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Int. J. Pres. Ves. & Piping 18 (1985) 35-53

Example Calculations Illustrating Methods for Analyzing Ductile Flaw Stability in Nuclear Pressure

Vessels

J. G. Merkle

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

&

R. E. Johnson

US Nuclear Regulatory Commission, Washington, DC 20555, USA

(Received: 17 January, 1984)

ABSTRACT

This paper contains example calculations of ductile flaw instability stresses for hypothetical .)Claws in nuclear pressure vessels. For comparison, three different methods of estimating upper shelf toughness as a function of Charpy impact energy were used, namely a power law R-curve correlation, the Rolfe-Novak correlation, and the Paris Jso correlation. All three methods were used in LEFM calculations including a plastic zone size correction, and gave similar results, with the Paris Jso method being the most conservative at low Charpy upper shelf energy levels. Safety factors based on the tearing modulus ratio Tm,t/Tappl can exceed those based on load by considerable amounts and use of them at this time is not recommended. The use of resistance curve data obtained .from actual vessel material test specimens is recommended over the use of correlations. Furthermore, evaluation of a recently proposed modified crack extension adjustment procedure for R-curve data, which is not over- conservative, is recommended.

35 Int. J. Pres. Ves. & Piping 0308-0161/85/$03.30 Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

36 J. G. Merkle, R. E. Johnson

NOMENCLATURE

a

a o

o f

B b CVN c

E J Js0 Jc Ju

K K~ KI m

Pd Ri I~y

T Tappl Tv

T mal

t

W X

Y

/3 Aa (T

~70

Crack depth (in). Initial crack depth (in). Crack depth at fracture (in). Specimen thickness (in). Half the surface length of a part-through surface crack (in). Charpy V-notch impact energy (ft lb). Coefficient in the power law representation of the J-integral resistance curve (dimensionless). Modulus of elasticity (psi). The J-integral (in lb/in2). Value of the J-integral for which J/T= 50 in lb/in 2 (in lb/in2). Value of the J-integral at fracture (in lb/in2). Value of the J-integral to which Ernst's modified Aa correction has been applied (in lb/in2). Linear elastic stress intensity factor ((ksi) in1' 2). Fracture toughness ((ksi) inl~Z). Plane strain mode I elastic stress intensity factor ((ksi)in1 2). Exponent in the power law representation of the J-integral resistance curve (dimensionless). Design pressure (ksi). Inside radius (in). Plastic zone radius (in). Tearing modulus (dimensionless). Applied value of the tearing modulus (dimensionless). Value of the tearing modulus calculated after applying Ernst's modified Aa correction to the values of J (dimensionless). Tearing modulus of a material (dimensionless). Pressure vessel wall thickness (in). Specimen width (in). Correlating parameter for estimating the factors in a power law representation of a J-integral resistance curve (dimensionless). Geometric shape factor (dimensionless). Normalized J-integral (dimensionless). Constraint factor (dimensionless). Stable crack growth (in). Stress (ksi). Flow stress (ksi).

Ductile flaw stabiliO, in nuclear pressure vessels 37

o'f O'y s

Fracture stress (ksi). Tensile yield stress (ksi). Normalized multiplying factor in the power law representation of a J-integral resistance curve (dimensionless).

INTRODUCTION

This paper contains example calculations of ductile flaw instability stresses for hypothetical flaws in nuclear pressure vessels. These calculations were originally prepared for the ASME Section XI Working Group on Flaw Evaluation, which is part of the American Society of Mechanical Engineers' Boiler and Pressure Vessel Committee. The purpose of the calculations was to illustrate the results of applying the elastic-plastic fracture mechanics analysis procedures described in the United States Nuclear Regulatory Commission 'For Comment' report NUREG-0744, 'Resolution of the reactor vessel materials toughness safety issue', ~ to conditions that might be encountered in the evaluations of flaws in nuclear pressure vessels. For comparison, three different methods for estimating upper shelf toughness as a function of Charpy impact energy were used. The 'For Comment' version of Ref. 1 was used as the basis for the task. The calculations were done to illustrate the approach described on page 6-1, Ref. 1, reprinted, using italic for emphasis, as Appendix A herein. After having been edited to accommodate the comments that were received, Chapter 6 of NUREG- 0744, Revision 1, no longer includes the approach suggested in the earlier version (i.e. Appendix A).

