- Email: [email protected]
Failure internal pressure of spherical steel containments
Nuclear Engineering and Design 90 (1985) 209-222 North-Holland. Amsterdatn
209
FAILURE I N T E R N A L P R E S S U R E OF SPHERICAL STEEL C O N T A I N M E N T S G. S A N C H E Z
SARMIENTO
Empresa Nuclear Argentina de Centrales Elbctricas S.A., Av. L.N. Alem 712 (1001) Buenos Aires, Argentina
S.R. I D E L S O H N ,
A. C A R D O N A
a n d V. S O N Z O G N I
lnstituto de Desarrollo Tecnolbgico para la lndustria Quirnica, (INTEC) Casilla de Correo n" 91 (3000) Santa Fe, Argentina
Received 21 May 1985
An application of the British CEGB's R6 Failure Assessment Approach to the determination of failure internal pressure of nuclear power plant spherical steel containments is presented. The presence of hypothetical cracks both in the base metal and in the welding material of the containment, with geometrical idealizations according to the ASME Boiler and Pressure Vessel Code (Section XI), was taken into account in order to analyze the sensitivity of the failure assessment with the values of the material fracture properties. Calculations of the elastoplastic collapse load have been performed by means of the Finite Element System SAMCEF. The clean axisymmetric shell (neglecting the influence of nozzles and minor irregularities) and two major penetrations (personnel and emergency locks) have been taken separately into account. Large-strain elastoplastic behaviour of the material was considered in the Code, using lower bounds of true stress-true strain relations obtained by testing a collection of tensile specimens. Assuming the presence of cracks in non-perturbed regions, the reserve factor for test pressure and the failure internal pressure have been determined as a function of the flaw depth.
1. Introduction The stress analysis of a nuclear power plant steel containment under a design basis accident (mainly the typical loss of coolant accident), and other design and test load conditions are nowadays performed in a relatively straightforward manner [1-11]. A wide variety of applicable computational and analytical tools are available, ranging from hand calculations to detailed finiteelement computer codes [12]. The nature and origin of loads, and the probability of their occurrence both individually and in combination, are widely known [13-15] for any given kind of containment. Core meltdown accidents of the types considered in probabilistic risk assessment (PRA) [16], notwithstanding, have been predicted to lead to environments that are much more severe than those considered for the design basis and may challenge the integrity of the containment structure [17]. The reactor Safety Study (WASH-1400) [18] demonstrated, based on PRA, that core meltdown accidents are the dominant risk contributors to the public from Light Water Reactors (LWR).
One reason for this conclusion was that containment failure and subsequent radioactivity release is possible given by the occurrence of any of a number of physical processes, such as hydrogen combustion, steam explosion, overpressurization of containment, or melt through the containment base-mat [19,20]. The quantification of the internal overpressurization at which the structure will fail, is thus of basic importance in any PRA. In order to determine this "failure internal pressure" it is necessary to define one or more failure criteria for the structure: a limiting stress, strain or some other condition, for example the onset of stability [16]. Given the geometrical complexity of the containments, the definition of failure criteria and the associated failure pressures is far from straightforward. Also, because of the uncertainties in the conditions leading to failure, a specific failure pressure cannot be determined. What is needed for P R A is a density function describing the probability of failure as a function of loading (pressure). The shape of such a density function will vary with the containment design, level of analysis, and knowledge of the details of the actual
0 0 2 9 - 5 4 9 3 / 8 5 / $ 0 3 . 3 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
210
G. Shnchez Sarmiento et al. / Failure internal pressure
containment. In this context, the nominal failure pressure is often defined as that pressure for which the corresponding probability of failure would be 0.5 [17]. Tsai and Orr [21] give a general route for performing such a probabilistic failure assessment. According to these authors, failure criteria must be defined in terms of leakage exceeding certain predetermined limits. At pressures somewhat above the design one, the probability of failure normally would be very small and the corresponding leakage would be from a localized area, but as the containment pressure rises, the probability and the amount of leakage will increase. Uncertainties associated with the approches for the prediction of the containment resistance of all kinds of designs have been extensively reviewed by Greimann and Fanous [22]. A number of studies concerned with the determination of the failure pressure of steel containments in a probabilistic manner have been published by several authors, mainly the last cited ones and co-workers [23-27], SchuEller and co-workers [28,29] and Tsai and Orr [21] among others [30,31].
On the other hand, investigations about the assessment of the ultimate internal pressure of steel containments in a deterministic manner (with fixed material and geometrical data), have been published by several authors: Blejwas and Horschel [1,32-34], Jeschke [35], Hassmann et al. [36], D u n h a m et al. [37], Clauss [38], GOller et al. [10,39] etc. Because of the exhaustive non-destructive testing of the material especially at the weld seams, and in the absence of loads causing fatigue, initial cracks will rarely be present in a pressure containment vessel. However, in the context of a PRA of nuclear power plants, the effects that cracks produce cannot be disregarded, as recognized by SchuEller [29]. In spite of this fact, modes of failure of steel containment due to brittle or ductile fracture are seldom considered in calculations of its failure internal pressure. SchuEller et al. [28,29] made a reliability assessment of the containment of a PWR containing cracks, considering as the failure function that one known in the literature as the "R6 Failure Assessment Diagram" [40], which was developed by the
lock
',
Cramping ~oee\
Fig. 1. Typical spherical steel containment of a PWR or a PHWR-nuclear power plant.
G. S~mchezSarmientoet al. / Failureinternalpressure British Central Electricity Generating Board. An exponential distribution of cracks was assumed and the probability of failure was obtained by solving the respective equations by means of a Monte Carlo approach. In this paper, an application of the R6 approach is presented for a deterministic failure assessment of a spherical steel containment as commonly used for PWR or PHWR nuclear power plants the characteristics of which are shown in fig. 1. The presence of hypothetical cracks both in the base metal and in the welding material of the containment, with geometrical idealization according to the ASME Boiler and Pressure Vessel Code, Section XI [41] was taken into account. Despite the fact that this procedure is deterministic, it can provide a sensitivity analysis of the failure pressure on the values of the strength and fracture properties of the base and welding metals. Conversely, knowing the fixed values of these materials' parameters, one can determine by means of this approach the reserve factor (both on the internal pressure and on the flaw depth) for a given internal pressure, and then, bounds for the failure load. A probabilistic treatment of this procedure, based on the Advanced First Order Second Moment Method is now in progress by the first author and will be published in the near future [42]. 2. The CEGB's 116 failure assessment diagram approach
211
the linear elastic compliance factor; a is the flaw depth; Sf is the applied stress at failure and S 1 is the flaw stress in the plastic zone. The use of eq. (1) for predicting elastic-plastic failure in a variety of geometries has been validated by a large number of experimental and analytical results presented by Downling and Townley [43], by Chell and Milne [47], by Chell [48] and by Milne [49,50]. Harrison et al. [40] reformulated the "Two-Criteria Approach" of Downling and Townley stating the failure procedure known as the GEGB's R6 Failure Assessment Diagram. The coordinates (S~, K~) of an assessment point in this diagram (see fig. 2) are calculated by: Sr =
Stress (load) applied Plastic collapse stress (load) of cracked structures S(L)
sa[L,(~)]' Stress (load) applied LEFM failure stress (load)
K r -
K~ ( a , L ) Kit
(2)
Here, K 1 is the applied stress intensity factor, L is the applied load, and L l ( a ) is the failure load. Rearranging eq. (1), it is clear that at failure the points (K r, St) must verify the expression:
gr=Sr(-~lnsec(2Sr)] -l/2
(3)
In the pressure containment vessels of nuclear power plants, steel of extremely high ductility is used [35]. It implies that plasticity effects play an important role in the failure of such structures when they have preexistent cracks. After the development of several approaches for dealing with elastoplasticity in crack behaviour, Downling and Townley [43] suggested that metal structures containing a flaw could only operate within limits imposed by the linear elastic fracture mechanics and plastic limit theory ("Two-Criteria Approach"). Between these limits, some interpolation is necessary for structures operating near the failure condition. This interpolation is provided by "the strip yield model" of Bilby et al. [44], and their work was then taken in a modified form by Heald et al. [45]. From this model the resulting expression for failures of finite geometries can be written as [46]:
which is plotted as the failure assessment line in fig. 2. Points (S r, Kr) inside this curve indicate that the structure is safe, and the position of these points relative to
K2c=8Fy2alnsec(2-~l )
0,0
(l)
where Kic is the plane strain fracture toughness, Fy is
Kr
1,2 At failure, Kr = Sr
~
In sec
Sr
1,0
Reserve factor on pressure: Fp : ~OF
0,8
0,6 ftow stress, S I , / ~
in toaa, r i
0.4
/./~/ //
0,2
I~
/// l
0
AIKr,Sr ) Increasein I fracturetoughness,KIc
I l
l
0,2
l
l
0,4
l
l
l
0,6
l
I I ~ S
O.B
1,0
1,2
Fig. 2. The CEGB's R6 failure assessment diagram.
r
212
G. Sanchez Sarmiento et al. / Failure internal pressure
such a line defines how safe the structure is. The locus of this point as a function of load is a straight line through the origin (zero load) to the failure assessment line, so that the reserve factor on load, Fp, is easily evaluated as shown in fig. 2. Alternatively, Fp may be calculated from the following formula [51]: /
Other loci, as a function of Kl~ or yield stress Y, are also straight lines parallel to Sr and K r respectively, while the locus as a function of flaw size is in general curved [49] as indicated in the same figure. These properties facilitate a sensitivity analysis of the input data on the failure condition, by varying the factors systematically in ways justified by the status of these input data. It can be used to build up confidence in the analysis and help quantify the risks in the final judgement [52]. Valuable discussions of the advantageous characteristics of this procedure can be seen in refs. [49,53-59]. Okamura et al. [60] examined the validity of the R6 approach from the standpoint of a J-integral evaluation for a cracked member under bending and tension, comparing analytical results with tests for an eccentric tension specimen of A533B Class 1 steel and the finite element method. Milne [61,62] and Bloom [63-65] have shown that the R6 approach can be extended to account for stable crack growth (ductile tearing) beyond initiation. Combined thermal and pressure loads have also been taken into account by Milne in ref. [66]. Bloom has also proposed [63] an extension of the R6-procedure for the assessment of the structural integrity of nuclear pressure vessels, combining this approach with deformation plasticity solutions [67-69] obtained in the US General Electric Company.
3. Calculation of the elasto plastic collapse load The simulation of the elastoplastic behaviour of the structure until its collapse is reached, has been performed with the clean axisymmetric shell taken separately into account (neglecting the influence of nozzles and minor irregularities) and two major penetrations (personnel and emergency locks). The Finite Element System SAMCEE [70-72] was employed, in a version running in the Instituto de Desarrollo Tecnol6gico para la Industria Quimica, Santa Fe, Argentina [73].
3.1. Analysis of the clean sphere
The clean axisymmetric shell (fig. 3) was idealized by axisymmetric thin shell elements, obtained by degenerating an isoparametric volume element of axisymmetric geometry and quadrangular cross section. The displacement field is linear through its thickness, and of second order along its longitudinal coordinate 0. A great densification of the elements was taken in the clamping zone. The bedding material at the clamping zone (fig. 4) was represented by axisymmetric isoparametric volume elements with a quadratic displacement field. A total number of 33 elements of this kind and the former one was taken for this modelling. The following boundary conditions were imposed: (i) Between 0 = 50.720 and O = 52.76°, the component of the displacement in the radial direction was constrained, leaving its meridional component free (this is a conservative assumption in the absence of trusty data about the contact friction between the steel shell and the concrete structure).
