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International Journal of Pressure Vessels and Piping 87 (2010) 66–73

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International Journal of Pressure Vessels and Piping

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

Reference stress method for evaluation of failure assessment curveof cracked pipes in nuclear power plants

Masayuki Kamaya a,*, Hideo Machida b

a Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japanb TEPCO Systems Corp., Shibusawa City Place Eitai, 2-37-28 Eitai, Koto-ku, Tokyo 135-0034, Japan

a r t i c l e i n f o

Article history:Received 24 October 2008Accepted 9 February 2009

Keywords:Reference stressJ-integralFailure assessment curveTwo-parameter methodStructural integrity

* Corresponding author. Tel.: þ81 770 379114; fax:E-mail address: kamaya@inss.co.jp (M. Kamaya).

0308-0161/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.ijpvp.2009.11.002

a b s t r a c t

In order to obtain a precise failure assessment curve (FAC) in the R6 defect assessment procedure, it isnecessary to evaluate the J-value of cracked components. The reference stress method can be used forestimating J-values. However, the accuracy of estimation depends on the limit load used for evaluatingthe reference stress. In this study, the applicability of several limit load solutions was investigatedthrough comparison with the results of elastic-plastic finite element analyses (FEA). A pipe containinga circumferential surface crack was analyzed under pure bending load. Six materials used in nuclearpower plants were assumed. It was shown that the reference stress method is valid for FAC evaluation.The maximum non-conservativeness caused by using the reference stress method is less than 20%compared to the results obtained by FEA.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The CEGB’s defect assessment procedure (R6) [1] is usedextensively for assessing the integrity of cracked components innuclear power plants [2,3]. Two kinds of curves are employed forthe assessment. One represents the driving force of crackextension and is referred to as the failure assessment curve(FAC). The other relates to the resistance of unstable ductile crackextension and is derived from J-R curves and the stress intensityfactor of the cracked component. The R6 procedure providesthree options for the evaluation of FAC. Option 3 enables accurateassessment for unstable ductile crack extension. However, it isnecessary to evaluate the J-integral value (J-value) of the crackedcomponent. Since the J-value depends not only on the geometryof the crack and component but also on the stress-strain rela-tionship of the material, only limited solutions are available formaterials of which stress-strain relationship is approximated bythe Ramberg–Osgood law [4,5]. Therefore, a numerical calcula-tion such as finite element analysis (FEA) is required in Option 3.On the other hand, in Option 2, FAC is derived from the stress-strain relationship of the material. The basis of this option is theestimation of the J-value using the reference stress method. The

þ81 770 372009.

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reference stress method is valid for estimating J-values undervarious conditions [6–9]. However, the accuracy of the estima-tion is dependent on the limit load used in evaluating thereference stress [10,11] and there is no theoretical solution thatgives a precise J-value. Several attempts have been made to showrelevant limit load solutions [12]. Moreover, modification wasmade to a limit load solution in order to optimize the J-valueestimation [13,14]. However, it is not yet clear which solution isavailable for the R6 procedure. In the fitness-for-service code fornuclear power plant components of the Japan Society ofMechanical Engineers (JSME Code) [3], the failure assessmentprocedure based on the R6 procedure is prescribed. However, inJSME Code, Option 2 is not included and the J-value has to beevaluated in order to obtain FAC. Then, in order to apply thereference stress method to JSME Code, available limit loadsolutions for materials used in the nuclear power plant compo-nents were investigated.

In this study, the reference stress method was applied to theevaluation of FAC of pipes with circumferential surface crack underbending stress. For assessment of nuclear power plant components,FAC for various combinations of crack and pipe geometry wasassumed for six kinds of materials used in nuclear power plants.Four limit load solutions were used for the reference stress methodand the accuracy was quantified through comparison with theresults obtained by elastic-plastic FEA. Finally, the validity of thelimit load solutions were discussed.

Fig. 1. Geometry of pipe containing circumferential crack under bending load.

M. Kamaya, H. Machida / International Journal of Pressure Vessels and Piping 87 (2010) 66–73 67

2. FAC by reference stress method

2.1. Procedure for FAC evaluation

A pipe containing a circumferential surface crack under globalbending load was treated as shown in Fig. 1. FAC is obtained byconnecting points (Sr, Kr), and is defined as follows:

Sr ¼PPC

(1)

Kr ¼

ffiffiffiffiJeJ

s(2)

where P is the bending stress. PC is the collapse stress and is givenby the following equation in the JSME Code [3]:

PC ¼4sy

p

�cos

ba2t� a

2tsin b

�; (3)

sy is the 0.2% proof (yield) strength of the material. FAC has anupper value of Sr¼ sf/sy (cut-off value), where sf is flow stressdefined as the mean of sy and the ultimate strength. J in Eq. (2)denotes the J-value and Je is the elastic J-value evaluated underelastic calculation:

Fig. 2. Examples of fin

Je ¼K2�1� n2

�E

; (4)

where K, E and n denote the stress intensity factor (SIF), Young’smodulus and Poisson’s ratio, respectively.

