0.1090.153

0.225

/ 2 B

0.0320.037

0.0600.092

fi0.015

0.0180.031

0.049

I/a = 10, Rs/t = 4

alt

0

0.2

0.50.8

/oB

0.5160.577

0.7591.144

/ , B

0.069

0.0750.134

0.250

f?0.017

0.022

0.0510.103

/ 3 B

0.009

0.0100.027

0.056

I/a = 10, Ri/t =10

a//

0

0.2

0.50.8

/o0.516

0.5780.7531.123

/ i B

0.0690.075

0.1310.241

/2B

0.0170.022

0.0500.099

/ 3 B

0.0090.010

0.0260.053

Table K8. Geometry functions for a finite axial internal surfacecrack in a cylinder — intersection of crack with free surface.


K\ is given by

;=3 i——

(K10)

The stress state <r= O(M) is to be taken normal to the prospective crack plane in anuncracked cylinder. The coordinate u is defined in Fig. G6.

fi(i= 1 to 3) are geometry functions which are given in Table K9 for the deepest pointof the crack (/A). See Fig. G6.

Ref: [K2].

48

alt

0

0.10.2

0.30.4

0.5

0.60.7

0.75

alt

0

0.10.2

0.30.4

0.50.60.70.75

R;lt = 0.5

/ , A

2.0002.000

2.0002.000

2.000

2.000

2.0002.0002.000

hK

1.328

0.8900.8951.032

1.329

1.7962.4573.597

4.571

/ 3 A

0.220

0.1550.193

0.2520.210

0.093

-0.074

-0.618-1.272

Ri/t = 2

/ , A

2.0002.000

2.0002.0002.000

2.0002.0002.0002.000

/2A

1.340

LJ.5192.1192.9343.8204.6925.6976.9957.656

hK

0.2190.2120.3220.5511.066

1.8532.6003.224

3.733

Ri/t=l

/ , A

2.0002.0002.000

2.000

2.0002.000

2.0002.000

2.000

/2A

1.336

1.2711.5661.997

2.5013.072

3.8074.8775.552

/ 3 A

0.2200.184

0.2370.360

0.542

0.7620.892

0.8250.786

Ri/t = 4

/ i A

2.000

2.0002.0002.0002.0002.0002.0002.0002.000

/ 2 A

1.340

1.6592.4753.6154.982

6.4557.9779.51310.24

/ 3 A

0.2190.2170.358

0.7091.4992.936

5.0187.6379.134

Table K9. Geometry functions for an infinite axial internalsurface crack in a cylinder.


K\ is given by

3

i=0, Rt 11). (Kll)

Ti (/ = 0 to 3) are stress components which define the stress state a according to

3

f o r 0 < u < a . (K12)

a is to be taken normal to the prospective crack plane in an uncracked cylinder, o; isdetermined by fitting cr to Eq. (K12). The coordinate u is defined in Fig. G7.

fi(i = 0 to 3) are geometry functions which are given in Tables K10 and Kll for thedeepest point of the crack (/*), and at the intersection of the crack with the free surface(Z8), respectively. See Fig. G7.

49



l/a = 2,Rt/t = 4

alt

00.2

0.50.8

/oA

0.659

0.6560.697

0.736

ft0.471

0.459

0.4730.495

/ 2 A

0.3870.377

0.384

0.398

/ 3 A

0.3370.327

0.331

0.342

I/a = 2, Rt/t= 10

alt

00.2

0.5

0.8

/oA

0.6590.6530.6870.712

/ i A

0.4710.457

0.4700.487

/ 2 A

0.387

0.3760.3820.394

/ 3 A

0.3370.327

0.3300.340

lla = 5,Rilt = 4

alt

00.2

0.50.8

/oA

0.9390.964

1.1831.502

/ i A

0.5800.5960.6720.795

/2A

0.4340.4610.5000.568

/ 3 A

0.3530.3870.4100.455

//a = 5,jR///=10

alt

00.20.50.8

/oA

0.9390.9531.1391.361

/ i A

0.580

0.5910.6560.746

/ 2A

0.434

0.4590.4910.543

/ 3 A

0.3530.3860.4050.439

lla = 10, Rilt = 4

alt

00.2

0.5

0.8

/oA

1.0531.107

1.562

2.390

/ , A

0.6060.6580.8201.122

/ 2 A

0.4430.499

0.5840.745

/ 3 A

0.3570.413

0.4650.568

50

I/a = 10, Rilt= 10

alt

0

0.2

0.50.8

/oA

1.0531.092

1.508

2.188

/ i A

0.6060.652

0.7991.047

/2A

0.4430.4960.5710.704

/ 3 A

0.3570.411

0.4570.541

Table K10. Geometry functions for a finite axial external surfacecrack in a cylinder — deepest point of crack.

l/a = 2,Ri/t = 4

alt

0

0.20.50.8

/oB

0.7160.741

0.8190.954

/ i B

0.1180.130

0.1550.192

/2B

0.041

0.0490.0610.078

/ 3 B

0.022

0.0260.0330.041

I/a = 2, Ri/t= 10

a/f

00.20.50.8

/oB

0.7160.7360.807

0.926

/ i B

0.118

0.1290.1500.182

f?0.041

0.0480.0590.072

/ 3 B

0.022

0.0250.031

0.038

l/a = 5,Ri/t = 4

alt

00.20.50.8

/oB

0.673

0.6900.8641.217

A*0.104

0.1130.1700.277

/ 2B

0.032

0.0390.0680.117

/ 3 B

0.015

0.0190.0360.064

lla = 5,Ri/t =10a//

00.2

0.5

0.8

/oB

0.6730.685

0.8561.198

f?0.104

0.1110.167

0.269

/ 2B

0.032

0.039

0.0660.112

/ 3 B

0.0150.019

0.035

0.061

lla = 10, R(/t = 4

alt

00.2

0.50.8

/oB

0.5160.583

0.7481.105

/ i B

0.069

0.076

0.1280.230

fB

Ji

0.0170.022

0.047

0.092

/ 3 B

0.0090.010

0.024

0.049

51

lla = 10, R,/t = 10

alt

00.2

0.5

0.8

ft0.516

0.583

0.7681.202

/ , B

0.0690.0760.1350.264

f?0.017

0.022

0.051

0.109

ff0.0090.010

0.027

0.059

Table Kl 1. Geometry functions for a finite axial external surfacecrack in a cylinder — intersection of crack with free surface.


K\ is given by

(K13)

The stress state a = o(u) is to be taken normal to the prospective crack plane in anuncracked cylinder. The coordinate u is defined in Fig. G8.

fi (/ = 1 to 4) are geometry functions which are given in Table K12 for the deepest pointof the crack (/*). See Fig. G8.

Ref.: [K2].

alt

00.10.20.30.4

0.50.60.7

0.75

R;lt = 0.5

/ i A

2.0002.0002.0002.0002.0002.000

2.0002.0002.000

fA

0.9011.3591.9332.614

3.408

4.3215.4597.1458.355

/ 3 A

1.4011.3761.3871.4221.541

1.799

2.1012.1872.112

fA

J4-0.620-0.585-0.549-0.510

-0.481-0.472

-0.456-0.361-0.265

Ri/t = 1

/ ,A

2.0002.0002.0002.0002.000

2.0002.0002.0002.000

/ 2 A

0.9011.3311.9672.766

3.7084.787

6.0557.7268.853

/ 3 A

1.4011.3651.3691.484

1.7592.2382.904

3.6013.901

fA

-0.620-0.584

-0.543-0.512

-0.505-0.528-0.577

-0.605-0.590

52

alt

00.10.2

0.30.4

0.50.60.7

0.75

R,/t = 2

ft2.0002.000

2.000

2.0002.000

2.000

2.0002.0002.000

ft0.9011.330

2.0863.0954.307

5.6437.103

8.97610.28

ft1.4011.370

1.4031.5802.054

3.004

4.3765.735

6.243

ft-0.620-0.585

-0.542

-0.510-0.524

-0.625-0.802

-0.949-0.963

Ri/t = 4

ft2.0002.0002.000

2.000

2.0002.000

2.0002.000

2.000

ft0.9001.335

2.2193.464

4.993

6.8238.984

11.1011.80

/ 3 A

1.4001.382

1.4161.658

2.4123.794

6.05110.07

13.08

ft-0.620-0.587-0.535

-0.501

-0.549-0.704

-1.011-1.674

-2.229

Table K12. Geometry functions for an infinite axial externalsurface crack in a cylinder.


K\ is given by

abfb(l/t, Rf/t)). (K14)

am and <?(, are the membrane and bending stress components, respectively, which definethe stress state a according to

a = a(u) = am + cyb 1 -2u

for 0 < u < t. (K15)

<ris to be taken normal to the prospective crack plane in an uncracked cylinder. <jm andG\) are determined by fitting <rto Eq. (K15). The coordinate u is defined in Fig. G9.

fm snAfb are geometry functions which are given in Table K13 for the intersections ofthe crack with the free surface at u = 0 (f1), and at u = t (jP). See Fig. G9.


Ref: [K6].

53

lit

024

6

81015

20

25

Rjlt= 10

/ m

1.000

0.966

1.020

1.1651.3791.643

2.4483.4014.441

fA

Jb1.000

0.973

0.9310.887

0.8450.8050.721

0.6530.597

J m

1.0001.176

1.4681.782

2.0882.371

2.9573.364

3.640

ft-1.000

-1.003-0.996-0.984

-0.970-0.954

-0.913

-0.876

-0.840

J m

1.000

0.9700.975

1.035

1.1401.281

1.749

2.3282.982

Rilt

ft1.0000.984

0.9560.924

0.8930.862

0.7920.731

0.681

= 20

J m

1.0001.105

1.2951.514

1.7391.961

2.4692.887

3.215

ft-1.000

-1.003

-1.001-0.995

-0.986-0.976

-0.948-0.917

-0.891

Table K13. Geometry functions for an axial through-thicknesscrack in a cylinder.

54



K\ is given by

jjia/t,l I a, R{ It) + abgfbg(a/ t,l I a, R( 11)). (K16)i=0

G[ (/ = 0 to 3) are stress components which define the axisymmetrical stress state aaccording to

forO<«<a , (K17)

and <jbg is the global bending stress. <rand abg are to be taken normal to the prospectivecrack plane in an uncracked cylinder, o; is determined by fitting crto Eq. (K17). Thecoordinate u is defined in Fig. G10.

fi(i = 0 to 3) andfbg are geometry functions which are given in Tables K14 and K15 forthe deepest point of the crack (f^), and at the intersection of the crack with the freesurface (Z8), respectively. See Fig. G10.

