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Fracture and safety analysis of nuclear pressure vessels
~,~pi~rr;ng
Frucrrwr Mrchcmics.
1973. Vol. 4. pp. 43 l-446.
Pergamon Press.
Printed in Great Britain
FRACTURE AND SAFETY ANALYSIS NUCLEAR PRESSURE VESSELS-t
OF
G. BARTHOLOMe, M. MIKSCH, G. NEUBRBCH, and G. VASOUKIS Siemens Aktiengesellschaft, Reactor Technology, E%Iangen, West Germany AbWnct- An evaluation of the safety factors of nuclrar reactor prcssuze msds for pressurized water rcactars to resist fracture is an important feature in ensuring nuckar reactfx safety. The safety analysis has to demonstrate that there is no danger of failure at all. By ~~JIUSof brittle and ductile fractwe mechanics the safety factors against critical stress, critical cmck sixe and critical stress intensity factor will be sbwn. The linear, elastic and plastic stress distributions for plane stress pnd pka strain umditions are derived for the different modes of fracture. A critical analysis of crack gtuwtb and of stress corrosion cracking extends the use of frachue mechanics. A comparison between stress intahty lkc%n~obtainul by analytical computation and by the Finite Element Method for the case ofnacrlcacy caecOoliqissbown. Some tech&al applications as the estimation of critical crack sizes, the evaluation of operating performance and undercladding cracks are briefly discussed.
INTRODUCTION THE MOST important feature in the application of nucltar energy is the safety of the nuclear power plant. The safety analysis of the whole primary circuit cspccially has to prove the intcgity of the nuclear reactor pressure vessel. This is done by using a fraeturc mechanics approach. The results of that approa& show that thcrc is a high safety factor against brittle and ductile fracture of the nuclear reactor pressure vessel. BASIC THEORY OF FRACTURE MECHANICS The two principal types of possible failure of a technical pressure-rctainingcomponent are brittle and ductile fracture. A safety evahration must wnccrn both these possibilities. B&/e fracture mechanics. (Linear elastic fmcture mechanics). As the basic theory of brittle fracture mechanics is shown in detail in other papers at this Confcrencc, only the main principles arc demonstrated to give a picture of the necessary information one has to have. In 1920 GrifIith [ 11developed a eritcrion for brittle fracture for ideal brittle mat&al using an energy balance of the elastic energy and the surface energy. As criteria for brittle fracture, the crack cxtcnsion force must bc greater than the critical crack extcnsion force, which is a material constant. This crack extension force must be derived for the plane strain and the plane stress condition. Plane stress is defiqed as a state of stress which is described in every volume clcment by a set of principal stresses, one of which is zero. Generahy, a plane stress condition exists in thin sections in which the stress normal to the surfaces is nearly zero. Plane strain is defined as a state of stress which results in a zero strain along a specific direction. The expressions for the two conditions are: (a) for plane stress,
tPresented at the Symposium on Fracture und Fatigue at the School of Engineering and Applied Science. George Washington University. Washington. D.C. May 3-5.1972. 431
432
G. BARTHOLOM6
et al.
and (b) for plane strain,
where g is the crack extension force, a the half crack length, u the applied stress, E, Young’s modulus, and YPoisson’s ratio.
The crack extension force, g, can be related to the stress intensity factor, K, which is the constant in the stress equations in the -on of the crack tip. Plane stress: g = KS/E
Plane straim
The stressequations inthe @on of the crack tip depend on the mode offiacture. The three modes of fmcture are (I) wedge opening, (II) forward shear and, (III) parallel shear. The crack extension forces for the dil&ent modes are as follows: ModesIandII plane stress
plane strain g1.n
_+(l-v2)
Mode III 8m
Kik =y(l+v).
BrittltfrackneoccurswhenthestrsssintensityfactorKisequaltoorgrtaterthanthe critical stress intensity factor K,. The critical stress intensity factor, K, is a material constant and a function of temperatme, load& rate, and neutron irradiation. Generally, the linear elastic stresses near the crack tip can be described [2]:
where i, k = x, y, z; n = hilure mode I, II, III; Ftk is a function of the angle; K. is a constant that is independent of r and cp;u is the stress in directions i, k; r is the distance from the crack tip; and y, is the angle. As an example, the stress equations are given for an infinite plate with a crack of the length a under biaxial load a:
Fractureand safety analysis ofnuclear pressurevesseis
433
surveyof the stress analysis of cfacks has been given by Paris and Sih 131for numerous cases and geometries. To evaluate rcA stresses in real structures it is necessary to take into account all kinds of ioadings. For any loading the crack extension force g is the sum of the crwk extension forces of the single modes: An excellent
For difBerentloadings of the same mode (stress i~~~sities caused by pnssurc, temperature, etc.) the stress intensity K is the sum of the single stress iutewities of the same mode; for example, for Mode I: K, 4 &(l)+&(2)+.
..
formulae shown above are strictly appiicable only in the linear elastic field. The influence of stresses near the yield stress can be evahWed approximately by a correction f&tor[4]:
The
Q = cbg-0.212 (3 #
is a cumpletc eliipticaIi&q@, expressed as
whefeaisthcbalf~klicasth,c~ecrackdepth,cr~grosss~ss,ando,a0.2per cent o&et tensile yield strws. Elastic--plastic fiactwe mechanics. In order to get an answer to the iaflueace of yi~~st#e~tip,ayitMiagrnodtlofloatlp~cbebaviotnatasherphatchwas dcvekq&. Details on the plastic analysis are given in other papers; procedures are given by wens [5I. The plastic xone size at the crack tip for plane stress is
wberc r,, is the plastic zone size, 1ythe stress intensity factor, and q, the yield strength.
From the displacement equations the crack opening displacement (COD) S can be caiculatedas
In the presence of large yielding the COD reaches the value
EFt.i.Vol.fNo.t-P
434
G. BARTHOLOMi.
er al.