In the reference stress method, J-value can be estimated by thefollowing equation:

J ¼E3ref

srefJe: (5)

sref is the reference stress evaluated by:

sref ¼�

PPL

�sy: (6)

The reference strain 3ref is the strain corresponding to the referencestress in the stress-strain relationship of the material. PL is the limit loaddiscussed in this study. By using Eq. (5), Kr can be expressed as follows:

Kr ¼

E3ref

sref

!�0:5

: (7)

ite element mesh.

Table 1Tensile properties and parameters for Ramberg–Osgood law.

Temperature(degree C)

Yieldstrength(MPa)

UltimateStrength(MPa)

Young’smodulus(GPa)

Parametersfor R–O’slaw

Ref.

a n

SCS14A 325 276 726 174 0.965 4.34 [17]SCS14

AW325 380 551 174 1.186 9.17 [17]

SUS304W 288 353 446 176 2.15 12.97 [19]SFVQ1A 325 413 577 180 1.47 7.91 [20]STPA24 310 245 482 181 1.64 5.98 [20]SUS316 RT 285 589 195 – – [18]

Fig. 4. Stress intensity factors for cracked pipe.

M. Kamaya, H. Machida / International Journal of Pressure Vessels and Piping 87 (2010) 66–7368

Therefore, we can obtain FAC without using the J-value. In the R6procedure, the following equation is used by incorporating a plas-ticity correction at the crack tip region:

Kr ¼

E3ref

srefþ

s3ref

2s2yE3ref

!�0:5

: (8)

2.2. Limit loads

In order to estimate the J-value by the reference stress method,the limit load must be defined in Eq. (6). Several limit load solutionshave been proposed. In this study, the following four equationswere used for the reference stress method:

(1) Global limit load (same as the collapse stress for Sr)

PL ¼ 4�

cosba� a

sin b

�(9)

sy p 2t 2t

(2) Optimized limit load [14]

PL ¼ g4R2

mt�

cosba� a

sin b

�(10a)

sy Zp 2t 2t

Fig. 3. Stress-strain relation used in finite element analyses.

g ¼ g1ða=tÞ2þg2ða=tÞ þ 1:04 (10b)

g1 ¼ 4:26ðb=pÞ2�1:35ðb=pÞ þ 0:8 (10c)

g2 ¼ �2:30ðb=pÞ2þ1:57ðb=pÞ � 0:77 (10d)

(3) Local limit load [15]

PL

sy¼ pR2

mtZp

1� a

t

1� at

1Mo

!(11a)

Mo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:26ðb=pÞ þ 47ðb=pÞ2�59ðb=pÞ3

q(11b)

Fig. 5. Effect of mesh size on FAC (SCS14A).

Fig. 6. FAC obtained by finite element analyses. (a) For various geometries (SCS14A).(b) For various materials.

Fig. 7. Comparison of FAC obtained by finite element analyses and reference stressmethod (for small crack). (a) SCS14A. (b) SUS316.

M. Kamaya, H. Machida / International Journal of Pressure Vessels and Piping 87 (2010) 66–73 69

(4) Limit load from R6 procedure [1]

PL ¼ 4R2mt��

1þ 1�

t�2

sin r� afcsin b

(12a)

sy Zp 12 Rm 2t

r ¼ p2

�1�

�1� t

2Rmþ a

2Rm

�bapt

(12b)

fc ¼ 1� t þ aRmþ 3t2 � 6at þ 4a2

12R2m

(12c)

3. Finite element analysis

In order to evaluate the accuracy of estimated J-values obtainedby the reference stress method, elastic-plastic FEA was carried outusing the general-purpose finite element program ABAQUS,Version 6.5 [16]. Fig. 2 shows an example of the finite element

model, which contains 7794 20-node solid elements and 40192nodes. The geometry of the analyzed model is shown in Fig. 1,where the length of pipe was set to L¼ 5(Riþ t) in order to suppressthe edge effect.

4. Geometrical and material conditions

Geometrical conditions were possible combinations of thefollowing:

Ri/t: 5, 10a/t: 0.3, 0.5, 0.7a/c: 0.2, 0.5

Pipe diameter was set to a constant value of 14B, because thesame FAC was obtained by FEA for various pipe diameters from 4B–36B under the same value of Ri/t.