Remarks: The cylinder should be long in the transverse direction to the crack so thatedge effects do not influence.

Refs.:[Kl]and[K7].

lla = 2 , Rift = 5

alt

0

0.20.40.60.8

/oA

0.6590.6650.6820.7000.729

/ i A

0.471

0.4600.4710.4810.506

/ 2A

0.387

0.3710.3810.3900.410

/ 3 A

0.3370.3160.3270.3350.352

/40.5490.5700.6000.6320.675

lla = 2 , Rflt = 10

alt

0

0.2

0.4

0.60.8

/oA

0.6590.6640.680

0.6960.714

/ i A

0.4710.459

0.4690.4780.497

hK

0.3870.370

0.3790.3870.403

/ 3 A

0.337

0.3150.325

0.3330.347

fbg

0.599

0.6130.636

0.6590.685

55

lla = 4, Rilt= 5

alt

00.20.4

0.60.8

/oA

0.886

0.8900.934

0.9911.066

/ i A

0.565

0.5560.5760.602

0.653

ft0.4300.424

0.4400.457

0.496

ft0.3520.347

0.3620.377

0.409

fbg

0.738

0.7610.817

0.8850.973

lla = 4, R,lt= 10

a/f

00.20.40.6

0.8

/oA

0.886

0.8950.947

1.0081.482

/ i A

0.5650.557

0.5800.6050.647

/2A

0.4300.424

0.441

0.4580.492

/ 3 A

0.352

0.3470.3630.377

0.406

fAJbg

0.8060.8250.883

0.9501.012

l/a = 8,Rj/t=5

alt

00.20.4

0.60.8

/•A

1.0251.0411.1421.274

1.463

/ i A

0.6000.6250.6660.7180.813

/2A

0.441

0.4690.4960.5270.589

/ 3 A

0.356

0.3810.4030.4270.471

fbg0.8540.8900.9951.1261.310

lla = 8, R-,lt = 10

a//

00.20.4

0.60.8

/oA

1.0251.0531.1801.3351.482

/ i A

0.600

0.6290.6780.7370.814

fAJi

0.4410.4710.502

0.5360.587

/ 3 A

0.3560.3820.407

0.4310.469

fA

Jbg0.9310.9701.0971.2531.402

lla = 16, Ri/t = 5

a//

00.20.4

0.60.8

/oA

1.0791.1301.294

1.521

1.899

/ i A

0.6350.6650.732

0.8200.987

/2A

0.4730.4930.5370.587

0.690

/ 3 A

0.3880.3980.433

0.4680.541

Jbg

0.8990.964

1.120

1.321

1.633

56

I/a = 16, Ri/t= 10

alt

00.20.4

0.6

0.8

/oA

1.0791.150

1.3661.643

1.972

/ i A

0.6350.6720.756

0.8591.002

/ 2 A

0.4730.498

0.5490.6060.694

/ 3 A

0.3880.4010.441

0.4790.541

0.9811.0591.267

1.531

1.842

I/a = 32, Rt/t = 5

alt

00.20.4

0.60.8

/oA

1.1011.180

1.521

1.7072.226

/ i A

0.6580.690

0.7750.9021.137

/2A

0.4990.5120.564

0.6380.783

/ 3 A

0.4130.414

0.4530.505

0.609

0.9181.004

1.1881.4301.794

I/a = 32, Ri/t = 10

alt

00.20.40.60.8

/•AJO

1.1011.2091.4901.8872.444

/ , A

0.6580.7010.8100.9581.187

/ 2 A

0.499

L0.5180.5820.665

0.799

fA

J30.4130.4180.4640.520

0.613

1.001

1.1121.3771.7372.219

Table K14. Geometry functions for a part circumferentialinternal surface crack in a cylinder—deepest point of crack.

l/a = 2,Ri/t = 5

alt

00.20.4

0.6

0.8

/oB

0.7180.7460.7740.882

0.876

/ , B

0.1170.1250.1330.147

0.161

f?0.0410.0460.0510.0580.064

ff0.0200.0230.0260.031

0.034

fbg0.5980.6250.652

0.696

0.746

11 a = 2,Rilt =10

alt

00.20.4

0.6

0.8

ft0.7160.747

0.778

0.8310.890

/ , B

0.1160.1250.134

0.148

0.163

f?0.0410.046

0.0510.0580.064

ff0.020

0.0230.0260.031

0.033

0.6520.682

0.712

0.7630.820

57

lla = 4,Ri/t = 5

alt

00.2

0.4

0.6

0.8

/o0.664

0.716

0.7680.8520.944

/ i B

0.0910.1080.125

0.1520.179

f?0.0290.039

0.0490.062

0.075

ff0.013

0.019

0.0250.033

0.040

Jbg

0.555

0.5990.643

0.712

0.788

I/a = 4, Ri/t= 10

alt

00.20.4

0.6

0.8

ff0.657

0.7190.781

0.8830.995

/ i B

0.089

0.1090.129

0.160

0.191

/ 2B

0.0300.0400.050

0.066

0.079

ff0.014

0.0200.026

0.0350.042

Jbg

0.598

0.6560.714

0.809

0.913

//a = 8,J?/// = 5

alt

0

0.20.4

0.60.8

ff0.541

0.5980.6550.7370.846

ff0.054

0.0720.0900.116

0.151

f?0.014

0.0230.032

0.0450.062

ff0.0040.010

0.0160.0230.033

Jbg

0.4610.496

0.5310.5760.634

I/a = S,R{/t= 10

alt

0

0.20.40.60.8

/oB

0.5270.602

0.6770.7880.927

/ , B

0.0470.0720.0970.1310.172

/ 2 B

0.010

0.0230.0360.0520.070

/ 3 B

0.0020.010

0.0180.0270.037

fbg

0.4810.5470.613

0.7100.829

I/a = 16, Rj/t = 5

a//

0

0.20.4

0.6

0.8

/oB

0.417

0.4470.477

0.5280.600

/ i B

0.027

0.0370.047

0.0620.085

f?0.0040.0090.014

0.0210.032

/ 3 B

0.0000.003

0.0060.0100.017

/ %0.3810.357

0.3330.292

0.236

58

I/a = 16, Rt/t= 10

alt

00.20.4

0.60.8

/oB

0.413

0.4550.497

0.5680.670

/ , B

0.025

0.0390.0530.0730.104

0.0030.0100.017

0.0260.041

f?0.0000.004

0.0080.013

0.021

/ «0.3870.411

0.4350.475

0.531

I/a = 32, R,/t = 5

alt

0

0.20.4

0.60.8

/oB

0.2760.2940.312

0.3310.348

/ i B

0.0100.014

0.0180.023

0.026

/2B

0.0000.0020.004

0.006

0.009

/ 3 B

0.0000.000

0.0010.0030.003

fbg

0.313

0.2000.0870.056

0.276

//a = 32,J?///=10alt

00.20.40.60.8

/oB

0.2750.2980.3210.352

0.389

/ , B

0.0090.0150.021

0.0280.038

f?0.0010.0030.0050.0090.012

f?0.0000.0000.0020.004

0.006

fi0.276

0.2580.2400.2000.139

Table K15. Geometry functions for a part circumferentialinternal surface crack in a cylinder — intersection of crack withfree surface.


K\ is given by

' I - (K18)

The stress state a = a{u) is to be taken normal to the prospective crack plane in anuncracked cylinder. The coordinate u is defined in Fig. G i l .

f(i= 1 to 3) are geometry functions which are given in Table K16 for the deepest pointof the crack (/*). See Fig. Gl 1.


Ref.: [K2].

59

alt

00.10.2

0.30.4

0.5

0.6

alt

00.1

0.20.30.4

0.50.6

alt

00.10.20.30.4

0.5

0.6

Ri/t=7l3

/ i A

2.0002.0002.000

2.0002.000

2.000

2.000

/ 2A

1.327

1.3371.5431.8802.321

2.879

3.720

/ 3 A

0.2180.2000.201

0.2280.293

0.373

0.282

/?,// = 5

/ i A

2.000

2.0002.0002.000

2.0002.0002.000

/ 2 A

1.336

1.460

1.8392.3592.9763.688

4.598

fA.

0.2180.2060.241

0.3530.5560.8371.086

Rilt= 10

/ i A

2.0002.0002.0002.0002.000

2.0002.000

/ 2A

1.346

1.5912.1832.9663.8764.888

5.970

/ 3 A

0.2190.211

0.2790.5180.9561.614

2.543

Table K16. Geometry functions for a complete circumferentialinternal surface crack in a cylinder.

60


K\ is given by

K} =-J^{£aifi(a/t,lfa,Rii=0

a, R; 11)). (K19)

a, (i = 0 to 3) are stress components which define the axisymmetrical stress state aaccording to

3

i=0

forO< w<a, (K20)

and <7bg is the global bending stress, a and a^g are to be taken normal to the prospectivecrack plane in an uncracked cylinder. a\ is determined by fitting a to Eq. (K20). Thecoordinate u is defined in Fig. G12.

fi (i = 0 to 3) and fbg are geometry functions which are given in Tables K17 and K18 forthe deepest point of the crack (/*), and at the intersection of the crack with the freesurface (Z8), respectively. See Fig. G12.