/
/
/
/
...(. /
--,
II
~drt
K Concrete*.210 ~mrn3 K1 --" 0.033L, KZ= 0.0376 K3= 0.014L,
i Fig. 3. Axisymmetricthin shell model for the clean sphere.
G. Sbncher Sarmiento
Fig. 4. Detail
of the
K1 = K2 =
0.0334 N /mm! 0.0376
K3 =
0.0144
213
et al. / Failure internal pressure
the sphere material from a collection of tensile specimens was used (fig. 5). Elastic isotropy and the Huber-von Mises elastoplasticity theory with isotropic strain-hardening were considered, also taking into account geometrical nonlinearities (large deformations). On the other hand, the elastic bedding material was assumed to have the bilinear elasto-plastic behaviour shown in fig. 5. Rupture in the circumferential direction (appearance of radial cracks) is assumed to occur when its equivalent strain exceeds 10%. The physical and geometrical data for the calculations are indicated in table 1. With these data, a nonlinear static analysis of the sphere subjected only to internal pressure was performed. The load was increased from zero by several discrete steps. The three initial increments were of 0.3 N/mm2; the fourth of 0.2 N/mm’; and then four increments each one of 0.05 N/mm2 were applied until the elastoplastic collapse was reached. Dimensionless meridional and hoop stresses at the outside and inside bounding surfaces calculated at the clamping zone and its neighbourhood are shown in fig. 6. Both stresses are divided by the asymptotic membrane stress far from irregularities, of value S, = P, R/2 t, where Pi is the internal relative pressure; R is the sphere mean radius and t is the shell thickness. The quasi-coincidence of the normalized stress for Pi = 0.9 and 1.1 N/mm2, as shown in fig. 6 indicates that the behaviour of the shell is almost linear elastic up
clamping zone.
(ii) Between $J = 50.72” and $I = 52.76”, equal radial displacements of the steel shell and of the elastic bedding was imposed. The second one was assumed to be fixed to the concrete bed. (iii) At the upper and bottom pole, null radial displacement was assumed. The minimum true stress-true strain relationship of
0 true ICQO.. 900 . .
Elastic
bedding
material
( 16’
N/mm2)
./
300. ./ 200 ” ‘00.
./ ,/
O/ 0 Fig.
5. True
0.02 stress
0.04
0.06
0.08
versus true strain curves
0.1 of the
0.12
0.11
0.16
0.18
o.2
%rue
containment steel and of the elastic bedding material used in the calculations.
214
G. Shnchez Sarmiento et al. / Failure internal pressure
to the second value of /° i. On the other hand, the relaxation of the stress concentration and of the bending stress by the plastic behaviour at the clamping zone are manifest in these plots for P~ greater than 1.1 N / m m 2. For the determination of the plastic analysis-collapse load (internal pressure that produces elasto-plastic collapse) we follow here a procedure dis,cussed by Berman [74] and based on rules of the A S M E Boiler and Pressure Vessel Code, Section III, Appendix II [14] for the experimental determination of collapse loads. The load must be plotted against the maximum principal strain or displacement as shown in fig. 7 herein. The collapse load is then determined by the intersection of the calculated curve with a line making an angle with the load axis whose tangent is twice the tangent of the angle that makes the linear stage line with the load axis. For the global containment (clean sphere) a collapse pressure of P, = 1.171 N / r a m 2 results.
Table 1 Geometrical and material data Material properties
Symbol Value
Young modulus Poisson ratio Yield point Fracture toughness of base metal Fracture toughness of welding metal
E ~, S~ Kl~
2.10 X 105 N / m m 2 0.3 544 N / m m 2 5.6 kN/mm 3/2
Kit
3.6 kN/mm 3/2
Geometrical parameters
Symbol Value (mm)
Sperical shell: Internal radius Thickness Personnel lock: Equivalent internal radius Longitude - Thickness Pad external radius Pad thickness Emergency lock: Equivalent internal radius - Longitude - Thickness Pad external radius Pad thickness -
-
-
-
115Lzt~ • ss~
Ri t
28000 30
ri L d rp S,,,
1787 2375 80 2833 50
ri L d rv SA
765 1350 50 1515 50
. /~ l/
~°
u°'"~ ~
100 ~ •
-
.
3.2. Analysis" of the personnel lock
The nozzle of the personnel lock (see basic dimensions in table 1) has been modelled by the finite element discretization shown in fig. 8. Axisymmetric. isoparametric volume elements with a quadratic dis-
I zl:
I~-R
e~.
-'
-
~
......
R = 0 9 Nknm 2
~ ....... ~........ ~ _ . . . . . . . . . . . . . . . . / - - .....
[-
ng~ I
I 2tcz "LIO- p ~ ~out
I 2t -p,R Go°u'
I"05~__--z-----_--'--T---T-~-----T_-__~L--_-_T_~_-__-_S-
7- ......
1,00-
; ............ ~- .......
-'-~"'
~
~
o o: 36
/
~ t.,4 /.,8 Meridiona[ coordinate
52 56 ~ (grades)
36
¢ - - - / ~
40 4t, L,B Meridional coordinate
52 56 i~ (grades)
Fig. 6. Dimensionless,mendional and hoop stressesat the outside and inside bounding surfaces of the spherical shell calculated at the damping zone and its neighbourhood, relative to the membrane stress of the free she]].
215
G. Srnchez Sarmiento et aL / Failure internal pressure
1.2 --
,.,.
.....
12 /
1o
lO
i f/J 0.8
°.,f/ /
I- / 0.2 I / ~
0, 8
.
,/
1.2 F
/
2
f/J }.- /
O2 I / I/
/
[mm l
1.o 0.8
o., r /
uppe, pole
Displacement
.....
~_ ~_.__j~__._~ /
2
f/j
o.,r/ Welding belween
personnel lock and its reinfOrcement"
Displacement
[mml
~- /
02 I / f/
Welding between
emergencyLock and its ;einforcemenf.
Displacement [mm]
Fig. 7. Comparison between the calculated displacements at three significant points and the respective collapse internal pressure deduced from them.
placement field have been used for the nozzle and for the pad. From the weld of the pad with the spherical shell, up to an angle of 30 ° with the nozzle axis, this shell was idealized by axisymmetric thin shell elements as used in the clean sphere. Boundary conditions of spherical symmetry .have been imposed at the edge of the spherical cap: null meridional component of the displacement; rotation constrained; and free radial displacement. The structure was subjected to the same discrete steps of internal pressure as done in the former analysis, and the total force coming from the enclosure of the lock has been circumferentially distributed along the bottom boundary in the axial direction. The calculated equivalent (von Mises) stresses at the outside and inside bounding surfaces of the pad and neighbouring spherical shell, are plotted in fig. 9 for four values of the internal pressure. Up to Pi = 1.1 N / r a m 2, the behaviour is practically linear elastic, but for greater values of this load, large plasticity effects are made evident. At the inside and outside bounding surfaces of the nozzle, the calculated equivalent stresses are as plotted
Fig. 8. Axisymmetric finite element model of the emergency nozzle.
216
G. Shnchez Sarmiento et aL / Failure internal pressure I
1
I
I
I
I
1
r
[
[
I
T
I
I
E "" 600 Z
/
u . . . . . . . . .
540
=. . . . . . . .
./¢
u~
-6 x
_
S
480
\
] \
c o
/ / I
\ 420
\
N/ram 2
0.90 1. 10 1. 20
360
1. 25
g uJ
300
I
1600
I
2000
P
I
E -'~
2400 Radia[ I
I
I
I
I
I
I
I
I
I
I
1
I
t
t 3200
I
I 3600
I
4000
2800 3200 3600 coordinate [mm]
I
4400
I
I
I 4000
I
I
t",,l
E E 600 //-.
F-
Z
-
o
_
o 540 +
/ /
; 480-
/
E
g ~ 420
360 cJ >
toW
300
I 1600
2000
~
t 2400
I
I 2800
I
I 4400
I
Fig. 9. Calculated equivalent (von Mises) stresses at the inside and outside bounding surfaces of the spherical shell and of the pad besides the personnel nozzle.
in fig. 10, for the same values of Pi a s in fig. 9. An (r, z)-map of the equivalent stress in the welding zone, corresponding t o Pi = 1.25 N / m m 2, can be observed in
fig. 11.
A n interesting conclusion can be drawn from the results shown in figs. 9, 10 and 11, as follows: in spite of the fact that in the elastic range the maximum equivalent
G. Shnchez Sarmiento et al. / Failure internal pressure
6oo~-
'
'
'
,~'
'
'
z s00
'
~
'
'
'
I
i __
217
',
o~
.
400
"
~\
° f
~300 o=
\
200
I00
.-" ~
0.00
I
-1290
',,,
\\
j ,'// /,/ ~,' /
",, ',,
\
,\
J,"Z I
-900
1
I
-600
I
I
t
I
I
_,r~ ~ . i . . . . , ,,, j/~ \ ',, 1 __
/.80
//' I . /)
_
,,'//!
a6o .~
I ,;
-1zoo -90o
I
-300 0.0 300 Axial Coordinate (ram)
-6o0
,"
I
900
1200
_ Pi = 090 NI
~ '.1 . . . . ', ............
I~o ,.zo
\ \ ',,, .....
/y
-3oo
I
600
/ J
1.2s I
", ~1
o.0
~o
~o0
goo
1zoo
Fig. 10. Calculated equivalent (von Mises) stresses at the inside and outside bounding surfaces of the personnel nozzle. stress appears at the nozzle internal surface, in the plastic range (Pi >-- 1.05 N / m m 2) the equivalent stress in the entire nozzle is not greater than the asymptotic membrane stress in the spherical cap, Sm ~s = Pi R / 2 t. At ei = 1.25 N / r a m 2, the value of Sr~S is 584 N / r a m 2, while in the nozzle the maximum equivalent stress is of 555 N / m m 2 (see fig. 11). The greatest stress concentration appears at the welding between the pad and the spherical shell. By plotting the displacement of a point in the welding between the nozzle and the pad (see fig. 7), a collapse pressure of P1 = 1.162 N / m m 2 results which is only a little lower than the value corresponding to the clean sphere. This fact is in agreement with the conclusions obtained in studies of the limit pressure of nozzles performed by other authors, for example in refs. [1,34,38]. 3.3. Analysis o f the emergency nozzle
For the nozzle of the emergency lock, the analysis was very similar to that of the former nozzle. In fig. 12 a
Fig. 11. Map of the equivalent (von Mises) stress in the zone of welding between the personnel nozzle and the pad of reinforcement (Pi 1.25 N/mm 2). =
map is shown of the equivalent stress in the welding between this nozzle and the pad, corresponding to an internal pressure of 1.35 N / m m 2. In fig. 7, the determination of the collapse pressure for this nozzle is also indicated. The resulting value is almost the same as in the former case.