Much of the following description of analytical procedures can be found in NUREG-0744, Appendix C, Section 3, 'Methods for vessel flaw analysis'. Values of specific input parameters used in the calculations are listed in Table 1. The parameter/~ listed in Table 1 is most familiar as part of the Irwin crack tip plastic zone radius formula

1 ry ---- ~ (K /ao) 2 (1)

where ry = crack tip plastic zone radius, fl = 2 for plane stress and 6 for plane strain, and K = stress intensity factor (which is assumed to be K~ when fl = 6).


TABLE 1 Parameter Values Used for Calculations

Vessel wall thickness Initial crack depth Flaw shape ratio Stress state parameter Geometry correction factor Flow stress Tensile elastic modulus

Note: a2/E = 270 psi.

t=8 in a o = t/4 = 2 in

a/2b= 1/6 /~=6 Y=I

ao = 90 ksi E=30 106 psi

In eqn (1), for consistency with eqn [26]* of Appendix B in Ref. 1, the expression for ry was written in terms of the flow stress, tr o, instead of the tensile yield stress, try S.

The factor Y in Table 1 was described in NUREG-0744, Appendix B (page B-12ff.) and Appendix C (page C-14ff.). Note, on page C-17 of Ref. 1, that the value Y= 1 was used for the surface crack problem addressed therein.

The flow stress, tr 0, is generally taken as half-way between the tensile yield and ultimate strengths. Here, the value of 90 ksi was chosen as a representative value for irradiated reactor pressure vessel steels.

Also, values of Charpy V-notch impact energy, CVN, were selected as required.

To be determined were values at fracture of failure stress, af; J-integral, J; and crack extension, Aa.

UPPER SHELF FRACTURE ANALYSIS PROCEDURES

Failure stresses were determined on the basis of LEFM, with a plastic zone size correction, using three different correlations between Charpy upper shelf impact energy and fracture toughness. First, a power-law fit to the R-curve, as described in NUREG-0744, Appendices C and D, was used to relate fracture mechanics and impact test parameters. Second, the correlation established by Rolfe and Novak 2 between toughness and Charpy energy was used. Third, the correlation between the J-integral corresponding to the ratio J/T = 50 in lb/in 2 and the Charpy upper shelf impact energy was used as described in NUREG-0744, Appendices B, D

* The equation numbers given in square brackets in this report are the same as those in NUREG-0744, ~For comment' (Ref. 1), Appendix C.

Ductile .flaw stability in nuclear pressure vessels 39

ff

I C_

Fig. I .

3000 I I I I I IRRADIATED O UNIRRADIATED

A533 WE LD V84 --~o / o o I

I I

A A533 WELD C88 J ooo _ -

1000 A533 WELD E 1 9 ~ ~ /

A302B~A533 WELD V86

~ ] ~ ~====~A302B

0 I I I I 0 20 40 60 80 100 120

C v (ft-lb) An attempted correlation of Jso values with Charpy upper shelf values (from

Ref. 1 ).

and H. The specific correlation for the third case is illustrated in Fig. 1 of this report, which is reprinted from NUREG-0744, 'For Comment', Appendix B.

Power law R-curve correlation

Referring to NUREG-0744, 'For Comment', Appendix C, the power-law representation for the R-curve was given as

[Aa\" J= 1000C[l~0 ] \ / [14] (2)

40 J . G. Merk le , R. E. Johnson

The coefficient c was fitted by the equation

c=-0 .114 CVN

It was found that by defining

the exponent m in eqn (2) could be fitted by

0.473x 3 D1 z . . . .