I/a = 2,Ri/t = 5

alt

00.20.4

0.6

0.8

/oA

0.6590.6610.6730.686

0.690

/ , A

0.4710.4550.4620.467

0.477

/ 2A

0.3870.3670.374

0.3780.387

/ 3 A

0.3370.3130.3210.3250.333

K0.6590.6450.642

0.6380.626

I/a = 2, Rilt= 10

alt

00.20.4

0.60.8

/oA

0.6590.662

0.676

0.6900.695

/ i A

0.4710.4560.464

0.4700.482

/ 2 A

0.387

0.3680.376

0.3810.392

/ 3 A

0.3370.313

0.322

0.3280.337

Jbg

0.6590.653

0.6590.664

0.660

61

lla = 4,/?,/* = 5

alt

0

0.2

0.4

0.60.8

/«A

0.886

0.9050.972

1.0601.133

/ , A

0.565

0.5600.586

0.6180.659

/2A

0.430

0.4250.443

0.4620.493

/ 3 A

0.3520.347

0.363

0.3780.403

0.8860.8850.932

0.9951.041

//a = 4,1?///=10

alt

0

0.20.4

0.6

0.8

/oA

0.886

0.9030.9691.051

1.108

/ i A

0.565

0.5590.5860.6160.654

/ 2 A

0.4300.425

0.4430.4620.491

/ 3 A

0.352

0.347

0.3630.3780.403

0.886

0.8910.9471.016

1.059

lla = 8, Rj/t= 5

alt

0

0.20.40.6

0.8

/oA

1.0251.0781.2531.502

1.773

/ , A

0.6000.6380.7020.7900.900

/ 2 A

0.441

0.4760.5130.561

0.625

/ 3 A

0.356

0.3860.4130.4460.490

fbg

1.0251.0551.2021.4131.631

//a = 8,J?f//=10

00.20.4

0.60.8

/oA

1.0251.0731.2461.4891.711

/ i A

0.6000.6370.7000.7860.880

ft0.441

0.4750.512

0.5590.616

/ 3 A

0.356~10.3860.4130.4450.484

fbg

1.0251.0601.2191.4431.640

lla = 16, Rilt = 5

a/f

00.20.4

0.60.8

/oA

1.0791.1861.4821.907

2.461

/ i A

0.635

0.6850.797

0.951

1.166

/2A

0.4730.504

0.5700.654

0.776

/ 3 A

0.3880.4060.454

0.5080.591

/41.0791.1621.4191.779

2.220

62

I/a = 16, R,/t= 10

alt

00.20.4

0.6

0.8

ft1.0791.182

1.491

1.9492.479

ft0.6350.684

0.8000.9621.165

ft0.4730.504

0.571

0.6580.772

ft0.3880.4050.454

0.5110.587

fbg1.079

1.168

1.4581.8832.363

I/a = 32, Ri/t = 5

ait

00.20.4

0.60.8

ft1.1011.252

1.5992.067

2.740

ft0.6580.7160.854

1.0361.313

ft0.4990.5250.607

0.7130.875

ft0.4130.4220.4820.555

0.666

1.1011.2251.525

1.9262.491

//a = 32,*///=10alt

00.20.40.6

0.8

ft1.1011.2521.6512.2433.011

/ i A

0.6580.7160.8691.0891.387

fAJ2

0.4990.5250.6140.7360.904

ft0.4130.4210.4850.566

0.678

1.1011.2371.6112.157

2.845

Table K17. Geometry functions for a part circumferentialexternal surface crack in a cylinder — deepest point of crack.

lfa = 2,Rjlt = 5

alt

00.20.40.60.8

/oB

0.7150.7480.781

0.8370.905

/ i B

0.1170.1250.1330.1470.163

f?0.0400.0450.0500.0570.063

fBJ3

0.0200.0230.0260.0300.033

0.7170.744

0.7710.8210.880

lla = 2, Ri/t= 10

alt

00.20.4

0.60.8

ft0.7130.748

0.7830.8410.912

/ i B

0.117

0.125

0.1330.1490.166

f?0.0410.046

0.0510.0580.064

f?0.0200.023

0.0260.0300.033

/ ;0.7130.7450.777

0.8320.898

63

I/a = 4, Ri/t = 5

alt

00.2

0.4

0.6

0.8

ff0.6540.724

0.794

0.915

1.059

/ , B

0.0880.1100.132

0.168

0.208

ff0.0280.0400.052

0.069

0.087

/ 3 B

0.013

0.0200.027

0.037

0.046

fBJbg

0.657

0.7190.781

0.8881.012

Ua = 4, Rilt= 10

alt

0

0.20.4

0.60.8

/<?0.649

0.7230.797

0.925

1.081

/ , B

0.087

0.110

0.1330.1720.215

/ 2 B

0.028

0.0400.0520.071

0.089

/ 3 B

0.013

0.0200.027

0.0380.048

fbg0.649

0.720

0.7910.9121.058

l/a = S,Ri/t = 5

alt

0

0.20.4

0.60.8

/oB

0.527

0.6100.6930.8180.972

/ i B

0.0470.074

0.101

0.1390.185

/ 2 B

0.0100.024

0.0380.0550.077

/ 3 B

0.0030.011

0.0190.0290.041

0.537

0.6030.6690.7620.868

lla = %,Rilt =10

alt

00.20.4

0.60.8

/oB

0.5180.6100.702

0.8561.060

/ , B

0.0430.074

0.1050.1520.211

/ 2 B

0.0090.024

0.0390.0620.088

/ 3 B

0.0020.011

0.0200.0330.047

/ *0.5210.607

0.6930.8341.019

lla = 16, Ri/t = 5

alt

00.20.4

0.6

0.8

/oB

0.425

0.4590.493

0.5290.542

A*0.0290.040

0.050

0.0580.057

/ 2B

0.004

0.010

0.016

0.0180.016

/ 3 B

0.001

0.004

0.007

0.0080.006

0.454

0.4430.432

0.3900.294

64

If a = 16, Rilt= 10

alt

00.20.4

0.60.8

/oB

0.4090.461

0.5130.589

0.671

/ , B

0.0230.040

0.057

0.0780.099

f?0.003

0.011

0.0190.0280.037

/ 3 B

0.0000.004

0.009

0.014

0.018

0.4170.455

0.4930.542

0.582

I/a = 32, Ri/t = 5

alt

00.2

0.4

0.60.8

/oB

0.3070.306

0.305

0.2990.292

/ i B

0.017

0.0160.014

0.0080.003

/ 2B

0.0050.003

0.0010.000

0.000

/ 3 B

0.0000.000

0.000

0.0000.000

7BJb%

0.3790.265

0.1510.024

0.255

I/a = 32, Ri/t= 10

a//

00.20.4

0.60.8

ft0.2990.3090.3190.3220.305

/ i B

0.021

0.0200.0190.0160.005

f?0.002

0.0030.0040.002

0.000

f?0.0000.0000.0000.0000.000

fi0.3230.2960.2690.2080.103

Table K18. Geometry functions for a part circumferentialexternal surface crack in a cylinder — intersection of crack withfree surface.


K\ is given by

a

3

(K21)

The stress state a = o(u) is to be taken normal to the prospective crack plane in anuncracked cylinder. The coordinate u is defined in Fig. G13.

fi(i=l to 3) are geometry functions which are given in Table K19 for the deepest pointof the crack (/*). See Fig. G13.


Ref: [K2].

65

alt

0

0.10.2

0.30.4

0.5

0.6

alt

0

0.10.20.30.4

0.5

0.6

alt

00.1

0.20.30.40.50.6

Rilt= 7/3

/ , A

2.000

2.0002.000

2.0002.0002.000

2.000

ft1.3591.642

2.1272.7273.4314.271

5.406

/ 3 A

0.220

0.2360.3070.447

0.6680.951

1.183

Rift = 5

AA

2.0002.0002.0002.000

2.000

2.0002.000

/2A

1.3621.6592.2202.904

3.701

4.6035.671

/ 3 A

0.2210.221

0.3030.5350.8571.3111.851

!?,/*= 10

/ i A

2.0002.0002.0002.0002.0002.0002.000

/ 2 A

1.364

1.6942.3753.2364.2525.334

6.606

/ 3 A

0.2200.2110.310

0.6301.1361.9722.902

Table K19. Geometry functions for a complete circumferentialexternal surface crack in a cylinder.

66


K\ is given by

(K22)

cjm, (Jb and (Tbg are the membrane, through-thickness bending and global bending stresscomponents, respectively. am, and ab define the axisymmetrical stress state a accordingto

2Mfor 0 < u < t. (K23)

<j and <Jbg are to be taken normal to the prospective crack plane in an uncrackedcylinder. <ym and ay are determined by fitting a to Eq. (K23). The coordinate u isdefined in Fig. G14.

fmfb andfhg are geometry functions which are given in Tables K20 and K21 for theintersections of the crack with the free surface at u - 0 (f^), and at u = t (Z8),respectively. See Fig. G14.


Refs.: [K8] and [K9].

1/nRi

00.10.2

0.30.40.50.60.7

0.80.9

1

R,/t = 5

J m

1.000

1.0491.1381.2571.4041.5851.8052.076

2.4102.825

3.341

A A

0.9570.6880.531

0.4190.3440.287

0.2460.2160.189

0.165

0.138

1.0001.0421.118

1.2181.3371.4731.627

1.799

1.9902.201

2.435

Rilt = 20/3

J m

1.0001.0531.152

1.2831.4461.6441.8872.186

2.5553.012

3.580

ft0.9290.6620.5020.3930.3220.2720.237

0.2100.188

0.1650.141

fbg

1.0001.0461.130

1.2401.3711.522

1.6911.881

2.0912.324

2.582

67

1/nRi

0

0.10.2

0.30.4

0.5

0.6

0.7

0.80.91

R,/t= 10fA

J m1.0001.062

1.1781.3301.520

1.752

2.0362.384

2.8153.3494.012

ft0.8880.623

0.4590.3560.285

0.252

0.2230.202

0.1810.1630.141

fAJbg

1.0001.054

1.1521.280

1.443

1.6091.807

2.0282.274

2.5462.847

Rilt = 20

J m

1.0001.090

1.2561.477

1.7512.086

2.4952.997

3.6194.389

5.346

ft0.8620.554

0.3860.297

0.249

0.2200.200

0.182

0.1660.1480.128

1.000

1.0781.220

1.4051.625

1.8792.1652.484

2.8383.2303.665

Table K20. Geometry functions for a circumferential through-thickness crack in a cylinder — intersection of crack with freesurface at u = 0.