4. Failure assessment of cracked spherical shell far from irregularities From the last analysis we can infer that the limit pressure of the containment in the complete absence of cracks is Pi = 1.16 N / m m 2. The CEGB's R6 Failure Assessment approach, as
218
G. Simchez Sarmiento et aL / Failure internal pressure _Kr 1.0
0.8
0.6
o
+= 6
\
i ° +:g
:+:o zx
a : 12
v
a : is
- •
a-',
-
0.4 Failure [ocation
564 ~ 5"/0 ~
0,0
i 0.2
,
1.2
I t=3Omm Flaw
model
i
I,S--- 7 ; I
// P, : 0.6 N/rnm 2 jr--"" ' l ~:/.-~[.,..~Pi = 0.B5 Nlmm ho
, 0.4
,F, 0.6
,I
, 0.8
.Kr
,
I 1,0
,
J S_r 1.2 --
.Q
~g ',~'/
-
tO
-.%'./" -
/ +?t.-
/-~,PT'.~"td"
~
O~ •
/] x ) ~, / / . ~ [
.¢'J~
~/Z/
-"///'/2/°
-
_
~,(2,.,/'/
_
0.~
0.6
556 0.~,
C/ / 1.,? , / / / // ( / J;L
O.2
55,~
II 0.+ Sr
0.0
o~
Fig. 12. Map of the equivalent (von Mises) stress in the zone of welding between the emergency nozzle and the pad of reinforcement (Pi = 1.35 N/mm2).
described in section 2, is now applied to the analyzed c o n t a i n m e n t with cracks located in regions far from irregularities and not interacting between them. A semielliptical model for the crack has been considered, as indicated in fig, 13, in order to follow the r e c o m m e n d a tions of the A S M E Code, Section XI, A p p e n d i x A [41]. We will consider two different ratios of the flaw d e p t h a to the flaw length l: a l l = 0.3 a n d 0.1. The ratio a / t will be increased from zero by steps of 0.1. Two values of the internal pressure will be taken into account: (a) the test pressure of Pi = 0 . 6 N / m m 2 ; and (b) Pi = 0.85 N / m m 2.
0.2
oA
0.6
OaS
t0
~2
Fig. 13. Loci of assessment points for two idealized flaw profiles and two different values of the internal pressure, as a function of the flaw depth. The flaw is assumed to be located at the base metal ( K l ~ = 5 . 6 N / m m 3/2) as well as at a weld (KI~ = 3.6 N/mm3/2).
3.6 k N / m m 3/2 for the welding metal. We adopt for the flow stress the value of the pure m e m b r a n e stress corres p o n d i n g to the collapse load Pi = 1.171 N / m m 2 calculated for the clean shell: = RP i 2t
28015 × 1.170 N / m m 2 = 547 N / m m 2. 2 × 30
This value is slightly greater than the yield point Sy = 544 N / m m 2. We suppose that the welding material has the same value of S.
4.1. Material properties o f the base and welding metals 4.2. Calculation of S r F r o m a collection of test data, the following lower b o u n d s result for the material fracture toughness: Kic = 5.6 k N / m m 3/2 for the base metal a n d KI~ =
For a nearly uniform stress field the collapse stress S l ( a / t ) can be taken [46,49] as S(1 - c / w ) , where S is
219
G. S[~nchezSarrniento et a L / Failure internalpressure Table 2 Failure analysis data and results for the internal pressure test (0.6 N / r a m 2)
all
Qo
a//t
a
Mm
c/w
Sr
Base metal
Welding metal K'. Fp
(mm)
Pc,+i, ( N / m m 2)
K'.
Fp
Pc,-+, ( N / m m 2)
0.1
1.103
0.1 0.2 0.3 0.4 0.5 0.1
3 6 9 12 15 3
1.12 1.20 1.32 1.51 1.77 1.10
0.052 0.131 0.224 0.325 0.422 0.029
0.540 0.589 0.660 0.758 0.886 0.527
0.255 0.386 0.520 0.687 0.900 0.206
1.85 1.64 1.38 1.13 0.908 1.90
1.11 0.982 0.829 0.678 0.545 1.14
0.164 0.248 0.334 0.441 0.579 0.132
1.85 1.70 1.51 1.30 1.09 1.90
1.11 1.02 0.904 0.778 0.653 1.14
0.3
1.630
0.2 0.3 0.4 0.5 0.6 0.7
6 9 12 15 18 21
1.105 1.13 1.16 1.22 1.27 1.36
0.077 0.151 0.232 0.317 0.406 0.507
0.555 0.603 0.667 0.750 0.862 1.039
0.292 0.366 0.434 0.510 0.582 0.673
1.79 1.62 1.45 1.27 1.11 0.930
1.07 0.973 0.868 0.765 0.666 0.558
0.188 0.235 0.279 0.328 0.374 0.433
1.80 1.66 1.50 1.33 1.16 0.962
1.08 0.995 0.899 0.799 0.695 0.577
the flow stress a d o p t e d in section 4.1, a n d the factor c / w ) corrects for the uncracked ligament and contains a flaw ellipticity. The values of c / w were o b t a i n e d from Milne [46] in terms of a / w a n d are indicated in table 2. Thus, for a given internal pressure:
(1 -
Sm Sr
= -~1
Pi R / 2 t =
S(1 - c / w )
metal as well as in the base metal for this internal pressure. The loci for any other internal pressure, such as 0.85 N / m m 2 in the diagrams, is directly o b t a i n e d by scaling the former ones. Flaw Depth Wall Thickness Ratio , a / t
Pi 0.854 1 - c/~-~"
(6)
0.0
0.1
02
I
I
0.3
03+
I
0,5
0.6
I
i
I
2.O
~
4.3. Calculation of Kf
i~
T h e A S M E Code, Section XI [41] provides the expression of k I for semielliptical flaws in shells or plates subjected to tension a n d bending. F o r the present purpose the following expression applies:
K I = SmMmcrl/2(a/Oo)l/2,
(7)
1.6
\,~
.
t'~
\ ~
1.2
/Failure
"~
\ ~
/
0.7
'
Flaw Model
~_
m E
0.9
0.8
z
where Q o ( a / l ) is the flaw shape p a r a m e t e r regardless of plasticity effects, given by [46]: Q0 = 1 + 4 . 5 9 3 ( a / l ) L65
(8)
a n d Mm(a/l; a / t ) is a correction factor for m e m b r a n e stress, given b y fig. A-3300-3 of ref. [41].
\ \
o.8 Base reefat 0.6
Welding
o.5 0.4
me+at
4.4. Failure analysis Table 2 contains the data and results of the failure analysis corresponding to the internal pressure test, o b t a i n e d by expressions (2) a n d (6)-(8). The points (Sr,Kr) determine the loci indicated in fig. 13 corres p o n d i n g to the increment of flaw d e p t h in the welding
Flaw
depth ,
a
[ mm ]
Fig. 14. Calculated reserve factor and the failure internal pressure as a function of the flaw depth. The flaw is assumed with two idealized profiles and to be located at the base metal as well as at a weld.
220
G. Shnchez Sarmiento et al. / Failure internal pressure
By means of expression (4) each value of the reserve factor Fp (on load) included in table 2 is obtained, as well as the critical internal pressure (Petit = F p × 0.6 N / m m 2 ) . They are plotted in fig. 14, in terms of flaw depth, for b o t h the base and welding metal a n d for both the ratios a l l considered.
5. Conclusions The goal of this paper has been to indicate the importance of considering crack p r o p a g a t i o n as a failure criterion, in order to calculate the failure internal pressure of metal c o n t a i n m e n t s in the context of a PRA. This failure criterion was not generally considered in the literature for these kinds of metallic structures. For this purpose, ~_ new application of the R6 Failure Assessment Diagram A p p r o a c h was shown. The presence of hypothetical initial cracks, with idealized geometries according to the A S M E Boiler a n d Pressure Vessel Code (Section XI), was taken into account in order to analyze the sensitivity of the failure pressure with the values of the material fracture properties. Nevertheless, the great relative spread of experimental data for the toughness and yield stress that these materials generally present, besides the stochastical character of the flaw dimensions, require this failure criterion to be treated in a probabilistic manner. A new formulation is now in progress by the first a u t h o r and will be published [42].
References [1] T.E. Blejwas and D.S. Horschel, Analysis of steel containment models, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, Vol. II, N U R E G / CP-0033; SAND 82-1659, 201-226 (October 1982). [2] P.F Gou and J.E. Love, Pressure carrying capability of the containment structural system of the Mark III Standard Plant, Proc, Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, Vol. II, NUREG/CP-0033; SAND82-1659, 263-318 (October 1982). [3] W. Gulden, B. G611er and R. Krieg, Analysis of the mechanical behaviour of LWR-containments under accident conditions, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-8, 1982, Vol. II, NUREG/CP-0033; SAND 82-1659, 319-336 (October 1982). [4] A. Andersen, S. Bantle and L.M. Habip, Stability analysis of spherical metal containment vessels, Proc. the Fifth Int. Conf. on Structural Mechanics in Reactor Technology, Paper J 6/6, Berlin (1979).
[5] A. Andersen and S. Bantle, Optimization of the containment nozzles, Proc. Fifth Int. Conf. on Structural Mechanics in Reactor Technology, Paper J 4/4, Berlin, (1979). [6] R. Krieg, B. G611er and G. Hailfinger, Dynamic stresses in spherical containments with pressure suppression system during steam condensation, Nucl. Engrg. Des. 64 (1981) 203-223. [7] F.S. Fanous and L.F. Greimann, Simplified dynamic analysis for internal locally loaded shells, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, Vol. II, N U R E G / C P 0033. SAND 82-1659, 337 364 (October 1982). [8] R. Krieg et al., Investigation program on PWR-steel-containment behavior under accident conditions, Kernforschungszentrum Karlsruhe, KfK 3589 (Oktober 1983). [9] B. GOller, Dynamic deformation of thin spherical shells based on analytical solutions, J. Sound and Vibration 73(4) (1980) 585-596. [10] B. G611er, Schalendynamisches Verhalten des Kugelformigen Containments eines Siedewasserreaktors bei Dampfkondensation im Druckabbausystem, Kernforschungszentrum Karlsruhe, KfK 2778 (May 1979). [11] G. Sfinchez Sarmiento and A.N. Bergmann, Thermal stresses at nozzles of nuclear steel containments under LOCA conditions, Conf. Structural Analysis and Design of Nuclear Power Plants, Porto Alegre, Brasil (October 3-5, 1984). [12] G. Sfinchez Sarmiento, A.N. Bergmann, A. Lotorto and H. Lux Wurm, Computational techniques for the stress analysis of nuclear steel containments (in Spanish). Revista de Ingenieria Estructural, Buenos Aires, Argentina, Vol. 1, no. 3 (June 1983) pp. 1 19. [13] L.I. Skundric, Metal containments: nature of loads and behaviour limits, Nucl, Engrg. Des. 46 (1978) 409-4t6. [14] American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section lII: Nuclear Power Plant Components, Division 1 (1984 Edition). [15] Kerntechnischer Ausschuss (KTA: Nuclear Safety Standards Commission): Reaktorsicherheitsbeh~ter aus Stahl. Teil: Auslegung, Konstruktion und Berechnung. KTA3401.2 (April 1980). [16] Probabilistic Risk Assessment Procedures Guide, N U R E G / C R 2300, Vol. 1 (Office of Nuclear Regulatory Research, US Nuclear Regulatory Commission, Washington D.C.). [17] P. Cybulskis, On the definition of containment failure, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7 9 1982, Vol. II, NUREG/CP-0033; SAND82-1659 (October 1982) pp. 65 76. [18] Reactor Safety Study - WASH 1400, NUREG-75/0114, Appendix VII, Nuclear Regulatory Commission (October 1975). [19] M.L. Corradini, D.V. Swenson, R.L. Woodfin and L.E. Voelker, An analysis of containment failure by a steam explosion following a postulated core meltdown in a light water reactor, Nuclear Engrg. Des. 66 (1981) 287-298.