14.42 + .v 3

CVNX]

-1 o/ [15] (3)

[16] (4)

[171 (5)

EJ v- , [42] (11)

o-;

Using the substitutions

From the derivation given on page C-10 of Ref. 1, it was shown that, at instability,

Aa=~l m)ao [24] (6)

and because (at fracture) af =a 0 +Aa, it follows that

Aa=maf [23] (7)

From eqn [29] on page C-13 of Ref. 1 and the relationship

K e = EJ (8)

it follows that for loads below gross section yield levels, the applied value of the stress intensity factor is given by

Yax/" ~a " K . . . . . . . . . . . . . . . . . . . (9)

1__ Y2

Also, the tearing modulus, 7", is defined as

Ductile flaw stability in nuclear pressure vessels 41

and = ( 1000c) / (a2 /E )

eqn (2) can be written y= q~(Aa) m

Also, by operating on eqn (13), eqn (10) can be written as

T = dy /da = m0(Aa)"- 1

Writing eqn (14) as

[44] (12)

[43] (13)

[46] (14)

T - mO(Aa)" (15) Aa

and substituting in accordance with eqns (7) and (13), the material value of the tearing modulus at instability is given by

T- my _ y [42] (16) maf af

Squaring both sides of eqn (9), then using eqn (8) and dividing both sides by a~ gives

y2( ff )2Tca E J _ \go / (17)

Y2 ( ' ) 2

Neglecting the variation of Y with a, applying eqn (10) to eqn (17) gives

E J T - aoZa (18)

and by using eqn (18)

K 2 T= a2 a (19)

Substituting eqn (19) into eqn (9) gives

T- \ao /

1 fl \go /

(20)

TABLE 2 Results of Example Calculations

(a) Power law R-cur~,e extrapolation

CV N c m Aa ,I; (13 Ih) (in) (in Ih,in 2 )

30 0.450 2 0" 136 2 0.315 5 385 35 0.6194 0.1638 0391 7 531 40 0.815 5 O- 195 5 0.485 9 708 45 1038 6 0-229 8 0.596 8 922 50 12885 0.2650 0721 0 I 182 55 1.5654 02990 0853 I 1 493 60 1.869 1 03302 0.9862 I 861 65 2.1998 0.3577 1.1138 2286 70 2.557 4 0380 9 1.230 6 2 768 75 2.941 9 0400 1.333 5 3 301

CVN T (0 Ih)

30 0.6154 35 0.822 7 40 1-055 1 45 1.3156 50 1.608 2 55 19378 60 2.307 6 65 2.7194 70 3.173 1 75 3.6674

(T I

(ksi)

39.2 45 I 50'8 56.3 61 8 67.3 72.8 78.3 837 89-0

(b) Rolfi" Novak

K c

(ksi #112)

107 126 146 166 188 212 236 262 288 315

CVN y K c cx~ (fi lh) (ksi in j:2) (ksi)

30 1.416 7 107 420 35 1.6944 117 45.7 40 1-972 2 126 49.2 45 2.2500 135 !52-3 50 2.527 8 143 155.3 [ 55 2.8056 151 i580i 60 3083 3 158 60.6 65 3-361 1 165 63. 70 3.638 9 172 i 65-4 75 3.9167 178 67 .6


CVN Jso (.It Ih) (in Ih/in 2)

30 320 35 380 40 440 45 520 50 620 55 710 60 820 65 960 70 1060 75 1180

TABLE 2--Contd. (c) Paris Jso

K, (ksi in t/2)

)' O'f (ksi)

98 1.1852 38.5 107 1'4074 41-8 I 15 I "629 6 44.9 125 1-925 9 48"6 36 2.296 3 52-8 46 2.629 6 56"3 57 3.037 0 60"2 70 3" 555 6 64-7 78 3.925 9 B 67.7 88 4.3704 71.1

Solving eqn (20) for (0"/0"0) 2 gives

1

At instability, using the values of Y and/3 given in Table 1, eqn (21) gives

0.0 (22) 0.r - /~n 1

For a specified value of CVN, the coefficient c can be determined from eqn (3). Substituting the resulting value ofc and the given value of the flow stress (a o =90ksi ) into eqn (4) produces the value of x, which when substituted into eqn (5) produces the value of the exponent m. Given the inital crack length, a o =2 in, and the calculated value of m, Aa can be calculated using eqn (6). For conditions at flaw instability, the calculated values of c, Aa and m, inserted into eqn (2) will yield the value of Jc. Remembering that af = a o + Aa, and that we were given 0.~/E = 270 psi, T at instability can be calculated from eqns (11) and (16). The fracture stress, of, then can be determined from eqn (22). Finally, from the relationship K=x/ -E~, the critical stress intensity factor can be determined.

44 J. G. Merkle. R. E. Johnson

All of the above-named parameters (c, m, Au, J,, T, of and K,) were calculated for assumed CVN values ranging from 30 to 75 ft lb and the results listed in Part (a) of Table 2.