VnRi

00.10.2

0.30.4

0.50.6

0.70.80.91

Ri/t=5

J m

1.0001.0491.1381.2571.4041.5851.805

2.0762.4102.8253.341

f?-0.964

-0.749-0.628-0.548-0.478-0.432

-0.400

-0.381-0.369-0.363-0.363

fbg

1.0001.0421.1181.2181.3371.4731.6271.7991.9902.2012.435

Ri/t = 20/3

J m

1.0001.0531.152

1.2831.4461.644

1.887

2.1862.5553.0123.580

f?-0.925-0.722-0.597-0.507-0.441

-0.398-0.368-0.347-0.337-0.330

-0.330

Jbg

1.0001.046

1.1301.2401.3711.522

1.6911.8812.0912.3242.582

68

1/nRj

00.1

0.2

0.30.4

0.50.60.7

0.8

0.91

/?,/*= 10

J m

1.0001.062

1.1781.330

1.5201.752

2.0362.384

2.815

3.3494.012

Jb-0.890-0.680

-0.545-0.457

-0.396

-0.356-0.330

-0.311-0.300-0.292-0.294

fi1.0001.054

1.152

1.2801.443

1.6091.8072.0282.274

2.5462.847

Ri/t = 20

J m

1.000

1.0901.256

1.477

1.751

2.0862.4952.997

3.6194.389

5.346

A*-0.843-0.620-0.474

-0.382

-0.328

-0.297-0.278

-0.263-0.253

-0.248-0.248

fbg

1.000

1.0781.220

1.405

1.6251.8792.1652.484

2.838

3.230

3.665

Table K21. Geometry functions for a circumferential through-thickness crack in a cylinder — intersection of crack with freesurface at u = t.

69


a) Through-thickness crack

K\ is given by

K x = - f i d i 2 ( ( j m f m { l l t , R j / O + (K24)

am and <Tb are the membrane and through-thickness bending stress components,respectively, which define the axisymmetrical stress state a according to

for 0 < u < t. (K25)

cris to be taken normal to the prospective crack plane in an uncracked sphere. am andare determined by fitting crto Eq. (K25). The coordinate u is defined in Fig. G15.

fm an&fb are geometry functions which are given in Table K22 for the intersections ofthe crack with the free surface at u = 0 (/*) and at u = t (J8). See Fig. Gl 5.

Ref: [K6].

lit

024

68

10

1520

Ri/t= 10

fA

J m1.000

0.9190.8940.944

1.0591.231

1.9152.968

ft1.0000.9930.9930.9971.0031.0111.031

1.050

J m

1.000

1.2401.6372.0832.5493.0164.1245.084

-1.000-1.031-1.074-1.111

-1.143-1.170-1.226-1.272

Ri/t = 20

J m

1.0000.9410.8970.8950.9321.003

1.3091.799

A A

1.0000.9950.992

0.9930.9961.0011.014

1.028

J m

1.0001.1441.401

1.7002.0202.351

3.1863.981

f?-1.000

-1.020-1.050-1.080-1.106-1.130-1.180-1.219

Table K22. Geometry functions for a through-thickness crack ina sphere.

70

References

[Kl] FETT T., MUNZ D. and J. NEUMANN, "Local stress intensity factors forsurface cracks in plates under power-shaped stress distributions." EngineeringFracture Mechanics. Vol. 36, No. 4, 647-651, 1990.

[K2] WU X. R., and A. J. CARLSSON, Weight functions and stress intensity factorsolutions. Pergamon Press, Oxford, U.K., 1991.

[K3] ZVEZDIN Y. I. et. al, "Handbook - Stress intensity and reduction factorscalculation." MR 125-01-90, Central Research Institute for Technology ofMachinery, Moscow, Russia, 1990.

[K4] SIH, G. C , PARIS, P. F. and F. ERDOGAN, "Stress intensity factors for planeextension and plate bending problems." Journal of Applied Mechanics. Vol. 29,pp. 306-312,1962.

[K5] RAJU, I. S., and J. C. NEWMAN, "Stress intensity factor influence coefficientsfor internal and external surface cracks in cylindrical vessels.". ASME PVP. Vol.58, pp. 37-48,1978.

[K6] ERDOGAN, F., and J. J. KIBLER, "Cylindrical and spherical shells withcracks." International Journal of Fracture Mechanics. Vol. 5, pp.229-237, 1969.

[K7] BERGMAN, M., "Stress intensity factors for circumferential surface cracks inpipes." Fatigue & Fracture of Engineering Materials & Structures. Vol. 18, No.10, pp.1155-1172,1995.

[K8] ZAHOOR, A., "Closed form expressions for fracture mechanics analysis ofcracked pipes." ASME PVP. Vol. 107, pp. 203-205, 1985.

[K9] SATTARI-FAR, I., "Stress intensity factors for circumferential through-thickness cracks in cylinders subjected to local bending". International Journalof Fracture. Vol. 53, pp. R9-R13,1992.

71

Appendix L. Limit load solutions



Lr is given by

~Lr = n\\-

where

,0.75

T T , <L2)

(L3)

See Fig. Gl. <jm and cr̂ are the membrane and bending stress components, respectively,which define the stress state ex according to

<N<f. (L4)

cris to be taken normal to the prospective crack plane in an uncracked plate. <jm and cr̂are determined by fitting <rto Eq. (L4). The coordinate u is defined in Fig. Gl.

Remarks: The solution is limited to alt < 0.8. Also, the plate should be large incomparison to the length of the crack so that edge effects do not influence.

Ref.: [LI].


Lr is given by

,2

3

where

„ a(L6)

72

See Fig. G2. am and c^ are the membrane and bending stress components, respectively,which define the stress state cr according to

- — I forO<«</. (L7)

<r is to be taken normal to the prospective crack plane in an uncracked plate. am and CFJ,are determined by fitting crto Eq. (L7). The coordinate u is defined in Fig. G2.

Remarks: The solution is limited to alt < 0.8. Also, the plate should be large in thetransverse direction to the crack so that edge effects do not influence.

Ref: [L2].

c) Embedded crack

Lr is given by

i(L8)

where

?nl(L9)

r = 5-7-7- (LIO)

See Fig. G3. crm and oj, are the membrane and bending stress components, respectively,which define the stress state cr according to

forO<w<r. (Lll)

cr is to be taken normal to the prospective crack plane in an uncracked plate. am and c%are determined by fitting crto Eq. (LI 1). The coordinate u is defined in Fig. Gl.

Remarks: The solution is limited to aft < 0.8 and elt > 0. Also, the plate should be largein comparison to the length of the crack so that edge effects do not influence.

Ref.: [L2].

73


Lr is given by

(L12)4

See Fig. G4. <jm and o& are the membrane and bending stress componentsrespectively, which define the stress state a according to

forO<u<t. (L13)

cr is to be taken normal to the prospective crack plane in an uncracked plate. <jm and at,are determined by fitting crto Eq. (LI3). The coordinate u is defined in Fig. G4.


74



Lr is given by

Lr = V h , (L14)

(l-Q oY

where

\J , (L15)

( L 1 6 )

See Fig. G5. am and <T& are the membrane and bending stress components, respectively,which define the stress state o according to

f 2u)cr = cr(u) = am + ab\ 1 1 forO<«</. (L17)

cr is to be taken normal to the prospective crack plane in an uncracked cylinder. <jm andOb are determined by fitting c to Eq. (LI 7). The coordinate u is defined in Fig. G5.

Remarks: The solution is limited to alt < 0.8. Also, the cylinder should be long incomparison to the length of the crack so that edge effects do not influence.

Ref.: [LI].


Lr is given by

Lr= ' ( 1 _ o 2 g '

where

G = ~. (L19)

75

See Fig. G6. crm and oi, are the membrane and bending stress components, respectively,which define the stress state cr according to

a = cr(u) = <jm + abl 1 - — I forO<«</ . (L20)

cris to be taken normal to the prospective crack plane in an uncracked cylinder. <jm andOb are determined by fitting crto Eq. (L20). The coordinate u is defined in Fig. G6.

Remarks: The solution is limited to alt < 0.8.

Ref.: [L2].


Lr is given by

- ~ , (L21){1-0 <?y

where

(L22)

£=~~rr- (L23)

See Fig. G7. crm and crb are the membrane and bending stress components, respectively,which define the stress state a according to

( 2u\a = a(u) = am + ab\l-— forO<«<r. (L24)

V t /

a is to be taken normal to the prospective crack plane in an uncracked cylinder. crm and<Jb are determined by fitting crto Eq. (L24). The coordinate u is defined in Fig. G7.

Remarks: The solution is limited to alt < 0.8. Also, the cylinder should be long incomparison to the length of the crack so that edge effects do not influence.

Ref.: [LI].

76


Lr is given by

,2

3

where

(L25)

(L26)

See Fig. G8. crm and oj> are the membrane and bending stress components, respectively,which define the stress state a according to

cr=<j(u) = am+crb\\- — \ forO<u<t. (L27)

a is to be taken normal to the prospective crack plane in an uncracked cylinder. crm and<Jb are determined by fitting crto Eq. (L27). The coordinate u is defined in Fig. G8.

Remarks: The solution is limited to alt < 0.8.

Ref.: [L2].


Lr is given by

aY

where

(L28)

R,t(L29)

See Fig. G9. crm is the membrane stress component, which defines the stress state craccording to

a = cr(u) = crm for 0 < u < t. (L30)

a is to be taken normal to the prospective crack plane in an uncracked cylinder. am isdetermined by fitting <rto Eq. (L30). The coordinate u is defined in Fig. G9.


Ref.: [L3].

77



Lr is given by

(L31)

where the parameters sm, s'm, sbg and s'bg are obtained by solving the equation system

Gy n t 7i

sbg 4 2 a .— = —sin /? - s i n aGy 71 71 r

^ =I (L32)

a =

^ m = ^ for 5 ^ = 0

for5m=0*te = ^

See Fig. G10. am and abg are the membrane and global bending stress components,respectively. am defines the axisymmetrical stress state a according to

a=cr(u) = for 0 < u < t. (L33)

G is to be taken normal to the prospective crack plane in an uncracked cylinder. am isdetermined by fitting <rto Eq. (L33). The coordinate u is defined in Fig. G10.

Remarks: The cylinder should be thin-walled. Also, the cylinder should be long in thetransverse direction to the crack so that edge effects do not influence.

Ref.: [L4].

78


Lr is given by

uj,2 ' (L34)

Sbg.


aY

sbg

K t

lit-a71 /

n

sin/? (L35)

- ahasm = 0

sm =

~ sbg for5m=0

See Fig. G i l . <jm and oyg are the membrane and global bending stress components,respectively. <ym defines the axisymmetrical stress state <yaccording to

a = a(u) = an for 0 < u < t. (L36)

a is to be taken normal to the prospective crack plane in an uncracked cylinder. am isdetermined by fitting a to Eq. (L36). The coordinate u is defined in Fig. G i l .


Ref: [L4].