Materials assumed in this study were cast stainless steel(SCS14A) [17], cast stainless steel weld metal (SCS14AW) [17],

M. Kamaya, H. Machida / International Journal of Pressure Vessels and Piping 87 (2010) 66–7370

stainless steel (SUS316) [18], stainless steel weld metal(SUS304 W) [19] and low-alloy steel for piping (SFVQ1A andSTPA24) [20]. Properties of the six materials are summarized inTable 1. Except for SUS316, the stress-strain relationship wasapproximated by the Ramberg–Osgood law, and is expressed bythe following equation:

3Esy¼ s

syþ a

�s

sy

�n

: (13)

The constants a and n are also shown in Table 1. In FEA, the stress-strain relationships shown in Fig. 4 were used. These curves consistof lines connecting the points obtained by Eq. (13), although thestrain at the yield strength is sy/E and a linear relationship isassumed below this strain.

The magnitude of applied load varied from zero up to the cut-offvalue. FEA was performed at every 5% of the cut-off load. Namely,FEA was performed 20 times for each geometrical and materialcondition.

Fig. 8. Comparison of FAC obtained by finite element analyses and reference stressmethod (for large crack). (a) SCS14A. (b) SUS316.

5. Results and discussion

5.1. Validity of FEA

Fig. 4 shows the comparison between SIF obtained by the elasticanalysis using the current finite element model and reference data(Ref. [21] and [22]). The analysis was carried out under conditionsof Ri/t¼ 10, a/t¼ 0.4 and a/c¼ 0.25. The agreement of the resultsimplies that the finite element model used in this study is valid forprecise analysis of the stress field near the crack tip. Furthermore,FAC for SCS14A was evaluated for different mesh size for theconditions of Ri/t¼ 10, a/t¼ 0.5 and a/c¼ 0.2 Fig. 5. The number ofelements and nodes for the current model are 17910 and 92156,whereas 45090 and 214446 for precise analysis, respectively.Almost identical FAC shows the well converged FAC was obtainedwith the current mesh size.

5.2. FAC by finite element analyses

Fig. 6 shows the FAC using the J-value obtained by FEA. Theelastic J-value, Je, was derived using stress intensity factorsobtained by elastic FEA of the same model. The shape of FAC wasdependent on the crack and pipe geometry. Kr seems to be larger asthe crack size increases when Sr is more than 1.0. In the range ofsuch large Sr, the material effect was significant and Kr decreased asthe constant n increased.

5.3. FAC by reference stress method

FAC obtained by the reference stress method is shown inFigs. 7 and 8 together with that by FEA. Since FAC obtained usingEq. (12) was almost the same as that obtained using Eq. (9) (globallimit load), the results from Eq. (12) were omitted in this study.The reference stress method gave reasonable agreement with FACobtained by FEA in most cases. In the R6 procedure, the distancebetween the point (Sr, Kr) and the origin of the diagram relates tofailure load due to unstable ductile crack extension. Smallerdistance to the origin implies greater conservativeness. Therefore,based on FAC by FEA, the reference stress method gives conser-vative results in the cases of Fig. 7(a) and Fig. 8(a) regardless of

Fig. 9. FAC obtained by reference stress method using poly-linear stress-strain curve(SCS14A).

Maximum value in all cases

Maximum

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5a

b

SCS1

4A

SCS1

4AW

SUS3

04W

SFVQ

1A

STPA

24

SUS3

16

K)fer(r/

)MEF(

Global LLOptimized LLLocal LL

Average1.3

Kr

Global LL

M. Kamaya, H. Machida / International Journal of Pressure Vessels and Piping 87 (2010) 66–73 71

the limit solution. As for the results for SUS316, assessment usingthe global and optimized limit load solutions possibly derivesnon-conservative evaluation depending on the geometricalconditions of the crack and pipe. The local limit load wasconservative in all cases (geometry and material), although itseemed excessively conservative in some cases as shown inFig. 8(a).

The non-conservativeness typically occurred between theposition of Sr¼ 0.7 and 1.0. In particular, in the case of SUS316, FACshowed discontinuous change near Sr¼ 1.0 and the reference stressmethod caused larger Kr compared to FEA. This was brought aboutby the stress-strain relationship, in which strain increases abruptlyafter the yield strength. It should be noted that, except for SUS316,Eq. (13) was used for the evaluation of reference stress and it showscontinuous change near the yield strength. For comparisonpurposes, FAC obtained using the poly-linear relationship of Fig. 3 isshown in Fig. 9 for SCS14A material. At the yield strength, the strainby Eq. (13) is (1þ a)sy/E, which is larger than sy/E assumed in Fig. 3.Although the difference in strain is minor, less than 0.5%, the J-valuenear the yield point is sensitive. Larger strain results in greater J-value and smaller Kr. In Eq. (8), the effect of yielding at the crack tipis compensated. FAC obtained using Eq. (7) instead of Eq. (8) also isshown in Fig. 9. The plasticity correction makes Kr small and seemsto contribute to accurate evaluation near Sr¼ 1.0. It is expected thatthe use of Eq. (13) brings about a similar effect to the plasticitycorrection.