Rolfe-Novak correlation

The correlation developed by Rolfe and Novak2 can be written as

(23)

where the second equality follows from eqns (8) and (11). Moreover, as a direct result of eqn (23)

Kc = o,& (24)

For the conditions under which the RolfeeNovak correlation is applicable, crack extension is neglected. Thus, at instability, combining eqns (9) and (24) gives

(25)

from which it follows that

(10 01. = --r---

J (26)

nuo J -+;

For specified values of CVN and oO, the parameter _r can be calculated from eqn (23). From eqn (24) K, can be determined and the fracture stress can be calculated using eqn (26). The results of determining values for the three parameters are listed in Part (b) of Table 2.

Paris J,, correlation

Given any CVN value for the upper shelf of a Charpy impact energy curve, the correlation shown in Fig. 1 (preferably the lower bound curve, for conservatism) can be used to determine the value of the J-integral corresponding to the ratio J/T =50in lb/in. Again using the elastic

Ductile [taw stability in nuclear pressure vessels" 45

approximation, K 2 = E J, with appropriate notation for the case at hand

K c =x/EJso (27)

Rewriting eqn (17) in compliance with the notation for this section,

J5o

The equation for the fracture stress, ~f, neglecting crack extension, is still eqn (26).

From Fig. 1 and eqns (26)-(28), values of Jso, Kc, 3' and af can be determined. The results of these calculations are listed in Part (c) of Table 2.

Application of results to nuclear vessels

Consider a pressure vessel with a design pressure

Pd = 2500 psi (29)

and a radius-to-thickness ratio of 10 (typical RPV design parameters). Applying a safety factor of 2 to the design pressure, for consistency with ASME code provisions, 3 and neglecting any contributions from thermal transients and other secondary or residual stresses, we calculate

2pd --=Ri 50 ksi (30) l

as the desired level of strength. Calculated failure stresses in excess of 50ksi have been highlighted by boxing them in Table 2. Because the associated values of CVN are on the order of 40 to 50 ftlb it can be concluded that the 50 ft lb minimum requirement presently used by the Nuclear Regulatory Commission (NRC) does provide an adequate safety margin against ductile flaw instability under design pressure loading. The results presented in Table 2 also illustrate that the three methods for ductile flaw instability analysis examined in this work produce numerically similar values. Furthermore, these results show that the elastic-plastic method described in NUREG-0744 can be applied to the analysis of a pressure vessel in a relatively easy, straightforward manner.


DISCUSSION

Safety factors

Consider the safety factors suggested in Chapter 6 of NUREG-0744, 'For Comment'* (see Appendix A). It was recommended that

Treat > 2 (31) 7~appl - -

If the conservative 50-to-1 linear J / T load line is used in evaluation,

Jso _ 50 or Tm, t = J50/50 (32) Tm~t

Note that the equation for T~ppl given on page B-20, Appendix B, NUREG-0744, ~For Comment', should be corrected by deleting the factor (a2a/E); otherwise it is a reasonable estimate of the plastic-zone size corrected tearing modulus. Thus

V:ro/ T.,pl - _l (~2 (33)

1 - 6 \O 'o )

Consider a steel vessel for which the upper shelf Charpy V-notch impact energy is 50 ft lb, which is the limit of acceptability according to the NRC regulations. For that case, Fig. 1 was used to determine

J5o = 620 in lb/in 2

Then

J5o _ 12'4 Tm"t- 50

Using the 'Paris Jso' result given in Table 2 for CVN = 50 ft lb, we note that

af = 528 ksi

* In response to several reviewers' comments, Chapter 6 of NUREG-0744 was rewritten extensively and the safety margins in the Rev. I version are very different from the original recommendations.


(incidentally, the most conservative value of af in Table 2 at the 50 ft lb energy level). Assuming an allowable accident-level pressure limit of twice the design pressure, let

Then

That results in

Therefore

p = 2Pd = 2(2500) = 5000 psi

a = 50 ksi

/50x~ 2

1 6\90)

Tm. , __ 12"4 - 12"2

rappl 1"02

which is well in excess of the safety factor of 2 on the tearing modulus suggested in NUREG-0744, 'For Comment'. But at the same time

af _ 52"8 - 1 "056

a 50

or, from the other viewpoint O" - - = 95 o /

/ O o'f

Thus it can be seen that although an acceptable safety .margin against fracture appears to exist in terms of the tearing modulus, the applied stress may be quite close to the calculated failure stress.