79


Lr is given by

•m

(L37)


aY

6 =

s b g 4 2 a .= — s n i p - sin or

<Ty 71 7t /

(L38)

a =

2(/?,+0

- sbg

for j f e g = 0

for s = 0

See Fig. G12. am and cr^g are the membrane and global bending stress components,respectively. am defines the axisymmetrical stress state <raccording to

a- = a(u) = an for 0 < u < t. (L39)

a is to be taken normal to the prospective crack plane in an uncracked cylinder. am isdetermined by fitting crto Eq. (L39). The coordinate u is defined in Fig. G12.


Ref: [L4].

80


Lr is given by

U(L40)

Mwhere the parameters sm, s'm, s^ and s\,g are obtained by solving the equation system

* , ^P an-/37C t 71

Sbg 2 It-a-——-sinjt?aY

(L41)

= 0

^ *5

r = 0

= 0

See Fig. G13. am and Obg are the membrane and global bending stress components,respectively. am defines the axisymmetrical stress state a according to

for 0 < u < t. (L42)

a is to be taken normal to the prospective crack plane in an uncracked cylinder. am isdetermined by fitting a to Eq. (L42). The coordinate u is defined in Fig. G13.


Ref.: [L4].

81


Lr is given by

Lr = (L43)

+Sbg

Ag.


sbg 4 2= —sin/?- —

- y 71 n

CTy n TC

2R,(L44)

for sbg = 0

for5 m =0

See Fig. G14. am and <jbg are the membrane and global bending stress components,respectively. am defines the axisymmetrical stress state a according to

forO<u<t. (L45)

a is to be taken normal to the prospective crack plane in an uncracked cylinder. <jm isdetermined by fitting crto Eq. (L45). The coordinate u is defined in Fig. G14.


Ref.: [L4].

82


Lr is given

Lr

where

e=

by

0

c

/

a)

rmTY

I

(*.

Through-thickness

l + yi + 8(/l/cos^

2

crack

)2

(L46)

(L47)

(L48)

See Fig. G15. <xm is the membrane stress components. crm defines the axisymmetricalstress state cr according to

a = a(u) = am for 0 < u < t. (L49)

cr is to be taken normal to the prospective crack plane in an uncracked sphere. erm isdetermined by fitting crto Eq. (L49). The coordinate u is defined in Fig. G15.

Remarks: The sphere should be thin-walled.

Ref.: [L5].

References

[LI] SATTARI-FAR, I., "Finite element analysis of limit loads for surface cracks inplates." The International Journal of Pressure Vessels and Piping. Vol. 57, pp.237-243, 1994.

[L2] WILLOUGHBY, A. A. and T. G. DAVEY, "Plastic collapse in part-wall flawsin plates." ASTM STP 1020. American Society for Testing and Materials,Philadelphia, U.S.A., pp. 390-409, 1989.

[L3] KIEFNER, J. F. MAXEY, W. A., EIBER, R. J. and A. R. DUFFY, "Failurestress levels of flaws in pressurized cylinders." ASTM STP 536. AmericanSociety for Testing and Materials, Philadelphia, U.S.A., pp. 461-481, 1973.

[L4] DELFIN, P., "Limit load solutions for cylinders with circumferential crackssubjected to tension and bending." SAQ/FoU-Report 96/05, SAQ Kontroll AB,Stockholm, Sweden, 1996.

[L5] BURDEKIN, F. M. and T. E. TAYLOR, "Fracture in spherical vessels." Journalof Mechanical Engineering and Science. Vol. 11, pp. 486-497, 1969.

84

Appendix M. Material data for nuclear applications

In order to perform fracture mechanics assessments according to this handbookknowledge about yield strength cry, ultimate tensile strength G\J and critical stressintensity factor Kcr is needed. In addition, crack growth calculations require 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 materials are lacking. In manycases the data given below are conservative estimates.

1. Yield strength, ultimate tensile strength

For many steels used in nuclear applications information about minimum levels of cryand <J\J as functions of the temperature can be found in ASME Sect. Ill, Appendices.For other materials minimum levels are in general specified by the respectivemanufacturer. In cases where higher values than the minimum specified can be verified,actual data may be used. In some cases, as for fracture assessments according to ASMESect. XI, Appendix C, only minimum yield stress data are allowed.

2. Fracture toughness

In cases when the fracture toughness could not be directly determined, Jjc-dataconverted according to Eq. (3), Chapter 2.5 has been used. In this case an elasticmodulus of 180000 MPa for stainless steels at 288 °C has been used and 195000 MPafor ferritic materials on the upper shelf region. Poisson's ratio has been set to 0.3 in allcases except for nickel base alloys.

a) Ferritic steel, pressure vessels

For the materials SA-533 Grade B Class 1, SA-508 Class 2 and SA-508 Class 3 thefracture toughness K\a (conservative fracture toughness value at crack arrest) is given asa function of the difference between actual temperature T and the nil ductility transitiontemperature RTNDT™ ASME Sect. XI, Appendix A, Fig. A-4200-1. For temperaturesabove the transition region higher values man 220 MPaVm are usually not assumed.This so called upper-shelf level is assumed to be reached for temperatures 102 °C aboveRTXDT- Note that neutron irradiation can decrease this level and also increase RTHDT-Fig. A-4200-1 corresponds to the analytic expression ( r in °C)

Kla = 29A3, + 1.355exp[0.0261(r- RTNDT) + 2.32] MPaVm. (Ml)

85

b) Ferritic steel, pipes

The following data are taken from [Ml]. They can be used in the absence of heatspecific data. They are also referenced by ASME Sect. XI, Appendix H and are intendedfor the following material categories:

Category 1: Seamless or welded carbon steel piping with a minimum yield strengthlower than or equal to 276 MPa (base material) and welds with electrodesof type E7015, E7016 or E7018 (basic electrodes with a yield strength ofthe order of 500 MPa, Charpy-V toughness 27 J at -29 °C and weldable inall welding positions).

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

The table below differentiates between temperatures above or below the upper-shelfregion. In addition, in [Ml] it is distinguished between circumferential and axial cracks.The fracture properties are often worse for cracks orientated along the texture direction(axial cracks) than for cracks orientated across the texture direction (circumferentialcracks). In the absence of heat specific data, the upper shelf temperature for class 1ferritic piping steels can be taken as 93 °C.

Materialcategory

1122

1,21,2

Temperatureinterval

T > upper shelfT< upper shelfT > upper shelfT < upper shelfT> upper shelfT < upper shelf

Crackorientation

CircumferentialCircumferentialCircumferentialCircumferential

AxialAxial

Jlc (kJ/m2)

1057.9

61.37.952.5

7.9

tfcr (MPaVm)

150421154210642

Table Ml. Fracture toughness data for carbon steel base metalsand weldments, Ref. [Ml].

c) Austenitic stainless steel, pipes

The following data are mainly taken from [M2]. The materials are base material of type304 or 316 and shielded metal arc welds (SMAW) or submerged arc welds (SAW). ForSAW, data are also taken from tests from the database [M3]. Together with [M2] thevalue of J\c for SAW at 288 °C in Table M2 is a mean between 8 different tests. Themean value for the corresponding fracture toughness of 122 MPaVm for SAW has astandard deviation of 21.5 MPaVm. For SMAW, data are more scarce. Some data can befound in [M2], [M4] and [M5]. In general the fracture toughness data for SMAW arebetter than for SAW.

86

Material typeBase material

SMAW

SMAWSAWSAW

Temperature

288 °C20 °C

288 °C20 °C

288 °C

Jic (kJ/m2)>620259168

9976

Kcr (MPaVm)

>350239182

148122

Table M2. Fracture toughness data for austenitic stainless steel,Refs. [M2] and [M3J.

The heat affected zone (HAZ) often shows better properties than the weld materialaccording to Ref. [M2]. Fracture toughness data for TIG-welds at 288 °C are onlyslightly below those of the base material.

It should be noted that for these kind of materials, the ./-resistance curve is often sosteep that it does not always intersect the blunting line so that a valid J\c can bedetermined according to ASTM E-813, see Ref. [M2]. According to [M2], at 2 mm ofstable crack growth, the J-resistance values are 385 and 206 kJ/m2 at 288 °C for SMAWand SAW, respectively.

Stainless steel materials are embrittled when subjected to a sufficiently high neutronirradiation. J-resistance curves for irradiated stainless steels can be found e. g. in [M6]and [M7].

d) Cast stainless steel

Cast stainless steels are normally ductile and have a fracture toughness of the same levelas the stainless steel base material in Table M2. However, cast duplex stainless steelsare subjected to thermal ageing which causes an embrittlement of the ferritic phase. Thedegree of embrittlement is determined mainly by the ferrite content, ageing time andexposure temperature. For a ferrite content of less than 10%, the risk of embrittlementshould be small for components aged at about 300 °C. A lower bound value for J\c of100 kJ/m2 has been measured for highly embrittled materials. A model to estimate J-resistance curves from knowledge of the impact energy is given in [M8]. Fracturetoughness data are also reported in [M9], [M10] and [Mil]. One of many literaturesurveys can be found in [Ml2].

e) Stainless steel cladding

Many ferritic pressure vessels and piping used for nuclear applications are cladded witha stainless steel layer. Fracture properties for these materials are scarce. J-resistancecurves for three-wire series-arc weld overlay cladding with combinations of type 304,308 and 309 stainless steel are reported in [Ml3]. Unirradiated fracture toughness data(mean values of three tests) are given in Table M3. Fracture toughness data for claddingare also reported in [Ml4] at 60 °C which is of the same order as those in [Ml3]. In[Ml 3], the influence from neutron irradiation on the fracture toughness properties of"thecladding is also quantified. At 2 mm of stable crack growth, a J-resistance value of 450kJ/m2 at 120 °C is reported in [Ml3] in the unirradiated condition, which also agreeswell with [M14] at 60 °C.