where t is the wall thickness. The designer has only to know the critical value 8, of the COD to evaluate the safety against fracture in the presence of large yielding. Attention, however, should be given to an exact elastic-plastic analysis of a structure with cracks. The use of more sophisticated calculating methods (e.g. the finite element method) has aheady given excellent results [6]. The relations described are a useful tool for the engineer to ensure the safety of components against brittle fracture. The application ofthe relations is detailed later in the paper. Ductilefiacture mechanics. In addition to the considerations of brittle fracture, the possibility of ductile fracture has to be evaluated. Components such as nuclear reactor pressure vessels operate in temperature regions where the allowable crack sixes for brittle fracnne come into such magnitudes that their influence on a possible ductile failure must be calculated. Baaed on experimental results, correlations between crack size and stress were estaMahed. One approach[7] is based on the observation that, for very tough ductile material, the failure caused by defects is controlled by largescale yielding. The critical hoop stress for a through crack is:
where m* is the flow stress [= O-5 (a,+~,,)], uy the ultimate strength, a, the yield strength, a the half crack length, R the radius of the cylinder, and t the wall thickness. This more or less empirical equation fits very well with the experimental results of many tests. For conservative calculations of the through crack another approach, using the mechanical model given in reference [8], results in the expression
Again it should be mentioned that these formulae only give an approximate but conservative solution, and further investigations and calcuhuions should be carried out with more detailed methods, for example with the &rite element method. An example for the critical stresses for brittle and ductile fmcture is given later under the heading ‘Safety evaluation”. Fatigue crack growth. The fracture mechanics correlations given above are also a good tool for evaluating the amount of stable slow fatigue crack propagation. The growth of an initial crack with a given crack length and crack geometry is dependent upon temper&me, environment, material, and applied stress range. In terms of fracture mechanics the crack growth can be expressed by
where da/dN is the crack growth (da) per cycle (dN), C,, a constant depending on material, frequency, mean load, etc., AK the stress intensity range, and n the exponent,
Fractureand safety analysis of nuclear pressure vessels
435
given by experimental results. The stress intensity range AK can be calculated using similar expressions to those for the stress intensity: AK - AVG. where Au is the applied stress range and a the actual crack size. The crack growth per cycle, da/dN, has to be determined experimentally, and is given in terms of the stress intensity range AK. Stress corrosion cracking. The influence of a corrosive environment upon the critical fracture toughness has to be examined. The critical stress intensity factor for stress corrosion cracking is a constant for a given material and environment. For nuclear reactor pressure vessels, stress corrosion cracking can be neglected as the pressure vessels are clad with an austenitic cladding which prevents this danger. Validity of type of approach. The most recent critical review of the developments in plane strain fracture mechanics [9] states that the above formulae are valid within the indicated boundaries. This review covers current progress in fracture testing, the influence of crack length and thickness, a WC analysis of the low energy fracture, and inch&s a commentary on present practice. A survey on the research for investigating the e&cts offlaws, var&tions of properties, and residual strest~on the stnMural r&ability is summari& in reference [lo]. Refmnce [ 111 discusses, in detail, the critical aspects of ductile &Sure mechanics; the use ofthe critical COD technique is detaikd in reference [123;and the application of notch ductility for the design and spec&ation of permissible defect sixes in welded metal structures is stated in reference [ 131. The application of fracture mechanics on nuclear reactor pressure vessels is extensively demonstrated in reference [ 141for the fohowing items: (a) metallurgy and materials; (b) forming, welding, and inspection; (c) large-scale testing; and (d) environmental factors and fatigue. CALCULATION METHODg In order to obtain immediate and inexpensive evaluations ofthe safety of pressure vessels against brittle fracture the application of analytical methods is recommendable. The reliability of these analytical methods should be proved by means of the &rite element method. More over, Unite elements are needed in very dii&ult cases for which no analytical equations are relevant (e.g. flaw in a pressure vessel nozzle). Analytical method. Well-known formulae for thick-walled pressure vessels result in the stress caused by the internal pressure and by the temperature distribution. Residual stresses are determined experimentally or by way of calculation. The components of these stresses perpendicular to the plane of the crack are needed for the computation of stress intensity factors KIP (caused by internal pressure), KIT (caused by temperature) and KIR (caused by residual stresses). If a stress distribution is constant along the crack the first equation in Fig. 1 can be used. But normally stresses are not constant and the second equation. is applicable. The safety against brittle fracture is assured if the sum of these three stress in-
436
G. BARTHOLOMk
M/‘t~-cimdlmkrm
et al.
dmfdwarlraia
Fig. 1. Fomw&e for cakuhtion of stress intensity Esctors K,.
tens&y fhctors is less than the criticalmaterialproperty,the fracturetoughness &,.
Finite element method. The application of the finite element method to a fracture mechanics analysis will be shown on the basis of an example: a hoop crack in the cyliudrical part of a nuclear pressure vessel during the emergency core cooling.The idealis&ion of the structure and the calculation of the stresses and displacements will be shown first,followed by a fracturemechanics approach. Culculurion of stresses. As the hoop crack is assumed to be in the middle of the cylindrical part (Fig. 2), only one part of the structure, that is the upper half, needs to be analysed (due to the symmetry). As both the geometry and the load are axisymmetrical- the temperature is radius dependent only-the ASKA J3lementTRIAX 6 is used for idealisation. This is an axisymmetric ring element with Wngular cross-section, 6 nodal points and 12 degrees of freedom and a quadraticaldisplacement field. Since the stresses and displacements near the plane of the crack should be known with high accuracy, in this region a very fine mesh is chosen, going over to largerelements in the upper part of the structure. As the plane of the crack (z = 4) is a plane of symmetry, the boundary conditions of symmetry have to be introduced: the axial dis-
Fig. 2. Cylindrical part of a reactor pressure vessel, finite element ideaiisation.
Fracture and safety analysis of nuciear pressurevessels
437
placements of all nodal points in the lowest line (from No. l-39) are restricted to be zero. The hoop crack is simulated by releasing a certain number of nodal points in this line. As we chose axisymmetric elements and only thermal loads were applied in particular no internal pressure-no additional boundary conditions have to be introduced at the inner and outer surfaces. At the upper boundary it would have been nice to have sliding boundary conditions. That means the axial displacements of the 5 nodal points in this line (No. 670-674) have the same value. But this has not yet been available with ASKA at the time of this computation. So we let the upper boundary free. As it can be seen in the literature [ 151, the influence of a disturbance in a hollow cylinder is unimportant after a length
m
x=3m
In this case X = 1500.
Figure 3 shows the deformed structure compared with the undeformed structure. Ba cause the undeformed structure is related to zero temperature level an additional expansionhasbeensuperposed. To reduce this expansion and to make the deflection more visible, the deformed structure has been moved. Furthermore the displacements both in r- and in Zdirection havebeenscaledbyafactor 10. Figure 4 shows the stress distribution in the plane of the crack-of course with no real crack there-resulting from analytical soliltion assuming the cylinder being infhritely long and integrating the d&rential equations of the stresses directly. The analytical solution gives v&es of the stresses which are a little bit smaller than those computed with the f&&e element method. Cufculutionof stress intensityjt.~fofs. In the piece&g chapter the computation of the undamaged cylindrical part of a reactor pressure vessel during the emergency core cooling was shown. Now the behaviour of the same cylinder under the same loadings, but provided with inhomogeneities which could perhaps cause brittle fracture, will be considered. The most dangerous defect is a single crack of finite length at the inner surface where the tension is a maximum. Therefore our finite element net has been provided
--
--d6iamed
Fig. 3. Dcfomed structure.
438
G. BARTHOLOMti
et al.