By way of another example, assume an internal pressure of 3000 psi. In fact, pressures much in excess of 3000psi are unlikely at high temperatures for a number of reasons, so the 3000psi value would be useful in analyzing a vessel with radius 10 times the wall thickness. Using the same equation as before,

(30V Tappi- n\90]'~/ =0'356 _1(3o V

1 6 \90)

48 J. G. Merkle. R. E. Johnson

and

Tma , _ 12.4 _ 34.8 Tappl 0"356

which is far in excess of the factor of 2 suggested in NUREG-0744, 'For Comment'. At the same time, however, a f /a=52.8 /30= 1.76, or a/af = 0"568. The conclusion drawn from this example is that there is a very large apparent safety margin in terms of the tearing modulus, but an actual safety margin still slightly less than 2 in terms of pressure, for a reasonable value of accident condition stress, with a conservative assumption of a pre-existing crack one-fourth of the way through the vessel wall in the most severe orientation (normal to the maximum principal stress). Thus, the use of safety margins expressed in terms of the tearing modulus ratio may be premature until such time as more extensive correlations have been established with the corresponding load ratios.

R-curve data

One of the principal factors in a ductile flaw instability analysis is the resistance curve of the material. In determining this curve from small specimen test data, it is necessary to make mathematical adjustments for the amount of crack extension taking place during the test, and for the inelastic rotation of the specimen arms (see attachments I and 2 to Appendix C of Ref. 1). These adjustments tend to reduce the calculated toughness values as well as the slope of the resistance curve. It has recently been pointed out by Ernst 4 that the adjustment procedure presently in use may include a degree of conservatism that increases with decreasing specimen size, as indicated by the J -T data shown in Fig. 2. Ernst 4 developed a modified crack extension adjustment procedure that tends to eliminate the apparent size effects shown in Fig. 2. Application of the modified adjustment results in an upward readjustment of the J -T data for the small specimens and, as shown in Fig. 3, substantial agreement with the larger specimen data. The possibility of relaxing the present restrictions on the toughness levels that can be reached without exceeding accuracy limits due to crack extension also was discussed by Ernst. 4 Since the data shown in Fig. 1 were obtained from relatively small specimens without Ernst's modified adjustment procedure, the correlation between

Ductile flaw stability in nuclear pressure vesseI.Y 49

.E

v

Fig. 2.

9000

8000

7000

6000

5000

4000

3000

2000

1000

I I I I

JVST A 508 C LASS 2A TEMP. 400F COMPACT SPECIMENS a/W ~ 0.6 B = PROPORTIONAL

(W/2)

-~o I'1

[]

[3

O

[] /k

/k A []

A -0 0 0

0 Z&O A

0 A

10T [ ] 4T A 1T O 1/2T

]

D

0 O[3 - -

o I I I I 0 50 100 150

Y J-integral-tearing modulus correlation for reactor vessel steel specimens (from

Ref. 4).


9000

8000

7000

6000

_ 5OOO

Fig. 3.

4000

3000

2000

1000

Q~

[]

[]

%

I I

JM VS T M

A 508 CLASS 2A TEMP 400F a/W = 0.6 B = PROPORTIONAL

(W/2)

10T [-I 4T /~ 1T O 1/2T

D

n

@n

D 0 A

OA A

A 0

O

rl

O@r l -

o I I 0 50 100 150

T M

The same test results as in Fig. 2, after modification (from Ref. 4).

Ductile.flaw stability in nuclear pressure vessels 51

J5o and CVN shown in Fig. 1 may contain some unnecessary conservatism. Further evaluations of the modified adjustment are recommended, as well as the use of actual pressure vessel material test data where possible.

CONCLUSIONS

The following conclusions were drawn from the work on application of the elastic plastic pressure vessel analysis presented in NUREG-0744.

1. The method presented in NUREG-0744 can be applied in a straightforward manner to pressure vessels to determine the pressure at ductile flaw instability.