G. Simchez Sarmiento et al. / Failure internal pressure
[20] D. Hippe and F.W. Heuser, Major aspects and current results of Phase B of the German Risk Study, GRS-Fachgesprach 1982, The Present State of Risk Investigations for Nuclear Power Plants., KOln, Fed. R. of Germany (November 3 and 4, 1982). [21] J.C. Tsai and R.S. Orr, Probabilistic failure modes and localizations in containments subjected to internal pressurization, Proc. Workshop on Containment Integrity, Arlington. Virginia, June 7-9, 1982, Vol. Ii, N U R E G / CP-0033, SAND82-1659 (1982) pp. 365-382. [22] L. Greimann and F. Fanous, On the uncertainties associated with containment analysis, Second Workshop on Containment Integrity. Washington, D.C., USA, June 13-15, 1984, N U R E G ICP-0056, SAND84-1514 (1984) pp. 229-244. [23] L. Greimann, F. Fanous, A. Sabri, D. Ketelaar, A. WoldTinsae and D. Bluhm, Reliability analysis of containment strength, NUREG/CR-1891, 1S-4753 (US Nuclear Regulatory Commission, August 1982). [24] L. Greimann, F. Fanous, A. Wold-Tinsae, D. Ketelaar, T. Lin and D. Bluhm, Reliability analysis of steel containment strength, NUREG/CR-1891 (June 1982). [25] L. Greimann, F. Fanous, D. Ketelaar, T. Lin and D. Bluhm, Probabilistic seismic resistance of steel containments, N U R E G / C R - 3 1 2 7 (January 1984). [26] F. Fanous and L. Greimann, Probabilistic seismic resistance of a Mark II1 Steel Containment, Symposium on Advances in Probabilistic Structural Mechanics, ASME, San Antonio, Texas (June 1984). [27] F. Fanous and L. Greimann, Fragility curves for steel containments with internal pressure, Second Workshop on Containment Integrity, Washington, D.C., USA, June 13-15, 1984, N U R E G LCP-0056, SAND84-1514 (1984) pp. 659-671. [28] G.I. SchuEller et al., Zuverlassigkeitsbeurteilung fi~r den Sicherheitseinschluss (SE) am Beispiel des Druckwasserreaktors, Report BMFT 150201 C, Institut fi~r Bauingenieurwesen III (Massivbau), Technische Universit~tt MiJnchen (September 1981). [29] G.I. Schiaeller, Impact of probability risk assessment on containment, Nucl. Engrg. Des. 80 (1984) 203-216. [30] A. Walser, Primary containment ultimate capacity of Zion Nuclear Power Plant for internal pressure load, Proc. Workshop on Containment Integrity, Arlington, Virginia June 7-9, 1982, Vol. II, NUREG/CP-0033; SAND82-1659 (1982) pp. 263-318. [31] P.F. Gon and J.E. Love, Pressure carrying capacity of the containment structural system of the Mark III Standard Plant, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, NUREG/CP-0033, SAND82-1659 (1982). [32] D.S. Horschel and T.E. Blejwas, An analytical investigation of the response of steel containment models to internal pressurization, Proc. 7th Int. Conf. on Structural Mechanics in Reactor Technology, Chicago, Illinois (August 22-26, 1983). [33] T.E. Blejwas, R . L Woodfin, A.W. Dennis and D.S.
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
221
Horschel, Containment Integrity Program FY82 Annual Report, N U R E G / C R - 3 1 3 1 / I , SAND83-0417 (March 1983). D.S. Horschel, Experimental and analytical results of steel containment tests, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15, 1984, NUREG/CP-0056, SAND84-1514, 621-634 (August 1984). J. Jeschke, Limit load analysis of actual spherical containments subjected to static internal pressure and temperature, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15, 1984, N U R E G / C P 0056; SAND84-1514, 287-304 (August 1984). K. Hassmann, G. Weckesser, J. Jeschke and D. Wi~nschmann, Bewertung einer einfachen MOglichkeit der Containmententl'/fftung zur Reduktion der Unfallfolgen der Freisetzungskategorie 6, Unfallablauf ohne zusatzliche Massnahmen, KWU Report R914/84/004 (1984). R.S. Dunham, Y.R. Rashid and Y.K. Tang, Recent results on the evaluation of the overpressure response of concrete and steel containments, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15 1984, N U R E G / C P - 0 0 5 6 ; SAND84-1514, 533 562 (August 1984), this issue, p. 193. D.B. Clauss, Analysis of a 1:8 scale steel containment model subject to internal static pressurization, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15, 1984. NUREG/CP-0056; SAND841514, 634-658 (August 1984), this issue, p. 223 and p. 241. B. Grller, R. Krieg and G. Messemer, Behaviour of spherical PWR-containments close to reinforced sections under excessive internal pressure, Proc. Second Workshop on Containment Integrity, Crystal City. Virginia, June 13 15, 1984, NUREG/CP-0056; SAND84-1514, 575-588 (August 1984). Harrison, K. Loosemore, 1. Milne and A.R. Dowling, Assessment of the integrity of structures containing defects, Report of the Central Electricity Generating Board No. R / H / R 6 (August 1976) Rev. 2 (April 1980). American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section XI: Rules for Inservice Inspection of Nuclear Power Plant Components, Division 1, Appendix A (Analysis of Flaw Indications) (1984 Edition). G. S~mchez Sarmiento and R.S. Carnicer, Assessment of the structural integrity of steel containments containing defects, to be presented at the 8th Int. Conference on Structural Mechanics in Reactor Technology Brussels, Belgium (August 19-23, 1985). A.R. Downling and C:H.A. Townley, The effects of defects on structural failures: a two-criteria approach, Int. J. Pressure Vessels and Piping 3 (1973) 77-107. B.A. Bilby, A.H. Cottrell and K.H. Swinden, The spread of plasticity from a notch, Proc. Royal Soc. of London A272 (1963) 304-314. P.T. Heald, G.M. Spink and P.J. Worthington, Post-yield fracture mechanics, Mater. Sci. Engrg. 10 (1972) 129.
222
G. Sanchez Sarmiento et al. / Failure internal pressure
[46] 1. Milne, K. Loosemore and R.P. Harrison, A procedure for assessing the significance of flaws in pressurized components, 1. Mech. E. Conf., Tolerance of flaws in pressurized components", Paper C106/78 (1978) pp. 317-324. [47] G.G. Chell and I. Milne, Mater. Sci. Engrg. 22 (1976) 349. [48] G.G. Chell. Mater. Sci. Engrg. 24 (1976) 233. [49] I. Milne, Failure assessment, in: Developments in Fracture Mechanics 1, ed. G.G. Chell (Applied Science, London, 1979) ch. 8. [50] I. Milne, Experimental validation of resistance curve analysis, Report of the Central Electricity Generating Board No. RD/L/2044N81 (June 1981). [51] G.G. Chell and I. Milne, A simple practical method for determining the ductile instability of cracked structures, Proc. CSNI Specialists Meeting on Plastic Tearing Instability, St. Louis, N U R E G CP-0010, CSNI Report no. 39 (1979). [52] I. Milne, R.P. Harrison and A.R. Dowling, Evaluation of flaw tolerance: use of a sensitivity analysis of the input data in contrast to rigid safety factors, 1. Mech. E. Conf. (1980) Paper C l l / 8 0 , pp. 25 32. [53] V. Kumar, M.D. German and C.F. Shih, An engineering approach for elastic-plastic fracture analysis, EPRI Topical Report NP 1931, Research Project 1237-1, Electric Power Research Institute, Palo Alto, CA, July 1981, Section 8 (Failure Assessment Diagram) pp. 8.1-8.17. [54] R.P. Harrison and I. Milne, Assessment of defects: the CEGB approach, Phil. Trans. R. Soc. Lond, A229 (1981) 145-153. [55] B.J.L. Darlaston, The CEGB two-criteria proposal, in: Developments in Pressure Vessel Technology 1 (Flaw Analysis), ed. R,W. Nichols (Applied Science, London. 1979) ch. 4. [56] A.R. Dowling, R.P. Harrison and I. Mitne, The CEGB approach to the assessment of the integrity of structures containing defects, Fitness for purpose validation of welded constructions, London, November 17-19 1981, Paper 34. [57] I. Milne, Application of the CEGB procedure to the analysis of HSST intermediate vessel tests V5 and V9, Fitness for purpose validation of welded constructions. London, November 17 19, 1981, Paper 35. [58] J.G. Collier, M.R, Hughes, C.J. Gardner, L.M. Davies and 1. Milne, Fracture assessment of a PWR pressure vessel, Int. Atomic Energy Agency, 1AEA-SM-269/103 (1983) pp. 189-209. [59] G.G. Chell, Elastic plastic fracture mechanics, in: Developments in Fracture Mechanics 1, ed. G.G. Chell (Applied Science, London, 1979) ch. 3. [60] H. Okamura, K. Kageyama and Y. Takahata, Lower-bound solutions and their application to the collapse load of a cracked member under axial force and bending moment, Elastic-Plastic Fracture: Second Symposium, Vol. II Fracture Resistance Curves and Engineering Applications, ASTM STP 803, eds. C.F. Shih and J.P. Gudas (American Society for Testing and Materials, 1983) pp. II-262-11-277.
[61] 1. Milne, Failure analysis in the presence of ductile crack growth, Mater. Sci. Engrg. 39 (1979) 65-79. [62] I. Milne, Calculating the load bearing capacity of a structure failing by ductile crack growth, Advances in Fracture Research, eds. D. Francois et al. 1751-1758, (Pergamon Press, Oxford, 1980) pp. 1751 I758. [63] J.M. Bloom, A procedure for the assessment of the structural integrity of nuclear pressure vessels, Trans. ASME J. Pressure Vessel Technology 105 (February 1983) 28-34. [64] J.M. Bloom, Prediction of ductile tearing of compact fracture specimens using the R-6 failure assessment diagram, Int. 3. Pressure Vessels and Piping (May-June. 1980) 215. [65] J.M. Bloom, Validation of a deformation plasticity failure assessment diagram approach to flaw evaluation, Elastic-Plastic Fracture: Second Symp., Vol. II Fracture Resistance Curves and Engineering Applications, ASTM STP 803, eds. C.F. Shih and J.P. Gudas (American Society for Testing and Materials, 1983) pp. 11. 206 II. 238. [66] 1. Milne, Evaluation of the defect tolerance of pressure vessels under combined thermal and pressure loads, Trans. 5th Int. Conf. on Structural Mechanics in Reactor Technology, Berlin (August 13 17, 1979) Paper G1/3. [67] C.F. Shih et al., Methodology for plastic fracture, Final Report to EPRI, Contract No. RP 60l-2, NP-1735 (General Electric Company, Schenectady, New York, August 1980). [68] C.F. Shih, V. Kumar and M.D. German, Studies on the failure assessment diagram using an estimation method and J-controlled crack growth approach, presented at the 2nd. Int. Symp. on Elastic-Plastic Fracture Mechanics, Philadelphia, October 6 9, 1981, to appear in ASTM STP: see also, Electric Power Research Institute Report NP 1931 (July 1981). [69] C.P. Shih and J.W. Hutchinson, Fully plastic solutions and large-scale yielding estimate for plane stress crack problems, ASME J. Engrg. Mater. Technol. 98 (October 1976) p. 289. [70] SAMCEF - User's Manual, Vols. 1, 2 and 3 (University of Liege, Belgium). [71] S. ldelsohn and G. Sander, On the deep, shallow or flat shell finite elements for the analysis of thin shell structures, Comput. Meth. Appl. Mechan. Engrg. 26, No. 3 (1981) 321-330. [72] M. Geradin, S. Idetsohn and M. Hogge, Nonlinear structural dynamics via Newton and Quasi-Newton methods, Nucl. Engrg. Des. 58 (1980) 339 348. [73] S. Idelsohn, El sistema SAMCEF y su desarrollo actual en el Instituto de Desarrollo Tecnol6gico para la lndustria Quimica, Anales del Ill Congreso Latinoamericano sobre M~:todos Computacionales para Ingenieria (Comp. by A.J. Ferrante). Buenos Aires, Argentina (May 1982). [74] I. Berman, Inelasticity and the ASME Code, Nucl. Engrg. Des. 46 (1978) 335-348.