_f_k 60
loo
150
2al -Wdthii
mm250
Fig. 4. Stress VIwallthickness(after 7 min).
with a hoop crack running all the way round the inner surface with variable crack depths of 0,20,40,60 and 80 mm. The crack has been produced at the lower wedge of the idea&cd structure by setting free the nodal points on the crack surface. For this evaluation only the axial stresses a, and the axial displacements u are important. Figure 5 shows the radial distribution of a, in the crack plane for several crack depths. For this example only, the situation 7 min after the b&uting of emergency core cooling, will be considered. Similar drawings could be produced from the axial distribution of CT~ above different crack tips or Born the displacements of the crack surfaces. The virtual elastic energy of the whole structure could be plotted vs the crack depth. From each of these four types of diagrams a stress intensity factor K, can be computed in four St ways which ought to give the same results (Fig. 6): (a) From the stress distribution in the plane of the crack, a&). (b) From the stress distribution perpendicular to the plane of the crack, cry(y). (c) From the displacements of the nodal points on the crack surface, u(r). (d) And from the change of the elastic energy W vs the change of the crack surface A. The determination of KI can be made by computing the arguments of the lim-functions,
--.-10.8% -m-.-
41.6%
Fig. 5. Cylinder with circumferential crack during themal
shock.
Fracture and safety analysis of nuclear pressure vessels
439
Fig. 6. Calculation of stress intensity factorK from results of finite element calculations.
K,*, for several nodal points, drawing them into graphs K?(x), K?(y) or K?(r), and by extrapolating the connecting curve vs X,y or r equal to 0. The easiest way to obtain Kris the energy method (compliance) which is compared in Fig. 7 with an analytical solution. The Merence between a critical material property, the fracture toughness KIE, and the stress intensity factor K, can be considered as the safety against brittle fractme. The dM&rence between the &rite element solution and the analytical solution can also be found in other studies [ 161. Now the extrapolation method ought to be used as well. We tried to obtain K,from the displacements and from the stresses. But -ties arise when extrapolating for K, because the points near the crack tip seem to aim at the origin. Therefore no extrapolation can be applied with a clear conscience. The main conclusion to be drawn so far would seem to be that the finite element mesh near the crack tip has not been line enough. But it will now be explained why even the finest finite element mesh cannot result in a reasonable K,-value if not a special ftacture mechanics element is used. Figure 8 shows again the computation of K,from the displacements: (a) At the top there is the equation for KI(I)and an admissible simpli8cation (II). sH!SsMmiQMoI N/mm* t
4c
Fig. 7. Cylinder with circumferential crack dwing thermal shock.
440
G. BARTHOLOM6
A,.
dii r-o
u(r.2)
et al.
v;
= A+Br+
Crz+D.z+E.zz+f.r.z;
Z*O:U(fZZ-O)*B.r+C2; Kr-Frnli
Al,
~.Oi
u(cr)=A+8.~
+C~r+D~+E.z+F.r.z;
u(r,z=o)=tW+C.r;
Fig. 8. Calculation of
K,
frcm~ crack
open@
disphement.
(b) An element with 6 nodal points requires an equation for the displacements u with6unknowns. (c) Equptioa III is the usual formula for u and equation IV is the special form in the gackplaaensprthecracktip. (d) Using the conventional formula for u toge&er with the extrapolationmethod it will be found that K, is exactly equal to 0. (e) Conclusion: This element is not suib&k enough for CaicuMons in fracture me&a&s by means of the extrapolationmethod. An element with a(dr -distribution) for the di@acements is needed urgently. (f) At the bottom there is an equation for u (VI), which would better be able to sati& fracturemechanics requirements. Figure 9 shows the corresponding treatment for the determination of KI from the stresses: ’ the conventional type of element results in a wrong K,, whereas the proposed Yam r element would provide a suitable solution.
u(x,zJ=A+Bx+Cx'+D.z+E.z'+F.x.z;
h-
= 0+2.E.z+
*
F.x;
c7~(z,x-o)-G,+H,.z; fi = !E,,
A2dG,
+M,.zb~
- Oj
u(x,zbA*BVji+C.x+D\jt+E~z+F.x~z;
62 = + I stD&*Etf.x; 4
(1, x=0)-G)
Fig.
9.
+
?
z
;
Calculation ofK, from stress distribution along sack tip.
Fracture and safety analysis of nuclear pressure vessels
441
TECHNICAL APPLICATION Estimation of permissible crack sizes for the specification of standards for inspection, hydrostatic pressure test, and for recurring inspections. with the @en methods
it is possible to specify allowable crack sizes. As will be shown later in the paper, the influence of crack growth is negligible. Thus the calculation of critical, allowable, and detectable crack sizes can be simpl.iGed.In accordance with the first equation of the Fig. 1 the critical crack size is proportional to
Standards for inspection: For a given material (ASOSC 12, stress equal to a yield strength of 400 N/mm*, fixture toughness of 5000 N/mm1’5, neutron dose of 5 X 10IRnvt, and temperature of 3OoC) the critical crack dimensions - length and depth-are ilhtstrated in Fig. 10. To obtain again a very conservative value of the allowable crack size, a factor of 5 is applied to the critical crack sire. This factor includes a factor of 2 on the fracture toughness (a - Kf,) and a factor of 1.25 on other parameters such as stress, crack geometry, and crack size. The detectable crack size has to be well below the allowable crack size which is demonstrated. Evaluation of the operating performarpce and life expectancy. To demonstrate the safety against brittle fracture during opera&n ofthe reactor pressure vessel, the operating stresses have to be below the critical sttesses for the specifkd allowable crack sizes. The very tzun~ervative superposition of membrane and thermal stresses during the operation is always outside the zone of caution defined by Porse [ 171.In addition, it can be seen from Fig. 11 that there exists a large safety margin against the critical crack size far inspection, hydrostatic pressure test, and recurring inspection. This statement is based on the assumption that no remarkabk crack growth occurs during the life of the reactor pressure vessel. This will be briefly demonstrated. In spite of the high standard of inspection during manufacture, which enables defects much less than the critical crack size for the hydrostatic pressure test to be detec&d and elimCriliieneLbprhaa
I
0
1
uloN.nnn-2 30 “C
8.1
5.10' nvl
)Mn*I
22 NiMoCr37
h
SlWNVf#+
Fig. 10. Critical, allowable and detectable crack size.
442
G. BARTHOLOe
ef al.
Fig. I 1. Brittle fracture diagam.
mated, this crack size is used as the initial crack size for the calculation of fatigue crack growth. For this purpose all specified cycles of load have to.be given with numbers of cycles and stress ranges Au, and the stress intensity range AK is thereby determined: AK= For each transient the crack growth is computed and multiplied with the number of the specified cycles. The sum of the integrated crack growth for all transients gives the total crack growth during the life of the component. For a typical nuclear power plant speci&tion, the total crack growth is below 05 mm. Emergency cure cooling. After a hypothetical double ended failure of a main coolant piping the reactor core must be cooled by the emergency core cooling system. For this cold water is fed into the reactor pressure vessel. This results in a high temperature gradient in the wail which produces thermal shock stresses. By meansof fixture mechanics it can be shown that the stress intensity factors produced by these stresses and by the presence of flaws is less the fracture toughness. The stress intensity during emergency core cooling is calculated in [ 181and plotted in Fig. 12 vs the temperature at the crack tip. The comparison with the fracture toughness shows that even much deeper cracks cannot cause brittle failure. Undercladding cracks Description of undercladding
cracks. The cracks found under the strip electrode cladding of reactor components display the geometry and distribution shown in Fig. 13. The causes and phenomenology of undercladding cracks are described in detail in the literature under [ 191. Here a typical undercladding crack is shown. The analysis of the crack revealed that the depth is less than 2.5 mm. The depth of the crack corresponds with that of the coarse-grain zone, as was proved by statistics and metallographic research. The length is less than 12 mm. Safety against brirrfe failure. The analysis of brittle failure is based on the Pellini concept, the crack instability and the crack arrest behaviour during hydrostatic tests, core emergency cooling and under unrealistic worst-case conditions. The measure-
Fracture andsafetyanalysisof nuclearpressurevessels
443
Emorgenqcue ccdmq
400 500 2m
100
Fii. 12. Stress intensityfactor& andfhcturc toughest& metercrack depth).