87

Temperature

20 °C120 °C288 °C

Jlc (kJ/m2)

157132

75

Kcr (MPaVm)

186167

122

Table M3. Fracture toughness data for unirradiated stainlesssteel cladding, Ref. [Ml3].

f) Nickel base alloys

Very few fracture toughness data for nickel base alloys are found in the literature. Alloy600 is normally very ductile and should have a fracture toughness of the same level asthe stainless steel base material in Table M2. For the weld material Alloy 182, roomtemperature data are published in [Ml5] and [Ml6]. In [Ml7] data at 285 °C is alsoreported. Table M4 summarizes the results. It should be noted that formally thethickness requirement according to ASTM E-813 was not able to fulfil in the testing in[Ml6] and [Ml 7]. The values of J\c have been determined as the value of the ./-integralat which the size requirements are barely fulfilled. This will be a lower bound of J\c .The ductility is very good for larger amounts of stable crack growth. According to[Ml 7], at 2 mm of stable crack growth, the ./-resistance value for Alloy 182 amounts to860kJ/m2at285°C.

Temperature

20 °C285 °C

Jic (kJ/m2)

180200

Kcr (MPaVm)

200205

Table M4. Fracture toughness data for Alloy 182, Refs. [Ml6]and [Ml7].

3. Crack growth data, fatigue

Fatigue crack growth data for nuclear applications according to the 1995 edition ofASME Sect. XI are presented below. Data for other materials and environments can befound in Ref. [Ml8].

Generally the fatigue crack growth rate is written as

-£r = C(AK1)H mm /cycle,

o/v(M2)

where C and n ait constants, and AATj is the range of applied stress intensity factorduring the load cycle, i.e.

AK{ =K{naK-K{n[n. (M3)

Beside the environment, C, and n may also depend on the so-called /?-value defined as

88

(M4)

Both R and AKi may be corrected with respect to a compressive phase during a loadcycle. Below K\ is input in MPaVm which according to Eq. M2 gives the growth rate inmm/cycle as output.

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 Sect. XI, Article A-4000 for cracks both inair and reactor water environment.

Values of C, n and AJq are given in Table M5 for air environment.

R

R<-2

-2<R<0

R>0

n

3.07

3.073.07

COmm/CMPaVm)")

3.785-10"9

3.785-10"9

9.734-10-8(2.88-J?)-307

AJTi (MPaVm)

(1 - R)K^

3r/-max

Ar - A ,

Table M5. Crack growth data for fatigue of ferritic steel in airenvironment.

For reactor water environment, R is not allowed to fall below 0. AK\ is entitled low orhigh depending on if AK\ falls below or exceeds the parameter K defined in Table M6.Values of C, n are given in Table M7.

0 <

0.25

0.6

R

R<

<R

0.25

<0.65

? < 1

K (MPaVm)19.49

J 3.75/? + 0.06 y25

19-49l26.9/?-5.725J13.23

Table M6. Definition of parameter K.

89

Low AiTi-values — AAi < K

if

0<R<0.250.25 <J?< 0.650.65 <R< 1.0

n

5.955.955.95

C (mm/(MPaVm)")

1.479-10"11

1.47910-n(26.9i?-5.725)

1.739-10"10

High Airrvalues — tsK\ > KR

0<tf<0.250.25 <R<0.650.65 </?< 1.0

n

1.951.951.95

C (mm/CMPaVm)")

2.135-10"6

2.135-10"6(3.75i? + 0.06)

5.337-10"6

Table M7. Crack growth data for fatigue of ferritic steel inreactor water environment.

The fatigue crack growth rate for reactor water environment is not allowed to fall belowthe one for air environment.

b) Austenitic stainless steel

Fatigue crack growth data for austenitic stainless steel including welds and castings inair environment are given in ASME Sect. XI, Article C-3000. n and C are definedaccording to

n = 3.3,

C= 1

(M5)

(M6)

where T is the temperature in °C and Q a scaling parameter which takes influence of thei?-ratio into account. Q is defined in Table M8.

RR<0

0<R< 0.790.79</?<l

Ql

1 + \.$R-43.35 + 57.97i?

Table M8. Parameter Q.

For austenitic stainless steel in reactor water environment, data can be found in Ref.[M18].

4. Crack growth data, stress corrosion

Recommended upper bound stress corrosion crack growth data for different materialsand environments can be found in Ref. [Ml9]. These are intended for use in Swedishnuclear power plants when more specific data are not available. Further data can befound in Ref. [M20] which contains a survey of stress corrosion crack growth rates indifferent materials in BWR environments.

90

References

[M1 ] NORRIS, D. M , "Evaluation of flaws in ferritic piping." Final Report EPRI NP-6045, EPRI Research Institute, U.S.A., 1988.

[M2] LANDES, J. D. and D. E. McCABE. "Toughness of austenitic stainless steelpipe welds." Topical Report EPRI NP-4768, EPRI Research Institute, U.S.A.,1986.

[M3] WILKOWSKI, G., "Database CIRCUMCK.WK1, Version 2.2. Database ofcircumferentially cracked pipe fracture experiments." BATTELLE, Columbus,U.S.A., 1995.

[M4] MILLS, J. W., "Fracture toughness of stainless steel welds." ASTM STP 969,ASTM, Philadelphia, U.S.A., pp. 330-355,1988.

[M5] GUDAS, J., P. and D. R. ANDERSON., "J-R characteristics of piping materialand welds." NUREG/CP0024, USNRC, Washington, DC, U.S.A., 1982.

[M6] MILLS, J. W., "Fracture toughness of irradiated stainless steel alloys." NuclearTechnology. Vol. 82, pp. 290-303,1988.

[M7] VALO, M., "Fracture toughness and tensile properties of irradiated austeniticstainless steel components removed from service." VTT-Report Dno REA 35/91,VTT, Finland, 1993.

[M8] CHOPRA, O. K. and W. J. SHACK, "Assessment of thermal embrittlement ofcast stainless steels." NUREG/CR-6177, USNRC, Washington D.C., U.S.A.,1994.

[M9] McDONALD, P. and J. W. SHECKHERD, "Fracture toughness characterizationof thermally embrittled cast duplex stainless steel." EPRI NP-5439, EPRIResearch Institute, U.S.A., 1987.

[M10] HISER, L., "Tensile and Jjf-cvarve characterization of thermally aged caststainless steels." NUREG/CR-5024, USNRC, Washington D.C., U.S.A., 1988.

[Mil] JANSSON, C , "Degradation of cast stainless steel elbows after 15 years inservice." International symposium on the contribution of materials investigationto the resolution of problems encountered in PWR plants. SFEN, Fontevraud II,France, 1990.

[M12] NORRING, K., "Degradation of cast stainless steels - a literature survey." SKIRapport 95:66 (in Swedish), Swedish Nuclear Power Inspectorate, Stockholm,Sweden, 1995.

[M13] HAGGAG, F. M., CORWIN, W. R. and R. K. NANSTAD, "Effects ofirradiation on the fracture properties of stainless steel weld overlay cladding.",Post-SMIRT Conference No 2, Monterey, U.S.A, 1989.

[Ml4] SATTARI-FAR, I., "Constraint effects on ductile crack growth in claddedcomponents subjected to uniaxial loading." Fatigue & Fracture of EngineeringMaterials & Structures. Vol. 18, No. 10, pp. 1051-1069, 1995.

91

[Ml5] YOSHIDA, K., KOJIMA, M., IIDA, M. and I. TAKAHASHI, "Fracturetoughness of weld metals in steel piping for nuclear power plants." TheInternational Journal of Pressure Vessels and Piping. Vol. 43, pp. 273-284,1990.

[Ml6] BERGENLID, U., "Fracture toughness of Alloy 182 at room temperature."Report No. M-91/29 (in Swedish), Studsvik AB, Studsvik, Sweden, 1991.

[Ml 7] BERGENLID, U., "Fracture toughness of Alloy 182 at 285 °C." Report No. M-91/95 (in Swedish), Studsvik AB, Studsvik, Sweden, 1991.

[Ml8] JANSSON, C. and U. MORIN, "Fatigue growth in reactor material." MD-02 (inSwedish), Vattenfall Energisystem, Stockholm, Sweden, 1995.

[Ml9] The Swedish Nuclear Power Inspectorate's Regulations Concerning StructuralComponents in Nuclear Installations. SKIFS 1994:1, Swedish NuclearInspectorate, Stockholm, Sweden, 1994.

[M20] U. MORIN and C , JANSSON, "Stress corrosion growth in BWR environment."MD-01 Rev. 2 (in Swedish), Sydkraft Konsult, Malmd, Sweden, 1996.

92

Appendix S. Safety factors for nuclear applications

For the choice of safety factors SFR and SFL, the objective has been to retain the safetymargins expressed in ASME, Sect. Ill and XI. The value of SFL during normal andupset conditions originates from the ASME code requirement that the primary generalmembrane stress Pm < Sm and the flow stress cy, at which net section plastic collapse isassumed to occur, can be estimated to 2ASm and 3.0Sm for ferritic and austeniticmaterials, respectively. Sm is defined in section 2.10 in the main document. SFK is thesafety factor on Kcr and corresponds to the safety factor SFj on J\c. They are relatedthrough SFJC = ^ISFj. More information about the development of these safety factorscan be found in Ref. [SI]. Tables SI to S3 show the safety factors aimed for ferriticsteel components, austenitic piping and ferritic piping, respectively. For austeniticcomponents other than pipes we recommend the same safety factors as for an axialcrack in an austenitic piping.

Type of load event

Normal/UpsetEmergency/Faulted

SFK SFL

2.4<«>1.2

(I) May be divided by 1.5 if the only primary stress is a local membranestress PL.

Table S1. Safety factors for ferritic steel components other than pipes.

Type of load eventNormal/Upset

Emergency /Faulted

Circumferential crackSFK

A/TO

-J2

SFL

2.770)1.39

Axial crackSFK

VTO

V2

SFL

3.0O1.5

(1) May be divided by 1.5 if the only primary stress is a local membrane stress/^.

Table S2. Safety factors for austenitic piping.

Type of load eventNormal/Upset

Emergency/Faulted

Circumferential crackSFK

yfTO

-J2

SFL

2.22™1.11

Axial crackSFK

V2

SFL

2.4<»>1.2

(1) May be divided by 1.5 if the only primary stress is a local membrane stress PL.

Table S3. Safety factors for ferritic piping.

References

[SI] BRICKSTAD, B. and M. BERGMAN, "Development of safety factors to beused for evaluation of cracked nuclear components." SAQ/FoU-Report 96/07,SAQ Kontroll AB, Stockholm, Sweden, 1996.