209
FAILURE I N T E R N A L P R E S S U R E OF SPHERICAL STEEL C O N T A I N M E N T S G. S A N C H E Z
SARMIENTO
Empresa Nuclear Argentina de Centrales Elbctricas S.A., Av. L.N. Alem 712 (1001) Buenos Aires, Argentina
S.R. I D E L S O H N ,
A. C A R D O N A
a n d V. S O N Z O G N I
lnstituto de Desarrollo Tecnolbgico para la lndustria Quirnica, (INTEC) Casilla de Correo n" 91 (3000) Santa Fe, Argentina
Received 21 May 1985
An application of the British CEGB's R6 Failure Assessment Approach to the determination of failure internal pressure of nuclear power plant spherical steel containments is presented. The presence of hypothetical cracks both in the base metal and in the welding material of the containment, with geometrical idealizations according to the ASME Boiler and Pressure Vessel Code (Section XI), was taken into account in order to analyze the sensitivity of the failure assessment with the values of the material fracture properties. Calculations of the elastoplastic collapse load have been performed by means of the Finite Element System SAMCEF. The clean axisymmetric shell (neglecting the influence of nozzles and minor irregularities) and two major penetrations (personnel and emergency locks) have been taken separately into account. Large-strain elastoplastic behaviour of the material was considered in the Code, using lower bounds of true stress-true strain relations obtained by testing a collection of tensile specimens. Assuming the presence of cracks in non-perturbed regions, the reserve factor for test pressure and the failure internal pressure have been determined as a function of the flaw depth.
1. Introduction The stress analysis of a nuclear power plant steel containment under a design basis accident (mainly the typical loss of coolant accident), and other design and test load conditions are nowadays performed in a relatively straightforward manner [1-11]. A wide variety of applicable computational and analytical tools are available, ranging from hand calculations to detailed finiteelement computer codes [12]. The nature and origin of loads, and the probability of their occurrence both individually and in combination, are widely known [13-15] for any given kind of containment. Core meltdown accidents of the types considered in probabilistic risk assessment (PRA) [16], notwithstanding, have been predicted to lead to environments that are much more severe than those considered for the design basis and may challenge the integrity of the containment structure [17]. The reactor Safety Study (WASH-1400) [18] demonstrated, based on PRA, that core meltdown accidents are the dominant risk contributors to the public from Light Water Reactors (LWR).
One reason for this conclusion was that containment failure and subsequent radioactivity release is possible given by the occurrence of any of a number of physical processes, such as hydrogen combustion, steam explosion, overpressurization of containment, or melt through the containment base-mat [19,20]. The quantification of the internal overpressurization at which the structure will fail, is thus of basic importance in any PRA. In order to determine this "failure internal pressure" it is necessary to define one or more failure criteria for the structure: a limiting stress, strain or some other condition, for example the onset of stability [16]. Given the geometrical complexity of the containments, the definition of failure criteria and the associated failure pressures is far from straightforward. Also, because of the uncertainties in the conditions leading to failure, a specific failure pressure cannot be determined. What is needed for P R A is a density function describing the probability of failure as a function of loading (pressure). The shape of such a density function will vary with the containment design, level of analysis, and knowledge of the details of the actual
0 0 2 9 - 5 4 9 3 / 8 5 / $ 0 3 . 3 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
210
G. Shnchez Sarmiento et al. / Failure internal pressure
containment. In this context, the nominal failure pressure is often defined as that pressure for which the corresponding probability of failure would be 0.5 [17]. Tsai and Orr [21] give a general route for performing such a probabilistic failure assessment. According to these authors, failure criteria must be defined in terms of leakage exceeding certain predetermined limits. At pressures somewhat above the design one, the probability of failure normally would be very small and the corresponding leakage would be from a localized area, but as the containment pressure rises, the probability and the amount of leakage will increase. Uncertainties associated with the approches for the prediction of the containment resistance of all kinds of designs have been extensively reviewed by Greimann and Fanous [22]. A number of studies concerned with the determination of the failure pressure of steel containments in a probabilistic manner have been published by several authors, mainly the last cited ones and co-workers [23-27], SchuEller and co-workers [28,29] and Tsai and Orr [21] among others [30,31].
On the other hand, investigations about the assessment of the ultimate internal pressure of steel containments in a deterministic manner (with fixed material and geometrical data), have been published by several authors: Blejwas and Horschel [1,32-34], Jeschke [35], Hassmann et al. [36], D u n h a m et al. [37], Clauss [38], GOller et al. [10,39] etc. Because of the exhaustive non-destructive testing of the material especially at the weld seams, and in the absence of loads causing fatigue, initial cracks will rarely be present in a pressure containment vessel. However, in the context of a PRA of nuclear power plants, the effects that cracks produce cannot be disregarded, as recognized by SchuEller [29]. In spite of this fact, modes of failure of steel containment due to brittle or ductile fracture are seldom considered in calculations of its failure internal pressure. SchuEller et al. [28,29] made a reliability assessment of the containment of a PWR containing cracks, considering as the failure function that one known in the literature as the "R6 Failure Assessment Diagram" [40], which was developed by the
lock
',
Cramping ~oee\
Fig. 1. Typical spherical steel containment of a PWR or a PHWR-nuclear power plant.
G. S~mchezSarmientoet al. / Failureinternalpressure British Central Electricity Generating Board. An exponential distribution of cracks was assumed and the probability of failure was obtained by solving the respective equations by means of a Monte Carlo approach. In this paper, an application of the R6 approach is presented for a deterministic failure assessment of a spherical steel containment as commonly used for PWR or PHWR nuclear power plants the characteristics of which are shown in fig. 1. The presence of hypothetical cracks both in the base metal and in the welding material of the containment, with geometrical idealization according to the ASME Boiler and Pressure Vessel Code, Section XI [41] was taken into account. Despite the fact that this procedure is deterministic, it can provide a sensitivity analysis of the failure pressure on the values of the strength and fracture properties of the base and welding metals. Conversely, knowing the fixed values of these materials' parameters, one can determine by means of this approach the reserve factor (both on the internal pressure and on the flaw depth) for a given internal pressure, and then, bounds for the failure load. A probabilistic treatment of this procedure, based on the Advanced First Order Second Moment Method is now in progress by the first author and will be published in the near future [42]. 2. The CEGB's 116 failure assessment diagram approach
211
the linear elastic compliance factor; a is the flaw depth; Sf is the applied stress at failure and S 1 is the flaw stress in the plastic zone. The use of eq. (1) for predicting elastic-plastic failure in a variety of geometries has been validated by a large number of experimental and analytical results presented by Downling and Townley [43], by Chell and Milne [47], by Chell [48] and by Milne [49,50]. Harrison et al. [40] reformulated the "Two-Criteria Approach" of Downling and Townley stating the failure procedure known as the GEGB's R6 Failure Assessment Diagram. The coordinates (S~, K~) of an assessment point in this diagram (see fig. 2) are calculated by: Sr =
Stress (load) applied Plastic collapse stress (load) of cracked structures S(L)
sa[L,(~)]' Stress (load) applied LEFM failure stress (load)
K r -
K~ ( a , L ) Kit
(2)
Here, K 1 is the applied stress intensity factor, L is the applied load, and L l ( a ) is the failure load. Rearranging eq. (1), it is clear that at failure the points (K r, St) must verify the expression:
gr=Sr(-~lnsec(2Sr)] -l/2
(3)
In the pressure containment vessels of nuclear power plants, steel of extremely high ductility is used [35]. It implies that plasticity effects play an important role in the failure of such structures when they have preexistent cracks. After the development of several approaches for dealing with elastoplasticity in crack behaviour, Downling and Townley [43] suggested that metal structures containing a flaw could only operate within limits imposed by the linear elastic fracture mechanics and plastic limit theory ("Two-Criteria Approach"). Between these limits, some interpolation is necessary for structures operating near the failure condition. This interpolation is provided by "the strip yield model" of Bilby et al. [44], and their work was then taken in a modified form by Heald et al. [45]. From this model the resulting expression for failures of finite geometries can be written as [46]:
which is plotted as the failure assessment line in fig. 2. Points (S r, Kr) inside this curve indicate that the structure is safe, and the position of these points relative to
K2c=8Fy2alnsec(2-~l )
0,0
(l)
where Kic is the plane strain fracture toughness, Fy is
Kr
1,2 At failure, Kr = Sr
~
In sec
Sr
1,0
Reserve factor on pressure: Fp : ~OF
0,8
0,6 ftow stress, S I , / ~
in toaa, r i
0.4
/./~/ //
0,2
I~
/// l
0
AIKr,Sr ) Increasein I fracturetoughness,KIc
I l
l
0,2
l
l
0,4
l
l
l
0,6
l
I I ~ S
O.B
1,0
1,2
Fig. 2. The CEGB's R6 failure assessment diagram.
r
212
G. Sanchez Sarmiento et al. / Failure internal pressure
such a line defines how safe the structure is. The locus of this point as a function of load is a straight line through the origin (zero load) to the failure assessment line, so that the reserve factor on load, Fp, is easily evaluated as shown in fig. 2. Alternatively, Fp may be calculated from the following formula [51]: /
Other loci, as a function of Kl~ or yield stress Y, are also straight lines parallel to Sr and K r respectively, while the locus as a function of flaw size is in general curved [49] as indicated in the same figure. These properties facilitate a sensitivity analysis of the input data on the failure condition, by varying the factors systematically in ways justified by the status of these input data. It can be used to build up confidence in the analysis and help quantify the risks in the final judgement [52]. Valuable discussions of the advantageous characteristics of this procedure can be seen in refs. [49,53-59]. Okamura et al. [60] examined the validity of the R6 approach from the standpoint of a J-integral evaluation for a cracked member under bending and tension, comparing analytical results with tests for an eccentric tension specimen of A533B Class 1 steel and the finite element method. Milne [61,62] and Bloom [63-65] have shown that the R6 approach can be extended to account for stable crack growth (ductile tearing) beyond initiation. Combined thermal and pressure loads have also been taken into account by Milne in ref. [66]. Bloom has also proposed [63] an extension of the R6-procedure for the assessment of the structural integrity of nuclear pressure vessels, combining this approach with deformation plasticity solutions [67-69] obtained in the US General Electric Company.
3. Calculation of the elasto plastic collapse load The simulation of the elastoplastic behaviour of the structure until its collapse is reached, has been performed with the clean axisymmetric shell taken separately into account (neglecting the influence of nozzles and minor irregularities) and two major penetrations (personnel and emergency locks). The Finite Element System SAMCEE [70-72] was employed, in a version running in the Instituto de Desarrollo Tecnol6gico para la Industria Quimica, Santa Fe, Argentina [73].
3.1. Analysis of the clean sphere
The clean axisymmetric shell (fig. 3) was idealized by axisymmetric thin shell elements, obtained by degenerating an isoparametric volume element of axisymmetric geometry and quadrangular cross section. The displacement field is linear through its thickness, and of second order along its longitudinal coordinate 0. A great densification of the elements was taken in the clamping zone. The bedding material at the clamping zone (fig. 4) was represented by axisymmetric isoparametric volume elements with a quadratic displacement field. A total number of 33 elements of this kind and the former one was taken for this modelling. The following boundary conditions were imposed: (i) Between 0 = 50.720 and O = 52.76°, the component of the displacement in the radial direction was constrained, leaving its meridional component free (this is a conservative assumption in the absence of trusty data about the contact friction between the steel shell and the concrete structure).
/
/
/
/
...(. /
--,
II
~drt
K Concrete*.210 ~mrn3 K1 --" 0.033L, KZ= 0.0376 K3= 0.014L,
i Fig. 3. Axisymmetricthin shell model for the clean sphere.