vs. temperatureat crack tip(para-
Fig 13. Heat athted zoneof A 508 Cl. 2 undera l-layer cladding
ments of the nil ductility transition temperature prove that the conditions imposed by the Pelhni concept are fGlled. Safety against instability of undercladding cracks can be proved by utilizing the iinear elastic fracture mechanics. For this purpose evidence must be produced that the stress intensity K, - u. Vu is smaller thait the fracture toughness K,,. Unrealistic worst-case con&ions. The stress intensity K, characteristic curve as shown in Fig. 14 results from the assumption of the maximum possible residual stresses, which are equal to the actual yield strength, and the internal pressure existing during the over-pressure test. A comparison with the existing fixture toughness K,, shows that crack instability can occur only if the “crack depth is greater than 15 mm”. It should be noted that the actual residual stresses are far below the actual yield strength [20,21]. The comparison and calculations made permit the conclusion that the undercladding cracks do not affect the operation and safety of nuclear power plants. SAFETY ANALYSIS Sqfety evaluation
analysis of a reactor pressure vessel has to demonstrate that there is no failure at all. By means of brittle and ductile fracture mechanics this safety is well established. ‘Safety” can be classified as safety against (a) critical stress, (b) critical crack size, The safety
danger of
G. BARTHOLOMe
444
.“.
0 _
-m
lil
er al.
ti FK
wbz
_,_
4bmm
sb Bummmd
-CWAbpma
J‘
Fig. 14. Stress intensity factor vs crack depth a.
and (c) critical stress intensity factor. To get these safety factors one has to perform a procedure using the previously described methods, including loading assumptions (specification), stress and strain analysis (calculation), materials analysis (properties), defect analysis (initial and recurring inspection), longtime behaviour of the material @ad&ion, creep, corrosion, f%igue), and fracture mechanics analysis. Safety against critical stress. The results of such a safety evahMon is demon strated in Fig. 15. Over the entire range of the operating temperature the operating stresses are below the critical’ stresses for brittle and ductile fracture. For a given crack size (inspection, hydrostatic pressure test, recurring inspection) the safety against the critical stress can be given in numbers. The fracture diagram (Fig. 15) is constructed for a reactor pressure vessel having the following data: Diameter Wail thickness Yield stress ultimate stress
5240 mm, 235 mm, 400 N/mm*, 600 N/n&,
Fig. 15. Fracture diagram (cylindrical Part).
Fracture and safety analysis of nuclear pressure vessels
445
Neutron exposure 5 X lo@ nvt Material A508C12 (22I&MoCr37), The diagnun also shows the transition between brittle and ductile fmcture bebaviour for a given crack size. To simplify the example, only the through crack is shown. Other crack geometries, such as surface cracks and inside cracks, can be easily calculated. S4fery against crizicul crack size. Having in mind the sizes in Fig. 10 for critical, allowable, and detectable cracks one can quantify the safety against critical crack size for all states of loading. Safefy aguinsr critical stress intensiryfucror. In cases where the conventional criteria for brittle fracture-as shown in Fig. 11 -cannot be proven, one can demonstrate by means of a hcture mechanics analysis that the calculated stress intensity factor for the worst crack size and stress condition has great safety against the fracture toughness.
CONCLUSIONS The safety of a nuclear power plant is the most important feature of the application of nuclear encrBy. This safety depends mainly on the integrity of the primary circuit of the reactor system. To qua&@ this safety the application of fmcture mechanics gives excellent results. The application of fracture mechanics requires an exact knowledge of the theory of brittle and ductile fhcture. The safety evaluation by means of hcture mechanics proves that there exists a very high degree of safety of the nuclear reactor pressure vessel against ihcture.
RRFERENCES 111 A.A.GriIBth,Phil.Trans.R.Soc,A221,163(1920). 121 H. J. G. Blauel, Materidprhfung 12, (No. 3). 69 (1970). [31 P.C.Pmis,tmdG.C.Sii.AS.T.M.STP381,30(1%5). [41 C.F.TiiyandJ.N.Mastws,AS.T.M.STfjsl,249(1%5). [Sl A. A. Web, Er. W&J. 221 May (1968). 161 C. Visser, S. E Gabriek W. Van Buren, W. A. Maxey, A. R D&y and T. A. Atterbmg, HSST Program, TR4; WCAP-7368 October ( 1969). PI R. J. Riir, BMI-1866. Uuly) (1969). 181 A.S.M.E. boiler and pressure wssel code, Sect. III: Nuclear vessels, Am. Sot. Mech. Engrs, New Yo* ( 1968). 191 W. F. Brown, Jr., Review of developments in plane strain &acture wughness testing, A.S.T.M. STf 463. (1970). Semiannual Progress Report for period ending February 28.1970, ORNLDOI F. j.‘WiiHSST-prognzm, 4590, (october) (1970). r111 Symp. Practical Fmcture Mechanics for Structural Steel R&y, (U.K.A.E.A.), April, (1969). WI RW.Nichdr,‘IheuKafaiticalrracLopcnisOdisplacemcattecbniquesforthesdcctionof~~ resistantmaterials,Symp. Practical Fracture Mechanics for Structural Steel, Ri&y, April, (1969). of permissiile defect sizes in welded metal [I31 A. A. Wells, Notch ductility, de&n and the spec&akn structures, Symp. Practical Fracture Mechanics for Structural Steel, Rialey, April. (1969). 1141 M. S. We&&r, The techndoey of pressure-ret&in8 steel wmponents, Nuclear Metall. 16. September, (1970). its D~~kkeasel. Design News, March. (1963). [ISI W. Griffel, St”wn Element Amdysis of structrual IntenE&y of a Reactor Pressun tl61 D. J. Ayres and W. F. Siddall, Fii Vessel dur@ Emer8eney Core Cool&, Combustion Ensineain& Inc., A-70-19-2, January, (1970). 1171 L. Parse, J. has. Engng, Trans. Am. Sot. me&. Engrs. (Series C), 743 (1964). [I81 G. BartholomC, Analysis of brittle failure of the reactor pressure vessels during emergency core cooling, Corlference on Reactors Bonn, pp. 323-326 (197 1).
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1191 H. Cetjak and W. Debray, Experience gained on austenitic cladding of nuclear reactor components, VGB Coy&ewe on Materials, Paper No. 3, (1971). [20] R. KC&, Residual S@CSSCS of the 6rst order in clad components,VGB Conference on Materials, Paper No. l(1971). on Materials, Contribution [2 11 K. Detcrt, Measuremen t of residual stnsscs in clad plates, VGB Coerence to PaperNo. l(1971). (Received 2 May 1972)
Frucrrwr Mrchcmics.