2. Safety factors in terms of the ratio Tmat/Tappl generally will not be the same as the corresponding safety factors expressed as stress or load ratios and, for the examples chosen, the tearing modulus ratio was shown to be quite large even though the corresponding stress ratio was rather small.

3. The modified crack extension adjustment suggested by Ernst, which appears to bring data from specimens of different sizes together, should be evaluated and, if validated, applied because of the significant increase in the indicated value of J at instability that may result, especially for actual vessel materials.

4. The use of the actual resistance curves for vessel materials is preferred over the use of conservative values obtained from the correlations between Jso and CVN.

5. Where relevant data are available for a vessel to be analyzed, the use of Aa values beyond the loading curve of slope J/T= 50 in lb/in 2 should be considered.

NOTE

Research sponsored by the Office of Nuclear Regulatory Research, United States Nuclear Regulatory Commission, under Interagency Agreements 40-551-75 and 40-552-75 with the US Department of Energy under Contract W-7405-eng-26 with the Union Carbide Corporation.

The work reported in this paper was done as an elaboration of a suggested approach to evaluation of flaws in nuclear reactor pressure


vessels using the elastic-plastic fracture mechanics relationships developed under the NRC Unresolved Safety Issue Task A-11. The flaw evaluation approach had been reported in the document NUREG-0744, 'For Comment'. One of the substantive comments received was that the NUREG report suffered from a lack of illustrative examples. The authors, both of whom serve on the ASME Code Subcommittee that plans to establish elastic-plastic fracture mechanics vessel margins, prepared this report to provide some example calculations and to serve as a basis for the committee work.

Although this paper has maintained the authors' version which references the 'For Comment' version, the reader is advised that the final report has been issued as NUREG-0744, Revision 1. With the issuance of that document, the NRC completed Task A-11 and it became a resolved issue. Responsibility for all implementation actions has been transferred to the Division of Licensing, Office of Nuclear Reactor Regulation of the NRC.

REFERENCES

I. Johnson, R., et al., Resolution of the reactor vessel materials toughness safety issue, N UREG-0744, "For Comment', U.S. Nuclear Regulatory Commission, Washington, D.C., September 1981.

2. Rolfe, S. T. and Novak, S. R., Slow bend K~c testing of medium-strength high- toughness steels, Review of Developments in Plane Strain Fracture Toughness Testing, ASTM-463 American Society for Testing and Materials, Philadelphia, 1970, pp. 124-59,

3. ASME Boiler and Pressure Vessel Code, Section Il i , Division 1, Non- mandatory Appendix G, American Society of Mechanical Engineers, New York, NY, 1980.

4. Ernst, H. A., Material resistance and instability beyond J-controlled crack growth, Elastic' Plastic' Fracture, Second Syrnposium, Vol. 1, Inelastic' Crack Analysis, ASTM STP 803, American Society for Testing and Materials, Philadelphia, 1983, pp. 191 213.

APPENDIX: LICENSING ASPECTS*

The staff concludes that the approach and methodology described in the following paragraphs provide an acceptable means for all commercial

* This appendix quotes the content of a page from NUREG-0744, ~For Comment'. (Sections of the text have been italicized for emphasis by the present author.)

Ductile .[taw stability in nuclear pressure vessels 53

nuclear power reactor licensees to meet the requirements of 10 CFR 50, Appendix G, with regard to the need to demonstrate adequate margins for continued operation when the requirements of Section V.B. (of Appendix G) cannot be satisfied.

First, all plants should establish J -T curves for all materials in the RPV, either from experimental J -R curves or by correlation with the lower bound J (at J /T=50)=f(USE) curve. Second, the USE at the plant-specific end of life (EOL) should be established in accordance with 10 CFR 50 and ASME Code. If the EOL USE>50ft-lb, the RPV is acceptable (other factors, detailed in 10 CFR and in the Code, remain in force). I f the EOL USEJlc. If the safety margin for some operating conditions is unacceptable, the licensee may opt to modify the plant system, plant operations, or both, to ensure that potentially damaging conditions are avoided.

The schedule for implementation of these actions for operating plants will be established concurrently with the issuing of this NUREG report in final form. With publication of NUREG-0744, the need to modify Appendix G to 10 CFR 50 (and possibly, Appendix H) will be established. The responsibility for these modifications and the timing of that task will be established by NRC.

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