In order to quantitatively evaluate the conservativeness or non-conservativeness of the reference stress method, the ratioQ¼ Kr(ref)/Kr(FEM) was evaluated (Fig. 10). The ratio of failure stressevaluated by the reference stress method to that by FEA isproportional to the parameter Q, when the critical points from thefailure assessment were on the same line which includes the originof the diagram. Therefore, based on this assumption, the referencestress method is conservative compared to FEA when Q< 1. Themaximum and mean value of Q among all geometrical and loadingconditions is shown in Fig. 11 for each material. As mentioned, FACobtained using the local limit load solution was always Q< 1 for allmaterial. The maximum Q is relatively large for SUS316. Unity of themean of Q implies that the J-value estimation by the referencestress method is reasonable. Due to excessive conservativeness ofthe local limit load solution, the mean value by the local limit load

Fig. 10. A schematic drawing of failure assessment curve. (a) Maximum value in allcases.

solution was relatively small. On the other hand, the global limitload solution exhibited the best estimation.

The position along the FAC corresponds to the failure mode(unstable ductile crack extension or plastic collapse). For ductilematerial, such as stainless steel, failure is expected to be evaluatedat a position of large Sr or over the cut-off value in FAC. Therefore,the maximum Q is not necessarily always less than unity for allranges of Sr, since the materials assumed in this study are ductileand have relatively large fracture toughness. When actually usingthe assessment procedure, the validity of the limit load should beconsidered at large Sr.

Averaged value of all cases

0.7

0.8

0.9

1.0

1.1

1.2

SCS1

4A

SCS1

4AW

SUS3

04W

SFVQ

1A

STPA

24

SUS3

16

K)fer(rK

r/

)MEF(

Optimized LLLocal LL

Fig. 11. Summary of difference between Kr value obtained by finite element analysesand the reference stress method. (a) Maximum value in all cases. (b) Averaged value ofall cases.

M. Kamaya, H. Machida / International Journal of Pressure Vessels and Piping 87 (2010) 66–7372

The position in the FAC can be represented by the parameterSC¼ Kr(FEM)/Sr(FEM), which corresponds to the inclination of theline connecting the origin and point (Sr(FEM), Kr(FEM)) on FAC asshown in Fig. 10. Fig. 12 shows the same figure as Fig. 11 obtainedusing data satisfying SC< 0.8. When we use the reference stressmethod only for the condition of SC< 0.8, the non-conservative-ness evaluated in Fig. 11 is reduced. Moreover, Q obtained usingEq. (7) was almost the same with that by Eq. (8) as shown inFig. 13. This implies that it is possible to use Eq. (7) instead ofEq. (8) when the critical point on the FAC in the failure assess-ment is under the condition of SC< 0.8.

In all cases, the maximum Q is less than 1.2. Therefore, we canconclude that the reference stress method is valid for failureassessment of nuclear power plant components. All limit load

Maximum

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

A41S

CS

WA41

SC

S

W403S

US

A1Q

VFS

42A

PTS

613S

US

K)fer(rK

r/

)M

EF(

Global LLOptimized LLLocal LL

Average

0.7

0.8

0.9

1.0

1.1

1.2

1.3

A41S

CS

WA41

SC

S

W403S

US

A1Q

VFS

42A

PTS

613S

US

K)fer(rK

r/

)M

EF(

Global LLOptimized LLLocal LL

a

b

Maximum value in all cases

Averaged value of all cases

Fig. 12. Summary of difference between Kr value obtained by finite element analysesand the reference stress method (for Kr(FEM)/Sr(FEM) < 0.8). (a) Maximum value in allcases. (b) Averaged value of all cases.

Fig. 13. Difference between Kr value obtained by finite element analyses and thereference stress method (for Kr(FEM)/Sr(FEM)< 0.8).

solutions shown in this study can be used for reference stressevaluation from an engineering point of view.

6. Conclusions

The reference stress method was applied to evaluate FAC in theR6 assessment procedure for cracked pipe under pure bending.The validity of four limit solutions was investigated by comparingthe results of elastic-plastic FEA assuming various materials used innuclear power plants. The conclusions obtained in this study are asfollows:

(1) The reference stress method can be applied to the R6 failureassessment procedure for cracked pipe in nuclear powerplants.

(2) The plasticity correction considered in Eq. (8) is effective forJ-value estimation under small-scale yielding conditions.Approximation of stress-strain relationship by the Ramberg–Osgood law can be used as an alternative for the correction.

(3) Plasticity correction is not necessary in an assessment ofunstable ductile crack extension and plastic collapse.

(4) The maximum non-conservativeness caused by the referencestress method is less than 20% compared to the resultsobtained by FEA, when the limit load solutions shown in thisstudy are used.

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