1973. Vol. 4. pp. 43 l-446.
Pergamon Press.
Printed in Great Britain
FRACTURE AND SAFETY ANALYSIS NUCLEAR PRESSURE VESSELS-t
OF
G. BARTHOLOMe, M. MIKSCH, G. NEUBRBCH, and G. VASOUKIS Siemens Aktiengesellschaft, Reactor Technology, E%Iangen, West Germany AbWnct- An evaluation of the safety factors of nuclrar reactor prcssuze msds for pressurized water rcactars to resist fracture is an important feature in ensuring nuckar reactfx safety. The safety analysis has to demonstrate that there is no danger of failure at all. By ~~JIUSof brittle and ductile fractwe mechanics the safety factors against critical stress, critical cmck sixe and critical stress intensity factor will be sbwn. The linear, elastic and plastic stress distributions for plane stress pnd pka strain umditions are derived for the different modes of fracture. A critical analysis of crack gtuwtb and of stress corrosion cracking extends the use of frachue mechanics. A comparison between stress intahty lkc%n~obtainul by analytical computation and by the Finite Element Method for the case ofnacrlcacy caecOoliqissbown. Some tech&al applications as the estimation of critical crack sizes, the evaluation of operating performance and undercladding cracks are briefly discussed.
INTRODUCTION THE MOST important feature in the application of nucltar energy is the safety of the nuclear power plant. The safety analysis of the whole primary circuit cspccially has to prove the intcgity of the nuclear reactor pressure vessel. This is done by using a fraeturc mechanics approach. The results of that approa& show that thcrc is a high safety factor against brittle and ductile fracture of the nuclear reactor pressure vessel. BASIC THEORY OF FRACTURE MECHANICS The two principal types of possible failure of a technical pressure-rctainingcomponent are brittle and ductile fracture. A safety evahration must wnccrn both these possibilities. B&/e fracture mechanics. (Linear elastic fmcture mechanics). As the basic theory of brittle fracture mechanics is shown in detail in other papers at this Confcrencc, only the main principles arc demonstrated to give a picture of the necessary information one has to have. In 1920 GrifIith [ 11developed a eritcrion for brittle fracture for ideal brittle mat&al using an energy balance of the elastic energy and the surface energy. As criteria for brittle fracture, the crack cxtcnsion force must bc greater than the critical crack extcnsion force, which is a material constant. This crack extension force must be derived for the plane strain and the plane stress condition. Plane stress is defiqed as a state of stress which is described in every volume clcment by a set of principal stresses, one of which is zero. Generahy, a plane stress condition exists in thin sections in which the stress normal to the surfaces is nearly zero. Plane strain is defined as a state of stress which results in a zero strain along a specific direction. The expressions for the two conditions are: (a) for plane stress,
tPresented at the Symposium on Fracture und Fatigue at the School of Engineering and Applied Science. George Washington University. Washington. D.C. May 3-5.1972. 431
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G. BARTHOLOM6
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and (b) for plane strain,
where g is the crack extension force, a the half crack length, u the applied stress, E, Young’s modulus, and YPoisson’s ratio.
The crack extension force, g, can be related to the stress intensity factor, K, which is the constant in the stress equations in the -on of the crack tip. Plane stress: g = KS/E
Plane straim
The stressequations inthe @on of the crack tip depend on the mode offiacture. The three modes of fmcture are (I) wedge opening, (II) forward shear and, (III) parallel shear. The crack extension forces for the dil&ent modes are as follows: ModesIandII plane stress
plane strain g1.n
_+(l-v2)
Mode III 8m
Kik =y(l+v).
BrittltfrackneoccurswhenthestrsssintensityfactorKisequaltoorgrtaterthanthe critical stress intensity factor K,. The critical stress intensity factor, K, is a material constant and a function of temperatme, load& rate, and neutron irradiation. Generally, the linear elastic stresses near the crack tip can be described [2]:
where i, k = x, y, z; n = hilure mode I, II, III; Ftk is a function of the angle; K. is a constant that is independent of r and cp;u is the stress in directions i, k; r is the distance from the crack tip; and y, is the angle. As an example, the stress equations are given for an infinite plate with a crack of the length a under biaxial load a:
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surveyof the stress analysis of cfacks has been given by Paris and Sih 131for numerous cases and geometries. To evaluate rcA stresses in real structures it is necessary to take into account all kinds of ioadings. For any loading the crack extension force g is the sum of the crwk extension forces of the single modes: An excellent
For difBerentloadings of the same mode (stress i~~~sities caused by pnssurc, temperature, etc.) the stress intensity K is the sum of the single stress iutewities of the same mode; for example, for Mode I: K, 4 &(l)+&(2)+.
..
formulae shown above are strictly appiicable only in the linear elastic field. The influence of stresses near the yield stress can be evahWed approximately by a correction f&tor[4]:
The
Q = cbg-0.212 (3 #
is a cumpletc eliipticaIi&q@, expressed as
whefeaisthcbalf~klicasth,c~ecrackdepth,cr~grosss~ss,ando,a0.2per cent o&et tensile yield strws. Elastic--plastic fiactwe mechanics. In order to get an answer to the iaflueace of yi~~st#e~tip,ayitMiagrnodtlofloatlp~cbebaviotnatasherphatchwas dcvekq&. Details on the plastic analysis are given in other papers; procedures are given by wens [5I. The plastic xone size at the crack tip for plane stress is
wberc r,, is the plastic zone size, 1ythe stress intensity factor, and q, the yield strength.