93

Appendix B. Background

This appendix gives the background to the current revision of the handbook and itsaccompanying PC-program SACC [Bl]. The objectives have been to remove some ofthe unnecessary conservatism that existed in the previous edition, to update thehandbook with respect to recent research and code requirements and finally to make thenew edition of the handbook and PC-program even more user-friendly. We willcontinue this work and as better knowledge and solutions become available, or specificuser demands are expressed new editions of the handbook and PC-program will bereleased. Some of the planned future work is also described below.

1. Assessment method

The method utilized in this handbook is based on the R6-method developed at NuclearElectric pic. [B2]. The third revision of the R6-metod contains three different optionsfor determination of the safe region in the failure assessment diagram. The option 1 is ageneral non-material specific failure assessment diagram intended to use as a firstapproach. If a more accurate failure assessment diagram is needed either option 2 can beused where the failure assessment curve is derived from the materials stress-strain curve,or option 3 where the failure assessment curve is based on complete numerical J-integral calculations for different levels of primary load. In addition, the R6-methodcontains three different types of analysis, categories, depending on how stable crackgrowth shall be considered. For category 1 no stable crack growth is allowed while forcategory 2 and 3 some amount of ductile tearing can be accounted for in the assessment.

In order to make the procedure safe and easy to use, the procedure has been restricted tothe option 1 and category 1 type of the R6-method. The option 1 type is however, notintended for materials with a discontinuous yield point, such as some carbon-manganesesteels at low temperatures. For such materials, the failure assessment curve exhibits asharp drop for Lr close to 1.0. In the procedure this has been handled ratherconservatively by restricting Lr to 1.0. This should be considered as a compromise whenapplying option 1 to a problem that is actually much better described by the option 2failure assessment curve. However, for nuclear applications the restriction is not severebut in a future release of the handbook the option 2 type failure assessment curve oughtto be included as an alternative to the option 1 type.

Ductile materials, e.g. stainless steel base and weld material, normally show asignificant raise of the 7-resistance curve after initiation. Setting fracture equal toinitiation without any account for possible stable crack growth, as in the category 1 typeanalysis, is rather strict especially if an adequate safety margin is used against fracture.A drawback caused by this is that deformation controlled stresses such as thermaltransient and weld residual stresses receive a much to large influence on the fractureassessment since these are not likely to cause unstable crack growth in ductile materials.Therefore, in the next release of the handbook we plan to include a method similar tothe category 2 type analysis where stable crack growth is accounted for. The amount ofductile tearing Aa must be limited to values where J still is likely to characterize thecrack-tip conditions.

2. Secondary stresses

The interaction between secondary and primary stresses on J is in the R6-methodhandled by the />factor. See Chapter 2.9. p is derived by Ainsworth and in this editionhis original work has been used to derive p [B3]. This reduces some of the conservatismin the previous edition of the handbook for low and moderate Z^-values. Compare Fig. 2in the previous and this edition if the handbook. However, p is still restricted to non-negative values. At SAQ research work is currently in progress on secondary stresses,

94

i.e. how weld residual stresses contribute to the fracture risk. A possible outcome of thiswork is that negative />values can be accounted for or that the secondary stresses aregiven a less weight in the safety assessment.

The R6-method prescribes that the elastic stress state is to be used when calculating thestress intensity factor for secondary stresses, Kf. For high secondary stresses, such asmay be induced by thermal transients, the actual ./-value is overestimated due to plasticrelaxation. The problem is further discussed in Ref. [B4]. In the computer programSACC a modified version of the method suggested by Budden [B5] has been built in.The secondary stress intensity factor used to determine Kr according to Eq. (9) is thencalculated by

Kj = max^ Kx(a)K2(a), jKx(a)K2{a)), (Bl)

where K\ and K2 are stress intensity factors calculated for the elastic and elastic-plasticstress state, respectively. The maximum value of the geometric mean of K\ and K2calculated for the actual crack size of interest a, and the actual crack size with a plasticzone correction according to Irwin

( B 2 )

is to replace Kf in Eq. (9). fi is set to 1 for plane stress and to 3 for plane strain. Incomp-arison to Ref. [B5], the elastic stress response to the elastic-plastic strain state has beenreplaced by the elastic stress state. This makes the method somewhat more easy to useand since the load is strain controlled the two stress states should be close to each other.Eq. (Bl) is to be considered as an estimate of an effective secondary stress intensityfactor that approximates J for high secondary stresses. Numerical evidence exists thatthe method works well, see Refs. [B4], [B6] and [B7]. Some minor non-conservatismmay exist, partly depending on how stresses are categorised as primary or secondary.However, since the method here is used for safety assessment where a safety margin ofthe order of 1.4 to 3.2 is used against initiation, this is can be neglected.

3. Safety assessment

The previous edition of the handbook contained a safety assessment system based onpartial coefficients. The nominal values of fracture toughness and yield strength werereplaced with design quantities according to

Vd — cr

<xy=~, (B4)

riwhere Y*L anc^ Ym a r e so-called partial coefficients. The assessment was then performedwith the design quantities instead of the nominal values. The size of the partialcoefficients was chosen to give the same overall safety margins against fractureinitiation and plastic collapse as applied in ASME Sect. XI and III. For normal andupset conditions, typical values of ym and y^ were VI0 and 1.5, respectively.

95

In the failure assessment diagram y^ and yym move the assessment point (Lr, Kr) to the

right with a factor yym and upwards with a factor ym. Since the failure assessment curve

drops for higher Lr-values a non-uniform safety margin to fracture initiation is obtaineddepending on the value of Lr. This can be further quantified by looking at the criterionfor fracture initiation based on the /-integral [B8].

J = Jic- (B5)

The R6 option 1 type failure assessment curve gives the following expressions for J andJ\c-

(l-v2)K? 1j A ^ ! J ( B 6 )E [fR6(Lr)-p]2

(\-v2)K2

where

(B7)

fR6 = (1 - 0A4L2 )[0.3 + 0.7exp(-0.65lJ)]. (B8)

With the partial coefficients ykm and y% applied, the criterion becomes

J=Jacc, (B9)

where, with the same definition of Jas above,

j _Q-v2)K?r[fR6(yymLr)-p]2

(Tm)2 E [fR6(Lr)-pf

The safety margin against fracture initiation is then given by

The result is shown in Fig. Bl for the case y^ = VlO, yym = 1.5 and p = 0. For low-

values of Lr, the desired safety margin becomes 10 to J-controlled initiation as inaccordance with the flaw assessment procedure in ASME Sect. XI, App. A. But for Lr-values exceeding 0.6, the margin increases dramatically with a maximum peak of about90.

96

SF

00.2 0.4 0.6 0.8

Fig. Bl. Increase of safety margin for initiation due to shape ofR6 option 1 type failure assessment curve.

The above behaviour seemed much too conservative in comparison to the flawassessment procedure in ASME Sect. XI, App. A. The presented example correspondsto a maximum safety factor of about 9.5 instead of 3.16 as in ASME Sect. XI, App. A.For that reason a new safety assessment procedure was developed as described in Ref.[B8] and presented in Chapter 2.11. The new safety assessment procedure gives auniform safety margin to fracture initiation.

As pointed out in Ref. [B8], we still believe a safety assessment procedure based onpartial coefficients to be a more preferable one as long as the partial coefficients reflectthe actual scatter of each input quantity. In the previous safety assessment procedure,the overall safety margins were taken from ASME Sect. XI and III and basically onlyapplied to a few input quantities, here the material properties. Since the overall marginsare quite large the uncertainty of the material properties were exaggerated which gavethe effect shown above. To resolve this, a project has been started at SAQ to develop aprobabilistic safety assessment procedure to be included in a future release of thehandbook. One outcome of this work may be a procedure based on several interactingpartial coefficients for defect size, loads, material properties etc. where the value of eachpartial coefficient is a function of the uncertainty of its corresponding input quantity.

4. Weld residual stresses

Appendix R contains guidelines on weld residual stresses. The appendix has beenrevised and for instance new recommendations for residual stresses for butt weldedpipes based on numerical calculations [B9] have been included. Currently there is a lotof activity in this field and in the next release of the handbook we hope to includeinformation on bi-metallic butt welds in pipes and nozzle attachment welds.

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5. Stress intensity factor and limit load solutions

Appendix K and L containing stress intensity factor and limit load solutions have beenextensively revised. The objective has been to improve the accuracy of the solutions.

For finite surface cracks in plates, finite axial surface cracks in cylinders and partcircumferential surface cracks in cylinders, stress intensity factor solutions for apolynomial stress distribution through the thickness of the order of 5 or 3 have beenincluded. This improve the accuracy for non-linear stress distributions which may arisefor thermal gradients and residual stresses. Also, new limit load solutions are introducedbased on numerical calculations and analytical work, see Ref. [BIO] and [B11 ]. Theseremove some of the conservatism connected to the previous solutions.

For infinite surface cracks in plates, infinite axial surface cracks in cylinders andcomplete circumferential surface cracks in cylinders, stress intensity factor solutionsbased on weight function technique have been included. By integration, K\ can bedetermined for an arbitrary stress distribution.

We are aware that the stress intensity factor solutions for through-thickness cracks arenot up to date. These are based on classical shell theory and in comparison to recentresults from solid finite element analysis work not very accurate. In the next edition ofthe handbook the current solutions will be replaced by improved ones currently beingderived at SAQ.

The corner crack geometries included in the previous edition of the handbook have beenomitted. The reason for this was that especially the limit load solutions were tooinaccurate and gave unrealistic results in the assessment. Such geometries are probablytoo complex to be handled well by simple engineering methods and may require a moredetailed analysis, e.g. finite element analysis. We will look further into this.

6. Fit of stress distribution for stress intensity factor calculation

As mentioned above, this edition of the handbook contains stress intensity factorsolutions for a polynomial stress distribution through the thickness of the order of up to5. A problem then arises how the actual stress distribution is fitted to a polynomial toestimate K\ as well as possible. Depending on type of situation, the followingalternatives are suggested:

a) For a smooth continuous stress distribution, the actual stress distribution over theextension of the crack is least-square fitted to a polynomial with an order within therange available. The order of the polynomial that gives the agreement with the actualstress is chosen.

b) For a discontinuous stress distribution such as may arise if dissimilar materialsthrough the thickness are present, an accurate least-square polynomial fit may not bepossible since the order of the polynomial is restricted. Instead a linearization isrecommended where the linear stress distribution is given the same resulting normalforce and moment as the actual stress distribution over the extension of the crack.