G. Sbncher Sarmiento
Fig. 4. Detail
of the
K1 = K2 =
0.0334 N /mm! 0.0376
K3 =
0.0144
213
et al. / Failure internal pressure
the sphere material from a collection of tensile specimens was used (fig. 5). Elastic isotropy and the Huber-von Mises elastoplasticity theory with isotropic strain-hardening were considered, also taking into account geometrical nonlinearities (large deformations). On the other hand, the elastic bedding material was assumed to have the bilinear elasto-plastic behaviour shown in fig. 5. Rupture in the circumferential direction (appearance of radial cracks) is assumed to occur when its equivalent strain exceeds 10%. The physical and geometrical data for the calculations are indicated in table 1. With these data, a nonlinear static analysis of the sphere subjected only to internal pressure was performed. The load was increased from zero by several discrete steps. The three initial increments were of 0.3 N/mm2; the fourth of 0.2 N/mm’; and then four increments each one of 0.05 N/mm2 were applied until the elastoplastic collapse was reached. Dimensionless meridional and hoop stresses at the outside and inside bounding surfaces calculated at the clamping zone and its neighbourhood are shown in fig. 6. Both stresses are divided by the asymptotic membrane stress far from irregularities, of value S, = P, R/2 t, where Pi is the internal relative pressure; R is the sphere mean radius and t is the shell thickness. The quasi-coincidence of the normalized stress for Pi = 0.9 and 1.1 N/mm2, as shown in fig. 6 indicates that the behaviour of the shell is almost linear elastic up
clamping zone.
(ii) Between $J = 50.72” and $I = 52.76”, equal radial displacements of the steel shell and of the elastic bedding was imposed. The second one was assumed to be fixed to the concrete bed. (iii) At the upper and bottom pole, null radial displacement was assumed. The minimum true stress-true strain relationship of
0 true ICQO.. 900 . .
Elastic
bedding
material
( 16’
N/mm2)
./
300. ./ 200 ” ‘00.
./ ,/
O/ 0 Fig.
5. True
0.02 stress
0.04
0.06
0.08
versus true strain curves
0.1 of the
0.12
0.11
0.16
0.18
o.2
%rue
containment steel and of the elastic bedding material used in the calculations.
214
G. Shnchez Sarmiento et al. / Failure internal pressure
to the second value of /° i. On the other hand, the relaxation of the stress concentration and of the bending stress by the plastic behaviour at the clamping zone are manifest in these plots for P~ greater than 1.1 N / m m 2. For the determination of the plastic analysis-collapse load (internal pressure that produces elasto-plastic collapse) we follow here a procedure dis,cussed by Berman [74] and based on rules of the A S M E Boiler and Pressure Vessel Code, Section III, Appendix II [14] for the experimental determination of collapse loads. The load must be plotted against the maximum principal strain or displacement as shown in fig. 7 herein. The collapse load is then determined by the intersection of the calculated curve with a line making an angle with the load axis whose tangent is twice the tangent of the angle that makes the linear stage line with the load axis. For the global containment (clean sphere) a collapse pressure of P, = 1.171 N / r a m 2 results.
Table 1 Geometrical and material data Material properties
Symbol Value
Young modulus Poisson ratio Yield point Fracture toughness of base metal Fracture toughness of welding metal
E ~, S~ Kl~
2.10 X 105 N / m m 2 0.3 544 N / m m 2 5.6 kN/mm 3/2
Kit
3.6 kN/mm 3/2
Geometrical parameters
Symbol Value (mm)
Sperical shell: Internal radius Thickness Personnel lock: Equivalent internal radius Longitude - Thickness Pad external radius Pad thickness Emergency lock: Equivalent internal radius - Longitude - Thickness Pad external radius Pad thickness -
-
-
-
115Lzt~ • ss~
Ri t
28000 30
ri L d rp S,,,
1787 2375 80 2833 50
ri L d rv SA
765 1350 50 1515 50
. /~ l/
~°
u°'"~ ~
100 ~ •
-
.
3.2. Analysis" of the personnel lock
The nozzle of the personnel lock (see basic dimensions in table 1) has been modelled by the finite element discretization shown in fig. 8. Axisymmetric. isoparametric volume elements with a quadratic dis-
I zl:
I~-R
e~.
-'
-
~
......
R = 0 9 Nknm 2
~ ....... ~........ ~ _ . . . . . . . . . . . . . . . . / - - .....
[-
ng~ I
I 2tcz "LIO- p ~ ~out
I 2t -p,R Go°u'
I"05~__--z-----_--'--T---T-~-----T_-__~L--_-_T_~_-__-_S-
7- ......
1,00-
; ............ ~- .......
-'-~"'
~
~
o o: 36
/
~ t.,4 /.,8 Meridiona[ coordinate
52 56 ~ (grades)
36
¢ - - - / ~
40 4t, L,B Meridional coordinate
52 56 i~ (grades)
Fig. 6. Dimensionless,mendional and hoop stressesat the outside and inside bounding surfaces of the spherical shell calculated at the damping zone and its neighbourhood, relative to the membrane stress of the free she]].
215
G. Srnchez Sarmiento et aL / Failure internal pressure
1.2 --
,.,.
.....
12 /
1o
lO
i f/J 0.8
°.,f/ /
I- / 0.2 I / ~
0, 8
.
,/
1.2 F
/
2
f/J }.- /
O2 I / I/
/
[mm l
1.o 0.8
o., r /
uppe, pole
Displacement
.....
~_ ~_.__j~__._~ /
2
f/j
o.,r/ Welding belween
personnel lock and its reinfOrcement"
Displacement
[mml
~- /
02 I / f/
Welding between
emergencyLock and its ;einforcemenf.
Displacement [mm]
Fig. 7. Comparison between the calculated displacements at three significant points and the respective collapse internal pressure deduced from them.
placement field have been used for the nozzle and for the pad. From the weld of the pad with the spherical shell, up to an angle of 30 ° with the nozzle axis, this shell was idealized by axisymmetric thin shell elements as used in the clean sphere. Boundary conditions of spherical symmetry .have been imposed at the edge of the spherical cap: null meridional component of the displacement; rotation constrained; and free radial displacement. The structure was subjected to the same discrete steps of internal pressure as done in the former analysis, and the total force coming from the enclosure of the lock has been circumferentially distributed along the bottom boundary in the axial direction. The calculated equivalent (von Mises) stresses at the outside and inside bounding surfaces of the pad and neighbouring spherical shell, are plotted in fig. 9 for four values of the internal pressure. Up to Pi = 1.1 N / r a m 2, the behaviour is practically linear elastic, but for greater values of this load, large plasticity effects are made evident. At the inside and outside bounding surfaces of the nozzle, the calculated equivalent stresses are as plotted
Fig. 8. Axisymmetric finite element model of the emergency nozzle.
216
G. Shnchez Sarmiento et aL / Failure internal pressure I
1
I
I
I
I
1
r
[
[
I
T
I
I
E "" 600 Z
/
u . . . . . . . . .
540
=. . . . . . . .
./¢
u~
-6 x
_
S
480
\
] \
c o
/ / I
\ 420
\
N/ram 2
0.90 1. 10 1. 20
360
1. 25
g uJ
300
I
1600
I
2000
P
I
E -'~
2400 Radia[ I
I
I
I
I
I
I
I
I
I
I
1
I
t
t 3200
I
I 3600
I
4000
2800 3200 3600 coordinate [mm]
I
4400
I
I
I 4000
I
I
t",,l
E E 600 //-.
F-
Z
-
o
_
o 540 +
/ /
; 480-
/
E
g ~ 420
360 cJ >
toW
300
I 1600
2000
~
t 2400
I
I 2800
I
I 4400
I
Fig. 9. Calculated equivalent (von Mises) stresses at the inside and outside bounding surfaces of the spherical shell and of the pad besides the personnel nozzle.
in fig. 10, for the same values of Pi a s in fig. 9. An (r, z)-map of the equivalent stress in the welding zone, corresponding t o Pi = 1.25 N / m m 2, can be observed in
fig. 11.
A n interesting conclusion can be drawn from the results shown in figs. 9, 10 and 11, as follows: in spite of the fact that in the elastic range the maximum equivalent
G. Shnchez Sarmiento et al. / Failure internal pressure
6oo~-
'
'
'
,~'
'
'
z s00
'
~
'
'
'
I
i __
217
',
o~
.
400
"
~\
° f
~300 o=
\
200
I00
.-" ~
0.00
I
-1290
',,,
\\
j ,'// /,/ ~,' /
",, ',,
\
,\
J,"Z I
-900
1
I
-600
I
I
t
I
I
_,r~ ~ . i . . . . , ,,, j/~ \ ',, 1 __
/.80
//' I . /)
_
,,'//!
a6o .~
I ,;
-1zoo -90o
I
-300 0.0 300 Axial Coordinate (ram)
-6o0
,"
I
900
1200
_ Pi = 090 NI
~ '.1 . . . . ', ............
I~o ,.zo
\ \ ',,, .....
/y
-3oo
I
600
/ J
1.2s I
", ~1
o.0
~o
~o0
goo
1zoo
Fig. 10. Calculated equivalent (von Mises) stresses at the inside and outside bounding surfaces of the personnel nozzle. stress appears at the nozzle internal surface, in the plastic range (Pi >-- 1.05 N / m m 2) the equivalent stress in the entire nozzle is not greater than the asymptotic membrane stress in the spherical cap, Sm ~s = Pi R / 2 t. At ei = 1.25 N / r a m 2, the value of Sr~S is 584 N / r a m 2, while in the nozzle the maximum equivalent stress is of 555 N / m m 2 (see fig. 11). The greatest stress concentration appears at the welding between the pad and the spherical shell. By plotting the displacement of a point in the welding between the nozzle and the pad (see fig. 7), a collapse pressure of P1 = 1.162 N / m m 2 results which is only a little lower than the value corresponding to the clean sphere. This fact is in agreement with the conclusions obtained in studies of the limit pressure of nozzles performed by other authors, for example in refs. [1,34,38]. 3.3. Analysis o f the emergency nozzle
For the nozzle of the emergency lock, the analysis was very similar to that of the former nozzle. In fig. 12 a
Fig. 11. Map of the equivalent (von Mises) stress in the zone of welding between the personnel nozzle and the pad of reinforcement (Pi 1.25 N/mm 2). =
map is shown of the equivalent stress in the welding between this nozzle and the pad, corresponding to an internal pressure of 1.35 N / m m 2. In fig. 7, the determination of the collapse pressure for this nozzle is also indicated. The resulting value is almost the same as in the former case.
4. Failure assessment of cracked spherical shell far from irregularities From the last analysis we can infer that the limit pressure of the containment in the complete absence of cracks is Pi = 1.16 N / m m 2. The CEGB's R6 Failure Assessment approach, as
218
G. Simchez Sarmiento et aL / Failure internal pressure _Kr 1.0
0.8
0.6
o
+= 6
\
i ° +:g
:+:o zx
a : 12
v
a : is
- •
a-',
-
0.4 Failure [ocation
564 ~ 5"/0 ~
0,0
i 0.2
,
1.2
I t=3Omm Flaw
model
i
I,S--- 7 ; I
// P, : 0.6 N/rnm 2 jr--"" ' l ~:/.-~[.,..~Pi = 0.B5 Nlmm ho
, 0.4
,F, 0.6
,I
, 0.8
.Kr
,
I 1,0
,
J S_r 1.2 --
.Q
~g ',~'/
-
tO
-.%'./" -
/ +?t.-
/-~,PT'.~"td"
~
O~ •
/] x ) ~, / / . ~ [
.¢'J~
~/Z/
-"///'/2/°
-
_
~,(2,.,/'/
_
0.~
0.6
556 0.~,
C/ / 1.,? , / / / // ( / J;L
O.2
55,~
II 0.+ Sr
0.0
o~
Fig. 12. Map of the equivalent (von Mises) stress in the zone of welding between the emergency nozzle and the pad of reinforcement (Pi = 1.35 N/mm2).
described in section 2, is now applied to the analyzed c o n t a i n m e n t with cracks located in regions far from irregularities and not interacting between them. A semielliptical model for the crack has been considered, as indicated in fig, 13, in order to follow the r e c o m m e n d a tions of the A S M E Code, Section XI, A p p e n d i x A [41]. We will consider two different ratios of the flaw d e p t h a to the flaw length l: a l l = 0.3 a n d 0.1. The ratio a / t will be increased from zero by steps of 0.1. Two values of the internal pressure will be taken into account: (a) the test pressure of Pi = 0 . 6 N / m m 2 ; and (b) Pi = 0.85 N / m m 2.