From the displacement equations the crack opening displacement (COD) S can be caiculatedas
In the presence of large yielding the COD reaches the value
EFt.i.Vol.fNo.t-P
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where t is the wall thickness. The designer has only to know the critical value 8, of the COD to evaluate the safety against fracture in the presence of large yielding. Attention, however, should be given to an exact elastic-plastic analysis of a structure with cracks. The use of more sophisticated calculating methods (e.g. the finite element method) has aheady given excellent results [6]. The relations described are a useful tool for the engineer to ensure the safety of components against brittle fracture. The application ofthe relations is detailed later in the paper. Ductilefiacture mechanics. In addition to the considerations of brittle fracture, the possibility of ductile fracture has to be evaluated. Components such as nuclear reactor pressure vessels operate in temperature regions where the allowable crack sixes for brittle fracnne come into such magnitudes that their influence on a possible ductile failure must be calculated. Baaed on experimental results, correlations between crack size and stress were estaMahed. One approach[7] is based on the observation that, for very tough ductile material, the failure caused by defects is controlled by largescale yielding. The critical hoop stress for a through crack is:
where m* is the flow stress [= O-5 (a,+~,,)], uy the ultimate strength, a, the yield strength, a the half crack length, R the radius of the cylinder, and t the wall thickness. This more or less empirical equation fits very well with the experimental results of many tests. For conservative calculations of the through crack another approach, using the mechanical model given in reference [8], results in the expression
Again it should be mentioned that these formulae only give an approximate but conservative solution, and further investigations and calcuhuions should be carried out with more detailed methods, for example with the &rite element method. An example for the critical stresses for brittle and ductile fmcture is given later under the heading ‘Safety evaluation”. Fatigue crack growth. The fracture mechanics correlations given above are also a good tool for evaluating the amount of stable slow fatigue crack propagation. The growth of an initial crack with a given crack length and crack geometry is dependent upon temper&me, environment, material, and applied stress range. In terms of fracture mechanics the crack growth can be expressed by
where da/dN is the crack growth (da) per cycle (dN), C,, a constant depending on material, frequency, mean load, etc., AK the stress intensity range, and n the exponent,
Fractureand safety analysis of nuclear pressure vessels
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given by experimental results. The stress intensity range AK can be calculated using similar expressions to those for the stress intensity: AK - AVG. where Au is the applied stress range and a the actual crack size. The crack growth per cycle, da/dN, has to be determined experimentally, and is given in terms of the stress intensity range AK. Stress corrosion cracking. The influence of a corrosive environment upon the critical fracture toughness has to be examined. The critical stress intensity factor for stress corrosion cracking is a constant for a given material and environment. For nuclear reactor pressure vessels, stress corrosion cracking can be neglected as the pressure vessels are clad with an austenitic cladding which prevents this danger. Validity of type of approach. The most recent critical review of the developments in plane strain fracture mechanics [9] states that the above formulae are valid within the indicated boundaries. This review covers current progress in fracture testing, the influence of crack length and thickness, a WC analysis of the low energy fracture, and inch&s a commentary on present practice. A survey on the research for investigating the e&cts offlaws, var&tions of properties, and residual strest~on the stnMural r&ability is summari& in reference [lo]. Refmnce [ 111 discusses, in detail, the critical aspects of ductile &Sure mechanics; the use ofthe critical COD technique is detaikd in reference [123;and the application of notch ductility for the design and spec&ation of permissible defect sixes in welded metal structures is stated in reference [ 131. The application of fracture mechanics on nuclear reactor pressure vessels is extensively demonstrated in reference [ 141for the fohowing items: (a) metallurgy and materials; (b) forming, welding, and inspection; (c) large-scale testing; and (d) environmental factors and fatigue. CALCULATION METHODg In order to obtain immediate and inexpensive evaluations ofthe safety of pressure vessels against brittle fracture the application of analytical methods is recommendable. The reliability of these analytical methods should be proved by means of the &rite element method. More over, Unite elements are needed in very dii&ult cases for which no analytical equations are relevant (e.g. flaw in a pressure vessel nozzle). Analytical method. Well-known formulae for thick-walled pressure vessels result in the stress caused by the internal pressure and by the temperature distribution. Residual stresses are determined experimentally or by way of calculation. The components of these stresses perpendicular to the plane of the crack are needed for the computation of stress intensity factors KIP (caused by internal pressure), KIT (caused by temperature) and KIR (caused by residual stresses). If a stress distribution is constant along the crack the first equation in Fig. 1 can be used. But normally stresses are not constant and the second equation. is applicable. The safety against brittle fracture is assured if the sum of these three stress in-
436
G. BARTHOLOMk
M/‘t~-cimdlmkrm
et al.
dmfdwarlraia
Fig. 1. Fomw&e for cakuhtion of stress intensity Esctors K,.
tens&y fhctors is less than the criticalmaterialproperty,the fracturetoughness &,.
Finite element method. The application of the finite element method to a fracture mechanics analysis will be shown on the basis of an example: a hoop crack in the cyliudrical part of a nuclear pressure vessel during the emergency core cooling.The idealis&ion of the structure and the calculation of the stresses and displacements will be shown first,followed by a fracturemechanics approach. Culculurion of stresses. As the hoop crack is assumed to be in the middle of the cylindrical part (Fig. 2), only one part of the structure, that is the upper half, needs to be analysed (due to the symmetry). As both the geometry and the load are axisymmetrical- the temperature is radius dependent only-the ASKA J3lementTRIAX 6 is used for idealisation. This is an axisymmetric ring element with Wngular cross-section, 6 nodal points and 12 degrees of freedom and a quadraticaldisplacement field. Since the stresses and displacements near the plane of the crack should be known with high accuracy, in this region a very fine mesh is chosen, going over to largerelements in the upper part of the structure. As the plane of the crack (z = 4) is a plane of symmetry, the boundary conditions of symmetry have to be introduced: the axial dis-
Fig. 2. Cylindrical part of a reactor pressure vessel, finite element ideaiisation.
Fracture and safety analysis of nuciear pressurevessels
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placements of all nodal points in the lowest line (from No. l-39) are restricted to be zero. The hoop crack is simulated by releasing a certain number of nodal points in this line. As we chose axisymmetric elements and only thermal loads were applied in particular no internal pressure-no additional boundary conditions have to be introduced at the inner and outer surfaces. At the upper boundary it would have been nice to have sliding boundary conditions. That means the axial displacements of the 5 nodal points in this line (No. 670-674) have the same value. But this has not yet been available with ASKA at the time of this computation. So we let the upper boundary free. As it can be seen in the literature [ 151, the influence of a disturbance in a hollow cylinder is unimportant after a length
m
x=3m
In this case X = 1500.
Figure 3 shows the deformed structure compared with the undeformed structure. Ba cause the undeformed structure is related to zero temperature level an additional expansionhasbeensuperposed. To reduce this expansion and to make the deflection more visible, the deformed structure has been moved. Furthermore the displacements both in r- and in Zdirection havebeenscaledbyafactor 10. Figure 4 shows the stress distribution in the plane of the crack-of course with no real crack there-resulting from analytical soliltion assuming the cylinder being infhritely long and integrating the d&rential equations of the stresses directly. The analytical solution gives v&es of the stresses which are a little bit smaller than those computed with the f&&e element method. Cufculutionof stress intensityjt.~fofs. In the piece&g chapter the computation of the undamaged cylindrical part of a reactor pressure vessel during the emergency core cooling was shown. Now the behaviour of the same cylinder under the same loadings, but provided with inhomogeneities which could perhaps cause brittle fracture, will be considered. The most dangerous defect is a single crack of finite length at the inner surface where the tension is a maximum. Therefore our finite element net has been provided
--
--d6iamed
Fig. 3. Dcfomed structure.
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_f_k 60
loo
150
2al -Wdthii
mm250
Fig. 4. Stress VIwallthickness(after 7 min).
with a hoop crack running all the way round the inner surface with variable crack depths of 0,20,40,60 and 80 mm. The crack has been produced at the lower wedge of the idea&cd structure by setting free the nodal points on the crack surface. For this evaluation only the axial stresses a, and the axial displacements u are important. Figure 5 shows the radial distribution of a, in the crack plane for several crack depths. For this example only, the situation 7 min after the b&uting of emergency core cooling, will be considered. Similar drawings could be produced from the axial distribution of CT~ above different crack tips or Born the displacements of the crack surfaces. The virtual elastic energy of the whole structure could be plotted vs the crack depth. From each of these four types of diagrams a stress intensity factor K, can be computed in four St ways which ought to give the same results (Fig. 6): (a) From the stress distribution in the plane of the crack, a&). (b) From the stress distribution perpendicular to the plane of the crack, cry(y). (c) From the displacements of the nodal points on the crack surface, u(r). (d) And from the change of the elastic energy W vs the change of the crack surface A. The determination of KI can be made by computing the arguments of the lim-functions,
--.-10.8% -m-.-
41.6%
Fig. 5. Cylinder with circumferential crack during themal
shock.