The PC-program SACC assists the user to select the best alternative. The programdisplays a graph where the actual and fitted stress distribution are shown. By testingdifferent alternatives and looking at the outcome on the graph the user can find outwhich alternative to select. When in doubt the fitting procedure that gives the mostconservative result in the assessment should be used.

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7. SACC

The procedure described in this handbook including calculation of crack growth due tofatigue and stress corrosion has been implemented in a modern Windows based PC-program called SACC [Bl]. Besides this, SACC contains options for assessment ofcracks according to the 1995 edition of the ASME Boiler and Pressure Vessel Code,Section XI. Appendices A, C and H for assessment of cracks in ferritic pressure vessels,austenitic piping and ferritic piping, respectively.

SACC is entirely written in Microsoft VisualBasic and can run under all MicrosoftWindows operating systems, i.e. Windows 3.x, Windows NT and Windows 95. Get incontact with SAQ Kontroll AB for more information..

References

[Bl] BERGMAN, M., "User's manual SACC Version 4.0." SAQ/FoU-Report 96/09,SAQ Kontroll AB, Stockholm, Sweden, 1996.

[B2] MILNE, I., AINSWORTH, R. A., DOWLING, A. R. and A. T. STEWART,"Assessment of the integrity of structures containing defects." The InternationalJournal of Pressure Vessels and Piping. Vol. 32, pp. 3-104, 1988.

[B3] AINSWORTH, R. A., "The treatment of thermal and residual stresses in fractureassessments." Engineering Fracture Mechanics. Vol. 24, pp. 65-76,1986.

[B4] BERGMAN, M., "Fracture mechanics analysis for secondary stresses - Part 1."SA/FoU-Report 94/03 (in Swedish), SAQ Kontroll AB, Stockholm, Sweden,1996.

[B5] BUDDEN, P., "Fracture assessments of combined thermal and mechanical loadsusing uncracked body stress analysis." CEGB Report RD/B/6158/R89, CentralElectricity Generating Board, Berkley, U.K., 1989.

[B6] SATTARI-FAR, I., "Constraint effects on behaviour of surface cracks in claddedreactor pressure vessels subjected to PTS transient." The International Journal ofPressure Vessels and Piping. Vol. 67, pp. 185-197, 1986.

[B7] HALLBACK, N. and F. NILSSON, "Fracture assessments for secondary loads."Fatigue & Fracture of Engineering Materials and Structures. Vol. 15, No. 2, pp.173-186,1992.

[B8] BRICKSTAD, B. and M. BERGMAN, "Development of safety factors to beused for evaluation of cracked nuclear components." SAQ/FoU-Report 96/07,SAQ Kontroll AB, Stockholm, Sweden, 1996.

[B9] BRICKSTAD, B. and L. JOSEFSON, "A parametric study of residual stresses inmultipass butt-welded stainless steel pipes." SAQ/FoU-Report 96/01, SAQKontroll AB, Stockholm, Sweden, 1996.

[B10] SATTARI-FAR, I., "Finite element analysis of limit loads for surface cracks inplates." The International Journal of Pressure Vessels and Piping. Vol. 57, pp.237-243,1994.

[Bl 1] DELFIN, P., "Limit load solutions for cylinders with circumferential crackssubjected to tension and bending." SAQ/FoU-Report 96/05, SAQ Kontroll AB,Stockholm, Sweden, 1996.

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Appendix X. Example problem

A defect has been discovered in a plate, see Fig XI. The plate is assumed to be acomponent in a nuclear power plant and made of a ferritic steel. The material has thefollowing measured properties at 20 and 150 °C.

Rp0.2(20 °C) = 300 MPa,

Rm(20 °C) = 490 MPa,

£k(150°C)=150MPaVm,

*P0.2(150 ° Q = 280 MPa,

/?m(150°C) = 490MPa.

The plate is loaded by a tensile load due to dead weight and a thermal gradient. A stressanalysis reveal that the nominal stress is 100 MPa due to dead weight and that thethermal gradient affects the plate with a bending stress of 180 MPa. The bending stressis tensile on the cracked side of the plate. The stresses act perpendicularly to the crackplane.

Perform a defect assessment to decide whether the crack can be accepted or not if theload event is categorised as normal. For reasons of simplicity, the temperature can beconsidered as constant and equal to 150 °C in the assessment.

36

40

Fig. XI. Plate with a defect (unit mm).

1. Solution

The defect assessment is perfonned according to the procedure described in Chapter 2of this handbook. A comparison is also done of the result on the same problem wherethe procedure according to the previous edition of the handbook [XI] has been used.

1) Characterization of defect

According to Appendix A, the defect is characterized as a semi-elliptical surface crackwith depth a = 9 mm and length 7 = 36 mm.

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2) Choice of geometry

A plate with a finite surface crack is selected, see Fig. Gl. The thickness of the plate is40 mm.

3) Determination of the stress state

The primary stress state consists of a membrane stress &£ = 100 MPa due to deadweight, and the secondary stress state of a bending stress cr5

b = 180 MPa due to thethermal gradient.

4) Determination of material data

According to Chapter 2.5 the yield strength ay, the ultimate tensile strength cry and thecritical stress intensity factor Kcr must be determined. This is done by putting ay = Rpo.2,

R d Kcr = K\c. Thus

£c,.(150oC)=150MPaVm,

oy(20 °C) = 300 MPa, <jy(\50 °C) = 280 MPa,

°C) = 490 MPa, cr^lSO °C) = 490 MPa.

5) Calculation of possible slow crack growth

No crack growth mechanisms need to be considered in this case.

6) Calculation of Kf and Kf

The stress intensity factors Kf and Kf are calculated according to Appendix K, Eqs.(Kl) and (K2). The primary and secondary stress are expressed according to Eq. (K2).

a* =100 = a £ ,

as = 180(1 - — ] = 1 8 0 - 3 6 0 - = 1 8 0 - 3 6 0 - - = 1 8 0 - 8 1 - = cr£ + af - .V t ) t at a u l a

Hence <?$ = 100 MPa, erg = 180 MPa and erf = -81 MPa. All other stress componentsare zero. With alt = 9/40 = 0.225 and I/a = 36/9 = 4, linear interpolation in Tables Kland K2 gives the required geometry functions.

/oA =0.896,

/ , A =0.573,

/oB =0.738,

/ , B = 0.123.

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

K[A = •Jna(agfA) = 15.07 MPaVm,

V*- 0.009 (180- 0.896 - 81 • 0.573) = 19.31 MPaVm.

K[ and Kj at the intersection of the crack with the free surface (point B) become

KfB = Vnfl (o"£/0B) = 12.41 MPaVm,

V* • 0.009 (180 • 0.738 - 81 • 0.123) = 20.66 MPaVm.

7) Calculation of Lr

Lr is calculated according to Appendix L, Eqs. (LI) - (L3). Eq. (L3) gives

a l 9 3 6

/(/ + 20 = 40(36 + 2-40)= °-0698-

Since the primary stress state in this case only consists of a membrane stress, Lrbecomes

Lr = {\Qa = (1-0.0698)-280 = °"384 '

8) Calculation of Kr

Kr is calculated according to Chapter 2.9. In the deepest point of the crack (point A) weobtain

AX

KlALr 19.31-0.384

Kj>A 15.07j

which according to Fig. 2 gives p= 0.036. Hence,

A K[A + KfA

A 15.07 + 1931

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Similarly we obtain at the intersection of the crack with the free surface (point B),

p= 0.041,

_Kcr 160

cr

The maximum value of Kr is used in the assessment and is obtained at the deepest pointof the crack.

9) Fracture assessment

The fracture assessment is described in Chapter 2.10. The non-critical region in thefailure assessment diagram is defined by

Kr <fR6 =(1-0.14 ^ ) [ 0 .3 + 0.7exp(-0.65 £*)],

Lr < L™ .

Since the component is nuclear and made of a ferritic material without a yield plateau(RpO.2 has been measured instead of Rei), L^ is given by

£max _ CT/ _ 2 - 4 ^ _ 2-4-163 = Hr aY <JY 280

where the allowable design stress Sm has been calculated according to Eq. (15). Thepoint (Lr, Kr) = (0.384, 0.251) is plotted in the failure assessment diagram. See Fig. X2.

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1.2

0 0.2 1.4 1.6

Fig. X2. Fracture assessment.

The point is situated within the non-critical region in the failure assessment diagram.Thus fracture is not to be expected. However, the defect may still not be acceptable withrespect to required safety demands to continue operation without repair or replacementof the component. This is investigated in the safety assessment.

10) Safety assessment

The safety assessment is described in Chapter 2.11. The acceptable region in the failureassessment diagram is defined by

v-accKr =Kf

K.cr

r-max

SFr

Recommended values for the safety factors SFj and SFL are found in Appendix S. For aferritic steel component under a normal load event, SFj= 10 and SFL = 2.4. Hence themaximum value of K™c (i.e. at the deepest point of the crack) becomes

15.07 + 19.31 0.036Kr = rzn + —T=T~ = y)J.Zo.

160 VlO

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The point (Lr, K°cc) = (0.384, 0.226) is plotted in the failure assessment diagram wherealso the acceptable region has been drawn. See Fig. X3.

Non-acceptable region

•j. (0.384,

Acceptable region

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6

Fig. X3. Safety assessment.

The point is situated within the acceptable region of the failure assessment diagram.Hence the defect is acceptable with respect to required safety demands and operationmay continue without repair or replacement of the component.

By repeating the above calculations for gradually increasing crack sizes until theassessment point falls on the line that limits the acceptable region it can be shown thatthe maximum acceptable crack depth is 16.6 mm if the crack aspect ratio IIa remainsconstant and equal to 4.

As a comparison the above problem has been solved using the safety assessmentprocedure according to the previous edition of the handbook [XI]. The maximumacceptable crack depth then became 12.9 mm. The difference is mainly due to the newsafety assessment procedure where we retain a uniform margin against fracture initiationas described in Appendix B.

References

[XI] BERGMAN, M., BRICKSTAD, B., DAHLBERG, L., NILSSON, F. and I.SATTARI-FAR, I., "A Procedure for safety assessment of components withcracks - Handbook." SA/FoU-Report 91/01, SAQ Kontroll AB, Stockholm,Sweden, 1991.

a procedure for safety assessment of components …stress intensity factor range, ... to fulfil the...

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