0.2
oA
0.6
OaS
t0
~2
Fig. 13. Loci of assessment points for two idealized flaw profiles and two different values of the internal pressure, as a function of the flaw depth. The flaw is assumed to be located at the base metal ( K l ~ = 5 . 6 N / m m 3/2) as well as at a weld (KI~ = 3.6 N/mm3/2).
3.6 k N / m m 3/2 for the welding metal. We adopt for the flow stress the value of the pure m e m b r a n e stress corres p o n d i n g to the collapse load Pi = 1.171 N / m m 2 calculated for the clean shell: = RP i 2t
28015 × 1.170 N / m m 2 = 547 N / m m 2. 2 × 30
This value is slightly greater than the yield point Sy = 544 N / m m 2. We suppose that the welding material has the same value of S.
4.1. Material properties o f the base and welding metals 4.2. Calculation of S r F r o m a collection of test data, the following lower b o u n d s result for the material fracture toughness: Kic = 5.6 k N / m m 3/2 for the base metal a n d KI~ =
For a nearly uniform stress field the collapse stress S l ( a / t ) can be taken [46,49] as S(1 - c / w ) , where S is
219
G. S[~nchezSarrniento et a L / Failure internalpressure Table 2 Failure analysis data and results for the internal pressure test (0.6 N / r a m 2)
all
Qo
a//t
a
Mm
c/w
Sr
Base metal
Welding metal K'. Fp
(mm)
Pc,+i, ( N / m m 2)
K'.
Fp
Pc,-+, ( N / m m 2)
0.1
1.103
0.1 0.2 0.3 0.4 0.5 0.1
3 6 9 12 15 3
1.12 1.20 1.32 1.51 1.77 1.10
0.052 0.131 0.224 0.325 0.422 0.029
0.540 0.589 0.660 0.758 0.886 0.527
0.255 0.386 0.520 0.687 0.900 0.206
1.85 1.64 1.38 1.13 0.908 1.90
1.11 0.982 0.829 0.678 0.545 1.14
0.164 0.248 0.334 0.441 0.579 0.132
1.85 1.70 1.51 1.30 1.09 1.90
1.11 1.02 0.904 0.778 0.653 1.14
0.3
1.630
0.2 0.3 0.4 0.5 0.6 0.7
6 9 12 15 18 21
1.105 1.13 1.16 1.22 1.27 1.36
0.077 0.151 0.232 0.317 0.406 0.507
0.555 0.603 0.667 0.750 0.862 1.039
0.292 0.366 0.434 0.510 0.582 0.673
1.79 1.62 1.45 1.27 1.11 0.930
1.07 0.973 0.868 0.765 0.666 0.558
0.188 0.235 0.279 0.328 0.374 0.433
1.80 1.66 1.50 1.33 1.16 0.962
1.08 0.995 0.899 0.799 0.695 0.577
the flow stress a d o p t e d in section 4.1, a n d the factor c / w ) corrects for the uncracked ligament and contains a flaw ellipticity. The values of c / w were o b t a i n e d from Milne [46] in terms of a / w a n d are indicated in table 2. Thus, for a given internal pressure:
(1 -
Sm Sr
= -~1
Pi R / 2 t =
S(1 - c / w )
metal as well as in the base metal for this internal pressure. The loci for any other internal pressure, such as 0.85 N / m m 2 in the diagrams, is directly o b t a i n e d by scaling the former ones. Flaw Depth Wall Thickness Ratio , a / t
Pi 0.854 1 - c/~-~"
(6)
0.0
0.1
02
I
I
0.3
03+
I
0,5
0.6
I
i
I
2.O
~
4.3. Calculation of Kf
i~
T h e A S M E Code, Section XI [41] provides the expression of k I for semielliptical flaws in shells or plates subjected to tension a n d bending. F o r the present purpose the following expression applies:
K I = SmMmcrl/2(a/Oo)l/2,
(7)
1.6
\,~
.
t'~
\ ~
1.2
/Failure
"~
\ ~
/
0.7
'
Flaw Model
~_
m E
0.9
0.8
z
where Q o ( a / l ) is the flaw shape p a r a m e t e r regardless of plasticity effects, given by [46]: Q0 = 1 + 4 . 5 9 3 ( a / l ) L65
(8)
a n d Mm(a/l; a / t ) is a correction factor for m e m b r a n e stress, given b y fig. A-3300-3 of ref. [41].
\ \
o.8 Base reefat 0.6
Welding
o.5 0.4
me+at
4.4. Failure analysis Table 2 contains the data and results of the failure analysis corresponding to the internal pressure test, o b t a i n e d by expressions (2) a n d (6)-(8). The points (Sr,Kr) determine the loci indicated in fig. 13 corres p o n d i n g to the increment of flaw d e p t h in the welding
Flaw
depth ,
a
[ mm ]
Fig. 14. Calculated reserve factor and the failure internal pressure as a function of the flaw depth. The flaw is assumed with two idealized profiles and to be located at the base metal as well as at a weld.
220
G. Shnchez Sarmiento et al. / Failure internal pressure
By means of expression (4) each value of the reserve factor Fp (on load) included in table 2 is obtained, as well as the critical internal pressure (Petit = F p × 0.6 N / m m 2 ) . They are plotted in fig. 14, in terms of flaw depth, for b o t h the base and welding metal a n d for both the ratios a l l considered.
5. Conclusions The goal of this paper has been to indicate the importance of considering crack p r o p a g a t i o n as a failure criterion, in order to calculate the failure internal pressure of metal c o n t a i n m e n t s in the context of a PRA. This failure criterion was not generally considered in the literature for these kinds of metallic structures. For this purpose, ~_ new application of the R6 Failure Assessment Diagram A p p r o a c h was shown. The presence of hypothetical initial cracks, with idealized geometries according to the A S M E Boiler a n d Pressure Vessel Code (Section XI), was taken into account in order to analyze the sensitivity of the failure pressure with the values of the material fracture properties. Nevertheless, the great relative spread of experimental data for the toughness and yield stress that these materials generally present, besides the stochastical character of the flaw dimensions, require this failure criterion to be treated in a probabilistic manner. A new formulation is now in progress by the first a u t h o r and will be published [42].
References [1] T.E. Blejwas and D.S. Horschel, Analysis of steel containment models, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, Vol. II, N U R E G / CP-0033; SAND 82-1659, 201-226 (October 1982). [2] P.F Gou and J.E. Love, Pressure carrying capability of the containment structural system of the Mark III Standard Plant, Proc, Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, Vol. II, NUREG/CP-0033; SAND82-1659, 263-318 (October 1982). [3] W. Gulden, B. G611er and R. Krieg, Analysis of the mechanical behaviour of LWR-containments under accident conditions, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-8, 1982, Vol. II, NUREG/CP-0033; SAND 82-1659, 319-336 (October 1982). [4] A. Andersen, S. Bantle and L.M. Habip, Stability analysis of spherical metal containment vessels, Proc. the Fifth Int. Conf. on Structural Mechanics in Reactor Technology, Paper J 6/6, Berlin (1979).
[5] A. Andersen and S. Bantle, Optimization of the containment nozzles, Proc. Fifth Int. Conf. on Structural Mechanics in Reactor Technology, Paper J 4/4, Berlin, (1979). [6] R. Krieg, B. G611er and G. Hailfinger, Dynamic stresses in spherical containments with pressure suppression system during steam condensation, Nucl. Engrg. Des. 64 (1981) 203-223. [7] F.S. Fanous and L.F. Greimann, Simplified dynamic analysis for internal locally loaded shells, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, Vol. II, N U R E G / C P 0033. SAND 82-1659, 337 364 (October 1982). [8] R. Krieg et al., Investigation program on PWR-steel-containment behavior under accident conditions, Kernforschungszentrum Karlsruhe, KfK 3589 (Oktober 1983). [9] B. GOller, Dynamic deformation of thin spherical shells based on analytical solutions, J. Sound and Vibration 73(4) (1980) 585-596. [10] B. G611er, Schalendynamisches Verhalten des Kugelformigen Containments eines Siedewasserreaktors bei Dampfkondensation im Druckabbausystem, Kernforschungszentrum Karlsruhe, KfK 2778 (May 1979). [11] G. Sfinchez Sarmiento and A.N. Bergmann, Thermal stresses at nozzles of nuclear steel containments under LOCA conditions, Conf. Structural Analysis and Design of Nuclear Power Plants, Porto Alegre, Brasil (October 3-5, 1984). [12] G. Sfinchez Sarmiento, A.N. Bergmann, A. Lotorto and H. Lux Wurm, Computational techniques for the stress analysis of nuclear steel containments (in Spanish). Revista de Ingenieria Estructural, Buenos Aires, Argentina, Vol. 1, no. 3 (June 1983) pp. 1 19. [13] L.I. Skundric, Metal containments: nature of loads and behaviour limits, Nucl, Engrg. Des. 46 (1978) 409-4t6. [14] American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section lII: Nuclear Power Plant Components, Division 1 (1984 Edition). [15] Kerntechnischer Ausschuss (KTA: Nuclear Safety Standards Commission): Reaktorsicherheitsbeh~ter aus Stahl. Teil: Auslegung, Konstruktion und Berechnung. KTA3401.2 (April 1980). [16] Probabilistic Risk Assessment Procedures Guide, N U R E G / C R 2300, Vol. 1 (Office of Nuclear Regulatory Research, US Nuclear Regulatory Commission, Washington D.C.). [17] P. Cybulskis, On the definition of containment failure, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7 9 1982, Vol. II, NUREG/CP-0033; SAND82-1659 (October 1982) pp. 65 76. [18] Reactor Safety Study - WASH 1400, NUREG-75/0114, Appendix VII, Nuclear Regulatory Commission (October 1975). [19] M.L. Corradini, D.V. Swenson, R.L. Woodfin and L.E. Voelker, An analysis of containment failure by a steam explosion following a postulated core meltdown in a light water reactor, Nuclear Engrg. Des. 66 (1981) 287-298.