Fracture and safety analysis of nuclear pressure vessels
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Fig. 6. Calculation of stress intensity factorK from results of finite element calculations.
K,*, for several nodal points, drawing them into graphs K?(x), K?(y) or K?(r), and by extrapolating the connecting curve vs X,y or r equal to 0. The easiest way to obtain Kris the energy method (compliance) which is compared in Fig. 7 with an analytical solution. The Merence between a critical material property, the fracture toughness KIE, and the stress intensity factor K, can be considered as the safety against brittle fractme. The dM&rence between the &rite element solution and the analytical solution can also be found in other studies [ 161. Now the extrapolation method ought to be used as well. We tried to obtain K,from the displacements and from the stresses. But -ties arise when extrapolating for K, because the points near the crack tip seem to aim at the origin. Therefore no extrapolation can be applied with a clear conscience. The main conclusion to be drawn so far would seem to be that the finite element mesh near the crack tip has not been line enough. But it will now be explained why even the finest finite element mesh cannot result in a reasonable K,-value if not a special ftacture mechanics element is used. Figure 8 shows again the computation of K,from the displacements: (a) At the top there is the equation for KI(I)and an admissible simpli8cation (II). sH!SsMmiQMoI N/mm* t
4c
Fig. 7. Cylinder with circumferential crack dwing thermal shock.
440
G. BARTHOLOM6
A,.
dii r-o
u(r.2)
et al.
v;
= A+Br+
Crz+D.z+E.zz+f.r.z;
Z*O:U(fZZ-O)*B.r+C2; Kr-Frnli
Al,
~.Oi
u(cr)=A+8.~
+C~r+D~+E.z+F.r.z;
u(r,z=o)=tW+C.r;
Fig. 8. Calculation of
K,
frcm~ crack
open@
disphement.
(b) An element with 6 nodal points requires an equation for the displacements u with6unknowns. (c) Equptioa III is the usual formula for u and equation IV is the special form in the gackplaaensprthecracktip. (d) Using the conventional formula for u toge&er with the extrapolationmethod it will be found that K, is exactly equal to 0. (e) Conclusion: This element is not suib&k enough for CaicuMons in fracture me&a&s by means of the extrapolationmethod. An element with a(dr -distribution) for the di@acements is needed urgently. (f) At the bottom there is an equation for u (VI), which would better be able to sati& fracturemechanics requirements. Figure 9 shows the corresponding treatment for the determination of KI from the stresses: ’ the conventional type of element results in a wrong K,, whereas the proposed Yam r element would provide a suitable solution.
u(x,zJ=A+Bx+Cx'+D.z+E.z'+F.x.z;
h-
= 0+2.E.z+
*
F.x;
c7~(z,x-o)-G,+H,.z; fi = !E,,
A2dG,
+M,.zb~
- Oj
u(x,zbA*BVji+C.x+D\jt+E~z+F.x~z;
62 = + I stD&*Etf.x; 4
(1, x=0)-G)
Fig.
9.
+
?
z
;
Calculation ofK, from stress distribution along sack tip.
Fracture and safety analysis of nuclear pressure vessels
441
TECHNICAL APPLICATION Estimation of permissible crack sizes for the specification of standards for inspection, hydrostatic pressure test, and for recurring inspections. with the @en methods
it is possible to specify allowable crack sizes. As will be shown later in the paper, the influence of crack growth is negligible. Thus the calculation of critical, allowable, and detectable crack sizes can be simpl.iGed.In accordance with the first equation of the Fig. 1 the critical crack size is proportional to
Standards for inspection: For a given material (ASOSC 12, stress equal to a yield strength of 400 N/mm*, fixture toughness of 5000 N/mm1’5, neutron dose of 5 X 10IRnvt, and temperature of 3OoC) the critical crack dimensions - length and depth-are ilhtstrated in Fig. 10. To obtain again a very conservative value of the allowable crack size, a factor of 5 is applied to the critical crack sire. This factor includes a factor of 2 on the fracture toughness (a - Kf,) and a factor of 1.25 on other parameters such as stress, crack geometry, and crack size. The detectable crack size has to be well below the allowable crack size which is demonstrated. Evaluation of the operating performarpce and life expectancy. To demonstrate the safety against brittle fracture during opera&n ofthe reactor pressure vessel, the operating stresses have to be below the critical sttesses for the specifkd allowable crack sizes. The very tzun~ervative superposition of membrane and thermal stresses during the operation is always outside the zone of caution defined by Porse [ 171.In addition, it can be seen from Fig. 11 that there exists a large safety margin against the critical crack size far inspection, hydrostatic pressure test, and recurring inspection. This statement is based on the assumption that no remarkabk crack growth occurs during the life of the reactor pressure vessel. This will be briefly demonstrated. In spite of the high standard of inspection during manufacture, which enables defects much less than the critical crack size for the hydrostatic pressure test to be detec&d and elimCriliieneLbprhaa
I
0
1
uloN.nnn-2 30 “C
8.1
5.10' nvl
)Mn*I
22 NiMoCr37
h
SlWNVf#+
Fig. 10. Critical, allowable and detectable crack size.
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Fig. I 1. Brittle fracture diagam.
mated, this crack size is used as the initial crack size for the calculation of fatigue crack growth. For this purpose all specified cycles of load have to.be given with numbers of cycles and stress ranges Au, and the stress intensity range AK is thereby determined: AK= For each transient the crack growth is computed and multiplied with the number of the specified cycles. The sum of the integrated crack growth for all transients gives the total crack growth during the life of the component. For a typical nuclear power plant speci&tion, the total crack growth is below 05 mm. Emergency cure cooling. After a hypothetical double ended failure of a main coolant piping the reactor core must be cooled by the emergency core cooling system. For this cold water is fed into the reactor pressure vessel. This results in a high temperature gradient in the wail which produces thermal shock stresses. By meansof fixture mechanics it can be shown that the stress intensity factors produced by these stresses and by the presence of flaws is less the fracture toughness. The stress intensity during emergency core cooling is calculated in [ 181and plotted in Fig. 12 vs the temperature at the crack tip. The comparison with the fracture toughness shows that even much deeper cracks cannot cause brittle failure. Undercladding cracks Description of undercladding
cracks. The cracks found under the strip electrode cladding of reactor components display the geometry and distribution shown in Fig. 13. The causes and phenomenology of undercladding cracks are described in detail in the literature under [ 191. Here a typical undercladding crack is shown. The analysis of the crack revealed that the depth is less than 2.5 mm. The depth of the crack corresponds with that of the coarse-grain zone, as was proved by statistics and metallographic research. The length is less than 12 mm. Safety against brirrfe failure. The analysis of brittle failure is based on the Pellini concept, the crack instability and the crack arrest behaviour during hydrostatic tests, core emergency cooling and under unrealistic worst-case conditions. The measure-
Fracture andsafetyanalysisof nuclearpressurevessels
443
Emorgenqcue ccdmq
400 500 2m
100
Fii. 12. Stress intensityfactor& andfhcturc toughest& metercrack depth).