G. Simchez Sarmiento et al. / Failure internal pressure
[20] D. Hippe and F.W. Heuser, Major aspects and current results of Phase B of the German Risk Study, GRS-Fachgesprach 1982, The Present State of Risk Investigations for Nuclear Power Plants., KOln, Fed. R. of Germany (November 3 and 4, 1982). [21] J.C. Tsai and R.S. Orr, Probabilistic failure modes and localizations in containments subjected to internal pressurization, Proc. Workshop on Containment Integrity, Arlington. Virginia, June 7-9, 1982, Vol. Ii, N U R E G / CP-0033, SAND82-1659 (1982) pp. 365-382. [22] L. Greimann and F. Fanous, On the uncertainties associated with containment analysis, Second Workshop on Containment Integrity. Washington, D.C., USA, June 13-15, 1984, N U R E G ICP-0056, SAND84-1514 (1984) pp. 229-244. [23] L. Greimann, F. Fanous, A. Sabri, D. Ketelaar, A. WoldTinsae and D. Bluhm, Reliability analysis of containment strength, NUREG/CR-1891, 1S-4753 (US Nuclear Regulatory Commission, August 1982). [24] L. Greimann, F. Fanous, A. Wold-Tinsae, D. Ketelaar, T. Lin and D. Bluhm, Reliability analysis of steel containment strength, NUREG/CR-1891 (June 1982). [25] L. Greimann, F. Fanous, D. Ketelaar, T. Lin and D. Bluhm, Probabilistic seismic resistance of steel containments, N U R E G / C R - 3 1 2 7 (January 1984). [26] F. Fanous and L. Greimann, Probabilistic seismic resistance of a Mark II1 Steel Containment, Symposium on Advances in Probabilistic Structural Mechanics, ASME, San Antonio, Texas (June 1984). [27] F. Fanous and L. Greimann, Fragility curves for steel containments with internal pressure, Second Workshop on Containment Integrity, Washington, D.C., USA, June 13-15, 1984, N U R E G LCP-0056, SAND84-1514 (1984) pp. 659-671. [28] G.I. SchuEller et al., Zuverlassigkeitsbeurteilung fi~r den Sicherheitseinschluss (SE) am Beispiel des Druckwasserreaktors, Report BMFT 150201 C, Institut fi~r Bauingenieurwesen III (Massivbau), Technische Universit~tt MiJnchen (September 1981). [29] G.I. Schiaeller, Impact of probability risk assessment on containment, Nucl. Engrg. Des. 80 (1984) 203-216. [30] A. Walser, Primary containment ultimate capacity of Zion Nuclear Power Plant for internal pressure load, Proc. Workshop on Containment Integrity, Arlington, Virginia June 7-9, 1982, Vol. II, NUREG/CP-0033; SAND82-1659 (1982) pp. 263-318. [31] P.F. Gon and J.E. Love, Pressure carrying capacity of the containment structural system of the Mark III Standard Plant, Proc. Workshop on Containment Integrity, Arlington, Virginia, June 7-9, 1982, NUREG/CP-0033, SAND82-1659 (1982). [32] D.S. Horschel and T.E. Blejwas, An analytical investigation of the response of steel containment models to internal pressurization, Proc. 7th Int. Conf. on Structural Mechanics in Reactor Technology, Chicago, Illinois (August 22-26, 1983). [33] T.E. Blejwas, R . L Woodfin, A.W. Dennis and D.S.
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
221
Horschel, Containment Integrity Program FY82 Annual Report, N U R E G / C R - 3 1 3 1 / I , SAND83-0417 (March 1983). D.S. Horschel, Experimental and analytical results of steel containment tests, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15, 1984, NUREG/CP-0056, SAND84-1514, 621-634 (August 1984). J. Jeschke, Limit load analysis of actual spherical containments subjected to static internal pressure and temperature, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15, 1984, N U R E G / C P 0056; SAND84-1514, 287-304 (August 1984). K. Hassmann, G. Weckesser, J. Jeschke and D. Wi~nschmann, Bewertung einer einfachen MOglichkeit der Containmententl'/fftung zur Reduktion der Unfallfolgen der Freisetzungskategorie 6, Unfallablauf ohne zusatzliche Massnahmen, KWU Report R914/84/004 (1984). R.S. Dunham, Y.R. Rashid and Y.K. Tang, Recent results on the evaluation of the overpressure response of concrete and steel containments, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15 1984, N U R E G / C P - 0 0 5 6 ; SAND84-1514, 533 562 (August 1984), this issue, p. 193. D.B. Clauss, Analysis of a 1:8 scale steel containment model subject to internal static pressurization, Proc. Second Workshop on Containment Integrity, Crystal City, Virginia, June 13-15, 1984. NUREG/CP-0056; SAND841514, 634-658 (August 1984), this issue, p. 223 and p. 241. B. Grller, R. Krieg and G. Messemer, Behaviour of spherical PWR-containments close to reinforced sections under excessive internal pressure, Proc. Second Workshop on Containment Integrity, Crystal City. Virginia, June 13 15, 1984, NUREG/CP-0056; SAND84-1514, 575-588 (August 1984). Harrison, K. Loosemore, 1. Milne and A.R. Dowling, Assessment of the integrity of structures containing defects, Report of the Central Electricity Generating Board No. R / H / R 6 (August 1976) Rev. 2 (April 1980). American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section XI: Rules for Inservice Inspection of Nuclear Power Plant Components, Division 1, Appendix A (Analysis of Flaw Indications) (1984 Edition). G. S~mchez Sarmiento and R.S. Carnicer, Assessment of the structural integrity of steel containments containing defects, to be presented at the 8th Int. Conference on Structural Mechanics in Reactor Technology Brussels, Belgium (August 19-23, 1985). A.R. Downling and C:H.A. Townley, The effects of defects on structural failures: a two-criteria approach, Int. J. Pressure Vessels and Piping 3 (1973) 77-107. B.A. Bilby, A.H. Cottrell and K.H. Swinden, The spread of plasticity from a notch, Proc. Royal Soc. of London A272 (1963) 304-314. P.T. Heald, G.M. Spink and P.J. Worthington, Post-yield fracture mechanics, Mater. Sci. Engrg. 10 (1972) 129.
222
G. Sanchez Sarmiento et al. / Failure internal pressure
[46] 1. Milne, K. Loosemore and R.P. Harrison, A procedure for assessing the significance of flaws in pressurized components, 1. Mech. E. Conf., Tolerance of flaws in pressurized components", Paper C106/78 (1978) pp. 317-324. [47] G.G. Chell and I. Milne, Mater. Sci. Engrg. 22 (1976) 349. [48] G.G. Chell. Mater. Sci. Engrg. 24 (1976) 233. [49] I. Milne, Failure assessment, in: Developments in Fracture Mechanics 1, ed. G.G. Chell (Applied Science, London, 1979) ch. 8. [50] I. Milne, Experimental validation of resistance curve analysis, Report of the Central Electricity Generating Board No. RD/L/2044N81 (June 1981). [51] G.G. Chell and I. Milne, A simple practical method for determining the ductile instability of cracked structures, Proc. CSNI Specialists Meeting on Plastic Tearing Instability, St. Louis, N U R E G CP-0010, CSNI Report no. 39 (1979). [52] I. Milne, R.P. Harrison and A.R. Dowling, Evaluation of flaw tolerance: use of a sensitivity analysis of the input data in contrast to rigid safety factors, 1. Mech. E. Conf. (1980) Paper C l l / 8 0 , pp. 25 32. [53] V. Kumar, M.D. German and C.F. Shih, An engineering approach for elastic-plastic fracture analysis, EPRI Topical Report NP 1931, Research Project 1237-1, Electric Power Research Institute, Palo Alto, CA, July 1981, Section 8 (Failure Assessment Diagram) pp. 8.1-8.17. [54] R.P. Harrison and I. Milne, Assessment of defects: the CEGB approach, Phil. Trans. R. Soc. Lond, A229 (1981) 145-153. [55] B.J.L. Darlaston, The CEGB two-criteria proposal, in: Developments in Pressure Vessel Technology 1 (Flaw Analysis), ed. R,W. Nichols (Applied Science, London. 1979) ch. 4. [56] A.R. Dowling, R.P. Harrison and I. Mitne, The CEGB approach to the assessment of the integrity of structures containing defects, Fitness for purpose validation of welded constructions, London, November 17-19 1981, Paper 34. [57] I. Milne, Application of the CEGB procedure to the analysis of HSST intermediate vessel tests V5 and V9, Fitness for purpose validation of welded constructions. London, November 17 19, 1981, Paper 35. [58] J.G. Collier, M.R, Hughes, C.J. Gardner, L.M. Davies and 1. Milne, Fracture assessment of a PWR pressure vessel, Int. Atomic Energy Agency, 1AEA-SM-269/103 (1983) pp. 189-209. [59] G.G. Chell, Elastic plastic fracture mechanics, in: Developments in Fracture Mechanics 1, ed. G.G. Chell (Applied Science, London, 1979) ch. 3. [60] H. Okamura, K. Kageyama and Y. Takahata, Lower-bound solutions and their application to the collapse load of a cracked member under axial force and bending moment, Elastic-Plastic Fracture: Second Symposium, Vol. II Fracture Resistance Curves and Engineering Applications, ASTM STP 803, eds. C.F. Shih and J.P. Gudas (American Society for Testing and Materials, 1983) pp. II-262-11-277.
[61] 1. Milne, Failure analysis in the presence of ductile crack growth, Mater. Sci. Engrg. 39 (1979) 65-79. [62] I. Milne, Calculating the load bearing capacity of a structure failing by ductile crack growth, Advances in Fracture Research, eds. D. Francois et al. 1751-1758, (Pergamon Press, Oxford, 1980) pp. 1751 I758. [63] J.M. Bloom, A procedure for the assessment of the structural integrity of nuclear pressure vessels, Trans. ASME J. Pressure Vessel Technology 105 (February 1983) 28-34. [64] J.M. Bloom, Prediction of ductile tearing of compact fracture specimens using the R-6 failure assessment diagram, Int. 3. Pressure Vessels and Piping (May-June. 1980) 215. [65] J.M. Bloom, Validation of a deformation plasticity failure assessment diagram approach to flaw evaluation, Elastic-Plastic Fracture: Second Symp., Vol. II Fracture Resistance Curves and Engineering Applications, ASTM STP 803, eds. C.F. Shih and J.P. Gudas (American Society for Testing and Materials, 1983) pp. 11. 206 II. 238. [66] 1. Milne, Evaluation of the defect tolerance of pressure vessels under combined thermal and pressure loads, Trans. 5th Int. Conf. on Structural Mechanics in Reactor Technology, Berlin (August 13 17, 1979) Paper G1/3. [67] C.F. Shih et al., Methodology for plastic fracture, Final Report to EPRI, Contract No. RP 60l-2, NP-1735 (General Electric Company, Schenectady, New York, August 1980). [68] C.F. Shih, V. Kumar and M.D. German, Studies on the failure assessment diagram using an estimation method and J-controlled crack growth approach, presented at the 2nd. Int. Symp. on Elastic-Plastic Fracture Mechanics, Philadelphia, October 6 9, 1981, to appear in ASTM STP: see also, Electric Power Research Institute Report NP 1931 (July 1981). [69] C.P. Shih and J.W. Hutchinson, Fully plastic solutions and large-scale yielding estimate for plane stress crack problems, ASME J. Engrg. Mater. Technol. 98 (October 1976) p. 289. [70] SAMCEF - User's Manual, Vols. 1, 2 and 3 (University of Liege, Belgium). [71] S. ldelsohn and G. Sander, On the deep, shallow or flat shell finite elements for the analysis of thin shell structures, Comput. Meth. Appl. Mechan. Engrg. 26, No. 3 (1981) 321-330. [72] M. Geradin, S. Idetsohn and M. Hogge, Nonlinear structural dynamics via Newton and Quasi-Newton methods, Nucl. Engrg. Des. 58 (1980) 339 348. [73] S. Idelsohn, El sistema SAMCEF y su desarrollo actual en el Instituto de Desarrollo Tecnol6gico para la lndustria Quimica, Anales del Ill Congreso Latinoamericano sobre M~:todos Computacionales para Ingenieria (Comp. by A.J. Ferrante). Buenos Aires, Argentina (May 1982). [74] I. Berman, Inelasticity and the ASME Code, Nucl. Engrg. Des. 46 (1978) 335-348.