vs. temperatureat crack tip(para-
Fig 13. Heat athted zoneof A 508 Cl. 2 undera l-layer cladding
ments of the nil ductility transition temperature prove that the conditions imposed by the Pelhni concept are fGlled. Safety against instability of undercladding cracks can be proved by utilizing the iinear elastic fracture mechanics. For this purpose evidence must be produced that the stress intensity K, - u. Vu is smaller thait the fracture toughness K,,. Unrealistic worst-case con&ions. The stress intensity K, characteristic curve as shown in Fig. 14 results from the assumption of the maximum possible residual stresses, which are equal to the actual yield strength, and the internal pressure existing during the over-pressure test. A comparison with the existing fixture toughness K,, shows that crack instability can occur only if the “crack depth is greater than 15 mm”. It should be noted that the actual residual stresses are far below the actual yield strength [20,21]. The comparison and calculations made permit the conclusion that the undercladding cracks do not affect the operation and safety of nuclear power plants. SAFETY ANALYSIS Sqfety evaluation
analysis of a reactor pressure vessel has to demonstrate that there is no failure at all. By means of brittle and ductile fracture mechanics this safety is well established. ‘Safety” can be classified as safety against (a) critical stress, (b) critical crack size, The safety
danger of
G. BARTHOLOMe
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.“.
0 _
-m
lil
er al.
ti FK
wbz
_,_
4bmm
sb Bummmd
-CWAbpma
J‘
Fig. 14. Stress intensity factor vs crack depth a.
and (c) critical stress intensity factor. To get these safety factors one has to perform a procedure using the previously described methods, including loading assumptions (specification), stress and strain analysis (calculation), materials analysis (properties), defect analysis (initial and recurring inspection), longtime behaviour of the material @ad&ion, creep, corrosion, f%igue), and fracture mechanics analysis. Safety against critical stress. The results of such a safety evahMon is demon strated in Fig. 15. Over the entire range of the operating temperature the operating stresses are below the critical’ stresses for brittle and ductile fracture. For a given crack size (inspection, hydrostatic pressure test, recurring inspection) the safety against the critical stress can be given in numbers. The fracture diagram (Fig. 15) is constructed for a reactor pressure vessel having the following data: Diameter Wail thickness Yield stress ultimate stress
5240 mm, 235 mm, 400 N/mm*, 600 N/n&,
Fig. 15. Fracture diagram (cylindrical Part).
Fracture and safety analysis of nuclear pressure vessels
445
Neutron exposure 5 X lo@ nvt Material A508C12 (22I&MoCr37), The diagnun also shows the transition between brittle and ductile fmcture bebaviour for a given crack size. To simplify the example, only the through crack is shown. Other crack geometries, such as surface cracks and inside cracks, can be easily calculated. S4fery against crizicul crack size. Having in mind the sizes in Fig. 10 for critical, allowable, and detectable cracks one can quantify the safety against critical crack size for all states of loading. Safefy aguinsr critical stress intensiryfucror. In cases where the conventional criteria for brittle fracture-as shown in Fig. 11 -cannot be proven, one can demonstrate by means of a hcture mechanics analysis that the calculated stress intensity factor for the worst crack size and stress condition has great safety against the fracture toughness.
CONCLUSIONS The safety of a nuclear power plant is the most important feature of the application of nuclear encrBy. This safety depends mainly on the integrity of the primary circuit of the reactor system. To qua&@ this safety the application of fmcture mechanics gives excellent results. The application of fracture mechanics requires an exact knowledge of the theory of brittle and ductile fhcture. The safety evaluation by means of hcture mechanics proves that there exists a very high degree of safety of the nuclear reactor pressure vessel against ihcture.
RRFERENCES 111 A.A.GriIBth,Phil.Trans.R.Soc,A221,163(1920). 121 H. J. G. Blauel, Materidprhfung 12, (No. 3). 69 (1970). [31 P.C.Pmis,tmdG.C.Sii.AS.T.M.STP381,30(1%5). [41 C.F.TiiyandJ.N.Mastws,AS.T.M.STfjsl,249(1%5). [Sl A. A. Web, Er. W&J. 221 May (1968). 161 C. Visser, S. E Gabriek W. Van Buren, W. A. Maxey, A. R D&y and T. A. Atterbmg, HSST Program, TR4; WCAP-7368 October ( 1969). PI R. J. Riir, BMI-1866. Uuly) (1969). 181 A.S.M.E. boiler and pressure wssel code, Sect. III: Nuclear vessels, Am. Sot. Mech. Engrs, New Yo* ( 1968). 191 W. F. Brown, Jr., Review of developments in plane strain &acture wughness testing, A.S.T.M. STf 463. (1970). Semiannual Progress Report for period ending February 28.1970, ORNLDOI F. j.‘WiiHSST-prognzm, 4590, (october) (1970). r111 Symp. Practical Fmcture Mechanics for Structural Steel R&y, (U.K.A.E.A.), April, (1969). WI RW.Nichdr,‘IheuKafaiticalrracLopcnisOdisplacemcattecbniquesforthesdcctionof~~ resistantmaterials,Symp. Practical Fracture Mechanics for Structural Steel, Ri&y, April, (1969). of permissiile defect sizes in welded metal [I31 A. A. Wells, Notch ductility, de&n and the spec&akn structures, Symp. Practical Fracture Mechanics for Structural Steel, Rialey, April. (1969). 1141 M. S. We&&r, The techndoey of pressure-ret&in8 steel wmponents, Nuclear Metall. 16. September, (1970). its D~~kkeasel. Design News, March. (1963). [ISI W. Griffel, St”wn Element Amdysis of structrual IntenE&y of a Reactor Pressun tl61 D. J. Ayres and W. F. Siddall, Fii Vessel dur@ Emer8eney Core Cool&, Combustion Ensineain& Inc., A-70-19-2, January, (1970). 1171 L. Parse, J. has. Engng, Trans. Am. Sot. me&. Engrs. (Series C), 743 (1964). [I81 G. BartholomC, Analysis of brittle failure of the reactor pressure vessels during emergency core cooling, Corlference on Reactors Bonn, pp. 323-326 (197 1).
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1191 H. Cetjak and W. Debray, Experience gained on austenitic cladding of nuclear reactor components, VGB Coy&ewe on Materials, Paper No. 3, (1971). [20] R. KC&, Residual S@CSSCS of the 6rst order in clad components,VGB Conference on Materials, Paper No. l(1971). on Materials, Contribution [2 11 K. Detcrt, Measuremen t of residual stnsscs in clad plates, VGB Coerence to PaperNo. l(1971). (Received 2 May 1972)